Properties

Label 912.6.a.y.1.3
Level $912$
Weight $6$
Character 912.1
Self dual yes
Analytic conductor $146.270$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 912.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(146.270043669\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} - 4725 x^{4} + 92430 x^{3} + 1610577 x^{2} - 16081740 x - 24661341\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 456)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.36301\) of defining polynomial
Character \(\chi\) \(=\) 912.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.00000 q^{3} -41.5770 q^{5} -98.3255 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q+9.00000 q^{3} -41.5770 q^{5} -98.3255 q^{7} +81.0000 q^{9} +628.684 q^{11} +154.682 q^{13} -374.193 q^{15} +1711.67 q^{17} -361.000 q^{19} -884.929 q^{21} -305.157 q^{23} -1396.35 q^{25} +729.000 q^{27} +3768.18 q^{29} -520.053 q^{31} +5658.16 q^{33} +4088.08 q^{35} -13114.9 q^{37} +1392.14 q^{39} +8264.15 q^{41} -6342.44 q^{43} -3367.74 q^{45} -27630.9 q^{47} -7139.10 q^{49} +15405.1 q^{51} +15998.2 q^{53} -26138.8 q^{55} -3249.00 q^{57} -43506.7 q^{59} -19745.5 q^{61} -7964.36 q^{63} -6431.24 q^{65} +54996.7 q^{67} -2746.42 q^{69} +76576.3 q^{71} -36253.1 q^{73} -12567.1 q^{75} -61815.7 q^{77} +48683.6 q^{79} +6561.00 q^{81} +98174.2 q^{83} -71166.4 q^{85} +33913.6 q^{87} +129077. q^{89} -15209.2 q^{91} -4680.48 q^{93} +15009.3 q^{95} +94431.1 q^{97} +50923.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 54 q^{3} - 65 q^{5} + 149 q^{7} + 486 q^{9} + O(q^{10}) \) \( 6 q + 54 q^{3} - 65 q^{5} + 149 q^{7} + 486 q^{9} + 203 q^{11} - 298 q^{13} - 585 q^{15} + 1319 q^{17} - 2166 q^{19} + 1341 q^{21} - 1234 q^{23} + 4589 q^{25} + 4374 q^{27} + 7356 q^{29} - 1632 q^{31} + 1827 q^{33} - 4383 q^{35} + 14204 q^{37} - 2682 q^{39} + 14734 q^{41} + 4693 q^{43} - 5265 q^{45} + 10955 q^{47} + 38561 q^{49} + 11871 q^{51} + 47500 q^{53} - 769 q^{55} - 19494 q^{57} + 61744 q^{59} - 81581 q^{61} + 12069 q^{63} - 59686 q^{65} + 45756 q^{67} - 11106 q^{69} + 10416 q^{71} - 54615 q^{73} + 41301 q^{75} - 29515 q^{77} + 145594 q^{79} + 39366 q^{81} + 160548 q^{83} - 53947 q^{85} + 66204 q^{87} - 97728 q^{89} + 418294 q^{91} - 14688 q^{93} + 23465 q^{95} - 760 q^{97} + 16443 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 9.00000 0.577350
\(4\) 0 0
\(5\) −41.5770 −0.743753 −0.371876 0.928282i \(-0.621285\pi\)
−0.371876 + 0.928282i \(0.621285\pi\)
\(6\) 0 0
\(7\) −98.3255 −0.758439 −0.379220 0.925307i \(-0.623808\pi\)
−0.379220 + 0.925307i \(0.623808\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 628.684 1.56657 0.783287 0.621660i \(-0.213542\pi\)
0.783287 + 0.621660i \(0.213542\pi\)
\(12\) 0 0
\(13\) 154.682 0.253853 0.126927 0.991912i \(-0.459489\pi\)
0.126927 + 0.991912i \(0.459489\pi\)
\(14\) 0 0
\(15\) −374.193 −0.429406
\(16\) 0 0
\(17\) 1711.67 1.43648 0.718239 0.695797i \(-0.244948\pi\)
0.718239 + 0.695797i \(0.244948\pi\)
\(18\) 0 0
\(19\) −361.000 −0.229416
\(20\) 0 0
\(21\) −884.929 −0.437885
\(22\) 0 0
\(23\) −305.157 −0.120283 −0.0601415 0.998190i \(-0.519155\pi\)
−0.0601415 + 0.998190i \(0.519155\pi\)
\(24\) 0 0
\(25\) −1396.35 −0.446832
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) 3768.18 0.832025 0.416012 0.909359i \(-0.363427\pi\)
0.416012 + 0.909359i \(0.363427\pi\)
\(30\) 0 0
\(31\) −520.053 −0.0971948 −0.0485974 0.998818i \(-0.515475\pi\)
−0.0485974 + 0.998818i \(0.515475\pi\)
\(32\) 0 0
\(33\) 5658.16 0.904462
\(34\) 0 0
\(35\) 4088.08 0.564091
\(36\) 0 0
\(37\) −13114.9 −1.57493 −0.787465 0.616360i \(-0.788607\pi\)
−0.787465 + 0.616360i \(0.788607\pi\)
\(38\) 0 0
\(39\) 1392.14 0.146562
\(40\) 0 0
\(41\) 8264.15 0.767783 0.383892 0.923378i \(-0.374584\pi\)
0.383892 + 0.923378i \(0.374584\pi\)
\(42\) 0 0
\(43\) −6342.44 −0.523100 −0.261550 0.965190i \(-0.584234\pi\)
−0.261550 + 0.965190i \(0.584234\pi\)
\(44\) 0 0
\(45\) −3367.74 −0.247918
\(46\) 0 0
\(47\) −27630.9 −1.82453 −0.912264 0.409602i \(-0.865668\pi\)
−0.912264 + 0.409602i \(0.865668\pi\)
\(48\) 0 0
\(49\) −7139.10 −0.424770
\(50\) 0 0
\(51\) 15405.1 0.829351
\(52\) 0 0
\(53\) 15998.2 0.782316 0.391158 0.920324i \(-0.372075\pi\)
0.391158 + 0.920324i \(0.372075\pi\)
\(54\) 0 0
\(55\) −26138.8 −1.16514
\(56\) 0 0
\(57\) −3249.00 −0.132453
