Properties

Label 147.6.a.h.1.2
Level $147$
Weight $6$
Character 147.1
Self dual yes
Analytic conductor $23.576$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.5764215125\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-6.44622\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.44622 q^{2} -9.00000 q^{3} -2.33867 q^{4} +36.0000 q^{5} -49.0160 q^{6} -187.016 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+5.44622 q^{2} -9.00000 q^{3} -2.33867 q^{4} +36.0000 q^{5} -49.0160 q^{6} -187.016 q^{8} +81.0000 q^{9} +196.064 q^{10} +184.430 q^{11} +21.0480 q^{12} +147.872 q^{13} -324.000 q^{15} -943.693 q^{16} +1968.38 q^{17} +441.144 q^{18} +1892.51 q^{19} -84.1920 q^{20} +1004.45 q^{22} +136.988 q^{23} +1683.14 q^{24} -1829.00 q^{25} +805.344 q^{26} -729.000 q^{27} -1259.58 q^{29} -1764.58 q^{30} +8969.02 q^{31} +844.949 q^{32} -1659.87 q^{33} +10720.3 q^{34} -189.432 q^{36} +12897.2 q^{37} +10307.0 q^{38} -1330.85 q^{39} -6732.58 q^{40} +8975.62 q^{41} +13538.9 q^{43} -431.321 q^{44} +2916.00 q^{45} +746.069 q^{46} -20046.1 q^{47} +8493.24 q^{48} -9961.14 q^{50} -17715.5 q^{51} -345.823 q^{52} +9334.33 q^{53} -3970.30 q^{54} +6639.49 q^{55} -17032.6 q^{57} -6859.96 q^{58} +8866.46 q^{59} +757.728 q^{60} -41148.3 q^{61} +48847.3 q^{62} +34800.0 q^{64} +5323.39 q^{65} -9040.03 q^{66} -55351.5 q^{67} -4603.39 q^{68} -1232.90 q^{69} -63866.8 q^{71} -15148.3 q^{72} +41299.3 q^{73} +70241.1 q^{74} +16461.0 q^{75} -4425.95 q^{76} -7248.09 q^{78} +16963.5 q^{79} -33973.0 q^{80} +6561.00 q^{81} +48883.2 q^{82} -101693. q^{83} +70861.8 q^{85} +73736.0 q^{86} +11336.2 q^{87} -34491.4 q^{88} +87102.5 q^{89} +15881.2 q^{90} -320.370 q^{92} -80721.2 q^{93} -109176. q^{94} +68130.4 q^{95} -7604.54 q^{96} -118107. q^{97} +14938.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 18 q^{3} + 37 q^{4} + 72 q^{5} + 27 q^{6} - 249 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 18 q^{3} + 37 q^{4} + 72 q^{5} + 27 q^{6} - 249 q^{8} + 162 q^{9} - 108 q^{10} + 480 q^{11} - 333 q^{12} + 1296 q^{13} - 648 q^{15} - 1679 q^{16} + 936 q^{17} - 243 q^{18} - 216 q^{19} + 1332 q^{20} - 1492 q^{22} - 504 q^{23} + 2241 q^{24} - 3658 q^{25} - 8892 q^{26} - 1458 q^{27} + 6372 q^{29} + 972 q^{30} + 9936 q^{31} + 9039 q^{32} - 4320 q^{33} + 19440 q^{34} + 2997 q^{36} + 11124 q^{37} + 28116 q^{38} - 11664 q^{39} - 8964 q^{40} + 20952 q^{41} - 6264 q^{43} + 11196 q^{44} + 5832 q^{45} + 6160 q^{46} + 7920 q^{47} + 15111 q^{48} + 5487 q^{50} - 8424 q^{51} + 44820 q^{52} + 2220 q^{53} + 2187 q^{54} + 17280 q^{55} + 1944 q^{57} - 71318 q^{58} + 29736 q^{59} - 11988 q^{60} - 17280 q^{61} + 40680 q^{62} - 10879 q^{64} + 46656 q^{65} + 13428 q^{66} - 20680 q^{67} - 45216 q^{68} + 4536 q^{69} - 92280 q^{71} - 20169 q^{72} + 56592 q^{73} + 85218 q^{74} + 32922 q^{75} - 87372 q^{76} + 80028 q^{78} - 56096 q^{79} - 60444 q^{80} + 13122 q^{81} - 52272 q^{82} - 71352 q^{83} + 33696 q^{85} + 240996 q^{86} - 57348 q^{87} - 52812 q^{88} + 123192 q^{89} - 8748 q^{90} - 25536 q^{92} - 89424 q^{93} - 345384 q^{94} - 7776 q^{95} - 81351 q^{96} + 35856 q^{97} + 38880 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.44622 0.962765 0.481383 0.876511i \(-0.340135\pi\)
0.481383 + 0.876511i \(0.340135\pi\)
\(3\) −9.00000 −0.577350
\(4\) −2.33867 −0.0730833
\(5\) 36.0000 0.643988 0.321994 0.946742i \(-0.395647\pi\)
0.321994 + 0.946742i \(0.395647\pi\)
\(6\) −49.0160 −0.555853
\(7\) 0 0
\(8\) −187.016 −1.03313
\(9\) 81.0000 0.333333
\(10\) 196.064 0.620009
\(11\) 184.430 0.459569 0.229784 0.973242i \(-0.426198\pi\)
0.229784 + 0.973242i \(0.426198\pi\)
\(12\) 21.0480 0.0421947
\(13\) 147.872 0.242676 0.121338 0.992611i \(-0.461281\pi\)
0.121338 + 0.992611i \(0.461281\pi\)
\(14\) 0 0
\(15\) −324.000 −0.371806
\(16\) −943.693 −0.921576
\(17\) 1968.38 1.65191 0.825957 0.563733i \(-0.190635\pi\)
0.825957 + 0.563733i \(0.190635\pi\)
\(18\) 441.144 0.320922
\(19\) 1892.51 1.20269 0.601346 0.798989i \(-0.294631\pi\)
0.601346 + 0.798989i \(0.294631\pi\)
\(20\) −84.1920 −0.0470647
\(21\) 0 0
\(22\) 1004.45 0.442457
\(23\) 136.988 0.0539963 0.0269982 0.999635i \(-0.491405\pi\)
0.0269982 + 0.999635i \(0.491405\pi\)
\(24\) 1683.14 0.596476
\(25\) −1829.00 −0.585280
\(26\) 805.344 0.233640
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −1259.58 −0.278120 −0.139060 0.990284i \(-0.544408\pi\)
−0.139060 + 0.990284i \(0.544408\pi\)
\(30\) −1764.58 −0.357962
\(31\) 8969.02 1.67626 0.838129 0.545472i \(-0.183650\pi\)
0.838129 + 0.545472i \(0.183650\pi\)
\(32\) 844.949 0.145866
\(33\) −1659.87 −0.265332
