Properties

Label 74.2.a.b.1.1
Level $74$
Weight $2$
Character 74.1
Self dual yes
Analytic conductor $0.591$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\chi\) \(=\) 74.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} +3.85410 q^{5} -1.61803 q^{6} -3.23607 q^{7} +1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} +3.85410 q^{5} -1.61803 q^{6} -3.23607 q^{7} +1.00000 q^{8} -0.381966 q^{9} +3.85410 q^{10} -1.38197 q^{11} -1.61803 q^{12} -2.85410 q^{13} -3.23607 q^{14} -6.23607 q^{15} +1.00000 q^{16} -4.47214 q^{17} -0.381966 q^{18} +4.47214 q^{19} +3.85410 q^{20} +5.23607 q^{21} -1.38197 q^{22} +2.85410 q^{23} -1.61803 q^{24} +9.85410 q^{25} -2.85410 q^{26} +5.47214 q^{27} -3.23607 q^{28} -9.32624 q^{29} -6.23607 q^{30} +7.38197 q^{31} +1.00000 q^{32} +2.23607 q^{33} -4.47214 q^{34} -12.4721 q^{35} -0.381966 q^{36} -1.00000 q^{37} +4.47214 q^{38} +4.61803 q^{39} +3.85410 q^{40} +9.61803 q^{41} +5.23607 q^{42} -5.23607 q^{43} -1.38197 q^{44} -1.47214 q^{45} +2.85410 q^{46} -1.23607 q^{47} -1.61803 q^{48} +3.47214 q^{49} +9.85410 q^{50} +7.23607 q^{51} -2.85410 q^{52} +0.472136 q^{53} +5.47214 q^{54} -5.32624 q^{55} -3.23607 q^{56} -7.23607 q^{57} -9.32624 q^{58} -4.76393 q^{59} -6.23607 q^{60} +10.6180 q^{61} +7.38197 q^{62} +1.23607 q^{63} +1.00000 q^{64} -11.0000 q^{65} +2.23607 q^{66} +1.09017 q^{67} -4.47214 q^{68} -4.61803 q^{69} -12.4721 q^{70} +2.94427 q^{71} -0.381966 q^{72} +7.09017 q^{73} -1.00000 q^{74} -15.9443 q^{75} +4.47214 q^{76} +4.47214 q^{77} +4.61803 q^{78} -8.56231 q^{79} +3.85410 q^{80} -7.70820 q^{81} +9.61803 q^{82} -14.4721 q^{83} +5.23607 q^{84} -17.2361 q^{85} -5.23607 q^{86} +15.0902 q^{87} -1.38197 q^{88} -1.52786 q^{89} -1.47214 q^{90} +9.23607 q^{91} +2.85410 q^{92} -11.9443 q^{93} -1.23607 q^{94} +17.2361 q^{95} -1.61803 q^{96} -0.472136 q^{97} +3.47214 q^{98} +0.527864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - 3 q^{9} + q^{10} - 5 q^{11} - q^{12} + q^{13} - 2 q^{14} - 8 q^{15} + 2 q^{16} - 3 q^{18} + q^{20} + 6 q^{21} - 5 q^{22} - q^{23} - q^{24} + 13 q^{25} + q^{26} + 2 q^{27} - 2 q^{28} - 3 q^{29} - 8 q^{30} + 17 q^{31} + 2 q^{32} - 16 q^{35} - 3 q^{36} - 2 q^{37} + 7 q^{39} + q^{40} + 17 q^{41} + 6 q^{42} - 6 q^{43} - 5 q^{44} + 6 q^{45} - q^{46} + 2 q^{47} - q^{48} - 2 q^{49} + 13 q^{50} + 10 q^{51} + q^{52} - 8 q^{53} + 2 q^{54} + 5 q^{55} - 2 q^{56} - 10 q^{57} - 3 q^{58} - 14 q^{59} - 8 q^{60} + 19 q^{61} + 17 q^{62} - 2 q^{63} + 2 q^{64} - 22 q^{65} - 9 q^{67} - 7 q^{69} - 16 q^{70} - 12 q^{71} - 3 q^{72} + 3 q^{73} - 2 q^{74} - 14 q^{75} + 7 q^{78} + 3 q^{79} + q^{80} - 2 q^{81} + 17 q^{82} - 20 q^{83} + 6 q^{84} - 30 q^{85} - 6 q^{86} + 19 q^{87} - 5 q^{88} - 12 q^{89} + 6 q^{90} + 14 q^{91} - q^{92} - 6 q^{93} + 2 q^{94} + 30 q^{95} - q^{96} + 8 q^{97} - 2 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.61803 −0.934172 −0.467086 0.884212i \(-0.654696\pi\)
−0.467086 + 0.884212i \(0.654696\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.85410 1.72361 0.861803 0.507242i \(-0.169335\pi\)
0.861803 + 0.507242i \(0.169335\pi\)
\(6\) −1.61803 −0.660560
\(7\) −3.23607 −1.22312 −0.611559 0.791199i \(-0.709457\pi\)
−0.611559 + 0.791199i \(0.709457\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.381966 −0.127322
\(10\) 3.85410 1.21877
\(11\) −1.38197 −0.416678 −0.208339 0.978057i \(-0.566806\pi\)
−0.208339 + 0.978057i \(0.566806\pi\)
\(12\) −1.61803 −0.467086
\(13\) −2.85410 −0.791585 −0.395793 0.918340i \(-0.629530\pi\)
−0.395793 + 0.918340i \(0.629530\pi\)
\(14\) −3.23607 −0.864876
\(15\) −6.23607 −1.61015
\(16\) 1.00000 0.250000
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) −0.381966 −0.0900303
\(19\) 4.47214 1.02598 0.512989 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) 3.85410 0.861803
\(21\) 5.23607 1.14260
\(22\) −1.38197 −0.294636
\(23\) 2.85410 0.595121 0.297561 0.954703i \(-0.403827\pi\)
0.297561 + 0.954703i \(0.403827\pi\)
\(24\) −1.61803 −0.330280
\(25\) 9.85410 1.97082
\(26\) −2.85410 −0.559735
\(27\) 5.47214 1.05311
\(28\) −3.23607 −0.611559
\(29\) −9.32624 −1.73184 −0.865919 0.500183i \(-0.833266\pi\)
−0.865919 + 0.500183i \(0.833266\pi\)
\(30\) −6.23607 −1.13855
\(31\) 7.38197 1.32584 0.662920 0.748690i \(-0.269317\pi\)
0.662920 + 0.748690i \(0.269317\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.23607 0.389249
\(34\) −4.47214 −0.766965
\(35\) −12.4721 −2.10818
\(36\) −0.381966 −0.0636610
\(37\) −1.00000 −0.164399
\(38\) 4.47214 0.725476
\(39\) 4.61803 0.739477
\(40\) 3.85410 0.609387
\(41\) 9.61803 1.50208 0.751042 0.660254i \(-0.229551\pi\)
0.751042 + 0.660254i \(0.229551\pi\)
\(42\) 5.23607 0.807943
\(43\) −5.23607 −0.798493 −0.399246 0.916844i \(-0.630728\pi\)
−0.399246 + 0.916844i \(0.630728\pi\)
\(44\) −1.38197 −0.208339
\(45\) −1.47214 −0.219453
\(46\) 2.85410 0.420814
\(47\) −1.23607 −0.180299 −0.0901495 0.995928i \(-0.528734\pi\)
−0.0901495 + 0.995928i \(0.528734\pi\)
\(48\) −1.61803 −0.233543
\(49\) 3.47214 0.496019
\(50\) 9.85410 1.39358
\(51\) 7.23607 1.01325
\(52\) −2.85410 −0.395793
\(53\) 0.472136 0.0648529 0.0324264 0.999474i \(-0.489677\pi\)
0.0324264 + 0.999474i \(0.489677\pi\)
\(54\) 5.47214 0.744663
\(55\) −5.32624 −0.718190
\(56\) −3.23607 −0.432438
