Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [31,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("31.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 31 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(15.9661109211\) |
| Analytic rank: | \(0\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{13} - 5 x^{12} - 5242 x^{11} + 18595 x^{10} + 10089855 x^{9} - 16319084 x^{8} - 8654161504 x^{7} + \cdots + 11\!\cdots\!96 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{5}\cdot 5 \) |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - 5 x^{12} - 5242 x^{11} + 18595 x^{10} + 10089855 x^{9} - 16319084 x^{8} - 8654161504 x^{7} + \cdots + 11\!\cdots\!96 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 808 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 17\!\cdots\!43 \nu^{12} + \cdots - 22\!\cdots\!92 ) / 42\!\cdots\!20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 61\!\cdots\!11 \nu^{12} + \cdots - 29\!\cdots\!80 ) / 12\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 36\!\cdots\!61 \nu^{12} + \cdots - 51\!\cdots\!88 ) / 10\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 46\!\cdots\!15 \nu^{12} + \cdots - 64\!\cdots\!16 ) / 12\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 10\!\cdots\!45 \nu^{12} + \cdots + 82\!\cdots\!40 ) / 16\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 31\!\cdots\!09 \nu^{12} + \cdots + 39\!\cdots\!40 ) / 42\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 69\!\cdots\!67 \nu^{12} + \cdots - 90\!\cdots\!48 ) / 85\!\cdots\!64 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 40\!\cdots\!21 \nu^{12} + \cdots - 58\!\cdots\!08 ) / 42\!\cdots\!32 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 45\!\cdots\!97 \nu^{12} + \cdots - 56\!\cdots\!72 ) / 42\!\cdots\!20 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 18\!\cdots\!43 \nu^{12} + \cdots + 21\!\cdots\!16 ) / 12\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 808 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \beta_{7} + 3 \beta_{6} - \beta_{4} + 23 \beta_{3} + \cdots + 1228 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 18 \beta_{12} + 38 \beta_{11} - 24 \beta_{10} - 3 \beta_{9} - 10 \beta_{8} + 26 \beta_{7} + \cdots + 1134219 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 691 \beta_{12} - 2628 \beta_{11} - 3814 \beta_{10} + 4328 \beta_{9} + 840 \beta_{8} + 2100 \beta_{7} + \cdots + 1456450 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 33512 \beta_{12} + 118181 \beta_{11} - 56986 \beta_{10} - 12992 \beta_{9} - 21352 \beta_{8} + \cdots + 1770547370 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1640268 \beta_{12} - 5037106 \beta_{11} - 6225708 \beta_{10} + 7915539 \beta_{9} + 2703066 \beta_{8} + \cdots + 740226961 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 48174333 \beta_{12} + 267671950 \beta_{11} - 103028718 \beta_{10} - 34706212 \beta_{9} + \cdots + 2854404086414 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2954355660 \beta_{12} - 8687146457 \beta_{11} - 9879453238 \beta_{10} + 13830541806 \beta_{9} + \cdots - 2099733009416 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 62889108902 \beta_{12} + 537481648226 \beta_{11} - 169453338280 \beta_{10} - 76550307243 \beta_{9} + \cdots + 46\!\cdots\!23 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 4866345025115 \beta_{12} - 14358952035008 \beta_{11} - 15650502156542 \beta_{10} + \cdots - 94\!\cdots\!74 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 76192052490708 \beta_{12} + \cdots + 77\!\cdots\!82 \)
|
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 |
|