\(58\) 0 0
\(59\) −43506.7 −1.62715 −0.813573 0.581463i \(-0.802480\pi\)
−0.813573 + 0.581463i \(0.802480\pi\)
\(60\) 0 0
\(61\) −19745.5 −0.679427 −0.339713 0.940529i \(-0.610330\pi\)
−0.339713 + 0.940529i \(0.610330\pi\)
\(62\) 0 0
\(63\) −7964.36 −0.252813
\(64\) 0 0
\(65\) −6431.24 −0.188804
\(66\) 0 0
\(67\) 54996.7 1.49675 0.748375 0.663276i \(-0.230834\pi\)
0.748375 + 0.663276i \(0.230834\pi\)
\(68\) 0 0
\(69\) −2746.42 −0.0694454
\(70\) 0 0
\(71\) 76576.3 1.80280 0.901402 0.432984i \(-0.142539\pi\)
0.901402 + 0.432984i \(0.142539\pi\)
\(72\) 0 0
\(73\) −36253.1 −0.796229 −0.398115 0.917336i \(-0.630335\pi\)
−0.398115 + 0.917336i \(0.630335\pi\)
\(74\) 0 0
\(75\) −12567.1 −0.257978
\(76\) 0 0
\(77\) −61815.7 −1.18815
\(78\) 0 0
\(79\) 48683.6 0.877638 0.438819 0.898576i \(-0.355397\pi\)
0.438819 + 0.898576i \(0.355397\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 98174.2 1.56424 0.782118 0.623130i \(-0.214139\pi\)
0.782118 + 0.623130i \(0.214139\pi\)
\(84\) 0 0
\(85\) −71166.4 −1.06838
\(86\) 0 0
\(87\) 33913.6 0.480370
\(88\) 0 0
\(89\) 129077. 1.72732 0.863659 0.504077i \(-0.168167\pi\)
0.863659 + 0.504077i \(0.168167\pi\)
\(90\) 0 0
\(91\) −15209.2 −0.192532
\(92\) 0 0
\(93\) −4680.48 −0.0561155
\(94\) 0 0
\(95\) 15009.3 0.170629
\(96\) 0 0
\(97\) 94431.1 1.01903 0.509514 0.860463i \(-0.329825\pi\)
0.509514 + 0.860463i \(0.329825\pi\)
\(98\) 0 0
\(99\) 50923.4 0.522191
\(100\) 0 0
\(101\) −59818.6 −0.583489 −0.291745 0.956496i \(-0.594236\pi\)
−0.291745 + 0.956496i \(0.594236\pi\)
\(102\) 0 0
\(103\) 60466.9 0.561597 0.280799 0.959767i \(-0.409401\pi\)
0.280799 + 0.959767i \(0.409401\pi\)
\(104\) 0 0
\(105\) 36792.7 0.325678
\(106\) 0 0
\(107\) 207839. 1.75496 0.877482 0.479610i \(-0.159222\pi\)
0.877482 + 0.479610i \(0.159222\pi\)
\(108\) 0 0
\(109\) −86244.5 −0.695289 −0.347644 0.937626i \(-0.613018\pi\)
−0.347644 + 0.937626i \(0.613018\pi\)
\(110\) 0 0
\(111\) −118034. −0.909286
\(112\) 0 0
\(113\) −74924.9 −0.551989 −0.275994 0.961159i \(-0.589007\pi\)
−0.275994 + 0.961159i \(0.589007\pi\)
\(114\) 0 0
\(115\) 12687.5 0.0894608
\(116\) 0 0
\(117\) 12529.3 0.0846178
\(118\) 0 0
\(119\) −168301. −1.08948
\(120\) 0 0
\(121\) 234193. 1.45415
\(122\) 0 0
\(123\) 74377.4 0.443280
\(124\) 0 0
\(125\) 187984. 1.07609
\(126\) 0 0
\(127\) 300875. 1.65530 0.827649 0.561246i \(-0.189678\pi\)
0.827649 + 0.561246i \(0.189678\pi\)
\(128\) 0 0
\(129\) −57081.9 −0.302012
\(130\) 0 0
\(131\) 288980. 1.47126 0.735629 0.677385i \(-0.236887\pi\)
0.735629 + 0.677385i \(0.236887\pi\)
\(132\) 0 0
\(133\) 35495.5 0.173998
\(134\) 0 0
\(135\) −30309.7 −0.143135
\(136\) 0 0
\(137\) 139752. 0.636145 0.318072 0.948066i \(-0.396965\pi\)
0.318072 + 0.948066i \(0.396965\pi\)
\(138\) 0 0
\(139\) 209297. 0.918810 0.459405 0.888227i \(-0.348063\pi\)
0.459405 + 0.888227i \(0.348063\pi\)
\(140\) 0 0
\(141\) −248678. −1.05339
\(142\) 0 0
\(143\) 97246.5 0.397680
\(144\) 0 0
\(145\) −156670. −0.618821
\(146\) 0 0
\(147\) −64251.9 −0.245241
\(148\) 0 0
\(149\) −377316. −1.39232 −0.696160 0.717887i \(-0.745110\pi\)
−0.696160 + 0.717887i \(0.745110\pi\)
\(150\) 0 0
\(151\) 388013. 1.38485 0.692427 0.721488i \(-0.256541\pi\)
0.692427 + 0.721488i \(0.256541\pi\)
\(152\) 0 0
\(153\) 138646. 0.478826
\(154\) 0 0
\(155\) 21622.3 0.0722889
\(156\) 0 0
\(157\) 193450. 0.626353 0.313176 0.949695i \(-0.398607\pi\)
0.313176 + 0.949695i \(0.398607\pi\)
\(158\) 0 0
\(159\) 143984. 0.451670
\(160\) 0 0
\(161\) 30004.7 0.0912274
\(162\) 0 0
\(163\) −103565. −0.305313 −0.152657 0.988279i \(-0.548783\pi\)
−0.152657 + 0.988279i \(0.548783\pi\)
\(164\) 0 0
\(165\) −235250. −0.672696
\(166\) 0 0
\(167\) −30833.1 −0.0855513 −0.0427757 0.999085i \(-0.513620\pi\)
−0.0427757 + 0.999085i \(0.513620\pi\)
\(168\) 0 0
\(169\) −347366. −0.935559
\(170\) 0 0
\(171\) −29241.0 −0.0764719
\(172\) 0 0
\(173\) −135090. −0.343169 −0.171584 0.985169i \(-0.554889\pi\)
−0.171584 + 0.985169i \(0.554889\pi\)
\(174\) 0 0
\(175\) 137297. 0.338895
\(176\) 0 0
\(177\) −391560. −0.939433
\(178\) 0 0
\(179\) −451653. −1.05359 −0.526796 0.849992i \(-0.676607\pi\)
−0.526796 + 0.849992i \(0.676607\pi\)
\(180\) 0 0
\(181\) 232211. 0.526849 0.263425 0.964680i \(-0.415148\pi\)
0.263425 + 0.964680i \(0.415148\pi\)
\(182\) 0 0
\(183\) −177709. −0.392267
