Properties

Label 495.2.a.c.1.2
Level $495$
Weight $2$
Character 495.1
Self dual yes
Analytic conductor $3.953$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.73205 q^{8} +O(q^{10})\) \(q+1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.73205 q^{8} +1.73205 q^{10} +1.00000 q^{11} +5.46410 q^{13} +3.46410 q^{14} -5.00000 q^{16} +5.46410 q^{19} +1.00000 q^{20} +1.73205 q^{22} -6.92820 q^{23} +1.00000 q^{25} +9.46410 q^{26} +2.00000 q^{28} +3.46410 q^{29} -10.9282 q^{31} -5.19615 q^{32} +2.00000 q^{35} -4.92820 q^{37} +9.46410 q^{38} -1.73205 q^{40} -3.46410 q^{41} -4.92820 q^{43} +1.00000 q^{44} -12.0000 q^{46} +6.92820 q^{47} -3.00000 q^{49} +1.73205 q^{50} +5.46410 q^{52} -0.928203 q^{53} +1.00000 q^{55} -3.46410 q^{56} +6.00000 q^{58} +6.92820 q^{59} +2.00000 q^{61} -18.9282 q^{62} +1.00000 q^{64} +5.46410 q^{65} +8.00000 q^{67} +3.46410 q^{70} -13.8564 q^{71} -8.39230 q^{73} -8.53590 q^{74} +5.46410 q^{76} +2.00000 q^{77} -6.53590 q^{79} -5.00000 q^{80} -6.00000 q^{82} -8.53590 q^{83} -8.53590 q^{86} -1.73205 q^{88} -0.928203 q^{89} +10.9282 q^{91} -6.92820 q^{92} +12.0000 q^{94} +5.46410 q^{95} -10.0000 q^{97} -5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + 2q^{5} + 4q^{7} + O(q^{10}) \) \( 2q + 2q^{4} + 2q^{5} + 4q^{7} + 2q^{11} + 4q^{13} - 10q^{16} + 4q^{19} + 2q^{20} + 2q^{25} + 12q^{26} + 4q^{28} - 8q^{31} + 4q^{35} + 4q^{37} + 12q^{38} + 4q^{43} + 2q^{44} - 24q^{46} - 6q^{49} + 4q^{52} + 12q^{53} + 2q^{55} + 12q^{58} + 4q^{61} - 24q^{62} + 2q^{64} + 4q^{65} + 16q^{67} + 4q^{73} - 24q^{74} + 4q^{76} + 4q^{77} - 20q^{79} - 10q^{80} - 12q^{82} - 24q^{83} - 24q^{86} + 12q^{89} + 8q^{91} + 24q^{94} + 4q^{95} - 20q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) 1.73205 0.547723
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.46410 1.51547 0.757735 0.652563i \(-0.226306\pi\)
0.757735 + 0.652563i \(0.226306\pi\)
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 5.46410 1.25355 0.626775 0.779200i \(-0.284374\pi\)
0.626775 + 0.779200i \(0.284374\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.73205 0.369274
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 9.46410 1.85606
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 3.46410 0.643268 0.321634 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) −10.9282 −1.96276 −0.981382 0.192068i \(-0.938481\pi\)
−0.981382 + 0.192068i \(0.938481\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −4.92820 −0.810192 −0.405096 0.914274i \(-0.632762\pi\)
−0.405096 + 0.914274i \(0.632762\pi\)
\(38\) 9.46410 1.53528
\(39\) 0 0
\(40\) −1.73205 −0.273861
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 0 0
\(43\) −4.92820 −0.751544 −0.375772 0.926712i \(-0.622622\pi\)
−0.375772 + 0.926712i \(0.622622\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −12.0000 −1.76930
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 1.73205 0.244949
\(51\) 0 0
\(52\) 5.46410 0.757735
\(53\) −0.928203 −0.127499 −0.0637493 0.997966i \(-0.520306\pi\)
−0.0637493 + 0.997966i \(0.520306\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −3.46410 −0.462910
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −18.9282 −2.40388
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 5.46410 0.677738
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 3.46410 0.414039
\(71\) −13.8564 −1.64445 −0.822226 0.569160i \(-0.807268\pi\)
−0.822226 + 0.569160i \(0.807268\pi\)
\(72\) 0 0
\(73\) −8.39230 −0.982245 −0.491122 0.871091i \(-0.663413\pi\)
−0.491122 + 0.871091i \(0.663413\pi\)
\(74\) −8.53590 −0.992278
\(75\) 0 0
\(76\) 5.46410 0.626775
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −6.53590 −0.735346 −0.367673 0.929955i \(-0.619845\pi\)
−0.367673 + 0.929955i \(0.619845\pi\)
\(80\) −5.00000 −0.559017
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −8.53590 −0.936937 −0.468468 0.883480i \(-0.655194\pi\)
−0.468468 + 0.883480i \(0.655194\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.53590 −0.920450
\(87\) 0 0
\(88\) −1.73205 −0.184637
\(89\) −0.928203 −0.0983893 −0.0491947 0.998789i \(-0.515665\pi\)
−0.0491947 + 0.998789i \(0.515665\pi\)
\(90\) 0 0
\(91\) 10.9282 1.14559
