Properties

Label 3330.2.a.bd.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.73205 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.73205 q^{7} +1.00000 q^{8} -1.00000 q^{10} +5.46410 q^{11} -5.46410 q^{13} -4.73205 q^{14} +1.00000 q^{16} -5.46410 q^{17} +6.19615 q^{19} -1.00000 q^{20} +5.46410 q^{22} +8.00000 q^{23} +1.00000 q^{25} -5.46410 q^{26} -4.73205 q^{28} -4.92820 q^{29} +0.732051 q^{31} +1.00000 q^{32} -5.46410 q^{34} +4.73205 q^{35} +1.00000 q^{37} +6.19615 q^{38} -1.00000 q^{40} +2.00000 q^{41} +6.92820 q^{43} +5.46410 q^{44} +8.00000 q^{46} +4.73205 q^{47} +15.3923 q^{49} +1.00000 q^{50} -5.46410 q^{52} +6.00000 q^{53} -5.46410 q^{55} -4.73205 q^{56} -4.92820 q^{58} +10.1962 q^{59} -4.92820 q^{61} +0.732051 q^{62} +1.00000 q^{64} +5.46410 q^{65} -3.66025 q^{67} -5.46410 q^{68} +4.73205 q^{70} -2.92820 q^{71} -0.928203 q^{73} +1.00000 q^{74} +6.19615 q^{76} -25.8564 q^{77} +8.73205 q^{79} -1.00000 q^{80} +2.00000 q^{82} +8.73205 q^{83} +5.46410 q^{85} +6.92820 q^{86} +5.46410 q^{88} +2.00000 q^{89} +25.8564 q^{91} +8.00000 q^{92} +4.73205 q^{94} -6.19615 q^{95} -2.00000 q^{97} +15.3923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} - 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{19} - 2 q^{20} + 4 q^{22} + 16 q^{23} + 2 q^{25} - 4 q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{35} + 2 q^{37} + 2 q^{38} - 2 q^{40} + 4 q^{41} + 4 q^{44} + 16 q^{46} + 6 q^{47} + 10 q^{49} + 2 q^{50} - 4 q^{52} + 12 q^{53} - 4 q^{55} - 6 q^{56} + 4 q^{58} + 10 q^{59} + 4 q^{61} - 2 q^{62} + 2 q^{64} + 4 q^{65} + 10 q^{67} - 4 q^{68} + 6 q^{70} + 8 q^{71} + 12 q^{73} + 2 q^{74} + 2 q^{76} - 24 q^{77} + 14 q^{79} - 2 q^{80} + 4 q^{82} + 14 q^{83} + 4 q^{85} + 4 q^{88} + 4 q^{89} + 24 q^{91} + 16 q^{92} + 6 q^{94} - 2 q^{95} - 4 q^{97} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) −4.73205 −1.78855 −0.894274 0.447521i \(-0.852307\pi\)
−0.894274 + 0.447521i \(0.852307\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 5.46410 1.64749 0.823744 0.566961i \(-0.191881\pi\)
0.823744 + 0.566961i \(0.191881\pi\)
\(12\) 0 0
\(13\) −5.46410 −1.51547 −0.757735 0.652563i \(-0.773694\pi\)
−0.757735 + 0.652563i \(0.773694\pi\)
\(14\) −4.73205 −1.26469
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.46410 −1.32524 −0.662620 0.748956i \(-0.730555\pi\)
−0.662620 + 0.748956i \(0.730555\pi\)
\(18\) 0 0
\(19\) 6.19615 1.42149 0.710747 0.703447i \(-0.248357\pi\)
0.710747 + 0.703447i \(0.248357\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 5.46410 1.16495
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −5.46410 −1.07160
\(27\) 0 0
\(28\) −4.73205 −0.894274
\(29\) −4.92820 −0.915144 −0.457572 0.889172i \(-0.651281\pi\)
−0.457572 + 0.889172i \(0.651281\pi\)
\(30\) 0 0
\(31\) 0.732051 0.131480 0.0657401 0.997837i \(-0.479059\pi\)
0.0657401 + 0.997837i \(0.479059\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −5.46410 −0.937086
\(35\) 4.73205 0.799863
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 6.19615 1.00515
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 6.92820 1.05654 0.528271 0.849076i \(-0.322841\pi\)
0.528271 + 0.849076i \(0.322841\pi\)
\(44\) 5.46410 0.823744
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 4.73205 0.690241 0.345120 0.938558i \(-0.387838\pi\)
0.345120 + 0.938558i \(0.387838\pi\)
\(48\) 0 0
\(49\) 15.3923 2.19890
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −5.46410 −0.757735
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −5.46410 −0.736779
\(56\) −4.73205 −0.632347
\(57\) 0 0
\(58\) −4.92820 −0.647105
\(59\) 10.1962 1.32743 0.663713 0.747987i \(-0.268980\pi\)
0.663713 + 0.747987i \(0.268980\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) 0.732051 0.0929705
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 5.46410 0.677738
\(66\) 0 0
\(67\) −3.66025 −0.447171 −0.223586 0.974684i \(-0.571776\pi\)
−0.223586 + 0.974684i \(0.571776\pi\)
\(68\) −5.46410 −0.662620
\(69\) 0 0
\(70\) 4.73205 0.565588
\(71\) −2.92820 −0.347514 −0.173757 0.984789i \(-0.555591\pi\)
−0.173757 + 0.984789i \(0.555591\pi\)
\(72\) 0 0
\(73\) −0.928203 −0.108638 −0.0543190 0.998524i \(-0.517299\pi\)
−0.0543190 + 0.998524i \(0.517299\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 6.19615 0.710747
