Properties

Label 430.2.a.c.1.1
Level $430$
Weight $2$
Character 430.1
Self dual yes
Analytic conductor $3.434$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [430,2,Mod(1,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.43356728692\)
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 \(\chi\) \(=\) 430.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.00000 q^{11} -2.00000 q^{12} +5.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -7.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -6.00000 q^{22} -6.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} +5.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} -3.00000 q^{29} +2.00000 q^{30} +5.00000 q^{31} +1.00000 q^{32} +12.0000 q^{33} -6.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -7.00000 q^{38} -10.0000 q^{39} -1.00000 q^{40} -3.00000 q^{41} +2.00000 q^{42} +1.00000 q^{43} -6.00000 q^{44} -1.00000 q^{45} -6.00000 q^{46} +12.0000 q^{47} -2.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +12.0000 q^{51} +5.00000 q^{52} +6.00000 q^{53} +4.00000 q^{54} +6.00000 q^{55} -1.00000 q^{56} +14.0000 q^{57} -3.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} -1.00000 q^{61} +5.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -5.00000 q^{65} +12.0000 q^{66} -13.0000 q^{67} -6.00000 q^{68} +12.0000 q^{69} +1.00000 q^{70} +12.0000 q^{71} +1.00000 q^{72} +11.0000 q^{73} +2.00000 q^{74} -2.00000 q^{75} -7.00000 q^{76} +6.00000 q^{77} -10.0000 q^{78} -1.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -3.00000 q^{82} +2.00000 q^{84} +6.00000 q^{85} +1.00000 q^{86} +6.00000 q^{87} -6.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} -5.00000 q^{91} -6.00000 q^{92} -10.0000 q^{93} +12.0000 q^{94} +7.00000 q^{95} -2.00000 q^{96} +8.00000 q^{97} -6.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See 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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.00000 −0.816497
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −2.00000 −0.577350
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.00000 0.436436
\(22\) −6.00000 −1.27920
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −2.00000 −0.408248
\(25\) 1.00000 0.200000
\(26\) 5.00000 0.980581
\(27\) 4.00000 0.769800
\(28\) −1.00000 −0.188982
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 2.00000 0.365148
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 1.00000 0.176777
\(33\) 12.0000 2.08893
\(34\) −6.00000 −1.02899
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −7.00000 −1.13555
\(39\) −10.0000 −1.60128
\(40\) −1.00000 −0.158114
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 2.00000 0.308607
\(43\) 1.00000 0.152499
\(44\) −6.00000 −0.904534
\(45\) −1.00000 −0.149071
\(46\) −6.00000 −0.884652
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −2.00000 −0.288675
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 12.0000 1.68034
\(52\) 5.00000 0.693375
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 4.00000 0.544331
\(55\) 6.00000 0.809040
\(56\) −1.00000 −0.133631
\(57\) 14.0000 1.85435
\(58\) −3.00000 −0.393919
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 5.00000 0.635001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −5.00000 −0.620174
\(66\) 12.0000 1.47710
\(67\) −13.0000 −1.58820 −0.794101 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) −6.00000 −0.727607
\(69\) 12.0000 1.44463
\(70\) 1.00000 0.119523
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 2.00000 0.232495
\(75\) −2.00000 −0.230940
\(76\) −7.00000 −0.802955
\(77\) 6.00000 0.683763
\(78\) −10.0000 −1.13228
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) −3.00000 −0.331295
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) 6.00000 0.650791
\(86\) 1.00000 0.107833
\(87\) 6.00000 0.643268
\(88\) −6.00000 −0.639602
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) −5.00000 −0.524142
\(92\) −6.00000 −0.625543
\(93\) −10.0000 −1.03695
\(94\) 12.0000 1.23771
\(95\) 7.00000 0.718185
\(96\) −2.00000 −0.204124
