Properties

Label 177.6.a.a.1.7
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $1$
Dimension $11$
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + 13849341 x^{3} - 23890558 x^{2} - 74443300 x - 14846072\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(4.20625\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.20625 q^{2} +9.00000 q^{3} -21.7200 q^{4} +26.7258 q^{5} +28.8562 q^{6} +39.0273 q^{7} -172.240 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+3.20625 q^{2} +9.00000 q^{3} -21.7200 q^{4} +26.7258 q^{5} +28.8562 q^{6} +39.0273 q^{7} -172.240 q^{8} +81.0000 q^{9} +85.6897 q^{10} -606.405 q^{11} -195.480 q^{12} -161.707 q^{13} +125.131 q^{14} +240.533 q^{15} +142.797 q^{16} -1651.20 q^{17} +259.706 q^{18} +882.094 q^{19} -580.485 q^{20} +351.246 q^{21} -1944.28 q^{22} +2923.63 q^{23} -1550.16 q^{24} -2410.73 q^{25} -518.474 q^{26} +729.000 q^{27} -847.672 q^{28} -7064.00 q^{29} +771.207 q^{30} -2253.17 q^{31} +5969.51 q^{32} -5457.64 q^{33} -5294.14 q^{34} +1043.04 q^{35} -1759.32 q^{36} -7561.88 q^{37} +2828.21 q^{38} -1455.37 q^{39} -4603.25 q^{40} -16708.0 q^{41} +1126.18 q^{42} -4502.50 q^{43} +13171.1 q^{44} +2164.79 q^{45} +9373.88 q^{46} +8408.79 q^{47} +1285.17 q^{48} -15283.9 q^{49} -7729.39 q^{50} -14860.8 q^{51} +3512.28 q^{52} +11049.2 q^{53} +2337.35 q^{54} -16206.7 q^{55} -6722.04 q^{56} +7938.85 q^{57} -22648.9 q^{58} +3481.00 q^{59} -5224.36 q^{60} +34738.9 q^{61} -7224.23 q^{62} +3161.21 q^{63} +14570.2 q^{64} -4321.76 q^{65} -17498.5 q^{66} -57227.0 q^{67} +35863.9 q^{68} +26312.7 q^{69} +3344.23 q^{70} +26.9044 q^{71} -13951.4 q^{72} +54481.3 q^{73} -24245.2 q^{74} -21696.6 q^{75} -19159.1 q^{76} -23666.3 q^{77} -4666.26 q^{78} +93190.6 q^{79} +3816.37 q^{80} +6561.00 q^{81} -53569.9 q^{82} -89600.1 q^{83} -7629.05 q^{84} -44129.6 q^{85} -14436.1 q^{86} -63576.0 q^{87} +104447. q^{88} -128157. q^{89} +6940.86 q^{90} -6311.00 q^{91} -63501.2 q^{92} -20278.6 q^{93} +26960.7 q^{94} +23574.7 q^{95} +53725.6 q^{96} -26665.5 q^{97} -49003.9 q^{98} -49118.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 6q^{2} + 99q^{3} + 150q^{4} - 192q^{5} - 54q^{6} - 371q^{7} - 621q^{8} + 891q^{9} + O(q^{10}) \) \( 11q - 6q^{2} + 99q^{3} + 150q^{4} - 192q^{5} - 54q^{6} - 371q^{7} - 621q^{8} + 891q^{9} - 399q^{10} - 698q^{11} + 1350q^{12} - 1556q^{13} - 1679q^{14} - 1728q^{15} - 2662q^{16} - 4793q^{17} - 486q^{18} - 3753q^{19} - 11023q^{20} - 3339q^{21} - 9534q^{22} - 7323q^{23} - 5589q^{24} + 7867q^{25} - 4844q^{26} + 8019q^{27} + 3650q^{28} - 15467q^{29} - 3591q^{30} - 5151q^{31} - 15368q^{32} - 6282q^{33} + 8452q^{34} - 23285q^{35} + 12150q^{36} + 8623q^{37} + 15205q^{38} - 14004q^{39} + 41530q^{40} - 6369q^{41} - 15111q^{42} - 20506q^{43} - 55632q^{44} - 15552q^{45} - 45191q^{46} - 47899q^{47} - 23958q^{48} - 10322q^{49} - 102147q^{50} - 43137q^{51} - 292q^{52} - 80048q^{53} - 4374q^{54} - 2114q^{55} - 108126q^{56} - 33777q^{57} - 58294q^{58} + 38291q^{59} - 99207q^{60} - 82527q^{61} - 67438q^{62} - 30051q^{63} - 51411q^{64} - 167646q^{65} - 85806q^{66} - 166976q^{67} - 136533q^{68} - 65907q^{69} + 76140q^{70} - 183560q^{71} - 50301q^{72} - 36809q^{73} - 116686q^{74} + 70803q^{75} + 55580q^{76} - 164885q^{77} - 43596q^{78} - 281518q^{79} - 32683q^{80} + 72171q^{81} + 178815q^{82} - 254691q^{83} + 32850q^{84} + 4763q^{85} + 349324q^{86} - 139203q^{87} + 251285q^{88} - 89687q^{89} - 32319q^{90} + 34897q^{91} - 20240q^{92} - 46359q^{93} + 96548q^{94} - 155113q^{95} - 138312q^{96} - 45828q^{97} + 465864q^{98} - 56538q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.20625 0.566790 0.283395 0.959003i \(-0.408539\pi\)
0.283395 + 0.959003i \(0.408539\pi\)
\(3\) 9.00000 0.577350
\(4\) −21.7200 −0.678749
\(5\) 26.7258 0.478086 0.239043 0.971009i \(-0.423166\pi\)
0.239043 + 0.971009i \(0.423166\pi\)
\(6\) 28.8562 0.327236
\(7\) 39.0273 0.301039 0.150520 0.988607i \(-0.451905\pi\)
0.150520 + 0.988607i \(0.451905\pi\)
\(8\) −172.240 −0.951498
\(9\) 81.0000 0.333333
\(10\) 85.6897 0.270974
\(11\) −606.405 −1.51106 −0.755528 0.655116i \(-0.772620\pi\)
−0.755528 + 0.655116i \(0.772620\pi\)
\(12\) −195.480 −0.391876
\(13\) −161.707 −0.265382 −0.132691 0.991157i \(-0.542362\pi\)
−0.132691 + 0.991157i \(0.542362\pi\)
\(14\) 125.131 0.170626
\(15\) 240.533 0.276023
\(16\) 142.797 0.139450
\(17\) −1651.20 −1.38572 −0.692861 0.721071i \(-0.743650\pi\)
−0.692861 + 0.721071i \(0.743650\pi\)
\(18\) 259.706 0.188930
\(19\) 882.094 0.560572 0.280286 0.959917i \(-0.409571\pi\)
0.280286 + 0.959917i \(0.409571\pi\)
\(20\) −580.485 −0.324501
\(21\) 351.246 0.173805
\(22\) −1944.28 −0.856451
\(23\) 2923.63 1.15240 0.576199 0.817309i \(-0.304535\pi\)
0.576199 + 0.817309i \(0.304535\pi\)
\(24\) −1550.16 −0.549348
\(25\) −2410.73 −0.771433
\(26\) −518.474 −0.150416
\(27\) 729.000 0.192450
\(28\) −847.672 −0.204330
\(29\) −7064.00 −1.55975 −0.779876 0.625934i \(-0.784718\pi\)
−0.779876 + 0.625934i \(0.784718\pi\)
\(30\) 771.207 0.156447
\(31\) −2253.17 −0.421105 −0.210553 0.977583i \(-0.567526\pi\)
−0.210553 + 0.977583i \(0.567526\pi\)
\(32\) 5969.51 1.03054
\(33\) −5457.64 −0.872409
\(34\) −5294.14 −0.785413
\(35\) 1043.04 0.143923
\(36\) −1759.32 −0.226250
\(37\) −7561.88 −0.908082 −0.454041 0.890981i \(-0.650018\pi\)
−0.454041 + 0.890981i \(0.650018\pi\)
\(38\) 2828.21 0.317726
\(39\) −1455.37 −0.153218
\(40\) −4603.25 −0.454898
\(41\) −16708.0 −1.55226 −0.776130 0.630574i \(-0.782820\pi\)
