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: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.9301.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} + x + 3 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 5x^{2} + x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 3 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.60312 | −1.45066 | 4.77625 | 0.450660 | 3.77625 | 0 | −7.22690 | −0.895586 | −1.17312 | ||||||||||||||||||||||||||||||
| 1.2 | −0.869986 | 2.57835 | −1.24312 | −3.57835 | −2.24312 | 0 | 2.82147 | 3.64787 | 3.11311 | |||||||||||||||||||||||||||||||
| 1.3 | 0.784476 | −3.03973 | −1.38460 | 2.03973 | −2.38460 | 0 | −2.65514 | 6.23998 | 1.60012 | |||||||||||||||||||||||||||||||
| 1.4 | 1.68863 | −0.0879544 | 0.851477 | −0.912046 | −0.148523 | 0 | −1.93943 | −2.99226 | −1.54011 | |||||||||||||||||||||||||||||||
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.e | 4 | |
| 3.b | odd | 2 | 1 | 7497.2.a.bl | 4 | ||
| 7.b | odd | 2 | 1 | 119.2.a.a | ✓ | 4 | |
| 7.c | even | 3 | 2 | 833.2.e.f | 8 | ||
| 7.d | odd | 6 | 2 | 833.2.e.e | 8 | ||
| 21.c | even | 2 | 1 | 1071.2.a.k | 4 | ||
| 28.d | even | 2 | 1 | 1904.2.a.s | 4 | ||
| 35.c | odd | 2 | 1 | 2975.2.a.k | 4 | ||
| 56.e | even | 2 | 1 | 7616.2.a.bn | 4 | ||
| 56.h | odd | 2 | 1 | 7616.2.a.bk | 4 | ||
| 119.d | odd | 2 | 1 | 2023.2.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 119.2.a.a | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 833.2.a.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 833.2.e.e | 8 | 7.d | odd | 6 | 2 | ||
| 833.2.e.f | 8 | 7.c | even | 3 | 2 | ||
| 1071.2.a.k | 4 | 21.c | even | 2 | 1 | ||
| 1904.2.a.s | 4 | 28.d | even | 2 | 1 | ||
| 2023.2.a.e | 4 | 119.d | odd | 2 | 1 | ||
| 2975.2.a.k | 4 | 35.c | odd | 2 | 1 | ||
| 7497.2.a.bl | 4 | 3.b | odd | 2 | 1 | ||
| 7616.2.a.bk | 4 | 56.h | odd | 2 | 1 | ||
| 7616.2.a.bn | 4 | 56.e | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(833))\):
|
\( T_{2}^{4} + T_{2}^{3} - 5T_{2}^{2} - T_{2} + 3 \)
|
|
\( T_{3}^{4} + 2T_{3}^{3} - 7T_{3}^{2} - 12T_{3} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} - 5 T^{2} + \cdots + 3 \)
|
| $3$ |
\( T^{4} + 2 T^{3} + \cdots - 1 \)
|
| $5$ |
\( T^{4} + 2 T^{3} + \cdots + 3 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 2 T^{3} + \cdots + 48 \)
|
| $13$ |
\( T^{4} + 8 T^{3} + \cdots - 368 \)
|
| $17$ |
\( (T - 1)^{4} \)
|
| $19$ |
\( T^{4} + 10 T^{3} + \cdots - 784 \)
|
| $23$ |
\( T^{4} + 6 T^{3} + \cdots - 240 \)
|
| $29$ |
\( T^{4} - 2 T^{3} + \cdots + 48 \)
|
| $31$ |
\( T^{4} + 12 T^{3} + \cdots - 917 \)
|
| $37$ |
\( T^{4} - 6 T^{3} + \cdots + 80 \)
|
| $41$ |
\( T^{4} + 12 T^{3} + \cdots - 237 \)
|
| $43$ |
\( T^{4} + 12 T^{3} + \cdots - 115 \)
|
| $47$ |
\( T^{4} + 2 T^{3} + \cdots + 1776 \)
|
| $53$ |
\( T^{4} + 26 T^{3} + \cdots + 801 \)
|
| $59$ |
\( T^{4} - 4 T^{3} + \cdots - 768 \)
|
| $61$ |
\( T^{4} + 12 T^{3} + \cdots + 6451 \)
|
| $67$ |
\( T^{4} + 12 T^{3} + \cdots + 1949 \)
|
| $71$ |
\( T^{4} + 14 T^{3} + \cdots - 3312 \)
|
| $73$ |
\( T^{4} + 20 T^{3} + \cdots + 131 \)
|
| $79$ |
\( T^{4} + 14 T^{3} + \cdots - 400 \)
|
| $83$ |
\( T^{4} - 28 T^{3} + \cdots + 1200 \)
|
| $89$ |
\( T^{4} - 10 T^{3} + \cdots + 720 \)
|
| $97$ |
\( T^{4} + 26 T^{3} + \cdots - 1901 \)
|
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