Properties

Label 177.6.a.b.1.11
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $1$
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + 3566264 x^{5} + 15496192 x^{4} - 53008480 x^{3} - 16576192 x^{2} + 120303168 x - 50564480\)
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 1.11
Root \(-8.78000\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.78000 q^{2} -9.00000 q^{3} +45.0883 q^{4} -17.5207 q^{5} -79.0200 q^{6} +12.9559 q^{7} +114.916 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+8.78000 q^{2} -9.00000 q^{3} +45.0883 q^{4} -17.5207 q^{5} -79.0200 q^{6} +12.9559 q^{7} +114.916 q^{8} +81.0000 q^{9} -153.831 q^{10} -434.226 q^{11} -405.795 q^{12} -637.700 q^{13} +113.753 q^{14} +157.686 q^{15} -433.869 q^{16} -850.752 q^{17} +711.180 q^{18} +1822.12 q^{19} -789.977 q^{20} -116.603 q^{21} -3812.51 q^{22} +715.863 q^{23} -1034.24 q^{24} -2818.03 q^{25} -5599.00 q^{26} -729.000 q^{27} +584.162 q^{28} +1133.86 q^{29} +1384.48 q^{30} -9559.39 q^{31} -7486.66 q^{32} +3908.04 q^{33} -7469.60 q^{34} -226.997 q^{35} +3652.16 q^{36} -107.111 q^{37} +15998.2 q^{38} +5739.30 q^{39} -2013.40 q^{40} -6614.88 q^{41} -1023.78 q^{42} -2995.66 q^{43} -19578.5 q^{44} -1419.17 q^{45} +6285.27 q^{46} +10180.6 q^{47} +3904.82 q^{48} -16639.1 q^{49} -24742.3 q^{50} +7656.77 q^{51} -28752.8 q^{52} +11009.5 q^{53} -6400.62 q^{54} +7607.93 q^{55} +1488.84 q^{56} -16399.1 q^{57} +9955.28 q^{58} -3481.00 q^{59} +7109.80 q^{60} +2294.87 q^{61} -83931.4 q^{62} +1049.43 q^{63} -51849.1 q^{64} +11172.9 q^{65} +34312.6 q^{66} +23569.8 q^{67} -38359.0 q^{68} -6442.76 q^{69} -1993.03 q^{70} +56039.7 q^{71} +9308.16 q^{72} +56572.1 q^{73} -940.434 q^{74} +25362.2 q^{75} +82156.5 q^{76} -5625.81 q^{77} +50391.0 q^{78} +65746.0 q^{79} +7601.67 q^{80} +6561.00 q^{81} -58078.6 q^{82} -13649.8 q^{83} -5257.45 q^{84} +14905.7 q^{85} -26301.9 q^{86} -10204.7 q^{87} -49899.4 q^{88} +77607.3 q^{89} -12460.3 q^{90} -8262.00 q^{91} +32277.1 q^{92} +86034.5 q^{93} +89386.0 q^{94} -31924.8 q^{95} +67380.0 q^{96} +41043.1 q^{97} -146092. q^{98} -35172.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.78000 1.55210 0.776049 0.630672i \(-0.217221\pi\)
0.776049 + 0.630672i \(0.217221\pi\)
\(3\) −9.00000 −0.577350
\(4\) 45.0883 1.40901
\(5\) −17.5207 −0.313419 −0.156710 0.987645i \(-0.550089\pi\)
−0.156710 + 0.987645i \(0.550089\pi\)
\(6\) −79.0200 −0.896105
\(7\) 12.9559 0.0999364 0.0499682 0.998751i \(-0.484088\pi\)
0.0499682 + 0.998751i \(0.484088\pi\)
\(8\) 114.916 0.634825
\(9\) 81.0000 0.333333
\(10\) −153.831 −0.486457
\(11\) −434.226 −1.08202 −0.541009 0.841017i \(-0.681958\pi\)
−0.541009 + 0.841017i \(0.681958\pi\)
\(12\) −405.795 −0.813493
\(13\) −637.700 −1.04655 −0.523273 0.852165i \(-0.675289\pi\)
−0.523273 + 0.852165i \(0.675289\pi\)
\(14\) 113.753 0.155111
\(15\) 157.686 0.180953
\(16\) −433.869 −0.423700
\(17\) −850.752 −0.713971 −0.356986 0.934110i \(-0.616196\pi\)
−0.356986 + 0.934110i \(0.616196\pi\)
\(18\) 711.180 0.517366
\(19\) 1822.12 1.15796 0.578980 0.815341i \(-0.303451\pi\)
0.578980 + 0.815341i \(0.303451\pi\)
\(20\) −789.977 −0.441611
\(21\) −116.603 −0.0576983
\(22\) −3812.51 −1.67940
\(23\) 715.863 0.282170 0.141085 0.989998i \(-0.454941\pi\)
0.141085 + 0.989998i \(0.454941\pi\)
\(24\) −1034.24 −0.366516
\(25\) −2818.03 −0.901768
\(26\) −5599.00 −1.62434
\(27\) −729.000 −0.192450
\(28\) 584.162 0.140811
\(29\) 1133.86 0.250360 0.125180 0.992134i \(-0.460049\pi\)
0.125180 + 0.992134i \(0.460049\pi\)
\(30\) 1384.48 0.280856
\(31\) −9559.39 −1.78659 −0.893297 0.449467i \(-0.851614\pi\)
−0.893297 + 0.449467i \(0.851614\pi\)
\(32\) −7486.66 −1.29245
\(33\) 3908.04 0.624703
\(34\) −7469.60 −1.10815
\(35\) −226.997 −0.0313220
\(36\) 3652.16 0.469670
\(37\) −107.111 −0.0128626 −0.00643132 0.999979i \(-0.502047\pi\)
−0.00643132 + 0.999979i \(0.502047\pi\)
\(38\) 15998.2 1.79727
\(39\) 5739.30 0.604223
\(40\) −2013.40 −0.198966
\(41\) −6614.88 −0.614557 −0.307279 0.951620i \(-0.599418\pi\)
−0.307279 + 0.951620i \(0.599418\pi\)
\(42\) −1023.78 −0.0895535
\(43\) −2995.66 −0.247070 −0.123535 0.992340i \(-0.539423\pi\)
−0.123535 + 0.992340i \(0.539423\pi\)
\(44\) −19578.5 −1.52457
\(45\) −1419.17 −0.104473
\(46\) 6285.27 0.437955
\(47\) 10180.6 0.672249 0.336125 0.941817i \(-0.390884\pi\)
0.336125 + 0.941817i \(0.390884\pi\)
\(48\) 3904.82 0.244623
\(49\) −16639.1 −0.990013
\(50\) −24742.3 −1.39963
\(51\) 7656.77 0.412212
\(52\) −28752.8 −1.47459
\(53\) 11009.5 0.538367 0.269183 0.963089i \(-0.413246\pi\)
0.269183 + 0.963089i \(0.413246\pi\)
\(54\) −6400.62 −0.298702
\(55\) 7607.93 0.339125
\(56\) 1488.84 0.0634421
\(57\) −16399.1 −0.668549
\(58\) 9955.28 0.388583
\(59\) −3481.00 −0.130189
\(60\) 7109.80 0.254964
\(61\) 2294.87 0.0789648 0.0394824 0.999220i \(-0.487429\pi\)
0.0394824 + 0.999220i \(0.487429\pi\)
\(62\) −83931.4 −2.77297
\(63\) 1049.43 0.0333121
\(64\) −51849.1 −1.58231
\(65\) 11172.9 0.328007
\(66\) 34312.6 0.969601
\(67\) 23569.8 0.641459 0.320729 0.947171i \(-0.396072\pi\)
0.320729 + 0.947171i \(0.396072\pi\)
\(68\) −38359.0 −1.00599
\(69\) −6442.76 −0.162911
\(70\) −1993.03 −0.0486148
\(71\) 56039.7 1.31932 0.659660 0.751564i \(-0.270700\pi\)
0.659660 + 0.751564i \(0.270700\pi\)
