Properties

Label 1045.6.a.h.1.15
Level $1045$
Weight $6$
Character 1045.1
Self dual yes
Analytic conductor $167.601$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(167.601091705\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 1045.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.77766 q^{2} -22.7115 q^{3} -24.2846 q^{4} +25.0000 q^{5} +63.0848 q^{6} -147.613 q^{7} +156.340 q^{8} +272.811 q^{9} +O(q^{10})\) \(q-2.77766 q^{2} -22.7115 q^{3} -24.2846 q^{4} +25.0000 q^{5} +63.0848 q^{6} -147.613 q^{7} +156.340 q^{8} +272.811 q^{9} -69.4415 q^{10} +121.000 q^{11} +551.539 q^{12} +639.175 q^{13} +410.019 q^{14} -567.787 q^{15} +342.849 q^{16} -659.486 q^{17} -757.777 q^{18} +361.000 q^{19} -607.115 q^{20} +3352.51 q^{21} -336.097 q^{22} +4675.18 q^{23} -3550.70 q^{24} +625.000 q^{25} -1775.41 q^{26} -677.054 q^{27} +3584.72 q^{28} +5066.41 q^{29} +1577.12 q^{30} +4496.85 q^{31} -5955.18 q^{32} -2748.09 q^{33} +1831.83 q^{34} -3690.32 q^{35} -6625.11 q^{36} +12358.5 q^{37} -1002.74 q^{38} -14516.6 q^{39} +3908.49 q^{40} +17965.0 q^{41} -9312.13 q^{42} +13840.7 q^{43} -2938.44 q^{44} +6820.28 q^{45} -12986.1 q^{46} -6956.38 q^{47} -7786.60 q^{48} +4982.57 q^{49} -1736.04 q^{50} +14977.9 q^{51} -15522.1 q^{52} +20615.9 q^{53} +1880.63 q^{54} +3025.00 q^{55} -23077.7 q^{56} -8198.84 q^{57} -14072.8 q^{58} +5808.65 q^{59} +13788.5 q^{60} -29010.9 q^{61} -12490.7 q^{62} -40270.4 q^{63} +5570.32 q^{64} +15979.4 q^{65} +7633.26 q^{66} -15642.2 q^{67} +16015.4 q^{68} -106180. q^{69} +10250.5 q^{70} +45233.9 q^{71} +42651.2 q^{72} -39182.9 q^{73} -34327.6 q^{74} -14194.7 q^{75} -8766.74 q^{76} -17861.2 q^{77} +40322.2 q^{78} -8183.99 q^{79} +8571.22 q^{80} -50916.2 q^{81} -49900.8 q^{82} +35339.3 q^{83} -81414.3 q^{84} -16487.2 q^{85} -38444.8 q^{86} -115066. q^{87} +18917.1 q^{88} -34162.3 q^{89} -18944.4 q^{90} -94350.5 q^{91} -113535. q^{92} -102130. q^{93} +19322.5 q^{94} +9025.00 q^{95} +135251. q^{96} -23325.9 q^{97} -13839.9 q^{98} +33010.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 24 q^{2} + 63 q^{3} + 690 q^{4} + 1000 q^{5} + 365 q^{6} + 839 q^{7} + 846 q^{8} + 3555 q^{9} + O(q^{10}) \) \( 40 q + 24 q^{2} + 63 q^{3} + 690 q^{4} + 1000 q^{5} + 365 q^{6} + 839 q^{7} + 846 q^{8} + 3555 q^{9} + 600 q^{10} + 4840 q^{11} + 2312 q^{12} + 2661 q^{13} + 395 q^{14} + 1575 q^{15} + 15974 q^{16} + 8249 q^{17} + 7225 q^{18} + 14440 q^{19} + 17250 q^{20} + 6845 q^{21} + 2904 q^{22} + 13948 q^{23} + 9740 q^{24} + 25000 q^{25} + 11581 q^{26} + 27864 q^{27} + 37879 q^{28} + 12965 q^{29} + 9125 q^{30} + 4411 q^{31} + 30751 q^{32} + 7623 q^{33} - 17739 q^{34} + 20975 q^{35} + 71345 q^{36} + 5729 q^{37} + 8664 q^{38} - 23560 q^{39} + 21150 q^{40} + 34059 q^{41} + 48528 q^{42} + 68593 q^{43} + 83490 q^{44} + 88875 q^{45} + 43829 q^{46} + 91592 q^{47} + 26539 q^{48} + 152447 q^{49} + 15000 q^{50} - 23170 q^{51} + 46798 q^{52} + 24361 q^{53} + 136436 q^{54} + 121000 q^{55} - 35393 q^{56} + 22743 q^{57} + 77722 q^{58} + 212881 q^{59} + 57800 q^{60} + 137627 q^{61} + 82606 q^{62} + 243832 q^{63} + 259580 q^{64} + 66525 q^{65} + 44165 q^{66} + 78752 q^{67} + 565000 q^{68} + 46134 q^{69} + 9875 q^{70} + 28888 q^{71} - 65574 q^{72} + 291074 q^{73} + 151963 q^{74} + 39375 q^{75} + 249090 q^{76} + 101519 q^{77} - 136222 q^{78} + 87079 q^{79} + 399350 q^{80} + 471360 q^{81} - 74882 q^{82} + 346989 q^{83} - 159196 q^{84} + 206225 q^{85} - 207742 q^{86} + 294612 q^{87} + 102366 q^{88} + 126718 q^{89} + 180625 q^{90} - 239900 q^{91} + 274196 q^{92} + 321654 q^{93} - 418108 q^{94} + 361000 q^{95} + 342154 q^{96} + 137404 q^{97} - 89356 q^{98} + 430155 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.77766 −0.491026 −0.245513 0.969393i \(-0.578956\pi\)
−0.245513 + 0.969393i \(0.578956\pi\)
\(3\) −22.7115 −1.45694 −0.728471 0.685077i \(-0.759769\pi\)
−0.728471 + 0.685077i \(0.759769\pi\)
\(4\) −24.2846 −0.758894
\(5\) 25.0000 0.447214
\(6\) 63.0848 0.715396
\(7\) −147.613 −1.13862 −0.569311 0.822123i \(-0.692790\pi\)
−0.569311 + 0.822123i \(0.692790\pi\)
\(8\) 156.340 0.863662
\(9\) 272.811 1.12268
\(10\) −69.4415 −0.219593
\(11\) 121.000 0.301511
\(12\) 551.539 1.10566
\(13\) 639.175 1.04897 0.524483 0.851421i \(-0.324259\pi\)
0.524483 + 0.851421i \(0.324259\pi\)
\(14\) 410.019 0.559092
\(15\) −567.787 −0.651564
\(16\) 342.849 0.334813
\(17\) −659.486 −0.553456 −0.276728 0.960948i \(-0.589250\pi\)
−0.276728 + 0.960948i \(0.589250\pi\)
\(18\) −757.777 −0.551265
\(19\) 361.000 0.229416
\(20\) −607.115 −0.339388
\(21\) 3352.51 1.65890
\(22\) −336.097 −0.148050
\(23\) 4675.18 1.84280 0.921401 0.388614i \(-0.127046\pi\)
0.921401 + 0.388614i \(0.127046\pi\)
\(24\) −3550.70 −1.25831
\(25\) 625.000 0.200000
\(26\) −1775.41 −0.515069
\(27\) −677.054 −0.178737
\(28\) 3584.72 0.864092
\(29\) 5066.41 1.11868 0.559340 0.828939i \(-0.311055\pi\)
0.559340 + 0.828939i \(0.311055\pi\)
\(30\) 1577.12 0.319935
\(31\) 4496.85 0.840435 0.420218 0.907423i \(-0.361954\pi\)
0.420218 + 0.907423i \(0.361954\pi\)
\(32\) −5955.18 −1.02806
\(33\) −2748.09 −0.439284
\(34\) 1831.83 0.271761
\(35\) −3690.32 −0.509207
\(36\) −6625.11 −0.851994
\(37\) 12358.5 1.48409 0.742045 0.670350i \(-0.233856\pi\)
0.742045 + 0.670350i \(0.233856\pi\)
\(38\) −1002.74 −0.112649
\(39\) −14516.6 −1.52828
\(40\) 3908.49 0.386241
\(41\) 17965.0 1.66905 0.834523 0.550973i \(-0.185743\pi\)
0.834523 + 0.550973i \(0.185743\pi\)
