Properties

Label 33.6.a.e.1.2
Level $33$
Weight $6$
Character 33.1
Self dual yes
Analytic conductor $5.293$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.29266605383\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 33.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.37228 q^{2} +9.00000 q^{3} +55.8397 q^{4} +0.277187 q^{5} +84.3505 q^{6} -105.081 q^{7} +223.432 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+9.37228 q^{2} +9.00000 q^{3} +55.8397 q^{4} +0.277187 q^{5} +84.3505 q^{6} -105.081 q^{7} +223.432 q^{8} +81.0000 q^{9} +2.59787 q^{10} -121.000 q^{11} +502.557 q^{12} +147.549 q^{13} -984.853 q^{14} +2.49468 q^{15} +307.198 q^{16} -1432.49 q^{17} +759.155 q^{18} +2033.18 q^{19} +15.4780 q^{20} -945.733 q^{21} -1134.05 q^{22} +828.853 q^{23} +2010.89 q^{24} -3124.92 q^{25} +1382.87 q^{26} +729.000 q^{27} -5867.71 q^{28} +4633.44 q^{29} +23.3809 q^{30} -9835.65 q^{31} -4270.67 q^{32} -1089.00 q^{33} -13425.7 q^{34} -29.1272 q^{35} +4523.01 q^{36} +7134.34 q^{37} +19055.6 q^{38} +1327.94 q^{39} +61.9324 q^{40} +18265.0 q^{41} -8863.68 q^{42} +13822.5 q^{43} -6756.60 q^{44} +22.4521 q^{45} +7768.24 q^{46} +22991.2 q^{47} +2764.78 q^{48} -5764.89 q^{49} -29287.7 q^{50} -12892.4 q^{51} +8239.08 q^{52} +14311.9 q^{53} +6832.39 q^{54} -33.5396 q^{55} -23478.6 q^{56} +18298.7 q^{57} +43425.9 q^{58} -7081.12 q^{59} +139.302 q^{60} -18470.2 q^{61} -92182.5 q^{62} -8511.60 q^{63} -49856.3 q^{64} +40.8986 q^{65} -10206.4 q^{66} +16229.5 q^{67} -79989.7 q^{68} +7459.68 q^{69} -272.988 q^{70} +28198.9 q^{71} +18098.0 q^{72} -39382.9 q^{73} +66865.1 q^{74} -28124.3 q^{75} +113532. q^{76} +12714.9 q^{77} +12445.8 q^{78} -41243.7 q^{79} +85.1513 q^{80} +6561.00 q^{81} +171185. q^{82} -23355.3 q^{83} -52809.4 q^{84} -397.067 q^{85} +129549. q^{86} +41701.0 q^{87} -27035.3 q^{88} -103803. q^{89} +210.428 q^{90} -15504.6 q^{91} +46282.9 q^{92} -88520.9 q^{93} +215480. q^{94} +563.572 q^{95} -38436.1 q^{96} -149289. q^{97} -54030.2 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 13q^{2} + 18q^{3} + 37q^{4} + 58q^{5} + 117q^{6} + 146q^{7} + 39q^{8} + 162q^{9} + O(q^{10}) \) \( 2q + 13q^{2} + 18q^{3} + 37q^{4} + 58q^{5} + 117q^{6} + 146q^{7} + 39q^{8} + 162q^{9} + 212q^{10} - 242q^{11} + 333q^{12} - 130q^{13} - 74q^{14} + 522q^{15} + 241q^{16} - 728q^{17} + 1053q^{18} - 828q^{19} - 1072q^{20} + 1314q^{21} - 1573q^{22} - 238q^{23} + 351q^{24} - 2918q^{25} + 376q^{26} + 1458q^{27} - 10598q^{28} + 696q^{29} + 1908q^{30} - 10480q^{31} + 1391q^{32} - 2178q^{33} - 10870q^{34} + 14464q^{35} + 2997q^{36} - 1908q^{37} + 8676q^{38} - 1170q^{39} - 10584q^{40} + 36484q^{41} - 666q^{42} + 9768q^{43} - 4477q^{44} + 4698q^{45} + 3898q^{46} + 43742q^{47} + 2169q^{48} + 40470q^{49} - 28537q^{50} - 6552q^{51} + 13468q^{52} - 12174q^{53} + 9477q^{54} - 7018q^{55} - 69786q^{56} - 7452q^{57} + 29142q^{58} - 2788q^{59} - 9648q^{60} - 25302q^{61} - 94520q^{62} + 11826q^{63} - 27199q^{64} - 15980q^{65} - 14157q^{66} - 40520q^{67} - 93262q^{68} - 2142q^{69} + 52304q^{70} + 31386q^{71} + 3159q^{72} - 46780q^{73} + 34062q^{74} - 26262q^{75} + 167436q^{76} - 17666q^{77} + 3384q^{78} - 16850q^{79} - 3736q^{80} + 13122q^{81} + 237278q^{82} + 79440q^{83} - 95382q^{84} + 40268q^{85} + 114840q^{86} + 6264q^{87} - 4719q^{88} - 54204q^{89} + 17172q^{90} - 85192q^{91} + 66382q^{92} - 94320q^{93} + 290758q^{94} - 164592q^{95} + 12519q^{96} - 241568q^{97} + 113697q^{98} - 19602q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.37228 1.65680 0.828400 0.560136i \(-0.189251\pi\)
0.828400 + 0.560136i \(0.189251\pi\)
\(3\) 9.00000 0.577350
\(4\) 55.8397 1.74499
\(5\) 0.277187 0.00495847 0.00247923 0.999997i \(-0.499211\pi\)
0.00247923 + 0.999997i \(0.499211\pi\)
\(6\) 84.3505 0.956554
\(7\) −105.081 −0.810552 −0.405276 0.914194i \(-0.632825\pi\)
−0.405276 + 0.914194i \(0.632825\pi\)
\(8\) 223.432 1.23430
\(9\) 81.0000 0.333333
\(10\) 2.59787 0.00821519
\(11\) −121.000 −0.301511
\(12\) 502.557 1.00747
\(13\) 147.549 0.242146 0.121073 0.992644i \(-0.461366\pi\)
0.121073 + 0.992644i \(0.461366\pi\)
\(14\) −984.853 −1.34292
\(15\) 2.49468 0.00286277
\(16\) 307.198 0.299998
\(17\) −1432.49 −1.20218 −0.601089 0.799182i \(-0.705266\pi\)
−0.601089 + 0.799182i \(0.705266\pi\)
\(18\) 759.155 0.552267
\(19\) 2033.18 1.29209 0.646045 0.763300i \(-0.276422\pi\)
0.646045 + 0.763300i \(0.276422\pi\)
\(20\) 15.4780 0.00865247
\(21\) −945.733 −0.467972
\(22\) −1134.05 −0.499544
\(23\) 828.853 0.326707 0.163353 0.986568i \(-0.447769\pi\)
0.163353 + 0.986568i \(0.447769\pi\)
\(24\) 2010.89 0.712623
\(25\) −3124.92 −0.999975
\(26\) 1382.87 0.401188
\(27\) 729.000 0.192450
\(28\) −5867.71 −1.41440
\(29\) 4633.44 1.02308 0.511539 0.859260i \(-0.329076\pi\)
0.511539 + 0.859260i \(0.329076\pi\)
\(30\) 23.3809 0.00474304
\(31\) −9835.65 −1.83823 −0.919113 0.393994i \(-0.871093\pi\)
−0.919113 + 0.393994i \(0.871093\pi\)
\(32\) −4270.67 −0.737261
\(33\) −1089.00 −0.174078
\(34\) −13425.7 −1.99177
\(35\) −29.1272 −0.00401910
\(36\) 4523.01 0.581663
\(37\) 7134.34 0.856741 0.428371 0.903603i \(-0.359088\pi\)
0.428371 + 0.903603i \(0.359088\pi\)
\(38\) 19055.6 2.14074
\(39\) 1327.94 0.139803
\(40\) 61.9324 0.00612023
\(41\) 18265.0 1.69691 0.848456 0.529265i \(-0.177532\pi\)
0.848456 + 0.529265i \(0.177532\pi\)
\(42\) −8863.68 −0.775337
\(43\) 13822.5 1.14003 0.570016 0.821634i \(-0.306937\pi\)
0.570016 + 0.821634i \(0.306937\pi\)
\(44\) −6756.60 −0.526134
