Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8034,2,Mod(1,8034)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8034, 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("8034.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8034.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: | \(64.1518129839\) |
| 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: |
\( x^{8} - x^{7} - 12x^{6} + 12x^{5} + 43x^{4} - 38x^{3} - 49x^{2} + 23x + 20 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 12x^{6} + 12x^{5} + 43x^{4} - 38x^{3} - 49x^{2} + 23x + 20 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - 11\nu^{4} + \nu^{3} + 32\nu^{2} - 5\nu - 17 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{7} + \nu^{6} - 32\nu^{5} - 6\nu^{4} + 92\nu^{3} + 7\nu^{2} - 53\nu - 11 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{7} + 2\nu^{6} - 21\nu^{5} - 18\nu^{4} + 60\nu^{3} + 43\nu^{2} - 37\nu - 25 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{7} + 2\nu^{6} - 21\nu^{5} - 18\nu^{4} + 60\nu^{3} + 44\nu^{2} - 37\nu - 28 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 11\nu^{5} + 33\nu^{4} + 29\nu^{3} - 95\nu^{2} - 2\nu + 46 \)
|
| \(\beta_{7}\) | \(=\) |
\( 2\nu^{7} + 3\nu^{6} - 21\nu^{5} - 29\nu^{4} + 62\nu^{3} + 76\nu^{2} - 47\nu - 45 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{5} - \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} - \beta_{6} + 6\beta_{5} - 6\beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{7} + 2\beta_{6} - 7\beta_{5} + 2\beta_{4} - 4\beta_{3} - 10\beta_{2} + 27\beta _1 - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -12\beta_{7} - 11\beta_{6} + 35\beta_{5} - 34\beta_{4} + 11\beta_{3} - 9\beta_{2} - 11\beta _1 + 64 \)
|
| \(\nu^{7}\) | \(=\) |
\( 78\beta_{7} + 23\beta_{6} - 46\beta_{5} + 23\beta_{4} - 44\beta_{3} - 75\beta_{2} + 154\beta _1 - 20 \)
|
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 | −3.53629 | 1.00000 | 0.738654 | 1.00000 | 1.00000 | −3.53629 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 1.00000 | 1.00000 | −3.08319 | 1.00000 | 1.60046 | 1.00000 | 1.00000 | −3.08319 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 1.00000 | 1.00000 | −2.24957 | 1.00000 | 3.57309 | 1.00000 | 1.00000 | −2.24957 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.00000 | 1.00000 | −1.15050 | 1.00000 | −2.62035 | 1.00000 | 1.00000 | −1.15050 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.00000 | 1.00000 | −0.612396 | 1.00000 | −3.48353 | 1.00000 | 1.00000 | −0.612396 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.00000 | 1.00000 | −0.216275 | 1.00000 | −0.843680 | 1.00000 | 1.00000 | −0.216275 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 1.00000 | 1.00000 | −0.176895 | 1.00000 | −2.27129 | 1.00000 | 1.00000 | −0.176895 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 1.00000 | 1.00000 | 3.02511 | 1.00000 | −2.69336 | 1.00000 | 1.00000 | 3.02511 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(13\) | \(-1\) |
| \(103\) | \(-1\) |
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 | 8034.2.a.p | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8034.2.a.p | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):
|
\( T_{5}^{8} + 8T_{5}^{7} + 13T_{5}^{6} - 47T_{5}^{5} - 189T_{5}^{4} - 237T_{5}^{3} - 125T_{5}^{2} - 27T_{5} - 2 \)
|
|
\( T_{7}^{8} + 6T_{7}^{7} - 6T_{7}^{6} - 94T_{7}^{5} - 117T_{7}^{4} + 228T_{7}^{3} + 408T_{7}^{2} - 81T_{7} - 199 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( (T - 1)^{8} \)
|
| $5$ |
\( T^{8} + 8 T^{7} + \cdots - 2 \)
|
| $7$ |
\( T^{8} + 6 T^{7} + \cdots - 199 \)
|
| $11$ |
\( T^{8} + 7 T^{7} + \cdots + 1565 \)
|
| $13$ |
\( (T - 1)^{8} \)
|
| $17$ |
\( T^{8} + 20 T^{7} + \cdots - 2827 \)
|
| $19$ |
\( T^{8} + 12 T^{7} + \cdots + 12034 \)
|
| $23$ |
\( T^{8} + 14 T^{7} + \cdots - 31420 \)
|
| $29$ |
\( T^{8} + 25 T^{7} + \cdots + 2650 \)
|
| $31$ |
\( T^{8} + 12 T^{7} + \cdots - 32146 \)
|
| $37$ |
\( T^{8} + 15 T^{7} + \cdots + 9670 \)
|
| $41$ |
\( T^{8} + 18 T^{7} + \cdots + 21098 \)
|
| $43$ |
\( T^{8} + 8 T^{7} + \cdots + 17666 \)
|
| $47$ |
\( T^{8} + 12 T^{7} + \cdots - 2152628 \)
|
| $53$ |
\( T^{8} + 25 T^{7} + \cdots + 266759 \)
|
| $59$ |
\( T^{8} + 9 T^{7} + \cdots + 111752 \)
|
| $61$ |
\( T^{8} + 2 T^{7} + \cdots - 73352 \)
|
| $67$ |
\( T^{8} + 8 T^{7} + \cdots - 183707 \)
|
| $71$ |
\( T^{8} + 13 T^{7} + \cdots + 87092 \)
|
| $73$ |
\( T^{8} + 2 T^{7} + \cdots - 16304777 \)
|
| $79$ |
\( T^{8} - T^{7} + \cdots - 551150 \)
|
| $83$ |
\( T^{8} + 6 T^{7} + \cdots - 42334 \)
|
| $89$ |
\( T^{8} + 17 T^{7} + \cdots - 423490 \)
|
| $97$ |
\( T^{8} - 19 T^{7} + \cdots + 118642 \)
|
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