Properties

Label 966.2.a.c.1.1
Level $966$
Weight $2$
Character 966.1
Self dual yes
Analytic conductor $7.714$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.71354883526\)
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 \(\chi\) \(=\) 966.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +2.00000 q^{30} +6.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +4.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +2.00000 q^{38} +4.00000 q^{39} -2.00000 q^{40} -10.0000 q^{41} +1.00000 q^{42} +8.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} -1.00000 q^{46} +10.0000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} -4.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -8.00000 q^{55} -1.00000 q^{56} +2.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} -2.00000 q^{60} -10.0000 q^{61} -6.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -8.00000 q^{65} -4.00000 q^{66} +8.00000 q^{67} -4.00000 q^{68} -1.00000 q^{69} -2.00000 q^{70} -4.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -2.00000 q^{76} -4.00000 q^{77} -4.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -6.00000 q^{83} -1.00000 q^{84} -8.00000 q^{85} -8.00000 q^{86} +6.00000 q^{87} +4.00000 q^{88} -2.00000 q^{90} -4.00000 q^{91} +1.00000 q^{92} -6.00000 q^{93} -10.0000 q^{94} -4.00000 q^{95} +1.00000 q^{96} -8.00000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.00000 −0.218218
\(22\) 4.00000 0.852803
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) 4.00000 0.685994
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 2.00000 0.324443
\(39\) 4.00000 0.640513
\(40\) −2.00000 −0.316228
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 1.00000 0.154303
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −4.00000 −0.603023
\(45\) 2.00000 0.298142
\(46\) −1.00000 −0.147442
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) −4.00000 −0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) −8.00000 −1.07872
\(56\) −1.00000 −0.133631
\(57\) 2.00000 0.264906
\(58\) 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
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −6.00000 −0.762001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −8.00000 −0.992278
\(66\) −4.00000 −0.492366
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −4.00000 −0.485071
\(69\) −1.00000 −0.120386
\(70\) −2.00000 −0.239046
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 2.00000 0.232495
\(75\) 1.00000 0.115470
\(76\) −2.00000 −0.229416
\(77\) −4.00000 −0.455842
\(78\) −4.00000 −0.452911
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −1.00000 −0.109109
\(85\) −8.00000 −0.867722
\(86\) −8.00000 −0.862662
\(87\) 6.00000 0.643268
\(88\) 4.00000 0.426401
\(89\) 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
\(93\) −6.00000 −0.622171
\(94\) −10.0000 −1.03142
\(95\) −4.00000 −0.410391
\(96\) 1.00000 0.102062
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −1.00000 −0.101015
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) 16.0000 1.59206 0.796030 0.605257i \(-0.206930\pi\)
0.796030 + 0.605257i \(0.206930\pi\)
\(102\) −4.00000 −0.396059
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 4.00000 0.392232
\(105\) −2.00000 −0.195180
\(106\) 6.00000 0.582772
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 8.00000 0.762770
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −2.00000 −0.187317
\(115\) 2.00000 0.186501
\(116\) −6.00000 −0.557086
\(117\) −4.00000 −0.369800
\(118\) 12.0000 1.10469
\(119\) −4.00000 −0.366679
\(120\) 2.00000 0.182574
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 10.0000 0.901670
\(124\) 6.00000 0.538816
\(125\) −12.0000 −1.07331
\(126\) −1.00000 −0.0890871
