Properties

Label 5070.2.a.bc.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.65685 q^{11} +1.00000 q^{12} +2.82843 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.82843 q^{17} -1.00000 q^{18} +2.82843 q^{19} -1.00000 q^{20} -2.82843 q^{21} -5.65685 q^{22} +8.48528 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -2.82843 q^{28} -3.17157 q^{29} +1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +5.65685 q^{33} +4.82843 q^{34} +2.82843 q^{35} +1.00000 q^{36} +0.343146 q^{37} -2.82843 q^{38} +1.00000 q^{40} -3.65685 q^{41} +2.82843 q^{42} -1.65685 q^{43} +5.65685 q^{44} -1.00000 q^{45} -8.48528 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -4.82843 q^{51} -9.31371 q^{53} -1.00000 q^{54} -5.65685 q^{55} +2.82843 q^{56} +2.82843 q^{57} +3.17157 q^{58} +13.6569 q^{59} -1.00000 q^{60} +6.00000 q^{61} +4.00000 q^{62} -2.82843 q^{63} +1.00000 q^{64} -5.65685 q^{66} +5.65685 q^{67} -4.82843 q^{68} +8.48528 q^{69} -2.82843 q^{70} -5.65685 q^{71} -1.00000 q^{72} -2.48528 q^{73} -0.343146 q^{74} +1.00000 q^{75} +2.82843 q^{76} -16.0000 q^{77} +13.6569 q^{79} -1.00000 q^{80} +1.00000 q^{81} +3.65685 q^{82} -17.6569 q^{83} -2.82843 q^{84} +4.82843 q^{85} +1.65685 q^{86} -3.17157 q^{87} -5.65685 q^{88} +4.34315 q^{89} +1.00000 q^{90} +8.48528 q^{92} -4.00000 q^{93} -8.00000 q^{94} -2.82843 q^{95} -1.00000 q^{96} +8.82843 q^{97} -1.00000 q^{98} +5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} - 2q^{15} + 2q^{16} - 4q^{17} - 2q^{18} - 2q^{20} - 2q^{24} + 2q^{25} + 2q^{27} - 12q^{29} + 2q^{30} - 8q^{31} - 2q^{32} + 4q^{34} + 2q^{36} + 12q^{37} + 2q^{40} + 4q^{41} + 8q^{43} - 2q^{45} + 16q^{47} + 2q^{48} + 2q^{49} - 2q^{50} - 4q^{51} + 4q^{53} - 2q^{54} + 12q^{58} + 16q^{59} - 2q^{60} + 12q^{61} + 8q^{62} + 2q^{64} - 4q^{68} - 2q^{72} + 12q^{73} - 12q^{74} + 2q^{75} - 32q^{77} + 16q^{79} - 2q^{80} + 2q^{81} - 4q^{82} - 24q^{83} + 4q^{85} - 8q^{86} - 12q^{87} + 20q^{89} + 2q^{90} - 8q^{93} - 16q^{94} - 2q^{96} + 12q^{97} - 2q^{98} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 5.65685 1.70561 0.852803 0.522233i \(-0.174901\pi\)
0.852803 + 0.522233i \(0.174901\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 2.82843 0.755929
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −4.82843 −1.17107 −0.585533 0.810649i \(-0.699115\pi\)
−0.585533 + 0.810649i \(0.699115\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.82843 −0.617213
\(22\) −5.65685 −1.20605
\(23\) 8.48528 1.76930 0.884652 0.466252i \(-0.154396\pi\)
0.884652 + 0.466252i \(0.154396\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.82843 −0.534522
\(29\) −3.17157 −0.588946 −0.294473 0.955660i \(-0.595144\pi\)
−0.294473 + 0.955660i \(0.595144\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.65685 0.984732
\(34\) 4.82843 0.828068
\(35\) 2.82843 0.478091
\(36\) 1.00000 0.166667
\(37\) 0.343146 0.0564128 0.0282064 0.999602i \(-0.491020\pi\)
0.0282064 + 0.999602i \(0.491020\pi\)
\(38\) −2.82843 −0.458831
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −3.65685 −0.571105 −0.285552 0.958363i \(-0.592177\pi\)
−0.285552 + 0.958363i \(0.592177\pi\)
\(42\) 2.82843 0.436436
\(43\) −1.65685 −0.252668 −0.126334 0.991988i \(-0.540321\pi\)
−0.126334 + 0.991988i \(0.540321\pi\)
\(44\) 5.65685 0.852803
\(45\) −1.00000 −0.149071
\(46\) −8.48528 −1.25109
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −4.82843 −0.676115
\(52\) 0 0
\(53\) −9.31371 −1.27934 −0.639668 0.768651i \(-0.720928\pi\)
−0.639668 + 0.768651i \(0.720928\pi\)
\(54\) −1.00000 −0.136083
\(55\) −5.65685 −0.762770
\(56\) 2.82843 0.377964
\(57\) 2.82843 0.374634
\(58\) 3.17157 0.416448
\(59\) 13.6569 1.77797 0.888985 0.457935i \(-0.151411\pi\)
0.888985 + 0.457935i \(0.151411\pi\)
\(60\) −1.00000 −0.129099
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 4.00000 0.508001
\(63\) −2.82843 −0.356348
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.65685 −0.696311
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) −4.82843 −0.585533
\(69\) 8.48528 1.02151
\(70\) −2.82843 −0.338062
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.48528 −0.290880 −0.145440 0.989367i \(-0.546460\pi\)
−0.145440 + 0.989367i \(0.546460\pi\)
\(74\) −0.343146 −0.0398899
\(75\) 1.00000 0.115470
\(76\) 2.82843 0.324443
\(77\) −16.0000 −1.82337
\(78\) 0 0
\(79\) 13.6569 1.53652 0.768258 0.640140i \(-0.221124\pi\)
0.768258 + 0.640140i \(0.221124\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 3.65685 0.403832
\(83\) −17.6569 −1.93809 −0.969046 0.246881i \(-0.920594\pi\)
−0.969046 + 0.246881i \(0.920594\pi\)
\(84\) −2.82843 −0.308607
\(85\) 4.82843 0.523716
\(86\) 1.65685 0.178663
\(87\) −3.17157 −0.340028
\(88\) −5.65685 −0.603023
\(89\) 4.34315 0.460373 0.230186 0.973147i \(-0.426066\pi\)
0.230186 + 0.973147i \(0.426066\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 8.48528 0.884652