\(72\) 9308.16 0.211608
\(73\) 56572.1 1.24250 0.621248 0.783614i \(-0.286626\pi\)
0.621248 + 0.783614i \(0.286626\pi\)
\(74\) −940.434 −0.0199641
\(75\) 25362.2 0.520636
\(76\) 82156.5 1.63158
\(77\) −5625.81 −0.108133
\(78\) 50391.0 0.937814
\(79\) 65746.0 1.18523 0.592614 0.805487i \(-0.298096\pi\)
0.592614 + 0.805487i \(0.298096\pi\)
\(80\) 7601.67 0.132796
\(81\) 6561.00 0.111111
\(82\) −58078.6 −0.953853
\(83\) −13649.8 −0.217486 −0.108743 0.994070i \(-0.534682\pi\)
−0.108743 + 0.994070i \(0.534682\pi\)
\(84\) −5257.45 −0.0812975
\(85\) 14905.7 0.223772
\(86\) −26301.9 −0.383478
\(87\) −10204.7 −0.144545
\(88\) −49899.4 −0.686892
\(89\) 77607.3 1.03855 0.519275 0.854607i \(-0.326202\pi\)
0.519275 + 0.854607i \(0.326202\pi\)
\(90\) −12460.3 −0.162152
\(91\) −8262.00 −0.104588
\(92\) 32277.1 0.397580
\(93\) 86034.5 1.03149
\(94\) 89386.0 1.04340
\(95\) −31924.8 −0.362927
\(96\) 67380.0 0.746196
\(97\) 41043.1 0.442906 0.221453 0.975171i \(-0.428920\pi\)
0.221453 + 0.975171i \(0.428920\pi\)
\(98\) −146092. −1.53660
\(99\) −35172.3 −0.360673
\(100\) −127060. −1.27060
\(101\) −175206. −1.70901 −0.854505 0.519444i \(-0.826139\pi\)
−0.854505 + 0.519444i \(0.826139\pi\)
\(102\) 67226.4 0.639793
\(103\) 6862.58 0.0637374 0.0318687 0.999492i \(-0.489854\pi\)
0.0318687 + 0.999492i \(0.489854\pi\)
\(104\) −73281.7 −0.664373
\(105\) 2042.97 0.0180838
\(106\) 96663.4 0.835598
\(107\) −231771. −1.95704 −0.978520 0.206152i \(-0.933906\pi\)
−0.978520 + 0.206152i \(0.933906\pi\)
\(108\) −32869.4 −0.271164
\(109\) 138570. 1.11713 0.558565 0.829461i \(-0.311352\pi\)
0.558565 + 0.829461i \(0.311352\pi\)
\(110\) 66797.6 0.526356
\(111\) 963.999 0.00742624
\(112\) −5621.17 −0.0423430
\(113\) 26310.1 0.193832 0.0969162 0.995293i \(-0.469102\pi\)
0.0969162 + 0.995293i \(0.469102\pi\)
\(114\) −143984. −1.03765
\(115\) −12542.4 −0.0884373
\(116\) 51123.8 0.352759
\(117\) −51653.7 −0.348848
\(118\) −30563.2 −0.202066
\(119\) −11022.3 −0.0713517
\(120\) 18120.6 0.114873
\(121\) 27501.6 0.170763
\(122\) 20149.0 0.122561
\(123\) 59533.9 0.354815
\(124\) −431017. −2.51733
\(125\) 104126. 0.596051
\(126\) 9214.00 0.0517037
\(127\) −328460. −1.80706 −0.903531 0.428522i \(-0.859035\pi\)
−0.903531 + 0.428522i \(0.859035\pi\)
\(128\) −215661. −1.16345
\(129\) 26960.9 0.142646
\(130\) 98098.3 0.509100
\(131\) 57771.7 0.294128 0.147064 0.989127i \(-0.453018\pi\)
0.147064 + 0.989127i \(0.453018\pi\)
\(132\) 176207. 0.880214
\(133\) 23607.3 0.115722
\(134\) 206943. 0.995608
\(135\) 12772.6 0.0603175
\(136\) −97764.7 −0.453247
\(137\) 116139. 0.528662 0.264331 0.964432i \(-0.414849\pi\)
0.264331 + 0.964432i \(0.414849\pi\)
\(138\) −56567.4 −0.252853
\(139\) −433033. −1.90101 −0.950504 0.310713i \(-0.899432\pi\)
−0.950504 + 0.310713i \(0.899432\pi\)
\(140\) −10234.9 −0.0441330
\(141\) −91625.8 −0.388123
\(142\) 492029. 2.04771
\(143\) 276906. 1.13238
\(144\) −35143.4 −0.141233
\(145\) −19866.0 −0.0784675
\(146\) 496703. 1.92848
\(147\) 149752. 0.571584
\(148\) −4829.46 −0.0181236
\(149\) 283234. 1.04515 0.522576 0.852593i \(-0.324971\pi\)
0.522576 + 0.852593i \(0.324971\pi\)
\(150\) 222680. 0.808079
\(151\) −118520. −0.423009 −0.211505 0.977377i \(-0.567836\pi\)
−0.211505 + 0.977377i \(0.567836\pi\)
\(152\) 209390. 0.735102
\(153\) −68910.9 −0.237990
\(154\) −49394.6 −0.167833
\(155\) 167487. 0.559953
\(156\) 258776. 0.851357
\(157\) −189854. −0.614712 −0.307356 0.951595i \(-0.599444\pi\)
−0.307356 + 0.951595i \(0.599444\pi\)
\(158\) 577250. 1.83959
\(159\) −99085.6 −0.310826
\(160\) 131171. 0.405078
\(161\) 9274.67 0.0281990
\(162\) 57605.6 0.172455
\(163\) 127496. 0.375861 0.187931 0.982182i \(-0.439822\pi\)
0.187931 + 0.982182i \(0.439822\pi\)
\(164\) −298254. −0.865917
\(165\) −68471.4 −0.195794
\(166\) −119845. −0.337559
\(167\) 201919. 0.560255 0.280128 0.959963i \(-0.409623\pi\)
0.280128 + 0.959963i \(0.409623\pi\)
\(168\) −13399.5 −0.0366283
\(169\) 35368.4 0.0952574
\(170\) 130872. 0.347317
\(171\) 147592. 0.385987
\(172\) −135069. −0.348125
\(173\) −299080. −0.759752 −0.379876 0.925037i \(-0.624033\pi\)
−0.379876 + 0.925037i \(0.624033\pi\)
\(174\) −89597.5 −0.224348
\(175\) −36510.2 −0.0901195
\(176\) 188397. 0.458451
\(177\) 31329.0 0.0751646
\(178\) 681392. 1.61193
\(179\) 659619. 1.53872 0.769362 0.638813i \(-0.220574\pi\)
0.769362 + 0.638813i \(0.220574\pi\)
\(180\) −63988.2 −0.147204
\(181\) −242677. −0.550595 −0.275297 0.961359i \(-0.588776\pi\)
−0.275297 + 0.961359i \(0.588776\pi\)
\(182\) −72540.3 −0.162331
\(183\) −20653.8 −0.0455903
\(184\) 82263.7 0.179128
\(185\) 1876.66 0.00403139
\(186\) 755382. 1.60097
\(187\) 369419. 0.772530
\(188\) 459028. 0.947207
\(189\) −9444.88 −0.0192328
\(190\) −280300. −0.563299
\(191\) 202851. 0.402341 0.201171 0.979556i \(-0.435525\pi\)
0.201171 + 0.979556i \(0.435525\pi\)
\(192\) 466642. 0.913546
\(193\) −240767. −0.465268 −0.232634 0.972564i \(-0.574734\pi\)
−0.232634 + 0.972564i \(0.574734\pi\)
\(194\) 360359. 0.687433
\(195\) −100556. −0.189375
\(196\) −750231. −1.39494
\(197\) −372003. −0.682937 −0.341469 0.939893i \(-0.610924\pi\)
−0.341469 + 0.939893i \(0.610924\pi\)
\(198\) −308813. −0.559800
\(199\) −927046. −1.65947 −0.829733 0.558160i \(-0.811508\pi\)
−0.829733 + 0.558160i \(0.811508\pi\)
