Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [845,2,Mod(654,845)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(845, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("845.654");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 845 = 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 845.l (of order \(6\), degree \(2\), 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: | \(6.74735897080\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 65) |
| 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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( 2\zeta_{12} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\zeta_{12}^{3} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( \beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) | \(677\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 654.1 |
|
−0.500000 | + | 0.866025i | −1.73205 | − | 1.00000i | 0.500000 | + | 0.866025i | 1.00000 | − | 2.00000i | 1.73205 | − | 1.00000i | 0 | −3.00000 | 0.500000 | + | 0.866025i | 1.23205 | + | 1.86603i | ||||||||||||||||
| 654.2 | −0.500000 | + | 0.866025i | 1.73205 | + | 1.00000i | 0.500000 | + | 0.866025i | 1.00000 | + | 2.00000i | −1.73205 | + | 1.00000i | 0 | −3.00000 | 0.500000 | + | 0.866025i | −2.23205 | − | 0.133975i | |||||||||||||||||
| 699.1 | −0.500000 | − | 0.866025i | −1.73205 | + | 1.00000i | 0.500000 | − | 0.866025i | 1.00000 | + | 2.00000i | 1.73205 | + | 1.00000i | 0 | −3.00000 | 0.500000 | − | 0.866025i | 1.23205 | − | 1.86603i | |||||||||||||||||
| 699.2 | −0.500000 | − | 0.866025i | 1.73205 | − | 1.00000i | 0.500000 | − | 0.866025i | 1.00000 | − | 2.00000i | −1.73205 | − | 1.00000i | 0 | −3.00000 | 0.500000 | − | 0.866025i | −2.23205 | + | 0.133975i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
| 65.d | even | 2 | 1 | inner |
| 65.l | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{2} \)
|
| $3$ |
\( T^{4} - 4T^{2} + 16 \)
|
| $5$ |
\( (T^{2} - 2 T + 5)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 4T^{2} + 16 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} - 36T^{2} + 1296 \)
|
| $23$ |
\( T^{4} - 36T^{2} + 1296 \)
|
| $29$ |
\( (T^{2} + 6 T + 36)^{2} \)
|
| $31$ |
\( (T^{2} + 36)^{2} \)
|
| $37$ |
\( (T^{2} - 6 T + 36)^{2} \)
|
| $41$ |
\( T^{4} - 64T^{2} + 4096 \)
|
| $43$ |
\( T^{4} - 36T^{2} + 1296 \)
|
| $47$ |
\( (T - 8)^{4} \)
|
| $53$ |
\( (T^{2} + 144)^{2} \)
|
| $59$ |
\( T^{4} - 4T^{2} + 16 \)
|
| $61$ |
\( (T^{2} + 6 T + 36)^{2} \)
|
| $67$ |
\( (T^{2} - 12 T + 144)^{2} \)
|
| $71$ |
\( T^{4} - 4T^{2} + 16 \)
|
| $73$ |
\( (T - 6)^{4} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( (T - 4)^{4} \)
|
| $89$ |
\( T^{4} - 64T^{2} + 4096 \)
|
| $97$ |
\( (T^{2} + 6 T + 36)^{2} \)
|
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