Properties

Label 135.1.d.a.134.1
Level $135$
Weight $1$
Character 135.134
Self dual yes
Analytic conductor $0.067$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -15
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] = [135,1,Mod(134,135)] 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("135.134"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, 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: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 135.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: \(0.0673737767055\)
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 3.1.135.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.135.1
Stark unit: Root of $x^{3} - 3x^{2} - 1$

Embedding invariants

Embedding label 134.1
Character \(\chi\) \(=\) 135.134

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{5} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{31} +1.00000 q^{34} +1.00000 q^{38} +1.00000 q^{40} +1.00000 q^{46} +2.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{53} -1.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} -1.00000 q^{79} -1.00000 q^{80} -1.00000 q^{83} -1.00000 q^{85} -2.00000 q^{94} -1.00000 q^{95} -1.00000 q^{98} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(0.666667\pi\)
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 1.00000
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000 1.00000
\(9\) 0 0
\(10\) −1.00000 −1.00000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 1.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 1.00000 1.00000
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 1.00000 1.00000
\(39\) 0 0
\(40\) 1.00000 1.00000
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.00000 1.00000
\(47\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) −1.00000 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−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 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 \(0.666667\pi\)
\(62\) 1.00000 1.00000
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) −1.00000 −1.00000
\(81\) 0 0
\(82\) 0 0
\(83\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 0 0
\(85\) −1.00000 −1.00000
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −2.00000 −2.00000
\(95\) −1.00000 −1.00000
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −1.00000 −1.00000
\(99\) 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 135.1.d.a.134.1 ✓ 1
3.2 odd 2 135.1.d.b.134.1 yes 1
4.3 odd 2 2160.1.c.b.1889.1 1
5.2 odd 4 675.1.c.c.26.1 2
5.3 odd 4 675.1.c.c.26.2 2
5.4 even 2 135.1.d.b.134.1 yes 1
9.2 odd 6 405.1.h.a.134.1 2
9.4 even 3 405.1.h.b.269.1 2
9.5 odd 6 405.1.h.a.269.1 2
9.7 even 3 405.1.h.b.134.1 2
12.11 even 2 2160.1.c.a.1889.1 1
15.2 even 4 675.1.c.c.26.2 2
15.8 even 4 675.1.c.c.26.1 2
15.14 odd 2 CM 135.1.d.a.134.1 ✓ 1
20.19 odd 2 2160.1.c.a.1889.1 1
27.2 odd 18 3645.1.n.e.1619.1 6
