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{"Title": "Even period polynomials of rational newforms", "Definition": "The primitive polynomial $P_f^+(X)\\in\\mathbb Z[X]$ with positive leading coefficient proportional to the even period polynomial of a rational-coefficient normalized newform $f=f_{N,k,i}$ with trivial character CITE{PP}.", "Parameters": {"N": {"type": "Z", "title": "exact level", "display": "$N$", "constraints": "$N\\geq1$"}, "k": {"type": "Z", "title": "weight", "display": "$k$", "constraints": "$k\\geq4$ is even and $S_k^{\\mathrm{new}}(\\Gamma_0(N))$ is nonzero"}, "i": {"type": "Z", "title": "embedded newform", "display": "$i$", "constraints": "$1\\leq i\\leq\\dim S_k^{\\mathrm{new}}(\\Gamma_0(N))$ in lexicographic order of all real embedded Fourier-coefficient sequences $(a_2,a_3,\\ldots)$; the form must have rational Hecke field and nonzero even period polynomial"}}, "Tags": ["modular form", "number theory", "polynomial"], "Keywords": ["even period polynomial", "primitive integral period polynomial", "Eichler integral", "rational newform", "Ramanujan Delta"], "Comments": {"comment-normalization": "The form lies in $S_k^{\\mathrm{new}}(\\Gamma_0(N))$. Write $f(\\tau)=\\sum_{n\\geq1}a_n e^{2\\pi\\mathrm{i}n\\tau}$ with $a_1=1$. The superscript $+$ means even powers of $X$, independently of the functional-equation sign. Rationality up to a nonzero period follows from CITE{PP}, Proposition 5.11. Clear denominators, divide by the positive gcd of the coefficients, and fix the leading sign. The constant term and all forced factors are retained; $X$ is not rescaled, inverted, or replaced by $X^2$.", "comment-indexing": "The index $i$ is assigned before the rational-field restriction, so gaps are retained. All embeddings of every Hecke field participate in the order; exact Fourier coefficients through a Sturm bound distinguish the forms. The attached roster records their coefficient fields and isolating intervals. This is the indexing of HREF{Zeros_of_period_polynomials_of_newforms}[the full period-polynomial zero table], which is currently a draft. At level one it agrees with the increasing-$a_2$ convention of HREF{Special_values_of_the_L-functions_of_level_one_cusp_forms}[the critical-value table]; HREF{Hecke_polynomials_of_level_one_cusp_forms}[the Hecke polynomials] identify these $a_2$ values.", "comment-parity": "The odd part $r_f^-(X)=(r_f(X)-r_f(-X))/2$ has a separate primitive shape. Neither primitive parity shape retains the relative transcendental scale between the two parts, so they do not by themselves recover HREF{Zeros_of_period_polynomials_of_newforms}[the full period-polynomial zeros]. Only the identity-coset polynomial is stored. At higher level, the complete period data also has other coset components CITE{PP}, Section 2.", "comment-pairing": "Pairings of period polynomials are related to HREF{Petersson_norms_of_level_one_cusp_forms}[level-one Petersson self-products] and HREF{Petersson_norms_of_higher-level_newforms}[higher-level Petersson self-products] CITE{PP}, Section 3. At higher level the pairing requires the full coset-valued data; this primitive even component alone does not determine the norm.", "comment-delta": "The form $(N,k,i)=(1,12,1)$ is HREF{Q-expansion_of_the_modular_discriminant}[the modular discriminant $\\Delta$], whose Fourier coefficients are the Ramanujan tau values. Its primitive even shape agrees with CITE{JMOS}, Example 1."}, "Formulas": {"formula-integral": "The identity-coset period polynomial is $r_f(X)=\\int_0^{\\mathrm{i}\\infty}f(\\tau)(\\tau-X)^{k-2}\\,d\\tau$, and its even part is $r_f^+(X)=(r_f(X)+r_f(-X))/2$ CITE{JMOS}.", "formula-coefficients": "For $w=k-2$, $r_f^+(X)=\\sum_{\\substack{0\\leq m\\leq w\\\\m\\text{ even}}}\\binom wm\\mathrm{i}^{1-m}\\frac{m!}{(2\\pi)^{m+1}}L(f,m+1)X^{w-m}$. Here $L(f,s)$ is the uncompleted arithmetic L-function; no conductor factor occurs CITE{JMOS}, equations (1.1)\u2013(1.4). HREF{Special_values_of_the_L-functions_of_level_one_cusp_forms}[The level-one critical-value table] contains the odd critical values for the level-one rows. HREF{Critical_L-values_of_higher-level_newforms}[The higher-level critical-value table] records the same arithmetic normalization at other weights.", "formula-reciprocal": "$P_f^+(X)=-w_N(f)N^{(k-2)/2}X^{k-2}P_f^+(1/(NX))$, where $w_N(f)\\in\\{-1,1\\}$ is fixed by $f(-1/(N\\tau))=w_N(f)N^{k/2}\\tau^k f(\\tau)$ CITE{PARI}. This follows from $\\Lambda_N(f,s)=\\varepsilon_f\\Lambda_N(f,k-s)$, with $\\Lambda_N(f,s)=(\\sqrt N/(2\\pi))^s\\Gamma(s)L(f,s)$ and $\\varepsilon_f=(-1)^{k/2}w_N(f)$.", "formula-manin": "At level one, with $P=P_f^+$ and $w=k-2$, $P(X)+X^wP(1-1/X)+(X-1)^wP(-1/(X-1))=0$ CITE{PP}, Section 2. Also $P(1)=P(-1)=0$."