Properties

Label 546.6.a.f.1.1
Level $546$
Weight $6$
Character 546.1
Self dual yes
Analytic conductor $87.570$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(87.5695656179\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -54.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -54.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -216.000 q^{10} -192.000 q^{11} +144.000 q^{12} +169.000 q^{13} +196.000 q^{14} -486.000 q^{15} +256.000 q^{16} -1422.00 q^{17} +324.000 q^{18} +1748.00 q^{19} -864.000 q^{20} +441.000 q^{21} -768.000 q^{22} -3792.00 q^{23} +576.000 q^{24} -209.000 q^{25} +676.000 q^{26} +729.000 q^{27} +784.000 q^{28} -954.000 q^{29} -1944.00 q^{30} -568.000 q^{31} +1024.00 q^{32} -1728.00 q^{33} -5688.00 q^{34} -2646.00 q^{35} +1296.00 q^{36} +7886.00 q^{37} +6992.00 q^{38} +1521.00 q^{39} -3456.00 q^{40} -14802.0 q^{41} +1764.00 q^{42} +4964.00 q^{43} -3072.00 q^{44} -4374.00 q^{45} -15168.0 q^{46} -18948.0 q^{47} +2304.00 q^{48} +2401.00 q^{49} -836.000 q^{50} -12798.0 q^{51} +2704.00 q^{52} -426.000 q^{53} +2916.00 q^{54} +10368.0 q^{55} +3136.00 q^{56} +15732.0 q^{57} -3816.00 q^{58} +34872.0 q^{59} -7776.00 q^{60} -25618.0 q^{61} -2272.00 q^{62} +3969.00 q^{63} +4096.00 q^{64} -9126.00 q^{65} -6912.00 q^{66} -67060.0 q^{67} -22752.0 q^{68} -34128.0 q^{69} -10584.0 q^{70} -28428.0 q^{71} +5184.00 q^{72} -22894.0 q^{73} +31544.0 q^{74} -1881.00 q^{75} +27968.0 q^{76} -9408.00 q^{77} +6084.00 q^{78} -1408.00 q^{79} -13824.0 q^{80} +6561.00 q^{81} -59208.0 q^{82} -17304.0 q^{83} +7056.00 q^{84} +76788.0 q^{85} +19856.0 q^{86} -8586.00 q^{87} -12288.0 q^{88} -93690.0 q^{89} -17496.0 q^{90} +8281.00 q^{91} -60672.0 q^{92} -5112.00 q^{93} -75792.0 q^{94} -94392.0 q^{95} +9216.00 q^{96} +16826.0 q^{97} +9604.00 q^{98} -15552.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) 9.00000 0.577350
\(4\) 16.0000 0.500000
\(5\) −54.0000 −0.965981 −0.482991 0.875625i \(-0.660450\pi\)
−0.482991 + 0.875625i \(0.660450\pi\)
\(6\) 36.0000 0.408248
\(7\) 49.0000 0.377964
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) −216.000 −0.683052
\(11\) −192.000 −0.478431 −0.239216 0.970966i \(-0.576890\pi\)
−0.239216 + 0.970966i \(0.576890\pi\)
\(12\) 144.000 0.288675
\(13\) 169.000 0.277350
\(14\) 196.000 0.267261
\(15\) −486.000 −0.557710
\(16\) 256.000 0.250000
\(17\) −1422.00 −1.19338 −0.596688 0.802473i \(-0.703517\pi\)
−0.596688 + 0.802473i \(0.703517\pi\)
\(18\) 324.000 0.235702
\(19\) 1748.00 1.11086 0.555428 0.831565i \(-0.312555\pi\)
0.555428 + 0.831565i \(0.312555\pi\)
\(20\) −864.000 −0.482991
\(21\) 441.000 0.218218
\(22\) −768.000 −0.338302
\(23\) −3792.00 −1.49468 −0.747341 0.664441i \(-0.768670\pi\)
−0.747341 + 0.664441i \(0.768670\pi\)
\(24\) 576.000 0.204124
\(25\) −209.000 −0.0668800
\(26\) 676.000 0.196116
\(27\) 729.000 0.192450
\(28\) 784.000 0.188982
\(29\) −954.000 −0.210646 −0.105323 0.994438i \(-0.533588\pi\)
−0.105323 + 0.994438i \(0.533588\pi\)
\(30\) −1944.00 −0.394360
\(31\) −568.000 −0.106156 −0.0530779 0.998590i \(-0.516903\pi\)
−0.0530779 + 0.998590i \(0.516903\pi\)
\(32\) 1024.00 0.176777
\(33\) −1728.00 −0.276222
\(34\) −5688.00 −0.843844
\(35\) −2646.00 −0.365107
\(36\) 1296.00 0.166667
\(37\) 7886.00 0.947005 0.473503 0.880792i \(-0.342989\pi\)
0.473503 + 0.880792i \(0.342989\pi\)
\(38\) 6992.00 0.785493
\(39\) 1521.00 0.160128
\(40\) −3456.00 −0.341526
\(41\) −14802.0 −1.37518 −0.687592 0.726097i \(-0.741332\pi\)
−0.687592 + 0.726097i \(0.741332\pi\)
\(42\) 1764.00 0.154303
\(43\) 4964.00 0.409412 0.204706 0.978823i \(-0.434376\pi\)
0.204706 + 0.978823i \(0.434376\pi\)
\(44\) −3072.00 −0.239216
\(45\) −4374.00 −0.321994
\(46\) −15168.0 −1.05690
\(47\) −18948.0 −1.25118 −0.625588 0.780153i \(-0.715141\pi\)
−0.625588 + 0.780153i \(0.715141\pi\)
\(48\) 2304.00 0.144338
\(49\) 2401.00 0.142857
\(50\) −836.000 −0.0472913
\(51\) −12798.0 −0.688996
\(52\) 2704.00 0.138675
\(53\) −426.000 −0.0208315 −0.0104157 0.999946i \(-0.503315\pi\)
−0.0104157 + 0.999946i \(0.503315\pi\)
\(54\) 2916.00 0.136083
\(55\) 10368.0 0.462156
\(56\) 3136.00 0.133631
\(57\) 15732.0 0.641353
\(58\) −3816.00 −0.148949
\(59\) 34872.0 1.30421 0.652104 0.758130i \(-0.273887\pi\)
0.652104 + 0.758130i \(0.273887\pi\)
\(60\) −7776.00 −0.278855
\(61\) −25618.0 −0.881497 −0.440748 0.897631i \(-0.645287\pi\)
−0.440748 + 0.897631i \(0.645287\pi\)
\(62\) −2272.00 −0.0750636
\(63\) 3969.00 0.125988
\(64\) 4096.00 0.125000
\(65\) −9126.00 −0.267915
\(66\) −6912.00 −0.195319
\(67\) −67060.0 −1.82506 −0.912529 0.409013i \(-0.865873\pi\)
−0.912529 + 0.409013i \(0.865873\pi\)
\(68\) −22752.0 −0.596688
\(69\) −34128.0 −0.862955
\(70\) −10584.0 −0.258169
\(71\) −28428.0 −0.669269 −0.334634 0.942348i \(-0.608613\pi\)
−0.334634 + 0.942348i \(0.608613\pi\)
\(72\) 5184.00 0.117851
\(73\) −22894.0 −0.502822 −0.251411 0.967880i \(-0.580895\pi\)
−0.251411 + 0.967880i \(0.580895\pi\)
\(74\) 31544.0 0.669634
\(75\) −1881.00 −0.0386132
\(76\) 27968.0 0.555428
\(77\) −9408.00 −0.180830
\(78\) 6084.00 0.113228
\(79\) −1408.00 −0.0253825 −0.0126913 0.999919i \(-0.504040\pi\)
−0.0126913 + 0.999919i \(0.504040\pi\)
\(80\) −13824.0 −0.241495
\(81\) 6561.00 0.111111
