Properties

Label 8550.2.a.ck.1.3
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.12489 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.12489 q^{7} -1.00000 q^{8} +2.64002 q^{11} +2.51514 q^{13} -4.12489 q^{14} +1.00000 q^{16} +0.515138 q^{17} +1.00000 q^{19} -2.64002 q^{22} -3.09461 q^{23} -2.51514 q^{26} +4.12489 q^{28} +7.79518 q^{29} +3.67030 q^{31} -1.00000 q^{32} -0.515138 q^{34} +10.2498 q^{37} -1.00000 q^{38} -8.88979 q^{41} +8.64002 q^{43} +2.64002 q^{44} +3.09461 q^{46} +4.96972 q^{47} +10.0147 q^{49} +2.51514 q^{52} -5.48486 q^{53} -4.12489 q^{56} -7.79518 q^{58} -3.15516 q^{59} -12.6400 q^{61} -3.67030 q^{62} +1.00000 q^{64} -7.40493 q^{67} +0.515138 q^{68} -11.1396 q^{71} -2.70436 q^{73} -10.2498 q^{74} +1.00000 q^{76} +10.8898 q^{77} +16.7493 q^{79} +8.88979 q^{82} -3.28005 q^{83} -8.64002 q^{86} -2.64002 q^{88} +7.60975 q^{89} +10.3747 q^{91} -3.09461 q^{92} -4.96972 q^{94} +3.93945 q^{97} -10.0147 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} + 4 q^{7} - 3 q^{8} + 8 q^{13} - 4 q^{14} + 3 q^{16} + 2 q^{17} + 3 q^{19} - 8 q^{26} + 4 q^{28} + 8 q^{29} + 4 q^{31} - 3 q^{32} - 2 q^{34} + 14 q^{37} - 3 q^{38} - 2 q^{41} + 18 q^{43} + 14 q^{47} - 3 q^{49} + 8 q^{52} - 16 q^{53} - 4 q^{56} - 8 q^{58} - 2 q^{59} - 30 q^{61} - 4 q^{62} + 3 q^{64} + 2 q^{67} + 2 q^{68} + 8 q^{71} + 10 q^{73} - 14 q^{74} + 3 q^{76} + 8 q^{77} + 2 q^{82} + 6 q^{83} - 18 q^{86} + 14 q^{89} + 6 q^{91} - 14 q^{94} + 10 q^{97} + 3 q^{98}+O(q^{100}) \) 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
\(5\) 0 0
\(6\) 0 0
\(7\) 4.12489 1.55906 0.779530 0.626365i \(-0.215458\pi\)
0.779530 + 0.626365i \(0.215458\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 2.64002 0.795997 0.397999 0.917386i \(-0.369705\pi\)
0.397999 + 0.917386i \(0.369705\pi\)
\(12\) 0 0
\(13\) 2.51514 0.697574 0.348787 0.937202i \(-0.386594\pi\)
0.348787 + 0.937202i \(0.386594\pi\)
\(14\) −4.12489 −1.10242
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.515138 0.124939 0.0624697 0.998047i \(-0.480102\pi\)
0.0624697 + 0.998047i \(0.480102\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −2.64002 −0.562855
\(23\) −3.09461 −0.645271 −0.322635 0.946523i \(-0.604569\pi\)
−0.322635 + 0.946523i \(0.604569\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.51514 −0.493259
\(27\) 0 0
\(28\) 4.12489 0.779530
\(29\) 7.79518 1.44753 0.723765 0.690047i \(-0.242410\pi\)
0.723765 + 0.690047i \(0.242410\pi\)
\(30\) 0 0
\(31\) 3.67030 0.659205 0.329603 0.944120i \(-0.393085\pi\)
0.329603 + 0.944120i \(0.393085\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −0.515138 −0.0883454
\(35\) 0 0
\(36\) 0 0
\(37\) 10.2498 1.68505 0.842526 0.538656i \(-0.181068\pi\)
0.842526 + 0.538656i \(0.181068\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −8.88979 −1.38835 −0.694176 0.719805i \(-0.744231\pi\)
−0.694176 + 0.719805i \(0.744231\pi\)
\(42\) 0 0
\(43\) 8.64002 1.31759 0.658796 0.752322i \(-0.271066\pi\)
0.658796 + 0.752322i \(0.271066\pi\)
\(44\) 2.64002 0.397999
\(45\) 0 0
\(46\) 3.09461 0.456275
\(47\) 4.96972 0.724909 0.362454 0.932002i \(-0.381939\pi\)
0.362454 + 0.932002i \(0.381939\pi\)
\(48\) 0 0
\(49\) 10.0147 1.43067
\(50\) 0 0
\(51\) 0 0
\(52\) 2.51514 0.348787
\(53\) −5.48486 −0.753404 −0.376702 0.926335i \(-0.622942\pi\)
−0.376702 + 0.926335i \(0.622942\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.12489 −0.551211
\(57\) 0 0
\(58\) −7.79518 −1.02356
\(59\) −3.15516 −0.410767 −0.205384 0.978682i \(-0.565844\pi\)
−0.205384 + 0.978682i \(0.565844\pi\)
\(60\) 0 0
\(61\) −12.6400 −1.61839 −0.809195 0.587541i \(-0.800096\pi\)
−0.809195 + 0.587541i \(0.800096\pi\)
\(62\) −3.67030 −0.466129
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −7.40493 −0.904656 −0.452328 0.891852i \(-0.649406\pi\)
−0.452328 + 0.891852i \(0.649406\pi\)
\(68\) 0.515138 0.0624697
\(69\) 0 0
\(70\) 0 0
\(71\) −11.1396 −1.32202 −0.661012 0.750376i \(-0.729873\pi\)
−0.661012 + 0.750376i \(0.729873\pi\)
\(72\) 0 0
\(73\) −2.70436 −0.316521 −0.158261 0.987397i \(-0.550589\pi\)
−0.158261 + 0.987397i \(0.550589\pi\)
\(74\) −10.2498 −1.19151
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 10.8898 1.24101
\(78\) 0 0
\(79\) 16.7493 1.88444 0.942222 0.334988i \(-0.108732\pi\)
0.942222 + 0.334988i \(0.108732\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 8.88979 0.981714