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) 8.00000 0.701646
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 4.00000 0.348155
\(133\) −2.00000 −0.173422
\(134\) −8.00000 −0.691095
\(135\) −2.00000 −0.172133
\(136\) 4.00000 0.342997
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 1.00000 0.0851257
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 2.00000 0.169031
\(141\) −10.0000 −0.842152
\(142\) 4.00000 0.335673
\(143\) 16.0000 1.33799
\(144\) 1.00000 0.0833333
\(145\) −12.0000 −0.996546
\(146\) 2.00000 0.165521
\(147\) −1.00000 −0.0824786
\(148\) −2.00000 −0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 2.00000 0.162221
\(153\) −4.00000 −0.323381
\(154\) 4.00000 0.322329
\(155\) 12.0000 0.963863
\(156\) 4.00000 0.320256
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 8.00000 0.636446
\(159\) 6.00000 0.475831
\(160\) −2.00000 −0.158114
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −10.0000 −0.780869
\(165\) 8.00000 0.622799
\(166\) 6.00000 0.465690
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 1.00000 0.0771517
\(169\) 3.00000 0.230769
\(170\) 8.00000 0.613572
\(171\) −2.00000 −0.152944
\(172\) 8.00000 0.609994
\(173\) 16.0000 1.21646 0.608229 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(174\) −6.00000 −0.454859
\(175\) −1.00000 −0.0755929
\(176\) −4.00000 −0.301511
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 2.00000 0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 4.00000 0.296500
\(183\) 10.0000 0.739221
\(184\) −1.00000 −0.0737210
\(185\) −4.00000 −0.294086
\(186\) 6.00000 0.439941
\(187\) 16.0000 1.17004
\(188\) 10.0000 0.729325
\(189\) −1.00000 −0.0727393
\(190\) 4.00000 0.290191
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 8.00000 0.574367
\(195\) 8.00000 0.572892
\(196\) 1.00000 0.0714286
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 4.00000 0.284268
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 1.00000 0.0707107
\(201\) −8.00000 −0.564276
\(202\) −16.0000 −1.12576
\(203\) −6.00000 −0.421117
\(204\) 4.00000 0.280056
\(205\) −20.0000 −1.39686
\(206\) −12.0000 −0.836080
\(207\) 1.00000 0.0695048
\(208\) −4.00000 −0.277350
\(209\) 8.00000 0.553372
\(210\) 2.00000 0.138013
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −6.00000 −0.412082
\(213\) 4.00000 0.274075
\(214\) 20.0000 1.36717
\(215\) 16.0000 1.09119
\(216\) 1.00000 0.0680414
\(217\) 6.00000 0.407307
\(218\) −2.00000 −0.135457
\(219\) 2.00000 0.135147
\(220\) −8.00000 −0.539360
\(221\) 16.0000 1.07628
\(222\) −2.00000 −0.134231
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 6.00000 0.399114
\(227\) −10.0000 −0.663723 −0.331862 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) 2.00000 0.132453
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −2.00000 −0.131876
\(231\) 4.00000 0.263181
\(232\) 6.00000 0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 4.00000 0.261488
\(235\) 20.0000 1.30466
\(236\) −12.0000 −0.781133
\(237\) 8.00000 0.519656
\(238\) 4.00000 0.259281
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) −2.00000 −0.129099
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 2.00000 0.127775
\(246\) −10.0000 −0.637577
\(247\) 8.00000 0.509028
\(248\) −6.00000 −0.381000
\(249\) 6.00000 0.380235
\(250\) 12.0000 0.758947
\(251\) 26.0000 1.64111 0.820553 0.571571i \(-0.193666\pi\)
0.820553 + 0.571571i \(0.193666\pi\)
\(252\) 1.00000 0.0629941
\(253\) −4.00000 −0.251478
\(254\) −20.0000 −1.25491
\(255\) 8.00000 0.500979
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 8.00000 0.498058
\(259\) −2.00000 −0.124274
\(260\) −8.00000 −0.496139
\(261\) −6.00000 −0.371391
\(262\) 12.0000 0.741362
