Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [348,4,Mod(1,348)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(348, 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("348.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 348 = 2^{2} \cdot 3 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 348.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: | \(20.5326646820\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - 2x^{3} - 151x^{2} + 8x + 444 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{4} - 2x^{3} - 151x^{2} + 8x + 444 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 3\nu^{2} - 154\nu - 384 ) / 36 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 3\nu^{2} + 148\nu - 84 ) / 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} + 6\beta_{2} + \beta _1 + 78 \)
|
| \(\nu^{3}\) | \(=\) |
\( -6\beta_{3} + 18\beta_{2} + 151\beta _1 + 150 \)
|
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 | −3.00000 | 0 | −9.27764 | 0 | 21.4779 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −6.61135 | 0 | −5.60269 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | 5.18211 | 0 | −7.90711 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.00000 | 0 | 19.7069 | 0 | −18.9681 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(29\) | \(1\) |
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 | 348.4.a.e | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1044.4.a.g | 4 | ||
| 4.b | odd | 2 | 1 | 1392.4.a.r | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 348.4.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1044.4.a.g | 4 | 3.b | odd | 2 | 1 | ||
| 1392.4.a.r | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 9T_{5}^{3} - 232T_{5}^{2} + 96T_{5} + 6264 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(348))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( T^{4} - 9 T^{3} + \cdots + 6264 \)
|
| $7$ |
\( T^{4} + 11 T^{3} + \cdots - 18048 \)
|
| $11$ |
\( T^{4} + 22 T^{3} + \cdots - 104040 \)
|
| $13$ |
\( T^{4} + 8 T^{3} + \cdots + 2960280 \)
|
| $17$ |
\( T^{4} - 7 T^{3} + \cdots + 15408234 \)
|
| $19$ |
\( T^{4} - T^{3} + \cdots + 5130480 \)
|
| $23$ |
\( T^{4} + 142 T^{3} + \cdots - 124167168 \)
|
| $29$ |
\( (T + 29)^{4} \)
|
| $31$ |
\( T^{4} + \cdots - 1632868416 \)
|
| $37$ |
\( T^{4} + 409 T^{3} + \cdots - 43107000 \)
|
| $41$ |
\( T^{4} + \cdots + 3116480328 \)
|
| $43$ |
\( T^{4} + 1151 T^{3} + \cdots + 823412544 \)
|
| $47$ |
\( T^{4} + 787 T^{3} + \cdots + 433337808 \)
|
| $53$ |
\( T^{4} + \cdots - 2525112144 \)
|
| $59$ |
\( T^{4} + \cdots + 18478066752 \)
|
| $61$ |
\( T^{4} + 946 T^{3} + \cdots + 720657680 \)
|
| $67$ |
\( T^{4} + 1320 T^{3} + \cdots - 557735088 \)
|
| $71$ |
\( T^{4} + \cdots - 166456034688 \)
|
| $73$ |
\( T^{4} + \cdots - 161909407344 \)
|
| $79$ |
\( T^{4} + \cdots - 77529031584 \)
|
| $83$ |
\( T^{4} + \cdots + 1683970560 \)
|
| $89$ |
\( T^{4} + \cdots - 37104909300 \)
|
| $97$ |
\( T^{4} + \cdots - 51396069888 \)
|
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