Properties

Label 3630.2.a.g.1.1
Level $3630$
Weight $2$
Character 3630.1
Self dual yes
Analytic conductor $28.986$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
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 \(\chi\) \(=\) 3630.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} +6.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +3.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +3.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +1.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} +1.00000 q^{42} -4.00000 q^{43} -1.00000 q^{45} -6.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} -3.00000 q^{51} -1.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} -1.00000 q^{57} -3.00000 q^{58} -6.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} +2.00000 q^{67} -3.00000 q^{68} +6.00000 q^{69} -1.00000 q^{70} -15.0000 q^{71} -1.00000 q^{72} -4.00000 q^{73} +1.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} +1.00000 q^{78} -10.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +3.00000 q^{83} -1.00000 q^{84} +3.00000 q^{85} +4.00000 q^{86} +3.00000 q^{87} +12.0000 q^{89} +1.00000 q^{90} +1.00000 q^{91} +6.00000 q^{92} -4.00000 q^{93} -6.00000 q^{94} +1.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} +6.00000 q^{98} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −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
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 1.00000 0.162221
\(39\) −1.00000 −0.160128
\(40\) 1.00000 0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 1.00000 0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) −6.00000 −0.884652
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) −1.00000 −0.141421
\(51\) −3.00000 −0.420084
\(52\) −1.00000 −0.138675
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −1.00000 −0.132453
\(58\) −3.00000 −0.393919
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 4.00000 0.508001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −3.00000 −0.363803
\(69\) 6.00000 0.722315
\(70\) −1.00000 −0.119523
\(71\) −15.0000 −1.78017 −0.890086 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 1.00000 0.116248
\(75\) 1.00000 0.115470
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) −1.00000 −0.109109
\(85\) 3.00000 0.325396
\(86\) 4.00000 0.431331
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 1.00000 0.105409
\(91\) 1.00000 0.104828
\(92\) 6.00000 0.625543
\(93\) −4.00000 −0.414781
\(94\) −6.00000 −0.618853
\(95\) 1.00000 0.102598
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 6.00000 0.606092
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 3.00000 0.297044
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 1.00000 0.0980581
\(105\) 1.00000 0.0975900
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) −1.00000 −0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 1.00000 0.0936586
\(115\) −6.00000 −0.559503
\(116\) 3.00000 0.278543
\(117\) −1.00000 −0.0924500
\(118\) 6.00000 0.552345
\(119\) 3.00000 0.275010
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −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
\(129\) −4.00000 −0.352180
\(130\) −1.00000 −0.0877058
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) −2.00000 −0.172774
\(135\) −1.00000 −0.0860663
\(136\) 3.00000 0.257248
\(137\) −21.0000 −1.79415 −0.897076 0.441877i \(-0.854313\pi\)
−0.897076 + 0.441877i \(0.854313\pi\)
\(138\) −6.00000 −0.510754
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 1.00000 0.0845154
\(141\) 6.00000 0.505291
\(142\) 15.0000 1.25877
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −3.00000 −0.249136
\(146\) 4.00000 0.331042
\(147\) −6.00000 −0.494872
\(148\) −1.00000 −0.0821995
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 1.00000 0.0811107
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) −1.00000 −0.0800641
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 10.0000 0.795557
