Properties

Label 69.6.a.b.1.2
Level $69$
Weight $6$
Character 69.1
Self dual yes
Analytic conductor $11.066$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0664835671\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.5333.1
Defining polynomial: \(x^{3} - x^{2} - 11 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.49331\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.29009 q^{2} +9.00000 q^{3} -30.3357 q^{4} +59.1123 q^{5} -11.6108 q^{6} -213.331 q^{7} +80.4187 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-1.29009 q^{2} +9.00000 q^{3} -30.3357 q^{4} +59.1123 q^{5} -11.6108 q^{6} -213.331 q^{7} +80.4187 q^{8} +81.0000 q^{9} -76.2602 q^{10} +126.528 q^{11} -273.021 q^{12} -884.825 q^{13} +275.217 q^{14} +532.011 q^{15} +866.994 q^{16} -1179.98 q^{17} -104.497 q^{18} -1866.19 q^{19} -1793.21 q^{20} -1919.98 q^{21} -163.233 q^{22} +529.000 q^{23} +723.768 q^{24} +369.263 q^{25} +1141.50 q^{26} +729.000 q^{27} +6471.55 q^{28} -6786.62 q^{29} -686.342 q^{30} -5146.34 q^{31} -3691.90 q^{32} +1138.75 q^{33} +1522.28 q^{34} -12610.5 q^{35} -2457.19 q^{36} +5137.07 q^{37} +2407.55 q^{38} -7963.42 q^{39} +4753.73 q^{40} +12482.7 q^{41} +2476.95 q^{42} +4198.66 q^{43} -3838.32 q^{44} +4788.10 q^{45} -682.458 q^{46} +23006.9 q^{47} +7802.94 q^{48} +28703.3 q^{49} -476.382 q^{50} -10619.8 q^{51} +26841.8 q^{52} +25175.4 q^{53} -940.476 q^{54} +7479.38 q^{55} -17155.8 q^{56} -16795.7 q^{57} +8755.36 q^{58} -37118.1 q^{59} -16138.9 q^{60} +26410.4 q^{61} +6639.24 q^{62} -17279.8 q^{63} -22980.9 q^{64} -52304.0 q^{65} -1469.10 q^{66} -54398.5 q^{67} +35795.4 q^{68} +4761.00 q^{69} +16268.7 q^{70} +35684.7 q^{71} +6513.91 q^{72} +33937.4 q^{73} -6627.29 q^{74} +3323.36 q^{75} +56612.0 q^{76} -26992.5 q^{77} +10273.5 q^{78} -76625.8 q^{79} +51250.0 q^{80} +6561.00 q^{81} -16103.8 q^{82} -96627.2 q^{83} +58244.0 q^{84} -69751.1 q^{85} -5416.65 q^{86} -61079.6 q^{87} +10175.2 q^{88} +30080.8 q^{89} -6177.08 q^{90} +188761. q^{91} -16047.6 q^{92} -46317.0 q^{93} -29680.9 q^{94} -110315. q^{95} -33227.1 q^{96} -14637.2 q^{97} -37029.9 q^{98} +10248.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 8q^{2} + 27q^{3} + 22q^{4} - 56q^{5} - 72q^{6} - 114q^{7} - 510q^{8} + 243q^{9} + O(q^{10}) \) \( 3q - 8q^{2} + 27q^{3} + 22q^{4} - 56q^{5} - 72q^{6} - 114q^{7} - 510q^{8} + 243q^{9} + 282q^{10} - 376q^{11} + 198q^{12} - 858q^{13} + 588q^{14} - 504q^{15} + 2738q^{16} - 2548q^{17} - 648q^{18} - 2846q^{19} - 4618q^{20} - 1026q^{21} - 5050q^{22} + 1587q^{23} - 4590q^{24} + 753q^{25} - 7788q^{26} + 2187q^{27} + 4736q^{28} - 16370q^{29} + 2538q^{30} - 14756q^{31} - 3878q^{32} - 3384q^{33} + 16520q^{34} - 18520q^{35} + 1782q^{36} + 15874q^{37} + 12438q^{38} - 7722q^{39} + 38270q^{40} + 12606q^{41} + 5292q^{42} + 3154q^{43} + 27114q^{44} - 4536q^{45} - 4232q^{46} + 29928q^{47} + 24642q^{48} + 4471q^{49} + 1452q^{50} - 22932q^{51} + 86856q^{52} - 44084q^{53} - 5832q^{54} + 38360q^{55} - 35704q^{56} - 25614q^{57} + 73316q^{58} - 29300q^{59} - 41562q^{60} + 54010q^{61} + 99908q^{62} - 9234q^{63} - 1582q^{64} - 51216q^{65} - 45450q^{66} + 43390q^{67} - 69840q^{68} + 14283q^{69} - 2476q^{70} + 23424q^{71} - 41310q^{72} - 91402q^{73} - 2294q^{74} + 6777q^{75} - 14274q^{76} - 97208q^{77} - 70092q^{78} - 49398q^{79} - 52626q^{80} + 19683q^{81} + 40152q^{82} - 103936q^{83} + 42624q^{84} + 5888q^{85} + 133634q^{86} - 147330q^{87} + 48898q^{88} + 96112q^{89} + 22842q^{90} + 129228q^{91} + 11638q^{92} - 132804q^{93} - 133688q^{94} - 55928q^{95} - 34902q^{96} - 135318q^{97} + 108440q^{98} - 30456q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.29009 −0.228058 −0.114029 0.993477i \(-0.536376\pi\)
−0.114029 + 0.993477i \(0.536376\pi\)
\(3\) 9.00000 0.577350
\(4\) −30.3357 −0.947990
\(5\) 59.1123 1.05743 0.528716 0.848799i \(-0.322674\pi\)
0.528716 + 0.848799i \(0.322674\pi\)
\(6\) −11.6108 −0.131669
\(7\) −213.331 −1.64554 −0.822772 0.568371i \(-0.807574\pi\)
−0.822772 + 0.568371i \(0.807574\pi\)
\(8\) 80.4187 0.444255
\(9\) 81.0000 0.333333
\(10\) −76.2602 −0.241156
\(11\) 126.528 0.315287 0.157643 0.987496i \(-0.449610\pi\)
0.157643 + 0.987496i \(0.449610\pi\)
\(12\) −273.021 −0.547322
\(13\) −884.825 −1.45211 −0.726054 0.687638i \(-0.758648\pi\)
−0.726054 + 0.687638i \(0.758648\pi\)
\(14\) 275.217 0.375280
\(15\) 532.011 0.610509
\(16\) 866.994 0.846674
\(17\) −1179.98 −0.990264 −0.495132 0.868818i \(-0.664880\pi\)
−0.495132 + 0.868818i \(0.664880\pi\)
\(18\) −104.497 −0.0760194
\(19\) −1866.19 −1.18596 −0.592981 0.805216i \(-0.702049\pi\)
−0.592981 + 0.805216i \(0.702049\pi\)
\(20\) −1793.21 −1.00244
\(21\) −1919.98 −0.950056
\(22\) −163.233 −0.0719037
\(23\) 529.000 0.208514
\(24\) 723.768 0.256491
\(25\) 369.263 0.118164
\(26\) 1141.50 0.331165
\(27\) 729.000 0.192450
\(28\) 6471.55 1.55996
\(29\) −6786.62 −1.49851 −0.749253 0.662284i \(-0.769587\pi\)
−0.749253 + 0.662284i \(0.769587\pi\)
\(30\) −686.342 −0.139232
\(31\) −5146.34 −0.961820 −0.480910 0.876770i \(-0.659694\pi\)
−0.480910 + 0.876770i \(0.659694\pi\)
\(32\) −3691.90 −0.637345
\(33\) 1138.75 0.182031
\(34\) 1522.28 0.225838
\(35\) −12610.5 −1.74005
\(36\) −2457.19 −0.315997
\(37\) 5137.07 0.616895 0.308448 0.951241i \(-0.400191\pi\)
0.308448 + 0.951241i \(0.400191\pi\)
\(38\) 2407.55 0.270468
\(39\) −7963.42 −0.838375
\(40\) 4753.73 0.469769
\(41\) 12482.7 1.15971 0.579855 0.814720i \(-0.303109\pi\)
0.579855 + 0.814720i \(0.303109\pi\)
