Properties

Label 6762.2.a.t.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -5.00000 q^{11} +1.00000 q^{12} +3.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +3.00000 q^{20} +5.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{27} -3.00000 q^{29} -3.00000 q^{30} -9.00000 q^{31} -1.00000 q^{32} -5.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -12.0000 q^{37} -3.00000 q^{40} +12.0000 q^{41} +6.00000 q^{43} -5.00000 q^{44} +3.00000 q^{45} -1.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} -4.00000 q^{51} +5.00000 q^{53} -1.00000 q^{54} -15.0000 q^{55} +3.00000 q^{58} -9.00000 q^{59} +3.00000 q^{60} -2.00000 q^{61} +9.00000 q^{62} +1.00000 q^{64} +5.00000 q^{66} -2.00000 q^{67} -4.00000 q^{68} +1.00000 q^{69} +6.00000 q^{71} -1.00000 q^{72} -14.0000 q^{73} +12.0000 q^{74} +4.00000 q^{75} +11.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} -17.0000 q^{83} -12.0000 q^{85} -6.00000 q^{86} -3.00000 q^{87} +5.00000 q^{88} +10.0000 q^{89} -3.00000 q^{90} +1.00000 q^{92} -9.00000 q^{93} -2.00000 q^{94} -1.00000 q^{96} -13.0000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 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
\(21\) 0 0
\(22\) 5.00000 1.06600
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) −3.00000 −0.547723
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.00000 −0.870388
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −12.0000 −1.97279 −0.986394 0.164399i \(-0.947432\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −5.00000 −0.753778
\(45\) 3.00000 0.447214
\(46\) −1.00000 −0.147442
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 5.00000 0.686803 0.343401 0.939189i \(-0.388421\pi\)
0.343401 + 0.939189i \(0.388421\pi\)
\(54\) −1.00000 −0.136083
\(55\) −15.0000 −2.02260
\(56\) 0 0
\(57\) 0 0
\(58\) 3.00000 0.393919
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 3.00000 0.387298
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 9.00000 1.14300
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.00000 0.615457
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −4.00000 −0.485071
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 12.0000 1.39497
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) −12.0000 −1.32518
\(83\) −17.0000 −1.86599 −0.932996 0.359886i \(-0.882816\pi\)
−0.932996 + 0.359886i \(0.882816\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) −6.00000 −0.646997
\(87\) −3.00000 −0.321634
\(88\) 5.00000 0.533002
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −9.00000 −0.933257
\(94\) −2.00000 −0.206284
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 4.00000 0.400000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 4.00000 0.396059
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −5.00000 −0.485643
\(107\) −19.0000 −1.83680 −0.918400 0.395654i \(-0.870518\pi\)
−0.918400 + 0.395654i \(0.870518\pi\)
\(108\) 1.00000 0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 15.0000 1.43019
\(111\) −12.0000 −1.13899
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 9.00000 0.828517
\(119\) 0 0
\(120\) −3.00000 −0.273861
\(121\) 14.0000 1.27273
\(122\) 2.00000 0.181071
\(123\) 12.0000 1.08200
\(124\) −9.00000 −0.808224
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −11.0000 −0.976092 −0.488046 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) −5.00000 −0.435194
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 3.00000 0.258199
\(136\) 4.00000 0.342997
