Properties

Label 38.6.a.c.1.2
Level $38$
Weight $6$
Character 38.1
Self dual yes
Analytic conductor $6.095$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.09458515289\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1441}) \)
Defining polynomial: \(x^{2} - x - 360\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-18.4803\) of defining polynomial
Character \(\chi\) \(=\) 38.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +20.4803 q^{3} +16.0000 q^{4} +34.4408 q^{5} -81.9210 q^{6} -18.9210 q^{7} -64.0000 q^{8} +176.441 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +20.4803 q^{3} +16.0000 q^{4} +34.4408 q^{5} -81.9210 q^{6} -18.9210 q^{7} -64.0000 q^{8} +176.441 q^{9} -137.763 q^{10} +349.480 q^{11} +327.684 q^{12} +711.599 q^{13} +75.6840 q^{14} +705.355 q^{15} +256.000 q^{16} +221.803 q^{17} -705.763 q^{18} +361.000 q^{19} +551.052 q^{20} -387.507 q^{21} -1397.92 q^{22} -662.468 q^{23} -1310.74 q^{24} -1938.83 q^{25} -2846.39 q^{26} -1363.15 q^{27} -302.736 q^{28} -7219.28 q^{29} -2821.42 q^{30} +5407.76 q^{31} -1024.00 q^{32} +7157.44 q^{33} -887.210 q^{34} -651.654 q^{35} +2823.05 q^{36} +1979.40 q^{37} -1444.00 q^{38} +14573.7 q^{39} -2204.21 q^{40} -3111.11 q^{41} +1550.03 q^{42} +318.049 q^{43} +5591.68 q^{44} +6076.75 q^{45} +2649.87 q^{46} -27240.5 q^{47} +5242.94 q^{48} -16449.0 q^{49} +7755.34 q^{50} +4542.57 q^{51} +11385.6 q^{52} -1114.63 q^{53} +5452.60 q^{54} +12036.4 q^{55} +1210.94 q^{56} +7393.37 q^{57} +28877.1 q^{58} -37904.9 q^{59} +11285.7 q^{60} +37469.2 q^{61} -21631.0 q^{62} -3338.44 q^{63} +4096.00 q^{64} +24508.0 q^{65} -28629.8 q^{66} -54955.3 q^{67} +3548.84 q^{68} -13567.5 q^{69} +2606.62 q^{70} -7177.04 q^{71} -11292.2 q^{72} +64746.1 q^{73} -7917.59 q^{74} -39707.8 q^{75} +5776.00 q^{76} -6612.52 q^{77} -58294.9 q^{78} +36104.4 q^{79} +8816.83 q^{80} -70792.8 q^{81} +12444.4 q^{82} -51782.7 q^{83} -6200.11 q^{84} +7639.05 q^{85} -1272.20 q^{86} -147853. q^{87} -22366.7 q^{88} +145254. q^{89} -24307.0 q^{90} -13464.2 q^{91} -10599.5 q^{92} +110752. q^{93} +108962. q^{94} +12433.1 q^{95} -20971.8 q^{96} +39512.8 q^{97} +65796.0 q^{98} +61662.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{2} + 3q^{3} + 32q^{4} - 45q^{5} - 12q^{6} + 114q^{7} - 128q^{8} + 239q^{9} + O(q^{10}) \) \( 2q - 8q^{2} + 3q^{3} + 32q^{4} - 45q^{5} - 12q^{6} + 114q^{7} - 128q^{8} + 239q^{9} + 180q^{10} + 661q^{11} + 48q^{12} + 1613q^{13} - 456q^{14} + 2094q^{15} + 512q^{16} + 64q^{17} - 956q^{18} + 722q^{19} - 720q^{20} - 2711q^{21} - 2644q^{22} - 3185q^{23} - 192q^{24} + 1247q^{25} - 6452q^{26} + 1791q^{27} + 1824q^{28} - 2481q^{29} - 8376q^{30} - 1180q^{31} - 2048q^{32} + 1712q^{33} - 256q^{34} - 11211q^{35} + 3824q^{36} + 10488q^{37} - 2888q^{38} - 1183q^{39} + 2880q^{40} + 16630q^{41} + 10844q^{42} + 11303q^{43} + 10576q^{44} + 1107q^{45} + 12740q^{46} - 12155q^{47} + 768q^{48} - 15588q^{49} - 4988q^{50} + 7301q^{51} + 25808q^{52} + 20585q^{53} - 7164q^{54} - 12711q^{55} - 7296q^{56} + 1083q^{57} + 9924q^{58} - 78581q^{59} + 33504q^{60} + 43621q^{61} + 4720q^{62} + 4977q^{63} + 8192q^{64} - 47100q^{65} - 6848q^{66} + 7805q^{67} + 1024q^{68} + 30527q^{69} + 44844q^{70} - 62488q^{71} - 15296q^{72} + 16218q^{73} - 41952q^{74} - 95397q^{75} + 11552q^{76} + 34795q^{77} + 4732q^{78} + 67122q^{79} - 11520q^{80} - 141130q^{81} - 66520q^{82} - 10714q^{83} - 43376q^{84} + 20175q^{85} - 45212q^{86} - 230679q^{87} - 42304q^{88} + 128188q^{89} - 4428q^{90} + 106351q^{91} - 50960q^{92} + 225908q^{93} + 48620q^{94} - 16245q^{95} - 3072q^{96} + 178558q^{97} + 62352q^{98} + 81151q^{99} + O(q^{100}) \)

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
\(3\) 20.4803 1.31381 0.656904 0.753974i \(-0.271865\pi\)
0.656904 + 0.753974i \(0.271865\pi\)
\(4\) 16.0000 0.500000
\(5\) 34.4408 0.616095 0.308048 0.951371i \(-0.400324\pi\)
0.308048 + 0.951371i \(0.400324\pi\)
\(6\) −81.9210 −0.929003
\(7\) −18.9210 −0.145948 −0.0729742 0.997334i \(-0.523249\pi\)
−0.0729742 + 0.997334i \(0.523249\pi\)
\(8\) −64.0000 −0.353553
\(9\) 176.441 0.726094
\(10\) −137.763 −0.435645
\(11\) 349.480 0.870845 0.435423 0.900226i \(-0.356599\pi\)
0.435423 + 0.900226i \(0.356599\pi\)
\(12\) 327.684 0.656904
\(13\) 711.599 1.16782 0.583911 0.811818i \(-0.301522\pi\)
0.583911 + 0.811818i \(0.301522\pi\)
\(14\) 75.6840 0.103201
\(15\) 705.355 0.809431
\(16\) 256.000 0.250000
\(17\) 221.803 0.186142 0.0930710 0.995659i \(-0.470332\pi\)
0.0930710 + 0.995659i \(0.470332\pi\)
\(18\) −705.763 −0.513426
\(19\) 361.000 0.229416
\(20\) 551.052 0.308048
\(21\) −387.507 −0.191748
\(22\) −1397.92 −0.615780
\(23\) −662.468 −0.261123 −0.130561 0.991440i \(-0.541678\pi\)
−0.130561 + 0.991440i \(0.541678\pi\)
\(24\) −1310.74 −0.464502
\(25\) −1938.83 −0.620427
\(26\) −2846.39 −0.825775
\(27\) −1363.15 −0.359861
\(28\) −302.736 −0.0729742
\(29\) −7219.28 −1.59404 −0.797019 0.603954i \(-0.793591\pi\)
−0.797019 + 0.603954i \(0.793591\pi\)
\(30\) −2821.42 −0.572354
\(31\) 5407.76 1.01068 0.505339 0.862921i \(-0.331367\pi\)
0.505339 + 0.862921i \(0.331367\pi\)
\(32\) −1024.00 −0.176777
\(33\) 7157.44 1.14412
\(34\) −887.210 −0.131622
\(35\) −651.654 −0.0899181
\(36\) 2823.05 0.363047
\(37\) 1979.40 0.237700 0.118850 0.992912i \(-0.462079\pi\)
0.118850 + 0.992912i \(0.462079\pi\)
\(38\) −1444.00 −0.162221
\(39\) 14573.7 1.53430
\(40\) −2204.21 −0.217822
\(41\) −3111.11 −0.289039 −0.144519 0.989502i \(-0.546164\pi\)
