Properties

Label 847.6.a.b.1.1
Level $847$
Weight $6$
Character 847.1
Self dual yes
Analytic conductor $135.845$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.845095382\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 847.1

$q$-expansion

\(f(q)\) \(=\) \(q+10.0000 q^{2} -14.0000 q^{3} +68.0000 q^{4} -56.0000 q^{5} -140.000 q^{6} +49.0000 q^{7} +360.000 q^{8} -47.0000 q^{9} +O(q^{10})\) \(q+10.0000 q^{2} -14.0000 q^{3} +68.0000 q^{4} -56.0000 q^{5} -140.000 q^{6} +49.0000 q^{7} +360.000 q^{8} -47.0000 q^{9} -560.000 q^{10} -952.000 q^{12} +140.000 q^{13} +490.000 q^{14} +784.000 q^{15} +1424.00 q^{16} +1722.00 q^{17} -470.000 q^{18} +98.0000 q^{19} -3808.00 q^{20} -686.000 q^{21} +1824.00 q^{23} -5040.00 q^{24} +11.0000 q^{25} +1400.00 q^{26} +4060.00 q^{27} +3332.00 q^{28} -3418.00 q^{29} +7840.00 q^{30} -7644.00 q^{31} +2720.00 q^{32} +17220.0 q^{34} -2744.00 q^{35} -3196.00 q^{36} -10398.0 q^{37} +980.000 q^{38} -1960.00 q^{39} -20160.0 q^{40} +17962.0 q^{41} -6860.00 q^{42} -10880.0 q^{43} +2632.00 q^{45} +18240.0 q^{46} +9324.00 q^{47} -19936.0 q^{48} +2401.00 q^{49} +110.000 q^{50} -24108.0 q^{51} +9520.00 q^{52} +2262.00 q^{53} +40600.0 q^{54} +17640.0 q^{56} -1372.00 q^{57} -34180.0 q^{58} -2730.00 q^{59} +53312.0 q^{60} -25648.0 q^{61} -76440.0 q^{62} -2303.00 q^{63} -18368.0 q^{64} -7840.00 q^{65} -48404.0 q^{67} +117096. q^{68} -25536.0 q^{69} -27440.0 q^{70} -58560.0 q^{71} -16920.0 q^{72} -68082.0 q^{73} -103980. q^{74} -154.000 q^{75} +6664.00 q^{76} -19600.0 q^{78} -31784.0 q^{79} -79744.0 q^{80} -45419.0 q^{81} +179620. q^{82} +20538.0 q^{83} -46648.0 q^{84} -96432.0 q^{85} -108800. q^{86} +47852.0 q^{87} -50582.0 q^{89} +26320.0 q^{90} +6860.00 q^{91} +124032. q^{92} +107016. q^{93} +93240.0 q^{94} -5488.00 q^{95} -38080.0 q^{96} -58506.0 q^{97} +24010.0 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.0000 1.76777 0.883883 0.467707i \(-0.154920\pi\)
0.883883 + 0.467707i \(0.154920\pi\)
\(3\) −14.0000 −0.898100 −0.449050 0.893507i \(-0.648238\pi\)
−0.449050 + 0.893507i \(0.648238\pi\)
\(4\) 68.0000 2.12500
\(5\) −56.0000 −1.00176 −0.500879 0.865517i \(-0.666990\pi\)
−0.500879 + 0.865517i \(0.666990\pi\)
\(6\) −140.000 −1.58763
\(7\) 49.0000 0.377964
\(8\) 360.000 1.98874
\(9\) −47.0000 −0.193416
\(10\) −560.000 −1.77088
\(11\) 0 0
\(12\) −952.000 −1.90846
\(13\) 140.000 0.229757 0.114879 0.993380i \(-0.463352\pi\)
0.114879 + 0.993380i \(0.463352\pi\)
\(14\) 490.000 0.668153
\(15\) 784.000 0.899680
\(16\) 1424.00 1.39062
\(17\) 1722.00 1.44514 0.722572 0.691296i \(-0.242960\pi\)
0.722572 + 0.691296i \(0.242960\pi\)
\(18\) −470.000 −0.341914
\(19\) 98.0000 0.0622791 0.0311395 0.999515i \(-0.490086\pi\)
0.0311395 + 0.999515i \(0.490086\pi\)
\(20\) −3808.00 −2.12874
\(21\) −686.000 −0.339450
\(22\) 0 0
\(23\) 1824.00 0.718961 0.359480 0.933153i \(-0.382954\pi\)
0.359480 + 0.933153i \(0.382954\pi\)
\(24\) −5040.00 −1.78609
\(25\) 11.0000 0.00352000
\(26\) 1400.00 0.406158
\(27\) 4060.00 1.07181
\(28\) 3332.00 0.803175
\(29\) −3418.00 −0.754705 −0.377352 0.926070i \(-0.623165\pi\)
−0.377352 + 0.926070i \(0.623165\pi\)
\(30\) 7840.00 1.59042
\(31\) −7644.00 −1.42862 −0.714310 0.699830i \(-0.753259\pi\)
−0.714310 + 0.699830i \(0.753259\pi\)
\(32\) 2720.00 0.469563
\(33\) 0 0
\(34\) 17220.0 2.55468
\(35\) −2744.00 −0.378629
\(36\) −3196.00 −0.411008
\(37\) −10398.0 −1.24866 −0.624332 0.781159i \(-0.714629\pi\)
−0.624332 + 0.781159i \(0.714629\pi\)
\(38\) 980.000 0.110095
\(39\) −1960.00 −0.206345
\(40\) −20160.0 −1.99223
\(41\) 17962.0 1.66876 0.834382 0.551186i \(-0.185825\pi\)
0.834382 + 0.551186i \(0.185825\pi\)
\(42\) −6860.00 −0.600069
\(43\) −10880.0 −0.897342 −0.448671 0.893697i \(-0.648102\pi\)
−0.448671 + 0.893697i \(0.648102\pi\)
\(44\) 0 0
\(45\) 2632.00 0.193756
\(46\) 18240.0 1.27096
\(47\) 9324.00 0.615684 0.307842 0.951438i \(-0.400393\pi\)
0.307842 + 0.951438i \(0.400393\pi\)
\(48\) −19936.0 −1.24892
\(49\) 2401.00 0.142857
\(50\) 110.000 0.00622254
\(51\) −24108.0 −1.29788
\(52\) 9520.00 0.488235
\(53\) 2262.00 0.110612 0.0553061 0.998469i \(-0.482387\pi\)
0.0553061 + 0.998469i \(0.482387\pi\)
\(54\) 40600.0 1.89471
\(55\) 0 0
\(56\) 17640.0 0.751672
\(57\) −1372.00 −0.0559329
\(58\) −34180.0 −1.33414
\(59\) −2730.00 −0.102102 −0.0510508 0.998696i \(-0.516257\pi\)
−0.0510508 + 0.998696i \(0.516257\pi\)
\(60\) 53312.0 1.91182
\(61\) −25648.0 −0.882529 −0.441264 0.897377i \(-0.645470\pi\)
−0.441264 + 0.897377i \(0.645470\pi\)
\(62\) −76440.0 −2.52547
\(63\) −2303.00 −0.0731042
\(64\) −18368.0 −0.560547
\(65\) −7840.00 −0.230161
\(66\) 0 0
\(67\) −48404.0 −1.31733 −0.658664 0.752437i \(-0.728878\pi\)
−0.658664 + 0.752437i \(0.728878\pi\)
\(68\) 117096. 3.07093
\(69\) −25536.0 −0.645699
\(70\) −27440.0 −0.669328
\(71\) −58560.0 −1.37865 −0.689327 0.724450i \(-0.742094\pi\)
−0.689327 + 0.724450i \(0.742094\pi\)
\(72\) −16920.0 −0.384653
\(73\) −68082.0 −1.49529 −0.747645 0.664099i \(-0.768815\pi\)
−0.747645 + 0.664099i \(0.768815\pi\)
\(74\) −103980. −2.20735
\(75\) −154.000 −0.00316131
\(76\) 6664.00 0.132343
\(77\) 0 0
\(78\) −19600.0 −0.364770
