Properties

Label 230.6.a.f.1.3
Level $230$
Weight $6$
Character 230.1
Self dual yes
Analytic conductor $36.888$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.8882785570\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 774 x^{3} - 197 x^{2} + 66287 x + 154128\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.48374\) of defining polynomial
Character \(\chi\) \(=\) 230.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} -2.48374 q^{3} +16.0000 q^{4} +25.0000 q^{5} +9.93494 q^{6} -252.275 q^{7} -64.0000 q^{8} -236.831 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -2.48374 q^{3} +16.0000 q^{4} +25.0000 q^{5} +9.93494 q^{6} -252.275 q^{7} -64.0000 q^{8} -236.831 q^{9} -100.000 q^{10} -739.487 q^{11} -39.7398 q^{12} +876.961 q^{13} +1009.10 q^{14} -62.0934 q^{15} +256.000 q^{16} -1363.74 q^{17} +947.324 q^{18} +1725.29 q^{19} +400.000 q^{20} +626.584 q^{21} +2957.95 q^{22} -529.000 q^{23} +158.959 q^{24} +625.000 q^{25} -3507.84 q^{26} +1191.77 q^{27} -4036.39 q^{28} +3068.97 q^{29} +248.374 q^{30} -8425.04 q^{31} -1024.00 q^{32} +1836.69 q^{33} +5454.97 q^{34} -6306.87 q^{35} -3789.30 q^{36} -4041.30 q^{37} -6901.15 q^{38} -2178.14 q^{39} -1600.00 q^{40} -1633.86 q^{41} -2506.33 q^{42} +8735.57 q^{43} -11831.8 q^{44} -5920.78 q^{45} +2116.00 q^{46} +18989.0 q^{47} -635.836 q^{48} +46835.5 q^{49} -2500.00 q^{50} +3387.18 q^{51} +14031.4 q^{52} +18955.0 q^{53} -4767.09 q^{54} -18487.2 q^{55} +16145.6 q^{56} -4285.16 q^{57} -12275.9 q^{58} -49726.2 q^{59} -993.494 q^{60} +19031.8 q^{61} +33700.1 q^{62} +59746.5 q^{63} +4096.00 q^{64} +21924.0 q^{65} -7346.76 q^{66} -9397.24 q^{67} -21819.9 q^{68} +1313.90 q^{69} +25227.5 q^{70} -28928.8 q^{71} +15157.2 q^{72} +2535.03 q^{73} +16165.2 q^{74} -1552.33 q^{75} +27604.6 q^{76} +186554. q^{77} +8712.56 q^{78} -17637.4 q^{79} +6400.00 q^{80} +54589.9 q^{81} +6535.44 q^{82} +87609.8 q^{83} +10025.3 q^{84} -34093.6 q^{85} -34942.3 q^{86} -7622.51 q^{87} +47327.2 q^{88} +33219.0 q^{89} +23683.1 q^{90} -221235. q^{91} -8464.00 q^{92} +20925.6 q^{93} -75956.0 q^{94} +43132.2 q^{95} +2543.35 q^{96} +41331.2 q^{97} -187342. q^{98} +175133. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 20 q^{2} + q^{3} + 80 q^{4} + 125 q^{5} - 4 q^{6} + 102 q^{7} - 320 q^{8} + 334 q^{9} + O(q^{10}) \) \( 5 q - 20 q^{2} + q^{3} + 80 q^{4} + 125 q^{5} - 4 q^{6} + 102 q^{7} - 320 q^{8} + 334 q^{9} - 500 q^{10} + 251 q^{11} + 16 q^{12} + 1743 q^{13} - 408 q^{14} + 25 q^{15} + 1280 q^{16} + 1944 q^{17} - 1336 q^{18} - 845 q^{19} + 2000 q^{20} + 4682 q^{21} - 1004 q^{22} - 2645 q^{23} - 64 q^{24} + 3125 q^{25} - 6972 q^{26} + 2428 q^{27} + 1632 q^{28} - 4021 q^{29} - 100 q^{30} - 15752 q^{31} - 5120 q^{32} + 2931 q^{33} - 7776 q^{34} + 2550 q^{35} + 5344 q^{36} - 3455 q^{37} + 3380 q^{38} - 16708 q^{39} - 8000 q^{40} - 11898 q^{41} - 18728 q^{42} + 6968 q^{43} + 4016 q^{44} + 8350 q^{45} + 10580 q^{46} + 13412 q^{47} + 256 q^{48} + 91041 q^{49} - 12500 q^{50} - 2115 q^{51} + 27888 q^{52} + 53029 q^{53} - 9712 q^{54} + 6275 q^{55} - 6528 q^{56} - 21730 q^{57} + 16084 q^{58} - 31223 q^{59} + 400 q^{60} + 71477 q^{61} + 63008 q^{62} + 262199 q^{63} + 20480 q^{64} + 43575 q^{65} - 11724 q^{66} + 76003 q^{67} + 31104 q^{68} - 529 q^{69} - 10200 q^{70} + 54418 q^{71} - 21376 q^{72} + 69418 q^{73} + 13820 q^{74} + 625 q^{75} - 13520 q^{76} + 283598 q^{77} + 66832 q^{78} + 105024 q^{79} + 32000 q^{80} + 102913 q^{81} + 47592 q^{82} + 89399 q^{83} + 74912 q^{84} + 48600 q^{85} - 27872 q^{86} + 276726 q^{87} - 16064 q^{88} + 96240 q^{89} - 33400 q^{90} + 59261 q^{91} - 42320 q^{92} + 84434 q^{93} - 53648 q^{94} - 21125 q^{95} - 1024 q^{96} + 216087 q^{97} - 364164 q^{98} + 386925 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) −2.48374 −0.159332 −0.0796659 0.996822i \(-0.525385\pi\)
−0.0796659 + 0.996822i \(0.525385\pi\)
\(4\) 16.0000 0.500000
\(5\) 25.0000 0.447214
\(6\) 9.93494 0.112665
\(7\) −252.275 −1.94594 −0.972968 0.230940i \(-0.925820\pi\)
−0.972968 + 0.230940i \(0.925820\pi\)
\(8\) −64.0000 −0.353553
\(9\) −236.831 −0.974613
\(10\) −100.000 −0.316228
\(11\) −739.487 −1.84268 −0.921338 0.388763i \(-0.872902\pi\)
−0.921338 + 0.388763i \(0.872902\pi\)
\(12\) −39.7398 −0.0796659
\(13\) 876.961 1.43920 0.719601 0.694387i \(-0.244325\pi\)
0.719601 + 0.694387i \(0.244325\pi\)
\(14\) 1009.10 1.37598
\(15\) −62.0934 −0.0712553
\(16\) 256.000 0.250000
\(17\) −1363.74 −1.14449 −0.572243 0.820084i \(-0.693926\pi\)
−0.572243 + 0.820084i \(0.693926\pi\)
\(18\) 947.324 0.689156
\(19\) 1725.29 1.09642 0.548211 0.836340i \(-0.315309\pi\)
0.548211 + 0.836340i \(0.315309\pi\)
\(20\) 400.000 0.223607
\(21\) 626.584 0.310049
\(22\) 2957.95 1.30297
\(23\) −529.000 −0.208514
\(24\) 158.959 0.0563323
\(25\) 625.000 0.200000
\(26\) −3507.84 −1.01767
\(27\) 1191.77 0.314619
\(28\) −4036.39 −0.972968
\(29\) 3068.97 0.677638 0.338819 0.940852i \(-0.389973\pi\)
0.338819 + 0.940852i \(0.389973\pi\)
\(30\) 248.374 0.0503851
\(31\) −8425.04 −1.57459 −0.787295 0.616576i \(-0.788519\pi\)
−0.787295 + 0.616576i \(0.788519\pi\)
\(32\) −1024.00 −0.176777
\(33\) 1836.69 0.293597
\(34\) 5454.97 0.809273
\(35\) −6306.87 −0.870249
\(36\) −3789.30 −0.487307
\(37\) −4041.30 −0.485308 −0.242654 0.970113i \(-0.578018\pi\)
−0.242654 + 0.970113i \(0.578018\pi\)
\(38\) −6901.15 −0.775287
\(39\) −2178.14 −0.229311
\(40\) −1600.00 −0.158114
\(41\) −1633.86 −0.151794 −0.0758971 0.997116i \(-0.524182\pi\)
