Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,5,Mod(53,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.53");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.g (of order \(6\), degree \(2\), 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: | \(33.4918680392\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{5} \) |
| Twist minimal: | no (minimal twist has level 108) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( 9\nu^{3} + 36\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -9\nu^{3} + 18\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 54 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{3} + \beta_{2} ) / 27 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).
| \(n\) | \(163\) | \(245\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
0 | 0 | 0 | −38.1838 | − | 22.0454i | 0 | 15.5000 | + | 26.8468i | 0 | 0 | 0 | ||||||||||||||||||||||||||
| 53.2 | 0 | 0 | 0 | 38.1838 | + | 22.0454i | 0 | 15.5000 | + | 26.8468i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 269.1 | 0 | 0 | 0 | −38.1838 | + | 22.0454i | 0 | 15.5000 | − | 26.8468i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 269.2 | 0 | 0 | 0 | 38.1838 | − | 22.0454i | 0 | 15.5000 | − | 26.8468i | 0 | 0 | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 324.5.g.e | 4 | |
| 3.b | odd | 2 | 1 | inner | 324.5.g.e | 4 | |
| 9.c | even | 3 | 1 | 108.5.c.b | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 324.5.g.e | 4 | |
| 9.d | odd | 6 | 1 | 108.5.c.b | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 324.5.g.e | 4 | |
| 36.f | odd | 6 | 1 | 432.5.e.f | 2 | ||
| 36.h | even | 6 | 1 | 432.5.e.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 108.5.c.b | ✓ | 2 | 9.c | even | 3 | 1 | |
| 108.5.c.b | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 324.5.g.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 324.5.g.e | 4 | 3.b | odd | 2 | 1 | inner | |
| 324.5.g.e | 4 | 9.c | even | 3 | 1 | inner | |
| 324.5.g.e | 4 | 9.d | odd | 6 | 1 | inner | |
| 432.5.e.f | 2 | 36.f | odd | 6 | 1 | ||
| 432.5.e.f | 2 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(324, [\chi])\):
|
\( T_{5}^{4} - 1944T_{5}^{2} + 3779136 \)
|
|
\( T_{7}^{2} - 31T_{7} + 961 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 1944 T^{2} + 3779136 \)
|
| $7$ |
\( (T^{2} - 31 T + 961)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 2361960000 \)
|
| $13$ |
\( (T^{2} - 241 T + 58081)^{2} \)
|
| $17$ |
\( (T^{2} + 48600)^{2} \)
|
| $19$ |
\( (T + 271)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 2361960000 \)
|
| $29$ |
\( T^{4} + \cdots + 37791360000 \)
|
| $31$ |
\( (T^{2} - 778 T + 605284)^{2} \)
|
| $37$ |
\( (T - 1079)^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 23619600000000 \)
|
| $43$ |
\( (T^{2} - 298 T + 88804)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 119574225000000 \)
|
| $53$ |
\( (T^{2} + 9525600)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 67459939560000 \)
|
| $61$ |
\( (T^{2} - 2641 T + 6974881)^{2} \)
|
| $67$ |
\( (T^{2} + 5609 T + 31460881)^{2} \)
|
| $71$ |
\( (T^{2} + 19440000)^{2} \)
|
| $73$ |
\( (T - 7199)^{4} \)
|
| $79$ |
\( (T^{2} + 329 T + 108241)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 3061100160000 \)
|
| $89$ |
\( (T^{2} + 66533400)^{2} \)
|
| $97$ |
\( (T^{2} - 15961 T + 254753521)^{2} \)
|
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