Properties

Label 6762.2.a.ci.1.2
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2624.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.77462\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.41421 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.41421 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.41421 q^{10} +2.29857 q^{11} -1.00000 q^{12} +1.54925 q^{13} +3.41421 q^{15} +1.00000 q^{16} +2.72082 q^{17} +1.00000 q^{18} -7.15442 q^{19} -3.41421 q^{20} +2.29857 q^{22} +1.00000 q^{23} -1.00000 q^{24} +6.65685 q^{25} +1.54925 q^{26} -1.00000 q^{27} -9.69564 q^{29} +3.41421 q^{30} -5.15442 q^{31} +1.00000 q^{32} -2.29857 q^{33} +2.72082 q^{34} +1.00000 q^{36} +2.72082 q^{37} -7.15442 q^{38} -1.54925 q^{39} -3.41421 q^{40} +6.01136 q^{41} +11.8193 q^{43} +2.29857 q^{44} -3.41421 q^{45} +1.00000 q^{46} +0.557278 q^{47} -1.00000 q^{48} +6.65685 q^{50} -2.72082 q^{51} +1.54925 q^{52} -3.31796 q^{53} -1.00000 q^{54} -7.84782 q^{55} +7.15442 q^{57} -9.69564 q^{58} +7.69564 q^{59} +3.41421 q^{60} +0.0113593 q^{61} -5.15442 q^{62} +1.00000 q^{64} -5.28946 q^{65} -2.29857 q^{66} +8.93604 q^{67} +2.72082 q^{68} -1.00000 q^{69} -9.25067 q^{71} +1.00000 q^{72} +8.09046 q^{73} +2.72082 q^{74} -6.65685 q^{75} -7.15442 q^{76} -1.54925 q^{78} -10.4062 q^{79} -3.41421 q^{80} +1.00000 q^{81} +6.01136 q^{82} -1.71279 q^{83} -9.28946 q^{85} +11.8193 q^{86} +9.69564 q^{87} +2.29857 q^{88} -16.0506 q^{89} -3.41421 q^{90} +1.00000 q^{92} +5.15442 q^{93} +0.557278 q^{94} +24.4267 q^{95} -1.00000 q^{96} +5.76446 q^{97} +2.29857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 8 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 8 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9} - 8 q^{10} + 4 q^{11} - 4 q^{12} - 12 q^{13} + 8 q^{15} + 4 q^{16} + 4 q^{17} + 4 q^{18} - 4 q^{19} - 8 q^{20} + 4 q^{22} + 4 q^{23} - 4 q^{24} + 4 q^{25} - 12 q^{26} - 4 q^{27} + 8 q^{29} + 8 q^{30} + 4 q^{31} + 4 q^{32} - 4 q^{33} + 4 q^{34} + 4 q^{36} + 4 q^{37} - 4 q^{38} + 12 q^{39} - 8 q^{40} + 8 q^{41} + 4 q^{43} + 4 q^{44} - 8 q^{45} + 4 q^{46} - 12 q^{47} - 4 q^{48} + 4 q^{50} - 4 q^{51} - 12 q^{52} + 4 q^{53} - 4 q^{54} - 8 q^{55} + 4 q^{57} + 8 q^{58} - 16 q^{59} + 8 q^{60} - 16 q^{61} + 4 q^{62} + 4 q^{64} + 16 q^{65} - 4 q^{66} + 20 q^{67} + 4 q^{68} - 4 q^{69} - 24 q^{71} + 4 q^{72} - 8 q^{73} + 4 q^{74} - 4 q^{75} - 4 q^{76} + 12 q^{78} - 32 q^{79} - 8 q^{80} + 4 q^{81} + 8 q^{82} + 4 q^{83} + 4 q^{86} - 8 q^{87} + 4 q^{88} - 20 q^{89} - 8 q^{90} + 4 q^{92} - 4 q^{93} - 12 q^{94} - 4 q^{96} - 4 q^{97} + 4 q^{99} + O(q^{100}) \)

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
\(5\) −3.41421 −1.52688 −0.763441 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.41421 −1.07967
\(11\) 2.29857 0.693046 0.346523 0.938042i \(-0.387362\pi\)
0.346523 + 0.938042i \(0.387362\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.54925 0.429683 0.214842 0.976649i \(-0.431076\pi\)
0.214842 + 0.976649i \(0.431076\pi\)
\(14\) 0 0
\(15\) 3.41421 0.881546
\(16\) 1.00000 0.250000
\(17\) 2.72082 0.659895 0.329948 0.943999i \(-0.392969\pi\)
0.329948 + 0.943999i \(0.392969\pi\)
\(18\) 1.00000 0.235702
\(19\) −7.15442 −1.64134 −0.820669 0.571404i \(-0.806399\pi\)
−0.820669 + 0.571404i \(0.806399\pi\)
\(20\) −3.41421 −0.763441
\(21\) 0 0
\(22\) 2.29857 0.490057
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 6.65685 1.33137
\(26\) 1.54925 0.303832
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −9.69564 −1.80043 −0.900217 0.435441i \(-0.856592\pi\)
−0.900217 + 0.435441i \(0.856592\pi\)
\(30\) 3.41421 0.623347
\(31\) −5.15442 −0.925762 −0.462881 0.886420i \(-0.653184\pi\)
−0.462881 + 0.886420i \(0.653184\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.29857 −0.400130
\(34\) 2.72082 0.466617
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.72082 0.447300 0.223650 0.974670i \(-0.428203\pi\)
0.223650 + 0.974670i \(0.428203\pi\)
\(38\) −7.15442 −1.16060
\(39\) −1.54925 −0.248078
\(40\) −3.41421 −0.539835
\(41\) 6.01136 0.938817 0.469408 0.882981i \(-0.344467\pi\)
0.469408 + 0.882981i \(0.344467\pi\)
\(42\) 0 0
\(43\) 11.8193 1.80243 0.901214 0.433374i \(-0.142677\pi\)
0.901214 + 0.433374i \(0.142677\pi\)
\(44\) 2.29857 0.346523
\(45\) −3.41421 −0.508961
\(46\) 1.00000 0.147442
\(47\) 0.557278 0.0812874 0.0406437 0.999174i \(-0.487059\pi\)
0.0406437 + 0.999174i \(0.487059\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 6.65685 0.941421
\(51\) −2.72082 −0.380991
\(52\) 1.54925 0.214842
\(53\) −3.31796 −0.455757 −0.227879 0.973690i \(-0.573179\pi\)
−0.227879 + 0.973690i \(0.573179\pi\)
\(54\) −1.00000 −0.136083
\(55\) −7.84782 −1.05820
\(56\) 0 0
\(57\) 7.15442 0.947627
\(58\) −9.69564 −1.27310
\(59\) 7.69564 1.00189 0.500943 0.865480i \(-0.332987\pi\)
0.500943 + 0.865480i \(0.332987\pi\)
\(60\) 3.41421 0.440773
\(61\) 0.0113593 0.00145441 0.000727205 1.00000i \(-0.499769\pi\)
