Properties

Label 87.1.d.a.86.1
Level $87$
Weight $1$
Character 87.86
Self dual yes
Analytic conductor $0.043$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -87
Inner twists $2$

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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] = [87,1,Mod(86,87)] 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("87.86"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 87.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.0434186560991\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.87.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.87.1
Stark unit: Root of $x^{3} - 2x^{2} - x - 1$

Embedding invariants

Embedding label 86.1
Character \(\chi\) \(=\) 87.86

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -1.00000 q^{21} +1.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{29} -1.00000 q^{33} +1.00000 q^{34} -1.00000 q^{39} +2.00000 q^{41} +1.00000 q^{42} -1.00000 q^{47} -1.00000 q^{48} -1.00000 q^{50} -1.00000 q^{51} -1.00000 q^{54} -1.00000 q^{56} -1.00000 q^{58} -1.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} -1.00000 q^{67} +1.00000 q^{72} +1.00000 q^{75} +1.00000 q^{77} +1.00000 q^{78} +1.00000 q^{81} -2.00000 q^{82} +1.00000 q^{87} -1.00000 q^{88} -1.00000 q^{89} +1.00000 q^{91} +1.00000 q^{94} -1.00000 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(59\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 1.00000 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −1.00000 −1.00000
\(7\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 1.00000 1.00000
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 1.00000 1.00000
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) −1.00000 −1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −1.00000 −1.00000
\(22\) 1.00000 1.00000
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 1.00000 1.00000
\(25\) 1.00000 1.00000
\(26\) 1.00000 1.00000
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) 1.00000 1.00000
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) −1.00000 −1.00000
\(34\) 1.00000 1.00000
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −1.00000 −1.00000
\(40\) 0 0
\(41\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(42\) 1.00000 1.00000
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) −1.00000 −1.00000
\(49\) 0 0
\(50\) −1.00000 −1.00000
\(51\) −1.00000 −1.00000
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −1.00000 −1.00000
\(55\) 0 0
\(56\) −1.00000 −1.00000
\(57\) 0 0
\(58\) −1.00000 −1.00000
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −1.00000 −1.00000
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 1.00000 1.00000
\(67\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 1.00000 1.00000
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.00000 1.00000
\(76\) 0 0
\(77\) 1.00000 1.00000
\(78\) 1.00000 1.00000
