Properties

Label 177.6.a.c.1.3
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3879361069\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + 15565376 x^{4} + 6775664 x^{3} - 75006848 x^{2} + 21512960 x + 49172480\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(7.96013\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.96013 q^{2} +9.00000 q^{3} +3.52316 q^{4} +65.6425 q^{5} -53.6412 q^{6} +119.026 q^{7} +169.726 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-5.96013 q^{2} +9.00000 q^{3} +3.52316 q^{4} +65.6425 q^{5} -53.6412 q^{6} +119.026 q^{7} +169.726 q^{8} +81.0000 q^{9} -391.238 q^{10} +237.620 q^{11} +31.7085 q^{12} +999.008 q^{13} -709.408 q^{14} +590.783 q^{15} -1124.33 q^{16} -1276.40 q^{17} -482.771 q^{18} +1426.51 q^{19} +231.269 q^{20} +1071.23 q^{21} -1416.25 q^{22} +546.573 q^{23} +1527.53 q^{24} +1183.94 q^{25} -5954.22 q^{26} +729.000 q^{27} +419.346 q^{28} -1547.67 q^{29} -3521.14 q^{30} -7569.31 q^{31} +1269.92 q^{32} +2138.58 q^{33} +7607.51 q^{34} +7813.14 q^{35} +285.376 q^{36} -1012.21 q^{37} -8502.16 q^{38} +8991.07 q^{39} +11141.2 q^{40} +13264.9 q^{41} -6384.67 q^{42} -21794.0 q^{43} +837.174 q^{44} +5317.05 q^{45} -3257.65 q^{46} +23038.8 q^{47} -10119.0 q^{48} -2639.91 q^{49} -7056.46 q^{50} -11487.6 q^{51} +3519.67 q^{52} +19051.5 q^{53} -4344.94 q^{54} +15598.0 q^{55} +20201.7 q^{56} +12838.6 q^{57} +9224.33 q^{58} -3481.00 q^{59} +2081.42 q^{60} -7235.44 q^{61} +45114.1 q^{62} +9641.07 q^{63} +28409.6 q^{64} +65577.4 q^{65} -12746.2 q^{66} +69873.8 q^{67} -4496.96 q^{68} +4919.16 q^{69} -46567.3 q^{70} +17826.0 q^{71} +13747.8 q^{72} +9089.91 q^{73} +6032.93 q^{74} +10655.5 q^{75} +5025.81 q^{76} +28282.9 q^{77} -53588.0 q^{78} +31488.7 q^{79} -73803.8 q^{80} +6561.00 q^{81} -79060.3 q^{82} -102025. q^{83} +3774.12 q^{84} -83786.1 q^{85} +129895. q^{86} -13929.0 q^{87} +40330.2 q^{88} +104185. q^{89} -31690.3 q^{90} +118908. q^{91} +1925.67 q^{92} -68123.8 q^{93} -137314. q^{94} +93639.5 q^{95} +11429.3 q^{96} -117345. q^{97} +15734.2 q^{98} +19247.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + O(q^{10}) \) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + 601q^{10} + 1480q^{11} + 1782q^{12} + 472q^{13} + 1065q^{14} + 1422q^{15} + 6370q^{16} + 1565q^{17} + 1782q^{18} + 3939q^{19} + 8033q^{20} + 3717q^{21} - 1738q^{22} + 7245q^{23} + 6507q^{24} + 9690q^{25} + 3764q^{26} + 8748q^{27} + 12154q^{28} + 10003q^{29} + 5409q^{30} + 7295q^{31} + 11628q^{32} + 13320q^{33} - 16344q^{34} + 11015q^{35} + 16038q^{36} + 6741q^{37} + 3035q^{38} + 4248q^{39} + 5572q^{40} + 34025q^{41} + 9585q^{42} - 6336q^{43} + 41168q^{44} + 12798q^{45} + 2345q^{46} + 66167q^{47} + 57330q^{48} + 28319q^{49} + 31173q^{50} + 14085q^{51} + 16440q^{52} + 62290q^{53} + 16038q^{54} + 55764q^{55} + 107306q^{56} + 35451q^{57} + 37952q^{58} - 41772q^{59} + 72297q^{60} + 68469q^{61} + 99190q^{62} + 33453q^{63} + 68525q^{64} + 80156q^{65} - 15642q^{66} + 113310q^{67} + 33887q^{68} + 65205q^{69} + 32034q^{70} + 84520q^{71} + 58563q^{72} + 135895q^{73} - 31962q^{74} + 87210q^{75} - 61848q^{76} - 3799q^{77} + 33876q^{78} + 14122q^{79} + 77609q^{80} + 78732q^{81} - 1501q^{82} + 114463q^{83} + 109386q^{84} - 101097q^{85} - 203536q^{86} + 90027q^{87} - 244967q^{88} + 189109q^{89} + 48681q^{90} - 168249q^{91} - 71946q^{92} + 65655q^{93} - 472284q^{94} + 21923q^{95} + 104652q^{96} - 76192q^{97} - 17544q^{98} + 119880q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.96013 −1.05361 −0.526806 0.849985i \(-0.676611\pi\)
−0.526806 + 0.849985i \(0.676611\pi\)
\(3\) 9.00000 0.577350
\(4\) 3.52316 0.110099
\(5\) 65.6425 1.17425 0.587125 0.809496i \(-0.300260\pi\)
0.587125 + 0.809496i \(0.300260\pi\)
\(6\) −53.6412 −0.608303
\(7\) 119.026 0.918111 0.459055 0.888408i \(-0.348188\pi\)
0.459055 + 0.888408i \(0.348188\pi\)
\(8\) 169.726 0.937611
\(9\) 81.0000 0.333333
\(10\) −391.238 −1.23720
\(11\) 237.620 0.592109 0.296054 0.955171i \(-0.404329\pi\)
0.296054 + 0.955171i \(0.404329\pi\)
\(12\) 31.7085 0.0635656
\(13\) 999.008 1.63950 0.819749 0.572723i \(-0.194113\pi\)
0.819749 + 0.572723i \(0.194113\pi\)
\(14\) −709.408 −0.967333
\(15\) 590.783 0.677953
\(16\) −1124.33 −1.09798
\(17\) −1276.40 −1.07118 −0.535592 0.844477i \(-0.679912\pi\)
−0.535592 + 0.844477i \(0.679912\pi\)
\(18\) −482.771 −0.351204
\(19\) 1426.51 0.906546 0.453273 0.891372i \(-0.350256\pi\)
0.453273 + 0.891372i \(0.350256\pi\)
\(20\) 231.269 0.129283
\(21\) 1071.23 0.530072
\(22\) −1416.25 −0.623853
\(23\) 546.573 0.215441 0.107721 0.994181i \(-0.465645\pi\)
0.107721 + 0.994181i \(0.465645\pi\)
\(24\) 1527.53 0.541330
\(25\) 1183.94 0.378862
\(26\) −5954.22 −1.72739
\(27\) 729.000 0.192450
\(28\) 419.346 0.101083
\(29\) −1547.67 −0.341731 −0.170865 0.985294i \(-0.554656\pi\)
−0.170865 + 0.985294i \(0.554656\pi\)
\(30\) −3521.14 −0.714300
\(31\) −7569.31 −1.41466 −0.707330 0.706883i \(-0.750101\pi\)
−0.707330 + 0.706883i \(0.750101\pi\)
\(32\) 1269.92 0.219231
\(33\) 2138.58 0.341854
\(34\) 7607.51 1.12861
\(35\) 7813.14 1.07809
\(36\) 285.376 0.0366996
\(37\) −1012.21 −0.121554 −0.0607769 0.998151i \(-0.519358\pi\)
−0.0607769 + 0.998151i \(0.519358\pi\)
\(38\) −8502.16 −0.955148
\(39\) 8991.07 0.946564
\(40\) 11141.2 1.10099
\(41\) 13264.9 1.23237 0.616187 0.787600i \(-0.288676\pi\)
0.616187 + 0.787600i \(0.288676\pi\)
\(42\) −6384.67 −0.558490
