Properties

Label 2151.2.a.j.1.16
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(-1.38158\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.38158 q^{2} -0.0912347 q^{4} +1.35155 q^{5} -1.50510 q^{7} -2.88921 q^{8} +O(q^{10})\) \(q+1.38158 q^{2} -0.0912347 q^{4} +1.35155 q^{5} -1.50510 q^{7} -2.88921 q^{8} +1.86728 q^{10} +5.20916 q^{11} -6.03908 q^{13} -2.07942 q^{14} -3.80921 q^{16} -2.23551 q^{17} -5.35678 q^{19} -0.123308 q^{20} +7.19688 q^{22} -2.95682 q^{23} -3.17331 q^{25} -8.34348 q^{26} +0.137318 q^{28} +0.529336 q^{29} -3.72398 q^{31} +0.515693 q^{32} -3.08854 q^{34} -2.03422 q^{35} +5.79583 q^{37} -7.40083 q^{38} -3.90492 q^{40} +4.00191 q^{41} +4.50457 q^{43} -0.475257 q^{44} -4.08509 q^{46} -11.3720 q^{47} -4.73466 q^{49} -4.38418 q^{50} +0.550974 q^{52} +8.22971 q^{53} +7.04045 q^{55} +4.34856 q^{56} +0.731321 q^{58} -5.35349 q^{59} -14.0242 q^{61} -5.14497 q^{62} +8.33088 q^{64} -8.16213 q^{65} -9.35409 q^{67} +0.203956 q^{68} -2.81045 q^{70} +12.5000 q^{71} +10.4873 q^{73} +8.00741 q^{74} +0.488725 q^{76} -7.84033 q^{77} -5.72701 q^{79} -5.14834 q^{80} +5.52897 q^{82} +4.70998 q^{83} -3.02141 q^{85} +6.22343 q^{86} -15.0504 q^{88} -14.4107 q^{89} +9.08944 q^{91} +0.269765 q^{92} -15.7113 q^{94} -7.23997 q^{95} -12.2286 q^{97} -6.54132 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + O(q^{10}) \) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + 4q^{10} - 12q^{11} - 4q^{13} - 20q^{14} + 22q^{16} - 24q^{17} - 4q^{19} - 40q^{20} - 6q^{22} - 12q^{23} + 22q^{25} - 30q^{26} - 12q^{28} - 24q^{29} - 4q^{31} - 28q^{32} + 8q^{34} - 20q^{35} - 10q^{37} - 26q^{38} + 6q^{40} - 66q^{41} + 8q^{43} - 36q^{44} - 12q^{46} - 28q^{47} + 18q^{49} - 28q^{50} - 18q^{52} - 28q^{53} - 4q^{55} - 60q^{56} - 54q^{59} - 4q^{61} - 20q^{62} + 22q^{64} - 42q^{65} + 12q^{67} - 12q^{68} + 20q^{70} - 36q^{71} + 14q^{73} - 50q^{76} - 8q^{77} - 12q^{79} - 88q^{80} - 8q^{82} - 20q^{83} + 4q^{85} - 18q^{86} - 10q^{88} - 130q^{89} - 6q^{91} + 46q^{92} - 26q^{94} - 2q^{97} - 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.38158 0.976925 0.488463 0.872585i \(-0.337558\pi\)
0.488463 + 0.872585i \(0.337558\pi\)
\(3\) 0 0
\(4\) −0.0912347 −0.0456174
\(5\) 1.35155 0.604432 0.302216 0.953239i \(-0.402274\pi\)
0.302216 + 0.953239i \(0.402274\pi\)
\(6\) 0 0
\(7\) −1.50510 −0.568876 −0.284438 0.958695i \(-0.591807\pi\)
−0.284438 + 0.958695i \(0.591807\pi\)
\(8\) −2.88921 −1.02149
\(9\) 0 0
\(10\) 1.86728 0.590485
\(11\) 5.20916 1.57062 0.785311 0.619102i \(-0.212503\pi\)
0.785311 + 0.619102i \(0.212503\pi\)
\(12\) 0 0
\(13\) −6.03908 −1.67494 −0.837470 0.546483i \(-0.815966\pi\)
−0.837470 + 0.546483i \(0.815966\pi\)
\(14\) −2.07942 −0.555749
\(15\) 0 0
\(16\) −3.80921 −0.952302
\(17\) −2.23551 −0.542191 −0.271096 0.962552i \(-0.587386\pi\)
−0.271096 + 0.962552i \(0.587386\pi\)
\(18\) 0 0
\(19\) −5.35678 −1.22893 −0.614465 0.788944i \(-0.710628\pi\)
−0.614465 + 0.788944i \(0.710628\pi\)
\(20\) −0.123308 −0.0275726
\(21\) 0 0
\(22\) 7.19688 1.53438
\(23\) −2.95682 −0.616541 −0.308270 0.951299i \(-0.599750\pi\)
−0.308270 + 0.951299i \(0.599750\pi\)
\(24\) 0 0
\(25\) −3.17331 −0.634662
\(26\) −8.34348 −1.63629
\(27\) 0 0
\(28\) 0.137318 0.0259506
\(29\) 0.529336 0.0982953 0.0491477 0.998792i \(-0.484350\pi\)
0.0491477 + 0.998792i \(0.484350\pi\)
\(30\) 0 0
\(31\) −3.72398 −0.668846 −0.334423 0.942423i \(-0.608541\pi\)
−0.334423 + 0.942423i \(0.608541\pi\)
\(32\) 0.515693 0.0911624
\(33\) 0 0
\(34\) −3.08854 −0.529680
\(35\) −2.03422 −0.343847
\(36\) 0 0
\(37\) 5.79583 0.952829 0.476415 0.879221i \(-0.341936\pi\)
0.476415 + 0.879221i \(0.341936\pi\)
\(38\) −7.40083 −1.20057
\(39\) 0 0
\(40\) −3.90492 −0.617422
\(41\) 4.00191 0.624994 0.312497 0.949919i \(-0.398835\pi\)
0.312497 + 0.949919i \(0.398835\pi\)
\(42\) 0 0
\(43\) 4.50457 0.686941 0.343470 0.939163i \(-0.388397\pi\)
0.343470 + 0.939163i \(0.388397\pi\)
\(44\) −0.475257 −0.0716476
\(45\) 0 0
\(46\) −4.08509 −0.602314
\(47\) −11.3720 −1.65878 −0.829388 0.558673i \(-0.811311\pi\)
−0.829388 + 0.558673i \(0.811311\pi\)
\(48\) 0 0
\(49\) −4.73466 −0.676381
\(50\) −4.38418 −0.620017
\(51\) 0 0
\(52\) 0.550974 0.0764064
\(53\) 8.22971 1.13044 0.565219 0.824941i \(-0.308792\pi\)
0.565219 + 0.824941i \(0.308792\pi\)
\(54\) 0 0
\(55\) 7.04045 0.949335
\(56\) 4.34856 0.581101
\(57\) 0 0
\(58\) 0.731321 0.0960272
\(59\) −5.35349 −0.696965 −0.348482 0.937315i \(-0.613303\pi\)
−0.348482 + 0.937315i \(0.613303\pi\)
\(60\) 0 0
\(61\) −14.0242 −1.79562 −0.897808 0.440386i \(-0.854841\pi\)
