Properties

Label 42.4.e.b
Level 4242
Weight 44
Character orbit 42.e
Analytic conductor 2.4782.478
Analytic rank 00
Dimension 22
Inner twists 22

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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] = [42,4,Mod(25,42)] 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("42.25"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: N N == 42=237 42 = 2 \cdot 3 \cdot 7
Weight: k k == 4 4
Character orbit: [χ][\chi] == 42.e (of order 33, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 2.478080220242.47808022024
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: yes
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+(2ζ6+2)q2+3ζ6q34ζ6q4+(15ζ6+15)q5+6q6+(7ζ6+14)q78q8+(9ζ69)q930ζ6q10+9ζ6q11+81q99+O(q100) q + ( - 2 \zeta_{6} + 2) q^{2} + 3 \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + ( - 15 \zeta_{6} + 15) q^{5} + 6 q^{6} + (7 \zeta_{6} + 14) q^{7} - 8 q^{8} + (9 \zeta_{6} - 9) q^{9} - 30 \zeta_{6} q^{10} + 9 \zeta_{6} q^{11} + \cdots - 81 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+2q2+3q34q4+15q5+12q6+35q716q89q930q10+9q11+12q12176q13+56q14+90q1516q16+84q17+18q18104q19+162q99+O(q100) 2 q + 2 q^{2} + 3 q^{3} - 4 q^{4} + 15 q^{5} + 12 q^{6} + 35 q^{7} - 16 q^{8} - 9 q^{9} - 30 q^{10} + 9 q^{11} + 12 q^{12} - 176 q^{13} + 56 q^{14} + 90 q^{15} - 16 q^{16} + 84 q^{17} + 18 q^{18} - 104 q^{19}+ \cdots - 162 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/42Z)×\left(\mathbb{Z}/42\mathbb{Z}\right)^\times.

nn 2929 3131
χ(n)\chi(n) 11 1+ζ6-1 + \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}
25.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 + 1.73205i 1.50000 2.59808i −2.00000 + 3.46410i 7.50000 + 12.9904i 6.00000 17.5000 6.06218i −8.00000 −4.50000 7.79423i −15.0000 + 25.9808i
37.1 1.00000 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i 7.50000 12.9904i 6.00000 17.5000 + 6.06218i −8.00000 −4.50000 + 7.79423i −15.0000 25.9808i
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 42.4.e.b 2
3.b odd 2 1 126.4.g.a 2
4.b odd 2 1 336.4.q.d 2
7.b odd 2 1 294.4.e.e 2
7.c even 3 1 inner 42.4.e.b 2
7.c even 3 1 294.4.a.a 1
7.d odd 6 1 294.4.a.g 1
7.d odd 6 1 294.4.e.e 2
21.c even 2 1 882.4.g.l 2
21.g even 6 1 882.4.a.h 1
21.g even 6 1 882.4.g.l 2
21.h odd 6 1 126.4.g.a 2
21.h odd 6 1 882.4.a.r 1
28.f even 6 1 2352.4.a.q 1
28.g odd 6 1 336.4.q.d 2
28.g odd 6 1 2352.4.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 1.a even 1 1 trivial
42.4.e.b 2 7.c even 3 1 inner
126.4.g.a 2 3.b odd 2 1
126.4.g.a 2 21.h odd 6 1
294.4.a.a 1 7.c even 3 1
294.4.a.g 1 7.d odd 6 1
294.4.e.e 2 7.b odd 2 1
294.4.e.e 2 7.d odd 6 1
336.4.q.d 2 4.b odd 2 1
336.4.q.d 2 28.g odd 6 1
882.4.a.h 1 21.g even 6 1
882.4.a.r 1 21.h odd 6 1
882.4.g.l 2 21.c even 2 1
882.4.g.l 2 21.g even 6 1
2352.4.a.q 1 28.f even 6 1
2352.4.a.u 1 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5215T5+225 T_{5}^{2} - 15T_{5} + 225 acting on S4new(42,[χ])S_{4}^{\mathrm{new}}(42, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T22T+4 T^{2} - 2T + 4 Copy content Toggle raw display
33 T23T+9 T^{2} - 3T + 9 Copy content Toggle raw display
55 T215T+225 T^{2} - 15T + 225 Copy content Toggle raw display
77 T235T+343 T^{2} - 35T + 343 Copy content Toggle raw display
1111 T29T+81 T^{2} - 9T + 81 Copy content Toggle raw display
1313 (T+88)2 (T + 88)^{2} Copy content Toggle raw display
1717 T284T+7056 T^{2} - 84T + 7056 Copy content Toggle raw display
1919 T2+104T+10816 T^{2} + 104T + 10816 Copy content Toggle raw display
2323 T284T+7056 T^{2} - 84T + 7056 Copy content Toggle raw display
2929 (T51)2 (T - 51)^{2} Copy content Toggle raw display
3131 T2+185T+34225 T^{2} + 185T + 34225 Copy content Toggle raw display
3737 T2+44T+1936 T^{2} + 44T + 1936 Copy content Toggle raw display
4141 (T+168)2 (T + 168)^{2} Copy content Toggle raw display
4343 (T326)2 (T - 326)^{2} Copy content Toggle raw display
4747 T2138T+19044 T^{2} - 138T + 19044 Copy content Toggle raw display
5353 T2+639T+408321 T^{2} + 639T + 408321 Copy content Toggle raw display
5959 T2+159T+25281 T^{2} + 159T + 25281 Copy content Toggle raw display
6161 T2+722T+521284 T^{2} + 722T + 521284 Copy content Toggle raw display
6767 T2166T+27556 T^{2} - 166T + 27556 Copy content Toggle raw display
7171 (T1086)2 (T - 1086)^{2} Copy content Toggle raw display
7373 T2+218T+47524 T^{2} + 218T + 47524 Copy content Toggle raw display
7979 T2583T+339889 T^{2} - 583T + 339889 Copy content Toggle raw display
8383 (T+597)2 (T + 597)^{2} Copy content Toggle raw display
8989 T21038T+1077444 T^{2} - 1038 T + 1077444 Copy content Toggle raw display
9797 (T+169)2 (T + 169)^{2} Copy content Toggle raw display
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