\(77\) −25.8564 −2.94661
\(78\) 0 0
\(79\) 8.73205 0.982432 0.491216 0.871038i \(-0.336552\pi\)
0.491216 + 0.871038i \(0.336552\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) 8.73205 0.958467 0.479234 0.877687i \(-0.340915\pi\)
0.479234 + 0.877687i \(0.340915\pi\)
\(84\) 0 0
\(85\) 5.46410 0.592665
\(86\) 6.92820 0.747087
\(87\) 0 0
\(88\) 5.46410 0.582475
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 25.8564 2.71049
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 4.73205 0.488074
\(95\) −6.19615 −0.635712
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 15.3923 1.55486
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 9.46410 0.941713 0.470857 0.882210i \(-0.343945\pi\)
0.470857 + 0.882210i \(0.343945\pi\)
\(102\) 0 0
\(103\) −6.53590 −0.644001 −0.322001 0.946739i \(-0.604355\pi\)
−0.322001 + 0.946739i \(0.604355\pi\)
\(104\) −5.46410 −0.535799
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −3.26795 −0.315925 −0.157962 0.987445i \(-0.550492\pi\)
−0.157962 + 0.987445i \(0.550492\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −5.46410 −0.520982
\(111\) 0 0
\(112\) −4.73205 −0.447137
\(113\) 10.5359 0.991134 0.495567 0.868570i \(-0.334960\pi\)
0.495567 + 0.868570i \(0.334960\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) −4.92820 −0.457572
\(117\) 0 0
\(118\) 10.1962 0.938632
\(119\) 25.8564 2.37025
\(120\) 0 0
\(121\) 18.8564 1.71422
\(122\) −4.92820 −0.446179
\(123\) 0 0
\(124\) 0.732051 0.0657401
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 3.66025 0.324795 0.162398 0.986725i \(-0.448077\pi\)
0.162398 + 0.986725i \(0.448077\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 5.46410 0.479233
\(131\) 18.5885 1.62408 0.812041 0.583601i \(-0.198357\pi\)
0.812041 + 0.583601i \(0.198357\pi\)
\(132\) 0 0
\(133\) −29.3205 −2.54241
\(134\) −3.66025 −0.316198
\(135\) 0 0
\(136\) −5.46410 −0.468543
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 6.92820 0.587643 0.293821 0.955860i \(-0.405073\pi\)
0.293821 + 0.955860i \(0.405073\pi\)
\(140\) 4.73205 0.399931
\(141\) 0 0
\(142\) −2.92820 −0.245729
\(143\) −29.8564 −2.49672
\(144\) 0 0
\(145\) 4.92820 0.409265
\(146\) −0.928203 −0.0768186
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) −4.39230 −0.359832 −0.179916 0.983682i \(-0.557583\pi\)
−0.179916 + 0.983682i \(0.557583\pi\)
\(150\) 0 0
\(151\) −12.3923 −1.00847 −0.504236 0.863566i \(-0.668226\pi\)
−0.504236 + 0.863566i \(0.668226\pi\)
\(152\) 6.19615 0.502574
\(153\) 0 0
\(154\) −25.8564 −2.08357
\(155\) −0.732051 −0.0587997
\(156\) 0 0
\(157\) 3.07180 0.245156 0.122578 0.992459i \(-0.460884\pi\)
0.122578 + 0.992459i \(0.460884\pi\)
\(158\) 8.73205 0.694685
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −37.8564 −2.98350
\(162\) 0 0
\(163\) −11.3205 −0.886691 −0.443345 0.896351i \(-0.646208\pi\)
−0.443345 + 0.896351i \(0.646208\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 8.73205 0.677739
\(167\) 1.46410 0.113296 0.0566478 0.998394i \(-0.481959\pi\)
0.0566478 + 0.998394i \(0.481959\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 5.46410 0.419077
\(171\) 0 0
\(172\) 6.92820 0.528271
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 0 0
\(175\) −4.73205 −0.357709
\(176\) 5.46410 0.411872
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) 0.339746 0.0253938 0.0126969 0.999919i \(-0.495958\pi\)
0.0126969 + 0.999919i \(0.495958\pi\)
\(180\) 0 0
\(181\) 5.46410 0.406143 0.203072 0.979164i \(-0.434908\pi\)
0.203072 + 0.979164i \(0.434908\pi\)
\(182\) 25.8564 1.91660
\(183\) 0 0
\(184\) 8.00000 0.589768
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −29.8564 −2.18332
\(188\) 4.73205 0.345120
\(189\) 0 0
\(190\) −6.19615 −0.449516
\(191\) 8.73205 0.631829 0.315915 0.948788i \(-0.397689\pi\)
0.315915 + 0.948788i \(0.397689\pi\)
\(192\) 0 0
\(193\) 15.8564 1.14137 0.570685 0.821169i \(-0.306678\pi\)
0.570685 + 0.821169i \(0.306678\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 15.3923 1.09945
\(197\) 22.7846 1.62334 0.811668 0.584119i \(-0.198560\pi\)
0.811668 + 0.584119i \(0.198560\pi\)
\(198\) 0 0
\(199\) −12.0526 −0.854383 −0.427192 0.904161i \(-0.640497\pi\)
−0.427192 + 0.904161i \(0.640497\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 9.46410 0.665892