27.4 even 9 3645.1.n.d.3239.1 6
27.5 odd 18 3645.1.n.e.809.1 6
27.7 even 9 3645.1.n.d.404.1 6
27.11 odd 18 3645.1.n.e.2834.1 6
27.13 even 9 3645.1.n.d.2024.1 6
27.14 odd 18 3645.1.n.e.2024.1 6
27.16 even 9 3645.1.n.d.2834.1 6
27.20 odd 18 3645.1.n.e.404.1 6
27.22 even 9 3645.1.n.d.809.1 6
27.23 odd 18 3645.1.n.e.3239.1 6
27.25 even 9 3645.1.n.d.1619.1 6
45.2 even 12 2025.1.j.c.701.2 4
45.4 even 6 405.1.h.a.269.1 2
45.7 odd 12 2025.1.j.c.701.1 4
45.13 odd 12 2025.1.j.c.26.1 4
45.14 odd 6 405.1.h.b.269.1 2
45.22 odd 12 2025.1.j.c.26.2 4
45.23 even 12 2025.1.j.c.26.2 4
45.29 odd 6 405.1.h.b.134.1 2
45.32 even 12 2025.1.j.c.26.1 4
45.34 even 6 405.1.h.a.134.1 2
45.38 even 12 2025.1.j.c.701.1 4
45.43 odd 12 2025.1.j.c.701.2 4
60.59 even 2 2160.1.c.b.1889.1 1
135.4 even 18 3645.1.n.e.3239.1 6
135.14 odd 18 3645.1.n.d.2024.1 6
135.29 odd 18 3645.1.n.d.1619.1 6
135.34 even 18 3645.1.n.e.404.1 6
135.49 even 18 3645.1.n.e.809.1 6
135.59 odd 18 3645.1.n.d.809.1 6
135.74 odd 18 3645.1.n.d.404.1 6
135.79 even 18 3645.1.n.e.1619.1 6
135.94 even 18 3645.1.n.e.2024.1 6
135.104 odd 18 3645.1.n.d.3239.1 6
135.119 odd 18 3645.1.n.d.2834.1 6
135.124 even 18 3645.1.n.e.2834.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.1.d.a.134.1 ✓ 1 1.1 even 1 trivial
135.1.d.a.134.1 ✓ 1 15.14 odd 2 CM
135.1.d.b.134.1 yes 1 3.2 odd 2
135.1.d.b.134.1 yes 1 5.4 even 2
405.1.h.a.134.1 2 9.2 odd 6
405.1.h.a.134.1 2 45.34 even 6
405.1.h.a.269.1 2 9.5 odd 6
405.1.h.a.269.1 2 45.4 even 6
405.1.h.b.134.1 2 9.7 even 3
405.1.h.b.134.1 2 45.29 odd 6
405.1.h.b.269.1 2 9.4 even 3
405.1.h.b.269.1 2 45.14 odd 6
675.1.c.c.26.1 2 5.2 odd 4
675.1.c.c.26.1 2 15.8 even 4
675.1.c.c.26.2 2 5.3 odd 4
675.1.c.c.26.2 2 15.2 even 4
2025.1.j.c.26.1 4 45.13 odd 12
2025.1.j.c.26.1 4 45.32 even 12
2025.1.j.c.26.2 4 45.22 odd 12
2025.1.j.c.26.2 4 45.23 even 12
2025.1.j.c.701.1 4 45.7 odd 12
2025.1.j.c.701.1 4 45.38 even 12
2025.1.j.c.701.2 4 45.2 even 12
2025.1.j.c.701.2 4 45.43 odd 12
2160.1.c.a.1889.1 1 12.11 even 2
2160.1.c.a.1889.1 1 20.19 odd 2
2160.1.c.b.1889.1 1 4.3 odd 2
2160.1.c.b.1889.1 1 60.59 even 2
3645.1.n.d.404.1 6 27.7 even 9
3645.1.n.d.404.1 6 135.74 odd 18
3645.1.n.d.809.1 6 27.22 even 9
3645.1.n.d.809.1 6 135.59 odd 18
3645.1.n.d.1619.1 6 27.25 even 9
3645.1.n.d.1619.1 6 135.29 odd 18
3645.1.n.d.2024.1 6 27.13 even 9
3645.1.n.d.2024.1 6 135.14 odd 18
3645.1.n.d.2834.1 6 27.16 even 9
3645.1.n.d.2834.1 6 135.119 odd 18
3645.1.n.d.3239.1 6 27.4 even 9
3645.1.n.d.3239.1 6 135.104 odd 18
3645.1.n.e.404.1 6 27.20 odd 18
3645.1.n.e.404.1 6 135.34 even 18
3645.1.n.e.809.1 6 27.5 odd 18
3645.1.n.e.809.1 6 135.49 even 18
3645.1.n.e.1619.1 6 27.2 odd 18
3645.1.n.e.1619.1 6 135.79 even 18
3645.1.n.e.2024.1 6 27.14 odd 18
3645.1.n.e.2024.1 6 135.94 even 18
3645.1.n.e.2834.1 6 27.11 odd 18
3645.1.n.e.2834.1 6 135.124 even 18
3645.1.n.e.3239.1 6 27.23 odd 18
3645.1.n.e.3239.1 6 135.4 even 18