\(184\) 0 0
\(185\) 545279. 1.17136
\(186\) 0 0
\(187\) 1.07610e6 2.25035
\(188\) 0 0
\(189\) −71679.3 −0.145962
\(190\) 0 0
\(191\) 319410. 0.633527 0.316763 0.948505i \(-0.397404\pi\)
0.316763 + 0.948505i \(0.397404\pi\)
\(192\) 0 0
\(193\) −151565. −0.292891 −0.146446 0.989219i \(-0.546783\pi\)
−0.146446 + 0.989219i \(0.546783\pi\)
\(194\) 0 0
\(195\) −57881.2 −0.109006
\(196\) 0 0
\(197\) −961377. −1.76493 −0.882466 0.470376i \(-0.844118\pi\)
−0.882466 + 0.470376i \(0.844118\pi\)
\(198\) 0 0
\(199\) −493703. −0.883757 −0.441879 0.897075i \(-0.645688\pi\)
−0.441879 + 0.897075i \(0.645688\pi\)
\(200\) 0 0
\(201\) 494970. 0.864149
\(202\) 0 0
\(203\) −370508. −0.631040
\(204\) 0 0
\(205\) −343599. −0.571041
\(206\) 0 0
\(207\) −24717.7 −0.0400943
\(208\) 0 0
\(209\) −226955. −0.359397
\(210\) 0 0
\(211\) −835416. −1.29180 −0.645902 0.763420i \(-0.723519\pi\)
−0.645902 + 0.763420i \(0.723519\pi\)
\(212\) 0 0
\(213\) 689187. 1.04085
\(214\) 0 0
\(215\) 263700. 0.389057
\(216\) 0 0
\(217\) 51134.4 0.0737164
\(218\) 0 0
\(219\) −326278. −0.459703
\(220\) 0 0
\(221\) 264766. 0.364655
\(222\) 0 0
\(223\) 574648. 0.773819 0.386910 0.922118i \(-0.373543\pi\)
0.386910 + 0.922118i \(0.373543\pi\)
\(224\) 0 0
\(225\) −113104. −0.148944
\(226\) 0 0
\(227\) 1.29201e6 1.66419 0.832094 0.554635i \(-0.187142\pi\)
0.832094 + 0.554635i \(0.187142\pi\)
\(228\) 0 0
\(229\) 1.36662e6 1.72210 0.861050 0.508520i \(-0.169807\pi\)
0.861050 + 0.508520i \(0.169807\pi\)
\(230\) 0 0
\(231\) −556341. −0.685980
\(232\) 0 0
\(233\) 941491. 1.13613 0.568063 0.822985i \(-0.307693\pi\)
0.568063 + 0.822985i \(0.307693\pi\)
\(234\) 0 0
\(235\) 1.14881e6 1.35700
\(236\) 0 0
\(237\) 438153. 0.506704
\(238\) 0 0
\(239\) 87383.8 0.0989547 0.0494773 0.998775i \(-0.484244\pi\)
0.0494773 + 0.998775i \(0.484244\pi\)
\(240\) 0 0
\(241\) 171635. 0.190355 0.0951775 0.995460i \(-0.469658\pi\)
0.0951775 + 0.995460i \(0.469658\pi\)
\(242\) 0 0
\(243\) 59049.0 0.0641500
\(244\) 0 0
\(245\) 296823. 0.315924
\(246\) 0 0
\(247\) −55840.4 −0.0582379
\(248\) 0 0
\(249\) 883568. 0.903112
\(250\) 0 0
\(251\) 459131. 0.459994 0.229997 0.973191i \(-0.426128\pi\)
0.229997 + 0.973191i \(0.426128\pi\)
\(252\) 0 0
\(253\) −191848. −0.188432
\(254\) 0 0
\(255\) −640497. −0.616832
\(256\) 0 0
\(257\) −1.35236e6 −1.27720 −0.638600 0.769538i \(-0.720486\pi\)
−0.638600 + 0.769538i \(0.720486\pi\)
\(258\) 0 0
\(259\) 1.28953e6 1.19449
\(260\) 0 0
\(261\) 305222. 0.277342
\(262\) 0 0
\(263\) −1.48689e6 −1.32553 −0.662766 0.748826i \(-0.730618\pi\)
−0.662766 + 0.748826i \(0.730618\pi\)
\(264\) 0 0
\(265\) −665159. −0.581850
\(266\) 0 0
\(267\) 1.16169e6 0.997267
\(268\) 0 0
\(269\) 266979. 0.224955 0.112478 0.993654i \(-0.464121\pi\)
0.112478 + 0.993654i \(0.464121\pi\)
\(270\) 0 0
\(271\) −1.10385e6 −0.913036 −0.456518 0.889714i \(-0.650904\pi\)
−0.456518 + 0.889714i \(0.650904\pi\)
\(272\) 0 0
\(273\) −136883. −0.111159
\(274\) 0 0
\(275\) −877863. −0.699995
\(276\) 0 0
\(277\) 162944. 0.127597 0.0637983 0.997963i \(-0.479679\pi\)
0.0637983 + 0.997963i \(0.479679\pi\)
\(278\) 0 0
\(279\) −42124.3 −0.0323983
\(280\) 0 0
\(281\) 428764. 0.323931 0.161966 0.986796i \(-0.448217\pi\)
0.161966 + 0.986796i \(0.448217\pi\)
\(282\) 0 0
\(283\) 397001. 0.294663 0.147331 0.989087i \(-0.452932\pi\)
0.147331 + 0.989087i \(0.452932\pi\)
\(284\) 0 0
\(285\) 135084. 0.0985125
\(286\) 0 0
\(287\) −812576. −0.582317
\(288\) 0 0
\(289\) 1.50997e6 1.06347
\(290\) 0 0
\(291\) 849880. 0.588336
\(292\) 0 0
\(293\) 1.80359e6 1.22735 0.613674 0.789559i \(-0.289691\pi\)
0.613674 + 0.789559i \(0.289691\pi\)
\(294\) 0 0
\(295\) 1.80888e6 1.21019
\(296\) 0 0
\(297\) 458311. 0.301487
\(298\) 0 0
\(299\) −47202.5 −0.0305342
\(300\) 0 0
\(301\) 623623. 0.396740
\(302\) 0 0
\(303\) −538367. −0.336878
\(304\) 0 0
\(305\) 820958. 0.505326
\(306\) 0 0
\(307\) 635443. 0.384796 0.192398 0.981317i \(-0.438374\pi\)
0.192398 + 0.981317i \(0.438374\pi\)
\(308\) 0 0
\(309\) 544202. 0.324238
\(310\) 0 0
\(311\) −2.28396e6 −1.33902 −0.669512 0.742802i \(-0.733497\pi\)
−0.669512 + 0.742802i \(0.733497\pi\)
\(312\) 0 0
\(313\) −2.91541e6 −1.68205 −0.841024 0.540998i \(-0.818046\pi\)
−0.841024 + 0.540998i \(0.818046\pi\)
\(314\) 0 0
\(315\) 331135. 0.188030
\(316\) 0 0
