Properties

Label 847.2.a.d.1.1
Level $847$
Weight $2$
Character 847.1
Self dual yes
Analytic conductor $6.763$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.76332905120\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 847.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.61803 q^{2} +0.618034 q^{3} +4.85410 q^{4} +1.00000 q^{5} -1.61803 q^{6} +1.00000 q^{7} -7.47214 q^{8} -2.61803 q^{9} +O(q^{10})\) \(q-2.61803 q^{2} +0.618034 q^{3} +4.85410 q^{4} +1.00000 q^{5} -1.61803 q^{6} +1.00000 q^{7} -7.47214 q^{8} -2.61803 q^{9} -2.61803 q^{10} +3.00000 q^{12} +3.23607 q^{13} -2.61803 q^{14} +0.618034 q^{15} +9.85410 q^{16} -8.09017 q^{17} +6.85410 q^{18} -6.23607 q^{19} +4.85410 q^{20} +0.618034 q^{21} -6.09017 q^{23} -4.61803 q^{24} -4.00000 q^{25} -8.47214 q^{26} -3.47214 q^{27} +4.85410 q^{28} -2.38197 q^{29} -1.61803 q^{30} +0.236068 q^{31} -10.8541 q^{32} +21.1803 q^{34} +1.00000 q^{35} -12.7082 q^{36} -2.47214 q^{37} +16.3262 q^{38} +2.00000 q^{39} -7.47214 q^{40} +11.1803 q^{41} -1.61803 q^{42} +7.56231 q^{43} -2.61803 q^{45} +15.9443 q^{46} -4.38197 q^{47} +6.09017 q^{48} +1.00000 q^{49} +10.4721 q^{50} -5.00000 q^{51} +15.7082 q^{52} -4.61803 q^{53} +9.09017 q^{54} -7.47214 q^{56} -3.85410 q^{57} +6.23607 q^{58} -0.0901699 q^{59} +3.00000 q^{60} -5.38197 q^{61} -0.618034 q^{62} -2.61803 q^{63} +8.70820 q^{64} +3.23607 q^{65} +7.32624 q^{67} -39.2705 q^{68} -3.76393 q^{69} -2.61803 q^{70} -4.90983 q^{71} +19.5623 q^{72} -9.76393 q^{73} +6.47214 q^{74} -2.47214 q^{75} -30.2705 q^{76} -5.23607 q^{78} +8.61803 q^{79} +9.85410 q^{80} +5.70820 q^{81} -29.2705 q^{82} -10.7082 q^{83} +3.00000 q^{84} -8.09017 q^{85} -19.7984 q^{86} -1.47214 q^{87} +0.145898 q^{89} +6.85410 q^{90} +3.23607 q^{91} -29.5623 q^{92} +0.145898 q^{93} +11.4721 q^{94} -6.23607 q^{95} -6.70820 q^{96} +7.00000 q^{97} -2.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - q^{3} + 3 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} - 6 q^{8} - 3 q^{9} + O(q^{10}) \) \( 2 q - 3 q^{2} - q^{3} + 3 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} - 6 q^{8} - 3 q^{9} - 3 q^{10} + 6 q^{12} + 2 q^{13} - 3 q^{14} - q^{15} + 13 q^{16} - 5 q^{17} + 7 q^{18} - 8 q^{19} + 3 q^{20} - q^{21} - q^{23} - 7 q^{24} - 8 q^{25} - 8 q^{26} + 2 q^{27} + 3 q^{28} - 7 q^{29} - q^{30} - 4 q^{31} - 15 q^{32} + 20 q^{34} + 2 q^{35} - 12 q^{36} + 4 q^{37} + 17 q^{38} + 4 q^{39} - 6 q^{40} - q^{42} - 5 q^{43} - 3 q^{45} + 14 q^{46} - 11 q^{47} + q^{48} + 2 q^{49} + 12 q^{50} - 10 q^{51} + 18 q^{52} - 7 q^{53} + 7 q^{54} - 6 q^{56} - q^{57} + 8 q^{58} + 11 q^{59} + 6 q^{60} - 13 q^{61} + q^{62} - 3 q^{63} + 4 q^{64} + 2 q^{65} - q^{67} - 45 q^{68} - 12 q^{69} - 3 q^{70} - 21 q^{71} + 19 q^{72} - 24 q^{73} + 4 q^{74} + 4 q^{75} - 27 q^{76} - 6 q^{78} + 15 q^{79} + 13 q^{80} - 2 q^{81} - 25 q^{82} - 8 q^{83} + 6 q^{84} - 5 q^{85} - 15 q^{86} + 6 q^{87} + 7 q^{89} + 7 q^{90} + 2 q^{91} - 39 q^{92} + 7 q^{93} + 14 q^{94} - 8 q^{95} + 14 q^{97} - 3 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.61803 −1.85123 −0.925615 0.378467i \(-0.876451\pi\)
−0.925615 + 0.378467i \(0.876451\pi\)
\(3\) 0.618034 0.356822 0.178411 0.983956i \(-0.442904\pi\)
0.178411 + 0.983956i \(0.442904\pi\)
\(4\) 4.85410 2.42705
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −1.61803 −0.660560
\(7\) 1.00000 0.377964
\(8\) −7.47214 −2.64180
\(9\) −2.61803 −0.872678
\(10\) −2.61803 −0.827895
\(11\) 0 0
\(12\) 3.00000 0.866025
\(13\) 3.23607 0.897524 0.448762 0.893651i \(-0.351865\pi\)
0.448762 + 0.893651i \(0.351865\pi\)
\(14\) −2.61803 −0.699699
\(15\) 0.618034 0.159576
\(16\) 9.85410 2.46353
\(17\) −8.09017 −1.96215 −0.981077 0.193617i \(-0.937978\pi\)
−0.981077 + 0.193617i \(0.937978\pi\)
\(18\) 6.85410 1.61553
\(19\) −6.23607 −1.43065 −0.715326 0.698791i \(-0.753722\pi\)
−0.715326 + 0.698791i \(0.753722\pi\)
\(20\) 4.85410 1.08541
\(21\) 0.618034 0.134866
\(22\) 0 0
\(23\) −6.09017 −1.26989 −0.634944 0.772558i \(-0.718977\pi\)
−0.634944 + 0.772558i \(0.718977\pi\)
\(24\) −4.61803 −0.942652
\(25\) −4.00000 −0.800000
\(26\) −8.47214 −1.66152
\(27\) −3.47214 −0.668213
\(28\) 4.85410 0.917339
\(29\) −2.38197 −0.442320 −0.221160 0.975238i \(-0.570984\pi\)
−0.221160 + 0.975238i \(0.570984\pi\)
\(30\) −1.61803 −0.295411
\(31\) 0.236068 0.0423991 0.0211995 0.999775i \(-0.493251\pi\)
0.0211995 + 0.999775i \(0.493251\pi\)
\(32\) −10.8541 −1.91875
\(33\) 0 0
\(34\) 21.1803 3.63240
\(35\) 1.00000 0.169031
\(36\) −12.7082 −2.11803
\(37\) −2.47214 −0.406417 −0.203208 0.979136i \(-0.565137\pi\)
−0.203208 + 0.979136i \(0.565137\pi\)
\(38\) 16.3262 2.64847
\(39\) 2.00000 0.320256
\(40\) −7.47214 −1.18145
\(41\) 11.1803 1.74608 0.873038 0.487652i \(-0.162147\pi\)
0.873038 + 0.487652i \(0.162147\pi\)
\(42\) −1.61803 −0.249668
\(43\) 7.56231 1.15324 0.576620 0.817012i \(-0.304371\pi\)
0.576620 + 0.817012i \(0.304371\pi\)
\(44\) 0 0
\(45\) −2.61803 −0.390273
\(46\) 15.9443 2.35085
\(47\) −4.38197 −0.639175 −0.319588 0.947557i \(-0.603544\pi\)
−0.319588 + 0.947557i \(0.603544\pi\)
\(48\) 6.09017 0.879040
\(49\) 1.00000 0.142857
\(50\) 10.4721 1.48098