−0.144519 + 0.989502i \(0.546164\pi\)
\(42\) 1550.03 0.135586
\(43\) 318.049 0.0262315 0.0131157 0.999914i \(-0.495825\pi\)
0.0131157 + 0.999914i \(0.495825\pi\)
\(44\) 5591.68 0.435423
\(45\) 6076.75 0.447343
\(46\) 2649.87 0.184642
\(47\) −27240.5 −1.79875 −0.899374 0.437181i \(-0.855977\pi\)
−0.899374 + 0.437181i \(0.855977\pi\)
\(48\) 5242.94 0.328452
\(49\) −16449.0 −0.978699
\(50\) 7755.34 0.438708
\(51\) 4542.57 0.244555
\(52\) 11385.6 0.583911
\(53\) −1114.63 −0.0545057 −0.0272528 0.999629i \(-0.508676\pi\)
−0.0272528 + 0.999629i \(0.508676\pi\)
\(54\) 5452.60 0.254460
\(55\) 12036.4 0.536523
\(56\) 1210.94 0.0516005
\(57\) 7393.37 0.301408
\(58\) 28877.1 1.12716
\(59\) −37904.9 −1.41764 −0.708820 0.705390i \(-0.750772\pi\)
−0.708820 + 0.705390i \(0.750772\pi\)
\(60\) 11285.7 0.404716
\(61\) 37469.2 1.28929 0.644644 0.764483i \(-0.277006\pi\)
0.644644 + 0.764483i \(0.277006\pi\)
\(62\) −21631.0 −0.714658
\(63\) −3338.44 −0.105972
\(64\) 4096.00 0.125000
\(65\) 24508.0 0.719490
\(66\) −28629.8 −0.809018
\(67\) −54955.3 −1.49562 −0.747812 0.663911i \(-0.768895\pi\)
−0.747812 + 0.663911i \(0.768895\pi\)
\(68\) 3548.84 0.0930710
\(69\) −13567.5 −0.343066
\(70\) 2606.62 0.0635817
\(71\) −7177.04 −0.168966 −0.0844830 0.996425i \(-0.526924\pi\)
−0.0844830 + 0.996425i \(0.526924\pi\)
\(72\) −11292.2 −0.256713
\(73\) 64746.1 1.42202 0.711011 0.703181i \(-0.248238\pi\)
0.711011 + 0.703181i \(0.248238\pi\)
\(74\) −7917.59 −0.168079
\(75\) −39707.8 −0.815122
\(76\) 5776.00 0.114708
\(77\) −6612.52 −0.127098
\(78\) −58294.9 −1.08491
\(79\) 36104.4 0.650866 0.325433 0.945565i \(-0.394490\pi\)
0.325433 + 0.945565i \(0.394490\pi\)
\(80\) 8816.83 0.154024
\(81\) −70792.8 −1.19888
\(82\) 12444.4 0.204381
\(83\) −51782.7 −0.825067 −0.412534 0.910942i \(-0.635356\pi\)
−0.412534 + 0.910942i \(0.635356\pi\)
\(84\) −6200.11 −0.0958741
\(85\) 7639.05 0.114681
\(86\) −1272.20 −0.0185485
\(87\) −147853. −2.09426
\(88\) −22366.7 −0.307890
\(89\) 145254. 1.94380 0.971900 0.235392i \(-0.0756375\pi\)
0.971900 + 0.235392i \(0.0756375\pi\)
\(90\) −24307.0 −0.316319
\(91\) −13464.2 −0.170442
\(92\) −10599.5 −0.130561
\(93\) 110752. 1.32784
\(94\) 108962. 1.27191
\(95\) 12433.1 0.141342
\(96\) −20971.8 −0.232251
\(97\) 39512.8 0.426391 0.213196 0.977010i \(-0.431613\pi\)
0.213196 + 0.977010i \(0.431613\pi\)
\(98\) 65796.0 0.692045
\(99\) 61662.6 0.632315
\(100\) −31021.3 −0.310213
\(101\) −16722.9 −0.163120 −0.0815602 0.996668i \(-0.525990\pi\)
−0.0815602 + 0.996668i \(0.525990\pi\)
\(102\) −18170.3 −0.172926
\(103\) 92664.4 0.860636 0.430318 0.902677i \(-0.358401\pi\)
0.430318 + 0.902677i \(0.358401\pi\)
\(104\) −45542.3 −0.412888
\(105\) −13346.0 −0.118135
\(106\) 4458.53 0.0385413
\(107\) −47156.3 −0.398180 −0.199090 0.979981i \(-0.563799\pi\)
−0.199090 + 0.979981i \(0.563799\pi\)
\(108\) −21810.4 −0.179930
\(109\) 52450.1 0.422844 0.211422 0.977395i \(-0.432191\pi\)
0.211422 + 0.977395i \(0.432191\pi\)
\(110\) −48145.5 −0.379379
\(111\) 40538.5 0.312292
\(112\) −4843.78 −0.0364871
\(113\) 150596. 1.10947 0.554737 0.832026i \(-0.312819\pi\)
0.554737 + 0.832026i \(0.312819\pi\)
\(114\) −29573.5 −0.213128
\(115\) −22815.9 −0.160877
\(116\) −115508. −0.797019
\(117\) 125555. 0.847948
\(118\) 151620. 1.00242
\(119\) −4196.73 −0.0271671
\(120\) −45142.7 −0.286177
\(121\) −38914.6 −0.241629
\(122\) −149877. −0.911664
\(123\) −63716.4 −0.379742
\(124\) 86524.2 0.505339
\(125\) −174402. −0.998337
\(126\) 13353.8 0.0749337
\(127\) 352240. 1.93789 0.968945 0.247276i \(-0.0795353\pi\)
0.968945 + 0.247276i \(0.0795353\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 6513.72 0.0344632
\(130\) −98032.0 −0.508756
\(131\) 19070.2 0.0970907 0.0485453 0.998821i \(-0.484541\pi\)
0.0485453 + 0.998821i \(0.484541\pi\)
\(132\) 114519. 0.572062
\(133\) −6830.49 −0.0334829
\(134\) 219821. 1.05757
\(135\) −46947.9 −0.221708
\(136\) −14195.4 −0.0658111
\(137\) −266677. −1.21390 −0.606952 0.794738i \(-0.707608\pi\)
−0.606952 + 0.794738i \(0.707608\pi\)
\(138\) 54270.0 0.242584
\(139\) −294888. −1.29456 −0.647278 0.762254i \(-0.724093\pi\)
−0.647278 + 0.762254i \(0.724093\pi\)
\(140\) −10426.5 −0.0449590
\(141\) −557892. −2.36321
\(142\) 28708.2 0.119477
\(143\) 248690. 1.01699
\(144\) 45168.8 0.181523
\(145\) −248637. −0.982079
\(146\) −258984. −1.00552
\(147\) −336880. −1.28582
\(148\) 31670.3 0.118850
\(149\) −103990. −0.383729 −0.191864 0.981421i \(-0.561453\pi\)
−0.191864 + 0.981421i \(0.561453\pi\)
\(150\) 158831. 0.576379
\(151\) −477343. −1.70368 −0.851840 0.523802i \(-0.824513\pi\)
−0.851840 + 0.523802i \(0.824513\pi\)
\(152\) −23104.0 −0.0811107
\(153\) 39135.0 0.135156
\(154\) 26450.1 0.0898722
\(155\) 186247. 0.622674
\(156\) 233180. 0.767148
\(157\) 434955. 1.40830 0.704151 0.710051i \(-0.251328\pi\)
0.704151 + 0.710051i \(0.251328\pi\)
\(158\) −144417. −0.460232
\(159\) −22827.9 −0.0716101
\(160\) −35267.3 −0.108911
\(161\) 12534.6 0.0381105
\(162\) 283171. 0.847737
\(163\) −255083. −0.751990 −0.375995 0.926622i \(-0.622699\pi\)
−0.375995 + 0.926622i \(0.622699\pi\)
\(164\) −49777.8 −0.144519
\(165\) 246508. 0.704889
\(166\) 207131. 0.583411
\(167\) 510501. 1.41646 0.708232 0.705980i \(-0.249493\pi\)
0.708232 + 0.705980i \(0.249493\pi\)
\(168\) 24800.5 0.0677932
\(169\) 135080. 0.363809
\(170\) −30556.2 −0.0810918
\(171\) 63695.1 0.166577
\(172\) 5088.78 0.0131157
\(173\) 773453. 1.96480 0.982401 0.186783i \(-0.0598062\pi\)
0.982401 + 0.186783i \(0.0598062\pi\)