\(317\) −876417. −0.489850 −0.244925 0.969542i \(-0.578763\pi\)
−0.244925 + 0.969542i \(0.578763\pi\)
\(318\) 0 0
\(319\) 2.36899e6 1.30343
\(320\) 0 0
\(321\) 1.87055e6 1.01323
\(322\) 0 0
\(323\) −617915. −0.329551
\(324\) 0 0
\(325\) −215991. −0.113430
\(326\) 0 0
\(327\) −776200. −0.401425
\(328\) 0 0
\(329\) 2.71682e6 1.38379
\(330\) 0 0
\(331\) −1.53007e6 −0.767614 −0.383807 0.923413i \(-0.625387\pi\)
−0.383807 + 0.923413i \(0.625387\pi\)
\(332\) 0 0
\(333\) −1.06231e6 −0.524976
\(334\) 0 0
\(335\) −2.28660e6 −1.11321
\(336\) 0 0
\(337\) −905811. −0.434473 −0.217237 0.976119i \(-0.569704\pi\)
−0.217237 + 0.976119i \(0.569704\pi\)
\(338\) 0 0
\(339\) −674324. −0.318691
\(340\) 0 0
\(341\) −326949. −0.152263
\(342\) 0 0
\(343\) 2.35451e6 1.08060
\(344\) 0 0
\(345\) 114188. 0.0516502
\(346\) 0 0
\(347\) 2.43581e6 1.08597 0.542987 0.839741i \(-0.317293\pi\)
0.542987 + 0.839741i \(0.317293\pi\)
\(348\) 0 0
\(349\) −2.48735e6 −1.09314 −0.546568 0.837415i \(-0.684066\pi\)
−0.546568 + 0.837415i \(0.684066\pi\)
\(350\) 0 0
\(351\) 112764. 0.0488541
\(352\) 0 0
\(353\) 3.35659e6 1.43371 0.716855 0.697222i \(-0.245581\pi\)
0.716855 + 0.697222i \(0.245581\pi\)
\(354\) 0 0
\(355\) −3.18382e6 −1.34084
\(356\) 0 0
\(357\) −1.51471e6 −0.629012
\(358\) 0 0
\(359\) 2.31960e6 0.949897 0.474948 0.880014i \(-0.342467\pi\)
0.474948 + 0.880014i \(0.342467\pi\)
\(360\) 0 0
\(361\) 130321. 0.0526316
\(362\) 0 0
\(363\) 2.10774e6 0.839556
\(364\) 0 0
\(365\) 1.50730e6 0.592198
\(366\) 0 0
\(367\) 4.82122e6 1.86850 0.934248 0.356625i \(-0.116073\pi\)
0.934248 + 0.356625i \(0.116073\pi\)
\(368\) 0 0
\(369\) 669396. 0.255928
\(370\) 0 0
\(371\) −1.57303e6 −0.593339
\(372\) 0 0
\(373\) 2.68655e6 0.999824 0.499912 0.866076i \(-0.333366\pi\)
0.499912 + 0.866076i \(0.333366\pi\)
\(374\) 0 0
\(375\) 1.69186e6 0.621278
\(376\) 0 0
\(377\) 582871. 0.211212
\(378\) 0 0
\(379\) −412645. −0.147563 −0.0737816 0.997274i \(-0.523507\pi\)
−0.0737816 + 0.997274i \(0.523507\pi\)
\(380\) 0 0
\(381\) 2.70787e6 0.955687
\(382\) 0 0
\(383\) 4.23505e6 1.47524 0.737619 0.675217i \(-0.235950\pi\)
0.737619 + 0.675217i \(0.235950\pi\)
\(384\) 0 0
\(385\) 2.57011e6 0.883691
\(386\) 0 0
\(387\) −513737. −0.174367
\(388\) 0 0
\(389\) 1.46665e6 0.491421 0.245711 0.969343i \(-0.420979\pi\)
0.245711 + 0.969343i \(0.420979\pi\)
\(390\) 0 0
\(391\) −522330. −0.172784
\(392\) 0 0
\(393\) 2.60082e6 0.849431
\(394\) 0 0
\(395\) −2.02412e6 −0.652745
\(396\) 0 0
\(397\) 3.48279e6 1.10905 0.554525 0.832167i \(-0.312900\pi\)
0.554525 + 0.832167i \(0.312900\pi\)
\(398\) 0 0
\(399\) 319459. 0.100458
\(400\) 0 0
\(401\) 1.82392e6 0.566429 0.283215 0.959057i \(-0.408599\pi\)
0.283215 + 0.959057i \(0.408599\pi\)
\(402\) 0 0
\(403\) −80443.1 −0.0246732
\(404\) 0 0
\(405\) −272787. −0.0826392
\(406\) 0 0
\(407\) −8.24514e6 −2.46724
\(408\) 0 0
\(409\) 4.64731e6 1.37371 0.686853 0.726797i \(-0.258992\pi\)
0.686853 + 0.726797i \(0.258992\pi\)
\(410\) 0 0
\(411\) 1.25777e6 0.367278
\(412\) 0 0
\(413\) 4.27782e6 1.23409
\(414\) 0 0
\(415\) −4.08179e6 −1.16340
\(416\) 0 0
\(417\) 1.88367e6 0.530475
\(418\) 0 0
\(419\) −794597. −0.221112 −0.110556 0.993870i \(-0.535263\pi\)
−0.110556 + 0.993870i \(0.535263\pi\)
\(420\) 0 0
\(421\) −3.05842e6 −0.840991 −0.420496 0.907295i \(-0.638144\pi\)
−0.420496 + 0.907295i \(0.638144\pi\)
\(422\) 0 0
\(423\) −2.23810e6 −0.608176
\(424\) 0 0
\(425\) −2.39010e6 −0.641864
\(426\) 0 0
\(427\) 1.94148e6 0.515304
\(428\) 0 0
\(429\) 875218. 0.229601
\(430\) 0 0
\(431\) 6.13429e6 1.59064 0.795318 0.606192i \(-0.207304\pi\)
0.795318 + 0.606192i \(0.207304\pi\)
\(432\) 0 0
\(433\) −2.92568e6 −0.749907 −0.374953 0.927044i \(-0.622341\pi\)
−0.374953 + 0.927044i \(0.622341\pi\)
\(434\) 0 0
\(435\) −1.41003e6 −0.357276
\(436\) 0 0
\(437\) 110162. 0.0275948
\(438\) 0 0
\(439\) −1.26416e6 −0.313068 −0.156534 0.987673i \(-0.550032\pi\)
−0.156534 + 0.987673i \(0.550032\pi\)
\(440\) 0 0
\(441\) −578267. −0.141590
\(442\) 0 0
\(443\) −4.51722e6 −1.09361 −0.546804 0.837261i \(-0.684156\pi\)
−0.546804 + 0.837261i \(0.684156\pi\)
\(444\) 0 0
\(445\) −5.36662e6 −1.28470
\(446\) 0 0
\(447\) −3.39584e6 −0.803856
\(448\) 0 0
\(449\) 3.59502e6 0.841562 0.420781 0.907162i \(-0.361756\pi\)
0.420781 + 0.907162i \(0.361756\pi\)
\(450\) 0 0
\(451\) 5.19554e6 1.20279
\(452\) 0 0
