Properties

Label 177.6.a.d.1.8
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3879361069\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 331 x^{11} - 41 x^{10} + 41990 x^{9} + 7229 x^{8} - 2592364 x^{7} - 312100 x^{6} + 81977088 x^{5} - 3773728 x^{4} - 1245415104 x^{3} + 453320896 x^{2} + 6872784896 x - 6400833792\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(2.69937\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.69937 q^{2} -9.00000 q^{3} -24.7134 q^{4} -80.3454 q^{5} -24.2943 q^{6} -162.093 q^{7} -153.090 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+2.69937 q^{2} -9.00000 q^{3} -24.7134 q^{4} -80.3454 q^{5} -24.2943 q^{6} -162.093 q^{7} -153.090 q^{8} +81.0000 q^{9} -216.882 q^{10} -426.976 q^{11} +222.421 q^{12} -106.245 q^{13} -437.550 q^{14} +723.108 q^{15} +377.581 q^{16} -864.470 q^{17} +218.649 q^{18} -676.226 q^{19} +1985.61 q^{20} +1458.84 q^{21} -1152.57 q^{22} +783.598 q^{23} +1377.81 q^{24} +3330.38 q^{25} -286.795 q^{26} -729.000 q^{27} +4005.88 q^{28} -2853.46 q^{29} +1951.94 q^{30} -1908.96 q^{31} +5918.13 q^{32} +3842.78 q^{33} -2333.52 q^{34} +13023.5 q^{35} -2001.79 q^{36} +568.886 q^{37} -1825.38 q^{38} +956.205 q^{39} +12300.1 q^{40} -5535.95 q^{41} +3937.95 q^{42} -6906.55 q^{43} +10552.0 q^{44} -6507.97 q^{45} +2115.22 q^{46} -2905.40 q^{47} -3398.23 q^{48} +9467.29 q^{49} +8989.92 q^{50} +7780.23 q^{51} +2625.68 q^{52} -29299.4 q^{53} -1967.84 q^{54} +34305.5 q^{55} +24815.0 q^{56} +6086.03 q^{57} -7702.54 q^{58} +3481.00 q^{59} -17870.5 q^{60} +15250.0 q^{61} -5153.00 q^{62} -13129.6 q^{63} +3892.61 q^{64} +8536.29 q^{65} +10373.1 q^{66} -37501.6 q^{67} +21364.0 q^{68} -7052.38 q^{69} +35155.1 q^{70} +32441.3 q^{71} -12400.3 q^{72} -60191.0 q^{73} +1535.63 q^{74} -29973.4 q^{75} +16711.8 q^{76} +69210.0 q^{77} +2581.15 q^{78} -54275.0 q^{79} -30336.9 q^{80} +6561.00 q^{81} -14943.6 q^{82} +59211.9 q^{83} -36052.9 q^{84} +69456.1 q^{85} -18643.3 q^{86} +25681.1 q^{87} +65365.9 q^{88} -58121.9 q^{89} -17567.4 q^{90} +17221.6 q^{91} -19365.4 q^{92} +17180.7 q^{93} -7842.76 q^{94} +54331.6 q^{95} -53263.1 q^{96} +174781. q^{97} +25555.7 q^{98} -34585.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 117q^{3} + 246q^{4} - 14q^{5} + 373q^{7} + 123q^{8} + 1053q^{9} + O(q^{10}) \) \( 13q - 117q^{3} + 246q^{4} - 14q^{5} + 373q^{7} + 123q^{8} + 1053q^{9} + 137q^{10} + 250q^{11} - 2214q^{12} + 1054q^{13} - 575q^{14} + 126q^{15} + 922q^{16} + 271q^{17} + 671q^{19} - 5491q^{20} - 3357q^{21} + 1094q^{22} + 3975q^{23} - 1107q^{24} + 15569q^{25} + 4622q^{26} - 9477q^{27} + 21214q^{28} - 10613q^{29} - 1233q^{30} + 25597q^{31} + 15966q^{32} - 2250q^{33} + 31796q^{34} + 6729q^{35} + 19926q^{36} + 17585q^{37} + 34903q^{38} - 9486q^{39} + 31382q^{40} + 12537q^{41} + 5175q^{42} + 26644q^{43} + 6654q^{44} - 1134q^{45} + 149005q^{46} + 52087q^{47} - 8298q^{48} + 95384q^{49} + 121821q^{50} - 2439q^{51} + 263630q^{52} + 20014q^{53} + 120932q^{55} + 126688q^{56} - 6039q^{57} + 86066q^{58} + 45253q^{59} + 49419q^{60} - 11667q^{61} + 164794q^{62} + 30213q^{63} + 151893q^{64} - 28674q^{65} - 9846q^{66} + 1106q^{67} - 4043q^{68} - 35775q^{69} + 56066q^{70} + 21230q^{71} + 9963q^{72} + 81131q^{73} + 102042q^{74} - 140121q^{75} + 73900q^{76} - 104655q^{77} - 41598q^{78} - 13470q^{79} - 191969q^{80} + 85293q^{81} + 79909q^{82} - 76149q^{83} - 190926q^{84} + 10035q^{85} - 321496q^{86} + 95517q^{87} - 276779q^{88} - 190205q^{89} + 11097q^{90} + 80601q^{91} + 45672q^{92} - 230373q^{93} + 36768q^{94} + 9875q^{95} - 143694q^{96} + 160850q^{97} - 116644q^{98} + 20250q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.69937 0.477186 0.238593 0.971120i \(-0.423314\pi\)
0.238593 + 0.971120i \(0.423314\pi\)
\(3\) −9.00000 −0.577350
\(4\) −24.7134 −0.772294
\(5\) −80.3454 −1.43726 −0.718631 0.695392i \(-0.755231\pi\)
−0.718631 + 0.695392i \(0.755231\pi\)
\(6\) −24.2943 −0.275503
\(7\) −162.093 −1.25032 −0.625159 0.780498i \(-0.714966\pi\)
−0.625159 + 0.780498i \(0.714966\pi\)
\(8\) −153.090 −0.845713
\(9\) 81.0000 0.333333
\(10\) −216.882 −0.685841
\(11\) −426.976 −1.06395 −0.531975 0.846760i \(-0.678550\pi\)
−0.531975 + 0.846760i \(0.678550\pi\)
\(12\) 222.421 0.445884
\(13\) −106.245 −0.174361 −0.0871807 0.996193i \(-0.527786\pi\)
−0.0871807 + 0.996193i \(0.527786\pi\)
\(14\) −437.550 −0.596634
\(15\) 723.108 0.829803
\(16\) 377.581 0.368732
\(17\) −864.470 −0.725483 −0.362742 0.931890i \(-0.618159\pi\)
−0.362742 + 0.931890i \(0.618159\pi\)
\(18\) 218.649 0.159062
\(19\) −676.226 −0.429742 −0.214871 0.976642i \(-0.568933\pi\)
−0.214871 + 0.976642i \(0.568933\pi\)
\(20\) 1985.61 1.10999
\(21\) 1458.84 0.721871
\(22\) −1152.57 −0.507702
\(23\) 783.598 0.308869 0.154434 0.988003i \(-0.450644\pi\)
0.154434 + 0.988003i \(0.450644\pi\)
\(24\) 1377.81 0.488273
\(25\) 3330.38 1.06572
\(26\) −286.795 −0.0832027
\(27\) −729.000 −0.192450
\(28\) 4005.88 0.965613
\(29\) −2853.46 −0.630052 −0.315026 0.949083i \(-0.602013\pi\)
−0.315026 + 0.949083i \(0.602013\pi\)
\(30\) 1951.94 0.395970
\(31\) −1908.96 −0.356774 −0.178387 0.983960i \(-0.557088\pi\)
−0.178387 + 0.983960i \(0.557088\pi\)
\(32\) 5918.13 1.02167
\(33\) 3842.78 0.614272
\(34\) −2333.52 −0.346190
\(35\) 13023.5 1.79703
\(36\) −2001.79 −0.257431
\(37\) 568.886 0.0683157 0.0341579 0.999416i \(-0.489125\pi\)
0.0341579 + 0.999416i \(0.489125\pi\)
\(38\) −1825.38 −0.205067
\(39\) 956.205 0.100668
\(40\) 12300.1 1.21551