\(92\) −6.92820 −0.722315
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 5.46410 0.560605
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −5.19615 −0.524891
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.3923 1.03407 0.517036 0.855963i \(-0.327035\pi\)
0.517036 + 0.855963i \(0.327035\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −9.46410 −0.928032
\(105\) 0 0
\(106\) −1.60770 −0.156153
\(107\) −8.53590 −0.825196 −0.412598 0.910913i \(-0.635379\pi\)
−0.412598 + 0.910913i \(0.635379\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 1.73205 0.165145
\(111\) 0 0
\(112\) −10.0000 −0.944911
\(113\) 12.9282 1.21618 0.608092 0.793867i \(-0.291935\pi\)
0.608092 + 0.793867i \(0.291935\pi\)
\(114\) 0 0
\(115\) −6.92820 −0.646058
\(116\) 3.46410 0.321634
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.46410 0.313625
\(123\) 0 0
\(124\) −10.9282 −0.981382
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.92820 0.792250 0.396125 0.918197i \(-0.370355\pi\)
0.396125 + 0.918197i \(0.370355\pi\)
\(128\) 12.1244 1.07165
\(129\) 0 0
\(130\) 9.46410 0.830057
\(131\) −18.9282 −1.65376 −0.826882 0.562375i \(-0.809888\pi\)
−0.826882 + 0.562375i \(0.809888\pi\)
\(132\) 0 0
\(133\) 10.9282 0.947595
\(134\) 13.8564 1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 12.3923 1.05110 0.525551 0.850762i \(-0.323859\pi\)
0.525551 + 0.850762i \(0.323859\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) −24.0000 −2.01404
\(143\) 5.46410 0.456931
\(144\) 0 0
\(145\) 3.46410 0.287678
\(146\) −14.5359 −1.20300
\(147\) 0 0
\(148\) −4.92820 −0.405096
\(149\) 15.4641 1.26687 0.633434 0.773796i \(-0.281645\pi\)
0.633434 + 0.773796i \(0.281645\pi\)
\(150\) 0 0
\(151\) −20.3923 −1.65950 −0.829751 0.558134i \(-0.811518\pi\)
−0.829751 + 0.558134i \(0.811518\pi\)
\(152\) −9.46410 −0.767640
\(153\) 0 0
\(154\) 3.46410 0.279145
\(155\) −10.9282 −0.877774
\(156\) 0 0
\(157\) −3.07180 −0.245156 −0.122578 0.992459i \(-0.539116\pi\)
−0.122578 + 0.992459i \(0.539116\pi\)
\(158\) −11.3205 −0.900611
\(159\) 0 0
\(160\) −5.19615 −0.410792
\(161\) −13.8564 −1.09204
\(162\) 0 0
\(163\) 9.85641 0.772013 0.386007 0.922496i \(-0.373854\pi\)
0.386007 + 0.922496i \(0.373854\pi\)
\(164\) −3.46410 −0.270501
\(165\) 0 0
\(166\) −14.7846 −1.14751
\(167\) 10.3923 0.804181 0.402090 0.915600i \(-0.368284\pi\)
0.402090 + 0.915600i \(0.368284\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) 0 0
\(172\) −4.92820 −0.375772
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) −5.00000 −0.376889
\(177\) 0 0
\(178\) −1.60770 −0.120502
\(179\) −6.92820 −0.517838 −0.258919 0.965899i \(-0.583366\pi\)
−0.258919 + 0.965899i \(0.583366\pi\)
\(180\) 0 0
\(181\) 15.8564 1.17860 0.589299 0.807915i \(-0.299404\pi\)
0.589299 + 0.807915i \(0.299404\pi\)
\(182\) 18.9282 1.40305
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) −4.92820 −0.362329
\(186\) 0 0
\(187\) 0 0
\(188\) 6.92820 0.505291
\(189\) 0 0
\(190\) 9.46410 0.686598
\(191\) 18.9282 1.36960 0.684798 0.728733i \(-0.259890\pi\)
0.684798 + 0.728733i \(0.259890\pi\)
\(192\) 0 0
\(193\) 24.3923 1.75580 0.877898 0.478847i \(-0.158945\pi\)
0.877898 + 0.478847i \(0.158945\pi\)
\(194\) −17.3205 −1.24354
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −24.7846 −1.75693 −0.878467 0.477803i \(-0.841433\pi\)
−0.878467 + 0.477803i \(0.841433\pi\)
\(200\) −1.73205 −0.122474
\(201\) 0 0
\(202\) 18.0000 1.26648
\(203\) 6.92820 0.486265
\(204\) 0 0
\(205\) −3.46410 −0.241943
\(206\) 13.8564 0.965422
\(207\) 0 0
\(208\) −27.3205 −1.89434
\(209\) 5.46410 0.377960
\(210\) 0 0
\(211\) −8.39230 −0.577750 −0.288875 0.957367i \(-0.593281\pi\)
−0.288875 + 0.957367i \(0.593281\pi\)
\(212\) −0.928203 −0.0637493
\(213\) 0 0
\(214\) −14.7846 −1.01066
\(215\) −4.92820 −0.336101
\(216\) 0 0
\(217\) −21.8564 −1.48371
\(218\) −17.3205 −1.17309
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) 0 0
\(222\) 0 0
\(223\) 9.85641 0.660034 0.330017 0.943975i \(-0.392946\pi\)
0.330017 + 0.943975i \(0.392946\pi\)
\(224\) −10.3923 −0.694365
\(225\) 0 0
\(226\) 22.3923 1.48951
\(227\) −15.4641 −1.02639 −0.513194 0.858272i \(-0.671538\pi\)