\(453\) 3.49212e6 0.799546
\(454\) 0 0
\(455\) 632355. 0.143196
\(456\) 0 0
\(457\) 2.48753e6 0.557158 0.278579 0.960413i \(-0.410137\pi\)
0.278579 + 0.960413i \(0.410137\pi\)
\(458\) 0 0
\(459\) 1.24781e6 0.276450
\(460\) 0 0
\(461\) −823421. −0.180455 −0.0902276 0.995921i \(-0.528759\pi\)
−0.0902276 + 0.995921i \(0.528759\pi\)
\(462\) 0 0
\(463\) −565940. −0.122693 −0.0613463 0.998117i \(-0.519539\pi\)
−0.0613463 + 0.998117i \(0.519539\pi\)
\(464\) 0 0
\(465\) 194600. 0.0417360
\(466\) 0 0
\(467\) −3.74802e6 −0.795260 −0.397630 0.917546i \(-0.630167\pi\)
−0.397630 + 0.917546i \(0.630167\pi\)
\(468\) 0 0
\(469\) −5.40757e6 −1.13519
\(470\) 0 0
\(471\) 1.74105e6 0.361625
\(472\) 0 0
\(473\) −3.98739e6 −0.819476
\(474\) 0 0
\(475\) 504082. 0.102510
\(476\) 0 0
\(477\) 1.29586e6 0.260772
\(478\) 0 0
\(479\) −5.42311e6 −1.07997 −0.539983 0.841676i \(-0.681569\pi\)
−0.539983 + 0.841676i \(0.681569\pi\)
\(480\) 0 0
\(481\) −2.02865e6 −0.399801
\(482\) 0 0
\(483\) 270043. 0.0526701
\(484\) 0 0
\(485\) −3.92617e6 −0.757904
\(486\) 0 0
\(487\) 185063. 0.0353587 0.0176794 0.999844i \(-0.494372\pi\)
0.0176794 + 0.999844i \(0.494372\pi\)
\(488\) 0 0
\(489\) −932088. −0.176273
\(490\) 0 0
\(491\) 7.49989e6 1.40395 0.701974 0.712203i \(-0.252302\pi\)
0.701974 + 0.712203i \(0.252302\pi\)
\(492\) 0 0
\(493\) 6.44989e6 1.19519
\(494\) 0 0
\(495\) −2.11725e6 −0.388381
\(496\) 0 0
\(497\) −7.52940e6 −1.36732
\(498\) 0 0
\(499\) 8.01308e6 1.44062 0.720308 0.693655i \(-0.244001\pi\)
0.720308 + 0.693655i \(0.244001\pi\)
\(500\) 0 0
\(501\) −277498. −0.0493931
\(502\) 0 0
\(503\) 9.11857e6 1.60697 0.803483 0.595327i \(-0.202978\pi\)
0.803483 + 0.595327i \(0.202978\pi\)
\(504\) 0 0
\(505\) 2.48708e6 0.433972
\(506\) 0 0
\(507\) −3.12630e6 −0.540145
\(508\) 0 0
\(509\) −4.81031e6 −0.822961 −0.411480 0.911419i \(-0.634988\pi\)
−0.411480 + 0.911419i \(0.634988\pi\)
\(510\) 0 0
\(511\) 3.56460e6 0.603892
\(512\) 0 0
\(513\) −263169. −0.0441511
\(514\) 0 0
\(515\) −2.51404e6 −0.417689
\(516\) 0 0
\(517\) −1.73711e7 −2.85826
\(518\) 0 0
\(519\) −1.21581e6 −0.198129
\(520\) 0 0
\(521\) 1.58430e6 0.255708 0.127854 0.991793i \(-0.459191\pi\)
0.127854 + 0.991793i \(0.459191\pi\)
\(522\) 0 0
\(523\) 2.68629e6 0.429436 0.214718 0.976676i \(-0.431117\pi\)
0.214718 + 0.976676i \(0.431117\pi\)
\(524\) 0 0
\(525\) 1.23567e6 0.195661
\(526\) 0 0
\(527\) −890161. −0.139618
\(528\) 0 0
\(529\) −6.34322e6 −0.985532
\(530\) 0 0
\(531\) −3.52404e6 −0.542382
\(532\) 0 0
\(533\) 1.27832e6 0.194904
\(534\) 0 0
\(535\) −8.64134e6 −1.30526
\(536\) 0 0
\(537\) −4.06488e6 −0.608292
\(538\) 0 0
\(539\) −4.48824e6 −0.665433
\(540\) 0 0
\(541\) −3.92760e6 −0.576945 −0.288473 0.957488i \(-0.593147\pi\)
−0.288473 + 0.957488i \(0.593147\pi\)
\(542\) 0 0
\(543\) 2.08990e6 0.304177
\(544\) 0 0
\(545\) 3.58579e6 0.517123
\(546\) 0 0
\(547\) 4.18962e6 0.598696 0.299348 0.954144i \(-0.403231\pi\)
0.299348 + 0.954144i \(0.403231\pi\)
\(548\) 0 0
\(549\) −1.59938e6 −0.226476
\(550\) 0 0
\(551\) −1.36031e6 −0.190880
\(552\) 0 0
\(553\) −4.78684e6 −0.665635
\(554\) 0 0
\(555\) 4.90751e6 0.676284
\(556\) 0 0
\(557\) 575462. 0.0785920 0.0392960 0.999228i \(-0.487488\pi\)
0.0392960 + 0.999228i \(0.487488\pi\)
\(558\) 0 0
\(559\) −981064. −0.132791
\(560\) 0 0
\(561\) 9.68493e6 1.29924
\(562\) 0 0
\(563\) −9.52296e6 −1.26620 −0.633098 0.774072i \(-0.718217\pi\)
−0.633098 + 0.774072i \(0.718217\pi\)
\(564\) 0 0
\(565\) 3.11516e6 0.410543
\(566\) 0 0
\(567\) −645113. −0.0842710
\(568\) 0 0
\(569\) −2.23080e6 −0.288855 −0.144427 0.989515i \(-0.546134\pi\)
−0.144427 + 0.989515i \(0.546134\pi\)
\(570\) 0 0
\(571\) 1.06169e7 1.36273 0.681364 0.731944i \(-0.261387\pi\)
0.681364 + 0.731944i \(0.261387\pi\)
\(572\) 0 0
\(573\) 2.87469e6 0.365767
\(574\) 0 0
\(575\) 426106. 0.0537463
\(576\) 0 0
\(577\) −4.22222e6 −0.527961 −0.263980 0.964528i \(-0.585035\pi\)
−0.263980 + 0.964528i \(0.585035\pi\)
\(578\) 0 0
\(579\) −1.36409e6 −0.169101
\(580\) 0 0
\(581\) −9.65303e6 −1.18638
\(582\) 0 0
\(583\) 1.00578e7 1.22556
\(584\) 0 0
\(585\) −520930. −0.0629347
\(586\) 0 0
\(587\) 6.01423e6 0.720419 0.360209 0.932871i \(-0.382705\pi\)
0.360209 + 0.932871i \(0.382705\pi\)
\(588\) 0 0
\(589\) 187739. 0.0222980
\(590\) 0 0
\(591\) −8.65239e6 −1.01898
\(592\) 0 0