\(43\) −21794.0 −1.79749 −0.898743 0.438476i \(-0.855518\pi\)
−0.898743 + 0.438476i \(0.855518\pi\)
\(44\) 837.174 0.0651905
\(45\) 5317.05 0.391417
\(46\) −3257.65 −0.226992
\(47\) 23038.8 1.52130 0.760650 0.649163i \(-0.224881\pi\)
0.760650 + 0.649163i \(0.224881\pi\)
\(48\) −10119.0 −0.633917
\(49\) −2639.91 −0.157072
\(50\) −7056.46 −0.399174
\(51\) −11487.6 −0.618449
\(52\) 3519.67 0.180507
\(53\) 19051.5 0.931624 0.465812 0.884884i \(-0.345762\pi\)
0.465812 + 0.884884i \(0.345762\pi\)
\(54\) −4344.94 −0.202768
\(55\) 15598.0 0.695283
\(56\) 20201.7 0.860831
\(57\) 12838.6 0.523394
\(58\) 9224.33 0.360052
\(59\) −3481.00 −0.130189
\(60\) 2081.42 0.0746418
\(61\) −7235.44 −0.248966 −0.124483 0.992222i \(-0.539727\pi\)
−0.124483 + 0.992222i \(0.539727\pi\)
\(62\) 45114.1 1.49050
\(63\) 9641.07 0.306037
\(64\) 28409.6 0.866992
\(65\) 65577.4 1.92518
\(66\) −12746.2 −0.360182
\(67\) 69873.8 1.90164 0.950818 0.309752i \(-0.100246\pi\)
0.950818 + 0.309752i \(0.100246\pi\)
\(68\) −4496.96 −0.117936
\(69\) 4919.16 0.124385
\(70\) −46567.3 −1.13589
\(71\) 17826.0 0.419671 0.209836 0.977737i \(-0.432707\pi\)
0.209836 + 0.977737i \(0.432707\pi\)
\(72\) 13747.8 0.312537
\(73\) 9089.91 0.199642 0.0998212 0.995005i \(-0.468173\pi\)
0.0998212 + 0.995005i \(0.468173\pi\)
\(74\) 6032.93 0.128071
\(75\) 10655.5 0.218736
\(76\) 5025.81 0.0998096
\(77\) 28282.9 0.543622
\(78\) −53588.0 −0.997312
\(79\) 31488.7 0.567658 0.283829 0.958875i \(-0.408395\pi\)
0.283829 + 0.958875i \(0.408395\pi\)
\(80\) −73803.8 −1.28930
\(81\) 6561.00 0.111111
\(82\) −79060.3 −1.29845
\(83\) −102025. −1.62560 −0.812798 0.582546i \(-0.802056\pi\)
−0.812798 + 0.582546i \(0.802056\pi\)
\(84\) 3774.12 0.0583602
\(85\) −83786.1 −1.25784
\(86\) 129895. 1.89385
\(87\) −13929.0 −0.197298
\(88\) 40330.2 0.555168
\(89\) 104185. 1.39422 0.697111 0.716963i \(-0.254468\pi\)
0.697111 + 0.716963i \(0.254468\pi\)
\(90\) −31690.3 −0.412401
\(91\) 118908. 1.50524
\(92\) 1925.67 0.0237198
\(93\) −68123.8 −0.816755
\(94\) −137314. −1.60286
\(95\) 93639.5 1.06451
\(96\) 11429.3 0.126573
\(97\) −117345. −1.26629 −0.633147 0.774032i \(-0.718237\pi\)
−0.633147 + 0.774032i \(0.718237\pi\)
\(98\) 15734.2 0.165493
\(99\) 19247.2 0.197370
\(100\) 4171.22 0.0417122
\(101\) 109234. 1.06550 0.532751 0.846272i \(-0.321158\pi\)
0.532751 + 0.846272i \(0.321158\pi\)
\(102\) 68467.6 0.651605
\(103\) −162469. −1.50896 −0.754481 0.656322i \(-0.772111\pi\)
−0.754481 + 0.656322i \(0.772111\pi\)
\(104\) 169557. 1.53721
\(105\) 70318.3 0.622436
\(106\) −113550. −0.981570
\(107\) −9555.85 −0.0806882 −0.0403441 0.999186i \(-0.512845\pi\)
−0.0403441 + 0.999186i \(0.512845\pi\)
\(108\) 2568.38 0.0211885
\(109\) −80849.6 −0.651796 −0.325898 0.945405i \(-0.605667\pi\)
−0.325898 + 0.945405i \(0.605667\pi\)
\(110\) −92966.1 −0.732559
\(111\) −9109.93 −0.0701791
\(112\) −133824. −1.00806
\(113\) −131306. −0.967359 −0.483679 0.875245i \(-0.660700\pi\)
−0.483679 + 0.875245i \(0.660700\pi\)
\(114\) −76519.5 −0.551455
\(115\) 35878.5 0.252982
\(116\) −5452.70 −0.0376241
\(117\) 80919.7 0.546499
\(118\) 20747.2 0.137169
\(119\) −151924. −0.983466
\(120\) 100271. 0.635656
\(121\) −104588. −0.649407
\(122\) 43124.1 0.262314
\(123\) 119384. 0.711512
\(124\) −26667.9 −0.155752
\(125\) −127416. −0.729371
\(126\) −57462.0 −0.322444
\(127\) −234331. −1.28920 −0.644600 0.764520i \(-0.722976\pi\)
−0.644600 + 0.764520i \(0.722976\pi\)
\(128\) −209963. −1.13270
\(129\) −196146. −1.03778
\(130\) −390850. −2.02839
\(131\) 135156. 0.688108 0.344054 0.938950i \(-0.388200\pi\)
0.344054 + 0.938950i \(0.388200\pi\)
\(132\) 7534.57 0.0376377
\(133\) 169791. 0.832309
\(134\) −416457. −2.00359
\(135\) 47853.4 0.225984
\(136\) −216638. −1.00435
\(137\) −133343. −0.606971 −0.303486 0.952836i \(-0.598150\pi\)
−0.303486 + 0.952836i \(0.598150\pi\)
\(138\) −29318.8 −0.131054
\(139\) 197409. 0.866622 0.433311 0.901244i \(-0.357345\pi\)
0.433311 + 0.901244i \(0.357345\pi\)
\(140\) 27527.0 0.118697
\(141\) 207349. 0.878323
\(142\) −106246. −0.442171
\(143\) 237384. 0.970761
\(144\) −91070.6 −0.365992
\(145\) −101593. −0.401277
\(146\) −54177.1 −0.210346
\(147\) −23759.2 −0.0906857
\(148\) −3566.20 −0.0133829
\(149\) 94205.2 0.347624 0.173812 0.984779i \(-0.444392\pi\)
0.173812 + 0.984779i \(0.444392\pi\)
\(150\) −63508.1 −0.230463
\(151\) −32019.8 −0.114281 −0.0571407 0.998366i \(-0.518198\pi\)
−0.0571407 + 0.998366i \(0.518198\pi\)
\(152\) 242115. 0.849987
\(153\) −103388. −0.357062
\(154\) −168570. −0.572766
\(155\) −496869. −1.66116
\(156\) 31677.0 0.104216
\(157\) 487099. 1.57713 0.788565 0.614951i \(-0.210824\pi\)
0.788565 + 0.614951i \(0.210824\pi\)
\(158\) −187677. −0.598091
\(159\) 171464. 0.537873
\(160\) 83361.0 0.257432
\(161\) 65056.2 0.197799
\(162\) −39104.4 −0.117068
\(163\) 349125. 1.02923 0.514615 0.857421i \(-0.327935\pi\)
0.514615 + 0.857421i \(0.327935\pi\)
\(164\) 46734.2 0.135683
\(165\) 140382. 0.401422
\(166\) 608084. 1.71275
\(167\) 68091.2 0.188930 0.0944648 0.995528i \(-0.469886\pi\)
0.0944648 + 0.995528i \(0.469886\pi\)
\(168\) 181815. 0.497001
\(169\) 626725. 1.68795
\(170\) 499376. 1.32527
\(171\) 115547. 0.302182
\(172\) −76783.7 −0.197901
\(173\) 15052.9 0.0382388 0.0191194 0.999817i \(-0.493914\pi\)
0.0191194 + 0.999817i \(0.493914\pi\)
\(174\) 83018.9 0.207876
\(175\) 140920. 0.347837
\(176\) −267163. −0.650122
\(177\) −31329.0 −0.0751646
\(178\) −620959. −1.46897