\(45\) 22.4521 0.00165282
\(46\) 7768.24 0.541288
\(47\) 22991.2 1.51816 0.759079 0.650999i \(-0.225650\pi\)
0.759079 + 0.650999i \(0.225650\pi\)
\(48\) 2764.78 0.173204
\(49\) −5764.89 −0.343005
\(50\) −29287.7 −1.65676
\(51\) −12892.4 −0.694078
\(52\) 8239.08 0.422542
\(53\) 14311.9 0.699856 0.349928 0.936777i \(-0.386206\pi\)
0.349928 + 0.936777i \(0.386206\pi\)
\(54\) 6832.39 0.318851
\(55\) −33.5396 −0.00149503
\(56\) −23478.6 −1.00046
\(57\) 18298.7 0.745988
\(58\) 43425.9 1.69504
\(59\) −7081.12 −0.264833 −0.132416 0.991194i \(-0.542274\pi\)
−0.132416 + 0.991194i \(0.542274\pi\)
\(60\) 139.302 0.00499551
\(61\) −18470.2 −0.635547 −0.317774 0.948167i \(-0.602935\pi\)
−0.317774 + 0.948167i \(0.602935\pi\)
\(62\) −92182.5 −3.04557
\(63\) −8511.60 −0.270184
\(64\) −49856.3 −1.52149
\(65\) 40.8986 0.00120067
\(66\) −10206.4 −0.288412
\(67\) 16229.5 0.441690 0.220845 0.975309i \(-0.429119\pi\)
0.220845 + 0.975309i \(0.429119\pi\)
\(68\) −79989.7 −2.09779
\(69\) 7459.68 0.188624
\(70\) −272.988 −0.00665884
\(71\) 28198.9 0.663875 0.331938 0.943301i \(-0.392298\pi\)
0.331938 + 0.943301i \(0.392298\pi\)
\(72\) 18098.0 0.411433
\(73\) −39382.9 −0.864968 −0.432484 0.901642i \(-0.642363\pi\)
−0.432484 + 0.901642i \(0.642363\pi\)
\(74\) 66865.1 1.41945
\(75\) −28124.3 −0.577336
\(76\) 113532. 2.25468
\(77\) 12714.9 0.244391
\(78\) 12445.8 0.231626
\(79\) −41243.7 −0.743515 −0.371758 0.928330i \(-0.621245\pi\)
−0.371758 + 0.928330i \(0.621245\pi\)
\(80\) 85.1513 0.00148753
\(81\) 6561.00 0.111111
\(82\) 171185. 2.81145
\(83\) −23355.3 −0.372126 −0.186063 0.982538i \(-0.559573\pi\)
−0.186063 + 0.982538i \(0.559573\pi\)
\(84\) −52809.4 −0.816607
\(85\) −397.067 −0.00596096
\(86\) 129549. 1.88880
\(87\) 41701.0 0.590675
\(88\) −27035.3 −0.372155
\(89\) −103803. −1.38911 −0.694555 0.719440i \(-0.744399\pi\)
−0.694555 + 0.719440i \(0.744399\pi\)
\(90\) 210.428 0.00273840
\(91\) −15504.6 −0.196272
\(92\) 46282.9 0.570099
\(93\) −88520.9 −1.06130
\(94\) 215480. 2.51528
\(95\) 563.572 0.00640678
\(96\) −38436.1 −0.425658
\(97\) −149289. −1.61101 −0.805503 0.592592i \(-0.798105\pi\)
−0.805503 + 0.592592i \(0.798105\pi\)
\(98\) −54030.2 −0.568292
\(99\) −9801.00 −0.100504
\(100\) −174495. −1.74495
\(101\) 62410.1 0.608768 0.304384 0.952549i \(-0.401549\pi\)
0.304384 + 0.952549i \(0.401549\pi\)
\(102\) −120831. −1.14995
\(103\) −90047.2 −0.836329 −0.418165 0.908371i \(-0.637326\pi\)
−0.418165 + 0.908371i \(0.637326\pi\)
\(104\) 32967.1 0.298881
\(105\) −262.145 −0.00232043
\(106\) 134136. 1.15952
\(107\) 120458. 1.01713 0.508566 0.861023i \(-0.330176\pi\)
0.508566 + 0.861023i \(0.330176\pi\)
\(108\) 40707.1 0.335823
\(109\) −91854.0 −0.740511 −0.370256 0.928930i \(-0.620730\pi\)
−0.370256 + 0.928930i \(0.620730\pi\)
\(110\) −314.343 −0.00247697
\(111\) 64209.1 0.494640
\(112\) −32280.8 −0.243164
\(113\) −29619.6 −0.218214 −0.109107 0.994030i \(-0.534799\pi\)
−0.109107 + 0.994030i \(0.534799\pi\)
\(114\) 171500. 1.23595
\(115\) 229.747 0.00161996
\(116\) 258730. 1.78526
\(117\) 11951.5 0.0807153
\(118\) −66366.2 −0.438775
\(119\) 150528. 0.974428
\(120\) 557.391 0.00353352
\(121\) 14641.0 0.0909091
\(122\) −173108. −1.05298
\(123\) 164385. 0.979713
\(124\) −549219. −3.20768
\(125\) −1732.40 −0.00991681
\(126\) −79773.1 −0.447641
\(127\) 271128. 1.49165 0.745823 0.666145i \(-0.232057\pi\)
0.745823 + 0.666145i \(0.232057\pi\)
\(128\) −330606. −1.78355
\(129\) 124403. 0.658197
\(130\) 383.313 0.00198928
\(131\) −223990. −1.14038 −0.570191 0.821512i \(-0.693131\pi\)
−0.570191 + 0.821512i \(0.693131\pi\)
\(132\) −60809.4 −0.303764
\(133\) −213650. −1.04731
\(134\) 152107. 0.731792
\(135\) 202.069 0.000954258 0
\(136\) −320064. −1.48385
\(137\) −348229. −1.58513 −0.792563 0.609789i \(-0.791254\pi\)
−0.792563 + 0.609789i \(0.791254\pi\)
\(138\) 69914.2 0.312513
\(139\) 333802. 1.46538 0.732692 0.680560i \(-0.238264\pi\)
0.732692 + 0.680560i \(0.238264\pi\)
\(140\) −1626.45 −0.00701328
\(141\) 206921. 0.876509
\(142\) 264288. 1.09991
\(143\) −17853.4 −0.0730098
\(144\) 24883.1 0.0999994
\(145\) 1284.33 0.00507290
\(146\) −369107. −1.43308
\(147\) −51884.0 −0.198034
\(148\) 398379. 1.49500
\(149\) 310249. 1.14484 0.572419 0.819961i \(-0.306005\pi\)
0.572419 + 0.819961i \(0.306005\pi\)
\(150\) −263589. −0.956531
\(151\) −54751.5 −0.195413 −0.0977066 0.995215i \(-0.531151\pi\)
−0.0977066 + 0.995215i \(0.531151\pi\)
\(152\) 454278. 1.59482
\(153\) −116032. −0.400726
\(154\) 119167. 0.404907
\(155\) −2726.31 −0.00911478
\(156\) 74151.7 0.243955
\(157\) 221556. 0.717354 0.358677 0.933462i \(-0.383228\pi\)
0.358677 + 0.933462i \(0.383228\pi\)
\(158\) −386547. −1.23186
\(159\) 128807. 0.404062
\(160\) −1183.77 −0.00365569
\(161\) −87097.1 −0.264813
\(162\) 61491.5 0.184089
\(163\) −225239. −0.664010 −0.332005 0.943278i \(-0.607725\pi\)
−0.332005 + 0.943278i \(0.607725\pi\)
\(164\) 1.01991e6 2.96109
\(165\) −301.856 −0.000863158 0
\(166\) −218892. −0.616539
\(167\) 186796. 0.518294 0.259147 0.965838i \(-0.416559\pi\)
0.259147 + 0.965838i \(0.416559\pi\)
\(168\) −211307. −0.577618
\(169\) −349522. −0.941365
\(170\) −3721.42 −0.00987613
\(171\) 164688. 0.430697
\(172\) 771846. 1.98934
\(173\) −713662. −1.81292 −0.906458 0.422296i \(-0.861224\pi\)
−0.906458 + 0.422296i \(0.861224\pi\)
\(174\) 390833. 0.978630
\(175\) 328371. 0.810532
\(176\) −37171.0 −0.0904529
\(177\) −63730.1 −0.152901
\(178\) −972875. −2.30148
\(179\) 789450. 1.84159 0.920793 0.390051i \(-0.127543\pi\)
