Properties

Label 342.6.a.i.1.1
Level $342$
Weight $6$
Character 342.1
Self dual yes
Analytic conductor $54.851$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 342.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.8512663760\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1441}) \)
Defining polynomial: \(x^{2} - x - 360\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(19.4803\) of defining polynomial
Character \(\chi\) \(=\) 342.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} +16.0000 q^{4} -34.4408 q^{5} -18.9210 q^{7} +64.0000 q^{8} +O(q^{10})\) \(q+4.00000 q^{2} +16.0000 q^{4} -34.4408 q^{5} -18.9210 q^{7} +64.0000 q^{8} -137.763 q^{10} -349.480 q^{11} +711.599 q^{13} -75.6840 q^{14} +256.000 q^{16} -221.803 q^{17} +361.000 q^{19} -551.052 q^{20} -1397.92 q^{22} +662.468 q^{23} -1938.83 q^{25} +2846.39 q^{26} -302.736 q^{28} +7219.28 q^{29} +5407.76 q^{31} +1024.00 q^{32} -887.210 q^{34} +651.654 q^{35} +1979.40 q^{37} +1444.00 q^{38} -2204.21 q^{40} +3111.11 q^{41} +318.049 q^{43} -5591.68 q^{44} +2649.87 q^{46} +27240.5 q^{47} -16449.0 q^{49} -7755.34 q^{50} +11385.6 q^{52} +1114.63 q^{53} +12036.4 q^{55} -1210.94 q^{56} +28877.1 q^{58} +37904.9 q^{59} +37469.2 q^{61} +21631.0 q^{62} +4096.00 q^{64} -24508.0 q^{65} -54955.3 q^{67} -3548.84 q^{68} +2606.62 q^{70} +7177.04 q^{71} +64746.1 q^{73} +7917.59 q^{74} +5776.00 q^{76} +6612.52 q^{77} +36104.4 q^{79} -8816.83 q^{80} +12444.4 q^{82} +51782.7 q^{83} +7639.05 q^{85} +1272.20 q^{86} -22366.7 q^{88} -145254. q^{89} -13464.2 q^{91} +10599.5 q^{92} +108962. q^{94} -12433.1 q^{95} +39512.8 q^{97} -65796.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{2} + 32q^{4} + 45q^{5} + 114q^{7} + 128q^{8} + O(q^{10}) \) \( 2q + 8q^{2} + 32q^{4} + 45q^{5} + 114q^{7} + 128q^{8} + 180q^{10} - 661q^{11} + 1613q^{13} + 456q^{14} + 512q^{16} - 64q^{17} + 722q^{19} + 720q^{20} - 2644q^{22} + 3185q^{23} + 1247q^{25} + 6452q^{26} + 1824q^{28} + 2481q^{29} - 1180q^{31} + 2048q^{32} - 256q^{34} + 11211q^{35} + 10488q^{37} + 2888q^{38} + 2880q^{40} - 16630q^{41} + 11303q^{43} - 10576q^{44} + 12740q^{46} + 12155q^{47} - 15588q^{49} + 4988q^{50} + 25808q^{52} - 20585q^{53} - 12711q^{55} + 7296q^{56} + 9924q^{58} + 78581q^{59} + 43621q^{61} - 4720q^{62} + 8192q^{64} + 47100q^{65} + 7805q^{67} - 1024q^{68} + 44844q^{70} + 62488q^{71} + 16218q^{73} + 41952q^{74} + 11552q^{76} - 34795q^{77} + 67122q^{79} + 11520q^{80} - 66520q^{82} + 10714q^{83} + 20175q^{85} + 45212q^{86} - 42304q^{88} - 128188q^{89} + 106351q^{91} + 50960q^{92} + 48620q^{94} + 16245q^{95} + 178558q^{97} - 62352q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) −34.4408 −0.616095 −0.308048 0.951371i \(-0.599676\pi\)
−0.308048 + 0.951371i \(0.599676\pi\)
\(6\) 0 0
\(7\) −18.9210 −0.145948 −0.0729742 0.997334i \(-0.523249\pi\)
−0.0729742 + 0.997334i \(0.523249\pi\)
\(8\) 64.0000 0.353553
\(9\) 0 0
\(10\) −137.763 −0.435645
\(11\) −349.480 −0.870845 −0.435423 0.900226i \(-0.643401\pi\)
−0.435423 + 0.900226i \(0.643401\pi\)
\(12\) 0 0
\(13\) 711.599 1.16782 0.583911 0.811818i \(-0.301522\pi\)
0.583911 + 0.811818i \(0.301522\pi\)
\(14\) −75.6840 −0.103201
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −221.803 −0.186142 −0.0930710 0.995659i \(-0.529668\pi\)
−0.0930710 + 0.995659i \(0.529668\pi\)
\(18\) 0 0
\(19\) 361.000 0.229416
\(20\) −551.052 −0.308048
\(21\) 0 0
\(22\) −1397.92 −0.615780
\(23\) 662.468 0.261123 0.130561 0.991440i \(-0.458322\pi\)
0.130561 + 0.991440i \(0.458322\pi\)
\(24\) 0 0
\(25\) −1938.83 −0.620427
\(26\) 2846.39 0.825775
\(27\) 0 0
\(28\) −302.736 −0.0729742
\(29\) 7219.28 1.59404 0.797019 0.603954i \(-0.206409\pi\)
0.797019 + 0.603954i \(0.206409\pi\)
\(30\) 0 0
\(31\) 5407.76 1.01068 0.505339 0.862921i \(-0.331367\pi\)
0.505339 + 0.862921i \(0.331367\pi\)
\(32\) 1024.00 0.176777
\(33\) 0 0
\(34\) −887.210 −0.131622
\(35\) 651.654 0.0899181
\(36\) 0 0
\(37\) 1979.40 0.237700 0.118850 0.992912i \(-0.462079\pi\)
0.118850 + 0.992912i \(0.462079\pi\)
\(38\) 1444.00 0.162221
\(39\) 0 0
\(40\) −2204.21 −0.217822
\(41\) 3111.11 0.289039 0.144519 0.989502i \(-0.453836\pi\)
0.144519 + 0.989502i \(0.453836\pi\)
\(42\) 0 0
\(43\) 318.049 0.0262315 0.0131157 0.999914i \(-0.495825\pi\)
0.0131157 + 0.999914i \(0.495825\pi\)
\(44\) −5591.68 −0.435423
\(45\) 0 0
\(46\) 2649.87 0.184642
\(47\) 27240.5 1.79875 0.899374 0.437181i \(-0.144023\pi\)
0.899374 + 0.437181i \(0.144023\pi\)
\(48\) 0 0
\(49\) −16449.0 −0.978699
\(50\) −7755.34 −0.438708
\(51\) 0 0
\(52\) 11385.6 0.583911
\(53\) 1114.63 0.0545057 0.0272528 0.999629i \(-0.491324\pi\)
0.0272528 + 0.999629i \(0.491324\pi\)
\(54\) 0 0
\(55\) 12036.4 0.536523
