Properties

Label 6039.2.a.i.1.11
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-1.46794\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.46794 q^{2} +0.154851 q^{4} -3.84216 q^{5} +2.46223 q^{7} -2.70857 q^{8} +O(q^{10})\) \(q+1.46794 q^{2} +0.154851 q^{4} -3.84216 q^{5} +2.46223 q^{7} -2.70857 q^{8} -5.64006 q^{10} -1.00000 q^{11} -4.64696 q^{13} +3.61441 q^{14} -4.28572 q^{16} -5.70692 q^{17} +6.71196 q^{19} -0.594961 q^{20} -1.46794 q^{22} -2.93787 q^{23} +9.76218 q^{25} -6.82147 q^{26} +0.381279 q^{28} -5.67137 q^{29} +0.232595 q^{31} -0.874048 q^{32} -8.37742 q^{34} -9.46029 q^{35} +5.55034 q^{37} +9.85276 q^{38} +10.4068 q^{40} +4.35928 q^{41} -9.35832 q^{43} -0.154851 q^{44} -4.31262 q^{46} -10.7428 q^{47} -0.937404 q^{49} +14.3303 q^{50} -0.719586 q^{52} +11.3834 q^{53} +3.84216 q^{55} -6.66913 q^{56} -8.32523 q^{58} +13.2037 q^{59} -1.00000 q^{61} +0.341436 q^{62} +7.28839 q^{64} +17.8544 q^{65} +12.5899 q^{67} -0.883721 q^{68} -13.8871 q^{70} +0.299770 q^{71} -2.65468 q^{73} +8.14757 q^{74} +1.03935 q^{76} -2.46223 q^{77} -12.9557 q^{79} +16.4664 q^{80} +6.39917 q^{82} +7.42505 q^{83} +21.9269 q^{85} -13.7375 q^{86} +2.70857 q^{88} +3.40420 q^{89} -11.4419 q^{91} -0.454931 q^{92} -15.7698 q^{94} -25.7884 q^{95} -15.8330 q^{97} -1.37605 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + 6q^{10} - 13q^{11} + 13q^{13} - q^{14} + 18q^{16} - 17q^{17} + 14q^{19} + 7q^{20} + 2q^{22} - 7q^{23} + 18q^{25} + 10q^{26} + 19q^{28} + 6q^{29} + 27q^{31} - 5q^{32} + 6q^{34} - 14q^{35} + 10q^{37} - 2q^{38} + 8q^{40} - 3q^{41} + 29q^{43} - 16q^{44} - 24q^{46} - 8q^{47} + 8q^{49} + 27q^{50} + 37q^{52} + 24q^{53} + 3q^{55} - 24q^{56} - 5q^{58} - 13q^{59} - 13q^{61} - 39q^{62} + 47q^{64} + 11q^{65} + 44q^{67} + 8q^{68} - 12q^{70} - 3q^{71} + 48q^{73} + 22q^{74} + 47q^{76} - 11q^{77} - 17q^{79} + 26q^{80} + 56q^{82} - 50q^{83} + 8q^{85} - 18q^{86} + 9q^{88} + 15q^{89} + 47q^{91} - 14q^{92} + 45q^{94} + q^{95} + 27q^{97} - 47q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.46794 1.03799 0.518996 0.854777i \(-0.326306\pi\)
0.518996 + 0.854777i \(0.326306\pi\)
\(3\) 0 0
\(4\) 0.154851 0.0774254
\(5\) −3.84216 −1.71827 −0.859133 0.511753i \(-0.828996\pi\)
−0.859133 + 0.511753i \(0.828996\pi\)
\(6\) 0 0
\(7\) 2.46223 0.930637 0.465318 0.885143i \(-0.345940\pi\)
0.465318 + 0.885143i \(0.345940\pi\)
\(8\) −2.70857 −0.957624
\(9\) 0 0
\(10\) −5.64006 −1.78354
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.64696 −1.28884 −0.644418 0.764674i \(-0.722900\pi\)
−0.644418 + 0.764674i \(0.722900\pi\)
\(14\) 3.61441 0.965993
\(15\) 0 0
\(16\) −4.28572 −1.07143
\(17\) −5.70692 −1.38413 −0.692066 0.721834i \(-0.743299\pi\)
−0.692066 + 0.721834i \(0.743299\pi\)
\(18\) 0 0
\(19\) 6.71196 1.53983 0.769914 0.638147i \(-0.220299\pi\)
0.769914 + 0.638147i \(0.220299\pi\)
\(20\) −0.594961 −0.133037
\(21\) 0 0
\(22\) −1.46794 −0.312966
\(23\) −2.93787 −0.612588 −0.306294 0.951937i \(-0.599089\pi\)
−0.306294 + 0.951937i \(0.599089\pi\)
\(24\) 0 0
\(25\) 9.76218 1.95244
\(26\) −6.82147 −1.33780
\(27\) 0 0
\(28\) 0.381279 0.0720549
\(29\) −5.67137 −1.05315 −0.526573 0.850130i \(-0.676523\pi\)
−0.526573 + 0.850130i \(0.676523\pi\)
\(30\) 0 0
\(31\) 0.232595 0.0417754 0.0208877 0.999782i \(-0.493351\pi\)
0.0208877 + 0.999782i \(0.493351\pi\)
\(32\) −0.874048 −0.154511
\(33\) 0 0
\(34\) −8.37742 −1.43672
\(35\) −9.46029 −1.59908
\(36\) 0 0
\(37\) 5.55034 0.912470 0.456235 0.889859i \(-0.349198\pi\)
0.456235 + 0.889859i \(0.349198\pi\)
\(38\) 9.85276 1.59833
\(39\) 0 0
\(40\) 10.4068 1.64545
\(41\) 4.35928 0.680806 0.340403 0.940280i \(-0.389437\pi\)
0.340403 + 0.940280i \(0.389437\pi\)
\(42\) 0 0
\(43\) −9.35832 −1.42713 −0.713565 0.700589i \(-0.752921\pi\)
−0.713565 + 0.700589i \(0.752921\pi\)
\(44\) −0.154851 −0.0233446
\(45\) 0 0
\(46\) −4.31262 −0.635861
\(47\) −10.7428 −1.56699 −0.783497 0.621395i \(-0.786566\pi\)
−0.783497 + 0.621395i \(0.786566\pi\)
\(48\) 0 0
\(49\) −0.937404 −0.133915
\(50\) 14.3303 2.02661
\(51\) 0 0
\(52\) −0.719586 −0.0997886
\(53\) 11.3834 1.56363 0.781816 0.623510i \(-0.214294\pi\)
0.781816 + 0.623510i \(0.214294\pi\)
\(54\) 0 0
\(55\) 3.84216 0.518076
\(56\) −6.66913 −0.891200
\(57\) 0 0
\(58\) −8.32523 −1.09316
\(59\) 13.2037 1.71897 0.859485 0.511161i \(-0.170784\pi\)
0.859485 + 0.511161i \(0.170784\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) 0.341436 0.0433624
\(63\) 0 0
\(64\) 7.28839 0.911049
\(65\) 17.8544 2.21456
\(66\) 0 0
