Properties

Label 370.2.a.g.1.1
Level $370$
Weight $2$
Character 370.1
Self dual yes
Analytic conductor $2.954$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) Copy content Toggle raw display
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 1.1
Root \(2.59774\) of defining polynomial
Character \(\chi\) \(=\) 370.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.34596 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.34596 q^{6} -2.59774 q^{7} +1.00000 q^{8} +8.19547 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.34596 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.34596 q^{6} -2.59774 q^{7} +1.00000 q^{8} +8.19547 q^{9} -1.00000 q^{10} +4.74823 q^{11} -3.34596 q^{12} +6.69193 q^{13} -2.59774 q^{14} +3.34596 q^{15} +1.00000 q^{16} -0.748228 q^{17} +8.19547 q^{18} -3.34596 q^{19} -1.00000 q^{20} +8.69193 q^{21} +4.74823 q^{22} +1.49646 q^{23} -3.34596 q^{24} +1.00000 q^{25} +6.69193 q^{26} -17.3839 q^{27} -2.59774 q^{28} +3.94370 q^{29} +3.34596 q^{30} +7.79321 q^{31} +1.00000 q^{32} -15.8874 q^{33} -0.748228 q^{34} +2.59774 q^{35} +8.19547 q^{36} -1.00000 q^{37} -3.34596 q^{38} -22.3909 q^{39} -1.00000 q^{40} -6.44724 q^{41} +8.69193 q^{42} +1.94370 q^{43} +4.74823 q^{44} -8.19547 q^{45} +1.49646 q^{46} +1.84951 q^{47} -3.34596 q^{48} -0.251772 q^{49} +1.00000 q^{50} +2.50354 q^{51} +6.69193 q^{52} +10.4472 q^{53} -17.3839 q^{54} -4.74823 q^{55} -2.59774 q^{56} +11.1955 q^{57} +3.94370 q^{58} +5.84951 q^{59} +3.34596 q^{60} +7.94370 q^{61} +7.79321 q^{62} -21.2897 q^{63} +1.00000 q^{64} -6.69193 q^{65} -15.8874 q^{66} +1.84951 q^{67} -0.748228 q^{68} -5.00709 q^{69} +2.59774 q^{70} -3.88740 q^{71} +8.19547 q^{72} -7.49646 q^{73} -1.00000 q^{74} -3.34596 q^{75} -3.34596 q^{76} -12.3346 q^{77} -22.3909 q^{78} -16.5414 q^{79} -1.00000 q^{80} +33.5793 q^{81} -6.44724 q^{82} +15.2334 q^{83} +8.69193 q^{84} +0.748228 q^{85} +1.94370 q^{86} -13.1955 q^{87} +4.74823 q^{88} +6.00000 q^{89} -8.19547 q^{90} -17.3839 q^{91} +1.49646 q^{92} -26.0758 q^{93} +1.84951 q^{94} +3.34596 q^{95} -3.34596 q^{96} -10.4472 q^{97} -0.251772 q^{98} +38.9140 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - q^{7} + 3 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - q^{7} + 3 q^{8} + 11 q^{9} - 3 q^{10} + 11 q^{11} - q^{14} + 3 q^{16} + q^{17} + 11 q^{18} - 3 q^{20} + 6 q^{21} + 11 q^{22} - 2 q^{23} + 3 q^{25} - 12 q^{27} - q^{28} - 5 q^{29} + 3 q^{31} + 3 q^{32} - 14 q^{33} + q^{34} + q^{35} + 11 q^{36} - 3 q^{37} - 40 q^{39} - 3 q^{40} - 9 q^{41} + 6 q^{42} - 11 q^{43} + 11 q^{44} - 11 q^{45} - 2 q^{46} + 2 q^{47} - 4 q^{49} + 3 q^{50} + 14 q^{51} + 21 q^{53} - 12 q^{54} - 11 q^{55} - q^{56} + 20 q^{57} - 5 q^{58} + 14 q^{59} + 7 q^{61} + 3 q^{62} - 37 q^{63} + 3 q^{64} - 14 q^{66} + 2 q^{67} + q^{68} - 28 q^{69} + q^{70} + 22 q^{71} + 11 q^{72} - 16 q^{73} - 3 q^{74} + 7 q^{77} - 40 q^{78} - 26 q^{79} - 3 q^{80} + 47 q^{81} - 9 q^{82} + 2 q^{83} + 6 q^{84} - q^{85} - 11 q^{86} - 26 q^{87} + 11 q^{88} + 18 q^{89} - 11 q^{90} - 12 q^{91} - 2 q^{92} - 18 q^{93} + 2 q^{94} - 21 q^{97} - 4 q^{98} + 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.34596 −1.93179 −0.965896 0.258929i \(-0.916630\pi\)
−0.965896 + 0.258929i \(0.916630\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −3.34596 −1.36598
\(7\) −2.59774 −0.981852 −0.490926 0.871201i \(-0.663341\pi\)
−0.490926 + 0.871201i \(0.663341\pi\)
\(8\) 1.00000 0.353553
\(9\) 8.19547 2.73182
\(10\) −1.00000 −0.316228
\(11\) 4.74823 1.43164 0.715822 0.698282i \(-0.246052\pi\)
0.715822 + 0.698282i \(0.246052\pi\)
\(12\) −3.34596 −0.965896
\(13\) 6.69193 1.85601 0.928003 0.372572i \(-0.121524\pi\)
0.928003 + 0.372572i \(0.121524\pi\)
\(14\) −2.59774 −0.694274
\(15\) 3.34596 0.863924
\(16\) 1.00000 0.250000
\(17\) −0.748228 −0.181472 −0.0907360 0.995875i \(-0.528922\pi\)
−0.0907360 + 0.995875i \(0.528922\pi\)
\(18\) 8.19547 1.93169
\(19\) −3.34596 −0.767617 −0.383808 0.923413i \(-0.625388\pi\)
−0.383808 + 0.923413i \(0.625388\pi\)
\(20\) −1.00000 −0.223607
\(21\) 8.69193 1.89673
\(22\) 4.74823 1.01233
\(23\) 1.49646 0.312033 0.156016 0.987754i \(-0.450135\pi\)
0.156016 + 0.987754i \(0.450135\pi\)
\(24\) −3.34596 −0.682992
\(25\) 1.00000 0.200000
\(26\) 6.69193 1.31239
\(27\) −17.3839 −3.34552
\(28\) −2.59774 −0.490926
\(29\) 3.94370 0.732326 0.366163 0.930551i \(-0.380671\pi\)
0.366163 + 0.930551i \(0.380671\pi\)
\(30\) 3.34596 0.610887
\(31\) 7.79321 1.39970 0.699851 0.714289i \(-0.253250\pi\)
0.699851 + 0.714289i \(0.253250\pi\)
\(32\) 1.00000 0.176777
\(33\) −15.8874 −2.76564
\(34\) −0.748228 −0.128320
\(35\) 2.59774 0.439097
\(36\) 8.19547 1.36591
\(37\) −1.00000 −0.164399
\(38\) −3.34596 −0.542787
\(39\) −22.3909 −3.58542
\(40\) −1.00000 −0.158114
\(41\) −6.44724 −1.00689 −0.503445 0.864027i \(-0.667934\pi\)
−0.503445 + 0.864027i \(0.667934\pi\)
\(42\) 8.69193 1.34119
\(43\) 1.94370 0.296411 0.148206 0.988957i \(-0.452650\pi\)
0.148206 + 0.988957i \(0.452650\pi\)
\(44\) 4.74823 0.715822
\(45\) −8.19547 −1.22171
\(46\) 1.49646 0.220640
\(47\) 1.84951 0.269778 0.134889 0.990861i \(-0.456932\pi\)
0.134889 + 0.990861i \(0.456932\pi\)
\(48\) −3.34596 −0.482948
\(49\) −0.251772 −0.0359674
\(50\) 1.00000 0.141421
