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Properties

Label 1539.1.n.a.1322.1
Level $1539$
Weight $1$
Character 1539.1322
Analytic conductor $0.768$
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] = [1539,1,Mod(539,1539)] 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("1539.539"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1539, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1539 = 3^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1539.n (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: \(0.768061054442\)
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: \(\Z[a_1, \ldots, a_{4}]\)
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: $C_3\times S_3$
Artin field: Galois closure of 6.0.7105563.2

Embedding invariants

Embedding label 1322.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1539.1322
Dual form 1539.1.n.a.539.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{7} +(0.500000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{16} +1.00000 q^{19} +1.00000 q^{25} +(0.500000 + 0.866025i) q^{28} +(0.500000 + 0.866025i) q^{31} -1.00000 q^{37} +(0.500000 + 0.866025i) q^{43} +(0.500000 + 0.866025i) q^{52} -1.00000 q^{61} +1.00000 q^{64} +(0.500000 - 0.866025i) q^{67} +(0.500000 - 0.866025i) q^{73} +(-0.500000 + 0.866025i) q^{76} +(0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{91} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{4} + q^{7} + q^{13} - q^{16} + 2 q^{19} + 2 q^{25} + q^{28} + q^{31} - 2 q^{37} + q^{43} + q^{52} - 2 q^{61} + 2 q^{64} + q^{67} + q^{73} - q^{76} + q^{79} - q^{91} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1539\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\)

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 0
\(4\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(17\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) 1.00000 1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 1.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 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 0
\(37\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(53\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(72\) 0 0
\(73\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(74\) 0 0
\(75\) 0 0
\(76\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00000 − 1.73205i −1.00000 − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 − 0.866025i \(-0.666667\pi\)
\(98\) 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 1539.1.n.a.1322.1 2
3.2 odd 2 CM 1539.1.n.a.1322.1 2
9.2 odd 6 57.1.h.a.11.1 ✓ 2
9.4 even 3 1539.1.j.a.296.1 2
9.5 odd 6 1539.1.j.a.296.1 2
9.7 even 3 57.1.h.a.11.1 ✓ 2
19.7 even 3 1539.1.j.a.26.1 2
36.7 odd 6 912.1.bl.a.353.1 2
36.11 even 6 912.1.bl.a.353.1 2
45.2 even 12 1425.1.o.a.524.2 4
45.7 odd 12 1425.1.o.a.524.2 4