\(593\) −7.55743e6 −0.882546 −0.441273 0.897373i \(-0.645473\pi\)
−0.441273 + 0.897373i \(0.645473\pi\)
\(594\) 0 0
\(595\) 6.99747e6 0.810305
\(596\) 0 0
\(597\) −4.44333e6 −0.510237
\(598\) 0 0
\(599\) −8.94017e6 −1.01807 −0.509036 0.860745i \(-0.669998\pi\)
−0.509036 + 0.860745i \(0.669998\pi\)
\(600\) 0 0
\(601\) 1.20808e7 1.36430 0.682148 0.731214i \(-0.261046\pi\)
0.682148 + 0.731214i \(0.261046\pi\)
\(602\) 0 0
\(603\) 4.45473e6 0.498917
\(604\) 0 0
\(605\) −9.73705e6 −1.08153
\(606\) 0 0
\(607\) 3.46693e6 0.381921 0.190960 0.981598i \(-0.438840\pi\)
0.190960 + 0.981598i \(0.438840\pi\)
\(608\) 0 0
\(609\) −3.33457e6 −0.364331
\(610\) 0 0
\(611\) −4.27402e6 −0.463163
\(612\) 0 0
\(613\) 9.87062e6 1.06095 0.530473 0.847702i \(-0.322014\pi\)
0.530473 + 0.847702i \(0.322014\pi\)
\(614\) 0 0
\(615\) −3.09239e6 −0.329691
\(616\) 0 0
\(617\) −4.87481e6 −0.515519 −0.257759 0.966209i \(-0.582984\pi\)
−0.257759 + 0.966209i \(0.582984\pi\)
\(618\) 0 0
\(619\) 1.41975e6 0.148931 0.0744657 0.997224i \(-0.476275\pi\)
0.0744657 + 0.997224i \(0.476275\pi\)
\(620\) 0 0
\(621\) −222460. −0.0231485
\(622\) 0 0
\(623\) −1.26915e7 −1.31007
\(624\) 0 0
\(625\) −3.45224e6 −0.353510
\(626\) 0 0
\(627\) −2.04260e6 −0.207498
\(628\) 0 0
\(629\) −2.24485e7 −2.26235
\(630\) 0 0
\(631\) 8.49724e6 0.849580 0.424790 0.905292i \(-0.360348\pi\)
0.424790 + 0.905292i \(0.360348\pi\)
\(632\) 0 0
\(633\) −7.51875e6 −0.745824
\(634\) 0 0
\(635\) −1.25095e7 −1.23113
\(636\) 0 0
\(637\) −1.10429e6 −0.107829
\(638\) 0 0
\(639\) 6.20268e6 0.600935
\(640\) 0 0
\(641\) −1.70919e7 −1.64302 −0.821512 0.570191i \(-0.806869\pi\)
−0.821512 + 0.570191i \(0.806869\pi\)
\(642\) 0 0
\(643\) −9.06433e6 −0.864586 −0.432293 0.901733i \(-0.642295\pi\)
−0.432293 + 0.901733i \(0.642295\pi\)
\(644\) 0 0
\(645\) 2.37330e6 0.224622
\(646\) 0 0
\(647\) 2.66878e6 0.250641 0.125321 0.992116i \(-0.460004\pi\)
0.125321 + 0.992116i \(0.460004\pi\)
\(648\) 0 0
\(649\) −2.73520e7 −2.54904
\(650\) 0 0
\(651\) 460210. 0.0425602
\(652\) 0 0
\(653\) 6.42272e6 0.589435 0.294717 0.955584i \(-0.404774\pi\)
0.294717 + 0.955584i \(0.404774\pi\)
\(654\) 0 0
\(655\) −1.20149e7 −1.09425
\(656\) 0 0
\(657\) −2.93650e6 −0.265410
\(658\) 0 0
\(659\) 1.64838e7 1.47857 0.739286 0.673391i \(-0.235163\pi\)
0.739286 + 0.673391i \(0.235163\pi\)
\(660\) 0 0
\(661\) 9.28585e6 0.826643 0.413322 0.910585i \(-0.364369\pi\)
0.413322 + 0.910585i \(0.364369\pi\)
\(662\) 0 0
\(663\) 2.38289e6 0.210533
\(664\) 0 0
\(665\) −1.47580e6 −0.129411
\(666\) 0 0
\(667\) −1.14989e6 −0.100078
\(668\) 0 0
\(669\) 5.17183e6 0.446765
\(670\) 0 0
\(671\) −1.24137e7 −1.06437
\(672\) 0 0
\(673\) −1.92940e7 −1.64204 −0.821020 0.570900i \(-0.806594\pi\)
−0.821020 + 0.570900i \(0.806594\pi\)
\(674\) 0 0
\(675\) −1.01794e6 −0.0859928
\(676\) 0 0
\(677\) 1.88174e7 1.57793 0.788965 0.614439i \(-0.210617\pi\)
0.788965 + 0.614439i \(0.210617\pi\)
\(678\) 0 0
\(679\) −9.28498e6 −0.772870
\(680\) 0 0
\(681\) 1.16281e7 0.960819
\(682\) 0 0
\(683\) 1.19903e6 0.0983507 0.0491754 0.998790i \(-0.484341\pi\)
0.0491754 + 0.998790i \(0.484341\pi\)
\(684\) 0 0
\(685\) −5.81047e6 −0.473134
\(686\) 0 0
\(687\) 1.22996e7 0.994255
\(688\) 0 0
\(689\) 2.47464e6 0.198593
\(690\) 0 0
\(691\) −1.18698e7 −0.945688 −0.472844 0.881146i \(-0.656773\pi\)
−0.472844 + 0.881146i \(0.656773\pi\)
\(692\) 0 0
\(693\) −5.00707e6 −0.396050
\(694\) 0 0
\(695\) −8.70195e6 −0.683368
\(696\) 0 0
\(697\) 1.41455e7 1.10290
\(698\) 0 0
\(699\) 8.47342e6 0.655942
\(700\) 0 0
\(701\) −7.63069e6 −0.586501 −0.293250 0.956036i \(-0.594737\pi\)
−0.293250 + 0.956036i \(0.594737\pi\)
\(702\) 0 0
\(703\) 4.73448e6 0.361314
\(704\) 0 0
\(705\) 1.03393e7 0.783463
\(706\) 0 0
\(707\) 5.88169e6 0.442541
\(708\) 0 0
\(709\) −1.31856e7 −0.985109 −0.492555 0.870282i \(-0.663937\pi\)
−0.492555 + 0.870282i \(0.663937\pi\)
\(710\) 0 0
\(711\) 3.94337e6 0.292546
\(712\) 0 0
\(713\) 158698. 0.0116909
\(714\) 0 0
\(715\) −4.04322e6 −0.295776
\(716\) 0 0
\(717\) 786455. 0.0571315
\(718\) 0 0
\(719\) −6.45341e6 −0.465550 −0.232775 0.972531i \(-0.574781\pi\)
−0.232775 + 0.972531i \(0.574781\pi\)
\(720\) 0 0
\(721\) −5.94544e6 −0.425937
\(722\) 0 0
\(723\) 1.54472e6 0.109902
\(724\) 0 0
\(725\) −5.26169e6 −0.371775
\(726\) 0 0
\(727\) 3.06969e6 0.215406 0.107703 0.994183i \(-0.465650\pi\)