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −6.00000 −0.606092
\(99\) −6.00000 −0.603023
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 12.0000 1.18818
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 5.00000 0.490290
\(105\) −2.00000 −0.195180
\(106\) 6.00000 0.582772
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 4.00000 0.384900
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 6.00000 0.572078
\(111\) −4.00000 −0.379663
\(112\) −1.00000 −0.0944911
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 14.0000 1.31122
\(115\) 6.00000 0.559503
\(116\) −3.00000 −0.278543
\(117\) 5.00000 0.462250
\(118\) −12.0000 −1.10469
\(119\) 6.00000 0.550019
\(120\) 2.00000 0.182574
\(121\) 25.0000 2.27273
\(122\) −1.00000 −0.0905357
\(123\) 6.00000 0.541002
\(124\) 5.00000 0.449013
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.00000 −0.176090
\(130\) −5.00000 −0.438529
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 12.0000 1.04447
\(133\) 7.00000 0.606977
\(134\) −13.0000 −1.12303
\(135\) −4.00000 −0.344265
\(136\) −6.00000 −0.514496
\(137\) −15.0000 −1.28154 −0.640768 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) 12.0000 1.02151
\(139\) −22.0000 −1.86602 −0.933008 0.359856i \(-0.882826\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 1.00000 0.0845154
\(141\) −24.0000 −2.02116
\(142\) 12.0000 1.00702
\(143\) −30.0000 −2.50873
\(144\) 1.00000 0.0833333
\(145\) 3.00000 0.249136
\(146\) 11.0000 0.910366
\(147\) 12.0000 0.989743
\(148\) 2.00000 0.164399
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) −2.00000 −0.163299
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −7.00000 −0.567775
\(153\) −6.00000 −0.485071
\(154\) 6.00000 0.483494
\(155\) −5.00000 −0.401610
\(156\) −10.0000 −0.800641
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −1.00000 −0.0795557
\(159\) −12.0000 −0.951662
\(160\) −1.00000 −0.0790569
\(161\) 6.00000 0.472866
\(162\) −11.0000 −0.864242
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −3.00000 −0.234261
\(165\) −12.0000 −0.934199
\(166\) 0 0
\(167\) −6.00000 −0.464294 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 2.00000 0.154303
\(169\) 12.0000 0.923077
\(170\) 6.00000 0.460179
\(171\) −7.00000 −0.535303
\(172\) 1.00000 0.0762493
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) 6.00000 0.454859
\(175\) −1.00000 −0.0755929
\(176\) −6.00000 −0.452267
\(177\) 24.0000 1.80395
\(178\) 6.00000 0.449719
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −5.00000 −0.370625
\(183\) 2.00000 0.147844
\(184\) −6.00000 −0.442326
\(185\) −2.00000 −0.147043
\(186\) −10.0000 −0.733236
\(187\) 36.0000 2.63258
\(188\) 12.0000 0.875190
\(189\) −4.00000 −0.290957
\(190\) 7.00000 0.507833
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −2.00000 −0.144338
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 8.00000 0.574367
\(195\) 10.0000 0.716115
\(196\) −6.00000 −0.428571
\(197\) −15.0000 −1.06871 −0.534353 0.845262i \(-0.679445\pi\)
−0.534353 + 0.845262i \(0.679445\pi\)
\(198\) −6.00000 −0.426401
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) 26.0000 1.83390
\(202\) 6.00000 0.422159
\(203\) 3.00000 0.210559
\(204\) 12.0000 0.840168
\(205\) 3.00000 0.209529
\(206\) 14.0000 0.975426
\(207\) −6.00000 −0.417029
\(208\) 5.00000 0.346688
\(209\) 42.0000 2.90520
\(210\) −2.00000 −0.138013
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) −24.0000 −1.64445
\(214\) −9.00000 −0.615227
\(215\) −1.00000 −0.0681994
\(216\) 4.00000 0.272166
\(217\) −5.00000 −0.339422
\(218\) −16.0000 −1.08366
\(219\) −22.0000 −1.48662
\(220\) 6.00000 0.404520
\(221\) −30.0000 −2.01802
\(222\) −4.00000 −0.268462
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) −3.00000 −0.199557
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 14.0000 0.927173
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 6.00000 0.395628
\(231\) −12.0000 −0.789542
\(232\) −3.00000 −0.196960
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 5.00000 0.326860
