Properties

Label 5445.2.a.bj.1.1
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} + 4 x + 1\)
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 1.1
Root \(1.33826\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.82709 q^{2} +1.33826 q^{4} -1.00000 q^{5} +1.74724 q^{7} +1.20906 q^{8} +O(q^{10})\) \(q-1.82709 q^{2} +1.33826 q^{4} -1.00000 q^{5} +1.74724 q^{7} +1.20906 q^{8} +1.82709 q^{10} -5.35772 q^{13} -3.19236 q^{14} -4.88558 q^{16} -7.33070 q^{17} +6.01478 q^{19} -1.33826 q^{20} -8.35772 q^{23} +1.00000 q^{25} +9.78903 q^{26} +2.33826 q^{28} -5.53818 q^{29} -0.0213642 q^{31} +6.50828 q^{32} +13.3939 q^{34} -1.74724 q^{35} -1.12165 q^{37} -10.9896 q^{38} -1.20906 q^{40} +3.65983 q^{41} -1.96950 q^{43} +15.2703 q^{46} -3.81953 q^{47} -3.94716 q^{49} -1.82709 q^{50} -7.17002 q^{52} -1.67088 q^{53} +2.11251 q^{56} +10.1188 q^{58} -5.66332 q^{59} +9.27977 q^{61} +0.0390343 q^{62} -2.12007 q^{64} +5.35772 q^{65} +0.350489 q^{67} -9.81040 q^{68} +3.19236 q^{70} -4.53818 q^{71} +10.1395 q^{73} +2.04935 q^{74} +8.04935 q^{76} +2.56401 q^{79} +4.88558 q^{80} -6.68684 q^{82} +16.0805 q^{83} +7.33070 q^{85} +3.59845 q^{86} +14.5062 q^{89} -9.36121 q^{91} -11.1848 q^{92} +6.97864 q^{94} -6.01478 q^{95} -1.89972 q^{97} +7.21182 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} + q^{4} - 4q^{5} + 3q^{8} + O(q^{10}) \) \( 4q - q^{2} + q^{4} - 4q^{5} + 3q^{8} + q^{10} + 7q^{13} + 5q^{14} - 9q^{16} - 8q^{17} + 11q^{19} - q^{20} - 5q^{23} + 4q^{25} + 12q^{26} + 5q^{28} - 17q^{29} - 5q^{31} + 17q^{34} + 15q^{37} + q^{38} - 3q^{40} - 10q^{41} - 4q^{43} + 15q^{46} + 8q^{47} - 8q^{49} - q^{50} - 7q^{52} - 10q^{53} - 10q^{56} - 7q^{58} - 9q^{59} + 37q^{61} + 20q^{62} - 7q^{64} - 7q^{65} + 3q^{67} - 17q^{68} - 5q^{70} - 13q^{71} + 15q^{73} + 5q^{74} + 29q^{76} + 20q^{79} + 9q^{80} + 5q^{82} + 17q^{83} + 8q^{85} - 34q^{86} + 24q^{89} - 20q^{91} - 10q^{92} + 23q^{94} - 11q^{95} - 32q^{97} - 13q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.82709 −1.29195 −0.645974 0.763359i \(-0.723549\pi\)
−0.645974 + 0.763359i \(0.723549\pi\)
\(3\) 0 0
\(4\) 1.33826 0.669131
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.74724 0.660394 0.330197 0.943912i \(-0.392885\pi\)
0.330197 + 0.943912i \(0.392885\pi\)
\(8\) 1.20906 0.427466
\(9\) 0 0
\(10\) 1.82709 0.577777
\(11\) 0 0
\(12\) 0 0
\(13\) −5.35772 −1.48596 −0.742981 0.669312i \(-0.766589\pi\)
−0.742981 + 0.669312i \(0.766589\pi\)
\(14\) −3.19236 −0.853195
\(15\) 0 0
\(16\) −4.88558 −1.22139
\(17\) −7.33070 −1.77796 −0.888978 0.457949i \(-0.848584\pi\)
−0.888978 + 0.457949i \(0.848584\pi\)
\(18\) 0 0
\(19\) 6.01478 1.37989 0.689943 0.723864i \(-0.257636\pi\)
0.689943 + 0.723864i \(0.257636\pi\)
\(20\) −1.33826 −0.299244
\(21\) 0 0
\(22\) 0 0
\(23\) −8.35772 −1.74270 −0.871352 0.490658i \(-0.836756\pi\)
−0.871352 + 0.490658i \(0.836756\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 9.78903 1.91979
\(27\) 0 0
\(28\) 2.33826 0.441890
\(29\) −5.53818 −1.02841 −0.514207 0.857666i \(-0.671914\pi\)
−0.514207 + 0.857666i \(0.671914\pi\)
\(30\) 0 0
\(31\) −0.0213642 −0.00383712 −0.00191856 0.999998i \(-0.500611\pi\)
−0.00191856 + 0.999998i \(0.500611\pi\)
\(32\) 6.50828 1.15051
\(33\) 0 0
\(34\) 13.3939 2.29703
\(35\) −1.74724 −0.295337
\(36\) 0 0
\(37\) −1.12165 −0.184398 −0.0921988 0.995741i \(-0.529390\pi\)
−0.0921988 + 0.995741i \(0.529390\pi\)
\(38\) −10.9896 −1.78274
\(39\) 0 0
\(40\) −1.20906 −0.191169
\(41\) 3.65983 0.571569 0.285785 0.958294i \(-0.407746\pi\)
0.285785 + 0.958294i \(0.407746\pi\)
\(42\) 0 0
\(43\) −1.96950 −0.300346 −0.150173 0.988660i \(-0.547983\pi\)
−0.150173 + 0.988660i \(0.547983\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 15.2703 2.25148
\(47\) −3.81953 −0.557136 −0.278568 0.960416i \(-0.589860\pi\)
−0.278568 + 0.960416i \(0.589860\pi\)
\(48\) 0 0
\(49\) −3.94716 −0.563880
\(50\) −1.82709 −0.258390
\(51\) 0 0
\(52\) −7.17002 −0.994303
\(53\) −1.67088 −0.229512 −0.114756 0.993394i \(-0.536609\pi\)
−0.114756 + 0.993394i \(0.536609\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.11251 0.282296
\(57\) 0 0
\(58\) 10.1188 1.32866
\(59\) −5.66332 −0.737301 −0.368651 0.929568i \(-0.620180\pi\)
−0.368651 + 0.929568i \(0.620180\pi\)
\(60\) 0 0
\(61\) 9.27977 1.18815 0.594077 0.804408i \(-0.297518\pi\)
0.594077 + 0.804408i \(0.297518\pi\)
\(62\) 0.0390343 0.00495737
\(63\) 0 0
\(64\) −2.12007 −0.265008
\(65\) 5.35772 0.664543
\(66\) 0 0
\(67\) 0.350489 0.0428190 0.0214095 0.999771i \(-0.493185\pi\)
0.0214095 + 0.999771i \(0.493185\pi\)
