Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [833,2,Mod(1,833)]
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, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("833.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 833 = 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 833.a (trivial) |
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: | yes |
| Analytic conductor: | \(6.65153848837\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.453749.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 5x^{3} + 10x^{2} + x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 119) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 5x^{3} + 10x^{2} + x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 4\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + 13 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.32942 | 3.21594 | 3.42621 | 3.16902 | −7.49128 | 0 | −3.32226 | 7.34225 | −7.38200 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.40868 | −2.55982 | −0.0156267 | −1.76660 | 3.60597 | 0 | 2.83937 | 3.55270 | 2.48857 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.877834 | −0.907578 | −1.22941 | −3.03818 | −0.796703 | 0 | −2.83488 | −2.17630 | −2.66702 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.36800 | −0.569378 | 3.60742 | 4.15465 | −1.34829 | 0 | 3.80636 | −2.67581 | 9.83819 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.49227 | 2.82084 | 4.21140 | −2.51889 | 7.03030 | 0 | 5.51141 | 4.95716 | −6.27775 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
| \(17\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 833.2.a.g | 5 | |
| 3.b | odd | 2 | 1 | 7497.2.a.br | 5 | ||
| 7.b | odd | 2 | 1 | 119.2.a.b | ✓ | 5 | |
| 7.c | even | 3 | 2 | 833.2.e.h | 10 | ||
| 7.d | odd | 6 | 2 | 833.2.e.i | 10 | ||
| 21.c | even | 2 | 1 | 1071.2.a.m | 5 | ||
| 28.d | even | 2 | 1 | 1904.2.a.t | 5 | ||
| 35.c | odd | 2 | 1 | 2975.2.a.m | 5 | ||
| 56.e | even | 2 | 1 | 7616.2.a.bq | 5 | ||
| 56.h | odd | 2 | 1 | 7616.2.a.bt | 5 | ||
| 119.d | odd | 2 | 1 | 2023.2.a.j | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 119.2.a.b | ✓ | 5 | 7.b | odd | 2 | 1 | |
| 833.2.a.g | 5 | 1.a | even | 1 | 1 | trivial | |
| 833.2.e.h | 10 | 7.c | even | 3 | 2 | ||
| 833.2.e.i | 10 | 7.d | odd | 6 | 2 | ||
| 1071.2.a.m | 5 | 21.c | even | 2 | 1 | ||
| 1904.2.a.t | 5 | 28.d | even | 2 | 1 | ||
| 2023.2.a.j | 5 | 119.d | odd | 2 | 1 | ||
| 2975.2.a.m | 5 | 35.c | odd | 2 | 1 | ||
| 7497.2.a.br | 5 | 3.b | odd | 2 | 1 | ||
| 7616.2.a.bq | 5 | 56.e | even | 2 | 1 | ||
| 7616.2.a.bt | 5 | 56.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(833))\):
|
\( T_{2}^{5} - 2T_{2}^{4} - 8T_{2}^{3} + 14T_{2}^{2} + 14T_{2} - 17 \)
|
|
\( T_{3}^{5} - 2T_{3}^{4} - 11T_{3}^{3} + 12T_{3}^{2} + 31T_{3} + 12 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 17 \)
|
| $3$ |
\( T^{5} - 2 T^{4} + \cdots + 12 \)
|
| $5$ |
\( T^{5} - 23 T^{3} + \cdots + 178 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} + 2 T^{4} + \cdots - 192 \)
|
| $13$ |
\( T^{5} + 2 T^{4} + \cdots + 544 \)
|
| $17$ |
\( (T + 1)^{5} \)
|
| $19$ |
\( T^{5} + 6 T^{4} + \cdots + 64 \)
|
| $23$ |
\( T^{5} + 10 T^{4} + \cdots - 128 \)
|
| $29$ |
\( T^{5} + 8 T^{4} + \cdots + 2592 \)
|
| $31$ |
\( T^{5} - 33 T^{3} + \cdots + 16 \)
|
| $37$ |
\( T^{5} - 8 T^{4} + \cdots + 4384 \)
|
| $41$ |
\( T^{5} + 18 T^{4} + \cdots - 162 \)
|
| $43$ |
\( T^{5} - 8 T^{4} + \cdots - 1052 \)
|
| $47$ |
\( T^{5} - 10 T^{4} + \cdots + 2304 \)
|
| $53$ |
\( T^{5} - 4 T^{4} + \cdots + 138 \)
|
| $59$ |
\( T^{5} + 8 T^{4} + \cdots + 3072 \)
|
| $61$ |
\( T^{5} + 22 T^{4} + \cdots - 5542 \)
|
| $67$ |
\( T^{5} - 16 T^{4} + \cdots + 1868 \)
|
| $71$ |
\( T^{5} + 2 T^{4} + \cdots + 13696 \)
|
| $73$ |
\( T^{5} + 10 T^{4} + \cdots + 11118 \)
|
| $79$ |
\( T^{5} - 18 T^{4} + \cdots + 3072 \)
|
| $83$ |
\( T^{5} - 12 T^{4} + \cdots - 1984 \)
|
| $89$ |
\( T^{5} + 20 T^{4} + \cdots - 7456 \)
|
| $97$ |
\( T^{5} + 12 T^{4} + \cdots - 218 \)
|
| show more | |
| show less |