\(51\) 2.50354 0.350566
\(52\) 6.69193 0.928003
\(53\) 10.4472 1.43504 0.717520 0.696538i \(-0.245277\pi\)
0.717520 + 0.696538i \(0.245277\pi\)
\(54\) −17.3839 −2.36564
\(55\) −4.74823 −0.640251
\(56\) −2.59774 −0.347137
\(57\) 11.1955 1.48288
\(58\) 3.94370 0.517833
\(59\) 5.84951 0.761541 0.380770 0.924670i \(-0.375659\pi\)
0.380770 + 0.924670i \(0.375659\pi\)
\(60\) 3.34596 0.431962
\(61\) 7.94370 1.01709 0.508543 0.861036i \(-0.330184\pi\)
0.508543 + 0.861036i \(0.330184\pi\)
\(62\) 7.79321 0.989738
\(63\) −21.2897 −2.68225
\(64\) 1.00000 0.125000
\(65\) −6.69193 −0.830031
\(66\) −15.8874 −1.95560
\(67\) 1.84951 0.225953 0.112977 0.993598i \(-0.463961\pi\)
0.112977 + 0.993598i \(0.463961\pi\)
\(68\) −0.748228 −0.0907360
\(69\) −5.00709 −0.602783
\(70\) 2.59774 0.310489
\(71\) −3.88740 −0.461349 −0.230675 0.973031i \(-0.574093\pi\)
−0.230675 + 0.973031i \(0.574093\pi\)
\(72\) 8.19547 0.965845
\(73\) −7.49646 −0.877394 −0.438697 0.898635i \(-0.644560\pi\)
−0.438697 + 0.898635i \(0.644560\pi\)
\(74\) −1.00000 −0.116248
\(75\) −3.34596 −0.386359
\(76\) −3.34596 −0.383808
\(77\) −12.3346 −1.40566
\(78\) −22.3909 −2.53527
\(79\) −16.5414 −1.86106 −0.930528 0.366220i \(-0.880652\pi\)
−0.930528 + 0.366220i \(0.880652\pi\)
\(80\) −1.00000 −0.111803
\(81\) 33.5793 3.73104
\(82\) −6.44724 −0.711979
\(83\) 15.2334 1.67208 0.836039 0.548670i \(-0.184866\pi\)
0.836039 + 0.548670i \(0.184866\pi\)
\(84\) 8.69193 0.948367
\(85\) 0.748228 0.0811567
\(86\) 1.94370 0.209594
\(87\) −13.1955 −1.41470
\(88\) 4.74823 0.506163
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −8.19547 −0.863878
\(91\) −17.3839 −1.82232
\(92\) 1.49646 0.156016
\(93\) −26.0758 −2.70393
\(94\) 1.84951 0.190762
\(95\) 3.34596 0.343289
\(96\) −3.34596 −0.341496
\(97\) −10.4472 −1.06076 −0.530378 0.847761i \(-0.677950\pi\)
−0.530378 + 0.847761i \(0.677950\pi\)
\(98\) −0.251772 −0.0254328
\(99\) 38.9140 3.91100
\(100\) 1.00000 0.100000
\(101\) −12.1884 −1.21279 −0.606395 0.795164i \(-0.707385\pi\)
−0.606395 + 0.795164i \(0.707385\pi\)
\(102\) 2.50354 0.247888
\(103\) −1.30807 −0.128888 −0.0644442 0.997921i \(-0.520527\pi\)
−0.0644442 + 0.997921i \(0.520527\pi\)
\(104\) 6.69193 0.656197
\(105\) −8.69193 −0.848245
\(106\) 10.4472 1.01473
\(107\) −3.04498 −0.294369 −0.147185 0.989109i \(-0.547021\pi\)
−0.147185 + 0.989109i \(0.547021\pi\)
\(108\) −17.3839 −1.67276
\(109\) −1.44015 −0.137942 −0.0689709 0.997619i \(-0.521972\pi\)
−0.0689709 + 0.997619i \(0.521972\pi\)
\(110\) −4.74823 −0.452726
\(111\) 3.34596 0.317585
\(112\) −2.59774 −0.245463
\(113\) 11.1392 1.04788 0.523942 0.851754i \(-0.324461\pi\)
0.523942 + 0.851754i \(0.324461\pi\)
\(114\) 11.1955 1.04855
\(115\) −1.49646 −0.139545
\(116\) 3.94370 0.366163
\(117\) 54.8435 5.07028
\(118\) 5.84951 0.538491
\(119\) 1.94370 0.178179
\(120\) 3.34596 0.305443
\(121\) 11.5457 1.04961
\(122\) 7.94370 0.719189
\(123\) 21.5722 1.94510
\(124\) 7.79321 0.699851
\(125\) −1.00000 −0.0894427
\(126\) −21.2897 −1.89663
\(127\) −8.84242 −0.784638 −0.392319 0.919829i \(-0.628327\pi\)
−0.392319 + 0.919829i \(0.628327\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.50354 −0.572605
\(130\) −6.69193 −0.586921
\(131\) 6.15049 0.537371 0.268686 0.963228i \(-0.413411\pi\)
0.268686 + 0.963228i \(0.413411\pi\)
\(132\) −15.8874 −1.38282
\(133\) 8.69193 0.753686
\(134\) 1.84951 0.159773
\(135\) 17.3839 1.49616
\(136\) −0.748228 −0.0641600
\(137\) 10.8945 0.930779 0.465389 0.885106i \(-0.345914\pi\)
0.465389 + 0.885106i \(0.345914\pi\)
\(138\) −5.00709 −0.426232
\(139\) 1.04921 0.0889932 0.0444966 0.999010i \(-0.485832\pi\)
0.0444966 + 0.999010i \(0.485832\pi\)
\(140\) 2.59774 0.219549
\(141\) −6.18838 −0.521156
\(142\) −3.88740 −0.326223
\(143\) 31.7748 2.65714
\(144\) 8.19547 0.682956
\(145\) −3.94370 −0.327506
\(146\) −7.49646 −0.620411
\(147\) 0.842420 0.0694816
\(148\) −1.00000 −0.0821995
\(149\) −4.18838 −0.343126 −0.171563 0.985173i \(-0.554882\pi\)
−0.171563 + 0.985173i \(0.554882\pi\)
\(150\) −3.34596 −0.273197
\(151\) −6.69193 −0.544581 −0.272291 0.962215i \(-0.587781\pi\)
−0.272291 + 0.962215i \(0.587781\pi\)
\(152\) −3.34596 −0.271393
\(153\) −6.13208 −0.495749
\(154\) −12.3346 −0.993954
\(155\) −7.79321 −0.625965
\(156\) −22.3909 −1.79271
\(157\) −3.94370 −0.314741 −0.157371 0.987540i \(-0.550302\pi\)
−0.157371 + 0.987540i \(0.550302\pi\)
\(158\) −16.5414 −1.31597
\(159\) −34.9561 −2.77220
\(160\) −1.00000 −0.0790569
\(161\) −3.88740 −0.306370
\(162\) 33.5793 2.63824
\(163\) −12.1463 −0.951368 −0.475684 0.879616i \(-0.657799\pi\)
−0.475684 + 0.879616i \(0.657799\pi\)
\(164\) −6.44724 −0.503445
\(165\) 15.8874 1.23683
\(166\) 15.2334 1.18234
\(167\) 17.0829 1.32191 0.660956 0.750425i \(-0.270151\pi\)
0.660956 + 0.750425i \(0.270151\pi\)
\(168\) 8.69193 0.670597
\(169\) 31.7819 2.44476
\(170\) 0.748228 0.0573865
\(171\) −27.4217 −2.09699
\(172\) 1.94370 0.148206
\(173\) −15.3417 −1.16641 −0.583205 0.812325i \(-0.698202\pi\)
−0.583205 + 0.812325i \(0.698202\pi\)
\(174\) −13.1955 −1.00035
\(175\) −2.59774 −0.196370
\(176\) 4.74823 0.357911
\(177\) −19.5722 −1.47114
\(178\) 6.00000 0.449719
\(179\) 2.45148 0.183232 0.0916161 0.995794i \(-0.470797\pi\)
0.0916161 + 0.995794i \(0.470797\pi\)
