Properties

Label 1104.6.a.i.1.3
Level $1104$
Weight $6$
Character 1104.1
Self dual yes
Analytic conductor $177.064$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1104.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(177.063737074\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5333.1
Defining polynomial: \(x^{3} - x^{2} - 11 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.49331\) of defining polynomial
Character \(\chi\) \(=\) 1104.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.00000 q^{3} +59.1123 q^{5} +213.331 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q-9.00000 q^{3} +59.1123 q^{5} +213.331 q^{7} +81.0000 q^{9} -126.528 q^{11} -884.825 q^{13} -532.011 q^{15} -1179.98 q^{17} +1866.19 q^{19} -1919.98 q^{21} -529.000 q^{23} +369.263 q^{25} -729.000 q^{27} -6786.62 q^{29} +5146.34 q^{31} +1138.75 q^{33} +12610.5 q^{35} +5137.07 q^{37} +7963.42 q^{39} +12482.7 q^{41} -4198.66 q^{43} +4788.10 q^{45} -23006.9 q^{47} +28703.3 q^{49} +10619.8 q^{51} +25175.4 q^{53} -7479.38 q^{55} -16795.7 q^{57} +37118.1 q^{59} +26410.4 q^{61} +17279.8 q^{63} -52304.0 q^{65} +54398.5 q^{67} +4761.00 q^{69} -35684.7 q^{71} +33937.4 q^{73} -3323.36 q^{75} -26992.5 q^{77} +76625.8 q^{79} +6561.00 q^{81} +96627.2 q^{83} -69751.1 q^{85} +61079.6 q^{87} +30080.8 q^{89} -188761. q^{91} -46317.0 q^{93} +110315. q^{95} -14637.2 q^{97} -10248.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 27q^{3} - 56q^{5} + 114q^{7} + 243q^{9} + O(q^{10}) \) \( 3q - 27q^{3} - 56q^{5} + 114q^{7} + 243q^{9} + 376q^{11} - 858q^{13} + 504q^{15} - 2548q^{17} + 2846q^{19} - 1026q^{21} - 1587q^{23} + 753q^{25} - 2187q^{27} - 16370q^{29} + 14756q^{31} - 3384q^{33} + 18520q^{35} + 15874q^{37} + 7722q^{39} + 12606q^{41} - 3154q^{43} - 4536q^{45} - 29928q^{47} + 4471q^{49} + 22932q^{51} - 44084q^{53} - 38360q^{55} - 25614q^{57} + 29300q^{59} + 54010q^{61} + 9234q^{63} - 51216q^{65} - 43390q^{67} + 14283q^{69} - 23424q^{71} - 91402q^{73} - 6777q^{75} - 97208q^{77} + 49398q^{79} + 19683q^{81} + 103936q^{83} + 5888q^{85} + 147330q^{87} + 96112q^{89} - 129228q^{91} - 132804q^{93} + 55928q^{95} - 135318q^{97} + 30456q^{99} + O(q^{100}) \)

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
\(5\) 59.1123 1.05743 0.528716 0.848799i \(-0.322674\pi\)
0.528716 + 0.848799i \(0.322674\pi\)
\(6\) 0 0
\(7\) 213.331 1.64554 0.822772 0.568371i \(-0.192426\pi\)
0.822772 + 0.568371i \(0.192426\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −126.528 −0.315287 −0.157643 0.987496i \(-0.550390\pi\)
−0.157643 + 0.987496i \(0.550390\pi\)
\(12\) 0 0
\(13\) −884.825 −1.45211 −0.726054 0.687638i \(-0.758648\pi\)
−0.726054 + 0.687638i \(0.758648\pi\)
\(14\) 0 0
\(15\) −532.011 −0.610509
\(16\) 0 0
\(17\) −1179.98 −0.990264 −0.495132 0.868818i \(-0.664880\pi\)
−0.495132 + 0.868818i \(0.664880\pi\)
\(18\) 0 0
\(19\) 1866.19 1.18596 0.592981 0.805216i \(-0.297951\pi\)
0.592981 + 0.805216i \(0.297951\pi\)
\(20\) 0 0
\(21\) −1919.98 −0.950056
\(22\) 0 0
\(23\) −529.000 −0.208514
\(24\) 0 0
\(25\) 369.263 0.118164
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −6786.62 −1.49851 −0.749253 0.662284i \(-0.769587\pi\)
−0.749253 + 0.662284i \(0.769587\pi\)
\(30\) 0 0
\(31\) 5146.34 0.961820 0.480910 0.876770i \(-0.340306\pi\)
0.480910 + 0.876770i \(0.340306\pi\)
\(32\) 0 0
\(33\) 1138.75 0.182031
\(34\) 0 0
\(35\) 12610.5 1.74005
\(36\) 0 0
\(37\) 5137.07 0.616895 0.308448 0.951241i \(-0.400191\pi\)
0.308448 + 0.951241i \(0.400191\pi\)
\(38\) 0 0
\(39\) 7963.42 0.838375
\(40\) 0 0
\(41\) 12482.7 1.15971 0.579855 0.814720i \(-0.303109\pi\)
0.579855 + 0.814720i \(0.303109\pi\)
\(42\) 0 0
\(43\) −4198.66 −0.346290 −0.173145 0.984896i \(-0.555393\pi\)
−0.173145 + 0.984896i \(0.555393\pi\)
\(44\) 0 0
\(45\) 4788.10 0.352478
\(46\) 0 0
\(47\) −23006.9 −1.51919 −0.759596 0.650395i \(-0.774603\pi\)
−0.759596 + 0.650395i \(0.774603\pi\)
\(48\) 0 0
\(49\) 28703.3 1.70782
\(50\) 0 0
\(51\) 10619.8 0.571729
\(52\) 0 0
\(53\) 25175.4 1.23108 0.615541 0.788105i \(-0.288938\pi\)
0.615541 + 0.788105i \(0.288938\pi\)
\(54\) 0 0
\(55\) −7479.38 −0.333395
\(56\) 0 0
\(57\) −16795.7 −0.684716
\(58\) 0 0
\(59\) 37118.1 1.38821 0.694107 0.719872i \(-0.255800\pi\)
0.694107 + 0.719872i \(0.255800\pi\)
\(60\) 0 0
