Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4560,2,Mod(2431,4560)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4560, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4560.2431");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4560.d (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(36.4117833217\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.792772608.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 22x^{4} + 62x^{2} + 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} + 22x^{4} + 62x^{2} + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 26\nu^{3} + 114\nu ) / 26 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{4} - 39\nu^{2} - 46 ) / 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 26\nu^{2} + 75 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{5} + 65\nu^{3} + 160\nu ) / 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{4} + \beta_{3} - 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + 6\beta_{2} - 14\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -39\beta_{4} - 26\beta_{3} + 133 \)
|
| \(\nu^{5}\) | \(=\) |
\( 26\beta_{5} - 130\beta_{2} + 250\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times\).
| \(n\) | \(1141\) | \(1711\) | \(1921\) | \(2737\) | \(3041\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2431.1 |
|
0 | 1.00000 | 0 | 1.00000 | 0 | − | 4.69162i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 2431.2 | 0 | 1.00000 | 0 | 1.00000 | 0 | − | 2.36627i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 2431.3 | 0 | 1.00000 | 0 | 1.00000 | 0 | − | 0.624069i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 2431.4 | 0 | 1.00000 | 0 | 1.00000 | 0 | 0.624069i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 2431.5 | 0 | 1.00000 | 0 | 1.00000 | 0 | 2.36627i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 2431.6 | 0 | 1.00000 | 0 | 1.00000 | 0 | 4.69162i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 76.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4560.2.d.h | yes | 6 |
| 4.b | odd | 2 | 1 | 4560.2.d.f | ✓ | 6 | |
| 19.b | odd | 2 | 1 | 4560.2.d.f | ✓ | 6 | |
| 76.d | even | 2 | 1 | inner | 4560.2.d.h | yes | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4560.2.d.f | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 4560.2.d.f | ✓ | 6 | 19.b | odd | 2 | 1 | |
| 4560.2.d.h | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 4560.2.d.h | yes | 6 | 76.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4560, [\chi])\):
|
\( T_{7}^{6} + 28T_{7}^{4} + 134T_{7}^{2} + 48 \)
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|
\( T_{31}^{3} - 6T_{31}^{2} - 2T_{31} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T - 1)^{6} \)
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| $5$ |
\( (T - 1)^{6} \)
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| $7$ |
\( T^{6} + 28 T^{4} + \cdots + 48 \)
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| $11$ |
\( T^{6} + 22 T^{4} + \cdots + 12 \)
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| $13$ |
\( T^{6} + 28 T^{4} + \cdots + 48 \)
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| $17$ |
\( (T^{3} + 2 T^{2} + \cdots - 156)^{2} \)
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| $19$ |
\( T^{6} - 6 T^{5} + \cdots + 6859 \)
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| $23$ |
\( T^{6} + 34 T^{4} + \cdots + 768 \)
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| $29$ |
\( T^{6} + 166 T^{4} + \cdots + 164268 \)
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| $31$ |
\( (T^{3} - 6 T^{2} - 2 T + 4)^{2} \)
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| $37$ |
\( T^{6} + 52 T^{4} + \cdots + 1728 \)
|
| $41$ |
\( T^{6} + 182 T^{4} + \cdots + 137388 \)
|
| $43$ |
\( T^{6} + 52 T^{4} + \cdots + 1728 \)
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| $47$ |
\( T^{6} + 242 T^{4} + \cdots + 27648 \)
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| $53$ |
\( T^{6} + 34 T^{4} + \cdots + 768 \)
|
| $59$ |
\( (T^{3} + 8 T^{2} + \cdots - 768)^{2} \)
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| $61$ |
\( (T^{3} + 8 T^{2} + \cdots - 688)^{2} \)
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| $67$ |
\( (T^{3} - 10 T^{2} + \cdots + 128)^{2} \)
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| $71$ |
\( (T^{3} - 8 T^{2} + \cdots + 768)^{2} \)
|
| $73$ |
\( (T^{3} + 4 T^{2} - 60 T - 32)^{2} \)
|
| $79$ |
\( (T^{3} + 4 T^{2} - 20 T - 32)^{2} \)
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| $83$ |
\( T^{6} + 274 T^{4} + \cdots + 292032 \)
|
| $89$ |
\( T^{6} + 454 T^{4} + \cdots + 629292 \)
|
| $97$ |
\( T^{6} + 172 T^{4} + \cdots + 3888 \)
|
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