Properties

Label 3450.2.a.bs.1.3
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3368.1
Defining polynomial: \(x^{3} - x^{2} - 15 x + 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.76300\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.76300 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.76300 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.18291 q^{11} -1.00000 q^{12} +6.34310 q^{13} +3.76300 q^{14} +1.00000 q^{16} +2.58010 q^{17} +1.00000 q^{18} -6.34310 q^{19} -3.76300 q^{21} +3.18291 q^{22} +1.00000 q^{23} -1.00000 q^{24} +6.34310 q^{26} -1.00000 q^{27} +3.76300 q^{28} -1.00000 q^{29} +6.00000 q^{31} +1.00000 q^{32} -3.18291 q^{33} +2.58010 q^{34} +1.00000 q^{36} -2.58010 q^{37} -6.34310 q^{38} -6.34310 q^{39} +12.3431 q^{41} -3.76300 q^{42} -10.3431 q^{43} +3.18291 q^{44} +1.00000 q^{46} -3.34310 q^{47} -1.00000 q^{48} +7.16019 q^{49} -2.58010 q^{51} +6.34310 q^{52} -7.16019 q^{53} -1.00000 q^{54} +3.76300 q^{56} +6.34310 q^{57} -1.00000 q^{58} -4.36581 q^{59} -1.52601 q^{61} +6.00000 q^{62} +3.76300 q^{63} +1.00000 q^{64} -3.18291 q^{66} -10.6862 q^{67} +2.58010 q^{68} -1.00000 q^{69} +1.81709 q^{71} +1.00000 q^{72} -8.52601 q^{73} -2.58010 q^{74} -6.34310 q^{76} +11.9773 q^{77} -6.34310 q^{78} -3.18291 q^{79} +1.00000 q^{81} +12.3431 q^{82} +7.28901 q^{83} -3.76300 q^{84} -10.3431 q^{86} +1.00000 q^{87} +3.18291 q^{88} +4.58010 q^{89} +23.8691 q^{91} +1.00000 q^{92} -6.00000 q^{93} -3.34310 q^{94} -1.00000 q^{96} -11.5260 q^{97} +7.16019 q^{98} +3.18291 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - q^{7} + 3q^{8} + 3q^{9} + 3q^{11} - 3q^{12} + q^{13} - q^{14} + 3q^{16} + 2q^{17} + 3q^{18} - q^{19} + q^{21} + 3q^{22} + 3q^{23} - 3q^{24} + q^{26} - 3q^{27} - q^{28} - 3q^{29} + 18q^{31} + 3q^{32} - 3q^{33} + 2q^{34} + 3q^{36} - 2q^{37} - q^{38} - q^{39} + 19q^{41} + q^{42} - 13q^{43} + 3q^{44} + 3q^{46} + 8q^{47} - 3q^{48} + 10q^{49} - 2q^{51} + q^{52} - 10q^{53} - 3q^{54} - q^{56} + q^{57} - 3q^{58} + 20q^{61} + 18q^{62} - q^{63} + 3q^{64} - 3q^{66} + 4q^{67} + 2q^{68} - 3q^{69} + 12q^{71} + 3q^{72} - q^{73} - 2q^{74} - q^{76} + 31q^{77} - q^{78} - 3q^{79} + 3q^{81} + 19q^{82} - 15q^{83} + q^{84} - 13q^{86} + 3q^{87} + 3q^{88} + 8q^{89} + 29q^{91} + 3q^{92} - 18q^{93} + 8q^{94} - 3q^{96} - 10q^{97} + 10q^{98} + 3q^{99} + O(q^{100}) \)

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
\(7\) 3.76300 1.42228 0.711141 0.703050i \(-0.248179\pi\)
0.711141 + 0.703050i \(0.248179\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.18291 0.959682 0.479841 0.877355i \(-0.340694\pi\)
0.479841 + 0.877355i \(0.340694\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.34310 1.75926 0.879630 0.475659i \(-0.157790\pi\)
0.879630 + 0.475659i \(0.157790\pi\)
\(14\) 3.76300 1.00570
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.58010 0.625765 0.312883 0.949792i \(-0.398705\pi\)
0.312883 + 0.949792i \(0.398705\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.34310 −1.45521 −0.727603 0.685998i \(-0.759366\pi\)
−0.727603 + 0.685998i \(0.759366\pi\)
\(20\) 0 0
\(21\) −3.76300 −0.821155
\(22\) 3.18291 0.678598
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 6.34310 1.24398
\(27\) −1.00000 −0.192450
\(28\) 3.76300 0.711141
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.18291 −0.554073
\(34\) 2.58010 0.442483
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.58010 −0.424165 −0.212083 0.977252i \(-0.568025\pi\)
−0.212083 + 0.977252i \(0.568025\pi\)
\(38\) −6.34310 −1.02899
\(39\) −6.34310 −1.01571
\(40\) 0 0
\(41\) 12.3431 1.92767 0.963834 0.266503i \(-0.0858681\pi\)
0.963834 + 0.266503i \(0.0858681\pi\)
\(42\) −3.76300 −0.580644
\(43\) −10.3431 −1.57731 −0.788654 0.614837i \(-0.789222\pi\)
−0.788654 + 0.614837i \(0.789222\pi\)
\(44\) 3.18291 0.479841
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −3.34310 −0.487641 −0.243821 0.969820i \(-0.578401\pi\)
−0.243821 + 0.969820i \(0.578401\pi\)
\(48\) −1.00000 −0.144338
\(49\) 7.16019 1.02288
\(50\) 0 0
\(51\) −2.58010 −0.361286
\(52\) 6.34310 0.879630
\(53\) −7.16019 −0.983528 −0.491764 0.870728i \(-0.663648\pi\)
−0.491764 + 0.870728i \(0.663648\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 3.76300 0.502852
\(57\) 6.34310 0.840164
\(58\) −1.00000 −0.131306
\(59\) −4.36581 −0.568380 −0.284190 0.958768i \(-0.591725\pi\)
−0.284190 + 0.958768i \(0.591725\pi\)
\(60\) 0 0
\(61\) −1.52601 −0.195385 −0.0976926 0.995217i \(-0.531146\pi\)
−0.0976926 + 0.995217i \(0.531146\pi\)
\(62\) 6.00000 0.762001
\(63\) 3.76300 0.474094
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.18291 −0.391789
\(67\) −10.6862 −1.30553 −0.652764 0.757562i \(-0.726390\pi\)
−0.652764 + 0.757562i \(0.726390\pi\)
\(68\) 2.58010 0.312883