\(68\) −9.81040 −1.18969
\(69\) 0 0
\(70\) 3.19236 0.381560
\(71\) −4.53818 −0.538583 −0.269292 0.963059i \(-0.586790\pi\)
−0.269292 + 0.963059i \(0.586790\pi\)
\(72\) 0 0
\(73\) 10.1395 1.18674 0.593371 0.804929i \(-0.297797\pi\)
0.593371 + 0.804929i \(0.297797\pi\)
\(74\) 2.04935 0.238232
\(75\) 0 0
\(76\) 8.04935 0.923324
\(77\) 0 0
\(78\) 0 0
\(79\) 2.56401 0.288474 0.144237 0.989543i \(-0.453927\pi\)
0.144237 + 0.989543i \(0.453927\pi\)
\(80\) 4.88558 0.546224
\(81\) 0 0
\(82\) −6.68684 −0.738438
\(83\) 16.0805 1.76506 0.882532 0.470252i \(-0.155837\pi\)
0.882532 + 0.470252i \(0.155837\pi\)
\(84\) 0 0
\(85\) 7.33070 0.795127
\(86\) 3.59845 0.388031
\(87\) 0 0
\(88\) 0 0
\(89\) 14.5062 1.53765 0.768825 0.639459i \(-0.220841\pi\)
0.768825 + 0.639459i \(0.220841\pi\)
\(90\) 0 0
\(91\) −9.36121 −0.981321
\(92\) −11.1848 −1.16610
\(93\) 0 0
\(94\) 6.97864 0.719791
\(95\) −6.01478 −0.617104
\(96\) 0 0
\(97\) −1.89972 −0.192887 −0.0964435 0.995338i \(-0.530747\pi\)
−0.0964435 + 0.995338i \(0.530747\pi\)
\(98\) 7.21182 0.728504
\(99\) 0 0
\(100\) 1.33826 0.133826
\(101\) −6.71011 −0.667681 −0.333841 0.942630i \(-0.608345\pi\)
−0.333841 + 0.942630i \(0.608345\pi\)
\(102\) 0 0
\(103\) −4.24678 −0.418448 −0.209224 0.977868i \(-0.567094\pi\)
−0.209224 + 0.977868i \(0.567094\pi\)
\(104\) −6.47778 −0.635199
\(105\) 0 0
\(106\) 3.05284 0.296518
\(107\) 15.7230 1.52000 0.759999 0.649924i \(-0.225199\pi\)
0.759999 + 0.649924i \(0.225199\pi\)
\(108\) 0 0
\(109\) 4.23895 0.406018 0.203009 0.979177i \(-0.434928\pi\)
0.203009 + 0.979177i \(0.434928\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −8.53627 −0.806602
\(113\) 7.19643 0.676983 0.338492 0.940969i \(-0.390083\pi\)
0.338492 + 0.940969i \(0.390083\pi\)
\(114\) 0 0
\(115\) 8.35772 0.779361
\(116\) −7.41153 −0.688144
\(117\) 0 0
\(118\) 10.3474 0.952555
\(119\) −12.8085 −1.17415
\(120\) 0 0
\(121\) 0 0
\(122\) −16.9550 −1.53503
\(123\) 0 0
\(124\) −0.0285909 −0.00256754
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 12.6870 1.12579 0.562897 0.826527i \(-0.309687\pi\)
0.562897 + 0.826527i \(0.309687\pi\)
\(128\) −9.14301 −0.808136
\(129\) 0 0
\(130\) −9.78903 −0.858555
\(131\) 0.970102 0.0847582 0.0423791 0.999102i \(-0.486506\pi\)
0.0423791 + 0.999102i \(0.486506\pi\)
\(132\) 0 0
\(133\) 10.5093 0.911268
\(134\) −0.640375 −0.0553199
\(135\) 0 0
\(136\) −8.86324 −0.760016
\(137\) −4.22035 −0.360569 −0.180284 0.983615i \(-0.557702\pi\)
−0.180284 + 0.983615i \(0.557702\pi\)
\(138\) 0 0
\(139\) 8.84846 0.750516 0.375258 0.926920i \(-0.377554\pi\)
0.375258 + 0.926920i \(0.377554\pi\)
\(140\) −2.33826 −0.197619
\(141\) 0 0
\(142\) 8.29167 0.695821
\(143\) 0 0
\(144\) 0 0
\(145\) 5.53818 0.459921
\(146\) −18.5258 −1.53321
\(147\) 0 0
\(148\) −1.50106 −0.123386
\(149\) −9.55357 −0.782659 −0.391329 0.920251i \(-0.627985\pi\)
−0.391329 + 0.920251i \(0.627985\pi\)
\(150\) 0 0
\(151\) −6.75158 −0.549436 −0.274718 0.961525i \(-0.588584\pi\)
−0.274718 + 0.961525i \(0.588584\pi\)
\(152\) 7.27222 0.589855
\(153\) 0 0
\(154\) 0 0
\(155\) 0.0213642 0.00171601
\(156\) 0 0
\(157\) −12.6604 −1.01041 −0.505206 0.862999i \(-0.668584\pi\)
−0.505206 + 0.862999i \(0.668584\pi\)
\(158\) −4.68468 −0.372693
\(159\) 0 0
\(160\) −6.50828 −0.514525
\(161\) −14.6029 −1.15087
\(162\) 0 0
\(163\) −23.4058 −1.83328 −0.916640 0.399713i \(-0.869110\pi\)
−0.916640 + 0.399713i \(0.869110\pi\)
\(164\) 4.89781 0.382454
\(165\) 0 0
\(166\) −29.3805 −2.28037
\(167\) −3.05191 −0.236164 −0.118082 0.993004i \(-0.537675\pi\)
−0.118082 + 0.993004i \(0.537675\pi\)
\(168\) 0 0
\(169\) 15.7051 1.20809
\(170\) −13.3939 −1.02726
\(171\) 0 0
\(172\) −2.63570 −0.200971
\(173\) −16.0074 −1.21702 −0.608511 0.793545i \(-0.708233\pi\)
−0.608511 + 0.793545i \(0.708233\pi\)
\(174\) 0 0
\(175\) 1.74724 0.132079
\(176\) 0 0
\(177\) 0 0
\(178\) −26.5041 −1.98657
\(179\) 19.2991 1.44248 0.721241 0.692684i \(-0.243572\pi\)
0.721241 + 0.692684i \(0.243572\pi\)
\(180\) 0 0
\(181\) −12.1911 −0.906154 −0.453077 0.891471i \(-0.649674\pi\)
−0.453077 + 0.891471i \(0.649674\pi\)
\(182\) 17.1038 1.26782
\(183\) 0 0
\(184\) −10.1050 −0.744947
\(185\) 1.12165 0.0824652
\(186\) 0 0
\(187\) 0 0
\(188\) −5.11153 −0.372797
\(189\) 0 0
\(190\) 10.9896 0.797266
\(191\) 13.4244 0.971353 0.485676 0.874139i \(-0.338573\pi\)
0.485676 + 0.874139i \(0.338573\pi\)
\(192\) 0 0
\(193\) 7.57782 0.545463 0.272732 0.962090i \(-0.412073\pi\)
0.272732 + 0.962090i \(0.412073\pi\)
\(194\) 3.47096 0.249200
\(195\) 0 0
\(196\) −5.28233 −0.377309