\(67\) 12.5899 1.53810 0.769049 0.639190i \(-0.220730\pi\)
0.769049 + 0.639190i \(0.220730\pi\)
\(68\) −0.883721 −0.107167
\(69\) 0 0
\(70\) −13.8871 −1.65983
\(71\) 0.299770 0.0355761 0.0177881 0.999842i \(-0.494338\pi\)
0.0177881 + 0.999842i \(0.494338\pi\)
\(72\) 0 0
\(73\) −2.65468 −0.310707 −0.155353 0.987859i \(-0.549652\pi\)
−0.155353 + 0.987859i \(0.549652\pi\)
\(74\) 8.14757 0.947135
\(75\) 0 0
\(76\) 1.03935 0.119222
\(77\) −2.46223 −0.280598
\(78\) 0 0
\(79\) −12.9557 −1.45763 −0.728815 0.684710i \(-0.759929\pi\)
−0.728815 + 0.684710i \(0.759929\pi\)
\(80\) 16.4664 1.84100
\(81\) 0 0
\(82\) 6.39917 0.706670
\(83\) 7.42505 0.815005 0.407502 0.913204i \(-0.366400\pi\)
0.407502 + 0.913204i \(0.366400\pi\)
\(84\) 0 0
\(85\) 21.9269 2.37831
\(86\) −13.7375 −1.48135
\(87\) 0 0
\(88\) 2.70857 0.288735
\(89\) 3.40420 0.360845 0.180422 0.983589i \(-0.442254\pi\)
0.180422 + 0.983589i \(0.442254\pi\)
\(90\) 0 0
\(91\) −11.4419 −1.19944
\(92\) −0.454931 −0.0474299
\(93\) 0 0
\(94\) −15.7698 −1.62653
\(95\) −25.7884 −2.64583
\(96\) 0 0
\(97\) −15.8330 −1.60760 −0.803798 0.594902i \(-0.797191\pi\)
−0.803798 + 0.594902i \(0.797191\pi\)
\(98\) −1.37605 −0.139002
\(99\) 0 0
\(100\) 1.51168 0.151168
\(101\) −7.72797 −0.768962 −0.384481 0.923133i \(-0.625620\pi\)
−0.384481 + 0.923133i \(0.625620\pi\)
\(102\) 0 0
\(103\) 16.1339 1.58972 0.794860 0.606793i \(-0.207544\pi\)
0.794860 + 0.606793i \(0.207544\pi\)
\(104\) 12.5866 1.23422
\(105\) 0 0
\(106\) 16.7102 1.62304
\(107\) −1.14677 −0.110863 −0.0554314 0.998462i \(-0.517653\pi\)
−0.0554314 + 0.998462i \(0.517653\pi\)
\(108\) 0 0
\(109\) 11.7557 1.12600 0.562998 0.826458i \(-0.309648\pi\)
0.562998 + 0.826458i \(0.309648\pi\)
\(110\) 5.64006 0.537759
\(111\) 0 0
\(112\) −10.5525 −0.997113
\(113\) 16.4246 1.54509 0.772546 0.634959i \(-0.218983\pi\)
0.772546 + 0.634959i \(0.218983\pi\)
\(114\) 0 0
\(115\) 11.2878 1.05259
\(116\) −0.878216 −0.0815403
\(117\) 0 0
\(118\) 19.3822 1.78428
\(119\) −14.0518 −1.28812
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.46794 −0.132901
\(123\) 0 0
\(124\) 0.0360176 0.00323447
\(125\) −18.2970 −1.63654
\(126\) 0 0
\(127\) 11.9297 1.05859 0.529295 0.848438i \(-0.322457\pi\)
0.529295 + 0.848438i \(0.322457\pi\)
\(128\) 12.4470 1.10017
\(129\) 0 0
\(130\) 26.2091 2.29869
\(131\) −20.8633 −1.82283 −0.911417 0.411485i \(-0.865010\pi\)
−0.911417 + 0.411485i \(0.865010\pi\)
\(132\) 0 0
\(133\) 16.5264 1.43302
\(134\) 18.4812 1.59653
\(135\) 0 0
\(136\) 15.4576 1.32548
\(137\) 19.1597 1.63692 0.818460 0.574564i \(-0.194828\pi\)
0.818460 + 0.574564i \(0.194828\pi\)
\(138\) 0 0
\(139\) −2.58202 −0.219004 −0.109502 0.993987i \(-0.534926\pi\)
−0.109502 + 0.993987i \(0.534926\pi\)
\(140\) −1.46493 −0.123809
\(141\) 0 0
\(142\) 0.440044 0.0369277
\(143\) 4.64696 0.388598
\(144\) 0 0
\(145\) 21.7903 1.80959
\(146\) −3.89691 −0.322511
\(147\) 0 0
\(148\) 0.859474 0.0706483
\(149\) −9.45719 −0.774763 −0.387382 0.921919i \(-0.626620\pi\)
−0.387382 + 0.921919i \(0.626620\pi\)
\(150\) 0 0
\(151\) 6.16589 0.501773 0.250887 0.968016i \(-0.419278\pi\)
0.250887 + 0.968016i \(0.419278\pi\)
\(152\) −18.1798 −1.47458
\(153\) 0 0
\(154\) −3.61441 −0.291258
\(155\) −0.893668 −0.0717811
\(156\) 0 0
\(157\) 0.475244 0.0379286 0.0189643 0.999820i \(-0.493963\pi\)
0.0189643 + 0.999820i \(0.493963\pi\)
\(158\) −19.0182 −1.51301
\(159\) 0 0
\(160\) 3.35823 0.265491
\(161\) −7.23372 −0.570097
\(162\) 0 0
\(163\) 17.6623 1.38341 0.691707 0.722178i \(-0.256859\pi\)
0.691707 + 0.722178i \(0.256859\pi\)
\(164\) 0.675038 0.0527116
\(165\) 0 0
\(166\) 10.8995 0.845968
\(167\) 15.9865 1.23707 0.618536 0.785756i \(-0.287726\pi\)
0.618536 + 0.785756i \(0.287726\pi\)
\(168\) 0 0
\(169\) 8.59425 0.661096
\(170\) 32.1874 2.46866
\(171\) 0 0
\(172\) −1.44914 −0.110496
\(173\) 2.22261 0.168982 0.0844909 0.996424i \(-0.473074\pi\)
0.0844909 + 0.996424i \(0.473074\pi\)
\(174\) 0 0
\(175\) 24.0368 1.81701
\(176\) 4.28572 0.323049
\(177\) 0 0
\(178\) 4.99717 0.374553
\(179\) 24.5110 1.83204 0.916020 0.401132i \(-0.131383\pi\)
0.916020 + 0.401132i \(0.131383\pi\)
\(180\) 0 0
\(181\) −14.1308 −1.05034 −0.525168 0.850999i \(-0.675997\pi\)
−0.525168 + 0.850999i \(0.675997\pi\)
\(182\) −16.7960 −1.24501
\(183\) 0 0
\(184\) 7.95743 0.586629
\(185\) −21.3253 −1.56787
\(186\) 0 0
\(187\) 5.70692 0.417331
\(188\) −1.66353 −0.121325
\(189\) 0 0
\(190\) −37.8558 −2.74635
\(191\) 21.1658 1.53151 0.765753 0.643135i \(-0.222367\pi\)
