Properties

Label 966.2.a.k.1.1
Level $966$
Weight $2$
Character 966.1
Self dual yes
Analytic conductor $7.714$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -3.00000 q^{13} -1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +3.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -3.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +3.00000 q^{29} +3.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -9.00000 q^{37} -3.00000 q^{39} +3.00000 q^{40} +9.00000 q^{41} -1.00000 q^{42} -3.00000 q^{43} +4.00000 q^{44} +3.00000 q^{45} +1.00000 q^{46} -7.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} -4.00000 q^{51} -3.00000 q^{52} -4.00000 q^{53} +1.00000 q^{54} +12.0000 q^{55} -1.00000 q^{56} +3.00000 q^{58} +6.00000 q^{59} +3.00000 q^{60} +10.0000 q^{61} -6.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -9.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} -4.00000 q^{68} +1.00000 q^{69} -3.00000 q^{70} -6.00000 q^{71} +1.00000 q^{72} -8.00000 q^{73} -9.00000 q^{74} +4.00000 q^{75} -4.00000 q^{77} -3.00000 q^{78} +8.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +9.00000 q^{82} +4.00000 q^{83} -1.00000 q^{84} -12.0000 q^{85} -3.00000 q^{86} +3.00000 q^{87} +4.00000 q^{88} -14.0000 q^{89} +3.00000 q^{90} +3.00000 q^{91} +1.00000 q^{92} -6.00000 q^{93} -7.00000 q^{94} +1.00000 q^{96} -7.00000 q^{97} +1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) −1.00000 −0.267261
\(15\) 3.00000 0.774597
\(16\) 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
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 3.00000 0.670820
\(21\) −1.00000 −0.218218
\(22\) 4.00000 0.852803
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 4.00000 0.800000
\(26\) −3.00000 −0.588348
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 3.00000 0.547723
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) −4.00000 −0.685994
\(35\) −3.00000 −0.507093
\(36\) 1.00000 0.166667
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 3.00000 0.474342
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) −1.00000 −0.154303
\(43\) −3.00000 −0.457496 −0.228748 0.973486i \(-0.573463\pi\)
−0.228748 + 0.973486i \(0.573463\pi\)
\(44\) 4.00000 0.603023
\(45\) 3.00000 0.447214
\(46\) 1.00000 0.147442
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) −4.00000 −0.560112
\(52\) −3.00000 −0.416025
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 1.00000 0.136083
\(55\) 12.0000 1.61808
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 3.00000 0.393919
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 3.00000 0.387298
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −6.00000 −0.762001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −9.00000 −1.11631
\(66\) 4.00000 0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −4.00000 −0.485071
\(69\) 1.00000 0.120386
\(70\) −3.00000 −0.358569
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000 0.117851
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) −9.00000 −1.04623
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) −3.00000 −0.339683
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) 9.00000 0.993884
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −1.00000 −0.109109
\(85\) −12.0000 −1.30158
\(86\) −3.00000 −0.323498
\(87\) 3.00000 0.321634
\(88\) 4.00000 0.426401
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 3.00000 0.316228
\(91\) 3.00000 0.314485
\(92\) 1.00000 0.104257
\(93\) −6.00000 −0.622171
\(94\) −7.00000 −0.721995
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 1.00000 0.101015
\(99\) 4.00000 0.402015
\(100\) 4.00000 0.400000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −4.00000 −0.396059
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) −3.00000 −0.294174
\(105\) −3.00000 −0.292770
\(106\) −4.00000 −0.388514
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 1.00000 0.0962250
\(109\) 3.00000 0.287348 0.143674 0.989625i \(-0.454108\pi\)
0.143674 + 0.989625i \(0.454108\pi\)
\(110\) 12.0000 1.14416
\(111\) −9.00000 −0.854242
