Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1170,2,Mod(181,1170)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1170, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1170.181");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1170.b (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: | \(9.34249703649\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.3057647616.6 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 30x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| 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
\( x^{8} + 30x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 33\nu^{2} ) / 18 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{4} + 33\nu^{3} + 18\nu - 45 ) / 18 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} - 33\nu^{3} + 81\nu^{2} + 18\nu ) / 18 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 3\nu^{6} - 33\nu^{3} - 81\nu^{2} + 18\nu ) / 18 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + 3\nu^{4} + 33\nu^{3} + 18\nu + 45 ) / 18 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{7} + 3\nu^{5} - 147\nu^{3} + 81\nu ) / 18 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} - 3\nu^{5} - 147\nu^{3} - 81\nu ) / 18 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + 6\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{7} + 2\beta_{6} + 5\beta_{5} - 5\beta_{4} - 5\beta_{3} + 5\beta_{2} ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{5} - 3\beta_{2} - 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -12\beta_{7} + 12\beta_{6} - 27\beta_{5} - 27\beta_{4} - 27\beta_{3} - 27\beta_{2} ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -33\beta_{4} + 33\beta_{3} - 162\beta_1 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -66\beta_{7} - 66\beta_{6} - 147\beta_{5} + 147\beta_{4} + 147\beta_{3} - 147\beta_{2} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).
| \(n\) | \(911\) | \(937\) | \(1081\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 181.1 |
|
− | 1.00000i | 0 | −1.00000 | − | 1.00000i | 0 | − | 4.66883i | 1.00000i | 0 | −1.00000 | |||||||||||||||||||||||||||||||||||||||
| 181.2 | − | 1.00000i | 0 | −1.00000 | − | 1.00000i | 0 | − | 1.48393i | 1.00000i | 0 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 181.3 | − | 1.00000i | 0 | −1.00000 | − | 1.00000i | 0 | 1.48393i | 1.00000i | 0 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||
| 181.4 | − | 1.00000i | 0 | −1.00000 | − | 1.00000i | 0 | 4.66883i | 1.00000i | 0 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||
| 181.5 | 1.00000i | 0 | −1.00000 | 1.00000i | 0 | − | 4.66883i | − | 1.00000i | 0 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||
| 181.6 | 1.00000i | 0 | −1.00000 | 1.00000i | 0 | − | 1.48393i | − | 1.00000i | 0 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||
| 181.7 | 1.00000i | 0 | −1.00000 | 1.00000i | 0 | 1.48393i | − | 1.00000i | 0 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||||
| 181.8 | 1.00000i | 0 | −1.00000 | 1.00000i | 0 | 4.66883i | − | 1.00000i | 0 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
| 39.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1170.2.b.g | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 1170.2.b.g | ✓ | 8 |
| 13.b | even | 2 | 1 | inner | 1170.2.b.g | ✓ | 8 |
| 39.d | odd | 2 | 1 | inner | 1170.2.b.g | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1170.2.b.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1170.2.b.g | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 1170.2.b.g | ✓ | 8 | 13.b | even | 2 | 1 | inner |
| 1170.2.b.g | ✓ | 8 | 39.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):
|
\( T_{7}^{4} + 24T_{7}^{2} + 48 \)
|
|
\( T_{11}^{2} + 24 \)
|
|
\( T_{17}^{4} - 72T_{17}^{2} + 432 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{2} + 1)^{4} \)
|
| $7$ |
\( (T^{4} + 24 T^{2} + 48)^{2} \)
|
| $11$ |
\( (T^{2} + 24)^{4} \)
|
| $13$ |
\( (T^{4} + 4 T^{3} + \cdots + 169)^{2} \)
|
| $17$ |
\( (T^{4} - 72 T^{2} + 432)^{2} \)
|
| $19$ |
\( (T^{4} + 72 T^{2} + 1200)^{2} \)
|
| $23$ |
\( (T^{4} - 24 T^{2} + 48)^{2} \)
|
| $29$ |
\( (T^{4} - 72 T^{2} + 1200)^{2} \)
|
| $31$ |
\( (T^{4} + 48 T^{2} + 192)^{2} \)
|
| $37$ |
\( (T^{4} + 144 T^{2} + 1728)^{2} \)
|
| $41$ |
\( (T^{4} + 120 T^{2} + 144)^{2} \)
|
| $43$ |
\( (T + 4)^{8} \)
|
| $47$ |
\( (T^{2} + 96)^{4} \)
|
| $53$ |
\( (T^{4} - 48 T^{2} + 192)^{2} \)
|
| $59$ |
\( (T^{2} + 24)^{4} \)
|
| $61$ |
\( (T + 2)^{8} \)
|
| $67$ |
\( (T^{4} + 144 T^{2} + 4800)^{2} \)
|
| $71$ |
\( T^{8} \)
|
| $73$ |
\( (T^{4} + 216 T^{2} + 3888)^{2} \)
|
| $79$ |
\( (T^{2} - 8 T - 80)^{4} \)
|
| $83$ |
\( (T^{2} + 96)^{4} \)
|
| $89$ |
\( (T^{4} + 120 T^{2} + 144)^{2} \)
|
| $97$ |
\( (T^{4} + 408 T^{2} + 30000)^{2} \)
|
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