45.29 odd 6 1425.1.t.a.1151.1 2
45.34 even 6 1425.1.t.a.1151.1 2
45.38 even 12 1425.1.o.a.524.1 4
45.43 odd 12 1425.1.o.a.524.1 4
57.26 odd 6 1539.1.j.a.26.1 2
63.2 odd 6 2793.1.n.a.410.1 2
63.11 odd 6 2793.1.bi.b.1892.1 2
63.16 even 3 2793.1.n.a.410.1 2
63.20 even 6 2793.1.bf.a.638.1 2
63.25 even 3 2793.1.bi.b.1892.1 2
63.34 odd 6 2793.1.bf.a.638.1 2
63.38 even 6 2793.1.bi.a.1892.1 2
63.47 even 6 2793.1.n.b.410.1 2
63.52 odd 6 2793.1.bi.a.1892.1 2
63.61 odd 6 2793.1.n.b.410.1 2
72.11 even 6 3648.1.bl.a.2177.1 2
72.29 odd 6 3648.1.bl.b.2177.1 2
72.43 odd 6 3648.1.bl.a.2177.1 2
72.61 even 6 3648.1.bl.b.2177.1 2
171.2 even 18 1083.1.l.b.956.1 6
171.7 even 3 57.1.h.a.26.1 yes 2
171.11 odd 6 1083.1.b.b.362.1 1
171.16 even 9 1083.1.l.a.821.1 6
171.25 even 9 1083.1.l.a.776.1 6
171.29 even 18 1083.1.l.b.245.1 6
171.34 odd 18 1083.1.l.b.62.1 6
171.43 even 9 1083.1.l.a.389.1 6
171.47 odd 18 1083.1.l.a.245.1 6
171.52 odd 18 1083.1.l.b.389.1 6
171.56 even 6 1083.1.h.a.68.1 2
171.61 even 9 1083.1.l.a.62.1 6
171.65 even 6 1083.1.b.a.362.1 1
171.70 odd 18 1083.1.l.b.776.1 6
171.74 odd 18 1083.1.l.a.956.1 6
171.79 odd 18 1083.1.l.b.821.1 6
171.83 odd 6 57.1.h.a.26.1 yes 2
171.88 odd 6 1083.1.h.a.653.1 2
171.92 odd 18 1083.1.l.a.821.1 6
171.97 odd 18 1083.1.l.b.956.1 6
171.101 odd 18 1083.1.l.a.776.1 6
171.106 even 3 1083.1.b.b.362.1 1
171.110 even 18 1083.1.l.b.62.1 6
171.119 odd 18 1083.1.l.a.389.1 6
171.121 even 3 inner 1539.1.n.a.539.1 2
171.124 odd 18 1083.1.l.b.245.1 6
171.128 even 18 1083.1.l.b.389.1 6
171.137 odd 18 1083.1.l.a.62.1 6
171.140 odd 6 inner 1539.1.n.a.539.1 2
171.142 even 9 1083.1.l.a.245.1 6
171.146 even 18 1083.1.l.b.776.1 6
171.151 odd 6 1083.1.h.a.68.1 2
171.155 even 18 1083.1.l.b.821.1 6
171.160 odd 6 1083.1.b.a.362.1 1
171.164 even 6 1083.1.h.a.653.1 2
171.169 even 9 1083.1.l.a.956.1 6
684.7 odd 6 912.1.bl.a.881.1 2
684.83 even 6 912.1.bl.a.881.1 2
855.7 odd 12 1425.1.o.a.824.1 4
855.83 even 12 1425.1.o.a.824.2 4
855.178 odd 12 1425.1.o.a.824.2 4
855.254 odd 6 1425.1.t.a.26.1 2
855.349 even 6 1425.1.t.a.26.1 2
855.767 even 12 1425.1.o.a.824.1 4
1197.83 even 6 2793.1.bf.a.197.1 2
1197.178 odd 6 2793.1.n.b.1451.1 2
1197.254 odd 6 2793.1.bi.b.2762.1 2
1197.349 odd 6 2793.1.bf.a.197.1 2
1197.425 even 6 2793.1.bi.a.2762.1 2
1197.520 even 3 2793.1.bi.b.2762.1 2
1197.691 odd 6 2793.1.bi.a.2762.1 2
1197.767 odd 6 2793.1.n.a.1451.1 2
1197.1033 even 3 2793.1.n.a.1451.1 2
1197.1109 even 6 2793.1.n.b.1451.1 2
1368.83 even 6 3648.1.bl.a.1793.1 2
1368.349 even 6 3648.1.bl.b.1793.1 2
1368.691 odd 6 3648.1.bl.a.1793.1 2
1368.1109 odd 6 3648.1.bl.b.1793.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.1.h.a.11.1 ✓ 2 9.2 odd 6
57.1.h.a.11.1 ✓ 2 9.7 even 3
57.1.h.a.26.1 yes 2 171.7 even 3
57.1.h.a.26.1 yes 2 171.83 odd 6
912.1.bl.a.353.1 2 36.7 odd 6