−0.0758971 + 0.997116i \(0.524182\pi\)
\(42\) −2506.33 −0.219238
\(43\) 8735.57 0.720477 0.360238 0.932860i \(-0.382695\pi\)
0.360238 + 0.932860i \(0.382695\pi\)
\(44\) −11831.8 −0.921338
\(45\) −5920.78 −0.435860
\(46\) 2116.00 0.147442
\(47\) 18989.0 1.25388 0.626942 0.779066i \(-0.284306\pi\)
0.626942 + 0.779066i \(0.284306\pi\)
\(48\) −635.836 −0.0398329
\(49\) 46835.5 2.78667
\(50\) −2500.00 −0.141421
\(51\) 3387.18 0.182353
\(52\) 14031.4 0.719601
\(53\) 18955.0 0.926903 0.463452 0.886122i \(-0.346611\pi\)
0.463452 + 0.886122i \(0.346611\pi\)
\(54\) −4767.09 −0.222469
\(55\) −18487.2 −0.824069
\(56\) 16145.6 0.687992
\(57\) −4285.16 −0.174695
\(58\) −12275.9 −0.479162
\(59\) −49726.2 −1.85975 −0.929876 0.367873i \(-0.880086\pi\)
−0.929876 + 0.367873i \(0.880086\pi\)
\(60\) −993.494 −0.0356277
\(61\) 19031.8 0.654871 0.327436 0.944873i \(-0.393816\pi\)
0.327436 + 0.944873i \(0.393816\pi\)
\(62\) 33700.1 1.11340
\(63\) 59746.5 1.89654
\(64\) 4096.00 0.125000
\(65\) 21924.0 0.643631
\(66\) −7346.76 −0.207604
\(67\) −9397.24 −0.255749 −0.127874 0.991790i \(-0.540815\pi\)
−0.127874 + 0.991790i \(0.540815\pi\)
\(68\) −21819.9 −0.572243
\(69\) 1313.90 0.0332230
\(70\) 25227.5 0.615359
\(71\) −28928.8 −0.681058 −0.340529 0.940234i \(-0.610606\pi\)
−0.340529 + 0.940234i \(0.610606\pi\)
\(72\) 15157.2 0.344578
\(73\) 2535.03 0.0556770 0.0278385 0.999612i \(-0.491138\pi\)
0.0278385 + 0.999612i \(0.491138\pi\)
\(74\) 16165.2 0.343164
\(75\) −1552.33 −0.0318663
\(76\) 27604.6 0.548211
\(77\) 186554. 3.58573
\(78\) 8712.56 0.162147
\(79\) −17637.4 −0.317956 −0.158978 0.987282i \(-0.550820\pi\)
−0.158978 + 0.987282i \(0.550820\pi\)
\(80\) 6400.00 0.111803
\(81\) 54589.9 0.924485
\(82\) 6535.44 0.107335
\(83\) 87609.8 1.39591 0.697955 0.716142i \(-0.254094\pi\)
0.697955 + 0.716142i \(0.254094\pi\)
\(84\) 10025.3 0.155025
\(85\) −34093.6 −0.511829
\(86\) −34942.3 −0.509454
\(87\) −7622.51 −0.107969
\(88\) 47327.2 0.651484
\(89\) 33219.0 0.444541 0.222270 0.974985i \(-0.428653\pi\)
0.222270 + 0.974985i \(0.428653\pi\)
\(90\) 23683.1 0.308200
\(91\) −221235. −2.80060
\(92\) −8464.00 −0.104257
\(93\) 20925.6 0.250882
\(94\) −75956.0 −0.886630
\(95\) 43132.2 0.490334
\(96\) 2543.35 0.0281661
\(97\) 41331.2 0.446014 0.223007 0.974817i \(-0.428413\pi\)
0.223007 + 0.974817i \(0.428413\pi\)
\(98\) −187342. −1.97047
\(99\) 175133. 1.79590
\(100\) 10000.0 0.100000
\(101\) −51868.0 −0.505936 −0.252968 0.967475i \(-0.581407\pi\)
−0.252968 + 0.967475i \(0.581407\pi\)
\(102\) −13548.7 −0.128943
\(103\) 54330.3 0.504602 0.252301 0.967649i \(-0.418813\pi\)
0.252301 + 0.967649i \(0.418813\pi\)
\(104\) −56125.5 −0.508835
\(105\) 15664.6 0.138658
\(106\) −75820.1 −0.655420
\(107\) −106154. −0.896353 −0.448176 0.893945i \(-0.647926\pi\)
−0.448176 + 0.893945i \(0.647926\pi\)
\(108\) 19068.4 0.157309
\(109\) −173904. −1.40198 −0.700992 0.713169i \(-0.747259\pi\)
−0.700992 + 0.713169i \(0.747259\pi\)
\(110\) 73948.7 0.582705
\(111\) 10037.5 0.0773249
\(112\) −64582.3 −0.486484
\(113\) 262107. 1.93100 0.965501 0.260399i \(-0.0838541\pi\)
0.965501 + 0.260399i \(0.0838541\pi\)
\(114\) 17140.6 0.123528
\(115\) −13225.0 −0.0932505
\(116\) 49103.5 0.338819
\(117\) −207692. −1.40267
\(118\) 198905. 1.31504
\(119\) 344038. 2.22710
\(120\) 3973.98 0.0251926
\(121\) 385790. 2.39545
\(122\) −76127.3 −0.463064
\(123\) 4058.08 0.0241856
\(124\) −134801. −0.787295
\(125\) 15625.0 0.0894427
\(126\) −238986. −1.34105
\(127\) 227525. 1.25176 0.625878 0.779921i \(-0.284741\pi\)
0.625878 + 0.779921i \(0.284741\pi\)
\(128\) −16384.0 −0.0883883
\(129\) −21696.8 −0.114795
\(130\) −87696.1 −0.455116
\(131\) 205957. 1.04857 0.524286 0.851542i \(-0.324332\pi\)
0.524286 + 0.851542i \(0.324332\pi\)
\(132\) 29387.0 0.146798
\(133\) −435246. −2.13357
\(134\) 37589.0 0.180842
\(135\) 29794.3 0.140702
\(136\) 87279.6 0.404637
\(137\) −320648. −1.45958 −0.729788 0.683673i \(-0.760381\pi\)
−0.729788 + 0.683673i \(0.760381\pi\)
\(138\) −5255.58 −0.0234922
\(139\) 194736. 0.854887 0.427444 0.904042i \(-0.359414\pi\)
0.427444 + 0.904042i \(0.359414\pi\)
\(140\) −100910. −0.435125
\(141\) −47163.7 −0.199784
\(142\) 115715. 0.481581
\(143\) −648501. −2.65198
\(144\) −60628.8 −0.243653
\(145\) 76724.2 0.303049
\(146\) −10140.1 −0.0393696
\(147\) −116327. −0.444004
\(148\) −64660.8 −0.242654
\(149\) −393030. −1.45031 −0.725155 0.688586i \(-0.758232\pi\)
−0.725155 + 0.688586i \(0.758232\pi\)
\(150\) 6209.34 0.0225329
\(151\) −5358.24 −0.0191240 −0.00956202 0.999954i \(-0.503044\pi\)
−0.00956202 + 0.999954i \(0.503044\pi\)
\(152\) −110418. −0.387643
\(153\) 322977. 1.11543
\(154\) −746215. −2.53549
\(155\) −210626. −0.704178
\(156\) −34850.2 −0.114655
\(157\) 290851. 0.941721 0.470860 0.882208i \(-0.343944\pi\)
0.470860 + 0.882208i \(0.343944\pi\)
\(158\) 70549.7 0.224829
\(159\) −47079.3 −0.147685
\(160\) −25600.0 −0.0790569
\(161\) 133453. 0.405756
\(162\) −218360. −0.653709
\(163\) −163365. −0.481604 −0.240802 0.970574i \(-0.577410\pi\)
−0.240802 + 0.970574i \(0.577410\pi\)
\(164\) −26141.8 −0.0758971
\(165\) 45917.3 0.131300
\(166\) −350439. −0.987058
\(167\) −264211. −0.733094 −0.366547 0.930400i \(-0.619460\pi\)
−0.366547 + 0.930400i \(0.619460\pi\)
\(168\) −40101.4 −0.109619
\(169\) 397768. 1.07130
\(170\) 136374. 0.361918
\(171\) −408602. −1.06859
\(172\) 139769. 0.360238
\(173\) 341880. 0.868478 0.434239 0.900798i \(-0.357017\pi\)