\(42\) −9312.13 −0.814565
\(43\) 13840.7 1.14153 0.570764 0.821114i \(-0.306647\pi\)
0.570764 + 0.821114i \(0.306647\pi\)
\(44\) −2938.44 −0.228815
\(45\) 6820.28 0.502078
\(46\) −12986.1 −0.904863
\(47\) −6956.38 −0.459345 −0.229672 0.973268i \(-0.573765\pi\)
−0.229672 + 0.973268i \(0.573765\pi\)
\(48\) −7786.60 −0.487803
\(49\) 4982.57 0.296458
\(50\) −1736.04 −0.0982052
\(51\) 14977.9 0.806354
\(52\) −15522.1 −0.796054
\(53\) 20615.9 1.00812 0.504060 0.863669i \(-0.331839\pi\)
0.504060 + 0.863669i \(0.331839\pi\)
\(54\) 1880.63 0.0877644
\(55\) 3025.00 0.134840
\(56\) −23077.7 −0.983384
\(57\) −8198.84 −0.334245
\(58\) −14072.8 −0.549300
\(59\) 5808.65 0.217243 0.108621 0.994083i \(-0.465356\pi\)
0.108621 + 0.994083i \(0.465356\pi\)
\(60\) 13788.5 0.494468
\(61\) −29010.9 −0.998243 −0.499121 0.866532i \(-0.666344\pi\)
−0.499121 + 0.866532i \(0.666344\pi\)
\(62\) −12490.7 −0.412675
\(63\) −40270.4 −1.27831
\(64\) 5570.32 0.169993
\(65\) 15979.4 0.469112
\(66\) 7633.26 0.215700
\(67\) −15642.2 −0.425706 −0.212853 0.977084i \(-0.568276\pi\)
−0.212853 + 0.977084i \(0.568276\pi\)
\(68\) 16015.4 0.420015
\(69\) −106180. −2.68485
\(70\) 10250.5 0.250034
\(71\) 45233.9 1.06492 0.532462 0.846454i \(-0.321267\pi\)
0.532462 + 0.846454i \(0.321267\pi\)
\(72\) 42651.2 0.969616
\(73\) −39182.9 −0.860576 −0.430288 0.902692i \(-0.641588\pi\)
−0.430288 + 0.902692i \(0.641588\pi\)
\(74\) −34327.6 −0.728726
\(75\) −14194.7 −0.291388
\(76\) −8766.74 −0.174102
\(77\) −17861.2 −0.343307
\(78\) 40322.2 0.750426
\(79\) −8183.99 −0.147536 −0.0737679 0.997275i \(-0.523502\pi\)
−0.0737679 + 0.997275i \(0.523502\pi\)
\(80\) 8571.22 0.149733
\(81\) −50916.2 −0.862270
\(82\) −49900.8 −0.819545
\(83\) 35339.3 0.563071 0.281535 0.959551i \(-0.409156\pi\)
0.281535 + 0.959551i \(0.409156\pi\)
\(84\) −81414.3 −1.25893
\(85\) −16487.2 −0.247513
\(86\) −38444.8 −0.560520
\(87\) −115066. −1.62985
\(88\) 18917.1 0.260404
\(89\) −34162.3 −0.457164 −0.228582 0.973525i \(-0.573409\pi\)
−0.228582 + 0.973525i \(0.573409\pi\)
\(90\) −18944.4 −0.246533
\(91\) −94350.5 −1.19437
\(92\) −113535. −1.39849
\(93\) −102130. −1.22447
\(94\) 19322.5 0.225550
\(95\) 9025.00 0.102598
\(96\) 135251. 1.49783
\(97\) −23325.9 −0.251715 −0.125858 0.992048i \(-0.540168\pi\)
−0.125858 + 0.992048i \(0.540168\pi\)
\(98\) −13839.9 −0.145568
\(99\) 33010.1 0.338501
\(100\) −15177.9 −0.151779
\(101\) 152832. 1.49077 0.745386 0.666633i \(-0.232265\pi\)
0.745386 + 0.666633i \(0.232265\pi\)
\(102\) −41603.6 −0.395941
\(103\) 197206. 1.83158 0.915791 0.401654i \(-0.131565\pi\)
0.915791 + 0.401654i \(0.131565\pi\)
\(104\) 99928.4 0.905952
\(105\) 83812.7 0.741885
\(106\) −57263.9 −0.495013
\(107\) 108757. 0.918327 0.459164 0.888352i \(-0.348149\pi\)
0.459164 + 0.888352i \(0.348149\pi\)
\(108\) 16442.0 0.135642
\(109\) −112523. −0.907138 −0.453569 0.891221i \(-0.649850\pi\)
−0.453569 + 0.891221i \(0.649850\pi\)
\(110\) −8402.43 −0.0662099
\(111\) −280679. −2.16223
\(112\) −50608.9 −0.381225
\(113\) −148256. −1.09224 −0.546119 0.837707i \(-0.683895\pi\)
−0.546119 + 0.837707i \(0.683895\pi\)
\(114\) 22773.6 0.164123
\(115\) 116879. 0.824126
\(116\) −123036. −0.848959
\(117\) 174374. 1.17765
\(118\) −16134.5 −0.106672
\(119\) 97348.7 0.630177
\(120\) −88767.5 −0.562731
\(121\) 14641.0 0.0909091
\(122\) 80582.4 0.490163
\(123\) −408012. −2.43170
\(124\) −109204. −0.637801
\(125\) 15625.0 0.0894427
\(126\) 111858. 0.627681
\(127\) 210080. 1.15578 0.577891 0.816114i \(-0.303876\pi\)
0.577891 + 0.816114i \(0.303876\pi\)
\(128\) 175093. 0.944593
\(129\) −314343. −1.66314
\(130\) −44385.3 −0.230346
\(131\) −10568.2 −0.0538052 −0.0269026 0.999638i \(-0.508564\pi\)
−0.0269026 + 0.999638i \(0.508564\pi\)
\(132\) 66736.2 0.333370
\(133\) −53288.3 −0.261218
\(134\) 43448.7 0.209033
\(135\) −16926.4 −0.0799335
\(136\) −103104. −0.477999
\(137\) −44366.9 −0.201956 −0.100978 0.994889i \(-0.532197\pi\)
−0.100978 + 0.994889i \(0.532197\pi\)
\(138\) 294933. 1.31833
\(139\) 93755.3 0.411584 0.205792 0.978596i \(-0.434023\pi\)
0.205792 + 0.978596i \(0.434023\pi\)
\(140\) 89618.0 0.386434
\(141\) 157990. 0.669238
\(142\) −125645. −0.522905
\(143\) 77340.2 0.316275
\(144\) 93533.0 0.375888
\(145\) 126660. 0.500289
\(146\) 108837. 0.422565
\(147\) −113161. −0.431922
\(148\) −300120. −1.12627
\(149\) −459538. −1.69573 −0.847863 0.530215i \(-0.822111\pi\)
−0.847863 + 0.530215i \(0.822111\pi\)
\(150\) 39428.0 0.143079
\(151\) 100751. 0.359591 0.179796 0.983704i \(-0.442456\pi\)
0.179796 + 0.983704i \(0.442456\pi\)
\(152\) 56438.6 0.198138
\(153\) −179915. −0.621354
\(154\) 49612.3 0.168573
\(155\) 112421. 0.375854
\(156\) 352530. 1.15980
\(157\) −265063. −0.858224 −0.429112 0.903251i \(-0.641173\pi\)
−0.429112 + 0.903251i \(0.641173\pi\)
\(158\) 22732.4 0.0724439
\(159\) −468217. −1.46877
\(160\) −148880. −0.459764
\(161\) −690117. −2.09825
\(162\) 141428. 0.423397
\(163\) 423842. 1.24950 0.624749 0.780826i \(-0.285201\pi\)
0.624749 + 0.780826i \(0.285201\pi\)
\(164\) −436274. −1.26663
\(165\) −68702.2 −0.196454
\(166\) −98160.7 −0.276482
\(167\) 127790. 0.354572 0.177286 0.984159i \(-0.443268\pi\)
0.177286 + 0.984159i \(0.443268\pi\)
\(168\) 524129. 1.43273
\(169\) 37251.9 0.100330
\(170\) 45795.7 0.121535
\(171\) 98484.8 0.257560
\(172\) −336116. −0.866299
\(173\) −739707. −1.87908 −0.939539 0.342442i \(-0.888746\pi\)
−0.939539 + 0.342442i \(0.888746\pi\)
\(174\) 319614. 0.800299
