Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [833,2,Mod(50,833)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(833, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("833.50");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 833 = 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 833.b (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: | \(6.65153848837\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 15x^{8} + 67x^{6} + 108x^{4} + 58x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 119) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 15x^{8} + 67x^{6} + 108x^{4} + 58x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{9} + 11\nu^{7} + 13\nu^{5} - 79\nu^{3} - 96\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{8} + 54\nu^{6} + 187\nu^{4} + 149\nu^{2} - 9 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{9} - 54\nu^{7} - 187\nu^{5} - 149\nu^{3} + 14\nu ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{9} - 92\nu^{7} - 291\nu^{5} - 117\nu^{3} + 167\nu ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 9\nu^{9} + 124\nu^{7} + 452\nu^{5} + 429\nu^{3} + 41\nu ) / 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -9\nu^{8} - 124\nu^{6} - 452\nu^{4} - 424\nu^{2} - 16 ) / 5 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -11\nu^{8} - 151\nu^{6} - 543\nu^{4} - 481\nu^{2} - 4 ) / 5 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13\nu^{8} + 178\nu^{6} + 639\nu^{4} + 578\nu^{2} + 22 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} + \beta_{7} - \beta_{3} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{2} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{9} + 2\beta_{8} - 9\beta_{7} + 8\beta_{3} + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( -11\beta_{6} - 9\beta_{5} - 14\beta_{4} - 20\beta_{2} + 46\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 52\beta_{9} - 25\beta_{8} + 75\beta_{7} - 69\beta_{3} - 133 \)
|
| \(\nu^{7}\) | \(=\) |
\( 102\beta_{6} + 75\beta_{5} + 142\beta_{4} + 175\beta_{2} - 379\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -412\beta_{9} + 244\beta_{8} - 629\beta_{7} + 596\beta_{3} + 1068 \)
|
| \(\nu^{9}\) | \(=\) |
\( -900\beta_{6} - 629\beta_{5} - 1301\beta_{4} - 1502\beta_{2} + 3193\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/833\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(785\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 50.1 |
|
−2.31941 | − | 0.133516i | 3.37966 | − | 2.52580i | 0.309679i | 0 | −3.19999 | 2.98217 | 5.85836i | ||||||||||||||||||||||||||||||||||||||||||||||
| 50.2 | −2.31941 | 0.133516i | 3.37966 | 2.52580i | − | 0.309679i | 0 | −3.19999 | 2.98217 | − | 5.85836i | |||||||||||||||||||||||||||||||||||||||||||||||
| 50.3 | −1.11644 | − | 1.95156i | −0.753562 | − | 1.49881i | 2.17880i | 0 | 3.07419 | −0.808579 | 1.67333i | |||||||||||||||||||||||||||||||||||||||||||||||
| 50.4 | −1.11644 | 1.95156i | −0.753562 | 1.49881i | − | 2.17880i | 0 | 3.07419 | −0.808579 | − | 1.67333i | |||||||||||||||||||||||||||||||||||||||||||||||
| 50.5 | 0.548857 | − | 1.12498i | −1.69876 | 1.14072i | − | 0.617451i | 0 | −2.03009 | 1.73443 | 0.626093i | |||||||||||||||||||||||||||||||||||||||||||||||
