Properties

Label 177.6.a.c.1.6
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + 15565376 x^{4} + 6775664 x^{3} - 75006848 x^{2} + 21512960 x + 49172480\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.38446\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.615542 q^{2} +9.00000 q^{3} -31.6211 q^{4} +61.9372 q^{5} +5.53988 q^{6} -209.534 q^{7} -39.1615 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+0.615542 q^{2} +9.00000 q^{3} -31.6211 q^{4} +61.9372 q^{5} +5.53988 q^{6} -209.534 q^{7} -39.1615 q^{8} +81.0000 q^{9} +38.1250 q^{10} +1.45515 q^{11} -284.590 q^{12} +351.120 q^{13} -128.977 q^{14} +557.435 q^{15} +987.770 q^{16} +689.107 q^{17} +49.8589 q^{18} +2735.73 q^{19} -1958.52 q^{20} -1885.80 q^{21} +0.895706 q^{22} -1310.96 q^{23} -352.453 q^{24} +711.222 q^{25} +216.129 q^{26} +729.000 q^{27} +6625.69 q^{28} +508.995 q^{29} +343.125 q^{30} +4218.33 q^{31} +1861.18 q^{32} +13.0964 q^{33} +424.174 q^{34} -12977.9 q^{35} -2561.31 q^{36} +4022.89 q^{37} +1683.95 q^{38} +3160.08 q^{39} -2425.55 q^{40} +7333.96 q^{41} -1160.79 q^{42} +5250.80 q^{43} -46.0135 q^{44} +5016.92 q^{45} -806.951 q^{46} +25394.3 q^{47} +8889.93 q^{48} +27097.3 q^{49} +437.787 q^{50} +6201.96 q^{51} -11102.8 q^{52} -16562.5 q^{53} +448.730 q^{54} +90.1280 q^{55} +8205.64 q^{56} +24621.5 q^{57} +313.308 q^{58} -3481.00 q^{59} -17626.7 q^{60} -4267.66 q^{61} +2596.56 q^{62} -16972.2 q^{63} -30463.0 q^{64} +21747.4 q^{65} +8.06136 q^{66} +13390.6 q^{67} -21790.3 q^{68} -11798.6 q^{69} -7988.46 q^{70} +54112.6 q^{71} -3172.08 q^{72} -32575.3 q^{73} +2476.26 q^{74} +6401.00 q^{75} -86506.7 q^{76} -304.903 q^{77} +1945.16 q^{78} +59592.4 q^{79} +61179.7 q^{80} +6561.00 q^{81} +4514.36 q^{82} +94095.2 q^{83} +59631.2 q^{84} +42681.4 q^{85} +3232.09 q^{86} +4580.95 q^{87} -56.9858 q^{88} -136311. q^{89} +3088.12 q^{90} -73571.4 q^{91} +41454.0 q^{92} +37965.0 q^{93} +15631.3 q^{94} +169443. q^{95} +16750.6 q^{96} +135175. q^{97} +16679.6 q^{98} +117.867 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + O(q^{10}) \) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + 601q^{10} + 1480q^{11} + 1782q^{12} + 472q^{13} + 1065q^{14} + 1422q^{15} + 6370q^{16} + 1565q^{17} + 1782q^{18} + 3939q^{19} + 8033q^{20} + 3717q^{21} - 1738q^{22} + 7245q^{23} + 6507q^{24} + 9690q^{25} + 3764q^{26} + 8748q^{27} + 12154q^{28} + 10003q^{29} + 5409q^{30} + 7295q^{31} + 11628q^{32} + 13320q^{33} - 16344q^{34} + 11015q^{35} + 16038q^{36} + 6741q^{37} + 3035q^{38} + 4248q^{39} + 5572q^{40} + 34025q^{41} + 9585q^{42} - 6336q^{43} + 41168q^{44} + 12798q^{45} + 2345q^{46} + 66167q^{47} + 57330q^{48} + 28319q^{49} + 31173q^{50} + 14085q^{51} + 16440q^{52} + 62290q^{53} + 16038q^{54} + 55764q^{55} + 107306q^{56} + 35451q^{57} + 37952q^{58} - 41772q^{59} + 72297q^{60} + 68469q^{61} + 99190q^{62} + 33453q^{63} + 68525q^{64} + 80156q^{65} - 15642q^{66} + 113310q^{67} + 33887q^{68} + 65205q^{69} + 32034q^{70} + 84520q^{71} + 58563q^{72} + 135895q^{73} - 31962q^{74} + 87210q^{75} - 61848q^{76} - 3799q^{77} + 33876q^{78} + 14122q^{79} + 77609q^{80} + 78732q^{81} - 1501q^{82} + 114463q^{83} + 109386q^{84} - 101097q^{85} - 203536q^{86} + 90027q^{87} - 244967q^{88} + 189109q^{89} + 48681q^{90} - 168249q^{91} - 71946q^{92} + 65655q^{93} - 472284q^{94} + 21923q^{95} + 104652q^{96} - 76192q^{97} - 17544q^{98} + 119880q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.615542 0.108813 0.0544067 0.998519i \(-0.482673\pi\)
0.0544067 + 0.998519i \(0.482673\pi\)
\(3\) 9.00000 0.577350
\(4\) −31.6211 −0.988160
\(5\) 61.9372 1.10797 0.553984 0.832528i \(-0.313107\pi\)
0.553984 + 0.832528i \(0.313107\pi\)
\(6\) 5.53988 0.0628235
\(7\) −209.534 −1.61625 −0.808125 0.589011i \(-0.799518\pi\)
−0.808125 + 0.589011i \(0.799518\pi\)
\(8\) −39.1615 −0.216339
\(9\) 81.0000 0.333333
\(10\) 38.1250 0.120562
\(11\) 1.45515 0.00362599 0.00181299 0.999998i \(-0.499423\pi\)
0.00181299 + 0.999998i \(0.499423\pi\)
\(12\) −284.590 −0.570514
\(13\) 351.120 0.576231 0.288116 0.957596i \(-0.406971\pi\)
0.288116 + 0.957596i \(0.406971\pi\)
\(14\) −128.977 −0.175870
\(15\) 557.435 0.639685
\(16\) 987.770 0.964619
\(17\) 689.107 0.578315 0.289157 0.957282i \(-0.406625\pi\)
0.289157 + 0.957282i \(0.406625\pi\)
\(18\) 49.8589 0.0362712
\(19\) 2735.73 1.73856 0.869278 0.494323i \(-0.164584\pi\)
0.869278 + 0.494323i \(0.164584\pi\)
\(20\) −1958.52 −1.09485
\(21\) −1885.80 −0.933143
\(22\) 0.895706 0.000394556 0
\(23\) −1310.96 −0.516737 −0.258369 0.966046i \(-0.583185\pi\)
−0.258369 + 0.966046i \(0.583185\pi\)
\(24\) −352.453 −0.124903
\(25\) 711.222 0.227591
\(26\) 216.129 0.0627017
\(27\) 729.000 0.192450
\(28\) 6625.69 1.59711
\(29\) 508.995 0.112388 0.0561938 0.998420i \(-0.482104\pi\)
0.0561938 + 0.998420i \(0.482104\pi\)
\(30\) 343.125 0.0696064
\(31\) 4218.33 0.788381 0.394191 0.919029i \(-0.371025\pi\)
0.394191 + 0.919029i \(0.371025\pi\)
\(32\) 1861.18 0.321302
\(33\) 13.0964 0.00209346
\(34\) 424.174 0.0629285
\(35\) −12977.9 −1.79075
\(36\) −2561.31 −0.329387
\(37\) 4022.89 0.483096 0.241548 0.970389i \(-0.422345\pi\)
0.241548 + 0.970389i \(0.422345\pi\)
\(38\) 1683.95 0.189178
\(39\) 3160.08 0.332687
\(40\) −2425.55 −0.239696
\(41\) 7333.96 0.681364 0.340682 0.940179i \(-0.389342\pi\)
0.340682 + 0.940179i \(0.389342\pi\)
\(42\) −1160.79 −0.101538
\(43\) 5250.80 0.433066 0.216533 0.976275i \(-0.430525\pi\)
0.216533 + 0.976275i \(0.430525\pi\)
