Properties

Label 6762.2.a.cg.1.3
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.138768.1
Defining polynomial: \(x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.47903\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.47903 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.47903 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.47903 q^{10} +5.70448 q^{11} -1.00000 q^{12} +6.35023 q^{13} -1.47903 q^{15} +1.00000 q^{16} -4.12478 q^{17} -1.00000 q^{18} +3.39217 q^{19} +1.47903 q^{20} -5.70448 q^{22} +1.00000 q^{23} +1.00000 q^{24} -2.81247 q^{25} -6.35023 q^{26} -1.00000 q^{27} +2.39217 q^{29} +1.47903 q^{30} -7.81247 q^{31} -1.00000 q^{32} -5.70448 q^{33} +4.12478 q^{34} +1.00000 q^{36} +8.35023 q^{37} -3.39217 q^{38} -6.35023 q^{39} -1.47903 q^{40} +7.39217 q^{41} +2.00000 q^{43} +5.70448 q^{44} +1.47903 q^{45} -1.00000 q^{46} +10.2214 q^{47} -1.00000 q^{48} +2.81247 q^{50} +4.12478 q^{51} +6.35023 q^{52} +0.520971 q^{53} +1.00000 q^{54} +8.43709 q^{55} -3.39217 q^{57} -2.39217 q^{58} +7.37836 q^{59} -1.47903 q^{60} -5.04194 q^{61} +7.81247 q^{62} +1.00000 q^{64} +9.39217 q^{65} +5.70448 q^{66} +15.4090 q^{67} -4.12478 q^{68} -1.00000 q^{69} +9.87120 q^{71} -1.00000 q^{72} +8.92993 q^{73} -8.35023 q^{74} +2.81247 q^{75} +3.39217 q^{76} +6.35023 q^{78} -13.4301 q^{79} +1.47903 q^{80} +1.00000 q^{81} -7.39217 q^{82} -9.22143 q^{83} -6.10067 q^{85} -2.00000 q^{86} -2.39217 q^{87} -5.70448 q^{88} -6.43411 q^{89} -1.47903 q^{90} +1.00000 q^{92} +7.81247 q^{93} -10.2214 q^{94} +5.01712 q^{95} +1.00000 q^{96} -10.3640 q^{97} +5.70448 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 2q^{5} + 4q^{6} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 2q^{5} + 4q^{6} - 4q^{8} + 4q^{9} - 2q^{10} + 6q^{11} - 4q^{12} - 2q^{13} - 2q^{15} + 4q^{16} - 2q^{17} - 4q^{18} - 6q^{19} + 2q^{20} - 6q^{22} + 4q^{23} + 4q^{24} + 6q^{25} + 2q^{26} - 4q^{27} - 10q^{29} + 2q^{30} - 14q^{31} - 4q^{32} - 6q^{33} + 2q^{34} + 4q^{36} + 6q^{37} + 6q^{38} + 2q^{39} - 2q^{40} + 10q^{41} + 8q^{43} + 6q^{44} + 2q^{45} - 4q^{46} - 10q^{47} - 4q^{48} - 6q^{50} + 2q^{51} - 2q^{52} + 6q^{53} + 4q^{54} + 22q^{55} + 6q^{57} + 10q^{58} + 24q^{59} - 2q^{60} - 28q^{61} + 14q^{62} + 4q^{64} + 18q^{65} + 6q^{66} + 28q^{67} - 2q^{68} - 4q^{69} + 16q^{71} - 4q^{72} + 6q^{73} - 6q^{74} - 6q^{75} - 6q^{76} - 2q^{78} - 4q^{79} + 2q^{80} + 4q^{81} - 10q^{82} + 14q^{83} - 26q^{85} - 8q^{86} + 10q^{87} - 6q^{88} - 14q^{89} - 2q^{90} + 4q^{92} + 14q^{93} + 10q^{94} - 34q^{95} + 4q^{96} + 6q^{99} + O(q^{100}) \)

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
\(5\) 1.47903 0.661442 0.330721 0.943729i \(-0.392708\pi\)
0.330721 + 0.943729i \(0.392708\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.47903 −0.467710
\(11\) 5.70448 1.71996 0.859982 0.510324i \(-0.170474\pi\)
0.859982 + 0.510324i \(0.170474\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.35023 1.76124 0.880618 0.473826i \(-0.157128\pi\)
0.880618 + 0.473826i \(0.157128\pi\)
\(14\) 0 0
\(15\) −1.47903 −0.381884
\(16\) 1.00000 0.250000
\(17\) −4.12478 −1.00041 −0.500203 0.865908i \(-0.666741\pi\)
−0.500203 + 0.865908i \(0.666741\pi\)
\(18\) −1.00000 −0.235702
\(19\) 3.39217 0.778217 0.389109 0.921192i \(-0.372783\pi\)
0.389109 + 0.921192i \(0.372783\pi\)
\(20\) 1.47903 0.330721
\(21\) 0 0
\(22\) −5.70448 −1.21620
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) −2.81247 −0.562495
\(26\) −6.35023 −1.24538
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.39217 0.444215 0.222108 0.975022i \(-0.428706\pi\)
0.222108 + 0.975022i \(0.428706\pi\)
\(30\) 1.47903 0.270032
\(31\) −7.81247 −1.40316 −0.701581 0.712590i \(-0.747522\pi\)
−0.701581 + 0.712590i \(0.747522\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.70448 −0.993022
\(34\) 4.12478 0.707394
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.35023 1.37277 0.686385 0.727239i \(-0.259197\pi\)
0.686385 + 0.727239i \(0.259197\pi\)
\(38\) −3.39217 −0.550283
\(39\) −6.35023 −1.01685
\(40\) −1.47903 −0.233855
\(41\) 7.39217 1.15446 0.577232 0.816580i \(-0.304133\pi\)
0.577232 + 0.816580i \(0.304133\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 5.70448 0.859982
\(45\) 1.47903 0.220481
\(46\) −1.00000 −0.147442
\(47\) 10.2214 1.49095 0.745474 0.666534i \(-0.232223\pi\)
0.745474 + 0.666534i \(0.232223\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 2.81247 0.397744
\(51\) 4.12478 0.577585
\(52\) 6.35023 0.880618
\(53\) 0.520971 0.0715609 0.0357804 0.999360i \(-0.488608\pi\)
0.0357804 + 0.999360i \(0.488608\pi\)
\(54\) 1.00000 0.136083
\(55\) 8.43709 1.13766
\(56\) 0 0
\(57\) −3.39217 −0.449304
\(58\) −2.39217 −0.314107
\(59\) 7.37836 0.960581 0.480290 0.877110i \(-0.340531\pi\)
0.480290 + 0.877110i \(0.340531\pi\)
\(60\) −1.47903 −0.190942
\(61\) −5.04194 −0.645555 −0.322777 0.946475i \(-0.604616\pi\)
−0.322777 + 0.946475i \(0.604616\pi\)
\(62\) 7.81247 0.992185
