Properties

Label 2850.2.a.bm.1.3
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 2850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.35026 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.35026 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.962389 q^{11} -1.00000 q^{12} +1.61213 q^{13} +3.35026 q^{14} +1.00000 q^{16} +0.387873 q^{17} +1.00000 q^{18} +1.00000 q^{19} -3.35026 q^{21} -0.962389 q^{22} +0.962389 q^{23} -1.00000 q^{24} +1.61213 q^{26} -1.00000 q^{27} +3.35026 q^{28} +6.96239 q^{29} +3.35026 q^{31} +1.00000 q^{32} +0.962389 q^{33} +0.387873 q^{34} +1.00000 q^{36} +1.61213 q^{37} +1.00000 q^{38} -1.61213 q^{39} -9.27504 q^{41} -3.35026 q^{42} -6.18664 q^{43} -0.962389 q^{44} +0.962389 q^{46} -0.962389 q^{47} -1.00000 q^{48} +4.22425 q^{49} -0.387873 q^{51} +1.61213 q^{52} -6.00000 q^{53} -1.00000 q^{54} +3.35026 q^{56} -1.00000 q^{57} +6.96239 q^{58} +10.3127 q^{59} +11.9248 q^{61} +3.35026 q^{62} +3.35026 q^{63} +1.00000 q^{64} +0.962389 q^{66} +7.22425 q^{67} +0.387873 q^{68} -0.962389 q^{69} +7.22425 q^{71} +1.00000 q^{72} +3.22425 q^{73} +1.61213 q^{74} +1.00000 q^{76} -3.22425 q^{77} -1.61213 q^{78} -3.35026 q^{79} +1.00000 q^{81} -9.27504 q^{82} -15.0132 q^{83} -3.35026 q^{84} -6.18664 q^{86} -6.96239 q^{87} -0.962389 q^{88} +4.64974 q^{89} +5.40105 q^{91} +0.962389 q^{92} -3.35026 q^{93} -0.962389 q^{94} -1.00000 q^{96} +10.9624 q^{97} +4.22425 q^{98} -0.962389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 3q^{8} + 3q^{9} + 8q^{11} - 3q^{12} + 4q^{13} + 3q^{16} + 2q^{17} + 3q^{18} + 3q^{19} + 8q^{22} - 8q^{23} - 3q^{24} + 4q^{26} - 3q^{27} + 10q^{29} + 3q^{32} - 8q^{33} + 2q^{34} + 3q^{36} + 4q^{37} + 3q^{38} - 4q^{39} + 4q^{41} - 6q^{43} + 8q^{44} - 8q^{46} + 8q^{47} - 3q^{48} + 11q^{49} - 2q^{51} + 4q^{52} - 18q^{53} - 3q^{54} - 3q^{57} + 10q^{58} + 10q^{59} + 14q^{61} + 3q^{64} - 8q^{66} + 20q^{67} + 2q^{68} + 8q^{69} + 20q^{71} + 3q^{72} + 8q^{73} + 4q^{74} + 3q^{76} - 8q^{77} - 4q^{78} + 3q^{81} + 4q^{82} - 4q^{83} - 6q^{86} - 10q^{87} + 8q^{88} + 24q^{89} - 24q^{91} - 8q^{92} + 8q^{94} - 3q^{96} + 22q^{97} + 11q^{98} + 8q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 3.35026 1.26628 0.633140 0.774037i \(-0.281766\pi\)
0.633140 + 0.774037i \(0.281766\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.962389 −0.290171 −0.145086 0.989419i \(-0.546346\pi\)
−0.145086 + 0.989419i \(0.546346\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.61213 0.447124 0.223562 0.974690i \(-0.428232\pi\)
0.223562 + 0.974690i \(0.428232\pi\)
\(14\) 3.35026 0.895395
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.387873 0.0940731 0.0470365 0.998893i \(-0.485022\pi\)
0.0470365 + 0.998893i \(0.485022\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.35026 −0.731087
\(22\) −0.962389 −0.205182
\(23\) 0.962389 0.200672 0.100336 0.994954i \(-0.468008\pi\)
0.100336 + 0.994954i \(0.468008\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 1.61213 0.316164
\(27\) −1.00000 −0.192450
\(28\) 3.35026 0.633140
\(29\) 6.96239 1.29288 0.646442 0.762964i \(-0.276256\pi\)
0.646442 + 0.762964i \(0.276256\pi\)
\(30\) 0 0
\(31\) 3.35026 0.601725 0.300862 0.953668i \(-0.402726\pi\)
0.300862 + 0.953668i \(0.402726\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.962389 0.167530
\(34\) 0.387873 0.0665197
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.61213 0.265032 0.132516 0.991181i \(-0.457694\pi\)
0.132516 + 0.991181i \(0.457694\pi\)
\(38\) 1.00000 0.162221
\(39\) −1.61213 −0.258147
\(40\) 0 0
\(41\) −9.27504 −1.44852 −0.724259 0.689528i \(-0.757818\pi\)
−0.724259 + 0.689528i \(0.757818\pi\)
\(42\) −3.35026 −0.516957
\(43\) −6.18664 −0.943454 −0.471727 0.881745i \(-0.656369\pi\)
−0.471727 + 0.881745i \(0.656369\pi\)
\(44\) −0.962389 −0.145086
\(45\) 0 0
\(46\) 0.962389 0.141896
\(47\) −0.962389 −0.140379 −0.0701894 0.997534i \(-0.522360\pi\)
−0.0701894 + 0.997534i \(0.522360\pi\)
\(48\) −1.00000 −0.144338
\(49\) 4.22425 0.603465
\(50\) 0 0
\(51\) −0.387873 −0.0543131
\(52\) 1.61213 0.223562
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 3.35026 0.447698
\(57\) −1.00000 −0.132453
\(58\) 6.96239 0.914206
\(59\) 10.3127 1.34259 0.671296 0.741189i \(-0.265738\pi\)
0.671296 + 0.741189i \(0.265738\pi\)
\(60\) 0 0
\(61\) 11.9248 1.52681 0.763406 0.645919i \(-0.223526\pi\)
0.763406 + 0.645919i \(0.223526\pi\)
\(62\) 3.35026 0.425484
\(63\) 3.35026 0.422093
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.962389 0.118462
\(67\) 7.22425 0.882583 0.441292 0.897364i \(-0.354520\pi\)
0.441292 + 0.897364i \(0.354520\pi\)
\(68\) 0.387873 0.0470365
\(69\) −0.962389 −0.115858
\(70\) 0 0
\(71\) 7.22425 0.857361 0.428681 0.903456i \(-0.358979\pi\)
0.428681 + 0.903456i \(0.358979\pi\)
\(72\) 1.00000 0.117851