−0.513194 + 0.858272i \(0.671538\pi\)
\(228\) 0 0
\(229\) −23.8564 −1.57648 −0.788238 0.615371i \(-0.789006\pi\)
−0.788238 + 0.615371i \(0.789006\pi\)
\(230\) −12.0000 −0.791257
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) 6.92820 0.451946
\(236\) 6.92820 0.450988
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 0.143594 0.00924967 0.00462484 0.999989i \(-0.498528\pi\)
0.00462484 + 0.999989i \(0.498528\pi\)
\(242\) 1.73205 0.111340
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 29.8564 1.89972
\(248\) 18.9282 1.20194
\(249\) 0 0
\(250\) 1.73205 0.109545
\(251\) 1.85641 0.117175 0.0585877 0.998282i \(-0.481340\pi\)
0.0585877 + 0.998282i \(0.481340\pi\)
\(252\) 0 0
\(253\) −6.92820 −0.435572
\(254\) 15.4641 0.970304
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 19.8564 1.23861 0.619304 0.785151i \(-0.287415\pi\)
0.619304 + 0.785151i \(0.287415\pi\)
\(258\) 0 0
\(259\) −9.85641 −0.612447
\(260\) 5.46410 0.338869
\(261\) 0 0
\(262\) −32.7846 −2.02544
\(263\) −20.5359 −1.26630 −0.633149 0.774030i \(-0.718238\pi\)
−0.633149 + 0.774030i \(0.718238\pi\)
\(264\) 0 0
\(265\) −0.928203 −0.0570191
\(266\) 18.9282 1.16056
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) −19.8564 −1.21067 −0.605333 0.795972i \(-0.706960\pi\)
−0.605333 + 0.795972i \(0.706960\pi\)
\(270\) 0 0
\(271\) −11.6077 −0.705117 −0.352559 0.935790i \(-0.614688\pi\)
−0.352559 + 0.935790i \(0.614688\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 31.1769 1.88347
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 29.4641 1.77033 0.885163 0.465281i \(-0.154047\pi\)
0.885163 + 0.465281i \(0.154047\pi\)
\(278\) 21.4641 1.28733
\(279\) 0 0
\(280\) −3.46410 −0.207020
\(281\) −3.46410 −0.206651 −0.103325 0.994648i \(-0.532948\pi\)
−0.103325 + 0.994648i \(0.532948\pi\)
\(282\) 0 0
\(283\) −4.92820 −0.292951 −0.146476 0.989214i \(-0.546793\pi\)
−0.146476 + 0.989214i \(0.546793\pi\)
\(284\) −13.8564 −0.822226
\(285\) 0 0
\(286\) 9.46410 0.559624
\(287\) −6.92820 −0.408959
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −8.39230 −0.491122
\(293\) 13.8564 0.809500 0.404750 0.914427i \(-0.367359\pi\)
0.404750 + 0.914427i \(0.367359\pi\)
\(294\) 0 0
\(295\) 6.92820 0.403376
\(296\) 8.53590 0.496139
\(297\) 0 0
\(298\) 26.7846 1.55159
\(299\) −37.8564 −2.18929
\(300\) 0 0
\(301\) −9.85641 −0.568114
\(302\) −35.3205 −2.03247
\(303\) 0 0
\(304\) −27.3205 −1.56694
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 14.0000 0.799022 0.399511 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) 2.00000 0.113961
\(309\) 0 0
\(310\) −18.9282 −1.07505
\(311\) −5.07180 −0.287595 −0.143798 0.989607i \(-0.545931\pi\)
−0.143798 + 0.989607i \(0.545931\pi\)
\(312\) 0 0
\(313\) 20.9282 1.18293 0.591466 0.806330i \(-0.298549\pi\)
0.591466 + 0.806330i \(0.298549\pi\)
\(314\) −5.32051 −0.300254
\(315\) 0 0
\(316\) −6.53590 −0.367673
\(317\) 24.9282 1.40011 0.700054 0.714090i \(-0.253159\pi\)
0.700054 + 0.714090i \(0.253159\pi\)
\(318\) 0 0
\(319\) 3.46410 0.193952
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −24.0000 −1.33747
\(323\) 0 0
\(324\) 0 0
\(325\) 5.46410 0.303094
\(326\) 17.0718 0.945519
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 13.8564 0.763928
\(330\) 0 0
\(331\) 9.85641 0.541757 0.270879 0.962614i \(-0.412686\pi\)
0.270879 + 0.962614i \(0.412686\pi\)
\(332\) −8.53590 −0.468468
\(333\) 0 0
\(334\) 18.0000 0.984916
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 33.1769 1.80726 0.903631 0.428312i \(-0.140892\pi\)
0.903631 + 0.428312i \(0.140892\pi\)
\(338\) 29.1962 1.58806
\(339\) 0 0
\(340\) 0 0
\(341\) −10.9282 −0.591795
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 8.53590 0.460225
\(345\) 0 0
\(346\) 20.7846 1.11739
\(347\) −22.3923 −1.20208 −0.601041 0.799218i \(-0.705247\pi\)
−0.601041 + 0.799218i \(0.705247\pi\)
\(348\) 0 0
\(349\) −8.14359 −0.435917 −0.217958 0.975958i \(-0.569940\pi\)
−0.217958 + 0.975958i \(0.569940\pi\)
\(350\) 3.46410 0.185164
\(351\) 0 0
\(352\) −5.19615 −0.276956
\(353\) −12.9282 −0.688099 −0.344049 0.938952i \(-0.611799\pi\)
−0.344049 + 0.938952i \(0.611799\pi\)
\(354\) 0 0
\(355\) −13.8564 −0.735422
\(356\) −0.928203 −0.0491947