}, "Similar tables": [{"table": "HREF{Zeros_of_period_polynomials_of_newforms}[Zeros of period polynomials of newforms]", "relation": "The existing draft uses the same form indices and includes nonrational Hecke fields. Its full-polynomial zeros depend on both parity parts and their relative period scale."}, {"table": "HREF{Special_values_of_the_L-functions_of_level_one_cusp_forms}[Critical L-values of level-one cusp forms]", "relation": "Its odd critical values supply the coefficients of the level-one even period polynomials before primitive normalization."}, {"table": "HREF{Critical_L-values_of_higher-level_newforms}[Critical L-values of higher-level newforms]", "relation": "It gives arithmetic critical values at weights three, four and six, complementing the weight-twelve higher-level slice here."}, {"table": "HREF{Hecke_polynomials_of_level_one_cusp_forms}[Hecke polynomials of level-one cusp forms]", "relation": "Their roots identify the Fourier coefficient that orders the level-one forms."}, {"table": "HREF{Q-expansion_of_the_modular_discriminant}[Fourier coefficients of Delta]", "relation": "They define the form in the first row."}, {"table": "HREF{Petersson_norms_of_level_one_cusp_forms}[Petersson norms of level-one cusp forms]", "relation": "These self-products are related to pairings between the two parity parts with their period scales."}, {"table": "HREF{Petersson_norms_of_higher-level_newforms}[Petersson norms of higher-level newforms]", "relation": "At higher level the period pairing uses all coset components."}], "Links": {"Sage": {"title": "SageMath: exact rational period mappings on modular symbols", "url": "https://doc.sagemath.org/html/en/reference/modsym/sage/modular/modsym/space.html#sage.modular.modsym.space.ModularSymbolsSpace.rational_period_mapping"}, "PARI": {"title": "PARI/GP: period polynomials, newforms and Fricke operators", "url": "https://pari.math.u-bordeaux.fr/dochtml/html-stable/Modular_forms.html"}, "Stein": {"title": "Stein: dimension formulas for modular forms", "url": "https://wstein.org/books/modform/modform/dimension_formulas.html"}}, "References": {"JMOS": {"bib": "S. Jin, W. Ma, K. Ono and K. Soundararajan, Riemann hypothesis for period polynomials of modular forms, Proceedings of the National Academy of Sciences 113 (2016), 2603\u20132608.", "doi": "10.1073/pnas.1600569113", "url": "https://pmc.ncbi.nlm.nih.gov/articles/PMC4790983/"}, "PP": {"bib": "V. Pa\u015fol and A. A. Popa, Modular forms and period polynomials, Proceedings of the London Mathematical Society 107 (2013), 713\u2013743.", "arxiv": "1202.5802", "url": "https://arxiv.org/html/1202.5802"}}, "Programs": {"program-sage": {"language": "SageMath", "code": "import numberdb.sage\nfrom sage.modular.modsym.modsym import ModularSymbols\nfrom sage.rings.rational_field import QQ\nfrom sage.rings.integer_ring import ZZ\nfrom sage.rings.infinity import infinity\nfrom sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing\nfrom math import comb, gcd, lcm\n\n# The unique level-one, weight-twelve form is Delta.\nM = ModularSymbols(1, 12, sign=1)\nA = M.cuspidal_subspace().decomposition()[0]\nperiod = A.rational_period_mapping()\nR = PolynomialRing(QQ, 'X')\nX = R.gen()\np = sum(comb(10,j) * period(M.modular_symbol([j,0,infinity]))[0]\n * X**(10-j) for j in range(0,11,2))\nden = lcm(*(int(c.denominator()) for c in p.list()))\nc = [ZZ(den*a) for a in p.list()]\ng = gcd(*(int(a) for a in c))\nif c[-1] < 0:\n g = -g\nprint(PolynomialRing(ZZ,'X')([a//g for a in c]))"}}, "Display properties": {"number-header": "$P_{f_{N,k,i}}^+(X)$", "group parameters": [["N"], ["k"], ["i"]]}, "Data properties": {"type": "Z[]", "rigour": "exact", "complete": "no", "complete-note": "it holds all 18 rational-Hecke-field newforms in two slices: $N=1$ at every nonempty even weight $k\\leq26$ (six forms, at weights 12, 16, 18, 20, 22 and 26), and $k=12$ at every exact level $2\\leq N\\leq10$ (twelve forms). All their even parts are nonzero. The openly noncanonical bounds cover the first