0.765753 + 0.643135i \(0.222367\pi\)
\(192\) 0 0
\(193\) −10.1194 −0.728414 −0.364207 0.931318i \(-0.618660\pi\)
−0.364207 + 0.931318i \(0.618660\pi\)
\(194\) −23.2419 −1.66867
\(195\) 0 0
\(196\) −0.145158 −0.0103684
\(197\) −4.80836 −0.342582 −0.171291 0.985221i \(-0.554794\pi\)
−0.171291 + 0.985221i \(0.554794\pi\)
\(198\) 0 0
\(199\) 13.0978 0.928480 0.464240 0.885710i \(-0.346328\pi\)
0.464240 + 0.885710i \(0.346328\pi\)
\(200\) −26.4415 −1.86970
\(201\) 0 0
\(202\) −11.3442 −0.798176
\(203\) −13.9642 −0.980097
\(204\) 0 0
\(205\) −16.7491 −1.16980
\(206\) 23.6836 1.65011
\(207\) 0 0
\(208\) 19.9156 1.38090
\(209\) −6.71196 −0.464276
\(210\) 0 0
\(211\) −6.36135 −0.437933 −0.218967 0.975732i \(-0.570269\pi\)
−0.218967 + 0.975732i \(0.570269\pi\)
\(212\) 1.76273 0.121065
\(213\) 0 0
\(214\) −1.68340 −0.115075
\(215\) 35.9561 2.45219
\(216\) 0 0
\(217\) 0.572704 0.0388777
\(218\) 17.2567 1.16877
\(219\) 0 0
\(220\) 0.594961 0.0401123
\(221\) 26.5198 1.78392
\(222\) 0 0
\(223\) 22.5356 1.50910 0.754548 0.656244i \(-0.227856\pi\)
0.754548 + 0.656244i \(0.227856\pi\)
\(224\) −2.15211 −0.143794
\(225\) 0 0
\(226\) 24.1103 1.60379
\(227\) −27.1685 −1.80323 −0.901617 0.432535i \(-0.857619\pi\)
−0.901617 + 0.432535i \(0.857619\pi\)
\(228\) 0 0
\(229\) −24.6096 −1.62625 −0.813124 0.582090i \(-0.802235\pi\)
−0.813124 + 0.582090i \(0.802235\pi\)
\(230\) 16.5698 1.09258
\(231\) 0 0
\(232\) 15.3613 1.00852
\(233\) −24.0471 −1.57538 −0.787690 0.616071i \(-0.788723\pi\)
−0.787690 + 0.616071i \(0.788723\pi\)
\(234\) 0 0
\(235\) 41.2754 2.69251
\(236\) 2.04460 0.133092
\(237\) 0 0
\(238\) −20.6272 −1.33706
\(239\) −16.2704 −1.05245 −0.526224 0.850346i \(-0.676393\pi\)
−0.526224 + 0.850346i \(0.676393\pi\)
\(240\) 0 0
\(241\) 1.53799 0.0990706 0.0495353 0.998772i \(-0.484226\pi\)
0.0495353 + 0.998772i \(0.484226\pi\)
\(242\) 1.46794 0.0943628
\(243\) 0 0
\(244\) −0.154851 −0.00991330
\(245\) 3.60165 0.230101
\(246\) 0 0
\(247\) −31.1902 −1.98458
\(248\) −0.630001 −0.0400051
\(249\) 0 0
\(250\) −26.8590 −1.69871
\(251\) 22.8738 1.44378 0.721891 0.692006i \(-0.243273\pi\)
0.721891 + 0.692006i \(0.243273\pi\)
\(252\) 0 0
\(253\) 2.93787 0.184702
\(254\) 17.5121 1.09881
\(255\) 0 0
\(256\) 3.69471 0.230920
\(257\) 10.1026 0.630186 0.315093 0.949061i \(-0.397964\pi\)
0.315093 + 0.949061i \(0.397964\pi\)
\(258\) 0 0
\(259\) 13.6662 0.849178
\(260\) 2.76476 0.171463
\(261\) 0 0
\(262\) −30.6261 −1.89208
\(263\) 11.1552 0.687859 0.343929 0.938996i \(-0.388242\pi\)
0.343929 + 0.938996i \(0.388242\pi\)
\(264\) 0 0
\(265\) −43.7368 −2.68673
\(266\) 24.2598 1.48746
\(267\) 0 0
\(268\) 1.94955 0.119088
\(269\) −6.03952 −0.368236 −0.184118 0.982904i \(-0.558943\pi\)
−0.184118 + 0.982904i \(0.558943\pi\)
\(270\) 0 0
\(271\) 4.26042 0.258802 0.129401 0.991592i \(-0.458695\pi\)
0.129401 + 0.991592i \(0.458695\pi\)
\(272\) 24.4583 1.48300
\(273\) 0 0
\(274\) 28.1252 1.69911
\(275\) −9.76218 −0.588681
\(276\) 0 0
\(277\) 8.39998 0.504706 0.252353 0.967635i \(-0.418796\pi\)
0.252353 + 0.967635i \(0.418796\pi\)
\(278\) −3.79026 −0.227325
\(279\) 0 0
\(280\) 25.6239 1.53132
\(281\) −12.3489 −0.736671 −0.368335 0.929693i \(-0.620072\pi\)
−0.368335 + 0.929693i \(0.620072\pi\)
\(282\) 0 0
\(283\) 7.31966 0.435109 0.217554 0.976048i \(-0.430192\pi\)
0.217554 + 0.976048i \(0.430192\pi\)
\(284\) 0.0464196 0.00275450
\(285\) 0 0
\(286\) 6.82147 0.403362
\(287\) 10.7336 0.633583
\(288\) 0 0
\(289\) 15.5689 0.915820
\(290\) 31.9869 1.87833
\(291\) 0 0
\(292\) −0.411079 −0.0240566
\(293\) −16.8649 −0.985255 −0.492628 0.870240i \(-0.663964\pi\)
−0.492628 + 0.870240i \(0.663964\pi\)
\(294\) 0 0
\(295\) −50.7305 −2.95365
\(296\) −15.0335 −0.873803
\(297\) 0 0
\(298\) −13.8826 −0.804197
\(299\) 13.6522 0.789525
\(300\) 0 0
\(301\) −23.0424 −1.32814
\(302\) 9.05117 0.520836
\(303\) 0 0
\(304\) −28.7656 −1.64982
\(305\) 3.84216 0.220001
\(306\) 0 0
\(307\) 20.2128 1.15360 0.576802 0.816884i \(-0.304300\pi\)
0.576802 + 0.816884i \(0.304300\pi\)
\(308\) −0.381279 −0.0217254
\(309\) 0 0
\(310\) −1.31185 −0.0745082
\(311\) 27.5538 1.56243 0.781215 0.624262i \(-0.214600\pi\)
0.781215 + 0.624262i \(0.214600\pi\)
\(312\) 0 0
\(313\) 18.0023 1.01755 0.508776 0.860899i \(-0.330098\pi\)
0.508776 + 0.860899i \(0.330098\pi\)
\(314\) 0.697630 0.0393695
\(315\) 0 0
\(316\) −2.00620 −0.112858
\(317\) −29.5066 −1.65726 −0.828628 0.559799i \(-0.810878\pi\)
−0.828628 + 0.559799i \(0.810878\pi\)
\(318\) 0 0
\(319\) 5.67137 0.317536
