Properties

Label 6762.2.a.o.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} +3.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +3.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} +2.00000 q^{39} +6.00000 q^{41} +2.00000 q^{43} -3.00000 q^{44} +1.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} +5.00000 q^{50} -3.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{57} -3.00000 q^{58} +2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +3.00000 q^{66} +2.00000 q^{67} -3.00000 q^{68} -1.00000 q^{69} +3.00000 q^{71} -1.00000 q^{72} +11.0000 q^{73} -2.00000 q^{74} -5.00000 q^{75} +2.00000 q^{76} -2.00000 q^{78} +11.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -2.00000 q^{86} +3.00000 q^{87} +3.00000 q^{88} -6.00000 q^{89} -1.00000 q^{92} +2.00000 q^{93} +3.00000 q^{94} -1.00000 q^{96} +8.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 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
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −2.00000 −0.324443
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) −3.00000 −0.420084
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) −3.00000 −0.393919
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −3.00000 −0.363803
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) −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
\(75\) −5.00000 −0.577350
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 3.00000 0.321634
\(88\) 3.00000 0.319801
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 2.00000 0.207390
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) −5.00000 −0.500000
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 3.00000 0.297044
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −2.00000 −0.181071
\(123\) 6.00000 0.541002
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −3.00000 −0.261116
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) −21.0000 −1.79415 −0.897076 0.441877i \(-0.854313\pi\)
−0.897076 + 0.441877i \(0.854313\pi\)
\(138\) 1.00000 0.0851257
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) −3.00000 −0.251754
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −11.0000 −0.910366
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 5.00000 0.408248
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −2.00000 −0.162221
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 23.0000 1.83560 0.917800 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) −11.0000 −0.875113
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 23.0000 1.80150 0.900750 0.434339i \(-0.143018\pi\)
0.900750 + 0.434339i \(0.143018\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 2.00000 0.152499
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 11.0000 0.817624 0.408812 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) 9.00000 0.658145
\(188\) −3.00000 −0.218797
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 0 0
\(197\) 15.0000 1.06871 0.534353 0.845262i \(-0.320555\pi\)
0.534353 + 0.845262i \(0.320555\pi\)
\(198\) 3.00000 0.213201
\(199\) 17.0000 1.20510 0.602549 0.798082i \(-0.294152\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) 5.00000 0.353553
\(201\) 2.00000 0.141069
\(202\) 3.00000 0.211079
\(203\) 0 0
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) −5.00000 −0.348367
\(207\) −1.00000 −0.0695048
\(208\) 2.00000 0.138675
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −19.0000 −1.30801 −0.654007 0.756489i \(-0.726913\pi\)
−0.654007 + 0.756489i \(0.726913\pi\)
\(212\) −6.00000 −0.412082
\(213\) 3.00000 0.205557
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 1.00000 0.0677285
\(219\) 11.0000 0.743311
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) −2.00000 −0.134231
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) −18.0000 −1.19734
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 2.00000 0.132453
\(229\) −1.00000 −0.0660819 −0.0330409 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) 11.0000 0.714527
\(238\) 0 0
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 4.00000 0.254514
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) 0 0
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 0 0
\(253\) 3.00000 0.188608
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −2.00000 −0.124515
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 2.00000 0.122169
\(269\) 15.0000 0.914566 0.457283 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) 0 0
\(271\) −22.0000 −1.33640 −0.668202 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) 21.0000 1.26866
\(275\) 15.0000 0.904534
\(276\) −1.00000 −0.0601929
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 13.0000 0.779688
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −3.00000 −0.178965 −0.0894825 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) 3.00000 0.178647
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 3.00000 0.178017
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 11.0000 0.643726
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −3.00000 −0.174078
\(298\) −12.0000 −0.695141
\(299\) −2.00000 −0.115663
\(300\) −5.00000 −0.288675
\(301\) 0 0
\(302\) 4.00000 0.230174