−0.776130 + 0.630574i \(0.782820\pi\)
\(42\) 1126.18 0.0985110
\(43\) −4502.50 −0.371349 −0.185675 0.982611i \(-0.559447\pi\)
−0.185675 + 0.982611i \(0.559447\pi\)
\(44\) 13171.1 1.02563
\(45\) 2164.79 0.159362
\(46\) 9373.88 0.653168
\(47\) 8408.79 0.555251 0.277625 0.960689i \(-0.410453\pi\)
0.277625 + 0.960689i \(0.410453\pi\)
\(48\) 1285.17 0.0805116
\(49\) −15283.9 −0.909375
\(50\) −7729.39 −0.437241
\(51\) −14860.8 −0.800047
\(52\) 3512.28 0.180128
\(53\) 11049.2 0.540307 0.270154 0.962817i \(-0.412925\pi\)
0.270154 + 0.962817i \(0.412925\pi\)
\(54\) 2337.35 0.109079
\(55\) −16206.7 −0.722416
\(56\) −6722.04 −0.286438
\(57\) 7938.85 0.323646
\(58\) −22648.9 −0.884052
\(59\) 3481.00 0.130189
\(60\) −5224.36 −0.187351
\(61\) 34738.9 1.19534 0.597670 0.801742i \(-0.296093\pi\)
0.597670 + 0.801742i \(0.296093\pi\)
\(62\) −7224.23 −0.238678
\(63\) 3161.21 0.100346
\(64\) 14570.2 0.444648
\(65\) −4321.76 −0.126875
\(66\) −17498.5 −0.494472
\(67\) −57227.0 −1.55745 −0.778724 0.627366i \(-0.784133\pi\)
−0.778724 + 0.627366i \(0.784133\pi\)
\(68\) 35863.9 0.940558
\(69\) 26312.7 0.665338
\(70\) 3344.23 0.0815740
\(71\) 26.9044 0.000633398 0 0.000316699 1.00000i \(-0.499899\pi\)
0.000316699 1.00000i \(0.499899\pi\)
\(72\) −13951.4 −0.317166
\(73\) 54481.3 1.19658 0.598288 0.801281i \(-0.295848\pi\)
0.598288 + 0.801281i \(0.295848\pi\)
\(74\) −24245.2 −0.514692
\(75\) −21696.6 −0.445387
\(76\) −19159.1 −0.380488
\(77\) −23666.3 −0.454887
\(78\) −4666.26 −0.0868426
\(79\) 93190.6 1.67998 0.839990 0.542601i \(-0.182561\pi\)
0.839990 + 0.542601i \(0.182561\pi\)
\(80\) 3816.37 0.0666692
\(81\) 6561.00 0.111111
\(82\) −53569.9 −0.879805
\(83\) −89600.1 −1.42762 −0.713811 0.700339i \(-0.753032\pi\)
−0.713811 + 0.700339i \(0.753032\pi\)
\(84\) −7629.05 −0.117970
\(85\) −44129.6 −0.662495
\(86\) −14436.1 −0.210477
\(87\) −63576.0 −0.900524
\(88\) 104447. 1.43777
\(89\) −128157. −1.71502 −0.857508 0.514471i \(-0.827988\pi\)
−0.857508 + 0.514471i \(0.827988\pi\)
\(90\) 6940.86 0.0903248
\(91\) −6311.00 −0.0798904
\(92\) −63501.2 −0.782190
\(93\) −20278.6 −0.243125
\(94\) 26960.7 0.314710
\(95\) 23574.7 0.268002
\(96\) 53725.6 0.594981
\(97\) −26665.5 −0.287753 −0.143877 0.989596i \(-0.545957\pi\)
−0.143877 + 0.989596i \(0.545957\pi\)
\(98\) −49003.9 −0.515425
\(99\) −49118.8 −0.503686
\(100\) 52361.0 0.523610
\(101\) 153961. 1.50178 0.750890 0.660427i \(-0.229625\pi\)
0.750890 + 0.660427i \(0.229625\pi\)
\(102\) −47647.3 −0.453459
\(103\) −51991.3 −0.482878 −0.241439 0.970416i \(-0.577619\pi\)
−0.241439 + 0.970416i \(0.577619\pi\)
\(104\) 27852.4 0.252510
\(105\) 9387.33 0.0830939
\(106\) 35426.4 0.306241
\(107\) 215950. 1.82345 0.911725 0.410801i \(-0.134751\pi\)
0.911725 + 0.410801i \(0.134751\pi\)
\(108\) −15833.9 −0.130625
\(109\) 62561.7 0.504362 0.252181 0.967680i \(-0.418852\pi\)
0.252181 + 0.967680i \(0.418852\pi\)
\(110\) −51962.6 −0.409458
\(111\) −68056.9 −0.524282
\(112\) 5572.98 0.0419800
\(113\) 242539. 1.78684 0.893421 0.449220i \(-0.148298\pi\)
0.893421 + 0.449220i \(0.148298\pi\)
\(114\) 25453.9 0.183439
\(115\) 78136.5 0.550946
\(116\) 153430. 1.05868
\(117\) −13098.3 −0.0884606
\(118\) 11160.9 0.0737897
\(119\) −64441.7 −0.417157
\(120\) −41429.2 −0.262636
\(121\) 206675. 1.28329
\(122\) 111381. 0.677506
\(123\) −150372. −0.896197
\(124\) 48938.9 0.285825
\(125\) −147947. −0.846898
\(126\) 10135.6 0.0568753
\(127\) 4520.72 0.0248713 0.0124356 0.999923i \(-0.496042\pi\)
0.0124356 + 0.999923i \(0.496042\pi\)
\(128\) −144309. −0.778515
\(129\) −40522.5 −0.214399
\(130\) −13856.6 −0.0719117
\(131\) −201847. −1.02765 −0.513823 0.857896i \(-0.671771\pi\)
−0.513823 + 0.857896i \(0.671771\pi\)
\(132\) 118540. 0.592147
\(133\) 34425.8 0.168754
\(134\) −183484. −0.882746
\(135\) 19483.1 0.0920078
\(136\) 284401. 1.31851
\(137\) 128569. 0.585242 0.292621 0.956229i \(-0.405473\pi\)
0.292621 + 0.956229i \(0.405473\pi\)
\(138\) 84364.9 0.377107
\(139\) 264506. 1.16118 0.580588 0.814198i \(-0.302823\pi\)
0.580588 + 0.814198i \(0.302823\pi\)
\(140\) −22654.7 −0.0976875
\(141\) 75679.1 0.320574
\(142\) 86.2620 0.000359003 0
\(143\) 98060.0 0.401007
\(144\) 11566.6 0.0464834
\(145\) −188791. −0.745697
\(146\) 174680. 0.678207
\(147\) −137555. −0.525028
\(148\) 164244. 0.616360
\(149\) −310579. −1.14606 −0.573028 0.819536i \(-0.694231\pi\)
−0.573028 + 0.819536i \(0.694231\pi\)
\(150\) −69564.5 −0.252441
\(151\) 150211. 0.536118 0.268059 0.963402i \(-0.413618\pi\)
0.268059 + 0.963402i \(0.413618\pi\)
\(152\) −151932. −0.533383
\(153\) −133747. −0.461907
\(154\) −75880.1 −0.257826
\(155\) −60218.0 −0.201325
\(156\) 31610.5 0.103997
\(157\) 381735. 1.23598 0.617992 0.786185i \(-0.287947\pi\)
0.617992 + 0.786185i \(0.287947\pi\)
\(158\) 298792. 0.952196
\(159\) 99442.7 0.311947
\(160\) 159540. 0.492686
\(161\) 114101. 0.346917
\(162\) 21036.2 0.0629766
\(163\) −43367.4 −0.127848 −0.0639240 0.997955i \(-0.520362\pi\)
−0.0639240 + 0.997955i \(0.520362\pi\)
\(164\) 362897. 1.05359
\(165\) −145860. −0.417087
\(166\) −287280. −0.809161
\(167\) 227398. 0.630951 0.315475 0.948934i \(-0.397836\pi\)
0.315475 + 0.948934i \(0.397836\pi\)
\(168\) −60498.4 −0.165375
\(169\) −345144. −0.929572
\(170\) −141490. −0.375495
\(171\) 71449.7 0.186857
\(172\) 97794.2 0.252053
\(173\) −196716. −0.499716 −0.249858 0.968282i \(-0.580384\pi\)
−0.249858 + 0.968282i \(0.580384\pi\)
\(174\) −203840. −0.510408
\(175\) −94084.2 −0.232232