0.000727205 1.00000i \(0.499769\pi\)
\(62\) −5.15442 −0.654612
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −5.28946 −0.656076
\(66\) −2.29857 −0.282935
\(67\) 8.93604 1.09171 0.545855 0.837879i \(-0.316205\pi\)
0.545855 + 0.837879i \(0.316205\pi\)
\(68\) 2.72082 0.329948
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −9.25067 −1.09785 −0.548926 0.835871i \(-0.684963\pi\)
−0.548926 + 0.835871i \(0.684963\pi\)
\(72\) 1.00000 0.117851
\(73\) 8.09046 0.946917 0.473458 0.880816i \(-0.343005\pi\)
0.473458 + 0.880816i \(0.343005\pi\)
\(74\) 2.72082 0.316289
\(75\) −6.65685 −0.768667
\(76\) −7.15442 −0.820669
\(77\) 0 0
\(78\) −1.54925 −0.175418
\(79\) −10.4062 −1.17079 −0.585393 0.810749i \(-0.699060\pi\)
−0.585393 + 0.810749i \(0.699060\pi\)
\(80\) −3.41421 −0.381721
\(81\) 1.00000 0.111111
\(82\) 6.01136 0.663844
\(83\) −1.71279 −0.188003 −0.0940014 0.995572i \(-0.529966\pi\)
−0.0940014 + 0.995572i \(0.529966\pi\)
\(84\) 0 0
\(85\) −9.28946 −1.00758
\(86\) 11.8193 1.27451
\(87\) 9.69564 1.03948
\(88\) 2.29857 0.245029
\(89\) −16.0506 −1.70136 −0.850680 0.525684i \(-0.823809\pi\)
−0.850680 + 0.525684i \(0.823809\pi\)
\(90\) −3.41421 −0.359890
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 5.15442 0.534489
\(94\) 0.557278 0.0574788
\(95\) 24.4267 2.50613
\(96\) −1.00000 −0.102062
\(97\) 5.76446 0.585293 0.292646 0.956221i \(-0.405464\pi\)
0.292646 + 0.956221i \(0.405464\pi\)
\(98\) 0 0
\(99\) 2.29857 0.231015
\(100\) 6.65685 0.665685
\(101\) −13.8193 −1.37507 −0.687536 0.726150i \(-0.741308\pi\)
−0.687536 + 0.726150i \(0.741308\pi\)
\(102\) −2.72082 −0.269401
\(103\) −8.86721 −0.873712 −0.436856 0.899531i \(-0.643908\pi\)
−0.436856 + 0.899531i \(0.643908\pi\)
\(104\) 1.54925 0.151916
\(105\) 0 0
\(106\) −3.31796 −0.322269
\(107\) 13.1818 1.27434 0.637169 0.770724i \(-0.280105\pi\)
0.637169 + 0.770724i \(0.280105\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 3.76446 0.360570 0.180285 0.983614i \(-0.442298\pi\)
0.180285 + 0.983614i \(0.442298\pi\)
\(110\) −7.84782 −0.748260
\(111\) −2.72082 −0.258249
\(112\) 0 0
\(113\) −6.15218 −0.578749 −0.289374 0.957216i \(-0.593447\pi\)
−0.289374 + 0.957216i \(0.593447\pi\)
\(114\) 7.15442 0.670073
\(115\) −3.41421 −0.318377
\(116\) −9.69564 −0.900217
\(117\) 1.54925 0.143228
\(118\) 7.69564 0.708441
\(119\) 0 0
\(120\) 3.41421 0.311674
\(121\) −5.71656 −0.519688
\(122\) 0.0113593 0.00102842
\(123\) −6.01136 −0.542026
\(124\) −5.15442 −0.462881
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −13.0033 −1.15386 −0.576929 0.816794i \(-0.695749\pi\)
−0.576929 + 0.816794i \(0.695749\pi\)
\(128\) 1.00000 0.0883883
\(129\) −11.8193 −1.04063
\(130\) −5.28946 −0.463916
\(131\) 15.4804 1.35253 0.676265 0.736658i \(-0.263597\pi\)
0.676265 + 0.736658i \(0.263597\pi\)
\(132\) −2.29857 −0.200065
\(133\) 0 0
\(134\) 8.93604 0.771956
\(135\) 3.41421 0.293849
\(136\) 2.72082 0.233308
\(137\) −15.4416 −1.31927 −0.659634 0.751587i \(-0.729289\pi\)
−0.659634 + 0.751587i \(0.729289\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −5.17157 −0.438647 −0.219324 0.975652i \(-0.570385\pi\)
−0.219324 + 0.975652i \(0.570385\pi\)
\(140\) 0 0
\(141\) −0.557278 −0.0469313
\(142\) −9.25067 −0.776299
\(143\) 3.56105 0.297790
\(144\) 1.00000 0.0833333
\(145\) 33.1030 2.74905
\(146\) 8.09046 0.669571
\(147\) 0 0
\(148\) 2.72082 0.223650
\(149\) 6.70475 0.549275 0.274637 0.961548i \(-0.411442\pi\)
0.274637 + 0.961548i \(0.411442\pi\)
\(150\) −6.65685 −0.543530
\(151\) 8.13395 0.661931 0.330966 0.943643i \(-0.392626\pi\)
0.330966 + 0.943643i \(0.392626\pi\)
\(152\) −7.15442 −0.580300
\(153\) 2.72082 0.219965
\(154\) 0 0
\(155\) 17.5983 1.41353
\(156\) −1.54925 −0.124039
\(157\) 3.64396 0.290820 0.145410 0.989372i \(-0.453550\pi\)
0.145410 + 0.989372i \(0.453550\pi\)
\(158\) −10.4062 −0.827871
\(159\) 3.31796 0.263132
\(160\) −3.41421 −0.269917
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −13.5047 −1.05777 −0.528884 0.848694i \(-0.677389\pi\)
−0.528884 + 0.848694i \(0.677389\pi\)
\(164\) 6.01136 0.469408
\(165\) 7.84782 0.610952
\(166\) −1.71279 −0.132938
\(167\) −19.6009 −1.51676 −0.758382 0.651810i \(-0.774010\pi\)
−0.758382 + 0.651810i \(0.774010\pi\)
\(168\) 0 0
\(169\) −10.5998 −0.815372
\(170\) −9.28946 −0.712469
\(171\) −7.15442 −0.547112
\(172\) 11.8193 0.901214
\(173\) −14.6317 −1.11243 −0.556213 0.831040i \(-0.687746\pi\)
−0.556213 + 0.831040i \(0.687746\pi\)
\(174\) 9.69564 0.735024
\(175\) 0 0
\(176\) 2.29857 0.173261
\(177\) −7.69564 −0.578440
\(178\) −16.0506 −1.20304
\(179\) −6.28613 −0.469847 −0.234924 0.972014i \(-0.575484\pi\)
−0.234924 + 0.972014i \(0.575484\pi\)
\(180\) −3.41421 −0.254480
\(181\) −14.8398 −1.10303 −0.551516 0.834164i \(-0.685951\pi\)
−0.551516 + 0.834164i \(0.685951\pi\)
\(182\) 0 0
\(183\) −0.0113593 −0.000839704 0
\(184\) 1.00000 0.0737210
\(185\) −9.28946 −0.682974
\(186\) 5.15442 0.377941