\(83\) −3.28005 −0.360032 −0.180016 0.983664i \(-0.557615\pi\)
−0.180016 + 0.983664i \(0.557615\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.64002 −0.931678
\(87\) 0 0
\(88\) −2.64002 −0.281427
\(89\) 7.60975 0.806632 0.403316 0.915061i \(-0.367858\pi\)
0.403316 + 0.915061i \(0.367858\pi\)
\(90\) 0 0
\(91\) 10.3747 1.08756
\(92\) −3.09461 −0.322635
\(93\) 0 0
\(94\) −4.96972 −0.512588
\(95\) 0 0
\(96\) 0 0
\(97\) 3.93945 0.399990 0.199995 0.979797i \(-0.435907\pi\)
0.199995 + 0.979797i \(0.435907\pi\)
\(98\) −10.0147 −1.01164
\(99\) 0 0
\(100\) 0 0
\(101\) 13.7990 1.37305 0.686524 0.727107i \(-0.259136\pi\)
0.686524 + 0.727107i \(0.259136\pi\)
\(102\) 0 0
\(103\) 4.57947 0.451229 0.225614 0.974217i \(-0.427561\pi\)
0.225614 + 0.974217i \(0.427561\pi\)
\(104\) −2.51514 −0.246630
\(105\) 0 0
\(106\) 5.48486 0.532737
\(107\) 10.3747 1.00296 0.501478 0.865170i \(-0.332790\pi\)
0.501478 + 0.865170i \(0.332790\pi\)
\(108\) 0 0
\(109\) 3.01468 0.288754 0.144377 0.989523i \(-0.453882\pi\)
0.144377 + 0.989523i \(0.453882\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.12489 0.389765
\(113\) 19.2001 1.80620 0.903098 0.429435i \(-0.141287\pi\)
0.903098 + 0.429435i \(0.141287\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 7.79518 0.723765
\(117\) 0 0
\(118\) 3.15516 0.290456
\(119\) 2.12489 0.194788
\(120\) 0 0
\(121\) −4.03028 −0.366389
\(122\) 12.6400 1.14437
\(123\) 0 0
\(124\) 3.67030 0.329603
\(125\) 0 0
\(126\) 0 0
\(127\) 14.3103 1.26984 0.634918 0.772580i \(-0.281034\pi\)
0.634918 + 0.772580i \(0.281034\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 6.56009 0.573158 0.286579 0.958057i \(-0.407482\pi\)
0.286579 + 0.958057i \(0.407482\pi\)
\(132\) 0 0
\(133\) 4.12489 0.357673
\(134\) 7.40493 0.639689
\(135\) 0 0
\(136\) −0.515138 −0.0441727
\(137\) 6.45459 0.551452 0.275726 0.961236i \(-0.411082\pi\)
0.275726 + 0.961236i \(0.411082\pi\)
\(138\) 0 0
\(139\) −23.0596 −1.95589 −0.977946 0.208856i \(-0.933026\pi\)
−0.977946 + 0.208856i \(0.933026\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 11.1396 0.934812
\(143\) 6.64002 0.555267
\(144\) 0 0
\(145\) 0 0
\(146\) 2.70436 0.223814
\(147\) 0 0
\(148\) 10.2498 0.842526
\(149\) −16.0294 −1.31318 −0.656588 0.754249i \(-0.728001\pi\)
−0.656588 + 0.754249i \(0.728001\pi\)
\(150\) 0 0
\(151\) −14.3103 −1.16456 −0.582279 0.812989i \(-0.697839\pi\)
−0.582279 + 0.812989i \(0.697839\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −10.8898 −0.877525
\(155\) 0 0
\(156\) 0 0
\(157\) −18.0294 −1.43890 −0.719450 0.694544i \(-0.755606\pi\)
−0.719450 + 0.694544i \(0.755606\pi\)
\(158\) −16.7493 −1.33250
\(159\) 0 0
\(160\) 0 0
\(161\) −12.7649 −1.00602
\(162\) 0 0
\(163\) 2.70058 0.211525 0.105763 0.994391i \(-0.466272\pi\)
0.105763 + 0.994391i \(0.466272\pi\)
\(164\) −8.88979 −0.694176
\(165\) 0 0
\(166\) 3.28005 0.254581
\(167\) −8.95035 −0.692599 −0.346299 0.938124i \(-0.612562\pi\)
−0.346299 + 0.938124i \(0.612562\pi\)
\(168\) 0 0
\(169\) −6.67408 −0.513391
\(170\) 0 0
\(171\) 0 0
\(172\) 8.64002 0.658796
\(173\) −0.310323 −0.0235934 −0.0117967 0.999930i \(-0.503755\pi\)
−0.0117967 + 0.999930i \(0.503755\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.64002 0.198999
\(177\) 0 0
\(178\) −7.60975 −0.570375
\(179\) 1.52982 0.114344 0.0571720 0.998364i \(-0.481792\pi\)
0.0571720 + 0.998364i \(0.481792\pi\)
\(180\) 0 0
\(181\) −14.7493 −1.09631 −0.548154 0.836377i \(-0.684669\pi\)
−0.548154 + 0.836377i \(0.684669\pi\)
\(182\) −10.3747 −0.769021
\(183\) 0 0
\(184\) 3.09461 0.228138
\(185\) 0 0
\(186\) 0 0
\(187\) 1.35998 0.0994513
\(188\) 4.96972 0.362454
\(189\) 0 0
\(190\) 0 0
\(191\) −18.1249 −1.31147 −0.655735 0.754991i \(-0.727641\pi\)
−0.655735 + 0.754991i \(0.727641\pi\)
\(192\) 0 0
\(193\) −4.56009 −0.328243 −0.164121 0.986440i \(-0.552479\pi\)
−0.164121 + 0.986440i \(0.552479\pi\)
\(194\) −3.93945 −0.282836
\(195\) 0 0
\(196\) 10.0147 0.715334
\(197\) 2.14048 0.152503 0.0762515 0.997089i \(-0.475705\pi\)
0.0762515 + 0.997089i \(0.475705\pi\)
\(198\) 0 0
\(199\) 16.5639 1.17418 0.587091 0.809521i \(-0.300273\pi\)
0.587091 + 0.809521i \(0.300273\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −13.7990 −0.970892
\(203\) 32.1542 2.25679
\(204\) 0 0
\(205\) 0 0
\(206\) −4.57947 −0.319067
\(207\) 0 0