full-level cusp forms and add prime, composite and nonsquarefree level breadth before extending degree beyond 24. Weight twelve permits this level comparison with richer parity polynomials than the lowest weights. Every rational form in the selected spaces is included, retaining its index among all embeddings; level two has no weight-twelve newforms and level seven has no rational ones. The coefficient-field restriction permits exact integral polynomial values. The attached argument-selection rationale explains every index and both bounds", "rigour details": "Computed using exact rational modular symbols CITE{Sage}. The rational period map of each one-dimensional simple cuspidal new factor evaluates the even monomial moments on the path from zero to infinity. Binomial weighting gives a rational polynomial; clearing denominators, dividing by its coefficient gcd and making its leading coefficient positive gives the stored integer polynomial. No rational reconstruction from approximate periods is used.\n\nIndependent full and new dimension formulas CITE{Stein}, Propositions 6.1 and 6.4, agree in all 21 considered spaces, including the empty ones. A separate PARI newform decomposition agrees on all 33 embedded forms through the Sturm bounds, and all 33 indices agree exactly with the full-zero companion roster. For each rational form, Sage and PARI agree on the first 100 Fourier coefficients. The integer eta product for $\\Delta$ controls those coefficients before the comparisons.\n\nPARI period-polynomial computations CITE{PARI} at 160 and 220 decimal working digits agree with every exact coefficient ratio. At 220 working digits the largest absolute ratio error is less than $5.3\\times10^{-228}$. A separate split Mellin sum uses 450 Fourier terms at 150 working digits and 600 at 190; at 190 working digits its largest ratio error is less than $1.7\\times10^{-197}$. These numerical checks corroborate the exact construction; their analytic errors are not claimed to be certified.\n\nPrimitivity, positive leading coefficient, even parity and the reciprocal identity are checked exactly for every polynomial; the level-one three-term relation and roots $\\pm1$ are checked in every level-one row. The published $\\Delta$ parity shape supplies a nonzero normalization control. The attached generator, checks, complete form roster, argument-selection rationale and numerical evidence record the reproducible computation and coordinated skill manifest."}, "Numbers": {"1": {"12": {"1": {"number": "36*X^10 - 691*X^8 + 2073*X^6 - 2073*X^4 + 691*X^2 - 36", "comment": "This is the even period polynomial of HREF{Q-expansion_of_the_modular_discriminant}[the modular discriminant $\\Delta$] CITE{JMOS}, Example 1."}}, "16": {"1": "360*X^14 - 7234*X^12 + 25319*X^10 - 39787*X^8 + 39787*X^6 - 25319*X^4 + 7234*X^2 - 360"}, "18": {"1": "18000*X^16 - 350936*X^14 + 1096675*X^12 - 1140542*X^10 + 1140542*X^6 - 1096675*X^4 + 350936*X^2 - 18000"}, "20": {"1": "26460*X^18 - 523833*X^16 + 1746110*X^14 - 2444554*X^12 + 2269943*X^10 - 2269943*X^8 + 2444554*X^6 - 1746110*X^4 + 523833*X^2 - 26460"}, "22": {"1": "126000*X^20 - 2485856*X^18 + 8156715*X^16 - 10564888*X^14 + 6603055*X^12 - 6603055*X^8 + 10564888*X^6 - 8156715*X^4 + 2485856*X^2 - 126000"}, "26": {"1": "34398000*X^24 - 678984792*X^22 + 2233675745*X^20 - 2938319846*X^18 + 2062613685*X^16 - 850046852*X^14 + 850046852*X^10 - 2062613685*X^8 + 2938319846*X^6 - 2233675745*X^4 + 678984792*X^2 - 34398000"}}, "3": {"12": {"1": "2430*X^10 - 5913*X^8 + 3066*X^6 - 1022*X^4 + 219*X^2 - 10"}}, "4": {"12": {"1": "5120*X^10 - 6144*X^8 + 896*X^6 + 224*X^4 - 96*X^2 + 5"}}, "5": {"12": {"2": "131250*X^10 - 104625*X^8 + 10850*X^6 + 2170*X^4 - 837*X^2 + 42"}}, "6": {"12": {"1": "31104*X^10 - 16200*X^8 + 1350*X^6 - 225*X^4 + 75*X^2 - 4", "2": "272160*X^10 - 143208*X^8 + 9828*X^6 + 1638*X^4 - 663*X^2 + 35", "3": "7776*X^10 - 4536*X^8 + 588*X^6 - 98*X^4 + 21*X^2 - 1"}}, "8": {"12": {"2": "1081344*X^10 - 331776*X^8 + 14336*X^6 + 1792*X^4 - 648*X^2 + 33"}}, "9": {"12": {"1": "590490*X^10 - 129033*X^8 + 3402*X^6 + 378*X^4 - 177*X^2 + 10", "3": "236196*X^10 - 59049*X^8 + 2187*X^6 + 243*X^4 - 81*X^2 + 4"}}, "10": {"12": {"1": "5900000*X^10 - 1113000*X^8 + 28700*X^6 + 2870*X^4 - 1113*X^2 + 59", "2": "600000*X^10 - 117000*X^8 + 4550*X^6 - 455*X^4 + 117*X^2 - 6", "3": "1400000*X^10 - 285000*X^8 + 8750*X^6 + 875*X^4 - 285*X^2 + 14"}}}}