Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5456,2,Mod(1,5456)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5456, 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("5456.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5456 = 2^{4} \cdot 11 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5456.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: | \(43.5663793428\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 6x^{6} + 30x^{5} + 9x^{4} - 58x^{3} - 15x^{2} + 32x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 341) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 6x^{6} + 30x^{5} + 9x^{4} - 58x^{3} - 15x^{2} + 32x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 7\nu + 8 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{6} - 6\nu^{5} + 30\nu^{4} + 7\nu^{3} - 54\nu^{2} - \nu + 18 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 6\nu^{6} - 2\nu^{5} - 38\nu^{4} + 39\nu^{3} + 48\nu^{2} - 33\nu - 14 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 6\nu^{6} - 2\nu^{5} - 38\nu^{4} + 39\nu^{3} + 52\nu^{2} - 37\nu - 26 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 5\nu^{6} - \nu^{5} + 31\nu^{4} - 23\nu^{3} - 35\nu^{2} + 27\nu + 10 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{6} - 5\nu^{5} + 27\nu^{4} + 2\nu^{3} - 40\nu^{2} - 2\nu + 12 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} + 7\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} - 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 2\beta_{3} + 2\beta_{2} + 14\beta _1 + 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13\beta_{7} - 11\beta_{6} + 18\beta_{5} - 29\beta_{4} - 15\beta_{3} + 6\beta_{2} + 64\beta _1 + 36 \)
|
| \(\nu^{6}\) | \(=\) |
\( 37\beta_{7} - 29\beta_{6} + 88\beta_{5} - 115\beta_{4} - 43\beta_{3} + 32\beta_{2} + 171\beta _1 + 158 \)
|
| \(\nu^{7}\) | \(=\) |
\( 159\beta_{7} - 115\beta_{6} + 237\beta_{5} - 344\beta_{4} - 191\beta_{3} + 104\beta_{2} + 654\beta _1 + 401 \)
|
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 | −3.17990 | 0 | −0.963523 | 0 | −2.56579 | 0 | 7.11176 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.10098 | 0 | −3.87097 | 0 | 3.77591 | 0 | 1.41412 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.82232 | 0 | 1.34847 | 0 | 1.23584 | 0 | 0.320854 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.42477 | 0 | 0.129361 | 0 | −5.26936 | 0 | −0.970023 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.0240968 | 0 | 0.530938 | 0 | −2.05623 | 0 | −2.99942 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.699619 | 0 | 3.56228 | 0 | −1.21823 | 0 | −2.51053 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.45657 | 0 | −3.82388 | 0 | 1.64018 | 0 | −0.878397 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.34769 | 0 | −1.91268 | 0 | −2.54232 | 0 | 2.51163 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(31\) | \( +1 \) |
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 | 5456.2.a.ba | 8 | |
| 4.b | odd | 2 | 1 | 341.2.a.c | ✓ | 8 | |
| 12.b | even | 2 | 1 | 3069.2.a.k | 8 | ||
| 20.d | odd | 2 | 1 | 8525.2.a.g | 8 | ||
| 44.c | even | 2 | 1 | 3751.2.a.h | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 341.2.a.c | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 3069.2.a.k | 8 | 12.b | even | 2 | 1 | ||
| 3751.2.a.h | 8 | 44.c | even | 2 | 1 | ||
| 5456.2.a.ba | 8 | 1.a | even | 1 | 1 | trivial | |
| 8525.2.a.g | 8 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5456))\):
|
\( T_{3}^{8} + 4T_{3}^{7} - 6T_{3}^{6} - 34T_{3}^{5} - T_{3}^{4} + 74T_{3}^{3} + 19T_{3}^{2} - 42T_{3} + 1 \)
|
|
\( T_{5}^{8} + 5T_{5}^{7} - 12T_{5}^{6} - 77T_{5}^{5} - 11T_{5}^{4} + 176T_{5}^{3} + 35T_{5}^{2} - 77T_{5} + 9 \)
|
|
\( T_{7}^{8} + 7T_{7}^{7} - 8T_{7}^{6} - 127T_{7}^{5} - 137T_{7}^{4} + 434T_{7}^{3} + 657T_{7}^{2} - 393T_{7} - 659 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 4 T^{7} + \cdots + 1 \)
|
| $5$ |
\( T^{8} + 5 T^{7} + \cdots + 9 \)
|
| $7$ |
\( T^{8} + 7 T^{7} + \cdots - 659 \)
|
| $11$ |
\( (T - 1)^{8} \)
|
| $13$ |
\( T^{8} - 8 T^{7} + \cdots + 37 \)
|
| $17$ |
\( T^{8} - 44 T^{6} + \cdots + 219 \)
|
| $19$ |
\( T^{8} + 16 T^{7} + \cdots - 4996 \)
|
| $23$ |
\( T^{8} + 4 T^{7} + \cdots + 372 \)
|
| $29$ |
\( T^{8} - 8 T^{7} + \cdots - 31728 \)
|
| $31$ |
\( (T + 1)^{8} \)
|
| $37$ |
\( T^{8} - 8 T^{7} + \cdots - 91648 \)
|
| $41$ |
\( T^{8} - 3 T^{7} + \cdots + 28224 \)
|
| $43$ |
\( T^{8} + 27 T^{7} + \cdots + 181312 \)
|
| $47$ |
\( T^{8} - 21 T^{7} + \cdots + 852672 \)
|
| $53$ |
\( T^{8} + T^{7} + \cdots - 315456 \)
|
| $59$ |
\( T^{8} + 16 T^{7} + \cdots + 250368 \)
|
| $61$ |
\( T^{8} - 23 T^{7} + \cdots - 138205 \)
|
| $67$ |
\( T^{8} - 10 T^{7} + \cdots - 43339328 \)
|
| $71$ |
\( T^{8} - 3 T^{7} + \cdots + 60864 \)
|
| $73$ |
\( T^{8} - 8 T^{7} + \cdots - 442769 \)
|
| $79$ |
\( T^{8} + 32 T^{7} + \cdots - 67328192 \)
|
| $83$ |
\( T^{8} - 7 T^{7} + \cdots - 960000 \)
|
| $89$ |
\( T^{8} + 49 T^{7} + \cdots + 31996224 \)
|
| $97$ |
\( T^{8} + 18 T^{7} + \cdots - 377968 \)
|
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