\(93\) −4.00000 −0.414781
\(94\) −8.00000 −0.825137
\(95\) −2.82843 −0.290191
\(96\) −1.00000 −0.102062
\(97\) 8.82843 0.896391 0.448195 0.893936i \(-0.352067\pi\)
0.448195 + 0.893936i \(0.352067\pi\)
\(98\) −1.00000 −0.101015
\(99\) 5.65685 0.568535
\(100\) 1.00000 0.100000
\(101\) −12.1421 −1.20819 −0.604094 0.796913i \(-0.706465\pi\)
−0.604094 + 0.796913i \(0.706465\pi\)
\(102\) 4.82843 0.478086
\(103\) 9.65685 0.951518 0.475759 0.879576i \(-0.342173\pi\)
0.475759 + 0.879576i \(0.342173\pi\)
\(104\) 0 0
\(105\) 2.82843 0.276026
\(106\) 9.31371 0.904627
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) −3.17157 −0.303782 −0.151891 0.988397i \(-0.548536\pi\)
−0.151891 + 0.988397i \(0.548536\pi\)
\(110\) 5.65685 0.539360
\(111\) 0.343146 0.0325700
\(112\) −2.82843 −0.267261
\(113\) −10.4853 −0.986372 −0.493186 0.869924i \(-0.664168\pi\)
−0.493186 + 0.869924i \(0.664168\pi\)
\(114\) −2.82843 −0.264906
\(115\) −8.48528 −0.791257
\(116\) −3.17157 −0.294473
\(117\) 0 0
\(118\) −13.6569 −1.25722
\(119\) 13.6569 1.25192
\(120\) 1.00000 0.0912871
\(121\) 21.0000 1.90909
\(122\) −6.00000 −0.543214
\(123\) −3.65685 −0.329727
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 2.82843 0.251976
\(127\) −1.65685 −0.147022 −0.0735110 0.997294i \(-0.523420\pi\)
−0.0735110 + 0.997294i \(0.523420\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.65685 −0.145878
\(130\) 0 0
\(131\) 22.1421 1.93457 0.967284 0.253697i \(-0.0816467\pi\)
0.967284 + 0.253697i \(0.0816467\pi\)
\(132\) 5.65685 0.492366
\(133\) −8.00000 −0.693688
\(134\) −5.65685 −0.488678
\(135\) −1.00000 −0.0860663
\(136\) 4.82843 0.414034
\(137\) −5.31371 −0.453981 −0.226990 0.973897i \(-0.572889\pi\)
−0.226990 + 0.973897i \(0.572889\pi\)
\(138\) −8.48528 −0.722315
\(139\) 17.6569 1.49763 0.748817 0.662776i \(-0.230622\pi\)
0.748817 + 0.662776i \(0.230622\pi\)
\(140\) 2.82843 0.239046
\(141\) 8.00000 0.673722
\(142\) 5.65685 0.474713
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 3.17157 0.263385
\(146\) 2.48528 0.205683
\(147\) 1.00000 0.0824786
\(148\) 0.343146 0.0282064
\(149\) 7.65685 0.627274 0.313637 0.949543i \(-0.398453\pi\)
0.313637 + 0.949543i \(0.398453\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −2.82843 −0.229416
\(153\) −4.82843 −0.390355
\(154\) 16.0000 1.28932
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −17.3137 −1.38178 −0.690892 0.722958i \(-0.742782\pi\)
−0.690892 + 0.722958i \(0.742782\pi\)
\(158\) −13.6569 −1.08648
\(159\) −9.31371 −0.738625
\(160\) 1.00000 0.0790569
\(161\) −24.0000 −1.89146
\(162\) −1.00000 −0.0785674
\(163\) 11.3137 0.886158 0.443079 0.896483i \(-0.353886\pi\)
0.443079 + 0.896483i \(0.353886\pi\)
\(164\) −3.65685 −0.285552
\(165\) −5.65685 −0.440386
\(166\) 17.6569 1.37044
\(167\) 24.9706 1.93228 0.966140 0.258018i \(-0.0830694\pi\)
0.966140 + 0.258018i \(0.0830694\pi\)
\(168\) 2.82843 0.218218
\(169\) 0 0
\(170\) −4.82843 −0.370323
\(171\) 2.82843 0.216295
\(172\) −1.65685 −0.126334
\(173\) 13.3137 1.01222 0.506111 0.862468i \(-0.331083\pi\)
0.506111 + 0.862468i \(0.331083\pi\)
\(174\) 3.17157 0.240436
\(175\) −2.82843 −0.213809
\(176\) 5.65685 0.426401
\(177\) 13.6569 1.02651
\(178\) −4.34315 −0.325533
\(179\) 24.4853 1.83012 0.915058 0.403322i \(-0.132145\pi\)
0.915058 + 0.403322i \(0.132145\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 3.65685 0.271812 0.135906 0.990722i \(-0.456606\pi\)
0.135906 + 0.990722i \(0.456606\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) −8.48528 −0.625543
\(185\) −0.343146 −0.0252286
\(186\) 4.00000 0.293294
\(187\) −27.3137 −1.99738
\(188\) 8.00000 0.583460
\(189\) −2.82843 −0.205738
\(190\) 2.82843 0.205196
\(191\) −11.3137 −0.818631 −0.409316 0.912393i \(-0.634232\pi\)
−0.409316 + 0.912393i \(0.634232\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.4853 1.04267 0.521337 0.853351i \(-0.325434\pi\)
0.521337 + 0.853351i \(0.325434\pi\)
\(194\) −8.82843 −0.633844
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −9.31371 −0.663574 −0.331787 0.943354i \(-0.607652\pi\)
−0.331787 + 0.943354i \(0.607652\pi\)
\(198\) −5.65685 −0.402015
\(199\) 21.6569 1.53521 0.767607 0.640921i \(-0.221447\pi\)
0.767607 + 0.640921i \(0.221447\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 5.65685 0.399004
\(202\) 12.1421 0.854318
\(203\) 8.97056 0.629610
\(204\) −4.82843 −0.338058
\(205\) 3.65685 0.255406
\(206\) −9.65685 −0.672825
\(207\) 8.48528 0.589768
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) −2.82843 −0.195180
\(211\) 23.3137 1.60498 0.802491 0.596664i \(-0.203508\pi\)
0.802491 + 0.596664i \(0.203508\pi\)
\(212\) −9.31371 −0.639668
\(213\) −5.65685 −0.387601
\(214\) 4.00000 0.273434
\(215\) 1.65685 0.112997
\(216\) −1.00000 −0.0680414
\(217\) 11.3137 0.768025
\(218\) 3.17157 0.214806
\(219\) −2.48528 −0.167940
