Properties

Label 4029.2.a.k.1.5
Level 4029
Weight 2
Character 4029.1
Self dual yes
Analytic conductor 32.172
Analytic rank 0
Dimension 31
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4029.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1717269744\)
Analytic rank: \(0\)
Dimension: \(31\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) = 4029.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.22001 q^{2} +1.00000 q^{3} +2.92842 q^{4} -1.63428 q^{5} -2.22001 q^{6} -1.90265 q^{7} -2.06110 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.22001 q^{2} +1.00000 q^{3} +2.92842 q^{4} -1.63428 q^{5} -2.22001 q^{6} -1.90265 q^{7} -2.06110 q^{8} +1.00000 q^{9} +3.62811 q^{10} -4.46342 q^{11} +2.92842 q^{12} -1.29449 q^{13} +4.22389 q^{14} -1.63428 q^{15} -1.28119 q^{16} +1.00000 q^{17} -2.22001 q^{18} -6.67824 q^{19} -4.78587 q^{20} -1.90265 q^{21} +9.90882 q^{22} +5.41253 q^{23} -2.06110 q^{24} -2.32912 q^{25} +2.87378 q^{26} +1.00000 q^{27} -5.57176 q^{28} -0.385878 q^{29} +3.62811 q^{30} -4.16741 q^{31} +6.96645 q^{32} -4.46342 q^{33} -2.22001 q^{34} +3.10947 q^{35} +2.92842 q^{36} -6.13298 q^{37} +14.8257 q^{38} -1.29449 q^{39} +3.36842 q^{40} -9.88644 q^{41} +4.22389 q^{42} +2.00780 q^{43} -13.0708 q^{44} -1.63428 q^{45} -12.0158 q^{46} +3.04402 q^{47} -1.28119 q^{48} -3.37992 q^{49} +5.17066 q^{50} +1.00000 q^{51} -3.79083 q^{52} +11.1495 q^{53} -2.22001 q^{54} +7.29449 q^{55} +3.92156 q^{56} -6.67824 q^{57} +0.856651 q^{58} +3.07457 q^{59} -4.78587 q^{60} +11.5138 q^{61} +9.25168 q^{62} -1.90265 q^{63} -12.9032 q^{64} +2.11557 q^{65} +9.90882 q^{66} -10.5512 q^{67} +2.92842 q^{68} +5.41253 q^{69} -6.90303 q^{70} -9.15147 q^{71} -2.06110 q^{72} -3.90855 q^{73} +13.6152 q^{74} -2.32912 q^{75} -19.5567 q^{76} +8.49233 q^{77} +2.87378 q^{78} +1.00000 q^{79} +2.09382 q^{80} +1.00000 q^{81} +21.9480 q^{82} +9.32483 q^{83} -5.57176 q^{84} -1.63428 q^{85} -4.45732 q^{86} -0.385878 q^{87} +9.19958 q^{88} -2.90621 q^{89} +3.62811 q^{90} +2.46297 q^{91} +15.8502 q^{92} -4.16741 q^{93} -6.75775 q^{94} +10.9141 q^{95} +6.96645 q^{96} -0.111051 q^{97} +7.50345 q^{98} -4.46342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + O(q^{10}) \) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + 5q^{10} + 26q^{11} + 34q^{12} + 7q^{13} + 19q^{14} + 11q^{15} + 40q^{16} + 31q^{17} + 4q^{18} + 32q^{19} + 23q^{20} + 4q^{21} + 2q^{22} + 29q^{23} + 12q^{24} + 32q^{25} + 13q^{26} + 31q^{27} - 13q^{28} + 25q^{29} + 5q^{30} + 22q^{31} + 28q^{32} + 26q^{33} + 4q^{34} + 20q^{35} + 34q^{36} - 4q^{37} + 19q^{38} + 7q^{39} - 3q^{40} + 33q^{41} + 19q^{42} + 6q^{43} + 30q^{44} + 11q^{45} - 11q^{46} + 23q^{47} + 40q^{48} + 31q^{49} + 6q^{50} + 31q^{51} - 7q^{52} + 12q^{53} + 4q^{54} + 40q^{56} + 32q^{57} + 9q^{58} + 27q^{59} + 23q^{60} - 4q^{61} + 25q^{62} + 4q^{63} + 10q^{64} + 54q^{65} + 2q^{66} + 34q^{68} + 29q^{69} - 59q^{70} + 35q^{71} + 12q^{72} + 5q^{73} + 48q^{74} + 32q^{75} + 32q^{76} + 42q^{77} + 13q^{78} + 31q^{79} + 24q^{80} + 31q^{81} + 5q^{82} + 67q^{83} - 13q^{84} + 11q^{85} - 20q^{86} + 25q^{87} - 7q^{88} + 22q^{89} + 5q^{90} + 16q^{91} + 57q^{92} + 22q^{93} + 45q^{94} + 73q^{95} + 28q^{96} - 13q^{97} - 19q^{98} + 26q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.22001 −1.56978 −0.784890 0.619635i \(-0.787281\pi\)
−0.784890 + 0.619635i \(0.787281\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.92842 1.46421
\(5\) −1.63428 −0.730873 −0.365437 0.930836i \(-0.619080\pi\)
−0.365437 + 0.930836i \(0.619080\pi\)
\(6\) −2.22001 −0.906313
\(7\) −1.90265 −0.719134 −0.359567 0.933119i \(-0.617076\pi\)
−0.359567 + 0.933119i \(0.617076\pi\)
\(8\) −2.06110 −0.728710
\(9\) 1.00000 0.333333
\(10\) 3.62811 1.14731
\(11\) −4.46342 −1.34577 −0.672886 0.739746i \(-0.734946\pi\)
−0.672886 + 0.739746i \(0.734946\pi\)
\(12\) 2.92842 0.845363
\(13\) −1.29449 −0.359028 −0.179514 0.983755i \(-0.557453\pi\)
−0.179514 + 0.983755i \(0.557453\pi\)
\(14\) 4.22389 1.12888
\(15\) −1.63428 −0.421970
\(16\) −1.28119 −0.320296
\(17\) 1.00000 0.242536
\(18\) −2.22001 −0.523260
\(19\) −6.67824 −1.53209 −0.766047 0.642785i \(-0.777779\pi\)
−0.766047 + 0.642785i \(0.777779\pi\)
\(20\) −4.78587 −1.07015
\(21\) −1.90265 −0.415192
\(22\) 9.90882 2.11257
\(23\) 5.41253 1.12859 0.564295 0.825573i \(-0.309148\pi\)
0.564295 + 0.825573i \(0.309148\pi\)
\(24\) −2.06110 −0.420721
\(25\) −2.32912 −0.465824
\(26\) 2.87378 0.563596
\(27\) 1.00000 0.192450
\(28\) −5.57176 −1.05296
\(29\) −0.385878 −0.0716557 −0.0358278 0.999358i \(-0.511407\pi\)
−0.0358278 + 0.999358i \(0.511407\pi\)
\(30\) 3.62811 0.662400
\(31\) −4.16741 −0.748490 −0.374245 0.927330i \(-0.622098\pi\)
−0.374245 + 0.927330i \(0.622098\pi\)
\(32\) 6.96645 1.23151
\(33\) −4.46342 −0.776982
\(34\) −2.22001 −0.380728
\(35\) 3.10947 0.525596
\(36\) 2.92842 0.488070
\(37\) −6.13298 −1.00826 −0.504128 0.863629i \(-0.668186\pi\)
−0.504128 + 0.863629i \(0.668186\pi\)
\(38\) 14.8257 2.40505
\(39\) −1.29449 −0.207285
\(40\) 3.36842 0.532595
\(41\) −9.88644 −1.54400 −0.772002 0.635621i \(-0.780744\pi\)
−0.772002 + 0.635621i \(0.780744\pi\)
\(42\) 4.22389 0.651761
\(43\) 2.00780 0.306187 0.153093 0.988212i \(-0.451077\pi\)
0.153093 + 0.988212i \(0.451077\pi\)