| 50.6 | 0.548857 | 1.12498i | −1.69876 | − | 1.14072i | 0.617451i | 0 | −2.03009 | 1.73443 | − | 0.626093i | |||||||||||||||||||||||||||||||||||||||||||||||
| 50.7 | 1.43442 | − | 2.92340i | 0.0575680 | − | 1.74859i | − | 4.19339i | 0 | −2.78627 | −5.54624 | − | 2.50822i | |||||||||||||||||||||||||||||||||||||||||||||
| 50.8 | 1.43442 | 2.92340i | 0.0575680 | 1.74859i | 4.19339i | 0 | −2.78627 | −5.54624 | 2.50822i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 50.9 | 2.45257 | − | 1.16695i | 4.01509 | 2.64860i | − | 2.86203i | 0 | 4.94216 | 1.63822 | 6.49588i | |||||||||||||||||||||||||||||||||||||||||||||||
| 50.10 | 2.45257 | 1.16695i | 4.01509 | − | 2.64860i | 2.86203i | 0 | 4.94216 | 1.63822 | − | 6.49588i | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 833.2.b.c | 10 | |
| 7.b | odd | 2 | 1 | 833.2.b.d | 10 | ||
| 7.c | even | 3 | 2 | 119.2.j.a | ✓ | 20 | |
| 7.d | odd | 6 | 2 | 833.2.j.a | 20 | ||
| 17.b | even | 2 | 1 | inner | 833.2.b.c | 10 | |
| 119.d | odd | 2 | 1 | 833.2.b.d | 10 | ||
| 119.h | odd | 6 | 2 | 833.2.j.a | 20 | ||
| 119.j | even | 6 | 2 | 119.2.j.a | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 119.2.j.a | ✓ | 20 | 7.c | even | 3 | 2 | |
| 119.2.j.a | ✓ | 20 | 119.j | even | 6 | 2 | |
| 833.2.b.c | 10 | 1.a | even | 1 | 1 | trivial | |
| 833.2.b.c | 10 | 17.b | even | 2 | 1 | inner | |
| 833.2.b.d | 10 | 7.b | odd | 2 | 1 | ||
| 833.2.b.d | 10 | 119.d | odd | 2 | 1 | ||
| 833.2.j.a | 20 | 7.d | odd | 6 | 2 | ||
| 833.2.j.a | 20 | 119.h | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(833, [\chi])\):
|
\( T_{2}^{5} - T_{2}^{4} - 7T_{2}^{3} + 6T_{2}^{2} + 8T_{2} - 5 \)
|
|
\( T_{13}^{5} + 4T_{13}^{4} - 24T_{13}^{3} - 59T_{13}^{2} + 119T_{13} + 245 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{5} - T^{4} - 7 T^{3} + \cdots - 5)^{2} \)
|
| $3$ |
\( T^{10} + 15 T^{8} + \cdots + 1 \)
|
| $5$ |
\( T^{10} + 20 T^{8} + \cdots + 400 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( T^{10} + 44 T^{8} + \cdots + 2401 \)
|
| $13$ |
\( (T^{5} + 4 T^{4} + \cdots + 245)^{2} \)
|
| $17$ |
\( T^{10} - 2 T^{9} + \cdots + 1419857 \)
|
| $19$ |
\( (T^{5} - 5 T^{4} + \cdots - 392)^{2} \)
|
| $23$ |
\( T^{10} + 156 T^{8} + \cdots + 614656 \)
|
| $29$ |
\( T^{10} + 175 T^{8} + \cdots + 24010000 \)
|
| $31$ |
\( T^{10} + 92 T^{8} + \cdots + 232324 \)
|
| $37$ |
\( T^{10} + 149 T^{8} + \cdots + 3841600 \)
|
| $41$ |
\( T^{10} + 182 T^{8} + \cdots + 2972176 \)
|
| $43$ |
\( (T^{5} + 13 T^{4} + \cdots - 1892)^{2} \)
|
| $47$ |
\( (T^{5} + 15 T^{4} + \cdots - 1568)^{2} \)
|
| $53$ |
\( (T^{5} + 20 T^{4} + \cdots - 179)^{2} \)
|
| $59$ |
\( (T^{5} - 7 T^{4} + \cdots - 3724)^{2} \)
|
| $61$ |
\( T^{10} + 320 T^{8} + \cdots + 25600 \)
|
| $67$ |
\( (T^{5} + 6 T^{4} - 5 T^{3} + \cdots + 16)^{2} \)
|
| $71$ |
\( T^{10} + \cdots + 782824441 \)
|
| $73$ |
\( T^{10} + 147 T^{8} + \cdots + 6130576 \)
|
| $79$ |
\( T^{10} + 421 T^{8} + \cdots + 866761 \)
|
| $83$ |
\( (T^{5} - 24 T^{4} + \cdots + 7448)^{2} \)
|
| $89$ |
\( (T^{5} + 8 T^{4} + \cdots - 392)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 736579600 \)
|
| show more | |
| show less |