Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3780,1,Mod(307,3780)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3780, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 4, 3, 6]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3780.307");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3780 = 2^{2} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3780.ee (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: | \(1.88646574775\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1260) |
| Projective image: | \(S_{4}\) |
| Projective field: | Galois closure of 4.2.1134000.1 |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3780\mathbb{Z}\right)^\times\).
| \(n\) | \(757\) | \(1081\) | \(1541\) | \(1891\) |
| \(\chi(n)\) | \(-\zeta_{24}^{6}\) | \(-1\) | \(\zeta_{24}^{8}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 307.1 |
|
0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.965926 | + | 0.258819i | 0 | 0.258819 | − | 0.965926i | −1.00000 | 0 | −0.707107 | − | 0.707107i | ||||||||||||||||||||||||||||||||
| 307.2 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0.965926 | − | 0.258819i | 0 | −0.258819 | + | 0.965926i | −1.00000 | 0 | 0.707107 | + | 0.707107i | |||||||||||||||||||||||||||||||||
| 1063.1 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.258819 | − | 0.965926i | 0 | 0.965926 | + | 0.258819i | −1.00000 | 0 | 0.707107 | − | 0.707107i | |||||||||||||||||||||||||||||||||
| 1063.2 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0.258819 | + | 0.965926i | 0 | −0.965926 | − | 0.258819i | −1.00000 | 0 | −0.707107 | + | 0.707107i | |||||||||||||||||||||||||||||||||
| 2827.1 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.258819 | + | 0.965926i | 0 | 0.965926 | − | 0.258819i | −1.00000 | 0 | 0.707107 | + | 0.707107i | |||||||||||||||||||||||||||||||||
| 2827.2 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | 0.258819 | − | 0.965926i | 0 | −0.965926 | + | 0.258819i | −1.00000 | 0 | −0.707107 | − | 0.707107i | |||||||||||||||||||||||||||||||||
| 3583.1 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.965926 | − | 0.258819i | 0 | 0.258819 | + | 0.965926i | −1.00000 | 0 | −0.707107 | + | 0.707107i | |||||||||||||||||||||||||||||||||
| 3583.2 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | 0.965926 | + | 0.258819i | 0 | −0.258819 | − | 0.965926i | −1.00000 | 0 | 0.707107 | − | 0.707107i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 20.e | even | 4 | 1 | inner |
| 63.l | odd | 6 | 1 | inner |
| 140.j | odd | 4 | 1 | inner |
| 180.x | even | 12 | 1 | inner |
| 1260.eb | odd | 12 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{43}^{4} + 2T_{43}^{3} + 2T_{43}^{2} + 4T_{43} + 4 \)
acting on \(S_{1}^{\mathrm{new}}(3780, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - T^{4} + 1 \)
|
| $7$ |
\( T^{8} - T^{4} + 1 \)
|
| $11$ |
\( (T^{4} - T^{2} + 1)^{2} \)
|
| $13$ |
\( T^{8} - T^{4} + 1 \)
|
| $17$ |
\( (T^{4} + 1)^{2} \)
|
| $19$ |
\( (T^{2} + 2)^{4} \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} \)
|
| $41$ |
\( T^{8} \)
|
| $43$ |
\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \)
|
| $47$ |
\( T^{8} - T^{4} + 1 \)
|
| $53$ |
\( (T^{2} + 2 T + 2)^{4} \)
|
| $59$ |
\( T^{8} \)
|
| $61$ |
\( (T^{4} - 2 T^{2} + 4)^{2} \)
|
| $67$ |
\( T^{8} \)
|
| $71$ |
\( (T^{2} + 1)^{4} \)
|
| $73$ |
\( (T^{4} + 1)^{2} \)
|
| $79$ |
\( (T^{2} - T + 1)^{4} \)
|
| $83$ |
\( T^{8} - T^{4} + 1 \)
|
| $89$ |
\( T^{8} \)
|
| $97$ |
\( T^{8} - T^{4} + 1 \)
|
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