\(320\) −28.0032 −1.56542
\(321\) 0 0
\(322\) −10.6187 −0.591756
\(323\) −38.3046 −2.13133
\(324\) 0 0
\(325\) −45.3645 −2.51637
\(326\) 25.9271 1.43597
\(327\) 0 0
\(328\) −11.8074 −0.651956
\(329\) −26.4512 −1.45830
\(330\) 0 0
\(331\) 17.8450 0.980852 0.490426 0.871483i \(-0.336841\pi\)
0.490426 + 0.871483i \(0.336841\pi\)
\(332\) 1.14977 0.0631021
\(333\) 0 0
\(334\) 23.4672 1.28407
\(335\) −48.3723 −2.64286
\(336\) 0 0
\(337\) −15.1235 −0.823832 −0.411916 0.911222i \(-0.635140\pi\)
−0.411916 + 0.911222i \(0.635140\pi\)
\(338\) 12.6159 0.686212
\(339\) 0 0
\(340\) 3.39540 0.184141
\(341\) −0.232595 −0.0125957
\(342\) 0 0
\(343\) −19.5437 −1.05526
\(344\) 25.3477 1.36665
\(345\) 0 0
\(346\) 3.26266 0.175402
\(347\) −11.2684 −0.604920 −0.302460 0.953162i \(-0.597808\pi\)
−0.302460 + 0.953162i \(0.597808\pi\)
\(348\) 0 0
\(349\) 6.44520 0.345004 0.172502 0.985009i \(-0.444815\pi\)
0.172502 + 0.985009i \(0.444815\pi\)
\(350\) 35.2845 1.88604
\(351\) 0 0
\(352\) 0.874048 0.0465869
\(353\) 22.9363 1.22078 0.610389 0.792102i \(-0.291013\pi\)
0.610389 + 0.792102i \(0.291013\pi\)
\(354\) 0 0
\(355\) −1.15176 −0.0611292
\(356\) 0.527143 0.0279385
\(357\) 0 0
\(358\) 35.9807 1.90164
\(359\) −11.2684 −0.594724 −0.297362 0.954765i \(-0.596107\pi\)
−0.297362 + 0.954765i \(0.596107\pi\)
\(360\) 0 0
\(361\) 26.0504 1.37107
\(362\) −20.7432 −1.09024
\(363\) 0 0
\(364\) −1.77179 −0.0928669
\(365\) 10.1997 0.533877
\(366\) 0 0
\(367\) 10.1541 0.530039 0.265020 0.964243i \(-0.414622\pi\)
0.265020 + 0.964243i \(0.414622\pi\)
\(368\) 12.5909 0.656346
\(369\) 0 0
\(370\) −31.3042 −1.62743
\(371\) 28.0286 1.45517
\(372\) 0 0
\(373\) −24.8297 −1.28563 −0.642817 0.766020i \(-0.722234\pi\)
−0.642817 + 0.766020i \(0.722234\pi\)
\(374\) 8.37742 0.433186
\(375\) 0 0
\(376\) 29.0976 1.50059
\(377\) 26.3546 1.35733
\(378\) 0 0
\(379\) 12.2701 0.630271 0.315136 0.949047i \(-0.397950\pi\)
0.315136 + 0.949047i \(0.397950\pi\)
\(380\) −3.99335 −0.204855
\(381\) 0 0
\(382\) 31.0702 1.58969
\(383\) 22.3129 1.14014 0.570068 0.821597i \(-0.306917\pi\)
0.570068 + 0.821597i \(0.306917\pi\)
\(384\) 0 0
\(385\) 9.46029 0.482141
\(386\) −14.8547 −0.756087
\(387\) 0 0
\(388\) −2.45175 −0.124469
\(389\) −24.6635 −1.25049 −0.625245 0.780429i \(-0.715001\pi\)
−0.625245 + 0.780429i \(0.715001\pi\)
\(390\) 0 0
\(391\) 16.7662 0.847903
\(392\) 2.53902 0.128240
\(393\) 0 0
\(394\) −7.05839 −0.355597
\(395\) 49.7779 2.50460
\(396\) 0 0
\(397\) −3.99794 −0.200651 −0.100325 0.994955i \(-0.531988\pi\)
−0.100325 + 0.994955i \(0.531988\pi\)
\(398\) 19.2268 0.963754
\(399\) 0 0
\(400\) −41.8380 −2.09190
\(401\) −1.45172 −0.0724952 −0.0362476 0.999343i \(-0.511540\pi\)
−0.0362476 + 0.999343i \(0.511540\pi\)
\(402\) 0 0
\(403\) −1.08086 −0.0538416
\(404\) −1.19668 −0.0595372
\(405\) 0 0
\(406\) −20.4987 −1.01733
\(407\) −5.55034 −0.275120
\(408\) 0 0
\(409\) 13.2091 0.653147 0.326574 0.945172i \(-0.394106\pi\)
0.326574 + 0.945172i \(0.394106\pi\)
\(410\) −24.5866 −1.21425
\(411\) 0 0
\(412\) 2.49835 0.123085
\(413\) 32.5105 1.59974
\(414\) 0 0
\(415\) −28.5282 −1.40039
\(416\) 4.06167 0.199140
\(417\) 0 0
\(418\) −9.85276 −0.481914
\(419\) 12.6761 0.619267 0.309634 0.950856i \(-0.399794\pi\)
0.309634 + 0.950856i \(0.399794\pi\)
\(420\) 0 0
\(421\) −4.36098 −0.212541 −0.106271 0.994337i \(-0.533891\pi\)
−0.106271 + 0.994337i \(0.533891\pi\)
\(422\) −9.33809 −0.454571
\(423\) 0 0
\(424\) −30.8328 −1.49737
\(425\) −55.7120 −2.70243
\(426\) 0 0
\(427\) −2.46223 −0.119156
\(428\) −0.177579 −0.00858359
\(429\) 0 0
\(430\) 52.7815 2.54535
\(431\) 1.72411 0.0830477 0.0415238 0.999138i \(-0.486779\pi\)
0.0415238 + 0.999138i \(0.486779\pi\)
\(432\) 0 0
\(433\) −6.68214 −0.321123 −0.160561 0.987026i \(-0.551330\pi\)
−0.160561 + 0.987026i \(0.551330\pi\)
\(434\) 0.840696 0.0403547
\(435\) 0 0
\(436\) 1.82039 0.0871807
\(437\) −19.7189 −0.943281
\(438\) 0 0
\(439\) −4.62223 −0.220607 −0.110303 0.993898i \(-0.535182\pi\)
−0.110303 + 0.993898i \(0.535182\pi\)
\(440\) −10.4068 −0.496123
\(441\) 0 0
\(442\) 38.9296 1.85169
\(443\) −24.7151 −1.17425 −0.587125 0.809497i \(-0.699740\pi\)
−0.587125 + 0.809497i \(0.699740\pi\)
\(444\) 0 0
\(445\) −13.0795 −0.620027
\(446\) 33.0810 1.56643
\(447\) 0 0
\(448\) 17.9457 0.847856
\(449\) 17.0015 0.802350 0.401175 0.916002i \(-0.368602\pi\)
0.401175 + 0.916002i \(0.368602\pi\)
\(450\) 0 0
\(451\) −4.35928 −0.205271
\(452\) 2.54335 0.119629
\(453\) 0 0
\(454\) −39.8817 −1.87174
\(455\) 43.9616 2.06095
