Properties

Label 207.6.a.c.1.2
Level $207$
Weight $6$
Character 207.1
Self dual yes
Analytic conductor $33.199$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 207.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.1994507013\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5333.1
Defining polynomial: \(x^{3} - x^{2} - 11 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.49331\) of defining polynomial
Character \(\chi\) \(=\) 207.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.29009 q^{2} -30.3357 q^{4} -59.1123 q^{5} -213.331 q^{7} -80.4187 q^{8} +O(q^{10})\) \(q+1.29009 q^{2} -30.3357 q^{4} -59.1123 q^{5} -213.331 q^{7} -80.4187 q^{8} -76.2602 q^{10} -126.528 q^{11} -884.825 q^{13} -275.217 q^{14} +866.994 q^{16} +1179.98 q^{17} -1866.19 q^{19} +1793.21 q^{20} -163.233 q^{22} -529.000 q^{23} +369.263 q^{25} -1141.50 q^{26} +6471.55 q^{28} +6786.62 q^{29} -5146.34 q^{31} +3691.90 q^{32} +1522.28 q^{34} +12610.5 q^{35} +5137.07 q^{37} -2407.55 q^{38} +4753.73 q^{40} -12482.7 q^{41} +4198.66 q^{43} +3838.32 q^{44} -682.458 q^{46} -23006.9 q^{47} +28703.3 q^{49} +476.382 q^{50} +26841.8 q^{52} -25175.4 q^{53} +7479.38 q^{55} +17155.8 q^{56} +8755.36 q^{58} +37118.1 q^{59} +26410.4 q^{61} -6639.24 q^{62} -22980.9 q^{64} +52304.0 q^{65} -54398.5 q^{67} -35795.4 q^{68} +16268.7 q^{70} -35684.7 q^{71} +33937.4 q^{73} +6627.29 q^{74} +56612.0 q^{76} +26992.5 q^{77} -76625.8 q^{79} -51250.0 q^{80} -16103.8 q^{82} +96627.2 q^{83} -69751.1 q^{85} +5416.65 q^{86} +10175.2 q^{88} -30080.8 q^{89} +188761. q^{91} +16047.6 q^{92} -29680.9 q^{94} +110315. q^{95} -14637.2 q^{97} +37029.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8} + O(q^{10}) \) \( 3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8} + 282 q^{10} + 376 q^{11} - 858 q^{13} - 588 q^{14} + 2738 q^{16} + 2548 q^{17} - 2846 q^{19} + 4618 q^{20} - 5050 q^{22} - 1587 q^{23} + 753 q^{25} + 7788 q^{26} + 4736 q^{28} + 16370 q^{29} - 14756 q^{31} + 3878 q^{32} + 16520 q^{34} + 18520 q^{35} + 15874 q^{37} - 12438 q^{38} + 38270 q^{40} - 12606 q^{41} + 3154 q^{43} - 27114 q^{44} - 4232 q^{46} - 29928 q^{47} + 4471 q^{49} - 1452 q^{50} + 86856 q^{52} + 44084 q^{53} + 38360 q^{55} + 35704 q^{56} + 73316 q^{58} + 29300 q^{59} + 54010 q^{61} - 99908 q^{62} - 1582 q^{64} + 51216 q^{65} + 43390 q^{67} + 69840 q^{68} - 2476 q^{70} - 23424 q^{71} - 91402 q^{73} + 2294 q^{74} - 14274 q^{76} + 97208 q^{77} - 49398 q^{79} + 52626 q^{80} + 40152 q^{82} + 103936 q^{83} + 5888 q^{85} - 133634 q^{86} + 48898 q^{88} - 96112 q^{89} + 129228 q^{91} - 11638 q^{92} - 133688 q^{94} + 55928 q^{95} - 135318 q^{97} - 108440 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.29009 0.228058 0.114029 0.993477i \(-0.463624\pi\)
0.114029 + 0.993477i \(0.463624\pi\)
\(3\) 0 0
\(4\) −30.3357 −0.947990
\(5\) −59.1123 −1.05743 −0.528716 0.848799i \(-0.677326\pi\)
−0.528716 + 0.848799i \(0.677326\pi\)
\(6\) 0 0
\(7\) −213.331 −1.64554 −0.822772 0.568371i \(-0.807574\pi\)
−0.822772 + 0.568371i \(0.807574\pi\)
\(8\) −80.4187 −0.444255
\(9\) 0 0
\(10\) −76.2602 −0.241156
\(11\) −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 \(0.758648\pi\)
\(14\) −275.217 −0.375280
\(15\) 0 0
\(16\) 866.994 0.846674
\(17\) 1179.98 0.990264 0.495132 0.868818i \(-0.335120\pi\)
0.495132 + 0.868818i \(0.335120\pi\)
\(18\) 0 0
\(19\) −1866.19 −1.18596 −0.592981 0.805216i \(-0.702049\pi\)
−0.592981 + 0.805216i \(0.702049\pi\)
\(20\) 1793.21 1.00244
\(21\) 0 0
\(22\) −163.233 −0.0719037
\(23\) −529.000 −0.208514
\(24\) 0 0
\(25\) 369.263 0.118164
\(26\) −1141.50 −0.331165
\(27\) 0 0
\(28\) 6471.55 1.55996
\(29\) 6786.62 1.49851 0.749253 0.662284i \(-0.230413\pi\)
0.749253 + 0.662284i \(0.230413\pi\)
\(30\) 0 0
\(31\) −5146.34 −0.961820 −0.480910 0.876770i \(-0.659694\pi\)
−0.480910 + 0.876770i \(0.659694\pi\)
\(32\) 3691.90 0.637345
\(33\) 0 0
\(34\) 1522.28 0.225838
\(35\) 12610.5 1.74005
\(36\) 0 0
\(37\) 5137.07 0.616895 0.308448 0.951241i \(-0.400191\pi\)
0.308448 + 0.951241i \(0.400191\pi\)
\(38\) −2407.55 −0.270468
\(39\) 0 0
\(40\) 4753.73 0.469769
\(41\) −12482.7 −1.15971 −0.579855 0.814720i \(-0.696891\pi\)
−0.579855 + 0.814720i \(0.696891\pi\)