\(200\) −323835. −0.572465
\(201\) −212128. −0.370346
\(202\) −1.53830e6 −2.65255
\(203\) 14690.2 0.0250200
\(204\) 345231. 0.580810
\(205\) 115897. 0.192614
\(206\) 60253.4 0.0989267
\(207\) 57984.9 0.0940565
\(208\) 276678. 0.443421
\(209\) −791214. −1.25293
\(210\) 17937.3 0.0280678
\(211\) −372955. −0.576701 −0.288350 0.957525i \(-0.593107\pi\)
−0.288350 + 0.957525i \(0.593107\pi\)
\(212\) 496400. 0.758564
\(213\) −504357. −0.761710
\(214\) −2.03495e6 −3.03752
\(215\) 52485.9 0.0774366
\(216\) −83773.4 −0.122172
\(217\) −123851. −0.178546
\(218\) 1.21665e6 1.73390
\(219\) −509149. −0.717355
\(220\) 343029. 0.477831
\(221\) 542525. 0.747203
\(222\) 8463.91 0.0115263
\(223\) −632099. −0.851183 −0.425592 0.904915i \(-0.639934\pi\)
−0.425592 + 0.904915i \(0.639934\pi\)
\(224\) −96996.7 −0.129163
\(225\) −228260. −0.300589
\(226\) 231003. 0.300847
\(227\) −890035. −1.14642 −0.573208 0.819410i \(-0.694301\pi\)
−0.573208 + 0.819410i \(0.694301\pi\)
\(228\) −739409. −0.941992
\(229\) 259318. 0.326771 0.163386 0.986562i \(-0.447758\pi\)
0.163386 + 0.986562i \(0.447758\pi\)
\(230\) −110122. −0.137263
\(231\) 50632.3 0.0624306
\(232\) 130298. 0.158934
\(233\) −476260. −0.574717 −0.287358 0.957823i \(-0.592777\pi\)
−0.287358 + 0.957823i \(0.592777\pi\)
\(234\) −453519. −0.541447
\(235\) −178372. −0.210696
\(236\) −156952. −0.183438
\(237\) −591714. −0.684291
\(238\) −96775.7 −0.110745
\(239\) −400093. −0.453071 −0.226535 0.974003i \(-0.572740\pi\)
−0.226535 + 0.974003i \(0.572740\pi\)
\(240\) −68415.0 −0.0766696
\(241\) 66354.4 0.0735914 0.0367957 0.999323i \(-0.488285\pi\)
0.0367957 + 0.999323i \(0.488285\pi\)
\(242\) 241464. 0.265041
\(243\) −59049.0 −0.0641500
\(244\) 103472. 0.111262
\(245\) 291529. 0.310289
\(246\) 522707. 0.550707
\(247\) −1.16197e6 −1.21186
\(248\) −1.09852e6 −1.13417
\(249\) 122848. 0.125565
\(250\) 914224. 0.925129
\(251\) −378287. −0.378998 −0.189499 0.981881i \(-0.560686\pi\)
−0.189499 + 0.981881i \(0.560686\pi\)
\(252\) 47317.1 0.0469371
\(253\) −310846. −0.305313
\(254\) −2.88388e6 −2.80474
\(255\) −134152. −0.129195
\(256\) −234337. −0.223481
\(257\) −6758.28 −0.00638268 −0.00319134 0.999995i \(-0.501016\pi\)
−0.00319134 + 0.999995i \(0.501016\pi\)
\(258\) 236717. 0.221401
\(259\) −1387.72 −0.00128545
\(260\) 503769. 0.462166
\(261\) 91842.6 0.0834532
\(262\) 507235. 0.456516
\(263\) 1.53754e6 1.37068 0.685342 0.728222i \(-0.259653\pi\)
0.685342 + 0.728222i \(0.259653\pi\)
\(264\) 449094. 0.396577
\(265\) −192894. −0.168734
\(266\) 207272. 0.179613
\(267\) −698465. −0.599607
\(268\) 1.06272e6 0.903822
\(269\) 760328. 0.640649 0.320325 0.947308i \(-0.396208\pi\)
0.320325 + 0.947308i \(0.396208\pi\)
\(270\) 112143. 0.0936188
\(271\) −621403. −0.513985 −0.256992 0.966413i \(-0.582732\pi\)
−0.256992 + 0.966413i \(0.582732\pi\)
\(272\) 369115. 0.302510
\(273\) 74358.0 0.0603839
\(274\) 1.01970e6 0.820536
\(275\) 1.22366e6 0.975730
\(276\) −290493. −0.229543
\(277\) −317260. −0.248437 −0.124219 0.992255i \(-0.539642\pi\)
−0.124219 + 0.992255i \(0.539642\pi\)
\(278\) −3.80203e6 −2.95055
\(279\) −774310. −0.595531
\(280\) −26085.4 −0.0198840
\(281\) −368043. −0.278057 −0.139028 0.990288i \(-0.544398\pi\)
−0.139028 + 0.990288i \(0.544398\pi\)
\(282\) −804474. −0.602406
\(283\) −1.66530e6 −1.23602 −0.618010 0.786170i \(-0.712061\pi\)
−0.618010 + 0.786170i \(0.712061\pi\)
\(284\) 2.52674e6 1.85894
\(285\) 287323. 0.209536
\(286\) 2.43124e6 1.75757
\(287\) −85701.9 −0.0614166
\(288\) −606420. −0.430816
\(289\) −696078. −0.490245
\(290\) −174423. −0.121789
\(291\) −369388. −0.255712
\(292\) 2.55074e6 1.75069
\(293\) 1.24407e6 0.846598 0.423299 0.905990i \(-0.360872\pi\)
0.423299 + 0.905990i \(0.360872\pi\)
\(294\) 1.31482e6 0.887155
\(295\) 60989.4 0.0408037
\(296\) −12308.7 −0.00816552
\(297\) 316551. 0.208234
\(298\) 2.48679e6 1.62218
\(299\) −456506. −0.295303
\(300\) 1.14354e6 0.733582
\(301\) −38811.5 −0.0246913
\(302\) −1.04061e6 −0.656552
\(303\) 1.57685e6 0.986697
\(304\) −790562. −0.490628
\(305\) −40207.6 −0.0247491
\(306\) −605038. −0.369385
\(307\) 1.88311e6 1.14033 0.570165 0.821530i \(-0.306879\pi\)
0.570165 + 0.821530i \(0.306879\pi\)
\(308\) −253658. −0.152361
\(309\) −61763.2 −0.0367988
\(310\) 1.47053e6 0.869102
\(311\) −3.10201e6 −1.81862 −0.909310 0.416119i \(-0.863390\pi\)
−0.909310 + 0.416119i \(0.863390\pi\)
\(312\) 659535. 0.383576
\(313\) −2.33451e6 −1.34690 −0.673449 0.739234i \(-0.735188\pi\)
−0.673449 + 0.739234i \(0.735188\pi\)
\(314\) −1.66692e6 −0.954094
\(315\) −18386.7 −0.0104407
\(316\) 2.96438e6 1.67000
\(317\) −103565. −0.0578846 −0.0289423 0.999581i \(-0.509214\pi\)
−0.0289423 + 0.999581i \(0.509214\pi\)
\(318\) −869971. −0.482433
\(319\) −492352. −0.270894
\(320\) 908430. 0.495926
\(321\) 2.08594e6 1.12990
\(322\) 81431.6 0.0437676
\(323\) −1.55018e6 −0.826751
\(324\) 295825. 0.156557
\(325\) 1.79706e6 0.943742
\(326\) 1.11942e6 0.583374
\(327\) −1.24713e6 −0.644975
\(328\) −760152. −0.390136
\(329\) 131900. 0.0671822
\(330\) −601179. −0.303892
\(331\) 1.75769e6 0.881804 0.440902 0.897555i \(-0.354659\pi\)
0.440902 + 0.897555i \(0.354659\pi\)
\(332\) −615446. −0.306439
\(333\) −8675.99 −0.00428754
\(334\) 1.77285e6 0.869572