\(159\) 6.00000 0.475831
\(160\) 1.00000 0.0790569
\(161\) −6.00000 −0.472866
\(162\) −1.00000 −0.0785674
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 1.00000 0.0771517
\(169\) −12.0000 −0.923077
\(170\) −3.00000 −0.230089
\(171\) −1.00000 −0.0764719
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −3.00000 −0.227429
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) −12.0000 −0.899438
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −1.00000 −0.0741249
\(183\) −10.0000 −0.739221
\(184\) −6.00000 −0.442326
\(185\) 1.00000 0.0735215
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) −1.00000 −0.0727393
\(190\) −1.00000 −0.0725476
\(191\) −9.00000 −0.651217 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) 1.00000 0.0721688
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −8.00000 −0.574367
\(195\) 1.00000 0.0716115
\(196\) −6.00000 −0.428571
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 2.00000 0.141069
\(202\) 3.00000 0.211079
\(203\) −3.00000 −0.210559
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) 1.00000 0.0696733
\(207\) 6.00000 0.417029
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 6.00000 0.412082
\(213\) −15.0000 −1.02778
\(214\) 12.0000 0.820303
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) 4.00000 0.271538
\(218\) 16.0000 1.08366
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 1.00000 0.0671156
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 1.00000 0.0653720
\(235\) −6.00000 −0.391397
\(236\) −6.00000 −0.390567
\(237\) −10.0000 −0.649570
\(238\) −3.00000 −0.194461
\(239\) −21.0000 −1.35838 −0.679189 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −19.0000 −1.22390 −0.611949 0.790897i \(-0.709614\pi\)
−0.611949 + 0.790897i \(0.709614\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) 4.00000 0.254000
\(249\) 3.00000 0.190117
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 3.00000 0.187867
\(256\) 1.00000 0.0625000
\(257\) −3.00000 −0.187135 −0.0935674 0.995613i \(-0.529827\pi\)
−0.0935674 + 0.995613i \(0.529827\pi\)
\(258\) 4.00000 0.249029
\(259\) 1.00000 0.0621370
\(260\) 1.00000 0.0620174
\(261\) 3.00000 0.185695
\(262\) 6.00000 0.370681
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −1.00000 −0.0613139
\(267\) 12.0000 0.734388
\(268\) 2.00000 0.122169
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 1.00000 0.0608581
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −3.00000 −0.181902
\(273\) 1.00000 0.0605228
\(274\) 21.0000 1.26866
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −5.00000 −0.299880
\(279\) −4.00000 −0.239474
\(280\) −1.00000 −0.0597614
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) −6.00000 −0.357295
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −15.0000 −0.890086
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) 3.00000 0.176166
\(291\) 8.00000 0.468968
\(292\) −4.00000 −0.234082
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 6.00000 0.349927
\(295\) 6.00000 0.349334
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) −6.00000 −0.346989
\(300\) 1.00000 0.0577350
\(301\) 4.00000 0.230556
\(302\) −2.00000 −0.115087
\(303\) −3.00000 −0.172345
\(304\) −1.00000 −0.0573539
\(305\) 10.0000 0.572598
\(306\) 3.00000 0.171499
\(307\) 32.0000 1.82634 0.913168 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) −4.00000 −0.227185
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 1.00000 0.0566139
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 7.00000 0.395033
\(315\) 1.00000 0.0563436
\(316\) −10.0000 −0.562544
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −12.0000 −0.669775
\(322\) 6.00000 0.334367
\(323\) 3.00000 0.166924
\(324\) 1.00000 0.0555556
\(325\) −1.00000 −0.0554700
\(326\) −14.0000 −0.775388
\(327\) −16.0000 −0.884802
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 3.00000 0.164646
\(333\) −1.00000 −0.0547997
\(334\) −12.0000 −0.656611