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 9.39217 1.16496
\(66\) 5.70448 0.702173
\(67\) 15.4090 1.88250 0.941252 0.337706i \(-0.109651\pi\)
0.941252 + 0.337706i \(0.109651\pi\)
\(68\) −4.12478 −0.500203
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 9.87120 1.17150 0.585748 0.810493i \(-0.300801\pi\)
0.585748 + 0.810493i \(0.300801\pi\)
\(72\) −1.00000 −0.117851
\(73\) 8.92993 1.04517 0.522584 0.852588i \(-0.324968\pi\)
0.522584 + 0.852588i \(0.324968\pi\)
\(74\) −8.35023 −0.970694
\(75\) 2.81247 0.324757
\(76\) 3.39217 0.389109
\(77\) 0 0
\(78\) 6.35023 0.719022
\(79\) −13.4301 −1.51100 −0.755502 0.655147i \(-0.772607\pi\)
−0.755502 + 0.655147i \(0.772607\pi\)
\(80\) 1.47903 0.165360
\(81\) 1.00000 0.111111
\(82\) −7.39217 −0.816329
\(83\) −9.22143 −1.01218 −0.506092 0.862480i \(-0.668910\pi\)
−0.506092 + 0.862480i \(0.668910\pi\)
\(84\) 0 0
\(85\) −6.10067 −0.661710
\(86\) −2.00000 −0.215666
\(87\) −2.39217 −0.256468
\(88\) −5.70448 −0.608099
\(89\) −6.43411 −0.682015 −0.341007 0.940061i \(-0.610768\pi\)
−0.341007 + 0.940061i \(0.610768\pi\)
\(90\) −1.47903 −0.155903
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 7.81247 0.810116
\(94\) −10.2214 −1.05426
\(95\) 5.01712 0.514746
\(96\) 1.00000 0.102062
\(97\) −10.3640 −1.05231 −0.526154 0.850389i \(-0.676367\pi\)
−0.526154 + 0.850389i \(0.676367\pi\)
\(98\) 0 0
\(99\) 5.70448 0.573322
\(100\) −2.81247 −0.281247
\(101\) 2.81247 0.279852 0.139926 0.990162i \(-0.455314\pi\)
0.139926 + 0.990162i \(0.455314\pi\)
\(102\) −4.12478 −0.408414
\(103\) −12.5589 −1.23746 −0.618732 0.785602i \(-0.712353\pi\)
−0.618732 + 0.785602i \(0.712353\pi\)
\(104\) −6.35023 −0.622691
\(105\) 0 0
\(106\) −0.520971 −0.0506012
\(107\) −5.22143 −0.504775 −0.252387 0.967626i \(-0.581216\pi\)
−0.252387 + 0.967626i \(0.581216\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −8.47501 −0.811759 −0.405879 0.913927i \(-0.633035\pi\)
−0.405879 + 0.913927i \(0.633035\pi\)
\(110\) −8.43709 −0.804445
\(111\) −8.35023 −0.792569
\(112\) 0 0
\(113\) 0.725281 0.0682287 0.0341144 0.999418i \(-0.489139\pi\)
0.0341144 + 0.999418i \(0.489139\pi\)
\(114\) 3.39217 0.317706
\(115\) 1.47903 0.137920
\(116\) 2.39217 0.222108
\(117\) 6.35023 0.587079
\(118\) −7.37836 −0.679233
\(119\) 0 0
\(120\) 1.47903 0.135016
\(121\) 21.5411 1.95828
\(122\) 5.04194 0.456476
\(123\) −7.39217 −0.666530
\(124\) −7.81247 −0.701581
\(125\) −11.5549 −1.03350
\(126\) 0 0
\(127\) 0.0138109 0.00122552 0.000612760 1.00000i \(-0.499805\pi\)
0.000612760 1.00000i \(0.499805\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) −9.39217 −0.823748
\(131\) −14.4961 −1.26653 −0.633267 0.773933i \(-0.718287\pi\)
−0.633267 + 0.773933i \(0.718287\pi\)
\(132\) −5.70448 −0.496511
\(133\) 0 0
\(134\) −15.4090 −1.33113
\(135\) −1.47903 −0.127295
\(136\) 4.12478 0.353697
\(137\) 8.02411 0.685546 0.342773 0.939418i \(-0.388634\pi\)
0.342773 + 0.939418i \(0.388634\pi\)
\(138\) 1.00000 0.0851257
\(139\) 14.2214 1.20625 0.603123 0.797648i \(-0.293923\pi\)
0.603123 + 0.797648i \(0.293923\pi\)
\(140\) 0 0
\(141\) −10.2214 −0.860800
\(142\) −9.87120 −0.828373
\(143\) 36.2247 3.02926
\(144\) 1.00000 0.0833333
\(145\) 3.53809 0.293822
\(146\) −8.92993 −0.739046
\(147\) 0 0
\(148\) 8.35023 0.686385
\(149\) −23.2835 −1.90746 −0.953728 0.300671i \(-0.902789\pi\)
−0.953728 + 0.300671i \(0.902789\pi\)
\(150\) −2.81247 −0.229638
\(151\) −6.01381 −0.489397 −0.244698 0.969599i \(-0.578689\pi\)
−0.244698 + 0.969599i \(0.578689\pi\)
\(152\) −3.39217 −0.275141
\(153\) −4.12478 −0.333469
\(154\) 0 0
\(155\) −11.5549 −0.928110
\(156\) −6.35023 −0.508425
\(157\) −17.8672 −1.42596 −0.712978 0.701186i \(-0.752654\pi\)
−0.712978 + 0.701186i \(0.752654\pi\)
\(158\) 13.4301 1.06844
\(159\) −0.520971 −0.0413157
\(160\) −1.47903 −0.116927
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 13.8712 1.08648 0.543238 0.839579i \(-0.317198\pi\)
0.543238 + 0.839579i \(0.317198\pi\)
\(164\) 7.39217 0.577232
\(165\) −8.43709 −0.656826
\(166\) 9.22143 0.715722
\(167\) 5.07551 0.392755 0.196377 0.980528i \(-0.437082\pi\)
0.196377 + 0.980528i \(0.437082\pi\)
\(168\) 0 0
\(169\) 27.3254 2.10195
\(170\) 6.10067 0.467900
\(171\) 3.39217 0.259406
\(172\) 2.00000 0.152499
\(173\) 2.92993 0.222758 0.111379 0.993778i \(-0.464473\pi\)
0.111379 + 0.993778i \(0.464473\pi\)
\(174\) 2.39217 0.181350
\(175\) 0 0
\(176\) 5.70448 0.429991
\(177\) −7.37836 −0.554592
\(178\) 6.43411 0.482257
\(179\) −7.94374 −0.593743 −0.296871 0.954917i \(-0.595943\pi\)
−0.296871 + 0.954917i \(0.595943\pi\)
\(180\) 1.47903 0.110240
\(181\) −7.39950 −0.550000 −0.275000 0.961444i \(-0.588678\pi\)
−0.275000 + 0.961444i \(0.588678\pi\)
\(182\) 0 0
\(183\) 5.04194 0.372711
\(184\) −1.00000 −0.0737210
\(185\) 12.3502 0.908007
\(186\) −7.81247 −0.572838
\(187\) −23.5297 −1.72066
\(188\) 10.2214 0.745474
\(189\) 0 0
\(190\) −5.01712 −0.363980
\(191\) 20.1765 1.45992 0.729961 0.683489i \(-0.239538\pi\)
