Properties

Label 3630.2.a.bo.1.2
Level $3630$
Weight $2$
Character 3630.1
Self dual yes
Analytic conductor $28.986$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3630.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -0.267949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -0.267949 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -4.46410 q^{13} -0.267949 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.73205 q^{17} +1.00000 q^{18} -7.00000 q^{19} -1.00000 q^{20} -0.267949 q^{21} -6.73205 q^{23} +1.00000 q^{24} +1.00000 q^{25} -4.46410 q^{26} +1.00000 q^{27} -0.267949 q^{28} -5.00000 q^{29} -1.00000 q^{30} +6.19615 q^{31} +1.00000 q^{32} -1.73205 q^{34} +0.267949 q^{35} +1.00000 q^{36} -7.19615 q^{37} -7.00000 q^{38} -4.46410 q^{39} -1.00000 q^{40} +4.19615 q^{41} -0.267949 q^{42} +2.19615 q^{43} -1.00000 q^{45} -6.73205 q^{46} +4.92820 q^{47} +1.00000 q^{48} -6.92820 q^{49} +1.00000 q^{50} -1.73205 q^{51} -4.46410 q^{52} +0.732051 q^{53} +1.00000 q^{54} -0.267949 q^{56} -7.00000 q^{57} -5.00000 q^{58} -11.6603 q^{59} -1.00000 q^{60} -15.1244 q^{61} +6.19615 q^{62} -0.267949 q^{63} +1.00000 q^{64} +4.46410 q^{65} +4.53590 q^{67} -1.73205 q^{68} -6.73205 q^{69} +0.267949 q^{70} -2.46410 q^{71} +1.00000 q^{72} +9.66025 q^{73} -7.19615 q^{74} +1.00000 q^{75} -7.00000 q^{76} -4.46410 q^{78} +10.3923 q^{79} -1.00000 q^{80} +1.00000 q^{81} +4.19615 q^{82} -9.53590 q^{83} -0.267949 q^{84} +1.73205 q^{85} +2.19615 q^{86} -5.00000 q^{87} +9.46410 q^{89} -1.00000 q^{90} +1.19615 q^{91} -6.73205 q^{92} +6.19615 q^{93} +4.92820 q^{94} +7.00000 q^{95} +1.00000 q^{96} +8.73205 q^{97} -6.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} + 2q^{8} + 2q^{9} - 2q^{10} + 2q^{12} - 2q^{13} - 4q^{14} - 2q^{15} + 2q^{16} + 2q^{18} - 14q^{19} - 2q^{20} - 4q^{21} - 10q^{23} + 2q^{24} + 2q^{25} - 2q^{26} + 2q^{27} - 4q^{28} - 10q^{29} - 2q^{30} + 2q^{31} + 2q^{32} + 4q^{35} + 2q^{36} - 4q^{37} - 14q^{38} - 2q^{39} - 2q^{40} - 2q^{41} - 4q^{42} - 6q^{43} - 2q^{45} - 10q^{46} - 4q^{47} + 2q^{48} + 2q^{50} - 2q^{52} - 2q^{53} + 2q^{54} - 4q^{56} - 14q^{57} - 10q^{58} - 6q^{59} - 2q^{60} - 6q^{61} + 2q^{62} - 4q^{63} + 2q^{64} + 2q^{65} + 16q^{67} - 10q^{69} + 4q^{70} + 2q^{71} + 2q^{72} + 2q^{73} - 4q^{74} + 2q^{75} - 14q^{76} - 2q^{78} - 2q^{80} + 2q^{81} - 2q^{82} - 26q^{83} - 4q^{84} - 6q^{86} - 10q^{87} + 12q^{89} - 2q^{90} - 8q^{91} - 10q^{92} + 2q^{93} - 4q^{94} + 14q^{95} + 2q^{96} + 14q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −0.267949 −0.101275 −0.0506376 0.998717i \(-0.516125\pi\)
−0.0506376 + 0.998717i \(0.516125\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) −4.46410 −1.23812 −0.619060 0.785344i \(-0.712486\pi\)
−0.619060 + 0.785344i \(0.712486\pi\)
\(14\) −0.267949 −0.0716124
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −1.73205 −0.420084 −0.210042 0.977692i \(-0.567360\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 1.00000 0.235702
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −1.00000 −0.223607
\(21\) −0.267949 −0.0584713
\(22\) 0 0
\(23\) −6.73205 −1.40373 −0.701865 0.712310i \(-0.747649\pi\)
−0.701865 + 0.712310i \(0.747649\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −4.46410 −0.875482
\(27\) 1.00000 0.192450
\(28\) −0.267949 −0.0506376
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) −1.00000 −0.182574
\(31\) 6.19615 1.11286 0.556431 0.830894i \(-0.312170\pi\)
0.556431 + 0.830894i \(0.312170\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.73205 −0.297044
\(35\) 0.267949 0.0452917
\(36\) 1.00000 0.166667
\(37\) −7.19615 −1.18304 −0.591520 0.806290i \(-0.701472\pi\)
−0.591520 + 0.806290i \(0.701472\pi\)
\(38\) −7.00000 −1.13555
\(39\) −4.46410 −0.714828
\(40\) −1.00000 −0.158114
\(41\) 4.19615 0.655329 0.327664 0.944794i \(-0.393738\pi\)
0.327664 + 0.944794i \(0.393738\pi\)
\(42\) −0.267949 −0.0413455
\(43\) 2.19615 0.334910 0.167455 0.985880i \(-0.446445\pi\)
0.167455 + 0.985880i \(0.446445\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) −6.73205 −0.992587
\(47\) 4.92820 0.718852 0.359426 0.933174i \(-0.382972\pi\)
0.359426 + 0.933174i \(0.382972\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.92820 −0.989743
\(50\) 1.00000 0.141421
\(51\) −1.73205 −0.242536
\(52\) −4.46410 −0.619060
\(53\) 0.732051 0.100555 0.0502775 0.998735i \(-0.483989\pi\)
0.0502775 + 0.998735i \(0.483989\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −0.267949 −0.0358062
\(57\) −7.00000 −0.927173
\(58\) −5.00000 −0.656532
\(59\) −11.6603 −1.51804 −0.759018 0.651070i \(-0.774321\pi\)
−0.759018 + 0.651070i \(0.774321\pi\)
\(60\) −1.00000 −0.129099
\(61\) −15.1244 −1.93648 −0.968238 0.250032i \(-0.919559\pi\)
−0.968238 + 0.250032i \(0.919559\pi\)
\(62\) 6.19615 0.786912
\(63\) −0.267949 −0.0337584
\(64\) 1.00000 0.125000
\(65\) 4.46410 0.553704
\(66\) 0 0
\(67\) 4.53590 0.554148 0.277074 0.960849i \(-0.410635\pi\)
0.277074 + 0.960849i \(0.410635\pi\)