\(44\) −46.0135 −0.00358305
\(45\) 5016.92 0.369322
\(46\) −806.951 −0.0562280
\(47\) 25394.3 1.67684 0.838420 0.545024i \(-0.183479\pi\)
0.838420 + 0.545024i \(0.183479\pi\)
\(48\) 8889.93 0.556923
\(49\) 27097.3 1.61227
\(50\) 437.787 0.0247650
\(51\) 6201.96 0.333890
\(52\) −11102.8 −0.569408
\(53\) −16562.5 −0.809911 −0.404956 0.914336i \(-0.632713\pi\)
−0.404956 + 0.914336i \(0.632713\pi\)
\(54\) 448.730 0.0209412
\(55\) 90.1280 0.00401747
\(56\) 8205.64 0.349657
\(57\) 24621.5 1.00376
\(58\) 313.308 0.0122293
\(59\) −3481.00 −0.130189
\(60\) −17626.7 −0.632111
\(61\) −4267.66 −0.146847 −0.0734235 0.997301i \(-0.523392\pi\)
−0.0734235 + 0.997301i \(0.523392\pi\)
\(62\) 2596.56 0.0857865
\(63\) −16972.2 −0.538750
\(64\) −30463.0 −0.929657
\(65\) 21747.4 0.638445
\(66\) 8.06136 0.000227797 0
\(67\) 13390.6 0.364429 0.182215 0.983259i \(-0.441673\pi\)
0.182215 + 0.983259i \(0.441673\pi\)
\(68\) −21790.3 −0.571467
\(69\) −11798.6 −0.298338
\(70\) −7988.46 −0.194858
\(71\) 54112.6 1.27395 0.636976 0.770884i \(-0.280185\pi\)
0.636976 + 0.770884i \(0.280185\pi\)
\(72\) −3172.08 −0.0721129
\(73\) −32575.3 −0.715453 −0.357726 0.933826i \(-0.616448\pi\)
−0.357726 + 0.933826i \(0.616448\pi\)
\(74\) 2476.26 0.0525674
\(75\) 6401.00 0.131400
\(76\) −86506.7 −1.71797
\(77\) −304.903 −0.00586050
\(78\) 1945.16 0.0362009
\(79\) 59592.4 1.07429 0.537147 0.843489i \(-0.319502\pi\)
0.537147 + 0.843489i \(0.319502\pi\)
\(80\) 61179.7 1.06877
\(81\) 6561.00 0.111111
\(82\) 4514.36 0.0741415
\(83\) 94095.2 1.49924 0.749622 0.661867i \(-0.230236\pi\)
0.749622 + 0.661867i \(0.230236\pi\)
\(84\) 59631.2 0.922094
\(85\) 42681.4 0.640754
\(86\) 3232.09 0.0471235
\(87\) 4580.95 0.0648870
\(88\) −56.9858 −0.000784441 0
\(89\) −136311. −1.82414 −0.912068 0.410040i \(-0.865515\pi\)
−0.912068 + 0.410040i \(0.865515\pi\)
\(90\) 3088.12 0.0401873
\(91\) −73571.4 −0.931334
\(92\) 41454.0 0.510619
\(93\) 37965.0 0.455172
\(94\) 15631.3 0.182463
\(95\) 169443. 1.92626
\(96\) 16750.6 0.185504
\(97\) 135175. 1.45870 0.729350 0.684141i \(-0.239823\pi\)
0.729350 + 0.684141i \(0.239823\pi\)
\(98\) 16679.6 0.175436
\(99\) 117.867 0.00120866
\(100\) −22489.6 −0.224896
\(101\) −25496.8 −0.248704 −0.124352 0.992238i \(-0.539685\pi\)
−0.124352 + 0.992238i \(0.539685\pi\)
\(102\) 3817.57 0.0363318
\(103\) 62339.3 0.578987 0.289493 0.957180i \(-0.406513\pi\)
0.289493 + 0.957180i \(0.406513\pi\)
\(104\) −13750.4 −0.124661
\(105\) −116801. −1.03389
\(106\) −10194.9 −0.0881293
\(107\) −16258.7 −0.137286 −0.0686429 0.997641i \(-0.521867\pi\)
−0.0686429 + 0.997641i \(0.521867\pi\)
\(108\) −23051.8 −0.190171
\(109\) −209396. −1.68812 −0.844059 0.536250i \(-0.819840\pi\)
−0.844059 + 0.536250i \(0.819840\pi\)
\(110\) 55.4776 0.000437155 0
\(111\) 36206.0 0.278916
\(112\) −206971. −1.55907
\(113\) −143342. −1.05603 −0.528017 0.849234i \(-0.677064\pi\)
−0.528017 + 0.849234i \(0.677064\pi\)
\(114\) 15155.6 0.109222
\(115\) −81197.2 −0.572528
\(116\) −16095.0 −0.111057
\(117\) 28440.7 0.192077
\(118\) −2142.70 −0.0141663
\(119\) −144391. −0.934702
\(120\) −21830.0 −0.138389
\(121\) −161049. −0.999987
\(122\) −2626.92 −0.0159789
\(123\) 66005.7 0.393386
\(124\) −133388. −0.779046
\(125\) −149503. −0.855804
\(126\) −10447.1 −0.0586233
\(127\) 79745.4 0.438729 0.219365 0.975643i \(-0.429602\pi\)
0.219365 + 0.975643i \(0.429602\pi\)
\(128\) −78309.0 −0.422461
\(129\) 47257.2 0.250031
\(130\) 13386.4 0.0694714
\(131\) −5118.41 −0.0260589 −0.0130295 0.999915i \(-0.504148\pi\)
−0.0130295 + 0.999915i \(0.504148\pi\)
\(132\) −414.121 −0.00206868
\(133\) −573227. −2.80994
\(134\) 8242.49 0.0396548
\(135\) 45152.3 0.213228
\(136\) −26986.4 −0.125112
\(137\) −237403. −1.08065 −0.540324 0.841457i \(-0.681698\pi\)
−0.540324 + 0.841457i \(0.681698\pi\)
\(138\) −7262.56 −0.0324632
\(139\) −60191.4 −0.264239 −0.132120 0.991234i \(-0.542178\pi\)
−0.132120 + 0.991234i \(0.542178\pi\)
\(140\) 410377. 1.76955
\(141\) 228549. 0.968125
\(142\) 33308.6 0.138623
\(143\) 510.932 0.00208941
\(144\) 80009.4 0.321540
\(145\) 31525.7 0.124522
\(146\) −20051.4 −0.0778509
\(147\) 243876. 0.930842
\(148\) −127208. −0.477376
\(149\) −226647. −0.836343 −0.418171 0.908368i \(-0.637329\pi\)
−0.418171 + 0.908368i \(0.637329\pi\)
\(150\) 3940.09 0.0142981
\(151\) −294492. −1.05107 −0.525534 0.850773i \(-0.676134\pi\)
−0.525534 + 0.850773i \(0.676134\pi\)
\(152\) −107135. −0.376117
\(153\) 55817.7 0.192772
\(154\) −187.681 −0.000637702 0
\(155\) 261272. 0.873500
\(156\) −99925.1 −0.328748
\(157\) 527580. 1.70820 0.854101 0.520108i \(-0.174108\pi\)
0.854101 + 0.520108i \(0.174108\pi\)
\(158\) 36681.6 0.116898
\(159\) −149063. −0.467602
\(160\) 115276. 0.355992
\(161\) 274690. 0.835177
\(162\) 4038.57 0.0120904
\(163\) 138581. 0.408540 0.204270 0.978915i \(-0.434518\pi\)
0.204270 + 0.978915i \(0.434518\pi\)
\(164\) −231908. −0.673296
\(165\) 811.152 0.00231949
\(166\) 57919.5 0.163138
\(167\) 201512. 0.559126 0.279563 0.960127i \(-0.409810\pi\)
0.279563 + 0.960127i \(0.409810\pi\)
\(168\) 73850.8 0.201875
\(169\) −248008. −0.667958
\(170\) 26272.2 0.0697227
\(171\) 221594. 0.579519
\(172\) −166036. −0.427939
\(173\) 463291. 1.17690 0.588449 0.808535i \(-0.299739\pi\)
0.588449 + 0.808535i \(0.299739\pi\)
\(174\) 2819.77 0.00706058
\(175\) −149025. −0.367844
\(176\) 1437.35 0.00349770
\(177\) −31329.0 −0.0751646
\(178\) −83905.4 −0.198490
