Properties

Label 1083.1.l.a.62.1
Level $1083$
Weight $1$
Character 1083.62
Analytic conductor $0.540$
Analytic rank $0$
Dimension $6$
Projective image $D_{3}$
CM discriminant -3
Inner twists $12$

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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] = [1083,1,Mod(62,1083)] 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("1083.62"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1083, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 16])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1083 = 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1083.l (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,3,0,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(12)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.540487408682\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Projective image: \(D_{3}\)
Projective field: Galois closure of \(\Q(\sqrt[3]{19})\)
Artin image: $S_3\times C_9$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{54} - \cdots)\)

Embedding invariants

Embedding label 62.1
Root \(-0.173648 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 1083.62
Dual form 1083.1.l.a.821.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.173648 - 0.984808i) q^{3} +(0.766044 - 0.642788i) q^{4} +(0.500000 + 0.866025i) q^{7} +(-0.939693 - 0.342020i) q^{9} +(-0.500000 - 0.866025i) q^{12} +(-0.173648 - 0.984808i) q^{13} +(0.173648 - 0.984808i) q^{16} +(0.939693 - 0.342020i) q^{21} +(0.173648 + 0.984808i) q^{25} +(-0.500000 + 0.866025i) q^{27} +(0.939693 + 0.342020i) q^{28} +(0.500000 + 0.866025i) q^{31} +(-0.939693 + 0.342020i) q^{36} -1.00000 q^{37} -1.00000 q^{39} +(-0.766044 - 0.642788i) q^{43} +(-0.939693 - 0.342020i) q^{48} +(-0.766044 - 0.642788i) q^{52} +(-0.766044 + 0.642788i) q^{61} +(-0.173648 - 0.984808i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(0.939693 + 0.342020i) q^{67} +(-0.173648 + 0.984808i) q^{73} +1.00000 q^{75} +(-0.173648 + 0.984808i) q^{79} +(0.766044 + 0.642788i) q^{81} +(0.500000 - 0.866025i) q^{84} +(0.766044 - 0.642788i) q^{91} +(0.939693 - 0.342020i) q^{93} +(-1.87939 + 0.684040i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{7} - 3 q^{12} - 3 q^{27} + 3 q^{31} - 6 q^{37} - 6 q^{39} - 3 q^{64} + 6 q^{75} + 3 q^{84}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1083\mathbb{Z}\right)^\times\).