\(174\) 591411. 1.48087
\(175\) 36684.7 0.0905503
\(176\) 89466.9 0.217711
\(177\) −776303. −1.86251
\(178\) −581014. −1.37447
\(179\) −477664. −1.11427 −0.557135 0.830422i \(-0.688099\pi\)
−0.557135 + 0.830422i \(0.688099\pi\)
\(180\) 97228.1 0.223671
\(181\) 729114. 1.65424 0.827121 0.562024i \(-0.189977\pi\)
0.827121 + 0.562024i \(0.189977\pi\)
\(182\) 53856.7 0.120521
\(183\) 767379. 1.69388
\(184\) 42397.9 0.0923209
\(185\) 68171.9 0.146446
\(186\) −443009. −0.938924
\(187\) 77515.6 0.162101
\(188\) −435848. −0.899374
\(189\) 25792.2 0.0525211
\(190\) −49732.5 −0.0999438
\(191\) −285584. −0.566435 −0.283218 0.959056i \(-0.591402\pi\)
−0.283218 + 0.959056i \(0.591402\pi\)
\(192\) 83887.1 0.164226
\(193\) −53690.2 −0.103753 −0.0518766 0.998654i \(-0.516520\pi\)
−0.0518766 + 0.998654i \(0.516520\pi\)
\(194\) −158051. −0.301504
\(195\) 501930. 0.945272
\(196\) −263184. −0.489350
\(197\) 14538.9 0.0266910 0.0133455 0.999911i \(-0.495752\pi\)
0.0133455 + 0.999911i \(0.495752\pi\)
\(198\) −246650. −0.447114
\(199\) −143700. −0.257232 −0.128616 0.991695i \(-0.541053\pi\)
−0.128616 + 0.991695i \(0.541053\pi\)
\(200\) 124085. 0.219354
\(201\) −1.12550e6 −1.96496
\(202\) 66891.6 0.115343
\(203\) 136596. 0.232647
\(204\) 72681.2 0.122277
\(205\) −107149. −0.178075
\(206\) −370658. −0.608562
\(207\) −116886. −0.189600
\(208\) 182169. 0.291956
\(209\) 126162. 0.199786
\(210\) 53384.2 0.0835342
\(211\) 83653.6 0.129354 0.0646768 0.997906i \(-0.479398\pi\)
0.0646768 + 0.997906i \(0.479398\pi\)
\(212\) −17834.1 −0.0272528
\(213\) −146988. −0.221989
\(214\) 188625. 0.281556
\(215\) 10953.8 0.0161611
\(216\) 87241.6 0.127230
\(217\) −102320. −0.147507
\(218\) −209800. −0.298996
\(219\) 1.32602e6 1.86826
\(220\) 192582. 0.268262
\(221\) 157834. 0.217381
\(222\) −162154. −0.220824
\(223\) −665414. −0.896045 −0.448022 0.894022i \(-0.647872\pi\)
−0.448022 + 0.894022i \(0.647872\pi\)
\(224\) 19375.1 0.0258003
\(225\) −342089. −0.450488
\(226\) −602384. −0.784517
\(227\) 1.15382e6 1.48619 0.743096 0.669185i \(-0.233357\pi\)
0.743096 + 0.669185i \(0.233357\pi\)
\(228\) 118294. 0.150704
\(229\) −433691. −0.546502 −0.273251 0.961943i \(-0.588099\pi\)
−0.273251 + 0.961943i \(0.588099\pi\)
\(230\) 91263.5 0.113757
\(231\) −135426. −0.166983
\(232\) 462034. 0.563578
\(233\) 387173. 0.467214 0.233607 0.972331i \(-0.424947\pi\)
0.233607 + 0.972331i \(0.424947\pi\)
\(234\) −502220. −0.599590
\(235\) −938183. −1.10820
\(236\) −606479. −0.708820
\(237\) 739426. 0.855114
\(238\) 16786.9 0.0192100
\(239\) −622463. −0.704886 −0.352443 0.935833i \(-0.614649\pi\)
−0.352443 + 0.935833i \(0.614649\pi\)
\(240\) 180571. 0.202358
\(241\) −371454. −0.411967 −0.205984 0.978555i \(-0.566039\pi\)
−0.205984 + 0.978555i \(0.566039\pi\)
\(242\) 155658. 0.170857
\(243\) −1.11861e6 −1.21524
\(244\) 599507. 0.644644
\(245\) −566516. −0.602972
\(246\) 254865. 0.268518
\(247\) 256887. 0.267917
\(248\) −346097. −0.357329
\(249\) −1.06052e6 −1.08398
\(250\) 697609. 0.705931
\(251\) 376098. 0.376805 0.188403 0.982092i \(-0.439669\pi\)
0.188403 + 0.982092i \(0.439669\pi\)
\(252\) −53415.0 −0.0529861
\(253\) −231519. −0.227398
\(254\) −1.40896e6 −1.37030
\(255\) 156450. 0.150669
\(256\) 65536.0 0.0625000
\(257\) 1.64382e6 1.55247 0.776233 0.630446i \(-0.217128\pi\)
0.776233 + 0.630446i \(0.217128\pi\)
\(258\) −26054.9 −0.0243691
\(259\) −37452.2 −0.0346919
\(260\) 392128. 0.359745
\(261\) −1.27378e6 −1.15742
\(262\) −76280.9 −0.0686535
\(263\) 775299. 0.691162 0.345581 0.938389i \(-0.387682\pi\)
0.345581 + 0.938389i \(0.387682\pi\)
\(264\) −458076. −0.404509
\(265\) −38388.8 −0.0335807
\(266\) 27321.9 0.0236760
\(267\) 2.97483e6 2.55378
\(268\) −879284. −0.747812
\(269\) 334851. 0.282144 0.141072 0.989999i \(-0.454945\pi\)
0.141072 + 0.989999i \(0.454945\pi\)
\(270\) 187792. 0.156771
\(271\) 893377. 0.738944 0.369472 0.929242i \(-0.379539\pi\)
0.369472 + 0.929242i \(0.379539\pi\)
\(272\) 56781.4 0.0465355
\(273\) −275750. −0.223928
\(274\) 1.06671e6 0.858360
\(275\) −677584. −0.540296
\(276\) −217080. −0.171533
\(277\) −1.13462e6 −0.888483 −0.444242 0.895907i \(-0.646527\pi\)
−0.444242 + 0.895907i \(0.646527\pi\)
\(278\) 1.17955e6 0.915389
\(279\) 954149. 0.733847
\(280\) 41705.9 0.0317908
\(281\) −1.98111e6 −1.49672 −0.748362 0.663290i \(-0.769159\pi\)
−0.748362 + 0.663290i \(0.769159\pi\)
\(282\) 2.23157e6 1.67104
\(283\) 1.32192e6 0.981161 0.490580 0.871396i \(-0.336785\pi\)
0.490580 + 0.871396i \(0.336785\pi\)
\(284\) −114833. −0.0844830
\(285\) 254633. 0.185696
\(286\) −994759. −0.719122
\(287\) 58865.4 0.0421847
\(288\) −180675. −0.128356
\(289\) −1.37066e6 −0.965351
\(290\) 994550. 0.694435
\(291\) 809232. 0.560197
\(292\) 1.03594e6 0.711011
\(293\) 1.35808e6 0.924179 0.462089 0.886833i \(-0.347100\pi\)
0.462089 + 0.886833i \(0.347100\pi\)
\(294\) 1.34752e6 0.909215
\(295\) −1.30547e6 −0.873401
\(296\) −126681. −0.0840395
\(297\) −476394. −0.313383
\(298\) 415959. 0.271337
\(299\) −471411. −0.304945
\(300\) −635325. −0.407561
\(301\) −6017.81 −0.00382844
\(302\) 1.90937e6 1.20468
\(303\) −342489. −0.214309
\(304\) 92416.0 0.0573539
\(305\) 1.29047e6 0.794324
\(306\) −156540. −0.0955701
\(307\) −1.23842e6 −0.749930 −0.374965 0.927039i \(-0.622345\pi\)
−0.374965 + 0.927039i \(0.622345\pi\)
\(308\) −105800. −0.0635492
\(309\) 1.89779e6 1.13071
\(310\) −744989. −0.440297
\(311\) −1.78565e6 −1.04688 −0.523439 0.852063i \(-0.675351\pi\)
