Properties

Label 3330.2.a.bk.1.2
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.23544108.1
Defining polynomial: \(x^{5} - x^{4} - 20 x^{3} + 39 x^{2} + 9 x - 36\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.76376\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.19175 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.19175 q^{7} -1.00000 q^{8} +1.00000 q^{10} +3.64179 q^{11} +4.86897 q^{13} +1.19175 q^{14} +1.00000 q^{16} -3.19175 q^{17} +2.27506 q^{19} -1.00000 q^{20} -3.64179 q^{22} +2.86897 q^{23} +1.00000 q^{25} -4.86897 q^{26} -1.19175 q^{28} -6.06072 q^{29} +5.36672 q^{31} -1.00000 q^{32} +3.19175 q^{34} +1.19175 q^{35} +1.00000 q^{37} -2.27506 q^{38} +1.00000 q^{40} +8.75471 q^{41} -6.75471 q^{43} +3.64179 q^{44} -2.86897 q^{46} -4.13955 q^{47} -5.57973 q^{49} -1.00000 q^{50} +4.86897 q^{52} -5.88574 q^{53} -3.64179 q^{55} +1.19175 q^{56} +6.06072 q^{58} -7.28357 q^{59} +14.5443 q^{61} -5.36672 q^{62} +1.00000 q^{64} -4.86897 q^{65} -4.90007 q^{67} -3.19175 q^{68} -1.19175 q^{70} -5.38798 q^{71} +11.8026 q^{73} -1.00000 q^{74} +2.27506 q^{76} -4.34010 q^{77} +7.38798 q^{79} -1.00000 q^{80} -8.75471 q^{82} -4.41461 q^{83} +3.19175 q^{85} +6.75471 q^{86} -3.64179 q^{88} -9.63597 q^{89} -5.80259 q^{91} +2.86897 q^{92} +4.13955 q^{94} -2.27506 q^{95} +14.8943 q^{97} +5.57973 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{2} + 5q^{4} - 5q^{5} + 3q^{7} - 5q^{8} + O(q^{10}) \) \( 5q - 5q^{2} + 5q^{4} - 5q^{5} + 3q^{7} - 5q^{8} + 5q^{10} - 5q^{11} + 5q^{13} - 3q^{14} + 5q^{16} - 7q^{17} - q^{19} - 5q^{20} + 5q^{22} - 5q^{23} + 5q^{25} - 5q^{26} + 3q^{28} - 2q^{29} + 16q^{31} - 5q^{32} + 7q^{34} - 3q^{35} + 5q^{37} + q^{38} + 5q^{40} - 2q^{41} + 12q^{43} - 5q^{44} + 5q^{46} - 6q^{47} + 16q^{49} - 5q^{50} + 5q^{52} - 3q^{53} + 5q^{55} - 3q^{56} + 2q^{58} + 10q^{59} + 16q^{61} - 16q^{62} + 5q^{64} - 5q^{65} + 4q^{67} - 7q^{68} + 3q^{70} + 8q^{71} - 3q^{73} - 5q^{74} - q^{76} - 3q^{77} + 2q^{79} - 5q^{80} + 2q^{82} + 5q^{83} + 7q^{85} - 12q^{86} + 5q^{88} + 7q^{89} + 33q^{91} - 5q^{92} + 6q^{94} + q^{95} + 14q^{97} - 16q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.19175 −0.450439 −0.225220 0.974308i \(-0.572310\pi\)
−0.225220 + 0.974308i \(0.572310\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 3.64179 1.09804 0.549020 0.835809i \(-0.315001\pi\)
0.549020 + 0.835809i \(0.315001\pi\)
\(12\) 0 0
\(13\) 4.86897 1.35041 0.675204 0.737631i \(-0.264056\pi\)
0.675204 + 0.737631i \(0.264056\pi\)
\(14\) 1.19175 0.318509
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.19175 −0.774113 −0.387057 0.922056i \(-0.626508\pi\)
−0.387057 + 0.922056i \(0.626508\pi\)
\(18\) 0 0
\(19\) 2.27506 0.521935 0.260968 0.965348i \(-0.415958\pi\)
0.260968 + 0.965348i \(0.415958\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −3.64179 −0.776431
\(23\) 2.86897 0.598221 0.299110 0.954219i \(-0.403310\pi\)
0.299110 + 0.954219i \(0.403310\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.86897 −0.954883
\(27\) 0 0
\(28\) −1.19175 −0.225220
\(29\) −6.06072 −1.12545 −0.562723 0.826645i \(-0.690246\pi\)
−0.562723 + 0.826645i \(0.690246\pi\)
\(30\) 0 0
\(31\) 5.36672 0.963892 0.481946 0.876201i \(-0.339930\pi\)
0.481946 + 0.876201i \(0.339930\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.19175 0.547381
\(35\) 1.19175 0.201443
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −2.27506 −0.369064
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 8.75471 1.36726 0.683628 0.729831i \(-0.260401\pi\)
0.683628 + 0.729831i \(0.260401\pi\)
\(42\) 0 0
\(43\) −6.75471 −1.03008 −0.515042 0.857165i \(-0.672224\pi\)
−0.515042 + 0.857165i \(0.672224\pi\)
\(44\) 3.64179 0.549020
\(45\) 0 0
\(46\) −2.86897 −0.423006
\(47\) −4.13955 −0.603815 −0.301907 0.953337i \(-0.597623\pi\)
−0.301907 + 0.953337i \(0.597623\pi\)
\(48\) 0 0
\(49\) −5.57973 −0.797105
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 4.86897 0.675204
\(53\) −5.88574 −0.808469 −0.404234 0.914655i \(-0.632462\pi\)
−0.404234 + 0.914655i \(0.632462\pi\)
\(54\) 0 0
\(55\) −3.64179 −0.491058
\(56\) 1.19175 0.159254
\(57\) 0 0
\(58\) 6.06072 0.795811
\(59\) −7.28357 −0.948240 −0.474120 0.880460i \(-0.657234\pi\)
−0.474120 + 0.880460i \(0.657234\pi\)
\(60\) 0 0
\(61\) 14.5443 1.86221 0.931104 0.364755i \(-0.118847\pi\)
0.931104 + 0.364755i \(0.118847\pi\)
\(62\) −5.36672 −0.681575
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.86897 −0.603921
\(66\) 0 0
\(67\) −4.90007 −0.598639 −0.299320 0.954153i \(-0.596760\pi\)