\(42\) 0 0
\(43\) 4198.66 0.346290 0.173145 0.984896i \(-0.444607\pi\)
0.173145 + 0.984896i \(0.444607\pi\)
\(44\) 3838.32 0.298889
\(45\) 0 0
\(46\) −682.458 −0.0475534
\(47\) −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
\(50\) 476.382 0.0269483
\(51\) 0 0
\(52\) 26841.8 1.37658
\(53\) −25175.4 −1.23108 −0.615541 0.788105i \(-0.711062\pi\)
−0.615541 + 0.788105i \(0.711062\pi\)
\(54\) 0 0
\(55\) 7479.38 0.333395
\(56\) 17155.8 0.731041
\(57\) 0 0
\(58\) 8755.36 0.341746
\(59\) 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 \(0.349861\pi\)
\(62\) −6639.24 −0.219351
\(63\) 0 0
\(64\) −22980.9 −0.701322
\(65\) 52304.0 1.53551
\(66\) 0 0
\(67\) −54398.5 −1.48047 −0.740236 0.672347i \(-0.765286\pi\)
−0.740236 + 0.672347i \(0.765286\pi\)
\(68\) −35795.4 −0.938760
\(69\) 0 0
\(70\) 16268.7 0.396833
\(71\) −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 \(0.378437\pi\)
\(74\) 6627.29 0.140688
\(75\) 0 0
\(76\) 56612.0 1.12428
\(77\) 26992.5 0.518819
\(78\) 0 0
\(79\) −76625.8 −1.38136 −0.690681 0.723160i \(-0.742689\pi\)
−0.690681 + 0.723160i \(0.742689\pi\)
\(80\) −51250.0 −0.895300
\(81\) 0 0
\(82\) −16103.8 −0.264481
\(83\) 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
\(86\) 5416.65 0.0789741
\(87\) 0 0
\(88\) 10175.2 0.140068
\(89\) −30080.8 −0.402545 −0.201273 0.979535i \(-0.564508\pi\)
−0.201273 + 0.979535i \(0.564508\pi\)
\(90\) 0 0
\(91\) 188761. 2.38951
\(92\) 16047.6 0.197669
\(93\) 0 0
\(94\) −29680.9 −0.346464
\(95\) 110315. 1.25408
\(96\) 0 0
\(97\) −14637.2 −0.157953 −0.0789766 0.996876i \(-0.525165\pi\)
−0.0789766 + 0.996876i \(0.525165\pi\)
\(98\) 37029.9 0.389482
\(99\) 0 0
\(100\) −11201.8 −0.112018
\(101\) 37693.7 0.367676 0.183838 0.982957i \(-0.441148\pi\)
0.183838 + 0.982957i \(0.441148\pi\)
\(102\) 0 0
\(103\) 125252. 1.16330 0.581649 0.813440i \(-0.302408\pi\)
0.581649 + 0.813440i \(0.302408\pi\)
\(104\) 71156.5 0.645106
\(105\) 0 0
\(106\) −32478.6 −0.280758
\(107\) −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 \(0.583257\pi\)
\(110\) 9649.08 0.0760333
\(111\) 0 0
\(112\) −184957. −1.39324
\(113\) 25226.7 0.185851 0.0929253 0.995673i \(-0.470378\pi\)
0.0929253 + 0.995673i \(0.470378\pi\)
\(114\) 0 0
\(115\) 31270.4 0.220490
\(116\) −205877. −1.42057
\(117\) 0 0
\(118\) 47885.8 0.316593
\(119\) −251726. −1.62952
\(120\) 0 0
\(121\) −145042. −0.900594
\(122\) 34071.8 0.207250
\(123\) 0 0
\(124\) 156118. 0.911796
\(125\) 162898. 0.932482
\(126\) 0 0
\(127\) 35783.9 0.196870 0.0984348 0.995144i \(-0.468616\pi\)
0.0984348 + 0.995144i \(0.468616\pi\)
\(128\) −147788. −0.797288
\(129\) 0 0
\(130\) 67477.0 0.350185
\(131\) −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
\(134\) −70179.1 −0.337634
\(135\) 0 0
\(136\) −94892.2 −0.439930
\(137\) 92816.7 0.422498 0.211249 0.977432i \(-0.432247\pi\)
0.211249 + 0.977432i \(0.432247\pi\)
\(138\) 0 0
\(139\) −417118. −1.83114 −0.915570 0.402159i \(-0.868260\pi\)
−0.915570 + 0.402159i \(0.868260\pi\)
\(140\) −382548. −1.64955
\(141\) 0 0
\(142\) −46036.5 −0.191594
\(143\) 111955. 0.457831
\(144\) 0 0
\(145\) −401173. −1.58457
\(146\) 43782.3 0.169987
\(147\) 0 0
\(148\) −155836. −0.584810
\(149\) −445936. −1.64553 −0.822766 0.568380i \(-0.807570\pi\)
−0.822766 + 0.568380i \(0.807570\pi\)
\(150\) 0 0
\(151\) −260458. −0.929597 −0.464799 0.885416i \(-0.653873\pi\)
−0.464799 + 0.885416i \(0.653873\pi\)
\(152\) 150076. 0.526869
\(153\) 0 0
\(154\) 34822.7 0.118321
\(155\) 304212. 1.01706
\(156\) 0 0
\(157\) 46067.4 0.149157 0.0745787 0.997215i \(-0.476239\pi\)
0.0745787 + 0.997215i \(0.476239\pi\)
\(158\) −98854.3 −0.315031
\(159\) 0 0
\(160\) −218237. −0.673950
\(161\) 112852. 0.343120
\(162\) 0 0
\(163\) 188377. 0.555341 0.277671 0.960676i \(-0.410438\pi\)
0.277671 + 0.960676i \(0.410438\pi\)
\(164\) 378671. 1.09939
\(165\) 0 0
\(166\) 124658. 0.351115
\(167\) −203214. −0.563849 −0.281924 0.959437i \(-0.590973\pi\)
−0.281924 + 0.959437i \(0.590973\pi\)
\(168\) 0 0
\(169\) 411622. 1.10862
\(170\) −89985.3 −0.238808
\(171\) 0 0
\(172\) −127369. −0.328279
\(173\) −158051. −0.401497 −0.200749 0.979643i \(-0.564337\pi\)
−0.200749 + 0.979643i \(0.564337\pi\)
\(174\) 0 0