\(73\) 3.22425 0.377370 0.188685 0.982038i \(-0.439577\pi\)
0.188685 + 0.982038i \(0.439577\pi\)
\(74\) 1.61213 0.187406
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −3.22425 −0.367438
\(78\) −1.61213 −0.182537
\(79\) −3.35026 −0.376934 −0.188467 0.982080i \(-0.560352\pi\)
−0.188467 + 0.982080i \(0.560352\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −9.27504 −1.02426
\(83\) −15.0132 −1.64791 −0.823955 0.566655i \(-0.808237\pi\)
−0.823955 + 0.566655i \(0.808237\pi\)
\(84\) −3.35026 −0.365544
\(85\) 0 0
\(86\) −6.18664 −0.667123
\(87\) −6.96239 −0.746446
\(88\) −0.962389 −0.102591
\(89\) 4.64974 0.492871 0.246436 0.969159i \(-0.420741\pi\)
0.246436 + 0.969159i \(0.420741\pi\)
\(90\) 0 0
\(91\) 5.40105 0.566184
\(92\) 0.962389 0.100336
\(93\) −3.35026 −0.347406
\(94\) −0.962389 −0.0992628
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 10.9624 1.11306 0.556531 0.830827i \(-0.312132\pi\)
0.556531 + 0.830827i \(0.312132\pi\)
\(98\) 4.22425 0.426714
\(99\) −0.962389 −0.0967237
\(100\) 0 0
\(101\) 2.72496 0.271144 0.135572 0.990768i \(-0.456713\pi\)
0.135572 + 0.990768i \(0.456713\pi\)
\(102\) −0.387873 −0.0384052
\(103\) −0.574515 −0.0566087 −0.0283043 0.999599i \(-0.509011\pi\)
−0.0283043 + 0.999599i \(0.509011\pi\)
\(104\) 1.61213 0.158082
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −10.7005 −1.03446 −0.517229 0.855847i \(-0.673037\pi\)
−0.517229 + 0.855847i \(0.673037\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.1260 0.969896 0.484948 0.874543i \(-0.338839\pi\)
0.484948 + 0.874543i \(0.338839\pi\)
\(110\) 0 0
\(111\) −1.61213 −0.153016
\(112\) 3.35026 0.316570
\(113\) −20.5501 −1.93319 −0.966594 0.256311i \(-0.917493\pi\)
−0.966594 + 0.256311i \(0.917493\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 6.96239 0.646442
\(117\) 1.61213 0.149041
\(118\) 10.3127 0.949356
\(119\) 1.29948 0.119123
\(120\) 0 0
\(121\) −10.0738 −0.915801
\(122\) 11.9248 1.07962
\(123\) 9.27504 0.836302
\(124\) 3.35026 0.300862
\(125\) 0 0
\(126\) 3.35026 0.298465
\(127\) −1.35026 −0.119816 −0.0599082 0.998204i \(-0.519081\pi\)
−0.0599082 + 0.998204i \(0.519081\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.18664 0.544703
\(130\) 0 0
\(131\) 12.4387 1.08677 0.543385 0.839483i \(-0.317142\pi\)
0.543385 + 0.839483i \(0.317142\pi\)
\(132\) 0.962389 0.0837652
\(133\) 3.35026 0.290505
\(134\) 7.22425 0.624080
\(135\) 0 0
\(136\) 0.387873 0.0332598
\(137\) 18.1622 1.55170 0.775851 0.630916i \(-0.217321\pi\)
0.775851 + 0.630916i \(0.217321\pi\)
\(138\) −0.962389 −0.0819240
\(139\) 8.77575 0.744349 0.372175 0.928163i \(-0.378612\pi\)
0.372175 + 0.928163i \(0.378612\pi\)
\(140\) 0 0
\(141\) 0.962389 0.0810477
\(142\) 7.22425 0.606246
\(143\) −1.55149 −0.129742
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 3.22425 0.266841
\(147\) −4.22425 −0.348411
\(148\) 1.61213 0.132516
\(149\) 15.9756 1.30877 0.654385 0.756162i \(-0.272928\pi\)
0.654385 + 0.756162i \(0.272928\pi\)
\(150\) 0 0
\(151\) 18.4241 1.49933 0.749665 0.661818i \(-0.230215\pi\)
0.749665 + 0.661818i \(0.230215\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0.387873 0.0313577
\(154\) −3.22425 −0.259818
\(155\) 0 0
\(156\) −1.61213 −0.129073
\(157\) 13.7889 1.10048 0.550238 0.835008i \(-0.314537\pi\)
0.550238 + 0.835008i \(0.314537\pi\)
\(158\) −3.35026 −0.266533
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 3.22425 0.254107
\(162\) 1.00000 0.0785674
\(163\) 3.73813 0.292793 0.146397 0.989226i \(-0.453232\pi\)
0.146397 + 0.989226i \(0.453232\pi\)
\(164\) −9.27504 −0.724259
\(165\) 0 0
\(166\) −15.0132 −1.16525
\(167\) −15.4763 −1.19759 −0.598795 0.800902i \(-0.704354\pi\)
−0.598795 + 0.800902i \(0.704354\pi\)
\(168\) −3.35026 −0.258478
\(169\) −10.4010 −0.800081
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) −6.18664 −0.471727
\(173\) 1.47627 0.112239 0.0561194 0.998424i \(-0.482127\pi\)
0.0561194 + 0.998424i \(0.482127\pi\)
\(174\) −6.96239 −0.527817
\(175\) 0 0
\(176\) −0.962389 −0.0725428
\(177\) −10.3127 −0.775146
\(178\) 4.64974 0.348513
\(179\) −14.3127 −1.06978 −0.534889 0.844922i \(-0.679647\pi\)
−0.534889 + 0.844922i \(0.679647\pi\)
\(180\) 0 0
\(181\) −8.82653 −0.656071 −0.328035 0.944665i \(-0.606387\pi\)
−0.328035 + 0.944665i \(0.606387\pi\)
\(182\) 5.40105 0.400352
\(183\) −11.9248 −0.881505
\(184\) 0.962389 0.0709482
\(185\) 0 0
\(186\) −3.35026 −0.245653
\(187\) −0.373285 −0.0272973
\(188\) −0.962389 −0.0701894
\(189\) −3.35026 −0.243696
\(190\) 0 0
\(191\) 2.31265 0.167338 0.0836688 0.996494i \(-0.473336\pi\)
0.0836688 + 0.996494i \(0.473336\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −7.58769 −0.546174 −0.273087 0.961989i \(-0.588045\pi\)
−0.273087 + 0.961989i \(0.588045\pi\)
\(194\) 10.9624 0.787054
\(195\) 0 0
\(196\) 4.22425 0.301732