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 20.7846 1.09697 0.548485 0.836160i \(-0.315205\pi\)
0.548485 + 0.836160i \(0.315205\pi\)
\(360\) 0 0
\(361\) 10.8564 0.571390
\(362\) 27.4641 1.44348
\(363\) 0 0
\(364\) 10.9282 0.572793
\(365\) −8.39230 −0.439273
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 34.6410 1.80579
\(369\) 0 0
\(370\) −8.53590 −0.443760
\(371\) −1.85641 −0.0963798
\(372\) 0 0
\(373\) −20.3923 −1.05587 −0.527937 0.849284i \(-0.677034\pi\)
−0.527937 + 0.849284i \(0.677034\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 18.9282 0.974852
\(378\) 0 0
\(379\) −17.8564 −0.917222 −0.458611 0.888637i \(-0.651653\pi\)
−0.458611 + 0.888637i \(0.651653\pi\)
\(380\) 5.46410 0.280302
\(381\) 0 0
\(382\) 32.7846 1.67741
\(383\) 13.8564 0.708029 0.354015 0.935240i \(-0.384816\pi\)
0.354015 + 0.935240i \(0.384816\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 42.2487 2.15040
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) −11.0718 −0.561362 −0.280681 0.959801i \(-0.590560\pi\)
−0.280681 + 0.959801i \(0.590560\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 5.19615 0.262445
\(393\) 0 0
\(394\) 20.7846 1.04711
\(395\) −6.53590 −0.328857
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −42.9282 −2.15180
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 7.85641 0.392330 0.196165 0.980571i \(-0.437151\pi\)
0.196165 + 0.980571i \(0.437151\pi\)
\(402\) 0 0
\(403\) −59.7128 −2.97451
\(404\) 10.3923 0.517036
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) −4.92820 −0.244282
\(408\) 0 0
\(409\) −6.78461 −0.335477 −0.167739 0.985831i \(-0.553646\pi\)
−0.167739 + 0.985831i \(0.553646\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) 13.8564 0.681829
\(414\) 0 0
\(415\) −8.53590 −0.419011
\(416\) −28.3923 −1.39205
\(417\) 0 0
\(418\) 9.46410 0.462904
\(419\) −30.9282 −1.51094 −0.755471 0.655182i \(-0.772592\pi\)
−0.755471 + 0.655182i \(0.772592\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −14.5359 −0.707596
\(423\) 0 0
\(424\) 1.60770 0.0780766
\(425\) 0 0
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) −8.53590 −0.412598
\(429\) 0 0
\(430\) −8.53590 −0.411638
\(431\) −8.78461 −0.423140 −0.211570 0.977363i \(-0.567858\pi\)
−0.211570 + 0.977363i \(0.567858\pi\)
\(432\) 0 0
\(433\) 0.143594 0.00690067 0.00345033 0.999994i \(-0.498902\pi\)
0.00345033 + 0.999994i \(0.498902\pi\)
\(434\) −37.8564 −1.81717
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −37.8564 −1.81092
\(438\) 0 0
\(439\) 33.1769 1.58345 0.791724 0.610879i \(-0.209184\pi\)
0.791724 + 0.610879i \(0.209184\pi\)
\(440\) −1.73205 −0.0825723
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −0.928203 −0.0440011
\(446\) 17.0718 0.808373
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) 26.7846 1.26404 0.632022 0.774950i \(-0.282225\pi\)
0.632022 + 0.774950i \(0.282225\pi\)
\(450\) 0 0
\(451\) −3.46410 −0.163118
\(452\) 12.9282 0.608092
\(453\) 0 0
\(454\) −26.7846 −1.25706
\(455\) 10.9282 0.512322
\(456\) 0 0
\(457\) 12.3923 0.579688 0.289844 0.957074i \(-0.406397\pi\)
0.289844 + 0.957074i \(0.406397\pi\)
\(458\) −41.3205 −1.93078
\(459\) 0 0
\(460\) −6.92820 −0.323029
\(461\) −36.2487 −1.68827 −0.844135 0.536130i \(-0.819886\pi\)
−0.844135 + 0.536130i \(0.819886\pi\)
\(462\) 0 0
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) −17.3205 −0.804084
\(465\) 0 0
\(466\) −20.7846 −0.962828
\(467\) −5.07180 −0.234695 −0.117347 0.993091i \(-0.537439\pi\)
−0.117347 + 0.993091i \(0.537439\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 12.0000 0.553519
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) −4.92820 −0.226599
\(474\) 0 0
\(475\) 5.46410 0.250710
\(476\) 0 0
\(477\) 0 0
\(478\) 20.7846 0.950666
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −26.9282 −1.22782
\(482\) 0.248711 0.0113285
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) −31.7128 −1.43704 −0.718522 0.695504i \(-0.755181\pi\)
−0.718522 + 0.695504i \(0.755181\pi\)
\(488\) −3.46410 −0.156813
\(489\) 0 0
\(490\) −5.19615 −0.234738
\(491\) 30.9282 1.39577 0.697885 0.716210i \(-0.254125\pi\)