\(42\) 2476.95 0.216668
\(43\) 4198.66 0.346290 0.173145 0.984896i \(-0.444607\pi\)
0.173145 + 0.984896i \(0.444607\pi\)
\(44\) −3838.32 −0.298889
\(45\) 4788.10 0.352478
\(46\) −682.458 −0.0475534
\(47\) 23006.9 1.51919 0.759596 0.650395i \(-0.225397\pi\)
0.759596 + 0.650395i \(0.225397\pi\)
\(48\) 7802.94 0.488827
\(49\) 28703.3 1.70782
\(50\) −476.382 −0.0269483
\(51\) −10619.8 −0.571729
\(52\) 26841.8 1.37658
\(53\) 25175.4 1.23108 0.615541 0.788105i \(-0.288938\pi\)
0.615541 + 0.788105i \(0.288938\pi\)
\(54\) −940.476 −0.0438898
\(55\) 7479.38 0.333395
\(56\) −17155.8 −0.731041
\(57\) −16795.7 −0.684716
\(58\) 8755.36 0.341746
\(59\) −37118.1 −1.38821 −0.694107 0.719872i \(-0.744200\pi\)
−0.694107 + 0.719872i \(0.744200\pi\)
\(60\) −16138.9 −0.578756
\(61\) 26410.4 0.908761 0.454381 0.890808i \(-0.349861\pi\)
0.454381 + 0.890808i \(0.349861\pi\)
\(62\) 6639.24 0.219351
\(63\) −17279.8 −0.548515
\(64\) −22980.9 −0.701322
\(65\) −52304.0 −1.53551
\(66\) −1469.10 −0.0415136
\(67\) −54398.5 −1.48047 −0.740236 0.672347i \(-0.765286\pi\)
−0.740236 + 0.672347i \(0.765286\pi\)
\(68\) 35795.4 0.938760
\(69\) 4761.00 0.120386
\(70\) 16268.7 0.396833
\(71\) 35684.7 0.840110 0.420055 0.907499i \(-0.362011\pi\)
0.420055 + 0.907499i \(0.362011\pi\)
\(72\) 6513.91 0.148085
\(73\) 33937.4 0.745369 0.372684 0.927958i \(-0.378437\pi\)
0.372684 + 0.927958i \(0.378437\pi\)
\(74\) −6627.29 −0.140688
\(75\) 3323.36 0.0682220
\(76\) 56612.0 1.12428
\(77\) −26992.5 −0.518819
\(78\) 10273.5 0.191198
\(79\) −76625.8 −1.38136 −0.690681 0.723160i \(-0.742689\pi\)
−0.690681 + 0.723160i \(0.742689\pi\)
\(80\) 51250.0 0.895300
\(81\) 6561.00 0.111111
\(82\) −16103.8 −0.264481
\(83\) −96627.2 −1.53959 −0.769794 0.638293i \(-0.779641\pi\)
−0.769794 + 0.638293i \(0.779641\pi\)
\(84\) 58244.0 0.900643
\(85\) −69751.1 −1.04714
\(86\) −5416.65 −0.0789741
\(87\) −61079.6 −0.865163
\(88\) 10175.2 0.140068
\(89\) 30080.8 0.402545 0.201273 0.979535i \(-0.435492\pi\)
0.201273 + 0.979535i \(0.435492\pi\)
\(90\) −6177.08 −0.0803854
\(91\) 188761. 2.38951
\(92\) −16047.6 −0.197669
\(93\) −46317.0 −0.555307
\(94\) −29680.9 −0.346464
\(95\) −110315. −1.25408
\(96\) −33227.1 −0.367972
\(97\) −14637.2 −0.157953 −0.0789766 0.996876i \(-0.525165\pi\)
−0.0789766 + 0.996876i \(0.525165\pi\)
\(98\) −37029.9 −0.389482
\(99\) 10248.8 0.105096
\(100\) −11201.8 −0.112018
\(101\) −37693.7 −0.367676 −0.183838 0.982957i \(-0.558852\pi\)
−0.183838 + 0.982957i \(0.558852\pi\)
\(102\) 13700.5 0.130388
\(103\) 125252. 1.16330 0.581649 0.813440i \(-0.302408\pi\)
0.581649 + 0.813440i \(0.302408\pi\)
\(104\) −71156.5 −0.645106
\(105\) −113495. −1.00462
\(106\) −32478.6 −0.280758
\(107\) 28002.7 0.236451 0.118225 0.992987i \(-0.462279\pi\)
0.118225 + 0.992987i \(0.462279\pi\)
\(108\) −22114.7 −0.182441
\(109\) −64151.4 −0.517178 −0.258589 0.965987i \(-0.583257\pi\)
−0.258589 + 0.965987i \(0.583257\pi\)
\(110\) −9649.08 −0.0760333
\(111\) 46233.6 0.356165
\(112\) −184957. −1.39324
\(113\) −25226.7 −0.185851 −0.0929253 0.995673i \(-0.529622\pi\)
−0.0929253 + 0.995673i \(0.529622\pi\)
\(114\) 21668.0 0.156155
\(115\) 31270.4 0.220490
\(116\) 205877. 1.42057
\(117\) −71670.8 −0.484036
\(118\) 47885.8 0.316593
\(119\) 251726. 1.62952
\(120\) 42783.6 0.271222
\(121\) −145042. −0.900594
\(122\) −34071.8 −0.207250
\(123\) 112344. 0.669559
\(124\) 156118. 0.911796
\(125\) −162898. −0.932482
\(126\) 22292.6 0.125093
\(127\) 35783.9 0.196870 0.0984348 0.995144i \(-0.468616\pi\)
0.0984348 + 0.995144i \(0.468616\pi\)
\(128\) 147788. 0.797288
\(129\) 37787.9 0.199930
\(130\) 67477.0 0.350185
\(131\) 78119.6 0.397724 0.198862 0.980028i \(-0.436275\pi\)
0.198862 + 0.980028i \(0.436275\pi\)
\(132\) −34544.9 −0.172563
\(133\) 398116. 1.95155
\(134\) 70179.1 0.337634
\(135\) 43092.9 0.203503
\(136\) −94892.2 −0.439930
\(137\) −92816.7 −0.422498 −0.211249 0.977432i \(-0.567753\pi\)
−0.211249 + 0.977432i \(0.567753\pi\)
\(138\) −6142.12 −0.0274550
\(139\) −417118. −1.83114 −0.915570 0.402159i \(-0.868260\pi\)
−0.915570 + 0.402159i \(0.868260\pi\)
\(140\) 382548. 1.64955
\(141\) 207062. 0.877106
\(142\) −46036.5 −0.191594
\(143\) −111955. −0.457831
\(144\) 70226.5 0.282225
\(145\) −401173. −1.58457
\(146\) −43782.3 −0.169987
\(147\) 258330. 0.986009
\(148\) −155836. −0.584810
\(149\) 445936. 1.64553 0.822766 0.568380i \(-0.192430\pi\)
0.822766 + 0.568380i \(0.192430\pi\)
\(150\) −4287.44 −0.0155586
\(151\) −260458. −0.929597 −0.464799 0.885416i \(-0.653873\pi\)
−0.464799 + 0.885416i \(0.653873\pi\)
\(152\) −150076. −0.526869
\(153\) −95578.1 −0.330088
\(154\) 34822.7 0.118321
\(155\) −304212. −1.01706
\(156\) 241576. 0.794771
\(157\) 46067.4 0.149157 0.0745787 0.997215i \(-0.476239\pi\)
0.0745787 + 0.997215i \(0.476239\pi\)
\(158\) 98854.3 0.315031
\(159\) 226579. 0.710765
\(160\) −218237. −0.673950
\(161\) −112852. −0.343120
\(162\) −8464.29 −0.0253398
\(163\) 188377. 0.555341 0.277671 0.960676i \(-0.410438\pi\)
0.277671 + 0.960676i \(0.410438\pi\)
\(164\) −378671. −1.09939
\(165\) 67314.4 0.192485
\(166\) 124658. 0.351115
\(167\) 203214. 0.563849 0.281924 0.959437i \(-0.409027\pi\)
0.281924 + 0.959437i \(0.409027\pi\)
\(168\) −154403. −0.422067
\(169\) 411622. 1.10862
\(170\) 89985.3 0.238808
\(171\) −151161. −0.395321
\(172\) −127369. −0.328279
\(173\) 158051. 0.401497 0.200749 0.979643i \(-0.435663\pi\)
0.200749 + 0.979643i \(0.435663\pi\)
\(174\) 78798.2 0.197307