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −6.00000 −0.508913 −0.254457 0.967084i \(-0.581897\pi\)
−0.254457 + 0.967084i \(0.581897\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −9.00000 −0.747409
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) −12.0000 −0.986394
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) −4.00000 −0.326599
\(151\) −9.00000 −0.732410 −0.366205 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) −27.0000 −2.16869
\(156\) 0 0
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) −11.0000 −0.875113
\(159\) 5.00000 0.396526
\(160\) −3.00000 −0.237171
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 12.0000 0.937043
\(165\) −15.0000 −1.16775
\(166\) 17.0000 1.31946
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 12.0000 0.920358
\(171\) 0 0
\(172\) 6.00000 0.457496
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 3.00000 0.227429
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) −9.00000 −0.676481
\(178\) −10.0000 −0.749532
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 3.00000 0.223607
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −1.00000 −0.0737210
\(185\) −36.0000 −2.64677
\(186\) 9.00000 0.659912
\(187\) 20.0000 1.46254
\(188\) 2.00000 0.145865
\(189\) 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
\(193\) 13.0000 0.935760 0.467880 0.883792i \(-0.345018\pi\)
0.467880 + 0.883792i \(0.345018\pi\)
\(194\) 13.0000 0.933346
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 5.00000 0.355335
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) −4.00000 −0.282843
\(201\) −2.00000 −0.141069
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 36.0000 2.51435
\(206\) −8.00000 −0.557386
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 6.00000 0.413057 0.206529 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(212\) 5.00000 0.343401
\(213\) 6.00000 0.411113
\(214\) 19.0000 1.29881
\(215\) 18.0000 1.22759
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) −14.0000 −0.946032
\(220\) −15.0000 −1.01130
\(221\) 0 0
\(222\) 12.0000 0.805387
\(223\) 13.0000 0.870544 0.435272 0.900299i \(-0.356652\pi\)
0.435272 + 0.900299i \(0.356652\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) −12.0000 −0.798228
\(227\) 5.00000 0.331862 0.165931 0.986137i \(-0.446937\pi\)
0.165931 + 0.986137i \(0.446937\pi\)
\(228\) 0 0
\(229\) −24.0000 −1.58596 −0.792982 0.609245i \(-0.791473\pi\)
−0.792982 + 0.609245i \(0.791473\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) 20.0000 1.31024 0.655122 0.755523i \(-0.272617\pi\)
0.655122 + 0.755523i \(0.272617\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) −9.00000 −0.585850
\(237\) 11.0000 0.714527
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 3.00000 0.193649
\(241\) 19.0000 1.22390 0.611949 0.790897i \(-0.290386\pi\)
0.611949 + 0.790897i \(0.290386\pi\)
\(242\) −14.0000 −0.899954
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −12.0000 −0.765092
\(247\) 0 0
\(248\) 9.00000 0.571501
\(249\) −17.0000 −1.07733
\(250\) 3.00000 0.189737
\(251\) −5.00000 −0.315597 −0.157799 0.987471i \(-0.550440\pi\)
−0.157799 + 0.987471i \(0.550440\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) 11.0000 0.690201
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) 10.0000 0.623783 0.311891 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) −6.00000 −0.373544
\(259\) 0 0
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) −3.00000 −0.185341