\(57\) −7.23607 −0.958441
\(58\) −9.32624 −1.22460
\(59\) −4.76393 −0.620211 −0.310106 0.950702i \(-0.600364\pi\)
−0.310106 + 0.950702i \(0.600364\pi\)
\(60\) −6.23607 −0.805073
\(61\) 10.6180 1.35950 0.679750 0.733444i \(-0.262088\pi\)
0.679750 + 0.733444i \(0.262088\pi\)
\(62\) 7.38197 0.937511
\(63\) 1.23607 0.155730
\(64\) 1.00000 0.125000
\(65\) −11.0000 −1.36438
\(66\) 2.23607 0.275241
\(67\) 1.09017 0.133185 0.0665927 0.997780i \(-0.478787\pi\)
0.0665927 + 0.997780i \(0.478787\pi\)
\(68\) −4.47214 −0.542326
\(69\) −4.61803 −0.555946
\(70\) −12.4721 −1.49071
\(71\) 2.94427 0.349421 0.174710 0.984620i \(-0.444101\pi\)
0.174710 + 0.984620i \(0.444101\pi\)
\(72\) −0.381966 −0.0450151
\(73\) 7.09017 0.829842 0.414921 0.909858i \(-0.363809\pi\)
0.414921 + 0.909858i \(0.363809\pi\)
\(74\) −1.00000 −0.116248
\(75\) −15.9443 −1.84109
\(76\) 4.47214 0.512989
\(77\) 4.47214 0.509647
\(78\) 4.61803 0.522889
\(79\) −8.56231 −0.963335 −0.481667 0.876354i \(-0.659969\pi\)
−0.481667 + 0.876354i \(0.659969\pi\)
\(80\) 3.85410 0.430902
\(81\) −7.70820 −0.856467
\(82\) 9.61803 1.06213
\(83\) −14.4721 −1.58852 −0.794262 0.607576i \(-0.792142\pi\)
−0.794262 + 0.607576i \(0.792142\pi\)
\(84\) 5.23607 0.571302
\(85\) −17.2361 −1.86951
\(86\) −5.23607 −0.564620
\(87\) 15.0902 1.61784
\(88\) −1.38197 −0.147318
\(89\) −1.52786 −0.161953 −0.0809766 0.996716i \(-0.525804\pi\)
−0.0809766 + 0.996716i \(0.525804\pi\)
\(90\) −1.47214 −0.155177
\(91\) 9.23607 0.968203
\(92\) 2.85410 0.297561
\(93\) −11.9443 −1.23856
\(94\) −1.23607 −0.127491
\(95\) 17.2361 1.76838
\(96\) −1.61803 −0.165140
\(97\) −0.472136 −0.0479381 −0.0239691 0.999713i \(-0.507630\pi\)
−0.0239691 + 0.999713i \(0.507630\pi\)
\(98\) 3.47214 0.350739
\(99\) 0.527864 0.0530523
\(100\) 9.85410 0.985410
\(101\) 3.52786 0.351036 0.175518 0.984476i \(-0.443840\pi\)
0.175518 + 0.984476i \(0.443840\pi\)
\(102\) 7.23607 0.716477
\(103\) 17.2705 1.70171 0.850857 0.525397i \(-0.176083\pi\)
0.850857 + 0.525397i \(0.176083\pi\)
\(104\) −2.85410 −0.279868
\(105\) 20.1803 1.96940
\(106\) 0.472136 0.0458579
\(107\) −7.32624 −0.708254 −0.354127 0.935197i \(-0.615222\pi\)
−0.354127 + 0.935197i \(0.615222\pi\)
\(108\) 5.47214 0.526557
\(109\) 2.94427 0.282010 0.141005 0.990009i \(-0.454967\pi\)
0.141005 + 0.990009i \(0.454967\pi\)
\(110\) −5.32624 −0.507837
\(111\) 1.61803 0.153577
\(112\) −3.23607 −0.305780
\(113\) −6.94427 −0.653262 −0.326631 0.945152i \(-0.605913\pi\)
−0.326631 + 0.945152i \(0.605913\pi\)
\(114\) −7.23607 −0.677720
\(115\) 11.0000 1.02576
\(116\) −9.32624 −0.865919
\(117\) 1.09017 0.100786
\(118\) −4.76393 −0.438555
\(119\) 14.4721 1.32666
\(120\) −6.23607 −0.569273
\(121\) −9.09017 −0.826379
\(122\) 10.6180 0.961312
\(123\) −15.5623 −1.40321
\(124\) 7.38197 0.662920
\(125\) 18.7082 1.67331
\(126\) 1.23607 0.110118
\(127\) −8.47214 −0.751780 −0.375890 0.926664i \(-0.622663\pi\)
−0.375890 + 0.926664i \(0.622663\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.47214 0.745930
\(130\) −11.0000 −0.964764
\(131\) −22.6525 −1.97916 −0.989578 0.143998i \(-0.954004\pi\)
−0.989578 + 0.143998i \(0.954004\pi\)
\(132\) 2.23607 0.194625
\(133\) −14.4721 −1.25489
\(134\) 1.09017 0.0941763
\(135\) 21.0902 1.81515
\(136\) −4.47214 −0.383482
\(137\) 3.67376 0.313871 0.156935 0.987609i \(-0.449839\pi\)
0.156935 + 0.987609i \(0.449839\pi\)
\(138\) −4.61803 −0.393113
\(139\) 4.85410 0.411720 0.205860 0.978581i \(-0.434001\pi\)
0.205860 + 0.978581i \(0.434001\pi\)
\(140\) −12.4721 −1.05409
\(141\) 2.00000 0.168430
\(142\) 2.94427 0.247078
\(143\) 3.94427 0.329837
\(144\) −0.381966 −0.0318305
\(145\) −35.9443 −2.98501
\(146\) 7.09017 0.586787
\(147\) −5.61803 −0.463368
\(148\) −1.00000 −0.0821995
\(149\) 16.1803 1.32555 0.662773 0.748821i \(-0.269380\pi\)
0.662773 + 0.748821i \(0.269380\pi\)
\(150\) −15.9443 −1.30184
\(151\) 4.29180 0.349261 0.174631 0.984634i \(-0.444127\pi\)
0.174631 + 0.984634i \(0.444127\pi\)
\(152\) 4.47214 0.362738
\(153\) 1.70820 0.138100
\(154\) 4.47214 0.360375
\(155\) 28.4508 2.28523
\(156\) 4.61803 0.369739
\(157\) −16.4721 −1.31462 −0.657310 0.753620i \(-0.728306\pi\)
−0.657310 + 0.753620i \(0.728306\pi\)
\(158\) −8.56231 −0.681180
\(159\) −0.763932 −0.0605838
\(160\) 3.85410 0.304694
\(161\) −9.23607 −0.727904
\(162\) −7.70820 −0.605614
\(163\) −3.52786 −0.276324 −0.138162 0.990410i \(-0.544119\pi\)
−0.138162 + 0.990410i \(0.544119\pi\)
\(164\) 9.61803 0.751042
\(165\) 8.61803 0.670913
\(166\) −14.4721 −1.12326
\(167\) 13.8541 1.07206 0.536031 0.844198i \(-0.319923\pi\)
0.536031 + 0.844198i \(0.319923\pi\)
\(168\) 5.23607 0.403971
\(169\) −4.85410 −0.373392
\(170\) −17.2361 −1.32195
\(171\) −1.70820 −0.130630
\(172\) −5.23607 −0.399246
\(173\) −0.472136 −0.0358958 −0.0179479 0.999839i \(-0.505713\pi\)
−0.0179479 + 0.999839i \(0.505713\pi\)
\(174\) 15.0902 1.14398
\(175\) −31.8885 −2.41055
\(176\) −1.38197 −0.104170
\(177\) 7.70820 0.579384
\(178\) −1.52786 −0.114518
\(179\) −12.6525 −0.945690 −0.472845 0.881146i \(-0.656773\pi\)
−0.472845 + 0.881146i \(0.656773\pi\)
\(180\) −1.47214 −0.109727
\(181\) 14.4721 1.07571 0.537853 0.843039i \(-0.319236\pi\)
0.537853 + 0.843039i \(0.319236\pi\)
\(182\) 9.23607 0.684623