\(335\) −2.00000 −0.109272
\(336\) −1.00000 −0.0545545
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 12.0000 0.652714
\(339\) −6.00000 −0.325875
\(340\) 3.00000 0.162698
\(341\) 0 0
\(342\) 1.00000 0.0540738
\(343\) 13.0000 0.701934
\(344\) 4.00000 0.215666
\(345\) −6.00000 −0.323029
\(346\) 6.00000 0.322562
\(347\) −3.00000 −0.161048 −0.0805242 0.996753i \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) 3.00000 0.160817
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 1.00000 0.0534522
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 6.00000 0.318896
\(355\) 15.0000 0.796117
\(356\) 12.0000 0.635999
\(357\) 3.00000 0.158777
\(358\) 18.0000 0.951330
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 1.00000 0.0527046
\(361\) −18.0000 −0.947368
\(362\) 22.0000 1.15629
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 4.00000 0.209370
\(366\) 10.0000 0.522708
\(367\) −13.0000 −0.678594 −0.339297 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) −6.00000 −0.311504
\(372\) −4.00000 −0.207390
\(373\) −31.0000 −1.60512 −0.802560 0.596572i \(-0.796529\pi\)
−0.802560 + 0.596572i \(0.796529\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −6.00000 −0.309426
\(377\) −3.00000 −0.154508
\(378\) 1.00000 0.0514344
\(379\) 11.0000 0.565032 0.282516 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) 1.00000 0.0512989
\(381\) −16.0000 −0.819705
\(382\) 9.00000 0.460480
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) −4.00000 −0.203331
\(388\) 8.00000 0.406138
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) −1.00000 −0.0506370
\(391\) −18.0000 −0.910299
\(392\) 6.00000 0.303046
\(393\) −6.00000 −0.302660
\(394\) −12.0000 −0.604551
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) −25.0000 −1.25471 −0.627357 0.778732i \(-0.715863\pi\)
−0.627357 + 0.778732i \(0.715863\pi\)
\(398\) −20.0000 −1.00251
\(399\) 1.00000 0.0500626
\(400\) 1.00000 0.0500000
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 4.00000 0.199254
\(404\) −3.00000 −0.149256
\(405\) −1.00000 −0.0496904
\(406\) 3.00000 0.148888
\(407\) 0 0
\(408\) 3.00000 0.148522
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) −21.0000 −1.03585
\(412\) −1.00000 −0.0492665
\(413\) 6.00000 0.295241
\(414\) −6.00000 −0.294884
\(415\) −3.00000 −0.147264
\(416\) 1.00000 0.0490290
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 1.00000 0.0487950
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 13.0000 0.632830
\(423\) 6.00000 0.291730
\(424\) −6.00000 −0.291386
\(425\) −3.00000 −0.145521
\(426\) 15.0000 0.726752
\(427\) 10.0000 0.483934
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 1.00000 0.0481125
\(433\) −28.0000 −1.34559 −0.672797 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) −4.00000 −0.192006
\(435\) −3.00000 −0.143839
\(436\) −16.0000 −0.766261
\(437\) −6.00000 −0.287019
\(438\) 4.00000 0.191127
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −3.00000 −0.142695
\(443\) −27.0000 −1.28281 −0.641404 0.767203i \(-0.721648\pi\)
−0.641404 + 0.767203i \(0.721648\pi\)
\(444\) −1.00000 −0.0474579
\(445\) −12.0000 −0.568855
\(446\) 19.0000 0.899676
\(447\) 18.0000 0.851371
\(448\) −1.00000 −0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 2.00000 0.0939682
\(454\) −12.0000 −0.563188
\(455\) −1.00000 −0.0468807
\(456\) 1.00000 0.0468293
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −8.00000 −0.373815
\(459\) −3.00000 −0.140028
\(460\) −6.00000 −0.279751
\(461\) −9.00000 −0.419172 −0.209586 0.977790i \(-0.567212\pi\)
−0.209586 + 0.977790i \(0.567212\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 3.00000 0.139272
\(465\) 4.00000 0.185496
\(466\) −6.00000 −0.277945
\(467\) −9.00000 −0.416470 −0.208235 0.978079i \(-0.566772\pi\)
−0.208235 + 0.978079i \(0.566772\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −2.00000 −0.0923514
\(470\) 6.00000 0.276759
\(471\) −7.00000 −0.322543