0.434239 + 0.900798i \(0.357017\pi\)
\(174\) 30490.0 0.0763457
\(175\) −157672. −0.389187
\(176\) −189309. −0.460669
\(177\) 123507. 0.296317
\(178\) −132876. −0.314338
\(179\) 311766. 0.727271 0.363636 0.931541i \(-0.381535\pi\)
0.363636 + 0.931541i \(0.381535\pi\)
\(180\) −94732.4 −0.217930
\(181\) −263189. −0.597132 −0.298566 0.954389i \(-0.596508\pi\)
−0.298566 + 0.954389i \(0.596508\pi\)
\(182\) 884940. 1.98032
\(183\) −47270.0 −0.104342
\(184\) 33856.0 0.0737210
\(185\) −101033. −0.217036
\(186\) −83702.2 −0.177400
\(187\) 1.00847e6 2.10891
\(188\) 303824. 0.626942
\(189\) −300654. −0.612228
\(190\) −172529. −0.346719
\(191\) −755404. −1.49829 −0.749145 0.662406i \(-0.769535\pi\)
−0.749145 + 0.662406i \(0.769535\pi\)
\(192\) −10173.4 −0.0199165
\(193\) −773822. −1.49537 −0.747683 0.664056i \(-0.768834\pi\)
−0.747683 + 0.664056i \(0.768834\pi\)
\(194\) −165325. −0.315379
\(195\) −54453.5 −0.102551
\(196\) 749368. 1.39333
\(197\) 465100. 0.853848 0.426924 0.904288i \(-0.359597\pi\)
0.426924 + 0.904288i \(0.359597\pi\)
\(198\) −700534. −1.26989
\(199\) −601491. −1.07671 −0.538353 0.842720i \(-0.680953\pi\)
−0.538353 + 0.842720i \(0.680953\pi\)
\(200\) −40000.0 −0.0707107
\(201\) 23340.3 0.0407489
\(202\) 207472. 0.357751
\(203\) −774223. −1.31864
\(204\) 54194.8 0.0911764
\(205\) −40846.5 −0.0678844
\(206\) −217321. −0.356807
\(207\) 125284. 0.203221
\(208\) 224502. 0.359801
\(209\) −1.27583e6 −2.02035
\(210\) −62658.4 −0.0980462
\(211\) −6024.40 −0.00931553 −0.00465777 0.999989i \(-0.501483\pi\)
−0.00465777 + 0.999989i \(0.501483\pi\)
\(212\) 303280. 0.463452
\(213\) 71851.4 0.108514
\(214\) 424618. 0.633817
\(215\) 218389. 0.322207
\(216\) −76273.5 −0.111234
\(217\) 2.12542e6 3.06405
\(218\) 695615. 0.991352
\(219\) −6296.34 −0.00887111
\(220\) −295795. −0.412035
\(221\) −1.19595e6 −1.64715
\(222\) −40150.1 −0.0546770
\(223\) 1.29198e6 1.73978 0.869891 0.493244i \(-0.164189\pi\)
0.869891 + 0.493244i \(0.164189\pi\)
\(224\) 258329. 0.343996
\(225\) −148019. −0.194923
\(226\) −1.04843e6 −1.36542
\(227\) 64294.3 0.0828148 0.0414074 0.999142i \(-0.486816\pi\)
0.0414074 + 0.999142i \(0.486816\pi\)
\(228\) −68562.5 −0.0873473
\(229\) 74653.2 0.0940718 0.0470359 0.998893i \(-0.485022\pi\)
0.0470359 + 0.998893i \(0.485022\pi\)
\(230\) 52900.0 0.0659380
\(231\) −463350. −0.571320
\(232\) −196414. −0.239581
\(233\) −163107. −0.196826 −0.0984129 0.995146i \(-0.531377\pi\)
−0.0984129 + 0.995146i \(0.531377\pi\)
\(234\) 830767. 0.991835
\(235\) 474725. 0.560754
\(236\) −795619. −0.929876
\(237\) 43806.7 0.0506605
\(238\) −1.37615e6 −1.57479
\(239\) −458112. −0.518773 −0.259386 0.965774i \(-0.583520\pi\)
−0.259386 + 0.965774i \(0.583520\pi\)
\(240\) −15895.9 −0.0178138
\(241\) 1.74728e6 1.93785 0.968924 0.247360i \(-0.0795631\pi\)
0.968924 + 0.247360i \(0.0795631\pi\)
\(242\) −1.54316e6 −1.69384
\(243\) −425188. −0.461918
\(244\) 304509. 0.327436
\(245\) 1.17089e6 1.24624
\(246\) −16232.3 −0.0171018
\(247\) 1.51301e6 1.57797
\(248\) 539202. 0.556702
\(249\) −217600. −0.222413
\(250\) −62500.0 −0.0632456
\(251\) −509792. −0.510751 −0.255375 0.966842i \(-0.582199\pi\)
−0.255375 + 0.966842i \(0.582199\pi\)
\(252\) 955944. 0.948268
\(253\) 391189. 0.384224
\(254\) −910100. −0.885126
\(255\) 84679.4 0.0815507
\(256\) 65536.0 0.0625000
\(257\) 253793. 0.239688 0.119844 0.992793i \(-0.461761\pi\)
0.119844 + 0.992793i \(0.461761\pi\)
\(258\) 86787.4 0.0811722
\(259\) 1.01952e6 0.944378
\(260\) 350784. 0.321816
\(261\) −726827. −0.660435
\(262\) −823827. −0.741452
\(263\) −245617. −0.218962 −0.109481 0.993989i \(-0.534919\pi\)
−0.109481 + 0.993989i \(0.534919\pi\)
\(264\) −117548. −0.103802
\(265\) 473875. 0.414524
\(266\) 1.74099e6 1.50866
\(267\) −82507.2 −0.0708294
\(268\) −150356. −0.127874
\(269\) −1.30972e6 −1.10356 −0.551782 0.833989i \(-0.686052\pi\)
−0.551782 + 0.833989i \(0.686052\pi\)
\(270\) −119177. −0.0994911
\(271\) 335307. 0.277345 0.138672 0.990338i \(-0.455717\pi\)
0.138672 + 0.990338i \(0.455717\pi\)
\(272\) −349118. −0.286121
\(273\) 549490. 0.446224
\(274\) 1.28259e6 1.03208
\(275\) −462179. −0.368535
\(276\) 21022.3 0.0166115
\(277\) 1.24277e6 0.973173 0.486586 0.873633i \(-0.338242\pi\)
0.486586 + 0.873633i \(0.338242\pi\)
\(278\) −778943. −0.604497
\(279\) 1.99531e6 1.53462
\(280\) 403639. 0.307680
\(281\) −549120. −0.414860 −0.207430 0.978250i \(-0.566510\pi\)
−0.207430 + 0.978250i \(0.566510\pi\)
\(282\) 188655. 0.141268
\(283\) −687675. −0.510407 −0.255204 0.966887i \(-0.582143\pi\)
−0.255204 + 0.966887i \(0.582143\pi\)
\(284\) −462860. −0.340529
\(285\) −107129. −0.0781258
\(286\) 2.59401e6 1.87524
\(287\) 412182. 0.295382
\(288\) 242515. 0.172289
\(289\) 439938. 0.309847
\(290\) −306897. −0.214288
\(291\) −102656. −0.0710641
\(292\) 40560.4 0.0278385
\(293\) −1.30317e6 −0.886813 −0.443406 0.896321i \(-0.646230\pi\)
−0.443406 + 0.896321i \(0.646230\pi\)
\(294\) 465308. 0.313959
\(295\) −1.24315e6 −0.831706
\(296\) 258643. 0.171582
\(297\) −881301. −0.579740
\(298\) 1.57212e6 1.02552
\(299\) −463912. −0.300095
\(300\) −24837.4 −0.0159332
\(301\) −2.20376e6 −1.40200
\(302\) 21433.0 0.0135227
\(303\) 128826. 0.0806117
\(304\) 441674. 0.274105
\(305\) 475796. 0.292867
\(306\) −1.29191e6 −0.788729
\(307\) 2.67886e6 1.62220 0.811099 0.584909i \(-0.198870\pi\)
0.811099 + 0.584909i \(0.198870\pi\)
\(308\) 2.98486e6 1.79286
\(309\) −134942. −0.0803991
\(310\) 842504. 0.497929