\(41\) −5535.95 −0.514319 −0.257160 0.966369i \(-0.582787\pi\)
−0.257160 + 0.966369i \(0.582787\pi\)
\(42\) 3937.95 0.344467
\(43\) −6906.55 −0.569626 −0.284813 0.958583i \(-0.591932\pi\)
−0.284813 + 0.958583i \(0.591932\pi\)
\(44\) 10552.0 0.821682
\(45\) −6507.97 −0.479087
\(46\) 2115.22 0.147388
\(47\) −2905.40 −0.191850 −0.0959250 0.995389i \(-0.530581\pi\)
−0.0959250 + 0.995389i \(0.530581\pi\)
\(48\) −3398.23 −0.212887
\(49\) 9467.29 0.563294
\(50\) 8989.92 0.508547
\(51\) 7780.23 0.418858
\(52\) 2625.68 0.134658
\(53\) −29299.4 −1.43275 −0.716373 0.697717i \(-0.754199\pi\)
−0.716373 + 0.697717i \(0.754199\pi\)
\(54\) −1967.84 −0.0918344
\(55\) 34305.5 1.52918
\(56\) 24815.0 1.05741
\(57\) 6086.03 0.248112
\(58\) −7702.54 −0.300652
\(59\) 3481.00 0.130189
\(60\) −17870.5 −0.640852
\(61\) 15250.0 0.524741 0.262370 0.964967i \(-0.415496\pi\)
0.262370 + 0.964967i \(0.415496\pi\)
\(62\) −5153.00 −0.170248
\(63\) −13129.6 −0.416773
\(64\) 3892.61 0.118793
\(65\) 8536.29 0.250603
\(66\) 10373.1 0.293122
\(67\) −37501.6 −1.02062 −0.510309 0.859991i \(-0.670469\pi\)
−0.510309 + 0.859991i \(0.670469\pi\)
\(68\) 21364.0 0.560286
\(69\) −7052.38 −0.178325
\(70\) 35155.1 0.857519
\(71\) 32441.3 0.763752 0.381876 0.924214i \(-0.375278\pi\)
0.381876 + 0.924214i \(0.375278\pi\)
\(72\) −12400.3 −0.281904
\(73\) −60191.0 −1.32198 −0.660989 0.750396i \(-0.729863\pi\)
−0.660989 + 0.750396i \(0.729863\pi\)
\(74\) 1535.63 0.0325993
\(75\) −29973.4 −0.615294
\(76\) 16711.8 0.331887
\(77\) 69210.0 1.33028
\(78\) 2581.15 0.0480371
\(79\) −54275.0 −0.978436 −0.489218 0.872162i \(-0.662718\pi\)
−0.489218 + 0.872162i \(0.662718\pi\)
\(80\) −30336.9 −0.529964
\(81\) 6561.00 0.111111
\(82\) −14943.6 −0.245426
\(83\) 59211.9 0.943439 0.471720 0.881749i \(-0.343633\pi\)
0.471720 + 0.881749i \(0.343633\pi\)
\(84\) −36052.9 −0.557497
\(85\) 69456.1 1.04271
\(86\) −18643.3 −0.271817
\(87\) 25681.1 0.363761
\(88\) 65365.9 0.899797
\(89\) −58121.9 −0.777794 −0.388897 0.921281i \(-0.627144\pi\)
−0.388897 + 0.921281i \(0.627144\pi\)
\(90\) −17567.4 −0.228614
\(91\) 17221.6 0.218007
\(92\) −19365.4 −0.238537
\(93\) 17180.7 0.205984
\(94\) −7842.76 −0.0915480
\(95\) 54331.6 0.617652
\(96\) −53263.1 −0.589860
\(97\) 174781. 1.88610 0.943051 0.332647i \(-0.107942\pi\)
0.943051 + 0.332647i \(0.107942\pi\)
\(98\) 25555.7 0.268796
\(99\) −34585.0 −0.354650
\(100\) −82304.9 −0.823049
\(101\) −178436. −1.74052 −0.870258 0.492596i \(-0.836048\pi\)
−0.870258 + 0.492596i \(0.836048\pi\)
\(102\) 21001.7 0.199873
\(103\) 126990. 1.17944 0.589722 0.807607i \(-0.299237\pi\)
0.589722 + 0.807607i \(0.299237\pi\)
\(104\) 16265.1 0.147460
\(105\) −117211. −1.03752
\(106\) −79090.0 −0.683686
\(107\) −70204.3 −0.592795 −0.296397 0.955065i \(-0.595785\pi\)
−0.296397 + 0.955065i \(0.595785\pi\)
\(108\) 18016.1 0.148628
\(109\) −50071.5 −0.403668 −0.201834 0.979420i \(-0.564690\pi\)
−0.201834 + 0.979420i \(0.564690\pi\)
\(110\) 92603.3 0.729700
\(111\) −5119.97 −0.0394421
\(112\) −61203.4 −0.461032
\(113\) −103197. −0.760279 −0.380139 0.924929i \(-0.624124\pi\)
−0.380139 + 0.924929i \(0.624124\pi\)
\(114\) 16428.5 0.118395
\(115\) −62958.5 −0.443925
\(116\) 70518.7 0.486585
\(117\) −8605.85 −0.0581204
\(118\) 9396.51 0.0621243
\(119\) 140125. 0.907085
\(120\) −110701. −0.701776
\(121\) 21257.2 0.131991
\(122\) 41165.3 0.250399
\(123\) 49823.6 0.296942
\(124\) 47177.0 0.275535
\(125\) −16501.1 −0.0944577
\(126\) −35441.6 −0.198878
\(127\) −95263.9 −0.524106 −0.262053 0.965053i \(-0.584400\pi\)
−0.262053 + 0.965053i \(0.584400\pi\)
\(128\) −178872. −0.964980
\(129\) 62159.0 0.328874
\(130\) 23042.6 0.119584
\(131\) −283090. −1.44127 −0.720636 0.693313i \(-0.756150\pi\)
−0.720636 + 0.693313i \(0.756150\pi\)
\(132\) −94968.2 −0.474399
\(133\) 109612. 0.537314
\(134\) −101231. −0.487024
\(135\) 58571.8 0.276601
\(136\) 132342. 0.613551
\(137\) −46840.7 −0.213217 −0.106609 0.994301i \(-0.533999\pi\)
−0.106609 + 0.994301i \(0.533999\pi\)
\(138\) −19037.0 −0.0850943
\(139\) 352418. 1.54711 0.773555 0.633729i \(-0.218477\pi\)
0.773555 + 0.633729i \(0.218477\pi\)
\(140\) −321854. −1.38784
\(141\) 26148.6 0.110765
\(142\) 87571.0 0.364451
\(143\) 45364.0 0.185512
\(144\) 30584.1 0.122911
\(145\) 229262. 0.905550
\(146\) −162478. −0.630829
\(147\) −85205.6 −0.325218
\(148\) −14059.1 −0.0527598
\(149\) −170739. −0.630040 −0.315020 0.949085i \(-0.602011\pi\)
−0.315020 + 0.949085i \(0.602011\pi\)
\(150\) −80909.3 −0.293610
\(151\) 51532.5 0.183924 0.0919620 0.995763i \(-0.470686\pi\)
0.0919620 + 0.995763i \(0.470686\pi\)
\(152\) 103524. 0.363438
\(153\) −70022.1 −0.241828
\(154\) 186823. 0.634789
\(155\) 153376. 0.512778
\(156\) −23631.1 −0.0777449
\(157\) −63321.5 −0.205023 −0.102511 0.994732i \(-0.532688\pi\)
−0.102511 + 0.994732i \(0.532688\pi\)
\(158\) −146508. −0.466896
\(159\) 263695. 0.827197
\(160\) −475494. −1.46840
\(161\) −127016. −0.386184
\(162\) 17710.6 0.0530206
\(163\) 248901. 0.733765 0.366882 0.930267i \(-0.380425\pi\)
0.366882 + 0.930267i \(0.380425\pi\)
\(164\) 136812. 0.397205
\(165\) −308750. −0.882870
\(166\) 159835. 0.450196
\(167\) −693685. −1.92474 −0.962368 0.271750i \(-0.912398\pi\)
−0.962368 + 0.271750i \(0.912398\pi\)
\(168\) −223335. −0.610496
\(169\) −360005. −0.969598
\(170\) 187488. 0.497566
\(171\) −54774.3 −0.143247
\(172\) 170684. 0.439919
\(173\) −311888. −0.792288 −0.396144 0.918188i \(-0.629652\pi\)
−0.396144 + 0.918188i \(0.629652\pi\)
