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: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{633}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 158 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 38) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{633})\). We also show the integral \(q\)-expansion of the trace form.
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 | −3.23924 | 0 | −312.472 | 0 | 766.767 | 0 | −2176.51 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 72.2392 | 0 | 467.472 | 0 | 1471.23 | 0 | 3031.51 | 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.d | 2 | |
| 4.b | odd | 2 | 1 | 38.8.a.d | ✓ | 2 | |
| 12.b | even | 2 | 1 | 342.8.a.g | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.8.a.d | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 304.8.a.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 342.8.a.g | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 69T_{3} - 234 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(304))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 69T - 234 \)
|
| $5$ |
\( T^{2} - 155T - 146072 \)
|
| $7$ |
\( T^{2} - 2238 T + 1128093 \)
|
| $11$ |
\( T^{2} - 3295 T - 894002 \)
|
| $13$ |
\( T^{2} + 13427 T + 13167724 \)
|
| $17$ |
\( T^{2} + 32256 T + 253133559 \)
|
| $19$ |
\( (T - 6859)^{2} \)
|
| $23$ |
\( T^{2} + \cdots - 1087606952 \)
|
| $29$ |
\( T^{2} + \cdots - 1552615514 \)
|
| $31$ |
\( T^{2} + \cdots + 9936311536 \)
|
| $37$ |
\( T^{2} + \cdots - 6156318428 \)
|
| $41$ |
\( T^{2} + \cdots - 393872642768 \)
|
| $43$ |
\( T^{2} + \cdots - 268023173408 \)
|
| $47$ |
\( T^{2} + \cdots + 474895142800 \)
|
| $53$ |
\( T^{2} + \cdots + 79392286228 \)
|
| $59$ |
\( T^{2} + \cdots - 1837525132614 \)
|
| $61$ |
\( T^{2} + \cdots - 359679230318 \)
|
| $67$ |
\( T^{2} + \cdots - 1107042934188 \)
|
| $71$ |
\( T^{2} + \cdots - 10503963815108 \)
|
| $73$ |
\( T^{2} + \cdots - 4520032488687 \)
|
| $79$ |
\( T^{2} + \cdots - 5162668519808 \)
|
| $83$ |
\( T^{2} + \cdots + 23086028557656 \)
|
| $89$ |
\( T^{2} + \cdots - 29977064763600 \)
|
| $97$ |
\( T^{2} + \cdots - 88352296135184 \)
|
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