\(187\) 6.25400 0.457338
\(188\) 0.557278 0.0406437
\(189\) 0 0
\(190\) 24.4267 1.77210
\(191\) −18.3492 −1.32770 −0.663849 0.747866i \(-0.731078\pi\)
−0.663849 + 0.747866i \(0.731078\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −0.979075 −0.0704753 −0.0352377 0.999379i \(-0.511219\pi\)
−0.0352377 + 0.999379i \(0.511219\pi\)
\(194\) 5.76446 0.413864
\(195\) 5.28946 0.378786
\(196\) 0 0
\(197\) 4.65353 0.331550 0.165775 0.986164i \(-0.446987\pi\)
0.165775 + 0.986164i \(0.446987\pi\)
\(198\) 2.29857 0.163352
\(199\) −1.46436 −0.103805 −0.0519027 0.998652i \(-0.516529\pi\)
−0.0519027 + 0.998652i \(0.516529\pi\)
\(200\) 6.65685 0.470711
\(201\) −8.93604 −0.630299
\(202\) −13.8193 −0.972323
\(203\) 0 0
\(204\) −2.72082 −0.190495
\(205\) −20.5241 −1.43346
\(206\) −8.86721 −0.617808
\(207\) 1.00000 0.0695048
\(208\) 1.54925 0.107421
\(209\) −16.4450 −1.13752
\(210\) 0 0
\(211\) −12.9075 −0.888591 −0.444295 0.895880i \(-0.646546\pi\)
−0.444295 + 0.895880i \(0.646546\pi\)
\(212\) −3.31796 −0.227879
\(213\) 9.25067 0.633846
\(214\) 13.1818 0.901093
\(215\) −40.3536 −2.75210
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 3.76446 0.254962
\(219\) −8.09046 −0.546703
\(220\) −7.84782 −0.529100
\(221\) 4.21522 0.283546
\(222\) −2.72082 −0.182609
\(223\) −14.4293 −0.966261 −0.483130 0.875548i \(-0.660500\pi\)
−0.483130 + 0.875548i \(0.660500\pi\)
\(224\) 0 0
\(225\) 6.65685 0.443790
\(226\) −6.15218 −0.409237
\(227\) 14.4133 0.956643 0.478322 0.878185i \(-0.341245\pi\)
0.478322 + 0.878185i \(0.341245\pi\)
\(228\) 7.15442 0.473813
\(229\) 1.26357 0.0834988 0.0417494 0.999128i \(-0.486707\pi\)
0.0417494 + 0.999128i \(0.486707\pi\)
\(230\) −3.41421 −0.225127
\(231\) 0 0
\(232\) −9.69564 −0.636550
\(233\) −16.7166 −1.09514 −0.547569 0.836760i \(-0.684447\pi\)
−0.547569 + 0.836760i \(0.684447\pi\)
\(234\) 1.54925 0.101277
\(235\) −1.90267 −0.124116
\(236\) 7.69564 0.500943
\(237\) 10.4062 0.675954
\(238\) 0 0
\(239\) −22.6923 −1.46784 −0.733922 0.679234i \(-0.762312\pi\)
−0.733922 + 0.679234i \(0.762312\pi\)
\(240\) 3.41421 0.220387
\(241\) −20.6705 −1.33150 −0.665751 0.746174i \(-0.731889\pi\)
−0.665751 + 0.746174i \(0.731889\pi\)
\(242\) −5.71656 −0.367475
\(243\) −1.00000 −0.0641500
\(244\) 0.0113593 0.000727205 0
\(245\) 0 0
\(246\) −6.01136 −0.383270
\(247\) −11.0840 −0.705256
\(248\) −5.15442 −0.327306
\(249\) 1.71279 0.108543
\(250\) −5.65685 −0.357771
\(251\) −27.6397 −1.74460 −0.872301 0.488969i \(-0.837373\pi\)
−0.872301 + 0.488969i \(0.837373\pi\)
\(252\) 0 0
\(253\) 2.29857 0.144510
\(254\) −13.0033 −0.815901
\(255\) 9.28946 0.581728
\(256\) 1.00000 0.0625000
\(257\) 11.9383 0.744689 0.372345 0.928095i \(-0.378554\pi\)
0.372345 + 0.928095i \(0.378554\pi\)
\(258\) −11.8193 −0.735838
\(259\) 0 0
\(260\) −5.28946 −0.328038
\(261\) −9.69564 −0.600145
\(262\) 15.4804 0.956384
\(263\) −8.68629 −0.535620 −0.267810 0.963472i \(-0.586300\pi\)
−0.267810 + 0.963472i \(0.586300\pi\)
\(264\) −2.29857 −0.141467
\(265\) 11.3282 0.695888
\(266\) 0 0
\(267\) 16.0506 0.982280
\(268\) 8.93604 0.545855
\(269\) 20.6134 1.25682 0.628412 0.777881i \(-0.283705\pi\)
0.628412 + 0.777881i \(0.283705\pi\)
\(270\) 3.41421 0.207782
\(271\) 3.53789 0.214911 0.107456 0.994210i \(-0.465730\pi\)
0.107456 + 0.994210i \(0.465730\pi\)
\(272\) 2.72082 0.164974
\(273\) 0 0
\(274\) −15.4416 −0.932863
\(275\) 15.3013 0.922701
\(276\) −1.00000 −0.0601929
\(277\) 24.4898 1.47145 0.735724 0.677282i \(-0.236842\pi\)
0.735724 + 0.677282i \(0.236842\pi\)
\(278\) −5.17157 −0.310170
\(279\) −5.15442 −0.308587
\(280\) 0 0
\(281\) −10.1582 −0.605987 −0.302994 0.952993i \(-0.597986\pi\)
−0.302994 + 0.952993i \(0.597986\pi\)
\(282\) −0.557278 −0.0331854
\(283\) 16.6042 0.987020 0.493510 0.869740i \(-0.335714\pi\)
0.493510 + 0.869740i \(0.335714\pi\)
\(284\) −9.25067 −0.548926
\(285\) −24.4267 −1.44691
\(286\) 3.56105 0.210570
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −9.59715 −0.564538
\(290\) 33.1030 1.94387
\(291\) −5.76446 −0.337919
\(292\) 8.09046 0.473458
\(293\) −24.9737 −1.45898 −0.729491 0.683991i \(-0.760243\pi\)
−0.729491 + 0.683991i \(0.760243\pi\)
\(294\) 0 0
\(295\) −26.2745 −1.52976
\(296\) 2.72082 0.158144
\(297\) −2.29857 −0.133377
\(298\) 6.70475 0.388396
\(299\) 1.54925 0.0895952
\(300\) −6.65685 −0.384334
\(301\) 0 0
\(302\) 8.13395 0.468056
\(303\) 13.8193 0.793899
\(304\) −7.15442 −0.410334
\(305\) −0.0387831 −0.00222071
\(306\) 2.72082 0.155539
\(307\) 32.1239 1.83341 0.916704 0.399567i \(-0.130840\pi\)
0.916704 + 0.399567i \(0.130840\pi\)
\(308\) 0 0
\(309\) 8.86721 0.504438
\(310\) 17.5983 0.999516
\(311\) −29.0571 −1.64768 −0.823838 0.566825i \(-0.808172\pi\)
−0.823838 + 0.566825i \(0.808172\pi\)
\(312\) −1.54925 −0.0877088
\(313\) 1.74174 0.0984492 0.0492246 0.998788i \(-0.484325\pi\)
0.0492246 + 0.998788i \(0.484325\pi\)
\(314\) 3.64396 0.205641
\(315\) 0 0
\(316\) −10.4062 −0.585393
