Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2312,4,Mod(1,2312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2312.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2312 = 2^{3} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2312.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: | \(136.412415933\) |
| 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: |
\( x^{6} - x^{5} - 92x^{4} + 123x^{3} + 2120x^{2} - 3573x - 261 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| 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
\( x^{6} - x^{5} - 92x^{4} + 123x^{3} + 2120x^{2} - 3573x - 261 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} + 35\nu^{3} - 142\nu^{2} + 561\nu - 507 ) / 51 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + \nu - 31 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 47\nu^{2} + 151\nu - 9 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{5} + 6\nu^{4} + 121\nu^{3} - 335\nu^{2} - 1224\nu + 1689 ) / 51 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta _1 + 31 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{3} - 2\beta_{2} + 45\beta _1 - 22 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{5} + 3\beta_{4} + 50\beta_{3} - 6\beta_{2} - 63\beta _1 + 1400 \)
|
| \(\nu^{5}\) | \(=\) |
\( 44\beta_{5} + 9\beta_{4} + 43\beta_{3} - 139\beta_{2} + 2089\beta _1 - 1479 \)
|
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 | −7.86357 | 0 | −12.8315 | 0 | 9.16275 | 0 | 34.8357 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −7.22873 | 0 | 20.1017 | 0 | −12.5412 | 0 | 25.2545 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.81785 | 0 | 4.22994 | 0 | 16.4587 | 0 | −19.0597 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.929859 | 0 | −11.7267 | 0 | −35.7679 | 0 | −26.1354 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 5.84285 | 0 | −13.3245 | 0 | 23.1160 | 0 | 7.13888 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 5.99716 | 0 | 10.5510 | 0 | −7.42839 | 0 | 8.96594 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(17\) | \(1\) |
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 | 2312.4.a.f | ✓ | 6 |
| 17.b | even | 2 | 1 | 2312.4.a.j | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2312.4.a.f | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2312.4.a.j | yes | 6 | 17.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 7T_{3}^{5} - 72T_{3}^{4} - 461T_{3}^{3} + 1224T_{3}^{2} + 7087T_{3} + 5219 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2312))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 7 T^{5} + \cdots + 5219 \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots - 1798731 \)
|
| $7$ |
\( T^{6} + 7 T^{5} + \cdots - 11616095 \)
|
| $11$ |
\( T^{6} + 72 T^{5} + \cdots - 127272981 \)
|
| $13$ |
\( T^{6} - 15 T^{5} + \cdots + 333721067 \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( T^{6} + \cdots - 845069773383 \)
|
| $23$ |
\( T^{6} + \cdots - 1268932834873 \)
|
| $29$ |
\( T^{6} + \cdots + 44210248371 \)
|
| $31$ |
\( T^{6} + \cdots + 2884448721537 \)
|
| $37$ |
\( T^{6} + \cdots + 20654708215488 \)
|
| $41$ |
\( T^{6} + \cdots - 38047968485824 \)
|
| $43$ |
\( T^{6} + \cdots - 3594420425007 \)
|
| $47$ |
\( T^{6} + \cdots - 61791151753809 \)
|
| $53$ |
\( T^{6} + \cdots - 81\!\cdots\!69 \)
|
| $59$ |
\( T^{6} + \cdots + 660969074991296 \)
|
| $61$ |
\( T^{6} + \cdots + 11\!\cdots\!23 \)
|
| $67$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $71$ |
\( T^{6} + \cdots + 158233687966272 \)
|
| $73$ |
\( T^{6} + \cdots + 23\!\cdots\!15 \)
|
| $79$ |
\( T^{6} + \cdots - 12\!\cdots\!28 \)
|
| $83$ |
\( T^{6} + \cdots - 11\!\cdots\!01 \)
|
| $89$ |
\( T^{6} + \cdots + 284179645674688 \)
|
| $97$ |
\( T^{6} + \cdots - 33\!\cdots\!83 \)
|
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