Properties

Label 666.2.a.j.1.2
Level $666$
Weight $2$
Character 666.1
Self dual yes
Analytic conductor $5.318$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 666.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.30278 q^{5} -2.60555 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.30278 q^{5} -2.60555 q^{7} +1.00000 q^{8} +2.30278 q^{10} +2.30278 q^{11} +1.30278 q^{13} -2.60555 q^{14} +1.00000 q^{16} +6.00000 q^{17} +2.00000 q^{19} +2.30278 q^{20} +2.30278 q^{22} -3.90833 q^{23} +0.302776 q^{25} +1.30278 q^{26} -2.60555 q^{28} +3.90833 q^{29} -0.302776 q^{31} +1.00000 q^{32} +6.00000 q^{34} -6.00000 q^{35} +1.00000 q^{37} +2.00000 q^{38} +2.30278 q^{40} -9.90833 q^{41} +0.605551 q^{43} +2.30278 q^{44} -3.90833 q^{46} -4.60555 q^{47} -0.211103 q^{49} +0.302776 q^{50} +1.30278 q^{52} +6.00000 q^{53} +5.30278 q^{55} -2.60555 q^{56} +3.90833 q^{58} -10.6056 q^{59} +7.51388 q^{61} -0.302776 q^{62} +1.00000 q^{64} +3.00000 q^{65} -3.51388 q^{67} +6.00000 q^{68} -6.00000 q^{70} -6.00000 q^{71} -12.3028 q^{73} +1.00000 q^{74} +2.00000 q^{76} -6.00000 q^{77} +9.11943 q^{79} +2.30278 q^{80} -9.90833 q^{82} -2.78890 q^{83} +13.8167 q^{85} +0.605551 q^{86} +2.30278 q^{88} +9.21110 q^{89} -3.39445 q^{91} -3.90833 q^{92} -4.60555 q^{94} +4.60555 q^{95} -16.4222 q^{97} -0.211103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8} + q^{10} + q^{11} - q^{13} + 2 q^{14} + 2 q^{16} + 12 q^{17} + 4 q^{19} + q^{20} + q^{22} + 3 q^{23} - 3 q^{25} - q^{26} + 2 q^{28} - 3 q^{29} + 3 q^{31} + 2 q^{32} + 12 q^{34} - 12 q^{35} + 2 q^{37} + 4 q^{38} + q^{40} - 9 q^{41} - 6 q^{43} + q^{44} + 3 q^{46} - 2 q^{47} + 14 q^{49} - 3 q^{50} - q^{52} + 12 q^{53} + 7 q^{55} + 2 q^{56} - 3 q^{58} - 14 q^{59} - 3 q^{61} + 3 q^{62} + 2 q^{64} + 6 q^{65} + 11 q^{67} + 12 q^{68} - 12 q^{70} - 12 q^{71} - 21 q^{73} + 2 q^{74} + 4 q^{76} - 12 q^{77} - 7 q^{79} + q^{80} - 9 q^{82} - 20 q^{83} + 6 q^{85} - 6 q^{86} + q^{88} + 4 q^{89} - 14 q^{91} + 3 q^{92} - 2 q^{94} + 2 q^{95} - 4 q^{97} + 14 q^{98}+O(q^{100}) \) 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
\(5\) 2.30278 1.02983 0.514916 0.857240i \(-0.327823\pi\)
0.514916 + 0.857240i \(0.327823\pi\)
\(6\) 0 0
\(7\) −2.60555 −0.984806 −0.492403 0.870367i \(-0.663881\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.30278 0.728202
\(11\) 2.30278 0.694313 0.347156 0.937807i \(-0.387147\pi\)
0.347156 + 0.937807i \(0.387147\pi\)
\(12\) 0 0
\(13\) 1.30278 0.361325 0.180662 0.983545i \(-0.442176\pi\)
0.180662 + 0.983545i \(0.442176\pi\)
\(14\) −2.60555 −0.696363
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 2.30278 0.514916
\(21\) 0 0
\(22\) 2.30278 0.490953
\(23\) −3.90833 −0.814942 −0.407471 0.913218i \(-0.633589\pi\)
−0.407471 + 0.913218i \(0.633589\pi\)
\(24\) 0 0
\(25\) 0.302776 0.0605551
\(26\) 1.30278 0.255495
\(27\) 0 0
\(28\) −2.60555 −0.492403
\(29\) 3.90833 0.725758 0.362879 0.931836i \(-0.381794\pi\)
0.362879 + 0.931836i \(0.381794\pi\)
\(30\) 0 0
\(31\) −0.302776 −0.0543801 −0.0271901 0.999630i \(-0.508656\pi\)
−0.0271901 + 0.999630i \(0.508656\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 2.30278 0.364101
\(41\) −9.90833 −1.54742 −0.773710 0.633540i \(-0.781601\pi\)
−0.773710 + 0.633540i \(0.781601\pi\)
\(42\) 0 0
\(43\) 0.605551 0.0923457 0.0461729 0.998933i \(-0.485297\pi\)
0.0461729 + 0.998933i \(0.485297\pi\)
\(44\) 2.30278 0.347156
\(45\) 0 0
\(46\) −3.90833 −0.576251
\(47\) −4.60555 −0.671789 −0.335894 0.941900i \(-0.609039\pi\)
−0.335894 + 0.941900i \(0.609039\pi\)
\(48\) 0 0
\(49\) −0.211103 −0.0301575
\(50\) 0.302776 0.0428189
\(51\) 0 0
\(52\) 1.30278 0.180662
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 5.30278 0.715026
\(56\) −2.60555 −0.348181
\(57\) 0 0
\(58\) 3.90833 0.513188
\(59\) −10.6056 −1.38073 −0.690363 0.723464i \(-0.742549\pi\)
−0.690363 + 0.723464i \(0.742549\pi\)
\(60\) 0 0
\(61\) 7.51388 0.962054 0.481027 0.876706i \(-0.340264\pi\)
0.481027 + 0.876706i \(0.340264\pi\)
\(62\) −0.302776 −0.0384525
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −3.51388 −0.429289 −0.214644 0.976692i \(-0.568859\pi\)
−0.214644 + 0.976692i \(0.568859\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) −6.00000 −0.717137
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −12.3028 −1.43993 −0.719965 0.694010i \(-0.755842\pi\)
−0.719965 + 0.694010i \(0.755842\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 9.11943 1.02602 0.513008 0.858384i \(-0.328531\pi\)