\(56\) −1210.94 −0.0516005
\(57\) 0 0
\(58\) 28877.1 1.12716
\(59\) 37904.9 1.41764 0.708820 0.705390i \(-0.249228\pi\)
0.708820 + 0.705390i \(0.249228\pi\)
\(60\) 0 0
\(61\) 37469.2 1.28929 0.644644 0.764483i \(-0.277006\pi\)
0.644644 + 0.764483i \(0.277006\pi\)
\(62\) 21631.0 0.714658
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) −24508.0 −0.719490
\(66\) 0 0
\(67\) −54955.3 −1.49562 −0.747812 0.663911i \(-0.768895\pi\)
−0.747812 + 0.663911i \(0.768895\pi\)
\(68\) −3548.84 −0.0930710
\(69\) 0 0
\(70\) 2606.62 0.0635817
\(71\) 7177.04 0.168966 0.0844830 0.996425i \(-0.473076\pi\)
0.0844830 + 0.996425i \(0.473076\pi\)
\(72\) 0 0
\(73\) 64746.1 1.42202 0.711011 0.703181i \(-0.248238\pi\)
0.711011 + 0.703181i \(0.248238\pi\)
\(74\) 7917.59 0.168079
\(75\) 0 0
\(76\) 5776.00 0.114708
\(77\) 6612.52 0.127098
\(78\) 0 0
\(79\) 36104.4 0.650866 0.325433 0.945565i \(-0.394490\pi\)
0.325433 + 0.945565i \(0.394490\pi\)
\(80\) −8816.83 −0.154024
\(81\) 0 0
\(82\) 12444.4 0.204381
\(83\) 51782.7 0.825067 0.412534 0.910942i \(-0.364644\pi\)
0.412534 + 0.910942i \(0.364644\pi\)
\(84\) 0 0
\(85\) 7639.05 0.114681
\(86\) 1272.20 0.0185485
\(87\) 0 0
\(88\) −22366.7 −0.307890
\(89\) −145254. −1.94380 −0.971900 0.235392i \(-0.924363\pi\)
−0.971900 + 0.235392i \(0.924363\pi\)
\(90\) 0 0
\(91\) −13464.2 −0.170442
\(92\) 10599.5 0.130561
\(93\) 0 0
\(94\) 108962. 1.27191
\(95\) −12433.1 −0.141342
\(96\) 0 0
\(97\) 39512.8 0.426391 0.213196 0.977010i \(-0.431613\pi\)
0.213196 + 0.977010i \(0.431613\pi\)
\(98\) −65796.0 −0.692045
\(99\) 0 0
\(100\) −31021.3 −0.310213
\(101\) 16722.9 0.163120 0.0815602 0.996668i \(-0.474010\pi\)
0.0815602 + 0.996668i \(0.474010\pi\)
\(102\) 0 0
\(103\) 92664.4 0.860636 0.430318 0.902677i \(-0.358401\pi\)
0.430318 + 0.902677i \(0.358401\pi\)
\(104\) 45542.3 0.412888
\(105\) 0 0
\(106\) 4458.53 0.0385413
\(107\) 47156.3 0.398180 0.199090 0.979981i \(-0.436201\pi\)
0.199090 + 0.979981i \(0.436201\pi\)
\(108\) 0 0
\(109\) 52450.1 0.422844 0.211422 0.977395i \(-0.432191\pi\)
0.211422 + 0.977395i \(0.432191\pi\)
\(110\) 48145.5 0.379379
\(111\) 0 0
\(112\) −4843.78 −0.0364871
\(113\) −150596. −1.10947 −0.554737 0.832026i \(-0.687181\pi\)
−0.554737 + 0.832026i \(0.687181\pi\)
\(114\) 0 0
\(115\) −22815.9 −0.160877
\(116\) 115508. 0.797019
\(117\) 0 0
\(118\) 151620. 1.00242
\(119\) 4196.73 0.0271671
\(120\) 0 0
\(121\) −38914.6 −0.241629
\(122\) 149877. 0.911664
\(123\) 0 0
\(124\) 86524.2 0.505339
\(125\) 174402. 0.998337
\(126\) 0 0
\(127\) 352240. 1.93789 0.968945 0.247276i \(-0.0795353\pi\)
0.968945 + 0.247276i \(0.0795353\pi\)
\(128\) 16384.0 0.0883883
\(129\) 0 0
\(130\) −98032.0 −0.508756
\(131\) −19070.2 −0.0970907 −0.0485453 0.998821i \(-0.515459\pi\)
−0.0485453 + 0.998821i \(0.515459\pi\)
\(132\) 0 0
\(133\) −6830.49 −0.0334829
\(134\) −219821. −1.05757
\(135\) 0 0
\(136\) −14195.4 −0.0658111
\(137\) 266677. 1.21390 0.606952 0.794738i \(-0.292392\pi\)
0.606952 + 0.794738i \(0.292392\pi\)
\(138\) 0 0
\(139\) −294888. −1.29456 −0.647278 0.762254i \(-0.724093\pi\)
−0.647278 + 0.762254i \(0.724093\pi\)
\(140\) 10426.5 0.0449590
\(141\) 0 0
\(142\) 28708.2 0.119477
\(143\) −248690. −1.01699
\(144\) 0 0
\(145\) −248637. −0.982079
\(146\) 258984. 1.00552
\(147\) 0 0
\(148\) 31670.3 0.118850
\(149\) 103990. 0.383729 0.191864 0.981421i \(-0.438547\pi\)
0.191864 + 0.981421i \(0.438547\pi\)
\(150\) 0 0
\(151\) −477343. −1.70368 −0.851840 0.523802i \(-0.824513\pi\)
−0.851840 + 0.523802i \(0.824513\pi\)
\(152\) 23104.0 0.0811107
\(153\) 0 0
\(154\) 26450.1 0.0898722
\(155\) −186247. −0.622674
\(156\) 0 0
\(157\) 434955. 1.40830 0.704151 0.710051i \(-0.251328\pi\)
0.704151 + 0.710051i \(0.251328\pi\)
\(158\) 144417. 0.460232
\(159\) 0 0
\(160\) −35267.3 −0.108911
\(161\) −12534.6 −0.0381105
\(162\) 0 0
\(163\) −255083. −0.751990 −0.375995 0.926622i \(-0.622699\pi\)
−0.375995 + 0.926622i \(0.622699\pi\)
\(164\) 49777.8 0.144519
\(165\) 0 0
\(166\) 207131. 0.583411
\(167\) −510501. −1.41646 −0.708232 0.705980i \(-0.750507\pi\)
−0.708232 + 0.705980i \(0.750507\pi\)
\(168\) 0 0
\(169\) 135080. 0.363809
\(170\) 30556.2 0.0810918
\(171\) 0 0
\(172\) 5088.78 0.0131157
\(173\) −773453. −1.96480 −0.982401 0.186783i \(-0.940194\pi\)
−0.982401 + 0.186783i \(0.940194\pi\)
\(174\) 0 0
\(175\) 36684.7 0.0905503
\(176\) −89466.9 −0.217711
\(177\) 0 0
\(178\) −581014. −1.37447
\(179\) 477664. 1.11427 0.557135 0.830422i \(-0.311901\pi\)
0.557135 + 0.830422i \(0.311901\pi\)
\(180\) 0 0
\(181\) 729114. 1.65424 0.827121 0.562024i \(-0.189977\pi\)
0.827121 + 0.562024i \(0.189977\pi\)
\(182\) −53856.7 −0.120521
\(183\) 0 0
\(184\) 42397.9 0.0923209
\(185\) −68171.9 −0.146446
\(186\) 0 0
