Properties

Label 378.2.e.a
Level 378378
Weight 22
Character orbit 378.e
Analytic conductor 3.0183.018
Analytic rank 11
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] = [378,2,Mod(37,378)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(378, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 2])) 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("378.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 378=2337 378 = 2 \cdot 3^{3} \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 378.e (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] 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: 3.018345196403.01834519640
Analytic rank: 11
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,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 126)
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) == qq2+q4+(3ζ63)q5+(2ζ63)q7q8+(3ζ6+3)q103ζ6q115ζ6q13+(2ζ6+3)q14+q16+(3ζ6+3)q17++(8ζ65)q98+O(q100) q - q^{2} + q^{4} + (3 \zeta_{6} - 3) q^{5} + (2 \zeta_{6} - 3) q^{7} - q^{8} + ( - 3 \zeta_{6} + 3) q^{10} - 3 \zeta_{6} q^{11} - 5 \zeta_{6} q^{13} + ( - 2 \zeta_{6} + 3) q^{14} + q^{16} + ( - 3 \zeta_{6} + 3) q^{17} + \cdots + (8 \zeta_{6} - 5) q^{98} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q2+2q43q54q72q8+3q103q115q13+4q14+2q16+3q175q193q20+3q223q234q25+5q264q283q29+2q98+O(q100) 2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 4 q^{7} - 2 q^{8} + 3 q^{10} - 3 q^{11} - 5 q^{13} + 4 q^{14} + 2 q^{16} + 3 q^{17} - 5 q^{19} - 3 q^{20} + 3 q^{22} - 3 q^{23} - 4 q^{25} + 5 q^{26} - 4 q^{28} - 3 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/378Z)×\left(\mathbb{Z}/378\mathbb{Z}\right)^\times.