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 5.00000 0.307729
\(265\) 15.0000 0.921443
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) −2.00000 −0.122169
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) −3.00000 −0.182574
\(271\) −13.0000 −0.789694 −0.394847 0.918747i \(-0.629202\pi\)
−0.394847 + 0.918747i \(0.629202\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −20.0000 −1.20605
\(276\) 1.00000 0.0601929
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 6.00000 0.359856
\(279\) −9.00000 −0.538816
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) −2.00000 −0.119098
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 9.00000 0.528498
\(291\) −13.0000 −0.762073
\(292\) −14.0000 −0.819288
\(293\) 33.0000 1.92788 0.963940 0.266119i \(-0.0857413\pi\)
0.963940 + 0.266119i \(0.0857413\pi\)
\(294\) 0 0
\(295\) −27.0000 −1.57200
\(296\) 12.0000 0.697486
\(297\) −5.00000 −0.290129
\(298\) 14.0000 0.810998
\(299\) 0 0
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) 9.00000 0.517892
\(303\) −2.00000 −0.114897
\(304\) 0 0
\(305\) −6.00000 −0.343559
\(306\) 4.00000 0.228665
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 27.0000 1.53350
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) 0 0
\(313\) −11.0000 −0.621757 −0.310878 0.950450i \(-0.600623\pi\)
−0.310878 + 0.950450i \(0.600623\pi\)
\(314\) 8.00000 0.451466
\(315\) 0 0
\(316\) 11.0000 0.618798
\(317\) 5.00000 0.280828 0.140414 0.990093i \(-0.455157\pi\)
0.140414 + 0.990093i \(0.455157\pi\)
\(318\) −5.00000 −0.280386
\(319\) 15.0000 0.839839
\(320\) 3.00000 0.167705
\(321\) −19.0000 −1.06048
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) 6.00000 0.331801
\(328\) −12.0000 −0.662589
\(329\) 0 0
\(330\) 15.0000 0.825723
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) −17.0000 −0.932996
\(333\) −12.0000 −0.657596
\(334\) 2.00000 0.109435
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) 21.0000 1.14394 0.571971 0.820274i \(-0.306179\pi\)
0.571971 + 0.820274i \(0.306179\pi\)
\(338\) 13.0000 0.707107
\(339\) 12.0000 0.651751
\(340\) −12.0000 −0.650791
\(341\) 45.0000 2.43689
\(342\) 0 0
\(343\) 0 0
\(344\) −6.00000 −0.323498
\(345\) 3.00000 0.161515
\(346\) 6.00000 0.322562
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) −3.00000 −0.160817
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) 4.00000 0.212899 0.106449 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 9.00000 0.478345
\(355\) 18.0000 0.955341
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) −3.00000 −0.158114
\(361\) −19.0000 −1.00000
\(362\) 8.00000 0.420471
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) −42.0000 −2.19838
\(366\) 2.00000 0.104542
\(367\) 25.0000 1.30499 0.652495 0.757793i \(-0.273722\pi\)
0.652495 + 0.757793i \(0.273722\pi\)
\(368\) 1.00000 0.0521286
\(369\) 12.0000 0.624695
\(370\) 36.0000 1.87155
\(371\) 0 0
\(372\) −9.00000 −0.466628
\(373\) −8.00000 −0.414224 −0.207112 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) −20.0000 −1.03418
\(375\) −3.00000 −0.154919
\(376\) −2.00000 −0.103142
\(377\) 0 0
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) −11.0000 −0.563547
\(382\) −12.0000 −0.613973
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −13.0000 −0.661683
\(387\) 6.00000 0.304997
\(388\) −13.0000 −0.659975
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) 3.00000 0.151330
\(394\) 2.00000 0.100759
\(395\) 33.0000 1.66041
\(396\) −5.00000 −0.251259