\(183\) −17.1803 −1.27001
\(184\) 2.85410 0.210407
\(185\) −3.85410 −0.283359
\(186\) −11.9443 −0.875797
\(187\) 6.18034 0.451951
\(188\) −1.23607 −0.0901495
\(189\) −17.7082 −1.28808
\(190\) 17.2361 1.25044
\(191\) −7.09017 −0.513027 −0.256513 0.966541i \(-0.582574\pi\)
−0.256513 + 0.966541i \(0.582574\pi\)
\(192\) −1.61803 −0.116772
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −0.472136 −0.0338974
\(195\) 17.7984 1.27457
\(196\) 3.47214 0.248010
\(197\) −7.52786 −0.536338 −0.268169 0.963372i \(-0.586419\pi\)
−0.268169 + 0.963372i \(0.586419\pi\)
\(198\) 0.527864 0.0375137
\(199\) 3.05573 0.216615 0.108307 0.994117i \(-0.465457\pi\)
0.108307 + 0.994117i \(0.465457\pi\)
\(200\) 9.85410 0.696790
\(201\) −1.76393 −0.124418
\(202\) 3.52786 0.248220
\(203\) 30.1803 2.11824
\(204\) 7.23607 0.506626
\(205\) 37.0689 2.58900
\(206\) 17.2705 1.20329
\(207\) −1.09017 −0.0757720
\(208\) −2.85410 −0.197896
\(209\) −6.18034 −0.427503
\(210\) 20.1803 1.39258
\(211\) 11.2705 0.775894 0.387947 0.921682i \(-0.373184\pi\)
0.387947 + 0.921682i \(0.373184\pi\)
\(212\) 0.472136 0.0324264
\(213\) −4.76393 −0.326419
\(214\) −7.32624 −0.500811
\(215\) −20.1803 −1.37629
\(216\) 5.47214 0.372332
\(217\) −23.8885 −1.62166
\(218\) 2.94427 0.199411
\(219\) −11.4721 −0.775215
\(220\) −5.32624 −0.359095
\(221\) 12.7639 0.858595
\(222\) 1.61803 0.108595
\(223\) 14.1803 0.949586 0.474793 0.880098i \(-0.342523\pi\)
0.474793 + 0.880098i \(0.342523\pi\)
\(224\) −3.23607 −0.216219
\(225\) −3.76393 −0.250929
\(226\) −6.94427 −0.461926
\(227\) −4.29180 −0.284857 −0.142428 0.989805i \(-0.545491\pi\)
−0.142428 + 0.989805i \(0.545491\pi\)
\(228\) −7.23607 −0.479220
\(229\) −23.1246 −1.52812 −0.764059 0.645147i \(-0.776796\pi\)
−0.764059 + 0.645147i \(0.776796\pi\)
\(230\) 11.0000 0.725319
\(231\) −7.23607 −0.476098
\(232\) −9.32624 −0.612298
\(233\) −6.56231 −0.429911 −0.214955 0.976624i \(-0.568961\pi\)
−0.214955 + 0.976624i \(0.568961\pi\)
\(234\) 1.09017 0.0712666
\(235\) −4.76393 −0.310765
\(236\) −4.76393 −0.310106
\(237\) 13.8541 0.899921
\(238\) 14.4721 0.938089
\(239\) 9.85410 0.637409 0.318704 0.947854i \(-0.396752\pi\)
0.318704 + 0.947854i \(0.396752\pi\)
\(240\) −6.23607 −0.402536
\(241\) −1.52786 −0.0984184 −0.0492092 0.998788i \(-0.515670\pi\)
−0.0492092 + 0.998788i \(0.515670\pi\)
\(242\) −9.09017 −0.584338
\(243\) −3.94427 −0.253025
\(244\) 10.6180 0.679750
\(245\) 13.3820 0.854942
\(246\) −15.5623 −0.992216
\(247\) −12.7639 −0.812150
\(248\) 7.38197 0.468755
\(249\) 23.4164 1.48395
\(250\) 18.7082 1.18321
\(251\) −20.9443 −1.32199 −0.660995 0.750390i \(-0.729866\pi\)
−0.660995 + 0.750390i \(0.729866\pi\)
\(252\) 1.23607 0.0778650
\(253\) −3.94427 −0.247974
\(254\) −8.47214 −0.531589
\(255\) 27.8885 1.74645
\(256\) 1.00000 0.0625000
\(257\) −1.05573 −0.0658545 −0.0329273 0.999458i \(-0.510483\pi\)
−0.0329273 + 0.999458i \(0.510483\pi\)
\(258\) 8.47214 0.527452
\(259\) 3.23607 0.201079
\(260\) −11.0000 −0.682191
\(261\) 3.56231 0.220501
\(262\) −22.6525 −1.39947
\(263\) −13.2361 −0.816171 −0.408085 0.912944i \(-0.633803\pi\)
−0.408085 + 0.912944i \(0.633803\pi\)
\(264\) 2.23607 0.137620
\(265\) 1.81966 0.111781
\(266\) −14.4721 −0.887344
\(267\) 2.47214 0.151292
\(268\) 1.09017 0.0665927
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 21.0902 1.28351
\(271\) 12.9443 0.786309 0.393154 0.919473i \(-0.371384\pi\)
0.393154 + 0.919473i \(0.371384\pi\)
\(272\) −4.47214 −0.271163
\(273\) −14.9443 −0.904468
\(274\) 3.67376 0.221940
\(275\) −13.6180 −0.821198
\(276\) −4.61803 −0.277973
\(277\) 16.7984 1.00932 0.504658 0.863319i \(-0.331619\pi\)
0.504658 + 0.863319i \(0.331619\pi\)
\(278\) 4.85410 0.291130
\(279\) −2.81966 −0.168809
\(280\) −12.4721 −0.745353
\(281\) 29.8885 1.78300 0.891501 0.453020i \(-0.149653\pi\)
0.891501 + 0.453020i \(0.149653\pi\)
\(282\) 2.00000 0.119098
\(283\) 6.76393 0.402074 0.201037 0.979584i \(-0.435569\pi\)
0.201037 + 0.979584i \(0.435569\pi\)
\(284\) 2.94427 0.174710
\(285\) −27.8885 −1.65197
\(286\) 3.94427 0.233230
\(287\) −31.1246 −1.83723
\(288\) −0.381966 −0.0225076
\(289\) 3.00000 0.176471
\(290\) −35.9443 −2.11072
\(291\) 0.763932 0.0447825
\(292\) 7.09017 0.414921
\(293\) −12.6525 −0.739166 −0.369583 0.929198i \(-0.620499\pi\)
−0.369583 + 0.929198i \(0.620499\pi\)
\(294\) −5.61803 −0.327650
\(295\) −18.3607 −1.06900
\(296\) −1.00000 −0.0581238
\(297\) −7.56231 −0.438809
\(298\) 16.1803 0.937302
\(299\) −8.14590 −0.471089
\(300\) −15.9443 −0.920543
\(301\) 16.9443 0.976652
\(302\) 4.29180 0.246965
\(303\) −5.70820 −0.327928
\(304\) 4.47214 0.256495
\(305\) 40.9230 2.34324
\(306\) 1.70820 0.0976515
\(307\) −12.8541 −0.733622 −0.366811 0.930295i \(-0.619550\pi\)
−0.366811 + 0.930295i \(0.619550\pi\)
\(308\) 4.47214 0.254824
\(309\) −27.9443 −1.58969
\(310\) 28.4508 1.61590
\(311\) −27.0344 −1.53298 −0.766491 0.642255i \(-0.777999\pi\)
−0.766491 + 0.642255i \(0.777999\pi\)
\(312\) 4.61803 0.261445
\(313\) 28.1803 1.59285 0.796423 0.604739i \(-0.206723\pi\)
0.796423 + 0.604739i \(0.206723\pi\)
\(314\) −16.4721 −0.929576
\(315\) 4.76393 0.268417
\(316\) −8.56231 −0.481667
\(317\) 20.9443 1.17635 0.588174 0.808735i \(-0.299847\pi\)
0.588174 + 0.808735i \(0.299847\pi\)