−0.299320 + 0.954153i \(0.596760\pi\)
\(68\) −3.19175 −0.387057
\(69\) 0 0
\(70\) −1.19175 −0.142441
\(71\) −5.38798 −0.639436 −0.319718 0.947513i \(-0.603588\pi\)
−0.319718 + 0.947513i \(0.603588\pi\)
\(72\) 0 0
\(73\) 11.8026 1.38139 0.690694 0.723147i \(-0.257305\pi\)
0.690694 + 0.723147i \(0.257305\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 2.27506 0.260968
\(77\) −4.34010 −0.494600
\(78\) 0 0
\(79\) 7.38798 0.831213 0.415606 0.909545i \(-0.363569\pi\)
0.415606 + 0.909545i \(0.363569\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −8.75471 −0.966796
\(83\) −4.41461 −0.484566 −0.242283 0.970206i \(-0.577896\pi\)
−0.242283 + 0.970206i \(0.577896\pi\)
\(84\) 0 0
\(85\) 3.19175 0.346194
\(86\) 6.75471 0.728379
\(87\) 0 0
\(88\) −3.64179 −0.388216
\(89\) −9.63597 −1.02141 −0.510705 0.859756i \(-0.670616\pi\)
−0.510705 + 0.859756i \(0.670616\pi\)
\(90\) 0 0
\(91\) −5.80259 −0.608277
\(92\) 2.86897 0.299110
\(93\) 0 0
\(94\) 4.13955 0.426962
\(95\) −2.27506 −0.233416
\(96\) 0 0
\(97\) 14.8943 1.51228 0.756141 0.654409i \(-0.227082\pi\)
0.756141 + 0.654409i \(0.227082\pi\)
\(98\) 5.57973 0.563638
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 7.63194 0.759406 0.379703 0.925108i \(-0.376026\pi\)
0.379703 + 0.925108i \(0.376026\pi\)
\(102\) 0 0
\(103\) −2.13955 −0.210816 −0.105408 0.994429i \(-0.533615\pi\)
−0.105408 + 0.994429i \(0.533615\pi\)
\(104\) −4.86897 −0.477441
\(105\) 0 0
\(106\) 5.88574 0.571674
\(107\) 14.9904 1.44918 0.724588 0.689182i \(-0.242030\pi\)
0.724588 + 0.689182i \(0.242030\pi\)
\(108\) 0 0
\(109\) 0.253804 0.0243101 0.0121550 0.999926i \(-0.496131\pi\)
0.0121550 + 0.999926i \(0.496131\pi\)
\(110\) 3.64179 0.347231
\(111\) 0 0
\(112\) −1.19175 −0.112610
\(113\) 13.3443 1.25533 0.627663 0.778486i \(-0.284012\pi\)
0.627663 + 0.778486i \(0.284012\pi\)
\(114\) 0 0
\(115\) −2.86897 −0.267532
\(116\) −6.06072 −0.562723
\(117\) 0 0
\(118\) 7.28357 0.670507
\(119\) 3.80377 0.348691
\(120\) 0 0
\(121\) 2.26261 0.205692
\(122\) −14.5443 −1.31678
\(123\) 0 0
\(124\) 5.36672 0.481946
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −7.21891 −0.640575 −0.320288 0.947320i \(-0.603780\pi\)
−0.320288 + 0.947320i \(0.603780\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.86897 0.427037
\(131\) 20.0506 1.75183 0.875913 0.482468i \(-0.160260\pi\)
0.875913 + 0.482468i \(0.160260\pi\)
\(132\) 0 0
\(133\) −2.71131 −0.235100
\(134\) 4.90007 0.423302
\(135\) 0 0
\(136\) 3.19175 0.273690
\(137\) −9.52753 −0.813992 −0.406996 0.913430i \(-0.633424\pi\)
−0.406996 + 0.913430i \(0.633424\pi\)
\(138\) 0 0
\(139\) 20.7323 1.75849 0.879244 0.476372i \(-0.158048\pi\)
0.879244 + 0.476372i \(0.158048\pi\)
\(140\) 1.19175 0.100721
\(141\) 0 0
\(142\) 5.38798 0.452149
\(143\) 17.7317 1.48280
\(144\) 0 0
\(145\) 6.06072 0.503315
\(146\) −11.8026 −0.976789
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 9.17758 0.751857 0.375928 0.926649i \(-0.377324\pi\)
0.375928 + 0.926649i \(0.377324\pi\)
\(150\) 0 0
\(151\) 17.6024 1.43246 0.716232 0.697862i \(-0.245865\pi\)
0.716232 + 0.697862i \(0.245865\pi\)
\(152\) −2.27506 −0.184532
\(153\) 0 0
\(154\) 4.34010 0.349735
\(155\) −5.36672 −0.431066
\(156\) 0 0
\(157\) 6.89857 0.550566 0.275283 0.961363i \(-0.411228\pi\)
0.275283 + 0.961363i \(0.411228\pi\)
\(158\) −7.38798 −0.587756
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −3.41909 −0.269462
\(162\) 0 0
\(163\) −10.8194 −0.847438 −0.423719 0.905794i \(-0.639276\pi\)
−0.423719 + 0.905794i \(0.639276\pi\)
\(164\) 8.75471 0.683628
\(165\) 0 0
\(166\) 4.41461 0.342640
\(167\) −2.86897 −0.222007 −0.111004 0.993820i \(-0.535407\pi\)
−0.111004 + 0.993820i \(0.535407\pi\)
\(168\) 0 0
\(169\) 10.7068 0.823602
\(170\) −3.19175 −0.244796
\(171\) 0 0
\(172\) −6.75471 −0.515042
\(173\) −18.3571 −1.39567 −0.697833 0.716260i \(-0.745852\pi\)
−0.697833 + 0.716260i \(0.745852\pi\)
\(174\) 0 0
\(175\) −1.19175 −0.0900878
\(176\) 3.64179 0.274510
\(177\) 0 0
\(178\) 9.63597 0.722246
\(179\) 23.1259 1.72851 0.864256 0.503052i \(-0.167790\pi\)
0.864256 + 0.503052i \(0.167790\pi\)
\(180\) 0 0
\(181\) 14.9001 1.10751 0.553757 0.832678i \(-0.313194\pi\)
0.553757 + 0.832678i \(0.313194\pi\)
\(182\) 5.80259 0.430117
\(183\) 0 0
\(184\) −2.86897 −0.211503
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −11.6237 −0.850007
\(188\) −4.13955 −0.301907
\(189\) 0 0
\(190\) 2.27506 0.165050
\(191\) −8.31766 −0.601845 −0.300922 0.953649i \(-0.597295\pi\)
−0.300922 + 0.953649i \(0.597295\pi\)
