Properties

Label 230.2.a.c.1.2
Level $230$
Weight $2$
Character 230.1
Self dual yes
Analytic conductor $1.837$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
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 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 230.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.61803 q^{6} -0.618034 q^{7} +1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.61803 q^{6} -0.618034 q^{7} +1.00000 q^{8} -0.381966 q^{9} +1.00000 q^{10} -2.85410 q^{11} +1.61803 q^{12} -7.09017 q^{13} -0.618034 q^{14} +1.61803 q^{15} +1.00000 q^{16} +6.09017 q^{17} -0.381966 q^{18} +1.85410 q^{19} +1.00000 q^{20} -1.00000 q^{21} -2.85410 q^{22} +1.00000 q^{23} +1.61803 q^{24} +1.00000 q^{25} -7.09017 q^{26} -5.47214 q^{27} -0.618034 q^{28} -9.23607 q^{29} +1.61803 q^{30} +9.09017 q^{31} +1.00000 q^{32} -4.61803 q^{33} +6.09017 q^{34} -0.618034 q^{35} -0.381966 q^{36} +6.47214 q^{37} +1.85410 q^{38} -11.4721 q^{39} +1.00000 q^{40} +3.32624 q^{41} -1.00000 q^{42} -2.85410 q^{44} -0.381966 q^{45} +1.00000 q^{46} -3.70820 q^{47} +1.61803 q^{48} -6.61803 q^{49} +1.00000 q^{50} +9.85410 q^{51} -7.09017 q^{52} +0.472136 q^{53} -5.47214 q^{54} -2.85410 q^{55} -0.618034 q^{56} +3.00000 q^{57} -9.23607 q^{58} +1.70820 q^{59} +1.61803 q^{60} -9.32624 q^{61} +9.09017 q^{62} +0.236068 q^{63} +1.00000 q^{64} -7.09017 q^{65} -4.61803 q^{66} +14.4721 q^{67} +6.09017 q^{68} +1.61803 q^{69} -0.618034 q^{70} -4.09017 q^{71} -0.381966 q^{72} +3.23607 q^{73} +6.47214 q^{74} +1.61803 q^{75} +1.85410 q^{76} +1.76393 q^{77} -11.4721 q^{78} +1.52786 q^{79} +1.00000 q^{80} -7.70820 q^{81} +3.32624 q^{82} -6.94427 q^{83} -1.00000 q^{84} +6.09017 q^{85} -14.9443 q^{87} -2.85410 q^{88} -10.4721 q^{89} -0.381966 q^{90} +4.38197 q^{91} +1.00000 q^{92} +14.7082 q^{93} -3.70820 q^{94} +1.85410 q^{95} +1.61803 q^{96} +12.3820 q^{97} -6.61803 q^{98} +1.09017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{8} - 3 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{8} - 3 q^{9} + 2 q^{10} + q^{11} + q^{12} - 3 q^{13} + q^{14} + q^{15} + 2 q^{16} + q^{17} - 3 q^{18} - 3 q^{19} + 2 q^{20} - 2 q^{21} + q^{22} + 2 q^{23} + q^{24} + 2 q^{25} - 3 q^{26} - 2 q^{27} + q^{28} - 14 q^{29} + q^{30} + 7 q^{31} + 2 q^{32} - 7 q^{33} + q^{34} + q^{35} - 3 q^{36} + 4 q^{37} - 3 q^{38} - 14 q^{39} + 2 q^{40} - 9 q^{41} - 2 q^{42} + q^{44} - 3 q^{45} + 2 q^{46} + 6 q^{47} + q^{48} - 11 q^{49} + 2 q^{50} + 13 q^{51} - 3 q^{52} - 8 q^{53} - 2 q^{54} + q^{55} + q^{56} + 6 q^{57} - 14 q^{58} - 10 q^{59} + q^{60} - 3 q^{61} + 7 q^{62} - 4 q^{63} + 2 q^{64} - 3 q^{65} - 7 q^{66} + 20 q^{67} + q^{68} + q^{69} + q^{70} + 3 q^{71} - 3 q^{72} + 2 q^{73} + 4 q^{74} + q^{75} - 3 q^{76} + 8 q^{77} - 14 q^{78} + 12 q^{79} + 2 q^{80} - 2 q^{81} - 9 q^{82} + 4 q^{83} - 2 q^{84} + q^{85} - 12 q^{87} + q^{88} - 12 q^{89} - 3 q^{90} + 11 q^{91} + 2 q^{92} + 16 q^{93} + 6 q^{94} - 3 q^{95} + q^{96} + 27 q^{97} - 11 q^{98} - 9 q^{99} + O(q^{100}) \)

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
\(3\) 1.61803 0.934172 0.467086 0.884212i \(-0.345304\pi\)
0.467086 + 0.884212i \(0.345304\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.61803 0.660560
\(7\) −0.618034 −0.233595 −0.116797 0.993156i \(-0.537263\pi\)
−0.116797 + 0.993156i \(0.537263\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.381966 −0.127322
\(10\) 1.00000 0.316228
\(11\) −2.85410 −0.860544 −0.430272 0.902699i \(-0.641582\pi\)
−0.430272 + 0.902699i \(0.641582\pi\)
\(12\) 1.61803 0.467086
\(13\) −7.09017 −1.96646 −0.983230 0.182372i \(-0.941623\pi\)
−0.983230 + 0.182372i \(0.941623\pi\)
\(14\) −0.618034 −0.165177
\(15\) 1.61803 0.417775
\(16\) 1.00000 0.250000
\(17\) 6.09017 1.47708 0.738542 0.674208i \(-0.235515\pi\)
0.738542 + 0.674208i \(0.235515\pi\)
\(18\) −0.381966 −0.0900303
\(19\) 1.85410 0.425360 0.212680 0.977122i \(-0.431781\pi\)
0.212680 + 0.977122i \(0.431781\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) −2.85410 −0.608497
\(23\) 1.00000 0.208514
\(24\) 1.61803 0.330280
\(25\) 1.00000 0.200000
\(26\) −7.09017 −1.39050
\(27\) −5.47214 −1.05311
\(28\) −0.618034 −0.116797
\(29\) −9.23607 −1.71509 −0.857547 0.514405i \(-0.828013\pi\)
−0.857547 + 0.514405i \(0.828013\pi\)
\(30\) 1.61803 0.295411
\(31\) 9.09017 1.63264 0.816321 0.577598i \(-0.196010\pi\)
0.816321 + 0.577598i \(0.196010\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.61803 −0.803897
\(34\) 6.09017 1.04446
\(35\) −0.618034 −0.104467
\(36\) −0.381966 −0.0636610
\(37\) 6.47214 1.06401 0.532006 0.846740i \(-0.321438\pi\)
0.532006 + 0.846740i \(0.321438\pi\)
\(38\) 1.85410 0.300775
\(39\) −11.4721 −1.83701
\(40\) 1.00000 0.158114
\(41\) 3.32624 0.519471 0.259736 0.965680i \(-0.416365\pi\)
0.259736 + 0.965680i \(0.416365\pi\)
\(42\) −1.00000 −0.154303
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −2.85410 −0.430272
\(45\) −0.381966 −0.0569401
\(46\) 1.00000 0.147442
\(47\) −3.70820 −0.540897 −0.270449 0.962734i \(-0.587172\pi\)
−0.270449 + 0.962734i \(0.587172\pi\)
\(48\) 1.61803 0.233543
\(49\) −6.61803 −0.945433
\(50\) 1.00000 0.141421
\(51\) 9.85410 1.37985
\(52\) −7.09017 −0.983230