\(176\) −86592.7 −0.210717
\(177\) 31329.0 0.0751646
\(178\) −410904. −0.972053
\(179\) −51846.3 −0.120944 −0.0604722 0.998170i \(-0.519261\pi\)
−0.0604722 + 0.998170i \(0.519261\pi\)
\(180\) −47019.3 −0.108167
\(181\) 692422. 1.57099 0.785496 0.618866i \(-0.212408\pi\)
0.785496 + 0.618866i \(0.212408\pi\)
\(182\) −20234.6 −0.0452810
\(183\) 312650. 0.690130
\(184\) −503565. −1.09651
\(185\) −202098. −0.434142
\(186\) −65018.1 −0.137801
\(187\) 1.00129e6 2.09390
\(188\) −182639. −0.376876
\(189\) 28450.9 0.0579350
\(190\) 75586.4 0.151901
\(191\) −964737. −1.91349 −0.956743 0.290933i \(-0.906034\pi\)
−0.956743 + 0.290933i \(0.906034\pi\)
\(192\) 131132. 0.256717
\(193\) −305452. −0.590268 −0.295134 0.955456i \(-0.595364\pi\)
−0.295134 + 0.955456i \(0.595364\pi\)
\(194\) −85496.1 −0.163096
\(195\) −38895.9 −0.0732516
\(196\) 331965. 0.617238
\(197\) −31988.0 −0.0587249 −0.0293624 0.999569i \(-0.509348\pi\)
−0.0293624 + 0.999569i \(0.509348\pi\)
\(198\) −157487. −0.285484
\(199\) 428137. 0.766390 0.383195 0.923668i \(-0.374824\pi\)
0.383195 + 0.923668i \(0.374824\pi\)
\(200\) 415223. 0.734017
\(201\) −515043. −0.899193
\(202\) 493636. 0.851193
\(203\) −275689. −0.469547
\(204\) 322775. 0.543032
\(205\) −446535. −0.742114
\(206\) −166697. −0.273691
\(207\) 236814. 0.384133
\(208\) −23091.3 −0.0370075
\(209\) −534906. −0.847055
\(210\) 30098.1 0.0470968
\(211\) −1.07522e6 −1.66261 −0.831306 0.555815i \(-0.812406\pi\)
−0.831306 + 0.555815i \(0.812406\pi\)
\(212\) −239988. −0.366733
\(213\) 242.139 0.000365693 0
\(214\) 692389. 1.03351
\(215\) −120333. −0.177537
\(216\) −125563. −0.183116
\(217\) −87935.3 −0.126769
\(218\) 200588. 0.285867
\(219\) 490332. 0.690843
\(220\) 352009. 0.490339
\(221\) 267010. 0.367746
\(222\) −218207. −0.297157
\(223\) 541635. 0.729365 0.364682 0.931132i \(-0.381178\pi\)
0.364682 + 0.931132i \(0.381178\pi\)
\(224\) 232974. 0.310232
\(225\) −195269. −0.257144
\(226\) 777641. 1.01276
\(227\) −503824. −0.648954 −0.324477 0.945894i \(-0.605188\pi\)
−0.324477 + 0.945894i \(0.605188\pi\)
\(228\) −172432. −0.219675
\(229\) 236193. 0.297631 0.148815 0.988865i \(-0.452454\pi\)
0.148815 + 0.988865i \(0.452454\pi\)
\(230\) 250525. 0.312271
\(231\) −212997. −0.262629
\(232\) 1.21670e6 1.48410
\(233\) −1.35136e6 −1.63073 −0.815366 0.578946i \(-0.803464\pi\)
−0.815366 + 0.578946i \(0.803464\pi\)
\(234\) −41996.4 −0.0501386
\(235\) 224732. 0.265458
\(236\) −75607.3 −0.0883656
\(237\) 838715. 0.969937
\(238\) −206616. −0.236440
\(239\) −1.00537e6 −1.13849 −0.569245 0.822168i \(-0.692765\pi\)
−0.569245 + 0.822168i \(0.692765\pi\)
\(240\) 34347.3 0.0384915
\(241\) −968910. −1.07458 −0.537292 0.843396i \(-0.680553\pi\)
−0.537292 + 0.843396i \(0.680553\pi\)
\(242\) 662653. 0.727357
\(243\) 59049.0 0.0641500
\(244\) −754528. −0.811336
\(245\) −408474. −0.434760
\(246\) −482129. −0.507955
\(247\) −142641. −0.148766
\(248\) 388086. 0.400681
\(249\) −806401. −0.824238
\(250\) −474355. −0.480013
\(251\) −1.74198e6 −1.74525 −0.872626 0.488390i \(-0.837584\pi\)
−0.872626 + 0.488390i \(0.837584\pi\)
\(252\) −68661.4 −0.0681101
\(253\) −1.77290e6 −1.74134
\(254\) 14494.5 0.0140968
\(255\) −397166. −0.382492
\(256\) −928936. −0.885902
\(257\) −696669. −0.657951 −0.328976 0.944338i \(-0.606703\pi\)
−0.328976 + 0.944338i \(0.606703\pi\)
\(258\) −129925. −0.121519
\(259\) −295120. −0.273369
\(260\) 93868.6 0.0861166
\(261\) −572184. −0.519918
\(262\) −647171. −0.582459
\(263\) −894241. −0.797196 −0.398598 0.917126i \(-0.630503\pi\)
−0.398598 + 0.917126i \(0.630503\pi\)
\(264\) 940021. 0.830095
\(265\) 295299. 0.258314
\(266\) 110377. 0.0956481
\(267\) −1.15341e6 −0.990164
\(268\) 1.24297e6 1.05712
\(269\) 346103. 0.291625 0.145812 0.989312i \(-0.453420\pi\)
0.145812 + 0.989312i \(0.453420\pi\)
\(270\) 62467.8 0.0521491
\(271\) 476055. 0.393762 0.196881 0.980427i \(-0.436919\pi\)
0.196881 + 0.980427i \(0.436919\pi\)
\(272\) −235786. −0.193239
\(273\) −56799.0 −0.0461247
\(274\) 412224. 0.331709
\(275\) 1.46188e6 1.16568
\(276\) −571511. −0.451598
\(277\) −1.33986e6 −1.04920 −0.524600 0.851349i \(-0.675785\pi\)
−0.524600 + 0.851349i \(0.675785\pi\)
\(278\) 848071. 0.658143
\(279\) −182507. −0.140368
\(280\) −179652. −0.136942
\(281\) 767258. 0.579663 0.289831 0.957078i \(-0.406401\pi\)
0.289831 + 0.957078i \(0.406401\pi\)
\(282\) 242646. 0.181698
\(283\) 391631. 0.290677 0.145339 0.989382i \(-0.453573\pi\)
0.145339 + 0.989382i \(0.453573\pi\)
\(284\) −584.362 −0.000429919 0
\(285\) 212172. 0.154731
\(286\) 314405. 0.227287
\(287\) −652067. −0.467291
\(288\) 483530. 0.343512
\(289\) 1.30659e6 0.920227
\(290\) −605312. −0.422653
\(291\) −239989. −0.166134
\(292\) −1.18333e6 −0.812175
\(293\) 1.78575e6 1.21521 0.607606 0.794239i \(-0.292130\pi\)
0.607606 + 0.794239i \(0.292130\pi\)
\(294\) −441035. −0.297581
\(295\) 93032.7 0.0622415
\(296\) 1.30245e6 0.864039
\(297\) −442069. −0.290803
\(298\) −995791. −0.649573
\(299\) −472772. −0.305826
\(300\) 471249. 0.302306
\(301\) −175720. −0.111791
\(302\) 481615. 0.303866
\(303\) 1.38565e6 0.867053
\(304\) 125960. 0.0781718
\(305\) 928426. 0.571476
\(306\) −428825. −0.261804
\(307\) −927343. −0.561558 −0.280779 0.959772i \(-0.590593\pi\)
−0.280779 + 0.959772i \(0.590593\pi\)
\(308\) 514032. 0.308755
\(309\) −467922. −0.278790
\(310\) −193074. −0.114109
\(311\) 766407. 0.449323 0.224661 0.974437i \(-0.427872\pi\)
0.224661 + 0.974437i \(0.427872\pi\)