0.920793 + 0.390051i \(0.127543\pi\)
\(180\) 1253.72 0.00288416
\(181\) 40330.7 0.0915039 0.0457519 0.998953i \(-0.485432\pi\)
0.0457519 + 0.998953i \(0.485432\pi\)
\(182\) −145314. −0.325184
\(183\) −166232. −0.366934
\(184\) 185192. 0.403254
\(185\) 1977.55 0.00424812
\(186\) −829642. −1.75836
\(187\) 173331. 0.362470
\(188\) 1.28382e6 2.64917
\(189\) −76604.4 −0.155991
\(190\) 5281.95 0.0106148
\(191\) 133736. 0.265256 0.132628 0.991166i \(-0.457658\pi\)
0.132628 + 0.991166i \(0.457658\pi\)
\(192\) −448707. −0.878435
\(193\) −462266. −0.893303 −0.446652 0.894708i \(-0.647384\pi\)
−0.446652 + 0.894708i \(0.647384\pi\)
\(194\) −1.39917e6 −2.66912
\(195\) 368.087 0.000693209 0
\(196\) −321910. −0.598541
\(197\) 96432.5 0.177035 0.0885173 0.996075i \(-0.471787\pi\)
0.0885173 + 0.996075i \(0.471787\pi\)
\(198\) −91857.7 −0.166515
\(199\) −1.08264e6 −1.93800 −0.968999 0.247066i \(-0.920533\pi\)
−0.968999 + 0.247066i \(0.920533\pi\)
\(200\) −698208. −1.23427
\(201\) 146065. 0.255010
\(202\) 584925. 1.00861
\(203\) −486889. −0.829258
\(204\) −719907. −1.21116
\(205\) 5062.81 0.00841409
\(206\) −843948. −1.38563
\(207\) 67137.1 0.108902
\(208\) 45326.7 0.0726434
\(209\) −246015. −0.389580
\(210\) −2456.89 −0.00384448
\(211\) 875588. 1.35392 0.676961 0.736019i \(-0.263297\pi\)
0.676961 + 0.736019i \(0.263297\pi\)
\(212\) 799174. 1.22124
\(213\) 253790. 0.383289
\(214\) 1.12897e6 1.68518
\(215\) 3831.43 0.00565281
\(216\) 162882. 0.237541
\(217\) 1.03354e6 1.48998
\(218\) −860881. −1.22688
\(219\) −354446. −0.499390
\(220\) −1872.84 −0.00260882
\(221\) −211362. −0.291103
\(222\) 601786. 0.819520
\(223\) 988042. 1.33049 0.665247 0.746623i \(-0.268326\pi\)
0.665247 + 0.746623i \(0.268326\pi\)
\(224\) 448769. 0.597589
\(225\) −253119. −0.333325
\(226\) −277603. −0.361537
\(227\) 67066.0 0.0863849 0.0431925 0.999067i \(-0.486247\pi\)
0.0431925 + 0.999067i \(0.486247\pi\)
\(228\) 1.02179e6 1.30174
\(229\) −662131. −0.834363 −0.417182 0.908823i \(-0.636982\pi\)
−0.417182 + 0.908823i \(0.636982\pi\)
\(230\) 2153.25 0.00268396
\(231\) 114434. 0.141099
\(232\) 1.03526e6 1.26278
\(233\) 1.37961e6 1.66482 0.832411 0.554159i \(-0.186960\pi\)
0.832411 + 0.554159i \(0.186960\pi\)
\(234\) 112012. 0.133729
\(235\) 6372.85 0.00752773
\(236\) −395407. −0.462130
\(237\) −371193. −0.429269
\(238\) 1.41079e6 1.61443
\(239\) 423791. 0.479907 0.239954 0.970784i \(-0.422868\pi\)
0.239954 + 0.970784i \(0.422868\pi\)
\(240\) 766.362 0.000858827 0
\(241\) −1.34802e6 −1.49504 −0.747521 0.664238i \(-0.768756\pi\)
−0.747521 + 0.664238i \(0.768756\pi\)
\(242\) 137220. 0.150618
\(243\) 59049.0 0.0641500
\(244\) −1.03137e6 −1.10902
\(245\) −1597.95 −0.00170078
\(246\) 1.54066e6 1.62319
\(247\) 299994. 0.312874
\(248\) −2.19760e6 −2.26892
\(249\) −210198. −0.214847
\(250\) −16236.5 −0.0164302
\(251\) −1.45914e6 −1.46188 −0.730940 0.682442i \(-0.760918\pi\)
−0.730940 + 0.682442i \(0.760918\pi\)
\(252\) −475285. −0.471468
\(253\) −100291. −0.0985057
\(254\) 2.54109e6 2.47136
\(255\) −3573.60 −0.00344156
\(256\) −1.50313e6 −1.43349
\(257\) 548503. 0.518020 0.259010 0.965875i \(-0.416604\pi\)
0.259010 + 0.965875i \(0.416604\pi\)
\(258\) 1.16594e6 1.09050
\(259\) −749687. −0.694434
\(260\) 2283.76 0.00209516
\(261\) 375309. 0.341026
\(262\) −2.09930e6 −1.88939
\(263\) 905299. 0.807054 0.403527 0.914968i \(-0.367784\pi\)
0.403527 + 0.914968i \(0.367784\pi\)
\(264\) −243317. −0.214864
\(265\) 3967.08 0.00347021
\(266\) −2.00239e6 −1.73518
\(267\) −934231. −0.802003
\(268\) 906248. 0.770744
\(269\) −164041. −0.138220 −0.0691100 0.997609i \(-0.522016\pi\)
−0.0691100 + 0.997609i \(0.522016\pi\)
\(270\) 1893.85 0.00158101
\(271\) 989636. 0.818563 0.409282 0.912408i \(-0.365779\pi\)
0.409282 + 0.912408i \(0.365779\pi\)
\(272\) −440058. −0.360651
\(273\) −139542. −0.113318
\(274\) −3.26370e6 −2.62624
\(275\) 378116. 0.301504
\(276\) 416546. 0.329147
\(277\) 727149. 0.569409 0.284704 0.958615i \(-0.408105\pi\)
0.284704 + 0.958615i \(0.408105\pi\)
\(278\) 3.12848e6 2.42785
\(279\) −796688. −0.612742
\(280\) −6507.94 −0.00496077
\(281\) −379100. −0.286410 −0.143205 0.989693i \(-0.545741\pi\)
−0.143205 + 0.989693i \(0.545741\pi\)
\(282\) 1.93932e6 1.45220
\(283\) −695371. −0.516120 −0.258060 0.966129i \(-0.583083\pi\)
−0.258060 + 0.966129i \(0.583083\pi\)
\(284\) 1.57462e6 1.15846
\(285\) 5072.14 0.00369896
\(286\) −167327. −0.120963
\(287\) −1.91931e6 −1.37544
\(288\) −345925. −0.245754
\(289\) 632167. 0.445233
\(290\) 12037.1 0.00840479
\(291\) −1.34360e6 −0.930115
\(292\) −2.19913e6 −1.50936
\(293\) 268373. 0.182629 0.0913146 0.995822i \(-0.470893\pi\)
0.0913146 + 0.995822i \(0.470893\pi\)
\(294\) −486272. −0.328103
\(295\) −1962.79 −0.00131316
\(296\) 1.59404e6 1.05747
\(297\) −88209.0 −0.0580259
\(298\) 2.90774e6 1.89677
\(299\) 122296. 0.0791107
\(300\) −1.57045e6 −1.00745
\(301\) −1.45249e6 −0.924055
\(302\) −513147. −0.323761
\(303\) 561691. 0.351472
\(304\) 624591. 0.387625
\(305\) −5119.71 −0.00315134
\(306\) −1.08748e6 −0.663923
\(307\) 1.78841e6 1.08298 0.541491 0.840706i \(-0.317860\pi\)
0.541491 + 0.840706i \(0.317860\pi\)
\(308\) 709993. 0.426459
\(309\) −810425. −0.482855
\(310\) −25551.8 −0.0151014
\(311\) 362727. 0.212657 0.106328 0.994331i \(-0.466091\pi\)
0.106328 + 0.994331i \(0.466091\pi\)
\(312\) 296704. 0.172559
\(313\) −1.14382e6 −0.659926 −0.329963 0.943994i \(-0.607036\pi\)
−0.329963 + 0.943994i \(0.607036\pi\)