\(174\) 69322.9 0.173581
\(175\) −539832. −1.33249
\(176\) −161218. −0.392312
\(177\) −31329.0 −0.0751646
\(178\) −156893. −0.371152
\(179\) −566672. −1.32190 −0.660951 0.750429i \(-0.729847\pi\)
−0.660951 + 0.750429i \(0.729847\pi\)
\(180\) 160834. 0.369996
\(181\) −166371. −0.377468 −0.188734 0.982028i \(-0.560438\pi\)
−0.188734 + 0.982028i \(0.560438\pi\)
\(182\) 46487.5 0.104030
\(183\) −137250. −0.302959
\(184\) −119961. −0.261214
\(185\) −45707.3 −0.0981876
\(186\) 46377.0 0.0982925
\(187\) 369108. 0.771878
\(188\) 71802.4 0.148165
\(189\) 118166. 0.240624
\(190\) 146661. 0.294734
\(191\) 57779.2 0.114601 0.0573005 0.998357i \(-0.481751\pi\)
0.0573005 + 0.998357i \(0.481751\pi\)
\(192\) −35033.5 −0.0685852
\(193\) 1.00421e6 1.94058 0.970290 0.241946i \(-0.0777855\pi\)
0.970290 + 0.241946i \(0.0777855\pi\)
\(194\) 471799. 0.900021
\(195\) −76826.6 −0.144686
\(196\) −233969. −0.435029
\(197\) 964560. 1.77078 0.885388 0.464853i \(-0.153893\pi\)
0.885388 + 0.464853i \(0.153893\pi\)
\(198\) −93357.8 −0.169234
\(199\) 623260. 1.11567 0.557836 0.829951i \(-0.311632\pi\)
0.557836 + 0.829951i \(0.311632\pi\)
\(200\) −509849. −0.901294
\(201\) 337514. 0.589254
\(202\) −481664. −0.830550
\(203\) 462527. 0.787765
\(204\) −192276. −0.323482
\(205\) 444788. 0.739211
\(206\) 342793. 0.562813
\(207\) 63471.5 0.102956
\(208\) −40116.1 −0.0642925
\(209\) 288732. 0.457224
\(210\) −316396. −0.495089
\(211\) 800296. 1.23750 0.618749 0.785589i \(-0.287640\pi\)
0.618749 + 0.785589i \(0.287640\pi\)
\(212\) 724088. 1.10650
\(213\) −291972. −0.440952
\(214\) −189507. −0.282873
\(215\) 554909. 0.818702
\(216\) 111603. 0.162758
\(217\) 309431. 0.446081
\(218\) −135161. −0.192624
\(219\) 541719. 0.763244
\(220\) −847806. −1.18097
\(221\) 91845.6 0.126496
\(222\) −13820.7 −0.0188212
\(223\) −195058. −0.262664 −0.131332 0.991338i \(-0.541925\pi\)
−0.131332 + 0.991338i \(0.541925\pi\)
\(224\) −959289. −1.27741
\(225\) 269761. 0.355240
\(226\) −278568. −0.362794
\(227\) −787763. −1.01468 −0.507342 0.861745i \(-0.669372\pi\)
−0.507342 + 0.861745i \(0.669372\pi\)
\(228\) −150407. −0.191615
\(229\) 1.12310e6 1.41524 0.707620 0.706593i \(-0.249769\pi\)
0.707620 + 0.706593i \(0.249769\pi\)
\(230\) −169948. −0.211835
\(231\) −622890. −0.768035
\(232\) 436837. 0.532843
\(233\) 1.17683e6 1.42012 0.710061 0.704140i \(-0.248667\pi\)
0.710061 + 0.704140i \(0.248667\pi\)
\(234\) −23230.4 −0.0277342
\(235\) 233436. 0.275739
\(236\) −86027.4 −0.100544
\(237\) 488475. 0.564900
\(238\) 378249. 0.432848
\(239\) 46985.5 0.0532070 0.0266035 0.999646i \(-0.491531\pi\)
0.0266035 + 0.999646i \(0.491531\pi\)
\(240\) 273032. 0.305975
\(241\) 1.64514e6 1.82457 0.912286 0.409553i \(-0.134315\pi\)
0.912286 + 0.409553i \(0.134315\pi\)
\(242\) 57381.1 0.0629840
\(243\) −59049.0 −0.0641500
\(244\) −376879. −0.405254
\(245\) −760653. −0.809601
\(246\) 134492. 0.141697
\(247\) 71845.6 0.0749304
\(248\) 292244. 0.301729
\(249\) −532907. −0.544695
\(250\) −44542.5 −0.0450739
\(251\) 26758.5 0.0268088 0.0134044 0.999910i \(-0.495733\pi\)
0.0134044 + 0.999910i \(0.495733\pi\)
\(252\) 324476. 0.321871
\(253\) −334577. −0.328621
\(254\) −257153. −0.250096
\(255\) −625105. −0.602009
\(256\) −607406. −0.579268
\(257\) 1.46732e6 1.38577 0.692887 0.721046i \(-0.256338\pi\)
0.692887 + 0.721046i \(0.256338\pi\)
\(258\) 167790. 0.156934
\(259\) −92212.6 −0.0854164
\(260\) −210961. −0.193539
\(261\) −231130. −0.210017
\(262\) −764164. −0.687754
\(263\) 972533. 0.866991 0.433496 0.901156i \(-0.357280\pi\)
0.433496 + 0.901156i \(0.357280\pi\)
\(264\) −588293. −0.519498
\(265\) 2.35407e6 2.05923
\(266\) 295883. 0.256399
\(267\) 523097. 0.449060
\(268\) 926792. 0.788216
\(269\) −440178. −0.370892 −0.185446 0.982654i \(-0.559373\pi\)
−0.185446 + 0.982654i \(0.559373\pi\)
\(270\) 158107. 0.131990
\(271\) 350066. 0.289552 0.144776 0.989464i \(-0.453754\pi\)
0.144776 + 0.989464i \(0.453754\pi\)
\(272\) −326408. −0.267509
\(273\) −154995. −0.125866
\(274\) −126440. −0.101744
\(275\) −1.42199e6 −1.13387
\(276\) 174288. 0.137720
\(277\) −144244. −0.112953 −0.0564766 0.998404i \(-0.517987\pi\)
−0.0564766 + 0.998404i \(0.517987\pi\)
\(278\) 951306. 0.738259
\(279\) −154626. −0.118925
\(280\) −1.99377e6 −1.51977
\(281\) −818381. −0.618287 −0.309143 0.951015i \(-0.600042\pi\)
−0.309143 + 0.951015i \(0.600042\pi\)
\(282\) 70584.8 0.0528553
\(283\) 1.15317e6 0.855909 0.427954 0.903800i \(-0.359234\pi\)
0.427954 + 0.903800i \(0.359234\pi\)
\(284\) −801735. −0.589841
\(285\) −488985. −0.356601
\(286\) 122454. 0.0885236
\(287\) 897341. 0.643062
\(288\) 479368. 0.340556
\(289\) −672549. −0.473674
\(290\) 618863. 0.432115
\(291\) −1.57303e6 −1.08894
\(292\) 1.48752e6 1.02096
\(293\) −1.43446e6 −0.976153 −0.488077 0.872801i \(-0.662301\pi\)
−0.488077 + 0.872801i \(0.662301\pi\)
\(294\) −230001. −0.155189
\(295\) −279682. −0.187116
\(296\) −87091.0 −0.0577755
\(297\) 311265. 0.204757
\(298\) −460889. −0.300646
\(299\) −83253.4 −0.0538548
\(300\) 740745. 0.475188
\(301\) 1.11951e6 0.712214
\(302\) 139105. 0.0877659
\(303\) 1.60592e6 1.00489
\(304\) −255330. −0.158459
\(305\) −1.22526e6 −0.754189
\(306\) −189015. −0.115397
\(307\) −1.40814e6 −0.852705 −0.426353 0.904557i \(-0.640202\pi\)
−0.426353 + 0.904557i \(0.640202\pi\)
\(308\) −1.71041e6 −1.02736
\(309\) −1.14291e6 −0.680952
\(310\) 414020. 0.244690
\(311\) −405966. −0.238007 −0.119003 0.992894i \(-0.537970\pi\)
−0.119003 + 0.992894i \(0.537970\pi\)