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) 10.0000 0.459315
\(475\) −1.00000 −0.0458831
\(476\) 3.00000 0.137505
\(477\) 6.00000 0.274721
\(478\) 21.0000 0.960518
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 1.00000 0.0456435
\(481\) 1.00000 0.0455961
\(482\) 19.0000 0.865426
\(483\) −6.00000 −0.273009
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) −1.00000 −0.0453609
\(487\) 23.0000 1.04223 0.521115 0.853487i \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) 10.0000 0.452679
\(489\) 14.0000 0.633102
\(490\) −6.00000 −0.271052
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) −1.00000 −0.0449921
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 15.0000 0.672842
\(498\) −3.00000 −0.134433
\(499\) −13.0000 −0.581960 −0.290980 0.956729i \(-0.593981\pi\)
−0.290980 + 0.956729i \(0.593981\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 1.00000 0.0445435
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) −16.0000 −0.709885
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) −3.00000 −0.132842
\(511\) 4.00000 0.176950
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 3.00000 0.132324
\(515\) 1.00000 0.0440653
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −1.00000 −0.0439375
\(519\) −6.00000 −0.263371
\(520\) −1.00000 −0.0438529
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −3.00000 −0.131306
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) −6.00000 −0.262111
\(525\) −1.00000 −0.0436436
\(526\) −24.0000 −1.04645
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 6.00000 0.260623
\(531\) −6.00000 −0.260378
\(532\) 1.00000 0.0433555
\(533\) 0 0
\(534\) −12.0000 −0.519291
\(535\) 12.0000 0.518805
\(536\) −2.00000 −0.0863868
\(537\) −18.0000 −0.776757
\(538\) −9.00000 −0.388018
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) −2.00000 −0.0859074
\(543\) −22.0000 −0.944110
\(544\) 3.00000 0.128624
\(545\) 16.0000 0.685365
\(546\) −1.00000 −0.0427960
\(547\) 26.0000 1.11168 0.555840 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(548\) −21.0000 −0.897076
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −3.00000 −0.127804
\(552\) −6.00000 −0.255377
\(553\) 10.0000 0.425243
\(554\) 10.0000 0.424859
\(555\) 1.00000 0.0424476
\(556\) 5.00000 0.212047
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 4.00000 0.169334
\(559\) 4.00000 0.169182
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −30.0000 −1.26547
\(563\) 9.00000 0.379305 0.189652 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) 6.00000 0.252646
\(565\) 6.00000 0.252422
\(566\) 4.00000 0.168133
\(567\) −1.00000 −0.0419961
\(568\) 15.0000 0.629386
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) −1.00000 −0.0418854
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) −9.00000 −0.375980
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 8.00000 0.332756
\(579\) −4.00000 −0.166234
\(580\) −3.00000 −0.124568
\(581\) −3.00000 −0.124461
\(582\) −8.00000 −0.331611
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) 1.00000 0.0413449
\(586\) 24.0000 0.991431
\(587\) 33.0000 1.36206 0.681028 0.732257i \(-0.261533\pi\)
0.681028 + 0.732257i \(0.261533\pi\)
\(588\) −6.00000 −0.247436
\(589\) 4.00000 0.164817
\(590\) −6.00000 −0.247016
\(591\) 12.0000 0.493614
\(592\) −1.00000 −0.0410997
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 18.0000 0.737309
\(597\) 20.0000 0.818546
\(598\) 6.00000 0.245358
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −4.00000 −0.163028
\(603\) 2.00000 0.0814463
\(604\) 2.00000 0.0813788
\(605\) 0 0
\(606\) 3.00000 0.121867
\(607\) 41.0000 1.66414 0.832069 0.554672i \(-0.187156\pi\)
0.832069 + 0.554672i \(0.187156\pi\)
\(608\) 1.00000 0.0405554
\(609\) −3.00000 −0.121566
\(610\) −10.0000 −0.404888
\(611\) −6.00000 −0.242734
\(612\) −3.00000 −0.121268
\(613\) 23.0000 0.928961 0.464481 0.885583i \(-0.346241\pi\)
0.464481 + 0.885583i \(0.346241\pi\)
\(614\) −32.0000 −1.29141
\(615\) 0 0