\(175\) −92258.1 −0.227724
\(176\) 41484.7 0.100950
\(177\) −131923. −0.316510
\(178\) 94891.3 0.224479
\(179\) −158041. −0.368670 −0.184335 0.982863i \(-0.559013\pi\)
−0.184335 + 0.982863i \(0.559013\pi\)
\(180\) −165628. −0.381023
\(181\) 98306.9 0.223042 0.111521 0.993762i \(-0.464428\pi\)
0.111521 + 0.993762i \(0.464428\pi\)
\(182\) 262074. 0.586469
\(183\) 658880. 1.45438
\(184\) 730915. 1.59156
\(185\) 308962. 0.663705
\(186\) 283683. 0.601244
\(187\) −79797.8 −0.166873
\(188\) 168933. 0.348594
\(189\) 99941.9 0.203514
\(190\) −25068.4 −0.0503782
\(191\) −739015. −1.46578 −0.732892 0.680345i \(-0.761830\pi\)
−0.732892 + 0.680345i \(0.761830\pi\)
\(192\) −126510. −0.247670
\(193\) −57053.8 −0.110253 −0.0551266 0.998479i \(-0.517556\pi\)
−0.0551266 + 0.998479i \(0.517556\pi\)
\(194\) 64791.5 0.123599
\(195\) −362915. −0.683469
\(196\) −121000. −0.224980
\(197\) 389659. 0.715351 0.357675 0.933846i \(-0.383569\pi\)
0.357675 + 0.933846i \(0.383569\pi\)
\(198\) −91691.0 −0.166213
\(199\) 541641. 0.969569 0.484784 0.874634i \(-0.338898\pi\)
0.484784 + 0.874634i \(0.338898\pi\)
\(200\) 97712.2 0.172732
\(201\) 355257. 0.620229
\(202\) −424516. −0.732007
\(203\) −747868. −1.27375
\(204\) −363732. −0.611937
\(205\) 449126. 0.746420
\(206\) −547771. −0.899354
\(207\) 1.27544e6 2.06888
\(208\) 219140. 0.351208
\(209\) 43681.0 0.0691714
\(210\) −232803. −0.364285
\(211\) 1.05480e6 1.63104 0.815520 0.578729i \(-0.196451\pi\)
0.815520 + 0.578729i \(0.196451\pi\)
\(212\) −500648. −0.765056
\(213\) −1.02733e6 −1.55153
\(214\) −302090. −0.450922
\(215\) 346017. 0.510507
\(216\) −105850. −0.154368
\(217\) −663793. −0.956937
\(218\) 312550. 0.445428
\(219\) 889901. 1.25381
\(220\) −73460.9 −0.102329
\(221\) −421527. −0.580557
\(222\) 779631. 1.06171
\(223\) 645294. 0.868951 0.434475 0.900684i \(-0.356934\pi\)
0.434475 + 0.900684i \(0.356934\pi\)
\(224\) 879062. 1.17058
\(225\) 170507. 0.224536
\(226\) 411806. 0.536317
\(227\) 397053. 0.511427 0.255714 0.966753i \(-0.417690\pi\)
0.255714 + 0.966753i \(0.417690\pi\)
\(228\) 199106. 0.253657
\(229\) 433047. 0.545690 0.272845 0.962058i \(-0.412035\pi\)
0.272845 + 0.962058i \(0.412035\pi\)
\(230\) −324652. −0.404667
\(231\) 405653. 0.500179
\(232\) 792081. 0.966161
\(233\) 775837. 0.936226 0.468113 0.883669i \(-0.344934\pi\)
0.468113 + 0.883669i \(0.344934\pi\)
\(234\) −484352. −0.578258
\(235\) −173909. −0.205425
\(236\) −141061. −0.164864
\(237\) 185871. 0.214951
\(238\) −270402. −0.309433
\(239\) −16141.9 −0.0182794 −0.00913968 0.999958i \(-0.502909\pi\)
−0.00913968 + 0.999958i \(0.502909\pi\)
\(240\) −194665. −0.218152
\(241\) 1.58738e6 1.76051 0.880255 0.474500i \(-0.157371\pi\)
0.880255 + 0.474500i \(0.157371\pi\)
\(242\) −40667.7 −0.0446387
\(243\) 1.32091e6 1.43501
\(244\) 704517. 0.757560
\(245\) 124564. 0.132580
\(246\) 1.13332e6 1.19403
\(247\) 230742. 0.240649
\(248\) 703036. 0.725852
\(249\) −802608. −0.820362
\(250\) −43401.0 −0.0439187
\(251\) −1.39816e6 −1.40079 −0.700393 0.713758i \(-0.746992\pi\)
−0.700393 + 0.713758i \(0.746992\pi\)
\(252\) 977951. 0.970099
\(253\) 565697. 0.555626
\(254\) −583532. −0.567519
\(255\) 374448. 0.360612
\(256\) −664601. −0.633812
\(257\) 1.62648e6 1.53608 0.768042 0.640400i \(-0.221231\pi\)
0.768042 + 0.640400i \(0.221231\pi\)
\(258\) 873137. 0.816645
\(259\) −1.82427e6 −1.68982
\(260\) −388053. −0.356006
\(261\) 1.38217e6 1.25592
\(262\) 29355.0 0.0264197
\(263\) 516461. 0.460413 0.230207 0.973142i \(-0.426060\pi\)
0.230207 + 0.973142i \(0.426060\pi\)
\(264\) −429635. −0.379393
\(265\) 515397. 0.450845
\(266\) 148017. 0.128265
\(267\) 775876. 0.666061
\(268\) 379864. 0.323066
\(269\) −340562. −0.286956 −0.143478 0.989653i \(-0.545829\pi\)
−0.143478 + 0.989653i \(0.545829\pi\)
\(270\) 47015.7 0.0392494
\(271\) −934573. −0.773019 −0.386509 0.922285i \(-0.626319\pi\)
−0.386509 + 0.922285i \(0.626319\pi\)
\(272\) −226104. −0.185305
\(273\) 2.14284e6 1.74013
\(274\) 123236. 0.0991657
\(275\) 75625.0 0.0603023
\(276\) 2.57854e6 2.03752
\(277\) 228804. 0.179169 0.0895846 0.995979i \(-0.471446\pi\)
0.0895846 + 0.995979i \(0.471446\pi\)
\(278\) −260420. −0.202099
\(279\) 1.22679e6 0.943540
\(280\) −576943. −0.439783
\(281\) 592380. 0.447543 0.223771 0.974642i \(-0.428163\pi\)
0.223771 + 0.974642i \(0.428163\pi\)
\(282\) −438842. −0.328613
\(283\) 1.98654e6 1.47445 0.737226 0.675647i \(-0.236135\pi\)
0.737226 + 0.675647i \(0.236135\pi\)
\(284\) −1.09849e6 −0.808164
\(285\) −204971. −0.149479
\(286\) −214825. −0.155299
\(287\) −2.65187e6 −1.90041
\(288\) −1.62464e6 −1.15419
\(289\) −984935. −0.693686
\(290\) −351819. −0.245655
\(291\) 529766. 0.366734
\(292\) 951541. 0.653086
\(293\) 1.14978e6 0.782433 0.391216 0.920299i \(-0.372054\pi\)
0.391216 + 0.920299i \(0.372054\pi\)
\(294\) 314324. 0.212085
\(295\) 145216. 0.0971538
\(296\) 1.93212e6 1.28175
\(297\) −81923.6 −0.0538912
\(298\) 1.27644e6 0.832645
\(299\) 2.98826e6 1.93304
\(300\) 344712. 0.221133
\(301\) −2.04307e6 −1.29977
\(302\) −279853. −0.176569
\(303\) −3.47104e6 −2.17197
\(304\) 123768. 0.0768114
\(305\) −725272. −0.446428
\(306\) 499743. 0.305101
\(307\) −2.53376e6 −1.53433 −0.767165 0.641449i \(-0.778333\pi\)
−0.767165 + 0.641449i \(0.778333\pi\)
\(308\) 433751. 0.260534
\(309\) −4.47883e6 −2.66851
\(310\) −312268. −0.184554
\(311\) −502306. −0.294488 −0.147244 0.989100i \(-0.547040\pi\)
−0.147244 + 0.989100i \(0.547040\pi\)
\(312\) −2.26952e6 −1.31992