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 32.0000 1.59800 0.799002 0.601329i \(-0.205362\pi\)
0.799002 + 0.601329i \(0.205362\pi\)
\(402\) 2.00000 0.0997509
\(403\) 0 0
\(404\) −2.00000 −0.0995037
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) 60.0000 2.97409
\(408\) 4.00000 0.198030
\(409\) 7.00000 0.346128 0.173064 0.984911i \(-0.444633\pi\)
0.173064 + 0.984911i \(0.444633\pi\)
\(410\) −36.0000 −1.77791
\(411\) −6.00000 −0.295958
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −51.0000 −2.50349
\(416\) 0 0
\(417\) −6.00000 −0.293821
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −6.00000 −0.292075
\(423\) 2.00000 0.0972433
\(424\) −5.00000 −0.242821
\(425\) −16.0000 −0.776114
\(426\) −6.00000 −0.290701
\(427\) 0 0
\(428\) −19.0000 −0.918400
\(429\) 0 0
\(430\) −18.0000 −0.868037
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 6.00000 0.287348
\(437\) 0 0
\(438\) 14.0000 0.668946
\(439\) −13.0000 −0.620456 −0.310228 0.950662i \(-0.600405\pi\)
−0.310228 + 0.950662i \(0.600405\pi\)
\(440\) 15.0000 0.715097
\(441\) 0 0
\(442\) 0 0
\(443\) −29.0000 −1.37783 −0.688916 0.724841i \(-0.741913\pi\)
−0.688916 + 0.724841i \(0.741913\pi\)
\(444\) −12.0000 −0.569495
\(445\) 30.0000 1.42214
\(446\) −13.0000 −0.615568
\(447\) −14.0000 −0.662177
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) −4.00000 −0.188562
\(451\) −60.0000 −2.82529
\(452\) 12.0000 0.564433
\(453\) −9.00000 −0.422857
\(454\) −5.00000 −0.234662
\(455\) 0 0
\(456\) 0 0
\(457\) −21.0000 −0.982339 −0.491169 0.871064i \(-0.663430\pi\)
−0.491169 + 0.871064i \(0.663430\pi\)
\(458\) 24.0000 1.12145
\(459\) −4.00000 −0.186704
\(460\) 3.00000 0.139876
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −3.00000 −0.139272
\(465\) −27.0000 −1.25210
\(466\) −20.0000 −0.926482
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −6.00000 −0.276759
\(471\) −8.00000 −0.368621
\(472\) 9.00000 0.414259
\(473\) −30.0000 −1.37940
\(474\) −11.0000 −0.505247
\(475\) 0 0
\(476\) 0 0
\(477\) 5.00000 0.228934
\(478\) 24.0000 1.09773
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) −3.00000 −0.136931
\(481\) 0 0
\(482\) −19.0000 −0.865426
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) −39.0000 −1.77090
\(486\) −1.00000 −0.0453609
\(487\) −23.0000 −1.04223 −0.521115 0.853487i \(-0.674484\pi\)
−0.521115 + 0.853487i \(0.674484\pi\)
\(488\) 2.00000 0.0905357
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 27.0000 1.21849 0.609246 0.792981i \(-0.291472\pi\)
0.609246 + 0.792981i \(0.291472\pi\)
\(492\) 12.0000 0.541002
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) −15.0000 −0.674200
\(496\) −9.00000 −0.404112
\(497\) 0 0
\(498\) 17.0000 0.761788
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) −3.00000 −0.134164
\(501\) −2.00000 −0.0893534
\(502\) 5.00000 0.223161
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 5.00000 0.222277
\(507\) −13.0000 −0.577350
\(508\) −11.0000 −0.488046
\(509\) −31.0000 −1.37405 −0.687025 0.726633i \(-0.741084\pi\)
−0.687025 + 0.726633i \(0.741084\pi\)
\(510\) 12.0000 0.531369
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −10.0000 −0.441081
\(515\) 24.0000 1.05757
\(516\) 6.00000 0.264135
\(517\) −10.0000 −0.439799
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 3.00000 0.131306
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 3.00000 0.131056
\(525\) 0 0
\(526\) −18.0000 −0.784837
\(527\) 36.0000 1.56818
\(528\) −5.00000 −0.217597