\(51\) −5.00000 −0.700140
\(52\) 15.7082 2.17834
\(53\) −4.61803 −0.634336 −0.317168 0.948369i \(-0.602732\pi\)
−0.317168 + 0.948369i \(0.602732\pi\)
\(54\) 9.09017 1.23702
\(55\) 0 0
\(56\) −7.47214 −0.998506
\(57\) −3.85410 −0.510488
\(58\) 6.23607 0.818836
\(59\) −0.0901699 −0.0117391 −0.00586956 0.999983i \(-0.501868\pi\)
−0.00586956 + 0.999983i \(0.501868\pi\)
\(60\) 3.00000 0.387298
\(61\) −5.38197 −0.689090 −0.344545 0.938770i \(-0.611967\pi\)
−0.344545 + 0.938770i \(0.611967\pi\)
\(62\) −0.618034 −0.0784904
\(63\) −2.61803 −0.329841
\(64\) 8.70820 1.08853
\(65\) 3.23607 0.401385
\(66\) 0 0
\(67\) 7.32624 0.895042 0.447521 0.894273i \(-0.352307\pi\)
0.447521 + 0.894273i \(0.352307\pi\)
\(68\) −39.2705 −4.76225
\(69\) −3.76393 −0.453124
\(70\) −2.61803 −0.312915
\(71\) −4.90983 −0.582690 −0.291345 0.956618i \(-0.594103\pi\)
−0.291345 + 0.956618i \(0.594103\pi\)
\(72\) 19.5623 2.30544
\(73\) −9.76393 −1.14278 −0.571391 0.820678i \(-0.693596\pi\)
−0.571391 + 0.820678i \(0.693596\pi\)
\(74\) 6.47214 0.752371
\(75\) −2.47214 −0.285458
\(76\) −30.2705 −3.47227
\(77\) 0 0
\(78\) −5.23607 −0.592868
\(79\) 8.61803 0.969605 0.484802 0.874624i \(-0.338892\pi\)
0.484802 + 0.874624i \(0.338892\pi\)
\(80\) 9.85410 1.10172
\(81\) 5.70820 0.634245
\(82\) −29.2705 −3.23239
\(83\) −10.7082 −1.17538 −0.587689 0.809087i \(-0.699962\pi\)
−0.587689 + 0.809087i \(0.699962\pi\)
\(84\) 3.00000 0.327327
\(85\) −8.09017 −0.877502
\(86\) −19.7984 −2.13491
\(87\) −1.47214 −0.157830
\(88\) 0 0
\(89\) 0.145898 0.0154652 0.00773258 0.999970i \(-0.497539\pi\)
0.00773258 + 0.999970i \(0.497539\pi\)
\(90\) 6.85410 0.722486
\(91\) 3.23607 0.339232
\(92\) −29.5623 −3.08208
\(93\) 0.145898 0.0151289
\(94\) 11.4721 1.18326
\(95\) −6.23607 −0.639807
\(96\) −6.70820 −0.684653
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) −2.61803 −0.264461
\(99\) 0 0
\(100\) −19.4164 −1.94164
\(101\) 9.79837 0.974975 0.487487 0.873130i \(-0.337914\pi\)
0.487487 + 0.873130i \(0.337914\pi\)
\(102\) 13.0902 1.29612
\(103\) 2.14590 0.211442 0.105721 0.994396i \(-0.466285\pi\)
0.105721 + 0.994396i \(0.466285\pi\)
\(104\) −24.1803 −2.37108
\(105\) 0.618034 0.0603139
\(106\) 12.0902 1.17430
\(107\) −10.7082 −1.03520 −0.517601 0.855622i \(-0.673175\pi\)
−0.517601 + 0.855622i \(0.673175\pi\)
\(108\) −16.8541 −1.62179
\(109\) −10.4721 −1.00305 −0.501524 0.865144i \(-0.667227\pi\)
−0.501524 + 0.865144i \(0.667227\pi\)
\(110\) 0 0
\(111\) −1.52786 −0.145018
\(112\) 9.85410 0.931125
\(113\) 6.85410 0.644780 0.322390 0.946607i \(-0.395514\pi\)
0.322390 + 0.946607i \(0.395514\pi\)
\(114\) 10.0902 0.945031
\(115\) −6.09017 −0.567911
\(116\) −11.5623 −1.07353
\(117\) −8.47214 −0.783249
\(118\) 0.236068 0.0217318
\(119\) −8.09017 −0.741625
\(120\) −4.61803 −0.421567
\(121\) 0 0
\(122\) 14.0902 1.27566
\(123\) 6.90983 0.623038
\(124\) 1.14590 0.102905
\(125\) −9.00000 −0.804984
\(126\) 6.85410 0.610612
\(127\) −14.9443 −1.32609 −0.663045 0.748580i \(-0.730736\pi\)
−0.663045 + 0.748580i \(0.730736\pi\)
\(128\) −1.09017 −0.0963583
\(129\) 4.67376 0.411502
\(130\) −8.47214 −0.743055
\(131\) 1.05573 0.0922394 0.0461197 0.998936i \(-0.485314\pi\)
0.0461197 + 0.998936i \(0.485314\pi\)
\(132\) 0 0
\(133\) −6.23607 −0.540736
\(134\) −19.1803 −1.65693
\(135\) −3.47214 −0.298834
\(136\) 60.4508 5.18362
\(137\) 0.326238 0.0278724 0.0139362 0.999903i \(-0.495564\pi\)
0.0139362 + 0.999903i \(0.495564\pi\)
\(138\) 9.85410 0.838837
\(139\) −5.94427 −0.504187 −0.252093 0.967703i \(-0.581119\pi\)
−0.252093 + 0.967703i \(0.581119\pi\)
\(140\) 4.85410 0.410246
\(141\) −2.70820 −0.228072
\(142\) 12.8541 1.07869
\(143\) 0 0
\(144\) −25.7984 −2.14986
\(145\) −2.38197 −0.197812
\(146\) 25.5623 2.11555
\(147\) 0.618034 0.0509746
\(148\) −12.0000 −0.986394
\(149\) −2.14590 −0.175799 −0.0878994 0.996129i \(-0.528015\pi\)
−0.0878994 + 0.996129i \(0.528015\pi\)
\(150\) 6.47214 0.528448
\(151\) 17.9443 1.46028 0.730142 0.683295i \(-0.239454\pi\)
0.730142 + 0.683295i \(0.239454\pi\)
\(152\) 46.5967 3.77950
\(153\) 21.1803 1.71233
\(154\) 0 0
\(155\) 0.236068 0.0189614
\(156\) 9.70820 0.777278
\(157\) −15.8885 −1.26804 −0.634022 0.773315i \(-0.718597\pi\)
−0.634022 + 0.773315i \(0.718597\pi\)
\(158\) −22.5623 −1.79496
\(159\) −2.85410 −0.226345
\(160\) −10.8541 −0.858092
\(161\) −6.09017 −0.479973
\(162\) −14.9443 −1.17413
\(163\) 4.70820 0.368775 0.184387 0.982854i \(-0.440970\pi\)
0.184387 + 0.982854i \(0.440970\pi\)
\(164\) 54.2705 4.23781
\(165\) 0 0
\(166\) 28.0344 2.17589
\(167\) −2.47214 −0.191300 −0.0956498 0.995415i \(-0.530493\pi\)
−0.0956498 + 0.995415i \(0.530493\pi\)
\(168\) −4.61803 −0.356289
\(169\) −2.52786 −0.194451
\(170\) 21.1803 1.62446
\(171\) 16.3262 1.24850
\(172\) 36.7082 2.79897
\(173\) −17.6180 −1.33947 −0.669737 0.742598i \(-0.733593\pi\)
−0.669737 + 0.742598i \(0.733593\pi\)
\(174\) 3.85410 0.292179
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −0.0557281 −0.00418878
\(178\) −0.381966 −0.0286296
\(179\) −8.23607 −0.615593 −0.307796 0.951452i \(-0.599592\pi\)
−0.307796 + 0.951452i \(0.599592\pi\)
\(180\) −12.7082 −0.947214