\(179\) 716413. 1.67121 0.835604 0.549332i \(-0.185118\pi\)
0.835604 + 0.549332i \(0.185118\pi\)
\(180\) 18732.8 0.0430945
\(181\) −702418. −1.59367 −0.796837 0.604194i \(-0.793495\pi\)
−0.796837 + 0.604194i \(0.793495\pi\)
\(182\) −708704. −1.58594
\(183\) −65118.9 −0.143741
\(184\) 92767.5 0.202000
\(185\) −66444.3 −0.142734
\(186\) 406027. 0.860543
\(187\) −303298. −0.634258
\(188\) 81169.3 0.167493
\(189\) 86769.6 0.176691
\(190\) −558104. −1.12158
\(191\) −28509.8 −0.0565472 −0.0282736 0.999600i \(-0.509001\pi\)
−0.0282736 + 0.999600i \(0.509001\pi\)
\(192\) 255686. 0.500558
\(193\) −310039. −0.599133 −0.299567 0.954075i \(-0.596842\pi\)
−0.299567 + 0.954075i \(0.596842\pi\)
\(194\) 699390. 1.33418
\(195\) 590197. 1.11150
\(196\) −9300.85 −0.0172935
\(197\) 664034. 1.21906 0.609530 0.792763i \(-0.291358\pi\)
0.609530 + 0.792763i \(0.291358\pi\)
\(198\) −114716. −0.207951
\(199\) 326015. 0.583586 0.291793 0.956481i \(-0.405748\pi\)
0.291793 + 0.956481i \(0.405748\pi\)
\(200\) 200946. 0.355225
\(201\) 628864. 1.09791
\(202\) −651048. −1.12263
\(203\) −184213. −0.313747
\(204\) −40472.7 −0.0680905
\(205\) 870739. 1.44712
\(206\) 968338. 1.58986
\(207\) 44272.4 0.0718137
\(208\) −1.12321e6 −1.80013
\(209\) 338967. 0.536774
\(210\) −419106. −0.655807
\(211\) −775314. −1.19887 −0.599434 0.800424i \(-0.704608\pi\)
−0.599434 + 0.800424i \(0.704608\pi\)
\(212\) 67121.7 0.102571
\(213\) 160434. 0.242297
\(214\) 56954.1 0.0850141
\(215\) −1.43061e6 −2.11070
\(216\) 123730. 0.180443
\(217\) −900942. −1.29882
\(218\) 481874. 0.686740
\(219\) 81809.2 0.115264
\(220\) 54954.2 0.0765499
\(221\) −1.27513e6 −1.75620
\(222\) 54296.4 0.0739415
\(223\) −855406. −1.15189 −0.575944 0.817489i \(-0.695365\pi\)
−0.575944 + 0.817489i \(0.695365\pi\)
\(224\) 151153. 0.201279
\(225\) 95899.4 0.126287
\(226\) 782599. 1.01922
\(227\) 255022. 0.328483 0.164241 0.986420i \(-0.447482\pi\)
0.164241 + 0.986420i \(0.447482\pi\)
\(228\) 45232.3 0.0576251
\(229\) 504531. 0.635769 0.317884 0.948129i \(-0.397028\pi\)
0.317884 + 0.948129i \(0.397028\pi\)
\(230\) −213840. −0.266545
\(231\) 254546. 0.313860
\(232\) −262680. −0.320410
\(233\) 315787. 0.381070 0.190535 0.981680i \(-0.438978\pi\)
0.190535 + 0.981680i \(0.438978\pi\)
\(234\) −482292. −0.575798
\(235\) 1.51232e6 1.78638
\(236\) −12264.1 −0.0143336
\(237\) 283398. 0.327738
\(238\) 905488. 1.03619
\(239\) −1.31913e6 −1.49380 −0.746901 0.664935i \(-0.768459\pi\)
−0.746901 + 0.664935i \(0.768459\pi\)
\(240\) −664234. −0.744377
\(241\) −101157. −0.112190 −0.0560950 0.998425i \(-0.517865\pi\)
−0.0560950 + 0.998425i \(0.517865\pi\)
\(242\) 623356. 0.684223
\(243\) 59049.0 0.0641500
\(244\) −25491.6 −0.0274109
\(245\) −173291. −0.184442
\(246\) −711542. −0.749658
\(247\) 1.42509e6 1.48628
\(248\) −1.28471e6 −1.32640
\(249\) −918227. −0.938538
\(250\) 759415. 0.768474
\(251\) 1.78020e6 1.78354 0.891771 0.452487i \(-0.149463\pi\)
0.891771 + 0.452487i \(0.149463\pi\)
\(252\) 33967.1 0.0336943
\(253\) 129877. 0.127565
\(254\) 1.39664e6 1.35832
\(255\) −754075. −0.726213
\(256\) 342297. 0.326440
\(257\) −381509. −0.360307 −0.180153 0.983639i \(-0.557659\pi\)
−0.180153 + 0.983639i \(0.557659\pi\)
\(258\) 1.16905e6 1.09342
\(259\) −120479. −0.111600
\(260\) 231040. 0.211960
\(261\) −125361. −0.113910
\(262\) −805547. −0.724999
\(263\) −225601. −0.201118 −0.100559 0.994931i \(-0.532063\pi\)
−0.100559 + 0.994931i \(0.532063\pi\)
\(264\) 362972. 0.320526
\(265\) 1.25059e6 1.09396
\(266\) −1.01197e6 −0.876931
\(267\) 937669. 0.804954
\(268\) 246177. 0.209368
\(269\) −562259. −0.473757 −0.236879 0.971539i \(-0.576124\pi\)
−0.236879 + 0.971539i \(0.576124\pi\)
\(270\) −285213. −0.238100
\(271\) −1.36135e6 −1.12602 −0.563012 0.826449i \(-0.690358\pi\)
−0.563012 + 0.826449i \(0.690358\pi\)
\(272\) 1.43509e6 1.17614
\(273\) 1.07017e6 0.869051
\(274\) 794741. 0.639512
\(275\) 281329. 0.224327
\(276\) 17331.0 0.0136946
\(277\) −2.03987e6 −1.59736 −0.798681 0.601755i \(-0.794468\pi\)
−0.798681 + 0.601755i \(0.794468\pi\)
\(278\) −1.17658e6 −0.913083
\(279\) −613114. −0.471554
\(280\) 1.32609e6 1.01083
\(281\) −667020. −0.503933 −0.251967 0.967736i \(-0.581077\pi\)
−0.251967 + 0.967736i \(0.581077\pi\)
\(282\) −1.23583e6 −0.925411
\(283\) 1.41041e6 1.04684 0.523418 0.852076i \(-0.324657\pi\)
0.523418 + 0.852076i \(0.324657\pi\)
\(284\) 62804.0 0.0462053
\(285\) 842756. 0.614596
\(286\) −1.41484e6 −1.02281
\(287\) 1.57886e6 1.13146
\(288\) 102864. 0.0730771
\(289\) 209339. 0.147437
\(290\) 605508. 0.422790
\(291\) −1.05610e6 −0.731095
\(292\) 32025.2 0.0219804
\(293\) 928970. 0.632168 0.316084 0.948731i \(-0.397632\pi\)
0.316084 + 0.948731i \(0.397632\pi\)
\(294\) 141608. 0.0955476
\(295\) −228502. −0.152874
\(296\) −171799. −0.113970
\(297\) 173225. 0.113951
\(298\) −561475. −0.366261
\(299\) 546031. 0.353215
\(300\) 37541.0 0.0240826
\(301\) −2.59404e6 −1.65029
\(302\) 190842. 0.120408
\(303\) 983105. 0.615167
\(304\) −1.60386e6 −0.995366
\(305\) −474952. −0.292348
\(306\) 616208. 0.376204
\(307\) −756619. −0.458175 −0.229087 0.973406i \(-0.573574\pi\)
−0.229087 + 0.973406i \(0.573574\pi\)
\(308\) 99645.1 0.0598521
\(309\) −1.46222e6 −0.871199
\(310\) 2.96140e6 1.75022
\(311\) −1.45073e6 −0.850521 −0.425260 0.905071i \(-0.639818\pi\)
−0.425260 + 0.905071i \(0.639818\pi\)
\(312\) 1.52602e6 0.887509
\(313\) −401791. −0.231814 −0.115907 0.993260i \(-0.536977\pi\)
