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Properties

Label 136.1.j.a.123.1
Level $136$
Weight $1$
Character 136.123
Analytic conductor $0.068$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -8
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] = [136,1,Mod(115,136)] 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("136.115"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(136, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 136.j (of order \(4\), degree \(2\), 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: \(0.0678728417181\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of \(\Q(\sqrt{17 +4 \sqrt{-34}})\)
Artin image: $C_4\wr C_2$
Artin field: Galois closure of 8.0.20123648.1

Embedding invariants

Embedding label 123.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 136.123
Dual form 136.1.j.a.115.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +(-1.00000 + 1.00000i) q^{3} -1.00000 q^{4} +(-1.00000 - 1.00000i) q^{6} -1.00000i q^{8} -1.00000i q^{9} +(1.00000 + 1.00000i) q^{11} +(1.00000 - 1.00000i) q^{12} +1.00000 q^{16} +1.00000i q^{17} +1.00000 q^{18} -2.00000i q^{19} +(-1.00000 + 1.00000i) q^{22} +(1.00000 + 1.00000i) q^{24} -1.00000i q^{25} +1.00000i q^{32} -2.00000 q^{33} -1.00000 q^{34} +1.00000i q^{36} +2.00000 q^{38} +(-1.00000 - 1.00000i) q^{41} +(-1.00000 - 1.00000i) q^{44} +(-1.00000 + 1.00000i) q^{48} +1.00000i q^{49} +1.00000 q^{50} +(-1.00000 - 1.00000i) q^{51} +(2.00000 + 2.00000i) q^{57} -1.00000 q^{64} -2.00000i q^{66} -1.00000i q^{68} -1.00000 q^{72} +(-1.00000 + 1.00000i) q^{73} +(1.00000 + 1.00000i) q^{75} +2.00000i q^{76} +1.00000 q^{81} +(1.00000 - 1.00000i) q^{82} +(1.00000 - 1.00000i) q^{88} +(-1.00000 - 1.00000i) q^{96} +(1.00000 - 1.00000i) q^{97} -1.00000 q^{98} +(1.00000 - 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{4} - 2 q^{6} + 2 q^{11} + 2 q^{12} + 2 q^{16} + 2 q^{18} - 2 q^{22} + 2 q^{24} - 4 q^{33} - 2 q^{34} + 4 q^{38} - 2 q^{41} - 2 q^{44} - 2 q^{48} + 2 q^{50} - 2 q^{51} + 4 q^{57} - 2 q^{64}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 1.00000i
\(3\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(4\) −1.00000 −1.00000
\(5\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) −1.00000 − 1.00000i −1.00000 − 1.00000i
\(7\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) − 1.00000i − 1.00000i
\(9\) − 1.00000i − 1.00000i
\(10\) 0 0
\(11\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 − 1.00000i 1.00000 − 1.00000i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 1.00000i 1.00000i
\(18\) 1.00000 1.00000
\(19\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(25\) − 1.00000i − 1.00000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 0 0
\(31\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(32\) 1.00000i 1.00000i
\(33\) −2.00000 −2.00000
\(34\) −1.00000 −1.00000
\(35\) 0 0
\(36\) 1.00000i 1.00000i
\(37\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 2.00000 2.00000
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 − 1.00000i −1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −1.00000 − 1.00000i −1.00000 − 1.00000i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(49\) 1.00000i 1.00000i
\(50\) 1.00000 1.00000
\(51\) −1.00000 − 1.00000i −1.00000 − 1.00000i
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000 + 2.00000i 2.00000 + 2.00000i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) − 2.00000i − 2.00000i
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) − 1.00000i − 1.00000i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(72\) −1.00000 −1.00000
\(73\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(76\) 2.00000i 2.00000i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 1.00000 − 1.00000i 1.00000 − 1.00000i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 1.00000 − 1.00000i 1.00000 − 1.00000i