\(53\) 0.472136 0.0648529 0.0324264 0.999474i \(-0.489677\pi\)
0.0324264 + 0.999474i \(0.489677\pi\)
\(54\) −5.47214 −0.744663
\(55\) −2.85410 −0.384847
\(56\) −0.618034 −0.0825883
\(57\) 3.00000 0.397360
\(58\) −9.23607 −1.21276
\(59\) 1.70820 0.222389 0.111195 0.993799i \(-0.464532\pi\)
0.111195 + 0.993799i \(0.464532\pi\)
\(60\) 1.61803 0.208887
\(61\) −9.32624 −1.19410 −0.597051 0.802203i \(-0.703661\pi\)
−0.597051 + 0.802203i \(0.703661\pi\)
\(62\) 9.09017 1.15445
\(63\) 0.236068 0.0297418
\(64\) 1.00000 0.125000
\(65\) −7.09017 −0.879427
\(66\) −4.61803 −0.568441
\(67\) 14.4721 1.76805 0.884026 0.467437i \(-0.154823\pi\)
0.884026 + 0.467437i \(0.154823\pi\)
\(68\) 6.09017 0.738542
\(69\) 1.61803 0.194788
\(70\) −0.618034 −0.0738692
\(71\) −4.09017 −0.485414 −0.242707 0.970100i \(-0.578035\pi\)
−0.242707 + 0.970100i \(0.578035\pi\)
\(72\) −0.381966 −0.0450151
\(73\) 3.23607 0.378753 0.189377 0.981905i \(-0.439353\pi\)
0.189377 + 0.981905i \(0.439353\pi\)
\(74\) 6.47214 0.752371
\(75\) 1.61803 0.186834
\(76\) 1.85410 0.212680
\(77\) 1.76393 0.201019
\(78\) −11.4721 −1.29896
\(79\) 1.52786 0.171898 0.0859491 0.996300i \(-0.472608\pi\)
0.0859491 + 0.996300i \(0.472608\pi\)
\(80\) 1.00000 0.111803
\(81\) −7.70820 −0.856467
\(82\) 3.32624 0.367322
\(83\) −6.94427 −0.762233 −0.381116 0.924527i \(-0.624460\pi\)
−0.381116 + 0.924527i \(0.624460\pi\)
\(84\) −1.00000 −0.109109
\(85\) 6.09017 0.660572
\(86\) 0 0
\(87\) −14.9443 −1.60219
\(88\) −2.85410 −0.304248
\(89\) −10.4721 −1.11004 −0.555022 0.831836i \(-0.687290\pi\)
−0.555022 + 0.831836i \(0.687290\pi\)
\(90\) −0.381966 −0.0402628
\(91\) 4.38197 0.459355
\(92\) 1.00000 0.104257
\(93\) 14.7082 1.52517
\(94\) −3.70820 −0.382472
\(95\) 1.85410 0.190227
\(96\) 1.61803 0.165140
\(97\) 12.3820 1.25720 0.628599 0.777730i \(-0.283629\pi\)
0.628599 + 0.777730i \(0.283629\pi\)
\(98\) −6.61803 −0.668522
\(99\) 1.09017 0.109566
\(100\) 1.00000 0.100000
\(101\) −0.291796 −0.0290348 −0.0145174 0.999895i \(-0.504621\pi\)
−0.0145174 + 0.999895i \(0.504621\pi\)
\(102\) 9.85410 0.975701
\(103\) 16.5623 1.63193 0.815966 0.578100i \(-0.196205\pi\)
0.815966 + 0.578100i \(0.196205\pi\)
\(104\) −7.09017 −0.695248
\(105\) −1.00000 −0.0975900
\(106\) 0.472136 0.0458579
\(107\) 18.1803 1.75756 0.878780 0.477227i \(-0.158358\pi\)
0.878780 + 0.477227i \(0.158358\pi\)
\(108\) −5.47214 −0.526557
\(109\) −11.5623 −1.10747 −0.553734 0.832694i \(-0.686798\pi\)
−0.553734 + 0.832694i \(0.686798\pi\)
\(110\) −2.85410 −0.272128
\(111\) 10.4721 0.993971
\(112\) −0.618034 −0.0583987
\(113\) 1.05573 0.0993145 0.0496573 0.998766i \(-0.484187\pi\)
0.0496573 + 0.998766i \(0.484187\pi\)
\(114\) 3.00000 0.280976
\(115\) 1.00000 0.0932505
\(116\) −9.23607 −0.857547
\(117\) 2.70820 0.250374
\(118\) 1.70820 0.157253
\(119\) −3.76393 −0.345039
\(120\) 1.61803 0.147706
\(121\) −2.85410 −0.259464
\(122\) −9.32624 −0.844358
\(123\) 5.38197 0.485276
\(124\) 9.09017 0.816321
\(125\) 1.00000 0.0894427
\(126\) 0.236068 0.0210306
\(127\) 16.1803 1.43577 0.717886 0.696160i \(-0.245110\pi\)
0.717886 + 0.696160i \(0.245110\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −7.09017 −0.621849
\(131\) 2.94427 0.257242 0.128621 0.991694i \(-0.458945\pi\)
0.128621 + 0.991694i \(0.458945\pi\)
\(132\) −4.61803 −0.401948
\(133\) −1.14590 −0.0993620
\(134\) 14.4721 1.25020
\(135\) −5.47214 −0.470966
\(136\) 6.09017 0.522228
\(137\) −10.3262 −0.882230 −0.441115 0.897451i \(-0.645417\pi\)
−0.441115 + 0.897451i \(0.645417\pi\)
\(138\) 1.61803 0.137736
\(139\) 12.7639 1.08262 0.541311 0.840822i \(-0.317928\pi\)
0.541311 + 0.840822i \(0.317928\pi\)
\(140\) −0.618034 −0.0522334
\(141\) −6.00000 −0.505291
\(142\) −4.09017 −0.343239
\(143\) 20.2361 1.69223
\(144\) −0.381966 −0.0318305
\(145\) −9.23607 −0.767014
\(146\) 3.23607 0.267819
\(147\) −10.7082 −0.883198
\(148\) 6.47214 0.532006
\(149\) −7.85410 −0.643433 −0.321717 0.946836i \(-0.604260\pi\)
−0.321717 + 0.946836i \(0.604260\pi\)
\(150\) 1.61803 0.132112
\(151\) −2.56231 −0.208517 −0.104259 0.994550i \(-0.533247\pi\)
−0.104259 + 0.994550i \(0.533247\pi\)
\(152\) 1.85410 0.150388
\(153\) −2.32624 −0.188065
\(154\) 1.76393 0.142142
\(155\) 9.09017 0.730140
\(156\) −11.4721 −0.918506
\(157\) −3.70820 −0.295947 −0.147973 0.988991i \(-0.547275\pi\)
−0.147973 + 0.988991i \(0.547275\pi\)
\(158\) 1.52786 0.121550
\(159\) 0.763932 0.0605838
\(160\) 1.00000 0.0790569
\(161\) −0.618034 −0.0487079
\(162\) −7.70820 −0.605614
\(163\) 1.38197 0.108244 0.0541220 0.998534i \(-0.482764\pi\)
0.0541220 + 0.998534i \(0.482764\pi\)
\(164\) 3.32624 0.259736
\(165\) −4.61803 −0.359513
\(166\) −6.94427 −0.538980
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 37.2705 2.86696
\(170\) 6.09017 0.467095
\(171\) −0.708204 −0.0541577
\(172\) 0 0
\(173\) −1.43769 −0.109306 −0.0546529 0.998505i \(-0.517405\pi\)
−0.0546529 + 0.998505i \(0.517405\pi\)
\(174\) −14.9443 −1.13292
\(175\) −0.618034 −0.0467190
\(176\) −2.85410 −0.215136
\(177\) 2.76393 0.207750
\(178\) −10.4721 −0.784920
\(179\) 2.18034 0.162966 0.0814831 0.996675i \(-0.474034\pi\)
0.0814831 + 0.996675i \(0.474034\pi\)
\(180\) −0.381966 −0.0284701
