Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [864,3,Mod(161,864)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(864, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("864.161");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 864 = 2^{5} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 864.e (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: | \(23.5422948407\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 4x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 4x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 14\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{3} + 4\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 4 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + 2\beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{2} + 2\beta_1 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(353\) | \(703\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 161.1 |
|
0 | 0 | 0 | − | 4.47214i | 0 | −3.32456 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 161.2 | 0 | 0 | 0 | − | 4.47214i | 0 | 9.32456 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 161.3 | 0 | 0 | 0 | 4.47214i | 0 | −3.32456 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 161.4 | 0 | 0 | 0 | 4.47214i | 0 | 9.32456 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
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 | 864.3.e.c | yes | 4 |
| 3.b | odd | 2 | 1 | inner | 864.3.e.c | yes | 4 |
| 4.b | odd | 2 | 1 | 864.3.e.a | ✓ | 4 | |
| 8.b | even | 2 | 1 | 1728.3.e.t | 4 | ||
| 8.d | odd | 2 | 1 | 1728.3.e.q | 4 | ||
| 12.b | even | 2 | 1 | 864.3.e.a | ✓ | 4 | |
| 24.f | even | 2 | 1 | 1728.3.e.q | 4 | ||
| 24.h | odd | 2 | 1 | 1728.3.e.t | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 864.3.e.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 864.3.e.a | ✓ | 4 | 12.b | even | 2 | 1 | |
| 864.3.e.c | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 864.3.e.c | yes | 4 | 3.b | odd | 2 | 1 | inner |
| 1728.3.e.q | 4 | 8.d | odd | 2 | 1 | ||
| 1728.3.e.q | 4 | 24.f | even | 2 | 1 | ||
| 1728.3.e.t | 4 | 8.b | even | 2 | 1 | ||
| 1728.3.e.t | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(864, [\chi])\):
|
\( T_{5}^{2} + 20 \)
|
|
\( T_{7}^{2} - 6T_{7} - 31 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 20)^{2} \)
|
| $7$ |
\( (T^{2} - 6 T - 31)^{2} \)
|
| $11$ |
\( T^{4} + 424 T^{2} + 21904 \)
|
| $13$ |
\( (T^{2} + 10 T - 335)^{2} \)
|
| $17$ |
\( T^{4} + 936 T^{2} + 11664 \)
|
| $19$ |
\( (T^{2} + 18 T - 79)^{2} \)
|
| $23$ |
\( (T^{2} + 180)^{2} \)
|
| $29$ |
\( T^{4} + 2464 T^{2} + 1149184 \)
|
| $31$ |
\( (T^{2} - 12 T - 604)^{2} \)
|
| $37$ |
\( (T^{2} - 14 T - 311)^{2} \)
|
| $41$ |
\( T^{4} + 6304 T^{2} + 719104 \)
|
| $43$ |
\( (T^{2} + 36 T - 2236)^{2} \)
|
| $47$ |
\( T^{4} + 3496 T^{2} + 2226064 \)
|
| $53$ |
\( T^{4} + 9376 T^{2} + 20502784 \)
|
| $59$ |
\( T^{4} + 4840 T^{2} + 672400 \)
|
| $61$ |
\( (T^{2} + 42 T + 81)^{2} \)
|
| $67$ |
\( (T^{2} + 90 T + 1385)^{2} \)
|
| $71$ |
\( T^{4} + 12160 T^{2} + 102400 \)
|
| $73$ |
\( (T^{2} + 66 T - 351)^{2} \)
|
| $79$ |
\( (T^{2} - 78 T + 1481)^{2} \)
|
| $83$ |
\( T^{4} + 10656 T^{2} + 15116544 \)
|
| $89$ |
\( T^{4} + 16360 T^{2} + 38688400 \)
|
| $97$ |
\( (T^{2} - 86 T + 409)^{2} \)
|
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