\(312\) −146386. −0.0851359
\(313\) −2.28785e6 −1.31998 −0.659990 0.751275i \(-0.729439\pi\)
−0.659990 + 0.751275i \(0.729439\pi\)
\(314\) −170928. −0.0978339
\(315\) 1.05490e6 0.599011
\(316\) 1.34132e6 0.755640
\(317\) −1.72021e6 −0.961462 −0.480731 0.876868i \(-0.659629\pi\)
−0.480731 + 0.876868i \(0.659629\pi\)
\(318\) 711810. 0.394726
\(319\) 1.21836e6 0.670344
\(320\) −312753. −0.170737
\(321\) 631839. 0.342250
\(322\) −342864. −0.184281
\(323\) 584577. 0.311771
\(324\) −162145. −0.0858104
\(325\) −353836. −0.185820
\(326\) 671875. 0.350142
\(327\) 450643. 0.233058
\(328\) 847501. 0.434966
\(329\) 470947. 0.239873
\(330\) −833429. −0.421293
\(331\) −1.23423e6 −0.619193 −0.309597 0.950868i \(-0.600194\pi\)
−0.309597 + 0.950868i \(0.600194\pi\)
\(332\) −1.46333e6 −0.728612
\(333\) 46079.7 0.0227719
\(334\) −1.87251e6 −0.918456
\(335\) 3.01308e6 1.46689
\(336\) 550831. 0.266177
\(337\) −1.65283e6 −0.792781 −0.396391 0.918082i \(-0.629737\pi\)
−0.396391 + 0.918082i \(0.629737\pi\)
\(338\) −971787. −0.462678
\(339\) 928777. 0.438947
\(340\) −1.71650e6 −0.805278
\(341\) 815082. 0.379590
\(342\) −147856. −0.0683556
\(343\) 1.18972e6 0.546021
\(344\) 1.05733e6 0.481740
\(345\) 566626. 0.256300
\(346\) −841900. −0.378068
\(347\) −1708.59 −0.000761752 0 −0.000380876 1.00000i \(-0.500121\pi\)
−0.000380876 1.00000i \(0.500121\pi\)
\(348\) −634668. −0.280930
\(349\) −4.09764e6 −1.80082 −0.900410 0.435042i \(-0.856734\pi\)
−0.900410 + 0.435042i \(0.856734\pi\)
\(350\) −1.45721e6 −0.635845
\(351\) 77452.6 0.0335559
\(352\) −2.52690e6 −1.08700
\(353\) −4.02008e6 −1.71711 −0.858555 0.512721i \(-0.828637\pi\)
−0.858555 + 0.512721i \(0.828637\pi\)
\(354\) −84568.6 −0.0358675
\(355\) −2.60651e6 −1.09771
\(356\) 1.43639e6 0.600686
\(357\) −1.26112e6 −0.523706
\(358\) −1.52966e6 −0.630793
\(359\) −1.72475e6 −0.706300 −0.353150 0.935567i \(-0.614889\pi\)
−0.353150 + 0.935567i \(0.614889\pi\)
\(360\) 996309. 0.405170
\(361\) −2.01882e6 −0.815322
\(362\) −449096. −0.180122
\(363\) −191315. −0.0762048
\(364\) −425605. −0.168366
\(365\) 4.83607e6 1.90003
\(366\) −370488. −0.144568
\(367\) −2.48879e6 −0.964547 −0.482274 0.876021i \(-0.660189\pi\)
−0.482274 + 0.876021i \(0.660189\pi\)
\(368\) 295872. 0.113890
\(369\) −448412. −0.171440
\(370\) −123381. −0.0468537
\(371\) 4.74924e6 1.79139
\(372\) −424593. −0.159080
\(373\) 4.73910e6 1.76370 0.881848 0.471534i \(-0.156299\pi\)
0.881848 + 0.471534i \(0.156299\pi\)
\(374\) 996358. 0.368329
\(375\) 148510. 0.0545352
\(376\) 444789. 0.162250
\(377\) 303166. 0.109857
\(378\) 318974. 0.114822
\(379\) −1.94813e6 −0.696658 −0.348329 0.937372i \(-0.613251\pi\)
−0.348329 + 0.937372i \(0.613251\pi\)
\(380\) −1.34272e6 −0.477009
\(381\) 857376. 0.302593
\(382\) 155967. 0.0546859
\(383\) 4.90807e6 1.70967 0.854837 0.518896i \(-0.173657\pi\)
0.854837 + 0.518896i \(0.173657\pi\)
\(384\) 1.60985e6 0.557132
\(385\) −5.56070e6 −1.91195
\(386\) 2.71074e6 0.926017
\(387\) −559431. −0.189875
\(388\) −4.31944e6 −1.45663
\(389\) −3.64955e6 −1.22283 −0.611414 0.791311i \(-0.709399\pi\)
−0.611414 + 0.791311i \(0.709399\pi\)
\(390\) −207384. −0.0690419
\(391\) −677397. −0.224079
\(392\) −1.44935e6 −0.476385
\(393\) 2.54781e6 0.832119
\(394\) 2.60370e6 0.844989
\(395\) 4.36075e6 1.40627
\(396\) 854714. 0.273894
\(397\) −3.33365e6 −1.06156 −0.530779 0.847510i \(-0.678101\pi\)
−0.530779 + 0.847510i \(0.678101\pi\)
\(398\) 1.68241e6 0.532382
\(399\) −986506. −0.310218
\(400\) 1.25749e6 0.392965
\(401\) 184478. 0.0572906 0.0286453 0.999590i \(-0.490881\pi\)
0.0286453 + 0.999590i \(0.490881\pi\)
\(402\) 911076. 0.281183
\(403\) 202818. 0.0622077
\(404\) 4.40975e6 1.34419
\(405\) −527146. −0.159696
\(406\) 1.24853e6 0.375910
\(407\) −242900. −0.0726845
\(408\) −1.19108e6 −0.354234
\(409\) −1.84713e6 −0.545996 −0.272998 0.962015i \(-0.588015\pi\)
−0.272998 + 0.962015i \(0.588015\pi\)
\(410\) 1.20065e6 0.352741
\(411\) 421567. 0.123101
\(412\) −3.13836e6 −0.910877
\(413\) −564247. −0.162778
\(414\) 171333. 0.0491292
\(415\) −4.75740e6 −1.35597
\(416\) −628771. −0.178139
\(417\) −3.17176e6 −0.893224
\(418\) 779394. 0.218181
\(419\) 3.58683e6 0.998105 0.499052 0.866572i \(-0.333682\pi\)
0.499052 + 0.866572i \(0.333682\pi\)
\(420\) 2.89669e6 0.801269
\(421\) 1.55188e6 0.426729 0.213364 0.976973i \(-0.431558\pi\)
0.213364 + 0.976973i \(0.431558\pi\)
\(422\) 2.16030e6 0.590516
\(423\) −235338. −0.0639500
\(424\) 4.48546e6 1.21169
\(425\) −2.87901e6 −0.773163
\(426\) −788139. −0.210416
\(427\) −2.47192e6 −0.656092
\(428\) 1.73499e6 0.457812
\(429\) −408276. −0.107105
\(430\) 1.49791e6 0.390673
\(431\) 3.66890e6 0.951356 0.475678 0.879619i \(-0.342203\pi\)
0.475678 + 0.879619i \(0.342203\pi\)
\(432\) −275257. −0.0709624
\(433\) −17511.1 −0.00448842 −0.00224421 0.999997i \(-0.500714\pi\)
−0.00224421 + 0.999997i \(0.500714\pi\)
\(434\) 835268. 0.212864
\(435\) −2.06336e6 −0.522819
\(436\) 1.23744e6 0.311750
\(437\) −529889. −0.132734
\(438\) 1.46230e6 0.364209
\(439\) 6.13137e6 1.51844 0.759218 0.650837i \(-0.225582\pi\)
0.759218 + 0.650837i \(0.225582\pi\)
\(440\) −5.25185e6 −1.29324
\(441\) 766850. 0.187765
\(442\) 247925. 0.0603622
\(443\) −2.69068e6 −0.651408 −0.325704 0.945472i \(-0.605601\pi\)
−0.325704 + 0.945472i \(0.605601\pi\)
\(444\) 126532. 0.0304609
\(445\) 4.66983e6 1.11789
\(446\) −526532. −0.125340
\(447\) 1.53665e6 0.363754
\(448\) −630966. −0.148529