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −4.00000 −0.246183
\(265\) −12.0000 −0.737154
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 2.00000 0.121716
\(271\) −26.0000 −1.57939 −0.789694 0.613501i \(-0.789761\pi\)
−0.789694 + 0.613501i \(0.789761\pi\)
\(272\) −4.00000 −0.242536
\(273\) 4.00000 0.242091
\(274\) −6.00000 −0.362473
\(275\) 4.00000 0.241209
\(276\) −1.00000 −0.0601929
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 16.0000 0.959616
\(279\) 6.00000 0.359211
\(280\) −2.00000 −0.119523
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) 10.0000 0.595491
\(283\) −18.0000 −1.06999 −0.534994 0.844856i \(-0.679686\pi\)
−0.534994 + 0.844856i \(0.679686\pi\)
\(284\) −4.00000 −0.237356
\(285\) 4.00000 0.236940
\(286\) −16.0000 −0.946100
\(287\) −10.0000 −0.590281
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 12.0000 0.704664
\(291\) 8.00000 0.468968
\(292\) −2.00000 −0.117041
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 1.00000 0.0583212
\(295\) −24.0000 −1.39733
\(296\) 2.00000 0.116248
\(297\) 4.00000 0.232104
\(298\) −6.00000 −0.347571
\(299\) −4.00000 −0.231326
\(300\) 1.00000 0.0577350
\(301\) 8.00000 0.461112
\(302\) −4.00000 −0.230174
\(303\) −16.0000 −0.919176
\(304\) −2.00000 −0.114708
\(305\) −20.0000 −1.14520
\(306\) 4.00000 0.228665
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −4.00000 −0.227921
\(309\) −12.0000 −0.682656
\(310\) −12.0000 −0.681554
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) −4.00000 −0.226455
\(313\) −20.0000 −1.13047 −0.565233 0.824931i \(-0.691214\pi\)
−0.565233 + 0.824931i \(0.691214\pi\)
\(314\) −10.0000 −0.564333
\(315\) 2.00000 0.112687
\(316\) −8.00000 −0.450035
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −6.00000 −0.336463
\(319\) 24.0000 1.34374
\(320\) 2.00000 0.111803
\(321\) 20.0000 1.11629
\(322\) −1.00000 −0.0557278
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) −12.0000 −0.664619
\(327\) −2.00000 −0.110600
\(328\) 10.0000 0.552158
\(329\) 10.0000 0.551318
\(330\) −8.00000 −0.440386
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −6.00000 −0.329293
\(333\) −2.00000 −0.109599
\(334\) 18.0000 0.984916
\(335\) 16.0000 0.874173
\(336\) −1.00000 −0.0545545
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) −3.00000 −0.163178
\(339\) 6.00000 0.325875
\(340\) −8.00000 −0.433861
\(341\) −24.0000 −1.29967
\(342\) 2.00000 0.108148
\(343\) 1.00000 0.0539949
\(344\) −8.00000 −0.431331
\(345\) −2.00000 −0.107676
\(346\) −16.0000 −0.860165
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 6.00000 0.321634
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 1.00000 0.0534522
\(351\) 4.00000 0.213504
\(352\) 4.00000 0.213201
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) −12.0000 −0.637793
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) −4.00000 −0.211407
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) −2.00000 −0.105409
\(361\) −15.0000 −0.789474
\(362\) 2.00000 0.105118
\(363\) −5.00000 −0.262432
\(364\) −4.00000 −0.209657
\(365\) −4.00000 −0.209370
\(366\) −10.0000 −0.522708
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 1.00000 0.0521286
\(369\) −10.0000 −0.520579
\(370\) 4.00000 0.207950
\(371\) −6.00000 −0.311504
\(372\) −6.00000 −0.311086
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) −16.0000 −0.827340
\(375\) 12.0000 0.619677
\(376\) −10.0000 −0.515711
\(377\) 24.0000 1.23606
\(378\) 1.00000 0.0514344
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) −4.00000 −0.205196
\(381\) −20.0000 −1.02463
\(382\) −16.0000 −0.818631
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 1.00000 0.0510310
\(385\) −8.00000 −0.407718
\(386\) −6.00000 −0.305392