\(197\) 6.89118 0.490976 0.245488 0.969400i \(-0.421052\pi\)
0.245488 + 0.969400i \(0.421052\pi\)
\(198\) 0 0
\(199\) 0.639398 0.0453257 0.0226629 0.999743i \(-0.492786\pi\)
0.0226629 + 0.999743i \(0.492786\pi\)
\(200\) 1.20906 0.0854932
\(201\) 0 0
\(202\) 12.2600 0.862610
\(203\) −9.67652 −0.679159
\(204\) 0 0
\(205\) −3.65983 −0.255614
\(206\) 7.75926 0.540613
\(207\) 0 0
\(208\) 26.1755 1.81495
\(209\) 0 0
\(210\) 0 0
\(211\) −3.82149 −0.263082 −0.131541 0.991311i \(-0.541992\pi\)
−0.131541 + 0.991311i \(0.541992\pi\)
\(212\) −2.23607 −0.153574
\(213\) 0 0
\(214\) −28.7273 −1.96376
\(215\) 1.96950 0.134319
\(216\) 0 0
\(217\) −0.0373284 −0.00253401
\(218\) −7.74496 −0.524555
\(219\) 0 0
\(220\) 0 0
\(221\) 39.2758 2.64198
\(222\) 0 0
\(223\) −26.7201 −1.78931 −0.894657 0.446755i \(-0.852580\pi\)
−0.894657 + 0.446755i \(0.852580\pi\)
\(224\) 11.3715 0.759792
\(225\) 0 0
\(226\) −13.1485 −0.874627
\(227\) −12.2373 −0.812219 −0.406110 0.913824i \(-0.633115\pi\)
−0.406110 + 0.913824i \(0.633115\pi\)
\(228\) 0 0
\(229\) −6.68664 −0.441865 −0.220933 0.975289i \(-0.570910\pi\)
−0.220933 + 0.975289i \(0.570910\pi\)
\(230\) −15.2703 −1.00689
\(231\) 0 0
\(232\) −6.69598 −0.439612
\(233\) 6.51742 0.426970 0.213485 0.976946i \(-0.431518\pi\)
0.213485 + 0.976946i \(0.431518\pi\)
\(234\) 0 0
\(235\) 3.81953 0.249159
\(236\) −7.57900 −0.493351
\(237\) 0 0
\(238\) 23.4023 1.51694
\(239\) 10.8419 0.701303 0.350651 0.936506i \(-0.385960\pi\)
0.350651 + 0.936506i \(0.385960\pi\)
\(240\) 0 0
\(241\) 27.1803 1.75084 0.875420 0.483363i \(-0.160585\pi\)
0.875420 + 0.483363i \(0.160585\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 12.4188 0.795030
\(245\) 3.94716 0.252175
\(246\) 0 0
\(247\) −32.2255 −2.05046
\(248\) −0.0258305 −0.00164024
\(249\) 0 0
\(250\) 1.82709 0.115555
\(251\) −5.54443 −0.349961 −0.174981 0.984572i \(-0.555986\pi\)
−0.174981 + 0.984572i \(0.555986\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −23.1804 −1.45447
\(255\) 0 0
\(256\) 20.9452 1.30908
\(257\) 18.9439 1.18169 0.590845 0.806785i \(-0.298794\pi\)
0.590845 + 0.806785i \(0.298794\pi\)
\(258\) 0 0
\(259\) −1.95979 −0.121775
\(260\) 7.17002 0.444666
\(261\) 0 0
\(262\) −1.77247 −0.109503
\(263\) −3.20873 −0.197859 −0.0989293 0.995094i \(-0.531542\pi\)
−0.0989293 + 0.995094i \(0.531542\pi\)
\(264\) 0 0
\(265\) 1.67088 0.102641
\(266\) −19.2014 −1.17731
\(267\) 0 0
\(268\) 0.469045 0.0286515
\(269\) 3.24423 0.197804 0.0989020 0.995097i \(-0.468467\pi\)
0.0989020 + 0.995097i \(0.468467\pi\)
\(270\) 0 0
\(271\) −2.02977 −0.123300 −0.0616499 0.998098i \(-0.519636\pi\)
−0.0616499 + 0.998098i \(0.519636\pi\)
\(272\) 35.8147 2.17159
\(273\) 0 0
\(274\) 7.71096 0.465836
\(275\) 0 0
\(276\) 0 0
\(277\) 11.3166 0.679947 0.339973 0.940435i \(-0.389582\pi\)
0.339973 + 0.940435i \(0.389582\pi\)
\(278\) −16.1669 −0.969628
\(279\) 0 0
\(280\) −2.11251 −0.126247
\(281\) 10.3177 0.615503 0.307751 0.951467i \(-0.400424\pi\)
0.307751 + 0.951467i \(0.400424\pi\)
\(282\) 0 0
\(283\) −2.10589 −0.125182 −0.0625910 0.998039i \(-0.519936\pi\)
−0.0625910 + 0.998039i \(0.519936\pi\)
\(284\) −6.07327 −0.360382
\(285\) 0 0
\(286\) 0 0
\(287\) 6.39459 0.377461
\(288\) 0 0
\(289\) 36.7392 2.16113
\(290\) −10.1188 −0.594194
\(291\) 0 0
\(292\) 13.5693 0.794085
\(293\) 23.3755 1.36561 0.682805 0.730601i \(-0.260760\pi\)
0.682805 + 0.730601i \(0.260760\pi\)
\(294\) 0 0
\(295\) 5.66332 0.329731
\(296\) −1.35614 −0.0788238
\(297\) 0 0
\(298\) 17.4552 1.01115
\(299\) 44.7783 2.58959
\(300\) 0 0
\(301\) −3.44118 −0.198347
\(302\) 12.3357 0.709843
\(303\) 0 0
\(304\) −29.3857 −1.68539
\(305\) −9.27977 −0.531358
\(306\) 0 0
\(307\) −32.0518 −1.82929 −0.914646 0.404256i \(-0.867531\pi\)
−0.914646 + 0.404256i \(0.867531\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −0.0390343 −0.00221700
\(311\) −1.63433 −0.0926743 −0.0463371 0.998926i \(-0.514755\pi\)
−0.0463371 + 0.998926i \(0.514755\pi\)
\(312\) 0 0
\(313\) 30.4130 1.71904 0.859522 0.511099i \(-0.170761\pi\)
0.859522 + 0.511099i \(0.170761\pi\)
\(314\) 23.1318 1.30540
\(315\) 0 0
\(316\) 3.43132 0.193027
\(317\) 29.4340 1.65318 0.826590 0.562805i \(-0.190278\pi\)
0.826590 + 0.562805i \(0.190278\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 2.12007 0.118515
\(321\) 0 0
\(322\) 26.6809 1.48687
\(323\) −44.0926 −2.45338
\(324\) 0 0
\(325\) −5.35772 −0.297193
\(326\) 42.7645 2.36850
\(327\) 0 0
\(328\) 4.42494 0.244327
\(329\) −6.67364 −0.367929
\(330\) 0 0
\(331\) −22.7902 −1.25266 −0.626332 0.779557i \(-0.715444\pi\)