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 − 1.00000i −1.00000 − 1.00000i
\(97\) 1.00000 − 1.00000i 1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(98\) −1.00000 −1.00000
\(99\) 1.00000 − 1.00000i 1.00000 − 1.00000i
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 136.1.j.a.123.1 yes 2
3.2 odd 2 1224.1.s.a.667.1 2
4.3 odd 2 544.1.n.a.463.1 2
5.2 odd 4 3400.1.bc.a.2299.1 2
5.3 odd 4 3400.1.bc.b.2299.1 2
5.4 even 2 3400.1.y.a.3251.1 2
8.3 odd 2 CM 136.1.j.a.123.1 yes 2
8.5 even 2 544.1.n.a.463.1 2
17.2 even 8 2312.1.e.a.1155.2 2
17.3 odd 16 2312.1.p.e.155.1 8
17.4 even 4 2312.1.j.b.251.1 2
17.5 odd 16 2312.1.p.e.1579.1 8
17.6 odd 16 2312.1.p.e.1555.1 8
17.7 odd 16 2312.1.p.e.179.2 8
17.8 even 8 2312.1.f.b.579.2 2
17.9 even 8 2312.1.f.b.579.1 2
17.10 odd 16 2312.1.p.e.179.1 8
17.11 odd 16 2312.1.p.e.1555.2 8
17.12 odd 16 2312.1.p.e.1579.2 8
17.13 even 4 inner 136.1.j.a.115.1 ✓ 2
17.14 odd 16 2312.1.p.e.155.2 8
17.15 even 8 2312.1.e.a.1155.1 2
17.16 even 2 2312.1.j.b.1483.1 2
24.11 even 2 1224.1.s.a.667.1 2
40.3 even 4 3400.1.bc.b.2299.1 2
40.19 odd 2 3400.1.y.a.3251.1 2
40.27 even 4 3400.1.bc.a.2299.1 2
51.47 odd 4 1224.1.s.a.523.1 2
68.47 odd 4 544.1.n.a.47.1 2
85.13 odd 4 3400.1.bc.a.2699.1 2
85.47 odd 4 3400.1.bc.b.2699.1 2
85.64 even 4 3400.1.y.a.251.1 2
136.3 even 16 2312.1.p.e.155.1 8
136.11 even 16 2312.1.p.e.1555.2 8
136.13 even 4 544.1.n.a.47.1 2
136.19 odd 8 2312.1.e.a.1155.2 2
136.27 even 16 2312.1.p.e.179.1 8
136.43 odd 8 2312.1.f.b.579.1 2
136.59 odd 8 2312.1.f.b.579.2 2
136.67 odd 2 2312.1.j.b.1483.1 2
136.75 even 16 2312.1.p.e.179.2 8
136.83 odd 8 2312.1.e.a.1155.1 2
136.91 even 16 2312.1.p.e.1555.1 8
136.99 even 16 2312.1.p.e.155.2 8
136.107 even 16 2312.1.p.e.1579.1 8
136.115 odd 4 inner 136.1.j.a.115.1 ✓ 2
136.123 odd 4 2312.1.j.b.251.1 2
136.131 even 16 2312.1.p.e.1579.2 8
408.251 even 4 1224.1.s.a.523.1 2
680.387 even 4 3400.1.bc.b.2699.1 2
680.523 even 4 3400.1.bc.a.2699.1 2
680.659 odd 4 3400.1.y.a.251.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.1.j.a.115.1 ✓ 2 17.13 even 4 inner
136.1.j.a.115.1 ✓ 2 136.115 odd 4 inner
136.1.j.a.123.1 yes 2 1.1 even 1 trivial
136.1.j.a.123.1 yes 2 8.3 odd 2 CM
544.1.n.a.47.1 2 68.47 odd 4
544.1.n.a.47.1 2 136.13 even 4
544.1.n.a.463.1 2 4.3 odd 2
544.1.n.a.463.1 2 8.5 even 2
1224.1.s.a.523.1 2 51.47 odd 4
1224.1.s.a.523.1 2 408.251 even 4
1224.1.s.a.667.1 2 3.2 odd 2
1224.1.s.a.667.1 2 24.11 even 2
2312.1.e.a.1155.1 2 17.15 even 8
2312.1.e.a.1155.1 2 136.83 odd 8
2312.1.e.a.1155.2 2 17.2 even 8
2312.1.e.a.1155.2 2 136.19 odd 8
2312.1.f.b.579.1 2 17.9 even 8
2312.1.f.b.579.1 2 136.43 odd 8
2312.1.f.b.579.2 2 17.8 even 8
2312.1.f.b.579.2 2 136.59 odd 8
2312.1.j.b.251.1 2 17.4 even 4
2312.1.j.b.251.1 2 136.123 odd 4
2312.1.j.b.1483.1 2 17.16 even 2
2312.1.j.b.1483.1 2 136.67 odd 2
2312.1.p.e.155.1 8 17.3 odd 16
2312.1.p.e.155.1 8 136.3 even 16
2312.1.p.e.155.2 8 17.14 odd 16
2312.1.p.e.155.2 8 136.99 even 16
2312.1.p.e.179.1 8 17.10 odd 16
2312.1.p.e.179.1 8 136.27 even 16
2312.1.p.e.179.2 8 17.7 odd 16
2312.1.p.e.179.2 8 136.75 even 16
2312.1.p.e.1555.1 8 17.6 odd 16
2312.1.p.e.1555.1 8 136.91 even 16
2312.1.p.e.1555.2 8 17.11 odd 16
2312.1.p.e.1555.2 8 136.11 even 16
2312.1.p.e.1579.1 8 17.5 odd 16
2312.1.p.e.1579.1 8 136.107 even 16
2312.1.p.e.1579.2 8 17.12 odd 16
2312.1.p.e.1579.2 8 136.131 even 16
3400.1.y.a.251.1 2 85.64 even 4
3400.1.y.a.251.1 2 680.659 odd 4
3400.1.y.a.3251.1 2 5.4 even 2
3400.1.y.a.3251.1 2 40.19 odd 2
3400.1.bc.a.2299.1 2 5.2 odd 4
3400.1.bc.a.2299.1 2 40.27 even 4
3400.1.bc.a.2699.1 2 85.13 odd 4
3400.1.bc.a.2699.1 2 680.523 even 4
3400.1.bc.b.2299.1 2 5.3 odd 4
3400.1.bc.b.2299.1 2 40.3 even 4
3400.1.bc.b.2699.1 2 85.47 odd 4
3400.1.bc.b.2699.1 2 680.387 even 4