Odd period polynomials of rational newforms

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Polynomials

$N$

$k$

$i$

$P_{f_{N,k,i}}^-(X)$

4*X^9 - 25*X^7 + 42*X^5 - 25*X^3 + 4*X

comment: This is the odd period polynomial of the modular discriminant $\Delta$ [1], Example 1.

36*X^13 - 245*X^11 + 539*X^9 - 660*X^7 + 539*X^5 - 245*X^3 + 36*X

24*X^15 - 154*X^13 + 273*X^11 - 143*X^9 - 143*X^7 + 273*X^5 - 154*X^3 + 24*X

108*X^17 - 715*X^15 + 1452*X^13 - 1560*X^11 + 1430*X^9 - 1560*X^7 + 1452*X^5 - 715*X^3 + 108*X

360*X^19 - 2366*X^17 + 4641*X^15 - 4182*X^13 + 1547*X^11 + 1547*X^9 - 4182*X^7 + 4641*X^5 - 2366*X^3 + 360*X

1080*X^23 - 7106*X^21 + 14025*X^19 - 13167*X^17 + 7106*X^15 - 1938*X^13 - 1938*X^11 + 7106*X^9 - 13167*X^7 + 14025*X^5 - 7106*X^3 + 1080*X

81*X^9 - 72*X^7 + 28*X^5 - 8*X^3 + X

256*X^9 - 96*X^7 + 6*X^3 - X

3125*X^9 - 800*X^7 + 32*X^3 - 5*X

6480*X^9 - 1080*X^7 + 81*X^5 - 30*X^3 + 5*X

1296*X^9 - 216*X^7 + 6*X^3 - X

6480*X^9 - 1368*X^7 + 168*X^5 - 38*X^3 + 5*X

20480*X^9 - 2048*X^7 + 32*X^3 - 5*X

19683*X^9 - 1296*X^7 + 16*X^3 - 3*X

26244*X^9 - 2187*X^7 + 27*X^3 - 4*X

10000*X^9 - 600*X^7 + 6*X^3 - X

370000*X^9 - 24700*X^7 + 1050*X^5 - 247*X^3 + 37*X

110000*X^9 - 7500*X^7 + 75*X^3 - 11*X

Definition

For a normalized rational-coefficient newform $f=f_{N,k,i}$ with trivial character, $P_f^-(X)$ is the primitive polynomial in $\mathbb Z[X]$ proportional to its nonzero odd period polynomial, with positive leading coefficient [2].

Parameters

$N$

— exact level ($N\geq1$)

$k$

— weight ($k\geq4$ is even and $S_k^{\mathrm{new}}(\Gamma_0(N))$ is nonzero)

$i$

— embedded newform ($1\leq i\leq\dim S_k^{\mathrm{new}}(\Gamma_0(N))$ in lexicographic order of all real embedded Fourier-coefficient sequences $(a_2,a_3,\ldots)$; the form must have rational Hecke field and nonzero odd period polynomial)

Formulas

(1)

The identity-coset period polynomial is $r_f(X)=\int_0^{\mathrm{i}\infty}f(\tau)(\tau-X)^{k-2}\,d\tau$, and its odd part is $r_f^-(X)=(r_f(X)-r_f(-X))/2$ [1].

(2)

For $w=k-2$, $r_f^-(X)=\sum_{\substack{1\leq m\leq w-1\\m\text{ odd}}}\binom wm\mathrm{i}^{1-m}\frac{m!}{(2\pi)^{m+1}}L(f,m+1)X^{w-m}$. Here $L(f,s)$ is the uncompleted arithmetic L-function; no conductor factor occurs [1], equations (1.1)–(1.4). The level-one critical-value table contains the even critical values for the level-one rows. The higher-level critical-value table records the same arithmetic normalization at other weights.

(3)

$P_f^-(X)=w_N(f)N^{(k-2)/2}X^{k-2}P_f^-(1/(NX))$, where $w_N(f)\in\{-1,1\}$ is fixed by $f(-1/(N\tau))=w_N(f)N^{k/2}\tau^k f(\tau)$ [4]. 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 $\varepsilon_f=(-1)^{k/2}w_N(f)$.

(4)

At level one, with $P=P_f^-$ and $w=k-2$, $P(X)+X^wP(1-1/X)+(X-1)^wP(-1/(X-1))=0$ [2], Section 2. Also $P(1)=P(-1)=0$.