\(203\) 23.3205 1.63678
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) −6.53590 −0.455378
\(207\) 0 0
\(208\) −5.46410 −0.378867
\(209\) 33.8564 2.34190
\(210\) 0 0
\(211\) −17.8564 −1.22929 −0.614643 0.788806i \(-0.710700\pi\)
−0.614643 + 0.788806i \(0.710700\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −3.26795 −0.223392
\(215\) −6.92820 −0.472500
\(216\) 0 0
\(217\) −3.46410 −0.235159
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) −5.46410 −0.368390
\(221\) 29.8564 2.00836
\(222\) 0 0
\(223\) 16.0526 1.07496 0.537479 0.843277i \(-0.319377\pi\)
0.537479 + 0.843277i \(0.319377\pi\)
\(224\) −4.73205 −0.316173
\(225\) 0 0
\(226\) 10.5359 0.700838
\(227\) −24.3923 −1.61897 −0.809487 0.587138i \(-0.800255\pi\)
−0.809487 + 0.587138i \(0.800255\pi\)
\(228\) 0 0
\(229\) −11.8564 −0.783493 −0.391747 0.920073i \(-0.628129\pi\)
−0.391747 + 0.920073i \(0.628129\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) −4.92820 −0.323552
\(233\) −28.9282 −1.89515 −0.947575 0.319534i \(-0.896474\pi\)
−0.947575 + 0.319534i \(0.896474\pi\)
\(234\) 0 0
\(235\) −4.73205 −0.308685
\(236\) 10.1962 0.663713
\(237\) 0 0
\(238\) 25.8564 1.67602
\(239\) 20.7321 1.34104 0.670522 0.741889i \(-0.266070\pi\)
0.670522 + 0.741889i \(0.266070\pi\)
\(240\) 0 0
\(241\) 4.92820 0.317453 0.158727 0.987323i \(-0.449261\pi\)
0.158727 + 0.987323i \(0.449261\pi\)
\(242\) 18.8564 1.21214
\(243\) 0 0
\(244\) −4.92820 −0.315496
\(245\) −15.3923 −0.983378
\(246\) 0 0
\(247\) −33.8564 −2.15423
\(248\) 0.732051 0.0464853
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −19.2679 −1.21618 −0.608091 0.793867i \(-0.708064\pi\)
−0.608091 + 0.793867i \(0.708064\pi\)
\(252\) 0 0
\(253\) 43.7128 2.74820
\(254\) 3.66025 0.229665
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.5359 1.15624 0.578119 0.815953i \(-0.303787\pi\)
0.578119 + 0.815953i \(0.303787\pi\)
\(258\) 0 0
\(259\) −4.73205 −0.294035
\(260\) 5.46410 0.338869
\(261\) 0 0
\(262\) 18.5885 1.14840
\(263\) 8.05256 0.496542 0.248271 0.968691i \(-0.420138\pi\)
0.248271 + 0.968691i \(0.420138\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −29.3205 −1.79776
\(267\) 0 0
\(268\) −3.66025 −0.223586
\(269\) −20.3923 −1.24334 −0.621670 0.783279i \(-0.713546\pi\)
−0.621670 + 0.783279i \(0.713546\pi\)
\(270\) 0 0
\(271\) −24.7846 −1.50556 −0.752779 0.658273i \(-0.771287\pi\)
−0.752779 + 0.658273i \(0.771287\pi\)
\(272\) −5.46410 −0.331310
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 5.46410 0.329498
\(276\) 0 0
\(277\) 22.2487 1.33680 0.668398 0.743804i \(-0.266980\pi\)
0.668398 + 0.743804i \(0.266980\pi\)
\(278\) 6.92820 0.415526
\(279\) 0 0
\(280\) 4.73205 0.282794
\(281\) 8.92820 0.532612 0.266306 0.963889i \(-0.414197\pi\)
0.266306 + 0.963889i \(0.414197\pi\)
\(282\) 0 0
\(283\) 16.3923 0.974421 0.487211 0.873284i \(-0.338014\pi\)
0.487211 + 0.873284i \(0.338014\pi\)
\(284\) −2.92820 −0.173757
\(285\) 0 0
\(286\) −29.8564 −1.76545
\(287\) −9.46410 −0.558648
\(288\) 0 0
\(289\) 12.8564 0.756259
\(290\) 4.92820 0.289394
\(291\) 0 0
\(292\) −0.928203 −0.0543190
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) −10.1962 −0.593643
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −4.39230 −0.254439
\(299\) −43.7128 −2.52798
\(300\) 0 0
\(301\) −32.7846 −1.88967
\(302\) −12.3923 −0.713097
\(303\) 0 0
\(304\) 6.19615 0.355374
\(305\) 4.92820 0.282188
\(306\) 0 0
\(307\) 18.5885 1.06090 0.530450 0.847716i \(-0.322023\pi\)
0.530450 + 0.847716i \(0.322023\pi\)
\(308\) −25.8564 −1.47331
\(309\) 0 0
\(310\) −0.732051 −0.0415777
\(311\) −2.87564 −0.163063 −0.0815314 0.996671i \(-0.525981\pi\)
−0.0815314 + 0.996671i \(0.525981\pi\)
\(312\) 0 0
\(313\) −23.8564 −1.34844 −0.674222 0.738529i \(-0.735521\pi\)
−0.674222 + 0.738529i \(0.735521\pi\)
\(314\) 3.07180 0.173352
\(315\) 0 0
\(316\) 8.73205 0.491216
\(317\) 4.14359 0.232727 0.116364 0.993207i \(-0.462876\pi\)
0.116364 + 0.993207i \(0.462876\pi\)
\(318\) 0 0
\(319\) −26.9282 −1.50769
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −37.8564 −2.10966
\(323\) −33.8564 −1.88382
\(324\) 0 0
\(325\) −5.46410 −0.303094
\(326\) −11.3205 −0.626985
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) −22.3923 −1.23453
\(330\) 0 0
\(331\) −33.1244 −1.82068 −0.910340 0.413862i \(-0.864180\pi\)
