Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2304,4,Mod(1,2304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2304, 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("2304.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2304 = 2^{8} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2304.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: | \(135.940400653\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.9792.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 7x^{2} + 2x + 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{10} \) |
| Twist minimal: | no (minimal twist has level 384) |
| 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 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
\( x^{4} - 2x^{3} - 7x^{2} + 2x + 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( -2\nu^{3} + 10\nu^{2} - 6\nu - 20 \)
|
| \(\beta_{2}\) | \(=\) |
\( 16\nu^{3} - 48\nu^{2} - 48\nu + 64 \)
|
| \(\beta_{3}\) | \(=\) |
\( -12\nu^{3} + 44\nu^{2} + 28\nu - 80 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{3} + \beta_{2} - 4\beta _1 + 16 ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 72 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 24\beta_{3} + 17\beta_{2} - 24\beta _1 + 352 ) / 32 \)
|
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 | 0 | 0 | −17.4288 | 0 | 2.99032 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −5.67763 | 0 | 33.0917 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 5.67763 | 0 | −33.0917 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 17.4288 | 0 | −2.99032 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2304.4.a.by | 4 | |
| 3.b | odd | 2 | 1 | 768.4.a.v | 4 | ||
| 4.b | odd | 2 | 1 | 2304.4.a.cb | 4 | ||
| 8.b | even | 2 | 1 | 2304.4.a.cb | 4 | ||
| 8.d | odd | 2 | 1 | inner | 2304.4.a.by | 4 | |
| 12.b | even | 2 | 1 | 768.4.a.u | 4 | ||
| 16.e | even | 4 | 2 | 1152.4.d.p | 8 | ||
| 16.f | odd | 4 | 2 | 1152.4.d.p | 8 | ||
| 24.f | even | 2 | 1 | 768.4.a.v | 4 | ||
| 24.h | odd | 2 | 1 | 768.4.a.u | 4 | ||
| 48.i | odd | 4 | 2 | 384.4.d.f | ✓ | 8 | |
| 48.k | even | 4 | 2 | 384.4.d.f | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.4.d.f | ✓ | 8 | 48.i | odd | 4 | 2 | |
| 384.4.d.f | ✓ | 8 | 48.k | even | 4 | 2 | |
| 768.4.a.u | 4 | 12.b | even | 2 | 1 | ||
| 768.4.a.u | 4 | 24.h | odd | 2 | 1 | ||
| 768.4.a.v | 4 | 3.b | odd | 2 | 1 | ||
| 768.4.a.v | 4 | 24.f | even | 2 | 1 | ||
| 1152.4.d.p | 8 | 16.e | even | 4 | 2 | ||
| 1152.4.d.p | 8 | 16.f | odd | 4 | 2 | ||
| 2304.4.a.by | 4 | 1.a | even | 1 | 1 | trivial | |
| 2304.4.a.by | 4 | 8.d | odd | 2 | 1 | inner | |
| 2304.4.a.cb | 4 | 4.b | odd | 2 | 1 | ||
| 2304.4.a.cb | 4 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2304))\):
|
\( T_{5}^{4} - 336T_{5}^{2} + 9792 \)
|
|
\( T_{7}^{4} - 1104T_{7}^{2} + 9792 \)
|
|
\( T_{11}^{2} + 24T_{11} - 368 \)
|
|
\( T_{13}^{4} - 8640T_{13}^{2} + 12690432 \)
|
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\( T_{17}^{2} + 60T_{17} - 3708 \)
|
|
\( T_{19}^{2} - 24T_{19} - 4464 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 336T^{2} + 9792 \)
|
| $7$ |
\( T^{4} - 1104 T^{2} + 9792 \)
|
| $11$ |
\( (T^{2} + 24 T - 368)^{2} \)
|
| $13$ |
\( T^{4} - 8640 T^{2} + 12690432 \)
|
| $17$ |
\( (T^{2} + 60 T - 3708)^{2} \)
|
| $19$ |
\( (T^{2} - 24 T - 4464)^{2} \)
|
| $23$ |
\( T^{4} - 53568 T^{2} + 621831168 \)
|
| $29$ |
\( T^{4} - 41040 T^{2} + 419577408 \)
|
| $31$ |
\( T^{4} - 55248 T^{2} + 461095488 \)
|
| $37$ |
\( T^{4} + \cdots + 2487324672 \)
|
| $41$ |
\( (T^{2} + 204 T - 104796)^{2} \)
|
| $43$ |
\( (T^{2} - 264 T + 12816)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 21332616192 \)
|
| $53$ |
\( T^{4} - 87888 T^{2} + 806557248 \)
|
| $59$ |
\( (T^{2} + 936 T + 87952)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 270509248512 \)
|
| $67$ |
\( (T^{2} - 1176 T + 272016)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 297070322688 \)
|
| $73$ |
\( (T^{2} - 484 T - 236348)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 1904357952 \)
|
| $83$ |
\( (T^{2} + 1704 T + 577936)^{2} \)
|
| $89$ |
\( (T^{2} - 1836 T + 676836)^{2} \)
|
| $97$ |
\( (T^{2} + 740 T + 118468)^{2} \)
|
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