\(n\) \(362\) \(724\)
\(\chi(n)\) \(-1\) \(e\left(\frac{8}{9}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(3\) 0.173648 − 0.984808i 0.173648 − 0.984808i
\(4\) 0.766044 − 0.642788i 0.766044 − 0.642788i
\(5\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 0 0
\(9\) −0.939693 − 0.342020i −0.939693 − 0.342020i
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(13\) −0.173648 − 0.984808i −0.173648 − 0.984808i −0.939693 − 0.342020i \(-0.888889\pi\)
0.766044 − 0.642788i \(-0.222222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.173648 − 0.984808i 0.173648 − 0.984808i
\(17\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) 0.939693 − 0.342020i 0.939693 − 0.342020i
\(22\) 0 0
\(23\) 0 0 0.766044 − 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(24\) 0 0
\(25\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(26\) 0 0
\(27\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(28\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(29\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(30\) 0 0
\(31\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(37\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) −1.00000 −1.00000
\(40\) 0 0
\(41\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(42\) 0 0
\(43\) −0.766044 − 0.642788i −0.766044 − 0.642788i 0.173648 − 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(48\) −0.939693 − 0.342020i −0.939693 − 0.342020i
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −0.766044 − 0.642788i −0.766044 − 0.642788i
\(53\) 0 0 0.766044 − 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(60\) 0 0
\(61\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 − 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(62\) 0 0
\(63\) −0.173648 − 0.984808i −0.173648 − 0.984808i
\(64\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 − 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(72\) 0 0
\(73\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(74\) 0 0
\(75\) 1.00000 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(80\) 0 0
\(81\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(82\) 0 0
\(83\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(84\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.173648 − 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(90\) 0 0
\(91\) 0.766044 − 0.642788i 0.766044 − 0.642788i
\(92\) 0 0
\(93\) 0.939693 − 0.342020i 0.939693 − 0.342020i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.87939 + 0.684040i −1.87939 + 0.684040i −0.939693 + 0.342020i \(0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1083.1.l.a.62.1 6
3.2 odd 2 CM 1083.1.l.a.62.1 6
19.2 odd 18 1083.1.b.a.362.1 1
19.3 odd 18 1083.1.h.a.653.1 2
19.4 even 9 inner 1083.1.l.a.821.1 6
19.5 even 9 57.1.h.a.11.1 ✓ 2
19.6 even 9 inner 1083.1.l.a.389.1 6
19.7 even 3 inner 1083.1.l.a.245.1 6
19.8 odd 6 1083.1.l.b.776.1 6
19.9 even 9 inner 1083.1.l.a.956.1 6
19.10 odd 18 1083.1.l.b.956.1 6
19.11 even 3 inner 1083.1.l.a.776.1 6
19.12 odd 6 1083.1.l.b.245.1 6
19.13 odd 18 1083.1.l.b.389.1 6
19.14 odd 18 1083.1.h.a.68.1 2
19.15 odd 18 1083.1.l.b.821.1 6
19.16 even 9 57.1.h.a.26.1 yes 2
19.17 even 9 1083.1.b.b.362.1 1
19.18 odd 2 1083.1.l.b.62.1 6
57.2 even 18 1083.1.b.a.362.1 1
57.5 odd 18 57.1.h.a.11.1 ✓ 2
57.8 even 6 1083.1.l.b.776.1 6
57.11 odd 6 inner 1083.1.l.a.776.1 6
57.14 even 18 1083.1.h.a.68.1 2
57.17 odd 18 1083.1.b.b.362.1 1
57.23 odd 18 inner 1083.1.l.a.821.1 6
57.26 odd 6 inner 1083.1.l.a.245.1 6
57.29 even 18 1083.1.l.b.956.1 6
57.32 even 18 1083.1.l.b.389.1 6
57.35 odd 18 57.1.h.a.26.1 yes 2
57.41 even 18 1083.1.h.a.653.1 2
57.44 odd 18 inner 1083.1.l.a.389.1 6
57.47 odd 18 inner 1083.1.l.a.956.1 6
57.50 even 6 1083.1.l.b.245.1 6
57.53 even 18 1083.1.l.b.821.1 6
57.56 even 2 1083.1.l.b.62.1 6
76.35 odd 18 912.1.bl.a.881.1 2
76.43 odd 18 912.1.bl.a.353.1 2
95.24 even 18 1425.1.t.a.1151.1 2
95.43 odd 36 1425.1.o.a.524.1 4
95.54 even 18 1425.1.t.a.26.1 2
95.62 odd 36 1425.1.o.a.524.2 4
95.73 odd 36 1425.1.o.a.824.2 4
95.92 odd 36 1425.1.o.a.824.1 4
133.5 odd 18 2793.1.n.b.410.1 2
133.16 even 9 2793.1.bi.b.2762.1 2
133.24 odd 18 2793.1.bi.a.1892.1 2
133.54 odd 18 2793.1.bi.a.2762.1 2
133.62 odd 18 2793.1.bf.a.638.1 2
133.73 odd 18 2793.1.n.b.1451.1 2
133.81 even 9 2793.1.bi.b.1892.1 2
133.100 even 9 2793.1.n.a.410.1 2