\(197\) 18.8119 1.34030 0.670148 0.742228i \(-0.266231\pi\)
0.670148 + 0.742228i \(0.266231\pi\)
\(198\) −0.962389 −0.0683940
\(199\) −9.40105 −0.666423 −0.333211 0.942852i \(-0.608132\pi\)
−0.333211 + 0.942852i \(0.608132\pi\)
\(200\) 0 0
\(201\) −7.22425 −0.509560
\(202\) 2.72496 0.191728
\(203\) 23.3258 1.63715
\(204\) −0.387873 −0.0271566
\(205\) 0 0
\(206\) −0.574515 −0.0400284
\(207\) 0.962389 0.0668906
\(208\) 1.61213 0.111781
\(209\) −0.962389 −0.0665698
\(210\) 0 0
\(211\) 4.77575 0.328776 0.164388 0.986396i \(-0.447435\pi\)
0.164388 + 0.986396i \(0.447435\pi\)
\(212\) −6.00000 −0.412082
\(213\) −7.22425 −0.494998
\(214\) −10.7005 −0.731473
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 11.2243 0.761952
\(218\) 10.1260 0.685820
\(219\) −3.22425 −0.217875
\(220\) 0 0
\(221\) 0.625301 0.0420623
\(222\) −1.61213 −0.108199
\(223\) 24.6761 1.65243 0.826216 0.563353i \(-0.190489\pi\)
0.826216 + 0.563353i \(0.190489\pi\)
\(224\) 3.35026 0.223849
\(225\) 0 0
\(226\) −20.5501 −1.36697
\(227\) −15.4763 −1.02720 −0.513598 0.858031i \(-0.671688\pi\)
−0.513598 + 0.858031i \(0.671688\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 21.3258 1.40925 0.704625 0.709580i \(-0.251115\pi\)
0.704625 + 0.709580i \(0.251115\pi\)
\(230\) 0 0
\(231\) 3.22425 0.212140
\(232\) 6.96239 0.457103
\(233\) 9.01317 0.590473 0.295236 0.955424i \(-0.404602\pi\)
0.295236 + 0.955424i \(0.404602\pi\)
\(234\) 1.61213 0.105388
\(235\) 0 0
\(236\) 10.3127 0.671296
\(237\) 3.35026 0.217623
\(238\) 1.29948 0.0842326
\(239\) −0.135857 −0.00878787 −0.00439393 0.999990i \(-0.501399\pi\)
−0.00439393 + 0.999990i \(0.501399\pi\)
\(240\) 0 0
\(241\) −25.8496 −1.66512 −0.832558 0.553938i \(-0.813125\pi\)
−0.832558 + 0.553938i \(0.813125\pi\)
\(242\) −10.0738 −0.647569
\(243\) −1.00000 −0.0641500
\(244\) 11.9248 0.763406
\(245\) 0 0
\(246\) 9.27504 0.591355
\(247\) 1.61213 0.102577
\(248\) 3.35026 0.212742
\(249\) 15.0132 0.951421
\(250\) 0 0
\(251\) −10.1114 −0.638227 −0.319114 0.947716i \(-0.603385\pi\)
−0.319114 + 0.947716i \(0.603385\pi\)
\(252\) 3.35026 0.211047
\(253\) −0.926192 −0.0582292
\(254\) −1.35026 −0.0847230
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −26.9986 −1.68413 −0.842063 0.539380i \(-0.818659\pi\)
−0.842063 + 0.539380i \(0.818659\pi\)
\(258\) 6.18664 0.385164
\(259\) 5.40105 0.335605
\(260\) 0 0
\(261\) 6.96239 0.430961
\(262\) 12.4387 0.768463
\(263\) −15.0376 −0.927259 −0.463629 0.886029i \(-0.653453\pi\)
−0.463629 + 0.886029i \(0.653453\pi\)
\(264\) 0.962389 0.0592309
\(265\) 0 0
\(266\) 3.35026 0.205418
\(267\) −4.64974 −0.284559
\(268\) 7.22425 0.441292
\(269\) −4.51388 −0.275216 −0.137608 0.990487i \(-0.543941\pi\)
−0.137608 + 0.990487i \(0.543941\pi\)
\(270\) 0 0
\(271\) 15.8496 0.962792 0.481396 0.876503i \(-0.340130\pi\)
0.481396 + 0.876503i \(0.340130\pi\)
\(272\) 0.387873 0.0235183
\(273\) −5.40105 −0.326886
\(274\) 18.1622 1.09722
\(275\) 0 0
\(276\) −0.962389 −0.0579290
\(277\) −10.3127 −0.619627 −0.309814 0.950797i \(-0.600267\pi\)
−0.309814 + 0.950797i \(0.600267\pi\)
\(278\) 8.77575 0.526334
\(279\) 3.35026 0.200575
\(280\) 0 0
\(281\) 24.3488 1.45253 0.726265 0.687415i \(-0.241254\pi\)
0.726265 + 0.687415i \(0.241254\pi\)
\(282\) 0.962389 0.0573094
\(283\) −26.2882 −1.56267 −0.781336 0.624111i \(-0.785461\pi\)
−0.781336 + 0.624111i \(0.785461\pi\)
\(284\) 7.22425 0.428681
\(285\) 0 0
\(286\) −1.55149 −0.0917417
\(287\) −31.0738 −1.83423
\(288\) 1.00000 0.0589256
\(289\) −16.8496 −0.991150
\(290\) 0 0
\(291\) −10.9624 −0.642627
\(292\) 3.22425 0.188685
\(293\) 13.0738 0.763780 0.381890 0.924208i \(-0.375273\pi\)
0.381890 + 0.924208i \(0.375273\pi\)
\(294\) −4.22425 −0.246363
\(295\) 0 0
\(296\) 1.61213 0.0937030
\(297\) 0.962389 0.0558435
\(298\) 15.9756 0.925439
\(299\) 1.55149 0.0897251
\(300\) 0 0
\(301\) −20.7269 −1.19468
\(302\) 18.4241 1.06019
\(303\) −2.72496 −0.156545
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0.387873 0.0221732
\(307\) 7.07381 0.403724 0.201862 0.979414i \(-0.435301\pi\)
0.201862 + 0.979414i \(0.435301\pi\)
\(308\) −3.22425 −0.183719
\(309\) 0.574515 0.0326830
\(310\) 0 0
\(311\) 26.3127 1.49205 0.746027 0.665916i \(-0.231959\pi\)
0.746027 + 0.665916i \(0.231959\pi\)
\(312\) −1.61213 −0.0912687
\(313\) 18.7005 1.05702 0.528508 0.848928i \(-0.322752\pi\)
0.528508 + 0.848928i \(0.322752\pi\)
\(314\) 13.7889 0.778154
\(315\) 0 0
\(316\) −3.35026 −0.188467
\(317\) −26.4749 −1.48698 −0.743488 0.668749i \(-0.766830\pi\)
−0.743488 + 0.668749i \(0.766830\pi\)
\(318\) 6.00000 0.336463
\(319\) −6.70052 −0.375157
\(320\) 0 0
\(321\) 10.7005 0.597245
\(322\) 3.22425 0.179681
\(323\) 0.387873 0.0215818
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 3.73813 0.207036
\(327\) −10.1260 −0.559970
\(328\) −9.27504 −0.512128
\(329\) −3.22425 −0.177759
\(330\) 0 0
