Properties

Label 177.6.a.c.1.12
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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 1.12
Root \(-8.57072\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+10.5707 q^{2} +9.00000 q^{3} +79.7401 q^{4} -45.7686 q^{5} +95.1365 q^{6} -1.90644 q^{7} +504.647 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+10.5707 q^{2} +9.00000 q^{3} +79.7401 q^{4} -45.7686 q^{5} +95.1365 q^{6} -1.90644 q^{7} +504.647 q^{8} +81.0000 q^{9} -483.807 q^{10} +342.480 q^{11} +717.661 q^{12} +540.430 q^{13} -20.1524 q^{14} -411.917 q^{15} +2782.80 q^{16} +551.410 q^{17} +856.228 q^{18} +1655.52 q^{19} -3649.59 q^{20} -17.1580 q^{21} +3620.26 q^{22} -1804.91 q^{23} +4541.83 q^{24} -1030.23 q^{25} +5712.73 q^{26} +729.000 q^{27} -152.020 q^{28} +2646.30 q^{29} -4354.26 q^{30} -2433.93 q^{31} +13267.5 q^{32} +3082.32 q^{33} +5828.80 q^{34} +87.2551 q^{35} +6458.95 q^{36} -11640.7 q^{37} +17500.0 q^{38} +4863.87 q^{39} -23097.0 q^{40} -13695.8 q^{41} -181.372 q^{42} -13668.0 q^{43} +27309.4 q^{44} -3707.26 q^{45} -19079.2 q^{46} -5338.31 q^{47} +25045.2 q^{48} -16803.4 q^{49} -10890.3 q^{50} +4962.69 q^{51} +43093.9 q^{52} +9601.76 q^{53} +7706.05 q^{54} -15674.8 q^{55} -962.080 q^{56} +14899.7 q^{57} +27973.3 q^{58} -3481.00 q^{59} -32846.3 q^{60} +4251.86 q^{61} -25728.4 q^{62} -154.422 q^{63} +51197.5 q^{64} -24734.7 q^{65} +32582.4 q^{66} +12481.8 q^{67} +43969.5 q^{68} -16244.2 q^{69} +922.349 q^{70} +57354.7 q^{71} +40876.4 q^{72} +21803.3 q^{73} -123051. q^{74} -9272.11 q^{75} +132011. q^{76} -652.918 q^{77} +51414.6 q^{78} -57192.4 q^{79} -127365. q^{80} +6561.00 q^{81} -144774. q^{82} -80517.2 q^{83} -1368.18 q^{84} -25237.3 q^{85} -144481. q^{86} +23816.7 q^{87} +172832. q^{88} -60390.4 q^{89} -39188.4 q^{90} -1030.30 q^{91} -143924. q^{92} -21905.3 q^{93} -56429.7 q^{94} -75770.8 q^{95} +119408. q^{96} +152618. q^{97} -177624. q^{98} +27740.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + O(q^{10}) \) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + 601q^{10} + 1480q^{11} + 1782q^{12} + 472q^{13} + 1065q^{14} + 1422q^{15} + 6370q^{16} + 1565q^{17} + 1782q^{18} + 3939q^{19} + 8033q^{20} + 3717q^{21} - 1738q^{22} + 7245q^{23} + 6507q^{24} + 9690q^{25} + 3764q^{26} + 8748q^{27} + 12154q^{28} + 10003q^{29} + 5409q^{30} + 7295q^{31} + 11628q^{32} + 13320q^{33} - 16344q^{34} + 11015q^{35} + 16038q^{36} + 6741q^{37} + 3035q^{38} + 4248q^{39} + 5572q^{40} + 34025q^{41} + 9585q^{42} - 6336q^{43} + 41168q^{44} + 12798q^{45} + 2345q^{46} + 66167q^{47} + 57330q^{48} + 28319q^{49} + 31173q^{50} + 14085q^{51} + 16440q^{52} + 62290q^{53} + 16038q^{54} + 55764q^{55} + 107306q^{56} + 35451q^{57} + 37952q^{58} - 41772q^{59} + 72297q^{60} + 68469q^{61} + 99190q^{62} + 33453q^{63} + 68525q^{64} + 80156q^{65} - 15642q^{66} + 113310q^{67} + 33887q^{68} + 65205q^{69} + 32034q^{70} + 84520q^{71} + 58563q^{72} + 135895q^{73} - 31962q^{74} + 87210q^{75} - 61848q^{76} - 3799q^{77} + 33876q^{78} + 14122q^{79} + 77609q^{80} + 78732q^{81} - 1501q^{82} + 114463q^{83} + 109386q^{84} - 101097q^{85} - 203536q^{86} + 90027q^{87} - 244967q^{88} + 189109q^{89} + 48681q^{90} - 168249q^{91} - 71946q^{92} + 65655q^{93} - 472284q^{94} + 21923q^{95} + 104652q^{96} - 76192q^{97} - 17544q^{98} + 119880q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.5707 1.86866 0.934328 0.356413i \(-0.116000\pi\)
0.934328 + 0.356413i \(0.116000\pi\)
\(3\) 9.00000 0.577350
\(4\) 79.7401 2.49188
\(5\) −45.7686 −0.818734 −0.409367 0.912370i \(-0.634250\pi\)
−0.409367 + 0.912370i \(0.634250\pi\)
\(6\) 95.1365 1.07887
\(7\) −1.90644 −0.0147054 −0.00735272 0.999973i \(-0.502340\pi\)
−0.00735272 + 0.999973i \(0.502340\pi\)
\(8\) 504.647 2.78781
\(9\) 81.0000 0.333333
\(10\) −483.807 −1.52993
\(11\) 342.480 0.853402 0.426701 0.904393i \(-0.359676\pi\)
0.426701 + 0.904393i \(0.359676\pi\)
\(12\) 717.661 1.43869
\(13\) 540.430 0.886912 0.443456 0.896296i \(-0.353752\pi\)
0.443456 + 0.896296i \(0.353752\pi\)
\(14\) −20.1524 −0.0274794
\(15\) −411.917 −0.472696
\(16\) 2782.80 2.71758
\(17\) 551.410 0.462756 0.231378 0.972864i \(-0.425677\pi\)
0.231378 + 0.972864i \(0.425677\pi\)
\(18\) 856.228 0.622886
\(19\) 1655.52 1.05208 0.526042 0.850459i \(-0.323675\pi\)
0.526042 + 0.850459i \(0.323675\pi\)
\(20\) −3649.59 −2.04019
\(21\) −17.1580 −0.00849019
\(22\) 3620.26 1.59472
\(23\) −1804.91 −0.711437 −0.355719 0.934593i \(-0.615764\pi\)
−0.355719 + 0.934593i \(0.615764\pi\)
\(24\) 4541.83 1.60954
\(25\) −1030.23 −0.329675
\(26\) 5712.73 1.65733
\(27\) 729.000 0.192450
\(28\) −152.020 −0.0366442
\(29\) 2646.30 0.584311 0.292156 0.956371i \(-0.405628\pi\)
0.292156 + 0.956371i \(0.405628\pi\)
\(30\) −4354.26 −0.883307
\(31\) −2433.93 −0.454887 −0.227443 0.973791i \(-0.573037\pi\)
−0.227443 + 0.973791i \(0.573037\pi\)
\(32\) 13267.5 2.29042
\(33\) 3082.32 0.492712
\(34\) 5828.80 0.864732
\(35\) 87.2551 0.0120398
\(36\) 6458.95 0.830626
\(37\) −11640.7 −1.39790 −0.698949 0.715171i \(-0.746349\pi\)
−0.698949 + 0.715171i \(0.746349\pi\)
\(38\) 17500.0 1.96598
\(39\) 4863.87 0.512059
\(40\) −23097.0 −2.28247
\(41\) −13695.8 −1.27241 −0.636205 0.771520i \(-0.719497\pi\)
−0.636205 + 0.771520i \(0.719497\pi\)
\(42\) −181.372 −0.0158653
\(43\) −13668.0 −1.12729 −0.563643 0.826019i \(-0.690601\pi\)
−0.563643 + 0.826019i \(0.690601\pi\)
\(44\) 27309.4 2.12657