\(180\) −8.19547 −0.610854
\(181\) 13.1955 0.980812 0.490406 0.871494i \(-0.336849\pi\)
0.490406 + 0.871494i \(0.336849\pi\)
\(182\) −17.3839 −1.28858
\(183\) −26.5793 −1.96480
\(184\) 1.49646 0.110320
\(185\) 1.00000 0.0735215
\(186\) −26.0758 −1.91197
\(187\) −3.55276 −0.259803
\(188\) 1.84951 0.134889
\(189\) 45.1586 3.28481
\(190\) 3.34596 0.242742
\(191\) −21.7790 −1.57588 −0.787938 0.615755i \(-0.788851\pi\)
−0.787938 + 0.615755i \(0.788851\pi\)
\(192\) −3.34596 −0.241474
\(193\) −12.9929 −0.935250 −0.467625 0.883927i \(-0.654890\pi\)
−0.467625 + 0.883927i \(0.654890\pi\)
\(194\) −10.4472 −0.750068
\(195\) 22.3909 1.60345
\(196\) −0.251772 −0.0179837
\(197\) 0.616147 0.0438986 0.0219493 0.999759i \(-0.493013\pi\)
0.0219493 + 0.999759i \(0.493013\pi\)
\(198\) 38.9140 2.76549
\(199\) −1.54852 −0.109772 −0.0548859 0.998493i \(-0.517480\pi\)
−0.0548859 + 0.998493i \(0.517480\pi\)
\(200\) 1.00000 0.0707107
\(201\) −6.18838 −0.436495
\(202\) −12.1884 −0.857572
\(203\) −10.2447 −0.719036
\(204\) 2.50354 0.175283
\(205\) 6.44724 0.450295
\(206\) −1.30807 −0.0911378
\(207\) 12.2642 0.852418
\(208\) 6.69193 0.464002
\(209\) −15.8874 −1.09895
\(210\) −8.69193 −0.599800
\(211\) 20.9366 1.44134 0.720668 0.693280i \(-0.243835\pi\)
0.720668 + 0.693280i \(0.243835\pi\)
\(212\) 10.4472 0.717520
\(213\) 13.0071 0.891231
\(214\) −3.04498 −0.208150
\(215\) −1.94370 −0.132559
\(216\) −17.3839 −1.18282
\(217\) −20.2447 −1.37430
\(218\) −1.44015 −0.0975396
\(219\) 25.0829 1.69494
\(220\) −4.74823 −0.320125
\(221\) −5.00709 −0.336813
\(222\) 3.34596 0.224566
\(223\) −2.71034 −0.181498 −0.0907488 0.995874i \(-0.528926\pi\)
−0.0907488 + 0.995874i \(0.528926\pi\)
\(224\) −2.59774 −0.173568
\(225\) 8.19547 0.546365
\(226\) 11.1392 0.740966
\(227\) 26.1321 1.73445 0.867224 0.497919i \(-0.165902\pi\)
0.867224 + 0.497919i \(0.165902\pi\)
\(228\) 11.1955 0.741438
\(229\) −24.7677 −1.63670 −0.818348 0.574723i \(-0.805110\pi\)
−0.818348 + 0.574723i \(0.805110\pi\)
\(230\) −1.49646 −0.0986734
\(231\) 41.2713 2.71545
\(232\) 3.94370 0.258916
\(233\) −12.2783 −0.804381 −0.402190 0.915556i \(-0.631751\pi\)
−0.402190 + 0.915556i \(0.631751\pi\)
\(234\) 54.8435 3.58523
\(235\) −1.84951 −0.120649
\(236\) 5.84951 0.380770
\(237\) 55.3470 3.59518
\(238\) 1.94370 0.125991
\(239\) −17.1771 −1.11109 −0.555546 0.831486i \(-0.687491\pi\)
−0.555546 + 0.831486i \(0.687491\pi\)
\(240\) 3.34596 0.215981
\(241\) −15.2713 −0.983708 −0.491854 0.870678i \(-0.663681\pi\)
−0.491854 + 0.870678i \(0.663681\pi\)
\(242\) 11.5457 0.742184
\(243\) −60.2036 −3.86206
\(244\) 7.94370 0.508543
\(245\) 0.251772 0.0160851
\(246\) 21.5722 1.37540
\(247\) −22.3909 −1.42470
\(248\) 7.79321 0.494869
\(249\) −50.9703 −3.23011
\(250\) −1.00000 −0.0632456
\(251\) 10.0379 0.633586 0.316793 0.948495i \(-0.397394\pi\)
0.316793 + 0.948495i \(0.397394\pi\)
\(252\) −21.2897 −1.34112
\(253\) 7.10552 0.446720
\(254\) −8.84242 −0.554823
\(255\) −2.50354 −0.156778
\(256\) 1.00000 0.0625000
\(257\) −20.9703 −1.30809 −0.654045 0.756456i \(-0.726929\pi\)
−0.654045 + 0.756456i \(0.726929\pi\)
\(258\) −6.50354 −0.404893
\(259\) 2.59774 0.161415
\(260\) −6.69193 −0.415016
\(261\) 32.3205 2.00059
\(262\) 6.15049 0.379979
\(263\) 22.3725 1.37955 0.689775 0.724024i \(-0.257710\pi\)
0.689775 + 0.724024i \(0.257710\pi\)
\(264\) −15.8874 −0.977802
\(265\) −10.4472 −0.641769
\(266\) 8.69193 0.532936
\(267\) −20.0758 −1.22862
\(268\) 1.84951 0.112977
\(269\) −4.18838 −0.255370 −0.127685 0.991815i \(-0.540755\pi\)
−0.127685 + 0.991815i \(0.540755\pi\)
\(270\) 17.3839 1.05795
\(271\) −12.1126 −0.735788 −0.367894 0.929868i \(-0.619921\pi\)
−0.367894 + 0.929868i \(0.619921\pi\)
\(272\) −0.748228 −0.0453680
\(273\) 58.1657 3.52035
\(274\) 10.8945 0.658160
\(275\) 4.74823 0.286329
\(276\) −5.00709 −0.301391
\(277\) −9.30807 −0.559268 −0.279634 0.960107i \(-0.590213\pi\)
−0.279634 + 0.960107i \(0.590213\pi\)
\(278\) 1.04921 0.0629277
\(279\) 63.8690 3.82374
\(280\) 2.59774 0.155244
\(281\) 19.2713 1.14963 0.574813 0.818285i \(-0.305075\pi\)
0.574813 + 0.818285i \(0.305075\pi\)
\(282\) −6.18838 −0.368513
\(283\) −13.6848 −0.813479 −0.406740 0.913544i \(-0.633334\pi\)
−0.406740 + 0.913544i \(0.633334\pi\)
\(284\) −3.88740 −0.230675
\(285\) −11.1955 −0.663162
\(286\) 31.7748 1.87888
\(287\) 16.7482 0.988617
\(288\) 8.19547 0.482923
\(289\) −16.4402 −0.967068
\(290\) −3.94370 −0.231582
\(291\) 34.9561 2.04916
\(292\) −7.49646 −0.438697
\(293\) −10.4472 −0.610334 −0.305167 0.952299i \(-0.598712\pi\)
−0.305167 + 0.952299i \(0.598712\pi\)
\(294\) 0.842420 0.0491309
\(295\) −5.84951 −0.340571
\(296\) −1.00000 −0.0581238
\(297\) −82.5425 −4.78960
\(298\) −4.18838 −0.242627
\(299\) 10.0142 0.579135
\(300\) −3.34596 −0.193179
\(301\) −5.04921 −0.291032
\(302\) −6.69193 −0.385077
\(303\) 40.7819 2.34286
\(304\) −3.34596 −0.191904
\(305\) −7.94370 −0.454855
\(306\) −6.13208 −0.350548
\(307\) −4.95502 −0.282798 −0.141399 0.989953i \(-0.545160\pi\)
−0.141399 + 0.989953i \(0.545160\pi\)
\(308\) −12.3346 −0.702831
\(309\) 4.37677 0.248985
\(310\) −7.79321 −0.442624
\(311\) 3.68060 0.208708 0.104354 0.994540i \(-0.466723\pi\)
0.104354 + 0.994540i \(0.466723\pi\)
\(312\) −22.3909 −1.26764