\(181\) 19.4164 1.44321 0.721605 0.692305i \(-0.243405\pi\)
0.721605 + 0.692305i \(0.243405\pi\)
\(182\) −8.47214 −0.627996
\(183\) −3.32624 −0.245883
\(184\) 45.5066 3.35479
\(185\) −2.47214 −0.181755
\(186\) −0.381966 −0.0280071
\(187\) 0 0
\(188\) −21.2705 −1.55131
\(189\) −3.47214 −0.252561
\(190\) 16.3262 1.18443
\(191\) 20.2361 1.46423 0.732115 0.681181i \(-0.238533\pi\)
0.732115 + 0.681181i \(0.238533\pi\)
\(192\) 5.38197 0.388410
\(193\) 0.326238 0.0234831 0.0117416 0.999931i \(-0.496262\pi\)
0.0117416 + 0.999931i \(0.496262\pi\)
\(194\) −18.3262 −1.31575
\(195\) 2.00000 0.143223
\(196\) 4.85410 0.346722
\(197\) −17.7082 −1.26166 −0.630829 0.775922i \(-0.717285\pi\)
−0.630829 + 0.775922i \(0.717285\pi\)
\(198\) 0 0
\(199\) −3.76393 −0.266818 −0.133409 0.991061i \(-0.542592\pi\)
−0.133409 + 0.991061i \(0.542592\pi\)
\(200\) 29.8885 2.11344
\(201\) 4.52786 0.319371
\(202\) −25.6525 −1.80490
\(203\) −2.38197 −0.167181
\(204\) −24.2705 −1.69928
\(205\) 11.1803 0.780869
\(206\) −5.61803 −0.391427
\(207\) 15.9443 1.10820
\(208\) 31.8885 2.21107
\(209\) 0 0
\(210\) −1.61803 −0.111655
\(211\) 13.3820 0.921253 0.460626 0.887594i \(-0.347625\pi\)
0.460626 + 0.887594i \(0.347625\pi\)
\(212\) −22.4164 −1.53957
\(213\) −3.03444 −0.207917
\(214\) 28.0344 1.91639
\(215\) 7.56231 0.515745
\(216\) 25.9443 1.76528
\(217\) 0.236068 0.0160253
\(218\) 27.4164 1.85687
\(219\) −6.03444 −0.407770
\(220\) 0 0
\(221\) −26.1803 −1.76108
\(222\) 4.00000 0.268462
\(223\) −27.0344 −1.81036 −0.905180 0.425028i \(-0.860264\pi\)
−0.905180 + 0.425028i \(0.860264\pi\)
\(224\) −10.8541 −0.725220
\(225\) 10.4721 0.698142
\(226\) −17.9443 −1.19364
\(227\) 13.0344 0.865126 0.432563 0.901604i \(-0.357609\pi\)
0.432563 + 0.901604i \(0.357609\pi\)
\(228\) −18.7082 −1.23898
\(229\) 11.2361 0.742500 0.371250 0.928533i \(-0.378929\pi\)
0.371250 + 0.928533i \(0.378929\pi\)
\(230\) 15.9443 1.05133
\(231\) 0 0
\(232\) 17.7984 1.16852
\(233\) −2.58359 −0.169257 −0.0846284 0.996413i \(-0.526970\pi\)
−0.0846284 + 0.996413i \(0.526970\pi\)
\(234\) 22.1803 1.44997
\(235\) −4.38197 −0.285848
\(236\) −0.437694 −0.0284915
\(237\) 5.32624 0.345976
\(238\) 21.1803 1.37292
\(239\) −5.90983 −0.382275 −0.191138 0.981563i \(-0.561218\pi\)
−0.191138 + 0.981563i \(0.561218\pi\)
\(240\) 6.09017 0.393119
\(241\) −17.2705 −1.11249 −0.556246 0.831018i \(-0.687759\pi\)
−0.556246 + 0.831018i \(0.687759\pi\)
\(242\) 0 0
\(243\) 13.9443 0.894525
\(244\) −26.1246 −1.67246
\(245\) 1.00000 0.0638877
\(246\) −18.0902 −1.15339
\(247\) −20.1803 −1.28404
\(248\) −1.76393 −0.112010
\(249\) −6.61803 −0.419401
\(250\) 23.5623 1.49021
\(251\) 23.0000 1.45175 0.725874 0.687828i \(-0.241436\pi\)
0.725874 + 0.687828i \(0.241436\pi\)
\(252\) −12.7082 −0.800542
\(253\) 0 0
\(254\) 39.1246 2.45490
\(255\) −5.00000 −0.313112
\(256\) −14.5623 −0.910144
\(257\) 11.5623 0.721237 0.360618 0.932713i \(-0.382566\pi\)
0.360618 + 0.932713i \(0.382566\pi\)
\(258\) −12.2361 −0.761784
\(259\) −2.47214 −0.153611
\(260\) 15.7082 0.974181
\(261\) 6.23607 0.386003
\(262\) −2.76393 −0.170756
\(263\) 23.1246 1.42592 0.712962 0.701202i \(-0.247353\pi\)
0.712962 + 0.701202i \(0.247353\pi\)
\(264\) 0 0
\(265\) −4.61803 −0.283684
\(266\) 16.3262 1.00103
\(267\) 0.0901699 0.00551831
\(268\) 35.5623 2.17231
\(269\) 10.1459 0.618606 0.309303 0.950963i \(-0.399904\pi\)
0.309303 + 0.950963i \(0.399904\pi\)
\(270\) 9.09017 0.553210
\(271\) 19.7984 1.20267 0.601333 0.798999i \(-0.294637\pi\)
0.601333 + 0.798999i \(0.294637\pi\)
\(272\) −79.7214 −4.83382
\(273\) 2.00000 0.121046
\(274\) −0.854102 −0.0515982
\(275\) 0 0
\(276\) −18.2705 −1.09976
\(277\) −6.03444 −0.362574 −0.181287 0.983430i \(-0.558026\pi\)
−0.181287 + 0.983430i \(0.558026\pi\)
\(278\) 15.5623 0.933365
\(279\) −0.618034 −0.0370007
\(280\) −7.47214 −0.446546
\(281\) −2.81966 −0.168207 −0.0841034 0.996457i \(-0.526803\pi\)
−0.0841034 + 0.996457i \(0.526803\pi\)
\(282\) 7.09017 0.422213
\(283\) −5.94427 −0.353350 −0.176675 0.984269i \(-0.556534\pi\)
−0.176675 + 0.984269i \(0.556534\pi\)
\(284\) −23.8328 −1.41422
\(285\) −3.85410 −0.228297
\(286\) 0 0
\(287\) 11.1803 0.659955
\(288\) 28.4164 1.67445
\(289\) 48.4508 2.85005
\(290\) 6.23607 0.366195
\(291\) 4.32624 0.253609
\(292\) −47.3951 −2.77359
\(293\) −11.0000 −0.642627 −0.321313 0.946973i \(-0.604124\pi\)
−0.321313 + 0.946973i \(0.604124\pi\)
\(294\) −1.61803 −0.0943657
\(295\) −0.0901699 −0.00524990
\(296\) 18.4721 1.07367
\(297\) 0 0
\(298\) 5.61803 0.325444
\(299\) −19.7082 −1.13975
\(300\) −12.0000 −0.692820
\(301\) 7.56231 0.435884
\(302\) −46.9787 −2.70332
\(303\) 6.05573 0.347892
\(304\) −61.4508 −3.52445
\(305\) −5.38197 −0.308170
\(306\) −55.4508 −3.16991
\(307\) −0.819660 −0.0467805 −0.0233902 0.999726i \(-0.507446\pi\)
−0.0233902 + 0.999726i \(0.507446\pi\)
\(308\) 0 0
\(309\) 1.32624 0.0754470
\(310\) −0.618034 −0.0351020
\(311\) 9.00000 0.510343 0.255172 0.966896i \(-0.417868\pi\)
0.255172 + 0.966896i \(0.417868\pi\)
\(312\) −14.9443 −0.846053
\(313\) 2.67376 0.151130 0.0755650 0.997141i \(-0.475924\pi\)
0.0755650 + 0.997141i \(0.475924\pi\)
\(314\) 41.5967 2.34744