\(318\) −0.763932 −0.0428392
\(319\) 12.8885 0.721620
\(320\) 3.85410 0.215451
\(321\) 11.8541 0.661631
\(322\) −9.23607 −0.514706
\(323\) −20.0000 −1.11283
\(324\) −7.70820 −0.428234
\(325\) −28.1246 −1.56007
\(326\) −3.52786 −0.195390
\(327\) −4.76393 −0.263446
\(328\) 9.61803 0.531067
\(329\) 4.00000 0.220527
\(330\) 8.61803 0.474407
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −14.4721 −0.794262
\(333\) 0.381966 0.0209316
\(334\) 13.8541 0.758063
\(335\) 4.20163 0.229559
\(336\) 5.23607 0.285651
\(337\) 12.0344 0.655558 0.327779 0.944754i \(-0.393700\pi\)
0.327779 + 0.944754i \(0.393700\pi\)
\(338\) −4.85410 −0.264028
\(339\) 11.2361 0.610259
\(340\) −17.2361 −0.934757
\(341\) −10.2016 −0.552449
\(342\) −1.70820 −0.0923691
\(343\) 11.4164 0.616428
\(344\) −5.23607 −0.282310
\(345\) −17.7984 −0.958232
\(346\) −0.472136 −0.0253822
\(347\) 17.2361 0.925281 0.462640 0.886546i \(-0.346902\pi\)
0.462640 + 0.886546i \(0.346902\pi\)
\(348\) 15.0902 0.808918
\(349\) −10.1803 −0.544941 −0.272471 0.962164i \(-0.587841\pi\)
−0.272471 + 0.962164i \(0.587841\pi\)
\(350\) −31.8885 −1.70451
\(351\) −15.6180 −0.833629
\(352\) −1.38197 −0.0736590
\(353\) 16.2918 0.867125 0.433562 0.901124i \(-0.357256\pi\)
0.433562 + 0.901124i \(0.357256\pi\)
\(354\) 7.70820 0.409686
\(355\) 11.3475 0.602264
\(356\) −1.52786 −0.0809766
\(357\) −23.4164 −1.23933
\(358\) −12.6525 −0.668704
\(359\) −4.47214 −0.236030 −0.118015 0.993012i \(-0.537653\pi\)
−0.118015 + 0.993012i \(0.537653\pi\)
\(360\) −1.47214 −0.0775884
\(361\) 1.00000 0.0526316
\(362\) 14.4721 0.760639
\(363\) 14.7082 0.771980
\(364\) 9.23607 0.484102
\(365\) 27.3262 1.43032
\(366\) −17.1803 −0.898031
\(367\) 13.1246 0.685099 0.342550 0.939500i \(-0.388710\pi\)
0.342550 + 0.939500i \(0.388710\pi\)
\(368\) 2.85410 0.148780
\(369\) −3.67376 −0.191248
\(370\) −3.85410 −0.200365
\(371\) −1.52786 −0.0793227
\(372\) −11.9443 −0.619282
\(373\) 27.7082 1.43468 0.717338 0.696725i \(-0.245360\pi\)
0.717338 + 0.696725i \(0.245360\pi\)
\(374\) 6.18034 0.319578
\(375\) −30.2705 −1.56316
\(376\) −1.23607 −0.0637453
\(377\) 26.6180 1.37090
\(378\) −17.7082 −0.910812
\(379\) 28.0902 1.44290 0.721448 0.692469i \(-0.243477\pi\)
0.721448 + 0.692469i \(0.243477\pi\)
\(380\) 17.2361 0.884192
\(381\) 13.7082 0.702293
\(382\) −7.09017 −0.362765
\(383\) 17.8885 0.914062 0.457031 0.889451i \(-0.348913\pi\)
0.457031 + 0.889451i \(0.348913\pi\)
\(384\) −1.61803 −0.0825700
\(385\) 17.2361 0.878431
\(386\) 4.00000 0.203595
\(387\) 2.00000 0.101666
\(388\) −0.472136 −0.0239691
\(389\) −6.85410 −0.347517 −0.173758 0.984788i \(-0.555591\pi\)
−0.173758 + 0.984788i \(0.555591\pi\)
\(390\) 17.7984 0.901256
\(391\) −12.7639 −0.645500
\(392\) 3.47214 0.175369
\(393\) 36.6525 1.84887
\(394\) −7.52786 −0.379248
\(395\) −33.0000 −1.66041
\(396\) 0.527864 0.0265262
\(397\) 20.6525 1.03652 0.518259 0.855224i \(-0.326580\pi\)
0.518259 + 0.855224i \(0.326580\pi\)
\(398\) 3.05573 0.153170
\(399\) 23.4164 1.17229
\(400\) 9.85410 0.492705
\(401\) −4.76393 −0.237899 −0.118950 0.992900i \(-0.537953\pi\)
−0.118950 + 0.992900i \(0.537953\pi\)
\(402\) −1.76393 −0.0879769
\(403\) −21.0689 −1.04952
\(404\) 3.52786 0.175518
\(405\) −29.7082 −1.47621
\(406\) 30.1803 1.49783
\(407\) 1.38197 0.0685015
\(408\) 7.23607 0.358239
\(409\) −26.1803 −1.29453 −0.647267 0.762263i \(-0.724088\pi\)
−0.647267 + 0.762263i \(0.724088\pi\)
\(410\) 37.0689 1.83070
\(411\) −5.94427 −0.293209
\(412\) 17.2705 0.850857
\(413\) 15.4164 0.758592
\(414\) −1.09017 −0.0535789
\(415\) −55.7771 −2.73799
\(416\) −2.85410 −0.139934
\(417\) −7.85410 −0.384617
\(418\) −6.18034 −0.302290
\(419\) −10.5623 −0.516002 −0.258001 0.966145i \(-0.583064\pi\)
−0.258001 + 0.966145i \(0.583064\pi\)
\(420\) 20.1803 0.984700
\(421\) −31.0344 −1.51253 −0.756263 0.654268i \(-0.772977\pi\)
−0.756263 + 0.654268i \(0.772977\pi\)
\(422\) 11.2705 0.548640
\(423\) 0.472136 0.0229560
\(424\) 0.472136 0.0229289
\(425\) −44.0689 −2.13765
\(426\) −4.76393 −0.230813
\(427\) −34.3607 −1.66283
\(428\) −7.32624 −0.354127
\(429\) −6.38197 −0.308124
\(430\) −20.1803 −0.973182
\(431\) 12.3607 0.595393 0.297696 0.954661i \(-0.403782\pi\)
0.297696 + 0.954661i \(0.403782\pi\)
\(432\) 5.47214 0.263278
\(433\) −20.6738 −0.993518 −0.496759 0.867889i \(-0.665477\pi\)
−0.496759 + 0.867889i \(0.665477\pi\)
\(434\) −23.8885 −1.14669
\(435\) 58.1591 2.78851
\(436\) 2.94427 0.141005
\(437\) 12.7639 0.610582
\(438\) −11.4721 −0.548160
\(439\) 7.79837 0.372196 0.186098 0.982531i \(-0.440416\pi\)
0.186098 + 0.982531i \(0.440416\pi\)
\(440\) −5.32624 −0.253918
\(441\) −1.32624 −0.0631542
\(442\) 12.7639 0.607118
\(443\) 15.2705 0.725524 0.362762 0.931882i \(-0.381834\pi\)
0.362762 + 0.931882i \(0.381834\pi\)
\(444\) 1.61803 0.0767885
\(445\) −5.88854 −0.279144
\(446\) 14.1803 0.671459
\(447\) −26.1803 −1.23829
\(448\) −3.23607 −0.152890
\(449\) −28.4721 −1.34368 −0.671842 0.740695i \(-0.734496\pi\)
−0.671842 + 0.740695i \(0.734496\pi\)
\(450\) −3.76393 −0.177433
\(451\) −13.2918 −0.625886
\(452\) −6.94427 −0.326631
\(453\) −6.94427 −0.326270
\(454\) −4.29180 −0.201424
\(455\) 35.5967 1.66880
\(456\) −7.23607 −0.338860
\(457\) −24.7639 −1.15841 −0.579204 0.815183i \(-0.696637\pi\)