\(312\) 250671. 0.145787
\(313\) 226189. 0.130500 0.0652501 0.997869i \(-0.479215\pi\)
0.0652501 + 0.997869i \(0.479215\pi\)
\(314\) 1.22394e6 0.700543
\(315\) 84486.0 0.0479743
\(316\) −2.02410e6 −1.14029
\(317\) 1.08057e6 0.603955 0.301978 0.953315i \(-0.402353\pi\)
0.301978 + 0.953315i \(0.402353\pi\)
\(318\) 318838. 0.176808
\(319\) 4.28364e6 2.35687
\(320\) 389401. 0.212580
\(321\) 1.94355e6 1.05277
\(322\) 365837. 0.196629
\(323\) −1.45651e6 −0.776797
\(324\) −142505. −0.0754166
\(325\) 389833. 0.204724
\(326\) −139047. −0.0724630
\(327\) 563055. 0.291193
\(328\) 2.87777e6 1.47697
\(329\) 328172. 0.167152
\(330\) −467663. −0.236401
\(331\) −2.89963e6 −1.45470 −0.727350 0.686267i \(-0.759248\pi\)
−0.727350 + 0.686267i \(0.759248\pi\)
\(332\) 1.94611e6 0.968997
\(333\) −612512. −0.302694
\(334\) 729094. 0.357617
\(335\) −1.52944e6 −0.744595
\(336\) 50156.8 0.0242371
\(337\) −204495. −0.0980863 −0.0490432 0.998797i \(-0.515617\pi\)
−0.0490432 + 0.998797i \(0.515617\pi\)
\(338\) −1.10662e6 −0.526872
\(339\) 2.18285e6 1.03163
\(340\) 958494. 0.449668
\(341\) 1.36634e6 0.636314
\(342\) 229085. 0.105909
\(343\) −1.25242e6 −0.574797
\(344\) 775509. 0.353338
\(345\) 703228. 0.318089
\(346\) −630719. −0.283234
\(347\) 1.14557e6 0.510739 0.255370 0.966843i \(-0.417803\pi\)
0.255370 + 0.966843i \(0.417803\pi\)
\(348\) 1.38087e6 0.611230
\(349\) 3.14511e6 1.38220 0.691101 0.722758i \(-0.257126\pi\)
0.691101 + 0.722758i \(0.257126\pi\)
\(350\) −301657. −0.131627
\(351\) −117885. −0.0510728
\(352\) −3.61994e6 −1.55720
\(353\) −2.35284e6 −1.00497 −0.502487 0.864584i \(-0.667582\pi\)
−0.502487 + 0.864584i \(0.667582\pi\)
\(354\) 100449. 0.0426025
\(355\) 719.042 0.000302819 0
\(356\) 2.78357e6 1.16407
\(357\) −579975. −0.240846
\(358\) −166232. −0.0685500
\(359\) 198560. 0.0813122 0.0406561 0.999173i \(-0.487055\pi\)
0.0406561 + 0.999173i \(0.487055\pi\)
\(360\) −372863. −0.151633
\(361\) −1.69801e6 −0.685759
\(362\) 2.22007e6 0.890423
\(363\) 1.86008e6 0.740909
\(364\) 137075. 0.0542255
\(365\) 1.45606e6 0.572067
\(366\) 1.00243e6 0.391158
\(367\) 3.33716e6 1.29334 0.646669 0.762771i \(-0.276162\pi\)
0.646669 + 0.762771i \(0.276162\pi\)
\(368\) 417485. 0.160702
\(369\) −1.35335e6 −0.517420
\(370\) −647975. −0.246067
\(371\) 431220. 0.162654
\(372\) 440450. 0.165021
\(373\) 3.49695e6 1.30142 0.650710 0.759326i \(-0.274471\pi\)
0.650710 + 0.759326i \(0.274471\pi\)
\(374\) 3.21039e6 1.18680
\(375\) −1.33152e6 −0.488957
\(376\) −1.44833e6 −0.528320
\(377\) 1.14230e6 0.413930
\(378\) 91220.6 0.0328370
\(379\) −4.62014e6 −1.65218 −0.826089 0.563539i \(-0.809439\pi\)
−0.826089 + 0.563539i \(0.809439\pi\)
\(380\) −512042. −0.181906
\(381\) 40686.4 0.0143594
\(382\) −3.09318e6 −1.08454
\(383\) −3.77589e6 −1.31529 −0.657645 0.753328i \(-0.728447\pi\)
−0.657645 + 0.753328i \(0.728447\pi\)
\(384\) −1.29878e6 −0.449476
\(385\) −632502. −0.217475
\(386\) −979353. −0.334558
\(387\) −364703. −0.123783
\(388\) 579174. 0.195312
\(389\) 1.44066e6 0.482710 0.241355 0.970437i \(-0.422408\pi\)
0.241355 + 0.970437i \(0.422408\pi\)
\(390\) −124710. −0.0415182
\(391\) −4.82749e6 −1.59691
\(392\) 2.63249e6 0.865269
\(393\) −1.81662e6 −0.593312
\(394\) −102562. −0.0332846
\(395\) 2.49060e6 0.803176
\(396\) 1.06686e6 0.341876
\(397\) 4.04765e6 1.28892 0.644461 0.764637i \(-0.277082\pi\)
0.644461 + 0.764637i \(0.277082\pi\)
\(398\) 1.37271e6 0.434382
\(399\) 309832. 0.0974302
\(400\) −344245. −0.107576
\(401\) −187555. −0.0582462 −0.0291231 0.999576i \(-0.509271\pi\)
−0.0291231 + 0.999576i \(0.509271\pi\)
\(402\) −1.65135e6 −0.509653
\(403\) 364355. 0.111754
\(404\) −3.34402e6 −1.01933
\(405\) 175348. 0.0531207
\(406\) −883926. −0.266134
\(407\) 4.58556e6 1.37216
\(408\) 2.55961e6 0.761243
\(409\) −3.25079e6 −0.960906 −0.480453 0.877021i \(-0.659528\pi\)
−0.480453 + 0.877021i \(0.659528\pi\)
\(410\) −1.43170e6 −0.420623
\(411\) 1.15712e6 0.337889
\(412\) 1.12925e6 0.327753
\(413\) 135854. 0.0391920
\(414\) 759284. 0.217723
\(415\) −2.39464e6 −0.682526
\(416\) −965313. −0.273486
\(417\) 2.38055e6 0.670405
\(418\) −1.71504e6 −0.480102
\(419\) 3.65412e6 1.01683 0.508414 0.861113i \(-0.330232\pi\)
0.508414 + 0.861113i \(0.330232\pi\)
\(420\) −203893. −0.0563999
\(421\) −5.36572e6 −1.47544 −0.737722 0.675105i \(-0.764098\pi\)
−0.737722 + 0.675105i \(0.764098\pi\)
\(422\) −3.44742e6 −0.942351
\(423\) 681112. 0.185084
\(424\) −1.90311e6 −0.514101
\(425\) 3.98059e6 1.06899
\(426\) 776.358 0.000207271 0
\(427\) 1.35576e6 0.359844
\(428\) −4.69043e6 −1.23767
\(429\) 882540. 0.231522
\(430\) −385818. −0.100626
\(431\) −6.87518e6 −1.78275 −0.891377 0.453264i \(-0.850260\pi\)
−0.891377 + 0.453264i \(0.850260\pi\)
\(432\) 104099. 0.0268372
\(433\) 40489.0 0.0103781 0.00518904 0.999987i \(-0.498348\pi\)
0.00518904 + 0.999987i \(0.498348\pi\)
\(434\) −281942. −0.0718515
\(435\) −1.69912e6 −0.430528
\(436\) −1.35884e6 −0.342335
\(437\) 2.57892e6 0.646002
\(438\) 1.57212e6 0.391563
\(439\) −4.44389e6 −1.10053 −0.550265 0.834990i \(-0.685473\pi\)
−0.550265 + 0.834990i \(0.685473\pi\)
\(440\) 2.79143e6 0.687377
\(441\) −1.23799e6 −0.303125
\(442\) 856101. 0.208434
\(443\) −278084. −0.0673236 −0.0336618 0.999433i \(-0.510717\pi\)
−0.0336618 + 0.999433i \(0.510717\pi\)
\(444\) 1.47819e6 0.355856
\(445\) −3.42511e6 −0.819925
\(446\) 1.73662e6 0.413397
\(447\) −2.79521e6 −0.661676
\(448\) 568636. 0.133856
\(449\) 1.07874e6 0.252523 0.126262 0.991997i \(-0.459702\pi\)