\(192\) 0 0
\(193\) −21.5445 −1.55081 −0.775405 0.631464i \(-0.782454\pi\)
−0.775405 + 0.631464i \(0.782454\pi\)
\(194\) −14.8943 −1.06934
\(195\) 0 0
\(196\) −5.57973 −0.398552
\(197\) 3.76904 0.268533 0.134266 0.990945i \(-0.457132\pi\)
0.134266 + 0.990945i \(0.457132\pi\)
\(198\) 0 0
\(199\) −5.92914 −0.420306 −0.210153 0.977669i \(-0.567396\pi\)
−0.210153 + 0.977669i \(0.567396\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −7.63194 −0.536981
\(203\) 7.22286 0.506945
\(204\) 0 0
\(205\) −8.75471 −0.611455
\(206\) 2.13955 0.149069
\(207\) 0 0
\(208\) 4.86897 0.337602
\(209\) 8.28529 0.573105
\(210\) 0 0
\(211\) −3.51059 −0.241679 −0.120840 0.992672i \(-0.538559\pi\)
−0.120840 + 0.992672i \(0.538559\pi\)
\(212\) −5.88574 −0.404234
\(213\) 0 0
\(214\) −14.9904 −1.02472
\(215\) 6.75471 0.460667
\(216\) 0 0
\(217\) −6.39579 −0.434175
\(218\) −0.253804 −0.0171898
\(219\) 0 0
\(220\) −3.64179 −0.245529
\(221\) −15.5405 −1.04537
\(222\) 0 0
\(223\) 19.8322 1.32806 0.664031 0.747705i \(-0.268844\pi\)
0.664031 + 0.747705i \(0.268844\pi\)
\(224\) 1.19175 0.0796271
\(225\) 0 0
\(226\) −13.3443 −0.887649
\(227\) 12.9608 0.860238 0.430119 0.902772i \(-0.358472\pi\)
0.430119 + 0.902772i \(0.358472\pi\)
\(228\) 0 0
\(229\) −23.2384 −1.53564 −0.767818 0.640668i \(-0.778657\pi\)
−0.767818 + 0.640668i \(0.778657\pi\)
\(230\) 2.86897 0.189174
\(231\) 0 0
\(232\) 6.06072 0.397905
\(233\) 8.88196 0.581876 0.290938 0.956742i \(-0.406033\pi\)
0.290938 + 0.956742i \(0.406033\pi\)
\(234\) 0 0
\(235\) 4.13955 0.270034
\(236\) −7.28357 −0.474120
\(237\) 0 0
\(238\) −3.80377 −0.246562
\(239\) 25.3778 1.64156 0.820778 0.571247i \(-0.193540\pi\)
0.820778 + 0.571247i \(0.193540\pi\)
\(240\) 0 0
\(241\) −10.2429 −0.659801 −0.329900 0.944016i \(-0.607015\pi\)
−0.329900 + 0.944016i \(0.607015\pi\)
\(242\) −2.26261 −0.145446
\(243\) 0 0
\(244\) 14.5443 0.931104
\(245\) 5.57973 0.356476
\(246\) 0 0
\(247\) 11.0772 0.704825
\(248\) −5.36672 −0.340787
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 10.2258 0.645449 0.322725 0.946493i \(-0.395401\pi\)
0.322725 + 0.946493i \(0.395401\pi\)
\(252\) 0 0
\(253\) 10.4482 0.656870
\(254\) 7.21891 0.452955
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0481 −1.12581 −0.562906 0.826521i \(-0.690317\pi\)
−0.562906 + 0.826521i \(0.690317\pi\)
\(258\) 0 0
\(259\) −1.19175 −0.0740517
\(260\) −4.86897 −0.301960
\(261\) 0 0
\(262\) −20.0506 −1.23873
\(263\) −13.5658 −0.836503 −0.418252 0.908331i \(-0.637357\pi\)
−0.418252 + 0.908331i \(0.637357\pi\)
\(264\) 0 0
\(265\) 5.88574 0.361558
\(266\) 2.71131 0.166241
\(267\) 0 0
\(268\) −4.90007 −0.299320
\(269\) 18.3629 1.11961 0.559804 0.828625i \(-0.310876\pi\)
0.559804 + 0.828625i \(0.310876\pi\)
\(270\) 0 0
\(271\) −0.279091 −0.0169536 −0.00847678 0.999964i \(-0.502698\pi\)
−0.00847678 + 0.999964i \(0.502698\pi\)
\(272\) −3.19175 −0.193528
\(273\) 0 0
\(274\) 9.52753 0.575579
\(275\) 3.64179 0.219608
\(276\) 0 0
\(277\) 24.9195 1.49727 0.748635 0.662982i \(-0.230710\pi\)
0.748635 + 0.662982i \(0.230710\pi\)
\(278\) −20.7323 −1.24344
\(279\) 0 0
\(280\) −1.19175 −0.0712207
\(281\) −2.25695 −0.134638 −0.0673191 0.997731i \(-0.521445\pi\)
−0.0673191 + 0.997731i \(0.521445\pi\)
\(282\) 0 0
\(283\) 4.31884 0.256728 0.128364 0.991727i \(-0.459027\pi\)
0.128364 + 0.991727i \(0.459027\pi\)
\(284\) −5.38798 −0.319718
\(285\) 0 0
\(286\) −17.7317 −1.04850
\(287\) −10.4334 −0.615865
\(288\) 0 0
\(289\) −6.81273 −0.400749
\(290\) −6.06072 −0.355897
\(291\) 0 0
\(292\) 11.8026 0.690694
\(293\) −7.43138 −0.434146 −0.217073 0.976155i \(-0.569651\pi\)
−0.217073 + 0.976155i \(0.569651\pi\)
\(294\) 0 0
\(295\) 7.28357 0.424066
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) −9.17758 −0.531643
\(299\) 13.9689 0.807842
\(300\) 0 0
\(301\) 8.04992 0.463990
\(302\) −17.6024 −1.01291
\(303\) 0 0
\(304\) 2.27506 0.130484
\(305\) −14.5443 −0.832804
\(306\) 0 0
\(307\) 3.28357 0.187403 0.0937017 0.995600i \(-0.470130\pi\)
0.0937017 + 0.995600i \(0.470130\pi\)
\(308\) −4.34010 −0.247300
\(309\) 0 0
\(310\) 5.36672 0.304809
\(311\) 11.8438 0.671603 0.335801 0.941933i \(-0.390993\pi\)
0.335801 + 0.941933i \(0.390993\pi\)
\(312\) 0 0
\(313\) 15.4653 0.874151 0.437076 0.899425i \(-0.356014\pi\)
0.437076 + 0.899425i \(0.356014\pi\)
\(314\) −6.89857 −0.389309
\(315\) 0 0
\(316\) 7.38798 0.415606
\(317\) 16.4553 0.924223 0.462112 0.886822i \(-0.347092\pi\)
0.462112 + 0.886822i \(0.347092\pi\)
\(318\) 0 0
\(319\) −22.0718 −1.23579
