Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [354,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("354.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 354 = 2 \cdot 3 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(56.7758722138\) |
| 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: |
\( x^{8} - 17732 x^{6} - 152272 x^{5} + 93277609 x^{4} + 1554240404 x^{3} - 156444406614 x^{2} + \cdots + 6279664243680 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{8} - 17732 x^{6} - 152272 x^{5} + 93277609 x^{4} + 1554240404 x^{3} - 156444406614 x^{2} + \cdots + 6279664243680 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 34359867950801 \nu^{7} + \cdots - 11\!\cdots\!60 ) / 40\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 213080717119766 \nu^{7} + \cdots - 63\!\cdots\!40 ) / 12\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 241173404580701 \nu^{7} + \cdots - 40\!\cdots\!40 ) / 12\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 86456910449438 \nu^{7} + \cdots + 89\!\cdots\!80 ) / 40\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 230696835595757 \nu^{7} + \cdots + 18\!\cdots\!20 ) / 61\!\cdots\!50 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 480161363957732 \nu^{7} + \cdots + 11\!\cdots\!20 ) / 12\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{7} + \beta_{6} + 3\beta_{5} + 4\beta_{4} - \beta_{3} + 10\beta_{2} + 14\beta _1 + 4431 \)
|
| \(\nu^{3}\) | \(=\) |
\( 37\beta_{7} + 114\beta_{6} + 122\beta_{5} - 274\beta_{4} + 141\beta_{3} + 640\beta_{2} + 7088\beta _1 + 56889 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 16896 \beta_{7} + 14543 \beta_{6} + 45544 \beta_{5} + 28672 \beta_{4} - 22748 \beta_{3} + \cdots + 31946788 \)
|
| \(\nu^{5}\) | \(=\) |
\( 188865 \beta_{7} + 1915400 \beta_{6} + 2055010 \beta_{5} - 3641590 \beta_{4} + 1402125 \beta_{3} + \cdots + 713420925 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 146841062 \beta_{7} + 173726981 \beta_{6} + 528049738 \beta_{5} + 209247184 \beta_{4} + \cdots + 279133727996 \)
|
| \(\nu^{7}\) | \(=\) |
\( 162735387 \beta_{7} + 23261647354 \beta_{6} + 27210948742 \beta_{5} - 39136997954 \beta_{4} + \cdots + 8575274969179 \)
|
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 |
|
4.00000 | 9.00000 | 16.0000 | −92.1518 | 36.0000 | −250.043 | 64.0000 | 81.0000 | −368.607 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | 9.00000 | 16.0000 | −53.5126 | 36.0000 | 183.368 | 64.0000 | 81.0000 | −214.050 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 9.00000 | 16.0000 | −53.0121 | 36.0000 | −109.285 | 64.0000 | 81.0000 | −212.049 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 9.00000 | 16.0000 | 10.4159 | 36.0000 | 172.112 | 64.0000 | 81.0000 | 41.6636 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | 9.00000 | 16.0000 | 43.4107 | 36.0000 | −6.75144 | 64.0000 | 81.0000 | 173.643 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.00000 | 9.00000 | 16.0000 | 62.6379 | 36.0000 | −69.5352 | 64.0000 | 81.0000 | 250.552 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.00000 | 9.00000 | 16.0000 | 71.0243 | 36.0000 | 223.194 | 64.0000 | 81.0000 | 284.097 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.00000 | 9.00000 | 16.0000 | 107.188 | 36.0000 | 37.9412 | 64.0000 | 81.0000 | 428.751 | |||||||||||||||||||||||||||||||||||||||||||
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 | 354.6.a.i | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1062.6.a.k | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 354.6.a.i | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1062.6.a.k | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 96 T_{5}^{7} - 13700 T_{5}^{6} + 1332208 T_{5}^{5} + 47291689 T_{5}^{4} + \cdots - 56366121500208 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(354))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{8} \)
|
| $3$ |
\( (T - 9)^{8} \)
|
| $5$ |
\( T^{8} + \cdots - 56366121500208 \)
|
| $7$ |
\( T^{8} + \cdots + 34\!\cdots\!00 \)
|
| $11$ |
\( T^{8} + \cdots - 22\!\cdots\!24 \)
|
| $13$ |
\( T^{8} + \cdots - 62\!\cdots\!12 \)
|
| $17$ |
\( T^{8} + \cdots + 39\!\cdots\!96 \)
|
| $19$ |
\( T^{8} + \cdots - 30\!\cdots\!12 \)
|
| $23$ |
\( T^{8} + \cdots + 27\!\cdots\!48 \)
|
| $29$ |
\( T^{8} + \cdots + 16\!\cdots\!48 \)
|
| $31$ |
\( T^{8} + \cdots - 34\!\cdots\!08 \)
|
| $37$ |
\( T^{8} + \cdots + 19\!\cdots\!72 \)
|
| $41$ |
\( T^{8} + \cdots - 15\!\cdots\!08 \)
|
| $43$ |
\( T^{8} + \cdots + 16\!\cdots\!60 \)
|
| $47$ |
\( T^{8} + \cdots + 21\!\cdots\!24 \)
|
| $53$ |
\( T^{8} + \cdots - 60\!\cdots\!44 \)
|
| $59$ |
\( (T - 3481)^{8} \)
|
| $61$ |
\( T^{8} + \cdots + 19\!\cdots\!20 \)
|
| $67$ |
\( T^{8} + \cdots + 61\!\cdots\!68 \)
|
| $71$ |
\( T^{8} + \cdots - 97\!\cdots\!92 \)
|
| $73$ |
\( T^{8} + \cdots - 10\!\cdots\!16 \)
|
| $79$ |
\( T^{8} + \cdots - 23\!\cdots\!12 \)
|
| $83$ |
\( T^{8} + \cdots - 87\!\cdots\!76 \)
|
| $89$ |
\( T^{8} + \cdots - 89\!\cdots\!32 \)
|
| $97$ |
\( T^{8} + \cdots - 16\!\cdots\!92 \)
|
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