\(220\) −5.65685 −0.381385
\(221\) 0 0
\(222\) −0.343146 −0.0230304
\(223\) −5.17157 −0.346314 −0.173157 0.984894i \(-0.555397\pi\)
−0.173157 + 0.984894i \(0.555397\pi\)
\(224\) 2.82843 0.188982
\(225\) 1.00000 0.0666667
\(226\) 10.4853 0.697471
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 2.82843 0.187317
\(229\) 24.1421 1.59536 0.797679 0.603083i \(-0.206061\pi\)
0.797679 + 0.603083i \(0.206061\pi\)
\(230\) 8.48528 0.559503
\(231\) −16.0000 −1.05272
\(232\) 3.17157 0.208224
\(233\) 22.4853 1.47306 0.736530 0.676405i \(-0.236463\pi\)
0.736530 + 0.676405i \(0.236463\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 13.6569 0.888985
\(237\) 13.6569 0.887108
\(238\) −13.6569 −0.885242
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 17.3137 1.11527 0.557637 0.830085i \(-0.311708\pi\)
0.557637 + 0.830085i \(0.311708\pi\)
\(242\) −21.0000 −1.34993
\(243\) 1.00000 0.0641500
\(244\) 6.00000 0.384111
\(245\) −1.00000 −0.0638877
\(246\) 3.65685 0.233153
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) −17.6569 −1.11896
\(250\) 1.00000 0.0632456
\(251\) 5.17157 0.326427 0.163213 0.986591i \(-0.447814\pi\)
0.163213 + 0.986591i \(0.447814\pi\)
\(252\) −2.82843 −0.178174
\(253\) 48.0000 3.01773
\(254\) 1.65685 0.103960
\(255\) 4.82843 0.302368
\(256\) 1.00000 0.0625000
\(257\) 0.828427 0.0516759 0.0258379 0.999666i \(-0.491775\pi\)
0.0258379 + 0.999666i \(0.491775\pi\)
\(258\) 1.65685 0.103151
\(259\) −0.970563 −0.0603078
\(260\) 0 0
\(261\) −3.17157 −0.196315
\(262\) −22.1421 −1.36795
\(263\) −0.485281 −0.0299237 −0.0149619 0.999888i \(-0.504763\pi\)
−0.0149619 + 0.999888i \(0.504763\pi\)
\(264\) −5.65685 −0.348155
\(265\) 9.31371 0.572137
\(266\) 8.00000 0.490511
\(267\) 4.34315 0.265796
\(268\) 5.65685 0.345547
\(269\) 2.48528 0.151530 0.0757651 0.997126i \(-0.475860\pi\)
0.0757651 + 0.997126i \(0.475860\pi\)
\(270\) 1.00000 0.0608581
\(271\) 15.3137 0.930242 0.465121 0.885247i \(-0.346011\pi\)
0.465121 + 0.885247i \(0.346011\pi\)
\(272\) −4.82843 −0.292766
\(273\) 0 0
\(274\) 5.31371 0.321013
\(275\) 5.65685 0.341121
\(276\) 8.48528 0.510754
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −17.6569 −1.05899
\(279\) −4.00000 −0.239474
\(280\) −2.82843 −0.169031
\(281\) −19.6569 −1.17263 −0.586315 0.810083i \(-0.699422\pi\)
−0.586315 + 0.810083i \(0.699422\pi\)
\(282\) −8.00000 −0.476393
\(283\) −6.34315 −0.377061 −0.188530 0.982067i \(-0.560372\pi\)
−0.188530 + 0.982067i \(0.560372\pi\)
\(284\) −5.65685 −0.335673
\(285\) −2.82843 −0.167542
\(286\) 0 0
\(287\) 10.3431 0.610537
\(288\) −1.00000 −0.0589256
\(289\) 6.31371 0.371395
\(290\) −3.17157 −0.186241
\(291\) 8.82843 0.517532
\(292\) −2.48528 −0.145440
\(293\) −28.6274 −1.67243 −0.836216 0.548401i \(-0.815237\pi\)
−0.836216 + 0.548401i \(0.815237\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −13.6569 −0.795133
\(296\) −0.343146 −0.0199449
\(297\) 5.65685 0.328244
\(298\) −7.65685 −0.443550
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 4.68629 0.270113
\(302\) 12.0000 0.690522
\(303\) −12.1421 −0.697547
\(304\) 2.82843 0.162221
\(305\) −6.00000 −0.343559
\(306\) 4.82843 0.276023
\(307\) 10.3431 0.590315 0.295157 0.955449i \(-0.404628\pi\)
0.295157 + 0.955449i \(0.404628\pi\)
\(308\) −16.0000 −0.911685
\(309\) 9.65685 0.549359
\(310\) −4.00000 −0.227185
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 2.97056 0.167906 0.0839531 0.996470i \(-0.473245\pi\)
0.0839531 + 0.996470i \(0.473245\pi\)
\(314\) 17.3137 0.977069
\(315\) 2.82843 0.159364
\(316\) 13.6569 0.768258
\(317\) −2.68629 −0.150877 −0.0754386 0.997150i \(-0.524036\pi\)
−0.0754386 + 0.997150i \(0.524036\pi\)
\(318\) 9.31371 0.522287
\(319\) −17.9411 −1.00451
\(320\) −1.00000 −0.0559017
\(321\) −4.00000 −0.223258
\(322\) 24.0000 1.33747
\(323\) −13.6569 −0.759888
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −11.3137 −0.626608
\(327\) −3.17157 −0.175388
\(328\) 3.65685 0.201916
\(329\) −22.6274 −1.24749
\(330\) 5.65685 0.311400
\(331\) −8.48528 −0.466393 −0.233197 0.972430i \(-0.574919\pi\)
−0.233197 + 0.972430i \(0.574919\pi\)
\(332\) −17.6569 −0.969046
\(333\) 0.343146 0.0188043
\(334\) −24.9706 −1.36633
\(335\) −5.65685 −0.309067
\(336\) −2.82843 −0.154303
\(337\) −22.9706 −1.25129 −0.625643 0.780109i \(-0.715163\pi\)
−0.625643 + 0.780109i \(0.715163\pi\)
\(338\) 0 0
\(339\) −10.4853 −0.569482
\(340\) 4.82843 0.261858
\(341\) −22.6274 −1.22534
\(342\) −2.82843 −0.152944
\(343\) 16.9706 0.916324
\(344\) 1.65685 0.0893316
\(345\) −8.48528 −0.456832
\(346\) −13.3137 −0.715749
\(347\) 1.65685 0.0889446 0.0444723 0.999011i \(-0.485839\pi\)
0.0444723 + 0.999011i \(0.485839\pi\)
\(348\) −3.17157 −0.170014
\(349\) 16.1421 0.864069 0.432034 0.901857i \(-0.357796\pi\)
0.432034 + 0.901857i \(0.357796\pi\)
\(350\) 2.82843 0.151186
\(351\) 0 0
\(352\) −5.65685 −0.301511