\(187\) 77515.6 0.162101
\(188\) 435848. 0.899374
\(189\) 0 0
\(190\) −49732.5 −0.0999438
\(191\) 285584. 0.566435 0.283218 0.959056i \(-0.408598\pi\)
0.283218 + 0.959056i \(0.408598\pi\)
\(192\) 0 0
\(193\) −53690.2 −0.103753 −0.0518766 0.998654i \(-0.516520\pi\)
−0.0518766 + 0.998654i \(0.516520\pi\)
\(194\) 158051. 0.301504
\(195\) 0 0
\(196\) −263184. −0.489350
\(197\) −14538.9 −0.0266910 −0.0133455 0.999911i \(-0.504248\pi\)
−0.0133455 + 0.999911i \(0.504248\pi\)
\(198\) 0 0
\(199\) −143700. −0.257232 −0.128616 0.991695i \(-0.541053\pi\)
−0.128616 + 0.991695i \(0.541053\pi\)
\(200\) −124085. −0.219354
\(201\) 0 0
\(202\) 66891.6 0.115343
\(203\) −136596. −0.232647
\(204\) 0 0
\(205\) −107149. −0.178075
\(206\) 370658. 0.608562
\(207\) 0 0
\(208\) 182169. 0.291956
\(209\) −126162. −0.199786
\(210\) 0 0
\(211\) 83653.6 0.129354 0.0646768 0.997906i \(-0.479398\pi\)
0.0646768 + 0.997906i \(0.479398\pi\)
\(212\) 17834.1 0.0272528
\(213\) 0 0
\(214\) 188625. 0.281556
\(215\) −10953.8 −0.0161611
\(216\) 0 0
\(217\) −102320. −0.147507
\(218\) 209800. 0.298996
\(219\) 0 0
\(220\) 192582. 0.268262
\(221\) −157834. −0.217381
\(222\) 0 0
\(223\) −665414. −0.896045 −0.448022 0.894022i \(-0.647872\pi\)
−0.448022 + 0.894022i \(0.647872\pi\)
\(224\) −19375.1 −0.0258003
\(225\) 0 0
\(226\) −602384. −0.784517
\(227\) −1.15382e6 −1.48619 −0.743096 0.669185i \(-0.766643\pi\)
−0.743096 + 0.669185i \(0.766643\pi\)
\(228\) 0 0
\(229\) −433691. −0.546502 −0.273251 0.961943i \(-0.588099\pi\)
−0.273251 + 0.961943i \(0.588099\pi\)
\(230\) −91263.5 −0.113757
\(231\) 0 0
\(232\) 462034. 0.563578
\(233\) −387173. −0.467214 −0.233607 0.972331i \(-0.575053\pi\)
−0.233607 + 0.972331i \(0.575053\pi\)
\(234\) 0 0
\(235\) −938183. −1.10820
\(236\) 606479. 0.708820
\(237\) 0 0
\(238\) 16786.9 0.0192100
\(239\) 622463. 0.704886 0.352443 0.935833i \(-0.385351\pi\)
0.352443 + 0.935833i \(0.385351\pi\)
\(240\) 0 0
\(241\) −371454. −0.411967 −0.205984 0.978555i \(-0.566039\pi\)
−0.205984 + 0.978555i \(0.566039\pi\)
\(242\) −155658. −0.170857
\(243\) 0 0
\(244\) 599507. 0.644644
\(245\) 566516. 0.602972
\(246\) 0 0
\(247\) 256887. 0.267917
\(248\) 346097. 0.357329
\(249\) 0 0
\(250\) 697609. 0.705931
\(251\) −376098. −0.376805 −0.188403 0.982092i \(-0.560331\pi\)
−0.188403 + 0.982092i \(0.560331\pi\)
\(252\) 0 0
\(253\) −231519. −0.227398
\(254\) 1.40896e6 1.37030
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) −1.64382e6 −1.55247 −0.776233 0.630446i \(-0.782872\pi\)
−0.776233 + 0.630446i \(0.782872\pi\)
\(258\) 0 0
\(259\) −37452.2 −0.0346919
\(260\) −392128. −0.359745
\(261\) 0 0
\(262\) −76280.9 −0.0686535
\(263\) −775299. −0.691162 −0.345581 0.938389i \(-0.612318\pi\)
−0.345581 + 0.938389i \(0.612318\pi\)
\(264\) 0 0
\(265\) −38388.8 −0.0335807
\(266\) −27321.9 −0.0236760
\(267\) 0 0
\(268\) −879284. −0.747812
\(269\) −334851. −0.282144 −0.141072 0.989999i \(-0.545055\pi\)
−0.141072 + 0.989999i \(0.545055\pi\)
\(270\) 0 0
\(271\) 893377. 0.738944 0.369472 0.929242i \(-0.379539\pi\)
0.369472 + 0.929242i \(0.379539\pi\)
\(272\) −56781.4 −0.0465355
\(273\) 0 0
\(274\) 1.06671e6 0.858360
\(275\) 677584. 0.540296
\(276\) 0 0
\(277\) −1.13462e6 −0.888483 −0.444242 0.895907i \(-0.646527\pi\)
−0.444242 + 0.895907i \(0.646527\pi\)
\(278\) −1.17955e6 −0.915389
\(279\) 0 0
\(280\) 41705.9 0.0317908
\(281\) 1.98111e6 1.49672 0.748362 0.663290i \(-0.230841\pi\)
0.748362 + 0.663290i \(0.230841\pi\)
\(282\) 0 0
\(283\) 1.32192e6 0.981161 0.490580 0.871396i \(-0.336785\pi\)
0.490580 + 0.871396i \(0.336785\pi\)
\(284\) 114833. 0.0844830
\(285\) 0 0
\(286\) −994759. −0.719122
\(287\) −58865.4 −0.0421847
\(288\) 0 0
\(289\) −1.37066e6 −0.965351
\(290\) −994550. −0.694435
\(291\) 0 0
\(292\) 1.03594e6 0.711011
\(293\) −1.35808e6 −0.924179 −0.462089 0.886833i \(-0.652900\pi\)
−0.462089 + 0.886833i \(0.652900\pi\)
\(294\) 0 0
\(295\) −1.30547e6 −0.873401
\(296\) 126681. 0.0840395
\(297\) 0 0
\(298\) 415959. 0.271337
\(299\) 471411. 0.304945
\(300\) 0 0
\(301\) −6017.81 −0.00382844
\(302\) −1.90937e6 −1.20468
\(303\) 0 0
\(304\) 92416.0 0.0573539
\(305\) −1.29047e6 −0.794324
\(306\) 0 0
\(307\) −1.23842e6 −0.749930 −0.374965 0.927039i \(-0.622345\pi\)
−0.374965 + 0.927039i \(0.622345\pi\)
\(308\) 105800. 0.0635492
\(309\) 0 0
\(310\) −744989. −0.440297
\(311\) 1.78565e6 1.04688 0.523439 0.852063i \(-0.324649\pi\)
0.523439 + 0.852063i \(0.324649\pi\)
\(312\) 0 0
\(313\) −3.01608e6 −1.74013 −0.870066 0.492935i \(-0.835924\pi\)
−0.870066 + 0.492935i \(0.835924\pi\)
\(314\) 1.73982e6 0.995819
\(315\) 0 0
\(316\) 577670. 0.325433
\(317\) −515839. −0.288314 −0.144157 0.989555i \(-0.546047\pi\)
−0.144157 + 0.989555i \(0.546047\pi\)