\(315\) −2.61803 −0.147510
\(316\) 41.8328 2.35328
\(317\) −24.5623 −1.37956 −0.689778 0.724021i \(-0.742292\pi\)
−0.689778 + 0.724021i \(0.742292\pi\)
\(318\) 7.47214 0.419017
\(319\) 0 0
\(320\) 8.70820 0.486803
\(321\) −6.61803 −0.369383
\(322\) 15.9443 0.888540
\(323\) 50.4508 2.80716
\(324\) 27.7082 1.53934
\(325\) −12.9443 −0.718019
\(326\) −12.3262 −0.682687
\(327\) −6.47214 −0.357910
\(328\) −83.5410 −4.61278
\(329\) −4.38197 −0.241586
\(330\) 0 0
\(331\) −16.1803 −0.889352 −0.444676 0.895692i \(-0.646681\pi\)
−0.444676 + 0.895692i \(0.646681\pi\)
\(332\) −51.9787 −2.85270
\(333\) 6.47214 0.354671
\(334\) 6.47214 0.354140
\(335\) 7.32624 0.400275
\(336\) 6.09017 0.332246
\(337\) 8.05573 0.438823 0.219412 0.975632i \(-0.429586\pi\)
0.219412 + 0.975632i \(0.429586\pi\)
\(338\) 6.61803 0.359974
\(339\) 4.23607 0.230072
\(340\) −39.2705 −2.12974
\(341\) 0 0
\(342\) −42.7426 −2.31126
\(343\) 1.00000 0.0539949
\(344\) −56.5066 −3.04663
\(345\) −3.76393 −0.202643
\(346\) 46.1246 2.47967
\(347\) 33.6525 1.80656 0.903280 0.429052i \(-0.141152\pi\)
0.903280 + 0.429052i \(0.141152\pi\)
\(348\) −7.14590 −0.383060
\(349\) 8.20163 0.439023 0.219511 0.975610i \(-0.429554\pi\)
0.219511 + 0.975610i \(0.429554\pi\)
\(350\) 10.4721 0.559759
\(351\) −11.2361 −0.599737
\(352\) 0 0
\(353\) 12.9098 0.687121 0.343560 0.939131i \(-0.388367\pi\)
0.343560 + 0.939131i \(0.388367\pi\)
\(354\) 0.145898 0.00775439
\(355\) −4.90983 −0.260587
\(356\) 0.708204 0.0375347
\(357\) −5.00000 −0.264628
\(358\) 21.5623 1.13960
\(359\) −13.3820 −0.706273 −0.353137 0.935572i \(-0.614885\pi\)
−0.353137 + 0.935572i \(0.614885\pi\)
\(360\) 19.5623 1.03102
\(361\) 19.8885 1.04677
\(362\) −50.8328 −2.67171
\(363\) 0 0
\(364\) 15.7082 0.823334
\(365\) −9.76393 −0.511068
\(366\) 8.70820 0.455185
\(367\) −20.8328 −1.08746 −0.543732 0.839259i \(-0.682989\pi\)
−0.543732 + 0.839259i \(0.682989\pi\)
\(368\) −60.0132 −3.12840
\(369\) −29.2705 −1.52376
\(370\) 6.47214 0.336470
\(371\) −4.61803 −0.239756
\(372\) 0.708204 0.0367187
\(373\) −35.5623 −1.84135 −0.920673 0.390334i \(-0.872359\pi\)
−0.920673 + 0.390334i \(0.872359\pi\)
\(374\) 0 0
\(375\) −5.56231 −0.287236
\(376\) 32.7426 1.68857
\(377\) −7.70820 −0.396993
\(378\) 9.09017 0.467548
\(379\) −26.0689 −1.33907 −0.669534 0.742781i \(-0.733506\pi\)
−0.669534 + 0.742781i \(0.733506\pi\)
\(380\) −30.2705 −1.55284
\(381\) −9.23607 −0.473178
\(382\) −52.9787 −2.71063
\(383\) 14.5967 0.745859 0.372929 0.927860i \(-0.378353\pi\)
0.372929 + 0.927860i \(0.378353\pi\)
\(384\) −0.673762 −0.0343828
\(385\) 0 0
\(386\) −0.854102 −0.0434726
\(387\) −19.7984 −1.00641
\(388\) 33.9787 1.72501
\(389\) −11.8197 −0.599281 −0.299640 0.954052i \(-0.596867\pi\)
−0.299640 + 0.954052i \(0.596867\pi\)
\(390\) −5.23607 −0.265139
\(391\) 49.2705 2.49172
\(392\) −7.47214 −0.377400
\(393\) 0.652476 0.0329131
\(394\) 46.3607 2.33562
\(395\) 8.61803 0.433620
\(396\) 0 0
\(397\) −23.1803 −1.16339 −0.581694 0.813408i \(-0.697610\pi\)
−0.581694 + 0.813408i \(0.697610\pi\)
\(398\) 9.85410 0.493941
\(399\) −3.85410 −0.192946
\(400\) −39.4164 −1.97082
\(401\) −26.4721 −1.32196 −0.660978 0.750406i \(-0.729858\pi\)
−0.660978 + 0.750406i \(0.729858\pi\)
\(402\) −11.8541 −0.591229
\(403\) 0.763932 0.0380542
\(404\) 47.5623 2.36631
\(405\) 5.70820 0.283643
\(406\) 6.23607 0.309491
\(407\) 0 0
\(408\) 37.3607 1.84963
\(409\) −4.29180 −0.212216 −0.106108 0.994355i \(-0.533839\pi\)
−0.106108 + 0.994355i \(0.533839\pi\)
\(410\) −29.2705 −1.44557
\(411\) 0.201626 0.00994548
\(412\) 10.4164 0.513180
\(413\) −0.0901699 −0.00443697
\(414\) −41.7426 −2.05154
\(415\) −10.7082 −0.525645
\(416\) −35.1246 −1.72213
\(417\) −3.67376 −0.179905
\(418\) 0 0
\(419\) −5.88854 −0.287674 −0.143837 0.989601i \(-0.545944\pi\)
−0.143837 + 0.989601i \(0.545944\pi\)
\(420\) 3.00000 0.146385
\(421\) 5.23607 0.255190 0.127595 0.991826i \(-0.459274\pi\)
0.127595 + 0.991826i \(0.459274\pi\)
\(422\) −35.0344 −1.70545
\(423\) 11.4721 0.557794
\(424\) 34.5066 1.67579
\(425\) 32.3607 1.56972
\(426\) 7.94427 0.384901
\(427\) −5.38197 −0.260452
\(428\) −51.9787 −2.51249
\(429\) 0 0
\(430\) −19.7984 −0.954762
\(431\) −7.14590 −0.344206 −0.172103 0.985079i \(-0.555056\pi\)
−0.172103 + 0.985079i \(0.555056\pi\)
\(432\) −34.2148 −1.64616
\(433\) −38.0344 −1.82782 −0.913909 0.405918i \(-0.866952\pi\)
−0.913909 + 0.405918i \(0.866952\pi\)
\(434\) −0.618034 −0.0296666
\(435\) −1.47214 −0.0705835
\(436\) −50.8328 −2.43445
\(437\) 37.9787 1.81677
\(438\) 15.7984 0.754876
\(439\) 5.61803 0.268134 0.134067 0.990972i \(-0.457196\pi\)
0.134067 + 0.990972i \(0.457196\pi\)
\(440\) 0 0
\(441\) −2.61803 −0.124668
\(442\) 68.5410 3.26016
\(443\) −8.09017 −0.384376 −0.192188 0.981358i \(-0.561558\pi\)
−0.192188 + 0.981358i \(0.561558\pi\)
\(444\) −7.41641 −0.351967
\(445\) 0.145898 0.00691623
\(446\) 70.7771 3.35139
\(447\) −1.32624 −0.0627289
\(448\) 8.70820 0.411424
\(449\) −3.32624 −0.156975 −0.0784874 0.996915i \(-0.525009\pi\)
−0.0784874 + 0.996915i \(0.525009\pi\)
\(450\) −27.4164 −1.29242
\(451\) 0 0
\(452\) 33.2705 1.56491
\(453\) 11.0902 0.521062
