Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [266,2,Mod(197,266)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(266, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([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("266.197");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 266 = 2 \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 266.f (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: | \(2.12402069377\) |
| 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: |
\( x^{8} - x^{7} + 11x^{6} - 6x^{5} + 104x^{4} - 72x^{3} + 104x^{2} + 32x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{8} - x^{7} + 11x^{6} - 6x^{5} + 104x^{4} - 72x^{3} + 104x^{2} + 32x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -25\nu^{7} - 465\nu^{6} - 245\nu^{5} - 4652\nu^{4} - 4168\nu^{3} - 41488\nu^{2} - 9264\nu - 3152 ) / 9144 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 245\nu^{7} - 15\nu^{6} + 2401\nu^{5} + 784\nu^{4} + 26216\nu^{3} + 3332\nu^{2} + 1176\nu + 8944 ) / 27432 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -559\nu^{7} + 804\nu^{6} - 6164\nu^{5} + 5755\nu^{4} - 57352\nu^{3} + 66464\nu^{2} - 54804\nu + 10720 ) / 27432 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -185\nu^{7} - 12\nu^{6} - 1813\nu^{5} - 592\nu^{4} - 17813\nu^{3} - 2516\nu^{2} - 888\nu + 1364 ) / 4572 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -670\nu^{7} + 111\nu^{6} - 6566\nu^{5} - 2144\nu^{4} - 63925\nu^{3} - 9112\nu^{2} - 3216\nu - 62528 ) / 13716 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 170\nu^{7} - 267\nu^{6} + 2047\nu^{5} - 2123\nu^{4} + 19046\nu^{3} - 22072\nu^{2} + 32820\nu - 3560 ) / 3048 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - \beta_{6} + 5\beta_{4} + \beta_{2} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 10\beta_{3} + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -9\beta_{7} + 11\beta_{5} - 45\beta_{4} - 11\beta_{2} - 2\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{7} - 9\beta_{6} + 5\beta_{4} - 98\beta_{3} - 13\beta_{2} - 98\beta _1 - 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( 85\beta_{6} - 111\beta_{5} - 38\beta_{3} + 433 \)
|
| \(\nu^{7}\) | \(=\) |
\( -73\beta_{7} - 149\beta_{5} + 27\beta_{4} + 149\beta_{2} + 962\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/266\mathbb{Z}\right)^\times\).
| \(n\) | \(115\) | \(211\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 197.1 |
|
0.500000 | + | 0.866025i | −1.51459 | − | 2.62334i | −0.500000 | + | 0.866025i | 1.68446 | + | 2.91758i | 1.51459 | − | 2.62334i | 1.00000 | −1.00000 | −3.08794 | + | 5.34848i | −1.68446 | + | 2.91758i | ||||||||||||||||||||||||||||
| 197.2 | 0.500000 | + | 0.866025i | −0.176135 | − | 0.305076i | −0.500000 | + | 0.866025i | −2.16259 | − | 3.74571i | 0.176135 | − | 0.305076i | 1.00000 | −1.00000 | 1.43795 | − | 2.49061i | 2.16259 | − | 3.74571i | |||||||||||||||||||||||||||||
| 197.3 | 0.500000 | + | 0.866025i | 0.582831 | + | 1.00949i | −0.500000 | + | 0.866025i | 0.775050 | + | 1.34243i | −0.582831 | + | 1.00949i | 1.00000 | −1.00000 | 0.820616 | − | 1.42135i | −0.775050 | + | 1.34243i | |||||||||||||||||||||||||||||