\(235\) −12.0000 −0.782794
\(236\) −12.0000 −0.781133
\(237\) 2.00000 0.129914
\(238\) 6.00000 0.388922
\(239\) 3.00000 0.194054 0.0970269 0.995282i \(-0.469067\pi\)
0.0970269 + 0.995282i \(0.469067\pi\)
\(240\) 2.00000 0.129099
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 25.0000 1.60706
\(243\) 10.0000 0.641500
\(244\) −1.00000 −0.0640184
\(245\) 6.00000 0.383326
\(246\) 6.00000 0.382546
\(247\) −35.0000 −2.22700
\(248\) 5.00000 0.317500
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 36.0000 2.26330
\(254\) −16.0000 −1.00393
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) −3.00000 −0.187135 −0.0935674 0.995613i \(-0.529827\pi\)
−0.0935674 + 0.995613i \(0.529827\pi\)
\(258\) −2.00000 −0.124515
\(259\) −2.00000 −0.124274
\(260\) −5.00000 −0.310087
\(261\) −3.00000 −0.185695
\(262\) −12.0000 −0.741362
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 12.0000 0.738549
\(265\) −6.00000 −0.368577
\(266\) 7.00000 0.429198
\(267\) −12.0000 −0.734388
\(268\) −13.0000 −0.794101
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) −4.00000 −0.243432
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) −6.00000 −0.363803
\(273\) 10.0000 0.605228
\(274\) −15.0000 −0.906183
\(275\) −6.00000 −0.361814
\(276\) 12.0000 0.722315
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −22.0000 −1.31947
\(279\) 5.00000 0.299342
\(280\) 1.00000 0.0597614
\(281\) 9.00000 0.536895 0.268447 0.963294i \(-0.413489\pi\)
0.268447 + 0.963294i \(0.413489\pi\)
\(282\) −24.0000 −1.42918
\(283\) 5.00000 0.297219 0.148610 0.988896i \(-0.452520\pi\)
0.148610 + 0.988896i \(0.452520\pi\)
\(284\) 12.0000 0.712069
\(285\) −14.0000 −0.829288
\(286\) −30.0000 −1.77394
\(287\) 3.00000 0.177084
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 3.00000 0.176166
\(291\) −16.0000 −0.937937
\(292\) 11.0000 0.643726
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 12.0000 0.699854
\(295\) 12.0000 0.698667
\(296\) 2.00000 0.116248
\(297\) −24.0000 −1.39262
\(298\) 3.00000 0.173785
\(299\) −30.0000 −1.73494
\(300\) −2.00000 −0.115470
\(301\) −1.00000 −0.0576390
\(302\) −10.0000 −0.575435
\(303\) −12.0000 −0.689382
\(304\) −7.00000 −0.401478
\(305\) 1.00000 0.0572598
\(306\) −6.00000 −0.342997
\(307\) 29.0000 1.65512 0.827559 0.561379i \(-0.189729\pi\)
0.827559 + 0.561379i \(0.189729\pi\)
\(308\) 6.00000 0.341882
\(309\) −28.0000 −1.59286
\(310\) −5.00000 −0.283981
\(311\) −15.0000 −0.850572 −0.425286 0.905059i \(-0.639826\pi\)
−0.425286 + 0.905059i \(0.639826\pi\)
\(312\) −10.0000 −0.566139
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 14.0000 0.790066
\(315\) 1.00000 0.0563436
\(316\) −1.00000 −0.0562544
\(317\) −15.0000 −0.842484 −0.421242 0.906948i \(-0.638406\pi\)
−0.421242 + 0.906948i \(0.638406\pi\)
\(318\) −12.0000 −0.672927
\(319\) 18.0000 1.00781
\(320\) −1.00000 −0.0559017
\(321\) 18.0000 1.00466
\(322\) 6.00000 0.334367
\(323\) 42.0000 2.33694
\(324\) −11.0000 −0.611111
\(325\) 5.00000 0.277350
\(326\) −16.0000 −0.886158
\(327\) 32.0000 1.76960
\(328\) −3.00000 −0.165647
\(329\) −12.0000 −0.661581
\(330\) −12.0000 −0.660578
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) −6.00000 −0.328305
\(335\) 13.0000 0.710266
\(336\) 2.00000 0.109109
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 12.0000 0.652714
\(339\) 6.00000 0.325875
\(340\) 6.00000 0.325396
\(341\) −30.0000 −1.62459
\(342\) −7.00000 −0.378517
\(343\) 13.0000 0.701934
\(344\) 1.00000 0.0539164
\(345\) −12.0000 −0.646058
\(346\) −15.0000 −0.806405
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 6.00000 0.321634
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 20.0000 1.06752
\(352\) −6.00000 −0.319801
\(353\) −12.0000 −0.638696 −0.319348 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 24.0000 1.27559
\(355\) −12.0000 −0.636894
\(356\) 6.00000 0.317999
\(357\) −12.0000 −0.635107
\(358\) 3.00000 0.158555
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 30.0000 1.57895
\(362\) 2.00000 0.105118
\(363\) −50.0000 −2.62432
\(364\) −5.00000 −0.262071
