Properties

Label 2940.2.q.n
Level 29402940
Weight 22
Character orbit 2940.q
Analytic conductor 23.47623.476
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] = [2940,2,Mod(361,2940)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2940, 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("2940.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 2940=223572 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 2940.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: 23.476018194323.4760181943
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: yes
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+ζ6q5ζ6q9+(6ζ6+6)q11+q15+(6ζ66)q174ζ6q196ζ6q23+(ζ61)q25q27+6q99+O(q100) q + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{5} - \zeta_{6} q^{9} + ( - 6 \zeta_{6} + 6) q^{11} + q^{15} + (6 \zeta_{6} - 6) q^{17} - 4 \zeta_{6} q^{19} - 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - q^{27} + \cdots - 6 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+q3+q5q9+6q11+2q156q174q196q23q252q274q29+8q316q33+2q37+20q4124q43+q458q47+6q51+12q99+O(q100) 2 q + q^{3} + q^{5} - q^{9} + 6 q^{11} + 2 q^{15} - 6 q^{17} - 4 q^{19} - 6 q^{23} - q^{25} - 2 q^{27} - 4 q^{29} + 8 q^{31} - 6 q^{33} + 2 q^{37} + 20 q^{41} - 24 q^{43} + q^{45} - 8 q^{47} + 6 q^{51}+ \cdots - 12 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/2940Z)×\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times.

nn 10811081 11771177 14711471 19611961
χ(n)\chi(n) ζ6-\zeta_{6} 11 11 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}
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 0 −0.500000 0.866025i 0
961.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 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 2940.2.q.n 2
7.b odd 2 1 2940.2.q.d 2
7.c even 3 1 2940.2.a.a 1
7.c even 3 1 inner 2940.2.q.n 2
7.d odd 6 1 2940.2.a.j yes 1
7.d odd 6 1 2940.2.q.d 2
21.g even 6 1 8820.2.a.o 1
21.h odd 6 1 8820.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2940.2.a.a 1 7.c even 3 1
2940.2.a.j yes 1 7.d odd 6 1
2940.2.q.d 2 7.b odd 2 1
2940.2.q.d 2 7.d odd 6 1
2940.2.q.n 2 1.a even 1 1 trivial
2940.2.q.n 2 7.c even 3 1 inner
8820.2.a.o 1 21.g even 6 1
8820.2.a.bc 1 21.h 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(2940,[χ])S_{2}^{\mathrm{new}}(2940, [\chi]):

T1126T11+36 T_{11}^{2} - 6T_{11} + 36 Copy content Toggle raw display
T13 T_{13} Copy content Toggle raw display
T172+6T17+36 T_{17}^{2} + 6T_{17} + 36 Copy content Toggle raw display
T3128T31+64 T_{31}^{2} - 8T_{31} + 64 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 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T26T+36 T^{2} - 6T + 36 Copy content Toggle raw display
1313 T2 T^{2} Copy content Toggle raw display
1717 T2+6T+36 T^{2} + 6T + 36 Copy content Toggle raw display
1919 T2+4T+16 T^{2} + 4T + 16 Copy content Toggle raw display
2323 T2+6T+36 T^{2} + 6T + 36 Copy content Toggle raw display
2929 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
3131 T28T+64 T^{2} - 8T + 64 Copy content Toggle raw display
3737 T22T+4 T^{2} - 2T + 4 Copy content Toggle raw display
4141 (T10)2 (T - 10)^{2} Copy content Toggle raw display
4343 (T+12)2 (T + 12)^{2} Copy content Toggle raw display
4747 T2+8T+64 T^{2} + 8T + 64 Copy content Toggle raw display
5353 T22T+4 T^{2} - 2T + 4 Copy content Toggle raw display
5959 T2+4T+16 T^{2} + 4T + 16 Copy content Toggle raw display
6161 T28T+64 T^{2} - 8T + 64 Copy content Toggle raw display
6767 T216T+256 T^{2} - 16T + 256 Copy content Toggle raw display
7171 (T+10)2 (T + 10)^{2} Copy content Toggle raw display
7373 T2 T^{2} Copy content Toggle raw display
7979 T2+4T+16 T^{2} + 4T + 16 Copy content Toggle raw display
8383 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
8989 T26T+36 T^{2} - 6T + 36 Copy content Toggle raw display
9797 (T+8)2 (T + 8)^{2} Copy content Toggle raw display
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