Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [961,1,Mod(115,961)]
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([17]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("961.115");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 961 = 31^{2} \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 961.h (of order \(30\), 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: | \(0.479601477140\) |
| 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) |
| Projective image: | \(D_{3}\) |
| Projective field: | Galois closure of 3.1.31.1 |
| Artin image: | $S_3\times C_{15}$ |
| Artin field: | Galois closure of \(\mathbb{Q}[x]/(x^{45} - \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-\zeta_{30}^{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 115.1 |
|
−0.309017 | + | 0.951057i | 0 | 0 | 0.500000 | + | 0.866025i | 0 | −0.913545 | + | 0.406737i | −0.809017 | + | 0.587785i | 0.913545 | + | 0.406737i | −0.978148 | + | 0.207912i | ||||||||||||||||||||||||||||||
| 117.1 | −0.309017 | − | 0.951057i | 0 | 0 | 0.500000 | − | 0.866025i | 0 | −0.913545 | − | 0.406737i | −0.809017 | − | 0.587785i | 0.913545 | − | 0.406737i | −0.978148 | − | 0.207912i | |||||||||||||||||||||||||||||||
| 145.1 | 0.809017 | + | 0.587785i | 0 | 0 | 0.500000 | + | 0.866025i | 0 | 0.978148 | + | 0.207912i | 0.309017 | − | 0.951057i | −0.978148 | + | 0.207912i | −0.104528 | + | 0.994522i | |||||||||||||||||||||||||||||||
| 229.1 | 0.809017 | + | 0.587785i | 0 | 0 | 0.500000 | − | 0.866025i | 0 | −0.669131 | + | 0.743145i | 0.309017 | − | 0.951057i | 0.669131 | + | 0.743145i | 0.913545 | − | 0.406737i | |||||||||||||||||||||||||||||||
| 414.1 | −0.309017 | − | 0.951057i | 0 | 0 | 0.500000 | + | 0.866025i | 0 | 0.104528 | + | 0.994522i | −0.809017 | − | 0.587785i | −0.104528 | + | 0.994522i | 0.669131 | − | 0.743145i | |||||||||||||||||||||||||||||||
| 513.1 | −0.309017 | + | 0.951057i | 0 | 0 | 0.500000 | − | 0.866025i | 0 | 0.104528 | − | 0.994522i | −0.809017 | + | 0.587785i | −0.104528 | − | 0.994522i | 0.669131 | + | 0.743145i | |||||||||||||||||||||||||||||||
| 623.1 | 0.809017 | − | 0.587785i | 0 | 0 | 0.500000 | − | 0.866025i | 0 | 0.978148 | − | 0.207912i | 0.309017 | + | 0.951057i | −0.978148 | − | 0.207912i | −0.104528 | − | 0.994522i | |||||||||||||||||||||||||||||||
| 726.1 | 0.809017 | − | 0.587785i | 0 | 0 | 0.500000 | + | 0.866025i | 0 | −0.669131 | − | 0.743145i | 0.309017 | + | 0.951057i | 0.669131 | − | 0.743145i | 0.913545 | + | 0.406737i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-31}) \) |
| 31.c | even | 3 | 1 | inner |
| 31.d | even | 5 | 3 | inner |
| 31.e | odd | 6 | 1 | inner |
| 31.f | odd | 10 | 3 | inner |
| 31.g | even | 15 | 3 | inner |
| 31.h | odd | 30 | 3 | inner |
Twists
Hecke kernels
This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(961, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{2} - T + 1)^{4} \)
|
| $7$ |
\( T^{8} + T^{7} - T^{5} + \cdots + 1 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} + T^{7} - T^{5} + \cdots + 1 \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} \)
|
| $41$ |
\( T^{8} + T^{7} - T^{5} + \cdots + 1 \)
|
| $43$ |
\( T^{8} \)
|
| $47$ |
\( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \)
|
| $53$ |
\( T^{8} \)
|
| $59$ |
\( T^{8} + T^{7} - T^{5} + \cdots + 1 \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( (T^{2} + 2 T + 4)^{4} \)
|
| $71$ |
\( T^{8} + T^{7} - T^{5} + \cdots + 1 \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( T^{8} \)
|
| $83$ |
\( T^{8} \)
|
| $89$ |
\( T^{8} \)
|
| $97$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
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