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 1.81709 0.215649 0.107825 0.994170i \(-0.465611\pi\)
0.107825 + 0.994170i \(0.465611\pi\)
\(72\) 1.00000 0.117851
\(73\) −8.52601 −0.997894 −0.498947 0.866633i \(-0.666280\pi\)
−0.498947 + 0.866633i \(0.666280\pi\)
\(74\) −2.58010 −0.299930
\(75\) 0 0
\(76\) −6.34310 −0.727603
\(77\) 11.9773 1.36494
\(78\) −6.34310 −0.718215
\(79\) −3.18291 −0.358105 −0.179052 0.983840i \(-0.557303\pi\)
−0.179052 + 0.983840i \(0.557303\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 12.3431 1.36307
\(83\) 7.28901 0.800073 0.400036 0.916499i \(-0.368997\pi\)
0.400036 + 0.916499i \(0.368997\pi\)
\(84\) −3.76300 −0.410577
\(85\) 0 0
\(86\) −10.3431 −1.11533
\(87\) 1.00000 0.107211
\(88\) 3.18291 0.339299
\(89\) 4.58010 0.485489 0.242745 0.970090i \(-0.421952\pi\)
0.242745 + 0.970090i \(0.421952\pi\)
\(90\) 0 0
\(91\) 23.8691 2.50216
\(92\) 1.00000 0.104257
\(93\) −6.00000 −0.622171
\(94\) −3.34310 −0.344814
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −11.5260 −1.17029 −0.585144 0.810929i \(-0.698962\pi\)
−0.585144 + 0.810929i \(0.698962\pi\)
\(98\) 7.16019 0.723289
\(99\) 3.18291 0.319894
\(100\) 0 0
\(101\) −7.70891 −0.767066 −0.383533 0.923527i \(-0.625293\pi\)
−0.383533 + 0.923527i \(0.625293\pi\)
\(102\) −2.58010 −0.255468
\(103\) −3.39719 −0.334735 −0.167368 0.985895i \(-0.553527\pi\)
−0.167368 + 0.985895i \(0.553527\pi\)
\(104\) 6.34310 0.621992
\(105\) 0 0
\(106\) −7.16019 −0.695459
\(107\) −6.68620 −0.646379 −0.323190 0.946334i \(-0.604755\pi\)
−0.323190 + 0.946334i \(0.604755\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 5.78572 0.554171 0.277086 0.960845i \(-0.410631\pi\)
0.277086 + 0.960845i \(0.410631\pi\)
\(110\) 0 0
\(111\) 2.58010 0.244892
\(112\) 3.76300 0.355570
\(113\) 0.580097 0.0545709 0.0272855 0.999628i \(-0.491314\pi\)
0.0272855 + 0.999628i \(0.491314\pi\)
\(114\) 6.34310 0.594086
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 6.34310 0.586420
\(118\) −4.36581 −0.401906
\(119\) 9.70891 0.890015
\(120\) 0 0
\(121\) −0.869107 −0.0790097
\(122\) −1.52601 −0.138158
\(123\) −12.3431 −1.11294
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) 3.76300 0.335235
\(127\) 14.3204 1.27073 0.635364 0.772212i \(-0.280850\pi\)
0.635364 + 0.772212i \(0.280850\pi\)
\(128\) 1.00000 0.0883883
\(129\) 10.3431 0.910659
\(130\) 0 0
\(131\) 15.8918 1.38847 0.694237 0.719746i \(-0.255742\pi\)
0.694237 + 0.719746i \(0.255742\pi\)
\(132\) −3.18291 −0.277036
\(133\) −23.8691 −2.06971
\(134\) −10.6862 −0.923147
\(135\) 0 0
\(136\) 2.58010 0.221241
\(137\) 15.4199 1.31741 0.658706 0.752401i \(-0.271104\pi\)
0.658706 + 0.752401i \(0.271104\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 3.02271 0.256383 0.128192 0.991749i \(-0.459083\pi\)
0.128192 + 0.991749i \(0.459083\pi\)
\(140\) 0 0
\(141\) 3.34310 0.281540
\(142\) 1.81709 0.152487
\(143\) 20.1895 1.68833
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −8.52601 −0.705617
\(147\) −7.16019 −0.590563
\(148\) −2.58010 −0.212083
\(149\) 16.6862 1.36699 0.683493 0.729957i \(-0.260460\pi\)
0.683493 + 0.729957i \(0.260460\pi\)
\(150\) 0 0
\(151\) −7.16019 −0.582688 −0.291344 0.956618i \(-0.594102\pi\)
−0.291344 + 0.956618i \(0.594102\pi\)
\(152\) −6.34310 −0.514493
\(153\) 2.58010 0.208588
\(154\) 11.9773 0.965157
\(155\) 0 0
\(156\) −6.34310 −0.507854
\(157\) −22.2122 −1.77273 −0.886364 0.462990i \(-0.846777\pi\)
−0.886364 + 0.462990i \(0.846777\pi\)
\(158\) −3.18291 −0.253218
\(159\) 7.16019 0.567840
\(160\) 0 0
\(161\) 3.76300 0.296566
\(162\) 1.00000 0.0785674
\(163\) 8.68620 0.680356 0.340178 0.940361i \(-0.389513\pi\)
0.340178 + 0.940361i \(0.389513\pi\)
\(164\) 12.3431 0.963834
\(165\) 0 0
\(166\) 7.28901 0.565737
\(167\) 25.2349 1.95274 0.976368 0.216113i \(-0.0693380\pi\)
0.976368 + 0.216113i \(0.0693380\pi\)
\(168\) −3.76300 −0.290322
\(169\) 27.2349 2.09499
\(170\) 0 0
\(171\) −6.34310 −0.485069
\(172\) −10.3431 −0.788654
\(173\) −0.817094 −0.0621225 −0.0310612 0.999517i \(-0.509889\pi\)
−0.0310612 + 0.999517i \(0.509889\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) 3.18291 0.239921
\(177\) 4.36581 0.328155
\(178\) 4.58010 0.343293
\(179\) −3.16019 −0.236204 −0.118102 0.993001i \(-0.537681\pi\)
−0.118102 + 0.993001i \(0.537681\pi\)
\(180\) 0 0
\(181\) 3.37448 0.250823 0.125411 0.992105i \(-0.459975\pi\)
0.125411 + 0.992105i \(0.459975\pi\)
\(182\) 23.8691 1.76930
\(183\) 1.52601 0.112806
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) 8.21221 0.600536
\(188\) −3.34310 −0.243821
\(189\) −3.76300 −0.273718
\(190\) 0 0
\(191\) −11.8691 −0.858818 −0.429409 0.903110i \(-0.641278\pi\)
−0.429409 + 0.903110i \(0.641278\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.97729 0.214310 0.107155 0.994242i \(-0.465826\pi\)