\(61\) 26410.4 0.908761 0.454381 0.890808i \(-0.349861\pi\)
0.454381 + 0.890808i \(0.349861\pi\)
\(62\) 0 0
\(63\) 17279.8 0.548515
\(64\) 0 0
\(65\) −52304.0 −1.53551
\(66\) 0 0
\(67\) 54398.5 1.48047 0.740236 0.672347i \(-0.234714\pi\)
0.740236 + 0.672347i \(0.234714\pi\)
\(68\) 0 0
\(69\) 4761.00 0.120386
\(70\) 0 0
\(71\) −35684.7 −0.840110 −0.420055 0.907499i \(-0.637989\pi\)
−0.420055 + 0.907499i \(0.637989\pi\)
\(72\) 0 0
\(73\) 33937.4 0.745369 0.372684 0.927958i \(-0.378437\pi\)
0.372684 + 0.927958i \(0.378437\pi\)
\(74\) 0 0
\(75\) −3323.36 −0.0682220
\(76\) 0 0
\(77\) −26992.5 −0.518819
\(78\) 0 0
\(79\) 76625.8 1.38136 0.690681 0.723160i \(-0.257311\pi\)
0.690681 + 0.723160i \(0.257311\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 96627.2 1.53959 0.769794 0.638293i \(-0.220359\pi\)
0.769794 + 0.638293i \(0.220359\pi\)
\(84\) 0 0
\(85\) −69751.1 −1.04714
\(86\) 0 0
\(87\) 61079.6 0.865163
\(88\) 0 0
\(89\) 30080.8 0.402545 0.201273 0.979535i \(-0.435492\pi\)
0.201273 + 0.979535i \(0.435492\pi\)
\(90\) 0 0
\(91\) −188761. −2.38951
\(92\) 0 0
\(93\) −46317.0 −0.555307
\(94\) 0 0
\(95\) 110315. 1.25408
\(96\) 0 0
\(97\) −14637.2 −0.157953 −0.0789766 0.996876i \(-0.525165\pi\)
−0.0789766 + 0.996876i \(0.525165\pi\)
\(98\) 0 0
\(99\) −10248.8 −0.105096
\(100\) 0 0
\(101\) −37693.7 −0.367676 −0.183838 0.982957i \(-0.558852\pi\)
−0.183838 + 0.982957i \(0.558852\pi\)
\(102\) 0 0
\(103\) −125252. −1.16330 −0.581649 0.813440i \(-0.697592\pi\)
−0.581649 + 0.813440i \(0.697592\pi\)
\(104\) 0 0
\(105\) −113495. −1.00462
\(106\) 0 0
\(107\) −28002.7 −0.236451 −0.118225 0.992987i \(-0.537721\pi\)
−0.118225 + 0.992987i \(0.537721\pi\)
\(108\) 0 0
\(109\) −64151.4 −0.517178 −0.258589 0.965987i \(-0.583257\pi\)
−0.258589 + 0.965987i \(0.583257\pi\)
\(110\) 0 0
\(111\) −46233.6 −0.356165
\(112\) 0 0
\(113\) −25226.7 −0.185851 −0.0929253 0.995673i \(-0.529622\pi\)
−0.0929253 + 0.995673i \(0.529622\pi\)
\(114\) 0 0
\(115\) −31270.4 −0.220490
\(116\) 0 0
\(117\) −71670.8 −0.484036
\(118\) 0 0
\(119\) −251726. −1.62952
\(120\) 0 0
\(121\) −145042. −0.900594
\(122\) 0 0
\(123\) −112344. −0.669559
\(124\) 0 0
\(125\) −162898. −0.932482
\(126\) 0 0
\(127\) −35783.9 −0.196870 −0.0984348 0.995144i \(-0.531384\pi\)
−0.0984348 + 0.995144i \(0.531384\pi\)
\(128\) 0 0
\(129\) 37787.9 0.199930
\(130\) 0 0
\(131\) −78119.6 −0.397724 −0.198862 0.980028i \(-0.563725\pi\)
−0.198862 + 0.980028i \(0.563725\pi\)
\(132\) 0 0
\(133\) 398116. 1.95155
\(134\) 0 0
\(135\) −43092.9 −0.203503
\(136\) 0 0
\(137\) −92816.7 −0.422498 −0.211249 0.977432i \(-0.567753\pi\)
−0.211249 + 0.977432i \(0.567753\pi\)
\(138\) 0 0
\(139\) 417118. 1.83114 0.915570 0.402159i \(-0.131740\pi\)
0.915570 + 0.402159i \(0.131740\pi\)
\(140\) 0 0
\(141\) 207062. 0.877106
\(142\) 0 0
\(143\) 111955. 0.457831
\(144\) 0 0
\(145\) −401173. −1.58457
\(146\) 0 0
\(147\) −258330. −0.986009
\(148\) 0 0
\(149\) 445936. 1.64553 0.822766 0.568380i \(-0.192430\pi\)
0.822766 + 0.568380i \(0.192430\pi\)
\(150\) 0 0
\(151\) 260458. 0.929597 0.464799 0.885416i \(-0.346127\pi\)
0.464799 + 0.885416i \(0.346127\pi\)
\(152\) 0 0
\(153\) −95578.1 −0.330088
\(154\) 0 0
\(155\) 304212. 1.01706
\(156\) 0 0
\(157\) 46067.4 0.149157 0.0745787 0.997215i \(-0.476239\pi\)
0.0745787 + 0.997215i \(0.476239\pi\)
\(158\) 0 0
\(159\) −226579. −0.710765
\(160\) 0 0
\(161\) −112852. −0.343120
\(162\) 0 0
\(163\) −188377. −0.555341 −0.277671 0.960676i \(-0.589562\pi\)
−0.277671 + 0.960676i \(0.589562\pi\)
\(164\) 0 0
\(165\) 67314.4 0.192485
\(166\) 0 0
\(167\) −203214. −0.563849 −0.281924 0.959437i \(-0.590973\pi\)
−0.281924 + 0.959437i \(0.590973\pi\)
\(168\) 0 0
\(169\) 411622. 1.10862
\(170\) 0 0
\(171\) 151161. 0.395321
\(172\) 0 0
\(173\) 158051. 0.401497 0.200749 0.979643i \(-0.435663\pi\)
0.200749 + 0.979643i \(0.435663\pi\)
\(174\) 0 0
\(175\) 78775.3 0.194444
\(176\) 0 0
\(177\) −334063. −0.801485
\(178\) 0 0
\(179\) −166444. −0.388271 −0.194136 0.980975i \(-0.562190\pi\)
−0.194136 + 0.980975i \(0.562190\pi\)
\(180\) 0 0
\(181\) 185620. 0.421141 0.210571 0.977579i \(-0.432468\pi\)
0.210571 + 0.977579i \(0.432468\pi\)
\(182\) 0 0
\(183\) −237693. −0.524673
\(184\) 0 0
\(185\) 303664. 0.652325
\(186\) 0 0
\(187\) 149300. 0.312217
\(188\) 0 0
\(189\) −155519. −0.316685
\(190\) 0 0
