Properties

Label 8550.2.a.ck.1.1
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: \(x^{3} - x^{2} - 6 x - 2\)
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 1.1
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.761557 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.761557 q^{7} -1.00000 q^{8} +0.864641 q^{11} +5.62620 q^{13} +0.761557 q^{14} +1.00000 q^{16} +3.62620 q^{17} +1.00000 q^{19} -0.864641 q^{22} +8.01395 q^{23} -5.62620 q^{26} -0.761557 q^{28} +7.35548 q^{29} +8.11704 q^{31} -1.00000 q^{32} -3.62620 q^{34} +0.476886 q^{37} -1.00000 q^{38} +2.65847 q^{41} +6.86464 q^{43} +0.864641 q^{44} -8.01395 q^{46} -1.25240 q^{47} -6.42003 q^{49} +5.62620 q^{52} -2.37380 q^{53} +0.761557 q^{56} -7.35548 q^{58} -4.49084 q^{59} -10.8646 q^{61} -8.11704 q^{62} +1.00000 q^{64} +1.03228 q^{67} +3.62620 q^{68} +10.1816 q^{71} +16.4017 q^{73} -0.476886 q^{74} +1.00000 q^{76} -0.658473 q^{77} -12.5693 q^{79} -2.65847 q^{82} +0.270718 q^{83} -6.86464 q^{86} -0.864641 q^{88} -0.387755 q^{89} -4.28467 q^{91} +8.01395 q^{92} +1.25240 q^{94} -8.50479 q^{97} +6.42003 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} + 4q^{7} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} + 4q^{7} - 3q^{8} + 8q^{13} - 4q^{14} + 3q^{16} + 2q^{17} + 3q^{19} - 8q^{26} + 4q^{28} + 8q^{29} + 4q^{31} - 3q^{32} - 2q^{34} + 14q^{37} - 3q^{38} - 2q^{41} + 18q^{43} + 14q^{47} - 3q^{49} + 8q^{52} - 16q^{53} - 4q^{56} - 8q^{58} - 2q^{59} - 30q^{61} - 4q^{62} + 3q^{64} + 2q^{67} + 2q^{68} + 8q^{71} + 10q^{73} - 14q^{74} + 3q^{76} + 8q^{77} + 2q^{82} + 6q^{83} - 18q^{86} + 14q^{89} + 6q^{91} - 14q^{94} + 10q^{97} + 3q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −0.761557 −0.287842 −0.143921 0.989589i \(-0.545971\pi\)
−0.143921 + 0.989589i \(0.545971\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 0.864641 0.260699 0.130350 0.991468i \(-0.458390\pi\)
0.130350 + 0.991468i \(0.458390\pi\)
\(12\) 0 0
\(13\) 5.62620 1.56043 0.780213 0.625514i \(-0.215111\pi\)
0.780213 + 0.625514i \(0.215111\pi\)
\(14\) 0.761557 0.203535
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.62620 0.879482 0.439741 0.898125i \(-0.355070\pi\)
0.439741 + 0.898125i \(0.355070\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −0.864641 −0.184342
\(23\) 8.01395 1.67102 0.835512 0.549472i \(-0.185171\pi\)
0.835512 + 0.549472i \(0.185171\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.62620 −1.10339
\(27\) 0 0
\(28\) −0.761557 −0.143921
\(29\) 7.35548 1.36588 0.682939 0.730475i \(-0.260701\pi\)
0.682939 + 0.730475i \(0.260701\pi\)
\(30\) 0 0
\(31\) 8.11704 1.45786 0.728931 0.684587i \(-0.240017\pi\)
0.728931 + 0.684587i \(0.240017\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.62620 −0.621888
\(35\) 0 0
\(36\) 0 0
\(37\) 0.476886 0.0783995 0.0391998 0.999231i \(-0.487519\pi\)
0.0391998 + 0.999231i \(0.487519\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 2.65847 0.415184 0.207592 0.978216i \(-0.433437\pi\)
0.207592 + 0.978216i \(0.433437\pi\)
\(42\) 0 0
\(43\) 6.86464 1.04685 0.523424 0.852072i \(-0.324654\pi\)
0.523424 + 0.852072i \(0.324654\pi\)
\(44\) 0.864641 0.130350
\(45\) 0 0
\(46\) −8.01395 −1.18159
\(47\) −1.25240 −0.182681 −0.0913404 0.995820i \(-0.529115\pi\)
−0.0913404 + 0.995820i \(0.529115\pi\)
\(48\) 0 0
\(49\) −6.42003 −0.917147
\(50\) 0 0
\(51\) 0 0
\(52\) 5.62620 0.780213
\(53\) −2.37380 −0.326067 −0.163033 0.986621i \(-0.552128\pi\)
−0.163033 + 0.986621i \(0.552128\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.761557 0.101767
\(57\) 0 0
\(58\) −7.35548 −0.965822
\(59\) −4.49084 −0.584657 −0.292329 0.956318i \(-0.594430\pi\)
−0.292329 + 0.956318i \(0.594430\pi\)
\(60\) 0 0
\(61\) −10.8646 −1.39107 −0.695537 0.718490i \(-0.744834\pi\)
−0.695537 + 0.718490i \(0.744834\pi\)
\(62\) −8.11704 −1.03086
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.03228 0.126113 0.0630563 0.998010i \(-0.479915\pi\)
0.0630563 + 0.998010i \(0.479915\pi\)
\(68\) 3.62620 0.439741
\(69\) 0 0
\(70\) 0 0
\(71\) 10.1816 1.20833 0.604166 0.796858i \(-0.293506\pi\)
0.604166 + 0.796858i \(0.293506\pi\)
\(72\) 0 0
\(73\) 16.4017 1.91967 0.959837 0.280557i \(-0.0905191\pi\)
0.959837 + 0.280557i \(0.0905191\pi\)
\(74\) −0.476886 −0.0554368
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −0.658473 −0.0750400
\(78\) 0 0