0.107155 + 0.994242i \(0.465826\pi\)
\(194\) −11.5260 −0.827519
\(195\) 0 0
\(196\) 7.16019 0.511442
\(197\) 24.8918 1.77347 0.886734 0.462279i \(-0.152968\pi\)
0.886734 + 0.462279i \(0.152968\pi\)
\(198\) 3.18291 0.226199
\(199\) 22.4492 1.59138 0.795691 0.605703i \(-0.207108\pi\)
0.795691 + 0.605703i \(0.207108\pi\)
\(200\) 0 0
\(201\) 10.6862 0.753746
\(202\) −7.70891 −0.542397
\(203\) −3.76300 −0.264111
\(204\) −2.58010 −0.180643
\(205\) 0 0
\(206\) −3.39719 −0.236693
\(207\) 1.00000 0.0695048
\(208\) 6.34310 0.439815
\(209\) −20.1895 −1.39654
\(210\) 0 0
\(211\) 19.5553 1.34624 0.673121 0.739532i \(-0.264953\pi\)
0.673121 + 0.739532i \(0.264953\pi\)
\(212\) −7.16019 −0.491764
\(213\) −1.81709 −0.124505
\(214\) −6.68620 −0.457059
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 22.5780 1.53270
\(218\) 5.78572 0.391858
\(219\) 8.52601 0.576134
\(220\) 0 0
\(221\) 16.3658 1.10088
\(222\) 2.58010 0.173165
\(223\) −28.8984 −1.93518 −0.967590 0.252525i \(-0.918739\pi\)
−0.967590 + 0.252525i \(0.918739\pi\)
\(224\) 3.76300 0.251426
\(225\) 0 0
\(226\) 0.580097 0.0385875
\(227\) −17.6321 −1.17028 −0.585142 0.810931i \(-0.698961\pi\)
−0.585142 + 0.810931i \(0.698961\pi\)
\(228\) 6.34310 0.420082
\(229\) −11.8918 −0.785834 −0.392917 0.919574i \(-0.628534\pi\)
−0.392917 + 0.919574i \(0.628534\pi\)
\(230\) 0 0
\(231\) −11.9773 −0.788048
\(232\) −1.00000 −0.0656532
\(233\) −9.02930 −0.591529 −0.295765 0.955261i \(-0.595574\pi\)
−0.295765 + 0.955261i \(0.595574\pi\)
\(234\) 6.34310 0.414661
\(235\) 0 0
\(236\) −4.36581 −0.284190
\(237\) 3.18291 0.206752
\(238\) 9.70891 0.629335
\(239\) −20.0747 −1.29853 −0.649263 0.760564i \(-0.724923\pi\)
−0.649263 + 0.760564i \(0.724923\pi\)
\(240\) 0 0
\(241\) −23.0066 −1.48198 −0.740992 0.671514i \(-0.765645\pi\)
−0.740992 + 0.671514i \(0.765645\pi\)
\(242\) −0.869107 −0.0558683
\(243\) −1.00000 −0.0641500
\(244\) −1.52601 −0.0976926
\(245\) 0 0
\(246\) −12.3431 −0.786967
\(247\) −40.2349 −2.56009
\(248\) 6.00000 0.381000
\(249\) −7.28901 −0.461922
\(250\) 0 0
\(251\) −10.9459 −0.690900 −0.345450 0.938437i \(-0.612274\pi\)
−0.345450 + 0.938437i \(0.612274\pi\)
\(252\) 3.76300 0.237047
\(253\) 3.18291 0.200108
\(254\) 14.3204 0.898541
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.3204 1.01804 0.509019 0.860755i \(-0.330008\pi\)
0.509019 + 0.860755i \(0.330008\pi\)
\(258\) 10.3431 0.643933
\(259\) −9.70891 −0.603282
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 15.8918 0.981800
\(263\) −29.0066 −1.78862 −0.894311 0.447445i \(-0.852334\pi\)
−0.894311 + 0.447445i \(0.852334\pi\)
\(264\) −3.18291 −0.195894
\(265\) 0 0
\(266\) −23.8691 −1.46351
\(267\) −4.58010 −0.280297
\(268\) −10.6862 −0.652764
\(269\) −26.1895 −1.59680 −0.798401 0.602126i \(-0.794320\pi\)
−0.798401 + 0.602126i \(0.794320\pi\)
\(270\) 0 0
\(271\) 6.36581 0.386696 0.193348 0.981130i \(-0.438065\pi\)
0.193348 + 0.981130i \(0.438065\pi\)
\(272\) 2.58010 0.156441
\(273\) −23.8691 −1.44462
\(274\) 15.4199 0.931550
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 9.50329 0.570998 0.285499 0.958379i \(-0.407841\pi\)
0.285499 + 0.958379i \(0.407841\pi\)
\(278\) 3.02271 0.181290
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 27.2663 1.62657 0.813285 0.581865i \(-0.197677\pi\)
0.813285 + 0.581865i \(0.197677\pi\)
\(282\) 3.34310 0.199079
\(283\) 10.3204 0.613483 0.306742 0.951793i \(-0.400761\pi\)
0.306742 + 0.951793i \(0.400761\pi\)
\(284\) 1.81709 0.107825
\(285\) 0 0
\(286\) 20.1895 1.19383
\(287\) 46.4471 2.74169
\(288\) 1.00000 0.0589256
\(289\) −10.3431 −0.608418
\(290\) 0 0
\(291\) 11.5260 0.675666
\(292\) −8.52601 −0.498947
\(293\) −8.36581 −0.488736 −0.244368 0.969683i \(-0.578580\pi\)
−0.244368 + 0.969683i \(0.578580\pi\)
\(294\) −7.16019 −0.417591
\(295\) 0 0
\(296\) −2.58010 −0.149965
\(297\) −3.18291 −0.184691
\(298\) 16.6862 0.966606
\(299\) 6.34310 0.366831
\(300\) 0 0
\(301\) −38.9211 −2.24338
\(302\) −7.16019 −0.412023
\(303\) 7.70891 0.442865
\(304\) −6.34310 −0.363802
\(305\) 0 0
\(306\) 2.58010 0.147494
\(307\) −33.7089 −1.92387 −0.961935 0.273280i \(-0.911892\pi\)
−0.961935 + 0.273280i \(0.911892\pi\)
\(308\) 11.9773 0.682469
\(309\) 3.39719 0.193259
\(310\) 0 0
\(311\) −1.86252 −0.105614 −0.0528069 0.998605i \(-0.516817\pi\)
−0.0528069 + 0.998605i \(0.516817\pi\)
\(312\) −6.34310 −0.359107
\(313\) 26.8984 1.52039 0.760194 0.649696i \(-0.225104\pi\)
0.760194 + 0.649696i \(0.225104\pi\)
\(314\) −22.2122 −1.25351
\(315\) 0 0
\(316\) −3.18291 −0.179052
\(317\) 11.0000 0.617822 0.308911 0.951091i \(-0.400036\pi\)
0.308911 + 0.951091i \(0.400036\pi\)
\(318\) 7.16019 0.401524
\(319\) −3.18291 −0.178209
\(320\) 0 0
\(321\) 6.68620 0.373187
\(322\) 3.76300 0.209704
\(323\) −16.3658 −0.910618