\(112\) −1.00000 −0.0944911
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 3.00000 0.278543
\(117\) −3.00000 −0.277350
\(118\) 6.00000 0.552345
\(119\) 4.00000 0.366679
\(120\) 3.00000 0.273861
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 9.00000 0.811503
\(124\) −6.00000 −0.538816
\(125\) −3.00000 −0.268328
\(126\) −1.00000 −0.0890871
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 1.00000 0.0883883
\(129\) −3.00000 −0.264135
\(130\) −9.00000 −0.789352
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 3.00000 0.258199
\(136\) −4.00000 −0.342997
\(137\) −15.0000 −1.28154 −0.640768 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) 1.00000 0.0851257
\(139\) 9.00000 0.763370 0.381685 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) −3.00000 −0.253546
\(141\) −7.00000 −0.589506
\(142\) −6.00000 −0.503509
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) 9.00000 0.747409
\(146\) −8.00000 −0.662085
\(147\) 1.00000 0.0824786
\(148\) −9.00000 −0.739795
\(149\) 16.0000 1.31077 0.655386 0.755295i \(-0.272506\pi\)
0.655386 + 0.755295i \(0.272506\pi\)
\(150\) 4.00000 0.326599
\(151\) 15.0000 1.22068 0.610341 0.792139i \(-0.291032\pi\)
0.610341 + 0.792139i \(0.291032\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) −4.00000 −0.322329
\(155\) −18.0000 −1.44579
\(156\) −3.00000 −0.240192
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 8.00000 0.636446
\(159\) −4.00000 −0.317221
\(160\) 3.00000 0.237171
\(161\) −1.00000 −0.0788110
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 9.00000 0.702782
\(165\) 12.0000 0.934199
\(166\) 4.00000 0.310460
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −4.00000 −0.307692
\(170\) −12.0000 −0.920358
\(171\) 0 0
\(172\) −3.00000 −0.228748
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 3.00000 0.227429
\(175\) −4.00000 −0.302372
\(176\) 4.00000 0.301511
\(177\) 6.00000 0.450988
\(178\) −14.0000 −1.04934
\(179\) 19.0000 1.42013 0.710063 0.704138i \(-0.248666\pi\)
0.710063 + 0.704138i \(0.248666\pi\)
\(180\) 3.00000 0.223607
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 3.00000 0.222375
\(183\) 10.0000 0.739221
\(184\) 1.00000 0.0737210
\(185\) −27.0000 −1.98508
\(186\) −6.00000 −0.439941
\(187\) −16.0000 −1.17004
\(188\) −7.00000 −0.510527
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) −17.0000 −1.22369 −0.611843 0.790979i \(-0.709572\pi\)
−0.611843 + 0.790979i \(0.709572\pi\)
\(194\) −7.00000 −0.502571
\(195\) −9.00000 −0.644503
\(196\) 1.00000 0.0714286
\(197\) −23.0000 −1.63868 −0.819341 0.573306i \(-0.805660\pi\)
−0.819341 + 0.573306i \(0.805660\pi\)
\(198\) 4.00000 0.284268
\(199\) −5.00000 −0.354441 −0.177220 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) 4.00000 0.282843
\(201\) 4.00000 0.282138
\(202\) −14.0000 −0.985037
\(203\) −3.00000 −0.210559
\(204\) −4.00000 −0.280056
\(205\) 27.0000 1.88576
\(206\) 5.00000 0.348367
\(207\) 1.00000 0.0695048
\(208\) −3.00000 −0.208013
\(209\) 0 0
\(210\) −3.00000 −0.207020
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −4.00000 −0.274721
\(213\) −6.00000 −0.411113
\(214\) 8.00000 0.546869
\(215\) −9.00000 −0.613795
\(216\) 1.00000 0.0680414
\(217\) 6.00000 0.407307
\(218\) 3.00000 0.203186
\(219\) −8.00000 −0.540590
\(220\) 12.0000 0.809040
\(221\) 12.0000 0.807207
\(222\) −9.00000 −0.604040
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 4.00000 0.266667
\(226\) 9.00000 0.598671
\(227\) 11.0000 0.730096 0.365048 0.930989i \(-0.381053\pi\)
0.365048 + 0.930989i \(0.381053\pi\)
\(228\) 0 0
\(229\) 12.0000 0.792982 0.396491 0.918039i \(-0.370228\pi\)
0.396491 + 0.918039i \(0.370228\pi\)
\(230\) 3.00000 0.197814
\(231\) −4.00000 −0.263181
\(232\) 3.00000 0.196960
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) −3.00000 −0.196116
\(235\) −21.0000 −1.36989
\(236\) 6.00000 0.390567
\(237\) 8.00000 0.519656
\(238\) 4.00000 0.259281
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 3.00000 0.193649
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) 3.00000 0.191663