\(79\) −31784.0 −0.572982 −0.286491 0.958083i \(-0.592489\pi\)
−0.286491 + 0.958083i \(0.592489\pi\)
\(80\) −79744.0 −1.39307
\(81\) −45419.0 −0.769175
\(82\) 179620. 2.94999
\(83\) 20538.0 0.327237 0.163619 0.986524i \(-0.447683\pi\)
0.163619 + 0.986524i \(0.447683\pi\)
\(84\) −46648.0 −0.721331
\(85\) −96432.0 −1.44768
\(86\) −108800. −1.58629
\(87\) 47852.0 0.677801
\(88\) 0 0
\(89\) −50582.0 −0.676894 −0.338447 0.940985i \(-0.609902\pi\)
−0.338447 + 0.940985i \(0.609902\pi\)
\(90\) 26320.0 0.342515
\(91\) 6860.00 0.0868402
\(92\) 124032. 1.52779
\(93\) 107016. 1.28304
\(94\) 93240.0 1.08839
\(95\) −5488.00 −0.0623886
\(96\) −38080.0 −0.421715
\(97\) −58506.0 −0.631351 −0.315676 0.948867i \(-0.602231\pi\)
−0.315676 + 0.948867i \(0.602231\pi\)
\(98\) 24010.0 0.252538
\(99\) 0 0
\(100\) 748.000 0.00748000
\(101\) −38696.0 −0.377453 −0.188726 0.982030i \(-0.560436\pi\)
−0.188726 + 0.982030i \(0.560436\pi\)
\(102\) −241080. −2.29436
\(103\) 53060.0 0.492804 0.246402 0.969168i \(-0.420752\pi\)
0.246402 + 0.969168i \(0.420752\pi\)
\(104\) 50400.0 0.456927
\(105\) 38416.0 0.340047
\(106\) 22620.0 0.195537
\(107\) 146324. 1.23554 0.617769 0.786360i \(-0.288037\pi\)
0.617769 + 0.786360i \(0.288037\pi\)
\(108\) 276080. 2.27759
\(109\) −92898.0 −0.748928 −0.374464 0.927241i \(-0.622173\pi\)
−0.374464 + 0.927241i \(0.622173\pi\)
\(110\) 0 0
\(111\) 145572. 1.12143
\(112\) 69776.0 0.525607
\(113\) −83354.0 −0.614088 −0.307044 0.951695i \(-0.599340\pi\)
−0.307044 + 0.951695i \(0.599340\pi\)
\(114\) −13720.0 −0.0988762
\(115\) −102144. −0.720225
\(116\) −232424. −1.60375
\(117\) −6580.00 −0.0444387
\(118\) −27300.0 −0.180492
\(119\) 84378.0 0.546213
\(120\) 282240. 1.78923
\(121\) 0 0
\(122\) −256480. −1.56011
\(123\) −251468. −1.49872
\(124\) −519792. −3.03582
\(125\) 174384. 0.998232
\(126\) −23030.0 −0.129231
\(127\) −60384.0 −0.332210 −0.166105 0.986108i \(-0.553119\pi\)
−0.166105 + 0.986108i \(0.553119\pi\)
\(128\) −270720. −1.46048
\(129\) 152320. 0.805903
\(130\) −78400.0 −0.406872
\(131\) 61586.0 0.313548 0.156774 0.987635i \(-0.449891\pi\)
0.156774 + 0.987635i \(0.449891\pi\)
\(132\) 0 0
\(133\) 4802.00 0.0235393
\(134\) −484040. −2.32873
\(135\) −227360. −1.07369
\(136\) 619920. 2.87401
\(137\) −204462. −0.930703 −0.465352 0.885126i \(-0.654072\pi\)
−0.465352 + 0.885126i \(0.654072\pi\)
\(138\) −255360. −1.14145
\(139\) 35406.0 0.155432 0.0777159 0.996976i \(-0.475237\pi\)
0.0777159 + 0.996976i \(0.475237\pi\)
\(140\) −186592. −0.804587
\(141\) −130536. −0.552946
\(142\) −585600. −2.43714
\(143\) 0 0
\(144\) −66928.0 −0.268969
\(145\) 191408. 0.756032
\(146\) −680820. −2.64332
\(147\) −33614.0 −0.128300
\(148\) −707064. −2.65341
\(149\) 20226.0 0.0746353 0.0373177 0.999303i \(-0.488119\pi\)
0.0373177 + 0.999303i \(0.488119\pi\)
\(150\) −1540.00 −0.00558847
\(151\) −70904.0 −0.253063 −0.126531 0.991963i \(-0.540384\pi\)
−0.126531 + 0.991963i \(0.540384\pi\)
\(152\) 35280.0 0.123857
\(153\) −80934.0 −0.279513
\(154\) 0 0
\(155\) 428064. 1.43113
\(156\) −133280. −0.438484
\(157\) 293524. 0.950374 0.475187 0.879885i \(-0.342380\pi\)
0.475187 + 0.879885i \(0.342380\pi\)
\(158\) −317840. −1.01290
\(159\) −31668.0 −0.0993408
\(160\) −152320. −0.470389
\(161\) 89376.0 0.271742
\(162\) −454190. −1.35972
\(163\) 13192.0 0.0388903 0.0194452 0.999811i \(-0.493810\pi\)
0.0194452 + 0.999811i \(0.493810\pi\)
\(164\) 1.22142e6 3.54612
\(165\) 0 0
\(166\) 205380. 0.578479
\(167\) −493612. −1.36960 −0.684801 0.728730i \(-0.740111\pi\)
−0.684801 + 0.728730i \(0.740111\pi\)
\(168\) −246960. −0.675077
\(169\) −351693. −0.947212
\(170\) −964320. −2.55917
\(171\) −4606.00 −0.0120457
\(172\) −739840. −1.90685
\(173\) −240716. −0.611490 −0.305745 0.952113i \(-0.598906\pi\)
−0.305745 + 0.952113i \(0.598906\pi\)
\(174\) 478520. 1.19819
\(175\) 539.000 0.00133043
\(176\) 0 0
\(177\) 38220.0 0.0916975
\(178\) −505820. −1.19659
\(179\) 294932. 0.688001 0.344001 0.938969i \(-0.388218\pi\)
0.344001 + 0.938969i \(0.388218\pi\)
\(180\) 178976. 0.411731
\(181\) −336980. −0.764553 −0.382277 0.924048i \(-0.624860\pi\)
−0.382277 + 0.924048i \(0.624860\pi\)
\(182\) 68600.0 0.153513
\(183\) 359072. 0.792600
\(184\) 656640. 1.42982
\(185\) 582288. 1.25086
\(186\) 1.07016e6 2.26812
\(187\) 0 0
\(188\) 634032. 1.30833
\(189\) 198940. 0.405105
\(190\) −54880.0 −0.110288
\(191\) 358264. 0.710591 0.355296 0.934754i \(-0.384380\pi\)
0.355296 + 0.934754i \(0.384380\pi\)
\(192\) 257152. 0.503427
\(193\) 989554. 1.91226 0.956128 0.292948i \(-0.0946362\pi\)
0.956128 + 0.292948i \(0.0946362\pi\)
\(194\) −585060. −1.11608
\(195\) 109760. 0.206708
\(196\) 163268. 0.303571
\(197\) 990050. 1.81757 0.908786 0.417263i \(-0.137011\pi\)
0.908786 + 0.417263i \(0.137011\pi\)
\(198\) 0 0
\(199\) −840756. −1.50500 −0.752501 0.658591i \(-0.771153\pi\)
−0.752501 + 0.658591i \(0.771153\pi\)
\(200\) 3960.00 0.00700036
\(201\) 677656. 1.18309
\(202\) −386960. −0.667249
\(203\) −167482. −0.285252
\(204\) −1.63934e6 −2.75800
\(205\) −1.00587e6 −1.67170
\(206\) 530600. 0.871163
\(207\) −85728.0 −0.139058
\(208\) 199360. 0.319506
\(209\) 0 0
\(210\) 384160. 0.601124
\(211\) −1.15073e6 −1.77938 −0.889689 0.456568i \(-0.849079\pi\)