\(181\) −12.1459 −0.902797 −0.451399 0.892322i \(-0.649075\pi\)
−0.451399 + 0.892322i \(0.649075\pi\)
\(182\) 4.38197 0.324813
\(183\) −15.0902 −1.11550
\(184\) 1.00000 0.0737210
\(185\) 6.47214 0.475841
\(186\) 14.7082 1.07846
\(187\) −17.3820 −1.27110
\(188\) −3.70820 −0.270449
\(189\) 3.38197 0.246002
\(190\) 1.85410 0.134511
\(191\) −13.7082 −0.991891 −0.495945 0.868354i \(-0.665178\pi\)
−0.495945 + 0.868354i \(0.665178\pi\)
\(192\) 1.61803 0.116772
\(193\) 0.763932 0.0549890 0.0274945 0.999622i \(-0.491247\pi\)
0.0274945 + 0.999622i \(0.491247\pi\)
\(194\) 12.3820 0.888973
\(195\) −11.4721 −0.821537
\(196\) −6.61803 −0.472717
\(197\) −22.5623 −1.60750 −0.803749 0.594969i \(-0.797164\pi\)
−0.803749 + 0.594969i \(0.797164\pi\)
\(198\) 1.09017 0.0774750
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 1.00000 0.0707107
\(201\) 23.4164 1.65167
\(202\) −0.291796 −0.0205307
\(203\) 5.70820 0.400637
\(204\) 9.85410 0.689925
\(205\) 3.32624 0.232315
\(206\) 16.5623 1.15395
\(207\) −0.381966 −0.0265485
\(208\) −7.09017 −0.491615
\(209\) −5.29180 −0.366041
\(210\) −1.00000 −0.0690066
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 0.472136 0.0324264
\(213\) −6.61803 −0.453460
\(214\) 18.1803 1.24278
\(215\) 0 0
\(216\) −5.47214 −0.372332
\(217\) −5.61803 −0.381377
\(218\) −11.5623 −0.783098
\(219\) 5.23607 0.353821
\(220\) −2.85410 −0.192424
\(221\) −43.1803 −2.90462
\(222\) 10.4721 0.702844
\(223\) −20.9443 −1.40253 −0.701266 0.712900i \(-0.747382\pi\)
−0.701266 + 0.712900i \(0.747382\pi\)
\(224\) −0.618034 −0.0412941
\(225\) −0.381966 −0.0254644
\(226\) 1.05573 0.0702260
\(227\) −18.7639 −1.24541 −0.622703 0.782458i \(-0.713965\pi\)
−0.622703 + 0.782458i \(0.713965\pi\)
\(228\) 3.00000 0.198680
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 1.00000 0.0659380
\(231\) 2.85410 0.187786
\(232\) −9.23607 −0.606378
\(233\) 6.29180 0.412189 0.206095 0.978532i \(-0.433925\pi\)
0.206095 + 0.978532i \(0.433925\pi\)
\(234\) 2.70820 0.177041
\(235\) −3.70820 −0.241897
\(236\) 1.70820 0.111195
\(237\) 2.47214 0.160582
\(238\) −3.76393 −0.243979
\(239\) −20.3607 −1.31702 −0.658511 0.752571i \(-0.728814\pi\)
−0.658511 + 0.752571i \(0.728814\pi\)
\(240\) 1.61803 0.104444
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −2.85410 −0.183469
\(243\) 3.94427 0.253025
\(244\) −9.32624 −0.597051
\(245\) −6.61803 −0.422811
\(246\) 5.38197 0.343142
\(247\) −13.1459 −0.836453
\(248\) 9.09017 0.577226
\(249\) −11.2361 −0.712057
\(250\) 1.00000 0.0632456
\(251\) 6.14590 0.387926 0.193963 0.981009i \(-0.437866\pi\)
0.193963 + 0.981009i \(0.437866\pi\)
\(252\) 0.236068 0.0148709
\(253\) −2.85410 −0.179436
\(254\) 16.1803 1.01524
\(255\) 9.85410 0.617088
\(256\) 1.00000 0.0625000
\(257\) −7.81966 −0.487777 −0.243888 0.969803i \(-0.578423\pi\)
−0.243888 + 0.969803i \(0.578423\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) −7.09017 −0.439714
\(261\) 3.52786 0.218369
\(262\) 2.94427 0.181898
\(263\) 20.7426 1.27905 0.639523 0.768772i \(-0.279132\pi\)
0.639523 + 0.768772i \(0.279132\pi\)
\(264\) −4.61803 −0.284220
\(265\) 0.472136 0.0290031
\(266\) −1.14590 −0.0702595
\(267\) −16.9443 −1.03697
\(268\) 14.4721 0.884026
\(269\) −14.1803 −0.864591 −0.432295 0.901732i \(-0.642296\pi\)
−0.432295 + 0.901732i \(0.642296\pi\)
\(270\) −5.47214 −0.333024
\(271\) −30.3262 −1.84219 −0.921094 0.389341i \(-0.872703\pi\)
−0.921094 + 0.389341i \(0.872703\pi\)
\(272\) 6.09017 0.369271
\(273\) 7.09017 0.429117
\(274\) −10.3262 −0.623831
\(275\) −2.85410 −0.172109
\(276\) 1.61803 0.0973942
\(277\) 29.4164 1.76746 0.883730 0.467996i \(-0.155024\pi\)
0.883730 + 0.467996i \(0.155024\pi\)
\(278\) 12.7639 0.765530
\(279\) −3.47214 −0.207871
\(280\) −0.618034 −0.0369346
\(281\) 22.7639 1.35798 0.678991 0.734146i \(-0.262417\pi\)
0.678991 + 0.734146i \(0.262417\pi\)
\(282\) −6.00000 −0.357295
\(283\) −26.9443 −1.60167 −0.800835 0.598885i \(-0.795611\pi\)
−0.800835 + 0.598885i \(0.795611\pi\)
\(284\) −4.09017 −0.242707
\(285\) 3.00000 0.177705
\(286\) 20.2361 1.19658
\(287\) −2.05573 −0.121346
\(288\) −0.381966 −0.0225076
\(289\) 20.0902 1.18177
\(290\) −9.23607 −0.542361
\(291\) 20.0344 1.17444
\(292\) 3.23607 0.189377
\(293\) −19.8885 −1.16190 −0.580951 0.813939i \(-0.697319\pi\)
−0.580951 + 0.813939i \(0.697319\pi\)
\(294\) −10.7082 −0.624515
\(295\) 1.70820 0.0994555
\(296\) 6.47214 0.376185
\(297\) 15.6180 0.906250
\(298\) −7.85410 −0.454976
\(299\) −7.09017 −0.410035
\(300\) 1.61803 0.0934172
\(301\) 0 0
\(302\) −2.56231 −0.147444
\(303\) −0.472136 −0.0271235
\(304\) 1.85410 0.106340
\(305\) −9.32624 −0.534019
\(306\) −2.32624 −0.132982
\(307\) −28.4508 −1.62378 −0.811888 0.583813i \(-0.801560\pi\)
−0.811888 + 0.583813i \(0.801560\pi\)
\(308\) 1.76393 0.100509
\(309\) 26.7984 1.52451
\(310\) 9.09017 0.516287
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) −11.4721 −0.649482
\(313\) 12.7984 0.723407 0.361703 0.932293i \(-0.382195\pi\)
0.361703 + 0.932293i \(0.382195\pi\)
\(314\) −3.70820 −0.209266
\(315\) 0.236068 0.0133009
\(316\) 1.52786 0.0859491
\(317\) −11.0902 −0.622886 −0.311443 0.950265i \(-0.600812\pi\)
−0.311443 + 0.950265i \(0.600812\pi\)