−39.5719 | −144.293 | 1053.93 | 928.453 | 5709.93 | −9223.06 | −21445.3 | 1137.42 | −36740.6 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −39.3570 | 12.5385 | 1036.98 | 1961.05 | −493.477 | 10580.2 | −20661.5 | −19525.8 | −77180.9 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −28.4867 | 39.2249 | 299.492 | −964.404 | −1117.39 | 2039.16 | 6053.64 | −18144.4 | 27472.7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −17.6893 | 263.113 | −199.090 | 2258.82 | −4654.28 | 2957.53 | 12578.7 | 49545.6 | −39956.9 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −8.02282 | −78.0439 | −447.634 | −1906.63 | 626.132 | −6900.57 | 7698.97 | −13592.2 | 15296.6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −6.90725 | 215.702 | −464.290 | −2173.79 | −1489.91 | 5935.19 | 6743.48 | 26844.4 | 15014.9 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −5.21228 | −25.5640 | −484.832 | 2714.21 | 133.247 | −8507.25 | 5195.77 | −19029.5 | −14147.2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 11.8588 | −237.489 | −371.369 | −1260.42 | −2816.34 | −2281.76 | −10475.7 | 36718.3 | −14947.1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 13.5794 | −103.523 | −327.600 | 48.5116 | −1405.78 | 4389.71 | −11401.3 | −8965.99 | 658.758 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 25.8209 | 183.508 | 154.717 | 1221.05 | 4738.33 | 4305.12 | −9225.36 | 13992.1 | 31528.6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 40.3169 | 35.5503 | 1113.45 | 1432.02 | 1433.28 | 2424.14 | 24248.6 | −18419.2 | 57734.4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 41.7733 | 248.324 | 1233.01 | −1644.27 | 10373.3 | −5367.98 | 30118.7 | 41981.8 | −68686.5 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 42.8980 | −255.048 | 1328.24 | −911.595 | −10941.0 | 8758.55 | 35015.2 | 45366.3 | −39105.7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 31.10.a.b | ✓ | 13 |
| 3.b | odd | 2 | 1 | 279.10.a.f | 13 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 31.10.a.b | ✓ | 13 | 1.a | even | 1 | 1 | trivial |
| 279.10.a.f | 13 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{13} - 31 T_{2}^{12} - 4810 T_{2}^{11} + 130311 T_{2}^{10} + 8584955 T_{2}^{9} + \cdots + 68\!\cdots\!04 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(31))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} + \cdots + 68\!\cdots\!04 \)
|
| $3$ |
\( T^{13} + \cdots - 81\!\cdots\!96 \)
|
| $5$ |
\( T^{13} + \cdots - 71\!\cdots\!00 \)
|
| $7$ |
\( T^{13} + \cdots + 10\!\cdots\!44 \)
|
| $11$ |
\( T^{13} + \cdots + 29\!\cdots\!04 \)
|
| $13$ |
\( T^{13} + \cdots + 35\!\cdots\!40 \)
|
| $17$ |
\( T^{13} + \cdots + 10\!\cdots\!96 \)
|
| $19$ |
\( T^{13} + \cdots + 74\!\cdots\!00 \)
|
| $23$ |
\( T^{13} + \cdots - 10\!\cdots\!76 \)
|
| $29$ |
\( T^{13} + \cdots + 17\!\cdots\!00 \)
|
| $31$ |
\( (T - 923521)^{13} \)
|
| $37$ |
\( T^{13} + \cdots + 69\!\cdots\!92 \)
|
| $41$ |
\( T^{13} + \cdots + 15\!\cdots\!76 \)
|
| $43$ |
\( T^{13} + \cdots - 48\!\cdots\!08 \)
|
| $47$ |
\( T^{13} + \cdots + 11\!\cdots\!68 \)
|
| $53$ |
\( T^{13} + \cdots + 17\!\cdots\!84 \)
|
| $59$ |
\( T^{13} + \cdots + 10\!\cdots\!00 \)
|
| $61$ |
\( T^{13} + \cdots + 95\!\cdots\!60 \)
|
| $67$ |
\( T^{13} + \cdots + 40\!\cdots\!84 \)
|
| $71$ |
\( T^{13} + \cdots + 54\!\cdots\!40 \)
|
| $73$ |
\( T^{13} + \cdots - 79\!\cdots\!36 \)
|
| $79$ |
\( T^{13} + \cdots - 40\!\cdots\!00 \)
|
| $83$ |
\( T^{13} + \cdots + 65\!\cdots\!36 \)
|
| $89$ |
\( T^{13} + \cdots + 35\!\cdots\!00 \)
|
| $97$ |
\( T^{13} + \cdots - 27\!\cdots\!92 \)
|
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