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Fourier coefficients of Maass forms of level 1
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Numbers
$R'$
$p$ 
$a_p$
9.533
2:
-1.0683335512235708066692862262769493600995441885504273112316
comment: LMFDB form 1.0.1.1.1 has odd symmetry and Fricke sign $+1$.
9.533
3:
-0.45619735450611838399314693202628717476966799190780255845593
9.533
5:
-0.29067255498507017568811748671868486651561632689532160072424
9.533
7:
-0.74494161214757989511169493132824604813500846259709982901176
9.533
11:
0.16616359661443977158538644796830403298769553233447629191395
9.533
13:
-0.58668852786270913083589873021510784021344893703490756880720
9.533
17:
0.57069580247220558238175226153168398445447741391594607492364
9.533
19:
-0.98193858651185920908429138056228985336467362003183598786446
9.533
23:
0.66296895859247565567379238914872160184294233135171299994123
9.533
29:
-1.0486885639818629514118350281045500224015617504231327398136
9.533
31:
0.78626844138397801418820308604618711376330494311259079635927
9.533
37:
-0.44819811991104455654080884212741306986828263971495055842585
9.533
41:
-1.1982526381660355239351779720071196839172572786949649033829
9.533
43:
1.5620309163830235728552806926276684558164054284975501666968
9.533
47:
-0.60315223950014029951083653422561859533336017443191157389161
12.17
2:
0.28925187146239926777326978031023465869345189218549058491243
comment: LMFDB form 1.0.1.2.1 has odd symmetry and Fricke sign $+1$.
12.17
3:
-1.2018587610118135635720280848661542988329796613941581377837
12.17
5:
0.039552707287414083275434069088972612663662550092628440522876
12.17
7:
0.4481331044912988447852656570094478366190577716935394118859
12.17
11:
-0.69145078339493173537348708854518903357400216955738177360266
12.17
13:
-0.80278000333233626075868116054637890233730129080299116740992
12.17
17:
-1.0376753734633174787702167379512072784721179838093687736194
12.17
19:
0.63717880716794656028474108322459448381778019146883624023485
12.17
23:
0.5087986923694297064590505481649289063295567212790433004147
12.17
29:
0.77999688121453209038717532781158217997955096354927668479030
12.17
31:
-1.5980135837087028805021234997152353399173358773551203267000
12.17
37:
-0.42921516095681464357365519704806000981584249364840417624221
12.17
41:
-0.54852418770590534328763385233219522910693546286242071207067
12.17
43:
-0.48935502226390572367807190498103653181843366415270031327151
12.17
47:
0.81686800589602004494422126292519068470935023876231493903423
13.77
2:
1.5493044779412962245074692628707670789215570288503878814865
comment: LMFDB form 1.0.1.3.1 has even symmetry and Fricke sign $+1$.
13.77
3:
0.24689977245398089801144105099610194900802480980710640281770
13.77
5:
0.7370603853483010863787784833901814807636983816482293617358
13.77
7:
-0.26142007576521614229988724642698652062543719805509375834486
13.77
11:
-0.95356465261777747676757370966216796597157371565462690649141
13.77
13:
0.27882702916232517218757032222422814440922538895395761847550
13.77
17:
1.307341714533658624115955967656608132997708504181108374253
13.77
19:
0.092558582508212260763691323643293819229525620438798375239196
13.77
23:
1.1380685214066112254447480215506244047775772938845987715637
13.77
29:
0.75211384546608752127000954814585721489582103707872417592967
13.77
31:
0.024851953513185519788786884991201652079099096228274924872134
13.77
37:
0.19926565559081418260496619182438029591653041163379802080036
13.77
41:
-0.30403299675582781313973532290719532172723684998778380557959
13.77
43:
0.78323936351627267407050722380323492448948402923286072806968
13.77
47:
0.36056841049481212577821887972575145670050983527724163735439
14.35
2:
-0.23091519120146070371421938003141946799818474686518027375116
comment: LMFDB form 1.0.1.4.1 has odd symmetry and Fricke sign $+1$.