0.107703 + 0.994183i \(0.465650\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.08562e7 −0.751422
\(732\) 0 0
\(733\) 9.72176e6 0.668321 0.334160 0.942516i \(-0.391547\pi\)
0.334160 + 0.942516i \(0.391547\pi\)
\(734\) 0 0
\(735\) 2.67141e6 0.182399
\(736\) 0 0
\(737\) 3.45755e7 2.34477
\(738\) 0 0
\(739\) 1.22412e7 0.824539 0.412270 0.911062i \(-0.364736\pi\)
0.412270 + 0.911062i \(0.364736\pi\)
\(740\) 0 0
\(741\) −502563. −0.0336237
\(742\) 0 0
\(743\) 8.20715e6 0.545407 0.272703 0.962098i \(-0.412082\pi\)
0.272703 + 0.962098i \(0.412082\pi\)
\(744\) 0 0
\(745\) 1.56877e7 1.03554
\(746\) 0 0
\(747\) 7.95211e6 0.521412
\(748\) 0 0
\(749\) −2.04359e7 −1.33103
\(750\) 0 0
\(751\) −2.79527e7 −1.80852 −0.904262 0.426978i \(-0.859578\pi\)
−0.904262 + 0.426978i \(0.859578\pi\)
\(752\) 0 0
\(753\) 4.13218e6 0.265578
\(754\) 0 0
\(755\) −1.61324e7 −1.02999
\(756\) 0 0
\(757\) 1.14432e7 0.725786 0.362893 0.931831i \(-0.381789\pi\)
0.362893 + 0.931831i \(0.381789\pi\)
\(758\) 0 0
\(759\) −1.72663e6 −0.108791
\(760\) 0 0
\(761\) −1.37450e7 −0.860366 −0.430183 0.902742i \(-0.641551\pi\)
−0.430183 + 0.902742i \(0.641551\pi\)
\(762\) 0 0
\(763\) 8.48003e6 0.527334
\(764\) 0 0
\(765\) −5.76448e6 −0.356128
\(766\) 0 0
\(767\) −6.72973e6 −0.413056
\(768\) 0 0
\(769\) 721713. 0.0440098 0.0220049 0.999758i \(-0.492995\pi\)
0.0220049 + 0.999758i \(0.492995\pi\)
\(770\) 0 0
\(771\) −1.21712e7 −0.737392
\(772\) 0 0
\(773\) −2.34603e7 −1.41216 −0.706081 0.708131i \(-0.749539\pi\)
−0.706081 + 0.708131i \(0.749539\pi\)
\(774\) 0 0
\(775\) 726175. 0.0434297
\(776\) 0 0
\(777\) 1.16058e7 0.689638
\(778\) 0 0
\(779\) −2.98336e6 −0.176142
\(780\) 0 0
\(781\) 4.81423e7 2.82423
\(782\) 0 0
\(783\) 2.74700e6 0.160123
\(784\) 0 0
\(785\) −8.04307e6 −0.465852
\(786\) 0 0
\(787\) 2.21725e7 1.27608 0.638041 0.770003i \(-0.279745\pi\)
0.638041 + 0.770003i \(0.279745\pi\)
\(788\) 0 0
\(789\) −1.33820e7 −0.765296
\(790\) 0 0
\(791\) 7.36703e6 0.418650
\(792\) 0 0
\(793\) −3.05428e6 −0.172475
\(794\) 0 0
\(795\) −5.98643e6 −0.335931
\(796\) 0 0
\(797\) 1.35348e7 0.754753 0.377376 0.926060i \(-0.376826\pi\)
0.377376 + 0.926060i \(0.376826\pi\)
\(798\) 0 0
\(799\) −4.72952e7 −2.62090
\(800\) 0 0
\(801\) 1.04552e7 0.575773
\(802\) 0 0
\(803\) −2.27918e7 −1.24735
\(804\) 0 0
\(805\) −1.24751e6 −0.0678506
\(806\) 0 0
\(807\) 2.40281e6 0.129878
\(808\) 0 0
\(809\) −2.83463e7 −1.52274 −0.761370 0.648318i \(-0.775473\pi\)
−0.761370 + 0.648318i \(0.775473\pi\)
\(810\) 0 0
\(811\) 2.24175e7 1.19684 0.598418 0.801184i \(-0.295796\pi\)
0.598418 + 0.801184i \(0.295796\pi\)
\(812\) 0 0
\(813\) −9.93468e6 −0.527142
\(814\) 0 0
\(815\) 4.30594e6 0.227078
\(816\) 0 0
\(817\) 2.28962e6 0.120007
\(818\) 0 0
\(819\) −1.23195e6 −0.0641774
\(820\) 0 0
\(821\) −1.83074e7 −0.947915 −0.473957 0.880548i \(-0.657175\pi\)
−0.473957 + 0.880548i \(0.657175\pi\)
\(822\) 0 0
\(823\) 9.89646e6 0.509308 0.254654 0.967032i \(-0.418038\pi\)
0.254654 + 0.967032i \(0.418038\pi\)
\(824\) 0 0
\(825\) −7.90077e6 −0.404142
\(826\) 0 0
\(827\) 1.25001e7 0.635548 0.317774 0.948166i \(-0.397065\pi\)
0.317774 + 0.948166i \(0.397065\pi\)
\(828\) 0 0
\(829\) −1.39087e7 −0.702911 −0.351456 0.936205i \(-0.614313\pi\)
−0.351456 + 0.936205i \(0.614313\pi\)
\(830\) 0 0
\(831\) 1.46650e6 0.0736679
\(832\) 0 0
\(833\) −1.22198e7 −0.610172
\(834\) 0 0
\(835\) 1.28195e6 0.0636290
\(836\) 0 0
\(837\) −379119. −0.0187052
\(838\) 0 0
\(839\) −6.61780e6 −0.324570 −0.162285 0.986744i \(-0.551886\pi\)
−0.162285 + 0.986744i \(0.551886\pi\)
\(840\) 0 0
\(841\) −6.31199e6 −0.307734
\(842\) 0 0
\(843\) 3.85888e6 0.187022
\(844\) 0 0
\(845\) 1.44425e7 0.695824
\(846\) 0 0
\(847\) −2.30271e7 −1.10289
\(848\) 0 0
\(849\) 3.57300e6 0.170124
\(850\) 0 0
\(851\) 4.00211e6 0.189437
\(852\) 0 0
\(853\) 3.18740e7 1.49990 0.749952 0.661492i \(-0.230076\pi\)
0.749952 + 0.661492i \(0.230076\pi\)
\(854\) 0 0
\(855\) 1.21575e6 0.0568762
\(856\) 0 0
\(857\) 2.29506e7 1.06743 0.533717 0.845663i \(-0.320795\pi\)
0.533717 + 0.845663i \(0.320795\pi\)
\(858\) 0 0
\(859\) −8.05123e6 −0.372288 −0.186144 0.982522i \(-0.559599\pi\)
−0.186144 + 0.982522i \(0.559599\pi\)
\(860\) 0 0
\(861\) −7.31319e6 −0.336201
\(862\) 0 0
\(863\) −3.62319e7 −1.65601 −0.828007 0.560718i \(-0.810525\pi\)