| 197.4 | 0.500000 | + | 0.866025i | 1.60789 | + | 2.78495i | −0.500000 | + | 0.866025i | −0.796924 | − | 1.38031i | −1.60789 | + | 2.78495i | 1.00000 | −1.00000 | −3.67062 | + | 6.35771i | 0.796924 | − | 1.38031i | |||||||||||||||||||||||||||||
| 239.1 | 0.500000 | − | 0.866025i | −1.51459 | + | 2.62334i | −0.500000 | − | 0.866025i | 1.68446 | − | 2.91758i | 1.51459 | + | 2.62334i | 1.00000 | −1.00000 | −3.08794 | − | 5.34848i | −1.68446 | − | 2.91758i | |||||||||||||||||||||||||||||
| 239.2 | 0.500000 | − | 0.866025i | −0.176135 | + | 0.305076i | −0.500000 | − | 0.866025i | −2.16259 | + | 3.74571i | 0.176135 | + | 0.305076i | 1.00000 | −1.00000 | 1.43795 | + | 2.49061i | 2.16259 | + | 3.74571i | |||||||||||||||||||||||||||||
| 239.3 | 0.500000 | − | 0.866025i | 0.582831 | − | 1.00949i | −0.500000 | − | 0.866025i | 0.775050 | − | 1.34243i | −0.582831 | − | 1.00949i | 1.00000 | −1.00000 | 0.820616 | + | 1.42135i | −0.775050 | − | 1.34243i | |||||||||||||||||||||||||||||
| 239.4 | 0.500000 | − | 0.866025i | 1.60789 | − | 2.78495i | −0.500000 | − | 0.866025i | −0.796924 | + | 1.38031i | −1.60789 | − | 2.78495i | 1.00000 | −1.00000 | −3.67062 | − | 6.35771i | 0.796924 | + | 1.38031i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 266.2.f.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 2394.2.o.v | 8 | ||
| 19.c | even | 3 | 1 | inner | 266.2.f.d | ✓ | 8 |
| 19.c | even | 3 | 1 | 5054.2.a.w | 4 | ||
| 19.d | odd | 6 | 1 | 5054.2.a.x | 4 | ||
| 57.h | odd | 6 | 1 | 2394.2.o.v | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 266.2.f.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 266.2.f.d | ✓ | 8 | 19.c | even | 3 | 1 | inner |
| 2394.2.o.v | 8 | 3.b | odd | 2 | 1 | ||
| 2394.2.o.v | 8 | 57.h | odd | 6 | 1 | ||
| 5054.2.a.w | 4 | 19.c | even | 3 | 1 | ||
| 5054.2.a.x | 4 | 19.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - T_{3}^{7} + 11T_{3}^{6} - 6T_{3}^{5} + 104T_{3}^{4} - 72T_{3}^{3} + 104T_{3}^{2} + 32T_{3} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(266, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{4} \)
|
| $3$ |
\( T^{8} - T^{7} + \cdots + 16 \)
|
| $5$ |
\( T^{8} + T^{7} + \cdots + 1296 \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( (T^{4} - 7 T^{3} + 8 T^{2} + \cdots - 12)^{2} \)
|
| $13$ |
\( T^{8} - 5 T^{7} + \cdots + 9604 \)
|
| $17$ |
\( T^{8} + 2 T^{7} + \cdots + 576 \)
|
| $19$ |
\( T^{8} - 6 T^{7} + \cdots + 130321 \)
|
| $23$ |
\( T^{8} + 5 T^{7} + \cdots + 213444 \)
|
| $29$ |
\( T^{8} + 4 T^{7} + \cdots + 2304 \)
|
| $31$ |
\( (T^{4} + 6 T^{3} + \cdots - 328)^{2} \)
|
| $37$ |
\( (T^{4} - 10 T^{3} + \cdots + 288)^{2} \)
|
| $41$ |
\( T^{8} - 17 T^{7} + \cdots + 17424 \)
|
| $43$ |
\( T^{8} + 18 T^{7} + \cdots + 256 \)
|
| $47$ |
\( T^{8} - 4 T^{7} + \cdots + 2304 \)
|
| $53$ |
\( T^{8} - 10 T^{7} + \cdots + 5184 \)
|
| $59$ |
\( T^{8} - 20 T^{7} + \cdots + 103041 \)
|
| $61$ |
\( T^{8} + 9 T^{7} + \cdots + 6512704 \)
|
| $67$ |
\( T^{8} - 7 T^{7} + \cdots + 256 \)
|
| $71$ |
\( T^{8} + 21 T^{7} + \cdots + 4356 \)
|
| $73$ |
\( T^{8} + 21 T^{7} + \cdots + 70425664 \)
|
| $79$ |
\( T^{8} + 8 T^{7} + \cdots + 30294016 \)
|
| $83$ |
\( (T^{4} + 12 T^{3} + \cdots - 11061)^{2} \)
|
| $89$ |
\( T^{8} + 34 T^{7} + \cdots + 4460544 \)
|
| $97$ |
\( T^{8} + 5 T^{7} + \cdots + 150544 \)
|
| show more | |
| show less |