\(68\) −1.73205 −0.210042
\(69\) −6.73205 −0.810444
\(70\) 0.267949 0.0320261
\(71\) −2.46410 −0.292435 −0.146218 0.989252i \(-0.546710\pi\)
−0.146218 + 0.989252i \(0.546710\pi\)
\(72\) 1.00000 0.117851
\(73\) 9.66025 1.13065 0.565324 0.824869i \(-0.308751\pi\)
0.565324 + 0.824869i \(0.308751\pi\)
\(74\) −7.19615 −0.836536
\(75\) 1.00000 0.115470
\(76\) −7.00000 −0.802955
\(77\) 0 0
\(78\) −4.46410 −0.505460
\(79\) 10.3923 1.16923 0.584613 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 4.19615 0.463388
\(83\) −9.53590 −1.04670 −0.523350 0.852118i \(-0.675318\pi\)
−0.523350 + 0.852118i \(0.675318\pi\)
\(84\) −0.267949 −0.0292357
\(85\) 1.73205 0.187867
\(86\) 2.19615 0.236817
\(87\) −5.00000 −0.536056
\(88\) 0 0
\(89\) 9.46410 1.00319 0.501596 0.865102i \(-0.332746\pi\)
0.501596 + 0.865102i \(0.332746\pi\)
\(90\) −1.00000 −0.105409
\(91\) 1.19615 0.125391
\(92\) −6.73205 −0.701865
\(93\) 6.19615 0.642511
\(94\) 4.92820 0.508305
\(95\) 7.00000 0.718185
\(96\) 1.00000 0.102062
\(97\) 8.73205 0.886605 0.443303 0.896372i \(-0.353807\pi\)
0.443303 + 0.896372i \(0.353807\pi\)
\(98\) −6.92820 −0.699854
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 1.53590 0.152828 0.0764138 0.997076i \(-0.475653\pi\)
0.0764138 + 0.997076i \(0.475653\pi\)
\(102\) −1.73205 −0.171499
\(103\) 3.39230 0.334254 0.167127 0.985935i \(-0.446551\pi\)
0.167127 + 0.985935i \(0.446551\pi\)
\(104\) −4.46410 −0.437741
\(105\) 0.267949 0.0261492
\(106\) 0.732051 0.0711031
\(107\) −6.39230 −0.617967 −0.308984 0.951067i \(-0.599989\pi\)
−0.308984 + 0.951067i \(0.599989\pi\)
\(108\) 1.00000 0.0962250
\(109\) −8.39230 −0.803837 −0.401919 0.915675i \(-0.631656\pi\)
−0.401919 + 0.915675i \(0.631656\pi\)
\(110\) 0 0
\(111\) −7.19615 −0.683029
\(112\) −0.267949 −0.0253188
\(113\) 13.8564 1.30350 0.651751 0.758433i \(-0.274035\pi\)
0.651751 + 0.758433i \(0.274035\pi\)
\(114\) −7.00000 −0.655610
\(115\) 6.73205 0.627767
\(116\) −5.00000 −0.464238
\(117\) −4.46410 −0.412706
\(118\) −11.6603 −1.07341
\(119\) 0.464102 0.0425441
\(120\) −1.00000 −0.0912871
\(121\) 0 0
\(122\) −15.1244 −1.36929
\(123\) 4.19615 0.378354
\(124\) 6.19615 0.556431
\(125\) −1.00000 −0.0894427
\(126\) −0.267949 −0.0238708
\(127\) −6.53590 −0.579967 −0.289984 0.957032i \(-0.593650\pi\)
−0.289984 + 0.957032i \(0.593650\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.19615 0.193360
\(130\) 4.46410 0.391528
\(131\) 3.12436 0.272976 0.136488 0.990642i \(-0.456418\pi\)
0.136488 + 0.990642i \(0.456418\pi\)
\(132\) 0 0
\(133\) 1.87564 0.162639
\(134\) 4.53590 0.391842
\(135\) −1.00000 −0.0860663
\(136\) −1.73205 −0.148522
\(137\) 3.53590 0.302092 0.151046 0.988527i \(-0.451736\pi\)
0.151046 + 0.988527i \(0.451736\pi\)
\(138\) −6.73205 −0.573070
\(139\) −7.00000 −0.593732 −0.296866 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0.267949 0.0226458
\(141\) 4.92820 0.415030
\(142\) −2.46410 −0.206783
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 5.00000 0.415227
\(146\) 9.66025 0.799488
\(147\) −6.92820 −0.571429
\(148\) −7.19615 −0.591520
\(149\) −12.3923 −1.01522 −0.507609 0.861588i \(-0.669470\pi\)
−0.507609 + 0.861588i \(0.669470\pi\)
\(150\) 1.00000 0.0816497
\(151\) −9.80385 −0.797826 −0.398913 0.916989i \(-0.630612\pi\)
−0.398913 + 0.916989i \(0.630612\pi\)
\(152\) −7.00000 −0.567775
\(153\) −1.73205 −0.140028
\(154\) 0 0
\(155\) −6.19615 −0.497687
\(156\) −4.46410 −0.357414
\(157\) 2.66025 0.212311 0.106156 0.994350i \(-0.466146\pi\)
0.106156 + 0.994350i \(0.466146\pi\)
\(158\) 10.3923 0.826767
\(159\) 0.732051 0.0580554
\(160\) −1.00000 −0.0790569
\(161\) 1.80385 0.142163
\(162\) 1.00000 0.0785674
\(163\) −2.19615 −0.172016 −0.0860080 0.996294i \(-0.527411\pi\)
−0.0860080 + 0.996294i \(0.527411\pi\)
\(164\) 4.19615 0.327664
\(165\) 0 0
\(166\) −9.53590 −0.740129
\(167\) 13.4641 1.04188 0.520942 0.853592i \(-0.325581\pi\)
0.520942 + 0.853592i \(0.325581\pi\)
\(168\) −0.267949 −0.0206727
\(169\) 6.92820 0.532939
\(170\) 1.73205 0.132842
\(171\) −7.00000 −0.535303
\(172\) 2.19615 0.167455
\(173\) −6.33975 −0.482002 −0.241001 0.970525i \(-0.577476\pi\)
−0.241001 + 0.970525i \(0.577476\pi\)
\(174\) −5.00000 −0.379049
\(175\) −0.267949 −0.0202551
\(176\) 0 0
\(177\) −11.6603 −0.876438
\(178\) 9.46410 0.709364
\(179\) 18.5885 1.38937 0.694683 0.719316i \(-0.255545\pi\)
0.694683 + 0.719316i \(0.255545\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −9.80385 −0.728714 −0.364357 0.931259i \(-0.618711\pi\)
−0.364357 + 0.931259i \(0.618711\pi\)
\(182\) 1.19615 0.0886647
\(183\) −15.1244 −1.11802
\(184\) −6.73205 −0.496293
\(185\) 7.19615 0.529072
\(186\) 6.19615 0.454324
\(187\) 0 0
\(188\) 4.92820 0.359426
\(189\) −0.267949 −0.0194904
\(190\) 7.00000 0.507833
\(191\) 13.9282 1.00781 0.503905 0.863759i \(-0.331896\pi\)
0.503905 + 0.863759i \(0.331896\pi\)
\(192\) 1.00000 0.0721688
\(193\) −23.3205 −1.67865 −0.839323 0.543632i \(-0.817049\pi\)