0.513008 + 0.858384i \(0.328531\pi\)
\(80\) 2.30278 0.257458
\(81\) 0 0
\(82\) −9.90833 −1.09419
\(83\) −2.78890 −0.306121 −0.153061 0.988217i \(-0.548913\pi\)
−0.153061 + 0.988217i \(0.548913\pi\)
\(84\) 0 0
\(85\) 13.8167 1.49863
\(86\) 0.605551 0.0652983
\(87\) 0 0
\(88\) 2.30278 0.245477
\(89\) 9.21110 0.976375 0.488187 0.872739i \(-0.337658\pi\)
0.488187 + 0.872739i \(0.337658\pi\)
\(90\) 0 0
\(91\) −3.39445 −0.355835
\(92\) −3.90833 −0.407471
\(93\) 0 0
\(94\) −4.60555 −0.475026
\(95\) 4.60555 0.472520
\(96\) 0 0
\(97\) −16.4222 −1.66742 −0.833711 0.552201i \(-0.813788\pi\)
−0.833711 + 0.552201i \(0.813788\pi\)
\(98\) −0.211103 −0.0213246
\(99\) 0 0
\(100\) 0.302776 0.0302776
\(101\) 12.4222 1.23606 0.618028 0.786156i \(-0.287932\pi\)
0.618028 + 0.786156i \(0.287932\pi\)
\(102\) 0 0
\(103\) −0.302776 −0.0298334 −0.0149167 0.999889i \(-0.504748\pi\)
−0.0149167 + 0.999889i \(0.504748\pi\)
\(104\) 1.30278 0.127748
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −0.697224 −0.0674032 −0.0337016 0.999432i \(-0.510730\pi\)
−0.0337016 + 0.999432i \(0.510730\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 5.30278 0.505600
\(111\) 0 0
\(112\) −2.60555 −0.246201
\(113\) 3.21110 0.302075 0.151038 0.988528i \(-0.451739\pi\)
0.151038 + 0.988528i \(0.451739\pi\)
\(114\) 0 0
\(115\) −9.00000 −0.839254
\(116\) 3.90833 0.362879
\(117\) 0 0
\(118\) −10.6056 −0.976320
\(119\) −15.6333 −1.43310
\(120\) 0 0
\(121\) −5.69722 −0.517929
\(122\) 7.51388 0.680275
\(123\) 0 0
\(124\) −0.302776 −0.0271901
\(125\) −10.8167 −0.967471
\(126\) 0 0
\(127\) −19.2111 −1.70471 −0.852355 0.522964i \(-0.824826\pi\)
−0.852355 + 0.522964i \(0.824826\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 3.00000 0.263117
\(131\) −10.6056 −0.926611 −0.463306 0.886199i \(-0.653337\pi\)
−0.463306 + 0.886199i \(0.653337\pi\)
\(132\) 0 0
\(133\) −5.21110 −0.451860
\(134\) −3.51388 −0.303553
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −0.908327 −0.0776036 −0.0388018 0.999247i \(-0.512354\pi\)
−0.0388018 + 0.999247i \(0.512354\pi\)
\(138\) 0 0
\(139\) −1.90833 −0.161862 −0.0809311 0.996720i \(-0.525789\pi\)
−0.0809311 + 0.996720i \(0.525789\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 3.00000 0.250873
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) −12.3028 −1.01818
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) −19.8167 −1.62344 −0.811722 0.584044i \(-0.801469\pi\)
−0.811722 + 0.584044i \(0.801469\pi\)
\(150\) 0 0
\(151\) −20.6056 −1.67686 −0.838428 0.545012i \(-0.816525\pi\)
−0.838428 + 0.545012i \(0.816525\pi\)
\(152\) 2.00000 0.162221
\(153\) 0 0
\(154\) −6.00000 −0.483494
\(155\) −0.697224 −0.0560024
\(156\) 0 0
\(157\) −7.21110 −0.575509 −0.287754 0.957704i \(-0.592909\pi\)
−0.287754 + 0.957704i \(0.592909\pi\)
\(158\) 9.11943 0.725503
\(159\) 0 0
\(160\) 2.30278 0.182050
\(161\) 10.1833 0.802560
\(162\) 0 0
\(163\) 8.42221 0.659678 0.329839 0.944037i \(-0.393006\pi\)
0.329839 + 0.944037i \(0.393006\pi\)
\(164\) −9.90833 −0.773710
\(165\) 0 0
\(166\) −2.78890 −0.216460
\(167\) 5.51388 0.426677 0.213338 0.976978i \(-0.431566\pi\)
0.213338 + 0.976978i \(0.431566\pi\)
\(168\) 0 0
\(169\) −11.3028 −0.869444
\(170\) 13.8167 1.05969
\(171\) 0 0
\(172\) 0.605551 0.0461729
\(173\) 8.78890 0.668207 0.334104 0.942536i \(-0.391566\pi\)
0.334104 + 0.942536i \(0.391566\pi\)
\(174\) 0 0
\(175\) −0.788897 −0.0596350
\(176\) 2.30278 0.173578
\(177\) 0 0
\(178\) 9.21110 0.690401
\(179\) 13.8167 1.03271 0.516353 0.856376i \(-0.327289\pi\)
0.516353 + 0.856376i \(0.327289\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) −3.39445 −0.251613
\(183\) 0 0
\(184\) −3.90833 −0.288126
\(185\) 2.30278 0.169303
\(186\) 0 0
\(187\) 13.8167 1.01037
\(188\) −4.60555 −0.335894
\(189\) 0 0
\(190\) 4.60555 0.334122
\(191\) 5.51388 0.398970 0.199485 0.979901i \(-0.436073\pi\)
0.199485 + 0.979901i \(0.436073\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −16.4222 −1.17905
\(195\) 0 0
\(196\) −0.211103 −0.0150788
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 26.4222 1.87302 0.936510 0.350640i \(-0.114036\pi\)
0.936510 + 0.350640i \(0.114036\pi\)
\(200\) 0.302776 0.0214095
\(201\) 0 0
\(202\) 12.4222 0.874023
\(203\) −10.1833 −0.714731
\(204\) 0 0
\(205\) −22.8167 −1.59358
\(206\) −0.302776 −0.0210954
\(207\) 0 0
\(208\) 1.30278 0.0903312
\(209\) 4.60555 0.318573