\(45\) −3707.26 −0.272911
\(46\) −19079.2 −1.32943
\(47\) −5338.31 −0.352500 −0.176250 0.984345i \(-0.556397\pi\)
−0.176250 + 0.984345i \(0.556397\pi\)
\(48\) 25045.2 1.56900
\(49\) −16803.4 −0.999784
\(50\) −10890.3 −0.616050
\(51\) 4962.69 0.267172
\(52\) 43093.9 2.21008
\(53\) 9601.76 0.469527 0.234764 0.972052i \(-0.424568\pi\)
0.234764 + 0.972052i \(0.424568\pi\)
\(54\) 7706.05 0.359623
\(55\) −15674.8 −0.698709
\(56\) −962.080 −0.0409960
\(57\) 14899.7 0.607421
\(58\) 27973.3 1.09188
\(59\) −3481.00 −0.130189
\(60\) −32846.3 −1.17790
\(61\) 4251.86 0.146303 0.0731517 0.997321i \(-0.476694\pi\)
0.0731517 + 0.997321i \(0.476694\pi\)
\(62\) −25728.4 −0.850027
\(63\) −154.422 −0.00490181
\(64\) 51197.5 1.56242
\(65\) −24734.7 −0.726145
\(66\) 32582.4 0.920710
\(67\) 12481.8 0.339695 0.169848 0.985470i \(-0.445672\pi\)
0.169848 + 0.985470i \(0.445672\pi\)
\(68\) 43969.5 1.15313
\(69\) −16244.2 −0.410749
\(70\) 922.349 0.0224983
\(71\) 57354.7 1.35028 0.675139 0.737690i \(-0.264084\pi\)
0.675139 + 0.737690i \(0.264084\pi\)
\(72\) 40876.4 0.929270
\(73\) 21803.3 0.478867 0.239433 0.970913i \(-0.423038\pi\)
0.239433 + 0.970913i \(0.423038\pi\)
\(74\) −123051. −2.61219
\(75\) −9272.11 −0.190338
\(76\) 132011. 2.62167
\(77\) −652.918 −0.0125497
\(78\) 51414.6 0.956863
\(79\) −57192.4 −1.03103 −0.515514 0.856881i \(-0.672399\pi\)
−0.515514 + 0.856881i \(0.672399\pi\)
\(80\) −127365. −2.22498
\(81\) 6561.00 0.111111
\(82\) −144774. −2.37770
\(83\) −80517.2 −1.28290 −0.641451 0.767164i \(-0.721667\pi\)
−0.641451 + 0.767164i \(0.721667\pi\)
\(84\) −1368.18 −0.0211565
\(85\) −25237.3 −0.378874
\(86\) −144481. −2.10651
\(87\) 23816.7 0.337352
\(88\) 172832. 2.37912
\(89\) −60390.4 −0.808151 −0.404076 0.914726i \(-0.632407\pi\)
−0.404076 + 0.914726i \(0.632407\pi\)
\(90\) −39188.4 −0.509977
\(91\) −1030.30 −0.0130424
\(92\) −143924. −1.77282
\(93\) −21905.3 −0.262629
\(94\) −56429.7 −0.658701
\(95\) −75770.8 −0.861377
\(96\) 119408. 1.32237
\(97\) 152618. 1.64693 0.823465 0.567366i \(-0.192038\pi\)
0.823465 + 0.567366i \(0.192038\pi\)
\(98\) −177624. −1.86825
\(99\) 27740.9 0.284467
\(100\) −82151.0 −0.821510
\(101\) 126188. 1.23087 0.615436 0.788187i \(-0.288980\pi\)
0.615436 + 0.788187i \(0.288980\pi\)
\(102\) 52459.2 0.499254
\(103\) 46370.9 0.430678 0.215339 0.976539i \(-0.430914\pi\)
0.215339 + 0.976539i \(0.430914\pi\)
\(104\) 272726. 2.47254
\(105\) 785.296 0.00695120
\(106\) 101497. 0.877386
\(107\) 206073. 1.74005 0.870025 0.493008i \(-0.164103\pi\)
0.870025 + 0.493008i \(0.164103\pi\)
\(108\) 58130.5 0.479562
\(109\) 64673.6 0.521387 0.260694 0.965422i \(-0.416049\pi\)
0.260694 + 0.965422i \(0.416049\pi\)
\(110\) −165694. −1.30565
\(111\) −104767. −0.807077
\(112\) −5305.25 −0.0399632
\(113\) −40561.7 −0.298827 −0.149414 0.988775i \(-0.547739\pi\)
−0.149414 + 0.988775i \(0.547739\pi\)
\(114\) 157500. 1.13506
\(115\) 82608.4 0.582478
\(116\) 211016. 1.45603
\(117\) 43774.8 0.295637
\(118\) −36796.7 −0.243278
\(119\) −1051.23 −0.00680503
\(120\) −207873. −1.31779
\(121\) −43758.3 −0.271705
\(122\) 44945.2 0.273391
\(123\) −123262. −0.734627
\(124\) −194082. −1.13352
\(125\) 190179. 1.08865
\(126\) −1632.35 −0.00915981
\(127\) −183521. −1.00966 −0.504832 0.863217i \(-0.668446\pi\)
−0.504832 + 0.863217i \(0.668446\pi\)
\(128\) 116634. 0.629215
\(129\) −123012. −0.650839
\(130\) −261464. −1.35692
\(131\) −105193. −0.535560 −0.267780 0.963480i \(-0.586290\pi\)
−0.267780 + 0.963480i \(0.586290\pi\)
\(132\) 245785. 1.22778
\(133\) −3156.15 −0.0154714
\(134\) 131941. 0.634774
\(135\) −33365.3 −0.157565
\(136\) 278268. 1.29008
\(137\) −413867. −1.88391 −0.941954 0.335741i \(-0.891013\pi\)
−0.941954 + 0.335741i \(0.891013\pi\)
\(138\) −171713. −0.767548
\(139\) −126624. −0.555875 −0.277938 0.960599i \(-0.589651\pi\)
−0.277938 + 0.960599i \(0.589651\pi\)
\(140\) 6957.73 0.0300018
\(141\) −48044.8 −0.203516
\(142\) 606281. 2.52321
\(143\) 185086. 0.756893
\(144\) 225407. 0.905860
\(145\) −121117. −0.478395
\(146\) 230476. 0.894838
\(147\) −151230. −0.577225
\(148\) −928233. −3.48339
\(149\) 250638. 0.924872 0.462436 0.886653i \(-0.346975\pi\)
0.462436 + 0.886653i \(0.346975\pi\)
\(150\) −98012.9 −0.355676
\(151\) 28034.2 0.100057 0.0500283 0.998748i \(-0.484069\pi\)
0.0500283 + 0.998748i \(0.484069\pi\)
\(152\) 835454. 2.93301
\(153\) 44664.2 0.154252
\(154\) −6901.81 −0.0234510
\(155\) 111397. 0.372431
\(156\) 387845. 1.27599
\(157\) −69003.5 −0.223420 −0.111710 0.993741i \(-0.535633\pi\)
−0.111710 + 0.993741i \(0.535633\pi\)
\(158\) −604565. −1.92664
\(159\) 86415.8 0.271082
\(160\) −607236. −1.87524
\(161\) 3440.96 0.0104620
\(162\) 69354.5 0.207629
\(163\) −241717. −0.712588 −0.356294 0.934374i \(-0.615960\pi\)
−0.356294 + 0.934374i \(0.615960\pi\)
\(164\) −1.09210e6 −3.17069
\(165\) −141074. −0.403400
\(166\) −851124. −2.39730
\(167\) −83610.7 −0.231991 −0.115995 0.993250i \(-0.537006\pi\)
−0.115995 + 0.993250i \(0.537006\pi\)
\(168\) −8658.72 −0.0236690
\(169\) −79228.9 −0.213386
\(170\) −266776. −0.707986
\(171\) 134097. 0.350695
\(172\) −1.08989e6 −2.80906
\(173\) 561408. 1.42614 0.713071 0.701091i \(-0.247304\pi\)
0.713071 + 0.701091i \(0.247304\pi\)
\(174\) 251760. 0.630395
\(175\) 1964.08 0.00484802
\(176\) 953055. 2.31919
\(177\) −31329.0 −0.0751646
\(178\) −638370. −1.51016
\(179\) −765988. −1.78685 −0.893427 0.449208i \(-0.851706\pi\)