\(34\) 10720.3 1.59041
\(35\) 0 0
\(36\) −189.432 −0.0243611
\(37\) 12897.2 1.54879 0.774393 0.632705i \(-0.218055\pi\)
0.774393 + 0.632705i \(0.218055\pi\)
\(38\) 10307.0 1.15791
\(39\) −1330.85 −0.140109
\(40\) −6732.58 −0.665321
\(41\) 8975.62 0.833882 0.416941 0.908934i \(-0.363102\pi\)
0.416941 + 0.908934i \(0.363102\pi\)
\(42\) 0 0
\(43\) 13538.9 1.11664 0.558320 0.829626i \(-0.311446\pi\)
0.558320 + 0.829626i \(0.311446\pi\)
\(44\) −431.321 −0.0335868
\(45\) 2916.00 0.214663
\(46\) 746.069 0.0519858
\(47\) −20046.1 −1.32369 −0.661845 0.749641i \(-0.730226\pi\)
−0.661845 + 0.749641i \(0.730226\pi\)
\(48\) 8493.24 0.532072
\(49\) 0 0
\(50\) −9961.14 −0.563487
\(51\) −17715.5 −0.953733
\(52\) −345.823 −0.0177356
\(53\) 9334.33 0.456450 0.228225 0.973608i \(-0.426708\pi\)
0.228225 + 0.973608i \(0.426708\pi\)
\(54\) −3970.30 −0.185284
\(55\) 6639.49 0.295956
\(56\) 0 0
\(57\) −17032.6 −0.694375
\(58\) −6859.96 −0.267764
\(59\) 8866.46 0.331605 0.165802 0.986159i \(-0.446979\pi\)
0.165802 + 0.986159i \(0.446979\pi\)
\(60\) 757.728 0.0271728
\(61\) −41148.3 −1.41588 −0.707942 0.706271i \(-0.750376\pi\)
−0.707942 + 0.706271i \(0.750376\pi\)
\(62\) 48847.3 1.61384
\(63\) 0 0
\(64\) 34800.0 1.06201
\(65\) 5323.39 0.156281
\(66\) −9040.03 −0.255452
\(67\) −55351.5 −1.50641 −0.753204 0.657787i \(-0.771493\pi\)
−0.753204 + 0.657787i \(0.771493\pi\)
\(68\) −4603.39 −0.120727
\(69\) −1232.90 −0.0311748
\(70\) 0 0
\(71\) −63866.8 −1.50359 −0.751794 0.659398i \(-0.770811\pi\)
−0.751794 + 0.659398i \(0.770811\pi\)
\(72\) −15148.3 −0.344376
\(73\) 41299.3 0.907060 0.453530 0.891241i \(-0.350165\pi\)
0.453530 + 0.891241i \(0.350165\pi\)
\(74\) 70241.1 1.49112
\(75\) 16461.0 0.337912
\(76\) −4425.95 −0.0878968
\(77\) 0 0
\(78\) −7248.09 −0.134892
\(79\) 16963.5 0.305808 0.152904 0.988241i \(-0.451138\pi\)
0.152904 + 0.988241i \(0.451138\pi\)
\(80\) −33973.0 −0.593483
\(81\) 6561.00 0.111111
\(82\) 48883.2 0.802833
\(83\) −101693. −1.62030 −0.810150 0.586223i \(-0.800614\pi\)
−0.810150 + 0.586223i \(0.800614\pi\)
\(84\) 0 0
\(85\) 70861.8 1.06381
\(86\) 73736.0 1.07506
\(87\) 11336.2 0.160572
\(88\) −34491.4 −0.474793
\(89\) 87102.5 1.16562 0.582808 0.812610i \(-0.301954\pi\)
0.582808 + 0.812610i \(0.301954\pi\)
\(90\) 15881.2 0.206670
\(91\) 0 0
\(92\) −320.370 −0.00394623
\(93\) −80721.2 −0.967788
\(94\) −109176. −1.27440
\(95\) 68130.4 0.774519
\(96\) −7604.54 −0.0842160
\(97\) −118107. −1.27452 −0.637258 0.770650i \(-0.719932\pi\)
−0.637258 + 0.770650i \(0.719932\pi\)
\(98\) 0 0
\(99\) 14938.8 0.153190
\(100\) 4277.42 0.0427742
\(101\) 22232.7 0.216865 0.108432 0.994104i \(-0.465417\pi\)
0.108432 + 0.994104i \(0.465417\pi\)
\(102\) −96482.3 −0.918221
\(103\) 135931. 1.26248 0.631239 0.775588i \(-0.282547\pi\)
0.631239 + 0.775588i \(0.282547\pi\)
\(104\) −27654.4 −0.250716
\(105\) 0 0
\(106\) 50836.8 0.439454
\(107\) 117626. 0.993218 0.496609 0.867974i \(-0.334578\pi\)
0.496609 + 0.867974i \(0.334578\pi\)
\(108\) 1704.89 0.0140649
\(109\) −34664.2 −0.279457 −0.139729 0.990190i \(-0.544623\pi\)
−0.139729 + 0.990190i \(0.544623\pi\)
\(110\) 36160.1 0.284937
\(111\) −116075. −0.894192
\(112\) 0 0
\(113\) −26584.5 −0.195854 −0.0979269 0.995194i \(-0.531221\pi\)
−0.0979269 + 0.995194i \(0.531221\pi\)
\(114\) −92763.4 −0.668520
\(115\) 4931.58 0.0347730
\(116\) 2945.74 0.0203259
\(117\) 11977.6 0.0808921
\(118\) 48288.7 0.319257
\(119\) 0 0
\(120\) 60593.2 0.384123
\(121\) −127036. −0.788797
\(122\) −224103. −1.36316
\(123\) −80780.5 −0.481442
\(124\) −20975.6 −0.122507
\(125\) −178344. −1.02090
\(126\) 0 0
\(127\) −137111. −0.754334 −0.377167 0.926145i \(-0.623102\pi\)
−0.377167 + 0.926145i \(0.623102\pi\)
\(128\) 162490. 0.876600
\(129\) −121850. −0.644693
\(130\) 28992.4 0.150462
\(131\) −54089.2 −0.275380 −0.137690 0.990475i \(-0.543968\pi\)
−0.137690 + 0.990475i \(0.543968\pi\)
\(132\) 3881.89 0.0193913
\(133\) 0 0
\(134\) −301457. −1.45032
\(135\) −26244.0 −0.123935
\(136\) −368119. −1.70664
\(137\) 422849. 1.92479 0.962395 0.271653i \(-0.0875703\pi\)
0.962395 + 0.271653i \(0.0875703\pi\)
\(138\) −6714.62 −0.0300140
\(139\) −9913.38 −0.0435196 −0.0217598 0.999763i \(-0.506927\pi\)
−0.0217598 + 0.999763i \(0.506927\pi\)
\(140\) 0 0
\(141\) 180415. 0.764233
\(142\) −347833. −1.44760
\(143\) 27272.1 0.111526
\(144\) −76439.2 −0.307192
\(145\) −45345.0 −0.179106
\(146\) 224925. 0.873285
\(147\) 0 0
\(148\) −30162.3 −0.113190
\(149\) 505462. 1.86519 0.932595 0.360925i \(-0.117539\pi\)
0.932595 + 0.360925i \(0.117539\pi\)
\(150\) 89650.3 0.325329
