Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8085,2,Mod(1,8085)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8085, 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("8085.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8085.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: | \(64.5590500342\) |
| 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: |
\( x^{7} - 12x^{5} + 37x^{3} - 2x^{2} - 26x - 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{7} - 12x^{5} + 37x^{3} - 2x^{2} - 26x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} + \nu^{4} - 10\nu^{3} - 9\nu^{2} + 18\nu + 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 11\nu^{4} + 27\nu^{2} - 4\nu - 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{6} - \nu^{5} + 11\nu^{4} + 9\nu^{3} - 26\nu^{2} - 9\nu + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 8\beta_{2} + \beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{6} - \beta_{5} + 9\beta_{3} + \beta_{2} + 41\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{6} + 12\beta_{5} + 11\beta_{4} + 11\beta_{3} + 61\beta_{2} + 15\beta _1 + 114 \)
|
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 |
|
−2.79651 | 1.00000 | 5.82048 | −1.00000 | −2.79651 | 0 | −10.6840 | 1.00000 | 2.79651 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.59343 | 1.00000 | 0.539031 | −1.00000 | −1.59343 | 0 | 2.32796 | 1.00000 | 1.59343 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.30931 | 1.00000 | −0.285710 | −1.00000 | −1.30931 | 0 | 2.99270 | 1.00000 | 1.30931 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.260640 | 1.00000 | −1.93207 | −1.00000 | 0.260640 | 0 | −1.02485 | 1.00000 | −0.260640 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.724859 | 1.00000 | −1.47458 | −1.00000 | 0.724859 | 0 | −2.51858 | 1.00000 | −0.724859 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.02288 | 1.00000 | 2.09206 | −1.00000 | 2.02288 | 0 | 0.186217 | 1.00000 | −2.02288 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.69087 | 1.00000 | 5.24079 | −1.00000 | 2.69087 | 0 | 8.72056 | 1.00000 | −2.69087 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 8085.2.a.cd | yes | 7 |
| 7.b | odd | 2 | 1 | 8085.2.a.cb | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8085.2.a.cb | ✓ | 7 | 7.b | odd | 2 | 1 | |
| 8085.2.a.cd | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):
|
\( T_{2}^{7} - 12T_{2}^{5} + 37T_{2}^{3} + 2T_{2}^{2} - 26T_{2} + 6 \)
|
|
\( T_{13}^{7} - 4T_{13}^{6} - 60T_{13}^{5} + 246T_{13}^{4} + 784T_{13}^{3} - 3432T_{13}^{2} + 864T_{13} + 128 \)
|
|
\( T_{17}^{7} + 7T_{17}^{6} - 45T_{17}^{5} - 247T_{17}^{4} + 744T_{17}^{3} + 1656T_{17}^{2} - 2000T_{17} - 1712 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 12 T^{5} + \cdots + 6 \)
|
| $3$ |
\( (T - 1)^{7} \)
|
| $5$ |
\( (T + 1)^{7} \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( (T - 1)^{7} \)
|
| $13$ |
\( T^{7} - 4 T^{6} + \cdots + 128 \)
|
| $17$ |
\( T^{7} + 7 T^{6} + \cdots - 1712 \)
|
| $19$ |
\( T^{7} - 3 T^{6} + \cdots + 209664 \)
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| $23$ |
\( T^{7} - 13 T^{6} + \cdots - 11072 \)
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| $29$ |
\( T^{7} - 7 T^{6} + \cdots + 576 \)
|
| $31$ |
\( T^{7} + 14 T^{6} + \cdots - 128 \)
|
| $37$ |
\( T^{7} - 14 T^{6} + \cdots - 12576 \)
|
| $41$ |
\( T^{7} - 120 T^{5} + \cdots + 75872 \)
|
| $43$ |
\( T^{7} - 27 T^{6} + \cdots + 6144 \)
|
| $47$ |
\( T^{7} - 10 T^{6} + \cdots + 350208 \)
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| $53$ |
\( T^{7} - 7 T^{6} + \cdots + 768 \)
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| $59$ |
\( T^{7} + 7 T^{6} + \cdots - 96 \)
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| $61$ |
\( T^{7} - 15 T^{6} + \cdots + 246944 \)
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| $67$ |
\( T^{7} - 18 T^{6} + \cdots - 744704 \)
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| $71$ |
\( T^{7} - 28 T^{6} + \cdots + 2048 \)
|
| $73$ |
\( T^{7} - 10 T^{6} + \cdots + 133632 \)
|
| $79$ |
\( T^{7} - 214 T^{5} + \cdots + 256 \)
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| $83$ |
\( T^{7} - 11 T^{6} + \cdots + 3836224 \)
|
| $89$ |
\( T^{7} + 21 T^{6} + \cdots - 869264 \)
|
| $97$ |
\( T^{7} - 17 T^{6} + \cdots - 7184 \)
|
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