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 3.41909 0.190538
\(323\) −7.26143 −0.404037
\(324\) 0 0
\(325\) 4.86897 0.270082
\(326\) 10.8194 0.599229
\(327\) 0 0
\(328\) −8.75471 −0.483398
\(329\) 4.93330 0.271982
\(330\) 0 0
\(331\) 35.4204 1.94688 0.973442 0.228935i \(-0.0735243\pi\)
0.973442 + 0.228935i \(0.0735243\pi\)
\(332\) −4.41461 −0.242283
\(333\) 0 0
\(334\) 2.86897 0.156983
\(335\) 4.90007 0.267720
\(336\) 0 0
\(337\) −4.80675 −0.261840 −0.130920 0.991393i \(-0.541793\pi\)
−0.130920 + 0.991393i \(0.541793\pi\)
\(338\) −10.7068 −0.582374
\(339\) 0 0
\(340\) 3.19175 0.173097
\(341\) 19.5445 1.05839
\(342\) 0 0
\(343\) 14.9919 0.809486
\(344\) 6.75471 0.364189
\(345\) 0 0
\(346\) 18.3571 0.986885
\(347\) −18.5049 −0.993397 −0.496698 0.867923i \(-0.665454\pi\)
−0.496698 + 0.867923i \(0.665454\pi\)
\(348\) 0 0
\(349\) −33.7975 −1.80914 −0.904568 0.426328i \(-0.859807\pi\)
−0.904568 + 0.426328i \(0.859807\pi\)
\(350\) 1.19175 0.0637017
\(351\) 0 0
\(352\) −3.64179 −0.194108
\(353\) 37.4657 1.99410 0.997049 0.0767613i \(-0.0244579\pi\)
0.997049 + 0.0767613i \(0.0244579\pi\)
\(354\) 0 0
\(355\) 5.38798 0.285964
\(356\) −9.63597 −0.510705
\(357\) 0 0
\(358\) −23.1259 −1.22224
\(359\) −14.6886 −0.775233 −0.387617 0.921821i \(-0.626702\pi\)
−0.387617 + 0.921821i \(0.626702\pi\)
\(360\) 0 0
\(361\) −13.8241 −0.727584
\(362\) −14.9001 −0.783130
\(363\) 0 0
\(364\) −5.80259 −0.304138
\(365\) −11.8026 −0.617776
\(366\) 0 0
\(367\) 16.1175 0.841326 0.420663 0.907217i \(-0.361797\pi\)
0.420663 + 0.907217i \(0.361797\pi\)
\(368\) 2.86897 0.149555
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) 7.01433 0.364166
\(372\) 0 0
\(373\) 3.25002 0.168280 0.0841399 0.996454i \(-0.473186\pi\)
0.0841399 + 0.996454i \(0.473186\pi\)
\(374\) 11.6237 0.601046
\(375\) 0 0
\(376\) 4.13955 0.213481
\(377\) −29.5094 −1.51981
\(378\) 0 0
\(379\) −33.7639 −1.73434 −0.867168 0.498016i \(-0.834062\pi\)
−0.867168 + 0.498016i \(0.834062\pi\)
\(380\) −2.27506 −0.116708
\(381\) 0 0
\(382\) 8.31766 0.425569
\(383\) 14.3859 0.735087 0.367544 0.930006i \(-0.380199\pi\)
0.367544 + 0.930006i \(0.380199\pi\)
\(384\) 0 0
\(385\) 4.34010 0.221192
\(386\) 21.5445 1.09659
\(387\) 0 0
\(388\) 14.8943 0.756141
\(389\) 25.5445 1.29516 0.647578 0.761999i \(-0.275782\pi\)
0.647578 + 0.761999i \(0.275782\pi\)
\(390\) 0 0
\(391\) −9.15702 −0.463090
\(392\) 5.57973 0.281819
\(393\) 0 0
\(394\) −3.76904 −0.189881
\(395\) −7.38798 −0.371730
\(396\) 0 0
\(397\) −4.81219 −0.241517 −0.120759 0.992682i \(-0.538533\pi\)
−0.120759 + 0.992682i \(0.538533\pi\)
\(398\) 5.92914 0.297201
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 8.45191 0.422068 0.211034 0.977479i \(-0.432317\pi\)
0.211034 + 0.977479i \(0.432317\pi\)
\(402\) 0 0
\(403\) 26.1304 1.30165
\(404\) 7.63194 0.379703
\(405\) 0 0
\(406\) −7.22286 −0.358464
\(407\) 3.64179 0.180517
\(408\) 0 0
\(409\) −5.51209 −0.272555 −0.136278 0.990671i \(-0.543514\pi\)
−0.136278 + 0.990671i \(0.543514\pi\)
\(410\) 8.75471 0.432364
\(411\) 0 0
\(412\) −2.13955 −0.105408
\(413\) 8.68020 0.427125
\(414\) 0 0
\(415\) 4.41461 0.216705
\(416\) −4.86897 −0.238721
\(417\) 0 0
\(418\) −8.28529 −0.405247
\(419\) 0.853529 0.0416976 0.0208488 0.999783i \(-0.493363\pi\)
0.0208488 + 0.999783i \(0.493363\pi\)
\(420\) 0 0
\(421\) 23.5275 1.14666 0.573331 0.819324i \(-0.305651\pi\)
0.573331 + 0.819324i \(0.305651\pi\)
\(422\) 3.51059 0.170893
\(423\) 0 0
\(424\) 5.88574 0.285837
\(425\) −3.19175 −0.154823
\(426\) 0 0
\(427\) −17.3332 −0.838811
\(428\) 14.9904 0.724588
\(429\) 0 0
\(430\) −6.75471 −0.325741
\(431\) 34.1849 1.64663 0.823315 0.567585i \(-0.192122\pi\)
0.823315 + 0.567585i \(0.192122\pi\)
\(432\) 0 0
\(433\) −0.747535 −0.0359242 −0.0179621 0.999839i \(-0.505718\pi\)
−0.0179621 + 0.999839i \(0.505718\pi\)
\(434\) 6.39579 0.307008
\(435\) 0 0
\(436\) 0.253804 0.0121550
\(437\) 6.52707 0.312232
\(438\) 0 0
\(439\) 14.4504 0.689682 0.344841 0.938661i \(-0.387933\pi\)
0.344841 + 0.938661i \(0.387933\pi\)
\(440\) 3.64179 0.173615
\(441\) 0 0
\(442\) 15.5405 0.739187
\(443\) 34.8974 1.65803 0.829013 0.559230i \(-0.188903\pi\)
0.829013 + 0.559230i \(0.188903\pi\)
\(444\) 0 0
\(445\) 9.63597 0.456789
\(446\) −19.8322 −0.939082
\(447\) 0 0
\(448\) −1.19175 −0.0563049
\(449\) 7.54564 0.356101 0.178050 0.984021i \(-0.443021\pi\)
0.178050 + 0.984021i \(0.443021\pi\)
\(450\) 0 0
\(451\) 31.8828 1.50130
\(452\) 13.3443 0.627663
\(453\) 0 0
\(454\) −12.9608 −0.608280
\(455\) 5.80259 0.272030
\(456\) 0 0