−0.523439 + 0.852063i \(0.675351\pi\)
\(312\) −932718. −0.542455
\(313\) −3.01608e6 −1.74013 −0.870066 0.492935i \(-0.835924\pi\)
−0.870066 + 0.492935i \(0.835924\pi\)
\(314\) −1.73982e6 −0.995819
\(315\) −114978. −0.0652889
\(316\) 577670. 0.325433
\(317\) 515839. 0.288314 0.144157 0.989555i \(-0.453953\pi\)
0.144157 + 0.989555i \(0.453953\pi\)
\(318\) 91311.8 0.0506360
\(319\) −2.52300e6 −1.38816
\(320\) 141069. 0.0770119
\(321\) −965772. −0.523133
\(322\) −50138.2 −0.0269482
\(323\) 80070.7 0.0427039
\(324\) −1.13268e6 −0.599441
\(325\) −1.37967e6 −0.724548
\(326\) 1.02033e6 0.531737
\(327\) 1.07419e6 0.555536
\(328\) 199111. 0.102191
\(329\) 515417. 0.262524
\(330\) −986031. −0.498432
\(331\) −2.25449e6 −1.13104 −0.565521 0.824734i \(-0.691325\pi\)
−0.565521 + 0.824734i \(0.691325\pi\)
\(332\) −828523. −0.412534
\(333\) 349246. 0.172592
\(334\) −2.04200e6 −1.00159
\(335\) −1.89270e6 −0.921446
\(336\) −99201.8 −0.0479371
\(337\) 2.40261e6 1.15242 0.576208 0.817303i \(-0.304532\pi\)
0.576208 + 0.817303i \(0.304532\pi\)
\(338\) −540319. −0.257252
\(339\) 3.08424e6 1.45764
\(340\) 122225. 0.0573406
\(341\) 1.88991e6 0.880145
\(342\) −254780. −0.117788
\(343\) 629237. 0.288788
\(344\) −20355.1 −0.00927423
\(345\) −467275. −0.211361
\(346\) −3.09381e6 −1.38933
\(347\) −684693. −0.305262 −0.152631 0.988283i \(-0.548775\pi\)
−0.152631 + 0.988283i \(0.548775\pi\)
\(348\) −2.36564e6 −1.04713
\(349\) 2.19857e6 0.966220 0.483110 0.875560i \(-0.339507\pi\)
0.483110 + 0.875560i \(0.339507\pi\)
\(350\) −146739. −0.0640287
\(351\) −970016. −0.420253
\(352\) −357868. −0.153945
\(353\) 2.03446e6 0.868987 0.434493 0.900675i \(-0.356927\pi\)
0.434493 + 0.900675i \(0.356927\pi\)
\(354\) 3.10521e6 1.31699
\(355\) −247183. −0.104099
\(356\) 2.32406e6 0.971900
\(357\) −85950.1 −0.0356924
\(358\) 1.91066e6 0.787907
\(359\) 2.30592e6 0.944298 0.472149 0.881519i \(-0.343478\pi\)
0.472149 + 0.881519i \(0.343478\pi\)
\(360\) −388912. −0.158160
\(361\) 130321. 0.0526316
\(362\) −2.91646e6 −1.16973
\(363\) −796980. −0.317454
\(364\) −215427. −0.0852209
\(365\) 2.22990e6 0.876101
\(366\) −3.06952e6 −1.19775
\(367\) 2.64475e6 1.02499 0.512495 0.858690i \(-0.328721\pi\)
0.512495 + 0.858690i \(0.328721\pi\)
\(368\) −169592. −0.0652807
\(369\) −548927. −0.209869
\(370\) −272688. −0.103553
\(371\) 21090.0 0.00795502
\(372\) 1.77204e6 0.663919
\(373\) 173405. 0.0645342 0.0322671 0.999479i \(-0.489727\pi\)
0.0322671 + 0.999479i \(0.489727\pi\)
\(374\) −310062. −0.114623
\(375\) −3.57180e6 −1.31162
\(376\) 1.74339e6 0.635953
\(377\) −5.13723e6 −1.86155
\(378\) −103169. −0.0371380
\(379\) 2.23408e6 0.798915 0.399458 0.916752i \(-0.369198\pi\)
0.399458 + 0.916752i \(0.369198\pi\)
\(380\) 198930. 0.0706709
\(381\) 7.21396e6 2.54602
\(382\) 1.14234e6 0.400530
\(383\) −4.44976e6 −1.55003 −0.775014 0.631944i \(-0.782257\pi\)
−0.775014 + 0.631944i \(0.782257\pi\)
\(384\) −335548. −0.116125
\(385\) −227740. −0.0783047
\(386\) 214761. 0.0733646
\(387\) 56116.8 0.0190465
\(388\) 632204. 0.213196
\(389\) 2.43083e6 0.814480 0.407240 0.913321i \(-0.366491\pi\)
0.407240 + 0.913321i \(0.366491\pi\)
\(390\) −2.00772e6 −0.668408
\(391\) −146937. −0.0486059
\(392\) 1.05274e6 0.346022
\(393\) 390563. 0.127559
\(394\) −58155.5 −0.0188734
\(395\) 1.24346e6 0.400996
\(396\) 986601. 0.316158
\(397\) 2.61205e6 0.831775 0.415887 0.909416i \(-0.363471\pi\)
0.415887 + 0.909416i \(0.363471\pi\)
\(398\) 574800. 0.181890
\(399\) −139890. −0.0439901
\(400\) −496342. −0.155107
\(401\) 1.42660e6 0.443038 0.221519 0.975156i \(-0.428899\pi\)
0.221519 + 0.975156i \(0.428899\pi\)
\(402\) 4.50199e6 1.38944
\(403\) 3.84816e6 1.18029
\(404\) −267566. −0.0815602
\(405\) −2.43816e6 −0.738625
\(406\) −546384. −0.164507
\(407\) 691760. 0.207000
\(408\) −290725. −0.0864632
\(409\) 4.73321e6 1.39910 0.699548 0.714586i \(-0.253385\pi\)
0.699548 + 0.714586i \(0.253385\pi\)
\(410\) 428596. 0.125918
\(411\) −5.46162e6 −1.59484
\(412\) 1.48263e6 0.430318
\(413\) 717200. 0.206902
\(414\) 467545. 0.134067
\(415\) −1.78344e6 −0.508320
\(416\) −728677. −0.206444
\(417\) −6.03939e6 −1.70080
\(418\) −504649. −0.141270
\(419\) −357759. −0.0995531 −0.0497766 0.998760i \(-0.515851\pi\)
−0.0497766 + 0.998760i \(0.515851\pi\)
\(420\) −213537. −0.0590676
\(421\) 652504. 0.179423 0.0897115 0.995968i \(-0.471405\pi\)
0.0897115 + 0.995968i \(0.471405\pi\)
\(422\) −334614. −0.0914668
\(423\) −4.80633e6 −1.30606
\(424\) 71336.4 0.0192707
\(425\) −430038. −0.115487
\(426\) 587950. 0.156970
\(427\) −708955. −0.188169
\(428\) −754500. −0.199090
\(429\) 5.09323e6 1.33613
\(430\) −43815.4 −0.0114276
\(431\) −6.25148e6 −1.62103 −0.810513 0.585721i \(-0.800812\pi\)
−0.810513 + 0.585721i \(0.800812\pi\)
\(432\) −348966. −0.0899651
\(433\) 4.45832e6 1.14275 0.571375 0.820689i \(-0.306410\pi\)
0.571375 + 0.820689i \(0.306410\pi\)
\(434\) 409281. 0.104303
\(435\) −5.09216e6 −1.29026
\(436\) 839201. 0.211422
\(437\) −239151. −0.0599057
\(438\) −5.30406e6 −1.32106
\(439\) 5.00652e6 1.23986 0.619932 0.784655i \(-0.287160\pi\)
0.619932 + 0.784655i \(0.287160\pi\)
\(440\) −770327. −0.189690
\(441\) −2.90227e6 −0.710627
\(442\) −631338. −0.153711
\(443\) 715908. 0.173320 0.0866599 0.996238i \(-0.472381\pi\)
0.0866599 + 0.996238i \(0.472381\pi\)
\(444\) 648617. 0.156146
\(445\) 5.00264e6 1.19757
\(446\) 2.66166e6 0.633599
\(447\) −2.12974e6 −0.504147
\(448\) −77500.5 −0.0182435
\(449\) −4.83183e6 −1.13109 −0.565544 0.824718i \(-0.691334\pi\)