0.697885 + 0.716210i \(0.254125\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 51.7128 2.32667
\(495\) 0 0
\(496\) 54.6410 2.45345
\(497\) −27.7128 −1.24309
\(498\) 0 0
\(499\) 28.7846 1.28858 0.644288 0.764783i \(-0.277154\pi\)
0.644288 + 0.764783i \(0.277154\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 3.21539 0.143510
\(503\) −31.1769 −1.39011 −0.695055 0.718957i \(-0.744620\pi\)
−0.695055 + 0.718957i \(0.744620\pi\)
\(504\) 0 0
\(505\) 10.3923 0.462451
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) 8.92820 0.396125
\(509\) −19.8564 −0.880120 −0.440060 0.897968i \(-0.645043\pi\)
−0.440060 + 0.897968i \(0.645043\pi\)
\(510\) 0 0
\(511\) −16.7846 −0.742507
\(512\) 8.66025 0.382733
\(513\) 0 0
\(514\) 34.3923 1.51698
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 6.92820 0.304702
\(518\) −17.0718 −0.750092
\(519\) 0 0
\(520\) −9.46410 −0.415028
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −22.0000 −0.961993 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) −18.9282 −0.826882
\(525\) 0 0
\(526\) −35.5692 −1.55089
\(527\) 0 0
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) −1.60770 −0.0698338
\(531\) 0 0
\(532\) 10.9282 0.473798
\(533\) −18.9282 −0.819871
\(534\) 0 0
\(535\) −8.53590 −0.369039
\(536\) −13.8564 −0.598506
\(537\) 0 0
\(538\) −34.3923 −1.48276
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 27.8564 1.19764 0.598820 0.800883i \(-0.295636\pi\)
0.598820 + 0.800883i \(0.295636\pi\)
\(542\) −20.1051 −0.863589
\(543\) 0 0
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) 1.73205 0.0738549
\(551\) 18.9282 0.806369
\(552\) 0 0
\(553\) −13.0718 −0.555869
\(554\) 51.0333 2.16820
\(555\) 0 0
\(556\) 12.3923 0.525551
\(557\) −3.21539 −0.136240 −0.0681202 0.997677i \(-0.521700\pi\)
−0.0681202 + 0.997677i \(0.521700\pi\)
\(558\) 0 0
\(559\) −26.9282 −1.13894
\(560\) −10.0000 −0.422577
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −10.3923 −0.437983 −0.218992 0.975727i \(-0.570277\pi\)
−0.218992 + 0.975727i \(0.570277\pi\)
\(564\) 0 0
\(565\) 12.9282 0.543894
\(566\) −8.53590 −0.358791
\(567\) 0 0
\(568\) 24.0000 1.00702
\(569\) −5.32051 −0.223047 −0.111524 0.993762i \(-0.535573\pi\)
−0.111524 + 0.993762i \(0.535573\pi\)
\(570\) 0 0
\(571\) 3.60770 0.150977 0.0754887 0.997147i \(-0.475948\pi\)
0.0754887 + 0.997147i \(0.475948\pi\)
\(572\) 5.46410 0.228466
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) −6.92820 −0.288926
\(576\) 0 0
\(577\) −18.7846 −0.782014 −0.391007 0.920388i \(-0.627873\pi\)
−0.391007 + 0.920388i \(0.627873\pi\)
\(578\) −29.4449 −1.22474
\(579\) 0 0
\(580\) 3.46410 0.143839
\(581\) −17.0718 −0.708257
\(582\) 0 0
\(583\) −0.928203 −0.0384422
\(584\) 14.5359 0.601500
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 18.9282 0.781251 0.390625 0.920550i \(-0.372259\pi\)
0.390625 + 0.920550i \(0.372259\pi\)
\(588\) 0 0
\(589\) −59.7128 −2.46042
\(590\) 12.0000 0.494032
\(591\) 0 0
\(592\) 24.6410 1.01274
\(593\) −8.78461 −0.360741 −0.180370 0.983599i \(-0.557730\pi\)
−0.180370 + 0.983599i \(0.557730\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.4641 0.633434
\(597\) 0 0
\(598\) −65.5692 −2.68132
\(599\) 37.8564 1.54677 0.773385 0.633936i \(-0.218562\pi\)
0.773385 + 0.633936i \(0.218562\pi\)
\(600\) 0 0
\(601\) −32.6410 −1.33145 −0.665727 0.746195i \(-0.731879\pi\)
−0.665727 + 0.746195i \(0.731879\pi\)
\(602\) −17.0718 −0.695794
\(603\) 0 0
\(604\) −20.3923 −0.829751
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −18.7846 −0.762444 −0.381222 0.924484i \(-0.624497\pi\)
−0.381222 + 0.924484i \(0.624497\pi\)
\(608\) −28.3923 −1.15146
\(609\) 0 0
\(610\) 3.46410 0.140257
\(611\) 37.8564 1.53151
\(612\) 0 0
\(613\) −20.3923 −0.823637 −0.411819 0.911266i \(-0.635106\pi\)
−0.411819 + 0.911266i \(0.635106\pi\)
\(614\) 24.2487 0.978598
\(615\) 0 0
\(616\) −3.46410 −0.139573
\(617\) 36.9282 1.48667 0.743337 0.668917i \(-0.233242\pi\)
0.743337 + 0.668917i \(0.233242\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) −10.9282 −0.438887
\(621\) 0 0
\(622\) −8.78461 −0.352231
\(623\) −1.85641 −0.0743754
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 36.2487 1.44879
\(627\) 0 0
\(628\) −3.07180 −0.122578
