Properties

Label 966.2.a.b.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +6.00000 q^{19} -2.00000 q^{20} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -2.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} +2.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -6.00000 q^{38} -4.00000 q^{39} +2.00000 q^{40} -2.00000 q^{41} -1.00000 q^{42} +12.0000 q^{43} -2.00000 q^{45} +1.00000 q^{46} +10.0000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} +1.00000 q^{56} -6.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} -14.0000 q^{61} +10.0000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -8.00000 q^{65} -12.0000 q^{67} +1.00000 q^{69} -2.00000 q^{70} -12.0000 q^{71} -1.00000 q^{72} +14.0000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +6.00000 q^{76} +4.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +2.00000 q^{83} +1.00000 q^{84} -12.0000 q^{86} +6.00000 q^{87} -12.0000 q^{89} +2.00000 q^{90} -4.00000 q^{91} -1.00000 q^{92} +10.0000 q^{93} -10.0000 q^{94} -12.0000 q^{95} +1.00000 q^{96} +12.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −2.00000 −0.447214
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −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
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −6.00000 −0.973329
\(39\) −4.00000 −0.640513
\(40\) 2.00000 0.316228
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −1.00000 −0.154303
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) −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
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −6.00000 −0.794719
\(58\) 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
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 10.0000 1.27000
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) −2.00000 −0.239046
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 2.00000 0.210819
\(91\) −4.00000 −0.419314
\(92\) −1.00000 −0.104257
\(93\) 10.0000 1.03695
\(94\) −10.0000 −1.03142
\(95\) −12.0000 −1.23117
\(96\) 1.00000 0.102062
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −4.00000 −0.392232
\(105\) −2.00000 −0.195180
\(106\) 10.0000 0.971286
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) −1.00000 −0.0944911
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 6.00000 0.561951
\(115\) 2.00000 0.186501
\(116\) −6.00000 −0.557086
\(117\) 4.00000 0.369800
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) −11.0000 −1.00000
\(122\) 14.0000 1.26750
\(123\) 2.00000 0.180334
\(124\) −10.0000 −0.898027
\(125\) 12.0000 1.07331
\(126\) 1.00000 0.0890871
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −12.0000 −1.05654
\(130\) 8.00000 0.701646
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 12.0000 1.03664
\(135\) 2.00000 0.172133
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 2.00000 0.169031
\(141\) −10.0000 −0.842152
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 12.0000 0.996546
\(146\) −14.0000 −1.15865
\(147\) −1.00000 −0.0824786
\(148\) −6.00000 −0.493197
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) −6.00000 −0.486664
\(153\) 0 0
\(154\) 0 0
\(155\) 20.0000 1.60644
\(156\) −4.00000 −0.320256
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 2.00000 0.158114
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) 14.0000 1.08335 0.541676 0.840587i \(-0.317790\pi\)
0.541676 + 0.840587i \(0.317790\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 12.0000 0.914991
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) −6.00000 −0.454859
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 12.0000 0.899438
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −2.00000 −0.149071
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 4.00000 0.296500
\(183\) 14.0000 1.03491
\(184\) 1.00000 0.0737210
\(185\) 12.0000 0.882258
\(186\) −10.0000 −0.733236
\(187\) 0 0
\(188\) 10.0000 0.729325
\(189\) 1.00000 0.0727393
\(190\) 12.0000 0.870572
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −12.0000 −0.861550
\(195\) 8.00000 0.572892
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 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
