Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1682,4,Mod(1,1682)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1682, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1682.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1682 = 2 \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1682.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: | \(99.2412126297\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{8} - x^{7} - 148x^{6} + 84x^{5} + 6219x^{4} - 1629x^{3} - 53561x^{2} + 49748x + 34571 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 148x^{6} + 84x^{5} + 6219x^{4} - 1629x^{3} - 53561x^{2} + 49748x + 34571 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1723 \nu^{7} + 3602 \nu^{6} - 234110 \nu^{5} - 508406 \nu^{4} + 8597423 \nu^{3} + 18084554 \nu^{2} + \cdots - 38527455 ) / 7967232 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3981 \nu^{7} - 5090 \nu^{6} - 578770 \nu^{5} + 135526 \nu^{4} + 22834297 \nu^{3} + 18149030 \nu^{2} + \cdots - 77507785 ) / 7967232 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 5125 \nu^{7} - 6766 \nu^{6} + 621826 \nu^{5} + 312714 \nu^{4} - 21403153 \nu^{3} + \cdots - 259587103 ) / 7967232 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5325 \nu^{7} + 20894 \nu^{6} - 653138 \nu^{5} - 2117914 \nu^{4} + 20891321 \nu^{3} + \cdots - 59565833 ) / 7967232 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 14667 \nu^{7} - 10354 \nu^{6} + 2264350 \nu^{5} + 3050774 \nu^{4} - 94939135 \nu^{3} + \cdots + 318642991 ) / 7967232 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 29677 \nu^{7} + 24670 \nu^{6} - 4309138 \nu^{5} - 5406042 \nu^{4} + 170096473 \nu^{3} + \cdots - 268997033 ) / 7967232 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 38 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{7} - 4\beta_{6} - 4\beta_{5} - 4\beta_{4} + \beta_{3} + 33\beta_{2} + 65\beta _1 + 34 \)
|
| \(\nu^{4}\) | \(=\) |
\( -70\beta_{7} - 76\beta_{6} + 98\beta_{5} - \beta_{4} + 151\beta_{3} - 96\beta_{2} + 195\beta _1 + 2381 \)
|
| \(\nu^{5}\) | \(=\) |
\( -341\beta_{7} - 284\beta_{6} - 396\beta_{5} - 319\beta_{4} + 138\beta_{3} + 3412\beta_{2} + 4485\beta _1 + 4333 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 4999 \beta_{7} - 5746 \beta_{6} + 8043 \beta_{5} - 83 \beta_{4} + 10464 \beta_{3} - 12091 \beta_{2} + \cdots + 161782 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 26082 \beta_{7} - 18546 \beta_{6} - 32240 \beta_{5} - 23506 \beta_{4} + 15449 \beta_{3} + \cdots + 406951 \)
|
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 |
|
2.00000 | −10.1475 | 4.00000 | 19.5630 | −20.2950 | −0.331021 | 8.00000 | 75.9718 | 39.1261 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −7.96606 | 4.00000 | −17.5061 | −15.9321 | 22.9272 | 8.00000 | 36.4582 | −35.0122 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.57586 | 4.00000 | −14.8075 | −7.15173 | 23.8268 | 8.00000 | −14.2132 | −29.6149 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | −1.52214 | 4.00000 | −0.0622592 | −3.04429 | −12.6896 | 8.00000 | −24.6831 | −0.124518 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | −1.14983 | 4.00000 | −0.621349 | −2.29966 | −22.1290 | 8.00000 | −25.6779 | −1.24270 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 4.41475 | 4.00000 | 17.9952 | 8.82951 | 35.6546 | 8.00000 | −7.50995 | 35.9905 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.00000 | 6.78303 | 4.00000 | 9.81007 | 13.5661 | 12.3971 | 8.00000 | 19.0094 | 19.6201 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.00000 | 8.16362 | 4.00000 | −4.37112 | 16.3272 | −27.6561 | 8.00000 | 39.6447 | −8.74224 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(29\) | \(-1\) |
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 | 1682.4.a.n | yes | 8 |
| 29.b | even | 2 | 1 | 1682.4.a.m | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1682.4.a.m | ✓ | 8 | 29.b | even | 2 | 1 | |
| 1682.4.a.n | yes | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 5T_{3}^{7} - 145T_{3}^{6} - 547T_{3}^{5} + 6124T_{3}^{4} + 17330T_{3}^{3} - 60189T_{3}^{2} - 189714T_{3} - 123676 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1682))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{8} \)
|
| $3$ |
\( T^{8} + 5 T^{7} + \cdots - 123676 \)
|
| $5$ |
\( T^{8} - 10 T^{7} + \cdots - 151380 \)
|
| $7$ |
\( T^{8} - 32 T^{7} + \cdots + 620736404 \)
|
| $11$ |
\( T^{8} + \cdots - 3383748171 \)
|
| $13$ |
\( T^{8} + \cdots + 2470251372084 \)
|
| $17$ |
\( T^{8} + \cdots - 3771002809116 \)
|
| $19$ |
\( T^{8} + \cdots + 126646360232571 \)
|
| $23$ |
\( T^{8} + \cdots - 5103738396 \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( T^{8} + \cdots + 18\!\cdots\!96 \)
|
| $37$ |
\( T^{8} + \cdots - 92\!\cdots\!20 \)
|
| $41$ |
\( T^{8} + \cdots + 82\!\cdots\!24 \)
|
| $43$ |
\( T^{8} + \cdots - 59\!\cdots\!76 \)
|
| $47$ |
\( T^{8} + \cdots - 28\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots - 99\!\cdots\!64 \)
|
| $59$ |
\( T^{8} + \cdots + 21\!\cdots\!81 \)
|
| $61$ |
\( T^{8} + \cdots - 20\!\cdots\!20 \)
|
| $67$ |
\( T^{8} + \cdots + 55\!\cdots\!05 \)
|
| $71$ |
\( T^{8} + \cdots - 30\!\cdots\!36 \)
|
| $73$ |
\( T^{8} + \cdots + 18\!\cdots\!49 \)
|
| $79$ |
\( T^{8} + \cdots + 55\!\cdots\!84 \)
|
| $83$ |
\( T^{8} + \cdots - 70\!\cdots\!41 \)
|
| $89$ |
\( T^{8} + \cdots + 24\!\cdots\!61 \)
|
| $97$ |
\( T^{8} + \cdots - 58\!\cdots\!96 \)
|
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