\(303\) −3.00000 −0.172345
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) 11.0000 0.627803 0.313902 0.949456i \(-0.398364\pi\)
0.313902 + 0.949456i \(0.398364\pi\)
\(308\) 0 0
\(309\) 5.00000 0.284440
\(310\) 0 0
\(311\) −3.00000 −0.170114 −0.0850572 0.996376i \(-0.527107\pi\)
−0.0850572 + 0.996376i \(0.527107\pi\)
\(312\) −2.00000 −0.113228
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −23.0000 −1.29797
\(315\) 0 0
\(316\) 11.0000 0.618798
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 6.00000 0.336463
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 1.00000 0.0555556
\(325\) −10.0000 −0.554700
\(326\) −23.0000 −1.27385
\(327\) −1.00000 −0.0553001
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 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
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 9.00000 0.489535
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) −2.00000 −0.108148
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) 15.0000 0.806405
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 3.00000 0.160817
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 3.00000 0.159901
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −6.00000 −0.317110
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −11.0000 −0.578147
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) −13.0000 −0.673114 −0.336557 0.941663i \(-0.609263\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) −9.00000 −0.465379
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) −12.0000 −0.613973
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 2.00000 0.101666
\(388\) 8.00000 0.406138
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) 0 0
\(394\) −15.0000 −0.755689
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) −17.0000 −0.852133
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −15.0000 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 4.00000 0.199254
\(404\) −3.00000 −0.149256
\(405\) 0 0
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 3.00000 0.148522
\(409\) 29.0000 1.43396 0.716979 0.697095i \(-0.245524\pi\)
0.716979 + 0.697095i \(0.245524\pi\)
\(410\) 0 0
\(411\) −21.0000 −1.03585
\(412\) 5.00000 0.246332
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −13.0000 −0.636613
\(418\) 6.00000 0.293470
\(419\) −21.0000 −1.02592 −0.512959 0.858413i \(-0.671451\pi\)
−0.512959 + 0.858413i \(0.671451\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) 19.0000 0.924906
\(423\) −3.00000 −0.145865
\(424\) 6.00000 0.291386
\(425\) 15.0000 0.727607
\(426\) −3.00000 −0.145350
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.00000 −0.0478913
\(437\) −2.00000 −0.0956730
\(438\) −11.0000 −0.525600
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.00000 0.285391
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 10.0000 0.473514
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 5.00000 0.235702
\(451\) −18.0000 −0.847587
\(452\) 18.0000 0.846649
\(453\) −4.00000 −0.187936
\(454\) 3.00000 0.140797
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 20.0000 0.935561 0.467780 0.883845i \(-0.345054\pi\)
0.467780 + 0.883845i \(0.345054\pi\)
\(458\) 1.00000 0.0467269
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) 0 0
\(467\) −21.0000 −0.971764 −0.485882 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 23.0000 1.05978
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) −11.0000 −0.505247
\(475\) −10.0000 −0.458831
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −15.0000 −0.686084
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 23.0000 1.04010
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) −9.00000 −0.405340
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) 0 0
\(499\) 35.0000 1.56682 0.783408 0.621508i \(-0.213480\pi\)
0.783408 + 0.621508i \(0.213480\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −3.00000 −0.133897
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −3.00000 −0.133366
\(507\) −9.00000 −0.399704
\(508\) 2.00000 0.0887357
\(509\) −21.0000 −0.930809 −0.465404 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) 9.00000 0.395820
\(518\) 0 0
\(519\) −15.0000 −0.658427
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) −3.00000 −0.131306
\(523\) −22.0000 −0.961993 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −6.00000 −0.261364
\(528\) −3.00000 −0.130558
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) 6.00000 0.258919
\(538\) −15.0000 −0.646696
\(539\) 0 0
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 22.0000 0.944981
\(543\) 11.0000 0.472055
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) 17.0000 0.726868 0.363434 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(548\) −21.0000 −0.897076
\(549\) 2.00000 0.0853579
\(550\) −15.0000 −0.639602
\(551\) 6.00000 0.255609
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −13.0000 −0.551323
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 9.00000 0.379980
\(562\) 3.00000 0.126547
\(563\) −9.00000 −0.379305 −0.189652 0.981851i \(-0.560736\pi\)
−0.189652 + 0.981851i \(0.560736\pi\)
\(564\) −3.00000 −0.126323
\(565\) 0 0
