Properties

Label 833.2.e.b
Level 833833
Weight 22
Character orbit 833.e
Analytic conductor 6.6526.652
Analytic rank 00
Dimension 22
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [833,2,Mod(18,833)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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 N == 833=7217 833 = 7^{2} \cdot 17
Weight: k k == 2 2
Character orbit: [χ][\chi] == 833.e (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,0,1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 6.651538488376.65153848837
Analytic rank: 00
Dimension: 22
Coefficient field: Q(3)\Q(\sqrt{-3})
Copy content comment:defining polynomial
 
Copy content 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]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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+ζ6q2+(ζ6+1)q4+2ζ6q5+3q8+3ζ6q9+(2ζ62)q102q13+ζ6q16+(ζ61)q17+(3ζ63)q18++2q97+O(q100) q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + 2 \zeta_{6} q^{5} + 3 q^{8} + 3 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{10} - 2 q^{13} + \zeta_{6} q^{16} + (\zeta_{6} - 1) q^{17} + (3 \zeta_{6} - 3) q^{18}+ \cdots + 2 q^{97}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+q2+q4+2q5+6q8+3q92q104q13+q16q173q18+4q19+4q204q23+q252q26+12q294q31+5q322q34++4q97+O(q100) 2 q + q^{2} + q^{4} + 2 q^{5} + 6 q^{8} + 3 q^{9} - 2 q^{10} - 4 q^{13} + q^{16} - q^{17} - 3 q^{18} + 4 q^{19} + 4 q^{20} - 4 q^{23} + q^{25} - 2 q^{26} + 12 q^{29} - 4 q^{31} + 5 q^{32} - 2 q^{34}+ \cdots + 4 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

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

nn 5252 785785
χ(n)\chi(n) ζ6-\zeta_{6} 11

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
18.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000 + 1.73205i 0 0 3.00000 1.50000 + 2.59808i −1.00000 + 1.73205i
324.1 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 1.73205i 0 0 3.00000 1.50000 2.59808i −1.00000 1.73205i
nn: e.g. 2-40 or 990-1000
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.b 2
7.b odd 2 1 833.2.e.a 2
7.c even 3 1 17.2.a.a 1
7.c even 3 1 inner 833.2.e.b 2
7.d odd 6 1 833.2.a.a 1
7.d odd 6 1 833.2.e.a 2
21.g even 6 1 7497.2.a.l 1
21.h odd 6 1 153.2.a.c 1
28.g odd 6 1 272.2.a.b 1
35.j even 6 1 425.2.a.d 1
35.l odd 12 2 425.2.b.b 2
56.k odd 6 1 1088.2.a.h 1
56.p even 6 1 1088.2.a.i 1
77.h odd 6 1 2057.2.a.e 1
84.n even 6 1 2448.2.a.o 1
91.r even 6 1 2873.2.a.c 1
105.o odd 6 1 3825.2.a.d 1
119.j even 6 1 289.2.a.a 1
119.n even 12 2 289.2.b.a 2
119.q even 24 4 289.2.c.a 4
119.t odd 48 8 289.2.d.d 8
133.r odd 6 1 6137.2.a.b 1
140.p odd 6 1 6800.2.a.n 1
161.f odd 6 1 8993.2.a.a 1
168.s odd 6 1 9792.2.a.n 1
168.v even 6 1 9792.2.a.i 1
357.q odd 6 1 2601.2.a.g 1
476.o odd 6 1 4624.2.a.d 1
595.z even 6 1 7225.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 7.c even 3 1
153.2.a.c 1 21.h odd 6 1
272.2.a.b 1 28.g odd 6 1
289.2.a.a 1 119.j even 6 1
289.2.b.a 2 119.n even 12 2
289.2.c.a 4 119.q even 24 4
289.2.d.d 8 119.t odd 48 8
425.2.a.d 1 35.j even 6 1
425.2.b.b 2 35.l odd 12 2
833.2.a.a 1 7.d odd 6 1
833.2.e.a 2 7.b odd 2 1
833.2.e.a 2 7.d odd 6 1
833.2.e.b 2 1.a even 1 1 trivial
833.2.e.b 2 7.c even 3 1 inner
1088.2.a.h 1 56.k odd 6 1
1088.2.a.i 1 56.p even 6 1
2057.2.a.e 1 77.h odd 6 1
2448.2.a.o 1 84.n even 6 1
2601.2.a.g 1 357.q odd 6 1
2873.2.a.c 1 91.r even 6 1
3825.2.a.d 1 105.o odd 6 1
4624.2.a.d 1 476.o odd 6 1
6137.2.a.b 1 133.r odd 6 1
6800.2.a.n 1 140.p odd 6 1
7225.2.a.g 1 595.z even 6 1
7497.2.a.l 1 21.g even 6 1
8993.2.a.a 1 161.f odd 6 1
9792.2.a.i 1 168.v even 6 1
9792.2.a.n 1 168.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(833,[χ])S_{2}^{\mathrm{new}}(833, [\chi]):

T22T2+1 T_{2}^{2} - T_{2} + 1 Copy content Toggle raw display
T3 T_{3} Copy content Toggle raw display
T522T5+4 T_{5}^{2} - 2T_{5} + 4 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 T22T+4 T^{2} - 2T + 4 Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T2 T^{2} Copy content Toggle raw display
1313 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
1717 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
1919 T24T+16 T^{2} - 4T + 16 Copy content Toggle raw display
2323 T2+4T+16 T^{2} + 4T + 16 Copy content Toggle raw display
2929 (T6)2 (T - 6)^{2} Copy content Toggle raw display
3131 T2+4T+16 T^{2} + 4T + 16 Copy content Toggle raw display
3737 T22T+4 T^{2} - 2T + 4 Copy content Toggle raw display
4141 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
4343 (T4)2 (T - 4)^{2} Copy content Toggle raw display
4747 T2 T^{2} Copy content Toggle raw display
5353 T2+6T+36 T^{2} + 6T + 36 Copy content Toggle raw display
5959 T212T+144 T^{2} - 12T + 144 Copy content Toggle raw display
6161 T210T+100 T^{2} - 10T + 100 Copy content Toggle raw display
6767 T2+4T+16 T^{2} + 4T + 16 Copy content Toggle raw display
7171 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
7373 T26T+36 T^{2} - 6T + 36 Copy content Toggle raw display
7979 T2+12T+144 T^{2} + 12T + 144 Copy content Toggle raw display
8383 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
8989 T2+10T+100 T^{2} + 10T + 100 Copy content Toggle raw display
9797 (T2)2 (T - 2)^{2} Copy content Toggle raw display
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