\(331\) −30.7005 −1.68745 −0.843727 0.536773i \(-0.819643\pi\)
−0.843727 + 0.536773i \(0.819643\pi\)
\(332\) −15.0132 −0.823955
\(333\) 1.61213 0.0883440
\(334\) −15.4763 −0.846824
\(335\) 0 0
\(336\) −3.35026 −0.182772
\(337\) −6.81194 −0.371070 −0.185535 0.982638i \(-0.559402\pi\)
−0.185535 + 0.982638i \(0.559402\pi\)
\(338\) −10.4010 −0.565742
\(339\) 20.5501 1.11613
\(340\) 0 0
\(341\) −3.22425 −0.174603
\(342\) 1.00000 0.0540738
\(343\) −9.29948 −0.502125
\(344\) −6.18664 −0.333561
\(345\) 0 0
\(346\) 1.47627 0.0793648
\(347\) −18.3879 −0.987113 −0.493556 0.869714i \(-0.664303\pi\)
−0.493556 + 0.869714i \(0.664303\pi\)
\(348\) −6.96239 −0.373223
\(349\) 31.1490 1.66737 0.833685 0.552241i \(-0.186227\pi\)
0.833685 + 0.552241i \(0.186227\pi\)
\(350\) 0 0
\(351\) −1.61213 −0.0860490
\(352\) −0.962389 −0.0512955
\(353\) −7.61213 −0.405153 −0.202576 0.979266i \(-0.564931\pi\)
−0.202576 + 0.979266i \(0.564931\pi\)
\(354\) −10.3127 −0.548111
\(355\) 0 0
\(356\) 4.64974 0.246436
\(357\) −1.29948 −0.0687756
\(358\) −14.3127 −0.756447
\(359\) −35.3112 −1.86366 −0.931828 0.362901i \(-0.881786\pi\)
−0.931828 + 0.362901i \(0.881786\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −8.82653 −0.463912
\(363\) 10.0738 0.528738
\(364\) 5.40105 0.283092
\(365\) 0 0
\(366\) −11.9248 −0.623318
\(367\) −31.9756 −1.66911 −0.834555 0.550924i \(-0.814275\pi\)
−0.834555 + 0.550924i \(0.814275\pi\)
\(368\) 0.962389 0.0501680
\(369\) −9.27504 −0.482839
\(370\) 0 0
\(371\) −20.1016 −1.04362
\(372\) −3.35026 −0.173703
\(373\) −26.4894 −1.37157 −0.685786 0.727804i \(-0.740541\pi\)
−0.685786 + 0.727804i \(0.740541\pi\)
\(374\) −0.373285 −0.0193021
\(375\) 0 0
\(376\) −0.962389 −0.0496314
\(377\) 11.2243 0.578078
\(378\) −3.35026 −0.172319
\(379\) −19.3258 −0.992701 −0.496350 0.868122i \(-0.665327\pi\)
−0.496350 + 0.868122i \(0.665327\pi\)
\(380\) 0 0
\(381\) 1.35026 0.0691760
\(382\) 2.31265 0.118325
\(383\) −3.37470 −0.172439 −0.0862195 0.996276i \(-0.527479\pi\)
−0.0862195 + 0.996276i \(0.527479\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −7.58769 −0.386203
\(387\) −6.18664 −0.314485
\(388\) 10.9624 0.556531
\(389\) −11.3503 −0.575481 −0.287741 0.957708i \(-0.592904\pi\)
−0.287741 + 0.957708i \(0.592904\pi\)
\(390\) 0 0
\(391\) 0.373285 0.0188778
\(392\) 4.22425 0.213357
\(393\) −12.4387 −0.627447
\(394\) 18.8119 0.947732
\(395\) 0 0
\(396\) −0.962389 −0.0483618
\(397\) 18.8364 0.945371 0.472685 0.881231i \(-0.343285\pi\)
0.472685 + 0.881231i \(0.343285\pi\)
\(398\) −9.40105 −0.471232
\(399\) −3.35026 −0.167723
\(400\) 0 0
\(401\) −4.12601 −0.206043 −0.103022 0.994679i \(-0.532851\pi\)
−0.103022 + 0.994679i \(0.532851\pi\)
\(402\) −7.22425 −0.360313
\(403\) 5.40105 0.269045
\(404\) 2.72496 0.135572
\(405\) 0 0
\(406\) 23.3258 1.15764
\(407\) −1.55149 −0.0769046
\(408\) −0.387873 −0.0192026
\(409\) 2.52373 0.124790 0.0623952 0.998052i \(-0.480126\pi\)
0.0623952 + 0.998052i \(0.480126\pi\)
\(410\) 0 0
\(411\) −18.1622 −0.895875
\(412\) −0.574515 −0.0283043
\(413\) 34.5501 1.70010
\(414\) 0.962389 0.0472988
\(415\) 0 0
\(416\) 1.61213 0.0790410
\(417\) −8.77575 −0.429750
\(418\) −0.962389 −0.0470720
\(419\) 7.51247 0.367008 0.183504 0.983019i \(-0.441256\pi\)
0.183504 + 0.983019i \(0.441256\pi\)
\(420\) 0 0
\(421\) 3.67750 0.179230 0.0896152 0.995976i \(-0.471436\pi\)
0.0896152 + 0.995976i \(0.471436\pi\)
\(422\) 4.77575 0.232480
\(423\) −0.962389 −0.0467929
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −7.22425 −0.350016
\(427\) 39.9511 1.93337
\(428\) −10.7005 −0.517229
\(429\) 1.55149 0.0749068
\(430\) 0 0
\(431\) −10.3272 −0.497446 −0.248723 0.968575i \(-0.580011\pi\)
−0.248723 + 0.968575i \(0.580011\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −31.5877 −1.51801 −0.759004 0.651086i \(-0.774314\pi\)
−0.759004 + 0.651086i \(0.774314\pi\)
\(434\) 11.2243 0.538781
\(435\) 0 0
\(436\) 10.1260 0.484948
\(437\) 0.962389 0.0460373
\(438\) −3.22425 −0.154061
\(439\) −38.1524 −1.82091 −0.910456 0.413605i \(-0.864269\pi\)
−0.910456 + 0.413605i \(0.864269\pi\)
\(440\) 0 0
\(441\) 4.22425 0.201155
\(442\) 0.625301 0.0297425
\(443\) −16.3127 −0.775037 −0.387519 0.921862i \(-0.626668\pi\)
−0.387519 + 0.921862i \(0.626668\pi\)
\(444\) −1.61213 −0.0765082
\(445\) 0 0
\(446\) 24.6761 1.16845
\(447\) −15.9756 −0.755618
\(448\) 3.35026 0.158285
\(449\) −37.5271 −1.77101 −0.885506 0.464629i \(-0.846188\pi\)
−0.885506 + 0.464629i \(0.846188\pi\)
\(450\) 0 0
\(451\) 8.92619 0.420318
\(452\) −20.5501 −0.966594
\(453\) −18.4241 −0.865638
\(454\) −15.4763 −0.726337
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 21.3258 0.996490
\(459\) −0.387873 −0.0181044
\(460\) 0 0
\(461\) 12.3780 0.576502 0.288251 0.957555i \(-0.406926\pi\)
0.288251 + 0.957555i \(0.406926\pi\)
\(462\) 3.22425 0.150006