−0.889689 + 0.456568i \(0.849079\pi\)
\(212\) 153816. 0.235051
\(213\) 819840. 1.23817
\(214\) 1.46324e6 2.18414
\(215\) 609280. 0.898919
\(216\) 1.46160e6 2.13154
\(217\) −374556. −0.539967
\(218\) −928980. −1.32393
\(219\) 953148. 1.34292
\(220\) 0 0
\(221\) 241080. 0.332032
\(222\) 1.45572e6 1.98242
\(223\) −824264. −1.10995 −0.554976 0.831866i \(-0.687273\pi\)
−0.554976 + 0.831866i \(0.687273\pi\)
\(224\) 133280. 0.177478
\(225\) −517.000 −0.000680823 0
\(226\) −833540. −1.08556
\(227\) −74382.0 −0.0958083 −0.0479042 0.998852i \(-0.515254\pi\)
−0.0479042 + 0.998852i \(0.515254\pi\)
\(228\) −93296.0 −0.118857
\(229\) 1.13196e6 1.42640 0.713199 0.700961i \(-0.247245\pi\)
0.713199 + 0.700961i \(0.247245\pi\)
\(230\) −1.02144e6 −1.27319
\(231\) 0 0
\(232\) −1.23048e6 −1.50091
\(233\) 198726. 0.239809 0.119904 0.992785i \(-0.461741\pi\)
0.119904 + 0.992785i \(0.461741\pi\)
\(234\) −65800.0 −0.0785572
\(235\) −522144. −0.616766
\(236\) −185640. −0.216966
\(237\) 444976. 0.514595
\(238\) 843780. 0.965577
\(239\) −482904. −0.546847 −0.273424 0.961894i \(-0.588156\pi\)
−0.273424 + 0.961894i \(0.588156\pi\)
\(240\) 1.11642e6 1.25112
\(241\) −805910. −0.893807 −0.446904 0.894582i \(-0.647473\pi\)
−0.446904 + 0.894582i \(0.647473\pi\)
\(242\) 0 0
\(243\) −350714. −0.381011
\(244\) −1.74406e6 −1.87537
\(245\) −134456. −0.143108
\(246\) −2.51468e6 −2.64938
\(247\) 13720.0 0.0143091
\(248\) −2.75184e6 −2.84115
\(249\) −287532. −0.293892
\(250\) 1.74384e6 1.76464
\(251\) 430738. 0.431548 0.215774 0.976443i \(-0.430773\pi\)
0.215774 + 0.976443i \(0.430773\pi\)
\(252\) −156604. −0.155347
\(253\) 0 0
\(254\) −603840. −0.587270
\(255\) 1.35005e6 1.30017
\(256\) −2.11942e6 −2.02124
\(257\) −1.17691e6 −1.11150 −0.555751 0.831349i \(-0.687569\pi\)
−0.555751 + 0.831349i \(0.687569\pi\)
\(258\) 1.52320e6 1.42465
\(259\) −509502. −0.471951
\(260\) −533120. −0.489093
\(261\) 160646. 0.145972
\(262\) 615860. 0.554279
\(263\) −1.29098e6 −1.15088 −0.575438 0.817845i \(-0.695169\pi\)
−0.575438 + 0.817845i \(0.695169\pi\)
\(264\) 0 0
\(265\) −126672. −0.110807
\(266\) 48020.0 0.0416119
\(267\) 708148. 0.607919
\(268\) −3.29147e6 −2.79932
\(269\) −1.27756e6 −1.07646 −0.538232 0.842797i \(-0.680907\pi\)
−0.538232 + 0.842797i \(0.680907\pi\)
\(270\) −2.27360e6 −1.89804
\(271\) −1.65054e6 −1.36522 −0.682612 0.730781i \(-0.739156\pi\)
−0.682612 + 0.730781i \(0.739156\pi\)
\(272\) 2.45213e6 2.00965
\(273\) −96040.0 −0.0779912
\(274\) −2.04462e6 −1.64527
\(275\) 0 0
\(276\) −1.73645e6 −1.37211
\(277\) 1.06409e6 0.833257 0.416628 0.909077i \(-0.363212\pi\)
0.416628 + 0.909077i \(0.363212\pi\)
\(278\) 354060. 0.274767
\(279\) 359268. 0.276317
\(280\) −987840. −0.752994
\(281\) 22342.0 0.0168794 0.00843969 0.999964i \(-0.497314\pi\)
0.00843969 + 0.999964i \(0.497314\pi\)
\(282\) −1.30536e6 −0.977479
\(283\) 2.49574e6 1.85239 0.926196 0.377042i \(-0.123059\pi\)
0.926196 + 0.377042i \(0.123059\pi\)
\(284\) −3.98208e6 −2.92964
\(285\) 76832.0 0.0560312
\(286\) 0 0
\(287\) 880138. 0.630734
\(288\) −127840. −0.0908208
\(289\) 1.54543e6 1.08844
\(290\) 1.91408e6 1.33649
\(291\) 819084. 0.567017
\(292\) −4.62958e6 −3.17749
\(293\) 1.93178e6 1.31458 0.657291 0.753637i \(-0.271702\pi\)
0.657291 + 0.753637i \(0.271702\pi\)
\(294\) −336140. −0.226805
\(295\) 152880. 0.102281
\(296\) −3.74328e6 −2.48326
\(297\) 0 0
\(298\) 202260. 0.131938
\(299\) 255360. 0.165187
\(300\) −10472.0 −0.00671779
\(301\) −533120. −0.339163
\(302\) −709040. −0.447356
\(303\) 541744. 0.338991
\(304\) 139552. 0.0866068
\(305\) 1.43629e6 0.884081
\(306\) −809340. −0.494114
\(307\) 459074. 0.277995 0.138997 0.990293i \(-0.455612\pi\)
0.138997 + 0.990293i \(0.455612\pi\)
\(308\) 0 0
\(309\) −742840. −0.442587
\(310\) 4.28064e6 2.52991
\(311\) 667128. 0.391118 0.195559 0.980692i \(-0.437348\pi\)
0.195559 + 0.980692i \(0.437348\pi\)
\(312\) −705600. −0.410367
\(313\) −111034. −0.0640612 −0.0320306 0.999487i \(-0.510197\pi\)
−0.0320306 + 0.999487i \(0.510197\pi\)
\(314\) 2.93524e6 1.68004
\(315\) 128968. 0.0732328
\(316\) −2.16131e6 −1.21759
\(317\) −68778.0 −0.0384416 −0.0192208 0.999815i \(-0.506119\pi\)
−0.0192208 + 0.999815i \(0.506119\pi\)
\(318\) −316680. −0.175611
\(319\) 0 0
\(320\) 1.02861e6 0.561533
\(321\) −2.04854e6 −1.10964
\(322\) 893760. 0.480376
\(323\) 168756. 0.0900022
\(324\) −3.08849e6 −1.63450
\(325\) 1540.00 0.000808746 0
\(326\) 131920. 0.0687490
\(327\) 1.30057e6 0.672613
\(328\) 6.46632e6 3.31874
\(329\) 456876. 0.232707
\(330\) 0 0
\(331\) −564448. −0.283174 −0.141587 0.989926i \(-0.545221\pi\)
−0.141587 + 0.989926i \(0.545221\pi\)
\(332\) 1.39658e6 0.695379
\(333\) 488706. 0.241511
\(334\) −4.93612e6 −2.42114
\(335\) 2.71062e6 1.31965
\(336\) −976864. −0.472048
\(337\) −2.07729e6 −0.996376 −0.498188 0.867069i \(-0.666001\pi\)
−0.498188 + 0.867069i \(0.666001\pi\)
\(338\) −3.51693e6 −1.67445
\(339\) 1.16696e6 0.551512
\(340\) −6.55738e6 −3.07633
\(341\) 0 0
\(342\) −46060.0 −0.0212941
\(343\) 117649. 0.0539949
\(344\) −3.91680e6 −1.78458
\(345\) 1.43002e6 0.646834
\(346\) −2.40716e6 −1.08097
\(347\) 53248.0 0.0237399 0.0118700 0.999930i \(-0.496222\pi\)