\(79\) −12.5693 −1.41416 −0.707081 0.707133i \(-0.749988\pi\)
−0.707081 + 0.707133i \(0.749988\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.65847 −0.293579
\(83\) 0.270718 0.0297152 0.0148576 0.999890i \(-0.495271\pi\)
0.0148576 + 0.999890i \(0.495271\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.86464 −0.740233
\(87\) 0 0
\(88\) −0.864641 −0.0921710
\(89\) −0.387755 −0.0411020 −0.0205510 0.999789i \(-0.506542\pi\)
−0.0205510 + 0.999789i \(0.506542\pi\)
\(90\) 0 0
\(91\) −4.28467 −0.449156
\(92\) 8.01395 0.835512
\(93\) 0 0
\(94\) 1.25240 0.129175
\(95\) 0 0
\(96\) 0 0
\(97\) −8.50479 −0.863531 −0.431765 0.901986i \(-0.642109\pi\)
−0.431765 + 0.901986i \(0.642109\pi\)
\(98\) 6.42003 0.648521
\(99\) 0 0
\(100\) 0 0
\(101\) −16.4157 −1.63342 −0.816710 0.577049i \(-0.804204\pi\)
−0.816710 + 0.577049i \(0.804204\pi\)
\(102\) 0 0
\(103\) −9.64015 −0.949872 −0.474936 0.880020i \(-0.657529\pi\)
−0.474936 + 0.880020i \(0.657529\pi\)
\(104\) −5.62620 −0.551694
\(105\) 0 0
\(106\) 2.37380 0.230564
\(107\) −4.28467 −0.414215 −0.207107 0.978318i \(-0.566405\pi\)
−0.207107 + 0.978318i \(0.566405\pi\)
\(108\) 0 0
\(109\) −13.4200 −1.28541 −0.642703 0.766116i \(-0.722187\pi\)
−0.642703 + 0.766116i \(0.722187\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.761557 −0.0719604
\(113\) 10.3232 0.971125 0.485563 0.874202i \(-0.338615\pi\)
0.485563 + 0.874202i \(0.338615\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 7.35548 0.682939
\(117\) 0 0
\(118\) 4.49084 0.413415
\(119\) −2.76156 −0.253152
\(120\) 0 0
\(121\) −10.2524 −0.932036
\(122\) 10.8646 0.983638
\(123\) 0 0
\(124\) 8.11704 0.728931
\(125\) 0 0
\(126\) 0 0
\(127\) 16.9817 1.50688 0.753440 0.657517i \(-0.228393\pi\)
0.753440 + 0.657517i \(0.228393\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −0.541436 −0.0473055 −0.0236528 0.999720i \(-0.507530\pi\)
−0.0236528 + 0.999720i \(0.507530\pi\)
\(132\) 0 0
\(133\) −0.761557 −0.0660354
\(134\) −1.03228 −0.0891750
\(135\) 0 0
\(136\) −3.62620 −0.310944
\(137\) −2.87859 −0.245935 −0.122967 0.992411i \(-0.539241\pi\)
−0.122967 + 0.992411i \(0.539241\pi\)
\(138\) 0 0
\(139\) 3.58767 0.304302 0.152151 0.988357i \(-0.451380\pi\)
0.152151 + 0.988357i \(0.451380\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −10.1816 −0.854420
\(143\) 4.86464 0.406802
\(144\) 0 0
\(145\) 0 0
\(146\) −16.4017 −1.35742
\(147\) 0 0
\(148\) 0.476886 0.0391998
\(149\) 16.8401 1.37959 0.689796 0.724004i \(-0.257700\pi\)
0.689796 + 0.724004i \(0.257700\pi\)
\(150\) 0 0
\(151\) −16.9817 −1.38195 −0.690975 0.722879i \(-0.742818\pi\)
−0.690975 + 0.722879i \(0.742818\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 0.658473 0.0530613
\(155\) 0 0
\(156\) 0 0
\(157\) 14.8401 1.18437 0.592183 0.805804i \(-0.298266\pi\)
0.592183 + 0.805804i \(0.298266\pi\)
\(158\) 12.5693 0.999963
\(159\) 0 0
\(160\) 0 0
\(161\) −6.10308 −0.480990
\(162\) 0 0
\(163\) 13.3694 1.04717 0.523587 0.851972i \(-0.324593\pi\)
0.523587 + 0.851972i \(0.324593\pi\)
\(164\) 2.65847 0.207592
\(165\) 0 0
\(166\) −0.270718 −0.0210118
\(167\) −9.84632 −0.761931 −0.380966 0.924589i \(-0.624408\pi\)
−0.380966 + 0.924589i \(0.624408\pi\)
\(168\) 0 0
\(169\) 18.6541 1.43493
\(170\) 0 0
\(171\) 0 0
\(172\) 6.86464 0.523424
\(173\) −2.98168 −0.226693 −0.113346 0.993556i \(-0.536157\pi\)
−0.113346 + 0.993556i \(0.536157\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.864641 0.0651748
\(177\) 0 0
\(178\) 0.387755 0.0290635
\(179\) −11.7938 −0.881512 −0.440756 0.897627i \(-0.645290\pi\)
−0.440756 + 0.897627i \(0.645290\pi\)
\(180\) 0 0
\(181\) 14.5693 1.08293 0.541465 0.840723i \(-0.317870\pi\)
0.541465 + 0.840723i \(0.317870\pi\)
\(182\) 4.28467 0.317601
\(183\) 0 0
\(184\) −8.01395 −0.590796
\(185\) 0 0
\(186\) 0 0
\(187\) 3.13536 0.229280
\(188\) −1.25240 −0.0913404
\(189\) 0 0
\(190\) 0 0
\(191\) −13.2384 −0.957900 −0.478950 0.877842i \(-0.658983\pi\)
−0.478950 + 0.877842i \(0.658983\pi\)
\(192\) 0 0
\(193\) 2.54144 0.182937 0.0914683 0.995808i \(-0.470844\pi\)
0.0914683 + 0.995808i \(0.470844\pi\)
\(194\) 8.50479 0.610609
\(195\) 0 0
\(196\) −6.42003 −0.458574
\(197\) 19.9109 1.41859 0.709295 0.704911i \(-0.249013\pi\)
0.709295 + 0.704911i \(0.249013\pi\)
\(198\) 0 0
\(199\) −20.3126 −1.43992 −0.719960 0.694015i \(-0.755840\pi\)
−0.719960 + 0.694015i \(0.755840\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 16.4157 1.15500