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 8.68620 0.481084
\(327\) −5.78572 −0.319951
\(328\) 12.3431 0.681534
\(329\) −12.5801 −0.693563
\(330\) 0 0
\(331\) −6.18291 −0.339843 −0.169922 0.985458i \(-0.554351\pi\)
−0.169922 + 0.985458i \(0.554351\pi\)
\(332\) 7.28901 0.400036
\(333\) −2.58010 −0.141388
\(334\) 25.2349 1.38079
\(335\) 0 0
\(336\) −3.76300 −0.205289
\(337\) 15.1602 0.825828 0.412914 0.910770i \(-0.364511\pi\)
0.412914 + 0.910770i \(0.364511\pi\)
\(338\) 27.2349 1.48138
\(339\) −0.580097 −0.0315065
\(340\) 0 0
\(341\) 19.0974 1.03418
\(342\) −6.34310 −0.342996
\(343\) 0.602810 0.0325487
\(344\) −10.3431 −0.557663
\(345\) 0 0
\(346\) −0.817094 −0.0439272
\(347\) −29.8918 −1.60468 −0.802338 0.596869i \(-0.796411\pi\)
−0.802338 + 0.596869i \(0.796411\pi\)
\(348\) 1.00000 0.0536056
\(349\) 7.02930 0.376270 0.188135 0.982143i \(-0.439756\pi\)
0.188135 + 0.982143i \(0.439756\pi\)
\(350\) 0 0
\(351\) −6.34310 −0.338570
\(352\) 3.18291 0.169649
\(353\) −26.2349 −1.39634 −0.698172 0.715930i \(-0.746003\pi\)
−0.698172 + 0.715930i \(0.746003\pi\)
\(354\) 4.36581 0.232040
\(355\) 0 0
\(356\) 4.58010 0.242745
\(357\) −9.70891 −0.513850
\(358\) −3.16019 −0.167021
\(359\) 19.5033 1.02934 0.514672 0.857387i \(-0.327914\pi\)
0.514672 + 0.857387i \(0.327914\pi\)
\(360\) 0 0
\(361\) 21.2349 1.11763
\(362\) 3.37448 0.177359
\(363\) 0.869107 0.0456163
\(364\) 23.8691 1.25108
\(365\) 0 0
\(366\) 1.52601 0.0797656
\(367\) −33.5033 −1.74886 −0.874429 0.485154i \(-0.838764\pi\)
−0.874429 + 0.485154i \(0.838764\pi\)
\(368\) 1.00000 0.0521286
\(369\) 12.3431 0.642556
\(370\) 0 0
\(371\) −26.9438 −1.39885
\(372\) −6.00000 −0.311086
\(373\) 10.1515 0.525626 0.262813 0.964847i \(-0.415350\pi\)
0.262813 + 0.964847i \(0.415350\pi\)
\(374\) 8.21221 0.424643
\(375\) 0 0
\(376\) −3.34310 −0.172407
\(377\) −6.34310 −0.326686
\(378\) −3.76300 −0.193548
\(379\) 20.5780 1.05702 0.528511 0.848926i \(-0.322751\pi\)
0.528511 + 0.848926i \(0.322751\pi\)
\(380\) 0 0
\(381\) −14.3204 −0.733656
\(382\) −11.8691 −0.607276
\(383\) −1.22833 −0.0627648 −0.0313824 0.999507i \(-0.509991\pi\)
−0.0313824 + 0.999507i \(0.509991\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 2.97729 0.151540
\(387\) −10.3431 −0.525769
\(388\) −11.5260 −0.585144
\(389\) 35.3724 1.79345 0.896726 0.442586i \(-0.145939\pi\)
0.896726 + 0.442586i \(0.145939\pi\)
\(390\) 0 0
\(391\) 2.58010 0.130481
\(392\) 7.16019 0.361644
\(393\) −15.8918 −0.801636
\(394\) 24.8918 1.25403
\(395\) 0 0
\(396\) 3.18291 0.159947
\(397\) −31.0066 −1.55618 −0.778088 0.628155i \(-0.783810\pi\)
−0.778088 + 0.628155i \(0.783810\pi\)
\(398\) 22.4492 1.12528
\(399\) 23.8691 1.19495
\(400\) 0 0
\(401\) 15.2056 0.759332 0.379666 0.925124i \(-0.376039\pi\)
0.379666 + 0.925124i \(0.376039\pi\)
\(402\) 10.6862 0.532979
\(403\) 38.0586 1.89583
\(404\) −7.70891 −0.383533
\(405\) 0 0
\(406\) −3.76300 −0.186755
\(407\) −8.21221 −0.407064
\(408\) −2.58010 −0.127734
\(409\) −2.57143 −0.127149 −0.0635746 0.997977i \(-0.520250\pi\)
−0.0635746 + 0.997977i \(0.520250\pi\)
\(410\) 0 0
\(411\) −15.4199 −0.760608
\(412\) −3.39719 −0.167368
\(413\) −16.4286 −0.808397
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 6.34310 0.310996
\(417\) −3.02271 −0.148023
\(418\) −20.1895 −0.987500
\(419\) −19.0768 −0.931963 −0.465981 0.884795i \(-0.654299\pi\)
−0.465981 + 0.884795i \(0.654299\pi\)
\(420\) 0 0
\(421\) −15.8918 −0.774520 −0.387260 0.921971i \(-0.626578\pi\)
−0.387260 + 0.921971i \(0.626578\pi\)
\(422\) 19.5553 0.951937
\(423\) −3.34310 −0.162547
\(424\) −7.16019 −0.347730
\(425\) 0 0
\(426\) −1.81709 −0.0880385
\(427\) −5.74237 −0.277893
\(428\) −6.68620 −0.323190
\(429\) −20.1895 −0.974758
\(430\) 0 0
\(431\) −24.2122 −1.16626 −0.583130 0.812379i \(-0.698172\pi\)
−0.583130 + 0.812379i \(0.698172\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 12.2122 0.586881 0.293441 0.955977i \(-0.405200\pi\)
0.293441 + 0.955977i \(0.405200\pi\)
\(434\) 22.5780 1.08378
\(435\) 0 0
\(436\) 5.78572 0.277086
\(437\) −6.34310 −0.303432
\(438\) 8.52601 0.407388
\(439\) −0.794381 −0.0379137 −0.0189569 0.999820i \(-0.506035\pi\)
−0.0189569 + 0.999820i \(0.506035\pi\)
\(440\) 0 0
\(441\) 7.16019 0.340962
\(442\) 16.3658 0.778442
\(443\) −14.2122 −0.675242 −0.337621 0.941282i \(-0.609622\pi\)
−0.337621 + 0.941282i \(0.609622\pi\)
\(444\) 2.58010 0.122446
\(445\) 0 0
\(446\) −28.8984 −1.36838
\(447\) −16.6862 −0.789230
\(448\) 3.76300 0.177785
\(449\) 1.05201 0.0496476 0.0248238 0.999692i \(-0.492098\pi\)
0.0248238 + 0.999692i \(0.492098\pi\)
\(450\) 0 0
\(451\) 39.2869 1.84995
\(452\) 0.580097 0.0272855
\(453\) 7.16019 0.336415
\(454\) −17.6321 −0.827516
\(455\) 0 0
\(456\) 6.34310 0.297043
\(457\) 0.686200 0.0320991 0.0160495 0.999871i \(-0.494891\pi\)
