Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [117,2,Mod(5,117)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(117, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([10, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("117.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 117.z (of order \(12\), degree \(4\), 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: | \(0.934249703649\) |
| Analytic rank: | \(1\) |
| 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: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(92\) |
| \(\chi(n)\) | \(\zeta_{12}^{3}\) | \(\zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
−0.633975 | + | 2.36603i | −1.50000 | − | 0.866025i | −3.46410 | − | 2.00000i | −2.36603 | + | 0.633975i | 3.00000 | − | 3.00000i | 0.732051 | − | 2.73205i | 3.46410 | − | 3.46410i | 1.50000 | + | 2.59808i | − | 6.00000i | |||||||||||||
| 47.1 | −0.633975 | − | 2.36603i | −1.50000 | + | 0.866025i | −3.46410 | + | 2.00000i | −2.36603 | − | 0.633975i | 3.00000 | + | 3.00000i | 0.732051 | + | 2.73205i | 3.46410 | + | 3.46410i | 1.50000 | − | 2.59808i | 6.00000i | |||||||||||||||
| 83.1 | −2.36603 | + | 0.633975i | −1.50000 | + | 0.866025i | 3.46410 | − | 2.00000i | −0.633975 | + | 2.36603i | 3.00000 | − | 3.00000i | −2.73205 | + | 0.732051i | −3.46410 | + | 3.46410i | 1.50000 | − | 2.59808i | − | 6.00000i | ||||||||||||||
| 86.1 | −2.36603 | − | 0.633975i | −1.50000 | − | 0.866025i | 3.46410 | + | 2.00000i | −0.633975 | − | 2.36603i | 3.00000 | + | 3.00000i | −2.73205 | − | 0.732051i | −3.46410 | − | 3.46410i | 1.50000 | + | 2.59808i | 6.00000i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.d | odd | 6 | 1 | inner |
| 13.d | odd | 4 | 1 | inner |
| 117.z | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 117.2.z.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 351.2.bc.a | 4 | ||
| 9.c | even | 3 | 1 | 351.2.bc.a | 4 | ||
| 9.d | odd | 6 | 1 | inner | 117.2.z.a | ✓ | 4 |
| 13.d | odd | 4 | 1 | inner | 117.2.z.a | ✓ | 4 |
| 39.f | even | 4 | 1 | 351.2.bc.a | 4 | ||
| 117.y | odd | 12 | 1 | 351.2.bc.a | 4 | ||
| 117.z | even | 12 | 1 | inner | 117.2.z.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 117.2.z.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 117.2.z.a | ✓ | 4 | 9.d | odd | 6 | 1 | inner |
| 117.2.z.a | ✓ | 4 | 13.d | odd | 4 | 1 | inner |
| 117.2.z.a | ✓ | 4 | 117.z | even | 12 | 1 | inner |
| 351.2.bc.a | 4 | 3.b | odd | 2 | 1 | ||
| 351.2.bc.a | 4 | 9.c | even | 3 | 1 | ||
| 351.2.bc.a | 4 | 39.f | even | 4 | 1 | ||
| 351.2.bc.a | 4 | 117.y | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 6T_{2}^{3} + 18T_{2}^{2} + 36T_{2} + 36 \)
acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
|
| $3$ |
\( (T^{2} + 3 T + 3)^{2} \)
|
| $5$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
|
| $7$ |
\( T^{4} + 4 T^{3} + \cdots + 64 \)
|
| $11$ |
\( T^{4} + 12 T^{3} + \cdots + 576 \)
|
| $13$ |
\( T^{4} + 6 T^{3} + \cdots + 169 \)
|
| $17$ |
\( (T^{2} - 3)^{2} \)
|
| $19$ |
\( (T^{2} + 8 T + 32)^{2} \)
|
| $23$ |
\( T^{4} + 3T^{2} + 9 \)
|
| $29$ |
\( (T^{2} + 6 T + 12)^{2} \)
|
| $31$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
|
| $37$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $41$ |
\( T^{4} + 18 T^{3} + \cdots + 2916 \)
|
| $43$ |
\( T^{4} - 81T^{2} + 6561 \)
|
| $47$ |
\( T^{4} - 6 T^{3} + \cdots + 36 \)
|
| $53$ |
\( (T^{2} + 3)^{2} \)
|
| $59$ |
\( T^{4} + 18 T^{3} + \cdots + 2916 \)
|
| $61$ |
\( (T^{2} + 3 T + 9)^{2} \)
|
| $67$ |
\( T^{4} + 16 T^{3} + \cdots + 16384 \)
|
| $71$ |
\( T^{4} + 576 \)
|
| $73$ |
\( (T^{2} + 14 T + 98)^{2} \)
|
| $79$ |
\( (T^{2} + 3 T + 9)^{2} \)
|
| $83$ |
\( T^{4} - 6 T^{3} + \cdots + 36 \)
|
| $89$ |
\( T^{4} + 46656 \)
|
| $97$ |
\( T^{4} + 20 T^{3} + \cdots + 40000 \)
|
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