0.126262 + 0.991997i \(0.459702\pi\)
\(450\) −626081. −0.145747
\(451\) 1.01318e7 2.34555
\(452\) −5.26795e6 −1.21282
\(453\) 1.35190e6 0.309528
\(454\) −1.61538e6 −0.367821
\(455\) −168667. −0.0381945
\(456\) −1.36738e6 −0.307949
\(457\) −2.34460e6 −0.525144 −0.262572 0.964912i \(-0.584571\pi\)
−0.262572 + 0.964912i \(0.584571\pi\)
\(458\) 757292. 0.168694
\(459\) −1.20372e6 −0.266682
\(460\) −1.69712e6 −0.373954
\(461\) −8.15879e6 −1.78802 −0.894012 0.448043i \(-0.852121\pi\)
−0.894012 + 0.448043i \(0.852121\pi\)
\(462\) −682921. −0.148856
\(463\) 5.42287e6 1.17565 0.587823 0.808990i \(-0.299985\pi\)
0.587823 + 0.808990i \(0.299985\pi\)
\(464\) −1.00872e6 −0.217508
\(465\) −541962. −0.116235
\(466\) −4.33281e6 −0.924282
\(467\) 8.21137e6 1.74230 0.871151 0.491015i \(-0.163374\pi\)
0.871151 + 0.491015i \(0.163374\pi\)
\(468\) 284495. 0.0600426
\(469\) −2.23341e6 −0.468853
\(470\) 720547. 0.150459
\(471\) 3.43561e6 0.713595
\(472\) −599566. −0.123874
\(473\) 2.73034e6 0.561130
\(474\) 2.68913e6 0.549751
\(475\) −2.12649e6 −0.432444
\(476\) 1.39967e6 0.283145
\(477\) 894985. 0.180102
\(478\) −3.22345e6 −0.645285
\(479\) −5.34027e6 −1.06347 −0.531734 0.846911i \(-0.678459\pi\)
−0.531734 + 0.846911i \(0.678459\pi\)
\(480\) 1.43586e6 0.284452
\(481\) 1.22281e6 0.240989
\(482\) −3.10656e6 −0.609064
\(483\) 1.02691e6 0.200293
\(484\) −4.48899e6 −0.871034
\(485\) −712658. −0.137571
\(486\) 189326. 0.0363596
\(487\) 2.52006e6 0.481492 0.240746 0.970588i \(-0.422608\pi\)
0.240746 + 0.970588i \(0.422608\pi\)
\(488\) −5.98341e6 −1.13736
\(489\) −390306. −0.0738131
\(490\) −1.30967e6 −0.246417
\(491\) −8.33075e6 −1.55948 −0.779741 0.626102i \(-0.784649\pi\)
−0.779741 + 0.626102i \(0.784649\pi\)
\(492\) 3.26607e6 0.608293
\(493\) 1.16640e7 2.16138
\(494\) −457343. −0.0843188
\(495\) −1.31274e6 −0.240805
\(496\) −321746. −0.0587232
\(497\) 1050.00 0.000190678 0
\(498\) −2.58552e6 −0.467169
\(499\) 9.76824e6 1.75616 0.878082 0.478510i \(-0.158823\pi\)
0.878082 + 0.478510i \(0.158823\pi\)
\(500\) 3.21341e6 0.574832
\(501\) 2.04658e6 0.364280
\(502\) −5.58521e6 −0.989190
\(503\) 2.49233e6 0.439223 0.219612 0.975587i \(-0.429521\pi\)
0.219612 + 0.975587i \(0.429521\pi\)
\(504\) −544485. −0.0954794
\(505\) 4.11473e6 0.717981
\(506\) −5.68436e6 −0.986974
\(507\) −3.10629e6 −0.536689
\(508\) −98189.9 −0.0168814
\(509\) −7.02509e6 −1.20187 −0.600935 0.799298i \(-0.705205\pi\)
−0.600935 + 0.799298i \(0.705205\pi\)
\(510\) −1.27341e6 −0.216792
\(511\) 2.12626e6 0.360216
\(512\) 1.63948e6 0.276395
\(513\) 643047. 0.107882
\(514\) −2.23369e6 −0.372920
\(515\) −1.38951e6 −0.230858
\(516\) 880148. 0.145523
\(517\) −5.09913e6 −0.839015
\(518\) −946226. −0.154942
\(519\) −1.77044e6 −0.288511
\(520\) 744379. 0.120722
\(521\) −5.27900e6 −0.852034 −0.426017 0.904715i \(-0.640084\pi\)
−0.426017 + 0.904715i \(0.640084\pi\)
\(522\) −1.83456e6 −0.294684
\(523\) −291703. −0.0466322 −0.0233161 0.999728i \(-0.507422\pi\)
−0.0233161 + 0.999728i \(0.507422\pi\)
\(524\) 4.38411e6 0.697514
\(525\) −846758. −0.134079
\(526\) −2.86716e6 −0.451843
\(527\) 3.72043e6 0.583535
\(528\) −779334. −0.121658
\(529\) 2.11127e6 0.328023
\(530\) 946802. 0.146410
\(531\) 281961. 0.0433963
\(532\) −747727. −0.114542
\(533\) 2.70180e6 0.411941
\(534\) −3.69813e6 −0.561215
\(535\) 5.77145e6 0.871767
\(536\) 9.85674e6 1.48191
\(537\) −466617. −0.0698273
\(538\) 1.10969e6 0.165290
\(539\) 9.26821e6 1.37412
\(540\) −423173. −0.0624502
\(541\) 1.81999e6 0.267347 0.133673 0.991025i \(-0.457323\pi\)
0.133673 + 0.991025i \(0.457323\pi\)
\(542\) 1.52635e6 0.223180
\(543\) 6.23179e6 0.907013
\(544\) −9.85682e6 −1.42804
\(545\) 1.67201e6 0.241128
\(546\) −182112. −0.0261430
\(547\) −9.72974e6 −1.39038 −0.695189 0.718827i \(-0.744679\pi\)
−0.695189 + 0.718827i \(0.744679\pi\)
\(548\) −2.79252e6 −0.397232
\(549\) 2.81385e6 0.398447
\(550\) 4.68714e6 0.660695
\(551\) −6.23112e6 −0.874353
\(552\) −4.53208e6 −0.633068
\(553\) 3.63698e6 0.505740
\(554\) −4.29591e6 −0.594676
\(555\) −1.81888e6 −0.250652
\(556\) −5.74506e6 −0.788147
\(557\) −3.95131e6 −0.539638 −0.269819 0.962911i \(-0.586964\pi\)
−0.269819 + 0.962911i \(0.586964\pi\)
\(558\) −585163. −0.0795594
\(559\) 728087. 0.0985494
\(560\) 148942. 0.0200701
\(561\) 9.01163e6 1.20892
\(562\) 2.46002e6 0.328547
\(563\) −907456. −0.120658 −0.0603288 0.998179i \(-0.519215\pi\)
−0.0603288 + 0.998179i \(0.519215\pi\)
\(564\) −1.64375e6 −0.217589
\(565\) 6.48207e6 0.854265
\(566\) 1.25567e6 0.164753
\(567\) 256058. 0.0334488
\(568\) −4633.99 −0.000602677 0
\(569\) −4.32375e6 −0.559861 −0.279931 0.960020i \(-0.590311\pi\)
−0.279931 + 0.960020i \(0.590311\pi\)
\(570\) 680277. 0.0876999
\(571\) −1.76582e6 −0.226650 −0.113325 0.993558i \(-0.536150\pi\)
−0.113325 + 0.993558i \(0.536150\pi\)
\(572\) −2.12986e6 −0.272183
\(573\) −8.68263e6 −1.10475
\(574\) −2.09069e6 −0.264856
\(575\) −7.04808e6 −0.888999
\(576\) 1.18019e6 0.148216
\(577\) 4.97841e6 0.622516 0.311258 0.950325i \(-0.399250\pi\)
0.311258 + 0.950325i \(0.399250\pi\)
\(578\) 4.18925e6 0.521575
\(579\) −2.74906e6 −0.340791
\(580\) 4.10054e6 0.506141
\(581\) −3.49685e6 −0.429770
\(582\) −769465. −0.0941633
\(583\) −6.70028e6 −0.816435
\(584\) −9.38383e6 −1.13854
\(585\) −350063. −0.0422918
\(586\) 5.72556e6 0.688769
\(587\) 2.55026e6 0.305484 0.152742 0.988266i \(-0.451190\pi\)
0.152742 + 0.988266i \(0.451190\pi\)
\(588\) 2.98769e6 0.356362
