Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8304,2,Mod(1,8304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8304, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8304.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8304 = 2^{4} \cdot 3 \cdot 173 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8304.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: | \(66.3077738385\) |
| 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: |
\( x^{6} - x^{5} - 22x^{4} + 27x^{3} + 126x^{2} - 158x - 94 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 4152) |
| 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
\( x^{6} - x^{5} - 22x^{4} + 27x^{3} + 126x^{2} - 158x - 94 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 11\nu^{3} + 5\nu^{2} - \nu + 6 ) / 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 22\nu^{3} + 5\nu^{2} + 109\nu - 38 ) / 11 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 22\nu^{3} - 6\nu^{2} + 109\nu + 50 ) / 11 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 6\nu^{5} + 11\nu^{4} - 99\nu^{3} - 124\nu^{2} + 401\nu + 234 ) / 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 10\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 14\beta_{4} + 11\beta_{3} - 3\beta_{2} - 7\beta _1 + 82 \)
|
| \(\nu^{5}\) | \(=\) |
\( 5\beta_{4} - 16\beta_{3} + 22\beta_{2} + 111\beta _1 - 90 \)
|
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 | −1.00000 | 0 | −3.07108 | 0 | 3.07108 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −2.87922 | 0 | 2.87922 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | −2.15351 | 0 | 2.15351 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | 0.452912 | 0 | −0.452912 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | 2.92636 | 0 | −2.92636 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.00000 | 0 | 3.72455 | 0 | −3.72455 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(173\) | \( +1 \) |
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 | 8304.2.a.bc | 6 | |
| 4.b | odd | 2 | 1 | 4152.2.a.l | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4152.2.a.l | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 8304.2.a.bc | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8304))\):
|
\( T_{5}^{6} + T_{5}^{5} - 22T_{5}^{4} - 27T_{5}^{3} + 126T_{5}^{2} + 158T_{5} - 94 \)
|
|
\( T_{7}^{6} - T_{7}^{5} - 22T_{7}^{4} + 27T_{7}^{3} + 126T_{7}^{2} - 158T_{7} - 94 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( T^{6} + T^{5} + \cdots - 94 \)
|
| $7$ |
\( T^{6} - T^{5} + \cdots - 94 \)
|
| $11$ |
\( T^{6} - 9 T^{5} + \cdots - 338 \)
|
| $13$ |
\( T^{6} - T^{5} + \cdots + 86 \)
|
| $17$ |
\( T^{6} - 7 T^{5} + \cdots + 10321 \)
|
| $19$ |
\( T^{6} + 16 T^{5} + \cdots - 101 \)
|
| $23$ |
\( T^{6} - 11 T^{5} + \cdots + 2826 \)
|
| $29$ |
\( T^{6} + 5 T^{5} + \cdots - 24655 \)
|
| $31$ |
\( (T + 3)^{6} \)
|
| $37$ |
\( T^{6} + 13 T^{5} + \cdots + 52376 \)
|
| $41$ |
\( T^{6} + 21 T^{5} + \cdots + 25150 \)
|
| $43$ |
\( T^{6} + 12 T^{5} + \cdots - 2704 \)
|
| $47$ |
\( T^{6} - 4 T^{5} + \cdots + 10 \)
|
| $53$ |
\( T^{6} - 3 T^{5} + \cdots - 102208 \)
|
| $59$ |
\( T^{6} + 9 T^{5} + \cdots - 4744 \)
|
| $61$ |
\( T^{6} - 21 T^{5} + \cdots - 52267 \)
|
| $67$ |
\( T^{6} + 41 T^{5} + \cdots - 87328 \)
|
| $71$ |
\( T^{6} + 10 T^{5} + \cdots - 3845 \)
|
| $73$ |
\( T^{6} - 40 T^{5} + \cdots - 20513 \)
|
| $79$ |
\( T^{6} + 23 T^{5} + \cdots + 681362 \)
|
| $83$ |
\( T^{6} + 12 T^{5} + \cdots + 11233 \)
|
| $89$ |
\( T^{6} - 46 T^{5} + \cdots - 621442 \)
|
| $97$ |
\( T^{6} + 7 T^{5} + \cdots - 202208 \)
|
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