Properties

Label 2151.2.a.k.1.5
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
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: \(0\)
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 1.5
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 \(-0.662442\pi\)
−0.488463 + 0.872585i \(0.662442\pi\)
\(3\) 0 0
\(4\) −0.0912347 −0.0456174
\(5\) −1.35155 −0.604432 −0.302216 0.953239i \(-0.597726\pi\)
−0.302216 + 0.953239i \(0.597726\pi\)
\(6\) 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 \(-0.787497\pi\)
−0.785311 + 0.619102i \(0.787497\pi\)
\(12\) 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 \(-0.412614\pi\)
0.271096 + 0.962552i \(0.412614\pi\)
\(18\) 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 \(-0.400250\pi\)
0.308270 + 0.951299i \(0.400250\pi\)
\(24\) 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 \(-0.515650\pi\)
−0.0491477 + 0.998792i \(0.515650\pi\)
\(30\) 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 \(-0.601165\pi\)
−0.312497 + 0.949919i \(0.601165\pi\)
\(42\) 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 \(-0.188689\pi\)
0.829388 + 0.558673i \(0.188689\pi\)
\(48\) 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 \(-0.691208\pi\)
−0.565219 + 0.824941i \(0.691208\pi\)
\(54\) 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 \(-0.386697\pi\)
0.348482 + 0.937315i \(0.386697\pi\)
\(60\) 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 \(-0.765997\pi\)
−0.741736 + 0.670692i \(0.765997\pi\)
\(72\) 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 \(-0.583226\pi\)
−0.258494 + 0.966013i \(0.583226\pi\)
\(84\) 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 \(-0.223348\pi\)
0.763766 + 0.645493i \(0.223348\pi\)
\(90\) 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 \(-0.272885\pi\)
0.654486 + 0.756074i \(0.272885\pi\)
\(102\) 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 \(-0.358437\pi\)
0.430217 + 0.902725i \(0.358437\pi\)
\(108\) 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 \(-0.499342\pi\)
0.00206721 + 0.999998i \(0.499342\pi\)
\(114\) 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 \(-0.471375\pi\)
0.0898055 + 0.995959i \(0.471375\pi\)
\(132\) 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 \(-0.337038\pi\)
0.489887 + 0.871786i \(0.337038\pi\)
\(138\) 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 \(-0.291331\pi\)
0.609597 + 0.792711i \(0.291331\pi\)
\(150\) 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 \(-0.798019\pi\)
−0.805344 + 0.592808i \(0.798019\pi\)
\(168\) 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 \(-0.700416\pi\)
−0.588841 + 0.808249i \(0.700416\pi\)
\(174\) 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 \(-0.430497\pi\)
0.216619 + 0.976256i \(0.430497\pi\)
\(180\) 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 \(-0.675116\pi\)
−0.522810 + 0.852449i \(0.675116\pi\)
\(192\) 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 \(-0.700703\pi\)
−0.589571 + 0.807717i \(0.700703\pi\)
\(198\) 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 \(-0.542683\pi\)
−0.133690 + 0.991023i \(0.542683\pi\)
\(228\) 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 \(-0.779197\pi\)
−0.768904 + 0.639365i \(0.779197\pi\)
\(234\) 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 \(-0.469615\pi\)
0.0953129 + 0.995447i \(0.469615\pi\)
\(252\) 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 \(-0.362020\pi\)
0.420029 + 0.907511i \(0.362020\pi\)
\(258\) 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 \(-0.433843\pi\)
0.206345 + 0.978479i \(0.433843\pi\)
\(264\) 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 \(-0.246379\pi\)
0.715106 + 0.699016i \(0.246379\pi\)
\(270\) 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 \(-0.388039\pi\)
0.344529 + 0.938776i \(0.388039\pi\)
\(282\) 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 \(-0.465034\pi\)
0.109627 + 0.993973i \(0.465034\pi\)
\(294\) 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 \(-0.273321\pi\)
0.653450 + 0.756970i \(0.273321\pi\)
\(312\) 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 \(-0.548639\pi\)
−0.152209 + 0.988348i \(0.548639\pi\)
\(318\) 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 \(-0.579779\pi\)
−0.248017 + 0.968756i \(0.579779\pi\)
\(348\) 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 \(-0.404776\pi\)
0.294712 + 0.955586i \(0.404776\pi\)
\(354\) 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 \(-0.590909\pi\)
−0.281732 + 0.959493i \(0.590909\pi\)
\(360\) 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 \(-0.509527\pi\)
−0.0299253 + 0.999552i \(0.509527\pi\)
\(384\) 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 \(-0.838985\pi\)
−0.874766 + 0.484546i \(0.838985\pi\)
\(390\) 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 \(-0.319139\pi\)
0.538109 + 0.842875i \(0.319139\pi\)
\(402\) 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 \(-0.535513\pi\)
−0.111336 + 0.993783i \(0.535513\pi\)
\(420\) 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 \(-0.453190\pi\)
0.146528 + 0.989206i \(0.453190\pi\)
\(432\) 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 \(-0.795213\pi\)
−0.800087 + 0.599884i \(0.795213\pi\)
