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Properties

Label 1083.1.h.a.653.1
Level $1083$
Weight $1$
Character 1083.653
Analytic conductor $0.540$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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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(68,1083)] 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("1083.68"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1083, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) 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.h (of order \(6\), degree \(2\), not 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: \(0.540487408682\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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: $C_6\times S_3$
Artin field: Galois closure of 12.0.12381017456889.1

Embedding invariants

Embedding label 653.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1083.653
Dual form 1083.1.h.a.68.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} -1.00000 q^{7} +(-0.500000 + 0.866025i) q^{9} -1.00000 q^{12} +(-0.500000 + 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 - 0.866025i) q^{21} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{27} +(0.500000 - 0.866025i) q^{28} +1.00000 q^{31} +(-0.500000 - 0.866025i) q^{36} +1.00000 q^{37} -1.00000 q^{39} +(0.500000 + 0.866025i) q^{43} +(0.500000 - 0.866025i) q^{48} +(-0.500000 - 0.866025i) q^{52} +(0.500000 - 0.866025i) q^{61} +(0.500000 - 0.866025i) q^{63} +1.00000 q^{64} +(-0.500000 + 0.866025i) q^{67} +(0.500000 + 0.866025i) q^{73} -1.00000 q^{75} +(-0.500000 - 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} +1.00000 q^{84} +(0.500000 - 0.866025i) q^{91} +(0.500000 + 0.866025i) q^{93} +(1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{4} - 2 q^{7} - q^{9} - 2 q^{12} - q^{13} - q^{16} - q^{21} - q^{25} - 2 q^{27} + q^{28} + 2 q^{31} - q^{36} + 2 q^{37} - 2 q^{39} + q^{43} + q^{48} - q^{52} + q^{61} + q^{63}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(362\) \(724\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(4\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(5\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −1.00000 −1.00000
\(13\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(17\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(22\) 0 0
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) −1.00000 −1.00000
\(28\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(37\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) −1.00000 −1.00000
\(40\) 0 0
\(41\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(53\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(62\) 0 0
\(63\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 0 0
\(75\) −1.00000 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 1.00000 1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(92\) 0 0