−0.626332 + 0.779557i \(0.715444\pi\)
\(332\) 21.5199 1.18106
\(333\) 0 0
\(334\) 5.57611 0.305111
\(335\) −0.350489 −0.0191492
\(336\) 0 0
\(337\) −14.4901 −0.789325 −0.394662 0.918826i \(-0.629138\pi\)
−0.394662 + 0.918826i \(0.629138\pi\)
\(338\) −28.6947 −1.56078
\(339\) 0 0
\(340\) 9.81040 0.532043
\(341\) 0 0
\(342\) 0 0
\(343\) −19.1273 −1.03278
\(344\) −2.38124 −0.128388
\(345\) 0 0
\(346\) 29.2470 1.57233
\(347\) −4.38728 −0.235522 −0.117761 0.993042i \(-0.537572\pi\)
−0.117761 + 0.993042i \(0.537572\pi\)
\(348\) 0 0
\(349\) 14.4061 0.771140 0.385570 0.922679i \(-0.374005\pi\)
0.385570 + 0.922679i \(0.374005\pi\)
\(350\) −3.19236 −0.170639
\(351\) 0 0
\(352\) 0 0
\(353\) 6.01065 0.319914 0.159957 0.987124i \(-0.448864\pi\)
0.159957 + 0.987124i \(0.448864\pi\)
\(354\) 0 0
\(355\) 4.53818 0.240862
\(356\) 19.4130 1.02889
\(357\) 0 0
\(358\) −35.2612 −1.86361
\(359\) 2.45715 0.129683 0.0648417 0.997896i \(-0.479346\pi\)
0.0648417 + 0.997896i \(0.479346\pi\)
\(360\) 0 0
\(361\) 17.1776 0.904085
\(362\) 22.2742 1.17070
\(363\) 0 0
\(364\) −12.5277 −0.656632
\(365\) −10.1395 −0.530727
\(366\) 0 0
\(367\) 1.47990 0.0772500 0.0386250 0.999254i \(-0.487702\pi\)
0.0386250 + 0.999254i \(0.487702\pi\)
\(368\) 40.8323 2.12853
\(369\) 0 0
\(370\) −2.04935 −0.106541
\(371\) −2.91942 −0.151569
\(372\) 0 0
\(373\) −24.3331 −1.25992 −0.629961 0.776627i \(-0.716929\pi\)
−0.629961 + 0.776627i \(0.716929\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −4.61803 −0.238157
\(377\) 29.6720 1.52819
\(378\) 0 0
\(379\) 15.9311 0.818328 0.409164 0.912461i \(-0.365820\pi\)
0.409164 + 0.912461i \(0.365820\pi\)
\(380\) −8.04935 −0.412923
\(381\) 0 0
\(382\) −24.5275 −1.25494
\(383\) 3.91994 0.200300 0.100150 0.994972i \(-0.468068\pi\)
0.100150 + 0.994972i \(0.468068\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −13.8454 −0.704710
\(387\) 0 0
\(388\) −2.54232 −0.129067
\(389\) 35.2910 1.78933 0.894663 0.446742i \(-0.147416\pi\)
0.894663 + 0.446742i \(0.147416\pi\)
\(390\) 0 0
\(391\) 61.2679 3.09845
\(392\) −4.77234 −0.241040
\(393\) 0 0
\(394\) −12.5908 −0.634316
\(395\) −2.56401 −0.129009
\(396\) 0 0
\(397\) 23.2231 1.16553 0.582767 0.812639i \(-0.301970\pi\)
0.582767 + 0.812639i \(0.301970\pi\)
\(398\) −1.16824 −0.0585585
\(399\) 0 0
\(400\) −4.88558 −0.244279
\(401\) −5.69460 −0.284375 −0.142187 0.989840i \(-0.545414\pi\)
−0.142187 + 0.989840i \(0.545414\pi\)
\(402\) 0 0
\(403\) 0.114463 0.00570182
\(404\) −8.97989 −0.446766
\(405\) 0 0
\(406\) 17.6799 0.877438
\(407\) 0 0
\(408\) 0 0
\(409\) 25.4530 1.25857 0.629284 0.777176i \(-0.283348\pi\)
0.629284 + 0.777176i \(0.283348\pi\)
\(410\) 6.68684 0.330239
\(411\) 0 0
\(412\) −5.68331 −0.279996
\(413\) −9.89517 −0.486909
\(414\) 0 0
\(415\) −16.0805 −0.789361
\(416\) −34.8695 −1.70962
\(417\) 0 0
\(418\) 0 0
\(419\) 13.4850 0.658786 0.329393 0.944193i \(-0.393156\pi\)
0.329393 + 0.944193i \(0.393156\pi\)
\(420\) 0 0
\(421\) 2.94967 0.143758 0.0718791 0.997413i \(-0.477100\pi\)
0.0718791 + 0.997413i \(0.477100\pi\)
\(422\) 6.98220 0.339888
\(423\) 0 0
\(424\) −2.02018 −0.0981088
\(425\) −7.33070 −0.355591
\(426\) 0 0
\(427\) 16.2140 0.784649
\(428\) 21.0415 1.01708
\(429\) 0 0
\(430\) −3.59845 −0.173533
\(431\) −11.5227 −0.555031 −0.277515 0.960721i \(-0.589511\pi\)
−0.277515 + 0.960721i \(0.589511\pi\)
\(432\) 0 0
\(433\) 11.1638 0.536500 0.268250 0.963349i \(-0.413555\pi\)
0.268250 + 0.963349i \(0.413555\pi\)
\(434\) 0.0682023 0.00327382
\(435\) 0 0
\(436\) 5.67283 0.271679
\(437\) −50.2698 −2.40473
\(438\) 0 0
\(439\) 0.993624 0.0474231 0.0237115 0.999719i \(-0.492452\pi\)
0.0237115 + 0.999719i \(0.492452\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −71.7605 −3.41330
\(443\) 10.9825 0.521796 0.260898 0.965366i \(-0.415981\pi\)
0.260898 + 0.965366i \(0.415981\pi\)
\(444\) 0 0
\(445\) −14.5062 −0.687658
\(446\) 48.8201 2.31170
\(447\) 0 0
\(448\) −3.70426 −0.175010
\(449\) −29.0203 −1.36955 −0.684776 0.728753i \(-0.740100\pi\)
−0.684776 + 0.728753i \(0.740100\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 9.63070 0.452990
\(453\) 0 0
\(454\) 22.3587 1.04935
\(455\) 9.36121 0.438860
\(456\) 0 0
\(457\) −3.55670 −0.166375 −0.0831877 0.996534i \(-0.526510\pi\)
−0.0831877 + 0.996534i \(0.526510\pi\)
\(458\) 12.2171 0.570867
\(459\) 0 0
\(460\) 11.1848 0.521494
\(461\) −25.1829 −1.17288 −0.586441 0.809992i \(-0.699472\pi\)
−0.586441 + 0.809992i \(0.699472\pi\)
\(462\) 0 0
\(463\) −17.8743 −0.830690 −0.415345 0.909664i \(-0.636339\pi\)
