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: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 13x^{5} - 8x^{4} + 24x^{3} + 6x^{2} - 13x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2076) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 13x^{5} - 8x^{4} + 24x^{3} + 6x^{2} - 13x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 10\nu^{2} - \nu + 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 13\nu^{4} - 20\nu^{3} + 16\nu^{2} + 19\nu - 10 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 13\nu^{4} + 9\nu^{3} - 25\nu^{2} - 14\nu + 12 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{6} - \nu^{5} + 25\nu^{4} + 28\nu^{3} - 27\nu^{2} - 18\nu + 8 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{6} - 2\nu^{5} + 25\nu^{4} + 41\nu^{3} - 20\nu^{2} - 37\nu + 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 2\beta_{3} + \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{6} - \beta_{5} + \beta_{4} + 3\beta_{3} + \beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{6} - \beta_{5} + \beta_{4} + 23\beta_{3} + 13\beta_{2} + 8\beta _1 + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( 31\beta_{6} - 11\beta_{5} + 13\beta_{4} + 53\beta_{3} + 20\beta_{2} + 72\beta _1 + 47 \)
|
| \(\nu^{6}\) | \(=\) |
\( 149\beta_{6} - 22\beta_{5} + 20\beta_{4} + 276\beta_{3} + 153\beta_{2} + 153\beta _1 + 293 \)
|
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 | −2.48191 | 0 | 1.43522 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −1.13640 | 0 | −1.30202 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | −1.06012 | 0 | 3.11693 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | 1.89785 | 0 | −4.20025 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | 2.27660 | 0 | −1.79036 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.00000 | 0 | 2.38904 | 0 | −0.737178 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −1.00000 | 0 | 4.11493 | 0 | 2.47765 | 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.bf | 7 | |
| 4.b | odd | 2 | 1 | 2076.2.a.g | ✓ | 7 | |
| 12.b | even | 2 | 1 | 6228.2.a.m | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2076.2.a.g | ✓ | 7 | 4.b | odd | 2 | 1 | |
| 6228.2.a.m | 7 | 12.b | even | 2 | 1 | ||
| 8304.2.a.bf | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8304))\):
|
\( T_{5}^{7} - 6T_{5}^{6} - 2T_{5}^{5} + 56T_{5}^{4} - 38T_{5}^{3} - 138T_{5}^{2} + 76T_{5} + 127 \)
|
|
\( T_{7}^{7} + T_{7}^{6} - 20T_{7}^{5} - 9T_{7}^{4} + 96T_{7}^{3} + 52T_{7}^{2} - 120T_{7} - 80 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( (T + 1)^{7} \)
|
| $5$ |
\( T^{7} - 6 T^{6} + \cdots + 127 \)
|
| $7$ |
\( T^{7} + T^{6} + \cdots - 80 \)
|
| $11$ |
\( T^{7} + 4 T^{6} + \cdots + 43 \)
|
| $13$ |
\( T^{7} - 4 T^{6} + \cdots + 58 \)
|
| $17$ |
\( T^{7} - 7 T^{6} + \cdots - 18 \)
|
| $19$ |
\( T^{7} + 9 T^{6} + \cdots - 144 \)
|
| $23$ |
\( T^{7} + 17 T^{6} + \cdots - 144 \)
|
| $29$ |
\( T^{7} - 6 T^{6} + \cdots - 4768 \)
|
| $31$ |
\( T^{7} + 10 T^{6} + \cdots + 9351 \)
|
| $37$ |
\( T^{7} - 6 T^{6} + \cdots - 42265 \)
|
| $41$ |
\( T^{7} - 7 T^{6} + \cdots + 18416 \)
|
| $43$ |
\( T^{7} + 11 T^{6} + \cdots - 3816 \)
|
| $47$ |
\( T^{7} + 22 T^{6} + \cdots + 3312 \)
|
| $53$ |
\( T^{7} - 20 T^{6} + \cdots + 8269 \)
|
| $59$ |
\( T^{7} + 18 T^{6} + \cdots - 52252 \)
|
| $61$ |
\( T^{7} - 217 T^{5} + \cdots - 68976 \)
|
| $67$ |
\( T^{7} + 8 T^{6} + \cdots + 67329 \)
|
| $71$ |
\( T^{7} + 14 T^{6} + \cdots - 276493 \)
|
| $73$ |
\( T^{7} + 14 T^{6} + \cdots + 3905599 \)
|
| $79$ |
\( T^{7} + T^{6} + \cdots + 60624 \)
|
| $83$ |
\( T^{7} + 7 T^{6} + \cdots - 585184 \)
|
| $89$ |
\( T^{7} + 6 T^{6} + \cdots - 826240 \)
|
| $97$ |
\( T^{7} + 17 T^{6} + \cdots + 4688 \)
|
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