\(387\) 8.00000 0.406663
\(388\) −8.00000 −0.406138
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) −8.00000 −0.405096
\(391\) −4.00000 −0.202289
\(392\) −1.00000 −0.0505076
\(393\) 12.0000 0.605320
\(394\) 26.0000 1.30986
\(395\) −16.0000 −0.805047
\(396\) −4.00000 −0.201008
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 12.0000 0.601506
\(399\) 2.00000 0.100125
\(400\) −1.00000 −0.0500000
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 8.00000 0.399004
\(403\) −24.0000 −1.19553
\(404\) 16.0000 0.796030
\(405\) 2.00000 0.0993808
\(406\) 6.00000 0.297775
\(407\) 8.00000 0.396545
\(408\) −4.00000 −0.198030
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 20.0000 0.987730
\(411\) −6.00000 −0.295958
\(412\) 12.0000 0.591198
\(413\) −12.0000 −0.590481
\(414\) −1.00000 −0.0491473
\(415\) −12.0000 −0.589057
\(416\) 4.00000 0.196116
\(417\) 16.0000 0.783523
\(418\) −8.00000 −0.391293
\(419\) −22.0000 −1.07477 −0.537385 0.843337i \(-0.680588\pi\)
−0.537385 + 0.843337i \(0.680588\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −4.00000 −0.194717
\(423\) 10.0000 0.486217
\(424\) 6.00000 0.291386
\(425\) 4.00000 0.194029
\(426\) −4.00000 −0.193801
\(427\) −10.0000 −0.483934
\(428\) −20.0000 −0.966736
\(429\) −16.0000 −0.772487
\(430\) −16.0000 −0.771589
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) −6.00000 −0.288009
\(435\) 12.0000 0.575356
\(436\) 2.00000 0.0957826
\(437\) −2.00000 −0.0956730
\(438\) −2.00000 −0.0955637
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 8.00000 0.381385
\(441\) 1.00000 0.0476190
\(442\) −16.0000 −0.761042
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 14.0000 0.662919
\(447\) −6.00000 −0.283790
\(448\) 1.00000 0.0472456
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 1.00000 0.0471405
\(451\) 40.0000 1.88353
\(452\) −6.00000 −0.282216
\(453\) −4.00000 −0.187936
\(454\) 10.0000 0.469323
\(455\) −8.00000 −0.375046
\(456\) −2.00000 −0.0936586
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −6.00000 −0.280362
\(459\) 4.00000 0.186704
\(460\) 2.00000 0.0932505
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) −4.00000 −0.186097
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −6.00000 −0.278543
\(465\) −12.0000 −0.556487
\(466\) −6.00000 −0.277945
\(467\) 22.0000 1.01804 0.509019 0.860755i \(-0.330008\pi\)
0.509019 + 0.860755i \(0.330008\pi\)
\(468\) −4.00000 −0.184900
\(469\) 8.00000 0.369406
\(470\) −20.0000 −0.922531
\(471\) −10.0000 −0.460776
\(472\) 12.0000 0.552345
\(473\) −32.0000 −1.47136
\(474\) −8.00000 −0.367452
\(475\) 2.00000 0.0917663
\(476\) −4.00000 −0.183340
\(477\) −6.00000 −0.274721
\(478\) −20.0000 −0.914779
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 2.00000 0.0912871
\(481\) 8.00000 0.364769
\(482\) −12.0000 −0.546585
\(483\) −1.00000 −0.0455016
\(484\) 5.00000 0.227273
\(485\) −16.0000 −0.726523
\(486\) 1.00000 0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 10.0000 0.452679
\(489\) −12.0000 −0.542659
\(490\) −2.00000 −0.0903508
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 10.0000 0.450835
\(493\) 24.0000 1.08091
\(494\) −8.00000 −0.359937
\(495\) −8.00000 −0.359573
\(496\) 6.00000 0.269408
\(497\) −4.00000 −0.179425
\(498\) −6.00000 −0.268866
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) −12.0000 −0.536656
\(501\) 18.0000 0.804181
\(502\) −26.0000 −1.16044
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 32.0000 1.42398
\(506\) 4.00000 0.177822
\(507\) −3.00000 −0.133235
\(508\) 20.0000 0.887357
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) −8.00000 −0.354246
\(511\) −2.00000 −0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) −2.00000 −0.0882162