\(589\) −1.98751e6 −0.236060
\(590\) 298286. 0.0352779
\(591\) −287892. −0.0339048
\(592\) −1.07981e6 −0.126632
\(593\) 5.95905e6 0.695889 0.347945 0.937515i \(-0.386880\pi\)
0.347945 + 0.937515i \(0.386880\pi\)
\(594\) −1.41738e6 −0.164824
\(595\) −1.72226e6 −0.199437
\(596\) 6.74576e6 0.777885
\(597\) 3.85323e6 0.442475
\(598\) −1.51582e6 −0.173339
\(599\) 7.34100e6 0.835966 0.417983 0.908455i \(-0.362737\pi\)
0.417983 + 0.908455i \(0.362737\pi\)
\(600\) 3.73701e6 0.423785
\(601\) 5.04285e6 0.569495 0.284748 0.958603i \(-0.408090\pi\)
0.284748 + 0.958603i \(0.408090\pi\)
\(602\) −563403. −0.0633619
\(603\) −4.63538e6 −0.519149
\(604\) −3.26259e6 −0.363890
\(605\) 5.52358e6 0.613524
\(606\) 4.44272e6 0.491437
\(607\) 1.78269e7 1.96383 0.981916 0.189318i \(-0.0606277\pi\)
0.981916 + 0.189318i \(0.0606277\pi\)
\(608\) 5.26567e6 0.577690
\(609\) −2.48120e6 −0.271093
\(610\) 2.97676e6 0.323906
\(611\) −1.35976e6 −0.147353
\(612\) 2.90498e6 0.313519
\(613\) 1.10977e7 1.19284 0.596421 0.802672i \(-0.296589\pi\)
0.596421 + 0.802672i \(0.296589\pi\)
\(614\) −2.97329e6 −0.318285
\(615\) −4.01881e6 −0.428460
\(616\) 4.07628e6 0.432824
\(617\) 6.00790e6 0.635345 0.317673 0.948200i \(-0.397099\pi\)
0.317673 + 0.948200i \(0.397099\pi\)
\(618\) −1.50027e6 −0.158015
\(619\) 1.79364e6 0.188151 0.0940757 0.995565i \(-0.470010\pi\)
0.0940757 + 0.995565i \(0.470010\pi\)
\(620\) 1.30793e6 0.136649
\(621\) 2.13133e6 0.221779
\(622\) 2.45729e6 0.254672
\(623\) −5.00163e6 −0.516287
\(624\) −207822. −0.0213663
\(625\) 3.57952e6 0.366543
\(626\) 725218. 0.0739661
\(627\) −4.81415e6 −0.489048
\(628\) −8.29127e6 −0.838923
\(629\) 1.24861e7 1.25835
\(630\) 270883. 0.0271913
\(631\) −1.00827e7 −1.00810 −0.504052 0.863673i \(-0.668158\pi\)
−0.504052 + 0.863673i \(0.668158\pi\)
\(632\) −1.60511e7 −1.59850
\(633\) −9.67697e6 −0.959909
\(634\) 3.46457e6 0.342316
\(635\) 120820. 0.0118906
\(636\) −2.15989e6 −0.211734
\(637\) 2.47151e6 0.241332
\(638\) 1.37344e7 1.33585
\(639\) 2179.25 0.000211133 0
\(640\) −3.85677e6 −0.372198
\(641\) −5.75277e6 −0.553009 −0.276505 0.961013i \(-0.589176\pi\)
−0.276505 + 0.961013i \(0.589176\pi\)
\(642\) 6.23150e6 0.596699
\(643\) −5.46072e6 −0.520862 −0.260431 0.965492i \(-0.583865\pi\)
−0.260431 + 0.965492i \(0.583865\pi\)
\(644\) −2.47828e6 −0.235470
\(645\) −1.08300e6 −0.102501
\(646\) −4.66993e6 −0.440280
\(647\) 2.48477e6 0.233360 0.116680 0.993170i \(-0.462775\pi\)
0.116680 + 0.993170i \(0.462775\pi\)
\(648\) −1.13006e6 −0.105722
\(649\) −2.11089e6 −0.196723
\(650\) 1.24990e6 0.116036
\(651\) −791417. −0.0731902
\(652\) 941939. 0.0867768
\(653\) −9.47172e6 −0.869253 −0.434626 0.900611i \(-0.643120\pi\)
−0.434626 + 0.900611i \(0.643120\pi\)
\(654\) 1.80529e6 0.165045
\(655\) −5.39453e6 −0.491304
\(656\) −2.38585e6 −0.216463
\(657\) 4.41298e6 0.398859
\(658\) 1.05220e6 0.0947402
\(659\) 9.34926e6 0.838617 0.419309 0.907844i \(-0.362273\pi\)
0.419309 + 0.907844i \(0.362273\pi\)
\(660\) 3.16808e6 0.283097
\(661\) 4.30563e6 0.383295 0.191647 0.981464i \(-0.438617\pi\)
0.191647 + 0.981464i \(0.438617\pi\)
\(662\) −9.29694e6 −0.824509
\(663\) 2.40309e6 0.212318
\(664\) 1.54327e7 1.35838
\(665\) 920057. 0.0806790
\(666\) −1.96386e6 −0.171564
\(667\) −2.06525e7 −1.79746
\(668\) −4.93908e6 −0.428258
\(669\) 4.87472e6 0.421099
\(670\) −4.90376e6 −0.422029
\(671\) −2.10658e7 −1.80623
\(672\) 2.09676e6 0.179113
\(673\) 1.43291e7 1.21950 0.609748 0.792595i \(-0.291271\pi\)
0.609748 + 0.792595i \(0.291271\pi\)
\(674\) −655662. −0.0555943
\(675\) −1.75742e6 −0.148462
\(676\) 7.49652e6 0.630947
\(677\) −6.86042e6 −0.575279 −0.287640 0.957739i \(-0.592871\pi\)
−0.287640 + 0.957739i \(0.592871\pi\)
\(678\) 6.99877e6 0.584719
\(679\) −1.04068e6 −0.0866250
\(680\) 7.60086e6 0.630363
\(681\) −4.53441e6 −0.374674
\(682\) 4.38081e6 0.360656
\(683\) −2.00520e7 −1.64478 −0.822388 0.568927i \(-0.807359\pi\)
−0.822388 + 0.568927i \(0.807359\pi\)
\(684\) −1.55189e6 −0.126829
\(685\) 3.43612e6 0.279796
\(686\) −4.01557e6 −0.325789
\(687\) 2.12573e6 0.171837
\(688\) −642943. −0.0517847
\(689\) −1.78674e6 −0.143388
\(690\) 2.25472e6 0.180290
\(691\) 1.21795e7 0.970365 0.485182 0.874413i \(-0.338753\pi\)
0.485182 + 0.874413i \(0.338753\pi\)
\(692\) 4.27266e6 0.339182
\(693\) −1.91697e6 −0.151629
\(694\) 3.67299e6 0.289482
\(695\) 7.06914e6 0.555142
\(696\) 1.09503e7 0.856846
\(697\) 2.75881e7 2.15100
\(698\) 1.00840e7 0.783418
\(699\) −1.21623e7 −0.941503
\(700\) 2.04351e6 0.157627
\(701\) 1.23445e7 0.948809 0.474404 0.880307i \(-0.342663\pi\)
0.474404 + 0.880307i \(0.342663\pi\)
\(702\) −377967. −0.0289475
\(703\) −6.67029e6 −0.509045
\(704\) −8.83544e6 −0.671888
\(705\) 2.02259e6 0.153262
\(706\) −7.54378e6 −0.569609
\(707\) 6.00867e6 0.452095
\(708\) −680465. −0.0510179
\(709\) 2.21727e7 1.65654 0.828272 0.560327i \(-0.189324\pi\)
0.828272 + 0.560327i \(0.189324\pi\)
\(710\) 2305.42 0.000171635 0
\(711\) 7.54844e6 0.559994
\(712\) 2.20737e7 1.63183
\(713\) −6.58745e6 −0.485281
\(714\) −1.85954e6 −0.136509
\(715\) 2.62074e6 0.191716
\(716\) 1.12610e6 0.0820909
\(717\) −9.04830e6 −0.657308
\(718\) 636632. 0.0460869
\(719\) −2.40050e6 −0.173173 −0.0865864 0.996244i \(-0.527596\pi\)
−0.0865864 + 0.996244i \(0.527596\pi\)
\(720\) 309126. 0.0222231
\(721\) −2.02908e6 −0.145365
\(722\) −5.44423e6 −0.388681
\(723\) −8.72019e6 −0.620412
\(724\) −1.50394e7 −1.06631
\(725\) 1.70294e7 1.20325
\(726\) 5.96387e6 0.419940