−0.910340 + 0.413862i \(0.864180\pi\)
\(332\) 8.73205 0.479234
\(333\) 0 0
\(334\) 1.46410 0.0801121
\(335\) 3.66025 0.199981
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 16.8564 0.916868
\(339\) 0 0
\(340\) 5.46410 0.296333
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) −39.7128 −2.14429
\(344\) 6.92820 0.373544
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) −30.9282 −1.66031 −0.830156 0.557530i \(-0.811749\pi\)
−0.830156 + 0.557530i \(0.811749\pi\)
\(348\) 0 0
\(349\) −15.3205 −0.820088 −0.410044 0.912066i \(-0.634487\pi\)
−0.410044 + 0.912066i \(0.634487\pi\)
\(350\) −4.73205 −0.252939
\(351\) 0 0
\(352\) 5.46410 0.291238
\(353\) 11.8564 0.631053 0.315526 0.948917i \(-0.397819\pi\)
0.315526 + 0.948917i \(0.397819\pi\)
\(354\) 0 0
\(355\) 2.92820 0.155413
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 0.339746 0.0179561
\(359\) 12.3923 0.654041 0.327020 0.945017i \(-0.393955\pi\)
0.327020 + 0.945017i \(0.393955\pi\)
\(360\) 0 0
\(361\) 19.3923 1.02065
\(362\) 5.46410 0.287187
\(363\) 0 0
\(364\) 25.8564 1.35524
\(365\) 0.928203 0.0485844
\(366\) 0 0
\(367\) −4.33975 −0.226533 −0.113266 0.993565i \(-0.536131\pi\)
−0.113266 + 0.993565i \(0.536131\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) −28.3923 −1.47406
\(372\) 0 0
\(373\) −10.7846 −0.558406 −0.279203 0.960232i \(-0.590070\pi\)
−0.279203 + 0.960232i \(0.590070\pi\)
\(374\) −29.8564 −1.54384
\(375\) 0 0
\(376\) 4.73205 0.244037
\(377\) 26.9282 1.38687
\(378\) 0 0
\(379\) 32.3923 1.66388 0.831940 0.554865i \(-0.187230\pi\)
0.831940 + 0.554865i \(0.187230\pi\)
\(380\) −6.19615 −0.317856
\(381\) 0 0
\(382\) 8.73205 0.446771
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) 25.8564 1.31776
\(386\) 15.8564 0.807070
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 11.8564 0.601144 0.300572 0.953759i \(-0.402822\pi\)
0.300572 + 0.953759i \(0.402822\pi\)
\(390\) 0 0
\(391\) −43.7128 −2.21065
\(392\) 15.3923 0.777429
\(393\) 0 0
\(394\) 22.7846 1.14787
\(395\) −8.73205 −0.439357
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −12.0526 −0.604140
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 32.9282 1.64436 0.822178 0.569230i \(-0.192759\pi\)
0.822178 + 0.569230i \(0.192759\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 9.46410 0.470857
\(405\) 0 0
\(406\) 23.3205 1.15738
\(407\) 5.46410 0.270845
\(408\) 0 0
\(409\) 3.07180 0.151891 0.0759453 0.997112i \(-0.475803\pi\)
0.0759453 + 0.997112i \(0.475803\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) −6.53590 −0.322001
\(413\) −48.2487 −2.37416
\(414\) 0 0
\(415\) −8.73205 −0.428640
\(416\) −5.46410 −0.267900
\(417\) 0 0
\(418\) 33.8564 1.65597
\(419\) −14.2487 −0.696095 −0.348048 0.937477i \(-0.613155\pi\)
−0.348048 + 0.937477i \(0.613155\pi\)
\(420\) 0 0
\(421\) −0.143594 −0.00699832 −0.00349916 0.999994i \(-0.501114\pi\)
−0.00349916 + 0.999994i \(0.501114\pi\)
\(422\) −17.8564 −0.869236
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −5.46410 −0.265048
\(426\) 0 0
\(427\) 23.3205 1.12856
\(428\) −3.26795 −0.157962
\(429\) 0 0
\(430\) −6.92820 −0.334108
\(431\) 2.19615 0.105785 0.0528925 0.998600i \(-0.483156\pi\)
0.0528925 + 0.998600i \(0.483156\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −3.46410 −0.166282
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 49.5692 2.37122
\(438\) 0 0
\(439\) 13.5167 0.645115 0.322558 0.946550i \(-0.395457\pi\)
0.322558 + 0.946550i \(0.395457\pi\)
\(440\) −5.46410 −0.260491
\(441\) 0 0
\(442\) 29.8564 1.42012
\(443\) 14.8756 0.706763 0.353382 0.935479i \(-0.385032\pi\)
0.353382 + 0.935479i \(0.385032\pi\)
\(444\) 0 0
\(445\) −2.00000 −0.0948091
\(446\) 16.0526 0.760111
\(447\) 0 0
\(448\) −4.73205 −0.223568
\(449\) 21.7128 1.02469 0.512345 0.858779i \(-0.328777\pi\)
0.512345 + 0.858779i \(0.328777\pi\)
\(450\) 0 0
\(451\) 10.9282 0.514589
\(452\) 10.5359 0.495567
\(453\) 0 0
\(454\) −24.3923 −1.14479
\(455\) −25.8564 −1.21217
\(456\) 0 0
\(457\) −31.8564 −1.49018 −0.745090 0.666964i \(-0.767593\pi\)
−0.745090 + 0.666964i \(0.767593\pi\)
\(458\) −11.8564 −0.554013
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) −14.7846 −0.688588 −0.344294 0.938862i \(-0.611882\pi\)
−0.344294 + 0.938862i \(0.611882\pi\)
\(462\) 0 0