\(210\) 0 0
\(211\) 10.3028 0.709272 0.354636 0.935004i \(-0.384605\pi\)
0.354636 + 0.935004i \(0.384605\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −0.697224 −0.0476613
\(215\) 1.39445 0.0951006
\(216\) 0 0
\(217\) 0.788897 0.0535538
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 5.30278 0.357513
\(221\) 7.81665 0.525805
\(222\) 0 0
\(223\) −5.81665 −0.389512 −0.194756 0.980852i \(-0.562391\pi\)
−0.194756 + 0.980852i \(0.562391\pi\)
\(224\) −2.60555 −0.174091
\(225\) 0 0
\(226\) 3.21110 0.213599
\(227\) −13.8167 −0.917044 −0.458522 0.888683i \(-0.651621\pi\)
−0.458522 + 0.888683i \(0.651621\pi\)
\(228\) 0 0
\(229\) 24.6056 1.62598 0.812990 0.582277i \(-0.197838\pi\)
0.812990 + 0.582277i \(0.197838\pi\)
\(230\) −9.00000 −0.593442
\(231\) 0 0
\(232\) 3.90833 0.256594
\(233\) −8.51388 −0.557763 −0.278881 0.960326i \(-0.589964\pi\)
−0.278881 + 0.960326i \(0.589964\pi\)
\(234\) 0 0
\(235\) −10.6056 −0.691830
\(236\) −10.6056 −0.690363
\(237\) 0 0
\(238\) −15.6333 −1.01336
\(239\) 17.5139 1.13288 0.566439 0.824103i \(-0.308321\pi\)
0.566439 + 0.824103i \(0.308321\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −5.69722 −0.366231
\(243\) 0 0
\(244\) 7.51388 0.481027
\(245\) −0.486122 −0.0310572
\(246\) 0 0
\(247\) 2.60555 0.165787
\(248\) −0.302776 −0.0192263
\(249\) 0 0
\(250\) −10.8167 −0.684105
\(251\) 21.2111 1.33883 0.669416 0.742887i \(-0.266544\pi\)
0.669416 + 0.742887i \(0.266544\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) −19.2111 −1.20541
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −3.21110 −0.200303 −0.100152 0.994972i \(-0.531933\pi\)
−0.100152 + 0.994972i \(0.531933\pi\)
\(258\) 0 0
\(259\) −2.60555 −0.161901
\(260\) 3.00000 0.186052
\(261\) 0 0
\(262\) −10.6056 −0.655213
\(263\) −13.8167 −0.851971 −0.425986 0.904730i \(-0.640073\pi\)
−0.425986 + 0.904730i \(0.640073\pi\)
\(264\) 0 0
\(265\) 13.8167 0.848750
\(266\) −5.21110 −0.319513
\(267\) 0 0
\(268\) −3.51388 −0.214644
\(269\) 21.2111 1.29326 0.646632 0.762802i \(-0.276177\pi\)
0.646632 + 0.762802i \(0.276177\pi\)
\(270\) 0 0
\(271\) −22.4222 −1.36205 −0.681026 0.732259i \(-0.738466\pi\)
−0.681026 + 0.732259i \(0.738466\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −0.908327 −0.0548740
\(275\) 0.697224 0.0420442
\(276\) 0 0
\(277\) 0.119429 0.00717582 0.00358791 0.999994i \(-0.498858\pi\)
0.00358791 + 0.999994i \(0.498858\pi\)
\(278\) −1.90833 −0.114454
\(279\) 0 0
\(280\) −6.00000 −0.358569
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) 24.6056 1.46265 0.731324 0.682030i \(-0.238903\pi\)
0.731324 + 0.682030i \(0.238903\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) 25.8167 1.52391
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 9.00000 0.528498
\(291\) 0 0
\(292\) −12.3028 −0.719965
\(293\) −11.0278 −0.644248 −0.322124 0.946697i \(-0.604397\pi\)
−0.322124 + 0.946697i \(0.604397\pi\)
\(294\) 0 0
\(295\) −24.4222 −1.42192
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −19.8167 −1.14795
\(299\) −5.09167 −0.294459
\(300\) 0 0
\(301\) −1.57779 −0.0909426
\(302\) −20.6056 −1.18572
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 17.3028 0.990754
\(306\) 0 0
\(307\) 17.9083 1.02208 0.511041 0.859556i \(-0.329260\pi\)
0.511041 + 0.859556i \(0.329260\pi\)
\(308\) −6.00000 −0.341882
\(309\) 0 0
\(310\) −0.697224 −0.0395997
\(311\) −15.9083 −0.902078 −0.451039 0.892504i \(-0.648947\pi\)
−0.451039 + 0.892504i \(0.648947\pi\)
\(312\) 0 0
\(313\) −9.02776 −0.510279 −0.255139 0.966904i \(-0.582121\pi\)
−0.255139 + 0.966904i \(0.582121\pi\)
\(314\) −7.21110 −0.406946
\(315\) 0 0
\(316\) 9.11943 0.513008
\(317\) −9.21110 −0.517347 −0.258674 0.965965i \(-0.583285\pi\)
−0.258674 + 0.965965i \(0.583285\pi\)
\(318\) 0 0
\(319\) 9.00000 0.503903
\(320\) 2.30278 0.128729
\(321\) 0 0
\(322\) 10.1833 0.567496
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 0.394449 0.0218801
\(326\) 8.42221 0.466463
\(327\) 0 0
\(328\) −9.90833 −0.547096
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −13.2111 −0.726148 −0.363074 0.931760i \(-0.618273\pi\)
−0.363074 + 0.931760i \(0.618273\pi\)
\(332\) −2.78890 −0.153061
\(333\) 0 0
\(334\) 5.51388 0.301706
\(335\) −8.09167 −0.442095
\(336\) 0 0
\(337\) 6.11943 0.333347 0.166673 0.986012i \(-0.446697\pi\)
0.166673 + 0.986012i \(0.446697\pi\)
\(338\) −11.3028 −0.614790
\(339\) 0 0
\(340\) 13.8167 0.749313
\(341\) −0.697224 −0.0377568