\(151\) −193103. −0.689201 −0.344601 0.938749i \(-0.611986\pi\)
−0.344601 + 0.938749i \(0.611986\pi\)
\(152\) −353930. −1.24253
\(153\) 159439. 0.550638
\(154\) 0 0
\(155\) 322885. 1.07949
\(156\) 3112.41 0.0102397
\(157\) 264923. 0.857770 0.428885 0.903359i \(-0.358907\pi\)
0.428885 + 0.903359i \(0.358907\pi\)
\(158\) 92387.1 0.294421
\(159\) −84008.9 −0.263532
\(160\) 30418.1 0.0939362
\(161\) 0 0
\(162\) 35732.7 0.106974
\(163\) −539093. −1.58926 −0.794629 0.607095i \(-0.792335\pi\)
−0.794629 + 0.607095i \(0.792335\pi\)
\(164\) −20991.0 −0.0609429
\(165\) −59755.4 −0.170871
\(166\) −553842. −1.55997
\(167\) 218748. 0.606949 0.303475 0.952840i \(-0.401853\pi\)
0.303475 + 0.952840i \(0.401853\pi\)
\(168\) 0 0
\(169\) −349427. −0.941108
\(170\) 385929. 1.02420
\(171\) 153293. 0.400898
\(172\) −31663.0 −0.0816078
\(173\) 590201. 1.49929 0.749644 0.661842i \(-0.230225\pi\)
0.749644 + 0.661842i \(0.230225\pi\)
\(174\) 61739.7 0.154593
\(175\) 0 0
\(176\) −174046. −0.423527
\(177\) −79798.2 −0.191452
\(178\) 474380. 1.12222
\(179\) −217352. −0.507026 −0.253513 0.967332i \(-0.581586\pi\)
−0.253513 + 0.967332i \(0.581586\pi\)
\(180\) −6819.55 −0.0156882
\(181\) 188109. 0.426790 0.213395 0.976966i \(-0.431548\pi\)
0.213395 + 0.976966i \(0.431548\pi\)
\(182\) 0 0
\(183\) 370335. 0.817461
\(184\) −25619.0 −0.0557851
\(185\) 464300. 0.997399
\(186\) −439626. −0.931753
\(187\) 363029. 0.759168
\(188\) 46881.2 0.0967396
\(189\) 0 0
\(190\) 371053. 0.745680
\(191\) 173523. 0.344170 0.172085 0.985082i \(-0.444950\pi\)
0.172085 + 0.985082i \(0.444950\pi\)
\(192\) −313200. −0.613152
\(193\) 117781. 0.227606 0.113803 0.993503i \(-0.463697\pi\)
0.113803 + 0.993503i \(0.463697\pi\)
\(194\) −643236. −1.22706
\(195\) −47910.5 −0.0902287
\(196\) 0 0
\(197\) 224734. 0.412575 0.206288 0.978491i \(-0.433862\pi\)
0.206288 + 0.978491i \(0.433862\pi\)
\(198\) 81360.3 0.147486
\(199\) 740273. 1.32513 0.662567 0.749003i \(-0.269467\pi\)
0.662567 + 0.749003i \(0.269467\pi\)
\(200\) 342052. 0.604669
\(201\) 498164. 0.869725
\(202\) 121084. 0.208790
\(203\) 0 0
\(204\) 41430.5 0.0697020
\(205\) 323122. 0.537010
\(206\) 740308. 1.21547
\(207\) 11096.1 0.0179988
\(208\) −139546. −0.223645
\(209\) 349036. 0.552720
\(210\) 0 0
\(211\) −705896. −1.09153 −0.545763 0.837939i \(-0.683760\pi\)
−0.545763 + 0.837939i \(0.683760\pi\)
\(212\) −21829.9 −0.0333589
\(213\) 574801. 0.868097
\(214\) 640618. 0.956236
\(215\) 487402. 0.719102
\(216\) 136335. 0.198825
\(217\) 0 0
\(218\) −188789. −0.269052
\(219\) −371694. −0.523691
\(220\) −15527.5 −0.0216295
\(221\) 291069. 0.400881
\(222\) −632170. −0.860897
\(223\) 42214.3 0.0568456 0.0284228 0.999596i \(-0.490952\pi\)
0.0284228 + 0.999596i \(0.490952\pi\)
\(224\) 0 0
\(225\) −148149. −0.195093
\(226\) −144785. −0.188561
\(227\) 742403. 0.956258 0.478129 0.878290i \(-0.341315\pi\)
0.478129 + 0.878290i \(0.341315\pi\)
\(228\) 39833.6 0.0507472
\(229\) 941236. 1.18607 0.593034 0.805177i \(-0.297930\pi\)
0.593034 + 0.805177i \(0.297930\pi\)
\(230\) 26858.5 0.0334782
\(231\) 0 0
\(232\) 235562. 0.287333
\(233\) −512331. −0.618246 −0.309123 0.951022i \(-0.600035\pi\)
−0.309123 + 0.951022i \(0.600035\pi\)
\(234\) 65232.9 0.0778801
\(235\) −721661. −0.852440
\(236\) −20735.7 −0.0242348
\(237\) −152672. −0.176558
\(238\) 0 0
\(239\) −115323. −0.130593 −0.0652966 0.997866i \(-0.520799\pi\)
−0.0652966 + 0.997866i \(0.520799\pi\)
\(240\) 305757. 0.342648
\(241\) −909864. −1.00910 −0.504549 0.863383i \(-0.668341\pi\)
−0.504549 + 0.863383i \(0.668341\pi\)
\(242\) −691869. −0.759426
\(243\) −59049.0 −0.0641500
\(244\) 96232.2 0.103477
\(245\) 0 0
\(246\) −439949. −0.463516
\(247\) 279850. 0.291865
\(248\) −1.67735e6 −1.73179
\(249\) 915236. 0.935481
\(250\) −971301. −0.982888
\(251\) −321264. −0.321868 −0.160934 0.986965i \(-0.551451\pi\)
−0.160934 + 0.986965i \(0.551451\pi\)
\(252\) 0 0
\(253\) 25264.8 0.0248150
\(254\) −746738. −0.726246
\(255\) −637756. −0.614192
\(256\) −228642. −0.218050
\(257\) −556492. −0.525565 −0.262782 0.964855i \(-0.584640\pi\)
−0.262782 + 0.964855i \(0.584640\pi\)
\(258\) −663624. −0.620688
\(259\) 0 0
\(260\) −12449.6 −0.0114215
\(261\) −102026. −0.0927065
\(262\) −294582. −0.265126
\(263\) −233379. −0.208052 −0.104026 0.994575i \(-0.533173\pi\)
−0.104026 + 0.994575i \(0.533173\pi\)
\(264\) 310423. 0.274122
\(265\) 336036. 0.293948
\(266\) 0 0
\(267\) −783923. −0.672969
\(268\) 129449. 0.110093
\(269\) −1.47857e6 −1.24584 −0.622919 0.782286i \(-0.714053\pi\)
−0.622919 + 0.782286i \(0.714053\pi\)
\(270\) −142931. −0.119321