\(318\) 0.763932 0.0428392
\(319\) 26.3607 1.47591
\(320\) 1.00000 0.0559017
\(321\) 29.4164 1.64186
\(322\) −0.618034 −0.0344417
\(323\) 11.2918 0.628292
\(324\) −7.70820 −0.428234
\(325\) −7.09017 −0.393292
\(326\) 1.38197 0.0765400
\(327\) −18.7082 −1.03457
\(328\) 3.32624 0.183661
\(329\) 2.29180 0.126351
\(330\) −4.61803 −0.254214
\(331\) 19.2361 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(332\) −6.94427 −0.381116
\(333\) −2.47214 −0.135472
\(334\) −8.00000 −0.437741
\(335\) 14.4721 0.790697
\(336\) −1.00000 −0.0545545
\(337\) 13.6738 0.744857 0.372429 0.928061i \(-0.378525\pi\)
0.372429 + 0.928061i \(0.378525\pi\)
\(338\) 37.2705 2.02725
\(339\) 1.70820 0.0927769
\(340\) 6.09017 0.330286
\(341\) −25.9443 −1.40496
\(342\) −0.708204 −0.0382953
\(343\) 8.41641 0.454443
\(344\) 0 0
\(345\) 1.61803 0.0871120
\(346\) −1.43769 −0.0772909
\(347\) −6.38197 −0.342602 −0.171301 0.985219i \(-0.554797\pi\)
−0.171301 + 0.985219i \(0.554797\pi\)
\(348\) −14.9443 −0.801097
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −0.618034 −0.0330353
\(351\) 38.7984 2.07090
\(352\) −2.85410 −0.152124
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 2.76393 0.146901
\(355\) −4.09017 −0.217084
\(356\) −10.4721 −0.555022
\(357\) −6.09017 −0.322326
\(358\) 2.18034 0.115235
\(359\) 26.3607 1.39126 0.695632 0.718399i \(-0.255125\pi\)
0.695632 + 0.718399i \(0.255125\pi\)
\(360\) −0.381966 −0.0201314
\(361\) −15.5623 −0.819069
\(362\) −12.1459 −0.638374
\(363\) −4.61803 −0.242384
\(364\) 4.38197 0.229677
\(365\) 3.23607 0.169384
\(366\) −15.0902 −0.788776
\(367\) 6.47214 0.337843 0.168921 0.985630i \(-0.445972\pi\)
0.168921 + 0.985630i \(0.445972\pi\)
\(368\) 1.00000 0.0521286
\(369\) −1.27051 −0.0661401
\(370\) 6.47214 0.336470
\(371\) −0.291796 −0.0151493
\(372\) 14.7082 0.762585
\(373\) 20.1803 1.04490 0.522449 0.852670i \(-0.325018\pi\)
0.522449 + 0.852670i \(0.325018\pi\)
\(374\) −17.3820 −0.898800
\(375\) 1.61803 0.0835549
\(376\) −3.70820 −0.191236
\(377\) 65.4853 3.37266
\(378\) 3.38197 0.173950
\(379\) −22.4508 −1.15322 −0.576611 0.817019i \(-0.695625\pi\)
−0.576611 + 0.817019i \(0.695625\pi\)
\(380\) 1.85410 0.0951134
\(381\) 26.1803 1.34126
\(382\) −13.7082 −0.701373
\(383\) −17.8885 −0.914062 −0.457031 0.889451i \(-0.651087\pi\)
−0.457031 + 0.889451i \(0.651087\pi\)
\(384\) 1.61803 0.0825700
\(385\) 1.76393 0.0898983
\(386\) 0.763932 0.0388831
\(387\) 0 0
\(388\) 12.3820 0.628599
\(389\) 21.3262 1.08128 0.540642 0.841253i \(-0.318182\pi\)
0.540642 + 0.841253i \(0.318182\pi\)
\(390\) −11.4721 −0.580914
\(391\) 6.09017 0.307993
\(392\) −6.61803 −0.334261
\(393\) 4.76393 0.240309
\(394\) −22.5623 −1.13667
\(395\) 1.52786 0.0768752
\(396\) 1.09017 0.0547831
\(397\) 7.32624 0.367693 0.183847 0.982955i \(-0.441145\pi\)
0.183847 + 0.982955i \(0.441145\pi\)
\(398\) 2.00000 0.100251
\(399\) −1.85410 −0.0928212
\(400\) 1.00000 0.0500000
\(401\) −1.70820 −0.0853036 −0.0426518 0.999090i \(-0.513581\pi\)
−0.0426518 + 0.999090i \(0.513581\pi\)
\(402\) 23.4164 1.16790
\(403\) −64.4508 −3.21053
\(404\) −0.291796 −0.0145174
\(405\) −7.70820 −0.383024
\(406\) 5.70820 0.283293
\(407\) −18.4721 −0.915630
\(408\) 9.85410 0.487851
\(409\) −30.2148 −1.49402 −0.747012 0.664810i \(-0.768512\pi\)
−0.747012 + 0.664810i \(0.768512\pi\)
\(410\) 3.32624 0.164271
\(411\) −16.7082 −0.824155
\(412\) 16.5623 0.815966
\(413\) −1.05573 −0.0519490
\(414\) −0.381966 −0.0187726
\(415\) −6.94427 −0.340881
\(416\) −7.09017 −0.347624
\(417\) 20.6525 1.01136
\(418\) −5.29180 −0.258830
\(419\) 14.4721 0.707010 0.353505 0.935433i \(-0.384990\pi\)
0.353505 + 0.935433i \(0.384990\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 13.7426 0.669776 0.334888 0.942258i \(-0.391302\pi\)
0.334888 + 0.942258i \(0.391302\pi\)
\(422\) 14.0000 0.681509
\(423\) 1.41641 0.0688681
\(424\) 0.472136 0.0229289
\(425\) 6.09017 0.295417
\(426\) −6.61803 −0.320645
\(427\) 5.76393 0.278936
\(428\) 18.1803 0.878780
\(429\) 32.7426 1.58083
\(430\) 0 0
\(431\) 3.34752 0.161245 0.0806223 0.996745i \(-0.474309\pi\)
0.0806223 + 0.996745i \(0.474309\pi\)
\(432\) −5.47214 −0.263278
\(433\) 8.50658 0.408800 0.204400 0.978887i \(-0.434476\pi\)
0.204400 + 0.978887i \(0.434476\pi\)
\(434\) −5.61803 −0.269674
\(435\) −14.9443 −0.716523
\(436\) −11.5623 −0.553734
\(437\) 1.85410 0.0886937
\(438\) 5.23607 0.250189
\(439\) −13.3820 −0.638686 −0.319343 0.947639i \(-0.603462\pi\)
−0.319343 + 0.947639i \(0.603462\pi\)
\(440\) −2.85410 −0.136064
\(441\) 2.52786 0.120374
\(442\) −43.1803 −2.05388
\(443\) 25.0902 1.19207 0.596035 0.802958i \(-0.296742\pi\)
0.596035 + 0.802958i \(0.296742\pi\)
\(444\) 10.4721 0.496986
\(445\) −10.4721 −0.496427
\(446\) −20.9443 −0.991740
\(447\) −12.7082 −0.601077
\(448\) −0.618034 −0.0291994
\(449\) −1.56231 −0.0737298 −0.0368649 0.999320i \(-0.511737\pi\)
−0.0368649 + 0.999320i \(0.511737\pi\)
\(450\) −0.381966 −0.0180061
\(451\) −9.49342 −0.447028
\(452\) 1.05573 0.0496573
\(453\) −4.14590 −0.194791
\(454\) −18.7639 −0.880635
\(455\) 4.38197 0.205430
\(456\) 3.00000 0.140488
\(457\) −37.7771 −1.76714 −0.883569 0.468301i \(-0.844866\pi\)
−0.883569 + 0.468301i \(0.844866\pi\)