\(616\) 0 0
\(617\) −45.0000 −1.81163 −0.905816 0.423672i \(-0.860741\pi\)
−0.905816 + 0.423672i \(0.860741\pi\)
\(618\) 1.00000 0.0402259
\(619\) −25.0000 −1.00483 −0.502417 0.864625i \(-0.667556\pi\)
−0.502417 + 0.864625i \(0.667556\pi\)
\(620\) 4.00000 0.160644
\(621\) 6.00000 0.240772
\(622\) 24.0000 0.962312
\(623\) −12.0000 −0.480770
\(624\) −1.00000 −0.0400320
\(625\) 1.00000 0.0400000
\(626\) −8.00000 −0.319744
\(627\) 0 0
\(628\) −7.00000 −0.279330
\(629\) 3.00000 0.119618
\(630\) −1.00000 −0.0398410
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) 10.0000 0.397779
\(633\) −13.0000 −0.516704
\(634\) −18.0000 −0.714871
\(635\) 16.0000 0.634941
\(636\) 6.00000 0.237915
\(637\) 6.00000 0.237729
\(638\) 0 0
\(639\) −15.0000 −0.593391
\(640\) 1.00000 0.0395285
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 12.0000 0.473602
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) −6.00000 −0.236433
\(645\) 4.00000 0.157500
\(646\) −3.00000 −0.118033
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 1.00000 0.0392232
\(651\) 4.00000 0.156772
\(652\) 14.0000 0.548282
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 16.0000 0.625650
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 6.00000 0.233904
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) 8.00000 0.311164 0.155582 0.987823i \(-0.450275\pi\)
0.155582 + 0.987823i \(0.450275\pi\)
\(662\) 19.0000 0.738456
\(663\) 3.00000 0.116510
\(664\) −3.00000 −0.116423
\(665\) −1.00000 −0.0387783
\(666\) 1.00000 0.0387492
\(667\) 18.0000 0.696963
\(668\) 12.0000 0.464294
\(669\) −19.0000 −0.734582
\(670\) 2.00000 0.0772667
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) 44.0000 1.69608 0.848038 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) −32.0000 −1.23259
\(675\) 1.00000 0.0384900
\(676\) −12.0000 −0.461538
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 6.00000 0.230429
\(679\) −8.00000 −0.307012
\(680\) −3.00000 −0.115045
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 15.0000 0.573959 0.286980 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 21.0000 0.802369
\(686\) −13.0000 −0.496342
\(687\) 8.00000 0.305219
\(688\) −4.00000 −0.152499
\(689\) −6.00000 −0.228582
\(690\) 6.00000 0.228416
\(691\) −37.0000 −1.40755 −0.703773 0.710425i \(-0.748503\pi\)
−0.703773 + 0.710425i \(0.748503\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 3.00000 0.113878
\(695\) −5.00000 −0.189661
\(696\) −3.00000 −0.113715
\(697\) 0 0
\(698\) −14.0000 −0.529908
\(699\) 6.00000 0.226941
\(700\) −1.00000 −0.0377964
\(701\) −9.00000 −0.339925 −0.169963 0.985451i \(-0.554365\pi\)
−0.169963 + 0.985451i \(0.554365\pi\)
\(702\) 1.00000 0.0377426
\(703\) 1.00000 0.0377157
\(704\) 0 0
\(705\) −6.00000 −0.225973
\(706\) −6.00000 −0.225813
\(707\) 3.00000 0.112827
\(708\) −6.00000 −0.225494
\(709\) 32.0000 1.20179 0.600893 0.799330i \(-0.294812\pi\)
0.600893 + 0.799330i \(0.294812\pi\)
\(710\) −15.0000 −0.562940
\(711\) −10.0000 −0.375029
\(712\) −12.0000 −0.449719
\(713\) −24.0000 −0.898807
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) −18.0000 −0.672692
\(717\) −21.0000 −0.784259
\(718\) −24.0000 −0.895672
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 1.00000 0.0372419
\(722\) 18.0000 0.669891
\(723\) −19.0000 −0.706618
\(724\) −22.0000 −0.817624
\(725\) 3.00000 0.111417
\(726\) 0 0
\(727\) 17.0000 0.630495 0.315248 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) 12.0000 0.443836
\(732\) −10.0000 −0.369611
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 13.0000 0.479839
\(735\) 6.00000 0.221313
\(736\) −6.00000 −0.221163
\(737\) 0 0
\(738\) 0 0
\(739\) 5.00000 0.183928 0.0919640 0.995762i \(-0.470686\pi\)
0.0919640 + 0.995762i \(0.470686\pi\)
\(740\) 1.00000 0.0367607
\(741\) 1.00000 0.0367359
\(742\) 6.00000 0.220267
\(743\) 18.0000 0.660356 0.330178 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 4.00000 0.146647