\(271\) 177762. 0.147033 0.0735165 0.997294i \(-0.476578\pi\)
0.0735165 + 0.997294i \(0.476578\pi\)
\(272\) −1.85755e6 −1.52236
\(273\) 0 0
\(274\) 2.30293e6 1.85312
\(275\) −337323. −0.268976
\(276\) 2883.33 0.00227836
\(277\) −1.22252e6 −0.957318 −0.478659 0.878001i \(-0.658877\pi\)
−0.478659 + 0.878001i \(0.658877\pi\)
\(278\) −53990.5 −0.0418991
\(279\) 726491. 0.558753
\(280\) 0 0
\(281\) 177799. 0.134327 0.0671636 0.997742i \(-0.478605\pi\)
0.0671636 + 0.997742i \(0.478605\pi\)
\(282\) 982582. 0.735776
\(283\) −1.09052e6 −0.809406 −0.404703 0.914448i \(-0.632625\pi\)
−0.404703 + 0.914448i \(0.632625\pi\)
\(284\) 149363. 0.109887
\(285\) −613174. −0.447169
\(286\) 148530. 0.107374
\(287\) 0 0
\(288\) 68440.8 0.0486221
\(289\) 2.45468e6 1.72882
\(290\) −246959. −0.172437
\(291\) 1.06296e6 0.735843
\(292\) −96585.3 −0.0662909
\(293\) 545742. 0.371380 0.185690 0.982608i \(-0.440548\pi\)
0.185690 + 0.982608i \(0.440548\pi\)
\(294\) 0 0
\(295\) 319193. 0.213549
\(296\) −2.41198e6 −1.60009
\(297\) −134450. −0.0884440
\(298\) 2.75286e6 1.79574
\(299\) 20256.8 0.0131036
\(300\) −38496.8 −0.0246957
\(301\) 0 0
\(302\) −1.05168e6 −0.663539
\(303\) −200095. −0.125207
\(304\) −1.78595e6 −1.10837
\(305\) −1.48134e6 −0.911811
\(306\) 868341. 0.530135
\(307\) −3.29000e6 −1.99228 −0.996138 0.0878052i \(-0.972015\pi\)
−0.996138 + 0.0878052i \(0.972015\pi\)
\(308\) 0 0
\(309\) −1.22338e6 −0.728892
\(310\) 1.75850e6 1.03929
\(311\) −699712. −0.410221 −0.205111 0.978739i \(-0.565755\pi\)
−0.205111 + 0.978739i \(0.565755\pi\)
\(312\) 248890. 0.144751
\(313\) 623219. 0.359567 0.179783 0.983706i \(-0.442460\pi\)
0.179783 + 0.983706i \(0.442460\pi\)
\(314\) 1.44283e6 0.825831
\(315\) 0 0
\(316\) −39672.0 −0.0223494
\(317\) −639244. −0.357288 −0.178644 0.983914i \(-0.557171\pi\)
−0.178644 + 0.983914i \(0.557171\pi\)
\(318\) −457531. −0.253719
\(319\) −232305. −0.127815
\(320\) 1.25280e6 0.683922
\(321\) −1.05864e6 −0.573435
\(322\) 0 0
\(323\) 3.72519e6 1.98675
\(324\) −15344.0 −0.00812037
\(325\) −270458. −0.142034
\(326\) −2.93602e6 −1.53008
\(327\) 311978. 0.161345
\(328\) −1.67858e6 −0.861506
\(329\) 0 0
\(330\) −325441. −0.164508
\(331\) 1.40963e6 0.707190 0.353595 0.935399i \(-0.384959\pi\)
0.353595 + 0.935399i \(0.384959\pi\)
\(332\) 237826. 0.118417
\(333\) 1.04467e6 0.516262
\(334\) 1.19135e6 0.584350
\(335\) −1.99265e6 −0.970108
\(336\) 0 0
\(337\) −1.55677e6 −0.746704 −0.373352 0.927690i \(-0.621792\pi\)
−0.373352 + 0.927690i \(0.621792\pi\)
\(338\) −1.90306e6 −0.906066
\(339\) 239260. 0.113076
\(340\) −165722. −0.0777469
\(341\) 1.65416e6 0.770356
\(342\) 834870. 0.385970
\(343\) 0 0
\(344\) −2.53200e6 −1.15363
\(345\) −44384.3 −0.0200762
\(346\) 3.21437e6 1.44346
\(347\) 248297. 0.110700 0.0553500 0.998467i \(-0.482373\pi\)
0.0553500 + 0.998467i \(0.482373\pi\)
\(348\) −26511.7 −0.0117352
\(349\) 1.86169e6 0.818171 0.409086 0.912496i \(-0.365848\pi\)
0.409086 + 0.912496i \(0.365848\pi\)
\(350\) 0 0
\(351\) −107799. −0.0467031
\(352\) 155834. 0.0670356
\(353\) 1.77128e6 0.756574 0.378287 0.925688i \(-0.376513\pi\)
0.378287 + 0.925688i \(0.376513\pi\)
\(354\) −434599. −0.184323
\(355\) −2.29920e6 −0.968292
\(356\) −203704. −0.0851871
\(357\) 0 0
\(358\) −1.18375e6 −0.488147
\(359\) −4.47560e6 −1.83280 −0.916401 0.400262i \(-0.868919\pi\)
−0.916401 + 0.400262i \(0.868919\pi\)
\(360\) −545339. −0.221774
\(361\) 1.10550e6 0.446469
\(362\) 1.02449e6 0.410898
\(363\) 1.14333e6 0.455412
\(364\) 0 0
\(365\) 1.48678e6 0.584135
\(366\) 2.01693e6 0.787023
\(367\) 3.17097e6 1.22893 0.614465 0.788944i \(-0.289372\pi\)
0.614465 + 0.788944i \(0.289372\pi\)
\(368\) −129275. −0.0497617
\(369\) 727025. 0.277961
\(370\) 2.52868e6 0.960261
\(371\) 0 0
\(372\) 188780. 0.0707292
\(373\) −2.31179e6 −0.860353 −0.430177 0.902745i \(-0.641549\pi\)
−0.430177 + 0.902745i \(0.641549\pi\)
\(374\) 1.97714e6 0.730900
\(375\) 1.60510e6 0.589417
\(376\) 3.74895e6 1.36754
\(377\) −186257. −0.0674931
\(378\) 0 0
\(379\) −591840. −0.211644 −0.105822 0.994385i \(-0.533747\pi\)
−0.105822 + 0.994385i \(0.533747\pi\)
\(380\) −159334. −0.0566044
\(381\) 1.23400e6 0.435515
\(382\) 945042. 0.331354
\(383\) −3.59766e6 −1.25321 −0.626605 0.779337i \(-0.715556\pi\)
−0.626605 + 0.779337i \(0.715556\pi\)
\(384\) −1.46241e6 −0.506105
\(385\) 0 0
\(386\) 641464. 0.219131
\(387\) 1.09665e6 0.372213
\(388\) 276212. 0.0931459
\(389\) −2.24746e6 −0.753040 −0.376520 0.926409i \(-0.622879\pi\)
−0.376520 + 0.926409i \(0.622879\pi\)
\(390\) −260931. −0.0868690
\(391\) 269646. 0.0891973
\(392\) 0 0
\(393\) 486803. 0.158991