nn 2929 325325
χ(n)\chi(n) ζ6-\zeta_{6} ζ6-\zeta_{6}

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}
37.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 0 1.00000 −1.50000 2.59808i 0 −2.00000 1.73205i −1.00000 0 1.50000 + 2.59808i
235.1 −1.00000 0 1.00000 −1.50000 + 2.59808i 0 −2.00000 + 1.73205i −1.00000 0 1.50000 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
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.e.a 2
3.b odd 2 1 126.2.e.b 2
4.b odd 2 1 3024.2.q.a 2
7.b odd 2 1 2646.2.e.e 2
7.c even 3 1 378.2.h.b 2
7.c even 3 1 2646.2.f.e 2
7.d odd 6 1 2646.2.f.i 2
7.d odd 6 1 2646.2.h.f 2
9.c even 3 1 378.2.h.b 2
9.c even 3 1 1134.2.g.f 2
9.d odd 6 1 126.2.h.a yes 2
9.d odd 6 1 1134.2.g.d 2
12.b even 2 1 1008.2.q.e 2
21.c even 2 1 882.2.e.h 2
21.g even 6 1 882.2.f.a 2
21.g even 6 1 882.2.h.e 2
21.h odd 6 1 126.2.h.a yes 2
21.h odd 6 1 882.2.f.e 2
28.g odd 6 1 3024.2.t.f 2
36.f odd 6 1 3024.2.t.f 2
36.h even 6 1 1008.2.t.c 2
63.g even 3 1 1134.2.g.f 2
63.g even 3 1 2646.2.f.e 2
63.h even 3 1 inner 378.2.e.a 2
63.h even 3 1 7938.2.a.o 1
63.i even 6 1 882.2.e.h 2
63.i even 6 1 7938.2.a.bd 1
63.j odd 6 1 126.2.e.b 2
63.j odd 6 1 7938.2.a.r 1
63.k odd 6 1 2646.2.f.i 2
63.l odd 6 1 2646.2.h.f 2
63.n odd 6 1 882.2.f.e 2
63.n odd 6 1 1134.2.g.d 2
63.o even 6 1 882.2.h.e 2
63.s even 6 1 882.2.f.a 2
63.t odd 6 1 2646.2.e.e 2
63.t odd 6 1 7938.2.a.c 1
84.n even 6 1 1008.2.t.c 2
252.u odd 6 1 3024.2.q.a 2
252.bb even 6 1 1008.2.q.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.b 2 3.b odd 2 1
126.2.e.b 2 63.j odd 6 1
126.2.h.a yes 2 9.d odd 6 1
126.2.h.a yes 2 21.h odd 6 1
378.2.e.a 2 1.a even 1 1 trivial
378.2.e.a 2 63.h even 3 1 inner
378.2.h.b 2 7.c even 3 1
378.2.h.b 2 9.c even 3 1
882.2.e.h 2 21.c even 2 1
882.2.e.h 2 63.i even 6 1
882.2.f.a 2 21.g even 6 1
882.2.f.a 2 63.s even 6 1
882.2.f.e 2 21.h odd 6 1
882.2.f.e 2 63.n odd 6 1
882.2.h.e 2 21.g even 6 1
882.2.h.e 2 63.o even 6 1
1008.2.q.e 2 12.b even 2 1
1008.2.q.e 2 252.bb even 6 1
1008.2.t.c 2 36.h even 6 1
1008.2.t.c 2 84.n even 6 1
1134.2.g.d 2 9.d odd 6 1
1134.2.g.d 2 63.n odd 6 1
1134.2.g.f 2 9.c even 3 1
1134.2.g.f 2 63.g even 3 1
2646.2.e.e 2 7.b odd 2 1
2646.2.e.e 2 63.t odd 6 1
2646.2.f.e 2 7.c even 3 1
2646.2.f.e 2 63.g even 3 1
2646.2.f.i 2 7.d odd 6 1
2646.2.f.i 2 63.k odd 6 1
2646.2.h.f 2 7.d odd 6 1
2646.2.h.f 2 63.l odd 6 1
3024.2.q.a 2 4.b odd 2 1
3024.2.q.a 2 252.u odd 6 1
3024.2.t.f 2 28.g odd 6 1
3024.2.t.f 2 36.f odd 6 1
7938.2.a.c 1 63.t odd 6 1
7938.2.a.o 1 63.h even 3 1
7938.2.a.r 1 63.j odd 6 1
7938.2.a.bd 1 63.i even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T52+3T5+9 T_{5}^{2} + 3T_{5} + 9 acting on S2new(378,[χ])S_{2}^{\mathrm{new}}(378, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 T2+3T+9 T^{2} + 3T + 9 Copy content Toggle raw display
77 T2+4T+7 T^{2} + 4T + 7 Copy content Toggle raw display
1111 T2+3T+9 T^{2} + 3T + 9 Copy content Toggle raw display
1313 T2+5T+25 T^{2} + 5T + 25 Copy content Toggle raw display
1717 T23T+9 T^{2} - 3T + 9 Copy content Toggle raw display
1919 T2+5T+25 T^{2} + 5T + 25 Copy content Toggle raw display
2323 T2+3T+9 T^{2} + 3T + 9 Copy content Toggle raw display
2929 T2+3T+9 T^{2} + 3T + 9 Copy content Toggle raw display
3131 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
3737 T27T+49 T^{2} - 7T + 49 Copy content Toggle raw display
4141 T2+9T+81 T^{2} + 9T + 81 Copy content Toggle raw display
4343 T2+11T+121 T^{2} + 11T + 121 Copy content Toggle raw display
4747 T2 T^{2} Copy content Toggle raw display
5353 T2+3T+9 T^{2} + 3T + 9 Copy content Toggle raw display
5959 (T+12)2 (T + 12)^{2} Copy content Toggle raw display
6161 (T2)2 (T - 2)^{2} Copy content Toggle raw display
6767 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
7171 T2 T^{2} Copy content Toggle raw display
7373 T2+11T+121 T^{2} + 11T + 121 Copy content Toggle raw display
7979 (T8)2 (T - 8)^{2} Copy content Toggle raw display
8383 T23T+9 T^{2} - 3T + 9 Copy content Toggle raw display
8989 T215T+225 T^{2} - 15T + 225 Copy content Toggle raw display
9797 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
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