0.0160495 + 0.999871i \(0.494891\pi\)
\(458\) −11.8918 −0.555668
\(459\) −2.58010 −0.120429
\(460\) 0 0
\(461\) −29.5780 −1.37759 −0.688793 0.724958i \(-0.741859\pi\)
−0.688793 + 0.724958i \(0.741859\pi\)
\(462\) −11.9773 −0.557234
\(463\) 12.0000 0.557687 0.278844 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) −9.02930 −0.418274
\(467\) 29.2890 1.35533 0.677667 0.735369i \(-0.262991\pi\)
0.677667 + 0.735369i \(0.262991\pi\)
\(468\) 6.34310 0.293210
\(469\) −40.2122 −1.85683
\(470\) 0 0
\(471\) 22.2122 1.02348
\(472\) −4.36581 −0.200953
\(473\) −32.9211 −1.51371
\(474\) 3.18291 0.146196
\(475\) 0 0
\(476\) 9.70891 0.445007
\(477\) −7.16019 −0.327843
\(478\) −20.0747 −0.918197
\(479\) 15.9145 0.727154 0.363577 0.931564i \(-0.381555\pi\)
0.363577 + 0.931564i \(0.381555\pi\)
\(480\) 0 0
\(481\) −16.3658 −0.746217
\(482\) −23.0066 −1.04792
\(483\) −3.76300 −0.171223
\(484\) −0.869107 −0.0395049
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −38.9438 −1.76471 −0.882357 0.470581i \(-0.844044\pi\)
−0.882357 + 0.470581i \(0.844044\pi\)
\(488\) −1.52601 −0.0690791
\(489\) −8.68620 −0.392804
\(490\) 0 0
\(491\) 27.4806 1.24018 0.620091 0.784530i \(-0.287096\pi\)
0.620091 + 0.784530i \(0.287096\pi\)
\(492\) −12.3431 −0.556470
\(493\) −2.58010 −0.116202
\(494\) −40.2349 −1.81025
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 6.83773 0.306714
\(498\) −7.28901 −0.326628
\(499\) 6.97729 0.312346 0.156173 0.987730i \(-0.450084\pi\)
0.156173 + 0.987730i \(0.450084\pi\)
\(500\) 0 0
\(501\) −25.2349 −1.12741
\(502\) −10.9459 −0.488540
\(503\) −7.65690 −0.341404 −0.170702 0.985323i \(-0.554604\pi\)
−0.170702 + 0.985323i \(0.554604\pi\)
\(504\) 3.76300 0.167617
\(505\) 0 0
\(506\) 3.18291 0.141497
\(507\) −27.2349 −1.20955
\(508\) 14.3204 0.635364
\(509\) −34.6073 −1.53394 −0.766971 0.641681i \(-0.778237\pi\)
−0.766971 + 0.641681i \(0.778237\pi\)
\(510\) 0 0
\(511\) −32.0834 −1.41929
\(512\) 1.00000 0.0441942
\(513\) 6.34310 0.280055
\(514\) 16.3204 0.719861
\(515\) 0 0
\(516\) 10.3431 0.455330
\(517\) −10.6408 −0.467981
\(518\) −9.70891 −0.426585
\(519\) 0.817094 0.0358664
\(520\) 0 0
\(521\) 25.4199 1.11367 0.556833 0.830624i \(-0.312016\pi\)
0.556833 + 0.830624i \(0.312016\pi\)
\(522\) −1.00000 −0.0437688
\(523\) −25.8691 −1.13118 −0.565589 0.824688i \(-0.691351\pi\)
−0.565589 + 0.824688i \(0.691351\pi\)
\(524\) 15.8918 0.694237
\(525\) 0 0
\(526\) −29.0066 −1.26475
\(527\) 15.4806 0.674345
\(528\) −3.18291 −0.138518
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −4.36581 −0.189460
\(532\) −23.8691 −1.03486
\(533\) 78.2935 3.39127
\(534\) −4.58010 −0.198200
\(535\) 0 0
\(536\) −10.6862 −0.461574
\(537\) 3.16019 0.136372
\(538\) −26.1895 −1.12911
\(539\) 22.7902 0.981645
\(540\) 0 0
\(541\) 34.6007 1.48760 0.743801 0.668401i \(-0.233021\pi\)
0.743801 + 0.668401i \(0.233021\pi\)
\(542\) 6.36581 0.273435
\(543\) −3.37448 −0.144813
\(544\) 2.58010 0.110621
\(545\) 0 0
\(546\) −23.8691 −1.02150
\(547\) 3.55531 0.152014 0.0760070 0.997107i \(-0.475783\pi\)
0.0760070 + 0.997107i \(0.475783\pi\)
\(548\) 15.4199 0.658706
\(549\) −1.52601 −0.0651284
\(550\) 0 0
\(551\) 6.34310 0.270225
\(552\) −1.00000 −0.0425628
\(553\) −11.9773 −0.509326
\(554\) 9.50329 0.403756
\(555\) 0 0
\(556\) 3.02271 0.128192
\(557\) −15.2642 −0.646766 −0.323383 0.946268i \(-0.604820\pi\)
−0.323383 + 0.946268i \(0.604820\pi\)
\(558\) 6.00000 0.254000
\(559\) −65.6073 −2.77489
\(560\) 0 0
\(561\) −8.21221 −0.346720
\(562\) 27.2663 1.15016
\(563\) −13.5487 −0.571010 −0.285505 0.958377i \(-0.592161\pi\)
−0.285505 + 0.958377i \(0.592161\pi\)
\(564\) 3.34310 0.140770
\(565\) 0 0
\(566\) 10.3204 0.433798
\(567\) 3.76300 0.158031
\(568\) 1.81709 0.0762436
\(569\) −18.4740 −0.774470 −0.387235 0.921981i \(-0.626570\pi\)
−0.387235 + 0.921981i \(0.626570\pi\)
\(570\) 0 0
\(571\) −9.89182 −0.413960 −0.206980 0.978345i \(-0.566363\pi\)
−0.206980 + 0.978345i \(0.566363\pi\)
\(572\) 20.1895 0.844165
\(573\) 11.8691 0.495839
\(574\) 46.4471 1.93867
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −29.5033 −1.22824 −0.614119 0.789213i \(-0.710489\pi\)
−0.614119 + 0.789213i \(0.710489\pi\)
\(578\) −10.3431 −0.430216
\(579\) −2.97729 −0.123732
\(580\) 0 0
\(581\) 27.4286 1.13793
\(582\) 11.5260 0.477768
\(583\) −22.7902 −0.943875
\(584\) −8.52601 −0.352809
\(585\) 0 0
\(586\) −8.36581 −0.345589
\(587\) 14.4740 0.597406 0.298703 0.954346i \(-0.403446\pi\)
0.298703 + 0.954346i \(0.403446\pi\)
\(588\) −7.16019 −0.295281
\(589\) −38.0586 −1.56818
\(590\) 0 0
\(591\) −24.8918 −1.02391
\(592\) −2.58010 −0.106041
\(593\) 5.86911 0.241015 0.120508 0.992712i \(-0.461548\pi\)
0.120508 + 0.992712i \(0.461548\pi\)
\(594\) −3.18291 −0.130596
\(595\) 0 0
\(596\) 16.6862 0.683493