\(246\) 9.00000 0.573819
\(247\) 0 0
\(248\) −6.00000 −0.381000
\(249\) 4.00000 0.253490
\(250\) −3.00000 −0.189737
\(251\) 19.0000 1.19927 0.599635 0.800274i \(-0.295313\pi\)
0.599635 + 0.800274i \(0.295313\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 4.00000 0.251478
\(254\) 7.00000 0.439219
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) −26.0000 −1.62184 −0.810918 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) −3.00000 −0.186772
\(259\) 9.00000 0.559233
\(260\) −9.00000 −0.558156
\(261\) 3.00000 0.185695
\(262\) −6.00000 −0.370681
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 4.00000 0.246183
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 4.00000 0.244339
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 3.00000 0.182574
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −4.00000 −0.242536
\(273\) 3.00000 0.181568
\(274\) −15.0000 −0.906183
\(275\) 16.0000 0.964836
\(276\) 1.00000 0.0601929
\(277\) 20.0000 1.20168 0.600842 0.799368i \(-0.294832\pi\)
0.600842 + 0.799368i \(0.294832\pi\)
\(278\) 9.00000 0.539784
\(279\) −6.00000 −0.359211
\(280\) −3.00000 −0.179284
\(281\) 23.0000 1.37206 0.686032 0.727571i \(-0.259351\pi\)
0.686032 + 0.727571i \(0.259351\pi\)
\(282\) −7.00000 −0.416844
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) −9.00000 −0.531253
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 9.00000 0.528498
\(291\) −7.00000 −0.410347
\(292\) −8.00000 −0.468165
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 1.00000 0.0583212
\(295\) 18.0000 1.04800
\(296\) −9.00000 −0.523114
\(297\) 4.00000 0.232104
\(298\) 16.0000 0.926855
\(299\) −3.00000 −0.173494
\(300\) 4.00000 0.230940
\(301\) 3.00000 0.172917
\(302\) 15.0000 0.863153
\(303\) −14.0000 −0.804279
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) −4.00000 −0.228665
\(307\) 15.0000 0.856095 0.428048 0.903756i \(-0.359202\pi\)
0.428048 + 0.903756i \(0.359202\pi\)
\(308\) −4.00000 −0.227921
\(309\) 5.00000 0.284440
\(310\) −18.0000 −1.02233
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) −3.00000 −0.169842
\(313\) 34.0000 1.92179 0.960897 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) −8.00000 −0.451466
\(315\) −3.00000 −0.169031
\(316\) 8.00000 0.450035
\(317\) 5.00000 0.280828 0.140414 0.990093i \(-0.455157\pi\)
0.140414 + 0.990093i \(0.455157\pi\)
\(318\) −4.00000 −0.224309
\(319\) 12.0000 0.671871
\(320\) 3.00000 0.167705
\(321\) 8.00000 0.446516
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −12.0000 −0.665640
\(326\) 16.0000 0.886158
\(327\) 3.00000 0.165900
\(328\) 9.00000 0.496942
\(329\) 7.00000 0.385922
\(330\) 12.0000 0.660578
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 4.00000 0.219529
\(333\) −9.00000 −0.493197
\(334\) −20.0000 −1.09435
\(335\) 12.0000 0.655630
\(336\) −1.00000 −0.0545545
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) −4.00000 −0.217571
\(339\) 9.00000 0.488813
\(340\) −12.0000 −0.650791
\(341\) −24.0000 −1.29967
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −3.00000 −0.161749
\(345\) 3.00000 0.161515
\(346\) −6.00000 −0.322562
\(347\) 23.0000 1.23470 0.617352 0.786687i \(-0.288205\pi\)
0.617352 + 0.786687i \(0.288205\pi\)
\(348\) 3.00000 0.160817
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) −4.00000 −0.213809
\(351\) −3.00000 −0.160128
\(352\) 4.00000 0.213201
\(353\) −29.0000 −1.54351 −0.771757 0.635917i \(-0.780622\pi\)
−0.771757 + 0.635917i \(0.780622\pi\)
\(354\) 6.00000 0.318896
\(355\) −18.0000 −0.955341
\(356\) −14.0000 −0.741999
\(357\) 4.00000 0.211702
\(358\) 19.0000 1.00418
\(359\) −1.00000 −0.0527780 −0.0263890 0.999652i \(-0.508401\pi\)
−0.0263890 + 0.999652i \(0.508401\pi\)
\(360\) 3.00000 0.158114
\(361\) −19.0000 −1.00000
\(362\) −2.00000 −0.105118
\(363\) 5.00000 0.262432
\(364\) 3.00000 0.157243
\(365\) −24.0000 −1.25622
\(366\) 10.0000 0.522708
\(367\) 31.0000 1.61819 0.809093 0.587680i \(-0.199959\pi\)
0.809093 + 0.587680i \(0.199959\pi\)
\(368\) 1.00000 0.0521286
\(369\) 9.00000 0.468521