−0.565544 + 0.824718i \(0.691334\pi\)
\(450\) 1.36836e6 0.318543
\(451\) −1.08727e6 −0.251708
\(452\) 2.40954e6 0.554737
\(453\) −9.77610e6 −2.23831
\(454\) −4.61530e6 −1.05090
\(455\) −463716. −0.105008
\(456\) −473176. −0.106564
\(457\) −6.44410e6 −1.44335 −0.721675 0.692232i \(-0.756628\pi\)
−0.721675 + 0.692232i \(0.756628\pi\)
\(458\) 1.73477e6 0.386436
\(459\) −302350. −0.0669851
\(460\) −365054. −0.0804383
\(461\) −4.10108e6 −0.898764 −0.449382 0.893340i \(-0.648356\pi\)
−0.449382 + 0.893340i \(0.648356\pi\)
\(462\) 541704. 0.118075
\(463\) 8.98683e6 1.94829 0.974146 0.225919i \(-0.0725386\pi\)
0.974146 + 0.225919i \(0.0725386\pi\)
\(464\) −1.84814e6 −0.398510
\(465\) 3.81439e6 0.818075
\(466\) −1.54869e6 −0.330370
\(467\) −8.84409e6 −1.87655 −0.938276 0.345886i \(-0.887578\pi\)
−0.938276 + 0.345886i \(0.887578\pi\)
\(468\) 2.00888e6 0.423974
\(469\) 1.03981e6 0.218284
\(470\) 3.75273e6 0.783615
\(471\) 8.90800e6 1.85024
\(472\) 2.42592e6 0.501211
\(473\) 111152. 0.0228436
\(474\) −2.95771e6 −0.604657
\(475\) −699919. −0.142336
\(476\) −67147.7 −0.0135836
\(477\) −196667. −0.0395762
\(478\) 2.48985e6 0.498430
\(479\) 2.89320e6 0.576155 0.288077 0.957607i \(-0.406984\pi\)
0.288077 + 0.957607i \(0.406984\pi\)
\(480\) −722284. −0.143089
\(481\) 1.40854e6 0.277591
\(482\) 1.48582e6 0.291305
\(483\) 256711. 0.0500699
\(484\) −622633. −0.120814
\(485\) 1.36085e6 0.262697
\(486\) 4.47443e6 0.859305
\(487\) −3.18166e6 −0.607900 −0.303950 0.952688i \(-0.598306\pi\)
−0.303950 + 0.952688i \(0.598306\pi\)
\(488\) −2.39803e6 −0.455832
\(489\) −5.22416e6 −0.987971
\(490\) 2.26606e6 0.426365
\(491\) −4.45509e6 −0.833975 −0.416987 0.908912i \(-0.636914\pi\)
−0.416987 + 0.908912i \(0.636914\pi\)
\(492\) −1.01946e6 −0.189871
\(493\) −1.60125e6 −0.296717
\(494\) −1.02755e6 −0.189446
\(495\) 2.12371e6 0.389566
\(496\) 1.38439e6 0.252670
\(497\) 135797. 0.0246603
\(498\) 4.24209e6 0.766490
\(499\) 9.31472e6 1.67463 0.837315 0.546721i \(-0.184124\pi\)
0.837315 + 0.546721i \(0.184124\pi\)
\(500\) −2.79044e6 −0.499168
\(501\) 1.04552e7 1.86096
\(502\) −1.50439e6 −0.266442
\(503\) −2.04835e6 −0.360980 −0.180490 0.983577i \(-0.557768\pi\)
−0.180490 + 0.983577i \(0.557768\pi\)
\(504\) 213660. 0.0374668
\(505\) −575949. −0.100498
\(506\) 926077. 0.160794
\(507\) 2.76647e6 0.477976
\(508\) 5.63584e6 0.968945
\(509\) −166211. −0.0284358 −0.0142179 0.999899i \(-0.504526\pi\)
−0.0142179 + 0.999899i \(0.504526\pi\)
\(510\) −625798. −0.106539
\(511\) −1.22506e6 −0.207542
\(512\) −262144. −0.0441942
\(513\) −492097. −0.0825577
\(514\) −6.57529e6 −1.09776
\(515\) 3.19143e6 0.530234
\(516\) 104220. 0.0172316
\(517\) −9.52001e6 −1.56643
\(518\) 149809. 0.0245309
\(519\) 1.58405e7 2.58137
\(520\) −1.56851e6 −0.254378
\(521\) 2.64602e6 0.427069 0.213535 0.976935i \(-0.431502\pi\)
0.213535 + 0.976935i \(0.431502\pi\)
\(522\) 5.09510e6 0.818421
\(523\) −7.02287e6 −1.12269 −0.561346 0.827581i \(-0.689716\pi\)
−0.561346 + 0.827581i \(0.689716\pi\)
\(524\) 305124. 0.0485453
\(525\) 751312. 0.118966
\(526\) −3.10120e6 −0.488726
\(527\) 1.19945e6 0.188130
\(528\) 1.83231e6 0.286031
\(529\) −5.99748e6 −0.931815
\(530\) 153555. 0.0237451
\(531\) −6.68798e6 −1.02934
\(532\) −109288. −0.0167414
\(533\) −2.21386e6 −0.337546
\(534\) −1.18993e7 −1.80580
\(535\) −1.62410e6 −0.245317
\(536\) 3.51714e6 0.528783
\(537\) −9.78268e6 −1.46394
\(538\) −1.33940e6 −0.199506
\(539\) −5.74860e6 −0.852295
\(540\) −751167. −0.110854
\(541\) 4.35066e6 0.639090 0.319545 0.947571i \(-0.396470\pi\)
0.319545 + 0.947571i \(0.396470\pi\)
\(542\) −3.57351e6 −0.522512
\(543\) 1.49324e7 2.17336
\(544\) −227126. −0.0329056
\(545\) 1.80642e6 0.260512
\(546\) 1.10300e6 0.158341
\(547\) 9.77794e6 1.39727 0.698633 0.715480i \(-0.253792\pi\)
0.698633 + 0.715480i \(0.253792\pi\)
\(548\) −4.26684e6 −0.606952
\(549\) 6.61110e6 0.936144
\(550\) 2.71034e6 0.382047
\(551\) −2.60616e6 −0.365698
\(552\) 868320. 0.121292
\(553\) −683131. −0.0949929
\(554\) 4.53846e6 0.628252
\(555\) 1.39618e6 0.192401
\(556\) −4.71822e6 −0.647278
\(557\) 6.69323e6 0.914108 0.457054 0.889439i \(-0.348905\pi\)
0.457054 + 0.889439i \(0.348905\pi\)
\(558\) −3.81660e6 −0.518909
\(559\) 226323. 0.0306337
\(560\) −166823. −0.0224795
\(561\) 1.58754e6 0.212969
\(562\) 7.92442e6 1.05834
\(563\) 7.54034e6 1.00258 0.501291 0.865279i \(-0.332859\pi\)
0.501291 + 0.865279i \(0.332859\pi\)
\(564\) −8.92627e6 −1.18161
\(565\) 5.18664e6 0.683542
\(566\) −5.28769e6 −0.693785
\(567\) 1.33947e6 0.174975
\(568\) 459331. 0.0597385
\(569\) 1.34726e6 0.174450 0.0872252 0.996189i \(-0.472200\pi\)
0.0872252 + 0.996189i \(0.472200\pi\)
\(570\) −1.01853e6 −0.131307
\(571\) 2.05762e6 0.264104 0.132052 0.991243i \(-0.457843\pi\)
0.132052 + 0.991243i \(0.457843\pi\)
\(572\) 3.97904e6 0.508496
\(573\) −5.84883e6 −0.744188
\(574\) −235462. −0.0298291
\(575\) 1.28441e6 0.162008
\(576\) 722701. 0.0907617
\(577\) 7.88561e6 0.986043 0.493021 0.870017i \(-0.335892\pi\)
0.493021 + 0.870017i \(0.335892\pi\)
\(578\) 5.48264e6 0.682606
\(579\) −1.09959e6 −0.136312
\(580\) −3.97820e6 −0.491040
\(581\) 979781. 0.120417
\(582\) −3.23693e6 −0.396119
\(583\) −389542. −0.0474660
\(584\) −4.14375e6 −0.502761
\(585\) 4.32421e6 0.522417
\(586\) −5.43232e6 −0.653493
\(587\) 1.90439e6 0.228119 0.114059 0.993474i \(-0.463615\pi\)
0.114059 + 0.993474i \(0.463615\pi\)
\(588\) −5.39007e6 −0.642912
\(589\) 1.95220e6 0.231866
\(590\) 5.22190e6 0.617588