\(457\) −5.70645 −0.266936 −0.133468 0.991053i \(-0.542611\pi\)
−0.133468 + 0.991053i \(0.542611\pi\)
\(458\) 23.2384 1.08586
\(459\) 0 0
\(460\) −2.86897 −0.133766
\(461\) −18.0324 −0.839851 −0.419926 0.907559i \(-0.637944\pi\)
−0.419926 + 0.907559i \(0.637944\pi\)
\(462\) 0 0
\(463\) −25.3869 −1.17983 −0.589914 0.807466i \(-0.700839\pi\)
−0.589914 + 0.807466i \(0.700839\pi\)
\(464\) −6.06072 −0.281362
\(465\) 0 0
\(466\) −8.88196 −0.411449
\(467\) −23.6153 −1.09279 −0.546393 0.837529i \(-0.684000\pi\)
−0.546393 + 0.837529i \(0.684000\pi\)
\(468\) 0 0
\(469\) 5.83966 0.269651
\(470\) −4.13955 −0.190943
\(471\) 0 0
\(472\) 7.28357 0.335254
\(473\) −24.5992 −1.13107
\(474\) 0 0
\(475\) 2.27506 0.104387
\(476\) 3.80377 0.174345
\(477\) 0 0
\(478\) −25.3778 −1.16076
\(479\) 34.7981 1.58997 0.794983 0.606632i \(-0.207480\pi\)
0.794983 + 0.606632i \(0.207480\pi\)
\(480\) 0 0
\(481\) 4.86897 0.222006
\(482\) 10.2429 0.466550
\(483\) 0 0
\(484\) 2.26261 0.102846
\(485\) −14.8943 −0.676313
\(486\) 0 0
\(487\) −8.65612 −0.392246 −0.196123 0.980579i \(-0.562835\pi\)
−0.196123 + 0.980579i \(0.562835\pi\)
\(488\) −14.5443 −0.658390
\(489\) 0 0
\(490\) −5.57973 −0.252067
\(491\) −12.0632 −0.544407 −0.272203 0.962240i \(-0.587752\pi\)
−0.272203 + 0.962240i \(0.587752\pi\)
\(492\) 0 0
\(493\) 19.3443 0.871223
\(494\) −11.0772 −0.498387
\(495\) 0 0
\(496\) 5.36672 0.240973
\(497\) 6.42113 0.288027
\(498\) 0 0
\(499\) 1.14647 0.0513231 0.0256616 0.999671i \(-0.491831\pi\)
0.0256616 + 0.999671i \(0.491831\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −10.2258 −0.456402
\(503\) −6.98795 −0.311577 −0.155789 0.987790i \(-0.549792\pi\)
−0.155789 + 0.987790i \(0.549792\pi\)
\(504\) 0 0
\(505\) −7.63194 −0.339617
\(506\) −10.4482 −0.464477
\(507\) 0 0
\(508\) −7.21891 −0.320288
\(509\) 27.9592 1.23927 0.619634 0.784891i \(-0.287281\pi\)
0.619634 + 0.784891i \(0.287281\pi\)
\(510\) 0 0
\(511\) −14.0657 −0.622232
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 18.0481 0.796069
\(515\) 2.13955 0.0942796
\(516\) 0 0
\(517\) −15.0753 −0.663013
\(518\) 1.19175 0.0523625
\(519\) 0 0
\(520\) 4.86897 0.213518
\(521\) −29.5052 −1.29265 −0.646323 0.763064i \(-0.723694\pi\)
−0.646323 + 0.763064i \(0.723694\pi\)
\(522\) 0 0
\(523\) 13.0588 0.571022 0.285511 0.958375i \(-0.407837\pi\)
0.285511 + 0.958375i \(0.407837\pi\)
\(524\) 20.0506 0.875913
\(525\) 0 0
\(526\) 13.5658 0.591497
\(527\) −17.1292 −0.746162
\(528\) 0 0
\(529\) −14.7690 −0.642132
\(530\) −5.88574 −0.255660
\(531\) 0 0
\(532\) −2.71131 −0.117550
\(533\) 42.6264 1.84635
\(534\) 0 0
\(535\) −14.9904 −0.648091
\(536\) 4.90007 0.211651
\(537\) 0 0
\(538\) −18.3629 −0.791683
\(539\) −20.3202 −0.875253
\(540\) 0 0
\(541\) −23.5134 −1.01092 −0.505461 0.862850i \(-0.668678\pi\)
−0.505461 + 0.862850i \(0.668678\pi\)
\(542\) 0.279091 0.0119880
\(543\) 0 0
\(544\) 3.19175 0.136845
\(545\) −0.253804 −0.0108718
\(546\) 0 0
\(547\) −33.5166 −1.43307 −0.716533 0.697553i \(-0.754272\pi\)
−0.716533 + 0.697553i \(0.754272\pi\)
\(548\) −9.52753 −0.406996
\(549\) 0 0
\(550\) −3.64179 −0.155286
\(551\) −13.7885 −0.587410
\(552\) 0 0
\(553\) −8.80463 −0.374411
\(554\) −24.9195 −1.05873
\(555\) 0 0
\(556\) 20.7323 0.879244
\(557\) 11.6308 0.492815 0.246407 0.969166i \(-0.420750\pi\)
0.246407 + 0.969166i \(0.420750\pi\)
\(558\) 0 0
\(559\) −32.8884 −1.39103
\(560\) 1.19175 0.0503606
\(561\) 0 0
\(562\) 2.25695 0.0952036
\(563\) −2.37336 −0.100025 −0.0500125 0.998749i \(-0.515926\pi\)
−0.0500125 + 0.998749i \(0.515926\pi\)
\(564\) 0 0
\(565\) −13.3443 −0.561398
\(566\) −4.31884 −0.181534
\(567\) 0 0
\(568\) 5.38798 0.226075
\(569\) −3.83093 −0.160601 −0.0803005 0.996771i \(-0.525588\pi\)
−0.0803005 + 0.996771i \(0.525588\pi\)
\(570\) 0 0
\(571\) 3.85678 0.161401 0.0807007 0.996738i \(-0.474284\pi\)
0.0807007 + 0.996738i \(0.474284\pi\)
\(572\) 17.7317 0.741401
\(573\) 0 0
\(574\) 10.4334 0.435483
\(575\) 2.86897 0.119644
\(576\) 0 0
\(577\) −15.2565 −0.635136 −0.317568 0.948235i \(-0.602866\pi\)
−0.317568 + 0.948235i \(0.602866\pi\)
\(578\) 6.81273 0.283372
\(579\) 0 0
\(580\) 6.06072 0.251658
\(581\) 5.26111 0.218268
\(582\) 0 0
\(583\) −21.4346 −0.887731
\(584\) −11.8026 −0.488395
\(585\) 0 0
\(586\) 7.43138 0.306988
\(587\) −20.0442 −0.827312 −0.413656 0.910433i \(-0.635748\pi\)
−0.413656 + 0.910433i \(0.635748\pi\)
\(588\) 0 0
\(589\) 12.2096 0.503089
\(590\) −7.28357 −0.299860
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −38.0414 −1.56217 −0.781087 0.624422i \(-0.785334\pi\)
