Properties

Label 58.2.a.b.1.1
Level $58$
Weight $2$
Character 58.1
Self dual yes
Analytic conductor $0.463$
Analytic rank $0$
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] = [58,2,Mod(1,58)] 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("58.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.a (trivial)

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: \(0.463132331723\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 58.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +8.00000 q^{17} -2.00000 q^{18} +1.00000 q^{20} +2.00000 q^{21} -3.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} +5.00000 q^{27} -2.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} -3.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} +8.00000 q^{34} -2.00000 q^{35} -2.00000 q^{36} +8.00000 q^{37} +1.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} +2.00000 q^{42} -11.0000 q^{43} -3.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} +13.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -4.00000 q^{50} -8.00000 q^{51} -1.00000 q^{52} -11.0000 q^{53} +5.00000 q^{54} -3.00000 q^{55} -2.00000 q^{56} -1.00000 q^{58} -1.00000 q^{60} -8.00000 q^{61} -3.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +3.00000 q^{66} -12.0000 q^{67} +8.00000 q^{68} -4.00000 q^{69} -2.00000 q^{70} +2.00000 q^{71} -2.00000 q^{72} +4.00000 q^{73} +8.00000 q^{74} +4.00000 q^{75} +6.00000 q^{77} +1.00000 q^{78} +15.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} +2.00000 q^{84} +8.00000 q^{85} -11.0000 q^{86} +1.00000 q^{87} -3.00000 q^{88} -10.0000 q^{89} -2.00000 q^{90} +2.00000 q^{91} +4.00000 q^{92} +3.00000 q^{93} +13.0000 q^{94} -1.00000 q^{96} -2.00000 q^{97} -3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −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
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −2.00000 −0.534522
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 8.00000 1.94029 0.970143 − 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) −2.00000 −0.471405
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.00000 0.436436
\(22\) −3.00000 −0.639602
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) −1.00000 −0.196116
\(27\) 5.00000 0.962250
\(28\) −2.00000 −0.377964
\(29\) −1.00000 −0.185695
\(30\) −1.00000 −0.182574
\(31\) −3.00000 −0.538816 −0.269408 − 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.00000 0.522233
\(34\) 8.00000 1.37199
\(35\) −2.00000 −0.338062
\(36\) −2.00000 −0.333333
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 1.00000 0.158114
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 2.00000 0.308607
\(43\) −11.0000 −1.67748 −0.838742 − 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) −3.00000 −0.452267
\(45\) −2.00000 −0.298142
\(46\) 4.00000 0.589768
\(47\) 13.0000 1.89624 0.948122 − 0.317905i \(-0.102979\pi\)
0.948122 + 0.317905i \(0.102979\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −4.00000 −0.565685
\(51\) −8.00000 −1.12022
\(52\) −1.00000 −0.138675
\(53\) −11.0000 −1.51097 −0.755483 − 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) 5.00000 0.680414
\(55\) −3.00000 −0.404520
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −0.129099
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −3.00000 −0.381000
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 3.00000 0.369274
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 8.00000 0.970143
\(69\) −4.00000 −0.481543
\(70\) −2.00000 −0.239046
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −2.00000 −0.235702
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 8.00000 0.929981
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 1.00000 0.113228
\(79\) 15.0000 1.68763 0.843816 − 0.536633i \(-0.180304\pi\)
0.843816 + 0.536633i \(0.180304\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 2.00000 0.218218
\(85\) 8.00000 0.867722
\(86\) −11.0000 −1.18616
\(87\) 1.00000 0.107211
\(88\) −3.00000 −0.319801
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −2.00000 −0.210819
\(91\) 2.00000 0.209657
\(92\) 4.00000 0.417029
\(93\) 3.00000 0.311086
\(94\) 13.0000 1.34085
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −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 58.2.a.b.1.1 ✓ 1
3.2 odd 2 522.2.a.b.1.1 1
4.3 odd 2 464.2.a.e.1.1 1
5.2 odd 4 1450.2.b.b.349.2 2
5.3 odd 4 1450.2.b.b.349.1 2
5.4 even 2 1450.2.a.c.1.1 1
7.6 odd 2 2842.2.a.e.1.1 1
8.3 odd 2 1856.2.a.f.1.1 1
8.5 even 2 1856.2.a.k.1.1 1
11.10 odd 2 7018.2.a.a.1.1 1
12.11 even 2 4176.2.a.n.1.1 1
13.12 even 2 9802.2.a.a.1.1 1
29.12 odd 4 1682.2.b.a.1681.2 2
29.17 odd 4 1682.2.b.a.1681.1 2
29.28 even 2 1682.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.b.1.1 ✓ 1 1.1 even 1 trivial
464.2.a.e.1.1 1 4.3 odd 2
522.2.a.b.1.1 1 3.2 odd 2
1450.2.a.c.1.1 1 5.4 even 2
1450.2.b.b.349.1 2 5.3 odd 4
1450.2.b.b.349.2 2 5.2 odd 4
1682.2.a.d.1.1 1 29.28 even 2
1682.2.b.a.1681.1 2 29.17 odd 4
1682.2.b.a.1681.2 2 29.12 odd 4
1856.2.a.f.1.1 1 8.3 odd 2
1856.2.a.k.1.1 1 8.5 even 2
2842.2.a.e.1.1 1 7.6 odd 2
4176.2.a.n.1.1 1 12.11 even 2
7018.2.a.a.1.1 1 11.10 odd 2
9802.2.a.a.1.1 1 13.12 even 2