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 4.00000 0.278693
\(207\) −1.00000 −0.0695048
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 2.00000 0.138013
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) −10.0000 −0.686803
\(213\) 12.0000 0.822226
\(214\) −8.00000 −0.546869
\(215\) −24.0000 −1.63679
\(216\) 1.00000 0.0680414
\(217\) 10.0000 0.678844
\(218\) 10.0000 0.677285
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) 0 0
\(222\) −6.00000 −0.402694
\(223\) 18.0000 1.20537 0.602685 0.797980i \(-0.294098\pi\)
0.602685 + 0.797980i \(0.294098\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 14.0000 0.931266
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) −6.00000 −0.397360
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) −4.00000 −0.261488
\(235\) −20.0000 −1.30466
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 2.00000 0.129099
\(241\) 24.0000 1.54598 0.772988 0.634421i \(-0.218761\pi\)
0.772988 + 0.634421i \(0.218761\pi\)
\(242\) 11.0000 0.707107
\(243\) −1.00000 −0.0641500
\(244\) −14.0000 −0.896258
\(245\) −2.00000 −0.127775
\(246\) −2.00000 −0.127515
\(247\) 24.0000 1.52708
\(248\) 10.0000 0.635001
\(249\) −2.00000 −0.126745
\(250\) −12.0000 −0.758947
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 12.0000 0.747087
\(259\) 6.00000 0.372822
\(260\) −8.00000 −0.496139
\(261\) −6.00000 −0.371391
\(262\) −4.00000 −0.247121
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 20.0000 1.22859
\(266\) 6.00000 0.367884
\(267\) 12.0000 0.734388
\(268\) −12.0000 −0.733017
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) −2.00000 −0.121716
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) −10.0000 −0.598684
\(280\) −2.00000 −0.119523
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 10.0000 0.595491
\(283\) −18.0000 −1.06999 −0.534994 0.844856i \(-0.679686\pi\)
−0.534994 + 0.844856i \(0.679686\pi\)
\(284\) −12.0000 −0.712069
\(285\) 12.0000 0.710819
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) −12.0000 −0.704664
\(291\) −12.0000 −0.703452
\(292\) 14.0000 0.819288
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 1.00000 0.0583212
\(295\) 24.0000 1.39733
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) −4.00000 −0.231326
\(300\) 1.00000 0.0577350
\(301\) −12.0000 −0.691669
\(302\) −12.0000 −0.690522
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 28.0000 1.60328
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) −20.0000 −1.13592
\(311\) 10.0000 0.567048 0.283524 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 4.00000 0.226455
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 2.00000 0.112867
\(315\) 2.00000 0.112687
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −10.0000 −0.560772
\(319\) 0 0
\(320\) −2.00000 −0.111803
\(321\) −8.00000 −0.446516
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 12.0000 0.664619
\(327\) 10.0000 0.553001
\(328\) 2.00000 0.110432
\(329\) −10.0000 −0.551318
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 2.00000 0.109764
\(333\) −6.00000 −0.328798
\(334\) −14.0000 −0.766046
\(335\) 24.0000 1.31126
\(336\) 1.00000 0.0545545
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −3.00000 −0.163178
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 0 0
\(342\) −6.00000 −0.324443
\(343\) −1.00000 −0.0539949
\(344\) −12.0000 −0.646997
\(345\) −2.00000 −0.107676
\(346\) 24.0000 1.29025
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 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
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) −12.0000 −0.637793
\(355\) 24.0000 1.27379
\(356\) −12.0000 −0.635999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 2.00000 0.105409
\(361\) 17.0000 0.894737
\(362\) −2.00000 −0.105118
\(363\) 11.0000 0.577350
\(364\) −4.00000 −0.209657
\(365\) −28.0000 −1.46559
\(366\) −14.0000 −0.731792
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −2.00000 −0.104116
\(370\) −12.0000 −0.623850
\(371\) 10.0000 0.519174
\(372\) 10.0000 0.518476