−0.897808 + 0.440386i \(0.854841\pi\)
\(62\) −5.14497 −0.653412
\(63\) 0 0
\(64\) 8.33088 1.04136
\(65\) −8.16213 −1.01239
\(66\) 0 0
\(67\) −9.35409 −1.14278 −0.571392 0.820677i \(-0.693596\pi\)
−0.571392 + 0.820677i \(0.693596\pi\)
\(68\) 0.203956 0.0247333
\(69\) 0 0
\(70\) −2.81045 −0.335913
\(71\) 12.5000 1.48347 0.741736 0.670692i \(-0.234003\pi\)
0.741736 + 0.670692i \(0.234003\pi\)
\(72\) 0 0
\(73\) 10.4873 1.22745 0.613725 0.789520i \(-0.289670\pi\)
0.613725 + 0.789520i \(0.289670\pi\)
\(74\) 8.00741 0.930843
\(75\) 0 0
\(76\) 0.488725 0.0560606
\(77\) −7.84033 −0.893488
\(78\) 0 0
\(79\) −5.72701 −0.644339 −0.322169 0.946682i \(-0.604412\pi\)
−0.322169 + 0.946682i \(0.604412\pi\)
\(80\) −5.14834 −0.575602
\(81\) 0 0
\(82\) 5.52897 0.610572
\(83\) 4.70998 0.516987 0.258494 0.966013i \(-0.416774\pi\)
0.258494 + 0.966013i \(0.416774\pi\)
\(84\) 0 0
\(85\) −3.02141 −0.327718
\(86\) 6.22343 0.671090
\(87\) 0 0
\(88\) −15.0504 −1.60437
\(89\) −14.4107 −1.52753 −0.763766 0.645493i \(-0.776652\pi\)
−0.763766 + 0.645493i \(0.776652\pi\)
\(90\) 0 0
\(91\) 9.08944 0.952832
\(92\) 0.269765 0.0281250
\(93\) 0 0
\(94\) −15.7113 −1.62050
\(95\) −7.23997 −0.742805
\(96\) 0 0
\(97\) −12.2286 −1.24162 −0.620812 0.783959i \(-0.713197\pi\)
−0.620812 + 0.783959i \(0.713197\pi\)
\(98\) −6.54132 −0.660773
\(99\) 0 0
\(100\) 0.289516 0.0289516
\(101\) −13.1550 −1.30897 −0.654486 0.756074i \(-0.727115\pi\)
−0.654486 + 0.756074i \(0.727115\pi\)
\(102\) 0 0
\(103\) 5.79068 0.570572 0.285286 0.958442i \(-0.407911\pi\)
0.285286 + 0.958442i \(0.407911\pi\)
\(104\) 17.4482 1.71093
\(105\) 0 0
\(106\) 11.3700 1.10435
\(107\) −8.90040 −0.860434 −0.430217 0.902725i \(-0.641563\pi\)
−0.430217 + 0.902725i \(0.641563\pi\)
\(108\) 0 0
\(109\) 9.44148 0.904330 0.452165 0.891934i \(-0.350652\pi\)
0.452165 + 0.891934i \(0.350652\pi\)
\(110\) 9.72696 0.927429
\(111\) 0 0
\(112\) 5.73325 0.541741
\(113\) −0.0439495 −0.00413442 −0.00206721 0.999998i \(-0.500658\pi\)
−0.00206721 + 0.999998i \(0.500658\pi\)
\(114\) 0 0
\(115\) −3.99630 −0.372657
\(116\) −0.0482939 −0.00448397
\(117\) 0 0
\(118\) −7.39628 −0.680883
\(119\) 3.36468 0.308439
\(120\) 0 0
\(121\) 16.1354 1.46685
\(122\) −19.3756 −1.75418
\(123\) 0 0
\(124\) 0.339756 0.0305110
\(125\) −11.0466 −0.988042
\(126\) 0 0
\(127\) 1.52724 0.135520 0.0677602 0.997702i \(-0.478415\pi\)
0.0677602 + 0.997702i \(0.478415\pi\)
\(128\) 10.4784 0.926169
\(129\) 0 0
\(130\) −11.2766 −0.989027
\(131\) −2.05574 −0.179611 −0.0898055 0.995959i \(-0.528625\pi\)
−0.0898055 + 0.995959i \(0.528625\pi\)
\(132\) 0 0
\(133\) 8.06251 0.699108
\(134\) −12.9234 −1.11641
\(135\) 0 0
\(136\) 6.45886 0.553843
\(137\) −11.4680 −0.979774 −0.489887 0.871786i \(-0.662962\pi\)
−0.489887 + 0.871786i \(0.662962\pi\)
\(138\) 0 0
\(139\) −16.4276 −1.39337 −0.696686 0.717376i \(-0.745343\pi\)
−0.696686 + 0.717376i \(0.745343\pi\)
\(140\) 0.185592 0.0156854
\(141\) 0 0
\(142\) 17.2697 1.44924
\(143\) −31.4586 −2.63070
\(144\) 0 0
\(145\) 0.715426 0.0594129
\(146\) 14.4891 1.19913
\(147\) 0 0
\(148\) −0.528781 −0.0434656
\(149\) −14.8822 −1.21919 −0.609597 0.792711i \(-0.708669\pi\)
−0.609597 + 0.792711i \(0.708669\pi\)
\(150\) 0 0
\(151\) 14.5748 1.18608 0.593040 0.805173i \(-0.297928\pi\)
0.593040 + 0.805173i \(0.297928\pi\)
\(152\) 15.4769 1.25534
\(153\) 0 0
\(154\) −10.8320 −0.872871
\(155\) −5.03315 −0.404272
\(156\) 0 0
\(157\) 3.59395 0.286829 0.143414 0.989663i \(-0.454192\pi\)
0.143414 + 0.989663i \(0.454192\pi\)
\(158\) −7.91232 −0.629471
\(159\) 0 0
\(160\) 0.696985 0.0551015
\(161\) 4.45033 0.350735
\(162\) 0 0
\(163\) 10.5991 0.830188 0.415094 0.909779i \(-0.363749\pi\)
0.415094 + 0.909779i \(0.363749\pi\)
\(164\) −0.365113 −0.0285106
\(165\) 0 0
\(166\) 6.50721 0.505058
\(167\) 20.8147 1.61069 0.805344 0.592808i \(-0.201981\pi\)
0.805344 + 0.592808i \(0.201981\pi\)
\(168\) 0 0
\(169\) 23.4705 1.80542
\(170\) −4.17432 −0.320156
\(171\) 0 0
\(172\) −0.410973 −0.0313364
\(173\) 15.4900 1.17768 0.588841 0.808249i \(-0.299584\pi\)
0.588841 + 0.808249i \(0.299584\pi\)
\(174\) 0 0
\(175\) 4.77616 0.361043
\(176\) −19.8428 −1.49571
\(177\) 0 0
\(178\) −19.9096 −1.49228
\(179\) −5.79634 −0.433239 −0.216619 0.976256i \(-0.569503\pi\)
−0.216619 + 0.976256i \(0.569503\pi\)
\(180\) 0 0
\(181\) 17.6304 1.31046 0.655230 0.755429i \(-0.272572\pi\)
0.655230 + 0.755429i \(0.272572\pi\)
\(182\) 12.5578 0.930846
\(183\) 0 0
\(184\) 8.54289 0.629790
\(185\) 7.83337 0.575921
\(186\) 0 0
