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Properties

Label 833.2.e.h
Level $833$
Weight $2$
Character orbit 833.e
Analytic conductor $6.652$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [833,2,Mod(18,833)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(833, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("833.18");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.e (of order \(3\), degree \(2\), not 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 9x^{8} - 10x^{7} + 44x^{6} - 49x^{5} + 99x^{4} - 20x^{3} + 31x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 119)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{8} - \beta_{7} + \beta_{3}) q^{3} + ( - \beta_{8} + 2 \beta_{5} + \beta_{3}) q^{4} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \cdots - 1) q^{5} + (2 \beta_{9} - \beta_{8} + 2 \beta_{4} + \cdots - 1) q^{6}+ \cdots + ( - 4 \beta_{9} + 2 \beta_{8} + \cdots - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} - 2 q^{3} - 10 q^{4} + 2 q^{6} + 12 q^{8} - 11 q^{9} + 4 q^{10} + 2 q^{11} - 22 q^{12} - 4 q^{13} + 16 q^{15} - 4 q^{16} + 5 q^{17} + 18 q^{18} + 6 q^{19} + 38 q^{20} + 12 q^{22} + 10 q^{23}+ \cdots - 124 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 9x^{8} - 10x^{7} + 44x^{6} - 49x^{5} + 99x^{4} - 20x^{3} + 31x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4212 \nu^{9} + 3675 \nu^{8} - 15575 \nu^{7} - 15678 \nu^{6} - 74025 \nu^{5} + 88200 \nu^{4} + \cdots + 2078070 ) / 773879 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4749 \nu^{9} + 22333 \nu^{8} - 57798 \nu^{7} + 111303 \nu^{6} - 118188 \nu^{5} + 535992 \nu^{4} + \cdots + 37908 ) / 773879 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19451 \nu^{9} + 168550 \nu^{8} - 235263 \nu^{7} + 1002431 \nu^{6} - 631225 \nu^{5} + \cdots + 1479609 ) / 2321637 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28988 \nu^{9} + 23213 \nu^{8} + 85878 \nu^{7} + 451846 \nu^{6} + 416857 \nu^{5} + 2104870 \nu^{4} + \cdots + 188058 ) / 2321637 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24776 \nu^{9} - 26888 \nu^{8} - 70303 \nu^{7} - 436168 \nu^{6} - 342832 \nu^{5} - 2193070 \nu^{4} + \cdots - 2266128 ) / 773879 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27063 \nu^{9} + 58338 \nu^{8} - 247242 \nu^{7} + 286205 \nu^{6} - 1175094 \nu^{5} + 1400112 \nu^{4} + \cdots + 86964 ) / 773879 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 108252 \nu^{9} + 233352 \nu^{8} - 988968 \nu^{7} + 1144820 \nu^{6} - 4700376 \nu^{5} + \cdots + 347856 ) / 773879 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 329980 \nu^{9} + 541460 \nu^{8} - 2773827 \nu^{7} + 2469163 \nu^{6} - 13670405 \nu^{5} + \cdots + 681951 ) / 2321637 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 3\beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - 4\beta_{7} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{6} - 13\beta_{5} - \beta_{4} + \beta_{3} - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} - 7\beta_{8} + 19\beta_{7} + \beta_{5} + 7\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{9} - 9\beta_{8} + \beta_{7} + 9\beta_{4} - 24\beta_{2} + \beta _1 + 60 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{6} - 6\beta_{5} + 18\beta_{4} - 42\beta_{3} + 93\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -60\beta_{9} + 61\beta_{8} - 13\beta_{7} + 117\beta_{6} + 285\beta_{5} - 61\beta_{3} + 117\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -121\beta_{9} + 238\beta_{8} - 462\beta_{7} - 121\beta_{4} + 14\beta_{2} - 462\beta _1 + 20 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/833\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(785\)
\(\chi(n)\) \(-1 - \beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
0.304720 + 0.527790i
−1.08840 − 1.88516i
1.16091 + 2.01076i
−0.272099 − 0.471289i
0.894862 + 1.54995i
0.304720 − 0.527790i
−1.08840 + 1.88516i
1.16091 − 2.01076i
−0.272099 + 0.471289i
0.894862 − 1.54995i
−1.24613 − 2.15837i −1.41042 + 2.44292i −2.10570 + 3.64718i 1.25944 + 2.18142i 7.03030 0 5.51141 −2.47858 − 4.29302i 3.13887 − 5.43669i
18.2 −1.18400 − 2.05075i 0.284689 − 0.493096i −1.80371 + 3.12411i −2.07732 − 3.59803i −1.34829 0 3.80636 1.33790 + 2.31732i −4.91910 + 8.52013i