0.729961 + 0.683489i \(0.239538\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 12.8880 0.927697 0.463849 0.885914i \(-0.346468\pi\)
0.463849 + 0.885914i \(0.346468\pi\)
\(194\) 10.3640 0.744095
\(195\) −9.39217 −0.672587
\(196\) 0 0
\(197\) −9.80410 −0.698513 −0.349257 0.937027i \(-0.613566\pi\)
−0.349257 + 0.937027i \(0.613566\pi\)
\(198\) −5.70448 −0.405400
\(199\) 3.31561 0.235038 0.117519 0.993071i \(-0.462506\pi\)
0.117519 + 0.993071i \(0.462506\pi\)
\(200\) 2.81247 0.198872
\(201\) −15.4090 −1.08686
\(202\) −2.81247 −0.197885
\(203\) 0 0
\(204\) 4.12478 0.288792
\(205\) 10.9332 0.763610
\(206\) 12.5589 0.875020
\(207\) 1.00000 0.0695048
\(208\) 6.35023 0.440309
\(209\) 19.3506 1.33851
\(210\) 0 0
\(211\) −17.9638 −1.23668 −0.618340 0.785910i \(-0.712195\pi\)
−0.618340 + 0.785910i \(0.712195\pi\)
\(212\) 0.520971 0.0357804
\(213\) −9.87120 −0.676363
\(214\) 5.22143 0.356930
\(215\) 2.95806 0.201738
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 8.47501 0.574000
\(219\) −8.92993 −0.603428
\(220\) 8.43709 0.568828
\(221\) −26.1933 −1.76195
\(222\) 8.35023 0.560431
\(223\) −1.22947 −0.0823313 −0.0411657 0.999152i \(-0.513107\pi\)
−0.0411657 + 0.999152i \(0.513107\pi\)
\(224\) 0 0
\(225\) −2.81247 −0.187498
\(226\) −0.725281 −0.0482450
\(227\) 16.4130 1.08937 0.544684 0.838642i \(-0.316650\pi\)
0.544684 + 0.838642i \(0.316650\pi\)
\(228\) −3.39217 −0.224652
\(229\) −25.4163 −1.67956 −0.839778 0.542931i \(-0.817315\pi\)
−0.839778 + 0.542931i \(0.817315\pi\)
\(230\) −1.47903 −0.0975243
\(231\) 0 0
\(232\) −2.39217 −0.157054
\(233\) 0.551566 0.0361343 0.0180671 0.999837i \(-0.494249\pi\)
0.0180671 + 0.999837i \(0.494249\pi\)
\(234\) −6.35023 −0.415127
\(235\) 15.1178 0.986176
\(236\) 7.37836 0.480290
\(237\) 13.4301 0.872378
\(238\) 0 0
\(239\) −11.2634 −0.728567 −0.364283 0.931288i \(-0.618686\pi\)
−0.364283 + 0.931288i \(0.618686\pi\)
\(240\) −1.47903 −0.0954709
\(241\) −28.3284 −1.82479 −0.912396 0.409309i \(-0.865770\pi\)
−0.912396 + 0.409309i \(0.865770\pi\)
\(242\) −21.5411 −1.38471
\(243\) −1.00000 −0.0641500
\(244\) −5.04194 −0.322777
\(245\) 0 0
\(246\) 7.39217 0.471308
\(247\) 21.5411 1.37063
\(248\) 7.81247 0.496093
\(249\) 9.22143 0.584384
\(250\) 11.5549 0.730794
\(251\) −7.83659 −0.494641 −0.247320 0.968934i \(-0.579550\pi\)
−0.247320 + 0.968934i \(0.579550\pi\)
\(252\) 0 0
\(253\) 5.70448 0.358637
\(254\) −0.0138109 −0.000866573 0
\(255\) 6.10067 0.382039
\(256\) 1.00000 0.0625000
\(257\) 3.72562 0.232398 0.116199 0.993226i \(-0.462929\pi\)
0.116199 + 0.993226i \(0.462929\pi\)
\(258\) 2.00000 0.124515
\(259\) 0 0
\(260\) 9.39217 0.582478
\(261\) 2.39217 0.148072
\(262\) 14.4961 0.895575
\(263\) 28.8661 1.77996 0.889981 0.455998i \(-0.150717\pi\)
0.889981 + 0.455998i \(0.150717\pi\)
\(264\) 5.70448 0.351086
\(265\) 0.770531 0.0473334
\(266\) 0 0
\(267\) 6.43411 0.393761
\(268\) 15.4090 0.941252
\(269\) −0.482002 −0.0293882 −0.0146941 0.999892i \(-0.504677\pi\)
−0.0146941 + 0.999892i \(0.504677\pi\)
\(270\) 1.47903 0.0900108
\(271\) −4.89602 −0.297412 −0.148706 0.988881i \(-0.547511\pi\)
−0.148706 + 0.988881i \(0.547511\pi\)
\(272\) −4.12478 −0.250102
\(273\) 0 0
\(274\) −8.02411 −0.484754
\(275\) −16.0437 −0.967471
\(276\) −1.00000 −0.0601929
\(277\) 15.5264 0.932891 0.466446 0.884550i \(-0.345534\pi\)
0.466446 + 0.884550i \(0.345534\pi\)
\(278\) −14.2214 −0.852945
\(279\) −7.81247 −0.467721
\(280\) 0 0
\(281\) 13.2003 0.787463 0.393732 0.919225i \(-0.371184\pi\)
0.393732 + 0.919225i \(0.371184\pi\)
\(282\) 10.2214 0.608677
\(283\) −2.75672 −0.163870 −0.0819350 0.996638i \(-0.526110\pi\)
−0.0819350 + 0.996638i \(0.526110\pi\)
\(284\) 9.87120 0.585748
\(285\) −5.01712 −0.297188
\(286\) −36.2247 −2.14201
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 0.0138109 0.000812406 0
\(290\) −3.53809 −0.207764
\(291\) 10.3640 0.607551
\(292\) 8.92993 0.522584
\(293\) 0.0976957 0.00570745 0.00285372 0.999996i \(-0.499092\pi\)
0.00285372 + 0.999996i \(0.499092\pi\)
\(294\) 0 0
\(295\) 10.9128 0.635368
\(296\) −8.35023 −0.485347
\(297\) −5.70448 −0.331007
\(298\) 23.2835 1.34877
\(299\) 6.35023 0.367243
\(300\) 2.81247 0.162378
\(301\) 0 0
\(302\) 6.01381 0.346056
\(303\) −2.81247 −0.161572
\(304\) 3.39217 0.194554
\(305\) −7.45718 −0.426997
\(306\) 4.12478 0.235798
\(307\) −24.9467 −1.42378 −0.711892 0.702289i \(-0.752161\pi\)
−0.711892 + 0.702289i \(0.752161\pi\)
\(308\) 0 0
\(309\) 12.5589 0.714450
\(310\) 11.5549 0.656273
\(311\) 26.4878 1.50198 0.750992 0.660311i \(-0.229576\pi\)
0.750992 + 0.660311i \(0.229576\pi\)
\(312\) 6.35023 0.359511
\(313\) −19.2382 −1.08741 −0.543704 0.839277i \(-0.682979\pi\)
−0.543704 + 0.839277i \(0.682979\pi\)
\(314\) 17.8672 1.00830
\(315\) 0 0
\(316\) −13.4301 −0.755502
\(317\) 25.3784 1.42539 0.712695 0.701474i \(-0.247474\pi\)
0.712695 + 0.701474i \(0.247474\pi\)
\(318\) 0.520971 0.0292146
\(319\) 13.6461 0.764034
