Properties

Label 585.2.a.k.1.2
Level $585$
Weight $2$
Character 585.1
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 585.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.73205 q^{8} +O(q^{10})\) \(q+1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.73205 q^{8} +1.73205 q^{10} +4.73205 q^{11} +1.00000 q^{13} +3.46410 q^{14} -5.00000 q^{16} +3.46410 q^{17} -6.19615 q^{19} +1.00000 q^{20} +8.19615 q^{22} -1.26795 q^{23} +1.00000 q^{25} +1.73205 q^{26} +2.00000 q^{28} +2.53590 q^{29} +10.1962 q^{31} -5.19615 q^{32} +6.00000 q^{34} +2.00000 q^{35} -4.00000 q^{37} -10.7321 q^{38} -1.73205 q^{40} -3.46410 q^{41} -0.196152 q^{43} +4.73205 q^{44} -2.19615 q^{46} -6.00000 q^{47} -3.00000 q^{49} +1.73205 q^{50} +1.00000 q^{52} -10.3923 q^{53} +4.73205 q^{55} -3.46410 q^{56} +4.39230 q^{58} -9.12436 q^{59} -8.39230 q^{61} +17.6603 q^{62} +1.00000 q^{64} +1.00000 q^{65} +6.39230 q^{67} +3.46410 q^{68} +3.46410 q^{70} -4.73205 q^{71} -4.00000 q^{73} -6.92820 q^{74} -6.19615 q^{76} +9.46410 q^{77} -8.39230 q^{79} -5.00000 q^{80} -6.00000 q^{82} +6.00000 q^{83} +3.46410 q^{85} -0.339746 q^{86} -8.19615 q^{88} +12.9282 q^{89} +2.00000 q^{91} -1.26795 q^{92} -10.3923 q^{94} -6.19615 q^{95} +2.00000 q^{97} -5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{11} + 2 q^{13} - 10 q^{16} - 2 q^{19} + 2 q^{20} + 6 q^{22} - 6 q^{23} + 2 q^{25} + 4 q^{28} + 12 q^{29} + 10 q^{31} + 12 q^{34} + 4 q^{35} - 8 q^{37} - 18 q^{38} + 10 q^{43} + 6 q^{44} + 6 q^{46} - 12 q^{47} - 6 q^{49} + 2 q^{52} + 6 q^{55} - 12 q^{58} + 6 q^{59} + 4 q^{61} + 18 q^{62} + 2 q^{64} + 2 q^{65} - 8 q^{67} - 6 q^{71} - 8 q^{73} - 2 q^{76} + 12 q^{77} + 4 q^{79} - 10 q^{80} - 12 q^{82} + 12 q^{83} - 18 q^{86} - 6 q^{88} + 12 q^{89} + 4 q^{91} - 6 q^{92} - 2 q^{95} + 4 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) 1.73205 0.547723
\(11\) 4.73205 1.42677 0.713384 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) −6.19615 −1.42149 −0.710747 0.703447i \(-0.751643\pi\)
−0.710747 + 0.703447i \(0.751643\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 8.19615 1.74743
\(23\) −1.26795 −0.264386 −0.132193 0.991224i \(-0.542202\pi\)
−0.132193 + 0.991224i \(0.542202\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.73205 0.339683
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 2.53590 0.470905 0.235452 0.971886i \(-0.424343\pi\)
0.235452 + 0.971886i \(0.424343\pi\)
\(30\) 0 0
\(31\) 10.1962 1.83128 0.915642 0.401996i \(-0.131683\pi\)
0.915642 + 0.401996i \(0.131683\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −10.7321 −1.74097
\(39\) 0 0
\(40\) −1.73205 −0.273861
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 0 0
\(43\) −0.196152 −0.0299130 −0.0149565 0.999888i \(-0.504761\pi\)
−0.0149565 + 0.999888i \(0.504761\pi\)
\(44\) 4.73205 0.713384
\(45\) 0 0
\(46\) −2.19615 −0.323805
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 1.73205 0.244949
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −10.3923 −1.42749 −0.713746 0.700404i \(-0.753003\pi\)
−0.713746 + 0.700404i \(0.753003\pi\)
\(54\) 0 0
\(55\) 4.73205 0.638070
\(56\) −3.46410 −0.462910
\(57\) 0 0
\(58\) 4.39230 0.576738
\(59\) −9.12436 −1.18789 −0.593945 0.804506i \(-0.702430\pi\)
−0.593945 + 0.804506i \(0.702430\pi\)
\(60\) 0 0
\(61\) −8.39230 −1.07452 −0.537262 0.843415i \(-0.680541\pi\)
−0.537262 + 0.843415i \(0.680541\pi\)
\(62\) 17.6603 2.24285
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 6.39230 0.780944 0.390472 0.920615i \(-0.372312\pi\)
0.390472 + 0.920615i \(0.372312\pi\)
\(68\) 3.46410 0.420084
\(69\) 0 0
\(70\) 3.46410 0.414039
\(71\) −4.73205 −0.561591 −0.280796 0.959768i \(-0.590598\pi\)
−0.280796 + 0.959768i \(0.590598\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −6.92820 −0.805387
\(75\) 0 0
\(76\) −6.19615 −0.710747
\(77\) 9.46410 1.07853
\(78\) 0 0
\(79\) −8.39230 −0.944208 −0.472104 0.881543i \(-0.656505\pi\)
−0.472104 + 0.881543i \(0.656505\pi\)
\(80\) −5.00000 −0.559017
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 3.46410 0.375735
\(86\) −0.339746 −0.0366357
\(87\) 0 0
\(88\) −8.19615 −0.873713
\(89\) 12.9282 1.37039 0.685193 0.728361i \(-0.259718\pi\)
0.685193 + 0.728361i \(0.259718\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −1.26795 −0.132193
\(93\) 0 0
\(94\) −10.3923 −1.07188
\(95\) −6.19615 −0.635712
