Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6336,2,Mod(2177,6336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6336, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6336.2177");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6336.b (of order \(2\), degree \(1\), not 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: | \(50.5932147207\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.12781568.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 8x^{4} + 17x^{2} + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 792) |
| 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
\( x^{6} + 8x^{4} + 17x^{2} + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 5\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 6\nu^{3} + 7\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} + 5\nu^{2} + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 8\nu^{3} + 13\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} + 7\nu^{2} + 9 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - \beta_{3} - 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{4} - 5\beta_{2} - 3\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -5\beta_{5} + 7\beta_{3} + 24 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -23\beta_{4} + 27\beta_{2} + 11\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times\).
| \(n\) | \(1729\) | \(3521\) | \(4159\) | \(4357\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2177.1 |
|
0 | 0 | 0 | − | 3.92933i | 0 | 2.07986i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||
| 2177.2 | 0 | 0 | 0 | − | 3.52039i | 0 | − | 4.72761i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||
| 2177.3 | 0 | 0 | 0 | − | 0.408946i | 0 | 1.15061i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 2177.4 | 0 | 0 | 0 | 0.408946i | 0 | − | 1.15061i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 2177.5 | 0 | 0 | 0 | 3.52039i | 0 | 4.72761i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 2177.6 | 0 | 0 | 0 | 3.92933i | 0 | − | 2.07986i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 33.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6336.2.b.w | 6 | |
| 3.b | odd | 2 | 1 | 6336.2.b.z | 6 | ||
| 4.b | odd | 2 | 1 | 6336.2.b.y | 6 | ||
| 8.b | even | 2 | 1 | 1584.2.b.g | 6 | ||
| 8.d | odd | 2 | 1 | 792.2.b.a | ✓ | 6 | |
| 11.b | odd | 2 | 1 | 6336.2.b.z | 6 | ||
| 12.b | even | 2 | 1 | 6336.2.b.x | 6 | ||
| 24.f | even | 2 | 1 | 792.2.b.b | yes | 6 | |
| 24.h | odd | 2 | 1 | 1584.2.b.f | 6 | ||
| 33.d | even | 2 | 1 | inner | 6336.2.b.w | 6 | |
| 44.c | even | 2 | 1 | 6336.2.b.x | 6 | ||
| 88.b | odd | 2 | 1 | 1584.2.b.f | 6 | ||
| 88.g | even | 2 | 1 | 792.2.b.b | yes | 6 | |
| 132.d | odd | 2 | 1 | 6336.2.b.y | 6 | ||
| 264.m | even | 2 | 1 | 1584.2.b.g | 6 | ||
| 264.p | odd | 2 | 1 | 792.2.b.a | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 792.2.b.a | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 792.2.b.a | ✓ | 6 | 264.p | odd | 2 | 1 | |
| 792.2.b.b | yes | 6 | 24.f | even | 2 | 1 | |
| 792.2.b.b | yes | 6 | 88.g | even | 2 | 1 | |
| 1584.2.b.f | 6 | 24.h | odd | 2 | 1 | ||
| 1584.2.b.f | 6 | 88.b | odd | 2 | 1 | ||
| 1584.2.b.g | 6 | 8.b | even | 2 | 1 | ||
| 1584.2.b.g | 6 | 264.m | even | 2 | 1 | ||
| 6336.2.b.w | 6 | 1.a | even | 1 | 1 | trivial | |
| 6336.2.b.w | 6 | 33.d | even | 2 | 1 | inner | |
| 6336.2.b.x | 6 | 12.b | even | 2 | 1 | ||
| 6336.2.b.x | 6 | 44.c | even | 2 | 1 | ||
| 6336.2.b.y | 6 | 4.b | odd | 2 | 1 | ||
| 6336.2.b.y | 6 | 132.d | odd | 2 | 1 | ||
| 6336.2.b.z | 6 | 3.b | odd | 2 | 1 | ||
| 6336.2.b.z | 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 \(S_{2}^{\mathrm{new}}(6336, [\chi])\):
|
\( T_{5}^{6} + 28T_{5}^{4} + 196T_{5}^{2} + 32 \)
|
|
\( T_{7}^{6} + 28T_{7}^{4} + 132T_{7}^{2} + 128 \)
|
|
\( T_{13}^{6} + 34T_{13}^{4} + 160T_{13}^{2} + 128 \)
|
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\( T_{17}^{3} + 2T_{17}^{2} - 16T_{17} - 16 \)
|
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\( T_{31}^{3} - 6T_{31}^{2} - 16T_{31} + 32 \)
|
|
\( T_{83}^{3} - 22T_{83}^{2} + 96T_{83} + 256 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 28 T^{4} + \cdots + 32 \)
|
| $7$ |
\( T^{6} + 28 T^{4} + \cdots + 128 \)
|
| $11$ |
\( T^{6} + 2 T^{5} + \cdots + 1331 \)
|
| $13$ |
\( T^{6} + 34 T^{4} + \cdots + 128 \)
|
| $17$ |
\( (T^{3} + 2 T^{2} - 16 T - 16)^{2} \)
|
| $19$ |
\( T^{6} + 30 T^{4} + \cdots + 8 \)
|
| $23$ |
\( T^{6} + 82 T^{4} + \cdots + 15488 \)
|
| $29$ |
\( (T^{3} + 4 T^{2} + \cdots - 256)^{2} \)
|
| $31$ |
\( (T^{3} - 6 T^{2} - 16 T + 32)^{2} \)
|
| $37$ |
\( (T^{3} + 6 T^{2} - 16 T - 64)^{2} \)
|
| $41$ |
\( (T^{3} + 2 T^{2} + \cdots + 304)^{2} \)
|
| $43$ |
\( T^{6} + 110 T^{4} + \cdots + 2888 \)
|
| $47$ |
\( T^{6} + 82 T^{4} + \cdots + 128 \)
|
| $53$ |
\( T^{6} + 52 T^{4} + \cdots + 512 \)
|
| $59$ |
\( T^{6} + 160 T^{4} + \cdots + 8192 \)
|
| $61$ |
\( T^{6} + 210 T^{4} + \cdots + 128 \)
|
| $67$ |
\( (T^{3} - 16 T^{2} + \cdots + 256)^{2} \)
|
| $71$ |
\( T^{6} + 34 T^{4} + \cdots + 128 \)
|
| $73$ |
\( T^{6} + 456 T^{4} + \cdots + 3442688 \)
|
| $79$ |
\( T^{6} + 388 T^{4} + \cdots + 2048288 \)
|
| $83$ |
\( (T^{3} - 22 T^{2} + \cdots + 256)^{2} \)
|
| $89$ |
\( (T^{2} + 18)^{3} \)
|
| $97$ |
\( (T^{3} - 8 T^{2} + \cdots + 848)^{2} \)
|
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