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: | \(6\) |
| Coefficient field: | 6.6.2803712.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 6x^{4} + 8x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 6x^{4} + 8x^{2} - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 4\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 6\nu^{3} - 7\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 5\beta_{2} + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 5\beta_{5} + 6\beta_{4} + 11\beta_1 \)
|
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 |
|
−2.05288 | 1.00000 | 2.21432 | −1.00000 | −2.05288 | 0 | −0.439973 | 1.00000 | 2.05288 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.20864 | 1.00000 | −0.539189 | −1.00000 | −1.20864 | 0 | 3.06897 | 1.00000 | 1.20864 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.569973 | 1.00000 | −1.67513 | −1.00000 | −0.569973 | 0 | 2.09473 | 1.00000 | 0.569973 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.569973 | 1.00000 | −1.67513 | −1.00000 | 0.569973 | 0 | −2.09473 | 1.00000 | −0.569973 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.20864 | 1.00000 | −0.539189 | −1.00000 | 1.20864 | 0 | −3.06897 | 1.00000 | −1.20864 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.05288 | 1.00000 | 2.21432 | −1.00000 | 2.05288 | 0 | 0.439973 | 1.00000 | −2.05288 | |||||||||||||||||||||||||||||||||||||
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.by | yes | 6 |
| 7.b | odd | 2 | 1 | 8085.2.a.bw | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8085.2.a.bw | ✓ | 6 | 7.b | odd | 2 | 1 | |
| 8085.2.a.by | yes | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):
|
\( T_{2}^{6} - 6T_{2}^{4} + 8T_{2}^{2} - 2 \)
|
|
\( T_{13}^{6} + 8T_{13}^{5} + 8T_{13}^{4} - 48T_{13}^{3} - 58T_{13}^{2} + 52T_{13} + 62 \)
|
|
\( T_{17}^{6} + 6T_{17}^{5} - 7T_{17}^{4} - 48T_{17}^{3} + 67T_{17}^{2} + 10T_{17} - 25 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 6 T^{4} + \cdots - 2 \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T - 1)^{6} \)
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| $13$ |
\( T^{6} + 8 T^{5} + \cdots + 62 \)
|
| $17$ |
\( T^{6} + 6 T^{5} + \cdots - 25 \)
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| $19$ |
\( T^{6} - 6 T^{5} + \cdots + 449 \)
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| $23$ |
\( T^{6} + 14 T^{5} + \cdots - 4097 \)
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| $29$ |
\( T^{6} + 6 T^{5} + \cdots - 431 \)
|
| $31$ |
\( T^{6} - 12 T^{5} + \cdots + 626 \)
|
| $37$ |
\( T^{6} + 12 T^{5} + \cdots - 5266 \)
|
| $41$ |
\( T^{6} - 8 T^{5} + \cdots + 2978 \)
|
| $43$ |
\( T^{6} + 30 T^{5} + \cdots - 42641 \)
|
| $47$ |
\( T^{6} - 116 T^{4} + \cdots - 562 \)
|
| $53$ |
\( T^{6} + 2 T^{5} + \cdots - 61297 \)
|
| $59$ |
\( T^{6} - 2 T^{5} + \cdots + 9001 \)
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| $61$ |
\( T^{6} - 10 T^{5} + \cdots + 2329 \)
|
| $67$ |
\( T^{6} + 24 T^{5} + \cdots - 133600 \)
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| $71$ |
\( T^{6} + 4 T^{5} + \cdots - 38158 \)
|
| $73$ |
\( T^{6} + 32 T^{5} + \cdots - 467912 \)
|
| $79$ |
\( T^{6} + 20 T^{5} + \cdots - 37342 \)
|
| $83$ |
\( T^{6} - 6 T^{5} + \cdots - 1433 \)
|
| $89$ |
\( T^{6} - 6 T^{5} + \cdots - 665351 \)
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| $97$ |
\( T^{6} + 34 T^{5} + \cdots + 823975 \)
|
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