\(313\) 978417. 0.564499 0.282250 0.959341i \(-0.408919\pi\)
0.282250 + 0.959341i \(0.408919\pi\)
\(314\) 736256. 0.421410
\(315\) −1.00676e6 −0.571676
\(316\) 198745. 0.111964
\(317\) 661270. 0.369599 0.184799 0.982776i \(-0.440836\pi\)
0.184799 + 0.982776i \(0.440836\pi\)
\(318\) 1.30055e6 0.721205
\(319\) 613036. 0.337294
\(320\) 139258. 0.0760231
\(321\) −2.47003e6 −1.33795
\(322\) 1.91691e6 1.03030
\(323\) −238075. −0.126972
\(324\) 1.23648e6 0.654371
\(325\) 399484. 0.209793
\(326\) −1.17729e6 −0.613535
\(327\) 2.55555e6 1.32165
\(328\) 2.80864e6 1.44149
\(329\) 1.02685e6 0.523019
\(330\) 190831. 0.0964640
\(331\) −837776. −0.420299 −0.210149 0.977669i \(-0.567395\pi\)
−0.210149 + 0.977669i \(0.567395\pi\)
\(332\) −858201. −0.427311
\(333\) 3.37153e6 1.66616
\(334\) −354956. −0.174104
\(335\) −391054. −0.190382
\(336\) 1.14940e6 0.555423
\(337\) 1.06270e6 0.509726 0.254863 0.966977i \(-0.417970\pi\)
0.254863 + 0.966977i \(0.417970\pi\)
\(338\) −103473. −0.0492647
\(339\) 3.36712e6 1.59133
\(340\) 400384. 0.187836
\(341\) 544119. 0.253401
\(342\) −273557. −0.126469
\(343\) 1.74544e6 0.801068
\(344\) 2.16385e6 0.985895
\(345\) −2.65450e6 −1.20070
\(346\) 2.05466e6 0.922676
\(347\) −2.77621e6 −1.23774 −0.618870 0.785494i \(-0.712409\pi\)
−0.618870 + 0.785494i \(0.712409\pi\)
\(348\) 2.79432e6 1.23688
\(349\) 3.96795e6 1.74382 0.871911 0.489664i \(-0.162881\pi\)
0.871911 + 0.489664i \(0.162881\pi\)
\(350\) 256262. 0.111818
\(351\) −432756. −0.187489
\(352\) −720577. −0.309973
\(353\) 411422. 0.175732 0.0878660 0.996132i \(-0.471995\pi\)
0.0878660 + 0.996132i \(0.471995\pi\)
\(354\) 366437. 0.155414
\(355\) 1.13085e6 0.476248
\(356\) 829618. 0.346939
\(357\) −2.21093e6 −0.918131
\(358\) 438985. 0.181026
\(359\) 1.47089e6 0.602343 0.301172 0.953570i \(-0.402622\pi\)
0.301172 + 0.953570i \(0.402622\pi\)
\(360\) 1.06628e6 0.433625
\(361\) 130321. 0.0526316
\(362\) −273063. −0.109520
\(363\) −332519. −0.132449
\(364\) 2.29126e6 0.906404
\(365\) −979572. −0.384861
\(366\) −1.83014e6 −0.714139
\(367\) −3.81255e6 −1.47758 −0.738790 0.673936i \(-0.764602\pi\)
−0.738790 + 0.673936i \(0.764602\pi\)
\(368\) 1.60288e6 0.616994
\(369\) 4.90106e6 1.87380
\(370\) −858191. −0.325896
\(371\) −3.04317e6 −1.14787
\(372\) 2.48019e6 0.929239
\(373\) 131393. 0.0488990 0.0244495 0.999701i \(-0.492217\pi\)
0.0244495 + 0.999701i \(0.492217\pi\)
\(374\) 221651. 0.0819391
\(375\) −354867. −0.130313
\(376\) −1.08756e6 −0.396719
\(377\) 3.23833e6 1.17346
\(378\) −277605. −0.0999304
\(379\) −993434. −0.355256 −0.177628 0.984098i \(-0.556842\pi\)
−0.177628 + 0.984098i \(0.556842\pi\)
\(380\) −219168. −0.0778608
\(381\) −4.77123e6 −1.68391
\(382\) 2.05273e6 0.719738
\(383\) 5.31329e6 1.85083 0.925416 0.378953i \(-0.123716\pi\)
0.925416 + 0.378953i \(0.123716\pi\)
\(384\) −3.97663e6 −1.37622
\(385\) −446529. −0.153532
\(386\) 158476. 0.0541372
\(387\) 3.77590e6 1.28157
\(388\) 566461. 0.191025
\(389\) −1.80123e6 −0.603526 −0.301763 0.953383i \(-0.597575\pi\)
−0.301763 + 0.953383i \(0.597575\pi\)
\(390\) 1.00806e6 0.335601
\(391\) −3.08322e6 −1.01991
\(392\) 778972. 0.256039
\(393\) 240020. 0.0783910
\(394\) −1.08234e6 −0.351256
\(395\) −204600. −0.0659800
\(396\) −801638. −0.256886
\(397\) 3.13187e6 0.997304 0.498652 0.866802i \(-0.333829\pi\)
0.498652 + 0.866802i \(0.333829\pi\)
\(398\) −1.50449e6 −0.476083
\(399\) 1.21025e6 0.380579
\(400\) 214280. 0.0669626
\(401\) −3.59188e6 −1.11548 −0.557739 0.830017i \(-0.688331\pi\)
−0.557739 + 0.830017i \(0.688331\pi\)
\(402\) −986783. −0.304549
\(403\) 2.87428e6 0.881588
\(404\) −3.71147e6 −1.13134
\(405\) −1.27290e6 −0.385619
\(406\) 2.07732e6 0.625445
\(407\) 1.49537e6 0.447470
\(408\) 2.34164e6 0.696417
\(409\) −751091. −0.222016 −0.111008 0.993820i \(-0.535408\pi\)
−0.111008 + 0.993820i \(0.535408\pi\)
\(410\) −1.24752e6 −0.366512
\(411\) 1.00764e6 0.294239
\(412\) −4.78906e6 −1.38998
\(413\) −857431. −0.247357
\(414\) −3.54274e6 −1.01587
\(415\) 883483. 0.251813
\(416\) −3.80641e6 −1.07840
\(417\) −2.12932e6 −0.599654
\(418\) −121331. −0.0339650
\(419\) 1.14562e6 0.318790 0.159395 0.987215i \(-0.449046\pi\)
0.159395 + 0.987215i \(0.449046\pi\)
\(420\) −2.03536e6 −0.563012
\(421\) −6.27013e6 −1.72413 −0.862067 0.506794i \(-0.830830\pi\)
−0.862067 + 0.506794i \(0.830830\pi\)
\(422\) −2.92988e6 −0.800883
\(423\) −1.89778e6 −0.515697
\(424\) 3.22308e6 0.870675
\(425\) −412179. −0.110691
\(426\) 2.85357e6 0.761842
\(427\) 4.28238e6 1.13662
\(428\) −2.64112e6 −0.696913
\(429\) −1.75651e6 −0.460795
\(430\) −961119. −0.250672
\(431\) −7.00508e6 −1.81643 −0.908217 0.418499i \(-0.862556\pi\)
−0.908217 + 0.418499i \(0.862556\pi\)
\(432\) −232127. −0.0598435
\(433\) −3.02004e6 −0.774093 −0.387046 0.922060i \(-0.626505\pi\)
−0.387046 + 0.922060i \(0.626505\pi\)
\(434\) 1.84379e6 0.469881
\(435\) −2.87664e6 −0.728891
\(436\) 2.73257e6 0.688422
\(437\) 1.68774e6 0.422768
\(438\) −2.47184e6 −0.615653
\(439\) 5.26645e6 1.30424 0.652119 0.758116i \(-0.273880\pi\)
0.652119 + 0.758116i \(0.273880\pi\)
\(440\) 472927. 0.116456
\(441\) 1.35930e6 0.332827
\(442\) 1.17086e6 0.285069
\(443\) −4.33126e6 −1.04859 −0.524294 0.851537i \(-0.675671\pi\)
−0.524294 + 0.851537i \(0.675671\pi\)
\(444\) 6.81617e6 1.64090
\(445\) −854057. −0.204450
\(446\) −1.79241e6 −0.426677
\(447\) 1.04368e7 2.47057
\(448\) −822252. −0.193557
\(449\) −3.07642e6 −0.720162 −0.360081 0.932921i \(-0.617251\pi\)