\(454\) −34.1246 −1.60155
\(455\) 3.23607 0.151709
\(456\) 28.7984 1.34861
\(457\) −6.85410 −0.320621 −0.160311 0.987067i \(-0.551250\pi\)
−0.160311 + 0.987067i \(0.551250\pi\)
\(458\) −29.4164 −1.37454
\(459\) 28.0902 1.31114
\(460\) −29.5623 −1.37835
\(461\) 19.5066 0.908512 0.454256 0.890871i \(-0.349905\pi\)
0.454256 + 0.890871i \(0.349905\pi\)
\(462\) 0 0
\(463\) 17.7639 0.825560 0.412780 0.910831i \(-0.364558\pi\)
0.412780 + 0.910831i \(0.364558\pi\)
\(464\) −23.4721 −1.08967
\(465\) 0.145898 0.00676586
\(466\) 6.76393 0.313333
\(467\) 33.5066 1.55050 0.775250 0.631655i \(-0.217624\pi\)
0.775250 + 0.631655i \(0.217624\pi\)
\(468\) −41.1246 −1.90099
\(469\) 7.32624 0.338294
\(470\) 11.4721 0.529170
\(471\) −9.81966 −0.452466
\(472\) 0.673762 0.0310124
\(473\) 0 0
\(474\) −13.9443 −0.640482
\(475\) 24.9443 1.14452
\(476\) −39.2705 −1.79996
\(477\) 12.0902 0.553571
\(478\) 15.4721 0.707679
\(479\) 26.2361 1.19876 0.599378 0.800466i \(-0.295415\pi\)
0.599378 + 0.800466i \(0.295415\pi\)
\(480\) −6.70820 −0.306186
\(481\) −8.00000 −0.364769
\(482\) 45.2148 2.05948
\(483\) −3.76393 −0.171265
\(484\) 0 0
\(485\) 7.00000 0.317854
\(486\) −36.5066 −1.65597
\(487\) 16.5967 0.752070 0.376035 0.926605i \(-0.377287\pi\)
0.376035 + 0.926605i \(0.377287\pi\)
\(488\) 40.2148 1.82044
\(489\) 2.90983 0.131587
\(490\) −2.61803 −0.118271
\(491\) 28.8541 1.30217 0.651084 0.759006i \(-0.274315\pi\)
0.651084 + 0.759006i \(0.274315\pi\)
\(492\) 33.5410 1.51215
\(493\) 19.2705 0.867900
\(494\) 52.8328 2.37706
\(495\) 0 0
\(496\) 2.32624 0.104451
\(497\) −4.90983 −0.220236
\(498\) 17.3262 0.776407
\(499\) 1.09017 0.0488027 0.0244014 0.999702i \(-0.492232\pi\)
0.0244014 + 0.999702i \(0.492232\pi\)
\(500\) −43.6869 −1.95374
\(501\) −1.52786 −0.0682599
\(502\) −60.2148 −2.68752
\(503\) 18.8328 0.839714 0.419857 0.907590i \(-0.362080\pi\)
0.419857 + 0.907590i \(0.362080\pi\)
\(504\) 19.5623 0.871374
\(505\) 9.79837 0.436022
\(506\) 0 0
\(507\) −1.56231 −0.0693844
\(508\) −72.5410 −3.21849
\(509\) 31.7426 1.40697 0.703484 0.710711i \(-0.251627\pi\)
0.703484 + 0.710711i \(0.251627\pi\)
\(510\) 13.0902 0.579642
\(511\) −9.76393 −0.431931
\(512\) 40.3050 1.78124
\(513\) 21.6525 0.955980
\(514\) −30.2705 −1.33517
\(515\) 2.14590 0.0945596
\(516\) 22.6869 0.998736
\(517\) 0 0
\(518\) 6.47214 0.284369
\(519\) −10.8885 −0.477954
\(520\) −24.1803 −1.06038
\(521\) −7.76393 −0.340144 −0.170072 0.985432i \(-0.554400\pi\)
−0.170072 + 0.985432i \(0.554400\pi\)
\(522\) −16.3262 −0.714580
\(523\) −3.56231 −0.155769 −0.0778844 0.996962i \(-0.524817\pi\)
−0.0778844 + 0.996962i \(0.524817\pi\)
\(524\) 5.12461 0.223870
\(525\) −2.47214 −0.107893
\(526\) −60.5410 −2.63971
\(527\) −1.90983 −0.0831935
\(528\) 0 0
\(529\) 14.0902 0.612616
\(530\) 12.0902 0.525163
\(531\) 0.236068 0.0102445
\(532\) −30.2705 −1.31239
\(533\) 36.1803 1.56714
\(534\) −0.236068 −0.0102157
\(535\) −10.7082 −0.462956
\(536\) −54.7426 −2.36452
\(537\) −5.09017 −0.219657
\(538\) −26.5623 −1.14518
\(539\) 0 0
\(540\) −16.8541 −0.725285
\(541\) −12.7082 −0.546368 −0.273184 0.961962i \(-0.588077\pi\)
−0.273184 + 0.961962i \(0.588077\pi\)
\(542\) −51.8328 −2.22641
\(543\) 12.0000 0.514969
\(544\) 87.8115 3.76489
\(545\) −10.4721 −0.448577
\(546\) −5.23607 −0.224083
\(547\) −35.2705 −1.50806 −0.754029 0.656841i \(-0.771892\pi\)
−0.754029 + 0.656841i \(0.771892\pi\)
\(548\) 1.58359 0.0676477
\(549\) 14.0902 0.601354
\(550\) 0 0
\(551\) 14.8541 0.632806
\(552\) 28.1246 1.19706
\(553\) 8.61803 0.366476
\(554\) 15.7984 0.671209
\(555\) −1.52786 −0.0648542
\(556\) −28.8541 −1.22369
\(557\) −14.2361 −0.603202 −0.301601 0.953434i \(-0.597521\pi\)
−0.301601 + 0.953434i \(0.597521\pi\)
\(558\) 1.61803 0.0684968
\(559\) 24.4721 1.03506
\(560\) 9.85410 0.416412
\(561\) 0 0
\(562\) 7.38197 0.311389
\(563\) 14.9787 0.631278 0.315639 0.948879i \(-0.397781\pi\)
0.315639 + 0.948879i \(0.397781\pi\)
\(564\) −13.1459 −0.553542
\(565\) 6.85410 0.288354
\(566\) 15.5623 0.654133
\(567\) 5.70820 0.239722
\(568\) 36.6869 1.53935
\(569\) −25.3050 −1.06084 −0.530419 0.847735i \(-0.677966\pi\)
−0.530419 + 0.847735i \(0.677966\pi\)
\(570\) 10.0902 0.422631
\(571\) −28.3050 −1.18453 −0.592263 0.805745i \(-0.701765\pi\)
−0.592263 + 0.805745i \(0.701765\pi\)
\(572\) 0 0
\(573\) 12.5066 0.522470
\(574\) −29.2705 −1.22173
\(575\) 24.3607 1.01591
\(576\) −22.7984 −0.949932
\(577\) −21.6525 −0.901404 −0.450702 0.892674i \(-0.648826\pi\)
−0.450702 + 0.892674i \(0.648826\pi\)
\(578\) −126.846 −5.27610
\(579\) 0.201626 0.00837930
\(580\) −11.5623 −0.480099
\(581\) −10.7082 −0.444251
\(582\) −11.3262 −0.469488
\(583\) 0 0
\(584\) 72.9574 3.01900
\(585\) −8.47214 −0.350280
\(586\) 28.7984 1.18965
\(587\) 1.00000 0.0412744 0.0206372 0.999787i \(-0.493431\pi\)
0.0206372 + 0.999787i \(0.493431\pi\)
\(588\) 3.00000 0.123718
\(589\) −1.47214 −0.0606583
\(590\) 0.236068 0.00971876
\(591\) −10.9443 −0.450187
\(592\) −24.3607 −1.00122
\(593\) −29.1246 −1.19600 −0.598002 0.801494i \(-0.704039\pi\)
−0.598002 + 0.801494i \(0.704039\pi\)
\(594\) 0 0
\(595\) −8.09017 −0.331665
\(596\) −10.4164 −0.426673