\(318\) 0 0
\(319\) −2.52300e6 −1.38816
\(320\) −141069. −0.0770119
\(321\) 0 0
\(322\) −50138.2 −0.0269482
\(323\) −80070.7 −0.0427039
\(324\) 0 0
\(325\) −1.37967e6 −0.724548
\(326\) −1.02033e6 −0.531737
\(327\) 0 0
\(328\) 199111. 0.102191
\(329\) −515417. −0.262524
\(330\) 0 0
\(331\) −2.25449e6 −1.13104 −0.565521 0.824734i \(-0.691325\pi\)
−0.565521 + 0.824734i \(0.691325\pi\)
\(332\) 828523. 0.412534
\(333\) 0 0
\(334\) −2.04200e6 −1.00159
\(335\) 1.89270e6 0.921446
\(336\) 0 0
\(337\) 2.40261e6 1.15242 0.576208 0.817303i \(-0.304532\pi\)
0.576208 + 0.817303i \(0.304532\pi\)
\(338\) 540319. 0.257252
\(339\) 0 0
\(340\) 122225. 0.0573406
\(341\) −1.88991e6 −0.880145
\(342\) 0 0
\(343\) 629237. 0.288788
\(344\) 20355.1 0.00927423
\(345\) 0 0
\(346\) −3.09381e6 −1.38933
\(347\) 684693. 0.305262 0.152631 0.988283i \(-0.451225\pi\)
0.152631 + 0.988283i \(0.451225\pi\)
\(348\) 0 0
\(349\) 2.19857e6 0.966220 0.483110 0.875560i \(-0.339507\pi\)
0.483110 + 0.875560i \(0.339507\pi\)
\(350\) 146739. 0.0640287
\(351\) 0 0
\(352\) −357868. −0.153945
\(353\) −2.03446e6 −0.868987 −0.434493 0.900675i \(-0.643073\pi\)
−0.434493 + 0.900675i \(0.643073\pi\)
\(354\) 0 0
\(355\) −247183. −0.104099
\(356\) −2.32406e6 −0.971900
\(357\) 0 0
\(358\) 1.91066e6 0.787907
\(359\) −2.30592e6 −0.944298 −0.472149 0.881519i \(-0.656522\pi\)
−0.472149 + 0.881519i \(0.656522\pi\)
\(360\) 0 0
\(361\) 130321. 0.0526316
\(362\) 2.91646e6 1.16973
\(363\) 0 0
\(364\) −215427. −0.0852209
\(365\) −2.22990e6 −0.876101
\(366\) 0 0
\(367\) 2.64475e6 1.02499 0.512495 0.858690i \(-0.328721\pi\)
0.512495 + 0.858690i \(0.328721\pi\)
\(368\) 169592. 0.0652807
\(369\) 0 0
\(370\) −272688. −0.103553
\(371\) −21090.0 −0.00795502
\(372\) 0 0
\(373\) 173405. 0.0645342 0.0322671 0.999479i \(-0.489727\pi\)
0.0322671 + 0.999479i \(0.489727\pi\)
\(374\) 310062. 0.114623
\(375\) 0 0
\(376\) 1.74339e6 0.635953
\(377\) 5.13723e6 1.86155
\(378\) 0 0
\(379\) 2.23408e6 0.798915 0.399458 0.916752i \(-0.369198\pi\)
0.399458 + 0.916752i \(0.369198\pi\)
\(380\) −198930. −0.0706709
\(381\) 0 0
\(382\) 1.14234e6 0.400530
\(383\) 4.44976e6 1.55003 0.775014 0.631944i \(-0.217743\pi\)
0.775014 + 0.631944i \(0.217743\pi\)
\(384\) 0 0
\(385\) −227740. −0.0783047
\(386\) −214761. −0.0733646
\(387\) 0 0
\(388\) 632204. 0.213196
\(389\) −2.43083e6 −0.814480 −0.407240 0.913321i \(-0.633509\pi\)
−0.407240 + 0.913321i \(0.633509\pi\)
\(390\) 0 0
\(391\) −146937. −0.0486059
\(392\) −1.05274e6 −0.346022
\(393\) 0 0
\(394\) −58155.5 −0.0188734
\(395\) −1.24346e6 −0.400996
\(396\) 0 0
\(397\) 2.61205e6 0.831775 0.415887 0.909416i \(-0.363471\pi\)
0.415887 + 0.909416i \(0.363471\pi\)
\(398\) −574800. −0.181890
\(399\) 0 0
\(400\) −496342. −0.155107
\(401\) −1.42660e6 −0.443038 −0.221519 0.975156i \(-0.571101\pi\)
−0.221519 + 0.975156i \(0.571101\pi\)
\(402\) 0 0
\(403\) 3.84816e6 1.18029
\(404\) 267566. 0.0815602
\(405\) 0 0
\(406\) −546384. −0.164507
\(407\) −691760. −0.207000
\(408\) 0 0
\(409\) 4.73321e6 1.39910 0.699548 0.714586i \(-0.253385\pi\)
0.699548 + 0.714586i \(0.253385\pi\)
\(410\) −428596. −0.125918
\(411\) 0 0
\(412\) 1.48263e6 0.430318
\(413\) −717200. −0.206902
\(414\) 0 0
\(415\) −1.78344e6 −0.508320
\(416\) 728677. 0.206444
\(417\) 0 0
\(418\) −504649. −0.141270
\(419\) 357759. 0.0995531 0.0497766 0.998760i \(-0.484149\pi\)
0.0497766 + 0.998760i \(0.484149\pi\)
\(420\) 0 0
\(421\) 652504. 0.179423 0.0897115 0.995968i \(-0.471405\pi\)
0.0897115 + 0.995968i \(0.471405\pi\)
\(422\) 334614. 0.0914668
\(423\) 0 0
\(424\) 71336.4 0.0192707
\(425\) 430038. 0.115487
\(426\) 0 0
\(427\) −708955. −0.188169
\(428\) 754500. 0.199090
\(429\) 0 0
\(430\) −43815.4 −0.0114276
\(431\) 6.25148e6 1.62103 0.810513 0.585721i \(-0.199188\pi\)
0.810513 + 0.585721i \(0.199188\pi\)
\(432\) 0 0
\(433\) 4.45832e6 1.14275 0.571375 0.820689i \(-0.306410\pi\)
0.571375 + 0.820689i \(0.306410\pi\)
\(434\) −409281. −0.104303
\(435\) 0 0
\(436\) 839201. 0.211422
\(437\) 239151. 0.0599057
\(438\) 0 0
\(439\) 5.00652e6 1.23986 0.619932 0.784655i \(-0.287160\pi\)
0.619932 + 0.784655i \(0.287160\pi\)
\(440\) 770327. 0.189690
\(441\) 0 0
\(442\) −631338. −0.153711
\(443\) −715908. −0.173320 −0.0866599 0.996238i \(-0.527619\pi\)
−0.0866599 + 0.996238i \(0.527619\pi\)
\(444\) 0 0
\(445\) 5.00264e6 1.19757
\(446\) −2.66166e6 −0.633599
\(447\) 0 0
\(448\) −77500.5 −0.0182435
\(449\) 4.83183e6 1.13109 0.565544 0.824718i \(-0.308666\pi\)
0.565544 + 0.824718i \(0.308666\pi\)
\(450\) 0 0
\(451\) −1.08727e6 −0.251708
\(452\) −2.40954e6 −0.554737
\(453\) 0 0
\(454\) −4.61530e6 −1.05090
\(455\) 463716. 0.105008
\(456\) 0 0
\(457\) −6.44410e6 −1.44335 −0.721675 0.692232i \(-0.756628\pi\)