\(44\) −13.0708 −1.97050
\(45\) −1.63428 −0.243624
\(46\) −12.0158 −1.77164
\(47\) 3.04402 0.444017 0.222008 0.975045i \(-0.428739\pi\)
0.222008 + 0.975045i \(0.428739\pi\)
\(48\) −1.28119 −0.184923
\(49\) −3.37992 −0.482846
\(50\) 5.17066 0.731242
\(51\) 1.00000 0.140028
\(52\) −3.79083 −0.525693
\(53\) 11.1495 1.53151 0.765754 0.643133i \(-0.222366\pi\)
0.765754 + 0.643133i \(0.222366\pi\)
\(54\) −2.22001 −0.302104
\(55\) 7.29449 0.983589
\(56\) 3.92156 0.524040
\(57\) −6.67824 −0.884555
\(58\) 0.856651 0.112484
\(59\) 3.07457 0.400275 0.200137 0.979768i \(-0.435861\pi\)
0.200137 + 0.979768i \(0.435861\pi\)
\(60\) −4.78587 −0.617853
\(61\) 11.5138 1.47420 0.737099 0.675785i \(-0.236195\pi\)
0.737099 + 0.675785i \(0.236195\pi\)
\(62\) 9.25168 1.17496
\(63\) −1.90265 −0.239711
\(64\) −12.9032 −1.61290
\(65\) 2.11557 0.262404
\(66\) 9.90882 1.21969
\(67\) −10.5512 −1.28904 −0.644519 0.764588i \(-0.722942\pi\)
−0.644519 + 0.764588i \(0.722942\pi\)
\(68\) 2.92842 0.355123
\(69\) 5.41253 0.651592
\(70\) −6.90303 −0.825070
\(71\) −9.15147 −1.08608 −0.543040 0.839707i \(-0.682727\pi\)
−0.543040 + 0.839707i \(0.682727\pi\)
\(72\) −2.06110 −0.242903
\(73\) −3.90855 −0.457461 −0.228730 0.973490i \(-0.573457\pi\)
−0.228730 + 0.973490i \(0.573457\pi\)
\(74\) 13.6152 1.58274
\(75\) −2.32912 −0.268944
\(76\) −19.5567 −2.24331
\(77\) 8.49233 0.967791
\(78\) 2.87378 0.325392
\(79\) 1.00000 0.112509
\(80\) 2.09382 0.234096
\(81\) 1.00000 0.111111
\(82\) 21.9480 2.42375
\(83\) 9.32483 1.02353 0.511766 0.859125i \(-0.328991\pi\)
0.511766 + 0.859125i \(0.328991\pi\)
\(84\) −5.57176 −0.607929
\(85\) −1.63428 −0.177263
\(86\) −4.45732 −0.480646
\(87\) −0.385878 −0.0413704
\(88\) 9.19958 0.980678
\(89\) −2.90621 −0.308058 −0.154029 0.988066i \(-0.549225\pi\)
−0.154029 + 0.988066i \(0.549225\pi\)
\(90\) 3.62811 0.382437
\(91\) 2.46297 0.258189
\(92\) 15.8502 1.65249
\(93\) −4.16741 −0.432141
\(94\) −6.75775 −0.697009
\(95\) 10.9141 1.11977
\(96\) 6.96645 0.711010
\(97\) −0.111051 −0.0112755 −0.00563777 0.999984i \(-0.501795\pi\)
−0.00563777 + 0.999984i \(0.501795\pi\)
\(98\) 7.50345 0.757963
\(99\) −4.46342 −0.448591
\(100\) −6.82066 −0.682066
\(101\) −4.06378 −0.404361 −0.202181 0.979348i \(-0.564803\pi\)
−0.202181 + 0.979348i \(0.564803\pi\)
\(102\) −2.22001 −0.219813
\(103\) 2.71058 0.267082 0.133541 0.991043i \(-0.457365\pi\)
0.133541 + 0.991043i \(0.457365\pi\)
\(104\) 2.66809 0.261628
\(105\) 3.10947 0.303453
\(106\) −24.7521 −2.40413
\(107\) −14.6277 −1.41411 −0.707056 0.707157i \(-0.749977\pi\)
−0.707056 + 0.707157i \(0.749977\pi\)
\(108\) 2.92842 0.281788
\(109\) −1.86669 −0.178797 −0.0893983 0.995996i \(-0.528494\pi\)
−0.0893983 + 0.995996i \(0.528494\pi\)
\(110\) −16.1938 −1.54402
\(111\) −6.13298 −0.582117
\(112\) 2.43765 0.230336
\(113\) 14.1312 1.32935 0.664674 0.747133i \(-0.268570\pi\)
0.664674 + 0.747133i \(0.268570\pi\)
\(114\) 14.8257 1.38856
\(115\) −8.84560 −0.824856
\(116\) −1.13001 −0.104919
\(117\) −1.29449 −0.119676
\(118\) −6.82556 −0.628343
\(119\) −1.90265 −0.174416
\(120\) 3.36842 0.307494
\(121\) 8.92215 0.811104
\(122\) −25.5608 −2.31417
\(123\) −9.88644 −0.891431
\(124\) −12.2039 −1.09595
\(125\) 11.9779 1.07133
\(126\) 4.22389 0.376294
\(127\) −4.97589 −0.441539 −0.220769 0.975326i \(-0.570857\pi\)
−0.220769 + 0.975326i \(0.570857\pi\)
\(128\) 14.7122 1.30039
\(129\) 2.00780 0.176777
\(130\) −4.69657 −0.411917
\(131\) 17.1743 1.50052 0.750261 0.661142i \(-0.229928\pi\)
0.750261 + 0.661142i \(0.229928\pi\)
\(132\) −13.0708 −1.13767
\(133\) 12.7064 1.10178
\(134\) 23.4238 2.02351
\(135\) −1.63428 −0.140657
\(136\) −2.06110 −0.176738
\(137\) 6.69309 0.571829 0.285915 0.958255i \(-0.407703\pi\)
0.285915 + 0.958255i \(0.407703\pi\)
\(138\) −12.0158 −1.02286
\(139\) 14.9696 1.26970 0.634852 0.772634i \(-0.281061\pi\)
0.634852 + 0.772634i \(0.281061\pi\)
\(140\) 9.10583 0.769583
\(141\) 3.04402 0.256353
\(142\) 20.3163 1.70491
\(143\) 5.77788 0.483170
\(144\) −1.28119 −0.106765
\(145\) 0.630633 0.0523712
\(146\) 8.67700 0.718113
\(147\) −3.37992 −0.278771
\(148\) −17.9600 −1.47630
\(149\) −7.16380 −0.586882 −0.293441 0.955977i \(-0.594800\pi\)
−0.293441 + 0.955977i \(0.594800\pi\)
\(150\) 5.17066 0.422183
\(151\) 21.9170 1.78358 0.891790 0.452450i \(-0.149450\pi\)
0.891790 + 0.452450i \(0.149450\pi\)
\(152\) 13.7645 1.11645
\(153\) 1.00000 0.0808452
\(154\) −18.8530 −1.51922
\(155\) 6.81073 0.547051
\(156\) −3.79083 −0.303509
\(157\) −17.8477 −1.42440 −0.712200 0.701977i \(-0.752301\pi\)
−0.712200 + 0.701977i \(0.752301\pi\)
\(158\) −2.22001 −0.176614
\(159\) 11.1495 0.884217
\(160\) −11.3851 −0.900074
\(161\) −10.2981 −0.811607
\(162\) −2.22001 −0.174420
\(163\) 9.39482 0.735859 0.367930 0.929854i \(-0.380067\pi\)
0.367930 + 0.929854i \(0.380067\pi\)
\(164\) −28.9517 −2.26075
\(165\) 7.29449 0.567875
\(166\) −20.7012 −1.60672
\(167\) −4.89296 −0.378628 −0.189314 0.981917i \(-0.560626\pi\)
−0.189314 + 0.981917i \(0.560626\pi\)
\(168\) 3.92156 0.302555
\(169\) −11.3243 −0.871099
\(170\) 3.62811 0.278264
\(171\) −6.67824 −0.510698
\(172\) 5.87968 0.448322
\(173\) 22.1087 1.68090 0.840448 0.541891i \(-0.182292\pi\)
0.840448 + 0.541891i \(0.182292\pi\)