\(208\) 2.51514 0.174393
\(209\) 2.64002 0.182614
\(210\) 0 0
\(211\) −15.2838 −1.05218 −0.526091 0.850428i \(-0.676343\pi\)
−0.526091 + 0.850428i \(0.676343\pi\)
\(212\) −5.48486 −0.376702
\(213\) 0 0
\(214\) −10.3747 −0.709197
\(215\) 0 0
\(216\) 0 0
\(217\) 15.1396 1.02774
\(218\) −3.01468 −0.204180
\(219\) 0 0
\(220\) 0 0
\(221\) 1.29564 0.0871544
\(222\) 0 0
\(223\) −3.75023 −0.251134 −0.125567 0.992085i \(-0.540075\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(224\) −4.12489 −0.275606
\(225\) 0 0
\(226\) −19.2001 −1.27717
\(227\) 24.1542 1.60317 0.801587 0.597878i \(-0.203989\pi\)
0.801587 + 0.597878i \(0.203989\pi\)
\(228\) 0 0
\(229\) −13.0596 −0.863005 −0.431503 0.902112i \(-0.642016\pi\)
−0.431503 + 0.902112i \(0.642016\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −7.79518 −0.511779
\(233\) 6.43899 0.421832 0.210916 0.977504i \(-0.432355\pi\)
0.210916 + 0.977504i \(0.432355\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.15516 −0.205384
\(237\) 0 0
\(238\) −2.12489 −0.137736
\(239\) −22.8742 −1.47961 −0.739804 0.672822i \(-0.765082\pi\)
−0.739804 + 0.672822i \(0.765082\pi\)
\(240\) 0 0
\(241\) 4.96972 0.320128 0.160064 0.987107i \(-0.448830\pi\)
0.160064 + 0.987107i \(0.448830\pi\)
\(242\) 4.03028 0.259076
\(243\) 0 0
\(244\) −12.6400 −0.809195
\(245\) 0 0
\(246\) 0 0
\(247\) 2.51514 0.160034
\(248\) −3.67030 −0.233064
\(249\) 0 0
\(250\) 0 0
\(251\) −15.9201 −1.00487 −0.502433 0.864616i \(-0.667562\pi\)
−0.502433 + 0.864616i \(0.667562\pi\)
\(252\) 0 0
\(253\) −8.16984 −0.513634
\(254\) −14.3103 −0.897910
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 1.04965 0.0654756 0.0327378 0.999464i \(-0.489577\pi\)
0.0327378 + 0.999464i \(0.489577\pi\)
\(258\) 0 0
\(259\) 42.2791 2.62710
\(260\) 0 0
\(261\) 0 0
\(262\) −6.56009 −0.405284
\(263\) 11.9394 0.736218 0.368109 0.929783i \(-0.380005\pi\)
0.368109 + 0.929783i \(0.380005\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.12489 −0.252913
\(267\) 0 0
\(268\) −7.40493 −0.452328
\(269\) 15.9394 0.971845 0.485923 0.874002i \(-0.338484\pi\)
0.485923 + 0.874002i \(0.338484\pi\)
\(270\) 0 0
\(271\) 1.46548 0.0890218 0.0445109 0.999009i \(-0.485827\pi\)
0.0445109 + 0.999009i \(0.485827\pi\)
\(272\) 0.515138 0.0312348
\(273\) 0 0
\(274\) −6.45459 −0.389936
\(275\) 0 0
\(276\) 0 0
\(277\) 32.4995 1.95271 0.976354 0.216177i \(-0.0693590\pi\)
0.976354 + 0.216177i \(0.0693590\pi\)
\(278\) 23.0596 1.38303
\(279\) 0 0
\(280\) 0 0
\(281\) −1.04965 −0.0626171 −0.0313085 0.999510i \(-0.509967\pi\)
−0.0313085 + 0.999510i \(0.509967\pi\)
\(282\) 0 0
\(283\) −9.34060 −0.555241 −0.277620 0.960691i \(-0.589546\pi\)
−0.277620 + 0.960691i \(0.589546\pi\)
\(284\) −11.1396 −0.661012
\(285\) 0 0
\(286\) −6.64002 −0.392633
\(287\) −36.6694 −2.16453
\(288\) 0 0
\(289\) −16.7346 −0.984390
\(290\) 0 0
\(291\) 0 0
\(292\) −2.70436 −0.158261
\(293\) −25.5748 −1.49409 −0.747047 0.664771i \(-0.768529\pi\)
−0.747047 + 0.664771i \(0.768529\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −10.2498 −0.595756
\(297\) 0 0
\(298\) 16.0294 0.928556
\(299\) −7.78337 −0.450124
\(300\) 0 0
\(301\) 35.6391 2.05420
\(302\) 14.3103 0.823467
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) 9.43991 0.538764 0.269382 0.963033i \(-0.413181\pi\)
0.269382 + 0.963033i \(0.413181\pi\)
\(308\) 10.8898 0.620504
\(309\) 0 0
\(310\) 0 0
\(311\) 17.4655 0.990377 0.495188 0.868786i \(-0.335099\pi\)
0.495188 + 0.868786i \(0.335099\pi\)
\(312\) 0 0
\(313\) −10.4546 −0.590928 −0.295464 0.955354i \(-0.595474\pi\)
−0.295464 + 0.955354i \(0.595474\pi\)
\(314\) 18.0294 1.01746
\(315\) 0 0
\(316\) 16.7493 0.942222
\(317\) −4.33348 −0.243393 −0.121696 0.992567i \(-0.538833\pi\)
−0.121696 + 0.992567i \(0.538833\pi\)
\(318\) 0 0
\(319\) 20.5795 1.15223
\(320\) 0 0
\(321\) 0 0
\(322\) 12.7649 0.711361
\(323\) 0.515138 0.0286630
\(324\) 0 0
\(325\) 0 0
\(326\) −2.70058 −0.149571
\(327\) 0 0
\(328\) 8.88979 0.490857
\(329\) 20.4995 1.13018
\(330\) 0 0
\(331\) −4.56387 −0.250853 −0.125427 0.992103i \(-0.540030\pi\)
−0.125427 + 0.992103i \(0.540030\pi\)
\(332\) −3.28005 −0.180016
\(333\) 0 0
\(334\) 8.95035 0.489741
\(335\) 0 0
\(336\) 0 0
\(337\) 13.0109 0.708749 0.354374 0.935104i \(-0.384694\pi\)
0.354374 + 0.935104i \(0.384694\pi\)
\(338\) 6.67408 0.363022
\(339\) 0 0
\(340\) 0 0