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) −10.0000 −0.515711
\(377\) −24.0000 −1.23606
\(378\) −1.00000 −0.0514344
\(379\) 24.0000 1.23280 0.616399 0.787434i \(-0.288591\pi\)
0.616399 + 0.787434i \(0.288591\pi\)
\(380\) −12.0000 −0.615587
\(381\) 12.0000 0.614779
\(382\) 0 0
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 12.0000 0.609994
\(388\) 12.0000 0.609208
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) −8.00000 −0.405096
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) −4.00000 −0.201773
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) −12.0000 −0.602263 −0.301131 0.953583i \(-0.597364\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) −20.0000 −1.00251
\(399\) 6.00000 0.300376
\(400\) −1.00000 −0.0500000
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) −12.0000 −0.598506
\(403\) −40.0000 −1.99254
\(404\) 0 0
\(405\) −2.00000 −0.0993808
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −4.00000 −0.197546
\(411\) 18.0000 0.887875
\(412\) −4.00000 −0.197066
\(413\) 12.0000 0.590481
\(414\) 1.00000 0.0491473
\(415\) −4.00000 −0.196352
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) 0 0
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 28.0000 1.36302
\(423\) 10.0000 0.486217
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) 14.0000 0.677507
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 24.0000 1.15738
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) −10.0000 −0.480015
\(435\) −12.0000 −0.575356
\(436\) −10.0000 −0.478913
\(437\) −6.00000 −0.287019
\(438\) 14.0000 0.668946
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 6.00000 0.284747
\(445\) 24.0000 1.13771
\(446\) −18.0000 −0.852325
\(447\) −18.0000 −0.851371
\(448\) −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
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) −12.0000 −0.563809
\(454\) −14.0000 −0.657053
\(455\) 8.00000 0.375046
\(456\) 6.00000 0.280976
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −6.00000 −0.278543
\(465\) −20.0000 −0.927478
\(466\) −22.0000 −1.01913
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 4.00000 0.184900
\(469\) 12.0000 0.554109
\(470\) 20.0000 0.922531
\(471\) 2.00000 0.0921551
\(472\) 12.0000 0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) −20.0000 −0.914779
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) −2.00000 −0.0912871
\(481\) −24.0000 −1.09431
\(482\) −24.0000 −1.09317
\(483\) −1.00000 −0.0455016
\(484\) −11.0000 −0.500000
\(485\) −24.0000 −1.08978
\(486\) 1.00000 0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 14.0000 0.633750
\(489\) 12.0000 0.542659
\(490\) 2.00000 0.0903508
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 2.00000 0.0901670
\(493\) 0 0
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 12.0000 0.538274
\(498\) 2.00000 0.0896221
\(499\) −44.0000 −1.96971 −0.984855 0.173379i \(-0.944532\pi\)
−0.984855 + 0.173379i \(0.944532\pi\)
\(500\) 12.0000 0.536656
\(501\) −14.0000 −0.625474
\(502\) −18.0000 −0.803379
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) −12.0000 −0.532414
\(509\) −16.0000 −0.709188 −0.354594 0.935020i \(-0.615381\pi\)
−0.354594 + 0.935020i \(0.615381\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) −1.00000 −0.0441942
\(513\) −6.00000 −0.264906
\(514\) 6.00000 0.264649
\(515\) 8.00000 0.352522
\(516\) −12.0000 −0.528271
\(517\) 0 0
\(518\) −6.00000 −0.263625
\(519\) 24.0000 1.05348
\(520\) 8.00000 0.350823
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) 6.00000 0.262613
\(523\) 38.0000 1.66162 0.830812 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) 4.00000 0.174741
\(525\) −1.00000 −0.0436436
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −20.0000 −0.868744
\(531\) −12.0000 −0.520756
\(532\) −6.00000 −0.260133
\(533\) −8.00000 −0.346518
\(534\) −12.0000 −0.519291
\(535\) −16.0000 −0.691740
\(536\) 12.0000 0.518321
\(537\) 4.00000 0.172613
\(538\) 24.0000 1.03471
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 2.00000 0.0859074
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) 20.0000 0.856706
\(546\) −4.00000 −0.171184