\(458\) 10.0000 0.467269
\(459\) −33.3262 −1.55554
\(460\) 1.00000 0.0466252
\(461\) −39.2361 −1.82741 −0.913703 0.406383i \(-0.866790\pi\)
−0.913703 + 0.406383i \(0.866790\pi\)
\(462\) 2.85410 0.132785
\(463\) 2.00000 0.0929479 0.0464739 0.998920i \(-0.485202\pi\)
0.0464739 + 0.998920i \(0.485202\pi\)
\(464\) −9.23607 −0.428774
\(465\) 14.7082 0.682077
\(466\) 6.29180 0.291462
\(467\) −17.1246 −0.792433 −0.396216 0.918157i \(-0.629677\pi\)
−0.396216 + 0.918157i \(0.629677\pi\)
\(468\) 2.70820 0.125187
\(469\) −8.94427 −0.413008
\(470\) −3.70820 −0.171047
\(471\) −6.00000 −0.276465
\(472\) 1.70820 0.0786265
\(473\) 0 0
\(474\) 2.47214 0.113549
\(475\) 1.85410 0.0850720
\(476\) −3.76393 −0.172520
\(477\) −0.180340 −0.00825720
\(478\) −20.3607 −0.931276
\(479\) −31.8885 −1.45702 −0.728512 0.685033i \(-0.759788\pi\)
−0.728512 + 0.685033i \(0.759788\pi\)
\(480\) 1.61803 0.0738528
\(481\) −45.8885 −2.09234
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) −2.85410 −0.129732
\(485\) 12.3820 0.562236
\(486\) 3.94427 0.178916
\(487\) −19.8197 −0.898115 −0.449057 0.893503i \(-0.648240\pi\)
−0.449057 + 0.893503i \(0.648240\pi\)
\(488\) −9.32624 −0.422179
\(489\) 2.23607 0.101118
\(490\) −6.61803 −0.298972
\(491\) 6.18034 0.278915 0.139457 0.990228i \(-0.455464\pi\)
0.139457 + 0.990228i \(0.455464\pi\)
\(492\) 5.38197 0.242638
\(493\) −56.2492 −2.53334
\(494\) −13.1459 −0.591462
\(495\) 1.09017 0.0489995
\(496\) 9.09017 0.408161
\(497\) 2.52786 0.113390
\(498\) −11.2361 −0.503500
\(499\) 12.3607 0.553340 0.276670 0.960965i \(-0.410769\pi\)
0.276670 + 0.960965i \(0.410769\pi\)
\(500\) 1.00000 0.0447214
\(501\) −12.9443 −0.578307
\(502\) 6.14590 0.274305
\(503\) 36.3262 1.61971 0.809853 0.586632i \(-0.199547\pi\)
0.809853 + 0.586632i \(0.199547\pi\)
\(504\) 0.236068 0.0105153
\(505\) −0.291796 −0.0129848
\(506\) −2.85410 −0.126880
\(507\) 60.3050 2.67824
\(508\) 16.1803 0.717886
\(509\) 36.6525 1.62459 0.812296 0.583245i \(-0.198217\pi\)
0.812296 + 0.583245i \(0.198217\pi\)
\(510\) 9.85410 0.436347
\(511\) −2.00000 −0.0884748
\(512\) 1.00000 0.0441942
\(513\) −10.1459 −0.447952
\(514\) −7.81966 −0.344910
\(515\) 16.5623 0.729822
\(516\) 0 0
\(517\) 10.5836 0.465466
\(518\) −4.00000 −0.175750
\(519\) −2.32624 −0.102111
\(520\) −7.09017 −0.310925
\(521\) −15.5279 −0.680288 −0.340144 0.940373i \(-0.610476\pi\)
−0.340144 + 0.940373i \(0.610476\pi\)
\(522\) 3.52786 0.154410
\(523\) −26.0000 −1.13690 −0.568450 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) 2.94427 0.128621
\(525\) −1.00000 −0.0436436
\(526\) 20.7426 0.904422
\(527\) 55.3607 2.41155
\(528\) −4.61803 −0.200974
\(529\) 1.00000 0.0434783
\(530\) 0.472136 0.0205083
\(531\) −0.652476 −0.0283150
\(532\) −1.14590 −0.0496810
\(533\) −23.5836 −1.02152
\(534\) −16.9443 −0.733250
\(535\) 18.1803 0.786005
\(536\) 14.4721 0.625101
\(537\) 3.52786 0.152239
\(538\) −14.1803 −0.611358
\(539\) 18.8885 0.813587
\(540\) −5.47214 −0.235483
\(541\) 22.8328 0.981659 0.490830 0.871256i \(-0.336694\pi\)
0.490830 + 0.871256i \(0.336694\pi\)
\(542\) −30.3262 −1.30262
\(543\) −19.6525 −0.843368
\(544\) 6.09017 0.261114
\(545\) −11.5623 −0.495275
\(546\) 7.09017 0.303431
\(547\) 27.9230 1.19390 0.596950 0.802278i \(-0.296379\pi\)
0.596950 + 0.802278i \(0.296379\pi\)
\(548\) −10.3262 −0.441115
\(549\) 3.56231 0.152036
\(550\) −2.85410 −0.121699
\(551\) −17.1246 −0.729533
\(552\) 1.61803 0.0688681
\(553\) −0.944272 −0.0401545
\(554\) 29.4164 1.24978
\(555\) 10.4721 0.444517
\(556\) 12.7639 0.541311
\(557\) −22.8328 −0.967457 −0.483729 0.875218i \(-0.660718\pi\)
−0.483729 + 0.875218i \(0.660718\pi\)
\(558\) −3.47214 −0.146987
\(559\) 0 0
\(560\) −0.618034 −0.0261167
\(561\) −28.1246 −1.18742
\(562\) 22.7639 0.960239
\(563\) −13.8885 −0.585332 −0.292666 0.956215i \(-0.594542\pi\)
−0.292666 + 0.956215i \(0.594542\pi\)
\(564\) −6.00000 −0.252646
\(565\) 1.05573 0.0444148
\(566\) −26.9443 −1.13255
\(567\) 4.76393 0.200066
\(568\) −4.09017 −0.171620
\(569\) −2.00000 −0.0838444 −0.0419222 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) 3.00000 0.125656
\(571\) 15.9787 0.668688 0.334344 0.942451i \(-0.391485\pi\)
0.334344 + 0.942451i \(0.391485\pi\)
\(572\) 20.2361 0.846113
\(573\) −22.1803 −0.926597
\(574\) −2.05573 −0.0858044
\(575\) 1.00000 0.0417029
\(576\) −0.381966 −0.0159153
\(577\) −3.52786 −0.146867 −0.0734335 0.997300i \(-0.523396\pi\)
−0.0734335 + 0.997300i \(0.523396\pi\)
\(578\) 20.0902 0.835641
\(579\) 1.23607 0.0513692
\(580\) −9.23607 −0.383507
\(581\) 4.29180 0.178054
\(582\) 20.0344 0.830454
\(583\) −1.34752 −0.0558087
\(584\) 3.23607 0.133909
\(585\) 2.70820 0.111970
\(586\) −19.8885 −0.821588
\(587\) 13.6180 0.562076 0.281038 0.959697i \(-0.409321\pi\)
0.281038 + 0.959697i \(0.409321\pi\)
\(588\) −10.7082 −0.441599
\(589\) 16.8541 0.694461
\(590\) 1.70820 0.0703256
\(591\) −36.5066 −1.50168
\(592\) 6.47214 0.266003
\(593\) 39.2361 1.61123 0.805616 0.592438i \(-0.201834\pi\)
0.805616 + 0.592438i \(0.201834\pi\)
\(594\) 15.6180 0.640816
\(595\) −3.76393 −0.154306
\(596\) −7.85410 −0.321717
\(597\) 3.23607 0.132443
\(598\) −7.09017 −0.289939