\(529\) 1.00000 0.0434783
\(530\) −15.0000 −0.651558
\(531\) −9.00000 −0.390567
\(532\) 0 0
\(533\) 0 0
\(534\) −10.0000 −0.432742
\(535\) −57.0000 −2.46432
\(536\) 2.00000 0.0863868
\(537\) −20.0000 −0.863064
\(538\) −9.00000 −0.388018
\(539\) 0 0
\(540\) 3.00000 0.129099
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 13.0000 0.558398
\(543\) −8.00000 −0.343313
\(544\) 4.00000 0.171499
\(545\) 18.0000 0.771035
\(546\) 0 0
\(547\) −44.0000 −1.88130 −0.940652 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) −6.00000 −0.256307
\(549\) −2.00000 −0.0853579
\(550\) 20.0000 0.852803
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −8.00000 −0.339887
\(555\) −36.0000 −1.52811
\(556\) −6.00000 −0.254457
\(557\) 3.00000 0.127114 0.0635570 0.997978i \(-0.479756\pi\)
0.0635570 + 0.997978i \(0.479756\pi\)
\(558\) 9.00000 0.381000
\(559\) 0 0
\(560\) 0 0
\(561\) 20.0000 0.844401
\(562\) −2.00000 −0.0843649
\(563\) 39.0000 1.64365 0.821827 0.569737i \(-0.192955\pi\)
0.821827 + 0.569737i \(0.192955\pi\)
\(564\) 2.00000 0.0842152
\(565\) 36.0000 1.51453
\(566\) −6.00000 −0.252199
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) 10.0000 0.418487 0.209243 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) 39.0000 1.62359 0.811796 0.583942i \(-0.198490\pi\)
0.811796 + 0.583942i \(0.198490\pi\)
\(578\) 1.00000 0.0415945
\(579\) 13.0000 0.540262
\(580\) −9.00000 −0.373705
\(581\) 0 0
\(582\) 13.0000 0.538867
\(583\) −25.0000 −1.03539
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) −33.0000 −1.36322
\(587\) −15.0000 −0.619116 −0.309558 0.950881i \(-0.600181\pi\)
−0.309558 + 0.950881i \(0.600181\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 27.0000 1.11157
\(591\) −2.00000 −0.0822690
\(592\) −12.0000 −0.493197
\(593\) 8.00000 0.328521 0.164260 0.986417i \(-0.447476\pi\)
0.164260 + 0.986417i \(0.447476\pi\)
\(594\) 5.00000 0.205152
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) −20.0000 −0.818546
\(598\) 0 0
\(599\) −22.0000 −0.898896 −0.449448 0.893307i \(-0.648379\pi\)
−0.449448 + 0.893307i \(0.648379\pi\)
\(600\) −4.00000 −0.163299
\(601\) 11.0000 0.448699 0.224350 0.974509i \(-0.427974\pi\)
0.224350 + 0.974509i \(0.427974\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) −9.00000 −0.366205
\(605\) 42.0000 1.70754
\(606\) 2.00000 0.0812444
\(607\) −15.0000 −0.608831 −0.304416 0.952539i \(-0.598461\pi\)
−0.304416 + 0.952539i \(0.598461\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 6.00000 0.242933
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) −12.0000 −0.484281
\(615\) 36.0000 1.45166
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) −8.00000 −0.321807
\(619\) −46.0000 −1.84890 −0.924448 0.381308i \(-0.875474\pi\)
−0.924448 + 0.381308i \(0.875474\pi\)
\(620\) −27.0000 −1.08435
\(621\) 1.00000 0.0401286
\(622\) −20.0000 −0.801927
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 11.0000 0.439648
\(627\) 0 0
\(628\) −8.00000 −0.319235
\(629\) 48.0000 1.91389
\(630\) 0 0
\(631\) 37.0000 1.47295 0.736473 0.676467i \(-0.236490\pi\)
0.736473 + 0.676467i \(0.236490\pi\)
\(632\) −11.0000 −0.437557
\(633\) 6.00000 0.238479
\(634\) −5.00000 −0.198575
\(635\) −33.0000 −1.30957
\(636\) 5.00000 0.198263
\(637\) 0 0
\(638\) −15.0000 −0.593856
\(639\) 6.00000 0.237356
\(640\) −3.00000 −0.118585
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 19.0000 0.749870
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 18.0000 0.708749