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −5.19615 −0.524891
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0.928203 0.0923597 0.0461798 0.998933i \(-0.485295\pi\)
0.0461798 + 0.998933i \(0.485295\pi\)
\(102\) 0 0
\(103\) −0.196152 −0.0193275 −0.00966374 0.999953i \(-0.503076\pi\)
−0.00966374 + 0.999953i \(0.503076\pi\)
\(104\) −1.73205 −0.169842
\(105\) 0 0
\(106\) −18.0000 −1.74831
\(107\) −17.6603 −1.70728 −0.853641 0.520862i \(-0.825610\pi\)
−0.853641 + 0.520862i \(0.825610\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 8.19615 0.781472
\(111\) 0 0
\(112\) −10.0000 −0.944911
\(113\) −8.53590 −0.802990 −0.401495 0.915861i \(-0.631509\pi\)
−0.401495 + 0.915861i \(0.631509\pi\)
\(114\) 0 0
\(115\) −1.26795 −0.118237
\(116\) 2.53590 0.235452
\(117\) 0 0
\(118\) −15.8038 −1.45486
\(119\) 6.92820 0.635107
\(120\) 0 0
\(121\) 11.3923 1.03566
\(122\) −14.5359 −1.31602
\(123\) 0 0
\(124\) 10.1962 0.915642
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 16.1962 1.43718 0.718588 0.695436i \(-0.244789\pi\)
0.718588 + 0.695436i \(0.244789\pi\)
\(128\) 12.1244 1.07165
\(129\) 0 0
\(130\) 1.73205 0.151911
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −12.3923 −1.07455
\(134\) 11.0718 0.956458
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −0.928203 −0.0793018 −0.0396509 0.999214i \(-0.512625\pi\)
−0.0396509 + 0.999214i \(0.512625\pi\)
\(138\) 0 0
\(139\) 12.3923 1.05110 0.525551 0.850762i \(-0.323859\pi\)
0.525551 + 0.850762i \(0.323859\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) −8.19615 −0.687806
\(143\) 4.73205 0.395714
\(144\) 0 0
\(145\) 2.53590 0.210595
\(146\) −6.92820 −0.573382
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 7.85641 0.643622 0.321811 0.946804i \(-0.395708\pi\)
0.321811 + 0.946804i \(0.395708\pi\)
\(150\) 0 0
\(151\) −1.80385 −0.146795 −0.0733975 0.997303i \(-0.523384\pi\)
−0.0733975 + 0.997303i \(0.523384\pi\)
\(152\) 10.7321 0.870484
\(153\) 0 0
\(154\) 16.3923 1.32093
\(155\) 10.1962 0.818975
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −14.5359 −1.15641
\(159\) 0 0
\(160\) −5.19615 −0.410792
\(161\) −2.53590 −0.199857
\(162\) 0 0
\(163\) −14.3923 −1.12729 −0.563646 0.826016i \(-0.690602\pi\)
−0.563646 + 0.826016i \(0.690602\pi\)
\(164\) −3.46410 −0.270501
\(165\) 0 0
\(166\) 10.3923 0.806599
\(167\) 0.928203 0.0718265 0.0359133 0.999355i \(-0.488566\pi\)
0.0359133 + 0.999355i \(0.488566\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) −0.196152 −0.0149565
\(173\) −8.53590 −0.648972 −0.324486 0.945890i \(-0.605191\pi\)
−0.324486 + 0.945890i \(0.605191\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) −23.6603 −1.78346
\(177\) 0 0
\(178\) 22.3923 1.67837
\(179\) 18.9282 1.41476 0.707380 0.706833i \(-0.249877\pi\)
0.707380 + 0.706833i \(0.249877\pi\)
\(180\) 0 0
\(181\) 0.392305 0.0291598 0.0145799 0.999894i \(-0.495359\pi\)
0.0145799 + 0.999894i \(0.495359\pi\)
\(182\) 3.46410 0.256776
\(183\) 0 0
\(184\) 2.19615 0.161903
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 16.3923 1.19872
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) −10.7321 −0.778585
\(191\) 5.07180 0.366982 0.183491 0.983021i \(-0.441260\pi\)
0.183491 + 0.983021i \(0.441260\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 3.46410 0.248708
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 12.9282 0.921096 0.460548 0.887635i \(-0.347653\pi\)
0.460548 + 0.887635i \(0.347653\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −1.73205 −0.122474
\(201\) 0 0
\(202\) 1.60770 0.113117
\(203\) 5.07180 0.355970
\(204\) 0 0
\(205\) −3.46410 −0.241943
\(206\) −0.339746 −0.0236712
\(207\) 0 0
\(208\) −5.00000 −0.346688
\(209\) −29.3205 −2.02814
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −10.3923 −0.713746
\(213\) 0 0
\(214\) −30.5885 −2.09098
\(215\) −0.196152 −0.0133775
\(216\) 0 0
\(217\) 20.3923 1.38432
\(218\) 3.46410 0.234619
\(219\) 0 0
\(220\) 4.73205 0.319035
\(221\) 3.46410 0.233021
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −10.3923 −0.694365
\(225\) 0 0
\(226\) −14.7846 −0.983458
\(227\) 3.46410 0.229920 0.114960 0.993370i \(-0.463326\pi\)
0.114960 + 0.993370i \(0.463326\pi\)
\(228\) 0 0
\(229\) 6.39230 0.422415 0.211208 0.977441i \(-0.432260\pi\)
0.211208 + 0.977441i \(0.432260\pi\)
\(230\) −2.19615 −0.144810
\(231\) 0 0
\(232\) −4.39230 −0.288369