\(370\) −27.0000 −1.40366
\(371\) 4.00000 0.207670
\(372\) −6.00000 −0.311086
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −16.0000 −0.827340
\(375\) −3.00000 −0.154919
\(376\) −7.00000 −0.360997
\(377\) −9.00000 −0.463524
\(378\) −1.00000 −0.0514344
\(379\) −25.0000 −1.28416 −0.642082 0.766636i \(-0.721929\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) 0 0
\(381\) 7.00000 0.358621
\(382\) −12.0000 −0.613973
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) −12.0000 −0.611577
\(386\) −17.0000 −0.865277
\(387\) −3.00000 −0.152499
\(388\) −7.00000 −0.355371
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) −9.00000 −0.455733
\(391\) −4.00000 −0.202289
\(392\) 1.00000 0.0505076
\(393\) −6.00000 −0.302660
\(394\) −23.0000 −1.15872
\(395\) 24.0000 1.20757
\(396\) 4.00000 0.201008
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) −5.00000 −0.250627
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 38.0000 1.89763 0.948815 0.315833i \(-0.102284\pi\)
0.948815 + 0.315833i \(0.102284\pi\)
\(402\) 4.00000 0.199502
\(403\) 18.0000 0.896644
\(404\) −14.0000 −0.696526
\(405\) 3.00000 0.149071
\(406\) −3.00000 −0.148888
\(407\) −36.0000 −1.78445
\(408\) −4.00000 −0.198030
\(409\) 16.0000 0.791149 0.395575 0.918434i \(-0.370545\pi\)
0.395575 + 0.918434i \(0.370545\pi\)
\(410\) 27.0000 1.33343
\(411\) −15.0000 −0.739895
\(412\) 5.00000 0.246332
\(413\) −6.00000 −0.295241
\(414\) 1.00000 0.0491473
\(415\) 12.0000 0.589057
\(416\) −3.00000 −0.147087
\(417\) 9.00000 0.440732
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) −3.00000 −0.146385
\(421\) −21.0000 −1.02348 −0.511739 0.859141i \(-0.670998\pi\)
−0.511739 + 0.859141i \(0.670998\pi\)
\(422\) −12.0000 −0.584151
\(423\) −7.00000 −0.340352
\(424\) −4.00000 −0.194257
\(425\) −16.0000 −0.776114
\(426\) −6.00000 −0.290701
\(427\) −10.0000 −0.483934
\(428\) 8.00000 0.386695
\(429\) −12.0000 −0.579365
\(430\) −9.00000 −0.434019
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 1.00000 0.0481125
\(433\) 23.0000 1.10531 0.552655 0.833410i \(-0.313615\pi\)
0.552655 + 0.833410i \(0.313615\pi\)
\(434\) 6.00000 0.288009
\(435\) 9.00000 0.431517
\(436\) 3.00000 0.143674
\(437\) 0 0
\(438\) −8.00000 −0.382255
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 12.0000 0.572078
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) 25.0000 1.18779 0.593893 0.804544i \(-0.297590\pi\)
0.593893 + 0.804544i \(0.297590\pi\)
\(444\) −9.00000 −0.427121
\(445\) −42.0000 −1.99099
\(446\) −14.0000 −0.662919
\(447\) 16.0000 0.756774
\(448\) −1.00000 −0.0472456
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 4.00000 0.188562
\(451\) 36.0000 1.69517
\(452\) 9.00000 0.423324
\(453\) 15.0000 0.704761
\(454\) 11.0000 0.516256
\(455\) 9.00000 0.421927
\(456\) 0 0
\(457\) 24.0000 1.12267 0.561336 0.827588i \(-0.310287\pi\)
0.561336 + 0.827588i \(0.310287\pi\)
\(458\) 12.0000 0.560723
\(459\) −4.00000 −0.186704
\(460\) 3.00000 0.139876
\(461\) −22.0000 −1.02464 −0.512321 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) −4.00000 −0.186097
\(463\) 13.0000 0.604161 0.302081 0.953282i \(-0.402319\pi\)
0.302081 + 0.953282i \(0.402319\pi\)
\(464\) 3.00000 0.139272
\(465\) −18.0000 −0.834730
\(466\) −10.0000 −0.463241
\(467\) 13.0000 0.601568 0.300784 0.953692i \(-0.402752\pi\)
0.300784 + 0.953692i \(0.402752\pi\)
\(468\) −3.00000 −0.138675
\(469\) −4.00000 −0.184703
\(470\) −21.0000 −0.968658
\(471\) −8.00000 −0.368621
\(472\) 6.00000 0.276172
\(473\) −12.0000 −0.551761
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) −4.00000 −0.183147
\(478\) −12.0000 −0.548867
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 3.00000 0.136931
\(481\) 27.0000 1.23109
\(482\) −5.00000 −0.227744
\(483\) −1.00000 −0.0455016
\(484\) 5.00000 0.227273
\(485\) −21.0000 −0.953561
\(486\) 1.00000 0.0453609
\(487\) 13.0000 0.589086 0.294543 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(488\) 10.0000 0.452679
\(489\) 16.0000 0.723545
\(490\) 3.00000 0.135526
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 9.00000 0.405751