\(597\) −22.4492 −0.918785
\(598\) 6.34310 0.259389
\(599\) −3.95457 −0.161580 −0.0807898 0.996731i \(-0.525744\pi\)
−0.0807898 + 0.996731i \(0.525744\pi\)
\(600\) 0 0
\(601\) 9.23492 0.376700 0.188350 0.982102i \(-0.439686\pi\)
0.188350 + 0.982102i \(0.439686\pi\)
\(602\) −38.9211 −1.58631
\(603\) −10.6862 −0.435176
\(604\) −7.16019 −0.291344
\(605\) 0 0
\(606\) 7.70891 0.313153
\(607\) 48.7448 1.97849 0.989245 0.146266i \(-0.0467257\pi\)
0.989245 + 0.146266i \(0.0467257\pi\)
\(608\) −6.34310 −0.257247
\(609\) 3.76300 0.152485
\(610\) 0 0
\(611\) −21.2056 −0.857888
\(612\) 2.58010 0.104294
\(613\) 3.67754 0.148534 0.0742671 0.997238i \(-0.476338\pi\)
0.0742671 + 0.997238i \(0.476338\pi\)
\(614\) −33.7089 −1.36038
\(615\) 0 0
\(616\) 11.9773 0.482579
\(617\) 0.794381 0.0319806 0.0159903 0.999872i \(-0.494910\pi\)
0.0159903 + 0.999872i \(0.494910\pi\)
\(618\) 3.39719 0.136655
\(619\) −4.57802 −0.184006 −0.0920031 0.995759i \(-0.529327\pi\)
−0.0920031 + 0.995759i \(0.529327\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −1.86252 −0.0746802
\(623\) 17.2349 0.690502
\(624\) −6.34310 −0.253927
\(625\) 0 0
\(626\) 26.8984 1.07508
\(627\) 20.1895 0.806291
\(628\) −22.2122 −0.886364
\(629\) −6.65690 −0.265428
\(630\) 0 0
\(631\) −31.1808 −1.24129 −0.620645 0.784092i \(-0.713129\pi\)
−0.620645 + 0.784092i \(0.713129\pi\)
\(632\) −3.18291 −0.126609
\(633\) −19.5553 −0.777254
\(634\) 11.0000 0.436866
\(635\) 0 0
\(636\) 7.16019 0.283920
\(637\) 45.4178 1.79952
\(638\) −3.18291 −0.126012
\(639\) 1.81709 0.0718831
\(640\) 0 0
\(641\) 46.7295 1.84571 0.922853 0.385152i \(-0.125851\pi\)
0.922853 + 0.385152i \(0.125851\pi\)
\(642\) 6.68620 0.263883
\(643\) −4.81709 −0.189968 −0.0949838 0.995479i \(-0.530280\pi\)
−0.0949838 + 0.995479i \(0.530280\pi\)
\(644\) 3.76300 0.148283
\(645\) 0 0
\(646\) −16.3658 −0.643904
\(647\) −19.4967 −0.766495 −0.383247 0.923646i \(-0.625194\pi\)
−0.383247 + 0.923646i \(0.625194\pi\)
\(648\) 1.00000 0.0392837
\(649\) −13.8960 −0.545465
\(650\) 0 0
\(651\) −22.5780 −0.884902
\(652\) 8.68620 0.340178
\(653\) 30.5260 1.19457 0.597287 0.802027i \(-0.296245\pi\)
0.597287 + 0.802027i \(0.296245\pi\)
\(654\) −5.78572 −0.226239
\(655\) 0 0
\(656\) 12.3431 0.481917
\(657\) −8.52601 −0.332631
\(658\) −12.5801 −0.490423
\(659\) −2.49463 −0.0971769 −0.0485885 0.998819i \(-0.515472\pi\)
−0.0485885 + 0.998819i \(0.515472\pi\)
\(660\) 0 0
\(661\) −16.7337 −0.650866 −0.325433 0.945565i \(-0.605510\pi\)
−0.325433 + 0.945565i \(0.605510\pi\)
\(662\) −6.18291 −0.240305
\(663\) −16.3658 −0.635596
\(664\) 7.28901 0.282868
\(665\) 0 0
\(666\) −2.58010 −0.0999767
\(667\) −1.00000 −0.0387202
\(668\) 25.2349 0.976368
\(669\) 28.8984 1.11728
\(670\) 0 0
\(671\) −4.85714 −0.187508
\(672\) −3.76300 −0.145161
\(673\) −2.73821 −0.105550 −0.0527752 0.998606i \(-0.516807\pi\)
−0.0527752 + 0.998606i \(0.516807\pi\)
\(674\) 15.1602 0.583949
\(675\) 0 0
\(676\) 27.2349 1.04750
\(677\) −38.1040 −1.46446 −0.732228 0.681059i \(-0.761520\pi\)
−0.732228 + 0.681059i \(0.761520\pi\)
\(678\) −0.580097 −0.0222785
\(679\) −43.3724 −1.66448
\(680\) 0 0
\(681\) 17.6321 0.675664
\(682\) 19.0974 0.731279
\(683\) 24.6234 0.942190 0.471095 0.882082i \(-0.343859\pi\)
0.471095 + 0.882082i \(0.343859\pi\)
\(684\) −6.34310 −0.242534
\(685\) 0 0
\(686\) 0.602810 0.0230154
\(687\) 11.8918 0.453701
\(688\) −10.3431 −0.394327
\(689\) −45.4178 −1.73028
\(690\) 0 0
\(691\) 43.2935 1.64696 0.823482 0.567343i \(-0.192029\pi\)
0.823482 + 0.567343i \(0.192029\pi\)
\(692\) −0.817094 −0.0310612
\(693\) 11.9773 0.454980
\(694\) −29.8918 −1.13468
\(695\) 0 0
\(696\) 1.00000 0.0379049
\(697\) 31.8464 1.20627
\(698\) 7.02930 0.266063
\(699\) 9.02930 0.341520
\(700\) 0 0
\(701\) 9.00659 0.340174 0.170087 0.985429i \(-0.445595\pi\)
0.170087 + 0.985429i \(0.445595\pi\)
\(702\) −6.34310 −0.239405
\(703\) 16.3658 0.617248
\(704\) 3.18291 0.119960
\(705\) 0 0
\(706\) −26.2349 −0.987364
\(707\) −29.0087 −1.09098
\(708\) 4.36581 0.164077
\(709\) 22.3183 0.838182 0.419091 0.907944i \(-0.362349\pi\)
0.419091 + 0.907944i \(0.362349\pi\)
\(710\) 0 0
\(711\) −3.18291 −0.119368
\(712\) 4.58010 0.171646
\(713\) 6.00000 0.224702
\(714\) −9.70891 −0.363347
\(715\) 0 0
\(716\) −3.16019 −0.118102
\(717\) 20.0747 0.749704
\(718\) 19.5033 0.727856
\(719\) −32.1829 −1.20022 −0.600110 0.799918i \(-0.704877\pi\)
−0.600110 + 0.799918i \(0.704877\pi\)
\(720\) 0 0
\(721\) −12.7836 −0.476088
\(722\) 21.2349 0.790282
\(723\) 23.0066 0.855624
\(724\) 3.37448 0.125411
\(725\) 0 0
\(726\) 0.869107 0.0322556
\(727\) −32.3658 −1.20038 −0.600191 0.799857i \(-0.704909\pi\)
−0.600191 + 0.799857i \(0.704909\pi\)
\(728\) 23.8691 0.884648
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −26.6862 −0.987025
\(732\) 1.52601 0.0564028