\(320\) 1.47903 0.0826802
\(321\) 5.22143 0.291432
\(322\) 0 0
\(323\) −13.9920 −0.778533
\(324\) 1.00000 0.0555556
\(325\) −17.8599 −0.990686
\(326\) −13.8712 −0.768255
\(327\) 8.47501 0.468669
\(328\) −7.39217 −0.408164
\(329\) 0 0
\(330\) 8.43709 0.464446
\(331\) 2.84273 0.156251 0.0781254 0.996944i \(-0.475107\pi\)
0.0781254 + 0.996944i \(0.475107\pi\)
\(332\) −9.22143 −0.506092
\(333\) 8.35023 0.457590
\(334\) −5.07551 −0.277720
\(335\) 22.7903 1.24517
\(336\) 0 0
\(337\) 26.7793 1.45876 0.729380 0.684109i \(-0.239809\pi\)
0.729380 + 0.684109i \(0.239809\pi\)
\(338\) −27.3254 −1.48631
\(339\) −0.725281 −0.0393919
\(340\) −6.10067 −0.330855
\(341\) −44.5661 −2.41339
\(342\) −3.39217 −0.183428
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) −1.47903 −0.0796282
\(346\) −2.92993 −0.157514
\(347\) −5.62495 −0.301963 −0.150982 0.988537i \(-0.548243\pi\)
−0.150982 + 0.988537i \(0.548243\pi\)
\(348\) −2.39217 −0.128234
\(349\) 20.8011 1.11346 0.556729 0.830694i \(-0.312056\pi\)
0.556729 + 0.830694i \(0.312056\pi\)
\(350\) 0 0
\(351\) −6.35023 −0.338950
\(352\) −5.70448 −0.304050
\(353\) −34.7092 −1.84738 −0.923692 0.383135i \(-0.874844\pi\)
−0.923692 + 0.383135i \(0.874844\pi\)
\(354\) 7.37836 0.392155
\(355\) 14.5998 0.774876
\(356\) −6.43411 −0.341007
\(357\) 0 0
\(358\) 7.94374 0.419840
\(359\) −5.78401 −0.305268 −0.152634 0.988283i \(-0.548776\pi\)
−0.152634 + 0.988283i \(0.548776\pi\)
\(360\) −1.47903 −0.0779517
\(361\) −7.49317 −0.394378
\(362\) 7.39950 0.388909
\(363\) −21.5411 −1.13061
\(364\) 0 0
\(365\) 13.2076 0.691318
\(366\) −5.04194 −0.263547
\(367\) 0.669863 0.0349666 0.0174833 0.999847i \(-0.494435\pi\)
0.0174833 + 0.999847i \(0.494435\pi\)
\(368\) 1.00000 0.0521286
\(369\) 7.39217 0.384821
\(370\) −12.3502 −0.642058
\(371\) 0 0
\(372\) 7.81247 0.405058
\(373\) −9.80008 −0.507429 −0.253714 0.967279i \(-0.581652\pi\)
−0.253714 + 0.967279i \(0.581652\pi\)
\(374\) 23.5297 1.21669
\(375\) 11.5549 0.596691
\(376\) −10.2214 −0.527130
\(377\) 15.1908 0.782368
\(378\) 0 0
\(379\) −0.742065 −0.0381173 −0.0190587 0.999818i \(-0.506067\pi\)
−0.0190587 + 0.999818i \(0.506067\pi\)
\(380\) 5.01712 0.257373
\(381\) −0.0138109 −0.000707554 0
\(382\) −20.1765 −1.03232
\(383\) 30.4114 1.55395 0.776975 0.629531i \(-0.216753\pi\)
0.776975 + 0.629531i \(0.216753\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −12.8880 −0.655981
\(387\) 2.00000 0.101666
\(388\) −10.3640 −0.526154
\(389\) −0.674930 −0.0342203 −0.0171102 0.999854i \(-0.505447\pi\)
−0.0171102 + 0.999854i \(0.505447\pi\)
\(390\) 9.39217 0.475591
\(391\) −4.12478 −0.208599
\(392\) 0 0
\(393\) 14.4961 0.731234
\(394\) 9.80410 0.493923
\(395\) −19.8635 −0.999441
\(396\) 5.70448 0.286661
\(397\) 8.42607 0.422893 0.211446 0.977390i \(-0.432183\pi\)
0.211446 + 0.977390i \(0.432183\pi\)
\(398\) −3.31561 −0.166197
\(399\) 0 0
\(400\) −2.81247 −0.140624
\(401\) 35.1192 1.75377 0.876885 0.480701i \(-0.159618\pi\)
0.876885 + 0.480701i \(0.159618\pi\)
\(402\) 15.4090 0.768529
\(403\) −49.6110 −2.47130
\(404\) 2.81247 0.139926
\(405\) 1.47903 0.0734935
\(406\) 0 0
\(407\) 47.6337 2.36111
\(408\) −4.12478 −0.204207
\(409\) 19.7281 0.975491 0.487745 0.872986i \(-0.337819\pi\)
0.487745 + 0.872986i \(0.337819\pi\)
\(410\) −10.9332 −0.539954
\(411\) −8.02411 −0.395800
\(412\) −12.5589 −0.618732
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −13.6388 −0.669500
\(416\) −6.35023 −0.311346
\(417\) −14.2214 −0.696426
\(418\) −19.3506 −0.946467
\(419\) 7.26739 0.355035 0.177518 0.984118i \(-0.443193\pi\)
0.177518 + 0.984118i \(0.443193\pi\)
\(420\) 0 0
\(421\) 3.82277 0.186311 0.0931553 0.995652i \(-0.470305\pi\)
0.0931553 + 0.995652i \(0.470305\pi\)
\(422\) 17.9638 0.874465
\(423\) 10.2214 0.496983
\(424\) −0.520971 −0.0253006
\(425\) 11.6008 0.562723
\(426\) 9.87120 0.478261
\(427\) 0 0
\(428\) −5.22143 −0.252387
\(429\) −36.2247 −1.74895
\(430\) −2.95806 −0.142650
\(431\) −1.97238 −0.0950061 −0.0475031 0.998871i \(-0.515126\pi\)
−0.0475031 + 0.998871i \(0.515126\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −31.3929 −1.50865 −0.754323 0.656504i \(-0.772035\pi\)
−0.754323 + 0.656504i \(0.772035\pi\)
\(434\) 0 0
\(435\) −3.53809 −0.169638
\(436\) −8.47501 −0.405879
\(437\) 3.39217 0.162270
\(438\) 8.92993 0.426688
\(439\) −35.7205 −1.70485 −0.852424 0.522851i \(-0.824869\pi\)
−0.852424 + 0.522851i \(0.824869\pi\)
\(440\) −8.43709 −0.402222
\(441\) 0 0
\(442\) 26.1933 1.24589
\(443\) −0.114479 −0.00543908 −0.00271954 0.999996i \(-0.500866\pi\)
−0.00271954 + 0.999996i \(0.500866\pi\)
\(444\) −8.35023 −0.396284
\(445\) −9.51624 −0.451113
\(446\) 1.22947 0.0582170
\(447\) 23.2835 1.10127
\(448\) 0 0
\(449\) −9.26006 −0.437009 −0.218505 0.975836i \(-0.570118\pi\)
−0.218505 + 0.975836i \(0.570118\pi\)
\(450\) 2.81247 0.132581
\(451\) 42.1685 1.98564
\(452\) 0.725281 0.0341144
\(453\) 6.01381 0.282553
\(454\) −16.4130 −0.770299
\(455\) 0 0