\(203\) −5.60162 −0.393157
\(204\) 0 0
\(205\) 0 0
\(206\) 9.64015 0.671661
\(207\) 0 0
\(208\) 5.62620 0.390107
\(209\) 0.864641 0.0598085
\(210\) 0 0
\(211\) 18.0419 1.24205 0.621026 0.783790i \(-0.286716\pi\)
0.621026 + 0.783790i \(0.286716\pi\)
\(212\) −2.37380 −0.163033
\(213\) 0 0
\(214\) 4.28467 0.292894
\(215\) 0 0
\(216\) 0 0
\(217\) −6.18159 −0.419634
\(218\) 13.4200 0.908919
\(219\) 0 0
\(220\) 0 0
\(221\) 20.4017 1.37237
\(222\) 0 0
\(223\) −13.5231 −0.905575 −0.452787 0.891619i \(-0.649570\pi\)
−0.452787 + 0.891619i \(0.649570\pi\)
\(224\) 0.761557 0.0508837
\(225\) 0 0
\(226\) −10.3232 −0.686689
\(227\) −13.6016 −0.902771 −0.451386 0.892329i \(-0.649070\pi\)
−0.451386 + 0.892329i \(0.649070\pi\)
\(228\) 0 0
\(229\) 13.5877 0.897898 0.448949 0.893557i \(-0.351798\pi\)
0.448949 + 0.893557i \(0.351798\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −7.35548 −0.482911
\(233\) −25.5510 −1.67390 −0.836952 0.547277i \(-0.815664\pi\)
−0.836952 + 0.547277i \(0.815664\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.49084 −0.292329
\(237\) 0 0
\(238\) 2.76156 0.179005
\(239\) 11.3309 0.732935 0.366468 0.930431i \(-0.380567\pi\)
0.366468 + 0.930431i \(0.380567\pi\)
\(240\) 0 0
\(241\) −1.25240 −0.0806739 −0.0403370 0.999186i \(-0.512843\pi\)
−0.0403370 + 0.999186i \(0.512843\pi\)
\(242\) 10.2524 0.659049
\(243\) 0 0
\(244\) −10.8646 −0.695537
\(245\) 0 0
\(246\) 0 0
\(247\) 5.62620 0.357986
\(248\) −8.11704 −0.515432
\(249\) 0 0
\(250\) 0 0
\(251\) −10.5939 −0.668682 −0.334341 0.942452i \(-0.608514\pi\)
−0.334341 + 0.942452i \(0.608514\pi\)
\(252\) 0 0
\(253\) 6.92919 0.435635
\(254\) −16.9817 −1.06553
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0.153681 0.00958637 0.00479319 0.999989i \(-0.498474\pi\)
0.00479319 + 0.999989i \(0.498474\pi\)
\(258\) 0 0
\(259\) −0.363176 −0.0225666
\(260\) 0 0
\(261\) 0 0
\(262\) 0.541436 0.0334501
\(263\) −0.504792 −0.0311268 −0.0155634 0.999879i \(-0.504954\pi\)
−0.0155634 + 0.999879i \(0.504954\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0.761557 0.0466941
\(267\) 0 0
\(268\) 1.03228 0.0630563
\(269\) 3.49521 0.213107 0.106553 0.994307i \(-0.466019\pi\)
0.106553 + 0.994307i \(0.466019\pi\)
\(270\) 0 0
\(271\) 5.47252 0.332432 0.166216 0.986089i \(-0.446845\pi\)
0.166216 + 0.986089i \(0.446845\pi\)
\(272\) 3.62620 0.219871
\(273\) 0 0
\(274\) 2.87859 0.173902
\(275\) 0 0
\(276\) 0 0
\(277\) 12.9538 0.778317 0.389158 0.921171i \(-0.372766\pi\)
0.389158 + 0.921171i \(0.372766\pi\)
\(278\) −3.58767 −0.215174
\(279\) 0 0
\(280\) 0 0
\(281\) −0.153681 −0.00916785 −0.00458393 0.999989i \(-0.501459\pi\)
−0.00458393 + 0.999989i \(0.501459\pi\)
\(282\) 0 0
\(283\) −18.2341 −1.08390 −0.541952 0.840410i \(-0.682314\pi\)
−0.541952 + 0.840410i \(0.682314\pi\)
\(284\) 10.1816 0.604166
\(285\) 0 0
\(286\) −4.86464 −0.287652
\(287\) −2.02458 −0.119507
\(288\) 0 0
\(289\) −3.85069 −0.226511
\(290\) 0 0
\(291\) 0 0
\(292\) 16.4017 0.959837
\(293\) −2.03853 −0.119092 −0.0595462 0.998226i \(-0.518965\pi\)
−0.0595462 + 0.998226i \(0.518965\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.476886 −0.0277184
\(297\) 0 0
\(298\) −16.8401 −0.975519
\(299\) 45.0881 2.60751
\(300\) 0 0
\(301\) −5.22782 −0.301326
\(302\) 16.9817 0.977186
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) 16.5414 0.944070 0.472035 0.881580i \(-0.343520\pi\)
0.472035 + 0.881580i \(0.343520\pi\)
\(308\) −0.658473 −0.0375200
\(309\) 0 0
\(310\) 0 0
\(311\) 21.4725 1.21759 0.608797 0.793326i \(-0.291652\pi\)
0.608797 + 0.793326i \(0.291652\pi\)
\(312\) 0 0
\(313\) −1.12141 −0.0633856 −0.0316928 0.999498i \(-0.510090\pi\)
−0.0316928 + 0.999498i \(0.510090\pi\)
\(314\) −14.8401 −0.837473
\(315\) 0 0
\(316\) −12.5693 −0.707081
\(317\) 29.8882 1.67869 0.839344 0.543601i \(-0.182940\pi\)
0.839344 + 0.543601i \(0.182940\pi\)
\(318\) 0 0
\(319\) 6.35985 0.356083
\(320\) 0 0
\(321\) 0 0
\(322\) 6.10308 0.340112
\(323\) 3.62620 0.201767
\(324\) 0 0
\(325\) 0 0
\(326\) −13.3694 −0.740464
\(327\) 0 0
\(328\) −2.65847 −0.146790
\(329\) 0.953771 0.0525831
\(330\) 0 0
\(331\) 32.3126 1.77606 0.888030 0.459786i \(-0.152074\pi\)
0.888030 + 0.459786i \(0.152074\pi\)
\(332\) 0.270718 0.0148576
\(333\) 0 0
\(334\) 9.84632 0.538767
\(335\) 0 0
\(336\) 0 0
\(337\) 26.3511 1.43544 0.717718 0.696334i \(-0.245187\pi\)