\(353\) 17.3137 0.921516 0.460758 0.887526i \(-0.347578\pi\)
0.460758 + 0.887526i \(0.347578\pi\)
\(354\) −13.6569 −0.725854
\(355\) 5.65685 0.300235
\(356\) 4.34315 0.230186
\(357\) 13.6569 0.722797
\(358\) −24.4853 −1.29409
\(359\) −28.2843 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(360\) 1.00000 0.0527046
\(361\) −11.0000 −0.578947
\(362\) −3.65685 −0.192200
\(363\) 21.0000 1.10221
\(364\) 0 0
\(365\) 2.48528 0.130086
\(366\) −6.00000 −0.313625
\(367\) 14.3431 0.748706 0.374353 0.927286i \(-0.377865\pi\)
0.374353 + 0.927286i \(0.377865\pi\)
\(368\) 8.48528 0.442326
\(369\) −3.65685 −0.190368
\(370\) 0.343146 0.0178393
\(371\) 26.3431 1.36767
\(372\) −4.00000 −0.207390
\(373\) −25.3137 −1.31069 −0.655347 0.755328i \(-0.727478\pi\)
−0.655347 + 0.755328i \(0.727478\pi\)
\(374\) 27.3137 1.41236
\(375\) −1.00000 −0.0516398
\(376\) −8.00000 −0.412568
\(377\) 0 0
\(378\) 2.82843 0.145479
\(379\) 24.4853 1.25772 0.628862 0.777517i \(-0.283521\pi\)
0.628862 + 0.777517i \(0.283521\pi\)
\(380\) −2.82843 −0.145095
\(381\) −1.65685 −0.0848832
\(382\) 11.3137 0.578860
\(383\) 18.3431 0.937291 0.468645 0.883386i \(-0.344742\pi\)
0.468645 + 0.883386i \(0.344742\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 16.0000 0.815436
\(386\) −14.4853 −0.737281
\(387\) −1.65685 −0.0842226
\(388\) 8.82843 0.448195
\(389\) 10.4853 0.531625 0.265812 0.964025i \(-0.414360\pi\)
0.265812 + 0.964025i \(0.414360\pi\)
\(390\) 0 0
\(391\) −40.9706 −2.07197
\(392\) −1.00000 −0.0505076
\(393\) 22.1421 1.11692
\(394\) 9.31371 0.469218
\(395\) −13.6569 −0.687151
\(396\) 5.65685 0.284268
\(397\) 26.2843 1.31917 0.659585 0.751630i \(-0.270732\pi\)
0.659585 + 0.751630i \(0.270732\pi\)
\(398\) −21.6569 −1.08556
\(399\) −8.00000 −0.400501
\(400\) 1.00000 0.0500000
\(401\) −6.97056 −0.348093 −0.174047 0.984737i \(-0.555684\pi\)
−0.174047 + 0.984737i \(0.555684\pi\)
\(402\) −5.65685 −0.282138
\(403\) 0 0
\(404\) −12.1421 −0.604094
\(405\) −1.00000 −0.0496904
\(406\) −8.97056 −0.445202
\(407\) 1.94113 0.0962180
\(408\) 4.82843 0.239043
\(409\) −7.65685 −0.378607 −0.189304 0.981919i \(-0.560623\pi\)
−0.189304 + 0.981919i \(0.560623\pi\)
\(410\) −3.65685 −0.180599
\(411\) −5.31371 −0.262106
\(412\) 9.65685 0.475759
\(413\) −38.6274 −1.90073
\(414\) −8.48528 −0.417029
\(415\) 17.6569 0.866741
\(416\) 0 0
\(417\) 17.6569 0.864660
\(418\) −16.0000 −0.782586
\(419\) −5.17157 −0.252648 −0.126324 0.991989i \(-0.540318\pi\)
−0.126324 + 0.991989i \(0.540318\pi\)
\(420\) 2.82843 0.138013
\(421\) −4.14214 −0.201875 −0.100938 0.994893i \(-0.532184\pi\)
−0.100938 + 0.994893i \(0.532184\pi\)
\(422\) −23.3137 −1.13489
\(423\) 8.00000 0.388973
\(424\) 9.31371 0.452314
\(425\) −4.82843 −0.234213
\(426\) 5.65685 0.274075
\(427\) −16.9706 −0.821263
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) −1.65685 −0.0799006
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.00000 0.0481125
\(433\) 10.9706 0.527212 0.263606 0.964630i \(-0.415088\pi\)
0.263606 + 0.964630i \(0.415088\pi\)
\(434\) −11.3137 −0.543075
\(435\) 3.17157 0.152065
\(436\) −3.17157 −0.151891
\(437\) 24.0000 1.14808
\(438\) 2.48528 0.118751
\(439\) −22.6274 −1.07995 −0.539974 0.841682i \(-0.681566\pi\)
−0.539974 + 0.841682i \(0.681566\pi\)
\(440\) 5.65685 0.269680
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 41.6569 1.97918 0.989588 0.143926i \(-0.0459728\pi\)
0.989588 + 0.143926i \(0.0459728\pi\)
\(444\) 0.343146 0.0162850
\(445\) −4.34315 −0.205885
\(446\) 5.17157 0.244881
\(447\) 7.65685 0.362157
\(448\) −2.82843 −0.133631
\(449\) 30.2843 1.42920 0.714602 0.699532i \(-0.246608\pi\)
0.714602 + 0.699532i \(0.246608\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −20.6863 −0.974079
\(452\) −10.4853 −0.493186
\(453\) −12.0000 −0.563809
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) −2.82843 −0.132453
\(457\) −15.1716 −0.709696 −0.354848 0.934924i \(-0.615467\pi\)
−0.354848 + 0.934924i \(0.615467\pi\)
\(458\) −24.1421 −1.12809
\(459\) −4.82843 −0.225372
\(460\) −8.48528 −0.395628
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 16.0000 0.744387
\(463\) −35.7990 −1.66372 −0.831860 0.554985i \(-0.812724\pi\)
−0.831860 + 0.554985i \(0.812724\pi\)
\(464\) −3.17157 −0.147237
\(465\) 4.00000 0.185496
\(466\) −22.4853 −1.04161
\(467\) −15.3137 −0.708634 −0.354317 0.935125i \(-0.615287\pi\)
−0.354317 + 0.935125i \(0.615287\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 8.00000 0.369012
\(471\) −17.3137 −0.797774
\(472\) −13.6569 −0.628608
\(473\) −9.37258 −0.430952
\(474\) −13.6569 −0.627280
\(475\) 2.82843 0.129777
\(476\) 13.6569 0.625961
\(477\) −9.31371 −0.426445
\(478\) 16.0000 0.731823
\(479\) −11.3137 −0.516937 −0.258468 0.966020i \(-0.583218\pi\)
−0.258468 + 0.966020i \(0.583218\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −17.3137 −0.788618
