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Characteristic values $a_n(q)$ of the Mathieu equation
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Numbers
$q$
$n$ 
$a_n(q)$
1/8
0:
-0.007799196358604526975374352136154036687299217285944618073902414755944902274201255594856326456442118010
1/8
1:
1.123016207901962324772205965014995647229602215489752564890920445657653934519515945023831824204850055
1/8
2:
4.006496989408539224054940851445548715345866468416493841378128021761740108209568779747354672414261116
1/8
3:
9.001007225417282011530820218239159585324229475546631322260705783097666490151547246364939606283649645
1/8
4:
16.00052095567822193071331413010718671651264326245962592193332457260372295224081623732654838570843591
1/8
5:
25.00032552450949576505236913927824608398532734378236020534714162851653939865283404219567443827035283
1/8
6:
36.00022321532585753742402234937581960155900792909294605664444589368051454565738835596126008962915245
1/8
7:
49.00016276080299647395838400643405085763972213985424587093825523310460576585067759066933020406595809
1/8
8:
64.00012400810279821400211867735691597149469470653001440976943268948226940160907983152134443027320650
1/8
9:
81.00009765632973109771613734375274761477679217811522926127752952605918015215657698426550206113470221
1/8
10:
100.0000789141829403169320119350109821568071705137135091419676632280470749313782303613267566461537861
1/6
0:
-0.01384695937541069230913686781821567005807446350334052775958613592523128255727382539213764898561916270
1/6
1:
1.163121644308183993212416629397528254787160597983779486699831488205918741081525202673880832849272288
1/6
2:
4.011531753889660437491929638533479367063157861881008877839668221189157043047466319172655228874786660
1/6
3:
9.001808897815730552995078206301032313781000045821347448974584591775663309964460973641823165027332651
1/6
4:
16.00092631257652583689189452746174361489006621860839181425739312848779463317752638289413024981882453
1/6
5:
25.00057871554057153351110251901899873840991413851653357253687984260791575592188823942398764698601095
1/6
6:
36.00039682868487210628184476175898760766449054317608664099859420427255614139230980673463612148972554
1/6
7:
49.00028935307285361623467821208348685598696563443130383271762719801234352614715960624246806917985825
1/6
8:
64.00022045907935310025308870269805387844150202864009546555323353288298458598516970814359690698346372
1/6
9:
81.00017361136310126765133655237464380784351741423204396565272293459802621416263988859521557107335023
1/6
10:
100.0001402919382018640209334666559483232915047978174194923114837958194180326106278337067394080628316
1/4
0:
-0.03103939547561732443850972818046737539757165614326449774963639995021558869432776803890860695284211942
1/4
1:
1.241941128242915144822310574778417251143947502895552363661845172294416462656788053099985600397654233
1/4
2:
4.025829084645603241713504935214025145567842519723189396201885495121768613564145164347843808109114114
1/4
3:
9.004152551546934780305101076204705133108115175136640144582830803966452045287290070123169646485238334
1/4
4:
16.00208529046719562998287970766353836898830958515672743518879794912515643689003450994109339303954003
1/4
5:
25.00130214546980228095721811268235655120523469079383051686019242407844892096160294851718352785604581
1/4
6:
36.00089287379843422726407677439950789278598660106615939604456343686406261386338761520001607237104588
1/4
7:
49.00065104784812144953869393158610105145720513527046311675748578099868179994552050789250399115465619
1/4
8:
64.00049603440671169350384368118283820868634066238671972153266908703000920360646459657100190959169656
1/4
9:
81.00039062627570760767623083270127588410325945344605255307350560255720770509476236944833261116462754
1/4
10:
100.0003156572300786741051138129095999243060123395834625326565516466033262729748840241128585629677041
1/3
0:
-0.05489713165705532397208929046553048388199291020363317459371287221127002619167143592324859627587682703
1/3
1:
1.318859047372450481101981856143870462360212664333034024528671384801159887947317368139545015402675080
1/3
2:
4.045631623751022898800951665065229441120222675967236000648978928138445313268010922929268909480210249
1/3
3:
9.007529612620503569383236397802036970003287600125066395457842546753472502363157350899099999371212154
1/3
4:
16.00370988795024875124699149268680262795388260258098603953352042506438170481651069817657009528203061
1/3
5:
25.00231501819326737677755280372201980081450814224017120101088981829681545512004810243270613947702041
1/3
6:
36.00158735427068683434051386358055952347281089421093629263496629858528301904040385672478327233301357
1/3
7:
49.00115742694448214799362815994418176414726936684097183140336195423560639531790770907102186367649896
1/3
8:
64.00088184262434661927940710220521540776855484222989849701339196235758433203090145015477039883970838
1/3
9:
81.00069444847634338528475938304091240346274226194924463440053095854098497504917430524115409425152705
1/3
10:
100.0005611693277466834983004152301835782364824099806315602140431442770733203539001957647042338317712
1/2
0:
-0.1217655449410826959442311040306376080721390230870514597566306137582620584666945364398653581765744783
1/2
1:
1.466766842516055774308238490098876885672560272473706374801003835403979645481582209215219603032449048
1/2
2:
4.100900595560480626440489824331561294060149376829491562485697274596914740203961592175034639444236598
1/2
3:
9.017606927797508666664672353869651055518659076593238410461923888085305282370989282757671476733208647
1/2
4:
16.00836462272300738824811359125159844797828806634741006833461607254723097522414026718343797181198691
1/2
5:
25.00520943382713712344600382449307492019434013603038899438761993078037365668246726417982741223002943
1/2
6:
36.00357169591063031522519472098600157826951290061620214287557145119814920299897428595007208469172654
1/2
7:
49.00260426558293770570452311256412501982966133175699699667414622316508313294605670985440301641832012
1/2
8:
64.00198416955728547894957882632918159284841419964743794961426865223166697116552114809355514010631432
1/2
9:
81.00156252041196460903368770896855035525739007695094904942107875428041959401138305353594010065502937
1/2
10:
100.0012626368935912164629661292596375992217717242222594399942103301654472828506854924219377716681311
2/3
0:
-0.2124005104094995389671529484997631125405249250426444602041374247159456785568344242256481786752596075
2/3
1:
1.606399696540880406614692369105349425438523078543963823020239796498686227064032728528529358187407275
2/3
2:
4.175263630201390381953877508274660135196871476241305273268717100366953363378167786856093349417167337
2/3
3:
9.032485081711299342905061057510380835731770920397834662621299735563007707350637783543316357239644632
2/3
4:
16.01491362279389969326212485498269275757615022247610173456574243064354598918530585584276887613376999
2/3
5:
25.00926296153305112953697873812228144705810074738599230247696243174645163683164332346575591990620301
2/3
6:
36.00635005405744662193394640315572068338741659314752419473427763633599028533593304697205426191573510
2/3
7:
49.00462994230187251654942702605449337080449466984507673485392095561813624926511794405081226062366022
2/3
8:
64.00352747142038203690130627338056619694974359726319656469893204365880324697744095430203338501077288
2/3
9:
81.00277784229177336152803983274550303768989688313554170165147495318286938605183348371810012772789729
2/3
10:
100.0022447025111270151661913757484682675975646907954086552092384954508441500243270783170954668992615
3/4
0:
-0.2658780338622578242246874193029283710942463359104779685914119589920423639830744343271071492832135847
3/4
1:
1.672975785748911760705060333906773537329322075936555722749928042968285781545703500029926566558645058
3/4
2:
4.218843182034351974712133128742127625952898001865100486989098198726916690417925364649394934128375071
3/4
3:
9.041861200197145093399549015222023810540640841073730622020484507830412479628815156735031026216192109
3/4
4:
16.01890819093569817104314048741779769894987028882988477061319836455566607274756082998729311235357524
3/4
5:
25.01172485988327497791459171903478520830748098106865277566744188460993839358597359425623564065309128
3/4
6:
36.00803707485707859277809771814896675200421678473563126489594465172233940553112388420948691121164378
3/4
7:
49.00585987589035357128113202536491786103447178907620513218109307803113846447188475393727970536183593
3/4
8:
64.00446450126025576622229374811920149178428956250849758522389232786220693932795450985633366619269843
3/4
9:
81.00351572834099638076999952687683898325935052307310980306417332845686452122319186604904300165971002
3/4
10:
100.0028409629119514764923762086561121903604704404094245090594535189410645717813257986123589067199302
1
0:
-0.4551386041074135482326331875288858669165198421966654710317486675822261235164160470159425623418250589
1
1:
1.859108072514363472329917512841423300127410341317695081625775769043966104074314694144409579202762489
1
2:
4.371300982735085661171120918199422244868492771912378157766245915462346356167914760113086656176209254
1
3:
9.078368847203101992542227389838886773846035067565821921475351184485590771532228511305502042807537785
1
4:
16.03383234035951396115310292247387218351311829540466158126228671529568438517272883259055193928865027
1
5:
