Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8750,2,Mod(1,8750)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8750, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8750.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8750 = 2 \cdot 5^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8750.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: | \(69.8691017686\) |
| 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: |
\( x^{12} - 2 x^{11} - 18 x^{10} + 44 x^{9} + 55 x^{8} - 178 x^{7} - 13 x^{6} + 202 x^{5} - 25 x^{4} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | no (minimal twist has level 350) |
| 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
\( x^{12} - 2 x^{11} - 18 x^{10} + 44 x^{9} + 55 x^{8} - 178 x^{7} - 13 x^{6} + 202 x^{5} - 25 x^{4} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 25892 \nu^{11} - 38618 \nu^{10} - 443323 \nu^{9} + 833189 \nu^{8} + 1117861 \nu^{7} - 2298013 \nu^{6} + \cdots - 552240 ) / 161711 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 63243 \nu^{11} + 208709 \nu^{10} + 1013802 \nu^{9} - 4278484 \nu^{8} - 628953 \nu^{7} + \cdots - 335778 ) / 161711 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 23940 \nu^{11} + 55881 \nu^{10} + 414836 \nu^{9} - 1195407 \nu^{8} - 966276 \nu^{7} + \cdots - 120712 ) / 14701 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 287967 \nu^{11} + 652484 \nu^{10} + 5067008 \nu^{9} - 14035312 \nu^{8} - 13175555 \nu^{7} + \cdots - 953891 ) / 161711 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 338185 \nu^{11} + 814348 \nu^{10} + 5819654 \nu^{9} - 17310340 \nu^{8} - 12750657 \nu^{7} + \cdots - 1778775 ) / 161711 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 387690 \nu^{11} + 799335 \nu^{10} + 6873952 \nu^{9} - 17451386 \nu^{8} - 19208396 \nu^{7} + \cdots - 1780932 ) / 161711 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 402001 \nu^{11} - 829761 \nu^{10} - 7088188 \nu^{9} + 18045430 \nu^{8} + 19179059 \nu^{7} + \cdots + 1499057 ) / 161711 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 49368 \nu^{11} - 103784 \nu^{10} - 871270 \nu^{9} + 2256405 \nu^{8} + 2358633 \nu^{7} + \cdots + 269722 ) / 14701 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 567003 \nu^{11} + 1246092 \nu^{10} + 9976996 \nu^{9} - 26935009 \nu^{8} - 26168836 \nu^{7} + \cdots - 3192921 ) / 161711 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 652847 \nu^{11} - 1400510 \nu^{10} - 11456111 \nu^{9} + 30272387 \nu^{8} + 29819978 \nu^{7} + \cdots + 2576283 ) / 161711 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 682483 \nu^{11} + 1552786 \nu^{10} + 11939131 \nu^{9} - 33361875 \nu^{8} - 29878743 \nu^{7} + \cdots - 3348149 ) / 161711 \)
|
| \(\nu\) | \(=\) |
\( ( - \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \cdots + 1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} - 8 \beta_{8} + 4 \beta_{7} - 11 \beta_{6} - 10 \beta_{5} + \cdots + 19 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 3 \beta_{11} - 2 \beta_{10} + 5 \beta_{9} + 8 \beta_{8} + 3 \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 104 \beta_{11} - 17 \beta_{10} - 38 \beta_{9} - 122 \beta_{8} + 51 \beta_{7} - 134 \beta_{6} + \cdots + 191 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 244 \beta_{11} - 83 \beta_{10} + 308 \beta_{9} + 582 \beta_{8} - 66 \beta_{7} + 279 \beta_{6} + \cdots - 441 ) / 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( 308 \beta_{11} - 19 \beta_{10} - 134 \beta_{9} - 378 \beta_{8} + 136 \beta_{7} - 360 \beta_{6} + \cdots + 506 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 4021 \beta_{11} - 867 \beta_{10} + 4197 \beta_{9} + 8593 \beta_{8} - 1414 \beta_{7} + 4736 \beta_{6} + \cdots - 7424 ) / 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 22306 \beta_{11} - 368 \beta_{10} - 11172 \beta_{9} - 29318 \beta_{8} + 9429 \beta_{7} - 25501 \beta_{6} + \cdots + 36199 ) / 5 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 13004 \beta_{11} - 2042 \beta_{10} + 11952 \beta_{9} + 25529 \beta_{8} - 4913 \beta_{7} + \cdots - 23613 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 324714 \beta_{11} + 3143 \beta_{10} - 179998 \beta_{9} - 452612 \beta_{8} + 134481 \beta_{7} + \cdots + 531516 ) / 5 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 1031219 \beta_{11} - 128643 \beta_{10} + 869308 \beta_{9} + 1905942 \beta_{8} - 400136 \beta_{7} + \cdots - 1839646 ) / 5 \)