\(591\) 297760. 0.0350669
\(592\) 506726. 0.0594249
\(593\) −1.35835e7 −1.58626 −0.793132 0.609050i \(-0.791551\pi\)
−0.793132 + 0.609050i \(0.791551\pi\)
\(594\) 1.90558e6 0.221595
\(595\) −144539. −0.0167375
\(596\) −1.66384e6 −0.191864
\(597\) −2.94301e6 −0.337953
\(598\) 1.88564e6 0.215629
\(599\) −1.04361e7 −1.18842 −0.594212 0.804309i \(-0.702536\pi\)
−0.594212 + 0.804309i \(0.702536\pi\)
\(600\) 2.54130e6 0.288189
\(601\) −1.17196e7 −1.32351 −0.661753 0.749722i \(-0.730187\pi\)
−0.661753 + 0.749722i \(0.730187\pi\)
\(602\) 24071.2 0.00270712
\(603\) −9.69635e6 −1.08596
\(604\) −7.63749e6 −0.851840
\(605\) −1.34025e6 −0.148866
\(606\) 1.36996e6 0.151539
\(607\) −7.53524e6 −0.830091 −0.415045 0.909801i \(-0.636234\pi\)
−0.415045 + 0.909801i \(0.636234\pi\)
\(608\) −369664. −0.0405554
\(609\) 2.79752e6 0.305654
\(610\) −5.16187e6 −0.561672
\(611\) −1.93843e7 −2.10062
\(612\) 626160. 0.0675782
\(613\) −4.37292e6 −0.470025 −0.235012 0.971992i \(-0.575513\pi\)
−0.235012 + 0.971992i \(0.575513\pi\)
\(614\) 4.95366e6 0.530280
\(615\) −2.19444e6 −0.233957
\(616\) 423201. 0.0449361
\(617\) −7.50595e6 −0.793767 −0.396883 0.917869i \(-0.629908\pi\)
−0.396883 + 0.917869i \(0.629908\pi\)
\(618\) −7.59116e6 −0.799534
\(619\) −1.30877e7 −1.37289 −0.686447 0.727180i \(-0.740831\pi\)
−0.686447 + 0.727180i \(0.740831\pi\)
\(620\) 2.97996e6 0.311337
\(621\) 903043. 0.0939679
\(622\) 7.14261e6 0.740254
\(623\) −2.74834e6 −0.283695
\(624\) 3.73087e6 0.383574
\(625\) 52309.5 0.00535649
\(626\) 1.20643e7 1.23046
\(627\) 2.58384e6 0.262480
\(628\) 6.95929e6 0.704151
\(629\) 439035. 0.0442459
\(630\) 459913. 0.0461662
\(631\) 8.92096e6 0.891945 0.445972 0.895047i \(-0.352858\pi\)
0.445972 + 0.895047i \(0.352858\pi\)
\(632\) −2.31068e6 −0.230116
\(633\) 1.71325e6 0.169946
\(634\) −2.06336e6 −0.203869
\(635\) 1.21314e7 1.19392
\(636\) −365247. −0.0358050
\(637\) −1.17051e7 −1.14295
\(638\) 1.00920e7 0.981578
\(639\) −1.26632e6 −0.122685
\(640\) −564277. −0.0544556
\(641\) 1.45438e7 1.39808 0.699041 0.715082i \(-0.253611\pi\)
0.699041 + 0.715082i \(0.253611\pi\)
\(642\) 3.86309e6 0.369911
\(643\) 1.45757e7 1.39028 0.695140 0.718874i \(-0.255342\pi\)
0.695140 + 0.718874i \(0.255342\pi\)
\(644\) 200553. 0.0190552
\(645\) 224338. 0.0212326
\(646\) −320283. −0.0301962
\(647\) −2.24522e6 −0.210862 −0.105431 0.994427i \(-0.533622\pi\)
−0.105431 + 0.994427i \(0.533622\pi\)
\(648\) 4.53074e6 0.423869
\(649\) −1.32470e7 −1.23454
\(650\) 5.51869e6 0.512333
\(651\) −2.09555e6 −0.193796
\(652\) −4.08132e6 −0.375995
\(653\) 1.54461e7 1.41754 0.708770 0.705440i \(-0.249251\pi\)
0.708770 + 0.705440i \(0.249251\pi\)
\(654\) −4.29676e6 −0.392823
\(655\) 656793. 0.0598171
\(656\) −796445. −0.0722597
\(657\) 1.14238e7 1.03252
\(658\) −2.06167e6 −0.185633
\(659\) 8.68940e6 0.779429 0.389714 0.920936i \(-0.372574\pi\)
0.389714 + 0.920936i \(0.372574\pi\)
\(660\) 3.94412e6 0.352445
\(661\) −2.03442e7 −1.81108 −0.905538 0.424266i \(-0.860532\pi\)
−0.905538 + 0.424266i \(0.860532\pi\)
\(662\) 9.01797e6 0.799767
\(663\) 3.23249e6 0.285597
\(664\) 3.31409e6 0.291705
\(665\) −235247. −0.0206286
\(666\) −1.39698e6 −0.122041
\(667\) 4.78254e6 0.416240
\(668\) 8.16802e6 0.708232
\(669\) −1.36278e7 −1.17723
\(670\) 7.57080e6 0.651561
\(671\) 1.30947e7 1.12277
\(672\) 396807. 0.0338966
\(673\) 1.71139e7 1.45650 0.728251 0.685310i \(-0.240333\pi\)
0.728251 + 0.685310i \(0.240333\pi\)
\(674\) −9.61045e6 −0.814881
\(675\) 2.64292e6 0.223267
\(676\) 2.16128e6 0.181905
\(677\) 2.25299e6 0.188924 0.0944621 0.995528i \(-0.469887\pi\)
0.0944621 + 0.995528i \(0.469887\pi\)
\(678\) −1.23370e7 −1.03071
\(679\) −747622. −0.0622311
\(680\) −488899. −0.0405459
\(681\) 2.36306e7 1.95257
\(682\) −7.55962e6 −0.622356
\(683\) 5.33481e6 0.437590 0.218795 0.975771i \(-0.429787\pi\)
0.218795 + 0.975771i \(0.429787\pi\)
\(684\) 1.01912e6 0.0832887
\(685\) −9.18457e6 −0.747881
\(686\) −2.51695e6 −0.204204
\(687\) −8.88211e6 −0.718000
\(688\) 81420.5 0.00655787
\(689\) −793171. −0.0636530
\(690\) 1.86910e6 0.149455
\(691\) 8.08495e6 0.644143 0.322071 0.946715i \(-0.395621\pi\)
0.322071 + 0.946715i \(0.395621\pi\)
\(692\) 1.23753e7 0.982401
\(693\) −1.16672e6 −0.0922854
\(694\) 2.73877e6 0.215853
\(695\) −1.01562e7 −0.797569
\(696\) 9.46257e6 0.740433
\(697\) −690053. −0.0538022
\(698\) −8.79427e6 −0.683221
\(699\) 7.92941e6 0.613829
\(700\) 586955. 0.0452751
\(701\) 7.88116e6 0.605752 0.302876 0.953030i \(-0.402053\pi\)
0.302876 + 0.953030i \(0.402053\pi\)
\(702\) 3.88006e6 0.297164
\(703\) 714562. 0.0545320
\(704\) 1.43147e6 0.108856
\(705\) −1.92142e7 −1.45596
\(706\) −8.13786e6 −0.614467
\(707\) 316414. 0.0238071
\(708\) −1.24208e7 −0.931254
\(709\) −2.07701e7 −1.55175 −0.775876 0.630885i \(-0.782692\pi\)
−0.775876 + 0.630885i \(0.782692\pi\)
\(710\) 988731. 0.0736092
\(711\) 6.37028e6 0.472590
\(712\) −9.29623e6 −0.687237
\(713\) −3.58247e6 −0.263911
\(714\) 343800. 0.0252383
\(715\) 8.56506e6 0.626564
\(716\) −7.64263e6 −0.557135
\(717\) −1.27482e7 −0.926086
\(718\) −9.22370e6 −0.667719
\(719\) 1.03826e7 0.749004 0.374502 0.927226i \(-0.377814\pi\)
0.374502 + 0.927226i \(0.377814\pi\)
\(720\) 1.55565e6 0.111836
\(721\) −1.75330e6 −0.125608
\(722\) −521284. −0.0372161
\(723\) −7.60748e6 −0.541246
\(724\) 1.16658e7 0.827121
\(725\) 1.39970e7 0.988985
\(726\) 3.18792e6 0.224474
\(727\) −6.52486e6 −0.457863 −0.228931 0.973443i \(-0.573523\pi\)
−0.228931 + 0.973443i \(0.573523\pi\)
