Properties

Label 832.2.i.g
Level 832832
Weight 22
Character orbit 832.i
Analytic conductor 6.6446.644
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [832,2,Mod(321,832)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(832, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("832.321");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 832=2613 832 = 2^{6} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 832.i (of order 33, degree 22, 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: 6.643553448176.64355344817
Analytic rank: 00
Dimension: 22
Coefficient field: Q(3)\Q(\sqrt{-3})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x+1 x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a primitive root of unity ζ6\zeta_{6}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(ζ6+1)q32q5+ζ6q7+2ζ6q9+(ζ6+1)q11+(4ζ61)q13+(2ζ62)q153ζ6q17+7ζ6q19++2q99+O(q100) q + ( - \zeta_{6} + 1) q^{3} - 2 q^{5} + \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{11} + (4 \zeta_{6} - 1) q^{13} + (2 \zeta_{6} - 2) q^{15} - 3 \zeta_{6} q^{17} + 7 \zeta_{6} q^{19} + \cdots + 2 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+q34q5+q7+2q9+q11+2q132q153q17+7q19+2q21q232q25+10q27+3q29+16q31q332q35q37+7q39++4q99+O(q100) 2 q + q^{3} - 4 q^{5} + q^{7} + 2 q^{9} + q^{11} + 2 q^{13} - 2 q^{15} - 3 q^{17} + 7 q^{19} + 2 q^{21} - q^{23} - 2 q^{25} + 10 q^{27} + 3 q^{29} + 16 q^{31} - q^{33} - 2 q^{35} - q^{37} + 7 q^{39}+ \cdots + 4 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/832Z)×\left(\mathbb{Z}/832\mathbb{Z}\right)^\times.

nn 261261 703703 769769
χ(n)\chi(n) 11 11 ζ6-\zeta_{6}

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
321.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 −2.00000 0 0.500000 + 0.866025i 0 1.00000 + 1.73205i 0
705.1 0 0.500000 + 0.866025i 0 −2.00000 0 0.500000 0.866025i 0 1.00000 1.73205i 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 832.2.i.g 2
4.b odd 2 1 832.2.i.d 2
8.b even 2 1 104.2.i.a 2
8.d odd 2 1 208.2.i.c 2
13.c even 3 1 inner 832.2.i.g 2
24.f even 2 1 1872.2.t.d 2
24.h odd 2 1 936.2.t.c 2
52.j odd 6 1 832.2.i.d 2
104.e even 2 1 1352.2.i.a 2
104.j odd 4 2 1352.2.o.b 4
104.n odd 6 1 208.2.i.c 2
104.n odd 6 1 2704.2.a.e 1
104.p odd 6 1 2704.2.a.c 1
104.r even 6 1 104.2.i.a 2
104.r even 6 1 1352.2.a.c 1
104.s even 6 1 1352.2.a.a 1
104.s even 6 1 1352.2.i.a 2
104.u even 12 2 2704.2.f.c 2
104.x odd 12 2 1352.2.f.a 2
104.x odd 12 2 1352.2.o.b 4
312.bh odd 6 1 936.2.t.c 2
312.bn even 6 1 1872.2.t.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.i.a 2 8.b even 2 1
104.2.i.a 2 104.r even 6 1
208.2.i.c 2 8.d odd 2 1
208.2.i.c 2 104.n odd 6 1
832.2.i.d 2 4.b odd 2 1
832.2.i.d 2 52.j odd 6 1
832.2.i.g 2 1.a even 1 1 trivial
832.2.i.g 2 13.c even 3 1 inner
936.2.t.c 2 24.h odd 2 1
936.2.t.c 2 312.bh odd 6 1
1352.2.a.a 1 104.s even 6 1
1352.2.a.c 1 104.r even 6 1
1352.2.f.a 2 104.x odd 12 2
1352.2.i.a 2 104.e even 2 1
1352.2.i.a 2 104.s even 6 1
1352.2.o.b 4 104.j odd 4 2
1352.2.o.b 4 104.x odd 12 2
1872.2.t.d 2 24.f even 2 1
1872.2.t.d 2 312.bn even 6 1
2704.2.a.c 1 104.p odd 6 1
2704.2.a.e 1 104.n odd 6 1
2704.2.f.c 2 104.u even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(832,[χ])S_{2}^{\mathrm{new}}(832, [\chi]):

T32T3+1 T_{3}^{2} - T_{3} + 1 Copy content Toggle raw display
T5+2 T_{5} + 2 Copy content Toggle raw display
T72T7+1 T_{7}^{2} - T_{7} + 1 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
55 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
77 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
1111 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
1313 T22T+13 T^{2} - 2T + 13 Copy content Toggle raw display
1717 T2+3T+9 T^{2} + 3T + 9 Copy content Toggle raw display
1919 T27T+49 T^{2} - 7T + 49 Copy content Toggle raw display
2323 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
2929 T23T+9 T^{2} - 3T + 9 Copy content Toggle raw display
3131 (T8)2 (T - 8)^{2} Copy content Toggle raw display
3737 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
4141 T2+11T+121 T^{2} + 11T + 121 Copy content Toggle raw display
4343 T211T+121 T^{2} - 11T + 121 Copy content Toggle raw display
4747 (T12)2 (T - 12)^{2} Copy content Toggle raw display
5353 (T6)2 (T - 6)^{2} Copy content Toggle raw display
5959 T2+9T+81 T^{2} + 9T + 81 Copy content Toggle raw display
6161 T2+9T+81 T^{2} + 9T + 81 Copy content Toggle raw display
6767 T2+3T+9 T^{2} + 3T + 9 Copy content Toggle raw display
7171 T25T+25 T^{2} - 5T + 25 Copy content Toggle raw display
7373 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
7979 (T+12)2 (T + 12)^{2} Copy content Toggle raw display
8383 (T4)2 (T - 4)^{2} Copy content Toggle raw display
8989 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
9797 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
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