\(175\) −78775.3 −0.194444
\(176\) 109699. 0.266945
\(177\) −334063. −0.801485
\(178\) −38807.0 −0.0918037
\(179\) 166444. 0.388271 0.194136 0.980975i \(-0.437810\pi\)
0.194136 + 0.980975i \(0.437810\pi\)
\(180\) −145250. −0.334145
\(181\) 185620. 0.421141 0.210571 0.977579i \(-0.432468\pi\)
0.210571 + 0.977579i \(0.432468\pi\)
\(182\) −243519. −0.544947
\(183\) 237693. 0.524673
\(184\) 42541.5 0.0926335
\(185\) 303664. 0.652325
\(186\) 59753.2 0.126642
\(187\) −149300. −0.312217
\(188\) −697928. −1.44018
\(189\) −155519. −0.316685
\(190\) 142316. 0.286002
\(191\) −769733. −1.52671 −0.763356 0.645978i \(-0.776450\pi\)
−0.763356 + 0.645978i \(0.776450\pi\)
\(192\) −206828. −0.404908
\(193\) −384145. −0.742338 −0.371169 0.928565i \(-0.621043\pi\)
−0.371169 + 0.928565i \(0.621043\pi\)
\(194\) 18883.3 0.0360225
\(195\) −470736. −0.886525
\(196\) −870734. −1.61899
\(197\) 266048. 0.488421 0.244211 0.969722i \(-0.421471\pi\)
0.244211 + 0.969722i \(0.421471\pi\)
\(198\) −13221.9 −0.0239679
\(199\) −706070. −1.26391 −0.631953 0.775007i \(-0.717746\pi\)
−0.631953 + 0.775007i \(0.717746\pi\)
\(200\) 29695.6 0.0524949
\(201\) −489587. −0.854751
\(202\) 48628.3 0.0838515
\(203\) 1.44780e6 2.46586
\(204\) 322158. 0.541994
\(205\) 737882. 1.22631
\(206\) −161586. −0.265300
\(207\) 42849.0 0.0695048
\(208\) −767138. −1.22946
\(209\) −236125. −0.373918
\(210\) 146418. 0.229112
\(211\) −290636. −0.449410 −0.224705 0.974427i \(-0.572142\pi\)
−0.224705 + 0.974427i \(0.572142\pi\)
\(212\) −763712. −1.16705
\(213\) 321162. 0.485038
\(214\) −36126.1 −0.0539245
\(215\) 248192. 0.366178
\(216\) 58625.2 0.0854969
\(217\) 1.09788e6 1.58272
\(218\) 82761.1 0.117947
\(219\) 305436. 0.430339
\(220\) −226892. −0.316055
\(221\) 1.04407e6 1.43797
\(222\) −59645.6 −0.0812262
\(223\) −574915. −0.774179 −0.387089 0.922042i \(-0.626519\pi\)
−0.387089 + 0.922042i \(0.626519\pi\)
\(224\) 787598. 1.04878
\(225\) 29910.3 0.0393880
\(226\) 32544.7 0.0423847
\(227\) −1.37363e6 −1.76931 −0.884655 0.466246i \(-0.845606\pi\)
−0.884655 + 0.466246i \(0.845606\pi\)
\(228\) 509508. 0.649103
\(229\) 870725. 1.09722 0.548608 0.836080i \(-0.315158\pi\)
0.548608 + 0.836080i \(0.315158\pi\)
\(230\) −40341.7 −0.0502845
\(231\) −242932. −0.299540
\(232\) −545771. −0.665718
\(233\) −1.61732e6 −1.95166 −0.975832 0.218522i \(-0.929876\pi\)
−0.975832 + 0.218522i \(0.929876\pi\)
\(234\) 92461.9 0.110388
\(235\) 1.35999e6 1.60644
\(236\) 1.12600e6 1.31601
\(237\) −689633. −0.797530
\(238\) −324750. −0.371626
\(239\) −1.65281e6 −1.87167 −0.935833 0.352444i \(-0.885351\pi\)
−0.935833 + 0.352444i \(0.885351\pi\)
\(240\) 461250. 0.516902
\(241\) 928466. 1.02973 0.514865 0.857271i \(-0.327842\pi\)
0.514865 + 0.857271i \(0.327842\pi\)
\(242\) 187117. 0.205388
\(243\) 59049.0 0.0641500
\(244\) −801176. −0.861496
\(245\) 1.69672e6 1.80590
\(246\) −144935. −0.152698
\(247\) 1.65125e6 1.72215
\(248\) −413862. −0.427293
\(249\) −869645. −0.888881
\(250\) 210153. 0.212660
\(251\) −909693. −0.911403 −0.455702 0.890133i \(-0.650612\pi\)
−0.455702 + 0.890133i \(0.650612\pi\)
\(252\) 524196. 0.519986
\(253\) 66933.5 0.0657419
\(254\) −46164.5 −0.0448977
\(255\) −627760. −0.604565
\(256\) 544729. 0.519494
\(257\) 52666.7 0.0497397 0.0248699 0.999691i \(-0.492083\pi\)
0.0248699 + 0.999691i \(0.492083\pi\)
\(258\) −48749.9 −0.0455957
\(259\) −1.09590e6 −1.01513
\(260\) 1.58668e6 1.45564
\(261\) −549716. −0.499502
\(262\) −100781. −0.0907042
\(263\) 464764. 0.414327 0.207164 0.978306i \(-0.433577\pi\)
0.207164 + 0.978306i \(0.433577\pi\)
\(264\) 91577.1 0.0808681
\(265\) 1.48818e6 1.30179
\(266\) −513606. −0.445068
\(267\) 270727. 0.232410
\(268\) 1.65022e6 1.40347
\(269\) 968420. 0.815987 0.407993 0.912985i \(-0.366229\pi\)
0.407993 + 0.912985i \(0.366229\pi\)
\(270\) −55593.7 −0.0464105
\(271\) 2.29262e6 1.89630 0.948152 0.317817i \(-0.102950\pi\)
0.948152 + 0.317817i \(0.102950\pi\)
\(272\) −1.02303e6 −0.838431
\(273\) 1.69885e6 1.37958
\(274\) 119742. 0.0963541
\(275\) 46722.2 0.0372556
\(276\) −144428. −0.114125
\(277\) 290954. 0.227837 0.113919 0.993490i \(-0.463660\pi\)
0.113919 + 0.993490i \(0.463660\pi\)
\(278\) 538120. 0.417606
\(279\) −416853. −0.320607
\(280\) −1.01412e6 −0.773027
\(281\) −391978. −0.296139 −0.148070 0.988977i \(-0.547306\pi\)
−0.148070 + 0.988977i \(0.547306\pi\)
\(282\) −267128. −0.200031
\(283\) 732768. 0.543877 0.271938 0.962315i \(-0.412335\pi\)
0.271938 + 0.962315i \(0.412335\pi\)
\(284\) −1.08252e6 −0.796415
\(285\) −992831. −0.724041
\(286\) 144433. 0.104412
\(287\) −2.66295e6 −1.90835
\(288\) −299044. −0.212448
\(289\) −27511.7 −0.0193764
\(290\) 517549. 0.361374
\(291\) −131735. −0.0911944
\(292\) −1.02951e6 −0.706602
\(293\) 1.62587e6 1.10641 0.553207 0.833044i \(-0.313404\pi\)
0.553207 + 0.833044i \(0.313404\pi\)
\(294\) −333269. −0.224867
\(295\) −2.19414e6 −1.46794
\(296\) 413117. 0.274059
\(297\) 92239.1 0.0606770
\(298\) −575298. −0.375277
\(299\) −468072. −0.302785
\(300\) −100816. −0.0646738
\(301\) −895706. −0.569835
\(302\) 336014. 0.212002
\(303\) −339243. −0.212278
\(304\) −1.61797e6 −1.00412
\(305\) 1.56118e6 0.960954
\(306\) 123304. 0.0752793
\(307\) 1.24581e6 0.754406 0.377203 0.926131i \(-0.376886\pi\)
0.377203 + 0.926131i \(0.376886\pi\)
\(308\) 818834. 0.491835
\(309\) 1.12727e6 0.671631
\(310\) 392461. 0.231949
\(311\) −3.26852e6 −1.91624 −0.958119 0.286369i \(-0.907552\pi\)
−0.958119 + 0.286369i \(0.907552\pi\)