\(394\) 1.22395e6 0.397213
\(395\) 610687. 0.196936
\(396\) −34937.0 −0.0111956
\(397\) −4.58670e6 −1.46058 −0.730288 0.683139i \(-0.760614\pi\)
−0.730288 + 0.683139i \(0.760614\pi\)
\(398\) 4.03169e6 1.27579
\(399\) 0 0
\(400\) 1.72602e6 0.539380
\(401\) 6.00432e6 1.86468 0.932338 0.361589i \(-0.117766\pi\)
0.932338 + 0.361589i \(0.117766\pi\)
\(402\) 2.71311e6 0.837341
\(403\) 1.32627e6 0.406788
\(404\) −51994.9 −0.0158492
\(405\) 236196. 0.0715542
\(406\) 0 0
\(407\) 2.37864e6 0.711774
\(408\) 3.31307e6 0.985328
\(409\) −3.55340e6 −1.05035 −0.525177 0.850993i \(-0.676001\pi\)
−0.525177 + 0.850993i \(0.676001\pi\)
\(410\) 1.75980e6 0.517014
\(411\) −3.80564e6 −1.11128
\(412\) −317896. −0.0922661
\(413\) 0 0
\(414\) 60431.6 0.0173286
\(415\) −3.66094e6 −1.04345
\(416\) 124944. 0.0353983
\(417\) 89220.4 0.0251260
\(418\) 1.90093e6 0.532139
\(419\) 2.01375e6 0.560365 0.280182 0.959947i \(-0.409605\pi\)
0.280182 + 0.959947i \(0.409605\pi\)
\(420\) 0 0
\(421\) −5.89987e6 −1.62232 −0.811161 0.584823i \(-0.801164\pi\)
−0.811161 + 0.584823i \(0.801164\pi\)
\(422\) −3.84446e6 −1.05088
\(423\) −1.62374e6 −0.441230
\(424\) −1.74567e6 −0.471571
\(425\) −3.60017e6 −0.966832
\(426\) 3.13049e6 0.835774
\(427\) 0 0
\(428\) −275088. −0.0725877
\(429\) −245449. −0.0643898
\(430\) 2.65450e6 0.692327
\(431\) −3.81048e6 −0.988066 −0.494033 0.869443i \(-0.664478\pi\)
−0.494033 + 0.869443i \(0.664478\pi\)
\(432\) 687952. 0.177357
\(433\) 6.59449e6 1.69029 0.845146 0.534536i \(-0.179513\pi\)
0.845146 + 0.534536i \(0.179513\pi\)
\(434\) 0 0
\(435\) 408105. 0.103407
\(436\) 81068.1 0.0204237
\(437\) 259252. 0.0649410
\(438\) −2.02433e6 −0.504192
\(439\) −4.55028e6 −1.12688 −0.563438 0.826158i \(-0.690522\pi\)
−0.563438 + 0.826158i \(0.690522\pi\)
\(440\) −1.24169e6 −0.305761
\(441\) 0 0
\(442\) 1.58523e6 0.385954
\(443\) 5.11537e6 1.23842 0.619210 0.785225i \(-0.287453\pi\)
0.619210 + 0.785225i \(0.287453\pi\)
\(444\) 271460. 0.0653505
\(445\) 3.13569e6 0.750643
\(446\) 229908. 0.0547290
\(447\) −4.54916e6 −1.07687
\(448\) 0 0
\(449\) 5.95600e6 1.39424 0.697122 0.716953i \(-0.254464\pi\)
0.697122 + 0.716953i \(0.254464\pi\)
\(450\) −806852. −0.187829
\(451\) 1.65537e6 0.383226
\(452\) 62172.2 0.0143136
\(453\) 1.73792e6 0.397911
\(454\) 4.04329e6 0.920652
\(455\) 0 0
\(456\) 3.18537e6 0.717378
\(457\) −2.59834e6 −0.581976 −0.290988 0.956727i \(-0.593984\pi\)
−0.290988 + 0.956727i \(0.593984\pi\)
\(458\) 5.12618e6 1.14191
\(459\) −1.43495e6 −0.317911
\(460\) −11533.3 −0.00254132
\(461\) 4.51513e6 0.989505 0.494752 0.869034i \(-0.335259\pi\)
0.494752 + 0.869034i \(0.335259\pi\)
\(462\) 0 0
\(463\) −5.55129e6 −1.20349 −0.601744 0.798689i \(-0.705527\pi\)
−0.601744 + 0.798689i \(0.705527\pi\)
\(464\) 1.18866e6 0.256308
\(465\) −2.90596e6 −0.623244
\(466\) −2.79027e6 −0.595225
\(467\) −7.95350e6 −1.68759 −0.843794 0.536668i \(-0.819683\pi\)
−0.843794 + 0.536668i \(0.819683\pi\)
\(468\) −28011.7 −0.00591187
\(469\) 0 0
\(470\) −3.93033e6 −0.820699
\(471\) −2.38431e6 −0.495233
\(472\) −1.65817e6 −0.342590
\(473\) 2.49699e6 0.513173
\(474\) −831484. −0.169984
\(475\) −3.46140e6 −0.703912
\(476\) 0 0
\(477\) 756080. 0.152150
\(478\) −628074. −0.125731
\(479\) −6.56370e6 −1.30710 −0.653552 0.756882i \(-0.726722\pi\)
−0.653552 + 0.756882i \(0.726722\pi\)
\(480\) −273763. −0.0542341
\(481\) 1.90714e6 0.375854
\(482\) −4.95532e6 −0.971525
\(483\) 0 0
\(484\) 297096. 0.0576479
\(485\) −4.25185e6 −0.820773
\(486\) −321594. −0.0617614
\(487\) 4.66370e6 0.891062 0.445531 0.895266i \(-0.353015\pi\)
0.445531 + 0.895266i \(0.353015\pi\)
\(488\) 7.69539e6 1.46279
\(489\) 4.85183e6 0.917558
\(490\) 0 0
\(491\) −917227. −0.171701 −0.0858506 0.996308i \(-0.527361\pi\)
−0.0858506 + 0.996308i \(0.527361\pi\)
\(492\) 188919. 0.0351854
\(493\) −2.47934e6 −0.459430
\(494\) 1.52412e6 0.280998
\(495\) 537799. 0.0986522
\(496\) −8.46401e6 −1.54480
\(497\) 0 0
\(498\) 4.98458e6 0.900648
\(499\) 313436. 0.0563504 0.0281752 0.999603i \(-0.491030\pi\)
0.0281752 + 0.999603i \(0.491030\pi\)
\(500\) 417087. 0.0746108
\(501\) −1.96873e6 −0.350422
\(502\) −1.74967e6 −0.309883
\(503\) −3.74110e6 −0.659294 −0.329647 0.944104i \(-0.606930\pi\)
−0.329647 + 0.944104i \(0.606930\pi\)
\(504\) 0 0
\(505\) 800378. 0.139658
\(506\) 137598. 0.0238910
\(507\) 3.14484e6 0.543349
\(508\) 320657. 0.0551292
\(509\) 1.03017e7 1.76244 0.881220 0.472707i \(-0.156723\pi\)
0.881220 + 0.472707i \(0.156723\pi\)
\(510\) −3.47336e6 −0.591323
\(511\) 0 0
\(512\) −6.44492e6 −1.08653
\(513\) −1.37964e6 −0.231458
