Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [480,2,Mod(241,480)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(480, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("480.241");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 480 = 2^{5} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 480.k (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: | \(3.83281929702\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.399424.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 120) |
| 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
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 2\nu + 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7\nu^{5} - 4\nu^{4} + 13\nu^{3} - 16\nu^{2} + 6\nu - 32 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{3} + 3\beta_{2} + \beta _1 - 2 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} - 4\beta_{2} + 2\beta _1 + 5 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{3} - 11\beta_{2} - \beta _1 + 6 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4\beta_{5} + \beta_{4} + 3\beta_{3} + 8\beta_{2} - 2\beta _1 + 7 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(161\) | \(421\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 241.1 |
|
0 | − | 1.00000i | 0 | 1.00000i | 0 | −4.68585 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 241.2 | 0 | − | 1.00000i | 0 | 1.00000i | 0 | −0.941367 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 241.3 | 0 | − | 1.00000i | 0 | 1.00000i | 0 | 3.62721 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 241.4 | 0 | 1.00000i | 0 | − | 1.00000i | 0 | −4.68585 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 241.5 | 0 | 1.00000i | 0 | − | 1.00000i | 0 | −0.941367 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 241.6 | 0 | 1.00000i | 0 | − | 1.00000i | 0 | 3.62721 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} + 2T_{7}^{2} - 16T_{7} - 16 \)
acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 1)^{3} \)
|
| $5$ |
\( (T^{2} + 1)^{3} \)
|
| $7$ |
\( (T^{3} + 2 T^{2} - 16 T - 16)^{2} \)
|
| $11$ |
\( T^{6} + 64 T^{4} + \cdots + 4096 \)
|
| $13$ |
\( T^{6} + 56 T^{4} + \cdots + 256 \)
|
| $17$ |
\( (T^{3} - 6 T^{2} - 16 T + 32)^{2} \)
|
| $19$ |
\( T^{6} + 40 T^{4} + \cdots + 256 \)
|
| $23$ |
\( (T^{3} - 4 T^{2} - 12 T + 16)^{2} \)
|
| $29$ |
\( (T^{2} + 4)^{3} \)
|
| $31$ |
\( (T^{3} - 6 T^{2} - 16 T + 64)^{2} \)
|
| $37$ |
\( T^{6} + 136 T^{4} + \cdots + 65536 \)
|
| $41$ |
\( (T^{3} + 10 T^{2} + \cdots - 232)^{2} \)
|
| $43$ |
\( T^{6} + 144 T^{4} + \cdots + 16384 \)
|
| $47$ |
\( (T^{3} + 4 T^{2} + \cdots - 496)^{2} \)
|
| $53$ |
\( (T^{2} + 4)^{3} \)
|
| $59$ |
\( T^{6} + 80 T^{4} + \cdots + 1024 \)
|
| $61$ |
\( T^{6} + 256 T^{4} + \cdots + 262144 \)
|
| $67$ |
\( (T^{2} + 16)^{3} \)
|
| $71$ |
\( (T^{3} - 4 T^{2} - 112 T - 64)^{2} \)
|
| $73$ |
\( (T + 6)^{6} \)
|
| $79$ |
\( (T^{3} + 18 T^{2} + \cdots + 64)^{2} \)
|
| $83$ |
\( T^{6} + 224 T^{4} + \cdots + 65536 \)
|
| $89$ |
\( (T^{3} + 14 T^{2} + \cdots - 184)^{2} \)
|
| $97$ |
\( (T^{3} - 18 T^{2} + \cdots + 328)^{2} \)
|
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