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) −9.12436 −0.593945
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) 14.1962 0.918273 0.459136 0.888366i \(-0.348159\pi\)
0.459136 + 0.888366i \(0.348159\pi\)
\(240\) 0 0
\(241\) −2.39230 −0.154102 −0.0770510 0.997027i \(-0.524550\pi\)
−0.0770510 + 0.997027i \(0.524550\pi\)
\(242\) 19.7321 1.26842
\(243\) 0 0
\(244\) −8.39230 −0.537262
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) −6.19615 −0.394252
\(248\) −17.6603 −1.12143
\(249\) 0 0
\(250\) 1.73205 0.109545
\(251\) 21.4641 1.35480 0.677401 0.735614i \(-0.263106\pi\)
0.677401 + 0.735614i \(0.263106\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 28.0526 1.76017
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −19.8564 −1.23861 −0.619304 0.785151i \(-0.712585\pi\)
−0.619304 + 0.785151i \(0.712585\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) 0 0
\(263\) −1.26795 −0.0781851 −0.0390925 0.999236i \(-0.512447\pi\)
−0.0390925 + 0.999236i \(0.512447\pi\)
\(264\) 0 0
\(265\) −10.3923 −0.638394
\(266\) −21.4641 −1.31605
\(267\) 0 0
\(268\) 6.39230 0.390472
\(269\) 19.8564 1.21067 0.605333 0.795972i \(-0.293040\pi\)
0.605333 + 0.795972i \(0.293040\pi\)
\(270\) 0 0
\(271\) 30.9808 1.88195 0.940974 0.338480i \(-0.109913\pi\)
0.940974 + 0.338480i \(0.109913\pi\)
\(272\) −17.3205 −1.05021
\(273\) 0 0
\(274\) −1.60770 −0.0971244
\(275\) 4.73205 0.285353
\(276\) 0 0
\(277\) −26.3923 −1.58576 −0.792880 0.609378i \(-0.791419\pi\)
−0.792880 + 0.609378i \(0.791419\pi\)
\(278\) 21.4641 1.28733
\(279\) 0 0
\(280\) −3.46410 −0.207020
\(281\) −22.3923 −1.33581 −0.667906 0.744245i \(-0.732809\pi\)
−0.667906 + 0.744245i \(0.732809\pi\)
\(282\) 0 0
\(283\) 32.5885 1.93718 0.968591 0.248658i \(-0.0799895\pi\)
0.968591 + 0.248658i \(0.0799895\pi\)
\(284\) −4.73205 −0.280796
\(285\) 0 0
\(286\) 8.19615 0.484649
\(287\) −6.92820 −0.408959
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 4.39230 0.257925
\(291\) 0 0
\(292\) −4.00000 −0.234082
\(293\) 5.07180 0.296298 0.148149 0.988965i \(-0.452669\pi\)
0.148149 + 0.988965i \(0.452669\pi\)
\(294\) 0 0
\(295\) −9.12436 −0.531241
\(296\) 6.92820 0.402694
\(297\) 0 0
\(298\) 13.6077 0.788273
\(299\) −1.26795 −0.0733274
\(300\) 0 0
\(301\) −0.392305 −0.0226121
\(302\) −3.12436 −0.179786
\(303\) 0 0
\(304\) 30.9808 1.77687
\(305\) −8.39230 −0.480542
\(306\) 0 0
\(307\) −18.7846 −1.07209 −0.536047 0.844188i \(-0.680083\pi\)
−0.536047 + 0.844188i \(0.680083\pi\)
\(308\) 9.46410 0.539267
\(309\) 0 0
\(310\) 17.6603 1.00304
\(311\) 16.3923 0.929522 0.464761 0.885436i \(-0.346140\pi\)
0.464761 + 0.885436i \(0.346140\pi\)
\(312\) 0 0
\(313\) −14.3923 −0.813501 −0.406751 0.913539i \(-0.633338\pi\)
−0.406751 + 0.913539i \(0.633338\pi\)
\(314\) −17.3205 −0.977453
\(315\) 0 0
\(316\) −8.39230 −0.472104
\(317\) −24.0000 −1.34797 −0.673987 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −4.39230 −0.244774
\(323\) −21.4641 −1.19429
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) −24.9282 −1.38065
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 2.58846 0.142274 0.0711372 0.997467i \(-0.477337\pi\)
0.0711372 + 0.997467i \(0.477337\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) 1.60770 0.0879692
\(335\) 6.39230 0.349249
\(336\) 0 0
\(337\) −26.3923 −1.43768 −0.718840 0.695175i \(-0.755327\pi\)
−0.718840 + 0.695175i \(0.755327\pi\)
\(338\) 1.73205 0.0942111
\(339\) 0 0
\(340\) 3.46410 0.187867
\(341\) 48.2487 2.61281
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0.339746 0.0183179
\(345\) 0 0
\(346\) −14.7846 −0.794826
\(347\) 5.66025 0.303858 0.151929 0.988391i \(-0.451451\pi\)
0.151929 + 0.988391i \(0.451451\pi\)
\(348\) 0 0
\(349\) −14.3923 −0.770402 −0.385201 0.922833i \(-0.625868\pi\)
−0.385201 + 0.922833i \(0.625868\pi\)
\(350\) 3.46410 0.185164
\(351\) 0 0
\(352\) −24.5885 −1.31057
\(353\) 27.7128 1.47500 0.737502 0.675345i \(-0.236005\pi\)
0.737502 + 0.675345i \(0.236005\pi\)
\(354\) 0 0
\(355\) −4.73205 −0.251151
\(356\) 12.9282 0.685193
\(357\) 0 0
\(358\) 32.7846 1.73272
\(359\) 2.19615 0.115908 0.0579542 0.998319i \(-0.481542\pi\)
0.0579542 + 0.998319i \(0.481542\pi\)
\(360\) 0 0
\(361\) 19.3923 1.02065
\(362\) 0.679492 0.0357133
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) 11.8038 0.616156 0.308078 0.951361i \(-0.400314\pi\)
0.308078 + 0.951361i \(0.400314\pi\)