\(483\) −24.0000 −1.09204
\(484\) 21.0000 0.954545
\(485\) −8.82843 −0.400878
\(486\) −1.00000 −0.0453609
\(487\) 0.485281 0.0219902 0.0109951 0.999940i \(-0.496500\pi\)
0.0109951 + 0.999940i \(0.496500\pi\)
\(488\) −6.00000 −0.271607
\(489\) 11.3137 0.511624
\(490\) 1.00000 0.0451754
\(491\) −9.85786 −0.444879 −0.222440 0.974946i \(-0.571402\pi\)
−0.222440 + 0.974946i \(0.571402\pi\)
\(492\) −3.65685 −0.164864
\(493\) 15.3137 0.689695
\(494\) 0 0
\(495\) −5.65685 −0.254257
\(496\) −4.00000 −0.179605
\(497\) 16.0000 0.717698
\(498\) 17.6569 0.791223
\(499\) 16.4853 0.737983 0.368991 0.929433i \(-0.379703\pi\)
0.368991 + 0.929433i \(0.379703\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 24.9706 1.11560
\(502\) −5.17157 −0.230819
\(503\) −40.4853 −1.80515 −0.902575 0.430533i \(-0.858326\pi\)
−0.902575 + 0.430533i \(0.858326\pi\)
\(504\) 2.82843 0.125988
\(505\) 12.1421 0.540318
\(506\) −48.0000 −2.13386
\(507\) 0 0
\(508\) −1.65685 −0.0735110
\(509\) 14.6863 0.650958 0.325479 0.945549i \(-0.394474\pi\)
0.325479 + 0.945549i \(0.394474\pi\)
\(510\) −4.82843 −0.213806
\(511\) 7.02944 0.310964
\(512\) −1.00000 −0.0441942
\(513\) 2.82843 0.124878
\(514\) −0.828427 −0.0365404
\(515\) −9.65685 −0.425532
\(516\) −1.65685 −0.0729389
\(517\) 45.2548 1.99031
\(518\) 0.970563 0.0426441
\(519\) 13.3137 0.584407
\(520\) 0 0
\(521\) −6.97056 −0.305386 −0.152693 0.988274i \(-0.548795\pi\)
−0.152693 + 0.988274i \(0.548795\pi\)
\(522\) 3.17157 0.138816
\(523\) 34.6274 1.51415 0.757076 0.653327i \(-0.226627\pi\)
0.757076 + 0.653327i \(0.226627\pi\)
\(524\) 22.1421 0.967284
\(525\) −2.82843 −0.123443
\(526\) 0.485281 0.0211593
\(527\) 19.3137 0.841318
\(528\) 5.65685 0.246183
\(529\) 49.0000 2.13043
\(530\) −9.31371 −0.404562
\(531\) 13.6569 0.592657
\(532\) −8.00000 −0.346844
\(533\) 0 0
\(534\) −4.34315 −0.187946
\(535\) 4.00000 0.172935
\(536\) −5.65685 −0.244339
\(537\) 24.4853 1.05662
\(538\) −2.48528 −0.107148
\(539\) 5.65685 0.243658
\(540\) −1.00000 −0.0430331
\(541\) 2.48528 0.106851 0.0534253 0.998572i \(-0.482986\pi\)
0.0534253 + 0.998572i \(0.482986\pi\)
\(542\) −15.3137 −0.657780
\(543\) 3.65685 0.156931
\(544\) 4.82843 0.207017
\(545\) 3.17157 0.135855
\(546\) 0 0
\(547\) 23.3137 0.996822 0.498411 0.866941i \(-0.333917\pi\)
0.498411 + 0.866941i \(0.333917\pi\)
\(548\) −5.31371 −0.226990
\(549\) 6.00000 0.256074
\(550\) −5.65685 −0.241209
\(551\) −8.97056 −0.382159
\(552\) −8.48528 −0.361158
\(553\) −38.6274 −1.64260
\(554\) −26.0000 −1.10463
\(555\) −0.343146 −0.0145657
\(556\) 17.6569 0.748817
\(557\) −33.3137 −1.41155 −0.705774 0.708437i \(-0.749400\pi\)
−0.705774 + 0.708437i \(0.749400\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) 2.82843 0.119523
\(561\) −27.3137 −1.15319
\(562\) 19.6569 0.829174
\(563\) −41.6569 −1.75563 −0.877814 0.479002i \(-0.840998\pi\)
−0.877814 + 0.479002i \(0.840998\pi\)
\(564\) 8.00000 0.336861
\(565\) 10.4853 0.441119
\(566\) 6.34315 0.266622
\(567\) −2.82843 −0.118783
\(568\) 5.65685 0.237356
\(569\) 20.3431 0.852829 0.426415 0.904528i \(-0.359776\pi\)
0.426415 + 0.904528i \(0.359776\pi\)
\(570\) 2.82843 0.118470
\(571\) −12.9706 −0.542801 −0.271401 0.962466i \(-0.587487\pi\)
−0.271401 + 0.962466i \(0.587487\pi\)
\(572\) 0 0
\(573\) −11.3137 −0.472637
\(574\) −10.3431 −0.431715
\(575\) 8.48528 0.353861
\(576\) 1.00000 0.0416667
\(577\) −27.4558 −1.14300 −0.571501 0.820601i \(-0.693639\pi\)
−0.571501 + 0.820601i \(0.693639\pi\)
\(578\) −6.31371 −0.262616
\(579\) 14.4853 0.601988
\(580\) 3.17157 0.131692
\(581\) 49.9411 2.07191
\(582\) −8.82843 −0.365950
\(583\) −52.6863 −2.18204
\(584\) 2.48528 0.102842
\(585\) 0 0
\(586\) 28.6274 1.18259
\(587\) 42.6274 1.75942 0.879711 0.475509i \(-0.157736\pi\)
0.879711 + 0.475509i \(0.157736\pi\)
\(588\) 1.00000 0.0412393
\(589\) −11.3137 −0.466173
\(590\) 13.6569 0.562244
\(591\) −9.31371 −0.383115
\(592\) 0.343146 0.0141032
\(593\) 11.6569 0.478690 0.239345 0.970935i \(-0.423067\pi\)
0.239345 + 0.970935i \(0.423067\pi\)
\(594\) −5.65685 −0.232104
\(595\) −13.6569 −0.559876
\(596\) 7.65685 0.313637
\(597\) 21.6569 0.886356
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 6.68629 0.272740 0.136370 0.990658i \(-0.456456\pi\)
0.136370 + 0.990658i \(0.456456\pi\)
\(602\) −4.68629 −0.190999
\(603\) 5.65685 0.230365
\(604\) −12.0000 −0.488273
\(605\) −21.0000 −0.853771
\(606\) 12.1421 0.493241
\(607\) −4.97056 −0.201749 −0.100874 0.994899i \(-0.532164\pi\)
−0.100874 + 0.994899i \(0.532164\pi\)
\(608\) −2.82843 −0.114708
\(609\) 8.97056 0.363506
\(610\) 6.00000 0.242933
\(611\) 0 0
\(612\) −4.82843 −0.195178
\(613\) 34.2843 1.38473 0.692364 0.721548i \(-0.256569\pi\)
0.692364 + 0.721548i \(0.256569\pi\)
\(614\) −10.3431 −0.417415
\(615\) 3.65685 0.147459
\(616\) 16.0000 0.644658
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) −9.65685 −0.388456
