Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6760,2,Mod(1,6760)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6760, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("6760.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6760 = 2^{3} \cdot 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6760.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: | \(53.9788717664\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{12} - 30 x^{10} - 2 x^{9} + 331 x^{8} + 40 x^{7} - 1670 x^{6} - 246 x^{5} + 3988 x^{4} + 482 x^{3} + \cdots + 1807 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 30 x^{10} - 2 x^{9} + 331 x^{8} + 40 x^{7} - 1670 x^{6} - 246 x^{5} + 3988 x^{4} + 482 x^{3} + \cdots + 1807 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 635 \nu^{11} + 83767 \nu^{10} - 67817 \nu^{9} - 2392204 \nu^{8} + 2008142 \nu^{7} + \cdots - 119599178 ) / 3353444 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 21271 \nu^{11} - 35189 \nu^{10} + 635489 \nu^{9} + 1047887 \nu^{8} - 6827256 \nu^{7} + \cdots + 50727730 ) / 1676722 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 38061 \nu^{11} + 104349 \nu^{10} - 1099977 \nu^{9} - 3042683 \nu^{8} + 11180917 \nu^{7} + \cdots - 140942877 ) / 1676722 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 38061 \nu^{11} + 104349 \nu^{10} - 1099977 \nu^{9} - 3042683 \nu^{8} + 11180917 \nu^{7} + \cdots - 132559267 ) / 1676722 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 51059 \nu^{11} - 113949 \nu^{10} + 1455866 \nu^{9} + 3396674 \nu^{8} - 14470519 \nu^{7} + \cdots + 153361262 ) / 1676722 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 104349 \nu^{11} - 41853 \nu^{10} + 2966561 \nu^{9} + 1417274 \nu^{8} - 29856031 \nu^{7} + \cdots + 68776227 ) / 1676722 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 139538 \nu^{11} - 44494 \nu^{10} + 3971906 \nu^{9} + 1630719 \nu^{8} - 40073633 \nu^{7} + \cdots + 107212924 ) / 1676722 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 325811 \nu^{11} - 101984 \nu^{10} - 9250197 \nu^{9} + 2092226 \nu^{8} + 93138255 \nu^{7} + \cdots - 11421115 ) / 3353444 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 360708 \nu^{11} + 263959 \nu^{10} - 10327655 \nu^{9} - 8307241 \nu^{8} + 104716290 \nu^{7} + \cdots - 369082222 ) / 3353444 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 214105 \nu^{11} + 379107 \nu^{10} - 6225710 \nu^{9} - 11296722 \nu^{8} + 64342865 \nu^{7} + \cdots - 526700386 ) / 1676722 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - 2\beta_{2} + 7\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + 2\beta_{8} - 2\beta_{7} - \beta_{6} - 15\beta_{5} + 11\beta_{4} + 2\beta_{3} + 38 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} + 15 \beta_{7} + 13 \beta_{6} - 14 \beta_{5} + \cdots + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 19 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} + 38 \beta_{8} - 36 \beta_{7} - 17 \beta_{6} - 179 \beta_{5} + \cdots + 341 \)
|
| \(\nu^{7}\) | \(=\) |
\( 174 \beta_{11} + 34 \beta_{10} + 34 \beta_{9} + 57 \beta_{8} + 175 \beta_{7} + 153 \beta_{6} - 182 \beta_{5} + \cdots + 99 \)
|
| \(\nu^{8}\) | \(=\) |
\( 265 \beta_{11} - 88 \beta_{10} + 96 \beta_{9} + 552 \beta_{8} - 502 \beta_{7} - 209 \beta_{6} + \cdots + 3290 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2089 \beta_{11} + 418 \beta_{10} + 434 \beta_{9} + 801 \beta_{8} + 1876 \beta_{7} + 1766 \beta_{6} + \cdots + 860 \)
|
| \(\nu^{10}\) | \(=\) |