Comments

(5)

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 odd powers of $X$, independently of the functional-equation sign. Rationality up to a nonzero period follows from [2], Proposition 5.11. Clear denominators, divide by the positive gcd of the coefficients, and fix the leading sign. All forced factors, including $X$, are retained; $X$ is not rescaled, inverted, or replaced by $X^2$.

(6)

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 the full period-polynomial zero table, which is currently a draft. At level one it agrees with the increasing-$a_2$ convention of the critical-value table; the Hecke polynomials identify these $a_2$ values.

(7)

The primitive even period polynomials form a separate companion table, currently a draft. Neither primitive parity shape retains the relative transcendental scale between the two parts, so the two shapes alone do not recover the full period-polynomial zeros. Only the identity-coset polynomial is stored. At higher level, the complete period data also has other coset components [2], Section 2.

(8)

Pairings of period polynomials are related to level-one Petersson self-products and higher-level Petersson self-products [2], Section 3. At higher level the pairing requires the full coset-valued data; this primitive odd component alone does not determine the norm.

(9)

The form $(N,k,i)=(1,12,1)$ is the modular discriminant $\Delta$, whose Fourier coefficients are the Ramanujan tau values. Its primitive odd shape agrees with [1], Example 1.

Programs

(P1)

SageMath

import numberdb.sage
from sage.modular.modsym.modsym import ModularSymbols
from sage.rings.rational_field import QQ
from sage.rings.integer_ring import ZZ
from sage.rings.infinity import infinity
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from math import comb, gcd, lcm

# The unique level-one, weight-twelve form is Delta.
M = ModularSymbols(1, 12, sign=-1)
A = M.cuspidal_subspace().decomposition()[0]
period = A.rational_period_mapping()
R = PolynomialRing(QQ, 'X')
X = R.gen()
p = sum(comb(10,j) * period(M.modular_symbol([j,0,infinity]))[0]
        * X**(10-j) for j in range(1,10,2))
den = lcm(*(int(c.denominator()) for c in p.list()))
c = [ZZ(den*a) for a in p.list()]
g = gcd(*(int(a) for a in c))
if c[-1] < 0:
    g = -g
print(PolynomialRing(ZZ,'X')([a//g for a in c]))

References

[1]

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), 2603–2608. (doi) https://pmc.ncbi.nlm.nih.gov/articles/PMC4790983/

[2]

V. Paşol and A. A. Popa, Modular forms and period polynomials, Proceedings of the London Mathematical Society 107 (2013), 713–743. (arXiv) https://arxiv.org/html/1202.5802

Links

[3]

SageMath: exact rational period mappings on modular symbols

[4]

PARI/GP: period polynomials, newforms and Fricke operators

[5]

Stein: dimension formulas for modular forms

Similar tables

Even period polynomials of rational newforms — The existing draft stores the primitive even shapes for the same rational forms and indices. The relative period scale between the parity parts is omitted from both tables.

Zeros of period polynomials of newforms — 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.

Critical L-values of level-one cusp forms — Its even critical values supply the coefficients of the level-one odd period polynomials before primitive normalization.

Critical L-values of higher-level newforms — It gives arithmetic critical values at weights three, four and six, complementing the weight-twelve higher-level slice here.

Hecke polynomials of level-one cusp forms — Their roots identify the Fourier coefficient that orders the level-one forms.

Fourier coefficients of Delta — They are the Fourier coefficients of $f_{1,12,1}=\Delta$.

Petersson norms of level-one cusp forms — These self-products are related to pairings between the two parity parts with their period scales.

Petersson norms of higher-level newforms — At higher level the period pairing uses all coset components.

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (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 odd parts are nonzero. The openly noncanonical bounds cover the first full-level cusp forms and add prime, composite and nonsquarefree level breadth before extending weight beyond 26. 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)

How they were obtained:

Computed using exact rational modular symbols [3]. The rational period map of each one-dimensional simple cuspidal new factor in the minus sign space evaluates the odd 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.

Independent full and new dimension formulas [5], 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.

PARI period-polynomial computations [4] 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 $6.6\times10^{-229}$. 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 $1.7\times10^{-198}$. Three nonzero exponential-integral controls test the implemented Mellin weights first. These numerical checks corroborate the exact construction; their analytic errors are not claimed to be certified.

Primitivity, positive leading coefficient, odd 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.