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: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 5 x^{19} - 2030 x^{18} + 8100 x^{17} + 1744106 x^{16} - 5171970 x^{15} - 824233578 x^{14} + \cdots + 21\!\cdots\!56 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{10} \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 5 x^{19} - 2030 x^{18} + 8100 x^{17} + 1744106 x^{16} - 5171970 x^{15} - 824233578 x^{14} + \cdots + 21\!\cdots\!56 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 204 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 27\!\cdots\!27 \nu^{19} + \cdots - 10\!\cdots\!84 ) / 13\!\cdots\!64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 27\!\cdots\!27 \nu^{19} + \cdots + 10\!\cdots\!64 ) / 13\!\cdots\!64 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 65\!\cdots\!19 \nu^{19} + \cdots + 23\!\cdots\!84 ) / 26\!\cdots\!28 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 13\!\cdots\!15 \nu^{19} + \cdots + 48\!\cdots\!16 ) / 52\!\cdots\!56 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 91\!\cdots\!09 \nu^{19} + \cdots + 16\!\cdots\!92 ) / 26\!\cdots\!28 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 29\!\cdots\!43 \nu^{19} + \cdots - 11\!\cdots\!72 ) / 75\!\cdots\!08 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 38\!\cdots\!43 \nu^{19} + \cdots - 38\!\cdots\!40 ) / 75\!\cdots\!08 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 16\!\cdots\!00 \nu^{19} + \cdots + 68\!\cdots\!04 ) / 32\!\cdots\!16 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 14\!\cdots\!97 \nu^{19} + \cdots + 58\!\cdots\!92 ) / 26\!\cdots\!28 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 33\!\cdots\!27 \nu^{19} + \cdots + 24\!\cdots\!72 ) / 52\!\cdots\!56 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 48\!\cdots\!27 \nu^{19} + \cdots - 69\!\cdots\!48 ) / 63\!\cdots\!56 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 57\!\cdots\!97 \nu^{19} + \cdots - 19\!\cdots\!56 ) / 52\!\cdots\!56 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 11\!\cdots\!77 \nu^{19} + \cdots + 91\!\cdots\!80 ) / 93\!\cdots\!76 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 19\!\cdots\!05 \nu^{19} + \cdots - 14\!\cdots\!76 ) / 13\!\cdots\!64 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 12\!\cdots\!05 \nu^{19} + \cdots + 55\!\cdots\!56 ) / 75\!\cdots\!08 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 27\!\cdots\!49 \nu^{19} + \cdots + 80\!\cdots\!96 ) / 13\!\cdots\!64 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 32\!\cdots\!57 \nu^{19} + \cdots - 19\!\cdots\!04 ) / 13\!\cdots\!64 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 204 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 319\beta _1 + 234 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{19} + \beta_{18} + \beta_{17} + 3 \beta_{14} + 2 \beta_{12} + 3 \beta_{11} + 4 \beta_{10} + \cdots + 65195 \)
|
| \(\nu^{5}\) | \(=\) |
\( 54 \beta_{19} + 23 \beta_{18} - 9 \beta_{17} + 22 \beta_{16} + 58 \beta_{15} + 5 \beta_{14} + \cdots + 137891 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 1666 \beta_{19} + 146 \beta_{18} + 780 \beta_{17} - 286 \beta_{16} - 392 \beta_{15} + 1706 \beta_{14} + \cdots + 23199690 \)
|
| \(\nu^{7}\) | \(=\) |
\( 42654 \beta_{19} + 18870 \beta_{18} - 9536 \beta_{17} + 21748 \beta_{16} + 49308 \beta_{15} + \cdots + 69731776 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 1016220 \beta_{19} - 126587 \beta_{18} + 442703 \beta_{17} - 209044 \beta_{16} - 382784 \beta_{15} + \cdots + 8733527887 \)
|
| \(\nu^{9}\) | \(=\) |
\( 24342924 \beta_{19} + 10943883 \beta_{18} - 6356815 \beta_{17} + 14603954 \beta_{16} + \cdots + 33122554947 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 544182108 \beta_{19} - 134462974 \beta_{18} + 223244518 \beta_{17} - 109450046 \beta_{16} + \cdots + 3400716558722 \)
|
| \(\nu^{11}\) | \(=\) |
\( 12275052252 \beta_{19} + 5537082198 \beta_{18} - 3513879794 \beta_{17} + 8321781424 \beta_{16} + \cdots + 15164595735236 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 272237627318 \beta_{19} - 88024319683 \beta_{18} + 106027613841 \beta_{17} - 49908478852 \beta_{16} + \cdots + 13\!\cdots\!51 \)
|
| \(\nu^{13}\) | \(=\) |
\( 5817351576210 \beta_{19} + 2607534045859 \beta_{18} - 1766863261513 \beta_{17} + 4337138022126 \beta_{16} + \cdots + 67\!\cdots\!95 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 130880602627462 \beta_{19} - 48816891000246 \beta_{18} + 48609038710712 \beta_{17} + \cdots + 54\!\cdots\!06 \)
|
| \(\nu^{15}\) | \(=\) |
\( 26\!\cdots\!86 \beta_{19} + \cdots + 29\!\cdots\!28 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 61\!\cdots\!80 \beta_{19} + \cdots + 22\!\cdots\!27 \)