\(82\) −59208.0 −0.972402
\(83\) −17304.0 −0.275709 −0.137855 0.990452i \(-0.544021\pi\)
−0.137855 + 0.990452i \(0.544021\pi\)
\(84\) 7056.00 0.109109
\(85\) 76788.0 1.15278
\(86\) 19856.0 0.289498
\(87\) −8586.00 −0.121617
\(88\) −12288.0 −0.169151
\(89\) −93690.0 −1.25377 −0.626886 0.779111i \(-0.715671\pi\)
−0.626886 + 0.779111i \(0.715671\pi\)
\(90\) −17496.0 −0.227684
\(91\) 8281.00 0.104828
\(92\) −60672.0 −0.747341
\(93\) −5112.00 −0.0612891
\(94\) −75792.0 −0.884716
\(95\) −94392.0 −1.07307
\(96\) 9216.00 0.102062
\(97\) 16826.0 0.181573 0.0907866 0.995870i \(-0.471062\pi\)
0.0907866 + 0.995870i \(0.471062\pi\)
\(98\) 9604.00 0.101015
\(99\) −15552.0 −0.159477
\(100\) −3344.00 −0.0334400
\(101\) −129090. −1.25918 −0.629592 0.776926i \(-0.716778\pi\)
−0.629592 + 0.776926i \(0.716778\pi\)
\(102\) −51192.0 −0.487194
\(103\) −5536.00 −0.0514166 −0.0257083 0.999669i \(-0.508184\pi\)
−0.0257083 + 0.999669i \(0.508184\pi\)
\(104\) 10816.0 0.0980581
\(105\) −23814.0 −0.210794
\(106\) −1704.00 −0.0147301
\(107\) −201996. −1.70562 −0.852812 0.522218i \(-0.825105\pi\)
−0.852812 + 0.522218i \(0.825105\pi\)
\(108\) 11664.0 0.0962250
\(109\) 32510.0 0.262090 0.131045 0.991376i \(-0.458167\pi\)
0.131045 + 0.991376i \(0.458167\pi\)
\(110\) 41472.0 0.326793
\(111\) 70974.0 0.546754
\(112\) 12544.0 0.0944911
\(113\) −36918.0 −0.271983 −0.135992 0.990710i \(-0.543422\pi\)
−0.135992 + 0.990710i \(0.543422\pi\)
\(114\) 62928.0 0.453505
\(115\) 204768. 1.44383
\(116\) −15264.0 −0.105323
\(117\) 13689.0 0.0924500
\(118\) 139488. 0.922214
\(119\) −69678.0 −0.451054
\(120\) −31104.0 −0.197180
\(121\) −124187. −0.771104
\(122\) −102472. −0.623312
\(123\) −133218. −0.793963
\(124\) −9088.00 −0.0530779
\(125\) 180036. 1.03059
\(126\) 15876.0 0.0890871
\(127\) −169792. −0.934131 −0.467066 0.884223i \(-0.654689\pi\)
−0.467066 + 0.884223i \(0.654689\pi\)
\(128\) 16384.0 0.0883883
\(129\) 44676.0 0.236374
\(130\) −36504.0 −0.189445
\(131\) 177660. 0.904506 0.452253 0.891890i \(-0.350620\pi\)
0.452253 + 0.891890i \(0.350620\pi\)
\(132\) −27648.0 −0.138111
\(133\) 85652.0 0.419864
\(134\) −268240. −1.29051
\(135\) −39366.0 −0.185903
\(136\) −91008.0 −0.421922
\(137\) 141582. 0.644476 0.322238 0.946659i \(-0.395565\pi\)
0.322238 + 0.946659i \(0.395565\pi\)
\(138\) −136512. −0.610201
\(139\) 282356. 1.23954 0.619769 0.784784i \(-0.287226\pi\)
0.619769 + 0.784784i \(0.287226\pi\)
\(140\) −42336.0 −0.182553
\(141\) −170532. −0.722367
\(142\) −113712. −0.473244
\(143\) −32448.0 −0.132693
\(144\) 20736.0 0.0833333
\(145\) 51516.0 0.203480
\(146\) −91576.0 −0.355549
\(147\) 21609.0 0.0824786
\(148\) 126176. 0.473503
\(149\) 203682. 0.751600 0.375800 0.926701i \(-0.377368\pi\)
0.375800 + 0.926701i \(0.377368\pi\)
\(150\) −7524.00 −0.0273036
\(151\) 114368. 0.408190 0.204095 0.978951i \(-0.434575\pi\)
0.204095 + 0.978951i \(0.434575\pi\)
\(152\) 111872. 0.392747
\(153\) −115182. −0.397792
\(154\) −37632.0 −0.127866
\(155\) 30672.0 0.102545
\(156\) 24336.0 0.0800641
\(157\) 48734.0 0.157791 0.0788956 0.996883i \(-0.474861\pi\)
0.0788956 + 0.996883i \(0.474861\pi\)
\(158\) −5632.00 −0.0179482
\(159\) −3834.00 −0.0120271
\(160\) −55296.0 −0.170763
\(161\) −185808. −0.564937
\(162\) 26244.0 0.0785674
\(163\) −18124.0 −0.0534300 −0.0267150 0.999643i \(-0.508505\pi\)
−0.0267150 + 0.999643i \(0.508505\pi\)
\(164\) −236832. −0.687592
\(165\) 93312.0 0.266826
\(166\) −69216.0 −0.194956
\(167\) 261300. 0.725017 0.362509 0.931980i \(-0.381920\pi\)
0.362509 + 0.931980i \(0.381920\pi\)
\(168\) 28224.0 0.0771517
\(169\) 28561.0 0.0769231
\(170\) 307152. 0.815138
\(171\) 141588. 0.370285
\(172\) 79424.0 0.204706
\(173\) 550710. 1.39897 0.699484 0.714648i \(-0.253413\pi\)
0.699484 + 0.714648i \(0.253413\pi\)
\(174\) −34344.0 −0.0859959
\(175\) −10241.0 −0.0252783
\(176\) −49152.0 −0.119608
\(177\) 313848. 0.752985
\(178\) −374760. −0.886550
\(179\) −264444. −0.616881 −0.308440 0.951244i \(-0.599807\pi\)
−0.308440 + 0.951244i \(0.599807\pi\)
\(180\) −69984.0 −0.160997
\(181\) −31930.0 −0.0724440 −0.0362220 0.999344i \(-0.511532\pi\)
−0.0362220 + 0.999344i \(0.511532\pi\)
\(182\) 33124.0 0.0741249
\(183\) −230562. −0.508932
\(184\) −242688. −0.528450
\(185\) −425844. −0.914790
\(186\) −20448.0 −0.0433380
\(187\) 273024. 0.570948
\(188\) −303168. −0.625588
\(189\) 35721.0 0.0727393
\(190\) −377568. −0.758772
\(191\) −210240. −0.416996 −0.208498 0.978023i \(-0.566857\pi\)
−0.208498 + 0.978023i \(0.566857\pi\)
\(192\) 36864.0 0.0721688
\(193\) −488062. −0.943152 −0.471576 0.881825i \(-0.656315\pi\)
−0.471576 + 0.881825i \(0.656315\pi\)
\(194\) 67304.0 0.128392
\(195\) −82134.0 −0.154681
\(196\) 38416.0 0.0714286
\(197\) −165558. −0.303938 −0.151969 0.988385i \(-0.548561\pi\)
−0.151969 + 0.988385i \(0.548561\pi\)
\(198\) −62208.0 −0.112767
\(199\) −800152. −1.43232 −0.716160 0.697937i \(-0.754102\pi\)
−0.716160 + 0.697937i \(0.754102\pi\)
\(200\) −13376.0 −0.0236457
\(201\) −603540. −1.05370
\(202\) −516360. −0.890377
\(203\) −46746.0 −0.0796167
\(204\) −204768. −0.344498
\(205\) 799308. 1.32840
\(206\) −22144.0 −0.0363570
\(207\) −307152. −0.498227
\(208\) 43264.0 0.0693375
\(209\) −335616. −0.531468
\(210\) −95256.0 −0.149054