\(312\) −640408. −0.372452
\(313\) 1.70161e6 0.981744 0.490872 0.871232i \(-0.336678\pi\)
0.490872 + 0.871232i \(0.336678\pi\)
\(314\) −59431.2 −0.0340166
\(315\) −1.02145e6 −0.580018
\(316\) 2.32450e6 1.30952
\(317\) −908317. −0.507679 −0.253840 0.967246i \(-0.581694\pi\)
−0.253840 + 0.967246i \(0.581694\pi\)
\(318\) −292307. −0.162096
\(319\) −858699. −0.472459
\(320\) −1.35845e6 −0.741601
\(321\) 252025. 0.136515
\(322\) 145590. 0.0782512
\(323\) 2.20206e6 1.17442
\(324\) −199032. −0.105332
\(325\) −326733. −0.171587
\(326\) −243024. −0.126650
\(327\) −577362. −0.298593
\(328\) 1.00384e6 0.515206
\(329\) −4.90809e6 −2.49990
\(330\) −86841.7 −0.0438979
\(331\) −2.14345e6 −1.07533 −0.537666 0.843158i \(-0.680694\pi\)
−0.537666 + 0.843158i \(0.680694\pi\)
\(332\) 2.93125e6 1.45951
\(333\) 416103. 0.205632
\(334\) −262165. −0.128590
\(335\) −3.21562e6 −1.56550
\(336\) −1.66461e6 −0.804387
\(337\) 817498. 0.392114 0.196057 0.980593i \(-0.437186\pi\)
0.196057 + 0.980593i \(0.437186\pi\)
\(338\) −531030. −0.252829
\(339\) −227040. −0.107301
\(340\) 2.11595e6 0.992676
\(341\) −651157. −0.303249
\(342\) 195012. 0.0901561
\(343\) −2.53785e6 −1.16475
\(344\) 337651. 0.153841
\(345\) 281434. 0.127300
\(346\) −203900. −0.0915647
\(347\) 3.20207e6 1.42760 0.713801 0.700349i \(-0.246972\pi\)
0.713801 + 0.700349i \(0.246972\pi\)
\(348\) 1.85289e6 0.820165
\(349\) −3.23916e6 −1.42354 −0.711768 0.702415i \(-0.752105\pi\)
−0.711768 + 0.702415i \(0.752105\pi\)
\(350\) 101627. 0.0443446
\(351\) −645037. −0.279458
\(352\) −467130. −0.200947
\(353\) −1.72676e6 −0.737555 −0.368777 0.929518i \(-0.620224\pi\)
−0.368777 + 0.929518i \(0.620224\pi\)
\(354\) 430972. 0.182785
\(355\) 2.10940e6 0.888360
\(356\) −912522. −0.381609
\(357\) 2.26554e6 0.940806
\(358\) −214728. −0.0885484
\(359\) 3.12338e6 1.27905 0.639527 0.768769i \(-0.279130\pi\)
0.639527 + 0.768769i \(0.279130\pi\)
\(360\) 385052. 0.156590
\(361\) 1.00655e6 0.406507
\(362\) −239466. −0.0960446
\(363\) −1.30537e6 −0.519958
\(364\) −5.72619e6 −2.26523
\(365\) 2.00612e6 0.788177
\(366\) −306646. −0.119656
\(367\) 1.17031e6 0.453560 0.226780 0.973946i \(-0.427180\pi\)
0.226780 + 0.973946i \(0.427180\pi\)
\(368\) 458640. 0.176544
\(369\) 1.01110e6 0.386570
\(370\) −391754. −0.148768
\(371\) −5.37070e6 −2.02580
\(372\) 1.40506e6 0.526425
\(373\) −692891. −0.257865 −0.128933 0.991653i \(-0.541155\pi\)
−0.128933 + 0.991653i \(0.541155\pi\)
\(374\) 192611. 0.0712037
\(375\) −1.46608e6 −0.538369
\(376\) 1.85018e6 0.674908
\(377\) 6.00497e6 2.17599
\(378\) 200633. 0.0722226
\(379\) 2.12163e6 0.758702 0.379351 0.925253i \(-0.376147\pi\)
0.379351 + 0.925253i \(0.376147\pi\)
\(380\) 3.34646e6 1.18885
\(381\) 322055. 0.113663
\(382\) 993026. 0.348179
\(383\) −287079. −0.100001 −0.0500005 0.998749i \(-0.515922\pi\)
−0.0500005 + 0.998749i \(0.515922\pi\)
\(384\) 1.33009e6 0.460314
\(385\) −1.59559e6 −0.548616
\(386\) 495582. 0.169296
\(387\) 340091. 0.115430
\(388\) 444029. 0.149738
\(389\) 679811. 0.227779 0.113890 0.993493i \(-0.463669\pi\)
0.113890 + 0.993493i \(0.463669\pi\)
\(390\) 607293. 0.202179
\(391\) −624208. −0.206484
\(392\) 2.30828e6 0.758706
\(393\) 703076. 0.229626
\(394\) −343226. −0.111388
\(395\) −4.52953e6 −1.46070
\(396\) −310904. −0.0996295
\(397\) −5.39917e6 −1.71930 −0.859648 0.510886i \(-0.829317\pi\)
−0.859648 + 0.510886i \(0.829317\pi\)
\(398\) 910894. 0.288244
\(399\) 3.58304e6 1.12673
\(400\) 320148. 0.100046
\(401\) −2.17808e6 −0.676416 −0.338208 0.941071i \(-0.609821\pi\)
−0.338208 + 0.941071i \(0.609821\pi\)
\(402\) 631612. 0.194933
\(403\) 4.55361e6 1.39667
\(404\) 1.14346e6 0.348553
\(405\) 387836. 0.117493
\(406\) −1.86779e6 −0.562359
\(407\) 649985. 0.194499
\(408\) −854030. −0.253993
\(409\) −356567. −0.105398 −0.0526991 0.998610i \(-0.516782\pi\)
−0.0526991 + 0.998610i \(0.516782\pi\)
\(410\) −951934. −0.279671
\(411\) −835350. −0.243929
\(412\) −3.79960e6 −1.10279
\(413\) 7.91847e6 2.28437
\(414\) −55279.1 −0.0158511
\(415\) −5.71186e6 −1.62801
\(416\) 3.26668e6 0.925495
\(417\) −3.75406e6 −1.05721
\(418\) 304623. 0.0852751
\(419\) 3.82634e6 1.06475 0.532376 0.846508i \(-0.321299\pi\)
0.532376 + 0.846508i \(0.321299\pi\)
\(420\) 3.44293e6 0.952369
\(421\) −3.00814e6 −0.827167 −0.413584 0.910466i \(-0.635723\pi\)
−0.413584 + 0.910466i \(0.635723\pi\)
\(422\) 374947. 0.102492
\(423\) 1.86356e6 0.506397
\(424\) 2.02457e6 0.546914
\(425\) −435721. −0.117014
\(426\) −414329. −0.110617
\(427\) −5.63416e6 −1.49541
\(428\) −849481. −0.224153
\(429\) −1.00760e6 −0.264329
\(430\) −320191. −0.0835098
\(431\) 577809. 0.149827 0.0749136 0.997190i \(-0.476132\pi\)
0.0749136 + 0.997190i \(0.476132\pi\)
\(432\) 632038. 0.162942
\(433\) 7.21909e6 1.85039 0.925194 0.379494i \(-0.123902\pi\)
0.925194 + 0.379494i \(0.123902\pi\)
\(434\) −1.41636e6 −0.360952
\(435\) −3.61055e6 −0.914852
\(436\) 1.94607e6 0.490279
\(437\) −987212. −0.247290
\(438\) −394041. −0.0981422
\(439\) −760935. −0.188446 −0.0942229 0.995551i \(-0.530037\pi\)
−0.0942229 + 0.995551i \(0.530037\pi\)
\(440\) 601482. 0.148112
\(441\) 2.32497e6 0.569273
\(442\) −1.34695e6 −0.327941
\(443\) −6.96892e6 −1.68716 −0.843580 0.537003i \(-0.819556\pi\)
−0.843580 + 0.537003i \(0.819556\pi\)
\(444\) −1.40253e6 −0.337640
\(445\) 1.77815e6 0.425665
\(446\) 741692. 0.176558
\(447\) 4.01342e6 0.950049
\(448\) 4.90255e6 1.15406
\(449\) 5.48897e6 1.28492 0.642458 0.766321i \(-0.277915\pi\)