−0.839323 + 0.543632i \(0.817049\pi\)
\(194\) 8.73205 0.626925
\(195\) 4.46410 0.319681
\(196\) −6.92820 −0.494872
\(197\) −2.53590 −0.180675 −0.0903376 0.995911i \(-0.528795\pi\)
−0.0903376 + 0.995911i \(0.528795\pi\)
\(198\) 0 0
\(199\) −15.3205 −1.08604 −0.543021 0.839719i \(-0.682720\pi\)
−0.543021 + 0.839719i \(0.682720\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.53590 0.319938
\(202\) 1.53590 0.108065
\(203\) 1.33975 0.0940317
\(204\) −1.73205 −0.121268
\(205\) −4.19615 −0.293072
\(206\) 3.39230 0.236353
\(207\) −6.73205 −0.467910
\(208\) −4.46410 −0.309530
\(209\) 0 0
\(210\) 0.267949 0.0184903
\(211\) −27.3923 −1.88576 −0.942882 0.333127i \(-0.891896\pi\)
−0.942882 + 0.333127i \(0.891896\pi\)
\(212\) 0.732051 0.0502775
\(213\) −2.46410 −0.168837
\(214\) −6.39230 −0.436969
\(215\) −2.19615 −0.149776
\(216\) 1.00000 0.0680414
\(217\) −1.66025 −0.112705
\(218\) −8.39230 −0.568399
\(219\) 9.66025 0.652779
\(220\) 0 0
\(221\) 7.73205 0.520114
\(222\) −7.19615 −0.482974
\(223\) −19.2487 −1.28899 −0.644495 0.764609i \(-0.722932\pi\)
−0.644495 + 0.764609i \(0.722932\pi\)
\(224\) −0.267949 −0.0179031
\(225\) 1.00000 0.0666667
\(226\) 13.8564 0.921714
\(227\) 20.5359 1.36302 0.681508 0.731811i \(-0.261325\pi\)
0.681508 + 0.731811i \(0.261325\pi\)
\(228\) −7.00000 −0.463586
\(229\) 21.8564 1.44431 0.722156 0.691730i \(-0.243151\pi\)
0.722156 + 0.691730i \(0.243151\pi\)
\(230\) 6.73205 0.443898
\(231\) 0 0
\(232\) −5.00000 −0.328266
\(233\) 22.9282 1.50208 0.751038 0.660259i \(-0.229553\pi\)
0.751038 + 0.660259i \(0.229553\pi\)
\(234\) −4.46410 −0.291827
\(235\) −4.92820 −0.321481
\(236\) −11.6603 −0.759018
\(237\) 10.3923 0.675053
\(238\) 0.464102 0.0300832
\(239\) 7.19615 0.465480 0.232740 0.972539i \(-0.425231\pi\)
0.232740 + 0.972539i \(0.425231\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −11.1962 −0.721208 −0.360604 0.932719i \(-0.617429\pi\)
−0.360604 + 0.932719i \(0.617429\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −15.1244 −0.968238
\(245\) 6.92820 0.442627
\(246\) 4.19615 0.267537
\(247\) 31.2487 1.98831
\(248\) 6.19615 0.393456
\(249\) −9.53590 −0.604313
\(250\) −1.00000 −0.0632456
\(251\) −3.32051 −0.209589 −0.104794 0.994494i \(-0.533418\pi\)
−0.104794 + 0.994494i \(0.533418\pi\)
\(252\) −0.267949 −0.0168792
\(253\) 0 0
\(254\) −6.53590 −0.410099
\(255\) 1.73205 0.108465
\(256\) 1.00000 0.0625000
\(257\) −21.7846 −1.35889 −0.679443 0.733728i \(-0.737779\pi\)
−0.679443 + 0.733728i \(0.737779\pi\)
\(258\) 2.19615 0.136726
\(259\) 1.92820 0.119813
\(260\) 4.46410 0.276852
\(261\) −5.00000 −0.309492
\(262\) 3.12436 0.193023
\(263\) 6.73205 0.415116 0.207558 0.978223i \(-0.433448\pi\)
0.207558 + 0.978223i \(0.433448\pi\)
\(264\) 0 0
\(265\) −0.732051 −0.0449695
\(266\) 1.87564 0.115003
\(267\) 9.46410 0.579194
\(268\) 4.53590 0.277074
\(269\) −29.9808 −1.82796 −0.913980 0.405760i \(-0.867007\pi\)
−0.913980 + 0.405760i \(0.867007\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 3.85641 0.234260 0.117130 0.993117i \(-0.462631\pi\)
0.117130 + 0.993117i \(0.462631\pi\)
\(272\) −1.73205 −0.105021
\(273\) 1.19615 0.0723944
\(274\) 3.53590 0.213611
\(275\) 0 0
\(276\) −6.73205 −0.405222
\(277\) 29.8564 1.79390 0.896949 0.442134i \(-0.145779\pi\)
0.896949 + 0.442134i \(0.145779\pi\)
\(278\) −7.00000 −0.419832
\(279\) 6.19615 0.370954
\(280\) 0.267949 0.0160130
\(281\) −21.1244 −1.26017 −0.630087 0.776525i \(-0.716981\pi\)
−0.630087 + 0.776525i \(0.716981\pi\)
\(282\) 4.92820 0.293470
\(283\) −12.3923 −0.736646 −0.368323 0.929698i \(-0.620068\pi\)
−0.368323 + 0.929698i \(0.620068\pi\)
\(284\) −2.46410 −0.146218
\(285\) 7.00000 0.414644
\(286\) 0 0
\(287\) −1.12436 −0.0663686
\(288\) 1.00000 0.0589256
\(289\) −14.0000 −0.823529
\(290\) 5.00000 0.293610
\(291\) 8.73205 0.511882
\(292\) 9.66025 0.565324
\(293\) 1.07180 0.0626150 0.0313075 0.999510i \(-0.490033\pi\)
0.0313075 + 0.999510i \(0.490033\pi\)
\(294\) −6.92820 −0.404061
\(295\) 11.6603 0.678886
\(296\) −7.19615 −0.418268
\(297\) 0 0
\(298\) −12.3923 −0.717867
\(299\) 30.0526 1.73798
\(300\) 1.00000 0.0577350
\(301\) −0.588457 −0.0339181
\(302\) −9.80385 −0.564148
\(303\) 1.53590 0.0882351
\(304\) −7.00000 −0.401478
\(305\) 15.1244 0.866018
\(306\) −1.73205 −0.0990148
\(307\) −14.5885 −0.832607 −0.416304 0.909226i \(-0.636675\pi\)
−0.416304 + 0.909226i \(0.636675\pi\)
\(308\) 0 0
\(309\) 3.39230 0.192981
\(310\) −6.19615 −0.351918
\(311\) 26.9282 1.52696 0.763479 0.645832i \(-0.223490\pi\)
0.763479 + 0.645832i \(0.223490\pi\)
\(312\) −4.46410 −0.252730
\(313\) 22.1962 1.25460 0.627300 0.778777i \(-0.284160\pi\)
0.627300 + 0.778777i \(0.284160\pi\)
\(314\) 2.66025 0.150127
\(315\) 0.267949 0.0150972
\(316\) 10.3923 0.584613
\(317\) −2.19615 −0.123348 −0.0616741 0.998096i \(-0.519644\pi\)
−0.0616741 + 0.998096i \(0.519644\pi\)
\(318\) 0.732051 0.0410514
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −6.39230 −0.356784
\(322\) 1.80385 0.100524