\(187\) −11.6451 −0.851577
\(188\) 1.03752 0.0756690
\(189\) 0 0
\(190\) −10.0026 −0.725665
\(191\) 14.4507 1.04562 0.522810 0.852449i \(-0.324884\pi\)
0.522810 + 0.852449i \(0.324884\pi\)
\(192\) 0 0
\(193\) 16.9861 1.22268 0.611342 0.791367i \(-0.290630\pi\)
0.611342 + 0.791367i \(0.290630\pi\)
\(194\) −16.8948 −1.21297
\(195\) 0 0
\(196\) 0.431966 0.0308547
\(197\) 16.5500 1.17914 0.589571 0.807717i \(-0.299297\pi\)
0.589571 + 0.807717i \(0.299297\pi\)
\(198\) 0 0
\(199\) −21.6857 −1.53726 −0.768631 0.639693i \(-0.779062\pi\)
−0.768631 + 0.639693i \(0.779062\pi\)
\(200\) 9.16835 0.648300
\(201\) 0 0
\(202\) −18.1747 −1.27877
\(203\) −0.796706 −0.0559178
\(204\) 0 0
\(205\) 5.40879 0.377767
\(206\) 8.00029 0.557406
\(207\) 0 0
\(208\) 23.0041 1.59505
\(209\) −27.9044 −1.93018
\(210\) 0 0
\(211\) −2.61046 −0.179711 −0.0898556 0.995955i \(-0.528641\pi\)
−0.0898556 + 0.995955i \(0.528641\pi\)
\(212\) −0.750835 −0.0515676
\(213\) 0 0
\(214\) −12.2966 −0.840580
\(215\) 6.08816 0.415209
\(216\) 0 0
\(217\) 5.60497 0.380490
\(218\) 13.0442 0.883462
\(219\) 0 0
\(220\) −0.642334 −0.0433061
\(221\) 13.5004 0.908138
\(222\) 0 0
\(223\) 17.8225 1.19348 0.596741 0.802434i \(-0.296462\pi\)
0.596741 + 0.802434i \(0.296462\pi\)
\(224\) −0.776170 −0.0518601
\(225\) 0 0
\(226\) −0.0607198 −0.00403902
\(227\) 4.02849 0.267380 0.133690 0.991023i \(-0.457317\pi\)
0.133690 + 0.991023i \(0.457317\pi\)
\(228\) 0 0
\(229\) 3.30206 0.218206 0.109103 0.994030i \(-0.465202\pi\)
0.109103 + 0.994030i \(0.465202\pi\)
\(230\) −5.52121 −0.364058
\(231\) 0 0
\(232\) −1.52936 −0.100408
\(233\) 23.4736 1.53781 0.768904 0.639365i \(-0.220803\pi\)
0.768904 + 0.639365i \(0.220803\pi\)
\(234\) 0 0
\(235\) −15.3698 −1.00262
\(236\) 0.488424 0.0317937
\(237\) 0 0
\(238\) 4.64857 0.301322
\(239\) −1.00000 −0.0646846
\(240\) 0 0
\(241\) 16.9690 1.09307 0.546534 0.837437i \(-0.315947\pi\)
0.546534 + 0.837437i \(0.315947\pi\)
\(242\) 22.2923 1.43301
\(243\) 0 0
\(244\) 1.27950 0.0819113
\(245\) −6.39914 −0.408826
\(246\) 0 0
\(247\) 32.3501 2.05839
\(248\) 10.7593 0.683219
\(249\) 0 0
\(250\) −15.2618 −0.965243
\(251\) −3.02008 −0.190626 −0.0953129 0.995447i \(-0.530385\pi\)
−0.0953129 + 0.995447i \(0.530385\pi\)
\(252\) 0 0
\(253\) −15.4026 −0.968352
\(254\) 2.11000 0.132393
\(255\) 0 0
\(256\) −2.18501 −0.136563
\(257\) −13.4671 −0.840058 −0.420029 0.907511i \(-0.637980\pi\)
−0.420029 + 0.907511i \(0.637980\pi\)
\(258\) 0 0
\(259\) −8.72333 −0.542041
\(260\) 0.744670 0.0461825
\(261\) 0 0
\(262\) −2.84017 −0.175467
\(263\) −6.69271 −0.412690 −0.206345 0.978479i \(-0.566157\pi\)
−0.206345 + 0.978479i \(0.566157\pi\)
\(264\) 0 0
\(265\) 11.1229 0.683273
\(266\) 11.1390 0.682977
\(267\) 0 0
\(268\) 0.853418 0.0521308
\(269\) −23.4572 −1.43021 −0.715106 0.699016i \(-0.753621\pi\)
−0.715106 + 0.699016i \(0.753621\pi\)
\(270\) 0 0
\(271\) 2.79891 0.170022 0.0850108 0.996380i \(-0.472908\pi\)
0.0850108 + 0.996380i \(0.472908\pi\)
\(272\) 8.51553 0.516330
\(273\) 0 0
\(274\) −15.8439 −0.957166
\(275\) −16.5303 −0.996813
\(276\) 0 0
\(277\) 1.33556 0.0802459 0.0401230 0.999195i \(-0.487225\pi\)
0.0401230 + 0.999195i \(0.487225\pi\)
\(278\) −22.6961 −1.36122
\(279\) 0 0
\(280\) 5.87730 0.351236
\(281\) −11.5507 −0.689057 −0.344529 0.938776i \(-0.611961\pi\)
−0.344529 + 0.938776i \(0.611961\pi\)
\(282\) 0 0
\(283\) 23.6047 1.40315 0.701576 0.712595i \(-0.252480\pi\)
0.701576 + 0.712595i \(0.252480\pi\)
\(284\) −1.14043 −0.0676721
\(285\) 0 0
\(286\) −43.4625 −2.56999
\(287\) −6.02329 −0.355544
\(288\) 0 0
\(289\) −12.0025 −0.706029
\(290\) 0.988418 0.0580419
\(291\) 0 0
\(292\) −0.956809 −0.0559930
\(293\) −3.75303 −0.219254 −0.109627 0.993973i \(-0.534966\pi\)
−0.109627 + 0.993973i \(0.534966\pi\)
\(294\) 0 0
\(295\) −7.23552 −0.421268
\(296\) −16.7454 −0.973305
\(297\) 0 0
\(298\) −20.5609 −1.19106
\(299\) 17.8565 1.03267
\(300\) 0 0
\(301\) −6.77985 −0.390784
\(302\) 20.1362 1.15871
\(303\) 0 0
\(304\) 20.4051 1.17031
\(305\) −18.9545 −1.08533
\(306\) 0 0
\(307\) −1.83405 −0.104675 −0.0523373 0.998629i \(-0.516667\pi\)
−0.0523373 + 0.998629i \(0.516667\pi\)
\(308\) 0.715310 0.0407586
\(309\) 0 0
\(310\) −6.95370 −0.394943
\(311\) −23.0474 −1.30690 −0.653450 0.756970i \(-0.726679\pi\)
−0.653450 + 0.756970i \(0.726679\pi\)
\(312\) 0 0
\(313\) −12.5384 −0.708712 −0.354356 0.935111i \(-0.615300\pi\)
−0.354356 + 0.935111i \(0.615300\pi\)
\(314\) 4.96533 0.280210
\(315\) 0 0
\(316\) 0.522502 0.0293930
\(317\) 5.42002 0.304419 0.152209 0.988348i \(-0.451361\pi\)