\(368\) 6.33975 0.330482
\(369\) 0 0
\(370\) −6.92820 −0.360180
\(371\) −20.7846 −1.07908
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 28.3923 1.46813
\(375\) 0 0
\(376\) 10.3923 0.535942
\(377\) 2.53590 0.130605
\(378\) 0 0
\(379\) 18.9808 0.974976 0.487488 0.873130i \(-0.337913\pi\)
0.487488 + 0.873130i \(0.337913\pi\)
\(380\) −6.19615 −0.317856
\(381\) 0 0
\(382\) 8.78461 0.449460
\(383\) −12.9282 −0.660600 −0.330300 0.943876i \(-0.607150\pi\)
−0.330300 + 0.943876i \(0.607150\pi\)
\(384\) 0 0
\(385\) 9.46410 0.482335
\(386\) −17.3205 −0.881591
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −4.39230 −0.222128
\(392\) 5.19615 0.262445
\(393\) 0 0
\(394\) 22.3923 1.12811
\(395\) −8.39230 −0.422263
\(396\) 0 0
\(397\) 28.7846 1.44466 0.722329 0.691549i \(-0.243072\pi\)
0.722329 + 0.691549i \(0.243072\pi\)
\(398\) 34.6410 1.73640
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 36.9282 1.84411 0.922053 0.387063i \(-0.126510\pi\)
0.922053 + 0.387063i \(0.126510\pi\)
\(402\) 0 0
\(403\) 10.1962 0.507907
\(404\) 0.928203 0.0461798
\(405\) 0 0
\(406\) 8.78461 0.435973
\(407\) −18.9282 −0.938236
\(408\) 0 0
\(409\) −17.6077 −0.870644 −0.435322 0.900275i \(-0.643366\pi\)
−0.435322 + 0.900275i \(0.643366\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) −0.196152 −0.00966374
\(413\) −18.2487 −0.897960
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) −5.19615 −0.254762
\(417\) 0 0
\(418\) −50.7846 −2.48396
\(419\) 2.53590 0.123887 0.0619434 0.998080i \(-0.480270\pi\)
0.0619434 + 0.998080i \(0.480270\pi\)
\(420\) 0 0
\(421\) −30.7846 −1.50035 −0.750175 0.661239i \(-0.770031\pi\)
−0.750175 + 0.661239i \(0.770031\pi\)
\(422\) 13.8564 0.674519
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) 3.46410 0.168034
\(426\) 0 0
\(427\) −16.7846 −0.812264
\(428\) −17.6603 −0.853641
\(429\) 0 0
\(430\) −0.339746 −0.0163840
\(431\) 25.5167 1.22909 0.614547 0.788880i \(-0.289339\pi\)
0.614547 + 0.788880i \(0.289339\pi\)
\(432\) 0 0
\(433\) 34.7846 1.67164 0.835821 0.549002i \(-0.184992\pi\)
0.835821 + 0.549002i \(0.184992\pi\)
\(434\) 35.3205 1.69544
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 7.85641 0.375823
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) −8.19615 −0.390736
\(441\) 0 0
\(442\) 6.00000 0.285391
\(443\) 16.9808 0.806780 0.403390 0.915028i \(-0.367832\pi\)
0.403390 + 0.915028i \(0.367832\pi\)
\(444\) 0 0
\(445\) 12.9282 0.612856
\(446\) 3.46410 0.164030
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −20.5359 −0.969149 −0.484574 0.874750i \(-0.661026\pi\)
−0.484574 + 0.874750i \(0.661026\pi\)
\(450\) 0 0
\(451\) −16.3923 −0.771883
\(452\) −8.53590 −0.401495
\(453\) 0 0
\(454\) 6.00000 0.281594
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 10.7846 0.504483 0.252241 0.967664i \(-0.418832\pi\)
0.252241 + 0.967664i \(0.418832\pi\)
\(458\) 11.0718 0.517351
\(459\) 0 0
\(460\) −1.26795 −0.0591184
\(461\) 3.46410 0.161339 0.0806696 0.996741i \(-0.474294\pi\)
0.0806696 + 0.996741i \(0.474294\pi\)
\(462\) 0 0
\(463\) −2.39230 −0.111180 −0.0555899 0.998454i \(-0.517704\pi\)
−0.0555899 + 0.998454i \(0.517704\pi\)
\(464\) −12.6795 −0.588631
\(465\) 0 0
\(466\) 10.3923 0.481414
\(467\) −27.8038 −1.28661 −0.643304 0.765611i \(-0.722437\pi\)
−0.643304 + 0.765611i \(0.722437\pi\)
\(468\) 0 0
\(469\) 12.7846 0.590338
\(470\) −10.3923 −0.479361
\(471\) 0 0
\(472\) 15.8038 0.727431
\(473\) −0.928203 −0.0426788
\(474\) 0 0
\(475\) −6.19615 −0.284299
\(476\) 6.92820 0.317554
\(477\) 0 0
\(478\) 24.5885 1.12465
\(479\) −35.6603 −1.62936 −0.814679 0.579912i \(-0.803087\pi\)
−0.814679 + 0.579912i \(0.803087\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −4.14359 −0.188736
\(483\) 0 0
\(484\) 11.3923 0.517832
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) −26.3923 −1.19595 −0.597975 0.801515i \(-0.704028\pi\)
−0.597975 + 0.801515i \(0.704028\pi\)
\(488\) 14.5359 0.658009
\(489\) 0 0
\(490\) −5.19615 −0.234738
\(491\) 2.53590 0.114443 0.0572217 0.998361i \(-0.481776\pi\)
0.0572217 + 0.998361i \(0.481776\pi\)
\(492\) 0 0
\(493\) 8.78461 0.395639
\(494\) −10.7321 −0.482858
\(495\) 0 0
\(496\) −50.9808 −2.28910
\(497\) −9.46410 −0.424523
\(498\) 0 0
\(499\) −38.9808 −1.74502 −0.872509 0.488598i \(-0.837509\pi\)
−0.872509 + 0.488598i \(0.837509\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 37.1769 1.65929