\(323\) 12.1244 0.674617
\(324\) 1.00000 0.0555556
\(325\) −4.46410 −0.247624
\(326\) −2.19615 −0.121634
\(327\) −8.39230 −0.464096
\(328\) 4.19615 0.231694
\(329\) −1.32051 −0.0728020
\(330\) 0 0
\(331\) 30.1244 1.65578 0.827892 0.560887i \(-0.189540\pi\)
0.827892 + 0.560887i \(0.189540\pi\)
\(332\) −9.53590 −0.523350
\(333\) −7.19615 −0.394347
\(334\) 13.4641 0.736723
\(335\) −4.53590 −0.247823
\(336\) −0.267949 −0.0146178
\(337\) 19.8038 1.07878 0.539392 0.842055i \(-0.318654\pi\)
0.539392 + 0.842055i \(0.318654\pi\)
\(338\) 6.92820 0.376845
\(339\) 13.8564 0.752577
\(340\) 1.73205 0.0939336
\(341\) 0 0
\(342\) −7.00000 −0.378517
\(343\) 3.73205 0.201512
\(344\) 2.19615 0.118409
\(345\) 6.73205 0.362441
\(346\) −6.33975 −0.340827
\(347\) 7.53590 0.404548 0.202274 0.979329i \(-0.435167\pi\)
0.202274 + 0.979329i \(0.435167\pi\)
\(348\) −5.00000 −0.268028
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) −0.267949 −0.0143225
\(351\) −4.46410 −0.238276
\(352\) 0 0
\(353\) −25.8564 −1.37620 −0.688099 0.725617i \(-0.741554\pi\)
−0.688099 + 0.725617i \(0.741554\pi\)
\(354\) −11.6603 −0.619736
\(355\) 2.46410 0.130781
\(356\) 9.46410 0.501596
\(357\) 0.464102 0.0245629
\(358\) 18.5885 0.982430
\(359\) −31.1769 −1.64545 −0.822727 0.568436i \(-0.807549\pi\)
−0.822727 + 0.568436i \(0.807549\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 30.0000 1.57895
\(362\) −9.80385 −0.515279
\(363\) 0 0
\(364\) 1.19615 0.0626954
\(365\) −9.66025 −0.505641
\(366\) −15.1244 −0.790563
\(367\) −17.0000 −0.887393 −0.443696 0.896177i \(-0.646333\pi\)
−0.443696 + 0.896177i \(0.646333\pi\)
\(368\) −6.73205 −0.350932
\(369\) 4.19615 0.218443
\(370\) 7.19615 0.374110
\(371\) −0.196152 −0.0101837
\(372\) 6.19615 0.321256
\(373\) 3.39230 0.175647 0.0878234 0.996136i \(-0.472009\pi\)
0.0878234 + 0.996136i \(0.472009\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 4.92820 0.254153
\(377\) 22.3205 1.14956
\(378\) −0.267949 −0.0137818
\(379\) −18.1244 −0.930986 −0.465493 0.885052i \(-0.654123\pi\)
−0.465493 + 0.885052i \(0.654123\pi\)
\(380\) 7.00000 0.359092
\(381\) −6.53590 −0.334844
\(382\) 13.9282 0.712629
\(383\) −31.8564 −1.62779 −0.813893 0.581015i \(-0.802656\pi\)
−0.813893 + 0.581015i \(0.802656\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −23.3205 −1.18698
\(387\) 2.19615 0.111637
\(388\) 8.73205 0.443303
\(389\) 35.7128 1.81071 0.905356 0.424654i \(-0.139604\pi\)
0.905356 + 0.424654i \(0.139604\pi\)
\(390\) 4.46410 0.226049
\(391\) 11.6603 0.589684
\(392\) −6.92820 −0.349927
\(393\) 3.12436 0.157603
\(394\) −2.53590 −0.127757
\(395\) −10.3923 −0.522894
\(396\) 0 0
\(397\) −23.9808 −1.20356 −0.601780 0.798662i \(-0.705542\pi\)
−0.601780 + 0.798662i \(0.705542\pi\)
\(398\) −15.3205 −0.767948
\(399\) 1.87564 0.0938997
\(400\) 1.00000 0.0500000
\(401\) 31.2679 1.56145 0.780723 0.624877i \(-0.214851\pi\)
0.780723 + 0.624877i \(0.214851\pi\)
\(402\) 4.53590 0.226230
\(403\) −27.6603 −1.37786
\(404\) 1.53590 0.0764138
\(405\) −1.00000 −0.0496904
\(406\) 1.33975 0.0664905
\(407\) 0 0
\(408\) −1.73205 −0.0857493
\(409\) 0.928203 0.0458967 0.0229483 0.999737i \(-0.492695\pi\)
0.0229483 + 0.999737i \(0.492695\pi\)
\(410\) −4.19615 −0.207233
\(411\) 3.53590 0.174413
\(412\) 3.39230 0.167127
\(413\) 3.12436 0.153739
\(414\) −6.73205 −0.330862
\(415\) 9.53590 0.468099
\(416\) −4.46410 −0.218871
\(417\) −7.00000 −0.342791
\(418\) 0 0
\(419\) 28.2487 1.38004 0.690020 0.723790i \(-0.257602\pi\)
0.690020 + 0.723790i \(0.257602\pi\)
\(420\) 0.267949 0.0130746
\(421\) 1.66025 0.0809158 0.0404579 0.999181i \(-0.487118\pi\)
0.0404579 + 0.999181i \(0.487118\pi\)
\(422\) −27.3923 −1.33344
\(423\) 4.92820 0.239617
\(424\) 0.732051 0.0355515
\(425\) −1.73205 −0.0840168
\(426\) −2.46410 −0.119386
\(427\) 4.05256 0.196117
\(428\) −6.39230 −0.308984
\(429\) 0 0
\(430\) −2.19615 −0.105908
\(431\) 37.4449 1.80366 0.901828 0.432096i \(-0.142226\pi\)
0.901828 + 0.432096i \(0.142226\pi\)
\(432\) 1.00000 0.0481125
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) −1.66025 −0.0796947
\(435\) 5.00000 0.239732
\(436\) −8.39230 −0.401919
\(437\) 47.1244 2.25426
\(438\) 9.66025 0.461585
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 0 0
\(441\) −6.92820 −0.329914
\(442\) 7.73205 0.367776
\(443\) 21.1962 1.00706 0.503530 0.863978i \(-0.332034\pi\)
0.503530 + 0.863978i \(0.332034\pi\)
\(444\) −7.19615 −0.341514
\(445\) −9.46410 −0.448641
\(446\) −19.2487 −0.911453
\(447\) −12.3923 −0.586136
\(448\) −0.267949 −0.0126594
\(449\) −29.3205 −1.38372 −0.691860 0.722032i \(-0.743209\pi\)
−0.691860 + 0.722032i \(0.743209\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 13.8564 0.651751
\(453\) −9.80385 −0.460625
\(454\) 20.5359 0.963797
\(455\) −1.19615 −0.0560765
\(456\) −7.00000 −0.327805
\(457\) 24.7321 1.15692 0.578458 0.815712i \(-0.303655\pi\)
0.578458 + 0.815712i \(0.303655\pi\)
\(458\) 21.8564 1.02128
\(459\) −1.73205 −0.0808452