18.3 −0.438917 − 0.760227i 0.453789 − 0.785986i 0.614704 − 1.06470i 1.51909 + 2.63114i −0.796703 0 −2.83488 1.08815 + 1.88473i 1.33351 − 2.30971i
18.4 0.704339 + 1.21995i 1.27991 − 2.21687i 0.00781334 − 0.0135331i 0.883300 + 1.52992i 3.60597 0 2.83937 −1.77635 − 3.07673i −1.24428 + 2.15516i
18.5 1.16471 + 2.01734i −1.60797 + 2.78508i −1.71311 + 2.96719i −1.58451 − 2.74445i −7.49128 0 −3.32226 −3.67113 − 6.35858i 3.69100 − 6.39300i
324.1 −1.24613 + 2.15837i −1.41042 − 2.44292i −2.10570 − 3.64718i 1.25944 − 2.18142i 7.03030 0 5.51141 −2.47858 + 4.29302i 3.13887 + 5.43669i
324.2 −1.18400 + 2.05075i 0.284689 + 0.493096i −1.80371 − 3.12411i −2.07732 + 3.59803i −1.34829 0 3.80636 1.33790 − 2.31732i −4.91910 − 8.52013i
324.3 −0.438917 + 0.760227i 0.453789 + 0.785986i 0.614704 + 1.06470i 1.51909 − 2.63114i −0.796703 0 −2.83488 1.08815 − 1.88473i 1.33351 + 2.30971i
324.4 0.704339 − 1.21995i 1.27991 + 2.21687i 0.00781334 + 0.0135331i 0.883300 − 1.52992i 3.60597 0 2.83937 −1.77635 + 3.07673i −1.24428 − 2.15516i
324.5 1.16471 − 2.01734i −1.60797 − 2.78508i −1.71311 − 2.96719i −1.58451 + 2.74445i −7.49128 0 −3.32226 −3.67113 + 6.35858i 3.69100 + 6.39300i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 833.2.e.h 10
7.b odd 2 1 833.2.e.i 10
7.c even 3 1 833.2.a.g 5
7.c even 3 1 inner 833.2.e.h 10
7.d odd 6 1 119.2.a.b ✓ 5
7.d odd 6 1 833.2.e.i 10
21.g even 6 1 1071.2.a.m 5
21.h odd 6 1 7497.2.a.br 5
28.f even 6 1 1904.2.a.t 5
35.i odd 6 1 2975.2.a.m 5
56.j odd 6 1 7616.2.a.bt 5
56.m even 6 1 7616.2.a.bq 5
119.h odd 6 1 2023.2.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.2.a.b ✓ 5 7.d odd 6 1
833.2.a.g 5 7.c even 3 1
833.2.e.h 10 1.a even 1 1 trivial
833.2.e.h 10 7.c even 3 1 inner
833.2.e.i 10 7.b odd 2 1
833.2.e.i 10 7.d odd 6 1
1071.2.a.m 5 21.g even 6 1
1904.2.a.t 5 28.f even 6 1
2023.2.a.j 5 119.h odd 6 1
2975.2.a.m 5 35.i odd 6 1
7497.2.a.br 5 21.h odd 6 1
7616.2.a.bq 5 56.m even 6 1
7616.2.a.bt 5 56.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(833, [\chi])\):

\( T_{2}^{10} + 2 T_{2}^{9} + 12 T_{2}^{8} + 12 T_{2}^{7} + 78 T_{2}^{6} + 73 T_{2}^{5} + 274 T_{2}^{4} + \cdots + 289 \) Copy content Toggle raw display
\( T_{3}^{10} + 2 T_{3}^{9} + 15 T_{3}^{8} + 2 T_{3}^{7} + 114 T_{3}^{6} - 4 T_{3}^{5} + 509 T_{3}^{4} + \cdots + 144 \) Copy content Toggle raw display
\( T_{5}^{10} + 23 T_{5}^{8} - 36 T_{5}^{7} + 398 T_{5}^{6} - 592 T_{5}^{5} + 3337 T_{5}^{4} - 5830 T_{5}^{3} + \cdots + 31684 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 2 T^{9} + \cdots + 289 \) Copy content Toggle raw display
$3$ \( T^{10} + 2 T^{9} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( T^{10} + 23 T^{8} + \cdots + 31684 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 2 T^{9} + \cdots + 36864 \) Copy content Toggle raw display
$13$ \( (T^{5} + 2 T^{4} + \cdots + 544)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$19$ \( T^{10} - 6 T^{9} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( T^{10} - 10 T^{9} + \cdots + 16384 \) Copy content Toggle raw display
$29$ \( (T^{5} + 8 T^{4} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 33 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{10} + 8 T^{9} + \cdots + 19219456 \) Copy content Toggle raw display
$41$ \( (T^{5} + 18 T^{4} + \cdots - 162)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 8 T^{4} + \cdots - 1052)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 10 T^{9} + \cdots + 5308416 \) Copy content Toggle raw display
$53$ \( T^{10} + 4 T^{9} + \cdots + 19044 \) Copy content Toggle raw display
$59$ \( T^{10} - 8 T^{9} + \cdots + 9437184 \) Copy content Toggle raw display
$61$ \( T^{10} - 22 T^{9} + \cdots + 30713764 \) Copy content Toggle raw display
$67$ \( T^{10} + 16 T^{9} + \cdots + 3489424 \) Copy content Toggle raw display
$71$ \( (T^{5} + 2 T^{4} + \cdots + 13696)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 123609924 \) Copy content Toggle raw display
$79$ \( T^{10} + 18 T^{9} + \cdots + 9437184 \) Copy content Toggle raw display
$83$ \( (T^{5} - 12 T^{4} + \cdots - 1984)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 20 T^{9} + \cdots + 55591936 \) Copy content Toggle raw display
$97$ \( (T^{5} + 12 T^{4} + \cdots - 218)^{2} \) Copy content Toggle raw display
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