\(317\) −12.7796 −0.717774 −0.358887 0.933381i \(-0.616844\pi\)
−0.358887 + 0.933381i \(0.616844\pi\)
\(318\) 3.31796 0.186062
\(319\) −22.2861 −1.24778
\(320\) −3.41421 −0.190860
\(321\) −13.1818 −0.735739
\(322\) 0 0
\(323\) −19.4659 −1.08311
\(324\) 1.00000 0.0555556
\(325\) 10.3131 0.572068
\(326\) −13.5047 −0.747955
\(327\) −3.76446 −0.208175
\(328\) 6.01136 0.331922
\(329\) 0 0
\(330\) 7.84782 0.432008
\(331\) 6.34315 0.348651 0.174325 0.984688i \(-0.444226\pi\)
0.174325 + 0.984688i \(0.444226\pi\)
\(332\) −1.71279 −0.0940014
\(333\) 2.72082 0.149100
\(334\) −19.6009 −1.07251
\(335\) −30.5095 −1.66691
\(336\) 0 0
\(337\) 15.8866 0.865398 0.432699 0.901538i \(-0.357561\pi\)
0.432699 + 0.901538i \(0.357561\pi\)
\(338\) −10.5998 −0.576555
\(339\) 6.15218 0.334141
\(340\) −9.28946 −0.503791
\(341\) −11.8478 −0.641595
\(342\) −7.15442 −0.386867
\(343\) 0 0
\(344\) 11.8193 0.637255
\(345\) 3.41421 0.183815
\(346\) −14.6317 −0.786604
\(347\) −28.7699 −1.54445 −0.772224 0.635350i \(-0.780856\pi\)
−0.772224 + 0.635350i \(0.780856\pi\)
\(348\) 9.69564 0.519741
\(349\) −14.6638 −0.784935 −0.392468 0.919766i \(-0.628379\pi\)
−0.392468 + 0.919766i \(0.628379\pi\)
\(350\) 0 0
\(351\) −1.54925 −0.0826926
\(352\) 2.29857 0.122514
\(353\) 12.6079 0.671049 0.335525 0.942031i \(-0.391086\pi\)
0.335525 + 0.942031i \(0.391086\pi\)
\(354\) −7.69564 −0.409019
\(355\) 31.5838 1.67629
\(356\) −16.0506 −0.850680
\(357\) 0 0
\(358\) −6.28613 −0.332232
\(359\) −13.5047 −0.712749 −0.356375 0.934343i \(-0.615987\pi\)
−0.356375 + 0.934343i \(0.615987\pi\)
\(360\) −3.41421 −0.179945
\(361\) 32.1858 1.69399
\(362\) −14.8398 −0.779962
\(363\) 5.71656 0.300042
\(364\) 0 0
\(365\) −27.6226 −1.44583
\(366\) −0.0113593 −0.000593760 0
\(367\) −3.04364 −0.158877 −0.0794385 0.996840i \(-0.525313\pi\)
−0.0794385 + 0.996840i \(0.525313\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.01136 0.312939
\(370\) −9.28946 −0.482936
\(371\) 0 0
\(372\) 5.15442 0.267244
\(373\) −37.2752 −1.93003 −0.965017 0.262187i \(-0.915556\pi\)
−0.965017 + 0.262187i \(0.915556\pi\)
\(374\) 6.25400 0.323387
\(375\) 5.65685 0.292119
\(376\) 0.557278 0.0287394
\(377\) −15.0209 −0.773617
\(378\) 0 0
\(379\) 22.6720 1.16458 0.582291 0.812980i \(-0.302156\pi\)
0.582291 + 0.812980i \(0.302156\pi\)
\(380\) 24.4267 1.25306
\(381\) 13.0033 0.666181
\(382\) −18.3492 −0.938825
\(383\) 11.7278 0.599261 0.299630 0.954055i \(-0.403137\pi\)
0.299630 + 0.954055i \(0.403137\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −0.979075 −0.0498336
\(387\) 11.8193 0.600809
\(388\) 5.76446 0.292646
\(389\) 7.49223 0.379871 0.189936 0.981797i \(-0.439172\pi\)
0.189936 + 0.981797i \(0.439172\pi\)
\(390\) 5.28946 0.267842
\(391\) 2.72082 0.137598
\(392\) 0 0
\(393\) −15.4804 −0.780884
\(394\) 4.65353 0.234441
\(395\) 35.5289 1.78765
\(396\) 2.29857 0.115508
\(397\) 22.3616 1.12230 0.561148 0.827715i \(-0.310360\pi\)
0.561148 + 0.827715i \(0.310360\pi\)
\(398\) −1.46436 −0.0734015
\(399\) 0 0
\(400\) 6.65685 0.332843
\(401\) −4.09580 −0.204534 −0.102267 0.994757i \(-0.532610\pi\)
−0.102267 + 0.994757i \(0.532610\pi\)
\(402\) −8.93604 −0.445689
\(403\) −7.98547 −0.397785
\(404\) −13.8193 −0.687536
\(405\) −3.41421 −0.169654
\(406\) 0 0
\(407\) 6.25400 0.309999
\(408\) −2.72082 −0.134701
\(409\) −19.5336 −0.965876 −0.482938 0.875655i \(-0.660430\pi\)
−0.482938 + 0.875655i \(0.660430\pi\)
\(410\) −20.5241 −1.01361
\(411\) 15.4416 0.761680
\(412\) −8.86721 −0.436856
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 5.84782 0.287058
\(416\) 1.54925 0.0759580
\(417\) 5.17157 0.253253
\(418\) −16.4450 −0.804350
\(419\) 5.44055 0.265788 0.132894 0.991130i \(-0.457573\pi\)
0.132894 + 0.991130i \(0.457573\pi\)
\(420\) 0 0
\(421\) 37.5149 1.82837 0.914183 0.405301i \(-0.132833\pi\)
0.914183 + 0.405301i \(0.132833\pi\)
\(422\) −12.9075 −0.628329
\(423\) 0.557278 0.0270958
\(424\) −3.31796 −0.161135
\(425\) 18.1121 0.878566
\(426\) 9.25067 0.448197
\(427\) 0 0
\(428\) 13.1818 0.637169
\(429\) −3.56105 −0.171929
\(430\) −40.3536 −1.94603
\(431\) 18.0303 0.868488 0.434244 0.900795i \(-0.357016\pi\)
0.434244 + 0.900795i \(0.357016\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 25.4031 1.22079 0.610397 0.792096i \(-0.291010\pi\)
0.610397 + 0.792096i \(0.291010\pi\)
\(434\) 0 0
\(435\) −33.1030 −1.58717
\(436\) 3.76446 0.180285
\(437\) −7.15442 −0.342243
\(438\) −8.09046 −0.386577
\(439\) 4.19205 0.200076 0.100038 0.994984i \(-0.468104\pi\)
0.100038 + 0.994984i \(0.468104\pi\)
\(440\) −7.84782 −0.374130
\(441\) 0 0
\(442\) 4.21522 0.200497
\(443\) −24.0836 −1.14425 −0.572123 0.820168i \(-0.693880\pi\)
−0.572123 + 0.820168i \(0.693880\pi\)
\(444\) −2.72082 −0.129124
\(445\) 54.8001 2.59778
\(446\) −14.4293 −0.683249
\(447\) −6.70475 −0.317124
\(448\) 0 0
\(449\) 5.89262 0.278090 0.139045 0.990286i \(-0.455597\pi\)
0.139045 + 0.990286i \(0.455597\pi\)
\(450\) 6.65685 0.313807