|
| \(\nu^{17}\) | \(=\) |
\( 11\!\cdots\!96 \beta_{19} + \cdots + 13\!\cdots\!87 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 28\!\cdots\!16 \beta_{19} + \cdots + 92\!\cdots\!66 \)
|
| \(\nu^{19}\) | \(=\) |
\( 52\!\cdots\!76 \beta_{19} + \cdots + 56\!\cdots\!56 \)
|
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 |
|
−21.7228 | 0 | 343.879 | −94.6859 | 0 | 9.52145 | −4689.48 | 0 | 2056.84 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −19.8910 | 0 | 267.654 | 251.805 | 0 | 1321.81 | −2777.86 | 0 | −5008.66 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −18.8992 | 0 | 229.179 | −133.272 | 0 | −1246.69 | −1912.20 | 0 | 2518.73 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −17.8581 | 0 | 190.912 | −442.906 | 0 | 709.006 | −1123.49 | 0 | 7909.46 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −15.8561 | 0 | 123.416 | −526.730 | 0 | −688.359 | 72.6920 | 0 | 8351.87 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −13.5679 | 0 | 56.0873 | 282.512 | 0 | 1725.92 | 975.703 | 0 | −3833.08 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −9.83316 | 0 | −31.3089 | 293.494 | 0 | −1278.62 | 1566.51 | 0 | −2885.97 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −9.40933 | 0 | −39.4645 | −111.581 | 0 | 495.831 | 1575.73 | 0 | 1049.90 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −2.24637 | 0 | −122.954 | −88.0729 | 0 | −912.978 | 563.734 | 0 | 197.844 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.71238 | 0 | −125.068 | −428.213 | 0 | 1130.68 | 433.347 | 0 | 733.263 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1.39209 | 0 | −126.062 | 490.725 | 0 | −130.215 | 353.678 | 0 | −683.135 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −1.14130 | 0 | −126.697 | 138.226 | 0 | 96.2730 | 290.687 | 0 | −157.758 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 6.17931 | 0 | −89.8162 | 363.961 | 0 | −1297.18 | −1345.95 | 0 | 2249.02 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 11.0947 | 0 | −4.90769 | −390.109 | 0 | −178.013 | −1474.57 | 0 | −4328.15 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 12.9169 | 0 | 38.8474 | −307.859 | 0 | 413.047 | −1151.58 | 0 | −3976.60 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 13.0162 | 0 | 41.4217 | 296.099 | 0 | −1240.98 | −1126.92 | 0 | 3854.08 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 17.8512 | 0 | 190.664 | −401.017 | 0 | 1236.34 | 1118.63 | 0 | −7158.62 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 17.9664 | 0 | 194.792 | 475.104 | 0 | 943.326 | 1200.01 | 0 | 8535.92 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 19.5214 | 0 | 253.086 | 89.3612 | 0 | 1245.27 | 2441.86 | 0 | 1744.46 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.20 | 19.9835 | 0 | 271.341 | −326.840 | 0 | −1314.01 | 2864.47 | 0 | −6531.42 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.f | 20 | |
| 3.b | odd | 2 | 1 | 59.8.a.b | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 59.8.a.b | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 531.8.a.f | 20 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} + 15 T_{2}^{19} - 1935 T_{2}^{18} - 28155 T_{2}^{17} + 1571216 T_{2}^{16} + \cdots + 25\!\cdots\!16 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(531))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 25\!\cdots\!16 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + \cdots - 12\!\cdots\!00 \)
|
| $7$ |
\( T^{20} + \cdots - 24\!\cdots\!96 \)
|
| $11$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $13$ |
\( T^{20} + \cdots + 21\!\cdots\!76 \)
|
| $17$ |
\( T^{20} + \cdots - 13\!\cdots\!96 \)
|
| $19$ |
\( T^{20} + \cdots + 32\!\cdots\!16 \)
|
| $23$ |
\( T^{20} + \cdots - 10\!\cdots\!16 \)
|
| $29$ |
\( T^{20} + \cdots + 49\!\cdots\!24 \)
|
| $31$ |
\( T^{20} + \cdots - 13\!\cdots\!00 \)
|
| $37$ |
\( T^{20} + \cdots + 45\!\cdots\!84 \)
|
| $41$ |
\( T^{20} + \cdots + 13\!\cdots\!46 \)
|
| $43$ |
\( T^{20} + \cdots - 74\!\cdots\!76 \)
|
| $47$ |
\( T^{20} + \cdots - 43\!\cdots\!64 \)
|
| $53$ |
\( T^{20} + \cdots + 15\!\cdots\!96 \)
|
| $59$ |
\( (T - 205379)^{20} \)
|
| $61$ |
\( T^{20} + \cdots - 64\!\cdots\!56 \)
|
| $67$ |
\( T^{20} + \cdots + 49\!\cdots\!96 \)
|
| $71$ |
\( T^{20} + \cdots + 22\!\cdots\!96 \)
|
| $73$ |
\( T^{20} + \cdots - 12\!\cdots\!64 \)
|
| $79$ |
\( T^{20} + \cdots - 23\!\cdots\!04 \)
|
| $83$ |
\( T^{20} + \cdots - 10\!\cdots\!44 \)
|
| $89$ |
\( T^{20} + \cdots + 11\!\cdots\!44 \)
|
| $97$ |
\( T^{20} + \cdots - 21\!\cdots\!16 \)
|
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