0.717718 + 0.696334i \(0.245187\pi\)
\(338\) −18.6541 −1.01465
\(339\) 0 0
\(340\) 0 0
\(341\) 7.01832 0.380063
\(342\) 0 0
\(343\) 10.2201 0.551835
\(344\) −6.86464 −0.370117
\(345\) 0 0
\(346\) 2.98168 0.160296
\(347\) 2.77551 0.148997 0.0744986 0.997221i \(-0.476264\pi\)
0.0744986 + 0.997221i \(0.476264\pi\)
\(348\) 0 0
\(349\) 11.5510 0.618312 0.309156 0.951011i \(-0.399953\pi\)
0.309156 + 0.951011i \(0.399953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.864641 −0.0460855
\(353\) 8.40171 0.447178 0.223589 0.974684i \(-0.428223\pi\)
0.223589 + 0.974684i \(0.428223\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −0.387755 −0.0205510
\(357\) 0 0
\(358\) 11.7938 0.623323
\(359\) −22.7895 −1.20278 −0.601391 0.798955i \(-0.705387\pi\)
−0.601391 + 0.798955i \(0.705387\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −14.5693 −0.765748
\(363\) 0 0
\(364\) −4.28467 −0.224578
\(365\) 0 0
\(366\) 0 0
\(367\) 4.06455 0.212168 0.106084 0.994357i \(-0.466169\pi\)
0.106084 + 0.994357i \(0.466169\pi\)
\(368\) 8.01395 0.417756
\(369\) 0 0
\(370\) 0 0
\(371\) 1.80779 0.0938556
\(372\) 0 0
\(373\) −18.4017 −0.952804 −0.476402 0.879227i \(-0.658059\pi\)
−0.476402 + 0.879227i \(0.658059\pi\)
\(374\) −3.13536 −0.162126
\(375\) 0 0
\(376\) 1.25240 0.0645874
\(377\) 41.3834 2.13135
\(378\) 0 0
\(379\) 1.23844 0.0636145 0.0318073 0.999494i \(-0.489874\pi\)
0.0318073 + 0.999494i \(0.489874\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 13.2384 0.677338
\(383\) −16.8646 −0.861743 −0.430871 0.902413i \(-0.641794\pi\)
−0.430871 + 0.902413i \(0.641794\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.54144 −0.129356
\(387\) 0 0
\(388\) −8.50479 −0.431765
\(389\) −8.59392 −0.435729 −0.217865 0.975979i \(-0.569909\pi\)
−0.217865 + 0.975979i \(0.569909\pi\)
\(390\) 0 0
\(391\) 29.0602 1.46964
\(392\) 6.42003 0.324261
\(393\) 0 0
\(394\) −19.9109 −1.00310
\(395\) 0 0
\(396\) 0 0
\(397\) 16.0558 0.805818 0.402909 0.915240i \(-0.367999\pi\)
0.402909 + 0.915240i \(0.367999\pi\)
\(398\) 20.3126 1.01818
\(399\) 0 0
\(400\) 0 0
\(401\) 14.8925 0.743698 0.371849 0.928293i \(-0.378724\pi\)
0.371849 + 0.928293i \(0.378724\pi\)
\(402\) 0 0
\(403\) 45.6681 2.27489
\(404\) −16.4157 −0.816710
\(405\) 0 0
\(406\) 5.60162 0.278004
\(407\) 0.412335 0.0204387
\(408\) 0 0
\(409\) −18.3511 −0.907404 −0.453702 0.891153i \(-0.649897\pi\)
−0.453702 + 0.891153i \(0.649897\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −9.64015 −0.474936
\(413\) 3.42003 0.168289
\(414\) 0 0
\(415\) 0 0
\(416\) −5.62620 −0.275847
\(417\) 0 0
\(418\) −0.864641 −0.0422910
\(419\) −34.7509 −1.69769 −0.848847 0.528639i \(-0.822703\pi\)
−0.848847 + 0.528639i \(0.822703\pi\)
\(420\) 0 0
\(421\) −40.1589 −1.95722 −0.978612 0.205713i \(-0.934049\pi\)
−0.978612 + 0.205713i \(0.934049\pi\)
\(422\) −18.0419 −0.878264
\(423\) 0 0
\(424\) 2.37380 0.115282
\(425\) 0 0
\(426\) 0 0
\(427\) 8.27405 0.400409
\(428\) −4.28467 −0.207107
\(429\) 0 0
\(430\) 0 0
\(431\) −34.9571 −1.68382 −0.841912 0.539615i \(-0.818570\pi\)
−0.841912 + 0.539615i \(0.818570\pi\)
\(432\) 0 0
\(433\) 1.13536 0.0545619 0.0272809 0.999628i \(-0.491315\pi\)
0.0272809 + 0.999628i \(0.491315\pi\)
\(434\) 6.18159 0.296726
\(435\) 0 0
\(436\) −13.4200 −0.642703
\(437\) 8.01395 0.383359
\(438\) 0 0
\(439\) −6.80009 −0.324551 −0.162275 0.986746i \(-0.551883\pi\)
−0.162275 + 0.986746i \(0.551883\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −20.4017 −0.970410
\(443\) −38.0679 −1.80866 −0.904330 0.426835i \(-0.859629\pi\)
−0.904330 + 0.426835i \(0.859629\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 13.5231 0.640338
\(447\) 0 0
\(448\) −0.761557 −0.0359802
\(449\) 18.5414 0.875024 0.437512 0.899212i \(-0.355860\pi\)
0.437512 + 0.899212i \(0.355860\pi\)
\(450\) 0 0
\(451\) 2.29862 0.108238
\(452\) 10.3232 0.485563
\(453\) 0 0
\(454\) 13.6016 0.638356
\(455\) 0 0
\(456\) 0 0
\(457\) 16.3738 0.765934 0.382967 0.923762i \(-0.374902\pi\)
0.382967 + 0.923762i \(0.374902\pi\)
\(458\) −13.5877 −0.634910
\(459\) 0 0
\(460\) 0 0
\(461\) −1.70470 −0.0793959 −0.0396979 0.999212i \(-0.512640\pi\)
−0.0396979 + 0.999212i \(0.512640\pi\)
\(462\) 0 0
\(463\) −10.0279 −0.466036 −0.233018 0.972472i \(-0.574860\pi\)
−0.233018 + 0.972472i \(0.574860\pi\)
\(464\) 7.35548 0.341470
\(465\) 0 0
\(466\) 25.5510 1.18363