\(451\) 13.8175 0.650643
\(452\) −6.15218 −0.289374
\(453\) −8.13395 −0.382166
\(454\) 14.4133 0.676449
\(455\) 0 0
\(456\) 7.15442 0.335037
\(457\) −9.40285 −0.439847 −0.219923 0.975517i \(-0.570581\pi\)
−0.219923 + 0.975517i \(0.570581\pi\)
\(458\) 1.26357 0.0590426
\(459\) −2.72082 −0.126997
\(460\) −3.41421 −0.159189
\(461\) −13.5720 −0.632109 −0.316055 0.948741i \(-0.602358\pi\)
−0.316055 + 0.948741i \(0.602358\pi\)
\(462\) 0 0
\(463\) 32.7759 1.52323 0.761613 0.648033i \(-0.224408\pi\)
0.761613 + 0.648033i \(0.224408\pi\)
\(464\) −9.69564 −0.450109
\(465\) −17.5983 −0.816102
\(466\) −16.7166 −0.774380
\(467\) −1.53789 −0.0711649 −0.0355824 0.999367i \(-0.511329\pi\)
−0.0355824 + 0.999367i \(0.511329\pi\)
\(468\) 1.54925 0.0716139
\(469\) 0 0
\(470\) −1.90267 −0.0877634
\(471\) −3.64396 −0.167905
\(472\) 7.69564 0.354220
\(473\) 27.1675 1.24917
\(474\) 10.4062 0.477972
\(475\) −47.6260 −2.18523
\(476\) 0 0
\(477\) −3.31796 −0.151919
\(478\) −22.6923 −1.03792
\(479\) 25.6547 1.17219 0.586096 0.810241i \(-0.300664\pi\)
0.586096 + 0.810241i \(0.300664\pi\)
\(480\) 3.41421 0.155837
\(481\) 4.21522 0.192197
\(482\) −20.6705 −0.941513
\(483\) 0 0
\(484\) −5.71656 −0.259844
\(485\) −19.6811 −0.893673
\(486\) −1.00000 −0.0453609
\(487\) −17.8151 −0.807277 −0.403639 0.914919i \(-0.632255\pi\)
−0.403639 + 0.914919i \(0.632255\pi\)
\(488\) 0.0113593 0.000514212 0
\(489\) 13.5047 0.610702
\(490\) 0 0
\(491\) −39.3973 −1.77797 −0.888987 0.457931i \(-0.848590\pi\)
−0.888987 + 0.457931i \(0.848590\pi\)
\(492\) −6.01136 −0.271013
\(493\) −26.3801 −1.18810
\(494\) −11.0840 −0.498691
\(495\) −7.84782 −0.352733
\(496\) −5.15442 −0.231440
\(497\) 0 0
\(498\) 1.71279 0.0767518
\(499\) 20.2297 0.905608 0.452804 0.891610i \(-0.350424\pi\)
0.452804 + 0.891610i \(0.350424\pi\)
\(500\) −5.65685 −0.252982
\(501\) 19.6009 0.875705
\(502\) −27.6397 −1.23362
\(503\) 12.9791 0.578708 0.289354 0.957222i \(-0.406559\pi\)
0.289354 + 0.957222i \(0.406559\pi\)
\(504\) 0 0
\(505\) 47.1821 2.09957
\(506\) 2.29857 0.102184
\(507\) 10.5998 0.470755
\(508\) −13.0033 −0.576929
\(509\) −15.9151 −0.705425 −0.352712 0.935732i \(-0.614741\pi\)
−0.352712 + 0.935732i \(0.614741\pi\)
\(510\) 9.28946 0.411344
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 7.15442 0.315876
\(514\) 11.9383 0.526575
\(515\) 30.2745 1.33406
\(516\) −11.8193 −0.520316
\(517\) 1.28094 0.0563359
\(518\) 0 0
\(519\) 14.6317 0.642259
\(520\) −5.28946 −0.231958
\(521\) −1.16732 −0.0511411 −0.0255705 0.999673i \(-0.508140\pi\)
−0.0255705 + 0.999673i \(0.508140\pi\)
\(522\) −9.69564 −0.424367
\(523\) −16.5412 −0.723297 −0.361648 0.932315i \(-0.617786\pi\)
−0.361648 + 0.932315i \(0.617786\pi\)
\(524\) 15.4804 0.676265
\(525\) 0 0
\(526\) −8.68629 −0.378740
\(527\) −14.0243 −0.610906
\(528\) −2.29857 −0.100033
\(529\) 1.00000 0.0434783
\(530\) 11.3282 0.492067
\(531\) 7.69564 0.333962
\(532\) 0 0
\(533\) 9.31307 0.403394
\(534\) 16.0506 0.694577
\(535\) −45.0056 −1.94576
\(536\) 8.93604 0.385978
\(537\) 6.28613 0.271266
\(538\) 20.6134 0.888708
\(539\) 0 0
\(540\) 3.41421 0.146924
\(541\) −32.1400 −1.38181 −0.690903 0.722948i \(-0.742787\pi\)
−0.690903 + 0.722948i \(0.742787\pi\)
\(542\) 3.53789 0.151965
\(543\) 14.8398 0.636836
\(544\) 2.72082 0.116654
\(545\) −12.8527 −0.550548
\(546\) 0 0
\(547\) 38.7371 1.65628 0.828140 0.560522i \(-0.189399\pi\)
0.828140 + 0.560522i \(0.189399\pi\)
\(548\) −15.4416 −0.659634
\(549\) 0.0113593 0.000484803 0
\(550\) 15.3013 0.652448
\(551\) 69.3667 2.95512
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) 24.4898 1.04047
\(555\) 9.28946 0.394315
\(556\) −5.17157 −0.219324
\(557\) 19.1879 0.813016 0.406508 0.913647i \(-0.366746\pi\)
0.406508 + 0.913647i \(0.366746\pi\)
\(558\) −5.15442 −0.218204
\(559\) 18.3110 0.774473
\(560\) 0 0
\(561\) −6.25400 −0.264044
\(562\) −10.1582 −0.428498
\(563\) −9.61852 −0.405372 −0.202686 0.979244i \(-0.564967\pi\)
−0.202686 + 0.979244i \(0.564967\pi\)
\(564\) −0.557278 −0.0234656
\(565\) 21.0049 0.883681
\(566\) 16.6042 0.697929
\(567\) 0 0
\(568\) −9.25067 −0.388150
\(569\) −10.0570 −0.421612 −0.210806 0.977528i \(-0.567609\pi\)
−0.210806 + 0.977528i \(0.567609\pi\)
\(570\) −24.4267 −1.02312
\(571\) −14.4552 −0.604933 −0.302466 0.953160i \(-0.597810\pi\)
−0.302466 + 0.953160i \(0.597810\pi\)
\(572\) 3.56105 0.148895
\(573\) 18.3492 0.766547
\(574\) 0 0
\(575\) 6.65685 0.277610
\(576\) 1.00000 0.0416667
\(577\) −34.4456 −1.43399 −0.716995 0.697078i \(-0.754483\pi\)
−0.716995 + 0.697078i \(0.754483\pi\)
\(578\) −9.59715 −0.399189
\(579\) 0.979075 0.0406890
\(580\) 33.1030 1.37453
\(581\) 0 0
\(582\) −5.76446 −0.238945
\(583\) −7.62658 −0.315861
\(584\) 8.09046 0.334786
\(585\) −5.28946 −0.218692
\(586\) −24.9737 −1.03166
\(587\) −0.135481 −0.00559192 −0.00279596 0.999996i \(-0.500890\pi\)
−0.00279596 + 0.999996i \(0.500890\pi\)
\(588\) 0 0
\(589\) 36.8769 1.51949