\(191\) 769733. 1.52671 0.763356 0.645978i \(-0.223550\pi\)
0.763356 + 0.645978i \(0.223550\pi\)
\(192\) 0 0
\(193\) −384145. −0.742338 −0.371169 0.928565i \(-0.621043\pi\)
−0.371169 + 0.928565i \(0.621043\pi\)
\(194\) 0 0
\(195\) 470736. 0.886525
\(196\) 0 0
\(197\) 266048. 0.488421 0.244211 0.969722i \(-0.421471\pi\)
0.244211 + 0.969722i \(0.421471\pi\)
\(198\) 0 0
\(199\) 706070. 1.26391 0.631953 0.775007i \(-0.282254\pi\)
0.631953 + 0.775007i \(0.282254\pi\)
\(200\) 0 0
\(201\) −489587. −0.854751
\(202\) 0 0
\(203\) −1.44780e6 −2.46586
\(204\) 0 0
\(205\) 737882. 1.22631
\(206\) 0 0
\(207\) −42849.0 −0.0695048
\(208\) 0 0
\(209\) −236125. −0.373918
\(210\) 0 0
\(211\) 290636. 0.449410 0.224705 0.974427i \(-0.427858\pi\)
0.224705 + 0.974427i \(0.427858\pi\)
\(212\) 0 0
\(213\) 321162. 0.485038
\(214\) 0 0
\(215\) −248192. −0.366178
\(216\) 0 0
\(217\) 1.09788e6 1.58272
\(218\) 0 0
\(219\) −305436. −0.430339
\(220\) 0 0
\(221\) 1.04407e6 1.43797
\(222\) 0 0
\(223\) 574915. 0.774179 0.387089 0.922042i \(-0.373481\pi\)
0.387089 + 0.922042i \(0.373481\pi\)
\(224\) 0 0
\(225\) 29910.3 0.0393880
\(226\) 0 0
\(227\) 1.37363e6 1.76931 0.884655 0.466246i \(-0.154394\pi\)
0.884655 + 0.466246i \(0.154394\pi\)
\(228\) 0 0
\(229\) 870725. 1.09722 0.548608 0.836080i \(-0.315158\pi\)
0.548608 + 0.836080i \(0.315158\pi\)
\(230\) 0 0
\(231\) 242932. 0.299540
\(232\) 0 0
\(233\) −1.61732e6 −1.95166 −0.975832 0.218522i \(-0.929876\pi\)
−0.975832 + 0.218522i \(0.929876\pi\)
\(234\) 0 0
\(235\) −1.35999e6 −1.60644
\(236\) 0 0
\(237\) −689633. −0.797530
\(238\) 0 0
\(239\) 1.65281e6 1.87167 0.935833 0.352444i \(-0.114649\pi\)
0.935833 + 0.352444i \(0.114649\pi\)
\(240\) 0 0
\(241\) 928466. 1.02973 0.514865 0.857271i \(-0.327842\pi\)
0.514865 + 0.857271i \(0.327842\pi\)
\(242\) 0 0
\(243\) −59049.0 −0.0641500
\(244\) 0 0
\(245\) 1.69672e6 1.80590
\(246\) 0 0
\(247\) −1.65125e6 −1.72215
\(248\) 0 0
\(249\) −869645. −0.888881
\(250\) 0 0
\(251\) 909693. 0.911403 0.455702 0.890133i \(-0.349388\pi\)
0.455702 + 0.890133i \(0.349388\pi\)
\(252\) 0 0
\(253\) 66933.5 0.0657419
\(254\) 0 0
\(255\) 627760. 0.604565
\(256\) 0 0
\(257\) 52666.7 0.0497397 0.0248699 0.999691i \(-0.492083\pi\)
0.0248699 + 0.999691i \(0.492083\pi\)
\(258\) 0 0
\(259\) 1.09590e6 1.01513
\(260\) 0 0
\(261\) −549716. −0.499502
\(262\) 0 0
\(263\) −464764. −0.414327 −0.207164 0.978306i \(-0.566423\pi\)
−0.207164 + 0.978306i \(0.566423\pi\)
\(264\) 0 0
\(265\) 1.48818e6 1.30179
\(266\) 0 0
\(267\) −270727. −0.232410
\(268\) 0 0
\(269\) 968420. 0.815987 0.407993 0.912985i \(-0.366229\pi\)
0.407993 + 0.912985i \(0.366229\pi\)
\(270\) 0 0
\(271\) −2.29262e6 −1.89630 −0.948152 0.317817i \(-0.897050\pi\)
−0.948152 + 0.317817i \(0.897050\pi\)
\(272\) 0 0
\(273\) 1.69885e6 1.37958
\(274\) 0 0
\(275\) −46722.2 −0.0372556
\(276\) 0 0
\(277\) 290954. 0.227837 0.113919 0.993490i \(-0.463660\pi\)
0.113919 + 0.993490i \(0.463660\pi\)
\(278\) 0 0
\(279\) 416853. 0.320607
\(280\) 0 0
\(281\) −391978. −0.296139 −0.148070 0.988977i \(-0.547306\pi\)
−0.148070 + 0.988977i \(0.547306\pi\)
\(282\) 0 0
\(283\) −732768. −0.543877 −0.271938 0.962315i \(-0.587665\pi\)
−0.271938 + 0.962315i \(0.587665\pi\)
\(284\) 0 0
\(285\) −992831. −0.724041
\(286\) 0 0
\(287\) 2.66295e6 1.90835
\(288\) 0 0
\(289\) −27511.7 −0.0193764
\(290\) 0 0
\(291\) 131735. 0.0911944
\(292\) 0 0
\(293\) 1.62587e6 1.10641 0.553207 0.833044i \(-0.313404\pi\)
0.553207 + 0.833044i \(0.313404\pi\)
\(294\) 0 0
\(295\) 2.19414e6 1.46794
\(296\) 0 0
\(297\) 92239.1 0.0606770
\(298\) 0 0
\(299\) 468072. 0.302785
\(300\) 0 0
\(301\) −895706. −0.569835
\(302\) 0 0
\(303\) 339243. 0.212278
\(304\) 0 0
\(305\) 1.56118e6 0.960954
\(306\) 0 0
\(307\) −1.24581e6 −0.754406 −0.377203 0.926131i \(-0.623114\pi\)
−0.377203 + 0.926131i \(0.623114\pi\)
\(308\) 0 0
\(309\) 1.12727e6 0.671631
\(310\) 0 0
\(311\) 3.26852e6 1.91624 0.958119 0.286369i \(-0.0924483\pi\)
0.958119 + 0.286369i \(0.0924483\pi\)
\(312\) 0 0
\(313\) 1.70161e6 0.981744 0.490872 0.871232i \(-0.336678\pi\)
0.490872 + 0.871232i \(0.336678\pi\)
\(314\) 0 0
\(315\) 1.02145e6 0.580018
\(316\) 0 0
\(317\) −908317. −0.507679 −0.253840 0.967246i \(-0.581694\pi\)
−0.253840 + 0.967246i \(0.581694\pi\)
\(318\) 0 0
\(319\) 858699. 0.472459
\(320\) 0 0