\(211\) −382060. −0.590780 −0.295390 0.955377i \(-0.595450\pi\)
−0.295390 + 0.955377i \(0.595450\pi\)
\(212\) −6816.00 −0.0104157
\(213\) −255852. −0.386402
\(214\) −807984. −1.20606
\(215\) −268056. −0.395484
\(216\) 46656.0 0.0680414
\(217\) −27832.0 −0.0401232
\(218\) 130040. 0.185326
\(219\) −206046. −0.290305
\(220\) 165888. 0.231078
\(221\) −240318. −0.330983
\(222\) 283896. 0.386613
\(223\) −303016. −0.408041 −0.204020 0.978967i \(-0.565401\pi\)
−0.204020 + 0.978967i \(0.565401\pi\)
\(224\) 50176.0 0.0668153
\(225\) −16929.0 −0.0222933
\(226\) −147672. −0.192321
\(227\) 383376. 0.493810 0.246905 0.969040i \(-0.420586\pi\)
0.246905 + 0.969040i \(0.420586\pi\)
\(228\) 251712. 0.320676
\(229\) 226070. 0.284875 0.142437 0.989804i \(-0.454506\pi\)
0.142437 + 0.989804i \(0.454506\pi\)
\(230\) 819072. 1.02095
\(231\) −84672.0 −0.104402
\(232\) −61056.0 −0.0744746
\(233\) 1.16387e6 1.40448 0.702241 0.711939i \(-0.252183\pi\)
0.702241 + 0.711939i \(0.252183\pi\)
\(234\) 54756.0 0.0653720
\(235\) 1.02319e6 1.20861
\(236\) 557952. 0.652104
\(237\) −12672.0 −0.0146546
\(238\) −278712. −0.318943
\(239\) −661332. −0.748902 −0.374451 0.927247i \(-0.622169\pi\)
−0.374451 + 0.927247i \(0.622169\pi\)
\(240\) −124416. −0.139427
\(241\) −662854. −0.735149 −0.367574 0.929994i \(-0.619812\pi\)
−0.367574 + 0.929994i \(0.619812\pi\)
\(242\) −496748. −0.545253
\(243\) 59049.0 0.0641500
\(244\) −409888. −0.440748
\(245\) −129654. −0.137997
\(246\) −532872. −0.561416
\(247\) 295412. 0.308096
\(248\) −36352.0 −0.0375318
\(249\) −155736. −0.159181
\(250\) 720144. 0.728734
\(251\) −221100. −0.221516 −0.110758 0.993847i \(-0.535328\pi\)
−0.110758 + 0.993847i \(0.535328\pi\)
\(252\) 63504.0 0.0629941
\(253\) 728064. 0.715102
\(254\) −679168. −0.660531
\(255\) 691092. 0.665557
\(256\) 65536.0 0.0625000
\(257\) 1.12972e6 1.06694 0.533469 0.845820i \(-0.320888\pi\)
0.533469 + 0.845820i \(0.320888\pi\)
\(258\) 178704. 0.167142
\(259\) 386414. 0.357934
\(260\) −146016. −0.133958
\(261\) −77274.0 −0.0702154
\(262\) 710640. 0.639582
\(263\) 1.01544e6 0.905242 0.452621 0.891703i \(-0.350489\pi\)
0.452621 + 0.891703i \(0.350489\pi\)
\(264\) −110592. −0.0976594
\(265\) 23004.0 0.0201228
\(266\) 342608. 0.296889
\(267\) −843210. −0.723865
\(268\) −1.07296e6 −0.912529
\(269\) 1.39282e6 1.17359 0.586793 0.809737i \(-0.300390\pi\)
0.586793 + 0.809737i \(0.300390\pi\)
\(270\) −157464. −0.131453
\(271\) −1.48739e6 −1.23028 −0.615138 0.788420i \(-0.710900\pi\)
−0.615138 + 0.788420i \(0.710900\pi\)
\(272\) −364032. −0.298344
\(273\) 74529.0 0.0605228
\(274\) 566328. 0.455713
\(275\) 40128.0 0.0319975
\(276\) −546048. −0.431477
\(277\) 195782. 0.153311 0.0766555 0.997058i \(-0.475576\pi\)
0.0766555 + 0.997058i \(0.475576\pi\)
\(278\) 1.12942e6 0.876486
\(279\) −46008.0 −0.0353853
\(280\) −169344. −0.129085
\(281\) −1.50376e6 −1.13609 −0.568046 0.822997i \(-0.692300\pi\)
−0.568046 + 0.822997i \(0.692300\pi\)
\(282\) −682128. −0.510791
\(283\) 365660. 0.271401 0.135700 0.990750i \(-0.456672\pi\)
0.135700 + 0.990750i \(0.456672\pi\)
\(284\) −454848. −0.334634
\(285\) −849528. −0.619535
\(286\) −129792. −0.0938281
\(287\) −725298. −0.519771
\(288\) 82944.0 0.0589256
\(289\) 602227. 0.424146
\(290\) 206064. 0.143882
\(291\) 151434. 0.104831
\(292\) −366304. −0.251411
\(293\) 1.87936e6 1.27891 0.639457 0.768827i \(-0.279159\pi\)
0.639457 + 0.768827i \(0.279159\pi\)
\(294\) 86436.0 0.0583212
\(295\) −1.88309e6 −1.25984
\(296\) 504704. 0.334817
\(297\) −139968. −0.0920741
\(298\) 814728. 0.531462
\(299\) −640848. −0.414550
\(300\) −30096.0 −0.0193066
\(301\) 243236. 0.154743
\(302\) 457472. 0.288634
\(303\) −1.16181e6 −0.726990
\(304\) 447488. 0.277714
\(305\) 1.38337e6 0.851509
\(306\) −460728. −0.281281
\(307\) 1.18737e6 0.719020 0.359510 0.933141i \(-0.382944\pi\)
0.359510 + 0.933141i \(0.382944\pi\)
\(308\) −150528. −0.0904150
\(309\) −49824.0 −0.0296854
\(310\) 122688. 0.0725100
\(311\) −539808. −0.316474 −0.158237 0.987401i \(-0.550581\pi\)
−0.158237 + 0.987401i \(0.550581\pi\)
\(312\) 97344.0 0.0566139
\(313\) 3.30535e6 1.90703 0.953514 0.301348i \(-0.0974367\pi\)
0.953514 + 0.301348i \(0.0974367\pi\)
\(314\) 194936. 0.111575
\(315\) −214326. −0.121702
\(316\) −22528.0 −0.0126913
\(317\) 1.92640e6 1.07671 0.538355 0.842718i \(-0.319046\pi\)
0.538355 + 0.842718i \(0.319046\pi\)
\(318\) −15336.0 −0.00850441
\(319\) 183168. 0.100780
\(320\) −221184. −0.120748
\(321\) −1.81796e6 −0.984743
\(322\) −743232. −0.399471
\(323\) −2.48566e6 −1.32567
\(324\) 104976. 0.0555556
\(325\) −35321.0 −0.0185492
\(326\) −72496.0 −0.0377807
\(327\) 292590. 0.151318
\(328\) −947328. −0.486201
\(329\) −928452. −0.472900
\(330\) 373248. 0.188674
\(331\) −967612. −0.485435 −0.242718 0.970097i \(-0.578039\pi\)
−0.242718 + 0.970097i \(0.578039\pi\)
\(332\) −276864. −0.137855
\(333\) 638766. 0.315668
\(334\) 1.04520e6 0.512664
\(335\) 3.62124e6 1.76297
\(336\) 112896. 0.0545545
\(337\) −763966. −0.366437 −0.183218 0.983072i \(-0.558652\pi\)
−0.183218 + 0.983072i \(0.558652\pi\)
\(338\) 114244. 0.0543928
\(339\) −332262. −0.157030
\(340\) 1.22861e6 0.576389
\(341\) 109056. 0.0507883
\(342\) 566352. 0.261831
\(343\) 117649. 0.0539949