−0.415345 + 0.909664i \(0.636339\pi\)
\(464\) 27.0572 1.25610
\(465\) 0 0
\(466\) −11.9079 −0.551624
\(467\) 25.2252 1.16728 0.583642 0.812011i \(-0.301627\pi\)
0.583642 + 0.812011i \(0.301627\pi\)
\(468\) 0 0
\(469\) 0.612387 0.0282774
\(470\) −6.97864 −0.321900
\(471\) 0 0
\(472\) −6.84727 −0.315171
\(473\) 0 0
\(474\) 0 0
\(475\) 6.01478 0.275977
\(476\) −17.1411 −0.785661
\(477\) 0 0
\(478\) −19.8091 −0.906047
\(479\) 5.23574 0.239227 0.119613 0.992821i \(-0.461834\pi\)
0.119613 + 0.992821i \(0.461834\pi\)
\(480\) 0 0
\(481\) 6.00947 0.274008
\(482\) −49.6610 −2.26199
\(483\) 0 0
\(484\) 0 0
\(485\) 1.89972 0.0862617
\(486\) 0 0
\(487\) 9.04508 0.409872 0.204936 0.978775i \(-0.434301\pi\)
0.204936 + 0.978775i \(0.434301\pi\)
\(488\) 11.2198 0.507895
\(489\) 0 0
\(490\) −7.21182 −0.325797
\(491\) 34.8112 1.57101 0.785505 0.618856i \(-0.212403\pi\)
0.785505 + 0.618856i \(0.212403\pi\)
\(492\) 0 0
\(493\) 40.5988 1.82848
\(494\) 58.8789 2.64909
\(495\) 0 0
\(496\) 0.104377 0.00468664
\(497\) −7.92928 −0.355677
\(498\) 0 0
\(499\) 19.5992 0.877380 0.438690 0.898638i \(-0.355443\pi\)
0.438690 + 0.898638i \(0.355443\pi\)
\(500\) −1.33826 −0.0598489
\(501\) 0 0
\(502\) 10.1302 0.452132
\(503\) −18.4160 −0.821129 −0.410564 0.911832i \(-0.634668\pi\)
−0.410564 + 0.911832i \(0.634668\pi\)
\(504\) 0 0
\(505\) 6.71011 0.298596
\(506\) 0 0
\(507\) 0 0
\(508\) 16.9786 0.753303
\(509\) −1.23838 −0.0548901 −0.0274451 0.999623i \(-0.508737\pi\)
−0.0274451 + 0.999623i \(0.508737\pi\)
\(510\) 0 0
\(511\) 17.7162 0.783717
\(512\) −19.9828 −0.883126
\(513\) 0 0
\(514\) −34.6123 −1.52668
\(515\) 4.24678 0.187136
\(516\) 0 0
\(517\) 0 0
\(518\) 3.58071 0.157327
\(519\) 0 0
\(520\) 6.47778 0.284070
\(521\) −33.1850 −1.45386 −0.726931 0.686711i \(-0.759054\pi\)
−0.726931 + 0.686711i \(0.759054\pi\)
\(522\) 0 0
\(523\) 13.1508 0.575046 0.287523 0.957774i \(-0.407168\pi\)
0.287523 + 0.957774i \(0.407168\pi\)
\(524\) 1.29825 0.0567143
\(525\) 0 0
\(526\) 5.86264 0.255623
\(527\) 0.156615 0.00682224
\(528\) 0 0
\(529\) 46.8514 2.03702
\(530\) −3.05284 −0.132607
\(531\) 0 0
\(532\) 14.0641 0.609758
\(533\) −19.6083 −0.849331
\(534\) 0 0
\(535\) −15.7230 −0.679764
\(536\) 0.423761 0.0183037
\(537\) 0 0
\(538\) −5.92750 −0.255553
\(539\) 0 0
\(540\) 0 0
\(541\) 32.6519 1.40381 0.701907 0.712269i \(-0.252332\pi\)
0.701907 + 0.712269i \(0.252332\pi\)
\(542\) 3.70858 0.159297
\(543\) 0 0
\(544\) −47.7103 −2.04556
\(545\) −4.23895 −0.181577
\(546\) 0 0
\(547\) 27.7048 1.18457 0.592285 0.805728i \(-0.298226\pi\)
0.592285 + 0.805728i \(0.298226\pi\)
\(548\) −5.64793 −0.241268
\(549\) 0 0
\(550\) 0 0
\(551\) −33.3110 −1.41909
\(552\) 0 0
\(553\) 4.47994 0.190506
\(554\) −20.6764 −0.878456
\(555\) 0 0
\(556\) 11.8415 0.502193
\(557\) 23.4252 0.992558 0.496279 0.868163i \(-0.334699\pi\)
0.496279 + 0.868163i \(0.334699\pi\)
\(558\) 0 0
\(559\) 10.5520 0.446303
\(560\) 8.53627 0.360723
\(561\) 0 0
\(562\) −18.8514 −0.795198
\(563\) −22.0856 −0.930796 −0.465398 0.885101i \(-0.654089\pi\)
−0.465398 + 0.885101i \(0.654089\pi\)
\(564\) 0 0
\(565\) −7.19643 −0.302756
\(566\) 3.84765 0.161729
\(567\) 0 0
\(568\) −5.48692 −0.230226
\(569\) 13.7351 0.575805 0.287902 0.957660i \(-0.407042\pi\)
0.287902 + 0.957660i \(0.407042\pi\)
\(570\) 0 0
\(571\) −26.5201 −1.10983 −0.554915 0.831907i \(-0.687249\pi\)
−0.554915 + 0.831907i \(0.687249\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −11.6835 −0.487660
\(575\) −8.35772 −0.348541
\(576\) 0 0
\(577\) 5.68765 0.236780 0.118390 0.992967i \(-0.462227\pi\)
0.118390 + 0.992967i \(0.462227\pi\)
\(578\) −67.1259 −2.79207
\(579\) 0 0
\(580\) 7.41153 0.307747
\(581\) 28.0965 1.16564
\(582\) 0 0
\(583\) 0 0
\(584\) 12.2593 0.507292
\(585\) 0 0
\(586\) −42.7091 −1.76430
\(587\) −13.5460 −0.559104 −0.279552 0.960131i \(-0.590186\pi\)
−0.279552 + 0.960131i \(0.590186\pi\)
\(588\) 0 0
\(589\) −0.128501 −0.00529479
\(590\) −10.3474 −0.425996
\(591\) 0 0
\(592\) 5.47990 0.225222
\(593\) −21.1786 −0.869702 −0.434851 0.900502i \(-0.643199\pi\)
−0.434851 + 0.900502i \(0.643199\pi\)
\(594\) 0 0
\(595\) 12.8085 0.525097
\(596\) −12.7852 −0.523701
\(597\) 0 0
\(598\) −81.8140 −3.34562
\(599\) 33.8404 1.38268 0.691341 0.722528i \(-0.257020\pi\)
0.691341 + 0.722528i \(0.257020\pi\)
\(600\) 0 0
\(601\) 29.3177 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(602\) 6.28736 0.256254
\(603\) 0 0
\(604\) −9.03538 −0.367644
\(605\) 0 0
\(606\) 0 0
\(607\) 7.17486 0.291219 0.145609 0.989342i \(-0.453486\pi\)