0.152209 + 0.988348i \(0.451361\pi\)
\(318\) 0 0
\(319\) 2.75740 0.154385
\(320\) 11.2596 0.629432
\(321\) 0 0
\(322\) 6.14848 0.342642
\(323\) 11.9752 0.666315
\(324\) 0 0
\(325\) 19.1639 1.06302
\(326\) 14.6436 0.811031
\(327\) 0 0
\(328\) −11.5624 −0.638425
\(329\) 17.1160 0.943637
\(330\) 0 0
\(331\) −13.7273 −0.754523 −0.377261 0.926107i \(-0.623134\pi\)
−0.377261 + 0.926107i \(0.623134\pi\)
\(332\) −0.429714 −0.0235836
\(333\) 0 0
\(334\) 28.7572 1.57352
\(335\) −12.6425 −0.690736
\(336\) 0 0
\(337\) −14.7765 −0.804926 −0.402463 0.915436i \(-0.631846\pi\)
−0.402463 + 0.915436i \(0.631846\pi\)
\(338\) 32.4264 1.76376
\(339\) 0 0
\(340\) 0.275658 0.0149496
\(341\) −19.3988 −1.05050
\(342\) 0 0
\(343\) 17.6619 0.953652
\(344\) −13.0147 −0.701703
\(345\) 0 0
\(346\) 21.4007 1.15051
\(347\) 9.24009 0.496034 0.248017 0.968756i \(-0.420221\pi\)
0.248017 + 0.968756i \(0.420221\pi\)
\(348\) 0 0
\(349\) −33.5592 −1.79639 −0.898193 0.439602i \(-0.855119\pi\)
−0.898193 + 0.439602i \(0.855119\pi\)
\(350\) 6.59864 0.352712
\(351\) 0 0
\(352\) 2.68633 0.143182
\(353\) −11.0743 −0.589423 −0.294712 0.955586i \(-0.595224\pi\)
−0.294712 + 0.955586i \(0.595224\pi\)
\(354\) 0 0
\(355\) 16.8943 0.896658
\(356\) 1.31476 0.0696820
\(357\) 0 0
\(358\) −8.00811 −0.423242
\(359\) 10.6761 0.563464 0.281732 0.959493i \(-0.409091\pi\)
0.281732 + 0.959493i \(0.409091\pi\)
\(360\) 0 0
\(361\) 9.69513 0.510270
\(362\) 24.3579 1.28022
\(363\) 0 0
\(364\) −0.829273 −0.0434657
\(365\) 14.1742 0.741910
\(366\) 0 0
\(367\) 20.7721 1.08430 0.542148 0.840283i \(-0.317611\pi\)
0.542148 + 0.840283i \(0.317611\pi\)
\(368\) 11.2632 0.587133
\(369\) 0 0
\(370\) 10.8224 0.562631
\(371\) −12.3866 −0.643078
\(372\) 0 0
\(373\) −37.5360 −1.94354 −0.971770 0.235931i \(-0.924186\pi\)
−0.971770 + 0.235931i \(0.924186\pi\)
\(374\) −16.0887 −0.831927
\(375\) 0 0
\(376\) 32.8561 1.69442
\(377\) −3.19671 −0.164639
\(378\) 0 0
\(379\) −14.7017 −0.755174 −0.377587 0.925974i \(-0.623246\pi\)
−0.377587 + 0.925974i \(0.623246\pi\)
\(380\) 0.660537 0.0338848
\(381\) 0 0
\(382\) 19.9649 1.02149
\(383\) 1.17130 0.0598507 0.0299253 0.999552i \(-0.490473\pi\)
0.0299253 + 0.999552i \(0.490473\pi\)
\(384\) 0 0
\(385\) −10.5966 −0.540053
\(386\) 23.4676 1.19447
\(387\) 0 0
\(388\) 1.11567 0.0566396
\(389\) 34.5061 1.74953 0.874766 0.484546i \(-0.161015\pi\)
0.874766 + 0.484546i \(0.161015\pi\)
\(390\) 0 0
\(391\) 6.61002 0.334283
\(392\) 13.6794 0.690916
\(393\) 0 0
\(394\) 22.8652 1.15193
\(395\) −7.74035 −0.389459
\(396\) 0 0
\(397\) 3.05365 0.153258 0.0766292 0.997060i \(-0.475584\pi\)
0.0766292 + 0.997060i \(0.475584\pi\)
\(398\) −29.9606 −1.50179
\(399\) 0 0
\(400\) 12.0878 0.604389
\(401\) −21.5513 −1.07622 −0.538109 0.842875i \(-0.680861\pi\)
−0.538109 + 0.842875i \(0.680861\pi\)
\(402\) 0 0
\(403\) 22.4894 1.12028
\(404\) 1.20019 0.0597118
\(405\) 0 0
\(406\) −1.10071 −0.0546275
\(407\) 30.1914 1.49653
\(408\) 0 0
\(409\) −2.09326 −0.103505 −0.0517526 0.998660i \(-0.516481\pi\)
−0.0517526 + 0.998660i \(0.516481\pi\)
\(410\) 7.47268 0.369050
\(411\) 0 0
\(412\) −0.528311 −0.0260280
\(413\) 8.05755 0.396486
\(414\) 0 0
\(415\) 6.36578 0.312484
\(416\) −3.11431 −0.152692
\(417\) 0 0
\(418\) −38.5521 −1.88565
\(419\) 4.55800 0.222673 0.111336 0.993783i \(-0.464487\pi\)
0.111336 + 0.993783i \(0.464487\pi\)
\(420\) 0 0
\(421\) 31.1765 1.51945 0.759724 0.650245i \(-0.225334\pi\)
0.759724 + 0.650245i \(0.225334\pi\)
\(422\) −3.60656 −0.175564
\(423\) 0 0
\(424\) −23.7774 −1.15473
\(425\) 7.09397 0.344108
\(426\) 0 0
\(427\) 21.1079 1.02148
\(428\) 0.812026 0.0392507
\(429\) 0 0
\(430\) 8.41129 0.405628
\(431\) −6.08401 −0.293056 −0.146528 0.989206i \(-0.546810\pi\)
−0.146528 + 0.989206i \(0.546810\pi\)
\(432\) 0 0
\(433\) 7.69823 0.369953 0.184977 0.982743i \(-0.440779\pi\)
0.184977 + 0.982743i \(0.440779\pi\)
\(434\) 7.74371 0.371710
\(435\) 0 0
\(436\) −0.861391 −0.0412531
\(437\) 15.8391 0.757686
\(438\) 0 0
\(439\) −11.9627 −0.570950 −0.285475 0.958386i \(-0.592151\pi\)
−0.285475 + 0.958386i \(0.592151\pi\)
\(440\) −20.3413 −0.969736
\(441\) 0 0
\(442\) 18.6520 0.887183
\(443\) 33.6798 1.60017 0.800087 0.599884i \(-0.204787\pi\)
0.800087 + 0.599884i \(0.204787\pi\)
\(444\) 0 0
\(445\) −19.4768 −0.923290
\(446\) 24.6232 1.16594
\(447\) 0 0
\(448\) −12.5388 −0.592405
\(449\) −8.14181 −0.384236 −0.192118 0.981372i \(-0.561536\pi\)
−0.192118 + 0.981372i \(0.561536\pi\)
\(450\) 0 0
\(451\) 20.8466 0.981629
\(452\) 0.00400972 0.000188601 0