0.0118700 + 0.999930i \(0.496222\pi\)
\(348\) 3.25394e6 1.44033
\(349\) 2.27200e6 0.998494 0.499247 0.866460i \(-0.333610\pi\)
0.499247 + 0.866460i \(0.333610\pi\)
\(350\) 5390.00 0.00235190
\(351\) 568400. 0.246256
\(352\) 0 0
\(353\) 4.00645e6 1.71129 0.855644 0.517565i \(-0.173162\pi\)
0.855644 + 0.517565i \(0.173162\pi\)
\(354\) 382200. 0.162100
\(355\) 3.27936e6 1.38108
\(356\) −3.43958e6 −1.43840
\(357\) −1.18129e6 −0.490554
\(358\) 2.94932e6 1.21623
\(359\) −73784.0 −0.0302152 −0.0151076 0.999886i \(-0.504809\pi\)
−0.0151076 + 0.999886i \(0.504809\pi\)
\(360\) 947520. 0.385329
\(361\) −2.46650e6 −0.996121
\(362\) −3.36980e6 −1.35155
\(363\) 0 0
\(364\) 466480. 0.184535
\(365\) 3.81259e6 1.49792
\(366\) 3.59072e6 1.40113
\(367\) 1.40431e6 0.544250 0.272125 0.962262i \(-0.412274\pi\)
0.272125 + 0.962262i \(0.412274\pi\)
\(368\) 2.59738e6 0.999805
\(369\) −844214. −0.322765
\(370\) 5.82288e6 2.21123
\(371\) 110838. 0.0418075
\(372\) 7.27709e6 2.72647
\(373\) 1.60323e6 0.596657 0.298329 0.954463i \(-0.403571\pi\)
0.298329 + 0.954463i \(0.403571\pi\)
\(374\) 0 0
\(375\) −2.44138e6 −0.896513
\(376\) 3.35664e6 1.22443
\(377\) −478520. −0.173399
\(378\) 1.98940e6 0.716131
\(379\) −4.77012e6 −1.70581 −0.852906 0.522064i \(-0.825162\pi\)
−0.852906 + 0.522064i \(0.825162\pi\)
\(380\) −373184. −0.132576
\(381\) 845376. 0.298358
\(382\) 3.58264e6 1.25616
\(383\) −2.23079e6 −0.777072 −0.388536 0.921434i \(-0.627019\pi\)
−0.388536 + 0.921434i \(0.627019\pi\)
\(384\) 3.79008e6 1.31166
\(385\) 0 0
\(386\) 9.89554e6 3.38042
\(387\) 511360. 0.173560
\(388\) −3.97841e6 −1.34162
\(389\) 4.84024e6 1.62178 0.810892 0.585196i \(-0.198982\pi\)
0.810892 + 0.585196i \(0.198982\pi\)
\(390\) 1.09760e6 0.365412
\(391\) 3.14093e6 1.03900
\(392\) 864360. 0.284105
\(393\) −862204. −0.281597
\(394\) 9.90050e6 3.21304
\(395\) 1.77990e6 0.573989
\(396\) 0 0
\(397\) 995820. 0.317106 0.158553 0.987350i \(-0.449317\pi\)
0.158553 + 0.987350i \(0.449317\pi\)
\(398\) −8.40756e6 −2.66049
\(399\) −67228.0 −0.0211406
\(400\) 15664.0 0.00489500
\(401\) −3.31605e6 −1.02982 −0.514909 0.857245i \(-0.672174\pi\)
−0.514909 + 0.857245i \(0.672174\pi\)
\(402\) 6.77656e6 2.09143
\(403\) −1.07016e6 −0.328236
\(404\) −2.63133e6 −0.802087
\(405\) 2.54346e6 0.770527
\(406\) −1.67482e6 −0.504258
\(407\) 0 0
\(408\) −8.67888e6 −2.58115
\(409\) −3.07273e6 −0.908274 −0.454137 0.890932i \(-0.650052\pi\)
−0.454137 + 0.890932i \(0.650052\pi\)
\(410\) −1.00587e7 −2.95517
\(411\) 2.86247e6 0.835865
\(412\) 3.60808e6 1.04721
\(413\) −133770. −0.0385908
\(414\) −857280. −0.245823
\(415\) −1.15013e6 −0.327813
\(416\) 380800. 0.107886
\(417\) −495684. −0.139593
\(418\) 0 0
\(419\) 2.81438e6 0.783154 0.391577 0.920145i \(-0.371930\pi\)
0.391577 + 0.920145i \(0.371930\pi\)
\(420\) 2.61229e6 0.722600
\(421\) 3.05802e6 0.840883 0.420441 0.907320i \(-0.361875\pi\)
0.420441 + 0.907320i \(0.361875\pi\)
\(422\) −1.15073e7 −3.14552
\(423\) −438228. −0.119083
\(424\) 814320. 0.219979
\(425\) 18942.0 0.00508690
\(426\) 8.19840e6 2.18880
\(427\) −1.25675e6 −0.333565
\(428\) 9.95003e6 2.62552
\(429\) 0 0
\(430\) 6.09280e6 1.58908
\(431\) −1.93750e6 −0.502398 −0.251199 0.967936i \(-0.580825\pi\)
−0.251199 + 0.967936i \(0.580825\pi\)
\(432\) 5.78144e6 1.49048
\(433\) 3.94790e6 1.01192 0.505961 0.862557i \(-0.331138\pi\)
0.505961 + 0.862557i \(0.331138\pi\)
\(434\) −3.74556e6 −0.954536
\(435\) −2.67971e6 −0.678993
\(436\) −6.31706e6 −1.59147
\(437\) 178752. 0.0447762
\(438\) 9.53148e6 2.37397
\(439\) 7.41770e6 1.83700 0.918498 0.395426i \(-0.129403\pi\)
0.918498 + 0.395426i \(0.129403\pi\)
\(440\) 0 0
\(441\) −112847. −0.0276308
\(442\) 2.41080e6 0.586956
\(443\) 1.40269e6 0.339589 0.169794 0.985480i \(-0.445690\pi\)
0.169794 + 0.985480i \(0.445690\pi\)
\(444\) 9.89890e6 2.38303
\(445\) 2.83259e6 0.678085
\(446\) −8.24264e6 −1.96214
\(447\) −283164. −0.0670300
\(448\) −900032. −0.211867
\(449\) −590574. −0.138248 −0.0691239 0.997608i \(-0.522020\pi\)
−0.0691239 + 0.997608i \(0.522020\pi\)
\(450\) −5170.00 −0.00120354
\(451\) 0 0
\(452\) −5.66807e6 −1.30494
\(453\) 992656. 0.227276
\(454\) −743820. −0.169367
\(455\) −384160. −0.0869929
\(456\) −493920. −0.111236
\(457\) 2.90484e6 0.650627 0.325313 0.945606i \(-0.394530\pi\)
0.325313 + 0.945606i \(0.394530\pi\)
\(458\) 1.13196e7 2.52154
\(459\) 6.99132e6 1.54891
\(460\) −6.94579e6 −1.53048
\(461\) 922684. 0.202209 0.101105 0.994876i \(-0.467762\pi\)
0.101105 + 0.994876i \(0.467762\pi\)
\(462\) 0 0
\(463\) 7.18235e6 1.55709 0.778546 0.627588i \(-0.215958\pi\)
0.778546 + 0.627588i \(0.215958\pi\)
\(464\) −4.86723e6 −1.04951
\(465\) −5.99290e6 −1.28530
\(466\) 1.98726e6 0.423926
\(467\) −612570. −0.129976 −0.0649881 0.997886i \(-0.520701\pi\)
−0.0649881 + 0.997886i \(0.520701\pi\)
\(468\) −447440. −0.0944322
\(469\) −2.37180e6 −0.497904
\(470\) −5.22144e6 −1.09030
\(471\) −4.10934e6 −0.853531
\(472\) −982800. −0.203053
\(473\) 0 0
\(474\) 4.44976e6 0.909684
\(475\) 1078.00 0.000219222 0
\(476\) 5.73770e6 1.16070
\(477\) −106314. −0.0213941
\(478\) −4.82904e6 −0.966699
\(479\) −2.60330e6 −0.518424 −0.259212 0.965820i \(-0.583463\pi\)