\(179\) −235004. −0.548205 −0.274103 0.961700i \(-0.588381\pi\)
−0.274103 + 0.961700i \(0.588381\pi\)
\(180\) −158640. −0.364949
\(181\) 345261. 0.783341 0.391670 0.920106i \(-0.371897\pi\)
0.391670 + 0.920106i \(0.371897\pi\)
\(182\) −45286.3 −0.101342
\(183\) −38408.9 −0.0847822
\(184\) 51339.1 0.111790
\(185\) 249167. 0.535255
\(186\) 23369.0 0.0495288
\(187\) 1002.75 0.00209696
\(188\) −802996. −1.65699
\(189\) −152750. −0.311048
\(190\) 104300. 0.209603
\(191\) 973228. 1.93033 0.965164 0.261646i \(-0.0842654\pi\)
0.965164 + 0.261646i \(0.0842654\pi\)
\(192\) −274167. −0.536738
\(193\) −493451. −0.953567 −0.476783 0.879021i \(-0.658197\pi\)
−0.476783 + 0.879021i \(0.658197\pi\)
\(194\) 83205.7 0.158726
\(195\) 195726. 0.368607
\(196\) −856848. −1.59318
\(197\) 339146. 0.622616 0.311308 0.950309i \(-0.399233\pi\)
0.311308 + 0.950309i \(0.399233\pi\)
\(198\) 72.5522 0.000131519 0
\(199\) −589702. −1.05560 −0.527801 0.849368i \(-0.676983\pi\)
−0.527801 + 0.849368i \(0.676983\pi\)
\(200\) −27852.5 −0.0492368
\(201\) 120516. 0.210403
\(202\) −15694.4 −0.0270623
\(203\) −106652. −0.181646
\(204\) −196113. −0.329937
\(205\) 454245. 0.754929
\(206\) 38372.4 0.0630015
\(207\) −106188. −0.172246
\(208\) 346825. 0.555844
\(209\) 3980.89 0.00630398
\(210\) −71896.2 −0.112501
\(211\) 779769. 1.20576 0.602879 0.797833i \(-0.294020\pi\)
0.602879 + 0.797833i \(0.294020\pi\)
\(212\) 523726. 0.800322
\(213\) 487014. 0.735516
\(214\) −10007.9 −0.0149385
\(215\) 325220. 0.479823
\(216\) −28548.7 −0.0416344
\(217\) −883882. −1.27422
\(218\) −128892. −0.183690
\(219\) −293177. −0.413067
\(220\) −2849.95 −0.00396991
\(221\) 241959. 0.333243
\(222\) 22286.3 0.0303498
\(223\) 148653. 0.200176 0.100088 0.994979i \(-0.468088\pi\)
0.100088 + 0.994979i \(0.468088\pi\)
\(224\) −389980. −0.519305
\(225\) 57609.0 0.0758637
\(226\) −88233.2 −0.114911
\(227\) −705507. −0.908733 −0.454367 0.890815i \(-0.650134\pi\)
−0.454367 + 0.890815i \(0.650134\pi\)
\(228\) −778561. −0.991871
\(229\) 221449. 0.279052 0.139526 0.990218i \(-0.455442\pi\)
0.139526 + 0.990218i \(0.455442\pi\)
\(230\) −49980.3 −0.0622987
\(231\) −2744.13 −0.00338356
\(232\) −19933.0 −0.0243138
\(233\) −303806. −0.366612 −0.183306 0.983056i \(-0.558680\pi\)
−0.183306 + 0.983056i \(0.558680\pi\)
\(234\) 17506.4 0.0209006
\(235\) 1.57285e6 1.85788
\(236\) 110073. 0.128647
\(237\) 536332. 0.620244
\(238\) −88878.8 −0.101708
\(239\) 1.12989e6 1.27950 0.639751 0.768583i \(-0.279038\pi\)
0.639751 + 0.768583i \(0.279038\pi\)
\(240\) 550618. 0.617052
\(241\) −1.30930e6 −1.45210 −0.726049 0.687643i \(-0.758645\pi\)
−0.726049 + 0.687643i \(0.758645\pi\)
\(242\) −99132.3 −0.108812
\(243\) 59049.0 0.0641500
\(244\) 134948. 0.145108
\(245\) 1.67834e6 1.78634
\(246\) 40629.2 0.0428056
\(247\) 960568. 1.00181
\(248\) −165196. −0.170557
\(249\) 846856. 0.865588
\(250\) −92025.2 −0.0931230
\(251\) −948490. −0.950273 −0.475136 0.879912i \(-0.657601\pi\)
−0.475136 + 0.879912i \(0.657601\pi\)
\(252\) 536681. 0.532371
\(253\) −1907.64 −0.00187368
\(254\) 49086.7 0.0477396
\(255\) 384133. 0.369939
\(256\) 926614. 0.883688
\(257\) 1.82976e6 1.72807 0.864036 0.503431i \(-0.167929\pi\)
0.864036 + 0.503431i \(0.167929\pi\)
\(258\) 29088.8 0.0272067
\(259\) −842930. −0.780804
\(260\) −687676. −0.630886
\(261\) 41228.6 0.0374625
\(262\) −3150.60 −0.00283556
\(263\) 1.23847e6 1.10407 0.552033 0.833822i \(-0.313852\pi\)
0.552033 + 0.833822i \(0.313852\pi\)
\(264\) −512.872 −0.000452897 0
\(265\) −1.02584e6 −0.897355
\(266\) −352845. −0.305760
\(267\) −1.22680e6 −1.05316
\(268\) −423426. −0.360115
\(269\) 1.19822e6 1.00962 0.504808 0.863232i \(-0.331563\pi\)
0.504808 + 0.863232i \(0.331563\pi\)
\(270\) 27793.1 0.0232021
\(271\) −907388. −0.750533 −0.375267 0.926917i \(-0.622449\pi\)
−0.375267 + 0.926917i \(0.622449\pi\)
\(272\) 680679. 0.557854
\(273\) −662142. −0.537706
\(274\) −146131. −0.117589
\(275\) 1034.94 0.000825243 0
\(276\) 373086. 0.294806
\(277\) 196862. 0.154157 0.0770783 0.997025i \(-0.475441\pi\)
0.0770783 + 0.997025i \(0.475441\pi\)
\(278\) −37050.3 −0.0287528
\(279\) 341685. 0.262794
\(280\) 508235. 0.387409
\(281\) 2.31407e6 1.74828 0.874140 0.485674i \(-0.161426\pi\)
0.874140 + 0.485674i \(0.161426\pi\)
\(282\) 140681. 0.105345
\(283\) 42804.6 0.0317705 0.0158852 0.999874i \(-0.494943\pi\)
0.0158852 + 0.999874i \(0.494943\pi\)
\(284\) −1.71110e6 −1.25887
\(285\) 1.52499e6 1.11213
\(286\) 314.500 0.000227356 0
\(287\) −1.53671e6 −1.10125
\(288\) 150756. 0.107101
\(289\) −944988. −0.665552
\(290\) 19405.4 0.0135496
\(291\) 1.21657e6 0.842181
\(292\) 1.03007e6 0.706981
\(293\) −943192. −0.641846 −0.320923 0.947105i \(-0.603993\pi\)
−0.320923 + 0.947105i \(0.603993\pi\)
\(294\) 150116. 0.101288
\(295\) −215604. −0.144245
\(296\) −157542. −0.104512
\(297\) 1060.80 0.000697821 0
\(298\) −139511. −0.0910054
\(299\) −460304. −0.297760
\(300\) −202407. −0.129844
\(301\) −1.10022e6 −0.699944
\(302\) −181272. −0.114370
\(303\) −229471. −0.143589
\(304\) 2.70227e6 1.67704
\(305\) −264327. −0.162702
\(306\) 34358.1 0.0209762
\(307\) −1.05157e6 −0.636784 −0.318392 0.947959i \(-0.603143\pi\)
−0.318392 + 0.947959i \(0.603143\pi\)
\(308\) 9641.37 0.00579111
\(309\) 561053. 0.334278
\(310\) 160824. 0.0950486
\(311\) −616781. −0.361601 −0.180801 0.983520i \(-0.557869\pi\)
−0.180801 + 0.983520i \(0.557869\pi\)
\(312\) −123753. −0.0719731
\(313\) 1.22803e6 0.708516 0.354258 0.935148i \(-0.384733\pi\)