\(590\) −26.2745 −1.08171
\(591\) −4.65353 −0.191421
\(592\) 2.72082 0.111825
\(593\) −28.1286 −1.15510 −0.577552 0.816354i \(-0.695992\pi\)
−0.577552 + 0.816354i \(0.695992\pi\)
\(594\) −2.29857 −0.0943116
\(595\) 0 0
\(596\) 6.70475 0.274637
\(597\) 1.46436 0.0599321
\(598\) 1.54925 0.0633534
\(599\) −6.31102 −0.257861 −0.128931 0.991654i \(-0.541154\pi\)
−0.128931 + 0.991654i \(0.541154\pi\)
\(600\) −6.65685 −0.271765
\(601\) 27.0450 1.10319 0.551593 0.834113i \(-0.314020\pi\)
0.551593 + 0.834113i \(0.314020\pi\)
\(602\) 0 0
\(603\) 8.93604 0.363904
\(604\) 8.13395 0.330966
\(605\) 19.5176 0.793502
\(606\) 13.8193 0.561371
\(607\) 42.1683 1.71156 0.855778 0.517343i \(-0.173079\pi\)
0.855778 + 0.517343i \(0.173079\pi\)
\(608\) −7.15442 −0.290150
\(609\) 0 0
\(610\) −0.0387831 −0.00157028
\(611\) 0.863361 0.0349278
\(612\) 2.72082 0.109983
\(613\) 26.4325 1.06760 0.533800 0.845611i \(-0.320764\pi\)
0.533800 + 0.845611i \(0.320764\pi\)
\(614\) 32.1239 1.29642
\(615\) 20.5241 0.827610
\(616\) 0 0
\(617\) −24.8833 −1.00176 −0.500881 0.865516i \(-0.666991\pi\)
−0.500881 + 0.865516i \(0.666991\pi\)
\(618\) 8.86721 0.356692
\(619\) 34.1496 1.37259 0.686293 0.727325i \(-0.259237\pi\)
0.686293 + 0.727325i \(0.259237\pi\)
\(620\) 17.5983 0.706765
\(621\) −1.00000 −0.0401286
\(622\) −29.0571 −1.16508
\(623\) 0 0
\(624\) −1.54925 −0.0620195
\(625\) −13.9706 −0.558823
\(626\) 1.74174 0.0696141
\(627\) 16.4450 0.656749
\(628\) 3.64396 0.145410
\(629\) 7.40285 0.295171
\(630\) 0 0
\(631\) −24.5783 −0.978446 −0.489223 0.872159i \(-0.662720\pi\)
−0.489223 + 0.872159i \(0.662720\pi\)
\(632\) −10.4062 −0.413936
\(633\) 12.9075 0.513028
\(634\) −12.7796 −0.507543
\(635\) 44.3961 1.76181
\(636\) 3.31796 0.131566
\(637\) 0 0
\(638\) −22.2861 −0.882316
\(639\) −9.25067 −0.365951
\(640\) −3.41421 −0.134959
\(641\) −27.5998 −1.09013 −0.545064 0.838394i \(-0.683495\pi\)
−0.545064 + 0.838394i \(0.683495\pi\)
\(642\) −13.1818 −0.520246
\(643\) −4.81128 −0.189738 −0.0948691 0.995490i \(-0.530243\pi\)
−0.0948691 + 0.995490i \(0.530243\pi\)
\(644\) 0 0
\(645\) 40.3536 1.58892
\(646\) −19.4659 −0.765875
\(647\) −32.4514 −1.27580 −0.637899 0.770120i \(-0.720196\pi\)
−0.637899 + 0.770120i \(0.720196\pi\)
\(648\) 1.00000 0.0392837
\(649\) 17.6890 0.694353
\(650\) 10.3131 0.404513
\(651\) 0 0
\(652\) −13.5047 −0.528884
\(653\) 11.5192 0.450781 0.225391 0.974268i \(-0.427634\pi\)
0.225391 + 0.974268i \(0.427634\pi\)
\(654\) −3.76446 −0.147202
\(655\) −52.8535 −2.06516
\(656\) 6.01136 0.234704
\(657\) 8.09046 0.315639
\(658\) 0 0
\(659\) 33.2288 1.29441 0.647205 0.762316i \(-0.275938\pi\)
0.647205 + 0.762316i \(0.275938\pi\)
\(660\) 7.84782 0.305476
\(661\) 11.6358 0.452579 0.226290 0.974060i \(-0.427340\pi\)
0.226290 + 0.974060i \(0.427340\pi\)
\(662\) 6.34315 0.246533
\(663\) −4.21522 −0.163705
\(664\) −1.71279 −0.0664690
\(665\) 0 0
\(666\) 2.72082 0.105430
\(667\) −9.69564 −0.375417
\(668\) −19.6009 −0.758382
\(669\) 14.4293 0.557871
\(670\) −30.5095 −1.17869
\(671\) 0.0261102 0.00100797
\(672\) 0 0
\(673\) −20.2745 −0.781526 −0.390763 0.920491i \(-0.627789\pi\)
−0.390763 + 0.920491i \(0.627789\pi\)
\(674\) 15.8866 0.611929
\(675\) −6.65685 −0.256222
\(676\) −10.5998 −0.407686
\(677\) 2.87408 0.110460 0.0552300 0.998474i \(-0.482411\pi\)
0.0552300 + 0.998474i \(0.482411\pi\)
\(678\) 6.15218 0.236273
\(679\) 0 0
\(680\) −9.28946 −0.356234
\(681\) −14.4133 −0.552318
\(682\) −11.8478 −0.453676
\(683\) 32.5710 1.24630 0.623148 0.782104i \(-0.285853\pi\)
0.623148 + 0.782104i \(0.285853\pi\)
\(684\) −7.15442 −0.273556
\(685\) 52.7210 2.01437
\(686\) 0 0
\(687\) −1.26357 −0.0482081
\(688\) 11.8193 0.450607
\(689\) −5.14034 −0.195831
\(690\) 3.41421 0.129977
\(691\) −11.8966 −0.452570 −0.226285 0.974061i \(-0.572658\pi\)
−0.226285 + 0.974061i \(0.572658\pi\)
\(692\) −14.6317 −0.556213
\(693\) 0 0
\(694\) −28.7699 −1.09209
\(695\) 17.6569 0.669763
\(696\) 9.69564 0.367512
\(697\) 16.3558 0.619521
\(698\) −14.6638 −0.555033
\(699\) 16.7166 0.632278
\(700\) 0 0
\(701\) 49.0439 1.85236 0.926181 0.377080i \(-0.123072\pi\)
0.926181 + 0.377080i \(0.123072\pi\)
\(702\) −1.54925 −0.0584725
\(703\) −19.4659 −0.734170
\(704\) 2.29857 0.0866307
\(705\) 1.90267 0.0716586
\(706\) 12.6079 0.474503
\(707\) 0 0
\(708\) −7.69564 −0.289220
\(709\) 40.4789 1.52022 0.760108 0.649797i \(-0.225146\pi\)
0.760108 + 0.649797i \(0.225146\pi\)
\(710\) 31.5838 1.18532
\(711\) −10.4062 −0.390262
\(712\) −16.0506 −0.601521
\(713\) −5.15442 −0.193035
\(714\) 0 0
\(715\) −12.1582 −0.454691
\(716\) −6.28613 −0.234924
\(717\) 22.6923 0.847460
\(718\) −13.5047 −0.503990
\(719\) 8.70946 0.324808 0.162404 0.986724i \(-0.448075\pi\)
0.162404 + 0.986724i \(0.448075\pi\)
\(720\) −3.41421 −0.127240
\(721\) 0 0
\(722\) 32.1858 1.19783
\(723\) 20.6705 0.768743
\(724\) −14.8398 −0.551516
\(725\) −64.5424 −2.39705
\(726\) 5.71656 0.212162