\(503\) −19.5167 −0.870205 −0.435102 0.900381i \(-0.643288\pi\)
−0.435102 + 0.900381i \(0.643288\pi\)
\(504\) 0 0
\(505\) 0.928203 0.0413045
\(506\) −10.3923 −0.461994
\(507\) 0 0
\(508\) 16.1962 0.718588
\(509\) −39.4641 −1.74922 −0.874608 0.484831i \(-0.838881\pi\)
−0.874608 + 0.484831i \(0.838881\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 8.66025 0.382733
\(513\) 0 0
\(514\) −34.3923 −1.51698
\(515\) −0.196152 −0.00864351
\(516\) 0 0
\(517\) −28.3923 −1.24869
\(518\) −13.8564 −0.608816
\(519\) 0 0
\(520\) −1.73205 −0.0759555
\(521\) 28.3923 1.24389 0.621945 0.783061i \(-0.286343\pi\)
0.621945 + 0.783061i \(0.286343\pi\)
\(522\) 0 0
\(523\) −24.1962 −1.05802 −0.529012 0.848614i \(-0.677437\pi\)
−0.529012 + 0.848614i \(0.677437\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −2.19615 −0.0957568
\(527\) 35.3205 1.53859
\(528\) 0 0
\(529\) −21.3923 −0.930100
\(530\) −18.0000 −0.781870
\(531\) 0 0
\(532\) −12.3923 −0.537275
\(533\) −3.46410 −0.150047
\(534\) 0 0
\(535\) −17.6603 −0.763519
\(536\) −11.0718 −0.478229
\(537\) 0 0
\(538\) 34.3923 1.48276
\(539\) −14.1962 −0.611472
\(540\) 0 0
\(541\) −26.3923 −1.13469 −0.567347 0.823479i \(-0.692030\pi\)
−0.567347 + 0.823479i \(0.692030\pi\)
\(542\) 53.6603 2.30491
\(543\) 0 0
\(544\) −18.0000 −0.771744
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −12.1962 −0.521470 −0.260735 0.965410i \(-0.583965\pi\)
−0.260735 + 0.965410i \(0.583965\pi\)
\(548\) −0.928203 −0.0396509
\(549\) 0 0
\(550\) 8.19615 0.349485
\(551\) −15.7128 −0.669388
\(552\) 0 0
\(553\) −16.7846 −0.713754
\(554\) −45.7128 −1.94215
\(555\) 0 0
\(556\) 12.3923 0.525551
\(557\) −1.85641 −0.0786585 −0.0393292 0.999226i \(-0.512522\pi\)
−0.0393292 + 0.999226i \(0.512522\pi\)
\(558\) 0 0
\(559\) −0.196152 −0.00829636
\(560\) −10.0000 −0.422577
\(561\) 0 0
\(562\) −38.7846 −1.63603
\(563\) 22.0526 0.929405 0.464702 0.885467i \(-0.346161\pi\)
0.464702 + 0.885467i \(0.346161\pi\)
\(564\) 0 0
\(565\) −8.53590 −0.359108
\(566\) 56.4449 2.37255
\(567\) 0 0
\(568\) 8.19615 0.343903
\(569\) 2.53590 0.106310 0.0531552 0.998586i \(-0.483072\pi\)
0.0531552 + 0.998586i \(0.483072\pi\)
\(570\) 0 0
\(571\) 36.3923 1.52297 0.761485 0.648182i \(-0.224470\pi\)
0.761485 + 0.648182i \(0.224470\pi\)
\(572\) 4.73205 0.197857
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) −1.26795 −0.0528771
\(576\) 0 0
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) −8.66025 −0.360219
\(579\) 0 0
\(580\) 2.53590 0.105297
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) −49.1769 −2.03670
\(584\) 6.92820 0.286691
\(585\) 0 0
\(586\) 8.78461 0.362889
\(587\) 8.53590 0.352314 0.176157 0.984362i \(-0.443633\pi\)
0.176157 + 0.984362i \(0.443633\pi\)
\(588\) 0 0
\(589\) −63.1769 −2.60316
\(590\) −15.8038 −0.650634
\(591\) 0 0
\(592\) 20.0000 0.821995
\(593\) −26.7846 −1.09991 −0.549956 0.835194i \(-0.685356\pi\)
−0.549956 + 0.835194i \(0.685356\pi\)
\(594\) 0 0
\(595\) 6.92820 0.284029
\(596\) 7.85641 0.321811
\(597\) 0 0
\(598\) −2.19615 −0.0898074
\(599\) 7.60770 0.310842 0.155421 0.987848i \(-0.450327\pi\)
0.155421 + 0.987848i \(0.450327\pi\)
\(600\) 0 0
\(601\) 43.5692 1.77723 0.888613 0.458658i \(-0.151670\pi\)
0.888613 + 0.458658i \(0.151670\pi\)
\(602\) −0.679492 −0.0276940
\(603\) 0 0
\(604\) −1.80385 −0.0733975
\(605\) 11.3923 0.463163
\(606\) 0 0
\(607\) 24.9808 1.01394 0.506969 0.861964i \(-0.330766\pi\)
0.506969 + 0.861964i \(0.330766\pi\)
\(608\) 32.1962 1.30573
\(609\) 0 0
\(610\) −14.5359 −0.588541
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −32.5359 −1.31304
\(615\) 0 0
\(616\) −16.3923 −0.660465
\(617\) −33.7128 −1.35723 −0.678613 0.734496i \(-0.737419\pi\)
−0.678613 + 0.734496i \(0.737419\pi\)
\(618\) 0 0
\(619\) 6.98076 0.280581 0.140290 0.990110i \(-0.455196\pi\)
0.140290 + 0.990110i \(0.455196\pi\)
\(620\) 10.1962 0.409487
\(621\) 0 0
\(622\) 28.3923 1.13843
\(623\) 25.8564 1.03592
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −24.9282 −0.996331
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) −13.8564 −0.552491
\(630\) 0 0
\(631\) 5.80385 0.231048 0.115524 0.993305i \(-0.463145\pi\)
0.115524 + 0.993305i \(0.463145\pi\)
\(632\) 14.5359 0.578207
\(633\) 0 0
\(634\) −41.5692 −1.65092
\(635\) 16.1962 0.642725
\(636\) 0 0
\(637\) −3.00000 −0.118864
\(638\) 20.7846 0.822871
\(639\) 0 0
