Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [354,8,Mod(1,354)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(354, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("354.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 354 = 2 \cdot 3 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 354.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: | \(110.584299021\) |
| 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: |
\( x^{8} - x^{7} - 103558 x^{6} + 5805883 x^{5} + 2559087821 x^{4} - 196601024266 x^{3} + \cdots - 22\!\cdots\!40 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{7}\cdot 5 \) |
| 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
\( x^{8} - x^{7} - 103558 x^{6} + 5805883 x^{5} + 2559087821 x^{4} - 196601024266 x^{3} + \cdots - 22\!\cdots\!40 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 40\!\cdots\!18 \nu^{7} + \cdots - 35\!\cdots\!60 ) / 18\!\cdots\!40 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 10\!\cdots\!01 \nu^{7} + \cdots - 13\!\cdots\!60 ) / 36\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17\!\cdots\!50 \nu^{7} + \cdots + 46\!\cdots\!56 ) / 36\!\cdots\!88 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 11\!\cdots\!55 \nu^{7} + \cdots - 95\!\cdots\!44 ) / 24\!\cdots\!92 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 29\!\cdots\!19 \nu^{7} + \cdots - 68\!\cdots\!40 ) / 61\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 71\!\cdots\!89 \nu^{7} + \cdots + 10\!\cdots\!30 ) / 92\!\cdots\!70 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 12\!\cdots\!16 \nu^{7} + \cdots - 11\!\cdots\!88 ) / 12\!\cdots\!96 \)
|
| \(\nu\) | \(=\) |
\( ( -14\beta_{7} + 3\beta_{6} - 2\beta_{5} + 4\beta_{4} - 16\beta_{3} - 12\beta_{2} - 8\beta _1 + 9 ) / 90 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6044 \beta_{7} - 1328 \beta_{6} + 2447 \beta_{5} + 566 \beta_{4} + 11711 \beta_{3} + 332 \beta_{2} + \cdots + 6990631 ) / 270 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 913904 \beta_{7} + 73783 \beta_{6} - 234227 \beta_{5} + 69334 \beta_{4} - 1407711 \beta_{3} + \cdots - 192529676 ) / 90 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 651379252 \beta_{7} - 90115579 \beta_{6} + 214880806 \beta_{5} + 48289978 \beta_{4} + \cdots + 377657536253 ) / 270 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 223698357058 \beta_{7} + 14051070376 \beta_{6} - 66076732819 \beta_{5} + 2795744528 \beta_{4} + \cdots - 66919763035637 ) / 270 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 12278943963646 \beta_{7} - 1362318816295 \beta_{6} + 3886489516477 \beta_{5} + 679818370000 \beta_{4} + \cdots + 55\!\cdots\!40 ) / 54 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 19\!\cdots\!46 \beta_{7} + \cdots - 66\!\cdots\!19 ) / 270 \)
|
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 |
|
8.00000 | −27.0000 | 64.0000 | −497.674 | −216.000 | −1339.98 | 512.000 | 729.000 | −3981.39 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | −27.0000 | 64.0000 | −490.703 | −216.000 | 975.190 | 512.000 | 729.000 | −3925.63 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | −27.0000 | 64.0000 | −365.356 | −216.000 | −311.570 | 512.000 | 729.000 | −2922.85 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | −27.0000 | 64.0000 | −11.9651 | −216.000 | 1695.87 | 512.000 | 729.000 | −95.7205 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 8.00000 | −27.0000 | 64.0000 | −6.85357 | −216.000 | 554.074 | 512.000 | 729.000 | −54.8286 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 8.00000 | −27.0000 | 64.0000 | 188.524 | −216.000 | −784.370 | 512.000 | 729.000 | 1508.19 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8.00000 | −27.0000 | 64.0000 | 272.880 | −216.000 | −1263.34 | 512.000 | 729.000 | 2183.04 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 8.00000 | −27.0000 | 64.0000 | 319.148 | −216.000 | 134.129 | 512.000 | 729.000 | 2553.18 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(59\) | \(1\) |
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 | 354.8.a.d | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 354.8.a.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 592 T_{5}^{7} - 241770 T_{5}^{6} - 135397930 T_{5}^{5} + 25927370375 T_{5}^{4} + \cdots - 12\!\cdots\!00 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(354))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{8} \)
|
| $3$ |
\( (T + 27)^{8} \)
|
| $5$ |
\( T^{8} + \cdots - 12\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 50\!\cdots\!60 \)
|
| $11$ |
\( T^{8} + \cdots - 31\!\cdots\!60 \)
|
| $13$ |
\( T^{8} + \cdots + 43\!\cdots\!50 \)
|
| $17$ |
\( T^{8} + \cdots - 13\!\cdots\!28 \)
|
| $19$ |
\( T^{8} + \cdots + 30\!\cdots\!48 \)
|
| $23$ |
\( T^{8} + \cdots + 71\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots + 70\!\cdots\!16 \)
|
| $31$ |
\( T^{8} + \cdots + 42\!\cdots\!20 \)
|
| $37$ |
\( T^{8} + \cdots - 69\!\cdots\!50 \)
|
| $41$ |
\( T^{8} + \cdots - 14\!\cdots\!72 \)
|
| $43$ |
\( T^{8} + \cdots + 11\!\cdots\!16 \)
|
| $47$ |
\( T^{8} + \cdots + 67\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 93\!\cdots\!76 \)
|
| $59$ |
\( (T + 205379)^{8} \)
|
| $61$ |
\( T^{8} + \cdots - 10\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots + 35\!\cdots\!80 \)
|
| $71$ |
\( T^{8} + \cdots - 18\!\cdots\!50 \)
|
| $73$ |
\( T^{8} + \cdots + 88\!\cdots\!44 \)
|
| $79$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 69\!\cdots\!36 \)
|
| $89$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 69\!\cdots\!40 \)
|
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