Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [464,3,Mod(175,464)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(464, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("464.175");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 464 = 2^{4} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 464.d (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: | \(12.6430842663\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.30599805184.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 3x^{6} + 13x^{4} + 27x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 13 \) |
| 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
\( x^{8} + 3x^{6} + 13x^{4} + 27x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{5} + 8\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 9\nu^{6} + 48\nu^{5} - 135\nu^{4} + 283\nu^{3} - 288\nu^{2} + 1044\nu - 810 ) / 189 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{6} + 6\nu^{4} + 8\nu^{2} + 27 ) / 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 15\nu^{6} + 6\nu^{5} - 27\nu^{4} + 31\nu^{3} - 276\nu^{2} + 120\nu - 288 ) / 63 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\nu^{7} - 3\nu^{6} + 24\nu^{5} + 45\nu^{4} + 89\nu^{3} + 96\nu^{2} + 396\nu + 270 ) / 189 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{7} - 6\nu^{5} - 26\nu^{3} ) / 27 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -8\nu^{7} - 2\nu^{6} - 6\nu^{5} + 30\nu^{4} + 4\nu^{3} + 64\nu^{2} - 36\nu + 180 ) / 63 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} + 13\beta_{6} + 14\beta_{5} + 4\beta_{2} ) / 52 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{7} + 10\beta_{5} - 13\beta_{4} - 13\beta_{3} + \beta_{2} - 39 ) / 52 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 20\beta_{7} - 39\beta_{6} - 7\beta_{5} + 11\beta_{2} - 13\beta_1 ) / 52 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 22\beta_{7} + 43\beta_{5} + 26\beta_{4} + 39\beta_{3} - 23\beta_{2} - 221 ) / 52 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{7} - 26\beta_{6} - 28\beta_{5} - 8\beta_{2} + 39\beta_1 ) / 26 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -6\beta_{7} - 13\beta_{5} - 2\beta_{4} + 13\beta_{3} + 5\beta_{2} + 9 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -272\beta_{7} - 39\beta_{6} + 259\beta_{5} - 95\beta_{2} - 65\beta_1 ) / 52 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).
| \(n\) | \(117\) | \(175\) | \(321\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 175.1 |
|
0 | − | 5.53698i | 0 | 3.68880 | 0 | − | 3.31120i | 0 | −21.6582 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 175.2 | 0 | − | 3.87448i | 0 | −4.36380 | 0 | 11.3638i | 0 | −6.01161 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 175.3 | 0 | − | 2.84818i | 0 | −1.68880 | 0 | − | 8.68880i | 0 | 0.887870 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 175.4 | 0 | − | 1.48932i | 0 | 6.36380 | 0 | − | 0.636202i | 0 | 6.78194 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 175.5 | 0 | 1.48932i | 0 | 6.36380 | 0 | 0.636202i | 0 | 6.78194 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 175.6 | 0 | 2.84818i | 0 | −1.68880 | 0 | 8.68880i | 0 | 0.887870 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 175.7 | 0 | 3.87448i | 0 | −4.36380 | 0 | − | 11.3638i | 0 | −6.01161 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 175.8 | 0 | 5.53698i | 0 | 3.68880 | 0 | 3.31120i | 0 | −21.6582 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 464.3.d.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 464.3.d.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 464.3.d.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 464.3.d.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 56T_{3}^{6} + 950T_{3}^{4} + 5576T_{3}^{2} + 8281 \)
acting on \(S_{3}^{\mathrm{new}}(464, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 56 T^{6} + \cdots + 8281 \)
|
| $5$ |
\( (T^{4} - 4 T^{3} + \cdots + 173)^{2} \)
|
| $7$ |
\( T^{8} + 216 T^{6} + \cdots + 43264 \)
|
| $11$ |
\( T^{8} + 472 T^{6} + \cdots + 105625 \)
|
| $13$ |
\( (T^{4} - 254 T^{2} + \cdots - 4635)^{2} \)
|
| $17$ |
\( (T^{4} - 16 T^{3} + \cdots - 40752)^{2} \)
|
| $19$ |
\( T^{8} + 1448 T^{6} + \cdots + 592240896 \)
|
| $23$ |
\( T^{8} + \cdots + 10686597376 \)
|
| $29$ |
\( (T^{2} - 29)^{4} \)
|
| $31$ |
\( T^{8} + \cdots + 3706730032369 \)
|
| $37$ |
\( (T^{4} - 80 T^{3} + \cdots - 342784)^{2} \)
|
| $41$ |
\( (T^{4} - 48 T^{3} + \cdots + 2659280)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 345502962025 \)
|
| $47$ |
\( T^{8} + \cdots + 1041659060689 \)
|
| $53$ |
\( (T^{4} + 120 T^{3} + \cdots + 1428541)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 405371609344 \)
|
| $61$ |
\( (T^{4} + 8 T^{3} + \cdots - 148464)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 321410889220096 \)
|
| $71$ |
\( T^{8} + \cdots + 570399541504 \)
|
| $73$ |
\( (T^{4} - 80 T^{3} + \cdots - 237312)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 862125223556961 \)
|
| $83$ |
\( T^{8} + \cdots + 82957704204544 \)
|
| $89$ |
\( (T^{4} - 136 T^{3} + \cdots + 77605840)^{2} \)
|
| $97$ |
\( (T^{4} + 152 T^{3} + \cdots - 54985840)^{2} \)
|
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