Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7254,2,Mod(1,7254)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7254, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("7254.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7254 = 2 \cdot 3^{2} \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7254.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: | \(57.9234816262\) |
| 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: |
\( x^{7} - 2x^{6} - 13x^{5} + 31x^{4} + 15x^{3} - 45x^{2} - 14x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{7} - 2x^{6} - 13x^{5} + 31x^{4} + 15x^{3} - 45x^{2} - 14x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 12\nu^{4} + 30\nu^{3} + 2\nu^{2} - 28\nu + 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 12\nu^{3} + 18\nu^{2} + 9\nu - 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 4\nu^{5} - 10\nu^{4} + 54\nu^{3} - 32\nu^{2} - 44\nu + 13 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 5\nu^{5} + 9\nu^{4} - 67\nu^{3} + 49\nu^{2} + 61\nu - 22 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{6} - 8\nu^{5} - 32\nu^{4} + 114\nu^{3} - 50\nu^{2} - 86\nu + 33 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{5} - 3\beta_{4} + \beta_{2} + 9\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 10\beta_{4} + 11\beta_{3} - 13\beta_{2} - 18\beta _1 + 35 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} - 24\beta_{5} - 44\beta_{4} - 6\beta_{3} + 17\beta_{2} + 99\beta _1 - 63 \)
|
| \(\nu^{6}\) | \(=\) |
\( 14\beta_{6} + 12\beta_{5} + 120\beta_{4} + 118\beta_{3} - 148\beta_{2} - 258\beta _1 + 375 \)
|
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 | 0 | 1.00000 | −3.45475 | 0 | 0.266571 | 1.00000 | 0 | −3.45475 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | −2.30001 | 0 | 2.71547 | 1.00000 | 0 | −2.30001 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | −2.23287 | 0 | −4.08675 | 1.00000 | 0 | −2.23287 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | 0.430208 | 0 | −2.41882 | 1.00000 | 0 | 0.430208 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0 | 1.00000 | 0.824459 | 0 | 4.84416 | 1.00000 | 0 | 0.824459 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0 | 1.00000 | 3.75511 | 0 | 2.45942 | 1.00000 | 0 | 3.75511 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 0 | 1.00000 | 3.97785 | 0 | 1.21995 | 1.00000 | 0 | 3.97785 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(13\) | \( +1 \) |
| \(31\) | \( +1 \) |
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 | 7254.2.a.br | yes | 7 |
| 3.b | odd | 2 | 1 | 7254.2.a.bo | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7254.2.a.bo | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 7254.2.a.br | yes | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7254))\):
|
\( T_{5}^{7} - T_{5}^{6} - 26T_{5}^{5} + 9T_{5}^{4} + 194T_{5}^{3} + 39T_{5}^{2} - 271T_{5} + 94 \)
|
|
\( T_{7}^{7} - 5T_{7}^{6} - 18T_{7}^{5} + 109T_{7}^{4} - 16T_{7}^{3} - 449T_{7}^{2} + 509T_{7} - 104 \)
|
|
\( T_{11}^{7} - 7T_{11}^{6} + 83T_{11}^{4} - 112T_{11}^{3} - 211T_{11}^{2} + 401T_{11} - 74 \)
|
|
\( T_{17}^{7} - 3T_{17}^{6} - 60T_{17}^{5} + 175T_{17}^{4} + 748T_{17}^{3} - 1923T_{17}^{2} - 1915T_{17} + 2598 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} - T^{6} + \cdots + 94 \)
|
| $7$ |
\( T^{7} - 5 T^{6} + \cdots - 104 \)
|
| $11$ |
\( T^{7} - 7 T^{6} + \cdots - 74 \)
|
| $13$ |
\( (T + 1)^{7} \)
|
| $17$ |
\( T^{7} - 3 T^{6} + \cdots + 2598 \)
|
| $19$ |
\( T^{7} - 5 T^{6} + \cdots - 1572 \)
|
| $23$ |
\( T^{7} - 8 T^{6} + \cdots - 6528 \)
|
| $29$ |
\( T^{7} + T^{6} + \cdots - 842 \)
|
| $31$ |
\( (T + 1)^{7} \)
|
| $37$ |
\( T^{7} - 3 T^{6} + \cdots + 102 \)
|
| $41$ |
\( T^{7} - 15 T^{6} + \cdots - 35334 \)
|
| $43$ |
\( T^{7} - T^{6} + \cdots - 14176 \)
|
| $47$ |
\( T^{7} + 4 T^{6} + \cdots + 51472 \)
|
| $53$ |
\( T^{7} - 34 T^{6} + \cdots + 175968 \)
|
| $59$ |
\( T^{7} - 12 T^{6} + \cdots + 99072 \)
|
| $61$ |
\( T^{7} + 6 T^{6} + \cdots - 1048 \)
|
| $67$ |
\( T^{7} - 19 T^{6} + \cdots + 1772800 \)
|
| $71$ |
\( T^{7} + 6 T^{6} + \cdots - 2133232 \)
|
| $73$ |
\( T^{7} - 13 T^{6} + \cdots + 41798 \)
|
| $79$ |
\( T^{7} + T^{6} + \cdots + 82816 \)
|
| $83$ |
\( T^{7} + T^{6} + \cdots - 19208286 \)
|
| $89$ |
\( T^{7} + 8 T^{6} + \cdots - 3456 \)
|
| $97$ |
\( T^{7} - 28 T^{6} + \cdots - 2970376 \)
|
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