\(335\) −412958. −0.201045
\(336\) 50590.6 0.0244468
\(337\) −620309. −0.297532 −0.148766 0.988872i \(-0.547530\pi\)
−0.148766 + 0.988872i \(0.547530\pi\)
\(338\) 310534. 0.147849
\(339\) −236791. −0.111909
\(340\) 672075. 0.315297
\(341\) 4.15094e6 1.93313
\(342\) 1.29586e6 0.599090
\(343\) −433326. −0.198875
\(344\) −344247. −0.156846
\(345\) 112881. 0.0510593
\(346\) −2.62592e6 −1.17921
\(347\) 495722. 0.221011 0.110506 0.993876i \(-0.464753\pi\)
0.110506 + 0.993876i \(0.464753\pi\)
\(348\) −460115. −0.203666
\(349\) −1.71145e6 −0.752142 −0.376071 0.926591i \(-0.622725\pi\)
−0.376071 + 0.926591i \(0.622725\pi\)
\(350\) −320559. −0.139874
\(351\) 464883. 0.201408
\(352\) 3.25091e6 1.39845
\(353\) −2.42249e6 −1.03472 −0.517362 0.855767i \(-0.673086\pi\)
−0.517362 + 0.855767i \(0.673086\pi\)
\(354\) 275069. 0.116663
\(355\) −981853. −0.413500
\(356\) 3.49918e6 1.46333
\(357\) 99200.6 0.0411949
\(358\) 5.79146e6 2.38825
\(359\) 2.09761e6 0.858989 0.429495 0.903069i \(-0.358692\pi\)
0.429495 + 0.903069i \(0.358692\pi\)
\(360\) −163085. −0.0663221
\(361\) 844035. 0.340873
\(362\) −2.13070e6 −0.854577
\(363\) −247514. −0.0985901
\(364\) −372520. −0.147366
\(365\) −991180. −0.389422
\(366\) −181341. −0.0707607
\(367\) −4.92708e6 −1.90952 −0.954760 0.297376i \(-0.903888\pi\)
−0.954760 + 0.297376i \(0.903888\pi\)
\(368\) −310590. −0.119555
\(369\) −535805. −0.204852
\(370\) 16477.0 0.00625712
\(371\) 142638. 0.0538024
\(372\) 3.87915e6 1.45338
\(373\) 1.78212e6 0.663231 0.331616 0.943415i \(-0.392406\pi\)
0.331616 + 0.943415i \(0.392406\pi\)
\(374\) 3.24350e6 1.19904
\(375\) −937132. −0.344130
\(376\) 1.16991e6 0.426761
\(377\) −723062. −0.262013
\(378\) −82926.0 −0.0298512
\(379\) 399928. 0.143016 0.0715079 0.997440i \(-0.477219\pi\)
0.0715079 + 0.997440i \(0.477219\pi\)
\(380\) −1.43944e6 −0.511368
\(381\) 2.95614e6 1.04331
\(382\) 1.78103e6 0.624473
\(383\) 1.37113e6 0.477618 0.238809 0.971067i \(-0.423243\pi\)
0.238809 + 0.971067i \(0.423243\pi\)
\(384\) 1.94095e6 0.671718
\(385\) 98567.9 0.0338909
\(386\) −2.11393e6 −0.722142
\(387\) −242648. −0.0823568
\(388\) 1.85057e6 0.624059
\(389\) 3.29557e6 1.10422 0.552111 0.833771i \(-0.313822\pi\)
0.552111 + 0.833771i \(0.313822\pi\)
\(390\) −882884. −0.293929
\(391\) −609022. −0.201461
\(392\) −1.91210e6 −0.628485
\(393\) −519945. −0.169815
\(394\) −3.26618e6 −1.05999
\(395\) −1.15191e6 −0.371473
\(396\) −1.58586e6 −0.508192
\(397\) 2.44883e6 0.779798 0.389899 0.920858i \(-0.372510\pi\)
0.389899 + 0.920858i \(0.372510\pi\)
\(398\) −8.13946e6 −2.57566
\(399\) −212466. −0.0668124
\(400\) 1.22265e6 0.382079
\(401\) 4.98383e6 1.54775 0.773877 0.633336i \(-0.218315\pi\)
0.773877 + 0.633336i \(0.218315\pi\)
\(402\) −1.86248e6 −0.574814
\(403\) 6.09602e6 1.86975
\(404\) −7.89973e6 −2.40801
\(405\) −114953. −0.0348243
\(406\) 128980. 0.0388336
\(407\) 46510.4 0.0139176
\(408\) 879882. 0.261682
\(409\) −1.36543e6 −0.403611 −0.201805 0.979426i \(-0.564681\pi\)
−0.201805 + 0.979426i \(0.564681\pi\)
\(410\) 1.01758e6 0.298956
\(411\) −1.04525e6 −0.305223
\(412\) 309422. 0.0898066
\(413\) −45099.6 −0.0130106
\(414\) 509107. 0.145985
\(415\) 239153. 0.0681641
\(416\) 4.77425e6 1.35261
\(417\) 3.89730e6 1.09755
\(418\) −6.94686e6 −1.94468
\(419\) 2.31238e6 0.643465 0.321732 0.946831i \(-0.395735\pi\)
0.321732 + 0.946831i \(0.395735\pi\)
\(420\) 92114.1 0.0254802
\(421\) 3.96201e6 1.08946 0.544728 0.838613i \(-0.316633\pi\)
0.544728 + 0.838613i \(0.316633\pi\)
\(422\) −3.27455e6 −0.895097
\(423\) 824632. 0.224083
\(424\) 1.26516e6 0.341768
\(425\) 2.39744e6 0.643837
\(426\) −4.42826e6 −1.18225
\(427\) 29732.2 0.00789146
\(428\) −1.04502e7 −2.75749
\(429\) −2.49216e6 −0.653781
\(430\) 460826. 0.120189
\(431\) −3.23219e6 −0.838115 −0.419057 0.907960i \(-0.637639\pi\)
−0.419057 + 0.907960i \(0.637639\pi\)
\(432\) 316290. 0.0815411
\(433\) −2.25277e6 −0.577426 −0.288713 0.957416i \(-0.593227\pi\)
−0.288713 + 0.957416i \(0.593227\pi\)
\(434\) −1.08741e6 −0.277121
\(435\) 178794. 0.0453032
\(436\) 6.24790e6 1.57405
\(437\) 1.30439e6 0.326741
\(438\) −4.47032e6 −1.11341
\(439\) −6.68557e6 −1.65568 −0.827841 0.560963i \(-0.810431\pi\)
−0.827841 + 0.560963i \(0.810431\pi\)
\(440\) 874270. 0.215285
\(441\) −1.34777e6 −0.330004
\(442\) 4.76337e6 1.15973
\(443\) −6.04955e6 −1.46458 −0.732291 0.680992i \(-0.761549\pi\)
−0.732291 + 0.680992i \(0.761549\pi\)
\(444\) 43465.1 0.0104637
\(445\) −1.35973e6 −0.325501
\(446\) −5.54983e6 −1.32112
\(447\) −2.54910e6 −0.603418
\(448\) −671753. −0.158130
\(449\) −514041. −0.120332 −0.0601661 0.998188i \(-0.519163\pi\)
−0.0601661 + 0.998188i \(0.519163\pi\)
\(450\) −2.00412e6 −0.466545
\(451\) 2.87235e6 0.664962
\(452\) 1.18628e6 0.273112
\(453\) 1.06668e6 0.244225
\(454\) −7.81450e6 −1.77935
\(455\) 144756. 0.0327799
\(456\) −1.88451e6 −0.424411
\(457\) 7.09850e6 1.58992 0.794962 0.606660i \(-0.207491\pi\)
0.794962 + 0.606660i \(0.207491\pi\)
\(458\) 2.27681e6 0.507182
\(459\) 620198. 0.137404
\(460\) −565515. −0.124609
\(461\) 538077. 0.117921 0.0589606 0.998260i \(-0.481221\pi\)
0.0589606 + 0.998260i \(0.481221\pi\)
\(462\) 444551. 0.0968985
\(463\) −5.69755e6 −1.23520 −0.617598 0.786494i \(-0.711894\pi\)
−0.617598 + 0.786494i \(0.711894\pi\)