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) −6.00000 −0.269408
\(497\) 6.00000 0.269137
\(498\) 4.00000 0.179244
\(499\) 34.0000 1.52205 0.761025 0.648723i \(-0.224697\pi\)
0.761025 + 0.648723i \(0.224697\pi\)
\(500\) −3.00000 −0.134164
\(501\) −20.0000 −0.893534
\(502\) 19.0000 0.848012
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −42.0000 −1.86898
\(506\) 4.00000 0.177822
\(507\) −4.00000 −0.177646
\(508\) 7.00000 0.310575
\(509\) 44.0000 1.95027 0.975133 0.221621i \(-0.0711348\pi\)
0.975133 + 0.221621i \(0.0711348\pi\)
\(510\) −12.0000 −0.531369
\(511\) 8.00000 0.353899
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −26.0000 −1.14681
\(515\) 15.0000 0.660979
\(516\) −3.00000 −0.132068
\(517\) −28.0000 −1.23144
\(518\) 9.00000 0.395437
\(519\) −6.00000 −0.263371
\(520\) −9.00000 −0.394676
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 3.00000 0.131306
\(523\) 42.0000 1.83653 0.918266 0.395964i \(-0.129590\pi\)
0.918266 + 0.395964i \(0.129590\pi\)
\(524\) −6.00000 −0.262111
\(525\) −4.00000 −0.174574
\(526\) 21.0000 0.915644
\(527\) 24.0000 1.04546
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) −12.0000 −0.521247
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) −27.0000 −1.16950
\(534\) −14.0000 −0.605839
\(535\) 24.0000 1.03761
\(536\) 4.00000 0.172774
\(537\) 19.0000 0.819911
\(538\) −6.00000 −0.258678
\(539\) 4.00000 0.172292
\(540\) 3.00000 0.129099
\(541\) −28.0000 −1.20381 −0.601907 0.798566i \(-0.705592\pi\)
−0.601907 + 0.798566i \(0.705592\pi\)
\(542\) 20.0000 0.859074
\(543\) −2.00000 −0.0858282
\(544\) −4.00000 −0.171499
\(545\) 9.00000 0.385518
\(546\) 3.00000 0.128388
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) −15.0000 −0.640768
\(549\) 10.0000 0.426790
\(550\) 16.0000 0.682242
\(551\) 0 0
\(552\) 1.00000 0.0425628
\(553\) −8.00000 −0.340195
\(554\) 20.0000 0.849719
\(555\) −27.0000 −1.14609
\(556\) 9.00000 0.381685
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) −6.00000 −0.254000
\(559\) 9.00000 0.380659
\(560\) −3.00000 −0.126773
\(561\) −16.0000 −0.675521
\(562\) 23.0000 0.970196
\(563\) 15.0000 0.632175 0.316087 0.948730i \(-0.397631\pi\)
0.316087 + 0.948730i \(0.397631\pi\)
\(564\) −7.00000 −0.294753
\(565\) 27.0000 1.13590
\(566\) −6.00000 −0.252199
\(567\) −1.00000 −0.0419961
\(568\) −6.00000 −0.251754
\(569\) −13.0000 −0.544988 −0.272494 0.962157i \(-0.587849\pi\)
−0.272494 + 0.962157i \(0.587849\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −12.0000 −0.501745
\(573\) −12.0000 −0.501307
\(574\) −9.00000 −0.375653
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −17.0000 −0.706496
\(580\) 9.00000 0.373705
\(581\) −4.00000 −0.165948
\(582\) −7.00000 −0.290159
\(583\) −16.0000 −0.662652
\(584\) −8.00000 −0.331042
\(585\) −9.00000 −0.372104
\(586\) 6.00000 0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) 18.0000 0.741048
\(591\) −23.0000 −0.946094
\(592\) −9.00000 −0.369898
\(593\) 29.0000 1.19089 0.595444 0.803397i \(-0.296976\pi\)
0.595444 + 0.803397i \(0.296976\pi\)
\(594\) 4.00000 0.164122
\(595\) 12.0000 0.491952
\(596\) 16.0000 0.655386
\(597\) −5.00000 −0.204636
\(598\) −3.00000 −0.122679
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 4.00000 0.163299
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 3.00000 0.122271
\(603\) 4.00000 0.162893
\(604\) 15.0000 0.610341
\(605\) 15.0000 0.609837
\(606\) −14.0000 −0.568711
\(607\) 30.0000 1.21766 0.608831 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(608\) 0 0
\(609\) −3.00000 −0.121566
\(610\) 30.0000 1.21466
\(611\) 21.0000 0.849569
\(612\) −4.00000 −0.161690
\(613\) 25.0000 1.00974 0.504870 0.863195i \(-0.331540\pi\)
0.504870 + 0.863195i \(0.331540\pi\)
\(614\) 15.0000 0.605351
\(615\) 27.0000 1.08875
\(616\) −4.00000 −0.161165
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 5.00000 0.201129
\(619\) 2.00000 0.0803868 0.0401934 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) −18.0000 −0.722897
\(621\) 1.00000 0.0401286
\(622\) −28.0000 −1.12270