\(456\) 3.39217 0.158853
\(457\) 5.83730 0.273057 0.136529 0.990636i \(-0.456405\pi\)
0.136529 + 0.990636i \(0.456405\pi\)
\(458\) 25.4163 1.18762
\(459\) 4.12478 0.192528
\(460\) 1.47903 0.0689601
\(461\) 23.1660 1.07895 0.539474 0.842002i \(-0.318623\pi\)
0.539474 + 0.842002i \(0.318623\pi\)
\(462\) 0 0
\(463\) 0.140148 0.00651322 0.00325661 0.999995i \(-0.498963\pi\)
0.00325661 + 0.999995i \(0.498963\pi\)
\(464\) 2.39217 0.111054
\(465\) 11.5549 0.535844
\(466\) −0.551566 −0.0255508
\(467\) 15.5170 0.718039 0.359019 0.933330i \(-0.383111\pi\)
0.359019 + 0.933330i \(0.383111\pi\)
\(468\) 6.35023 0.293539
\(469\) 0 0
\(470\) −15.1178 −0.697332
\(471\) 17.8672 0.823276
\(472\) −7.37836 −0.339617
\(473\) 11.4090 0.524584
\(474\) −13.4301 −0.616865
\(475\) −9.54039 −0.437743
\(476\) 0 0
\(477\) 0.520971 0.0238536
\(478\) 11.2634 0.515175
\(479\) 15.3922 0.703286 0.351643 0.936134i \(-0.385623\pi\)
0.351643 + 0.936134i \(0.385623\pi\)
\(480\) 1.47903 0.0675081
\(481\) 53.0259 2.41777
\(482\) 28.3284 1.29032
\(483\) 0 0
\(484\) 21.5411 0.979139
\(485\) −15.3287 −0.696041
\(486\) 1.00000 0.0453609
\(487\) −0.554539 −0.0251285 −0.0125643 0.999921i \(-0.503999\pi\)
−0.0125643 + 0.999921i \(0.503999\pi\)
\(488\) 5.04194 0.228238
\(489\) −13.8712 −0.627277
\(490\) 0 0
\(491\) 43.1966 1.94944 0.974718 0.223440i \(-0.0717286\pi\)
0.974718 + 0.223440i \(0.0717286\pi\)
\(492\) −7.39217 −0.333265
\(493\) −9.86718 −0.444395
\(494\) −21.5411 −0.969178
\(495\) 8.43709 0.379219
\(496\) −7.81247 −0.350790
\(497\) 0 0
\(498\) −9.22143 −0.413222
\(499\) −20.8943 −0.935356 −0.467678 0.883899i \(-0.654909\pi\)
−0.467678 + 0.883899i \(0.654909\pi\)
\(500\) −11.5549 −0.516750
\(501\) −5.07551 −0.226757
\(502\) 7.83659 0.349764
\(503\) −10.1657 −0.453265 −0.226633 0.973980i \(-0.572772\pi\)
−0.226633 + 0.973980i \(0.572772\pi\)
\(504\) 0 0
\(505\) 4.15973 0.185106
\(506\) −5.70448 −0.253595
\(507\) −27.3254 −1.21356
\(508\) 0.0138109 0.000612760 0
\(509\) −13.7319 −0.608656 −0.304328 0.952567i \(-0.598432\pi\)
−0.304328 + 0.952567i \(0.598432\pi\)
\(510\) −6.10067 −0.270142
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −3.39217 −0.149768
\(514\) −3.72562 −0.164330
\(515\) −18.5750 −0.818511
\(516\) −2.00000 −0.0880451
\(517\) 58.3079 2.56438
\(518\) 0 0
\(519\) −2.92993 −0.128609
\(520\) −9.39217 −0.411874
\(521\) −3.62495 −0.158812 −0.0794059 0.996842i \(-0.525302\pi\)
−0.0794059 + 0.996842i \(0.525302\pi\)
\(522\) −2.39217 −0.104702
\(523\) −28.9420 −1.26554 −0.632772 0.774338i \(-0.718083\pi\)
−0.632772 + 0.774338i \(0.718083\pi\)
\(524\) −14.4961 −0.633267
\(525\) 0 0
\(526\) −28.8661 −1.25862
\(527\) 32.2247 1.40373
\(528\) −5.70448 −0.248256
\(529\) 1.00000 0.0434783
\(530\) −0.770531 −0.0334697
\(531\) 7.37836 0.320194
\(532\) 0 0
\(533\) 46.9420 2.03328
\(534\) −6.43411 −0.278431
\(535\) −7.72264 −0.333879
\(536\) −15.4090 −0.665565
\(537\) 7.94374 0.342798
\(538\) 0.482002 0.0207806
\(539\) 0 0
\(540\) −1.47903 −0.0636473
\(541\) −33.0171 −1.41952 −0.709758 0.704445i \(-0.751196\pi\)
−0.709758 + 0.704445i \(0.751196\pi\)
\(542\) 4.89602 0.210302
\(543\) 7.39950 0.317543
\(544\) 4.12478 0.176848
\(545\) −12.5348 −0.536931
\(546\) 0 0
\(547\) 36.6891 1.56871 0.784357 0.620310i \(-0.212993\pi\)
0.784357 + 0.620310i \(0.212993\pi\)
\(548\) 8.02411 0.342773
\(549\) −5.04194 −0.215185
\(550\) 16.0437 0.684105
\(551\) 8.11465 0.345696
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −15.5264 −0.659654
\(555\) −12.3502 −0.524238
\(556\) 14.2214 0.603123
\(557\) 30.9362 1.31081 0.655405 0.755278i \(-0.272498\pi\)
0.655405 + 0.755278i \(0.272498\pi\)
\(558\) 7.81247 0.330728
\(559\) 12.7005 0.537172
\(560\) 0 0
\(561\) 23.5297 0.993425
\(562\) −13.2003 −0.556821
\(563\) 0.553123 0.0233113 0.0116557 0.999932i \(-0.496290\pi\)
0.0116557 + 0.999932i \(0.496290\pi\)
\(564\) −10.2214 −0.430400
\(565\) 1.07271 0.0451293
\(566\) 2.75672 0.115874
\(567\) 0 0
\(568\) −9.87120 −0.414186
\(569\) 2.38167 0.0998447 0.0499224 0.998753i \(-0.484103\pi\)
0.0499224 + 0.998753i \(0.484103\pi\)
\(570\) 5.01712 0.210144
\(571\) 26.5662 1.11176 0.555881 0.831262i \(-0.312381\pi\)
0.555881 + 0.831262i \(0.312381\pi\)
\(572\) 36.2247 1.51463
\(573\) −20.1765 −0.842886
\(574\) 0 0
\(575\) −2.81247 −0.117288
\(576\) 1.00000 0.0416667
\(577\) −5.72315 −0.238258 −0.119129 0.992879i \(-0.538010\pi\)
−0.119129 + 0.992879i \(0.538010\pi\)
\(578\) −0.0138109 −0.000574458 0
\(579\) −12.8880 −0.535606
\(580\) 3.53809 0.146911
\(581\) 0 0
\(582\) −10.3640 −0.429603
\(583\) 2.97187 0.123082
\(584\) −8.92993 −0.369523
\(585\) 9.39217 0.388318
\(586\) −0.0976957 −0.00403577
\(587\) 19.2323 0.793801 0.396900 0.917862i \(-0.370086\pi\)
0.396900 + 0.917862i \(0.370086\pi\)
\(588\) 0 0
\(589\) −26.5012 −1.09196
\(590\) −10.9128 −0.449273
\(591\) 9.80410 0.403287
\(592\) 8.35023 0.343192
\(593\) −20.6858 −0.849464 −0.424732 0.905319i \(-0.639632\pi\)