−0.579204 + 0.815183i \(0.696637\pi\)
\(458\) −23.1246 −1.08054
\(459\) −24.4721 −1.14226
\(460\) 11.0000 0.512878
\(461\) −38.9443 −1.81382 −0.906908 0.421329i \(-0.861564\pi\)
−0.906908 + 0.421329i \(0.861564\pi\)
\(462\) −7.23607 −0.336652
\(463\) −4.56231 −0.212028 −0.106014 0.994365i \(-0.533809\pi\)
−0.106014 + 0.994365i \(0.533809\pi\)
\(464\) −9.32624 −0.432960
\(465\) −46.0344 −2.13480
\(466\) −6.56231 −0.303993
\(467\) −24.3607 −1.12728 −0.563639 0.826021i \(-0.690599\pi\)
−0.563639 + 0.826021i \(0.690599\pi\)
\(468\) 1.09017 0.0503931
\(469\) −3.52786 −0.162902
\(470\) −4.76393 −0.219744
\(471\) 26.6525 1.22808
\(472\) −4.76393 −0.219278
\(473\) 7.23607 0.332715
\(474\) 13.8541 0.636340
\(475\) 44.0689 2.02202
\(476\) 14.4721 0.663329
\(477\) −0.180340 −0.00825720
\(478\) 9.85410 0.450716
\(479\) 36.5623 1.67057 0.835287 0.549814i \(-0.185301\pi\)
0.835287 + 0.549814i \(0.185301\pi\)
\(480\) −6.23607 −0.284636
\(481\) 2.85410 0.130136
\(482\) −1.52786 −0.0695923
\(483\) 14.9443 0.679988
\(484\) −9.09017 −0.413190
\(485\) −1.81966 −0.0826265
\(486\) −3.94427 −0.178916
\(487\) 37.3050 1.69045 0.845224 0.534412i \(-0.179467\pi\)
0.845224 + 0.534412i \(0.179467\pi\)
\(488\) 10.6180 0.480656
\(489\) 5.70820 0.258134
\(490\) 13.3820 0.604536
\(491\) −28.4508 −1.28397 −0.641984 0.766718i \(-0.721889\pi\)
−0.641984 + 0.766718i \(0.721889\pi\)
\(492\) −15.5623 −0.701603
\(493\) 41.7082 1.87844
\(494\) −12.7639 −0.574276
\(495\) 2.03444 0.0914414
\(496\) 7.38197 0.331460
\(497\) −9.52786 −0.427383
\(498\) 23.4164 1.04931
\(499\) 10.2918 0.460724 0.230362 0.973105i \(-0.426009\pi\)
0.230362 + 0.973105i \(0.426009\pi\)
\(500\) 18.7082 0.836656
\(501\) −22.4164 −1.00149
\(502\) −20.9443 −0.934789
\(503\) 19.0902 0.851189 0.425594 0.904914i \(-0.360065\pi\)
0.425594 + 0.904914i \(0.360065\pi\)
\(504\) 1.23607 0.0550588
\(505\) 13.5967 0.605047
\(506\) −3.94427 −0.175344
\(507\) 7.85410 0.348813
\(508\) −8.47214 −0.375890
\(509\) −17.7082 −0.784902 −0.392451 0.919773i \(-0.628373\pi\)
−0.392451 + 0.919773i \(0.628373\pi\)
\(510\) 27.8885 1.23493
\(511\) −22.9443 −1.01499
\(512\) 1.00000 0.0441942
\(513\) 24.4721 1.08047
\(514\) −1.05573 −0.0465662
\(515\) 66.5623 2.93309
\(516\) 8.47214 0.372965
\(517\) 1.70820 0.0751267
\(518\) 3.23607 0.142185
\(519\) 0.763932 0.0335329
\(520\) −11.0000 −0.482382
\(521\) −1.41641 −0.0620540 −0.0310270 0.999519i \(-0.509878\pi\)
−0.0310270 + 0.999519i \(0.509878\pi\)
\(522\) 3.56231 0.155918
\(523\) 11.8197 0.516838 0.258419 0.966033i \(-0.416799\pi\)
0.258419 + 0.966033i \(0.416799\pi\)
\(524\) −22.6525 −0.989578
\(525\) 51.5967 2.25187
\(526\) −13.2361 −0.577120
\(527\) −33.0132 −1.43808
\(528\) 2.23607 0.0973124
\(529\) −14.8541 −0.645831
\(530\) 1.81966 0.0790410
\(531\) 1.81966 0.0789665
\(532\) −14.4721 −0.627447
\(533\) −27.4508 −1.18903
\(534\) 2.47214 0.106980
\(535\) −28.2361 −1.22075
\(536\) 1.09017 0.0470882
\(537\) 20.4721 0.883438
\(538\) 4.00000 0.172452
\(539\) −4.79837 −0.206681
\(540\) 21.0902 0.907576
\(541\) −10.3262 −0.443960 −0.221980 0.975051i \(-0.571252\pi\)
−0.221980 + 0.975051i \(0.571252\pi\)
\(542\) 12.9443 0.556004
\(543\) −23.4164 −1.00489
\(544\) −4.47214 −0.191741
\(545\) 11.3475 0.486075
\(546\) −14.9443 −0.639556
\(547\) −16.0689 −0.687056 −0.343528 0.939142i \(-0.611622\pi\)
−0.343528 + 0.939142i \(0.611622\pi\)
\(548\) 3.67376 0.156935
\(549\) −4.05573 −0.173094
\(550\) −13.6180 −0.580675
\(551\) −41.7082 −1.77683
\(552\) −4.61803 −0.196557
\(553\) 27.7082 1.17827
\(554\) 16.7984 0.713695
\(555\) 6.23607 0.264706
\(556\) 4.85410 0.205860
\(557\) 19.5623 0.828882 0.414441 0.910076i \(-0.363977\pi\)
0.414441 + 0.910076i \(0.363977\pi\)
\(558\) −2.81966 −0.119366
\(559\) 14.9443 0.632075
\(560\) −12.4721 −0.527044
\(561\) −10.0000 −0.422200
\(562\) 29.8885 1.26077
\(563\) 7.88854 0.332462 0.166231 0.986087i \(-0.446840\pi\)
0.166231 + 0.986087i \(0.446840\pi\)
\(564\) 2.00000 0.0842152
\(565\) −26.7639 −1.12597
\(566\) 6.76393 0.284309
\(567\) 24.9443 1.04756
\(568\) 2.94427 0.123539
\(569\) 13.8885 0.582238 0.291119 0.956687i \(-0.405972\pi\)
0.291119 + 0.956687i \(0.405972\pi\)
\(570\) −27.8885 −1.16812
\(571\) −10.5623 −0.442019 −0.221009 0.975272i \(-0.570935\pi\)
−0.221009 + 0.975272i \(0.570935\pi\)
\(572\) 3.94427 0.164918
\(573\) 11.4721 0.479255
\(574\) −31.1246 −1.29912
\(575\) 28.1246 1.17288
\(576\) −0.381966 −0.0159153
\(577\) 10.6525 0.443468 0.221734 0.975107i \(-0.428828\pi\)
0.221734 + 0.975107i \(0.428828\pi\)
\(578\) 3.00000 0.124784
\(579\) −6.47214 −0.268973
\(580\) −35.9443 −1.49250
\(581\) 46.8328 1.94295
\(582\) 0.763932 0.0316660
\(583\) −0.652476 −0.0270228
\(584\) 7.09017 0.293393
\(585\) 4.20163 0.173716
\(586\) −12.6525 −0.522669
\(587\) −13.0557 −0.538868 −0.269434 0.963019i \(-0.586837\pi\)
−0.269434 + 0.963019i \(0.586837\pi\)
\(588\) −5.61803 −0.231684
\(589\) 33.0132 1.36028
\(590\) −18.3607 −0.755897
\(591\) 12.1803 0.501032
\(592\) −1.00000 −0.0410997
\(593\) 17.5623 0.721197 0.360599 0.932721i \(-0.382572\pi\)
0.360599 + 0.932721i \(0.382572\pi\)
\(594\) −7.56231 −0.310285
\(595\) 55.7771 2.28664
\(596\) 16.1803 0.662773
\(597\) −4.94427 −0.202356
\(598\) −8.14590 −0.333111
