Properties

Label 136.1.j.a
Level 136136
Weight 11
Character orbit 136.j
Analytic conductor 0.0680.068
Analytic rank 00
Dimension 22
Projective image D4D_{4}
CM discriminant -8
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [136,1,Mod(115,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.115");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 136=2317 136 = 2^{3} \cdot 17
Weight: k k == 1 1
Character orbit: [χ][\chi] == 136.j (of order 44, 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: 0.06787284171810.0678728417181
Analytic rank: 00
Dimension: 22
Coefficient field: Q(i)\Q(i)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2+1 x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Projective image: D4D_{4}
Projective field: Galois closure of 4.0.314432.1
Artin image: C4C2C_4\wr C_2
Artin field: Galois closure of 8.0.20123648.1

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) == qiq2+(i1)q3q4+(i1)q6+iq8+iq9+(i+1)q11+(i+1)q12+q16iq17+q18+2iq19+(i1)q22+(i+1)q24+iq25++(i+1)q99+O(q100) q - i q^{2} + ( - i - 1) q^{3} - q^{4} + (i - 1) q^{6} + i q^{8} + i q^{9} + ( - i + 1) q^{11} + (i + 1) q^{12} + q^{16} - i q^{17} + q^{18} + 2 i q^{19} + ( - i - 1) q^{22} + ( - i + 1) q^{24} + i q^{25} + \cdots + (i + 1) q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q32q42q6+2q11+2q12+2q16+2q182q22+2q244q332q34+4q382q412q442q48+2q502q51+4q572q64++2q99+O(q100) 2 q - 2 q^{3} - 2 q^{4} - 2 q^{6} + 2 q^{11} + 2 q^{12} + 2 q^{16} + 2 q^{18} - 2 q^{22} + 2 q^{24} - 4 q^{33} - 2 q^{34} + 4 q^{38} - 2 q^{41} - 2 q^{44} - 2 q^{48} + 2 q^{50} - 2 q^{51} + 4 q^{57} - 2 q^{64}+ \cdots + 2 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/136Z)×\left(\mathbb{Z}/136\mathbb{Z}\right)^\times.

nn 6969 103103 105105
χ(n)\chi(n) 1-1 1-1 ii

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}
115.1
1.00000i
1.00000i
1.00000i −1.00000 1.00000i −1.00000 0 −1.00000 + 1.00000i 0 1.00000i 1.00000i 0
123.1 1.00000i −1.00000 + 1.00000i −1.00000 0 −1.00000 1.00000i 0 1.00000i 1.00000i 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
8.d odd 2 1 CM by Q(2)\Q(\sqrt{-2})
17.c even 4 1 inner
136.j odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.1.j.a 2
3.b odd 2 1 1224.1.s.a 2
4.b odd 2 1 544.1.n.a 2
5.b even 2 1 3400.1.y.a 2
5.c odd 4 1 3400.1.bc.a 2
5.c odd 4 1 3400.1.bc.b 2
8.b even 2 1 544.1.n.a 2
8.d odd 2 1 CM 136.1.j.a 2
17.b even 2 1 2312.1.j.b 2
17.c even 4 1 inner 136.1.j.a 2
17.c even 4 1 2312.1.j.b 2
17.d even 8 2 2312.1.e.a 2
17.d even 8 2 2312.1.f.b 2
17.e odd 16 8 2312.1.p.e 8
24.f even 2 1 1224.1.s.a 2
40.e odd 2 1 3400.1.y.a 2
40.k even 4 1 3400.1.bc.a 2
40.k even 4 1 3400.1.bc.b 2
51.f odd 4 1 1224.1.s.a 2
68.f odd 4 1 544.1.n.a 2
85.f odd 4 1 3400.1.bc.a 2
85.i odd 4 1 3400.1.bc.b 2
85.j even 4 1 3400.1.y.a 2
136.e odd 2 1 2312.1.j.b 2
136.i even 4 1 544.1.n.a 2
136.j odd 4 1 inner 136.1.j.a 2
136.j odd 4 1 2312.1.j.b 2
136.p odd 8 2 2312.1.e.a 2
136.p odd 8 2 2312.1.f.b 2
136.s even 16 8 2312.1.p.e 8
408.q even 4 1 1224.1.s.a 2
680.t even 4 1 3400.1.bc.a 2
680.bc odd 4 1 3400.1.y.a 2
680.bl even 4 1 3400.1.bc.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.j.a 2 1.a even 1 1 trivial
136.1.j.a 2 8.d odd 2 1 CM
136.1.j.a 2 17.c even 4 1 inner
136.1.j.a 2 136.j odd 4 1 inner
544.1.n.a 2 4.b odd 2 1
544.1.n.a 2 8.b even 2 1
544.1.n.a 2 68.f odd 4 1
544.1.n.a 2 136.i even 4 1
1224.1.s.a 2 3.b odd 2 1
1224.1.s.a 2 24.f even 2 1
1224.1.s.a 2 51.f odd 4 1
1224.1.s.a 2 408.q even 4 1
2312.1.e.a 2 17.d even 8 2
2312.1.e.a 2 136.p odd 8 2
2312.1.f.b 2 17.d even 8 2
2312.1.f.b 2 136.p odd 8 2
2312.1.j.b 2 17.b even 2 1
2312.1.j.b 2 17.c even 4 1
2312.1.j.b 2 136.e odd 2 1
2312.1.j.b 2 136.j odd 4 1
2312.1.p.e 8 17.e odd 16 8
2312.1.p.e 8 136.s even 16 8
3400.1.y.a 2 5.b even 2 1
3400.1.y.a 2 40.e odd 2 1
3400.1.y.a 2 85.j even 4 1
3400.1.y.a 2 680.bc odd 4 1
3400.1.bc.a 2 5.c odd 4 1
3400.1.bc.a 2 40.k even 4 1
3400.1.bc.a 2 85.f odd 4 1
3400.1.bc.a 2 680.t even 4 1
3400.1.bc.b 2 5.c odd 4 1
3400.1.bc.b 2 40.k even 4 1
3400.1.bc.b 2 85.i odd 4 1
3400.1.bc.b 2 680.bl even 4 1

Hecke kernels

This newform subspace is the entire newspace S1new(136,[χ])S_{1}^{\mathrm{new}}(136, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+1 T^{2} + 1 Copy content Toggle raw display
33 T2+2T+2 T^{2} + 2T + 2 Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T22T+2 T^{2} - 2T + 2 Copy content Toggle raw display
1313 T2 T^{2} Copy content Toggle raw display
1717 T2+1 T^{2} + 1 Copy content Toggle raw display
1919 T2+4 T^{2} + 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 T2 T^{2} Copy content Toggle raw display
3737 T2 T^{2} Copy content Toggle raw display
4141 T2+2T+2 T^{2} + 2T + 2 Copy content Toggle raw display
4343 T2 T^{2} 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^{2} Copy content Toggle raw display
6767 T2 T^{2} Copy content Toggle raw display
7171 T2 T^{2} Copy content Toggle raw display
7373 T2+2T+2 T^{2} + 2T + 2 Copy content Toggle raw display
7979 T2 T^{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 T22T+2 T^{2} - 2T + 2 Copy content Toggle raw display
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