−0.424732 + 0.905319i \(0.639632\pi\)
\(594\) 5.70448 0.234058
\(595\) 0 0
\(596\) −23.2835 −0.953728
\(597\) −3.31561 −0.135699
\(598\) −6.35023 −0.259680
\(599\) −19.6195 −0.801633 −0.400816 0.916158i \(-0.631273\pi\)
−0.400816 + 0.916158i \(0.631273\pi\)
\(600\) −2.81247 −0.114819
\(601\) −8.88798 −0.362548 −0.181274 0.983433i \(-0.558022\pi\)
−0.181274 + 0.983433i \(0.558022\pi\)
\(602\) 0 0
\(603\) 15.4090 0.627501
\(604\) −6.01381 −0.244698
\(605\) 31.8599 1.29529
\(606\) 2.81247 0.114249
\(607\) −11.7789 −0.478091 −0.239046 0.971008i \(-0.576835\pi\)
−0.239046 + 0.971008i \(0.576835\pi\)
\(608\) −3.39217 −0.137571
\(609\) 0 0
\(610\) 7.45718 0.301932
\(611\) 64.9084 2.62591
\(612\) −4.12478 −0.166734
\(613\) 24.7186 0.998376 0.499188 0.866494i \(-0.333632\pi\)
0.499188 + 0.866494i \(0.333632\pi\)
\(614\) 24.9467 1.00677
\(615\) −10.9332 −0.440871
\(616\) 0 0
\(617\) −19.0580 −0.767247 −0.383623 0.923490i \(-0.625324\pi\)
−0.383623 + 0.923490i \(0.625324\pi\)
\(618\) −12.5589 −0.505193
\(619\) −19.5096 −0.784158 −0.392079 0.919932i \(-0.628244\pi\)
−0.392079 + 0.919932i \(0.628244\pi\)
\(620\) −11.5549 −0.464055
\(621\) −1.00000 −0.0401286
\(622\) −26.4878 −1.06206
\(623\) 0 0
\(624\) −6.35023 −0.254213
\(625\) −3.02762 −0.121105
\(626\) 19.2382 0.768914
\(627\) −19.3506 −0.772787
\(628\) −17.8672 −0.712978
\(629\) −34.4429 −1.37333
\(630\) 0 0
\(631\) −46.6443 −1.85688 −0.928441 0.371481i \(-0.878850\pi\)
−0.928441 + 0.371481i \(0.878850\pi\)
\(632\) 13.4301 0.534220
\(633\) 17.9638 0.713998
\(634\) −25.3784 −1.00790
\(635\) 0.0204267 0.000810610 0
\(636\) −0.520971 −0.0206579
\(637\) 0 0
\(638\) −13.6461 −0.540254
\(639\) 9.87120 0.390499
\(640\) −1.47903 −0.0584637
\(641\) −26.5341 −1.04803 −0.524016 0.851708i \(-0.675567\pi\)
−0.524016 + 0.851708i \(0.675567\pi\)
\(642\) −5.22143 −0.206073
\(643\) 1.53826 0.0606632 0.0303316 0.999540i \(-0.490344\pi\)
0.0303316 + 0.999540i \(0.490344\pi\)
\(644\) 0 0
\(645\) −2.95806 −0.116473
\(646\) 13.9920 0.550506
\(647\) 37.0620 1.45706 0.728529 0.685014i \(-0.240204\pi\)
0.728529 + 0.685014i \(0.240204\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 42.0897 1.65216
\(650\) 17.8599 0.700521
\(651\) 0 0
\(652\) 13.8712 0.543238
\(653\) 8.19083 0.320532 0.160266 0.987074i \(-0.448765\pi\)
0.160266 + 0.987074i \(0.448765\pi\)
\(654\) −8.47501 −0.331399
\(655\) −21.4402 −0.837739
\(656\) 7.39217 0.288616
\(657\) 8.92993 0.348390
\(658\) 0 0
\(659\) 24.2174 0.943376 0.471688 0.881765i \(-0.343645\pi\)
0.471688 + 0.881765i \(0.343645\pi\)
\(660\) −8.43709 −0.328413
\(661\) −37.5928 −1.46219 −0.731095 0.682276i \(-0.760990\pi\)
−0.731095 + 0.682276i \(0.760990\pi\)
\(662\) −2.84273 −0.110486
\(663\) 26.1933 1.01726
\(664\) 9.22143 0.357861
\(665\) 0 0
\(666\) −8.35023 −0.323565
\(667\) 2.39217 0.0926252
\(668\) 5.07551 0.196377
\(669\) 1.22947 0.0475340
\(670\) −22.7903 −0.880466
\(671\) −28.7616 −1.11033
\(672\) 0 0
\(673\) 4.15939 0.160333 0.0801664 0.996781i \(-0.474455\pi\)
0.0801664 + 0.996781i \(0.474455\pi\)
\(674\) −26.7793 −1.03150
\(675\) 2.81247 0.108252
\(676\) 27.3254 1.05098
\(677\) 12.3452 0.474463 0.237232 0.971453i \(-0.423760\pi\)
0.237232 + 0.971453i \(0.423760\pi\)
\(678\) 0.725281 0.0278543
\(679\) 0 0
\(680\) 6.10067 0.233950
\(681\) −16.4130 −0.628947
\(682\) 44.5661 1.70652
\(683\) 41.1463 1.57442 0.787209 0.616686i \(-0.211525\pi\)
0.787209 + 0.616686i \(0.211525\pi\)
\(684\) 3.39217 0.129703
\(685\) 11.8679 0.453449
\(686\) 0 0
\(687\) 25.4163 0.969692
\(688\) 2.00000 0.0762493
\(689\) 3.30829 0.126036
\(690\) 1.47903 0.0563057
\(691\) −6.16237 −0.234428 −0.117214 0.993107i \(-0.537396\pi\)
−0.117214 + 0.993107i \(0.537396\pi\)
\(692\) 2.92993 0.111379
\(693\) 0 0
\(694\) 5.62495 0.213520
\(695\) 21.0339 0.797861
\(696\) 2.39217 0.0906750
\(697\) −30.4911 −1.15493
\(698\) −20.8011 −0.787334
\(699\) −0.551566 −0.0208621
\(700\) 0 0
\(701\) −27.6447 −1.04413 −0.522063 0.852907i \(-0.674837\pi\)
−0.522063 + 0.852907i \(0.674837\pi\)
\(702\) 6.35023 0.239674
\(703\) 28.3254 1.06831
\(704\) 5.70448 0.214996
\(705\) −15.1178 −0.569369
\(706\) 34.7092 1.30630
\(707\) 0 0
\(708\) −7.37836 −0.277296
\(709\) 25.2426 0.948005 0.474002 0.880524i \(-0.342809\pi\)
0.474002 + 0.880524i \(0.342809\pi\)
\(710\) −14.5998 −0.547920
\(711\) −13.4301 −0.503668
\(712\) 6.43411 0.241129
\(713\) −7.81247 −0.292579
\(714\) 0 0
\(715\) 53.5774 2.00368
\(716\) −7.94374 −0.296871
\(717\) 11.2634 0.420638
\(718\) 5.78401 0.215857
\(719\) 39.4075 1.46965 0.734826 0.678255i \(-0.237264\pi\)
0.734826 + 0.678255i \(0.237264\pi\)
\(720\) 1.47903 0.0551201
\(721\) 0 0
\(722\) 7.49317 0.278867
\(723\) 28.3284 1.05354
\(724\) −7.39950 −0.275000
\(725\) −6.72792 −0.249869
\(726\) 21.5411 0.799464
\(727\) 33.9963 1.26085 0.630427 0.776248i \(-0.282880\pi\)
0.630427 + 0.776248i \(0.282880\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −13.2076 −0.488836