\(464\) −491946. −0.106077
\(465\) −1.50738e6 −0.323289
\(466\) −4.18156e6 −0.892017
\(467\) 6.84726e6 1.45286 0.726431 0.687239i \(-0.241178\pi\)
0.726431 + 0.687239i \(0.241178\pi\)
\(468\) −2.32898e6 −0.491531
\(469\) 305369. 0.0641051
\(470\) −1.56610e6 −0.327021
\(471\) 1.70869e6 0.354904
\(472\) −400021. −0.0826471
\(473\) 1.30079e6 0.267335
\(474\) −5.19525e6 −1.06209
\(475\) −5.13479e6 −1.04421
\(476\) −496977. −0.100535
\(477\) 891770. 0.179456
\(478\) −3.51281e6 −0.703210
\(479\) 2.13269e6 0.424706 0.212353 0.977193i \(-0.431887\pi\)
0.212353 + 0.977193i \(0.431887\pi\)
\(480\) −1.18054e6 −0.233872
\(481\) 68304.7 0.0134613
\(482\) 582591. 0.114221
\(483\) −83472.0 −0.0162807
\(484\) 1.24000e6 0.240607
\(485\) −719103. −0.138815
\(486\) −518450. −0.0995672
\(487\) 1.44011e6 0.275152 0.137576 0.990491i \(-0.456069\pi\)
0.137576 + 0.990491i \(0.456069\pi\)
\(488\) 263716. 0.0501288
\(489\) −1.14746e6 −0.217004
\(490\) 2.55962e6 0.481599
\(491\) 193566. 0.0362347 0.0181174 0.999836i \(-0.494233\pi\)
0.0181174 + 0.999836i \(0.494233\pi\)
\(492\) 2.68428e6 0.499938
\(493\) −964633. −0.178750
\(494\) −1.02021e7 −1.88092
\(495\) 616243. 0.113042
\(496\) 4.14752e6 0.756979
\(497\) 726047. 0.131848
\(498\) 1.07861e6 0.194890
\(499\) 1.09525e7 1.96908 0.984539 0.175165i \(-0.0560460\pi\)
0.984539 + 0.175165i \(0.0560460\pi\)
\(500\) 4.69486e6 0.839842
\(501\) −1.81727e6 −0.323464
\(502\) −3.32136e6 −0.588243
\(503\) −1.09972e6 −0.193804 −0.0969022 0.995294i \(-0.530893\pi\)
−0.0969022 + 0.995294i \(0.530893\pi\)
\(504\) 120596. 0.0211474
\(505\) 3.06972e6 0.535636
\(506\) −2.72923e6 −0.473875
\(507\) −318316. −0.0549969
\(508\) −1.48097e7 −2.54617
\(509\) 5.13320e6 0.878201 0.439100 0.898438i \(-0.355297\pi\)
0.439100 + 0.898438i \(0.355297\pi\)
\(510\) −1.17785e6 −0.200523
\(511\) 732944. 0.124171
\(512\) 4.84369e6 0.816586
\(513\) −1.32833e6 −0.222850
\(514\) −59337.6 −0.00990655
\(515\) −120237. −0.0199765
\(516\) 1.21562e6 0.200990
\(517\) −4.42070e6 −0.727386
\(518\) −12184.2 −0.00199514
\(519\) 2.69172e6 0.438643
\(520\) 1.28394e6 0.208227
\(521\) 9.05184e6 1.46097 0.730487 0.682926i \(-0.239293\pi\)
0.730487 + 0.682926i \(0.239293\pi\)
\(522\) 806378. 0.129528
\(523\) 974287. 0.155752 0.0778758 0.996963i \(-0.475186\pi\)
0.0778758 + 0.996963i \(0.475186\pi\)
\(524\) 2.60483e6 0.414430
\(525\) 328592. 0.0520305
\(526\) 1.34996e7 2.12744
\(527\) 8.13267e6 1.27558
\(528\) −1.69558e6 −0.264687
\(529\) −5.92388e6 −0.920380
\(530\) −1.69361e6 −0.261892
\(531\) −281961. −0.0433963
\(532\) 1.06441e6 0.163054
\(533\) 4.21831e6 0.643162
\(534\) −6.13252e6 −0.930649
\(535\) 4.06078e6 0.613374
\(536\) 2.70854e6 0.407214
\(537\) −5.93657e6 −0.888383
\(538\) 6.67568e6 0.994351
\(539\) 7.22516e6 1.07121
\(540\) 575894. 0.0849880
\(541\) 6.38853e6 0.938443 0.469222 0.883080i \(-0.344535\pi\)
0.469222 + 0.883080i \(0.344535\pi\)
\(542\) −5.45592e6 −0.797755
\(543\) 2.18409e6 0.317886
\(544\) 6.36930e6 0.922771
\(545\) −2.42784e6 −0.350130
\(546\) 652863. 0.0937218
\(547\) −4.31658e6 −0.616838 −0.308419 0.951251i \(-0.599800\pi\)
−0.308419 + 0.951251i \(0.599800\pi\)
\(548\) 5.23653e6 0.744891
\(549\) 185884. 0.0263216
\(550\) 1.07437e7 1.51443
\(551\) 2.06603e6 0.289907
\(552\) −740374. −0.103420
\(553\) 851801. 0.118447
\(554\) −2.78554e6 −0.385599
\(555\) −16889.9 −0.00232753
\(556\) −1.95247e7 −2.67854
\(557\) 6.96213e6 0.950833 0.475416 0.879761i \(-0.342297\pi\)
0.475416 + 0.879761i \(0.342297\pi\)
\(558\) −6.79844e6 −0.924323
\(559\) 1.91033e6 0.258570
\(560\) 98486.7 0.0132711
\(561\) −3.32477e6 −0.446020
\(562\) −3.23142e6 −0.431571
\(563\) −442629. −0.0588530 −0.0294265 0.999567i \(-0.509368\pi\)
−0.0294265 + 0.999567i \(0.509368\pi\)
\(564\) −4.13125e6 −0.546870
\(565\) −460970. −0.0607508
\(566\) −1.46213e7 −1.91843
\(567\) 85003.9 0.0111040
\(568\) 6.43983e6 0.837537
\(569\) 4.56709e6 0.591369 0.295685 0.955286i \(-0.404452\pi\)
0.295685 + 0.955286i \(0.404452\pi\)
\(570\) 2.52270e6 0.325221
\(571\) −1.20733e7 −1.54966 −0.774831 0.632169i \(-0.782165\pi\)
−0.774831 + 0.632169i \(0.782165\pi\)
\(572\) 1.24852e7 1.59554
\(573\) −1.82566e6 −0.232292
\(574\) −752463. −0.0953247
\(575\) −2.01732e6 −0.254452
\(576\) −4.19978e6 −0.527436
\(577\) −3.82492e6 −0.478280 −0.239140 0.970985i \(-0.576866\pi\)
−0.239140 + 0.970985i \(0.576866\pi\)
\(578\) −6.11156e6 −0.760908
\(579\) 2.16690e6 0.268623
\(580\) −895723. −0.110562
\(581\) −176846. −0.0217347
\(582\) −3.24323e6 −0.396890
\(583\) −4.78062e6 −0.582522
\(584\) 6.50101e6 0.788767
\(585\) 905007. 0.109336
\(586\) 1.09230e7 1.31400
\(587\) −1.24579e7 −1.49227 −0.746136 0.665793i \(-0.768093\pi\)
−0.746136 + 0.665793i \(0.768093\pi\)
\(588\) 6.75208e6 0.805368
\(589\) −1.74184e7 −2.06880
\(590\) 535487. 0.0633314
\(591\) 3.34803e6 0.394294
\(592\) 46472.1 0.00544990
\(593\) 8.12876e6 0.949265 0.474633 0.880184i \(-0.342581\pi\)
0.474633 + 0.880184i \(0.342581\pi\)
\(594\) 2.77932e6 0.323200
\(595\) 193118. 0.0223630
\(596\) 1.27705e7 1.47263
\(597\) 8.34341e6 0.958094
\(598\) −4.00812e6 −0.458340
\(599\) −2.18535e6 −0.248860 −0.124430 0.992228i \(-0.539710\pi\)
−0.124430 + 0.992228i \(0.539710\pi\)
\(600\) 2.91452e6 0.330513
\(601\) −3.56350e6 −0.402430 −0.201215 0.979547i \(-0.564489\pi\)
