Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [104,3,Mod(33,104)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(104, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 0, 11]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("104.33");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 104 = 2^{3} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 104.v (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: | \(2.83379474935\) |
| 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: | \(\Z[a_1, \ldots, a_{9}]\) |
| 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 \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).
| \(n\) | \(41\) | \(53\) | \(79\) |
| \(\chi(n)\) | \(\zeta_{12}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.1 |
|
0 | −0.732051 | + | 1.26795i | 0 | −1.09808 | − | 1.09808i | 0 | 10.9282 | + | 2.92820i | 0 | 3.42820 | + | 5.93782i | 0 | ||||||||||||||||||||||
| 41.1 | 0 | −0.732051 | − | 1.26795i | 0 | −1.09808 | + | 1.09808i | 0 | 10.9282 | − | 2.92820i | 0 | 3.42820 | − | 5.93782i | 0 | |||||||||||||||||||||||
| 89.1 | 0 | 2.73205 | + | 4.73205i | 0 | 4.09808 | + | 4.09808i | 0 | −2.92820 | − | 10.9282i | 0 | −10.4282 | + | 18.0622i | 0 | |||||||||||||||||||||||
| 97.1 | 0 | 2.73205 | − | 4.73205i | 0 | 4.09808 | − | 4.09808i | 0 | −2.92820 | + | 10.9282i | 0 | −10.4282 | − | 18.0622i | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.f | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 104.3.v.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 208.3.bd.b | 4 | ||
| 13.f | odd | 12 | 1 | inner | 104.3.v.b | ✓ | 4 |
| 52.l | even | 12 | 1 | 208.3.bd.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.3.v.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 104.3.v.b | ✓ | 4 | 13.f | odd | 12 | 1 | inner |
| 208.3.bd.b | 4 | 4.b | odd | 2 | 1 | ||
| 208.3.bd.b | 4 | 52.l | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 4T_{3}^{3} + 24T_{3}^{2} + 32T_{3} + 64 \)
acting on \(S_{3}^{\mathrm{new}}(104, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 64 \)
|
| $5$ |
\( T^{4} - 6 T^{3} + \cdots + 81 \)
|
| $7$ |
\( T^{4} - 16 T^{3} + \cdots + 16384 \)
|
| $11$ |
\( T^{4} + 8 T^{3} + \cdots + 30976 \)
|
| $13$ |
\( T^{4} - 169 T^{2} + 28561 \)
|
| $17$ |
\( T^{4} - T^{2} + 1 \)
|
| $19$ |
\( T^{4} - 44 T^{3} + \cdots + 141376 \)
|
| $23$ |
\( T^{4} - 12 T^{3} + \cdots + 7744 \)
|
| $29$ |
\( T^{4} - 48 T^{3} + \cdots + 251001 \)
|
| $31$ |
\( T^{4} + 24 T^{3} + \cdots + 576 \)
|
| $37$ |
\( T^{4} + 26 T^{3} + \cdots + 28561 \)
|
| $41$ |
\( T^{4} + 116 T^{3} + \cdots + 2819041 \)
|
| $43$ |
\( T^{4} + 168 T^{3} + \cdots + 5234944 \)
|
| $47$ |
\( T^{4} - 80 T^{3} + \cdots + 341056 \)
|
| $53$ |
\( (T^{2} - 106 T + 2041)^{2} \)
|
| $59$ |
\( T^{4} + 168 T^{3} + \cdots + 14017536 \)
|
| $61$ |
\( T^{4} - 22 T^{3} + \cdots + 418609 \)
|
| $67$ |
\( T^{4} + 84 T^{3} + \cdots + 2143296 \)
|
| $71$ |
\( T^{4} - 60 T^{3} + \cdots + 788544 \)
|
| $73$ |
\( T^{4} - 126 T^{3} + \cdots + 15752961 \)
|
| $79$ |
\( (T^{2} - 8 T - 4784)^{2} \)
|
| $83$ |
\( T^{4} - 56 T^{3} + \cdots + 30976 \)
|
| $89$ |
\( T^{4} + 190 T^{3} + \cdots + 37185604 \)
|
| $97$ |
\( T^{4} - 222 T^{3} + \cdots + 115218756 \)
|
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