\(463\) −32.4504 −1.50810 −0.754049 0.656818i \(-0.771902\pi\)
−0.754049 + 0.656818i \(0.771902\pi\)
\(464\) 6.96239 0.323221
\(465\) 0 0
\(466\) 9.01317 0.417527
\(467\) −7.53690 −0.348766 −0.174383 0.984678i \(-0.555793\pi\)
−0.174383 + 0.984678i \(0.555793\pi\)
\(468\) 1.61213 0.0745206
\(469\) 24.2031 1.11760
\(470\) 0 0
\(471\) −13.7889 −0.635360
\(472\) 10.3127 0.474678
\(473\) 5.95395 0.273763
\(474\) 3.35026 0.153883
\(475\) 0 0
\(476\) 1.29948 0.0595614
\(477\) −6.00000 −0.274721
\(478\) −0.135857 −0.00621396
\(479\) 15.2097 0.694947 0.347474 0.937690i \(-0.387040\pi\)
0.347474 + 0.937690i \(0.387040\pi\)
\(480\) 0 0
\(481\) 2.59895 0.118502
\(482\) −25.8496 −1.17741
\(483\) −3.22425 −0.146709
\(484\) −10.0738 −0.457900
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 1.19982 0.0543689 0.0271844 0.999630i \(-0.491346\pi\)
0.0271844 + 0.999630i \(0.491346\pi\)
\(488\) 11.9248 0.539809
\(489\) −3.73813 −0.169044
\(490\) 0 0
\(491\) −5.11283 −0.230739 −0.115369 0.993323i \(-0.536805\pi\)
−0.115369 + 0.993323i \(0.536805\pi\)
\(492\) 9.27504 0.418151
\(493\) 2.70052 0.121625
\(494\) 1.61213 0.0725330
\(495\) 0 0
\(496\) 3.35026 0.150431
\(497\) 24.2031 1.08566
\(498\) 15.0132 0.672756
\(499\) 14.2981 0.640069 0.320035 0.947406i \(-0.396305\pi\)
0.320035 + 0.947406i \(0.396305\pi\)
\(500\) 0 0
\(501\) 15.4763 0.691429
\(502\) −10.1114 −0.451295
\(503\) −11.5125 −0.513316 −0.256658 0.966502i \(-0.582621\pi\)
−0.256658 + 0.966502i \(0.582621\pi\)
\(504\) 3.35026 0.149233
\(505\) 0 0
\(506\) −0.926192 −0.0411742
\(507\) 10.4010 0.461927
\(508\) −1.35026 −0.0599082
\(509\) −31.9610 −1.41665 −0.708323 0.705889i \(-0.750548\pi\)
−0.708323 + 0.705889i \(0.750548\pi\)
\(510\) 0 0
\(511\) 10.8021 0.477857
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −26.9986 −1.19086
\(515\) 0 0
\(516\) 6.18664 0.272352
\(517\) 0.926192 0.0407339
\(518\) 5.40105 0.237308
\(519\) −1.47627 −0.0648010
\(520\) 0 0
\(521\) −33.2750 −1.45781 −0.728903 0.684617i \(-0.759969\pi\)
−0.728903 + 0.684617i \(0.759969\pi\)
\(522\) 6.96239 0.304735
\(523\) 9.29948 0.406638 0.203319 0.979113i \(-0.434827\pi\)
0.203319 + 0.979113i \(0.434827\pi\)
\(524\) 12.4387 0.543385
\(525\) 0 0
\(526\) −15.0376 −0.655671
\(527\) 1.29948 0.0566061
\(528\) 0.962389 0.0418826
\(529\) −22.0738 −0.959731
\(530\) 0 0
\(531\) 10.3127 0.447531
\(532\) 3.35026 0.145252
\(533\) −14.9525 −0.647666
\(534\) −4.64974 −0.201214
\(535\) 0 0
\(536\) 7.22425 0.312040
\(537\) 14.3127 0.617636
\(538\) −4.51388 −0.194607
\(539\) −4.06537 −0.175108
\(540\) 0 0
\(541\) 28.5501 1.22746 0.613732 0.789515i \(-0.289668\pi\)
0.613732 + 0.789515i \(0.289668\pi\)
\(542\) 15.8496 0.680797
\(543\) 8.82653 0.378783
\(544\) 0.387873 0.0166299
\(545\) 0 0
\(546\) −5.40105 −0.231143
\(547\) 28.4749 1.21750 0.608748 0.793363i \(-0.291672\pi\)
0.608748 + 0.793363i \(0.291672\pi\)
\(548\) 18.1622 0.775851
\(549\) 11.9248 0.508937
\(550\) 0 0
\(551\) 6.96239 0.296608
\(552\) −0.962389 −0.0409620
\(553\) −11.2243 −0.477304
\(554\) −10.3127 −0.438143
\(555\) 0 0
\(556\) 8.77575 0.372175
\(557\) 4.88717 0.207076 0.103538 0.994626i \(-0.466984\pi\)
0.103538 + 0.994626i \(0.466984\pi\)
\(558\) 3.35026 0.141828
\(559\) −9.97365 −0.421841
\(560\) 0 0
\(561\) 0.373285 0.0157601
\(562\) 24.3488 1.02709
\(563\) −30.8021 −1.29815 −0.649077 0.760723i \(-0.724845\pi\)
−0.649077 + 0.760723i \(0.724845\pi\)
\(564\) 0.962389 0.0405239
\(565\) 0 0
\(566\) −26.2882 −1.10498
\(567\) 3.35026 0.140698
\(568\) 7.22425 0.303123
\(569\) −24.1260 −1.01141 −0.505707 0.862705i \(-0.668768\pi\)
−0.505707 + 0.862705i \(0.668768\pi\)
\(570\) 0 0
\(571\) −5.67276 −0.237398 −0.118699 0.992930i \(-0.537872\pi\)
−0.118699 + 0.992930i \(0.537872\pi\)
\(572\) −1.55149 −0.0648712
\(573\) −2.31265 −0.0966124
\(574\) −31.0738 −1.29700
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 32.0000 1.33218 0.666089 0.745873i \(-0.267967\pi\)
0.666089 + 0.745873i \(0.267967\pi\)
\(578\) −16.8496 −0.700849
\(579\) 7.58769 0.315334
\(580\) 0 0
\(581\) −50.2981 −2.08672
\(582\) −10.9624 −0.454406
\(583\) 5.77433 0.239148
\(584\) 3.22425 0.133421
\(585\) 0 0
\(586\) 13.0738 0.540074
\(587\) 13.6121 0.561833 0.280916 0.959732i \(-0.409362\pi\)
0.280916 + 0.959732i \(0.409362\pi\)
\(588\) −4.22425 −0.174205
\(589\) 3.35026 0.138045
\(590\) 0 0
\(591\) −18.8119 −0.773820
\(592\) 1.61213 0.0662580
\(593\) −23.4617 −0.963456 −0.481728 0.876321i \(-0.659991\pi\)
−0.481728 + 0.876321i \(0.659991\pi\)
\(594\) 0.962389 0.0394873
\(595\) 0 0
\(596\) 15.9756 0.654385
\(597\) 9.40105 0.384759
\(598\) 1.55149 0.0634452
\(599\) 10.0263 0.409665 0.204833 0.978797i \(-0.434335\pi\)
0.204833 + 0.978797i \(0.434335\pi\)
\(600\) 0 0
\(601\) 11.7743 0.480285 0.240143 0.970738i \(-0.422806\pi\)