\(314\) 2.07648e6 1.18851
\(315\) −2359.30 −0.00133970
\(316\) −2.30303e6 −1.29743
\(317\) 3.43750e6 1.92130 0.960649 0.277766i \(-0.0895941\pi\)
0.960649 + 0.277766i \(0.0895941\pi\)
\(318\) 1.20722e6 0.669451
\(319\) −560647. −0.308470
\(320\) −13819.5 −0.00754428
\(321\) 1.08412e6 0.587241
\(322\) −816298. −0.438742
\(323\) −2.91251e6 −1.55332
\(324\) 366364. 0.193888
\(325\) −461079. −0.242140
\(326\) −2.11100e6 −1.10013
\(327\) −826686. −0.427534
\(328\) 4.08098e6 2.09450
\(329\) −2.41595e6 −1.23055
\(330\) −2829.08 −0.00143008
\(331\) 1.11428e6 0.559015 0.279507 0.960144i \(-0.409829\pi\)
0.279507 + 0.960144i \(0.409829\pi\)
\(332\) −1.30415e6 −0.649356
\(333\) 577882. 0.285580
\(334\) 1.75070e6 0.858710
\(335\) 4498.59 0.00219010
\(336\) −290528. −0.140391
\(337\) −2.25804e6 −1.08307 −0.541535 0.840678i \(-0.682157\pi\)
−0.541535 + 0.840678i \(0.682157\pi\)
\(338\) −3.27582e6 −1.55965
\(339\) −266576. −0.125986
\(340\) −22172.1 −0.0104018
\(341\) 1.19011e6 0.554246
\(342\) 1.54350e6 0.713578
\(343\) 2.37189e6 1.08858
\(344\) 3.08840e6 1.40714
\(345\) 2067.72 0.000935287 0
\(346\) −6.68865e6 −3.00364
\(347\) −1.17055e6 −0.521873 −0.260937 0.965356i \(-0.584031\pi\)
−0.260937 + 0.965356i \(0.584031\pi\)
\(348\) 2.32857e6 1.03072
\(349\) 1.21704e6 0.534863 0.267432 0.963577i \(-0.413825\pi\)
0.267432 + 0.963577i \(0.413825\pi\)
\(350\) 3.07759e6 1.34289
\(351\) 107563. 0.0466010
\(352\) 516752. 0.222293
\(353\) 286188. 0.122240 0.0611201 0.998130i \(-0.480533\pi\)
0.0611201 + 0.998130i \(0.480533\pi\)
\(354\) −597296. −0.253327
\(355\) 7816.37 0.00329180
\(356\) −5.79635e6 −2.42398
\(357\) 1.35475e6 0.562586
\(358\) 7.39895e6 3.05114
\(359\) 1.19066e6 0.487588 0.243794 0.969827i \(-0.421608\pi\)
0.243794 + 0.969827i \(0.421608\pi\)
\(360\) 5016.52 0.00204008
\(361\) 1.65774e6 0.669495
\(362\) 377991. 0.151604
\(363\) 131769. 0.0524864
\(364\) −865774. −0.342492
\(365\) −10916.4 −0.00428892
\(366\) −1.55797e6 −0.607936
\(367\) 3.11120e6 1.20577 0.602884 0.797829i \(-0.294018\pi\)
0.602884 + 0.797829i \(0.294018\pi\)
\(368\) 254622. 0.0980114
\(369\) 1.47946e6 0.565638
\(370\) 18534.1 0.00703830
\(371\) −1.50392e6 −0.567270
\(372\) −4.94297e6 −1.85196
\(373\) 3.58536e6 1.33432 0.667160 0.744914i \(-0.267510\pi\)
0.667160 + 0.744914i \(0.267510\pi\)
\(374\) 1.62451e6 0.600541
\(375\) −15591.6 −0.00572547
\(376\) 5.13697e6 1.87386
\(377\) 683659. 0.247734
\(378\) −717958. −0.258446
\(379\) 1.87822e6 0.671660 0.335830 0.941923i \(-0.390983\pi\)
0.335830 + 0.941923i \(0.390983\pi\)
\(380\) 31469.6 0.0111798
\(381\) 2.44015e6 0.861202
\(382\) 1.25341e6 0.439476
\(383\) −662654. −0.230829 −0.115414 0.993317i \(-0.536820\pi\)
−0.115414 + 0.993317i \(0.536820\pi\)
\(384\) −2.97545e6 −1.02973
\(385\) 3524.39 0.00121180
\(386\) −4.33249e6 −1.48003
\(387\) 1.11963e6 0.380010
\(388\) −8.33622e6 −2.81119
\(389\) −5.08370e6 −1.70336 −0.851678 0.524065i \(-0.824415\pi\)
−0.851678 + 0.524065i \(0.824415\pi\)
\(390\) 3449.82 0.00114851
\(391\) −1.18732e6 −0.392760
\(392\) −1.28806e6 −0.423371
\(393\) −2.01591e6 −0.658400
\(394\) 903793. 0.293311
\(395\) −11432.2 −0.00368670
\(396\) −547284. −0.175378
\(397\) −518385. −0.165073 −0.0825366 0.996588i \(-0.526302\pi\)
−0.0825366 + 0.996588i \(0.526302\pi\)
\(398\) −1.01468e7 −3.21088
\(399\) −1.92285e6 −0.604662
\(400\) −959971. −0.299991
\(401\) 1.12538e6 0.349493 0.174747 0.984613i \(-0.444089\pi\)
0.174747 + 0.984613i \(0.444089\pi\)
\(402\) 1.36896e6 0.422500
\(403\) −1.45124e6 −0.445119
\(404\) 3.48496e6 1.06229
\(405\) 1818.62 0.000550941 0
\(406\) −4.56326e6 −1.37392
\(407\) −863256. −0.258317
\(408\) −2.88057e6 −0.856700
\(409\) 3.15394e6 0.932279 0.466139 0.884711i \(-0.345645\pi\)
0.466139 + 0.884711i \(0.345645\pi\)
\(410\) 47450.1 0.0139405
\(411\) −3.13406e6 −0.915173
\(412\) −5.02821e6 −1.45939
\(413\) 744094. 0.214661
\(414\) 629228. 0.180429
\(415\) −6473.78 −0.00184518
\(416\) −630133. −0.178525
\(417\) 3.00421e6 0.846040
\(418\) −2.30572e6 −0.645456
\(419\) 801315. 0.222981 0.111491 0.993765i \(-0.464438\pi\)
0.111491 + 0.993765i \(0.464438\pi\)
\(420\) −14638.1 −0.00404912
\(421\) −1.85449e6 −0.509940 −0.254970 0.966949i \(-0.582066\pi\)
−0.254970 + 0.966949i \(0.582066\pi\)
\(422\) 8.20626e6 2.24318
\(423\) 1.86229e6 0.506052
\(424\) 3.19775e6 0.863832
\(425\) 4.47642e6 1.20215
\(426\) 2.37859e6 0.635033
\(427\) 1.94088e6 0.515144
\(428\) 6.72635e6 1.77488
\(429\) −160681. −0.0421522
\(430\) 35909.2 0.00936558
\(431\) 2.46004e6 0.637895 0.318947 0.947772i \(-0.396671\pi\)
0.318947 + 0.947772i \(0.396671\pi\)
\(432\) 223948. 0.0577347
\(433\) 3.12223e6 0.800287 0.400144 0.916452i \(-0.368960\pi\)
0.400144 + 0.916452i \(0.368960\pi\)
\(434\) 9.68667e6 2.46860
\(435\) 11559.0 0.00292884
\(436\) −5.12910e6 −1.29218
\(437\) 1.68521e6 0.422134
\(438\) −3.32197e6 −0.827389
\(439\) −6.09275e6 −1.50887 −0.754436 0.656374i \(-0.772089\pi\)
−0.754436 + 0.656374i \(0.772089\pi\)
\(440\) −7493.82 −0.00184532
\(441\) −466956. −0.114335
\(442\) −1.98094e6 −0.482299
\(443\) −7.93457e6 −1.92094 −0.960470 0.278382i \(-0.910202\pi\)
−0.960470 + 0.278382i \(0.910202\pi\)
\(444\) 3.58541e6 0.863141
\(445\) −28772.9 −0.00688786
\(446\) 9.26021e6 2.20436
\(447\) 2.79224e6 0.660973
\(448\) 5.23897e6 1.23325
\(449\) 4.61011e6 1.07918 0.539592 0.841926i \(-0.318578\pi\)
0.539592 + 0.841926i \(0.318578\pi\)
\(450\) −2.37230e6 −0.552253
\(451\) −2.21006e6 −0.511638
