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Properties

Label 833.2.o.c
Level $833$
Weight $2$
Character orbit 833.o
Analytic conductor $6.652$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [833,2,Mod(30,833)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(833, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("833.30");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.o (of order \(12\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 119)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_1 q^{3} + \beta_{4} q^{4} + ( - \beta_{4} + \beta_{2} - 1) q^{5} + (\beta_{7} + 1) q^{6} - 3 \beta_{5} q^{8} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_1 q^{3} + \beta_{4} q^{4} + ( - \beta_{4} + \beta_{2} - 1) q^{5} + (\beta_{7} + 1) q^{6} - 3 \beta_{5} q^{8} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots + 1) q^{9}+ \cdots + ( - 2 \beta_{6} + \beta_{5} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 4 q^{4} - 4 q^{5} + 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 4 q^{4} - 4 q^{5} + 4 q^{6} + 4 q^{10} + 10 q^{11} + 2 q^{12} + 4 q^{16} + 16 q^{17} + 12 q^{18} + 8 q^{20} - 20 q^{22} + 12 q^{23} - 6 q^{24} + 44 q^{27} + 16 q^{29} - 4 q^{30} - 8 q^{31} + 12 q^{33} + 8 q^{34} + 4 q^{38} - 22 q^{39} + 12 q^{40} + 16 q^{41} + 10 q^{44} + 12 q^{45} + 12 q^{46} + 4 q^{47} + 4 q^{48} + 24 q^{50} - 6 q^{51} - 22 q^{54} - 40 q^{55} + 48 q^{57} + 8 q^{58} - 20 q^{61} - 16 q^{62} - 56 q^{64} + 4 q^{67} + 16 q^{68} + 24 q^{69} - 12 q^{71} + 36 q^{72} - 20 q^{73} + 6 q^{75} - 44 q^{78} - 2 q^{79} + 4 q^{80} + 8 q^{81} - 8 q^{82} - 24 q^{85} + 8 q^{86} + 30 q^{88} + 56 q^{89} - 24 q^{90} - 24 q^{92} + 4 q^{95} - 10 q^{96} + 8 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 144\nu + 45 ) / 144 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 32\nu^{4} - 16\nu^{2} + 144\nu - 45 ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 8\nu^{3} + 81\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 9\nu^{6} + 16\nu^{5} + 80\nu^{3} + 81\nu + 117 ) / 144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25\nu^{7} - 27\nu^{6} + 80\nu^{5} + 256\nu^{3} + 405\nu - 351 ) / 432 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} - 4\beta_{4} + \beta_{3} - \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{4} - 5\beta_{3} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + 30\beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{7} + 8\beta_{6} + 8\beta_{5} - 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 96\beta_{5} - 13\beta_{3} - 96\beta_{2} - 13\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(52\) \(785\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1
−0.396143 − 1.68614i
1.26217 + 1.18614i
−0.396143 + 1.68614i
1.26217 − 1.18614i
0.396143 − 1.68614i
−1.26217 + 1.18614i
0.396143 + 1.68614i
−1.26217 − 1.18614i
−0.866025 − 0.500000i −0.789997 + 2.94831i −0.500000 − 0.866025i −1.36603 + 0.366025i 2.15831 − 2.15831i 0 3.00000i −5.47036 − 3.15831i 1.36603 + 0.366025i
30.2 −0.866025 − 0.500000i 0.423972 − 1.58228i −0.500000 − 0.866025i −1.36603 + 0.366025i −1.15831 + 1.15831i 0 3.00000i 0.274205 + 0.158312i 1.36603 + 0.366025i
361.1 −0.866025 + 0.500000i −0.789997 − 2.94831i −0.500000 + 0.866025i −1.36603 − 0.366025i 2.15831 + 2.15831i 0 − 3.00000i −5.47036 + 3.15831i 1.36603 − 0.366025i
361.2 −0.866025 + 0.500000i 0.423972 + 1.58228i −0.500000 + 0.866025i −1.36603 − 0.366025i −1.15831 − 1.15831i 0 − 3.00000i 0.274205 − 0.158312i 1.36603 − 0.366025i
557.1 0.866025 − 0.500000i −1.58228 + 0.423972i −0.500000 + 0.866025i 0.366025 − 1.36603i −1.15831 + 1.15831i 0 3.00000i −0.274205 + 0.158312i −0.366025 − 1.36603i
557.2 0.866025 − 0.500000i 2.94831 − 0.789997i −0.500000 + 0.866025i 0.366025 − 1.36603i 2.15831 − 2.15831i 0 3.00000i 5.47036 − 3.15831i −0.366025 − 1.36603i
667.1 0.866025 + 0.500000i −1.58228 − 0.423972i −0.500000 − 0.866025i 0.366025 + 1.36603i −1.15831 − 1.15831i 0 − 3.00000i −0.274205 − 0.158312i −0.366025 + 1.36603i
667.2 0.866025 + 0.500000i 2.94831 + 0.789997i −0.500000 − 0.866025i 0.366025 + 1.36603i 2.15831 + 2.15831i 0 − 3.00000i 5.47036 + 3.15831i −0.366025 + 1.36603i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 30.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
17.c even 4 1 inner
119.n even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 833.2.o.c 8
7.b odd 2 1 119.2.n.a ✓ 8
7.c even 3 1 833.2.g.c 4
7.c even 3 1 inner 833.2.o.c 8
7.d odd 6 1 119.2.n.a ✓ 8
7.d odd 6 1 833.2.g.d 4
17.c even 4 1 inner 833.2.o.c 8
119.f odd 4 1 119.2.n.a ✓ 8
119.m odd 12 1 119.2.n.a ✓ 8
119.m odd 12 1 833.2.g.d 4
119.n even 12 1 833.2.g.c 4
119.n even 12 1 inner 833.2.o.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.2.n.a ✓ 8 7.b odd 2 1
119.2.n.a ✓ 8 7.d odd 6 1
119.2.n.a ✓ 8 119.f odd 4 1
119.2.n.a ✓ 8 119.m odd 12 1
833.2.g.c 4 7.c even 3 1
833.2.g.c 4 119.n even 12 1
833.2.g.d 4 7.d odd 6 1
833.2.g.d 4 119.m odd 12 1
833.2.o.c 8 1.a even 1 1 trivial
833.2.o.c 8 7.c even 3 1 inner
833.2.o.c 8 17.c even 4 1 inner
833.2.o.c 8 119.n even 12 1 inner

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} - 2T_{3}^{7} + 2T_{3}^{6} - 24T_{3}^{5} - T_{3}^{4} + 120T_{3}^{3} + 50T_{3}^{2} + 250T_{3} + 625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 10 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$13$ \( (T^{2} - 11)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 8 T^{3} + \cdots + 289)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 24 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$23$ \( (T^{4} - 6 T^{3} + \cdots + 324)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots + 196)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 8 T^{7} + \cdots + 38416 \) Copy content Toggle raw display
$37$ \( T^{8} - 7744 T^{4} + 59969536 \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} + \cdots + 196)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 96 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 2 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 106 T^{6} + \cdots + 1500625 \) Copy content Toggle raw display
$59$ \( T^{8} - 120 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$61$ \( T^{8} + 20 T^{7} + \cdots + 614656 \) Copy content Toggle raw display
$67$ \( (T^{4} - 2 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 6 T^{3} + 18 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 10 T^{3} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 2 T^{7} + \cdots + 5764801 \) Copy content Toggle raw display
$83$ \( (T^{4} + 96 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 28 T^{3} + \cdots + 34225)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} + \cdots + 38416)^{2} \) Copy content Toggle raw display
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