\(619\) 29.1716 1.17250 0.586252 0.810129i \(-0.300603\pi\)
0.586252 + 0.810129i \(0.300603\pi\)
\(620\) 4.00000 0.160644
\(621\) 8.48528 0.340503
\(622\) 24.0000 0.962312
\(623\) −12.2843 −0.492159
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −2.97056 −0.118728
\(627\) 16.0000 0.638978
\(628\) −17.3137 −0.690892
\(629\) −1.65685 −0.0660631
\(630\) −2.82843 −0.112687
\(631\) 22.3431 0.889467 0.444733 0.895663i \(-0.353298\pi\)
0.444733 + 0.895663i \(0.353298\pi\)
\(632\) −13.6569 −0.543240
\(633\) 23.3137 0.926637
\(634\) 2.68629 0.106686
\(635\) 1.65685 0.0657503
\(636\) −9.31371 −0.369313
\(637\) 0 0
\(638\) 17.9411 0.710296
\(639\) −5.65685 −0.223782
\(640\) 1.00000 0.0395285
\(641\) 40.6274 1.60469 0.802343 0.596863i \(-0.203586\pi\)
0.802343 + 0.596863i \(0.203586\pi\)
\(642\) 4.00000 0.157867
\(643\) 39.5980 1.56159 0.780796 0.624786i \(-0.214814\pi\)
0.780796 + 0.624786i \(0.214814\pi\)
\(644\) −24.0000 −0.945732
\(645\) 1.65685 0.0652386
\(646\) 13.6569 0.537322
\(647\) −8.48528 −0.333591 −0.166795 0.985992i \(-0.553342\pi\)
−0.166795 + 0.985992i \(0.553342\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 77.2548 3.03252
\(650\) 0 0
\(651\) 11.3137 0.443419
\(652\) 11.3137 0.443079
\(653\) 14.2843 0.558987 0.279493 0.960148i \(-0.409834\pi\)
0.279493 + 0.960148i \(0.409834\pi\)
\(654\) 3.17157 0.124018
\(655\) −22.1421 −0.865165
\(656\) −3.65685 −0.142776
\(657\) −2.48528 −0.0969601
\(658\) 22.6274 0.882109
\(659\) −24.4853 −0.953811 −0.476906 0.878955i \(-0.658242\pi\)
−0.476906 + 0.878955i \(0.658242\pi\)
\(660\) −5.65685 −0.220193
\(661\) −20.1421 −0.783438 −0.391719 0.920085i \(-0.628120\pi\)
−0.391719 + 0.920085i \(0.628120\pi\)
\(662\) 8.48528 0.329790
\(663\) 0 0
\(664\) 17.6569 0.685219
\(665\) 8.00000 0.310227
\(666\) −0.343146 −0.0132966
\(667\) −26.9117 −1.04202
\(668\) 24.9706 0.966140
\(669\) −5.17157 −0.199945
\(670\) 5.65685 0.218543
\(671\) 33.9411 1.31028
\(672\) 2.82843 0.109109
\(673\) −12.6274 −0.486751 −0.243376 0.969932i \(-0.578255\pi\)
−0.243376 + 0.969932i \(0.578255\pi\)
\(674\) 22.9706 0.884793
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 23.6569 0.909207 0.454603 0.890694i \(-0.349781\pi\)
0.454603 + 0.890694i \(0.349781\pi\)
\(678\) 10.4853 0.402685
\(679\) −24.9706 −0.958282
\(680\) −4.82843 −0.185162
\(681\) −4.00000 −0.153280
\(682\) 22.6274 0.866449
\(683\) −22.3431 −0.854937 −0.427468 0.904030i \(-0.640594\pi\)
−0.427468 + 0.904030i \(0.640594\pi\)
\(684\) 2.82843 0.108148
\(685\) 5.31371 0.203026
\(686\) −16.9706 −0.647939
\(687\) 24.1421 0.921080
\(688\) −1.65685 −0.0631670
\(689\) 0 0
\(690\) 8.48528 0.323029
\(691\) −11.7990 −0.448855 −0.224427 0.974491i \(-0.572051\pi\)
−0.224427 + 0.974491i \(0.572051\pi\)
\(692\) 13.3137 0.506111
\(693\) −16.0000 −0.607790
\(694\) −1.65685 −0.0628933
\(695\) −17.6569 −0.669763
\(696\) 3.17157 0.120218
\(697\) 17.6569 0.668801
\(698\) −16.1421 −0.610989
\(699\) 22.4853 0.850471
\(700\) −2.82843 −0.106904
\(701\) −28.1421 −1.06291 −0.531457 0.847085i \(-0.678355\pi\)
−0.531457 + 0.847085i \(0.678355\pi\)
\(702\) 0 0
\(703\) 0.970563 0.0366055
\(704\) 5.65685 0.213201
\(705\) −8.00000 −0.301297
\(706\) −17.3137 −0.651610
\(707\) 34.3431 1.29161
\(708\) 13.6569 0.513256
\(709\) 12.8284 0.481782 0.240891 0.970552i \(-0.422560\pi\)
0.240891 + 0.970552i \(0.422560\pi\)
\(710\) −5.65685 −0.212298
\(711\) 13.6569 0.512172
\(712\) −4.34315 −0.162766
\(713\) −33.9411 −1.27111
\(714\) −13.6569 −0.511095
\(715\) 0 0
\(716\) 24.4853 0.915058
\(717\) −16.0000 −0.597531
\(718\) 28.2843 1.05556
\(719\) −18.3431 −0.684084 −0.342042 0.939685i \(-0.611118\pi\)
−0.342042 + 0.939685i \(0.611118\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −27.3137 −1.01722
\(722\) 11.0000 0.409378
\(723\) 17.3137 0.643904
\(724\) 3.65685 0.135906
\(725\) −3.17157 −0.117789
\(726\) −21.0000 −0.779383
\(727\) 21.9411 0.813751 0.406876 0.913484i \(-0.366618\pi\)
0.406876 + 0.913484i \(0.366618\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.48528 −0.0919844
\(731\) 8.00000 0.295891
\(732\) 6.00000 0.221766
\(733\) 11.6569 0.430556 0.215278 0.976553i \(-0.430934\pi\)
0.215278 + 0.976553i \(0.430934\pi\)
\(734\) −14.3431 −0.529415
\(735\) −1.00000 −0.0368856
\(736\) −8.48528 −0.312772
\(737\) 32.0000 1.17874
\(738\) 3.65685 0.134611
\(739\) −14.1421 −0.520227 −0.260113 0.965578i \(-0.583760\pi\)
−0.260113 + 0.965578i \(0.583760\pi\)
\(740\) −0.343146 −0.0126143
\(741\) 0 0
\(742\) −26.3431 −0.967087
\(743\) −20.2843 −0.744158 −0.372079 0.928201i \(-0.621355\pi\)
−0.372079 + 0.928201i \(0.621355\pi\)
\(744\) 4.00000 0.146647
\(745\) −7.65685 −0.280525
\(746\) 25.3137 0.926801
\(747\) −17.6569 −0.646031
\(748\) −27.3137 −0.998688
\(749\) 11.3137 0.413394
\(750\) 1.00000 0.0365148
\(751\) −11.3137 −0.412843 −0.206422 0.978463i \(-0.566182\pi\)