\(456\) 0 0
\(457\) −31.3936 −1.46853 −0.734265 0.678863i \(-0.762473\pi\)
−0.734265 + 0.678863i \(0.762473\pi\)
\(458\) −36.1254 −1.68803
\(459\) 0 0
\(460\) 1.74792 0.0814971
\(461\) −4.61978 −0.215165 −0.107582 0.994196i \(-0.534311\pi\)
−0.107582 + 0.994196i \(0.534311\pi\)
\(462\) 0 0
\(463\) 27.6857 1.28666 0.643331 0.765588i \(-0.277552\pi\)
0.643331 + 0.765588i \(0.277552\pi\)
\(464\) 24.3059 1.12837
\(465\) 0 0
\(466\) −35.2998 −1.63523
\(467\) −3.83606 −0.177512 −0.0887558 0.996053i \(-0.528289\pi\)
−0.0887558 + 0.996053i \(0.528289\pi\)
\(468\) 0 0
\(469\) 30.9992 1.43141
\(470\) 60.5899 2.79480
\(471\) 0 0
\(472\) −35.7630 −1.64613
\(473\) 9.35832 0.430296
\(474\) 0 0
\(475\) 65.5233 3.00641
\(476\) −2.17593 −0.0997335
\(477\) 0 0
\(478\) −23.8841 −1.09243
\(479\) −19.8463 −0.906801 −0.453400 0.891307i \(-0.649789\pi\)
−0.453400 + 0.891307i \(0.649789\pi\)
\(480\) 0 0
\(481\) −25.7922 −1.17602
\(482\) 2.25768 0.102834
\(483\) 0 0
\(484\) 0.154851 0.00703867
\(485\) 60.8329 2.76228
\(486\) 0 0
\(487\) −18.7149 −0.848054 −0.424027 0.905650i \(-0.639384\pi\)
−0.424027 + 0.905650i \(0.639384\pi\)
\(488\) 2.70857 0.122611
\(489\) 0 0
\(490\) 5.28701 0.238843
\(491\) 33.1187 1.49463 0.747313 0.664472i \(-0.231344\pi\)
0.747313 + 0.664472i \(0.231344\pi\)
\(492\) 0 0
\(493\) 32.3661 1.45769
\(494\) −45.7854 −2.05998
\(495\) 0 0
\(496\) −0.996839 −0.0447594
\(497\) 0.738103 0.0331085
\(498\) 0 0
\(499\) 3.36699 0.150727 0.0753635 0.997156i \(-0.475988\pi\)
0.0753635 + 0.997156i \(0.475988\pi\)
\(500\) −2.83331 −0.126709
\(501\) 0 0
\(502\) 33.5774 1.49863
\(503\) −15.7130 −0.700609 −0.350305 0.936636i \(-0.613922\pi\)
−0.350305 + 0.936636i \(0.613922\pi\)
\(504\) 0 0
\(505\) 29.6921 1.32128
\(506\) 4.31262 0.191719
\(507\) 0 0
\(508\) 1.84732 0.0819617
\(509\) 7.97241 0.353371 0.176685 0.984267i \(-0.443462\pi\)
0.176685 + 0.984267i \(0.443462\pi\)
\(510\) 0 0
\(511\) −6.53644 −0.289155
\(512\) −19.4704 −0.860480
\(513\) 0 0
\(514\) 14.8301 0.654127
\(515\) −61.9889 −2.73156
\(516\) 0 0
\(517\) 10.7428 0.472467
\(518\) 20.0612 0.881439
\(519\) 0 0
\(520\) −48.3598 −2.12072
\(521\) 19.8637 0.870244 0.435122 0.900372i \(-0.356705\pi\)
0.435122 + 0.900372i \(0.356705\pi\)
\(522\) 0 0
\(523\) 10.5930 0.463197 0.231599 0.972811i \(-0.425604\pi\)
0.231599 + 0.972811i \(0.425604\pi\)
\(524\) −3.23070 −0.141134
\(525\) 0 0
\(526\) 16.3752 0.713991
\(527\) −1.32740 −0.0578226
\(528\) 0 0
\(529\) −14.3689 −0.624736
\(530\) −64.2031 −2.78880
\(531\) 0 0
\(532\) 2.55913 0.110952
\(533\) −20.2574 −0.877446
\(534\) 0 0
\(535\) 4.40608 0.190492
\(536\) −34.1006 −1.47292
\(537\) 0 0
\(538\) −8.86566 −0.382226
\(539\) 0.937404 0.0403768
\(540\) 0 0
\(541\) 0.967556 0.0415985 0.0207993 0.999784i \(-0.493379\pi\)
0.0207993 + 0.999784i \(0.493379\pi\)
\(542\) 6.25404 0.268634
\(543\) 0 0
\(544\) 4.98812 0.213864
\(545\) −45.1674 −1.93476
\(546\) 0 0
\(547\) 2.94593 0.125959 0.0629794 0.998015i \(-0.479940\pi\)
0.0629794 + 0.998015i \(0.479940\pi\)
\(548\) 2.96689 0.126739
\(549\) 0 0
\(550\) −14.3303 −0.611046
\(551\) −38.0660 −1.62167
\(552\) 0 0
\(553\) −31.9000 −1.35653
\(554\) 12.3307 0.523880
\(555\) 0 0
\(556\) −0.399828 −0.0169565
\(557\) −0.0922439 −0.00390850 −0.00195425 0.999998i \(-0.500622\pi\)
−0.00195425 + 0.999998i \(0.500622\pi\)
\(558\) 0 0
\(559\) 43.4877 1.83934
\(560\) 40.5442 1.71330
\(561\) 0 0
\(562\) −18.1274 −0.764658
\(563\) −16.5637 −0.698079 −0.349039 0.937108i \(-0.613492\pi\)
−0.349039 + 0.937108i \(0.613492\pi\)
\(564\) 0 0
\(565\) −63.1057 −2.65488
\(566\) 10.7448 0.451639
\(567\) 0 0
\(568\) −0.811948 −0.0340686
\(569\) 44.9171 1.88302 0.941512 0.336980i \(-0.109406\pi\)
0.941512 + 0.336980i \(0.109406\pi\)
\(570\) 0 0
\(571\) 11.9694 0.500903 0.250452 0.968129i \(-0.419421\pi\)
0.250452 + 0.968129i \(0.419421\pi\)
\(572\) 0.719586 0.0300874
\(573\) 0 0
\(574\) 15.7563 0.657653
\(575\) −28.6800 −1.19604
\(576\) 0 0
\(577\) 1.73895 0.0723935 0.0361967 0.999345i \(-0.488476\pi\)
0.0361967 + 0.999345i \(0.488476\pi\)
\(578\) 22.8543 0.950613
\(579\) 0 0
\(580\) 3.37424 0.140108
\(581\) 18.2822 0.758474
\(582\) 0 0
\(583\) −11.3834 −0.471453
\(584\) 7.19039 0.297540
\(585\) 0 0
\(586\) −24.7566 −1.02269
\(587\) −34.0791 −1.40660 −0.703298 0.710895i \(-0.748290\pi\)
−0.703298 + 0.710895i \(0.748290\pi\)
\(588\) 0 0
\(589\) 1.56117 0.0643269
\(590\) −74.4694 −3.06586
\(591\) 0 0
\(592\) −23.7872 −0.977648
\(593\) 19.1334 0.785714 0.392857 0.919600i \(-0.371487\pi\)
