Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9196,2,Mod(1,9196)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9196, 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("9196.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9196.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: | \(73.4304296988\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.34963625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 10x^{4} + 13x^{3} + 27x^{2} - 6x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 836) |
| 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
\( x^{6} - 2x^{5} - 10x^{4} + 13x^{3} + 27x^{2} - 6x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} - 4\nu^{2} + \nu + 2 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 10\nu^{3} - \nu^{2} + 19\nu - 4 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 10\nu^{3} + 7\nu^{2} - 25\nu - 20 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 6\nu^{4} + 10\nu^{3} - 41\nu^{2} - 37\nu + 10 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 7\beta_{4} + 8\beta_{3} + 10\beta _1 + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( 11\beta_{4} - 3\beta_{3} + 10\beta_{2} + 52\beta _1 + 28 \)
|
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 | −2.89822 | 0 | 2.44421 | 0 | 0.245210 | 0 | 5.39968 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.75406 | 0 | −2.20821 | 0 | −1.49389 | 0 | 4.58484 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.471235 | 0 | −3.53686 | 0 | 2.29934 | 0 | −2.77794 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.311022 | 0 | 1.88331 | 0 | −3.69109 | 0 | −2.90327 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.44304 | 0 | −4.29314 | 0 | 3.18498 | 0 | −0.917645 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.36946 | 0 | −1.28931 | 0 | −4.54455 | 0 | 2.61432 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(-1\) |
| \(19\) | \(1\) |
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 | 9196.2.a.i | 6 | |
| 11.b | odd | 2 | 1 | 9196.2.a.j | 6 | ||
| 11.d | odd | 10 | 2 | 836.2.j.a | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 836.2.j.a | ✓ | 12 | 11.d | odd | 10 | 2 | |
| 9196.2.a.i | 6 | 1.a | even | 1 | 1 | trivial | |
| 9196.2.a.j | 6 | 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9196))\):
|
\( T_{3}^{6} + 2T_{3}^{5} - 10T_{3}^{4} - 13T_{3}^{3} + 27T_{3}^{2} + 6T_{3} - 4 \)
|
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\( T_{5}^{6} + 7T_{5}^{5} + T_{5}^{4} - 69T_{5}^{3} - 74T_{5}^{2} + 160T_{5} + 199 \)
|
|
\( T_{7}^{6} + 4T_{7}^{5} - 18T_{7}^{4} - 59T_{7}^{3} + 91T_{7}^{2} + 165T_{7} - 45 \)
|
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\( T_{13}^{6} + T_{13}^{5} - 20T_{13}^{4} - 6T_{13}^{3} + 37T_{13}^{2} + 28T_{13} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 2 T^{5} + \cdots - 4 \)
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| $5$ |
\( T^{6} + 7 T^{5} + \cdots + 199 \)
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| $7$ |
\( T^{6} + 4 T^{5} + \cdots - 45 \)
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| $11$ |
\( T^{6} \)
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| $13$ |
\( T^{6} + T^{5} - 20 T^{4} + \cdots + 4 \)
|
| $17$ |
\( T^{6} - 9 T^{5} + \cdots + 16 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} + 12 T^{5} + \cdots + 81 \)
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| $29$ |
\( T^{6} - 8 T^{5} + \cdots - 1436 \)
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| $31$ |
\( T^{6} - 7 T^{5} + \cdots - 1900 \)
|
| $37$ |
\( T^{6} - 24 T^{5} + \cdots - 144 \)
|
| $41$ |
\( T^{6} - 15 T^{5} + \cdots + 19804 \)
|
| $43$ |
\( T^{6} - 23 T^{5} + \cdots + 69975 \)
|
| $47$ |
\( T^{6} + 10 T^{5} + \cdots - 2339 \)
|
| $53$ |
\( T^{6} - 16 T^{5} + \cdots + 210836 \)
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| $59$ |
\( T^{6} + 27 T^{5} + \cdots - 8516 \)
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| $61$ |
\( T^{6} - 7 T^{5} + \cdots + 3681 \)
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| $67$ |
\( T^{6} + 11 T^{5} + \cdots - 4 \)
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| $71$ |
\( T^{6} + 16 T^{5} + \cdots - 135920 \)
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| $73$ |
\( T^{6} + 6 T^{5} + \cdots + 22756 \)
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| $79$ |
\( T^{6} + 34 T^{5} + \cdots + 20736 \)
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| $83$ |
\( T^{6} - 2 T^{5} + \cdots - 35951 \)
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| $89$ |
\( T^{6} - 13 T^{5} + \cdots - 35044 \)
|
| $97$ |
\( T^{6} + 13 T^{5} + \cdots - 28656 \)
|
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