−0.206422 + 0.978463i \(0.566182\pi\)
\(752\) 8.00000 0.291730
\(753\) 5.17157 0.188463
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) −2.82843 −0.102869
\(757\) −47.9411 −1.74245 −0.871225 0.490884i \(-0.836674\pi\)
−0.871225 + 0.490884i \(0.836674\pi\)
\(758\) −24.4853 −0.889345
\(759\) 48.0000 1.74229
\(760\) 2.82843 0.102598
\(761\) −16.3431 −0.592439 −0.296219 0.955120i \(-0.595726\pi\)
−0.296219 + 0.955120i \(0.595726\pi\)
\(762\) 1.65685 0.0600215
\(763\) 8.97056 0.324756
\(764\) −11.3137 −0.409316
\(765\) 4.82843 0.174572
\(766\) −18.3431 −0.662765
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) −16.0000 −0.576600
\(771\) 0.828427 0.0298351
\(772\) 14.4853 0.521337
\(773\) 30.6863 1.10371 0.551855 0.833940i \(-0.313920\pi\)
0.551855 + 0.833940i \(0.313920\pi\)
\(774\) 1.65685 0.0595544
\(775\) −4.00000 −0.143684
\(776\) −8.82843 −0.316922
\(777\) −0.970563 −0.0348187
\(778\) −10.4853 −0.375916
\(779\) −10.3431 −0.370582
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 40.9706 1.46510
\(783\) −3.17157 −0.113343
\(784\) 1.00000 0.0357143
\(785\) 17.3137 0.617953
\(786\) −22.1421 −0.789784
\(787\) 24.0000 0.855508 0.427754 0.903895i \(-0.359305\pi\)
0.427754 + 0.903895i \(0.359305\pi\)
\(788\) −9.31371 −0.331787
\(789\) −0.485281 −0.0172765
\(790\) 13.6569 0.485889
\(791\) 29.6569 1.05448
\(792\) −5.65685 −0.201008
\(793\) 0 0
\(794\) −26.2843 −0.932794
\(795\) 9.31371 0.330323
\(796\) 21.6569 0.767607
\(797\) −28.6274 −1.01404 −0.507018 0.861936i \(-0.669252\pi\)
−0.507018 + 0.861936i \(0.669252\pi\)
\(798\) 8.00000 0.283197
\(799\) −38.6274 −1.36654
\(800\) −1.00000 −0.0353553
\(801\) 4.34315 0.153458
\(802\) 6.97056 0.246139
\(803\) −14.0589 −0.496127
\(804\) 5.65685 0.199502
\(805\) 24.0000 0.845889
\(806\) 0 0
\(807\) 2.48528 0.0874860
\(808\) 12.1421 0.427159
\(809\) −9.31371 −0.327453 −0.163726 0.986506i \(-0.552351\pi\)
−0.163726 + 0.986506i \(0.552351\pi\)
\(810\) 1.00000 0.0351364
\(811\) 30.1421 1.05843 0.529217 0.848487i \(-0.322486\pi\)
0.529217 + 0.848487i \(0.322486\pi\)
\(812\) 8.97056 0.314805
\(813\) 15.3137 0.537075
\(814\) −1.94113 −0.0680364
\(815\) −11.3137 −0.396302
\(816\) −4.82843 −0.169029
\(817\) −4.68629 −0.163953
\(818\) 7.65685 0.267716
\(819\) 0 0
\(820\) 3.65685 0.127703
\(821\) 22.2843 0.777726 0.388863 0.921296i \(-0.372868\pi\)
0.388863 + 0.921296i \(0.372868\pi\)
\(822\) 5.31371 0.185337
\(823\) −19.0294 −0.663324 −0.331662 0.943398i \(-0.607609\pi\)
−0.331662 + 0.943398i \(0.607609\pi\)
\(824\) −9.65685 −0.336412
\(825\) 5.65685 0.196946
\(826\) 38.6274 1.34402
\(827\) 1.65685 0.0576145 0.0288072 0.999585i \(-0.490829\pi\)
0.0288072 + 0.999585i \(0.490829\pi\)
\(828\) 8.48528 0.294884
\(829\) −30.6863 −1.06578 −0.532889 0.846185i \(-0.678894\pi\)
−0.532889 + 0.846185i \(0.678894\pi\)
\(830\) −17.6569 −0.612878
\(831\) 26.0000 0.901930
\(832\) 0 0
\(833\) −4.82843 −0.167295
\(834\) −17.6569 −0.611407
\(835\) −24.9706 −0.864142
\(836\) 16.0000 0.553372
\(837\) −4.00000 −0.138260
\(838\) 5.17157 0.178649
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −2.82843 −0.0975900
\(841\) −18.9411 −0.653142
\(842\) 4.14214 0.142747
\(843\) −19.6569 −0.677018
\(844\) 23.3137 0.802491
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) −59.3970 −2.04090
\(848\) −9.31371 −0.319834
\(849\) −6.34315 −0.217696
\(850\) 4.82843 0.165614
\(851\) 2.91169 0.0998114
\(852\) −5.65685 −0.193801
\(853\) 18.2843 0.626042 0.313021 0.949746i \(-0.398659\pi\)
0.313021 + 0.949746i \(0.398659\pi\)
\(854\) 16.9706 0.580721
\(855\) −2.82843 −0.0967302
\(856\) 4.00000 0.136717
\(857\) −15.1716 −0.518251 −0.259126 0.965844i \(-0.583434\pi\)
−0.259126 + 0.965844i \(0.583434\pi\)
\(858\) 0 0
\(859\) −29.9411 −1.02158 −0.510789 0.859706i \(-0.670647\pi\)
−0.510789 + 0.859706i \(0.670647\pi\)
\(860\) 1.65685 0.0564983
\(861\) 10.3431 0.352493
\(862\) −16.0000 −0.544962
\(863\) −28.2843 −0.962808 −0.481404 0.876499i \(-0.659873\pi\)
−0.481404 + 0.876499i \(0.659873\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −13.3137 −0.452680
\(866\) −10.9706 −0.372795
\(867\) 6.31371 0.214425
\(868\) 11.3137 0.384012
\(869\) 77.2548 2.62069
\(870\) −3.17157 −0.107526
\(871\) 0 0
\(872\) 3.17157 0.107403
\(873\) 8.82843 0.298797
\(874\) −24.0000 −0.811812
\(875\) 2.82843 0.0956183
\(876\) −2.48528 −0.0839699
\(877\) −39.2548 −1.32554 −0.662771 0.748822i \(-0.730620\pi\)
−0.662771 + 0.748822i \(0.730620\pi\)
\(878\) 22.6274 0.763638
\(879\) −28.6274 −0.965579
\(880\) −5.65685 −0.190693
\(881\) 46.2843 1.55936 0.779678 0.626180i \(-0.215383\pi\)
0.779678 + 0.626180i \(0.215383\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 8.68629 0.292317 0.146158 0.989261i \(-0.453309\pi\)
0.146158 + 0.989261i \(0.453309\pi\)
\(884\) 0 0
\(885\) −13.6569 −0.459070
\(886\) −41.6569 −1.39949