\(597\) −2.32624 −0.0952066
\(598\) 51.5967 2.10995
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 18.4721 0.754122
\(601\) 0.291796 0.0119026 0.00595130 0.999982i \(-0.498106\pi\)
0.00595130 + 0.999982i \(0.498106\pi\)
\(602\) −19.7984 −0.806921
\(603\) −19.1803 −0.781084
\(604\) 87.1033 3.54418
\(605\) 0 0
\(606\) −15.8541 −0.644029
\(607\) 44.7984 1.81831 0.909155 0.416458i \(-0.136729\pi\)
0.909155 + 0.416458i \(0.136729\pi\)
\(608\) 67.6869 2.74507
\(609\) −1.47214 −0.0596540
\(610\) 14.0902 0.570494
\(611\) −14.1803 −0.573675
\(612\) 102.812 4.15591
\(613\) 34.8885 1.40914 0.704568 0.709637i \(-0.251141\pi\)
0.704568 + 0.709637i \(0.251141\pi\)
\(614\) 2.14590 0.0866014
\(615\) 6.90983 0.278631
\(616\) 0 0
\(617\) −17.4164 −0.701158 −0.350579 0.936533i \(-0.614015\pi\)
−0.350579 + 0.936533i \(0.614015\pi\)
\(618\) −3.47214 −0.139670
\(619\) −26.2918 −1.05676 −0.528378 0.849009i \(-0.677200\pi\)
−0.528378 + 0.849009i \(0.677200\pi\)
\(620\) 1.14590 0.0460204
\(621\) 21.1459 0.848556
\(622\) −23.5623 −0.944762
\(623\) 0.145898 0.00584528
\(624\) 19.7082 0.788960
\(625\) 11.0000 0.440000
\(626\) −7.00000 −0.279776
\(627\) 0 0
\(628\) −77.1246 −3.07761
\(629\) 20.0000 0.797452
\(630\) 6.85410 0.273074
\(631\) −35.8328 −1.42648 −0.713241 0.700919i \(-0.752773\pi\)
−0.713241 + 0.700919i \(0.752773\pi\)
\(632\) −64.3951 −2.56150
\(633\) 8.27051 0.328723
\(634\) 64.3050 2.55388
\(635\) −14.9443 −0.593045
\(636\) −13.8541 −0.549351
\(637\) 3.23607 0.128218
\(638\) 0 0
\(639\) 12.8541 0.508500
\(640\) −1.09017 −0.0430928
\(641\) 0.347524 0.0137264 0.00686319 0.999976i \(-0.497815\pi\)
0.00686319 + 0.999976i \(0.497815\pi\)
\(642\) 17.3262 0.683812
\(643\) 28.4164 1.12063 0.560317 0.828278i \(-0.310679\pi\)
0.560317 + 0.828278i \(0.310679\pi\)
\(644\) −29.5623 −1.16492
\(645\) 4.67376 0.184029
\(646\) −132.082 −5.19670
\(647\) −16.1803 −0.636115 −0.318057 0.948071i \(-0.603030\pi\)
−0.318057 + 0.948071i \(0.603030\pi\)
\(648\) −42.6525 −1.67555
\(649\) 0 0
\(650\) 33.8885 1.32922
\(651\) 0.145898 0.00571819
\(652\) 22.8541 0.895036
\(653\) 13.7639 0.538624 0.269312 0.963053i \(-0.413204\pi\)
0.269312 + 0.963053i \(0.413204\pi\)
\(654\) 16.9443 0.662573
\(655\) 1.05573 0.0412507
\(656\) 110.172 4.30150
\(657\) 25.5623 0.997281
\(658\) 11.4721 0.447230
\(659\) 22.5279 0.877561 0.438780 0.898594i \(-0.355411\pi\)
0.438780 + 0.898594i \(0.355411\pi\)
\(660\) 0 0
\(661\) −14.4377 −0.561561 −0.280781 0.959772i \(-0.590593\pi\)
−0.280781 + 0.959772i \(0.590593\pi\)
\(662\) 42.3607 1.64639
\(663\) −16.1803 −0.628392
\(664\) 80.0132 3.10511
\(665\) −6.23607 −0.241824
\(666\) −16.9443 −0.656577
\(667\) 14.5066 0.561697
\(668\) −12.0000 −0.464294
\(669\) −16.7082 −0.645976
\(670\) −19.1803 −0.741001
\(671\) 0 0
\(672\) −6.70820 −0.258775
\(673\) −33.4164 −1.28811 −0.644054 0.764980i \(-0.722749\pi\)
−0.644054 + 0.764980i \(0.722749\pi\)
\(674\) −21.0902 −0.812363
\(675\) 13.8885 0.534570
\(676\) −12.2705 −0.471943
\(677\) −2.72949 −0.104903 −0.0524514 0.998623i \(-0.516703\pi\)
−0.0524514 + 0.998623i \(0.516703\pi\)
\(678\) −11.0902 −0.425915
\(679\) 7.00000 0.268635
\(680\) 60.4508 2.31818
\(681\) 8.05573 0.308696
\(682\) 0 0
\(683\) 38.5066 1.47341 0.736707 0.676213i \(-0.236380\pi\)
0.736707 + 0.676213i \(0.236380\pi\)
\(684\) 79.2492 3.03017
\(685\) 0.326238 0.0124649
\(686\) −2.61803 −0.0999570
\(687\) 6.94427 0.264940
\(688\) 74.5197 2.84104
\(689\) −14.9443 −0.569331
\(690\) 9.85410 0.375139
\(691\) −48.5410 −1.84659 −0.923294 0.384095i \(-0.874514\pi\)
−0.923294 + 0.384095i \(0.874514\pi\)
\(692\) −85.5197 −3.25097
\(693\) 0 0
\(694\) −88.1033 −3.34436
\(695\) −5.94427 −0.225479
\(696\) 11.0000 0.416954
\(697\) −90.4508 −3.42607
\(698\) −21.4721 −0.812732
\(699\) −1.59675 −0.0603945
\(700\) −19.4164 −0.733871
\(701\) −19.5066 −0.736753 −0.368377 0.929677i \(-0.620086\pi\)
−0.368377 + 0.929677i \(0.620086\pi\)
\(702\) 29.4164 1.11025
\(703\) 15.4164 0.581441
\(704\) 0 0
\(705\) −2.70820 −0.101997
\(706\) −33.7984 −1.27202
\(707\) 9.79837 0.368506
\(708\) −0.270510 −0.0101664
\(709\) 22.0344 0.827521 0.413760 0.910386i \(-0.364215\pi\)
0.413760 + 0.910386i \(0.364215\pi\)
\(710\) 12.8541 0.482406
\(711\) −22.5623 −0.846153
\(712\) −1.09017 −0.0408558
\(713\) −1.43769 −0.0538421
\(714\) 13.0902 0.489887
\(715\) 0 0
\(716\) −39.9787 −1.49407
\(717\) −3.65248 −0.136404
\(718\) 35.0344 1.30747
\(719\) 32.8885 1.22654 0.613268 0.789875i \(-0.289855\pi\)
0.613268 + 0.789875i \(0.289855\pi\)
\(720\) −25.7984 −0.961449
\(721\) 2.14590 0.0799174
\(722\) −52.0689 −1.93780
\(723\) −10.6738 −0.396961
\(724\) 94.2492 3.50274
\(725\) 9.52786 0.353856
\(726\) 0 0
\(727\) 43.4508 1.61150 0.805751 0.592254i \(-0.201762\pi\)
0.805751 + 0.592254i \(0.201762\pi\)
\(728\) −24.1803 −0.896183
\(729\) −8.50658 −0.315058
\(730\) 25.5623 0.946103
\(731\) −61.1803 −2.26284
\(732\) −16.1459 −0.596770
\(733\) 35.0557 1.29481 0.647406 0.762145i \(-0.275854\pi\)
0.647406 + 0.762145i \(0.275854\pi\)
\(734\) 54.5410 2.01315
\(735\) 0.618034 0.0227965
\(736\) 66.1033 2.43660
\(737\) 0 0
\(738\) 76.6312 2.82083