\(341\) 9.68968 0.524725
\(342\) 0 0
\(343\) 12.4352 0.671438
\(344\) −8.64002 −0.465839
\(345\) 0 0
\(346\) 0.310323 0.0166831
\(347\) −13.2195 −0.709660 −0.354830 0.934931i \(-0.615461\pi\)
−0.354830 + 0.934931i \(0.615461\pi\)
\(348\) 0 0
\(349\) −20.4390 −1.09407 −0.547037 0.837108i \(-0.684244\pi\)
−0.547037 + 0.837108i \(0.684244\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.64002 −0.140714
\(353\) −10.7044 −0.569735 −0.284868 0.958567i \(-0.591950\pi\)
−0.284868 + 0.958567i \(0.591950\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.60975 0.403316
\(357\) 0 0
\(358\) −1.52982 −0.0808534
\(359\) 4.31410 0.227690 0.113845 0.993499i \(-0.463683\pi\)
0.113845 + 0.993499i \(0.463683\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 14.7493 0.775207
\(363\) 0 0
\(364\) 10.3747 0.543780
\(365\) 0 0
\(366\) 0 0
\(367\) −12.8099 −0.668669 −0.334335 0.942454i \(-0.608512\pi\)
−0.334335 + 0.942454i \(0.608512\pi\)
\(368\) −3.09461 −0.161318
\(369\) 0 0
\(370\) 0 0
\(371\) −22.6244 −1.17460
\(372\) 0 0
\(373\) 0.704357 0.0364702 0.0182351 0.999834i \(-0.494195\pi\)
0.0182351 + 0.999834i \(0.494195\pi\)
\(374\) −1.35998 −0.0703227
\(375\) 0 0
\(376\) −4.96972 −0.256294
\(377\) 19.6060 1.00976
\(378\) 0 0
\(379\) 6.12489 0.314614 0.157307 0.987550i \(-0.449719\pi\)
0.157307 + 0.987550i \(0.449719\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 18.1249 0.927350
\(383\) −18.6400 −0.952461 −0.476230 0.879321i \(-0.657997\pi\)
−0.476230 + 0.879321i \(0.657997\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.56009 0.232103
\(387\) 0 0
\(388\) 3.93945 0.199995
\(389\) −13.9201 −0.705776 −0.352888 0.935666i \(-0.614800\pi\)
−0.352888 + 0.935666i \(0.614800\pi\)
\(390\) 0 0
\(391\) −1.59415 −0.0806197
\(392\) −10.0147 −0.505818
\(393\) 0 0
\(394\) −2.14048 −0.107836
\(395\) 0 0
\(396\) 0 0
\(397\) −28.3784 −1.42427 −0.712136 0.702041i \(-0.752272\pi\)
−0.712136 + 0.702041i \(0.752272\pi\)
\(398\) −16.5639 −0.830272
\(399\) 0 0
\(400\) 0 0
\(401\) −5.54920 −0.277114 −0.138557 0.990354i \(-0.544246\pi\)
−0.138557 + 0.990354i \(0.544246\pi\)
\(402\) 0 0
\(403\) 9.23131 0.459844
\(404\) 13.7990 0.686524
\(405\) 0 0
\(406\) −32.1542 −1.59579
\(407\) 27.0596 1.34130
\(408\) 0 0
\(409\) −5.01090 −0.247773 −0.123886 0.992296i \(-0.539536\pi\)
−0.123886 + 0.992296i \(0.539536\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.57947 0.225614
\(413\) −13.0147 −0.640411
\(414\) 0 0
\(415\) 0 0
\(416\) −2.51514 −0.123315
\(417\) 0 0
\(418\) −2.64002 −0.129128
\(419\) 15.8889 0.776222 0.388111 0.921613i \(-0.373128\pi\)
0.388111 + 0.921613i \(0.373128\pi\)
\(420\) 0 0
\(421\) −2.38647 −0.116310 −0.0581548 0.998308i \(-0.518522\pi\)
−0.0581548 + 0.998308i \(0.518522\pi\)
\(422\) 15.2838 0.744005
\(423\) 0 0
\(424\) 5.48486 0.266368
\(425\) 0 0
\(426\) 0 0
\(427\) −52.1386 −2.52317
\(428\) 10.3747 0.501478
\(429\) 0 0
\(430\) 0 0
\(431\) 2.35906 0.113632 0.0568160 0.998385i \(-0.481905\pi\)
0.0568160 + 0.998385i \(0.481905\pi\)
\(432\) 0 0
\(433\) −0.640023 −0.0307576 −0.0153788 0.999882i \(-0.504895\pi\)
−0.0153788 + 0.999882i \(0.504895\pi\)
\(434\) −15.1396 −0.726722
\(435\) 0 0
\(436\) 3.01468 0.144377
\(437\) −3.09461 −0.148035
\(438\) 0 0
\(439\) −25.4499 −1.21466 −0.607328 0.794451i \(-0.707759\pi\)
−0.607328 + 0.794451i \(0.707759\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.29564 −0.0616275
\(443\) 35.6685 1.69466 0.847330 0.531067i \(-0.178209\pi\)
0.847330 + 0.531067i \(0.178209\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.75023 0.177578
\(447\) 0 0
\(448\) 4.12489 0.194883
\(449\) 11.4399 0.539883 0.269941 0.962877i \(-0.412996\pi\)
0.269941 + 0.962877i \(0.412996\pi\)
\(450\) 0 0
\(451\) −23.4693 −1.10512
\(452\) 19.2001 0.903098
\(453\) 0 0
\(454\) −24.1542 −1.13361
\(455\) 0 0
\(456\) 0 0
\(457\) 19.4849 0.911463 0.455732 0.890117i \(-0.349378\pi\)
0.455732 + 0.890117i \(0.349378\pi\)
\(458\) 13.0596 0.610237
\(459\) 0 0
\(460\) 0 0
\(461\) 29.3893 1.36880 0.684399 0.729108i \(-0.260065\pi\)
0.684399 + 0.729108i \(0.260065\pi\)
\(462\) 0 0
\(463\) 12.1892 0.566481 0.283241 0.959049i \(-0.408591\pi\)
0.283241 + 0.959049i \(0.408591\pi\)
\(464\) 7.79518 0.361882
\(465\) 0 0
\(466\) −6.43899 −0.298280
\(467\) 17.8889 0.827799 0.413899 0.910323i \(-0.364167\pi\)
0.413899 + 0.910323i \(0.364167\pi\)
