Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [31,8,Mod(1,31)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("31.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 31 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 31.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(9.68393579001\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 538x^{5} + 2328x^{4} + 78000x^{3} - 344224x^{2} - 3123712x + 13256192 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{7} - 3x^{6} - 538x^{5} + 2328x^{4} + 78000x^{3} - 344224x^{2} - 3123712x + 13256192 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -59\nu^{6} - 283\nu^{5} + 26546\nu^{4} + 59520\nu^{3} - 2552464\nu^{2} - 2024288\nu + 8975488 ) / 599040 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -59\nu^{6} - 283\nu^{5} + 26546\nu^{4} + 59520\nu^{3} - 3151504\nu^{2} - 3821408\nu + 102425728 ) / 599040 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -15\nu^{6} - 79\nu^{5} + 8138\nu^{4} + 40064\nu^{3} - 1142096\nu^{2} - 5211360\nu + 35560832 ) / 149760 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -17\nu^{6} + 271\nu^{5} + 13078\nu^{4} - 106240\nu^{3} - 2434992\nu^{2} + 7815136\nu + 95324544 ) / 299520 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 9\nu^{6} + 193\nu^{5} - 5486\nu^{4} - 80240\nu^{3} + 1134704\nu^{2} + 6626208\nu - 58026368 ) / 149760 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} - 3\beta _1 + 156 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{6} - 3\beta_{5} + 10\beta_{4} + 4\beta_{3} - 10\beta_{2} + 241\beta _1 - 404 \)
|
| \(\nu^{4}\) | \(=\) |
\( -60\beta_{6} + 49\beta_{5} - 50\beta_{4} - 412\beta_{3} + 398\beta_{2} - 1647\beta _1 + 37512 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2324\beta_{6} - 955\beta_{5} + 4270\beta_{4} + 2924\beta_{3} - 5298\beta_{2} + 71429\beta _1 - 230096 \)
|
| \(\nu^{6}\) | \(=\) |
\( -34108\beta_{6} + 23601\beta_{5} - 32890\beta_{4} - 152100\beta_{3} + 140982\beta_{2} - 745055\beta _1 + 10977216 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−17.7787 | −70.5395 | 188.081 | 66.6445 | 1254.10 | 686.409 | −1068.16 | 2788.82 | −1184.85 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −15.4504 | 45.8636 | 110.714 | −341.008 | −708.609 | 1239.47 | 267.071 | −83.5342 | 5268.71 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −10.4367 | −22.3960 | −19.0748 | 387.877 | 233.741 | 57.7427 | 1534.98 | −1685.42 | −4048.17 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −6.13721 | 75.6319 | −90.3346 | −111.619 | −464.169 | −1653.58 | 1339.97 | 3533.19 | 685.031 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 6.14320 | −6.66852 | −90.2611 | 156.240 | −40.9660 | 1.41436 | −1340.82 | −2142.53 | 959.813 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 9.42458 | 23.4932 | −39.1772 | −412.782 | 221.414 | −505.664 | −1575.58 | −1635.07 | −3890.30 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 17.2352 | −59.3847 | 169.052 | −175.353 | −1023.51 | −657.787 | 707.545 | 1339.55 | −3022.24 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(31\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 31.8.a.a | ✓ | 7 |
| 3.b | odd | 2 | 1 | 279.8.a.b | 7 | ||
| 4.b | odd | 2 | 1 | 496.8.a.e | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 31.8.a.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 279.8.a.b | 7 | 3.b | odd | 2 | 1 | ||
| 496.8.a.e | 7 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 17T_{2}^{6} - 418T_{2}^{5} - 7248T_{2}^{4} + 38896T_{2}^{3} + 714704T_{2}^{2} - 927328T_{2} - 17556864 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(31))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 17 T^{6} + \cdots - 17556864 \)
|
| $3$ |
\( T^{7} + \cdots - 50982628608 \)
|
| $5$ |
\( T^{7} + \cdots - 11\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots + 38\!\cdots\!84 \)
|
| $11$ |
\( T^{7} + \cdots + 20\!\cdots\!12 \)
|
| $13$ |
\( T^{7} + \cdots + 64\!\cdots\!32 \)
|
| $17$ |
\( T^{7} + \cdots + 11\!\cdots\!36 \)
|
| $19$ |
\( T^{7} + \cdots - 15\!\cdots\!40 \)
|
| $23$ |
\( T^{7} + \cdots - 77\!\cdots\!72 \)
|
| $29$ |
\( T^{7} + \cdots - 89\!\cdots\!40 \)
|
| $31$ |
\( (T - 29791)^{7} \)
|
| $37$ |
\( T^{7} + \cdots + 10\!\cdots\!56 \)
|
| $41$ |
\( T^{7} + \cdots + 15\!\cdots\!92 \)
|
| $43$ |
\( T^{7} + \cdots - 11\!\cdots\!32 \)
|
| $47$ |
\( T^{7} + \cdots + 45\!\cdots\!76 \)
|
| $53$ |
\( T^{7} + \cdots - 18\!\cdots\!28 \)
|
| $59$ |
\( T^{7} + \cdots - 23\!\cdots\!40 \)
|
| $61$ |
\( T^{7} + \cdots - 30\!\cdots\!92 \)
|
| $67$ |
\( T^{7} + \cdots - 13\!\cdots\!36 \)
|
| $71$ |
\( T^{7} + \cdots - 22\!\cdots\!12 \)
|
| $73$ |
\( T^{7} + \cdots + 12\!\cdots\!08 \)
|
| $79$ |
\( T^{7} + \cdots + 28\!\cdots\!80 \)
|
| $83$ |
\( T^{7} + \cdots - 13\!\cdots\!68 \)
|
| $89$ |
\( T^{7} + \cdots + 13\!\cdots\!20 \)
|
| $97$ |
\( T^{7} + \cdots + 11\!\cdots\!04 \)
|
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