\(365\) −11.0000 −0.575766
\(366\) 2.00000 0.104542
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −6.00000 −0.312772
\(369\) −3.00000 −0.156174
\(370\) −2.00000 −0.103975
\(371\) −6.00000 −0.311504
\(372\) −10.0000 −0.518476
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 36.0000 1.86152
\(375\) 2.00000 0.103280
\(376\) 12.0000 0.618853
\(377\) −15.0000 −0.772539
\(378\) −4.00000 −0.205738
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 7.00000 0.359092
\(381\) 32.0000 1.63941
\(382\) 0 0
\(383\) 15.0000 0.766464 0.383232 0.923652i \(-0.374811\pi\)
0.383232 + 0.923652i \(0.374811\pi\)
\(384\) −2.00000 −0.102062
\(385\) −6.00000 −0.305788
\(386\) −4.00000 −0.203595
\(387\) 1.00000 0.0508329
\(388\) 8.00000 0.406138
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 10.0000 0.506370
\(391\) 36.0000 1.82060
\(392\) −6.00000 −0.303046
\(393\) 24.0000 1.21064
\(394\) −15.0000 −0.755689
\(395\) 1.00000 0.0503155
\(396\) −6.00000 −0.301511
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 20.0000 1.00251
\(399\) −14.0000 −0.700877
\(400\) 1.00000 0.0500000
\(401\) −9.00000 −0.449439 −0.224719 0.974424i \(-0.572147\pi\)
−0.224719 + 0.974424i \(0.572147\pi\)
\(402\) 26.0000 1.29676
\(403\) 25.0000 1.24534
\(404\) 6.00000 0.298511
\(405\) 11.0000 0.546594
\(406\) 3.00000 0.148888
\(407\) −12.0000 −0.594818
\(408\) 12.0000 0.594089
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 3.00000 0.148159
\(411\) 30.0000 1.47979
\(412\) 14.0000 0.689730
\(413\) 12.0000 0.590481
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) 5.00000 0.245145
\(417\) 44.0000 2.15469
\(418\) 42.0000 2.05429
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 17.0000 0.828529 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) −4.00000 −0.194717
\(423\) 12.0000 0.583460
\(424\) 6.00000 0.291386
\(425\) −6.00000 −0.291043
\(426\) −24.0000 −1.16280
\(427\) 1.00000 0.0483934
\(428\) −9.00000 −0.435031
\(429\) 60.0000 2.89683
\(430\) −1.00000 −0.0482243
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 4.00000 0.192450
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) −5.00000 −0.240008
\(435\) −6.00000 −0.287678
\(436\) −16.0000 −0.766261
\(437\) 42.0000 2.00913
\(438\) −22.0000 −1.05120
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 6.00000 0.286039
\(441\) −6.00000 −0.285714
\(442\) −30.0000 −1.42695
\(443\) −39.0000 −1.85295 −0.926473 0.376361i \(-0.877175\pi\)
−0.926473 + 0.376361i \(0.877175\pi\)
\(444\) −4.00000 −0.189832
\(445\) −6.00000 −0.284427
\(446\) 8.00000 0.378811
\(447\) −6.00000 −0.283790
\(448\) −1.00000 −0.0472456
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 1.00000 0.0471405
\(451\) 18.0000 0.847587
\(452\) −3.00000 −0.141108
\(453\) 20.0000 0.939682
\(454\) 0 0
\(455\) 5.00000 0.234404
\(456\) 14.0000 0.655610
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −22.0000 −1.02799
\(459\) −24.0000 −1.12022
\(460\) 6.00000 0.279751
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) −12.0000 −0.558291
\(463\) 5.00000 0.232370 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) −3.00000 −0.139272
\(465\) 10.0000 0.463739
\(466\) −6.00000 −0.277945
\(467\) −30.0000 −1.38823 −0.694117 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 5.00000 0.231125
\(469\) 13.0000 0.600284
\(470\) −12.0000 −0.553519
\(471\) −28.0000 −1.29017
\(472\) −12.0000 −0.552345
\(473\) −6.00000 −0.275880
\(474\) 2.00000 0.0918630
\(475\) −7.00000 −0.321182
\(476\) 6.00000 0.275010
\(477\) 6.00000 0.274721
\(478\) 3.00000 0.137217
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 2.00000 0.0912871
\(481\) 10.0000 0.455961
\(482\) −28.0000 −1.27537
\(483\) −12.0000 −0.546019
\(484\) 25.0000 1.13636
\(485\) −8.00000 −0.363261
\(486\) 10.0000 0.453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 32.0000 1.44709
\(490\) 6.00000 0.271052
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 6.00000 0.270501
\(493\) 18.0000 0.810679
\(494\) −35.0000 −1.57472
\(495\) 6.00000 0.269680
\(496\) 5.00000 0.224507