\(453\) 0 0
\(454\) 5.56569 0.261211
\(455\) 12.2849 0.575923
\(456\) 0 0
\(457\) −18.2011 −0.851411 −0.425705 0.904862i \(-0.639974\pi\)
−0.425705 + 0.904862i \(0.639974\pi\)
\(458\) 4.56206 0.213171
\(459\) 0 0
\(460\) 0.364602 0.0169996
\(461\) −35.8143 −1.66804 −0.834018 0.551737i \(-0.813965\pi\)
−0.834018 + 0.551737i \(0.813965\pi\)
\(462\) 0 0
\(463\) 17.6283 0.819255 0.409628 0.912253i \(-0.365659\pi\)
0.409628 + 0.912253i \(0.365659\pi\)
\(464\) −2.01635 −0.0936068
\(465\) 0 0
\(466\) 32.4307 1.50232
\(467\) 35.8370 1.65834 0.829169 0.558999i \(-0.188814\pi\)
0.829169 + 0.558999i \(0.188814\pi\)
\(468\) 0 0
\(469\) 14.0789 0.650102
\(470\) −21.2347 −0.979482
\(471\) 0 0
\(472\) 15.4674 0.711943
\(473\) 23.4651 1.07892
\(474\) 0 0
\(475\) 16.9987 0.779955
\(476\) −0.306975 −0.0140702
\(477\) 0 0
\(478\) −1.38158 −0.0631920
\(479\) 18.6108 0.850351 0.425176 0.905111i \(-0.360212\pi\)
0.425176 + 0.905111i \(0.360212\pi\)
\(480\) 0 0
\(481\) −35.0015 −1.59593
\(482\) 23.4440 1.06785
\(483\) 0 0
\(484\) −1.47211 −0.0669139
\(485\) −16.5276 −0.750478
\(486\) 0 0
\(487\) 13.7084 0.621187 0.310594 0.950543i \(-0.399472\pi\)
0.310594 + 0.950543i \(0.399472\pi\)
\(488\) 40.5189 1.83420
\(489\) 0 0
\(490\) −8.84093 −0.399393
\(491\) −26.8768 −1.21293 −0.606466 0.795109i \(-0.707413\pi\)
−0.606466 + 0.795109i \(0.707413\pi\)
\(492\) 0 0
\(493\) −1.18334 −0.0532949
\(494\) 44.6942 2.01089
\(495\) 0 0
\(496\) 14.1854 0.636943
\(497\) −18.8137 −0.843911
\(498\) 0 0
\(499\) −35.7292 −1.59946 −0.799729 0.600361i \(-0.795024\pi\)
−0.799729 + 0.600361i \(0.795024\pi\)
\(500\) 1.00784 0.0450719
\(501\) 0 0
\(502\) −4.17249 −0.186227
\(503\) 26.3328 1.17412 0.587062 0.809542i \(-0.300285\pi\)
0.587062 + 0.809542i \(0.300285\pi\)
\(504\) 0 0
\(505\) −17.7797 −0.791185
\(506\) −21.2799 −0.946007
\(507\) 0 0
\(508\) −0.139337 −0.00618209
\(509\) 42.2347 1.87202 0.936011 0.351972i \(-0.114489\pi\)
0.936011 + 0.351972i \(0.114489\pi\)
\(510\) 0 0
\(511\) −15.7845 −0.698266
\(512\) −23.9756 −1.05958
\(513\) 0 0
\(514\) −18.6060 −0.820673
\(515\) 7.82640 0.344872
\(516\) 0 0
\(517\) −59.2386 −2.60531
\(518\) −12.0520 −0.529534
\(519\) 0 0
\(520\) 23.5821 1.03414
\(521\) −13.8354 −0.606142 −0.303071 0.952968i \(-0.598012\pi\)
−0.303071 + 0.952968i \(0.598012\pi\)
\(522\) 0 0
\(523\) −22.9910 −1.00533 −0.502663 0.864483i \(-0.667646\pi\)
−0.502663 + 0.864483i \(0.667646\pi\)
\(524\) 0.187555 0.00819338
\(525\) 0 0
\(526\) −9.24652 −0.403167
\(527\) 8.32499 0.362642
\(528\) 0 0
\(529\) −14.2572 −0.619878
\(530\) 15.3672 0.667507
\(531\) 0 0
\(532\) −0.735581 −0.0318915
\(533\) −24.1679 −1.04683
\(534\) 0 0
\(535\) −12.0294 −0.520074
\(536\) 27.0259 1.16734
\(537\) 0 0
\(538\) −32.4080 −1.39721
\(539\) −24.6636 −1.06234
\(540\) 0 0
\(541\) −41.7557 −1.79522 −0.897610 0.440791i \(-0.854698\pi\)
−0.897610 + 0.440791i \(0.854698\pi\)
\(542\) 3.86692 0.166098
\(543\) 0 0
\(544\) −1.15284 −0.0494275
\(545\) 12.7606 0.546606
\(546\) 0 0
\(547\) −33.2023 −1.41963 −0.709814 0.704389i \(-0.751221\pi\)
−0.709814 + 0.704389i \(0.751221\pi\)
\(548\) 1.04628 0.0446947
\(549\) 0 0
\(550\) −22.8379 −0.973812
\(551\) −2.83554 −0.120798
\(552\) 0 0
\(553\) 8.61974 0.366549
\(554\) 1.84518 0.0783943
\(555\) 0 0
\(556\) 1.49877 0.0635620
\(557\) −25.6256 −1.08579 −0.542895 0.839800i \(-0.682672\pi\)
−0.542895 + 0.839800i \(0.682672\pi\)
\(558\) 0 0
\(559\) −27.2035 −1.15058
\(560\) 7.74878 0.327446
\(561\) 0 0
\(562\) −15.9582 −0.673157
\(563\) −8.59177 −0.362100 −0.181050 0.983474i \(-0.557950\pi\)
−0.181050 + 0.983474i \(0.557950\pi\)
\(564\) 0 0
\(565\) −0.0594000 −0.00249898
\(566\) 32.6118 1.37077
\(567\) 0 0
\(568\) −36.1150 −1.51535
\(569\) 2.27120 0.0952138 0.0476069 0.998866i \(-0.484841\pi\)
0.0476069 + 0.998866i \(0.484841\pi\)
\(570\) 0 0
\(571\) −38.3368 −1.60434 −0.802172 0.597093i \(-0.796322\pi\)
−0.802172 + 0.597093i \(0.796322\pi\)
\(572\) 2.87011 0.120005
\(573\) 0 0
\(574\) −8.32166 −0.347340
\(575\) 9.38291 0.391295
\(576\) 0 0
\(577\) 22.0162 0.916547 0.458274 0.888811i \(-0.348468\pi\)
0.458274 + 0.888811i \(0.348468\pi\)
\(578\) −16.5824 −0.689737
\(579\) 0 0
\(580\) −0.0652717 −0.00271026
\(581\) −7.08900 −0.294101
\(582\) 0 0
\(583\) 42.8699 1.77549
\(584\) −30.3001 −1.25383
\(585\) 0 0
\(586\) −5.18511 −0.214195
\(587\) 28.8423 1.19045 0.595224 0.803560i \(-0.297063\pi\)
0.595224 + 0.803560i \(0.297063\pi\)
\(588\) 0 0
\(589\) 19.9485 0.821965
\(590\) −9.99645 −0.411547
\(591\) 0 0