\(739\) −13.6180 −0.500947 −0.250474 0.968123i \(-0.580586\pi\)
−0.250474 + 0.968123i \(0.580586\pi\)
\(740\) −12.0000 −0.441129
\(741\) −12.4721 −0.458175
\(742\) 12.0902 0.443844
\(743\) −26.0902 −0.957156 −0.478578 0.878045i \(-0.658848\pi\)
−0.478578 + 0.878045i \(0.658848\pi\)
\(744\) −1.09017 −0.0399676
\(745\) −2.14590 −0.0786196
\(746\) 93.1033 3.40875
\(747\) 28.0344 1.02573
\(748\) 0 0
\(749\) −10.7082 −0.391269
\(750\) 14.5623 0.531740
\(751\) −21.3050 −0.777429 −0.388714 0.921358i \(-0.627081\pi\)
−0.388714 + 0.921358i \(0.627081\pi\)
\(752\) −43.1803 −1.57462
\(753\) 14.2148 0.518015
\(754\) 20.1803 0.734925
\(755\) 17.9443 0.653059
\(756\) −16.8541 −0.612978
\(757\) −19.2361 −0.699147 −0.349573 0.936909i \(-0.613673\pi\)
−0.349573 + 0.936909i \(0.613673\pi\)
\(758\) 68.2492 2.47892
\(759\) 0 0
\(760\) 46.5967 1.69024
\(761\) −47.3050 −1.71480 −0.857402 0.514648i \(-0.827923\pi\)
−0.857402 + 0.514648i \(0.827923\pi\)
\(762\) 24.1803 0.875961
\(763\) −10.4721 −0.379117
\(764\) 98.2279 3.55376
\(765\) 21.1803 0.765777
\(766\) −38.2148 −1.38076
\(767\) −0.291796 −0.0105361
\(768\) −9.00000 −0.324760
\(769\) 28.4377 1.02549 0.512745 0.858541i \(-0.328629\pi\)
0.512745 + 0.858541i \(0.328629\pi\)
\(770\) 0 0
\(771\) 7.14590 0.257353
\(772\) 1.58359 0.0569947
\(773\) 35.7639 1.28634 0.643170 0.765724i \(-0.277619\pi\)
0.643170 + 0.765724i \(0.277619\pi\)
\(774\) 51.8328 1.86309
\(775\) −0.944272 −0.0339192
\(776\) −52.3050 −1.87764
\(777\) −1.52786 −0.0548118
\(778\) 30.9443 1.10941
\(779\) −69.7214 −2.49803
\(780\) 9.70820 0.347609
\(781\) 0 0
\(782\) −128.992 −4.61274
\(783\) 8.27051 0.295564
\(784\) 9.85410 0.351932
\(785\) −15.8885 −0.567086
\(786\) −1.70820 −0.0609296
\(787\) 0.437694 0.0156021 0.00780105 0.999970i \(-0.497517\pi\)
0.00780105 + 0.999970i \(0.497517\pi\)
\(788\) −85.9574 −3.06211
\(789\) 14.2918 0.508801
\(790\) −22.5623 −0.802731
\(791\) 6.85410 0.243704
\(792\) 0 0
\(793\) −17.4164 −0.618475
\(794\) 60.6869 2.15370
\(795\) −2.85410 −0.101225
\(796\) −18.2705 −0.647581
\(797\) 27.7639 0.983449 0.491724 0.870751i \(-0.336367\pi\)
0.491724 + 0.870751i \(0.336367\pi\)
\(798\) 10.0902 0.357188
\(799\) 35.4508 1.25416
\(800\) 43.4164 1.53500
\(801\) −0.381966 −0.0134961
\(802\) 69.3050 2.44724
\(803\) 0 0
\(804\) 21.9787 0.775129
\(805\) −6.09017 −0.214650
\(806\) −2.00000 −0.0704470
\(807\) 6.27051 0.220732
\(808\) −73.2148 −2.57569
\(809\) −5.50658 −0.193601 −0.0968005 0.995304i \(-0.530861\pi\)
−0.0968005 + 0.995304i \(0.530861\pi\)
\(810\) −14.9443 −0.525088
\(811\) 54.8328 1.92544 0.962720 0.270499i \(-0.0871886\pi\)
0.962720 + 0.270499i \(0.0871886\pi\)
\(812\) −11.5623 −0.405757
\(813\) 12.2361 0.429138
\(814\) 0 0
\(815\) 4.70820 0.164921
\(816\) −49.2705 −1.72481
\(817\) −47.1591 −1.64989
\(818\) 11.2361 0.392860
\(819\) −8.47214 −0.296040
\(820\) 54.2705 1.89521
\(821\) −50.6656 −1.76824 −0.884121 0.467257i \(-0.845242\pi\)
−0.884121 + 0.467257i \(0.845242\pi\)
\(822\) −0.527864 −0.0184114
\(823\) 3.88854 0.135546 0.0677731 0.997701i \(-0.478411\pi\)
0.0677731 + 0.997701i \(0.478411\pi\)
\(824\) −16.0344 −0.558586
\(825\) 0 0
\(826\) 0.236068 0.00821386
\(827\) 29.5967 1.02918 0.514590 0.857436i \(-0.327944\pi\)
0.514590 + 0.857436i \(0.327944\pi\)
\(828\) 77.3951 2.68967
\(829\) −1.02129 −0.0354707 −0.0177354 0.999843i \(-0.505646\pi\)
−0.0177354 + 0.999843i \(0.505646\pi\)
\(830\) 28.0344 0.973090
\(831\) −3.72949 −0.129375
\(832\) 28.1803 0.976978
\(833\) −8.09017 −0.280308
\(834\) 9.61803 0.333045
\(835\) −2.47214 −0.0855518
\(836\) 0 0
\(837\) −0.819660 −0.0283316
\(838\) 15.4164 0.532551
\(839\) −18.5967 −0.642031 −0.321016 0.947074i \(-0.604024\pi\)
−0.321016 + 0.947074i \(0.604024\pi\)
\(840\) −4.61803 −0.159337
\(841\) −23.3262 −0.804353
\(842\) −13.7082 −0.472416
\(843\) −1.74265 −0.0600199
\(844\) 64.9574 2.23593
\(845\) −2.52786 −0.0869612
\(846\) −30.0344 −1.03261
\(847\) 0 0
\(848\) −45.5066 −1.56270
\(849\) −3.67376 −0.126083
\(850\) −84.7214 −2.90592
\(851\) 15.0557 0.516104
\(852\) −14.7295 −0.504624
\(853\) −46.0902 −1.57810 −0.789049 0.614331i \(-0.789426\pi\)
−0.789049 + 0.614331i \(0.789426\pi\)
\(854\) 14.0902 0.482156
\(855\) 16.3262 0.558346
\(856\) 80.0132 2.73479
\(857\) −3.90983 −0.133557 −0.0667786 0.997768i \(-0.521272\pi\)
−0.0667786 + 0.997768i \(0.521272\pi\)
\(858\) 0 0
\(859\) 1.43769 0.0490535 0.0245267 0.999699i \(-0.492192\pi\)
0.0245267 + 0.999699i \(0.492192\pi\)
\(860\) 36.7082 1.25174
\(861\) 6.90983 0.235486
\(862\) 18.7082 0.637204
\(863\) −30.0557 −1.02311 −0.511554 0.859251i \(-0.670930\pi\)
−0.511554 + 0.859251i \(0.670930\pi\)
\(864\) 37.6869 1.28213
\(865\) −17.6180 −0.599031
\(866\) 99.5755 3.38371
\(867\) 29.9443 1.01696
\(868\) 1.14590 0.0388943
\(869\) 0 0
\(870\) 3.85410 0.130666
\(871\) 23.7082 0.803322
\(872\) 78.2492 2.64985
\(873\) −18.3262 −0.620249
\(874\) −99.4296 −3.36326
\(875\) −9.00000 −0.304256
\(876\) −29.2918 −0.989678
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) −14.7082 −0.496378
\(879\) −6.79837 −0.229303
\(880\) 0 0
\(881\) 20.8541 0.702593 0.351296 0.936264i \(-0.385741\pi\)