0.354258 + 0.935148i \(0.384733\pi\)
\(314\) 324747. 0.185875
\(315\) −1.05121e6 −0.596917
\(316\) −1.88438e6 −1.06157
\(317\) −2.46036e6 −1.37515 −0.687577 0.726112i \(-0.741325\pi\)
−0.687577 + 0.726112i \(0.741325\pi\)
\(318\) −91754.5 −0.0508815
\(319\) 740.664 0.000407516 0
\(320\) −1.88679e6 −1.03003
\(321\) −146328. −0.0792620
\(322\) 169083. 0.0908785
\(323\) 1.88521e6 1.00543
\(324\) −207466. −0.109796
\(325\) 249724. 0.131145
\(326\) 85302.3 0.0444546
\(327\) −1.88457e6 −0.974635
\(328\) −287209. −0.147405
\(329\) −5.32096e6 −2.71019
\(330\) 499.298 0.000252392 0
\(331\) 58317.0 0.0292567 0.0146283 0.999893i \(-0.495343\pi\)
0.0146283 + 0.999893i \(0.495343\pi\)
\(332\) −2.97539e6 −1.48149
\(333\) 325854. 0.161032
\(334\) 124039. 0.0608405
\(335\) 829378. 0.403776
\(336\) −1.86274e6 −0.900127
\(337\) 2.05912e6 0.987658 0.493829 0.869559i \(-0.335597\pi\)
0.493829 + 0.869559i \(0.335597\pi\)
\(338\) −152659. −0.0726828
\(339\) −1.29008e6 −0.609702
\(340\) −1.34963e6 −0.633167
\(341\) 6138.30 0.00285866
\(342\) 136400. 0.0630595
\(343\) −2.15617e6 −0.989574
\(344\) −205629. −0.0936890
\(345\) −730775. −0.330549
\(346\) 285175. 0.128062
\(347\) −3.05058e6 −1.36006 −0.680030 0.733184i \(-0.738033\pi\)
−0.680030 + 0.733184i \(0.738033\pi\)
\(348\) −144855. −0.0641187
\(349\) −2.54878e6 −1.12013 −0.560065 0.828449i \(-0.689224\pi\)
−0.560065 + 0.828449i \(0.689224\pi\)
\(350\) −91731.2 −0.0400264
\(351\) 255966. 0.110896
\(352\) 2708.30 0.00116504
\(353\) −139978. −0.0597891 −0.0298945 0.999553i \(-0.509517\pi\)
−0.0298945 + 0.999553i \(0.509517\pi\)
\(354\) −19284.3 −0.00817892
\(355\) 3.35159e6 1.41150
\(356\) 4.31032e6 1.80254
\(357\) −1.29952e6 −0.539650
\(358\) −144655. −0.0596521
\(359\) −587724. −0.240679 −0.120339 0.992733i \(-0.538398\pi\)
−0.120339 + 0.992733i \(0.538398\pi\)
\(360\) −196470. −0.0798987
\(361\) 5.00810e6 2.02258
\(362\) 212522. 0.0852380
\(363\) −1.44944e6 −0.577343
\(364\) 2.32641e6 0.920307
\(365\) −2.01762e6 −0.792698
\(366\) −23642.3 −0.00922544
\(367\) 360316. 0.139643 0.0698214 0.997560i \(-0.477757\pi\)
0.0698214 + 0.997560i \(0.477757\pi\)
\(368\) −1.29493e6 −0.498455
\(369\) 594051. 0.227121
\(370\) 153372. 0.0582429
\(371\) 3.47041e6 1.30902
\(372\) −1.20049e6 −0.449783
\(373\) 5.26924e6 1.96099 0.980497 0.196535i \(-0.0629690\pi\)
0.980497 + 0.196535i \(0.0629690\pi\)
\(374\) 617.237 0.000228178 0
\(375\) −1.34552e6 −0.494098
\(376\) −994478. −0.362765
\(377\) 178718. 0.0647612
\(378\) −94024.1 −0.0338462
\(379\) 3.43445e6 1.22817 0.614086 0.789239i \(-0.289525\pi\)
0.614086 + 0.789239i \(0.289525\pi\)
\(380\) −5.35799e6 −1.90346
\(381\) 717709. 0.253300
\(382\) 599062. 0.210046
\(383\) −3.55421e6 −1.23807 −0.619036 0.785363i \(-0.712476\pi\)
−0.619036 + 0.785363i \(0.712476\pi\)
\(384\) −704781. −0.243908
\(385\) −18884.8 −0.00649324
\(386\) −303740. −0.103761
\(387\) 425315. 0.144355
\(388\) −4.27437e6 −1.44143
\(389\) 4.52272e6 1.51539 0.757697 0.652606i \(-0.226324\pi\)
0.757697 + 0.652606i \(0.226324\pi\)
\(390\) 120478. 0.0401094
\(391\) −903392. −0.298837
\(392\) −1.06117e6 −0.348795
\(393\) −46065.7 −0.0150451
\(394\) 208758. 0.0677491
\(395\) 3.69099e6 1.19028
\(396\) −3727.09 −0.00119435
\(397\) 407622. 0.129802 0.0649011 0.997892i \(-0.479327\pi\)
0.0649011 + 0.997892i \(0.479327\pi\)
\(398\) −362987. −0.114864
\(399\) −5.15904e6 −1.62232
\(400\) 702524. 0.219539
\(401\) 3.00347e6 0.932743 0.466372 0.884589i \(-0.345561\pi\)
0.466372 + 0.884589i \(0.345561\pi\)
\(402\) 74182.4 0.0228947
\(403\) 1.48114e6 0.454290
\(404\) 806237. 0.245759
\(405\) 406370. 0.123107
\(406\) −65648.5 −0.0197656
\(407\) 5853.91 0.00175170
\(408\) −242878. −0.0722333
\(409\) −4.77645e6 −1.41188 −0.705938 0.708274i \(-0.749474\pi\)
−0.705938 + 0.708274i \(0.749474\pi\)
\(410\) 279607. 0.0821464
\(411\) −2.13662e6 −0.623912
\(412\) −1.97124e6 −0.572131
\(413\) 729387. 0.210418
\(414\) −65363.0 −0.0187427
\(415\) 5.82799e6 1.66111
\(416\) 653497. 0.185144
\(417\) −541722. −0.152559
\(418\) 2450.41 0.000685958 0
\(419\) −2.95949e6 −0.823535 −0.411768 0.911289i \(-0.635088\pi\)
−0.411768 + 0.911289i \(0.635088\pi\)
\(420\) 3.69339e6 1.02165
\(421\) 1.17118e6 0.322047 0.161024 0.986951i \(-0.448520\pi\)
0.161024 + 0.986951i \(0.448520\pi\)
\(422\) 479981. 0.131203
\(423\) 2.05694e6 0.558947
\(424\) 648614. 0.175215
\(425\) 490108. 0.131619
\(426\) 299777. 0.0800341
\(427\) 894218. 0.237342
\(428\) 514117. 0.135660
\(429\) 4598.39 0.00120632
\(430\) 200187. 0.0522112
\(431\) −2.43477e6 −0.631341 −0.315671 0.948869i \(-0.602229\pi\)
−0.315671 + 0.948869i \(0.602229\pi\)
\(432\) 720084. 0.185641
\(433\) −229760. −0.0588917 −0.0294459 0.999566i \(-0.509374\pi\)
−0.0294459 + 0.999566i \(0.509374\pi\)
\(434\) −544066. −0.138652
\(435\) 283732. 0.0718927
\(436\) 6.62134e6 1.66813
\(437\) −3.58643e6 −0.898377
\(438\) −180463. −0.0449472
\(439\) −2.63343e6 −0.652168 −0.326084 0.945341i \(-0.605729\pi\)
−0.326084 + 0.945341i \(0.605729\pi\)
\(440\) −3529.54 −0.000869134 0
\(441\) 2.19489e6 0.537422
\(442\) 148936. 0.0362613
\(443\) −5.30123e6 −1.28342 −0.641708 0.766949i \(-0.721774\pi\)
−0.641708 + 0.766949i \(0.721774\pi\)
\(444\) −1.14487e6 −0.275613
\(445\) −8.44275e6 −2.02108
\(446\) 91502.4 0.0217819
\(447\) −2.03982e6 −0.482863
\(448\) 6.38302e6 1.50256
\(449\) 2.32172e6 0.543494 0.271747 0.962369i \(-0.412399\pi\)
0.271747 + 0.962369i \(0.412399\pi\)