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −18.0000 −0.768922
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) −12.0000 −0.509372
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 10.0000 0.423334
\(559\) 48.0000 2.03018
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) −22.0000 −0.927189 −0.463595 0.886047i \(-0.653441\pi\)
−0.463595 + 0.886047i \(0.653441\pi\)
\(564\) −10.0000 −0.421076
\(565\) 28.0000 1.17797
\(566\) 18.0000 0.756596
\(567\) −1.00000 −0.0419961
\(568\) 12.0000 0.503509
\(569\) 46.0000 1.92842 0.964210 0.265139i \(-0.0854179\pi\)
0.964210 + 0.265139i \(0.0854179\pi\)
\(570\) −12.0000 −0.502625
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 17.0000 0.707107
\(579\) 10.0000 0.415586
\(580\) 12.0000 0.498273
\(581\) −2.00000 −0.0829740
\(582\) 12.0000 0.497416
\(583\) 0 0
\(584\) −14.0000 −0.579324
\(585\) −8.00000 −0.330759
\(586\) 18.0000 0.743573
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −60.0000 −2.47226
\(590\) −24.0000 −0.988064
\(591\) −6.00000 −0.246807
\(592\) −6.00000 −0.246598
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) −20.0000 −0.818546
\(598\) 4.00000 0.163572
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 12.0000 0.489083
\(603\) −12.0000 −0.488678
\(604\) 12.0000 0.488273
\(605\) 22.0000 0.894427
\(606\) 0 0
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) −6.00000 −0.243332
\(609\) −6.00000 −0.243132
\(610\) −28.0000 −1.13369
\(611\) 40.0000 1.61823
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −16.0000 −0.645707
\(615\) −4.00000 −0.161296
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) −4.00000 −0.160904
\(619\) −6.00000 −0.241160 −0.120580 0.992704i \(-0.538475\pi\)
−0.120580 + 0.992704i \(0.538475\pi\)
\(620\) 20.0000 0.803219
\(621\) 1.00000 0.0401286
\(622\) −10.0000 −0.400963
\(623\) 12.0000 0.480770
\(624\) −4.00000 −0.160128
\(625\) −19.0000 −0.760000
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 0 0
\(630\) −2.00000 −0.0796819
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) 28.0000 1.11290
\(634\) 2.00000 0.0794301
\(635\) 24.0000 0.952411
\(636\) 10.0000 0.396526
\(637\) 4.00000 0.158486
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 2.00000 0.0790569
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 8.00000 0.315735
\(643\) −46.0000 −1.81406 −0.907031 0.421063i \(-0.861657\pi\)
−0.907031 + 0.421063i \(0.861657\pi\)
\(644\) 1.00000 0.0394055
\(645\) 24.0000 0.944999
\(646\) 0 0
\(647\) 38.0000 1.49393 0.746967 0.664861i \(-0.231509\pi\)
0.746967 + 0.664861i \(0.231509\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) −10.0000 −0.391931
\(652\) −12.0000 −0.469956
\(653\) −46.0000 −1.80012 −0.900060 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) −10.0000 −0.391031
\(655\) −8.00000 −0.312586
\(656\) −2.00000 −0.0780869
\(657\) 14.0000 0.546192
\(658\) 10.0000 0.389841
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) −2.00000 −0.0776151
\(665\) 12.0000 0.465340
\(666\) 6.00000 0.232495
\(667\) 6.00000 0.232321
\(668\) 14.0000 0.541676
\(669\) −18.0000 −0.695920
\(670\) −24.0000 −0.927201
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −14.0000 −0.539260
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) −14.0000 −0.537667
\(679\) −12.0000 −0.460518
\(680\) 0 0
\(681\) −14.0000 −0.536481
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 6.00000 0.229416
\(685\) 36.0000 1.37549
\(686\) 1.00000 0.0381802
\(687\) 6.00000 0.228914
\(688\) 12.0000 0.457496
\(689\) −40.0000 −1.52388
\(690\) 2.00000 0.0761387
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) −24.0000 −0.912343
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) 16.0000 0.605609
\(699\) −22.0000 −0.832116
\(700\) 1.00000 0.0377964
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 4.00000 0.150970
\(703\) −36.0000 −1.35777
\(704\) 0 0
\(705\) 20.0000 0.753244
\(706\) 26.0000 0.978523
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) −24.0000 −0.900704
\(711\) 0 0
\(712\) 12.0000 0.449719
\(713\) 10.0000 0.374503
\(714\) 0 0
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) −20.0000 −0.746914