0.642458 + 0.766321i \(0.277915\pi\)
\(450\) −38587.0 −0.00898275
\(451\) 1.57942e6 0.365641
\(452\) 765268. 0.176184
\(453\) −2.34412e6 −0.536703
\(454\) 1.77210e6 0.403506
\(455\) 1.11581e7 2.52675
\(456\) −1.35069e6 −0.304188
\(457\) −2.49968e6 −0.559879 −0.279940 0.960018i \(-0.590314\pi\)
−0.279940 + 0.960018i \(0.590314\pi\)
\(458\) −1.12331e6 −0.250229
\(459\) −860203. −0.190576
\(460\) −948608. −0.209022
\(461\) −8.10339e6 −1.77588 −0.887942 0.459956i \(-0.847865\pi\)
−0.887942 + 0.459956i \(0.847865\pi\)
\(462\) 313405. 0.0683125
\(463\) −4.71421e6 −1.02201 −0.511007 0.859577i \(-0.670727\pi\)
−0.511007 + 0.859577i \(0.670727\pi\)
\(464\) −5.88396e6 −1.26875
\(465\) −2.73791e6 −0.587200
\(466\) 2.08649e6 0.445093
\(467\) −1.03346e6 −0.219281 −0.109641 0.993971i \(-0.534970\pi\)
−0.109641 + 0.993971i \(0.534970\pi\)
\(468\) 2.17418e6 0.458861
\(469\) 1.16049e7 2.43618
\(470\) −1.75451e6 −0.366362
\(471\) 414607. 0.0861161
\(472\) −2.98499e6 −0.616720
\(473\) 531249. 0.109181
\(474\) 889689. 0.181883
\(475\) −689113. −0.140138
\(476\) −7.63628e6 −1.54477
\(477\) 2.03921e6 0.410360
\(478\) 2.13228e6 0.426849
\(479\) 1.13020e6 0.225069 0.112534 0.993648i \(-0.464103\pi\)
0.112534 + 0.993648i \(0.464103\pi\)
\(480\) −1.96413e6 −0.389105
\(481\) −4.54541e6 −0.895798
\(482\) −1.19781e6 −0.234838
\(483\) −1.01567e6 −0.198100
\(484\) 4.39993e6 0.853754
\(485\) −865238. −0.167025
\(486\) −76178.6 −0.0146299
\(487\) −793366. −0.151583 −0.0757916 0.997124i \(-0.524148\pi\)
−0.0757916 + 0.997124i \(0.524148\pi\)
\(488\) 2.12389e6 0.403721
\(489\) 1.69540e6 0.320626
\(490\) −2.18892e6 −0.411851
\(491\) −3.66618e6 −0.686294 −0.343147 0.939282i \(-0.611493\pi\)
−0.343147 + 0.939282i \(0.611493\pi\)
\(492\) −3.40804e6 −0.634735
\(493\) 8.00805e6 1.48392
\(494\) −2.13026e6 −0.392749
\(495\) 605829. 0.111132
\(496\) −4.46184e6 −0.814348
\(497\) −7.61267e6 −1.38244
\(498\) 1.12192e6 0.202717
\(499\) −5.08658e6 −0.914480 −0.457240 0.889343i \(-0.651162\pi\)
−0.457240 + 0.889343i \(0.651162\pi\)
\(500\) 4.94162e6 0.883983
\(501\) 1.82893e6 0.325538
\(502\) 1.17359e6 0.207853
\(503\) 6.49233e6 1.14414 0.572072 0.820203i \(-0.306140\pi\)
0.572072 + 0.820203i \(0.306140\pi\)
\(504\) −1.38962e6 −0.243680
\(505\) −2.22816e6 −0.388793
\(506\) −86350.3 −0.0149930
\(507\) 3.70460e6 0.640061
\(508\) −1.08553e6 −0.186630
\(509\) −5.58035e6 −0.954700 −0.477350 0.878713i \(-0.658403\pi\)
−0.477350 + 0.878713i \(0.658403\pi\)
\(510\) 809868. 0.137876
\(511\) −7.23991e6 −1.22654
\(512\) −5.43197e6 −0.915762
\(513\) −1.36045e6 −0.228239
\(514\) −67944.9 −0.0113435
\(515\) 7.40393e6 1.23011
\(516\) −1.14632e6 −0.189532
\(517\) 2.91102e6 0.478981
\(518\) 1.41381e6 0.231508
\(519\) 1.42246e6 0.231804
\(520\) −4.20622e6 −0.682156
\(521\) −4.02579e6 −0.649766 −0.324883 0.945754i \(-0.605325\pi\)
−0.324883 + 0.945754i \(0.605325\pi\)
\(522\) 709184. 0.113915
\(523\) −9.41059e6 −1.50440 −0.752198 0.658937i \(-0.771007\pi\)
−0.752198 + 0.658937i \(0.771007\pi\)
\(524\) −2.36981e6 −0.377038
\(525\) −708978. −0.112262
\(526\) −599588. −0.0944906
\(527\) 6.07256e6 0.952457
\(528\) 987293. 0.154121
\(529\) 279841. 0.0434783
\(530\) −1.91988e6 −0.296883
\(531\) −3.00657e6 −0.462738
\(532\) −1.20771e7 −1.85005
\(533\) −1.10450e7 −1.68402
\(534\) −349263. −0.0530029
\(535\) 1.65531e6 0.250031
\(536\) −4.37466e6 −0.657707
\(537\) 1.49800e6 0.224169
\(538\) −1.24935e6 −0.186092
\(539\) 3.63178e6 0.538453
\(540\) −1.30725e6 −0.192919
\(541\) −2.68147e6 −0.393894 −0.196947 0.980414i \(-0.563103\pi\)
−0.196947 + 0.980414i \(0.563103\pi\)
\(542\) −2.95768e6 −0.432467
\(543\) 1.67058e6 0.243146
\(544\) 4.35636e6 0.631141
\(545\) −3.79214e6 −0.546881
\(546\) −2.19167e6 −0.314625
\(547\) 8.74676e6 1.24991 0.624955 0.780660i \(-0.285117\pi\)
0.624955 + 0.780660i \(0.285117\pi\)
\(548\) 2.81566e6 0.400524
\(549\) 2.13924e6 0.302920
\(550\) −60275.8 −0.00849643
\(551\) 1.26651e7 1.77717
\(552\) 382873. 0.0534820
\(553\) 1.63467e7 2.27309
\(554\) −375357. −0.0519601
\(555\) 2.73298e6 0.376620
\(556\) 1.26535e7 1.73590
\(557\) 2.07905e6 0.283940 0.141970 0.989871i \(-0.454656\pi\)
0.141970 + 0.989871i \(0.454656\pi\)
\(558\) 537779. 0.0731170
\(559\) −3.71508e6 −0.502850
\(560\) −1.09332e7 −1.47326
\(561\) −1.34370e6 −0.180259
\(562\) 505687. 0.0675369
\(563\) 8.52157e6 1.13305 0.566524 0.824045i \(-0.308288\pi\)
0.566524 + 0.824045i \(0.308288\pi\)
\(564\) −6.28135e6 −0.831487
\(565\) −1.49121e6 −0.196524
\(566\) −945338. −0.124035
\(567\) −1.39967e6 −0.182838
\(568\) 2.86972e6 0.373223
\(569\) 6.58526e6 0.852692 0.426346 0.904560i \(-0.359801\pi\)
0.426346 + 0.904560i \(0.359801\pi\)
\(570\) 1.28084e6 0.165123
\(571\) −6.31614e6 −0.810702 −0.405351 0.914161i \(-0.632851\pi\)
−0.405351 + 0.914161i \(0.632851\pi\)
\(572\) 3.39624e6 0.434019
\(573\) −6.92760e6 −0.881447
\(574\) 3.43545e6 0.435216
\(575\) 195340. 0.0246389
\(576\) −1.86145e6 −0.233774
\(577\) 4.48677e6 0.561040 0.280520 0.959848i \(-0.409493\pi\)
0.280520 + 0.959848i \(0.409493\pi\)
\(578\) 35492.6 0.00441894
\(579\) −3.45730e6 −0.428589
\(580\) 1.21698e7 1.50216
\(581\) 2.06136e7 2.53346
\(582\) 169950. 0.0207976
\(583\) 3.18540e6 0.388144
\(584\) 2.72920e6 0.331134
\(585\) −4.23663e6 −0.511836
\(586\) −2.09752e6 −0.252326
\(587\) −7.42328e6 −0.889203 −0.444601 0.895729i \(-0.646655\pi\)
−0.444601 + 0.895729i \(0.646655\pi\)
\(588\) −7.83660e6 −0.934726
\(589\) 9.60402e6 1.14068