\(731\) −8.24956 −0.305121
\(732\) 5.04194 0.186356
\(733\) −2.49179 −0.0920365 −0.0460182 0.998941i \(-0.514653\pi\)
−0.0460182 + 0.998941i \(0.514653\pi\)
\(734\) −0.669863 −0.0247251
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 87.9000 3.23784
\(738\) −7.39217 −0.272110
\(739\) 47.2976 1.73987 0.869935 0.493166i \(-0.164160\pi\)
0.869935 + 0.493166i \(0.164160\pi\)
\(740\) 12.3502 0.454003
\(741\) −21.5411 −0.791331
\(742\) 0 0
\(743\) 44.5435 1.63414 0.817072 0.576536i \(-0.195596\pi\)
0.817072 + 0.576536i \(0.195596\pi\)
\(744\) −7.81247 −0.286419
\(745\) −34.4369 −1.26167
\(746\) 9.80008 0.358806
\(747\) −9.22143 −0.337394
\(748\) −23.5297 −0.860332
\(749\) 0 0
\(750\) −11.5549 −0.421924
\(751\) −13.5105 −0.493004 −0.246502 0.969142i \(-0.579281\pi\)
−0.246502 + 0.969142i \(0.579281\pi\)
\(752\) 10.2214 0.372737
\(753\) 7.83659 0.285581
\(754\) −15.1908 −0.553218
\(755\) −8.89460 −0.323708
\(756\) 0 0
\(757\) −24.9895 −0.908259 −0.454129 0.890936i \(-0.650050\pi\)
−0.454129 + 0.890936i \(0.650050\pi\)
\(758\) 0.742065 0.0269530
\(759\) −5.70448 −0.207059
\(760\) −5.01712 −0.181990
\(761\) −2.53860 −0.0920242 −0.0460121 0.998941i \(-0.514651\pi\)
−0.0460121 + 0.998941i \(0.514651\pi\)
\(762\) 0.0138109 0.000500316 0
\(763\) 0 0
\(764\) 20.1765 0.729961
\(765\) −6.10067 −0.220570
\(766\) −30.4114 −1.09881
\(767\) 46.8543 1.69181
\(768\) −1.00000 −0.0360844
\(769\) −7.59435 −0.273859 −0.136930 0.990581i \(-0.543723\pi\)
−0.136930 + 0.990581i \(0.543723\pi\)
\(770\) 0 0
\(771\) −3.72562 −0.134175
\(772\) 12.8880 0.463849
\(773\) −21.5687 −0.775772 −0.387886 0.921707i \(-0.626795\pi\)
−0.387886 + 0.921707i \(0.626795\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 21.9724 0.789271
\(776\) 10.3640 0.372047
\(777\) 0 0
\(778\) 0.674930 0.0241974
\(779\) 25.0755 0.898423
\(780\) −9.39217 −0.336294
\(781\) 56.3100 2.01493
\(782\) 4.12478 0.147502
\(783\) −2.39217 −0.0854892
\(784\) 0 0
\(785\) −26.4261 −0.943187
\(786\) −14.4961 −0.517060
\(787\) −45.2192 −1.61189 −0.805944 0.591991i \(-0.798342\pi\)
−0.805944 + 0.591991i \(0.798342\pi\)
\(788\) −9.80410 −0.349257
\(789\) −28.8661 −1.02766
\(790\) 19.8635 0.706711
\(791\) 0 0
\(792\) −5.70448 −0.202700
\(793\) −32.0175 −1.13697
\(794\) −8.42607 −0.299030
\(795\) −0.770531 −0.0273279
\(796\) 3.31561 0.117519
\(797\) 26.6028 0.942318 0.471159 0.882048i \(-0.343836\pi\)
0.471159 + 0.882048i \(0.343836\pi\)
\(798\) 0 0
\(799\) −42.1611 −1.49155
\(800\) 2.81247 0.0994360
\(801\) −6.43411 −0.227338
\(802\) −35.1192 −1.24010
\(803\) 50.9406 1.79765
\(804\) −15.4090 −0.543432
\(805\) 0 0
\(806\) 49.6110 1.74747
\(807\) 0.482002 0.0169673
\(808\) −2.81247 −0.0989425
\(809\) −40.5582 −1.42595 −0.712975 0.701190i \(-0.752653\pi\)
−0.712975 + 0.701190i \(0.752653\pi\)
\(810\) −1.47903 −0.0519678
\(811\) 17.7894 0.624672 0.312336 0.949972i \(-0.398889\pi\)
0.312336 + 0.949972i \(0.398889\pi\)
\(812\) 0 0
\(813\) 4.89602 0.171711
\(814\) −47.6337 −1.66956
\(815\) 20.5159 0.718641
\(816\) 4.12478 0.144396
\(817\) 6.78434 0.237354
\(818\) −19.7281 −0.689776
\(819\) 0 0
\(820\) 10.9332 0.381805
\(821\) 3.44006 0.120059 0.0600294 0.998197i \(-0.480881\pi\)
0.0600294 + 0.998197i \(0.480881\pi\)
\(822\) 8.02411 0.279873
\(823\) 7.28489 0.253935 0.126968 0.991907i \(-0.459476\pi\)
0.126968 + 0.991907i \(0.459476\pi\)
\(824\) 12.5589 0.437510
\(825\) 16.0437 0.558570
\(826\) 0 0
\(827\) −25.4665 −0.885556 −0.442778 0.896631i \(-0.646007\pi\)
−0.442778 + 0.896631i \(0.646007\pi\)
\(828\) 1.00000 0.0347524
\(829\) 0.00661582 0.000229777 0 0.000114889 1.00000i \(-0.499963\pi\)
0.000114889 1.00000i \(0.499963\pi\)
\(830\) 13.6388 0.473408
\(831\) −15.5264 −0.538605
\(832\) 6.35023 0.220155
\(833\) 0 0
\(834\) 14.2214 0.492448
\(835\) 7.50683 0.259784
\(836\) 19.3506 0.669253
\(837\) 7.81247 0.270039
\(838\) −7.26739 −0.251048
\(839\) 14.8092 0.511269 0.255635 0.966773i \(-0.417716\pi\)
0.255635 + 0.966773i \(0.417716\pi\)
\(840\) 0 0
\(841\) −23.2775 −0.802673
\(842\) −3.82277 −0.131742
\(843\) −13.2003 −0.454642
\(844\) −17.9638 −0.618340
\(845\) 40.4151 1.39032
\(846\) −10.2214 −0.351420
\(847\) 0 0
\(848\) 0.520971 0.0178902
\(849\) 2.75672 0.0946104
\(850\) −11.6008 −0.397905
\(851\) 8.35023 0.286242
\(852\) −9.87120 −0.338182
\(853\) −13.5264 −0.463135 −0.231568 0.972819i \(-0.574385\pi\)
−0.231568 + 0.972819i \(0.574385\pi\)
\(854\) 0 0
\(855\) 5.01712 0.171582
\(856\) 5.22143 0.178465
\(857\) 8.62741 0.294707 0.147353 0.989084i \(-0.452925\pi\)
0.147353 + 0.989084i \(0.452925\pi\)
\(858\) 36.2247 1.23669
\(859\) 24.3074 0.829359 0.414680 0.909967i \(-0.363894\pi\)
0.414680 + 0.909967i \(0.363894\pi\)
\(860\) 2.95806 0.100869
\(861\) 0 0
\(862\) 1.97238 0.0671795
\(863\) −14.7796 −0.503104 −0.251552 0.967844i \(-0.580941\pi\)
−0.251552 + 0.967844i \(0.580941\pi\)
\(864\) 1.00000 0.0340207
\(865\) 4.33345 0.147342
\(866\) 31.3929 1.06677