\(174\) 0.856651 0.0649425
\(175\) 4.43150 0.334990
\(176\) 5.71847 0.431046
\(177\) 3.07457 0.231099
\(178\) 6.45180 0.483583
\(179\) −15.7268 −1.17547 −0.587737 0.809052i \(-0.699981\pi\)
−0.587737 + 0.809052i \(0.699981\pi\)
\(180\) −4.78587 −0.356718
\(181\) 15.6878 1.16606 0.583031 0.812450i \(-0.301867\pi\)
0.583031 + 0.812450i \(0.301867\pi\)
\(182\) −5.46781 −0.405301
\(183\) 11.5138 0.851128
\(184\) −11.1558 −0.822415
\(185\) 10.0230 0.736907
\(186\) 9.25168 0.678366
\(187\) −4.46342 −0.326398
\(188\) 8.91419 0.650134
\(189\) −1.90265 −0.138397
\(190\) −24.2294 −1.75779
\(191\) 19.6849 1.42435 0.712176 0.702001i \(-0.247710\pi\)
0.712176 + 0.702001i \(0.247710\pi\)
\(192\) −12.9032 −0.931206
\(193\) 4.07050 0.293001 0.146500 0.989211i \(-0.453199\pi\)
0.146500 + 0.989211i \(0.453199\pi\)
\(194\) 0.246534 0.0177001
\(195\) 2.11557 0.151499
\(196\) −9.89785 −0.706989
\(197\) −20.3233 −1.44797 −0.723987 0.689814i \(-0.757692\pi\)
−0.723987 + 0.689814i \(0.757692\pi\)
\(198\) 9.90882 0.704189
\(199\) −10.5630 −0.748789 −0.374395 0.927269i \(-0.622149\pi\)
−0.374395 + 0.927269i \(0.622149\pi\)
\(200\) 4.80056 0.339451
\(201\) −10.5512 −0.744227
\(202\) 9.02162 0.634759
\(203\) 0.734190 0.0515300
\(204\) 2.92842 0.205031
\(205\) 16.1572 1.12847
\(206\) −6.01751 −0.419260
\(207\) 5.41253 0.376197
\(208\) 1.65849 0.114995
\(209\) 29.8078 2.06185
\(210\) −6.90303 −0.476354
\(211\) 19.0212 1.30947 0.654735 0.755858i \(-0.272780\pi\)
0.654735 + 0.755858i \(0.272780\pi\)
\(212\) 32.6506 2.24245
\(213\) −9.15147 −0.627048
\(214\) 32.4736 2.21985
\(215\) −3.28131 −0.223783
\(216\) −2.06110 −0.140240
\(217\) 7.92913 0.538264
\(218\) 4.14406 0.280671
\(219\) −3.90855 −0.264115
\(220\) 21.3614 1.44018
\(221\) −1.29449 −0.0870771
\(222\) 13.6152 0.913795
\(223\) 8.55625 0.572969 0.286485 0.958085i \(-0.407513\pi\)
0.286485 + 0.958085i \(0.407513\pi\)
\(224\) −13.2547 −0.885617
\(225\) −2.32912 −0.155275
\(226\) −31.3713 −2.08679
\(227\) 8.88220 0.589532 0.294766 0.955569i \(-0.404758\pi\)
0.294766 + 0.955569i \(0.404758\pi\)
\(228\) −19.5567 −1.29518
\(229\) −29.3221 −1.93766 −0.968830 0.247725i \(-0.920317\pi\)
−0.968830 + 0.247725i \(0.920317\pi\)
\(230\) 19.6373 1.29484
\(231\) 8.49233 0.558754
\(232\) 0.795334 0.0522162
\(233\) 2.85833 0.187256 0.0936278 0.995607i \(-0.470154\pi\)
0.0936278 + 0.995607i \(0.470154\pi\)
\(234\) 2.87378 0.187865
\(235\) −4.97479 −0.324520
\(236\) 9.00363 0.586087
\(237\) 1.00000 0.0649570
\(238\) 4.22389 0.273794
\(239\) −24.0944 −1.55854 −0.779270 0.626689i \(-0.784410\pi\)
−0.779270 + 0.626689i \(0.784410\pi\)
\(240\) 2.09382 0.135155
\(241\) 0.357116 0.0230039 0.0115019 0.999934i \(-0.496339\pi\)
0.0115019 + 0.999934i \(0.496339\pi\)
\(242\) −19.8072 −1.27326
\(243\) 1.00000 0.0641500
\(244\) 33.7174 2.15854
\(245\) 5.52375 0.352899
\(246\) 21.9480 1.39935
\(247\) 8.64495 0.550065
\(248\) 8.58947 0.545432
\(249\) 9.32483 0.590937
\(250\) −26.5909 −1.68176
\(251\) −9.94112 −0.627478 −0.313739 0.949509i \(-0.601582\pi\)
−0.313739 + 0.949509i \(0.601582\pi\)
\(252\) −5.57176 −0.350988
\(253\) −24.1584 −1.51883
\(254\) 11.0465 0.693119
\(255\) −1.63428 −0.102343
\(256\) −6.85486 −0.428429
\(257\) −30.7848 −1.92030 −0.960150 0.279485i \(-0.909836\pi\)
−0.960150 + 0.279485i \(0.909836\pi\)
\(258\) −4.45732 −0.277501
\(259\) 11.6689 0.725071
\(260\) 6.19528 0.384215
\(261\) −0.385878 −0.0238852
\(262\) −38.1269 −2.35549
\(263\) −14.0747 −0.867885 −0.433943 0.900941i \(-0.642878\pi\)
−0.433943 + 0.900941i \(0.642878\pi\)
\(264\) 9.19958 0.566195
\(265\) −18.2215 −1.11934
\(266\) −28.2082 −1.72955
\(267\) −2.90621 −0.177857
\(268\) −30.8985 −1.88743
\(269\) −29.9446 −1.82575 −0.912877 0.408235i \(-0.866144\pi\)
−0.912877 + 0.408235i \(0.866144\pi\)
\(270\) 3.62811 0.220800
\(271\) 22.7835 1.38400 0.692000 0.721898i \(-0.256730\pi\)
0.692000 + 0.721898i \(0.256730\pi\)
\(272\) −1.28119 −0.0776833
\(273\) 2.46297 0.149066
\(274\) −14.8587 −0.897646
\(275\) 10.3959 0.626894
\(276\) 15.8502 0.954068
\(277\) 26.0724 1.56654 0.783269 0.621682i \(-0.213551\pi\)
0.783269 + 0.621682i \(0.213551\pi\)
\(278\) −33.2326 −1.99316
\(279\) −4.16741 −0.249497
\(280\) −6.40893 −0.383007
\(281\) 31.1424 1.85780 0.928898 0.370335i \(-0.120757\pi\)
0.928898 + 0.370335i \(0.120757\pi\)
\(282\) −6.75775 −0.402418
\(283\) −4.82456 −0.286791 −0.143395 0.989665i \(-0.545802\pi\)
−0.143395 + 0.989665i \(0.545802\pi\)
\(284\) −26.7994 −1.59025
\(285\) 10.9141 0.646497
\(286\) −12.8269 −0.758472
\(287\) 18.8104 1.11035
\(288\) 6.96645 0.410502
\(289\) 1.00000 0.0588235
\(290\) −1.40001 −0.0822113
\(291\) −0.111051 −0.00650993
\(292\) −11.4459 −0.669820
\(293\) 14.7399 0.861115 0.430557 0.902563i \(-0.358317\pi\)
0.430557 + 0.902563i \(0.358317\pi\)
\(294\) 7.50345 0.437610
\(295\) −5.02471 −0.292550
\(296\) 12.6407 0.734726
\(297\) −4.46342 −0.258994
\(298\) 15.9037 0.921275
\(299\) −7.00649 −0.405196
\(300\) −6.82066 −0.393791
\(301\) −3.82014 −0.220189
\(302\) −48.6558 −2.79983
\(303\) −4.06378 −0.233458
\(304\) 8.55607 0.490724
\(305\) −18.8169 −1.07745
\(306\) −2.22001 −0.126909
\(307\) −9.51663 −0.543143 −0.271571 0.962418i \(-0.587543\pi\)
−0.271571 + 0.962418i \(0.587543\pi\)