\(590\) 2.83064e6 0.334776
\(591\) 2.39443e6 0.281990
\(592\) 4.45381e6 0.522309
\(593\) 7.02182e6 0.819998 0.409999 0.912086i \(-0.365529\pi\)
0.409999 + 0.912086i \(0.365529\pi\)
\(594\) −118997. −0.0138379
\(595\) 1.48801e7 1.72311
\(596\) −1.35278e7 −1.55995
\(597\) −6.35463e6 −0.729717
\(598\) 603856. 0.0690527
\(599\) 5.02165e6 0.571846 0.285923 0.958253i \(-0.407700\pi\)
0.285923 + 0.958253i \(0.407700\pi\)
\(600\) 267261. 0.0303080
\(601\) 4.66921e6 0.527300 0.263650 0.964618i \(-0.415074\pi\)
0.263650 + 0.964618i \(0.415074\pi\)
\(602\) 1.15554e6 0.129955
\(603\) −4.40628e6 −0.493491
\(604\) 7.90116e6 0.881248
\(605\) −8.57374e6 −0.952318
\(606\) 437655. 0.0484117
\(607\) −1.48942e7 −1.64076 −0.820380 0.571819i \(-0.806238\pi\)
−0.820380 + 0.571819i \(0.806238\pi\)
\(608\) 6.88977e6 0.755868
\(609\) 1.30302e7 1.42366
\(610\) −2.01406e6 −0.219153
\(611\) −2.03570e7 −2.20603
\(612\) 2.89943e6 0.312920
\(613\) 1.09408e7 1.17598 0.587989 0.808869i \(-0.299920\pi\)
0.587989 + 0.808869i \(0.299920\pi\)
\(614\) −1.60721e6 −0.172048
\(615\) 6.64093e6 0.708013
\(616\) −2.17070e6 −0.230488
\(617\) −6.42199e6 −0.679136 −0.339568 0.940582i \(-0.610281\pi\)
−0.339568 + 0.940582i \(0.610281\pi\)
\(618\) −1.45428e6 −0.153171
\(619\) −3.05345e6 −0.320305 −0.160152 0.987092i \(-0.551199\pi\)
−0.160152 + 0.987092i \(0.551199\pi\)
\(620\) 9.22847e6 0.964163
\(621\) 385641. 0.0401286
\(622\) 4.21668e6 0.437014
\(623\) −6.41719e6 −0.662406
\(624\) −6.90424e6 −0.709830
\(625\) −1.07832e7 −1.10420
\(626\) −2.19523e6 −0.223895
\(627\) −2.12513e6 −0.215882
\(628\) −1.39749e6 −0.141400
\(629\) −6.06162e6 −0.610889
\(630\) 1.31777e6 0.132278
\(631\) 1.05012e7 1.04994 0.524971 0.851120i \(-0.324076\pi\)
0.524971 + 0.851120i \(0.324076\pi\)
\(632\) −6.16215e6 −0.613677
\(633\) −2.61572e6 −0.259467
\(634\) 1.17181e6 0.115780
\(635\) 2.11527e6 0.208176
\(636\) −6.87341e6 −0.673798
\(637\) −2.53974e7 −2.47994
\(638\) 1.10780e6 0.107748
\(639\) 2.89046e6 0.280037
\(640\) 8.73610e6 0.843078
\(641\) −1.57928e7 −1.51815 −0.759073 0.651006i \(-0.774347\pi\)
−0.759073 + 0.651006i \(0.774347\pi\)
\(642\) −325135. −0.0311333
\(643\) 551159. 0.0525714 0.0262857 0.999654i \(-0.491632\pi\)
0.0262857 + 0.999654i \(0.491632\pi\)
\(644\) 3.42345e6 0.325274
\(645\) 2.23373e6 0.211413
\(646\) −2.84085e6 −0.267835
\(647\) 1.12498e7 1.05653 0.528267 0.849078i \(-0.322842\pi\)
0.528267 + 0.849078i \(0.322842\pi\)
\(648\) 527627. 0.0493616
\(649\) −4.69649e6 −0.437685
\(650\) 421515. 0.0391318
\(651\) 9.88088e6 0.913783
\(652\) −5.71456e6 −0.526458
\(653\) −2.56046e6 −0.234982 −0.117491 0.993074i \(-0.537485\pi\)
−0.117491 + 0.993074i \(0.537485\pi\)
\(654\) 744850. 0.0680965
\(655\) 4.61783e6 0.420566
\(656\) 1.08224e7 0.981895
\(657\) 2.74893e6 0.248456
\(658\) 6.33188e6 0.570122
\(659\) −734143. −0.0658518 −0.0329259 0.999458i \(-0.510483\pi\)
−0.0329259 + 0.999458i \(0.510483\pi\)
\(660\) −2.04203e6 −0.182474
\(661\) 1.92354e7 1.71237 0.856185 0.516670i \(-0.172828\pi\)
0.856185 + 0.516670i \(0.172828\pi\)
\(662\) 2.76524e6 0.245238
\(663\) 9.39666e6 0.830213
\(664\) −7.77064e6 −0.683969
\(665\) 2.35336e7 2.06364
\(666\) −536810. −0.0468960
\(667\) −3.59012e6 −0.312460
\(668\) −6.16464e6 −0.534523
\(669\) −5.17423e6 −0.446972
\(670\) 4.14845e6 0.357025
\(671\) 3.34166e6 0.286520
\(672\) 7.08838e6 0.605514
\(673\) 1.20397e7 1.02465 0.512326 0.858791i \(-0.328784\pi\)
0.512326 + 0.858791i \(0.328784\pi\)
\(674\) −1.05465e6 −0.0894247
\(675\) 269192. 0.0227407
\(676\) −1.24868e7 −1.05096
\(677\) 99259.2 0.00832337 0.00416168 0.999991i \(-0.498675\pi\)
0.00416168 + 0.999991i \(0.498675\pi\)
\(678\) 292902. 0.0244708
\(679\) 3.12257e6 0.259919
\(680\) −5.60929e6 −0.465196
\(681\) −1.23626e7 −1.02151
\(682\) 840052. 0.0691585
\(683\) 9.22059e6 0.756323 0.378161 0.925740i \(-0.376556\pi\)
0.378161 + 0.925740i \(0.376556\pi\)
\(684\) 4.58557e6 0.374760
\(685\) −5.48661e6 −0.446763
\(686\) 3.27406e6 0.265630
\(687\) 7.83652e6 0.633478
\(688\) 3.64021e6 0.293194
\(689\) −2.22758e7 −1.78766
\(690\) −363075. −0.0290318
\(691\) −1.77887e7 −1.41726 −0.708629 0.705581i \(-0.750686\pi\)
−0.708629 + 0.705581i \(0.750686\pi\)
\(692\) −4.79459e6 −0.380615
\(693\) −2.18639e6 −0.172940
\(694\) −4.13096e6 −0.325576
\(695\) −2.46568e7 −1.93631
\(696\) −4.91194e6 −0.384353
\(697\) −1.47293e7 −1.14842
\(698\) 4.17881e6 0.324649
\(699\) −1.45558e7 −1.12679
\(700\) 2.38970e6 0.184331
\(701\) 1.60390e7 1.23277 0.616384 0.787446i \(-0.288597\pi\)
0.616384 + 0.787446i \(0.288597\pi\)
\(702\) 832157. 0.0637327
\(703\) −9.58673e6 −0.731614
\(704\) −2.90774e6 −0.221118
\(705\) 1.22399e7 0.927480
\(706\) 2.22767e6 0.168205
\(707\) 8.04125e6 0.605028
\(708\) 1.01340e7 0.759800
\(709\) −1.64673e7 −1.23029 −0.615145 0.788414i \(-0.710903\pi\)
−0.615145 + 0.788414i \(0.710903\pi\)
\(710\) −2.72132e6 −0.202598
\(711\) −6.20669e6 −0.460454
\(712\) 2.41906e6 0.178833
\(713\) −2.72241e6 −0.200553
\(714\) −2.92275e6 −0.214558
\(715\) −6.61794e6 −0.484125
\(716\) −5.04919e6 −0.368077
\(717\) −1.48753e7 −1.08061
\(718\) −4.02945e6 −0.291698
\(719\) 5.48538e6 0.395717 0.197859 0.980231i \(-0.436601\pi\)
0.197859 + 0.980231i \(0.436601\pi\)
\(720\) 4.15125e6 0.298433
\(721\) −2.67202e7 −1.91426
\(722\) −1.29854e6 −0.0927071
\(723\) 8.35619e6 0.594515
\(724\) −5.63090e6 −0.399237
\(725\) −2.50604e6 −0.177070
\(726\) 1.68405e6 0.118581