\(450\) 35460.8 0.00825500
\(451\) 10672.0 0.00247062
\(452\) 4.53264e6 1.04353
\(453\) −2.65043e6 −0.606834
\(454\) −434269. −0.0988824
\(455\) −4.55681e6 −1.03189
\(456\) −964216. −0.217151
\(457\) −3.05394e6 −0.684022 −0.342011 0.939696i \(-0.611108\pi\)
−0.342011 + 0.939696i \(0.611108\pi\)
\(458\) 136311. 0.0303647
\(459\) 502359. 0.111297
\(460\) 2.56755e6 0.565749
\(461\) 4.33222e6 0.949419 0.474710 0.880143i \(-0.342553\pi\)
0.474710 + 0.880143i \(0.342553\pi\)
\(462\) −1689.13 −0.000368177 0
\(463\) −6.38768e6 −1.38481 −0.692406 0.721508i \(-0.743449\pi\)
−0.692406 + 0.721508i \(0.743449\pi\)
\(464\) 502770. 0.108411
\(465\) 2.35144e6 0.504316
\(466\) −187006. −0.0398924
\(467\) 595087. 0.126266 0.0631332 0.998005i \(-0.479891\pi\)
0.0631332 + 0.998005i \(0.479891\pi\)
\(468\) −899326. −0.189803
\(469\) −2.80578e6 −0.589009
\(470\) 968158. 0.202163
\(471\) 4.74822e6 0.986230
\(472\) 136321. 0.0281649
\(473\) 7640.71 0.00157029
\(474\) 330135. 0.0674909
\(475\) 1.94571e6 0.395680
\(476\) 4.56581e6 0.923635
\(477\) −1.34157e6 −0.269970
\(478\) 695494. 0.139227
\(479\) 854216. 0.170110 0.0850548 0.996376i \(-0.472893\pi\)
0.0850548 + 0.996376i \(0.472893\pi\)
\(480\) 1.03749e6 0.205532
\(481\) 1.41251e6 0.278375
\(482\) −805928. −0.158008
\(483\) 2.47221e6 0.482190
\(484\) 5.09254e6 0.988147
\(485\) 8.37235e6 1.61619
\(486\) 36347.1 0.00698039
\(487\) −6.99300e6 −1.33611 −0.668054 0.744113i \(-0.732872\pi\)
−0.668054 + 0.744113i \(0.732872\pi\)
\(488\) 167128. 0.0317687
\(489\) 1.24723e6 0.235871
\(490\) 1.03309e6 0.194378
\(491\) 6.08880e6 1.13980 0.569899 0.821715i \(-0.306982\pi\)
0.569899 + 0.821715i \(0.306982\pi\)
\(492\) −2.08717e6 −0.388728
\(493\) 350752. 0.0649954
\(494\) 591270. 0.109010
\(495\) 7300.37 0.00133916
\(496\) 4.16674e6 0.760487
\(497\) −1.13384e7 −2.05902
\(498\) 521276. 0.0941877
\(499\) 1.04489e7 1.87853 0.939264 0.343196i \(-0.111510\pi\)
0.939264 + 0.343196i \(0.111510\pi\)
\(500\) 4.72744e6 0.845671
\(501\) 1.81361e6 0.322812
\(502\) −583835. −0.103403
\(503\) 3.92622e6 0.691918 0.345959 0.938250i \(-0.387554\pi\)
0.345959 + 0.938250i \(0.387554\pi\)
\(504\) 664657. 0.116552
\(505\) −1.57920e6 −0.275556
\(506\) −1174.23 −0.000203882 0
\(507\) −2.23207e6 −0.385646
\(508\) −2.52164e6 −0.433534
\(509\) −5.65513e6 −0.967493 −0.483746 0.875208i \(-0.660724\pi\)
−0.483746 + 0.875208i \(0.660724\pi\)
\(510\) 236450. 0.0402544
\(511\) 6.82561e6 1.15635
\(512\) 3.07626e6 0.518618
\(513\) 1.99435e6 0.334585
\(514\) 1.12629e6 0.188037
\(515\) 3.86112e6 0.641498
\(516\) −1.49433e6 −0.247071
\(517\) 36952.5 0.00608020
\(518\) −518859. −0.0849620
\(519\) 4.16962e6 0.679482
\(520\) −851659. −0.138120
\(521\) −1.00145e7 −1.61634 −0.808172 0.588947i \(-0.799543\pi\)
−0.808172 + 0.588947i \(0.799543\pi\)
\(522\) 25377.9 0.00407643
\(523\) −9.06776e6 −1.44959 −0.724796 0.688964i \(-0.758066\pi\)
−0.724796 + 0.688964i \(0.758066\pi\)
\(524\) 161850. 0.0257504
\(525\) −1.34123e6 −0.212375
\(526\) 762328. 0.120137
\(527\) 2.90688e6 0.455933
\(528\) 12936.2 0.00201940
\(529\) −4.71773e6 −0.732983
\(530\) −631447. −0.0976443
\(531\) −281961. −0.0433963
\(532\) 1.81261e7 2.77667
\(533\) 2.57510e6 0.392623
\(534\) −755148. −0.114599
\(535\) −1.00702e6 −0.152108
\(536\) −524396. −0.0788402
\(537\) −2.11504e6 −0.316506
\(538\) 737555. 0.109860
\(539\) 39430.7 0.00584605
\(540\) −1.42776e6 −0.210704
\(541\) 1.13999e7 1.67459 0.837296 0.546750i \(-0.184135\pi\)
0.837296 + 0.546750i \(0.184135\pi\)
\(542\) −558536. −0.0816681
\(543\) 3.10735e6 0.452262
\(544\) 1.28255e6 0.185814
\(545\) −1.29694e7 −1.87038
\(546\) −407576. −0.0585096
\(547\) 9.29969e6 1.32892 0.664462 0.747322i \(-0.268661\pi\)
0.664462 + 0.747322i \(0.268661\pi\)
\(548\) 7.50694e6 1.06785
\(549\) −345680. −0.0489490
\(550\) 637.046 8.97975e−5 0
\(551\) 1.39247e6 0.195392
\(552\) 462052. 0.0645421
\(553\) −1.24866e7 −1.73633
\(554\) 121177. 0.0167743
\(555\) 2.24250e6 0.309029
\(556\) 1.90332e6 0.261110
\(557\) −5.55080e6 −0.758084 −0.379042 0.925379i \(-0.623746\pi\)
−0.379042 + 0.925379i \(0.623746\pi\)
\(558\) 210321. 0.0285955
\(559\) 1.84366e6 0.249546
\(560\) −1.28192e7 −1.72739
\(561\) 9024.79 0.00121068
\(562\) 1.42441e6 0.190236
\(563\) 361373. 0.0480490 0.0240245 0.999711i \(-0.492352\pi\)
0.0240245 + 0.999711i \(0.492352\pi\)
\(564\) −7.22697e6 −0.956662
\(565\) −8.87823e6 −1.17005
\(566\) 26348.0 0.00345706
\(567\) −1.37475e6 −0.179583
\(568\) −2.11913e6 −0.275605
\(569\) 6.95770e6 0.900917 0.450459 0.892797i \(-0.351261\pi\)
0.450459 + 0.892797i \(0.351261\pi\)
\(570\) 938696. 0.121015
\(571\) −1.46705e7 −1.88302 −0.941511 0.336982i \(-0.890594\pi\)
−0.941511 + 0.336982i \(0.890594\pi\)
\(572\) −16156.2 −0.00206467
\(573\) 8.75905e6 1.11448
\(574\) −945911. −0.119831
\(575\) −932384. −0.117605
\(576\) −2.46750e6 −0.309886
\(577\) −1.16193e7 −1.45291 −0.726457 0.687212i \(-0.758834\pi\)
−0.726457 + 0.687212i \(0.758834\pi\)
\(578\) −581680. −0.0724210
\(579\) −4.44106e6 −0.550542
\(580\) −996879. −0.123047
\(581\) −1.97161e7 −2.42315
\(582\) 748851. 0.0916406
\(583\) −24101.0 −0.00293673
\(584\) 1.27570e6 0.154780
\(585\) 1.76154e6 0.212815
\(586\) −580574. −0.0698415
\(587\) 1.26219e7 1.51192 0.755959 0.654619i \(-0.227171\pi\)
0.755959 + 0.654619i \(0.227171\pi\)
\(588\) −7.71163e6 −0.919820
\(589\) 1.15402e7 1.37064
\(590\) −132713. −0.0156958
\(591\) 3.05231e6 0.359468
