Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,3,Mod(163,324)]
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([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.163");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.d (of order \(2\), degree \(1\), 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: | \(8.82836056527\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.389136420864.4 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 5x^{6} + 24x^{4} + 80x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 5x^{6} + 24x^{4} + 80x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + \nu^{3} + 4\nu ) / 16 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - \nu^{5} - 20\nu^{3} + 64\nu ) / 64 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 5\nu^{5} + 24\nu^{3} + 80\nu ) / 64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + \nu^{2} + 8 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} - \nu^{4} + 12\nu^{2} + 16 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 5\nu^{4} + 24\nu^{2} + 64 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{7} - 7\nu^{5} - 16\nu ) / 64 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{5} - \beta_{4} - 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{7} + 3\beta_{3} - 3\beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{6} - \beta_{5} + 9\beta_{4} - 13 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -2\beta_{7} - 7\beta_{3} - \beta_{2} + 37\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 13\beta_{6} - 19\beta_{5} - 21\beta_{4} + 9 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -38\beta_{7} + 11\beta_{3} - 3\beta_{2} - 81\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).
| \(n\) | \(163\) | \(245\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 163.1 |
|
−1.52274 | − | 1.29664i | 0 | 0.637459 | + | 3.94888i | 7.82300 | 0 | − | 12.3894i | 4.14959 | − | 6.83966i | 0 | −11.9124 | − | 10.1436i | |||||||||||||||||||||||||||||||||
| 163.2 | −1.52274 | + | 1.29664i | 0 | 0.637459 | − | 3.94888i | 7.82300 | 0 | 12.3894i | 4.14959 | + | 6.83966i | 0 | −11.9124 | + | 10.1436i | |||||||||||||||||||||||||||||||||||
| 163.3 | −0.656712 | − | 1.88911i | 0 | −3.13746 | + | 2.48120i | 0.894797 | 0 | 1.58166i | 6.74766 | + | 4.29756i | 0 | −0.587624 | − | 1.69037i | |||||||||||||||||||||||||||||||||||
| 163.4 | −0.656712 | + | 1.88911i | 0 | −3.13746 | − | 2.48120i | 0.894797 | 0 | − | 1.58166i | 6.74766 | − | 4.29756i | 0 | −0.587624 | + | 1.69037i | ||||||||||||||||||||||||||||||||||
| 163.5 | 0.656712 | − | 1.88911i | 0 | −3.13746 | − | 2.48120i | −0.894797 | 0 | − | 1.58166i | −6.74766 | + | 4.29756i | 0 | −0.587624 | + | 1.69037i | ||||||||||||||||||||||||||||||||||
| 163.6 | 0.656712 | + | 1.88911i | 0 | −3.13746 | + | 2.48120i | −0.894797 | 0 | 1.58166i | −6.74766 | − | 4.29756i | 0 | −0.587624 | − | 1.69037i | |||||||||||||||||||||||||||||||||||
| 163.7 | 1.52274 | − | 1.29664i | 0 | 0.637459 | − | 3.94888i | −7.82300 | 0 | 12.3894i | −4.14959 | − | 6.83966i | 0 | −11.9124 | + | 10.1436i | |||||||||||||||||||||||||||||||||||
| 163.8 | 1.52274 | + | 1.29664i | 0 | 0.637459 | + | 3.94888i | −7.82300 | 0 | − | 12.3894i | −4.14959 | + | 6.83966i | 0 | −11.9124 | − | 10.1436i | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 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 | 324.3.d.h | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 324.3.d.h | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 324.3.d.h | ✓ | 8 |
| 9.c | even | 3 | 2 | 324.3.f.s | 16 | ||
| 9.d | odd | 6 | 2 | 324.3.f.s | 16 | ||
| 12.b | even | 2 | 1 | inner | 324.3.d.h | ✓ | 8 |
| 36.f | odd | 6 | 2 | 324.3.f.s | 16 | ||
| 36.h | even | 6 | 2 | 324.3.f.s | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 324.3.d.h | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 324.3.d.h | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 324.3.d.h | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 324.3.d.h | ✓ | 8 | 12.b | even | 2 | 1 | inner |
| 324.3.f.s | 16 | 9.c | even | 3 | 2 | ||
| 324.3.f.s | 16 | 9.d | odd | 6 | 2 | ||
| 324.3.f.s | 16 | 36.f | odd | 6 | 2 | ||
| 324.3.f.s | 16 | 36.h | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 62T_{5}^{2} + 49 \)
acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 5 T^{6} + \cdots + 256 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} - 62 T^{2} + 49)^{2} \)
|
| $7$ |
\( (T^{4} + 156 T^{2} + 384)^{2} \)
|
| $11$ |
\( (T^{4} + 276 T^{2} + 18816)^{2} \)
|
| $13$ |
\( (T^{2} + 8 T - 41)^{4} \)
|
| $17$ |
\( (T^{4} - 758 T^{2} + 52441)^{2} \)
|
| $19$ |
\( (T^{4} + 828 T^{2} + 169344)^{2} \)
|
| $23$ |
\( (T^{4} + 756 T^{2} + 384)^{2} \)
|
| $29$ |
\( (T^{4} - 422 T^{2} + 29929)^{2} \)
|
| $31$ |
\( (T^{4} + 2544 T^{2} + 301056)^{2} \)
|
| $37$ |
\( (T^{2} + 56 T + 727)^{4} \)
|
| $41$ |
\( (T^{4} - 176 T^{2} + 4096)^{2} \)
|
| $43$ |
\( (T^{4} + 10812 T^{2} + 28201344)^{2} \)
|
| $47$ |
\( (T^{4} + 5376 T^{2} + 6291456)^{2} \)
|
| $53$ |
\( (T^{4} - 5744 T^{2} + 861184)^{2} \)
|
| $59$ |
\( (T^{4} + 1344 T^{2} + 393216)^{2} \)
|
| $61$ |
\( (T^{2} + 8 T - 2777)^{4} \)
|
| $67$ |
\( (T^{4} + 5052 T^{2} + 322944)^{2} \)
|
| $71$ |
\( (T^{4} + 12852 T^{2} + 39567744)^{2} \)
|
| $73$ |
\( (T^{2} - 94 T + 1981)^{4} \)
|
| $79$ |
\( (T^{4} + 9468 T^{2} + 921984)^{2} \)
|
| $83$ |
\( (T^{4} + 1104 T^{2} + 301056)^{2} \)
|
| $89$ |
\( (T^{4} - 15926 T^{2} + 52374169)^{2} \)
|
| $97$ |
\( (T^{2} - 232 T + 13228)^{4} \)
|
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