\(342\) 0 0
\(343\) 18.7889 1.01451
\(344\) 0.605551 0.0326491
\(345\) 0 0
\(346\) 8.78890 0.472494
\(347\) −10.1833 −0.546671 −0.273335 0.961919i \(-0.588127\pi\)
−0.273335 + 0.961919i \(0.588127\pi\)
\(348\) 0 0
\(349\) 28.2389 1.51159 0.755796 0.654807i \(-0.227250\pi\)
0.755796 + 0.654807i \(0.227250\pi\)
\(350\) −0.788897 −0.0421683
\(351\) 0 0
\(352\) 2.30278 0.122738
\(353\) −10.1833 −0.542005 −0.271002 0.962579i \(-0.587355\pi\)
−0.271002 + 0.962579i \(0.587355\pi\)
\(354\) 0 0
\(355\) −13.8167 −0.733312
\(356\) 9.21110 0.488187
\(357\) 0 0
\(358\) 13.8167 0.730233
\(359\) −3.21110 −0.169476 −0.0847378 0.996403i \(-0.527005\pi\)
−0.0847378 + 0.996403i \(0.527005\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 20.0000 1.05118
\(363\) 0 0
\(364\) −3.39445 −0.177917
\(365\) −28.3305 −1.48289
\(366\) 0 0
\(367\) 3.81665 0.199228 0.0996139 0.995026i \(-0.468239\pi\)
0.0996139 + 0.995026i \(0.468239\pi\)
\(368\) −3.90833 −0.203736
\(369\) 0 0
\(370\) 2.30278 0.119716
\(371\) −15.6333 −0.811641
\(372\) 0 0
\(373\) −17.8167 −0.922511 −0.461256 0.887267i \(-0.652601\pi\)
−0.461256 + 0.887267i \(0.652601\pi\)
\(374\) 13.8167 0.714442
\(375\) 0 0
\(376\) −4.60555 −0.237513
\(377\) 5.09167 0.262235
\(378\) 0 0
\(379\) 24.3305 1.24978 0.624888 0.780715i \(-0.285145\pi\)
0.624888 + 0.780715i \(0.285145\pi\)
\(380\) 4.60555 0.236260
\(381\) 0 0
\(382\) 5.51388 0.282115
\(383\) 36.8444 1.88266 0.941331 0.337486i \(-0.109576\pi\)
0.941331 + 0.337486i \(0.109576\pi\)
\(384\) 0 0
\(385\) −13.8167 −0.704162
\(386\) −4.00000 −0.203595
\(387\) 0 0
\(388\) −16.4222 −0.833711
\(389\) 37.1194 1.88203 0.941015 0.338365i \(-0.109874\pi\)
0.941015 + 0.338365i \(0.109874\pi\)
\(390\) 0 0
\(391\) −23.4500 −1.18592
\(392\) −0.211103 −0.0106623
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 21.0000 1.05662
\(396\) 0 0
\(397\) 6.18335 0.310333 0.155167 0.987888i \(-0.450409\pi\)
0.155167 + 0.987888i \(0.450409\pi\)
\(398\) 26.4222 1.32443
\(399\) 0 0
\(400\) 0.302776 0.0151388
\(401\) 7.81665 0.390345 0.195173 0.980769i \(-0.437473\pi\)
0.195173 + 0.980769i \(0.437473\pi\)
\(402\) 0 0
\(403\) −0.394449 −0.0196489
\(404\) 12.4222 0.618028
\(405\) 0 0
\(406\) −10.1833 −0.505391
\(407\) 2.30278 0.114144
\(408\) 0 0
\(409\) 31.0278 1.53422 0.767112 0.641513i \(-0.221693\pi\)
0.767112 + 0.641513i \(0.221693\pi\)
\(410\) −22.8167 −1.12683
\(411\) 0 0
\(412\) −0.302776 −0.0149167
\(413\) 27.6333 1.35975
\(414\) 0 0
\(415\) −6.42221 −0.315254
\(416\) 1.30278 0.0638738
\(417\) 0 0
\(418\) 4.60555 0.225265
\(419\) −36.1472 −1.76591 −0.882953 0.469462i \(-0.844448\pi\)
−0.882953 + 0.469462i \(0.844448\pi\)
\(420\) 0 0
\(421\) −3.72498 −0.181544 −0.0907722 0.995872i \(-0.528934\pi\)
−0.0907722 + 0.995872i \(0.528934\pi\)
\(422\) 10.3028 0.501531
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 1.81665 0.0881207
\(426\) 0 0
\(427\) −19.5778 −0.947436
\(428\) −0.697224 −0.0337016
\(429\) 0 0
\(430\) 1.39445 0.0672463
\(431\) −9.21110 −0.443683 −0.221842 0.975083i \(-0.571207\pi\)
−0.221842 + 0.975083i \(0.571207\pi\)
\(432\) 0 0
\(433\) 34.9361 1.67892 0.839461 0.543421i \(-0.182871\pi\)
0.839461 + 0.543421i \(0.182871\pi\)
\(434\) 0.788897 0.0378683
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −7.81665 −0.373921
\(438\) 0 0
\(439\) 30.3305 1.44760 0.723799 0.690011i \(-0.242394\pi\)
0.723799 + 0.690011i \(0.242394\pi\)
\(440\) 5.30278 0.252800
\(441\) 0 0
\(442\) 7.81665 0.371800
\(443\) −32.7250 −1.55481 −0.777405 0.629000i \(-0.783465\pi\)
−0.777405 + 0.629000i \(0.783465\pi\)
\(444\) 0 0
\(445\) 21.2111 1.00550
\(446\) −5.81665 −0.275427
\(447\) 0 0
\(448\) −2.60555 −0.123101
\(449\) 15.2111 0.717856 0.358928 0.933365i \(-0.383142\pi\)
0.358928 + 0.933365i \(0.383142\pi\)
\(450\) 0 0
\(451\) −22.8167 −1.07439
\(452\) 3.21110 0.151038
\(453\) 0 0
\(454\) −13.8167 −0.648448
\(455\) −7.81665 −0.366450
\(456\) 0 0
\(457\) −2.60555 −0.121883 −0.0609413 0.998141i \(-0.519410\pi\)
−0.0609413 + 0.998141i \(0.519410\pi\)
\(458\) 24.6056 1.14974
\(459\) 0 0
\(460\) −9.00000 −0.419627
\(461\) −12.4222 −0.578560 −0.289280 0.957245i \(-0.593416\pi\)
−0.289280 + 0.957245i \(0.593416\pi\)
\(462\) 0 0
\(463\) 26.6972 1.24073 0.620363 0.784315i \(-0.286985\pi\)
0.620363 + 0.784315i \(0.286985\pi\)
\(464\) 3.90833 0.181440
\(465\) 0 0
\(466\) −8.51388 −0.394398
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 9.15559 0.422766