\( 3334 \beta_{11} - 1364 \beta_{10} + 1576 \beta_{9} + 7256 \beta_{8} - 6418 \beta_{7} - 2284 \beta_{6} + \cdots + 33042 \)
|
| \(\nu^{11}\) | \(=\) |
\( 24593 \beta_{11} + 4572 \beta_{10} + 5088 \beta_{9} + 10196 \beta_{8} + 19393 \beta_{7} + 20209 \beta_{6} + \cdots + 7663 \)
|
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 |
|
0 | −3.37017 | 0 | 1.00000 | 0 | −0.137901 | 0 | 8.35804 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.99569 | 0 | 1.00000 | 0 | 3.99418 | 0 | 5.97417 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.43833 | 0 | 1.00000 | 0 | −0.670431 | 0 | 2.94547 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.34467 | 0 | 1.00000 | 0 | −4.22858 | 0 | −1.19187 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.12058 | 0 | 1.00000 | 0 | 4.65434 | 0 | −1.74429 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.07031 | 0 | 1.00000 | 0 | 0.924154 | 0 | −1.85443 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.04943 | 0 | 1.00000 | 0 | 1.46925 | 0 | −1.89869 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.35049 | 0 | 1.00000 | 0 | −3.90896 | 0 | −1.17618 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.50558 | 0 | 1.00000 | 0 | −1.83468 | 0 | −0.733222 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.20843 | 0 | 1.00000 | 0 | 2.37010 | 0 | 1.87717 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.93490 | 0 | 1.00000 | 0 | 4.85036 | 0 | 5.61365 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 3.29092 | 0 | 1.00000 | 0 | −0.481821 | 0 | 7.83018 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(13\) | \(1\) |
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 | 6760.2.a.bp | yes | 12 |
| 13.b | even | 2 | 1 | 6760.2.a.bo | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6760.2.a.bo | ✓ | 12 | 13.b | even | 2 | 1 | |
| 6760.2.a.bp | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6760))\):
|
\( T_{3}^{12} - 30 T_{3}^{10} + 2 T_{3}^{9} + 331 T_{3}^{8} - 40 T_{3}^{7} - 1670 T_{3}^{6} + 246 T_{3}^{5} + \cdots + 1807 \)
|
|
\( T_{7}^{12} - 7 T_{7}^{11} - 29 T_{7}^{10} + 266 T_{7}^{9} + 110 T_{7}^{8} - 3059 T_{7}^{7} + 2031 T_{7}^{6} + \cdots + 392 \)
|
|
\( T_{11}^{12} - 14 T_{11}^{11} + 13 T_{11}^{10} + 582 T_{11}^{9} - 2002 T_{11}^{8} - 6772 T_{11}^{7} + \cdots + 129856 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} - 30 T^{10} + \cdots + 1807 \)
|
| $5$ |
\( (T - 1)^{12} \)
|
| $7$ |
\( T^{12} - 7 T^{11} + \cdots + 392 \)
|
| $11$ |
\( T^{12} - 14 T^{11} + \cdots + 129856 \)
|
| $13$ |
\( T^{12} \)
|
| $17$ |
\( T^{12} + 2 T^{11} + \cdots + 215552 \)
|
| $19$ |
\( T^{12} - 5 T^{11} + \cdots - 2648512 \)
|
| $23$ |
\( T^{12} + 10 T^{11} + \cdots + 30366568 \)
|
| $29$ |
\( T^{12} - 12 T^{11} + \cdots + 52839704 \)
|
| $31$ |
\( T^{12} - 5 T^{11} + \cdots + 272896 \)
|
| $37$ |
\( T^{12} + \cdots - 239453696 \)
|
| $41$ |
\( T^{12} + 3 T^{11} + \cdots - 66197111 \)
|
| $43$ |
\( T^{12} - 2 T^{11} + \cdots - 24181793 \)
|
| $47$ |
\( T^{12} - 10 T^{11} + \cdots + 61490584 \)
|
| $53$ |
\( T^{12} + \cdots - 119842304 \)
|
| $59$ |
\( T^{12} + \cdots + 11790900736 \)
|
| $61$ |
\( T^{12} - 65 T^{11} + \cdots - 19246424 \)
|
| $67$ |
\( T^{12} - 8 T^{11} + \cdots + 23657341 \)
|
| $71$ |
\( T^{12} + \cdots + 1944557056 \)
|
| $73$ |
\( T^{12} - 9 T^{11} + \cdots - 727552 \)
|
| $79$ |
\( T^{12} + \cdots - 208272896 \)
|
| $83$ |
\( T^{12} + \cdots + 2815821091 \)
|
| $89$ |
\( T^{12} + \cdots + 5994828671 \)
|
| $97$ |
\( T^{12} + \cdots + 1588780544 \)
|
| show more | |
| show less |