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 45.0000 1.76640
\(650\) 0 0
\(651\) 0 0
\(652\) −20.0000 −0.783260
\(653\) −17.0000 −0.665261 −0.332631 0.943057i \(-0.607936\pi\)
−0.332631 + 0.943057i \(0.607936\pi\)
\(654\) −6.00000 −0.234619
\(655\) 9.00000 0.351659
\(656\) 12.0000 0.468521
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) −15.0000 −0.583874
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 16.0000 0.621858
\(663\) 0 0
\(664\) 17.0000 0.659728
\(665\) 0 0
\(666\) 12.0000 0.464991
\(667\) −3.00000 −0.116160
\(668\) −2.00000 −0.0773823
\(669\) 13.0000 0.502609
\(670\) 6.00000 0.231800
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) 13.0000 0.501113 0.250557 0.968102i \(-0.419386\pi\)
0.250557 + 0.968102i \(0.419386\pi\)
\(674\) −21.0000 −0.808890
\(675\) 4.00000 0.153960
\(676\) −13.0000 −0.500000
\(677\) 31.0000 1.19143 0.595713 0.803197i \(-0.296869\pi\)
0.595713 + 0.803197i \(0.296869\pi\)
\(678\) −12.0000 −0.460857
\(679\) 0 0
\(680\) 12.0000 0.460179
\(681\) 5.00000 0.191600
\(682\) −45.0000 −1.72314
\(683\) −43.0000 −1.64535 −0.822675 0.568512i \(-0.807519\pi\)
−0.822675 + 0.568512i \(0.807519\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) −24.0000 −0.915657
\(688\) 6.00000 0.228748
\(689\) 0 0
\(690\) −3.00000 −0.114208
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) −18.0000 −0.682779
\(696\) 3.00000 0.113715
\(697\) −48.0000 −1.81813
\(698\) −22.0000 −0.832712
\(699\) 20.0000 0.756469
\(700\) 0 0
\(701\) 9.00000 0.339925 0.169963 0.985451i \(-0.445635\pi\)
0.169963 + 0.985451i \(0.445635\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −5.00000 −0.188445
\(705\) 6.00000 0.225973
\(706\) −4.00000 −0.150542
\(707\) 0 0
\(708\) −9.00000 −0.338241
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) −18.0000 −0.675528
\(711\) 11.0000 0.412532
\(712\) −10.0000 −0.374766
\(713\) −9.00000 −0.337053
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) −24.0000 −0.896296
\(718\) 10.0000 0.373197
\(719\) 14.0000 0.522112 0.261056 0.965324i \(-0.415929\pi\)
0.261056 + 0.965324i \(0.415929\pi\)
\(720\) 3.00000 0.111803
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) 19.0000 0.706618
\(724\) −8.00000 −0.297318
\(725\) −12.0000 −0.445669
\(726\) −14.0000 −0.519589
\(727\) 7.00000 0.259616 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 42.0000 1.55449
\(731\) −24.0000 −0.887672
\(732\) −2.00000 −0.0739221
\(733\) −46.0000 −1.69905 −0.849524 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) −25.0000 −0.922767
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 10.0000 0.368355
\(738\) −12.0000 −0.441726
\(739\) 2.00000 0.0735712 0.0367856 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) −36.0000 −1.32339
\(741\) 0 0
\(742\) 0 0
\(743\) −26.0000 −0.953847 −0.476924 0.878945i \(-0.658248\pi\)
−0.476924 + 0.878945i \(0.658248\pi\)
\(744\) 9.00000 0.329956
\(745\) −42.0000 −1.53876
\(746\) 8.00000 0.292901
\(747\) −17.0000 −0.621997
\(748\) 20.0000 0.731272
\(749\) 0 0
\(750\) 3.00000 0.109545
\(751\) 5.00000 0.182453 0.0912263 0.995830i \(-0.470921\pi\)
0.0912263 + 0.995830i \(0.470921\pi\)
\(752\) 2.00000 0.0729325
\(753\) −5.00000 −0.182210
\(754\) 0 0
\(755\) −27.0000 −0.982631
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 16.0000 0.581146
\(759\) −5.00000 −0.181489
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 11.0000 0.398488
\(763\) 0 0
\(764\) 12.0000 0.434145
\(765\) −12.0000 −0.433861
\(766\) 18.0000 0.650366
