Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [946,2,Mod(1,946)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(946, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("946.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 946 = 2 \cdot 11 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 946.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: | \(7.55384803121\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 11x^{5} + 31x^{4} + 39x^{3} - 91x^{2} - 48x + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 11x^{5} + 31x^{4} + 39x^{3} - 91x^{2} - 48x + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 5\nu^{5} - 19\nu^{4} - 57\nu^{3} + 79\nu^{2} + 157\nu - 56 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{6} + \nu^{5} + 41\nu^{4} - 5\nu^{3} - 141\nu^{2} - 7\nu + 72 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} + \nu^{5} + 41\nu^{4} + 11\nu^{3} - 173\nu^{2} - 87\nu + 168 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 11\nu^{5} + 13\nu^{4} + 87\nu^{3} - 129\nu^{2} - 147\nu + 168 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 2\beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} - \beta_{4} + 2\beta_{3} + 11\beta_{2} + 11\beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + 13\beta_{5} - 11\beta_{4} + 5\beta_{3} + 27\beta_{2} + 53\beta _1 + 28 \)
|
| \(\nu^{6}\) | \(=\) |
\( 14\beta_{6} + 30\beta_{5} - 21\beta_{4} + 29\beta_{3} + 109\beta_{2} + 107\beta _1 + 170 \)
|
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 |
|
1.00000 | −2.35403 | 1.00000 | −2.03689 | −2.35403 | −2.66006 | 1.00000 | 2.54146 | −2.03689 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.82849 | 1.00000 | 2.89393 | −1.82849 | 4.36216 | 1.00000 | 0.343373 | 2.89393 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.23095 | 1.00000 | 3.67721 | −1.23095 | −4.90006 | 1.00000 | −1.48475 | 3.67721 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0.750731 | 1.00000 | 1.13207 | 0.750731 | 2.59042 | 1.00000 | −2.43640 | 1.13207 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.11676 | 1.00000 | 2.64074 | 2.11676 | 0.778817 | 1.00000 | 1.48068 | 2.64074 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.47575 | 1.00000 | −2.56726 | 2.47575 | 2.09328 | 1.00000 | 3.12936 | −2.56726 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 3.07023 | 1.00000 | 2.26020 | 3.07023 | −4.26455 | 1.00000 | 6.42629 | 2.26020 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(-1\) |
| \(43\) | \(-1\) |
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 | 946.2.a.k | ✓ | 7 |
| 3.b | odd | 2 | 1 | 8514.2.a.bl | 7 | ||
| 4.b | odd | 2 | 1 | 7568.2.a.bg | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 946.2.a.k | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 7568.2.a.bg | 7 | 4.b | odd | 2 | 1 | ||
| 8514.2.a.bl | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(946))\):
|
\( T_{3}^{7} - 3T_{3}^{6} - 11T_{3}^{5} + 31T_{3}^{4} + 39T_{3}^{3} - 91T_{3}^{2} - 48T_{3} + 64 \)
|
|
\( T_{5}^{7} - 8T_{5}^{6} + 9T_{5}^{5} + 72T_{5}^{4} - 185T_{5}^{3} - 70T_{5}^{2} + 542T_{5} - 376 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots + 64 \)
|
| $5$ |
\( T^{7} - 8 T^{6} + \cdots - 376 \)
|
| $7$ |
\( T^{7} + 2 T^{6} + \cdots + 1024 \)
|
| $11$ |
\( (T - 1)^{7} \)
|
| $13$ |
\( T^{7} - 4 T^{6} + \cdots - 112 \)
|
| $17$ |
\( T^{7} - 10 T^{6} + \cdots - 2232 \)
|
| $19$ |
\( T^{7} + 2 T^{6} + \cdots + 11240 \)
|
| $23$ |
\( T^{7} + 4 T^{6} + \cdots - 118336 \)
|
| $29$ |
\( T^{7} - 5 T^{6} + \cdots + 300 \)
|
| $31$ |
\( T^{7} + 2 T^{6} + \cdots - 4112 \)
|
| $37$ |
\( T^{7} - 6 T^{6} + \cdots - 37836 \)
|
| $41$ |
\( T^{7} - 4 T^{6} + \cdots - 4144 \)
|
| $43$ |
\( (T - 1)^{7} \)
|
| $47$ |
\( T^{7} + 6 T^{6} + \cdots - 144 \)
|
| $53$ |
\( T^{7} - T^{6} + \cdots - 8016 \)
|
| $59$ |
\( T^{7} + 4 T^{6} + \cdots + 135360 \)
|
| $61$ |
\( T^{7} - 9 T^{6} + \cdots + 27072 \)
|
| $67$ |
\( T^{7} + 6 T^{6} + \cdots + 576 \)
|
| $71$ |
\( T^{7} - 304 T^{5} + \cdots + 72192 \)
|
| $73$ |
\( T^{7} - 13 T^{6} + \cdots - 51872 \)
|
| $79$ |
\( T^{7} + 31 T^{6} + \cdots - 17970 \)
|
| $83$ |
\( T^{7} + 11 T^{6} + \cdots - 756224 \)
|
| $89$ |
\( T^{7} - 10 T^{6} + \cdots + 11918080 \)
|
| $97$ |
\( T^{7} - 23 T^{6} + \cdots + 719292 \)
|
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