14.35
3:
0.69559498626441516958228535583197344930055344457808205851956
14.35
5:
-1.2982845988718940207661436930726846146864468412296364587875
14.35
7:
-0.48328198177351983524151012378472019698921213770271109484519
14.35
11:
0.1774980728856124362259565683360284940671806637248990615877
14.35
13:
0.62544354221988362098806633826490765073245877267461214202497
14.35
17:
-0.15155949893373666705511187006422452515278796674236227218014
14.35
19:
-0.98026479651556768114350971699858158720552939928436664601374
14.35
23:
-0.56801529373101016417310397443630557255213239283224723393268
14.35
29:
0.3813445524533840284774692778272245189517198436255018568262
14.35
31:
-0.12958423119254204551535045698408052853915189536363885871070
14.35
37:
-1.4860236750319200796174966609717978885743668267281118392947
14.35
41:
1.2936373589005672955978445720282604081982016295499207750355
14.35
43:
-1.3888699728811002553546059284070247166490105959293855926940
14.35
47:
0.053102429275706550123544119993985820265539791924098325958351
16.13
2:
1.1618555924158509937981715228583145424133853466373153810290
comment: LMFDB form 1.0.1.5.1 has odd symmetry and Fricke sign $+1$.
16.13
3:
-1.2819725611533323123224532131324097770506209614859371186857
16.13
5:
-0.75680641385830019630133459826188995980321012353877419266793
16.13
7:
-0.2985191165978981871287397702969865024289601408382775529412
16.13
11:
0.76409070445339806163719189406597942949069687276728135892912
16.13
13:
0.16260231764807469479333696890484271699868439029015341271250
16.13
17:
0.4164174325265319057977010493112093084057358189859708646164
16.13
19:
-0.75567638408309873041814244821405037460569073983154505506336
16.13
23:
-1.6016444877022552381130307865872415386555102210083408923329
16.13
29:
-1.4843721310824877578983886047926422523171092451935608880997
16.13
31:
0.95937599106496661039055458713556933917895150261660635948259
16.13
37:
0.40047380747031914925994877544573616855827191566904457980385
16.13
41:
-0.39041127567383526771799331438222165264866588576308484585132
16.13
43:
0.15805251280315974036205178652218030327938432148839588437663
16.13
47:
1.561732882121532129443091665606811247530395651627526209008
16.64
2:
-1.540227825716921697381751613939752551825698624979828448337
comment: LMFDB form 1.0.1.6.1 has odd symmetry and Fricke sign $+1$.
16.64
3:
0.97749259146855253061084424761089434757594750071002435534513
16.64
5:
-0.10524236262770334743192453077738720148058752349827246341803
16.64
7:
-0.6926408590851301853714120242912182284149851842144175811535
16.64
11:
-0.76020127595254165553244022597455378682179134344732460357127
16.64
13:
-1.570437062046106795750248111397511535044850572654288263094
16.64
17:
0.28188982261736816654946804627807560420949854258242749463084
16.64
19:
0.1040460905156093628877139620429861275613400971558822950983
16.64
23:
-0.14387947063516627376206476627439766635393277308837743459153
16.64
29:
0.79124408131888788593745880825721549852341381912370053402979
16.64
31:
-0.75717551364058719452634426371721949246084690418099719108231
16.64
37:
1.2725598641206614058346344766048422289187834098442675038081
16.64
41:
0.5474899861320420283334676635950511481037414004274646471827
16.64
43:
-0.23542298308600077902187424622302669075687459291995370579154
16.64
47:
0.56574380216105501548365696440576610101045468085671987392573
17.73
2:
-0.76545805660033816923256966396140998293857959287192527309848
comment: LMFDB form 1.0.1.7.1 has even symmetry and Fricke sign $+1$.