\(515\) 24.0000 1.05757
\(516\) −8.00000 −0.352180
\(517\) −40.0000 −1.75920
\(518\) 2.00000 0.0878750
\(519\) −16.0000 −0.702322
\(520\) 8.00000 0.350823
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 6.00000 0.262613
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) −12.0000 −0.524222
\(525\) 1.00000 0.0436436
\(526\) −16.0000 −0.697633
\(527\) −24.0000 −1.04546
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) 12.0000 0.521247
\(531\) −12.0000 −0.520756
\(532\) −2.00000 −0.0867110
\(533\) 40.0000 1.73259
\(534\) 0 0
\(535\) −40.0000 −1.72935
\(536\) −8.00000 −0.345547
\(537\) −4.00000 −0.172613
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) −2.00000 −0.0860663
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 26.0000 1.11680
\(543\) 2.00000 0.0858282
\(544\) 4.00000 0.171499
\(545\) 4.00000 0.171341
\(546\) −4.00000 −0.171184
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 6.00000 0.256307
\(549\) −10.0000 −0.426790
\(550\) −4.00000 −0.170561
\(551\) 12.0000 0.511217
\(552\) 1.00000 0.0425628
\(553\) −8.00000 −0.340195
\(554\) −14.0000 −0.594803
\(555\) 4.00000 0.169791
\(556\) −16.0000 −0.678551
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) −6.00000 −0.254000
\(559\) −32.0000 −1.35346
\(560\) 2.00000 0.0845154
\(561\) −16.0000 −0.675521
\(562\) −14.0000 −0.590554
\(563\) −22.0000 −0.927189 −0.463595 0.886047i \(-0.653441\pi\)
−0.463595 + 0.886047i \(0.653441\pi\)
\(564\) −10.0000 −0.421076
\(565\) −12.0000 −0.504844
\(566\) 18.0000 0.756596
\(567\) 1.00000 0.0419961
\(568\) 4.00000 0.167836
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) −4.00000 −0.167542
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 16.0000 0.668994
\(573\) −16.0000 −0.668410
\(574\) 10.0000 0.417392
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 1.00000 0.0415945
\(579\) −6.00000 −0.249351
\(580\) −12.0000 −0.498273
\(581\) −6.00000 −0.248922
\(582\) −8.00000 −0.331611
\(583\) 24.0000 0.993978
\(584\) 2.00000 0.0827606
\(585\) −8.00000 −0.330759
\(586\) 6.00000 0.247858
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −12.0000 −0.494451
\(590\) 24.0000 0.988064
\(591\) 26.0000 1.06950
\(592\) −2.00000 −0.0821995
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) −4.00000 −0.164122
\(595\) −8.00000 −0.327968
\(596\) 6.00000 0.245770
\(597\) 12.0000 0.491127
\(598\) 4.00000 0.163572
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) −8.00000 −0.326056
\(603\) 8.00000 0.325785
\(604\) 4.00000 0.162758
\(605\) 10.0000 0.406558
\(606\) 16.0000 0.649956
\(607\) −2.00000 −0.0811775 −0.0405887 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 2.00000 0.0811107
\(609\) 6.00000 0.243132
\(610\) 20.0000 0.809776
\(611\) −40.0000 −1.61823
\(612\) −4.00000 −0.161690
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 0 0
\(615\) 20.0000 0.806478
\(616\) 4.00000 0.161165
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 12.0000 0.482711
\(619\) −14.0000 −0.562708 −0.281354 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) 12.0000 0.481932
\(621\) −1.00000 −0.0401286
\(622\) 30.0000 1.20289
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) −19.0000 −0.760000
\(626\) 20.0000 0.799361
\(627\) −8.00000 −0.319489
\(628\) 10.0000 0.399043
\(629\) 8.00000 0.318981
\(630\) −2.00000 −0.0796819
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 8.00000 0.318223
\(633\) −4.00000 −0.158986
\(634\) −30.0000 −1.19145
\(635\) 40.0000 1.58735
\(636\) 6.00000 0.237915
\(637\) −4.00000 −0.158486
\(638\) −24.0000 −0.950169
\(639\) −4.00000 −0.158238
\(640\) −2.00000 −0.0790569
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) −20.0000 −0.789337