\(467\) −32.7509 −1.51553 −0.757766 0.652526i \(-0.773709\pi\)
−0.757766 + 0.652526i \(0.773709\pi\)
\(468\) 0 0
\(469\) −0.786137 −0.0363004
\(470\) 0 0
\(471\) 0 0
\(472\) 4.49084 0.206708
\(473\) 5.93545 0.272912
\(474\) 0 0
\(475\) 0 0
\(476\) −2.76156 −0.126576
\(477\) 0 0
\(478\) −11.3309 −0.518263
\(479\) −27.2803 −1.24647 −0.623234 0.782035i \(-0.714182\pi\)
−0.623234 + 0.782035i \(0.714182\pi\)
\(480\) 0 0
\(481\) 2.68305 0.122337
\(482\) 1.25240 0.0570451
\(483\) 0 0
\(484\) −10.2524 −0.466018
\(485\) 0 0
\(486\) 0 0
\(487\) −11.0741 −0.501817 −0.250908 0.968011i \(-0.580729\pi\)
−0.250908 + 0.968011i \(0.580729\pi\)
\(488\) 10.8646 0.491819
\(489\) 0 0
\(490\) 0 0
\(491\) −8.11704 −0.366317 −0.183158 0.983083i \(-0.558632\pi\)
−0.183158 + 0.983083i \(0.558632\pi\)
\(492\) 0 0
\(493\) 26.6724 1.20127
\(494\) −5.62620 −0.253135
\(495\) 0 0
\(496\) 8.11704 0.364466
\(497\) −7.75386 −0.347808
\(498\) 0 0
\(499\) 0.295298 0.0132193 0.00660967 0.999978i \(-0.497896\pi\)
0.00660967 + 0.999978i \(0.497896\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 10.5939 0.472830
\(503\) −19.6016 −0.873993 −0.436996 0.899463i \(-0.643958\pi\)
−0.436996 + 0.899463i \(0.643958\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.92919 −0.308040
\(507\) 0 0
\(508\) 16.9817 0.753440
\(509\) 1.79383 0.0795102 0.0397551 0.999209i \(-0.487342\pi\)
0.0397551 + 0.999209i \(0.487342\pi\)
\(510\) 0 0
\(511\) −12.4908 −0.552562
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −0.153681 −0.00677859
\(515\) 0 0
\(516\) 0 0
\(517\) −1.08287 −0.0476247
\(518\) 0.363176 0.0159570
\(519\) 0 0
\(520\) 0 0
\(521\) 2.61850 0.114719 0.0573593 0.998354i \(-0.481732\pi\)
0.0573593 + 0.998354i \(0.481732\pi\)
\(522\) 0 0
\(523\) 14.9956 0.655713 0.327857 0.944728i \(-0.393674\pi\)
0.327857 + 0.944728i \(0.393674\pi\)
\(524\) −0.541436 −0.0236528
\(525\) 0 0
\(526\) 0.504792 0.0220100
\(527\) 29.4340 1.28216
\(528\) 0 0
\(529\) 41.2234 1.79232
\(530\) 0 0
\(531\) 0 0
\(532\) −0.761557 −0.0330177
\(533\) 14.9571 0.647864
\(534\) 0 0
\(535\) 0 0
\(536\) −1.03228 −0.0445875
\(537\) 0 0
\(538\) −3.49521 −0.150689
\(539\) −5.55102 −0.239099
\(540\) 0 0
\(541\) 3.40608 0.146439 0.0732194 0.997316i \(-0.476673\pi\)
0.0732194 + 0.997316i \(0.476673\pi\)
\(542\) −5.47252 −0.235065
\(543\) 0 0
\(544\) −3.62620 −0.155472
\(545\) 0 0
\(546\) 0 0
\(547\) 4.74760 0.202993 0.101496 0.994836i \(-0.467637\pi\)
0.101496 + 0.994836i \(0.467637\pi\)
\(548\) −2.87859 −0.122967
\(549\) 0 0
\(550\) 0 0
\(551\) 7.35548 0.313354
\(552\) 0 0
\(553\) 9.57227 0.407054
\(554\) −12.9538 −0.550353
\(555\) 0 0
\(556\) 3.58767 0.152151
\(557\) −43.0462 −1.82393 −0.911964 0.410271i \(-0.865434\pi\)
−0.911964 + 0.410271i \(0.865434\pi\)
\(558\) 0 0
\(559\) 38.6218 1.63353
\(560\) 0 0
\(561\) 0 0
\(562\) 0.153681 0.00648265
\(563\) −17.0096 −0.716869 −0.358434 0.933555i \(-0.616689\pi\)
−0.358434 + 0.933555i \(0.616689\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 18.2341 0.766435
\(567\) 0 0
\(568\) −10.1816 −0.427210
\(569\) 19.7572 0.828264 0.414132 0.910217i \(-0.364085\pi\)
0.414132 + 0.910217i \(0.364085\pi\)
\(570\) 0 0
\(571\) −11.3973 −0.476964 −0.238482 0.971147i \(-0.576650\pi\)
−0.238482 + 0.971147i \(0.576650\pi\)
\(572\) 4.86464 0.203401
\(573\) 0 0
\(574\) 2.02458 0.0845043
\(575\) 0 0
\(576\) 0 0
\(577\) −18.3372 −0.763386 −0.381693 0.924289i \(-0.624659\pi\)
−0.381693 + 0.924289i \(0.624659\pi\)
\(578\) 3.85069 0.160167
\(579\) 0 0
\(580\) 0 0
\(581\) −0.206167 −0.00855327
\(582\) 0 0
\(583\) −2.05249 −0.0850053
\(584\) −16.4017 −0.678708
\(585\) 0 0
\(586\) 2.03853 0.0842110
\(587\) 11.9475 0.493127 0.246563 0.969127i \(-0.420699\pi\)
0.246563 + 0.969127i \(0.420699\pi\)
\(588\) 0 0
\(589\) 8.11704 0.334457
\(590\) 0 0
\(591\) 0 0
\(592\) 0.476886 0.0195999
\(593\) 24.3911 1.00162 0.500811 0.865557i \(-0.333035\pi\)
0.500811 + 0.865557i \(0.333035\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 16.8401 0.689796
\(597\) 0 0
\(598\) −45.0881 −1.84379
\(599\) 21.0708 0.860930 0.430465 0.902607i \(-0.358350\pi\)
0.430465 + 0.902607i \(0.358350\pi\)
\(600\) 0 0
\(601\) 32.3878 1.32112 0.660562 0.750771i \(-0.270318\pi\)
0.660562 + 0.750771i \(0.270318\pi\)
\(602\) 5.22782 0.213070
\(603\) 0 0
\(604\) −16.9817 −0.690975
\(605\) 0 0
\(606\) 0 0