\(344\) 317696. 0.144749
\(345\) 1.84291e6 0.833598
\(346\) 2.20284e6 0.989220
\(347\) −3.32893e6 −1.48416 −0.742081 0.670310i \(-0.766161\pi\)
−0.742081 + 0.670310i \(0.766161\pi\)
\(348\) −137376. −0.0608083
\(349\) 2.02761e6 0.891087 0.445543 0.895260i \(-0.353011\pi\)
0.445543 + 0.895260i \(0.353011\pi\)
\(350\) −40964.0 −0.0178744
\(351\) 123201. 0.0533761
\(352\) −196608. −0.0845755
\(353\) 2.94337e6 1.25721 0.628605 0.777725i \(-0.283626\pi\)
0.628605 + 0.777725i \(0.283626\pi\)
\(354\) 1.25539e6 0.532441
\(355\) 1.53511e6 0.646501
\(356\) −1.49904e6 −0.626886
\(357\) −627102. −0.260416
\(358\) −1.05778e6 −0.436200
\(359\) 4.25456e6 1.74228 0.871142 0.491031i \(-0.163380\pi\)
0.871142 + 0.491031i \(0.163380\pi\)
\(360\) −279936. −0.113842
\(361\) 579405. 0.233999
\(362\) −127720. −0.0512257
\(363\) −1.11768e6 −0.445197
\(364\) 132496. 0.0524142
\(365\) 1.23628e6 0.485717
\(366\) −922248. −0.359869
\(367\) −3.30786e6 −1.28198 −0.640992 0.767548i \(-0.721477\pi\)
−0.640992 + 0.767548i \(0.721477\pi\)
\(368\) −970752. −0.373670
\(369\) −1.19896e6 −0.458395
\(370\) −1.70338e6 −0.646854
\(371\) −20874.0 −0.00787356
\(372\) −81792.0 −0.0306446
\(373\) 2.86357e6 1.06570 0.532852 0.846209i \(-0.321120\pi\)
0.532852 + 0.846209i \(0.321120\pi\)
\(374\) 1.09210e6 0.403721
\(375\) 1.62032e6 0.595009
\(376\) −1.21267e6 −0.442358
\(377\) −161226. −0.0584227
\(378\) 142884. 0.0514344
\(379\) 1.41360e6 0.505507 0.252754 0.967531i \(-0.418664\pi\)
0.252754 + 0.967531i \(0.418664\pi\)
\(380\) −1.51027e6 −0.536533
\(381\) −1.52813e6 −0.539321
\(382\) −840960. −0.294861
\(383\) −4.39148e6 −1.52973 −0.764864 0.644191i \(-0.777194\pi\)
−0.764864 + 0.644191i \(0.777194\pi\)
\(384\) 147456. 0.0510310
\(385\) 508032. 0.174678
\(386\) −1.95225e6 −0.666909
\(387\) 402084. 0.136471
\(388\) 269216. 0.0907866
\(389\) −1.33886e6 −0.448601 −0.224301 0.974520i \(-0.572010\pi\)
−0.224301 + 0.974520i \(0.572010\pi\)
\(390\) −328536. −0.109376
\(391\) 5.39222e6 1.78372
\(392\) 153664. 0.0505076
\(393\) 1.59894e6 0.522217
\(394\) −662232. −0.214916
\(395\) 76032.0 0.0245191
\(396\) −248832. −0.0797385
\(397\) 3.38994e6 1.07948 0.539742 0.841831i \(-0.318522\pi\)
0.539742 + 0.841831i \(0.318522\pi\)
\(398\) −3.20061e6 −1.01280
\(399\) 770868. 0.242408
\(400\) −53504.0 −0.0167200
\(401\) −2.84695e6 −0.884134 −0.442067 0.896982i \(-0.645755\pi\)
−0.442067 + 0.896982i \(0.645755\pi\)
\(402\) −2.41416e6 −0.745077
\(403\) −95992.0 −0.0294423
\(404\) −2.06544e6 −0.629592
\(405\) −354294. −0.107331
\(406\) −186984. −0.0562975
\(407\) −1.51411e6 −0.453077
\(408\) −819072. −0.243597
\(409\) 811970. 0.240011 0.120006 0.992773i \(-0.461709\pi\)
0.120006 + 0.992773i \(0.461709\pi\)
\(410\) 3.19723e6 0.939322
\(411\) 1.27424e6 0.372088
\(412\) −88576.0 −0.0257083
\(413\) 1.70873e6 0.492944
\(414\) −1.22861e6 −0.352300
\(415\) 934416. 0.266330
\(416\) 173056. 0.0490290
\(417\) 2.54120e6 0.715648
\(418\) −1.34246e6 −0.375804
\(419\) 4.41160e6 1.22761 0.613805 0.789457i \(-0.289638\pi\)
0.613805 + 0.789457i \(0.289638\pi\)
\(420\) −381024. −0.105397
\(421\) 5.62170e6 1.54583 0.772917 0.634508i \(-0.218797\pi\)
0.772917 + 0.634508i \(0.218797\pi\)
\(422\) −1.52824e6 −0.417744
\(423\) −1.53479e6 −0.417059
\(424\) −27264.0 −0.00736504
\(425\) 297198. 0.0798130
\(426\) −1.02341e6 −0.273228
\(427\) −1.25528e6 −0.333174
\(428\) −3.23194e6 −0.852812
\(429\) −292032. −0.0766103
\(430\) −1.07222e6 −0.279650
\(431\) 3.35442e6 0.869810 0.434905 0.900476i \(-0.356782\pi\)
0.434905 + 0.900476i \(0.356782\pi\)
\(432\) 186624. 0.0481125
\(433\) −691486. −0.177241 −0.0886204 0.996065i \(-0.528246\pi\)
−0.0886204 + 0.996065i \(0.528246\pi\)
\(434\) −111328. −0.0283714
\(435\) 463644. 0.117479
\(436\) 520160. 0.131045
\(437\) −6.62842e6 −1.66037
\(438\) −824184. −0.205276
\(439\) 7.77128e6 1.92456 0.962280 0.272063i \(-0.0877058\pi\)
0.962280 + 0.272063i \(0.0877058\pi\)
\(440\) 663552. 0.163397
\(441\) 194481. 0.0476190
\(442\) −961272. −0.234040
\(443\) −1.39940e6 −0.338793 −0.169396 0.985548i \(-0.554182\pi\)
−0.169396 + 0.985548i \(0.554182\pi\)
\(444\) 1.13558e6 0.273377
\(445\) 5.05926e6 1.21112
\(446\) −1.21206e6 −0.288528
\(447\) 1.83314e6 0.433937
\(448\) 200704. 0.0472456
\(449\) 246846. 0.0577844 0.0288922 0.999583i \(-0.490802\pi\)
0.0288922 + 0.999583i \(0.490802\pi\)
\(450\) −67716.0 −0.0157638
\(451\) 2.84198e6 0.657931
\(452\) −590688. −0.135992
\(453\) 1.02931e6 0.235668
\(454\) 1.53350e6 0.349177
\(455\) −447174. −0.101262
\(456\) 1.00685e6 0.226752
\(457\) −2.55912e6 −0.573192 −0.286596 0.958052i \(-0.592524\pi\)
−0.286596 + 0.958052i \(0.592524\pi\)
\(458\) 904280. 0.201437
\(459\) −1.03664e6 −0.229665
\(460\) 3.27629e6 0.721917
\(461\) −8.34037e6 −1.82782 −0.913909 0.405920i \(-0.866951\pi\)
−0.913909 + 0.405920i \(0.866951\pi\)
\(462\) −338688. −0.0738235
\(463\) −3.17066e6 −0.687380 −0.343690 0.939083i \(-0.611677\pi\)
−0.343690 + 0.939083i \(0.611677\pi\)
\(464\) −244224. −0.0526615
\(465\) 276048. 0.0592042
\(466\) 4.65550e6 0.993119
\(467\) 7.53467e6 1.59872 0.799359 0.600854i \(-0.205173\pi\)
0.799359 + 0.600854i \(0.205173\pi\)
\(468\) 219024. 0.0462250
\(469\) −3.28594e6 −0.689807
