Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [961,2,Mod(235,961)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(961, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([26]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("961.235");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 961 = 31^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 961.g (of order \(15\), degree \(8\), 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: | \(7.67362363425\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{15})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 31) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(1 - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 235.1 |
|
0.500000 | − | 0.363271i | 1.12920 | − | 0.502754i | −0.500000 | + | 1.53884i | −0.500000 | − | 0.866025i | 0.381966 | − | 0.661585i | −2.83448 | − | 3.14801i | 0.690983 | + | 2.12663i | −0.985051 | + | 1.09401i | −0.564602 | − | 0.251377i | ||||||||||||||||||||||||
| 338.1 | 0.500000 | − | 0.363271i | −0.129204 | + | 1.22930i | −0.500000 | + | 1.53884i | −0.500000 | + | 0.866025i | 0.381966 | + | 0.661585i | 4.14350 | − | 0.880728i | 0.690983 | + | 2.12663i | 1.43997 | + | 0.306074i | 0.0646021 | + | 0.614648i | |||||||||||||||||||||||||
| 448.1 | 0.500000 | − | 1.53884i | −2.16535 | + | 2.40487i | −0.500000 | − | 0.363271i | −0.500000 | + | 0.866025i | 2.61803 | + | 4.53457i | −0.0246758 | + | 0.234775i | 1.80902 | − | 1.31433i | −0.781051 | − | 7.43120i | 1.08268 | + | 1.20243i | |||||||||||||||||||||||||
| 547.1 | 0.500000 | + | 1.53884i | −2.16535 | − | 2.40487i | −0.500000 | + | 0.363271i | −0.500000 | − | 0.866025i | 2.61803 | − | 4.53457i | −0.0246758 | − | 0.234775i | 1.80902 | + | 1.31433i | −0.781051 | + | 7.43120i | 1.08268 | − | 1.20243i | |||||||||||||||||||||||||
| 732.1 | 0.500000 | + | 0.363271i | 1.12920 | + | 0.502754i | −0.500000 | − | 1.53884i | −0.500000 | + | 0.866025i | 0.381966 | + | 0.661585i | −2.83448 | + | 3.14801i | 0.690983 | − | 2.12663i | −0.985051 | − | 1.09401i | −0.564602 | + | 0.251377i | |||||||||||||||||||||||||
| 816.1 | 0.500000 | + | 0.363271i | −0.129204 | − | 1.22930i | −0.500000 | − | 1.53884i | −0.500000 | − | 0.866025i | 0.381966 | − | 0.661585i | 4.14350 | + | 0.880728i | 0.690983 | − | 2.12663i | 1.43997 | − | 0.306074i | 0.0646021 | − | 0.614648i | |||||||||||||||||||||||||
| 844.1 | 0.500000 | + | 1.53884i | 3.16535 | − | 0.672816i | −0.500000 | + | 0.363271i | −0.500000 | + | 0.866025i | 2.61803 | + | 4.53457i | 0.215659 | + | 0.0960175i | 1.80902 | + | 1.31433i | 6.82614 | − | 3.03919i | −1.58268 | − | 0.336408i | |||||||||||||||||||||||||
| 846.1 | 0.500000 | − | 1.53884i | 3.16535 | + | 0.672816i | −0.500000 | − | 0.363271i | −0.500000 | − | 0.866025i | 2.61803 | − | 4.53457i | 0.215659 | − | 0.0960175i | 1.80902 | − | 1.31433i | 6.82614 | + | 3.03919i | −1.58268 | + | 0.336408i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.c | even | 3 | 1 | inner |
| 31.d | even | 5 | 1 | inner |
| 31.g | even | 15 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(961, [\chi])\):
|
\( T_{2}^{4} - 2T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 1 \)
|
|
\( T_{3}^{8} - 4T_{3}^{7} - 16T_{3}^{5} + 144T_{3}^{4} - 256T_{3}^{3} + 320T_{3}^{2} - 384T_{3} + 256 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 256 \)
|
| $5$ |
\( (T^{2} + T + 1)^{4} \)
|
| $7$ |
\( T^{8} - 3 T^{7} + \cdots + 1 \)
|
| $11$ |
\( T^{8} - 2 T^{7} + \cdots + 256 \)
|
| $13$ |
\( T^{8} - 4 T^{7} + \cdots + 256 \)
|
| $17$ |
\( T^{8} + 2 T^{7} + \cdots + 256 \)
|
| $19$ |
\( T^{8} - 5 T^{7} + \cdots + 625 \)
|
| $23$ |
\( (T^{4} + 14 T^{3} + \cdots + 1936)^{2} \)
|
| $29$ |
\( (T^{4} + 10 T^{3} + \cdots + 400)^{2} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( (T^{2} - 2 T + 4)^{4} \)
|
| $41$ |
\( T^{8} - 7 T^{7} + \cdots + 5764801 \)
|
| $43$ |
\( T^{8} + 6 T^{7} + \cdots + 256 \)
|
| $47$ |
\( (T^{4} - 12 T^{3} + \cdots + 256)^{2} \)
|
| $53$ |
\( T^{8} + 16 T^{7} + \cdots + 65536 \)
|
| $59$ |
\( T^{8} + 5 T^{7} + \cdots + 625 \)
|
| $61$ |
\( (T^{2} + 6 T - 116)^{4} \)
|
| $67$ |
\( (T^{2} + 8 T + 64)^{4} \)
|
| $71$ |
\( T^{8} + 23 T^{7} + \cdots + 214358881 \)
|
| $73$ |
\( T^{8} - 14 T^{7} + \cdots + 256 \)
|
| $79$ |
\( T^{8} + 20 T^{7} + \cdots + 160000 \)
|
| $83$ |
\( T^{8} - 14 T^{7} + \cdots + 3748096 \)
|
| $89$ |
\( (T^{4} + 20 T^{3} + \cdots + 400)^{2} \)
|
| $97$ |
\( (T^{4} - 27 T^{3} + \cdots + 961)^{2} \)
|
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