\(463\) 18.9282 0.879668 0.439834 0.898079i \(-0.355037\pi\)
0.439834 + 0.898079i \(0.355037\pi\)
\(464\) −4.92820 −0.228786
\(465\) 0 0
\(466\) −28.9282 −1.34007
\(467\) 25.1769 1.16505 0.582524 0.812813i \(-0.302065\pi\)
0.582524 + 0.812813i \(0.302065\pi\)
\(468\) 0 0
\(469\) 17.3205 0.799787
\(470\) −4.73205 −0.218273
\(471\) 0 0
\(472\) 10.1962 0.469316
\(473\) 37.8564 1.74064
\(474\) 0 0
\(475\) 6.19615 0.284299
\(476\) 25.8564 1.18513
\(477\) 0 0
\(478\) 20.7321 0.948262
\(479\) −4.05256 −0.185166 −0.0925831 0.995705i \(-0.529512\pi\)
−0.0925831 + 0.995705i \(0.529512\pi\)
\(480\) 0 0
\(481\) −5.46410 −0.249142
\(482\) 4.92820 0.224474
\(483\) 0 0
\(484\) 18.8564 0.857109
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) 4.39230 0.199034 0.0995172 0.995036i \(-0.468270\pi\)
0.0995172 + 0.995036i \(0.468270\pi\)
\(488\) −4.92820 −0.223089
\(489\) 0 0
\(490\) −15.3923 −0.695353
\(491\) 14.9282 0.673700 0.336850 0.941558i \(-0.390638\pi\)
0.336850 + 0.941558i \(0.390638\pi\)
\(492\) 0 0
\(493\) 26.9282 1.21279
\(494\) −33.8564 −1.52327
\(495\) 0 0
\(496\) 0.732051 0.0328701
\(497\) 13.8564 0.621545
\(498\) 0 0
\(499\) 9.41154 0.421319 0.210659 0.977560i \(-0.432439\pi\)
0.210659 + 0.977560i \(0.432439\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −19.2679 −0.859971
\(503\) 18.9282 0.843967 0.421983 0.906604i \(-0.361334\pi\)
0.421983 + 0.906604i \(0.361334\pi\)
\(504\) 0 0
\(505\) −9.46410 −0.421147
\(506\) 43.7128 1.94327
\(507\) 0 0
\(508\) 3.66025 0.162398
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) 4.39230 0.194304
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.5359 0.817583
\(515\) 6.53590 0.288006
\(516\) 0 0
\(517\) 25.8564 1.13716
\(518\) −4.73205 −0.207914
\(519\) 0 0
\(520\) 5.46410 0.239617
\(521\) −26.5359 −1.16256 −0.581279 0.813704i \(-0.697448\pi\)
−0.581279 + 0.813704i \(0.697448\pi\)
\(522\) 0 0
\(523\) 36.7846 1.60848 0.804239 0.594306i \(-0.202573\pi\)
0.804239 + 0.594306i \(0.202573\pi\)
\(524\) 18.5885 0.812041
\(525\) 0 0
\(526\) 8.05256 0.351108
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) −29.3205 −1.27121
\(533\) −10.9282 −0.473353
\(534\) 0 0
\(535\) 3.26795 0.141286
\(536\) −3.66025 −0.158099
\(537\) 0 0
\(538\) −20.3923 −0.879175
\(539\) 84.1051 3.62266
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −24.7846 −1.06459
\(543\) 0 0
\(544\) −5.46410 −0.234271
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 0 0
\(550\) 5.46410 0.232990
\(551\) −30.5359 −1.30087
\(552\) 0 0
\(553\) −41.3205 −1.75713
\(554\) 22.2487 0.945257
\(555\) 0 0
\(556\) 6.92820 0.293821
\(557\) −44.1051 −1.86879 −0.934397 0.356234i \(-0.884061\pi\)
−0.934397 + 0.356234i \(0.884061\pi\)
\(558\) 0 0
\(559\) −37.8564 −1.60116
\(560\) 4.73205 0.199966
\(561\) 0 0
\(562\) 8.92820 0.376614
\(563\) −30.9282 −1.30347 −0.651734 0.758447i \(-0.725958\pi\)
−0.651734 + 0.758447i \(0.725958\pi\)
\(564\) 0 0
\(565\) −10.5359 −0.443249
\(566\) 16.3923 0.689020
\(567\) 0 0
\(568\) −2.92820 −0.122865
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) −29.8564 −1.24836
\(573\) 0 0
\(574\) −9.46410 −0.395024
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) −24.3923 −1.01546 −0.507732 0.861515i \(-0.669516\pi\)
−0.507732 + 0.861515i \(0.669516\pi\)
\(578\) 12.8564 0.534756
\(579\) 0 0
\(580\) 4.92820 0.204633
\(581\) −41.3205 −1.71426
\(582\) 0 0
\(583\) 32.7846 1.35780
\(584\) −0.928203 −0.0384093
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −44.7846 −1.84846 −0.924229 0.381838i \(-0.875291\pi\)
−0.924229 + 0.381838i \(0.875291\pi\)
\(588\) 0 0
\(589\) 4.53590 0.186898
\(590\) −10.1962 −0.419769
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) 34.7846 1.42843 0.714216 0.699925i \(-0.246783\pi\)
0.714216 + 0.699925i \(0.246783\pi\)
\(594\) 0 0
\(595\) −25.8564 −1.06001
\(596\) −4.39230 −0.179916
\(597\) 0 0
\(598\) −43.7128 −1.78755
\(599\) −9.46410 −0.386693 −0.193346 0.981131i \(-0.561934\pi\)
−0.193346 + 0.981131i \(0.561934\pi\)
\(600\) 0 0
\(601\) 27.6077 1.12614 0.563071 0.826409i \(-0.309620\pi\)
0.563071 + 0.826409i \(0.309620\pi\)
\(602\) −32.7846 −1.33620
\(603\) 0 0
\(604\) −12.3923 −0.504236
\(605\) −18.8564 −0.766622
\(606\) 0 0
\(607\) 0.784610 0.0318463 0.0159232 0.999873i \(-0.494931\pi\)
