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: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 23x^{4} - 32x^{3} - 16x^{2} + 17x - 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 23x^{4} - 32x^{3} - 16x^{2} + 17x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 4\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 7\nu^{4} + 17\nu^{3} + 16\nu^{2} - 14\nu - 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{7} - 4\nu^{6} - 4\nu^{5} + 23\nu^{4} + 2\nu^{3} - 27\nu^{2} + 3\nu + 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 17\nu^{4} + 17\nu^{3} - 16\nu^{2} - 6\nu + 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 5\nu^{6} + 27\nu^{4} - 21\nu^{3} - 28\nu^{2} + 28\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} + 9\beta_{2} + 12\beta _1 + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 3\beta_{6} - 4\beta_{5} - 2\beta_{4} + 12\beta_{3} + 24\beta_{2} + 46\beta _1 + 37 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3\beta_{7} + 16\beta_{6} - 19\beta_{5} - 12\beta_{4} + 33\beta_{3} + 85\beta_{2} + 118\beta _1 + 141 \)
|
| \(\nu^{7}\) | \(=\) |
\( 16\beta_{7} + 53\beta_{6} - 68\beta_{5} - 33\beta_{4} + 132\beta_{3} + 252\beta_{2} + 392\beta _1 + 370 \)
|
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 | −3.14203 | 1.00000 | −1.00000 | −3.14203 | −3.42333 | 1.00000 | 6.87232 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.62474 | 1.00000 | −1.00000 | −2.62474 | 3.44910 | 1.00000 | 3.88926 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.936264 | 1.00000 | −1.00000 | −0.936264 | −3.94603 | 1.00000 | −2.12341 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.677086 | 1.00000 | −1.00000 | −0.677086 | 0.767746 | 1.00000 | −2.54155 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −0.0630085 | 1.00000 | −1.00000 | −0.0630085 | −2.56294 | 1.00000 | −2.99603 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.20960 | 1.00000 | −1.00000 | 1.20960 | 1.90541 | 1.00000 | −1.53686 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 1.29392 | 1.00000 | −1.00000 | 1.29392 | −0.0375670 | 1.00000 | −1.32576 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 1.93960 | 1.00000 | −1.00000 | 1.93960 | −0.152382 | 1.00000 | 0.762032 | −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.bb | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8030.2.a.bb | ✓ | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):
|
\( T_{3}^{8} + 3T_{3}^{7} - 8T_{3}^{6} - 20T_{3}^{5} + 23T_{3}^{4} + 32T_{3}^{3} - 16T_{3}^{2} - 17T_{3} - 1 \)
|
|
\( T_{7}^{8} + 4T_{7}^{7} - 17T_{7}^{6} - 66T_{7}^{5} + 72T_{7}^{4} + 222T_{7}^{3} - 135T_{7}^{2} - 32T_{7} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} + 3 T^{7} + \cdots - 1 \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( T^{8} + 4 T^{7} + \cdots - 1 \)
|
| $11$ |
\( (T - 1)^{8} \)
|
| $13$ |
\( T^{8} + 7 T^{7} + \cdots - 26669 \)
|
| $17$ |
\( T^{8} - 2 T^{7} + \cdots - 1697 \)
|
| $19$ |
\( T^{8} + T^{7} + \cdots + 68732 \)
|
| $23$ |
\( T^{8} + 16 T^{7} + \cdots + 6863 \)
|
| $29$ |
\( T^{8} + 5 T^{7} + \cdots + 17343 \)
|
| $31$ |
\( T^{8} + 22 T^{7} + \cdots + 1487424 \)
|
| $37$ |
\( T^{8} + 9 T^{7} + \cdots + 7501 \)
|
| $41$ |
\( T^{8} - 6 T^{7} + \cdots - 54324 \)
|
| $43$ |
\( T^{8} - 15 T^{7} + \cdots + 44816 \)
|
| $47$ |
\( T^{8} + 7 T^{7} + \cdots - 432 \)
|
| $53$ |
\( T^{8} + 11 T^{7} + \cdots - 108 \)
|
| $59$ |
\( T^{8} + 11 T^{7} + \cdots - 438404 \)
|
| $61$ |
\( T^{8} + 22 T^{7} + \cdots + 124252 \)
|
| $67$ |
\( T^{8} - 21 T^{7} + \cdots + 178083 \)
|
| $71$ |
\( T^{8} + 28 T^{7} + \cdots + 23067 \)
|
| $73$ |
\( (T + 1)^{8} \)
|
| $79$ |
\( T^{8} + 28 T^{7} + \cdots - 51391024 \)
|
| $83$ |
\( T^{8} + 5 T^{7} + \cdots + 4088009 \)
|
| $89$ |
\( T^{8} + 17 T^{7} + \cdots + 3008961 \)
|
| $97$ |
\( T^{8} - T^{7} + \cdots - 63221041 \)
|
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