Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [304,8,Mod(1,304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(304, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("304.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 304.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: | \(94.9650477472\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 8376x^{4} + 135458x^{3} + 16275767x^{2} - 280013424x - 6276171312 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 76) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 8376x^{4} + 135458x^{3} + 16275767x^{2} - 280013424x - 6276171312 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -23\nu^{5} + 1337\nu^{4} + 331031\nu^{3} - 3798869\nu^{2} - 655626372\nu - 394340616 ) / 12622824 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 51\nu^{5} + 2117\nu^{4} - 357983\nu^{3} - 9662021\nu^{2} + 548036696\nu + 11111121348 ) / 2103804 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -269\nu^{5} - 12312\nu^{4} + 1628074\nu^{3} + 41365824\nu^{2} - 2202338405\nu - 33289998684 ) / 2103804 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -617\nu^{5} - 30195\nu^{4} + 3641083\nu^{3} + 101646951\nu^{2} - 4929275990\nu - 76626597048 ) / 4207608 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - \beta_{3} - 3\beta_{2} - 20\beta _1 + 2800 \)
|
| \(\nu^{3}\) | \(=\) |
\( 78\beta_{5} - 90\beta_{4} + 12\beta_{3} + 198\beta_{2} + 4321\beta _1 - 60834 \)
|
| \(\nu^{4}\) | \(=\) |
\( -9331\beta_{5} + 10219\beta_{4} - 4033\beta_{3} - 19821\beta_{2} - 212696\beta _1 + 12452470 \)
|
| \(\nu^{5}\) | \(=\) |
\( 745380\beta_{5} - 866472\beta_{4} + 103440\beta_{3} + 1644228\beta_{2} + 24624409\beta _1 - 631311360 \)
|
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 | −91.8914 | 0 | 432.300 | 0 | 1482.77 | 0 | 6257.03 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −53.2258 | 0 | 88.7692 | 0 | −1094.09 | 0 | 645.987 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −20.7370 | 0 | −501.411 | 0 | 1008.02 | 0 | −1756.98 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 37.4724 | 0 | 534.866 | 0 | −1051.36 | 0 | −782.818 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 42.5941 | 0 | 51.7645 | 0 | 1212.43 | 0 | −372.745 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 45.7878 | 0 | −327.289 | 0 | 7.23734 | 0 | −90.4805 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 304.8.a.i | 6 | |
| 4.b | odd | 2 | 1 | 76.8.a.b | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.8.a.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 304.8.a.i | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 40T_{3}^{5} - 7711T_{3}^{4} - 93190T_{3}^{3} + 16686996T_{3}^{2} - 43655400T_{3} - 7412317344 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(304))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 7412317344 \)
|
| $5$ |
\( T^{6} + \cdots + 174360864384000 \)
|
| $7$ |
\( T^{6} + \cdots + 15\!\cdots\!84 \)
|
| $11$ |
\( T^{6} + \cdots - 47\!\cdots\!56 \)
|
| $13$ |
\( T^{6} + \cdots - 15\!\cdots\!56 \)
|
| $17$ |
\( T^{6} + \cdots + 81\!\cdots\!86 \)
|
| $19$ |
\( (T + 6859)^{6} \)
|
| $23$ |
\( T^{6} + \cdots - 10\!\cdots\!64 \)
|
| $29$ |
\( T^{6} + \cdots - 19\!\cdots\!56 \)
|
| $31$ |
\( T^{6} + \cdots - 59\!\cdots\!24 \)
|
| $37$ |
\( T^{6} + \cdots + 12\!\cdots\!96 \)
|
| $41$ |
\( T^{6} + \cdots + 23\!\cdots\!84 \)
|
| $43$ |
\( T^{6} + \cdots - 15\!\cdots\!44 \)
|
| $47$ |
\( T^{6} + \cdots + 10\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots - 69\!\cdots\!76 \)
|
| $59$ |
\( T^{6} + \cdots - 86\!\cdots\!56 \)
|
| $61$ |
\( T^{6} + \cdots - 23\!\cdots\!64 \)
|
| $67$ |
\( T^{6} + \cdots - 13\!\cdots\!76 \)
|
| $71$ |
\( T^{6} + \cdots + 13\!\cdots\!04 \)
|
| $73$ |
\( T^{6} + \cdots + 96\!\cdots\!26 \)
|
| $79$ |
\( T^{6} + \cdots + 29\!\cdots\!76 \)
|
| $83$ |
\( T^{6} + \cdots - 65\!\cdots\!64 \)
|
| $89$ |
\( T^{6} + \cdots + 18\!\cdots\!76 \)
|
| $97$ |
\( T^{6} + \cdots + 48\!\cdots\!44 \)
|
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