\(313\) −0.992912 −0.0561227 −0.0280614 0.999606i \(-0.508933\pi\)
−0.0280614 + 0.999606i \(0.508933\pi\)
\(314\) −3.94370 −0.222556
\(315\) 21.2897 1.19954
\(316\) −16.5414 −0.930528
\(317\) −6.55985 −0.368438 −0.184219 0.982885i \(-0.558976\pi\)
−0.184219 + 0.982885i \(0.558976\pi\)
\(318\) −34.9561 −1.96024
\(319\) 18.7256 1.04843
\(320\) −1.00000 −0.0559017
\(321\) 10.1884 0.568660
\(322\) −3.88740 −0.216636
\(323\) 2.50354 0.139301
\(324\) 33.5793 1.86552
\(325\) 6.69193 0.371201
\(326\) −12.1463 −0.672719
\(327\) 4.81870 0.266475
\(328\) −6.44724 −0.355989
\(329\) −4.80453 −0.264882
\(330\) 15.8874 0.874572
\(331\) −18.0379 −0.991452 −0.495726 0.868479i \(-0.665098\pi\)
−0.495726 + 0.868479i \(0.665098\pi\)
\(332\) 15.2334 0.836039
\(333\) −8.19547 −0.449109
\(334\) 17.0829 0.934733
\(335\) −1.84951 −0.101049
\(336\) 8.69193 0.474183
\(337\) −14.8945 −0.811354 −0.405677 0.914016i \(-0.632964\pi\)
−0.405677 + 0.914016i \(0.632964\pi\)
\(338\) 31.7819 1.72871
\(339\) −37.2713 −2.02430
\(340\) 0.748228 0.0405784
\(341\) 37.0039 2.00387
\(342\) −27.4217 −1.48280
\(343\) 18.8382 1.01717
\(344\) 1.94370 0.104797
\(345\) 5.00709 0.269573
\(346\) −15.3417 −0.824776
\(347\) −9.38385 −0.503752 −0.251876 0.967760i \(-0.581047\pi\)
−0.251876 + 0.967760i \(0.581047\pi\)
\(348\) −13.1955 −0.707351
\(349\) −32.9703 −1.76486 −0.882429 0.470446i \(-0.844093\pi\)
−0.882429 + 0.470446i \(0.844093\pi\)
\(350\) −2.59774 −0.138855
\(351\) −116.331 −6.20931
\(352\) 4.74823 0.253081
\(353\) 27.4260 1.45974 0.729869 0.683587i \(-0.239581\pi\)
0.729869 + 0.683587i \(0.239581\pi\)
\(354\) −19.5722 −1.04025
\(355\) 3.88740 0.206322
\(356\) 6.00000 0.317999
\(357\) −6.50354 −0.344204
\(358\) 2.45148 0.129565
\(359\) −25.0829 −1.32382 −0.661912 0.749582i \(-0.730255\pi\)
−0.661912 + 0.749582i \(0.730255\pi\)
\(360\) −8.19547 −0.431939
\(361\) −7.80453 −0.410765
\(362\) 13.1955 0.693539
\(363\) −38.6314 −2.02762
\(364\) −17.3839 −0.911161
\(365\) 7.49646 0.392382
\(366\) −26.5793 −1.38932
\(367\) 2.48513 0.129723 0.0648614 0.997894i \(-0.479339\pi\)
0.0648614 + 0.997894i \(0.479339\pi\)
\(368\) 1.49646 0.0780082
\(369\) −52.8382 −2.75065
\(370\) 1.00000 0.0519875
\(371\) −27.1392 −1.40900
\(372\) −26.0758 −1.35197
\(373\) −35.1586 −1.82045 −0.910223 0.414119i \(-0.864090\pi\)
−0.910223 + 0.414119i \(0.864090\pi\)
\(374\) −3.55276 −0.183709
\(375\) 3.34596 0.172785
\(376\) 1.84951 0.0953810
\(377\) 26.3909 1.35920
\(378\) 45.1586 2.32271
\(379\) 24.6778 1.26761 0.633805 0.773492i \(-0.281492\pi\)
0.633805 + 0.773492i \(0.281492\pi\)
\(380\) 3.34596 0.171644
\(381\) 29.5864 1.51576
\(382\) −21.7790 −1.11431
\(383\) −13.3839 −0.683883 −0.341941 0.939721i \(-0.611084\pi\)
−0.341941 + 0.939721i \(0.611084\pi\)
\(384\) −3.34596 −0.170748
\(385\) 12.3346 0.628631
\(386\) −12.9929 −0.661322
\(387\) 15.9295 0.809743
\(388\) −10.4472 −0.530378
\(389\) 18.2220 0.923894 0.461947 0.886908i \(-0.347151\pi\)
0.461947 + 0.886908i \(0.347151\pi\)
\(390\) 22.3909 1.13381
\(391\) −1.11969 −0.0566252
\(392\) −0.251772 −0.0127164
\(393\) −20.5793 −1.03809
\(394\) 0.616147 0.0310410
\(395\) 16.5414 0.832290
\(396\) 38.9140 1.95550
\(397\) −2.22521 −0.111680 −0.0558399 0.998440i \(-0.517784\pi\)
−0.0558399 + 0.998440i \(0.517784\pi\)
\(398\) −1.54852 −0.0776204
\(399\) −29.0829 −1.45596
\(400\) 1.00000 0.0500000
\(401\) 12.3909 0.618774 0.309387 0.950936i \(-0.399876\pi\)
0.309387 + 0.950936i \(0.399876\pi\)
\(402\) −6.18838 −0.308648
\(403\) 52.1516 2.59785
\(404\) −12.1884 −0.606395
\(405\) −33.5793 −1.66857
\(406\) −10.2447 −0.508435
\(407\) −4.74823 −0.235361
\(408\) 2.50354 0.123944
\(409\) 7.27125 0.359540 0.179770 0.983709i \(-0.442465\pi\)
0.179770 + 0.983709i \(0.442465\pi\)
\(410\) 6.44724 0.318407
\(411\) −36.4525 −1.79807
\(412\) −1.30807 −0.0644442
\(413\) −15.1955 −0.747720
\(414\) 12.2642 0.602751
\(415\) −15.2334 −0.747776
\(416\) 6.69193 0.328099
\(417\) −3.51063 −0.171916
\(418\) −15.8874 −0.777078
\(419\) 28.4809 1.39138 0.695691 0.718341i \(-0.255098\pi\)
0.695691 + 0.718341i \(0.255098\pi\)
\(420\) −8.69193 −0.424123
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 20.9366 1.01918
\(423\) 15.1576 0.736987
\(424\) 10.4472 0.507363
\(425\) −0.748228 −0.0362944
\(426\) 13.0071 0.630195
\(427\) −20.6356 −0.998628
\(428\) −3.04498 −0.147185
\(429\) −106.317 −5.13305
\(430\) −1.94370 −0.0937335
\(431\) 1.51487 0.0729686 0.0364843 0.999334i \(-0.488384\pi\)
0.0364843 + 0.999334i \(0.488384\pi\)
\(432\) −17.3839 −0.836381
\(433\) 24.9929 1.20108 0.600541 0.799594i \(-0.294952\pi\)
0.600541 + 0.799594i \(0.294952\pi\)
\(434\) −20.2447 −0.971776
\(435\) 13.1955 0.632674
\(436\) −1.44015 −0.0689709
\(437\) −5.00709 −0.239521
\(438\) 25.0829 1.19851
\(439\) −5.10128 −0.243471 −0.121735 0.992563i \(-0.538846\pi\)
−0.121735 + 0.992563i \(0.538846\pi\)
\(440\) −4.74823 −0.226363
\(441\) −2.06339 −0.0982566
\(442\) −5.00709 −0.238163
\(443\) 25.7369 1.22280 0.611399 0.791323i \(-0.290607\pi\)
0.611399 + 0.791323i \(0.290607\pi\)
\(444\) 3.34596 0.158792
\(445\) −6.00000 −0.284427
\(446\) −2.71034 −0.128338
\(447\) 14.0142 0.662848
\(448\) −2.59774 −0.122731
\(449\) −17.7748 −0.838844 −0.419422 0.907791i \(-0.637767\pi\)
