Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [425,4,Mod(324,425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("425.324");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 425 = 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 425.b (of order \(2\), degree \(1\), 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: | \(25.0758117524\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 85) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{12}^{2} - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( ( \beta_{2} + 1 ) / 2 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(326\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 324.1 |
|
− | 3.73205i | 2.73205i | −5.92820 | 0 | 10.1962 | 16.5885i | − | 7.73205i | 19.5359 | 0 | ||||||||||||||||||||||||||||
| 324.2 | − | 0.267949i | − | 0.732051i | 7.92820 | 0 | −0.196152 | − | 14.5885i | − | 4.26795i | 26.4641 | 0 | |||||||||||||||||||||||||||
| 324.3 | 0.267949i | 0.732051i | 7.92820 | 0 | −0.196152 | 14.5885i | 4.26795i | 26.4641 | 0 | |||||||||||||||||||||||||||||||
| 324.4 | 3.73205i | − | 2.73205i | −5.92820 | 0 | 10.1962 | − | 16.5885i | 7.73205i | 19.5359 | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 425.4.b.e | 4 | |
| 5.b | even | 2 | 1 | inner | 425.4.b.e | 4 | |
| 5.c | odd | 4 | 1 | 85.4.a.d | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 425.4.a.e | 2 | ||
| 15.e | even | 4 | 1 | 765.4.a.i | 2 | ||
| 20.e | even | 4 | 1 | 1360.4.a.m | 2 | ||
| 85.g | odd | 4 | 1 | 1445.4.a.i | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.4.a.d | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 425.4.a.e | 2 | 5.c | odd | 4 | 1 | ||
| 425.4.b.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 425.4.b.e | 4 | 5.b | even | 2 | 1 | inner | |
| 765.4.a.i | 2 | 15.e | even | 4 | 1 | ||
| 1360.4.a.m | 2 | 20.e | even | 4 | 1 | ||
| 1445.4.a.i | 2 | 85.g | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):
|
\( T_{2}^{4} + 14T_{2}^{2} + 1 \)
|
|
\( T_{3}^{4} + 8T_{3}^{2} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 14T^{2} + 1 \)
|
| $3$ |
\( T^{4} + 8T^{2} + 4 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 488 T^{2} + 58564 \)
|
| $11$ |
\( (T^{2} + 38 T - 146)^{2} \)
|
| $13$ |
\( T^{4} + 7904 T^{2} + 565504 \)
|
| $17$ |
\( (T^{2} + 289)^{2} \)
|
| $19$ |
\( (T^{2} - 180 T + 7992)^{2} \)
|
| $23$ |
\( T^{4} + 62088 T^{2} + 909746244 \)
|
| $29$ |
\( (T^{2} - 216 T + 5316)^{2} \)
|
| $31$ |
\( (T^{2} - 70 T - 58418)^{2} \)
|
| $37$ |
\( T^{4} + 129368 T^{2} + 7463824 \)
|
| $41$ |
\( (T^{2} - 344 T - 30908)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 6517655824 \)
|
| $47$ |
\( T^{4} + 60936 T^{2} + 905047056 \)
|
| $53$ |
\( T^{4} + 38024 T^{2} + 30958096 \)
|
| $59$ |
\( (T^{2} - 1436 T + 502456)^{2} \)
|
| $61$ |
\( (T^{2} + 76 T - 214028)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 80717355664 \)
|
| $71$ |
\( (T^{2} + 1510 T + 527542)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 87003761296 \)
|
| $79$ |
\( (T^{2} - 958 T - 343466)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 23887557136 \)
|
| $89$ |
\( (T^{2} - 204 T - 1641288)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 98754319504 \)
|
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