\(727\) −3.88814 −0.144203 −0.0721015 0.997397i \(-0.522971\pi\)
−0.0721015 + 0.997397i \(0.522971\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −27.6226 −1.02236
\(731\) 32.1582 1.18941
\(732\) −0.0113593 −0.000419852 0
\(733\) 14.0158 0.517687 0.258844 0.965919i \(-0.416659\pi\)
0.258844 + 0.965919i \(0.416659\pi\)
\(734\) −3.04364 −0.112343
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 20.5401 0.756605
\(738\) 6.01136 0.221281
\(739\) −38.5528 −1.41819 −0.709094 0.705114i \(-0.750896\pi\)
−0.709094 + 0.705114i \(0.750896\pi\)
\(740\) −9.28946 −0.341487
\(741\) 11.0840 0.407179
\(742\) 0 0
\(743\) −1.86605 −0.0684588 −0.0342294 0.999414i \(-0.510898\pi\)
−0.0342294 + 0.999414i \(0.510898\pi\)
\(744\) 5.15442 0.188970
\(745\) −22.8915 −0.838678
\(746\) −37.2752 −1.36474
\(747\) −1.71279 −0.0626676
\(748\) 6.25400 0.228669
\(749\) 0 0
\(750\) 5.65685 0.206559
\(751\) 9.90420 0.361409 0.180705 0.983537i \(-0.442162\pi\)
0.180705 + 0.983537i \(0.442162\pi\)
\(752\) 0.557278 0.0203218
\(753\) 27.6397 1.00725
\(754\) −15.0209 −0.547030
\(755\) −27.7710 −1.01069
\(756\) 0 0
\(757\) 32.6059 1.18508 0.592541 0.805541i \(-0.298125\pi\)
0.592541 + 0.805541i \(0.298125\pi\)
\(758\) 22.6720 0.823484
\(759\) −2.29857 −0.0834329
\(760\) 24.4267 0.886051
\(761\) −23.8204 −0.863489 −0.431744 0.901996i \(-0.642102\pi\)
−0.431744 + 0.901996i \(0.642102\pi\)
\(762\) 13.0033 0.471061
\(763\) 0 0
\(764\) −18.3492 −0.663849
\(765\) −9.28946 −0.335861
\(766\) 11.7278 0.423741
\(767\) 11.9224 0.430494
\(768\) −1.00000 −0.0360844
\(769\) −20.8608 −0.752259 −0.376130 0.926567i \(-0.622745\pi\)
−0.376130 + 0.926567i \(0.622745\pi\)
\(770\) 0 0
\(771\) −11.9383 −0.429947
\(772\) −0.979075 −0.0352377
\(773\) −50.2919 −1.80887 −0.904437 0.426606i \(-0.859709\pi\)
−0.904437 + 0.426606i \(0.859709\pi\)
\(774\) 11.8193 0.424836
\(775\) −34.3122 −1.23253
\(776\) 5.76446 0.206932
\(777\) 0 0
\(778\) 7.49223 0.268609
\(779\) −43.0078 −1.54091
\(780\) 5.28946 0.189393
\(781\) −21.2633 −0.760862
\(782\) 2.72082 0.0972963
\(783\) 9.69564 0.346494
\(784\) 0 0
\(785\) −12.4413 −0.444048
\(786\) −15.4804 −0.552168
\(787\) 2.23020 0.0794979 0.0397490 0.999210i \(-0.487344\pi\)
0.0397490 + 0.999210i \(0.487344\pi\)
\(788\) 4.65353 0.165775
\(789\) 8.68629 0.309240
\(790\) 35.5289 1.26406
\(791\) 0 0
\(792\) 2.29857 0.0816762
\(793\) 0.0175984 0.000624936 0
\(794\) 22.3616 0.793584
\(795\) −11.3282 −0.401771
\(796\) −1.46436 −0.0519027
\(797\) 10.3151 0.365379 0.182690 0.983171i \(-0.441520\pi\)
0.182690 + 0.983171i \(0.441520\pi\)
\(798\) 0 0
\(799\) 1.51625 0.0536412
\(800\) 6.65685 0.235355
\(801\) −16.0506 −0.567120
\(802\) −4.09580 −0.144628
\(803\) 18.5965 0.656257
\(804\) −8.93604 −0.315150
\(805\) 0 0
\(806\) −7.98547 −0.281276
\(807\) −20.6134 −0.725627
\(808\) −13.8193 −0.486162
\(809\) −23.7763 −0.835929 −0.417965 0.908463i \(-0.637256\pi\)
−0.417965 + 0.908463i \(0.637256\pi\)
\(810\) −3.41421 −0.119963
\(811\) 7.77627 0.273062 0.136531 0.990636i \(-0.456405\pi\)
0.136531 + 0.990636i \(0.456405\pi\)
\(812\) 0 0
\(813\) −3.53789 −0.124079
\(814\) 6.25400 0.219203
\(815\) 46.1078 1.61509
\(816\) −2.72082 −0.0952477
\(817\) −84.5604 −2.95839
\(818\) −19.5336 −0.682977
\(819\) 0 0
\(820\) −20.5241 −0.716731
\(821\) −31.9557 −1.11526 −0.557630 0.830090i \(-0.688289\pi\)
−0.557630 + 0.830090i \(0.688289\pi\)
\(822\) 15.4416 0.538589
\(823\) −35.1918 −1.22671 −0.613354 0.789808i \(-0.710180\pi\)
−0.613354 + 0.789808i \(0.710180\pi\)
\(824\) −8.86721 −0.308904
\(825\) −15.3013 −0.532722
\(826\) 0 0
\(827\) −31.8868 −1.10881 −0.554407 0.832246i \(-0.687055\pi\)
−0.554407 + 0.832246i \(0.687055\pi\)
\(828\) 1.00000 0.0347524
\(829\) 3.80631 0.132199 0.0660994 0.997813i \(-0.478945\pi\)
0.0660994 + 0.997813i \(0.478945\pi\)
\(830\) 5.84782 0.202981
\(831\) −24.4898 −0.849541
\(832\) 1.54925 0.0537104
\(833\) 0 0
\(834\) 5.17157 0.179077
\(835\) 66.9217 2.31592
\(836\) −16.4450 −0.568761
\(837\) 5.15442 0.178163
\(838\) 5.44055 0.187941
\(839\) −9.29547 −0.320915 −0.160458 0.987043i \(-0.551297\pi\)
−0.160458 + 0.987043i \(0.551297\pi\)
\(840\) 0 0
\(841\) 65.0054 2.24156
\(842\) 37.5149 1.29285
\(843\) 10.1582 0.349867
\(844\) −12.9075 −0.444295
\(845\) 36.1901 1.24498
\(846\) 0.557278 0.0191596
\(847\) 0 0
\(848\) −3.31796 −0.113939
\(849\) −16.6042 −0.569856
\(850\) 18.1121 0.621240
\(851\) 2.72082 0.0932685
\(852\) 9.25067 0.316923
\(853\) 23.4213 0.801931 0.400965 0.916093i \(-0.368675\pi\)
0.400965 + 0.916093i \(0.368675\pi\)
\(854\) 0 0
\(855\) 24.4267 0.835377
\(856\) 13.1818 0.450546
\(857\) −14.6467 −0.500320 −0.250160 0.968204i \(-0.580483\pi\)
−0.250160 + 0.968204i \(0.580483\pi\)
\(858\) −3.56105 −0.121572
\(859\) −16.0142 −0.546398 −0.273199 0.961958i \(-0.588082\pi\)
−0.273199 + 0.961958i \(0.588082\pi\)
\(860\) −40.3536 −1.37605
\(861\) 0 0
\(862\) 18.0303 0.614113
\(863\) −19.9994 −0.680786 −0.340393 0.940283i \(-0.610560\pi\)