\(592\) 3.97369e6 0.466004
\(593\) 8.70200e6 1.01621 0.508103 0.861296i \(-0.330347\pi\)
0.508103 + 0.861296i \(0.330347\pi\)
\(594\) 652.970 7.59324e−5 0
\(595\) −8.94319e6 −1.03562
\(596\) 7.16683e6 0.826440
\(597\) −5.30732e6 −0.609452
\(598\) −283336. −0.0324003
\(599\) 314477. 0.0358115 0.0179057 0.999840i \(-0.494300\pi\)
0.0179057 + 0.999840i \(0.494300\pi\)
\(600\) −250673. −0.0284269
\(601\) −1.33321e7 −1.50561 −0.752803 0.658246i \(-0.771299\pi\)
−0.752803 + 0.658246i \(0.771299\pi\)
\(602\) −677231. −0.0761633
\(603\) 1.08464e6 0.121476
\(604\) 9.31216e6 1.03862
\(605\) −9.97492e6 −1.10795
\(606\) −141249. −0.0156244
\(607\) 4.65927e6 0.513270 0.256635 0.966508i \(-0.417386\pi\)
0.256635 + 0.966508i \(0.417386\pi\)
\(608\) 5.09168e6 0.558602
\(609\) −959864. −0.104874
\(610\) −162704. −0.0177041
\(611\) 8.91644e6 0.966248
\(612\) −1.76502e6 −0.190489
\(613\) 3.95032e6 0.424601 0.212301 0.977204i \(-0.431904\pi\)
0.212301 + 0.977204i \(0.431904\pi\)
\(614\) −647286. −0.0692907
\(615\) 4.08821e6 0.435858
\(616\) 11940.4 0.00126785
\(617\) 9.77175e6 1.03338 0.516689 0.856173i \(-0.327164\pi\)
0.516689 + 0.856173i \(0.327164\pi\)
\(618\) 345352. 0.0363740
\(619\) 1.29372e7 1.35710 0.678552 0.734553i \(-0.262608\pi\)
0.678552 + 0.734553i \(0.262608\pi\)
\(620\) −8.26170e6 −0.863158
\(621\) −955690. −0.0994461
\(622\) −379655. −0.0393471
\(623\) 2.85618e7 2.94826
\(624\) 3.12143e6 0.320916
\(625\) −1.14824e7 −1.17579
\(626\) 755907. 0.0770961
\(627\) 35828.1 0.00363961
\(628\) −1.66827e7 −1.68798
\(629\) 2.77220e6 0.279382
\(630\) −647066. −0.0649527
\(631\) −1.78566e7 −1.78536 −0.892680 0.450690i \(-0.851178\pi\)
−0.892680 + 0.450690i \(0.851178\pi\)
\(632\) −2.33373e6 −0.232411
\(633\) 7.01792e6 0.696144
\(634\) −1.51446e6 −0.149635
\(635\) 4.93921e6 0.486097
\(636\) 4.71353e6 0.462066
\(637\) 9.51441e6 0.929038
\(638\) 455.910 4.43432e−5 0
\(639\) 4.38312e6 0.424650
\(640\) −4.85025e6 −0.468073
\(641\) −1.43521e7 −1.37966 −0.689828 0.723973i \(-0.742314\pi\)
−0.689828 + 0.723973i \(0.742314\pi\)
\(642\) −90071.0 −0.00862477
\(643\) −1.09103e6 −0.104066 −0.0520329 0.998645i \(-0.516570\pi\)
−0.0520329 + 0.998645i \(0.516570\pi\)
\(644\) −8.68601e6 −0.825288
\(645\) 2.92698e6 0.277026
\(646\) 1.16043e6 0.109405
\(647\) −6.11094e6 −0.573914 −0.286957 0.957943i \(-0.592644\pi\)
−0.286957 + 0.957943i \(0.592644\pi\)
\(648\) −256938. −0.0240376
\(649\) −5065.38 −0.000472063 0
\(650\) 153716. 0.0142704
\(651\) −7.95494e6 −0.735672
\(652\) −4.38208e6 −0.403702
\(653\) 9.26459e6 0.850243 0.425122 0.905136i \(-0.360231\pi\)
0.425122 + 0.905136i \(0.360231\pi\)
\(654\) −1.16003e6 −0.106053
\(655\) −317020. −0.0288724
\(656\) 7.24427e6 0.657256
\(657\) −2.63860e6 −0.238484
\(658\) −3.27528e6 −0.294906
\(659\) 1.26244e7 1.13239 0.566196 0.824271i \(-0.308415\pi\)
0.566196 + 0.824271i \(0.308415\pi\)
\(660\) −25649.5 −0.00229203
\(661\) 6.62521e6 0.589788 0.294894 0.955530i \(-0.404716\pi\)
0.294894 + 0.955530i \(0.404716\pi\)
\(662\) 35896.5 0.00318352
\(663\) 2.17763e6 0.192398
\(664\) −3.68490e6 −0.324344
\(665\) −3.55041e7 −3.11332
\(666\) 200577. 0.0175225
\(667\) −667272. −0.0580748
\(668\) −6.37203e6 −0.552506
\(669\) 1.33788e6 0.115572
\(670\) 510517. 0.0439363
\(671\) −6210.08 −0.000532465 0
\(672\) −3.50982e6 −0.299821
\(673\) −4.38372e6 −0.373083 −0.186541 0.982447i \(-0.559728\pi\)
−0.186541 + 0.982447i \(0.559728\pi\)
\(674\) 1.26747e6 0.107470
\(675\) 518481. 0.0437999
\(676\) 7.84229e6 0.660049
\(677\) 1.24543e7 1.04435 0.522177 0.852837i \(-0.325120\pi\)
0.522177 + 0.852837i \(0.325120\pi\)
\(678\) −794099. −0.0663438
\(679\) −2.83236e7 −2.35762
\(680\) −1.67147e6 −0.138620
\(681\) −6.34956e6 −0.524657
\(682\) 3778.38 0.000311061 0
\(683\) −5.99410e6 −0.491668 −0.245834 0.969312i \(-0.579062\pi\)
−0.245834 + 0.969312i \(0.579062\pi\)
\(684\) −7.00704e6 −0.572657
\(685\) −1.47041e7 −1.19732
\(686\) −1.32722e6 −0.107679
\(687\) 1.99304e6 0.161111
\(688\) 5.18658e6 0.417744
\(689\) −5.81544e6 −0.466696
\(690\) −449823. −0.0359682
\(691\) −1.41412e7 −1.12666 −0.563329 0.826232i \(-0.690480\pi\)
−0.563329 + 0.826232i \(0.690480\pi\)
\(692\) −1.46498e7 −1.16296
\(693\) −24697.1 −0.00195350
\(694\) −1.87776e6 −0.147993
\(695\) −3.72809e6 −0.292768
\(696\) −179397. −0.0140376
\(697\) 5.05388e6 0.394043
\(698\) −1.56888e6 −0.121885
\(699\) −2.73426e6 −0.211664
\(700\) 4.71234e6 0.363489
\(701\) 1.48140e7 1.13861 0.569307 0.822125i \(-0.307212\pi\)
0.569307 + 0.822125i \(0.307212\pi\)
\(702\) 157558. 0.0120670
\(703\) 1.10055e7 0.839890
\(704\) −44328.3 −0.00337092
\(705\) 1.41557e7 1.07265
\(706\) −86162.1 −0.00650586
\(707\) 5.34244e6 0.401968
\(708\) 990658. 0.0742746
\(709\) −7.25689e6 −0.542170 −0.271085 0.962555i \(-0.587382\pi\)
−0.271085 + 0.962555i \(0.587382\pi\)
\(710\) 2.06304e6 0.153590
\(711\) 4.82699e6 0.358098
\(712\) 5.33815e6 0.394631
\(713\) −5.53006e6 −0.407386
\(714\) −799909. −0.0587212
\(715\) 31645.7 0.00231499
\(716\) 7.43109e6 0.541714
\(717\) 1.01690e7 0.738720
\(718\) −361769. −0.0261891
\(719\) −4.63110e6 −0.334089 −0.167044 0.985949i \(-0.553422\pi\)
−0.167044 + 0.985949i \(0.553422\pi\)
\(720\) 4.95556e6 0.356255
\(721\) −1.30622e7 −0.935787
\(722\) 3.08270e6 0.220084
\(723\) −1.17837e7 −0.838369
\(724\) −1.09175e7 −0.774066
\(725\) 362009. 0.0255784
\(726\) −892191. −0.0628227
\(727\) −1.27291e7 −0.893224 −0.446612 0.894728i \(-0.647369\pi\)