\(514\) −3.03078e6 −0.505995
\(515\) 4.89350e6 0.813021
\(516\) 284967. 0.0471163
\(517\) −3.69711e6 −0.608326
\(518\) 0 0
\(519\) −5.31181e6 −0.865614
\(520\) −995560. −0.161458
\(521\) −6.57325e6 −1.06093 −0.530464 0.847707i \(-0.677982\pi\)
−0.530464 + 0.847707i \(0.677982\pi\)
\(522\) −555657. −0.0892546
\(523\) 8.36532e6 1.33730 0.668650 0.743578i \(-0.266873\pi\)
0.668650 + 0.743578i \(0.266873\pi\)
\(524\) 126497. 0.0201257
\(525\) 0 0
\(526\) −1.27103e6 −0.200306
\(527\) 1.76545e7 2.76904
\(528\) 1.56641e6 0.244524
\(529\) −6.41758e6 −0.997084
\(530\) 1.83013e6 0.283003
\(531\) 718184. 0.110535
\(532\) 0 0
\(533\) 1.32724e6 0.202364
\(534\) −4.26942e6 −0.647911
\(535\) 4.23454e6 0.639620
\(536\) 1.03516e7 1.55631
\(537\) 1.95617e6 0.292732
\(538\) −8.05263e6 −1.19945
\(539\) 0 0
\(540\) 61376.0 0.00905761
\(541\) −8.06623e6 −1.18489 −0.592444 0.805612i \(-0.701837\pi\)
−0.592444 + 0.805612i \(0.701837\pi\)
\(542\) 968129. 0.141558
\(543\) −1.69298e6 −0.246407
\(544\) 1.66318e6 0.240959
\(545\) −1.24791e6 −0.179967
\(546\) 0 0
\(547\) 3.90775e6 0.558416 0.279208 0.960231i \(-0.409928\pi\)
0.279208 + 0.960231i \(0.409928\pi\)
\(548\) −988902. −0.140670
\(549\) −3.33301e6 −0.471961
\(550\) −1.83714e6 −0.258961
\(551\) −2.38377e6 −0.334492
\(552\) 230571. 0.0322075
\(553\) 0 0
\(554\) −6.65811e6 −0.921672
\(555\) −4.17870e6 −0.575849
\(556\) 23184.1 0.00318056
\(557\) −1.15862e7 −1.58235 −0.791177 0.611588i \(-0.790531\pi\)
−0.791177 + 0.611588i \(0.790531\pi\)
\(558\) 3.95663e6 0.537948
\(559\) 2.00203e6 0.270982
\(560\) 0 0
\(561\) −3.26727e6 −0.438306
\(562\) 968334. 0.129326
\(563\) 1.91176e6 0.254192 0.127096 0.991890i \(-0.459434\pi\)
0.127096 + 0.991890i \(0.459434\pi\)
\(564\) −421931. −0.0558526
\(565\) −957041. −0.126127
\(566\) −5.93920e6 −0.779268
\(567\) 0 0
\(568\) 1.19441e7 1.55340
\(569\) 8.16817e6 1.05766 0.528828 0.848729i \(-0.322632\pi\)
0.528828 + 0.848729i \(0.322632\pi\)
\(570\) −3.33948e6 −0.430519
\(571\) −7.46593e6 −0.958283 −0.479141 0.877738i \(-0.659052\pi\)
−0.479141 + 0.877738i \(0.659052\pi\)
\(572\) −63780.3 −0.00815072
\(573\) −1.56170e6 −0.198706
\(574\) 0 0
\(575\) −250552. −0.0316030
\(576\) 2.81880e6 0.354004
\(577\) −6.88438e6 −0.860845 −0.430423 0.902627i \(-0.641635\pi\)
−0.430423 + 0.902627i \(0.641635\pi\)
\(578\) 1.33687e7 1.66445
\(579\) −1.06003e6 −0.131408
\(580\) 106047. 0.0130896
\(581\) 0 0
\(582\) 5.78912e6 0.708444
\(583\) 1.72153e6 0.209770
\(584\) −7.72363e6 −0.937108
\(585\) 431195. 0.0520935
\(586\) 2.97223e6 0.357551
\(587\) −8.91086e6 −1.06739 −0.533696 0.845676i \(-0.679197\pi\)
−0.533696 + 0.845676i \(0.679197\pi\)
\(588\) 0 0
\(589\) 1.69740e7 2.01602
\(590\) 1.73839e6 0.205598
\(591\) −2.02261e6 −0.238200
\(592\) −1.21710e7 −1.42732
\(593\) 1.23430e7 1.44140 0.720700 0.693247i \(-0.243820\pi\)
0.720700 + 0.693247i \(0.243820\pi\)
\(594\) −732243. −0.0851508
\(595\) 0 0
\(596\) −1.18211e6 −0.136314
\(597\) −6.66246e6 −0.765066
\(598\) 110323. 0.0126157
\(599\) 9.05732e6 1.03141 0.515707 0.856765i \(-0.327529\pi\)
0.515707 + 0.856765i \(0.327529\pi\)
\(600\) −3.07847e6 −0.349106
\(601\) 7.41700e6 0.837611 0.418805 0.908076i \(-0.362449\pi\)
0.418805 + 0.908076i \(0.362449\pi\)
\(602\) 0 0
\(603\) −4.48347e6 −0.502136
\(604\) 451603. 0.0503691
\(605\) −4.57331e6 −0.507975
\(606\) −1.08976e6 −0.120545
\(607\) −5.77813e6 −0.636525 −0.318263 0.948003i \(-0.603099\pi\)
−0.318263 + 0.948003i \(0.603099\pi\)
\(608\) 1.59908e6 0.175432
\(609\) 0 0
\(610\) −8.06770e6 −0.877860
\(611\) −2.96426e6 −0.321228
\(612\) −372875. −0.0402425
\(613\) 6.29264e6 0.676366 0.338183 0.941080i \(-0.390188\pi\)
0.338183 + 0.941080i \(0.390188\pi\)
\(614\) −1.79180e7 −1.91809
\(615\) −2.90810e6 −0.310043
\(616\) 0 0
\(617\) −9.79133e6 −1.03545 −0.517725 0.855547i \(-0.673221\pi\)
−0.517725 + 0.855547i \(0.673221\pi\)
\(618\) −6.66277e6 −0.701752
\(619\) −1.32677e7 −1.39178 −0.695889 0.718150i \(-0.744989\pi\)
−0.695889 + 0.718150i \(0.744989\pi\)
\(620\) −755120. −0.0788927
\(621\) −99864.6 −0.0103916
\(622\) −3.81079e6 −0.394947
\(623\) 0 0
\(624\) 1.25591e6 0.129121
\(625\) −704759. −0.0721673
\(626\) 3.39419e6 0.346178
\(627\) −3.14133e6 −0.319113
\(628\) −619567. −0.0626886
\(629\) 2.53867e7 2.55846
\(630\) 0 0
\(631\) 4.41233e6 0.441158 0.220579 0.975369i \(-0.429205\pi\)
0.220579 + 0.975369i \(0.429205\pi\)
\(632\) −3.17245e6 −0.315938
\(633\) 6.35306e6 0.630193
\(634\) −3.48146e6 −0.343984
\(635\) −4.93600e6 −0.485782
\(636\) 196469. 0.0192598
\(637\) 0 0
\(638\) −1.26518e6 −0.123056
\(639\) −5.17321e6 −0.501196
