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Properties

Label 2280.1.el.b
Level 22802280
Weight 11
Character orbit 2280.el
Analytic conductor 1.1381.138
Analytic rank 00
Dimension 1212
Projective image D18D_{18}
CM discriminant -15
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] = [2280,1,Mod(149,2280)] 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("2280.149"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2280, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 9, 9, 9, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: N N == 2280=23⋅3⋅5⋅19 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19
Weight: k k == 1 1
Character orbit: [χ][\chi] == 2280.el (of order 1818, degree 66, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 1.137868228801.13786822880
Analytic rank: 00
Dimension: 1212
Relative dimension: 22 over Q(ζ18)\Q(\zeta_{18})
Coefficient field: Q(ζ36)\Q(\zeta_{36})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x12−x6+1 x^{12} - x^{6} + 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
Projective image: D18D_{18}
Projective field: Galois closure of Q[x]/(x18−⋯ )\mathbb{Q}[x]/(x^{18} - \cdots)

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−ζ3615q2+ζ3617q3−ζ3612q4−ζ365q5+ζ3614q6−ζ369q8−ζ3616q9−ζ362q10+ζ3611q12+⋯−ζ363q98+O(q100) q - \zeta_{36}^{15} q^{2} + \zeta_{36}^{17} q^{3} - \zeta_{36}^{12} q^{4} - \zeta_{36}^{5} q^{5} + \zeta_{36}^{14} q^{6} - \zeta_{36}^{9} q^{8} - \zeta_{36}^{16} q^{9} - \zeta_{36}^{2} q^{10} + \zeta_{36}^{11} q^{12} + \cdots - \zeta_{36}^{3} q^{98} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 12q+6q4−6q16+6q34−6q49−6q51−6q54−12q64+18q69−12q85−12q90−12q94+O(q100) 12 q + 6 q^{4} - 6 q^{16} + 6 q^{34} - 6 q^{49} - 6 q^{51} - 6 q^{54} - 12 q^{64} + 18 q^{69} - 12 q^{85} - 12 q^{90} - 12 q^{94}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/2280Z)×\left(\mathbb{Z}/2280\mathbb{Z}\right)^\times.

nn 457457 761761 11411141 17111711 19211921
χ(n)\chi(n) −1-1 −1-1 −1-1 11 ζ364\zeta_{36}^{4}