0.0159232 + 0.999873i \(0.494931\pi\)
\(608\) 6.19615 0.251287
\(609\) 0 0
\(610\) 4.92820 0.199537
\(611\) −25.8564 −1.04604
\(612\) 0 0
\(613\) 16.9282 0.683724 0.341862 0.939750i \(-0.388943\pi\)
0.341862 + 0.939750i \(0.388943\pi\)
\(614\) 18.5885 0.750169
\(615\) 0 0
\(616\) −25.8564 −1.04178
\(617\) 0.928203 0.0373681 0.0186840 0.999825i \(-0.494052\pi\)
0.0186840 + 0.999825i \(0.494052\pi\)
\(618\) 0 0
\(619\) −17.8564 −0.717710 −0.358855 0.933393i \(-0.616833\pi\)
−0.358855 + 0.933393i \(0.616833\pi\)
\(620\) −0.732051 −0.0293999
\(621\) 0 0
\(622\) −2.87564 −0.115303
\(623\) −9.46410 −0.379171
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −23.8564 −0.953494
\(627\) 0 0
\(628\) 3.07180 0.122578
\(629\) −5.46410 −0.217868
\(630\) 0 0
\(631\) 30.9808 1.23332 0.616662 0.787228i \(-0.288484\pi\)
0.616662 + 0.787228i \(0.288484\pi\)
\(632\) 8.73205 0.347342
\(633\) 0 0
\(634\) 4.14359 0.164563
\(635\) −3.66025 −0.145253
\(636\) 0 0
\(637\) −84.1051 −3.33237
\(638\) −26.9282 −1.06610
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 6.53590 0.258152 0.129076 0.991635i \(-0.458799\pi\)
0.129076 + 0.991635i \(0.458799\pi\)
\(642\) 0 0
\(643\) 34.5359 1.36196 0.680981 0.732301i \(-0.261553\pi\)
0.680981 + 0.732301i \(0.261553\pi\)
\(644\) −37.8564 −1.49175
\(645\) 0 0
\(646\) −33.8564 −1.33206
\(647\) 35.7128 1.40402 0.702008 0.712169i \(-0.252287\pi\)
0.702008 + 0.712169i \(0.252287\pi\)
\(648\) 0 0
\(649\) 55.7128 2.18692
\(650\) −5.46410 −0.214320
\(651\) 0 0
\(652\) −11.3205 −0.443345
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) −18.5885 −0.726311
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) −22.3923 −0.872943
\(659\) 6.92820 0.269884 0.134942 0.990853i \(-0.456915\pi\)
0.134942 + 0.990853i \(0.456915\pi\)
\(660\) 0 0
\(661\) −31.0718 −1.20855 −0.604276 0.796775i \(-0.706538\pi\)
−0.604276 + 0.796775i \(0.706538\pi\)
\(662\) −33.1244 −1.28741
\(663\) 0 0
\(664\) 8.73205 0.338869
\(665\) 29.3205 1.13700
\(666\) 0 0
\(667\) −39.4256 −1.52657
\(668\) 1.46410 0.0566478
\(669\) 0 0
\(670\) 3.66025 0.141408
\(671\) −26.9282 −1.03955
\(672\) 0 0
\(673\) −32.9282 −1.26929 −0.634644 0.772804i \(-0.718853\pi\)
−0.634644 + 0.772804i \(0.718853\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) 4.14359 0.159251 0.0796256 0.996825i \(-0.474628\pi\)
0.0796256 + 0.996825i \(0.474628\pi\)
\(678\) 0 0
\(679\) 9.46410 0.363199
\(680\) 5.46410 0.209539
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) 4.78461 0.183078 0.0915390 0.995801i \(-0.470821\pi\)
0.0915390 + 0.995801i \(0.470821\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) −39.7128 −1.51624
\(687\) 0 0
\(688\) 6.92820 0.264135
\(689\) −32.7846 −1.24899
\(690\) 0 0
\(691\) −0.392305 −0.0149240 −0.00746199 0.999972i \(-0.502375\pi\)
−0.00746199 + 0.999972i \(0.502375\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) −30.9282 −1.17402
\(695\) −6.92820 −0.262802
\(696\) 0 0
\(697\) −10.9282 −0.413935
\(698\) −15.3205 −0.579890
\(699\) 0 0
\(700\) −4.73205 −0.178855
\(701\) 11.8564 0.447810 0.223905 0.974611i \(-0.428119\pi\)
0.223905 + 0.974611i \(0.428119\pi\)
\(702\) 0 0
\(703\) 6.19615 0.233692
\(704\) 5.46410 0.205936
\(705\) 0 0
\(706\) 11.8564 0.446222
\(707\) −44.7846 −1.68430
\(708\) 0 0
\(709\) 43.8564 1.64706 0.823531 0.567271i \(-0.192001\pi\)
0.823531 + 0.567271i \(0.192001\pi\)
\(710\) 2.92820 0.109894
\(711\) 0 0
\(712\) 2.00000 0.0749532
\(713\) 5.85641 0.219324
\(714\) 0 0
\(715\) 29.8564 1.11657
\(716\) 0.339746 0.0126969
\(717\) 0 0
\(718\) 12.3923 0.462477
\(719\) 12.3923 0.462155 0.231077 0.972935i \(-0.425775\pi\)
0.231077 + 0.972935i \(0.425775\pi\)
\(720\) 0 0
\(721\) 30.9282 1.15183
\(722\) 19.3923 0.721707
\(723\) 0 0
\(724\) 5.46410 0.203072
\(725\) −4.92820 −0.183029
\(726\) 0 0
\(727\) 8.78461 0.325803 0.162902 0.986642i \(-0.447915\pi\)
0.162902 + 0.986642i \(0.447915\pi\)
\(728\) 25.8564 0.958302
\(729\) 0 0
\(730\) 0.928203 0.0343543
\(731\) −37.8564 −1.40017
\(732\) 0 0
\(733\) 27.8564 1.02890 0.514450 0.857520i \(-0.327996\pi\)
0.514450 + 0.857520i \(0.327996\pi\)
\(734\) −4.33975 −0.160183
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) −20.0000 −0.736709
\(738\) 0 0
\(739\) 6.92820 0.254858 0.127429 0.991848i \(-0.459327\pi\)