17.73
3:
-0.97777890747614303810850512609759902825095233728204061645335
17.73
5:
-1.0152735225259196208958533987625369177969914205359320686079
17.73
7:
1.1808208356059652990821815717534682015863922355225510365418
17.73
11:
-0.62048772617512068374540303350066933001504177996011168646228
17.73
13:
0.26528869893032460448965274424218563953882211240336512877432
17.73
17:
-0.13574040867822474105957407245795322024828137827887397881238
17.73
19:
0.1769710409503025815614113869495579215708312682244653919542
17.73
23:
1.0013733555558193389238588535128582089691799992754397237132
17.73
29:
-0.1101599541994124000216153019470542759969111431761163859516
17.73
31:
1.2758684886969307712496097408105358195396742976007320177410
17.73
37:
1.3923186985185721327261751066640729868971429425520473421596
17.73
41:
0.34471623925489672433989566437708410879302299041908510487997
17.73
43:
-0.35123461715473772545744701168330697268018065668741049037602
17.73
47:
-0.13337745164333139364820808807733567987907398800265836341781
18.18
2:
0.3740633467235858858961866665803541061859413242589207376162
comment: LMFDB form 1.0.1.8.1 has odd symmetry and Fricke sign $+1$.
18.18
3:
0.10195869761957854389068314007760206154137946677034340458302
18.18
5:
0.63733082934618812558788617921125847965264268689356881654819
18.18
7:
-1.5420992232720381199968529733122349199274945299705763665316
18.18
11:
-0.41171570679312535131853822598406044792662774754264283104234
18.18
13:
0.47870372967875754498907620242453336069870878178890623770190
18.18
17:
-0.5485574149972439801665127469075879282481594059618504264039
18.18
19:
1.0309841579899739731980287887747269688997270038221118281913
18.18
23:
0.6634257705923329082068824684470769227882736672386372563732
18.18
29:
-1.2708224959446018140734238017901960221722842774335140614329
18.18
31:
-0.22396875991098728945495253264814253523747908640993475142293
18.18
37:
-0.10874155650469024553742623781376862161684001907918195144406
18.18
41:
-0.41909753978231638844530326411453871712148078795444552001841
18.18
43:
0.8481508653777888483444583690077689524514756204116623016526
18.18
47:
-1.0863016664143442497718152171982777548191126837971113554629
19.42
2:
-0.6927619764024025062649537931473836203816163940089463456289
comment: LMFDB form 1.0.1.9.1 has even symmetry and Fricke sign $+1$.
19.42
3:
1.5623543021149369783968190828050280803380182545845677451725
19.42
5:
-0.038411698226421721089834114448487491941858816202755645753325
19.42
7:
0.3129529806962077238817424417707576711672585932797372128586
19.42
11:
1.1536335544840807341114207335438611354650675675916598021605
19.42
13:
0.75897142565658636532262774769574684301043415061026107879619
19.42
17:
0.84438100981917211119388474119662957097921110584110744095340
19.42
19:
0.39321289698130854002984202358089325435331095692203842680288
19.42
23:
-0.35799874280956788290167306365225105838279925162579387357506
19.42
29:
0.22470357186308468801086700594051736231613062588488900887141
19.42
31:
-0.33028611704282498880626292237866740935274430217031808153712
19.42
37:
0.40396539773123137687400553966758719718677474334079450716780
19.42
41:
-1.379815224860671878003643341966808440895837143952820227527
19.42
43:
1.6447244719125496023494362579666300297679864799669313669758
19.42
47:
-0.018395946939323111169623898720556623539170937740662114374043
19.48
2:
-1.7001880323693752049475700121143829472882534160677173481581
comment: LMFDB form 1.0.1.10.1 has odd symmetry and Fricke sign $+1$.