−0.360081 + 0.932921i \(0.617251\pi\)
\(450\) −473611. −0.110253
\(451\) 2.17377e6 0.503236
\(452\) 3.60035e6 0.828893
\(453\) −2.28821e6 −0.523903
\(454\) −1.10288e6 −0.251124
\(455\) −2.35876e6 −0.534141
\(456\) −1.28180e6 −0.288675
\(457\) −5.20085e6 −1.16489 −0.582444 0.812871i \(-0.697903\pi\)
−0.582444 + 0.812871i \(0.697903\pi\)
\(458\) −1.20286e6 −0.267948
\(459\) 446508. 0.0989231
\(460\) −2.83837e6 −0.625424
\(461\) −7.02031e6 −1.53852 −0.769261 0.638934i \(-0.779376\pi\)
−0.769261 + 0.638934i \(0.779376\pi\)
\(462\) −1.12677e6 −0.245601
\(463\) −1.26341e6 −0.273900 −0.136950 0.990578i \(-0.543730\pi\)
−0.136950 + 0.990578i \(0.543730\pi\)
\(464\) 1.73701e6 0.374549
\(465\) −2.55325e6 −0.547598
\(466\) −2.15501e6 −0.459711
\(467\) 3.77999e6 0.802045 0.401022 0.916068i \(-0.368655\pi\)
0.401022 + 0.916068i \(0.368655\pi\)
\(468\) −4.23460e6 −0.893713
\(469\) 2.30899e6 0.484718
\(470\) 483062. 0.100869
\(471\) 6.01998e6 1.25038
\(472\) 908121. 0.187624
\(473\) 1.67472e6 0.344184
\(474\) −516286. −0.105547
\(475\) 225625. 0.0458831
\(476\) −2.36407e6 −0.478237
\(477\) 5.62424e6 1.13180
\(478\) 44836.8 0.00897563
\(479\) −277404. −0.0552426 −0.0276213 0.999618i \(-0.508793\pi\)
−0.0276213 + 0.999618i \(0.508793\pi\)
\(480\) 3.38128e6 0.669850
\(481\) 7.89922e6 1.55676
\(482\) −4.40921e6 −0.864456
\(483\) 1.56736e7 3.05703
\(484\) −355551. −0.0689903
\(485\) −583148. −0.112570
\(486\) −3.66903e6 −0.704629
\(487\) 733481. 0.140141 0.0700707 0.997542i \(-0.477678\pi\)
0.0700707 + 0.997542i \(0.477678\pi\)
\(488\) −4.53555e6 −0.862144
\(489\) −9.62609e6 −1.82044
\(490\) −345997. −0.0651002
\(491\) −2.33920e6 −0.437888 −0.218944 0.975737i \(-0.570261\pi\)
−0.218944 + 0.975737i \(0.570261\pi\)
\(492\) 9.90842e6 1.84540
\(493\) −3.34123e6 −0.619140
\(494\) −640924. −0.118165
\(495\) 825254. 0.151382
\(496\) 1.54174e6 0.281389
\(497\) −6.67711e6 −1.21254
\(498\) 2.22937e6 0.402819
\(499\) −4.49956e6 −0.808944 −0.404472 0.914550i \(-0.632545\pi\)
−0.404472 + 0.914550i \(0.632545\pi\)
\(500\) −379447. −0.0678775
\(501\) −2.90229e6 −0.516590
\(502\) 3.88361e6 0.687822
\(503\) −2.62082e6 −0.461868 −0.230934 0.972969i \(-0.574178\pi\)
−0.230934 + 0.972969i \(0.574178\pi\)
\(504\) −6.29586e6 −1.10402
\(505\) 3.82080e6 0.666693
\(506\) −1.57131e6 −0.272827
\(507\) −846045. −0.146175
\(508\) −5.10172e6 −0.877116
\(509\) 4.58014e6 0.783582 0.391791 0.920054i \(-0.371856\pi\)
0.391791 + 0.920054i \(0.371856\pi\)
\(510\) −1.04009e6 −0.177070
\(511\) 5.78390e6 0.979870
\(512\) −3.75695e6 −0.633375
\(513\) −244417. −0.0410050
\(514\) −4.51780e6 −0.754256
\(515\) 4.93014e6 0.819109
\(516\) 7.63368e6 1.26215
\(517\) −841722. −0.138498
\(518\) 5.06720e6 0.829743
\(519\) 1.67998e7 2.73771
\(520\) 2.49821e6 0.405154
\(521\) −7.30964e6 −1.17978 −0.589891 0.807483i \(-0.700829\pi\)
−0.589891 + 0.807483i \(0.700829\pi\)
\(522\) −3.83921e6 −0.616688
\(523\) −3.94203e6 −0.630181 −0.315091 0.949062i \(-0.602035\pi\)
−0.315091 + 0.949062i \(0.602035\pi\)
\(524\) 256645. 0.0408324
\(525\) 2.09532e6 0.331781
\(526\) −1.43455e6 −0.226075
\(527\) −2.96561e6 −0.465144
\(528\) −942179. −0.147078
\(529\) 1.54209e7 2.39592
\(530\) −1.43160e6 −0.221376
\(531\) 1.58466e6 0.243894
\(532\) 1.29408e6 0.198236
\(533\) 1.14828e7 1.75077
\(534\) −2.15512e6 −0.327053
\(535\) 2.71892e6 0.410688
\(536\) −2.44549e6 −0.367666
\(537\) 3.58935e6 0.537131
\(538\) 945967. 0.140903
\(539\) 602891. 0.0893854
\(540\) 411050. 0.0606611
\(541\) 4.42219e6 0.649598 0.324799 0.945783i \(-0.394703\pi\)
0.324799 + 0.945783i \(0.394703\pi\)
\(542\) 2.59593e6 0.379572
\(543\) −2.23269e6 −0.324960
\(544\) 3.92736e6 0.568989
\(545\) −2.81307e6 −0.405685
\(546\) −5.95208e6 −0.854451
\(547\) 1.36377e7 1.94883 0.974414 0.224760i \(-0.0721597\pi\)
0.974414 + 0.224760i \(0.0721597\pi\)
\(548\) 1.07743e6 0.153263
\(549\) −7.91449e6 −1.12071
\(550\) −210061. −0.0296100
\(551\) 1.82898e6 0.256643
\(552\) −1.66002e7 −2.31881
\(553\) 1.20806e6 0.167987
\(554\) −635539. −0.0879767
\(555\) −7.01697e6 −0.966980
\(556\) −2.27681e6 −0.312349
\(557\) −1.22341e7 −1.67084 −0.835421 0.549611i \(-0.814776\pi\)
−0.835421 + 0.549611i \(0.814776\pi\)
\(558\) −3.40761e6 −0.463302
\(559\) 8.84663e6 1.19742
\(560\) −1.26522e6 −0.170489
\(561\) 1.81233e6 0.243125
\(562\) −1.64543e6 −0.219755
\(563\) −7.67718e6 −1.02078 −0.510388 0.859944i \(-0.670498\pi\)
−0.510388 + 0.859944i \(0.670498\pi\)
\(564\) −3.83671e6 −0.507881
\(565\) −3.70641e6 −0.488464
\(566\) −5.51793e6 −0.723994
\(567\) 7.51589e6 0.981799
\(568\) 7.07185e6 0.919734
\(569\) 1.41750e7 1.83545 0.917727 0.397212i \(-0.130022\pi\)
0.917727 + 0.397212i \(0.130022\pi\)
\(570\) 569340. 0.0733981
\(571\) −7.51681e6 −0.964814 −0.482407 0.875947i \(-0.660237\pi\)
−0.482407 + 0.875947i \(0.660237\pi\)
\(572\) −1.87818e6 −0.240019
\(573\) 1.67841e7 2.13556
\(574\) 7.36600e6 0.933151
\(575\) 2.92199e6 0.368560
\(576\) 1.51965e6 0.190847
\(577\) −9.67827e6 −1.21020 −0.605102 0.796148i \(-0.706868\pi\)
−0.605102 + 0.796148i \(0.706868\pi\)
\(578\) 2.73582e6 0.340618
\(579\) 1.29578e6 0.160633
\(580\) −3.07589e6 −0.379666
\(581\) −5.21654e6 −0.641124
\(582\) −1.47151e6 −0.180076
\(583\) 2.49452e6 0.303960
\(584\) −6.12584e6 −0.743247
\(585\) 4.35935e6 0.526662
\(586\) −3.19371e6 −0.384195
\(587\) 7.43801e6 0.890967 0.445483 0.895290i \(-0.353032\pi\)
0.445483 + 0.895290i \(0.353032\pi\)
\(588\) 2.74808e6 0.327783