0.127429 + 0.991848i \(0.459327\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) −28.3923 −1.04231
\(743\) −17.1244 −0.628232 −0.314116 0.949385i \(-0.601708\pi\)
−0.314116 + 0.949385i \(0.601708\pi\)
\(744\) 0 0
\(745\) 4.39230 0.160922
\(746\) −10.7846 −0.394853
\(747\) 0 0
\(748\) −29.8564 −1.09166
\(749\) 15.4641 0.565046
\(750\) 0 0
\(751\) −20.3923 −0.744126 −0.372063 0.928208i \(-0.621349\pi\)
−0.372063 + 0.928208i \(0.621349\pi\)
\(752\) 4.73205 0.172560
\(753\) 0 0
\(754\) 26.9282 0.980667
\(755\) 12.3923 0.451002
\(756\) 0 0
\(757\) 1.71281 0.0622532 0.0311266 0.999515i \(-0.490090\pi\)
0.0311266 + 0.999515i \(0.490090\pi\)
\(758\) 32.3923 1.17654
\(759\) 0 0
\(760\) −6.19615 −0.224758
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) −9.46410 −0.342623
\(764\) 8.73205 0.315915
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) −55.7128 −2.01167
\(768\) 0 0
\(769\) 7.07180 0.255016 0.127508 0.991838i \(-0.459302\pi\)
0.127508 + 0.991838i \(0.459302\pi\)
\(770\) 25.8564 0.931800
\(771\) 0 0
\(772\) 15.8564 0.570685
\(773\) −2.78461 −0.100155 −0.0500777 0.998745i \(-0.515947\pi\)
−0.0500777 + 0.998745i \(0.515947\pi\)
\(774\) 0 0
\(775\) 0.732051 0.0262960
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 11.8564 0.425073
\(779\) 12.3923 0.444000
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) −43.7128 −1.56317
\(783\) 0 0
\(784\) 15.3923 0.549725
\(785\) −3.07180 −0.109637
\(786\) 0 0
\(787\) 22.1962 0.791207 0.395604 0.918421i \(-0.370535\pi\)
0.395604 + 0.918421i \(0.370535\pi\)
\(788\) 22.7846 0.811668
\(789\) 0 0
\(790\) −8.73205 −0.310672
\(791\) −49.8564 −1.77269
\(792\) 0 0
\(793\) 26.9282 0.956249
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) −12.0526 −0.427192
\(797\) −10.5359 −0.373201 −0.186600 0.982436i \(-0.559747\pi\)
−0.186600 + 0.982436i \(0.559747\pi\)
\(798\) 0 0
\(799\) −25.8564 −0.914734
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 32.9282 1.16274
\(803\) −5.07180 −0.178980
\(804\) 0 0
\(805\) 37.8564 1.33426
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 9.46410 0.332946
\(809\) −43.5692 −1.53181 −0.765906 0.642952i \(-0.777709\pi\)
−0.765906 + 0.642952i \(0.777709\pi\)
\(810\) 0 0
\(811\) −41.8564 −1.46978 −0.734889 0.678188i \(-0.762766\pi\)
−0.734889 + 0.678188i \(0.762766\pi\)
\(812\) 23.3205 0.818389
\(813\) 0 0
\(814\) 5.46410 0.191517
\(815\) 11.3205 0.396540
\(816\) 0 0
\(817\) 42.9282 1.50187
\(818\) 3.07180 0.107403
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) 48.4449 1.68868 0.844341 0.535806i \(-0.179992\pi\)
0.844341 + 0.535806i \(0.179992\pi\)
\(824\) −6.53590 −0.227689
\(825\) 0 0
\(826\) −48.2487 −1.67879
\(827\) 16.3923 0.570016 0.285008 0.958525i \(-0.408004\pi\)
0.285008 + 0.958525i \(0.408004\pi\)
\(828\) 0 0
\(829\) −51.5692 −1.79107 −0.895537 0.444988i \(-0.853208\pi\)
−0.895537 + 0.444988i \(0.853208\pi\)
\(830\) −8.73205 −0.303094
\(831\) 0 0
\(832\) −5.46410 −0.189434
\(833\) −84.1051 −2.91407
\(834\) 0 0
\(835\) −1.46410 −0.0506673
\(836\) 33.8564 1.17095
\(837\) 0 0
\(838\) −14.2487 −0.492214
\(839\) 8.78461 0.303278 0.151639 0.988436i \(-0.451545\pi\)
0.151639 + 0.988436i \(0.451545\pi\)
\(840\) 0 0
\(841\) −4.71281 −0.162511
\(842\) −0.143594 −0.00494856
\(843\) 0 0
\(844\) −17.8564 −0.614643
\(845\) −16.8564 −0.579878
\(846\) 0 0
\(847\) −89.2295 −3.06596
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) −5.46410 −0.187417
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 23.3205 0.798011
\(855\) 0 0
\(856\) −3.26795 −0.111696
\(857\) 31.8564 1.08819 0.544097 0.839022i \(-0.316872\pi\)
0.544097 + 0.839022i \(0.316872\pi\)
\(858\) 0 0
\(859\) 44.4449 1.51644 0.758220 0.651999i \(-0.226069\pi\)
0.758220 + 0.651999i \(0.226069\pi\)
\(860\) −6.92820 −0.236250
\(861\) 0 0
\(862\) 2.19615 0.0748012
\(863\) 20.4449 0.695951 0.347976 0.937504i \(-0.386869\pi\)
0.347976 + 0.937504i \(0.386869\pi\)
\(864\) 0 0
\(865\) 10.0000 0.340010
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) −3.46410 −0.117579
\(869\) 47.7128 1.61855
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) 49.5692 1.67670
\(875\) 4.73205 0.159973
\(876\) 0 0
\(877\) −9.21539 −0.311182 −0.155591 0.987822i \(-0.549728\pi\)
−0.155591 + 0.987822i \(0.549728\pi\)
\(878\) 13.5167 0.456165