\(728\) 861707. 0.0602603
\(729\) −5.70674e6 −0.397712
\(730\) −8.91962e6 −0.619497
\(731\) 70544.1 0.00488278
\(732\) 1.22781e7 0.846939
\(733\) −2.46101e6 −0.169182 −0.0845908 0.996416i \(-0.526958\pi\)
−0.0845908 + 0.996416i \(0.526958\pi\)
\(734\) −1.05790e7 −0.724778
\(735\) −1.16024e7 −0.792189
\(736\) 678367. 0.0461605
\(737\) −1.92058e7 −1.30246
\(738\) 2.19571e6 0.148400
\(739\) −7.33586e6 −0.494128 −0.247064 0.968999i \(-0.579466\pi\)
−0.247064 + 0.968999i \(0.579466\pi\)
\(740\) 1.09075e6 0.0732228
\(741\) 5.26111e6 0.351991
\(742\) −84359.9 −0.00562505
\(743\) 2.44394e7 1.62412 0.812060 0.583574i \(-0.198346\pi\)
0.812060 + 0.583574i \(0.198346\pi\)
\(744\) −7.08815e6 −0.469462
\(745\) −3.58148e6 −0.236414
\(746\) −693621. −0.0456326
\(747\) −9.13658e6 −0.599076
\(748\) 1.24025e6 0.0810504
\(749\) 892244. 0.0581138
\(750\) 1.42872e7 0.927458
\(751\) −2.11169e6 −0.136625 −0.0683126 0.997664i \(-0.521762\pi\)
−0.0683126 + 0.997664i \(0.521762\pi\)
\(752\) −6.97356e6 −0.449687
\(753\) 7.70259e6 0.495050
\(754\) 2.05489e7 1.31632
\(755\) −1.64401e7 −1.04963
\(756\) 412675. 0.0262605
\(757\) −1.47627e7 −0.936325 −0.468162 0.883642i \(-0.655084\pi\)
−0.468162 + 0.883642i \(0.655084\pi\)
\(758\) −8.93632e6 −0.564919
\(759\) −4.74157e6 −0.298757
\(760\) −795719. −0.0499719
\(761\) 2.67074e7 1.67174 0.835871 0.548926i \(-0.184963\pi\)
0.835871 + 0.548926i \(0.184963\pi\)
\(762\) −2.88558e7 −1.80031
\(763\) −992408. −0.0617133
\(764\) −4.56934e6 −0.283218
\(765\) 1.34784e6 0.0832692
\(766\) 1.77990e7 1.09604
\(767\) −2.69731e7 −1.65555
\(768\) 1.34219e6 0.0821131
\(769\) 3.05104e6 0.186051 0.0930256 0.995664i \(-0.470346\pi\)
0.0930256 + 0.995664i \(0.470346\pi\)
\(770\) 910961. 0.0553698
\(771\) 3.36659e7 2.03964
\(772\) −859043. −0.0518766
\(773\) −2.18723e7 −1.31657 −0.658287 0.752767i \(-0.728719\pi\)
−0.658287 + 0.752767i \(0.728719\pi\)
\(774\) −224467. −0.0134679
\(775\) −1.04847e7 −0.627052
\(776\) −2.52882e6 −0.150752
\(777\) −767030. −0.0455785
\(778\) −9.72331e6 −0.575924
\(779\) −1.12311e6 −0.0663100
\(780\) 8.03088e6 0.472636
\(781\) −2.50823e6 −0.147143
\(782\) 587748. 0.0343696
\(783\) 9.84096e6 0.573632
\(784\) −4.21094e6 −0.244675
\(785\) 1.49802e7 0.867647
\(786\) −1.56225e6 −0.0901975
\(787\) −1.26837e7 −0.729975 −0.364987 0.931013i \(-0.618927\pi\)
−0.364987 + 0.931013i \(0.618927\pi\)
\(788\) 232622. 0.0133455
\(789\) 1.58783e7 0.908055
\(790\) −4.97385e6 −0.283547
\(791\) −2.84943e6 −0.161926
\(792\) −3.94640e6 −0.223557
\(793\) 2.66630e7 1.50566
\(794\) −1.04482e7 −0.588154
\(795\) −786212. −0.0441186
\(796\) −2.29920e6 −0.128616
\(797\) 7.28989e6 0.406514 0.203257 0.979125i \(-0.434847\pi\)
0.203257 + 0.979125i \(0.434847\pi\)
\(798\) 559560. 0.0311057
\(799\) −6.04201e6 −0.334822
\(800\) 1.98537e6 0.109677
\(801\) 2.56286e7 1.41138
\(802\) −5.70640e6 −0.313275
\(803\) 2.26275e7 1.23836
\(804\) −1.80080e7 −0.982482
\(805\) 431700. 0.0234797
\(806\) −1.53926e7 −0.834593
\(807\) 6.85783e6 0.370683
\(808\) 1.07027e6 0.0576717
\(809\) −2.72506e7 −1.46388 −0.731939 0.681370i \(-0.761385\pi\)
−0.731939 + 0.681370i \(0.761385\pi\)
\(810\) 9.75263e6 0.522287
\(811\) 1.45954e7 0.779229 0.389615 0.920978i \(-0.372608\pi\)
0.389615 + 0.920978i \(0.372608\pi\)
\(812\) 2.18554e6 0.116324
\(813\) 1.82966e7 0.970831
\(814\) −2.76704e6 −0.146371
\(815\) −8.78524e6 −0.463297
\(816\) 1.16290e6 0.0611387
\(817\) 114816. 0.00601791
\(818\) −1.89328e7 −0.989310
\(819\) −2.37563e6 −0.123757
\(820\) −1.71439e6 −0.0890377
\(821\) 4.16570e6 0.215690 0.107845 0.994168i \(-0.465605\pi\)
0.107845 + 0.994168i \(0.465605\pi\)
\(822\) 2.18465e7 1.12772
\(823\) −9.74487e6 −0.501506 −0.250753 0.968051i \(-0.580678\pi\)
−0.250753 + 0.968051i \(0.580678\pi\)
\(824\) −5.93052e6 −0.304281
\(825\) −1.38771e7 −0.709845
\(826\) −2.86880e6 −0.146302
\(827\) 1.04970e6 0.0533707 0.0266854 0.999644i \(-0.491505\pi\)
0.0266854 + 0.999644i \(0.491505\pi\)
\(828\) −1.87018e6 −0.0947999
\(829\) 1.87001e7 0.945056 0.472528 0.881316i \(-0.343342\pi\)
0.472528 + 0.881316i \(0.343342\pi\)
\(830\) 7.13374e6 0.359436
\(831\) −2.32372e7 −1.16730
\(832\) 2.91471e6 0.145978
\(833\) −3.64843e6 −0.182177
\(834\) 2.41576e7 1.20265
\(835\) 1.75820e7 0.872676
\(836\) 2.01860e6 0.0998928
\(837\) −7.37159e6 −0.363703
\(838\) 1.43103e6 0.0703947
\(839\) 1.96094e7 0.961742 0.480871 0.876791i \(-0.340320\pi\)
0.480871 + 0.876791i \(0.340320\pi\)
\(840\) 854146. 0.0417671
\(841\) 3.16068e7 1.54096
\(842\) −2.61002e6 −0.126871
\(843\) −4.05735e7 −1.96641
\(844\) 1.33846e6 0.0646768
\(845\) 4.65225e6 0.224141
\(846\) 1.92253e7 0.923523
\(847\) 736303. 0.0352653
\(848\) −285346. −0.0136264
\(849\) 2.70733e7 1.28906
\(850\) 1.72015e6 0.0816620
\(851\) −1.31129e6 −0.0620688
\(852\) −2.35180e6 −0.110995
\(853\) −5.00288e6 −0.235422 −0.117711 0.993048i \(-0.537556\pi\)
−0.117711 + 0.993048i \(0.537556\pi\)
\(854\) 2.83582e6 0.133056
\(855\) 2.19371e6 0.102627
\(856\) 3.01800e6 0.140778
\(857\) 1.36218e7 0.633553 0.316776 0.948500i \(-0.397400\pi\)
0.316776 + 0.948500i \(0.397400\pi\)
\(858\) −2.03729e7 −0.944789
\(859\) 1.88600e7 0.872084 0.436042 0.899926i \(-0.356380\pi\)
0.436042 + 0.899926i \(0.356380\pi\)
\(860\) 175262. 0.00808054
\(861\) 1.20558e6 0.0554227
\(862\) 2.50059e7 1.14624
\(863\) −3.62744e7 −1.65796 −0.828979 0.559280i \(-0.811078\pi\)
−0.828979 + 0.559280i \(0.811078\pi\)
\(864\) 1.39587e6 0.0636150
\(865\) 2.66383e7 1.21050