−0.893427 + 0.449208i \(0.851706\pi\)
\(180\) −295617. −0.680062
\(181\) 161608. 0.366662 0.183331 0.983051i \(-0.441312\pi\)
0.183331 + 0.983051i \(0.441312\pi\)
\(182\) −10891.0 −0.0243718
\(183\) 38266.7 0.0844683
\(184\) −910845. −1.98335
\(185\) 532780. 1.14451
\(186\) −231555. −0.490764
\(187\) 188847. 0.394917
\(188\) −425677. −0.878387
\(189\) −1389.79 −0.00283006
\(190\) −800952. −1.60962
\(191\) −241767. −0.479528 −0.239764 0.970831i \(-0.577070\pi\)
−0.239764 + 0.970831i \(0.577070\pi\)
\(192\) 460777. 0.902065
\(193\) −121285. −0.234376 −0.117188 0.993110i \(-0.537388\pi\)
−0.117188 + 0.993110i \(0.537388\pi\)
\(194\) 1.61328e6 3.07755
\(195\) −222612. −0.419240
\(196\) −1.33990e6 −2.49134
\(197\) 140688. 0.258281 0.129141 0.991626i \(-0.458778\pi\)
0.129141 + 0.991626i \(0.458778\pi\)
\(198\) 293241. 0.531572
\(199\) −141272. −0.252885 −0.126442 0.991974i \(-0.540356\pi\)
−0.126442 + 0.991974i \(0.540356\pi\)
\(200\) −519905. −0.919071
\(201\) 112336. 0.196123
\(202\) 1.33389e6 2.30008
\(203\) −5045.01 −0.00859255
\(204\) 395725. 0.665761
\(205\) 626837. 1.04177
\(206\) 490174. 0.804789
\(207\) −146198. −0.237146
\(208\) 1.50391e6 2.41026
\(209\) 566983. 0.897851
\(210\) 8301.14 0.0129894
\(211\) 194478. 0.300722 0.150361 0.988631i \(-0.451956\pi\)
0.150361 + 0.988631i \(0.451956\pi\)
\(212\) 765645. 1.17001
\(213\) 516193. 0.779584
\(214\) 2.17834e6 3.25155
\(215\) 625566. 0.922947
\(216\) 367888. 0.536514
\(217\) 4640.14 0.00668931
\(218\) 683646. 0.974294
\(219\) 196230. 0.276474
\(220\) −1.24991e6 −1.74110
\(221\) 297998. 0.410424
\(222\) −1.10746e6 −1.50815
\(223\) −623897. −0.840138 −0.420069 0.907492i \(-0.637994\pi\)
−0.420069 + 0.907492i \(0.637994\pi\)
\(224\) −25293.7 −0.0336816
\(225\) −83449.0 −0.109892
\(226\) −428767. −0.558406
\(227\) 715141. 0.921143 0.460572 0.887623i \(-0.347644\pi\)
0.460572 + 0.887623i \(0.347644\pi\)
\(228\) 1.18810e6 1.51362
\(229\) 196467. 0.247572 0.123786 0.992309i \(-0.460496\pi\)
0.123786 + 0.992309i \(0.460496\pi\)
\(230\) 873230. 1.08845
\(231\) −5876.26 −0.00724555
\(232\) 1.33545e6 1.62895
\(233\) 91445.6 0.110350 0.0551751 0.998477i \(-0.482428\pi\)
0.0551751 + 0.998477i \(0.482428\pi\)
\(234\) 462731. 0.552445
\(235\) 244327. 0.288603
\(236\) −277575. −0.324415
\(237\) −514732. −0.595265
\(238\) −11112.3 −0.0127163
\(239\) 1.15180e6 1.30431 0.652155 0.758086i \(-0.273865\pi\)
0.652155 + 0.758086i \(0.273865\pi\)
\(240\) −1.14629e6 −1.28459
\(241\) −816973. −0.906077 −0.453038 0.891491i \(-0.649660\pi\)
−0.453038 + 0.891491i \(0.649660\pi\)
\(242\) −462557. −0.507723
\(243\) 59049.0 0.0641500
\(244\) 339044. 0.364570
\(245\) 769067. 0.818557
\(246\) −1.30297e6 −1.37277
\(247\) 894692. 0.933106
\(248\) −1.22828e6 −1.26814
\(249\) −724655. −0.740683
\(250\) 2.01033e6 2.03431
\(251\) 1.45693e6 1.45967 0.729837 0.683622i \(-0.239596\pi\)
0.729837 + 0.683622i \(0.239596\pi\)
\(252\) −12313.6 −0.0122147
\(253\) −618147. −0.607142
\(254\) −1.93995e6 −1.88672
\(255\) −227135. −0.218743
\(256\) −405417. −0.386636
\(257\) 1.74236e6 1.64553 0.822765 0.568381i \(-0.192430\pi\)
0.822765 + 0.568381i \(0.192430\pi\)
\(258\) −1.30033e6 −1.21619
\(259\) 22192.3 0.0205567
\(260\) −1.97235e6 −1.80947
\(261\) 214350. 0.194770
\(262\) −1.11196e6 −1.00078
\(263\) −1.50324e6 −1.34011 −0.670054 0.742312i \(-0.733729\pi\)
−0.670054 + 0.742312i \(0.733729\pi\)
\(264\) 1.55549e6 1.37359
\(265\) −439459. −0.384418
\(266\) −33362.8 −0.0289107
\(267\) −543513. −0.466586
\(268\) 995298. 0.846479
\(269\) 1.76367e6 1.48606 0.743030 0.669258i \(-0.233388\pi\)
0.743030 + 0.669258i \(0.233388\pi\)
\(270\) −352695. −0.294436
\(271\) 1.13163e6 0.936012 0.468006 0.883725i \(-0.344973\pi\)
0.468006 + 0.883725i \(0.344973\pi\)
\(272\) 1.53446e6 1.25758
\(273\) −9272.67 −0.00753005
\(274\) −4.37488e6 −3.52038
\(275\) −352835. −0.281345
\(276\) −1.29532e6 −1.02354
\(277\) 1.24787e6 0.977173 0.488586 0.872516i \(-0.337513\pi\)
0.488586 + 0.872516i \(0.337513\pi\)
\(278\) −1.33850e6 −1.03874
\(279\) −197148. −0.151629
\(280\) 44033.1 0.0335648
\(281\) −1.94155e6 −1.46684 −0.733421 0.679775i \(-0.762077\pi\)
−0.733421 + 0.679775i \(0.762077\pi\)
\(282\) −507868. −0.380301
\(283\) 2.66714e6 1.97961 0.989807 0.142419i \(-0.0454879\pi\)
0.989807 + 0.142419i \(0.0454879\pi\)
\(284\) 4.57347e6 3.36473
\(285\) −681937. −0.497316
\(286\) 1.95650e6 1.41437
\(287\) 26110.2 0.0187114
\(288\) 1.07467e6 0.763472
\(289\) −1.11580e6 −0.785857
\(290\) −1.28030e6 −0.893956
\(291\) 1.37356e6 0.950856
\(292\) 1.73860e6 1.19328
\(293\) 1.33165e6 0.906195 0.453098 0.891461i \(-0.350319\pi\)
0.453098 + 0.891461i \(0.350319\pi\)
\(294\) −1.59861e6 −1.07864
\(295\) 159321. 0.106590
\(296\) −5.87446e6 −3.89707
\(297\) 249668. 0.164237
\(298\) 2.64943e6 1.72827
\(299\) −975428. −0.630983
\(300\) −739359. −0.474299
\(301\) 26057.2 0.0165772
\(302\) 296341. 0.186971
\(303\) 1.13569e6 0.710644
\(304\) 4.60699e6 2.85912
\(305\) −194602. −0.119783
\(306\) 472133. 0.288244
\(307\) −630471. −0.381785 −0.190893 0.981611i \(-0.561138\pi\)
−0.190893 + 0.981611i \(0.561138\pi\)
\(308\) −52063.8 −0.0312722
\(309\) 417338. 0.248652
\(310\) 1.17755e6 0.695946
\(311\) −734743. −0.430759 −0.215380 0.976530i \(-0.569099\pi\)
−0.215380 + 0.976530i \(0.569099\pi\)
\(312\) 2.45454e6 1.42752
\(313\) 2.25797e6 1.30274 0.651368 0.758762i \(-0.274195\pi\)
0.651368 + 0.758762i \(0.274195\pi\)