\(308\) 24.8691 1.41705
\(309\) 2.71058 0.154200
\(310\) −15.1199 −0.858750
\(311\) 10.6710 0.605098 0.302549 0.953134i \(-0.402162\pi\)
0.302549 + 0.953134i \(0.402162\pi\)
\(312\) 2.66809 0.151051
\(313\) −20.8019 −1.17579 −0.587897 0.808936i \(-0.700044\pi\)
−0.587897 + 0.808936i \(0.700044\pi\)
\(314\) 39.6219 2.23600
\(315\) 3.10947 0.175199
\(316\) 2.92842 0.164737
\(317\) 17.2323 0.967861 0.483930 0.875106i \(-0.339209\pi\)
0.483930 + 0.875106i \(0.339209\pi\)
\(318\) −24.7521 −1.38803
\(319\) 1.72234 0.0964323
\(320\) 21.0874 1.17882
\(321\) −14.6277 −0.816438
\(322\) 22.8619 1.27405
\(323\) −6.67824 −0.371587
\(324\) 2.92842 0.162690
\(325\) 3.01504 0.167244
\(326\) −20.8566 −1.15514
\(327\) −1.86669 −0.103228
\(328\) 20.3770 1.12513
\(329\) −5.79171 −0.319307
\(330\) −16.1938 −0.891440
\(331\) 26.6575 1.46523 0.732614 0.680645i \(-0.238300\pi\)
0.732614 + 0.680645i \(0.238300\pi\)
\(332\) 27.3070 1.49867
\(333\) −6.13298 −0.336085
\(334\) 10.8624 0.594363
\(335\) 17.2437 0.942124
\(336\) 2.43765 0.132985
\(337\) −12.8716 −0.701163 −0.350582 0.936532i \(-0.614016\pi\)
−0.350582 + 0.936532i \(0.614016\pi\)
\(338\) 25.1400 1.36743
\(339\) 14.1312 0.767500
\(340\) −4.78587 −0.259550
\(341\) 18.6009 1.00730
\(342\) 14.8257 0.801684
\(343\) 19.7494 1.06637
\(344\) −4.13828 −0.223121
\(345\) −8.84560 −0.476231
\(346\) −49.0815 −2.63864
\(347\) 19.9735 1.07223 0.536117 0.844144i \(-0.319891\pi\)
0.536117 + 0.844144i \(0.319891\pi\)
\(348\) −1.13001 −0.0605751
\(349\) −16.2872 −0.871833 −0.435916 0.899987i \(-0.643576\pi\)
−0.435916 + 0.899987i \(0.643576\pi\)
\(350\) −9.83796 −0.525861
\(351\) −1.29449 −0.0690950
\(352\) −31.0942 −1.65733
\(353\) −14.3591 −0.764258 −0.382129 0.924109i \(-0.624809\pi\)
−0.382129 + 0.924109i \(0.624809\pi\)
\(354\) −6.82556 −0.362774
\(355\) 14.9561 0.793786
\(356\) −8.51062 −0.451062
\(357\) −1.90265 −0.100699
\(358\) 34.9135 1.84524
\(359\) 9.07260 0.478834 0.239417 0.970917i \(-0.423044\pi\)
0.239417 + 0.970917i \(0.423044\pi\)
\(360\) 3.36842 0.177532
\(361\) 25.5989 1.34731
\(362\) −34.8269 −1.83046
\(363\) 8.92215 0.468291
\(364\) 7.21262 0.378044
\(365\) 6.38767 0.334346
\(366\) −25.5608 −1.33608
\(367\) −22.8469 −1.19260 −0.596299 0.802762i \(-0.703363\pi\)
−0.596299 + 0.802762i \(0.703363\pi\)
\(368\) −6.93445 −0.361483
\(369\) −9.88644 −0.514668
\(370\) −22.2511 −1.15678
\(371\) −21.2137 −1.10136
\(372\) −12.2039 −0.632745
\(373\) −17.2152 −0.891369 −0.445684 0.895190i \(-0.647040\pi\)
−0.445684 + 0.895190i \(0.647040\pi\)
\(374\) 9.90882 0.512373
\(375\) 11.9779 0.618534
\(376\) −6.27405 −0.323559
\(377\) 0.499517 0.0257264
\(378\) 4.22389 0.217254
\(379\) −2.54211 −0.130580 −0.0652898 0.997866i \(-0.520797\pi\)
−0.0652898 + 0.997866i \(0.520797\pi\)
\(380\) 31.9612 1.63957
\(381\) −4.97589 −0.254923
\(382\) −43.7007 −2.23592
\(383\) 30.8515 1.57644 0.788220 0.615394i \(-0.211003\pi\)
0.788220 + 0.615394i \(0.211003\pi\)
\(384\) 14.7122 0.750780
\(385\) −13.8789 −0.707332
\(386\) −9.03653 −0.459947
\(387\) 2.00780 0.102062
\(388\) −0.325205 −0.0165098
\(389\) 9.85490 0.499663 0.249832 0.968289i \(-0.419625\pi\)
0.249832 + 0.968289i \(0.419625\pi\)
\(390\) −4.69657 −0.237820
\(391\) 5.41253 0.273723
\(392\) 6.96637 0.351855
\(393\) 17.1743 0.866326
\(394\) 45.1178 2.27300
\(395\) −1.63428 −0.0822297
\(396\) −13.0708 −0.656832
\(397\) 18.2723 0.917061 0.458531 0.888679i \(-0.348376\pi\)
0.458531 + 0.888679i \(0.348376\pi\)
\(398\) 23.4499 1.17544
\(399\) 12.7064 0.636113
\(400\) 2.98404 0.149202
\(401\) 10.6339 0.531030 0.265515 0.964107i \(-0.414458\pi\)
0.265515 + 0.964107i \(0.414458\pi\)
\(402\) 23.4238 1.16827
\(403\) 5.39469 0.268729
\(404\) −11.9005 −0.592071
\(405\) −1.63428 −0.0812081
\(406\) −1.62991 −0.0808909
\(407\) 27.3741 1.35688
\(408\) −2.06110 −0.102040
\(409\) 20.1190 0.994822 0.497411 0.867515i \(-0.334284\pi\)
0.497411 + 0.867515i \(0.334284\pi\)
\(410\) −35.8691 −1.77145
\(411\) 6.69309 0.330146
\(412\) 7.93774 0.391064
\(413\) −5.84982 −0.287851
\(414\) −12.0158 −0.590546
\(415\) −15.2394 −0.748072
\(416\) −9.01803 −0.442145
\(417\) 14.9696 0.733064
\(418\) −66.1735 −3.23665
\(419\) 18.7028 0.913690 0.456845 0.889546i \(-0.348979\pi\)
0.456845 + 0.889546i \(0.348979\pi\)
\(420\) 9.10583 0.444319
\(421\) 12.0322 0.586415 0.293208 0.956049i \(-0.405277\pi\)
0.293208 + 0.956049i \(0.405277\pi\)
\(422\) −42.2271 −2.05558
\(423\) 3.04402 0.148006
\(424\) −22.9804 −1.11603
\(425\) −2.32912 −0.112979
\(426\) 20.3163 0.984328
\(427\) −21.9068 −1.06015
\(428\) −42.8361 −2.07056
\(429\) 5.77788 0.278959
\(430\) 7.28453 0.351291
\(431\) 32.4481 1.56297 0.781484 0.623925i \(-0.214463\pi\)
0.781484 + 0.623925i \(0.214463\pi\)
\(432\) −1.28119 −0.0616411
\(433\) −33.4347 −1.60677 −0.803384 0.595461i \(-0.796969\pi\)
−0.803384 + 0.595461i \(0.796969\pi\)
\(434\) −17.6027 −0.844957
\(435\) 0.630633 0.0302365
\(436\) −5.46646 −0.261796
\(437\) −36.1462 −1.72911
\(438\) 8.67700 0.414603
\(439\) 13.4602 0.642421 0.321210 0.947008i \(-0.395910\pi\)
0.321210 + 0.947008i \(0.395910\pi\)
\(440\) −15.0347 −0.716751
\(441\) −3.37992 −0.160949
\(442\) 2.87378 0.136692
\(443\) 24.6866 1.17289 0.586447 0.809988i \(-0.300526\pi\)
