Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [864,3,Mod(593,864)]
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, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("864.593");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 864 = 2^{5} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 864.h (of order \(2\), degree \(1\), 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: | \(23.5422948407\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.121670000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 4x^{4} - 6x^{3} + 16x^{2} - 16x + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 216) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 4x^{4} - 6x^{3} + 16x^{2} - 16x + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} + \nu^{3} + 2\nu - 4 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 4\nu^{3} + 6\nu^{2} + 8 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 12\nu^{3} - 10\nu^{2} + 32\nu - 32 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 2\nu^{3} + 10\nu^{2} + 12\nu + 16 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{5} - 7\nu^{4} - 34\nu^{2} + 72\nu - 64 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{3} + 4\beta_{2} + 4 ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + 3\beta_{4} - \beta_{3} + 4\beta_{2} + 8\beta _1 - 20 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{5} + \beta_{4} + 5\beta_{3} - 12\beta_{2} + 8\beta _1 + 12 ) / 16 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{5} + 3\beta_{4} + 7\beta_{3} - 4\beta_{2} - 56\beta _1 - 44 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5\beta_{5} + 17\beta_{4} - 19\beta_{3} - 60\beta_{2} - 40\beta _1 - 84 ) / 16 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 593.1 |
|
0 | 0 | 0 | −5.18465 | 0 | −3.67337 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 593.2 | 0 | 0 | 0 | −5.18465 | 0 | −3.67337 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 593.3 | 0 | 0 | 0 | 2.38717 | 0 | −2.13992 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 593.4 | 0 | 0 | 0 | 2.38717 | 0 | −2.13992 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 593.5 | 0 | 0 | 0 | 3.79748 | 0 | 10.8133 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 593.6 | 0 | 0 | 0 | 3.79748 | 0 | 10.8133 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 24.h | 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.h.d | 6 | |
| 3.b | odd | 2 | 1 | 864.3.h.c | 6 | ||
| 4.b | odd | 2 | 1 | 216.3.h.d | yes | 6 | |
| 8.b | even | 2 | 1 | 864.3.h.c | 6 | ||
| 8.d | odd | 2 | 1 | 216.3.h.c | ✓ | 6 | |
| 12.b | even | 2 | 1 | 216.3.h.c | ✓ | 6 | |
| 24.f | even | 2 | 1 | 216.3.h.d | yes | 6 | |
| 24.h | odd | 2 | 1 | inner | 864.3.h.d | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 216.3.h.c | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 216.3.h.c | ✓ | 6 | 12.b | even | 2 | 1 | |
| 216.3.h.d | yes | 6 | 4.b | odd | 2 | 1 | |
| 216.3.h.d | yes | 6 | 24.f | even | 2 | 1 | |
| 864.3.h.c | 6 | 3.b | odd | 2 | 1 | ||
| 864.3.h.c | 6 | 8.b | even | 2 | 1 | ||
| 864.3.h.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 864.3.h.d | 6 | 24.h | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - T_{5}^{2} - 23T_{5} + 47 \)
acting on \(S_{3}^{\mathrm{new}}(864, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} - T^{2} - 23 T + 47)^{2} \)
|
| $7$ |
\( (T^{3} - 5 T^{2} - 55 T - 85)^{2} \)
|
| $11$ |
\( (T^{3} - 5 T^{2} - 105 T + 45)^{2} \)
|
| $13$ |
\( T^{6} + 680 T^{4} + \cdots + 5299200 \)
|
| $17$ |
\( T^{6} + 1336 T^{4} + \cdots + 17169408 \)
|
| $19$ |
\( T^{6} + 1304 T^{4} + \cdots + 30523392 \)
|
| $23$ |
\( T^{6} + 2056 T^{4} + \cdots + 245035008 \)
|
| $29$ |
\( (T^{3} + 50 T^{2} + \cdots + 2880)^{2} \)
|
| $31$ |
\( (T^{3} + 3 T^{2} + \cdots + 15971)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 1716940800 \)
|
| $41$ |
\( T^{6} + 7000 T^{4} + \cdots + 529920000 \)
|
| $43$ |
\( T^{6} + 6080 T^{4} + \cdots + 339148800 \)
|
| $47$ |
\( T^{6} + \cdots + 6867763200 \)
|
| $53$ |
\( (T^{3} - T^{2} + \cdots - 32553)^{2} \)
|
| $59$ |
\( (T^{3} + 10 T^{2} + \cdots + 190360)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 89006211072 \)
|
| $67$ |
\( T^{6} + \cdots + 27471052800 \)
|
| $71$ |
\( T^{6} + \cdots + 3863116800 \)
|
| $73$ |
\( (T^{3} - 65 T^{2} + \cdots + 983025)^{2} \)
|
| $79$ |
\( (T^{3} - 38 T^{2} + \cdots + 323744)^{2} \)
|
| $83$ |
\( (T^{3} + 19 T^{2} + \cdots - 378683)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 61809868800 \)
|
| $97$ |
\( (T^{3} + 35 T^{2} + \cdots + 193825)^{2} \)
|
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