\(468\) 0 0
\(469\) −30.5445 −1.41041
\(470\) 0 0
\(471\) 0 0
\(472\) 3.15516 0.145228
\(473\) 22.8099 1.04880
\(474\) 0 0
\(475\) 0 0
\(476\) 2.12489 0.0973940
\(477\) 0 0
\(478\) 22.8742 1.04624
\(479\) 1.15894 0.0529534 0.0264767 0.999649i \(-0.491571\pi\)
0.0264767 + 0.999649i \(0.491571\pi\)
\(480\) 0 0
\(481\) 25.7796 1.17545
\(482\) −4.96972 −0.226365
\(483\) 0 0
\(484\) −4.03028 −0.183194
\(485\) 0 0
\(486\) 0 0
\(487\) 30.6888 1.39064 0.695320 0.718700i \(-0.255263\pi\)
0.695320 + 0.718700i \(0.255263\pi\)
\(488\) 12.6400 0.572187
\(489\) 0 0
\(490\) 0 0
\(491\) −3.67030 −0.165638 −0.0828192 0.996565i \(-0.526392\pi\)
−0.0828192 + 0.996565i \(0.526392\pi\)
\(492\) 0 0
\(493\) 4.01560 0.180853
\(494\) −2.51514 −0.113161
\(495\) 0 0
\(496\) 3.67030 0.164801
\(497\) −45.9494 −2.06111
\(498\) 0 0
\(499\) 31.3893 1.40518 0.702590 0.711595i \(-0.252027\pi\)
0.702590 + 0.711595i \(0.252027\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 15.9201 0.710548
\(503\) 18.1542 0.809458 0.404729 0.914437i \(-0.367366\pi\)
0.404729 + 0.914437i \(0.367366\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8.16984 0.363194
\(507\) 0 0
\(508\) 14.3103 0.634918
\(509\) −11.5298 −0.511050 −0.255525 0.966802i \(-0.582248\pi\)
−0.255525 + 0.966802i \(0.582248\pi\)
\(510\) 0 0
\(511\) −11.1552 −0.493475
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −1.04965 −0.0462982
\(515\) 0 0
\(516\) 0 0
\(517\) 13.1202 0.577025
\(518\) −42.2791 −1.85764
\(519\) 0 0
\(520\) 0 0
\(521\) 42.5895 1.86588 0.932939 0.360035i \(-0.117235\pi\)
0.932939 + 0.360035i \(0.117235\pi\)
\(522\) 0 0
\(523\) 1.21571 0.0531594 0.0265797 0.999647i \(-0.491538\pi\)
0.0265797 + 0.999647i \(0.491538\pi\)
\(524\) 6.56009 0.286579
\(525\) 0 0
\(526\) −11.9394 −0.520585
\(527\) 1.89071 0.0823607
\(528\) 0 0
\(529\) −13.4234 −0.583626
\(530\) 0 0
\(531\) 0 0
\(532\) 4.12489 0.178836
\(533\) −22.3591 −0.968478
\(534\) 0 0
\(535\) 0 0
\(536\) 7.40493 0.319844
\(537\) 0 0
\(538\) −15.9394 −0.687198
\(539\) 26.4390 1.13881
\(540\) 0 0
\(541\) −1.92007 −0.0825503 −0.0412751 0.999148i \(-0.513142\pi\)
−0.0412751 + 0.999148i \(0.513142\pi\)
\(542\) −1.46548 −0.0629480
\(543\) 0 0
\(544\) −0.515138 −0.0220864
\(545\) 0 0
\(546\) 0 0
\(547\) 10.9697 0.469032 0.234516 0.972112i \(-0.424650\pi\)
0.234516 + 0.972112i \(0.424650\pi\)
\(548\) 6.45459 0.275726
\(549\) 0 0
\(550\) 0 0
\(551\) 7.79518 0.332086
\(552\) 0 0
\(553\) 69.0890 2.93796
\(554\) −32.4995 −1.38077
\(555\) 0 0
\(556\) −23.0596 −0.977946
\(557\) −23.5005 −0.995746 −0.497873 0.867250i \(-0.665886\pi\)
−0.497873 + 0.867250i \(0.665886\pi\)
\(558\) 0 0
\(559\) 21.7309 0.919117
\(560\) 0 0
\(561\) 0 0
\(562\) 1.04965 0.0442770
\(563\) 7.87890 0.332056 0.166028 0.986121i \(-0.446906\pi\)
0.166028 + 0.986121i \(0.446906\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 9.34060 0.392615
\(567\) 0 0
\(568\) 11.1396 0.467406
\(569\) 1.09083 0.0457299 0.0228650 0.999739i \(-0.492721\pi\)
0.0228650 + 0.999739i \(0.492721\pi\)
\(570\) 0 0
\(571\) 21.4886 0.899272 0.449636 0.893212i \(-0.351554\pi\)
0.449636 + 0.893212i \(0.351554\pi\)
\(572\) 6.64002 0.277633
\(573\) 0 0
\(574\) 36.6694 1.53055
\(575\) 0 0
\(576\) 0 0
\(577\) −16.1055 −0.670481 −0.335241 0.942133i \(-0.608818\pi\)
−0.335241 + 0.942133i \(0.608818\pi\)
\(578\) 16.7346 0.696069
\(579\) 0 0
\(580\) 0 0
\(581\) −13.5298 −0.561311
\(582\) 0 0
\(583\) −14.4802 −0.599707
\(584\) 2.70436 0.111907
\(585\) 0 0
\(586\) 25.5748 1.05648
\(587\) −0.480164 −0.0198185 −0.00990925 0.999951i \(-0.503154\pi\)
−0.00990925 + 0.999951i \(0.503154\pi\)
\(588\) 0 0
\(589\) 3.67030 0.151232
\(590\) 0 0
\(591\) 0 0
\(592\) 10.2498 0.421263
\(593\) −40.4683 −1.66184 −0.830918 0.556395i \(-0.812184\pi\)
−0.830918 + 0.556395i \(0.812184\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −16.0294 −0.656588
\(597\) 0 0
\(598\) 7.78337 0.318286
\(599\) 36.1698 1.47786 0.738930 0.673782i \(-0.235331\pi\)
0.738930 + 0.673782i \(0.235331\pi\)
\(600\) 0 0
\(601\) 24.3903 0.994899 0.497450 0.867493i \(-0.334270\pi\)
0.497450 + 0.867493i \(0.334270\pi\)
\(602\) −35.6391 −1.45254
\(603\) 0 0
\(604\) −14.3103 −0.582279
\(605\) 0 0
\(606\) 0 0
\(607\) 21.6897 0.880357 0.440178 0.897910i \(-0.354915\pi\)
0.440178 + 0.897910i \(0.354915\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 12.4995 0.505677
