Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7524,2,Mod(1,7524)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7524, 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("7524.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7524 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7524.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: | \(60.0794424808\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.171777.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 13x^{2} + 24x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2508) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 13x^{2} + 24x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 2\nu^{2} - 9\nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + \nu^{2} - 10\nu + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} - \beta _1 + 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} - \beta_{2} + 11\beta _1 - 11 \)
|
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 | 0 | 0 | −3.10619 | 0 | 3.64841 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −2.67113 | 0 | 1.13495 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 1.75414 | 0 | −2.92298 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 3.02318 | 0 | 3.13961 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
| \(19\) | \(1\) |
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 | 7524.2.a.n | 4 | |
| 3.b | odd | 2 | 1 | 2508.2.a.i | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2508.2.a.i | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 7524.2.a.n | 4 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7524))\):
|
\( T_{5}^{4} + T_{5}^{3} - 14T_{5}^{2} - 9T_{5} + 44 \)
|
|
\( T_{7}^{4} - 5T_{7}^{3} - 4T_{7}^{2} + 43T_{7} - 38 \)
|
|
\( T_{17}^{4} + 5T_{17}^{3} - 29T_{17}^{2} - 156T_{17} - 159 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + T^{3} + \cdots + 44 \)
|
| $7$ |
\( T^{4} - 5 T^{3} + \cdots - 38 \)
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| $11$ |
\( (T + 1)^{4} \)
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| $13$ |
\( T^{4} - 19 T^{2} + \cdots - 13 \)
|
| $17$ |
\( T^{4} + 5 T^{3} + \cdots - 159 \)
|
| $19$ |
\( (T + 1)^{4} \)
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| $23$ |
\( T^{4} + 3 T^{3} + \cdots + 182 \)
|
| $29$ |
\( T^{4} + 14 T^{3} + \cdots - 956 \)
|
| $31$ |
\( T^{4} - 8 T^{3} + \cdots + 352 \)
|
| $37$ |
\( T^{4} + 13 T^{3} + \cdots + 18 \)
|
| $41$ |
\( T^{4} - T^{3} + \cdots + 422 \)
|
| $43$ |
\( T^{4} - 18 T^{3} + \cdots - 288 \)
|
| $47$ |
\( T^{4} + 14 T^{3} + \cdots - 502 \)
|
| $53$ |
\( T^{4} + T^{3} + \cdots - 39 \)
|
| $59$ |
\( T^{4} + 18 T^{3} + \cdots - 753 \)
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| $61$ |
\( T^{4} + 3 T^{3} + \cdots + 2942 \)
|
| $67$ |
\( T^{4} + 17 T^{3} + \cdots - 1234 \)
|
| $71$ |
\( T^{4} - 5 T^{3} + \cdots - 199 \)
|
| $73$ |
\( T^{4} - 12 T^{3} + \cdots + 2756 \)
|
| $79$ |
\( T^{4} + 11 T^{3} + \cdots - 1461 \)
|
| $83$ |
\( T^{4} + 6 T^{3} + \cdots + 549 \)
|
| $89$ |
\( T^{4} - T^{3} + \cdots - 63 \)
|
| $97$ |
\( T^{4} - 223 T^{2} + \cdots + 578 \)
|
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