−0.419422 + 0.907791i \(0.637767\pi\)
\(450\) 8.19547 0.386338
\(451\) −30.6130 −1.44151
\(452\) 11.1392 0.523942
\(453\) 22.3909 1.05202
\(454\) 26.1321 1.22644
\(455\) 17.3839 0.814968
\(456\) 11.1955 0.524276
\(457\) 34.3346 1.60611 0.803053 0.595907i \(-0.203207\pi\)
0.803053 + 0.595907i \(0.203207\pi\)
\(458\) −24.7677 −1.15732
\(459\) 13.0071 0.607119
\(460\) −1.49646 −0.0697726
\(461\) −16.9508 −0.789477 −0.394738 0.918794i \(-0.629165\pi\)
−0.394738 + 0.918794i \(0.629165\pi\)
\(462\) 41.2713 1.92011
\(463\) 10.9929 0.510884 0.255442 0.966824i \(-0.417779\pi\)
0.255442 + 0.966824i \(0.417779\pi\)
\(464\) 3.94370 0.183082
\(465\) 26.0758 1.20924
\(466\) −12.2783 −0.568783
\(467\) 37.4175 1.73148 0.865738 0.500498i \(-0.166850\pi\)
0.865738 + 0.500498i \(0.166850\pi\)
\(468\) 54.8435 2.53514
\(469\) −4.80453 −0.221853
\(470\) −1.84951 −0.0853114
\(471\) 13.1955 0.608015
\(472\) 5.84951 0.269245
\(473\) 9.22912 0.424356
\(474\) 55.3470 2.54217
\(475\) −3.34596 −0.153523
\(476\) 1.94370 0.0890893
\(477\) 85.6201 3.92027
\(478\) −17.1771 −0.785660
\(479\) −0.465654 −0.0212763 −0.0106381 0.999943i \(-0.503386\pi\)
−0.0106381 + 0.999943i \(0.503386\pi\)
\(480\) 3.34596 0.152722
\(481\) −6.69193 −0.305126
\(482\) −15.2713 −0.695586
\(483\) 13.0071 0.591843
\(484\) 11.5457 0.524803
\(485\) 10.4472 0.474385
\(486\) −60.2036 −2.73089
\(487\) 34.9561 1.58401 0.792006 0.610514i \(-0.209037\pi\)
0.792006 + 0.610514i \(0.209037\pi\)
\(488\) 7.94370 0.359594
\(489\) 40.6409 1.83785
\(490\) 0.251772 0.0113739
\(491\) −24.7819 −1.11839 −0.559195 0.829036i \(-0.688890\pi\)
−0.559195 + 0.829036i \(0.688890\pi\)
\(492\) 21.5722 0.972552
\(493\) −2.95079 −0.132897
\(494\) −22.3909 −1.00742
\(495\) −38.9140 −1.74905
\(496\) 7.79321 0.349925
\(497\) 10.0984 0.452976
\(498\) −50.9703 −2.28403
\(499\) −2.33888 −0.104702 −0.0523512 0.998629i \(-0.516672\pi\)
−0.0523512 + 0.998629i \(0.516672\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −57.1586 −2.55366
\(502\) 10.0379 0.448013
\(503\) −18.6161 −0.830053 −0.415026 0.909809i \(-0.636228\pi\)
−0.415026 + 0.909809i \(0.636228\pi\)
\(504\) −21.2897 −0.948317
\(505\) 12.1884 0.542376
\(506\) 7.10552 0.315879
\(507\) −106.341 −4.72277
\(508\) −8.84242 −0.392319
\(509\) 6.89448 0.305593 0.152796 0.988258i \(-0.451172\pi\)
0.152796 + 0.988258i \(0.451172\pi\)
\(510\) −2.50354 −0.110859
\(511\) 19.4738 0.861471
\(512\) 1.00000 0.0441942
\(513\) 58.1657 2.56808
\(514\) −20.9703 −0.924959
\(515\) 1.30807 0.0576406
\(516\) −6.50354 −0.286303
\(517\) 8.78188 0.386227
\(518\) 2.59774 0.114138
\(519\) 51.3329 2.25326
\(520\) −6.69193 −0.293460
\(521\) 4.86083 0.212957 0.106478 0.994315i \(-0.466042\pi\)
0.106478 + 0.994315i \(0.466042\pi\)
\(522\) 32.3205 1.41463
\(523\) 22.7677 0.995562 0.497781 0.867303i \(-0.334148\pi\)
0.497781 + 0.867303i \(0.334148\pi\)
\(524\) 6.15049 0.268686
\(525\) 8.69193 0.379347
\(526\) 22.3725 0.975489
\(527\) −5.83110 −0.254007
\(528\) −15.8874 −0.691410
\(529\) −20.7606 −0.902636
\(530\) −10.4472 −0.453799
\(531\) 47.9395 2.08040
\(532\) 8.69193 0.376843
\(533\) −43.1445 −1.86879
\(534\) −20.0758 −0.868764
\(535\) 3.04498 0.131646
\(536\) 1.84951 0.0798865
\(537\) −8.20256 −0.353967
\(538\) −4.18838 −0.180574
\(539\) −1.19547 −0.0514926
\(540\) 17.3839 0.748082
\(541\) 8.99291 0.386636 0.193318 0.981136i \(-0.438075\pi\)
0.193318 + 0.981136i \(0.438075\pi\)
\(542\) −12.1126 −0.520281
\(543\) −44.1516 −1.89472
\(544\) −0.748228 −0.0320800
\(545\) 1.44015 0.0616894
\(546\) 58.1657 2.48926
\(547\) −18.8382 −0.805463 −0.402731 0.915318i \(-0.631939\pi\)
−0.402731 + 0.915318i \(0.631939\pi\)
\(548\) 10.8945 0.465389
\(549\) 65.1023 2.77850
\(550\) 4.74823 0.202465
\(551\) −13.1955 −0.562146
\(552\) −5.00709 −0.213116
\(553\) 42.9703 1.82728
\(554\) −9.30807 −0.395462
\(555\) −3.34596 −0.142028
\(556\) 1.04921 0.0444966
\(557\) 3.69901 0.156732 0.0783661 0.996925i \(-0.475030\pi\)
0.0783661 + 0.996925i \(0.475030\pi\)
\(558\) 63.8690 2.70379
\(559\) 13.0071 0.550141
\(560\) 2.59774 0.109774
\(561\) 11.8874 0.501886
\(562\) 19.2713 0.812909
\(563\) −14.3488 −0.604730 −0.302365 0.953192i \(-0.597776\pi\)
−0.302365 + 0.953192i \(0.597776\pi\)
\(564\) −6.18838 −0.260578
\(565\) −11.1392 −0.468628
\(566\) −13.6848 −0.575217
\(567\) −87.2302 −3.66332
\(568\) −3.88740 −0.163112
\(569\) 15.9858 0.670161 0.335080 0.942190i \(-0.391237\pi\)
0.335080 + 0.942190i \(0.391237\pi\)
\(570\) −11.1955 −0.468927
\(571\) −30.2079 −1.26416 −0.632080 0.774903i \(-0.717799\pi\)
−0.632080 + 0.774903i \(0.717799\pi\)
\(572\) 31.7748 1.32857
\(573\) 72.8718 3.04426
\(574\) 16.7482 0.699058
\(575\) 1.49646 0.0624065
\(576\) 8.19547 0.341478
\(577\) −19.1813 −0.798528 −0.399264 0.916836i \(-0.630734\pi\)
−0.399264 + 0.916836i \(0.630734\pi\)
\(578\) −16.4402 −0.683820
\(579\) 43.4738 1.80671
\(580\) −3.94370 −0.163753
\(581\) −39.5722 −1.64173
\(582\) 34.9561 1.44898
\(583\) 49.6059 2.05447
\(584\) −7.49646 −0.310206
\(585\) −54.8435 −2.26750
\(586\) −10.4472 −0.431572
\(587\) −26.6130 −1.09844 −0.549218 0.835679i \(-0.685074\pi\)
−0.549218 + 0.835679i \(0.685074\pi\)
\(588\) 0.842420 0.0347408
\(589\) −26.0758 −1.07443
\(590\) −5.84951 −0.240820
\(591\) −2.06160 −0.0848031
