Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [531,8,Mod(1,531)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(531, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("531.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 531 = 3^{2} \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 531.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: | \(165.876448532\) |
| Analytic rank: | \(1\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - 5 x^{13} - 1169 x^{12} + 5113 x^{11} + 509966 x^{10} - 1844082 x^{9} - 104172650 x^{8} + \cdots - 143083653176 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 59) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 5 x^{13} - 1169 x^{12} + 5113 x^{11} + 509966 x^{10} - 1844082 x^{9} - 104172650 x^{8} + \cdots - 143083653176 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 168 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 44\!\cdots\!91 \nu^{13} + \cdots + 13\!\cdots\!36 ) / 17\!\cdots\!76 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 94\!\cdots\!13 \nu^{13} + \cdots - 24\!\cdots\!40 ) / 17\!\cdots\!76 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 39\!\cdots\!31 \nu^{13} + \cdots - 54\!\cdots\!36 ) / 17\!\cdots\!76 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 66\!\cdots\!89 \nu^{13} + \cdots - 12\!\cdots\!68 ) / 17\!\cdots\!76 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 67\!\cdots\!33 \nu^{13} + \cdots - 36\!\cdots\!56 ) / 17\!\cdots\!76 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 75\!\cdots\!71 \nu^{13} + \cdots - 13\!\cdots\!36 ) / 17\!\cdots\!76 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 22\!\cdots\!23 \nu^{13} + \cdots + 20\!\cdots\!08 ) / 44\!\cdots\!44 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 10\!\cdots\!55 \nu^{13} + \cdots - 74\!\cdots\!68 ) / 17\!\cdots\!76 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 12\!\cdots\!25 \nu^{13} + \cdots - 38\!\cdots\!60 ) / 17\!\cdots\!76 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 16\!\cdots\!87 \nu^{13} + \cdots + 55\!\cdots\!48 ) / 17\!\cdots\!76 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 25\!\cdots\!09 \nu^{13} + \cdots - 11\!\cdots\!56 ) / 17\!\cdots\!76 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 168 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{10} + \beta_{8} - 3\beta_{7} - 2\beta_{4} - 16\beta_{3} - \beta_{2} + 297\beta _1 + 60 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 14 \beta_{13} - 9 \beta_{12} - 21 \beta_{11} + 8 \beta_{10} + 9 \beta_{9} + 6 \beta_{8} + \cdots + 50279 \)
|
| \(\nu^{5}\) | \(=\) |
\( 70 \beta_{13} + 147 \beta_{12} - 133 \beta_{11} - 1034 \beta_{10} + \beta_{9} + 457 \beta_{8} + \cdots + 23391 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 8900 \beta_{13} - 8804 \beta_{12} - 9908 \beta_{11} + 4496 \beta_{10} + 4240 \beta_{9} + \cdots + 17707160 \)
|
| \(\nu^{7}\) | \(=\) |
\( 53292 \beta_{13} + 88366 \beta_{12} - 86082 \beta_{11} - 443138 \beta_{10} + 23114 \beta_{9} + \cdots + 11623570 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 4391658 \beta_{13} - 5431319 \beta_{12} - 3672843 \beta_{11} + 1803432 \beta_{10} + 1589487 \beta_{9} + \cdots + 6684735793 \)
|
| \(\nu^{9}\) | \(=\) |
\( 29485426 \beta_{13} + 39207577 \beta_{12} - 42925871 \beta_{11} - 181448570 \beta_{10} + \cdots + 5998187573 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 1984576400 \beta_{13} - 2845578842 \beta_{12} - 1243198034 \beta_{11} + 641846176 \beta_{10} + \cdots + 2610417674202 \)
|
| \(\nu^{11}\) | \(=\) |
\( 14454971544 \beta_{13} + 15492444388 \beta_{12} - 19655125884 \beta_{11} - 73529474802 \beta_{10} + \cdots + 2897583067704 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 863853025078 \beta_{13} - 1380963457325 \beta_{12} - 398154438793 \beta_{11} + 214629149880 \beta_{10} + \cdots + 10\!\cdots\!67 \)