\(321\) 252025. 0.136515
\(322\) 0 0
\(323\) −2.20206e6 −1.17442
\(324\) 0 0
\(325\) −326733. −0.171587
\(326\) 0 0
\(327\) 577362. 0.298593
\(328\) 0 0
\(329\) −4.90809e6 −2.49990
\(330\) 0 0
\(331\) 2.14345e6 1.07533 0.537666 0.843158i \(-0.319306\pi\)
0.537666 + 0.843158i \(0.319306\pi\)
\(332\) 0 0
\(333\) 416103. 0.205632
\(334\) 0 0
\(335\) 3.21562e6 1.56550
\(336\) 0 0
\(337\) 817498. 0.392114 0.196057 0.980593i \(-0.437186\pi\)
0.196057 + 0.980593i \(0.437186\pi\)
\(338\) 0 0
\(339\) 227040. 0.107301
\(340\) 0 0
\(341\) −651157. −0.303249
\(342\) 0 0
\(343\) 2.53785e6 1.16475
\(344\) 0 0
\(345\) 281434. 0.127300
\(346\) 0 0
\(347\) −3.20207e6 −1.42760 −0.713801 0.700349i \(-0.753028\pi\)
−0.713801 + 0.700349i \(0.753028\pi\)
\(348\) 0 0
\(349\) −3.23916e6 −1.42354 −0.711768 0.702415i \(-0.752105\pi\)
−0.711768 + 0.702415i \(0.752105\pi\)
\(350\) 0 0
\(351\) 645037. 0.279458
\(352\) 0 0
\(353\) −1.72676e6 −0.737555 −0.368777 0.929518i \(-0.620224\pi\)
−0.368777 + 0.929518i \(0.620224\pi\)
\(354\) 0 0
\(355\) −2.10940e6 −0.888360
\(356\) 0 0
\(357\) 2.26554e6 0.940806
\(358\) 0 0
\(359\) −3.12338e6 −1.27905 −0.639527 0.768769i \(-0.720870\pi\)
−0.639527 + 0.768769i \(0.720870\pi\)
\(360\) 0 0
\(361\) 1.00655e6 0.406507
\(362\) 0 0
\(363\) 1.30537e6 0.519958
\(364\) 0 0
\(365\) 2.00612e6 0.788177
\(366\) 0 0
\(367\) −1.17031e6 −0.453560 −0.226780 0.973946i \(-0.572820\pi\)
−0.226780 + 0.973946i \(0.572820\pi\)
\(368\) 0 0
\(369\) 1.01110e6 0.386570
\(370\) 0 0
\(371\) 5.37070e6 2.02580
\(372\) 0 0
\(373\) −692891. −0.257865 −0.128933 0.991653i \(-0.541155\pi\)
−0.128933 + 0.991653i \(0.541155\pi\)
\(374\) 0 0
\(375\) 1.46608e6 0.538369
\(376\) 0 0
\(377\) 6.00497e6 2.17599
\(378\) 0 0
\(379\) −2.12163e6 −0.758702 −0.379351 0.925253i \(-0.623853\pi\)
−0.379351 + 0.925253i \(0.623853\pi\)
\(380\) 0 0
\(381\) 322055. 0.113663
\(382\) 0 0
\(383\) 287079. 0.100001 0.0500005 0.998749i \(-0.484078\pi\)
0.0500005 + 0.998749i \(0.484078\pi\)
\(384\) 0 0
\(385\) −1.59559e6 −0.548616
\(386\) 0 0
\(387\) −340091. −0.115430
\(388\) 0 0
\(389\) 679811. 0.227779 0.113890 0.993493i \(-0.463669\pi\)
0.113890 + 0.993493i \(0.463669\pi\)
\(390\) 0 0
\(391\) 624208. 0.206484
\(392\) 0 0
\(393\) 703076. 0.229626
\(394\) 0 0
\(395\) 4.52953e6 1.46070
\(396\) 0 0
\(397\) −5.39917e6 −1.71930 −0.859648 0.510886i \(-0.829317\pi\)
−0.859648 + 0.510886i \(0.829317\pi\)
\(398\) 0 0
\(399\) −3.58304e6 −1.12673
\(400\) 0 0
\(401\) −2.17808e6 −0.676416 −0.338208 0.941071i \(-0.609821\pi\)
−0.338208 + 0.941071i \(0.609821\pi\)
\(402\) 0 0
\(403\) −4.55361e6 −1.39667
\(404\) 0 0
\(405\) 387836. 0.117493
\(406\) 0 0
\(407\) −649985. −0.194499
\(408\) 0 0
\(409\) −356567. −0.105398 −0.0526991 0.998610i \(-0.516782\pi\)
−0.0526991 + 0.998610i \(0.516782\pi\)
\(410\) 0 0
\(411\) 835350. 0.243929
\(412\) 0 0
\(413\) 7.91847e6 2.28437
\(414\) 0 0
\(415\) 5.71186e6 1.62801
\(416\) 0 0
\(417\) −3.75406e6 −1.05721
\(418\) 0 0
\(419\) −3.82634e6 −1.06475 −0.532376 0.846508i \(-0.678701\pi\)
−0.532376 + 0.846508i \(0.678701\pi\)
\(420\) 0 0
\(421\) −3.00814e6 −0.827167 −0.413584 0.910466i \(-0.635723\pi\)
−0.413584 + 0.910466i \(0.635723\pi\)
\(422\) 0 0
\(423\) −1.86356e6 −0.506397
\(424\) 0 0
\(425\) −435721. −0.117014
\(426\) 0 0
\(427\) 5.63416e6 1.49541
\(428\) 0 0
\(429\) −1.00760e6 −0.264329
\(430\) 0 0
\(431\) −577809. −0.149827 −0.0749136 0.997190i \(-0.523868\pi\)
−0.0749136 + 0.997190i \(0.523868\pi\)
\(432\) 0 0
\(433\) 7.21909e6 1.85039 0.925194 0.379494i \(-0.123902\pi\)
0.925194 + 0.379494i \(0.123902\pi\)
\(434\) 0 0
\(435\) 3.61055e6 0.914852
\(436\) 0 0
\(437\) −987212. −0.247290
\(438\) 0 0
\(439\) 760935. 0.188446 0.0942229 0.995551i \(-0.469963\pi\)
0.0942229 + 0.995551i \(0.469963\pi\)
\(440\) 0 0
\(441\) 2.32497e6 0.569273
\(442\) 0 0
\(443\) 6.96892e6 1.68716 0.843580 0.537003i \(-0.180444\pi\)
0.843580 + 0.537003i \(0.180444\pi\)
\(444\) 0 0
\(445\) 1.77815e6 0.425665
\(446\) 0 0
\(447\) −4.01342e6 −0.950049
\(448\) 0 0
\(449\) 5.48897e6 1.28492 0.642458 0.766321i \(-0.277915\pi\)
0.642458 + 0.766321i \(0.277915\pi\)
\(450\) 0 0
\(451\) −1.57942e6 −0.365641
\(452\) 0 0
\(453\) −2.34412e6 −0.536703
\(454\) 0 0
\(455\) −1.11581e7 −2.52675
\(456\) 0 0
\(457\) −2.49968e6 −0.559879 −0.279940 0.960018i \(-0.590314\pi\)
