Properties

Label 126.4.g.b
Level 126126
Weight 44
Character orbit 126.g
Analytic conductor 7.4347.434
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,4,Mod(37,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 126=2327 126 = 2 \cdot 3^{2} \cdot 7
Weight: k k == 4 4
Character orbit: [χ][\chi] == 126.g (of order 33, degree 22, minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 7.434240660727.43424066072
Analytic rank: 00
Dimension: 22
Coefficient field: Q(3)\Q(\sqrt{-3})
comment: defining polynomial
 
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 42)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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+(2ζ62)q24ζ6q4+(6ζ66)q5+(21ζ6+7)q7+8q812ζ6q1030ζ6q11+53q13+(14ζ6+28)q14+(16ζ616)q16++(784ζ6+490)q98+O(q100) q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + (6 \zeta_{6} - 6) q^{5} + ( - 21 \zeta_{6} + 7) q^{7} + 8 q^{8} - 12 \zeta_{6} q^{10} - 30 \zeta_{6} q^{11} + 53 q^{13} + (14 \zeta_{6} + 28) q^{14} + (16 \zeta_{6} - 16) q^{16} + \cdots + ( - 784 \zeta_{6} + 490) q^{98} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q24q46q57q7+16q812q1030q11+106q13+70q1416q1684q17+97q19+48q20+120q22+84q23+89q25106q26++196q98+O(q100) 2 q - 2 q^{2} - 4 q^{4} - 6 q^{5} - 7 q^{7} + 16 q^{8} - 12 q^{10} - 30 q^{11} + 106 q^{13} + 70 q^{14} - 16 q^{16} - 84 q^{17} + 97 q^{19} + 48 q^{20} + 120 q^{22} + 84 q^{23} + 89 q^{25} - 106 q^{26}+ \cdots + 196 q^{98}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/126Z)×\left(\mathbb{Z}/126\mathbb{Z}\right)^\times.

nn 2929 7373
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.

comment: embeddings in the coefficient field
 
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 + 1.73205i 0 −2.00000 3.46410i −3.00000 + 5.19615i 0 −3.50000 18.1865i 8.00000 0 −6.00000 10.3923i
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −3.00000 5.19615i 0 −3.50000 + 18.1865i 8.00000 0 −6.00000 + 10.3923i
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 126.4.g.b 2
3.b odd 2 1 42.4.e.a 2
7.b odd 2 1 882.4.g.g 2
7.c even 3 1 inner 126.4.g.b 2
7.c even 3 1 882.4.a.o 1
7.d odd 6 1 882.4.a.l 1
7.d odd 6 1 882.4.g.g 2
12.b even 2 1 336.4.q.f 2
21.c even 2 1 294.4.e.i 2
21.g even 6 1 294.4.a.c 1
21.g even 6 1 294.4.e.i 2
21.h odd 6 1 42.4.e.a 2
21.h odd 6 1 294.4.a.d 1
84.j odd 6 1 2352.4.a.bf 1
84.n even 6 1 336.4.q.f 2
84.n even 6 1 2352.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 3.b odd 2 1
42.4.e.a 2 21.h odd 6 1
126.4.g.b 2 1.a even 1 1 trivial
126.4.g.b 2 7.c even 3 1 inner
294.4.a.c 1 21.g even 6 1
294.4.a.d 1 21.h odd 6 1
294.4.e.i 2 21.c even 2 1
294.4.e.i 2 21.g even 6 1
336.4.q.f 2 12.b even 2 1
336.4.q.f 2 84.n even 6 1
882.4.a.l 1 7.d odd 6 1
882.4.a.o 1 7.c even 3 1
882.4.g.g 2 7.b odd 2 1
882.4.g.g 2 7.d odd 6 1
2352.4.a.f 1 84.n even 6 1
2352.4.a.bf 1 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T52+6T5+36 T_{5}^{2} + 6T_{5} + 36 acting on S4new(126,[χ])S_{4}^{\mathrm{new}}(126, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 T2+6T+36 T^{2} + 6T + 36 Copy content Toggle raw display
77 T2+7T+343 T^{2} + 7T + 343 Copy content Toggle raw display
1111 T2+30T+900 T^{2} + 30T + 900 Copy content Toggle raw display
1313 (T53)2 (T - 53)^{2} Copy content Toggle raw display
1717 T2+84T+7056 T^{2} + 84T + 7056 Copy content Toggle raw display
1919 T297T+9409 T^{2} - 97T + 9409 Copy content Toggle raw display
2323 T284T+7056 T^{2} - 84T + 7056 Copy content Toggle raw display
2929 (T180)2 (T - 180)^{2} Copy content Toggle raw display
3131 T2+179T+32041 T^{2} + 179T + 32041 Copy content Toggle raw display
3737 T2145T+21025 T^{2} - 145T + 21025 Copy content Toggle raw display
4141 (T+126)2 (T + 126)^{2} Copy content Toggle raw display
4343 (T+325)2 (T + 325)^{2} Copy content Toggle raw display
4747 T2+366T+133956 T^{2} + 366T + 133956 Copy content Toggle raw display
5353 T2+768T+589824 T^{2} + 768T + 589824 Copy content Toggle raw display
5959 T2+264T+69696 T^{2} + 264T + 69696 Copy content Toggle raw display
6161 T2+818T+669124 T^{2} + 818T + 669124 Copy content Toggle raw display
6767 T2523T+273529 T^{2} - 523T + 273529 Copy content Toggle raw display
7171 (T342)2 (T - 342)^{2} Copy content Toggle raw display
7373 T243T+1849 T^{2} - 43T + 1849 Copy content Toggle raw display
7979 T21171T+1371241 T^{2} - 1171 T + 1371241 Copy content Toggle raw display
8383 (T810)2 (T - 810)^{2} Copy content Toggle raw display
8989 T2+600T+360000 T^{2} + 600T + 360000 Copy content Toggle raw display
9797 (T386)2 (T - 386)^{2} Copy content Toggle raw display
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