\(311\) 2.66680e6 1.56347 0.781734 0.623612i \(-0.214335\pi\)
0.781734 + 0.623612i \(0.214335\pi\)
\(312\) 139401. 0.0810736
\(313\) 935722. 0.539866 0.269933 0.962879i \(-0.412998\pi\)
0.269933 + 0.962879i \(0.412998\pi\)
\(314\) −1.16341e6 −0.665897
\(315\) 1.49366e6 0.848156
\(316\) −282199. −0.158978
\(317\) −1.45385e6 −0.812592 −0.406296 0.913742i \(-0.633180\pi\)
−0.406296 + 0.913742i \(0.633180\pi\)
\(318\) 188317. 0.104429
\(319\) −2.26946e6 −1.24867
\(320\) 102400. 0.0559017
\(321\) 263660. 0.142817
\(322\) −533813. −0.286913
\(323\) −2.35285e6 −1.25484
\(324\) 873438. 0.462242
\(325\) 548101. 0.287841
\(326\) 653461. 0.340546
\(327\) 431931. 0.223380
\(328\) 104567. 0.0536674
\(329\) −4.79045e6 −2.43998
\(330\) −183669. −0.0928434
\(331\) −13841.3 −0.00694395 −0.00347198 0.999994i \(-0.501105\pi\)
−0.00347198 + 0.999994i \(0.501105\pi\)
\(332\) 1.40176e6 0.697955
\(333\) 957106. 0.472987
\(334\) 1.05684e6 0.518376
\(335\) −234931. −0.114374
\(336\) 160405. 0.0775123
\(337\) −1.93165e6 −0.926520 −0.463260 0.886222i \(-0.653320\pi\)
−0.463260 + 0.886222i \(0.653320\pi\)
\(338\) −1.59107e6 −0.757527
\(339\) −651005. −0.307670
\(340\) −545497. −0.255915
\(341\) 6.23020e6 2.90146
\(342\) 1.63441e6 0.755605
\(343\) −7.57543e6 −3.47674
\(344\) −559076. −0.254727
\(345\) 32847.4 0.0148578
\(346\) −1.36752e6 −0.614107
\(347\) −2.44241e6 −1.08892 −0.544458 0.838788i \(-0.683265\pi\)
−0.544458 + 0.838788i \(0.683265\pi\)
\(348\) −121960. −0.0539846
\(349\) 956707. 0.420451 0.210226 0.977653i \(-0.432580\pi\)
0.210226 + 0.977653i \(0.432580\pi\)
\(350\) 630687. 0.275197
\(351\) 1.04514e6 0.452800
\(352\) 757235. 0.325742
\(353\) 3.71100e6 1.58509 0.792545 0.609813i \(-0.208756\pi\)
0.792545 + 0.609813i \(0.208756\pi\)
\(354\) −494027. −0.209528
\(355\) −723219. −0.304578
\(356\) 531504. 0.222270
\(357\) −854499. −0.354847
\(358\) −1.24706e6 −0.514258
\(359\) −1.33403e6 −0.546298 −0.273149 0.961972i \(-0.588065\pi\)
−0.273149 + 0.961972i \(0.588065\pi\)
\(360\) 378930. 0.154100
\(361\) 500518. 0.202140
\(362\) 1.05275e6 0.422236
\(363\) −958200. −0.381671
\(364\) −3.53976e6 −1.40030
\(365\) 63375.7 0.0248995
\(366\) 189080. 0.0737808
\(367\) 2.92197e6 1.13243 0.566214 0.824258i \(-0.308408\pi\)
0.566214 + 0.824258i \(0.308408\pi\)
\(368\) −135424. −0.0521286
\(369\) 386949. 0.147941
\(370\) 404130. 0.153468
\(371\) −4.78187e6 −1.80369
\(372\) 334809. 0.125441
\(373\) −3.06510e6 −1.14070 −0.570352 0.821401i \(-0.693193\pi\)
−0.570352 + 0.821401i \(0.693193\pi\)
\(374\) −4.03388e6 −1.49123
\(375\) −38808.4 −0.0142511
\(376\) −1.21530e6 −0.443315
\(377\) 2.69137e6 0.975258
\(378\) 1.20262e6 0.432910
\(379\) 296537. 0.106043 0.0530214 0.998593i \(-0.483115\pi\)
0.0530214 + 0.998593i \(0.483115\pi\)
\(380\) 690115. 0.245167
\(381\) −565112. −0.199445
\(382\) 3.02162e6 1.05945
\(383\) 2.25387e6 0.785113 0.392556 0.919728i \(-0.371591\pi\)
0.392556 + 0.919728i \(0.371591\pi\)
\(384\) 40693.5 0.0140831
\(385\) 4.66385e6 1.60359
\(386\) 3.09529e6 1.05738
\(387\) −2.06885e6 −0.702186
\(388\) 661299. 0.223007
\(389\) −956682. −0.320548 −0.160274 0.987073i \(-0.551238\pi\)
−0.160274 + 0.987073i \(0.551238\pi\)
\(390\) 217814. 0.0725144
\(391\) 721420. 0.238642
\(392\) −2.99747e6 −0.985236
\(393\) −511542. −0.167071
\(394\) −1.86040e6 −0.603762
\(395\) −440935. −0.142194
\(396\) 2.80214e6 0.897948
\(397\) 2.14436e6 0.682843 0.341421 0.939910i \(-0.389092\pi\)
0.341421 + 0.939910i \(0.389092\pi\)
\(398\) 2.40597e6 0.761345
\(399\) 1.08104e6 0.339945
\(400\) 160000. 0.0500000
\(401\) 702874. 0.218281 0.109141 0.994026i \(-0.465190\pi\)
0.109141 + 0.994026i \(0.465190\pi\)
\(402\) −93361.0 −0.0288138
\(403\) −7.38843e6 −2.26615
\(404\) −829888. −0.252968
\(405\) 1.36475e6 0.413442
\(406\) 3.09689e6 0.932419
\(407\) 2.98849e6 0.894264
\(408\) −216779. −0.0644715
\(409\) 2.71607e6 0.802848 0.401424 0.915892i \(-0.368515\pi\)
0.401424 + 0.915892i \(0.368515\pi\)
\(410\) 163386. 0.0480015
\(411\) 796404. 0.232557
\(412\) 869284. 0.252301
\(413\) 1.25447e7 3.61896
\(414\) −501135. −0.143699
\(415\) 2.19025e6 0.624270
\(416\) −898008. −0.254418
\(417\) −483672. −0.136211
\(418\) 5.10331e6 1.42860
\(419\) −3.78764e6 −1.05398 −0.526992 0.849870i \(-0.676680\pi\)
−0.526992 + 0.849870i \(0.676680\pi\)
\(420\) 250633. 0.0693291
\(421\) −1.72773e6 −0.475085 −0.237542 0.971377i \(-0.576342\pi\)
−0.237542 + 0.971377i \(0.576342\pi\)
\(422\) 24097.6 0.00658707
\(423\) −4.49719e6 −1.22205
\(424\) −1.21312e6 −0.327710
\(425\) −852339. −0.228897
\(426\) −287406. −0.0767311
\(427\) −4.80125e6 −1.27434
\(428\) −1.69847e6 −0.448176
\(429\) 1.61071e6 0.422545
\(430\) −873557. −0.227835
\(431\) −5.04905e6 −1.30923 −0.654615 0.755962i \(-0.727169\pi\)
−0.654615 + 0.755962i \(0.727169\pi\)
\(432\) 305094. 0.0786546
\(433\) 37457.0 0.00960093 0.00480047 0.999988i \(-0.498472\pi\)
0.00480047 + 0.999988i \(0.498472\pi\)
\(434\) −8.50169e6 −2.16661
\(435\) −190563. −0.0482853
\(436\) −2.78246e6 −0.700992
\(437\) −912677. −0.228620
\(438\) 25185.4 0.00627282
\(439\) −99673.6 −0.0246842 −0.0123421 0.999924i \(-0.503929\pi\)
−0.0123421 + 0.999924i \(0.503929\pi\)
\(440\) 1.18318e6 0.291353
\(441\) −1.10921e7 −2.71592
\(442\) 4.78380e6 1.16471
\(443\) −4.60309e6 −1.11440 −0.557199 0.830379i \(-0.688124\pi\)
−0.557199 + 0.830379i \(0.688124\pi\)
\(444\) 160600. 0.0386624
\(445\) 830475. 0.198805
\(446\) −5.16793e6 −1.23021
\(447\) 976184. 0.231080