\(175\) −78775.3 −0.194444
\(176\) −109699. −0.266945
\(177\) 0 0
\(178\) −38807.0 −0.0918037
\(179\) −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 \(0.432468\pi\)
\(182\) 243519. 0.544947
\(183\) 0 0
\(184\) 42541.5 0.0926335
\(185\) −303664. −0.652325
\(186\) 0 0
\(187\) −149300. −0.312217
\(188\) 697928. 1.44018
\(189\) 0 0
\(190\) 142316. 0.286002
\(191\) 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 \(0.621043\pi\)
\(194\) −18883.3 −0.0360225
\(195\) 0 0
\(196\) −870734. −1.61899
\(197\) −266048. −0.488421 −0.244211 0.969722i \(-0.578529\pi\)
−0.244211 + 0.969722i \(0.578529\pi\)
\(198\) 0 0
\(199\) −706070. −1.26391 −0.631953 0.775007i \(-0.717746\pi\)
−0.631953 + 0.775007i \(0.717746\pi\)
\(200\) −29695.6 −0.0524949
\(201\) 0 0
\(202\) 48628.3 0.0838515
\(203\) −1.44780e6 −2.46586
\(204\) 0 0
\(205\) 737882. 1.22631
\(206\) 161586. 0.265300
\(207\) 0 0
\(208\) −767138. −1.22946
\(209\) 236125. 0.373918
\(210\) 0 0
\(211\) −290636. −0.449410 −0.224705 0.974427i \(-0.572142\pi\)
−0.224705 + 0.974427i \(0.572142\pi\)
\(212\) 763712. 1.16705
\(213\) 0 0
\(214\) −36126.1 −0.0539245
\(215\) −248192. −0.366178
\(216\) 0 0
\(217\) 1.09788e6 1.58272
\(218\) −82761.1 −0.117947
\(219\) 0 0
\(220\) −226892. −0.316055
\(221\) −1.04407e6 −1.43797
\(222\) 0 0
\(223\) −574915. −0.774179 −0.387089 0.922042i \(-0.626519\pi\)
−0.387089 + 0.922042i \(0.626519\pi\)
\(224\) −787598. −1.04878
\(225\) 0 0
\(226\) 32544.7 0.0423847
\(227\) 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 \(0.315158\pi\)
\(230\) 40341.7 0.0502845
\(231\) 0 0
\(232\) −545771. −0.665718
\(233\) 1.61732e6 1.95166 0.975832 0.218522i \(-0.0701235\pi\)
0.975832 + 0.218522i \(0.0701235\pi\)
\(234\) 0 0
\(235\) 1.35999e6 1.60644
\(236\) −1.12600e6 −1.31601
\(237\) 0 0
\(238\) −324750. −0.371626
\(239\) 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 \(0.327842\pi\)
\(242\) −187117. −0.205388
\(243\) 0 0
\(244\) −801176. −0.861496
\(245\) −1.69672e6 −1.80590
\(246\) 0 0
\(247\) 1.65125e6 1.72215
\(248\) 413862. 0.427293
\(249\) 0 0
\(250\) 210153. 0.212660
\(251\) 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
\(254\) 46164.5 0.0448977
\(255\) 0 0
\(256\) 544729. 0.519494
\(257\) −52666.7 −0.0497397 −0.0248699 0.999691i \(-0.507917\pi\)
−0.0248699 + 0.999691i \(0.507917\pi\)
\(258\) 0 0
\(259\) −1.09590e6 −1.01513
\(260\) −1.58668e6 −1.45564
\(261\) 0 0
\(262\) −100781. −0.0907042
\(263\) −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
\(266\) 513606. 0.445068
\(267\) 0 0
\(268\) 1.65022e6 1.40347
\(269\) −968420. −0.815987 −0.407993 0.912985i \(-0.633771\pi\)
−0.407993 + 0.912985i \(0.633771\pi\)
\(270\) 0 0
\(271\) 2.29262e6 1.89630 0.948152 0.317817i \(-0.102950\pi\)
0.948152 + 0.317817i \(0.102950\pi\)
\(272\) 1.02303e6 0.838431
\(273\) 0 0
\(274\) 119742. 0.0963541
\(275\) −46722.2 −0.0372556
\(276\) 0 0
\(277\) 290954. 0.227837 0.113919 0.993490i \(-0.463660\pi\)
0.113919 + 0.993490i \(0.463660\pi\)
\(278\) −538120. −0.417606
\(279\) 0 0
\(280\) −1.01412e6 −0.773027
\(281\) 391978. 0.296139 0.148070 0.988977i \(-0.452694\pi\)
0.148070 + 0.988977i \(0.452694\pi\)
\(282\) 0 0
\(283\) 732768. 0.543877 0.271938 0.962315i \(-0.412335\pi\)
0.271938 + 0.962315i \(0.412335\pi\)
\(284\) 1.08252e6 0.796415
\(285\) 0 0
\(286\) 144433. 0.104412
\(287\) 2.66295e6 1.90835
\(288\) 0 0
\(289\) −27511.7 −0.0193764
\(290\) −517549. −0.361374
\(291\) 0 0
\(292\) −1.02951e6 −0.706602
\(293\) −1.62587e6 −1.10641 −0.553207 0.833044i \(-0.686596\pi\)
−0.553207 + 0.833044i \(0.686596\pi\)
\(294\) 0 0
\(295\) −2.19414e6 −1.46794
\(296\) −413117. −0.274059
\(297\) 0 0
\(298\) −575298. −0.375277
\(299\) 468072. 0.302785
\(300\) 0 0
\(301\) −895706. −0.569835
\(302\) −336014. −0.212002
\(303\) 0 0
\(304\) −1.61797e6 −1.00412
\(305\) −1.56118e6 −0.960954
\(306\) 0 0
\(307\) 1.24581e6 0.754406 0.377203 0.926131i \(-0.376886\pi\)
0.377203 + 0.926131i \(0.376886\pi\)
\(308\) −818834. −0.491835
\(309\) 0 0
\(310\) 392461. 0.231949
\(311\) 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 \(0.336678\pi\)
\(314\) 59431.2 0.0340166
\(315\) 0 0
\(316\) 2.32450e6 1.30952
\(317\) 908317. 0.507679 0.253840 0.967246i \(-0.418306\pi\)