\(745\) −18.0000 −0.659469
\(746\) 31.0000 1.13499
\(747\) 3.00000 0.109764
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 1.00000 0.0365148
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 3.00000 0.109254
\(755\) −2.00000 −0.0727875
\(756\) −1.00000 −0.0363696
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −11.0000 −0.399538
\(759\) 0 0
\(760\) −1.00000 −0.0362738
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) 16.0000 0.579619
\(763\) 16.0000 0.579239
\(764\) −9.00000 −0.325609
\(765\) 3.00000 0.108465
\(766\) 18.0000 0.650366
\(767\) 6.00000 0.216647
\(768\) 1.00000 0.0360844
\(769\) −49.0000 −1.76699 −0.883493 0.468445i \(-0.844814\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 0 0
\(771\) −3.00000 −0.108042
\(772\) −4.00000 −0.143963
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 4.00000 0.143777
\(775\) −4.00000 −0.143684
\(776\) −8.00000 −0.287183
\(777\) 1.00000 0.0358748
\(778\) −18.0000 −0.645331
\(779\) 0 0
\(780\) 1.00000 0.0358057
\(781\) 0 0
\(782\) 18.0000 0.643679
\(783\) 3.00000 0.107211
\(784\) −6.00000 −0.214286
\(785\) 7.00000 0.249841
\(786\) 6.00000 0.214013
\(787\) −46.0000 −1.63972 −0.819861 0.572562i \(-0.805950\pi\)
−0.819861 + 0.572562i \(0.805950\pi\)
\(788\) 12.0000 0.427482
\(789\) 24.0000 0.854423
\(790\) −10.0000 −0.355784
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 25.0000 0.887217
\(795\) −6.00000 −0.212798
\(796\) 20.0000 0.708881
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) −1.00000 −0.0353996
\(799\) −18.0000 −0.636794
\(800\) −1.00000 −0.0353553
\(801\) 12.0000 0.423999
\(802\) 24.0000 0.847469
\(803\) 0 0
\(804\) 2.00000 0.0705346
\(805\) 6.00000 0.211472
\(806\) −4.00000 −0.140894
\(807\) 9.00000 0.316815
\(808\) 3.00000 0.105540
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 1.00000 0.0351364
\(811\) 11.0000 0.386262 0.193131 0.981173i \(-0.438136\pi\)
0.193131 + 0.981173i \(0.438136\pi\)
\(812\) −3.00000 −0.105279
\(813\) 2.00000 0.0701431
\(814\) 0 0
\(815\) −14.0000 −0.490399
\(816\) −3.00000 −0.105021
\(817\) 4.00000 0.139942
\(818\) 22.0000 0.769212
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 21.0000 0.732459
\(823\) −31.0000 −1.08059 −0.540296 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) 1.00000 0.0348367
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 21.0000 0.730242 0.365121 0.930960i \(-0.381028\pi\)
0.365121 + 0.930960i \(0.381028\pi\)
\(828\) 6.00000 0.208514
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 3.00000 0.104132
\(831\) −10.0000 −0.346896
\(832\) −1.00000 −0.0346688
\(833\) 18.0000 0.623663
\(834\) −5.00000 −0.173136
\(835\) −12.0000 −0.415277
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) 6.00000 0.207267
\(839\) −27.0000 −0.932144 −0.466072 0.884747i \(-0.654331\pi\)
−0.466072 + 0.884747i \(0.654331\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −20.0000 −0.689655
\(842\) −8.00000 −0.275698
\(843\) 30.0000 1.03325
\(844\) −13.0000 −0.447478
\(845\) 12.0000 0.412813
\(846\) −6.00000 −0.206284
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −4.00000 −0.137280
\(850\) 3.00000 0.102899
\(851\) −6.00000 −0.205677
\(852\) −15.0000 −0.513892
\(853\) 23.0000 0.787505 0.393753 0.919216i \(-0.371177\pi\)
0.393753 + 0.919216i \(0.371177\pi\)
\(854\) −10.0000 −0.342193
\(855\) 1.00000 0.0341993
\(856\) 12.0000 0.410152
\(857\) 21.0000 0.717346 0.358673 0.933463i \(-0.383229\pi\)
0.358673 + 0.933463i \(0.383229\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) −21.0000 −0.715263
\(863\) 30.0000 1.02121 0.510606 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) 28.0000 0.951479
\(867\) −8.00000 −0.271694
\(868\) 4.00000 0.135769
\(869\) 0 0
\(870\) 3.00000 0.101710
\(871\) −2.00000 −0.0677674
\(872\) 16.0000 0.541828
\(873\) 8.00000 0.270759
\(874\) 6.00000 0.202953
\(875\) 1.00000 0.0338062
\(876\) −4.00000 −0.135147
\(877\) 53.0000 1.78968 0.894841 0.446384i \(-0.147289\pi\)