−0.279940 + 0.960018i \(0.590314\pi\)
\(458\) 0 0
\(459\) 860203. 0.190576
\(460\) 0 0
\(461\) −8.10339e6 −1.77588 −0.887942 0.459956i \(-0.847865\pi\)
−0.887942 + 0.459956i \(0.847865\pi\)
\(462\) 0 0
\(463\) 4.71421e6 1.02201 0.511007 0.859577i \(-0.329273\pi\)
0.511007 + 0.859577i \(0.329273\pi\)
\(464\) 0 0
\(465\) −2.73791e6 −0.587200
\(466\) 0 0
\(467\) 1.03346e6 0.219281 0.109641 0.993971i \(-0.465030\pi\)
0.109641 + 0.993971i \(0.465030\pi\)
\(468\) 0 0
\(469\) 1.16049e7 2.43618
\(470\) 0 0
\(471\) −414607. −0.0861161
\(472\) 0 0
\(473\) 531249. 0.109181
\(474\) 0 0
\(475\) 689113. 0.140138
\(476\) 0 0
\(477\) 2.03921e6 0.410360
\(478\) 0 0
\(479\) −1.13020e6 −0.225069 −0.112534 0.993648i \(-0.535897\pi\)
−0.112534 + 0.993648i \(0.535897\pi\)
\(480\) 0 0
\(481\) −4.54541e6 −0.895798
\(482\) 0 0
\(483\) 1.01567e6 0.198100
\(484\) 0 0
\(485\) −865238. −0.167025
\(486\) 0 0
\(487\) 793366. 0.151583 0.0757916 0.997124i \(-0.475852\pi\)
0.0757916 + 0.997124i \(0.475852\pi\)
\(488\) 0 0
\(489\) 1.69540e6 0.320626
\(490\) 0 0
\(491\) 3.66618e6 0.686294 0.343147 0.939282i \(-0.388507\pi\)
0.343147 + 0.939282i \(0.388507\pi\)
\(492\) 0 0
\(493\) 8.00805e6 1.48392
\(494\) 0 0
\(495\) −605829. −0.111132
\(496\) 0 0
\(497\) −7.61267e6 −1.38244
\(498\) 0 0
\(499\) 5.08658e6 0.914480 0.457240 0.889343i \(-0.348838\pi\)
0.457240 + 0.889343i \(0.348838\pi\)
\(500\) 0 0
\(501\) 1.82893e6 0.325538
\(502\) 0 0
\(503\) −6.49233e6 −1.14414 −0.572072 0.820203i \(-0.693860\pi\)
−0.572072 + 0.820203i \(0.693860\pi\)
\(504\) 0 0
\(505\) −2.22816e6 −0.388793
\(506\) 0 0
\(507\) −3.70460e6 −0.640061
\(508\) 0 0
\(509\) −5.58035e6 −0.954700 −0.477350 0.878713i \(-0.658403\pi\)
−0.477350 + 0.878713i \(0.658403\pi\)
\(510\) 0 0
\(511\) 7.23991e6 1.22654
\(512\) 0 0
\(513\) −1.36045e6 −0.228239
\(514\) 0 0
\(515\) −7.40393e6 −1.23011
\(516\) 0 0
\(517\) 2.91102e6 0.478981
\(518\) 0 0
\(519\) −1.42246e6 −0.231804
\(520\) 0 0
\(521\) −4.02579e6 −0.649766 −0.324883 0.945754i \(-0.605325\pi\)
−0.324883 + 0.945754i \(0.605325\pi\)
\(522\) 0 0
\(523\) 9.41059e6 1.50440 0.752198 0.658937i \(-0.228993\pi\)
0.752198 + 0.658937i \(0.228993\pi\)
\(524\) 0 0
\(525\) −708978. −0.112262
\(526\) 0 0
\(527\) −6.07256e6 −0.952457
\(528\) 0 0
\(529\) 279841. 0.0434783
\(530\) 0 0
\(531\) 3.00657e6 0.462738
\(532\) 0 0
\(533\) −1.10450e7 −1.68402
\(534\) 0 0
\(535\) −1.65531e6 −0.250031
\(536\) 0 0
\(537\) 1.49800e6 0.224169
\(538\) 0 0
\(539\) −3.63178e6 −0.538453
\(540\) 0 0
\(541\) −2.68147e6 −0.393894 −0.196947 0.980414i \(-0.563103\pi\)
−0.196947 + 0.980414i \(0.563103\pi\)
\(542\) 0 0
\(543\) −1.67058e6 −0.243146
\(544\) 0 0
\(545\) −3.79214e6 −0.546881
\(546\) 0 0
\(547\) −8.74676e6 −1.24991 −0.624955 0.780660i \(-0.714883\pi\)
−0.624955 + 0.780660i \(0.714883\pi\)
\(548\) 0 0
\(549\) 2.13924e6 0.302920
\(550\) 0 0
\(551\) −1.26651e7 −1.77717
\(552\) 0 0
\(553\) 1.63467e7 2.27309
\(554\) 0 0
\(555\) −2.73298e6 −0.376620
\(556\) 0 0
\(557\) 2.07905e6 0.283940 0.141970 0.989871i \(-0.454656\pi\)
0.141970 + 0.989871i \(0.454656\pi\)
\(558\) 0 0
\(559\) 3.71508e6 0.502850
\(560\) 0 0
\(561\) −1.34370e6 −0.180259
\(562\) 0 0
\(563\) −8.52157e6 −1.13305 −0.566524 0.824045i \(-0.691712\pi\)
−0.566524 + 0.824045i \(0.691712\pi\)
\(564\) 0 0
\(565\) −1.49121e6 −0.196524
\(566\) 0 0
\(567\) 1.39967e6 0.182838
\(568\) 0 0
\(569\) 6.58526e6 0.852692 0.426346 0.904560i \(-0.359801\pi\)
0.426346 + 0.904560i \(0.359801\pi\)
\(570\) 0 0
\(571\) 6.31614e6 0.810702 0.405351 0.914161i \(-0.367149\pi\)
0.405351 + 0.914161i \(0.367149\pi\)
\(572\) 0 0
\(573\) −6.92760e6 −0.881447
\(574\) 0 0
\(575\) −195340. −0.0246389
\(576\) 0 0
\(577\) 4.48677e6 0.561040 0.280520 0.959848i \(-0.409493\pi\)
0.280520 + 0.959848i \(0.409493\pi\)
\(578\) 0 0
\(579\) 3.45730e6 0.428589
\(580\) 0 0
\(581\) 2.06136e7 2.53346
\(582\) 0 0
\(583\) −3.18540e6 −0.388144
\(584\) 0 0
\(585\) −4.23663e6 −0.511836
\(586\) 0 0
\(587\) 7.42328e6 0.889203 0.444601 0.895729i \(-0.353345\pi\)
0.444601 + 0.895729i \(0.353345\pi\)
\(588\) 0 0
\(589\) 9.60402e6 1.14068
\(590\) 0 0
\(591\) −2.39443e6 −0.281990
\(592\) 0 0
\(593\) 7.02182e6 0.819998 0.409999 0.912086i \(-0.365529\pi\)
0.409999 + 0.912086i \(0.365529\pi\)
\(594\) 0 0
\(595\) −1.48801e7 −1.72311