0.253840 + 0.967246i \(0.418306\pi\)
\(318\) 0 0
\(319\) −858699. −0.472459
\(320\) 1.35845e6 0.741601
\(321\) 0 0
\(322\) 145590. 0.0782512
\(323\) −2.20206e6 −1.17442
\(324\) 0 0
\(325\) −326733. −0.171587
\(326\) 243024. 0.126650
\(327\) 0 0
\(328\) 1.00384e6 0.515206
\(329\) 4.90809e6 2.49990
\(330\) 0 0
\(331\) −2.14345e6 −1.07533 −0.537666 0.843158i \(-0.680694\pi\)
−0.537666 + 0.843158i \(0.680694\pi\)
\(332\) −2.93125e6 −1.45951
\(333\) 0 0
\(334\) −262165. −0.128590
\(335\) 3.21562e6 1.56550
\(336\) 0 0
\(337\) 817498. 0.392114 0.196057 0.980593i \(-0.437186\pi\)
0.196057 + 0.980593i \(0.437186\pi\)
\(338\) 531030. 0.252829
\(339\) 0 0
\(340\) 2.11595e6 0.992676
\(341\) 651157. 0.303249
\(342\) 0 0
\(343\) −2.53785e6 −1.16475
\(344\) −337651. −0.153841
\(345\) 0 0
\(346\) −203900. −0.0915647
\(347\) −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 \(0.752105\pi\)
\(350\) −101627. −0.0443446
\(351\) 0 0
\(352\) −467130. −0.200947
\(353\) 1.72676e6 0.737555 0.368777 0.929518i \(-0.379776\pi\)
0.368777 + 0.929518i \(0.379776\pi\)
\(354\) 0 0
\(355\) 2.10940e6 0.888360
\(356\) 912522. 0.381609
\(357\) 0 0
\(358\) −214728. −0.0885484
\(359\) −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
\(362\) 239466. 0.0960446
\(363\) 0 0
\(364\) −5.72619e6 −2.26523
\(365\) −2.00612e6 −0.788177
\(366\) 0 0
\(367\) 1.17031e6 0.453560 0.226780 0.973946i \(-0.427180\pi\)
0.226780 + 0.973946i \(0.427180\pi\)
\(368\) −458640. −0.176544
\(369\) 0 0
\(370\) −391754. −0.148768
\(371\) 5.37070e6 2.02580
\(372\) 0 0
\(373\) −692891. −0.257865 −0.128933 0.991653i \(-0.541155\pi\)
−0.128933 + 0.991653i \(0.541155\pi\)
\(374\) −192611. −0.0712037
\(375\) 0 0
\(376\) 1.85018e6 0.674908
\(377\) −6.00497e6 −2.17599
\(378\) 0 0
\(379\) 2.12163e6 0.758702 0.379351 0.925253i \(-0.376147\pi\)
0.379351 + 0.925253i \(0.376147\pi\)
\(380\) −3.34646e6 −1.18885
\(381\) 0 0
\(382\) 993026. 0.348179
\(383\) 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
\(386\) −495582. −0.169296
\(387\) 0 0
\(388\) 444029. 0.149738
\(389\) −679811. −0.227779 −0.113890 0.993493i \(-0.536331\pi\)
−0.113890 + 0.993493i \(0.536331\pi\)
\(390\) 0 0
\(391\) −624208. −0.206484
\(392\) −2.30828e6 −0.758706
\(393\) 0 0
\(394\) −343226. −0.111388
\(395\) 4.52953e6 1.46070
\(396\) 0 0
\(397\) −5.39917e6 −1.71930 −0.859648 0.510886i \(-0.829317\pi\)
−0.859648 + 0.510886i \(0.829317\pi\)
\(398\) −910894. −0.288244
\(399\) 0 0
\(400\) 320148. 0.100046
\(401\) 2.17808e6 0.676416 0.338208 0.941071i \(-0.390179\pi\)
0.338208 + 0.941071i \(0.390179\pi\)
\(402\) 0 0
\(403\) 4.55361e6 1.39667
\(404\) −1.14346e6 −0.348553
\(405\) 0 0
\(406\) −1.86779e6 −0.562359
\(407\) −649985. −0.194499
\(408\) 0 0
\(409\) −356567. −0.105398 −0.0526991 0.998610i \(-0.516782\pi\)
−0.0526991 + 0.998610i \(0.516782\pi\)
\(410\) 951934. 0.279671
\(411\) 0 0
\(412\) −3.79960e6 −1.10279
\(413\) −7.91847e6 −2.28437
\(414\) 0 0
\(415\) −5.71186e6 −1.62801
\(416\) −3.26668e6 −0.925495
\(417\) 0 0
\(418\) 304623. 0.0852751
\(419\) −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 \(0.635723\pi\)
\(422\) −374947. −0.102492
\(423\) 0 0
\(424\) 2.02457e6 0.546914
\(425\) 435721. 0.117014
\(426\) 0 0
\(427\) −5.63416e6 −1.49541
\(428\) 849481. 0.224153
\(429\) 0 0
\(430\) −320191. −0.0835098
\(431\) −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 \(0.123902\pi\)
\(434\) 1.41636e6 0.360952
\(435\) 0 0
\(436\) 1.94607e6 0.490279
\(437\) 987212. 0.247290
\(438\) 0 0
\(439\) −760935. −0.188446 −0.0942229 0.995551i \(-0.530037\pi\)
−0.0942229 + 0.995551i \(0.530037\pi\)
\(440\) −601482. −0.148112
\(441\) 0 0
\(442\) −1.34695e6 −0.327941
\(443\) 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
\(446\) −741692. −0.176558
\(447\) 0 0
\(448\) 4.90255e6 1.15406
\(449\) −5.48897e6 −1.28492 −0.642458 0.766321i \(-0.722085\pi\)
−0.642458 + 0.766321i \(0.722085\pi\)
\(450\) 0 0
\(451\) 1.57942e6 0.365641
\(452\) −765268. −0.176184
\(453\) 0 0
\(454\) 1.77210e6 0.403506
\(455\) −1.11581e7 −2.52675
\(456\) 0 0
\(457\) −2.49968e6 −0.559879 −0.279940 0.960018i \(-0.590314\pi\)
−0.279940 + 0.960018i \(0.590314\pi\)
