Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8030,2,Mod(1,8030)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8030, 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("8030.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8030.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: | \(64.1198728231\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{7} - x^{6} - 11x^{5} + 17x^{4} + 9x^{3} - 15x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{7} - x^{6} - 11x^{5} + 17x^{4} + 9x^{3} - 15x^{2} + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 10\nu^{3} + 7\nu^{2} + 8\nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 12\nu^{4} + 5\nu^{3} + 24\nu^{2} - 8 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{6} - \nu^{5} - 22\nu^{4} + 22\nu^{3} + 23\nu^{2} - 6\nu - 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} + 2\nu^{5} + 34\nu^{4} - 39\nu^{3} - 42\nu^{2} + 28\nu + 8 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{6} - \nu^{5} - 22\nu^{4} + 23\nu^{3} + 25\nu^{2} - 13\nu - 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} - 2\beta_{5} - 2\beta_{3} + 2\beta_{2} + 7\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10\beta_{6} + 11\beta_{5} - 8\beta_{4} + 9\beta_{3} - 10\beta_{2} - 8\beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( -17\beta_{6} - 27\beta_{5} + 7\beta_{4} - 27\beta_{3} + 29\beta_{2} + 62\beta _1 - 49 \)
|
| \(\nu^{6}\) | \(=\) |
\( 101\beta_{6} + 118\beta_{5} - 72\beta_{4} + 96\beta_{3} - 106\beta_{2} - 131\beta _1 + 239 \)
|
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 | −1.93518 | 1.00000 | 1.00000 | 1.93518 | −4.53467 | −1.00000 | 0.744937 | −1.00000 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.61566 | 1.00000 | 1.00000 | 1.61566 | 3.03677 | −1.00000 | −0.389633 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.37595 | 1.00000 | 1.00000 | 1.37595 | −1.05505 | −1.00000 | −1.10676 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −0.550071 | 1.00000 | 1.00000 | 0.550071 | −3.11640 | −1.00000 | −2.69742 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.75674 | 1.00000 | 1.00000 | −1.75674 | 2.75784 | −1.00000 | 0.0861198 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.82268 | 1.00000 | 1.00000 | −2.82268 | 0.448361 | −1.00000 | 4.96752 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 2.89745 | 1.00000 | 1.00000 | −2.89745 | −4.53686 | −1.00000 | 5.39524 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(11\) | \(1\) |
| \(73\) | \(-1\) |
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 | 8030.2.a.ba | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8030.2.a.ba | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):
|
\( T_{3}^{7} - 2T_{3}^{6} - 12T_{3}^{5} + 14T_{3}^{4} + 54T_{3}^{3} - 13T_{3}^{2} - 82T_{3} - 34 \)
|
|
\( T_{7}^{7} + 7T_{7}^{6} - 10T_{7}^{5} - 128T_{7}^{4} - 27T_{7}^{3} + 615T_{7}^{2} + 308T_{7} - 254 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{7} \)
|
| $3$ |
\( T^{7} - 2 T^{6} + \cdots - 34 \)
|
| $5$ |
\( (T - 1)^{7} \)
|
| $7$ |
\( T^{7} + 7 T^{6} + \cdots - 254 \)
|
| $11$ |
\( (T + 1)^{7} \)
|
| $13$ |
\( T^{7} + 11 T^{6} + \cdots + 2804 \)
|
| $17$ |
\( T^{7} + 7 T^{6} + \cdots + 407 \)
|
| $19$ |
\( T^{7} - 57 T^{5} + \cdots + 211 \)
|
| $23$ |
\( T^{7} + 4 T^{6} + \cdots - 2 \)
|
| $29$ |
\( T^{7} - 4 T^{6} + \cdots - 517 \)
|
| $31$ |
\( T^{7} - 3 T^{6} + \cdots + 1264 \)
|
| $37$ |
\( T^{7} + 22 T^{6} + \cdots - 1343 \)
|
| $41$ |
\( T^{7} + 8 T^{6} + \cdots + 7346 \)
|
| $43$ |
\( T^{7} + 19 T^{6} + \cdots + 9244 \)
|
| $47$ |
\( T^{7} - 3 T^{6} + \cdots - 183728 \)
|
| $53$ |
\( T^{7} - 245 T^{5} + \cdots + 479378 \)
|
| $59$ |
\( T^{7} - 23 T^{6} + \cdots + 136264 \)
|
| $61$ |
\( T^{7} + 25 T^{6} + \cdots - 253432 \)
|
| $67$ |
\( T^{7} - 27 T^{6} + \cdots - 3358624 \)
|
| $71$ |
\( T^{7} - 11 T^{6} + \cdots + 197744 \)
|
| $73$ |
\( (T - 1)^{7} \)
|
| $79$ |
\( T^{7} - 34 T^{6} + \cdots - 986924 \)
|
| $83$ |
\( T^{7} + 27 T^{6} + \cdots + 1648352 \)
|
| $89$ |
\( T^{7} + 16 T^{6} + \cdots - 10837 \)
|
| $97$ |
\( T^{7} + 25 T^{6} + \cdots - 15161024 \)
|
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