0.240143 + 0.970738i \(0.422806\pi\)
\(602\) −20.7269 −0.844764
\(603\) 7.22425 0.294194
\(604\) 18.4241 0.749665
\(605\) 0 0
\(606\) −2.72496 −0.110694
\(607\) −38.4993 −1.56264 −0.781319 0.624132i \(-0.785453\pi\)
−0.781319 + 0.624132i \(0.785453\pi\)
\(608\) 1.00000 0.0405554
\(609\) −23.3258 −0.945210
\(610\) 0 0
\(611\) −1.55149 −0.0627667
\(612\) 0.387873 0.0156788
\(613\) 7.61213 0.307451 0.153725 0.988114i \(-0.450873\pi\)
0.153725 + 0.988114i \(0.450873\pi\)
\(614\) 7.07381 0.285476
\(615\) 0 0
\(616\) −3.22425 −0.129909
\(617\) 29.5369 1.18911 0.594555 0.804055i \(-0.297328\pi\)
0.594555 + 0.804055i \(0.297328\pi\)
\(618\) 0.574515 0.0231104
\(619\) −22.5501 −0.906364 −0.453182 0.891418i \(-0.649711\pi\)
−0.453182 + 0.891418i \(0.649711\pi\)
\(620\) 0 0
\(621\) −0.962389 −0.0386193
\(622\) 26.3127 1.05504
\(623\) 15.5778 0.624113
\(624\) −1.61213 −0.0645367
\(625\) 0 0
\(626\) 18.7005 0.747423
\(627\) 0.962389 0.0384341
\(628\) 13.7889 0.550238
\(629\) 0.625301 0.0249324
\(630\) 0 0
\(631\) −37.9248 −1.50976 −0.754881 0.655862i \(-0.772305\pi\)
−0.754881 + 0.655862i \(0.772305\pi\)
\(632\) −3.35026 −0.133266
\(633\) −4.77575 −0.189819
\(634\) −26.4749 −1.05145
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 6.81003 0.269823
\(638\) −6.70052 −0.265276
\(639\) 7.22425 0.285787
\(640\) 0 0
\(641\) 6.67609 0.263690 0.131845 0.991270i \(-0.457910\pi\)
0.131845 + 0.991270i \(0.457910\pi\)
\(642\) 10.7005 0.422316
\(643\) 25.2605 0.996175 0.498087 0.867127i \(-0.334036\pi\)
0.498087 + 0.867127i \(0.334036\pi\)
\(644\) 3.22425 0.127053
\(645\) 0 0
\(646\) 0.387873 0.0152607
\(647\) 16.9135 0.664939 0.332469 0.943114i \(-0.392118\pi\)
0.332469 + 0.943114i \(0.392118\pi\)
\(648\) 1.00000 0.0392837
\(649\) −9.92478 −0.389582
\(650\) 0 0
\(651\) −11.2243 −0.439913
\(652\) 3.73813 0.146397
\(653\) 24.1114 0.943553 0.471776 0.881718i \(-0.343613\pi\)
0.471776 + 0.881718i \(0.343613\pi\)
\(654\) −10.1260 −0.395958
\(655\) 0 0
\(656\) −9.27504 −0.362129
\(657\) 3.22425 0.125790
\(658\) −3.22425 −0.125694
\(659\) −20.3879 −0.794199 −0.397099 0.917776i \(-0.629983\pi\)
−0.397099 + 0.917776i \(0.629983\pi\)
\(660\) 0 0
\(661\) 17.6023 0.684649 0.342325 0.939582i \(-0.388786\pi\)
0.342325 + 0.939582i \(0.388786\pi\)
\(662\) −30.7005 −1.19321
\(663\) −0.625301 −0.0242847
\(664\) −15.0132 −0.582624
\(665\) 0 0
\(666\) 1.61213 0.0624686
\(667\) 6.70052 0.259445
\(668\) −15.4763 −0.598795
\(669\) −24.6761 −0.954033
\(670\) 0 0
\(671\) −11.4763 −0.443036
\(672\) −3.35026 −0.129239
\(673\) 44.3634 1.71008 0.855042 0.518558i \(-0.173531\pi\)
0.855042 + 0.518558i \(0.173531\pi\)
\(674\) −6.81194 −0.262386
\(675\) 0 0
\(676\) −10.4010 −0.400040
\(677\) 8.70052 0.334388 0.167194 0.985924i \(-0.446529\pi\)
0.167194 + 0.985924i \(0.446529\pi\)
\(678\) 20.5501 0.789221
\(679\) 36.7269 1.40945
\(680\) 0 0
\(681\) 15.4763 0.593052
\(682\) −3.22425 −0.123463
\(683\) −37.8759 −1.44928 −0.724641 0.689127i \(-0.757994\pi\)
−0.724641 + 0.689127i \(0.757994\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) −9.29948 −0.355056
\(687\) −21.3258 −0.813631
\(688\) −6.18664 −0.235864
\(689\) −9.67276 −0.368503
\(690\) 0 0
\(691\) −0.775746 −0.0295108 −0.0147554 0.999891i \(-0.504697\pi\)
−0.0147554 + 0.999891i \(0.504697\pi\)
\(692\) 1.47627 0.0561194
\(693\) −3.22425 −0.122479
\(694\) −18.3879 −0.697994
\(695\) 0 0
\(696\) −6.96239 −0.263909
\(697\) −3.59754 −0.136266
\(698\) 31.1490 1.17901
\(699\) −9.01317 −0.340910
\(700\) 0 0
\(701\) 42.3752 1.60049 0.800245 0.599674i \(-0.204703\pi\)
0.800245 + 0.599674i \(0.204703\pi\)
\(702\) −1.61213 −0.0608458
\(703\) 1.61213 0.0608025
\(704\) −0.962389 −0.0362714
\(705\) 0 0
\(706\) −7.61213 −0.286486
\(707\) 9.12933 0.343344
\(708\) −10.3127 −0.387573
\(709\) 36.2784 1.36246 0.681231 0.732068i \(-0.261445\pi\)
0.681231 + 0.732068i \(0.261445\pi\)
\(710\) 0 0
\(711\) −3.35026 −0.125645
\(712\) 4.64974 0.174256
\(713\) 3.22425 0.120749
\(714\) −1.29948 −0.0486317
\(715\) 0 0
\(716\) −14.3127 −0.534889
\(717\) 0.135857 0.00507368
\(718\) −35.3112 −1.31780
\(719\) −42.5355 −1.58631 −0.793153 0.609022i \(-0.791562\pi\)
−0.793153 + 0.609022i \(0.791562\pi\)
\(720\) 0 0
\(721\) −1.92478 −0.0716824
\(722\) 1.00000 0.0372161
\(723\) 25.8496 0.961355
\(724\) −8.82653 −0.328035
\(725\) 0 0
\(726\) 10.0738 0.373874
\(727\) −0.378024 −0.0140201 −0.00701007 0.999975i \(-0.502231\pi\)
−0.00701007 + 0.999975i \(0.502231\pi\)
\(728\) 5.40105 0.200176
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −2.39963 −0.0887536
\(732\) −11.9248 −0.440752
\(733\) 26.0118 0.960766 0.480383 0.877059i \(-0.340498\pi\)
0.480383 + 0.877059i \(0.340498\pi\)
\(734\) −31.9756 −1.18024
\(735\) 0 0
\(736\) 0.962389 0.0354741
\(737\) −6.95254 −0.256100
\(738\) −9.27504 −0.341419