0.586447 + 0.809988i \(0.300526\pi\)
\(444\) −17.9600 −0.852342
\(445\) 4.74957 0.225151
\(446\) −18.9949 −0.899436
\(447\) −7.16380 −0.338836
\(448\) 24.5502 1.15989
\(449\) 33.3851 1.57554 0.787769 0.615971i \(-0.211236\pi\)
0.787769 + 0.615971i \(0.211236\pi\)
\(450\) 5.17066 0.243747
\(451\) 44.1274 2.07788
\(452\) 41.3821 1.94645
\(453\) 21.9170 1.02975
\(454\) −19.7185 −0.925437
\(455\) −4.02519 −0.188704
\(456\) 13.7645 0.644584
\(457\) 14.2678 0.667418 0.333709 0.942676i \(-0.391700\pi\)
0.333709 + 0.942676i \(0.391700\pi\)
\(458\) 65.0953 3.04170
\(459\) 1.00000 0.0466760
\(460\) −25.9036 −1.20776
\(461\) −8.54505 −0.397983 −0.198991 0.980001i \(-0.563767\pi\)
−0.198991 + 0.980001i \(0.563767\pi\)
\(462\) −18.8530 −0.877122
\(463\) 6.34967 0.295094 0.147547 0.989055i \(-0.452862\pi\)
0.147547 + 0.989055i \(0.452862\pi\)
\(464\) 0.494381 0.0229511
\(465\) 6.81073 0.315840
\(466\) −6.34551 −0.293950
\(467\) 5.83527 0.270024 0.135012 0.990844i \(-0.456893\pi\)
0.135012 + 0.990844i \(0.456893\pi\)
\(468\) −3.79083 −0.175231
\(469\) 20.0753 0.926991
\(470\) 11.0441 0.509425
\(471\) −17.8477 −0.822378
\(472\) −6.33700 −0.291684
\(473\) −8.96166 −0.412057
\(474\) −2.22001 −0.101968
\(475\) 15.5544 0.713687
\(476\) −5.57176 −0.255381
\(477\) 11.1495 0.510503
\(478\) 53.4898 2.44657
\(479\) 10.9464 0.500155 0.250077 0.968226i \(-0.419544\pi\)
0.250077 + 0.968226i \(0.419544\pi\)
\(480\) −11.3851 −0.519658
\(481\) 7.93911 0.361992
\(482\) −0.792799 −0.0361110
\(483\) −10.2981 −0.468582
\(484\) 26.1278 1.18763
\(485\) 0.181489 0.00824099
\(486\) −2.22001 −0.100701
\(487\) −7.31707 −0.331568 −0.165784 0.986162i \(-0.553015\pi\)
−0.165784 + 0.986162i \(0.553015\pi\)
\(488\) −23.7312 −1.07426
\(489\) 9.39482 0.424849
\(490\) −12.2628 −0.553975
\(491\) −11.9953 −0.541339 −0.270670 0.962672i \(-0.587245\pi\)
−0.270670 + 0.962672i \(0.587245\pi\)
\(492\) −28.9517 −1.30524
\(493\) −0.385878 −0.0173791
\(494\) −19.1918 −0.863481
\(495\) 7.29449 0.327863
\(496\) 5.33923 0.239738
\(497\) 17.4120 0.781036
\(498\) −20.7012 −0.927641
\(499\) 14.1153 0.631889 0.315945 0.948778i \(-0.397679\pi\)
0.315945 + 0.948778i \(0.397679\pi\)
\(500\) 35.0762 1.56866
\(501\) −4.89296 −0.218601
\(502\) 22.0693 0.985002
\(503\) 23.3365 1.04052 0.520261 0.854007i \(-0.325835\pi\)
0.520261 + 0.854007i \(0.325835\pi\)
\(504\) 3.92156 0.174680
\(505\) 6.64137 0.295537
\(506\) 53.6318 2.38422
\(507\) −11.3243 −0.502929
\(508\) −14.5715 −0.646506
\(509\) 32.1813 1.42641 0.713206 0.700954i \(-0.247242\pi\)
0.713206 + 0.700954i \(0.247242\pi\)
\(510\) 3.62811 0.160656
\(511\) 7.43660 0.328976
\(512\) −14.2066 −0.627850
\(513\) −6.67824 −0.294852
\(514\) 68.3423 3.01445
\(515\) −4.42986 −0.195203
\(516\) 5.87968 0.258839
\(517\) −13.5868 −0.597545
\(518\) −25.9050 −1.13820
\(519\) 22.1087 0.970466
\(520\) −4.36041 −0.191217
\(521\) 11.4869 0.503251 0.251625 0.967825i \(-0.419035\pi\)
0.251625 + 0.967825i \(0.419035\pi\)
\(522\) 0.856651 0.0374946
\(523\) 3.23156 0.141306 0.0706531 0.997501i \(-0.477492\pi\)
0.0706531 + 0.997501i \(0.477492\pi\)
\(524\) 50.2935 2.19708
\(525\) 4.43150 0.193407
\(526\) 31.2460 1.36239
\(527\) −4.16741 −0.181535
\(528\) 5.71847 0.248865
\(529\) 6.29545 0.273715
\(530\) 40.4518 1.75712
\(531\) 3.07457 0.133425
\(532\) 37.2096 1.61324
\(533\) 12.7979 0.554341
\(534\) 6.45180 0.279197
\(535\) 23.9058 1.03354
\(536\) 21.7472 0.939336
\(537\) −15.7268 −0.678660
\(538\) 66.4771 2.86603
\(539\) 15.0860 0.649802
\(540\) −4.78587 −0.205951
\(541\) −5.01331 −0.215539 −0.107770 0.994176i \(-0.534371\pi\)
−0.107770 + 0.994176i \(0.534371\pi\)
\(542\) −50.5795 −2.17258
\(543\) 15.6878 0.673226
\(544\) 6.96645 0.298684
\(545\) 3.05070 0.130678
\(546\) −5.46781 −0.234000
\(547\) −22.2531 −0.951472 −0.475736 0.879588i \(-0.657818\pi\)
−0.475736 + 0.879588i \(0.657818\pi\)
\(548\) 19.6002 0.837279
\(549\) 11.5138 0.491399
\(550\) −23.0789 −0.984086
\(551\) 2.57698 0.109783
\(552\) −11.1558 −0.474821
\(553\) −1.90265 −0.0809089
\(554\) −57.8809 −2.45912
\(555\) 10.0230 0.425453
\(556\) 43.8373 1.85912
\(557\) −0.355109 −0.0150464 −0.00752322 0.999972i \(-0.502395\pi\)
−0.00752322 + 0.999972i \(0.502395\pi\)
\(558\) 9.25168 0.391655
\(559\) −2.59909 −0.109930
\(560\) −3.98380 −0.168346
\(561\) −4.46342 −0.188446
\(562\) −69.1362 −2.91633
\(563\) −38.8358 −1.63673 −0.818367 0.574697i \(-0.805120\pi\)
−0.818367 + 0.574697i \(0.805120\pi\)
\(564\) 8.91419 0.375355
\(565\) −23.0943 −0.971585
\(566\) 10.7106 0.450198
\(567\) −1.90265 −0.0799038
\(568\) 18.8621 0.791437
\(569\) 8.40024 0.352156 0.176078 0.984376i \(-0.443659\pi\)
0.176078 + 0.984376i \(0.443659\pi\)
\(570\) −24.2294 −1.01486
\(571\) 15.9096 0.665797 0.332899 0.942963i \(-0.391973\pi\)
0.332899 + 0.942963i \(0.391973\pi\)
\(572\) 16.9201 0.707464
\(573\) 19.6849 0.822350
\(574\) −41.7593 −1.74300
\(575\) −12.6064 −0.525725
\(576\) −12.9032 −0.537632
\(577\) 10.4062 0.433216 0.216608 0.976259i \(-0.430501\pi\)
0.216608 + 0.976259i \(0.430501\pi\)
\(578\) −2.22001 −0.0923400
\(579\) 4.07050 0.169164
\(580\) 1.84676 0.0766825
\(581\) −17.7419 −0.736057
\(582\) 0.246534 0.0102192
\(583\) −49.7652 −2.06106
\(584\) 8.05592 0.333356
\(585\) 2.11557 0.0874680