\(448\) −1.03332e6 −0.243242
\(449\) 3.74524e6 0.876725 0.438363 0.898798i \(-0.355559\pi\)
0.438363 + 0.898798i \(0.355559\pi\)
\(450\) 592078. 0.137831
\(451\) 1.20822e6 0.279707
\(452\) 4.19371e6 0.965501
\(453\) 13308.5 0.00304707
\(454\) −257177. −0.0585589
\(455\) −5.53088e6 −1.25246
\(456\) 274250. 0.0617639
\(457\) 499745. 0.111933 0.0559665 0.998433i \(-0.482176\pi\)
0.0559665 + 0.998433i \(0.482176\pi\)
\(458\) −298613. −0.0665188
\(459\) −1.62527e6 −0.360076
\(460\) −211600. −0.0466252
\(461\) 1.73935e6 0.381183 0.190592 0.981669i \(-0.438959\pi\)
0.190592 + 0.981669i \(0.438959\pi\)
\(462\) 1.85340e6 0.403984
\(463\) 866807. 0.187919 0.0939594 0.995576i \(-0.470048\pi\)
0.0939594 + 0.995576i \(0.470048\pi\)
\(464\) 785656. 0.169409
\(465\) 523139. 0.112198
\(466\) 652427. 0.139177
\(467\) −1.52340e6 −0.323238 −0.161619 0.986853i \(-0.551672\pi\)
−0.161619 + 0.986853i \(0.551672\pi\)
\(468\) −3.32307e6 −0.701333
\(469\) 2.37069e6 0.497671
\(470\) −1.89890e6 −0.396513
\(471\) −722398. −0.150046
\(472\) 3.18247e6 0.657522
\(473\) −6.45984e6 −1.32760
\(474\) −175227. −0.0358224
\(475\) 1.07830e6 0.219284
\(476\) 5.50461e6 1.11355
\(477\) −4.48914e6 −0.903373
\(478\) 1.83245e6 0.366828
\(479\) 9.23081e6 1.83824 0.919118 0.393983i \(-0.128903\pi\)
0.919118 + 0.393983i \(0.128903\pi\)
\(480\) 63583.6 0.0125963
\(481\) −3.54407e6 −0.698456
\(482\) −6.98911e6 −1.37026
\(483\) −331463. −0.0646498
\(484\) 6.17264e6 1.19773
\(485\) 1.03328e6 0.199463
\(486\) 1.70075e6 0.326626
\(487\) −1.22749e6 −0.234529 −0.117264 0.993101i \(-0.537413\pi\)
−0.117264 + 0.993101i \(0.537413\pi\)
\(488\) −1.21804e6 −0.231532
\(489\) 405756. 0.0767348
\(490\) −4.68355e6 −0.881222
\(491\) −2.99377e6 −0.560421 −0.280210 0.959939i \(-0.590404\pi\)
−0.280210 + 0.959939i \(0.590404\pi\)
\(492\) 64929.2 0.0120928
\(493\) −4.18529e6 −0.775546
\(494\) −6.05204e6 −1.11580
\(495\) 4.37834e6 0.803149
\(496\) −2.15681e6 −0.393648
\(497\) 7.29799e6 1.32529
\(498\) 870398. 0.157270
\(499\) −123550. −0.0222121 −0.0111061 0.999938i \(-0.503535\pi\)
−0.0111061 + 0.999938i \(0.503535\pi\)
\(500\) 250000. 0.0447214
\(501\) 656230. 0.116805
\(502\) 2.03917e6 0.361155
\(503\) 2.10833e6 0.371551 0.185775 0.982592i \(-0.440520\pi\)
0.185775 + 0.982592i \(0.440520\pi\)
\(504\) −3.82377e6 −0.670527
\(505\) −1.29670e6 −0.226262
\(506\) −1.56475e6 −0.271688
\(507\) −987950. −0.170693
\(508\) 3.64040e6 0.625878
\(509\) 1.57302e6 0.269116 0.134558 0.990906i \(-0.457038\pi\)
0.134558 + 0.990906i \(0.457038\pi\)
\(510\) −338718. −0.0576650
\(511\) −639523. −0.108344
\(512\) −262144. −0.0441942
\(513\) 2.05615e6 0.344954
\(514\) −1.01517e6 −0.169485
\(515\) 1.35826e6 0.225665
\(516\) −347150. −0.0573974
\(517\) −1.40421e7 −2.31050
\(518\) −4.07807e6 −0.667776
\(519\) −849140. −0.138376
\(520\) −1.40314e6 −0.227558
\(521\) −1.90859e6 −0.308049 −0.154024 0.988067i \(-0.549223\pi\)
−0.154024 + 0.988067i \(0.549223\pi\)
\(522\) 2.90731e6 0.466998
\(523\) 7.33852e6 1.17315 0.586576 0.809894i \(-0.300475\pi\)
0.586576 + 0.809894i \(0.300475\pi\)
\(524\) 3.29531e6 0.524286
\(525\) 391615. 0.0620099
\(526\) 982467. 0.154829
\(527\) 1.14896e7 1.80210
\(528\) 470193. 0.0733992
\(529\) 279841. 0.0434783
\(530\) −1.89550e6 −0.293113
\(531\) 1.17767e7 1.81254
\(532\) −6.96394e6 −1.06678
\(533\) −1.43283e6 −0.218463
\(534\) 330029. 0.0500840
\(535\) −2.65386e6 −0.400861
\(536\) 601423. 0.0904208
\(537\) −774345. −0.115877
\(538\) 5.23887e6 0.780337
\(539\) −3.46343e7 −5.13492
\(540\) 476709. 0.0703508
\(541\) 1.09412e7 1.60721 0.803606 0.595162i \(-0.202912\pi\)
0.803606 + 0.595162i \(0.202912\pi\)
\(542\) −1.34123e6 −0.196112
\(543\) 653691. 0.0951421
\(544\) 1.39647e6 0.202318
\(545\) −4.34760e6 −0.626986
\(546\) −2.19796e6 −0.315528
\(547\) 7.37797e6 1.05431 0.527155 0.849769i \(-0.323259\pi\)
0.527155 + 0.849769i \(0.323259\pi\)
\(548\) −5.13036e6 −0.729788
\(549\) −4.50733e6 −0.638246
\(550\) 1.84872e6 0.260594
\(551\) 5.29485e6 0.742976
\(552\) −84089.4 −0.0117461
\(553\) 4.44947e6 0.618722
\(554\) −4.97106e6 −0.688137
\(555\) 250938. 0.0345807
\(556\) 3.11577e6 0.427444
\(557\) 7.81229e6 1.06694 0.533471 0.845819i \(-0.320887\pi\)
0.533471 + 0.845819i \(0.320887\pi\)
\(558\) −7.98124e6 −1.08514
\(559\) 7.66076e6 1.03691
\(560\) −1.61456e6 −0.217562
\(561\) −2.50477e6 −0.336017
\(562\) 2.19648e6 0.293350
\(563\) −9.88567e6 −1.31442 −0.657212 0.753706i \(-0.728264\pi\)
−0.657212 + 0.753706i \(0.728264\pi\)
\(564\) −754619. −0.0998918
\(565\) 6.55268e6 0.863570
\(566\) 2.75070e6 0.360913
\(567\) −1.37716e7 −1.79899
\(568\) 1.85144e6 0.240790
\(569\) 3.37792e6 0.437390 0.218695 0.975793i \(-0.429820\pi\)
0.218695 + 0.975793i \(0.429820\pi\)
\(570\) 428516. 0.0552433
\(571\) 1.44970e7 1.86075 0.930375 0.366610i \(-0.119482\pi\)
0.930375 + 0.366610i \(0.119482\pi\)
\(572\) −1.03760e7 −1.32599
\(573\) 1.87622e6 0.238725
\(574\) −1.64873e6 −0.208866
\(575\) −330625. −0.0417029
\(576\) −970060. −0.121827
\(577\) −2.74177e6 −0.342840 −0.171420 0.985198i \(-0.554836\pi\)
−0.171420 + 0.985198i \(0.554836\pi\)
\(578\) −1.75975e6 −0.219095
\(579\) 1.92197e6 0.238259
\(580\) 1.22759e6 0.151524
\(581\) −2.21017e7 −2.71635
\(582\) 410623. 0.0502499
\(583\) −1.40170e7 −1.70798
\(584\) −162242. −0.0196848
\(585\) −5.19229e6 −0.627291
\(586\) 5.21268e6 0.627071
\(587\) 2.70287e6 0.323765 0.161882 0.986810i \(-0.448243\pi\)
0.161882 + 0.986810i \(0.448243\pi\)