\(93\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) 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.h.a.653.1 2
3.2 odd 2 CM 1083.1.h.a.653.1 2
19.2 odd 18 1083.1.l.a.389.1 6
19.3 odd 18 1083.1.l.a.956.1 6
19.4 even 9 1083.1.l.b.245.1 6
19.5 even 9 1083.1.l.b.821.1 6
19.6 even 9 1083.1.l.b.62.1 6
19.7 even 3 1083.1.b.a.362.1 1
19.8 odd 6 57.1.h.a.11.1 ✓ 2
19.9 even 9 1083.1.l.b.776.1 6
19.10 odd 18 1083.1.l.a.776.1 6
19.11 even 3 inner 1083.1.h.a.68.1 2
19.12 odd 6 1083.1.b.b.362.1 1
19.13 odd 18 1083.1.l.a.62.1 6
19.14 odd 18 1083.1.l.a.821.1 6
19.15 odd 18 1083.1.l.a.245.1 6
19.16 even 9 1083.1.l.b.956.1 6
19.17 even 9 1083.1.l.b.389.1 6
19.18 odd 2 57.1.h.a.26.1 yes 2
57.2 even 18 1083.1.l.a.389.1 6
57.5 odd 18 1083.1.l.b.821.1 6
57.8 even 6 57.1.h.a.11.1 ✓ 2
57.11 odd 6 inner 1083.1.h.a.68.1 2
57.14 even 18 1083.1.l.a.821.1 6
57.17 odd 18 1083.1.l.b.389.1 6
57.23 odd 18 1083.1.l.b.245.1 6
57.26 odd 6 1083.1.b.a.362.1 1
57.29 even 18 1083.1.l.a.776.1 6
57.32 even 18 1083.1.l.a.62.1 6
57.35 odd 18 1083.1.l.b.956.1 6
57.41 even 18 1083.1.l.a.956.1 6
57.44 odd 18 1083.1.l.b.62.1 6
57.47 odd 18 1083.1.l.b.776.1 6
57.50 even 6 1083.1.b.b.362.1 1
57.53 even 18 1083.1.l.a.245.1 6
57.56 even 2 57.1.h.a.26.1 yes 2
76.27 even 6 912.1.bl.a.353.1 2
76.75 even 2 912.1.bl.a.881.1 2
95.8 even 12 1425.1.o.a.524.1 4
95.18 even 4 1425.1.o.a.824.2 4
95.27 even 12 1425.1.o.a.524.2 4
95.37 even 4 1425.1.o.a.824.1 4
95.84 odd 6 1425.1.t.a.1151.1 2
95.94 odd 2 1425.1.t.a.26.1 2
133.18 odd 6 2793.1.n.a.1451.1 2
133.27 even 6 2793.1.bf.a.638.1 2
133.37 odd 6 2793.1.bi.b.2762.1 2
133.46 odd 6 2793.1.bi.b.1892.1 2
133.65 odd 6 2793.1.n.a.410.1 2
133.75 even 6 2793.1.bi.a.2762.1 2
133.94 even 6 2793.1.n.b.1451.1 2
133.103 even 6 2793.1.n.b.410.1 2
133.122 even 6 2793.1.bi.a.1892.1 2
133.132 even 2 2793.1.bf.a.197.1 2
152.27 even 6 3648.1.bl.a.2177.1 2
152.37 odd 2 3648.1.bl.b.1793.1 2
152.75 even 2 3648.1.bl.a.1793.1 2
152.141 odd 6 3648.1.bl.b.2177.1 2
171.56 even 6 1539.1.n.a.539.1 2
171.65 even 6 1539.1.j.a.296.1 2
171.94 odd 6 1539.1.j.a.26.1 2
171.103 odd 6 1539.1.n.a.1322.1 2
171.113 even 6 1539.1.j.a.26.1 2
171.122 even 6 1539.1.n.a.1322.1 2
171.151 odd 6 1539.1.n.a.539.1 2
171.160 odd 6 1539.1.j.a.296.1 2
228.179 odd 6 912.1.bl.a.353.1 2
228.227 odd 2 912.1.bl.a.881.1 2
285.8 odd 12 1425.1.o.a.524.1 4
285.113 odd 4 1425.1.o.a.824.2 4
285.122 odd 12 1425.1.o.a.524.2 4
285.179 even 6 1425.1.t.a.1151.1 2
285.227 odd 4 1425.1.o.a.824.1 4
285.284 even 2 1425.1.t.a.26.1 2
399.65 even 6 2793.1.n.a.410.1 2
399.122 odd 6 2793.1.bi.a.1892.1 2
399.170 even 6 2793.1.bi.b.2762.1 2
399.179 even 6 2793.1.bi.b.1892.1 2
399.227 odd 6 2793.1.n.b.1451.1 2
399.236 odd 6 2793.1.n.b.410.1 2
399.284 even 6 2793.1.n.a.1451.1 2
399.293 odd 6 2793.1.bf.a.638.1 2
399.341 odd 6 2793.1.bi.a.2762.1 2
399.398 odd 2 2793.1.bf.a.197.1 2
456.179 odd 6 3648.1.bl.a.2177.1 2
456.227 odd 2 3648.1.bl.a.1793.1 2
456.293 even 6 3648.1.bl.b.2177.1 2