\(733\) 34.6841 1.28109 0.640544 0.767922i \(-0.278709\pi\)
0.640544 + 0.767922i \(0.278709\pi\)
\(734\) −33.5033 −1.23663
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −34.0132 −1.25289
\(738\) 12.3431 0.454356
\(739\) 45.1441 1.66065 0.830326 0.557278i \(-0.188154\pi\)
0.830326 + 0.557278i \(0.188154\pi\)
\(740\) 0 0
\(741\) 40.2349 1.47807
\(742\) −26.9438 −0.989139
\(743\) −3.02930 −0.111134 −0.0555671 0.998455i \(-0.517697\pi\)
−0.0555671 + 0.998455i \(0.517697\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0 0
\(746\) 10.1515 0.371674
\(747\) 7.28901 0.266691
\(748\) 8.21221 0.300268
\(749\) −25.1602 −0.919333
\(750\) 0 0
\(751\) −12.3410 −0.450330 −0.225165 0.974321i \(-0.572292\pi\)
−0.225165 + 0.974321i \(0.572292\pi\)
\(752\) −3.34310 −0.121910
\(753\) 10.9459 0.398891
\(754\) −6.34310 −0.231002
\(755\) 0 0
\(756\) −3.76300 −0.136859
\(757\) 6.00208 0.218149 0.109075 0.994034i \(-0.465211\pi\)
0.109075 + 0.994034i \(0.465211\pi\)
\(758\) 20.5780 0.747427
\(759\) −3.18291 −0.115532
\(760\) 0 0
\(761\) −26.2977 −0.953290 −0.476645 0.879096i \(-0.658147\pi\)
−0.476645 + 0.879096i \(0.658147\pi\)
\(762\) −14.3204 −0.518773
\(763\) 21.7717 0.788187
\(764\) −11.8691 −0.429409
\(765\) 0 0
\(766\) −1.22833 −0.0443814
\(767\) −27.6928 −0.999929
\(768\) −1.00000 −0.0360844
\(769\) −45.8918 −1.65490 −0.827451 0.561538i \(-0.810210\pi\)
−0.827451 + 0.561538i \(0.810210\pi\)
\(770\) 0 0
\(771\) −16.3204 −0.587764
\(772\) 2.97729 0.107155
\(773\) 37.5846 1.35182 0.675912 0.736982i \(-0.263750\pi\)
0.675912 + 0.736982i \(0.263750\pi\)
\(774\) −10.3431 −0.371775
\(775\) 0 0
\(776\) −11.5260 −0.413760
\(777\) 9.70891 0.348305
\(778\) 35.3724 1.26816
\(779\) −78.2935 −2.80516
\(780\) 0 0
\(781\) 5.78364 0.206955
\(782\) 2.58010 0.0922641
\(783\) 1.00000 0.0357371
\(784\) 7.16019 0.255721
\(785\) 0 0
\(786\) −15.8918 −0.566842
\(787\) −20.9211 −0.745757 −0.372879 0.927880i \(-0.621629\pi\)
−0.372879 + 0.927880i \(0.621629\pi\)
\(788\) 24.8918 0.886734
\(789\) 29.0066 1.03266
\(790\) 0 0
\(791\) 2.18291 0.0776152
\(792\) 3.18291 0.113100
\(793\) −9.67961 −0.343733
\(794\) −31.0066 −1.10038
\(795\) 0 0
\(796\) 22.4492 0.795691
\(797\) 46.5326 1.64827 0.824135 0.566394i \(-0.191662\pi\)
0.824135 + 0.566394i \(0.191662\pi\)
\(798\) 23.8691 0.844957
\(799\) −8.62552 −0.305149
\(800\) 0 0
\(801\) 4.58010 0.161830
\(802\) 15.2056 0.536929
\(803\) −27.1375 −0.957661
\(804\) 10.6862 0.376873
\(805\) 0 0
\(806\) 38.0586 1.34056
\(807\) 26.1895 0.921914
\(808\) −7.70891 −0.271199
\(809\) 26.1895 0.920774 0.460387 0.887718i \(-0.347711\pi\)
0.460387 + 0.887718i \(0.347711\pi\)
\(810\) 0 0
\(811\) 10.4112 0.365588 0.182794 0.983151i \(-0.441486\pi\)
0.182794 + 0.983151i \(0.441486\pi\)
\(812\) −3.76300 −0.132056
\(813\) −6.36581 −0.223259
\(814\) −8.21221 −0.287838
\(815\) 0 0
\(816\) −2.58010 −0.0903215
\(817\) 65.6073 2.29531
\(818\) −2.57143 −0.0899080
\(819\) 23.8691 0.834054
\(820\) 0 0
\(821\) 32.8171 1.14532 0.572662 0.819791i \(-0.305911\pi\)
0.572662 + 0.819791i \(0.305911\pi\)
\(822\) −15.4199 −0.537831
\(823\) −20.5780 −0.717305 −0.358652 0.933471i \(-0.616764\pi\)
−0.358652 + 0.933471i \(0.616764\pi\)
\(824\) −3.39719 −0.118347
\(825\) 0 0
\(826\) −16.4286 −0.571623
\(827\) 21.1982 0.737132 0.368566 0.929602i \(-0.379849\pi\)
0.368566 + 0.929602i \(0.379849\pi\)
\(828\) 1.00000 0.0347524
\(829\) −3.29109 −0.114304 −0.0571521 0.998365i \(-0.518202\pi\)
−0.0571521 + 0.998365i \(0.518202\pi\)
\(830\) 0 0
\(831\) −9.50329 −0.329666
\(832\) 6.34310 0.219907
\(833\) 18.4740 0.640086
\(834\) −3.02271 −0.104668
\(835\) 0 0
\(836\) −20.1895 −0.698268
\(837\) −6.00000 −0.207390
\(838\) −19.0768 −0.658997
\(839\) −20.0813 −0.693284 −0.346642 0.937998i \(-0.612678\pi\)
−0.346642 + 0.937998i \(0.612678\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −15.8918 −0.547668
\(843\) −27.2663 −0.939101
\(844\) 19.5553 0.673121
\(845\) 0 0
\(846\) −3.34310 −0.114938
\(847\) −3.27045 −0.112374
\(848\) −7.16019 −0.245882
\(849\) −10.3204 −0.354195
\(850\) 0 0
\(851\) −2.58010 −0.0884446
\(852\) −1.81709 −0.0622526
\(853\) −32.9211 −1.12720 −0.563599 0.826049i \(-0.690584\pi\)
−0.563599 + 0.826049i \(0.690584\pi\)
\(854\) −5.74237 −0.196500
\(855\) 0 0
\(856\) −6.68620 −0.228530
\(857\) 46.5780 1.59107 0.795537 0.605904i \(-0.207189\pi\)
0.795537 + 0.605904i \(0.207189\pi\)
\(858\) −20.1895 −0.689258
\(859\) −42.0132 −1.43347 −0.716736 0.697345i \(-0.754365\pi\)
−0.716736 + 0.697345i \(0.754365\pi\)
\(860\) 0 0
\(861\) −46.4471 −1.58291
\(862\) −24.2122 −0.824671
\(863\) −15.5553 −0.529509 −0.264754 0.964316i \(-0.585291\pi\)
−0.264754 + 0.964316i \(0.585291\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 12.2122 0.414988
\(867\) 10.3431 0.351270