\(727\) −2.75484e7 −1.93313 −0.966563 0.256431i \(-0.917453\pi\)
−0.966563 + 0.256431i \(0.917453\pi\)
\(728\) 1.51799e7 1.06155
\(729\) 531441. 0.0370370
\(730\) −2.58807e6 −0.179750
\(731\) −4.95432e6 −0.342918
\(732\) −7.21058e6 −0.497385
\(733\) 1.51233e7 1.03965 0.519825 0.854273i \(-0.325997\pi\)
0.519825 + 0.854273i \(0.325997\pi\)
\(734\) −1.50980e6 −0.103438
\(735\) 1.52705e7 1.04264
\(736\) −1.95301e6 −0.132896
\(737\) −6.88295e6 −0.466773
\(738\) −1.30441e6 −0.0881604
\(739\) −1.56004e7 −1.05081 −0.525405 0.850852i \(-0.676086\pi\)
−0.525405 + 0.850852i \(0.676086\pi\)
\(740\) −9.21185e6 −0.618397
\(741\) 1.48612e7 0.994281
\(742\) 6.92870e6 0.462000
\(743\) 1.77279e6 0.117811 0.0589053 0.998264i \(-0.481239\pi\)
0.0589053 + 0.998264i \(0.481239\pi\)
\(744\) −3.72476e6 −0.246698
\(745\) 2.63603e7 1.74004
\(746\) 893892. 0.0588082
\(747\) −7.82681e6 −0.513196
\(748\) 4.52913e6 0.295979
\(749\) −5.97386e6 −0.389090
\(750\) 1.89138e6 0.122779
\(751\) 5.07089e6 0.328083 0.164042 0.986453i \(-0.447547\pi\)
0.164042 + 0.986453i \(0.447547\pi\)
\(752\) 1.99468e7 1.28626
\(753\) −8.18724e6 −0.526199
\(754\) −7.74696e6 −0.496253
\(755\) −1.53962e7 −0.982987
\(756\) 4.71776e6 0.300214
\(757\) −733561. −0.0465261 −0.0232631 0.999729i \(-0.507406\pi\)
−0.0232631 + 0.999729i \(0.507406\pi\)
\(758\) −2.73709e6 −0.173028
\(759\) 602401. 0.0379561
\(760\) −8.87135e6 −0.557129
\(761\) 2.90077e6 0.181573 0.0907866 0.995870i \(-0.471062\pi\)
0.0907866 + 0.995870i \(0.471062\pi\)
\(762\) −415481. −0.0259217
\(763\) 1.36855e7 0.851039
\(764\) 2.33504e7 1.44731
\(765\) −5.64984e6 −0.349046
\(766\) 370358. 0.0228060
\(767\) 3.28431e7 2.01584
\(768\) 4.90256e6 0.299930
\(769\) −6.02882e6 −0.367635 −0.183817 0.982960i \(-0.558846\pi\)
−0.183817 + 0.982960i \(0.558846\pi\)
\(770\) 2.05845e6 0.125116
\(771\) 474001. 0.0287173
\(772\) 1.16533e7 0.703729
\(773\) 1.48827e7 0.895844 0.447922 0.894073i \(-0.352164\pi\)
0.447922 + 0.894073i \(0.352164\pi\)
\(774\) −438749. −0.0263247
\(775\) −1.90035e6 −0.113653
\(776\) −1.17710e6 −0.0701715
\(777\) −9.86309e6 −0.586085
\(778\) −877019. −0.0519469
\(779\) −2.32951e7 −1.37537
\(780\) 1.42801e7 0.840417
\(781\) 4.51512e6 0.264876
\(782\) 805285. 0.0470904
\(783\) −4.94745e6 −0.288388
\(784\) 2.48856e7 1.44596
\(785\) 2.72315e6 0.157724
\(786\) −907033. −0.0523681
\(787\) 2.19846e7 1.26526 0.632631 0.774453i \(-0.281975\pi\)
0.632631 + 0.774453i \(0.281975\pi\)
\(788\) −8.07075e6 −0.463018
\(789\) 4.18288e6 0.239212
\(790\) 5.84350e6 0.333124
\(791\) 5.38164e6 0.305825
\(792\) 824194. 0.0466892
\(793\) −2.33685e7 −1.31962
\(794\) 6.96542e6 0.392099
\(795\) 1.33936e7 0.751586
\(796\) 2.14191e7 1.19817
\(797\) −2.92578e6 −0.163153 −0.0815765 0.996667i \(-0.525995\pi\)
−0.0815765 + 0.996667i \(0.525995\pi\)
\(798\) −4.62245e6 −0.256960
\(799\) −2.71476e7 −1.50440
\(800\) −1.36328e6 −0.0753113
\(801\) 2.43655e6 0.134182
\(802\) 2.80993e6 0.154262
\(803\) 4.29404e6 0.235005
\(804\) 1.48519e7 0.810295
\(805\) −6.67096e6 −0.362826
\(806\) −5.87457e6 −0.318521
\(807\) 8.71578e6 0.471110
\(808\) −3.03128e6 −0.163342
\(809\) −2.36363e6 −0.126972 −0.0634860 0.997983i \(-0.520222\pi\)
−0.0634860 + 0.997983i \(0.520222\pi\)
\(810\) −500343. −0.0267951
\(811\) −1.40226e7 −0.748648 −0.374324 0.927298i \(-0.622125\pi\)
−0.374324 + 0.927298i \(0.622125\pi\)
\(812\) −4.39200e7 −2.33761
\(813\) 2.06335e7 1.09483
\(814\) −838540. −0.0443570
\(815\) 1.11354e7 0.587236
\(816\) −9.20729e6 −0.484068
\(817\) −7.83548e6 −0.410686
\(818\) 460004. 0.0240369
\(819\) 1.52896e7 0.796503
\(820\) −2.23841e7 −1.16253
\(821\) 3.57170e7 1.84934 0.924670 0.380769i \(-0.124341\pi\)
0.924670 + 0.380769i \(0.124341\pi\)
\(822\) 1.07768e6 0.0556300
\(823\) −1.15100e7 −0.592344 −0.296172 0.955135i \(-0.595710\pi\)
−0.296172 + 0.955135i \(0.595710\pi\)
\(824\) 1.00726e7 0.516801
\(825\) 420499. 0.0215095
\(826\) −1.02155e7 −0.520968
\(827\) −1.24864e7 −0.634854 −0.317427 0.948283i \(-0.602819\pi\)
−0.317427 + 0.948283i \(0.602819\pi\)
\(828\) −1.29985e6 −0.0658898
\(829\) −1.13398e7 −0.573087 −0.286544 0.958067i \(-0.592506\pi\)
−0.286544 + 0.958067i \(0.592506\pi\)
\(830\) 7.36882e6 0.371281
\(831\) 2.61858e6 0.131542
\(832\) 2.03341e7 1.01840
\(833\) −3.38692e7 −1.69119
\(834\) 4.84308e6 0.241105
\(835\) 1.20125e7 0.596232
\(836\) 7.16302e6 0.354471
\(837\) −3.75168e6 −0.185102
\(838\) −4.93633e6 −0.242825
\(839\) 2.91430e7 1.42932 0.714659 0.699473i \(-0.246582\pi\)
0.714659 + 0.699473i \(0.246582\pi\)
\(840\) −9.12709e6 −0.446307
\(841\) 2.55471e7 1.24552
\(842\) 3.88078e6 0.188642
\(843\) −3.52780e6 −0.170976
\(844\) 8.81663e6 0.426036
\(845\) 2.43319e7 1.17229
\(846\) −2.40416e6 −0.115488
\(847\) 3.09419e7 1.48197
\(848\) 2.18269e7 1.04232
\(849\) 6.59491e6 0.314007
\(850\) 562120. 0.0266859
\(851\) 2.71751e6 0.128632
\(852\) −9.74267e6 −0.459811
\(853\) −9.77420e6 −0.459948 −0.229974 0.973197i \(-0.573864\pi\)
−0.229974 + 0.973197i \(0.573864\pi\)
\(854\) 7.26858e6 0.341040
\(855\) −8.93548e6 −0.418025
\(856\) 2.25194e6 0.105044
\(857\) −2.33262e7 −1.08491 −0.542453 0.840086i \(-0.682504\pi\)
−0.542453 + 0.840086i \(0.682504\pi\)
\(858\) 1.29989e6 0.0602823
\(859\) −1.14340e7 −0.528708 −0.264354 0.964426i \(-0.585159\pi\)
−0.264354 + 0.964426i \(0.585159\pi\)
\(860\) −7.52908e6 −0.347133
\(861\) −2.39666e7 −1.10179
\(862\) −745426. −0.0341693
\(863\) −7.70458e6 −0.352145 −0.176073 0.984377i \(-0.556339\pi\)
