Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [784,2,Mod(165,784)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(784, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("784.165");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 784 = 2^{4} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 784.x (of order \(12\), degree \(4\), 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: | \(6.26027151847\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 112) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(687\) | \(689\) |
| \(\chi(n)\) | \(\zeta_{12}^{3}\) | \(1\) | \(-1 + \zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 165.1 |
|
−1.36603 | + | 0.366025i | 0.133975 | + | 0.500000i | 1.73205 | − | 1.00000i | −0.232051 | + | 0.866025i | −0.366025 | − | 0.633975i | 0 | −2.00000 | + | 2.00000i | 2.36603 | − | 1.36603i | − | 1.26795i | |||||||||||||||
| 373.1 | 0.366025 | − | 1.36603i | 1.86603 | + | 0.500000i | −1.73205 | − | 1.00000i | 3.23205 | − | 0.866025i | 1.36603 | − | 2.36603i | 0 | −2.00000 | + | 2.00000i | 0.633975 | + | 0.366025i | − | 4.73205i | ||||||||||||||||
| 557.1 | 0.366025 | + | 1.36603i | 1.86603 | − | 0.500000i | −1.73205 | + | 1.00000i | 3.23205 | + | 0.866025i | 1.36603 | + | 2.36603i | 0 | −2.00000 | − | 2.00000i | 0.633975 | − | 0.366025i | 4.73205i | |||||||||||||||||
| 765.1 | −1.36603 | − | 0.366025i | 0.133975 | − | 0.500000i | 1.73205 | + | 1.00000i | −0.232051 | − | 0.866025i | −0.366025 | + | 0.633975i | 0 | −2.00000 | − | 2.00000i | 2.36603 | + | 1.36603i | 1.26795i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 112.w | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 784.2.x.a | 4 | |
| 7.b | odd | 2 | 1 | 112.2.w.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | 784.2.m.d | 4 | ||
| 7.c | even | 3 | 1 | 784.2.x.h | 4 | ||
| 7.d | odd | 6 | 1 | 112.2.w.b | yes | 4 | |
| 7.d | odd | 6 | 1 | 784.2.m.e | 4 | ||
| 16.e | even | 4 | 1 | 784.2.x.h | 4 | ||
| 28.d | even | 2 | 1 | 448.2.ba.b | 4 | ||
| 28.f | even | 6 | 1 | 448.2.ba.a | 4 | ||
| 56.e | even | 2 | 1 | 896.2.ba.a | 4 | ||
| 56.h | odd | 2 | 1 | 896.2.ba.d | 4 | ||
| 56.j | odd | 6 | 1 | 896.2.ba.b | 4 | ||
| 56.m | even | 6 | 1 | 896.2.ba.c | 4 | ||
| 112.j | even | 4 | 1 | 448.2.ba.a | 4 | ||
| 112.j | even | 4 | 1 | 896.2.ba.c | 4 | ||
| 112.l | odd | 4 | 1 | 112.2.w.b | yes | 4 | |
| 112.l | odd | 4 | 1 | 896.2.ba.b | 4 | ||
| 112.v | even | 12 | 1 | 448.2.ba.b | 4 | ||
| 112.v | even | 12 | 1 | 896.2.ba.a | 4 | ||
| 112.w | even | 12 | 1 | 784.2.m.d | 4 | ||
| 112.w | even | 12 | 1 | inner | 784.2.x.a | 4 | |
| 112.x | odd | 12 | 1 | 112.2.w.a | ✓ | 4 | |
| 112.x | odd | 12 | 1 | 784.2.m.e | 4 | ||
| 112.x | odd | 12 | 1 | 896.2.ba.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 112.2.w.a | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 112.2.w.a | ✓ | 4 | 112.x | odd | 12 | 1 | |
| 112.2.w.b | yes | 4 | 7.d | odd | 6 | 1 | |
| 112.2.w.b | yes | 4 | 112.l | odd | 4 | 1 | |
| 448.2.ba.a | 4 | 28.f | even | 6 | 1 | ||
| 448.2.ba.a | 4 | 112.j | even | 4 | 1 | ||
| 448.2.ba.b | 4 | 28.d | even | 2 | 1 | ||
| 448.2.ba.b | 4 | 112.v | even | 12 | 1 | ||
| 784.2.m.d | 4 | 7.c | even | 3 | 1 | ||
| 784.2.m.d | 4 | 112.w | even | 12 | 1 | ||
| 784.2.m.e | 4 | 7.d | odd | 6 | 1 | ||
| 784.2.m.e | 4 | 112.x | odd | 12 | 1 | ||
| 784.2.x.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 784.2.x.a | 4 | 112.w | even | 12 | 1 | inner | |
| 784.2.x.h | 4 | 7.c | even | 3 | 1 | ||
| 784.2.x.h | 4 | 16.e | even | 4 | 1 | ||
| 896.2.ba.a | 4 | 56.e | even | 2 | 1 | ||
| 896.2.ba.a | 4 | 112.v | even | 12 | 1 | ||
| 896.2.ba.b | 4 | 56.j | odd | 6 | 1 | ||
| 896.2.ba.b | 4 | 112.l | odd | 4 | 1 | ||
| 896.2.ba.c | 4 | 56.m | even | 6 | 1 | ||
| 896.2.ba.c | 4 | 112.j | even | 4 | 1 | ||
| 896.2.ba.d | 4 | 56.h | odd | 2 | 1 | ||
| 896.2.ba.d | 4 | 112.x | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):
|
\( T_{3}^{4} - 4T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1 \)
|
|
\( T_{5}^{4} - 6T_{5}^{3} + 9T_{5}^{2} + 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $5$ |
\( T^{4} - 6 T^{3} + \cdots + 9 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 8 T^{3} + \cdots + 169 \)
|
| $13$ |
\( T^{4} + 8 T^{3} + \cdots + 4 \)
|
| $17$ |
\( T^{4} - 6 T^{3} + \cdots + 9 \)
|
| $19$ |
\( T^{4} - 8 T^{3} + \cdots + 169 \)
|
| $23$ |
\( T^{4} + 12 T^{3} + \cdots + 121 \)
|
| $29$ |
\( T^{4} + 8 T^{3} + \cdots + 4 \)
|
| $31$ |
\( T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $37$ |
\( T^{4} - 22 T^{3} + \cdots + 169 \)
|
| $41$ |
\( T^{4} + 104T^{2} + 1936 \)
|
| $43$ |
\( T^{4} - 12 T^{3} + \cdots + 36 \)
|
| $47$ |
\( T^{4} - 12 T^{3} + \cdots + 1089 \)
|
| $53$ |
\( T^{4} - 2 T^{3} + \cdots + 2209 \)
|
| $59$ |
\( T^{4} + 28 T^{3} + \cdots + 14641 \)
|
| $61$ |
\( T^{4} - 14 T^{3} + \cdots + 1 \)
|
| $67$ |
\( T^{4} + 12 T^{3} + \cdots + 1521 \)
|
| $71$ |
\( T^{4} + 56T^{2} + 16 \)
|
| $73$ |
\( T^{4} - 18 T^{3} + \cdots + 529 \)
|
| $79$ |
\( T^{4} - 16 T^{3} + \cdots + 121 \)
|
| $83$ |
\( T^{4} + 20 T^{3} + \cdots + 676 \)
|
| $89$ |
\( (T^{2} + 9 T + 27)^{2} \)
|
| $97$ |
\( (T^{2} + 8 T - 32)^{2} \)
|
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