\(592\) −1.00000 −0.0410997
\(593\) 26.2642 1.07854 0.539270 0.842133i \(-0.318700\pi\)
0.539270 + 0.842133i \(0.318700\pi\)
\(594\) −82.5425 −3.38676
\(595\) −1.94370 −0.0796839
\(596\) −4.18838 −0.171563
\(597\) 5.18130 0.212056
\(598\) 10.0142 0.409510
\(599\) −38.2415 −1.56251 −0.781253 0.624215i \(-0.785419\pi\)
−0.781253 + 0.624215i \(0.785419\pi\)
\(600\) −3.34596 −0.136598
\(601\) 30.2447 1.23371 0.616853 0.787078i \(-0.288407\pi\)
0.616853 + 0.787078i \(0.288407\pi\)
\(602\) −5.04921 −0.205791
\(603\) 15.1576 0.617264
\(604\) −6.69193 −0.272291
\(605\) −11.5457 −0.469398
\(606\) 40.7819 1.65665
\(607\) −5.90157 −0.239537 −0.119769 0.992802i \(-0.538215\pi\)
−0.119769 + 0.992802i \(0.538215\pi\)
\(608\) −3.34596 −0.135697
\(609\) 34.2783 1.38903
\(610\) −7.94370 −0.321631
\(611\) 12.3768 0.500710
\(612\) −6.13208 −0.247875
\(613\) 6.15473 0.248587 0.124294 0.992245i \(-0.460334\pi\)
0.124294 + 0.992245i \(0.460334\pi\)
\(614\) −4.95502 −0.199968
\(615\) −21.5722 −0.869877
\(616\) −12.3346 −0.496977
\(617\) 16.5035 0.664408 0.332204 0.943208i \(-0.392208\pi\)
0.332204 + 0.943208i \(0.392208\pi\)
\(618\) 4.37677 0.176059
\(619\) 9.04921 0.363719 0.181859 0.983325i \(-0.441788\pi\)
0.181859 + 0.983325i \(0.441788\pi\)
\(620\) −7.79321 −0.312983
\(621\) −26.0142 −1.04391
\(622\) 3.68060 0.147579
\(623\) −15.5864 −0.624456
\(624\) −22.3909 −0.896355
\(625\) 1.00000 0.0400000
\(626\) −0.992912 −0.0396848
\(627\) 53.1586 2.12295
\(628\) −3.94370 −0.157371
\(629\) 0.748228 0.0298338
\(630\) 21.2897 0.848200
\(631\) −0.507780 −0.0202144 −0.0101072 0.999949i \(-0.503217\pi\)
−0.0101072 + 0.999949i \(0.503217\pi\)
\(632\) −16.5414 −0.657983
\(633\) −70.0531 −2.78436
\(634\) −6.55985 −0.260525
\(635\) 8.84242 0.350901
\(636\) −34.9561 −1.38610
\(637\) −1.68484 −0.0667558
\(638\) 18.7256 0.741353
\(639\) −31.8590 −1.26032
\(640\) −1.00000 −0.0395285
\(641\) −40.6356 −1.60501 −0.802505 0.596645i \(-0.796500\pi\)
−0.802505 + 0.596645i \(0.796500\pi\)
\(642\) 10.1884 0.402103
\(643\) −9.53011 −0.375831 −0.187915 0.982185i \(-0.560173\pi\)
−0.187915 + 0.982185i \(0.560173\pi\)
\(644\) −3.88740 −0.153185
\(645\) 6.50354 0.256077
\(646\) 2.50354 0.0985006
\(647\) −21.0071 −0.825874 −0.412937 0.910760i \(-0.635497\pi\)
−0.412937 + 0.910760i \(0.635497\pi\)
\(648\) 33.5793 1.31912
\(649\) 27.7748 1.09026
\(650\) 6.69193 0.262479
\(651\) 67.7380 2.65486
\(652\) −12.1463 −0.475684
\(653\) −45.5496 −1.78249 −0.891247 0.453519i \(-0.850168\pi\)
−0.891247 + 0.453519i \(0.850168\pi\)
\(654\) 4.81870 0.188426
\(655\) −6.15049 −0.240320
\(656\) −6.44724 −0.251723
\(657\) −61.4370 −2.39689
\(658\) −4.80453 −0.187300
\(659\) 9.38385 0.365543 0.182772 0.983155i \(-0.441493\pi\)
0.182772 + 0.983155i \(0.441493\pi\)
\(660\) 15.8874 0.618416
\(661\) 4.05630 0.157772 0.0788859 0.996884i \(-0.474864\pi\)
0.0788859 + 0.996884i \(0.474864\pi\)
\(662\) −18.0379 −0.701062
\(663\) 16.7535 0.650653
\(664\) 15.2334 0.591169
\(665\) −8.69193 −0.337058
\(666\) −8.19547 −0.317568
\(667\) 5.90157 0.228510
\(668\) 17.0829 0.660956
\(669\) 9.06869 0.350616
\(670\) −1.84951 −0.0714527
\(671\) 37.7185 1.45611
\(672\) 8.69193 0.335298
\(673\) 7.60906 0.293308 0.146654 0.989188i \(-0.453150\pi\)
0.146654 + 0.989188i \(0.453150\pi\)
\(674\) −14.8945 −0.573714
\(675\) −17.3839 −0.669105
\(676\) 31.7819 1.22238
\(677\) 28.7677 1.10563 0.552816 0.833303i \(-0.313553\pi\)
0.552816 + 0.833303i \(0.313553\pi\)
\(678\) −37.2713 −1.43139
\(679\) 27.1392 1.04151
\(680\) 0.748228 0.0286932
\(681\) −87.4370 −3.35059
\(682\) 37.0039 1.41695
\(683\) 0.0704767 0.00269671 0.00134836 0.999999i \(-0.499571\pi\)
0.00134836 + 0.999999i \(0.499571\pi\)
\(684\) −27.4217 −1.04850
\(685\) −10.8945 −0.416257
\(686\) 18.8382 0.719245
\(687\) 82.8718 3.16176
\(688\) 1.94370 0.0741028
\(689\) 69.9122 2.66344
\(690\) 5.00709 0.190617
\(691\) 10.6498 0.405137 0.202569 0.979268i \(-0.435071\pi\)
0.202569 + 0.979268i \(0.435071\pi\)
\(692\) −15.3417 −0.583205
\(693\) −101.088 −3.84002
\(694\) −9.38385 −0.356206
\(695\) −1.04921 −0.0397990
\(696\) −13.1955 −0.500173
\(697\) 4.82401 0.182722
\(698\) −32.9703 −1.24794
\(699\) 41.0829 1.55390
\(700\) −2.59774 −0.0981852
\(701\) −7.98582 −0.301620 −0.150810 0.988563i \(-0.548188\pi\)
−0.150810 + 0.988563i \(0.548188\pi\)
\(702\) −116.331 −4.39065
\(703\) 3.34596 0.126195
\(704\) 4.74823 0.178956
\(705\) 6.18838 0.233068
\(706\) 27.4260 1.03219
\(707\) 31.6622 1.19078
\(708\) −19.5722 −0.735570
\(709\) −37.2149 −1.39764 −0.698818 0.715299i \(-0.746290\pi\)
−0.698818 + 0.715299i \(0.746290\pi\)
\(710\) 3.88740 0.145891
\(711\) −135.565 −5.08408
\(712\) 6.00000 0.224860
\(713\) 11.6622 0.436752
\(714\) −6.50354 −0.243389
\(715\) −31.7748 −1.18831
\(716\) 2.45148 0.0916161
\(717\) 57.4738 2.14640
\(718\) −25.0829 −0.936084
\(719\) 30.1742 1.12531 0.562654 0.826692i \(-0.309780\pi\)
0.562654 + 0.826692i \(0.309780\pi\)
\(720\) −8.19547 −0.305427
\(721\) 3.39803 0.126549
\(722\) −7.80453 −0.290455
\(723\) 51.0970 1.90032
\(724\) 13.1955 0.490406
\(725\) 3.94370 0.146465
\(726\) −38.6314 −1.43375
\(727\) −18.7677 −0.696056 −0.348028 0.937484i \(-0.613149\pi\)
−0.348028 + 0.937484i \(0.613149\pi\)
