Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9405,2,Mod(1,9405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9405, 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("9405.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9405.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: | \(75.0993031010\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.905177.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 7x^{4} + 9x^{3} + 7x^{2} - 9x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 3135) |
| 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
\( x^{6} - x^{5} - 7x^{4} + 9x^{3} + 7x^{2} - 9x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 5\nu^{2} - 2\nu + 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{5} + 6\nu^{3} - 3\nu^{2} - 3\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 6\nu^{3} + 3\nu^{2} + 5\nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + 7\nu^{3} - \nu^{2} - 8\nu - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( 5\nu^{5} + 2\nu^{4} - 32\nu^{3} + \nu^{2} + 35\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 4\beta_{4} - 2\beta_{3} - \beta_{2} - 2\beta _1 + 11 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{5} - 2\beta_{4} + 7\beta_{3} + 4\beta_{2} + 2\beta _1 - 3 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7\beta_{5} + 24\beta_{4} - 20\beta_{3} - 9\beta_{2} - 10\beta _1 + 49 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -15\beta_{5} - 36\beta_{4} + 84\beta_{3} + 41\beta_{2} + 30\beta _1 - 61 ) / 4 \)
|
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 |
|
−2.03545 | 0 | 2.14306 | −1.00000 | 0 | −1.92553 | −0.291185 | 0 | 2.03545 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.63929 | 0 | 0.687261 | −1.00000 | 0 | 1.85467 | 2.15196 | 0 | 1.63929 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.157201 | 0 | −1.97529 | −1.00000 | 0 | −2.76108 | 0.624919 | 0 | 0.157201 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.392307 | 0 | −1.84610 | −1.00000 | 0 | 3.88627 | −1.50885 | 0 | −0.392307 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.95914 | 0 | 1.83822 | −1.00000 | 0 | −0.644737 | −0.316941 | 0 | −1.95914 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.48049 | 0 | 4.15284 | −1.00000 | 0 | 2.59041 | 5.34010 | 0 | −2.48049 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 9405.2.a.x | 6 | |
| 3.b | odd | 2 | 1 | 3135.2.a.p | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3135.2.a.p | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 9405.2.a.x | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):
|
\( T_{2}^{6} - T_{2}^{5} - 8T_{2}^{4} + 5T_{2}^{3} + 16T_{2}^{2} - 4T_{2} - 1 \)
|
|
\( T_{7}^{6} - 3T_{7}^{5} - 14T_{7}^{4} + 33T_{7}^{3} + 56T_{7}^{2} - 80T_{7} - 64 \)
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\( T_{13}^{6} + 4T_{13}^{5} - 23T_{13}^{4} - 89T_{13}^{3} + 50T_{13}^{2} + 260T_{13} + 104 \)
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\( T_{17}^{6} - 4T_{17}^{5} - 35T_{17}^{4} + 65T_{17}^{3} + 378T_{17}^{2} - 4T_{17} - 568 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} - 8 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T + 1)^{6} \)
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| $7$ |
\( T^{6} - 3 T^{5} + \cdots - 64 \)
|
| $11$ |
\( (T - 1)^{6} \)
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| $13$ |
\( T^{6} + 4 T^{5} + \cdots + 104 \)
|
| $17$ |
\( T^{6} - 4 T^{5} + \cdots - 568 \)
|
| $19$ |
\( (T - 1)^{6} \)
|
| $23$ |
\( T^{6} - 39 T^{4} + \cdots + 64 \)
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| $29$ |
\( T^{6} - 21 T^{5} + \cdots + 26552 \)
|
| $31$ |
\( T^{6} + 16 T^{5} + \cdots - 64 \)
|
| $37$ |
\( T^{6} + T^{5} + \cdots - 3928 \)
|
| $41$ |
\( T^{6} - 31 T^{5} + \cdots - 33832 \)
|
| $43$ |
\( T^{6} - 6 T^{5} + \cdots - 1856 \)
|
| $47$ |
\( T^{6} - 4 T^{5} + \cdots - 22016 \)
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| $53$ |
\( T^{6} - 9 T^{5} + \cdots + 104 \)
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| $59$ |
\( T^{6} - 24 T^{5} + \cdots + 1856 \)
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| $61$ |
\( T^{6} + 11 T^{5} + \cdots + 2248 \)
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| $67$ |
\( T^{6} + 11 T^{5} + \cdots + 1856 \)
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| $71$ |
\( T^{6} - 23 T^{5} + \cdots + 930112 \)
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| $73$ |
\( T^{6} + 6 T^{5} + \cdots - 4544 \)
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| $79$ |
\( T^{6} - 4 T^{5} + \cdots + 194048 \)
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| $83$ |
\( T^{6} + 17 T^{5} + \cdots + 129856 \)
|
| $89$ |
\( T^{6} - 37 T^{5} + \cdots - 1448 \)
|
| $97$ |
\( T^{6} + 26 T^{5} + \cdots - 18472 \)
|
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