−0.201215 + 0.979547i \(0.564489\pi\)
\(602\) −340765. −0.0383234
\(603\) 1.90915e6 0.213820
\(604\) −5.34388e6 −0.596025
\(605\) −481845. −0.0535204
\(606\) 1.38447e7 1.53145
\(607\) 1.07019e6 0.117893 0.0589464 0.998261i \(-0.481226\pi\)
0.0589464 + 0.998261i \(0.481226\pi\)
\(608\) −1.36416e7 −1.49660
\(609\) −132212. −0.0144453
\(610\) −353023. −0.0384130
\(611\) −6.49220e6 −0.703540
\(612\) −3.10708e6 −0.335331
\(613\) −1.33424e7 −1.43411 −0.717057 0.697015i \(-0.754511\pi\)
−0.717057 + 0.697015i \(0.754511\pi\)
\(614\) 1.65337e7 1.76991
\(615\) −1.04307e6 −0.111206
\(616\) −646493. −0.0686455
\(617\) −791902. −0.0837449 −0.0418724 0.999123i \(-0.513332\pi\)
−0.0418724 + 0.999123i \(0.513332\pi\)
\(618\) −542281. −0.0571154
\(619\) 9.53469e6 1.00018 0.500092 0.865972i \(-0.333299\pi\)
0.500092 + 0.865972i \(0.333299\pi\)
\(620\) 7.55170e6 0.788979
\(621\) −521864. −0.0543035
\(622\) −2.72356e7 −2.82268
\(623\) 1.00547e6 0.103789
\(624\) −2.49010e6 −0.256009
\(625\) 6.98198e6 0.714955
\(626\) −2.04970e7 −2.09052
\(627\) 7.12093e6 0.723382
\(628\) −8.56022e6 −0.866136
\(629\) 91124.9 0.00918355
\(630\) −161435. −0.0162049
\(631\) −1.67624e6 −0.167595 −0.0837977 0.996483i \(-0.526705\pi\)
−0.0837977 + 0.996483i \(0.526705\pi\)
\(632\) 7.55524e6 0.752412
\(633\) 3.35660e6 0.332958
\(634\) −909296. −0.0898426
\(635\) 5.75484e6 0.566368
\(636\) −4.46760e6 −0.437957
\(637\) 1.06108e7 1.03609
\(638\) −4.32285e6 −0.420454
\(639\) 4.53922e6 0.439773
\(640\) 3.77853e6 0.364647
\(641\) −4.71863e6 −0.453598 −0.226799 0.973942i \(-0.572826\pi\)
−0.226799 + 0.973942i \(0.572826\pi\)
\(642\) 1.83145e7 1.75371
\(643\) −1.25582e7 −1.19785 −0.598923 0.800807i \(-0.704404\pi\)
−0.598923 + 0.800807i \(0.704404\pi\)
\(644\) 418179. 0.0397327
\(645\) −472373. −0.0447080
\(646\) −1.36105e7 −1.28320
\(647\) 1.79155e6 0.168256 0.0841278 0.996455i \(-0.473190\pi\)
0.0841278 + 0.996455i \(0.473190\pi\)
\(648\) 753961. 0.0705361
\(649\) 1.51154e6 0.140867
\(650\) 1.57781e7 1.46478
\(651\) 1.11466e6 0.103083
\(652\) 5.74859e6 0.529593
\(653\) 1.71252e7 1.57164 0.785821 0.618454i \(-0.212241\pi\)
0.785821 + 0.618454i \(0.212241\pi\)
\(654\) −1.09498e7 −1.00106
\(655\) −1.01220e6 −0.0921854
\(656\) 2.86999e6 0.260388
\(657\) 4.58234e6 0.414165
\(658\) 1.15808e6 0.104273
\(659\) −1.46762e7 −1.31643 −0.658216 0.752829i \(-0.728689\pi\)
−0.658216 + 0.752829i \(0.728689\pi\)
\(660\) −3.08726e6 −0.275876
\(661\) −1.49380e7 −1.32981 −0.664904 0.746929i \(-0.731528\pi\)
−0.664904 + 0.746929i \(0.731528\pi\)
\(662\) 1.54325e7 1.36865
\(663\) −4.88272e6 −0.431398
\(664\) −1.56857e6 −0.138065
\(665\) −413616. −0.0362696
\(666\) −76175.2 −0.00665469
\(667\) 811688. 0.0706438
\(668\) 9.10419e6 0.789406
\(669\) 5.68889e6 0.491431
\(670\) −3.62577e6 −0.312042
\(671\) −996493. −0.0854413
\(672\) 872971. 0.0745721
\(673\) 8.30140e6 0.706502 0.353251 0.935529i \(-0.385076\pi\)
0.353251 + 0.935529i \(0.385076\pi\)
\(674\) −5.44631e6 −0.461799
\(675\) 2.05434e6 0.173545
\(676\) 1.59470e6 0.134219
\(677\) −287812. −0.0241345 −0.0120672 0.999927i \(-0.503841\pi\)
−0.0120672 + 0.999927i \(0.503841\pi\)
\(678\) −2.07902e6 −0.173694
\(679\) 531752. 0.0442624
\(680\) 1.71290e6 0.142056
\(681\) 8.01031e6 0.661884
\(682\) 3.64452e7 3.00040
\(683\) −2.12554e7 −1.74348 −0.871742 0.489965i \(-0.837010\pi\)
−0.871742 + 0.489965i \(0.837010\pi\)
\(684\) 6.65468e6 0.543860
\(685\) −2.03484e6 −0.165693
\(686\) −3.80460e6 −0.308673
\(687\) −2.33386e6 −0.188662
\(688\) 1.29972e6 0.104684
\(689\) −7.02076e6 −0.563425
\(690\) 991099. 0.0792491
\(691\) −2.41473e6 −0.192386 −0.0961932 0.995363i \(-0.530667\pi\)
−0.0961932 + 0.995363i \(0.530667\pi\)
\(692\) −1.34850e7 −1.07050
\(693\) −455691. −0.0360443
\(694\) 4.35243e6 0.343031
\(695\) 7.58702e6 0.595812
\(696\) −1.17268e6 −0.0917608
\(697\) 5.62762e6 0.438776
\(698\) −1.50265e7 −1.16740
\(699\) 4.28634e6 0.331813
\(700\) −1.64618e6 −0.126979
\(701\) 1.06511e7 0.818654 0.409327 0.912388i \(-0.365764\pi\)
0.409327 + 0.912388i \(0.365764\pi\)
\(702\) 4.08167e6 0.312605
\(703\) −195170. −0.0148944
\(704\) 2.25142e7 1.71209
\(705\) 1.60534e6 0.121645
\(706\) −2.12694e7 −1.60599
\(707\) −2.26995e6 −0.170792
\(708\) 1.41257e6 0.105908
\(709\) 1.04662e7 0.781943 0.390971 0.920403i \(-0.372139\pi\)
0.390971 + 0.920403i \(0.372139\pi\)
\(710\) −8.62067e6 −0.641793
\(711\) 5.32543e6 0.395076
\(712\) 8.91828e6 0.659297
\(713\) −6.84321e6 −0.504122
\(714\) 870981. 0.0639386
\(715\) −4.85158e6 −0.354910
\(716\) 2.97411e7 2.16808
\(717\) 3.60083e6 0.261580
\(718\) 1.84170e7 1.33324
\(719\) −1.40608e7 −1.01435 −0.507175 0.861843i \(-0.669310\pi\)
−0.507175 + 0.861843i \(0.669310\pi\)
\(720\) 615735. 0.0442652
\(721\) 88911.1 0.00636968
\(722\) 7.41063e6 0.529069
\(723\) −597189. −0.0424880
\(724\) −1.09419e7 −0.775793
\(725\) −3.19525e6 −0.225766
\(726\) −2.17317e6 −0.153022
\(727\) 4.54577e6 0.318986 0.159493 0.987199i \(-0.449014\pi\)
0.159493 + 0.987199i \(0.449014\pi\)
\(728\) −949432. −0.0663950
\(729\) 531441. 0.0370370
\(730\) −8.70256e6 −0.604421
\(731\) 2.54856e6 0.176401
\(732\) −931247. −0.0642373
\(733\) −6.72753e6 −0.462483 −0.231241 0.972896i \(-0.574279\pi\)
−0.231241 + 0.972896i \(0.574279\pi\)
