Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [242,8,Mod(1,242)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(242, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("242.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 242 = 2 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 242.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: | \(75.5971761672\) |
| 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: |
\( x^{4} - 2x^{3} - 935x^{2} - 10836x - 31788 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 11 \) |
| 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
\( x^{4} - 2x^{3} - 935x^{2} - 10836x - 31788 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 24\nu^{2} + 739\nu - 2082 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 8\nu^{2} - 899\nu - 5334 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} - 32\nu^{2} - 2353\nu - 12426 ) / 12 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{3} - 10\beta_{2} + \beta _1 + 44 ) / 88 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 40\beta_{3} - 34\beta_{2} + 43\beta _1 + 41228 ) / 88 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 356\beta_{3} - 746\beta_{2} + 113\beta _1 + 76252 ) / 8 \)
|
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 | −89.8122 | 64.0000 | 433.883 | −718.498 | −1000.54 | 512.000 | 5879.24 | 3471.06 | ||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | −20.0001 | 64.0000 | −236.039 | −160.001 | 377.921 | 512.000 | −1787.00 | −1888.31 | |||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 47.3597 | 64.0000 | 499.352 | 378.877 | 1391.56 | 512.000 | 55.9387 | 3994.81 | |||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | 78.4527 | 64.0000 | −293.196 | 627.621 | −1424.94 | 512.000 | 3967.82 | −2345.56 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 242.8.a.m | yes | 4 |
| 11.b | odd | 2 | 1 | 242.8.a.l | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 242.8.a.l | ✓ | 4 | 11.b | odd | 2 | 1 | |
| 242.8.a.m | yes | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(242))\):
|
\( T_{3}^{4} - 16T_{3}^{3} - 8304T_{3}^{2} + 182016T_{3} + 6673968 \)
|
|
\( T_{7}^{4} + 656T_{7}^{3} - 2340240T_{7}^{2} - 1247214848T_{7} + 749784850352 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{4} \)
|
| $3$ |
\( T^{4} - 16 T^{3} + \cdots + 6673968 \)
|
| $5$ |
\( T^{4} + \cdots + 14994056025 \)
|
| $7$ |
\( T^{4} + \cdots + 749784850352 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + \cdots - 147342277215255 \)
|
| $17$ |
\( T^{4} + \cdots - 93\!\cdots\!19 \)
|
| $19$ |
\( T^{4} + \cdots + 26\!\cdots\!16 \)
|
| $23$ |
\( T^{4} + \cdots + 27\!\cdots\!64 \)
|
| $29$ |
\( T^{4} + \cdots - 13\!\cdots\!63 \)
|
| $31$ |
\( T^{4} + \cdots - 13\!\cdots\!84 \)
|
| $37$ |
\( T^{4} + \cdots - 12\!\cdots\!39 \)
|
| $41$ |
\( T^{4} + \cdots + 23\!\cdots\!49 \)
|
| $43$ |
\( T^{4} + \cdots + 40\!\cdots\!48 \)
|
| $47$ |
\( T^{4} + \cdots + 13\!\cdots\!72 \)
|
| $53$ |
\( T^{4} + \cdots - 72\!\cdots\!15 \)
|
| $59$ |
\( T^{4} + \cdots - 41\!\cdots\!60 \)
|
| $61$ |
\( T^{4} + \cdots + 29\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots + 54\!\cdots\!80 \)
|
| $73$ |
\( T^{4} + \cdots + 18\!\cdots\!08 \)
|
| $79$ |
\( T^{4} + \cdots - 19\!\cdots\!12 \)
|
| $83$ |
\( T^{4} + \cdots + 12\!\cdots\!88 \)
|
| $89$ |
\( T^{4} + \cdots - 44\!\cdots\!95 \)
|
| $97$ |
\( T^{4} + \cdots - 78\!\cdots\!59 \)
|
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