|
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 |
|
1.00000 | −3.38121 | 1.00000 | 0 | −3.38121 | −1.00000 | 1.00000 | 8.43257 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.41090 | 1.00000 | 0 | −2.41090 | −1.00000 | 1.00000 | 2.81243 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −2.14996 | 1.00000 | 0 | −2.14996 | −1.00000 | 1.00000 | 1.62233 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.94863 | 1.00000 | 0 | −1.94863 | −1.00000 | 1.00000 | 0.797146 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −1.42985 | 1.00000 | 0 | −1.42985 | −1.00000 | 1.00000 | −0.955535 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −0.348449 | 1.00000 | 0 | −0.348449 | −1.00000 | 1.00000 | −2.87858 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | −0.205506 | 1.00000 | 0 | −0.205506 | −1.00000 | 1.00000 | −2.95777 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 0.527915 | 1.00000 | 0 | 0.527915 | −1.00000 | 1.00000 | −2.72131 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 1.55409 | 1.00000 | 0 | 1.55409 | −1.00000 | 1.00000 | −0.584813 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | 1.65875 | 1.00000 | 0 | 1.65875 | −1.00000 | 1.00000 | −0.248565 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.00000 | 1.80405 | 1.00000 | 0 | 1.80405 | −1.00000 | 1.00000 | 0.254581 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.00000 | 2.32970 | 1.00000 | 0 | 2.32970 | −1.00000 | 1.00000 | 2.42751 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
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 | 8750.2.a.bb | 12 | |
| 5.b | even | 2 | 1 | 8750.2.a.z | 12 | ||
| 25.f | odd | 20 | 2 | 350.2.m.a | ✓ | 24 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.2.m.a | ✓ | 24 | 25.f | odd | 20 | 2 | |
| 8750.2.a.z | 12 | 5.b | even | 2 | 1 | ||
| 8750.2.a.bb | 12 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8750))\):
|
\( T_{3}^{12} + 4 T_{3}^{11} - 13 T_{3}^{10} - 58 T_{3}^{9} + 60 T_{3}^{8} + 308 T_{3}^{7} - 118 T_{3}^{6} + \cdots - 20 \)
|
|
\( T_{11}^{12} + 4 T_{11}^{11} - 39 T_{11}^{10} - 116 T_{11}^{9} + 506 T_{11}^{8} + 1350 T_{11}^{7} + \cdots - 8100 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{12} \)
|
| $3$ |
\( T^{12} + 4 T^{11} + \cdots - 20 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( (T + 1)^{12} \)
|
| $11$ |
\( T^{12} + 4 T^{11} + \cdots - 8100 \)
|
| $13$ |
\( T^{12} + 2 T^{11} + \cdots - 919 \)
|
| $17$ |
\( T^{12} - 72 T^{10} + \cdots - 359 \)
|
| $19$ |
\( T^{12} + 24 T^{11} + \cdots + 380 \)
|
| $23$ |
\( T^{12} + 10 T^{11} + \cdots + 197500 \)
|
| $29$ |
\( T^{12} + 4 T^{11} + \cdots - 90184579 \)
|
| $31$ |
\( T^{12} + 30 T^{11} + \cdots - 383924 \)
|
| $37$ |
\( T^{12} - 14 T^{11} + \cdots - 510395 \)
|
| $41$ |
\( T^{12} + 24 T^{11} + \cdots - 41916139 \)
|
| $43$ |
\( T^{12} - 12 T^{11} + \cdots - 70667924 \)
|
| $47$ |
\( T^{12} + 2 T^{11} + \cdots + 826816 \)
|
| $53$ |
\( T^{12} + 6 T^{11} + \cdots - 39539 \)
|
| $59$ |
\( T^{12} + \cdots - 1011066964 \)
|
| $61$ |
\( T^{12} + \cdots + 13816135661 \)
|
| $67$ |
\( T^{12} + \cdots - 33983959124 \)
|
| $71$ |
\( T^{12} + 14 T^{11} + \cdots - 29857600 \)
|
| $73$ |
\( T^{12} + \cdots + 3240797321 \)
|
| $79$ |
\( T^{12} + \cdots - 301300564 \)
|
| $83$ |
\( T^{12} + \cdots + 24391953856 \)
|
| $89$ |
\( T^{12} + \cdots + 4036368605 \)
|
| $97$ |
\( T^{12} + \cdots + 608838525 \)
|
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