\(460\) 6.73205 0.313883
\(461\) 15.3923 0.716891 0.358446 0.933551i \(-0.383307\pi\)
0.358446 + 0.933551i \(0.383307\pi\)
\(462\) 0 0
\(463\) 24.2487 1.12693 0.563467 0.826139i \(-0.309467\pi\)
0.563467 + 0.826139i \(0.309467\pi\)
\(464\) −5.00000 −0.232119
\(465\) −6.19615 −0.287340
\(466\) 22.9282 1.06213
\(467\) 39.1962 1.81378 0.906891 0.421366i \(-0.138449\pi\)
0.906891 + 0.421366i \(0.138449\pi\)
\(468\) −4.46410 −0.206353
\(469\) −1.21539 −0.0561215
\(470\) −4.92820 −0.227321
\(471\) 2.66025 0.122578
\(472\) −11.6603 −0.536707
\(473\) 0 0
\(474\) 10.3923 0.477334
\(475\) −7.00000 −0.321182
\(476\) 0.464102 0.0212721
\(477\) 0.732051 0.0335183
\(478\) 7.19615 0.329144
\(479\) 25.9808 1.18709 0.593546 0.804800i \(-0.297728\pi\)
0.593546 + 0.804800i \(0.297728\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 32.1244 1.46474
\(482\) −11.1962 −0.509971
\(483\) 1.80385 0.0820779
\(484\) 0 0
\(485\) −8.73205 −0.396502
\(486\) 1.00000 0.0453609
\(487\) −40.3205 −1.82710 −0.913548 0.406730i \(-0.866669\pi\)
−0.913548 + 0.406730i \(0.866669\pi\)
\(488\) −15.1244 −0.684647
\(489\) −2.19615 −0.0993134
\(490\) 6.92820 0.312984
\(491\) 4.73205 0.213554 0.106777 0.994283i \(-0.465947\pi\)
0.106777 + 0.994283i \(0.465947\pi\)
\(492\) 4.19615 0.189177
\(493\) 8.66025 0.390038
\(494\) 31.2487 1.40595
\(495\) 0 0
\(496\) 6.19615 0.278215
\(497\) 0.660254 0.0296164
\(498\) −9.53590 −0.427314
\(499\) −16.8038 −0.752244 −0.376122 0.926570i \(-0.622743\pi\)
−0.376122 + 0.926570i \(0.622743\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 13.4641 0.601532
\(502\) −3.32051 −0.148202
\(503\) −41.9090 −1.86863 −0.934314 0.356451i \(-0.883987\pi\)
−0.934314 + 0.356451i \(0.883987\pi\)
\(504\) −0.267949 −0.0119354
\(505\) −1.53590 −0.0683466
\(506\) 0 0
\(507\) 6.92820 0.307692
\(508\) −6.53590 −0.289984
\(509\) −15.8564 −0.702823 −0.351411 0.936221i \(-0.614298\pi\)
−0.351411 + 0.936221i \(0.614298\pi\)
\(510\) 1.73205 0.0766965
\(511\) −2.58846 −0.114507
\(512\) 1.00000 0.0441942
\(513\) −7.00000 −0.309058
\(514\) −21.7846 −0.960878
\(515\) −3.39230 −0.149483
\(516\) 2.19615 0.0966802
\(517\) 0 0
\(518\) 1.92820 0.0847204
\(519\) −6.33975 −0.278284
\(520\) 4.46410 0.195764
\(521\) 12.5885 0.551510 0.275755 0.961228i \(-0.411072\pi\)
0.275755 + 0.961228i \(0.411072\pi\)
\(522\) −5.00000 −0.218844
\(523\) 27.1244 1.18607 0.593033 0.805178i \(-0.297931\pi\)
0.593033 + 0.805178i \(0.297931\pi\)
\(524\) 3.12436 0.136488
\(525\) −0.267949 −0.0116943
\(526\) 6.73205 0.293531
\(527\) −10.7321 −0.467495
\(528\) 0 0
\(529\) 22.3205 0.970457
\(530\) −0.732051 −0.0317983
\(531\) −11.6603 −0.506012
\(532\) 1.87564 0.0813195
\(533\) −18.7321 −0.811375
\(534\) 9.46410 0.409552
\(535\) 6.39230 0.276363
\(536\) 4.53590 0.195921
\(537\) 18.5885 0.802151
\(538\) −29.9808 −1.29256
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −27.2679 −1.17234 −0.586170 0.810188i \(-0.699365\pi\)
−0.586170 + 0.810188i \(0.699365\pi\)
\(542\) 3.85641 0.165647
\(543\) −9.80385 −0.420723
\(544\) −1.73205 −0.0742611
\(545\) 8.39230 0.359487
\(546\) 1.19615 0.0511906
\(547\) −3.85641 −0.164888 −0.0824440 0.996596i \(-0.526273\pi\)
−0.0824440 + 0.996596i \(0.526273\pi\)
\(548\) 3.53590 0.151046
\(549\) −15.1244 −0.645492
\(550\) 0 0
\(551\) 35.0000 1.49105
\(552\) −6.73205 −0.286535
\(553\) −2.78461 −0.118414
\(554\) 29.8564 1.26848
\(555\) 7.19615 0.305460
\(556\) −7.00000 −0.296866
\(557\) 2.67949 0.113534 0.0567669 0.998387i \(-0.481921\pi\)
0.0567669 + 0.998387i \(0.481921\pi\)
\(558\) 6.19615 0.262304
\(559\) −9.80385 −0.414659
\(560\) 0.267949 0.0113229
\(561\) 0 0
\(562\) −21.1244 −0.891077
\(563\) −12.4641 −0.525299 −0.262650 0.964891i \(-0.584596\pi\)
−0.262650 + 0.964891i \(0.584596\pi\)
\(564\) 4.92820 0.207515
\(565\) −13.8564 −0.582943
\(566\) −12.3923 −0.520887
\(567\) −0.267949 −0.0112528
\(568\) −2.46410 −0.103391
\(569\) −6.33975 −0.265776 −0.132888 0.991131i \(-0.542425\pi\)
−0.132888 + 0.991131i \(0.542425\pi\)
\(570\) 7.00000 0.293198
\(571\) −11.4641 −0.479758 −0.239879 0.970803i \(-0.577108\pi\)
−0.239879 + 0.970803i \(0.577108\pi\)
\(572\) 0 0
\(573\) 13.9282 0.581859
\(574\) −1.12436 −0.0469297
\(575\) −6.73205 −0.280746
\(576\) 1.00000 0.0416667
\(577\) 45.0333 1.87476 0.937381 0.348306i \(-0.113243\pi\)
0.937381 + 0.348306i \(0.113243\pi\)
\(578\) −14.0000 −0.582323
\(579\) −23.3205 −0.969167
\(580\) 5.00000 0.207614
\(581\) 2.55514 0.106005
\(582\) 8.73205 0.361955
\(583\) 0 0
\(584\) 9.66025 0.399744
\(585\) 4.46410 0.184568
\(586\) 1.07180 0.0442755
\(587\) −26.1244 −1.07827 −0.539134 0.842220i \(-0.681248\pi\)
−0.539134 + 0.842220i \(0.681248\pi\)
\(588\) −6.92820 −0.285714
\(589\) −43.3731 −1.78716
\(590\) 11.6603 0.480045
\(591\) −2.53590 −0.104313
\(592\) −7.19615 −0.295760
\(593\) 6.78461 0.278611 0.139305 0.990249i \(-0.455513\pi\)
0.139305 + 0.990249i \(0.455513\pi\)
\(594\) 0 0
\(595\) −0.464102 −0.0190263