\(607\) 19.0183 0.771930 0.385965 0.922513i \(-0.373869\pi\)
0.385965 + 0.922513i \(0.373869\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −7.04623 −0.285060
\(612\) 0 0
\(613\) −23.5756 −0.952210 −0.476105 0.879389i \(-0.657952\pi\)
−0.476105 + 0.879389i \(0.657952\pi\)
\(614\) −16.5414 −0.667558
\(615\) 0 0
\(616\) 0.658473 0.0265307
\(617\) 26.2707 1.05762 0.528810 0.848740i \(-0.322639\pi\)
0.528810 + 0.848740i \(0.322639\pi\)
\(618\) 0 0
\(619\) 11.4985 0.462165 0.231083 0.972934i \(-0.425773\pi\)
0.231083 + 0.972934i \(0.425773\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −21.4725 −0.860969
\(623\) 0.295298 0.0118309
\(624\) 0 0
\(625\) 0 0
\(626\) 1.12141 0.0448204
\(627\) 0 0
\(628\) 14.8401 0.592183
\(629\) 1.72928 0.0689510
\(630\) 0 0
\(631\) −45.8130 −1.82379 −0.911893 0.410427i \(-0.865380\pi\)
−0.911893 + 0.410427i \(0.865380\pi\)
\(632\) 12.5693 0.499982
\(633\) 0 0
\(634\) −29.8882 −1.18701
\(635\) 0 0
\(636\) 0 0
\(637\) −36.1204 −1.43114
\(638\) −6.35985 −0.251789
\(639\) 0 0
\(640\) 0 0
\(641\) −1.36943 −0.0540894 −0.0270447 0.999634i \(-0.508610\pi\)
−0.0270447 + 0.999634i \(0.508610\pi\)
\(642\) 0 0
\(643\) −24.7389 −0.975606 −0.487803 0.872954i \(-0.662201\pi\)
−0.487803 + 0.872954i \(0.662201\pi\)
\(644\) −6.10308 −0.240495
\(645\) 0 0
\(646\) −3.62620 −0.142671
\(647\) 6.82611 0.268362 0.134181 0.990957i \(-0.457160\pi\)
0.134181 + 0.990957i \(0.457160\pi\)
\(648\) 0 0
\(649\) −3.88296 −0.152420
\(650\) 0 0
\(651\) 0 0
\(652\) 13.3694 0.523587
\(653\) −6.91713 −0.270688 −0.135344 0.990799i \(-0.543214\pi\)
−0.135344 + 0.990799i \(0.543214\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.65847 0.103796
\(657\) 0 0
\(658\) −0.953771 −0.0371819
\(659\) −9.44461 −0.367910 −0.183955 0.982935i \(-0.558890\pi\)
−0.183955 + 0.982935i \(0.558890\pi\)
\(660\) 0 0
\(661\) −22.1955 −0.863306 −0.431653 0.902040i \(-0.642070\pi\)
−0.431653 + 0.902040i \(0.642070\pi\)
\(662\) −32.3126 −1.25586
\(663\) 0 0
\(664\) −0.270718 −0.0105059
\(665\) 0 0
\(666\) 0 0
\(667\) 58.9465 2.28242
\(668\) −9.84632 −0.380966
\(669\) 0 0
\(670\) 0 0
\(671\) −9.39401 −0.362652
\(672\) 0 0
\(673\) 12.2986 0.474077 0.237039 0.971500i \(-0.423823\pi\)
0.237039 + 0.971500i \(0.423823\pi\)
\(674\) −26.3511 −1.01501
\(675\) 0 0
\(676\) 18.6541 0.717466
\(677\) 35.9527 1.38178 0.690888 0.722962i \(-0.257220\pi\)
0.690888 + 0.722962i \(0.257220\pi\)
\(678\) 0 0
\(679\) 6.47689 0.248560
\(680\) 0 0
\(681\) 0 0
\(682\) −7.01832 −0.268745
\(683\) 9.00958 0.344742 0.172371 0.985032i \(-0.444857\pi\)
0.172371 + 0.985032i \(0.444857\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −10.2201 −0.390206
\(687\) 0 0
\(688\) 6.86464 0.261712
\(689\) −13.3555 −0.508803
\(690\) 0 0
\(691\) −9.11078 −0.346590 −0.173295 0.984870i \(-0.555441\pi\)
−0.173295 + 0.984870i \(0.555441\pi\)
\(692\) −2.98168 −0.113346
\(693\) 0 0
\(694\) −2.77551 −0.105357
\(695\) 0 0
\(696\) 0 0
\(697\) 9.64015 0.365147
\(698\) −11.5510 −0.437213
\(699\) 0 0
\(700\) 0 0
\(701\) 14.7476 0.557009 0.278505 0.960435i \(-0.410161\pi\)
0.278505 + 0.960435i \(0.410161\pi\)
\(702\) 0 0
\(703\) 0.476886 0.0179861
\(704\) 0.864641 0.0325874
\(705\) 0 0
\(706\) −8.40171 −0.316202
\(707\) 12.5015 0.470166
\(708\) 0 0
\(709\) −8.63389 −0.324253 −0.162126 0.986770i \(-0.551835\pi\)
−0.162126 + 0.986770i \(0.551835\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.387755 0.0145317
\(713\) 65.0496 2.43613
\(714\) 0 0
\(715\) 0 0
\(716\) −11.7938 −0.440756
\(717\) 0 0
\(718\) 22.7895 0.850495
\(719\) 38.2759 1.42745 0.713726 0.700425i \(-0.247006\pi\)
0.713726 + 0.700425i \(0.247006\pi\)
\(720\) 0 0
\(721\) 7.34153 0.273413
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 14.5693 0.541465
\(725\) 0 0
\(726\) 0 0
\(727\) −31.1893 −1.15675 −0.578373 0.815772i \(-0.696312\pi\)
−0.578373 + 0.815772i \(0.696312\pi\)
\(728\) 4.28467 0.158800
\(729\) 0 0
\(730\) 0 0
\(731\) 24.8925 0.920684
\(732\) 0 0
\(733\) −13.9634 −0.515748 −0.257874 0.966179i \(-0.583022\pi\)
−0.257874 + 0.966179i \(0.583022\pi\)
\(734\) −4.06455 −0.150025
\(735\) 0 0
\(736\) −8.01395 −0.295398
\(737\) 0.892548 0.0328774
\(738\) 0 0
\(739\) 9.02165 0.331867 0.165933 0.986137i \(-0.446936\pi\)
0.165933 + 0.986137i \(0.446936\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.80779 −0.0663659
\(743\) 15.0342 0.551550 0.275775 0.961222i \(-0.411066\pi\)