\(588\) −1.86123e6 −0.222002
\(589\) −1.45356e7 −1.72641
\(590\) 4.97262e6 0.588105
\(591\) −1.15518e6 −0.136045
\(592\) −1.03457e6 −0.121327
\(593\) 5.21163e6 0.608607 0.304303 0.952575i \(-0.401576\pi\)
0.304303 + 0.952575i \(0.401576\pi\)
\(594\) 3.52520e6 0.409938
\(595\) 8.60095e6 0.995987
\(596\) −6.28849e6 −0.725155
\(597\) 1.49395e6 0.171553
\(598\) 1.85565e6 0.212199
\(599\) 1.51225e7 1.72210 0.861049 0.508523i \(-0.169808\pi\)
0.861049 + 0.508523i \(0.169808\pi\)
\(600\) 99349.4 0.0112665
\(601\) 1.54889e7 1.74918 0.874592 0.484860i \(-0.161129\pi\)
0.874592 + 0.484860i \(0.161129\pi\)
\(602\) 8.81505e6 0.991365
\(603\) 2.22556e6 0.249256
\(604\) −85731.8 −0.00956202
\(605\) 9.64475e6 1.07128
\(606\) −515305. −0.0570011
\(607\) −1.32452e7 −1.45910 −0.729551 0.683926i \(-0.760271\pi\)
−0.729551 + 0.683926i \(0.760271\pi\)
\(608\) −1.76669e6 −0.193822
\(609\) 1.92297e6 0.210101
\(610\) −1.90318e6 −0.207088
\(611\) 1.66526e7 1.80459
\(612\) 5.16763e6 0.557715
\(613\) 9.10746e6 0.978918 0.489459 0.872026i \(-0.337194\pi\)
0.489459 + 0.872026i \(0.337194\pi\)
\(614\) −1.07154e7 −1.14707
\(615\) 101452. 0.0108161
\(616\) −1.19394e7 −1.26775
\(617\) −2.14311e6 −0.226637 −0.113319 0.993559i \(-0.536148\pi\)
−0.113319 + 0.993559i \(0.536148\pi\)
\(618\) 539768. 0.0568507
\(619\) 1.74838e7 1.83404 0.917021 0.398839i \(-0.130587\pi\)
0.917021 + 0.398839i \(0.130587\pi\)
\(620\) −3.37001e6 −0.352089
\(621\) −630448. −0.0656025
\(622\) −1.06672e7 −1.10554
\(623\) −8.38031e6 −0.865048
\(624\) −557604. −0.0573277
\(625\) 390625. 0.0400000
\(626\) −3.74289e6 −0.381743
\(627\) 3.16882e6 0.321906
\(628\) 4.65362e6 0.470860
\(629\) 5.51130e6 0.555427
\(630\) −5.97465e6 −0.599737
\(631\) 4.20504e6 0.420433 0.210216 0.977655i \(-0.432583\pi\)
0.210216 + 0.977655i \(0.432583\pi\)
\(632\) 1.12879e6 0.112414
\(633\) 14963.0 0.00148426
\(634\) 5.81541e6 0.574589
\(635\) 5.68813e6 0.559803
\(636\) −753268. −0.0738426
\(637\) 4.10729e7 4.01058
\(638\) 9.07785e6 0.882940
\(639\) 6.85123e6 0.663768
\(640\) −409600. −0.0395285
\(641\) 7.83415e6 0.753090 0.376545 0.926398i \(-0.377112\pi\)
0.376545 + 0.926398i \(0.377112\pi\)
\(642\) −1.05464e6 −0.100987
\(643\) −1.18578e7 −1.13103 −0.565517 0.824737i \(-0.691323\pi\)
−0.565517 + 0.824737i \(0.691323\pi\)
\(644\) 2.13525e6 0.202878
\(645\) −542421. −0.0513378
\(646\) 9.41139e6 0.887304
\(647\) 5.26318e6 0.494297 0.247148 0.968978i \(-0.420506\pi\)
0.247148 + 0.968978i \(0.420506\pi\)
\(648\) −3.49375e6 −0.326855
\(649\) 3.67719e7 3.42692
\(650\) −2.19240e6 −0.203534
\(651\) −5.27899e6 −0.488201
\(652\) −2.61384e6 −0.240802
\(653\) 1.64840e7 1.51279 0.756397 0.654113i \(-0.226958\pi\)
0.756397 + 0.654113i \(0.226958\pi\)
\(654\) −1.72772e6 −0.157954
\(655\) 5.14892e6 0.468935
\(656\) −418268. −0.0379486
\(657\) −600373. −0.0542635
\(658\) 1.91618e7 1.72533
\(659\) −1.73924e7 −1.56008 −0.780038 0.625732i \(-0.784800\pi\)
−0.780038 + 0.625732i \(0.784800\pi\)
\(660\) 734676. 0.0656502
\(661\) 1.03745e7 0.923553 0.461776 0.886996i \(-0.347212\pi\)
0.461776 + 0.886996i \(0.347212\pi\)
\(662\) 55365.2 0.00491012
\(663\) 2.97042e6 0.262443
\(664\) −5.60703e6 −0.493529
\(665\) −1.08812e7 −0.954160
\(666\) −3.82842e6 −0.334453
\(667\) −1.62348e6 −0.141297
\(668\) −4.22737e6 −0.366547
\(669\) −3.20895e6 −0.277202
\(670\) 939724. 0.0808748
\(671\) −1.40738e7 −1.20672
\(672\) −641622. −0.0548095
\(673\) −2.69576e6 −0.229426 −0.114713 0.993399i \(-0.536595\pi\)
−0.114713 + 0.993399i \(0.536595\pi\)
\(674\) 7.72662e6 0.655148
\(675\) 744858. 0.0629237
\(676\) 6.36429e6 0.535652
\(677\) −229429. −0.0192387 −0.00961937 0.999954i \(-0.503062\pi\)
−0.00961937 + 0.999954i \(0.503062\pi\)
\(678\) 2.60402e6 0.217555
\(679\) −1.04268e7 −0.867914
\(680\) 2.18199e6 0.180959
\(681\) −159690. −0.0131950
\(682\) −2.49208e7 −2.05164
\(683\) 1.09516e7 0.898313 0.449157 0.893453i \(-0.351725\pi\)
0.449157 + 0.893453i \(0.351725\pi\)
\(684\) −6.53763e6 −0.534293
\(685\) −8.01619e6 −0.652742
\(686\) 3.03017e7 2.45843
\(687\) −185419. −0.0149886
\(688\) 2.23631e6 0.180119
\(689\) 1.66228e7 1.33400
\(690\) −131390. −0.0105060
\(691\) −2.33754e7 −1.86236 −0.931181 0.364558i \(-0.881220\pi\)
−0.931181 + 0.364558i \(0.881220\pi\)
\(692\) 5.47008e6 0.434239
\(693\) −4.41817e7 −3.49470
\(694\) 9.76963e6 0.769980
\(695\) 4.86839e6 0.382317
\(696\) 487841. 0.0381729
\(697\) 2.22817e6 0.173726
\(698\) −3.82683e6 −0.297304
\(699\) 405114. 0.0313606
\(700\) −2.52275e6 −0.194594
\(701\) −285635. −0.0219542 −0.0109771 0.999940i \(-0.503494\pi\)
−0.0109771 + 0.999940i \(0.503494\pi\)
\(702\) −4.18056e6 −0.320178
\(703\) −6.97241e6 −0.532102
\(704\) −3.02894e6 −0.230334
\(705\) −1.17909e6 −0.0893459
\(706\) −1.48440e7 −1.12083
\(707\) 1.30850e7 0.984520
\(708\) 1.97611e6 0.148159
\(709\) 7.04217e6 0.526127 0.263064 0.964778i \(-0.415267\pi\)
0.263064 + 0.964778i \(0.415267\pi\)
\(710\) 2.89288e6 0.215369
\(711\) 4.17709e6 0.309884
\(712\) −2.12602e6 −0.157169
\(713\) 4.45684e6 0.328325
\(714\) 3.41800e6 0.250915
\(715\) −1.62125e7 −1.18600
\(716\) 4.98826e6 0.363636
\(717\) 1.13783e6 0.0826569
\(718\) 5.33612e6 0.386291
\(719\) 1.60147e7 1.15531 0.577653 0.816283i \(-0.303969\pi\)
0.577653 + 0.816283i \(0.303969\pi\)
\(720\) −1.51572e6 −0.108965
\(721\) −1.37061e7 −0.981923
\(722\) −2.00207e6 −0.142934
\(723\) −4.33978e6 −0.308761
\(724\) −4.21102e6 −0.298566
\(725\) 1.91811e6 0.135528