−0.828007 + 0.560718i \(0.810525\pi\)
\(864\) 0 0
\(865\) 5.61664e6 0.255233
\(866\) 0 0
\(867\) 1.35898e7 0.613994
\(868\) 0 0
\(869\) 3.06066e7 1.37488
\(870\) 0 0
\(871\) 8.50702e6 0.379955
\(872\) 0 0
\(873\) 7.64892e6 0.339676
\(874\) 0 0
\(875\) −1.84836e7 −0.816145
\(876\) 0 0
\(877\) −1.03367e7 −0.453817 −0.226909 0.973916i \(-0.572862\pi\)
−0.226909 + 0.973916i \(0.572862\pi\)
\(878\) 0 0
\(879\) 1.62323e7 0.708610
\(880\) 0 0
\(881\) −4.24457e7 −1.84244 −0.921221 0.389040i \(-0.872807\pi\)
−0.921221 + 0.389040i \(0.872807\pi\)
\(882\) 0 0
\(883\) 2.98731e7 1.28937 0.644686 0.764448i \(-0.276988\pi\)
0.644686 + 0.764448i \(0.276988\pi\)
\(884\) 0 0
\(885\) 1.62799e7 0.698706
\(886\) 0 0
\(887\) −1.57379e6 −0.0671643 −0.0335822 0.999436i \(-0.510692\pi\)
−0.0335822 + 0.999436i \(0.510692\pi\)
\(888\) 0 0
\(889\) −2.95836e7 −1.25544
\(890\) 0 0
\(891\) 4.12480e6 0.174064
\(892\) 0 0
\(893\) 9.97476e6 0.418576
\(894\) 0 0
\(895\) 1.87784e7 0.783612
\(896\) 0 0
\(897\) −424822. −0.0176289
\(898\) 0 0
\(899\) −1.95965e6 −0.0808685
\(900\) 0 0
\(901\) 2.73838e7 1.12378
\(902\) 0 0
\(903\) 5.61261e6 0.229058
\(904\) 0 0
\(905\) −9.65465e6 −0.391846
\(906\) 0 0
\(907\) −1.81071e7 −0.730855 −0.365428 0.930840i \(-0.619077\pi\)
−0.365428 + 0.930840i \(0.619077\pi\)
\(908\) 0 0
\(909\) −4.84531e6 −0.194496
\(910\) 0 0
\(911\) −2.17268e7 −0.867362 −0.433681 0.901066i \(-0.642786\pi\)
−0.433681 + 0.901066i \(0.642786\pi\)
\(912\) 0 0
\(913\) 6.17206e7 2.45049
\(914\) 0 0
\(915\) 7.38862e6 0.291750
\(916\) 0 0
\(917\) −2.84140e7 −1.11586
\(918\) 0 0
\(919\) −1.45453e7 −0.568112 −0.284056 0.958808i \(-0.591680\pi\)
−0.284056 + 0.958808i \(0.591680\pi\)
\(920\) 0 0
\(921\) 5.71899e6 0.222162
\(922\) 0 0
\(923\) 1.18450e7 0.457648
\(924\) 0 0
\(925\) 1.83130e7 0.703728
\(926\) 0 0
\(927\) 4.89782e6 0.187199
\(928\) 0 0
\(929\) −3.79575e7 −1.44297 −0.721487 0.692428i \(-0.756541\pi\)
−0.721487 + 0.692428i \(0.756541\pi\)
\(930\) 0 0
\(931\) 2.57722e6 0.0974489
\(932\) 0 0
\(933\) −2.05557e7 −0.773085
\(934\) 0 0
\(935\) −4.47412e7 −1.67370
\(936\) 0 0
\(937\) 2.91623e7 1.08511 0.542554 0.840021i \(-0.317458\pi\)
0.542554 + 0.840021i \(0.317458\pi\)
\(938\) 0 0
\(939\) −2.62387e7 −0.971130
\(940\) 0 0
\(941\) 2.32132e7 0.854596 0.427298 0.904111i \(-0.359466\pi\)
0.427298 + 0.904111i \(0.359466\pi\)
\(942\) 0 0
\(943\) −2.52187e6 −0.0923513
\(944\) 0 0
\(945\) 2.98021e6 0.108559
\(946\) 0 0
\(947\) −2.89625e7 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(948\) 0 0
\(949\) −5.60772e6 −0.202125
\(950\) 0 0
\(951\) −7.88776e6 −0.282815
\(952\) 0 0
\(953\) 2.73878e7 0.976845 0.488422 0.872607i \(-0.337573\pi\)
0.488422 + 0.872607i \(0.337573\pi\)
\(954\) 0 0
\(955\) −1.32801e7 −0.471187
\(956\) 0 0
\(957\) 2.13209e7 0.752535
\(958\) 0 0
\(959\) −1.37412e7 −0.482477
\(960\) 0 0
\(961\) −2.83587e7 −0.990553
\(962\) 0 0
\(963\) 1.68350e7 0.584988
\(964\) 0 0
\(965\) 6.30164e6 0.217839
\(966\) 0 0
\(967\) −3.83778e7 −1.31982 −0.659908 0.751346i \(-0.729405\pi\)
−0.659908 + 0.751346i \(0.729405\pi\)
\(968\) 0 0
\(969\) −5.56123e6 −0.190266
\(970\) 0 0
\(971\) 1.73911e7 0.591940 0.295970 0.955197i \(-0.404357\pi\)
0.295970 + 0.955197i \(0.404357\pi\)
\(972\) 0 0
\(973\) −2.05792e7 −0.696862
\(974\) 0 0
\(975\) −1.94392e6 −0.0654887
\(976\) 0 0
\(977\) 2.73441e7 0.916490 0.458245 0.888826i \(-0.348478\pi\)
0.458245 + 0.888826i \(0.348478\pi\)
\(978\) 0 0
\(979\) 8.11484e7 2.70597
\(980\) 0 0
\(981\) −6.98580e6 −0.231763
\(982\) 0 0
\(983\) −2.65136e7 −0.875155 −0.437578 0.899181i \(-0.644164\pi\)
−0.437578 + 0.899181i \(0.644164\pi\)
\(984\) 0 0
\(985\) 3.99712e7 1.31267
\(986\) 0 0
\(987\) 2.44514e7 0.798934
\(988\) 0 0
\(989\) 1.93544e6 0.0629201
\(990\) 0 0
\(991\) −1.75748e7 −0.568469 −0.284235 0.958755i \(-0.591739\pi\)
−0.284235 + 0.958755i \(0.591739\pi\)
\(992\) 0 0
\(993\) −1.37707e7 −0.443182
\(994\) 0 0
\(995\) 2.05267e7 0.657297
\(996\) 0 0
\(997\) 3.84097e7 1.22378 0.611889 0.790944i \(-0.290410\pi\)
0.611889 + 0.790944i \(0.290410\pi\)
\(998\) 0 0
\(999\) −9.56077e6 −0.303095
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 912.6.a.y.1.3 6
4.3 odd 2 456.6.a.f.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.6.a.f.1.3 6 4.3 odd 2
912.6.a.y.1.3 6 1.1 even 1 trivial