\(596\) −12.3923 −0.507609
\(597\) −15.3205 −0.627027
\(598\) 30.0526 1.22894
\(599\) −8.24871 −0.337033 −0.168517 0.985699i \(-0.553898\pi\)
−0.168517 + 0.985699i \(0.553898\pi\)
\(600\) 1.00000 0.0408248
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −0.588457 −0.0239837
\(603\) 4.53590 0.184716
\(604\) −9.80385 −0.398913
\(605\) 0 0
\(606\) 1.53590 0.0623916
\(607\) −3.87564 −0.157308 −0.0786538 0.996902i \(-0.525062\pi\)
−0.0786538 + 0.996902i \(0.525062\pi\)
\(608\) −7.00000 −0.283887
\(609\) 1.33975 0.0542892
\(610\) 15.1244 0.612367
\(611\) −22.0000 −0.890025
\(612\) −1.73205 −0.0700140
\(613\) 22.4641 0.907317 0.453658 0.891176i \(-0.350119\pi\)
0.453658 + 0.891176i \(0.350119\pi\)
\(614\) −14.5885 −0.588742
\(615\) −4.19615 −0.169205
\(616\) 0 0
\(617\) 35.1051 1.41328 0.706639 0.707574i \(-0.250210\pi\)
0.706639 + 0.707574i \(0.250210\pi\)
\(618\) 3.39230 0.136459
\(619\) −36.1244 −1.45196 −0.725980 0.687716i \(-0.758614\pi\)
−0.725980 + 0.687716i \(0.758614\pi\)
\(620\) −6.19615 −0.248843
\(621\) −6.73205 −0.270148
\(622\) 26.9282 1.07972
\(623\) −2.53590 −0.101599
\(624\) −4.46410 −0.178707
\(625\) 1.00000 0.0400000
\(626\) 22.1962 0.887137
\(627\) 0 0
\(628\) 2.66025 0.106156
\(629\) 12.4641 0.496976
\(630\) 0.267949 0.0106754
\(631\) −24.9808 −0.994468 −0.497234 0.867616i \(-0.665651\pi\)
−0.497234 + 0.867616i \(0.665651\pi\)
\(632\) 10.3923 0.413384
\(633\) −27.3923 −1.08875
\(634\) −2.19615 −0.0872204
\(635\) 6.53590 0.259369
\(636\) 0.732051 0.0290277
\(637\) 30.9282 1.22542
\(638\) 0 0
\(639\) −2.46410 −0.0974784
\(640\) −1.00000 −0.0395285
\(641\) 4.14359 0.163662 0.0818311 0.996646i \(-0.473923\pi\)
0.0818311 + 0.996646i \(0.473923\pi\)
\(642\) −6.39230 −0.252284
\(643\) 3.21539 0.126803 0.0634013 0.997988i \(-0.479805\pi\)
0.0634013 + 0.997988i \(0.479805\pi\)
\(644\) 1.80385 0.0710816
\(645\) −2.19615 −0.0864734
\(646\) 12.1244 0.477026
\(647\) −33.4641 −1.31561 −0.657805 0.753188i \(-0.728515\pi\)
−0.657805 + 0.753188i \(0.728515\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −4.46410 −0.175096
\(651\) −1.66025 −0.0650705
\(652\) −2.19615 −0.0860080
\(653\) 39.5167 1.54641 0.773203 0.634158i \(-0.218653\pi\)
0.773203 + 0.634158i \(0.218653\pi\)
\(654\) −8.39230 −0.328165
\(655\) −3.12436 −0.122079
\(656\) 4.19615 0.163832
\(657\) 9.66025 0.376882
\(658\) −1.32051 −0.0514788
\(659\) −32.2487 −1.25623 −0.628116 0.778120i \(-0.716174\pi\)
−0.628116 + 0.778120i \(0.716174\pi\)
\(660\) 0 0
\(661\) 18.4449 0.717422 0.358711 0.933449i \(-0.383216\pi\)
0.358711 + 0.933449i \(0.383216\pi\)
\(662\) 30.1244 1.17082
\(663\) 7.73205 0.300288
\(664\) −9.53590 −0.370065
\(665\) −1.87564 −0.0727344
\(666\) −7.19615 −0.278845
\(667\) 33.6603 1.30333
\(668\) 13.4641 0.520942
\(669\) −19.2487 −0.744198
\(670\) −4.53590 −0.175237
\(671\) 0 0
\(672\) −0.267949 −0.0103364
\(673\) −30.9282 −1.19219 −0.596097 0.802912i \(-0.703283\pi\)
−0.596097 + 0.802912i \(0.703283\pi\)
\(674\) 19.8038 0.762816
\(675\) 1.00000 0.0384900
\(676\) 6.92820 0.266469
\(677\) 2.87564 0.110520 0.0552600 0.998472i \(-0.482401\pi\)
0.0552600 + 0.998472i \(0.482401\pi\)
\(678\) 13.8564 0.532152
\(679\) −2.33975 −0.0897912
\(680\) 1.73205 0.0664211
\(681\) 20.5359 0.786937
\(682\) 0 0
\(683\) 30.3731 1.16219 0.581097 0.813835i \(-0.302624\pi\)
0.581097 + 0.813835i \(0.302624\pi\)
\(684\) −7.00000 −0.267652
\(685\) −3.53590 −0.135100
\(686\) 3.73205 0.142490
\(687\) 21.8564 0.833874
\(688\) 2.19615 0.0837275
\(689\) −3.26795 −0.124499
\(690\) 6.73205 0.256285
\(691\) −32.5167 −1.23699 −0.618496 0.785788i \(-0.712258\pi\)
−0.618496 + 0.785788i \(0.712258\pi\)
\(692\) −6.33975 −0.241001
\(693\) 0 0
\(694\) 7.53590 0.286059
\(695\) 7.00000 0.265525
\(696\) −5.00000 −0.189525
\(697\) −7.26795 −0.275293
\(698\) −34.0000 −1.28692
\(699\) 22.9282 0.867224
\(700\) −0.267949 −0.0101275
\(701\) 45.2487 1.70902 0.854510 0.519435i \(-0.173857\pi\)
0.854510 + 0.519435i \(0.173857\pi\)
\(702\) −4.46410 −0.168487
\(703\) 50.3731 1.89986
\(704\) 0 0
\(705\) −4.92820 −0.185607
\(706\) −25.8564 −0.973119
\(707\) −0.411543 −0.0154777
\(708\) −11.6603 −0.438219
\(709\) −21.2679 −0.798735 −0.399367 0.916791i \(-0.630770\pi\)
−0.399367 + 0.916791i \(0.630770\pi\)
\(710\) 2.46410 0.0924761
\(711\) 10.3923 0.389742
\(712\) 9.46410 0.354682
\(713\) −41.7128 −1.56216
\(714\) 0.464102 0.0173686
\(715\) 0 0
\(716\) 18.5885 0.694683
\(717\) 7.19615 0.268745
\(718\) −31.1769 −1.16351
\(719\) −25.6077 −0.955006 −0.477503 0.878630i \(-0.658458\pi\)
−0.477503 + 0.878630i \(0.658458\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −0.908965 −0.0338516
\(722\) 30.0000 1.11648
\(723\) −11.1962 −0.416389
\(724\) −9.80385 −0.364357
\(725\) −5.00000 −0.185695
\(726\) 0 0
\(727\) −48.8564 −1.81198 −0.905992 0.423295i \(-0.860873\pi\)
−0.905992 + 0.423295i \(0.860873\pi\)
\(728\) 1.19615 0.0443324
\(729\) 1.00000 0.0370370
\(730\) −9.66025 −0.357542
\(731\) −3.80385 −0.140690