−0.259212 + 0.965820i \(0.583463\pi\)
\(480\) 2.13248e6 0.422456
\(481\) −1.45572e6 −0.286890
\(482\) −8.05910e6 −1.58004
\(483\) −1.25126e6 −0.244051
\(484\) 0 0
\(485\) 3.27634e6 0.632461
\(486\) −3.50714e6 −0.673539
\(487\) 5.46309e6 1.04380 0.521898 0.853008i \(-0.325224\pi\)
0.521898 + 0.853008i \(0.325224\pi\)
\(488\) −9.23328e6 −1.75512
\(489\) −184688. −0.0349274
\(490\) −1.34456e6 −0.252982
\(491\) −1.64090e6 −0.307170 −0.153585 0.988135i \(-0.549082\pi\)
−0.153585 + 0.988135i \(0.549082\pi\)
\(492\) −1.70998e7 −3.18478
\(493\) −5.88580e6 −1.09066
\(494\) 137200. 0.0252951
\(495\) 0 0
\(496\) −1.08851e7 −1.98667
\(497\) −2.86944e6 −0.521082
\(498\) −2.87532e6 −0.519533
\(499\) 2.99796e6 0.538983 0.269491 0.963003i \(-0.413144\pi\)
0.269491 + 0.963003i \(0.413144\pi\)
\(500\) 1.18581e7 2.12124
\(501\) 6.91057e6 1.23004
\(502\) 4.30738e6 0.762876
\(503\) 6.89405e6 1.21494 0.607469 0.794343i \(-0.292185\pi\)
0.607469 + 0.794343i \(0.292185\pi\)
\(504\) −829080. −0.145385
\(505\) 2.16698e6 0.378117
\(506\) 0 0
\(507\) 4.92370e6 0.850691
\(508\) −4.10611e6 −0.705946
\(509\) 2.30476e6 0.394305 0.197152 0.980373i \(-0.436831\pi\)
0.197152 + 0.980373i \(0.436831\pi\)
\(510\) 1.35005e7 2.29839
\(511\) −3.33602e6 −0.565166
\(512\) −1.25312e7 −2.11260
\(513\) 397880. 0.0667511
\(514\) −1.17691e7 −1.96488
\(515\) −2.97136e6 −0.493671
\(516\) 1.03578e7 1.71254
\(517\) 0 0
\(518\) −5.09502e6 −0.834299
\(519\) 3.37002e6 0.549180
\(520\) −2.82240e6 −0.457731
\(521\) −1.20960e7 −1.95231 −0.976155 0.217073i \(-0.930349\pi\)
−0.976155 + 0.217073i \(0.930349\pi\)
\(522\) 1.60646e6 0.258044
\(523\) −5.48443e6 −0.876753 −0.438377 0.898791i \(-0.644446\pi\)
−0.438377 + 0.898791i \(0.644446\pi\)
\(524\) 4.18785e6 0.666289
\(525\) −7546.00 −0.00119486
\(526\) −1.29098e7 −2.03448
\(527\) −1.31630e7 −2.06456
\(528\) 0 0
\(529\) −3.10937e6 −0.483095
\(530\) −1.26672e6 −0.195880
\(531\) 128310. 0.0197480
\(532\) 326536. 0.0500210
\(533\) 2.51468e6 0.383411
\(534\) 7.08148e6 1.07466
\(535\) −8.19414e6 −1.23771
\(536\) −1.74254e7 −2.61982
\(537\) −4.12905e6 −0.617894
\(538\) −1.27756e7 −1.90294
\(539\) 0 0
\(540\) −1.54605e7 −2.28160
\(541\) 6.71799e6 0.986839 0.493420 0.869791i \(-0.335747\pi\)
0.493420 + 0.869791i \(0.335747\pi\)
\(542\) −1.65054e7 −2.41340
\(543\) 4.71772e6 0.686646
\(544\) 4.68384e6 0.678586
\(545\) 5.20229e6 0.750245
\(546\) −960400. −0.137870
\(547\) 5.00235e6 0.714835 0.357418 0.933945i \(-0.383657\pi\)
0.357418 + 0.933945i \(0.383657\pi\)
\(548\) −1.39034e7 −1.97774
\(549\) 1.20546e6 0.170695
\(550\) 0 0
\(551\) −334964. −0.0470023
\(552\) −9.19296e6 −1.28413
\(553\) −1.55742e6 −0.216567
\(554\) 1.06409e7 1.47300
\(555\) −8.15203e6 −1.12340
\(556\) 2.40761e6 0.330293
\(557\) −9.01961e6 −1.23183 −0.615913 0.787814i \(-0.711213\pi\)
−0.615913 + 0.787814i \(0.711213\pi\)
\(558\) 3.59268e6 0.488465
\(559\) −1.52320e6 −0.206171
\(560\) −3.90746e6 −0.526531
\(561\) 0 0
\(562\) 223420. 0.0298388
\(563\) −1.24051e7 −1.64941 −0.824707 0.565561i \(-0.808660\pi\)
−0.824707 + 0.565561i \(0.808660\pi\)
\(564\) −8.87645e6 −1.17501
\(565\) 4.66782e6 0.615167
\(566\) 2.49574e7 3.27460
\(567\) −2.22553e6 −0.290721
\(568\) −2.10816e7 −2.74178
\(569\) −6.48804e6 −0.840103 −0.420052 0.907500i \(-0.637988\pi\)
−0.420052 + 0.907500i \(0.637988\pi\)
\(570\) 768320. 0.0990501
\(571\) 1.02285e7 1.31287 0.656435 0.754382i \(-0.272064\pi\)
0.656435 + 0.754382i \(0.272064\pi\)
\(572\) 0 0
\(573\) −5.01570e6 −0.638182
\(574\) 8.80138e6 1.11499
\(575\) 20064.0 0.00253074
\(576\) 863296. 0.108419
\(577\) 2.65338e6 0.331787 0.165894 0.986144i \(-0.446949\pi\)
0.165894 + 0.986144i \(0.446949\pi\)
\(578\) 1.54543e7 1.92411
\(579\) −1.38538e7 −1.71740
\(580\) 1.30157e7 1.60657
\(581\) 1.00636e6 0.123684
\(582\) 8.19084e6 1.00235
\(583\) 0 0
\(584\) −2.45095e7 −2.97374
\(585\) 368480. 0.0445168
\(586\) 1.93178e7 2.32387
\(587\) −1.43044e7 −1.71346 −0.856729 0.515766i \(-0.827507\pi\)
−0.856729 + 0.515766i \(0.827507\pi\)
\(588\) −2.28575e6 −0.272638
\(589\) −749112. −0.0889731
\(590\) 1.52880e6 0.180809
\(591\) −1.38607e7 −1.63236
\(592\) −1.48068e7 −1.73642
\(593\) 1.00265e7 1.17088 0.585442 0.810714i \(-0.300921\pi\)
0.585442 + 0.810714i \(0.300921\pi\)
\(594\) 0 0
\(595\) −4.72517e6 −0.547173
\(596\) 1.37537e6 0.158600
\(597\) 1.17706e7 1.35164
\(598\) 2.55360e6 0.292011
\(599\) −7.52292e6 −0.856681 −0.428341 0.903617i \(-0.640902\pi\)
−0.428341 + 0.903617i \(0.640902\pi\)
\(600\) −55440.0 −0.00628702
\(601\) −3.38625e6 −0.382413 −0.191207 0.981550i \(-0.561240\pi\)
−0.191207 + 0.981550i \(0.561240\pi\)
\(602\) −5.33120e6 −0.599562
\(603\) 2.27499e6 0.254792
\(604\) −4.82147e6 −0.537759
\(605\) 0 0
\(606\) 5.41744e6 0.599256
\(607\) 6.90861e6 0.761060 0.380530 0.924769i \(-0.375742\pi\)
0.380530 + 0.924769i \(0.375742\pi\)
\(608\) 266560. 0.0292439
\(609\) 2.34475e6 0.256185
\(610\) 1.43629e7 1.56285
\(611\) 1.30536e6 0.141458
\(612\) −5.50351e6 −0.593966
\(613\) 9.68896e6 1.04142 0.520710 0.853734i \(-0.325667\pi\)
0.520710 + 0.853734i \(0.325667\pi\)
\(614\) 4.59074e6 0.491430
\(615\) 1.40822e7 1.50135
\(616\) 0 0