25.02085434544858105321885332923390473871795259780880798928037652340677700614002341588557773473264703
1
6:
36.01429004604049922203489434681190541280766028060644661189175650238832013276422730013130353709670056
1
7:
49.01041825036485279618077257200961372365694752070295449051413140270824516896976883381858925789446006
1
8:
64.00793718925467418869817603308066535182183622326957198192325631647331426322701336074013750443234299
1
9:
81.00625032663259684324348148794582909327212528430737639890439786173206877812709196249076067328513169
1
10:
100.0050506751594644392114505162757105040259099215869168613957919988409052483841699338935298735106845
4/3
0:
-0.7634681675887358772327438815001853557943606438640445007645777015268912022165948533148540253232895731
4/3
1:
2.073749858679927278565493732320708140226083028013585056612344634227349494592993893554215318034412982
4/3
2:
4.613731568890988365111375330100006583942575669351738633126429440586346623293424049239630393782902481
4/3
3:
9.148392254974258107116485391304075377397815038555327512784063624009014528688221127222649098630160422
4/3
4:
16.06083084096889328123470616336161600144686741497152742291424729824809249292768475222419153006378265
4/3
5:
25.03711060492945616938426346145528389936444747152078646227528308457343591037114044915737658757410642
4/3
6:
36.02541069356661176650247097542518238214723685556881077390869511587137476894338059190447891194046952
4/3
7:
49.01852352787728672579655384568935971106208636618677033537025590123815794495049522536583715328068420
4/3
8:
64.01411150123775835088558088160840943980503344811686430914988057726070963958826938150305928081499405
4/3
9:
81.01111214356645247050903770361201402720270702026560417653359889335403436740695248865017989724471426
4/3
10:
100.0089792133179828705062248445049935564508660012696625807048855970143432744484188401900343198960799
3/2
0:
-0.9368184941236269317413715932774891733327321056206193351222878238757184130658197938006814403226025140
3/2
1:
2.165939910185351349896995577169831680180515512751946887065969221322899479822767091600646620069912841
3/2
2:
4.746779468115387393261751626866046237228385633448342803191475798689440481504196188703808961942256379
3/2
3:
9.193301047680609740478042470910558277710726356841266889322403625145539305202083273780790206657894961
3/2
4:
16.07751183234476395375959995518599672116566204801029356827045002455815148549281771899254545681267166
3/2
5:
25.04699856069667774493658052741258809868222506985589201637750905390973310230783894316652128282831329
3/2
6:
36.03216524377443418577194749867137993067451267543069265585800321310721000909380256346964978204917123
3/2
7:
49.02344552841276624999401100292785707391272188716237988570957213145777339056614845136230377228193686
3/2
8:
64.01786059330750215454829720124235665378075837409797684845475479636288045895464085307975776575558431
3/2
9:
81.01406415392525933435619909642048112242190087020459311626738546680612293259729980700468451505915041
3/2
10:
100.0113644976543585583330914920435999237734624619988499720362677871806304483385906299114865202655209
2
0:
-1.513956885056520254181643952263881021949068787817187068660392445849420212558570792864098704258150306
2
1:
2.379199880488686029803809752454532306373741290498460568853094811701863325584713744478579045495258950
2
2:
5.172665133358294193203854859202831839908089575366441002779336192962730845247857265703409480229956989
2
3:
9.370322483621104012944711601480567278968253843604621840262910116867044296749375724411199010262582904
2
4:
16.14120378558493654182861394362272251930212095099842290054792991854211602243066576162218168069652635
2
5:
25.08377778277331010228825984598053601350574832869307972155372554907983626763257478516319348624729283
2
6:
36.05721562419987781348859745443850957145990206399936000870896867980649721706596107302377636781968705
2
7:
49.04169209857535048543676175418884184373819242787653950619079656318296462281106688497226741603228755
2
8:
64.03175694273274356144049246757719621526795541503011938691430126934561730039657272096614010146212784
2
9:
81.02500522876272709810731686721618815944972842409414519780737787872027955401579252590283205892108709
2
10:
100.0202047428112256104570604168341337250735629702311511894310050540701818700544903396687563831079711
3
0:
-2.834391889904311013140306104313720144850025440786131097230803799579596211425068458227373315452320508
3
1:
2.519039087508437069770826574294387469657188428022183631821259297493951364865744872098885843648763861
3
2:
6.045196852234818808804060338035966230368595243731987201308613066151416144063773421413593080462054900
3
3:
9.915506290452133694224887677216734098835564742137273004745654186277577195310636986001759432792094019
