Properties

Label 1960.2.q.k
Level 19601960
Weight 22
Character orbit 1960.q
Analytic conductor 15.65115.651
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] = [1960,2,Mod(361,1960)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1960, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) 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("1960.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 1960=23572 1960 = 2^{3} \cdot 5 \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1960.q (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,-1,0,0,0,2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 15.650678796215.6506787962
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,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 280)
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+(ζ6+1)q3ζ6q5+2ζ6q9+(2ζ62)q11q15+(4ζ6+4)q172ζ6q19ζ6q23+(ζ61)q25+5q27+4q99+O(q100) q + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} - q^{15} + ( - 4 \zeta_{6} + 4) q^{17} - 2 \zeta_{6} q^{19} - \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 5 q^{27} + \cdots - 4 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+q3q5+2q92q112q15+4q172q19q23q25+10q27+18q29+4q31+2q334q372q41+18q43+2q454q51+10q53+8q99+O(q100) 2 q + q^{3} - q^{5} + 2 q^{9} - 2 q^{11} - 2 q^{15} + 4 q^{17} - 2 q^{19} - q^{23} - q^{25} + 10 q^{27} + 18 q^{29} + 4 q^{31} + 2 q^{33} - 4 q^{37} - 2 q^{41} + 18 q^{43} + 2 q^{45} - 4 q^{51} + 10 q^{53}+ \cdots - 8 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1960Z)×\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times.

nn 981981 10811081 11771177 14711471
χ(n)\chi(n) 11 ζ6-\zeta_{6} 11 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}
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 1.00000 + 1.73205i 0
961.1 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 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
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 1960.2.q.k 2
7.b odd 2 1 280.2.q.a 2
7.c even 3 1 1960.2.a.e 1
7.c even 3 1 inner 1960.2.q.k 2
7.d odd 6 1 280.2.q.a 2
7.d odd 6 1 1960.2.a.i 1
21.c even 2 1 2520.2.bi.a 2
21.g even 6 1 2520.2.bi.a 2
28.d even 2 1 560.2.q.h 2
28.f even 6 1 560.2.q.h 2
28.f even 6 1 3920.2.a.m 1
28.g odd 6 1 3920.2.a.y 1
35.c odd 2 1 1400.2.q.e 2
35.f even 4 2 1400.2.bh.b 4
35.i odd 6 1 1400.2.q.e 2
35.i odd 6 1 9800.2.a.r 1
35.j even 6 1 9800.2.a.bc 1
35.k even 12 2 1400.2.bh.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.a 2 7.b odd 2 1
280.2.q.a 2 7.d odd 6 1
560.2.q.h 2 28.d even 2 1
560.2.q.h 2 28.f even 6 1
1400.2.q.e 2 35.c odd 2 1
1400.2.q.e 2 35.i odd 6 1
1400.2.bh.b 4 35.f even 4 2
1400.2.bh.b 4 35.k even 12 2
1960.2.a.e 1 7.c even 3 1
1960.2.a.i 1 7.d odd 6 1
1960.2.q.k 2 1.a even 1 1 trivial
1960.2.q.k 2 7.c even 3 1 inner
2520.2.bi.a 2 21.c even 2 1
2520.2.bi.a 2 21.g even 6 1
3920.2.a.m 1 28.f even 6 1
3920.2.a.y 1 28.g odd 6 1
9800.2.a.r 1 35.i odd 6 1
9800.2.a.bc 1 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1960,[χ])S_{2}^{\mathrm{new}}(1960, [\chi]):

T32T3+1 T_{3}^{2} - T_{3} + 1 Copy content Toggle raw display
T112+2T11+4 T_{11}^{2} + 2T_{11} + 4 Copy content Toggle raw display
T13 T_{13} 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 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
1313 T2 T^{2} Copy content Toggle raw display
1717 T24T+16 T^{2} - 4T + 16 Copy content Toggle raw display
1919 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
2323 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
2929 (T9)2 (T - 9)^{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+1)2 (T + 1)^{2} Copy content Toggle raw display
4343 (T9)2 (T - 9)^{2} Copy content Toggle raw display
4747 T2 T^{2} Copy content Toggle raw display
5353 T210T+100 T^{2} - 10T + 100 Copy content Toggle raw display
5959 T2+10T+100 T^{2} + 10T + 100 Copy content Toggle raw display
6161 T29T+81 T^{2} - 9T + 81 Copy content Toggle raw display
6767 T2+5T+25 T^{2} + 5T + 25 Copy content Toggle raw display
7171 (T14)2 (T - 14)^{2} Copy content Toggle raw display
7373 T212T+144 T^{2} - 12T + 144 Copy content Toggle raw display
7979 T2+14T+196 T^{2} + 14T + 196 Copy content Toggle raw display
8383 (T+11)2 (T + 11)^{2} Copy content Toggle raw display
8989 T2+15T+225 T^{2} + 15T + 225 Copy content Toggle raw display
9797 (T18)2 (T - 18)^{2} Copy content Toggle raw display
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