Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [531,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("531.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 531 = 3^{2} \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(85.1638083207\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + \cdots - 50564480 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 177) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + \cdots - 50564480 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 49 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 70995163178963 \nu^{11} + \cdots - 44\!\cdots\!16 ) / 37\!\cdots\!24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 6070866593397 \nu^{11} + 12704329665294 \nu^{10} + \cdots + 25\!\cdots\!40 ) / 13\!\cdots\!08 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 276443349871579 \nu^{11} + \cdots + 75\!\cdots\!32 ) / 37\!\cdots\!24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 157574427250607 \nu^{11} - 62814114617374 \nu^{10} + \cdots - 11\!\cdots\!96 ) / 18\!\cdots\!12 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 14329992266831 \nu^{11} + 68921999442182 \nu^{10} + \cdots + 14\!\cdots\!96 ) / 13\!\cdots\!08 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 12\!\cdots\!15 \nu^{11} + 339054598749730 \nu^{10} + \cdots - 55\!\cdots\!44 ) / 37\!\cdots\!24 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13\!\cdots\!07 \nu^{11} + \cdots + 86\!\cdots\!16 ) / 37\!\cdots\!24 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 15\!\cdots\!05 \nu^{11} + \cdots + 41\!\cdots\!08 ) / 37\!\cdots\!24 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 11\!\cdots\!45 \nu^{11} + \cdots - 47\!\cdots\!92 ) / 18\!\cdots\!12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 49 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{11} - 2\beta_{10} - \beta_{8} - \beta_{5} + \beta_{4} + 85\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2 \beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{8} - 4 \beta_{7} - 15 \beta_{6} + 8 \beta_{5} + \cdots + 4204 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 174 \beta_{11} - 274 \beta_{10} - 74 \beta_{9} - 132 \beta_{8} + 16 \beta_{7} - 16 \beta_{6} + \cdots + 1067 \)
|
| \(\nu^{6}\) | \(=\) |
\( 241 \beta_{11} - 256 \beta_{10} - 16 \beta_{9} + 345 \beta_{8} - 700 \beta_{7} - 2934 \beta_{6} + \cdots + 408346 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 23672 \beta_{11} - 33084 \beta_{10} - 15203 \beta_{9} - 16221 \beta_{8} + 2168 \beta_{7} + \cdots + 152254 \)
|
| \(\nu^{8}\) | \(=\) |
\( 21552 \beta_{11} - 33848 \beta_{10} - 17550 \beta_{9} + 59270 \beta_{8} - 94636 \beta_{7} + \cdots + 42181637 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2994209 \beta_{11} - 3901586 \beta_{10} - 2315000 \beta_{9} - 1948793 \beta_{8} + 220864 \beta_{7} + \cdots + 17198384 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1651466 \beta_{11} - 4556482 \beta_{10} - 3545407 \beta_{9} + 8261609 \beta_{8} - 11753444 \beta_{7} + \cdots + 4534829964 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 366962750 \beta_{11} - 457488818 \beta_{10} - 314484546 \beta_{9} - 231380860 \beta_{8} + \cdots + 1913913907 \)
|
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 |
|
−10.7661 | 0 | 83.9098 | 60.5969 | 0 | −233.164 | −558.868 | 0 | −652.394 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.78000 | 0 | 45.0883 | 17.5207 | 0 | 12.9559 | −114.916 | 0 | −153.831 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −7.32600 | 0 | 21.6702 | −46.7845 | 0 | −85.8026 | 75.6759 | 0 | 342.743 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −5.27208 | 0 | −4.20512 | 17.5332 | 0 | 172.711 | 190.876 | 0 | −92.4366 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.65902 | 0 | −29.2477 | −59.9319 | 0 | −87.2478 | 101.611 | 0 | 99.4282 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.501674 | 0 | −31.7483 | 14.0506 | 0 | 117.234 | −31.9809 | 0 | 7.04881 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.62334 | 0 | −29.3648 | −103.513 | 0 | −137.577 | −99.6159 | 0 | −168.037 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.87969 | 0 | −23.7074 | 77.5334 | 0 | 45.2937 | −160.420 | 0 | 223.272 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 4.14510 | 0 | −14.8181 | 34.9684 | 0 | −110.249 | −194.066 | 0 | 144.948 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 8.80731 | 0 | 45.5687 | 88.4760 | 0 | −134.279 | 119.504 | 0 | 779.236 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 8.94324 | 0 | 47.9816 | −48.3356 | 0 | −32.5863 | 142.927 | 0 | −432.277 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 10.9029 | 0 | 86.8728 | −88.1143 | 0 | 61.7114 | 598.272 | 0 | −960.699 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 531.6.a.d | 12 | |
| 3.b | odd | 2 | 1 | 177.6.a.b | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 177.6.a.b | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 531.6.a.d | 12 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 4 T_{2}^{11} - 283 T_{2}^{10} + 1045 T_{2}^{9} + 27968 T_{2}^{8} - 94393 T_{2}^{7} + \cdots - 50564480 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(531))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 4 T^{11} + \cdots - 50564480 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots - 77\!\cdots\!84 \)
|
| $7$ |
\( T^{12} + \cdots - 84\!\cdots\!00 \)
|
| $11$ |
\( T^{12} + \cdots + 13\!\cdots\!80 \)
|
| $13$ |
\( T^{12} + \cdots + 62\!\cdots\!40 \)
|
| $17$ |
\( T^{12} + \cdots + 13\!\cdots\!28 \)
|
| $19$ |
\( T^{12} + \cdots - 24\!\cdots\!00 \)
|
| $23$ |
\( T^{12} + \cdots - 32\!\cdots\!44 \)
|
| $29$ |
\( T^{12} + \cdots - 18\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots - 10\!\cdots\!00 \)
|
| $37$ |
\( T^{12} + \cdots - 16\!\cdots\!04 \)
|
| $41$ |
\( T^{12} + \cdots - 36\!\cdots\!16 \)
|
| $43$ |
\( T^{12} + \cdots + 21\!\cdots\!84 \)
|
| $47$ |
\( T^{12} + \cdots + 12\!\cdots\!96 \)
|
| $53$ |
\( T^{12} + \cdots - 30\!\cdots\!92 \)
|
| $59$ |
\( (T - 3481)^{12} \)
|
| $61$ |
\( T^{12} + \cdots + 18\!\cdots\!60 \)
|
| $67$ |
\( T^{12} + \cdots - 60\!\cdots\!00 \)
|
| $71$ |
\( T^{12} + \cdots + 75\!\cdots\!56 \)
|
| $73$ |
\( T^{12} + \cdots - 88\!\cdots\!64 \)
|
| $79$ |
\( T^{12} + \cdots - 10\!\cdots\!40 \)
|
| $83$ |
\( T^{12} + \cdots - 15\!\cdots\!40 \)
|
| $89$ |
\( T^{12} + \cdots + 33\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots + 66\!\cdots\!40 \)
|
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