Properties

Label 1078.2.e.f
Level 10781078
Weight 22
Character orbit 1078.e
Analytic conductor 8.6088.608
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] = [1078,2,Mod(67,1078)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1078, 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("1078.67"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 1078=27211 1078 = 2 \cdot 7^{2} \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1078.e (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,3,-1,2,-6,0,2,-6,2,1,3,14,0,12,-1,2,-6,0,-4,0,-2,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 8.607873337898.60787333789
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 154)
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) == qζ6q2+(3ζ6+3)q3+(ζ61)q4+2ζ6q53q6+q86ζ6q9+(2ζ6+2)q10+(ζ6+1)q11+3ζ6q12+6q99+O(q100) q - \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + (\zeta_{6} - 1) q^{4} + 2 \zeta_{6} q^{5} - 3 q^{6} + q^{8} - 6 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{10} + ( - \zeta_{6} + 1) q^{11} + 3 \zeta_{6} q^{12} + \cdots - 6 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2qq2+3q3q4+2q56q6+2q86q9+2q10+q11+3q12+14q13+12q15q16+2q176q184q202q22+8q23+3q24+12q99+O(q100) 2 q - q^{2} + 3 q^{3} - q^{4} + 2 q^{5} - 6 q^{6} + 2 q^{8} - 6 q^{9} + 2 q^{10} + q^{11} + 3 q^{12} + 14 q^{13} + 12 q^{15} - q^{16} + 2 q^{17} - 6 q^{18} - 4 q^{20} - 2 q^{22} + 8 q^{23} + 3 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1078Z)×\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times.

nn 199199 981981
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}
67.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 1.50000 2.59808i −0.500000 + 0.866025i 1.00000 + 1.73205i −3.00000 0 1.00000 −3.00000 5.19615i 1.00000 1.73205i
177.1 −0.500000 + 0.866025i 1.50000 + 2.59808i −0.500000 0.866025i 1.00000 1.73205i −3.00000 0 1.00000 −3.00000 + 5.19615i 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 1078.2.e.f 2
7.b odd 2 1 154.2.e.a 2
7.c even 3 1 1078.2.a.g 1
7.c even 3 1 inner 1078.2.e.f 2
7.d odd 6 1 154.2.e.a 2
7.d odd 6 1 1078.2.a.m 1
21.c even 2 1 1386.2.k.o 2
21.g even 6 1 1386.2.k.o 2
21.g even 6 1 9702.2.a.i 1
21.h odd 6 1 9702.2.a.y 1
28.d even 2 1 1232.2.q.e 2
28.f even 6 1 1232.2.q.e 2
28.f even 6 1 8624.2.a.b 1
28.g odd 6 1 8624.2.a.be 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.a 2 7.b odd 2 1
154.2.e.a 2 7.d odd 6 1
1078.2.a.g 1 7.c even 3 1
1078.2.a.m 1 7.d odd 6 1
1078.2.e.f 2 1.a even 1 1 trivial
1078.2.e.f 2 7.c even 3 1 inner
1232.2.q.e 2 28.d even 2 1
1232.2.q.e 2 28.f even 6 1
1386.2.k.o 2 21.c even 2 1
1386.2.k.o 2 21.g even 6 1
8624.2.a.b 1 28.f even 6 1
8624.2.a.be 1 28.g odd 6 1
9702.2.a.i 1 21.g even 6 1
9702.2.a.y 1 21.h 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(1078,[χ])S_{2}^{\mathrm{new}}(1078, [\chi]):

T323T3+9 T_{3}^{2} - 3T_{3} + 9 Copy content Toggle raw display
T522T5+4 T_{5}^{2} - 2T_{5} + 4 Copy content Toggle raw display
T137 T_{13} - 7 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
33 T23T+9 T^{2} - 3T + 9 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 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
1313 (T7)2 (T - 7)^{2} Copy content Toggle raw display
1717 T22T+4 T^{2} - 2T + 4 Copy content Toggle raw display
1919 T2 T^{2} Copy content Toggle raw display
2323 T28T+64 T^{2} - 8T + 64 Copy content Toggle raw display
2929 (T+5)2 (T + 5)^{2} Copy content Toggle raw display
3131 T24T+16 T^{2} - 4T + 16 Copy content Toggle raw display
3737 T2+4T+16 T^{2} + 4T + 16 Copy content Toggle raw display
4141 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
4343 (T+8)2 (T + 8)^{2} Copy content Toggle raw display
4747 T22T+4 T^{2} - 2T + 4 Copy content Toggle raw display
5353 T26T+36 T^{2} - 6T + 36 Copy content Toggle raw display
5959 T23T+9 T^{2} - 3T + 9 Copy content Toggle raw display
6161 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
6767 T2+9T+81 T^{2} + 9T + 81 Copy content Toggle raw display
7171 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
7373 T24T+16 T^{2} - 4T + 16 Copy content Toggle raw display
7979 T2+9T+81 T^{2} + 9T + 81 Copy content Toggle raw display
8383 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
8989 T26T+36 T^{2} - 6T + 36 Copy content Toggle raw display
9797 (T+7)2 (T + 7)^{2} Copy content Toggle raw display
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