\(640\) 12.1244 0.479257
\(641\) −12.9282 −0.510633 −0.255317 0.966857i \(-0.582180\pi\)
−0.255317 + 0.966857i \(0.582180\pi\)
\(642\) 0 0
\(643\) −6.78461 −0.267559 −0.133779 0.991011i \(-0.542711\pi\)
−0.133779 + 0.991011i \(0.542711\pi\)
\(644\) −2.53590 −0.0999284
\(645\) 0 0
\(646\) −37.1769 −1.46271
\(647\) −22.0526 −0.866976 −0.433488 0.901159i \(-0.642717\pi\)
−0.433488 + 0.901159i \(0.642717\pi\)
\(648\) 0 0
\(649\) −43.1769 −1.69484
\(650\) 1.73205 0.0679366
\(651\) 0 0
\(652\) −14.3923 −0.563646
\(653\) −7.85641 −0.307445 −0.153722 0.988114i \(-0.549126\pi\)
−0.153722 + 0.988114i \(0.549126\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 17.3205 0.676252
\(657\) 0 0
\(658\) −20.7846 −0.810268
\(659\) 21.4641 0.836123 0.418061 0.908419i \(-0.362710\pi\)
0.418061 + 0.908419i \(0.362710\pi\)
\(660\) 0 0
\(661\) 10.7846 0.419473 0.209736 0.977758i \(-0.432739\pi\)
0.209736 + 0.977758i \(0.432739\pi\)
\(662\) 4.48334 0.174250
\(663\) 0 0
\(664\) −10.3923 −0.403300
\(665\) −12.3923 −0.480553
\(666\) 0 0
\(667\) −3.21539 −0.124500
\(668\) 0.928203 0.0359133
\(669\) 0 0
\(670\) 11.0718 0.427741
\(671\) −39.7128 −1.53310
\(672\) 0 0
\(673\) −14.3923 −0.554783 −0.277391 0.960757i \(-0.589470\pi\)
−0.277391 + 0.960757i \(0.589470\pi\)
\(674\) −45.7128 −1.76079
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −10.3923 −0.399409 −0.199704 0.979856i \(-0.563998\pi\)
−0.199704 + 0.979856i \(0.563998\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) 83.5692 3.20003
\(683\) −32.5359 −1.24495 −0.622476 0.782639i \(-0.713873\pi\)
−0.622476 + 0.782639i \(0.713873\pi\)
\(684\) 0 0
\(685\) −0.928203 −0.0354648
\(686\) −34.6410 −1.32260
\(687\) 0 0
\(688\) 0.980762 0.0373912
\(689\) −10.3923 −0.395915
\(690\) 0 0
\(691\) −47.7654 −1.81708 −0.908540 0.417797i \(-0.862802\pi\)
−0.908540 + 0.417797i \(0.862802\pi\)
\(692\) −8.53590 −0.324486
\(693\) 0 0
\(694\) 9.80385 0.372149
\(695\) 12.3923 0.470067
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) −24.9282 −0.943546
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 24.7846 0.934769
\(704\) 4.73205 0.178346
\(705\) 0 0
\(706\) 48.0000 1.80650
\(707\) 1.85641 0.0698174
\(708\) 0 0
\(709\) 30.3923 1.14141 0.570703 0.821156i \(-0.306671\pi\)
0.570703 + 0.821156i \(0.306671\pi\)
\(710\) −8.19615 −0.307596
\(711\) 0 0
\(712\) −22.3923 −0.839187
\(713\) −12.9282 −0.484165
\(714\) 0 0
\(715\) 4.73205 0.176969
\(716\) 18.9282 0.707380
\(717\) 0 0
\(718\) 3.80385 0.141958
\(719\) 25.8564 0.964281 0.482141 0.876094i \(-0.339859\pi\)
0.482141 + 0.876094i \(0.339859\pi\)
\(720\) 0 0
\(721\) −0.392305 −0.0146102
\(722\) 33.5885 1.25003
\(723\) 0 0
\(724\) 0.392305 0.0145799
\(725\) 2.53590 0.0941809
\(726\) 0 0
\(727\) 44.5885 1.65369 0.826847 0.562427i \(-0.190132\pi\)
0.826847 + 0.562427i \(0.190132\pi\)
\(728\) −3.46410 −0.128388
\(729\) 0 0
\(730\) −6.92820 −0.256424
\(731\) −0.679492 −0.0251319
\(732\) 0 0
\(733\) 38.0000 1.40356 0.701781 0.712393i \(-0.252388\pi\)
0.701781 + 0.712393i \(0.252388\pi\)
\(734\) 20.4449 0.754634
\(735\) 0 0
\(736\) 6.58846 0.242854
\(737\) 30.2487 1.11423
\(738\) 0 0
\(739\) −18.1962 −0.669356 −0.334678 0.942332i \(-0.608628\pi\)
−0.334678 + 0.942332i \(0.608628\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) −36.0000 −1.32160
\(743\) 16.1436 0.592251 0.296126 0.955149i \(-0.404305\pi\)
0.296126 + 0.955149i \(0.404305\pi\)
\(744\) 0 0
\(745\) 7.85641 0.287836
\(746\) −17.3205 −0.634149
\(747\) 0 0
\(748\) 16.3923 0.599362
\(749\) −35.3205 −1.29058
\(750\) 0 0
\(751\) 36.3923 1.32797 0.663987 0.747744i \(-0.268863\pi\)
0.663987 + 0.747744i \(0.268863\pi\)
\(752\) 30.0000 1.09399
\(753\) 0 0
\(754\) 4.39230 0.159958
\(755\) −1.80385 −0.0656487
\(756\) 0 0
\(757\) −2.39230 −0.0869498 −0.0434749 0.999055i \(-0.513843\pi\)
−0.0434749 + 0.999055i \(0.513843\pi\)
\(758\) 32.8756 1.19410
\(759\) 0 0
\(760\) 10.7321 0.389292
\(761\) 19.8564 0.719794 0.359897 0.932992i \(-0.382812\pi\)
0.359897 + 0.932992i \(0.382812\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 5.07180 0.183491
\(765\) 0 0
\(766\) −22.3923 −0.809067
\(767\) −9.12436 −0.329461
\(768\) 0 0
\(769\) 34.7846 1.25437 0.627183 0.778872i \(-0.284208\pi\)
0.627183 + 0.778872i \(0.284208\pi\)
\(770\) 16.3923 0.590738
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) −6.92820 −0.249190 −0.124595 0.992208i \(-0.539763\pi\)