\(640\) 5.84964e6 0.564520
\(641\) 8.13035e6 0.781564 0.390782 0.920483i \(-0.372205\pi\)
0.390782 + 0.920483i \(0.372205\pi\)
\(642\) −5.76557e6 −0.552083
\(643\) −3.12961e6 −0.298513 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(644\) 0 0
\(645\) −4.38661e6 −0.415174
\(646\) 2.02882e7 1.91277
\(647\) 1.34660e7 1.26467 0.632336 0.774695i \(-0.282096\pi\)
0.632336 + 0.774695i \(0.282096\pi\)
\(648\) −1.22701e6 −0.114792
\(649\) 1.63524e6 0.152395
\(650\) −1.47297e6 −0.136745
\(651\) 0 0
\(652\) 1.26076e6 0.116148
\(653\) 1.44772e7 1.32862 0.664312 0.747455i \(-0.268725\pi\)
0.664312 + 0.747455i \(0.268725\pi\)
\(654\) 1.69910e6 0.155337
\(655\) −1.94721e6 −0.177341
\(656\) −8.47023e6 −0.768485
\(657\) 3.34525e6 0.302353
\(658\) 0 0
\(659\) 432708. 0.0388134 0.0194067 0.999812i \(-0.493822\pi\)
0.0194067 + 0.999812i \(0.493822\pi\)
\(660\) 139748. 0.0124878
\(661\) −29965.2 −0.00266756 −0.00133378 0.999999i \(-0.500425\pi\)
−0.00133378 + 0.999999i \(0.500425\pi\)
\(662\) 7.67718e6 0.680858
\(663\) −2.61962e6 −0.231449
\(664\) 1.90182e7 1.67398
\(665\) 0 0
\(666\) 5.68953e6 0.497039
\(667\) −172548. −0.0150174
\(668\) −511578. −0.0443579
\(669\) −379928. −0.0328198
\(670\) −1.08524e7 −0.933986
\(671\) −7.58899e6 −0.650696
\(672\) 0 0
\(673\) −6.71329e6 −0.571344 −0.285672 0.958327i \(-0.592217\pi\)
−0.285672 + 0.958327i \(0.592217\pi\)
\(674\) −8.47849e6 −0.718901
\(675\) 1.33334e6 0.112637
\(676\) 817193. 0.0687793
\(677\) −4.45929e6 −0.373933 −0.186967 0.982366i \(-0.559866\pi\)
−0.186967 + 0.982366i \(0.559866\pi\)
\(678\) 1.30306e6 0.108866
\(679\) 0 0
\(680\) −1.32523e7 −1.09905
\(681\) −6.68163e6 −0.552096
\(682\) 9.00892e6 0.741672
\(683\) −1.79839e7 −1.47514 −0.737569 0.675272i \(-0.764026\pi\)
−0.737569 + 0.675272i \(0.764026\pi\)
\(684\) −358502. −0.0292989
\(685\) 1.52225e7 1.23954
\(686\) 0 0
\(687\) −8.47112e6 −0.684777
\(688\) −1.27766e7 −1.02907
\(689\) 1.38029e6 0.110770
\(690\) −241726. −0.0193286
\(691\) 2.50935e6 0.199925 0.0999624 0.994991i \(-0.468128\pi\)
0.0999624 + 0.994991i \(0.468128\pi\)
\(692\) −1.38028e6 −0.109573
\(693\) 0 0
\(694\) 1.35228e6 0.106578
\(695\) −356882. −0.0280261
\(696\) −2.12006e6 −0.165892
\(697\) 1.76675e7 1.37750
\(698\) 1.01392e7 0.787707
\(699\) 4.61098e6 0.356944
\(700\) 0 0
\(701\) 9.68649e6 0.744512 0.372256 0.928130i \(-0.378584\pi\)
0.372256 + 0.928130i \(0.378584\pi\)
\(702\) −587096. −0.0449641
\(703\) 2.44081e7 1.86271
\(704\) 6.41817e6 0.488067
\(705\) 6.49495e6 0.492156
\(706\) 9.64681e6 0.728403
\(707\) 0 0
\(708\) 186621. 0.0139919
\(709\) −9.52720e6 −0.711786 −0.355893 0.934527i \(-0.615823\pi\)
−0.355893 + 0.934527i \(0.615823\pi\)
\(710\) −1.25220e7 −0.932238
\(711\) 1.37405e6 0.101936
\(712\) −1.62896e7 −1.20423
\(713\) 1.22865e6 0.0905118
\(714\) 0 0
\(715\) 981794. 0.0718217
\(716\) 508313. 0.0370552
\(717\) 1.03791e6 0.0753980
\(718\) −2.43751e7 −1.76456
\(719\) 757176. 0.0546229 0.0273114 0.999627i \(-0.491305\pi\)
0.0273114 + 0.999627i \(0.491305\pi\)
\(720\) −2.75181e6 −0.197828
\(721\) 0 0
\(722\) 6.02081e6 0.429845
\(723\) 8.18877e6 0.582603
\(724\) −439925. −0.0311912
\(725\) 2.30378e6 0.162778
\(726\) 6.22682e6 0.438455
\(727\) −2.66570e7 −1.87058 −0.935288 0.353888i \(-0.884859\pi\)
−0.935288 + 0.353888i \(0.884859\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 8.09731e6 0.562385
\(731\) 2.66498e7 1.84459
\(732\) −866090. −0.0597427
\(733\) 2.70857e7 1.86200 0.931001 0.365017i \(-0.118937\pi\)
0.931001 + 0.365017i \(0.118937\pi\)
\(734\) 1.72698e7 1.18317
\(735\) 0 0
\(736\) 115748. 0.00787625
\(737\) −1.02085e7 −0.692298
\(738\) 3.95954e6 0.267611
\(739\) 2.19507e7 1.47855 0.739276 0.673402i \(-0.235168\pi\)
0.739276 + 0.673402i \(0.235168\pi\)
\(740\) −1.08584e6 −0.0728932
\(741\) −2.51865e6 −0.168508
\(742\) 0 0
\(743\) 5.77370e6 0.383691 0.191846 0.981425i \(-0.438553\pi\)
0.191846 + 0.981425i \(0.438553\pi\)
\(744\) 1.50962e7 0.999848
\(745\) 1.81966e7 1.20116
\(746\) −1.25905e7 −0.828318
\(747\) −8.23712e6 −0.540100
\(748\) −849005. −0.0554825
\(749\) 0 0
\(750\) 8.74171e6 0.567470
\(751\) −8.37749e6 −0.542018 −0.271009 0.962577i \(-0.587357\pi\)
−0.271009 + 0.962577i \(0.587357\pi\)
\(752\) 1.89174e7 1.21988
\(753\) 2.89138e6 0.185831
\(754\) −1.01440e6 −0.0649800
\(755\) −6.95170e6 −0.443837
\(756\) 0 0
\(757\) 1.18828e7 0.753665 0.376833 0.926281i \(-0.377013\pi\)
0.376833 + 0.926281i \(0.377013\pi\)
\(758\) −3.22329e6 −0.203764
\(759\) −227383. −0.0143270
\(760\) −1.27415e7 −0.800177
\(761\) 1.76940e7 1.10755 0.553777 0.832665i \(-0.313186\pi\)