\(592\) −22.0775 −0.907381
\(593\) −10.5494 −0.433213 −0.216607 0.976259i \(-0.569499\pi\)
−0.216607 + 0.976259i \(0.569499\pi\)
\(594\) 0 0
\(595\) 4.54753 0.186431
\(596\) 1.35777 0.0556164
\(597\) 0 0
\(598\) 24.6702 1.00884
\(599\) 7.21520 0.294805 0.147403 0.989077i \(-0.452909\pi\)
0.147403 + 0.989077i \(0.452909\pi\)
\(600\) 0 0
\(601\) 24.0334 0.980344 0.490172 0.871626i \(-0.336934\pi\)
0.490172 + 0.871626i \(0.336934\pi\)
\(602\) −9.36690 −0.381767
\(603\) 0 0
\(604\) −1.32973 −0.0541058
\(605\) 21.8078 0.886613
\(606\) 0 0
\(607\) 17.5172 0.711001 0.355500 0.934676i \(-0.384310\pi\)
0.355500 + 0.934676i \(0.384310\pi\)
\(608\) −2.76245 −0.112032
\(609\) 0 0
\(610\) −26.1871 −1.06029
\(611\) 68.6764 2.77835
\(612\) 0 0
\(613\) 23.5816 0.952453 0.476226 0.879323i \(-0.342004\pi\)
0.476226 + 0.879323i \(0.342004\pi\)
\(614\) −2.53388 −0.102259
\(615\) 0 0
\(616\) 22.6523 0.912689
\(617\) −26.2668 −1.05746 −0.528731 0.848789i \(-0.677332\pi\)
−0.528731 + 0.848789i \(0.677332\pi\)
\(618\) 0 0
\(619\) −11.7447 −0.472060 −0.236030 0.971746i \(-0.575846\pi\)
−0.236030 + 0.971746i \(0.575846\pi\)
\(620\) 0.459198 0.0184418
\(621\) 0 0
\(622\) −31.8419 −1.27674
\(623\) 21.6896 0.868975
\(624\) 0 0
\(625\) 0.936418 0.0374567
\(626\) −17.3228 −0.692359
\(627\) 0 0
\(628\) −0.327893 −0.0130844
\(629\) −12.9567 −0.516616
\(630\) 0 0
\(631\) 3.30347 0.131509 0.0657545 0.997836i \(-0.479055\pi\)
0.0657545 + 0.997836i \(0.479055\pi\)
\(632\) 16.5465 0.658186
\(633\) 0 0
\(634\) 7.48820 0.297394
\(635\) 2.06414 0.0819129
\(636\) 0 0
\(637\) 28.5930 1.13290
\(638\) 3.80957 0.150822
\(639\) 0 0
\(640\) 14.1621 0.559806
\(641\) −25.9601 −1.02536 −0.512680 0.858580i \(-0.671347\pi\)
−0.512680 + 0.858580i \(0.671347\pi\)
\(642\) 0 0
\(643\) −24.7989 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(644\) −0.406024 −0.0159996
\(645\) 0 0
\(646\) 16.5446 0.650940
\(647\) 35.1636 1.38242 0.691211 0.722653i \(-0.257077\pi\)
0.691211 + 0.722653i \(0.257077\pi\)
\(648\) 0 0
\(649\) −27.8872 −1.09467
\(650\) 26.4764 1.03849
\(651\) 0 0
\(652\) −0.967009 −0.0378710
\(653\) −43.2533 −1.69263 −0.846316 0.532681i \(-0.821184\pi\)
−0.846316 + 0.532681i \(0.821184\pi\)
\(654\) 0 0
\(655\) −2.77844 −0.108563
\(656\) −15.2441 −0.595183
\(657\) 0 0
\(658\) 23.6472 0.921863
\(659\) −20.3893 −0.794256 −0.397128 0.917763i \(-0.629993\pi\)
−0.397128 + 0.917763i \(0.629993\pi\)
\(660\) 0 0
\(661\) −41.5166 −1.61481 −0.807404 0.589999i \(-0.799128\pi\)
−0.807404 + 0.589999i \(0.799128\pi\)
\(662\) −18.9654 −0.737112
\(663\) 0 0
\(664\) −13.6081 −0.528097
\(665\) 10.8969 0.422564
\(666\) 0 0
\(667\) −1.56516 −0.0606031
\(668\) −1.89902 −0.0734754
\(669\) 0 0
\(670\) −17.4667 −0.674797
\(671\) −73.0544 −2.82023
\(672\) 0 0
\(673\) −29.0303 −1.11903 −0.559517 0.828819i \(-0.689013\pi\)
−0.559517 + 0.828819i \(0.689013\pi\)
\(674\) −20.4149 −0.786352
\(675\) 0 0
\(676\) −2.14133 −0.0823587
\(677\) 13.5700 0.521539 0.260769 0.965401i \(-0.416024\pi\)
0.260769 + 0.965401i \(0.416024\pi\)
\(678\) 0 0
\(679\) 18.4053 0.706330
\(680\) 8.72949 0.334761
\(681\) 0 0
\(682\) −26.8010 −1.02626
\(683\) −42.1926 −1.61446 −0.807228 0.590240i \(-0.799033\pi\)
−0.807228 + 0.590240i \(0.799033\pi\)
\(684\) 0 0
\(685\) −15.4995 −0.592207
\(686\) 24.4013 0.931646
\(687\) 0 0
\(688\) −17.1588 −0.654175
\(689\) −49.6999 −1.89342
\(690\) 0 0
\(691\) 43.0952 1.63942 0.819709 0.572781i \(-0.194135\pi\)
0.819709 + 0.572781i \(0.194135\pi\)
\(692\) −1.41322 −0.0537227
\(693\) 0 0
\(694\) 12.7659 0.484588
\(695\) −22.2028 −0.842199
\(696\) 0 0
\(697\) −8.94633 −0.338866
\(698\) −46.3648 −1.75493
\(699\) 0 0
\(700\) −0.435751 −0.0164698
\(701\) 47.5830 1.79718 0.898592 0.438785i \(-0.144591\pi\)
0.898592 + 0.438785i \(0.144591\pi\)
\(702\) 0 0
\(703\) −31.0470 −1.17096
\(704\) 43.3969 1.63558
\(705\) 0 0
\(706\) −15.3000 −0.575822
\(707\) 19.7996 0.744642
\(708\) 0 0
\(709\) 8.00508 0.300637 0.150319 0.988638i \(-0.451970\pi\)
0.150319 + 0.988638i \(0.451970\pi\)
\(710\) 23.3409 0.875968
\(711\) 0 0
\(712\) 41.6355 1.56036
\(713\) 11.0111 0.412371
\(714\) 0 0
\(715\) −42.5179 −1.59008
\(716\) 0.528827 0.0197632
\(717\) 0 0
\(718\) 14.7499 0.550462
\(719\) −3.65160 −0.136182 −0.0680908 0.997679i \(-0.521691\pi\)
−0.0680908 + 0.997679i \(0.521691\pi\)
\(720\) 0 0
\(721\) −8.71556 −0.324585
\(722\) 13.3946 0.498496
\(723\) 0 0
\(724\) −1.60851 −0.0597798
\(725\) −1.67975 −0.0623843
\(726\) 0 0
\(727\) 18.3105 0.679100 0.339550 0.940588i \(-0.389725\pi\)