\(589\) 1.62336e6 0.192809
\(590\) −403361. −0.0477050
\(591\) −8.84973e6 −1.04222
\(592\) 4.23708e6 0.496893
\(593\) −7.09183e6 −0.828174 −0.414087 0.910237i \(-0.635899\pi\)
−0.414087 + 0.910237i \(0.635899\pi\)
\(594\) 227556. 0.0264620
\(595\) 2.43372e6 0.281824
\(596\) 1.11597e7 1.28688
\(597\) −1.23015e7 −1.41261
\(598\) −8.30037e6 −0.949171
\(599\) 6.07424e6 0.691712 0.345856 0.938288i \(-0.387589\pi\)
0.345856 + 0.938288i \(0.387589\pi\)
\(600\) −2.21919e6 −0.251661
\(601\) 3.66384e6 0.413762 0.206881 0.978366i \(-0.433669\pi\)
0.206881 + 0.978366i \(0.433669\pi\)
\(602\) 5.67494e6 0.638220
\(603\) −4.26736e6 −0.477932
\(604\) −2.44671e6 −0.272891
\(605\) 366025. 0.0406558
\(606\) 9.64138e6 1.06649
\(607\) 1.10632e7 1.21873 0.609366 0.792889i \(-0.291424\pi\)
0.609366 + 0.792889i \(0.291424\pi\)
\(608\) −2.14982e6 −0.235854
\(609\) 1.69852e7 1.85578
\(610\) 2.01456e6 0.219208
\(611\) −4.44634e6 −0.481837
\(612\) 4.36917e6 0.471542
\(613\) −1.59812e7 −1.71775 −0.858873 0.512189i \(-0.828835\pi\)
−0.858873 + 0.512189i \(0.828835\pi\)
\(614\) 7.03792e6 0.753396
\(615\) −1.02003e7 −1.08749
\(616\) −2.79241e6 −0.296501
\(617\) −8.35800e6 −0.883872 −0.441936 0.897047i \(-0.645708\pi\)
−0.441936 + 0.897047i \(0.645708\pi\)
\(618\) 1.24407e7 1.31031
\(619\) −585900. −0.0614606 −0.0307303 0.999528i \(-0.509783\pi\)
−0.0307303 + 0.999528i \(0.509783\pi\)
\(620\) −2.73011e6 −0.285233
\(621\) −3.16535e6 −0.329377
\(622\) 1.39524e6 0.144601
\(623\) 5.04280e6 0.520537
\(624\) −4.97700e6 −0.511689
\(625\) 390625. 0.0400000
\(626\) −2.71771e6 −0.277184
\(627\) −992060. −0.100779
\(628\) 6.43696e6 0.651301
\(629\) −8.15024e6 −0.821379
\(630\) 2.79644e6 0.280708
\(631\) 1.18136e7 1.18116 0.590580 0.806979i \(-0.298899\pi\)
0.590580 + 0.806979i \(0.298899\pi\)
\(632\) −1.27948e6 −0.127421
\(633\) −2.39561e7 −2.37633
\(634\) −1.83678e6 −0.181483
\(635\) 5.25201e6 0.516882
\(636\) 1.13705e7 1.11464
\(637\) 3.18473e6 0.310974
\(638\) −1.70281e6 −0.165620
\(639\) 1.23403e7 1.19557
\(640\) 4.37733e6 0.422435
\(641\) −1.70903e7 −1.64287 −0.821437 0.570299i \(-0.806828\pi\)
−0.821437 + 0.570299i \(0.806828\pi\)
\(642\) 6.86091e6 0.656968
\(643\) −5.15697e6 −0.491889 −0.245945 0.969284i \(-0.579098\pi\)
−0.245945 + 0.969284i \(0.579098\pi\)
\(644\) 1.67592e7 1.59235
\(645\) −7.85857e6 −0.743779
\(646\) 661290. 0.0623463
\(647\) 1.65395e7 1.55332 0.776661 0.629919i \(-0.216912\pi\)
0.776661 + 0.629919i \(0.216912\pi\)
\(648\) −7.96022e6 −0.744710
\(649\) 702846. 0.0655011
\(650\) −1.10963e6 −0.103014
\(651\) 1.50757e7 1.39420
\(652\) −1.02928e7 −0.948236
\(653\) 2.10047e7 1.92767 0.963837 0.266492i \(-0.0858647\pi\)
0.963837 + 0.266492i \(0.0858647\pi\)
\(654\) −7.09846e6 −0.648963
\(655\) −264206. −0.0240624
\(656\) 6.15929e6 0.558819
\(657\) −1.06895e7 −0.966151
\(658\) −2.85224e6 −0.256816
\(659\) −1.76115e7 −1.57973 −0.789866 0.613280i \(-0.789850\pi\)
−0.789866 + 0.613280i \(0.789850\pi\)
\(660\) 1.66841e6 0.149088
\(661\) 1.20980e7 1.07699 0.538493 0.842630i \(-0.318994\pi\)
0.538493 + 0.842630i \(0.318994\pi\)
\(662\) 2.32706e6 0.206377
\(663\) 9.57351e6 0.845838
\(664\) 5.52494e6 0.486303
\(665\) −1.33221e6 −0.116820
\(666\) −9.36496e6 −0.818126
\(667\) 2.36864e7 2.06150
\(668\) −3.10332e6 −0.269082
\(669\) −1.46556e7 −1.26601
\(670\) 1.08622e6 0.0934823
\(671\) −3.51032e6 −0.300981
\(672\) −1.99648e7 −1.70546
\(673\) −1.67169e6 −0.142271 −0.0711357 0.997467i \(-0.522662\pi\)
−0.0711357 + 0.997467i \(0.522662\pi\)
\(674\) −2.95183e6 −0.250289
\(675\) −423159. −0.0357474
\(676\) −904647. −0.0761399
\(677\) −1.13500e7 −0.951753 −0.475877 0.879512i \(-0.657869\pi\)
−0.475877 + 0.879512i \(0.657869\pi\)
\(678\) −9.35273e6 −0.781383
\(679\) 3.44321e6 0.286608
\(680\) −2.57759e6 −0.213768
\(681\) −9.01766e6 −0.745120
\(682\) −1.51138e6 −0.124426
\(683\) −5.37535e6 −0.440915 −0.220458 0.975397i \(-0.570755\pi\)
−0.220458 + 0.975397i \(0.570755\pi\)
\(684\) −2.39166e6 −0.195461
\(685\) −1.10917e6 −0.0903176
\(686\) −4.84824e6 −0.393345
\(687\) −9.83514e6 −0.795039
\(688\) 4.74527e6 0.382199
\(689\) 1.31772e7 1.05748
\(690\) 7.37332e6 0.589576
\(691\) −1.10396e7 −0.879542 −0.439771 0.898110i \(-0.644940\pi\)
−0.439771 + 0.898110i \(0.644940\pi\)
\(692\) 1.79635e7 1.42602
\(693\) −4.87272e6 −0.385424
\(694\) 7.71138e6 0.607762
\(695\) 2.34388e6 0.184066
\(696\) −1.79893e7 −1.40764
\(697\) −1.18477e7 −0.923744
\(698\) −1.10216e7 −0.856262
\(699\) −1.76204e7 −1.36403
\(700\) 2.24045e6 0.172818
\(701\) −7.18378e6 −0.552151 −0.276075 0.961136i \(-0.589034\pi\)
−0.276075 + 0.961136i \(0.589034\pi\)
\(702\) 1.20205e6 0.0920619
\(703\) 4.46141e6 0.340474
\(704\) 674009. 0.0512547
\(705\) 3.94974e6 0.299292
\(706\) −1.14279e6 −0.0862890
\(707\) −2.25600e7 −1.69742
\(708\) 3.20369e6 0.240197
\(709\) −72529.0 −0.00541871 −0.00270936 0.999996i \(-0.500862\pi\)
−0.00270936 + 0.999996i \(0.500862\pi\)
\(710\) −3.14111e6 −0.233850
\(711\) −2.23268e6 −0.165635
\(712\) −5.34092e6 −0.394835
\(713\) 2.10236e7 1.54876
\(714\) 6.14122e6 0.450826
\(715\) 1.93350e6 0.141443
\(716\) 3.83797e6 0.279781
\(717\) 366607. 0.0266320
\(718\) −4.08563e6 −0.295766
\(719\) −8.52315e6 −0.614862 −0.307431 0.951570i \(-0.599469\pi\)
−0.307431 + 0.951570i \(0.599469\pi\)
\(720\) 2.33832e6 0.168102
\(721\) −2.91101e7 −2.08548
\(722\) −361988. −0.0258435
\(723\) −3.60518e7 −2.56496
\(724\) −2.38734e6 −0.169266
\(725\) 3.16651e6 0.223736