\(599\) −6.36068 −0.259890 −0.129945 0.991521i \(-0.541480\pi\)
−0.129945 + 0.991521i \(0.541480\pi\)
\(600\) −15.9443 −0.650922
\(601\) −24.6869 −1.00700 −0.503500 0.863995i \(-0.667955\pi\)
−0.503500 + 0.863995i \(0.667955\pi\)
\(602\) 16.9443 0.690597
\(603\) −0.416408 −0.0169574
\(604\) 4.29180 0.174631
\(605\) −35.0344 −1.42435
\(606\) −5.70820 −0.231880
\(607\) 35.0344 1.42200 0.711002 0.703190i \(-0.248242\pi\)
0.711002 + 0.703190i \(0.248242\pi\)
\(608\) 4.47214 0.181369
\(609\) −48.8328 −1.97881
\(610\) 40.9230 1.65692
\(611\) 3.52786 0.142722
\(612\) 1.70820 0.0690501
\(613\) −13.8197 −0.558171 −0.279085 0.960266i \(-0.590031\pi\)
−0.279085 + 0.960266i \(0.590031\pi\)
\(614\) −12.8541 −0.518749
\(615\) −59.9787 −2.41858
\(616\) 4.47214 0.180187
\(617\) 0.0901699 0.00363011 0.00181505 0.999998i \(-0.499422\pi\)
0.00181505 + 0.999998i \(0.499422\pi\)
\(618\) −27.9443 −1.12408
\(619\) −15.2705 −0.613774 −0.306887 0.951746i \(-0.599287\pi\)
−0.306887 + 0.951746i \(0.599287\pi\)
\(620\) 28.4508 1.14261
\(621\) 15.6180 0.626730
\(622\) −27.0344 −1.08398
\(623\) 4.94427 0.198088
\(624\) 4.61803 0.184869
\(625\) 22.8328 0.913313
\(626\) 28.1803 1.12631
\(627\) 10.0000 0.399362
\(628\) −16.4721 −0.657310
\(629\) 4.47214 0.178316
\(630\) 4.76393 0.189800
\(631\) −47.3951 −1.88677 −0.943385 0.331700i \(-0.892378\pi\)
−0.943385 + 0.331700i \(0.892378\pi\)
\(632\) −8.56231 −0.340590
\(633\) −18.2361 −0.724819
\(634\) 20.9443 0.831803
\(635\) −32.6525 −1.29577
\(636\) −0.763932 −0.0302919
\(637\) −9.90983 −0.392642
\(638\) 12.8885 0.510262
\(639\) −1.12461 −0.0444890
\(640\) 3.85410 0.152347
\(641\) 15.5066 0.612473 0.306236 0.951955i \(-0.400930\pi\)
0.306236 + 0.951955i \(0.400930\pi\)
\(642\) 11.8541 0.467844
\(643\) −28.7639 −1.13434 −0.567169 0.823601i \(-0.691962\pi\)
−0.567169 + 0.823601i \(0.691962\pi\)
\(644\) −9.23607 −0.363952
\(645\) 32.6525 1.28569
\(646\) −20.0000 −0.786889
\(647\) 30.0902 1.18297 0.591483 0.806317i \(-0.298543\pi\)
0.591483 + 0.806317i \(0.298543\pi\)
\(648\) −7.70820 −0.302807
\(649\) 6.58359 0.258429
\(650\) −28.1246 −1.10314
\(651\) 38.6525 1.51491
\(652\) −3.52786 −0.138162
\(653\) −38.2705 −1.49764 −0.748820 0.662773i \(-0.769379\pi\)
−0.748820 + 0.662773i \(0.769379\pi\)
\(654\) −4.76393 −0.186284
\(655\) −87.3050 −3.41129
\(656\) 9.61803 0.375521
\(657\) −2.70820 −0.105657
\(658\) 4.00000 0.155936
\(659\) 40.4508 1.57574 0.787871 0.615841i \(-0.211184\pi\)
0.787871 + 0.615841i \(0.211184\pi\)
\(660\) 8.61803 0.335457
\(661\) 17.3262 0.673913 0.336956 0.941520i \(-0.390603\pi\)
0.336956 + 0.941520i \(0.390603\pi\)
\(662\) 28.0000 1.08825
\(663\) −20.6525 −0.802076
\(664\) −14.4721 −0.561628
\(665\) −55.7771 −2.16294
\(666\) 0.381966 0.0148009
\(667\) −26.6180 −1.03065
\(668\) 13.8541 0.536031
\(669\) −22.9443 −0.887077
\(670\) 4.20163 0.162323
\(671\) −14.6738 −0.566474
\(672\) 5.23607 0.201986
\(673\) 11.1459 0.429643 0.214821 0.976653i \(-0.431083\pi\)
0.214821 + 0.976653i \(0.431083\pi\)
\(674\) 12.0344 0.463549
\(675\) 53.9230 2.07550
\(676\) −4.85410 −0.186696
\(677\) 34.6525 1.33180 0.665901 0.746040i \(-0.268047\pi\)
0.665901 + 0.746040i \(0.268047\pi\)
\(678\) 11.2361 0.431519
\(679\) 1.52786 0.0586340
\(680\) −17.2361 −0.660973
\(681\) 6.94427 0.266105
\(682\) −10.2016 −0.390640
\(683\) −8.58359 −0.328442 −0.164221 0.986424i \(-0.552511\pi\)
−0.164221 + 0.986424i \(0.552511\pi\)
\(684\) −1.70820 −0.0653148
\(685\) 14.1591 0.540990
\(686\) 11.4164 0.435880
\(687\) 37.4164 1.42752
\(688\) −5.23607 −0.199623
\(689\) −1.34752 −0.0513366
\(690\) −17.7984 −0.677573
\(691\) 35.7771 1.36102 0.680512 0.732737i \(-0.261757\pi\)
0.680512 + 0.732737i \(0.261757\pi\)
\(692\) −0.472136 −0.0179479
\(693\) −1.70820 −0.0648893
\(694\) 17.2361 0.654272
\(695\) 18.7082 0.709643
\(696\) 15.0902 0.571991
\(697\) −43.0132 −1.62924
\(698\) −10.1803 −0.385332
\(699\) 10.6180 0.401611
\(700\) −31.8885 −1.20527
\(701\) −42.9787 −1.62328 −0.811642 0.584155i \(-0.801426\pi\)
−0.811642 + 0.584155i \(0.801426\pi\)
\(702\) −15.6180 −0.589465
\(703\) −4.47214 −0.168670
\(704\) −1.38197 −0.0520848
\(705\) 7.70820 0.290308
\(706\) 16.2918 0.613150
\(707\) −11.4164 −0.429358
\(708\) 7.70820 0.289692
\(709\) −8.21478 −0.308513 −0.154256 0.988031i \(-0.549298\pi\)
−0.154256 + 0.988031i \(0.549298\pi\)
\(710\) 11.3475 0.425865
\(711\) 3.27051 0.122654
\(712\) −1.52786 −0.0572591
\(713\) 21.0689 0.789036
\(714\) −23.4164 −0.876337
\(715\) 15.2016 0.568509
\(716\) −12.6525 −0.472845
\(717\) −15.9443 −0.595450
\(718\) −4.47214 −0.166899
\(719\) 27.4164 1.02246 0.511230 0.859444i \(-0.329190\pi\)
0.511230 + 0.859444i \(0.329190\pi\)
\(720\) −1.47214 −0.0548633
\(721\) −55.8885 −2.08140
\(722\) 1.00000 0.0372161
\(723\) 2.47214 0.0919397
\(724\) 14.4721 0.537853
\(725\) −91.9017 −3.41314
\(726\) 14.7082 0.545873
\(727\) −29.8541 −1.10723 −0.553614 0.832774i \(-0.686752\pi\)
−0.553614 + 0.832774i \(0.686752\pi\)
\(728\) 9.23607 0.342311
\(729\) 29.5066 1.09284
\(730\) 27.3262 1.01139
\(731\) 23.4164 0.866087
\(732\) −17.1803 −0.635004
\(733\) 27.5279 1.01676 0.508382 0.861131i \(-0.330244\pi\)
0.508382 + 0.861131i \(0.330244\pi\)
\(734\) 13.1246 0.484438
\(735\) −21.6525 −0.798664
\(736\) 2.85410 0.105204