−0.721675 + 0.692232i \(0.756628\pi\)
\(458\) −1.73477e6 −0.386436
\(459\) 0 0
\(460\) −365054. −0.0804383
\(461\) 4.10108e6 0.898764 0.449382 0.893340i \(-0.351644\pi\)
0.449382 + 0.893340i \(0.351644\pi\)
\(462\) 0 0
\(463\) 8.98683e6 1.94829 0.974146 0.225919i \(-0.0725386\pi\)
0.974146 + 0.225919i \(0.0725386\pi\)
\(464\) 1.84814e6 0.398510
\(465\) 0 0
\(466\) −1.54869e6 −0.330370
\(467\) 8.84409e6 1.87655 0.938276 0.345886i \(-0.112422\pi\)
0.938276 + 0.345886i \(0.112422\pi\)
\(468\) 0 0
\(469\) 1.03981e6 0.218284
\(470\) −3.75273e6 −0.783615
\(471\) 0 0
\(472\) 2.42592e6 0.501211
\(473\) −111152. −0.0228436
\(474\) 0 0
\(475\) −699919. −0.142336
\(476\) 67147.7 0.0135836
\(477\) 0 0
\(478\) 2.48985e6 0.498430
\(479\) −2.89320e6 −0.576155 −0.288077 0.957607i \(-0.593016\pi\)
−0.288077 + 0.957607i \(0.593016\pi\)
\(480\) 0 0
\(481\) 1.40854e6 0.277591
\(482\) −1.48582e6 −0.291305
\(483\) 0 0
\(484\) −622633. −0.120814
\(485\) −1.36085e6 −0.262697
\(486\) 0 0
\(487\) −3.18166e6 −0.607900 −0.303950 0.952688i \(-0.598306\pi\)
−0.303950 + 0.952688i \(0.598306\pi\)
\(488\) 2.39803e6 0.455832
\(489\) 0 0
\(490\) 2.26606e6 0.426365
\(491\) 4.45509e6 0.833975 0.416987 0.908912i \(-0.363086\pi\)
0.416987 + 0.908912i \(0.363086\pi\)
\(492\) 0 0
\(493\) −1.60125e6 −0.296717
\(494\) 1.02755e6 0.189446
\(495\) 0 0
\(496\) 1.38439e6 0.252670
\(497\) −135797. −0.0246603
\(498\) 0 0
\(499\) 9.31472e6 1.67463 0.837315 0.546721i \(-0.184124\pi\)
0.837315 + 0.546721i \(0.184124\pi\)
\(500\) 2.79044e6 0.499168
\(501\) 0 0
\(502\) −1.50439e6 −0.266442
\(503\) 2.04835e6 0.360980 0.180490 0.983577i \(-0.442232\pi\)
0.180490 + 0.983577i \(0.442232\pi\)
\(504\) 0 0
\(505\) −575949. −0.100498
\(506\) −926077. −0.160794
\(507\) 0 0
\(508\) 5.63584e6 0.968945
\(509\) 166211. 0.0284358 0.0142179 0.999899i \(-0.495474\pi\)
0.0142179 + 0.999899i \(0.495474\pi\)
\(510\) 0 0
\(511\) −1.22506e6 −0.207542
\(512\) 262144. 0.0441942
\(513\) 0 0
\(514\) −6.57529e6 −1.09776
\(515\) −3.19143e6 −0.530234
\(516\) 0 0
\(517\) −9.52001e6 −1.56643
\(518\) −149809. −0.0245309
\(519\) 0 0
\(520\) −1.56851e6 −0.254378
\(521\) −2.64602e6 −0.427069 −0.213535 0.976935i \(-0.568498\pi\)
−0.213535 + 0.976935i \(0.568498\pi\)
\(522\) 0 0
\(523\) −7.02287e6 −1.12269 −0.561346 0.827581i \(-0.689716\pi\)
−0.561346 + 0.827581i \(0.689716\pi\)
\(524\) −305124. −0.0485453
\(525\) 0 0
\(526\) −3.10120e6 −0.488726
\(527\) −1.19945e6 −0.188130
\(528\) 0 0
\(529\) −5.99748e6 −0.931815
\(530\) −153555. −0.0237451
\(531\) 0 0
\(532\) −109288. −0.0167414
\(533\) 2.21386e6 0.337546
\(534\) 0 0
\(535\) −1.62410e6 −0.245317
\(536\) −3.51714e6 −0.528783
\(537\) 0 0
\(538\) −1.33940e6 −0.199506
\(539\) 5.74860e6 0.852295
\(540\) 0 0
\(541\) 4.35066e6 0.639090 0.319545 0.947571i \(-0.396470\pi\)
0.319545 + 0.947571i \(0.396470\pi\)
\(542\) 3.57351e6 0.522512
\(543\) 0 0
\(544\) −227126. −0.0329056
\(545\) −1.80642e6 −0.260512
\(546\) 0 0
\(547\) 9.77794e6 1.39727 0.698633 0.715480i \(-0.253792\pi\)
0.698633 + 0.715480i \(0.253792\pi\)
\(548\) 4.26684e6 0.606952
\(549\) 0 0
\(550\) 2.71034e6 0.382047
\(551\) 2.60616e6 0.365698
\(552\) 0 0
\(553\) −683131. −0.0949929
\(554\) −4.53846e6 −0.628252
\(555\) 0 0
\(556\) −4.71822e6 −0.647278
\(557\) −6.69323e6 −0.914108 −0.457054 0.889439i \(-0.651095\pi\)
−0.457054 + 0.889439i \(0.651095\pi\)
\(558\) 0 0
\(559\) 226323. 0.0306337
\(560\) 166823. 0.0224795
\(561\) 0 0
\(562\) 7.92442e6 1.05834
\(563\) −7.54034e6 −1.00258 −0.501291 0.865279i \(-0.667141\pi\)
−0.501291 + 0.865279i \(0.667141\pi\)
\(564\) 0 0
\(565\) 5.18664e6 0.683542
\(566\) 5.28769e6 0.693785
\(567\) 0 0
\(568\) 459331. 0.0597385
\(569\) −1.34726e6 −0.174450 −0.0872252 0.996189i \(-0.527800\pi\)
−0.0872252 + 0.996189i \(0.527800\pi\)
\(570\) 0 0
\(571\) 2.05762e6 0.264104 0.132052 0.991243i \(-0.457843\pi\)
0.132052 + 0.991243i \(0.457843\pi\)
\(572\) −3.97904e6 −0.508496
\(573\) 0 0
\(574\) −235462. −0.0298291
\(575\) −1.28441e6 −0.162008
\(576\) 0 0
\(577\) 7.88561e6 0.986043 0.493021 0.870017i \(-0.335892\pi\)
0.493021 + 0.870017i \(0.335892\pi\)
\(578\) −5.48264e6 −0.682606
\(579\) 0 0
\(580\) −3.97820e6 −0.491040
\(581\) −979781. −0.120417
\(582\) 0 0
\(583\) −389542. −0.0474660
\(584\) 4.14375e6 0.502761
\(585\) 0 0
\(586\) −5.43232e6 −0.653493
\(587\) −1.90439e6 −0.228119 −0.114059 0.993474i \(-0.536385\pi\)
−0.114059 + 0.993474i \(0.536385\pi\)
\(588\) 0 0
\(589\) 1.95220e6 0.231866
\(590\) −5.22190e6 −0.617588
\(591\) 0 0
\(592\) 506726. 0.0594249
\(593\) 1.35835e7 1.58626 0.793132 0.609050i \(-0.208449\pi\)
0.793132 + 0.609050i \(0.208449\pi\)
\(594\) 0 0
\(595\) −144539. −0.0167375