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 41.0000 1.83541 0.917706 0.397260i \(-0.130039\pi\)
0.917706 + 0.397260i \(0.130039\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 12.0000 0.536120
\(502\) 30.0000 1.33897
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −6.00000 −0.266996
\(506\) 36.0000 1.60040
\(507\) −24.0000 −1.06588
\(508\) −16.0000 −0.709885
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) −12.0000 −0.531369
\(511\) −11.0000 −0.486611
\(512\) 1.00000 0.0441942
\(513\) −28.0000 −1.23623
\(514\) −3.00000 −0.132324
\(515\) −14.0000 −0.616914
\(516\) −2.00000 −0.0880451
\(517\) −72.0000 −3.16656
\(518\) −2.00000 −0.0878750
\(519\) 30.0000 1.31685
\(520\) −5.00000 −0.219265
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) −3.00000 −0.131306
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −12.0000 −0.524222
\(525\) 2.00000 0.0872872
\(526\) 9.00000 0.392419
\(527\) −30.0000 −1.30682
\(528\) 12.0000 0.522233
\(529\) 13.0000 0.565217
\(530\) −6.00000 −0.260623
\(531\) −12.0000 −0.520756
\(532\) 7.00000 0.303488
\(533\) −15.0000 −0.649722
\(534\) −12.0000 −0.519291
\(535\) 9.00000 0.389104
\(536\) −13.0000 −0.561514
\(537\) −6.00000 −0.258919
\(538\) 0 0
\(539\) 36.0000 1.55063
\(540\) −4.00000 −0.172133
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 11.0000 0.472490
\(543\) −4.00000 −0.171656
\(544\) −6.00000 −0.257248
\(545\) 16.0000 0.685365
\(546\) 10.0000 0.427960
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −15.0000 −0.640768
\(549\) −1.00000 −0.0426790
\(550\) −6.00000 −0.255841
\(551\) 21.0000 0.894630
\(552\) 12.0000 0.510754
\(553\) 1.00000 0.0425243
\(554\) 8.00000 0.339887
\(555\) 4.00000 0.169791
\(556\) −22.0000 −0.933008
\(557\) −9.00000 −0.381342 −0.190671 0.981654i \(-0.561066\pi\)
−0.190671 + 0.981654i \(0.561066\pi\)
\(558\) 5.00000 0.211667
\(559\) 5.00000 0.211477
\(560\) 1.00000 0.0422577
\(561\) −72.0000 −3.03984
\(562\) 9.00000 0.379642
\(563\) −21.0000 −0.885044 −0.442522 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) −24.0000 −1.01058
\(565\) 3.00000 0.126211
\(566\) 5.00000 0.210166
\(567\) 11.0000 0.461957
\(568\) 12.0000 0.503509
\(569\) −27.0000 −1.13190 −0.565949 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) −14.0000 −0.586395
\(571\) 5.00000 0.209243 0.104622 0.994512i \(-0.466637\pi\)
0.104622 + 0.994512i \(0.466637\pi\)
\(572\) −30.0000 −1.25436
\(573\) 0 0
\(574\) 3.00000 0.125218
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) 19.0000 0.790296
\(579\) 8.00000 0.332469
\(580\) 3.00000 0.124568
\(581\) 0 0
\(582\) −16.0000 −0.663221
\(583\) −36.0000 −1.49097
\(584\) 11.0000 0.455183
\(585\) −5.00000 −0.206725
\(586\) 6.00000 0.247858
\(587\) 42.0000 1.73353 0.866763 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) 12.0000 0.494872
\(589\) −35.0000 −1.44215
\(590\) 12.0000 0.494032
\(591\) 30.0000 1.23404
\(592\) 2.00000 0.0821995
\(593\) 21.0000 0.862367 0.431183 0.902264i \(-0.358096\pi\)
0.431183 + 0.902264i \(0.358096\pi\)
\(594\) −24.0000 −0.984732
\(595\) −6.00000 −0.245976
\(596\) 3.00000 0.122885
\(597\) −40.0000 −1.63709
\(598\) −30.0000 −1.22679
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) −2.00000 −0.0816497
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) −1.00000 −0.0407570
\(603\) −13.0000 −0.529401
\(604\) −10.0000 −0.406894
\(605\) −25.0000 −1.01639
\(606\) −12.0000 −0.487467
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) −7.00000 −0.283887
\(609\) −6.00000 −0.243132
\(610\) 1.00000 0.0404888
\(611\) 60.0000 2.42734
\(612\) −6.00000 −0.242536
\(613\) 29.0000 1.17130 0.585649 0.810564i \(-0.300840\pi\)
0.585649 + 0.810564i \(0.300840\pi\)
\(614\) 29.0000 1.17034
\(615\) −6.00000 −0.241943
\(616\) 6.00000 0.241747
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) −28.0000 −1.12633
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) −5.00000 −0.200805
\(621\) −24.0000 −0.963087
\(622\) −15.0000 −0.601445
\(623\) −6.00000 −0.240385
\(624\) −10.0000 −0.400320