\(452\) −1.65395e6 −0.380781
\(453\) −492764. −0.112822
\(454\) 628562. 0.143123
\(455\) −4297.68 −0.000973208 0
\(456\) 4.08850e6 0.920773
\(457\) −8.05458e6 −1.80407 −0.902033 0.431667i \(-0.857926\pi\)
−0.902033 + 0.431667i \(0.857926\pi\)
\(458\) −6.20568e6 −1.38237
\(459\) −1.04428e6 −0.231359
\(460\) 12829.0 0.00282682
\(461\) 3.22431e6 0.706617 0.353309 0.935507i \(-0.385057\pi\)
0.353309 + 0.935507i \(0.385057\pi\)
\(462\) 1.07250e6 0.233773
\(463\) 1.29607e6 0.280979 0.140490 0.990082i \(-0.455132\pi\)
0.140490 + 0.990082i \(0.455132\pi\)
\(464\) 1.42339e6 0.306922
\(465\) −24536.8 −0.00526242
\(466\) 1.29301e7 2.75828
\(467\) −2.13902e6 −0.453860 −0.226930 0.973911i \(-0.572869\pi\)
−0.226930 + 0.973911i \(0.572869\pi\)
\(468\) 667365. 0.140847
\(469\) −1.70542e6 −0.358012
\(470\) 59728.2 0.0124720
\(471\) 1.99400e6 0.414165
\(472\) −1.58215e6 −0.326883
\(473\) −1.67253e6 −0.343732
\(474\) −3.47893e6 −0.711213
\(475\) −6.35354e6 −1.29206
\(476\) 8.40543e6 1.70037
\(477\) 1.15927e6 0.233285
\(478\) 3.97189e6 0.795111
\(479\) −4.77343e6 −0.950586 −0.475293 0.879828i \(-0.657658\pi\)
−0.475293 + 0.879828i \(0.657658\pi\)
\(480\) −10654.0 −0.00211061
\(481\) 1.05266e6 0.207457
\(482\) −1.26340e7 −2.47699
\(483\) −783873. −0.152890
\(484\) 817548. 0.158635
\(485\) −41380.8 −0.00798812
\(486\) 553424. 0.106284
\(487\) 7.98376e6 1.52541 0.762703 0.646749i \(-0.223872\pi\)
0.762703 + 0.646749i \(0.223872\pi\)
\(488\) −4.12684e6 −0.784456
\(489\) −2.02715e6 −0.383366
\(490\) −14976.4 −0.00281786
\(491\) −3.70031e6 −0.692682 −0.346341 0.938109i \(-0.612576\pi\)
−0.346341 + 0.938109i \(0.612576\pi\)
\(492\) 9.17919e6 1.70959
\(493\) −6.63736e6 −1.22992
\(494\) 2.81163e6 0.518371
\(495\) −2716.71 −0.000498345 0
\(496\) −3.02149e6 −0.551465
\(497\) −2.96318e6 −0.538105
\(498\) −1.97003e6 −0.355959
\(499\) −7.05137e6 −1.26772 −0.633858 0.773449i \(-0.718530\pi\)
−0.633858 + 0.773449i \(0.718530\pi\)
\(500\) −96736.4 −0.0173047
\(501\) 1.68116e6 0.299237
\(502\) −1.36754e7 −2.42204
\(503\) −7.77325e6 −1.36988 −0.684940 0.728599i \(-0.740172\pi\)
−0.684940 + 0.728599i \(0.740172\pi\)
\(504\) −1.90176e6 −0.333488
\(505\) 17299.3 0.00301856
\(506\) −939957. −0.163204
\(507\) −3.14570e6 −0.543498
\(508\) 1.51397e7 2.60290
\(509\) 4.11598e6 0.704173 0.352086 0.935968i \(-0.385472\pi\)
0.352086 + 0.935968i \(0.385472\pi\)
\(510\) −33492.8 −0.00570198
\(511\) 4.13841e6 0.701102
\(512\) −3.50836e6 −0.591465
\(513\) 1.48219e6 0.248663
\(514\) 5.14073e6 0.858256
\(515\) −24959.9 −0.00414691
\(516\) 6.94661e6 1.14855
\(517\) −2.78193e6 −0.457742
\(518\) −7.02628e6 −1.15054
\(519\) −6.42296e6 −1.04669
\(520\) 9138.05 0.00148199
\(521\) −2.56352e6 −0.413754 −0.206877 0.978367i \(-0.566330\pi\)
−0.206877 + 0.978367i \(0.566330\pi\)
\(522\) 3.51750e6 0.565012
\(523\) −3.69240e6 −0.590274 −0.295137 0.955455i \(-0.595365\pi\)
−0.295137 + 0.955455i \(0.595365\pi\)
\(524\) −1.25075e7 −1.98996
\(525\) 2.95534e6 0.467961
\(526\) 8.48471e6 1.33713
\(527\) 1.40895e7 2.20988
\(528\) −334539. −0.0522230
\(529\) −5.74935e6 −0.893263
\(530\) 37180.6 0.00574946
\(531\) −573570. −0.0882776
\(532\) −1.19301e7 −1.82754
\(533\) 2.69498e6 0.410901
\(534\) −8.75587e6 −1.32876
\(535\) 33389.4 0.00504341
\(536\) 3.62618e6 0.545177
\(537\) 7.10505e6 1.06324
\(538\) −1.53743e6 −0.229003
\(539\) 697552. 0.103420
\(540\) 11283.5 0.00166517
\(541\) −9.33636e6 −1.37146 −0.685732 0.727854i \(-0.740518\pi\)
−0.685732 + 0.727854i \(0.740518\pi\)
\(542\) 9.27515e6 1.35620
\(543\) 362977. 0.0528298
\(544\) 6.11769e6 0.886320
\(545\) −25460.7 −0.00367180
\(546\) −1.30782e6 −0.187745
\(547\) 1.30814e6 0.186933 0.0934664 0.995622i \(-0.470205\pi\)
0.0934664 + 0.995622i \(0.470205\pi\)
\(548\) −1.94450e7 −2.76603
\(549\) −1.49609e6 −0.211849
\(550\) 3.54381e6 0.499532
\(551\) 9.42064e6 1.32191
\(552\) 1.66673e6 0.232819
\(553\) 4.33395e6 0.602658
\(554\) 6.81505e6 0.943397
\(555\) 17797.9 0.00245266
\(556\) 1.86394e7 2.55708
\(557\) 6.95945e6 0.950467 0.475234 0.879860i \(-0.342364\pi\)
0.475234 + 0.879860i \(0.342364\pi\)
\(558\) −7.46678e6 −1.01519
\(559\) 2.03950e6 0.276054
\(560\) −8947.82 −0.00120572
\(561\) 1.55998e6 0.209272
\(562\) −3.55303e6 −0.474524
\(563\) 6.90198e6 0.917704 0.458852 0.888513i \(-0.348261\pi\)
0.458852 + 0.888513i \(0.348261\pi\)
\(564\) 1.15544e7 1.52950
\(565\) −8210.15 −0.00108201
\(566\) −6.51721e6 −0.855107
\(567\) −689439. −0.0900613
\(568\) 6.30054e6 0.819421
\(569\) 3.31261e6 0.428933 0.214466 0.976731i \(-0.431199\pi\)
0.214466 + 0.976731i \(0.431199\pi\)
\(570\) 47537.6 0.00612844
\(571\) 1.43930e7 1.84741 0.923703 0.383109i \(-0.125147\pi\)
0.923703 + 0.383109i \(0.125147\pi\)
\(572\) −996928. −0.127401
\(573\) 1.20362e6 0.153146
\(574\) −1.79883e7 −2.27882
\(575\) −2.59010e6 −0.326699
\(576\) −4.03836e6 −0.507165
\(577\) 3.31761e6 0.414845 0.207423 0.978251i \(-0.433492\pi\)
0.207423 + 0.978251i \(0.433492\pi\)
\(578\) 5.92484e6 0.737662
\(579\) −4.16040e6 −0.515749
\(580\) 71716.5 0.00885216
\(581\) 2.45421e6 0.301628
\(582\) −1.25926e7 −1.54101
\(583\) −1.73174e6 −0.211015
\(584\) −8.79939e6 −1.06763
\(585\) 3312.78 0.000400224 0
\(586\) 2.51527e6 0.302580
\(587\) 2.10735e6 0.252431 0.126215 0.992003i \(-0.459717\pi\)
0.126215 + 0.992003i \(0.459717\pi\)
\(588\) −2.89719e6 −0.345568
\(589\) −1.99977e7 −2.37515
\(590\) −18395.8 −0.00217565
\(591\) 867893. 0.102211
\(592\) 2.19166e6 0.257021
\(593\) 525039. 0.0613133 0.0306566 0.999530i \(-0.490240\pi\)