\(458\) 1.12331e6 0.250229
\(459\) 0 0
\(460\) −948608. −0.209022
\(461\) 8.10339e6 1.77588 0.887942 0.459956i \(-0.152135\pi\)
0.887942 + 0.459956i \(0.152135\pi\)
\(462\) 0 0
\(463\) −4.71421e6 −1.02201 −0.511007 0.859577i \(-0.670727\pi\)
−0.511007 + 0.859577i \(0.670727\pi\)
\(464\) 5.88396e6 1.26875
\(465\) 0 0
\(466\) 2.08649e6 0.445093
\(467\) 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
\(470\) 1.75451e6 0.366362
\(471\) 0 0
\(472\) −2.98499e6 −0.616720
\(473\) −531249. −0.109181
\(474\) 0 0
\(475\) −689113. −0.140138
\(476\) 7.63628e6 1.54477
\(477\) 0 0
\(478\) 2.13228e6 0.426849
\(479\) −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
\(482\) 1.19781e6 0.234838
\(483\) 0 0
\(484\) 4.39993e6 0.853754
\(485\) 865238. 0.167025
\(486\) 0 0
\(487\) −793366. −0.151583 −0.0757916 0.997124i \(-0.524148\pi\)
−0.0757916 + 0.997124i \(0.524148\pi\)
\(488\) −2.12389e6 −0.403721
\(489\) 0 0
\(490\) −2.18892e6 −0.411851
\(491\) 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
\(494\) 2.13026e6 0.392749
\(495\) 0 0
\(496\) −4.46184e6 −0.814348
\(497\) 7.61267e6 1.38244
\(498\) 0 0
\(499\) −5.08658e6 −0.914480 −0.457240 0.889343i \(-0.651162\pi\)
−0.457240 + 0.889343i \(0.651162\pi\)
\(500\) −4.94162e6 −0.883983
\(501\) 0 0
\(502\) 1.17359e6 0.207853
\(503\) −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
\(506\) 86350.3 0.0149930
\(507\) 0 0
\(508\) −1.08553e6 −0.186630
\(509\) 5.58035e6 0.954700 0.477350 0.878713i \(-0.341597\pi\)
0.477350 + 0.878713i \(0.341597\pi\)
\(510\) 0 0
\(511\) −7.23991e6 −1.22654
\(512\) 5.43197e6 0.915762
\(513\) 0 0
\(514\) −67944.9 −0.0113435
\(515\) −7.40393e6 −1.23011
\(516\) 0 0
\(517\) 2.91102e6 0.478981
\(518\) −1.41381e6 −0.231508
\(519\) 0 0
\(520\) −4.20622e6 −0.682156
\(521\) 4.02579e6 0.649766 0.324883 0.945754i \(-0.394675\pi\)
0.324883 + 0.945754i \(0.394675\pi\)
\(522\) 0 0
\(523\) −9.41059e6 −1.50440 −0.752198 0.658937i \(-0.771007\pi\)
−0.752198 + 0.658937i \(0.771007\pi\)
\(524\) 2.36981e6 0.377038
\(525\) 0 0
\(526\) −599588. −0.0944906
\(527\) −6.07256e6 −0.952457
\(528\) 0 0
\(529\) 279841. 0.0434783
\(530\) 1.91988e6 0.296883
\(531\) 0 0
\(532\) −1.20771e7 −1.85005
\(533\) 1.10450e7 1.68402
\(534\) 0 0
\(535\) 1.65531e6 0.250031
\(536\) 4.37466e6 0.657707
\(537\) 0 0
\(538\) −1.24935e6 −0.186092
\(539\) −3.63178e6 −0.538453
\(540\) 0 0
\(541\) −2.68147e6 −0.393894 −0.196947 0.980414i \(-0.563103\pi\)
−0.196947 + 0.980414i \(0.563103\pi\)
\(542\) 2.95768e6 0.432467
\(543\) 0 0
\(544\) 4.35636e6 0.631141
\(545\) 3.79214e6 0.546881
\(546\) 0 0
\(547\) 8.74676e6 1.24991 0.624955 0.780660i \(-0.285117\pi\)
0.624955 + 0.780660i \(0.285117\pi\)
\(548\) −2.81566e6 −0.400524
\(549\) 0 0
\(550\) −60275.8 −0.00849643
\(551\) −1.26651e7 −1.77717
\(552\) 0 0
\(553\) 1.63467e7 2.27309
\(554\) 375357. 0.0519601
\(555\) 0 0
\(556\) 1.26535e7 1.73590
\(557\) −2.07905e6 −0.283940 −0.141970 0.989871i \(-0.545344\pi\)
−0.141970 + 0.989871i \(0.545344\pi\)
\(558\) 0 0
\(559\) −3.71508e6 −0.502850
\(560\) 1.09332e7 1.47326
\(561\) 0 0
\(562\) 505687. 0.0675369
\(563\) −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
\(566\) 945338. 0.124035
\(567\) 0 0
\(568\) 2.86972e6 0.373223
\(569\) −6.58526e6 −0.852692 −0.426346 0.904560i \(-0.640199\pi\)
−0.426346 + 0.904560i \(0.640199\pi\)
\(570\) 0 0
\(571\) −6.31614e6 −0.810702 −0.405351 0.914161i \(-0.632851\pi\)
−0.405351 + 0.914161i \(0.632851\pi\)
\(572\) −3.39624e6 −0.434019
\(573\) 0 0
\(574\) 3.43545e6 0.435216
\(575\) −195340. −0.0246389
\(576\) 0 0
\(577\) 4.48677e6 0.561040 0.280520 0.959848i \(-0.409493\pi\)
0.280520 + 0.959848i \(0.409493\pi\)
\(578\) −35492.6 −0.00441894
\(579\) 0 0
\(580\) 1.21698e7 1.50216
\(581\) −2.06136e7 −2.53346
\(582\) 0 0
\(583\) 3.18540e6 0.388144
\(584\) −2.72920e6 −0.331134
\(585\) 0 0
\(586\) −2.09752e6 −0.252326
\(587\) 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
\(590\) −2.83064e6 −0.334776
\(591\) 0 0
\(592\) 4.45381e6 0.522309
\(593\) −7.02182e6 −0.819998 −0.409999 0.912086i \(-0.634471\pi\)
−0.409999 + 0.912086i \(0.634471\pi\)
\(594\) 0 0
\(595\) 1.48801e7 1.72311
\(596\) 1.35278e7 1.55995
\(597\) 0 0
\(598\) 603856. 0.0690527