\(586\) −32.7227 −1.35176
\(587\) 29.6498 1.22378 0.611888 0.790944i \(-0.290410\pi\)
0.611888 + 0.790944i \(0.290410\pi\)
\(588\) −9.89785 −0.408180
\(589\) 27.8310 1.14676
\(590\) 11.1549 0.459239
\(591\) −20.3233 −0.835988
\(592\) 7.85748 0.322941
\(593\) 44.7799 1.83889 0.919445 0.393219i \(-0.128639\pi\)
0.919445 + 0.393219i \(0.128639\pi\)
\(594\) 9.90882 0.406564
\(595\) 3.10947 0.127476
\(596\) −20.9786 −0.859319
\(597\) −10.5630 −0.432314
\(598\) 15.5544 0.636068
\(599\) −24.6038 −1.00529 −0.502643 0.864494i \(-0.667639\pi\)
−0.502643 + 0.864494i \(0.667639\pi\)
\(600\) 4.80056 0.195982
\(601\) 20.3776 0.831219 0.415609 0.909543i \(-0.363568\pi\)
0.415609 + 0.909543i \(0.363568\pi\)
\(602\) 8.48073 0.345649
\(603\) −10.5512 −0.429680
\(604\) 64.1822 2.61154
\(605\) −14.5813 −0.592814
\(606\) 9.02162 0.366478
\(607\) 4.01090 0.162797 0.0813987 0.996682i \(-0.474061\pi\)
0.0813987 + 0.996682i \(0.474061\pi\)
\(608\) −46.5236 −1.88678
\(609\) 0.734190 0.0297509
\(610\) 41.7736 1.69136
\(611\) −3.94047 −0.159414
\(612\) 2.92842 0.118374
\(613\) −19.0827 −0.770743 −0.385372 0.922761i \(-0.625927\pi\)
−0.385372 + 0.922761i \(0.625927\pi\)
\(614\) 21.1270 0.852615
\(615\) 16.1572 0.651523
\(616\) −17.5036 −0.705239
\(617\) −43.6562 −1.75753 −0.878767 0.477251i \(-0.841633\pi\)
−0.878767 + 0.477251i \(0.841633\pi\)
\(618\) −6.01751 −0.242060
\(619\) 21.0467 0.845937 0.422969 0.906144i \(-0.360988\pi\)
0.422969 + 0.906144i \(0.360988\pi\)
\(620\) 19.9447 0.800998
\(621\) 5.41253 0.217197
\(622\) −23.6897 −0.949870
\(623\) 5.52950 0.221535
\(624\) 1.65849 0.0663926
\(625\) −7.92958 −0.317183
\(626\) 46.1803 1.84574
\(627\) 29.8078 1.19041
\(628\) −52.2655 −2.08562
\(629\) −6.13298 −0.244538
\(630\) −6.90303 −0.275023
\(631\) −23.1984 −0.923513 −0.461757 0.887007i \(-0.652781\pi\)
−0.461757 + 0.887007i \(0.652781\pi\)
\(632\) −2.06110 −0.0819863
\(633\) 19.0212 0.756023
\(634\) −38.2557 −1.51933
\(635\) 8.13201 0.322709
\(636\) 32.6506 1.29468
\(637\) 4.37529 0.173355
\(638\) −3.82359 −0.151378
\(639\) −9.15147 −0.362026
\(640\) −24.0439 −0.950419
\(641\) 4.56317 0.180234 0.0901172 0.995931i \(-0.471276\pi\)
0.0901172 + 0.995931i \(0.471276\pi\)
\(642\) 32.4736 1.28163
\(643\) 38.5256 1.51930 0.759651 0.650331i \(-0.225370\pi\)
0.759651 + 0.650331i \(0.225370\pi\)
\(644\) −30.1573 −1.18836
\(645\) −3.28131 −0.129201
\(646\) 14.8257 0.583311
\(647\) −40.2058 −1.58065 −0.790326 0.612687i \(-0.790089\pi\)
−0.790326 + 0.612687i \(0.790089\pi\)
\(648\) −2.06110 −0.0809678
\(649\) −13.7231 −0.538679
\(650\) −6.69340 −0.262537
\(651\) 7.92913 0.310767
\(652\) 27.5120 1.07745
\(653\) 46.1148 1.80461 0.902305 0.431098i \(-0.141874\pi\)
0.902305 + 0.431098i \(0.141874\pi\)
\(654\) 4.14406 0.162046
\(655\) −28.0676 −1.09669
\(656\) 12.6664 0.494539
\(657\) −3.90855 −0.152487
\(658\) 12.8576 0.501243
\(659\) 8.17654 0.318513 0.159256 0.987237i \(-0.449090\pi\)
0.159256 + 0.987237i \(0.449090\pi\)
\(660\) 21.3614 0.831490
\(661\) −1.66625 −0.0648096 −0.0324048 0.999475i \(-0.510317\pi\)
−0.0324048 + 0.999475i \(0.510317\pi\)
\(662\) −59.1797 −2.30009
\(663\) −1.29449 −0.0502740
\(664\) −19.2194 −0.745859
\(665\) −20.7658 −0.805262
\(666\) 13.6152 0.527580
\(667\) −2.08857 −0.0808699
\(668\) −14.3286 −0.554392
\(669\) 8.55625 0.330804
\(670\) −38.2811 −1.47893
\(671\) −51.3912 −1.98393
\(672\) −13.2547 −0.511311
\(673\) −28.9598 −1.11632 −0.558159 0.829734i \(-0.688492\pi\)
−0.558159 + 0.829734i \(0.688492\pi\)
\(674\) 28.5751 1.10067
\(675\) −2.32912 −0.0896480
\(676\) −33.1623 −1.27547
\(677\) −27.2001 −1.04539 −0.522693 0.852521i \(-0.675073\pi\)
−0.522693 + 0.852521i \(0.675073\pi\)
\(678\) −31.3713 −1.20481
\(679\) 0.211291 0.00810862
\(680\) 3.36842 0.129173
\(681\) 8.88220 0.340367
\(682\) −41.2942 −1.58124
\(683\) −2.31369 −0.0885307 −0.0442654 0.999020i \(-0.514095\pi\)
−0.0442654 + 0.999020i \(0.514095\pi\)
\(684\) −19.5567 −0.747770
\(685\) −10.9384 −0.417935
\(686\) −43.8437 −1.67396
\(687\) −29.3221 −1.11871
\(688\) −2.57236 −0.0980704
\(689\) −14.4330 −0.549855
\(690\) 19.6373 0.747578
\(691\) −12.9395 −0.492243 −0.246122 0.969239i \(-0.579156\pi\)
−0.246122 + 0.969239i \(0.579156\pi\)
\(692\) 64.7438 2.46119
\(693\) 8.49233 0.322597
\(694\) −44.3413 −1.68317
\(695\) −24.4645 −0.927993
\(696\) 0.795334 0.0301471
\(697\) −9.88644 −0.374476
\(698\) 36.1576 1.36859
\(699\) 2.85833 0.108112
\(700\) 12.9773 0.490496
\(701\) −17.5815 −0.664043 −0.332021 0.943272i \(-0.607731\pi\)
−0.332021 + 0.943272i \(0.607731\pi\)
\(702\) 2.87378 0.108464
\(703\) 40.9575 1.54474
\(704\) 57.5923 2.17059
\(705\) −4.97479 −0.187362
\(706\) 31.8773 1.19972
\(707\) 7.73195 0.290790
\(708\) 9.00363 0.338377
\(709\) −24.5669 −0.922628 −0.461314 0.887237i \(-0.652622\pi\)
−0.461314 + 0.887237i \(0.652622\pi\)
\(710\) −33.2026 −1.24607
\(711\) 1.00000 0.0375029
\(712\) 5.99000 0.224485
\(713\) −22.5562 −0.844738
\(714\) 4.22389 0.158075
\(715\) −9.44268 −0.353136
\(716\) −46.0546 −1.72114
\(717\) −24.0944 −0.899823
\(718\) −20.1412 −0.751664
\(719\) −37.4207 −1.39556 −0.697778 0.716314i \(-0.745828\pi\)
−0.697778 + 0.716314i \(0.745828\pi\)
\(720\) 2.09382 0.0780320
\(721\) −5.15729 −0.192068
