Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,8,Mod(1,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("315.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 315.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: | \(98.4012830275\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 832x^{6} + 1124x^{5} + 214693x^{4} - 78223x^{3} - 18019982x^{2} - 2181234x + 146417544 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 832x^{6} + 1124x^{5} + 214693x^{4} - 78223x^{3} - 18019982x^{2} - 2181234x + 146417544 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 208 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu^{2} - 313\nu + 645 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 639 \nu^{7} + 32908 \nu^{6} + 417668 \nu^{5} - 21685840 \nu^{4} - 95289051 \nu^{3} + \cdots - 64106712280 ) / 10948192 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1555 \nu^{7} - 11548 \nu^{6} - 1239124 \nu^{5} + 5852672 \nu^{4} + 304847183 \nu^{3} + \cdots + 710236072 ) / 5474096 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4171 \nu^{7} - 82020 \nu^{6} - 2473564 \nu^{5} + 44144648 \nu^{4} + 381192495 \nu^{3} + \cdots + 12880491576 ) / 10948192 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 556 \nu^{7} + 3469 \nu^{6} + 359669 \nu^{5} - 1512467 \nu^{4} - 57142863 \nu^{3} + \cdots - 2058185000 ) / 1368524 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 208 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_{2} + 323\beta _1 + 395 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{7} - 2\beta_{6} - \beta_{5} - 4\beta_{4} + 7\beta_{3} + 459\beta_{2} + 1591\beta _1 + 67010 \)
|
| \(\nu^{5}\) | \(=\) |
\( -19\beta_{7} - 3\beta_{6} - 25\beta_{5} - 9\beta_{4} + 625\beta_{3} + 3838\beta_{2} + 121318\beta _1 + 320678 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 1431 \beta_{7} - 1379 \beta_{6} - 686 \beta_{5} - 2379 \beta_{4} + 5759 \beta_{3} + 207676 \beta_{2} + \cdots + 25111157 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 18240 \beta_{7} - 5104 \beta_{6} - 17732 \beta_{5} - 9784 \beta_{4} + 318419 \beta_{3} + \cdots + 200451213 \)
|
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 |
|
−21.2018 | 0 | 321.518 | −125.000 | 0 | 343.000 | −4102.93 | 0 | 2650.23 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −14.7627 | 0 | 89.9380 | −125.000 | 0 | 343.000 | 561.898 | 0 | 1845.34 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −11.3623 | 0 | 1.10101 | −125.000 | 0 | 343.000 | 1441.86 | 0 | 1420.28 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.91759 | 0 | −124.323 | −125.000 | 0 | 343.000 | 483.851 | 0 | 239.698 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 4.11340 | 0 | −111.080 | −125.000 | 0 | 343.000 | −983.432 | 0 | −514.175 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 13.9246 | 0 | 65.8934 | −125.000 | 0 | 343.000 | −864.807 | 0 | −1740.57 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 16.0418 | 0 | 129.338 | −125.000 | 0 | 343.000 | 21.4631 | 0 | −2005.22 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 20.1647 | 0 | 278.615 | −125.000 | 0 | 343.000 | 3037.10 | 0 | −2520.59 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 315.8.a.q | yes | 8 |
| 3.b | odd | 2 | 1 | 315.8.a.p | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 315.8.a.p | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 315.8.a.q | yes | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 5 T_{2}^{7} - 825 T_{2}^{6} + 3875 T_{2}^{5} + 207798 T_{2}^{4} - 775100 T_{2}^{3} + \cdots + 126353088 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 5 T^{7} + \cdots + 126353088 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T + 125)^{8} \)
|
| $7$ |
\( (T - 343)^{8} \)
|
| $11$ |
\( T^{8} + \cdots + 55\!\cdots\!00 \)
|
| $13$ |
\( T^{8} + \cdots - 19\!\cdots\!56 \)
|
| $17$ |
\( T^{8} + \cdots + 38\!\cdots\!52 \)
|
| $19$ |
\( T^{8} + \cdots + 94\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 33\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots + 78\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 22\!\cdots\!24 \)
|
| $37$ |
\( T^{8} + \cdots - 10\!\cdots\!84 \)
|
| $41$ |
\( T^{8} + \cdots + 21\!\cdots\!32 \)
|
| $43$ |
\( T^{8} + \cdots + 11\!\cdots\!44 \)
|
| $47$ |
\( T^{8} + \cdots - 25\!\cdots\!88 \)
|
| $53$ |
\( T^{8} + \cdots + 34\!\cdots\!08 \)
|
| $59$ |
\( T^{8} + \cdots - 29\!\cdots\!68 \)
|
| $61$ |
\( T^{8} + \cdots + 65\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots + 93\!\cdots\!36 \)
|
| $71$ |
\( T^{8} + \cdots - 25\!\cdots\!48 \)
|
| $73$ |
\( T^{8} + \cdots - 76\!\cdots\!96 \)
|
| $79$ |
\( T^{8} + \cdots + 91\!\cdots\!84 \)
|
| $83$ |
\( T^{8} + \cdots + 11\!\cdots\!00 \)
|
| $89$ |
\( T^{8} + \cdots - 99\!\cdots\!68 \)
|
| $97$ |
\( T^{8} + \cdots + 51\!\cdots\!24 \)
|
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