\(734\) −4.32597e7 −2.96376
\(735\) −2.62376e6 −0.179145
\(736\) −5.35942e6 −0.364690
\(737\) −1.02346e7 −0.694070
\(738\) −4.70437e6 −0.317951
\(739\) 7.96547e6 0.536538 0.268269 0.963344i \(-0.413548\pi\)
0.268269 + 0.963344i \(0.413548\pi\)
\(740\) 84615.3 0.00568028
\(741\) 1.04577e7 0.699667
\(742\) 1.25237e6 0.0835067
\(743\) 3.70267e6 0.246061 0.123031 0.992403i \(-0.460739\pi\)
0.123031 + 0.992403i \(0.460739\pi\)
\(744\) 9.88670e6 0.654815
\(745\) −4.96244e6 −0.327570
\(746\) 1.56470e7 1.02940
\(747\) −1.10563e6 −0.0724952
\(748\) 1.66565e7 1.08850
\(749\) −3.00281e6 −0.195580
\(750\) −8.22801e6 −0.534124
\(751\) −2.71608e7 −1.75728 −0.878642 0.477480i \(-0.841550\pi\)
−0.878642 + 0.477480i \(0.841550\pi\)
\(752\) −4.41706e6 −0.284832
\(753\) 3.40459e6 0.218815
\(754\) −6.34848e6 −0.406670
\(755\) 2.07655e6 0.132579
\(756\) −425854. −0.0270992
\(757\) 2.64905e7 1.68016 0.840078 0.542465i \(-0.182509\pi\)
0.840078 + 0.542465i \(0.182509\pi\)
\(758\) 3.51137e6 0.221975
\(759\) 2.79762e6 0.176272
\(760\) −3.66866e6 −0.230395
\(761\) −2.94938e7 −1.84616 −0.923080 0.384607i \(-0.874337\pi\)
−0.923080 + 0.384607i \(0.874337\pi\)
\(762\) 2.59549e7 1.61932
\(763\) 1.79531e6 0.111642
\(764\) 9.14623e6 0.566903
\(765\) 1.20737e6 0.0745908
\(766\) 1.20385e7 0.741310
\(767\) 2.21983e6 0.136249
\(768\) 2.10903e6 0.129027
\(769\) 9.58663e6 0.584588 0.292294 0.956329i \(-0.405581\pi\)
0.292294 + 0.956329i \(0.405581\pi\)
\(770\) 865426. 0.0526021
\(771\) 60824.5 0.00368504
\(772\) −1.08558e7 −0.655567
\(773\) 6.41914e6 0.386392 0.193196 0.981160i \(-0.438115\pi\)
0.193196 + 0.981160i \(0.438115\pi\)
\(774\) −2.13045e6 −0.127826
\(775\) 2.69386e7 1.61109
\(776\) 4.71649e6 0.281167
\(777\) 12489.5 0.000742152 0
\(778\) 2.89351e7 1.71386
\(779\) −1.20531e7 −0.711633
\(780\) −4.53392e6 −0.266832
\(781\) −2.43339e7 −1.42753
\(782\) −5.34721e6 −0.312687
\(783\) −826584. −0.0481817
\(784\) 7.21920e6 0.419468
\(785\) 3.32638e6 0.192663
\(786\) −4.56512e6 −0.263570
\(787\) 3.15258e7 1.81438 0.907191 0.420719i \(-0.138222\pi\)
0.907191 + 0.420719i \(0.138222\pi\)
\(788\) −1.67730e7 −0.962265
\(789\) −1.38379e7 −0.791364
\(790\) −1.01138e7 −0.576563
\(791\) 340872. 0.0193709
\(792\) −4.04185e6 −0.228964
\(793\) −1.46344e6 −0.0826402
\(794\) 2.15007e7 1.21032
\(795\) 1.73604e6 0.0974189
\(796\) −4.17990e7 −2.33821
\(797\) −2.66338e7 −1.48521 −0.742605 0.669730i \(-0.766410\pi\)
−0.742605 + 0.669730i \(0.766410\pi\)
\(798\) −1.86545e6 −0.103699
\(799\) −8.66120e6 −0.479967
\(800\) 2.10976e7 1.16549
\(801\) 6.28619e6 0.346183
\(802\) 4.37580e7 2.40227
\(803\) −2.45651e7 −1.34440
\(804\) −9.56451e6 −0.521822
\(805\) −162498. −0.00883811
\(806\) 5.35230e7 2.90204
\(807\) −6.84295e6 −0.369879
\(808\) −2.01338e7 −1.08492
\(809\) −4.02522e6 −0.216231 −0.108116 0.994138i \(-0.534482\pi\)
−0.108116 + 0.994138i \(0.534482\pi\)
\(810\) −1.00929e6 −0.0540508
\(811\) −2.10486e7 −1.12375 −0.561876 0.827222i \(-0.689920\pi\)
−0.561876 + 0.827222i \(0.689920\pi\)
\(812\) 662357. 0.0352535
\(813\) 5.59263e6 0.296749
\(814\) 408361. 0.0216015
\(815\) −2.23382e6 −0.117802
\(816\) −3.32203e6 −0.174654
\(817\) −5.45846e6 −0.286098
\(818\) −1.19885e7 −0.626444
\(819\) −669222. −0.0348627
\(820\) 5.22561e6 0.271395
\(821\) 7.16356e6 0.370912 0.185456 0.982653i \(-0.440624\pi\)
0.185456 + 0.982653i \(0.440624\pi\)
\(822\) −9.17733e6 −0.473737
\(823\) −1.93310e7 −0.994846 −0.497423 0.867508i \(-0.665720\pi\)
−0.497423 + 0.867508i \(0.665720\pi\)
\(824\) 788617. 0.0404621
\(825\) −1.10130e7 −0.563338
\(826\) −395974. −0.0201938
\(827\) 8.87409e6 0.451191 0.225595 0.974221i \(-0.427567\pi\)
0.225595 + 0.974221i \(0.427567\pi\)
\(828\) 2.61444e6 0.132527
\(829\) 8.50913e6 0.430030 0.215015 0.976611i \(-0.431020\pi\)
0.215015 + 0.976611i \(0.431020\pi\)
\(830\) 2.09976e6 0.105797
\(831\) 2.85534e6 0.143435
\(832\) 3.30642e7 1.65596
\(833\) 1.41558e7 0.706841
\(834\) 3.42182e7 1.70350
\(835\) −3.53776e6 −0.175595
\(836\) −3.56745e7 −1.76540
\(837\) 6.96879e6 0.343830
\(838\) 2.03027e7 0.998721
\(839\) −3.61604e7 −1.77349 −0.886744 0.462261i \(-0.847038\pi\)
−0.886744 + 0.462261i \(0.847038\pi\)
\(840\) 234769. 0.0114800
\(841\) −1.92255e7 −0.937320
\(842\) 3.47864e7 1.69094
\(843\) 3.31239e6 0.160536
\(844\) −1.68159e7 −0.812578
\(845\) −619678. −0.0298555
\(846\) 7.24026e6 0.347799
\(847\) 356308. 0.0170654
\(848\) −4.77668e6 −0.228106
\(849\) 1.49877e7 0.713617
\(850\) 2.10495e7 0.999298
\(851\) −76676.8 −0.00362944
\(852\) −2.27406e7 −1.07326
\(853\) 1.02908e7 0.484257 0.242128 0.970244i \(-0.422154\pi\)
0.242128 + 0.970244i \(0.422154\pi\)
\(854\) 261049. 0.0122483
\(855\) −2.58591e6 −0.120976
\(856\) −2.66341e7 −1.24238
\(857\) 1.01966e7 0.474244 0.237122 0.971480i \(-0.423796\pi\)
0.237122 + 0.971480i \(0.423796\pi\)
\(858\) −2.18811e7 −1.01473
\(859\) −3.60575e7 −1.66730 −0.833648 0.552296i \(-0.813752\pi\)
−0.833648 + 0.552296i \(0.813752\pi\)
\(860\) 2.36650e6 0.109109
\(861\) 771317. 0.0354589
\(862\) −2.83786e7 −1.30084
\(863\) −3.91789e7 −1.79071 −0.895354 0.445355i \(-0.853077\pi\)
−0.895354 + 0.445355i \(0.853077\pi\)
\(864\) 5.45778e6 0.248732
\(865\) 5.24008e6 0.238121
\(866\) −1.97793e7 −0.896222
\(867\) 6.26470e6 0.283043