\(643\) 10.0000 0.394362 0.197181 0.980367i \(-0.436821\pi\)
0.197181 + 0.980367i \(0.436821\pi\)
\(644\) 1.00000 0.0394055
\(645\) −16.0000 −0.629999
\(646\) −8.00000 −0.314756
\(647\) 14.0000 0.550397 0.275198 0.961387i \(-0.411256\pi\)
0.275198 + 0.961387i \(0.411256\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 48.0000 1.88416
\(650\) −4.00000 −0.156893
\(651\) −6.00000 −0.235159
\(652\) 12.0000 0.469956
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 2.00000 0.0782062
\(655\) −24.0000 −0.937758
\(656\) −10.0000 −0.390434
\(657\) −2.00000 −0.0780274
\(658\) −10.0000 −0.389841
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 8.00000 0.311400
\(661\) −50.0000 −1.94477 −0.972387 0.233373i \(-0.925024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) 4.00000 0.155464
\(663\) −16.0000 −0.621389
\(664\) 6.00000 0.232845
\(665\) −4.00000 −0.155113
\(666\) 2.00000 0.0774984
\(667\) −6.00000 −0.232321
\(668\) −18.0000 −0.696441
\(669\) 14.0000 0.541271
\(670\) −16.0000 −0.618134
\(671\) 40.0000 1.54418
\(672\) 1.00000 0.0385758
\(673\) 30.0000 1.15642 0.578208 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) 10.0000 0.385186
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) −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
\(680\) 8.00000 0.306786
\(681\) 10.0000 0.383201
\(682\) 24.0000 0.919007
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 12.0000 0.458496
\(686\) −1.00000 −0.0381802
\(687\) −6.00000 −0.228914
\(688\) 8.00000 0.304997
\(689\) 24.0000 0.914327
\(690\) 2.00000 0.0761387
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 16.0000 0.608229
\(693\) −4.00000 −0.151947
\(694\) −28.0000 −1.06287
\(695\) −32.0000 −1.21383
\(696\) −6.00000 −0.227429
\(697\) 40.0000 1.51511
\(698\) 16.0000 0.605609
\(699\) −6.00000 −0.226941
\(700\) −1.00000 −0.0377964
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −4.00000 −0.150970
\(703\) 4.00000 0.150863
\(704\) −4.00000 −0.150756
\(705\) −20.0000 −0.753244
\(706\) 2.00000 0.0752710
\(707\) 16.0000 0.601742
\(708\) 12.0000 0.450988
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 8.00000 0.300235
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) −4.00000 −0.149696
\(715\) 32.0000 1.19673
\(716\) 4.00000 0.149487
\(717\) −20.0000 −0.746914
\(718\) −16.0000 −0.597115
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 2.00000 0.0745356
\(721\) 12.0000 0.446903
\(722\) 15.0000 0.558242
\(723\) −12.0000 −0.446285
\(724\) −2.00000 −0.0743294
\(725\) 6.00000 0.222834
\(726\) 5.00000 0.185567
\(727\) 20.0000 0.741759 0.370879 0.928681i \(-0.379056\pi\)
0.370879 + 0.928681i \(0.379056\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) −32.0000 −1.18356
\(732\) 10.0000 0.369611
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −28.0000 −1.03350
\(735\) −2.00000 −0.0737711
\(736\) −1.00000 −0.0368605
\(737\) −32.0000 −1.17874
\(738\) 10.0000 0.368105
\(739\) 52.0000 1.91285 0.956425 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) −4.00000 −0.147043
\(741\) −8.00000 −0.293887
\(742\) 6.00000 0.220267
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 6.00000 0.219971
\(745\) 12.0000 0.439646
\(746\) −6.00000 −0.219676
\(747\) −6.00000 −0.219529
\(748\) 16.0000 0.585018
\(749\) −20.0000 −0.730784
\(750\) −12.0000 −0.438178
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 10.0000 0.364662
\(753\) −26.0000 −0.947493
\(754\) −24.0000 −0.874028
\(755\) 8.00000 0.291150
\(756\) −1.00000 −0.0363696
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 28.0000 1.01701
\(759\) 4.00000 0.145191
\(760\) 4.00000 0.145095
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 20.0000 0.724524
