Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1344,2,Mod(289,1344)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1344, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1344.289");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1344.bk (of order \(6\), degree \(2\), 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: | \(10.7318940317\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.49787136.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 15\nu^{4} + 5\nu^{2} + 12 ) / 20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 5\nu^{5} + 5\nu^{3} + 12\nu ) / 40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{6} - 5\nu^{4} - 15\nu^{2} - 36 ) / 20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 7\nu ) / 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 13\nu ) / 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -9\nu^{6} - 15\nu^{4} - 5\nu^{2} - 48 ) / 20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 11\nu^{7} + 25\nu^{5} + 55\nu^{3} + 132\nu ) / 40 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - 3\beta_{3} - 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} - \beta_{5} + 5\beta_{4} + 5\beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{3} + 3\beta_1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( \beta_{7} - 11\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -5\beta_{6} - 5\beta _1 - 9 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -7\beta_{5} - 13\beta_{4} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(449\) | \(577\) | \(1093\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{3}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 289.1 |
|
0 | −0.866025 | + | 0.500000i | 0 | −3.79129 | − | 2.18890i | 0 | 1.32288 | − | 2.29129i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||
| 289.2 | 0 | −0.866025 | + | 0.500000i | 0 | 0.791288 | + | 0.456850i | 0 | −1.32288 | + | 2.29129i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||
| 289.3 | 0 | 0.866025 | − | 0.500000i | 0 | −3.79129 | − | 2.18890i | 0 | −1.32288 | + | 2.29129i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||
| 289.4 | 0 | 0.866025 | − | 0.500000i | 0 | 0.791288 | + | 0.456850i | 0 | 1.32288 | − | 2.29129i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||
| 865.1 | 0 | −0.866025 | − | 0.500000i | 0 | −3.79129 | + | 2.18890i | 0 | 1.32288 | + | 2.29129i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||
| 865.2 | 0 | −0.866025 | − | 0.500000i | 0 | 0.791288 | − | 0.456850i | 0 | −1.32288 | − | 2.29129i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||
| 865.3 | 0 | 0.866025 | + | 0.500000i | 0 | −3.79129 | + | 2.18890i | 0 | −1.32288 | − | 2.29129i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||
| 865.4 | 0 | 0.866025 | + | 0.500000i | 0 | 0.791288 | − | 0.456850i | 0 | 1.32288 | + | 2.29129i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 56.k | odd | 6 | 1 | inner |
| 56.p | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1344.2.bk.i | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 1344.2.bk.i | ✓ | 8 |
| 7.c | even | 3 | 1 | 1344.2.bk.l | yes | 8 | |
| 8.b | even | 2 | 1 | 1344.2.bk.l | yes | 8 | |
| 8.d | odd | 2 | 1 | 1344.2.bk.l | yes | 8 | |
| 28.g | odd | 6 | 1 | 1344.2.bk.l | yes | 8 | |
| 56.k | odd | 6 | 1 | inner | 1344.2.bk.i | ✓ | 8 |
| 56.p | even | 6 | 1 | inner | 1344.2.bk.i | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1344.2.bk.i | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1344.2.bk.i | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 1344.2.bk.i | ✓ | 8 | 56.k | odd | 6 | 1 | inner |
| 1344.2.bk.i | ✓ | 8 | 56.p | even | 6 | 1 | inner |
| 1344.2.bk.l | yes | 8 | 7.c | even | 3 | 1 | |
| 1344.2.bk.l | yes | 8 | 8.b | even | 2 | 1 | |
| 1344.2.bk.l | yes | 8 | 8.d | odd | 2 | 1 | |
| 1344.2.bk.l | yes | 8 | 28.g | odd | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\):
|
\( T_{5}^{4} + 6T_{5}^{3} + 8T_{5}^{2} - 24T_{5} + 16 \)
|
|
\( T_{11}^{4} - 4T_{11}^{2} + 16 \)
|
|
\( T_{23}^{8} + 20T_{23}^{6} + 384T_{23}^{4} + 320T_{23}^{2} + 256 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} - T^{2} + 1)^{2} \)
|
| $5$ |
\( (T^{4} + 6 T^{3} + 8 T^{2} + \cdots + 16)^{2} \)
|
| $7$ |
\( (T^{4} + 7 T^{2} + 49)^{2} \)
|
| $11$ |
\( (T^{4} - 4 T^{2} + 16)^{2} \)
|
| $13$ |
\( (T^{2} + 27)^{4} \)
|
| $17$ |
\( (T^{4} + 6 T^{3} + \cdots + 144)^{2} \)
|
| $19$ |
\( T^{8} - 50 T^{6} + \cdots + 83521 \)
|
| $23$ |
\( T^{8} + 20 T^{6} + \cdots + 256 \)
|
| $29$ |
\( (T^{2} + 48)^{4} \)
|
| $31$ |
\( T^{8} + 38 T^{6} + \cdots + 625 \)
|
| $37$ |
\( (T^{4} - 18 T^{3} + 107 T^{2} + \cdots + 1)^{2} \)
|
| $41$ |
\( (T^{2} - 84)^{4} \)
|
| $43$ |
\( (T^{4} + 114 T^{2} + 225)^{2} \)
|
| $47$ |
\( T^{8} + 132 T^{6} + \cdots + 12960000 \)
|
| $53$ |
\( (T^{4} + 6 T^{3} + \cdots + 3600)^{2} \)
|
| $59$ |
\( T^{8} - 140 T^{6} + \cdots + 614656 \)
|
| $61$ |
\( (T^{4} - 12 T^{3} + \cdots + 256)^{2} \)
|
| $67$ |
\( T^{8} - 74 T^{6} + \cdots + 625 \)
|
| $71$ |
\( (T^{4} - 180 T^{2} + 1296)^{2} \)
|
| $73$ |
\( (T^{2} - 3 T + 9)^{4} \)
|
| $79$ |
\( T^{8} + 150 T^{6} + \cdots + 6765201 \)
|
| $83$ |
\( (T^{4} + 380 T^{2} + 35344)^{2} \)
|
| $89$ |
\( (T^{4} - 12 T^{3} + \cdots + 2304)^{2} \)
|
| $97$ |
\( (T - 2)^{8} \)
|
| show more | |
| show less |