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) −84.0000 −3.35464
\(628\) 14.0000 0.558661
\(629\) −12.0000 −0.478471
\(630\) 1.00000 0.0398410
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) −1.00000 −0.0397779
\(633\) 8.00000 0.317971
\(634\) −15.0000 −0.595726
\(635\) 16.0000 0.634941
\(636\) −12.0000 −0.475831
\(637\) −30.0000 −1.18864
\(638\) 18.0000 0.712627
\(639\) 12.0000 0.474713
\(640\) −1.00000 −0.0395285
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 18.0000 0.710403
\(643\) −31.0000 −1.22252 −0.611260 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) 6.00000 0.236433
\(645\) 2.00000 0.0787499
\(646\) 42.0000 1.65247
\(647\) 45.0000 1.76913 0.884566 0.466415i \(-0.154454\pi\)
0.884566 + 0.466415i \(0.154454\pi\)
\(648\) −11.0000 −0.432121
\(649\) 72.0000 2.82625
\(650\) 5.00000 0.196116
\(651\) 10.0000 0.391931
\(652\) −16.0000 −0.626608
\(653\) −12.0000 −0.469596 −0.234798 0.972044i \(-0.575443\pi\)
−0.234798 + 0.972044i \(0.575443\pi\)
\(654\) 32.0000 1.25130
\(655\) 12.0000 0.468879
\(656\) −3.00000 −0.117130
\(657\) 11.0000 0.429151
\(658\) −12.0000 −0.467809
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) −12.0000 −0.467099
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) 8.00000 0.310929
\(663\) 60.0000 2.33021
\(664\) 0 0
\(665\) −7.00000 −0.271448
\(666\) 2.00000 0.0774984
\(667\) 18.0000 0.696963
\(668\) −6.00000 −0.232147
\(669\) −16.0000 −0.618596
\(670\) 13.0000 0.502234
\(671\) 6.00000 0.231627
\(672\) 2.00000 0.0771517
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −22.0000 −0.847408
\(675\) 4.00000 0.153960
\(676\) 12.0000 0.461538
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 6.00000 0.230429
\(679\) −8.00000 −0.307012
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) −30.0000 −1.14876
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −7.00000 −0.267652
\(685\) 15.0000 0.573121
\(686\) 13.0000 0.496342
\(687\) 44.0000 1.67870
\(688\) 1.00000 0.0381246
\(689\) 30.0000 1.14291
\(690\) −12.0000 −0.456832
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −15.0000 −0.570214
\(693\) 6.00000 0.227921
\(694\) 12.0000 0.455514
\(695\) 22.0000 0.834508
\(696\) 6.00000 0.227429
\(697\) 18.0000 0.681799
\(698\) −10.0000 −0.378506
\(699\) 12.0000 0.453882
\(700\) −1.00000 −0.0377964
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 20.0000 0.754851
\(703\) −14.0000 −0.528020
\(704\) −6.00000 −0.226134
\(705\) 24.0000 0.903892
\(706\) −12.0000 −0.451626
\(707\) −6.00000 −0.225653
\(708\) 24.0000 0.901975
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) −12.0000 −0.450352
\(711\) −1.00000 −0.0375029
\(712\) 6.00000 0.224860
\(713\) −30.0000 −1.12351
\(714\) −12.0000 −0.449089
\(715\) 30.0000 1.12194
\(716\) 3.00000 0.112115
\(717\) −6.00000 −0.224074
\(718\) −15.0000 −0.559795
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −14.0000 −0.521387
\(722\) 30.0000 1.11648
\(723\) 56.0000 2.08266
\(724\) 2.00000 0.0743294
\(725\) −3.00000 −0.111417
\(726\) −50.0000 −1.85567
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) −5.00000 −0.185312
\(729\) 13.0000 0.481481
\(730\) −11.0000 −0.407128
\(731\) −6.00000 −0.221918
\(732\) 2.00000 0.0739221
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 8.00000 0.295285
\(735\) −12.0000 −0.442627
\(736\) −6.00000 −0.221163
\(737\) 78.0000 2.87317
\(738\) −3.00000 −0.110432
\(739\) 29.0000 1.06678 0.533391 0.845869i \(-0.320917\pi\)
0.533391 + 0.845869i \(0.320917\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 70.0000 2.57151
\(742\) −6.00000 −0.220267
\(743\) 21.0000 0.770415 0.385208 0.922830i \(-0.374130\pi\)
0.385208 + 0.922830i \(0.374130\pi\)
\(744\) −10.0000 −0.366618
\(745\) −3.00000 −0.109911
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 36.0000 1.31629
\(749\) 9.00000 0.328853
\(750\) 2.00000 0.0730297
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) 12.0000 0.437595
\(753\) −60.0000 −2.18652
\(754\) −15.0000 −0.546268
\(755\) 10.0000 0.363937
\(756\) −4.00000 −0.145479