\(623\) 14.0000 0.560898
\(624\) −3.00000 −0.120096
\(625\) −29.0000 −1.16000
\(626\) 34.0000 1.35891
\(627\) 0 0
\(628\) −8.00000 −0.319235
\(629\) 36.0000 1.43541
\(630\) −3.00000 −0.119523
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 8.00000 0.318223
\(633\) −12.0000 −0.476957
\(634\) 5.00000 0.198575
\(635\) 21.0000 0.833360
\(636\) −4.00000 −0.158610
\(637\) −3.00000 −0.118864
\(638\) 12.0000 0.475085
\(639\) −6.00000 −0.237356
\(640\) 3.00000 0.118585
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 8.00000 0.315735
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −9.00000 −0.354375
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) 24.0000 0.942082
\(650\) −12.0000 −0.470679
\(651\) 6.00000 0.235159
\(652\) 16.0000 0.626608
\(653\) 31.0000 1.21312 0.606562 0.795036i \(-0.292548\pi\)
0.606562 + 0.795036i \(0.292548\pi\)
\(654\) 3.00000 0.117309
\(655\) −18.0000 −0.703318
\(656\) 9.00000 0.351391
\(657\) −8.00000 −0.312110
\(658\) 7.00000 0.272888
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 12.0000 0.467099
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 8.00000 0.310929
\(663\) 12.0000 0.466041
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −9.00000 −0.348743
\(667\) 3.00000 0.116160
\(668\) −20.0000 −0.773823
\(669\) −14.0000 −0.541271
\(670\) 12.0000 0.463600
\(671\) 40.0000 1.54418
\(672\) −1.00000 −0.0385758
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) −12.0000 −0.462223
\(675\) 4.00000 0.153960
\(676\) −4.00000 −0.153846
\(677\) 46.0000 1.76792 0.883962 0.467559i \(-0.154866\pi\)
0.883962 + 0.467559i \(0.154866\pi\)
\(678\) 9.00000 0.345643
\(679\) 7.00000 0.268635
\(680\) −12.0000 −0.460179
\(681\) 11.0000 0.421521
\(682\) −24.0000 −0.919007
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) −45.0000 −1.71936
\(686\) −1.00000 −0.0381802
\(687\) 12.0000 0.457829
\(688\) −3.00000 −0.114374
\(689\) 12.0000 0.457164
\(690\) 3.00000 0.114208
\(691\) −5.00000 −0.190209 −0.0951045 0.995467i \(-0.530319\pi\)
−0.0951045 + 0.995467i \(0.530319\pi\)
\(692\) −6.00000 −0.228086
\(693\) −4.00000 −0.151947
\(694\) 23.0000 0.873068
\(695\) 27.0000 1.02417
\(696\) 3.00000 0.113715
\(697\) −36.0000 −1.36360
\(698\) −26.0000 −0.984115
\(699\) −10.0000 −0.378235
\(700\) −4.00000 −0.151186
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) −3.00000 −0.113228
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) −21.0000 −0.790906
\(706\) −29.0000 −1.09143
\(707\) 14.0000 0.526524
\(708\) 6.00000 0.225494
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) −18.0000 −0.675528
\(711\) 8.00000 0.300023
\(712\) −14.0000 −0.524672
\(713\) −6.00000 −0.224702
\(714\) 4.00000 0.149696
\(715\) −36.0000 −1.34632
\(716\) 19.0000 0.710063
\(717\) −12.0000 −0.448148
\(718\) −1.00000 −0.0373197
\(719\) −7.00000 −0.261056 −0.130528 0.991445i \(-0.541667\pi\)
−0.130528 + 0.991445i \(0.541667\pi\)
\(720\) 3.00000 0.111803
\(721\) −5.00000 −0.186210
\(722\) −19.0000 −0.707107
\(723\) −5.00000 −0.185952
\(724\) −2.00000 −0.0743294
\(725\) 12.0000 0.445669
\(726\) 5.00000 0.185567
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 3.00000 0.111187
\(729\) 1.00000 0.0370370
\(730\) −24.0000 −0.888280
\(731\) 12.0000 0.443836
\(732\) 10.0000 0.369611
\(733\) −16.0000 −0.590973 −0.295487 0.955347i \(-0.595482\pi\)
−0.295487 + 0.955347i \(0.595482\pi\)
\(734\) 31.0000 1.14423
\(735\) 3.00000 0.110657
\(736\) 1.00000 0.0368605
\(737\) 16.0000 0.589368
\(738\) 9.00000 0.331295
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) −27.0000 −0.992540
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) −6.00000 −0.219971
\(745\) 48.0000 1.75858
\(746\) 22.0000 0.805477
\(747\) 4.00000 0.146352
\(748\) −16.0000 −0.585018
\(749\) −8.00000 −0.292314
\(750\) −3.00000 −0.109545
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) −7.00000 −0.255264
\(753\) 19.0000 0.692398
\(754\) −9.00000 −0.327761
\(755\) 45.0000 1.63772
\(756\) −1.00000 −0.0363696
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) −25.0000 −0.908041
