Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3150,2,Mod(899,3150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3150.899");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3150.bp (of order \(6\), degree \(2\), 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: | \(25.1528766367\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 630) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(451\) | \(2801\) |
| \(\chi(n)\) | \(-1\) | \(1 - \zeta_{24}^{4}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 899.1 |
|
−0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0 | 0 | −2.63896 | + | 0.189469i | 1.00000 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 899.2 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0 | 0 | −0.189469 | − | 2.63896i | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 899.3 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0 | 0 | 0.189469 | + | 2.63896i | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 899.4 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0 | 0 | 2.63896 | − | 0.189469i | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1349.1 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 0 | 0 | −2.63896 | − | 0.189469i | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1349.2 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 0 | 0 | −0.189469 | + | 2.63896i | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1349.3 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 0 | 0 | 0.189469 | − | 2.63896i | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1349.4 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 0 | 0 | 2.63896 | + | 0.189469i | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 105.p | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3150.2.bp.a | 8 | |
| 3.b | odd | 2 | 1 | 3150.2.bp.f | 8 | ||
| 5.b | even | 2 | 1 | 3150.2.bp.d | 8 | ||
| 5.c | odd | 4 | 1 | 630.2.be.a | ✓ | 8 | |
| 5.c | odd | 4 | 1 | 3150.2.bf.b | 8 | ||
| 7.d | odd | 6 | 1 | 3150.2.bp.c | 8 | ||
| 15.d | odd | 2 | 1 | 3150.2.bp.c | 8 | ||
| 15.e | even | 4 | 1 | 630.2.be.b | yes | 8 | |
| 15.e | even | 4 | 1 | 3150.2.bf.c | 8 | ||
| 21.g | even | 6 | 1 | 3150.2.bp.d | 8 | ||
| 35.i | odd | 6 | 1 | 3150.2.bp.f | 8 | ||
| 35.k | even | 12 | 1 | 630.2.be.b | yes | 8 | |
| 35.k | even | 12 | 1 | 3150.2.bf.c | 8 | ||
| 35.k | even | 12 | 1 | 4410.2.b.b | 8 | ||
| 35.l | odd | 12 | 1 | 4410.2.b.e | 8 | ||
| 105.p | even | 6 | 1 | inner | 3150.2.bp.a | 8 | |
| 105.w | odd | 12 | 1 | 630.2.be.a | ✓ | 8 | |
| 105.w | odd | 12 | 1 | 3150.2.bf.b | 8 | ||
| 105.w | odd | 12 | 1 | 4410.2.b.e | 8 | ||
| 105.x | even | 12 | 1 | 4410.2.b.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 630.2.be.a | ✓ | 8 | 5.c | odd | 4 | 1 | |
| 630.2.be.a | ✓ | 8 | 105.w | odd | 12 | 1 | |
| 630.2.be.b | yes | 8 | 15.e | even | 4 | 1 | |
| 630.2.be.b | yes | 8 | 35.k | even | 12 | 1 | |
| 3150.2.bf.b | 8 | 5.c | odd | 4 | 1 | ||
| 3150.2.bf.b | 8 | 105.w | odd | 12 | 1 | ||
| 3150.2.bf.c | 8 | 15.e | even | 4 | 1 | ||
| 3150.2.bf.c | 8 | 35.k | even | 12 | 1 | ||
| 3150.2.bp.a | 8 | 1.a | even | 1 | 1 | trivial | |
| 3150.2.bp.a | 8 | 105.p | even | 6 | 1 | inner | |
| 3150.2.bp.c | 8 | 7.d | odd | 6 | 1 | ||
| 3150.2.bp.c | 8 | 15.d | odd | 2 | 1 | ||
| 3150.2.bp.d | 8 | 5.b | even | 2 | 1 | ||
| 3150.2.bp.d | 8 | 21.g | even | 6 | 1 | ||
| 3150.2.bp.f | 8 | 3.b | odd | 2 | 1 | ||
| 3150.2.bp.f | 8 | 35.i | odd | 6 | 1 | ||
| 4410.2.b.b | 8 | 35.k | even | 12 | 1 | ||
| 4410.2.b.b | 8 | 105.x | even | 12 | 1 | ||
| 4410.2.b.e | 8 | 35.l | odd | 12 | 1 | ||
| 4410.2.b.e | 8 | 105.w | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3150, [\chi])\):
|
\( T_{11}^{8} + 24 T_{11}^{7} + 260 T_{11}^{6} + 1632 T_{11}^{5} + 6447 T_{11}^{4} + 16320 T_{11}^{3} + \cdots + 9409 \)
|
|
\( T_{13}^{4} - 8T_{13}^{3} + 20T_{13}^{2} - 16T_{13} + 1 \)
|
|
\( T_{17}^{8} + 24T_{17}^{7} + 248T_{17}^{6} + 1344T_{17}^{5} + 3936T_{17}^{4} + 5376T_{17}^{3} + 1280T_{17}^{2} - 3072T_{17} + 1024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} - 94T^{4} + 2401 \)
|
| $11$ |
\( T^{8} + 24 T^{7} + \cdots + 9409 \)
|
| $13$ |
\( (T^{4} - 8 T^{3} + 20 T^{2} + \cdots + 1)^{2} \)
|
| $17$ |
\( T^{8} + 24 T^{7} + \cdots + 1024 \)
|
| $19$ |
\( T^{8} - 18 T^{6} + \cdots + 1 \)
|
| $23$ |
\( (T^{4} - 4 T^{3} + 15 T^{2} + \cdots + 1)^{2} \)
|
| $29$ |
\( (T^{4} + 80 T^{2} + 64)^{2} \)
|
| $31$ |
\( T^{8} - 24 T^{6} + \cdots + 256 \)
|
| $37$ |
\( T^{8} - 76 T^{6} + \cdots + 1329409 \)
|
| $41$ |
\( (T^{4} + 16 T^{3} + \cdots - 71)^{2} \)
|
| $43$ |
\( T^{8} + 48 T^{6} + \cdots + 1024 \)
|
| $47$ |
\( T^{8} + 12 T^{7} + \cdots + 330625 \)
|
| $53$ |
\( T^{8} + 4 T^{7} + \cdots + 2745649 \)
|
| $59$ |
\( T^{8} + 24 T^{7} + \cdots + 22090000 \)
|
| $61$ |
\( T^{8} - 120 T^{6} + \cdots + 2262016 \)
|
| $67$ |
\( T^{8} - 48 T^{7} + \cdots + 442008576 \)
|
| $71$ |
\( T^{8} + 288 T^{6} + \cdots + 1364224 \)
|
| $73$ |
\( (T^{4} + 12 T^{3} + \cdots + 16)^{2} \)
|
| $79$ |
\( T^{8} - 24 T^{7} + \cdots + 171295744 \)
|
| $83$ |
\( T^{8} + 432 T^{6} + \cdots + 58491904 \)
|
| $89$ |
\( T^{8} + 16 T^{7} + \cdots + 2062096 \)
|
| $97$ |
\( (T^{4} + 24 T^{3} + \cdots - 24704)^{2} \)
|
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