\(612\) 0 0
\(613\) −26.2304 −1.05944 −0.529718 0.848174i \(-0.677702\pi\)
−0.529718 + 0.848174i \(0.677702\pi\)
\(614\) −9.43991 −0.380964
\(615\) 0 0
\(616\) −10.8898 −0.438762
\(617\) 22.7200 0.914671 0.457335 0.889294i \(-0.348804\pi\)
0.457335 + 0.889294i \(0.348804\pi\)
\(618\) 0 0
\(619\) −32.9192 −1.32313 −0.661566 0.749887i \(-0.730108\pi\)
−0.661566 + 0.749887i \(0.730108\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −17.4655 −0.700302
\(623\) 31.3893 1.25759
\(624\) 0 0
\(625\) 0 0
\(626\) 10.4546 0.417849
\(627\) 0 0
\(628\) −18.0294 −0.719450
\(629\) 5.28005 0.210529
\(630\) 0 0
\(631\) 17.2876 0.688209 0.344104 0.938931i \(-0.388183\pi\)
0.344104 + 0.938931i \(0.388183\pi\)
\(632\) −16.7493 −0.666252
\(633\) 0 0
\(634\) 4.33348 0.172105
\(635\) 0 0
\(636\) 0 0
\(637\) 25.1883 0.997997
\(638\) −20.5795 −0.814749
\(639\) 0 0
\(640\) 0 0
\(641\) 9.29942 0.367305 0.183653 0.982991i \(-0.441208\pi\)
0.183653 + 0.982991i \(0.441208\pi\)
\(642\) 0 0
\(643\) −3.40115 −0.134128 −0.0670642 0.997749i \(-0.521363\pi\)
−0.0670642 + 0.997749i \(0.521363\pi\)
\(644\) −12.7649 −0.503008
\(645\) 0 0
\(646\) −0.515138 −0.0202678
\(647\) −14.9348 −0.587146 −0.293573 0.955937i \(-0.594844\pi\)
−0.293573 + 0.955937i \(0.594844\pi\)
\(648\) 0 0
\(649\) −8.32970 −0.326969
\(650\) 0 0
\(651\) 0 0
\(652\) 2.70058 0.105763
\(653\) −21.1202 −0.826497 −0.413248 0.910618i \(-0.635606\pi\)
−0.413248 + 0.910618i \(0.635606\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −8.88979 −0.347088
\(657\) 0 0
\(658\) −20.4995 −0.799155
\(659\) −27.6547 −1.07727 −0.538637 0.842538i \(-0.681061\pi\)
−0.538637 + 0.842538i \(0.681061\pi\)
\(660\) 0 0
\(661\) 10.2342 0.398063 0.199032 0.979993i \(-0.436220\pi\)
0.199032 + 0.979993i \(0.436220\pi\)
\(662\) 4.56387 0.177380
\(663\) 0 0
\(664\) 3.28005 0.127291
\(665\) 0 0
\(666\) 0 0
\(667\) −24.1231 −0.934048
\(668\) −8.95035 −0.346299
\(669\) 0 0
\(670\) 0 0
\(671\) −33.3700 −1.28823
\(672\) 0 0
\(673\) −13.4693 −0.519202 −0.259601 0.965716i \(-0.583591\pi\)
−0.259601 + 0.965716i \(0.583591\pi\)
\(674\) −13.0109 −0.501161
\(675\) 0 0
\(676\) −6.67408 −0.256695
\(677\) −15.1433 −0.582006 −0.291003 0.956722i \(-0.593989\pi\)
−0.291003 + 0.956722i \(0.593989\pi\)
\(678\) 0 0
\(679\) 16.2498 0.623609
\(680\) 0 0
\(681\) 0 0
\(682\) −9.68968 −0.371037
\(683\) −15.8789 −0.607589 −0.303795 0.952738i \(-0.598254\pi\)
−0.303795 + 0.952738i \(0.598254\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −12.4352 −0.474778
\(687\) 0 0
\(688\) 8.64002 0.329398
\(689\) −13.7952 −0.525555
\(690\) 0 0
\(691\) 27.3094 1.03890 0.519449 0.854501i \(-0.326137\pi\)
0.519449 + 0.854501i \(0.326137\pi\)
\(692\) −0.310323 −0.0117967
\(693\) 0 0
\(694\) 13.2195 0.501805
\(695\) 0 0
\(696\) 0 0
\(697\) −4.57947 −0.173460
\(698\) 20.4390 0.773627
\(699\) 0 0
\(700\) 0 0
\(701\) 20.9697 0.792016 0.396008 0.918247i \(-0.370395\pi\)
0.396008 + 0.918247i \(0.370395\pi\)
\(702\) 0 0
\(703\) 10.2498 0.386577
\(704\) 2.64002 0.0994996
\(705\) 0 0
\(706\) 10.7044 0.402864
\(707\) 56.9192 2.14067
\(708\) 0 0
\(709\) 37.5592 1.41056 0.705282 0.708927i \(-0.250820\pi\)
0.705282 + 0.708927i \(0.250820\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −7.60975 −0.285187
\(713\) −11.3581 −0.425366
\(714\) 0 0
\(715\) 0 0
\(716\) 1.52982 0.0571720
\(717\) 0 0
\(718\) −4.31410 −0.161001
\(719\) −3.94323 −0.147058 −0.0735288 0.997293i \(-0.523426\pi\)
−0.0735288 + 0.997293i \(0.523426\pi\)
\(720\) 0 0
\(721\) 18.8898 0.703493
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) −14.7493 −0.548154
\(725\) 0 0
\(726\) 0 0
\(727\) 33.2139 1.23183 0.615917 0.787811i \(-0.288786\pi\)
0.615917 + 0.787811i \(0.288786\pi\)
\(728\) −10.3747 −0.384510
\(729\) 0 0
\(730\) 0 0
\(731\) 4.45080 0.164619
\(732\) 0 0
\(733\) −8.62065 −0.318411 −0.159205 0.987245i \(-0.550893\pi\)
−0.159205 + 0.987245i \(0.550893\pi\)
\(734\) 12.8099 0.472821
\(735\) 0 0
\(736\) 3.09461 0.114069
\(737\) −19.5492 −0.720104
\(738\) 0 0
\(739\) −45.1689 −1.66157 −0.830783 0.556597i \(-0.812107\pi\)
−0.830783 + 0.556597i \(0.812107\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 22.6244 0.830569
\(743\) 24.7905 0.909475 0.454737 0.890626i \(-0.349733\pi\)