\(728\) −17.3839 −0.644288
\(729\) 100.701 3.72967
\(730\) 7.49646 0.277456
\(731\) −1.45433 −0.0537903
\(732\) −26.5793 −0.982400
\(733\) 26.1094 0.964374 0.482187 0.876068i \(-0.339843\pi\)
0.482187 + 0.876068i \(0.339843\pi\)
\(734\) 2.48513 0.0917279
\(735\) −0.842420 −0.0310731
\(736\) 1.49646 0.0551601
\(737\) 8.78188 0.323485
\(738\) −52.8382 −1.94500
\(739\) 42.6130 1.56754 0.783772 0.621049i \(-0.213293\pi\)
0.783772 + 0.621049i \(0.213293\pi\)
\(740\) 1.00000 0.0367607
\(741\) 74.9193 2.75223
\(742\) −27.1392 −0.996310
\(743\) −35.4922 −1.30208 −0.651042 0.759042i \(-0.725668\pi\)
−0.651042 + 0.759042i \(0.725668\pi\)
\(744\) −26.0758 −0.955984
\(745\) 4.18838 0.153450
\(746\) −35.1586 −1.28725
\(747\) 124.845 4.56782
\(748\) −3.55276 −0.129902
\(749\) 7.91005 0.289027
\(750\) 3.34596 0.122177
\(751\) 21.5722 0.787182 0.393591 0.919286i \(-0.371233\pi\)
0.393591 + 0.919286i \(0.371233\pi\)
\(752\) 1.84951 0.0674446
\(753\) −33.5864 −1.22396
\(754\) 26.3909 0.961101
\(755\) 6.69193 0.243544
\(756\) 45.1586 1.64240
\(757\) −23.7890 −0.864625 −0.432312 0.901724i \(-0.642302\pi\)
−0.432312 + 0.901724i \(0.642302\pi\)
\(758\) 24.6778 0.896336
\(759\) −23.7748 −0.862970
\(760\) 3.34596 0.121371
\(761\) −18.6724 −0.676876 −0.338438 0.940989i \(-0.609898\pi\)
−0.338438 + 0.940989i \(0.609898\pi\)
\(762\) 29.5864 1.07180
\(763\) 3.74114 0.135438
\(764\) −21.7790 −0.787938
\(765\) 6.13208 0.221706
\(766\) −13.3839 −0.483578
\(767\) 39.1445 1.41342
\(768\) −3.34596 −0.120737
\(769\) −2.48937 −0.0897689 −0.0448845 0.998992i \(-0.514292\pi\)
−0.0448845 + 0.998992i \(0.514292\pi\)
\(770\) 12.3346 0.444510
\(771\) 70.1657 2.52696
\(772\) −12.9929 −0.467625
\(773\) 0.950786 0.0341974 0.0170987 0.999854i \(-0.494557\pi\)
0.0170987 + 0.999854i \(0.494557\pi\)
\(774\) 15.9295 0.572575
\(775\) 7.79321 0.279940
\(776\) −10.4472 −0.375034
\(777\) −8.69193 −0.311821
\(778\) 18.2220 0.653292
\(779\) 21.5722 0.772906
\(780\) 22.3909 0.801724
\(781\) −18.4582 −0.660488
\(782\) −1.11969 −0.0400401
\(783\) −68.5567 −2.45002
\(784\) −0.251772 −0.00899185
\(785\) 3.94370 0.140757
\(786\) −20.5793 −0.734040
\(787\) −29.4359 −1.04928 −0.524639 0.851325i \(-0.675800\pi\)
−0.524639 + 0.851325i \(0.675800\pi\)
\(788\) 0.616147 0.0219493
\(789\) −74.8577 −2.66500
\(790\) 16.5414 0.588518
\(791\) −28.9366 −1.02887
\(792\) 38.9140 1.38275
\(793\) 53.1586 1.88772
\(794\) −2.22521 −0.0789696
\(795\) 34.9561 1.23976
\(796\) −1.54852 −0.0548859
\(797\) −14.1742 −0.502076 −0.251038 0.967977i \(-0.580772\pi\)
−0.251038 + 0.967977i \(0.580772\pi\)
\(798\) −29.0829 −1.02952
\(799\) −1.38385 −0.0489572
\(800\) 1.00000 0.0353553
\(801\) 49.1728 1.73744
\(802\) 12.3909 0.437539
\(803\) −35.5949 −1.25612
\(804\) −6.18838 −0.218247
\(805\) 3.88740 0.137013
\(806\) 52.1516 1.83696
\(807\) 14.0142 0.493322
\(808\) −12.1884 −0.428786
\(809\) −32.9929 −1.15997 −0.579985 0.814627i \(-0.696941\pi\)
−0.579985 + 0.814627i \(0.696941\pi\)
\(810\) −33.5793 −1.17986
\(811\) 40.5567 1.42414 0.712069 0.702110i \(-0.247758\pi\)
0.712069 + 0.702110i \(0.247758\pi\)
\(812\) −10.2447 −0.359518
\(813\) 40.5283 1.42139
\(814\) −4.74823 −0.166425
\(815\) 12.1463 0.425465
\(816\) 2.50354 0.0876416
\(817\) −6.50354 −0.227530
\(818\) 7.27125 0.254233
\(819\) −142.469 −4.97826
\(820\) 6.44724 0.225147
\(821\) −38.8661 −1.35644 −0.678219 0.734860i \(-0.737248\pi\)
−0.678219 + 0.734860i \(0.737248\pi\)
\(822\) −36.4525 −1.27143
\(823\) −49.6243 −1.72979 −0.864897 0.501949i \(-0.832616\pi\)
−0.864897 + 0.501949i \(0.832616\pi\)
\(824\) −1.30807 −0.0455689
\(825\) −15.8874 −0.553128
\(826\) −15.1955 −0.528718
\(827\) −30.0195 −1.04388 −0.521940 0.852982i \(-0.674791\pi\)
−0.521940 + 0.852982i \(0.674791\pi\)
\(828\) 12.2642 0.426209
\(829\) 10.5598 0.366759 0.183379 0.983042i \(-0.441296\pi\)
0.183379 + 0.983042i \(0.441296\pi\)
\(830\) −15.2334 −0.528758
\(831\) 31.1445 1.08039
\(832\) 6.69193 0.232001
\(833\) 0.188383 0.00652708
\(834\) −3.51063 −0.121563
\(835\) −17.0829 −0.591177
\(836\) −15.8874 −0.549477
\(837\) −135.476 −4.68273
\(838\) 28.4809 0.983856
\(839\) 29.3839 1.01444 0.507222 0.861816i \(-0.330673\pi\)
0.507222 + 0.861816i \(0.330673\pi\)
\(840\) −8.69193 −0.299900
\(841\) −13.4472 −0.463698
\(842\) 22.0000 0.758170
\(843\) −64.4809 −2.22084
\(844\) 20.9366 0.720668
\(845\) −31.7819 −1.09333
\(846\) 15.1576 0.521128
\(847\) −29.9926 −1.03056
\(848\) 10.4472 0.358760
\(849\) 45.7890 1.57147
\(850\) −0.748228 −0.0256640
\(851\) −1.49646 −0.0512979
\(852\) 13.0071 0.445615
\(853\) 6.22521 0.213147 0.106573 0.994305i \(-0.466012\pi\)
0.106573 + 0.994305i \(0.466012\pi\)
\(854\) −20.6356 −0.706137
\(855\) 27.4217 0.937804
\(856\) −3.04498 −0.104075
\(857\) −3.65119 −0.124722 −0.0623611 0.998054i \(-0.519863\pi\)
−0.0623611 + 0.998054i \(0.519863\pi\)
\(858\) −106.317 −3.62961
\(859\) −36.3162 −1.23909 −0.619547 0.784960i \(-0.712684\pi\)
−0.619547 + 0.784960i \(0.712684\pi\)
\(860\) −1.94370 −0.0662796
\(861\) −56.0390 −1.90980
\(862\) 1.51487 0.0515966
\(863\) 24.5751 0.836546 0.418273 0.908321i \(-0.362636\pi\)
0.418273 + 0.908321i \(0.362636\pi\)
\(864\) −17.3839 −0.591411
\(865\) 15.3417 0.521634
\(866\) 24.9929 0.849294
\(867\) 55.0082 1.86817