\(737\) −1.50658 −0.0554955
\(738\) −3.67376 −0.135233
\(739\) −10.9098 −0.401325 −0.200662 0.979660i \(-0.564309\pi\)
−0.200662 + 0.979660i \(0.564309\pi\)
\(740\) −3.85410 −0.141680
\(741\) 20.6525 0.758688
\(742\) −1.52786 −0.0560897
\(743\) −48.0689 −1.76348 −0.881738 0.471739i \(-0.843626\pi\)
−0.881738 + 0.471739i \(0.843626\pi\)
\(744\) −11.9443 −0.437898
\(745\) 62.3607 2.28472
\(746\) 27.7082 1.01447
\(747\) 5.52786 0.202254
\(748\) 6.18034 0.225976
\(749\) 23.7082 0.866279
\(750\) −30.2705 −1.10532
\(751\) −1.05573 −0.0385241 −0.0192620 0.999814i \(-0.506132\pi\)
−0.0192620 + 0.999814i \(0.506132\pi\)
\(752\) −1.23607 −0.0450748
\(753\) 33.8885 1.23497
\(754\) 26.6180 0.969372
\(755\) 16.5410 0.601989
\(756\) −17.7082 −0.644041
\(757\) −4.14590 −0.150685 −0.0753426 0.997158i \(-0.524005\pi\)
−0.0753426 + 0.997158i \(0.524005\pi\)
\(758\) 28.0902 1.02028
\(759\) 6.38197 0.231651
\(760\) 17.2361 0.625218
\(761\) −19.1459 −0.694038 −0.347019 0.937858i \(-0.612806\pi\)
−0.347019 + 0.937858i \(0.612806\pi\)
\(762\) 13.7082 0.496596
\(763\) −9.52786 −0.344932
\(764\) −7.09017 −0.256513
\(765\) 6.58359 0.238030
\(766\) 17.8885 0.646339
\(767\) 13.5967 0.490950
\(768\) −1.61803 −0.0583858
\(769\) 23.8885 0.861443 0.430721 0.902485i \(-0.358259\pi\)
0.430721 + 0.902485i \(0.358259\pi\)
\(770\) 17.2361 0.621145
\(771\) 1.70820 0.0615195
\(772\) 4.00000 0.143963
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 2.00000 0.0718885
\(775\) 72.7426 2.61299
\(776\) −0.472136 −0.0169487
\(777\) −5.23607 −0.187843
\(778\) −6.85410 −0.245731
\(779\) 43.0132 1.54111
\(780\) 17.7984 0.637284
\(781\) −4.06888 −0.145596
\(782\) −12.7639 −0.456437
\(783\) −51.0344 −1.82382
\(784\) 3.47214 0.124005
\(785\) −63.4853 −2.26589
\(786\) 36.6525 1.30735
\(787\) −25.5279 −0.909970 −0.454985 0.890499i \(-0.650355\pi\)
−0.454985 + 0.890499i \(0.650355\pi\)
\(788\) −7.52786 −0.268169
\(789\) 21.4164 0.762444
\(790\) −33.0000 −1.17409
\(791\) 22.4721 0.799017
\(792\) 0.527864 0.0187568
\(793\) −30.3050 −1.07616
\(794\) 20.6525 0.732929
\(795\) −2.94427 −0.104423
\(796\) 3.05573 0.108307
\(797\) 46.2705 1.63899 0.819493 0.573090i \(-0.194255\pi\)
0.819493 + 0.573090i \(0.194255\pi\)
\(798\) 23.4164 0.828932
\(799\) 5.52786 0.195562
\(800\) 9.85410 0.348395
\(801\) 0.583592 0.0206202
\(802\) −4.76393 −0.168220
\(803\) −9.79837 −0.345777
\(804\) −1.76393 −0.0622091
\(805\) −35.5967 −1.25462
\(806\) −21.0689 −0.742120
\(807\) −6.47214 −0.227830
\(808\) 3.52786 0.124110
\(809\) −13.1246 −0.461437 −0.230718 0.973021i \(-0.574108\pi\)
−0.230718 + 0.973021i \(0.574108\pi\)
\(810\) −29.7082 −1.04384
\(811\) −34.1033 −1.19753 −0.598765 0.800925i \(-0.704342\pi\)
−0.598765 + 0.800925i \(0.704342\pi\)
\(812\) 30.1803 1.05912
\(813\) −20.9443 −0.734548
\(814\) 1.38197 0.0484379
\(815\) −13.5967 −0.476273
\(816\) 7.23607 0.253313
\(817\) −23.4164 −0.819236
\(818\) −26.1803 −0.915374
\(819\) −3.52786 −0.123274
\(820\) 37.0689 1.29450
\(821\) −5.41641 −0.189034 −0.0945170 0.995523i \(-0.530131\pi\)
−0.0945170 + 0.995523i \(0.530131\pi\)
\(822\) −5.94427 −0.207330
\(823\) 1.88854 0.0658305 0.0329152 0.999458i \(-0.489521\pi\)
0.0329152 + 0.999458i \(0.489521\pi\)
\(824\) 17.2705 0.601647
\(825\) 22.0344 0.767141
\(826\) 15.4164 0.536405
\(827\) −2.06888 −0.0719421 −0.0359711 0.999353i \(-0.511452\pi\)
−0.0359711 + 0.999353i \(0.511452\pi\)
\(828\) −1.09017 −0.0378860
\(829\) 55.7984 1.93796 0.968979 0.247144i \(-0.0794920\pi\)
0.968979 + 0.247144i \(0.0794920\pi\)
\(830\) −55.7771 −1.93605
\(831\) −27.1803 −0.942876
\(832\) −2.85410 −0.0989482
\(833\) −15.5279 −0.538009
\(834\) −7.85410 −0.271965
\(835\) 53.3951 1.84781
\(836\) −6.18034 −0.213752
\(837\) 40.3951 1.39626
\(838\) −10.5623 −0.364869
\(839\) −36.6525 −1.26538 −0.632692 0.774404i \(-0.718050\pi\)
−0.632692 + 0.774404i \(0.718050\pi\)
\(840\) 20.1803 0.696288
\(841\) 57.9787 1.99927
\(842\) −31.0344 −1.06952
\(843\) −48.3607 −1.66563
\(844\) 11.2705 0.387947
\(845\) −18.7082 −0.643582
\(846\) 0.472136 0.0162324
\(847\) 29.4164 1.01076
\(848\) 0.472136 0.0162132
\(849\) −10.9443 −0.375606
\(850\) −44.0689 −1.51155
\(851\) −2.85410 −0.0978374
\(852\) −4.76393 −0.163210
\(853\) 29.7426 1.01837 0.509184 0.860657i \(-0.329947\pi\)
0.509184 + 0.860657i \(0.329947\pi\)
\(854\) −34.3607 −1.17580
\(855\) −6.58359 −0.225154
\(856\) −7.32624 −0.250406
\(857\) −9.05573 −0.309338 −0.154669 0.987966i \(-0.549431\pi\)
−0.154669 + 0.987966i \(0.549431\pi\)
\(858\) −6.38197 −0.217877
\(859\) 53.4164 1.82254 0.911272 0.411805i \(-0.135101\pi\)
0.911272 + 0.411805i \(0.135101\pi\)
\(860\) −20.1803 −0.688144
\(861\) 50.3607 1.71629
\(862\) 12.3607 0.421006
\(863\) 19.4164 0.660942 0.330471 0.943816i \(-0.392792\pi\)
0.330471 + 0.943816i \(0.392792\pi\)
\(864\) 5.47214 0.186166
\(865\) −1.81966 −0.0618703
\(866\) −20.6738 −0.702523
\(867\) −4.85410 −0.164854
\(868\) −23.8885 −0.810830
\(869\) 11.8328 0.401401
\(870\) 58.1591 1.97178
\(871\) −3.11146 −0.105428
\(872\) 2.94427 0.0997056
\(873\) 0.180340 0.00610358
\(874\) 12.7639 0.431746
\(875\) −60.5410 −2.04666
\(876\) −11.4721 −0.387608
\(877\) −16.8328 −0.568404 −0.284202 0.958764i \(-0.591729\pi\)