0.454737 + 0.890626i \(0.349733\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −0.704357 −0.0257883
\(747\) 0 0
\(748\) 1.35998 0.0497257
\(749\) 42.7943 1.56367
\(750\) 0 0
\(751\) −6.76869 −0.246993 −0.123497 0.992345i \(-0.539411\pi\)
−0.123497 + 0.992345i \(0.539411\pi\)
\(752\) 4.96972 0.181227
\(753\) 0 0
\(754\) −19.6060 −0.714007
\(755\) 0 0
\(756\) 0 0
\(757\) 45.2101 1.64319 0.821595 0.570072i \(-0.193085\pi\)
0.821595 + 0.570072i \(0.193085\pi\)
\(758\) −6.12489 −0.222466
\(759\) 0 0
\(760\) 0 0
\(761\) −19.8851 −0.720834 −0.360417 0.932791i \(-0.617366\pi\)
−0.360417 + 0.932791i \(0.617366\pi\)
\(762\) 0 0
\(763\) 12.4352 0.450185
\(764\) −18.1249 −0.655735
\(765\) 0 0
\(766\) 18.6400 0.673491
\(767\) −7.93567 −0.286540
\(768\) 0 0
\(769\) 8.07615 0.291233 0.145617 0.989341i \(-0.453483\pi\)
0.145617 + 0.989341i \(0.453483\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.56009 −0.164121
\(773\) −45.9456 −1.65255 −0.826275 0.563267i \(-0.809544\pi\)
−0.826275 + 0.563267i \(0.809544\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3.93945 −0.141418
\(777\) 0 0
\(778\) 13.9201 0.499059
\(779\) −8.88979 −0.318510
\(780\) 0 0
\(781\) −29.4087 −1.05233
\(782\) 1.59415 0.0570067
\(783\) 0 0
\(784\) 10.0147 0.357667
\(785\) 0 0
\(786\) 0 0
\(787\) −17.8439 −0.636067 −0.318034 0.948079i \(-0.603022\pi\)
−0.318034 + 0.948079i \(0.603022\pi\)
\(788\) 2.14048 0.0762515
\(789\) 0 0
\(790\) 0 0
\(791\) 79.1983 2.81597
\(792\) 0 0
\(793\) −31.7914 −1.12895
\(794\) 28.3784 1.00711
\(795\) 0 0
\(796\) 16.5639 0.587091
\(797\) 37.3326 1.32239 0.661194 0.750215i \(-0.270050\pi\)
0.661194 + 0.750215i \(0.270050\pi\)
\(798\) 0 0
\(799\) 2.56009 0.0905696
\(800\) 0 0
\(801\) 0 0
\(802\) 5.54920 0.195949
\(803\) −7.13957 −0.251950
\(804\) 0 0
\(805\) 0 0
\(806\) −9.23131 −0.325159
\(807\) 0 0
\(808\) −13.7990 −0.485446
\(809\) 38.6950 1.36044 0.680221 0.733007i \(-0.261884\pi\)
0.680221 + 0.733007i \(0.261884\pi\)
\(810\) 0 0
\(811\) 13.3444 0.468585 0.234292 0.972166i \(-0.424723\pi\)
0.234292 + 0.972166i \(0.424723\pi\)
\(812\) 32.1542 1.12839
\(813\) 0 0
\(814\) −27.0596 −0.948440
\(815\) 0 0
\(816\) 0 0
\(817\) 8.64002 0.302276
\(818\) 5.01090 0.175202
\(819\) 0 0
\(820\) 0 0
\(821\) 37.3482 1.30346 0.651730 0.758451i \(-0.274044\pi\)
0.651730 + 0.758451i \(0.274044\pi\)
\(822\) 0 0
\(823\) −51.5630 −1.79737 −0.898686 0.438593i \(-0.855477\pi\)
−0.898686 + 0.438593i \(0.855477\pi\)
\(824\) −4.57947 −0.159533
\(825\) 0 0
\(826\) 13.0147 0.452839
\(827\) 48.5639 1.68873 0.844366 0.535767i \(-0.179978\pi\)
0.844366 + 0.535767i \(0.179978\pi\)
\(828\) 0 0
\(829\) 0.325919 0.0113196 0.00565982 0.999984i \(-0.498198\pi\)
0.00565982 + 0.999984i \(0.498198\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.51514 0.0871967
\(833\) 5.15894 0.178747
\(834\) 0 0
\(835\) 0 0
\(836\) 2.64002 0.0913071
\(837\) 0 0
\(838\) −15.8889 −0.548872
\(839\) 17.1202 0.591055 0.295527 0.955334i \(-0.404505\pi\)
0.295527 + 0.955334i \(0.404505\pi\)
\(840\) 0 0
\(841\) 31.7649 1.09534
\(842\) 2.38647 0.0822432
\(843\) 0 0
\(844\) −15.2838 −0.526091
\(845\) 0 0
\(846\) 0 0
\(847\) −16.6244 −0.571222
\(848\) −5.48486 −0.188351
\(849\) 0 0
\(850\) 0 0
\(851\) −31.7190 −1.08731
\(852\) 0 0
\(853\) −24.9092 −0.852874 −0.426437 0.904517i \(-0.640231\pi\)
−0.426437 + 0.904517i \(0.640231\pi\)
\(854\) 52.1386 1.78415
\(855\) 0 0
\(856\) −10.3747 −0.354598
\(857\) 11.6509 0.397988 0.198994 0.980001i \(-0.436233\pi\)
0.198994 + 0.980001i \(0.436233\pi\)
\(858\) 0 0
\(859\) −5.35998 −0.182880 −0.0914400 0.995811i \(-0.529147\pi\)
−0.0914400 + 0.995811i \(0.529147\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −2.35906 −0.0803499
\(863\) 41.0790 1.39835 0.699173 0.714953i \(-0.253552\pi\)
0.699173 + 0.714953i \(0.253552\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0.640023 0.0217489
\(867\) 0 0
\(868\) 15.1396 0.513870
\(869\) 44.2186 1.50001
\(870\) 0 0
\(871\) −18.6244 −0.631065
\(872\) −3.01468 −0.102090
\(873\) 0 0
\(874\) 3.09461 0.104677
\(875\) 0 0
\(876\) 0 0
\(877\) 45.9532 1.55173 0.775865 0.630899i \(-0.217314\pi\)
0.775865 + 0.630899i \(0.217314\pi\)
\(878\) 25.4499 0.858892
\(879\) 0 0
\(880\) 0 0
\(881\) 6.90917 0.232776 0.116388 0.993204i \(-0.462868\pi\)
0.116388 + 0.993204i \(0.462868\pi\)
\(882\) 0 0
\(883\) 48.5213 1.63287 0.816437 0.577435i \(-0.195946\pi\)