912.1.bl.a.353.1 2 36.11 even 6
912.1.bl.a.881.1 2 684.7 odd 6
912.1.bl.a.881.1 2 684.83 even 6
1083.1.b.a.362.1 1 171.65 even 6
1083.1.b.a.362.1 1 171.160 odd 6
1083.1.b.b.362.1 1 171.11 odd 6
1083.1.b.b.362.1 1 171.106 even 3
1083.1.h.a.68.1 2 171.56 even 6
1083.1.h.a.68.1 2 171.151 odd 6
1083.1.h.a.653.1 2 171.88 odd 6
1083.1.h.a.653.1 2 171.164 even 6
1083.1.l.a.62.1 6 171.61 even 9
1083.1.l.a.62.1 6 171.137 odd 18
1083.1.l.a.245.1 6 171.47 odd 18
1083.1.l.a.245.1 6 171.142 even 9
1083.1.l.a.389.1 6 171.43 even 9
1083.1.l.a.389.1 6 171.119 odd 18
1083.1.l.a.776.1 6 171.25 even 9
1083.1.l.a.776.1 6 171.101 odd 18
1083.1.l.a.821.1 6 171.16 even 9
1083.1.l.a.821.1 6 171.92 odd 18
1083.1.l.a.956.1 6 171.74 odd 18
1083.1.l.a.956.1 6 171.169 even 9
1083.1.l.b.62.1 6 171.34 odd 18
1083.1.l.b.62.1 6 171.110 even 18
1083.1.l.b.245.1 6 171.29 even 18
1083.1.l.b.245.1 6 171.124 odd 18
1083.1.l.b.389.1 6 171.52 odd 18
1083.1.l.b.389.1 6 171.128 even 18
1083.1.l.b.776.1 6 171.70 odd 18
1083.1.l.b.776.1 6 171.146 even 18
1083.1.l.b.821.1 6 171.79 odd 18
1083.1.l.b.821.1 6 171.155 even 18
1083.1.l.b.956.1 6 171.2 even 18
1083.1.l.b.956.1 6 171.97 odd 18
1425.1.o.a.524.1 4 45.38 even 12
1425.1.o.a.524.1 4 45.43 odd 12
1425.1.o.a.524.2 4 45.2 even 12
1425.1.o.a.524.2 4 45.7 odd 12
1425.1.o.a.824.1 4 855.7 odd 12
1425.1.o.a.824.1 4 855.767 even 12
1425.1.o.a.824.2 4 855.83 even 12
1425.1.o.a.824.2 4 855.178 odd 12
1425.1.t.a.26.1 2 855.254 odd 6
1425.1.t.a.26.1 2 855.349 even 6
1425.1.t.a.1151.1 2 45.29 odd 6
1425.1.t.a.1151.1 2 45.34 even 6
1539.1.j.a.26.1 2 19.7 even 3
1539.1.j.a.26.1 2 57.26 odd 6
1539.1.j.a.296.1 2 9.4 even 3
1539.1.j.a.296.1 2 9.5 odd 6
1539.1.n.a.539.1 2 171.121 even 3 inner
1539.1.n.a.539.1 2 171.140 odd 6 inner
1539.1.n.a.1322.1 2 1.1 even 1 trivial
1539.1.n.a.1322.1 2 3.2 odd 2 CM
2793.1.n.a.410.1 2 63.2 odd 6
2793.1.n.a.410.1 2 63.16 even 3
2793.1.n.a.1451.1 2 1197.767 odd 6
2793.1.n.a.1451.1 2 1197.1033 even 3
2793.1.n.b.410.1 2 63.47 even 6
2793.1.n.b.410.1 2 63.61 odd 6
2793.1.n.b.1451.1 2 1197.178 odd 6
2793.1.n.b.1451.1 2 1197.1109 even 6
2793.1.bf.a.197.1 2 1197.83 even 6
2793.1.bf.a.197.1 2 1197.349 odd 6
2793.1.bf.a.638.1 2 63.20 even 6
2793.1.bf.a.638.1 2 63.34 odd 6
2793.1.bi.a.1892.1 2 63.38 even 6
2793.1.bi.a.1892.1 2 63.52 odd 6
2793.1.bi.a.2762.1 2 1197.425 even 6
2793.1.bi.a.2762.1 2 1197.691 odd 6
2793.1.bi.b.1892.1 2 63.11 odd 6
2793.1.bi.b.1892.1 2 63.25 even 3
2793.1.bi.b.2762.1 2 1197.254 odd 6
2793.1.bi.b.2762.1 2 1197.520 even 3
3648.1.bl.a.1793.1 2 1368.83 even 6
3648.1.bl.a.1793.1 2 1368.691 odd 6
3648.1.bl.a.2177.1 2 72.11 even 6
3648.1.bl.a.2177.1 2 72.43 odd 6
3648.1.bl.b.1793.1 2 1368.349 even 6
3648.1.bl.b.1793.1 2 1368.1109 odd 6
3648.1.bl.b.2177.1 2 72.29 odd 6
3648.1.bl.b.2177.1 2 72.61 even 6