−0.176073 + 0.984377i \(0.556339\pi\)
\(864\) −2.69139e6 −0.122657
\(865\) 9.34277e6 0.424556
\(866\) −9.31329e6 −0.421996
\(867\) −247605. −0.0111870
\(868\) −3.33048e7 −1.50040
\(869\) −9.69534e6 −0.435525
\(870\) 4.65794e6 0.208639
\(871\) 4.81332e7 2.14981
\(872\) −5.15897e6 −0.229759
\(873\) −1.18561e6 −0.0526511
\(874\) 1.27359e6 0.0563965
\(875\) 3.47513e7 1.53444
\(876\) −9.26562e6 −0.407957
\(877\) 1.42476e7 0.625520 0.312760 0.949832i \(-0.398746\pi\)
0.312760 + 0.949832i \(0.398746\pi\)
\(878\) 981676. 0.0429766
\(879\) 1.46328e7 0.638788
\(880\) 6.48457e6 0.282276
\(881\) −1.76114e7 −0.764459 −0.382230 0.924067i \(-0.624844\pi\)
−0.382230 + 0.924067i \(0.624844\pi\)
\(882\) −2.99942e6 −0.129827
\(883\) −9.40461e6 −0.405919 −0.202959 0.979187i \(-0.565056\pi\)
−0.202959 + 0.979187i \(0.565056\pi\)
\(884\) −3.16726e7 −1.36318
\(885\) −1.97472e7 −0.847517
\(886\) 8.99054e6 0.384770
\(887\) 1.38034e7 0.589086 0.294543 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(888\) 3.71805e6 0.158228
\(889\) −7.63383e6 −0.323958
\(890\) −2.29397e6 −0.0970762
\(891\) 830152. 0.0350319
\(892\) 1.74404e7 0.733913
\(893\) −4.29351e7 −1.80170
\(894\) −5.17768e6 −0.216666
\(895\) 9.83888e6 0.410571
\(896\) −3.15279e7 −1.31197
\(897\) −4.21265e6 −0.174813
\(898\) −7.08127e6 −0.293036
\(899\) 3.49262e7 1.44129
\(900\) −907348. −0.0373394
\(901\) −2.97064e7 −1.21910
\(902\) −2.03759e6 −0.0833874
\(903\) −8.06135e6 −0.328994
\(904\) −2.02870e6 −0.0825650
\(905\) 1.09724e7 0.445328
\(906\) 3.02413e6 0.122399
\(907\) 802157. 0.0323773 0.0161887 0.999869i \(-0.494847\pi\)
0.0161887 + 0.999869i \(0.494847\pi\)
\(908\) 4.16699e7 1.67729
\(909\) −3.05319e6 −0.122559
\(910\) −1.43950e7 −0.576245
\(911\) 4.80457e6 0.191804 0.0959022 0.995391i \(-0.469426\pi\)
0.0959022 + 0.995391i \(0.469426\pi\)
\(912\) −1.45617e7 −0.579731
\(913\) −1.22261e7 −0.485412
\(914\) 3.22482e6 0.127685
\(915\) 1.40506e7 0.554807
\(916\) −2.64140e7 −1.04015
\(917\) −1.66654e7 −0.654473
\(918\) 1.10974e6 0.0434625
\(919\) 2.84366e7 1.11068 0.555339 0.831624i \(-0.312588\pi\)
0.555339 + 0.831624i \(0.312588\pi\)
\(920\) 2.51472e6 0.0979537
\(921\) 1.12123e7 0.435556
\(922\) 1.04541e7 0.405005
\(923\) −3.15747e7 −1.21993
\(924\) 7.36951e6 0.283961
\(925\) 1.89693e6 0.0728948
\(926\) 6.08176e6 0.233078
\(927\) 1.01454e7 0.387766
\(928\) 2.50555e7 0.955066
\(929\) 5.06040e6 0.192374 0.0961868 0.995363i \(-0.469335\pi\)
0.0961868 + 0.995363i \(0.469335\pi\)
\(930\) 3.53215e6 0.133916
\(931\) −5.35657e7 −2.02541
\(932\) 4.90624e7 1.85016
\(933\) −2.94166e7 −1.10634
\(934\) 1.33326e6 0.0500089
\(935\) −8.82549e6 −0.330149
\(936\) −5.76367e6 −0.215035
\(937\) 1.27197e7 0.473291 0.236646 0.971596i \(-0.423952\pi\)
0.236646 + 0.971596i \(0.423952\pi\)
\(938\) −1.49714e7 −0.555591
\(939\) 1.53145e7 0.566810
\(940\) −4.12561e7 −1.52289
\(941\) −1.66122e7 −0.611581 −0.305790 0.952099i \(-0.598921\pi\)
−0.305790 + 0.952099i \(0.598921\pi\)
\(942\) −534881. −0.0196395
\(943\) 6.60335e6 0.241816
\(944\) −3.21812e7 −1.17536
\(945\) −9.19306e6 −0.334873
\(946\) −685360. −0.0248995
\(947\) 4.77751e7 1.73112 0.865558 0.500808i \(-0.166964\pi\)
0.865558 + 0.500808i \(0.166964\pi\)
\(948\) 2.09205e7 0.756050
\(949\) −3.00286e7 −1.08236
\(950\) 889018. 0.0319596
\(951\) −8.17485e6 −0.293109
\(952\) 2.02435e7 0.723924
\(953\) −3.77139e7 −1.34514 −0.672572 0.740032i \(-0.734810\pi\)
−0.672572 + 0.740032i \(0.734810\pi\)
\(954\) −2.63076e6 −0.0935860
\(955\) −4.55007e7 −1.61439
\(956\) 5.01391e7 1.77432
\(957\) −7.72829e6 −0.272775
\(958\) −1.45806e6 −0.0513287
\(959\) 1.98007e7 0.695239
\(960\) −1.22261e7 −0.428163
\(961\) −2.14436e6 −0.0749014
\(962\) 5.86399e6 0.204294
\(963\) 2.26822e6 0.0788169
\(964\) −2.81656e7 −0.976173
\(965\) −2.27077e7 −0.784973
\(966\) 1.31031e6 0.0451784
\(967\) −1.36407e7 −0.469105 −0.234553 0.972103i \(-0.575362\pi\)
−0.234553 + 0.972103i \(0.575362\pi\)
\(968\) −1.16641e7 −0.400093
\(969\) 1.98185e7 0.678050
\(970\) 1.11624e6 0.0380914
\(971\) −3.96602e7 −1.34992 −0.674958 0.737856i \(-0.735838\pi\)
−0.674958 + 0.737856i \(0.735838\pi\)
\(972\) −1.79129e6 −0.0608136
\(973\) 8.89843e7 3.01322
\(974\) 1.02351e6 0.0345698
\(975\) −2.94059e6 −0.0990658
\(976\) 2.28976e7 0.769424
\(977\) −9.70600e6 −0.325315 −0.162657 0.986683i \(-0.552007\pi\)
−0.162657 + 0.986683i \(0.552007\pi\)
\(978\) −2.18722e6 −0.0731214
\(979\) 3.80608e6 0.126917
\(980\) −5.14711e7 −1.71198
\(981\) −5.19626e6 −0.172393
\(982\) 4.72971e6 0.156515
\(983\) −2.88398e7 −0.951937 −0.475969 0.879462i \(-0.657902\pi\)
−0.475969 + 0.879462i \(0.657902\pi\)
\(984\) 9.03459e6 0.297455
\(985\) 1.57267e7 0.516473
\(986\) −1.03311e7 −0.338419
\(987\) −4.41728e7 −1.44332
\(988\) −5.00917e7 −1.63258
\(989\) 2.22109e6 0.0722064
\(990\) −781575. −0.0253444
\(991\) 3.94595e7 1.27634 0.638172 0.769894i \(-0.279691\pi\)
0.638172 + 0.769894i \(0.279691\pi\)
\(992\) 1.89998e7 0.613012
\(993\) −1.92910e7 −0.620844
\(994\) 9.82104e6 0.315276
\(995\) −4.17374e7 −1.33650
\(996\) 2.63813e7 0.842650
\(997\) 3.61342e7 1.15128 0.575639 0.817704i \(-0.304753\pi\)
0.575639 + 0.817704i \(0.304753\pi\)
\(998\) 6.56215e6 0.208555
\(999\) 3.74492e6 0.118722
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 69.6.a.b.1.2 3
3.2 odd 2 207.6.a.c.1.2 3
4.3 odd 2 1104.6.a.i.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.b.1.2 3 1.1 even 1 trivial
207.6.a.c.1.2 3 3.2 odd 2
1104.6.a.i.1.3 3 4.3 odd 2