\(596\) 0 0
\(597\) −6.35463e6 −0.729717
\(598\) 0 0
\(599\) −5.02165e6 −0.571846 −0.285923 0.958253i \(-0.592300\pi\)
−0.285923 + 0.958253i \(0.592300\pi\)
\(600\) 0 0
\(601\) 4.66921e6 0.527300 0.263650 0.964618i \(-0.415074\pi\)
0.263650 + 0.964618i \(0.415074\pi\)
\(602\) 0 0
\(603\) 4.40628e6 0.493491
\(604\) 0 0
\(605\) −8.57374e6 −0.952318
\(606\) 0 0
\(607\) 1.48942e7 1.64076 0.820380 0.571819i \(-0.193762\pi\)
0.820380 + 0.571819i \(0.193762\pi\)
\(608\) 0 0
\(609\) 1.30302e7 1.42366
\(610\) 0 0
\(611\) 2.03570e7 2.20603
\(612\) 0 0
\(613\) 1.09408e7 1.17598 0.587989 0.808869i \(-0.299920\pi\)
0.587989 + 0.808869i \(0.299920\pi\)
\(614\) 0 0
\(615\) −6.64093e6 −0.708013
\(616\) 0 0
\(617\) −6.42199e6 −0.679136 −0.339568 0.940582i \(-0.610281\pi\)
−0.339568 + 0.940582i \(0.610281\pi\)
\(618\) 0 0
\(619\) 3.05345e6 0.320305 0.160152 0.987092i \(-0.448801\pi\)
0.160152 + 0.987092i \(0.448801\pi\)
\(620\) 0 0
\(621\) 385641. 0.0401286
\(622\) 0 0
\(623\) 6.41719e6 0.662406
\(624\) 0 0
\(625\) −1.07832e7 −1.10420
\(626\) 0 0
\(627\) 2.12513e6 0.215882
\(628\) 0 0
\(629\) −6.06162e6 −0.610889
\(630\) 0 0
\(631\) −1.05012e7 −1.04994 −0.524971 0.851120i \(-0.675924\pi\)
−0.524971 + 0.851120i \(0.675924\pi\)
\(632\) 0 0
\(633\) −2.61572e6 −0.259467
\(634\) 0 0
\(635\) −2.11527e6 −0.208176
\(636\) 0 0
\(637\) −2.53974e7 −2.47994
\(638\) 0 0
\(639\) −2.89046e6 −0.280037
\(640\) 0 0
\(641\) −1.57928e7 −1.51815 −0.759073 0.651006i \(-0.774347\pi\)
−0.759073 + 0.651006i \(0.774347\pi\)
\(642\) 0 0
\(643\) −551159. −0.0525714 −0.0262857 0.999654i \(-0.508368\pi\)
−0.0262857 + 0.999654i \(0.508368\pi\)
\(644\) 0 0
\(645\) 2.23373e6 0.211413
\(646\) 0 0
\(647\) −1.12498e7 −1.05653 −0.528267 0.849078i \(-0.677158\pi\)
−0.528267 + 0.849078i \(0.677158\pi\)
\(648\) 0 0
\(649\) −4.69649e6 −0.437685
\(650\) 0 0
\(651\) −9.88088e6 −0.913783
\(652\) 0 0
\(653\) −2.56046e6 −0.234982 −0.117491 0.993074i \(-0.537485\pi\)
−0.117491 + 0.993074i \(0.537485\pi\)
\(654\) 0 0
\(655\) −4.61783e6 −0.420566
\(656\) 0 0
\(657\) 2.74893e6 0.248456
\(658\) 0 0
\(659\) 734143. 0.0658518 0.0329259 0.999458i \(-0.489517\pi\)
0.0329259 + 0.999458i \(0.489517\pi\)
\(660\) 0 0
\(661\) 1.92354e7 1.71237 0.856185 0.516670i \(-0.172828\pi\)
0.856185 + 0.516670i \(0.172828\pi\)
\(662\) 0 0
\(663\) −9.39666e6 −0.830213
\(664\) 0 0
\(665\) 2.35336e7 2.06364
\(666\) 0 0
\(667\) 3.59012e6 0.312460
\(668\) 0 0
\(669\) −5.17423e6 −0.446972
\(670\) 0 0
\(671\) −3.34166e6 −0.286520
\(672\) 0 0
\(673\) 1.20397e7 1.02465 0.512326 0.858791i \(-0.328784\pi\)
0.512326 + 0.858791i \(0.328784\pi\)
\(674\) 0 0
\(675\) −269192. −0.0227407
\(676\) 0 0
\(677\) 99259.2 0.00832337 0.00416168 0.999991i \(-0.498675\pi\)
0.00416168 + 0.999991i \(0.498675\pi\)
\(678\) 0 0
\(679\) −3.12257e6 −0.259919
\(680\) 0 0
\(681\) −1.23626e7 −1.02151
\(682\) 0 0
\(683\) −9.22059e6 −0.756323 −0.378161 0.925740i \(-0.623444\pi\)
−0.378161 + 0.925740i \(0.623444\pi\)
\(684\) 0 0
\(685\) −5.48661e6 −0.446763
\(686\) 0 0
\(687\) −7.83652e6 −0.633478
\(688\) 0 0
\(689\) −2.22758e7 −1.78766
\(690\) 0 0
\(691\) 1.77887e7 1.41726 0.708629 0.705581i \(-0.249314\pi\)
0.708629 + 0.705581i \(0.249314\pi\)
\(692\) 0 0
\(693\) −2.18639e6 −0.172940
\(694\) 0 0
\(695\) 2.46568e7 1.93631
\(696\) 0 0
\(697\) −1.47293e7 −1.14842
\(698\) 0 0
\(699\) 1.45558e7 1.12679
\(700\) 0 0
\(701\) 1.60390e7 1.23277 0.616384 0.787446i \(-0.288597\pi\)
0.616384 + 0.787446i \(0.288597\pi\)
\(702\) 0 0
\(703\) 9.58673e6 0.731614
\(704\) 0 0
\(705\) 1.22399e7 0.927480
\(706\) 0 0
\(707\) −8.04125e6 −0.605028
\(708\) 0 0
\(709\) −1.64673e7 −1.23029 −0.615145 0.788414i \(-0.710903\pi\)
−0.615145 + 0.788414i \(0.710903\pi\)
\(710\) 0 0
\(711\) 6.20669e6 0.460454
\(712\) 0 0
\(713\) −2.72241e6 −0.200553
\(714\) 0 0
\(715\) 6.61794e6 0.484125
\(716\) 0 0
\(717\) −1.48753e7 −1.08061
\(718\) 0 0
\(719\) −5.48538e6 −0.395717 −0.197859 0.980231i \(-0.563399\pi\)
−0.197859 + 0.980231i \(0.563399\pi\)
\(720\) 0 0
\(721\) −2.67202e7 −1.91426
\(722\) 0 0
\(723\) −8.35619e6 −0.594515
\(724\) 0 0
\(725\) −2.50604e6 −0.177070
\(726\) 0 0
\(727\) 2.75484e7 1.93313 0.966563 0.256431i \(-0.0825465\pi\)
0.966563 + 0.256431i \(0.0825465\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 4.95432e6 0.342918