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 2.00000 0.0726433
\(759\) −72.0000 −2.61343
\(760\) 7.00000 0.253917
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 32.0000 1.15924
\(763\) 16.0000 0.579239
\(764\) 0 0
\(765\) 6.00000 0.216930
\(766\) 15.0000 0.541972
\(767\) −60.0000 −2.16647
\(768\) −2.00000 −0.0721688
\(769\) −13.0000 −0.468792 −0.234396 0.972141i \(-0.575311\pi\)
−0.234396 + 0.972141i \(0.575311\pi\)
\(770\) −6.00000 −0.216225
\(771\) 6.00000 0.216085
\(772\) −4.00000 −0.143963
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 1.00000 0.0359443
\(775\) 5.00000 0.179605
\(776\) 8.00000 0.287183
\(777\) 4.00000 0.143499
\(778\) −6.00000 −0.215110
\(779\) 21.0000 0.752403
\(780\) 10.0000 0.358057
\(781\) −72.0000 −2.57636
\(782\) 36.0000 1.28736
\(783\) −12.0000 −0.428845
\(784\) −6.00000 −0.214286
\(785\) −14.0000 −0.499681
\(786\) 24.0000 0.856052
\(787\) −13.0000 −0.463400 −0.231700 0.972787i \(-0.574429\pi\)
−0.231700 + 0.972787i \(0.574429\pi\)
\(788\) −15.0000 −0.534353
\(789\) −18.0000 −0.640817
\(790\) 1.00000 0.0355784
\(791\) 3.00000 0.106668
\(792\) −6.00000 −0.213201
\(793\) −5.00000 −0.177555
\(794\) 2.00000 0.0709773
\(795\) 12.0000 0.425596
\(796\) 20.0000 0.708881
\(797\) 21.0000 0.743858 0.371929 0.928261i \(-0.378696\pi\)
0.371929 + 0.928261i \(0.378696\pi\)
\(798\) −14.0000 −0.495595
\(799\) −72.0000 −2.54718
\(800\) 1.00000 0.0353553
\(801\) 6.00000 0.212000
\(802\) −9.00000 −0.317801
\(803\) −66.0000 −2.32909
\(804\) 26.0000 0.916949
\(805\) −6.00000 −0.211472
\(806\) 25.0000 0.880587
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 45.0000 1.58212 0.791058 0.611741i \(-0.209531\pi\)
0.791058 + 0.611741i \(0.209531\pi\)
\(810\) 11.0000 0.386501
\(811\) 11.0000 0.386262 0.193131 0.981173i \(-0.438136\pi\)
0.193131 + 0.981173i \(0.438136\pi\)
\(812\) 3.00000 0.105279
\(813\) −22.0000 −0.771574
\(814\) −12.0000 −0.420600
\(815\) 16.0000 0.560456
\(816\) 12.0000 0.420084
\(817\) −7.00000 −0.244899
\(818\) −22.0000 −0.769212
\(819\) −5.00000 −0.174714
\(820\) 3.00000 0.104765
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 30.0000 1.04637
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 14.0000 0.487713
\(825\) 12.0000 0.417786
\(826\) 12.0000 0.417533
\(827\) 33.0000 1.14752 0.573761 0.819023i \(-0.305484\pi\)
0.573761 + 0.819023i \(0.305484\pi\)
\(828\) −6.00000 −0.208514
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) −16.0000 −0.555034
\(832\) 5.00000 0.173344
\(833\) 36.0000 1.24733
\(834\) 44.0000 1.52360
\(835\) 6.00000 0.207639
\(836\) 42.0000 1.45260
\(837\) 20.0000 0.691301
\(838\) −3.00000 −0.103633
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −20.0000 −0.689655
\(842\) 17.0000 0.585859
\(843\) −18.0000 −0.619953
\(844\) −4.00000 −0.137686
\(845\) −12.0000 −0.412813
\(846\) 12.0000 0.412568
\(847\) −25.0000 −0.859010
\(848\) 6.00000 0.206041
\(849\) −10.0000 −0.343199
\(850\) −6.00000 −0.205798
\(851\) −12.0000 −0.411355
\(852\) −24.0000 −0.822226
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 1.00000 0.0342193
\(855\) 7.00000 0.239395
\(856\) −9.00000 −0.307614
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 60.0000 2.04837
\(859\) 23.0000 0.784750 0.392375 0.919805i \(-0.371654\pi\)
0.392375 + 0.919805i \(0.371654\pi\)
\(860\) −1.00000 −0.0340997
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 4.00000 0.136083
\(865\) 15.0000 0.510015
\(866\) 11.0000 0.373795
\(867\) −38.0000 −1.29055
\(868\) −5.00000 −0.169711
\(869\) 6.00000 0.203536
\(870\) −6.00000 −0.203419
\(871\) −65.0000 −2.20244
\(872\) −16.0000 −0.541828
\(873\) 8.00000 0.270759
\(874\) 42.0000 1.42067
\(875\) 1.00000 0.0338062
\(876\) −22.0000 −0.743311
\(877\) 50.0000 1.68838 0.844190 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 8.00000 0.269987
\(879\) −12.0000 −0.404750
\(880\) 6.00000 0.202260
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) −6.00000 −0.202031
\(883\) −25.0000 −0.841317 −0.420658 0.907219i \(-0.638201\pi\)
−0.420658 + 0.907219i \(0.638201\pi\)
\(884\) −30.0000 −1.00901
