Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,6,Mod(1,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.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: | \(51.9643576194\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{3}, \sqrt{91})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 47x^{2} + 484 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{4} \) |
| Twist minimal: | yes |
| 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
\( x^{4} - 47x^{2} + 484 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 9\nu^{3} - 225\nu ) / 22 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -9\nu^{3} + 489\nu ) / 22 \)
|
| \(\beta_{3}\) | \(=\) |
\( 12\nu^{2} - 282 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 282 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 75\beta_{2} + 163\beta_1 ) / 36 \)
|
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 | −83.2171 | 0 | 143.136 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −31.2556 | 0 | −55.1363 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 31.2556 | 0 | −55.1363 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 83.2171 | 0 | 143.136 | 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 |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 324.6.a.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 324.6.a.c | ✓ | 4 |
| 9.c | even | 3 | 2 | 324.6.e.j | 8 | ||
| 9.d | odd | 6 | 2 | 324.6.e.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 324.6.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 324.6.a.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 324.6.e.j | 8 | 9.c | even | 3 | 2 | ||
| 324.6.e.j | 8 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 7902T_{5}^{2} + 6765201 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(324))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 7902 T^{2} + 6765201 \)
|
| $7$ |
\( (T^{2} - 88 T - 7892)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 3688375824 \)
|
| $13$ |
\( (T^{2} + 686 T - 363923)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 2032053101001 \)
|
| $19$ |
\( (T^{2} + 1472 T - 1119236)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 1466879477904 \)
|
| $29$ |
\( T^{4} + \cdots + 2250981106929 \)
|
| $31$ |
\( (T^{2} + 2072 T + 719488)^{2} \)
|
| $37$ |
\( (T^{2} - 838 T - 52197851)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 18437886372096 \)
|
| $43$ |
\( (T^{2} + 19160 T - 4547828)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 72\!\cdots\!96 \)
|
| $53$ |
\( T^{4} + \cdots + 21\!\cdots\!64 \)
|
| $59$ |
\( T^{4} + \cdots + 15\!\cdots\!84 \)
|
| $61$ |
\( (T^{2} + 60098 T + 405399901)^{2} \)
|
| $67$ |
\( (T^{2} + 53648 T + 693964348)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 11\!\cdots\!36 \)
|
| $73$ |
\( (T^{2} + 46190 T - 281794607)^{2} \)
|
| $79$ |
\( (T^{2} - 2512 T - 8154886052)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!64 \)
|
| $89$ |
\( T^{4} + \cdots + 13\!\cdots\!69 \)
|
| $97$ |
\( (T^{2} + 165932 T + 6826590628)^{2} \)
|
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