0.0306566 + 0.999530i \(0.490240\pi\)
\(594\) −826720. −0.0961373
\(595\) 41724.4 0.00483167
\(596\) 1.73242e7 1.99773
\(597\) −9.74380e6 −1.11890
\(598\) 1.14619e6 0.131071
\(599\) −7.02964e6 −0.800509 −0.400254 0.916404i \(-0.631078\pi\)
−0.400254 + 0.916404i \(0.631078\pi\)
\(600\) −6.28387e6 −0.712605
\(601\) −1.11158e6 −0.125532 −0.0627660 0.998028i \(-0.519992\pi\)
−0.0627660 + 0.998028i \(0.519992\pi\)
\(602\) −1.36132e7 −1.53097
\(603\) 1.31459e6 0.147230
\(604\) −3.05731e6 −0.340994
\(605\) 4058.29 0.000450770 0
\(606\) 5.26433e6 0.582320
\(607\) 1.53428e7 1.69018 0.845088 0.534627i \(-0.179548\pi\)
0.845088 + 0.534627i \(0.179548\pi\)
\(608\) −8.68307e6 −0.952608
\(609\) −4.38200e6 −0.478772
\(610\) −47983.3 −0.00522115
\(611\) 3.39232e6 0.367616
\(612\) −6.47916e6 −0.699263
\(613\) 4.78522e6 0.514341 0.257170 0.966366i \(-0.417210\pi\)
0.257170 + 0.966366i \(0.417210\pi\)
\(614\) 1.67615e7 1.79429
\(615\) 45565.3 0.00485788
\(616\) 2.84091e6 0.301651
\(617\) 1.69116e7 1.78843 0.894215 0.447639i \(-0.147735\pi\)
0.894215 + 0.447639i \(0.147735\pi\)
\(618\) −7.59553e6 −0.799994
\(619\) −8.65985e6 −0.908414 −0.454207 0.890896i \(-0.650077\pi\)
−0.454207 + 0.890896i \(0.650077\pi\)
\(620\) −152236. −0.0159052
\(621\) 604234. 0.0628747
\(622\) 3.39958e6 0.352330
\(623\) 1.09078e7 1.12595
\(624\) 407941. 0.0419407
\(625\) 9.76490e6 0.999926
\(626\) −1.07202e7 −1.09337
\(627\) −2.21414e6 −0.224924
\(628\) 1.23716e7 1.25178
\(629\) −1.02199e7 −1.02996
\(630\) −22112.0 −0.00221961
\(631\) 9.04083e6 0.903930 0.451965 0.892036i \(-0.350723\pi\)
0.451965 + 0.892036i \(0.350723\pi\)
\(632\) −9.21516e6 −0.917720
\(633\) 7.88029e6 0.781687
\(634\) 3.22172e7 3.18321
\(635\) 75153.2 0.00739627
\(636\) 7.19257e6 0.705084
\(637\) −850603. −0.0830574
\(638\) −5.25454e6 −0.511073
\(639\) 2.28411e6 0.221292
\(640\) −91639.5 −0.00884368
\(641\) −1.21113e7 −1.16424 −0.582122 0.813101i \(-0.697778\pi\)
−0.582122 + 0.813101i \(0.697778\pi\)
\(642\) 1.01607e7 0.972942
\(643\) 2.24955e6 0.214570 0.107285 0.994228i \(-0.465784\pi\)
0.107285 + 0.994228i \(0.465784\pi\)
\(644\) −4.86347e6 −0.462095
\(645\) 34482.8 0.00326365
\(646\) −2.72969e7 −2.57355
\(647\) 8.21206e6 0.771244 0.385622 0.922657i \(-0.373987\pi\)
0.385622 + 0.922657i \(0.373987\pi\)
\(648\) 1.46594e6 0.137144
\(649\) 856815. 0.0798501
\(650\) −4.32136e6 −0.401178
\(651\) 9.30190e6 0.860239
\(652\) −1.25773e7 −1.15869
\(653\) −1.15110e7 −1.05641 −0.528203 0.849118i \(-0.677134\pi\)
−0.528203 + 0.849118i \(0.677134\pi\)
\(654\) −7.74793e6 −0.708340
\(655\) −62087.1 −0.00565455
\(656\) 5.61097e6 0.509071
\(657\) −3.19001e6 −0.288323
\(658\) −2.26429e7 −2.03877
\(659\) −1.79339e7 −1.60865 −0.804323 0.594192i \(-0.797472\pi\)
−0.804323 + 0.594192i \(0.797472\pi\)
\(660\) −16855.6 −0.00150620
\(661\) −8.63046e6 −0.768299 −0.384150 0.923271i \(-0.625505\pi\)
−0.384150 + 0.923271i \(0.625505\pi\)
\(662\) 1.04433e7 0.926176
\(663\) −1.90226e6 −0.168068
\(664\) −5.21832e6 −0.459315
\(665\) −59220.9 −0.00519303
\(666\) 5.41607e6 0.473150
\(667\) 3.84044e6 0.334246
\(668\) 1.04306e7 0.904418
\(669\) 8.89238e6 0.768162
\(670\) 42162.1 0.00362857
\(671\) 2.23490e6 0.191625
\(672\) 4.03892e6 0.345018
\(673\) −1.92878e7 −1.64152 −0.820758 0.571276i \(-0.806449\pi\)
−0.820758 + 0.571276i \(0.806449\pi\)
\(674\) −2.11630e7 −1.79443
\(675\) −2.27807e6 −0.192445
\(676\) −1.95172e7 −1.64267
\(677\) 6.67400e6 0.559647 0.279824 0.960051i \(-0.409724\pi\)
0.279824 + 0.960051i \(0.409724\pi\)
\(678\) −2.49843e6 −0.208734
\(679\) 1.56875e7 1.30580
\(680\) −88717.4 −0.00735761
\(681\) 603594. 0.0498744
\(682\) 1.11541e7 0.918275
\(683\) −7.83722e6 −0.642851 −0.321426 0.946935i \(-0.604162\pi\)
−0.321426 + 0.946935i \(0.604162\pi\)
\(684\) 9.19611e6 0.751561
\(685\) −96524.6 −0.00785980
\(686\) 2.22300e7 1.80355
\(687\) −5.95918e6 −0.481720
\(688\) 4.24626e6 0.342007
\(689\) 2.11171e6 0.169467
\(690\) 19379.3 0.00154958
\(691\) −2.05838e7 −1.63995 −0.819974 0.572400i \(-0.806012\pi\)
−0.819974 + 0.572400i \(0.806012\pi\)
\(692\) −3.98507e7 −3.16352
\(693\) 1.02990e6 0.0814635
\(694\) −1.09707e7 −0.864640
\(695\) 92525.4 0.00726606
\(696\) 9.31734e6 0.729069
\(697\) −2.61644e7 −2.03999
\(698\) 1.14065e7 0.886162
\(699\) 1.24165e7 0.961185
\(700\) 1.83361e7 1.41437
\(701\) −8.88356e6 −0.682797 −0.341399 0.939919i \(-0.610901\pi\)
−0.341399 + 0.939919i \(0.610901\pi\)
\(702\) 1.00811e6 0.0772086
\(703\) 1.45054e7 1.10699
\(704\) 6.03261e6 0.458748
\(705\) 57355.7 0.00434614
\(706\) 2.68223e6 0.202528
\(707\) −6.55815e6 −0.493438
\(708\) −3.55866e6 −0.266811
\(709\) 6.33167e6 0.473045 0.236523 0.971626i \(-0.423992\pi\)
0.236523 + 0.971626i \(0.423992\pi\)
\(710\) 73257.2 0.00545386
\(711\) −3.34074e6 −0.247838
\(712\) −2.31930e7 −1.71458
\(713\) −8.15231e6 −0.600560
\(714\) 1.26971e7 0.932094
\(715\) −4948.73 −0.000362017 0
\(716\) 4.40826e7 3.21355
\(717\) 3.81412e6 0.277075
\(718\) 1.11592e7 0.807836
\(719\) 3.88936e6 0.280579 0.140290 0.990111i \(-0.455197\pi\)
0.140290 + 0.990111i \(0.455197\pi\)
\(720\) 6897.26 0.000495844 0
\(721\) 9.46229e6 0.677888
\(722\) 1.55368e7 1.10922
\(723\) −1.21322e7 −0.863163
\(724\) 2.25205e6 0.159673
\(725\) −1.44792e7 −1.02305
\(726\) 1.23498e6 0.0869595
\(727\) 2.34319e7 1.64426 0.822131 0.569299i \(-0.192785\pi\)
0.822131 + 0.569299i \(0.192785\pi\)
\(728\) −3.46423e6 −0.242258
\(729\) 531441. 0.0370370
\(730\) −102312. −0.00710588