\(449\) −4.36660e6 −1.02218 −0.511091 0.859527i \(-0.670758\pi\)
−0.511091 + 0.859527i \(0.670758\pi\)
\(450\) 728183. 0.169516
\(451\) 2.36372e6 0.547210
\(452\) 2.55036e6 0.587158
\(453\) −463792. −0.106189
\(454\) −2.12646e6 −0.484193
\(455\) −1.38368e6 −0.313333
\(456\) −931713. −0.209831
\(457\) −7.30210e6 −1.63553 −0.817763 0.575555i \(-0.804786\pi\)
−0.817763 + 0.575555i \(0.804786\pi\)
\(458\) 3.03167e6 0.675332
\(459\) 630198. 0.139619
\(460\) 1.55592e6 0.342841
\(461\) −7.01104e6 −1.53649 −0.768246 0.640155i \(-0.778870\pi\)
−0.768246 + 0.640155i \(0.778870\pi\)
\(462\) −1.68141e6 −0.366495
\(463\) 1.31573e6 0.285243 0.142621 0.989777i \(-0.454447\pi\)
0.142621 + 0.989777i \(0.454447\pi\)
\(464\) −1.07741e6 −0.232320
\(465\) −1.38039e6 −0.296053
\(466\) 3.17671e6 0.677662
\(467\) 3.13078e6 0.664293 0.332147 0.943228i \(-0.392227\pi\)
0.332147 + 0.943228i \(0.392227\pi\)
\(468\) 212680. 0.0448861
\(469\) 6.07876e6 1.27610
\(470\) 630129. 0.131578
\(471\) 569894. 0.118370
\(472\) −532908. −0.110102
\(473\) 2.94893e6 0.606054
\(474\) 1.31858e6 0.269562
\(475\) −2.25209e6 −0.457985
\(476\) −3.46296e6 −0.700536
\(477\) −2.37325e6 −0.477582
\(478\) 126831. 0.0253896
\(479\) 7.00448e6 1.39488 0.697441 0.716642i \(-0.254322\pi\)
0.697441 + 0.716642i \(0.254322\pi\)
\(480\) 4.27945e6 0.847782
\(481\) −60441.3 −0.0119116
\(482\) 4.44085e6 0.870660
\(483\) 1.14315e6 0.222963
\(484\) −525338. −0.101935
\(485\) −1.40429e7 −2.71082
\(486\) −159395. −0.0306115
\(487\) 2.09435e6 0.400154 0.200077 0.979780i \(-0.435881\pi\)
0.200077 + 0.979780i \(0.435881\pi\)
\(488\) −2.33463e6 −0.443780
\(489\) −2.24011e6 −0.423639
\(490\) −2.05328e6 −0.386330
\(491\) −4.28066e6 −0.801322 −0.400661 0.916226i \(-0.631219\pi\)
−0.400661 + 0.916226i \(0.631219\pi\)
\(492\) −1.23131e6 −0.229327
\(493\) 2.46673e6 0.457092
\(494\) 193938. 0.0357557
\(495\) 2.77875e6 0.509725
\(496\) −720789. −0.131554
\(497\) −5.25852e6 −0.954932
\(498\) −1.43851e6 −0.259921
\(499\) −5.62839e6 −1.01189 −0.505944 0.862566i \(-0.668856\pi\)
−0.505944 + 0.862566i \(0.668856\pi\)
\(500\) 407798. 0.0729491
\(501\) 6.24316e6 1.11125
\(502\) 72231.0 0.0127928
\(503\) −2.46000e6 −0.433526 −0.216763 0.976224i \(-0.569550\pi\)
−0.216763 + 0.976224i \(0.569550\pi\)
\(504\) 2.01001e6 0.352470
\(505\) 1.43365e7 2.50158
\(506\) −903148. −0.156813
\(507\) 3.24004e6 0.559798
\(508\) 2.35430e6 0.404764
\(509\) 7.82777e6 1.33919 0.669597 0.742724i \(-0.266467\pi\)
0.669597 + 0.742724i \(0.266467\pi\)
\(510\) −1.68739e6 −0.287270
\(511\) 9.75656e6 1.65289
\(512\) 4.08430e6 0.688562
\(513\) 492969. 0.0827039
\(514\) 3.96084e6 0.661272
\(515\) −1.02031e7 −1.69517
\(516\) −1.53616e6 −0.253987
\(517\) 1.24054e6 0.204119
\(518\) −248916. −0.0407595
\(519\) 2.80699e6 0.457428
\(520\) −1.30683e6 −0.211938
\(521\) −9.96359e6 −1.60813 −0.804066 0.594541i \(-0.797334\pi\)
−0.804066 + 0.594541i \(0.797334\pi\)
\(522\) −623906. −0.100217
\(523\) 4.30575e6 0.688327 0.344164 0.938910i \(-0.388163\pi\)
0.344164 + 0.938910i \(0.388163\pi\)
\(524\) 6.99611e6 1.11309
\(525\) 4.85849e6 0.769313
\(526\) 2.62522e6 0.413716
\(527\) 1.65024e6 0.258834
\(528\) 1.45096e6 0.226502
\(529\) −5.82232e6 −0.904600
\(530\) 6.35451e6 0.982636
\(531\) 281961. 0.0433963
\(532\) −2.70888e6 −0.414964
\(533\) 588167. 0.0896774
\(534\) 1.41203e6 0.214285
\(535\) 5.64059e6 0.852001
\(536\) 5.74114e6 0.863149
\(537\) 5.10005e6 0.763201
\(538\) −1.18820e6 −0.176984
\(539\) −4.04230e6 −0.599317
\(540\) −1.44751e6 −0.213617
\(541\) 340272. 0.0499843 0.0249922 0.999688i \(-0.492044\pi\)
0.0249922 + 0.999688i \(0.492044\pi\)
\(542\) 944959. 0.138170
\(543\) 1.49734e6 0.217931
\(544\) −5.11604e6 −0.741202
\(545\) 4.02301e6 0.580176
\(546\) −418388. −0.0600617
\(547\) −1.33663e7 −1.91004 −0.955021 0.296540i \(-0.904167\pi\)
−0.955021 + 0.296540i \(0.904167\pi\)
\(548\) 1.15759e6 0.164666
\(549\) 1.23525e6 0.174914
\(550\) −3.83848e6 −0.541068
\(551\) 1.92958e6 0.270760
\(552\) 1.07965e6 0.150812
\(553\) 8.79763e6 1.22336
\(554\) −389368. −0.0538996
\(555\) 411366. 0.0566886
\(556\) −8.70945e6 −1.19482
\(557\) −5.96149e6 −0.814174 −0.407087 0.913389i \(-0.633455\pi\)
−0.407087 + 0.913389i \(0.633455\pi\)
\(558\) −417393. −0.0567492
\(559\) 733787. 0.0993208
\(560\) 4.91741e6 0.662623
\(561\) −3.32197e6 −0.445644
\(562\) −2.20911e6 −0.295037
\(563\) 7.64380e6 1.01634 0.508169 0.861257i \(-0.330322\pi\)
0.508169 + 0.861257i \(0.330322\pi\)
\(564\) −646222. −0.0855428
\(565\) 8.29143e6 1.09272
\(566\) 3.11283e6 0.408427
\(567\) −1.06350e6 −0.138924
\(568\) −4.96645e6 −0.645915
\(569\) −2.06870e6 −0.267865 −0.133933 0.990990i \(-0.542761\pi\)
−0.133933 + 0.990990i \(0.542761\pi\)
\(570\) −1.31995e6 −0.170165
\(571\) 1.48686e6 0.190845 0.0954224 0.995437i \(-0.469580\pi\)
0.0954224 + 0.995437i \(0.469580\pi\)
\(572\) −1.12110e6 −0.143270
\(573\) −520013. −0.0661649
\(574\) 2.42226e6 0.306860
\(575\) 2.60968e6 0.329168
\(576\) 315301. 0.0395977
\(577\) −2.80446e6 −0.350679 −0.175340 0.984508i \(-0.556102\pi\)
−0.175340 + 0.984508i \(0.556102\pi\)
\(578\) −1.81546e6 −0.226030
\(579\) −9.03789e6 −1.12039
\(580\) −5.66585e6 −0.699351
\(581\) −9.59786e6 −1.17960
\(582\) −4.24619e6 −0.519627
\(583\) 1.25101e7 1.52437
\(584\) 9.21466e6 1.11801
\(585\) 691440. 0.0835343
\(586\) −3.87212e6 −0.465806
\(587\) −6.90944e6 −0.827652 −0.413826 0.910356i \(-0.635808\pi\)
−0.413826 + 0.910356i \(0.635808\pi\)
\(588\) 2.10572e6 0.251164
\(589\) 1.29089e6 0.153321
