Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [507,2,Mod(239,507)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("507.239");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 507 = 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 507.f (of order \(4\), 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: | \(4.04841538248\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.56070144.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 39) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{7} - \nu^{6} - 25\nu^{5} - 46\nu^{4} - 5\nu^{3} - 132\nu^{2} + 28\nu - 55 ) / 37 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -6\nu^{7} + 21\nu^{6} - 67\nu^{5} + 115\nu^{4} - 117\nu^{3} + 71\nu^{2} + 41\nu - 29 ) / 37 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{7} - 7\nu^{6} + 47\nu^{5} - 100\nu^{4} + 261\nu^{3} - 295\nu^{2} + 344\nu - 126 ) / 37 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{7} + 36\nu^{6} - 136\nu^{5} + 361\nu^{4} - 634\nu^{3} + 793\nu^{2} - 601\nu + 278 ) / 37 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -18\nu^{7} + 63\nu^{6} - 238\nu^{5} + 419\nu^{4} - 684\nu^{3} + 546\nu^{2} - 395\nu + 61 ) / 37 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -18\nu^{7} + 63\nu^{6} - 238\nu^{5} + 456\nu^{4} - 758\nu^{3} + 805\nu^{2} - 617\nu + 283 ) / 37 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 - 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} - \beta_{5} - 5\beta_{4} + 6\beta_{3} - \beta_{2} + 3\beta _1 - 3 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -4\beta_{6} - 3\beta_{5} - 12\beta_{4} + 13\beta_{3} - 3\beta_{2} - 8\beta _1 + 10 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 9\beta_{7} - \beta_{6} - 2\beta_{5} + 11\beta_{4} - 17\beta_{3} - 2\beta_{2} - 25\beta _1 + 19 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 5\beta_{7} + 12\beta_{6} + 2\beta_{5} + 32\beta_{4} - 42\beta_{3} + 7\beta _1 - 11 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -46\beta_{7} + 38\beta_{6} + 17\beta_{5} + 6\beta_{4} - 3\beta_{3} + 3\beta_{2} + 140\beta _1 - 96 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).
| \(n\) | \(170\) | \(340\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 239.1 |
|
−1.69293 | − | 1.69293i | −0.366025 | + | 1.69293i | 3.73205i | −1.69293 | − | 1.69293i | 3.48568 | − | 2.24637i | 1.00000 | + | 1.00000i | 2.93225 | − | 2.93225i | −2.73205 | − | 1.23931i | 5.73205i | ||||||||||||||||||||||||||||
| 239.2 | −1.06488 | − | 1.06488i | 1.36603 | + | 1.06488i | 0.267949i | −1.06488 | − | 1.06488i | −0.320682 | − | 2.58863i | 1.00000 | + | 1.00000i | −1.84443 | + | 1.84443i | 0.732051 | + | 2.90931i | 2.26795i | |||||||||||||||||||||||||||||
| 239.3 | 1.06488 | + | 1.06488i | 1.36603 | − | 1.06488i | 0.267949i | 1.06488 | + | 1.06488i | 2.58863 | + | 0.320682i | 1.00000 | + | 1.00000i | 1.84443 | − | 1.84443i | 0.732051 | − | 2.90931i | 2.26795i | |||||||||||||||||||||||||||||
| 239.4 | 1.69293 | + | 1.69293i | −0.366025 | − | 1.69293i | 3.73205i | 1.69293 | + | 1.69293i | 2.24637 | − | 3.48568i | 1.00000 | + | 1.00000i | −2.93225 | + | 2.93225i | −2.73205 | + | 1.23931i | 5.73205i | |||||||||||||||||||||||||||||
| 437.1 | −1.69293 | + | 1.69293i | −0.366025 | − | 1.69293i | − | 3.73205i | −1.69293 | + | 1.69293i | 3.48568 | + | 2.24637i | 1.00000 | − | 1.00000i | 2.93225 | + | 2.93225i | −2.73205 | + | 1.23931i | − | 5.73205i | |||||||||||||||||||||||||||
| 437.2 | −1.06488 | + | 1.06488i | 1.36603 | − | 1.06488i | − | 0.267949i | −1.06488 | + | 1.06488i | −0.320682 | + | 2.58863i | 1.00000 | − | 1.00000i | −1.84443 | − | 1.84443i | 0.732051 | − | 2.90931i | − | 2.26795i | |||||||||||||||||||||||||||
| 437.3 | 1.06488 | − | 1.06488i | 1.36603 | + | 1.06488i | − | 0.267949i | 1.06488 | − | 1.06488i | 2.58863 | − | 0.320682i | 1.00000 | − | 1.00000i | 1.84443 | + | 1.84443i | 0.732051 | + | 2.90931i | − | 2.26795i | |||||||||||||||||||||||||||
| 437.4 | 1.69293 | − | 1.69293i | −0.366025 | + | 1.69293i | − | 3.73205i | 1.69293 | − | 1.69293i | 2.24637 | + | 3.48568i | 1.00000 | − | 1.00000i | −2.93225 | − | 2.93225i | −2.73205 | − | 1.23931i | − | 5.73205i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.d | odd | 4 | 1 | inner |
| 39.f | even | 4 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):
|
\( T_{2}^{8} + 38T_{2}^{4} + 169 \)
|
|
\( T_{5}^{8} + 38T_{5}^{4} + 169 \)
|
|
\( T_{7}^{2} - 2T_{7} + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 38T^{4} + 169 \)
|
| $3$ |
\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 9)^{2} \)
|
| $5$ |
\( T^{8} + 38T^{4} + 169 \)
|
| $7$ |
\( (T^{2} - 2 T + 2)^{4} \)
|
| $11$ |
\( T^{8} + 296T^{4} + 2704 \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( (T^{4} - 30 T^{2} + 117)^{2} \)
|
| $19$ |
\( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 16)^{2} \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( (T^{4} + 82 T^{2} + 1573)^{2} \)
|
| $31$ |
\( (T^{4} - 4 T^{3} + \cdots + 484)^{2} \)
|
| $37$ |
\( (T^{4} + 2 T^{3} + \cdots + 1369)^{2} \)
|
| $41$ |
\( T^{8} + 998T^{4} + 169 \)
|
| $43$ |
\( (T^{4} + 72 T^{2} + 324)^{2} \)
|
| $47$ |
\( T^{8} + 9728 T^{4} + 11075584 \)
|
| $53$ |
\( (T^{4} + 22 T^{2} + 13)^{2} \)
|
| $59$ |
\( T^{8} + 608 T^{4} + 43264 \)
|
| $61$ |
\( (T + 7)^{8} \)
|
| $67$ |
\( (T^{4} - 4 T^{3} + \cdots + 2704)^{2} \)
|
| $71$ |
\( T^{8} + 608 T^{4} + 43264 \)
|
| $73$ |
\( (T^{4} + 14 T^{3} + \cdots + 121)^{2} \)
|
| $79$ |
\( (T - 2)^{8} \)
|
| $83$ |
\( T^{8} + 296T^{4} + 2704 \)
|
| $89$ |
\( T^{8} + 17768 T^{4} + 77228944 \)
|
| $97$ |
\( (T^{4} - 16 T^{3} + \cdots + 484)^{2} \)
|
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