\(629\) 0 0
\(630\) 0 0
\(631\) −21.0718 −0.838855 −0.419427 0.907789i \(-0.637769\pi\)
−0.419427 + 0.907789i \(0.637769\pi\)
\(632\) 11.3205 0.450306
\(633\) 0 0
\(634\) 43.1769 1.71477
\(635\) 8.92820 0.354305
\(636\) 0 0
\(637\) −16.3923 −0.649487
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) 12.1244 0.479257
\(641\) 12.9282 0.510633 0.255317 0.966857i \(-0.417820\pi\)
0.255317 + 0.966857i \(0.417820\pi\)
\(642\) 0 0
\(643\) 37.5692 1.48159 0.740793 0.671734i \(-0.234450\pi\)
0.740793 + 0.671734i \(0.234450\pi\)
\(644\) −13.8564 −0.546019
\(645\) 0 0
\(646\) 0 0
\(647\) −27.7128 −1.08950 −0.544752 0.838597i \(-0.683376\pi\)
−0.544752 + 0.838597i \(0.683376\pi\)
\(648\) 0 0
\(649\) 6.92820 0.271956
\(650\) 9.46410 0.371213
\(651\) 0 0
\(652\) 9.85641 0.386007
\(653\) −19.8564 −0.777041 −0.388521 0.921440i \(-0.627014\pi\)
−0.388521 + 0.921440i \(0.627014\pi\)
\(654\) 0 0
\(655\) −18.9282 −0.739586
\(656\) 17.3205 0.676252
\(657\) 0 0
\(658\) 24.0000 0.935617
\(659\) 15.7128 0.612084 0.306042 0.952018i \(-0.400995\pi\)
0.306042 + 0.952018i \(0.400995\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 17.0718 0.663514
\(663\) 0 0
\(664\) 14.7846 0.573754
\(665\) 10.9282 0.423778
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 10.3923 0.402090
\(669\) 0 0
\(670\) 13.8564 0.535320
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) −3.32051 −0.127996 −0.0639981 0.997950i \(-0.520385\pi\)
−0.0639981 + 0.997950i \(0.520385\pi\)
\(674\) 57.4641 2.21343
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) −8.78461 −0.337620 −0.168810 0.985649i \(-0.553992\pi\)
−0.168810 + 0.985649i \(0.553992\pi\)
\(678\) 0 0
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 0 0
\(682\) −18.9282 −0.724798
\(683\) 32.7846 1.25447 0.627234 0.778831i \(-0.284187\pi\)
0.627234 + 0.778831i \(0.284187\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) −34.6410 −1.32260
\(687\) 0 0
\(688\) 24.6410 0.939430
\(689\) −5.07180 −0.193220
\(690\) 0 0
\(691\) 47.7128 1.81508 0.907540 0.419965i \(-0.137958\pi\)
0.907540 + 0.419965i \(0.137958\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) −38.7846 −1.47224
\(695\) 12.3923 0.470067
\(696\) 0 0
\(697\) 0 0
\(698\) −14.1051 −0.533887
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) −39.4641 −1.49054 −0.745269 0.666764i \(-0.767679\pi\)
−0.745269 + 0.666764i \(0.767679\pi\)
\(702\) 0 0
\(703\) −26.9282 −1.01562
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −22.3923 −0.842746
\(707\) 20.7846 0.781686
\(708\) 0 0
\(709\) −11.8564 −0.445277 −0.222638 0.974901i \(-0.571467\pi\)
−0.222638 + 0.974901i \(0.571467\pi\)
\(710\) −24.0000 −0.900704
\(711\) 0 0
\(712\) 1.60770 0.0602509
\(713\) 75.7128 2.83547
\(714\) 0 0
\(715\) 5.46410 0.204346
\(716\) −6.92820 −0.258919
\(717\) 0 0
\(718\) 36.0000 1.34351
\(719\) 5.07180 0.189146 0.0945731 0.995518i \(-0.469851\pi\)
0.0945731 + 0.995518i \(0.469851\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 18.8038 0.699807
\(723\) 0 0
\(724\) 15.8564 0.589299
\(725\) 3.46410 0.128654
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −18.9282 −0.701526
\(729\) 0 0
\(730\) −14.5359 −0.537998
\(731\) 0 0
\(732\) 0 0
\(733\) 53.9615 1.99311 0.996557 0.0829082i \(-0.0264208\pi\)
0.996557 + 0.0829082i \(0.0264208\pi\)
\(734\) 34.6410 1.27862
\(735\) 0 0
\(736\) 36.0000 1.32698
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 17.4641 0.642427 0.321214 0.947007i \(-0.395909\pi\)
0.321214 + 0.947007i \(0.395909\pi\)
\(740\) −4.92820 −0.181164
\(741\) 0 0
\(742\) −3.21539 −0.118041
\(743\) −25.6077 −0.939455 −0.469728 0.882811i \(-0.655648\pi\)
−0.469728 + 0.882811i \(0.655648\pi\)
\(744\) 0 0
\(745\) 15.4641 0.566561
\(746\) −35.3205 −1.29318
\(747\) 0 0
\(748\) 0 0
\(749\) −17.0718 −0.623790
\(750\) 0 0
\(751\) 26.9282 0.982624 0.491312 0.870984i \(-0.336517\pi\)
0.491312 + 0.870984i \(0.336517\pi\)
\(752\) −34.6410 −1.26323
\(753\) 0 0
\(754\) 32.7846 1.19395
\(755\) −20.3923 −0.742152
\(756\) 0 0
\(757\) 34.7846 1.26427 0.632134 0.774859i \(-0.282179\pi\)
0.632134 + 0.774859i \(0.282179\pi\)
\(758\) −30.9282 −1.12336
\(759\) 0 0
\(760\) −9.46410 −0.343299