\(470\) −10.6056 −0.489198
\(471\) 0 0
\(472\) −10.6056 −0.488160
\(473\) 1.39445 0.0641168
\(474\) 0 0
\(475\) 0.605551 0.0277846
\(476\) −15.6333 −0.716551
\(477\) 0 0
\(478\) 17.5139 0.801066
\(479\) 13.1194 0.599442 0.299721 0.954027i \(-0.403106\pi\)
0.299721 + 0.954027i \(0.403106\pi\)
\(480\) 0 0
\(481\) 1.30278 0.0594015
\(482\) 8.00000 0.364390
\(483\) 0 0
\(484\) −5.69722 −0.258965
\(485\) −37.8167 −1.71717
\(486\) 0 0
\(487\) −37.2111 −1.68620 −0.843098 0.537760i \(-0.819271\pi\)
−0.843098 + 0.537760i \(0.819271\pi\)
\(488\) 7.51388 0.340137
\(489\) 0 0
\(490\) −0.486122 −0.0219607
\(491\) −17.7250 −0.799917 −0.399959 0.916533i \(-0.630975\pi\)
−0.399959 + 0.916533i \(0.630975\pi\)
\(492\) 0 0
\(493\) 23.4500 1.05613
\(494\) 2.60555 0.117229
\(495\) 0 0
\(496\) −0.302776 −0.0135950
\(497\) 15.6333 0.701250
\(498\) 0 0
\(499\) −42.2389 −1.89087 −0.945436 0.325809i \(-0.894363\pi\)
−0.945436 + 0.325809i \(0.894363\pi\)
\(500\) −10.8167 −0.483735
\(501\) 0 0
\(502\) 21.2111 0.946698
\(503\) 6.48612 0.289202 0.144601 0.989490i \(-0.453810\pi\)
0.144601 + 0.989490i \(0.453810\pi\)
\(504\) 0 0
\(505\) 28.6056 1.27293
\(506\) −9.00000 −0.400099
\(507\) 0 0
\(508\) −19.2111 −0.852355
\(509\) 4.18335 0.185424 0.0927118 0.995693i \(-0.470446\pi\)
0.0927118 + 0.995693i \(0.470446\pi\)
\(510\) 0 0
\(511\) 32.0555 1.41805
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −3.21110 −0.141636
\(515\) −0.697224 −0.0307234
\(516\) 0 0
\(517\) −10.6056 −0.466432
\(518\) −2.60555 −0.114481
\(519\) 0 0
\(520\) 3.00000 0.131559
\(521\) −33.6333 −1.47350 −0.736751 0.676164i \(-0.763641\pi\)
−0.736751 + 0.676164i \(0.763641\pi\)
\(522\) 0 0
\(523\) −18.2389 −0.797530 −0.398765 0.917053i \(-0.630561\pi\)
−0.398765 + 0.917053i \(0.630561\pi\)
\(524\) −10.6056 −0.463306
\(525\) 0 0
\(526\) −13.8167 −0.602435
\(527\) −1.81665 −0.0791347
\(528\) 0 0
\(529\) −7.72498 −0.335869
\(530\) 13.8167 0.600157
\(531\) 0 0
\(532\) −5.21110 −0.225930
\(533\) −12.9083 −0.559122
\(534\) 0 0
\(535\) −1.60555 −0.0694140
\(536\) −3.51388 −0.151776
\(537\) 0 0
\(538\) 21.2111 0.914476
\(539\) −0.486122 −0.0209387
\(540\) 0 0
\(541\) 25.9361 1.11508 0.557540 0.830150i \(-0.311745\pi\)
0.557540 + 0.830150i \(0.311745\pi\)
\(542\) −22.4222 −0.963116
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) 4.60555 0.197280
\(546\) 0 0
\(547\) −20.6056 −0.881030 −0.440515 0.897745i \(-0.645204\pi\)
−0.440515 + 0.897745i \(0.645204\pi\)
\(548\) −0.908327 −0.0388018
\(549\) 0 0
\(550\) 0.697224 0.0297297
\(551\) 7.81665 0.333001
\(552\) 0 0
\(553\) −23.7611 −1.01043
\(554\) 0.119429 0.00507407
\(555\) 0 0
\(556\) −1.90833 −0.0809311
\(557\) −11.5139 −0.487859 −0.243929 0.969793i \(-0.578436\pi\)
−0.243929 + 0.969793i \(0.578436\pi\)
\(558\) 0 0
\(559\) 0.788897 0.0333668
\(560\) −6.00000 −0.253546
\(561\) 0 0
\(562\) 12.0000 0.506189
\(563\) 28.0555 1.18240 0.591199 0.806525i \(-0.298655\pi\)
0.591199 + 0.806525i \(0.298655\pi\)
\(564\) 0 0
\(565\) 7.39445 0.311087
\(566\) 24.6056 1.03425
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −18.4222 −0.772299 −0.386150 0.922436i \(-0.626195\pi\)
−0.386150 + 0.922436i \(0.626195\pi\)
\(570\) 0 0
\(571\) −16.6972 −0.698757 −0.349379 0.936982i \(-0.613607\pi\)
−0.349379 + 0.936982i \(0.613607\pi\)
\(572\) 3.00000 0.125436
\(573\) 0 0
\(574\) 25.8167 1.07757
\(575\) −1.18335 −0.0493489
\(576\) 0 0
\(577\) 22.2389 0.925816 0.462908 0.886406i \(-0.346806\pi\)
0.462908 + 0.886406i \(0.346806\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) 9.00000 0.373705
\(581\) 7.26662 0.301470
\(582\) 0 0
\(583\) 13.8167 0.572227
\(584\) −12.3028 −0.509092
\(585\) 0 0
\(586\) −11.0278 −0.455552
\(587\) 45.6333 1.88349 0.941744 0.336330i \(-0.109186\pi\)
0.941744 + 0.336330i \(0.109186\pi\)
\(588\) 0 0
\(589\) −0.605551 −0.0249513
\(590\) −24.4222 −1.00545
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −18.4861 −0.759134 −0.379567 0.925164i \(-0.623927\pi\)
−0.379567 + 0.925164i \(0.623927\pi\)
\(594\) 0 0
\(595\) −36.0000 −1.47586
\(596\) −19.8167 −0.811722
\(597\) 0 0
\(598\) −5.09167 −0.208214
\(599\) 20.7889 0.849411 0.424706 0.905331i \(-0.360378\pi\)
0.424706 + 0.905331i \(0.360378\pi\)
\(600\) 0 0
\(601\) −24.3028 −0.991331 −0.495665 0.868514i \(-0.665076\pi\)
−0.495665 + 0.868514i \(0.665076\pi\)
\(602\) −1.57779 −0.0643061
\(603\) 0 0
\(604\) −20.6056 −0.838428