\(470\) 4.09277e6 0.854619
\(471\) 438606. 0.0911008
\(472\) 2.23181e6 0.461107
\(473\) −953088. −0.195876
\(474\) −50688.0 −0.0103624
\(475\) −365332. −0.0742940
\(476\) −1.11485e6 −0.225527
\(477\) −34506.0 −0.00694382
\(478\) −2.64533e6 −0.529553
\(479\) −3.18068e6 −0.633405 −0.316703 0.948525i \(-0.602576\pi\)
−0.316703 + 0.948525i \(0.602576\pi\)
\(480\) −497664. −0.0985901
\(481\) 1.33273e6 0.262652
\(482\) −2.65142e6 −0.519829
\(483\) −1.67227e6 −0.326166
\(484\) −1.98699e6 −0.385552
\(485\) −908604. −0.175396
\(486\) 236196. 0.0453609
\(487\) 2.09245e6 0.399790 0.199895 0.979817i \(-0.435940\pi\)
0.199895 + 0.979817i \(0.435940\pi\)
\(488\) −1.63955e6 −0.311656
\(489\) −163116. −0.0308478
\(490\) −518616. −0.0975789
\(491\) 1.21314e6 0.227095 0.113547 0.993533i \(-0.463779\pi\)
0.113547 + 0.993533i \(0.463779\pi\)
\(492\) −2.13149e6 −0.396981
\(493\) 1.35659e6 0.251380
\(494\) 1.18165e6 0.217857
\(495\) 839808. 0.154052
\(496\) −145408. −0.0265390
\(497\) −1.39297e6 −0.252960
\(498\) −622944. −0.112558
\(499\) −63316.0 −0.0113831 −0.00569157 0.999984i \(-0.501812\pi\)
−0.00569157 + 0.999984i \(0.501812\pi\)
\(500\) 2.88058e6 0.515293
\(501\) 2.35170e6 0.418589
\(502\) −884400. −0.156635
\(503\) 1.06032e6 0.186860 0.0934301 0.995626i \(-0.470217\pi\)
0.0934301 + 0.995626i \(0.470217\pi\)
\(504\) 254016. 0.0445435
\(505\) 6.97086e6 1.21635
\(506\) 2.91226e6 0.505654
\(507\) 257049. 0.0444116
\(508\) −2.71667e6 −0.467066
\(509\) −3.46136e6 −0.592178 −0.296089 0.955160i \(-0.595682\pi\)
−0.296089 + 0.955160i \(0.595682\pi\)
\(510\) 2.76437e6 0.470620
\(511\) −1.12181e6 −0.190049
\(512\) 262144. 0.0441942
\(513\) 1.27429e6 0.213784
\(514\) 4.51889e6 0.754438
\(515\) 298944. 0.0496674
\(516\) 714816. 0.118187
\(517\) 3.63802e6 0.598602
\(518\) 1.54566e6 0.253098
\(519\) 4.95639e6 0.807694
\(520\) −584064. −0.0947223
\(521\) 1.05956e7 1.71014 0.855072 0.518509i \(-0.173513\pi\)
0.855072 + 0.518509i \(0.173513\pi\)
\(522\) −309096. −0.0496498
\(523\) 2.49885e6 0.399472 0.199736 0.979850i \(-0.435992\pi\)
0.199736 + 0.979850i \(0.435992\pi\)
\(524\) 2.84256e6 0.452253
\(525\) −92169.0 −0.0145944
\(526\) 4.06176e6 0.640103
\(527\) 807696. 0.126684
\(528\) −442368. −0.0690556
\(529\) 7.94292e6 1.23407
\(530\) 92016.0 0.0142290
\(531\) 2.82463e6 0.434736
\(532\) 1.37043e6 0.209932
\(533\) −2.50154e6 −0.381407
\(534\) −3.37284e6 −0.511850
\(535\) 1.09078e7 1.64760
\(536\) −4.29184e6 −0.645255
\(537\) −2.38000e6 −0.356156
\(538\) 5.57129e6 0.829851
\(539\) −460992. −0.0683473
\(540\) −629856. −0.0929516
\(541\) −9.36147e6 −1.37515 −0.687577 0.726112i \(-0.741325\pi\)
−0.687577 + 0.726112i \(0.741325\pi\)
\(542\) −5.94957e6 −0.869936
\(543\) −287370. −0.0418256
\(544\) −1.45613e6 −0.210961
\(545\) −1.75554e6 −0.253174
\(546\) 298116. 0.0427960
\(547\) 1.00395e7 1.43465 0.717324 0.696740i \(-0.245367\pi\)
0.717324 + 0.696740i \(0.245367\pi\)
\(548\) 2.26531e6 0.322238
\(549\) −2.07506e6 −0.293832
\(550\) 160512. 0.0226256
\(551\) −1.66759e6 −0.233997
\(552\) −2.18419e6 −0.305101
\(553\) −68992.0 −0.00959369
\(554\) 783128. 0.108407
\(555\) −3.83260e6 −0.528154
\(556\) 4.51770e6 0.619769
\(557\) 3.85718e6 0.526783 0.263391 0.964689i \(-0.415159\pi\)
0.263391 + 0.964689i \(0.415159\pi\)
\(558\) −184032. −0.0250212
\(559\) 838916. 0.113550
\(560\) −677376. −0.0912767
\(561\) 2.45722e6 0.329637
\(562\) −6.01505e6 −0.803338
\(563\) 1.62056e6 0.215474 0.107737 0.994179i \(-0.465640\pi\)
0.107737 + 0.994179i \(0.465640\pi\)
\(564\) −2.72851e6 −0.361184
\(565\) 1.99357e6 0.262731
\(566\) 1.46264e6 0.191909
\(567\) 321489. 0.0419961
\(568\) −1.81939e6 −0.236622
\(569\) −4.09976e6 −0.530857 −0.265429 0.964131i \(-0.585513\pi\)
−0.265429 + 0.964131i \(0.585513\pi\)
\(570\) −3.39811e6 −0.438077
\(571\) −826660. −0.106105 −0.0530526 0.998592i \(-0.516895\pi\)
−0.0530526 + 0.998592i \(0.516895\pi\)
\(572\) −519168. −0.0663465
\(573\) −1.89216e6 −0.240753
\(574\) −2.90119e6 −0.367533
\(575\) 792528. 0.0999643
\(576\) 331776. 0.0416667
\(577\) −9.54662e6 −1.19374 −0.596871 0.802338i \(-0.703589\pi\)
−0.596871 + 0.802338i \(0.703589\pi\)
\(578\) 2.40891e6 0.299917
\(579\) −4.39256e6 −0.544529
\(580\) 824256. 0.101740
\(581\) −847896. −0.104208
\(582\) 605736. 0.0741269
\(583\) 81792.0 0.00996643
\(584\) −1.46522e6 −0.177775
\(585\) −739206. −0.0893050
\(586\) 7.51745e6 0.904329
\(587\) 8.65829e6 1.03714 0.518569 0.855036i \(-0.326465\pi\)
0.518569 + 0.855036i \(0.326465\pi\)
\(588\) 345744. 0.0412393
\(589\) −992864. −0.117924
\(590\) −7.53235e6 −0.890842
\(591\) −1.49002e6 −0.175479
\(592\) 2.01882e6 0.236751
\(593\) −1.07810e7 −1.25899 −0.629493 0.777006i \(-0.716737\pi\)
−0.629493 + 0.777006i \(0.716737\pi\)
\(594\) −559872. −0.0651062
\(595\) 3.76261e6 0.435709
\(596\) 3.25891e6 0.375800
\(597\) −7.20137e6 −0.826950
\(598\) −2.56339e6 −0.293131
\(599\) −7.08360e6 −0.806653 −0.403327 0.915056i \(-0.632146\pi\)
−0.403327 + 0.915056i \(0.632146\pi\)
\(600\) −120384. −0.0136518
\(601\) 1.59307e7 1.79907 0.899536 0.436846i \(-0.143905\pi\)
0.899536 + 0.436846i \(0.143905\pi\)
\(602\) 972944. 0.109420
\(603\) −5.43186e6 −0.608352
\(604\) 1.82989e6 0.204095
\(605\) 6.70610e6 0.744872