\(732\) 0 0
\(733\) 1.51233e7 1.03965 0.519825 0.854273i \(-0.325997\pi\)
0.519825 + 0.854273i \(0.325997\pi\)
\(734\) 0 0
\(735\) −1.52705e7 −1.04264
\(736\) 0 0
\(737\) −6.88295e6 −0.466773
\(738\) 0 0
\(739\) 1.56004e7 1.05081 0.525405 0.850852i \(-0.323914\pi\)
0.525405 + 0.850852i \(0.323914\pi\)
\(740\) 0 0
\(741\) 1.48612e7 0.994281
\(742\) 0 0
\(743\) −1.77279e6 −0.117811 −0.0589053 0.998264i \(-0.518761\pi\)
−0.0589053 + 0.998264i \(0.518761\pi\)
\(744\) 0 0
\(745\) 2.63603e7 1.74004
\(746\) 0 0
\(747\) 7.82681e6 0.513196
\(748\) 0 0
\(749\) −5.97386e6 −0.389090
\(750\) 0 0
\(751\) −5.07089e6 −0.328083 −0.164042 0.986453i \(-0.552453\pi\)
−0.164042 + 0.986453i \(0.552453\pi\)
\(752\) 0 0
\(753\) −8.18724e6 −0.526199
\(754\) 0 0
\(755\) 1.53962e7 0.982987
\(756\) 0 0
\(757\) −733561. −0.0465261 −0.0232631 0.999729i \(-0.507406\pi\)
−0.0232631 + 0.999729i \(0.507406\pi\)
\(758\) 0 0
\(759\) −602401. −0.0379561
\(760\) 0 0
\(761\) 2.90077e6 0.181573 0.0907866 0.995870i \(-0.471062\pi\)
0.0907866 + 0.995870i \(0.471062\pi\)
\(762\) 0 0
\(763\) −1.36855e7 −0.851039
\(764\) 0 0
\(765\) −5.64984e6 −0.349046
\(766\) 0 0
\(767\) −3.28431e7 −2.01584
\(768\) 0 0
\(769\) −6.02882e6 −0.367635 −0.183817 0.982960i \(-0.558846\pi\)
−0.183817 + 0.982960i \(0.558846\pi\)
\(770\) 0 0
\(771\) −474001. −0.0287173
\(772\) 0 0
\(773\) 1.48827e7 0.895844 0.447922 0.894073i \(-0.352164\pi\)
0.447922 + 0.894073i \(0.352164\pi\)
\(774\) 0 0
\(775\) 1.90035e6 0.113653
\(776\) 0 0
\(777\) −9.86309e6 −0.586085
\(778\) 0 0
\(779\) 2.32951e7 1.37537
\(780\) 0 0
\(781\) 4.51512e6 0.264876
\(782\) 0 0
\(783\) 4.94745e6 0.288388
\(784\) 0 0
\(785\) 2.72315e6 0.157724
\(786\) 0 0
\(787\) −2.19846e7 −1.26526 −0.632631 0.774453i \(-0.718025\pi\)
−0.632631 + 0.774453i \(0.718025\pi\)
\(788\) 0 0
\(789\) 4.18288e6 0.239212
\(790\) 0 0
\(791\) −5.38164e6 −0.305825
\(792\) 0 0
\(793\) −2.33685e7 −1.31962
\(794\) 0 0
\(795\) −1.33936e7 −0.751586
\(796\) 0 0
\(797\) −2.92578e6 −0.163153 −0.0815765 0.996667i \(-0.525995\pi\)
−0.0815765 + 0.996667i \(0.525995\pi\)
\(798\) 0 0
\(799\) 2.71476e7 1.50440
\(800\) 0 0
\(801\) 2.43655e6 0.134182
\(802\) 0 0
\(803\) −4.29404e6 −0.235005
\(804\) 0 0
\(805\) −6.67096e6 −0.362826
\(806\) 0 0
\(807\) −8.71578e6 −0.471110
\(808\) 0 0
\(809\) −2.36363e6 −0.126972 −0.0634860 0.997983i \(-0.520222\pi\)
−0.0634860 + 0.997983i \(0.520222\pi\)
\(810\) 0 0
\(811\) 1.40226e7 0.748648 0.374324 0.927298i \(-0.377875\pi\)
0.374324 + 0.927298i \(0.377875\pi\)
\(812\) 0 0
\(813\) 2.06335e7 1.09483
\(814\) 0 0
\(815\) −1.11354e7 −0.587236
\(816\) 0 0
\(817\) −7.83548e6 −0.410686
\(818\) 0 0
\(819\) −1.52896e7 −0.796503
\(820\) 0 0
\(821\) 3.57170e7 1.84934 0.924670 0.380769i \(-0.124341\pi\)
0.924670 + 0.380769i \(0.124341\pi\)
\(822\) 0 0
\(823\) 1.15100e7 0.592344 0.296172 0.955135i \(-0.404290\pi\)
0.296172 + 0.955135i \(0.404290\pi\)
\(824\) 0 0
\(825\) 420499. 0.0215095
\(826\) 0 0
\(827\) 1.24864e7 0.634854 0.317427 0.948283i \(-0.397181\pi\)
0.317427 + 0.948283i \(0.397181\pi\)
\(828\) 0 0
\(829\) −1.13398e7 −0.573087 −0.286544 0.958067i \(-0.592506\pi\)
−0.286544 + 0.958067i \(0.592506\pi\)
\(830\) 0 0
\(831\) −2.61858e6 −0.131542
\(832\) 0 0
\(833\) −3.38692e7 −1.69119
\(834\) 0 0
\(835\) −1.20125e7 −0.596232
\(836\) 0 0
\(837\) −3.75168e6 −0.185102
\(838\) 0 0
\(839\) −2.91430e7 −1.42932 −0.714659 0.699473i \(-0.753418\pi\)
−0.714659 + 0.699473i \(0.753418\pi\)
\(840\) 0 0
\(841\) 2.55471e7 1.24552
\(842\) 0 0
\(843\) 3.52780e6 0.170976
\(844\) 0 0
\(845\) 2.43319e7 1.17229
\(846\) 0 0
\(847\) −3.09419e7 −1.48197
\(848\) 0 0
\(849\) 6.59491e6 0.314007
\(850\) 0 0
\(851\) −2.71751e6 −0.128632
\(852\) 0 0
\(853\) −9.77420e6 −0.459948 −0.229974 0.973197i \(-0.573864\pi\)
−0.229974 + 0.973197i \(0.573864\pi\)
\(854\) 0 0
\(855\) 8.93548e6 0.418025
\(856\) 0 0
\(857\) −2.33262e7 −1.08491 −0.542453 0.840086i \(-0.682504\pi\)
−0.542453 + 0.840086i \(0.682504\pi\)
\(858\) 0 0
\(859\) 1.14340e7 0.528708 0.264354 0.964426i \(-0.414841\pi\)
0.264354 + 0.964426i \(0.414841\pi\)
\(860\) 0 0
\(861\) −2.39666e7 −1.10179
\(862\) 0 0
\(863\) 7.70458e6 0.352145 0.176073 0.984377i \(-0.443661\pi\)
0.176073 + 0.984377i \(0.443661\pi\)
\(864\) 0 0
\(865\) 9.34277e6 0.424556
\(866\) 0 0