\(590\) −754966. −0.0892888
\(591\) −8.68104e6 −1.02236
\(592\) 214801. 0.0251902
\(593\) −4.57203e6 −0.533915 −0.266958 0.963708i \(-0.586018\pi\)
−0.266958 + 0.963708i \(0.586018\pi\)
\(594\) 840220. 0.0977073
\(595\) −1.12584e7 −1.30372
\(596\) 4.21955e6 0.486576
\(597\) −5.60934e6 −0.644133
\(598\) −224732. −0.0256987
\(599\) 3.76968e6 0.429277 0.214638 0.976694i \(-0.431143\pi\)
0.214638 + 0.976694i \(0.431143\pi\)
\(600\) 4.58864e6 0.520362
\(601\) 4.52282e6 0.510768 0.255384 0.966840i \(-0.417798\pi\)
0.255384 + 0.966840i \(0.417798\pi\)
\(602\) 3.02196e6 0.339858
\(603\) −3.03763e6 −0.340206
\(604\) −1.27354e6 −0.142043
\(605\) −1.70792e6 −0.189705
\(606\) 4.33497e6 0.479518
\(607\) 5.43182e6 0.598376 0.299188 0.954194i \(-0.403284\pi\)
0.299188 + 0.954194i \(0.403284\pi\)
\(608\) −4.00199e6 −0.439053
\(609\) −4.16274e6 −0.454817
\(610\) −3.30744e6 −0.359888
\(611\) 308685. 0.0334512
\(612\) 1.73048e6 0.186762
\(613\) −4.97549e6 −0.534791 −0.267396 0.963587i \(-0.586163\pi\)
−0.267396 + 0.963587i \(0.586163\pi\)
\(614\) −3.80108e6 −0.406899
\(615\) −4.00309e6 −0.426784
\(616\) −1.05954e7 −1.12503
\(617\) −1.15456e7 −1.22097 −0.610483 0.792029i \(-0.709025\pi\)
−0.610483 + 0.792029i \(0.709025\pi\)
\(618\) −3.08514e6 −0.324940
\(619\) −7.93466e6 −0.832341 −0.416171 0.909287i \(-0.636628\pi\)
−0.416171 + 0.909287i \(0.636628\pi\)
\(620\) −3.79045e6 −0.396015
\(621\) −571243. −0.0594418
\(622\) −1.09585e6 −0.113573
\(623\) 9.42118e6 0.972490
\(624\) 361045. 0.0371193
\(625\) −9.08164e6 −0.929960
\(626\) −6.17576e6 −0.629875
\(627\) −2.59859e6 −0.263979
\(628\) 1.56489e6 0.158338
\(629\) −491784. −0.0495619
\(630\) 2.84757e6 0.285840
\(631\) 1.48499e6 0.148474 0.0742370 0.997241i \(-0.476348\pi\)
0.0742370 + 0.997241i \(0.476348\pi\)
\(632\) 8.30899e6 0.827476
\(633\) −7.20267e6 −0.714470
\(634\) −4.64347e6 −0.458796
\(635\) 7.65402e6 0.753278
\(636\) −6.51680e6 −0.638839
\(637\) −1.00585e6 −0.0982168
\(638\) 3.28880e6 0.319879
\(639\) 2.62774e6 0.254584
\(640\) 1.43716e7 1.38693
\(641\) −1.80343e7 −1.73362 −0.866811 0.498636i \(-0.833834\pi\)
−0.866811 + 0.498636i \(0.833834\pi\)
\(642\) 1.70557e6 0.163317
\(643\) 6.54380e6 0.624170 0.312085 0.950054i \(-0.398973\pi\)
0.312085 + 0.950054i \(0.398973\pi\)
\(644\) 3.13900e6 0.298248
\(645\) −4.99418e6 −0.472678
\(646\) 1.57799e6 0.148773
\(647\) −3.41879e6 −0.321078 −0.160539 0.987029i \(-0.551323\pi\)
−0.160539 + 0.987029i \(0.551323\pi\)
\(648\) −1.00443e6 −0.0939681
\(649\) −1.48630e6 −0.138515
\(650\) −955134. −0.0886709
\(651\) −2.78488e6 −0.257545
\(652\) −6.15118e6 −0.566682
\(653\) −4.85652e6 −0.445700 −0.222850 0.974853i \(-0.571536\pi\)
−0.222850 + 0.974853i \(0.571536\pi\)
\(654\) 1.21645e6 0.111212
\(655\) 2.27450e7 2.07149
\(656\) −2.09027e6 −0.189646
\(657\) −4.87547e6 −0.440659
\(658\) 1.27126e6 0.114464
\(659\) 2.71649e6 0.243665 0.121833 0.992551i \(-0.461123\pi\)
0.121833 + 0.992551i \(0.461123\pi\)
\(660\) 7.63025e6 0.681835
\(661\) 9.13072e6 0.812833 0.406417 0.913688i \(-0.366778\pi\)
0.406417 + 0.913688i \(0.366778\pi\)
\(662\) −3.33164e6 −0.295470
\(663\) −826610. −0.0730327
\(664\) −9.06478e6 −0.797879
\(665\) −8.80680e6 −0.772261
\(666\) 124386. 0.0108664
\(667\) −2.23597e6 −0.194603
\(668\) 1.71433e7 1.48646
\(669\) 1.75552e6 0.151649
\(670\) 8.13342e6 0.699981
\(671\) −6.51137e6 −0.558298
\(672\) 8.63360e6 0.737512
\(673\) 1.84820e7 1.57294 0.786470 0.617629i \(-0.211906\pi\)
0.786470 + 0.617629i \(0.211906\pi\)
\(674\) −4.46160e6 −0.378304
\(675\) −2.42784e6 −0.205098
\(676\) 8.89695e6 0.748815
\(677\) −1.19598e6 −0.100289 −0.0501444 0.998742i \(-0.515968\pi\)
−0.0501444 + 0.998742i \(0.515968\pi\)
\(678\) 2.50711e6 0.209459
\(679\) −2.83309e7 −2.35823
\(680\) −1.06331e7 −0.881833
\(681\) 7.08987e6 0.585828
\(682\) 2.20021e6 0.181135
\(683\) −3.97416e6 −0.325982 −0.162991 0.986628i \(-0.552114\pi\)
−0.162991 + 0.986628i \(0.552114\pi\)
\(684\) 1.35366e6 0.110629
\(685\) 3.76344e6 0.306449
\(686\) 3.21149e6 0.260553
\(687\) −1.01079e7 −0.817090
\(688\) −2.60778e6 −0.210039
\(689\) 3.11292e6 0.249816
\(690\) 1.52953e6 0.122303
\(691\) 2.02655e7 1.61459 0.807296 0.590147i \(-0.200930\pi\)
0.807296 + 0.590147i \(0.200930\pi\)
\(692\) 7.70781e6 0.611879
\(693\) 5.60601e6 0.443425
\(694\) −4612.11 −0.000363497 0
\(695\) −2.83152e7 −2.22360
\(696\) −3.93153e6 −0.307637
\(697\) 4.78566e6 0.373130
\(698\) −1.10611e7 −0.859326
\(699\) −1.05915e7 −0.819908
\(700\) 1.33411e7 1.02907
\(701\) −4.25157e6 −0.326779 −0.163390 0.986562i \(-0.552243\pi\)
−0.163390 + 0.986562i \(0.552243\pi\)
\(702\) 209073. 0.0160124
\(703\) −384695. −0.0293581
\(704\) −1.66205e6 −0.126390
\(705\) −2.10092e6 −0.159198
\(706\) −1.08517e7 −0.819380
\(707\) 2.89233e7 2.17620
\(708\) 774246. 0.0580492
\(709\) 1.60414e6 0.119847 0.0599234 0.998203i \(-0.480914\pi\)
0.0599234 + 0.998203i \(0.480914\pi\)
\(710\) −7.03593e6 −0.523812
\(711\) −4.39628e6 −0.326145
\(712\) 8.89791e6 0.657791
\(713\) −1.49586e6 −0.110196
\(714\) −3.40424e6 −0.249905
\(715\) −3.64479e6 −0.266629
\(716\) 1.40044e7 1.02090
\(717\) −422869. −0.0307191
\(718\) −4.65573e6 −0.337036
\(719\) 3.70513e6 0.267289 0.133645 0.991029i \(-0.457332\pi\)
0.133645 + 0.991029i \(0.457332\pi\)
\(720\) −2.45729e6 −0.176655
\(721\) −2.05843e7 −1.47468
\(722\) −5.44953e6 −0.389060
\(723\) −1.48063e7 −1.05342
\(724\) 4.11158e6 0.291516
\(725\) −9.50309e6 −0.671460
\(726\) −516430. −0.0363638