\(617\) −7.84742e6 −0.829877 −0.414939 0.909849i \(-0.636197\pi\)
−0.414939 + 0.909849i \(0.636197\pi\)
\(618\) −7.42840e6 −0.782391
\(619\) −1.01972e7 −1.06968 −0.534840 0.844953i \(-0.679628\pi\)
−0.534840 + 0.844953i \(0.679628\pi\)
\(620\) 2.91084e7 3.04115
\(621\) 7.40544e6 0.770587
\(622\) 6.67128e6 0.691406
\(623\) −2.47852e6 −0.255842
\(624\) −2.79104e6 −0.286949
\(625\) −9.79988e6 −1.00351
\(626\) −1.11034e6 −0.113245
\(627\) 0 0
\(628\) 1.99596e7 2.01954
\(629\) −1.79054e7 −1.80450
\(630\) 1.28968e6 0.129459
\(631\) −8.36258e6 −0.836116 −0.418058 0.908420i \(-0.637289\pi\)
−0.418058 + 0.908420i \(0.637289\pi\)
\(632\) −1.14422e7 −1.13951
\(633\) 1.61102e7 1.59806
\(634\) −687780. −0.0679558
\(635\) 3.38150e6 0.332794
\(636\) −2.15342e6 −0.211099
\(637\) 336140. 0.0328225
\(638\) 0 0
\(639\) 2.75232e6 0.266653
\(640\) 1.51603e7 1.46305
\(641\) 1.10283e6 0.106014 0.0530070 0.998594i \(-0.483119\pi\)
0.0530070 + 0.998594i \(0.483119\pi\)
\(642\) −2.04854e7 −1.96158
\(643\) 1.71354e7 1.63443 0.817217 0.576330i \(-0.195516\pi\)
0.817217 + 0.576330i \(0.195516\pi\)
\(644\) 6.07757e6 0.577451
\(645\) −8.52992e6 −0.807320
\(646\) 1.68756e6 0.159103
\(647\) −54964.0 −0.00516200 −0.00258100 0.999997i \(-0.500822\pi\)
−0.00258100 + 0.999997i \(0.500822\pi\)
\(648\) −1.63508e7 −1.52969
\(649\) 0 0
\(650\) 15400.0 0.00142968
\(651\) 5.24378e6 0.484945
\(652\) 897056. 0.0826420
\(653\) −485166. −0.0445254 −0.0222627 0.999752i \(-0.507087\pi\)
−0.0222627 + 0.999752i \(0.507087\pi\)
\(654\) 1.30057e7 1.18902
\(655\) −3.44882e6 −0.314099
\(656\) 2.55779e7 2.32063
\(657\) 3.19985e6 0.289212
\(658\) 4.56876e6 0.411371
\(659\) 2.72136e6 0.244103 0.122051 0.992524i \(-0.461053\pi\)
0.122051 + 0.992524i \(0.461053\pi\)
\(660\) 0 0
\(661\) −2.14525e6 −0.190974 −0.0954869 0.995431i \(-0.530441\pi\)
−0.0954869 + 0.995431i \(0.530441\pi\)
\(662\) −5.64448e6 −0.500586
\(663\) −3.37512e6 −0.298198
\(664\) 7.39368e6 0.650789
\(665\) −268912. −0.0235807
\(666\) 4.88706e6 0.426935
\(667\) −6.23443e6 −0.542603
\(668\) −3.35656e7 −2.91041
\(669\) 1.15397e7 0.996848
\(670\) 2.71062e7 2.33283
\(671\) 0 0
\(672\) −1.86592e6 −0.159393
\(673\) −2.92796e6 −0.249188 −0.124594 0.992208i \(-0.539763\pi\)
−0.124594 + 0.992208i \(0.539763\pi\)
\(674\) −2.07729e7 −1.76136
\(675\) 44660.0 0.00377276
\(676\) −2.39151e7 −2.01282
\(677\) 1.34992e7 1.13198 0.565988 0.824414i \(-0.308495\pi\)
0.565988 + 0.824414i \(0.308495\pi\)
\(678\) 1.16696e7 0.974945
\(679\) −2.86679e6 −0.238628
\(680\) −3.47155e7 −2.87906
\(681\) 1.04135e6 0.0860455
\(682\) 0 0
\(683\) −5.42972e6 −0.445375 −0.222688 0.974890i \(-0.571483\pi\)
−0.222688 + 0.974890i \(0.571483\pi\)
\(684\) −313208. −0.0255972
\(685\) 1.14499e7 0.932340
\(686\) 1.17649e6 0.0954504
\(687\) −1.58474e7 −1.28105
\(688\) −1.54931e7 −1.24787
\(689\) 316680. 0.0254140
\(690\) 1.43002e7 1.14345
\(691\) 2.08280e7 1.65940 0.829702 0.558207i \(-0.188510\pi\)
0.829702 + 0.558207i \(0.188510\pi\)
\(692\) −1.63687e7 −1.29942
\(693\) 0 0
\(694\) 532480. 0.0419667
\(695\) −1.98274e6 −0.155705
\(696\) 1.72267e7 1.34797
\(697\) 3.09306e7 2.41160
\(698\) 2.27200e7 1.76510
\(699\) −2.78216e6 −0.215372
\(700\) 36652.0 0.00282717
\(701\) −2.35141e7 −1.80731 −0.903655 0.428261i \(-0.859126\pi\)
−0.903655 + 0.428261i \(0.859126\pi\)
\(702\) 5.68400e6 0.435323
\(703\) −1.01900e6 −0.0777656
\(704\) 0 0
\(705\) 7.31002e6 0.553918
\(706\) 4.00645e7 3.02516
\(707\) −1.89610e6 −0.142664
\(708\) 2.59896e6 0.194857
\(709\) −1.95747e7 −1.46244 −0.731221 0.682140i \(-0.761049\pi\)
−0.731221 + 0.682140i \(0.761049\pi\)
\(710\) 3.27936e7 2.44142
\(711\) 1.49385e6 0.110824
\(712\) −1.82095e7 −1.34617
\(713\) −1.39427e7 −1.02712
\(714\) −1.18129e7 −0.867185
\(715\) 0 0
\(716\) 2.00554e7 1.46200
\(717\) 6.76066e6 0.491124
\(718\) −737840. −0.0534135
\(719\) −2.61152e7 −1.88396 −0.941978 0.335674i \(-0.891036\pi\)
−0.941978 + 0.335674i \(0.891036\pi\)
\(720\) 3.74797e6 0.269442
\(721\) 2.59994e6 0.186262
\(722\) −2.46650e7 −1.76091
\(723\) 1.12827e7 0.802729
\(724\) −2.29146e7 −1.62468
\(725\) −37598.0 −0.00265656
\(726\) 0 0
\(727\) 1.54126e7 1.08154 0.540768 0.841172i \(-0.318134\pi\)
0.540768 + 0.841172i \(0.318134\pi\)
\(728\) 2.46960e6 0.172702
\(729\) 1.59468e7 1.11136
\(730\) 3.81259e7 2.64797
\(731\) −1.87354e7 −1.29679
\(732\) 2.44169e7 1.68427
\(733\) 1.69868e7 1.16776 0.583878 0.811841i \(-0.301535\pi\)
0.583878 + 0.811841i \(0.301535\pi\)
\(734\) 1.40431e7 0.962107
\(735\) 1.88238e6 0.128526
\(736\) 4.96128e6 0.337597
\(737\) 0 0
\(738\) −8.44214e6 −0.570574
\(739\) −2.01511e6 −0.135734 −0.0678669 0.997694i \(-0.521619\pi\)
−0.0678669 + 0.997694i \(0.521619\pi\)
\(740\) 3.95956e7 2.65808
\(741\) −192080. −0.0128510
\(742\) 1.10838e6 0.0739059
\(743\) 1.51381e7 1.00600 0.503001 0.864286i \(-0.332229\pi\)
0.503001 + 0.864286i \(0.332229\pi\)
\(744\) 3.85258e7 2.55164
\(745\) −1.13266e6 −0.0747666
\(746\) 1.60323e7 1.05475
\(747\) −965286. −0.0632928
\(748\) 0 0
\(749\) 7.16988e6 0.466989
\(750\) −2.44138e7 −1.58483
\(751\) 7.21401e6 0.466742 0.233371 0.972388i \(-0.425024\pi\)
0.233371 + 0.972388i \(0.425024\pi\)
\(752\) 1.32774e7 0.856185