\(731\) −1.98006e7 −1.37052
\(732\) −9.28235e6 −0.640295
\(733\) 1.77978e7 1.22351 0.611753 0.791049i \(-0.290465\pi\)
0.611753 + 0.791049i \(0.290465\pi\)
\(734\) 2.91591e7 1.99772
\(735\) −14381.6 −0.000981946 0
\(736\) −3.53976e6 −0.240868
\(737\) −1.96376e6 −0.133174
\(738\) 1.38659e7 0.937149
\(739\) 3.15950e6 0.212817 0.106409 0.994322i \(-0.466065\pi\)
0.106409 + 0.994322i \(0.466065\pi\)
\(740\) 110425. 0.00741293
\(741\) 2.69994e6 0.180638
\(742\) −1.40952e7 −0.939853
\(743\) −4.37086e6 −0.290465 −0.145233 0.989398i \(-0.546393\pi\)
−0.145233 + 0.989398i \(0.546393\pi\)
\(744\) −1.97784e7 −1.30996
\(745\) 85996.8 0.00567664
\(746\) 3.36030e7 2.21070
\(747\) −1.89178e6 −0.124042
\(748\) 9.67875e6 0.632507
\(749\) −1.26579e7 −0.824438
\(750\) −146129. −0.00948597
\(751\) 1.72544e7 1.11635 0.558173 0.829724i \(-0.311502\pi\)
0.558173 + 0.829724i \(0.311502\pi\)
\(752\) 7.06285e6 0.455445
\(753\) −1.31322e7 −0.844017
\(754\) 6.40745e6 0.410447
\(755\) −15176.4 −0.000968950 0
\(756\) −4.27756e6 −0.272202
\(757\) −1.57084e7 −0.996307 −0.498153 0.867089i \(-0.665988\pi\)
−0.498153 + 0.867089i \(0.665988\pi\)
\(758\) 1.76033e7 1.11281
\(759\) −902621. −0.0568723
\(760\) 125920. 0.00790789
\(761\) 5.01006e6 0.313604 0.156802 0.987630i \(-0.449882\pi\)
0.156802 + 0.987630i \(0.449882\pi\)
\(762\) 2.28698e7 1.42684
\(763\) 9.65215e6 0.600223
\(764\) 7.46778e6 0.462869
\(765\) −32162.4 −0.00198699
\(766\) −6.21058e6 −0.382437
\(767\) −1.04481e6 −0.0641282
\(768\) −1.35282e7 −0.827629
\(769\) 3.47489e6 0.211897 0.105949 0.994372i \(-0.466212\pi\)
0.105949 + 0.994372i \(0.466212\pi\)
\(770\) 33031.6 0.00200772
\(771\) 4.93653e6 0.299079
\(772\) −2.58128e7 −1.55880
\(773\) 2.29786e7 1.38317 0.691584 0.722296i \(-0.256913\pi\)
0.691584 + 0.722296i \(0.256913\pi\)
\(774\) 1.04934e7 0.629602
\(775\) 3.07357e7 1.83818
\(776\) −3.33558e7 −1.98846
\(777\) −6.74718e6 −0.400931
\(778\) −4.76458e7 −2.82212
\(779\) 3.71361e7 2.19256
\(780\) 20553.9 0.00120964
\(781\) −3.41207e6 −0.200166
\(782\) −1.11279e7 −0.650724
\(783\) 3.37778e6 0.196892
\(784\) −1.77096e6 −0.102901
\(785\) 61412.3 0.00355698
\(786\) −1.88937e7 −1.09084
\(787\) −1.40930e7 −0.811087 −0.405544 0.914076i \(-0.632918\pi\)
−0.405544 + 0.914076i \(0.632918\pi\)
\(788\) 5.38476e6 0.308923
\(789\) 8.14769e6 0.465953
\(790\) −107146. −0.00610812
\(791\) 3.11247e6 0.176874
\(792\) −2.18986e6 −0.124052
\(793\) −2.72526e6 −0.153895
\(794\) −4.85845e6 −0.273493
\(795\) 35703.7 0.00200353
\(796\) −6.04545e7 −3.38178
\(797\) −1.08281e7 −0.603819 −0.301909 0.953337i \(-0.597624\pi\)
−0.301909 + 0.953337i \(0.597624\pi\)
\(798\) −1.80215e7 −1.00181
\(799\) −3.29346e7 −1.82510
\(800\) 1.33455e7 0.737243
\(801\) −8.40808e6 −0.463037
\(802\) 1.05474e7 0.579041
\(803\) 4.76533e6 0.260798
\(804\) 8.15623e6 0.444989
\(805\) −24142.2 −0.00131307
\(806\) −1.36014e7 −0.737474
\(807\) −1.47637e6 −0.0798013
\(808\) 1.39444e7 0.751402
\(809\) −2.27494e7 −1.22208 −0.611040 0.791600i \(-0.709248\pi\)
−0.611040 + 0.791600i \(0.709248\pi\)
\(810\) 17044.6 0.000912799 0
\(811\) −1.33958e7 −0.715179 −0.357590 0.933879i \(-0.616401\pi\)
−0.357590 + 0.933879i \(0.616401\pi\)
\(812\) −2.71877e7 −1.44705
\(813\) 8.90673e6 0.472598
\(814\) −8.09067e6 −0.427980
\(815\) −62433.2 −0.00329247
\(816\) −3.96052e6 −0.208222
\(817\) 2.81038e7 1.47302
\(818\) 2.95596e7 1.54460
\(819\) −1.25588e6 −0.0654240
\(820\) 282706. 0.0146825
\(821\) −1.77359e7 −0.918323 −0.459161 0.888353i \(-0.651850\pi\)
−0.459161 + 0.888353i \(0.651850\pi\)
\(822\) −2.93733e7 −1.51626
\(823\) −2.24560e7 −1.15567 −0.577834 0.816154i \(-0.696102\pi\)
−0.577834 + 0.816154i \(0.696102\pi\)
\(824\) −2.01194e7 −1.03228
\(825\) 3.40304e6 0.174073
\(826\) 6.97386e6 0.355650
\(827\) −1.97457e7 −1.00394 −0.501972 0.864884i \(-0.667392\pi\)
−0.501972 + 0.864884i \(0.667392\pi\)
\(828\) 3.74891e6 0.190033
\(829\) 2.61754e7 1.32284 0.661418 0.750017i \(-0.269955\pi\)
0.661418 + 0.750017i \(0.269955\pi\)
\(830\) −60674.1 −0.00305709
\(831\) 6.54434e6 0.328748
\(832\) −7.35624e6 −0.368424
\(833\) 8.25814e6 0.412354
\(834\) 2.81563e7 1.40172
\(835\) 51777.4 0.00256995
\(836\) −1.37374e7 −0.679812
\(837\) −7.17019e6 −0.353767
\(838\) 7.51015e6 0.369435
\(839\) −3.69992e6 −0.181463 −0.0907313 0.995875i \(-0.528920\pi\)
−0.0907313 + 0.995875i \(0.528920\pi\)
\(840\) −58571.5 −0.00286410
\(841\) 957652. 0.0466893
\(842\) −1.73808e7 −0.844869
\(843\) −3.41190e6 −0.165359
\(844\) 4.88925e7 2.36258
\(845\) −96883.0 −0.00466773
\(846\) 1.74539e7 0.838428
\(847\) −1.53850e6 −0.0736866
\(848\) 4.39660e6 0.209956
\(849\) −6.25834e6 −0.297982
\(850\) 4.19542e7 1.99172
\(851\) 5.91332e6 0.279903
\(852\) 1.41716e7 0.668834
\(853\) 2.90121e7 1.36523 0.682617 0.730776i \(-0.260842\pi\)
0.682617 + 0.730776i \(0.260842\pi\)
\(854\) 1.81905e7 0.853492
\(855\) 45649.3 0.00213559
\(856\) 2.69142e7 1.25544
\(857\) −3.60120e7 −1.67493 −0.837463 0.546494i \(-0.815962\pi\)
−0.837463 + 0.546494i \(0.815962\pi\)
\(858\) −1.50594e6 −0.0698378
\(859\) −2.14715e7 −0.992841 −0.496421 0.868082i \(-0.665353\pi\)
−0.496421 + 0.868082i \(0.665353\pi\)
\(860\) 213945. 0.00986409
\(861\) −1.72738e7 −0.794108
\(862\) 2.30562e7 1.05686
\(863\) 1.43154e7 0.654298 0.327149 0.944973i \(-0.393912\pi\)
0.327149 + 0.944973i \(0.393912\pi\)
\(864\) −3.11332e6 −0.141886
\(865\) −197818. −0.00898928
\(866\) 2.92625e7 1.32592
\(867\) 5.68950e6 0.257055
\(868\) 5.77128e7 2.60000