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 7.00000 0.253583
\(763\) −3.00000 −0.108607
\(764\) −12.0000 −0.434145
\(765\) −12.0000 −0.433861
\(766\) 0 0
\(767\) −18.0000 −0.649942
\(768\) 1.00000 0.0360844
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) −12.0000 −0.432450
\(771\) −26.0000 −0.936367
\(772\) −17.0000 −0.611843
\(773\) 19.0000 0.683383 0.341691 0.939812i \(-0.389000\pi\)
0.341691 + 0.939812i \(0.389000\pi\)
\(774\) −3.00000 −0.107833
\(775\) −24.0000 −0.862105
\(776\) −7.00000 −0.251285
\(777\) 9.00000 0.322873
\(778\) 18.0000 0.645331
\(779\) 0 0
\(780\) −9.00000 −0.322252
\(781\) −24.0000 −0.858788
\(782\) −4.00000 −0.143040
\(783\) 3.00000 0.107211
\(784\) 1.00000 0.0357143
\(785\) −24.0000 −0.856597
\(786\) −6.00000 −0.214013
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) −23.0000 −0.819341
\(789\) 21.0000 0.747620
\(790\) 24.0000 0.853882
\(791\) −9.00000 −0.320003
\(792\) 4.00000 0.142134
\(793\) −30.0000 −1.06533
\(794\) 34.0000 1.20661
\(795\) −12.0000 −0.425596
\(796\) −5.00000 −0.177220
\(797\) −49.0000 −1.73567 −0.867835 0.496853i \(-0.834489\pi\)
−0.867835 + 0.496853i \(0.834489\pi\)
\(798\) 0 0
\(799\) 28.0000 0.990569
\(800\) 4.00000 0.141421
\(801\) −14.0000 −0.494666
\(802\) 38.0000 1.34183
\(803\) −32.0000 −1.12926
\(804\) 4.00000 0.141069
\(805\) −3.00000 −0.105736
\(806\) 18.0000 0.634023
\(807\) −6.00000 −0.211210
\(808\) −14.0000 −0.492518
\(809\) −16.0000 −0.562530 −0.281265 0.959630i \(-0.590754\pi\)
−0.281265 + 0.959630i \(0.590754\pi\)
\(810\) 3.00000 0.105409
\(811\) 33.0000 1.15879 0.579393 0.815048i \(-0.303290\pi\)
0.579393 + 0.815048i \(0.303290\pi\)
\(812\) −3.00000 −0.105279
\(813\) 20.0000 0.701431
\(814\) −36.0000 −1.26180
\(815\) 48.0000 1.68137
\(816\) −4.00000 −0.140028
\(817\) 0 0
\(818\) 16.0000 0.559427
\(819\) 3.00000 0.104828
\(820\) 27.0000 0.942881
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) −15.0000 −0.523185
\(823\) 7.00000 0.244005 0.122002 0.992530i \(-0.461068\pi\)
0.122002 + 0.992530i \(0.461068\pi\)
\(824\) 5.00000 0.174183
\(825\) 16.0000 0.557048
\(826\) −6.00000 −0.208767
\(827\) −30.0000 −1.04320 −0.521601 0.853189i \(-0.674665\pi\)
−0.521601 + 0.853189i \(0.674665\pi\)
\(828\) 1.00000 0.0347524
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 12.0000 0.416526
\(831\) 20.0000 0.693792
\(832\) −3.00000 −0.104006
\(833\) −4.00000 −0.138592
\(834\) 9.00000 0.311645
\(835\) −60.0000 −2.07639
\(836\) 0 0
\(837\) −6.00000 −0.207390
\(838\) −4.00000 −0.138178
\(839\) 6.00000 0.207143 0.103572 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) −3.00000 −0.103510
\(841\) −20.0000 −0.689655
\(842\) −21.0000 −0.723708
\(843\) 23.0000 0.792162
\(844\) −12.0000 −0.413057
\(845\) −12.0000 −0.412813
\(846\) −7.00000 −0.240665
\(847\) −5.00000 −0.171802
\(848\) −4.00000 −0.137361
\(849\) −6.00000 −0.205919
\(850\) −16.0000 −0.548795
\(851\) −9.00000 −0.308516
\(852\) −6.00000 −0.205557
\(853\) 7.00000 0.239675 0.119838 0.992793i \(-0.461763\pi\)
0.119838 + 0.992793i \(0.461763\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) 33.0000 1.12726 0.563629 0.826028i \(-0.309405\pi\)
0.563629 + 0.826028i \(0.309405\pi\)
\(858\) −12.0000 −0.409673
\(859\) 13.0000 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) −9.00000 −0.306897
\(861\) −9.00000 −0.306719
\(862\) 21.0000 0.715263
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 1.00000 0.0340207
\(865\) −18.0000 −0.612018
\(866\) 23.0000 0.781572
\(867\) −1.00000 −0.0339618
\(868\) 6.00000 0.203653
\(869\) 32.0000 1.08553
\(870\) 9.00000 0.305129
\(871\) −12.0000 −0.406604
\(872\) 3.00000 0.101593
\(873\) −7.00000 −0.236914
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) −8.00000 −0.270295
\(877\) −36.0000 −1.21563 −0.607817 0.794077i \(-0.707955\pi\)
−0.607817 + 0.794077i \(0.707955\pi\)
\(878\) −10.0000 −0.337484
\(879\) 6.00000 0.202375
\(880\) 12.0000 0.404520
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 1.00000 0.0336718