\(763\) 2.00000 0.0724049
\(764\) 16.0000 0.578860
\(765\) −8.00000 −0.289241
\(766\) −32.0000 −1.15621
\(767\) 48.0000 1.73318
\(768\) −1.00000 −0.0360844
\(769\) −28.0000 −1.00971 −0.504853 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(770\) 8.00000 0.288300
\(771\) −2.00000 −0.0720282
\(772\) 6.00000 0.215945
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) −8.00000 −0.287554
\(775\) −6.00000 −0.215526
\(776\) 8.00000 0.287183
\(777\) 2.00000 0.0717496
\(778\) 6.00000 0.215110
\(779\) 20.0000 0.716574
\(780\) 8.00000 0.286446
\(781\) 16.0000 0.572525
\(782\) 4.00000 0.143040
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) 20.0000 0.713831
\(786\) −12.0000 −0.428026
\(787\) −6.00000 −0.213877 −0.106938 0.994266i \(-0.534105\pi\)
−0.106938 + 0.994266i \(0.534105\pi\)
\(788\) −26.0000 −0.926212
\(789\) −16.0000 −0.569615
\(790\) 16.0000 0.569254
\(791\) −6.00000 −0.213335
\(792\) 4.00000 0.142134
\(793\) 40.0000 1.42044
\(794\) −28.0000 −0.993683
\(795\) 12.0000 0.425596
\(796\) −12.0000 −0.425329
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) −2.00000 −0.0707992
\(799\) −40.0000 −1.41510
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 2.00000 0.0706225
\(803\) 8.00000 0.282314
\(804\) −8.00000 −0.282138
\(805\) 2.00000 0.0704907
\(806\) 24.0000 0.845364
\(807\) 0 0
\(808\) −16.0000 −0.562878
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) −6.00000 −0.210559
\(813\) 26.0000 0.911860
\(814\) −8.00000 −0.280400
\(815\) 24.0000 0.840683
\(816\) 4.00000 0.140028
\(817\) −16.0000 −0.559769
\(818\) 30.0000 1.04893
\(819\) −4.00000 −0.139771
\(820\) −20.0000 −0.698430
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 6.00000 0.209274
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) −12.0000 −0.418040
\(825\) −4.00000 −0.139262
\(826\) 12.0000 0.417533
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 1.00000 0.0347524
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 12.0000 0.416526
\(831\) −14.0000 −0.485655
\(832\) −4.00000 −0.138675
\(833\) −4.00000 −0.138592
\(834\) −16.0000 −0.554035
\(835\) −36.0000 −1.24583
\(836\) 8.00000 0.276686
\(837\) −6.00000 −0.207390
\(838\) 22.0000 0.759977
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 2.00000 0.0690066
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) −14.0000 −0.482186
\(844\) 4.00000 0.137686
\(845\) 6.00000 0.206406
\(846\) −10.0000 −0.343807
\(847\) 5.00000 0.171802
\(848\) −6.00000 −0.206041
\(849\) 18.0000 0.617758
\(850\) −4.00000 −0.137199
\(851\) −2.00000 −0.0685591
\(852\) 4.00000 0.137038
\(853\) 12.0000 0.410872 0.205436 0.978671i \(-0.434139\pi\)
0.205436 + 0.978671i \(0.434139\pi\)
\(854\) 10.0000 0.342193
\(855\) −4.00000 −0.136797
\(856\) 20.0000 0.683586
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 16.0000 0.546231
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 16.0000 0.545595
\(861\) 10.0000 0.340799
\(862\) −32.0000 −1.08992
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 1.00000 0.0340207
\(865\) 32.0000 1.08803
\(866\) 16.0000 0.543702
\(867\) 1.00000 0.0339618
\(868\) 6.00000 0.203653
\(869\) 32.0000 1.08553
\(870\) −12.0000 −0.406838
\(871\) −32.0000 −1.08428
\(872\) −2.00000 −0.0677285
\(873\) −8.00000 −0.270759
\(874\) 2.00000 0.0676510
\(875\) −12.0000 −0.405674
\(876\) 2.00000 0.0675737
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 14.0000 0.472477
\(879\) 6.00000 0.202375
\(880\) −8.00000 −0.269680
\(881\) 20.0000 0.673817 0.336909 0.941537i \(-0.390619\pi\)
0.336909 + 0.941537i \(0.390619\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 16.0000 0.538138
\(885\) 24.0000 0.806751
\(886\) 4.00000 0.134383
