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Properties

Label 138.2.a.a.1.1
Level $138$
Weight $2$
Character 138.1
Self dual yes
Analytic conductor $1.102$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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] = [138,2,Mod(1,138)] 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("138.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(138, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.10193554789\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 138.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -2.00000 q^{20} +2.00000 q^{21} +6.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +4.00000 q^{35} +1.00000 q^{36} +2.00000 q^{39} +2.00000 q^{40} +10.0000 q^{41} -2.00000 q^{42} -12.0000 q^{43} -6.00000 q^{44} -2.00000 q^{45} +1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +12.0000 q^{55} +2.00000 q^{56} -6.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} +4.00000 q^{61} -8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -6.00000 q^{66} -12.0000 q^{67} +1.00000 q^{69} -4.00000 q^{70} -1.00000 q^{72} -10.0000 q^{73} +1.00000 q^{75} +12.0000 q^{77} -2.00000 q^{78} -6.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +14.0000 q^{83} +2.00000 q^{84} +12.0000 q^{86} -6.00000 q^{87} +6.00000 q^{88} +2.00000 q^{90} +4.00000 q^{91} -1.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} -6.00000 q^{97} +3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −2.00000 −0.447214
\(21\) 2.00000 0.436436
\(22\) 6.00000 1.27920
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −2.00000 −0.365148
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 2.00000 0.316228
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) −2.00000 −0.308607
\(43\) −12.0000 −1.82998 −0.914991 − 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) −6.00000 −0.904534
\(45\) −2.00000 −0.298142
\(46\) 1.00000 0.147442
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.00000 0.136083
\(55\) 12.0000 1.61808
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −8.00000 −1.01600
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −6.00000 −0.738549
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) −4.00000 −0.478091
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) −2.00000 −0.226455
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 14.0000 1.53670 0.768350 − 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) −6.00000 −0.643268
\(88\) 6.00000 0.639602
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 2.00000 0.210819
\(91\) 4.00000 0.419314
\(92\) −1.00000 −0.104257
\(93\) −8.00000 −0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 3.00000 0.303046
\(99\) −6.00000 −0.603023
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 138.2.a.a.1.1 ✓ 1
3.2 odd 2 414.2.a.d.1.1 1
4.3 odd 2 1104.2.a.e.1.1 1
5.2 odd 4 3450.2.d.j.2899.1 2
5.3 odd 4 3450.2.d.j.2899.2 2
5.4 even 2 3450.2.a.y.1.1 1
7.6 odd 2 6762.2.a.q.1.1 1
8.3 odd 2 4416.2.a.m.1.1 1
8.5 even 2 4416.2.a.z.1.1 1
12.11 even 2 3312.2.a.n.1.1 1
23.22 odd 2 3174.2.a.b.1.1 1
69.68 even 2 9522.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
414.2.a.d.1.1 1 3.2 odd 2
1104.2.a.e.1.1 1 4.3 odd 2
3174.2.a.b.1.1 1 23.22 odd 2
3312.2.a.n.1.1 1 12.11 even 2
3450.2.a.y.1.1 1 5.4 even 2
3450.2.d.j.2899.1 2 5.2 odd 4
3450.2.d.j.2899.2 2 5.3 odd 4
4416.2.a.m.1.1 1 8.3 odd 2
4416.2.a.z.1.1 1 8.5 even 2
6762.2.a.q.1.1 1 7.6 odd 2
9522.2.a.i.1.1 1 69.68 even 2