Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [407,2,Mod(122,407)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(407, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("407.122");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 407 = 11 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 407.k (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: | \(3.24991136227\) |
| Analytic rank: | \(1\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(\left(\mathbb{Z}/407\mathbb{Z}\right)^\times\).
| \(n\) | \(112\) | \(298\) |
| \(\chi(n)\) | \(1\) | \(\zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 122.1 |
|
−2.36603 | + | 1.36603i | −1.00000 | + | 1.73205i | 2.73205 | − | 4.73205i | 0.232051 | + | 0.133975i | − | 5.46410i | −0.633975 | + | 1.09808i | 9.46410i | −0.500000 | − | 0.866025i | −0.732051 | |||||||||||||||||
| 122.2 | −0.633975 | + | 0.366025i | −1.00000 | + | 1.73205i | −0.732051 | + | 1.26795i | −3.23205 | − | 1.86603i | − | 1.46410i | −2.36603 | + | 4.09808i | − | 2.53590i | −0.500000 | − | 0.866025i | 2.73205 | |||||||||||||||||
| 397.1 | −2.36603 | − | 1.36603i | −1.00000 | − | 1.73205i | 2.73205 | + | 4.73205i | 0.232051 | − | 0.133975i | 5.46410i | −0.633975 | − | 1.09808i | − | 9.46410i | −0.500000 | + | 0.866025i | −0.732051 | ||||||||||||||||||
| 397.2 | −0.633975 | − | 0.366025i | −1.00000 | − | 1.73205i | −0.732051 | − | 1.26795i | −3.23205 | + | 1.86603i | 1.46410i | −2.36603 | − | 4.09808i | 2.53590i | −0.500000 | + | 0.866025i | 2.73205 | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.e | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 407.2.k.b | ✓ | 4 |
| 37.e | even | 6 | 1 | inner | 407.2.k.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 407.2.k.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 407.2.k.b | ✓ | 4 | 37.e | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 6T_{2}^{3} + 14T_{2}^{2} + 12T_{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(407, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 6 T^{3} + \cdots + 4 \)
|
| $3$ |
\( (T^{2} + 2 T + 4)^{2} \)
|
| $5$ |
\( T^{4} + 6 T^{3} + \cdots + 1 \)
|
| $7$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
|
| $11$ |
\( (T - 1)^{4} \)
|
| $13$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
|
| $17$ |
\( T^{4} - 16T^{2} + 256 \)
|
| $19$ |
\( (T^{2} + 6 T + 12)^{2} \)
|
| $23$ |
\( T^{4} + 32T^{2} + 64 \)
|
| $29$ |
\( T^{4} + 104T^{2} + 4 \)
|
| $31$ |
\( (T^{2} + 3)^{2} \)
|
| $37$ |
\( (T^{2} - 10 T + 37)^{2} \)
|
| $41$ |
\( T^{4} + 48T^{2} + 2304 \)
|
| $43$ |
\( (T^{2} + 12)^{2} \)
|
| $47$ |
\( (T + 3)^{4} \)
|
| $53$ |
\( T^{4} + 18 T^{3} + \cdots + 4761 \)
|
| $59$ |
\( T^{4} + 6 T^{3} + \cdots + 3721 \)
|
| $61$ |
\( T^{4} + 24 T^{3} + \cdots + 144 \)
|
| $67$ |
\( T^{4} - 18 T^{3} + \cdots + 1089 \)
|
| $71$ |
\( T^{4} + 108 T^{2} + 11664 \)
|
| $73$ |
\( (T^{2} + 12 T - 12)^{2} \)
|
| $79$ |
\( T^{4} + 24 T^{3} + \cdots + 144 \)
|
| $83$ |
\( T^{4} - 18 T^{3} + \cdots + 6084 \)
|
| $89$ |
\( T^{4} + 48 T^{3} + \cdots + 35344 \)
|
| $97$ |
\( (T^{2} + 36)^{2} \)
|
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