\(868\) 22.5780 0.766348
\(869\) −10.1309 −0.343667
\(870\) 0 0
\(871\) −67.7836 −2.29676
\(872\) 5.78572 0.195929
\(873\) −11.5260 −0.390096
\(874\) −6.34310 −0.214559
\(875\) 0 0
\(876\) 8.52601 0.288067
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) −0.794381 −0.0268091
\(879\) 8.36581 0.282172
\(880\) 0 0
\(881\) 10.4740 0.352878 0.176439 0.984312i \(-0.443542\pi\)
0.176439 + 0.984312i \(0.443542\pi\)
\(882\) 7.16019 0.241096
\(883\) −35.8757 −1.20731 −0.603657 0.797244i \(-0.706290\pi\)
−0.603657 + 0.797244i \(0.706290\pi\)
\(884\) 16.3658 0.550442
\(885\) 0 0
\(886\) −14.2122 −0.477468
\(887\) −7.29767 −0.245032 −0.122516 0.992467i \(-0.539096\pi\)
−0.122516 + 0.992467i \(0.539096\pi\)
\(888\) 2.58010 0.0865824
\(889\) 53.8877 1.80733
\(890\) 0 0
\(891\) 3.18291 0.106631
\(892\) −28.8984 −0.967590
\(893\) 21.2056 0.709619
\(894\) −16.6862 −0.558070
\(895\) 0 0
\(896\) 3.76300 0.125713
\(897\) −6.34310 −0.211790
\(898\) 1.05201 0.0351061
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) −18.4740 −0.615458
\(902\) 39.2869 1.30811
\(903\) 38.9211 1.29521
\(904\) 0.580097 0.0192937
\(905\) 0 0
\(906\) 7.16019 0.237882
\(907\) −40.7089 −1.35172 −0.675859 0.737031i \(-0.736227\pi\)
−0.675859 + 0.737031i \(0.736227\pi\)
\(908\) −17.6321 −0.585142
\(909\) −7.70891 −0.255689
\(910\) 0 0
\(911\) 22.7089 0.752380 0.376190 0.926543i \(-0.377234\pi\)
0.376190 + 0.926543i \(0.377234\pi\)
\(912\) 6.34310 0.210041
\(913\) 23.2002 0.767816
\(914\) 0.686200 0.0226975
\(915\) 0 0
\(916\) −11.8918 −0.392917
\(917\) 59.8010 1.97480
\(918\) −2.58010 −0.0851559
\(919\) −26.5801 −0.876796 −0.438398 0.898781i \(-0.644454\pi\)
−0.438398 + 0.898781i \(0.644454\pi\)
\(920\) 0 0
\(921\) 33.7089 1.11075
\(922\) −29.5780 −0.974100
\(923\) 11.5260 0.379383
\(924\) −11.9773 −0.394024
\(925\) 0 0
\(926\) 12.0000 0.394344
\(927\) −3.39719 −0.111578
\(928\) −1.00000 −0.0328266
\(929\) 36.5553 1.19934 0.599670 0.800247i \(-0.295298\pi\)
0.599670 + 0.800247i \(0.295298\pi\)
\(930\) 0 0
\(931\) −45.4178 −1.48851
\(932\) −9.02930 −0.295765
\(933\) 1.86252 0.0609761
\(934\) 29.2890 0.958366
\(935\) 0 0
\(936\) 6.34310 0.207331
\(937\) 19.3724 0.632869 0.316434 0.948614i \(-0.397514\pi\)
0.316434 + 0.948614i \(0.397514\pi\)
\(938\) −40.2122 −1.31298
\(939\) −26.8984 −0.877796
\(940\) 0 0
\(941\) −36.3204 −1.18401 −0.592005 0.805934i \(-0.701664\pi\)
−0.592005 + 0.805934i \(0.701664\pi\)
\(942\) 22.2122 0.723713
\(943\) 12.3431 0.401947
\(944\) −4.36581 −0.142095
\(945\) 0 0
\(946\) −32.9211 −1.07036
\(947\) −5.63419 −0.183086 −0.0915432 0.995801i \(-0.529180\pi\)
−0.0915432 + 0.995801i \(0.529180\pi\)
\(948\) 3.18291 0.103376
\(949\) −54.0813 −1.75555
\(950\) 0 0
\(951\) −11.0000 −0.356699
\(952\) 9.70891 0.314668
\(953\) 28.9005 0.936179 0.468089 0.883681i \(-0.344943\pi\)
0.468089 + 0.883681i \(0.344943\pi\)
\(954\) −7.16019 −0.231820
\(955\) 0 0
\(956\) −20.0747 −0.649263
\(957\) 3.18291 0.102889
\(958\) 15.9145 0.514175
\(959\) 58.0251 1.87373
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −16.3658 −0.527655
\(963\) −6.68620 −0.215460
\(964\) −23.0066 −0.740992
\(965\) 0 0
\(966\) −3.76300 −0.121073
\(967\) −12.3658 −0.397658 −0.198829 0.980034i \(-0.563714\pi\)
−0.198829 + 0.980034i \(0.563714\pi\)
\(968\) −0.869107 −0.0279341
\(969\) 16.3658 0.525746
\(970\) 0 0
\(971\) −33.9125 −1.08830 −0.544151 0.838987i \(-0.683148\pi\)
−0.544151 + 0.838987i \(0.683148\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 11.3745 0.364649
\(974\) −38.9438 −1.24784
\(975\) 0 0
\(976\) −1.52601 −0.0488463
\(977\) −41.3703 −1.32355 −0.661777 0.749701i \(-0.730197\pi\)
−0.661777 + 0.749701i \(0.730197\pi\)
\(978\) −8.68620 −0.277754
\(979\) 14.5780 0.465916
\(980\) 0 0
\(981\) 5.78572 0.184724
\(982\) 27.4806 0.876941
\(983\) −1.91453 −0.0610641 −0.0305320 0.999534i \(-0.509720\pi\)
−0.0305320 + 0.999534i \(0.509720\pi\)
\(984\) −12.3431 −0.393484
\(985\) 0 0
\(986\) −2.58010 −0.0821670
\(987\) 12.5801 0.400429
\(988\) −40.2349 −1.28004
\(989\) −10.3431 −0.328891
\(990\) 0 0
\(991\) 3.31380 0.105266 0.0526332 0.998614i \(-0.483239\pi\)
0.0526332 + 0.998614i \(0.483239\pi\)
\(992\) 6.00000 0.190500
\(993\) 6.18291 0.196209
\(994\) 6.83773 0.216880
\(995\) 0 0
\(996\) −7.28901 −0.230961
\(997\) 8.70891 0.275814 0.137907 0.990445i \(-0.455962\pi\)
0.137907 + 0.990445i \(0.455962\pi\)
\(998\) 6.97729 0.220862
\(999\) 2.58010 0.0816307
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 3450.2.a.bs.1.3 yes 3
5.2 odd 4 3450.2.d.ba.2899.6 6
5.3 odd 4 3450.2.d.ba.2899.1 6
5.4 even 2 3450.2.a.bp.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.bp.1.1 3 5.4 even 2
3450.2.a.bs.1.3 yes 3 1.1 even 1 trivial
3450.2.d.ba.2899.1 6 5.3 odd 4
3450.2.d.ba.2899.6 6 5.2 odd 4