\(887\) −23.5147 −0.789547 −0.394773 0.918778i \(-0.629177\pi\)
−0.394773 + 0.918778i \(0.629177\pi\)
\(888\) −0.343146 −0.0115152
\(889\) 4.68629 0.157173
\(890\) 4.34315 0.145583
\(891\) 5.65685 0.189512
\(892\) −5.17157 −0.173157
\(893\) 22.6274 0.757198
\(894\) −7.65685 −0.256084
\(895\) −24.4853 −0.818453
\(896\) 2.82843 0.0944911
\(897\) 0 0
\(898\) −30.2843 −1.01060
\(899\) 12.6863 0.423112
\(900\) 1.00000 0.0333333
\(901\) 44.9706 1.49819
\(902\) 20.6863 0.688778
\(903\) 4.68629 0.155950
\(904\) 10.4853 0.348735
\(905\) −3.65685 −0.121558
\(906\) 12.0000 0.398673
\(907\) −48.2843 −1.60325 −0.801626 0.597825i \(-0.796032\pi\)
−0.801626 + 0.597825i \(0.796032\pi\)
\(908\) −4.00000 −0.132745
\(909\) −12.1421 −0.402729
\(910\) 0 0
\(911\) −8.97056 −0.297208 −0.148604 0.988897i \(-0.547478\pi\)
−0.148604 + 0.988897i \(0.547478\pi\)
\(912\) 2.82843 0.0936586
\(913\) −99.8823 −3.30562
\(914\) 15.1716 0.501831
\(915\) −6.00000 −0.198354
\(916\) 24.1421 0.797679
\(917\) −62.6274 −2.06814
\(918\) 4.82843 0.159362
\(919\) −25.9411 −0.855719 −0.427859 0.903845i \(-0.640732\pi\)
−0.427859 + 0.903845i \(0.640732\pi\)
\(920\) 8.48528 0.279751
\(921\) 10.3431 0.340818
\(922\) 14.0000 0.461065
\(923\) 0 0
\(924\) −16.0000 −0.526361
\(925\) 0.343146 0.0112826
\(926\) 35.7990 1.17643
\(927\) 9.65685 0.317173
\(928\) 3.17157 0.104112
\(929\) −45.5980 −1.49602 −0.748011 0.663687i \(-0.768991\pi\)
−0.748011 + 0.663687i \(0.768991\pi\)
\(930\) −4.00000 −0.131165
\(931\) 2.82843 0.0926980
\(932\) 22.4853 0.736530
\(933\) −24.0000 −0.785725
\(934\) 15.3137 0.501080
\(935\) 27.3137 0.893254
\(936\) 0 0
\(937\) −28.6274 −0.935217 −0.467608 0.883936i \(-0.654884\pi\)
−0.467608 + 0.883936i \(0.654884\pi\)
\(938\) 16.0000 0.522419
\(939\) 2.97056 0.0969407
\(940\) −8.00000 −0.260931
\(941\) −21.0294 −0.685540 −0.342770 0.939419i \(-0.611365\pi\)
−0.342770 + 0.939419i \(0.611365\pi\)
\(942\) 17.3137 0.564111
\(943\) −31.0294 −1.01046
\(944\) 13.6569 0.444493
\(945\) 2.82843 0.0920087
\(946\) 9.37258 0.304729
\(947\) −41.6569 −1.35367 −0.676833 0.736137i \(-0.736648\pi\)
−0.676833 + 0.736137i \(0.736648\pi\)
\(948\) 13.6569 0.443554
\(949\) 0 0
\(950\) −2.82843 −0.0917663
\(951\) −2.68629 −0.0871090
\(952\) −13.6569 −0.442621
\(953\) −56.1421 −1.81862 −0.909311 0.416117i \(-0.863391\pi\)
−0.909311 + 0.416117i \(0.863391\pi\)
\(954\) 9.31371 0.301542
\(955\) 11.3137 0.366103
\(956\) −16.0000 −0.517477
\(957\) −17.9411 −0.579954
\(958\) 11.3137 0.365529
\(959\) 15.0294 0.485326
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 17.3137 0.557637
\(965\) −14.4853 −0.466298
\(966\) 24.0000 0.772187
\(967\) 24.4853 0.787394 0.393697 0.919240i \(-0.371196\pi\)
0.393697 + 0.919240i \(0.371196\pi\)
\(968\) −21.0000 −0.674966
\(969\) −13.6569 −0.438721
\(970\) 8.82843 0.283464
\(971\) 32.4853 1.04250 0.521251 0.853403i \(-0.325465\pi\)
0.521251 + 0.853403i \(0.325465\pi\)
\(972\) 1.00000 0.0320750
\(973\) −49.9411 −1.60104
\(974\) −0.485281 −0.0155494
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 19.6569 0.628878 0.314439 0.949278i \(-0.398184\pi\)
0.314439 + 0.949278i \(0.398184\pi\)
\(978\) −11.3137 −0.361773
\(979\) 24.5685 0.785214
\(980\) −1.00000 −0.0319438
\(981\) −3.17157 −0.101261
\(982\) 9.85786 0.314577
\(983\) 13.6569 0.435586 0.217793 0.975995i \(-0.430114\pi\)
0.217793 + 0.975995i \(0.430114\pi\)
\(984\) 3.65685 0.116576
\(985\) 9.31371 0.296759
\(986\) −15.3137 −0.487688
\(987\) −22.6274 −0.720239
\(988\) 0 0
\(989\) −14.0589 −0.447046
\(990\) 5.65685 0.179787
\(991\) 58.9117 1.87139 0.935696 0.352808i \(-0.114773\pi\)
0.935696 + 0.352808i \(0.114773\pi\)
\(992\) 4.00000 0.127000
\(993\) −8.48528 −0.269272
\(994\) −16.0000 −0.507489
\(995\) −21.6569 −0.686568
\(996\) −17.6569 −0.559479
\(997\) 38.6863 1.22521 0.612604 0.790390i \(-0.290122\pi\)
0.612604 + 0.790390i \(0.290122\pi\)
\(998\) −16.4853 −0.521832
\(999\) 0.343146 0.0108567
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 5070.2.a.bc.1.1 2
13.5 odd 4 5070.2.b.q.1351.3 4
13.8 odd 4 5070.2.b.q.1351.2 4
13.12 even 2 390.2.a.h.1.2 2
39.38 odd 2 1170.2.a.o.1.2 2
52.51 odd 2 3120.2.a.bc.1.1 2
65.12 odd 4 1950.2.e.o.1249.4 4
65.38 odd 4 1950.2.e.o.1249.1 4
65.64 even 2 1950.2.a.bd.1.1 2
156.155 even 2 9360.2.a.ch.1.1 2
195.38 even 4 5850.2.e.bk.5149.3 4
195.77 even 4 5850.2.e.bk.5149.2 4
195.194 odd 2 5850.2.a.cl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.2 2 13.12 even 2
1170.2.a.o.1.2 2 39.38 odd 2
1950.2.a.bd.1.1 2 65.64 even 2
1950.2.e.o.1249.1 4 65.38 odd 4
1950.2.e.o.1249.4 4 65.12 odd 4
3120.2.a.bc.1.1 2 52.51 odd 2
5070.2.a.bc.1.1 2 1.1 even 1 trivial
5070.2.b.q.1351.2 4 13.8 odd 4
5070.2.b.q.1351.3 4 13.5 odd 4
5850.2.a.cl.1.1 2 195.194 odd 2
5850.2.e.bk.5149.2 4 195.77 even 4
5850.2.e.bk.5149.3 4 195.38 even 4
9360.2.a.ch.1.1 2 156.155 even 2