\(718\) 0 0
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 4.00000 0.148968
\(722\) −17.0000 −0.632674
\(723\) −24.0000 −0.892570
\(724\) 2.00000 0.0743294
\(725\) 6.00000 0.222834
\(726\) −11.0000 −0.408248
\(727\) 36.0000 1.33517 0.667583 0.744535i \(-0.267329\pi\)
0.667583 + 0.744535i \(0.267329\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 28.0000 1.03633
\(731\) 0 0
\(732\) 14.0000 0.517455
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 20.0000 0.738213
\(735\) 2.00000 0.0737711
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 2.00000 0.0736210
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 12.0000 0.441129
\(741\) −24.0000 −0.881662
\(742\) −10.0000 −0.367112
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) −10.0000 −0.366618
\(745\) −36.0000 −1.31894
\(746\) 14.0000 0.512576
\(747\) 2.00000 0.0731762
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 12.0000 0.438178
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 10.0000 0.364662
\(753\) −18.0000 −0.655956
\(754\) 24.0000 0.874028
\(755\) −24.0000 −0.873449
\(756\) 1.00000 0.0363696
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −24.0000 −0.871719
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) −12.0000 −0.434714
\(763\) 10.0000 0.362024
\(764\) 0 0
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) −48.0000 −1.73318
\(768\) −1.00000 −0.0360844
\(769\) 16.0000 0.576975 0.288487 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −10.0000 −0.359908
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) −12.0000 −0.431331
\(775\) 10.0000 0.359211
\(776\) −12.0000 −0.430775
\(777\) −6.00000 −0.215249
\(778\) 18.0000 0.645331
\(779\) −12.0000 −0.429945
\(780\) 8.00000 0.286446
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) 4.00000 0.142766
\(786\) 4.00000 0.142675
\(787\) 2.00000 0.0712923 0.0356462 0.999364i \(-0.488651\pi\)
0.0356462 + 0.999364i \(0.488651\pi\)
\(788\) 6.00000 0.213741
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 0 0
\(793\) −56.0000 −1.98862
\(794\) 12.0000 0.425864
\(795\) −20.0000 −0.709327
\(796\) 20.0000 0.708881
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) −6.00000 −0.212398
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −12.0000 −0.423999
\(802\) −22.0000 −0.776847
\(803\) 0 0
\(804\) 12.0000 0.423207
\(805\) −2.00000 −0.0704907
\(806\) 40.0000 1.40894
\(807\) 24.0000 0.844840
\(808\) 0 0
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 2.00000 0.0702728
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 6.00000 0.210559
\(813\) 2.00000 0.0701431
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) 0 0
\(817\) 72.0000 2.51896
\(818\) 22.0000 0.769212
\(819\) −4.00000 −0.139771
\(820\) 4.00000 0.139686
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −18.0000 −0.627822
\(823\) 36.0000 1.25488 0.627441 0.778664i \(-0.284103\pi\)
0.627441 + 0.778664i \(0.284103\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −12.0000 −0.416777 −0.208389 0.978046i \(-0.566822\pi\)
−0.208389 + 0.978046i \(0.566822\pi\)
\(830\) 4.00000 0.138842
\(831\) 10.0000 0.346896
\(832\) 4.00000 0.138675
\(833\) 0 0
\(834\) 0 0
\(835\) −28.0000 −0.968980
\(836\) 0 0
\(837\) 10.0000 0.345651
\(838\) 14.0000 0.483622
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 2.00000 0.0690066
\(841\) 7.00000 0.241379
\(842\) 26.0000 0.896019
\(843\) 2.00000 0.0688837
\(844\) −28.0000 −0.963800
\(845\) −6.00000 −0.206406
\(846\) −10.0000 −0.343807
\(847\) 11.0000 0.377964
\(848\) −10.0000 −0.343401
\(849\) 18.0000 0.617758
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 12.0000 0.411113
\(853\) −12.0000 −0.410872 −0.205436 0.978671i \(-0.565861\pi\)
−0.205436 + 0.978671i \(0.565861\pi\)
\(854\) −14.0000 −0.479070
\(855\) −12.0000 −0.410391
\(856\) −8.00000 −0.273434
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) 56.0000 1.91070 0.955348 0.295484i \(-0.0954809\pi\)
0.955348 + 0.295484i \(0.0954809\pi\)
\(860\) −24.0000 −0.818393
\(861\) −2.00000 −0.0681598
\(862\) 24.0000 0.817443