−0.284202 + 0.958764i \(0.591729\pi\)
\(878\) 7.79837 0.263182
\(879\) 20.4721 0.690508
\(880\) −5.32624 −0.179547
\(881\) 19.7426 0.665147 0.332573 0.943077i \(-0.392083\pi\)
0.332573 + 0.943077i \(0.392083\pi\)
\(882\) −1.32624 −0.0446568
\(883\) −33.3050 −1.12080 −0.560400 0.828222i \(-0.689353\pi\)
−0.560400 + 0.828222i \(0.689353\pi\)
\(884\) 12.7639 0.429297
\(885\) 29.7082 0.998630
\(886\) 15.2705 0.513023
\(887\) 27.8885 0.936406 0.468203 0.883621i \(-0.344902\pi\)
0.468203 + 0.883621i \(0.344902\pi\)
\(888\) 1.61803 0.0542977
\(889\) 27.4164 0.919517
\(890\) −5.88854 −0.197384
\(891\) 10.6525 0.356871
\(892\) 14.1803 0.474793
\(893\) −5.52786 −0.184983
\(894\) −26.1803 −0.875602
\(895\) −48.7639 −1.63000
\(896\) −3.23607 −0.108109
\(897\) 13.1803 0.440079
\(898\) −28.4721 −0.950127
\(899\) −68.8460 −2.29614
\(900\) −3.76393 −0.125464
\(901\) −2.11146 −0.0703428
\(902\) −13.2918 −0.442568
\(903\) −27.4164 −0.912361
\(904\) −6.94427 −0.230963
\(905\) 55.7771 1.85409
\(906\) −6.94427 −0.230708
\(907\) −55.8885 −1.85575 −0.927874 0.372893i \(-0.878366\pi\)
−0.927874 + 0.372893i \(0.878366\pi\)
\(908\) −4.29180 −0.142428
\(909\) −1.34752 −0.0446946
\(910\) 35.5967 1.18002
\(911\) 13.8885 0.460148 0.230074 0.973173i \(-0.426103\pi\)
0.230074 + 0.973173i \(0.426103\pi\)
\(912\) −7.23607 −0.239610
\(913\) 20.0000 0.661903
\(914\) −24.7639 −0.819118
\(915\) −66.2148 −2.18899
\(916\) −23.1246 −0.764059
\(917\) 73.3050 2.42074
\(918\) −24.4721 −0.807701
\(919\) −5.88854 −0.194245 −0.0971226 0.995272i \(-0.530964\pi\)
−0.0971226 + 0.995272i \(0.530964\pi\)
\(920\) 11.0000 0.362659
\(921\) 20.7984 0.685330
\(922\) −38.9443 −1.28256
\(923\) −8.40325 −0.276596
\(924\) −7.23607 −0.238049
\(925\) −9.85410 −0.324001
\(926\) −4.56231 −0.149927
\(927\) −6.59675 −0.216666
\(928\) −9.32624 −0.306149
\(929\) −27.4508 −0.900633 −0.450317 0.892869i \(-0.648689\pi\)
−0.450317 + 0.892869i \(0.648689\pi\)
\(930\) −46.0344 −1.50953
\(931\) 15.5279 0.508905
\(932\) −6.56231 −0.214955
\(933\) 43.7426 1.43207
\(934\) −24.3607 −0.797106
\(935\) 23.8197 0.778986
\(936\) 1.09017 0.0356333
\(937\) −48.0476 −1.56965 −0.784823 0.619720i \(-0.787246\pi\)
−0.784823 + 0.619720i \(0.787246\pi\)
\(938\) −3.52786 −0.115189
\(939\) −45.5967 −1.48799
\(940\) −4.76393 −0.155382
\(941\) −26.1803 −0.853455 −0.426727 0.904380i \(-0.640334\pi\)
−0.426727 + 0.904380i \(0.640334\pi\)
\(942\) 26.6525 0.868385
\(943\) 27.4508 0.893923
\(944\) −4.76393 −0.155053
\(945\) −68.2492 −2.22015
\(946\) 7.23607 0.235265
\(947\) 18.8328 0.611984 0.305992 0.952034i \(-0.401012\pi\)
0.305992 + 0.952034i \(0.401012\pi\)
\(948\) 13.8541 0.449960
\(949\) −20.2361 −0.656891
\(950\) 44.0689 1.42978
\(951\) −33.8885 −1.09891
\(952\) 14.4721 0.469045
\(953\) 11.4508 0.370929 0.185465 0.982651i \(-0.440621\pi\)
0.185465 + 0.982651i \(0.440621\pi\)
\(954\) −0.180340 −0.00583872
\(955\) −27.3262 −0.884256
\(956\) 9.85410 0.318704
\(957\) −20.8541 −0.674117
\(958\) 36.5623 1.18127
\(959\) −11.8885 −0.383901
\(960\) −6.23607 −0.201268
\(961\) 23.4934 0.757852
\(962\) 2.85410 0.0920199
\(963\) 2.79837 0.0901763
\(964\) −1.52786 −0.0492092
\(965\) 15.4164 0.496272
\(966\) 14.9443 0.480824
\(967\) 45.2705 1.45580 0.727901 0.685683i \(-0.240496\pi\)
0.727901 + 0.685683i \(0.240496\pi\)
\(968\) −9.09017 −0.292169
\(969\) 32.3607 1.03957
\(970\) −1.81966 −0.0584258
\(971\) 42.3262 1.35831 0.679157 0.733993i \(-0.262346\pi\)
0.679157 + 0.733993i \(0.262346\pi\)
\(972\) −3.94427 −0.126513
\(973\) −15.7082 −0.503582
\(974\) 37.3050 1.19533
\(975\) 45.5066 1.45738
\(976\) 10.6180 0.339875
\(977\) 43.5279 1.39258 0.696290 0.717761i \(-0.254833\pi\)
0.696290 + 0.717761i \(0.254833\pi\)
\(978\) 5.70820 0.182528
\(979\) 2.11146 0.0674824
\(980\) 13.3820 0.427471
\(981\) −1.12461 −0.0359061
\(982\) −28.4508 −0.907903
\(983\) −31.7771 −1.01353 −0.506766 0.862084i \(-0.669159\pi\)
−0.506766 + 0.862084i \(0.669159\pi\)
\(984\) −15.5623 −0.496108
\(985\) −29.0132 −0.924436
\(986\) 41.7082 1.32826
\(987\) −6.47214 −0.206010
\(988\) −12.7639 −0.406075
\(989\) −14.9443 −0.475200
\(990\) 2.03444 0.0646588
\(991\) 33.1033 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(992\) 7.38197 0.234378
\(993\) −45.3050 −1.43771
\(994\) −9.52786 −0.302205
\(995\) 11.7771 0.373359
\(996\) 23.4164 0.741977
\(997\) −17.7771 −0.563006 −0.281503 0.959560i \(-0.590833\pi\)
−0.281503 + 0.959560i \(0.590833\pi\)
\(998\) 10.2918 0.325781
\(999\) −5.47214 −0.173131
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 74.2.a.b.1.1 2
3.2 odd 2 666.2.a.i.1.1 2
4.3 odd 2 592.2.a.g.1.2 2
5.2 odd 4 1850.2.b.j.149.4 4
5.3 odd 4 1850.2.b.j.149.1 4
5.4 even 2 1850.2.a.t.1.2 2
7.6 odd 2 3626.2.a.s.1.2 2
8.3 odd 2 2368.2.a.u.1.1 2
8.5 even 2 2368.2.a.y.1.2 2
11.10 odd 2 8954.2.a.j.1.1 2
12.11 even 2 5328.2.a.bc.1.1 2
37.36 even 2 2738.2.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.1 2 1.1 even 1 trivial
592.2.a.g.1.2 2 4.3 odd 2
666.2.a.i.1.1 2 3.2 odd 2
1850.2.a.t.1.2 2 5.4 even 2
1850.2.b.j.149.1 4 5.3 odd 4
1850.2.b.j.149.4 4 5.2 odd 4
2368.2.a.u.1.1 2 8.3 odd 2
2368.2.a.y.1.2 2 8.5 even 2
2738.2.a.g.1.1 2 37.36 even 2
3626.2.a.s.1.2 2 7.6 odd 2
5328.2.a.bc.1.1 2 12.11 even 2
8954.2.a.j.1.1 2 11.10 odd 2