0.392857 + 0.919600i \(0.371487\pi\)
\(594\) 0 0
\(595\) 53.9891 2.21334
\(596\) −1.46445 −0.0599863
\(597\) 0 0
\(598\) 20.0406 0.819520
\(599\) −11.7116 −0.478521 −0.239261 0.970955i \(-0.576905\pi\)
−0.239261 + 0.970955i \(0.576905\pi\)
\(600\) 0 0
\(601\) −20.3403 −0.829698 −0.414849 0.909890i \(-0.636166\pi\)
−0.414849 + 0.909890i \(0.636166\pi\)
\(602\) −33.8248 −1.37860
\(603\) 0 0
\(604\) 0.954793 0.0388500
\(605\) −3.84216 −0.156206
\(606\) 0 0
\(607\) 18.2830 0.742085 0.371042 0.928616i \(-0.379000\pi\)
0.371042 + 0.928616i \(0.379000\pi\)
\(608\) −5.86657 −0.237921
\(609\) 0 0
\(610\) 5.64006 0.228359
\(611\) 49.9213 2.01960
\(612\) 0 0
\(613\) −33.4060 −1.34925 −0.674627 0.738158i \(-0.735696\pi\)
−0.674627 + 0.738158i \(0.735696\pi\)
\(614\) 29.6712 1.19743
\(615\) 0 0
\(616\) 6.66913 0.268707
\(617\) 37.0969 1.49347 0.746733 0.665124i \(-0.231621\pi\)
0.746733 + 0.665124i \(0.231621\pi\)
\(618\) 0 0
\(619\) −35.5774 −1.42998 −0.714988 0.699137i \(-0.753568\pi\)
−0.714988 + 0.699137i \(0.753568\pi\)
\(620\) −0.138385 −0.00555768
\(621\) 0 0
\(622\) 40.4473 1.62179
\(623\) 8.38194 0.335815
\(624\) 0 0
\(625\) 21.4892 0.859568
\(626\) 26.4263 1.05621
\(627\) 0 0
\(628\) 0.0735919 0.00293663
\(629\) −31.6753 −1.26298
\(630\) 0 0
\(631\) 3.96808 0.157967 0.0789834 0.996876i \(-0.474833\pi\)
0.0789834 + 0.996876i \(0.474833\pi\)
\(632\) 35.0914 1.39586
\(633\) 0 0
\(634\) −43.3140 −1.72022
\(635\) −45.8358 −1.81894
\(636\) 0 0
\(637\) 4.35608 0.172594
\(638\) 8.32523 0.329599
\(639\) 0 0
\(640\) −47.8235 −1.89039
\(641\) 4.20491 0.166084 0.0830419 0.996546i \(-0.473536\pi\)
0.0830419 + 0.996546i \(0.473536\pi\)
\(642\) 0 0
\(643\) 29.2183 1.15226 0.576129 0.817359i \(-0.304563\pi\)
0.576129 + 0.817359i \(0.304563\pi\)
\(644\) −1.12015 −0.0441400
\(645\) 0 0
\(646\) −56.2289 −2.21230
\(647\) −8.11234 −0.318929 −0.159464 0.987204i \(-0.550977\pi\)
−0.159464 + 0.987204i \(0.550977\pi\)
\(648\) 0 0
\(649\) −13.2037 −0.518289
\(650\) −66.5923 −2.61197
\(651\) 0 0
\(652\) 2.73501 0.107111
\(653\) 23.5399 0.921189 0.460595 0.887611i \(-0.347636\pi\)
0.460595 + 0.887611i \(0.347636\pi\)
\(654\) 0 0
\(655\) 80.1600 3.13211
\(656\) −18.6827 −0.729436
\(657\) 0 0
\(658\) −38.8288 −1.51371
\(659\) 46.8281 1.82416 0.912082 0.410007i \(-0.134474\pi\)
0.912082 + 0.410007i \(0.134474\pi\)
\(660\) 0 0
\(661\) −45.2831 −1.76131 −0.880655 0.473758i \(-0.842897\pi\)
−0.880655 + 0.473758i \(0.842897\pi\)
\(662\) 26.1955 1.01812
\(663\) 0 0
\(664\) −20.1113 −0.780468
\(665\) −63.4971 −2.46231
\(666\) 0 0
\(667\) 16.6617 0.645145
\(668\) 2.47552 0.0957808
\(669\) 0 0
\(670\) −71.0077 −2.74327
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 42.7822 1.64913 0.824566 0.565765i \(-0.191419\pi\)
0.824566 + 0.565765i \(0.191419\pi\)
\(674\) −22.2005 −0.855130
\(675\) 0 0
\(676\) 1.33083 0.0511856
\(677\) −32.8918 −1.26414 −0.632068 0.774913i \(-0.717794\pi\)
−0.632068 + 0.774913i \(0.717794\pi\)
\(678\) 0 0
\(679\) −38.9845 −1.49609
\(680\) −59.3905 −2.27752
\(681\) 0 0
\(682\) −0.341436 −0.0130743
\(683\) 29.9574 1.14629 0.573144 0.819455i \(-0.305724\pi\)
0.573144 + 0.819455i \(0.305724\pi\)
\(684\) 0 0
\(685\) −73.6144 −2.81266
\(686\) −28.6891 −1.09535
\(687\) 0 0
\(688\) 40.1072 1.52907
\(689\) −52.8983 −2.01526
\(690\) 0 0
\(691\) −46.0051 −1.75011 −0.875057 0.484019i \(-0.839177\pi\)
−0.875057 + 0.484019i \(0.839177\pi\)
\(692\) 0.344173 0.0130835
\(693\) 0 0
\(694\) −16.5414 −0.627901
\(695\) 9.92054 0.376308
\(696\) 0 0
\(697\) −24.8781 −0.942324
\(698\) 9.46117 0.358111
\(699\) 0 0
\(700\) 3.72211 0.140683
\(701\) 4.97041 0.187730 0.0938650 0.995585i \(-0.470078\pi\)
0.0938650 + 0.995585i \(0.470078\pi\)
\(702\) 0 0
\(703\) 37.2536 1.40505
\(704\) −7.28839 −0.274692
\(705\) 0 0
\(706\) 33.6692 1.26716
\(707\) −19.0281 −0.715625
\(708\) 0 0
\(709\) −31.0750 −1.16704 −0.583522 0.812097i \(-0.698326\pi\)
−0.583522 + 0.812097i \(0.698326\pi\)
\(710\) −1.69072 −0.0634516
\(711\) 0 0
\(712\) −9.22052 −0.345554
\(713\) −0.683335 −0.0255911
\(714\) 0 0
\(715\) −17.8544 −0.667715
\(716\) 3.79555 0.141846
\(717\) 0 0
\(718\) −16.5414 −0.617318
\(719\) 37.4717 1.39746 0.698729 0.715386i \(-0.253749\pi\)
0.698729 + 0.715386i \(0.253749\pi\)
\(720\) 0 0
\(721\) 39.7254 1.47945
\(722\) 38.2404 1.42316
\(723\) 0 0
\(724\) −2.18817 −0.0813226
\(725\) −55.3649 −2.05620
\(726\) 0 0
\(727\) −7.56311 −0.280500 −0.140250 0.990116i \(-0.544791\pi\)
−0.140250 + 0.990116i \(0.544791\pi\)