456.341 even 2 3648.1.bl.b.1793.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.1.h.a.11.1 ✓ 2 19.8 odd 6
57.1.h.a.11.1 ✓ 2 57.8 even 6
57.1.h.a.26.1 yes 2 19.18 odd 2
57.1.h.a.26.1 yes 2 57.56 even 2
912.1.bl.a.353.1 2 76.27 even 6
912.1.bl.a.353.1 2 228.179 odd 6
912.1.bl.a.881.1 2 76.75 even 2
912.1.bl.a.881.1 2 228.227 odd 2
1083.1.b.a.362.1 1 19.7 even 3
1083.1.b.a.362.1 1 57.26 odd 6
1083.1.b.b.362.1 1 19.12 odd 6
1083.1.b.b.362.1 1 57.50 even 6
1083.1.h.a.68.1 2 19.11 even 3 inner
1083.1.h.a.68.1 2 57.11 odd 6 inner
1083.1.h.a.653.1 2 1.1 even 1 trivial
1083.1.h.a.653.1 2 3.2 odd 2 CM
1083.1.l.a.62.1 6 19.13 odd 18
1083.1.l.a.62.1 6 57.32 even 18
1083.1.l.a.245.1 6 19.15 odd 18
1083.1.l.a.245.1 6 57.53 even 18
1083.1.l.a.389.1 6 19.2 odd 18
1083.1.l.a.389.1 6 57.2 even 18
1083.1.l.a.776.1 6 19.10 odd 18
1083.1.l.a.776.1 6 57.29 even 18
1083.1.l.a.821.1 6 19.14 odd 18
1083.1.l.a.821.1 6 57.14 even 18
1083.1.l.a.956.1 6 19.3 odd 18
1083.1.l.a.956.1 6 57.41 even 18
1083.1.l.b.62.1 6 19.6 even 9
1083.1.l.b.62.1 6 57.44 odd 18
1083.1.l.b.245.1 6 19.4 even 9
1083.1.l.b.245.1 6 57.23 odd 18
1083.1.l.b.389.1 6 19.17 even 9
1083.1.l.b.389.1 6 57.17 odd 18
1083.1.l.b.776.1 6 19.9 even 9
1083.1.l.b.776.1 6 57.47 odd 18
1083.1.l.b.821.1 6 19.5 even 9
1083.1.l.b.821.1 6 57.5 odd 18
1083.1.l.b.956.1 6 19.16 even 9
1083.1.l.b.956.1 6 57.35 odd 18
1425.1.o.a.524.1 4 95.8 even 12
1425.1.o.a.524.1 4 285.8 odd 12
1425.1.o.a.524.2 4 95.27 even 12
1425.1.o.a.524.2 4 285.122 odd 12
1425.1.o.a.824.1 4 95.37 even 4
1425.1.o.a.824.1 4 285.227 odd 4
1425.1.o.a.824.2 4 95.18 even 4
1425.1.o.a.824.2 4 285.113 odd 4
1425.1.t.a.26.1 2 95.94 odd 2
1425.1.t.a.26.1 2 285.284 even 2
1425.1.t.a.1151.1 2 95.84 odd 6
1425.1.t.a.1151.1 2 285.179 even 6
1539.1.j.a.26.1 2 171.94 odd 6
1539.1.j.a.26.1 2 171.113 even 6
1539.1.j.a.296.1 2 171.65 even 6
1539.1.j.a.296.1 2 171.160 odd 6
1539.1.n.a.539.1 2 171.56 even 6
1539.1.n.a.539.1 2 171.151 odd 6
1539.1.n.a.1322.1 2 171.103 odd 6
1539.1.n.a.1322.1 2 171.122 even 6
2793.1.n.a.410.1 2 133.65 odd 6
2793.1.n.a.410.1 2 399.65 even 6
2793.1.n.a.1451.1 2 133.18 odd 6
2793.1.n.a.1451.1 2 399.284 even 6
2793.1.n.b.410.1 2 133.103 even 6
2793.1.n.b.410.1 2 399.236 odd 6
2793.1.n.b.1451.1 2 133.94 even 6
2793.1.n.b.1451.1 2 399.227 odd 6
2793.1.bf.a.197.1 2 133.132 even 2
2793.1.bf.a.197.1 2 399.398 odd 2
2793.1.bf.a.638.1 2 133.27 even 6
2793.1.bf.a.638.1 2 399.293 odd 6
2793.1.bi.a.1892.1 2 133.122 even 6
2793.1.bi.a.1892.1 2 399.122 odd 6
2793.1.bi.a.2762.1 2 133.75 even 6
2793.1.bi.a.2762.1 2 399.341 odd 6
2793.1.bi.b.1892.1 2 133.46 odd 6
2793.1.bi.b.1892.1 2 399.179 even 6
2793.1.bi.b.2762.1 2 133.37 odd 6
2793.1.bi.b.2762.1 2 399.170 even 6
3648.1.bl.a.1793.1 2 152.75 even 2
3648.1.bl.a.1793.1 2 456.227 odd 2
3648.1.bl.a.2177.1 2 152.27 even 6
3648.1.bl.a.2177.1 2 456.179 odd 6
3648.1.bl.b.1793.1 2 152.37 odd 2
3648.1.bl.b.1793.1 2 456.341 even 2
3648.1.bl.b.2177.1 2 152.141 odd 6
3648.1.bl.b.2177.1 2 456.293 even 6