\(726\) 923624. 0.0650360
\(727\) −1.22292e7 −0.858145 −0.429072 0.903270i \(-0.641159\pi\)
−0.429072 + 0.903270i \(0.641159\pi\)
\(728\) −1.47507e7 −1.03154
\(729\) −1.76271e7 −1.22846
\(730\) 2.72092e6 0.188977
\(731\) −9.12775e6 −0.631786
\(732\) −1.60006e7 −1.10372
\(733\) −1.56294e7 −1.07444 −0.537219 0.843443i \(-0.680525\pi\)
−0.537219 + 0.843443i \(0.680525\pi\)
\(734\) 1.05900e7 0.725530
\(735\) −2.82904e6 −0.193161
\(736\) −2.78415e7 −1.89452
\(737\) −1.89270e6 −0.128355
\(738\) −1.36135e7 −0.920086
\(739\) −8.31727e6 −0.560235 −0.280117 0.959966i \(-0.590373\pi\)
−0.280117 + 0.959966i \(0.590373\pi\)
\(740\) −7.50301e6 −0.503682
\(741\) −5.24050e6 −0.350612
\(742\) 8.45289e6 0.563632
\(743\) 2.91763e7 1.93891 0.969456 0.245266i \(-0.0788753\pi\)
0.969456 + 0.245266i \(0.0788753\pi\)
\(744\) −1.59670e7 −1.05752
\(745\) −1.14884e7 −0.758352
\(746\) −364965. −0.0240107
\(747\) 9.64096e6 0.632148
\(748\) 1.93786e6 0.126639
\(749\) −1.60539e7 −1.04563
\(750\) 985700. 0.0639870
\(751\) −2.28320e7 −1.47722 −0.738610 0.674133i \(-0.764517\pi\)
−0.738610 + 0.674133i \(0.764517\pi\)
\(752\) −2.38499e6 −0.153795
\(753\) 3.17542e7 2.04086
\(754\) −8.99497e6 −0.576198
\(755\) 2.51879e6 0.160814
\(756\) −2.42705e6 −0.154445
\(757\) 2.14668e7 1.36153 0.680767 0.732500i \(-0.261647\pi\)
0.680767 + 0.732500i \(0.261647\pi\)
\(758\) 2.75942e6 0.174440
\(759\) −1.28478e7 −0.809514
\(760\) 1.41096e6 0.0886099
\(761\) 7.29835e6 0.456839 0.228419 0.973563i \(-0.426644\pi\)
0.228419 + 0.973563i \(0.426644\pi\)
\(762\) 1.32529e7 0.826842
\(763\) 1.66098e7 1.03289
\(764\) 1.79467e7 1.11237
\(765\) −4.49788e6 −0.277878
\(766\) −1.47585e7 −0.908806
\(767\) 3.71274e6 0.227880
\(768\) 1.50941e7 0.923428
\(769\) −1.77905e7 −1.08486 −0.542428 0.840103i \(-0.682495\pi\)
−0.542428 + 0.840103i \(0.682495\pi\)
\(770\) 1.24031e6 0.0753880
\(771\) −3.69397e7 −2.23798
\(772\) 1.38553e6 0.0836705
\(773\) 2.07962e7 1.25180 0.625900 0.779903i \(-0.284732\pi\)
0.625900 + 0.779903i \(0.284732\pi\)
\(774\) −1.04882e7 −0.629284
\(775\) 2.81053e6 0.168087
\(776\) −3.64676e6 −0.217397
\(777\) 4.14318e7 2.46196
\(778\) 5.00322e6 0.296347
\(779\) 6.48538e6 0.382905
\(780\) 8.81325e6 0.518680
\(781\) 5.47330e6 0.321086
\(782\) 8.56413e6 0.500802
\(783\) −3.43024e6 −0.199949
\(784\) 1.70827e6 0.0992580
\(785\) −6.62658e6 −0.383809
\(786\) −666695. −0.0384920
\(787\) −1.93347e7 −1.11276 −0.556378 0.830929i \(-0.687809\pi\)
−0.556378 + 0.830929i \(0.687809\pi\)
\(788\) −9.46271e6 −0.542875
\(789\) −1.17296e7 −0.670796
\(790\) 568309. 0.0323979
\(791\) 2.18846e7 1.24365
\(792\) 5.16079e6 0.292350
\(793\) −1.85430e7 −1.04712
\(794\) −8.69927e6 −0.489702
\(795\) −1.17054e7 −0.656855
\(796\) −1.31535e7 −0.735800
\(797\) 2.30780e7 1.28692 0.643462 0.765478i \(-0.277497\pi\)
0.643462 + 0.765478i \(0.277497\pi\)
\(798\) −3.36168e6 −0.186874
\(799\) 4.58764e6 0.254227
\(800\) −3.72199e6 −0.205613
\(801\) −9.31985e6 −0.513249
\(802\) 9.97702e6 0.547728
\(803\) −4.74113e6 −0.259473
\(804\) −8.62727e6 −0.470688
\(805\) −1.72529e7 −0.938367
\(806\) −7.98377e6 −0.432883
\(807\) 7.73467e6 0.418079
\(808\) 2.38937e7 1.28752
\(809\) −2.12625e7 −1.14220 −0.571102 0.820879i \(-0.693484\pi\)
−0.571102 + 0.820879i \(0.693484\pi\)
\(810\) 3.53570e6 0.189349
\(811\) 8.66812e6 0.462778 0.231389 0.972861i \(-0.425673\pi\)
0.231389 + 0.972861i \(0.425673\pi\)
\(812\) 1.81617e7 0.966642
\(813\) 2.12255e7 1.12624
\(814\) −4.15364e6 −0.219719
\(815\) 1.05961e7 0.558792
\(816\) 5.13516e6 0.269978
\(817\) 4.99649e6 0.261885
\(818\) 2.08628e6 0.109016
\(819\) −2.57399e7 −1.34090
\(820\) −1.09068e7 −0.566453
\(821\) −4.48129e6 −0.232031 −0.116015 0.993247i \(-0.537012\pi\)
−0.116015 + 0.993247i \(0.537012\pi\)
\(822\) −2.79887e6 −0.144479
\(823\) 3.41551e7 1.75774 0.878872 0.477057i \(-0.158296\pi\)
0.878872 + 0.477057i \(0.158296\pi\)
\(824\) 3.08311e7 1.58187
\(825\) −1.71756e6 −0.0878569
\(826\) 2.38165e6 0.121459
\(827\) 493047. 0.0250683 0.0125341 0.999921i \(-0.496010\pi\)
0.0125341 + 0.999921i \(0.496010\pi\)
\(828\) −3.09736e7 −1.57006
\(829\) −7.83379e6 −0.395900 −0.197950 0.980212i \(-0.563428\pi\)
−0.197950 + 0.980212i \(0.563428\pi\)
\(830\) −2.45402e6 −0.123647
\(831\) −5.19647e6 −0.261039
\(832\) 3.56041e6 0.178317
\(833\) −3.28593e6 −0.164076
\(834\) 5.91453e6 0.294446
\(835\) 3.19474e6 0.158569
\(836\) −1.06078e6 −0.0524938
\(837\) −3.04461e6 −0.150217
\(838\) −3.18213e6 −0.156534
\(839\) −1.09993e7 −0.539459 −0.269729 0.962936i \(-0.586934\pi\)
−0.269729 + 0.962936i \(0.586934\pi\)
\(840\) 1.31032e7 0.640738
\(841\) 5.15739e6 0.251443
\(842\) 1.74163e7 0.846594
\(843\) −1.34538e7 −0.652044
\(844\) −2.56154e7 −1.23779
\(845\) 931297. 0.0448690
\(846\) 5.27138e6 0.253220
\(847\) −2.16120e6 −0.103511
\(848\) 7.06813e6 0.337532
\(849\) −4.51172e7 −2.14819
\(850\) 1.14489e6 0.0543523
\(851\) 5.77780e7 2.73488
\(852\) 2.49483e7 1.17745
\(853\) 1.18799e7 0.559035 0.279518 0.960141i \(-0.409825\pi\)
0.279518 + 0.960141i \(0.409825\pi\)
\(854\) −1.18950e7 −0.558110
\(855\) 2.46212e6 0.115184
\(856\) 1.70030e7 0.793125
\(857\) 1.87188e6 0.0870614 0.0435307 0.999052i \(-0.486139\pi\)
0.0435307 + 0.999052i \(0.486139\pi\)
\(858\) 4.87899e6 0.226262
\(859\) −1.32533e7 −0.612831 −0.306415 0.951898i \(-0.599130\pi\)
−0.306415 + 0.951898i \(0.599130\pi\)
\(860\) −8.40289e6 −0.387421
\(861\) 6.02279e7 2.76879
\(862\) 1.94577e7 0.891916