−0.115907 + 0.993260i \(0.536977\pi\)
\(314\) −2.90317e6 −1.66168
\(315\) 632864. 0.359364
\(316\) 110940. 0.0624985
\(317\) −1.81917e6 −1.01678 −0.508388 0.861128i \(-0.669759\pi\)
−0.508388 + 0.861128i \(0.669759\pi\)
\(318\) −1.02195e6 −0.566710
\(319\) −367758. −0.202342
\(320\) 1.86488e6 1.01807
\(321\) −86002.7 −0.0465853
\(322\) −387743. −0.208403
\(323\) −1.82079e6 −0.971078
\(324\) 23115.5 0.0122332
\(325\) 1.18277e6 0.621143
\(326\) −2.08083e6 −1.08441
\(327\) −727646. −0.376314
\(328\) 2.25139e6 1.15549
\(329\) 2.74220e6 1.39672
\(330\) −836695. −0.422943
\(331\) 3.27514e6 1.64309 0.821543 0.570147i \(-0.193114\pi\)
0.821543 + 0.570147i \(0.193114\pi\)
\(332\) −359451. −0.178976
\(333\) −81989.4 −0.0405179
\(334\) −405833. −0.199059
\(335\) 4.58669e6 2.23299
\(336\) −1.20441e6 −0.582006
\(337\) 1.40437e6 0.673606 0.336803 0.941575i \(-0.390654\pi\)
0.336803 + 0.941575i \(0.390654\pi\)
\(338\) −3.73536e6 −1.77845
\(339\) −1.18175e6 −0.558505
\(340\) −295192. −0.138486
\(341\) −1.79862e6 −0.837633
\(342\) −688675. −0.318383
\(343\) −2.31468e6 −1.06232
\(344\) −3.69900e6 −1.68534
\(345\) 322906. 0.146059
\(346\) −89717.1 −0.0402888
\(347\) 4.10924e6 1.83205 0.916025 0.401120i \(-0.131379\pi\)
0.916025 + 0.401120i \(0.131379\pi\)
\(348\) −49074.3 −0.0217223
\(349\) 65008.4 0.0285697 0.0142849 0.999898i \(-0.495453\pi\)
0.0142849 + 0.999898i \(0.495453\pi\)
\(350\) −839899. −0.366486
\(351\) 728277. 0.315521
\(352\) 301759. 0.129809
\(353\) 2.87105e6 1.22632 0.613160 0.789959i \(-0.289898\pi\)
0.613160 + 0.789959i \(0.289898\pi\)
\(354\) 186725. 0.0791943
\(355\) 1.17015e6 0.492799
\(356\) 367062. 0.153502
\(357\) −1.36732e6 −0.567805
\(358\) −4.26991e6 −1.76081
\(359\) −2.59406e6 −1.06229 −0.531146 0.847281i \(-0.678238\pi\)
−0.531146 + 0.847281i \(0.678238\pi\)
\(360\) 902439. 0.366996
\(361\) −441179. −0.178175
\(362\) 4.18651e6 1.67911
\(363\) −941289. −0.374935
\(364\) 418930. 0.165725
\(365\) 596685. 0.234430
\(366\) 388117. 0.151447
\(367\) 1.69904e6 0.658472 0.329236 0.944248i \(-0.393209\pi\)
0.329236 + 0.944248i \(0.393209\pi\)
\(368\) −614528. −0.236550
\(369\) 1.07445e6 0.410792
\(370\) 396017. 0.150387
\(371\) 2.26762e6 0.855334
\(372\) −240011. −0.0899237
\(373\) 1.30896e6 0.487142 0.243571 0.969883i \(-0.421681\pi\)
0.243571 + 0.969883i \(0.421681\pi\)
\(374\) 1.80770e6 0.668262
\(375\) −1.14674e6 −0.421103
\(376\) 3.91027e6 1.42639
\(377\) −1.54614e6 −0.560266
\(378\) −517158. −0.186163
\(379\) −2.85503e6 −1.02097 −0.510485 0.859887i \(-0.670534\pi\)
−0.510485 + 0.859887i \(0.670534\pi\)
\(380\) 329907. 0.117201
\(381\) −2.10898e6 −0.744320
\(382\) 169922. 0.0595788
\(383\) −1.29291e6 −0.450371 −0.225186 0.974316i \(-0.572299\pi\)
−0.225186 + 0.974316i \(0.572299\pi\)
\(384\) −1.88966e6 −0.653968
\(385\) 1.85656e6 0.638347
\(386\) 1.84788e6 0.631254
\(387\) −1.76531e6 −0.599162
\(388\) −413425. −0.139417
\(389\) −514685. −0.172452 −0.0862259 0.996276i \(-0.527481\pi\)
−0.0862259 + 0.996276i \(0.527481\pi\)
\(390\) −3.51765e6 −1.17109
\(391\) −697646. −0.230777
\(392\) −448061. −0.147273
\(393\) 1.21640e6 0.397280
\(394\) −3.95773e6 −1.28442
\(395\) 2.06700e6 0.666572
\(396\) 67811.1 0.0217302
\(397\) −3.92612e6 −1.25022 −0.625112 0.780535i \(-0.714947\pi\)
−0.625112 + 0.780535i \(0.714947\pi\)
\(398\) −1.94309e6 −0.614873
\(399\) 1.52812e6 0.480534
\(400\) −1.33114e6 −0.415982
\(401\) −2.41292e6 −0.749346 −0.374673 0.927157i \(-0.622245\pi\)
−0.374673 + 0.927157i \(0.622245\pi\)
\(402\) −3.74811e6 −1.15677
\(403\) −7.56181e6 −2.31933
\(404\) 384849. 0.117310
\(405\) 430681. 0.130472
\(406\) 1.09793e6 0.330567
\(407\) −240523. −0.0719730
\(408\) −1.94974e6 −0.579864
\(409\) 3.67029e6 1.08491 0.542454 0.840086i \(-0.317495\pi\)
0.542454 + 0.840086i \(0.317495\pi\)
\(410\) −5.18972e6 −1.52470
\(411\) −1.20009e6 −0.350435
\(412\) −572405. −0.166135
\(413\) −414328. −0.119528
\(414\) −263870. −0.0756638
\(415\) −6.69720e6 −1.90885
\(416\) 1.26866e6 0.359429
\(417\) 1.77668e6 0.500344
\(418\) −2.02029e6 −0.565551
\(419\) −1.90475e6 −0.530034 −0.265017 0.964244i \(-0.585378\pi\)
−0.265017 + 0.964244i \(0.585378\pi\)
\(420\) 247743. 0.0685295
\(421\) 4.91363e6 1.35113 0.675565 0.737300i \(-0.263900\pi\)
0.675565 + 0.737300i \(0.263900\pi\)
\(422\) 4.62097e6 1.26314
\(423\) 1.86614e6 0.507100
\(424\) 3.23354e6 0.873500
\(425\) −1.51118e6 −0.405831
\(426\) −956210. −0.255287
\(427\) −861202. −0.228578
\(428\) −33666.8 −0.00888367
\(429\) 2.13646e6 0.560469
\(430\) 8.52664e6 2.22386
\(431\) −3.65588e6 −0.947979 −0.473990 0.880530i \(-0.657187\pi\)
−0.473990 + 0.880530i \(0.657187\pi\)
\(432\) −819635. −0.211306
\(433\) −5.06298e6 −1.29774 −0.648869 0.760900i \(-0.724758\pi\)
−0.648869 + 0.760900i \(0.724758\pi\)
\(434\) 5.36973e6 1.36845
\(435\) −914338. −0.231677
\(436\) −284846. −0.0717619
\(437\) 779690. 0.195307
\(438\) −487594. −0.121443
\(439\) −3.94551e6 −0.977108 −0.488554 0.872534i \(-0.662475\pi\)
−0.488554 + 0.872534i \(0.662475\pi\)
\(440\) 2.64738e6 0.651905
\(441\) −213833. −0.0523574
\(442\) 7.59996e6 1.85036
\(443\) −3.37423e6 −0.816894 −0.408447 0.912782i \(-0.633930\pi\)
−0.408447 + 0.912782i \(0.633930\pi\)
\(444\) −32095.8 −0.00772663
\(445\) 6.83900e6 1.63716
\(446\) 5.09833e6 1.21364
\(447\) 847847. 0.200701
\(448\) 3.38147e6 0.795995
\(449\) −5.64111e6 −1.32053 −0.660266 0.751032i \(-0.729556\pi\)