−0.446612 + 0.894728i \(0.647369\pi\)
\(728\) 2.88116e6 0.201483
\(729\) 531441. 0.0370370
\(730\) −1.24193e6 −0.0862562
\(731\) 3.61836e6 0.250449
\(732\) 1.21453e6 0.0837783
\(733\) −7.43767e6 −0.511302 −0.255651 0.966769i \(-0.582290\pi\)
−0.255651 + 0.966769i \(0.582290\pi\)
\(734\) 221790. 0.0151950
\(735\) 1.51050e7 1.03134
\(736\) −2.43993e6 −0.166029
\(737\) 19485.4 0.00132142
\(738\) 365663. 0.0247138
\(739\) 1.96771e6 0.132541 0.0662705 0.997802i \(-0.478890\pi\)
0.0662705 + 0.997802i \(0.478890\pi\)
\(740\) −7.87892e6 −0.528917
\(741\) 8.64511e6 0.578396
\(742\) 2.13618e6 0.142439
\(743\) 1.30961e7 0.870300 0.435150 0.900358i \(-0.356695\pi\)
0.435150 + 0.900358i \(0.356695\pi\)
\(744\) −1.48676e6 −0.0984712
\(745\) −1.40379e7 −0.926640
\(746\) 3.24344e6 0.213383
\(747\) 7.62171e6 0.499748
\(748\) −31708.2 −0.00207213
\(749\) 3.40674e6 0.221888
\(750\) −828227. −0.0537646
\(751\) −2.49792e7 −1.61614 −0.808068 0.589089i \(-0.799487\pi\)
−0.808068 + 0.589089i \(0.799487\pi\)
\(752\) 2.50837e7 1.61751
\(753\) −8.53641e6 −0.548640
\(754\) 110008. 0.00704689
\(755\) −1.82400e7 −1.16455
\(756\) 4.83013e6 0.307365
\(757\) −553214. −0.0350876 −0.0175438 0.999846i \(-0.505585\pi\)
−0.0175438 + 0.999846i \(0.505585\pi\)
\(758\) 2.11405e6 0.133642
\(759\) −17168.8 −0.00108177
\(760\) −6.63565e6 −0.416725
\(761\) −6.91848e6 −0.433061 −0.216530 0.976276i \(-0.569474\pi\)
−0.216530 + 0.976276i \(0.569474\pi\)
\(762\) 441780. 0.0275625
\(763\) 4.38756e7 2.72842
\(764\) −3.07745e7 −1.90747
\(765\) 3.45719e6 0.213585
\(766\) −2.18776e6 −0.134719
\(767\) −1.22225e6 −0.0750189
\(768\) 8.33952e6 0.510197
\(769\) −1.29851e7 −0.791825 −0.395912 0.918288i \(-0.629572\pi\)
−0.395912 + 0.918288i \(0.629572\pi\)
\(770\) −11624.4 −0.000706552 0
\(771\) 1.64679e7 0.997702
\(772\) 1.56035e7 0.942276
\(773\) −5.40829e6 −0.325545 −0.162773 0.986664i \(-0.552044\pi\)
−0.162773 + 0.986664i \(0.552044\pi\)
\(774\) 261799. 0.0157078
\(775\) 3.00017e6 0.179429
\(776\) −5.29364e6 −0.315573
\(777\) −7.58637e6 −0.450798
\(778\) 2.78392e6 0.164895
\(779\) 2.00637e7 1.18459
\(780\) −6.18909e6 −0.364242
\(781\) 78742.0 0.00461933
\(782\) −556075. −0.0325175
\(783\) 371057. 0.0216290
\(784\) 2.67659e7 1.55522
\(785\) 3.26768e7 1.89263
\(786\) −28355.4 −0.00163711
\(787\) 4.47738e6 0.257684 0.128842 0.991665i \(-0.458874\pi\)
0.128842 + 0.991665i \(0.458874\pi\)
\(788\) −1.07242e7 −0.615244
\(789\) 1.11462e7 0.637433
\(790\) 2.27196e6 0.129519
\(791\) 3.00350e7 1.70682
\(792\) −4615.85 −0.000261480 0
\(793\) −1.49846e6 −0.0846178
\(794\) 250909. 0.0141242
\(795\) −9.23255e6 −0.518088
\(796\) 1.86470e7 1.04310
\(797\) −2.50186e7 −1.39514 −0.697570 0.716516i \(-0.745736\pi\)
−0.697570 + 0.716516i \(0.745736\pi\)
\(798\) −3.17561e6 −0.176530
\(799\) 1.74994e7 0.969742
\(800\) 1.32371e6 0.0731255
\(801\) −1.10412e7 −0.608045
\(802\) 1.84876e6 0.101495
\(803\) −47401.9 −0.00259422
\(804\) −3.81083e6 −0.207912
\(805\) 1.70136e7 0.925348
\(806\) 911703. 0.0494328
\(807\) 1.07840e7 0.582902
\(808\) 998492. 0.0538042
\(809\) −3.51699e7 −1.88929 −0.944646 0.328090i \(-0.893595\pi\)
−0.944646 + 0.328090i \(0.893595\pi\)
\(810\) 250138. 0.0133958
\(811\) −2.65046e6 −0.141504 −0.0707521 0.997494i \(-0.522540\pi\)
−0.0707521 + 0.997494i \(0.522540\pi\)
\(812\) 3.37244e6 0.179496
\(813\) −8.16649e6 −0.433321
\(814\) 3603.32 0.000190609 0
\(815\) 8.58332e6 0.452649
\(816\) 6.12611e6 0.322077
\(817\) 1.43648e7 0.752910
\(818\) −2.94010e6 −0.153631
\(819\) −5.95928e6 −0.310445
\(820\) −1.43637e7 −0.745990
\(821\) 9.39154e6 0.486272 0.243136 0.969992i \(-0.421824\pi\)
0.243136 + 0.969992i \(0.421824\pi\)
\(822\) −1.31518e6 −0.0678901
\(823\) −1.99606e6 −0.102725 −0.0513623 0.998680i \(-0.516356\pi\)
−0.0513623 + 0.998680i \(0.516356\pi\)
\(824\) −2.44130e6 −0.125257
\(825\) 9314.42 0.000476454 0
\(826\) 448968. 0.0228963
\(827\) 6.93283e6 0.352490 0.176245 0.984346i \(-0.443605\pi\)
0.176245 + 0.984346i \(0.443605\pi\)
\(828\) 3.35777e6 0.170206
\(829\) −3.01802e7 −1.52523 −0.762616 0.646851i \(-0.776085\pi\)
−0.762616 + 0.646851i \(0.776085\pi\)
\(830\) 3.58738e6 0.180751
\(831\) 1.77176e6 0.0890024
\(832\) −1.06962e7 −0.535697
\(833\) 1.86730e7 0.932397
\(834\) −333453. −0.0166004
\(835\) 1.24811e7 0.619493
\(836\) −125880. −0.00622934
\(837\) 3.07516e6 0.151724
\(838\) −1.82169e6 −0.0896117
\(839\) −1.09044e7 −0.534809 −0.267404 0.963584i \(-0.586166\pi\)
−0.267404 + 0.963584i \(0.586166\pi\)
\(840\) 4.57411e6 0.223671
\(841\) −2.02521e7 −0.987369
\(842\) 720913. 0.0350431
\(843\) 2.08266e7 1.00937
\(844\) −2.46572e7 −1.19148
\(845\) −1.53609e7 −0.740075
\(846\) 1.26613e6 0.0608210
\(847\) 3.37452e7 1.61623
\(848\) −1.63600e7 −0.781256
\(849\) 385241. 0.0183427
\(850\) 301682. 0.0143220
\(851\) −5.27384e6 −0.249634
\(852\) −1.53999e7 −0.726807
\(853\) 8.47613e6 0.398864 0.199432 0.979912i \(-0.436090\pi\)
0.199432 + 0.979912i \(0.436090\pi\)
\(854\) 550429. 0.0258260
\(855\) 1.37249e7 0.642088
\(856\) 636713. 0.0297002
\(857\) 2.96420e7 1.37865 0.689327 0.724450i \(-0.257906\pi\)
0.689327 + 0.724450i \(0.257906\pi\)
\(858\) 2830.50 0.000131264 0
\(859\) −58116.2 −0.00268729 −0.00134364 0.999999i \(-0.500428\pi\)
−0.00134364 + 0.999999i \(0.500428\pi\)
\(860\) −1.02838e7 −0.474142
\(861\) −1.38304e7 −0.635810
\(862\) −1.49870e6 −0.0686984
\(863\) 6.44367e6 0.294514 0.147257 0.989098i \(-0.452956\pi\)
