Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9282,2,Mod(1,9282)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9282, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("9282.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9282 = 2 \cdot 3 \cdot 7 \cdot 13 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9282.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: | \(74.1171431562\) |
| 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: |
\( x^{7} - 3x^{6} - 23x^{5} + 70x^{4} + 115x^{3} - 422x^{2} + 118x + 208 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| 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
\( x^{7} - 3x^{6} - 23x^{5} + 70x^{4} + 115x^{3} - 422x^{2} + 118x + 208 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -29\nu^{6} + 75\nu^{5} + 407\nu^{4} - 1134\nu^{3} + 343\nu^{2} + 2412\nu - 5116 ) / 2110 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{6} - 11\nu^{5} + 171\nu^{4} + 352\nu^{3} - 1125\nu^{2} - 2008\nu + 2112 ) / 422 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 189\nu^{6} - 125\nu^{5} - 4617\nu^{4} + 2734\nu^{3} + 26577\nu^{2} - 19212\nu - 12714 ) / 2110 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 189\nu^{6} - 125\nu^{5} - 4617\nu^{4} + 2734\nu^{3} + 28687\nu^{2} - 19212\nu - 29594 ) / 2110 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 273\nu^{6} - 415\nu^{5} - 6669\nu^{4} + 8638\nu^{3} + 40499\nu^{2} - 47444\nu - 27508 ) / 2110 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + 2\beta_{5} + 3\beta_{3} + 10\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{6} + 19\beta_{5} - 15\beta_{4} + 4\beta_{3} - 7\beta_{2} - 4\beta _1 + 100 \)
|
| \(\nu^{5}\) | \(=\) |
\( -29\beta_{6} + 49\beta_{5} + 4\beta_{4} + 60\beta_{3} + 116\beta _1 + 33 \)
|
| \(\nu^{6}\) | \(=\) |
\( -78\beta_{6} + 327\beta_{5} - 212\beta_{4} + 94\beta_{3} - 171\beta_{2} - 64\beta _1 + 1407 \)
|
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 | 1.00000 | 1.00000 | −3.71682 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | −3.71682 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 1.00000 | 1.00000 | −3.10233 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | −3.10233 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 1.00000 | 1.00000 | −0.555964 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | −0.555964 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.00000 | 1.00000 | 1.35432 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.35432 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.00000 | 1.00000 | 2.04736 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | 2.04736 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.00000 | 1.00000 | 2.81166 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | 2.81166 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 1.00000 | 1.00000 | 4.16178 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | 4.16178 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(13\) | \(1\) |
| \(17\) | \(-1\) |
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 | 9282.2.a.ce | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9282.2.a.ce | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9282))\):
|
\( T_{5}^{7} - 3T_{5}^{6} - 23T_{5}^{5} + 70T_{5}^{4} + 115T_{5}^{3} - 422T_{5}^{2} + 118T_{5} + 208 \)
|
|
\( T_{11}^{7} - 7T_{11}^{6} - 35T_{11}^{5} + 267T_{11}^{4} + 138T_{11}^{3} - 2172T_{11}^{2} + 3208T_{11} - 1408 \)
|
|
\( T_{19}^{7} + 2T_{19}^{6} - 57T_{19}^{5} + 35T_{19}^{4} + 539T_{19}^{3} - 762T_{19}^{2} + 208T_{19} - 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
|
| $3$ |
\( (T - 1)^{7} \)
|
| $5$ |
\( T^{7} - 3 T^{6} + \cdots + 208 \)
|
| $7$ |
\( (T + 1)^{7} \)
|
| $11$ |
\( T^{7} - 7 T^{6} + \cdots - 1408 \)
|
| $13$ |
\( (T + 1)^{7} \)
|
| $17$ |
\( (T - 1)^{7} \)
|
| $19$ |
\( T^{7} + 2 T^{6} + \cdots - 16 \)
|
| $23$ |
\( T^{7} - 12 T^{6} + \cdots - 9152 \)
|
| $29$ |
\( T^{7} - 18 T^{6} + \cdots - 57476 \)
|
| $31$ |
\( T^{7} + 6 T^{6} + \cdots - 31096 \)
|
| $37$ |
\( T^{7} - 5 T^{6} + \cdots + 103168 \)
|
| $41$ |
\( T^{7} - 10 T^{6} + \cdots + 4 \)
|
| $43$ |
\( T^{7} - 18 T^{6} + \cdots + 256 \)
|
| $47$ |
\( T^{7} - 3 T^{6} + \cdots - 1591502 \)
|
| $53$ |
\( T^{7} - 18 T^{6} + \cdots - 919024 \)
|
| $59$ |
\( T^{7} - 20 T^{6} + \cdots + 1471184 \)
|
| $61$ |
\( T^{7} - 19 T^{6} + \cdots + 24224 \)
|
| $67$ |
\( T^{7} + 16 T^{6} + \cdots - 70936 \)
|
| $71$ |
\( T^{7} - 5 T^{6} + \cdots + 258464 \)
|
| $73$ |
\( T^{7} - 2 T^{6} + \cdots + 2939648 \)
|
| $79$ |
\( T^{7} - 12 T^{6} + \cdots + 6656 \)
|
| $83$ |
\( T^{7} - 11 T^{6} + \cdots - 62072 \)
|
| $89$ |
\( T^{7} - 6 T^{6} + \cdots + 4184 \)
|
| $97$ |
\( T^{7} + 3 T^{6} + \cdots + 336176 \)
|
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