Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(19.1166188419\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1509.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 7x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 36) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 7x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{2} - 3\nu - 14 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + \beta _1 + 30 ) / 6 \)
|
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 | −13.8439 | 0 | 30.7080 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −4.89803 | 0 | −10.6545 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 12.7419 | 0 | −14.0535 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 324.4.a.c | 3 | |
| 3.b | odd | 2 | 1 | 324.4.a.d | 3 | ||
| 4.b | odd | 2 | 1 | 1296.4.a.v | 3 | ||
| 9.c | even | 3 | 2 | 36.4.e.a | ✓ | 6 | |
| 9.d | odd | 6 | 2 | 108.4.e.a | 6 | ||
| 12.b | even | 2 | 1 | 1296.4.a.w | 3 | ||
| 36.f | odd | 6 | 2 | 144.4.i.d | 6 | ||
| 36.h | even | 6 | 2 | 432.4.i.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 36.4.e.a | ✓ | 6 | 9.c | even | 3 | 2 | |
| 108.4.e.a | 6 | 9.d | odd | 6 | 2 | ||
| 144.4.i.d | 6 | 36.f | odd | 6 | 2 | ||
| 324.4.a.c | 3 | 1.a | even | 1 | 1 | trivial | |
| 324.4.a.d | 3 | 3.b | odd | 2 | 1 | ||
| 432.4.i.d | 6 | 36.h | even | 6 | 2 | ||
| 1296.4.a.v | 3 | 4.b | odd | 2 | 1 | ||
| 1296.4.a.w | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 6T_{5}^{2} - 171T_{5} - 864 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(324))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 6 T^{2} + \cdots - 864 \)
|
| $7$ |
\( T^{3} - 6 T^{2} + \cdots - 4598 \)
|
| $11$ |
\( T^{3} + 51 T^{2} + \cdots - 66231 \)
|
| $13$ |
\( T^{3} + 12 T^{2} + \cdots - 64466 \)
|
| $17$ |
\( T^{3} + 111 T^{2} + \cdots - 577476 \)
|
| $19$ |
\( T^{3} - 15 T^{2} + \cdots - 216368 \)
|
| $23$ |
\( T^{3} + 210 T^{2} + \cdots + 2322 \)
|
| $29$ |
\( T^{3} + 456 T^{2} + \cdots + 2879658 \)
|
| $31$ |
\( T^{3} + 48 T^{2} + \cdots - 3054788 \)
|
| $37$ |
\( T^{3} + 48 T^{2} + \cdots - 682352 \)
|
| $41$ |
\( T^{3} + 897 T^{2} + \cdots + 11796543 \)
|
| $43$ |
\( T^{3} + 129 T^{2} + \cdots - 1425149 \)
|
| $47$ |
\( T^{3} + 522 T^{2} + \cdots - 64558782 \)
|
| $53$ |
\( T^{3} + 1104 T^{2} + \cdots + 11853648 \)
|
| $59$ |
\( T^{3} + 453 T^{2} + \cdots - 96892713 \)
|
| $61$ |
\( T^{3} - 402 T^{2} + \cdots + 1209736 \)
|
| $67$ |
\( T^{3} - 213 T^{2} + \cdots - 3095063 \)
|
| $71$ |
\( T^{3} - 60 T^{2} + \cdots + 113211648 \)
|
| $73$ |
\( T^{3} - 375 T^{2} + \cdots + 158369284 \)
|
| $79$ |
\( T^{3} + 552 T^{2} + \cdots - 17848772 \)
|
| $83$ |
\( T^{3} - 612 T^{2} + \cdots + 3478788 \)
|
| $89$ |
\( T^{3} + 462 T^{2} + \cdots + 170122248 \)
|
| $97$ |
\( T^{3} + 93 T^{2} + \cdots + 86400523 \)
|
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