0.147257 + 0.989098i \(0.452956\pi\)
\(864\) 1.35680e6 0.0618346
\(865\) 2.86950e7 1.30396
\(866\) −141427. −0.00640821
\(867\) −8.50490e6 −0.384257
\(868\) 2.79493e7 1.25913
\(869\) 86715.9 0.00389538
\(870\) 174649. 0.00782289
\(871\) 4.70171e6 0.209996
\(872\) 8.20027e6 0.365205
\(873\) 1.09491e7 0.486233
\(874\) −2.20760e6 −0.0977555
\(875\) 3.13259e7 1.38319
\(876\) 9.27059e6 0.408176
\(877\) −1.86307e7 −0.817957 −0.408979 0.912544i \(-0.634115\pi\)
−0.408979 + 0.912544i \(0.634115\pi\)
\(878\) −1.62098e6 −0.0709647
\(879\) −8.48873e6 −0.370570
\(880\) 89025.7 0.00387533
\(881\) 6.95060e6 0.301705 0.150853 0.988556i \(-0.451798\pi\)
0.150853 + 0.988556i \(0.451798\pi\)
\(882\) 1.35104e6 0.0584787
\(883\) −1.75323e7 −0.756722 −0.378361 0.925658i \(-0.623512\pi\)
−0.378361 + 0.925658i \(0.623512\pi\)
\(884\) −7.65101e6 −0.329297
\(885\) −1.94043e6 −0.0832799
\(886\) −3.26313e6 −0.139653
\(887\) 4.14544e6 0.176914 0.0884568 0.996080i \(-0.471806\pi\)
0.0884568 + 0.996080i \(0.471806\pi\)
\(888\) −1.41788e6 −0.0603402
\(889\) −1.67093e7 −0.709096
\(890\) −5.19687e6 −0.219921
\(891\) 9547.24 0.000402887 0
\(892\) −4.70058e6 −0.197806
\(893\) 6.94719e7 2.91528
\(894\) −1.25560e6 −0.0525420
\(895\) −1.45555e7 −0.607393
\(896\) 1.64084e7 0.682803
\(897\) −4.14273e6 −0.171912
\(898\) 1.42912e6 0.0591395
\(899\) 2.14711e6 0.0886042
\(900\) −1.82166e6 −0.0749655
\(901\) −1.14134e7 −0.468384
\(902\) 6569.07 0.000268836 0
\(903\) −9.90198e6 −0.404113
\(904\) 5.61349e6 0.228461
\(905\) 2.13845e7 0.867916
\(906\) −1.63145e6 −0.0660318
\(907\) 6.52626e6 0.263418 0.131709 0.991288i \(-0.457953\pi\)
0.131709 + 0.991288i \(0.457953\pi\)
\(908\) 2.23089e7 0.897974
\(909\) −2.06524e6 −0.0829013
\(910\) −2.80491e6 −0.112283
\(911\) 4.12964e6 0.164860 0.0824302 0.996597i \(-0.473732\pi\)
0.0824302 + 0.996597i \(0.473732\pi\)
\(912\) 2.43204e7 0.968242
\(913\) 136923. 0.00543623
\(914\) −1.87983e6 −0.0744308
\(915\) −2.37894e6 −0.0939358
\(916\) −7.00248e6 −0.275748
\(917\) 1.07248e6 0.0421178
\(918\) 309223. 0.0121106
\(919\) −3.37200e7 −1.31704 −0.658520 0.752563i \(-0.728817\pi\)
−0.658520 + 0.752563i \(0.728817\pi\)
\(920\) 3.17980e6 0.123860
\(921\) −9.46413e6 −0.367648
\(922\) 2.66666e6 0.103310
\(923\) 1.90000e7 0.734090
\(924\) 86772.3 0.00334350
\(925\) 2.86117e6 0.109948
\(926\) −3.93188e6 −0.150686
\(927\) 5.04948e6 0.192996
\(928\) 947331. 0.0361104
\(929\) −2.85561e7 −1.08557 −0.542787 0.839870i \(-0.682631\pi\)
−0.542787 + 0.839870i \(0.682631\pi\)
\(930\) 1.44741e6 0.0548763
\(931\) 7.41309e7 2.80301
\(932\) 9.60669e6 0.362272
\(933\) −5.55103e6 −0.208771
\(934\) 366301. 0.0137395
\(935\) 62107.9 0.00232337
\(936\) −1.11378e6 −0.0415537
\(937\) 1.16758e7 0.434446 0.217223 0.976122i \(-0.430300\pi\)
0.217223 + 0.976122i \(0.430300\pi\)
\(938\) −1.72708e6 −0.0640922
\(939\) 1.10523e7 0.409062
\(940\) −4.97354e7 −1.83589
\(941\) −1.56740e7 −0.577040 −0.288520 0.957474i \(-0.593163\pi\)
−0.288520 + 0.957474i \(0.593163\pi\)
\(942\) 2.92273e6 0.107315
\(943\) −9.61453e6 −0.352086
\(944\) −3.43843e6 −0.125583
\(945\) −9.46092e6 −0.344630
\(946\) 4703.18 0.000170869 0
\(947\) −2.91932e7 −1.05781 −0.528904 0.848681i \(-0.677397\pi\)
−0.528904 + 0.848681i \(0.677397\pi\)
\(948\) −1.69594e7 −0.612900
\(949\) −1.14378e7 −0.412266
\(950\) 1.19767e6 0.0430553
\(951\) −2.21433e7 −0.793945
\(952\) 5.65457e6 0.202212
\(953\) −4.07979e7 −1.45514 −0.727571 0.686032i \(-0.759351\pi\)
−0.727571 + 0.686032i \(0.759351\pi\)
\(954\) −825790. −0.0293764
\(955\) 6.02790e7 2.13874
\(956\) −3.57283e7 −1.26435
\(957\) 6665.98 0.000235279 0
\(958\) 525806. 0.0185102
\(959\) 4.97439e7 1.74660
\(960\) −1.69812e7 −0.594688
\(961\) −1.08349e7 −0.378455
\(962\) 869462. 0.0302910
\(963\) −1.31695e6 −0.0457619
\(964\) 4.14015e7 1.43490
\(965\) −3.05630e7 −1.05652
\(966\) 1.52175e6 0.0524687
\(967\) −2.00499e7 −0.689519 −0.344760 0.938691i \(-0.612040\pi\)
−0.344760 + 0.938691i \(0.612040\pi\)
\(968\) 6.30691e6 0.216336
\(969\) 1.69669e7 0.580487
\(970\) 5.15353e6 0.175863
\(971\) −5.07323e7 −1.72678 −0.863388 0.504540i \(-0.831662\pi\)
−0.863388 + 0.504540i \(0.831662\pi\)
\(972\) −1.86719e6 −0.0633905
\(973\) 1.26121e7 0.427077
\(974\) −4.30449e6 −0.145386
\(975\) 2.24752e6 0.0757167
\(976\) −4.21546e6 −0.141651
\(977\) 4.01142e7 1.34450 0.672252 0.740322i \(-0.265327\pi\)
0.672252 + 0.740322i \(0.265327\pi\)
\(978\) 767721. 0.0256659
\(979\) −198354. −0.00661429
\(980\) −5.30708e7 −1.76519
\(981\) −1.69611e7 −0.562706
\(982\) 3.74791e6 0.124025
\(983\) −5.36333e7 −1.77032 −0.885159 0.465289i \(-0.845950\pi\)
−0.885159 + 0.465289i \(0.845950\pi\)
\(984\) −2.58488e6 −0.0851045
\(985\) 2.10057e7 0.689839
\(986\) 215903. 0.00707238
\(987\) −4.78887e7 −1.56473
\(988\) −3.03742e7 −0.989949
\(989\) −6.88359e6 −0.223782
\(990\) 4493.68 0.000145718 0
\(991\) 4.34550e7 1.40558 0.702790 0.711397i \(-0.251937\pi\)
0.702790 + 0.711397i \(0.251937\pi\)
\(992\) 7.85107e6 0.253308
\(993\) 524853. 0.0168913
\(994\) −6.97927e6 −0.224050
\(995\) −3.65245e7 −1.16957
\(996\) −2.67785e7 −0.855339
\(997\) 3.48232e7 1.10951 0.554754 0.832015i \(-0.312812\pi\)
0.554754 + 0.832015i \(0.312812\pi\)
\(998\) 6.43171e6 0.204409
\(999\) 2.93268e6 0.0929719
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.c.1.6 12
3.2 odd 2 531.6.a.c.1.7 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.c.1.6 12 1.1 even 1 trivial
531.6.a.c.1.7 12 3.2 odd 2