\(739\) −44.8773 −1.65084 −0.825419 0.564520i \(-0.809061\pi\)
−0.825419 + 0.564520i \(0.809061\pi\)
\(740\) 0 0
\(741\) −1.61213 −0.0592230
\(742\) −20.1016 −0.737952
\(743\) −4.67418 −0.171479 −0.0857394 0.996318i \(-0.527325\pi\)
−0.0857394 + 0.996318i \(0.527325\pi\)
\(744\) −3.35026 −0.122827
\(745\) 0 0
\(746\) −26.4894 −0.969847
\(747\) −15.0132 −0.549303
\(748\) −0.373285 −0.0136486
\(749\) −35.8496 −1.30991
\(750\) 0 0
\(751\) −6.57452 −0.239907 −0.119954 0.992779i \(-0.538275\pi\)
−0.119954 + 0.992779i \(0.538275\pi\)
\(752\) −0.962389 −0.0350947
\(753\) 10.1114 0.368481
\(754\) 11.2243 0.408763
\(755\) 0 0
\(756\) −3.35026 −0.121848
\(757\) −15.5633 −0.565656 −0.282828 0.959171i \(-0.591273\pi\)
−0.282828 + 0.959171i \(0.591273\pi\)
\(758\) −19.3258 −0.701946
\(759\) 0.926192 0.0336186
\(760\) 0 0
\(761\) −23.8759 −0.865501 −0.432750 0.901514i \(-0.642457\pi\)
−0.432750 + 0.901514i \(0.642457\pi\)
\(762\) 1.35026 0.0489148
\(763\) 33.9248 1.22816
\(764\) 2.31265 0.0836688
\(765\) 0 0
\(766\) −3.37470 −0.121933
\(767\) 16.6253 0.600305
\(768\) −1.00000 −0.0360844
\(769\) 30.4749 1.09895 0.549476 0.835510i \(-0.314827\pi\)
0.549476 + 0.835510i \(0.314827\pi\)
\(770\) 0 0
\(771\) 26.9986 0.972330
\(772\) −7.58769 −0.273087
\(773\) 16.3272 0.587250 0.293625 0.955921i \(-0.405138\pi\)
0.293625 + 0.955921i \(0.405138\pi\)
\(774\) −6.18664 −0.222374
\(775\) 0 0
\(776\) 10.9624 0.393527
\(777\) −5.40105 −0.193761
\(778\) −11.3503 −0.406927
\(779\) −9.27504 −0.332313
\(780\) 0 0
\(781\) −6.95254 −0.248781
\(782\) 0.373285 0.0133486
\(783\) −6.96239 −0.248815
\(784\) 4.22425 0.150866
\(785\) 0 0
\(786\) −12.4387 −0.443672
\(787\) 30.9525 1.10334 0.551669 0.834063i \(-0.313991\pi\)
0.551669 + 0.834063i \(0.313991\pi\)
\(788\) 18.8119 0.670148
\(789\) 15.0376 0.535353
\(790\) 0 0
\(791\) −68.8481 −2.44796
\(792\) −0.962389 −0.0341970
\(793\) 19.2243 0.682673
\(794\) 18.8364 0.668478
\(795\) 0 0
\(796\) −9.40105 −0.333211
\(797\) 14.8773 0.526982 0.263491 0.964662i \(-0.415126\pi\)
0.263491 + 0.964662i \(0.415126\pi\)
\(798\) −3.35026 −0.118598
\(799\) −0.373285 −0.0132059
\(800\) 0 0
\(801\) 4.64974 0.164290
\(802\) −4.12601 −0.145694
\(803\) −3.10299 −0.109502
\(804\) −7.22425 −0.254780
\(805\) 0 0
\(806\) 5.40105 0.190244
\(807\) 4.51388 0.158896
\(808\) 2.72496 0.0958638
\(809\) 25.4471 0.894672 0.447336 0.894366i \(-0.352373\pi\)
0.447336 + 0.894366i \(0.352373\pi\)
\(810\) 0 0
\(811\) −53.3522 −1.87345 −0.936724 0.350069i \(-0.886158\pi\)
−0.936724 + 0.350069i \(0.886158\pi\)
\(812\) 23.3258 0.818576
\(813\) −15.8496 −0.555868
\(814\) −1.55149 −0.0543798
\(815\) 0 0
\(816\) −0.387873 −0.0135783
\(817\) −6.18664 −0.216443
\(818\) 2.52373 0.0882402
\(819\) 5.40105 0.188728
\(820\) 0 0
\(821\) −27.9756 −0.976354 −0.488177 0.872745i \(-0.662338\pi\)
−0.488177 + 0.872745i \(0.662338\pi\)
\(822\) −18.1622 −0.633480
\(823\) −7.87399 −0.274470 −0.137235 0.990539i \(-0.543822\pi\)
−0.137235 + 0.990539i \(0.543822\pi\)
\(824\) −0.574515 −0.0200142
\(825\) 0 0
\(826\) 34.5501 1.20215
\(827\) 40.1016 1.39447 0.697234 0.716843i \(-0.254414\pi\)
0.697234 + 0.716843i \(0.254414\pi\)
\(828\) 0.962389 0.0334453
\(829\) 47.2262 1.64023 0.820116 0.572197i \(-0.193909\pi\)
0.820116 + 0.572197i \(0.193909\pi\)
\(830\) 0 0
\(831\) 10.3127 0.357742
\(832\) 1.61213 0.0558904
\(833\) 1.63847 0.0567698
\(834\) −8.77575 −0.303879
\(835\) 0 0
\(836\) −0.962389 −0.0332849
\(837\) −3.35026 −0.115802
\(838\) 7.51247 0.259514
\(839\) −3.89843 −0.134589 −0.0672944 0.997733i \(-0.521437\pi\)
−0.0672944 + 0.997733i \(0.521437\pi\)
\(840\) 0 0
\(841\) 19.4749 0.671547
\(842\) 3.67750 0.126735
\(843\) −24.3488 −0.838619
\(844\) 4.77575 0.164388
\(845\) 0 0
\(846\) −0.962389 −0.0330876
\(847\) −33.7499 −1.15966
\(848\) −6.00000 −0.206041
\(849\) 26.2882 0.902209
\(850\) 0 0
\(851\) 1.55149 0.0531845
\(852\) −7.22425 −0.247499
\(853\) 35.1900 1.20488 0.602441 0.798164i \(-0.294195\pi\)
0.602441 + 0.798164i \(0.294195\pi\)
\(854\) 39.9511 1.36710
\(855\) 0 0
\(856\) −10.7005 −0.365736
\(857\) −12.1768 −0.415951 −0.207976 0.978134i \(-0.566687\pi\)
−0.207976 + 0.978134i \(0.566687\pi\)
\(858\) 1.55149 0.0529671
\(859\) 57.3522 1.95683 0.978415 0.206648i \(-0.0662554\pi\)
0.978415 + 0.206648i \(0.0662554\pi\)
\(860\) 0 0
\(861\) 31.0738 1.05899
\(862\) −10.3272 −0.351747
\(863\) 14.0752 0.479126 0.239563 0.970881i \(-0.422996\pi\)
0.239563 + 0.970881i \(0.422996\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −31.5877 −1.07339
\(867\) 16.8496 0.572241
\(868\) 11.2243 0.380976
\(869\) 3.22425 0.109375
\(870\) 0 0
\(871\) 11.6464 0.394624
\(872\) 10.1260 0.342910
\(873\) 10.9624 0.371021
\(874\) 0.962389 0.0325533
\(875\) 0 0
\(876\) −3.22425 −0.108937
\(877\) −18.8627 −0.636949 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(878\) −38.1524 −1.28758