−0.660266 + 0.751032i \(0.729556\pi\)
\(450\) −571573. −0.133058
\(451\) 3.15200e6 0.729700
\(452\) −462611. −0.106505
\(453\) −288178. −0.0659804
\(454\) −1.51996e6 −0.346094
\(455\) 7.80539e6 1.76753
\(456\) 2.17903e6 0.490740
\(457\) −134096. −0.0300348 −0.0150174 0.999887i \(-0.504780\pi\)
−0.0150174 + 0.999887i \(0.504780\pi\)
\(458\) −3.00707e6 −0.669854
\(459\) −930495. −0.206150
\(460\) 126406. 0.0278530
\(461\) 4.00954e6 0.878703 0.439351 0.898315i \(-0.355208\pi\)
0.439351 + 0.898315i \(0.355208\pi\)
\(462\) −1.51713e6 −0.330687
\(463\) 8.02971e6 1.74079 0.870397 0.492351i \(-0.163862\pi\)
0.870397 + 0.492351i \(0.163862\pi\)
\(464\) 1.74009e6 0.375212
\(465\) −4.47182e6 −0.959074
\(466\) −1.88213e6 −0.401500
\(467\) −1.17884e6 −0.250129 −0.125065 0.992149i \(-0.539914\pi\)
−0.125065 + 0.992149i \(0.539914\pi\)
\(468\) 285093. 0.0601689
\(469\) 8.31677e6 1.74591
\(470\) −9.01365e6 −1.88216
\(471\) 4.38389e6 0.910557
\(472\) −590815. −0.122067
\(473\) −5.17869e6 −1.06431
\(474\) −1.68909e6 −0.345308
\(475\) 1.68890e6 0.343456
\(476\) −535254. −0.108278
\(477\) 1.54317e6 0.310541
\(478\) 7.86219e6 1.57389
\(479\) −7.03708e6 −1.40137 −0.700686 0.713470i \(-0.747123\pi\)
−0.700686 + 0.713470i \(0.747123\pi\)
\(480\) 750249. 0.148629
\(481\) −1.01121e6 −0.199287
\(482\) 602910. 0.118205
\(483\) 585506. 0.114199
\(484\) −368479. −0.0714990
\(485\) −7.70281e6 −1.48694
\(486\) −351940. −0.0675893
\(487\) −1.74859e6 −0.334093 −0.167046 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(488\) −1.22804e6 −0.233433
\(489\) 3.14213e6 0.594226
\(490\) 1.03284e6 0.194330
\(491\) 5.71036e6 1.06896 0.534478 0.845183i \(-0.320508\pi\)
0.534478 + 0.845183i \(0.320508\pi\)
\(492\) 420608. 0.0783366
\(493\) 1.97545e6 0.366057
\(494\) −8.49373e6 −1.56596
\(495\) 1.26344e6 0.231761
\(496\) 8.51040e6 1.55327
\(497\) 2.12176e6 0.385305
\(498\) 5.47275e6 0.988855
\(499\) −3.18564e6 −0.572724 −0.286362 0.958121i \(-0.592446\pi\)
−0.286362 + 0.958121i \(0.592446\pi\)
\(500\) −448907. −0.0803029
\(501\) 612821. 0.109079
\(502\) −1.06102e7 −1.87916
\(503\) 7.57762e6 1.33541 0.667703 0.744428i \(-0.267278\pi\)
0.667703 + 0.744428i \(0.267278\pi\)
\(504\) 1.63634e6 0.286944
\(505\) 7.17039e6 1.25116
\(506\) −774083. −0.134404
\(507\) 5.64052e6 0.974539
\(508\) −825586. −0.141939
\(509\) −6.23348e6 −1.06644 −0.533220 0.845977i \(-0.679018\pi\)
−0.533220 + 0.845977i \(0.679018\pi\)
\(510\) 4.49439e6 0.765147
\(511\) 1.08193e6 0.183294
\(512\) 4.67867e6 0.788764
\(513\) 1.03992e6 0.174465
\(514\) 2.27385e6 0.379624
\(515\) −1.06649e7 −1.77190
\(516\) −691053. −0.114258
\(517\) 5.47448e6 0.900775
\(518\) 718073. 0.117583
\(519\) 135476. 0.0220772
\(520\) 1.11302e7 1.80507
\(521\) −4.48711e6 −0.724224 −0.362112 0.932135i \(-0.617944\pi\)
−0.362112 + 0.932135i \(0.617944\pi\)
\(522\) 747170. 0.120017
\(523\) −6.66955e6 −1.06621 −0.533104 0.846050i \(-0.678975\pi\)
−0.533104 + 0.846050i \(0.678975\pi\)
\(524\) 476176. 0.0757599
\(525\) 1.26828e6 0.200824
\(526\) 1.34461e6 0.211901
\(527\) 9.66147e6 1.51536
\(528\) −2.40447e6 −0.375348
\(529\) −6.13760e6 −0.953585
\(530\) −7.45369e6 −1.15261
\(531\) −281961. −0.0433963
\(532\) 598200. 0.0916363
\(533\) 1.32517e7 2.02048
\(534\) −5.58863e6 −0.848110
\(535\) −627270. −0.0947481
\(536\) 1.18594e7 1.78299
\(537\) 6.44772e6 0.964873
\(538\) 3.35114e6 0.499157
\(539\) −627297. −0.0930039
\(540\) 168595. 0.0248806
\(541\) −5.56886e6 −0.818038 −0.409019 0.912526i \(-0.634129\pi\)
−0.409019 + 0.912526i \(0.634129\pi\)
\(542\) 8.11385e6 1.18639
\(543\) −6.32177e6 −0.920108
\(544\) −1.62093e6 −0.234837
\(545\) −5.30717e6 −0.765371
\(546\) −6.37834e6 −0.915643
\(547\) 1.15071e7 1.64437 0.822184 0.569222i \(-0.192756\pi\)
0.822184 + 0.569222i \(0.192756\pi\)
\(548\) −469788. −0.0668268
\(549\) −586070. −0.0829887
\(550\) −1.67676e6 −0.236354
\(551\) −2.20776e6 −0.309794
\(552\) 834908. 0.116625
\(553\) 3.74796e6 0.521173
\(554\) 1.21579e7 1.68300
\(555\) −597999. −0.0824077
\(556\) 695503. 0.0954140
\(557\) 3.88277e6 0.530277 0.265139 0.964210i \(-0.414582\pi\)
0.265139 + 0.964210i \(0.414582\pi\)
\(558\) 3.65424e6 0.496835
\(559\) −2.17724e7 −2.94697
\(560\) −8.78454e6 −1.18372
\(561\) −2.72968e6 −0.366189
\(562\) 3.97553e6 0.530950
\(563\) 981269. 0.130472 0.0652360 0.997870i \(-0.479220\pi\)
0.0652360 + 0.997870i \(0.479220\pi\)
\(564\) 730524. 0.0967023
\(565\) −8.61924e6 −1.13592
\(566\) −8.40621e6 −1.10296
\(567\) 780927. 0.102012
\(568\) 3.02554e6 0.393488
\(569\) −8.63190e6 −1.11770 −0.558850 0.829268i \(-0.688757\pi\)
−0.558850 + 0.829268i \(0.688757\pi\)
\(570\) −5.02293e6 −0.647545
\(571\) 1.06342e7 1.36494 0.682469 0.730915i \(-0.260906\pi\)
0.682469 + 0.730915i \(0.260906\pi\)
\(572\) 836344. 0.106880
\(573\) −256588. −0.0326475
\(574\) −9.41019e6 −1.19212
\(575\) 647112. 0.0816225
\(576\) 2.30118e6 0.288997
\(577\) 3.83248e6 0.479225 0.239613 0.970869i \(-0.422980\pi\)
0.239613 + 0.970869i \(0.422980\pi\)
\(578\) −1.24769e6 −0.155341
\(579\) −2.79035e6 −0.345910
\(580\) −357929. −0.0441801
\(581\) −1.21436e7 −1.49248
\(582\) 6.29451e6 0.770291
\(583\) 4.52703e6 0.551622
\(584\) 1.54279e6 0.187187
\(585\) 5.31177e6 0.641726
\(586\) −5.53678e6 −0.666060
\(587\) −3.74170e6 −0.448202 −0.224101 0.974566i \(-0.571945\pi\)
−0.224101 + 0.974566i \(0.571945\pi\)
\(588\) −83707.6 −0.00998439
\(589\) −1.07977e7 −1.28245
\(590\) 1.36190e6 0.161070