\(599\) −18.3820 −0.751067 −0.375533 0.926809i \(-0.622540\pi\)
−0.375533 + 0.926809i \(0.622540\pi\)
\(600\) 1.61803 0.0660560
\(601\) −33.2705 −1.35713 −0.678566 0.734539i \(-0.737398\pi\)
−0.678566 + 0.734539i \(0.737398\pi\)
\(602\) 0 0
\(603\) −5.52786 −0.225112
\(604\) −2.56231 −0.104259
\(605\) −2.85410 −0.116036
\(606\) −0.472136 −0.0191792
\(607\) 26.4721 1.07447 0.537235 0.843432i \(-0.319469\pi\)
0.537235 + 0.843432i \(0.319469\pi\)
\(608\) 1.85410 0.0751938
\(609\) 9.23607 0.374264
\(610\) −9.32624 −0.377608
\(611\) 26.2918 1.06365
\(612\) −2.32624 −0.0940326
\(613\) 19.3050 0.779720 0.389860 0.920874i \(-0.372523\pi\)
0.389860 + 0.920874i \(0.372523\pi\)
\(614\) −28.4508 −1.14818
\(615\) 5.38197 0.217022
\(616\) 1.76393 0.0710708
\(617\) 34.0902 1.37242 0.686209 0.727404i \(-0.259273\pi\)
0.686209 + 0.727404i \(0.259273\pi\)
\(618\) 26.7984 1.07799
\(619\) −2.79837 −0.112476 −0.0562381 0.998417i \(-0.517911\pi\)
−0.0562381 + 0.998417i \(0.517911\pi\)
\(620\) 9.09017 0.365070
\(621\) −5.47214 −0.219589
\(622\) 4.00000 0.160385
\(623\) 6.47214 0.259301
\(624\) −11.4721 −0.459253
\(625\) 1.00000 0.0400000
\(626\) 12.7984 0.511526
\(627\) −8.56231 −0.341946
\(628\) −3.70820 −0.147973
\(629\) 39.4164 1.57164
\(630\) 0.236068 0.00940517
\(631\) 42.0689 1.67474 0.837368 0.546640i \(-0.184093\pi\)
0.837368 + 0.546640i \(0.184093\pi\)
\(632\) 1.52786 0.0607752
\(633\) 22.6525 0.900355
\(634\) −11.0902 −0.440447
\(635\) 16.1803 0.642097
\(636\) 0.763932 0.0302919
\(637\) 46.9230 1.85916
\(638\) 26.3607 1.04363
\(639\) 1.56231 0.0618039
\(640\) 1.00000 0.0395285
\(641\) 0.360680 0.0142460 0.00712300 0.999975i \(-0.497733\pi\)
0.00712300 + 0.999975i \(0.497733\pi\)
\(642\) 29.4164 1.16097
\(643\) −8.29180 −0.326997 −0.163498 0.986544i \(-0.552278\pi\)
−0.163498 + 0.986544i \(0.552278\pi\)
\(644\) −0.618034 −0.0243540
\(645\) 0 0
\(646\) 11.2918 0.444270
\(647\) −36.2492 −1.42510 −0.712552 0.701619i \(-0.752461\pi\)
−0.712552 + 0.701619i \(0.752461\pi\)
\(648\) −7.70820 −0.302807
\(649\) −4.87539 −0.191376
\(650\) −7.09017 −0.278099
\(651\) −9.09017 −0.356272
\(652\) 1.38197 0.0541220
\(653\) −8.03444 −0.314412 −0.157206 0.987566i \(-0.550249\pi\)
−0.157206 + 0.987566i \(0.550249\pi\)
\(654\) −18.7082 −0.731549
\(655\) 2.94427 0.115042
\(656\) 3.32624 0.129868
\(657\) −1.23607 −0.0482236
\(658\) 2.29180 0.0893435
\(659\) −46.2492 −1.80161 −0.900807 0.434220i \(-0.857024\pi\)
−0.900807 + 0.434220i \(0.857024\pi\)
\(660\) −4.61803 −0.179757
\(661\) 18.6738 0.726325 0.363163 0.931726i \(-0.381697\pi\)
0.363163 + 0.931726i \(0.381697\pi\)
\(662\) 19.2361 0.747631
\(663\) −69.8673 −2.71342
\(664\) −6.94427 −0.269490
\(665\) −1.14590 −0.0444360
\(666\) −2.47214 −0.0957933
\(667\) −9.23607 −0.357622
\(668\) −8.00000 −0.309529
\(669\) −33.8885 −1.31021
\(670\) 14.4721 0.559107
\(671\) 26.6180 1.02758
\(672\) −1.00000 −0.0385758
\(673\) −10.9443 −0.421871 −0.210935 0.977500i \(-0.567651\pi\)
−0.210935 + 0.977500i \(0.567651\pi\)
\(674\) 13.6738 0.526694
\(675\) −5.47214 −0.210623
\(676\) 37.2705 1.43348
\(677\) −50.9443 −1.95795 −0.978974 0.203986i \(-0.934610\pi\)
−0.978974 + 0.203986i \(0.934610\pi\)
\(678\) 1.70820 0.0656032
\(679\) −7.65248 −0.293675
\(680\) 6.09017 0.233547
\(681\) −30.3607 −1.16342
\(682\) −25.9443 −0.993458
\(683\) −31.5623 −1.20770 −0.603849 0.797099i \(-0.706367\pi\)
−0.603849 + 0.797099i \(0.706367\pi\)
\(684\) −0.708204 −0.0270789
\(685\) −10.3262 −0.394545
\(686\) 8.41641 0.321340
\(687\) 16.1803 0.617318
\(688\) 0 0
\(689\) −3.34752 −0.127531
\(690\) 1.61803 0.0615975
\(691\) −29.2361 −1.11219 −0.556096 0.831118i \(-0.687701\pi\)
−0.556096 + 0.831118i \(0.687701\pi\)
\(692\) −1.43769 −0.0546529
\(693\) −0.673762 −0.0255941
\(694\) −6.38197 −0.242256
\(695\) 12.7639 0.484164
\(696\) −14.9443 −0.566461
\(697\) 20.2574 0.767302
\(698\) −2.00000 −0.0757011
\(699\) 10.1803 0.385056
\(700\) −0.618034 −0.0233595
\(701\) 43.3394 1.63691 0.818453 0.574573i \(-0.194832\pi\)
0.818453 + 0.574573i \(0.194832\pi\)
\(702\) 38.7984 1.46435
\(703\) 12.0000 0.452589
\(704\) −2.85410 −0.107568
\(705\) −6.00000 −0.225973
\(706\) 24.0000 0.903252
\(707\) 0.180340 0.00678238
\(708\) 2.76393 0.103875
\(709\) −26.0902 −0.979837 −0.489918 0.871768i \(-0.662973\pi\)
−0.489918 + 0.871768i \(0.662973\pi\)
\(710\) −4.09017 −0.153501
\(711\) −0.583592 −0.0218864
\(712\) −10.4721 −0.392460
\(713\) 9.09017 0.340430
\(714\) −6.09017 −0.227919
\(715\) 20.2361 0.756786
\(716\) 2.18034 0.0814831
\(717\) −32.9443 −1.23033
\(718\) 26.3607 0.983772
\(719\) 35.2705 1.31537 0.657684 0.753294i \(-0.271536\pi\)
0.657684 + 0.753294i \(0.271536\pi\)
\(720\) −0.381966 −0.0142350
\(721\) −10.2361 −0.381211
\(722\) −15.5623 −0.579169
\(723\) 0 0
\(724\) −12.1459 −0.451399
\(725\) −9.23607 −0.343019
\(726\) −4.61803 −0.171391
\(727\) 28.2016 1.04594 0.522970 0.852351i \(-0.324824\pi\)
0.522970 + 0.852351i \(0.324824\pi\)
\(728\) 4.38197 0.162406
\(729\) 29.5066 1.09284
\(730\) 3.23607 0.119772
\(731\) 0 0
\(732\) −15.0902 −0.557749
\(733\) −29.4164 −1.08652 −0.543260 0.839565i \(-0.682810\pi\)
−0.543260 + 0.839565i \(0.682810\pi\)
\(734\) 6.47214 0.238891
\(735\) −10.7082 −0.394978