\(722\) −56.8297 −2.11498
\(723\) 0.357116 0.0132813
\(724\) 45.9404 1.70736
\(725\) 0.898756 0.0333790
\(726\) −19.8072 −0.735115
\(727\) −13.6349 −0.505692 −0.252846 0.967507i \(-0.581367\pi\)
−0.252846 + 0.967507i \(0.581367\pi\)
\(728\) −5.07644 −0.188145
\(729\) 1.00000 0.0370370
\(730\) −14.1807 −0.524850
\(731\) 2.00780 0.0742611
\(732\) 33.7174 1.24623
\(733\) 14.8079 0.546944 0.273472 0.961880i \(-0.411828\pi\)
0.273472 + 0.961880i \(0.411828\pi\)
\(734\) 50.7203 1.87212
\(735\) 5.52375 0.203747
\(736\) 37.7061 1.38986
\(737\) 47.0946 1.73475
\(738\) 21.9480 0.807915
\(739\) −9.86864 −0.363024 −0.181512 0.983389i \(-0.558099\pi\)
−0.181512 + 0.983389i \(0.558099\pi\)
\(740\) 29.3516 1.07899
\(741\) 8.64495 0.317580
\(742\) 47.0945 1.72889
\(743\) 26.6119 0.976295 0.488148 0.872761i \(-0.337673\pi\)
0.488148 + 0.872761i \(0.337673\pi\)
\(744\) 8.58947 0.314905
\(745\) 11.7077 0.428936
\(746\) 38.2178 1.39925
\(747\) 9.32483 0.341178
\(748\) −13.0708 −0.477915
\(749\) 27.8314 1.01694
\(750\) −26.5909 −0.970962
\(751\) −32.8146 −1.19742 −0.598712 0.800965i \(-0.704320\pi\)
−0.598712 + 0.800965i \(0.704320\pi\)
\(752\) −3.89996 −0.142217
\(753\) −9.94112 −0.362274
\(754\) −1.10893 −0.0403848
\(755\) −35.8185 −1.30357
\(756\) −5.57176 −0.202643
\(757\) −42.5526 −1.54660 −0.773300 0.634040i \(-0.781395\pi\)
−0.773300 + 0.634040i \(0.781395\pi\)
\(758\) 5.64351 0.204981
\(759\) −24.1584 −0.876894
\(760\) −22.4952 −0.815985
\(761\) 20.5221 0.743924 0.371962 0.928248i \(-0.378685\pi\)
0.371962 + 0.928248i \(0.378685\pi\)
\(762\) 11.0465 0.400172
\(763\) 3.55166 0.128579
\(764\) 57.6458 2.08555
\(765\) −1.63428 −0.0590876
\(766\) −68.4906 −2.47466
\(767\) −3.98001 −0.143710
\(768\) −6.85486 −0.247353
\(769\) 13.8634 0.499928 0.249964 0.968255i \(-0.419581\pi\)
0.249964 + 0.968255i \(0.419581\pi\)
\(770\) 30.8111 1.11036
\(771\) −30.7848 −1.10869
\(772\) 11.9201 0.429015
\(773\) 16.6664 0.599447 0.299724 0.954026i \(-0.403106\pi\)
0.299724 + 0.954026i \(0.403106\pi\)
\(774\) −4.45732 −0.160215
\(775\) 9.70642 0.348665
\(776\) 0.228888 0.00821660
\(777\) 11.6689 0.418620
\(778\) −21.8779 −0.784361
\(779\) 66.0241 2.36556
\(780\) 6.19528 0.221827
\(781\) 40.8469 1.46162
\(782\) −12.0158 −0.429685
\(783\) −0.385878 −0.0137901
\(784\) 4.33031 0.154654
\(785\) 29.1681 1.04106
\(786\) −38.1269 −1.35994
\(787\) 38.5395 1.37379 0.686893 0.726759i \(-0.258974\pi\)
0.686893 + 0.726759i \(0.258974\pi\)
\(788\) −59.5152 −2.12014
\(789\) −14.0747 −0.501074
\(790\) 3.62811 0.129083
\(791\) −26.8867 −0.955980
\(792\) 9.19958 0.326893
\(793\) −14.9046 −0.529278
\(794\) −40.5646 −1.43958
\(795\) −18.2215 −0.646250
\(796\) −30.9329 −1.09639
\(797\) 49.0978 1.73913 0.869566 0.493817i \(-0.164399\pi\)
0.869566 + 0.493817i \(0.164399\pi\)
\(798\) −28.2082 −0.998558
\(799\) 3.04402 0.107690
\(800\) −16.2257 −0.573665
\(801\) −2.90621 −0.102686
\(802\) −23.6072 −0.833600
\(803\) 17.4455 0.615639
\(804\) −30.8985 −1.08971
\(805\) 16.8301 0.593182
\(806\) −11.9762 −0.421845
\(807\) −29.9446 −1.05410
\(808\) 8.37588 0.294662
\(809\) −27.4817 −0.966204 −0.483102 0.875564i \(-0.660490\pi\)
−0.483102 + 0.875564i \(0.660490\pi\)
\(810\) 3.62811 0.127479
\(811\) −20.6161 −0.723930 −0.361965 0.932192i \(-0.617894\pi\)
−0.361965 + 0.932192i \(0.617894\pi\)
\(812\) 2.15002 0.0754509
\(813\) 22.7835 0.799052
\(814\) −60.7706 −2.13001
\(815\) −15.3538 −0.537820
\(816\) −1.28119 −0.0448505
\(817\) −13.4086 −0.469106
\(818\) −44.6643 −1.56165
\(819\) 2.46297 0.0860631
\(820\) 47.3152 1.65232
\(821\) −52.2856 −1.82478 −0.912391 0.409321i \(-0.865766\pi\)
−0.912391 + 0.409321i \(0.865766\pi\)
\(822\) −14.8587 −0.518256
\(823\) −2.33779 −0.0814903 −0.0407451 0.999170i \(-0.512973\pi\)
−0.0407451 + 0.999170i \(0.512973\pi\)
\(824\) −5.58679 −0.194625
\(825\) 10.3959 0.361937
\(826\) 12.9866 0.451863
\(827\) 26.2013 0.911107 0.455553 0.890208i \(-0.349441\pi\)
0.455553 + 0.890208i \(0.349441\pi\)
\(828\) 15.8502 0.550831
\(829\) 22.0549 0.766000 0.383000 0.923748i \(-0.374891\pi\)
0.383000 + 0.923748i \(0.374891\pi\)
\(830\) 33.8315 1.17431
\(831\) 26.0724 0.904442
\(832\) 16.7031 0.579075
\(833\) −3.37992 −0.117107
\(834\) −33.2326 −1.15075
\(835\) 7.99647 0.276729
\(836\) 87.2899 3.01898
\(837\) −4.16741 −0.144047
\(838\) −41.5203 −1.43429
\(839\) −16.5348 −0.570843 −0.285422 0.958402i \(-0.592134\pi\)
−0.285422 + 0.958402i \(0.592134\pi\)
\(840\) −6.40893 −0.221129
\(841\) −28.8511 −0.994865
\(842\) −26.7116 −0.920543
\(843\) 31.1424 1.07260
\(844\) 55.7020 1.91734
\(845\) 18.5071 0.636663
\(846\) −6.75775 −0.232336
\(847\) −16.9757 −0.583293
\(848\) −14.2846 −0.490537
\(849\) −4.82456 −0.165579
\(850\) 5.17066 0.177352
\(851\) −33.1949 −1.13791
\(852\) −26.7994 −0.918131
\(853\) 43.3295 1.48357 0.741787 0.670636i \(-0.233979\pi\)
0.741787 + 0.670636i \(0.233979\pi\)
\(854\) 48.6332 1.66420
\(855\) 10.9141 0.373255
\(856\) 30.1492 1.03048
\(857\) 1.52979 0.0522565 0.0261283 0.999659i \(-0.491682\pi\)
0.0261283 + 0.999659i \(0.491682\pi\)
\(858\) −12.8269 −0.437904
\(859\) 5.96544 0.203538 0.101769 0.994808i \(-0.467550\pi\)
0.101769 + 0.994808i \(0.467550\pi\)
\(860\) −9.60906 −0.327666
\(861\) 18.8104 0.641058