0.339550 + 0.940588i \(0.389725\pi\)
\(728\) −26.2613 −0.973309
\(729\) 0 0
\(730\) 19.5828 0.724791
\(731\) −10.0700 −0.372453
\(732\) 0 0
\(733\) 45.3607 1.67543 0.837717 0.546104i \(-0.183890\pi\)
0.837717 + 0.546104i \(0.183890\pi\)
\(734\) 28.6983 1.05928
\(735\) 0 0
\(736\) −1.52481 −0.0562053
\(737\) −48.7270 −1.79488
\(738\) 0 0
\(739\) 33.4799 1.23158 0.615790 0.787911i \(-0.288837\pi\)
0.615790 + 0.787911i \(0.288837\pi\)
\(740\) −0.714675 −0.0262720
\(741\) 0 0
\(742\) −17.1130 −0.628239
\(743\) 13.6930 0.502348 0.251174 0.967942i \(-0.419183\pi\)
0.251174 + 0.967942i \(0.419183\pi\)
\(744\) 0 0
\(745\) −20.1140 −0.736921
\(746\) −51.8590 −1.89869
\(747\) 0 0
\(748\) 1.06244 0.0388467
\(749\) 13.3960 0.489480
\(750\) 0 0
\(751\) −27.4062 −1.00007 −0.500034 0.866006i \(-0.666679\pi\)
−0.500034 + 0.866006i \(0.666679\pi\)
\(752\) 43.3183 1.57965
\(753\) 0 0
\(754\) −4.41651 −0.160840
\(755\) 19.6986 0.716905
\(756\) 0 0
\(757\) −9.98983 −0.363087 −0.181543 0.983383i \(-0.558109\pi\)
−0.181543 + 0.983383i \(0.558109\pi\)
\(758\) −20.3116 −0.737749
\(759\) 0 0
\(760\) 20.9178 0.758768
\(761\) −38.8553 −1.40851 −0.704253 0.709949i \(-0.748718\pi\)
−0.704253 + 0.709949i \(0.748718\pi\)
\(762\) 0 0
\(763\) −14.2104 −0.514451
\(764\) −1.31841 −0.0476984
\(765\) 0 0
\(766\) 1.61825 0.0584696
\(767\) 32.3302 1.16737
\(768\) 0 0
\(769\) 19.8198 0.714720 0.357360 0.933967i \(-0.383677\pi\)
0.357360 + 0.933967i \(0.383677\pi\)
\(770\) −14.6401 −0.527592
\(771\) 0 0
\(772\) −1.54972 −0.0557756
\(773\) −29.0662 −1.04544 −0.522720 0.852505i \(-0.675083\pi\)
−0.522720 + 0.852505i \(0.675083\pi\)
\(774\) 0 0
\(775\) 11.8173 0.424491
\(776\) 35.3309 1.26831
\(777\) 0 0
\(778\) 47.6730 1.70916
\(779\) −21.4374 −0.768074
\(780\) 0 0
\(781\) 65.1143 2.32997
\(782\) 9.13227 0.326569
\(783\) 0 0
\(784\) 18.0353 0.644118
\(785\) 4.85741 0.173368
\(786\) 0 0
\(787\) −40.4799 −1.44295 −0.721476 0.692440i \(-0.756536\pi\)
−0.721476 + 0.692440i \(0.756536\pi\)
\(788\) −1.50994 −0.0537893
\(789\) 0 0
\(790\) −10.6939 −0.380472
\(791\) 0.0661485 0.00235197
\(792\) 0 0
\(793\) 84.6934 3.00755
\(794\) 4.21886 0.149722
\(795\) 0 0
\(796\) 1.97849 0.0701258
\(797\) −11.4938 −0.407132 −0.203566 0.979061i \(-0.565253\pi\)
−0.203566 + 0.979061i \(0.565253\pi\)
\(798\) 0 0
\(799\) 25.4222 0.899374
\(800\) −1.63645 −0.0578573
\(801\) 0 0
\(802\) −29.7748 −1.05139
\(803\) 54.6302 1.92786
\(804\) 0 0
\(805\) 6.01485 0.211995
\(806\) 31.0709 1.09443
\(807\) 0 0
\(808\) 38.0076 1.33710
\(809\) −51.3120 −1.80403 −0.902016 0.431702i \(-0.857913\pi\)
−0.902016 + 0.431702i \(0.857913\pi\)
\(810\) 0 0
\(811\) 32.5242 1.14208 0.571040 0.820922i \(-0.306540\pi\)
0.571040 + 0.820922i \(0.306540\pi\)
\(812\) 0.0726873 0.00255082
\(813\) 0 0
\(814\) 41.7119 1.46200
\(815\) 14.3253 0.501792
\(816\) 0 0
\(817\) −24.1300 −0.844203
\(818\) −2.89201 −0.101117
\(819\) 0 0
\(820\) −0.493470 −0.0172327
\(821\) −11.2602 −0.392983 −0.196492 0.980506i \(-0.562955\pi\)
−0.196492 + 0.980506i \(0.562955\pi\)
\(822\) 0 0
\(823\) −3.83791 −0.133781 −0.0668906 0.997760i \(-0.521308\pi\)
−0.0668906 + 0.997760i \(0.521308\pi\)
\(824\) −16.7305 −0.582834
\(825\) 0 0
\(826\) 11.1322 0.387337
\(827\) 22.1887 0.771577 0.385788 0.922587i \(-0.373930\pi\)
0.385788 + 0.922587i \(0.373930\pi\)
\(828\) 0 0
\(829\) −12.6872 −0.440645 −0.220323 0.975427i \(-0.570711\pi\)
−0.220323 + 0.975427i \(0.570711\pi\)
\(830\) 8.79484 0.305273
\(831\) 0 0
\(832\) −50.3109 −1.74422
\(833\) 10.5844 0.366728
\(834\) 0 0
\(835\) 28.1321 0.973552
\(836\) 2.54585 0.0880499
\(837\) 0 0
\(838\) 6.29725 0.217535
\(839\) 31.9752 1.10391 0.551953 0.833875i \(-0.313883\pi\)
0.551953 + 0.833875i \(0.313883\pi\)
\(840\) 0 0
\(841\) −28.7198 −0.990338
\(842\) 43.0728 1.48439
\(843\) 0 0
\(844\) 0.238164 0.00819795
\(845\) 31.7216 1.09126
\(846\) 0 0
\(847\) −24.2854 −0.834457
\(848\) −31.3487 −1.07652
\(849\) 0 0
\(850\) 9.80089 0.336168
\(851\) −17.1373 −0.587458
\(852\) 0 0
\(853\) −31.1065 −1.06507 −0.532533 0.846409i \(-0.678760\pi\)
−0.532533 + 0.846409i \(0.678760\pi\)
\(854\) 29.1623 0.997912
\(855\) 0 0
\(856\) 25.7151 0.878925
\(857\) 11.3097 0.386333 0.193166 0.981166i \(-0.438124\pi\)
0.193166 + 0.981166i \(0.438124\pi\)
\(858\) 0 0
\(859\) 32.1279 1.09619 0.548095 0.836416i \(-0.315353\pi\)
0.548095 + 0.836416i \(0.315353\pi\)
\(860\) −0.555452 −0.0189408
\(861\) 0 0
\(862\) −8.40555 −0.286294
\(863\) 15.1014 0.514056 0.257028 0.966404i \(-0.417257\pi\)
0.257028 + 0.966404i \(0.417257\pi\)