\(736\) 1.00000 0.0368605
\(737\) −41.3050 −1.52149
\(738\) −1.27051 −0.0467681
\(739\) −13.8885 −0.510898 −0.255449 0.966822i \(-0.582223\pi\)
−0.255449 + 0.966822i \(0.582223\pi\)
\(740\) 6.47214 0.237920
\(741\) −21.2705 −0.781392
\(742\) −0.291796 −0.0107122
\(743\) 33.6312 1.23381 0.616904 0.787038i \(-0.288387\pi\)
0.616904 + 0.787038i \(0.288387\pi\)
\(744\) 14.7082 0.539229
\(745\) −7.85410 −0.287752
\(746\) 20.1803 0.738855
\(747\) 2.65248 0.0970490
\(748\) −17.3820 −0.635548
\(749\) −11.2361 −0.410557
\(750\) 1.61803 0.0590822
\(751\) −47.0132 −1.71553 −0.857767 0.514038i \(-0.828149\pi\)
−0.857767 + 0.514038i \(0.828149\pi\)
\(752\) −3.70820 −0.135224
\(753\) 9.94427 0.362389
\(754\) 65.4853 2.38483
\(755\) −2.56231 −0.0932519
\(756\) 3.38197 0.123001
\(757\) −17.8885 −0.650170 −0.325085 0.945685i \(-0.605393\pi\)
−0.325085 + 0.945685i \(0.605393\pi\)
\(758\) −22.4508 −0.815452
\(759\) −4.61803 −0.167624
\(760\) 1.85410 0.0672553
\(761\) 46.8673 1.69894 0.849468 0.527640i \(-0.176923\pi\)
0.849468 + 0.527640i \(0.176923\pi\)
\(762\) 26.1803 0.948414
\(763\) 7.14590 0.258699
\(764\) −13.7082 −0.495945
\(765\) −2.32624 −0.0841053
\(766\) −17.8885 −0.646339
\(767\) −12.1115 −0.437319
\(768\) 1.61803 0.0583858
\(769\) −6.58359 −0.237410 −0.118705 0.992930i \(-0.537874\pi\)
−0.118705 + 0.992930i \(0.537874\pi\)
\(770\) 1.76393 0.0635677
\(771\) −12.6525 −0.455668
\(772\) 0.763932 0.0274945
\(773\) 28.9443 1.04105 0.520527 0.853845i \(-0.325736\pi\)
0.520527 + 0.853845i \(0.325736\pi\)
\(774\) 0 0
\(775\) 9.09017 0.326529
\(776\) 12.3820 0.444487
\(777\) −6.47214 −0.232187
\(778\) 21.3262 0.764583
\(779\) 6.16718 0.220962
\(780\) −11.4721 −0.410768
\(781\) 11.6738 0.417720
\(782\) 6.09017 0.217784
\(783\) 50.5410 1.80619
\(784\) −6.61803 −0.236358
\(785\) −3.70820 −0.132351
\(786\) 4.76393 0.169924
\(787\) −2.87539 −0.102497 −0.0512483 0.998686i \(-0.516320\pi\)
−0.0512483 + 0.998686i \(0.516320\pi\)
\(788\) −22.5623 −0.803749
\(789\) 33.5623 1.19485
\(790\) 1.52786 0.0543590
\(791\) −0.652476 −0.0231994
\(792\) 1.09017 0.0387375
\(793\) 66.1246 2.34815
\(794\) 7.32624 0.259998
\(795\) 0.763932 0.0270939
\(796\) 2.00000 0.0708881
\(797\) 13.7082 0.485569 0.242785 0.970080i \(-0.421939\pi\)
0.242785 + 0.970080i \(0.421939\pi\)
\(798\) −1.85410 −0.0656345
\(799\) −22.5836 −0.798950
\(800\) 1.00000 0.0353553
\(801\) 4.00000 0.141333
\(802\) −1.70820 −0.0603188
\(803\) −9.23607 −0.325934
\(804\) 23.4164 0.825833
\(805\) −0.618034 −0.0217828
\(806\) −64.4508 −2.27018
\(807\) −22.9443 −0.807677
\(808\) −0.291796 −0.0102653
\(809\) −46.7426 −1.64338 −0.821692 0.569932i \(-0.806970\pi\)
−0.821692 + 0.569932i \(0.806970\pi\)
\(810\) −7.70820 −0.270839
\(811\) 21.8197 0.766192 0.383096 0.923709i \(-0.374858\pi\)
0.383096 + 0.923709i \(0.374858\pi\)
\(812\) 5.70820 0.200319
\(813\) −49.0689 −1.72092
\(814\) −18.4721 −0.647448
\(815\) 1.38197 0.0484082
\(816\) 9.85410 0.344963
\(817\) 0 0
\(818\) −30.2148 −1.05644
\(819\) −1.67376 −0.0584860
\(820\) 3.32624 0.116157
\(821\) −33.0557 −1.15365 −0.576826 0.816867i \(-0.695709\pi\)
−0.576826 + 0.816867i \(0.695709\pi\)
\(822\) −16.7082 −0.582766
\(823\) 25.4164 0.885960 0.442980 0.896531i \(-0.353921\pi\)
0.442980 + 0.896531i \(0.353921\pi\)
\(824\) 16.5623 0.576975
\(825\) −4.61803 −0.160779
\(826\) −1.05573 −0.0367335
\(827\) 21.7082 0.754868 0.377434 0.926036i \(-0.376806\pi\)
0.377434 + 0.926036i \(0.376806\pi\)
\(828\) −0.381966 −0.0132742
\(829\) 18.9443 0.657962 0.328981 0.944337i \(-0.393295\pi\)
0.328981 + 0.944337i \(0.393295\pi\)
\(830\) −6.94427 −0.241039
\(831\) 47.5967 1.65111
\(832\) −7.09017 −0.245807
\(833\) −40.3050 −1.39648
\(834\) 20.6525 0.715137
\(835\) −8.00000 −0.276851
\(836\) −5.29180 −0.183021
\(837\) −49.7426 −1.71936
\(838\) 14.4721 0.499932
\(839\) 33.0132 1.13974 0.569870 0.821735i \(-0.306993\pi\)
0.569870 + 0.821735i \(0.306993\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 56.3050 1.94155
\(842\) 13.7426 0.473603
\(843\) 36.8328 1.26859
\(844\) 14.0000 0.481900
\(845\) 37.2705 1.28214
\(846\) 1.41641 0.0486971
\(847\) 1.76393 0.0606094
\(848\) 0.472136 0.0162132
\(849\) −43.5967 −1.49624
\(850\) 6.09017 0.208891
\(851\) 6.47214 0.221862
\(852\) −6.61803 −0.226730
\(853\) −10.7984 −0.369729 −0.184865 0.982764i \(-0.559185\pi\)
−0.184865 + 0.982764i \(0.559185\pi\)
\(854\) 5.76393 0.197238
\(855\) −0.708204 −0.0242201
\(856\) 18.1803 0.621391
\(857\) −6.58359 −0.224891 −0.112446 0.993658i \(-0.535868\pi\)
−0.112446 + 0.993658i \(0.535868\pi\)
\(858\) 32.7426 1.11782
\(859\) −24.0689 −0.821220 −0.410610 0.911811i \(-0.634684\pi\)
−0.410610 + 0.911811i \(0.634684\pi\)
\(860\) 0 0
\(861\) −3.32624 −0.113358
\(862\) 3.34752 0.114017
\(863\) 32.7639 1.11530 0.557649 0.830077i \(-0.311704\pi\)
0.557649 + 0.830077i \(0.311704\pi\)
\(864\) −5.47214 −0.186166
\(865\) −1.43769 −0.0488831
\(866\) 8.50658 0.289065
\(867\) 32.5066 1.10398
\(868\) −5.61803 −0.190688
\(869\) −4.36068 −0.147926
\(870\) −14.9443 −0.506658
\(871\) −102.610 −3.47680
\(872\) −11.5623 −0.391549
\(873\) −4.72949 −0.160069
\(874\) 1.85410 0.0627159
\(875\) −0.618034 −0.0208934