\(727\) −9.51466e6 −0.667663 −0.333831 0.942633i \(-0.608342\pi\)
−0.333831 + 0.942633i \(0.608342\pi\)
\(728\) 1.08700e6 0.0760155
\(729\) 531441. 0.0370370
\(730\) 4.66848e6 0.324242
\(731\) 7.43451e6 0.514587
\(732\) −6.79075e6 −0.468425
\(733\) 1.04964e6 0.0721572 0.0360786 0.999349i \(-0.488513\pi\)
0.0360786 + 0.999349i \(0.488513\pi\)
\(734\) 1.06998e7 0.733050
\(735\) −3.67627e6 −0.251009
\(736\) 1.74526e7 1.18759
\(737\) 3.47027e7 2.35339
\(738\) −4.33916e6 −0.293268
\(739\) −2.68604e6 −0.180926 −0.0904629 0.995900i \(-0.528835\pi\)
−0.0904629 + 0.995900i \(0.528835\pi\)
\(740\) 4.38955e6 0.294674
\(741\) −1.28377e6 −0.0858898
\(742\) 1.38260e6 0.0921905
\(743\) −2.09335e7 −1.39114 −0.695569 0.718460i \(-0.744847\pi\)
−0.695569 + 0.718460i \(0.744847\pi\)
\(744\) 3.49277e6 0.231333
\(745\) −8.30047e6 −0.547914
\(746\) 1.12121e7 0.737632
\(747\) −7.25761e6 −0.475874
\(748\) −2.17481e7 −1.42124
\(749\) 8.42794e6 0.548930
\(750\) −4.26919e6 −0.277136
\(751\) 2.59501e7 1.67895 0.839477 0.543395i \(-0.182861\pi\)
0.839477 + 0.543395i \(0.182861\pi\)
\(752\) 1.20075e6 0.0774298
\(753\) −1.56778e7 −1.00762
\(754\) 3.66250e6 0.234611
\(755\) 4.01452e6 0.256311
\(756\) −617953. −0.0393234
\(757\) −8.97503e6 −0.569241 −0.284620 0.958640i \(-0.591868\pi\)
−0.284620 + 0.958640i \(0.591868\pi\)
\(758\) −1.48133e7 −0.936438
\(759\) −1.59561e7 −1.00536
\(760\) −4.06050e6 −0.255003
\(761\) 1.20515e7 0.754361 0.377180 0.926140i \(-0.376894\pi\)
0.377180 + 0.926140i \(0.376894\pi\)
\(762\) 130451. 0.00813878
\(763\) 2.44161e6 0.151833
\(764\) 2.09541e7 1.29878
\(765\) −3.57450e6 −0.220832
\(766\) −1.21064e7 −0.745493
\(767\) −562903. −0.0345498
\(768\) −8.36042e6 −0.511476
\(769\) 1.32176e7 0.806001 0.403001 0.915200i \(-0.367967\pi\)
0.403001 + 0.915200i \(0.367967\pi\)
\(770\) −2.02796e6 −0.123263
\(771\) −6.27002e6 −0.379868
\(772\) 6.63440e6 0.400644
\(773\) −1.10628e7 −0.665914 −0.332957 0.942942i \(-0.608046\pi\)
−0.332957 + 0.942942i \(0.608046\pi\)
\(774\) −1.16933e6 −0.0701590
\(775\) 5.43179e6 0.324855
\(776\) 4.59285e6 0.273797
\(777\) −2.65608e6 −0.157829
\(778\) 4.61910e6 0.273595
\(779\) −1.47380e7 −0.870152
\(780\) 844818. 0.0497195
\(781\) −16314.9 −0.000957100 0
\(782\) −1.54781e7 −0.905109
\(783\) −5.14966e6 −0.300175
\(784\) −2.18249e6 −0.126813
\(785\) 1.02022e7 0.590907
\(786\) −5.82454e6 −0.336283
\(787\) −1.33119e7 −0.766132 −0.383066 0.923721i \(-0.625132\pi\)
−0.383066 + 0.923721i \(0.625132\pi\)
\(788\) 694779. 0.0398595
\(789\) −8.04817e6 −0.460261
\(790\) 7.98547e6 0.455232
\(791\) 9.46565e6 0.537910
\(792\) 8.46019e6 0.479256
\(793\) −5.61753e6 −0.317221
\(794\) 1.29778e7 0.730548
\(795\) 2.65769e6 0.149137
\(796\) −9.29912e6 −0.520186
\(797\) 1.61590e6 0.0901089 0.0450544 0.998985i \(-0.485654\pi\)
0.0450544 + 0.998985i \(0.485654\pi\)
\(798\) 993397. 0.0552224
\(799\) −1.38846e7 −0.769423
\(800\) −1.43909e7 −0.794991
\(801\) −1.03807e7 −0.571672
\(802\) −601347. −0.0330133
\(803\) −3.30377e7 −1.80809
\(804\) 1.11867e7 0.610327
\(805\) 3.04945e6 0.165856
\(806\) 1.16821e6 0.0633408
\(807\) 3.11492e6 0.168370
\(808\) −2.65181e7 −1.42894
\(809\) −1.48836e7 −0.799535 −0.399768 0.916617i \(-0.630909\pi\)
−0.399768 + 0.916617i \(0.630909\pi\)
\(810\) 562210. 0.0301083
\(811\) −3.19912e7 −1.70796 −0.853982 0.520303i \(-0.825819\pi\)
−0.853982 + 0.520303i \(0.825819\pi\)
\(812\) 5.98795e6 0.318705
\(813\) 4.28450e6 0.227339
\(814\) 1.47024e7 0.777728
\(815\) −1.15903e6 −0.0611224
\(816\) −2.12207e6 −0.111567
\(817\) −3.97163e6 −0.208168
\(818\) −1.04228e7 −0.544632
\(819\) −511191. −0.0266301
\(820\) 9.69873e6 0.503709
\(821\) −2.95946e7 −1.53234 −0.766169 0.642640i \(-0.777839\pi\)
−0.766169 + 0.642640i \(0.777839\pi\)
\(822\) 3.71002e6 0.191512
\(823\) 1.71026e7 0.880163 0.440081 0.897958i \(-0.354950\pi\)
0.440081 + 0.897958i \(0.354950\pi\)
\(824\) 8.95496e6 0.459458
\(825\) 1.31569e7 0.673005
\(826\) 435581. 0.0222136
\(827\) −2.24600e7 −1.14195 −0.570974 0.820968i \(-0.693434\pi\)
−0.570974 + 0.820968i \(0.693434\pi\)
\(828\) −5.14360e6 −0.260730
\(829\) −2.65000e7 −1.33924 −0.669622 0.742702i \(-0.733544\pi\)
−0.669622 + 0.742702i \(0.733544\pi\)
\(830\) −7.67780e6 −0.386849
\(831\) −1.20587e7 −0.605756
\(832\) −2.35611e6 −0.118001
\(833\) 2.52367e7 1.26014
\(834\) 7.63263e6 0.379979
\(835\) 6.07740e6 0.301649
\(836\) 1.16182e7 0.574938
\(837\) −1.64256e6 −0.0810417
\(838\) 1.17160e7 0.576328
\(839\) 3.38841e7 1.66185 0.830924 0.556386i \(-0.187813\pi\)
0.830924 + 0.556386i \(0.187813\pi\)
\(840\) −1.61687e6 −0.0790636
\(841\) 2.93890e7 1.43283
\(842\) −1.72038e7 −0.836266
\(843\) 6.90532e6 0.334669
\(844\) 2.33537e7 1.12850
\(845\) −9.22426e6 −0.444416
\(846\) 2.18381e6 0.104903
\(847\) 8.06598e6 0.386321
\(848\) 1.57779e6 0.0753459
\(849\) 3.52468e6 0.167823
\(850\) 1.27627e7 0.605894
\(851\) −2.21081e7 −1.04647
\(852\) −5259.26 −0.000248214 0
\(853\) −1.88452e6 −0.0886804 −0.0443402 0.999016i \(-0.514119\pi\)
−0.0443402 + 0.999016i \(0.514119\pi\)
\(854\) 4.34691e6 0.203956
\(855\) 1.90955e6 0.0893339
\(856\) −3.71951e7 −1.73501
\(857\) 1.95529e6 0.0909407 0.0454704 0.998966i \(-0.485521\pi\)
0.0454704 + 0.998966i \(0.485521\pi\)
\(858\) 2.82964e6 0.131224
\(859\) −1.15147e7 −0.532437 −0.266219 0.963913i \(-0.585774\pi\)
−0.266219 + 0.963913i \(0.585774\pi\)
\(860\) 2.61363e6 0.120503
\(861\) −5.86860e6 −0.269791
\(862\) −2.20435e7 −1.01045
\(863\) 2.60655e6 0.119135 0.0595674 0.998224i \(-0.481028\pi\)