−0.781087 + 0.624422i \(0.785334\pi\)
\(594\) 0 0
\(595\) −3.80377 −0.155939
\(596\) 9.17758 0.375928
\(597\) 0 0
\(598\) −13.9689 −0.571230
\(599\) −0.900073 −0.0367760 −0.0183880 0.999831i \(-0.505853\pi\)
−0.0183880 + 0.999831i \(0.505853\pi\)
\(600\) 0 0
\(601\) 17.8298 0.727291 0.363645 0.931537i \(-0.381532\pi\)
0.363645 + 0.931537i \(0.381532\pi\)
\(602\) −8.04992 −0.328090
\(603\) 0 0
\(604\) 17.6024 0.716232
\(605\) −2.26261 −0.0919881
\(606\) 0 0
\(607\) −38.6109 −1.56717 −0.783585 0.621285i \(-0.786611\pi\)
−0.783585 + 0.621285i \(0.786611\pi\)
\(608\) −2.27506 −0.0922659
\(609\) 0 0
\(610\) 14.5443 0.588882
\(611\) −20.1553 −0.815396
\(612\) 0 0
\(613\) 13.7108 0.553773 0.276886 0.960903i \(-0.410697\pi\)
0.276886 + 0.960903i \(0.410697\pi\)
\(614\) −3.28357 −0.132514
\(615\) 0 0
\(616\) 4.34010 0.174868
\(617\) 23.0875 0.929468 0.464734 0.885450i \(-0.346150\pi\)
0.464734 + 0.885450i \(0.346150\pi\)
\(618\) 0 0
\(619\) 28.4872 1.14500 0.572499 0.819905i \(-0.305974\pi\)
0.572499 + 0.819905i \(0.305974\pi\)
\(620\) −5.36672 −0.215533
\(621\) 0 0
\(622\) −11.8438 −0.474895
\(623\) 11.4837 0.460083
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −15.4653 −0.618118
\(627\) 0 0
\(628\) 6.89857 0.275283
\(629\) −3.19175 −0.127263
\(630\) 0 0
\(631\) 5.84975 0.232875 0.116437 0.993198i \(-0.462853\pi\)
0.116437 + 0.993198i \(0.462853\pi\)
\(632\) −7.38798 −0.293878
\(633\) 0 0
\(634\) −16.4553 −0.653525
\(635\) 7.21891 0.286474
\(636\) 0 0
\(637\) −27.1675 −1.07642
\(638\) 22.0718 0.873832
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 16.5884 0.655203 0.327601 0.944816i \(-0.393760\pi\)
0.327601 + 0.944816i \(0.393760\pi\)
\(642\) 0 0
\(643\) −23.3695 −0.921603 −0.460801 0.887503i \(-0.652438\pi\)
−0.460801 + 0.887503i \(0.652438\pi\)
\(644\) −3.41909 −0.134731
\(645\) 0 0
\(646\) 7.26143 0.285697
\(647\) −25.8618 −1.01673 −0.508366 0.861141i \(-0.669750\pi\)
−0.508366 + 0.861141i \(0.669750\pi\)
\(648\) 0 0
\(649\) −26.5252 −1.04121
\(650\) −4.86897 −0.190977
\(651\) 0 0
\(652\) −10.8194 −0.423719
\(653\) −34.9853 −1.36908 −0.684540 0.728976i \(-0.739997\pi\)
−0.684540 + 0.728976i \(0.739997\pi\)
\(654\) 0 0
\(655\) −20.0506 −0.783441
\(656\) 8.75471 0.341814
\(657\) 0 0
\(658\) −4.93330 −0.192320
\(659\) 21.3274 0.830796 0.415398 0.909640i \(-0.363642\pi\)
0.415398 + 0.909640i \(0.363642\pi\)
\(660\) 0 0
\(661\) −13.8420 −0.538390 −0.269195 0.963086i \(-0.586758\pi\)
−0.269195 + 0.963086i \(0.586758\pi\)
\(662\) −35.4204 −1.37665
\(663\) 0 0
\(664\) 4.41461 0.171320
\(665\) 2.71131 0.105140
\(666\) 0 0
\(667\) −17.3880 −0.673265
\(668\) −2.86897 −0.111004
\(669\) 0 0
\(670\) −4.90007 −0.189306
\(671\) 52.9672 2.04478
\(672\) 0 0
\(673\) −34.5866 −1.33322 −0.666608 0.745409i \(-0.732254\pi\)
−0.666608 + 0.745409i \(0.732254\pi\)
\(674\) 4.80675 0.185149
\(675\) 0 0
\(676\) 10.7068 0.411801
\(677\) −32.0401 −1.23140 −0.615700 0.787981i \(-0.711127\pi\)
−0.615700 + 0.787981i \(0.711127\pi\)
\(678\) 0 0
\(679\) −17.7502 −0.681191
\(680\) −3.19175 −0.122398
\(681\) 0 0
\(682\) −19.5445 −0.748396
\(683\) 25.2036 0.964391 0.482195 0.876064i \(-0.339840\pi\)
0.482195 + 0.876064i \(0.339840\pi\)
\(684\) 0 0
\(685\) 9.52753 0.364028
\(686\) −14.9919 −0.572393
\(687\) 0 0
\(688\) −6.75471 −0.257521
\(689\) −28.6575 −1.09176
\(690\) 0 0
\(691\) −21.5818 −0.821009 −0.410505 0.911859i \(-0.634647\pi\)
−0.410505 + 0.911859i \(0.634647\pi\)
\(692\) −18.3571 −0.697833
\(693\) 0 0
\(694\) 18.5049 0.702438
\(695\) −20.7323 −0.786420
\(696\) 0 0
\(697\) −27.9428 −1.05841
\(698\) 33.7975 1.27925
\(699\) 0 0
\(700\) −1.19175 −0.0450439
\(701\) −14.0622 −0.531123 −0.265561 0.964094i \(-0.585557\pi\)
−0.265561 + 0.964094i \(0.585557\pi\)
\(702\) 0 0
\(703\) 2.27506 0.0858056
\(704\) 3.64179 0.137255
\(705\) 0 0
\(706\) −37.4657 −1.41004
\(707\) −9.09536 −0.342066
\(708\) 0 0
\(709\) −3.83300 −0.143951 −0.0719756 0.997406i \(-0.522930\pi\)
−0.0719756 + 0.997406i \(0.522930\pi\)
\(710\) −5.38798 −0.202207
\(711\) 0 0
\(712\) 9.63597 0.361123
\(713\) 15.3969 0.576620
\(714\) 0 0
\(715\) −17.7317 −0.663129
\(716\) 23.1259 0.864256
\(717\) 0 0
\(718\) 14.6886 0.548173
\(719\) 40.5298 1.51151 0.755754 0.654856i \(-0.227271\pi\)
0.755754 + 0.654856i \(0.227271\pi\)
\(720\) 0 0
\(721\) 2.54980 0.0949596
\(722\) 13.8241 0.514479
\(723\) 0 0
\(724\) 14.9001 0.553757
\(725\) −6.06072 −0.225089
\(726\) 0 0
\(727\) 29.5871 1.09732 0.548662 0.836044i \(-0.315137\pi\)
0.548662 + 0.836044i \(0.315137\pi\)