\(726\) 3.83280e6 0.269882
\(727\) 1.09411e7 0.767758 0.383879 0.923383i \(-0.374588\pi\)
0.383879 + 0.923383i \(0.374588\pi\)
\(728\) 1.41590e7 0.990160
\(729\) −1.22093e7 −0.850886
\(730\) −253503. −0.0176066
\(731\) −1.19131e7 −0.824575
\(732\) −756321. −0.0521709
\(733\) 1.33642e7 0.918718 0.459359 0.888251i \(-0.348079\pi\)
0.459359 + 0.888251i \(0.348079\pi\)
\(734\) −1.16879e7 −0.800748
\(735\) −2.90818e6 −0.198565
\(736\) 541696. 0.0368605
\(737\) 6.94914e6 0.471262
\(738\) −1.54780e6 −0.104610
\(739\) 9.50621e6 0.640319 0.320159 0.947364i \(-0.396264\pi\)
0.320159 + 0.947364i \(0.396264\pi\)
\(740\) −1.61652e6 −0.108518
\(741\) −3.75792e6 −0.251421
\(742\) 1.91275e7 1.27540
\(743\) −9.84795e6 −0.654446 −0.327223 0.944947i \(-0.606113\pi\)
−0.327223 + 0.944947i \(0.606113\pi\)
\(744\) −1.33924e6 −0.0887002
\(745\) −9.82576e6 −0.648598
\(746\) 1.22604e7 0.806599
\(747\) −2.07487e7 −1.36047
\(748\) 1.61355e7 1.05446
\(749\) 2.67801e7 1.74425
\(750\) 155233. 0.0100770
\(751\) −1.90181e7 −1.23046 −0.615230 0.788348i \(-0.710937\pi\)
−0.615230 + 0.788348i \(0.710937\pi\)
\(752\) 4.86119e6 0.313471
\(753\) 1.26619e6 0.0813788
\(754\) −1.07655e7 −0.689612
\(755\) −133956. −0.00855253
\(756\) −4.81047e6 −0.306114
\(757\) −1.70495e7 −1.08136 −0.540681 0.841227i \(-0.681834\pi\)
−0.540681 + 0.841227i \(0.681834\pi\)
\(758\) −1.18615e6 −0.0749836
\(759\) −971609. −0.0612191
\(760\) −2.76046e6 −0.173359
\(761\) −2.75852e6 −0.172669 −0.0863346 0.996266i \(-0.527515\pi\)
−0.0863346 + 0.996266i \(0.527515\pi\)
\(762\) 2.26045e6 0.141029
\(763\) 4.38715e7 2.72817
\(764\) −1.20865e7 −0.749145
\(765\) 8.07442e6 0.498836
\(766\) −9.01548e6 −0.555158
\(767\) −4.36079e7 −2.67656
\(768\) −162774. −0.00995823
\(769\) 1.87467e7 1.14316 0.571581 0.820546i \(-0.306330\pi\)
0.571581 + 0.820546i \(0.306330\pi\)
\(770\) −1.86554e7 −1.13391
\(771\) −630354. −0.0381899
\(772\) −1.23811e7 −0.747683
\(773\) 1.37482e7 0.827553 0.413776 0.910379i \(-0.364210\pi\)
0.413776 + 0.910379i \(0.364210\pi\)
\(774\) 8.27542e6 0.496521
\(775\) −5.26565e6 −0.314918
\(776\) −2.64519e6 −0.157690
\(777\) −2.53221e6 −0.150469
\(778\) 3.82673e6 0.226662
\(779\) −2.81888e6 −0.166430
\(780\) −871256. −0.0512754
\(781\) 2.13924e7 1.25497
\(782\) −2.88568e6 −0.168745
\(783\) 3.65752e6 0.213197
\(784\) 1.19899e7 0.696667
\(785\) 7.27129e6 0.421150
\(786\) 2.04617e6 0.118137
\(787\) 6.89984e6 0.397102 0.198551 0.980091i \(-0.436376\pi\)
0.198551 + 0.980091i \(0.436376\pi\)
\(788\) 7.44160e6 0.426924
\(789\) 610047. 0.0348876
\(790\) 1.76374e6 0.100547
\(791\) −6.61230e7 −3.75761
\(792\) −1.12085e7 −0.634945
\(793\) 1.66902e7 0.942493
\(794\) −8.57742e6 −0.482843
\(795\) −1.17698e6 −0.0660468
\(796\) −9.62386e6 −0.538353
\(797\) 1.13532e7 0.633098 0.316549 0.948576i \(-0.397476\pi\)
0.316549 + 0.948576i \(0.397476\pi\)
\(798\) −4.32415e6 −0.240377
\(799\) −2.58961e7 −1.43505
\(800\) −640000. −0.0353553
\(801\) −7.86729e6 −0.433255
\(802\) −2.81150e6 −0.154348
\(803\) −1.87462e6 −0.102595
\(804\) 373444. 0.0203744
\(805\) 3.33633e6 0.181459
\(806\) 2.95537e7 1.60241
\(807\) 3.25299e6 0.175833
\(808\) 3.31955e6 0.178875
\(809\) 2.58652e7 1.38946 0.694728 0.719273i \(-0.255525\pi\)
0.694728 + 0.719273i \(0.255525\pi\)
\(810\) −5.45899e6 −0.292348
\(811\) 3.88499e6 0.207414 0.103707 0.994608i \(-0.466930\pi\)
0.103707 + 0.994608i \(0.466930\pi\)
\(812\) −1.23876e7 −0.659320
\(813\) −832815. −0.0441898
\(814\) −1.19540e7 −0.632340
\(815\) −4.08413e6 −0.215380
\(816\) 867117. 0.0455882
\(817\) 1.50714e7 0.789946
\(818\) −1.08643e7 −0.567699
\(819\) 5.23953e7 2.72950
\(820\) −653544. −0.0339422
\(821\) −1.98988e7 −1.03031 −0.515157 0.857096i \(-0.672266\pi\)
−0.515157 + 0.857096i \(0.672266\pi\)
\(822\) −3.18562e6 −0.164442
\(823\) 3.02715e7 1.55788 0.778941 0.627097i \(-0.215757\pi\)
0.778941 + 0.627097i \(0.215757\pi\)
\(824\) −3.47714e6 −0.178404
\(825\) 1.14793e6 0.0587193
\(826\) −5.01786e7 −2.55899
\(827\) 1.36026e7 0.691605 0.345803 0.938307i \(-0.387607\pi\)
0.345803 + 0.938307i \(0.387607\pi\)
\(828\) 2.00454e6 0.101610
\(829\) 2.08242e6 0.105240 0.0526202 0.998615i \(-0.483243\pi\)
0.0526202 + 0.998615i \(0.483243\pi\)
\(830\) −8.76098e6 −0.441426
\(831\) −3.08670e6 −0.155057
\(832\) 3.59203e6 0.179900
\(833\) −6.38716e7 −3.18930
\(834\) 1.93469e6 0.0963155
\(835\) −6.60527e6 −0.327850
\(836\) −2.04132e7 −1.01017
\(837\) −1.00407e7 −0.495395
\(838\) 1.51506e7 0.745279
\(839\) 1.26418e7 0.620016 0.310008 0.950734i \(-0.399668\pi\)
0.310008 + 0.950734i \(0.399668\pi\)
\(840\) −1.00253e6 −0.0490231
\(841\) −1.10926e7 −0.540807
\(842\) 6.91093e6 0.335936
\(843\) 1.36387e6 0.0661003
\(844\) −96390.4 −0.00465777
\(845\) 9.94420e6 0.479102
\(846\) 1.79887e7 0.864122
\(847\) −9.73250e7 −4.66140
\(848\) 4.85248e6 0.231726
\(849\) 1.70800e6 0.0813241
\(850\) 3.40936e6 0.161855
\(851\) 2.13785e6 0.101194
\(852\) 1.14962e6 0.0542571
\(853\) −2.77167e7 −1.30427 −0.652137 0.758101i \(-0.726127\pi\)
−0.652137 + 0.758101i \(0.726127\pi\)
\(854\) 1.92050e7 0.901093
\(855\) −1.02150e7 −0.477887
\(856\) 6.79389e6 0.316909
\(857\) −1.68037e7 −0.781541 −0.390770 0.920488i \(-0.627791\pi\)
−0.390770 + 0.920488i \(0.627791\pi\)
\(858\) −6.44282e6 −0.298784
\(859\) 6.32943e6 0.292672 0.146336 0.989235i \(-0.453252\pi\)
0.146336 + 0.989235i \(0.453252\pi\)
\(860\) 3.49423e6 0.161104
\(861\) −1.02375e6 −0.0470637
\(862\) 2.01962e7 0.925766