133.111 odd 18 2793.1.bf.a.197.1 2
133.130 even 9 2793.1.n.a.1451.1 2
152.5 even 18 3648.1.bl.b.2177.1 2
152.35 odd 18 3648.1.bl.a.1793.1 2
152.43 odd 18 3648.1.bl.a.2177.1 2
152.149 even 18 3648.1.bl.b.1793.1 2
171.5 odd 18 1539.1.n.a.1322.1 2
171.16 even 9 1539.1.n.a.539.1 2
171.43 even 9 1539.1.j.a.296.1 2
171.92 odd 18 1539.1.n.a.539.1 2
171.119 odd 18 1539.1.j.a.296.1 2
171.130 even 9 1539.1.j.a.26.1 2
171.149 odd 18 1539.1.j.a.26.1 2
171.157 even 9 1539.1.n.a.1322.1 2
228.35 even 18 912.1.bl.a.881.1 2
228.119 even 18 912.1.bl.a.353.1 2
285.62 even 36 1425.1.o.a.524.2 4
285.92 even 36 1425.1.o.a.824.1 4
285.119 odd 18 1425.1.t.a.1151.1 2
285.149 odd 18 1425.1.t.a.26.1 2
285.233 even 36 1425.1.o.a.524.1 4
285.263 even 36 1425.1.o.a.824.2 4
399.5 even 18 2793.1.n.b.410.1 2
399.62 even 18 2793.1.bf.a.638.1 2
399.149 odd 18 2793.1.bi.b.2762.1 2
399.206 even 18 2793.1.n.b.1451.1 2
399.233 odd 18 2793.1.n.a.410.1 2
399.263 odd 18 2793.1.n.a.1451.1 2
399.290 even 18 2793.1.bi.a.1892.1 2
399.320 even 18 2793.1.bi.a.2762.1 2
399.347 odd 18 2793.1.bi.b.1892.1 2
399.377 even 18 2793.1.bf.a.197.1 2
456.5 odd 18 3648.1.bl.b.2177.1 2
456.35 even 18 3648.1.bl.a.1793.1 2
456.149 odd 18 3648.1.bl.b.1793.1 2
456.347 even 18 3648.1.bl.a.2177.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.1.h.a.11.1 ✓ 2 19.5 even 9
57.1.h.a.11.1 ✓ 2 57.5 odd 18
57.1.h.a.26.1 yes 2 19.16 even 9
57.1.h.a.26.1 yes 2 57.35 odd 18
912.1.bl.a.353.1 2 76.43 odd 18
912.1.bl.a.353.1 2 228.119 even 18
912.1.bl.a.881.1 2 76.35 odd 18
912.1.bl.a.881.1 2 228.35 even 18
1083.1.b.a.362.1 1 19.2 odd 18
1083.1.b.a.362.1 1 57.2 even 18
1083.1.b.b.362.1 1 19.17 even 9
1083.1.b.b.362.1 1 57.17 odd 18
1083.1.h.a.68.1 2 19.14 odd 18
1083.1.h.a.68.1 2 57.14 even 18
1083.1.h.a.653.1 2 19.3 odd 18
1083.1.h.a.653.1 2 57.41 even 18
1083.1.l.a.62.1 6 1.1 even 1 trivial
1083.1.l.a.62.1 6 3.2 odd 2 CM
1083.1.l.a.245.1 6 19.7 even 3 inner
1083.1.l.a.245.1 6 57.26 odd 6 inner
1083.1.l.a.389.1 6 19.6 even 9 inner
1083.1.l.a.389.1 6 57.44 odd 18 inner
1083.1.l.a.776.1 6 19.11 even 3 inner
1083.1.l.a.776.1 6 57.11 odd 6 inner
1083.1.l.a.821.1 6 19.4 even 9 inner
1083.1.l.a.821.1 6 57.23 odd 18 inner
1083.1.l.a.956.1 6 19.9 even 9 inner
1083.1.l.a.956.1 6 57.47 odd 18 inner
1083.1.l.b.62.1 6 19.18 odd 2
1083.1.l.b.62.1 6 57.56 even 2
1083.1.l.b.245.1 6 19.12 odd 6
1083.1.l.b.245.1 6 57.50 even 6
1083.1.l.b.389.1 6 19.13 odd 18
1083.1.l.b.389.1 6 57.32 even 18
1083.1.l.b.776.1 6 19.8 odd 6
1083.1.l.b.776.1 6 57.8 even 6
1083.1.l.b.821.1 6 19.15 odd 18
1083.1.l.b.821.1 6 57.53 even 18
1083.1.l.b.956.1 6 19.10 odd 18
1083.1.l.b.956.1 6 57.29 even 18
1425.1.o.a.524.1 4 95.43 odd 36
1425.1.o.a.524.1 4 285.233 even 36
1425.1.o.a.524.2 4 95.62 odd 36
1425.1.o.a.524.2 4 285.62 even 36
1425.1.o.a.824.1 4 95.92 odd 36
1425.1.o.a.824.1 4 285.92 even 36
1425.1.o.a.824.2 4 95.73 odd 36
1425.1.o.a.824.2 4 285.263 even 36
1425.1.t.a.26.1 2 95.54 even 18
1425.1.t.a.26.1 2 285.149 odd 18
1425.1.t.a.1151.1 2 95.24 even 18
1425.1.t.a.1151.1 2 285.119 odd 18
1539.1.j.a.26.1 2 171.130 even 9
1539.1.j.a.26.1 2 171.149 odd 18
1539.1.j.a.296.1 2 171.43 even 9
1539.1.j.a.296.1 2 171.119 odd 18
1539.1.n.a.539.1 2 171.16 even 9
1539.1.n.a.539.1 2 171.92 odd 18
1539.1.n.a.1322.1 2 171.5 odd 18
1539.1.n.a.1322.1 2 171.157 even 9
2793.1.n.a.410.1 2 133.100 even 9
2793.1.n.a.410.1 2 399.233 odd 18
2793.1.n.a.1451.1 2 133.130 even 9
2793.1.n.a.1451.1 2 399.263 odd 18
2793.1.n.b.410.1 2 133.5 odd 18
2793.1.n.b.410.1 2 399.5 even 18
2793.1.n.b.1451.1 2 133.73 odd 18
2793.1.n.b.1451.1 2 399.206 even 18
2793.1.bf.a.197.1 2 133.111 odd 18
2793.1.bf.a.197.1 2 399.377 even 18
2793.1.bf.a.638.1 2 133.62 odd 18
2793.1.bf.a.638.1 2 399.62 even 18
2793.1.bi.a.1892.1 2 133.24 odd 18
2793.1.bi.a.1892.1 2 399.290 even 18
2793.1.bi.a.2762.1 2 133.54 odd 18
2793.1.bi.a.2762.1 2 399.320 even 18
2793.1.bi.b.1892.1 2 133.81 even 9
2793.1.bi.b.1892.1 2 399.347 odd 18
2793.1.bi.b.2762.1 2 133.16 even 9
2793.1.bi.b.2762.1 2 399.149 odd 18
3648.1.bl.a.1793.1 2 152.35 odd 18
3648.1.bl.a.1793.1 2 456.35 even 18
3648.1.bl.a.2177.1 2 152.43 odd 18
3648.1.bl.a.2177.1 2 456.347 even 18
3648.1.bl.b.1793.1 2 152.149 even 18
3648.1.bl.b.1793.1 2 456.149 odd 18
3648.1.bl.b.2177.1 2 152.5 even 18
3648.1.bl.b.2177.1 2 456.5 odd 18