Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [804,2,Mod(37,804)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(804, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("804.37");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 804 = 2^{2} \cdot 3 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 804.i (of order \(3\), 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: | \(6.41997232251\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 8x^{6} - 9x^{5} + 54x^{4} - 50x^{3} + 85x^{2} + 24x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - x^{7} + 8x^{6} - 9x^{5} + 54x^{4} - 50x^{3} + 85x^{2} + 24x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 315\nu^{7} + 154\nu^{6} + 2025\nu^{5} + 180\nu^{4} + 17278\nu^{3} + 4905\nu^{2} + 1485\nu + 9771 ) / 32593 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1867\nu^{7} - 8503\nu^{6} + 7346\nu^{5} - 59463\nu^{4} + 97026\nu^{3} - 292202\nu^{2} + 55363\nu + 13524 ) / 97779 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3257 \nu^{7} + 4202 \nu^{6} - 25594 \nu^{5} + 35388 \nu^{4} - 175338 \nu^{3} + 214684 \nu^{2} + \cdots - 73713 ) / 97779 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -1106\nu^{7} - 1265\nu^{6} - 7110\nu^{5} - 632\nu^{4} - 33142\nu^{3} - 17222\nu^{2} - 5214\nu + 13496 ) / 32593 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -1890\nu^{7} - 924\nu^{6} - 12150\nu^{5} - 1080\nu^{4} - 71075\nu^{3} - 29430\nu^{2} - 8910\nu - 123812 ) / 32593 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 6436 \nu^{7} - 10615 \nu^{6} + 64655 \nu^{5} - 84789 \nu^{4} + 442935 \nu^{3} - 542330 \nu^{2} + \cdots - 60795 ) / 97779 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - \beta_{6} + 4\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + 6\beta_{2} + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{7} + 7\beta_{5} - 23\beta_{4} - 7\beta_{3} + 9\beta _1 - 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{7} - 9\beta_{6} + 23\beta_{4} + \beta_{3} - 38\beta_{2} - 38\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 38\beta_{6} - 45\beta_{5} + 70\beta_{2} + 142 \)
|
| \(\nu^{7}\) | \(=\) |
\( -70\beta_{7} + 18\beta_{5} - 197\beta_{4} - 18\beta_{3} + 250\beta _1 - 197 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).
| \(n\) | \(269\) | \(337\) | \(403\) |
| \(\chi(n)\) | \(1\) | \(\beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
0 | 1.00000 | 0 | −3.96236 | 0 | −0.107790 | + | 0.186699i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 37.2 | 0 | 1.00000 | 0 | −2.18550 | 0 | 1.80395 | − | 3.12454i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 37.3 | 0 | 1.00000 | 0 | 0.238118 | 0 | −1.47881 | + | 2.56138i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 37.4 | 0 | 1.00000 | 0 | 2.90974 | 0 | −0.217351 | + | 0.376462i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 565.1 | 0 | 1.00000 | 0 | −3.96236 | 0 | −0.107790 | − | 0.186699i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 565.2 | 0 | 1.00000 | 0 | −2.18550 | 0 | 1.80395 | + | 3.12454i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 565.3 | 0 | 1.00000 | 0 | 0.238118 | 0 | −1.47881 | − | 2.56138i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 565.4 | 0 | 1.00000 | 0 | 2.90974 | 0 | −0.217351 | − | 0.376462i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 67.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 804.2.i.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | 2412.2.l.f | 8 | ||
| 67.c | even | 3 | 1 | inner | 804.2.i.e | ✓ | 8 |
| 201.g | odd | 6 | 1 | 2412.2.l.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 804.2.i.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 804.2.i.e | ✓ | 8 | 67.c | even | 3 | 1 | inner |
| 2412.2.l.f | 8 | 3.b | odd | 2 | 1 | ||
| 2412.2.l.f | 8 | 201.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\):
|
\( T_{5}^{4} + 3T_{5}^{3} - 10T_{5}^{2} - 23T_{5} + 6 \)
|
|
\( T_{7}^{8} + 11T_{7}^{6} + 14T_{7}^{5} + 122T_{7}^{4} + 77T_{7}^{3} + 38T_{7}^{2} + 7T_{7} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T - 1)^{8} \)
|
| $5$ |
\( (T^{4} + 3 T^{3} - 10 T^{2} + \cdots + 6)^{2} \)
|
| $7$ |
\( T^{8} + 11 T^{6} + \cdots + 1 \)
|
| $11$ |
\( T^{8} - 3 T^{7} + \cdots + 5184 \)
|
| $13$ |
\( T^{8} + 2 T^{7} + \cdots + 25 \)
|
| $17$ |
\( T^{8} + T^{7} + 8 T^{6} + \cdots + 9 \)
|
| $19$ |
\( T^{8} - 4 T^{7} + \cdots + 6889 \)
|
| $23$ |
\( (T^{2} - 3 T + 9)^{4} \)
|
| $29$ |
\( T^{8} + 9 T^{7} + \cdots + 225 \)
|
| $31$ |
\( T^{8} + T^{7} + \cdots + 4096 \)
|
| $37$ |
\( T^{8} - 14 T^{7} + \cdots + 4305625 \)
|
| $41$ |
\( T^{8} - 10 T^{7} + \cdots + 131769 \)
|
| $43$ |
\( (T^{4} - 4 T^{3} - 7 T^{2} + \cdots - 16)^{2} \)
|
| $47$ |
\( T^{8} - 16 T^{7} + \cdots + 4761 \)
|
| $53$ |
\( (T^{4} + 6 T^{3} + \cdots + 3438)^{2} \)
|
| $59$ |
\( (T^{4} - 33 T^{2} + \cdots + 60)^{2} \)
|
| $61$ |
\( T^{8} - 4 T^{7} + \cdots + 1100401 \)
|
| $67$ |
\( T^{8} - 20 T^{7} + \cdots + 20151121 \)
|
| $71$ |
\( T^{8} - 6 T^{7} + \cdots + 1782225 \)
|
| $73$ |
\( T^{8} + 13 T^{7} + \cdots + 7059649 \)
|
| $79$ |
\( T^{8} + 18 T^{7} + \cdots + 625 \)
|
| $83$ |
\( T^{8} + 15 T^{7} + \cdots + 106929 \)
|
| $89$ |
\( (T^{4} + 2 T^{3} + \cdots + 498)^{2} \)
|
| $97$ |
\( T^{8} - 30 T^{7} + \cdots + 15625 \)
|
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