\(867\) −0.0138109 −0.000469043 0
\(868\) 0 0
\(869\) −76.6117 −2.59887
\(870\) 3.53809 0.119952
\(871\) 97.8504 3.31553
\(872\) 8.47501 0.287000
\(873\) −10.3640 −0.350770
\(874\) −3.39217 −0.114742
\(875\) 0 0
\(876\) −8.92993 −0.301714
\(877\) −52.7092 −1.77986 −0.889932 0.456094i \(-0.849248\pi\)
−0.889932 + 0.456094i \(0.849248\pi\)
\(878\) 35.7205 1.20551
\(879\) −0.0976957 −0.00329520
\(880\) 8.43709 0.284414
\(881\) 35.2594 1.18792 0.593959 0.804495i \(-0.297564\pi\)
0.593959 + 0.804495i \(0.297564\pi\)
\(882\) 0 0
\(883\) 6.05906 0.203904 0.101952 0.994789i \(-0.467491\pi\)
0.101952 + 0.994789i \(0.467491\pi\)
\(884\) −26.1933 −0.880976
\(885\) −10.9128 −0.366830
\(886\) 0.114479 0.00384601
\(887\) −4.91527 −0.165039 −0.0825193 0.996589i \(-0.526297\pi\)
−0.0825193 + 0.996589i \(0.526297\pi\)
\(888\) 8.35023 0.280215
\(889\) 0 0
\(890\) 9.51624 0.318985
\(891\) 5.70448 0.191107
\(892\) −1.22947 −0.0411657
\(893\) 34.6728 1.16028
\(894\) −23.2835 −0.778715
\(895\) −11.7490 −0.392726
\(896\) 0 0
\(897\) −6.35023 −0.212028
\(898\) 9.26006 0.309012
\(899\) −18.6888 −0.623306
\(900\) −2.81247 −0.0937491
\(901\) −2.14889 −0.0715900
\(902\) −42.1685 −1.40406
\(903\) 0 0
\(904\) −0.725281 −0.0241225
\(905\) −10.9441 −0.363793
\(906\) −6.01381 −0.199795
\(907\) 34.1538 1.13406 0.567029 0.823698i \(-0.308093\pi\)
0.567029 + 0.823698i \(0.308093\pi\)
\(908\) 16.4130 0.544684
\(909\) 2.81247 0.0932839
\(910\) 0 0
\(911\) −52.3590 −1.73473 −0.867365 0.497672i \(-0.834188\pi\)
−0.867365 + 0.497672i \(0.834188\pi\)
\(912\) −3.39217 −0.112326
\(913\) −52.6034 −1.74092
\(914\) −5.83730 −0.193081
\(915\) 7.45718 0.246527
\(916\) −25.4163 −0.839778
\(917\) 0 0
\(918\) −4.12478 −0.136138
\(919\) 36.4464 1.20225 0.601127 0.799153i \(-0.294718\pi\)
0.601127 + 0.799153i \(0.294718\pi\)
\(920\) −1.47903 −0.0487621
\(921\) 24.9467 0.822022
\(922\) −23.1660 −0.762932
\(923\) 62.6844 2.06328
\(924\) 0 0
\(925\) −23.4848 −0.772175
\(926\) −0.140148 −0.00460554
\(927\) −12.5589 −0.412488
\(928\) −2.39217 −0.0785269
\(929\) −55.4453 −1.81910 −0.909551 0.415592i \(-0.863574\pi\)
−0.909551 + 0.415592i \(0.863574\pi\)
\(930\) −11.5549 −0.378899
\(931\) 0 0
\(932\) 0.551566 0.0180671
\(933\) −26.4878 −0.867171
\(934\) −15.5170 −0.507730
\(935\) −34.8011 −1.13812
\(936\) −6.35023 −0.207564
\(937\) −0.308796 −0.0100879 −0.00504396 0.999987i \(-0.501606\pi\)
−0.00504396 + 0.999987i \(0.501606\pi\)
\(938\) 0 0
\(939\) 19.2382 0.627815
\(940\) 15.1178 0.493088
\(941\) −15.5046 −0.505434 −0.252717 0.967540i \(-0.581324\pi\)
−0.252717 + 0.967540i \(0.581324\pi\)
\(942\) −17.8672 −0.582144
\(943\) 7.39217 0.240722
\(944\) 7.37836 0.240145
\(945\) 0 0
\(946\) −11.4090 −0.370937
\(947\) 58.6924 1.90725 0.953624 0.301002i \(-0.0973209\pi\)
0.953624 + 0.301002i \(0.0973209\pi\)
\(948\) 13.4301 0.436189
\(949\) 56.7071 1.84079
\(950\) 9.54039 0.309531
\(951\) −25.3784 −0.822950
\(952\) 0 0
\(953\) 28.6120 0.926835 0.463417 0.886140i \(-0.346623\pi\)
0.463417 + 0.886140i \(0.346623\pi\)
\(954\) −0.520971 −0.0168671
\(955\) 29.8416 0.965653
\(956\) −11.2634 −0.364283
\(957\) −13.6461 −0.441115
\(958\) −15.3922 −0.497298
\(959\) 0 0
\(960\) −1.47903 −0.0477354
\(961\) 30.0347 0.968863
\(962\) −53.0259 −1.70962
\(963\) −5.22143 −0.168258
\(964\) −28.3284 −0.912396
\(965\) 19.0617 0.613618
\(966\) 0 0
\(967\) −15.7059 −0.505067 −0.252534 0.967588i \(-0.581264\pi\)
−0.252534 + 0.967588i \(0.581264\pi\)
\(968\) −21.5411 −0.692356
\(969\) 13.9920 0.449487
\(970\) 15.3287 0.492175
\(971\) 16.7660 0.538047 0.269023 0.963134i \(-0.413299\pi\)
0.269023 + 0.963134i \(0.413299\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 0.554539 0.0177686
\(975\) 17.8599 0.571973
\(976\) −5.04194 −0.161389
\(977\) 17.3733 0.555821 0.277910 0.960607i \(-0.410358\pi\)
0.277910 + 0.960607i \(0.410358\pi\)
\(978\) 13.8712 0.443552
\(979\) −36.7033 −1.17304
\(980\) 0 0
\(981\) −8.47501 −0.270586
\(982\) −43.1966 −1.37846
\(983\) 24.0867 0.768246 0.384123 0.923282i \(-0.374504\pi\)
0.384123 + 0.923282i \(0.374504\pi\)
\(984\) 7.39217 0.235654
\(985\) −14.5005 −0.462026
\(986\) 9.86718 0.314235
\(987\) 0 0
\(988\) 21.5411 0.685313
\(989\) 2.00000 0.0635963
\(990\) −8.43709 −0.268148
\(991\) 56.6793 1.80048 0.900238 0.435398i \(-0.143392\pi\)
0.900238 + 0.435398i \(0.143392\pi\)
\(992\) 7.81247 0.248046
\(993\) −2.84273 −0.0902114
\(994\) 0 0
\(995\) 4.90389 0.155464
\(996\) 9.22143 0.292192
\(997\) 41.4037 1.31127 0.655634 0.755079i \(-0.272401\pi\)
0.655634 + 0.755079i \(0.272401\pi\)
\(998\) 20.8943 0.661396
\(999\) −8.35023 −0.264190
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 6762.2.a.cg.1.3 4
7.2 even 3 966.2.i.n.277.2 8
7.4 even 3 966.2.i.n.415.2 yes 8
7.6 odd 2 6762.2.a.ch.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.n.277.2 8 7.2 even 3
966.2.i.n.415.2 yes 8 7.4 even 3
6762.2.a.cg.1.3 4 1.1 even 1 trivial
6762.2.a.ch.1.2 4 7.6 odd 2