\(885\) −24.0000 −0.806751
\(886\) −39.0000 −1.31023
\(887\) 51.0000 1.71241 0.856206 0.516634i \(-0.172815\pi\)
0.856206 + 0.516634i \(0.172815\pi\)
\(888\) −4.00000 −0.134231
\(889\) 16.0000 0.536623
\(890\) −6.00000 −0.201120
\(891\) 66.0000 2.21108
\(892\) 8.00000 0.267860
\(893\) −84.0000 −2.81095
\(894\) −6.00000 −0.200670
\(895\) −3.00000 −0.100279
\(896\) −1.00000 −0.0334077
\(897\) 60.0000 2.00334
\(898\) 6.00000 0.200223
\(899\) −15.0000 −0.500278
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) 18.0000 0.599334
\(903\) 2.00000 0.0665558
\(904\) −3.00000 −0.0997785
\(905\) −2.00000 −0.0664822
\(906\) 20.0000 0.664455
\(907\) 17.0000 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 5.00000 0.165748
\(911\) −60.0000 −1.98789 −0.993944 0.109885i \(-0.964952\pi\)
−0.993944 + 0.109885i \(0.964952\pi\)
\(912\) 14.0000 0.463586
\(913\) 0 0
\(914\) 26.0000 0.860004
\(915\) −2.00000 −0.0661180
\(916\) −22.0000 −0.726900
\(917\) 12.0000 0.396275
\(918\) −24.0000 −0.792118
\(919\) 11.0000 0.362857 0.181428 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 6.00000 0.197814
\(921\) −58.0000 −1.91116
\(922\) −18.0000 −0.592798
\(923\) 60.0000 1.97492
\(924\) −12.0000 −0.394771
\(925\) 2.00000 0.0657596
\(926\) 5.00000 0.164310
\(927\) 14.0000 0.459820
\(928\) −3.00000 −0.0984798
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 10.0000 0.327913
\(931\) 42.0000 1.37649
\(932\) −6.00000 −0.196537
\(933\) 30.0000 0.982156
\(934\) −30.0000 −0.981630
\(935\) −36.0000 −1.17733
\(936\) 5.00000 0.163430
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 13.0000 0.424465
\(939\) 20.0000 0.652675
\(940\) −12.0000 −0.391397
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) −28.0000 −0.912289
\(943\) 18.0000 0.586161
\(944\) −12.0000 −0.390567
\(945\) 4.00000 0.130120
\(946\) −6.00000 −0.195077
\(947\) 57.0000 1.85225 0.926126 0.377215i \(-0.123118\pi\)
0.926126 + 0.377215i \(0.123118\pi\)
\(948\) 2.00000 0.0649570
\(949\) 55.0000 1.78538
\(950\) −7.00000 −0.227110
\(951\) 30.0000 0.972817
\(952\) 6.00000 0.194461
\(953\) −33.0000 −1.06897 −0.534487 0.845176i \(-0.679495\pi\)
−0.534487 + 0.845176i \(0.679495\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 3.00000 0.0970269
\(957\) −36.0000 −1.16371
\(958\) 0 0
\(959\) 15.0000 0.484375
\(960\) 2.00000 0.0645497
\(961\) −6.00000 −0.193548
\(962\) 10.0000 0.322413
\(963\) −9.00000 −0.290021
\(964\) −28.0000 −0.901819
\(965\) 4.00000 0.128765
\(966\) −12.0000 −0.386094
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 25.0000 0.803530
\(969\) −84.0000 −2.69847
\(970\) −8.00000 −0.256865
\(971\) 30.0000 0.962746 0.481373 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) 10.0000 0.320750
\(973\) 22.0000 0.705288
\(974\) −16.0000 −0.512673
\(975\) −10.0000 −0.320256
\(976\) −1.00000 −0.0320092
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 32.0000 1.02325
\(979\) −36.0000 −1.15056
\(980\) 6.00000 0.191663
\(981\) −16.0000 −0.510841
\(982\) 36.0000 1.14881
\(983\) 9.00000 0.287055 0.143528 0.989646i \(-0.454155\pi\)
0.143528 + 0.989646i \(0.454155\pi\)
\(984\) 6.00000 0.191273
\(985\) 15.0000 0.477940
\(986\) 18.0000 0.573237
\(987\) 24.0000 0.763928
\(988\) −35.0000 −1.11350
\(989\) −6.00000 −0.190789
\(990\) 6.00000 0.190693
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 5.00000 0.158750
\(993\) −16.0000 −0.507745
\(994\) −12.0000 −0.380617
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) 44.0000 1.39349 0.696747 0.717317i \(-0.254630\pi\)
0.696747 + 0.717317i \(0.254630\pi\)
\(998\) 41.0000 1.29783
\(999\) 8.00000 0.253109
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 430.2.a.c.1.1 1
3.2 odd 2 3870.2.a.h.1.1 1
4.3 odd 2 3440.2.a.d.1.1 1
5.2 odd 4 2150.2.b.c.1549.2 2
5.3 odd 4 2150.2.b.c.1549.1 2
5.4 even 2 2150.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
430.2.a.c.1.1 1 1.1 even 1 trivial
2150.2.a.f.1.1 1 5.4 even 2
2150.2.b.c.1549.1 2 5.3 odd 4
2150.2.b.c.1549.2 2 5.2 odd 4
3440.2.a.d.1.1 1 4.3 odd 2
3870.2.a.h.1.1 1 3.2 odd 2