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Properties

Label 156.1.o.a
Level 156156
Weight 11
Character orbit 156.o
Analytic conductor 0.0780.078
Analytic rank 00
Dimension 22
Projective image D3D_{3}
CM discriminant -3
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [156,1,Mod(29,156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(156, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("156.29");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 156=22⋅3⋅13 156 = 2^{2} \cdot 3 \cdot 13
Weight: k k == 1 1
Character orbit: [χ][\chi] == 156.o (of order 66, 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: 0.07785414197070.0778541419707
Analytic rank: 00
Dimension: 22
Coefficient field: Q(ζ6)\Q(\zeta_{6})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2−x+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
Projective image: D3D_{3}
Projective field: Galois closure of 3.1.2028.1
Artin image: C3×S3C_3\times S_3
Artin field: Galois closure of 6.0.73008.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The qq-expansion and trace form are shown below.

f(q)f(q) == q−ζ6q3−ζ62q7+ζ62q9+ζ62q13+2ζ62q19−q21+q25+q27−q31−2ζ6q37+q39−ζ62q43+⋯−ζ62q97+O(q100) q - \zeta_{6} q^{3} - \zeta_{6}^{2} q^{7} + \zeta_{6}^{2} q^{9} + \zeta_{6}^{2} q^{13} + 2 \zeta_{6}^{2} q^{19} - q^{21} + q^{25} + q^{27} - q^{31} - 2 \zeta_{6} q^{37} + q^{39} - \zeta_{6}^{2} q^{43} + \cdots - \zeta_{6}^{2} q^{97} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 2q−q3+q7−q9−q13−2q19−2q21+2q25+2q27−2q31−2q37+2q39+q43+4q57+q61+q63+q67−2q73−q75−2q79−q81+⋯+q97+O(q100) 2 q - q^{3} + q^{7} - q^{9} - q^{13} - 2 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} - 2 q^{31} - 2 q^{37} + 2 q^{39} + q^{43} + 4 q^{57} + q^{61} + q^{63} + q^{67} - 2 q^{73} - q^{75} - 2 q^{79} - q^{81}+ \cdots + q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/156Z)×\left(\mathbb{Z}/156\mathbb{Z}\right)^\times.

nn 5353 7979 145145
χ(n)\chi(n) −1-1 11 ζ62\zeta_{6}^{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.