\(728\) 5.80259 0.215058
\(729\) 0 0
\(730\) 11.8026 0.436833
\(731\) 21.5593 0.797401
\(732\) 0 0
\(733\) −30.5400 −1.12802 −0.564010 0.825768i \(-0.690742\pi\)
−0.564010 + 0.825768i \(0.690742\pi\)
\(734\) −16.1175 −0.594907
\(735\) 0 0
\(736\) −2.86897 −0.105751
\(737\) −17.8450 −0.657330
\(738\) 0 0
\(739\) −1.49357 −0.0549419 −0.0274709 0.999623i \(-0.508745\pi\)
−0.0274709 + 0.999623i \(0.508745\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) −7.01433 −0.257504
\(743\) 25.2442 0.926120 0.463060 0.886327i \(-0.346751\pi\)
0.463060 + 0.886327i \(0.346751\pi\)
\(744\) 0 0
\(745\) −9.17758 −0.336240
\(746\) −3.25002 −0.118992
\(747\) 0 0
\(748\) −11.6237 −0.425004
\(749\) −17.8648 −0.652766
\(750\) 0 0
\(751\) 8.44304 0.308091 0.154045 0.988064i \(-0.450770\pi\)
0.154045 + 0.988064i \(0.450770\pi\)
\(752\) −4.13955 −0.150954
\(753\) 0 0
\(754\) 29.5094 1.07467
\(755\) −17.6024 −0.640617
\(756\) 0 0
\(757\) 35.8169 1.30179 0.650894 0.759168i \(-0.274394\pi\)
0.650894 + 0.759168i \(0.274394\pi\)
\(758\) 33.7639 1.22636
\(759\) 0 0
\(760\) 2.27506 0.0825252
\(761\) −27.0428 −0.980299 −0.490150 0.871638i \(-0.663058\pi\)
−0.490150 + 0.871638i \(0.663058\pi\)
\(762\) 0 0
\(763\) −0.302471 −0.0109502
\(764\) −8.31766 −0.300922
\(765\) 0 0
\(766\) −14.3859 −0.519785
\(767\) −35.4635 −1.28051
\(768\) 0 0
\(769\) −42.3051 −1.52556 −0.762780 0.646658i \(-0.776166\pi\)
−0.762780 + 0.646658i \(0.776166\pi\)
\(770\) −4.34010 −0.156406
\(771\) 0 0
\(772\) −21.5445 −0.775405
\(773\) −11.5566 −0.415661 −0.207830 0.978165i \(-0.566640\pi\)
−0.207830 + 0.978165i \(0.566640\pi\)
\(774\) 0 0
\(775\) 5.36672 0.192778
\(776\) −14.8943 −0.534672
\(777\) 0 0
\(778\) −25.5445 −0.915813
\(779\) 19.9175 0.713618
\(780\) 0 0
\(781\) −19.6219 −0.702126
\(782\) 9.15702 0.327454
\(783\) 0 0
\(784\) −5.57973 −0.199276
\(785\) −6.89857 −0.246221
\(786\) 0 0
\(787\) 33.9099 1.20876 0.604379 0.796697i \(-0.293421\pi\)
0.604379 + 0.796697i \(0.293421\pi\)
\(788\) 3.76904 0.134266
\(789\) 0 0
\(790\) 7.38798 0.262853
\(791\) −15.9031 −0.565448
\(792\) 0 0
\(793\) 70.8157 2.51474
\(794\) 4.81219 0.170778
\(795\) 0 0
\(796\) −5.92914 −0.210153
\(797\) 20.6886 0.732827 0.366413 0.930452i \(-0.380586\pi\)
0.366413 + 0.930452i \(0.380586\pi\)
\(798\) 0 0
\(799\) 13.2124 0.467421
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −8.45191 −0.298447
\(803\) 42.9825 1.51682
\(804\) 0 0
\(805\) 3.41909 0.120507
\(806\) −26.1304 −0.920404
\(807\) 0 0
\(808\) −7.63194 −0.268491
\(809\) 44.3868 1.56056 0.780278 0.625433i \(-0.215078\pi\)
0.780278 + 0.625433i \(0.215078\pi\)
\(810\) 0 0
\(811\) −41.4929 −1.45701 −0.728506 0.685039i \(-0.759785\pi\)
−0.728506 + 0.685039i \(0.759785\pi\)
\(812\) 7.22286 0.253473
\(813\) 0 0
\(814\) −3.64179 −0.127645
\(815\) 10.8194 0.378986
\(816\) 0 0
\(817\) −15.3674 −0.537636
\(818\) 5.51209 0.192726
\(819\) 0 0
\(820\) −8.75471 −0.305728
\(821\) −21.9624 −0.766494 −0.383247 0.923646i \(-0.625194\pi\)
−0.383247 + 0.923646i \(0.625194\pi\)
\(822\) 0 0
\(823\) 31.5911 1.10120 0.550598 0.834770i \(-0.314400\pi\)
0.550598 + 0.834770i \(0.314400\pi\)
\(824\) 2.13955 0.0745346
\(825\) 0 0
\(826\) −8.68020 −0.302023
\(827\) −5.42271 −0.188566 −0.0942831 0.995545i \(-0.530056\pi\)
−0.0942831 + 0.995545i \(0.530056\pi\)
\(828\) 0 0
\(829\) −54.0226 −1.87628 −0.938141 0.346252i \(-0.887454\pi\)
−0.938141 + 0.346252i \(0.887454\pi\)
\(830\) −4.41461 −0.153233
\(831\) 0 0
\(832\) 4.86897 0.168801
\(833\) 17.8091 0.617049
\(834\) 0 0
\(835\) 2.86897 0.0992846
\(836\) 8.28529 0.286553
\(837\) 0 0
\(838\) −0.853529 −0.0294847
\(839\) 22.4803 0.776108 0.388054 0.921637i \(-0.373147\pi\)
0.388054 + 0.921637i \(0.373147\pi\)
\(840\) 0 0
\(841\) 7.73227 0.266630
\(842\) −23.5275 −0.810812
\(843\) 0 0
\(844\) −3.51059 −0.120840
\(845\) −10.7068 −0.368326
\(846\) 0 0
\(847\) −2.69646 −0.0926516
\(848\) −5.88574 −0.202117
\(849\) 0 0
\(850\) 3.19175 0.109476
\(851\) 2.86897 0.0983469
\(852\) 0 0
\(853\) 24.4482 0.837089 0.418545 0.908196i \(-0.362540\pi\)
0.418545 + 0.908196i \(0.362540\pi\)
\(854\) 17.3332 0.593129
\(855\) 0 0
\(856\) −14.9904 −0.512361
\(857\) −19.1295 −0.653452 −0.326726 0.945119i \(-0.605946\pi\)
−0.326726 + 0.945119i \(0.605946\pi\)
\(858\) 0 0
\(859\) −11.5662 −0.394634 −0.197317 0.980340i \(-0.563223\pi\)
−0.197317 + 0.980340i \(0.563223\pi\)
\(860\) 6.75471 0.230334
\(861\) 0 0
\(862\) −34.1849 −1.16434
\(863\) −14.2397 −0.484726 −0.242363 0.970186i \(-0.577922\pi\)
−0.242363 + 0.970186i \(0.577922\pi\)