\(444\) 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 \(-0.438464\pi\)
0.192118 + 0.981372i \(0.438464\pi\)
\(450\) 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 \(-0.186035\pi\)
0.834018 + 0.551737i \(0.186035\pi\)
\(462\) 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 \(-0.811186\pi\)
−0.829169 + 0.558999i \(0.811186\pi\)
\(468\) 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 \(-0.639788\pi\)
−0.425176 + 0.905111i \(0.639788\pi\)
\(480\) 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 \(-0.292587\pi\)
0.606466 + 0.795109i \(0.292587\pi\)
\(492\) 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 \(-0.699715\pi\)
−0.587062 + 0.809542i \(0.699715\pi\)
\(504\) 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 \(-0.885511\pi\)
−0.936011 + 0.351972i \(0.885511\pi\)
\(510\) 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 \(-0.401988\pi\)
0.303071 + 0.952968i \(0.401988\pi\)
\(522\) 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 \(-0.317328\pi\)
0.542895 + 0.839800i \(0.317328\pi\)
\(558\) 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 \(-0.442050\pi\)
0.181050 + 0.983474i \(0.442050\pi\)
\(564\) 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 \(-0.515159\pi\)
−0.0476069 + 0.998866i \(0.515159\pi\)
\(570\) 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 \(-0.702937\pi\)
−0.595224 + 0.803560i \(0.702937\pi\)
\(588\) 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 \(-0.430501\pi\)
0.216607 + 0.976259i \(0.430501\pi\)
\(594\) 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 \(-0.547091\pi\)
−0.147403 + 0.989077i \(0.547091\pi\)
\(600\) 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 \(-0.322668\pi\)
0.528731 + 0.848789i \(0.322668\pi\)
\(618\) 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 \(-0.328653\pi\)
0.512680 + 0.858580i \(0.328653\pi\)
\(642\) 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 \(-0.742923\pi\)
−0.691211 + 0.722653i \(0.742923\pi\)
\(648\) 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 \(-0.178816\pi\)
0.846316 + 0.532681i \(0.178816\pi\)
\(654\) 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 \(-0.370007\pi\)
0.397128 + 0.917763i \(0.370007\pi\)
\(660\) 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 \(-0.583976\pi\)
−0.260769 + 0.965401i \(0.583976\pi\)
\(678\) 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 \(-0.200967\pi\)
0.807228 + 0.590240i \(0.200967\pi\)
\(684\) 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 \(-0.855409\pi\)
−0.898592 + 0.438785i \(0.855409\pi\)
\(702\) 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 \(-0.478309\pi\)
0.0680908 + 0.997679i \(0.478309\pi\)
\(720\) 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 \(-0.580817\pi\)
−0.251174 + 0.967942i \(0.580817\pi\)
\(744\) 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 \(-0.251282\pi\)
0.704253 + 0.709949i \(0.251282\pi\)
\(762\) 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 \(-0.324917\pi\)
0.522720 + 0.852505i \(0.324917\pi\)
\(774\) 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 \(-0.434747\pi\)
0.203566 + 0.979061i \(0.434747\pi\)
\(798\) 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 \(-0.142087\pi\)
0.902016 + 0.431702i \(0.142087\pi\)
\(810\) 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 \(-0.437045\pi\)
0.196492 + 0.980506i \(0.437045\pi\)
\(822\) 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 \(-0.626070\pi\)
−0.385788 + 0.922587i \(0.626070\pi\)
\(828\) 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 \(-0.686117\pi\)
−0.551953 + 0.833875i \(0.686117\pi\)
\(840\) 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 \(-0.561876\pi\)
−0.193166 + 0.981166i \(0.561876\pi\)
\(858\) 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 \(-0.582743\pi\)
−0.257028 + 0.966404i \(0.582743\pi\)
\(864\) 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 \(-0.320128\pi\)
0.535487 + 0.844543i \(0.320128\pi\)
\(882\) 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 \(-0.479007\pi\)
0.0659029 + 0.997826i \(0.479007\pi\)
\(888\) 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 \(-0.419545\pi\)
0.250075 + 0.968226i \(0.419545\pi\)
\(912\) 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 \(-0.323814\pi\)
0.525671 + 0.850688i \(0.323814\pi\)
\(930\) 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 \(-0.521554\pi\)
−0.0676608 + 0.997708i \(0.521554\pi\)
\(942\) 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 \(-0.540086\pi\)
−0.125600 + 0.992081i \(0.540086\pi\)
\(948\) 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 \(-0.272184\pi\)
0.656149 + 0.754631i \(0.272184\pi\)
\(954\) 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 \(-0.431680\pi\)
0.212990 + 0.977054i \(0.431680\pi\)
\(972\) 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 \(-0.585927\pi\)
−0.266680 + 0.963785i \(0.585927\pi\)
\(978\) 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 \(-0.480036\pi\)
0.0626790 + 0.998034i \(0.480036\pi\)
\(984\) 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.k.1.5 yes 20
3.2 odd 2 2151.2.a.j.1.16 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.16 20 3.2 odd 2
2151.2.a.k.1.5 yes 20 1.1 even 1 trivial