Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1040,2,Mod(49,1040)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1040, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1040.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1040 = 2^{4} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1040.df (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: | \(8.30444181021\) |
| 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: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 65) |
| 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
\( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} + 5\nu^{4} - 5\nu^{2} + 20\nu - 12 ) / 20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + 5\nu^{5} - 5\nu^{3} - 12\nu ) / 40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 36 ) / 20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{7} + \nu^{6} - 5\nu^{4} + 5\nu^{2} + 6\nu + 12 ) / 20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{7} - 4\nu^{6} - 5\nu^{5} - 15\nu^{3} - 16\nu - 28 ) / 20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 9\nu^{7} - 8\nu^{6} + 15\nu^{5} + 25\nu^{3} + 48\nu - 56 ) / 40 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{3} - \beta_{2} - \beta _1 - 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} + 5\beta_{4} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -3\beta_{5} + 3\beta_{4} + 2\beta_{3} + 3\beta_{2} + 3\beta_1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} + \beta_{5} + 11\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -5\beta_{7} - 5\beta_{6} - 5\beta_{4} - 14 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -7\beta_{5} - 13\beta_{4} - 13\beta_{2} - 7\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(417\) | \(561\) | \(911\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | −0.866025 | + | 0.500000i | 0 | −2.18890 | + | 0.456850i | 0 | 0.866025 | − | 1.50000i | 0 | −1.00000 | + | 1.73205i | 0 | ||||||||||||||||||||||||||||||||||
| 49.2 | 0 | −0.866025 | + | 0.500000i | 0 | 0.456850 | − | 2.18890i | 0 | 0.866025 | − | 1.50000i | 0 | −1.00000 | + | 1.73205i | 0 | |||||||||||||||||||||||||||||||||||
| 49.3 | 0 | 0.866025 | − | 0.500000i | 0 | −0.456850 | − | 2.18890i | 0 | −0.866025 | + | 1.50000i | 0 | −1.00000 | + | 1.73205i | 0 | |||||||||||||||||||||||||||||||||||
| 49.4 | 0 | 0.866025 | − | 0.500000i | 0 | 2.18890 | + | 0.456850i | 0 | −0.866025 | + | 1.50000i | 0 | −1.00000 | + | 1.73205i | 0 | |||||||||||||||||||||||||||||||||||
| 849.1 | 0 | −0.866025 | − | 0.500000i | 0 | −2.18890 | − | 0.456850i | 0 | 0.866025 | + | 1.50000i | 0 | −1.00000 | − | 1.73205i | 0 | |||||||||||||||||||||||||||||||||||
| 849.2 | 0 | −0.866025 | − | 0.500000i | 0 | 0.456850 | + | 2.18890i | 0 | 0.866025 | + | 1.50000i | 0 | −1.00000 | − | 1.73205i | 0 | |||||||||||||||||||||||||||||||||||
| 849.3 | 0 | 0.866025 | + | 0.500000i | 0 | −0.456850 | + | 2.18890i | 0 | −0.866025 | − | 1.50000i | 0 | −1.00000 | − | 1.73205i | 0 | |||||||||||||||||||||||||||||||||||
| 849.4 | 0 | 0.866025 | + | 0.500000i | 0 | 2.18890 | − | 0.456850i | 0 | −0.866025 | − | 1.50000i | 0 | −1.00000 | − | 1.73205i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 13.e | even | 6 | 1 | inner |
| 65.l | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - T_{3}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} - T^{2} + 1)^{2} \)
|
| $5$ |
\( T^{8} - 34T^{4} + 625 \)
|
| $7$ |
\( (T^{4} + 3 T^{2} + 9)^{2} \)
|
| $11$ |
\( (T^{4} - 7 T^{2} + 49)^{2} \)
|
| $13$ |
\( (T^{4} - 22 T^{2} + 169)^{2} \)
|
| $17$ |
\( (T^{4} - 21 T^{2} + 441)^{2} \)
|
| $19$ |
\( (T^{2} - 3 T + 3)^{4} \)
|
| $23$ |
\( (T^{4} - 21 T^{2} + 441)^{2} \)
|
| $29$ |
\( (T^{4} + 21 T^{2} + 441)^{2} \)
|
| $31$ |
\( (T^{4} + 132 T^{2} + 3600)^{2} \)
|
| $37$ |
\( (T^{4} + 63 T^{2} + 3969)^{2} \)
|
| $41$ |
\( (T^{4} - 7 T^{2} + 49)^{2} \)
|
| $43$ |
\( T^{8} - 114 T^{6} + \cdots + 50625 \)
|
| $47$ |
\( (T^{4} - 80 T^{2} + 256)^{2} \)
|
| $53$ |
\( (T^{4} + 60 T^{2} + 144)^{2} \)
|
| $59$ |
\( (T^{4} - 30 T^{3} + \cdots + 2209)^{2} \)
|
| $61$ |
\( (T^{4} - 12 T^{3} + \cdots + 225)^{2} \)
|
| $67$ |
\( T^{8} + 222 T^{6} + \cdots + 50625 \)
|
| $71$ |
\( (T^{4} + 6 T^{3} + \cdots + 625)^{2} \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( (T + 6)^{8} \)
|
| $83$ |
\( (T^{4} - 164 T^{2} + 4624)^{2} \)
|
| $89$ |
\( (T^{4} - 24 T^{3} + \cdots + 1681)^{2} \)
|
| $97$ |
\( T^{8} + 150 T^{6} + \cdots + 6765201 \)
|
| show more | |
| show less |