\(727\) 9.76045e6 0.684910 0.342455 0.939534i \(-0.388742\pi\)
0.342455 + 0.939534i \(0.388742\pi\)
\(728\) −2.63647e6 −0.184371
\(729\) 531441. 0.0370370
\(730\) 1.30543e7 0.906666
\(731\) 5.97050e6 0.413254
\(732\) 3.39191e6 0.233973
\(733\) 6.56659e6 0.451419 0.225710 0.974195i \(-0.427530\pi\)
0.225710 + 0.974195i \(0.427530\pi\)
\(734\) −6.71817e6 −0.460268
\(735\) 6.84587e6 0.467424
\(736\) 4.63743e6 0.315561
\(737\) 1.60123e7 1.08589
\(738\) −1.21043e6 −0.0818086
\(739\) −2.02187e7 −1.36189 −0.680946 0.732333i \(-0.738431\pi\)
−0.680946 + 0.732333i \(0.738431\pi\)
\(740\) 1.12958e6 0.0758297
\(741\) −646611. −0.0432611
\(742\) 1.28200e7 0.854825
\(743\) 5.51074e6 0.366216 0.183108 0.983093i \(-0.441384\pi\)
0.183108 + 0.983093i \(0.441384\pi\)
\(744\) −2.63020e6 −0.174203
\(745\) 1.37181e7 0.905532
\(746\) 1.27926e7 0.841610
\(747\) 4.79616e6 0.314480
\(748\) −9.12190e6 −0.596117
\(749\) 1.13797e7 0.741182
\(750\) 400883. 0.0260234
\(751\) −1.06252e6 −0.0687445 −0.0343723 0.999409i \(-0.510943\pi\)
−0.0343723 + 0.999409i \(0.510943\pi\)
\(752\) −1.09703e6 −0.0707412
\(753\) −240826. −0.0154781
\(754\) 818356. 0.0524221
\(755\) −4.14040e6 −0.264347
\(756\) −2.92029e6 −0.185832
\(757\) −2.03351e7 −1.28976 −0.644878 0.764285i \(-0.723092\pi\)
−0.644878 + 0.764285i \(0.723092\pi\)
\(758\) −5.25872e6 −0.332435
\(759\) 3.01120e6 0.189729
\(760\) −8.31765e6 −0.522356
\(761\) −4.03059e6 −0.252294 −0.126147 0.992012i \(-0.540261\pi\)
−0.126147 + 0.992012i \(0.540261\pi\)
\(762\) 2.31437e6 0.144393
\(763\) 8.11626e6 0.504713
\(764\) −1.42792e6 −0.0885056
\(765\) 5.62595e6 0.347570
\(766\) 1.32487e7 0.815832
\(767\) −369839. −0.0226999
\(768\) 5.46666e6 0.334440
\(769\) 1.40981e7 0.859697 0.429848 0.902901i \(-0.358567\pi\)
0.429848 + 0.902901i \(0.358567\pi\)
\(770\) −1.50104e7 −0.912357
\(771\) −1.32059e7 −0.800077
\(772\) −2.48175e7 −1.49870
\(773\) 2.73980e7 1.64919 0.824593 0.565727i \(-0.191404\pi\)
0.824593 + 0.565727i \(0.191404\pi\)
\(774\) −1.51011e6 −0.0906058
\(775\) −6.35757e6 −0.380222
\(776\) −2.67573e7 −1.59510
\(777\) 829914. 0.0493152
\(778\) −9.85149e6 −0.583516
\(779\) 3.74355e6 0.221024
\(780\) 1.89865e6 0.111740
\(781\) −1.38516e7 −0.812594
\(782\) −1.82855e6 −0.106927
\(783\) 2.08017e6 0.121254
\(784\) 3.57467e6 0.207704
\(785\) 5.08759e6 0.294671
\(786\) 6.87748e6 0.397075
\(787\) −4.27413e6 −0.245986 −0.122993 0.992408i \(-0.539249\pi\)
−0.122993 + 0.992408i \(0.539249\pi\)
\(788\) −2.38376e7 −1.36756
\(789\) −8.75279e6 −0.500558
\(790\) 1.17713e7 0.671051
\(791\) 1.67276e7 0.950590
\(792\) 5.29464e6 0.299932
\(793\) −1.62023e6 −0.0914945
\(794\) −8.99875e6 −0.506560
\(795\) −2.11867e7 −1.18890
\(796\) −1.54029e7 −0.861626
\(797\) −1.91541e7 −1.06811 −0.534056 0.845449i \(-0.679333\pi\)
−0.534056 + 0.845449i \(0.679333\pi\)
\(798\) −2.66294e6 −0.148032
\(799\) 2.51163e6 0.139184
\(800\) 1.97096e7 1.08881
\(801\) −4.70787e6 −0.259265
\(802\) 497974. 0.0273383
\(803\) 2.57001e7 1.40652
\(804\) −8.34113e6 −0.455077
\(805\) 1.02052e7 0.555047
\(806\) 547481. 0.0296846
\(807\) 3.96160e6 0.214135
\(808\) 2.73168e7 1.47198
\(809\) 3.40439e6 0.182881 0.0914405 0.995811i \(-0.470853\pi\)
0.0914405 + 0.995811i \(0.470853\pi\)
\(810\) −1.42296e6 −0.0762045
\(811\) 110790. 0.00591490 0.00295745 0.999996i \(-0.499059\pi\)
0.00295745 + 0.999996i \(0.499059\pi\)
\(812\) −1.14306e7 −0.608386
\(813\) −3.15060e6 −0.167173
\(814\) −655678. −0.0346840
\(815\) −1.99980e7 −1.05461
\(816\) 2.93767e6 0.154446
\(817\) 4.67039e6 0.244792
\(818\) −4.98609e6 −0.260542
\(819\) 1.39495e6 0.0726690
\(820\) −1.09922e7 −0.570888
\(821\) −3.27368e6 −0.169503 −0.0847516 0.996402i \(-0.527010\pi\)
−0.0847516 + 0.996402i \(0.527010\pi\)
\(822\) 1.13796e6 0.0587421
\(823\) 3.40687e7 1.75330 0.876650 0.481128i \(-0.159773\pi\)
0.876650 + 0.481128i \(0.159773\pi\)
\(824\) −1.94410e7 −0.997471
\(825\) 1.27979e7 0.654642
\(826\) −1.52311e6 −0.0776751
\(827\) −8.58953e6 −0.436723 −0.218361 0.975868i \(-0.570071\pi\)
−0.218361 + 0.975868i \(0.570071\pi\)
\(828\) −1.56860e6 −0.0795125
\(829\) −3.09207e7 −1.56266 −0.781328 0.624121i \(-0.785457\pi\)
−0.781328 + 0.624121i \(0.785457\pi\)
\(830\) −1.28420e7 −0.647049
\(831\) 1.29820e6 0.0652135
\(832\) −413570. −0.0207129
\(833\) −8.18418e6 −0.408661
\(834\) −8.56176e6 −0.426234
\(835\) 5.57344e7 2.76635
\(836\) −7.13555e6 −0.353111
\(837\) 1.39164e6 0.0686613
\(838\) 9.68219e6 0.476281
\(839\) −1.62311e7 −0.796054 −0.398027 0.917374i \(-0.630305\pi\)
−0.398027 + 0.917374i \(0.630305\pi\)
\(840\) 1.79439e7 0.877442
\(841\) −1.23689e7 −0.603034
\(842\) 4.18909e6 0.203629
\(843\) 7.36543e6 0.356968
\(844\) −1.97780e7 −0.955712
\(845\) 2.89247e7 1.39357
\(846\) −635263. −0.0305160
\(847\) −3.44565e6 −0.165030
\(848\) −1.10629e7 −0.528299
\(849\) −1.03785e7 −0.494159
\(850\) −7.77151e6 −0.368942
\(851\) 445778. 0.0211006
\(852\) 7.21561e6 0.340545
\(853\) 2.71754e7 1.27880 0.639402 0.768873i \(-0.279182\pi\)
0.639402 + 0.768873i \(0.279182\pi\)
\(854\) −6.67263e6 −0.313078
\(855\) 4.40086e6 0.205884
\(856\) 1.07476e7 0.501334
\(857\) 3.35816e7 1.56188 0.780942 0.624603i \(-0.214739\pi\)
0.780942 + 0.624603i \(0.214739\pi\)
\(858\) −1.10209e6 −0.0511091
\(859\) 2.03318e6 0.0940140 0.0470070 0.998895i \(-0.485032\pi\)
0.0470070 + 0.998895i \(0.485032\pi\)
\(860\) −1.37137e7 −0.632278
\(861\) −8.07607e6 −0.371272
\(862\) 9.90373e6 0.453974
\(863\) −5.62968e6 −0.257310 −0.128655 0.991689i \(-0.541066\pi\)