3
4:
16.33872074599636992795855535181471890262235819752696749455612244834442917700728832870371409303980848
3
5:
25.19028553097009793032034270797276308941844030284844804022864099764936856429693105187680997841177847
3
6:
36.12896867478926369043540850262715706665106302092266698353222703747524088541479371265031482083329794
3
7:
49.09387980791561140519718734927394102744505620424477971194453391168539956199151093030514687374249187
3
8:
64.07148389981453948341949775412937950527677453621485343592981195253062950719964807634874687299825754
3
9:
81.05627649306505961363441086050231901870695771381427671375351040926583021556550638001538398694298419
3
10:
100.0454683359803443841200354794337049775666124620393946315010520925968516917295951391322549556351781
4
0:
-4.280518818302522853185949110214066425446412521962047240281209601420138549531089427279119608062170430
4
1:
2.318008170106524556942155346945909058357864792073953885312514316635356495510005791264953461543250515
4
2:
6.829074834566389522788131454511703275254781926809503319936209396371331054844014406826238746937377927
4
3:
10.67102710352055041525603120498526563237569891231465767356512397570783697530037929601595679457604783
4
4:
16.64981890681717919404676418529949166880083636114051362191806772983639510655330718375444466321384085
4
5:
25.34375763316195280264411047268140033586038402616212102279990284170859194367201483446809005533826168
4
6:
36.22995250746758808085721215125359955721432249575095538161297906060944326775928972099633779519946589
4
7:
49.16708282345133794924179724570106834403610639240533281841958891795920310055741885450335930455248965
4
8:
64.12715944968494166021242756009772969081511979415966888653508971490525401938212648819350556296799618
4
9:
81.10008383199694564951031935932191350816040879957629968032520160495610368718073971941914879105001123
4
10:
100.0808516981366363647298426255707936698455882395264330143271852796677423697232591633828243924965798
6
0:
-7.368830832026793797149345649787662734129545644359721173990882323169058069472043342624738625235613376
6
1:
1.214278164420449610802778659941113649023485171461866785782177147081797022067762724076107983215249793
6
2:
7.870064474775305097321220512573943666688710735013628982253578620983367256161953470635789382885967697
6
3:
12.46560068333770097036021956818097421219956588254863893345075366386391895395349322678883616983940984
6
4:
17.68878295456020039187106970848038995783695714086423955153138137929284399337825930251159848445699798
6
5:
25.81727199269119100241357415449762965796680028228051665985085226017007579437467932524803212079280854
6
6:
36.52302500965929189284482903924887750203770396433586586010828323191232160021553548371366105785802797
6
7:
49.37721830687154281846122757618543843431681825467906200099475266233097681808648290162163578742059898
6
8:
64.28661016154512517164877093881918113967509354647890867928707094712558993972445083043943227991900594
6
9:
81.22542592234079297646480252586953778893649086856816762912755137765628099306867034996724278657332104
6
10:
100.1820394673546403304578641980913893317905729410035396133071512631890592698035339139195782997587161
8
0:
-10.60672923555264798520235682968093760778469204533139425594202543196908169466702308839138470342683097
8
1:
-0.4359436013208289560517636084483093606927888366674576802517326019895604471284741658408291802415298352
8
2:
8.115238830263265859507801525085707597920212536041494633702869896314731710612947483074298755978239529
8
3:
14.18188036231634285867148974395923099240454868476373742135405393246355986914386502352668284016532488
8
4:
19.25270505942423998900507513098168265855105151248033807596673799102517845512598619596523487785792484
8
5:
26.57775329260188279907447933894863281440629791011221123953677020798430250520346525588706388720445762
8
6:
36.94908697167029500762277165272167575363046836416035700417621065414108417364177361326070753850212325
8
7:
49.67430779078868194069836884799656407020672835517209097767778183428898800795964050086569257171165076
8
8:
64.51081748710467843627516205966985449608317196462582632143767181296415994185698986231279446880425367
8
9:
81.40135352208632780874103812469657772370008319138151517366489899783635133836839227610185044034527067
8
10:
100.3239338218036670088308257959513311048085247988468250776379415414513881449265562938752733433974081
Definition
For real $q$, $a_n(q)$ is the value of $a$ for which Mathieu's equation $w''+(a-2q\cos 2z)w=0$ has the even periodic Mathieu solution $\mathrm{ce}_n(z,q)$, with $a_n(0)=n^2$ and with the branch of $a$ followed continuously in $q$ from $q=0$ [1].
