Properties

Label 1344.2.q.u
Level 13441344
Weight 22
Character orbit 1344.q
Analytic conductor 10.73210.732
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] = [1344,2,Mod(193,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1344=2637 1344 = 2^{6} \cdot 3 \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1344.q (of order 33, degree 22, not 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: 10.731894031710.7318940317
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,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 672)
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+(ζ6+1)q3+3ζ6q5+(3ζ61)q7ζ6q9+(ζ61)q11+4q13+3q15+(4ζ64)q17+(ζ6+2)q218ζ6q23++q99+O(q100) q + ( - \zeta_{6} + 1) q^{3} + 3 \zeta_{6} q^{5} + (3 \zeta_{6} - 1) q^{7} - \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{11} + 4 q^{13} + 3 q^{15} + (4 \zeta_{6} - 4) q^{17} + (\zeta_{6} + 2) q^{21} - 8 \zeta_{6} q^{23} + \cdots + q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+q3+3q5+q7q9q11+8q13+6q154q17+5q218q234q252q27+14q2911q31+q3312q35+4q37+4q398q41++2q99+O(q100) 2 q + q^{3} + 3 q^{5} + q^{7} - q^{9} - q^{11} + 8 q^{13} + 6 q^{15} - 4 q^{17} + 5 q^{21} - 8 q^{23} - 4 q^{25} - 2 q^{27} + 14 q^{29} - 11 q^{31} + q^{33} - 12 q^{35} + 4 q^{37} + 4 q^{39} - 8 q^{41}+ \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/1344Z)×\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times.

nn 127127 449449 577577 10931093
χ(n)\chi(n) 11 11 ζ6-\zeta_{6} 11

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}
193.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 1.50000 + 2.59808i 0 0.500000 + 2.59808i 0 −0.500000 0.866025i 0
961.1 0 0.500000 + 0.866025i 0 1.50000 2.59808i 0 0.500000 2.59808i 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
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 1344.2.q.u 2
4.b odd 2 1 1344.2.q.k 2
7.c even 3 1 inner 1344.2.q.u 2
7.c even 3 1 9408.2.a.e 1
7.d odd 6 1 9408.2.a.dc 1
8.b even 2 1 672.2.q.a 2
8.d odd 2 1 672.2.q.f yes 2
24.f even 2 1 2016.2.s.k 2
24.h odd 2 1 2016.2.s.n 2
28.f even 6 1 9408.2.a.bl 1
28.g odd 6 1 1344.2.q.k 2
28.g odd 6 1 9408.2.a.bt 1
56.j odd 6 1 4704.2.a.b 1
56.k odd 6 1 672.2.q.f yes 2
56.k odd 6 1 4704.2.a.o 1
56.m even 6 1 4704.2.a.s 1
56.p even 6 1 672.2.q.a 2
56.p even 6 1 4704.2.a.bf 1
168.s odd 6 1 2016.2.s.n 2
168.v even 6 1 2016.2.s.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.a 2 8.b even 2 1
672.2.q.a 2 56.p even 6 1
672.2.q.f yes 2 8.d odd 2 1
672.2.q.f yes 2 56.k odd 6 1
1344.2.q.k 2 4.b odd 2 1
1344.2.q.k 2 28.g odd 6 1
1344.2.q.u 2 1.a even 1 1 trivial
1344.2.q.u 2 7.c even 3 1 inner
2016.2.s.k 2 24.f even 2 1
2016.2.s.k 2 168.v even 6 1
2016.2.s.n 2 24.h odd 2 1
2016.2.s.n 2 168.s odd 6 1
4704.2.a.b 1 56.j odd 6 1
4704.2.a.o 1 56.k odd 6 1
4704.2.a.s 1 56.m even 6 1
4704.2.a.bf 1 56.p even 6 1
9408.2.a.e 1 7.c even 3 1
9408.2.a.bl 1 28.f even 6 1
9408.2.a.bt 1 28.g odd 6 1
9408.2.a.dc 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1344,[χ])S_{2}^{\mathrm{new}}(1344, [\chi]):

T523T5+9 T_{5}^{2} - 3T_{5} + 9 Copy content Toggle raw display
T112+T11+1 T_{11}^{2} + T_{11} + 1 Copy content Toggle raw display
T134 T_{13} - 4 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
55 T23T+9 T^{2} - 3T + 9 Copy content Toggle raw display
77 T2T+7 T^{2} - T + 7 Copy content Toggle raw display
1111 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
1313 (T4)2 (T - 4)^{2} Copy content Toggle raw display
1717 T2+4T+16 T^{2} + 4T + 16 Copy content Toggle raw display
1919 T2 T^{2} Copy content Toggle raw display
2323 T2+8T+64 T^{2} + 8T + 64 Copy content Toggle raw display
2929 (T7)2 (T - 7)^{2} Copy content Toggle raw display
3131 T2+11T+121 T^{2} + 11T + 121 Copy content Toggle raw display
3737 T24T+16 T^{2} - 4T + 16 Copy content Toggle raw display
4141 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
4343 (T2)2 (T - 2)^{2} Copy content Toggle raw display
4747 T22T+4 T^{2} - 2T + 4 Copy content Toggle raw display
5353 T2+11T+121 T^{2} + 11T + 121 Copy content Toggle raw display
5959 T27T+49 T^{2} - 7T + 49 Copy content Toggle raw display
6161 T210T+100 T^{2} - 10T + 100 Copy content Toggle raw display
6767 T210T+100 T^{2} - 10T + 100 Copy content Toggle raw display
7171 (T6)2 (T - 6)^{2} Copy content Toggle raw display
7373 T26T+36 T^{2} - 6T + 36 Copy content Toggle raw display
7979 T2+11T+121 T^{2} + 11T + 121 Copy content Toggle raw display
8383 (T+11)2 (T + 11)^{2} Copy content Toggle raw display
8989 T2+6T+36 T^{2} + 6T + 36 Copy content Toggle raw display
9797 (T7)2 (T - 7)^{2} Copy content Toggle raw display
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