\(606\) −4.64724e6 −0.514060
\(607\) 5.97212e6 0.657895 0.328948 0.944348i \(-0.393306\pi\)
0.328948 + 0.944348i \(0.393306\pi\)
\(608\) 1.78995e6 0.196373
\(609\) −420714. −0.0459667
\(610\) 5.53349e6 0.602108
\(611\) −3.20221e6 −0.347014
\(612\) −1.84291e6 −0.198896
\(613\) −877426. −0.0943103 −0.0471552 0.998888i \(-0.515016\pi\)
−0.0471552 + 0.998888i \(0.515016\pi\)
\(614\) 4.74949e6 0.508424
\(615\) 7.19377e6 0.766953
\(616\) −602112. −0.0639331
\(617\) −9.60568e6 −1.01582 −0.507908 0.861411i \(-0.669581\pi\)
−0.507908 + 0.861411i \(0.669581\pi\)
\(618\) −199296. −0.0209907
\(619\) 818276. 0.0858367 0.0429184 0.999079i \(-0.486334\pi\)
0.0429184 + 0.999079i \(0.486334\pi\)
\(620\) 490752. 0.0512723
\(621\) −2.76437e6 −0.287652
\(622\) −2.15923e6 −0.223781
\(623\) −4.59081e6 −0.473881
\(624\) 389376. 0.0400320
\(625\) −9.06882e6 −0.928647
\(626\) 1.32214e7 1.34847
\(627\) −3.02054e6 −0.306843
\(628\) 779744. 0.0788956
\(629\) −1.12139e7 −1.13013
\(630\) −857304. −0.0860565
\(631\) −2.73947e6 −0.273901 −0.136950 0.990578i \(-0.543730\pi\)
−0.136950 + 0.990578i \(0.543730\pi\)
\(632\) −90112.0 −0.00897408
\(633\) −3.43854e6 −0.341087
\(634\) 7.70561e6 0.761349
\(635\) 9.16877e6 0.902353
\(636\) −61344.0 −0.00601353
\(637\) 405769. 0.0396214
\(638\) 732672. 0.0712620
\(639\) −2.30267e6 −0.223090
\(640\) −884736. −0.0853815
\(641\) −1.39383e7 −1.33987 −0.669937 0.742418i \(-0.733679\pi\)
−0.669937 + 0.742418i \(0.733679\pi\)
\(642\) −7.27186e6 −0.696318
\(643\) −3.92770e6 −0.374637 −0.187319 0.982299i \(-0.559980\pi\)
−0.187319 + 0.982299i \(0.559980\pi\)
\(644\) −2.97293e6 −0.282468
\(645\) −2.41250e6 −0.228333
\(646\) −9.94262e6 −0.937389
\(647\) −1.85072e7 −1.73812 −0.869060 0.494706i \(-0.835276\pi\)
−0.869060 + 0.494706i \(0.835276\pi\)
\(648\) 419904. 0.0392837
\(649\) −6.69542e6 −0.623974
\(650\) −141284. −0.0131162
\(651\) −250488. −0.0231651
\(652\) −289984. −0.0267150
\(653\) −5.05771e6 −0.464163 −0.232082 0.972696i \(-0.574554\pi\)
−0.232082 + 0.972696i \(0.574554\pi\)
\(654\) 1.17036e6 0.106998
\(655\) −9.59364e6 −0.873736
\(656\) −3.78931e6 −0.343796
\(657\) −1.85441e6 −0.167607
\(658\) −3.71381e6 −0.334391
\(659\) −1.03811e7 −0.931173 −0.465587 0.885002i \(-0.654157\pi\)
−0.465587 + 0.885002i \(0.654157\pi\)
\(660\) 1.49299e6 0.133413
\(661\) −1.40680e7 −1.25236 −0.626180 0.779678i \(-0.715383\pi\)
−0.626180 + 0.779678i \(0.715383\pi\)
\(662\) −3.87045e6 −0.343255
\(663\) −2.16286e6 −0.191093
\(664\) −1.10746e6 −0.0974779
\(665\) −4.62521e6 −0.405581
\(666\) 2.55506e6 0.223211
\(667\) 3.61757e6 0.314849
\(668\) 4.18080e6 0.362509
\(669\) −2.72714e6 −0.235582
\(670\) 1.44850e7 1.24661
\(671\) 4.91866e6 0.421735
\(672\) 451584. 0.0385758
\(673\) 1.92279e7 1.63642 0.818211 0.574919i \(-0.194966\pi\)
0.818211 + 0.574919i \(0.194966\pi\)
\(674\) −3.05586e6 −0.259110
\(675\) −152361. −0.0128711
\(676\) 456976. 0.0384615
\(677\) −9.59831e6 −0.804865 −0.402433 0.915450i \(-0.631835\pi\)
−0.402433 + 0.915450i \(0.631835\pi\)
\(678\) −1.32905e6 −0.111037
\(679\) 824474. 0.0686282
\(680\) 4.91443e6 0.407569
\(681\) 3.45038e6 0.285102
\(682\) 436224. 0.0359127
\(683\) −1.88472e7 −1.54594 −0.772972 0.634440i \(-0.781231\pi\)
−0.772972 + 0.634440i \(0.781231\pi\)
\(684\) 2.26541e6 0.185143
\(685\) −7.64543e6 −0.622552
\(686\) 470596. 0.0381802
\(687\) 2.03463e6 0.164473
\(688\) 1.27078e6 0.102353
\(689\) −71994.0 −0.00577761
\(690\) 7.37165e6 0.589443
\(691\) −1.96469e7 −1.56531 −0.782654 0.622457i \(-0.786134\pi\)
−0.782654 + 0.622457i \(0.786134\pi\)
\(692\) 8.81136e6 0.699484
\(693\) −762048. −0.0602767
\(694\) −1.33157e7 −1.04946
\(695\) −1.52472e7 −1.19737
\(696\) −549504. −0.0429980
\(697\) 2.10484e7 1.64111
\(698\) 8.11042e6 0.630094
\(699\) 1.04749e7 0.810878
\(700\) −163856. −0.0126391
\(701\) −4.95664e6 −0.380972 −0.190486 0.981690i \(-0.561006\pi\)
−0.190486 + 0.981690i \(0.561006\pi\)
\(702\) 492804. 0.0377426
\(703\) 1.37847e7 1.05199
\(704\) −786432. −0.0598039
\(705\) 9.20873e6 0.697793
\(706\) 1.17735e7 0.888981
\(707\) −6.32541e6 −0.475927
\(708\) 5.02157e6 0.376492
\(709\) −8.40260e6 −0.627767 −0.313883 0.949462i \(-0.601630\pi\)
−0.313883 + 0.949462i \(0.601630\pi\)
\(710\) 6.14045e6 0.457145
\(711\) −114048. −0.00846084
\(712\) −5.99616e6 −0.443275
\(713\) 2.15386e6 0.158669
\(714\) −2.50841e6 −0.184142
\(715\) 1.75219e6 0.128179
\(716\) −4.23110e6 −0.308440
\(717\) −5.95199e6 −0.432379
\(718\) 1.70183e7 1.23198
\(719\) 7.19556e6 0.519090 0.259545 0.965731i \(-0.416427\pi\)
0.259545 + 0.965731i \(0.416427\pi\)
\(720\) −1.11974e6 −0.0804984
\(721\) −271264. −0.0194336
\(722\) 2.31762e6 0.165462
\(723\) −5.96569e6 −0.424438
\(724\) −510880. −0.0362220
\(725\) 199386. 0.0140880
\(726\) −4.47073e6 −0.314802
\(727\) 4.22593e6 0.296542 0.148271 0.988947i \(-0.452629\pi\)
0.148271 + 0.988947i \(0.452629\pi\)
\(728\) 529984. 0.0370625
\(729\) 531441. 0.0370370
\(730\) 4.94510e6 0.343454
\(731\) −7.05881e6 −0.488583
\(732\) −3.68899e6 −0.254466
\(733\) −7.64403e6 −0.525488 −0.262744 0.964866i \(-0.584627\pi\)
−0.262744 + 0.964866i \(0.584627\pi\)
\(734\) −1.32315e7 −0.906499
\(735\) −1.16689e6 −0.0796728
\(736\) −3.88301e6 −0.264225
