Properties

Label 63.3.m.c
Level 6363
Weight 33
Character orbit 63.m
Analytic conductor 1.7171.717
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] = [63,3,Mod(10,63)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("63.10"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: N N == 63=327 63 = 3^{2} \cdot 7
Weight: k k == 3 3
Character orbit: [χ][\chi] == 63.m (of order 66, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 1.716625665471.71662566547
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 21)
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 ζ6\zeta_{6}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+ζ6q2+(3ζ6+3)q4+(3ζ6+6)q5+(7ζ67)q7+7q8+(3ζ6+3)q10+(11ζ611)q11+(8ζ6+4)q137q14++(49ζ6+49)q98+O(q100) q + \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{4} + ( - 3 \zeta_{6} + 6) q^{5} + (7 \zeta_{6} - 7) q^{7} + 7 q^{8} + (3 \zeta_{6} + 3) q^{10} + (11 \zeta_{6} - 11) q^{11} + ( - 8 \zeta_{6} + 4) q^{13} - 7 q^{14}+ \cdots + ( - 49 \zeta_{6} + 49) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+q2+3q4+9q57q7+14q8+9q1011q1114q145q1642q176q1922q22+28q23+2q25+12q26+21q2850q2957q31++49q98+O(q100) 2 q + q^{2} + 3 q^{4} + 9 q^{5} - 7 q^{7} + 14 q^{8} + 9 q^{10} - 11 q^{11} - 14 q^{14} - 5 q^{16} - 42 q^{17} - 6 q^{19} - 22 q^{22} + 28 q^{23} + 2 q^{25} + 12 q^{26} + 21 q^{28} - 50 q^{29} - 57 q^{31}+ \cdots + 49 q^{98}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/63Z)×\left(\mathbb{Z}/63\mathbb{Z}\right)^\times.

nn 1010 2929
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}
10.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 1.50000 2.59808i 4.50000 2.59808i 0 −3.50000 + 6.06218i 7.00000 0 4.50000 + 2.59808i
19.1 0.500000 0.866025i 0 1.50000 + 2.59808i 4.50000 + 2.59808i 0 −3.50000 6.06218i 7.00000 0 4.50000 2.59808i
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.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.3.m.c 2
3.b odd 2 1 21.3.f.b 2
4.b odd 2 1 1008.3.cg.g 2
7.b odd 2 1 441.3.m.e 2
7.c even 3 1 441.3.d.b 2
7.c even 3 1 441.3.m.e 2
7.d odd 6 1 inner 63.3.m.c 2
7.d odd 6 1 441.3.d.b 2
12.b even 2 1 336.3.bh.a 2
15.d odd 2 1 525.3.o.g 2
15.e even 4 2 525.3.s.c 4
21.c even 2 1 147.3.f.c 2
21.g even 6 1 21.3.f.b 2
21.g even 6 1 147.3.d.b 2
21.h odd 6 1 147.3.d.b 2
21.h odd 6 1 147.3.f.c 2
28.f even 6 1 1008.3.cg.g 2
84.j odd 6 1 336.3.bh.a 2
84.j odd 6 1 2352.3.f.d 2
84.n even 6 1 2352.3.f.d 2
105.p even 6 1 525.3.o.g 2
105.w odd 12 2 525.3.s.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.b 2 3.b odd 2 1
21.3.f.b 2 21.g even 6 1
63.3.m.c 2 1.a even 1 1 trivial
63.3.m.c 2 7.d odd 6 1 inner
147.3.d.b 2 21.g even 6 1
147.3.d.b 2 21.h odd 6 1
147.3.f.c 2 21.c even 2 1
147.3.f.c 2 21.h odd 6 1
336.3.bh.a 2 12.b even 2 1
336.3.bh.a 2 84.j odd 6 1
441.3.d.b 2 7.c even 3 1
441.3.d.b 2 7.d odd 6 1
441.3.m.e 2 7.b odd 2 1
441.3.m.e 2 7.c even 3 1
525.3.o.g 2 15.d odd 2 1
525.3.o.g 2 105.p even 6 1
525.3.s.c 4 15.e even 4 2
525.3.s.c 4 105.w odd 12 2
1008.3.cg.g 2 4.b odd 2 1
1008.3.cg.g 2 28.f even 6 1
2352.3.f.d 2 84.j odd 6 1
2352.3.f.d 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T22T2+1 T_{2}^{2} - T_{2} + 1 acting on S3new(63,[χ])S_{3}^{\mathrm{new}}(63, [\chi]). 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 T29T+27 T^{2} - 9T + 27 Copy content Toggle raw display
77 T2+7T+49 T^{2} + 7T + 49 Copy content Toggle raw display
1111 T2+11T+121 T^{2} + 11T + 121 Copy content Toggle raw display
1313 T2+48 T^{2} + 48 Copy content Toggle raw display
1717 T2+42T+588 T^{2} + 42T + 588 Copy content Toggle raw display
1919 T2+6T+12 T^{2} + 6T + 12 Copy content Toggle raw display
2323 T228T+784 T^{2} - 28T + 784 Copy content Toggle raw display
2929 (T+25)2 (T + 25)^{2} Copy content Toggle raw display
3131 T2+57T+1083 T^{2} + 57T + 1083 Copy content Toggle raw display
3737 T258T+3364 T^{2} - 58T + 3364 Copy content Toggle raw display
4141 T2+12 T^{2} + 12 Copy content Toggle raw display
4343 (T26)2 (T - 26)^{2} Copy content Toggle raw display
4747 T2132T+5808 T^{2} - 132T + 5808 Copy content Toggle raw display
5353 T231T+961 T^{2} - 31T + 961 Copy content Toggle raw display
5959 T215T+75 T^{2} - 15T + 75 Copy content Toggle raw display
6161 T224T+192 T^{2} - 24T + 192 Copy content Toggle raw display
6767 T252T+2704 T^{2} - 52T + 2704 Copy content Toggle raw display
7171 (T+64)2 (T + 64)^{2} Copy content Toggle raw display
7373 T212T+48 T^{2} - 12T + 48 Copy content Toggle raw display
7979 T2+17T+289 T^{2} + 17T + 289 Copy content Toggle raw display
8383 T2+2883 T^{2} + 2883 Copy content Toggle raw display
8989 T2138T+6348 T^{2} - 138T + 6348 Copy content Toggle raw display
9797 T2+8427 T^{2} + 8427 Copy content Toggle raw display
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