Properties

Label 650.2.o.b
Level 650650
Weight 22
Character orbit 650.o
Analytic conductor 5.1905.190
Analytic rank 00
Dimension 44
Inner twists 44

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [650,2,Mod(399,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("650.399"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 2, names="a")
 
Level: N N == 650=25213 650 = 2 \cdot 5^{2} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 650.o (of order 66, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,-4,0,0,2,0,-6] 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: 5.190276131385.19027613138
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q(ζ12)\Q(\zeta_{12})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4x2+1 x^{4} - x^{2} + 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 130)
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

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 ζ12\zeta_{12}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+ζ12q22ζ12q3+ζ122q42ζ122q6+(ζ123ζ12)q7+ζ123q8+ζ122q9+(3ζ1223)q11+3q99+O(q100) q + \zeta_{12} q^{2} - 2 \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} - 2 \zeta_{12}^{2} q^{6} + (\zeta_{12}^{3} - \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + (3 \zeta_{12}^{2} - 3) q^{11} + \cdots - 3 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+2q44q6+2q96q114q142q16+10q19+8q21+4q244q2616q3124q342q36+8q3912q4112q4412q49+48q51+12q99+O(q100) 4 q + 2 q^{4} - 4 q^{6} + 2 q^{9} - 6 q^{11} - 4 q^{14} - 2 q^{16} + 10 q^{19} + 8 q^{21} + 4 q^{24} - 4 q^{26} - 16 q^{31} - 24 q^{34} - 2 q^{36} + 8 q^{39} - 12 q^{41} - 12 q^{44} - 12 q^{49} + 48 q^{51}+ \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/650Z)×\left(\mathbb{Z}/650\mathbb{Z}\right)^\times.

nn 2727 301301
χ(n)\chi(n) 1-1 ζ122-\zeta_{12}^{2}

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}
399.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i 1.73205 + 1.00000i 0.500000 + 0.866025i 0 −1.00000 1.73205i 0.866025 0.500000i 1.00000i 0.500000 + 0.866025i 0
399.2 0.866025 + 0.500000i −1.73205 1.00000i 0.500000 + 0.866025i 0 −1.00000 1.73205i −0.866025 + 0.500000i 1.00000i 0.500000 + 0.866025i 0
549.1 −0.866025 + 0.500000i 1.73205 1.00000i 0.500000 0.866025i 0 −1.00000 + 1.73205i 0.866025 + 0.500000i 1.00000i 0.500000 0.866025i 0
549.2 0.866025 0.500000i −1.73205 + 1.00000i 0.500000 0.866025i 0 −1.00000 + 1.73205i −0.866025 0.500000i 1.00000i 0.500000 0.866025i 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
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.o.b 4
5.b even 2 1 inner 650.2.o.b 4
5.c odd 4 1 130.2.e.b 2
5.c odd 4 1 650.2.e.a 2
13.c even 3 1 inner 650.2.o.b 4
15.e even 4 1 1170.2.i.f 2
20.e even 4 1 1040.2.q.c 2
65.f even 4 1 1690.2.l.i 4
65.h odd 4 1 1690.2.e.e 2
65.k even 4 1 1690.2.l.i 4
65.n even 6 1 inner 650.2.o.b 4
65.o even 12 1 1690.2.d.a 2
65.o even 12 1 1690.2.l.i 4
65.q odd 12 1 130.2.e.b 2
65.q odd 12 1 650.2.e.a 2
65.q odd 12 1 1690.2.a.a 1
65.q odd 12 1 8450.2.a.w 1
65.r odd 12 1 1690.2.a.g 1
65.r odd 12 1 1690.2.e.e 2
65.r odd 12 1 8450.2.a.k 1
65.t even 12 1 1690.2.d.a 2
65.t even 12 1 1690.2.l.i 4
195.bl even 12 1 1170.2.i.f 2
260.bj even 12 1 1040.2.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.b 2 5.c odd 4 1
130.2.e.b 2 65.q odd 12 1
650.2.e.a 2 5.c odd 4 1
650.2.e.a 2 65.q odd 12 1
650.2.o.b 4 1.a even 1 1 trivial
650.2.o.b 4 5.b even 2 1 inner
650.2.o.b 4 13.c even 3 1 inner
650.2.o.b 4 65.n even 6 1 inner
1040.2.q.c 2 20.e even 4 1
1040.2.q.c 2 260.bj even 12 1
1170.2.i.f 2 15.e even 4 1
1170.2.i.f 2 195.bl even 12 1
1690.2.a.a 1 65.q odd 12 1
1690.2.a.g 1 65.r odd 12 1
1690.2.d.a 2 65.o even 12 1
1690.2.d.a 2 65.t even 12 1
1690.2.e.e 2 65.h odd 4 1
1690.2.e.e 2 65.r odd 12 1
1690.2.l.i 4 65.f even 4 1
1690.2.l.i 4 65.k even 4 1
1690.2.l.i 4 65.o even 12 1
1690.2.l.i 4 65.t even 12 1
8450.2.a.k 1 65.r odd 12 1
8450.2.a.w 1 65.q odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(650,[χ])S_{2}^{\mathrm{new}}(650, [\chi]):

T344T32+16 T_{3}^{4} - 4T_{3}^{2} + 16 Copy content Toggle raw display
T74T72+1 T_{7}^{4} - T_{7}^{2} + 1 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
33 T44T2+16 T^{4} - 4T^{2} + 16 Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 T4T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
1111 (T2+3T+9)2 (T^{2} + 3 T + 9)^{2} Copy content Toggle raw display
1313 T4T2+169 T^{4} - T^{2} + 169 Copy content Toggle raw display
1717 T436T2+1296 T^{4} - 36T^{2} + 1296 Copy content Toggle raw display
1919 (T25T+25)2 (T^{2} - 5 T + 25)^{2} Copy content Toggle raw display
2323 T4 T^{4} Copy content Toggle raw display
2929 T4 T^{4} Copy content Toggle raw display
3131 (T+4)4 (T + 4)^{4} Copy content Toggle raw display
3737 T4121T2+14641 T^{4} - 121 T^{2} + 14641 Copy content Toggle raw display
4141 (T2+6T+36)2 (T^{2} + 6 T + 36)^{2} Copy content Toggle raw display
4343 T44T2+16 T^{4} - 4T^{2} + 16 Copy content Toggle raw display
4747 (T2+9)2 (T^{2} + 9)^{2} Copy content Toggle raw display
5353 (T2+81)2 (T^{2} + 81)^{2} Copy content Toggle raw display
5959 T4 T^{4} Copy content Toggle raw display
6161 (T2+8T+64)2 (T^{2} + 8 T + 64)^{2} Copy content Toggle raw display
6767 T4256T2+65536 T^{4} - 256 T^{2} + 65536 Copy content Toggle raw display
7171 (T2+6T+36)2 (T^{2} + 6 T + 36)^{2} Copy content Toggle raw display
7373 (T2+196)2 (T^{2} + 196)^{2} Copy content Toggle raw display
7979 (T16)4 (T - 16)^{4} Copy content Toggle raw display
8383 (T2+36)2 (T^{2} + 36)^{2} Copy content Toggle raw display
8989 (T29T+81)2 (T^{2} - 9 T + 81)^{2} Copy content Toggle raw display
9797 T4100T2+10000 T^{4} - 100 T^{2} + 10000 Copy content Toggle raw display
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