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −31.0000 −1.11789 −0.558944 0.829205i \(-0.688793\pi\)
−0.558944 + 0.829205i \(0.688793\pi\)
\(770\) 0 0
\(771\) 10.0000 0.360141
\(772\) 13.0000 0.467880
\(773\) 46.0000 1.65451 0.827253 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) −6.00000 −0.215666
\(775\) −36.0000 −1.29316
\(776\) 13.0000 0.466673
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) 0 0
\(780\) 0 0
\(781\) −30.0000 −1.07348
\(782\) 4.00000 0.143040
\(783\) −3.00000 −0.107211
\(784\) 0 0
\(785\) −24.0000 −0.856597
\(786\) −3.00000 −0.107006
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 18.0000 0.640817
\(790\) −33.0000 −1.17409
\(791\) 0 0
\(792\) 5.00000 0.177667
\(793\) 0 0
\(794\) 8.00000 0.283909
\(795\) 15.0000 0.531995
\(796\) −20.0000 −0.708881
\(797\) −49.0000 −1.73567 −0.867835 0.496853i \(-0.834489\pi\)
−0.867835 + 0.496853i \(0.834489\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) −4.00000 −0.141421
\(801\) 10.0000 0.353333
\(802\) −32.0000 −1.12996
\(803\) 70.0000 2.47025
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 0 0
\(807\) 9.00000 0.316815
\(808\) 2.00000 0.0703598
\(809\) 44.0000 1.54696 0.773479 0.633822i \(-0.218515\pi\)
0.773479 + 0.633822i \(0.218515\pi\)
\(810\) −3.00000 −0.105409
\(811\) 42.0000 1.47482 0.737410 0.675446i \(-0.236049\pi\)
0.737410 + 0.675446i \(0.236049\pi\)
\(812\) 0 0
\(813\) −13.0000 −0.455930
\(814\) −60.0000 −2.10300
\(815\) −60.0000 −2.10171
\(816\) −4.00000 −0.140028
\(817\) 0 0
\(818\) −7.00000 −0.244749
\(819\) 0 0
\(820\) 36.0000 1.25717
\(821\) 57.0000 1.98931 0.994657 0.103236i \(-0.0329198\pi\)
0.994657 + 0.103236i \(0.0329198\pi\)
\(822\) 6.00000 0.209274
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −8.00000 −0.278693
\(825\) −20.0000 −0.696311
\(826\) 0 0
\(827\) −33.0000 −1.14752 −0.573761 0.819023i \(-0.694516\pi\)
−0.573761 + 0.819023i \(0.694516\pi\)
\(828\) 1.00000 0.0347524
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 51.0000 1.77024
\(831\) 8.00000 0.277517
\(832\) 0 0
\(833\) 0 0
\(834\) 6.00000 0.207763
\(835\) −6.00000 −0.207639
\(836\) 0 0
\(837\) −9.00000 −0.311086
\(838\) 16.0000 0.552711
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 6.00000 0.206774
\(843\) 2.00000 0.0688837
\(844\) 6.00000 0.206529
\(845\) −39.0000 −1.34164
\(846\) −2.00000 −0.0687614
\(847\) 0 0
\(848\) 5.00000 0.171701
\(849\) 6.00000 0.205919
\(850\) 16.0000 0.548795
\(851\) −12.0000 −0.411355
\(852\) 6.00000 0.205557
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 19.0000 0.649407
\(857\) −54.0000 −1.84460 −0.922302 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) 22.0000 0.750630 0.375315 0.926897i \(-0.377534\pi\)
0.375315 + 0.926897i \(0.377534\pi\)
\(860\) 18.0000 0.613795
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) −14.0000 −0.476566 −0.238283 0.971196i \(-0.576585\pi\)
−0.238283 + 0.971196i \(0.576585\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.0000 −0.612018
\(866\) −14.0000 −0.475739
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −55.0000 −1.86575
\(870\) 9.00000 0.305129
\(871\) 0 0
\(872\) −6.00000 −0.203186
\(873\) −13.0000 −0.439983
\(874\) 0 0
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) −12.0000 −0.405211 −0.202606 0.979260i \(-0.564941\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) 13.0000 0.438729
\(879\) 33.0000 1.11306
\(880\) −15.0000 −0.505650
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) −27.0000 −0.907595