\(863\) −3.58594e7 −1.63899 −0.819494 0.573088i \(-0.805745\pi\)
−0.819494 + 0.573088i \(0.805745\pi\)
\(864\) −1.22038e6 −0.0556172
\(865\) 8.54701e6 0.388395
\(866\) −149828. −0.00678888
\(867\) −1.09269e6 −0.0493684
\(868\) 3.40068e7 1.53203
\(869\) 1.30426e7 0.585890
\(870\) 762251. 0.0341429
\(871\) −8.24101e6 −0.368074
\(872\) 1.11298e7 0.495676
\(873\) −9.78850e6 −0.434691
\(874\) 3.65071e6 0.161658
\(875\) −3.94179e6 −0.174050
\(876\) −100741. −0.00443555
\(877\) 757238. 0.0332455 0.0166228 0.999862i \(-0.494709\pi\)
0.0166228 + 0.999862i \(0.494709\pi\)
\(878\) 398694. 0.0174544
\(879\) 3.23673e6 0.141297
\(880\) −4.73272e6 −0.206017
\(881\) 3.19130e6 0.138525 0.0692625 0.997598i \(-0.477935\pi\)
0.0692625 + 0.997598i \(0.477935\pi\)
\(882\) 4.43684e7 1.92045
\(883\) 1.92363e7 0.830271 0.415136 0.909760i \(-0.363734\pi\)
0.415136 + 0.909760i \(0.363734\pi\)
\(884\) −1.91352e7 −0.823573
\(885\) 3.08767e6 0.132517
\(886\) 1.84124e7 0.787999
\(887\) −2.68483e7 −1.14580 −0.572898 0.819627i \(-0.694181\pi\)
−0.572898 + 0.819627i \(0.694181\pi\)
\(888\) −642402. −0.0273385
\(889\) −5.73988e7 −2.43584
\(890\) −3.32190e6 −0.140576
\(891\) −4.03685e7 −1.70353
\(892\) 2.06717e7 0.869891
\(893\) 3.27615e7 1.37479
\(894\) −3.90474e6 −0.163398
\(895\) 7.79415e6 0.325246
\(896\) 4.13327e6 0.171998
\(897\) 1.15224e6 0.0478146
\(898\) −1.49809e7 −0.619938
\(899\) −2.58562e7 −1.06700
\(900\) −2.36831e6 −0.0974613
\(901\) −2.58498e7 −1.06083
\(902\) −4.83287e6 −0.197783
\(903\) 5.47356e6 0.223383
\(904\) −1.67749e7 −0.682712
\(905\) −6.57971e6 −0.267046
\(906\) −53233.8 −0.00215460
\(907\) 7.62585e6 0.307801 0.153901 0.988086i \(-0.450816\pi\)
0.153901 + 0.988086i \(0.450816\pi\)
\(908\) 1.02871e6 0.0414074
\(909\) 1.22839e7 0.493092
\(910\) 2.21235e7 0.885626
\(911\) −3.05793e6 −0.122076 −0.0610381 0.998135i \(-0.519441\pi\)
−0.0610381 + 0.998135i \(0.519441\pi\)
\(912\) −1.09700e6 −0.0436737
\(913\) −6.47863e7 −2.57221
\(914\) −1.99898e6 −0.0791485
\(915\) −1.18175e6 −0.0466631
\(916\) 1.19445e6 0.0470359
\(917\) −5.19577e7 −2.04045
\(918\) 6.50109e6 0.254612
\(919\) −1.40082e7 −0.547134 −0.273567 0.961853i \(-0.588204\pi\)
−0.273567 + 0.961853i \(0.588204\pi\)
\(920\) 846400. 0.0329690
\(921\) −6.65358e6 −0.258468
\(922\) −6.95739e6 −0.269537
\(923\) −2.53694e7 −0.980180
\(924\) −7.41361e6 −0.285660
\(925\) −2.52581e6 −0.0970615
\(926\) −3.46723e6 −0.132879
\(927\) −1.28671e7 −0.491792
\(928\) −3.14262e6 −0.119791
\(929\) 1.68894e7 0.642061 0.321030 0.947069i \(-0.395971\pi\)
0.321030 + 0.947069i \(0.395971\pi\)
\(930\) −2.09256e6 −0.0793359
\(931\) 8.08047e7 3.05536
\(932\) −2.60971e6 −0.0984129
\(933\) −6.62362e6 −0.249110
\(934\) 6.09362e6 0.228564
\(935\) 2.52118e7 0.943135
\(936\) 1.32923e7 0.495917
\(937\) −7.56415e6 −0.281457 −0.140728 0.990048i \(-0.544944\pi\)
−0.140728 + 0.990048i \(0.544944\pi\)
\(938\) −9.48274e6 −0.351906
\(939\) −2.32409e6 −0.0860178
\(940\) 7.59560e6 0.280377
\(941\) −3.45946e7 −1.27360 −0.636802 0.771027i \(-0.719743\pi\)
−0.636802 + 0.771027i \(0.719743\pi\)
\(942\) 2.88959e6 0.106099
\(943\) 864312. 0.0316513
\(944\) −1.27299e7 −0.464938
\(945\) −7.51636e6 −0.273797
\(946\) 2.58394e7 0.938758
\(947\) 2.11440e7 0.766149 0.383074 0.923717i \(-0.374865\pi\)
0.383074 + 0.923717i \(0.374865\pi\)
\(948\) 700907. 0.0253303
\(949\) 2.22312e6 0.0801304
\(950\) −4.31322e6 −0.155057
\(951\) 3.61099e6 0.129472
\(952\) −2.20184e7 −0.787397
\(953\) −4.32490e6 −0.154257 −0.0771283 0.997021i \(-0.524575\pi\)
−0.0771283 + 0.997021i \(0.524575\pi\)
\(954\) 1.79565e7 0.638781
\(955\) −1.88851e7 −0.670056
\(956\) −7.32979e6 −0.259386
\(957\) 5.63675e6 0.198952
\(958\) −3.69232e7 −1.29983
\(959\) 8.08913e7 2.84024
\(960\) −254335. −0.00890691
\(961\) 4.23521e7 1.47933
\(962\) 1.41763e7 0.493883
\(963\) 2.51407e7 0.873597
\(964\) 2.79565e7 0.968924
\(965\) −1.93455e7 −0.668748
\(966\) 1.32585e6 0.0457143
\(967\) −1.38356e7 −0.475807 −0.237904 0.971289i \(-0.576460\pi\)
−0.237904 + 0.971289i \(0.576460\pi\)
\(968\) −2.46906e7 −0.846920
\(969\) 5.84385e6 0.199936
\(970\) −4.13312e6 −0.141042
\(971\) 3.62720e7 1.23459 0.617296 0.786731i \(-0.288228\pi\)
0.617296 + 0.786731i \(0.288228\pi\)
\(972\) −6.80301e6 −0.230959
\(973\) −4.91269e7 −1.66356
\(974\) 4.90997e6 0.165837
\(975\) −1.36134e6 −0.0458621
\(976\) 4.87215e6 0.163718
\(977\) −1.26207e7 −0.423006 −0.211503 0.977377i \(-0.567836\pi\)
−0.211503 + 0.977377i \(0.567836\pi\)
\(978\) −1.62302e6 −0.0542597
\(979\) −2.45650e7 −0.819144
\(980\) 1.87342e7 0.623118
\(981\) 4.11858e7 1.36639
\(982\) 1.19751e7 0.396277
\(983\) 3.30183e7 1.08986 0.544930 0.838481i \(-0.316556\pi\)
0.544930 + 0.838481i \(0.316556\pi\)
\(984\) −259717. −0.00855091
\(985\) 1.16275e7 0.381852
\(986\) 1.67411e7 0.548394
\(987\) 1.18982e7 0.388766
\(988\) 2.42082e7 0.788986
\(989\) −4.62112e6 −0.150230
\(990\) −1.75133e7 −0.567912
\(991\) 5.10495e7 1.65123 0.825614 0.564235i \(-0.190829\pi\)
0.825614 + 0.564235i \(0.190829\pi\)
\(992\) 8.62724e6 0.278351
\(993\) 34378.1 0.00110639
\(994\) −2.91920e7 −0.937125
\(995\) −1.50373e7 −0.481517
\(996\) −3.48159e6 −0.111206
\(997\) −8.22079e6 −0.261924 −0.130962 0.991387i \(-0.541807\pi\)
−0.130962 + 0.991387i \(0.541807\pi\)
\(998\) 494198. 0.0157063
\(999\) −4.81632e6 −0.152687
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 230.6.a.f.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.6.a.f.1.3 5 1.1 even 1 trivial