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 1.00000 0.0340207
\(865\) 48.0000 1.63205
\(866\) −4.00000 −0.135926
\(867\) 17.0000 0.577350
\(868\) 10.0000 0.339422
\(869\) 0 0
\(870\) 12.0000 0.406838
\(871\) −48.0000 −1.62642
\(872\) 10.0000 0.338643
\(873\) 12.0000 0.406138
\(874\) 6.00000 0.202953
\(875\) −12.0000 −0.405674
\(876\) −14.0000 −0.473016
\(877\) 10.0000 0.337676 0.168838 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) 22.0000 0.742464
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) −32.0000 −1.07811 −0.539054 0.842271i \(-0.681218\pi\)
−0.539054 + 0.842271i \(0.681218\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −12.0000 −0.403832 −0.201916 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) 0 0
\(885\) −24.0000 −0.806751
\(886\) −36.0000 −1.20944
\(887\) −10.0000 −0.335767 −0.167884 0.985807i \(-0.553693\pi\)
−0.167884 + 0.985807i \(0.553693\pi\)
\(888\) −6.00000 −0.201347
\(889\) 12.0000 0.402467
\(890\) −24.0000 −0.804482
\(891\) 0 0
\(892\) 18.0000 0.602685
\(893\) 60.0000 2.00782
\(894\) 18.0000 0.602010
\(895\) 8.00000 0.267411
\(896\) 1.00000 0.0334077
\(897\) 4.00000 0.133556
\(898\) −14.0000 −0.467186
\(899\) 60.0000 2.00111
\(900\) −1.00000 −0.0333333
\(901\) 0 0
\(902\) 0 0
\(903\) 12.0000 0.399335
\(904\) 14.0000 0.465633
\(905\) −4.00000 −0.132964
\(906\) 12.0000 0.398673
\(907\) −32.0000 −1.06254 −0.531271 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) 14.0000 0.464606
\(909\) 0 0
\(910\) −8.00000 −0.265197
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) −6.00000 −0.198680
\(913\) 0 0
\(914\) −26.0000 −0.860004
\(915\) −28.0000 −0.925651
\(916\) −6.00000 −0.198246
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) −2.00000 −0.0659380
\(921\) −16.0000 −0.527218
\(922\) 0 0
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 16.0000 0.525793
\(927\) −4.00000 −0.131377
\(928\) 6.00000 0.196960
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 20.0000 0.655826
\(931\) 6.00000 0.196642
\(932\) 22.0000 0.720634
\(933\) −10.0000 −0.327385
\(934\) −6.00000 −0.196326
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) 56.0000 1.82944 0.914720 0.404088i \(-0.132411\pi\)
0.914720 + 0.404088i \(0.132411\pi\)
\(938\) −12.0000 −0.391814
\(939\) 8.00000 0.261070
\(940\) −20.0000 −0.652328
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 2.00000 0.0651290
\(944\) −12.0000 −0.390567
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) 20.0000 0.649913 0.324956 0.945729i \(-0.394650\pi\)
0.324956 + 0.945729i \(0.394650\pi\)
\(948\) 0 0
\(949\) 56.0000 1.81784
\(950\) 6.00000 0.194666
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) −36.0000 −1.16311
\(959\) 18.0000 0.581250
\(960\) 2.00000 0.0645497
\(961\) 69.0000 2.22581
\(962\) 24.0000 0.773791
\(963\) 8.00000 0.257796
\(964\) 24.0000 0.772988
\(965\) 20.0000 0.643823
\(966\) 1.00000 0.0321745
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) 24.0000 0.770594
\(971\) −22.0000 −0.706014 −0.353007 0.935621i \(-0.614841\pi\)
−0.353007 + 0.935621i \(0.614841\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 4.00000 0.128103
\(976\) −14.0000 −0.448129
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) −12.0000 −0.383718
\(979\) 0 0
\(980\) −2.00000 −0.0638877
\(981\) −10.0000 −0.319275
\(982\) 20.0000 0.638226
\(983\) −32.0000 −1.02064 −0.510321 0.859984i \(-0.670473\pi\)
−0.510321 + 0.859984i \(0.670473\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −12.0000 −0.382352
\(986\) 0 0
\(987\) 10.0000 0.318304
\(988\) 24.0000 0.763542
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 10.0000 0.317500
\(993\) 20.0000 0.634681
\(994\) −12.0000 −0.380617
\(995\) −40.0000 −1.26809
\(996\) −2.00000 −0.0633724
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) 44.0000 1.39280
\(999\) 6.00000 0.189832
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.b.1.1 1
3.2 odd 2 2898.2.a.r.1.1 1
4.3 odd 2 7728.2.a.p.1.1 1
7.6 odd 2 6762.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.b.1.1 1 1.1 even 1 trivial
2898.2.a.r.1.1 1 3.2 odd 2
6762.2.a.r.1.1 1 7.6 odd 2
7728.2.a.p.1.1 1 4.3 odd 2