19.48
3:
-0.61456537658770353222137071142210533296820429264733668459926
19.48
5:
0.81982534416093697266593236975517560141966972657963791590816
19.48
7:
0.063568993750935177124642125607756656458264346074326838688814
19.48
11:
0.65973642364744539381352540136487520562567986317987583477026
19.48
13:
0.66233012247927235461986777013871950470540618612215298029593
19.48
17:
-0.67778111963241493398148990277404678830667044188765883495918
19.48
19:
-1.7344074845841628537734995888530719751933524207792861866911
19.48
23:
0.33416076268849829598857443319815497074939969904644122792261
19.48
29:
-0.58562395070246034570847656563207554727953003496813771740119
19.48
31:
-0.98865993315398397900688815545132013243036550090040748640457
19.48
37:
-1.2809808080512615503658051392621635625570937060357949864422
19.48
41:
0.23915019058867365130028770857370752169820938009988700217525
19.48
43:
-0.42913822131513632542303525108134525708572625690047881248273
19.48
47:
0.90751523633452057384561642310751539081229426324030241268626
Definition
Let $f$ be a weight $0$ Maass cusp form for $\mathrm{SL}_2(\mathbb{Z})$ that is a Hecke eigenform normalized by $a_1=1$. Its Laplace eigenvalue is $1/4+R^2$ with $R>0$. This table gives $a_p$ for primes $p$, with the coefficients $a_n$ defined by (1).
Parameters
$R'$
—   truncated spectral parameter ($R'$ is the truncated spectral-parameter label used for the corresponding row of T84)
$p$
—   prime ($p$ is prime)
Formulas
(1)
If $f$ is even, then $f(x+iy)=\sum_{n\geq1}a_n\sqrt{y}K_{iR}(2\pi n y)\cos(2\pi n x)$. If $f$ is odd, replace the cosine by $\sin(2\pi n x)$.
(2)
The coefficients define the Dirichlet series $L(f,s)=\sum_{n\geq1}a_n n^{-s}$.
(3)
For the normalized level-one Hecke eigenforms here, $a_{mn}=a_m a_n$ when $(m,n)=1$, and $a_{p^2}=a_p^2-1$ for each prime $p$.
Comments
(4)
The first parameter is a label, not an approximation to be used in a computation. It is the first five characters of the LMFDB spectral parameter string, matching the entry keys in the spectral-parameter table.
(5)
For even forms the Fourier expansion uses $\cos(2\pi n x)$, and for odd forms it uses $\sin(2\pi n x)$. Each entry with $p=2$ gives the LMFDB label, the symmetry and the Fricke sign for that form.
(6)
Composite coefficients are omitted. They are determined from the prime coefficients by the Hecke relations stated in (3) for these forms.
Programs
(P1)
Python
from urllib.request import urlopen

url = "https://www.lmfdb.org/ModularForm/GL2/Q/Maass/download_coefficients/1.0.1.1.1"
text = urlopen(url).read().decode()
body = text.split("\n", 2)[2].strip()
coefficients = body[1:-1].split(", ")
coefficients[1]       # a_2 with its source error
Links
Similar tables
Spectral parameter of Maass forms of level 1 —   stores the spectral parameters used to order and label the Maass forms here
Hecke polynomials of level one cusp forms —   stores Hecke eigenvalue data for holomorphic cusp forms on $\mathrm{SL}_2(\mathbb{Z})$
Hecke polynomials of weight 2 newforms —   stores Hecke eigenvalue data for holomorphic newforms of weight $2$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds $a_p$ for the first ten LMFDB rigorous Maass forms of level $1$, weight $0$ and trivial character, for every prime $p<50$)
How they were obtained:

The generator uses the LMFDB all-data downloads [6] as the source format and stores the centers and error bounds for the prime coefficient rows.

more

The LMFDB reliability statement says that each rigorous Maass form is proven to approximate a true Maass form and that the decimal coefficients are proven close to the coefficients of an actual Maass form [4]. The LMFDB completeness statement says that each database entry gives error intervals containing the data for exactly one Maass form [3]. The generator parses each decimal center and source radius as an exact rational number and returns a Sage interval widened by one unit in the last displayed decimal place, so that the stored ball contains the LMFDB interval after the center is rounded. It writes $60$ significant digits. Before the draft was created, the truncated spectral-parameter labels were compared with T84, and the source coefficients with composite indices were checked against the Hecke relations in (3) wherever the needed composite coefficient was among the $1000$ downloaded coefficients.



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