\(596\) 1.66384e6 0.191864
\(597\) 0 0
\(598\) 1.88564e6 0.215629
\(599\) 1.04361e7 1.18842 0.594212 0.804309i \(-0.297464\pi\)
0.594212 + 0.804309i \(0.297464\pi\)
\(600\) 0 0
\(601\) −1.17196e7 −1.32351 −0.661753 0.749722i \(-0.730187\pi\)
−0.661753 + 0.749722i \(0.730187\pi\)
\(602\) −24071.2 −0.00270712
\(603\) 0 0
\(604\) −7.63749e6 −0.851840
\(605\) 1.34025e6 0.148866
\(606\) 0 0
\(607\) −7.53524e6 −0.830091 −0.415045 0.909801i \(-0.636234\pi\)
−0.415045 + 0.909801i \(0.636234\pi\)
\(608\) 369664. 0.0405554
\(609\) 0 0
\(610\) −5.16187e6 −0.561672
\(611\) 1.93843e7 2.10062
\(612\) 0 0
\(613\) −4.37292e6 −0.470025 −0.235012 0.971992i \(-0.575513\pi\)
−0.235012 + 0.971992i \(0.575513\pi\)
\(614\) −4.95366e6 −0.530280
\(615\) 0 0
\(616\) 423201. 0.0449361
\(617\) 7.50595e6 0.793767 0.396883 0.917869i \(-0.370092\pi\)
0.396883 + 0.917869i \(0.370092\pi\)
\(618\) 0 0
\(619\) −1.30877e7 −1.37289 −0.686447 0.727180i \(-0.740831\pi\)
−0.686447 + 0.727180i \(0.740831\pi\)
\(620\) −2.97996e6 −0.311337
\(621\) 0 0
\(622\) 7.14261e6 0.740254
\(623\) 2.74834e6 0.283695
\(624\) 0 0
\(625\) 52309.5 0.00535649
\(626\) −1.20643e7 −1.23046
\(627\) 0 0
\(628\) 6.95929e6 0.704151
\(629\) −439035. −0.0442459
\(630\) 0 0
\(631\) 8.92096e6 0.891945 0.445972 0.895047i \(-0.352858\pi\)
0.445972 + 0.895047i \(0.352858\pi\)
\(632\) 2.31068e6 0.230116
\(633\) 0 0
\(634\) −2.06336e6 −0.203869
\(635\) −1.21314e7 −1.19392
\(636\) 0 0
\(637\) −1.17051e7 −1.14295
\(638\) −1.00920e7 −0.981578
\(639\) 0 0
\(640\) −564277. −0.0544556
\(641\) −1.45438e7 −1.39808 −0.699041 0.715082i \(-0.746389\pi\)
−0.699041 + 0.715082i \(0.746389\pi\)
\(642\) 0 0
\(643\) 1.45757e7 1.39028 0.695140 0.718874i \(-0.255342\pi\)
0.695140 + 0.718874i \(0.255342\pi\)
\(644\) −200553. −0.0190552
\(645\) 0 0
\(646\) −320283. −0.0301962
\(647\) 2.24522e6 0.210862 0.105431 0.994427i \(-0.466378\pi\)
0.105431 + 0.994427i \(0.466378\pi\)
\(648\) 0 0
\(649\) −1.32470e7 −1.23454
\(650\) −5.51869e6 −0.512333
\(651\) 0 0
\(652\) −4.08132e6 −0.375995
\(653\) −1.54461e7 −1.41754 −0.708770 0.705440i \(-0.750749\pi\)
−0.708770 + 0.705440i \(0.750749\pi\)
\(654\) 0 0
\(655\) 656793. 0.0598171
\(656\) 796445. 0.0722597
\(657\) 0 0
\(658\) −2.06167e6 −0.185633
\(659\) −8.68940e6 −0.779429 −0.389714 0.920936i \(-0.627426\pi\)
−0.389714 + 0.920936i \(0.627426\pi\)
\(660\) 0 0
\(661\) −2.03442e7 −1.81108 −0.905538 0.424266i \(-0.860532\pi\)
−0.905538 + 0.424266i \(0.860532\pi\)
\(662\) −9.01797e6 −0.799767
\(663\) 0 0
\(664\) 3.31409e6 0.291705
\(665\) 235247. 0.0206286
\(666\) 0 0
\(667\) 4.78254e6 0.416240
\(668\) −8.16802e6 −0.708232
\(669\) 0 0
\(670\) 7.57080e6 0.651561
\(671\) −1.30947e7 −1.12277
\(672\) 0 0
\(673\) 1.71139e7 1.45650 0.728251 0.685310i \(-0.240333\pi\)
0.728251 + 0.685310i \(0.240333\pi\)
\(674\) 9.61045e6 0.814881
\(675\) 0 0
\(676\) 2.16128e6 0.181905
\(677\) −2.25299e6 −0.188924 −0.0944621 0.995528i \(-0.530113\pi\)
−0.0944621 + 0.995528i \(0.530113\pi\)
\(678\) 0 0
\(679\) −747622. −0.0622311
\(680\) 488899. 0.0405459
\(681\) 0 0
\(682\) −7.55962e6 −0.622356
\(683\) −5.33481e6 −0.437590 −0.218795 0.975771i \(-0.570213\pi\)
−0.218795 + 0.975771i \(0.570213\pi\)
\(684\) 0 0
\(685\) −9.18457e6 −0.747881
\(686\) 2.51695e6 0.204204
\(687\) 0 0
\(688\) 81420.5 0.00655787
\(689\) 793171. 0.0636530
\(690\) 0 0
\(691\) 8.08495e6 0.644143 0.322071 0.946715i \(-0.395621\pi\)
0.322071 + 0.946715i \(0.395621\pi\)
\(692\) −1.23753e7 −0.982401
\(693\) 0 0
\(694\) 2.73877e6 0.215853
\(695\) 1.01562e7 0.797569
\(696\) 0 0
\(697\) −690053. −0.0538022
\(698\) 8.79427e6 0.683221
\(699\) 0 0
\(700\) 586955. 0.0452751
\(701\) −7.88116e6 −0.605752 −0.302876 0.953030i \(-0.597947\pi\)
−0.302876 + 0.953030i \(0.597947\pi\)
\(702\) 0 0
\(703\) 714562. 0.0545320
\(704\) −1.43147e6 −0.108856
\(705\) 0 0
\(706\) −8.13786e6 −0.614467
\(707\) −316414. −0.0238071
\(708\) 0 0
\(709\) −2.07701e7 −1.55175 −0.775876 0.630885i \(-0.782692\pi\)
−0.775876 + 0.630885i \(0.782692\pi\)
\(710\) −988731. −0.0736092
\(711\) 0 0
\(712\) −9.29623e6 −0.687237
\(713\) 3.58247e6 0.263911
\(714\) 0 0
\(715\) 8.56506e6 0.626564
\(716\) 7.64263e6 0.557135
\(717\) 0 0
\(718\) −9.22370e6 −0.667719
\(719\) −1.03826e7 −0.749004 −0.374502 0.927226i \(-0.622186\pi\)
−0.374502 + 0.927226i \(0.622186\pi\)
\(720\) 0 0
\(721\) −1.75330e6 −0.125608
\(722\) 521284. 0.0372161
\(723\) 0 0
\(724\) 1.16658e7 0.827121
\(725\) −1.39970e7 −0.988985
\(726\) 0 0
\(727\) −6.52486e6 −0.457863 −0.228931 0.973443i \(-0.573523\pi\)
−0.228931 + 0.973443i \(0.573523\pi\)
\(728\) −861707. −0.0602603
\(729\) 0 0
\(730\) −8.91962e6 −0.619497
\(731\) −70544.1 −0.00488278
\(732\) 0 0