\(605\) −13.1194 −0.533381
\(606\) 0 0
\(607\) −13.4861 −0.547385 −0.273692 0.961817i \(-0.588245\pi\)
−0.273692 + 0.961817i \(0.588245\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 17.3028 0.700569
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) −29.8167 −1.20428 −0.602142 0.798389i \(-0.705686\pi\)
−0.602142 + 0.798389i \(0.705686\pi\)
\(614\) 17.9083 0.722721
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) 42.5694 1.71378 0.856890 0.515500i \(-0.172394\pi\)
0.856890 + 0.515500i \(0.172394\pi\)
\(618\) 0 0
\(619\) −6.30278 −0.253330 −0.126665 0.991946i \(-0.540427\pi\)
−0.126665 + 0.991946i \(0.540427\pi\)
\(620\) −0.697224 −0.0280012
\(621\) 0 0
\(622\) −15.9083 −0.637866
\(623\) −24.0000 −0.961540
\(624\) 0 0
\(625\) −26.4222 −1.05689
\(626\) −9.02776 −0.360822
\(627\) 0 0
\(628\) −7.21110 −0.287754
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 14.6972 0.585087 0.292544 0.956252i \(-0.405498\pi\)
0.292544 + 0.956252i \(0.405498\pi\)
\(632\) 9.11943 0.362751
\(633\) 0 0
\(634\) −9.21110 −0.365820
\(635\) −44.2389 −1.75557
\(636\) 0 0
\(637\) −0.275019 −0.0108967
\(638\) 9.00000 0.356313
\(639\) 0 0
\(640\) 2.30278 0.0910252
\(641\) 20.5139 0.810249 0.405125 0.914261i \(-0.367228\pi\)
0.405125 + 0.914261i \(0.367228\pi\)
\(642\) 0 0
\(643\) −8.18335 −0.322720 −0.161360 0.986896i \(-0.551588\pi\)
−0.161360 + 0.986896i \(0.551588\pi\)
\(644\) 10.1833 0.401280
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −20.9361 −0.823082 −0.411541 0.911391i \(-0.635009\pi\)
−0.411541 + 0.911391i \(0.635009\pi\)
\(648\) 0 0
\(649\) −24.4222 −0.958655
\(650\) 0.394449 0.0154716
\(651\) 0 0
\(652\) 8.42221 0.329839
\(653\) 3.90833 0.152945 0.0764723 0.997072i \(-0.475634\pi\)
0.0764723 + 0.997072i \(0.475634\pi\)
\(654\) 0 0
\(655\) −24.4222 −0.954255
\(656\) −9.90833 −0.386855
\(657\) 0 0
\(658\) 12.0000 0.467809
\(659\) 16.8806 0.657574 0.328787 0.944404i \(-0.393360\pi\)
0.328787 + 0.944404i \(0.393360\pi\)
\(660\) 0 0
\(661\) −30.5139 −1.18685 −0.593426 0.804888i \(-0.702225\pi\)
−0.593426 + 0.804888i \(0.702225\pi\)
\(662\) −13.2111 −0.513464
\(663\) 0 0
\(664\) −2.78890 −0.108230
\(665\) −12.0000 −0.465340
\(666\) 0 0
\(667\) −15.2750 −0.591451
\(668\) 5.51388 0.213338
\(669\) 0 0
\(670\) −8.09167 −0.312609
\(671\) 17.3028 0.667966
\(672\) 0 0
\(673\) 20.6972 0.797819 0.398910 0.916990i \(-0.369389\pi\)
0.398910 + 0.916990i \(0.369389\pi\)
\(674\) 6.11943 0.235712
\(675\) 0 0
\(676\) −11.3028 −0.434722
\(677\) −14.2389 −0.547244 −0.273622 0.961837i \(-0.588222\pi\)
−0.273622 + 0.961837i \(0.588222\pi\)
\(678\) 0 0
\(679\) 42.7889 1.64209
\(680\) 13.8167 0.529844
\(681\) 0 0
\(682\) −0.697224 −0.0266981
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −2.09167 −0.0799187
\(686\) 18.7889 0.717363
\(687\) 0 0
\(688\) 0.605551 0.0230864
\(689\) 7.81665 0.297791
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 8.78890 0.334104
\(693\) 0 0
\(694\) −10.1833 −0.386555
\(695\) −4.39445 −0.166691
\(696\) 0 0
\(697\) −59.4500 −2.25183
\(698\) 28.2389 1.06886
\(699\) 0 0
\(700\) −0.788897 −0.0298175
\(701\) −40.1194 −1.51529 −0.757645 0.652667i \(-0.773650\pi\)
−0.757645 + 0.652667i \(0.773650\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 2.30278 0.0867891
\(705\) 0 0
\(706\) −10.1833 −0.383255
\(707\) −32.3667 −1.21727
\(708\) 0 0
\(709\) −41.3305 −1.55220 −0.776100 0.630609i \(-0.782805\pi\)
−0.776100 + 0.630609i \(0.782805\pi\)
\(710\) −13.8167 −0.518530
\(711\) 0 0
\(712\) 9.21110 0.345201
\(713\) 1.18335 0.0443167
\(714\) 0 0
\(715\) 6.90833 0.258357
\(716\) 13.8167 0.516353
\(717\) 0 0
\(718\) −3.21110 −0.119837
\(719\) 51.6333 1.92560 0.962799 0.270220i \(-0.0870963\pi\)
0.962799 + 0.270220i \(0.0870963\pi\)
\(720\) 0 0
\(721\) 0.788897 0.0293801
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) 20.0000 0.743294
\(725\) 1.18335 0.0439484
\(726\) 0 0
\(727\) 19.0917 0.708071 0.354035 0.935232i \(-0.384809\pi\)
0.354035 + 0.935232i \(0.384809\pi\)
\(728\) −3.39445 −0.125807
\(729\) 0 0
\(730\) −28.3305 −1.04856
\(731\) 3.63331 0.134383
\(732\) 0 0
\(733\) −13.6333 −0.503558 −0.251779 0.967785i \(-0.581016\pi\)
−0.251779 + 0.967785i \(0.581016\pi\)
\(734\) 3.81665 0.140875
\(735\) 0 0
\(736\) −3.90833 −0.144063
\(737\) −8.09167 −0.298061
\(738\) 0 0
\(739\) −2.66947 −0.0981980 −0.0490990 0.998794i \(-0.515635\pi\)
−0.0490990 + 0.998794i \(0.515635\pi\)