\(737\) 1.28755e7 0.873164
\(738\) −4.79585e6 −0.324134
\(739\) −2.62718e7 −1.76962 −0.884809 0.465954i \(-0.845711\pi\)
−0.884809 + 0.465954i \(0.845711\pi\)
\(740\) −6.81350e6 −0.457395
\(741\) 2.65871e6 0.177879
\(742\) −83496.0 −0.00556744
\(743\) −2.46314e7 −1.63688 −0.818441 0.574591i \(-0.805161\pi\)
−0.818441 + 0.574591i \(0.805161\pi\)
\(744\) −327168. −0.0216690
\(745\) −1.09988e7 −0.726032
\(746\) 1.14543e7 0.753566
\(747\) −1.40162e6 −0.0919031
\(748\) 4.36838e6 0.285474
\(749\) −9.89780e6 −0.644665
\(750\) 6.48130e6 0.420735
\(751\) 1.96305e7 1.27008 0.635041 0.772478i \(-0.280983\pi\)
0.635041 + 0.772478i \(0.280983\pi\)
\(752\) −4.85069e6 −0.312794
\(753\) −1.98990e6 −0.127892
\(754\) −644904. −0.0413111
\(755\) −6.17587e6 −0.394304
\(756\) 571536. 0.0363696
\(757\) −8.87151e6 −0.562676 −0.281338 0.959609i \(-0.590778\pi\)
−0.281338 + 0.959609i \(0.590778\pi\)
\(758\) 5.65438e6 0.357448
\(759\) 6.55258e6 0.412865
\(760\) −6.04109e6 −0.379386
\(761\) 1.97983e7 1.23927 0.619636 0.784890i \(-0.287280\pi\)
0.619636 + 0.784890i \(0.287280\pi\)
\(762\) −6.11251e6 −0.381358
\(763\) 1.59299e6 0.0990608
\(764\) −3.36384e6 −0.208498
\(765\) 6.21983e6 0.384260
\(766\) −1.75659e7 −1.08168
\(767\) 5.89337e6 0.361722
\(768\) 589824. 0.0360844
\(769\) −8.14829e6 −0.496879 −0.248439 0.968647i \(-0.579918\pi\)
−0.248439 + 0.968647i \(0.579918\pi\)
\(770\) 2.03213e6 0.123516
\(771\) 1.01675e7 0.615996
\(772\) −7.80899e6 −0.471576
\(773\) −7.69145e6 −0.462977 −0.231489 0.972838i \(-0.574360\pi\)
−0.231489 + 0.972838i \(0.574360\pi\)
\(774\) 1.60834e6 0.0964993
\(775\) 118712. 0.00709971
\(776\) 1.07686e6 0.0641958
\(777\) 3.47773e6 0.206654
\(778\) −5.35543e6 −0.317209
\(779\) −2.58739e7 −1.52763
\(780\) −1.31414e6 −0.0773404
\(781\) 5.45818e6 0.320199
\(782\) 2.15689e7 1.26128
\(783\) −695466. −0.0405389
\(784\) 614656. 0.0357143
\(785\) −2.63164e6 −0.152423
\(786\) 6.39576e6 0.369263
\(787\) −2.49837e7 −1.43787 −0.718935 0.695078i \(-0.755370\pi\)
−0.718935 + 0.695078i \(0.755370\pi\)
\(788\) −2.64893e6 −0.151969
\(789\) 9.13896e6 0.522642
\(790\) 304128. 0.0173376
\(791\) −1.80898e6 −0.102800
\(792\) −995328. −0.0563837
\(793\) −4.32944e6 −0.244483
\(794\) 1.35598e7 0.763310
\(795\) 207036. 0.0116179
\(796\) −1.28024e7 −0.716160
\(797\) 6.19328e6 0.345362 0.172681 0.984978i \(-0.444757\pi\)
0.172681 + 0.984978i \(0.444757\pi\)
\(798\) 3.08347e6 0.171409
\(799\) 2.69441e7 1.49312
\(800\) −214016. −0.0118228
\(801\) −7.58889e6 −0.417924
\(802\) −1.13878e7 −0.625177
\(803\) 4.39565e6 0.240566
\(804\) −9.65664e6 −0.526849
\(805\) 1.00336e7 0.545718
\(806\) −383968. −0.0208189
\(807\) 1.25354e7 0.677570
\(808\) −8.26176e6 −0.445189
\(809\) −5.09077e6 −0.273471 −0.136736 0.990608i \(-0.543661\pi\)
−0.136736 + 0.990608i \(0.543661\pi\)
\(810\) −1.41718e6 −0.0758947
\(811\) 2.11099e6 0.112703 0.0563513 0.998411i \(-0.482053\pi\)
0.0563513 + 0.998411i \(0.482053\pi\)
\(812\) −747936. −0.0398084
\(813\) −1.33865e7 −0.710300
\(814\) −6.05645e6 −0.320374
\(815\) 978696. 0.0516124
\(816\) −3.27629e6 −0.172249
\(817\) 8.67707e6 0.454797
\(818\) 3.24788e6 0.169714
\(819\) 670761. 0.0349428
\(820\) 1.27889e7 0.664201
\(821\) −2.92262e7 −1.51326 −0.756631 0.653842i \(-0.773156\pi\)
−0.756631 + 0.653842i \(0.773156\pi\)
\(822\) 5.09695e6 0.263106
\(823\) 1.08712e7 0.559472 0.279736 0.960077i \(-0.409753\pi\)
0.279736 + 0.960077i \(0.409753\pi\)
\(824\) −354304. −0.0181785
\(825\) 361152. 0.0184738
\(826\) 6.83491e6 0.348564
\(827\) 7.79011e6 0.396077 0.198039 0.980194i \(-0.436543\pi\)
0.198039 + 0.980194i \(0.436543\pi\)
\(828\) −4.91443e6 −0.249114
\(829\) −1.49523e7 −0.755650 −0.377825 0.925877i \(-0.623328\pi\)
−0.377825 + 0.925877i \(0.623328\pi\)
\(830\) 3.73766e6 0.188324
\(831\) 1.76204e6 0.0885142
\(832\) 692224. 0.0346688
\(833\) −3.41422e6 −0.170482
\(834\) 1.01648e7 0.506039
\(835\) −1.41102e7 −0.700353
\(836\) −5.36986e6 −0.265734
\(837\) −414072. −0.0204297
\(838\) 1.76464e7 0.868052
\(839\) 2.94083e7 1.44233 0.721166 0.692762i \(-0.243606\pi\)
0.721166 + 0.692762i \(0.243606\pi\)
\(840\) −1.52410e6 −0.0745271
\(841\) −1.96010e7 −0.955628
\(842\) 2.24868e7 1.09307
\(843\) −1.35339e7 −0.655923
\(844\) −6.11296e6 −0.295390
\(845\) −1.54229e6 −0.0743063
\(846\) −6.13915e6 −0.294905
\(847\) −6.08516e6 −0.291450
\(848\) −109056. −0.00520787
\(849\) 3.29094e6 0.156693
\(850\) 1.18879e6 0.0564363
\(851\) −2.99037e7 −1.41547
\(852\) −4.09363e6 −0.193201
\(853\) 2.59454e7 1.22092 0.610460 0.792047i \(-0.290985\pi\)
0.610460 + 0.792047i \(0.290985\pi\)
\(854\) −5.02113e6 −0.235590
\(855\) −7.64575e6 −0.357688
\(856\) −1.29277e7 −0.603029
\(857\) −2.71981e7 −1.26499 −0.632495 0.774565i \(-0.717969\pi\)
−0.632495 + 0.774565i \(0.717969\pi\)
\(858\) −1.16813e6 −0.0541717
\(859\) 1.55649e7 0.719718 0.359859 0.933007i \(-0.382825\pi\)
0.359859 + 0.933007i \(0.382825\pi\)
\(860\) −4.28890e6 −0.197742
\(861\) −6.52768e6 −0.300090
\(862\) 1.34177e7 0.615048
\(863\) 3.61264e7 1.65119 0.825596 0.564262i \(-0.190839\pi\)
0.825596 + 0.564262i \(0.190839\pi\)
\(864\) 746496. 0.0340207
\(865\) −2.97383e7 −1.35138
\(866\) −2.76594e6 −0.125328
\(867\) 5.42004e6 0.244881
\(868\) −445312. −0.0200616
