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: | \(1\) |
| 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: |
\( x^{7} - 9x^{5} + 23x^{3} - 14x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1155) |
| 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
\( x^{7} - 9x^{5} + 23x^{3} - 14x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 4\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 6\nu^{3} - \nu^{2} + 7\nu + 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 8\nu^{4} + 16\nu^{2} - \nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 6\beta_{3} + \beta_{2} + 17\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 8\beta_{4} + 8\beta_{3} + 24\beta_{2} + \beta _1 + 52 \)
|
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 |
|
−2.16614 | −1.00000 | 2.69216 | 1.00000 | 2.16614 | 0 | −1.49931 | 1.00000 | −2.16614 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.88643 | −1.00000 | 1.55863 | 1.00000 | 1.88643 | 0 | 0.832608 | 1.00000 | −1.88643 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.811884 | −1.00000 | −1.34084 | 1.00000 | 0.811884 | 0 | 2.71238 | 1.00000 | −0.811884 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.148154 | −1.00000 | −1.97805 | 1.00000 | 0.148154 | 0 | 0.589363 | 1.00000 | −0.148154 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.05922 | −1.00000 | −0.878050 | 1.00000 | −1.05922 | 0 | −3.04849 | 1.00000 | 1.05922 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.72027 | −1.00000 | 0.959331 | 1.00000 | −1.72027 | 0 | −1.79023 | 1.00000 | 1.72027 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.23312 | −1.00000 | 2.98682 | 1.00000 | −2.23312 | 0 | 2.20369 | 1.00000 | 2.23312 | ||||||||||||||||||||||||||||||||||||||||
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.ca | 7 | |
| 7.b | odd | 2 | 1 | 8085.2.a.cc | 7 | ||
| 7.c | even | 3 | 2 | 1155.2.q.i | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1155.2.q.i | ✓ | 14 | 7.c | even | 3 | 2 | |
| 8085.2.a.ca | 7 | 1.a | even | 1 | 1 | trivial | |
| 8085.2.a.cc | 7 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):
|
\( T_{2}^{7} - 9T_{2}^{5} + 23T_{2}^{3} - 14T_{2} - 2 \)
|
|
\( T_{13}^{7} + 7T_{13}^{6} - 21T_{13}^{5} - 231T_{13}^{4} - 165T_{13}^{3} + 1405T_{13}^{2} + 1225T_{13} - 2797 \)
|
|
\( T_{17}^{7} + 6T_{17}^{6} - 88T_{17}^{5} - 608T_{17}^{4} + 1568T_{17}^{3} + 16000T_{17}^{2} + 16576T_{17} - 31872 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 9 T^{5} + \cdots - 2 \)
|
| $3$ |
\( (T + 1)^{7} \)
|
| $5$ |
\( (T - 1)^{7} \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( (T - 1)^{7} \)
|
| $13$ |
\( T^{7} + 7 T^{6} + \cdots - 2797 \)
|
| $17$ |
\( T^{7} + 6 T^{6} + \cdots - 31872 \)
|
| $19$ |
\( T^{7} + 8 T^{6} + \cdots - 3584 \)
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| $23$ |
\( T^{7} + T^{6} + \cdots - 5392 \)
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| $29$ |
\( T^{7} + 9 T^{6} + \cdots + 3888 \)
|
| $31$ |
\( T^{7} + 20 T^{6} + \cdots - 5444 \)
|
| $37$ |
\( T^{7} - 6 T^{6} + \cdots + 96 \)
|
| $41$ |
\( T^{7} + 5 T^{6} + \cdots + 30768 \)
|
| $43$ |
\( T^{7} - 15 T^{6} + \cdots + 1690793 \)
|
| $47$ |
\( T^{7} + T^{6} + \cdots - 14192 \)
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| $53$ |
\( T^{7} - 13 T^{6} + \cdots - 144 \)
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| $59$ |
\( T^{7} + 38 T^{6} + \cdots - 333912 \)
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| $61$ |
\( T^{7} + 4 T^{6} + \cdots - 4096 \)
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| $67$ |
\( T^{7} - 22 T^{6} + \cdots - 431712 \)
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| $71$ |
\( T^{7} + 26 T^{6} + \cdots - 338 \)
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| $73$ |
\( T^{7} + 24 T^{6} + \cdots + 5775948 \)
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| $79$ |
\( T^{7} - 216 T^{5} + \cdots - 392192 \)
|
| $83$ |
\( T^{7} + 20 T^{6} + \cdots + 2145038 \)
|
| $89$ |
\( T^{7} + 34 T^{6} + \cdots - 408362 \)
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| $97$ |
\( T^{7} + 10 T^{6} + \cdots - 1952 \)
|
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