\(864\) 0 0
\(865\) 18.3571 0.624161
\(866\) 0.747535 0.0254023
\(867\) 0 0
\(868\) −6.39579 −0.217087
\(869\) 26.9055 0.912705
\(870\) 0 0
\(871\) −23.8583 −0.808407
\(872\) −0.253804 −0.00859490
\(873\) 0 0
\(874\) −6.52707 −0.220782
\(875\) 1.19175 0.0402885
\(876\) 0 0
\(877\) −36.8021 −1.24272 −0.621358 0.783526i \(-0.713419\pi\)
−0.621358 + 0.783526i \(0.713419\pi\)
\(878\) −14.4504 −0.487679
\(879\) 0 0
\(880\) −3.64179 −0.122765
\(881\) −5.78645 −0.194951 −0.0974753 0.995238i \(-0.531077\pi\)
−0.0974753 + 0.995238i \(0.531077\pi\)
\(882\) 0 0
\(883\) 8.30279 0.279411 0.139706 0.990193i \(-0.455384\pi\)
0.139706 + 0.990193i \(0.455384\pi\)
\(884\) −15.5405 −0.522684
\(885\) 0 0
\(886\) −34.8974 −1.17240
\(887\) −40.2125 −1.35020 −0.675101 0.737725i \(-0.735900\pi\)
−0.675101 + 0.737725i \(0.735900\pi\)
\(888\) 0 0
\(889\) 8.60314 0.288540
\(890\) −9.63597 −0.322998
\(891\) 0 0
\(892\) 19.8322 0.664031
\(893\) −9.41772 −0.315152
\(894\) 0 0
\(895\) −23.1259 −0.773014
\(896\) 1.19175 0.0398136
\(897\) 0 0
\(898\) −7.54564 −0.251801
\(899\) −32.5262 −1.08481
\(900\) 0 0
\(901\) 18.7858 0.625846
\(902\) −31.8828 −1.06158
\(903\) 0 0
\(904\) −13.3443 −0.443824
\(905\) −14.9001 −0.495295
\(906\) 0 0
\(907\) 17.9432 0.595795 0.297898 0.954598i \(-0.403715\pi\)
0.297898 + 0.954598i \(0.403715\pi\)
\(908\) 12.9608 0.430119
\(909\) 0 0
\(910\) −5.80259 −0.192354
\(911\) −10.0090 −0.331612 −0.165806 0.986158i \(-0.553023\pi\)
−0.165806 + 0.986158i \(0.553023\pi\)
\(912\) 0 0
\(913\) −16.0771 −0.532073
\(914\) 5.70645 0.188752
\(915\) 0 0
\(916\) −23.2384 −0.767818
\(917\) −23.8953 −0.789091
\(918\) 0 0
\(919\) 53.4439 1.76295 0.881476 0.472228i \(-0.156550\pi\)
0.881476 + 0.472228i \(0.156550\pi\)
\(920\) 2.86897 0.0945870
\(921\) 0 0
\(922\) 18.0324 0.593865
\(923\) −26.2339 −0.863499
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 25.3869 0.834265
\(927\) 0 0
\(928\) 6.06072 0.198953
\(929\) −17.5723 −0.576528 −0.288264 0.957551i \(-0.593078\pi\)
−0.288264 + 0.957551i \(0.593078\pi\)
\(930\) 0 0
\(931\) −12.6942 −0.416037
\(932\) 8.88196 0.290938
\(933\) 0 0
\(934\) 23.6153 0.772717
\(935\) 11.6237 0.380135
\(936\) 0 0
\(937\) −32.5641 −1.06382 −0.531912 0.846800i \(-0.678526\pi\)
−0.531912 + 0.846800i \(0.678526\pi\)
\(938\) −5.83966 −0.190672
\(939\) 0 0
\(940\) 4.13955 0.135017
\(941\) −17.0656 −0.556323 −0.278161 0.960534i \(-0.589725\pi\)
−0.278161 + 0.960534i \(0.589725\pi\)
\(942\) 0 0
\(943\) 25.1169 0.817920
\(944\) −7.28357 −0.237060
\(945\) 0 0
\(946\) 24.5992 0.799789
\(947\) 25.2036 0.819009 0.409504 0.912308i \(-0.365702\pi\)
0.409504 + 0.912308i \(0.365702\pi\)
\(948\) 0 0
\(949\) 57.4664 1.86544
\(950\) −2.27506 −0.0738128
\(951\) 0 0
\(952\) −3.80377 −0.123281
\(953\) −21.7560 −0.704747 −0.352374 0.935859i \(-0.614625\pi\)
−0.352374 + 0.935859i \(0.614625\pi\)
\(954\) 0 0
\(955\) 8.31766 0.269153
\(956\) 25.3778 0.820778
\(957\) 0 0
\(958\) −34.7981 −1.12428
\(959\) 11.3544 0.366654
\(960\) 0 0
\(961\) −2.19827 −0.0709119
\(962\) −4.86897 −0.156982
\(963\) 0 0
\(964\) −10.2429 −0.329900
\(965\) 21.5445 0.693544
\(966\) 0 0
\(967\) −12.3654 −0.397644 −0.198822 0.980036i \(-0.563712\pi\)
−0.198822 + 0.980036i \(0.563712\pi\)
\(968\) −2.26261 −0.0727230
\(969\) 0 0
\(970\) 14.8943 0.478226
\(971\) −50.2114 −1.61136 −0.805680 0.592351i \(-0.798200\pi\)
−0.805680 + 0.592351i \(0.798200\pi\)
\(972\) 0 0
\(973\) −24.7077 −0.792092
\(974\) 8.65612 0.277360
\(975\) 0 0
\(976\) 14.5443 0.465552
\(977\) −8.06724 −0.258094 −0.129047 0.991638i \(-0.541192\pi\)
−0.129047 + 0.991638i \(0.541192\pi\)
\(978\) 0 0
\(979\) −35.0921 −1.12155
\(980\) 5.57973 0.178238
\(981\) 0 0
\(982\) 12.0632 0.384954
\(983\) 42.2151 1.34645 0.673227 0.739436i \(-0.264908\pi\)
0.673227 + 0.739436i \(0.264908\pi\)
\(984\) 0 0
\(985\) −3.76904 −0.120092
\(986\) −19.3443 −0.616048
\(987\) 0 0
\(988\) 11.0772 0.352413
\(989\) −19.3790 −0.616217
\(990\) 0 0
\(991\) −16.9294 −0.537780 −0.268890 0.963171i \(-0.586657\pi\)
−0.268890 + 0.963171i \(0.586657\pi\)
\(992\) −5.36672 −0.170394
\(993\) 0 0
\(994\) −6.42113 −0.203666
\(995\) 5.92914 0.187966
\(996\) 0 0
\(997\) −8.75699 −0.277337 −0.138668 0.990339i \(-0.544282\pi\)
−0.138668 + 0.990339i \(0.544282\pi\)
\(998\) −1.14647 −0.0362909
\(999\) 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 3330.2.a.bk.1.2 5
3.2 odd 2 3330.2.a.bl.1.2 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3330.2.a.bk.1.2 5 1.1 even 1 trivial
3330.2.a.bl.1.2 yes 5 3.2 odd 2