\(314\) −729417. −0.417495
\(315\) 7067.66 0.00401328
\(316\) −4.56053e6 −2.56920
\(317\) 3.23249e6 1.80671 0.903356 0.428891i \(-0.141096\pi\)
0.903356 + 0.428891i \(0.141096\pi\)
\(318\) 913477. 0.506559
\(319\) 906306. 0.498652
\(320\) −2.34324e6 −1.27921
\(321\) 1.85466e6 1.00462
\(322\) 36373.4 0.0195499
\(323\) 912870. 0.486858
\(324\) 523175. 0.276875
\(325\) −556769. −0.292393
\(326\) −2.55513e6 −1.33158
\(327\) 582062. 0.301023
\(328\) −6.91154e6 −3.54724
\(329\) 10177.2 0.00518366
\(330\) −1.49125e6 −0.753816
\(331\) −2.44902e6 −1.22863 −0.614316 0.789060i \(-0.710568\pi\)
−0.614316 + 0.789060i \(0.710568\pi\)
\(332\) −6.42045e6 −3.19683
\(333\) −942899. −0.465966
\(334\) −883825. −0.433511
\(335\) −571274. −0.278120
\(336\) −47747.2 −0.0230728
\(337\) −2.48571e6 −1.19227 −0.596137 0.802883i \(-0.703298\pi\)
−0.596137 + 0.802883i \(0.703298\pi\)
\(338\) −837506. −0.398746
\(339\) −365056. −0.172528
\(340\) −2.01242e6 −0.944108
\(341\) −833572. −0.388201
\(342\) 1.41750e6 0.655328
\(343\) 64076.1 0.0294077
\(344\) −6.89753e6 −3.14266
\(345\) 743475. 0.336294
\(346\) 5.93448e6 2.66497
\(347\) 180776. 0.0805968 0.0402984 0.999188i \(-0.487169\pi\)
0.0402984 + 0.999188i \(0.487169\pi\)
\(348\) 1.89915e6 0.840641
\(349\) −3.84165e6 −1.68832 −0.844158 0.536094i \(-0.819899\pi\)
−0.844158 + 0.536094i \(0.819899\pi\)
\(350\) 20761.7 0.00905928
\(351\) 393973. 0.170686
\(352\) 4.54386e6 1.95465
\(353\) 3.36238e6 1.43618 0.718091 0.695949i \(-0.245016\pi\)
0.718091 + 0.695949i \(0.245016\pi\)
\(354\) −331170. −0.140457
\(355\) −2.62505e6 −1.10552
\(356\) −4.81554e6 −2.01382
\(357\) −9461.07 −0.00392889
\(358\) −8.09704e6 −3.33902
\(359\) 1.74122e6 0.713044 0.356522 0.934287i \(-0.383962\pi\)
0.356522 + 0.934287i \(0.383962\pi\)
\(360\) −1.87086e6 −0.760825
\(361\) 264647. 0.106880
\(362\) 1.70831e6 0.685166
\(363\) −393825. −0.156869
\(364\) −82156.0 −0.0325002
\(365\) −997906. −0.392064
\(366\) 404507. 0.157842
\(367\) −536023. −0.207739 −0.103870 0.994591i \(-0.533122\pi\)
−0.103870 + 0.994591i \(0.533122\pi\)
\(368\) −5.02272e6 −1.93339
\(369\) −1.10936e6 −0.424137
\(370\) 5.63186e6 2.13869
\(371\) −18305.2 −0.00690461
\(372\) −1.74673e6 −0.654440
\(373\) 2.75309e6 1.02459 0.512293 0.858810i \(-0.328796\pi\)
0.512293 + 0.858810i \(0.328796\pi\)
\(374\) 1.99625e6 0.737965
\(375\) 1.71161e6 0.628532
\(376\) −2.69396e6 −0.982702
\(377\) 1.43014e6 0.518233
\(378\) −14691.1 −0.00528842
\(379\) 1.19419e6 0.427048 0.213524 0.976938i \(-0.431506\pi\)
0.213524 + 0.976938i \(0.431506\pi\)
\(380\) −6.04198e6 −2.14645
\(381\) −1.65169e6 −0.582930
\(382\) −2.55565e6 −0.896073
\(383\) −1.80937e6 −0.630275 −0.315138 0.949046i \(-0.602051\pi\)
−0.315138 + 0.949046i \(0.602051\pi\)
\(384\) 1.04970e6 0.363277
\(385\) 29883.1 0.0102748
\(386\) −1.28207e6 −0.437969
\(387\) −1.10711e6 −0.375762
\(388\) 1.21697e7 4.10395
\(389\) 1.25898e6 0.421836 0.210918 0.977504i \(-0.432355\pi\)
0.210918 + 0.977504i \(0.432355\pi\)
\(390\) −2.35317e6 −0.783416
\(391\) −995247. −0.329222
\(392\) −8.47978e6 −2.78721
\(393\) −946736. −0.309206
\(394\) 1.48718e6 0.482639
\(395\) 2.61762e6 0.844138
\(396\) 2.21206e6 0.708858
\(397\) −4.79980e6 −1.52844 −0.764218 0.644958i \(-0.776875\pi\)
−0.764218 + 0.644958i \(0.776875\pi\)
\(398\) −1.49334e6 −0.472555
\(399\) −28405.3 −0.00893239
\(400\) −2.86694e6 −0.895919
\(401\) 753289. 0.233938 0.116969 0.993136i \(-0.462682\pi\)
0.116969 + 0.993136i \(0.462682\pi\)
\(402\) 1.18747e6 0.366487
\(403\) −1.31537e6 −0.403445
\(404\) 1.00622e7 3.06718
\(405\) −300288. −0.0909704
\(406\) −53329.4 −0.0160565
\(407\) −3.98672e6 −1.19297
\(408\) 2.50441e6 0.744826
\(409\) −3.77416e6 −1.11561 −0.557804 0.829973i \(-0.688356\pi\)
−0.557804 + 0.829973i \(0.688356\pi\)
\(410\) 6.62612e6 1.94670
\(411\) −3.72481e6 −1.08768
\(412\) 3.69762e6 1.07320
\(413\) 6636.32 0.00191449
\(414\) −1.54542e6 −0.443144
\(415\) 3.68516e6 1.05035
\(416\) 7.17016e6 2.03140
\(417\) −1.13961e6 −0.320935
\(418\) 5.99342e6 1.67777
\(419\) 4.57695e6 1.27362 0.636812 0.771019i \(-0.280253\pi\)
0.636812 + 0.771019i \(0.280253\pi\)
\(420\) 62619.6 0.0173216
\(421\) −5.69179e6 −1.56511 −0.782553 0.622584i \(-0.786083\pi\)
−0.782553 + 0.622584i \(0.786083\pi\)
\(422\) 2.05578e6 0.561946
\(423\) −432403. −0.117500
\(424\) 4.84550e6 1.30895
\(425\) −568082. −0.152559
\(426\) 5.45653e6 1.45677
\(427\) −8105.91 −0.00215145
\(428\) 1.64323e7 4.33599
\(429\) 1.66578e6 0.436992
\(430\) 6.61268e6 1.72467
\(431\) 5.14157e6 1.33322 0.666612 0.745405i \(-0.267744\pi\)
0.666612 + 0.745405i \(0.267744\pi\)
\(432\) 2.02866e6 0.522999
\(433\) 7.46920e6 1.91450 0.957248 0.289269i \(-0.0934123\pi\)
0.957248 + 0.289269i \(0.0934123\pi\)
\(434\) 49049.6 0.0125000
\(435\) −1.09006e6 −0.276202
\(436\) 5.15708e6 1.29923
\(437\) −2.98807e6 −0.748492
\(438\) 2.07429e6 0.516635
\(439\) −6.22735e6 −1.54221 −0.771103 0.636711i \(-0.780294\pi\)
−0.771103 + 0.636711i \(0.780294\pi\)
\(440\) −7.91027e6 −1.94787
\(441\) −1.36107e6 −0.333261
\(442\) 3.15006e6 0.766942
\(443\) −7.06367e6 −1.71010 −0.855049 0.518546i \(-0.826473\pi\)
−0.855049 + 0.518546i \(0.826473\pi\)
\(444\) −8.35409e6 −2.01114
\(445\) 2.76398e6 0.661661
\(446\) −6.59504e6 −1.56993
\(447\) 2.25574e6 0.533975
\(448\) −97604.9 −0.0229761
\(449\) 969905. 0.227046 0.113523 0.993535i \(-0.463786\pi\)
0.113523 + 0.993535i \(0.463786\pi\)
