Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8036,2,Mod(1,8036)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8036, 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("8036.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8036.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: | \(64.1677830643\) |
| 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: |
\( x^{8} - 14x^{6} - 4x^{5} + 60x^{4} + 31x^{3} - 75x^{2} - 60x - 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1148) |
| 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
\( x^{8} - 14x^{6} - 4x^{5} + 60x^{4} + 31x^{3} - 75x^{2} - 60x - 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} + 10\nu^{5} - 13\nu^{4} - 34\nu^{3} + 10\nu^{2} + 46\nu + 19 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{7} - 4\nu^{6} - 17\nu^{5} + 26\nu^{4} + 38\nu^{3} - 29\nu^{2} - 26\nu - 8 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} - 16\nu^{5} - 11\nu^{4} + 73\nu^{3} + 35\nu^{2} - 94\nu - 40 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{7} - 4\nu^{6} - 20\nu^{5} + 29\nu^{4} + 65\nu^{3} - 41\nu^{2} - 80\nu - 17 ) / 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{7} + \nu^{6} + 26\nu^{5} - 5\nu^{4} - 101\nu^{3} - 7\nu^{2} + 113\nu + 44 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + 2\beta_{6} + \beta_{5} + 3\beta_{3} + 8\beta_{2} + \beta _1 + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{7} + 10\beta_{6} + 10\beta_{5} + \beta_{4} + 12\beta_{3} + 13\beta_{2} + 28\beta _1 + 22 \)
|
| \(\nu^{6}\) | \(=\) |
\( 15\beta_{7} + 23\beta_{6} + 16\beta_{5} + 2\beta_{4} + 36\beta_{3} + 62\beta_{2} + 15\beta _1 + 149 \)
|
| \(\nu^{7}\) | \(=\) |
\( 83\beta_{7} + 86\beta_{6} + 85\beta_{5} + 14\beta_{4} + 116\beta_{3} + 126\beta_{2} + 173\beta _1 + 210 \)
|
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 |
|
0 | −2.28881 | 0 | −0.350035 | 0 | 0 | 0 | 2.23863 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.24904 | 0 | 1.47097 | 0 | 0 | 0 | 2.05817 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.33338 | 0 | −1.76687 | 0 | 0 | 0 | −1.22209 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.503623 | 0 | −2.23729 | 0 | 0 | 0 | −2.74636 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.308117 | 0 | 3.30124 | 0 | 0 | 0 | −2.90506 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.71291 | 0 | −2.15120 | 0 | 0 | 0 | −0.0659453 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.10300 | 0 | 2.93524 | 0 | 0 | 0 | 1.42260 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.86706 | 0 | −1.20206 | 0 | 0 | 0 | 5.22005 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(41\) | \(1\) |
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 | 8036.2.a.n | 8 | |
| 7.b | odd | 2 | 1 | 8036.2.a.m | 8 | ||
| 7.d | odd | 6 | 2 | 1148.2.i.d | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1148.2.i.d | ✓ | 16 | 7.d | odd | 6 | 2 | |
| 8036.2.a.m | 8 | 7.b | odd | 2 | 1 | ||
| 8036.2.a.n | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):
|
\( T_{3}^{8} - 14T_{3}^{6} - 4T_{3}^{5} + 60T_{3}^{4} + 31T_{3}^{3} - 75T_{3}^{2} - 60T_{3} - 11 \)
|
|
\( T_{5}^{8} - 18T_{5}^{6} - 12T_{5}^{5} + 98T_{5}^{4} + 117T_{5}^{3} - 115T_{5}^{2} - 196T_{5} - 51 \)
|
|
\( T_{11}^{8} + 8T_{11}^{7} + T_{11}^{6} - 94T_{11}^{5} - 131T_{11}^{4} + 154T_{11}^{3} + 265T_{11}^{2} + 77T_{11} - 3 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 14 T^{6} + \cdots - 11 \)
|
| $5$ |
\( T^{8} - 18 T^{6} + \cdots - 51 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} + 8 T^{7} + \cdots - 3 \)
|
| $13$ |
\( T^{8} - 7 T^{7} + \cdots - 634 \)
|
| $17$ |
\( T^{8} - T^{7} + \cdots + 3267 \)
|
| $19$ |
\( T^{8} + 4 T^{7} + \cdots + 561 \)
|
| $23$ |
\( T^{8} + 3 T^{7} + \cdots + 9 \)
|
| $29$ |
\( T^{8} + 4 T^{7} + \cdots + 486 \)
|
| $31$ |
\( T^{8} + 4 T^{7} + \cdots - 59729 \)
|
| $37$ |
\( T^{8} + 31 T^{7} + \cdots - 1244 \)
|
| $41$ |
\( (T + 1)^{8} \)
|
| $43$ |
\( T^{8} + 8 T^{7} + \cdots - 20588 \)
|
| $47$ |
\( T^{8} + 24 T^{7} + \cdots + 734691 \)
|
| $53$ |
\( T^{8} + T^{7} + \cdots + 73233 \)
|
| $59$ |
\( T^{8} + 4 T^{7} + \cdots - 11259 \)
|
| $61$ |
\( T^{8} - 4 T^{7} + \cdots + 72284 \)
|
| $67$ |
\( T^{8} - 196 T^{6} + \cdots + 2223124 \)
|
| $71$ |
\( T^{8} - 8 T^{7} + \cdots + 896976 \)
|
| $73$ |
\( T^{8} + 11 T^{7} + \cdots + 2731 \)
|
| $79$ |
\( T^{8} - 14 T^{7} + \cdots + 463761 \)
|
| $83$ |
\( T^{8} - 42 T^{7} + \cdots - 3922452 \)
|
| $89$ |
\( T^{8} - 11 T^{7} + \cdots + 72729 \)
|
| $97$ |
\( T^{8} - 16 T^{7} + \cdots - 69682 \)
|
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