Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [980,4,Mod(361,980)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(980, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("980.361");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 980.i (of order \(3\), 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: | \(57.8218718056\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-83})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 20x^{2} - 21x + 441 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 20x^{2} - 21x + 441 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 20\nu^{2} - 20\nu - 441 ) / 420 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 41\nu ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 20\nu - 41 ) / 20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 61\beta _1 + 62 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 40\beta_{3} - 20\beta_{2} + 20\beta _1 + 103 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) | \(491\) |
| \(\chi(n)\) | \(\beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 361.1 |
|
0 | −3.69493 | + | 6.39981i | 0 | −2.50000 | − | 4.33013i | 0 | 0 | 0 | −13.8051 | − | 23.9111i | 0 | ||||||||||||||||||||||||
| 361.2 | 0 | 4.19493 | − | 7.26584i | 0 | −2.50000 | − | 4.33013i | 0 | 0 | 0 | −21.6949 | − | 37.5767i | 0 | |||||||||||||||||||||||||
| 961.1 | 0 | −3.69493 | − | 6.39981i | 0 | −2.50000 | + | 4.33013i | 0 | 0 | 0 | −13.8051 | + | 23.9111i | 0 | |||||||||||||||||||||||||
| 961.2 | 0 | 4.19493 | + | 7.26584i | 0 | −2.50000 | + | 4.33013i | 0 | 0 | 0 | −21.6949 | + | 37.5767i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 980.4.i.v | 4 | |
| 7.b | odd | 2 | 1 | 980.4.i.t | 4 | ||
| 7.c | even | 3 | 1 | 980.4.a.p | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 980.4.i.v | 4 | |
| 7.d | odd | 6 | 1 | 980.4.a.s | yes | 2 | |
| 7.d | odd | 6 | 1 | 980.4.i.t | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 980.4.a.p | ✓ | 2 | 7.c | even | 3 | 1 | |
| 980.4.a.s | yes | 2 | 7.d | odd | 6 | 1 | |
| 980.4.i.t | 4 | 7.b | odd | 2 | 1 | ||
| 980.4.i.t | 4 | 7.d | odd | 6 | 1 | ||
| 980.4.i.v | 4 | 1.a | even | 1 | 1 | trivial | |
| 980.4.i.v | 4 | 7.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(980, [\chi])\):
|
\( T_{3}^{4} - T_{3}^{3} + 63T_{3}^{2} + 62T_{3} + 3844 \)
|
|
\( T_{11}^{4} - 47T_{11}^{3} + 3213T_{11}^{2} + 47188T_{11} + 1008016 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - T^{3} + \cdots + 3844 \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 47 T^{3} + \cdots + 1008016 \)
|
| $13$ |
\( (T^{2} + 109 T + 2410)^{2} \)
|
| $17$ |
\( T^{4} - 121 T^{3} + \cdots + 4426816 \)
|
| $19$ |
\( T^{4} + 156 T^{3} + \cdots + 4410000 \)
|
| $23$ |
\( T^{4} - 152 T^{3} + \cdots + 103225600 \)
|
| $29$ |
\( (T^{2} + 15 T - 2994)^{2} \)
|
| $31$ |
\( T^{4} + 142 T^{3} + \cdots + 51265600 \)
|
| $37$ |
\( T^{4} + \cdots + 11942118400 \)
|
| $41$ |
\( (T^{2} - 310 T + 17800)^{2} \)
|
| $43$ |
\( (T^{2} + 946 T + 223480)^{2} \)
|
| $47$ |
\( T^{4} - 131 T^{3} + \cdots + 17875984 \)
|
| $53$ |
\( T^{4} + \cdots + 66698227600 \)
|
| $59$ |
\( T^{4} + \cdots + 52523472400 \)
|
| $61$ |
\( T^{4} + \cdots + 39744409600 \)
|
| $67$ |
\( T^{4} + 478 T^{3} + \cdots + 220225600 \)
|
| $71$ |
\( (T^{2} - 1548 T + 598080)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 7293160000 \)
|
| $79$ |
\( T^{4} + 623 T^{3} + \cdots + 326452624 \)
|
| $83$ |
\( (T^{2} + 782 T - 656120)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 52569318400 \)
|
| $97$ |
\( (T^{2} + 2253 T + 1265952)^{2} \)
|
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