\(450\) −882116. −0.205350
\(451\) −4.69054e6 −1.08588
\(452\) −3.23440e6 −0.744642
\(453\) 252308. 0.0577677
\(454\) 7.55956e6 1.72130
\(455\) 47155.2 0.0106783
\(456\) 7.51908e6 1.69337
\(457\) −5.99930e6 −1.34372 −0.671862 0.740676i \(-0.734505\pi\)
−0.671862 + 0.740676i \(0.734505\pi\)
\(458\) 2.07680e6 0.462626
\(459\) 401978. 0.0890575
\(460\) 6.58720e6 1.45146
\(461\) −6.61011e6 −1.44863 −0.724313 0.689472i \(-0.757843\pi\)
−0.724313 + 0.689472i \(0.757843\pi\)
\(462\) −62116.3 −0.0135394
\(463\) −2.72859e6 −0.591542 −0.295771 0.955259i \(-0.595577\pi\)
−0.295771 + 0.955259i \(0.595577\pi\)
\(464\) 7.36413e6 1.58791
\(465\) 1.00258e6 0.215023
\(466\) 966646. 0.206207
\(467\) 1.22653e6 0.260248 0.130124 0.991498i \(-0.458462\pi\)
0.130124 + 0.991498i \(0.458462\pi\)
\(468\) 3.49061e6 0.736693
\(469\) −23795.8 −0.00499537
\(470\) 2.58271e6 0.539301
\(471\) −621032. −0.128992
\(472\) −1.75668e6 −0.362942
\(473\) −4.68102e6 −0.962028
\(474\) −5.44109e6 −1.11235
\(475\) −1.70557e6 −0.346846
\(476\) −83825.2 −0.0169573
\(477\) 777742. 0.156509
\(478\) 1.21753e7 2.43731
\(479\) 9.07091e6 1.80639 0.903196 0.429229i \(-0.141215\pi\)
0.903196 + 0.429229i \(0.141215\pi\)
\(480\) −5.46512e6 −1.08267
\(481\) −6.29099e6 −1.23981
\(482\) −8.63599e6 −1.69315
\(483\) 30968.6 0.00604024
\(484\) −3.48929e6 −0.677055
\(485\) −6.98510e6 −1.34840
\(486\) 624190. 0.119874
\(487\) 9.78320e6 1.86921 0.934606 0.355686i \(-0.115753\pi\)
0.934606 + 0.355686i \(0.115753\pi\)
\(488\) 2.14569e6 0.407866
\(489\) −2.17546e6 −0.411413
\(490\) 8.12959e6 1.52960
\(491\) 5.86427e6 1.09777 0.548883 0.835899i \(-0.315053\pi\)
0.548883 + 0.835899i \(0.315053\pi\)
\(492\) −9.82893e6 −1.83060
\(493\) 1.45920e6 0.270394
\(494\) 9.45754e6 1.74366
\(495\) −1.26966e6 −0.232903
\(496\) −6.77314e6 −1.23619
\(497\) −109343. −0.0198564
\(498\) −7.66012e6 −1.38408
\(499\) 1.94897e6 0.350391 0.175196 0.984534i \(-0.443944\pi\)
0.175196 + 0.984534i \(0.443944\pi\)
\(500\) 1.51649e7 2.71278
\(501\) −752496. −0.133940
\(502\) 1.54008e7 2.72763
\(503\) −2.91888e6 −0.514395 −0.257198 0.966359i \(-0.582799\pi\)
−0.257198 + 0.966359i \(0.582799\pi\)
\(504\) −77928.5 −0.0136653
\(505\) −5.77543e6 −1.00776
\(506\) −6.53426e6 −1.13454
\(507\) −713060. −0.123199
\(508\) −1.46340e7 −2.51596
\(509\) 4.92340e6 0.842307 0.421153 0.906989i \(-0.361625\pi\)
0.421153 + 0.906989i \(0.361625\pi\)
\(510\) −2.40098e6 −0.408756
\(511\) −41566.6 −0.00704195
\(512\) −8.01783e6 −1.35170
\(513\) 1.20687e6 0.202474
\(514\) 1.84180e7 3.07493
\(515\) −2.12233e6 −0.352611
\(516\) −9.80900e6 −1.62181
\(517\) −1.82826e6 −0.300824
\(518\) 234589. 0.0384134
\(519\) 5.05267e6 0.823384
\(520\) −1.24823e7 −2.02435
\(521\) −7.79371e6 −1.25791 −0.628955 0.777441i \(-0.716517\pi\)
−0.628955 + 0.777441i \(0.716517\pi\)
\(522\) 2.26584e6 0.363959
\(523\) 3.88156e6 0.620515 0.310257 0.950653i \(-0.399585\pi\)
0.310257 + 0.950653i \(0.399585\pi\)
\(524\) −8.38809e6 −1.33455
\(525\) 17676.7 0.00279900
\(526\) −1.58904e7 −2.50420
\(527\) −1.34209e6 −0.210502
\(528\) 8.57749e6 1.33898
\(529\) −3.17863e6 −0.493857
\(530\) −4.64540e6 −0.718345
\(531\) −281961. −0.0433963
\(532\) −251672. −0.0385527
\(533\) −7.40161e6 −1.12852
\(534\) −5.74533e6 −0.871890
\(535\) −9.43167e6 −1.42464
\(536\) 6.29890e6 0.947006
\(537\) −6.89389e6 −1.03164
\(538\) 1.86433e7 2.77694
\(539\) −5.75482e6 −0.853218
\(540\) −2.66055e6 −0.392634
\(541\) −4.62496e6 −0.679383 −0.339691 0.940537i \(-0.610323\pi\)
−0.339691 + 0.940537i \(0.610323\pi\)
\(542\) 1.19621e7 1.74908
\(543\) 1.45447e6 0.211692
\(544\) 7.31584e6 1.05990
\(545\) −2.96002e6 −0.426877
\(546\) −98018.8 −0.0140711
\(547\) −3.47224e6 −0.496182 −0.248091 0.968737i \(-0.579803\pi\)
−0.248091 + 0.968737i \(0.579803\pi\)
\(548\) −3.30018e7 −4.69447
\(549\) 344401. 0.0487678
\(550\) −3.72972e6 −0.525738
\(551\) 4.38100e6 0.614744
\(552\) −8.19760e6 −1.14509
\(553\) 109034. 0.0151617
\(554\) 1.31909e7 1.82600
\(555\) 4.79502e6 0.660781
\(556\) −1.00970e7 −1.38517
\(557\) 1.28284e7 1.75201 0.876004 0.482305i \(-0.160200\pi\)
0.876004 + 0.482305i \(0.160200\pi\)
\(558\) −2.08400e6 −0.283342
\(559\) −7.38660e6 −0.999804
\(560\) 242814. 0.0327192
\(561\) 1.69962e6 0.228006
\(562\) −2.05236e7 −2.74102
\(563\) −6.05947e6 −0.805682 −0.402841 0.915270i \(-0.631977\pi\)
−0.402841 + 0.915270i \(0.631977\pi\)
\(564\) −3.83109e6 −0.507137
\(565\) 1.85645e6 0.244660
\(566\) 2.81936e7 3.69922
\(567\) −12508.2 −0.00163394
\(568\) 2.89439e7 3.76432
\(569\) 1.02026e7 1.32108 0.660539 0.750791i \(-0.270328\pi\)
0.660539 + 0.750791i \(0.270328\pi\)
\(570\) −7.20857e6 −0.929313
\(571\) −7.93919e6 −1.01903 −0.509514 0.860462i \(-0.670175\pi\)
−0.509514 + 0.860462i \(0.670175\pi\)
\(572\) 1.47588e7 1.88609
\(573\) −2.17590e6 −0.276856
\(574\) 276004. 0.0349651
\(575\) 1.85948e6 0.234543
\(576\) 4.14699e6 0.520808
\(577\) −5.69036e6 −0.711542 −0.355771 0.934573i \(-0.615782\pi\)
−0.355771 + 0.934573i \(0.615782\pi\)
\(578\) −1.17949e7 −1.46850
\(579\) −1.09157e6 −0.135317
\(580\) −9.65792e6 −1.19210
\(581\) 153501. 0.0188656
\(582\) 1.45195e7 1.77682
\(583\) 3.28841e6 0.400696
\(584\) 1.10030e7 1.33499
\(585\) −2.00351e6 −0.242048
\(586\) 1.40765e7 1.69337
\(587\) 7.93842e6 0.950909 0.475454 0.879740i \(-0.342284\pi\)
0.475454 + 0.879740i \(0.342284\pi\)
\(588\) −1.20591e7 −1.43838
\(589\) −4.02941e6 −0.478579
\(590\) 1.68413e6 0.199180