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) −2.00000 −2.00000
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000 1.00000
\(88\) −1.00000 −1.00000
\(89\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) 1.00000 1.00000
\(92\) 0 0
\(93\) 0 0
\(94\) 1.00000 1.00000
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −1.00000 −1.00000
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 87.1.d.a.86.1 ✓ 1
3.2 odd 2 87.1.d.b.86.1 yes 1
4.3 odd 2 1392.1.i.a.1217.1 1
5.2 odd 4 2175.1.b.a.2174.1 2
5.3 odd 4 2175.1.b.a.2174.2 2
5.4 even 2 2175.1.h.b.1826.1 1
9.2 odd 6 2349.1.h.a.782.1 2
9.4 even 3 2349.1.h.b.1565.1 2
9.5 odd 6 2349.1.h.a.1565.1 2
9.7 even 3 2349.1.h.b.782.1 2
12.11 even 2 1392.1.i.b.1217.1 1
15.2 even 4 2175.1.b.b.2174.2 2
15.8 even 4 2175.1.b.b.2174.1 2
15.14 odd 2 2175.1.h.a.1826.1 1
29.2 odd 28 2523.1.j.b.605.1 12
29.3 odd 28 2523.1.j.b.1412.1 12
29.4 even 14 2523.1.h.a.2333.1 6
29.5 even 14 2523.1.h.a.236.1 6
29.6 even 14 2523.1.h.a.1037.1 6
29.7 even 7 2523.1.h.b.1952.1 6
29.8 odd 28 2523.1.j.b.1415.1 12
29.9 even 14 2523.1.h.a.1949.1 6
29.10 odd 28 2523.1.j.b.1031.1 12
29.11 odd 28 2523.1.j.b.1619.1 12
29.12 odd 4 2523.1.b.b.842.1 2
29.13 even 14 2523.1.h.a.1745.1 6
29.14 odd 28 2523.1.j.b.2327.2 12
29.15 odd 28 2523.1.j.b.2327.1 12
29.16 even 7 2523.1.h.b.1745.1 6
29.17 odd 4 2523.1.b.b.842.2 2
29.18 odd 28 2523.1.j.b.1619.2 12
29.19 odd 28 2523.1.j.b.1031.2 12
29.20 even 7 2523.1.h.b.1949.1 6
29.21 odd 28 2523.1.j.b.1415.2 12
29.22 even 14 2523.1.h.a.1952.1 6
29.23 even 7 2523.1.h.b.1037.1 6
29.24 even 7 2523.1.h.b.236.1 6
29.25 even 7 2523.1.h.b.2333.1 6
29.26 odd 28 2523.1.j.b.1412.2 12
29.27 odd 28 2523.1.j.b.605.2 12
29.28 even 2 87.1.d.b.86.1 yes 1
87.2 even 28 2523.1.j.b.605.2 12
87.5 odd 14 2523.1.h.b.236.1 6
87.8 even 28 2523.1.j.b.1415.2 12
87.11 even 28 2523.1.j.b.1619.2 12
87.14 even 28 2523.1.j.b.2327.1 12
87.17 even 4 2523.1.b.b.842.1 2
87.20 odd 14 2523.1.h.a.1949.1 6
87.23 odd 14 2523.1.h.a.1037.1 6
87.26 even 28 2523.1.j.b.1412.1 12
87.32 even 28 2523.1.j.b.1412.2 12
87.35 odd 14 2523.1.h.b.1037.1 6
87.38 odd 14 2523.1.h.b.1949.1 6
87.41 even 4 2523.1.b.b.842.2 2
87.44 even 28 2523.1.j.b.2327.2 12
87.47 even 28 2523.1.j.b.1619.1 12
87.50 even 28 2523.1.j.b.1415.1 12
87.53 odd 14 2523.1.h.a.236.1 6
87.56 even 28 2523.1.j.b.605.1 12
87.62 odd 14 2523.1.h.b.2333.1 6
87.65 odd 14 2523.1.h.a.1952.1 6
87.68 even 28 2523.1.j.b.1031.2 12
87.71 odd 14 2523.1.h.b.1745.1 6
87.74 odd 14 2523.1.h.a.1745.1 6
87.77 even 28 2523.1.j.b.1031.1 12
87.80 odd 14 2523.1.h.b.1952.1 6
87.83 odd 14 2523.1.h.a.2333.1 6
87.86 odd 2 CM 87.1.d.a.86.1 ✓ 1
116.115 odd 2 1392.1.i.b.1217.1 1
145.28 odd 4 2175.1.b.b.2174.1 2
145.57 odd 4 2175.1.b.b.2174.2 2
145.144 even 2 2175.1.h.a.1826.1 1
261.86 odd 6 2349.1.h.b.1565.1 2
261.115 even 6 2349.1.h.a.782.1 2
261.173 odd 6 2349.1.h.b.782.1 2