0.0595674 + 0.998224i \(0.481028\pi\)
\(864\) 4.35177e6 0.198327
\(865\) −5.25739e6 −0.238908
\(866\) 129818. 0.00588219
\(867\) 1.17593e7 0.531293
\(868\) 1.90995e6 0.0860445
\(869\) −5.65112e7 −2.53855
\(870\) −5.44781e6 −0.244019
\(871\) 9.25402e6 0.413319
\(872\) −1.07756e7 −0.479899
\(873\) −2.15990e6 −0.0959177
\(874\) 8.26865e6 0.366147
\(875\) −5.77397e6 −0.254950
\(876\) −1.06500e7 −0.468910
\(877\) 3.54109e7 1.55467 0.777335 0.629087i \(-0.216571\pi\)
0.777335 + 0.629087i \(0.216571\pi\)
\(878\) −1.42482e7 −0.623769
\(879\) 1.60718e7 0.701602
\(880\) −2.31426e6 −0.100741
\(881\) 1.78051e7 0.772867 0.386434 0.922317i \(-0.373707\pi\)
0.386434 + 0.922317i \(0.373707\pi\)
\(882\) −3.96931e6 −0.171808
\(883\) −3.70288e7 −1.59822 −0.799111 0.601183i \(-0.794696\pi\)
−0.799111 + 0.601183i \(0.794696\pi\)
\(884\) −5.79946e6 −0.249607
\(885\) 837294. 0.0359352
\(886\) −891607. −0.0381583
\(887\) −3.85936e7 −1.64705 −0.823524 0.567281i \(-0.807995\pi\)
−0.823524 + 0.567281i \(0.807995\pi\)
\(888\) 1.17221e7 0.498853
\(889\) 176431. 0.00748723
\(890\) −1.09817e7 −0.464725
\(891\) −3.97862e6 −0.167895
\(892\) −1.17643e7 −0.495056
\(893\) 7.41735e6 0.311258
\(894\) −8.96212e6 −0.375031
\(895\) −1.38564e6 −0.0578218
\(896\) −5.63197e6 −0.234364
\(897\) −4.25495e6 −0.176569
\(898\) 3.45871e6 0.143128
\(899\) 1.59164e7 0.656820
\(900\) 4.24124e6 0.174537
\(901\) −1.82444e7 −0.748716
\(902\) 3.24850e7 1.32943
\(903\) −1.58148e6 −0.0645424
\(904\) −4.17749e7 −1.70018
\(905\) 1.85056e7 0.751070
\(906\) 4.33453e6 0.175437
\(907\) −9.11662e6 −0.367973 −0.183986 0.982929i \(-0.558900\pi\)
−0.183986 + 0.982929i \(0.558900\pi\)
\(908\) 1.09430e7 0.440477
\(909\) 1.24708e7 0.500593
\(910\) −540787. −0.0216483
\(911\) −4.35218e7 −1.73745 −0.868723 0.495299i \(-0.835059\pi\)
−0.868723 + 0.495299i \(0.835059\pi\)
\(912\) 1.13364e6 0.0451325
\(913\) 5.43339e7 2.15722
\(914\) −7.51737e6 −0.297646
\(915\) 8.35583e6 0.329942
\(916\) −5.13010e6 −0.202017
\(917\) −7.87753e6 −0.309362
\(918\) −3.85943e6 −0.151153
\(919\) 2.99781e6 0.117089 0.0585444 0.998285i \(-0.481354\pi\)
0.0585444 + 0.998285i \(0.481354\pi\)
\(920\) −1.34582e7 −0.524224
\(921\) −8.34609e6 −0.324216
\(922\) −2.61591e7 −1.01343
\(923\) −4350.63 −0.000168092 0
\(924\) 4.62629e6 0.178260
\(925\) 1.82296e7 0.700525
\(926\) 1.73871e7 0.666344
\(927\) −4.21130e6 −0.160959
\(928\) −4.21686e7 −1.60738
\(929\) 2.19993e7 0.836313 0.418156 0.908375i \(-0.362676\pi\)
0.418156 + 0.908375i \(0.362676\pi\)
\(930\) −1.73766e6 −0.0658807
\(931\) −1.34818e7 −0.509770
\(932\) 2.93516e7 1.10686
\(933\) 6.89767e6 0.259417
\(934\) 2.63277e7 0.987519
\(935\) 2.67604e7 1.00107
\(936\) 2.25604e6 0.0841701
\(937\) 8.01794e6 0.298341 0.149171 0.988811i \(-0.452340\pi\)
0.149171 + 0.988811i \(0.452340\pi\)
\(938\) −7.16087e6 −0.265741
\(939\) 2.03570e6 0.0753443
\(940\) −4.88118e6 −0.180179
\(941\) 9.19445e6 0.338494 0.169247 0.985574i \(-0.445866\pi\)
0.169247 + 0.985574i \(0.445866\pi\)
\(942\) 1.10154e7 0.404459
\(943\) −4.88479e7 −1.78882
\(944\) 497076. 0.0181549
\(945\) 760374. 0.0276980
\(946\) 8.75414e6 0.318043
\(947\) −4.27420e6 −0.154874 −0.0774372 0.996997i \(-0.524674\pi\)
−0.0774372 + 0.996997i \(0.524674\pi\)
\(948\) −1.82169e7 −0.658344
\(949\) −8.81002e6 −0.317550
\(950\) −6.81806e6 −0.245105
\(951\) 9.72513e6 0.348694
\(952\) 1.10994e7 0.396924
\(953\) −5.03679e7 −1.79648 −0.898239 0.439507i \(-0.855153\pi\)
−0.898239 + 0.439507i \(0.855153\pi\)
\(954\) 2.86954e6 0.102080
\(955\) −2.57834e7 −0.914812
\(956\) 2.18365e7 0.772750
\(957\) 3.85528e7 1.36074
\(958\) −1.71222e7 −0.602763
\(959\) 5.01770e6 0.176181
\(960\) 3.50461e6 0.122733
\(961\) −2.35524e7 −0.822670
\(962\) 3.92063e6 0.136590
\(963\) 1.74920e7 0.607817
\(964\) 2.10447e7 0.729374
\(965\) −8.16345e6 −0.282199
\(966\) 3.29253e6 0.113524
\(967\) −1.89781e6 −0.0652659 −0.0326329 0.999467i \(-0.510389\pi\)
−0.0326329 + 0.999467i \(0.510389\pi\)
\(968\) −3.55977e7 −1.22105
\(969\) −1.31086e7 −0.448484
\(970\) −2.28496e6 −0.0779738
\(971\) 2.68041e7 0.912331 0.456166 0.889895i \(-0.349222\pi\)
0.456166 + 0.889895i \(0.349222\pi\)
\(972\) −1.28254e6 −0.0435418
\(973\) 1.03229e7 0.349560
\(974\) 8.07994e6 0.272905
\(975\) 3.50849e6 0.118198
\(976\) 4.96061e6 0.166690
\(977\) 4.27764e7 1.43373 0.716865 0.697212i \(-0.245576\pi\)
0.716865 + 0.697212i \(0.245576\pi\)
\(978\) −1.25142e6 −0.0418365
\(979\) 7.77151e7 2.59148
\(980\) 8.87205e6 0.295093
\(981\) 5.06750e6 0.168121
\(982\) −2.67104e7 −0.883898
\(983\) −3.97908e7 −1.31340 −0.656702 0.754150i \(-0.728049\pi\)
−0.656702 + 0.754150i \(0.728049\pi\)
\(984\) 2.59000e7 0.852730
\(985\) −854907. −0.0280756
\(986\) 3.73978e7 1.22505
\(987\) 2.95355e6 0.0965054
\(988\) 3.09816e6 0.100975
\(989\) −1.31636e7 −0.427943
\(990\) −4.20897e6 −0.136486
\(991\) 2.18022e7 0.705207 0.352604 0.935773i \(-0.385296\pi\)
0.352604 + 0.935773i \(0.385296\pi\)
\(992\) −1.34503e7 −0.433964
\(993\) −2.60967e7 −0.839871
\(994\) 3366.57 0.000108074 0
\(995\) 1.14423e7 0.366400
\(996\) 1.75150e7 0.559451
\(997\) −2.61869e7 −0.834346 −0.417173 0.908827i \(-0.636979\pi\)
−0.417173 + 0.908827i \(0.636979\pi\)
\(998\) 3.13194e7 0.995376
\(999\) −5.51261e6 −0.174761
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.a.1.7 11
3.2 odd 2 531.6.a.b.1.5 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.a.1.7 11 1.1 even 1 trivial
531.6.a.b.1.5 11 3.2 odd 2