−0.124595 + 0.992208i \(0.539763\pi\)
\(774\) 0 0
\(775\) 10.1962 0.366257
\(776\) −3.46410 −0.124354
\(777\) 0 0
\(778\) −10.3923 −0.372582
\(779\) 21.4641 0.769031
\(780\) 0 0
\(781\) −22.3923 −0.801260
\(782\) −7.60770 −0.272051
\(783\) 0 0
\(784\) 15.0000 0.535714
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) 31.5692 1.12532 0.562661 0.826688i \(-0.309778\pi\)
0.562661 + 0.826688i \(0.309778\pi\)
\(788\) 12.9282 0.460548
\(789\) 0 0
\(790\) −14.5359 −0.517164
\(791\) −17.0718 −0.607003
\(792\) 0 0
\(793\) −8.39230 −0.298019
\(794\) 49.8564 1.76934
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 40.6410 1.43958 0.719789 0.694193i \(-0.244238\pi\)
0.719789 + 0.694193i \(0.244238\pi\)
\(798\) 0 0
\(799\) −20.7846 −0.735307
\(800\) −5.19615 −0.183712
\(801\) 0 0
\(802\) 63.9615 2.25856
\(803\) −18.9282 −0.667962
\(804\) 0 0
\(805\) −2.53590 −0.0893787
\(806\) 17.6603 0.622056
\(807\) 0 0
\(808\) −1.60770 −0.0565585
\(809\) 2.53590 0.0891574 0.0445787 0.999006i \(-0.485805\pi\)
0.0445787 + 0.999006i \(0.485805\pi\)
\(810\) 0 0
\(811\) 17.8038 0.625178 0.312589 0.949889i \(-0.398804\pi\)
0.312589 + 0.949889i \(0.398804\pi\)
\(812\) 5.07180 0.177985
\(813\) 0 0
\(814\) −32.7846 −1.14910
\(815\) −14.3923 −0.504140
\(816\) 0 0
\(817\) 1.21539 0.0425211
\(818\) −30.4974 −1.06632
\(819\) 0 0
\(820\) −3.46410 −0.120972
\(821\) −28.6410 −0.999578 −0.499789 0.866147i \(-0.666589\pi\)
−0.499789 + 0.866147i \(0.666589\pi\)
\(822\) 0 0
\(823\) −15.4115 −0.537213 −0.268606 0.963250i \(-0.586563\pi\)
−0.268606 + 0.963250i \(0.586563\pi\)
\(824\) 0.339746 0.0118356
\(825\) 0 0
\(826\) −31.6077 −1.09977
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 0 0
\(829\) 0.392305 0.0136253 0.00681266 0.999977i \(-0.497831\pi\)
0.00681266 + 0.999977i \(0.497831\pi\)
\(830\) 10.3923 0.360722
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) −10.3923 −0.360072
\(834\) 0 0
\(835\) 0.928203 0.0321218
\(836\) −29.3205 −1.01407
\(837\) 0 0
\(838\) 4.39230 0.151730
\(839\) −0.339746 −0.0117293 −0.00586467 0.999983i \(-0.501867\pi\)
−0.00586467 + 0.999983i \(0.501867\pi\)
\(840\) 0 0
\(841\) −22.5692 −0.778249
\(842\) −53.3205 −1.83755
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 22.7846 0.782888
\(848\) 51.9615 1.78437
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 5.07180 0.173859
\(852\) 0 0
\(853\) 8.00000 0.273915 0.136957 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) −29.0718 −0.994816
\(855\) 0 0
\(856\) 30.5885 1.04549
\(857\) 35.5692 1.21502 0.607511 0.794311i \(-0.292168\pi\)
0.607511 + 0.794311i \(0.292168\pi\)
\(858\) 0 0
\(859\) −17.1769 −0.586069 −0.293034 0.956102i \(-0.594665\pi\)
−0.293034 + 0.956102i \(0.594665\pi\)
\(860\) −0.196152 −0.00668874
\(861\) 0 0
\(862\) 44.1962 1.50533
\(863\) 38.7846 1.32024 0.660122 0.751159i \(-0.270505\pi\)
0.660122 + 0.751159i \(0.270505\pi\)
\(864\) 0 0
\(865\) −8.53590 −0.290229
\(866\) 60.2487 2.04733
\(867\) 0 0
\(868\) 20.3923 0.692160
\(869\) −39.7128 −1.34716
\(870\) 0 0
\(871\) 6.39230 0.216595
\(872\) −3.46410 −0.117309
\(873\) 0 0
\(874\) 13.6077 0.460287
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 55.4256 1.87052
\(879\) 0 0
\(880\) −23.6603 −0.797587
\(881\) 47.3205 1.59427 0.797134 0.603802i \(-0.206348\pi\)
0.797134 + 0.603802i \(0.206348\pi\)
\(882\) 0 0
\(883\) 23.8038 0.801063 0.400532 0.916283i \(-0.368825\pi\)
0.400532 + 0.916283i \(0.368825\pi\)
\(884\) 3.46410 0.116510
\(885\) 0 0
\(886\) 29.4115 0.988100
\(887\) 47.9090 1.60863 0.804313 0.594206i \(-0.202534\pi\)
0.804313 + 0.594206i \(0.202534\pi\)
\(888\) 0 0
\(889\) 32.3923 1.08640
\(890\) 22.3923 0.750592
\(891\) 0 0
\(892\) 2.00000 0.0669650
\(893\) 37.1769 1.24408
\(894\) 0 0
\(895\) 18.9282 0.632700
\(896\) 24.2487 0.810093
\(897\) 0 0
\(898\) −35.5692 −1.18696
\(899\) 25.8564 0.862359
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) −28.3923 −0.945360
\(903\) 0 0
\(904\) 14.7846 0.491729
\(905\) 0.392305 0.0130407
\(906\) 0 0
\(907\) −53.7654 −1.78525 −0.892625 0.450800i \(-0.851139\pi\)
−0.892625 + 0.450800i \(0.851139\pi\)
\(908\) 3.46410 0.114960
\(909\) 0 0
\(910\) 3.46410 0.114834
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 28.3923 0.939648
\(914\) 18.6795 0.617863
\(915\) 0 0
\(916\) 6.39230 0.211208
\(917\) 0 0
\(918\) 0 0