\(728\) 30.9912 1.14861
\(729\) 0 0
\(730\) 14.9726 0.554159
\(731\) 53.4072 1.97534
\(732\) 0 0
\(733\) −12.6942 −0.468869 −0.234435 0.972132i \(-0.575324\pi\)
−0.234435 + 0.972132i \(0.575324\pi\)
\(734\) 14.9056 0.550176
\(735\) 0 0
\(736\) 2.56784 0.0946518
\(737\) −12.5899 −0.463754
\(738\) 0 0
\(739\) 13.3928 0.492662 0.246331 0.969186i \(-0.420775\pi\)
0.246331 + 0.969186i \(0.420775\pi\)
\(740\) −3.30223 −0.121393
\(741\) 0 0
\(742\) 41.1444 1.51046
\(743\) 3.77240 0.138396 0.0691980 0.997603i \(-0.477956\pi\)
0.0691980 + 0.997603i \(0.477956\pi\)
\(744\) 0 0
\(745\) 36.3360 1.33125
\(746\) −36.4486 −1.33448
\(747\) 0 0
\(748\) 0.883721 0.0323120
\(749\) −2.82362 −0.103173
\(750\) 0 0
\(751\) −9.51843 −0.347332 −0.173666 0.984805i \(-0.555561\pi\)
−0.173666 + 0.984805i \(0.555561\pi\)
\(752\) 46.0406 1.67893
\(753\) 0 0
\(754\) 38.6870 1.40890
\(755\) −23.6903 −0.862179
\(756\) 0 0
\(757\) −10.7916 −0.392229 −0.196114 0.980581i \(-0.562832\pi\)
−0.196114 + 0.980581i \(0.562832\pi\)
\(758\) 18.0117 0.654216
\(759\) 0 0
\(760\) 69.8497 2.53371
\(761\) 22.1374 0.802480 0.401240 0.915973i \(-0.368579\pi\)
0.401240 + 0.915973i \(0.368579\pi\)
\(762\) 0 0
\(763\) 28.9454 1.04789
\(764\) 3.27755 0.118577
\(765\) 0 0
\(766\) 32.7540 1.18345
\(767\) −61.3569 −2.21547
\(768\) 0 0
\(769\) 19.4521 0.701460 0.350730 0.936477i \(-0.385934\pi\)
0.350730 + 0.936477i \(0.385934\pi\)
\(770\) 13.8871 0.500458
\(771\) 0 0
\(772\) −1.56700 −0.0563977
\(773\) −39.4207 −1.41786 −0.708932 0.705277i \(-0.750823\pi\)
−0.708932 + 0.705277i \(0.750823\pi\)
\(774\) 0 0
\(775\) 2.27064 0.0815637
\(776\) 42.8848 1.53947
\(777\) 0 0
\(778\) −36.2046 −1.29800
\(779\) 29.2593 1.04832
\(780\) 0 0
\(781\) −0.299770 −0.0107266
\(782\) 24.6118 0.880116
\(783\) 0 0
\(784\) 4.01745 0.143480
\(785\) −1.82596 −0.0651714
\(786\) 0 0
\(787\) −37.7502 −1.34565 −0.672824 0.739802i \(-0.734919\pi\)
−0.672824 + 0.739802i \(0.734919\pi\)
\(788\) −0.744578 −0.0265245
\(789\) 0 0
\(790\) 73.0710 2.59975
\(791\) 40.4411 1.43792
\(792\) 0 0
\(793\) 4.64696 0.165018
\(794\) −5.86874 −0.208274
\(795\) 0 0
\(796\) 2.02821 0.0718879
\(797\) 20.3463 0.720705 0.360352 0.932816i \(-0.382656\pi\)
0.360352 + 0.932816i \(0.382656\pi\)
\(798\) 0 0
\(799\) 61.3082 2.16893
\(800\) −8.53261 −0.301673
\(801\) 0 0
\(802\) −2.13103 −0.0752494
\(803\) 2.65468 0.0936816
\(804\) 0 0
\(805\) 27.7931 0.979578
\(806\) −1.58664 −0.0558870
\(807\) 0 0
\(808\) 20.9318 0.736377
\(809\) 3.16766 0.111369 0.0556844 0.998448i \(-0.482266\pi\)
0.0556844 + 0.998448i \(0.482266\pi\)
\(810\) 0 0
\(811\) −14.3129 −0.502593 −0.251296 0.967910i \(-0.580857\pi\)
−0.251296 + 0.967910i \(0.580857\pi\)
\(812\) −2.16237 −0.0758844
\(813\) 0 0
\(814\) −8.14757 −0.285572
\(815\) −67.8612 −2.37707
\(816\) 0 0
\(817\) −62.8126 −2.19754
\(818\) 19.3902 0.677961
\(819\) 0 0
\(820\) −2.59360 −0.0905726
\(821\) 14.3547 0.500982 0.250491 0.968119i \(-0.419408\pi\)
0.250491 + 0.968119i \(0.419408\pi\)
\(822\) 0 0
\(823\) 21.5880 0.752509 0.376255 0.926516i \(-0.377212\pi\)
0.376255 + 0.926516i \(0.377212\pi\)
\(824\) −43.6998 −1.52235
\(825\) 0 0
\(826\) 47.7235 1.66051
\(827\) −27.7946 −0.966513 −0.483257 0.875479i \(-0.660546\pi\)
−0.483257 + 0.875479i \(0.660546\pi\)
\(828\) 0 0
\(829\) 18.2361 0.633368 0.316684 0.948531i \(-0.397431\pi\)
0.316684 + 0.948531i \(0.397431\pi\)
\(830\) −41.8777 −1.45360
\(831\) 0 0
\(832\) −33.8689 −1.17419
\(833\) 5.34969 0.185356
\(834\) 0 0
\(835\) −61.4227 −2.12562
\(836\) −1.03935 −0.0359467
\(837\) 0 0
\(838\) 18.6078 0.642794
\(839\) −25.3815 −0.876268 −0.438134 0.898910i \(-0.644360\pi\)
−0.438134 + 0.898910i \(0.644360\pi\)
\(840\) 0 0
\(841\) 3.16442 0.109118
\(842\) −6.40166 −0.220616
\(843\) 0 0
\(844\) −0.985060 −0.0339072
\(845\) −33.0205 −1.13594
\(846\) 0 0
\(847\) 2.46223 0.0846034
\(848\) −48.7861 −1.67532
\(849\) 0 0
\(850\) −81.7819 −2.80510
\(851\) −16.3062 −0.558968
\(852\) 0 0
\(853\) −50.9440 −1.74429 −0.872144 0.489249i \(-0.837271\pi\)
−0.872144 + 0.489249i \(0.837271\pi\)
\(854\) −3.61441 −0.123683
\(855\) 0 0
\(856\) 3.10612 0.106165
\(857\) −25.5195 −0.871729 −0.435865 0.900012i \(-0.643557\pi\)
−0.435865 + 0.900012i \(0.643557\pi\)
\(858\) 0 0
\(859\) 19.5049 0.665497 0.332748 0.943016i \(-0.392024\pi\)
0.332748 + 0.943016i \(0.392024\pi\)
\(860\) 5.56784 0.189862
\(861\) 0 0
\(862\) 2.53090 0.0862027
\(863\) 23.2335 0.790876 0.395438 0.918493i \(-0.370593\pi\)
0.395438 + 0.918493i \(0.370593\pi\)