\(740\) 2.30278 0.0846517
\(741\) 0 0
\(742\) −15.6333 −0.573917
\(743\) 29.4500 1.08041 0.540207 0.841532i \(-0.318346\pi\)
0.540207 + 0.841532i \(0.318346\pi\)
\(744\) 0 0
\(745\) −45.6333 −1.67188
\(746\) −17.8167 −0.652314
\(747\) 0 0
\(748\) 13.8167 0.505187
\(749\) 1.81665 0.0663791
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) −4.60555 −0.167947
\(753\) 0 0
\(754\) 5.09167 0.185428
\(755\) −47.4500 −1.72688
\(756\) 0 0
\(757\) 5.69722 0.207069 0.103535 0.994626i \(-0.466985\pi\)
0.103535 + 0.994626i \(0.466985\pi\)
\(758\) 24.3305 0.883725
\(759\) 0 0
\(760\) 4.60555 0.167061
\(761\) −16.8806 −0.611920 −0.305960 0.952044i \(-0.598977\pi\)
−0.305960 + 0.952044i \(0.598977\pi\)
\(762\) 0 0
\(763\) −5.21110 −0.188655
\(764\) 5.51388 0.199485
\(765\) 0 0
\(766\) 36.8444 1.33124
\(767\) −13.8167 −0.498890
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) −13.8167 −0.497918
\(771\) 0 0
\(772\) −4.00000 −0.143963
\(773\) −22.0555 −0.793282 −0.396641 0.917974i \(-0.629824\pi\)
−0.396641 + 0.917974i \(0.629824\pi\)
\(774\) 0 0
\(775\) −0.0916731 −0.00329299
\(776\) −16.4222 −0.589523
\(777\) 0 0
\(778\) 37.1194 1.33080
\(779\) −19.8167 −0.710005
\(780\) 0 0
\(781\) −13.8167 −0.494399
\(782\) −23.4500 −0.838569
\(783\) 0 0
\(784\) −0.211103 −0.00753938
\(785\) −16.6056 −0.592678
\(786\) 0 0
\(787\) 10.7889 0.384583 0.192291 0.981338i \(-0.438408\pi\)
0.192291 + 0.981338i \(0.438408\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 21.0000 0.747146
\(791\) −8.36669 −0.297485
\(792\) 0 0
\(793\) 9.78890 0.347614
\(794\) 6.18335 0.219439
\(795\) 0 0
\(796\) 26.4222 0.936510
\(797\) −22.3305 −0.790988 −0.395494 0.918469i \(-0.629427\pi\)
−0.395494 + 0.918469i \(0.629427\pi\)
\(798\) 0 0
\(799\) −27.6333 −0.977596
\(800\) 0.302776 0.0107047
\(801\) 0 0
\(802\) 7.81665 0.276016
\(803\) −28.3305 −0.999763
\(804\) 0 0
\(805\) 23.4500 0.826503
\(806\) −0.394449 −0.0138939
\(807\) 0 0
\(808\) 12.4222 0.437012
\(809\) 35.4500 1.24635 0.623177 0.782081i \(-0.285842\pi\)
0.623177 + 0.782081i \(0.285842\pi\)
\(810\) 0 0
\(811\) −7.14719 −0.250972 −0.125486 0.992095i \(-0.540049\pi\)
−0.125486 + 0.992095i \(0.540049\pi\)
\(812\) −10.1833 −0.357365
\(813\) 0 0
\(814\) 2.30278 0.0807122
\(815\) 19.3944 0.679358
\(816\) 0 0
\(817\) 1.21110 0.0423711
\(818\) 31.0278 1.08486
\(819\) 0 0
\(820\) −22.8167 −0.796792
\(821\) 3.21110 0.112068 0.0560341 0.998429i \(-0.482154\pi\)
0.0560341 + 0.998429i \(0.482154\pi\)
\(822\) 0 0
\(823\) 44.8444 1.56318 0.781589 0.623794i \(-0.214410\pi\)
0.781589 + 0.623794i \(0.214410\pi\)
\(824\) −0.302776 −0.0105477
\(825\) 0 0
\(826\) 27.6333 0.961486
\(827\) −34.6056 −1.20335 −0.601676 0.798740i \(-0.705500\pi\)
−0.601676 + 0.798740i \(0.705500\pi\)
\(828\) 0 0
\(829\) −27.7250 −0.962928 −0.481464 0.876466i \(-0.659895\pi\)
−0.481464 + 0.876466i \(0.659895\pi\)
\(830\) −6.42221 −0.222918
\(831\) 0 0
\(832\) 1.30278 0.0451656
\(833\) −1.26662 −0.0438856
\(834\) 0 0
\(835\) 12.6972 0.439406
\(836\) 4.60555 0.159286
\(837\) 0 0
\(838\) −36.1472 −1.24868
\(839\) 12.9722 0.447852 0.223926 0.974606i \(-0.428113\pi\)
0.223926 + 0.974606i \(0.428113\pi\)
\(840\) 0 0
\(841\) −13.7250 −0.473275
\(842\) −3.72498 −0.128371
\(843\) 0 0
\(844\) 10.3028 0.354636
\(845\) −26.0278 −0.895382
\(846\) 0 0
\(847\) 14.8444 0.510060
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 1.81665 0.0623107
\(851\) −3.90833 −0.133976
\(852\) 0 0
\(853\) 42.5416 1.45660 0.728299 0.685260i \(-0.240311\pi\)
0.728299 + 0.685260i \(0.240311\pi\)
\(854\) −19.5778 −0.669938
\(855\) 0 0
\(856\) −0.697224 −0.0238306
\(857\) −42.8444 −1.46354 −0.731769 0.681553i \(-0.761305\pi\)
−0.731769 + 0.681553i \(0.761305\pi\)
\(858\) 0 0
\(859\) 48.0555 1.63963 0.819816 0.572626i \(-0.194075\pi\)
0.819816 + 0.572626i \(0.194075\pi\)
\(860\) 1.39445 0.0475503
\(861\) 0 0
\(862\) −9.21110 −0.313731
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) 20.2389 0.688142
\(866\) 34.9361 1.18718
\(867\) 0 0
\(868\) 0.788897 0.0267769
\(869\) 21.0000 0.712376
\(870\) 0 0
\(871\) −4.57779 −0.155113
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) −7.81665 −0.264402
\(875\) 28.1833 0.952771
\(876\) 0 0
\(877\) −7.21110 −0.243502 −0.121751 0.992561i \(-0.538851\pi\)
−0.121751 + 0.992561i \(0.538851\pi\)
\(878\) 30.3305 1.02361
\(879\) 0 0
\(880\) 5.30278 0.178757
\(881\) 28.5416 0.961592 0.480796 0.876832i \(-0.340348\pi\)