\(863\) 8.61616e6 0.393810 0.196905 0.980423i \(-0.436911\pi\)
0.196905 + 0.980423i \(0.436911\pi\)
\(864\) 4.03198e6 0.183753
\(865\) −1.84927e7 −0.840349
\(866\) 8.38865e6 0.380099
\(867\) 2.23693e7 1.01066
\(868\) 1.61200e7 0.726214
\(869\) −990263. −0.0444837
\(870\) 7.99034e6 0.357904
\(871\) −9.99809e6 −0.446551
\(872\) −1.75917e7 −0.783461
\(873\) −6.36357e6 −0.282596
\(874\) −4.68797e6 −0.207590
\(875\) −2.30645e6 −0.101841
\(876\) −2.16109e7 −0.951508
\(877\) −3.19796e7 −1.40402 −0.702012 0.712165i \(-0.747715\pi\)
−0.702012 + 0.712165i \(0.747715\pi\)
\(878\) −1.46284e7 −0.640415
\(879\) −2.61133e7 −1.13996
\(880\) 1.03712e6 0.0451462
\(881\) 5.05038e6 0.219222 0.109611 0.993975i \(-0.465039\pi\)
0.109611 + 0.993975i \(0.465039\pi\)
\(882\) −3.77567e6 −0.163427
\(883\) 1.11703e7 0.482130 0.241065 0.970509i \(-0.422503\pi\)
0.241065 + 0.970509i \(0.422503\pi\)
\(884\) 1.02366e7 0.440581
\(885\) −3.29807e6 −0.141547
\(886\) 1.20308e7 0.514884
\(887\) 2.80528e7 1.19720 0.598600 0.801048i \(-0.295724\pi\)
0.598600 + 0.801048i \(0.295724\pi\)
\(888\) −4.38812e7 −1.86744
\(889\) −3.10106e7 −1.31600
\(890\) 2.37228e6 0.100390
\(891\) −6.16086e6 −0.259984
\(892\) −1.56707e7 −0.659441
\(893\) −2.51125e6 −0.105381
\(894\) −2.89899e7 −1.21312
\(895\) −3.95103e6 −0.164874
\(896\) −2.58460e7 −1.07553
\(897\) −6.78677e7 −2.81632
\(898\) 8.54527e6 0.353618
\(899\) 2.27829e7 0.940178
\(900\) −4.14069e6 −0.170399
\(901\) −1.35959e7 −0.557950
\(902\) −6.03799e6 −0.247102
\(903\) 4.64010e7 1.89369
\(904\) −2.31783e7 −0.943325
\(905\) 2.45767e6 0.0997476
\(906\) 6.35588e6 0.257250
\(907\) 3.38959e6 0.136814 0.0684068 0.997658i \(-0.478208\pi\)
0.0684068 + 0.997658i \(0.478208\pi\)
\(908\) −9.64227e6 −0.388119
\(909\) 4.16943e7 1.67366
\(910\) 6.55184e6 0.262277
\(911\) −3.70080e7 −1.47741 −0.738703 0.674031i \(-0.764561\pi\)
−0.738703 + 0.674031i \(0.764561\pi\)
\(912\) −2.81096e6 −0.111910
\(913\) 4.27606e6 0.169772
\(914\) 1.44462e7 0.571990
\(915\) 1.64720e7 0.650419
\(916\) −1.05164e7 −0.414121
\(917\) 1.56001e6 0.0612637
\(918\) −1.24025e6 −0.0485738
\(919\) 1.57795e7 0.616316 0.308158 0.951335i \(-0.400287\pi\)
0.308158 + 0.951335i \(0.400287\pi\)
\(920\) 1.82729e7 0.711766
\(921\) 5.75454e7 2.23543
\(922\) 1.95000e7 0.755454
\(923\) 2.89124e7 1.11707
\(924\) −9.85113e6 −0.379582
\(925\) 7.72404e6 0.296818
\(926\) 3.50933e6 0.134492
\(927\) 5.37999e7 2.05628
\(928\) −3.01714e7 −1.15007
\(929\) 1.99445e7 0.758201 0.379101 0.925355i \(-0.376233\pi\)
0.379101 + 0.925355i \(0.376233\pi\)
\(930\) 7.09207e6 0.268885
\(931\) 1.79871e6 0.0680121
\(932\) −1.88409e7 −0.710496
\(933\) 1.14081e7 0.429052
\(934\) −1.04995e7 −0.393825
\(935\) −1.99495e6 −0.0746281
\(936\) 2.72616e7 1.01709
\(937\) 3.56765e7 1.32749 0.663747 0.747957i \(-0.268965\pi\)
0.663747 + 0.747957i \(0.268965\pi\)
\(938\) −6.41358e6 −0.238009
\(939\) −2.22213e7 −0.822442
\(940\) 4.22332e6 0.155896
\(941\) 1.71799e7 0.632478 0.316239 0.948680i \(-0.397580\pi\)
0.316239 + 0.948680i \(0.397580\pi\)
\(942\) −1.67215e7 −0.613970
\(943\) 8.39897e7 3.07572
\(944\) 1.99149e6 0.0727357
\(945\) 2.49855e6 0.0910140
\(946\) −4.65182e6 −0.169003
\(947\) −3.05290e7 −1.10621 −0.553105 0.833112i \(-0.686557\pi\)
−0.553105 + 0.833112i \(0.686557\pi\)
\(948\) −4.51379e6 −0.163125
\(949\) −2.50447e7 −0.902715
\(950\) −626710. −0.0225298
\(951\) −1.50184e7 −0.538484
\(952\) 1.52194e7 0.544260
\(953\) −4.87656e7 −1.73933 −0.869663 0.493646i \(-0.835664\pi\)
−0.869663 + 0.493646i \(0.835664\pi\)
\(954\) −1.56222e7 −0.555741
\(955\) −1.84754e7 −0.655519
\(956\) 392000. 0.0138721
\(957\) −1.39230e7 −0.491418
\(958\) 770535. 0.0271256
\(959\) 6.54912e6 0.229952
\(960\) −3.16276e6 −0.110761
\(961\) −8.40747e6 −0.293668
\(962\) −2.19414e7 −0.764409
\(963\) 2.96701e7 1.03099
\(964\) −3.85489e7 −1.33604
\(965\) −1.42634e6 −0.0493067
\(966\) −4.35359e7 −1.50108
\(967\) 1.89646e7 0.652194 0.326097 0.945336i \(-0.394266\pi\)
0.326097 + 0.945336i \(0.394266\pi\)
\(968\) 2.28897e6 0.0785147
\(969\) 5.40702e6 0.184990
\(970\) 1.61979e6 0.0552750
\(971\) −4.65960e7 −1.58599 −0.792995 0.609228i \(-0.791480\pi\)
−0.792995 + 0.609228i \(0.791480\pi\)
\(972\) −3.20777e7 −1.08902
\(973\) −1.38395e7 −0.468639
\(974\) −2.03736e6 −0.0688131
\(975\) −9.07288e6 −0.305657
\(976\) −9.94634e6 −0.334225
\(977\) −1.52749e7 −0.511968 −0.255984 0.966681i \(-0.582399\pi\)
−0.255984 + 0.966681i \(0.582399\pi\)
\(978\) 2.67380e7 0.893885
\(979\) −4.13364e6 −0.137840
\(980\) −3.02499e6 −0.100614
\(981\) −3.06974e7 −1.01843
\(982\) 6.49750e6 0.215014
\(983\) 3.09412e7 1.02130 0.510649 0.859789i \(-0.329405\pi\)
0.510649 + 0.859789i \(0.329405\pi\)
\(984\) −6.37885e7 −2.10017
\(985\) 9.74148e6 0.319915
\(986\) 9.28080e6 0.304014
\(987\) −2.33213e7 −0.762009
\(988\) −5.60348e6 −0.182627
\(989\) 6.47077e7 2.10361
\(990\) −2.29228e6 −0.0743325
\(991\) −6.76991e6 −0.218977 −0.109489 0.993988i \(-0.534921\pi\)
−0.109489 + 0.993988i \(0.534921\pi\)
\(992\) −2.67796e7 −0.864021
\(993\) 1.90271e7 0.612351
\(994\) 1.85467e7 0.595390
\(995\) 1.35410e7 0.433604
\(996\) 1.94910e7 0.622567
\(997\) 1.52639e6 0.0486326 0.0243163 0.999704i \(-0.492259\pi\)
0.0243163 + 0.999704i \(0.492259\pi\)
\(998\) 1.24982e7 0.397212
\(999\) −8.36735e6 −0.265261
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 1045.6.a.h.1.15 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.6.a.h.1.15 40 1.1 even 1 trivial