\(591\) 1.26620e6 0.149119
\(592\) −3.23938e7 −3.79890
\(593\) 9.22345e6 1.07710 0.538551 0.842593i \(-0.318972\pi\)
0.538551 + 0.842593i \(0.318972\pi\)
\(594\) 2.63917e6 0.306903
\(595\) 48113.3 0.00557151
\(596\) 1.99859e7 2.30467
\(597\) −1.27145e6 −0.146003
\(598\) −1.03110e7 −1.17909
\(599\) 1.12236e7 1.27810 0.639049 0.769166i \(-0.279328\pi\)
0.639049 + 0.769166i \(0.279328\pi\)
\(600\) −4.67915e6 −0.530626
\(601\) −7.66886e6 −0.866053 −0.433027 0.901381i \(-0.642554\pi\)
−0.433027 + 0.901381i \(0.642554\pi\)
\(602\) 275444. 0.0309772
\(603\) 1.01102e6 0.113232
\(604\) 2.23545e6 0.249329
\(605\) 2.00276e6 0.222454
\(606\) 1.20050e7 1.32795
\(607\) −1.02486e7 −1.12899 −0.564496 0.825436i \(-0.690929\pi\)
−0.564496 + 0.825436i \(0.690929\pi\)
\(608\) 2.19646e7 2.40971
\(609\) −45405.1 −0.00496091
\(610\) −2.05708e6 −0.223834
\(611\) −2.88498e6 −0.312636
\(612\) 3.56153e6 0.384377
\(613\) −3.76554e6 −0.404740 −0.202370 0.979309i \(-0.564864\pi\)
−0.202370 + 0.979309i \(0.564864\pi\)
\(614\) −6.66453e6 −0.713425
\(615\) 5.64153e6 0.601464
\(616\) −329493. −0.0349860
\(617\) −1.26192e7 −1.33450 −0.667252 0.744832i \(-0.732530\pi\)
−0.667252 + 0.744832i \(0.732530\pi\)
\(618\) 4.41157e6 0.464645
\(619\) 1.15894e7 1.21572 0.607859 0.794045i \(-0.292028\pi\)
0.607859 + 0.794045i \(0.292028\pi\)
\(620\) 8.88285e6 0.928053
\(621\) −1.31578e6 −0.136916
\(622\) −7.76676e6 −0.804941
\(623\) 115131. 0.0118842
\(624\) 1.35352e7 1.39156
\(625\) −5.48476e6 −0.561639
\(626\) 2.38683e7 2.43437
\(627\) 5.10284e6 0.518374
\(628\) −5.50235e6 −0.556736
\(629\) −6.41881e6 −0.646886
\(630\) 74710.3 0.00749944
\(631\) 1.73656e7 1.73627 0.868134 0.496330i \(-0.165319\pi\)
0.868134 + 0.496330i \(0.165319\pi\)
\(632\) −2.88620e7 −2.87431
\(633\) 1.75031e6 0.173622
\(634\) 3.41697e7 3.37613
\(635\) 8.39952e6 0.826647
\(636\) 6.89081e6 0.675503
\(637\) −9.08104e6 −0.886721
\(638\) 9.58030e6 0.931810
\(639\) 4.64573e6 0.450093
\(640\) −5.33816e6 −0.515159
\(641\) 4.27049e6 0.410518 0.205259 0.978708i \(-0.434196\pi\)
0.205259 + 0.978708i \(0.434196\pi\)
\(642\) 1.96051e7 1.87729
\(643\) 1.02304e7 0.975806 0.487903 0.872898i \(-0.337762\pi\)
0.487903 + 0.872898i \(0.337762\pi\)
\(644\) 274382. 0.0260700
\(645\) 5.63009e6 0.532864
\(646\) 9.64969e6 0.909771
\(647\) −2.92187e6 −0.274410 −0.137205 0.990543i \(-0.543812\pi\)
−0.137205 + 0.990543i \(0.543812\pi\)
\(648\) 3.31099e6 0.309757
\(649\) −1.19217e6 −0.111104
\(650\) −5.88545e6 −0.546382
\(651\) 41761.2 0.00386208
\(652\) −1.92746e7 −1.77568
\(653\) −1.60697e7 −1.47477 −0.737387 0.675471i \(-0.763940\pi\)
−0.737387 + 0.675471i \(0.763940\pi\)
\(654\) 6.15281e6 0.562509
\(655\) 4.81453e6 0.438481
\(656\) −3.81127e7 −3.45788
\(657\) 1.76607e6 0.159622
\(658\) 107580. 0.00968649
\(659\) −4.97844e6 −0.446560 −0.223280 0.974754i \(-0.571676\pi\)
−0.223280 + 0.974754i \(0.571676\pi\)
\(660\) −1.12492e7 −1.00522
\(661\) 1.94882e7 1.73488 0.867438 0.497545i \(-0.165765\pi\)
0.867438 + 0.497545i \(0.165765\pi\)
\(662\) −2.58879e7 −2.29589
\(663\) 2.68198e6 0.236958
\(664\) −4.06328e7 −3.57648
\(665\) 144453. 0.0126669
\(666\) −9.96712e6 −0.870731
\(667\) −4.77634e6 −0.415701
\(668\) −6.66713e6 −0.578093
\(669\) −5.61507e6 −0.485054
\(670\) −6.03877e6 −0.519711
\(671\) 1.45618e6 0.124856
\(672\) −227643. −0.0194461
\(673\) 4.90084e6 0.417093 0.208546 0.978012i \(-0.433127\pi\)
0.208546 + 0.978012i \(0.433127\pi\)
\(674\) −2.62758e7 −2.22795
\(675\) −751041. −0.0634460
\(676\) −6.31772e6 −0.531733
\(677\) −1.45552e6 −0.122052 −0.0610261 0.998136i \(-0.519437\pi\)
−0.0610261 + 0.998136i \(0.519437\pi\)
\(678\) −3.85890e6 −0.322396
\(679\) −290956. −0.0242188
\(680\) −1.27359e7 −1.05623
\(681\) 6.43627e6 0.531822
\(682\) −8.81146e6 −0.725415
\(683\) −260979. −0.0214069 −0.0107034 0.999943i \(-0.503407\pi\)
−0.0107034 + 0.999943i \(0.503407\pi\)
\(684\) 1.06929e7 0.873888
\(685\) 1.89421e7 1.54242
\(686\) 677331. 0.0549529
\(687\) 1.76820e6 0.142935
\(688\) −3.80354e7 −3.06349
\(689\) 5.18907e6 0.416430
\(690\) 7.85907e6 0.628418
\(691\) 1.41924e7 1.13074 0.565369 0.824838i \(-0.308734\pi\)
0.565369 + 0.824838i \(0.308734\pi\)
\(692\) 4.47667e7 3.55377
\(693\) −52886.4 −0.00418322
\(694\) 1.91094e6 0.150608
\(695\) 5.79538e6 0.455114
\(696\) 1.20190e7 0.940474
\(697\) −7.55199e6 −0.588816
\(698\) −4.06090e7 −3.15488
\(699\) 823011. 0.0637107
\(700\) 156616. 0.0120807
\(701\) −4.62422e6 −0.355421 −0.177711 0.984083i \(-0.556869\pi\)
−0.177711 + 0.984083i \(0.556869\pi\)
\(702\) 4.16458e6 0.318954
\(703\) −1.92714e7 −1.47071
\(704\) 1.75341e7 1.33337
\(705\) 2.19894e6 0.166625
\(706\) 3.55428e7 2.68373
\(707\) −240569. −0.0181005
\(708\) −2.49818e6 −0.187301
\(709\) −3.19889e6 −0.238992 −0.119496 0.992835i \(-0.538128\pi\)
−0.119496 + 0.992835i \(0.538128\pi\)
\(710\) −2.77486e7 −2.06584
\(711\) −4.63259e6 −0.343676
\(712\) −3.04759e7 −2.25297
\(713\) 4.39303e6 0.323624
\(714\) −100010. −0.00734174
\(715\) −8.47115e6 −0.619694
\(716\) −6.10799e7 −4.45262
\(717\) 1.03662e7 0.753044
\(718\) 1.84059e7 1.33244
\(719\) 9.85622e6 0.711030 0.355515 0.934671i \(-0.384305\pi\)
0.355515 + 0.934671i \(0.384305\pi\)
\(720\) −1.03166e7 −0.741658
\(721\) −88403.4 −0.00633331
\(722\) 2.79751e6 0.199723
\(723\) −7.35275e6 −0.523124
\(724\) 1.28866e7 0.913678
\(725\) −2.72631e6 −0.192633
\(726\) −4.16301e6 −0.293134