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 20.0000 0.670778
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) −14.0000 −0.468755
\(893\) −20.0000 −0.669274
\(894\) 6.00000 0.200670
\(895\) 8.00000 0.267411
\(896\) −1.00000 −0.0334077
\(897\) 4.00000 0.133556
\(898\) −14.0000 −0.467186
\(899\) −36.0000 −1.20067
\(900\) −1.00000 −0.0333333
\(901\) 24.0000 0.799556
\(902\) −40.0000 −1.33185
\(903\) −8.00000 −0.266223
\(904\) 6.00000 0.199557
\(905\) −4.00000 −0.132964
\(906\) 4.00000 0.132891
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −10.0000 −0.331862
\(909\) 16.0000 0.530687
\(910\) 8.00000 0.265197
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 2.00000 0.0662266
\(913\) 24.0000 0.794284
\(914\) −18.0000 −0.595387
\(915\) 20.0000 0.661180
\(916\) 6.00000 0.198246
\(917\) −12.0000 −0.396275
\(918\) −4.00000 −0.132020
\(919\) 48.0000 1.58337 0.791687 0.610927i \(-0.209203\pi\)
0.791687 + 0.610927i \(0.209203\pi\)
\(920\) −2.00000 −0.0659380
\(921\) 0 0
\(922\) 24.0000 0.790398
\(923\) 16.0000 0.526646
\(924\) 4.00000 0.131590
\(925\) 2.00000 0.0657596
\(926\) 32.0000 1.05159
\(927\) 12.0000 0.394132
\(928\) 6.00000 0.196960
\(929\) −10.0000 −0.328089 −0.164045 0.986453i \(-0.552454\pi\)
−0.164045 + 0.986453i \(0.552454\pi\)
\(930\) 12.0000 0.393496
\(931\) −2.00000 −0.0655474
\(932\) 6.00000 0.196537
\(933\) 30.0000 0.982156
\(934\) −22.0000 −0.719862
\(935\) 32.0000 1.04651
\(936\) 4.00000 0.130744
\(937\) 4.00000 0.130674 0.0653372 0.997863i \(-0.479188\pi\)
0.0653372 + 0.997863i \(0.479188\pi\)
\(938\) −8.00000 −0.261209
\(939\) 20.0000 0.652675
\(940\) 20.0000 0.652328
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 10.0000 0.325818
\(943\) −10.0000 −0.325645
\(944\) −12.0000 −0.390567
\(945\) −2.00000 −0.0650600
\(946\) 32.0000 1.04041
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 8.00000 0.259828
\(949\) 8.00000 0.259691
\(950\) −2.00000 −0.0648886
\(951\) −30.0000 −0.972817
\(952\) 4.00000 0.129641
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 6.00000 0.194257
\(955\) 32.0000 1.03550
\(956\) 20.0000 0.646846
\(957\) −24.0000 −0.775810
\(958\) −4.00000 −0.129234
\(959\) 6.00000 0.193750
\(960\) −2.00000 −0.0645497
\(961\) 5.00000 0.161290
\(962\) −8.00000 −0.257930
\(963\) −20.0000 −0.644491
\(964\) 12.0000 0.386494
\(965\) 12.0000 0.386294
\(966\) 1.00000 0.0321745
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) −5.00000 −0.160706
\(969\) −8.00000 −0.256997
\(970\) 16.0000 0.513729
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −16.0000 −0.512936
\(974\) −16.0000 −0.512673
\(975\) −4.00000 −0.128103
\(976\) −10.0000 −0.320092
\(977\) −10.0000 −0.319928 −0.159964 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) 12.0000 0.383718
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) 2.00000 0.0638551
\(982\) 28.0000 0.893516
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −10.0000 −0.318788
\(985\) −52.0000 −1.65686
\(986\) −24.0000 −0.764316
\(987\) −10.0000 −0.318304
\(988\) 8.00000 0.254514
\(989\) 8.00000 0.254385
\(990\) 8.00000 0.254257
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) −6.00000 −0.190500
\(993\) 4.00000 0.126936
\(994\) 4.00000 0.126872
\(995\) −24.0000 −0.760851
\(996\) 6.00000 0.190117
\(997\) −52.0000 −1.64686 −0.823428 0.567420i \(-0.807941\pi\)
−0.823428 + 0.567420i \(0.807941\pi\)
\(998\) −28.0000 −0.886325
\(999\) 2.00000 0.0632772
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 966.2.a.c.1.1 1
3.2 odd 2 2898.2.a.l.1.1 1
4.3 odd 2 7728.2.a.t.1.1 1
7.6 odd 2 6762.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.c.1.1 1 1.1 even 1 trivial
2898.2.a.l.1.1 1 3.2 odd 2
6762.2.a.m.1.1 1 7.6 odd 2
7728.2.a.t.1.1 1 4.3 odd 2