0.145609 + 0.989342i \(0.453486\pi\)
\(608\) 39.1459 1.58758
\(609\) 0 0
\(610\) 16.9550 0.686487
\(611\) 20.4640 0.827884
\(612\) 0 0
\(613\) −27.5044 −1.11089 −0.555447 0.831552i \(-0.687453\pi\)
−0.555447 + 0.831552i \(0.687453\pi\)
\(614\) 58.5615 2.36335
\(615\) 0 0
\(616\) 0 0
\(617\) −24.0712 −0.969068 −0.484534 0.874772i \(-0.661011\pi\)
−0.484534 + 0.874772i \(0.661011\pi\)
\(618\) 0 0
\(619\) 26.9729 1.08413 0.542067 0.840335i \(-0.317642\pi\)
0.542067 + 0.840335i \(0.317642\pi\)
\(620\) 0.0285909 0.00114824
\(621\) 0 0
\(622\) 2.98607 0.119730
\(623\) 25.3457 1.01546
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −55.5673 −2.22092
\(627\) 0 0
\(628\) −16.9430 −0.676098
\(629\) 8.22246 0.327851
\(630\) 0 0
\(631\) 9.73627 0.387595 0.193797 0.981042i \(-0.437920\pi\)
0.193797 + 0.981042i \(0.437920\pi\)
\(632\) 3.10004 0.123313
\(633\) 0 0
\(634\) −53.7786 −2.13582
\(635\) −12.6870 −0.503470
\(636\) 0 0
\(637\) 21.1478 0.837904
\(638\) 0 0
\(639\) 0 0
\(640\) 9.14301 0.361409
\(641\) −25.1165 −0.992040 −0.496020 0.868311i \(-0.665206\pi\)
−0.496020 + 0.868311i \(0.665206\pi\)
\(642\) 0 0
\(643\) −26.9257 −1.06185 −0.530924 0.847420i \(-0.678155\pi\)
−0.530924 + 0.847420i \(0.678155\pi\)
\(644\) −19.5425 −0.770083
\(645\) 0 0
\(646\) 80.5612 3.16964
\(647\) 26.8011 1.05366 0.526829 0.849971i \(-0.323381\pi\)
0.526829 + 0.849971i \(0.323381\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 9.78903 0.383957
\(651\) 0 0
\(652\) −31.3230 −1.22670
\(653\) −28.7633 −1.12559 −0.562797 0.826595i \(-0.690275\pi\)
−0.562797 + 0.826595i \(0.690275\pi\)
\(654\) 0 0
\(655\) −0.970102 −0.0379050
\(656\) −17.8804 −0.698112
\(657\) 0 0
\(658\) 12.1933 0.475346
\(659\) −36.5327 −1.42311 −0.711556 0.702629i \(-0.752009\pi\)
−0.711556 + 0.702629i \(0.752009\pi\)
\(660\) 0 0
\(661\) −35.7750 −1.39149 −0.695743 0.718291i \(-0.744925\pi\)
−0.695743 + 0.718291i \(0.744925\pi\)
\(662\) 41.6398 1.61838
\(663\) 0 0
\(664\) 19.4422 0.754505
\(665\) −10.5093 −0.407532
\(666\) 0 0
\(667\) 46.2865 1.79222
\(668\) −4.08425 −0.158024
\(669\) 0 0
\(670\) 0.640375 0.0247398
\(671\) 0 0
\(672\) 0 0
\(673\) −11.2915 −0.435254 −0.217627 0.976032i \(-0.569832\pi\)
−0.217627 + 0.976032i \(0.569832\pi\)
\(674\) 26.4747 1.01977
\(675\) 0 0
\(676\) 21.0175 0.808367
\(677\) 43.8574 1.68558 0.842788 0.538246i \(-0.180913\pi\)
0.842788 + 0.538246i \(0.180913\pi\)
\(678\) 0 0
\(679\) −3.31926 −0.127381
\(680\) 8.86324 0.339890
\(681\) 0 0
\(682\) 0 0
\(683\) 18.5355 0.709241 0.354620 0.935010i \(-0.384610\pi\)
0.354620 + 0.935010i \(0.384610\pi\)
\(684\) 0 0
\(685\) 4.22035 0.161251
\(686\) 34.9473 1.33429
\(687\) 0 0
\(688\) 9.62214 0.366841
\(689\) 8.95208 0.341047
\(690\) 0 0
\(691\) −28.2055 −1.07299 −0.536494 0.843904i \(-0.680251\pi\)
−0.536494 + 0.843904i \(0.680251\pi\)
\(692\) −21.4221 −0.814347
\(693\) 0 0
\(694\) 8.01596 0.304282
\(695\) −8.84846 −0.335641
\(696\) 0 0
\(697\) −26.8291 −1.01623
\(698\) −26.3212 −0.996274
\(699\) 0 0
\(700\) 2.33826 0.0883780
\(701\) −0.315386 −0.0119120 −0.00595599 0.999982i \(-0.501896\pi\)
−0.00595599 + 0.999982i \(0.501896\pi\)
\(702\) 0 0
\(703\) −6.74647 −0.254448
\(704\) 0 0
\(705\) 0 0
\(706\) −10.9820 −0.413313
\(707\) −11.7242 −0.440933
\(708\) 0 0
\(709\) 39.0246 1.46560 0.732800 0.680444i \(-0.238213\pi\)
0.732800 + 0.680444i \(0.238213\pi\)
\(710\) −8.29167 −0.311181
\(711\) 0 0
\(712\) 17.5388 0.657294
\(713\) 0.178556 0.00668697
\(714\) 0 0
\(715\) 0 0
\(716\) 25.8272 0.965209
\(717\) 0 0
\(718\) −4.48943 −0.167544
\(719\) −28.4691 −1.06172 −0.530859 0.847460i \(-0.678131\pi\)
−0.530859 + 0.847460i \(0.678131\pi\)
\(720\) 0 0
\(721\) −7.42014 −0.276341
\(722\) −31.3851 −1.16803
\(723\) 0 0
\(724\) −16.3148 −0.606335
\(725\) −5.53818 −0.205683
\(726\) 0 0
\(727\) 45.8400 1.70011 0.850057 0.526691i \(-0.176568\pi\)
0.850057 + 0.526691i \(0.176568\pi\)
\(728\) −11.3182 −0.419482
\(729\) 0 0
\(730\) 18.5258 0.685672
\(731\) 14.4378 0.534002
\(732\) 0 0
\(733\) 37.9681 1.40239 0.701193 0.712972i \(-0.252651\pi\)
0.701193 + 0.712972i \(0.252651\pi\)
\(734\) −2.70391 −0.0998030
\(735\) 0 0
\(736\) −54.3944 −2.00500
\(737\) 0 0
\(738\) 0 0
\(739\) 21.1226 0.777008 0.388504 0.921447i \(-0.372992\pi\)
0.388504 + 0.921447i \(0.372992\pi\)
\(740\) 1.50106 0.0551800
\(741\) 0 0
\(742\) 5.33404 0.195819
\(743\) −7.98035 −0.292771 −0.146385 0.989228i \(-0.546764\pi\)
−0.146385 + 0.989228i \(0.546764\pi\)
\(744\) 0 0
\(745\) 9.55357 0.350016
\(746\) 44.4588 1.62775
\(747\) 0 0
\(748\) 0 0