\(727\) 1.24980e7 0.877011 0.438506 0.898728i \(-0.355508\pi\)
0.438506 + 0.898728i \(0.355508\pi\)
\(728\) −519936. −0.0363598
\(729\) 531441. 0.0370370
\(730\) −1.05486e7 −0.732634
\(731\) −7.53667e6 −0.521659
\(732\) 3.05139e6 0.210485
\(733\) 3.96280e6 0.272422 0.136211 0.990680i \(-0.456508\pi\)
0.136211 + 0.990680i \(0.456508\pi\)
\(734\) −5.66615e6 −0.388193
\(735\) 6.92160e6 0.472594
\(736\) −2.39467e7 −1.62949
\(737\) 4.27476e6 0.289897
\(738\) −1.17267e7 −0.792566
\(739\) 2.26893e6 0.152831 0.0764153 0.997076i \(-0.475653\pi\)
0.0764153 + 0.997076i \(0.475653\pi\)
\(740\) 4.24839e7 2.85197
\(741\) 8.05223e6 0.538729
\(742\) −193499. −0.0129023
\(743\) −3.99835e6 −0.265710 −0.132855 0.991135i \(-0.542415\pi\)
−0.132855 + 0.991135i \(0.542415\pi\)
\(744\) −1.10545e7 −0.732160
\(745\) −1.14714e7 −0.757224
\(746\) 2.91022e7 1.91460
\(747\) −6.52189e6 −0.427634
\(748\) 1.50587e7 0.984086
\(749\) −392866. −0.0255882
\(750\) 1.80930e7 1.17451
\(751\) 2.09254e7 1.35386 0.676931 0.736047i \(-0.263310\pi\)
0.676931 + 0.736047i \(0.263310\pi\)
\(752\) −1.48555e7 −0.957947
\(753\) 1.31124e7 0.842743
\(754\) 1.51176e7 0.968399
\(755\) −1.28309e6 −0.0819197
\(756\) −110822. −0.00705217
\(757\) 1.80041e7 1.14191 0.570955 0.820981i \(-0.306573\pi\)
0.570955 + 0.820981i \(0.306573\pi\)
\(758\) 1.26235e7 0.798007
\(759\) −5.56332e6 −0.350534
\(760\) −3.82376e7 −2.40135
\(761\) 5.73582e6 0.359033 0.179516 0.983755i \(-0.442547\pi\)
0.179516 + 0.983755i \(0.442547\pi\)
\(762\) −1.74596e7 −1.08930
\(763\) −123296. −0.00766723
\(764\) −1.92785e7 −1.19493
\(765\) −2.04422e6 −0.126291
\(766\) −1.91263e7 −1.17777
\(767\) −1.88124e6 −0.115466
\(768\) −3.64875e6 −0.223224
\(769\) 1.12378e7 0.685277 0.342639 0.939467i \(-0.388679\pi\)
0.342639 + 0.939467i \(0.388679\pi\)
\(770\) 315886. 0.0192001
\(771\) 1.56813e7 0.950048
\(772\) −9.67128e6 −0.584038
\(773\) −1.07942e6 −0.0649740 −0.0324870 0.999472i \(-0.510343\pi\)
−0.0324870 + 0.999472i \(0.510343\pi\)
\(774\) −1.17029e7 −0.702170
\(775\) 2.50752e6 0.149965
\(776\) 7.70181e7 4.59133
\(777\) 199731. 0.0118684
\(778\) 1.33083e7 0.788266
\(779\) −2.26736e7 −1.33868
\(780\) −1.77511e7 −1.04470
\(781\) 1.96429e7 1.15233
\(782\) −1.05205e7 −0.615203
\(783\) 1.92915e6 0.112451
\(784\) −4.67605e7 −2.71699
\(785\) 3.15819e6 0.182921
\(786\) −1.00077e7 −0.577799
\(787\) −1.39491e7 −0.802803 −0.401401 0.915902i \(-0.631477\pi\)
−0.401401 + 0.915902i \(0.631477\pi\)
\(788\) 1.12185e7 0.643606
\(789\) −1.35292e7 −0.773712
\(790\) 2.76701e7 1.57740
\(791\) 77328.5 0.00439439
\(792\) 1.39994e7 0.793041
\(793\) 2.29783e6 0.129758
\(794\) −5.07374e7 −2.85612
\(795\) −3.95513e6 −0.221944
\(796\) −1.12650e7 −0.630158
\(797\) −2.36684e7 −1.31984 −0.659922 0.751334i \(-0.729411\pi\)
−0.659922 + 0.751334i \(0.729411\pi\)
\(798\) −300265. −0.0166916
\(799\) −2.94359e6 −0.163121
\(800\) −1.36687e7 −0.755094
\(801\) −4.89162e6 −0.269384
\(802\) 7.96280e6 0.437150
\(803\) 7.46719e6 0.408666
\(804\) 8.95769e6 0.488715
\(805\) −157488. −0.00856559
\(806\) −1.39044e7 −0.753900
\(807\) 1.58730e7 0.857977
\(808\) 6.36802e7 3.43144
\(809\) 1.59899e7 0.858963 0.429481 0.903076i \(-0.358696\pi\)
0.429481 + 0.903076i \(0.358696\pi\)
\(810\) −3.17426e6 −0.169992
\(811\) 2.53790e7 1.35495 0.677473 0.735548i \(-0.263075\pi\)
0.677473 + 0.735548i \(0.263075\pi\)
\(812\) −402290. −0.0214116
\(813\) 1.01847e7 0.540407
\(814\) −4.21425e7 −2.22925
\(815\) 1.10631e7 0.583420
\(816\) 1.38102e7 0.726063
\(817\) −2.26277e7 −1.18600
\(818\) −3.98955e7 −2.08469
\(819\) −83454.0 −0.00434748
\(820\) 4.99841e7 2.59595
\(821\) 7.99354e6 0.413886 0.206943 0.978353i \(-0.433648\pi\)
0.206943 + 0.978353i \(0.433648\pi\)
\(822\) −3.93739e7 −2.03249
\(823\) 2.85727e7 1.47046 0.735228 0.677820i \(-0.237075\pi\)
0.735228 + 0.677820i \(0.237075\pi\)
\(824\) 2.34010e7 1.20065
\(825\) −3.17552e6 −0.162435
\(826\) 70150.7 0.00357752
\(827\) −2.01677e7 −1.02540 −0.512700 0.858568i \(-0.671355\pi\)
−0.512700 + 0.858568i \(0.671355\pi\)
\(828\) −1.16578e7 −0.590939
\(829\) −2.74741e7 −1.38847 −0.694236 0.719747i \(-0.744258\pi\)
−0.694236 + 0.719747i \(0.744258\pi\)
\(830\) 3.89548e7 1.96275
\(831\) 1.12309e7 0.564171
\(832\) 2.76686e7 1.38573
\(833\) −9.26554e6 −0.462656
\(834\) −1.20465e7 −0.599717
\(835\) 3.82674e6 0.189939
\(836\) 4.52113e7 2.23734
\(837\) −1.77433e6 −0.0875430
\(838\) 4.83817e7 2.37997
\(839\) −2.12558e6 −0.104249 −0.0521245 0.998641i \(-0.516599\pi\)
−0.0521245 + 0.998641i \(0.516599\pi\)
\(840\) 396298. 0.0193786
\(841\) −1.35082e7 −0.658581
\(842\) −6.01663e7 −2.92465
\(843\) −1.74740e7 −0.846881
\(844\) 1.55077e7 0.749363
\(845\) 3.62620e6 0.174707
\(846\) −4.57081e6 −0.219567
\(847\) 83422.6 0.00399554
\(848\) 2.67198e7 1.27598
\(849\) 2.40043e7 1.14293
\(850\) −6.00503e6 −0.285081
\(851\) 2.10105e7 0.994517
\(852\) 4.11613e7 1.94263
\(853\) −1.87729e7 −0.883405 −0.441702 0.897162i \(-0.645625\pi\)
−0.441702 + 0.897162i \(0.645625\pi\)
\(854\) −85685.3 −0.00402033
\(855\) −6.13744e6 −0.287126
\(856\) 1.03994e8 4.85093
\(857\) −1.59601e7 −0.742307 −0.371154 0.928572i \(-0.621038\pi\)
−0.371154 + 0.928572i \(0.621038\pi\)
\(858\) 1.76085e7 0.816589
\(859\) −1.46395e7 −0.676931 −0.338465 0.940979i \(-0.609908\pi\)
−0.338465 + 0.940979i \(0.609908\pi\)
\(860\) 4.98827e7 2.29987
\(861\) 234992. 0.0108030
\(862\) 5.43501e7 2.49134