\(876\) 5.23607 0.176910
\(877\) 18.7426 0.632894 0.316447 0.948610i \(-0.397510\pi\)
0.316447 + 0.948610i \(0.397510\pi\)
\(878\) −13.3820 −0.451619
\(879\) −32.1803 −1.08542
\(880\) −2.85410 −0.0962118
\(881\) 8.58359 0.289189 0.144594 0.989491i \(-0.453812\pi\)
0.144594 + 0.989491i \(0.453812\pi\)
\(882\) 2.52786 0.0851176
\(883\) 15.5623 0.523713 0.261857 0.965107i \(-0.415665\pi\)
0.261857 + 0.965107i \(0.415665\pi\)
\(884\) −43.1803 −1.45231
\(885\) 2.76393 0.0929086
\(886\) 25.0902 0.842921
\(887\) −5.16718 −0.173497 −0.0867485 0.996230i \(-0.527648\pi\)
−0.0867485 + 0.996230i \(0.527648\pi\)
\(888\) 10.4721 0.351422
\(889\) −10.0000 −0.335389
\(890\) −10.4721 −0.351027
\(891\) 22.0000 0.737028
\(892\) −20.9443 −0.701266
\(893\) −6.87539 −0.230076
\(894\) −12.7082 −0.425026
\(895\) 2.18034 0.0728807
\(896\) −0.618034 −0.0206471
\(897\) −11.4721 −0.383043
\(898\) −1.56231 −0.0521348
\(899\) −83.9574 −2.80014
\(900\) −0.381966 −0.0127322
\(901\) 2.87539 0.0957931
\(902\) −9.49342 −0.316096
\(903\) 0 0
\(904\) 1.05573 0.0351130
\(905\) −12.1459 −0.403743
\(906\) −4.14590 −0.137738
\(907\) 7.12461 0.236569 0.118284 0.992980i \(-0.462261\pi\)
0.118284 + 0.992980i \(0.462261\pi\)
\(908\) −18.7639 −0.622703
\(909\) 0.111456 0.00369677
\(910\) 4.38197 0.145261
\(911\) 36.0689 1.19502 0.597508 0.801863i \(-0.296158\pi\)
0.597508 + 0.801863i \(0.296158\pi\)
\(912\) 3.00000 0.0993399
\(913\) 19.8197 0.655935
\(914\) −37.7771 −1.24955
\(915\) −15.0902 −0.498866
\(916\) 10.0000 0.330409
\(917\) −1.81966 −0.0600905
\(918\) −33.3262 −1.09993
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 1.00000 0.0329690
\(921\) −46.0344 −1.51689
\(922\) −39.2361 −1.29217
\(923\) 29.0000 0.954547
\(924\) 2.85410 0.0938931
\(925\) 6.47214 0.212803
\(926\) 2.00000 0.0657241
\(927\) −6.32624 −0.207781
\(928\) −9.23607 −0.303189
\(929\) −3.52786 −0.115745 −0.0578727 0.998324i \(-0.518432\pi\)
−0.0578727 + 0.998324i \(0.518432\pi\)
\(930\) 14.7082 0.482301
\(931\) −12.2705 −0.402150
\(932\) 6.29180 0.206095
\(933\) 6.47214 0.211888
\(934\) −17.1246 −0.560334
\(935\) −17.3820 −0.568451
\(936\) 2.70820 0.0885204
\(937\) −36.7984 −1.20215 −0.601075 0.799192i \(-0.705261\pi\)
−0.601075 + 0.799192i \(0.705261\pi\)
\(938\) −8.94427 −0.292041
\(939\) 20.7082 0.675787
\(940\) −3.70820 −0.120948
\(941\) −22.4934 −0.733265 −0.366632 0.930366i \(-0.619489\pi\)
−0.366632 + 0.930366i \(0.619489\pi\)
\(942\) −6.00000 −0.195491
\(943\) 3.32624 0.108317
\(944\) 1.70820 0.0555973
\(945\) 3.38197 0.110015
\(946\) 0 0
\(947\) −54.6869 −1.77709 −0.888543 0.458793i \(-0.848282\pi\)
−0.888543 + 0.458793i \(0.848282\pi\)
\(948\) 2.47214 0.0802912
\(949\) −22.9443 −0.744803
\(950\) 1.85410 0.0601550
\(951\) −17.9443 −0.581883
\(952\) −3.76393 −0.121990
\(953\) 3.79837 0.123041 0.0615207 0.998106i \(-0.480405\pi\)
0.0615207 + 0.998106i \(0.480405\pi\)
\(954\) −0.180340 −0.00583872
\(955\) −13.7082 −0.443587
\(956\) −20.3607 −0.658511
\(957\) 42.6525 1.37876
\(958\) −31.8885 −1.03027
\(959\) 6.38197 0.206084
\(960\) 1.61803 0.0522218
\(961\) 51.6312 1.66552
\(962\) −45.8885 −1.47951
\(963\) −6.94427 −0.223776
\(964\) 0 0
\(965\) 0.763932 0.0245918
\(966\) −1.00000 −0.0321745
\(967\) −16.5410 −0.531923 −0.265962 0.963984i \(-0.585689\pi\)
−0.265962 + 0.963984i \(0.585689\pi\)
\(968\) −2.85410 −0.0917343
\(969\) 18.2705 0.586933
\(970\) 12.3820 0.397561
\(971\) 34.2705 1.09979 0.549896 0.835233i \(-0.314667\pi\)
0.549896 + 0.835233i \(0.314667\pi\)
\(972\) 3.94427 0.126513
\(973\) −7.88854 −0.252895
\(974\) −19.8197 −0.635063
\(975\) −11.4721 −0.367402
\(976\) −9.32624 −0.298526
\(977\) −23.5623 −0.753825 −0.376912 0.926249i \(-0.623014\pi\)
−0.376912 + 0.926249i \(0.623014\pi\)
\(978\) 2.23607 0.0715016
\(979\) 29.8885 0.955242
\(980\) −6.61803 −0.211405
\(981\) 4.41641 0.141005
\(982\) 6.18034 0.197223
\(983\) −14.2705 −0.455159 −0.227579 0.973760i \(-0.573081\pi\)
−0.227579 + 0.973760i \(0.573081\pi\)
\(984\) 5.38197 0.171571
\(985\) −22.5623 −0.718895
\(986\) −56.2492 −1.79134
\(987\) 3.70820 0.118033
\(988\) −13.1459 −0.418227
\(989\) 0 0
\(990\) 1.09017 0.0346479
\(991\) 27.5066 0.873775 0.436888 0.899516i \(-0.356081\pi\)
0.436888 + 0.899516i \(0.356081\pi\)
\(992\) 9.09017 0.288613
\(993\) 31.1246 0.987710
\(994\) 2.52786 0.0801790
\(995\) 2.00000 0.0634043
\(996\) −11.2361 −0.356028
\(997\) 57.1935 1.81134 0.905668 0.423987i \(-0.139370\pi\)
0.905668 + 0.423987i \(0.139370\pi\)
\(998\) 12.3607 0.391270
\(999\) −35.4164 −1.12053
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 230.2.a.c.1.2 2
3.2 odd 2 2070.2.a.u.1.1 2
4.3 odd 2 1840.2.a.l.1.1 2
5.2 odd 4 1150.2.b.i.599.3 4
5.3 odd 4 1150.2.b.i.599.2 4
5.4 even 2 1150.2.a.j.1.1 2
8.3 odd 2 7360.2.a.bn.1.2 2
8.5 even 2 7360.2.a.bh.1.1 2
20.19 odd 2 9200.2.a.bu.1.2 2
23.22 odd 2 5290.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.2 2 1.1 even 1 trivial
1150.2.a.j.1.1 2 5.4 even 2
1150.2.b.i.599.2 4 5.3 odd 4
1150.2.b.i.599.3 4 5.2 odd 4
1840.2.a.l.1.1 2 4.3 odd 2
2070.2.a.u.1.1 2 3.2 odd 2
5290.2.a.o.1.2 2 23.22 odd 2
7360.2.a.bh.1.1 2 8.5 even 2
7360.2.a.bn.1.2 2 8.3 odd 2
9200.2.a.bu.1.2 2 20.19 odd 2