Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [675,3,Mod(224,675)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("675.224");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 675.i (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: | \(18.3924178443\) |
| 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: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 9) |
| 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 \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).
| \(n\) | \(326\) | \(352\) |
| \(\chi(n)\) | \(1 - \zeta_{12}^{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 224.1 |
|
−0.866025 | + | 1.50000i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | −1.73205 | − | 1.00000i | −8.66025 | 0 | 0 | ||||||||||||||||||||||||
| 224.2 | 0.866025 | − | 1.50000i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | 1.73205 | + | 1.00000i | 8.66025 | 0 | 0 | |||||||||||||||||||||||||
| 449.1 | −0.866025 | − | 1.50000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | −1.73205 | + | 1.00000i | −8.66025 | 0 | 0 | |||||||||||||||||||||||||
| 449.2 | 0.866025 | + | 1.50000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 1.73205 | − | 1.00000i | 8.66025 | 0 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
| 45.h | odd | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 3T_{2}^{2} + 9 \)
acting on \(S_{3}^{\mathrm{new}}(675, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 3T^{2} + 9 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 4T^{2} + 16 \)
|
| $11$ |
\( (T^{2} - 3 T + 3)^{2} \)
|
| $13$ |
\( T^{4} - 16T^{2} + 256 \)
|
| $17$ |
\( (T^{2} - 243)^{2} \)
|
| $19$ |
\( (T + 11)^{4} \)
|
| $23$ |
\( T^{4} + 768 T^{2} + 589824 \)
|
| $29$ |
\( (T^{2} - 78 T + 2028)^{2} \)
|
| $31$ |
\( (T^{2} + 32 T + 1024)^{2} \)
|
| $37$ |
\( (T^{2} + 1156)^{2} \)
|
| $41$ |
\( (T^{2} - 21 T + 147)^{2} \)
|
| $43$ |
\( T^{4} - 3721 T^{2} + 13845841 \)
|
| $47$ |
\( T^{4} + 2352 T^{2} + 5531904 \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( (T^{2} - 87 T + 2523)^{2} \)
|
| $61$ |
\( (T^{2} + 56 T + 3136)^{2} \)
|
| $67$ |
\( T^{4} - 961 T^{2} + 923521 \)
|
| $71$ |
\( (T^{2} + 972)^{2} \)
|
| $73$ |
\( (T^{2} + 4225)^{2} \)
|
| $79$ |
\( (T^{2} - 38 T + 1444)^{2} \)
|
| $83$ |
\( T^{4} + 2352 T^{2} + 5531904 \)
|
| $89$ |
\( (T^{2} + 15552)^{2} \)
|
| $97$ |
\( T^{4} - 13225 T^{2} + 174900625 \)
|
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