Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,4,Mod(337,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.337");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 624.c (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: | \(36.8171918436\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.1362828.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 23x^{2} + 48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 39) |
| 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 in terms of a root \(\nu\) of
\( x^{4} + 23x^{2} + 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 19\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} + 46 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 46 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{2} - 19\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).
| \(n\) | \(79\) | \(145\) | \(209\) | \(469\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
0 | −3.00000 | 0 | − | 12.9118i | 0 | − | 16.7289i | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 337.2 | 0 | −3.00000 | 0 | − | 9.65841i | 0 | − | 22.3639i | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||
| 337.3 | 0 | −3.00000 | 0 | 9.65841i | 0 | 22.3639i | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 337.4 | 0 | −3.00000 | 0 | 12.9118i | 0 | 16.7289i | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 624.4.c.c | 4 | |
| 4.b | odd | 2 | 1 | 39.4.b.b | ✓ | 4 | |
| 12.b | even | 2 | 1 | 117.4.b.e | 4 | ||
| 13.b | even | 2 | 1 | inner | 624.4.c.c | 4 | |
| 52.b | odd | 2 | 1 | 39.4.b.b | ✓ | 4 | |
| 52.f | even | 4 | 2 | 507.4.a.l | 4 | ||
| 156.h | even | 2 | 1 | 117.4.b.e | 4 | ||
| 156.l | odd | 4 | 2 | 1521.4.a.w | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.4.b.b | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 39.4.b.b | ✓ | 4 | 52.b | odd | 2 | 1 | |
| 117.4.b.e | 4 | 12.b | even | 2 | 1 | ||
| 117.4.b.e | 4 | 156.h | even | 2 | 1 | ||
| 507.4.a.l | 4 | 52.f | even | 4 | 2 | ||
| 624.4.c.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 624.4.c.c | 4 | 13.b | even | 2 | 1 | inner | |
| 1521.4.a.w | 4 | 156.l | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 260T_{5}^{2} + 15552 \)
acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( T^{4} + 260 T^{2} + 15552 \)
|
| $7$ |
\( T^{4} + 780 T^{2} + 139968 \)
|
| $11$ |
\( T^{4} + 3152 T^{2} + 1572528 \)
|
| $13$ |
\( T^{4} + 12 T^{3} + \cdots + 4826809 \)
|
| $17$ |
\( (T^{2} - 48 T - 11556)^{2} \)
|
| $19$ |
\( T^{4} + 13884 T^{2} + 3048192 \)
|
| $23$ |
\( (T - 72)^{4} \)
|
| $29$ |
\( (T^{2} + 192 T - 2916)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 2389782528 \)
|
| $37$ |
\( T^{4} + \cdots + 2060577792 \)
|
| $41$ |
\( T^{4} + \cdots + 4431055872 \)
|
| $43$ |
\( (T^{2} - 644 T + 91552)^{2} \)
|
| $47$ |
\( T^{4} + 30128 T^{2} + 116663088 \)
|
| $53$ |
\( (T^{2} - 492 T - 133596)^{2} \)
|
| $59$ |
\( T^{4} + 62576 T^{2} + 457419312 \)
|
| $61$ |
\( (T^{2} - 144 T - 60868)^{2} \)
|
| $67$ |
\( T^{4} + 591948 T^{2} + 918330048 \)
|
| $71$ |
\( T^{4} + \cdots + 59484058032 \)
|
| $73$ |
\( T^{4} + \cdots + 179358354432 \)
|
| $79$ |
\( (T^{2} + 2160 T + 1144832)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 317023217328 \)
|
| $89$ |
\( T^{4} + \cdots + 78903164928 \)
|
| $97$ |
\( T^{4} + 137040 T^{2} + 725594112 \)
|
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