Properties

Label 2352.2.b.a
Level 23522352
Weight 22
Character orbit 2352.b
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(1567,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.1567");
 
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.b (of order 22, degree 11, 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,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

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 β=3\beta = \sqrt{-3}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qq32βq5+q92βq11+3βq13+2βq15+4βq177q197q25q275q31+2βq33+q373βq39+6βq41+2βq99+O(q100) q - q^{3} - 2 \beta q^{5} + q^{9} - 2 \beta q^{11} + 3 \beta q^{13} + 2 \beta q^{15} + 4 \beta q^{17} - 7 q^{19} - 7 q^{25} - q^{27} - 5 q^{31} + 2 \beta q^{33} + q^{37} - 3 \beta q^{39} + 6 \beta q^{41} + \cdots - 2 \beta q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q3+2q914q1914q252q2710q31+2q37+12q4724q55+14q57+36q65+14q75+2q81+12q83+48q85+10q93+O(q100) 2 q - 2 q^{3} + 2 q^{9} - 14 q^{19} - 14 q^{25} - 2 q^{27} - 10 q^{31} + 2 q^{37} + 12 q^{47} - 24 q^{55} + 14 q^{57} + 36 q^{65} + 14 q^{75} + 2 q^{81} + 12 q^{83} + 48 q^{85} + 10 q^{93}+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 1-1 11 1-1

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}
1567.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.00000 0 3.46410i 0 0 0 1.00000 0
1567.2 0 −1.00000 0 3.46410i 0 0 0 1.00000 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
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.b.a 2
3.b odd 2 1 7056.2.b.a 2
4.b odd 2 1 2352.2.b.h 2
7.b odd 2 1 2352.2.b.h 2
7.c even 3 1 336.2.bl.e yes 2
7.c even 3 1 2352.2.bl.l 2
7.d odd 6 1 336.2.bl.a 2
7.d odd 6 1 2352.2.bl.f 2
12.b even 2 1 7056.2.b.l 2
21.c even 2 1 7056.2.b.l 2
21.g even 6 1 1008.2.cs.n 2
21.h odd 6 1 1008.2.cs.m 2
28.d even 2 1 inner 2352.2.b.a 2
28.f even 6 1 336.2.bl.e yes 2
28.f even 6 1 2352.2.bl.l 2
28.g odd 6 1 336.2.bl.a 2
28.g odd 6 1 2352.2.bl.f 2
56.j odd 6 1 1344.2.bl.h 2
56.k odd 6 1 1344.2.bl.h 2
56.m even 6 1 1344.2.bl.d 2
56.p even 6 1 1344.2.bl.d 2
84.h odd 2 1 7056.2.b.a 2
84.j odd 6 1 1008.2.cs.m 2
84.n even 6 1 1008.2.cs.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bl.a 2 7.d odd 6 1
336.2.bl.a 2 28.g odd 6 1
336.2.bl.e yes 2 7.c even 3 1
336.2.bl.e yes 2 28.f even 6 1
1008.2.cs.m 2 21.h odd 6 1
1008.2.cs.m 2 84.j odd 6 1
1008.2.cs.n 2 21.g even 6 1
1008.2.cs.n 2 84.n even 6 1
1344.2.bl.d 2 56.m even 6 1
1344.2.bl.d 2 56.p even 6 1
1344.2.bl.h 2 56.j odd 6 1
1344.2.bl.h 2 56.k odd 6 1
2352.2.b.a 2 1.a even 1 1 trivial
2352.2.b.a 2 28.d even 2 1 inner
2352.2.b.h 2 4.b odd 2 1
2352.2.b.h 2 7.b odd 2 1
2352.2.bl.f 2 7.d odd 6 1
2352.2.bl.f 2 28.g odd 6 1
2352.2.bl.l 2 7.c even 3 1
2352.2.bl.l 2 28.f even 6 1
7056.2.b.a 2 3.b odd 2 1
7056.2.b.a 2 84.h odd 2 1
7056.2.b.l 2 12.b even 2 1
7056.2.b.l 2 21.c even 2 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]):

T52+12 T_{5}^{2} + 12 Copy content Toggle raw display
T112+12 T_{11}^{2} + 12 Copy content Toggle raw display
T132+27 T_{13}^{2} + 27 Copy content Toggle raw display
T19+7 T_{19} + 7 Copy content Toggle raw display
T31+5 T_{31} + 5 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
55 T2+12 T^{2} + 12 Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T2+12 T^{2} + 12 Copy content Toggle raw display
1313 T2+27 T^{2} + 27 Copy content Toggle raw display
1717 T2+48 T^{2} + 48 Copy content Toggle raw display
1919 (T+7)2 (T + 7)^{2} 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+5)2 (T + 5)^{2} Copy content Toggle raw display
3737 (T1)2 (T - 1)^{2} Copy content Toggle raw display
4141 T2+108 T^{2} + 108 Copy content Toggle raw display
4343 T2+3 T^{2} + 3 Copy content Toggle raw display
4747 (T6)2 (T - 6)^{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+3 T^{2} + 3 Copy content Toggle raw display
7171 T2+12 T^{2} + 12 Copy content Toggle raw display
7373 T2+75 T^{2} + 75 Copy content Toggle raw display
7979 T2+243 T^{2} + 243 Copy content Toggle raw display
8383 (T6)2 (T - 6)^{2} Copy content Toggle raw display
8989 T2+48 T^{2} + 48 Copy content Toggle raw display
9797 T2+48 T^{2} + 48 Copy content Toggle raw display
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