\(599\) −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 \(0.415074\pi\)
\(602\) −1.15554e6 −0.129955
\(603\) 0 0
\(604\) 7.90116e6 0.881248
\(605\) 8.57374e6 0.952318
\(606\) 0 0
\(607\) −1.48942e7 −1.64076 −0.820380 0.571819i \(-0.806238\pi\)
−0.820380 + 0.571819i \(0.806238\pi\)
\(608\) −6.88977e6 −0.755868
\(609\) 0 0
\(610\) −2.01406e6 −0.219153
\(611\) 2.03570e7 2.20603
\(612\) 0 0
\(613\) 1.09408e7 1.17598 0.587989 0.808869i \(-0.299920\pi\)
0.587989 + 0.808869i \(0.299920\pi\)
\(614\) 1.60721e6 0.172048
\(615\) 0 0
\(616\) −2.17070e6 −0.230488
\(617\) 6.42199e6 0.679136 0.339568 0.940582i \(-0.389719\pi\)
0.339568 + 0.940582i \(0.389719\pi\)
\(618\) 0 0
\(619\) −3.05345e6 −0.320305 −0.160152 0.987092i \(-0.551199\pi\)
−0.160152 + 0.987092i \(0.551199\pi\)
\(620\) −9.22847e6 −0.964163
\(621\) 0 0
\(622\) 4.21668e6 0.437014
\(623\) 6.41719e6 0.662406
\(624\) 0 0
\(625\) −1.07832e7 −1.10420
\(626\) 2.19523e6 0.223895
\(627\) 0 0
\(628\) −1.39749e6 −0.141400
\(629\) 6.06162e6 0.610889
\(630\) 0 0
\(631\) 1.05012e7 1.04994 0.524971 0.851120i \(-0.324076\pi\)
0.524971 + 0.851120i \(0.324076\pi\)
\(632\) 6.16215e6 0.613677
\(633\) 0 0
\(634\) 1.17181e6 0.115780
\(635\) −2.11527e6 −0.208176
\(636\) 0 0
\(637\) −2.53974e7 −2.47994
\(638\) −1.10780e6 −0.107748
\(639\) 0 0
\(640\) 8.73610e6 0.843078
\(641\) 1.57928e7 1.51815 0.759073 0.651006i \(-0.225653\pi\)
0.759073 + 0.651006i \(0.225653\pi\)
\(642\) 0 0
\(643\) 551159. 0.0525714 0.0262857 0.999654i \(-0.491632\pi\)
0.0262857 + 0.999654i \(0.491632\pi\)
\(644\) −3.42345e6 −0.325274
\(645\) 0 0
\(646\) −2.84085e6 −0.267835
\(647\) −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
\(650\) −421515. −0.0391318
\(651\) 0 0
\(652\) −5.71456e6 −0.526458
\(653\) 2.56046e6 0.234982 0.117491 0.993074i \(-0.462515\pi\)
0.117491 + 0.993074i \(0.462515\pi\)
\(654\) 0 0
\(655\) 4.61783e6 0.420566
\(656\) −1.08224e7 −0.981895
\(657\) 0 0
\(658\) 6.33188e6 0.570122
\(659\) 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 \(0.172828\pi\)
\(662\) −2.76524e6 −0.245238
\(663\) 0 0
\(664\) −7.77064e6 −0.683969
\(665\) −2.35336e7 −2.06364
\(666\) 0 0
\(667\) −3.59012e6 −0.312460
\(668\) 6.16464e6 0.534523
\(669\) 0 0
\(670\) 4.14845e6 0.357025
\(671\) −3.34166e6 −0.286520
\(672\) 0 0
\(673\) 1.20397e7 1.02465 0.512326 0.858791i \(-0.328784\pi\)
0.512326 + 0.858791i \(0.328784\pi\)
\(674\) 1.05465e6 0.0894247
\(675\) 0 0
\(676\) −1.24868e7 −1.05096
\(677\) −99259.2 −0.00832337 −0.00416168 0.999991i \(-0.501325\pi\)
−0.00416168 + 0.999991i \(0.501325\pi\)
\(678\) 0 0
\(679\) 3.12257e6 0.259919
\(680\) 5.60929e6 0.465196
\(681\) 0 0
\(682\) 840052. 0.0691585
\(683\) −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
\(686\) −3.27406e6 −0.265630
\(687\) 0 0
\(688\) 3.64021e6 0.293194
\(689\) 2.22758e7 1.78766
\(690\) 0 0
\(691\) −1.77887e7 −1.41726 −0.708629 0.705581i \(-0.750686\pi\)
−0.708629 + 0.705581i \(0.750686\pi\)
\(692\) 4.79459e6 0.380615
\(693\) 0 0
\(694\) −4.13096e6 −0.325576
\(695\) 2.46568e7 1.93631
\(696\) 0 0
\(697\) −1.47293e7 −1.14842
\(698\) −4.17881e6 −0.324649
\(699\) 0 0
\(700\) 2.38970e6 0.184331
\(701\) −1.60390e7 −1.23277 −0.616384 0.787446i \(-0.711403\pi\)
−0.616384 + 0.787446i \(0.711403\pi\)
\(702\) 0 0
\(703\) −9.58673e6 −0.731614
\(704\) 2.90774e6 0.221118
\(705\) 0 0
\(706\) 2.22767e6 0.168205
\(707\) −8.04125e6 −0.605028
\(708\) 0 0
\(709\) −1.64673e7 −1.23029 −0.615145 0.788414i \(-0.710903\pi\)
−0.615145 + 0.788414i \(0.710903\pi\)
\(710\) 2.72132e6 0.202598
\(711\) 0 0
\(712\) 2.41906e6 0.178833
\(713\) 2.72241e6 0.200553
\(714\) 0 0
\(715\) −6.61794e6 −0.484125
\(716\) 5.04919e6 0.368077
\(717\) 0 0
\(718\) −4.02945e6 −0.291698
\(719\) −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
\(722\) 1.29854e6 0.0927071
\(723\) 0 0
\(724\) −5.63090e6 −0.399237
\(725\) 2.50604e6 0.177070
\(726\) 0 0
\(727\) −2.75484e7 −1.93313 −0.966563 0.256431i \(-0.917453\pi\)
−0.966563 + 0.256431i \(0.917453\pi\)
\(728\) −1.51799e7 −1.06155
\(729\) 0 0
\(730\) −2.58807e6 −0.179750
\(731\) 4.95432e6 0.342918
\(732\) 0 0
\(733\) 1.51233e7 1.03965 0.519825 0.854273i \(-0.325997\pi\)
0.519825 + 0.854273i \(0.325997\pi\)