\(869\) 270336. 0.0121438
\(870\) 1.85458e6 0.0830704
\(871\) −1.13331e7 −0.506180
\(872\) 2.08064e6 0.0926629
\(873\) 1.36291e6 0.0605244
\(874\) −2.65137e7 −1.17406
\(875\) 8.82176e6 0.389525
\(876\) −3.29674e6 −0.145152
\(877\) −6.96990e6 −0.306004 −0.153002 0.988226i \(-0.548894\pi\)
−0.153002 + 0.988226i \(0.548894\pi\)
\(878\) 3.10851e7 1.36087
\(879\) 1.69143e7 0.738381
\(880\) 2.65421e6 0.115539
\(881\) −1.73080e7 −0.751291 −0.375645 0.926763i \(-0.622579\pi\)
−0.375645 + 0.926763i \(0.622579\pi\)
\(882\) 777924. 0.0336718
\(883\) −1.97666e7 −0.853158 −0.426579 0.904450i \(-0.640281\pi\)
−0.426579 + 0.904450i \(0.640281\pi\)
\(884\) −3.84509e6 −0.165491
\(885\) −1.69478e7 −0.727369
\(886\) −5.59762e6 −0.239562
\(887\) 500856. 0.0213749 0.0106874 0.999943i \(-0.496598\pi\)
0.0106874 + 0.999943i \(0.496598\pi\)
\(888\) 4.54234e6 0.193307
\(889\) −8.31981e6 −0.353068
\(890\) 2.02370e7 0.856391
\(891\) −1.25971e6 −0.0531590
\(892\) −4.84826e6 −0.204020
\(893\) −3.31211e7 −1.38988
\(894\) 7.33255e6 0.306840
\(895\) 1.42800e7 0.595895
\(896\) 802816. 0.0334077
\(897\) −5.76763e6 −0.239341
\(898\) 987384. 0.0408597
\(899\) 541872. 0.0223613
\(900\) −270864. −0.0111467
\(901\) 605772. 0.0248598
\(902\) 1.13679e7 0.465227
\(903\) 2.18912e6 0.0893410
\(904\) −2.36275e6 −0.0961606
\(905\) 1.72422e6 0.0699796
\(906\) 4.11725e6 0.166643
\(907\) 3.70905e7 1.49708 0.748539 0.663091i \(-0.230756\pi\)
0.748539 + 0.663091i \(0.230756\pi\)
\(908\) 6.13402e6 0.246905
\(909\) −1.04563e7 −0.419728
\(910\) −1.78870e6 −0.0716033
\(911\) 4.04155e7 1.61344 0.806719 0.590935i \(-0.201241\pi\)
0.806719 + 0.590935i \(0.201241\pi\)
\(912\) 4.02739e6 0.160338
\(913\) 3.32237e6 0.131908
\(914\) −1.02365e7 −0.405308
\(915\) 1.24503e7 0.491619
\(916\) 3.61712e6 0.142437
\(917\) 8.70534e6 0.341871
\(918\) −4.14655e6 −0.162398
\(919\) −1.25534e7 −0.490313 −0.245157 0.969483i \(-0.578839\pi\)
−0.245157 + 0.969483i \(0.578839\pi\)
\(920\) 1.31052e7 0.510473
\(921\) 1.06863e7 0.415126
\(922\) −3.33615e7 −1.29246
\(923\) −4.80433e6 −0.185622
\(924\) −1.35475e6 −0.0522011
\(925\) −1.64817e6 −0.0633357
\(926\) −1.26826e7 −0.486051
\(927\) −448416. −0.0171389
\(928\) −976896. −0.0372373
\(929\) −8.47147e6 −0.322047 −0.161024 0.986951i \(-0.551479\pi\)
−0.161024 + 0.986951i \(0.551479\pi\)
\(930\) 1.10419e6 0.0418637
\(931\) 4.19695e6 0.158694
\(932\) 1.86220e7 0.702241
\(933\) −4.85827e6 −0.182716
\(934\) 3.01387e7 1.13046
\(935\) −1.47433e7 −0.551525
\(936\) 876096. 0.0326860
\(937\) −2.13109e7 −0.792963 −0.396482 0.918043i \(-0.629769\pi\)
−0.396482 + 0.918043i \(0.629769\pi\)
\(938\) −1.31438e7 −0.487767
\(939\) 2.97482e7 1.10102
\(940\) 1.63711e7 0.604307
\(941\) −2.70629e7 −0.996325 −0.498162 0.867084i \(-0.665992\pi\)
−0.498162 + 0.867084i \(0.665992\pi\)
\(942\) 1.75442e6 0.0644180
\(943\) 5.61292e7 2.05546
\(944\) 8.92723e6 0.326052
\(945\) −1.92893e6 −0.0702648
\(946\) −3.81235e6 −0.138505
\(947\) −2.34466e7 −0.849582 −0.424791 0.905292i \(-0.639652\pi\)
−0.424791 + 0.905292i \(0.639652\pi\)
\(948\) −202752. −0.00732731
\(949\) −3.86909e6 −0.139458
\(950\) −1.46133e6 −0.0525338
\(951\) 1.73376e7 0.621639
\(952\) −4.45939e6 −0.159472
\(953\) −3.26567e7 −1.16477 −0.582385 0.812913i \(-0.697880\pi\)
−0.582385 + 0.812913i \(0.697880\pi\)
\(954\) −138024. −0.00491002
\(955\) 1.13530e7 0.402810
\(956\) −1.05813e7 −0.374451
\(957\) 1.64851e6 0.0581852
\(958\) −1.27227e7 −0.447885
\(959\) 6.93752e6 0.243589
\(960\) −1.99066e6 −0.0697137
\(961\) −2.83065e7 −0.988731
\(962\) 5.33094e6 0.185723
\(963\) −1.63617e7 −0.568541
\(964\) −1.06057e7 −0.367574
\(965\) 2.63553e7 0.911067
\(966\) −6.68909e6 −0.230634
\(967\) −4.74790e7 −1.63281 −0.816405 0.577480i \(-0.804036\pi\)
−0.816405 + 0.577480i \(0.804036\pi\)
\(968\) −7.94797e6 −0.272626
\(969\) −2.23709e7 −0.765375
\(970\) −3.63442e6 −0.124024
\(971\) 14748.0 0.000501978 0 0.000250989 1.00000i \(-0.499920\pi\)
0.000250989 1.00000i \(0.499920\pi\)
\(972\) 944784. 0.0320750
\(973\) 1.38354e7 0.468502
\(974\) 8.36979e6 0.282694
\(975\) −317889. −0.0107094
\(976\) −6.55821e6 −0.220374
\(977\) −4.16012e7 −1.39434 −0.697171 0.716905i \(-0.745558\pi\)
−0.697171 + 0.716905i \(0.745558\pi\)
\(978\) −652464. −0.0218127
\(979\) 1.79885e7 0.599843
\(980\) −2.07446e6 −0.0689987
\(981\) 2.63331e6 0.0873634
\(982\) 4.85256e6 0.160580
\(983\) 5.28488e7 1.74442 0.872211 0.489130i \(-0.162686\pi\)
0.872211 + 0.489130i \(0.162686\pi\)
\(984\) −8.52595e6 −0.280708
\(985\) 8.94013e6 0.293598
\(986\) 5.42635e6 0.177752
\(987\) −8.35607e6 −0.273029
\(988\) 4.72659e6 0.154048
\(989\) −1.88235e7 −0.611941
\(990\) 3.35923e6 0.108931
\(991\) 1.09734e6 0.0354940 0.0177470 0.999843i \(-0.494351\pi\)
0.0177470 + 0.999843i \(0.494351\pi\)
\(992\) −581632. −0.0187659
\(993\) −8.70851e6 −0.280266
\(994\) −5.57189e6 −0.178870
\(995\) 4.32082e7 1.38359
\(996\) −2.49178e6 −0.0795904
\(997\) 3.63025e7 1.15664 0.578320 0.815810i \(-0.303708\pi\)
0.578320 + 0.815810i \(0.303708\pi\)
\(998\) −253264. −0.00804910
\(999\) 5.74889e6 0.182251
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 546.6.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.6.a.f.1.1 1 1.1 even 1 trivial