\(761\) 32.5359 1.17943 0.589713 0.807613i \(-0.299241\pi\)
0.589713 + 0.807613i \(0.299241\pi\)
\(762\) 0 0
\(763\) −20.0000 −0.724049
\(764\) 18.9282 0.684798
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 37.8564 1.36692
\(768\) 0 0
\(769\) 50.4974 1.82098 0.910492 0.413527i \(-0.135703\pi\)
0.910492 + 0.413527i \(0.135703\pi\)
\(770\) 3.46410 0.124838
\(771\) 0 0
\(772\) 24.3923 0.877898
\(773\) 4.14359 0.149035 0.0745174 0.997220i \(-0.476258\pi\)
0.0745174 + 0.997220i \(0.476258\pi\)
\(774\) 0 0
\(775\) −10.9282 −0.392553
\(776\) 17.3205 0.621770
\(777\) 0 0
\(778\) −19.1769 −0.687526
\(779\) −18.9282 −0.678173
\(780\) 0 0
\(781\) −13.8564 −0.495821
\(782\) 0 0
\(783\) 0 0
\(784\) 15.0000 0.535714
\(785\) −3.07180 −0.109637
\(786\) 0 0
\(787\) 22.7846 0.812184 0.406092 0.913832i \(-0.366891\pi\)
0.406092 + 0.913832i \(0.366891\pi\)
\(788\) 12.0000 0.427482
\(789\) 0 0
\(790\) −11.3205 −0.402766
\(791\) 25.8564 0.919348
\(792\) 0 0
\(793\) 10.9282 0.388072
\(794\) 3.46410 0.122936
\(795\) 0 0
\(796\) −24.7846 −0.878467
\(797\) 52.6410 1.86464 0.932320 0.361634i \(-0.117781\pi\)
0.932320 + 0.361634i \(0.117781\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −5.19615 −0.183712
\(801\) 0 0
\(802\) 13.6077 0.480504
\(803\) −8.39230 −0.296158
\(804\) 0 0
\(805\) −13.8564 −0.488374
\(806\) −103.426 −3.64301
\(807\) 0 0
\(808\) −18.0000 −0.633238
\(809\) 15.4641 0.543689 0.271844 0.962341i \(-0.412366\pi\)
0.271844 + 0.962341i \(0.412366\pi\)
\(810\) 0 0
\(811\) 12.3923 0.435153 0.217576 0.976043i \(-0.430185\pi\)
0.217576 + 0.976043i \(0.430185\pi\)
\(812\) 6.92820 0.243132
\(813\) 0 0
\(814\) −8.53590 −0.299183
\(815\) 9.85641 0.345255
\(816\) 0 0
\(817\) −26.9282 −0.942099
\(818\) −11.7513 −0.410874
\(819\) 0 0
\(820\) −3.46410 −0.120972
\(821\) −20.5359 −0.716708 −0.358354 0.933586i \(-0.616662\pi\)
−0.358354 + 0.933586i \(0.616662\pi\)
\(822\) 0 0
\(823\) −33.5692 −1.17015 −0.585075 0.810979i \(-0.698935\pi\)
−0.585075 + 0.810979i \(0.698935\pi\)
\(824\) −13.8564 −0.482711
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) −22.3923 −0.778657 −0.389328 0.921099i \(-0.627293\pi\)
−0.389328 + 0.921099i \(0.627293\pi\)
\(828\) 0 0
\(829\) 29.7128 1.03197 0.515984 0.856598i \(-0.327426\pi\)
0.515984 + 0.856598i \(0.327426\pi\)
\(830\) −14.7846 −0.513181
\(831\) 0 0
\(832\) 5.46410 0.189434
\(833\) 0 0
\(834\) 0 0
\(835\) 10.3923 0.359641
\(836\) 5.46410 0.188980
\(837\) 0 0
\(838\) −53.5692 −1.85052
\(839\) −56.7846 −1.96042 −0.980211 0.197954i \(-0.936570\pi\)
−0.980211 + 0.197954i \(0.936570\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 3.46410 0.119381
\(843\) 0 0
\(844\) −8.39230 −0.288875
\(845\) 16.8564 0.579878
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 4.64102 0.159373
\(849\) 0 0
\(850\) 0 0
\(851\) 34.1436 1.17043
\(852\) 0 0
\(853\) 3.60770 0.123525 0.0617626 0.998091i \(-0.480328\pi\)
0.0617626 + 0.998091i \(0.480328\pi\)
\(854\) 6.92820 0.237078
\(855\) 0 0
\(856\) 14.7846 0.505328
\(857\) 37.8564 1.29315 0.646575 0.762850i \(-0.276201\pi\)
0.646575 + 0.762850i \(0.276201\pi\)
\(858\) 0 0
\(859\) −7.71281 −0.263158 −0.131579 0.991306i \(-0.542005\pi\)
−0.131579 + 0.991306i \(0.542005\pi\)
\(860\) −4.92820 −0.168050
\(861\) 0 0
\(862\) −15.2154 −0.518238
\(863\) 37.8564 1.28865 0.644324 0.764753i \(-0.277139\pi\)
0.644324 + 0.764753i \(0.277139\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0.248711 0.00845155
\(867\) 0 0
\(868\) −21.8564 −0.741855
\(869\) −6.53590 −0.221715
\(870\) 0 0
\(871\) 43.7128 1.48115
\(872\) 17.3205 0.586546
\(873\) 0 0
\(874\) −65.5692 −2.21791
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −34.2487 −1.15650 −0.578248 0.815861i \(-0.696264\pi\)
−0.578248 + 0.815861i \(0.696264\pi\)
\(878\) 57.4641 1.93932
\(879\) 0 0
\(880\) −5.00000 −0.168550
\(881\) 0.928203 0.0312720 0.0156360 0.999878i \(-0.495023\pi\)
0.0156360 + 0.999878i \(0.495023\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −20.7846 −0.698273
\(887\) −12.2487 −0.411271 −0.205636 0.978629i \(-0.565926\pi\)
−0.205636 + 0.978629i \(0.565926\pi\)
\(888\) 0 0
\(889\) 17.8564 0.598885
\(890\) −1.60770 −0.0538901