\(879\) 0 0
\(880\) −5.46410 −0.184195
\(881\) −12.6795 −0.427183 −0.213591 0.976923i \(-0.568516\pi\)
−0.213591 + 0.976923i \(0.568516\pi\)
\(882\) 0 0
\(883\) 14.2487 0.479507 0.239754 0.970834i \(-0.422933\pi\)
0.239754 + 0.970834i \(0.422933\pi\)
\(884\) 29.8564 1.00418
\(885\) 0 0
\(886\) 14.8756 0.499757
\(887\) −9.80385 −0.329181 −0.164590 0.986362i \(-0.552630\pi\)
−0.164590 + 0.986362i \(0.552630\pi\)
\(888\) 0 0
\(889\) −17.3205 −0.580911
\(890\) −2.00000 −0.0670402
\(891\) 0 0
\(892\) 16.0526 0.537479
\(893\) 29.3205 0.981173
\(894\) 0 0
\(895\) −0.339746 −0.0113565
\(896\) −4.73205 −0.158087
\(897\) 0 0
\(898\) 21.7128 0.724566
\(899\) −3.60770 −0.120323
\(900\) 0 0
\(901\) −32.7846 −1.09221
\(902\) 10.9282 0.363869
\(903\) 0 0
\(904\) 10.5359 0.350419
\(905\) −5.46410 −0.181633
\(906\) 0 0
\(907\) 54.2487 1.80130 0.900649 0.434546i \(-0.143091\pi\)
0.900649 + 0.434546i \(0.143091\pi\)
\(908\) −24.3923 −0.809487
\(909\) 0 0
\(910\) −25.8564 −0.857132
\(911\) −37.9090 −1.25598 −0.627990 0.778221i \(-0.716122\pi\)
−0.627990 + 0.778221i \(0.716122\pi\)
\(912\) 0 0
\(913\) 47.7128 1.57906
\(914\) −31.8564 −1.05372
\(915\) 0 0
\(916\) −11.8564 −0.391747
\(917\) −87.9615 −2.90475
\(918\) 0 0
\(919\) −16.7321 −0.551939 −0.275970 0.961166i \(-0.588999\pi\)
−0.275970 + 0.961166i \(0.588999\pi\)
\(920\) −8.00000 −0.263752
\(921\) 0 0
\(922\) −14.7846 −0.486905
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 18.9282 0.622019
\(927\) 0 0
\(928\) −4.92820 −0.161776
\(929\) −11.8564 −0.388996 −0.194498 0.980903i \(-0.562308\pi\)
−0.194498 + 0.980903i \(0.562308\pi\)
\(930\) 0 0
\(931\) 95.3731 3.12573
\(932\) −28.9282 −0.947575
\(933\) 0 0
\(934\) 25.1769 0.823814
\(935\) 29.8564 0.976409
\(936\) 0 0
\(937\) −23.8564 −0.779355 −0.389677 0.920951i \(-0.627413\pi\)
−0.389677 + 0.920951i \(0.627413\pi\)
\(938\) 17.3205 0.565535
\(939\) 0 0
\(940\) −4.73205 −0.154342
\(941\) −7.60770 −0.248004 −0.124002 0.992282i \(-0.539573\pi\)
−0.124002 + 0.992282i \(0.539573\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) 10.1962 0.331856
\(945\) 0 0
\(946\) 37.8564 1.23082
\(947\) −17.1769 −0.558175 −0.279087 0.960266i \(-0.590032\pi\)
−0.279087 + 0.960266i \(0.590032\pi\)
\(948\) 0 0
\(949\) 5.07180 0.164637
\(950\) 6.19615 0.201030
\(951\) 0 0
\(952\) 25.8564 0.838011
\(953\) −15.8564 −0.513639 −0.256820 0.966459i \(-0.582675\pi\)
−0.256820 + 0.966459i \(0.582675\pi\)
\(954\) 0 0
\(955\) −8.73205 −0.282563
\(956\) 20.7321 0.670522
\(957\) 0 0
\(958\) −4.05256 −0.130932
\(959\) 9.46410 0.305612
\(960\) 0 0
\(961\) −30.4641 −0.982713
\(962\) −5.46410 −0.176170
\(963\) 0 0
\(964\) 4.92820 0.158727
\(965\) −15.8564 −0.510436
\(966\) 0 0
\(967\) −44.3923 −1.42756 −0.713780 0.700370i \(-0.753018\pi\)
−0.713780 + 0.700370i \(0.753018\pi\)
\(968\) 18.8564 0.606068
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) −1.07180 −0.0343956 −0.0171978 0.999852i \(-0.505474\pi\)
−0.0171978 + 0.999852i \(0.505474\pi\)
\(972\) 0 0
\(973\) −32.7846 −1.05103
\(974\) 4.39230 0.140739
\(975\) 0 0
\(976\) −4.92820 −0.157748
\(977\) 34.0000 1.08776 0.543878 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) 0 0
\(979\) 10.9282 0.349267
\(980\) −15.3923 −0.491689
\(981\) 0 0
\(982\) 14.9282 0.476378
\(983\) −20.4449 −0.652090 −0.326045 0.945354i \(-0.605716\pi\)
−0.326045 + 0.945354i \(0.605716\pi\)
\(984\) 0 0
\(985\) −22.7846 −0.725978
\(986\) 26.9282 0.857569
\(987\) 0 0
\(988\) −33.8564 −1.07712
\(989\) 55.4256 1.76243
\(990\) 0 0
\(991\) 44.0526 1.39938 0.699688 0.714449i \(-0.253322\pi\)
0.699688 + 0.714449i \(0.253322\pi\)
\(992\) 0.732051 0.0232426
\(993\) 0 0
\(994\) 13.8564 0.439499
\(995\) 12.0526 0.382092
\(996\) 0 0
\(997\) −31.8564 −1.00890 −0.504451 0.863440i \(-0.668305\pi\)
−0.504451 + 0.863440i \(0.668305\pi\)
\(998\) 9.41154 0.297917
\(999\) 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.bd.1.1 2
3.2 odd 2 370.2.a.e.1.2 2
12.11 even 2 2960.2.a.q.1.1 2
15.2 even 4 1850.2.b.l.149.1 4
15.8 even 4 1850.2.b.l.149.4 4
15.14 odd 2 1850.2.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.e.1.2 2 3.2 odd 2
1850.2.a.x.1.1 2 15.14 odd 2
1850.2.b.l.149.1 4 15.2 even 4
1850.2.b.l.149.4 4 15.8 even 4
2960.2.a.q.1.1 2 12.11 even 2
3330.2.a.bd.1.1 2 1.1 even 1 trivial