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}
29.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 −0.500000 − 0.866025i 0 0 0 0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0
113.1 0 −0.500000 + 0.866025i 0 0 0 0.500000 + 0.866025i 0 −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
3.b odd 2 1 CM by Q(−3)\Q(\sqrt{-3})
13.c even 3 1 inner
39.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.1.o.a ✓ 2
3.b odd 2 1 CM 156.1.o.a ✓ 2
4.b odd 2 1 624.1.bs.a 2
5.b even 2 1 3900.1.bu.b 2
5.c odd 4 2 3900.1.cb.b 4
8.b even 2 1 2496.1.bs.b 2
8.d odd 2 1 2496.1.bs.a 2
12.b even 2 1 624.1.bs.a 2
13.b even 2 1 2028.1.o.a 2
13.c even 3 1 inner 156.1.o.a ✓ 2
13.c even 3 1 2028.1.d.a 1
13.d odd 4 2 2028.1.s.b 4
13.e even 6 1 2028.1.d.b 1
13.e even 6 1 2028.1.o.a 2
13.f odd 12 2 2028.1.g.b 2
13.f odd 12 2 2028.1.s.b 4
15.d odd 2 1 3900.1.bu.b 2
15.e even 4 2 3900.1.cb.b 4
24.f even 2 1 2496.1.bs.a 2
24.h odd 2 1 2496.1.bs.b 2
39.d odd 2 1 2028.1.o.a 2
39.f even 4 2 2028.1.s.b 4
39.h odd 6 1 2028.1.d.b 1
39.h odd 6 1 2028.1.o.a 2
39.i odd 6 1 inner 156.1.o.a ✓ 2
39.i odd 6 1 2028.1.d.a 1
39.k even 12 2 2028.1.g.b 2
39.k even 12 2 2028.1.s.b 4
52.j odd 6 1 624.1.bs.a 2
65.n even 6 1 3900.1.bu.b 2
65.q odd 12 2 3900.1.cb.b 4
104.n odd 6 1 2496.1.bs.a 2
104.r even 6 1 2496.1.bs.b 2
156.p even 6 1 624.1.bs.a 2
195.x odd 6 1 3900.1.bu.b 2
195.bl even 12 2 3900.1.cb.b 4
312.bh odd 6 1 2496.1.bs.b 2
312.bn even 6 1 2496.1.bs.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.1.o.a ✓ 2 1.a even 1 1 trivial
156.1.o.a ✓ 2 3.b odd 2 1 CM
156.1.o.a ✓ 2 13.c even 3 1 inner
156.1.o.a ✓ 2 39.i odd 6 1 inner
624.1.bs.a 2 4.b odd 2 1
624.1.bs.a 2 12.b even 2 1
624.1.bs.a 2 52.j odd 6 1
624.1.bs.a 2 156.p even 6 1
2028.1.d.a 1 13.c even 3 1
2028.1.d.a 1 39.i odd 6 1
2028.1.d.b 1 13.e even 6 1
2028.1.d.b 1 39.h odd 6 1
2028.1.g.b 2 13.f odd 12 2
2028.1.g.b 2 39.k even 12 2
2028.1.o.a 2 13.b even 2 1
2028.1.o.a 2 13.e even 6 1
2028.1.o.a 2 39.d odd 2 1
2028.1.o.a 2 39.h odd 6 1
2028.1.s.b 4 13.d odd 4 2
2028.1.s.b 4 13.f odd 12 2
2028.1.s.b 4 39.f even 4 2
2028.1.s.b 4 39.k even 12 2
2496.1.bs.a 2 8.d odd 2 1
2496.1.bs.a 2 24.f even 2 1
2496.1.bs.a 2 104.n odd 6 1
2496.1.bs.a 2 312.bn even 6 1
2496.1.bs.b 2 8.b even 2 1
2496.1.bs.b 2 24.h odd 2 1
2496.1.bs.b 2 104.r even 6 1
2496.1.bs.b 2 312.bh odd 6 1
3900.1.bu.b 2 5.b even 2 1
3900.1.bu.b 2 15.d odd 2 1
3900.1.bu.b 2 65.n even 6 1
3900.1.bu.b 2 195.x odd 6 1
3900.1.cb.b 4 5.c odd 4 2
3900.1.cb.b 4 15.e even 4 2
3900.1.cb.b 4 65.q odd 12 2
3900.1.cb.b 4 195.bl even 12 2

Hecke kernels

This newform subspace is the entire newspace S1new(156,[χ])S_{1}^{\mathrm{new}}(156, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2−T+1 T^{2} - T + 1 Copy content Toggle raw display
1111 T2 T^{2} Copy content Toggle raw display
1313 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
1717 T2 T^{2} Copy content Toggle raw display
1919 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
2323 T2 T^{2} Copy content Toggle raw display
2929 T2 T^{2} Copy content Toggle raw display
3131 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
3737 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
4141 T2 T^{2} Copy content Toggle raw display
4343 T2−T+1 T^{2} - T + 1 Copy content Toggle raw display
4747 T2 T^{2} Copy content Toggle raw display
5353 T2 T^{2} Copy content Toggle raw display
5959 T2 T^{2} Copy content Toggle raw display
6161 T2−T+1 T^{2} - T + 1 Copy content Toggle raw display
6767 T2−T+1 T^{2} - T + 1 Copy content Toggle raw display
7171 T2 T^{2} Copy content Toggle raw display
7373 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
7979 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
8383 T2 T^{2} Copy content Toggle raw display
8989 T2 T^{2} Copy content Toggle raw display
9797 T2−T+1 T^{2} - T + 1 Copy content Toggle raw display
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