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: | \(5\) |
| Coefficient field: | \(\Q(\zeta_{22})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1045) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{22} + \zeta_{22}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
|
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 |
|
−2.22871 | 0 | 2.96714 | −1.00000 | 0 | −4.42518 | −2.15546 | 0 | 2.22871 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.0881559 | 0 | −1.99223 | −1.00000 | 0 | −1.95185 | 0.351939 | 0 | 0.0881559 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.37279 | 0 | −0.115460 | −1.00000 | 0 | 2.43232 | −2.90407 | 0 | −1.37279 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.54620 | 0 | 0.390736 | −1.00000 | 0 | −4.16140 | −2.48825 | 0 | −1.54620 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.39788 | 0 | 3.74982 | −1.00000 | 0 | −2.89389 | 4.19584 | 0 | −2.39788 | ||||||||||||||||||||||||||||||||||
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.v | 5 | |
| 3.b | odd | 2 | 1 | 1045.2.a.d | ✓ | 5 | |
| 15.d | odd | 2 | 1 | 5225.2.a.j | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1045.2.a.d | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 5225.2.a.j | 5 | 15.d | odd | 2 | 1 | ||
| 9405.2.a.v | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):
|
\( T_{2}^{5} - 3T_{2}^{4} - 3T_{2}^{3} + 15T_{2}^{2} - 10T_{2} - 1 \)
|
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\( T_{7}^{5} + 11T_{7}^{4} + 33T_{7}^{3} - 22T_{7}^{2} - 231T_{7} - 253 \)
|
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\( T_{13}^{5} - T_{13}^{4} - 37T_{13}^{3} + 47T_{13}^{2} + 322T_{13} - 529 \)
|
|
\( T_{17}^{5} - 3T_{17}^{4} - 25T_{17}^{3} + 59T_{17}^{2} - 10T_{17} - 23 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 3 T^{4} + \cdots - 1 \)
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| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T + 1)^{5} \)
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| $7$ |
\( T^{5} + 11 T^{4} + \cdots - 253 \)
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| $11$ |
\( (T - 1)^{5} \)
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| $13$ |
\( T^{5} - T^{4} + \cdots - 529 \)
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| $17$ |
\( T^{5} - 3 T^{4} + \cdots - 23 \)
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| $19$ |
\( (T - 1)^{5} \)
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| $23$ |
\( T^{5} - 8 T^{4} + \cdots + 1 \)
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| $29$ |
\( T^{5} + 11 T^{4} + \cdots - 1441 \)
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| $31$ |
\( T^{5} + 5 T^{4} + \cdots + 10649 \)
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| $37$ |
\( T^{5} + 9 T^{4} + \cdots + 3917 \)
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| $41$ |
\( T^{5} + 15 T^{4} + \cdots - 593 \)
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| $43$ |
\( T^{5} + 13 T^{4} + \cdots + 28753 \)
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| $47$ |
\( T^{5} - 20 T^{4} + \cdots + 989 \)
|
| $53$ |
\( T^{5} - 5 T^{4} + \cdots - 3917 \)
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| $59$ |
\( T^{5} - 17 T^{4} + \cdots - 197 \)
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| $61$ |
\( T^{5} - 3 T^{4} + \cdots + 241 \)
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| $67$ |
\( T^{5} + 28 T^{4} + \cdots - 11881 \)
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| $71$ |
\( T^{5} - 6 T^{4} + \cdots + 23 \)
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| $73$ |
\( T^{5} + 16 T^{4} + \cdots + 23 \)
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| $79$ |
\( T^{5} - 3 T^{4} + \cdots - 15203 \)
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| $83$ |
\( T^{5} - 33 T^{4} + \cdots + 737 \)
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| $89$ |
\( T^{5} - 16 T^{4} + \cdots - 9791 \)
|
| $97$ |
\( T^{5} + 14 T^{4} + \cdots + 3323 \)
|
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