\(591\) 5.97631e6 0.703825
\(592\) 1.13806e6 0.133463
\(593\) −411565. −0.0480620 −0.0240310 0.999711i \(-0.507650\pi\)
−0.0240310 + 0.999711i \(0.507650\pi\)
\(594\) −1.03244e6 −0.120061
\(595\) −9.97269e6 −1.15483
\(596\) 331900. 0.0382729
\(597\) 2.93414e6 0.336934
\(598\) −3.25442e6 −0.372152
\(599\) 6.06004e6 0.690094 0.345047 0.938585i \(-0.387863\pi\)
0.345047 + 0.938585i \(0.387863\pi\)
\(600\) 1.80851e6 0.205089
\(601\) 6.69768e6 0.756376 0.378188 0.925729i \(-0.376547\pi\)
0.378188 + 0.925729i \(0.376547\pi\)
\(602\) 1.54608e7 1.73877
\(603\) 5.65978e6 0.633878
\(604\) −112811. −0.0125822
\(605\) −6.86540e6 −0.762566
\(606\) −5.85943e6 −0.648148
\(607\) −1.00236e6 −0.110421 −0.0552106 0.998475i \(-0.517583\pi\)
−0.0552106 + 0.998475i \(0.517583\pi\)
\(608\) 1.81155e6 0.198743
\(609\) −1.65791e6 −0.181142
\(610\) 2.83078e6 0.308022
\(611\) 2.30159e7 2.49417
\(612\) −364254. −0.0393121
\(613\) −1.24680e7 −1.34013 −0.670064 0.742303i \(-0.733733\pi\)
−0.670064 + 0.742303i \(0.733733\pi\)
\(614\) 4.50955e6 0.482739
\(615\) 7.83665e6 0.835492
\(616\) 4.80033e6 0.509705
\(617\) −7.98964e6 −0.844917 −0.422459 0.906382i \(-0.638833\pi\)
−0.422459 + 0.906382i \(0.638833\pi\)
\(618\) 8.71504e6 0.917906
\(619\) −8.83016e6 −0.926279 −0.463140 0.886285i \(-0.653277\pi\)
−0.463140 + 0.886285i \(0.653277\pi\)
\(620\) −1.75055e6 −0.182892
\(621\) 398452. 0.0414617
\(622\) 8.64653e6 0.896119
\(623\) 1.24007e7 1.28005
\(624\) −1.01089e7 −1.03931
\(625\) −1.20637e7 −1.23533
\(626\) 2.39473e6 0.244242
\(627\) 3.05070e6 0.309906
\(628\) 1.71613e6 0.173640
\(629\) 1.29199e6 0.130207
\(630\) −3.77195e6 −0.378630
\(631\) −8.18563e6 −0.818425 −0.409212 0.912439i \(-0.634196\pi\)
−0.409212 + 0.912439i \(0.634196\pi\)
\(632\) 5.34444e6 0.532242
\(633\) −6.97783e6 −0.692167
\(634\) 1.08425e7 1.07129
\(635\) −1.53821e7 −1.51384
\(636\) 604095. 0.0592192
\(637\) −2.63730e6 −0.257520
\(638\) 2.19189e6 0.213190
\(639\) 1.44391e6 0.139890
\(640\) −1.37825e7 −1.33008
\(641\) −1.08929e7 −1.04713 −0.523564 0.851986i \(-0.675398\pi\)
−0.523564 + 0.851986i \(0.675398\pi\)
\(642\) 512587. 0.0490829
\(643\) 4.87772e6 0.465254 0.232627 0.972566i \(-0.425268\pi\)
0.232627 + 0.972566i \(0.425268\pi\)
\(644\) 229204. 0.0217774
\(645\) −1.28755e7 −1.21861
\(646\) 1.08522e7 1.02314
\(647\) −5.09261e6 −0.478277 −0.239138 0.970985i \(-0.576865\pi\)
−0.239138 + 0.970985i \(0.576865\pi\)
\(648\) 1.11357e6 0.104179
\(649\) −827156. −0.0770860
\(650\) −7.04946e6 −0.654444
\(651\) −8.10848e6 −0.749871
\(652\) 1.23002e6 0.113317
\(653\) −1.57557e7 −1.44595 −0.722976 0.690873i \(-0.757226\pi\)
−0.722976 + 0.690873i \(0.757226\pi\)
\(654\) 4.33687e6 0.396489
\(655\) 8.87198e6 0.808011
\(656\) −1.49141e7 −1.35312
\(657\) 736283. 0.0665474
\(658\) −1.63439e7 −1.47160
\(659\) 1.52866e7 1.37119 0.685596 0.727983i \(-0.259542\pi\)
0.685596 + 0.727983i \(0.259542\pi\)
\(660\) 494588. 0.0441961
\(661\) −7.43781e6 −0.662127 −0.331064 0.943608i \(-0.607408\pi\)
−0.331064 + 0.943608i \(0.607408\pi\)
\(662\) −1.95203e7 −1.73117
\(663\) −1.14762e7 −1.01395
\(664\) −1.73163e7 −1.52418
\(665\) 1.11455e7 0.977339
\(666\) 488667. 0.0426902
\(667\) −845916. −0.0736229
\(668\) 239896. 0.0208009
\(669\) −7.69866e6 −0.665043
\(670\) −2.73373e7 −2.35271
\(671\) −1.71929e6 −0.147415
\(672\) 1.36038e6 0.116208
\(673\) −2.00568e7 −1.70697 −0.853483 0.521121i \(-0.825514\pi\)
−0.853483 + 0.521121i \(0.825514\pi\)
\(674\) −8.37021e6 −0.709719
\(675\) 863095. 0.0729120
\(676\) 2.20805e6 0.185841
\(677\) 2.08686e6 0.174993 0.0874965 0.996165i \(-0.472113\pi\)
0.0874965 + 0.996165i \(0.472113\pi\)
\(678\) 7.04339e6 0.588448
\(679\) −1.39670e7 −1.16260
\(680\) −1.42207e7 −1.17936
\(681\) 2.29520e6 0.189650
\(682\) 1.07200e7 0.882540
\(683\) 5.45861e6 0.447745 0.223872 0.974618i \(-0.428130\pi\)
0.223872 + 0.974618i \(0.428130\pi\)
\(684\) 407091. 0.0332699
\(685\) −8.75296e6 −0.712736
\(686\) 1.37958e7 1.11927
\(687\) 4.54078e6 0.367061
\(688\) 2.45036e7 1.97360
\(689\) 1.90326e7 1.52739
\(690\) −1.92456e6 −0.153890
\(691\) 655073. 0.0521908 0.0260954 0.999659i \(-0.491693\pi\)
0.0260954 + 0.999659i \(0.491693\pi\)
\(692\) 53033.7 0.00421004
\(693\) 2.29091e6 0.181207
\(694\) −2.44916e7 −1.93027
\(695\) 1.29584e7 1.01763
\(696\) −2.36412e6 −0.184989
\(697\) −1.69313e7 −1.32010
\(698\) −387459. −0.0301014
\(699\) 2.84208e6 0.220011
\(700\) 496482. 0.0382965
\(701\) −2.15464e7 −1.65607 −0.828037 0.560674i \(-0.810542\pi\)
−0.828037 + 0.560674i \(0.810542\pi\)
\(702\) −4.34063e6 −0.332437
\(703\) −1.44393e6 −0.110194
\(704\) 6.75069e6 0.513354
\(705\) 1.36109e7 1.03137
\(706\) −1.71118e7 −1.29207
\(707\) 1.30016e7 0.978248
\(708\) −110377. −0.00827553
\(709\) 1.65045e7 1.23307 0.616533 0.787329i \(-0.288537\pi\)
0.616533 + 0.787329i \(0.288537\pi\)
\(710\) −6.97423e6 −0.519219
\(711\) 2.55058e6 0.189219
\(712\) 1.76829e7 1.30724
\(713\) −4.13719e6 −0.304776
\(714\) 8.14939e6 0.598246
\(715\) 1.55825e7 1.13992
\(716\) 2.52404e6 0.183998
\(717\) −1.18722e7 −0.862447
\(718\) 1.54609e7 1.11924
\(719\) 3.65458e6 0.263642 0.131821 0.991274i \(-0.457918\pi\)
0.131821 + 0.991274i \(0.457918\pi\)
\(720\) −5.97811e6 −0.429766
\(721\) −1.93380e7 −1.38539
\(722\) 2.62948e6 0.187727
\(723\) −910415. −0.0647729
\(724\) −2.47473e6 −0.175462
\(725\) −1.83236e6 −0.129469
\(726\) 5.61021e6 0.395037
\(727\) 1.40192e7 0.983757 0.491878 0.870664i \(-0.336310\pi\)