0.816437 + 0.577435i \(0.195946\pi\)
\(884\) 1.29564 0.0435772
\(885\) 0 0
\(886\) −35.6685 −1.19831
\(887\) −23.7990 −0.799091 −0.399546 0.916713i \(-0.630832\pi\)
−0.399546 + 0.916713i \(0.630832\pi\)
\(888\) 0 0
\(889\) 59.0284 1.97975
\(890\) 0 0
\(891\) 0 0
\(892\) −3.75023 −0.125567
\(893\) 4.96972 0.166305
\(894\) 0 0
\(895\) 0 0
\(896\) −4.12489 −0.137803
\(897\) 0 0
\(898\) −11.4399 −0.381755
\(899\) 28.6107 0.954219
\(900\) 0 0
\(901\) −2.82546 −0.0941298
\(902\) 23.4693 0.781441
\(903\) 0 0
\(904\) −19.2001 −0.638586
\(905\) 0 0
\(906\) 0 0
\(907\) 17.9726 0.596770 0.298385 0.954446i \(-0.403552\pi\)
0.298385 + 0.954446i \(0.403552\pi\)
\(908\) 24.1542 0.801587
\(909\) 0 0
\(910\) 0 0
\(911\) −19.5592 −0.648024 −0.324012 0.946053i \(-0.605032\pi\)
−0.324012 + 0.946053i \(0.605032\pi\)
\(912\) 0 0
\(913\) −8.65940 −0.286584
\(914\) −19.4849 −0.644502
\(915\) 0 0
\(916\) −13.0596 −0.431503
\(917\) 27.0596 0.893588
\(918\) 0 0
\(919\) 35.4948 1.17087 0.585433 0.810720i \(-0.300924\pi\)
0.585433 + 0.810720i \(0.300924\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −29.3893 −0.967886
\(923\) −28.0175 −0.922209
\(924\) 0 0
\(925\) 0 0
\(926\) −12.1892 −0.400563
\(927\) 0 0
\(928\) −7.79518 −0.255889
\(929\) −17.9532 −0.589026 −0.294513 0.955648i \(-0.595157\pi\)
−0.294513 + 0.955648i \(0.595157\pi\)
\(930\) 0 0
\(931\) 10.0147 0.328218
\(932\) 6.43899 0.210916
\(933\) 0 0
\(934\) −17.8889 −0.585342
\(935\) 0 0
\(936\) 0 0
\(937\) −5.66652 −0.185117 −0.0925585 0.995707i \(-0.529505\pi\)
−0.0925585 + 0.995707i \(0.529505\pi\)
\(938\) 30.5445 0.997313
\(939\) 0 0
\(940\) 0 0
\(941\) 24.7044 0.805339 0.402670 0.915345i \(-0.368082\pi\)
0.402670 + 0.915345i \(0.368082\pi\)
\(942\) 0 0
\(943\) 27.5104 0.895863
\(944\) −3.15516 −0.102692
\(945\) 0 0
\(946\) −22.8099 −0.741613
\(947\) −13.5904 −0.441628 −0.220814 0.975316i \(-0.570871\pi\)
−0.220814 + 0.975316i \(0.570871\pi\)
\(948\) 0 0
\(949\) −6.80183 −0.220797
\(950\) 0 0
\(951\) 0 0
\(952\) −2.12489 −0.0688679
\(953\) −24.7375 −0.801326 −0.400663 0.916225i \(-0.631220\pi\)
−0.400663 + 0.916225i \(0.631220\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −22.8742 −0.739804
\(957\) 0 0
\(958\) −1.15894 −0.0374437
\(959\) 26.6244 0.859748
\(960\) 0 0
\(961\) −17.5289 −0.565448
\(962\) −25.7796 −0.831167
\(963\) 0 0
\(964\) 4.96972 0.160064
\(965\) 0 0
\(966\) 0 0
\(967\) −35.6897 −1.14770 −0.573851 0.818960i \(-0.694551\pi\)
−0.573851 + 0.818960i \(0.694551\pi\)
\(968\) 4.03028 0.129538
\(969\) 0 0
\(970\) 0 0
\(971\) −16.4995 −0.529495 −0.264748 0.964318i \(-0.585289\pi\)
−0.264748 + 0.964318i \(0.585289\pi\)
\(972\) 0 0
\(973\) −95.1184 −3.04935
\(974\) −30.6888 −0.983331
\(975\) 0 0
\(976\) −12.6400 −0.404597
\(977\) −49.8501 −1.59485 −0.797423 0.603420i \(-0.793804\pi\)
−0.797423 + 0.603420i \(0.793804\pi\)
\(978\) 0 0
\(979\) 20.0899 0.642076
\(980\) 0 0
\(981\) 0 0
\(982\) 3.67030 0.117124
\(983\) 5.19014 0.165540 0.0827698 0.996569i \(-0.473623\pi\)
0.0827698 + 0.996569i \(0.473623\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −4.01560 −0.127883
\(987\) 0 0
\(988\) 2.51514 0.0800172
\(989\) −26.7375 −0.850203
\(990\) 0 0
\(991\) −32.4272 −1.03008 −0.515042 0.857165i \(-0.672224\pi\)
−0.515042 + 0.857165i \(0.672224\pi\)
\(992\) −3.67030 −0.116532
\(993\) 0 0
\(994\) 45.9494 1.45743
\(995\) 0 0
\(996\) 0 0
\(997\) 10.1992 0.323012 0.161506 0.986872i \(-0.448365\pi\)
0.161506 + 0.986872i \(0.448365\pi\)
\(998\) −31.3893 −0.993612
\(999\) 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 8550.2.a.ck.1.3 3
3.2 odd 2 950.2.a.n.1.1 3
5.2 odd 4 1710.2.d.d.1369.1 6
5.3 odd 4 1710.2.d.d.1369.4 6
5.4 even 2 8550.2.a.cl.1.1 3
12.11 even 2 7600.2.a.bi.1.3 3
15.2 even 4 190.2.b.b.39.6 yes 6
15.8 even 4 190.2.b.b.39.1 6
15.14 odd 2 950.2.a.i.1.3 3
60.23 odd 4 1520.2.d.j.609.5 6
60.47 odd 4 1520.2.d.j.609.2 6
60.59 even 2 7600.2.a.cd.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.b.39.1 6 15.8 even 4
190.2.b.b.39.6 yes 6 15.2 even 4
950.2.a.i.1.3 3 15.14 odd 2
950.2.a.n.1.1 3 3.2 odd 2
1520.2.d.j.609.2 6 60.47 odd 4
1520.2.d.j.609.5 6 60.23 odd 4
1710.2.d.d.1369.1 6 5.2 odd 4
1710.2.d.d.1369.4 6 5.3 odd 4
7600.2.a.bi.1.3 3 12.11 even 2
7600.2.a.cd.1.1 3 60.59 even 2
8550.2.a.ck.1.3 3 1.1 even 1 trivial
8550.2.a.cl.1.1 3 5.4 even 2