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) −3.00000 −0.125877
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −6.00000 −0.250873
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 5.00000 0.208514
\(576\) 1.00000 0.0416667
\(577\) −31.0000 −1.29055 −0.645273 0.763952i \(-0.723257\pi\)
−0.645273 + 0.763952i \(0.723257\pi\)
\(578\) 8.00000 0.332756
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) −8.00000 −0.331611
\(583\) 18.0000 0.745484
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 15.0000 0.617018
\(592\) 2.00000 0.0821995
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 17.0000 0.695764
\(598\) 2.00000 0.0817861
\(599\) 15.0000 0.612883 0.306442 0.951889i \(-0.400862\pi\)
0.306442 + 0.951889i \(0.400862\pi\)
\(600\) 5.00000 0.204124
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) 3.00000 0.121867
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) −3.00000 −0.121268
\(613\) 5.00000 0.201948 0.100974 0.994889i \(-0.467804\pi\)
0.100974 + 0.994889i \(0.467804\pi\)
\(614\) −11.0000 −0.443924
\(615\) 0 0
\(616\) 0 0
\(617\) 33.0000 1.32853 0.664265 0.747497i \(-0.268745\pi\)
0.664265 + 0.747497i \(0.268745\pi\)
\(618\) −5.00000 −0.201129
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 3.00000 0.120289
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 25.0000 1.00000
\(626\) −8.00000 −0.319744
\(627\) −6.00000 −0.239617
\(628\) 23.0000 0.917800
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) −11.0000 −0.437557
\(633\) −19.0000 −0.755182
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 9.00000 0.356313
\(639\) 3.00000 0.118678
\(640\) 0 0
\(641\) −3.00000 −0.118493 −0.0592464 0.998243i \(-0.518870\pi\)
−0.0592464 + 0.998243i \(0.518870\pi\)
\(642\) −12.0000 −0.473602
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) −15.0000 −0.589711 −0.294855 0.955542i \(-0.595271\pi\)
−0.294855 + 0.955542i \(0.595271\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 10.0000 0.392232
\(651\) 0 0
\(652\) 23.0000 0.900750
\(653\) 3.00000 0.117399 0.0586995 0.998276i \(-0.481305\pi\)
0.0586995 + 0.998276i \(0.481305\pi\)
\(654\) 1.00000 0.0391031
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 11.0000 0.429151
\(658\) 0 0
\(659\) 3.00000 0.116863 0.0584317 0.998291i \(-0.481390\pi\)
0.0584317 + 0.998291i \(0.481390\pi\)
\(660\) 0 0
\(661\) 5.00000 0.194477 0.0972387 0.995261i \(-0.468999\pi\)
0.0972387 + 0.995261i \(0.468999\pi\)
\(662\) −8.00000 −0.310929
\(663\) −6.00000 −0.233021
\(664\) 0 0
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −3.00000 −0.116160
\(668\) 0 0
\(669\) −10.0000 −0.386622
\(670\) 0 0
\(671\) −6.00000 −0.231627
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) −14.0000 −0.539260
\(675\) −5.00000 −0.192450
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −18.0000 −0.691286
\(679\) 0 0
\(680\) 0 0
\(681\) −3.00000 −0.114960
\(682\) 6.00000 0.229752
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) 0 0
\(687\) −1.00000 −0.0381524
\(688\) 2.00000 0.0762493
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) −15.0000 −0.570214
\(693\) 0 0
\(694\) −6.00000 −0.227757
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) −18.0000 −0.681799
\(698\) 22.0000 0.832712
\(699\) 0 0
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 4.00000 0.150863
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 0 0
\(709\) −31.0000 −1.16423 −0.582115 0.813107i \(-0.697775\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) 11.0000 0.412532
\(712\) 6.00000 0.224860
\(713\) −2.00000 −0.0749006
\(714\) 0 0
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) 15.0000 0.560185
\(718\) 12.0000 0.447836
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) 2.00000 0.0743808
\(724\) 11.0000 0.408812
\(725\) −15.0000 −0.557086
\(726\) 2.00000 0.0742270
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.00000 −0.221918
\(732\) 2.00000 0.0739221
\(733\) 41.0000 1.51437 0.757185 0.653201i \(-0.226574\pi\)
0.757185 + 0.653201i \(0.226574\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −6.00000 −0.221013
\(738\) −6.00000 −0.220863
\(739\) −25.0000 −0.919640 −0.459820 0.888012i \(-0.652086\pi\)
−0.459820 + 0.888012i \(0.652086\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 0 0
\(746\) 13.0000 0.475964
\(747\) 0 0
\(748\) 9.00000 0.329073
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −3.00000 −0.109399
\(753\) 3.00000 0.109326
\(754\) −6.00000 −0.218507
\(755\) 0 0
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −20.0000 −0.726433
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) 48.0000 1.74000 0.869999 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 0 0
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 14.0000 0.503871
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) −2.00000 −0.0718885
\(775\) −10.0000 −0.359211
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) −24.0000 −0.860442
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −9.00000 −0.322045
\(782\) −3.00000 −0.107280
\(783\) 3.00000 0.107211
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) 15.0000 0.534353
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) 3.00000 0.106600