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}
149.1
0.342020 − 0.939693i
−0.342020 + 0.939693i
0.642788 − 0.766044i
−0.642788 + 0.766044i
0.342020 + 0.939693i
−0.342020 − 0.939693i
−0.984808 + 0.173648i
0.984808 − 0.173648i
0.642788 + 0.766044i
−0.642788 − 0.766044i
−0.984808 − 0.173648i
0.984808 + 0.173648i
−0.866025 − 0.500000i −0.342020 − 0.939693i 0.500000 + 0.866025i −0.984808 − 0.173648i −0.173648 + 0.984808i 0 − 1.00000i −0.766044 + 0.642788i 0.766044 + 0.642788i
149.2 0.866025 + 0.500000i 0.342020 + 0.939693i 0.500000 + 0.866025i 0.984808 + 0.173648i −0.173648 + 0.984808i 0 1.00000i −0.766044 + 0.642788i 0.766044 + 0.642788i
389.1 −0.866025 + 0.500000i −0.642788 − 0.766044i 0.500000 − 0.866025i 0.342020 − 0.939693i 0.939693 + 0.342020i 0 1.00000i −0.173648 + 0.984808i 0.173648 + 0.984808i
389.2 0.866025 − 0.500000i 0.642788 + 0.766044i 0.500000 − 0.866025i −0.342020 + 0.939693i 0.939693 + 0.342020i 0 − 1.00000i −0.173648 + 0.984808i 0.173648 + 0.984808i
1469.1 −0.866025 + 0.500000i −0.342020 + 0.939693i 0.500000 − 0.866025i −0.984808 + 0.173648i −0.173648 − 0.984808i 0 1.00000i −0.766044 − 0.642788i 0.766044 − 0.642788i
1469.2 0.866025 − 0.500000i 0.342020 − 0.939693i 0.500000 − 0.866025i 0.984808 − 0.173648i −0.173648 − 0.984808i 0 − 1.00000i −0.766044 − 0.642788i 0.766044 − 0.642788i
1829.1 −0.866025 − 0.500000i 0.984808 + 0.173648i 0.500000 + 0.866025i 0.642788 − 0.766044i −0.766044 − 0.642788i 0 − 1.00000i 0.939693 + 0.342020i −0.939693 + 0.342020i
1829.2 0.866025 + 0.500000i −0.984808 − 0.173648i 0.500000 + 0.866025i −0.642788 + 0.766044i −0.766044 − 0.642788i 0 1.00000i 0.939693 + 0.342020i −0.939693 + 0.342020i
2069.1 −0.866025 − 0.500000i −0.642788 + 0.766044i 0.500000 + 0.866025i 0.342020 + 0.939693i 0.939693 − 0.342020i 0 − 1.00000i −0.173648 − 0.984808i 0.173648 − 0.984808i
2069.2 0.866025 + 0.500000i 0.642788 − 0.766044i 0.500000 + 0.866025i −0.342020 − 0.939693i 0.939693 − 0.342020i 0 1.00000i −0.173648 − 0.984808i 0.173648 − 0.984808i
2189.1 −0.866025 + 0.500000i 0.984808 − 0.173648i 0.500000 − 0.866025i 0.642788 + 0.766044i −0.766044 + 0.642788i 0 1.00000i 0.939693 − 0.342020i −0.939693 − 0.342020i
2189.2 0.866025 − 0.500000i −0.984808 + 0.173648i 0.500000 − 0.866025i −0.642788 − 0.766044i −0.766044 + 0.642788i 0 − 1.00000i 0.939693 − 0.342020i −0.939693 − 0.342020i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by Q(−15)\Q(\sqrt{-15})
3.b odd 2 1 inner
5.b even 2 1 inner
152.t even 18 1 inner
456.bh odd 18 1 inner
760.cj even 18 1 inner
2280.el odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2280.1.el.b yes 12
3.b odd 2 1 inner 2280.1.el.b yes 12
5.b even 2 1 inner 2280.1.el.b yes 12
8.b even 2 1 2280.1.el.a ✓ 12
15.d odd 2 1 CM 2280.1.el.b yes 12
19.e even 9 1 2280.1.el.a ✓ 12
24.h odd 2 1 2280.1.el.a ✓ 12
40.f even 2 1 2280.1.el.a ✓ 12
57.l odd 18 1 2280.1.el.a ✓ 12
95.p even 18 1 2280.1.el.a ✓ 12
120.i odd 2 1 2280.1.el.a ✓ 12
152.t even 18 1 inner 2280.1.el.b yes 12
285.bd odd 18 1 2280.1.el.a ✓ 12
456.bh odd 18 1 inner 2280.1.el.b yes 12
760.cj even 18 1 inner 2280.1.el.b yes 12
2280.el odd 18 1 inner 2280.1.el.b yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.1.el.a ✓ 12 8.b even 2 1
2280.1.el.a ✓ 12 19.e even 9 1
2280.1.el.a ✓ 12 24.h odd 2 1
2280.1.el.a ✓ 12 40.f even 2 1
2280.1.el.a ✓ 12 57.l odd 18 1
2280.1.el.a ✓ 12 95.p even 18 1
2280.1.el.a ✓ 12 120.i odd 2 1
2280.1.el.a ✓ 12 285.bd odd 18 1
2280.1.el.b yes 12 1.a even 1 1 trivial
2280.1.el.b yes 12 3.b odd 2 1 inner
2280.1.el.b yes 12 5.b even 2 1 inner
2280.1.el.b yes 12 15.d odd 2 1 CM
2280.1.el.b yes 12 152.t even 18 1 inner
2280.1.el.b yes 12 456.bh odd 18 1 inner
2280.1.el.b yes 12 760.cj even 18 1 inner
2280.1.el.b yes 12 2280.el odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T616−9T613+27 T_{61}^{6} - 9T_{61}^{3} + 27 acting on S1new(2280,[χ])S_{1}^{\mathrm{new}}(2280, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4−T2+1)3 (T^{4} - T^{2} + 1)^{3} Copy content Toggle raw display
33 T12−T6+1 T^{12} - T^{6} + 1 Copy content Toggle raw display
55 T12−T6+1 T^{12} - T^{6} + 1 Copy content Toggle raw display
77 T12 T^{12} Copy content Toggle raw display
1111 T12 T^{12} Copy content Toggle raw display
1313 T12 T^{12} Copy content Toggle raw display
1717 T12+6T10+⋯+9 T^{12} + 6 T^{10} + \cdots + 9 Copy content Toggle raw display
1919 (T6−T3+1)2 (T^{6} - T^{3} + 1)^{2} Copy content Toggle raw display
2323 T12+27T6+729 T^{12} + 27T^{6} + 729 Copy content Toggle raw display
2929 T12 T^{12} Copy content Toggle raw display
3131 (T6+3T4−2T3+⋯+1)2 (T^{6} + 3 T^{4} - 2 T^{3} + \cdots + 1)^{2} Copy content Toggle raw display
3737 T12 T^{12} Copy content Toggle raw display
4141 T12 T^{12} Copy content Toggle raw display
4343 T12 T^{12} Copy content Toggle raw display
4747 T12−3T10+⋯+9 T^{12} - 3 T^{10} + \cdots + 9 Copy content Toggle raw display
5353 T12−3T10+⋯+1 T^{12} - 3 T^{10} + \cdots + 1 Copy content Toggle raw display
5959 T12 T^{12} Copy content Toggle raw display
6161 (T6−9T3+27)2 (T^{6} - 9 T^{3} + 27)^{2} Copy content Toggle raw display
6767 T12 T^{12} Copy content Toggle raw display
7171 T12 T^{12} Copy content Toggle raw display
7373 T12 T^{12} Copy content Toggle raw display
7979 (T6−T3+1)2 (T^{6} - T^{3} + 1)^{2} Copy content Toggle raw display
8383 T12−6T10+⋯+1 T^{12} - 6 T^{10} + \cdots + 1 Copy content Toggle raw display
8989 T12 T^{12} Copy content Toggle raw display
9797 T12 T^{12} Copy content Toggle raw display
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