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Properties

Label 2664.1.cy.a
Level 26642664
Weight 11
Character orbit 2664.cy
Analytic conductor 1.3301.330
Analytic rank 00
Dimension 88
Projective image S4S_{4}
CM/RM no
Inner twists 88

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2664,1,Mod(1099,2664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2664.1099"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2664, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 0, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: N N == 2664=23⋅32⋅37 2664 = 2^{3} \cdot 3^{2} \cdot 37
Weight: k k == 1 1
Character orbit: [χ][\chi] == 2664.cy (of order 66, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 1.329509193651.32950919365
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q(ζ24)\Q(\zeta_{24})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x8−x4+1 x^{8} - x^{4} + 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: S4S_{4}
Projective field: Galois closure of 4.2.262848.3

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

The qq-expansion and trace form are shown below.

f(q)f(q) == q−ζ247q2−ζ242q4+(ζ2411−ζ245)q5+ζ242q7+ζ249q8+(ζ246−1)q10+(−ζ249+ζ243)q11+⋯−q97+O(q100) q - \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{5} + \zeta_{24}^{2} q^{7} + \zeta_{24}^{9} q^{8} + (\zeta_{24}^{6} - 1) q^{10} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{11} + \cdots - q^{97} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−8q10+4q16−8q19−4q22+4q25−4q28+4q34+4q40+8q43−4q52−4q58+4q67−4q70+8q73−8q88+4q91−8q97+O(q100) 8 q - 8 q^{10} + 4 q^{16} - 8 q^{19} - 4 q^{22} + 4 q^{25} - 4 q^{28} + 4 q^{34} + 4 q^{40} + 8 q^{43} - 4 q^{52} - 4 q^{58} + 4 q^{67} - 4 q^{70} + 8 q^{73} - 8 q^{88} + 4 q^{91} - 8 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/2664Z)×\left(\mathbb{Z}/2664\mathbb{Z}\right)^\times.

nn 12971297 13331333 19991999 23692369
χ(n)\chi(n) ζ248\zeta_{24}^{8} −1-1 −1-1 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
1099.1
−0.258819 + 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.965926 − 0.258819i 0 0.866025 + 0.500000i 1.22474 + 0.707107i 0 −0.866025 − 0.500000i −0.707107 − 0.707107i 0 −1.00000 − 1.00000i
1099.2 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i 1.22474 + 0.707107i 0 0.866025 + 0.500000i 0.707107 − 0.707107i 0 −1.00000 + 1.00000i
1099.3 0.258819 − 0.965926i 0 −0.866025 − 0.500000i −1.22474 − 0.707107i 0 0.866025 + 0.500000i −0.707107 + 0.707107i 0 −1.00000 + 1.00000i
1099.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i −1.22474 − 0.707107i 0 −0.866025 − 0.500000i 0.707107 + 0.707107i 0 −1.00000 − 1.00000i
1675.1 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 1.22474 − 0.707107i 0 −0.866025 + 0.500000i −0.707107 + 0.707107i 0 −1.00000 + 1.00000i
1675.2 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 1.22474 − 0.707107i 0 0.866025 − 0.500000i 0.707107 + 0.707107i 0 −1.00000 − 1.00000i
1675.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −1.22474 + 0.707107i 0 0.866025 − 0.500000i −0.707107 − 0.707107i 0 −1.00000 − 1.00000i
1675.4 0.965926 − 0.258819i 0 0.866025 − 0.500000i −1.22474 + 0.707107i 0 −0.866025 + 0.500000i 0.707107 − 0.707107i 0 −1.00000 + 1.00000i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1099.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner
37.c even 3 1 inner
111.i odd 6 1 inner
296.p odd 6 1 inner
888.bd even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2664.1.cy.a ✓ 8
3.b odd 2 1 inner 2664.1.cy.a ✓ 8
8.d odd 2 1 inner 2664.1.cy.a ✓ 8
24.f even 2 1 inner 2664.1.cy.a ✓ 8
37.c even 3 1 inner 2664.1.cy.a ✓ 8
111.i odd 6 1 inner 2664.1.cy.a ✓ 8
296.p odd 6 1 inner 2664.1.cy.a ✓ 8
888.bd even 6 1 inner 2664.1.cy.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2664.1.cy.a ✓ 8 1.a even 1 1 trivial
2664.1.cy.a ✓ 8 3.b odd 2 1 inner
2664.1.cy.a ✓ 8 8.d odd 2 1 inner
2664.1.cy.a ✓ 8 24.f even 2 1 inner
2664.1.cy.a ✓ 8 37.c even 3 1 inner
2664.1.cy.a ✓ 8 111.i odd 6 1 inner
2664.1.cy.a ✓ 8 296.p odd 6 1 inner
2664.1.cy.a ✓ 8 888.bd even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace S1new(2664,[χ])S_{1}^{\mathrm{new}}(2664, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8−T4+1 T^{8} - T^{4} + 1 Copy content Toggle raw display
33 T8 T^{8} Copy content Toggle raw display
55 (T4−2T2+4)2 (T^{4} - 2 T^{2} + 4)^{2} Copy content Toggle raw display
77 (T4−T2+1)2 (T^{4} - T^{2} + 1)^{2} Copy content Toggle raw display
1111 (T2−2)4 (T^{2} - 2)^{4} Copy content Toggle raw display
1313 (T4−T2+1)2 (T^{4} - T^{2} + 1)^{2} Copy content Toggle raw display
1717 (T4+2T2+4)2 (T^{4} + 2 T^{2} + 4)^{2} Copy content Toggle raw display
1919 (T2+2T+4)4 (T^{2} + 2 T + 4)^{4} Copy content Toggle raw display
2323 T8 T^{8} Copy content Toggle raw display
2929 (T2+2)4 (T^{2} + 2)^{4} Copy content Toggle raw display
3131 (T2+1)4 (T^{2} + 1)^{4} Copy content Toggle raw display
3737 (T2+1)4 (T^{2} + 1)^{4} Copy content Toggle raw display
4141 T8 T^{8} Copy content Toggle raw display
4343 (T−1)8 (T - 1)^{8} Copy content Toggle raw display
4747 T8 T^{8} Copy content Toggle raw display
5353 (T4−2T2+4)2 (T^{4} - 2 T^{2} + 4)^{2} Copy content Toggle raw display
5959 T8 T^{8} Copy content Toggle raw display
6161 T8 T^{8} Copy content Toggle raw display
6767 (T2−T+1)4 (T^{2} - T + 1)^{4} Copy content Toggle raw display
7171 T8 T^{8} Copy content Toggle raw display
7373 (T−1)8 (T - 1)^{8} Copy content Toggle raw display
7979 (T4−T2+1)2 (T^{4} - T^{2} + 1)^{2} Copy content Toggle raw display
8383 (T4+2T2+4)2 (T^{4} + 2 T^{2} + 4)^{2} Copy content Toggle raw display
8989 (T4+2T2+4)2 (T^{4} + 2 T^{2} + 4)^{2} Copy content Toggle raw display
9797 (T+1)8 (T + 1)^{8} Copy content Toggle raw display
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