0.275775 + 0.961222i \(0.411066\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 18.4017 0.673734
\(747\) 0 0
\(748\) 3.13536 0.114640
\(749\) 3.26302 0.119228
\(750\) 0 0
\(751\) 29.6681 1.08260 0.541301 0.840829i \(-0.317932\pi\)
0.541301 + 0.840829i \(0.317932\pi\)
\(752\) −1.25240 −0.0456702
\(753\) 0 0
\(754\) −41.3834 −1.50709
\(755\) 0 0
\(756\) 0 0
\(757\) 10.5819 0.384604 0.192302 0.981336i \(-0.438405\pi\)
0.192302 + 0.981336i \(0.438405\pi\)
\(758\) −1.23844 −0.0449823
\(759\) 0 0
\(760\) 0 0
\(761\) 0.979789 0.0355173 0.0177587 0.999842i \(-0.494347\pi\)
0.0177587 + 0.999842i \(0.494347\pi\)
\(762\) 0 0
\(763\) 10.2201 0.369993
\(764\) −13.2384 −0.478950
\(765\) 0 0
\(766\) 16.8646 0.609344
\(767\) −25.2663 −0.912315
\(768\) 0 0
\(769\) 43.1772 1.55701 0.778505 0.627638i \(-0.215978\pi\)
0.778505 + 0.627638i \(0.215978\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.54144 0.0914683
\(773\) −37.5250 −1.34968 −0.674840 0.737964i \(-0.735787\pi\)
−0.674840 + 0.737964i \(0.735787\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.50479 0.305304
\(777\) 0 0
\(778\) 8.59392 0.308107
\(779\) 2.65847 0.0952497
\(780\) 0 0
\(781\) 8.80342 0.315011
\(782\) −29.0602 −1.03919
\(783\) 0 0
\(784\) −6.42003 −0.229287
\(785\) 0 0
\(786\) 0 0
\(787\) 22.5833 0.805008 0.402504 0.915418i \(-0.368140\pi\)
0.402504 + 0.915418i \(0.368140\pi\)
\(788\) 19.9109 0.709295
\(789\) 0 0
\(790\) 0 0
\(791\) −7.86171 −0.279530
\(792\) 0 0
\(793\) −61.1266 −2.17067
\(794\) −16.0558 −0.569799
\(795\) 0 0
\(796\) −20.3126 −0.719960
\(797\) −35.9806 −1.27450 −0.637250 0.770657i \(-0.719928\pi\)
−0.637250 + 0.770657i \(0.719928\pi\)
\(798\) 0 0
\(799\) −4.54144 −0.160664
\(800\) 0 0
\(801\) 0 0
\(802\) −14.8925 −0.525874
\(803\) 14.1816 0.500457
\(804\) 0 0
\(805\) 0 0
\(806\) −45.6681 −1.60859
\(807\) 0 0
\(808\) 16.4157 0.577501
\(809\) 0.955660 0.0335992 0.0167996 0.999859i \(-0.494652\pi\)
0.0167996 + 0.999859i \(0.494652\pi\)
\(810\) 0 0
\(811\) −7.53707 −0.264662 −0.132331 0.991206i \(-0.542246\pi\)
−0.132331 + 0.991206i \(0.542246\pi\)
\(812\) −5.60162 −0.196578
\(813\) 0 0
\(814\) −0.412335 −0.0144523
\(815\) 0 0
\(816\) 0 0
\(817\) 6.86464 0.240163
\(818\) 18.3511 0.641632
\(819\) 0 0
\(820\) 0 0
\(821\) −13.3082 −0.464460 −0.232230 0.972661i \(-0.574602\pi\)
−0.232230 + 0.972661i \(0.574602\pi\)
\(822\) 0 0
\(823\) 24.4050 0.850706 0.425353 0.905028i \(-0.360150\pi\)
0.425353 + 0.905028i \(0.360150\pi\)
\(824\) 9.64015 0.335831
\(825\) 0 0
\(826\) −3.42003 −0.118998
\(827\) 11.6874 0.406411 0.203206 0.979136i \(-0.434864\pi\)
0.203206 + 0.979136i \(0.434864\pi\)
\(828\) 0 0
\(829\) 25.6541 0.891004 0.445502 0.895281i \(-0.353025\pi\)
0.445502 + 0.895281i \(0.353025\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.62620 0.195053
\(833\) −23.2803 −0.806615
\(834\) 0 0
\(835\) 0 0
\(836\) 0.864641 0.0299042
\(837\) 0 0
\(838\) 34.7509 1.20045
\(839\) 2.91713 0.100710 0.0503552 0.998731i \(-0.483965\pi\)
0.0503552 + 0.998731i \(0.483965\pi\)
\(840\) 0 0
\(841\) 25.1031 0.865624
\(842\) 40.1589 1.38397
\(843\) 0 0
\(844\) 18.0419 0.621026
\(845\) 0 0
\(846\) 0 0
\(847\) 7.80779 0.268279
\(848\) −2.37380 −0.0815167
\(849\) 0 0
\(850\) 0 0
\(851\) 3.82174 0.131008
\(852\) 0 0
\(853\) −6.24281 −0.213750 −0.106875 0.994272i \(-0.534084\pi\)
−0.106875 + 0.994272i \(0.534084\pi\)
\(854\) −8.27405 −0.283132
\(855\) 0 0
\(856\) 4.28467 0.146447
\(857\) 23.2158 0.793035 0.396517 0.918027i \(-0.370219\pi\)
0.396517 + 0.918027i \(0.370219\pi\)
\(858\) 0 0
\(859\) −7.13536 −0.243455 −0.121728 0.992564i \(-0.538843\pi\)
−0.121728 + 0.992564i \(0.538843\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 34.9571 1.19064
\(863\) 7.31362 0.248959 0.124479 0.992222i \(-0.460274\pi\)
0.124479 + 0.992222i \(0.460274\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −1.13536 −0.0385811
\(867\) 0 0
\(868\) −6.18159 −0.209817
\(869\) −10.8680 −0.368671
\(870\) 0 0
\(871\) 5.80779 0.196789
\(872\) 13.4200 0.454460
\(873\) 0 0
\(874\) −8.01395 −0.271076
\(875\) 0 0
\(876\) 0 0
\(877\) −22.0173 −0.743471 −0.371735 0.928339i \(-0.621237\pi\)
−0.371735 + 0.928339i \(0.621237\pi\)
\(878\) 6.80009 0.229492
\(879\) 0 0
\(880\) 0 0
\(881\) −11.7572 −0.396110 −0.198055 0.980191i \(-0.563462\pi\)
−0.198055 + 0.980191i \(0.563462\pi\)
\(882\) 0 0
