Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [920,2,Mod(461,920)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(920, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("920.461");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 920 = 2^{3} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 920.f (of order \(2\), degree \(1\), 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: | \(7.34623698596\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.1871773696.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 31x^{4} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} + 31x^{4} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 40\nu^{2} ) / 63 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 12 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{6} - 97\nu^{2} ) / 63 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 40\nu^{3} + 63\nu ) / 63 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{7} + 9\nu^{5} - 97\nu^{3} + 171\nu ) / 189 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -4\nu^{7} - 9\nu^{5} - 97\nu^{3} - 171\nu ) / 189 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7\nu^{7} - 9\nu^{5} + 217\nu^{3} - 360\nu ) / 189 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} - \beta_{5} + \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{7} + 3\beta_{6} + 7\beta_{5} + 4\beta_{4} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{2} - 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 19\beta_{7} - 21\beta_{6} + 40\beta_{5} - 19\beta_{4} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -40\beta_{3} - 97\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -97\beta_{7} - 120\beta_{6} - 217\beta_{5} - 97\beta_{4} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/920\mathbb{Z}\right)^\times\).
| \(n\) | \(231\) | \(281\) | \(461\) | \(737\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 461.1 |
|
− | 1.41421i | − | 2.30278i | −2.00000 | 1.00000i | −3.25662 | −1.30278 | 2.82843i | −2.30278 | 1.41421 | ||||||||||||||||||||||||||||||||||||||||
| 461.2 | − | 1.41421i | − | 1.30278i | −2.00000 | − | 1.00000i | −1.84240 | 2.30278 | 2.82843i | 1.30278 | −1.41421 | ||||||||||||||||||||||||||||||||||||||||
| 461.3 | − | 1.41421i | 1.30278i | −2.00000 | 1.00000i | 1.84240 | 2.30278 | 2.82843i | 1.30278 | 1.41421 | ||||||||||||||||||||||||||||||||||||||||||
| 461.4 | − | 1.41421i | 2.30278i | −2.00000 | − | 1.00000i | 3.25662 | −1.30278 | 2.82843i | −2.30278 | −1.41421 | |||||||||||||||||||||||||||||||||||||||||
| 461.5 | 1.41421i | − | 2.30278i | −2.00000 | 1.00000i | 3.25662 | −1.30278 | − | 2.82843i | −2.30278 | −1.41421 | |||||||||||||||||||||||||||||||||||||||||
| 461.6 | 1.41421i | − | 1.30278i | −2.00000 | − | 1.00000i | 1.84240 | 2.30278 | − | 2.82843i | 1.30278 | 1.41421 | ||||||||||||||||||||||||||||||||||||||||
| 461.7 | 1.41421i | 1.30278i | −2.00000 | 1.00000i | −1.84240 | 2.30278 | − | 2.82843i | 1.30278 | −1.41421 | ||||||||||||||||||||||||||||||||||||||||||
| 461.8 | 1.41421i | 2.30278i | −2.00000 | − | 1.00000i | −3.25662 | −1.30278 | − | 2.82843i | −2.30278 | 1.41421 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 920.2.f.b | ✓ | 8 |
| 4.b | odd | 2 | 1 | 3680.2.f.b | 8 | ||
| 8.b | even | 2 | 1 | inner | 920.2.f.b | ✓ | 8 |
| 8.d | odd | 2 | 1 | 3680.2.f.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 920.2.f.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 920.2.f.b | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 3680.2.f.b | 8 | 4.b | odd | 2 | 1 | ||
| 3680.2.f.b | 8 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 7T_{3}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(920, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{4} \)
|
| $3$ |
\( (T^{4} + 7 T^{2} + 9)^{2} \)
|
| $5$ |
\( (T^{2} + 1)^{4} \)
|
| $7$ |
\( (T^{2} - T - 3)^{4} \)
|
| $11$ |
\( T^{8} + 58 T^{6} + \cdots + 10609 \)
|
| $13$ |
\( T^{8} + 82 T^{6} + \cdots + 152881 \)
|
| $17$ |
\( (T^{4} + 10 T^{3} + \cdots - 23)^{2} \)
|
| $19$ |
\( T^{8} + 90 T^{6} + \cdots + 36481 \)
|
| $23$ |
\( (T + 1)^{8} \)
|
| $29$ |
\( T^{8} + 88 T^{6} + \cdots + 256 \)
|
| $31$ |
\( (T^{4} + 2 T^{3} + \cdots - 439)^{2} \)
|
| $37$ |
\( T^{8} + 184 T^{6} + \cdots + 160000 \)
|
| $41$ |
\( (T^{4} - 6 T^{3} + \cdots + 113)^{2} \)
|
| $43$ |
\( T^{8} + 168 T^{6} + \cdots + 20736 \)
|
| $47$ |
\( (T^{4} - 4 T^{3} + \cdots + 3312)^{2} \)
|
| $53$ |
\( T^{8} + 328 T^{6} + \cdots + 39137536 \)
|
| $59$ |
\( (T^{4} + 56 T^{2} + 576)^{2} \)
|
| $61$ |
\( T^{8} + 126 T^{6} + \cdots + 194481 \)
|
| $67$ |
\( (T^{4} + 56 T^{2} + 576)^{2} \)
|
| $71$ |
\( (T^{4} + 10 T^{3} + \cdots - 23)^{2} \)
|
| $73$ |
\( (T^{4} - 160 T^{2} + \cdots + 368)^{2} \)
|
| $79$ |
\( (T^{4} + 12 T^{3} + \cdots + 2032)^{2} \)
|
| $83$ |
\( T^{8} + 320 T^{6} + \cdots + 3268864 \)
|
| $89$ |
\( (T^{4} - 248 T^{2} + 5184)^{2} \)
|
| $97$ |
\( (T^{4} - 6 T^{3} + \cdots + 1873)^{2} \)
|
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