0.480796 + 0.876832i \(0.340348\pi\)
\(882\) 0 0
\(883\) 26.4222 0.889178 0.444589 0.895735i \(-0.353350\pi\)
0.444589 + 0.895735i \(0.353350\pi\)
\(884\) 7.81665 0.262903
\(885\) 0 0
\(886\) −32.7250 −1.09942
\(887\) 0.422205 0.0141763 0.00708813 0.999975i \(-0.497744\pi\)
0.00708813 + 0.999975i \(0.497744\pi\)
\(888\) 0 0
\(889\) 50.0555 1.67881
\(890\) 21.2111 0.710998
\(891\) 0 0
\(892\) −5.81665 −0.194756
\(893\) −9.21110 −0.308238
\(894\) 0 0
\(895\) 31.8167 1.06351
\(896\) −2.60555 −0.0870454
\(897\) 0 0
\(898\) 15.2111 0.507601
\(899\) −1.18335 −0.0394668
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) −22.8167 −0.759711
\(903\) 0 0
\(904\) 3.21110 0.106800
\(905\) 46.0555 1.53094
\(906\) 0 0
\(907\) 26.0000 0.863316 0.431658 0.902037i \(-0.357929\pi\)
0.431658 + 0.902037i \(0.357929\pi\)
\(908\) −13.8167 −0.458522
\(909\) 0 0
\(910\) −7.81665 −0.259120
\(911\) 17.5778 0.582378 0.291189 0.956665i \(-0.405949\pi\)
0.291189 + 0.956665i \(0.405949\pi\)
\(912\) 0 0
\(913\) −6.42221 −0.212544
\(914\) −2.60555 −0.0861840
\(915\) 0 0
\(916\) 24.6056 0.812990
\(917\) 27.6333 0.912532
\(918\) 0 0
\(919\) −9.57779 −0.315942 −0.157971 0.987444i \(-0.550495\pi\)
−0.157971 + 0.987444i \(0.550495\pi\)
\(920\) −9.00000 −0.296721
\(921\) 0 0
\(922\) −12.4222 −0.409104
\(923\) −7.81665 −0.257288
\(924\) 0 0
\(925\) 0.302776 0.00995520
\(926\) 26.6972 0.877325
\(927\) 0 0
\(928\) 3.90833 0.128297
\(929\) 18.4861 0.606510 0.303255 0.952909i \(-0.401927\pi\)
0.303255 + 0.952909i \(0.401927\pi\)
\(930\) 0 0
\(931\) −0.422205 −0.0138372
\(932\) −8.51388 −0.278881
\(933\) 0 0
\(934\) 0 0
\(935\) 31.8167 1.04052
\(936\) 0 0
\(937\) −18.0917 −0.591029 −0.295515 0.955338i \(-0.595491\pi\)
−0.295515 + 0.955338i \(0.595491\pi\)
\(938\) 9.15559 0.298941
\(939\) 0 0
\(940\) −10.6056 −0.345915
\(941\) −13.8167 −0.450410 −0.225205 0.974311i \(-0.572305\pi\)
−0.225205 + 0.974311i \(0.572305\pi\)
\(942\) 0 0
\(943\) 38.7250 1.26106
\(944\) −10.6056 −0.345181
\(945\) 0 0
\(946\) 1.39445 0.0453374
\(947\) 3.63331 0.118067 0.0590333 0.998256i \(-0.481198\pi\)
0.0590333 + 0.998256i \(0.481198\pi\)
\(948\) 0 0
\(949\) −16.0278 −0.520283
\(950\) 0.605551 0.0196467
\(951\) 0 0
\(952\) −15.6333 −0.506678
\(953\) −49.7527 −1.61165 −0.805825 0.592154i \(-0.798278\pi\)
−0.805825 + 0.592154i \(0.798278\pi\)
\(954\) 0 0
\(955\) 12.6972 0.410873
\(956\) 17.5139 0.566439
\(957\) 0 0
\(958\) 13.1194 0.423870
\(959\) 2.36669 0.0764245
\(960\) 0 0
\(961\) −30.9083 −0.997043
\(962\) 1.30278 0.0420032
\(963\) 0 0
\(964\) 8.00000 0.257663
\(965\) −9.21110 −0.296516
\(966\) 0 0
\(967\) −6.72498 −0.216261 −0.108130 0.994137i \(-0.534486\pi\)
−0.108130 + 0.994137i \(0.534486\pi\)
\(968\) −5.69722 −0.183116
\(969\) 0 0
\(970\) −37.8167 −1.21422
\(971\) 22.5416 0.723395 0.361698 0.932295i \(-0.382197\pi\)
0.361698 + 0.932295i \(0.382197\pi\)
\(972\) 0 0
\(973\) 4.97224 0.159403
\(974\) −37.2111 −1.19232
\(975\) 0 0
\(976\) 7.51388 0.240513
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 21.2111 0.677910
\(980\) −0.486122 −0.0155286
\(981\) 0 0
\(982\) −17.7250 −0.565627
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) 13.8167 0.440235
\(986\) 23.4500 0.746799
\(987\) 0 0
\(988\) 2.60555 0.0828936
\(989\) −2.36669 −0.0752564
\(990\) 0 0
\(991\) 50.6972 1.61045 0.805225 0.592969i \(-0.202044\pi\)
0.805225 + 0.592969i \(0.202044\pi\)
\(992\) −0.302776 −0.00961314
\(993\) 0 0
\(994\) 15.6333 0.495858
\(995\) 60.8444 1.92890
\(996\) 0 0
\(997\) −52.4222 −1.66023 −0.830114 0.557594i \(-0.811725\pi\)
−0.830114 + 0.557594i \(0.811725\pi\)
\(998\) −42.2389 −1.33705
\(999\) 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 666.2.a.j.1.2 2
3.2 odd 2 74.2.a.a.1.2 2
4.3 odd 2 5328.2.a.bf.1.2 2
12.11 even 2 592.2.a.f.1.1 2
15.2 even 4 1850.2.b.i.149.1 4
15.8 even 4 1850.2.b.i.149.4 4
15.14 odd 2 1850.2.a.u.1.1 2
21.20 even 2 3626.2.a.a.1.1 2
24.5 odd 2 2368.2.a.s.1.1 2
24.11 even 2 2368.2.a.ba.1.2 2
33.32 even 2 8954.2.a.p.1.2 2
111.110 odd 2 2738.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 2 3.2 odd 2
592.2.a.f.1.1 2 12.11 even 2
666.2.a.j.1.2 2 1.1 even 1 trivial
1850.2.a.u.1.1 2 15.14 odd 2
1850.2.b.i.149.1 4 15.2 even 4
1850.2.b.i.149.4 4 15.8 even 4
2368.2.a.s.1.1 2 24.5 odd 2
2368.2.a.ba.1.2 2 24.11 even 2
2738.2.a.l.1.2 2 111.110 odd 2
3626.2.a.a.1.1 2 21.20 even 2
5328.2.a.bf.1.2 2 4.3 odd 2
8954.2.a.p.1.2 2 33.32 even 2