\(867\) 247605. 0.0111870
\(868\) 0 0
\(869\) −9.69534e6 −0.435525
\(870\) 0 0
\(871\) −4.81332e7 −2.14981
\(872\) 0 0
\(873\) −1.18561e6 −0.0526511
\(874\) 0 0
\(875\) −3.47513e7 −1.53444
\(876\) 0 0
\(877\) 1.42476e7 0.625520 0.312760 0.949832i \(-0.398746\pi\)
0.312760 + 0.949832i \(0.398746\pi\)
\(878\) 0 0
\(879\) −1.46328e7 −0.638788
\(880\) 0 0
\(881\) −1.76114e7 −0.764459 −0.382230 0.924067i \(-0.624844\pi\)
−0.382230 + 0.924067i \(0.624844\pi\)
\(882\) 0 0
\(883\) 9.40461e6 0.405919 0.202959 0.979187i \(-0.434944\pi\)
0.202959 + 0.979187i \(0.434944\pi\)
\(884\) 0 0
\(885\) −1.97472e7 −0.847517
\(886\) 0 0
\(887\) −1.38034e7 −0.589086 −0.294543 0.955638i \(-0.595167\pi\)
−0.294543 + 0.955638i \(0.595167\pi\)
\(888\) 0 0
\(889\) −7.63383e6 −0.323958
\(890\) 0 0
\(891\) −830152. −0.0350319
\(892\) 0 0
\(893\) −4.29351e7 −1.80170
\(894\) 0 0
\(895\) −9.83888e6 −0.410571
\(896\) 0 0
\(897\) −4.21265e6 −0.174813
\(898\) 0 0
\(899\) −3.49262e7 −1.44129
\(900\) 0 0
\(901\) −2.97064e7 −1.21910
\(902\) 0 0
\(903\) 8.06135e6 0.328994
\(904\) 0 0
\(905\) 1.09724e7 0.445328
\(906\) 0 0
\(907\) −802157. −0.0323773 −0.0161887 0.999869i \(-0.505153\pi\)
−0.0161887 + 0.999869i \(0.505153\pi\)
\(908\) 0 0
\(909\) −3.05319e6 −0.122559
\(910\) 0 0
\(911\) −4.80457e6 −0.191804 −0.0959022 0.995391i \(-0.530574\pi\)
−0.0959022 + 0.995391i \(0.530574\pi\)
\(912\) 0 0
\(913\) −1.22261e7 −0.485412
\(914\) 0 0
\(915\) −1.40506e7 −0.554807
\(916\) 0 0
\(917\) −1.66654e7 −0.654473
\(918\) 0 0
\(919\) −2.84366e7 −1.11068 −0.555339 0.831624i \(-0.687412\pi\)
−0.555339 + 0.831624i \(0.687412\pi\)
\(920\) 0 0
\(921\) 1.12123e7 0.435556
\(922\) 0 0
\(923\) 3.15747e7 1.21993
\(924\) 0 0
\(925\) 1.89693e6 0.0728948
\(926\) 0 0
\(927\) −1.01454e7 −0.387766
\(928\) 0 0
\(929\) 5.06040e6 0.192374 0.0961868 0.995363i \(-0.469335\pi\)
0.0961868 + 0.995363i \(0.469335\pi\)
\(930\) 0 0
\(931\) 5.35657e7 2.02541
\(932\) 0 0
\(933\) −2.94166e7 −1.10634
\(934\) 0 0
\(935\) 8.82549e6 0.330149
\(936\) 0 0
\(937\) 1.27197e7 0.473291 0.236646 0.971596i \(-0.423952\pi\)
0.236646 + 0.971596i \(0.423952\pi\)
\(938\) 0 0
\(939\) −1.53145e7 −0.566810
\(940\) 0 0
\(941\) −1.66122e7 −0.611581 −0.305790 0.952099i \(-0.598921\pi\)
−0.305790 + 0.952099i \(0.598921\pi\)
\(942\) 0 0
\(943\) −6.60335e6 −0.241816
\(944\) 0 0
\(945\) −9.19306e6 −0.334873
\(946\) 0 0
\(947\) −4.77751e7 −1.73112 −0.865558 0.500808i \(-0.833036\pi\)
−0.865558 + 0.500808i \(0.833036\pi\)
\(948\) 0 0
\(949\) −3.00286e7 −1.08236
\(950\) 0 0
\(951\) 8.17485e6 0.293109
\(952\) 0 0
\(953\) −3.77139e7 −1.34514 −0.672572 0.740032i \(-0.734810\pi\)
−0.672572 + 0.740032i \(0.734810\pi\)
\(954\) 0 0
\(955\) 4.55007e7 1.61439
\(956\) 0 0
\(957\) −7.72829e6 −0.272775
\(958\) 0 0
\(959\) −1.98007e7 −0.695239
\(960\) 0 0
\(961\) −2.14436e6 −0.0749014
\(962\) 0 0
\(963\) −2.26822e6 −0.0788169
\(964\) 0 0
\(965\) −2.27077e7 −0.784973
\(966\) 0 0
\(967\) 1.36407e7 0.469105 0.234553 0.972103i \(-0.424638\pi\)
0.234553 + 0.972103i \(0.424638\pi\)
\(968\) 0 0
\(969\) 1.98185e7 0.678050
\(970\) 0 0
\(971\) 3.96602e7 1.34992 0.674958 0.737856i \(-0.264162\pi\)
0.674958 + 0.737856i \(0.264162\pi\)
\(972\) 0 0
\(973\) 8.89843e7 3.01322
\(974\) 0 0
\(975\) 2.94059e6 0.0990658
\(976\) 0 0
\(977\) −9.70600e6 −0.325315 −0.162657 0.986683i \(-0.552007\pi\)
−0.162657 + 0.986683i \(0.552007\pi\)
\(978\) 0 0
\(979\) −3.80608e6 −0.126917
\(980\) 0 0
\(981\) −5.19626e6 −0.172393
\(982\) 0 0
\(983\) 2.88398e7 0.951937 0.475969 0.879462i \(-0.342098\pi\)
0.475969 + 0.879462i \(0.342098\pi\)
\(984\) 0 0
\(985\) 1.57267e7 0.516473
\(986\) 0 0
\(987\) 4.41728e7 1.44332
\(988\) 0 0
\(989\) 2.22109e6 0.0722064
\(990\) 0 0
\(991\) −3.94595e7 −1.27634 −0.638172 0.769894i \(-0.720309\pi\)
−0.638172 + 0.769894i \(0.720309\pi\)
\(992\) 0 0
\(993\) −1.92910e7 −0.620844
\(994\) 0 0
\(995\) 4.17374e7 1.33650
\(996\) 0 0
\(997\) 3.61342e7 1.15128 0.575639 0.817704i \(-0.304753\pi\)
0.575639 + 0.817704i \(0.304753\pi\)
\(998\) 0 0
\(999\) −3.74492e6 −0.118722
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 1104.6.a.i.1.3 3
4.3 odd 2 69.6.a.b.1.2 3
12.11 even 2 207.6.a.c.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.b.1.2 3 4.3 odd 2
207.6.a.c.1.2 3 12.11 even 2
1104.6.a.i.1.3 3 1.1 even 1 trivial