261.202 even 6 2349.1.h.a.1565.1 2
348.347 even 2 1392.1.i.a.1217.1 1
435.173 even 4 2175.1.b.a.2174.2 2
435.347 even 4 2175.1.b.a.2174.1 2
435.434 odd 2 2175.1.h.b.1826.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.1.d.a.86.1 ✓ 1 1.1 even 1 trivial
87.1.d.a.86.1 ✓ 1 87.86 odd 2 CM
87.1.d.b.86.1 yes 1 3.2 odd 2
87.1.d.b.86.1 yes 1 29.28 even 2
1392.1.i.a.1217.1 1 4.3 odd 2
1392.1.i.a.1217.1 1 348.347 even 2
1392.1.i.b.1217.1 1 12.11 even 2
1392.1.i.b.1217.1 1 116.115 odd 2
2175.1.b.a.2174.1 2 5.2 odd 4
2175.1.b.a.2174.1 2 435.347 even 4
2175.1.b.a.2174.2 2 5.3 odd 4
2175.1.b.a.2174.2 2 435.173 even 4
2175.1.b.b.2174.1 2 15.8 even 4
2175.1.b.b.2174.1 2 145.28 odd 4
2175.1.b.b.2174.2 2 15.2 even 4
2175.1.b.b.2174.2 2 145.57 odd 4
2175.1.h.a.1826.1 1 15.14 odd 2
2175.1.h.a.1826.1 1 145.144 even 2
2175.1.h.b.1826.1 1 5.4 even 2
2175.1.h.b.1826.1 1 435.434 odd 2
2349.1.h.a.782.1 2 9.2 odd 6
2349.1.h.a.782.1 2 261.115 even 6
2349.1.h.a.1565.1 2 9.5 odd 6
2349.1.h.a.1565.1 2 261.202 even 6
2349.1.h.b.782.1 2 9.7 even 3
2349.1.h.b.782.1 2 261.173 odd 6
2349.1.h.b.1565.1 2 9.4 even 3
2349.1.h.b.1565.1 2 261.86 odd 6
2523.1.b.b.842.1 2 29.12 odd 4
2523.1.b.b.842.1 2 87.17 even 4
2523.1.b.b.842.2 2 29.17 odd 4
2523.1.b.b.842.2 2 87.41 even 4
2523.1.h.a.236.1 6 29.5 even 14
2523.1.h.a.236.1 6 87.53 odd 14
2523.1.h.a.1037.1 6 29.6 even 14
2523.1.h.a.1037.1 6 87.23 odd 14
2523.1.h.a.1745.1 6 29.13 even 14
2523.1.h.a.1745.1 6 87.74 odd 14
2523.1.h.a.1949.1 6 29.9 even 14
2523.1.h.a.1949.1 6 87.20 odd 14
2523.1.h.a.1952.1 6 29.22 even 14
2523.1.h.a.1952.1 6 87.65 odd 14
2523.1.h.a.2333.1 6 29.4 even 14
2523.1.h.a.2333.1 6 87.83 odd 14
2523.1.h.b.236.1 6 29.24 even 7
2523.1.h.b.236.1 6 87.5 odd 14
2523.1.h.b.1037.1 6 29.23 even 7
2523.1.h.b.1037.1 6 87.35 odd 14
2523.1.h.b.1745.1 6 29.16 even 7
2523.1.h.b.1745.1 6 87.71 odd 14
2523.1.h.b.1949.1 6 29.20 even 7
2523.1.h.b.1949.1 6 87.38 odd 14
2523.1.h.b.1952.1 6 29.7 even 7
2523.1.h.b.1952.1 6 87.80 odd 14
2523.1.h.b.2333.1 6 29.25 even 7
2523.1.h.b.2333.1 6 87.62 odd 14
2523.1.j.b.605.1 12 29.2 odd 28
2523.1.j.b.605.1 12 87.56 even 28
2523.1.j.b.605.2 12 29.27 odd 28
2523.1.j.b.605.2 12 87.2 even 28
2523.1.j.b.1031.1 12 29.10 odd 28
2523.1.j.b.1031.1 12 87.77 even 28
2523.1.j.b.1031.2 12 29.19 odd 28
2523.1.j.b.1031.2 12 87.68 even 28
2523.1.j.b.1412.1 12 29.3 odd 28
2523.1.j.b.1412.1 12 87.26 even 28
2523.1.j.b.1412.2 12 29.26 odd 28
2523.1.j.b.1412.2 12 87.32 even 28
2523.1.j.b.1415.1 12 29.8 odd 28
2523.1.j.b.1415.1 12 87.50 even 28
2523.1.j.b.1415.2 12 29.21 odd 28
2523.1.j.b.1415.2 12 87.8 even 28
2523.1.j.b.1619.1 12 29.11 odd 28
2523.1.j.b.1619.1 12 87.47 even 28
2523.1.j.b.1619.2 12 29.18 odd 28
2523.1.j.b.1619.2 12 87.11 even 28
2523.1.j.b.2327.1 12 29.15 odd 28
2523.1.j.b.2327.1 12 87.14 even 28
2523.1.j.b.2327.2 12 29.14 odd 28
2523.1.j.b.2327.2 12 87.44 even 28