0.491878 + 0.870664i \(0.336310\pi\)
\(728\) 2.01817e7 1.41133
\(729\) 531441. 0.0370370
\(730\) −3.55632e6 −0.246998
\(731\) 2.78178e7 1.92544
\(732\) −229425. −0.0158257
\(733\) −2.41229e7 −1.65833 −0.829163 0.559007i \(-0.811183\pi\)
−0.829163 + 0.559007i \(0.811183\pi\)
\(734\) −1.01265e7 −0.693774
\(735\) −1.55962e6 −0.106488
\(736\) 694106. 0.0472315
\(737\) 1.66034e7 1.12597
\(738\) −6.40388e6 −0.432815
\(739\) −1.00291e7 −0.675537 −0.337768 0.941229i \(-0.609672\pi\)
−0.337768 + 0.941229i \(0.609672\pi\)
\(740\) −234094. −0.0157149
\(741\) 1.28258e7 0.858104
\(742\) −1.35153e7 −0.901190
\(743\) 4.51165e6 0.299822 0.149911 0.988699i \(-0.452101\pi\)
0.149911 + 0.988699i \(0.452101\pi\)
\(744\) −1.15624e7 −0.765798
\(745\) 6.18387e6 0.408197
\(746\) −7.80160e6 −0.513259
\(747\) −8.26404e6 −0.541865
\(748\) −1.06857e6 −0.0698310
\(749\) −1.13739e6 −0.0740807
\(750\) 6.83474e6 0.443679
\(751\) 1.58055e7 1.02260 0.511302 0.859401i \(-0.329163\pi\)
0.511302 + 0.859401i \(0.329163\pi\)
\(752\) −2.59031e7 −1.67035
\(753\) 1.60218e7 1.02973
\(754\) 9.21518e6 0.590304
\(755\) −2.10186e6 −0.134195
\(756\) 305703. 0.0194534
\(757\) 7.71419e6 0.489272 0.244636 0.969615i \(-0.421331\pi\)
0.244636 + 0.969615i \(0.421331\pi\)
\(758\) 1.70163e7 1.07571
\(759\) 1.16889e6 0.0736495
\(760\) 1.58930e7 0.998097
\(761\) −1.26944e7 −0.794603 −0.397302 0.917688i \(-0.630053\pi\)
−0.397302 + 0.917688i \(0.630053\pi\)
\(762\) 1.25698e7 0.784225
\(763\) −9.62316e6 −0.598421
\(764\) −100445. −0.00622578
\(765\) −6.78668e6 −0.419279
\(766\) 7.70590e6 0.474517
\(767\) −3.47755e6 −0.213444
\(768\) 3.08067e6 0.188470
\(769\) −3.29528e6 −0.200944 −0.100472 0.994940i \(-0.532035\pi\)
−0.100472 + 0.994940i \(0.532035\pi\)
\(770\) −1.10653e7 −0.672571
\(771\) −3.43358e6 −0.208023
\(772\) −1.09232e6 −0.0659639
\(773\) −5.51865e6 −0.332188 −0.166094 0.986110i \(-0.553116\pi\)
−0.166094 + 0.986110i \(0.553116\pi\)
\(774\) 1.05215e7 0.631284
\(775\) −8.96164e6 −0.535961
\(776\) −1.99164e7 −1.18729
\(777\) −1.08431e6 −0.0644322
\(778\) 3.06759e6 0.181697
\(779\) 1.89224e7 1.11720
\(780\) 2.07936e6 0.122375
\(781\) 4.23583e6 0.248491
\(782\) 4.15806e6 0.243150
\(783\) −1.12825e6 −0.0657661
\(784\) 2.96813e6 0.172462
\(785\) 3.19744e7 1.85194
\(786\) −7.24992e6 −0.418579
\(787\) 1.63062e7 0.938459 0.469230 0.883076i \(-0.344532\pi\)
0.469230 + 0.883076i \(0.344532\pi\)
\(788\) 2.33950e6 0.134217
\(789\) −2.03041e6 −0.116116
\(790\) −1.23196e7 −0.702309
\(791\) −1.56287e7 −0.888143
\(792\) 3.26675e6 0.185056
\(793\) −7.22826e6 −0.408179
\(794\) 2.34002e7 1.31725
\(795\) 1.12553e7 0.631597
\(796\) 1.14860e6 0.0642521
\(797\) 3.19084e7 1.77934 0.889671 0.456602i \(-0.150934\pi\)
0.889671 + 0.456602i \(0.150934\pi\)
\(798\) −9.10777e6 −0.506297
\(799\) −2.94067e7 −1.62959
\(800\) 1.50352e6 0.0830584
\(801\) 8.43902e6 0.464741
\(802\) 1.43813e7 0.789520
\(803\) 2.15995e6 0.118210
\(804\) 2.21559e6 0.120879
\(805\) 4.27045e6 0.232265
\(806\) 4.50694e7 2.44368
\(807\) −5.06033e6 −0.273524
\(808\) 1.85398e7 0.999025
\(809\) 1.70126e7 0.913900 0.456950 0.889492i \(-0.348942\pi\)
0.456950 + 0.889492i \(0.348942\pi\)
\(810\) −2.56691e6 −0.137467
\(811\) 1.05241e7 0.561865 0.280932 0.959728i \(-0.409356\pi\)
0.280932 + 0.959728i \(0.409356\pi\)
\(812\) −649010. −0.0345431
\(813\) −1.22522e7 −0.650110
\(814\) 1.43355e6 0.0758317
\(815\) 2.29175e7 1.20857
\(816\) 1.29158e7 0.679043
\(817\) −3.10892e7 −1.62950
\(818\) −2.18754e7 −1.14307
\(819\) 9.63151e6 0.501747
\(820\) 3.06775e6 0.159326
\(821\) 3.31358e7 1.71569 0.857847 0.513906i \(-0.171802\pi\)
0.857847 + 0.513906i \(0.171802\pi\)
\(822\) 7.15266e6 0.369223
\(823\) 2.83322e7 1.45808 0.729039 0.684472i \(-0.239967\pi\)
0.729039 + 0.684472i \(0.239967\pi\)
\(824\) −2.75752e7 −1.41482
\(825\) 2.53196e6 0.129516
\(826\) 2.46945e6 0.125936
\(827\) 8.10406e6 0.412039 0.206020 0.978548i \(-0.433949\pi\)
0.206020 + 0.978548i \(0.433949\pi\)
\(828\) 155979. 0.00790661
\(829\) −2.88681e7 −1.45892 −0.729461 0.684022i \(-0.760229\pi\)
−0.729461 + 0.684022i \(0.760229\pi\)
\(830\) 3.99162e7 2.01119
\(831\) −1.83588e7 −0.922237
\(832\) 2.83814e7 1.42143
\(833\) 3.36959e6 0.168253
\(834\) −1.05892e7 −0.527169
\(835\) 4.46968e6 0.221851
\(836\) 1.19423e6 0.0590981
\(837\) −5.51803e6 −0.272252
\(838\) 1.13526e7 0.558450
\(839\) −3.01221e7 −1.47734 −0.738671 0.674066i \(-0.764546\pi\)
−0.738671 + 0.674066i \(0.764546\pi\)
\(840\) 1.19348e7 0.583603
\(841\) −1.81159e7 −0.883220
\(842\) −2.92859e7 −1.42357
\(843\) −6.00318e6 −0.290946
\(844\) −2.73156e6 −0.131994
\(845\) 4.11398e7 1.98208
\(846\) −1.11224e7 −0.534287
\(847\) −1.24486e7 −0.596228
\(848\) −2.14202e7 −1.02290
\(849\) 1.26937e7 0.604391
\(850\) 9.00686e6 0.427589
\(851\) −553249. −0.0261877
\(852\) 565236. 0.0266766
\(853\) −3.03837e7 −1.42977 −0.714887 0.699240i \(-0.753522\pi\)
−0.714887 + 0.699240i \(0.753522\pi\)
\(854\) 5.13288e6 0.240833
\(855\) 7.58480e6 0.354837
\(856\) −1.62187e6 −0.0756541
\(857\) −3.42703e7 −1.59392 −0.796958 0.604035i \(-0.793559\pi\)
−0.796958 + 0.604035i \(0.793559\pi\)
\(858\) −1.27336e7 −0.590517
\(859\) −7.21405e6 −0.333577 −0.166789 0.985993i \(-0.553340\pi\)
−0.166789 + 0.985993i \(0.553340\pi\)
\(860\) −5.04028e6 −0.232385
\(861\) 1.42097e7 0.653247
\(862\) 2.17895e7 0.998802
\(863\) −3.87581e7 −1.77148 −0.885739 0.464184i \(-0.846348\pi\)