\(869\) 4.99049e6 0.224178
\(870\) 108334. 0.00485251
\(871\) 2.39464e6 0.106953
\(872\) −2.05231e7 −0.914013
\(873\) −1.20924e7 −0.537002
\(874\) 1.57943e7 0.699392
\(875\) 182043. 0.00803809
\(876\) −1.97921e7 −0.871430
\(877\) 2.25088e7 0.988221 0.494110 0.869399i \(-0.335494\pi\)
0.494110 + 0.869399i \(0.335494\pi\)
\(878\) −5.71030e7 −2.49990
\(879\) 2.41536e6 0.105441
\(880\) −10303.3 −0.000448508 0
\(881\) 8.39629e6 0.364458 0.182229 0.983256i \(-0.441669\pi\)
0.182229 + 0.983256i \(0.441669\pi\)
\(882\) −4.37644e6 −0.189431
\(883\) 1.08666e7 0.469021 0.234510 0.972114i \(-0.424651\pi\)
0.234510 + 0.972114i \(0.424651\pi\)
\(884\) −1.18024e7 −0.507971
\(885\) −17665.1 −0.000758156 0
\(886\) −7.43650e7 −3.18262
\(887\) 1.53488e7 0.655038 0.327519 0.944845i \(-0.393788\pi\)
0.327519 + 0.944845i \(0.393788\pi\)
\(888\) 1.43464e7 0.610533
\(889\) −2.84905e7 −1.20906
\(890\) −269668. −0.0114118
\(891\) −793881. −0.0335013
\(892\) 5.51719e7 2.32170
\(893\) 4.67453e7 1.96160
\(894\) 2.61696e7 1.09510
\(895\) 218825. 0.00913145
\(896\) 3.47405e7 1.44566
\(897\) 1.10067e6 0.0456746
\(898\) 4.32073e7 1.78799
\(899\) −4.55729e7 −1.88065
\(900\) −1.41341e7 −0.581649
\(901\) −2.05017e7 −0.841352
\(902\) −2.07133e7 −0.847683
\(903\) −1.30724e7 −0.533503
\(904\) −6.61796e6 −0.269341
\(905\) 11179.1 0.000453719 0
\(906\) −4.61832e6 −0.186923
\(907\) 7.93009e6 0.320081 0.160041 0.987110i \(-0.448838\pi\)
0.160041 + 0.987110i \(0.448838\pi\)
\(908\) 3.74494e6 0.150741
\(909\) 5.05522e6 0.202923
\(910\) −40279.1 −0.00161241
\(911\) −4.31907e7 −1.72423 −0.862114 0.506715i \(-0.830860\pi\)
−0.862114 + 0.506715i \(0.830860\pi\)
\(912\) 5.62131e6 0.223795
\(913\) 2.82599e6 0.112200
\(914\) −7.54898e7 −2.98898
\(915\) −46077.4 −0.00181943
\(916\) −3.69732e7 −1.45595
\(917\) 2.35372e7 0.924340
\(918\) −9.78733e6 −0.383316
\(919\) −2.17987e7 −0.851414 −0.425707 0.904861i \(-0.639975\pi\)
−0.425707 + 0.904861i \(0.639975\pi\)
\(920\) 51332.8 0.00199952
\(921\) 1.60957e7 0.625260
\(922\) 3.02191e7 1.17072
\(923\) 4.16072e6 0.160755
\(924\) 6.38994e6 0.246216
\(925\) −2.22943e7 −0.856720
\(926\) 1.21471e7 0.465527
\(927\) −7.29382e6 −0.278776
\(928\) −1.97879e7 −0.754276
\(929\) −1.33390e7 −0.507089 −0.253545 0.967324i \(-0.581596\pi\)
−0.253545 + 0.967324i \(0.581596\pi\)
\(930\) −229966. −0.00871879
\(931\) −1.17211e7 −0.443194
\(932\) 7.70371e7 2.90510
\(933\) 3.26454e6 0.122777
\(934\) −2.00475e7 −0.751956
\(935\) 48045.1 0.00179730
\(936\) 2.67034e6 0.0996269
\(937\) 1.33697e7 0.497476 0.248738 0.968571i \(-0.419984\pi\)
0.248738 + 0.968571i \(0.419984\pi\)
\(938\) −1.59836e7 −0.593155
\(939\) −1.02943e7 −0.381008
\(940\) 355858. 0.0131358
\(941\) 2.28926e7 0.842793 0.421397 0.906876i \(-0.361540\pi\)
0.421397 + 0.906876i \(0.361540\pi\)
\(942\) 1.86883e7 0.686189
\(943\) 1.51390e7 0.554393
\(944\) −2.17531e6 −0.0794494
\(945\) −21233.7 −0.000773475 0
\(946\) −1.56754e7 −0.569496
\(947\) 3.33079e6 0.120690 0.0603451 0.998178i \(-0.480780\pi\)
0.0603451 + 0.998178i \(0.480780\pi\)
\(948\) −2.07273e7 −0.749069
\(949\) −5.81089e6 −0.209449
\(950\) −5.95472e7 −2.14068
\(951\) 3.09375e7 1.10926
\(952\) 3.36328e7 1.20274
\(953\) 4.68461e7 1.67087 0.835433 0.549593i \(-0.185217\pi\)
0.835433 + 0.549593i \(0.185217\pi\)
\(954\) 1.08650e7 0.386508
\(955\) 37069.9 0.00131526
\(956\) 2.36644e7 0.837433
\(957\) −5.04582e6 −0.178095
\(958\) −4.47379e7 −1.57493
\(959\) 3.65924e7 1.28483
\(960\) −124376. −0.00435569
\(961\) 6.81109e7 2.37907
\(962\) 9.86586e6 0.343714
\(963\) 9.75712e6 0.339044
\(964\) −7.52730e7 −2.60883
\(965\) −128134. −0.00442942
\(966\) −7.34668e6 −0.253308
\(967\) −3.65887e7 −1.25829 −0.629146 0.777287i \(-0.716595\pi\)
−0.629146 + 0.777287i \(0.716595\pi\)
\(968\) 3.27127e6 0.112209
\(969\) −2.62126e7 −0.896811
\(970\) −387833. −0.0132347
\(971\) 1.62147e7 0.551901 0.275951 0.961172i \(-0.411007\pi\)
0.275951 + 0.961172i \(0.411007\pi\)
\(972\) 3.29728e6 0.111941
\(973\) −3.50764e7 −1.18777
\(974\) 7.48261e7 2.52729
\(975\) −4.14971e6 −0.139800
\(976\) −5.67403e6 −0.190663
\(977\) −1.50181e6 −0.0503360 −0.0251680 0.999683i \(-0.508012\pi\)
−0.0251680 + 0.999683i \(0.508012\pi\)
\(978\) −1.89990e7 −0.635161
\(979\) 1.25602e7 0.418832
\(980\) −89229.1 −0.00296784
\(981\) −7.44017e6 −0.246837
\(982\) −3.46803e7 −1.14764
\(983\) 3.72690e7 1.23017 0.615083 0.788462i \(-0.289122\pi\)
0.615083 + 0.788462i \(0.289122\pi\)
\(984\) 3.67288e7 1.20926
\(985\) 26729.8 0.000877820 0
\(986\) −6.22072e7 −2.03774
\(987\) −2.17435e7 −0.710456
\(988\) 1.67516e7 0.545962
\(989\) 1.14569e7 0.372456
\(990\) −25461.7 −0.000825658 0
\(991\) −3.20015e7 −1.03511 −0.517554 0.855651i \(-0.673157\pi\)
−0.517554 + 0.855651i \(0.673157\pi\)
\(992\) 4.20049e7 1.35525
\(993\) 1.00285e7 0.322747
\(994\) −2.77718e7 −0.891534
\(995\) −300095. −0.00960950
\(996\) −1.17374e7 −0.374906
\(997\) −5.59172e7 −1.78159 −0.890795 0.454406i \(-0.849852\pi\)
−0.890795 + 0.454406i \(0.849852\pi\)
\(998\) −6.60874e7 −2.10035
\(999\) 5.20094e6 0.164880
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 33.6.a.e.1.2 2
3.2 odd 2 99.6.a.d.1.1 2
4.3 odd 2 528.6.a.o.1.1 2
5.4 even 2 825.6.a.c.1.1 2
11.10 odd 2 363.6.a.f.1.1 2
33.32 even 2 1089.6.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.e.1.2 2 1.1 even 1 trivial
99.6.a.d.1.1 2 3.2 odd 2
363.6.a.f.1.1 2 11.10 odd 2
528.6.a.o.1.1 2 4.3 odd 2
825.6.a.c.1.1 2 5.4 even 2
1089.6.a.p.1.2 2 33.32 even 2