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: | \(1\) |
| 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: |
\( x^{4} - 8244x^{2} - 405980x - 5609007 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{6} \) |
| 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
\( x^{4} - 8244x^{2} - 405980x - 5609007 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 71\nu^{2} - 13936\nu - 316308 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{3} - 142\nu^{2} - 27980\nu - 632616 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -44\nu^{3} + 1688\nu^{2} + 298024\nu + 6439404 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + 14\beta_1 ) / 108 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} - 34\beta_{2} + 542\beta _1 + 222588 ) / 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 213\beta_{3} - 9382\beta_{2} + 136412\beta _1 + 32884380 ) / 108 \)
|
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 | −549.316 | 0 | −343.000 | −512.000 | 0 | 4394.53 | ||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | 0 | 64.0000 | −114.609 | 0 | −343.000 | −512.000 | 0 | 916.869 | |||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | 0 | 64.0000 | 99.4841 | 0 | −343.000 | −512.000 | 0 | −795.873 | |||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | 0 | 64.0000 | 436.440 | 0 | −343.000 | −512.000 | 0 | −3491.52 | |||||||||||||||||||||||||||||||
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.o | ✓ | 4 |
| 3.b | odd | 2 | 1 | 378.8.a.r | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 378.8.a.o | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 378.8.a.r | yes | 4 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 128T_{5}^{3} - 249438T_{5}^{2} - 4912960T_{5} + 2733491425 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(378))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 2733491425 \)
|
| $7$ |
\( (T + 343)^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 28681455494975 \)
|
| $13$ |
\( T^{4} + \cdots - 12\!\cdots\!52 \)
|
| $17$ |
\( T^{4} + \cdots + 20\!\cdots\!72 \)
|
| $19$ |
\( T^{4} + \cdots + 28\!\cdots\!85 \)
|
| $23$ |
\( T^{4} + \cdots + 39\!\cdots\!29 \)
|
| $29$ |
\( T^{4} + \cdots + 15\!\cdots\!92 \)
|
| $31$ |
\( T^{4} + \cdots - 13\!\cdots\!39 \)
|
| $37$ |
\( T^{4} + \cdots + 46\!\cdots\!01 \)
|
| $41$ |
\( T^{4} + \cdots - 43\!\cdots\!63 \)
|
| $43$ |
\( T^{4} + \cdots - 64\!\cdots\!20 \)
|
| $47$ |
\( T^{4} + \cdots - 15\!\cdots\!32 \)
|
| $53$ |
\( T^{4} + \cdots - 65\!\cdots\!48 \)
|
| $59$ |
\( T^{4} + \cdots - 88\!\cdots\!60 \)
|
| $61$ |
\( T^{4} + \cdots - 62\!\cdots\!44 \)
|
| $67$ |
\( T^{4} + \cdots - 46\!\cdots\!88 \)
|
| $71$ |
\( T^{4} + \cdots + 12\!\cdots\!85 \)
|
| $73$ |
\( T^{4} + \cdots - 44\!\cdots\!36 \)
|
| $79$ |
\( T^{4} + \cdots + 65\!\cdots\!48 \)
|
| $83$ |
\( T^{4} + \cdots + 18\!\cdots\!72 \)
|
| $89$ |
\( T^{4} + \cdots - 55\!\cdots\!15 \)
|
| $97$ |
\( T^{4} + \cdots - 23\!\cdots\!60 \)
|
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