Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,8,Mod(1,378)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(378, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 378.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: | \(118.081539633\) |
| 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: |
\( x^{4} - x^{3} - 4051x^{2} - 71519x + 526930 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{6} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 4051x^{2} - 71519x + 526930 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 50\nu^{2} - 3741\nu - 157030 ) / 448 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -19\nu^{3} + 394\nu^{2} + 72423\nu + 260626 ) / 448 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -15\nu^{3} + 594\nu^{2} + 33267\nu - 361670 ) / 448 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 4\beta _1 + 13 ) / 54 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 17\beta_{2} + 338\beta _1 + 109391 ) / 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3791\beta_{3} + 2891\beta_{2} + 22256\beta _1 + 3058703 ) / 54 \)
|
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 |
|
8.00000 | 0 | 64.0000 | −438.734 | 0 | −343.000 | 512.000 | 0 | −3509.87 | ||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | 0 | 64.0000 | −84.2838 | 0 | −343.000 | 512.000 | 0 | −674.270 | |||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 0 | 64.0000 | 54.6225 | 0 | −343.000 | 512.000 | 0 | 436.980 | |||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | 0 | 64.0000 | 368.395 | 0 | −343.000 | 512.000 | 0 | 2947.16 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
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 | 378.8.a.q | yes | 4 |
| 3.b | odd | 2 | 1 | 378.8.a.p | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 378.8.a.p | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 378.8.a.q | yes | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 100T_{5}^{3} - 164145T_{5}^{2} - 5117900T_{5} + 744100000 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(378))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 100 T^{3} + \cdots + 744100000 \)
|
| $7$ |
\( (T + 343)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 365615787147520 \)
|
| $13$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $17$ |
\( T^{4} + \cdots + 87\!\cdots\!25 \)
|
| $19$ |
\( T^{4} + \cdots - 13\!\cdots\!68 \)
|
| $23$ |
\( T^{4} + \cdots - 13\!\cdots\!08 \)
|
| $29$ |
\( T^{4} + \cdots + 33\!\cdots\!20 \)
|
| $31$ |
\( T^{4} + \cdots + 16\!\cdots\!40 \)
|
| $37$ |
\( T^{4} + \cdots + 11\!\cdots\!28 \)
|
| $41$ |
\( T^{4} + \cdots + 44\!\cdots\!84 \)
|
| $43$ |
\( T^{4} + \cdots - 52\!\cdots\!51 \)
|
| $47$ |
\( T^{4} + \cdots + 28\!\cdots\!12 \)
|
| $53$ |
\( T^{4} + \cdots + 95\!\cdots\!04 \)
|
| $59$ |
\( T^{4} + \cdots - 39\!\cdots\!63 \)
|
| $61$ |
\( T^{4} + \cdots + 39\!\cdots\!80 \)
|
| $67$ |
\( T^{4} + \cdots - 57\!\cdots\!88 \)
|
| $71$ |
\( T^{4} + \cdots - 20\!\cdots\!40 \)
|
| $73$ |
\( T^{4} + \cdots - 43\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 90\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 25\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots - 55\!\cdots\!60 \)
|
| $97$ |
\( T^{4} + \cdots + 50\!\cdots\!64 \)
|
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