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Properties

Label 1166.2.c.a
Level 11661166
Weight 22
Character orbit 1166.c
Analytic conductor 9.3119.311
Analytic rank 00
Dimension 22
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1166,2,Mod(529,1166)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1166, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1166.529"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 1166=2⋅11⋅53 1166 = 2 \cdot 11 \cdot 53
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1166.c (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 9.310556875689.31055687568
Analytic rank: 00
Dimension: 22
Coefficient field: Q(−1)\Q(\sqrt{-1})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2+1 x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

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)
 

Coefficients of the qq-expansion are expressed in terms of i=−1i = \sqrt{-1}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q−iq2+iq3−q4−iq5+q6−4q7+iq8+2q9−q10−q11−iq12−q13+4iq14+q15+q16+4q17−2iq18+6iq19+iq20+⋯−2q99+O(q100) q - i q^{2} + i q^{3} - q^{4} - i q^{5} + q^{6} - 4 q^{7} + i q^{8} + 2 q^{9} - q^{10} - q^{11} - i q^{12} - q^{13} + 4 i q^{14} + q^{15} + q^{16} + 4 q^{17} - 2 i q^{18} + 6 i q^{19} + i q^{20} + \cdots - 2 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 2q−2q4+2q6−8q7+4q9−2q10−2q11−2q13+2q15+2q16+8q17−2q24+8q25+8q28−4q29−4q36+24q37+12q38+2q40−8q42+⋯−4q99+O(q100) 2 q - 2 q^{4} + 2 q^{6} - 8 q^{7} + 4 q^{9} - 2 q^{10} - 2 q^{11} - 2 q^{13} + 2 q^{15} + 2 q^{16} + 8 q^{17} - 2 q^{24} + 8 q^{25} + 8 q^{28} - 4 q^{29} - 4 q^{36} + 24 q^{37} + 12 q^{38} + 2 q^{40} - 8 q^{42}+ \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/1166Z)×\left(\mathbb{Z}/1166\mathbb{Z}\right)^\times.

nn 849849 903903
χ(n)\chi(n) 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.

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}
529.1
1.00000i
− 1.00000i
− 1.00000i 1.00000i −1.00000 − 1.00000i 1.00000 −4.00000 1.00000i 2.00000 −1.00000
529.2 1.00000i − 1.00000i −1.00000 1.00000i 1.00000 −4.00000 − 1.00000i 2.00000 −1.00000
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
53.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1166.2.c.a ✓ 2
53.b even 2 1 inner 1166.2.c.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1166.2.c.a ✓ 2 1.a even 1 1 trivial
1166.2.c.a ✓ 2 53.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T32+1 T_{3}^{2} + 1 acting on S2new(1166,[χ])S_{2}^{\mathrm{new}}(1166, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+1 T^{2} + 1 Copy content Toggle raw display
33 T2+1 T^{2} + 1 Copy content Toggle raw display
55 T2+1 T^{2} + 1 Copy content Toggle raw display
77 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
1111 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
1313 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
1717 (T−4)2 (T - 4)^{2} Copy content Toggle raw display
1919 T2+36 T^{2} + 36 Copy content Toggle raw display
2323 T2+4 T^{2} + 4 Copy content Toggle raw display
2929 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
3131 T2+4 T^{2} + 4 Copy content Toggle raw display
3737 (T−12)2 (T - 12)^{2} Copy content Toggle raw display
4141 T2+81 T^{2} + 81 Copy content Toggle raw display
4343 (T−8)2 (T - 8)^{2} Copy content Toggle raw display
4747 (T+5)2 (T + 5)^{2} Copy content Toggle raw display
5353 T2+4T+53 T^{2} + 4T + 53 Copy content Toggle raw display
5959 (T−6)2 (T - 6)^{2} Copy content Toggle raw display
6161 T2+64 T^{2} + 64 Copy content Toggle raw display
6767 T2+49 T^{2} + 49 Copy content Toggle raw display
7171 T2 T^{2} Copy content Toggle raw display
7373 T2+4 T^{2} + 4 Copy content Toggle raw display
7979 T2+49 T^{2} + 49 Copy content Toggle raw display
8383 T2+36 T^{2} + 36 Copy content Toggle raw display
8989 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
9797 (T−2)2 (T - 2)^{2} Copy content Toggle raw display
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