0.351296 + 0.936264i \(0.385741\pi\)
\(882\) 6.85410 0.230790
\(883\) −46.6312 −1.56926 −0.784632 0.619961i \(-0.787148\pi\)
−0.784632 + 0.619961i \(0.787148\pi\)
\(884\) −127.082 −4.27423
\(885\) −0.0557281 −0.00187328
\(886\) 21.1803 0.711567
\(887\) 5.78522 0.194249 0.0971243 0.995272i \(-0.469036\pi\)
0.0971243 + 0.995272i \(0.469036\pi\)
\(888\) 11.4164 0.383110
\(889\) −14.9443 −0.501215
\(890\) −0.381966 −0.0128035
\(891\) 0 0
\(892\) −131.228 −4.39384
\(893\) 27.3262 0.914438
\(894\) 3.47214 0.116126
\(895\) −8.23607 −0.275301
\(896\) −1.09017 −0.0364200
\(897\) −12.1803 −0.406690
\(898\) 8.70820 0.290597
\(899\) −0.562306 −0.0187540
\(900\) 50.8328 1.69443
\(901\) 37.3607 1.24466
\(902\) 0 0
\(903\) 4.67376 0.155533
\(904\) −51.2148 −1.70338
\(905\) 19.4164 0.645423
\(906\) −29.0344 −0.964605
\(907\) −20.2361 −0.671928 −0.335964 0.941875i \(-0.609062\pi\)
−0.335964 + 0.941875i \(0.609062\pi\)
\(908\) 63.2705 2.09971
\(909\) −25.6525 −0.850839
\(910\) −8.47214 −0.280849
\(911\) −22.8197 −0.756049 −0.378025 0.925796i \(-0.623397\pi\)
−0.378025 + 0.925796i \(0.623397\pi\)
\(912\) −37.9787 −1.25760
\(913\) 0 0
\(914\) 17.9443 0.593544
\(915\) −3.32624 −0.109962
\(916\) 54.5410 1.80209
\(917\) 1.05573 0.0348632
\(918\) −73.5410 −2.42722
\(919\) 20.8885 0.689049 0.344525 0.938777i \(-0.388040\pi\)
0.344525 + 0.938777i \(0.388040\pi\)
\(920\) 45.5066 1.50031
\(921\) −0.506578 −0.0166923
\(922\) −51.0689 −1.68186
\(923\) −15.8885 −0.522978
\(924\) 0 0
\(925\) 9.88854 0.325133
\(926\) −46.5066 −1.52830
\(927\) −5.61803 −0.184520
\(928\) 25.8541 0.848702
\(929\) 24.0000 0.787414 0.393707 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) −0.381966 −0.0125252
\(931\) −6.23607 −0.204379
\(932\) −12.5410 −0.410795
\(933\) 5.56231 0.182102
\(934\) −87.7214 −2.87033
\(935\) 0 0
\(936\) 63.3050 2.06919
\(937\) 37.7214 1.23230 0.616152 0.787628i \(-0.288691\pi\)
0.616152 + 0.787628i \(0.288691\pi\)
\(938\) −19.1803 −0.626260
\(939\) 1.65248 0.0539265
\(940\) −21.2705 −0.693768
\(941\) −31.0689 −1.01282 −0.506408 0.862294i \(-0.669027\pi\)
−0.506408 + 0.862294i \(0.669027\pi\)
\(942\) 25.7082 0.837619
\(943\) −68.0902 −2.21732
\(944\) −0.888544 −0.0289196
\(945\) −3.47214 −0.112949
\(946\) 0 0
\(947\) 19.7082 0.640431 0.320215 0.947345i \(-0.396245\pi\)
0.320215 + 0.947345i \(0.396245\pi\)
\(948\) 25.8541 0.839702
\(949\) −31.5967 −1.02567
\(950\) −65.3050 −2.11877
\(951\) −15.1803 −0.492256
\(952\) 60.4508 1.95922
\(953\) −1.76393 −0.0571394 −0.0285697 0.999592i \(-0.509095\pi\)
−0.0285697 + 0.999592i \(0.509095\pi\)
\(954\) −31.6525 −1.02479
\(955\) 20.2361 0.654824
\(956\) −28.6869 −0.927801
\(957\) 0 0
\(958\) −68.6869 −2.21917
\(959\) 0.326238 0.0105348
\(960\) 5.38197 0.173702
\(961\) −30.9443 −0.998202
\(962\) 20.9443 0.675270
\(963\) 28.0344 0.903397
\(964\) −83.8328 −2.70007
\(965\) 0.326238 0.0105020
\(966\) 9.85410 0.317051
\(967\) −28.4508 −0.914918 −0.457459 0.889231i \(-0.651240\pi\)
−0.457459 + 0.889231i \(0.651240\pi\)
\(968\) 0 0
\(969\) 31.1803 1.00166
\(970\) −18.3262 −0.588420
\(971\) 52.2492 1.67676 0.838379 0.545088i \(-0.183504\pi\)
0.838379 + 0.545088i \(0.183504\pi\)
\(972\) 67.6869 2.17106
\(973\) −5.94427 −0.190565
\(974\) −43.4508 −1.39226
\(975\) −8.00000 −0.256205
\(976\) −53.0344 −1.69759
\(977\) −33.1803 −1.06153 −0.530767 0.847518i \(-0.678096\pi\)
−0.530767 + 0.847518i \(0.678096\pi\)
\(978\) −7.61803 −0.243598
\(979\) 0 0
\(980\) 4.85410 0.155059
\(981\) 27.4164 0.875339
\(982\) −75.5410 −2.41061
\(983\) −14.6180 −0.466243 −0.233121 0.972448i \(-0.574894\pi\)
−0.233121 + 0.972448i \(0.574894\pi\)
\(984\) −51.6312 −1.64594
\(985\) −17.7082 −0.564230
\(986\) −50.4508 −1.60668
\(987\) −2.70820 −0.0862031
\(988\) −97.9574 −3.11644
\(989\) −46.0557 −1.46449
\(990\) 0 0
\(991\) −34.2705 −1.08864 −0.544319 0.838878i \(-0.683212\pi\)
−0.544319 + 0.838878i \(0.683212\pi\)
\(992\) −2.56231 −0.0813533
\(993\) −10.0000 −0.317340
\(994\) 12.8541 0.407707
\(995\) −3.76393 −0.119325
\(996\) −32.1246 −1.01791
\(997\) 26.8673 0.850895 0.425447 0.904983i \(-0.360117\pi\)
0.425447 + 0.904983i \(0.360117\pi\)
\(998\) −2.85410 −0.0903450
\(999\) 8.58359 0.271573
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 847.2.a.d.1.1 2
3.2 odd 2 7623.2.a.bx.1.2 2
7.6 odd 2 5929.2.a.i.1.1 2
11.2 odd 10 847.2.f.c.323.1 4
11.3 even 5 847.2.f.d.372.1 4
11.4 even 5 847.2.f.d.148.1 4
11.5 even 5 847.2.f.l.729.1 4
11.6 odd 10 847.2.f.c.729.1 4
11.7 odd 10 847.2.f.j.148.1 4
11.8 odd 10 847.2.f.j.372.1 4
11.9 even 5 847.2.f.l.323.1 4
11.10 odd 2 847.2.a.h.1.2 yes 2
33.32 even 2 7623.2.a.t.1.1 2
77.76 even 2 5929.2.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.d.1.1 2 1.1 even 1 trivial
847.2.a.h.1.2 yes 2 11.10 odd 2
847.2.f.c.323.1 4 11.2 odd 10
847.2.f.c.729.1 4 11.6 odd 10
847.2.f.d.148.1 4 11.4 even 5
847.2.f.d.372.1 4 11.3 even 5
847.2.f.j.148.1 4 11.7 odd 10
847.2.f.j.372.1 4 11.8 odd 10
847.2.f.l.323.1 4 11.9 even 5
847.2.f.l.729.1 4 11.5 even 5
5929.2.a.i.1.1 2 7.6 odd 2
5929.2.a.s.1.2 2 77.76 even 2
7623.2.a.t.1.1 2 33.32 even 2
7623.2.a.bx.1.2 2 3.2 odd 2