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: |
\( x^{6} - x^{5} - 6373x^{4} - 12403x^{3} + 11165936x^{2} + 51537728x - 4683020288 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 152) |
| 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
\( x^{6} - x^{5} - 6373x^{4} - 12403x^{3} + 11165936x^{2} + 51537728x - 4683020288 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11\nu^{5} - 355\nu^{4} - 47023\nu^{3} + 1765319\nu^{2} + 31097480\nu - 1668447712 ) / 1844640 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 149\nu^{5} + 515\nu^{4} - 752737\nu^{3} + 711641\nu^{2} + 741593960\nu - 4789303168 ) / 7378560 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -37\nu^{5} + 2525\nu^{4} + 206081\nu^{3} - 12045433\nu^{2} - 247787800\nu + 10034105984 ) / 1844640 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -11\nu^{5} + 355\nu^{4} + 55807\nu^{3} - 1589639\nu^{2} - 57669080\nu + 1217178496 ) / 210816 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} + 2\beta_{2} + 5\beta _1 + 2124 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{5} + 20\beta_{4} - 20\beta_{3} + 170\beta_{2} + 2925\beta _1 + 8894 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4445\beta_{5} - 3923\beta_{4} + 5309\beta_{3} + 7720\beta_{2} + 25231\beta _1 + 6104270 \)
|
| \(\nu^{5}\) | \(=\) |
\( 68\beta_{5} + 119374\beta_{4} - 74644\beta_{3} + 822592\beta_{2} + 9688655\beta _1 + 45831688 \)
|
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 | −65.0909 | 0 | 165.215 | 0 | 774.961 | 0 | 2049.82 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −42.3527 | 0 | −290.965 | 0 | −951.495 | 0 | −393.248 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −38.1137 | 0 | −17.3107 | 0 | 1338.47 | 0 | −734.347 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 15.8260 | 0 | 353.595 | 0 | −621.598 | 0 | −1936.54 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 44.0444 | 0 | 148.297 | 0 | 341.431 | 0 | −247.091 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 56.6869 | 0 | −401.832 | 0 | 26.2293 | 0 | 1026.40 | 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.j | 6 | |
| 4.b | odd | 2 | 1 | 152.8.a.a | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.8.a.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 304.8.a.j | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 29T_{3}^{5} - 6023T_{3}^{4} - 137613T_{3}^{3} + 10032066T_{3}^{2} + 159095988T_{3} - 4151704248 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(304))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 4151704248 \)
|
| $5$ |
\( T^{6} + \cdots - 17534371424000 \)
|
| $7$ |
\( T^{6} + \cdots + 54\!\cdots\!39 \)
|
| $11$ |
\( T^{6} + \cdots - 96\!\cdots\!96 \)
|
| $13$ |
\( T^{6} + \cdots + 66\!\cdots\!40 \)
|
| $17$ |
\( T^{6} + \cdots - 15\!\cdots\!05 \)
|
| $19$ |
\( (T - 6859)^{6} \)
|
| $23$ |
\( T^{6} + \cdots - 20\!\cdots\!80 \)
|
| $29$ |
\( T^{6} + \cdots + 25\!\cdots\!44 \)
|
| $31$ |
\( T^{6} + \cdots - 50\!\cdots\!84 \)
|
| $37$ |
\( T^{6} + \cdots + 63\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots + 39\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots - 97\!\cdots\!60 \)
|
| $47$ |
\( T^{6} + \cdots + 15\!\cdots\!44 \)
|
| $53$ |
\( T^{6} + \cdots + 79\!\cdots\!60 \)
|
| $59$ |
\( T^{6} + \cdots + 31\!\cdots\!84 \)
|
| $61$ |
\( T^{6} + \cdots + 22\!\cdots\!60 \)
|
| $67$ |
\( T^{6} + \cdots - 34\!\cdots\!00 \)
|
| $71$ |
\( T^{6} + \cdots + 13\!\cdots\!12 \)
|
| $73$ |
\( T^{6} + \cdots + 10\!\cdots\!85 \)
|
| $79$ |
\( T^{6} + \cdots - 36\!\cdots\!16 \)
|
| $83$ |
\( T^{6} + \cdots - 10\!\cdots\!80 \)
|
| $89$ |
\( T^{6} + \cdots + 22\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 17\!\cdots\!32 \)
|
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