\(879\) −13.0738 −0.440969
\(880\) 0 0
\(881\) −19.1490 −0.645147 −0.322574 0.946544i \(-0.604548\pi\)
−0.322574 + 0.946544i \(0.604548\pi\)
\(882\) 4.22425 0.142238
\(883\) −42.9135 −1.44415 −0.722077 0.691812i \(-0.756813\pi\)
−0.722077 + 0.691812i \(0.756813\pi\)
\(884\) 0.625301 0.0210311
\(885\) 0 0
\(886\) −16.3127 −0.548034
\(887\) −22.7005 −0.762209 −0.381104 0.924532i \(-0.624456\pi\)
−0.381104 + 0.924532i \(0.624456\pi\)
\(888\) −1.61213 −0.0540994
\(889\) −4.52373 −0.151721
\(890\) 0 0
\(891\) −0.962389 −0.0322412
\(892\) 24.6761 0.826216
\(893\) −0.962389 −0.0322051
\(894\) −15.9756 −0.534303
\(895\) 0 0
\(896\) 3.35026 0.111924
\(897\) −1.55149 −0.0518028
\(898\) −37.5271 −1.25229
\(899\) 23.3258 0.777960
\(900\) 0 0
\(901\) −2.32724 −0.0775316
\(902\) 8.92619 0.297210
\(903\) 20.7269 0.689747
\(904\) −20.5501 −0.683485
\(905\) 0 0
\(906\) −18.4241 −0.612099
\(907\) −14.5501 −0.483127 −0.241564 0.970385i \(-0.577660\pi\)
−0.241564 + 0.970385i \(0.577660\pi\)
\(908\) −15.4763 −0.513598
\(909\) 2.72496 0.0903813
\(910\) 0 0
\(911\) −0.998585 −0.0330846 −0.0165423 0.999863i \(-0.505266\pi\)
−0.0165423 + 0.999863i \(0.505266\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 14.4485 0.478176
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) 21.3258 0.704625
\(917\) 41.6728 1.37616
\(918\) −0.387873 −0.0128017
\(919\) −33.7743 −1.11411 −0.557056 0.830475i \(-0.688069\pi\)
−0.557056 + 0.830475i \(0.688069\pi\)
\(920\) 0 0
\(921\) −7.07381 −0.233090
\(922\) 12.3780 0.407649
\(923\) 11.6464 0.383346
\(924\) 3.22425 0.106070
\(925\) 0 0
\(926\) −32.4504 −1.06639
\(927\) −0.574515 −0.0188696
\(928\) 6.96239 0.228552
\(929\) 14.8773 0.488109 0.244054 0.969762i \(-0.421522\pi\)
0.244054 + 0.969762i \(0.421522\pi\)
\(930\) 0 0
\(931\) 4.22425 0.138444
\(932\) 9.01317 0.295236
\(933\) −26.3127 −0.861437
\(934\) −7.53690 −0.246615
\(935\) 0 0
\(936\) 1.61213 0.0526940
\(937\) −22.2981 −0.728446 −0.364223 0.931312i \(-0.618665\pi\)
−0.364223 + 0.931312i \(0.618665\pi\)
\(938\) 24.2031 0.790261
\(939\) −18.7005 −0.610269
\(940\) 0 0
\(941\) −36.3634 −1.18541 −0.592707 0.805418i \(-0.701941\pi\)
−0.592707 + 0.805418i \(0.701941\pi\)
\(942\) −13.7889 −0.449267
\(943\) −8.92619 −0.290677
\(944\) 10.3127 0.335648
\(945\) 0 0
\(946\) 5.95395 0.193580
\(947\) −50.3390 −1.63580 −0.817899 0.575362i \(-0.804861\pi\)
−0.817899 + 0.575362i \(0.804861\pi\)
\(948\) 3.35026 0.108811
\(949\) 5.19791 0.168731
\(950\) 0 0
\(951\) 26.4749 0.858506
\(952\) 1.29948 0.0421163
\(953\) 16.3272 0.528891 0.264446 0.964401i \(-0.414811\pi\)
0.264446 + 0.964401i \(0.414811\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −0.135857 −0.00439393
\(957\) 6.70052 0.216597
\(958\) 15.2097 0.491402
\(959\) 60.8481 1.96489
\(960\) 0 0
\(961\) −19.7757 −0.637927
\(962\) 2.59895 0.0837936
\(963\) −10.7005 −0.344820
\(964\) −25.8496 −0.832558
\(965\) 0 0
\(966\) −3.22425 −0.103739
\(967\) −0.276454 −0.00889015 −0.00444507 0.999990i \(-0.501415\pi\)
−0.00444507 + 0.999990i \(0.501415\pi\)
\(968\) −10.0738 −0.323784
\(969\) −0.387873 −0.0124603
\(970\) 0 0
\(971\) 37.9102 1.21660 0.608298 0.793709i \(-0.291853\pi\)
0.608298 + 0.793709i \(0.291853\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 29.4010 0.942554
\(974\) 1.19982 0.0384446
\(975\) 0 0
\(976\) 11.9248 0.381703
\(977\) −28.3996 −0.908585 −0.454292 0.890853i \(-0.650108\pi\)
−0.454292 + 0.890853i \(0.650108\pi\)
\(978\) −3.73813 −0.119532
\(979\) −4.47486 −0.143017
\(980\) 0 0
\(981\) 10.1260 0.323299
\(982\) −5.11283 −0.163157
\(983\) 0.926192 0.0295409 0.0147705 0.999891i \(-0.495298\pi\)
0.0147705 + 0.999891i \(0.495298\pi\)
\(984\) 9.27504 0.295677
\(985\) 0 0
\(986\) 2.70052 0.0860022
\(987\) 3.22425 0.102629
\(988\) 1.61213 0.0512886
\(989\) −5.95395 −0.189325
\(990\) 0 0
\(991\) 30.5256 0.969679 0.484839 0.874603i \(-0.338878\pi\)
0.484839 + 0.874603i \(0.338878\pi\)
\(992\) 3.35026 0.106371
\(993\) 30.7005 0.974252
\(994\) 24.2031 0.767677
\(995\) 0 0
\(996\) 15.0132 0.475711
\(997\) −21.0132 −0.665494 −0.332747 0.943016i \(-0.607975\pi\)
−0.332747 + 0.943016i \(0.607975\pi\)
\(998\) 14.2981 0.452597
\(999\) −1.61213 −0.0510054
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 2850.2.a.bm.1.3 3
3.2 odd 2 8550.2.a.ce.1.3 3
5.2 odd 4 570.2.d.c.229.4 yes 6
5.3 odd 4 570.2.d.c.229.1 6
5.4 even 2 2850.2.a.bl.1.1 3
15.2 even 4 1710.2.d.f.1369.3 6
15.8 even 4 1710.2.d.f.1369.6 6
15.14 odd 2 8550.2.a.cq.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.d.c.229.1 6 5.3 odd 4
570.2.d.c.229.4 yes 6 5.2 odd 4
1710.2.d.f.1369.3 6 15.2 even 4
1710.2.d.f.1369.6 6 15.8 even 4
2850.2.a.bl.1.1 3 5.4 even 2
2850.2.a.bm.1.3 3 1.1 even 1 trivial
8550.2.a.ce.1.3 3 3.2 odd 2
8550.2.a.cq.1.1 3 15.14 odd 2