−0.128655 + 0.991689i \(0.541066\pi\)
\(864\) −4.31431e6 −0.196620
\(865\) 2.50587e7 1.13873
\(866\) −47268.9 −0.00214181
\(867\) 6.05294e6 0.273476
\(868\) −7.64709e6 −0.344506
\(869\) 2.31741e7 1.04101
\(870\) −5.56977e6 −0.249482
\(871\) 3.98436e6 0.177956
\(872\) 7.66546e6 0.341387
\(873\) 1.41573e7 0.628701
\(874\) −1.43037e6 −0.0633387
\(875\) 2.67472e6 0.118102
\(876\) −1.33877e7 −0.589449
\(877\) 3.64806e7 1.60163 0.800816 0.598910i \(-0.204399\pi\)
0.800816 + 0.598910i \(0.204399\pi\)
\(878\) 1.65508e7 0.724576
\(879\) 1.29101e7 0.563582
\(880\) 1.29531e7 0.563855
\(881\) 3.83390e7 1.66418 0.832091 0.554640i \(-0.187144\pi\)
0.832091 + 0.554640i \(0.187144\pi\)
\(882\) 2.07001e6 0.0895987
\(883\) −316951. −0.0136801 −0.00684006 0.999977i \(-0.502177\pi\)
−0.00684006 + 0.999977i \(0.502177\pi\)
\(884\) −2.26982e6 −0.0976923
\(885\) 2.51714e6 0.108031
\(886\) −7.26315e6 −0.310843
\(887\) 4.42907e7 1.89018 0.945092 0.326805i \(-0.105972\pi\)
0.945092 + 0.326805i \(0.105972\pi\)
\(888\) 783819. 0.0333567
\(889\) 1.54417e7 0.655299
\(890\) 1.26056e7 0.533443
\(891\) −2.80139e6 −0.118217
\(892\) 4.82054e6 0.202854
\(893\) 1.96471e6 0.0824460
\(894\) 4.14800e6 0.173578
\(895\) 4.55295e7 1.89992
\(896\) 2.89940e7 1.20653
\(897\) 749281. 0.0310931
\(898\) −1.17871e7 −0.487770
\(899\) 5.44715e6 0.224787
\(900\) −6.66670e6 −0.274350
\(901\) 2.53285e7 1.03943
\(902\) 6.38054e6 0.261121
\(903\) −1.00756e7 −0.411197
\(904\) 1.57985e7 0.642978
\(905\) 1.33671e7 0.542520
\(906\) −1.25195e6 −0.0506717
\(907\) −3.22962e7 −1.30357 −0.651783 0.758405i \(-0.725979\pi\)
−0.651783 + 0.758405i \(0.725979\pi\)
\(908\) 1.94683e7 0.783635
\(909\) −1.44533e7 −0.580172
\(910\) −3.73506e6 −0.149518
\(911\) 1.74012e6 0.0694679 0.0347340 0.999397i \(-0.488942\pi\)
0.0347340 + 0.999397i \(0.488942\pi\)
\(912\) 2.29797e6 0.0914866
\(913\) −2.52820e7 −1.00377
\(914\) −1.97111e7 −0.780449
\(915\) 1.10274e7 0.435431
\(916\) −2.77557e7 −1.09298
\(917\) 4.58870e7 1.80205
\(918\) 1.70114e6 0.0666244
\(919\) −2.88435e7 −1.12657 −0.563286 0.826262i \(-0.690463\pi\)
−0.563286 + 0.826262i \(0.690463\pi\)
\(920\) 9.63834e6 0.375433
\(921\) 1.26732e7 0.492310
\(922\) −1.89254e7 −0.733192
\(923\) −3.44673e6 −0.133169
\(924\) 1.53937e7 0.593149
\(925\) 1.89460e6 0.0728055
\(926\) 3.55164e6 0.136114
\(927\) 1.02862e7 0.393148
\(928\) −1.68871e7 −0.643703
\(929\) −2.74877e7 −1.04496 −0.522479 0.852652i \(-0.674993\pi\)
−0.522479 + 0.852652i \(0.674993\pi\)
\(930\) −3.72618e6 −0.141272
\(931\) −6.40203e6 −0.242071
\(932\) −2.90836e7 −1.09675
\(933\) 3.65370e6 0.137413
\(934\) 8.45112e6 0.316991
\(935\) −2.96561e7 −1.10939
\(936\) 1.31747e6 0.0491532
\(937\) −1.55562e7 −0.578834 −0.289417 0.957203i \(-0.593461\pi\)
−0.289417 + 0.957203i \(0.593461\pi\)
\(938\) 1.64088e7 0.608935
\(939\) 2.05907e7 0.762091
\(940\) −5.76899e6 −0.212951
\(941\) 3.24594e6 0.119499 0.0597497 0.998213i \(-0.480970\pi\)
0.0597497 + 0.998213i \(0.480970\pi\)
\(942\) 1.53835e6 0.0564845
\(943\) −4.33796e6 −0.158857
\(944\) 1.31436e6 0.0480048
\(945\) −9.49410e6 −0.345839
\(946\) 7.96025e6 0.289200
\(947\) −1.66618e6 −0.0603734 −0.0301867 0.999544i \(-0.509610\pi\)
−0.0301867 + 0.999544i \(0.509610\pi\)
\(948\) −1.20719e7 −0.436269
\(949\) 6.39499e6 0.230502
\(950\) −6.07922e6 −0.218544
\(951\) 1.54818e7 0.555100
\(952\) −2.14518e7 −0.767134
\(953\) 9.27492e6 0.330810 0.165405 0.986226i \(-0.447107\pi\)
0.165405 + 0.986226i \(0.447107\pi\)
\(954\) −6.40629e6 −0.227895
\(955\) −4.64229e6 −0.164711
\(956\) −1.16117e6 −0.0410914
\(957\) −1.09652e7 −0.387023
\(958\) 1.89077e7 0.665617
\(959\) 7.59258e6 0.266589
\(960\) 2.81478e6 0.0985748
\(961\) −2.49850e7 −0.872712
\(962\) −163153. −0.00568405
\(963\) −5.68655e6 −0.197598
\(964\) −4.06571e7 −1.40911
\(965\) −8.06837e7 −2.78912
\(966\) 3.08577e6 0.106395
\(967\) 3.90239e7 1.34204 0.671019 0.741440i \(-0.265857\pi\)
0.671019 + 0.741440i \(0.265857\pi\)
\(968\) −3.25428e6 −0.111626
\(969\) −5.26119e6 −0.180001
\(970\) −3.79069e7 −1.29357
\(971\) 1.52419e7 0.518791 0.259395 0.965771i \(-0.416477\pi\)
0.259395 + 0.965771i \(0.416477\pi\)
\(972\) 1.45930e6 0.0495427
\(973\) −5.71247e7 −1.93438
\(974\) 5.65343e6 0.190948
\(975\) 3.18452e6 0.107283
\(976\) 5.75810e6 0.193488
\(977\) −4.86169e7 −1.62949 −0.814744 0.579821i \(-0.803122\pi\)
−0.814744 + 0.579821i \(0.803122\pi\)
\(978\) −6.04687e6 −0.202155
\(979\) 2.48166e7 0.827535
\(980\) 1.87983e7 0.625250
\(981\) −4.05579e6 −0.134556
\(982\) −1.15551e7 −0.382379
\(983\) −5.24218e6 −0.173033 −0.0865163 0.996250i \(-0.527573\pi\)
−0.0865163 + 0.996250i \(0.527573\pi\)
\(984\) −7.62751e6 −0.251128
\(985\) −7.74979e7 −2.54507
\(986\) 6.65861e6 0.218118
\(987\) −4.23852e6 −0.138491
\(988\) −1.77555e6 −0.0578683
\(989\) −5.41196e6 −0.175940
\(990\) 7.50086e6 0.243233
\(991\) −2.53147e7 −0.818819 −0.409409 0.912351i \(-0.634265\pi\)
−0.409409 + 0.912351i \(0.634265\pi\)
\(992\) −1.12975e7 −0.364505
\(993\) 1.11081e7 0.357491
\(994\) −1.41947e7 −0.455680
\(995\) −5.00760e7 −1.60351
\(996\) 1.31700e7 0.420664
\(997\) −3.69734e7 −1.17802 −0.589008 0.808127i \(-0.700481\pi\)
−0.589008 + 0.808127i \(0.700481\pi\)
\(998\) −1.51931e7 −0.482859
\(999\) −414718. −0.0131474
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 177.6.a.d.1.8 13
3.2 odd 2 531.6.a.e.1.6 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.d.1.8 13 1.1 even 1 trivial
531.6.a.e.1.6 13 3.2 odd 2