\(862\) −72.0349 −2.45352
\(863\) −4.67341 −0.159085 −0.0795423 0.996831i \(-0.525346\pi\)
−0.0795423 + 0.996831i \(0.525346\pi\)
\(864\) 6.96645 0.237003
\(865\) −36.1319 −1.22852
\(866\) 74.2251 2.52227
\(867\) 1.00000 0.0339618
\(868\) 23.2198 0.788133
\(869\) −4.46342 −0.151411
\(870\) −1.40001 −0.0474647
\(871\) 13.6585 0.462801
\(872\) 3.84744 0.130291
\(873\) −0.111051 −0.00375851
\(874\) 80.2447 2.71432
\(875\) −22.7897 −0.770431
\(876\) −11.4459 −0.386721
\(877\) 26.2398 0.886055 0.443027 0.896508i \(-0.353904\pi\)
0.443027 + 0.896508i \(0.353904\pi\)
\(878\) −29.8817 −1.00846
\(879\) 14.7399 0.497165
\(880\) −9.34560 −0.315040
\(881\) −39.1655 −1.31952 −0.659760 0.751477i \(-0.729342\pi\)
−0.659760 + 0.751477i \(0.729342\pi\)
\(882\) 7.50345 0.252654
\(883\) −48.0021 −1.61540 −0.807700 0.589594i \(-0.799288\pi\)
−0.807700 + 0.589594i \(0.799288\pi\)
\(884\) −3.79083 −0.127499
\(885\) −5.02471 −0.168904
\(886\) −54.8043 −1.84119
\(887\) 15.5810 0.523158 0.261579 0.965182i \(-0.415757\pi\)
0.261579 + 0.965182i \(0.415757\pi\)
\(888\) 12.6407 0.424194
\(889\) 9.46737 0.317526
\(890\) −10.5441 −0.353438
\(891\) −4.46342 −0.149530
\(892\) 25.0563 0.838948
\(893\) −20.3287 −0.680275
\(894\) 15.9037 0.531899
\(895\) 25.7020 0.859122
\(896\) −27.9922 −0.935154
\(897\) −7.00649 −0.233940
\(898\) −74.1150 −2.47325
\(899\) 1.60811 0.0536335
\(900\) −6.82066 −0.227355
\(901\) 11.1495 0.371445
\(902\) −97.9630 −3.26181
\(903\) −3.82014 −0.127126
\(904\) −29.1258 −0.968710
\(905\) −25.6382 −0.852243
\(906\) −48.6558 −1.61648
\(907\) 6.98488 0.231929 0.115965 0.993253i \(-0.463004\pi\)
0.115965 + 0.993253i \(0.463004\pi\)
\(908\) 26.0108 0.863200
\(909\) −4.06378 −0.134787
\(910\) 8.93594 0.296223
\(911\) −45.3891 −1.50381 −0.751905 0.659272i \(-0.770865\pi\)
−0.751905 + 0.659272i \(0.770865\pi\)
\(912\) 8.55607 0.283320
\(913\) −41.6206 −1.37744
\(914\) −31.6745 −1.04770
\(915\) −18.8169 −0.622067
\(916\) −85.8676 −2.83715
\(917\) −32.6766 −1.07908
\(918\) −2.22001 −0.0732711
\(919\) 0.150106 0.00495153 0.00247577 0.999997i \(-0.499212\pi\)
0.00247577 + 0.999997i \(0.499212\pi\)
\(920\) 18.2317 0.601081
\(921\) −9.51663 −0.313584
\(922\) 18.9701 0.624746
\(923\) 11.8465 0.389933
\(924\) 24.8691 0.818134
\(925\) 14.2845 0.469670
\(926\) −14.0963 −0.463233
\(927\) 2.71058 0.0890273
\(928\) −2.68820 −0.0882444
\(929\) 34.2867 1.12491 0.562455 0.826828i \(-0.309857\pi\)
0.562455 + 0.826828i \(0.309857\pi\)
\(930\) −15.1199 −0.495799
\(931\) 22.5720 0.739766
\(932\) 8.37040 0.274182
\(933\) 10.6710 0.349353
\(934\) −12.9543 −0.423879
\(935\) 7.29449 0.238555
\(936\) 2.66809 0.0872092
\(937\) 58.5314 1.91214 0.956069 0.293143i \(-0.0947011\pi\)
0.956069 + 0.293143i \(0.0947011\pi\)
\(938\) −44.5673 −1.45517
\(939\) −20.8019 −0.678844
\(940\) −14.5683 −0.475166
\(941\) −7.10197 −0.231518 −0.115759 0.993277i \(-0.536930\pi\)
−0.115759 + 0.993277i \(0.536930\pi\)
\(942\) 39.6219 1.29095
\(943\) −53.5106 −1.74255
\(944\) −3.93909 −0.128206
\(945\) 3.10947 0.101151
\(946\) 19.8949 0.646840
\(947\) 20.7604 0.674623 0.337312 0.941393i \(-0.390482\pi\)
0.337312 + 0.941393i \(0.390482\pi\)
\(948\) 2.92842 0.0951108
\(949\) 5.05960 0.164241
\(950\) −34.5309 −1.12033
\(951\) 17.2323 0.558795
\(952\) 3.92156 0.127098
\(953\) 2.03780 0.0660108 0.0330054 0.999455i \(-0.489492\pi\)
0.0330054 + 0.999455i \(0.489492\pi\)
\(954\) −24.7521 −0.801377
\(955\) −32.1708 −1.04102
\(956\) −70.5587 −2.28203
\(957\) 1.72234 0.0556752
\(958\) −24.3011 −0.785134
\(959\) −12.7346 −0.411222
\(960\) 21.0874 0.680594
\(961\) −13.6327 −0.439763
\(962\) −17.6249 −0.568248
\(963\) −14.6277 −0.471371
\(964\) 1.04579 0.0336825
\(965\) −6.65234 −0.214146
\(966\) 22.8619 0.735570
\(967\) −45.6011 −1.46643 −0.733216 0.679996i \(-0.761982\pi\)
−0.733216 + 0.679996i \(0.761982\pi\)
\(968\) −18.3895 −0.591060
\(969\) −6.67824 −0.214536
\(970\) −0.402906 −0.0129365
\(971\) 16.9241 0.543121 0.271561 0.962421i \(-0.412460\pi\)
0.271561 + 0.962421i \(0.412460\pi\)
\(972\) 2.92842 0.0939292
\(973\) −28.4819 −0.913087
\(974\) 16.2439 0.520489
\(975\) 3.01504 0.0965584
\(976\) −14.7514 −0.472180
\(977\) 8.46580 0.270845 0.135422 0.990788i \(-0.456761\pi\)
0.135422 + 0.990788i \(0.456761\pi\)
\(978\) −20.8566 −0.666919
\(979\) 12.9717 0.414576
\(980\) 16.1759 0.516719
\(981\) −1.86669 −0.0595988
\(982\) 26.6296 0.849784
\(983\) −51.5719 −1.64489 −0.822444 0.568846i \(-0.807390\pi\)
−0.822444 + 0.568846i \(0.807390\pi\)
\(984\) 20.3770 0.649594
\(985\) 33.2140 1.05829
\(986\) 0.856651 0.0272813
\(987\) −5.79171 −0.184352
\(988\) 25.3161 0.805411
\(989\) 10.8673 0.345559
\(990\) −16.1938 −0.514673
\(991\) 15.8735 0.504237 0.252118 0.967696i \(-0.418873\pi\)
0.252118 + 0.967696i \(0.418873\pi\)
\(992\) −29.0321 −0.921769
\(993\) 26.6575 0.845949
\(994\) −38.6548 −1.22606
\(995\) 17.2629 0.547270
\(996\) 27.3070 0.865256
\(997\) 11.4346 0.362139 0.181069 0.983470i \(-0.442044\pi\)
0.181069 + 0.983470i \(0.442044\pi\)
\(998\) −31.3361 −0.991927
\(999\) −6.13298 −0.194039
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 4029.2.a.k.1.5 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4029.2.a.k.1.5 31 1.1 even 1 trivial