Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [980,4,Mod(1,980)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(980, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("980.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 980.a (trivial) |
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: | yes |
| Analytic conductor: | \(57.8218718056\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{22}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 22 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 140) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \sqrt{22}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −9.69042 | 0 | −5.00000 | 0 | 0 | 0 | 66.9042 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | −0.309584 | 0 | −5.00000 | 0 | 0 | 0 | −26.9042 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 980.4.a.n | 2 | |
| 7.b | odd | 2 | 1 | 980.4.a.u | 2 | ||
| 7.c | even | 3 | 2 | 980.4.i.w | 4 | ||
| 7.d | odd | 6 | 2 | 140.4.i.c | ✓ | 4 | |
| 21.g | even | 6 | 2 | 1260.4.s.e | 4 | ||
| 28.f | even | 6 | 2 | 560.4.q.l | 4 | ||
| 35.i | odd | 6 | 2 | 700.4.i.h | 4 | ||
| 35.k | even | 12 | 4 | 700.4.r.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.4.i.c | ✓ | 4 | 7.d | odd | 6 | 2 | |
| 560.4.q.l | 4 | 28.f | even | 6 | 2 | ||
| 700.4.i.h | 4 | 35.i | odd | 6 | 2 | ||
| 700.4.r.f | 8 | 35.k | even | 12 | 4 | ||
| 980.4.a.n | 2 | 1.a | even | 1 | 1 | trivial | |
| 980.4.a.u | 2 | 7.b | odd | 2 | 1 | ||
| 980.4.i.w | 4 | 7.c | even | 3 | 2 | ||
| 1260.4.s.e | 4 | 21.g | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(980))\):
|
\( T_{3}^{2} + 10T_{3} + 3 \)
|
|
\( T_{11}^{2} - 52T_{11} + 588 \)
|
|
\( T_{13}^{2} - 52T_{13} - 3636 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 10T + 3 \)
|
| $5$ |
\( (T + 5)^{2} \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 52T + 588 \)
|
| $13$ |
\( T^{2} - 52T - 3636 \)
|
| $17$ |
\( T^{2} - 76T - 2868 \)
|
| $19$ |
\( T^{2} + 176T + 7656 \)
|
| $23$ |
\( T^{2} - 102T + 2403 \)
|
| $29$ |
\( T^{2} + 306T + 20241 \)
|
| $31$ |
\( T^{2} + 244T + 11716 \)
|
| $37$ |
\( T^{2} + 176T - 42944 \)
|
| $41$ |
\( T^{2} - 130T - 80343 \)
|
| $43$ |
\( T^{2} - 114T - 195301 \)
|
| $47$ |
\( T^{2} + 716T + 119364 \)
|
| $53$ |
\( T^{2} - 808T + 152568 \)
|
| $59$ |
\( T^{2} - 1016 T + 226296 \)
|
| $61$ |
\( T^{2} - 222T - 182071 \)
|
| $67$ |
\( T^{2} - 134T - 57309 \)
|
| $71$ |
\( T^{2} - 296T - 447048 \)
|
| $73$ |
\( T^{2} + 724T - 363956 \)
|
| $79$ |
\( T^{2} - 1128 T + 170168 \)
|
| $83$ |
\( T^{2} + 338T - 740757 \)
|
| $89$ |
\( T^{2} - 326 T - 1038231 \)
|
| $97$ |
\( T^{2} - 1940 T + 912388 \)
|
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