\(733\) −2.46101e6 −0.169182 −0.0845908 0.996416i \(-0.526958\pi\)
−0.0845908 + 0.996416i \(0.526958\pi\)
\(734\) 1.05790e7 0.724778
\(735\) 0 0
\(736\) 678367. 0.0461605
\(737\) 1.92058e7 1.30246
\(738\) 0 0
\(739\) −7.33586e6 −0.494128 −0.247064 0.968999i \(-0.579466\pi\)
−0.247064 + 0.968999i \(0.579466\pi\)
\(740\) −1.09075e6 −0.0732228
\(741\) 0 0
\(742\) −84359.9 −0.00562505
\(743\) −2.44394e7 −1.62412 −0.812060 0.583574i \(-0.801654\pi\)
−0.812060 + 0.583574i \(0.801654\pi\)
\(744\) 0 0
\(745\) −3.58148e6 −0.236414
\(746\) 693621. 0.0456326
\(747\) 0 0
\(748\) 1.24025e6 0.0810504
\(749\) −892244. −0.0581138
\(750\) 0 0
\(751\) −2.11169e6 −0.136625 −0.0683126 0.997664i \(-0.521762\pi\)
−0.0683126 + 0.997664i \(0.521762\pi\)
\(752\) 6.97356e6 0.449687
\(753\) 0 0
\(754\) 2.05489e7 1.31632
\(755\) 1.64401e7 1.04963
\(756\) 0 0
\(757\) −1.47627e7 −0.936325 −0.468162 0.883642i \(-0.655084\pi\)
−0.468162 + 0.883642i \(0.655084\pi\)
\(758\) 8.93632e6 0.564919
\(759\) 0 0
\(760\) −795719. −0.0499719
\(761\) −2.67074e7 −1.67174 −0.835871 0.548926i \(-0.815037\pi\)
−0.835871 + 0.548926i \(0.815037\pi\)
\(762\) 0 0
\(763\) −992408. −0.0617133
\(764\) 4.56934e6 0.283218
\(765\) 0 0
\(766\) 1.77990e7 1.09604
\(767\) 2.69731e7 1.65555
\(768\) 0 0
\(769\) 3.05104e6 0.186051 0.0930256 0.995664i \(-0.470346\pi\)
0.0930256 + 0.995664i \(0.470346\pi\)
\(770\) −910961. −0.0553698
\(771\) 0 0
\(772\) −859043. −0.0518766
\(773\) 2.18723e7 1.31657 0.658287 0.752767i \(-0.271281\pi\)
0.658287 + 0.752767i \(0.271281\pi\)
\(774\) 0 0
\(775\) −1.04847e7 −0.627052
\(776\) 2.52882e6 0.150752
\(777\) 0 0
\(778\) −9.72331e6 −0.575924
\(779\) 1.12311e6 0.0663100
\(780\) 0 0
\(781\) −2.50823e6 −0.147143
\(782\) −587748. −0.0343696
\(783\) 0 0
\(784\) −4.21094e6 −0.244675
\(785\) −1.49802e7 −0.867647
\(786\) 0 0
\(787\) −1.26837e7 −0.729975 −0.364987 0.931013i \(-0.618927\pi\)
−0.364987 + 0.931013i \(0.618927\pi\)
\(788\) −232622. −0.0133455
\(789\) 0 0
\(790\) −4.97385e6 −0.283547
\(791\) 2.84943e6 0.161926
\(792\) 0 0
\(793\) 2.66630e7 1.50566
\(794\) 1.04482e7 0.588154
\(795\) 0 0
\(796\) −2.29920e6 −0.128616
\(797\) −7.28989e6 −0.406514 −0.203257 0.979125i \(-0.565153\pi\)
−0.203257 + 0.979125i \(0.565153\pi\)
\(798\) 0 0
\(799\) −6.04201e6 −0.334822
\(800\) −1.98537e6 −0.109677
\(801\) 0 0
\(802\) −5.70640e6 −0.313275
\(803\) −2.26275e7 −1.23836
\(804\) 0 0
\(805\) 431700. 0.0234797
\(806\) 1.53926e7 0.834593
\(807\) 0 0
\(808\) 1.07027e6 0.0576717
\(809\) 2.72506e7 1.46388 0.731939 0.681370i \(-0.238615\pi\)
0.731939 + 0.681370i \(0.238615\pi\)
\(810\) 0 0
\(811\) 1.45954e7 0.779229 0.389615 0.920978i \(-0.372608\pi\)
0.389615 + 0.920978i \(0.372608\pi\)
\(812\) −2.18554e6 −0.116324
\(813\) 0 0
\(814\) −2.76704e6 −0.146371
\(815\) 8.78524e6 0.463297
\(816\) 0 0
\(817\) 114816. 0.00601791
\(818\) 1.89328e7 0.989310
\(819\) 0 0
\(820\) −1.71439e6 −0.0890377
\(821\) −4.16570e6 −0.215690 −0.107845 0.994168i \(-0.534395\pi\)
−0.107845 + 0.994168i \(0.534395\pi\)
\(822\) 0 0
\(823\) −9.74487e6 −0.501506 −0.250753 0.968051i \(-0.580678\pi\)
−0.250753 + 0.968051i \(0.580678\pi\)
\(824\) 5.93052e6 0.304281
\(825\) 0 0
\(826\) −2.86880e6 −0.146302
\(827\) −1.04970e6 −0.0533707 −0.0266854 0.999644i \(-0.508495\pi\)
−0.0266854 + 0.999644i \(0.508495\pi\)
\(828\) 0 0
\(829\) 1.87001e7 0.945056 0.472528 0.881316i \(-0.343342\pi\)
0.472528 + 0.881316i \(0.343342\pi\)
\(830\) −7.13374e6 −0.359436
\(831\) 0 0
\(832\) 2.91471e6 0.145978
\(833\) 3.64843e6 0.182177
\(834\) 0 0
\(835\) 1.75820e7 0.872676
\(836\) −2.01860e6 −0.0998928
\(837\) 0 0
\(838\) 1.43103e6 0.0703947
\(839\) −1.96094e7 −0.961742 −0.480871 0.876791i \(-0.659680\pi\)
−0.480871 + 0.876791i \(0.659680\pi\)
\(840\) 0 0
\(841\) 3.16068e7 1.54096
\(842\) 2.61002e6 0.126871
\(843\) 0 0
\(844\) 1.33846e6 0.0646768
\(845\) −4.65225e6 −0.224141
\(846\) 0 0
\(847\) 736303. 0.0352653
\(848\) 285346. 0.0136264
\(849\) 0 0
\(850\) 1.72015e6 0.0816620
\(851\) 1.31129e6 0.0620688
\(852\) 0 0
\(853\) −5.00288e6 −0.235422 −0.117711 0.993048i \(-0.537556\pi\)
−0.117711 + 0.993048i \(0.537556\pi\)
\(854\) −2.83582e6 −0.133056
\(855\) 0 0
\(856\) 3.01800e6 0.140778
\(857\) −1.36218e7 −0.633553 −0.316776 0.948500i \(-0.602600\pi\)
−0.316776 + 0.948500i \(0.602600\pi\)
\(858\) 0 0
\(859\) 1.88600e7 0.872084 0.436042 0.899926i \(-0.356380\pi\)
0.436042 + 0.899926i \(0.356380\pi\)
\(860\) −175262. −0.00808054
\(861\) 0 0
\(862\) 2.50059e7 1.14624
\(863\) 3.62744e7 1.65796 0.828979 0.559280i \(-0.188922\pi\)
0.828979 + 0.559280i \(0.188922\pi\)
\(864\) 0 0
\(865\) 2.66383e7 1.21050
\(866\) 1.78333e7 0.808047
\(867\) 0 0
\(868\) −1.63712e6 −0.0737535