−0.340393 + 0.940283i \(0.610560\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 49.9557 1.69854
\(866\) 25.4031 0.863232
\(867\) 9.59715 0.325936
\(868\) 0 0
\(869\) −23.9194 −0.811409
\(870\) −33.1030 −1.12230
\(871\) 13.8441 0.469090
\(872\) 3.76446 0.127481
\(873\) 5.76446 0.195098
\(874\) −7.15442 −0.242002
\(875\) 0 0
\(876\) −8.09046 −0.273351
\(877\) 20.5613 0.694306 0.347153 0.937808i \(-0.387148\pi\)
0.347153 + 0.937808i \(0.387148\pi\)
\(878\) 4.19205 0.141475
\(879\) 24.9737 0.842343
\(880\) −7.84782 −0.264550
\(881\) 57.6246 1.94142 0.970712 0.240247i \(-0.0772285\pi\)
0.970712 + 0.240247i \(0.0772285\pi\)
\(882\) 0 0
\(883\) 17.3701 0.584550 0.292275 0.956334i \(-0.405588\pi\)
0.292275 + 0.956334i \(0.405588\pi\)
\(884\) 4.21522 0.141773
\(885\) 26.2745 0.883209
\(886\) −24.0836 −0.809104
\(887\) 35.3772 1.18785 0.593925 0.804520i \(-0.297578\pi\)
0.593925 + 0.804520i \(0.297578\pi\)
\(888\) −2.72082 −0.0913047
\(889\) 0 0
\(890\) 54.8001 1.83691
\(891\) 2.29857 0.0770051
\(892\) −14.4293 −0.483130
\(893\) −3.98700 −0.133420
\(894\) −6.70475 −0.224241
\(895\) 21.4622 0.717402
\(896\) 0 0
\(897\) −1.54925 −0.0517278
\(898\) 5.89262 0.196639
\(899\) 49.9754 1.66677
\(900\) 6.65685 0.221895
\(901\) −9.02758 −0.300752
\(902\) 13.8175 0.460074
\(903\) 0 0
\(904\) −6.15218 −0.204619
\(905\) 50.6662 1.68420
\(906\) −8.13395 −0.270232
\(907\) 55.0175 1.82682 0.913412 0.407036i \(-0.133438\pi\)
0.913412 + 0.407036i \(0.133438\pi\)
\(908\) 14.4133 0.478322
\(909\) −13.8193 −0.458358
\(910\) 0 0
\(911\) −7.43376 −0.246291 −0.123146 0.992389i \(-0.539298\pi\)
−0.123146 + 0.992389i \(0.539298\pi\)
\(912\) 7.15442 0.236907
\(913\) −3.93696 −0.130295
\(914\) −9.40285 −0.311019
\(915\) 0.0387831 0.00128213
\(916\) 1.26357 0.0417494
\(917\) 0 0
\(918\) −2.72082 −0.0898004
\(919\) −18.5159 −0.610782 −0.305391 0.952227i \(-0.598787\pi\)
−0.305391 + 0.952227i \(0.598787\pi\)
\(920\) −3.41421 −0.112563
\(921\) −32.1239 −1.05852
\(922\) −13.5720 −0.446969
\(923\) −14.3316 −0.471729
\(924\) 0 0
\(925\) 18.1121 0.595522
\(926\) 32.7759 1.07708
\(927\) −8.86721 −0.291237
\(928\) −9.69564 −0.318275
\(929\) 9.62943 0.315931 0.157966 0.987445i \(-0.449506\pi\)
0.157966 + 0.987445i \(0.449506\pi\)
\(930\) −17.5983 −0.577071
\(931\) 0 0
\(932\) −16.7166 −0.547569
\(933\) 29.0571 0.951286
\(934\) −1.53789 −0.0503212
\(935\) −21.3525 −0.698301
\(936\) 1.54925 0.0506387
\(937\) 23.5568 0.769567 0.384784 0.923007i \(-0.374276\pi\)
0.384784 + 0.923007i \(0.374276\pi\)
\(938\) 0 0
\(939\) −1.74174 −0.0568397
\(940\) −1.90267 −0.0620581
\(941\) −29.3496 −0.956771 −0.478386 0.878150i \(-0.658778\pi\)
−0.478386 + 0.878150i \(0.658778\pi\)
\(942\) −3.64396 −0.118727
\(943\) 6.01136 0.195757
\(944\) 7.69564 0.250472
\(945\) 0 0
\(946\) 27.1675 0.883293
\(947\) 23.1615 0.752649 0.376324 0.926488i \(-0.377188\pi\)
0.376324 + 0.926488i \(0.377188\pi\)
\(948\) 10.4062 0.337977
\(949\) 12.5341 0.406874
\(950\) −47.6260 −1.54519
\(951\) 12.7796 0.414407
\(952\) 0 0
\(953\) 15.2573 0.494233 0.247117 0.968986i \(-0.420517\pi\)
0.247117 + 0.968986i \(0.420517\pi\)
\(954\) −3.31796 −0.107423
\(955\) 62.6480 2.02724
\(956\) −22.6923 −0.733922
\(957\) 22.2861 0.720408
\(958\) 25.6547 0.828865
\(959\) 0 0
\(960\) 3.41421 0.110193
\(961\) −4.43192 −0.142965
\(962\) 4.21522 0.135904
\(963\) 13.1818 0.424779
\(964\) −20.6705 −0.665751
\(965\) 3.34277 0.107608
\(966\) 0 0
\(967\) −21.9247 −0.705052 −0.352526 0.935802i \(-0.614677\pi\)
−0.352526 + 0.935802i \(0.614677\pi\)
\(968\) −5.71656 −0.183737
\(969\) 19.4659 0.625334
\(970\) −19.6811 −0.631922
\(971\) −17.3574 −0.557026 −0.278513 0.960432i \(-0.589842\pi\)
−0.278513 + 0.960432i \(0.589842\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −17.8151 −0.570831
\(975\) −10.3131 −0.330284
\(976\) 0.0113593 0.000363602 0
\(977\) 32.8263 1.05021 0.525103 0.851039i \(-0.324027\pi\)
0.525103 + 0.851039i \(0.324027\pi\)
\(978\) 13.5047 0.431832
\(979\) −36.8935 −1.17912
\(980\) 0 0
\(981\) 3.76446 0.120190
\(982\) −39.3973 −1.25722
\(983\) 34.4371 1.09837 0.549186 0.835700i \(-0.314938\pi\)
0.549186 + 0.835700i \(0.314938\pi\)
\(984\) −6.01136 −0.191635
\(985\) −15.8881 −0.506238
\(986\) −26.3801 −0.840113
\(987\) 0 0
\(988\) −11.0840 −0.352628
\(989\) 11.8193 0.375832
\(990\) −7.84782 −0.249420
\(991\) 18.9045 0.600520 0.300260 0.953857i \(-0.402927\pi\)
0.300260 + 0.953857i \(0.402927\pi\)
\(992\) −5.15442 −0.163653
\(993\) −6.34315 −0.201294
\(994\) 0 0
\(995\) 4.99962 0.158499
\(996\) 1.71279 0.0542717
\(997\) −15.4989 −0.490855 −0.245427 0.969415i \(-0.578928\pi\)
−0.245427 + 0.969415i \(0.578928\pi\)
\(998\) 20.2297 0.640361
\(999\) −2.72082 −0.0860829
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.ci.1.2 4
7.6 odd 2 6762.2.a.ct.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.ci.1.2 4 1.1 even 1 trivial
6762.2.a.ct.1.4 yes 4 7.6 odd 2