Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,4,Mod(1,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, 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("930.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 930.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: | \(54.8717763053\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 226x^{2} + 1606x - 2280 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 226x^{2} + 1606x - 2280 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 15\nu^{2} - 106\nu - 630 ) / 30 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu^{2} - 176\nu + 510 ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} - 5\nu^{2} + 196\nu - 520 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{3} - 9\beta_{2} + 6\beta _1 + 221 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 211\beta_{3} + 241\beta_{2} - 30\beta _1 - 1949 ) / 2 \)
|
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 | 5.00000 | −6.00000 | −25.7949 | −8.00000 | 9.00000 | −10.0000 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 3.00000 | 4.00000 | 5.00000 | −6.00000 | −13.9863 | −8.00000 | 9.00000 | −10.0000 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 3.00000 | 4.00000 | 5.00000 | −6.00000 | 19.3106 | −8.00000 | 9.00000 | −10.0000 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 3.00000 | 4.00000 | 5.00000 | −6.00000 | 19.4706 | −8.00000 | 9.00000 | −10.0000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(31\) | \(1\) |
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 | 930.4.a.m | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 930.4.a.m | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + T_{7}^{3} - 806T_{7}^{2} + 966T_{7} + 135648 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(930))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( (T - 5)^{4} \)
|
| $7$ |
\( T^{4} + T^{3} + \cdots + 135648 \)
|
| $11$ |
\( T^{4} - 73 T^{3} + \cdots - 72000 \)
|
| $13$ |
\( T^{4} - 68 T^{3} + \cdots + 8786088 \)
|
| $17$ |
\( T^{4} - 54 T^{3} + \cdots + 7380480 \)
|
| $19$ |
\( T^{4} - 79 T^{3} + \cdots + 5824032 \)
|
| $23$ |
\( T^{4} - 39 T^{3} + \cdots + 8489200 \)
|
| $29$ |
\( T^{4} + 6 T^{3} + \cdots + 199888 \)
|
| $31$ |
\( (T + 31)^{4} \)
|
| $37$ |
\( T^{4} - 202 T^{3} + \cdots - 225057808 \)
|
| $41$ |
\( T^{4} - 162 T^{3} + \cdots + 118519200 \)
|
| $43$ |
\( T^{4} - 821 T^{3} + \cdots + 9953600 \)
|
| $47$ |
\( T^{4} + \cdots - 1234984320 \)
|
| $53$ |
\( T^{4} + \cdots - 10202456080 \)
|
| $59$ |
\( T^{4} + \cdots + 2410434000 \)
|
| $61$ |
\( T^{4} + \cdots - 53853138000 \)
|
| $67$ |
\( T^{4} + \cdots + 339499668688 \)
|
| $71$ |
\( T^{4} + \cdots + 73035432000 \)
|
| $73$ |
\( T^{4} + \cdots - 4356585516 \)
|
| $79$ |
\( T^{4} + \cdots - 46655324448 \)
|
| $83$ |
\( T^{4} + \cdots - 23729850048 \)
|
| $89$ |
\( T^{4} + \cdots + 410859615708 \)
|
| $97$ |
\( T^{4} + \cdots + 196480465680 \)
|
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