−0.885739 + 0.464184i \(0.846348\pi\)
\(864\) 925774. 0.0421911
\(865\) 988108. 0.0449019
\(866\) 3.01761e7 1.36731
\(867\) 1.88405e6 0.0851226
\(868\) −3.17416e6 −0.142998
\(869\) 7.48235e6 0.336115
\(870\) 5.44957e6 0.244098
\(871\) 6.98045e7 3.11773
\(872\) −1.37222e7 −0.611131
\(873\) −9.50493e6 −0.422098
\(874\) −4.64706e6 −0.205778
\(875\) −1.51657e7 −0.669644
\(876\) 288227. 0.0126904
\(877\) −4.06345e6 −0.178400 −0.0892002 0.996014i \(-0.528431\pi\)
−0.0892002 + 0.996014i \(0.528431\pi\)
\(878\) 2.35158e7 1.02949
\(879\) 8.36073e6 0.364982
\(880\) −1.75373e7 −0.763405
\(881\) −4.47103e7 −1.94074 −0.970372 0.241616i \(-0.922323\pi\)
−0.970372 + 0.241616i \(0.922323\pi\)
\(882\) 1.27447e6 0.0551644
\(883\) −6.18489e6 −0.266950 −0.133475 0.991052i \(-0.542614\pi\)
−0.133475 + 0.991052i \(0.542614\pi\)
\(884\) −4.49250e6 −0.193356
\(885\) −2.05652e6 −0.0882620
\(886\) 2.01109e7 0.860690
\(887\) 4.16533e7 1.77763 0.888813 0.458270i \(-0.151531\pi\)
0.888813 + 0.458270i \(0.151531\pi\)
\(888\) −1.54619e6 −0.0658007
\(889\) −2.78914e7 −1.18363
\(890\) −4.07613e7 −1.72494
\(891\) 1.55903e6 0.0657899
\(892\) −3.01374e6 −0.126822
\(893\) 3.28650e7 1.37913
\(894\) −5.05328e6 −0.211461
\(895\) 4.70272e7 1.96242
\(896\) −2.49909e7 −1.03995
\(897\) 4.91428e6 0.203929
\(898\) 3.36218e7 1.39133
\(899\) 1.17148e7 0.483433
\(900\) 337869. 0.0139041
\(901\) −2.43174e7 −0.997941
\(902\) −1.87863e7 −0.768821
\(903\) −2.33464e7 −0.952796
\(904\) −2.22860e7 −0.907006
\(905\) −4.61085e7 −1.87137
\(906\) 1.71758e6 0.0695178
\(907\) 2.48069e7 1.00128 0.500639 0.865656i \(-0.333098\pi\)
0.500639 + 0.865656i \(0.333098\pi\)
\(908\) 898483. 0.0361656
\(909\) 8.84794e6 0.355167
\(910\) −4.65212e7 −1.86229
\(911\) 2.87038e7 1.14589 0.572946 0.819593i \(-0.305800\pi\)
0.572946 + 0.819593i \(0.305800\pi\)
\(912\) −1.44348e7 −0.574675
\(913\) −2.42432e7 −0.962529
\(914\) 799228. 0.0316450
\(915\) −4.27457e6 −0.168787
\(916\) 1.77754e6 0.0699974
\(917\) 1.60870e7 0.631760
\(918\) 5.54587e6 0.217202
\(919\) 3.98850e7 1.55783 0.778917 0.627127i \(-0.215769\pi\)
0.778917 + 0.627127i \(0.215769\pi\)
\(920\) 6.08950e6 0.237198
\(921\) −6.80957e6 −0.264527
\(922\) −2.38974e7 −0.925812
\(923\) 1.78084e7 0.688050
\(924\) 896806. 0.0345556
\(925\) −1.19840e6 −0.0460521
\(926\) −4.78581e7 −1.83412
\(927\) −1.31600e7 −0.502987
\(928\) −1.96542e6 −0.0749180
\(929\) −3.09058e7 −1.17490 −0.587451 0.809260i \(-0.699868\pi\)
−0.587451 + 0.809260i \(0.699868\pi\)
\(930\) 2.66526e7 1.01049
\(931\) −3.76585e6 −0.142393
\(932\) 1.11257e6 0.0419553
\(933\) −1.30566e7 −0.491048
\(934\) 7.02606e6 0.263539
\(935\) −1.99093e7 −0.744777
\(936\) 1.37341e7 0.512403
\(937\) −3.92068e7 −1.45886 −0.729428 0.684058i \(-0.760214\pi\)
−0.729428 + 0.684058i \(0.760214\pi\)
\(938\) −4.95690e7 −1.83951
\(939\) −3.61612e6 −0.133838
\(940\) 5.32816e6 0.196679
\(941\) −4.21602e7 −1.55213 −0.776066 0.630652i \(-0.782788\pi\)
−0.776066 + 0.630652i \(0.782788\pi\)
\(942\) −2.61285e7 −0.959374
\(943\) 7.25021e6 0.265504
\(944\) 3.91379e6 0.142944
\(945\) 5.69578e6 0.207479
\(946\) 3.08657e7 1.12137
\(947\) 5.32912e7 1.93099 0.965497 0.260416i \(-0.0838596\pi\)
0.965497 + 0.260416i \(0.0838596\pi\)
\(948\) 998458. 0.0360835
\(949\) 9.08090e6 0.327313
\(950\) −1.00661e7 −0.361869
\(951\) −1.63725e7 −0.587036
\(952\) −2.57854e7 −0.922109
\(953\) 3.66273e7 1.30639 0.653194 0.757190i \(-0.273428\pi\)
0.653194 + 0.757190i \(0.273428\pi\)
\(954\) −9.19752e6 −0.327190
\(955\) −1.87146e6 −0.0664005
\(956\) −4.64751e6 −0.164466
\(957\) −3.30982e6 −0.116822
\(958\) 4.19419e7 1.47650
\(959\) −1.58712e7 −0.557267
\(960\) 1.67839e7 0.587780
\(961\) 2.86654e7 1.00127
\(962\) 6.02695e6 0.209971
\(963\) −774024. −0.0268961
\(964\) −356393. −0.0123520
\(965\) −2.03518e7 −0.703532
\(966\) −3.48969e6 −0.120322
\(967\) −3.44677e7 −1.18535 −0.592675 0.805442i \(-0.701928\pi\)
−0.592675 + 0.805442i \(0.701928\pi\)
\(968\) −1.77512e7 −0.608891
\(969\) −1.63871e7 −0.560652
\(970\) 4.59098e7 1.56666
\(971\) 9.21316e6 0.313589 0.156794 0.987631i \(-0.449884\pi\)
0.156794 + 0.987631i \(0.449884\pi\)
\(972\) 208039. 0.00706284
\(973\) 2.34967e7 0.795655
\(974\) 1.04219e7 0.352004
\(975\) 1.06449e7 0.358617
\(976\) 8.13501e6 0.273359
\(977\) 3.66542e6 0.122854 0.0614268 0.998112i \(-0.480435\pi\)
0.0614268 + 0.998112i \(0.480435\pi\)
\(978\) −1.87275e7 −0.626084
\(979\) 2.47566e7 0.825531
\(980\) −610531. −0.0203069
\(981\) −6.54881e6 −0.217265
\(982\) −3.40345e7 −1.12626
\(983\) −7.59961e6 −0.250846 −0.125423 0.992103i \(-0.540029\pi\)
−0.125423 + 0.992103i \(0.540029\pi\)
\(984\) 2.02625e7 0.667121
\(985\) 4.35889e7 1.43148
\(986\) −1.17739e7 −0.385682
\(987\) 2.46798e7 0.806398
\(988\) 5.02083e6 0.163638
\(989\) −1.19120e7 −0.387252
\(990\) −7.53025e6 −0.244186
\(991\) 3.53749e7 1.14422 0.572112 0.820175i \(-0.306124\pi\)
0.572112 + 0.820175i \(0.306124\pi\)
\(992\) −9.61245e6 −0.310138
\(993\) 2.94763e7 0.948636
\(994\) −1.26459e7 −0.405962
\(995\) 2.14005e7 0.685276
\(996\) −3.23506e6 −0.103332
\(997\) 3.58278e7 1.14152 0.570759 0.821118i \(-0.306649\pi\)
0.570759 + 0.821118i \(0.306649\pi\)
\(998\) 1.89868e7 0.603429
\(999\) −737904. −0.0233930
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 177.6.a.c.1.3 12
3.2 odd 2 531.6.a.c.1.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.c.1.3 12 1.1 even 1 trivial
531.6.a.c.1.10 12 3.2 odd 2