\(864\) 0 0
\(865\) 20.9355 0.711829
\(866\) 10.6357 0.361416
\(867\) 0 0
\(868\) −0.511368 −0.0173569
\(869\) −29.8329 −1.01201
\(870\) 0 0
\(871\) 56.4901 1.91410
\(872\) −27.2784 −0.923764
\(873\) 0 0
\(874\) 21.8830 0.740202
\(875\) 16.6263 0.562073
\(876\) 0 0
\(877\) −37.9980 −1.28310 −0.641551 0.767080i \(-0.721709\pi\)
−0.641551 + 0.767080i \(0.721709\pi\)
\(878\) −16.5275 −0.557775
\(879\) 0 0
\(880\) −26.8185 −0.904053
\(881\) −31.7883 −1.07097 −0.535487 0.844543i \(-0.679872\pi\)
−0.535487 + 0.844543i \(0.679872\pi\)
\(882\) 0 0
\(883\) −31.7099 −1.06712 −0.533562 0.845761i \(-0.679147\pi\)
−0.533562 + 0.845761i \(0.679147\pi\)
\(884\) −1.23171 −0.0414269
\(885\) 0 0
\(886\) 46.5313 1.56325
\(887\) −3.92551 −0.131806 −0.0659029 0.997826i \(-0.520993\pi\)
−0.0659029 + 0.997826i \(0.520993\pi\)
\(888\) 0 0
\(889\) −2.29865 −0.0770943
\(890\) −26.9088 −0.901985
\(891\) 0 0
\(892\) −1.62603 −0.0544435
\(893\) 60.9173 2.03852
\(894\) 0 0
\(895\) −7.83405 −0.261863
\(896\) −15.7711 −0.526875
\(897\) 0 0
\(898\) −11.2486 −0.375370
\(899\) −1.97124 −0.0657444
\(900\) 0 0
\(901\) −18.3976 −0.612914
\(902\) 28.8013 0.958978
\(903\) 0 0
\(904\) 0.126979 0.00422327
\(905\) 23.8285 0.792085
\(906\) 0 0
\(907\) 37.1647 1.23403 0.617016 0.786950i \(-0.288341\pi\)
0.617016 + 0.786950i \(0.288341\pi\)
\(908\) −0.367538 −0.0121972
\(909\) 0 0
\(910\) 16.9725 0.562633
\(911\) −15.0959 −0.500151 −0.250075 0.968226i \(-0.580455\pi\)
−0.250075 + 0.968226i \(0.580455\pi\)
\(912\) 0 0
\(913\) 24.5350 0.811991
\(914\) −25.1463 −0.831765
\(915\) 0 0
\(916\) −0.301262 −0.00995399
\(917\) 3.09410 0.102176
\(918\) 0 0
\(919\) −17.8896 −0.590123 −0.295062 0.955478i \(-0.595340\pi\)
−0.295062 + 0.955478i \(0.595340\pi\)
\(920\) 11.5462 0.380665
\(921\) 0 0
\(922\) −49.4803 −1.62955
\(923\) −75.4883 −2.48473
\(924\) 0 0
\(925\) −18.3920 −0.604724
\(926\) 24.3549 0.800351
\(927\) 0 0
\(928\) 0.272975 0.00896084
\(929\) −32.0444 −1.05134 −0.525671 0.850688i \(-0.676186\pi\)
−0.525671 + 0.850688i \(0.676186\pi\)
\(930\) 0 0
\(931\) 25.3626 0.831225
\(932\) −2.14161 −0.0701507
\(933\) 0 0
\(934\) 49.5117 1.62007
\(935\) −15.7390 −0.514721
\(936\) 0 0
\(937\) 11.6221 0.379677 0.189839 0.981815i \(-0.439204\pi\)
0.189839 + 0.981815i \(0.439204\pi\)
\(938\) 19.4511 0.635101
\(939\) 0 0
\(940\) 1.40226 0.0457368
\(941\) 4.15109 0.135322 0.0676608 0.997708i \(-0.478446\pi\)
0.0676608 + 0.997708i \(0.478446\pi\)
\(942\) 0 0
\(943\) −11.8330 −0.385334
\(944\) 20.3925 0.663721
\(945\) 0 0
\(946\) 32.4189 1.05403
\(947\) 7.73026 0.251200 0.125600 0.992081i \(-0.459914\pi\)
0.125600 + 0.992081i \(0.459914\pi\)
\(948\) 0 0
\(949\) −63.3339 −2.05590
\(950\) 23.4851 0.761957
\(951\) 0 0
\(952\) −9.72126 −0.315068
\(953\) −40.5116 −1.31230 −0.656149 0.754631i \(-0.727816\pi\)
−0.656149 + 0.754631i \(0.727816\pi\)
\(954\) 0 0
\(955\) 19.5309 0.632006
\(956\) 0.0912347 0.00295074
\(957\) 0 0
\(958\) 25.7124 0.830729
\(959\) 17.2605 0.557370
\(960\) 0 0
\(961\) −17.1320 −0.552645
\(962\) −48.3574 −1.55911
\(963\) 0 0
\(964\) −1.54816 −0.0498629
\(965\) 22.9575 0.739030
\(966\) 0 0
\(967\) −32.0195 −1.02968 −0.514839 0.857287i \(-0.672148\pi\)
−0.514839 + 0.857287i \(0.672148\pi\)
\(968\) −46.6185 −1.49837
\(969\) 0 0
\(970\) −22.8342 −0.733161
\(971\) −13.2739 −0.425980 −0.212990 0.977054i \(-0.568320\pi\)
−0.212990 + 0.977054i \(0.568320\pi\)
\(972\) 0 0
\(973\) 24.7253 0.792655
\(974\) 18.9393 0.606853
\(975\) 0 0
\(976\) 53.4211 1.70997
\(977\) 16.6713 0.533361 0.266680 0.963785i \(-0.414073\pi\)
0.266680 + 0.963785i \(0.414073\pi\)
\(978\) 0 0
\(979\) −75.0677 −2.39917
\(980\) 0.583824 0.0186496
\(981\) 0 0
\(982\) −37.1325 −1.18494
\(983\) −3.93033 −0.125358 −0.0626790 0.998034i \(-0.519964\pi\)
−0.0626790 + 0.998034i \(0.519964\pi\)
\(984\) 0 0
\(985\) 22.3682 0.712711
\(986\) −1.63488 −0.0520651
\(987\) 0 0
\(988\) −2.95145 −0.0938981
\(989\) −13.3192 −0.423527
\(990\) 0 0
\(991\) −48.9367 −1.55452 −0.777262 0.629177i \(-0.783392\pi\)
−0.777262 + 0.629177i \(0.783392\pi\)
\(992\) −1.92043 −0.0609736
\(993\) 0 0
\(994\) −25.9927 −0.824438
\(995\) −29.3094 −0.929171
\(996\) 0 0
\(997\) −37.6435 −1.19218 −0.596091 0.802917i \(-0.703280\pi\)
−0.596091 + 0.802917i \(0.703280\pi\)
\(998\) −49.3628 −1.56255
\(999\) 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 2151.2.a.j.1.16 20
3.2 odd 2 2151.2.a.k.1.5 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.16 20 1.1 even 1 trivial
2151.2.a.k.1.5 yes 20 3.2 odd 2