\(863\) −1.10645e7 −0.505716 −0.252858 0.967503i \(-0.581371\pi\)
−0.252858 + 0.967503i \(0.581371\pi\)
\(864\) 9.67202e6 0.440791
\(865\) −2.56948e7 −1.16763
\(866\) 7.89548e7 3.57754
\(867\) −1.00422e7 −0.453715
\(868\) 370005. 0.0166690
\(869\) −1.95873e7 −0.879882
\(870\) −1.15227e7 −0.516126
\(871\) 6.74552e6 0.301280
\(872\) 3.26373e7 1.45353
\(873\) 1.23620e7 0.548977
\(874\) −3.15860e7 −1.39867
\(875\) −362565. −0.0160091
\(876\) 1.56474e7 0.688939
\(877\) −3.21834e7 −1.41297 −0.706486 0.707727i \(-0.749721\pi\)
−0.706486 + 0.707727i \(0.749721\pi\)
\(878\) −6.58276e7 −2.88185
\(879\) 1.19849e7 0.523192
\(880\) −4.36200e7 −1.89880
\(881\) −2.44052e7 −1.05936 −0.529678 0.848199i \(-0.677687\pi\)
−0.529678 + 0.848199i \(0.677687\pi\)
\(882\) −1.43875e7 −0.622751
\(883\) 8.92253e6 0.385111 0.192556 0.981286i \(-0.438322\pi\)
0.192556 + 0.981286i \(0.438322\pi\)
\(884\) 2.37624e7 1.02273
\(885\) 1.43388e6 0.0615398
\(886\) −7.46681e7 −3.19559
\(887\) −1.51942e7 −0.648439 −0.324219 0.945982i \(-0.605102\pi\)
−0.324219 + 0.945982i \(0.605102\pi\)
\(888\) −5.28701e7 −2.24998
\(889\) 349872. 0.0148476
\(890\) 2.92173e7 1.23642
\(891\) 2.24701e6 0.0948225
\(892\) −4.97496e7 −2.09352
\(893\) −8.83767e6 −0.370859
\(894\) 2.38448e7 0.997816
\(895\) 3.50582e7 1.46296
\(896\) −222355. −0.00925288
\(897\) −8.77886e6 −0.364298
\(898\) 1.02526e7 0.424270
\(899\) −6.44090e6 −0.265795
\(900\) −6.65423e6 −0.273837
\(901\) 5.29450e6 0.217277
\(902\) −4.95823e7 −2.02913
\(903\) 234515. 0.00957087
\(904\) −2.04694e7 −0.833074
\(905\) −7.39657e6 −0.300199
\(906\) 2.66707e6 0.107948
\(907\) 1.89637e7 0.765431 0.382715 0.923866i \(-0.374989\pi\)
0.382715 + 0.923866i \(0.374989\pi\)
\(908\) 5.70254e7 2.29538
\(909\) 1.02212e7 0.410291
\(910\) 498465. 0.0199540
\(911\) −8.21481e6 −0.327945 −0.163973 0.986465i \(-0.552431\pi\)
−0.163973 + 0.986465i \(0.552431\pi\)
\(912\) 4.14629e7 1.65072
\(913\) −2.75755e7 −1.09483
\(914\) −6.34169e7 −2.51096
\(915\) −1.75141e6 −0.0691570
\(916\) 1.56663e7 0.616918
\(917\) 200544. 0.00787564
\(918\) 4.24919e6 0.166418
\(919\) 4.45960e7 1.74183 0.870917 0.491430i \(-0.163526\pi\)
0.870917 + 0.491430i \(0.163526\pi\)
\(920\) 4.16881e7 1.62384
\(921\) −5.67424e6 −0.220424
\(922\) −6.98736e7 −2.70698
\(923\) 3.09962e7 1.19758
\(924\) −468574. −0.0180550
\(925\) 1.19927e7 0.460852
\(926\) −2.88431e7 −1.10539
\(927\) 3.75605e6 0.143559
\(928\) 3.51098e7 1.33832
\(929\) 6.22237e6 0.236546 0.118273 0.992981i \(-0.462264\pi\)
0.118273 + 0.992981i \(0.462264\pi\)
\(930\) 1.05980e7 0.401805
\(931\) −2.78183e7 −1.05186
\(932\) 7.29189e6 0.274979
\(933\) −6.61269e6 −0.248699
\(934\) 1.29653e7 0.486314
\(935\) −8.64326e6 −0.323332
\(936\) 2.20908e7 0.824181
\(937\) −2.51600e7 −0.936185 −0.468093 0.883679i \(-0.655059\pi\)
−0.468093 + 0.883679i \(0.655059\pi\)
\(938\) −251538. −0.00933463
\(939\) 2.03217e7 0.752135
\(940\) 1.94827e7 0.719165
\(941\) −3.46345e7 −1.27507 −0.637537 0.770420i \(-0.720047\pi\)
−0.637537 + 0.770420i \(0.720047\pi\)
\(942\) −6.56475e6 −0.241041
\(943\) 2.47197e7 0.905241
\(944\) −9.68694e6 −0.353799
\(945\) 63609.0 0.00231707
\(946\) −4.94818e7 −1.79770
\(947\) 6.37603e6 0.231034 0.115517 0.993306i \(-0.463148\pi\)
0.115517 + 0.993306i \(0.463148\pi\)
\(948\) −4.10448e7 −1.48333
\(949\) 1.17831e7 0.424713
\(950\) −1.80291e7 −0.648136
\(951\) 2.90924e7 1.04311
\(952\) −530500. −0.0189711
\(953\) 2.62513e7 0.936309 0.468154 0.883647i \(-0.344919\pi\)
0.468154 + 0.883647i \(0.344919\pi\)
\(954\) 8.22129e6 0.292462
\(955\) 1.10653e7 0.392606
\(956\) 9.18443e7 3.25018
\(957\) 8.15675e6 0.287897
\(958\) 9.58860e7 3.37553
\(959\) 789013. 0.0277037
\(960\) −2.10891e7 −0.738551
\(961\) −2.27051e7 −0.793078
\(962\) −6.65003e7 −2.31679
\(963\) 1.66919e7 0.580016
\(964\) −6.51455e7 −2.25783
\(965\) 5.55105e6 0.191892
\(966\) 327361. 0.0112871
\(967\) 7.74673e6 0.266411 0.133205 0.991088i \(-0.457473\pi\)
0.133205 + 0.991088i \(0.457473\pi\)
\(968\) −2.20825e7 −0.757461
\(969\) 8.21583e6 0.281088
\(970\) −7.38375e7 −2.51969
\(971\) 3.17051e7 1.07915 0.539575 0.841938i \(-0.318585\pi\)
0.539575 + 0.841938i \(0.318585\pi\)
\(972\) 4.70857e6 0.159854
\(973\) 241400. 0.00817439
\(974\) 1.03415e8 3.49291
\(975\) −5.01092e6 −0.168813
\(976\) 1.18321e7 0.397591
\(977\) −2.61156e7 −0.875313 −0.437657 0.899142i \(-0.644191\pi\)
−0.437657 + 0.899142i \(0.644191\pi\)
\(978\) −2.29961e7 −0.768790
\(979\) −2.06825e7 −0.689678
\(980\) 6.13255e7 2.03974
\(981\) 5.23856e6 0.173796
\(982\) 6.19895e7 2.05135
\(983\) −7.25374e6 −0.239430 −0.119715 0.992808i \(-0.538198\pi\)
−0.119715 + 0.992808i \(0.538198\pi\)
\(984\) −6.22039e7 −2.04800
\(985\) −6.43911e6 −0.211464
\(986\) 1.54248e7 0.505273
\(987\) 91594.4 0.00299279
\(988\) 7.13428e7 2.32519
\(989\) 2.46696e7 0.801993
\(990\) −1.34212e7 −0.435216
\(991\) 2.63294e7 0.851640 0.425820 0.904808i \(-0.359986\pi\)
0.425820 + 0.904808i \(0.359986\pi\)
\(992\) −3.22922e7 −1.04188
\(993\) −2.20411e7 −0.709351
\(994\) −1.15584e6 −0.0371049
\(995\) 6.46581e6 0.207045
\(996\) −5.77840e7 −1.84569
\(997\) 4.01593e7 1.27952 0.639762 0.768573i \(-0.279033\pi\)
0.639762 + 0.768573i \(0.279033\pi\)
\(998\) 2.06020e7 0.654761
\(999\) −8.48609e6 −0.269026
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.1.12 12
3.2 odd 2 531.6.a.c.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.c.1.12 12 1.1 even 1 trivial
531.6.a.c.1.1 12 3.2 odd 2