\(883\) 55.6560 1.87297 0.936487 0.350703i \(-0.114057\pi\)
0.936487 + 0.350703i \(0.114057\pi\)
\(884\) 20.4017 0.686184
\(885\) 0 0
\(886\) 38.0679 1.27892
\(887\) 6.41566 0.215417 0.107708 0.994183i \(-0.465649\pi\)
0.107708 + 0.994183i \(0.465649\pi\)
\(888\) 0 0
\(889\) −12.9325 −0.433743
\(890\) 0 0
\(891\) 0 0
\(892\) −13.5231 −0.452787
\(893\) −1.25240 −0.0419098
\(894\) 0 0
\(895\) 0 0
\(896\) 0.761557 0.0254418
\(897\) 0 0
\(898\) −18.5414 −0.618736
\(899\) 59.7047 1.99126
\(900\) 0 0
\(901\) −8.60788 −0.286770
\(902\) −2.29862 −0.0765358
\(903\) 0 0
\(904\) −10.3232 −0.343345
\(905\) 0 0
\(906\) 0 0
\(907\) −57.1160 −1.89651 −0.948253 0.317516i \(-0.897151\pi\)
−0.948253 + 0.317516i \(0.897151\pi\)
\(908\) −13.6016 −0.451386
\(909\) 0 0
\(910\) 0 0
\(911\) 26.6339 0.882420 0.441210 0.897404i \(-0.354549\pi\)
0.441210 + 0.897404i \(0.354549\pi\)
\(912\) 0 0
\(913\) 0.234074 0.00774672
\(914\) −16.3738 −0.541597
\(915\) 0 0
\(916\) 13.5877 0.448949
\(917\) 0.412335 0.0136165
\(918\) 0 0
\(919\) 6.63246 0.218785 0.109392 0.993999i \(-0.465110\pi\)
0.109392 + 0.993999i \(0.465110\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.70470 0.0561414
\(923\) 57.2836 1.88551
\(924\) 0 0
\(925\) 0 0
\(926\) 10.0279 0.329537
\(927\) 0 0
\(928\) −7.35548 −0.241455
\(929\) 50.0173 1.64101 0.820507 0.571637i \(-0.193691\pi\)
0.820507 + 0.571637i \(0.193691\pi\)
\(930\) 0 0
\(931\) −6.42003 −0.210408
\(932\) −25.5510 −0.836952
\(933\) 0 0
\(934\) 32.7509 1.07164
\(935\) 0 0
\(936\) 0 0
\(937\) −39.8882 −1.30309 −0.651545 0.758610i \(-0.725879\pi\)
−0.651545 + 0.758610i \(0.725879\pi\)
\(938\) 0.786137 0.0256683
\(939\) 0 0
\(940\) 0 0
\(941\) 5.59829 0.182499 0.0912495 0.995828i \(-0.470914\pi\)
0.0912495 + 0.995828i \(0.470914\pi\)
\(942\) 0 0
\(943\) 21.3049 0.693782
\(944\) −4.49084 −0.146164
\(945\) 0 0
\(946\) −5.93545 −0.192978
\(947\) −12.7110 −0.413051 −0.206525 0.978441i \(-0.566216\pi\)
−0.206525 + 0.978441i \(0.566216\pi\)
\(948\) 0 0
\(949\) 92.2793 2.99551
\(950\) 0 0
\(951\) 0 0
\(952\) 2.76156 0.0895026
\(953\) 57.0129 1.84683 0.923415 0.383804i \(-0.125386\pi\)
0.923415 + 0.383804i \(0.125386\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 11.3309 0.366468
\(957\) 0 0
\(958\) 27.2803 0.881387
\(959\) 2.19221 0.0707903
\(960\) 0 0
\(961\) 34.8863 1.12536
\(962\) −2.68305 −0.0865051
\(963\) 0 0
\(964\) −1.25240 −0.0403370
\(965\) 0 0
\(966\) 0 0
\(967\) −33.0183 −1.06180 −0.530899 0.847435i \(-0.678146\pi\)
−0.530899 + 0.847435i \(0.678146\pi\)
\(968\) 10.2524 0.329524
\(969\) 0 0
\(970\) 0 0
\(971\) 3.04623 0.0977581 0.0488791 0.998805i \(-0.484435\pi\)
0.0488791 + 0.998805i \(0.484435\pi\)
\(972\) 0 0
\(973\) −2.73221 −0.0875907
\(974\) 11.0741 0.354838
\(975\) 0 0
\(976\) −10.8646 −0.347769
\(977\) −13.4465 −0.430192 −0.215096 0.976593i \(-0.569006\pi\)
−0.215096 + 0.976593i \(0.569006\pi\)
\(978\) 0 0
\(979\) −0.335269 −0.0107152
\(980\) 0 0
\(981\) 0 0
\(982\) 8.11704 0.259025
\(983\) 22.0646 0.703750 0.351875 0.936047i \(-0.385544\pi\)
0.351875 + 0.936047i \(0.385544\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −26.6724 −0.849423
\(987\) 0 0
\(988\) 5.62620 0.178993
\(989\) 55.0129 1.74931
\(990\) 0 0
\(991\) 51.9946 1.65166 0.825831 0.563917i \(-0.190706\pi\)
0.825831 + 0.563917i \(0.190706\pi\)
\(992\) −8.11704 −0.257716
\(993\) 0 0
\(994\) 7.75386 0.245938
\(995\) 0 0
\(996\) 0 0
\(997\) −37.7693 −1.19616 −0.598082 0.801435i \(-0.704070\pi\)
−0.598082 + 0.801435i \(0.704070\pi\)
\(998\) −0.295298 −0.00934749
\(999\) 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.1.1 3
3.2 odd 2 950.2.a.n.1.3 3
5.2 odd 4 1710.2.d.d.1369.3 6
5.3 odd 4 1710.2.d.d.1369.6 6
5.4 even 2 8550.2.a.cl.1.3 3
12.11 even 2 7600.2.a.bi.1.1 3
15.2 even 4 190.2.b.b.39.4 yes 6
15.8 even 4 190.2.b.b.39.3 6
15.14 odd 2 950.2.a.i.1.1 3
60.23 odd 4 1520.2.d.j.609.1 6
60.47 odd 4 1520.2.d.j.609.6 6
60.59 even 2 7600.2.a.cd.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.b.39.3 6 15.8 even 4
190.2.b.b.39.4 yes 6 15.2 even 4
950.2.a.i.1.1 3 15.14 odd 2
950.2.a.n.1.3 3 3.2 odd 2
1520.2.d.j.609.1 6 60.23 odd 4
1520.2.d.j.609.6 6 60.47 odd 4
1710.2.d.d.1369.3 6 5.2 odd 4
1710.2.d.d.1369.6 6 5.3 odd 4
7600.2.a.bi.1.1 3 12.11 even 2
7600.2.a.cd.1.3 3 60.59 even 2
8550.2.a.ck.1.1 3 1.1 even 1 trivial
8550.2.a.cl.1.3 3 5.4 even 2