\(753\) −6.03033e6 −0.387573
\(754\) −4.78520e6 −0.306529
\(755\) 3.97062e6 0.253508
\(756\) 1.35279e7 0.860848
\(757\) −1.09697e7 −0.695755 −0.347877 0.937540i \(-0.613097\pi\)
−0.347877 + 0.937540i \(0.613097\pi\)
\(758\) −4.77012e7 −3.01548
\(759\) 0 0
\(760\) −1.97568e6 −0.124075
\(761\) −1.92442e7 −1.20459 −0.602293 0.798275i \(-0.705746\pi\)
−0.602293 + 0.798275i \(0.705746\pi\)
\(762\) 8.45376e6 0.527427
\(763\) −4.55200e6 −0.283068
\(764\) 2.43620e7 1.51001
\(765\) 4.53230e6 0.280005
\(766\) −2.23079e7 −1.37368
\(767\) −382200. −0.0234586
\(768\) 2.96719e7 1.81528
\(769\) −8.21185e6 −0.500755 −0.250378 0.968148i \(-0.580555\pi\)
−0.250378 + 0.968148i \(0.580555\pi\)
\(770\) 0 0
\(771\) 1.64767e7 0.998241
\(772\) 6.72897e7 4.06355
\(773\) 1.86187e7 1.12073 0.560363 0.828247i \(-0.310662\pi\)
0.560363 + 0.828247i \(0.310662\pi\)
\(774\) 5.11360e6 0.306813
\(775\) −84084.0 −0.00502874
\(776\) −2.10622e7 −1.25559
\(777\) 7.13303e6 0.423859
\(778\) 4.84024e7 2.86694
\(779\) 1.76028e6 0.103929
\(780\) 7.46368e6 0.439255
\(781\) 0 0
\(782\) 3.14093e7 1.83671
\(783\) −1.38771e7 −0.808898
\(784\) 3.41902e6 0.198661
\(785\) −1.64373e7 −0.952045
\(786\) −8.62204e6 −0.497799
\(787\) −2.62501e7 −1.51075 −0.755377 0.655291i \(-0.772546\pi\)
−0.755377 + 0.655291i \(0.772546\pi\)
\(788\) 6.73234e7 3.86234
\(789\) 1.80737e7 1.03360
\(790\) 1.77990e7 1.01468
\(791\) −4.08435e6 −0.232103
\(792\) 0 0
\(793\) −3.59072e6 −0.202768
\(794\) 9.95820e6 0.560570
\(795\) 1.77341e6 0.0995155
\(796\) −5.71714e7 −3.19813
\(797\) −1.00373e7 −0.559720 −0.279860 0.960041i \(-0.590288\pi\)
−0.279860 + 0.960041i \(0.590288\pi\)
\(798\) −672280. −0.0373717
\(799\) 1.60559e7 0.889751
\(800\) 29920.0 0.00165286
\(801\) 2.37735e6 0.130922
\(802\) −3.31605e7 −1.82048
\(803\) 0 0
\(804\) 4.60806e7 2.51407
\(805\) −5.00506e6 −0.272220
\(806\) −1.07016e7 −0.580245
\(807\) 1.78858e7 0.966772
\(808\) −1.39306e7 −0.750655
\(809\) −1.40884e7 −0.756816 −0.378408 0.925639i \(-0.623528\pi\)
−0.378408 + 0.925639i \(0.623528\pi\)
\(810\) 2.54346e7 1.36211
\(811\) −1.81433e7 −0.968646 −0.484323 0.874889i \(-0.660934\pi\)
−0.484323 + 0.874889i \(0.660934\pi\)
\(812\) −1.13888e7 −0.606160
\(813\) 2.31076e7 1.22611
\(814\) 0 0
\(815\) −738752. −0.0389587
\(816\) −3.43298e7 −1.80487
\(817\) −1.06624e6 −0.0558856
\(818\) −3.07273e7 −1.60562
\(819\) −322420. −0.0167962
\(820\) −6.83993e7 −3.55236
\(821\) 2.13669e7 1.10633 0.553164 0.833072i \(-0.313420\pi\)
0.553164 + 0.833072i \(0.313420\pi\)
\(822\) 2.86247e7 1.47761
\(823\) 1.78017e7 0.916142 0.458071 0.888916i \(-0.348541\pi\)
0.458071 + 0.888916i \(0.348541\pi\)
\(824\) 1.91016e7 0.980058
\(825\) 0 0
\(826\) −1.33770e6 −0.0682195
\(827\) −1.62921e7 −0.828350 −0.414175 0.910197i \(-0.635930\pi\)
−0.414175 + 0.910197i \(0.635930\pi\)
\(828\) −5.82950e6 −0.295499
\(829\) −2.08499e6 −0.105370 −0.0526851 0.998611i \(-0.516778\pi\)
−0.0526851 + 0.998611i \(0.516778\pi\)
\(830\) −1.15013e7 −0.579497
\(831\) −1.48973e7 −0.748348
\(832\) −2.57152e6 −0.128790
\(833\) 4.13452e6 0.206449
\(834\) −4.95684e6 −0.246769
\(835\) 2.76423e7 1.37201
\(836\) 0 0
\(837\) −3.10346e7 −1.53120
\(838\) 2.81438e7 1.38443
\(839\) −2.27850e7 −1.11749 −0.558745 0.829340i \(-0.688717\pi\)
−0.558745 + 0.829340i \(0.688717\pi\)
\(840\) 1.38298e7 0.676264
\(841\) −8.82842e6 −0.430421
\(842\) 3.05802e7 1.48648
\(843\) −312788. −0.0151594
\(844\) −7.82498e7 −3.78118
\(845\) 1.96948e7 0.948877
\(846\) −4.38228e6 −0.210511
\(847\) 0 0
\(848\) 3.22109e6 0.153820
\(849\) −3.49403e7 −1.66363
\(850\) 189420. 0.00899246
\(851\) −1.89660e7 −0.897740
\(852\) 5.57491e7 2.63111
\(853\) 2.26975e7 1.06808 0.534042 0.845458i \(-0.320672\pi\)
0.534042 + 0.845458i \(0.320672\pi\)
\(854\) −1.25675e7 −0.589664
\(855\) 257936. 0.0120669
\(856\) 5.26766e7 2.45716
\(857\) −2.52900e7 −1.17624 −0.588120 0.808774i \(-0.700132\pi\)
−0.588120 + 0.808774i \(0.700132\pi\)
\(858\) 0 0
\(859\) −1.03947e7 −0.480652 −0.240326 0.970692i \(-0.577254\pi\)
−0.240326 + 0.970692i \(0.577254\pi\)
\(860\) 4.14310e7 1.91020
\(861\) −1.23219e7 −0.566462
\(862\) −1.93750e7 −0.888122
\(863\) 4.33399e7 1.98089 0.990447 0.137892i \(-0.0440327\pi\)
0.990447 + 0.137892i \(0.0440327\pi\)
\(864\) 1.10432e7 0.503281
\(865\) 1.34801e7 0.612566
\(866\) 3.94790e7 1.78884
\(867\) −2.16360e7 −0.977527
\(868\) −2.54698e7 −1.14743
\(869\) 0 0
\(870\) −2.67971e7 −1.20030
\(871\) −6.77656e6 −0.302666
\(872\) −3.34433e7 −1.48942
\(873\) 2.74978e6 0.122113
\(874\) 1.78752e6 0.0791539
\(875\) 8.54482e6 0.377296
\(876\) 6.48141e7 2.85370
\(877\) −3.71659e7 −1.63172 −0.815861 0.578248i \(-0.803736\pi\)
−0.815861 + 0.578248i \(0.803736\pi\)
\(878\) 7.41770e7 3.24738
\(879\) −2.70449e7 −1.18063
\(880\) 0 0
\(881\) 9.04785e6 0.392740 0.196370 0.980530i \(-0.437085\pi\)
0.196370 + 0.980530i \(0.437085\pi\)
\(882\) −1.12847e6 −0.0488448
\(883\) 3.29679e7 1.42295 0.711474 0.702712i \(-0.248028\pi\)
0.711474 + 0.702712i \(0.248028\pi\)
\(884\) 1.63934e7 0.705569
\(885\) −2.14032e6 −0.0918588
\(886\) 1.40269e7 0.600313
\(887\) 1.61099e7 0.687517 0.343758 0.939058i \(-0.388300\pi\)