\(883\) 54.0000 1.81724 0.908622 0.417619i \(-0.137135\pi\)
0.908622 + 0.417619i \(0.137135\pi\)
\(884\) 12.0000 0.403604
\(885\) 18.0000 0.605063
\(886\) 25.0000 0.839891
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) −9.00000 −0.302020
\(889\) −7.00000 −0.234772
\(890\) −42.0000 −1.40784
\(891\) 4.00000 0.134005
\(892\) −14.0000 −0.468755
\(893\) 0 0
\(894\) 16.0000 0.535120
\(895\) 57.0000 1.90530
\(896\) −1.00000 −0.0334077
\(897\) −3.00000 −0.100167
\(898\) 30.0000 1.00111
\(899\) −18.0000 −0.600334
\(900\) 4.00000 0.133333
\(901\) 16.0000 0.533037
\(902\) 36.0000 1.19867
\(903\) 3.00000 0.0998337
\(904\) 9.00000 0.299336
\(905\) −6.00000 −0.199447
\(906\) 15.0000 0.498342
\(907\) −53.0000 −1.75984 −0.879918 0.475125i \(-0.842403\pi\)
−0.879918 + 0.475125i \(0.842403\pi\)
\(908\) 11.0000 0.365048
\(909\) −14.0000 −0.464351
\(910\) 9.00000 0.298347
\(911\) −49.0000 −1.62344 −0.811721 0.584045i \(-0.801469\pi\)
−0.811721 + 0.584045i \(0.801469\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 24.0000 0.793849
\(915\) 30.0000 0.991769
\(916\) 12.0000 0.396491
\(917\) 6.00000 0.198137
\(918\) −4.00000 −0.132020
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 3.00000 0.0989071
\(921\) 15.0000 0.494267
\(922\) −22.0000 −0.724531
\(923\) 18.0000 0.592477
\(924\) −4.00000 −0.131590
\(925\) −36.0000 −1.18367
\(926\) 13.0000 0.427207
\(927\) 5.00000 0.164222
\(928\) 3.00000 0.0984798
\(929\) 9.00000 0.295280 0.147640 0.989041i \(-0.452832\pi\)
0.147640 + 0.989041i \(0.452832\pi\)
\(930\) −18.0000 −0.590243
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) −28.0000 −0.916679
\(934\) 13.0000 0.425373
\(935\) −48.0000 −1.56977
\(936\) −3.00000 −0.0980581
\(937\) 25.0000 0.816714 0.408357 0.912822i \(-0.366102\pi\)
0.408357 + 0.912822i \(0.366102\pi\)
\(938\) −4.00000 −0.130605
\(939\) 34.0000 1.10955
\(940\) −21.0000 −0.684944
\(941\) 3.00000 0.0977972 0.0488986 0.998804i \(-0.484429\pi\)
0.0488986 + 0.998804i \(0.484429\pi\)
\(942\) −8.00000 −0.260654
\(943\) 9.00000 0.293080
\(944\) 6.00000 0.195283
\(945\) −3.00000 −0.0975900
\(946\) −12.0000 −0.390154
\(947\) −31.0000 −1.00736 −0.503682 0.863889i \(-0.668022\pi\)
−0.503682 + 0.863889i \(0.668022\pi\)
\(948\) 8.00000 0.259828
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 5.00000 0.162136
\(952\) 4.00000 0.129641
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) −4.00000 −0.129505
\(955\) −36.0000 −1.16493
\(956\) −12.0000 −0.388108
\(957\) 12.0000 0.387905
\(958\) 30.0000 0.969256
\(959\) 15.0000 0.484375
\(960\) 3.00000 0.0968246
\(961\) 5.00000 0.161290
\(962\) 27.0000 0.870515
\(963\) 8.00000 0.257796
\(964\) −5.00000 −0.161039
\(965\) −51.0000 −1.64175
\(966\) −1.00000 −0.0321745
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) −21.0000 −0.674269
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 1.00000 0.0320750
\(973\) −9.00000 −0.288527
\(974\) 13.0000 0.416547
\(975\) −12.0000 −0.384308
\(976\) 10.0000 0.320092
\(977\) 13.0000 0.415907 0.207953 0.978139i \(-0.433320\pi\)
0.207953 + 0.978139i \(0.433320\pi\)
\(978\) 16.0000 0.511624
\(979\) −56.0000 −1.78977
\(980\) 3.00000 0.0958315
\(981\) 3.00000 0.0957826
\(982\) −12.0000 −0.382935
\(983\) −54.0000 −1.72233 −0.861166 0.508323i \(-0.830265\pi\)
−0.861166 + 0.508323i \(0.830265\pi\)
\(984\) 9.00000 0.286910
\(985\) −69.0000 −2.19852
\(986\) −12.0000 −0.382158
\(987\) 7.00000 0.222812
\(988\) 0 0
\(989\) −3.00000 −0.0953945
\(990\) 12.0000 0.381385
\(991\) −36.0000 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) −6.00000 −0.190500
\(993\) 8.00000 0.253872
\(994\) 6.00000 0.190308
\(995\) −15.0000 −0.475532
\(996\) 4.00000 0.126745
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) 34.0000 1.07625
\(999\) −9.00000 −0.284747
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.k.1.1 1
3.2 odd 2 2898.2.a.a.1.1 1
4.3 odd 2 7728.2.a.j.1.1 1
7.6 odd 2 6762.2.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.k.1.1 1 1.1 even 1 trivial
2898.2.a.a.1.1 1 3.2 odd 2
6762.2.a.y.1.1 1 7.6 odd 2
7728.2.a.j.1.1 1 4.3 odd 2