Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [576,8,Mod(1,576)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("576.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 576 = 2^{6} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 576.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: | \(179.933774679\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 579x^{2} - 126x + 3969 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 288) |
| 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
\( x^{4} - 2x^{3} - 579x^{2} - 126x + 3969 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -64\nu^{3} + 128\nu^{2} + 41088\nu + 4032 ) / 21 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 16\nu^{3} + 136\nu^{2} - 8592\nu - 50652 ) / 189 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -16\nu^{3} + 80\nu^{2} + 8160\nu - 11880 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( -4\beta_{3} + 8\beta_{2} + \beta _1 + 192 ) / 384 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 20\beta_{3} + 176\beta_{2} + \beta _1 + 55776 ) / 192 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -311\beta_{3} + 730\beta_{2} + 65\beta _1 + 46320 ) / 48 \)
|
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 | −376.408 | 0 | 329.200 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −138.727 | 0 | −1239.68 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 138.727 | 0 | 1239.68 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 376.408 | 0 | −329.200 | 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 |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 576.8.a.bs | 4 | |
| 3.b | odd | 2 | 1 | 576.8.a.bt | 4 | ||
| 4.b | odd | 2 | 1 | 576.8.a.bt | 4 | ||
| 8.b | even | 2 | 1 | 288.8.a.r | yes | 4 | |
| 8.d | odd | 2 | 1 | 288.8.a.q | ✓ | 4 | |
| 12.b | even | 2 | 1 | inner | 576.8.a.bs | 4 | |
| 24.f | even | 2 | 1 | 288.8.a.r | yes | 4 | |
| 24.h | odd | 2 | 1 | 288.8.a.q | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 288.8.a.q | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 288.8.a.q | ✓ | 4 | 24.h | odd | 2 | 1 | |
| 288.8.a.r | yes | 4 | 8.b | even | 2 | 1 | |
| 288.8.a.r | yes | 4 | 24.f | even | 2 | 1 | |
| 576.8.a.bs | 4 | 1.a | even | 1 | 1 | trivial | |
| 576.8.a.bs | 4 | 12.b | even | 2 | 1 | inner | |
| 576.8.a.bt | 4 | 3.b | odd | 2 | 1 | ||
| 576.8.a.bt | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(576))\):
|
\( T_{5}^{4} - 160928T_{5}^{2} + 2726713600 \)
|
|
\( T_{7}^{4} - 1645184T_{7}^{2} + 166548312064 \)
|
|
\( T_{11}^{2} + 320T_{11} - 26000384 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 2726713600 \)
|
| $7$ |
\( T^{4} + \cdots + 166548312064 \)
|
| $11$ |
\( (T^{2} + 320 T - 26000384)^{2} \)
|
| $13$ |
\( (T^{2} - 700 T - 103981436)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 47\!\cdots\!84 \)
|
| $19$ |
\( T^{4} + \cdots + 60\!\cdots\!84 \)
|
| $23$ |
\( (T^{2} + 7296 T - 2589290496)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 42\!\cdots\!84 \)
|
| $31$ |
\( T^{4} + \cdots + 45\!\cdots\!64 \)
|
| $37$ |
\( (T^{2} - 6580 T - 93279836)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 47\!\cdots\!64 \)
|
| $43$ |
\( T^{4} + \cdots + 61\!\cdots\!44 \)
|
| $47$ |
\( (T^{2} + 354944 T + 8072925184)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 69\!\cdots\!64 \)
|
| $59$ |
\( (T^{2} + \cdots - 1850580660224)^{2} \)
|
| $61$ |
\( (T^{2} - 132132 T - 747786221244)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 13\!\cdots\!84 \)
|
| $71$ |
\( (T^{2} + \cdots + 1182120607744)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots - 9837801827100)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 18\!\cdots\!24 \)
|
| $83$ |
\( (T^{2} + \cdots - 2514184068096)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots - 31526327944124)^{2} \)
|
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