\(919\) 9.17691 0.302718 0.151359 0.988479i \(-0.451635\pi\)
0.151359 + 0.988479i \(0.451635\pi\)
\(920\) 2.19615 0.0724050
\(921\) 0 0
\(922\) 6.00000 0.197599
\(923\) −4.73205 −0.155757
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −4.14359 −0.136167
\(927\) 0 0
\(928\) −13.1769 −0.432553
\(929\) −44.5359 −1.46118 −0.730588 0.682819i \(-0.760754\pi\)
−0.730588 + 0.682819i \(0.760754\pi\)
\(930\) 0 0
\(931\) 18.5885 0.609212
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −48.1577 −1.57577
\(935\) 16.3923 0.536086
\(936\) 0 0
\(937\) 34.7846 1.13636 0.568182 0.822903i \(-0.307647\pi\)
0.568182 + 0.822903i \(0.307647\pi\)
\(938\) 22.1436 0.723014
\(939\) 0 0
\(940\) −6.00000 −0.195698
\(941\) −31.1769 −1.01634 −0.508169 0.861257i \(-0.669678\pi\)
−0.508169 + 0.861257i \(0.669678\pi\)
\(942\) 0 0
\(943\) 4.39230 0.143033
\(944\) 45.6218 1.48486
\(945\) 0 0
\(946\) −1.60770 −0.0522707
\(947\) −40.6410 −1.32066 −0.660328 0.750978i \(-0.729583\pi\)
−0.660328 + 0.750978i \(0.729583\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) −10.7321 −0.348194
\(951\) 0 0
\(952\) −12.0000 −0.388922
\(953\) −0.928203 −0.0300675 −0.0150337 0.999887i \(-0.504786\pi\)
−0.0150337 + 0.999887i \(0.504786\pi\)
\(954\) 0 0
\(955\) 5.07180 0.164119
\(956\) 14.1962 0.459136
\(957\) 0 0
\(958\) −61.7654 −1.99555
\(959\) −1.85641 −0.0599465
\(960\) 0 0
\(961\) 72.9615 2.35360
\(962\) −6.92820 −0.223374
\(963\) 0 0
\(964\) −2.39230 −0.0770510
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) −50.3923 −1.62051 −0.810254 0.586079i \(-0.800671\pi\)
−0.810254 + 0.586079i \(0.800671\pi\)
\(968\) −19.7321 −0.634212
\(969\) 0 0
\(970\) 3.46410 0.111226
\(971\) −18.9282 −0.607435 −0.303717 0.952762i \(-0.598228\pi\)
−0.303717 + 0.952762i \(0.598228\pi\)
\(972\) 0 0
\(973\) 24.7846 0.794558
\(974\) −45.7128 −1.46473
\(975\) 0 0
\(976\) 41.9615 1.34316
\(977\) 15.7128 0.502697 0.251349 0.967897i \(-0.419126\pi\)
0.251349 + 0.967897i \(0.419126\pi\)
\(978\) 0 0
\(979\) 61.1769 1.95522
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) 4.39230 0.140164
\(983\) 34.3923 1.09694 0.548472 0.836169i \(-0.315210\pi\)
0.548472 + 0.836169i \(0.315210\pi\)
\(984\) 0 0
\(985\) 12.9282 0.411927
\(986\) 15.2154 0.484557
\(987\) 0 0
\(988\) −6.19615 −0.197126
\(989\) 0.248711 0.00790856
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −52.9808 −1.68214
\(993\) 0 0
\(994\) −16.3923 −0.519932
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) 33.6077 1.06437 0.532183 0.846629i \(-0.321372\pi\)
0.532183 + 0.846629i \(0.321372\pi\)
\(998\) −67.5167 −2.13720
\(999\) 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 585.2.a.k.1.2 2
3.2 odd 2 65.2.a.c.1.1 2
4.3 odd 2 9360.2.a.cm.1.1 2
5.2 odd 4 2925.2.c.v.2224.4 4
5.3 odd 4 2925.2.c.v.2224.1 4
5.4 even 2 2925.2.a.z.1.1 2
12.11 even 2 1040.2.a.h.1.1 2
13.12 even 2 7605.2.a.be.1.1 2
15.2 even 4 325.2.b.e.274.1 4
15.8 even 4 325.2.b.e.274.4 4
15.14 odd 2 325.2.a.g.1.2 2
21.20 even 2 3185.2.a.k.1.1 2
24.5 odd 2 4160.2.a.y.1.1 2
24.11 even 2 4160.2.a.bj.1.2 2
33.32 even 2 7865.2.a.h.1.2 2
39.2 even 12 845.2.m.a.316.1 4
39.5 even 4 845.2.c.e.506.4 4
39.8 even 4 845.2.c.e.506.2 4
39.11 even 12 845.2.m.c.316.1 4
39.17 odd 6 845.2.e.f.146.1 4
39.20 even 12 845.2.m.a.361.1 4
39.23 odd 6 845.2.e.f.191.1 4
39.29 odd 6 845.2.e.e.191.2 4
39.32 even 12 845.2.m.c.361.1 4
39.35 odd 6 845.2.e.e.146.2 4
39.38 odd 2 845.2.a.d.1.2 2
60.59 even 2 5200.2.a.ca.1.2 2
195.194 odd 2 4225.2.a.w.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.c.1.1 2 3.2 odd 2
325.2.a.g.1.2 2 15.14 odd 2
325.2.b.e.274.1 4 15.2 even 4
325.2.b.e.274.4 4 15.8 even 4
585.2.a.k.1.2 2 1.1 even 1 trivial
845.2.a.d.1.2 2 39.38 odd 2
845.2.c.e.506.2 4 39.8 even 4
845.2.c.e.506.4 4 39.5 even 4
845.2.e.e.146.2 4 39.35 odd 6
845.2.e.e.191.2 4 39.29 odd 6
845.2.e.f.146.1 4 39.17 odd 6
845.2.e.f.191.1 4 39.23 odd 6
845.2.m.a.316.1 4 39.2 even 12
845.2.m.a.361.1 4 39.20 even 12
845.2.m.c.316.1 4 39.11 even 12
845.2.m.c.361.1 4 39.32 even 12
1040.2.a.h.1.1 2 12.11 even 2
2925.2.a.z.1.1 2 5.4 even 2
2925.2.c.v.2224.1 4 5.3 odd 4
2925.2.c.v.2224.4 4 5.2 odd 4
3185.2.a.k.1.1 2 21.20 even 2
4160.2.a.y.1.1 2 24.5 odd 2
4160.2.a.bj.1.2 2 24.11 even 2
4225.2.a.w.1.1 2 195.194 odd 2
5200.2.a.ca.1.2 2 60.59 even 2
7605.2.a.be.1.1 2 13.12 even 2
7865.2.a.h.1.2 2 33.32 even 2
9360.2.a.cm.1.1 2 4.3 odd 2