\(793\) 4.00000 0.142044
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) 17.0000 0.602549
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) 5.00000 0.176777
\(801\) −6.00000 −0.212000
\(802\) 15.0000 0.529668
\(803\) −33.0000 −1.16454
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 15.0000 0.528025
\(808\) 3.00000 0.105540
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −7.00000 −0.245803 −0.122902 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(812\) 0 0
\(813\) −22.0000 −0.771574
\(814\) 6.00000 0.210300
\(815\) 0 0
\(816\) −3.00000 −0.105021
\(817\) 4.00000 0.139942
\(818\) −29.0000 −1.01396
\(819\) 0 0
\(820\) 0 0
\(821\) 33.0000 1.15171 0.575854 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) 21.0000 0.732459
\(823\) −22.0000 −0.766872 −0.383436 0.923567i \(-0.625259\pi\)
−0.383436 + 0.923567i \(0.625259\pi\)
\(824\) −5.00000 −0.174183
\(825\) 15.0000 0.522233
\(826\) 0 0
\(827\) −21.0000 −0.730242 −0.365121 0.930960i \(-0.618972\pi\)
−0.365121 + 0.930960i \(0.618972\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 13.0000 0.450153
\(835\) 0 0
\(836\) −6.00000 −0.207514
\(837\) 2.00000 0.0691301
\(838\) 21.0000 0.725433
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −17.0000 −0.585859
\(843\) −3.00000 −0.103325
\(844\) −19.0000 −0.654007
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 8.00000 0.274559
\(850\) −15.0000 −0.514496
\(851\) −2.00000 −0.0685591
\(852\) 3.00000 0.102778
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 6.00000 0.204837
\(859\) 11.0000 0.375315 0.187658 0.982235i \(-0.439910\pi\)
0.187658 + 0.982235i \(0.439910\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −30.0000 −1.02180
\(863\) 3.00000 0.102121 0.0510606 0.998696i \(-0.483740\pi\)
0.0510606 + 0.998696i \(0.483740\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −33.0000 −1.11945
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 1.00000 0.0338643
\(873\) 8.00000 0.270759
\(874\) 2.00000 0.0676510
\(875\) 0 0
\(876\) 11.0000 0.371656
\(877\) −4.00000 −0.135070 −0.0675352 0.997717i \(-0.521513\pi\)
−0.0675352 + 0.997717i \(0.521513\pi\)
\(878\) −8.00000 −0.269987
\(879\) 0 0
\(880\) 0 0
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 27.0000 0.906571 0.453286 0.891365i \(-0.350252\pi\)
0.453286 + 0.891365i \(0.350252\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) −10.0000 −0.334825
\(893\) −6.00000 −0.200782
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) −12.0000 −0.400445
\(899\) 6.00000 0.200111
\(900\) −5.00000 −0.166667
\(901\) 18.0000 0.599667
\(902\) 18.0000 0.599334
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 4.00000 0.132891
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −3.00000 −0.0995585
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 2.00000 0.0662266
\(913\) 0 0
\(914\) −20.0000 −0.661541
\(915\) 0 0
\(916\) −1.00000 −0.0330409
\(917\) 0 0
\(918\) 3.00000 0.0990148
\(919\) 29.0000 0.956622 0.478311 0.878191i \(-0.341249\pi\)
0.478311 + 0.878191i \(0.341249\pi\)
\(920\) 0 0
\(921\) 11.0000 0.362462
\(922\) 30.0000 0.987997
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) −32.0000 −1.05159
\(927\) 5.00000 0.164222
\(928\) −3.00000 −0.0984798
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −3.00000 −0.0982156
\(934\) 21.0000 0.687141
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) −23.0000 −0.749380
\(943\) −6.00000 −0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) −6.00000 −0.194974 −0.0974869 0.995237i \(-0.531080\pi\)
−0.0974869 + 0.995237i \(0.531080\pi\)
\(948\) 11.0000 0.357263
\(949\) 22.0000 0.714150
\(950\) 10.0000 0.324443
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 33.0000 1.06897 0.534487 0.845176i \(-0.320505\pi\)
0.534487 + 0.845176i \(0.320505\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 15.0000 0.485135
\(957\) −9.00000 −0.290929
\(958\) −30.0000 −0.969256
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −4.00000 −0.128965
\(963\) 12.0000 0.386695
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) 14.0000 0.450210 0.225105 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) 2.00000 0.0642824
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) 45.0000 1.44412 0.722059 0.691831i \(-0.243196\pi\)
0.722059 + 0.691831i \(0.243196\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −2.00000 −0.0640841
\(975\) −10.0000 −0.320256
\(976\) 2.00000 0.0640184
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −23.0000 −0.735459
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) −1.00000 −0.0319275
\(982\) 0 0
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 9.00000 0.286618
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −52.0000 −1.64686 −0.823428 0.567420i \(-0.807941\pi\)
−0.823428 + 0.567420i \(0.807941\pi\)
\(998\) −35.0000 −1.10791
\(999\) 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 6762.2.a.o.1.1 1
7.2 even 3 966.2.i.f.277.1 2
7.4 even 3 966.2.i.f.415.1 yes 2
7.6 odd 2 6762.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.f.277.1 2 7.2 even 3
966.2.i.f.415.1 yes 2 7.4 even 3
6762.2.a.e.1.1 1 7.6 odd 2
6762.2.a.o.1.1 1 1.1 even 1 trivial