Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [416,2,Mod(225,416)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([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("416.225");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 416 = 2^{5} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 416.w (of order \(6\), degree \(2\), 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: | \(3.32177672409\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| 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: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(287\) | \(353\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1 - \zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 225.1 |
|
0 | −0.866025 | + | 1.50000i | 0 | − | 3.46410i | 0 | −2.59808 | + | 1.50000i | 0 | 0 | 0 | |||||||||||||||||||||||||
| 225.2 | 0 | 0.866025 | − | 1.50000i | 0 | − | 3.46410i | 0 | 2.59808 | − | 1.50000i | 0 | 0 | 0 | ||||||||||||||||||||||||||
| 257.1 | 0 | −0.866025 | − | 1.50000i | 0 | 3.46410i | 0 | −2.59808 | − | 1.50000i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 257.2 | 0 | 0.866025 | + | 1.50000i | 0 | 3.46410i | 0 | 2.59808 | + | 1.50000i | 0 | 0 | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 13.e | even | 6 | 1 | inner |
| 52.i | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 416.2.w.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 416.2.w.a | ✓ | 4 |
| 8.b | even | 2 | 1 | 832.2.w.e | 4 | ||
| 8.d | odd | 2 | 1 | 832.2.w.e | 4 | ||
| 13.e | even | 6 | 1 | inner | 416.2.w.a | ✓ | 4 |
| 13.f | odd | 12 | 1 | 5408.2.a.u | 2 | ||
| 13.f | odd | 12 | 1 | 5408.2.a.z | 2 | ||
| 52.i | odd | 6 | 1 | inner | 416.2.w.a | ✓ | 4 |
| 52.l | even | 12 | 1 | 5408.2.a.u | 2 | ||
| 52.l | even | 12 | 1 | 5408.2.a.z | 2 | ||
| 104.p | odd | 6 | 1 | 832.2.w.e | 4 | ||
| 104.s | even | 6 | 1 | 832.2.w.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 416.2.w.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 416.2.w.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 416.2.w.a | ✓ | 4 | 13.e | even | 6 | 1 | inner |
| 416.2.w.a | ✓ | 4 | 52.i | odd | 6 | 1 | inner |
| 832.2.w.e | 4 | 8.b | even | 2 | 1 | ||
| 832.2.w.e | 4 | 8.d | odd | 2 | 1 | ||
| 832.2.w.e | 4 | 104.p | odd | 6 | 1 | ||
| 832.2.w.e | 4 | 104.s | even | 6 | 1 | ||
| 5408.2.a.u | 2 | 13.f | odd | 12 | 1 | ||
| 5408.2.a.u | 2 | 52.l | even | 12 | 1 | ||
| 5408.2.a.z | 2 | 13.f | odd | 12 | 1 | ||
| 5408.2.a.z | 2 | 52.l | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 3T_{3}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 3T^{2} + 9 \)
|
| $5$ |
\( (T^{2} + 12)^{2} \)
|
| $7$ |
\( T^{4} - 9T^{2} + 81 \)
|
| $11$ |
\( T^{4} - 25T^{2} + 625 \)
|
| $13$ |
\( (T^{2} + 2 T + 13)^{2} \)
|
| $17$ |
\( (T^{2} - 7 T + 49)^{2} \)
|
| $19$ |
\( T^{4} - 25T^{2} + 625 \)
|
| $23$ |
\( T^{4} + 27T^{2} + 729 \)
|
| $29$ |
\( (T^{2} - 5 T + 25)^{2} \)
|
| $31$ |
\( (T^{2} + 4)^{2} \)
|
| $37$ |
\( (T^{2} - 9 T + 27)^{2} \)
|
| $41$ |
\( (T^{2} + 3 T + 3)^{2} \)
|
| $43$ |
\( T^{4} + 27T^{2} + 729 \)
|
| $47$ |
\( (T^{2} + 16)^{2} \)
|
| $53$ |
\( (T - 4)^{4} \)
|
| $59$ |
\( T^{4} - 49T^{2} + 2401 \)
|
| $61$ |
\( (T^{2} + 3 T + 9)^{2} \)
|
| $67$ |
\( T^{4} - 9T^{2} + 81 \)
|
| $71$ |
\( T^{4} - 49T^{2} + 2401 \)
|
| $73$ |
\( (T^{2} + 12)^{2} \)
|
| $79$ |
\( (T^{2} - 12)^{2} \)
|
| $83$ |
\( (T^{2} + 196)^{2} \)
|
| $89$ |
\( (T^{2} - 3 T + 3)^{2} \)
|
| $97$ |
\( (T^{2} - 15 T + 75)^{2} \)
|
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