\(869\) −1.26178e7 −0.566804
\(870\) 0 0
\(871\) −3.91061e7 −1.74662
\(872\) 3.35680e6 0.149498
\(873\) 0 0
\(874\) 956603. 0.0423597
\(875\) −3.29987e6 −0.145706
\(876\) 0 0
\(877\) −1.47531e7 −0.647715 −0.323857 0.946106i \(-0.604980\pi\)
−0.323857 + 0.946106i \(0.604980\pi\)
\(878\) 2.00261e7 0.876717
\(879\) 0 0
\(880\) 3.08131e6 0.134131
\(881\) −1.41746e7 −0.615277 −0.307638 0.951503i \(-0.599539\pi\)
−0.307638 + 0.951503i \(0.599539\pi\)
\(882\) 0 0
\(883\) 2.05570e7 0.887276 0.443638 0.896206i \(-0.353688\pi\)
0.443638 + 0.896206i \(0.353688\pi\)
\(884\) −2.52535e6 −0.108690
\(885\) 0 0
\(886\) −2.86363e6 −0.122556
\(887\) 9.48962e6 0.404986 0.202493 0.979284i \(-0.435096\pi\)
0.202493 + 0.979284i \(0.435096\pi\)
\(888\) 0 0
\(889\) −6.66473e6 −0.282832
\(890\) 2.00106e7 0.846807
\(891\) 0 0
\(892\) −1.06466e7 −0.448022
\(893\) 9.83381e6 0.412661
\(894\) 0 0
\(895\) −1.64511e7 −0.686496
\(896\) −310002. −0.0129001
\(897\) 0 0
\(898\) 1.93273e7 0.799800
\(899\) 3.90401e7 1.61106
\(900\) 0 0
\(901\) −247228. −0.0101458
\(902\) −4.34909e6 −0.177984
\(903\) 0 0
\(904\) −9.63814e6 −0.392258
\(905\) −2.51112e7 −1.01917
\(906\) 0 0
\(907\) 1.30672e7 0.527428 0.263714 0.964601i \(-0.415053\pi\)
0.263714 + 0.964601i \(0.415053\pi\)
\(908\) −1.84612e7 −0.743096
\(909\) 0 0
\(910\) 1.85486e6 0.0742521
\(911\) 4.60549e7 1.83857 0.919284 0.393594i \(-0.128768\pi\)
0.919284 + 0.393594i \(0.128768\pi\)
\(912\) 0 0
\(913\) −1.80970e7 −0.718506
\(914\) −2.57764e7 −1.02060
\(915\) 0 0
\(916\) −6.93906e6 −0.273251
\(917\) 360828. 0.0141702
\(918\) 0 0
\(919\) −6.67836e6 −0.260844 −0.130422 0.991459i \(-0.541633\pi\)
−0.130422 + 0.991459i \(0.541633\pi\)
\(920\) −1.46022e6 −0.0568785
\(921\) 0 0
\(922\) 1.64043e7 0.635522
\(923\) 5.10717e6 0.197322
\(924\) 0 0
\(925\) −3.83772e6 −0.147475
\(926\) 3.59473e7 1.37765
\(927\) 0 0
\(928\) 7.39254e6 0.281789
\(929\) 84737.5 0.00322134 0.00161067 0.999999i \(-0.499487\pi\)
0.00161067 + 0.999999i \(0.499487\pi\)
\(930\) 0 0
\(931\) −5.93809e6 −0.224529
\(932\) −6.19477e6 −0.233607
\(933\) 0 0
\(934\) 3.53763e7 1.32692
\(935\) −2.66970e6 −0.0998695
\(936\) 0 0
\(937\) −3.12820e7 −1.16398 −0.581989 0.813196i \(-0.697725\pi\)
−0.581989 + 0.813196i \(0.697725\pi\)
\(938\) 4.15924e6 0.154350
\(939\) 0 0
\(940\) −1.50109e7 −0.554100
\(941\) −2.88841e7 −1.06337 −0.531685 0.846942i \(-0.678441\pi\)
−0.531685 + 0.846942i \(0.678441\pi\)
\(942\) 0 0
\(943\) 2.06101e6 0.0754746
\(944\) 9.70367e6 0.354410
\(945\) 0 0
\(946\) −444607. −0.0161528
\(947\) −1.99925e7 −0.724422 −0.362211 0.932096i \(-0.617978\pi\)
−0.362211 + 0.932096i \(0.617978\pi\)
\(948\) 0 0
\(949\) 4.60732e7 1.66067
\(950\) −2.79968e6 −0.100647
\(951\) 0 0
\(952\) 268591. 0.00960502
\(953\) −5.22187e7 −1.86249 −0.931244 0.364396i \(-0.881276\pi\)
−0.931244 + 0.364396i \(0.881276\pi\)
\(954\) 0 0
\(955\) −9.83573e6 −0.348978
\(956\) 9.95941e6 0.352443
\(957\) 0 0
\(958\) −1.15728e7 −0.407403
\(959\) −5.04580e6 −0.177167
\(960\) 0 0
\(961\) 614716. 0.0214717
\(962\) 5.63414e6 0.196286
\(963\) 0 0
\(964\) −5.94327e6 −0.205984
\(965\) 1.84913e6 0.0639218
\(966\) 0 0
\(967\) 2.66303e7 0.915818 0.457909 0.888999i \(-0.348598\pi\)
0.457909 + 0.888999i \(0.348598\pi\)
\(968\) −2.49053e6 −0.0854287
\(969\) 0 0
\(970\) −5.44340e6 −0.185755
\(971\) −1.04206e7 −0.354686 −0.177343 0.984149i \(-0.556750\pi\)
−0.177343 + 0.984149i \(0.556750\pi\)
\(972\) 0 0
\(973\) 5.57959e6 0.188938
\(974\) −1.27267e7 −0.429850
\(975\) 0 0
\(976\) 9.59212e6 0.322322
\(977\) −1.82795e7 −0.612673 −0.306336 0.951923i \(-0.599103\pi\)
−0.306336 + 0.951923i \(0.599103\pi\)
\(978\) 0 0
\(979\) 5.07633e7 1.69275
\(980\) 9.06425e6 0.301486
\(981\) 0 0
\(982\) 1.78204e7 0.589709
\(983\) −2.25159e7 −0.743199 −0.371600 0.928393i \(-0.621191\pi\)
−0.371600 + 0.928393i \(0.621191\pi\)
\(984\) 0 0
\(985\) 500730. 0.0164442
\(986\) −6.40502e6 −0.209811
\(987\) 0 0
\(988\) 4.11019e6 0.133958
\(989\) 210697. 0.00684964
\(990\) 0 0
\(991\) 3.42068e7 1.10644 0.553221 0.833034i \(-0.313398\pi\)
0.553221 + 0.833034i \(0.313398\pi\)
\(992\) 5.53755e6 0.178664
\(993\) 0 0
\(994\) −543187. −0.0174375
\(995\) 4.94914e6 0.158479
\(996\) 0 0
\(997\) −1.12682e7 −0.359018 −0.179509 0.983756i \(-0.557451\pi\)
−0.179509 + 0.983756i \(0.557451\pi\)
\(998\) 3.72589e7 1.18414
\(999\) 0 0
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 342.6.a.i.1.1 2
3.2 odd 2 38.6.a.c.1.2 2
12.11 even 2 304.6.a.f.1.1 2
15.14 odd 2 950.6.a.d.1.1 2
57.56 even 2 722.6.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.6.a.c.1.2 2 3.2 odd 2
304.6.a.f.1.1 2 12.11 even 2
342.6.a.i.1.1 2 1.1 even 1 trivial
722.6.a.c.1.1 2 57.56 even 2
950.6.a.d.1.1 2 15.14 odd 2