\(866\) −1.78333e7 −0.808047
\(867\) −2.80715e7 −1.26829
\(868\) −1.63712e6 −0.0737535
\(869\) 1.26178e7 0.566804
\(870\) 2.03686e7 0.912355
\(871\) −3.91061e7 −1.74662
\(872\) −3.35680e6 −0.149498
\(873\) 6.97166e6 0.309600
\(874\) 956603. 0.0423597
\(875\) 3.29987e6 0.145706
\(876\) 2.12163e7 0.934132
\(877\) −1.47531e7 −0.647715 −0.323857 0.946106i \(-0.604980\pi\)
−0.323857 + 0.946106i \(0.604980\pi\)
\(878\) −2.00261e7 −0.876717
\(879\) 2.78138e7 1.21419
\(880\) 3.08131e6 0.134131
\(881\) 1.41746e7 0.615277 0.307638 0.951503i \(-0.400461\pi\)
0.307638 + 0.951503i \(0.400461\pi\)
\(882\) 1.16091e7 0.502489
\(883\) 2.05570e7 0.887276 0.443638 0.896206i \(-0.353688\pi\)
0.443638 + 0.896206i \(0.353688\pi\)
\(884\) 2.52535e6 0.108690
\(885\) −2.67365e7 −1.14748
\(886\) −2.86363e6 −0.122556
\(887\) −9.48962e6 −0.404986 −0.202493 0.979284i \(-0.564904\pi\)
−0.202493 + 0.979284i \(0.564904\pi\)
\(888\) −2.59447e6 −0.110412
\(889\) −6.66473e6 −0.282832
\(890\) −2.00106e7 −0.846807
\(891\) −2.47407e7 −1.04404
\(892\) −1.06466e7 −0.448022
\(893\) −9.83381e6 −0.412661
\(894\) 8.51894e6 0.356485
\(895\) −1.64511e7 −0.686496
\(896\) 310002. 0.0129001
\(897\) −9.65462e6 −0.400640
\(898\) 1.93273e7 0.799800
\(899\) −3.90401e7 −1.61106
\(900\) −5.47343e6 −0.225244
\(901\) −247228. −0.0101458
\(902\) 4.34909e6 0.177984
\(903\) −123246. −0.00502984
\(904\) −9.63814e6 −0.392258
\(905\) 2.51112e7 1.01917
\(906\) 3.91044e7 1.58272
\(907\) 1.30672e7 0.527428 0.263714 0.964601i \(-0.415053\pi\)
0.263714 + 0.964601i \(0.415053\pi\)
\(908\) 1.84612e7 0.743096
\(909\) −2.95060e6 −0.118441
\(910\) 1.85486e6 0.0742521
\(911\) −4.60549e7 −1.83857 −0.919284 0.393594i \(-0.871232\pi\)
−0.919284 + 0.393594i \(0.871232\pi\)
\(912\) 1.89270e6 0.0753521
\(913\) −1.80970e7 −0.718506
\(914\) 2.57764e7 1.02060
\(915\) 2.64291e7 1.04359
\(916\) −6.93906e6 −0.273251
\(917\) −360828. −0.0141702
\(918\) 1.20940e6 0.0473656
\(919\) −6.67836e6 −0.260844 −0.130422 0.991459i \(-0.541633\pi\)
−0.130422 + 0.991459i \(0.541633\pi\)
\(920\) 1.46022e6 0.0568785
\(921\) −2.53631e7 −0.985264
\(922\) 1.64043e7 0.635522
\(923\) −5.10717e6 −0.197322
\(924\) −2.16682e6 −0.0834915
\(925\) −3.83772e6 −0.147475
\(926\) −3.59473e7 −1.37765
\(927\) 1.63498e7 0.624903
\(928\) 7.39254e6 0.281789
\(929\) −84737.5 −0.00322134 −0.00161067 0.999999i \(-0.500513\pi\)
−0.00161067 + 0.999999i \(0.500513\pi\)
\(930\) −1.52576e7 −0.578466
\(931\) −5.93809e6 −0.224529
\(932\) 6.19477e6 0.233607
\(933\) −3.65706e7 −1.37540
\(934\) 3.53763e7 1.32692
\(935\) 2.66970e6 0.0998695
\(936\) −8.03552e6 −0.299795
\(937\) −3.12820e7 −1.16398 −0.581989 0.813196i \(-0.697725\pi\)
−0.581989 + 0.813196i \(0.697725\pi\)
\(938\) −4.15924e6 −0.154350
\(939\) −6.17701e7 −2.28620
\(940\) −1.50109e7 −0.554100
\(941\) 2.88841e7 1.06337 0.531685 0.846942i \(-0.321559\pi\)
0.531685 + 0.846942i \(0.321559\pi\)
\(942\) −3.56320e7 −1.30832
\(943\) 2.06101e6 0.0754746
\(944\) −9.70367e6 −0.354410
\(945\) 888302. 0.0323580
\(946\) −444607. −0.0161528
\(947\) 1.99925e7 0.724422 0.362211 0.932096i \(-0.382022\pi\)
0.362211 + 0.932096i \(0.382022\pi\)
\(948\) 1.18308e7 0.427557
\(949\) 4.60732e7 1.66067
\(950\) 2.79968e6 0.100647
\(951\) 1.05645e7 0.378790
\(952\) 268591. 0.00960502
\(953\) 5.22187e7 1.86249 0.931244 0.364396i \(-0.118724\pi\)
0.931244 + 0.364396i \(0.118724\pi\)
\(954\) 786666. 0.0279846
\(955\) −9.83573e6 −0.348978
\(956\) −9.95941e6 −0.352443
\(957\) −5.16716e7 −1.82378
\(958\) −1.15728e7 −0.407403
\(959\) 5.04580e6 0.177167
\(960\) 2.88914e6 0.101179
\(961\) 614716. 0.0214717
\(962\) −5.63414e6 −0.196286
\(963\) −8.32029e6 −0.289116
\(964\) −5.94327e6 −0.205984
\(965\) −1.84913e6 −0.0639218
\(966\) −1.02684e6 −0.0354047
\(967\) 2.66303e7 0.915818 0.457909 0.888999i \(-0.348598\pi\)
0.457909 + 0.888999i \(0.348598\pi\)
\(968\) 2.49053e6 0.0854287
\(969\) 1.63987e6 0.0561047
\(970\) −5.44340e6 −0.185755
\(971\) 1.04206e7 0.354686 0.177343 0.984149i \(-0.443250\pi\)
0.177343 + 0.984149i \(0.443250\pi\)
\(972\) −1.78977e7 −0.607620
\(973\) 5.57959e6 0.188938
\(974\) 1.27267e7 0.429850
\(975\) −2.82560e7 −0.951918
\(976\) 9.59212e6 0.322322
\(977\) 1.82795e7 0.612673 0.306336 0.951923i \(-0.400897\pi\)
0.306336 + 0.951923i \(0.400897\pi\)
\(978\) 2.08966e7 0.698601
\(979\) 5.07633e7 1.69275
\(980\) −9.06425e6 −0.301486
\(981\) 9.25433e6 0.307024
\(982\) 1.78204e7 0.589709
\(983\) 2.25159e7 0.743199 0.371600 0.928393i \(-0.378809\pi\)
0.371600 + 0.928393i \(0.378809\pi\)
\(984\) 4.07785e6 0.134259
\(985\) 500730. 0.0164442
\(986\) 6.40502e6 0.209811
\(987\) 1.05559e7 0.344907
\(988\) 4.11019e6 0.133958
\(989\) −210697. −0.00684964
\(990\) −8.49482e6 −0.275465
\(991\) 3.42068e7 1.10644 0.553221 0.833034i \(-0.313398\pi\)
0.553221 + 0.833034i \(0.313398\pi\)
\(992\) −5.53755e6 −0.178664
\(993\) −4.61726e7 −1.48597
\(994\) −543187. −0.0174375
\(995\) −4.94914e6 −0.158479
\(996\) −1.69684e7 −0.541990
\(997\) −1.12682e7 −0.359018 −0.179509 0.983756i \(-0.557451\pi\)
−0.179509 + 0.983756i \(0.557451\pi\)
\(998\) −3.72589e7 −1.18414
\(999\) −2.69821e6 −0.0855387
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 38.6.a.c.1.2 2
3.2 odd 2 342.6.a.i.1.1 2
4.3 odd 2 304.6.a.f.1.1 2
5.4 even 2 950.6.a.d.1.1 2
19.18 odd 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 1.1 even 1 trivial
304.6.a.f.1.1 2 4.3 odd 2
342.6.a.i.1.1 2 3.2 odd 2
722.6.a.c.1.1 2 19.18 odd 2
950.6.a.d.1.1 2 5.4 even 2