\(868\) −5.58423e6 −0.251573
\(869\) −2.85487e7 −1.28244
\(870\) 1.56981e6 0.0703151
\(871\) −1.50305e7 −0.671316
\(872\) 1.59239e7 0.709181
\(873\) 3.32449e6 0.147635
\(874\) 1.14525e7 0.507135
\(875\) 1.34905e6 0.0595671
\(876\) −2.29567e7 −1.01076
\(877\) −3.90462e6 −0.171427 −0.0857137 0.996320i \(-0.527317\pi\)
−0.0857137 + 0.996320i \(0.527317\pi\)
\(878\) −5.86993e7 −2.56978
\(879\) −1.11967e7 −0.488783
\(880\) −3.30084e6 −0.143687
\(881\) 3.06209e7 1.32916 0.664582 0.747215i \(-0.268610\pi\)
0.664582 + 0.747215i \(0.268610\pi\)
\(882\) −1.18334e7 −0.512199
\(883\) 2.72635e7 1.17674 0.588368 0.808593i \(-0.299771\pi\)
0.588368 + 0.808593i \(0.299771\pi\)
\(884\) 2.44615e7 1.05282
\(885\) −548905. −0.0235580
\(886\) −5.31150e7 −2.27318
\(887\) 7.45001e6 0.317942 0.158971 0.987283i \(-0.449182\pi\)
0.158971 + 0.987283i \(0.449182\pi\)
\(888\) 110778. 0.00471436
\(889\) −4.25551e6 −0.180591
\(890\) −1.19384e7 −0.505210
\(891\) −2.84896e6 −0.120224
\(892\) −2.85003e7 −1.19933
\(893\) 1.85504e7 0.778438
\(894\) −2.23811e7 −0.936565
\(895\) −1.15570e7 −0.482266
\(896\) −2.79410e6 −0.116271
\(897\) 4.10855e6 0.170493
\(898\) −4.51327e6 −0.186767
\(899\) −1.08390e7 −0.447291
\(900\) −1.02919e7 −0.423534
\(901\) −9.36636e6 −0.384378
\(902\) 2.52193e7 1.03209
\(903\) 349304. 0.0142555
\(904\) 3.02344e6 0.123050
\(905\) 4.25186e6 0.172567
\(906\) 9.36546e6 0.379061
\(907\) 8.74302e6 0.352893 0.176447 0.984310i \(-0.443540\pi\)
0.176447 + 0.984310i \(0.443540\pi\)
\(908\) −4.01302e7 −1.61531
\(909\) −1.41916e7 −0.569670
\(910\) 1.27095e6 0.0508776
\(911\) 2.92020e7 1.16578 0.582891 0.812551i \(-0.301922\pi\)
0.582891 + 0.812551i \(0.301922\pi\)
\(912\) 7.11506e6 0.283264
\(913\) 5.92709e6 0.235323
\(914\) 6.23248e7 2.46772
\(915\) 361869. 0.0142889
\(916\) 1.16922e7 0.460424
\(917\) 748486. 0.0293941
\(918\) 5.44534e6 0.213264
\(919\) 3.40665e7 1.33057 0.665287 0.746588i \(-0.268309\pi\)
0.665287 + 0.746588i \(0.268309\pi\)
\(920\) −1.44132e6 −0.0561422
\(921\) −1.69480e7 −0.658370
\(922\) 4.72431e6 0.183025
\(923\) −3.57365e7 −1.38073
\(924\) 2.28293e6 0.0879654
\(925\) 301842. 0.0115991
\(926\) −5.00245e7 −1.91715
\(927\) 555869. 0.0212458
\(928\) −8.48882e6 −0.323577
\(929\) 7.22466e6 0.274649 0.137325 0.990526i \(-0.456150\pi\)
0.137325 + 0.990526i \(0.456150\pi\)
\(930\) −1.32348e7 −0.501776
\(931\) −3.03186e7 −1.14640
\(932\) −2.14737e7 −0.809782
\(933\) 2.79181e7 1.04998
\(934\) 6.01189e7 2.25499
\(935\) −6.47247e6 −0.242126
\(936\) −5.93581e6 −0.221458
\(937\) 2.02690e7 0.754194 0.377097 0.926174i \(-0.376922\pi\)
0.377097 + 0.926174i \(0.376922\pi\)
\(938\) 2.68114e6 0.0994974
\(939\) 2.10106e7 0.777632
\(940\) −8.04248e6 −0.296873
\(941\) −1.37966e7 −0.507923 −0.253962 0.967214i \(-0.581734\pi\)
−0.253962 + 0.967214i \(0.581734\pi\)
\(942\) 1.50023e7 0.550846
\(943\) −4.73534e6 −0.173409
\(944\) 1.51030e6 0.0551610
\(945\) 165480. 0.00602792
\(946\) 1.14210e7 0.414930
\(947\) 3.23524e7 1.17228 0.586139 0.810210i \(-0.300647\pi\)
0.586139 + 0.810210i \(0.300647\pi\)
\(948\) −2.66794e7 −0.964174
\(949\) −3.60760e7 −1.30033
\(950\) −4.50835e7 −1.62072
\(951\) 932081. 0.0334197
\(952\) −1.26663e6 −0.0452958
\(953\) −1.28170e7 −0.457146 −0.228573 0.973527i \(-0.573406\pi\)
−0.228573 + 0.973527i \(0.573406\pi\)
\(954\) 7.82974e6 0.278533
\(955\) −3.55409e6 −0.126101
\(956\) −1.80395e7 −0.638381
\(957\) 4.43117e6 0.156400
\(958\) 1.87250e7 0.659186
\(959\) 1.50470e6 0.0528326
\(960\) −8.17587e6 −0.286323
\(961\) 6.27527e7 2.19192
\(962\) 599715. 0.0208933
\(963\) −1.87735e7 −0.652347
\(964\) 2.99181e6 0.103691
\(965\) 4.21839e6 0.145824
\(966\) −732884. −0.0252693
\(967\) −4.53561e7 −1.55980 −0.779901 0.625903i \(-0.784731\pi\)
−0.779901 + 0.625903i \(0.784731\pi\)
\(968\) 3.16036e6 0.108405
\(969\) 1.39516e7 0.477325
\(970\) −6.31372e6 −0.215455
\(971\) −6.25506e6 −0.212904 −0.106452 0.994318i \(-0.533949\pi\)
−0.106452 + 0.994318i \(0.533949\pi\)
\(972\) −2.66242e6 −0.0903881
\(973\) −5.61035e6 −0.189980
\(974\) 1.26442e7 0.427063
\(975\) −1.61735e7 −0.544870
\(976\) −995672. −0.0334574
\(977\) 6.26249e6 0.209899 0.104950 0.994478i \(-0.466532\pi\)
0.104950 + 0.994478i \(0.466532\pi\)
\(978\) −1.00747e7 −0.336811
\(979\) −3.36991e7 −1.12373
\(980\) 1.31445e7 0.437200
\(981\) 1.12242e7 0.372377
\(982\) 1.69951e6 0.0562398
\(983\) 2.95453e7 0.975226 0.487613 0.873060i \(-0.337868\pi\)
0.487613 + 0.873060i \(0.337868\pi\)
\(984\) 6.84137e6 0.225245
\(985\) 6.51774e6 0.214046
\(986\) −8.46948e6 −0.277437
\(987\) −1.18710e6 −0.0387877
\(988\) −5.23912e7 −1.70752
\(989\) −2.14448e6 −0.0697158
\(990\) 5.41061e6 0.175452
\(991\) −2.48240e7 −0.802947 −0.401474 0.915871i \(-0.631502\pi\)
−0.401474 + 0.915871i \(0.631502\pi\)
\(992\) 7.15679e7 2.30908
\(993\) −1.58192e7 −0.509110
\(994\) 6.37469e6 0.204641
\(995\) 1.62425e7 0.520109
\(996\) 5.53901e6 0.176923
\(997\) −2.40550e7 −0.766421 −0.383211 0.923661i \(-0.625182\pi\)
−0.383211 + 0.923661i \(0.625182\pi\)
\(998\) 9.61631e7 3.05620
\(999\) 78083.9 0.00247541
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.b.1.11 12
3.2 odd 2 531.6.a.d.1.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.11 12 1.1 even 1 trivial
531.6.a.d.1.2 12 3.2 odd 2