\(749\) 27.4718 1.00380
\(750\) 0 0
\(751\) −9.62482 −0.351215 −0.175607 0.984460i \(-0.556189\pi\)
−0.175607 + 0.984460i \(0.556189\pi\)
\(752\) 18.6606 0.680483
\(753\) 0 0
\(754\) −54.2134 −1.97434
\(755\) 6.75158 0.245715
\(756\) 0 0
\(757\) −46.6153 −1.69426 −0.847130 0.531385i \(-0.821672\pi\)
−0.847130 + 0.531385i \(0.821672\pi\)
\(758\) −29.1076 −1.05724
\(759\) 0 0
\(760\) −7.27222 −0.263791
\(761\) 21.9943 0.797293 0.398646 0.917105i \(-0.369480\pi\)
0.398646 + 0.917105i \(0.369480\pi\)
\(762\) 0 0
\(763\) 7.40646 0.268132
\(764\) 17.9653 0.649962
\(765\) 0 0
\(766\) −7.16209 −0.258777
\(767\) 30.3424 1.09560
\(768\) 0 0
\(769\) 48.0330 1.73211 0.866057 0.499946i \(-0.166647\pi\)
0.866057 + 0.499946i \(0.166647\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.1411 0.364986
\(773\) 17.0851 0.614509 0.307255 0.951627i \(-0.400590\pi\)
0.307255 + 0.951627i \(0.400590\pi\)
\(774\) 0 0
\(775\) −0.0213642 −0.000767425 0
\(776\) −2.29687 −0.0824527
\(777\) 0 0
\(778\) −64.4799 −2.31172
\(779\) 22.0131 0.788700
\(780\) 0 0
\(781\) 0 0
\(782\) −111.942 −4.00304
\(783\) 0 0
\(784\) 19.2842 0.688720
\(785\) 12.6604 0.451870
\(786\) 0 0
\(787\) −22.4918 −0.801746 −0.400873 0.916134i \(-0.631293\pi\)
−0.400873 + 0.916134i \(0.631293\pi\)
\(788\) 9.22220 0.328527
\(789\) 0 0
\(790\) 4.68468 0.166674
\(791\) 12.5739 0.447076
\(792\) 0 0
\(793\) −49.7184 −1.76555
\(794\) −42.4307 −1.50581
\(795\) 0 0
\(796\) 0.855682 0.0303288
\(797\) 3.41778 0.121064 0.0605321 0.998166i \(-0.480720\pi\)
0.0605321 + 0.998166i \(0.480720\pi\)
\(798\) 0 0
\(799\) 27.9999 0.990564
\(800\) 6.50828 0.230103
\(801\) 0 0
\(802\) 10.4046 0.367398
\(803\) 0 0
\(804\) 0 0
\(805\) 14.6029 0.514685
\(806\) −0.209135 −0.00736646
\(807\) 0 0
\(808\) −8.11291 −0.285411
\(809\) 10.1185 0.355748 0.177874 0.984053i \(-0.443078\pi\)
0.177874 + 0.984053i \(0.443078\pi\)
\(810\) 0 0
\(811\) 25.3749 0.891034 0.445517 0.895273i \(-0.353020\pi\)
0.445517 + 0.895273i \(0.353020\pi\)
\(812\) −12.9497 −0.454446
\(813\) 0 0
\(814\) 0 0
\(815\) 23.4058 0.819868
\(816\) 0 0
\(817\) −11.8461 −0.414443
\(818\) −46.5049 −1.62600
\(819\) 0 0
\(820\) −4.89781 −0.171039
\(821\) −1.49915 −0.0523206 −0.0261603 0.999658i \(-0.508328\pi\)
−0.0261603 + 0.999658i \(0.508328\pi\)
\(822\) 0 0
\(823\) −19.1246 −0.666641 −0.333320 0.942814i \(-0.608169\pi\)
−0.333320 + 0.942814i \(0.608169\pi\)
\(824\) −5.13460 −0.178872
\(825\) 0 0
\(826\) 18.0794 0.629062
\(827\) 16.6305 0.578298 0.289149 0.957284i \(-0.406628\pi\)
0.289149 + 0.957284i \(0.406628\pi\)
\(828\) 0 0
\(829\) 25.6955 0.892442 0.446221 0.894923i \(-0.352769\pi\)
0.446221 + 0.894923i \(0.352769\pi\)
\(830\) 29.3805 1.01981
\(831\) 0 0
\(832\) 11.3587 0.393793
\(833\) 28.9355 1.00255
\(834\) 0 0
\(835\) 3.05191 0.105616
\(836\) 0 0
\(837\) 0 0
\(838\) −24.6383 −0.851117
\(839\) 28.3971 0.980376 0.490188 0.871617i \(-0.336928\pi\)
0.490188 + 0.871617i \(0.336928\pi\)
\(840\) 0 0
\(841\) 1.67145 0.0576363
\(842\) −5.38932 −0.185728
\(843\) 0 0
\(844\) −5.11415 −0.176036
\(845\) −15.7051 −0.540272
\(846\) 0 0
\(847\) 0 0
\(848\) 8.16320 0.280325
\(849\) 0 0
\(850\) 13.3939 0.459406
\(851\) 9.37441 0.321351
\(852\) 0 0
\(853\) 15.1072 0.517260 0.258630 0.965976i \(-0.416729\pi\)
0.258630 + 0.965976i \(0.416729\pi\)
\(854\) −29.6244 −1.01373
\(855\) 0 0
\(856\) 19.0100 0.649748
\(857\) −2.37040 −0.0809713 −0.0404856 0.999180i \(-0.512891\pi\)
−0.0404856 + 0.999180i \(0.512891\pi\)
\(858\) 0 0
\(859\) 45.0423 1.53682 0.768411 0.639956i \(-0.221048\pi\)
0.768411 + 0.639956i \(0.221048\pi\)
\(860\) 2.63570 0.0898768
\(861\) 0 0
\(862\) 21.0531 0.717071
\(863\) 25.0210 0.851724 0.425862 0.904788i \(-0.359971\pi\)
0.425862 + 0.904788i \(0.359971\pi\)
\(864\) 0 0
\(865\) 16.0074 0.544269
\(866\) −20.3974 −0.693130
\(867\) 0 0
\(868\) −0.0499551 −0.00169559
\(869\) 0 0
\(870\) 0 0
\(871\) −1.87782 −0.0636274
\(872\) 5.12514 0.173559
\(873\) 0 0
\(874\) 91.8476 3.10679
\(875\) −1.74724 −0.0590674
\(876\) 0 0
\(877\) −48.8755 −1.65041 −0.825203 0.564836i \(-0.808940\pi\)
−0.825203 + 0.564836i \(0.808940\pi\)
\(878\) −1.81544 −0.0612682
\(879\) 0 0
\(880\) 0 0
\(881\) −44.5530 −1.50103 −0.750515 0.660853i \(-0.770195\pi\)
−0.750515 + 0.660853i \(0.770195\pi\)
\(882\) 0 0
\(883\) −44.0547 −1.48256 −0.741279 0.671197i \(-0.765780\pi\)
−0.741279 + 0.671197i \(0.765780\pi\)
\(884\) 52.5613 1.76783
\(885\) 0 0
\(886\) −20.0661 −0.674133
\(887\) 12.1293 0.407261 0.203631 0.979048i \(-0.434726\pi\)
0.203631 + 0.979048i \(0.434726\pi\)