\(732\) −15.1244 −0.559012
\(733\) −36.7846 −1.35867 −0.679335 0.733828i \(-0.737732\pi\)
−0.679335 + 0.733828i \(0.737732\pi\)
\(734\) −17.0000 −0.627481
\(735\) 6.92820 0.255551
\(736\) −6.73205 −0.248147
\(737\) 0 0
\(738\) 4.19615 0.154463
\(739\) 19.0000 0.698926 0.349463 0.936950i \(-0.386364\pi\)
0.349463 + 0.936950i \(0.386364\pi\)
\(740\) 7.19615 0.264536
\(741\) 31.2487 1.14795
\(742\) −0.196152 −0.00720098
\(743\) 8.73205 0.320348 0.160174 0.987089i \(-0.448794\pi\)
0.160174 + 0.987089i \(0.448794\pi\)
\(744\) 6.19615 0.227162
\(745\) 12.3923 0.454019
\(746\) 3.39230 0.124201
\(747\) −9.53590 −0.348900
\(748\) 0 0
\(749\) 1.71281 0.0625848
\(750\) −1.00000 −0.0365148
\(751\) 15.6603 0.571451 0.285725 0.958312i \(-0.407765\pi\)
0.285725 + 0.958312i \(0.407765\pi\)
\(752\) 4.92820 0.179713
\(753\) −3.32051 −0.121006
\(754\) 22.3205 0.812865
\(755\) 9.80385 0.356799
\(756\) −0.267949 −0.00974522
\(757\) 15.4641 0.562052 0.281026 0.959700i \(-0.409325\pi\)
0.281026 + 0.959700i \(0.409325\pi\)
\(758\) −18.1244 −0.658306
\(759\) 0 0
\(760\) 7.00000 0.253917
\(761\) −2.24871 −0.0815157 −0.0407579 0.999169i \(-0.512977\pi\)
−0.0407579 + 0.999169i \(0.512977\pi\)
\(762\) −6.53590 −0.236771
\(763\) 2.24871 0.0814088
\(764\) 13.9282 0.503905
\(765\) 1.73205 0.0626224
\(766\) −31.8564 −1.15102
\(767\) 52.0526 1.87951
\(768\) 1.00000 0.0360844
\(769\) 31.8372 1.14808 0.574039 0.818828i \(-0.305376\pi\)
0.574039 + 0.818828i \(0.305376\pi\)
\(770\) 0 0
\(771\) −21.7846 −0.784554
\(772\) −23.3205 −0.839323
\(773\) −12.3397 −0.443830 −0.221915 0.975066i \(-0.571231\pi\)
−0.221915 + 0.975066i \(0.571231\pi\)
\(774\) 2.19615 0.0789391
\(775\) 6.19615 0.222572
\(776\) 8.73205 0.313462
\(777\) 1.92820 0.0691739
\(778\) 35.7128 1.28037
\(779\) −29.3731 −1.05240
\(780\) 4.46410 0.159840
\(781\) 0 0
\(782\) 11.6603 0.416970
\(783\) −5.00000 −0.178685
\(784\) −6.92820 −0.247436
\(785\) −2.66025 −0.0949485
\(786\) 3.12436 0.111442
\(787\) 27.8038 0.991100 0.495550 0.868579i \(-0.334967\pi\)
0.495550 + 0.868579i \(0.334967\pi\)
\(788\) −2.53590 −0.0903376
\(789\) 6.73205 0.239667
\(790\) −10.3923 −0.369742
\(791\) −3.71281 −0.132012
\(792\) 0 0
\(793\) 67.5167 2.39759
\(794\) −23.9808 −0.851045
\(795\) −0.732051 −0.0259632
\(796\) −15.3205 −0.543021
\(797\) −26.8756 −0.951984 −0.475992 0.879450i \(-0.657911\pi\)
−0.475992 + 0.879450i \(0.657911\pi\)
\(798\) 1.87564 0.0663971
\(799\) −8.53590 −0.301978
\(800\) 1.00000 0.0353553
\(801\) 9.46410 0.334398
\(802\) 31.2679 1.10411
\(803\) 0 0
\(804\) 4.53590 0.159969
\(805\) −1.80385 −0.0635773
\(806\) −27.6603 −0.974291
\(807\) −29.9808 −1.05537
\(808\) 1.53590 0.0540327
\(809\) 8.19615 0.288161 0.144081 0.989566i \(-0.453978\pi\)
0.144081 + 0.989566i \(0.453978\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −18.1769 −0.638278 −0.319139 0.947708i \(-0.603394\pi\)
−0.319139 + 0.947708i \(0.603394\pi\)
\(812\) 1.33975 0.0470159
\(813\) 3.85641 0.135250
\(814\) 0 0
\(815\) 2.19615 0.0769279
\(816\) −1.73205 −0.0606339
\(817\) −15.3731 −0.537836
\(818\) 0.928203 0.0324539
\(819\) 1.19615 0.0417969
\(820\) −4.19615 −0.146536
\(821\) 44.3923 1.54930 0.774651 0.632389i \(-0.217926\pi\)
0.774651 + 0.632389i \(0.217926\pi\)
\(822\) 3.53590 0.123329
\(823\) −36.3205 −1.26605 −0.633027 0.774130i \(-0.718188\pi\)
−0.633027 + 0.774130i \(0.718188\pi\)
\(824\) 3.39230 0.118177
\(825\) 0 0
\(826\) 3.12436 0.108710
\(827\) 21.0000 0.730242 0.365121 0.930960i \(-0.381028\pi\)
0.365121 + 0.930960i \(0.381028\pi\)
\(828\) −6.73205 −0.233955
\(829\) 10.5885 0.367752 0.183876 0.982949i \(-0.441135\pi\)
0.183876 + 0.982949i \(0.441135\pi\)
\(830\) 9.53590 0.330996
\(831\) 29.8564 1.03571
\(832\) −4.46410 −0.154765
\(833\) 12.0000 0.415775
\(834\) −7.00000 −0.242390
\(835\) −13.4641 −0.465944
\(836\) 0 0
\(837\) 6.19615 0.214170
\(838\) 28.2487 0.975836
\(839\) −22.4641 −0.775547 −0.387773 0.921755i \(-0.626756\pi\)
−0.387773 + 0.921755i \(0.626756\pi\)
\(840\) 0.267949 0.00924513
\(841\) −4.00000 −0.137931
\(842\) 1.66025 0.0572161
\(843\) −21.1244 −0.727561
\(844\) −27.3923 −0.942882
\(845\) −6.92820 −0.238337
\(846\) 4.92820 0.169435
\(847\) 0 0
\(848\) 0.732051 0.0251387
\(849\) −12.3923 −0.425303
\(850\) −1.73205 −0.0594089
\(851\) 48.4449 1.66067
\(852\) −2.46410 −0.0844187
\(853\) −28.7128 −0.983108 −0.491554 0.870847i \(-0.663571\pi\)
−0.491554 + 0.870847i \(0.663571\pi\)
\(854\) 4.05256 0.138676
\(855\) 7.00000 0.239395
\(856\) −6.39230 −0.218484
\(857\) −36.5167 −1.24739 −0.623693 0.781670i \(-0.714368\pi\)
−0.623693 + 0.781670i \(0.714368\pi\)
\(858\) 0 0
\(859\) −2.67949 −0.0914231 −0.0457115 0.998955i \(-0.514556\pi\)
−0.0457115 + 0.998955i \(0.514556\pi\)
\(860\) −2.19615 −0.0748882
\(861\) −1.12436 −0.0383179
\(862\) 37.4449 1.27538
\(863\) −11.4641 −0.390243 −0.195121 0.980779i \(-0.562510\pi\)
−0.195121 + 0.980779i \(0.562510\pi\)
\(864\) 1.00000 0.0340207
\(865\) 6.33975 0.215558