\(886\) 29.0000 0.974274
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 12.0000 0.402694
\(889\) 0 0
\(890\) −30.0000 −1.00560
\(891\) −5.00000 −0.167506
\(892\) 13.0000 0.435272
\(893\) 0 0
\(894\) 14.0000 0.468230
\(895\) −60.0000 −2.00558
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 27.0000 0.900500
\(900\) 4.00000 0.133333
\(901\) −20.0000 −0.666297
\(902\) 60.0000 1.99778
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) −24.0000 −0.797787
\(906\) 9.00000 0.299005
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 5.00000 0.165931
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −10.0000 −0.331315 −0.165657 0.986183i \(-0.552975\pi\)
−0.165657 + 0.986183i \(0.552975\pi\)
\(912\) 0 0
\(913\) 85.0000 2.81309
\(914\) 21.0000 0.694618
\(915\) −6.00000 −0.198354
\(916\) −24.0000 −0.792982
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 12.0000 0.395413
\(922\) −14.0000 −0.461065
\(923\) 0 0
\(924\) 0 0
\(925\) −48.0000 −1.57823
\(926\) 32.0000 1.05159
\(927\) 8.00000 0.262754
\(928\) 3.00000 0.0984798
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 27.0000 0.885365
\(931\) 0 0
\(932\) 20.0000 0.655122
\(933\) 20.0000 0.654771
\(934\) −28.0000 −0.916188
\(935\) 60.0000 1.96221
\(936\) 0 0
\(937\) 7.00000 0.228680 0.114340 0.993442i \(-0.463525\pi\)
0.114340 + 0.993442i \(0.463525\pi\)
\(938\) 0 0
\(939\) −11.0000 −0.358971
\(940\) 6.00000 0.195698
\(941\) 15.0000 0.488986 0.244493 0.969651i \(-0.421378\pi\)
0.244493 + 0.969651i \(0.421378\pi\)
\(942\) 8.00000 0.260654
\(943\) 12.0000 0.390774
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) 30.0000 0.975384
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 11.0000 0.357263
\(949\) 0 0
\(950\) 0 0
\(951\) 5.00000 0.162136
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) −5.00000 −0.161881
\(955\) 36.0000 1.16493
\(956\) −24.0000 −0.776215
\(957\) 15.0000 0.484881
\(958\) −18.0000 −0.581554
\(959\) 0 0
\(960\) 3.00000 0.0968246
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) −19.0000 −0.612266
\(964\) 19.0000 0.611949
\(965\) 39.0000 1.25545
\(966\) 0 0
\(967\) 7.00000 0.225105 0.112552 0.993646i \(-0.464097\pi\)
0.112552 + 0.993646i \(0.464097\pi\)
\(968\) −14.0000 −0.449977
\(969\) 0 0
\(970\) 39.0000 1.25221
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 23.0000 0.736968
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −62.0000 −1.98356 −0.991778 0.127971i \(-0.959153\pi\)
−0.991778 + 0.127971i \(0.959153\pi\)
\(978\) 20.0000 0.639529
\(979\) −50.0000 −1.59801
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) −27.0000 −0.861605
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −12.0000 −0.382546
\(985\) −6.00000 −0.191176
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) 6.00000 0.190789
\(990\) 15.0000 0.476731
\(991\) 51.0000 1.62007 0.810034 0.586383i \(-0.199448\pi\)
0.810034 + 0.586383i \(0.199448\pi\)
\(992\) 9.00000 0.285750
\(993\) −16.0000 −0.507745
\(994\) 0 0
\(995\) −60.0000 −1.90213
\(996\) −17.0000 −0.538666
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) −10.0000 −0.316544
\(999\) −12.0000 −0.379663
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.t.1.1 1
7.2 even 3 966.2.i.e.277.1 2
7.4 even 3 966.2.i.e.415.1 yes 2
7.6 odd 2 6762.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.e.277.1 2 7.2 even 3
966.2.i.e.415.1 yes 2 7.4 even 3
6762.2.a.a.1.1 1 7.6 odd 2
6762.2.a.t.1.1 1 1.1 even 1 trivial