Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5312,2,Mod(1,5312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5312, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("5312.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5312 = 2^{6} \cdot 83 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5312.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: | \(42.4165335537\) |
| 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: |
\( x^{8} - 3x^{7} - 10x^{6} + 31x^{5} + 24x^{4} - 94x^{3} + 11x^{2} + 70x - 28 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2656) |
| 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
\( x^{8} - 3x^{7} - 10x^{6} + 31x^{5} + 24x^{4} - 94x^{3} + 11x^{2} + 70x - 28 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 6\nu^{5} + 17\nu^{4} + 6\nu^{3} - 12\nu^{2} - \nu - 4 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 5\nu^{6} - 29\nu^{4} + 30\nu^{3} + 22\nu^{2} - 35\nu + 6 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 8\nu^{5} + 23\nu^{4} + 18\nu^{3} - 46\nu^{2} - 13\nu + 22 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 11\nu^{5} + 16\nu^{4} + 37\nu^{3} - 34\nu^{2} - 34\nu + 18 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 15\nu^{5} + 13\nu^{4} + 60\nu^{3} - 45\nu^{2} - 52\nu + 30 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} + 3\beta_{5} + 2\beta_{4} + \beta_{3} + 8\beta_{2} + 3\beta _1 + 25 \)
|
| \(\nu^{5}\) | \(=\) |
\( -9\beta_{7} + 6\beta_{6} + 14\beta_{5} + 12\beta_{4} + 4\beta_{3} + 13\beta_{2} + 39\beta _1 + 32 \)
|
| \(\nu^{6}\) | \(=\) |
\( -15\beta_{7} + 42\beta_{5} + 31\beta_{4} + 19\beta_{3} + 64\beta_{2} + 45\beta _1 + 189 \)
|
| \(\nu^{7}\) | \(=\) |
\( -76\beta_{7} + 30\beta_{6} + 153\beta_{5} + 125\beta_{4} + 66\beta_{3} + 140\beta_{2} + 283\beta _1 + 374 \)
|
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 | −3.11667 | 0 | −2.24507 | 0 | −2.12162 | 0 | 6.71362 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.25826 | 0 | 3.49837 | 0 | −5.02233 | 0 | 2.09975 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.35406 | 0 | −3.61598 | 0 | 0.0461395 | 0 | −1.16653 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.20048 | 0 | −1.42390 | 0 | 3.61476 | 0 | −1.55884 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.490450 | 0 | 2.94534 | 0 | −2.53969 | 0 | −2.75946 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.04095 | 0 | 0.288535 | 0 | −3.42576 | 0 | −1.91642 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.17938 | 0 | −1.30386 | 0 | 0.921839 | 0 | 1.74968 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.19959 | 0 | 2.85657 | 0 | −1.47334 | 0 | 1.83820 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(83\) | \(-1\) |
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 | 5312.2.a.bq | 8 | |
| 4.b | odd | 2 | 1 | 5312.2.a.bt | 8 | ||
| 8.b | even | 2 | 1 | 2656.2.a.r | yes | 8 | |
| 8.d | odd | 2 | 1 | 2656.2.a.q | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2656.2.a.q | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 2656.2.a.r | yes | 8 | 8.b | even | 2 | 1 | |
| 5312.2.a.bq | 8 | 1.a | even | 1 | 1 | trivial | |
| 5312.2.a.bt | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5312))\):
|
\( T_{3}^{8} + 3T_{3}^{7} - 10T_{3}^{6} - 31T_{3}^{5} + 24T_{3}^{4} + 94T_{3}^{3} + 11T_{3}^{2} - 70T_{3} - 28 \)
|
|
\( T_{5}^{8} - T_{5}^{7} - 25T_{5}^{6} + 16T_{5}^{5} + 198T_{5}^{4} - 14T_{5}^{3} - 552T_{5}^{2} - 288T_{5} + 128 \)
|
|
\( T_{7}^{8} + 10T_{7}^{7} + 18T_{7}^{6} - 104T_{7}^{5} - 431T_{7}^{4} - 381T_{7}^{3} + 325T_{7}^{2} + 441T_{7} - 21 \)
|
|
\( T_{11}^{8} + 15T_{11}^{7} + 46T_{11}^{6} - 243T_{11}^{5} - 1202T_{11}^{4} + 1092T_{11}^{3} + 7495T_{11}^{2} - 2198T_{11} - 7532 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 3 T^{7} + \cdots - 28 \)
|
| $5$ |
\( T^{8} - T^{7} + \cdots + 128 \)
|
| $7$ |
\( T^{8} + 10 T^{7} + \cdots - 21 \)
|
| $11$ |
\( T^{8} + 15 T^{7} + \cdots - 7532 \)
|
| $13$ |
\( T^{8} - 9 T^{7} + \cdots + 240 \)
|
| $17$ |
\( T^{8} + 2 T^{7} + \cdots + 1575 \)
|
| $19$ |
\( T^{8} + 15 T^{7} + \cdots - 11760 \)
|
| $23$ |
\( T^{8} - 103 T^{6} + \cdots - 27440 \)
|
| $29$ |
\( T^{8} - T^{7} + \cdots - 336 \)
|
| $31$ |
\( T^{8} - 8 T^{7} + \cdots - 69643 \)
|
| $37$ |
\( T^{8} - 9 T^{7} + \cdots + 293668 \)
|
| $41$ |
\( T^{8} - 12 T^{7} + \cdots - 4015036 \)
|
| $43$ |
\( T^{8} + 23 T^{7} + \cdots + 11200 \)
|
| $47$ |
\( T^{8} - 4 T^{7} + \cdots - 1443008 \)
|
| $53$ |
\( T^{8} - 41 T^{7} + \cdots + 86016 \)
|
| $59$ |
\( T^{8} + 37 T^{7} + \cdots - 632492 \)
|
| $61$ |
\( T^{8} + 19 T^{7} + \cdots - 11456 \)
|
| $67$ |
\( T^{8} + 39 T^{7} + \cdots + 2705920 \)
|
| $71$ |
\( T^{8} - 12 T^{7} + \cdots + 24743936 \)
|
| $73$ |
\( T^{8} - 8 T^{7} + \cdots - 86204608 \)
|
| $79$ |
\( T^{8} - 206 T^{6} + \cdots + 11200 \)
|
| $83$ |
\( (T - 1)^{8} \)
|
| $89$ |
\( T^{8} - 10 T^{7} + \cdots - 302336 \)
|
| $97$ |
\( T^{8} - 448 T^{6} + \cdots + 7105024 \)
|
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