|
| \(\nu^{13}\) | \(=\) |
\( 6679729613342 \beta_{13} + 5741064352015 \beta_{12} - 8649426088873 \beta_{11} - 29817592060970 \beta_{10} + \cdots + 13\!\cdots\!39 \)
|
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 |
|
−19.5748 | 0 | 255.172 | 72.4807 | 0 | −64.7916 | −2489.37 | 0 | −1418.79 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −18.0516 | 0 | 197.860 | 418.904 | 0 | −1086.58 | −1261.08 | 0 | −7561.89 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −11.6908 | 0 | 8.67453 | −56.7681 | 0 | −1471.33 | 1395.01 | 0 | 663.664 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −11.6389 | 0 | 7.46330 | −102.369 | 0 | 222.706 | 1402.91 | 0 | 1191.46 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −9.73528 | 0 | −33.2243 | 440.340 | 0 | 384.373 | 1569.56 | 0 | −4286.84 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −3.29939 | 0 | −117.114 | −344.524 | 0 | 1322.34 | 808.726 | 0 | 1136.72 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.520474 | 0 | −127.729 | −477.166 | 0 | −1635.27 | −133.100 | 0 | −248.352 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.04700 | 0 | −126.904 | 45.6259 | 0 | −399.043 | −266.885 | 0 | 47.7705 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 7.20543 | 0 | −76.0817 | 133.759 | 0 | 1302.79 | −1470.50 | 0 | 963.791 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 8.18182 | 0 | −61.0578 | −131.051 | 0 | −311.388 | −1546.84 | 0 | −1072.23 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 10.4062 | 0 | −19.7113 | 351.654 | 0 | 1011.39 | −1537.11 | 0 | 3659.38 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 15.1713 | 0 | 102.167 | −105.628 | 0 | −580.307 | −391.915 | 0 | −1602.51 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 18.9966 | 0 | 232.872 | 324.125 | 0 | −536.491 | 1992.22 | 0 | 6157.28 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 21.4619 | 0 | 332.613 | −139.384 | 0 | −548.392 | 4391.37 | 0 | −2991.44 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(59\) | \(1\) |
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 | 531.8.a.a | 14 | |
| 3.b | odd | 2 | 1 | 59.8.a.a | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 59.8.a.a | ✓ | 14 | 3.b | odd | 2 | 1 | |
| 531.8.a.a | 14 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} - 9 T_{2}^{13} - 1143 T_{2}^{12} + 8941 T_{2}^{11} + 488626 T_{2}^{10} - 3278040 T_{2}^{9} + \cdots + 3193476509696 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(531))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} + \cdots + 3193476509696 \)
|
| $3$ |
\( T^{14} \)
|
| $5$ |
\( T^{14} + \cdots - 17\!\cdots\!00 \)
|
| $7$ |
\( T^{14} + \cdots - 53\!\cdots\!53 \)
|
| $11$ |
\( T^{14} + \cdots + 11\!\cdots\!92 \)
|
| $13$ |
\( T^{14} + \cdots + 60\!\cdots\!96 \)
|
| $17$ |
\( T^{14} + \cdots - 11\!\cdots\!36 \)
|
| $19$ |
\( T^{14} + \cdots - 46\!\cdots\!40 \)
|
| $23$ |
\( T^{14} + \cdots + 18\!\cdots\!92 \)
|
| $29$ |
\( T^{14} + \cdots - 50\!\cdots\!20 \)
|
| $31$ |
\( T^{14} + \cdots - 35\!\cdots\!48 \)
|
| $37$ |
\( T^{14} + \cdots + 22\!\cdots\!24 \)
|
| $41$ |
\( T^{14} + \cdots + 83\!\cdots\!21 \)
|
| $43$ |
\( T^{14} + \cdots + 11\!\cdots\!72 \)
|
| $47$ |
\( T^{14} + \cdots + 18\!\cdots\!72 \)
|
| $53$ |
\( T^{14} + \cdots - 15\!\cdots\!48 \)
|
| $59$ |
\( (T + 205379)^{14} \)
|
| $61$ |
\( T^{14} + \cdots - 25\!\cdots\!64 \)
|
| $67$ |
\( T^{14} + \cdots + 84\!\cdots\!00 \)
|
| $71$ |
\( T^{14} + \cdots + 17\!\cdots\!72 \)
|
| $73$ |
\( T^{14} + \cdots + 60\!\cdots\!12 \)
|
| $79$ |
\( T^{14} + \cdots - 55\!\cdots\!05 \)
|
| $83$ |
\( T^{14} + \cdots - 69\!\cdots\!92 \)
|
| $89$ |
\( T^{14} + \cdots + 15\!\cdots\!80 \)
|
| $97$ |
\( T^{14} + \cdots + 11\!\cdots\!00 \)
|
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