\(868\) −20.2447 −0.687149
\(869\) −78.5425 −2.66437
\(870\) 13.1955 0.447368
\(871\) 12.3768 0.419371
\(872\) −1.44015 −0.0487698
\(873\) −85.6201 −2.89780
\(874\) −5.00709 −0.169367
\(875\) 2.59774 0.0878195
\(876\) 25.0829 0.847472
\(877\) 20.0563 0.677253 0.338627 0.940921i \(-0.390038\pi\)
0.338627 + 0.940921i \(0.390038\pi\)
\(878\) −5.10128 −0.172160
\(879\) 34.9561 1.17904
\(880\) −4.74823 −0.160063
\(881\) −15.0266 −0.506258 −0.253129 0.967433i \(-0.581460\pi\)
−0.253129 + 0.967433i \(0.581460\pi\)
\(882\) −2.06339 −0.0694779
\(883\) −6.62145 −0.222830 −0.111415 0.993774i \(-0.535538\pi\)
−0.111415 + 0.993774i \(0.535538\pi\)
\(884\) −5.00709 −0.168407
\(885\) 19.5722 0.657914
\(886\) 25.7369 0.864648
\(887\) −37.5538 −1.26093 −0.630467 0.776216i \(-0.717137\pi\)
−0.630467 + 0.776216i \(0.717137\pi\)
\(888\) 3.34596 0.112283
\(889\) 22.9703 0.770398
\(890\) −6.00000 −0.201120
\(891\) 159.442 5.34152
\(892\) −2.71034 −0.0907488
\(893\) −6.18838 −0.207086
\(894\) 14.0142 0.468704
\(895\) −2.45148 −0.0819439
\(896\) −2.59774 −0.0867842
\(897\) −33.5071 −1.11877
\(898\) −17.7748 −0.593153
\(899\) 30.7341 1.02504
\(900\) 8.19547 0.273182
\(901\) −7.81692 −0.260419
\(902\) −30.6130 −1.01930
\(903\) 16.8945 0.562213
\(904\) 11.1392 0.370483
\(905\) −13.1955 −0.438632
\(906\) 22.3909 0.743889
\(907\) 50.4667 1.67572 0.837860 0.545885i \(-0.183807\pi\)
0.837860 + 0.545885i \(0.183807\pi\)
\(908\) 26.1321 0.867224
\(909\) −99.8895 −3.31313
\(910\) 17.3839 0.576269
\(911\) 15.9395 0.528098 0.264049 0.964509i \(-0.414942\pi\)
0.264049 + 0.964509i \(0.414942\pi\)
\(912\) 11.1955 0.370719
\(913\) 72.3315 2.39382
\(914\) 34.3346 1.13569
\(915\) 26.5793 0.878685
\(916\) −24.7677 −0.818348
\(917\) −15.9774 −0.527619
\(918\) 13.0071 0.429298
\(919\) 32.5414 1.07344 0.536721 0.843760i \(-0.319663\pi\)
0.536721 + 0.843760i \(0.319663\pi\)
\(920\) −1.49646 −0.0493367
\(921\) 16.5793 0.546307
\(922\) −16.9508 −0.558244
\(923\) −26.0142 −0.856267
\(924\) 41.2713 1.35772
\(925\) −1.00000 −0.0328798
\(926\) 10.9929 0.361250
\(927\) −10.7203 −0.352100
\(928\) 3.94370 0.129458
\(929\) −34.1094 −1.11909 −0.559547 0.828799i \(-0.689025\pi\)
−0.559547 + 0.828799i \(0.689025\pi\)
\(930\) 26.0758 0.855059
\(931\) 0.842420 0.0276092
\(932\) −12.2783 −0.402190
\(933\) −12.3152 −0.403180
\(934\) 37.4175 1.22434
\(935\) 3.55276 0.116188
\(936\) 54.8435 1.79262
\(937\) 0.0141751 0.000463082 0 0.000231541 1.00000i \(-0.499926\pi\)
0.000231541 1.00000i \(0.499926\pi\)
\(938\) −4.80453 −0.156873
\(939\) 3.32225 0.108417
\(940\) −1.84951 −0.0603243
\(941\) 11.1813 0.364500 0.182250 0.983252i \(-0.441662\pi\)
0.182250 + 0.983252i \(0.441662\pi\)
\(942\) 13.1955 0.429932
\(943\) −9.64802 −0.314183
\(944\) 5.84951 0.190385
\(945\) −45.1586 −1.46901
\(946\) 9.22912 0.300065
\(947\) −53.6427 −1.74315 −0.871577 0.490259i \(-0.836902\pi\)
−0.871577 + 0.490259i \(0.836902\pi\)
\(948\) 55.3470 1.79759
\(949\) −50.1657 −1.62845
\(950\) −3.34596 −0.108557
\(951\) 21.9490 0.711745
\(952\) 1.94370 0.0629956
\(953\) −9.88740 −0.320284 −0.160142 0.987094i \(-0.551195\pi\)
−0.160142 + 0.987094i \(0.551195\pi\)
\(954\) 85.6201 2.77205
\(955\) 21.7790 0.704753
\(956\) −17.1771 −0.555546
\(957\) −62.6551 −2.02535
\(958\) −0.465654 −0.0150446
\(959\) −28.3010 −0.913886
\(960\) 3.34596 0.107991
\(961\) 29.7341 0.959163
\(962\) −6.69193 −0.215756
\(963\) −24.9550 −0.804164
\(964\) −15.2713 −0.491854
\(965\) 12.9929 0.418257
\(966\) 13.0071 0.418496
\(967\) 51.6254 1.66016 0.830080 0.557644i \(-0.188295\pi\)
0.830080 + 0.557644i \(0.188295\pi\)
\(968\) 11.5457 0.371092
\(969\) −8.37677 −0.269100
\(970\) 10.4472 0.335441
\(971\) 20.5598 0.659797 0.329898 0.944016i \(-0.392986\pi\)
0.329898 + 0.944016i \(0.392986\pi\)
\(972\) −60.2036 −1.93103
\(973\) −2.72558 −0.0873781
\(974\) 34.9561 1.12007
\(975\) −22.3909 −0.717084
\(976\) 7.94370 0.254272
\(977\) 38.4472 1.23004 0.615018 0.788513i \(-0.289149\pi\)
0.615018 + 0.788513i \(0.289149\pi\)
\(978\) 40.6409 1.29955
\(979\) 28.4894 0.910524
\(980\) 0.251772 0.00804256
\(981\) −11.8027 −0.376833
\(982\) −24.7819 −0.790822
\(983\) 11.8973 0.379466 0.189733 0.981836i \(-0.439238\pi\)
0.189733 + 0.981836i \(0.439238\pi\)
\(984\) 21.5722 0.687698
\(985\) −0.616147 −0.0196321
\(986\) −2.95079 −0.0939722
\(987\) 16.0758 0.511698
\(988\) −22.3909 −0.712351
\(989\) 2.90866 0.0924900
\(990\) −38.9140 −1.23677
\(991\) −19.7932 −0.628752 −0.314376 0.949299i \(-0.601795\pi\)
−0.314376 + 0.949299i \(0.601795\pi\)
\(992\) 7.79321 0.247435
\(993\) 60.3541 1.91528
\(994\) 10.0984 0.320303
\(995\) 1.54852 0.0490914
\(996\) −50.9703 −1.61505
\(997\) 5.88740 0.186456 0.0932279 0.995645i \(-0.470281\pi\)
0.0932279 + 0.995645i \(0.470281\pi\)
\(998\) −2.33888 −0.0740358
\(999\) 17.3839 0.550001
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 370.2.a.g.1.1 3
3.2 odd 2 3330.2.a.bg.1.1 3
4.3 odd 2 2960.2.a.u.1.3 3
5.2 odd 4 1850.2.b.o.149.6 6
5.3 odd 4 1850.2.b.o.149.1 6
5.4 even 2 1850.2.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.1 3 1.1 even 1 trivial
1850.2.a.z.1.3 3 5.4 even 2
1850.2.b.o.149.1 6 5.3 odd 4
1850.2.b.o.149.6 6 5.2 odd 4
2960.2.a.u.1.3 3 4.3 odd 2
3330.2.a.bg.1.1 3 3.2 odd 2