\(866\) 4.00000 0.135926
\(867\) −14.0000 −0.475465
\(868\) −1.66025 −0.0563527
\(869\) 0 0
\(870\) 5.00000 0.169516
\(871\) −20.2487 −0.686101
\(872\) −8.39230 −0.284199
\(873\) 8.73205 0.295535
\(874\) 47.1244 1.59401
\(875\) 0.267949 0.00905834
\(876\) 9.66025 0.326390
\(877\) 22.4641 0.758559 0.379279 0.925282i \(-0.376172\pi\)
0.379279 + 0.925282i \(0.376172\pi\)
\(878\) 14.0000 0.472477
\(879\) 1.07180 0.0361508
\(880\) 0 0
\(881\) −26.5885 −0.895788 −0.447894 0.894087i \(-0.647826\pi\)
−0.447894 + 0.894087i \(0.647826\pi\)
\(882\) −6.92820 −0.233285
\(883\) 26.0526 0.876738 0.438369 0.898795i \(-0.355556\pi\)
0.438369 + 0.898795i \(0.355556\pi\)
\(884\) 7.73205 0.260057
\(885\) 11.6603 0.391955
\(886\) 21.1962 0.712099
\(887\) 36.7846 1.23511 0.617553 0.786529i \(-0.288124\pi\)
0.617553 + 0.786529i \(0.288124\pi\)
\(888\) −7.19615 −0.241487
\(889\) 1.75129 0.0587363
\(890\) −9.46410 −0.317237
\(891\) 0 0
\(892\) −19.2487 −0.644495
\(893\) −34.4974 −1.15441
\(894\) −12.3923 −0.414461
\(895\) −18.5885 −0.621344
\(896\) −0.267949 −0.00895155
\(897\) 30.0526 1.00343
\(898\) −29.3205 −0.978438
\(899\) −30.9808 −1.03327
\(900\) 1.00000 0.0333333
\(901\) −1.26795 −0.0422415
\(902\) 0 0
\(903\) −0.588457 −0.0195826
\(904\) 13.8564 0.460857
\(905\) 9.80385 0.325891
\(906\) −9.80385 −0.325711
\(907\) −48.9282 −1.62463 −0.812317 0.583216i \(-0.801794\pi\)
−0.812317 + 0.583216i \(0.801794\pi\)
\(908\) 20.5359 0.681508
\(909\) 1.53590 0.0509425
\(910\) −1.19615 −0.0396521
\(911\) 45.1051 1.49440 0.747200 0.664600i \(-0.231398\pi\)
0.747200 + 0.664600i \(0.231398\pi\)
\(912\) −7.00000 −0.231793
\(913\) 0 0
\(914\) 24.7321 0.818064
\(915\) 15.1244 0.499996
\(916\) 21.8564 0.722156
\(917\) −0.837169 −0.0276457
\(918\) −1.73205 −0.0571662
\(919\) −36.0526 −1.18926 −0.594632 0.803998i \(-0.702702\pi\)
−0.594632 + 0.803998i \(0.702702\pi\)
\(920\) 6.73205 0.221949
\(921\) −14.5885 −0.480706
\(922\) 15.3923 0.506919
\(923\) 11.0000 0.362069
\(924\) 0 0
\(925\) −7.19615 −0.236608
\(926\) 24.2487 0.796862
\(927\) 3.39230 0.111418
\(928\) −5.00000 −0.164133
\(929\) −31.1244 −1.02116 −0.510578 0.859831i \(-0.670569\pi\)
−0.510578 + 0.859831i \(0.670569\pi\)
\(930\) −6.19615 −0.203180
\(931\) 48.4974 1.58944
\(932\) 22.9282 0.751038
\(933\) 26.9282 0.881590
\(934\) 39.1962 1.28254
\(935\) 0 0
\(936\) −4.46410 −0.145914
\(937\) −8.92820 −0.291672 −0.145836 0.989309i \(-0.546587\pi\)
−0.145836 + 0.989309i \(0.546587\pi\)
\(938\) −1.21539 −0.0396839
\(939\) 22.1962 0.724344
\(940\) −4.92820 −0.160740
\(941\) 19.9282 0.649641 0.324820 0.945776i \(-0.394696\pi\)
0.324820 + 0.945776i \(0.394696\pi\)
\(942\) 2.66025 0.0866758
\(943\) −28.2487 −0.919905
\(944\) −11.6603 −0.379509
\(945\) 0.267949 0.00871639
\(946\) 0 0
\(947\) −28.3731 −0.922001 −0.461000 0.887400i \(-0.652509\pi\)
−0.461000 + 0.887400i \(0.652509\pi\)
\(948\) 10.3923 0.337526
\(949\) −43.1244 −1.39988
\(950\) −7.00000 −0.227110
\(951\) −2.19615 −0.0712151
\(952\) 0.464102 0.0150416
\(953\) 30.7846 0.997211 0.498606 0.866829i \(-0.333846\pi\)
0.498606 + 0.866829i \(0.333846\pi\)
\(954\) 0.732051 0.0237010
\(955\) −13.9282 −0.450706
\(956\) 7.19615 0.232740
\(957\) 0 0
\(958\) 25.9808 0.839400
\(959\) −0.947441 −0.0305945
\(960\) −1.00000 −0.0322749
\(961\) 7.39230 0.238461
\(962\) 32.1244 1.03573
\(963\) −6.39230 −0.205989
\(964\) −11.1962 −0.360604
\(965\) 23.3205 0.750714
\(966\) 1.80385 0.0580378
\(967\) 15.1769 0.488057 0.244028 0.969768i \(-0.421531\pi\)
0.244028 + 0.969768i \(0.421531\pi\)
\(968\) 0 0
\(969\) 12.1244 0.389490
\(970\) −8.73205 −0.280369
\(971\) −60.3923 −1.93808 −0.969041 0.246901i \(-0.920588\pi\)
−0.969041 + 0.246901i \(0.920588\pi\)
\(972\) 1.00000 0.0320750
\(973\) 1.87564 0.0601304
\(974\) −40.3205 −1.29195
\(975\) −4.46410 −0.142966
\(976\) −15.1244 −0.484119
\(977\) 55.8564 1.78700 0.893502 0.449058i \(-0.148241\pi\)
0.893502 + 0.449058i \(0.148241\pi\)
\(978\) −2.19615 −0.0702252
\(979\) 0 0
\(980\) 6.92820 0.221313
\(981\) −8.39230 −0.267946
\(982\) 4.73205 0.151006
\(983\) −41.9090 −1.33669 −0.668344 0.743852i \(-0.732997\pi\)
−0.668344 + 0.743852i \(0.732997\pi\)
\(984\) 4.19615 0.133768
\(985\) 2.53590 0.0808004
\(986\) 8.66025 0.275799
\(987\) −1.32051 −0.0420322
\(988\) 31.2487 0.994154
\(989\) −14.7846 −0.470123
\(990\) 0 0
\(991\) −1.21539 −0.0386081 −0.0193041 0.999814i \(-0.506145\pi\)
−0.0193041 + 0.999814i \(0.506145\pi\)
\(992\) 6.19615 0.196728
\(993\) 30.1244 0.955968
\(994\) 0.660254 0.0209420
\(995\) 15.3205 0.485693
\(996\) −9.53590 −0.302157
\(997\) 48.7128 1.54275 0.771375 0.636381i \(-0.219569\pi\)
0.771375 + 0.636381i \(0.219569\pi\)
\(998\) −16.8038 −0.531917
\(999\) −7.19615 −0.227676
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 3630.2.a.bo.1.2 yes 2
11.10 odd 2 3630.2.a.bg.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3630.2.a.bg.1.1 2 11.10 odd 2
3630.2.a.bo.1.2 yes 2 1.1 even 1 trivial