\(891\) 0 0
\(892\) 9.85641 0.330017
\(893\) 37.8564 1.26682
\(894\) 0 0
\(895\) −6.92820 −0.231584
\(896\) 24.2487 0.810093
\(897\) 0 0
\(898\) 46.3923 1.54813
\(899\) −37.8564 −1.26258
\(900\) 0 0
\(901\) 0 0
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) −22.3923 −0.744757
\(905\) 15.8564 0.527085
\(906\) 0 0
\(907\) 18.1436 0.602448 0.301224 0.953553i \(-0.402605\pi\)
0.301224 + 0.953553i \(0.402605\pi\)
\(908\) −15.4641 −0.513194
\(909\) 0 0
\(910\) 18.9282 0.627464
\(911\) −18.9282 −0.627119 −0.313560 0.949568i \(-0.601522\pi\)
−0.313560 + 0.949568i \(0.601522\pi\)
\(912\) 0 0
\(913\) −8.53590 −0.282497
\(914\) 21.4641 0.709969
\(915\) 0 0
\(916\) −23.8564 −0.788238
\(917\) −37.8564 −1.25013
\(918\) 0 0
\(919\) −32.3923 −1.06852 −0.534262 0.845319i \(-0.679410\pi\)
−0.534262 + 0.845319i \(0.679410\pi\)
\(920\) 12.0000 0.395628
\(921\) 0 0
\(922\) −62.7846 −2.06770
\(923\) −75.7128 −2.49212
\(924\) 0 0
\(925\) −4.92820 −0.162038
\(926\) −48.4974 −1.59372
\(927\) 0 0
\(928\) −18.0000 −0.590879
\(929\) −2.78461 −0.0913601 −0.0456800 0.998956i \(-0.514545\pi\)
−0.0456800 + 0.998956i \(0.514545\pi\)
\(930\) 0 0
\(931\) −16.3923 −0.537236
\(932\) −12.0000 −0.393073
\(933\) 0 0
\(934\) −8.78461 −0.287441
\(935\) 0 0
\(936\) 0 0
\(937\) −20.3923 −0.666188 −0.333094 0.942894i \(-0.608093\pi\)
−0.333094 + 0.942894i \(0.608093\pi\)
\(938\) 27.7128 0.904855
\(939\) 0 0
\(940\) 6.92820 0.225973
\(941\) −27.4641 −0.895304 −0.447652 0.894208i \(-0.647740\pi\)
−0.447652 + 0.894208i \(0.647740\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) −34.6410 −1.12747
\(945\) 0 0
\(946\) −8.53590 −0.277526
\(947\) 18.9282 0.615084 0.307542 0.951535i \(-0.400494\pi\)
0.307542 + 0.951535i \(0.400494\pi\)
\(948\) 0 0
\(949\) −45.8564 −1.48856
\(950\) 9.46410 0.307056
\(951\) 0 0
\(952\) 0 0
\(953\) 3.21539 0.104157 0.0520784 0.998643i \(-0.483415\pi\)
0.0520784 + 0.998643i \(0.483415\pi\)
\(954\) 0 0
\(955\) 18.9282 0.612502
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −20.7846 −0.671520
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) 88.4256 2.85244
\(962\) −46.6410 −1.50377
\(963\) 0 0
\(964\) 0.143594 0.00462484
\(965\) 24.3923 0.785216
\(966\) 0 0
\(967\) 22.7846 0.732704 0.366352 0.930476i \(-0.380607\pi\)
0.366352 + 0.930476i \(0.380607\pi\)
\(968\) −1.73205 −0.0556702
\(969\) 0 0
\(970\) −17.3205 −0.556128
\(971\) −25.8564 −0.829772 −0.414886 0.909873i \(-0.636178\pi\)
−0.414886 + 0.909873i \(0.636178\pi\)
\(972\) 0 0
\(973\) 24.7846 0.794558
\(974\) −54.9282 −1.76001
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −47.5692 −1.52187 −0.760937 0.648826i \(-0.775260\pi\)
−0.760937 + 0.648826i \(0.775260\pi\)
\(978\) 0 0
\(979\) −0.928203 −0.0296655
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) 53.5692 1.70946
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) 0 0
\(988\) 29.8564 0.949859
\(989\) 34.1436 1.08570
\(990\) 0 0
\(991\) −7.21539 −0.229204 −0.114602 0.993411i \(-0.536559\pi\)
−0.114602 + 0.993411i \(0.536559\pi\)
\(992\) 56.7846 1.80291
\(993\) 0 0
\(994\) −48.0000 −1.52247
\(995\) −24.7846 −0.785725
\(996\) 0 0
\(997\) 27.6077 0.874344 0.437172 0.899378i \(-0.355980\pi\)
0.437172 + 0.899378i \(0.355980\pi\)
\(998\) 49.8564 1.57818
\(999\) 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 495.2.a.c.1.2 2
3.2 odd 2 165.2.a.b.1.1 2
4.3 odd 2 7920.2.a.bz.1.2 2
5.2 odd 4 2475.2.c.n.199.4 4
5.3 odd 4 2475.2.c.n.199.1 4
5.4 even 2 2475.2.a.r.1.1 2
11.10 odd 2 5445.2.a.s.1.1 2
12.11 even 2 2640.2.a.x.1.2 2
15.2 even 4 825.2.c.c.199.1 4
15.8 even 4 825.2.c.c.199.4 4
15.14 odd 2 825.2.a.e.1.2 2
21.20 even 2 8085.2.a.bd.1.1 2
33.32 even 2 1815.2.a.i.1.2 2
165.164 even 2 9075.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.1 2 3.2 odd 2
495.2.a.c.1.2 2 1.1 even 1 trivial
825.2.a.e.1.2 2 15.14 odd 2
825.2.c.c.199.1 4 15.2 even 4
825.2.c.c.199.4 4 15.8 even 4
1815.2.a.i.1.2 2 33.32 even 2
2475.2.a.r.1.1 2 5.4 even 2
2475.2.c.n.199.1 4 5.3 odd 4
2475.2.c.n.199.4 4 5.2 odd 4
2640.2.a.x.1.2 2 12.11 even 2
5445.2.a.s.1.1 2 11.10 odd 2
7920.2.a.bz.1.2 2 4.3 odd 2
8085.2.a.bd.1.1 2 21.20 even 2
9075.2.a.bh.1.1 2 165.164 even 2