0.343758 + 0.939058i \(0.388300\pi\)
\(888\) 5.24059e7 2.23022
\(889\) −2.95882e6 −0.125564
\(890\) 2.83259e7 1.19870
\(891\) 0 0
\(892\) −5.60500e7 −2.35865
\(893\) 913752. 0.0383442
\(894\) −2.83164e6 −0.118493
\(895\) −1.65162e7 −0.689211
\(896\) −1.32653e7 −0.552009
\(897\) −3.57504e6 −0.148354
\(898\) −5.90574e6 −0.244390
\(899\) 2.61272e7 1.07819
\(900\) −35156.0 −0.00144675
\(901\) 3.89516e6 0.159850
\(902\) 0 0
\(903\) 7.46368e6 0.304603
\(904\) −3.00074e7 −1.22126
\(905\) 1.88709e7 0.765898
\(906\) 9.92656e6 0.401771
\(907\) −4.47286e7 −1.80537 −0.902686 0.430300i \(-0.858408\pi\)
−0.902686 + 0.430300i \(0.858408\pi\)
\(908\) −5.05798e6 −0.203593
\(909\) 1.81871e6 0.0730053
\(910\) −3.84160e6 −0.153783
\(911\) −6.60518e6 −0.263687 −0.131844 0.991271i \(-0.542090\pi\)
−0.131844 + 0.991271i \(0.542090\pi\)
\(912\) −1.95373e6 −0.0777816
\(913\) 0 0
\(914\) 2.90484e7 1.15016
\(915\) −2.01080e7 −0.793993
\(916\) 7.69730e7 3.03110
\(917\) 3.01771e6 0.118510
\(918\) 6.99132e7 2.73812
\(919\) 3.08930e7 1.20662 0.603311 0.797506i \(-0.293848\pi\)
0.603311 + 0.797506i \(0.293848\pi\)
\(920\) −3.67718e7 −1.43234
\(921\) −6.42704e6 −0.249667
\(922\) 9.22684e6 0.357459
\(923\) −8.19840e6 −0.316756
\(924\) 0 0
\(925\) −114378. −0.00439530
\(926\) 7.18235e7 2.75258
\(927\) −2.49382e6 −0.0953160
\(928\) −9.29696e6 −0.354381
\(929\) −4.87215e6 −0.185217 −0.0926087 0.995703i \(-0.529521\pi\)
−0.0926087 + 0.995703i \(0.529521\pi\)
\(930\) −5.99290e7 −2.27211
\(931\) 235298. 0.00889701
\(932\) 1.35134e7 0.509593
\(933\) −9.33979e6 −0.351264
\(934\) −6.12570e6 −0.229767
\(935\) 0 0
\(936\) −2.36880e6 −0.0883769
\(937\) 3.25004e7 1.20932 0.604658 0.796485i \(-0.293310\pi\)
0.604658 + 0.796485i \(0.293310\pi\)
\(938\) −2.37180e7 −0.880177
\(939\) 1.55448e6 0.0575334
\(940\) −3.55058e7 −1.31063
\(941\) 2.64040e6 0.0972066 0.0486033 0.998818i \(-0.484523\pi\)
0.0486033 + 0.998818i \(0.484523\pi\)
\(942\) −4.10934e7 −1.50884
\(943\) 3.27627e7 1.19978
\(944\) −3.88752e6 −0.141985
\(945\) −1.11406e7 −0.405817
\(946\) 0 0
\(947\) −4.08179e7 −1.47903 −0.739513 0.673142i \(-0.764944\pi\)
−0.739513 + 0.673142i \(0.764944\pi\)
\(948\) 3.02584e7 1.09351
\(949\) −9.53148e6 −0.343554
\(950\) 10780.0 0.000387534 0
\(951\) 962892. 0.0345244
\(952\) 3.03761e7 1.08627
\(953\) 6.71983e6 0.239677 0.119838 0.992793i \(-0.461762\pi\)
0.119838 + 0.992793i \(0.461762\pi\)
\(954\) −1.06314e6 −0.0378198
\(955\) −2.00628e7 −0.711841
\(956\) −3.28375e7 −1.16205
\(957\) 0 0
\(958\) −2.60330e7 −0.916454
\(959\) −1.00186e7 −0.351773
\(960\) −1.44005e7 −0.504313
\(961\) 2.98016e7 1.04095
\(962\) −1.45572e7 −0.507154
\(963\) −6.87723e6 −0.238972
\(964\) −5.48019e7 −1.89934
\(965\) −5.54150e7 −1.91562
\(966\) −1.25126e7 −0.431426
\(967\) 2.78979e6 0.0959413 0.0479707 0.998849i \(-0.484725\pi\)
0.0479707 + 0.998849i \(0.484725\pi\)
\(968\) 0 0
\(969\) −2.36258e6 −0.0808310
\(970\) 3.27634e7 1.11804
\(971\) 3.33594e7 1.13545 0.567727 0.823217i \(-0.307823\pi\)
0.567727 + 0.823217i \(0.307823\pi\)
\(972\) −2.38486e7 −0.809648
\(973\) 1.73489e6 0.0587477
\(974\) 5.46309e7 1.84519
\(975\) −21560.0 −0.000726335 0
\(976\) −3.65228e7 −1.22727
\(977\) −7.60033e6 −0.254739 −0.127370 0.991855i \(-0.540653\pi\)
−0.127370 + 0.991855i \(0.540653\pi\)
\(978\) −1.84688e6 −0.0617435
\(979\) 0 0
\(980\) −9.14301e6 −0.304105
\(981\) 4.36621e6 0.144854
\(982\) −1.64090e7 −0.543004
\(983\) −5.79760e6 −0.191366 −0.0956829 0.995412i \(-0.530503\pi\)
−0.0956829 + 0.995412i \(0.530503\pi\)
\(984\) −9.05285e7 −2.98056
\(985\) −5.54428e7 −1.82077
\(986\) −5.88580e7 −1.92803
\(987\) −6.39626e6 −0.208994
\(988\) 932960. 0.0304068
\(989\) −1.98451e7 −0.645153
\(990\) 0 0
\(991\) 1.26825e7 0.410224 0.205112 0.978739i \(-0.434244\pi\)
0.205112 + 0.978739i \(0.434244\pi\)
\(992\) −2.07917e7 −0.670827
\(993\) 7.90227e6 0.254319
\(994\) −2.86944e7 −0.921152
\(995\) 4.70823e7 1.50765
\(996\) −1.95522e7 −0.624521
\(997\) −1.44400e7 −0.460077 −0.230039 0.973182i \(-0.573885\pi\)
−0.230039 + 0.973182i \(0.573885\pi\)
\(998\) 2.99796e7 0.952796
\(999\) −4.22159e7 −1.33833
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 847.6.a.b.1.1 1
11.10 odd 2 7.6.a.a.1.1 1
33.32 even 2 63.6.a.e.1.1 1
44.43 even 2 112.6.a.g.1.1 1
55.32 even 4 175.6.b.a.99.1 2
55.43 even 4 175.6.b.a.99.2 2
55.54 odd 2 175.6.a.b.1.1 1
77.10 even 6 49.6.c.b.30.1 2
77.32 odd 6 49.6.c.c.30.1 2
77.54 even 6 49.6.c.b.18.1 2
77.65 odd 6 49.6.c.c.18.1 2
77.76 even 2 49.6.a.a.1.1 1
88.21 odd 2 448.6.a.m.1.1 1
88.43 even 2 448.6.a.c.1.1 1
132.131 odd 2 1008.6.a.y.1.1 1
231.230 odd 2 441.6.a.k.1.1 1
308.307 odd 2 784.6.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 1 11.10 odd 2
49.6.a.a.1.1 1 77.76 even 2
49.6.c.b.18.1 2 77.54 even 6
49.6.c.b.30.1 2 77.10 even 6
49.6.c.c.18.1 2 77.65 odd 6
49.6.c.c.30.1 2 77.32 odd 6
63.6.a.e.1.1 1 33.32 even 2
112.6.a.g.1.1 1 44.43 even 2
175.6.a.b.1.1 1 55.54 odd 2
175.6.b.a.99.1 2 55.32 even 4
175.6.b.a.99.2 2 55.43 even 4
441.6.a.k.1.1 1 231.230 odd 2
448.6.a.c.1.1 1 88.43 even 2
448.6.a.m.1.1 1 88.21 odd 2
784.6.a.c.1.1 1 308.307 odd 2
847.6.a.b.1.1 1 1.1 even 1 trivial
1008.6.a.y.1.1 1 132.131 odd 2