Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [966,4,Mod(1,966)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(966, 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("966.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 966.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: | \(56.9958450655\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 267x^{3} + 1502x^{2} + 1857x + 198 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 267x^{3} + 1502x^{2} + 1857x + 198 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -8\nu^{4} - 11\nu^{3} + 2031\nu^{2} - 5365\nu - 19320 ) / 1086 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 13\nu^{4} - 50\nu^{3} - 3504\nu^{2} + 24533\nu + 6960 ) / 1086 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{4} + 133\nu^{3} + 2001\nu^{2} - 25369\nu - 77592 ) / 1086 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{4} - 13\nu^{3} - 1845\nu^{2} + 10690\nu + 8398 ) / 181 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - 2\beta_{2} + 2\beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{4} + 4\beta_{3} + 22\beta_{2} - 2\beta _1 + 434 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 127\beta_{4} - 6\beta_{3} - 298\beta_{2} + 184\beta _1 - 1138 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2797\beta_{4} + 1032\beta_{3} + 7746\beta_{2} - 2898\beta _1 + 102310 ) / 4 \)
|
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 |
|
−2.00000 | 3.00000 | 4.00000 | −17.6455 | −6.00000 | −7.00000 | −8.00000 | 9.00000 | 35.2910 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 3.00000 | 4.00000 | −8.30883 | −6.00000 | −7.00000 | −8.00000 | 9.00000 | 16.6177 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 3.00000 | 4.00000 | 3.69480 | −6.00000 | −7.00000 | −8.00000 | 9.00000 | −7.38960 | ||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 3.00000 | 4.00000 | 7.50994 | −6.00000 | −7.00000 | −8.00000 | 9.00000 | −15.0199 | ||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 3.00000 | 4.00000 | 14.7496 | −6.00000 | −7.00000 | −8.00000 | 9.00000 | −29.4991 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(23\) | \(-1\) |
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 | 966.4.a.n | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 966.4.a.n | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} - 334T_{5}^{3} + 795T_{5}^{2} + 17676T_{5} - 60004 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(966))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{5} \)
|
| $3$ |
\( (T - 3)^{5} \)
|
| $5$ |
\( T^{5} - 334 T^{3} + \cdots - 60004 \)
|
| $7$ |
\( (T + 7)^{5} \)
|
| $11$ |
\( T^{5} - 18 T^{4} + \cdots + 4096576 \)
|
| $13$ |
\( T^{5} + 22 T^{4} + \cdots + 39550252 \)
|
| $17$ |
\( T^{5} - 44 T^{4} + \cdots + 256399952 \)
|
| $19$ |
\( T^{5} + \cdots - 3376969856 \)
|
| $23$ |
\( (T - 23)^{5} \)
|
| $29$ |
\( T^{5} + 39 T^{4} + \cdots + 499949716 \)
|
| $31$ |
\( T^{5} + \cdots + 129027454688 \)
|
| $37$ |
\( T^{5} + \cdots - 1752755460964 \)
|
| $41$ |
\( T^{5} + \cdots + 1347225491444 \)
|
| $43$ |
\( T^{5} + \cdots + 198454918768 \)
|
| $47$ |
\( T^{5} + \cdots - 3919588628032 \)
|
| $53$ |
\( T^{5} + \cdots + 460114266352 \)
|
| $59$ |
\( T^{5} + \cdots - 88518485664 \)
|
| $61$ |
\( T^{5} + \cdots + 4528088303864 \)
|
| $67$ |
\( T^{5} + \cdots - 1352835082304 \)
|
| $71$ |
\( T^{5} + \cdots + 10575799339392 \)
|
| $73$ |
\( T^{5} + \cdots - 202562391842272 \)
|
| $79$ |
\( T^{5} + \cdots + 36500687712256 \)
|
| $83$ |
\( T^{5} + \cdots + 3555275628608 \)
|
| $89$ |
\( T^{5} + \cdots + 10886725290664 \)
|
| $97$ |
\( T^{5} + \cdots - 190187620864468 \)
|
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