Parameters
$q$
—   Mathieu parameter ($q>0$)
$n$
—   characteristic index (integer with $n\geq0$)
Formulas
(1)
$a_n(0)=n^2$.
(2)
$a_{2m}(-q)=a_{2m}(q)$.
Comments
(3)
The normalisation is DLMF's [1]: the oscillatory term is $2q\cos 2z$. Conventions using $2q\cos z$, $q\cos 2z$ or $16q\cos 2z$ give different values when $q\ne0$.
(4)
The companion characteristic values $b_n(q)$ belong to the odd Mathieu solutions $\mathrm{se}_n(z,q)$ and are not stored in this table. They give the remaining reflection, $a_{2m+1}(-q)=b_{2m+1}(q)$, and they interlace with the values here: $a_0(q)<b_1(q)<a_1(q)<\cdots$ for $q>0$.
(5)
Together with the companion curves $b_n(q)$, the curves $a_n(q)$ form the boundaries of the stability regions in the Ince-Strutt diagram [1].
Programs
(P1)
Python
from scipy import special

# a_4(1) in DLMF's normalisation
special.mathieu_a(4, 1.0)
Links
Similar tables
Eigenvalues of the imaginary cubic oscillator $-y''+ix^3y$ —   also stores real spectral values of a one-dimensional differential operator
Eigenvalues of the pure quartic, sextic, octic and decic oscillators $-y''+x^{2m}y$ —   also stores eigenvalues of one-dimensional Schrödinger operators, for polynomial potentials rather than the periodic potential $-2q\cos 2z$
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds every $0\leq n\leq10$ for $q\in\{1/8,1/6,1/4,1/3,1/2,2/3,3/4,1,4/3,3/2,2,3,4,6,8\}$ (165 entries). Small positive ratios and reciprocal coupling pairs cover weak through moderately stronger coupling and agree with the coupling choices in the even/odd function tables. Other $q$, higher orders and the companion $b_n(q)$ values are not included.)
How they were obtained:

The generator forms the finite symmetric Hill matrices obtained from the Fourier expansions of $\mathrm{ce}_{2m}$ in $\cos(2rz)$ and of $\mathrm{ce}_{2m+1}$ in $\cos((2r+1)z)$. It isolates eigenvalues by Sturm counts and bisection on the tridiagonal matrices, without using a dense numerical eigensolver. The stored 100-significant-digit values come from the agreement interval of truncation sizes $N=90$ and $N=120$ at 220 and 260 decimal working digits. Every stored entry was then checked against the stronger finite section $N=150$ at 320 decimal working digits. The worst spread among the candidate-generation finite sections was 6.68434263938e-129 at $(q,n)=(1/2,6)$. The worst center-to-center relative difference between the printed value and the stronger finite-section check was 4.42984972439e-100 at $(q,n)=(4,3)$, which is within the printed 100-digit interval. The finite-section comparison is not a rigorous tail bound for the infinite Hill operator, so the table remains agreement-checked rather than proven.

more

As controls, the generator checks the $q=0$ limit $a_n(0)=n^2$, the symmetry $a_{2m}(-q)=a_{2m}(q)$ for all stored even orders, interlacing with companion $b_n(q)$ values over the stored range, auxiliary $q=1/10$ values against the small-$q$ expansions from DLMF [2], all stored rows against SciPy's scipy.special.mathieu_a to double precision [3], and selected high-precision ODE shooting residuals on $[0,\pi/2]$ for ordinary and endpoint cases.



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