\(888\) 0 0
\(889\) 22.1673 0.743467
\(890\) 26.5041 0.888419
\(891\) 0 0
\(892\) −35.7585 −1.19728
\(893\) −22.9737 −0.768785
\(894\) 0 0
\(895\) −19.2991 −0.645098
\(896\) −15.9750 −0.533688
\(897\) 0 0
\(898\) 53.0227 1.76939
\(899\) 0.118319 0.00394615
\(900\) 0 0
\(901\) 12.2487 0.408063
\(902\) 0 0
\(903\) 0 0
\(904\) 8.70089 0.289387
\(905\) 12.1911 0.405244
\(906\) 0 0
\(907\) 32.8672 1.09134 0.545670 0.838000i \(-0.316275\pi\)
0.545670 + 0.838000i \(0.316275\pi\)
\(908\) −16.3767 −0.543481
\(909\) 0 0
\(910\) −17.1038 −0.566985
\(911\) 46.7054 1.54742 0.773710 0.633540i \(-0.218399\pi\)
0.773710 + 0.633540i \(0.218399\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 6.49842 0.214949
\(915\) 0 0
\(916\) −8.94847 −0.295666
\(917\) 1.69500 0.0559738
\(918\) 0 0
\(919\) 7.33826 0.242067 0.121033 0.992648i \(-0.461379\pi\)
0.121033 + 0.992648i \(0.461379\pi\)
\(920\) 10.1050 0.333150
\(921\) 0 0
\(922\) 46.0114 1.51530
\(923\) 24.3143 0.800314
\(924\) 0 0
\(925\) −1.12165 −0.0368795
\(926\) 32.6580 1.07321
\(927\) 0 0
\(928\) −36.0441 −1.18320
\(929\) −12.7285 −0.417609 −0.208804 0.977957i \(-0.566957\pi\)
−0.208804 + 0.977957i \(0.566957\pi\)
\(930\) 0 0
\(931\) −23.7413 −0.778090
\(932\) 8.72201 0.285699
\(933\) 0 0
\(934\) −46.0888 −1.50807
\(935\) 0 0
\(936\) 0 0
\(937\) 19.2533 0.628979 0.314490 0.949261i \(-0.398167\pi\)
0.314490 + 0.949261i \(0.398167\pi\)
\(938\) −1.11889 −0.0365330
\(939\) 0 0
\(940\) 5.11153 0.166720
\(941\) 41.2694 1.34535 0.672673 0.739940i \(-0.265146\pi\)
0.672673 + 0.739940i \(0.265146\pi\)
\(942\) 0 0
\(943\) −30.5878 −0.996076
\(944\) 27.6686 0.900536
\(945\) 0 0
\(946\) 0 0
\(947\) 50.3012 1.63457 0.817285 0.576233i \(-0.195478\pi\)
0.817285 + 0.576233i \(0.195478\pi\)
\(948\) 0 0
\(949\) −54.3247 −1.76345
\(950\) −10.9896 −0.356548
\(951\) 0 0
\(952\) −15.4862 −0.501910
\(953\) 9.02580 0.292374 0.146187 0.989257i \(-0.453300\pi\)
0.146187 + 0.989257i \(0.453300\pi\)
\(954\) 0 0
\(955\) −13.4244 −0.434402
\(956\) 14.5093 0.469263
\(957\) 0 0
\(958\) −9.56617 −0.309069
\(959\) −7.37396 −0.238118
\(960\) 0 0
\(961\) −30.9995 −0.999985
\(962\) −10.9798 −0.354004
\(963\) 0 0
\(964\) 36.3744 1.17154
\(965\) −7.57782 −0.243939
\(966\) 0 0
\(967\) −30.2503 −0.972785 −0.486392 0.873740i \(-0.661687\pi\)
−0.486392 + 0.873740i \(0.661687\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −3.47096 −0.111446
\(971\) −12.8285 −0.411686 −0.205843 0.978585i \(-0.565994\pi\)
−0.205843 + 0.978585i \(0.565994\pi\)
\(972\) 0 0
\(973\) 15.4604 0.495636
\(974\) −16.5262 −0.529533
\(975\) 0 0
\(976\) −45.3371 −1.45120
\(977\) 13.1592 0.420999 0.210500 0.977594i \(-0.432491\pi\)
0.210500 + 0.977594i \(0.432491\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 5.28233 0.168738
\(981\) 0 0
\(982\) −63.6033 −2.02966
\(983\) −22.4369 −0.715627 −0.357813 0.933793i \(-0.616478\pi\)
−0.357813 + 0.933793i \(0.616478\pi\)
\(984\) 0 0
\(985\) −6.89118 −0.219571
\(986\) −74.1776 −2.36230
\(987\) 0 0
\(988\) −43.1261 −1.37203
\(989\) 16.4605 0.523414
\(990\) 0 0
\(991\) 22.9146 0.727907 0.363953 0.931417i \(-0.381427\pi\)
0.363953 + 0.931417i \(0.381427\pi\)
\(992\) −0.139044 −0.00441466
\(993\) 0 0
\(994\) 14.4875 0.459516
\(995\) −0.639398 −0.0202703
\(996\) 0 0
\(997\) −16.8394 −0.533309 −0.266654 0.963792i \(-0.585918\pi\)
−0.266654 + 0.963792i \(0.585918\pi\)
\(998\) −35.8095 −1.13353
\(999\) 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 5445.2.a.bj.1.1 4
3.2 odd 2 1815.2.a.u.1.4 4
11.3 even 5 495.2.n.c.361.1 8
11.4 even 5 495.2.n.c.181.1 8
11.10 odd 2 5445.2.a.bq.1.4 4
15.14 odd 2 9075.2.a.co.1.1 4
33.14 odd 10 165.2.m.c.31.2 yes 8
33.26 odd 10 165.2.m.c.16.2 8
33.32 even 2 1815.2.a.q.1.1 4
165.14 odd 10 825.2.n.j.526.1 8
165.47 even 20 825.2.bx.e.724.4 16
165.59 odd 10 825.2.n.j.676.1 8
165.92 even 20 825.2.bx.e.49.1 16
165.113 even 20 825.2.bx.e.724.1 16
165.158 even 20 825.2.bx.e.49.4 16
165.164 even 2 9075.2.a.df.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.c.16.2 8 33.26 odd 10
165.2.m.c.31.2 yes 8 33.14 odd 10
495.2.n.c.181.1 8 11.4 even 5
495.2.n.c.361.1 8 11.3 even 5
825.2.n.j.526.1 8 165.14 odd 10
825.2.n.j.676.1 8 165.59 odd 10
825.2.bx.e.49.1 16 165.92 even 20
825.2.bx.e.49.4 16 165.158 even 20
825.2.bx.e.724.1 16 165.113 even 20
825.2.bx.e.724.4 16 165.47 even 20
1815.2.a.q.1.1 4 33.32 even 2
1815.2.a.u.1.4 4 3.2 odd 2
5445.2.a.bj.1.1 4 1.1 even 1 trivial
5445.2.a.bq.1.4 4 11.10 odd 2
9075.2.a.co.1.1 4 15.14 odd 2
9075.2.a.df.1.4 4 165.164 even 2