0.894841 + 0.446384i \(0.147289\pi\)
\(878\) 10.0000 0.337484
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) −12.0000 −0.404290 −0.202145 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) 6.00000 0.202031
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 3.00000 0.100901
\(885\) 6.00000 0.201688
\(886\) 27.0000 0.907083
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 1.00000 0.0335578
\(889\) 16.0000 0.536623
\(890\) 12.0000 0.402241
\(891\) 0 0
\(892\) −19.0000 −0.636167
\(893\) −6.00000 −0.200782
\(894\) −18.0000 −0.602010
\(895\) 18.0000 0.601674
\(896\) 1.00000 0.0334077
\(897\) −6.00000 −0.200334
\(898\) 6.00000 0.200223
\(899\) −12.0000 −0.400222
\(900\) 1.00000 0.0333333
\(901\) −18.0000 −0.599667
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 6.00000 0.199557
\(905\) 22.0000 0.731305
\(906\) −2.00000 −0.0664455
\(907\) 38.0000 1.26177 0.630885 0.775877i \(-0.282692\pi\)
0.630885 + 0.775877i \(0.282692\pi\)
\(908\) 12.0000 0.398234
\(909\) −3.00000 −0.0995037
\(910\) 1.00000 0.0331497
\(911\) 9.00000 0.298183 0.149092 0.988823i \(-0.452365\pi\)
0.149092 + 0.988823i \(0.452365\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 10.0000 0.330590
\(916\) 8.00000 0.264327
\(917\) 6.00000 0.198137
\(918\) 3.00000 0.0990148
\(919\) 2.00000 0.0659739 0.0329870 0.999456i \(-0.489498\pi\)
0.0329870 + 0.999456i \(0.489498\pi\)
\(920\) 6.00000 0.197814
\(921\) 32.0000 1.05444
\(922\) 9.00000 0.296399
\(923\) 15.0000 0.493731
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 4.00000 0.131448
\(927\) −1.00000 −0.0328443
\(928\) −3.00000 −0.0984798
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) −4.00000 −0.131165
\(931\) 6.00000 0.196642
\(932\) 6.00000 0.196537
\(933\) −24.0000 −0.785725
\(934\) 9.00000 0.294489
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 2.00000 0.0653023
\(939\) 8.00000 0.261070
\(940\) −6.00000 −0.195698
\(941\) 27.0000 0.880175 0.440087 0.897955i \(-0.354947\pi\)
0.440087 + 0.897955i \(0.354947\pi\)
\(942\) 7.00000 0.228072
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −15.0000 −0.487435 −0.243717 0.969846i \(-0.578367\pi\)
−0.243717 + 0.969846i \(0.578367\pi\)
\(948\) −10.0000 −0.324785
\(949\) 4.00000 0.129845
\(950\) 1.00000 0.0324443
\(951\) 18.0000 0.583690
\(952\) −3.00000 −0.0972306
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −6.00000 −0.194257
\(955\) 9.00000 0.291233
\(956\) −21.0000 −0.679189
\(957\) 0 0
\(958\) −15.0000 −0.484628
\(959\) 21.0000 0.678125
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) −1.00000 −0.0322413
\(963\) −12.0000 −0.386695
\(964\) −19.0000 −0.611949
\(965\) 4.00000 0.128765
\(966\) 6.00000 0.193047
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) 3.00000 0.0963739
\(970\) 8.00000 0.256865
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 1.00000 0.0320750
\(973\) −5.00000 −0.160293
\(974\) −23.0000 −0.736968
\(975\) −1.00000 −0.0320256
\(976\) −10.0000 −0.320092
\(977\) −54.0000 −1.72761 −0.863807 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) −14.0000 −0.447671
\(979\) 0 0
\(980\) 6.00000 0.191663
\(981\) −16.0000 −0.510841
\(982\) −12.0000 −0.382935
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) 9.00000 0.286618
\(987\) −6.00000 −0.190982
\(988\) 1.00000 0.0318142
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 14.0000 0.444725 0.222362 0.974964i \(-0.428623\pi\)
0.222362 + 0.974964i \(0.428623\pi\)
\(992\) 4.00000 0.127000
\(993\) −19.0000 −0.602947
\(994\) −15.0000 −0.475771
\(995\) −20.0000 −0.634043
\(996\) 3.00000 0.0950586
\(997\) −31.0000 −0.981780 −0.490890 0.871222i \(-0.663328\pi\)
−0.490890 + 0.871222i \(0.663328\pi\)
\(998\) 13.0000 0.411508
\(999\) −1.00000 −0.0316386
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 3630.2.a.g.1.1 1
11.10 odd 2 3630.2.a.v.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3630.2.a.g.1.1 1 1.1 even 1 trivial
3630.2.a.v.1.1 yes 1 11.10 odd 2