\(734\) 1.50980e6 0.103438
\(735\) 0 0
\(736\) −1.95301e6 −0.132896
\(737\) 6.88295e6 0.466773
\(738\) 0 0
\(739\) −1.56004e7 −1.05081 −0.525405 0.850852i \(-0.676086\pi\)
−0.525405 + 0.850852i \(0.676086\pi\)
\(740\) 9.21185e6 0.618397
\(741\) 0 0
\(742\) 6.92870e6 0.462000
\(743\) −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
\(746\) −893892. −0.0588082
\(747\) 0 0
\(748\) 4.52913e6 0.295979
\(749\) 5.97386e6 0.389090
\(750\) 0 0
\(751\) 5.07089e6 0.328083 0.164042 0.986453i \(-0.447547\pi\)
0.164042 + 0.986453i \(0.447547\pi\)
\(752\) −1.99468e7 −1.28626
\(753\) 0 0
\(754\) −7.74696e6 −0.496253
\(755\) 1.53962e7 0.982987
\(756\) 0 0
\(757\) −733561. −0.0465261 −0.0232631 0.999729i \(-0.507406\pi\)
−0.0232631 + 0.999729i \(0.507406\pi\)
\(758\) 2.73709e6 0.173028
\(759\) 0 0
\(760\) −8.87135e6 −0.557129
\(761\) −2.90077e6 −0.181573 −0.0907866 0.995870i \(-0.528938\pi\)
−0.0907866 + 0.995870i \(0.528938\pi\)
\(762\) 0 0
\(763\) 1.36855e7 0.851039
\(764\) −2.33504e7 −1.44731
\(765\) 0 0
\(766\) 370358. 0.0228060
\(767\) −3.28431e7 −2.01584
\(768\) 0 0
\(769\) −6.02882e6 −0.367635 −0.183817 0.982960i \(-0.558846\pi\)
−0.183817 + 0.982960i \(0.558846\pi\)
\(770\) −2.05845e6 −0.125116
\(771\) 0 0
\(772\) 1.16533e7 0.703729
\(773\) −1.48827e7 −0.895844 −0.447922 0.894073i \(-0.647836\pi\)
−0.447922 + 0.894073i \(0.647836\pi\)
\(774\) 0 0
\(775\) −1.90035e6 −0.113653
\(776\) 1.17710e6 0.0701715
\(777\) 0 0
\(778\) −877019. −0.0519469
\(779\) 2.32951e7 1.37537
\(780\) 0 0
\(781\) 4.51512e6 0.264876
\(782\) −805285. −0.0470904
\(783\) 0 0
\(784\) 2.48856e7 1.44596
\(785\) −2.72315e6 −0.157724
\(786\) 0 0
\(787\) 2.19846e7 1.26526 0.632631 0.774453i \(-0.281975\pi\)
0.632631 + 0.774453i \(0.281975\pi\)
\(788\) 8.07075e6 0.463018
\(789\) 0 0
\(790\) 5.84350e6 0.333124
\(791\) −5.38164e6 −0.305825
\(792\) 0 0
\(793\) −2.33685e7 −1.31962
\(794\) −6.96542e6 −0.392099
\(795\) 0 0
\(796\) 2.14191e7 1.19817
\(797\) 2.92578e6 0.163153 0.0815765 0.996667i \(-0.474005\pi\)
0.0815765 + 0.996667i \(0.474005\pi\)
\(798\) 0 0
\(799\) −2.71476e7 −1.50440
\(800\) 1.36328e6 0.0753113
\(801\) 0 0
\(802\) 2.80993e6 0.154262
\(803\) −4.29404e6 −0.235005
\(804\) 0 0
\(805\) −6.67096e6 −0.362826
\(806\) 5.87457e6 0.318521
\(807\) 0 0
\(808\) −3.03128e6 −0.163342
\(809\) 2.36363e6 0.126972 0.0634860 0.997983i \(-0.479778\pi\)
0.0634860 + 0.997983i \(0.479778\pi\)
\(810\) 0 0
\(811\) −1.40226e7 −0.748648 −0.374324 0.927298i \(-0.622125\pi\)
−0.374324 + 0.927298i \(0.622125\pi\)
\(812\) 4.39200e7 2.33761
\(813\) 0 0
\(814\) −838540. −0.0443570
\(815\) −1.11354e7 −0.587236
\(816\) 0 0
\(817\) −7.83548e6 −0.410686
\(818\) −460004. −0.0240369
\(819\) 0 0
\(820\) −2.23841e7 −1.16253
\(821\) −3.57170e7 −1.84934 −0.924670 0.380769i \(-0.875659\pi\)
−0.924670 + 0.380769i \(0.875659\pi\)
\(822\) 0 0
\(823\) −1.15100e7 −0.592344 −0.296172 0.955135i \(-0.595710\pi\)
−0.296172 + 0.955135i \(0.595710\pi\)
\(824\) −1.00726e7 −0.516801
\(825\) 0 0
\(826\) −1.02155e7 −0.520968
\(827\) 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 \(0.592506\pi\)
\(830\) −7.36882e6 −0.371281
\(831\) 0 0
\(832\) 2.03341e7 1.01840
\(833\) 3.38692e7 1.69119
\(834\) 0 0
\(835\) 1.20125e7 0.596232
\(836\) −7.16302e6 −0.354471
\(837\) 0 0
\(838\) −4.93633e6 −0.242825
\(839\) −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
\(842\) −3.88078e6 −0.188642
\(843\) 0 0
\(844\) 8.81663e6 0.426036
\(845\) −2.43319e7 −1.17229
\(846\) 0 0
\(847\) 3.09419e7 1.48197
\(848\) −2.18269e7 −1.04232
\(849\) 0 0
\(850\) 562120. 0.0266859
\(851\) −2.71751e6 −0.128632
\(852\) 0 0
\(853\) −9.77420e6 −0.459948 −0.229974 0.973197i \(-0.573864\pi\)
−0.229974 + 0.973197i \(0.573864\pi\)
\(854\) −7.26858e6 −0.341040
\(855\) 0 0
\(856\) 2.25194e6 0.105044
\(857\) 2.33262e7 1.08491 0.542453 0.840086i \(-0.317496\pi\)
0.542453 + 0.840086i \(0.317496\pi\)
\(858\) 0 0
\(859\) −1.14340e7 −0.528708 −0.264354 0.964426i \(-0.585159\pi\)
−0.264354 + 0.964426i \(0.585159\pi\)
\(860\) 7.52908e6 0.347133
\(861\) 0 0
\(862\) −745426. −0.0341693
\(863\) 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
\(866\) 9.31329e6 0.421996
\(867\) 0 0
