Properties

Label 2352.2.q.z
Level 23522352
Weight 22
Character orbit 2352.q
Analytic conductor 18.78118.781
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] = [2352,2,Mod(961,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 2352=24372 2352 = 2^{4} \cdot 3 \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 2352.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: 18.780814555418.7808145554
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 84)
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+4ζ6q5ζ6q9+(2ζ6+2)q11+6q13+4q15+(4ζ64)q17+4ζ6q19+2ζ6q23+(11ζ611)q25+2q99+O(q100) q + ( - \zeta_{6} + 1) q^{3} + 4 \zeta_{6} q^{5} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} + 6 q^{13} + 4 q^{15} + (4 \zeta_{6} - 4) q^{17} + 4 \zeta_{6} q^{19} + 2 \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} + \cdots - 2 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+q3+4q5q9+2q11+12q13+8q154q17+4q19+2q2311q252q274q292q332q37+6q39+8q43+4q4512q47+4q51+4q99+O(q100) 2 q + q^{3} + 4 q^{5} - q^{9} + 2 q^{11} + 12 q^{13} + 8 q^{15} - 4 q^{17} + 4 q^{19} + 2 q^{23} - 11 q^{25} - 2 q^{27} - 4 q^{29} - 2 q^{33} - 2 q^{37} + 6 q^{39} + 8 q^{43} + 4 q^{45} - 12 q^{47} + 4 q^{51}+ \cdots - 4 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/2352Z)×\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times.

nn 785785 14711471 17651765 22572257
χ(n)\chi(n) 11 11 11 ζ6-\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}
961.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 2.00000 3.46410i 0 0 0 −0.500000 + 0.866025i 0
1537.1 0 0.500000 0.866025i 0 2.00000 + 3.46410i 0 0 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 2352.2.q.z 2
4.b odd 2 1 588.2.i.d 2
7.b odd 2 1 2352.2.q.b 2
7.c even 3 1 2352.2.a.a 1
7.c even 3 1 inner 2352.2.q.z 2
7.d odd 6 1 336.2.a.f 1
7.d odd 6 1 2352.2.q.b 2
12.b even 2 1 1764.2.k.a 2
21.g even 6 1 1008.2.a.a 1
21.h odd 6 1 7056.2.a.cd 1
28.d even 2 1 588.2.i.e 2
28.f even 6 1 84.2.a.a 1
28.f even 6 1 588.2.i.e 2
28.g odd 6 1 588.2.a.d 1
28.g odd 6 1 588.2.i.d 2
35.i odd 6 1 8400.2.a.e 1
56.j odd 6 1 1344.2.a.a 1
56.k odd 6 1 9408.2.a.bn 1
56.m even 6 1 1344.2.a.k 1
56.p even 6 1 9408.2.a.df 1
84.h odd 2 1 1764.2.k.k 2
84.j odd 6 1 252.2.a.a 1
84.j odd 6 1 1764.2.k.k 2
84.n even 6 1 1764.2.a.k 1
84.n even 6 1 1764.2.k.a 2
112.v even 12 2 5376.2.c.q 2
112.x odd 12 2 5376.2.c.p 2
140.s even 6 1 2100.2.a.r 1
140.x odd 12 2 2100.2.k.i 2
168.ba even 6 1 4032.2.a.bn 1
168.be odd 6 1 4032.2.a.bm 1
252.n even 6 1 2268.2.j.a 2
252.r odd 6 1 2268.2.j.n 2
252.bj even 6 1 2268.2.j.a 2
252.bn odd 6 1 2268.2.j.n 2
420.be odd 6 1 6300.2.a.w 1
420.br even 12 2 6300.2.k.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.a.a 1 28.f even 6 1
252.2.a.a 1 84.j odd 6 1
336.2.a.f 1 7.d odd 6 1
588.2.a.d 1 28.g odd 6 1
588.2.i.d 2 4.b odd 2 1
588.2.i.d 2 28.g odd 6 1
588.2.i.e 2 28.d even 2 1
588.2.i.e 2 28.f even 6 1
1008.2.a.a 1 21.g even 6 1
1344.2.a.a 1 56.j odd 6 1
1344.2.a.k 1 56.m even 6 1
1764.2.a.k 1 84.n even 6 1
1764.2.k.a 2 12.b even 2 1
1764.2.k.a 2 84.n even 6 1
1764.2.k.k 2 84.h odd 2 1
1764.2.k.k 2 84.j odd 6 1
2100.2.a.r 1 140.s even 6 1
2100.2.k.i 2 140.x odd 12 2
2268.2.j.a 2 252.n even 6 1
2268.2.j.a 2 252.bj even 6 1
2268.2.j.n 2 252.r odd 6 1
2268.2.j.n 2 252.bn odd 6 1
2352.2.a.a 1 7.c even 3 1
2352.2.q.b 2 7.b odd 2 1
2352.2.q.b 2 7.d odd 6 1
2352.2.q.z 2 1.a even 1 1 trivial
2352.2.q.z 2 7.c even 3 1 inner
4032.2.a.bm 1 168.be odd 6 1
4032.2.a.bn 1 168.ba even 6 1
5376.2.c.p 2 112.x odd 12 2
5376.2.c.q 2 112.v even 12 2
6300.2.a.w 1 420.be odd 6 1
6300.2.k.g 2 420.br even 12 2
7056.2.a.cd 1 21.h odd 6 1
8400.2.a.e 1 35.i odd 6 1
9408.2.a.bn 1 56.k odd 6 1
9408.2.a.df 1 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(2352,[χ])S_{2}^{\mathrm{new}}(2352, [\chi]):

T524T5+16 T_{5}^{2} - 4T_{5} + 16 Copy content Toggle raw display
T1122T11+4 T_{11}^{2} - 2T_{11} + 4 Copy content Toggle raw display
T136 T_{13} - 6 Copy content Toggle raw display
T172+4T17+16 T_{17}^{2} + 4T_{17} + 16 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 T24T+16 T^{2} - 4T + 16 Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T22T+4 T^{2} - 2T + 4 Copy content Toggle raw display
1313 (T6)2 (T - 6)^{2} Copy content Toggle raw display
1717 T2+4T+16 T^{2} + 4T + 16 Copy content Toggle raw display
1919 T24T+16 T^{2} - 4T + 16 Copy content Toggle raw display
2323 T22T+4 T^{2} - 2T + 4 Copy content Toggle raw display
2929 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
3131 T2 T^{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 (T4)2 (T - 4)^{2} Copy content Toggle raw display
4747 T2+12T+144 T^{2} + 12T + 144 Copy content Toggle raw display
5353 T26T+36 T^{2} - 6T + 36 Copy content Toggle raw display
5959 T28T+64 T^{2} - 8T + 64 Copy content Toggle raw display
6161 T26T+36 T^{2} - 6T + 36 Copy content Toggle raw display
6767 T2+8T+64 T^{2} + 8T + 64 Copy content Toggle raw display
7171 (T+14)2 (T + 14)^{2} Copy content Toggle raw display
7373 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
7979 T212T+144 T^{2} - 12T + 144 Copy content Toggle raw display
8383 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
8989 T2 T^{2} Copy content Toggle raw display
9797 (T2)2 (T - 2)^{2} Copy content Toggle raw display
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