\(864\) 0 0
\(865\) −8.53962 −0.290356
\(866\) −9.80898 −0.333323
\(867\) 0 0
\(868\) 0.0886837 0.00301012
\(869\) 12.9557 0.439492
\(870\) 0 0
\(871\) −58.5047 −1.98235
\(872\) −31.8413 −1.07828
\(873\) 0 0
\(874\) −28.9461 −0.979117
\(875\) −45.0516 −1.52302
\(876\) 0 0
\(877\) 7.94166 0.268171 0.134085 0.990970i \(-0.457190\pi\)
0.134085 + 0.990970i \(0.457190\pi\)
\(878\) −6.78516 −0.228988
\(879\) 0 0
\(880\) −16.4664 −0.555083
\(881\) −8.19164 −0.275983 −0.137992 0.990433i \(-0.544065\pi\)
−0.137992 + 0.990433i \(0.544065\pi\)
\(882\) 0 0
\(883\) 16.4576 0.553844 0.276922 0.960892i \(-0.410686\pi\)
0.276922 + 0.960892i \(0.410686\pi\)
\(884\) 4.10662 0.138121
\(885\) 0 0
\(886\) −36.2803 −1.21886
\(887\) −13.3032 −0.446679 −0.223340 0.974741i \(-0.571696\pi\)
−0.223340 + 0.974741i \(0.571696\pi\)
\(888\) 0 0
\(889\) 29.3737 0.985162
\(890\) −19.1999 −0.643582
\(891\) 0 0
\(892\) 3.48966 0.116842
\(893\) −72.1050 −2.41290
\(894\) 0 0
\(895\) −94.1752 −3.14793
\(896\) 30.6475 1.02386
\(897\) 0 0
\(898\) 24.9572 0.832832
\(899\) −1.31913 −0.0439956
\(900\) 0 0
\(901\) −64.9642 −2.16427
\(902\) −6.39917 −0.213069
\(903\) 0 0
\(904\) −44.4871 −1.47962
\(905\) 54.2928 1.80476
\(906\) 0 0
\(907\) 6.59934 0.219128 0.109564 0.993980i \(-0.465055\pi\)
0.109564 + 0.993980i \(0.465055\pi\)
\(908\) −4.20706 −0.139616
\(909\) 0 0
\(910\) 64.5330 2.13925
\(911\) 45.1362 1.49543 0.747714 0.664021i \(-0.231152\pi\)
0.747714 + 0.664021i \(0.231152\pi\)
\(912\) 0 0
\(913\) −7.42505 −0.245733
\(914\) −46.0840 −1.52432
\(915\) 0 0
\(916\) −3.81082 −0.125913
\(917\) −51.3703 −1.69640
\(918\) 0 0
\(919\) 22.6447 0.746979 0.373490 0.927634i \(-0.378161\pi\)
0.373490 + 0.927634i \(0.378161\pi\)
\(920\) −30.5737 −1.00798
\(921\) 0 0
\(922\) −6.78156 −0.223339
\(923\) −1.39302 −0.0458518
\(924\) 0 0
\(925\) 54.1834 1.78154
\(926\) 40.6409 1.33554
\(927\) 0 0
\(928\) 4.95705 0.162723
\(929\) 11.5176 0.377879 0.188940 0.981989i \(-0.439495\pi\)
0.188940 + 0.981989i \(0.439495\pi\)
\(930\) 0 0
\(931\) −6.29181 −0.206206
\(932\) −3.72372 −0.121974
\(933\) 0 0
\(934\) −5.63111 −0.184255
\(935\) −21.9269 −0.717086
\(936\) 0 0
\(937\) −7.74043 −0.252869 −0.126434 0.991975i \(-0.540353\pi\)
−0.126434 + 0.991975i \(0.540353\pi\)
\(938\) 45.5050 1.48579
\(939\) 0 0
\(940\) 6.39153 0.208469
\(941\) 31.6675 1.03233 0.516166 0.856489i \(-0.327359\pi\)
0.516166 + 0.856489i \(0.327359\pi\)
\(942\) 0 0
\(943\) −12.8070 −0.417053
\(944\) −56.5872 −1.84176
\(945\) 0 0
\(946\) 13.7375 0.446643
\(947\) −32.2734 −1.04874 −0.524372 0.851489i \(-0.675700\pi\)
−0.524372 + 0.851489i \(0.675700\pi\)
\(948\) 0 0
\(949\) 12.3362 0.400450
\(950\) 96.1843 3.12063
\(951\) 0 0
\(952\) 38.0602 1.23354
\(953\) −32.3994 −1.04952 −0.524759 0.851251i \(-0.675845\pi\)
−0.524759 + 0.851251i \(0.675845\pi\)
\(954\) 0 0
\(955\) −81.3225 −2.63153
\(956\) −2.51949 −0.0814862
\(957\) 0 0
\(958\) −29.1332 −0.941251
\(959\) 47.1755 1.52338
\(960\) 0 0
\(961\) −30.9459 −0.998255
\(962\) −37.8614 −1.22070
\(963\) 0 0
\(964\) 0.238159 0.00767058
\(965\) 38.8805 1.25161
\(966\) 0 0
\(967\) −54.5762 −1.75505 −0.877526 0.479529i \(-0.840808\pi\)
−0.877526 + 0.479529i \(0.840808\pi\)
\(968\) −2.70857 −0.0870567
\(969\) 0 0
\(970\) 89.2990 2.86722
\(971\) −3.84654 −0.123441 −0.0617207 0.998093i \(-0.519659\pi\)
−0.0617207 + 0.998093i \(0.519659\pi\)
\(972\) 0 0
\(973\) −6.35755 −0.203814
\(974\) −27.4724 −0.880272
\(975\) 0 0
\(976\) 4.28572 0.137183
\(977\) 3.97466 0.127161 0.0635804 0.997977i \(-0.479748\pi\)
0.0635804 + 0.997977i \(0.479748\pi\)
\(978\) 0 0
\(979\) −3.40420 −0.108799
\(980\) 0.557719 0.0178157
\(981\) 0 0
\(982\) 48.6163 1.55141
\(983\) −3.16990 −0.101104 −0.0505521 0.998721i \(-0.516098\pi\)
−0.0505521 + 0.998721i \(0.516098\pi\)
\(984\) 0 0
\(985\) 18.4745 0.588646
\(986\) 47.5115 1.51307
\(987\) 0 0
\(988\) −4.82983 −0.153657
\(989\) 27.4935 0.874243
\(990\) 0 0
\(991\) 25.3534 0.805378 0.402689 0.915337i \(-0.368076\pi\)
0.402689 + 0.915337i \(0.368076\pi\)
\(992\) −0.203299 −0.00645476
\(993\) 0 0
\(994\) 1.08349 0.0343663
\(995\) −50.3239 −1.59537
\(996\) 0 0
\(997\) 5.71711 0.181063 0.0905314 0.995894i \(-0.471143\pi\)
0.0905314 + 0.995894i \(0.471143\pi\)
\(998\) 4.94254 0.156453
\(999\) 0 0
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 6039.2.a.i.1.11 13
3.2 odd 2 2013.2.a.e.1.3 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.3 13 3.2 odd 2
6039.2.a.i.1.11 13 1.1 even 1 trivial