\(868\) −3.33048e7 −1.50040
\(869\) 9.69534e6 0.435525
\(870\) 0 0
\(871\) 4.81332e7 2.14981
\(872\) 5.15897e6 0.229759
\(873\) 0 0
\(874\) 1.27359e6 0.0563965
\(875\) −3.47513e7 −1.53444
\(876\) 0 0
\(877\) 1.42476e7 0.625520 0.312760 0.949832i \(-0.398746\pi\)
0.312760 + 0.949832i \(0.398746\pi\)
\(878\) −981676. −0.0429766
\(879\) 0 0
\(880\) 6.48457e6 0.282276
\(881\) 1.76114e7 0.764459 0.382230 0.924067i \(-0.375156\pi\)
0.382230 + 0.924067i \(0.375156\pi\)
\(882\) 0 0
\(883\) −9.40461e6 −0.405919 −0.202959 0.979187i \(-0.565056\pi\)
−0.202959 + 0.979187i \(0.565056\pi\)
\(884\) 3.16726e7 1.36318
\(885\) 0 0
\(886\) 8.99054e6 0.384770
\(887\) −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
\(890\) 2.29397e6 0.0970762
\(891\) 0 0
\(892\) 1.74404e7 0.733913
\(893\) 4.29351e7 1.80170
\(894\) 0 0
\(895\) 9.83888e6 0.410571
\(896\) 3.15279e7 1.31197
\(897\) 0 0
\(898\) −7.08127e6 −0.293036
\(899\) −3.49262e7 −1.44129
\(900\) 0 0
\(901\) −2.97064e7 −1.21910
\(902\) 2.03759e6 0.0833874
\(903\) 0 0
\(904\) −2.02870e6 −0.0825650
\(905\) −1.09724e7 −0.445328
\(906\) 0 0
\(907\) 802157. 0.0323773 0.0161887 0.999869i \(-0.494847\pi\)
0.0161887 + 0.999869i \(0.494847\pi\)
\(908\) −4.16699e7 −1.67729
\(909\) 0 0
\(910\) −1.43950e7 −0.576245
\(911\) −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
\(914\) −3.22482e6 −0.127685
\(915\) 0 0
\(916\) −2.64140e7 −1.04015
\(917\) 1.66654e7 0.654473
\(918\) 0 0
\(919\) 2.84366e7 1.11068 0.555339 0.831624i \(-0.312588\pi\)
0.555339 + 0.831624i \(0.312588\pi\)
\(920\) −2.51472e6 −0.0979537
\(921\) 0 0
\(922\) 1.04541e7 0.405005
\(923\) 3.15747e7 1.21993
\(924\) 0 0
\(925\) 1.89693e6 0.0728948
\(926\) −6.08176e6 −0.233078
\(927\) 0 0
\(928\) 2.50555e7 0.955066
\(929\) −5.06040e6 −0.192374 −0.0961868 0.995363i \(-0.530665\pi\)
−0.0961868 + 0.995363i \(0.530665\pi\)
\(930\) 0 0
\(931\) −5.35657e7 −2.02541
\(932\) −4.90624e7 −1.85016
\(933\) 0 0
\(934\) 1.33326e6 0.0500089
\(935\) 8.82549e6 0.330149
\(936\) 0 0
\(937\) 1.27197e7 0.473291 0.236646 0.971596i \(-0.423952\pi\)
0.236646 + 0.971596i \(0.423952\pi\)
\(938\) 1.49714e7 0.555591
\(939\) 0 0
\(940\) −4.12561e7 −1.52289
\(941\) 1.66122e7 0.611581 0.305790 0.952099i \(-0.401079\pi\)
0.305790 + 0.952099i \(0.401079\pi\)
\(942\) 0 0
\(943\) 6.60335e6 0.241816
\(944\) 3.21812e7 1.17536
\(945\) 0 0
\(946\) −685360. −0.0248995
\(947\) −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
\(950\) −889018. −0.0319596
\(951\) 0 0
\(952\) 2.02435e7 0.723924
\(953\) 3.77139e7 1.34514 0.672572 0.740032i \(-0.265190\pi\)
0.672572 + 0.740032i \(0.265190\pi\)
\(954\) 0 0
\(955\) −4.55007e7 −1.61439
\(956\) −5.01391e7 −1.77432
\(957\) 0 0
\(958\) −1.45806e6 −0.0513287
\(959\) −1.98007e7 −0.695239
\(960\) 0 0
\(961\) −2.14436e6 −0.0749014
\(962\) −5.86399e6 −0.204294
\(963\) 0 0
\(964\) −2.81656e7 −0.976173
\(965\) 2.27077e7 0.784973
\(966\) 0 0
\(967\) −1.36407e7 −0.469105 −0.234553 0.972103i \(-0.575362\pi\)
−0.234553 + 0.972103i \(0.575362\pi\)
\(968\) 1.16641e7 0.400093
\(969\) 0 0
\(970\) 1.11624e6 0.0380914
\(971\) 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
\(974\) −1.02351e6 −0.0345698
\(975\) 0 0
\(976\) 2.28976e7 0.769424
\(977\) 9.70600e6 0.325315 0.162657 0.986683i \(-0.447993\pi\)
0.162657 + 0.986683i \(0.447993\pi\)
\(978\) 0 0
\(979\) 3.80608e6 0.126917
\(980\) 5.14711e7 1.71198
\(981\) 0 0
\(982\) 4.72971e6 0.156515
\(983\) 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
\(986\) 1.03311e7 0.338419
\(987\) 0 0
\(988\) −5.00917e7 −1.63258
\(989\) −2.22109e6 −0.0722064
\(990\) 0 0
\(991\) 3.94595e7 1.27634 0.638172 0.769894i \(-0.279691\pi\)
0.638172 + 0.769894i \(0.279691\pi\)
\(992\) −1.89998e7 −0.613012
\(993\) 0 0
\(994\) 9.82104e6 0.315276
\(995\) 4.17374e7 1.33650
\(996\) 0 0
\(997\) 3.61342e7 1.15128 0.575639 0.817704i \(-0.304753\pi\)
0.575639 + 0.817704i \(0.304753\pi\)
\(998\) −6.56215e6 −0.208555
\(999\) 0 0
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 207.6.a.c.1.2 3
3.2 odd 2 69.6.a.b.1.2 3
12.11 even 2 1104.6.a.i.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.b.1.2 3 3.2 odd 2
207.6.a.c.1.2 3 1.1 even 1 trivial
1104.6.a.i.1.3 3 12.11 even 2