0.553777 + 0.832665i \(0.313186\pi\)
\(762\) 6.72064e6 0.419298
\(763\) 0 0
\(764\) −405811. −0.0251531
\(765\) 5.73981e6 0.354604
\(766\) −1.95937e7 −1.20655
\(767\) 1.31110e6 0.0804726
\(768\) 2.05778e6 0.125891
\(769\) −4.12006e6 −0.251239 −0.125620 0.992078i \(-0.540092\pi\)
−0.125620 + 0.992078i \(0.540092\pi\)
\(770\) 0 0
\(771\) 5.00843e6 0.303435
\(772\) −275452. −0.0166342
\(773\) 1.26864e7 0.763642 0.381821 0.924236i \(-0.375297\pi\)
0.381821 + 0.924236i \(0.375297\pi\)
\(774\) 5.97262e6 0.358354
\(775\) −1.64043e7 −0.981080
\(776\) 2.20879e7 1.31674
\(777\) 0 0
\(778\) −1.22402e7 −0.725001
\(779\) 1.69865e7 1.00290
\(780\) 112047. 0.00659421
\(781\) −1.17790e7 −0.691002
\(782\) 1.46855e6 0.0858761
\(783\) 918235. 0.0535241
\(784\) 0 0
\(785\) 9.53723e6 0.552393
\(786\) 2.65124e6 0.153071
\(787\) 9.14809e6 0.526494 0.263247 0.964728i \(-0.415207\pi\)
0.263247 + 0.964728i \(0.415207\pi\)
\(788\) −525578. −0.0301524
\(789\) 2.10041e6 0.120119
\(790\) 3.32594e6 0.189603
\(791\) 0 0
\(792\) −2.79380e6 −0.158264
\(793\) −6.08468e6 −0.343602
\(794\) −2.49802e7 −1.40619
\(795\) −3.02432e6 −0.169711
\(796\) −1.73125e6 −0.0968451
\(797\) −1.10180e7 −0.614408 −0.307204 0.951644i \(-0.599393\pi\)
−0.307204 + 0.951644i \(0.599393\pi\)
\(798\) 0 0
\(799\) −3.94585e7 −2.18662
\(800\) −1.54541e6 −0.0853727
\(801\) 7.05530e6 0.388539
\(802\) 3.27009e7 1.79524
\(803\) 7.61684e6 0.416856
\(804\) −1.16504e6 −0.0635624
\(805\) 0 0
\(806\) 7.22315e6 0.391642
\(807\) 1.33071e7 0.719285
\(808\) −4.15788e6 −0.224049
\(809\) −3.10273e7 −1.66676 −0.833378 0.552703i \(-0.813596\pi\)
−0.833378 + 0.552703i \(0.813596\pi\)
\(810\) 1.28638e6 0.0688899
\(811\) −2.94456e7 −1.57206 −0.786028 0.618191i \(-0.787866\pi\)
−0.786028 + 0.618191i \(0.787866\pi\)
\(812\) 0 0
\(813\) −1.59985e6 −0.0848895
\(814\) 1.29546e7 0.685271
\(815\) −1.94073e7 −1.02346
\(816\) 1.67180e7 0.878937
\(817\) 2.56226e7 1.34297
\(818\) −1.93526e7 −1.01124
\(819\) 0 0
\(820\) −755675. −0.0392464
\(821\) −5.23311e6 −0.270958 −0.135479 0.990780i \(-0.543257\pi\)
−0.135479 + 0.990780i \(0.543257\pi\)
\(822\) −2.07263e7 −1.06990
\(823\) 1.41631e7 0.728884 0.364442 0.931226i \(-0.381260\pi\)
0.364442 + 0.931226i \(0.381260\pi\)
\(824\) −2.54212e7 −1.30430
\(825\) 3.03591e6 0.155294
\(826\) 0 0
\(827\) 2.98006e7 1.51517 0.757585 0.652737i \(-0.226379\pi\)
0.757585 + 0.652737i \(0.226379\pi\)
\(828\) −25950.0 −0.00131541
\(829\) −1.99304e7 −1.00723 −0.503617 0.863927i \(-0.667998\pi\)
−0.503617 + 0.863927i \(0.667998\pi\)
\(830\) −1.99383e7 −1.00460
\(831\) 1.10027e7 0.552708
\(832\) 5.14594e6 0.257725
\(833\) 0 0
\(834\) 485914. 0.0241905
\(835\) 7.87492e6 0.390868
\(836\) −816280. −0.0403946
\(837\) −6.53842e6 −0.322596
\(838\) 1.09673e7 0.539500
\(839\) −1.88103e7 −0.922552 −0.461276 0.887257i \(-0.652608\pi\)
−0.461276 + 0.887257i \(0.652608\pi\)
\(840\) 0 0
\(841\) −1.89246e7 −0.922650
\(842\) −3.21320e7 −1.56191
\(843\) −1.60019e6 −0.0775538
\(844\) 1.65085e6 0.0797724
\(845\) −1.25794e7 −0.606062
\(846\) −8.84324e6 −0.424801
\(847\) 0 0
\(848\) −8.80874e6 −0.420653
\(849\) 9.81466e6 0.467311
\(850\) −1.96073e7 −0.930833
\(851\) 1.76677e6 0.0836288
\(852\) −1.34427e6 −0.0634434
\(853\) 2.05980e7 0.969285 0.484643 0.874712i \(-0.338950\pi\)
0.484643 + 0.874712i \(0.338950\pi\)
\(854\) 0 0
\(855\) 5.51856e6 0.258173
\(856\) −2.19980e7 −1.02612
\(857\) −2.47572e7 −1.15146 −0.575731 0.817639i \(-0.695282\pi\)
−0.575731 + 0.817639i \(0.695282\pi\)
\(858\) −1.33677e6 −0.0619923
\(859\) −3.91065e7 −1.80828 −0.904141 0.427234i \(-0.859488\pi\)
−0.904141 + 0.427234i \(0.859488\pi\)
\(860\) −1.13987e6 −0.0525544
\(861\) 0 0
\(862\) −2.07527e7 −0.951275
\(863\) −3.93363e7 −1.79790 −0.898952 0.438048i \(-0.855670\pi\)
−0.898952 + 0.438048i \(0.855670\pi\)
\(864\) −615968. −0.0280720
\(865\) 2.12472e7 0.965522
\(866\) 3.59151e7 1.62735
\(867\) −2.20921e7 −0.998135
\(868\) 0 0
\(869\) 3.12859e6 0.140540
\(870\) 2.22263e6 0.0995563
\(871\) −8.18494e6 −0.365570
\(872\) 6.48277e6 0.288715
\(873\) −9.56665e6 −0.424839
\(874\) 1.41195e6 0.0625229
\(875\) 0 0
\(876\) 869268. 0.0382731
\(877\) 8.09253e6 0.355292 0.177646 0.984094i \(-0.443152\pi\)
0.177646 + 0.984094i \(0.443152\pi\)
\(878\) −2.47818e7 −1.08492
\(879\) −4.91168e6 −0.214416
\(880\) −6.26564e6 −0.272746
\(881\) −4.05755e6 −0.176126 −0.0880631 0.996115i \(-0.528068\pi\)
−0.0880631 + 0.996115i \(0.528068\pi\)
\(882\) 0 0
\(883\) 1.79813e7 0.776102 0.388051 0.921638i \(-0.373148\pi\)
0.388051 + 0.921638i \(0.373148\pi\)
\(884\) −680713. −0.0292977
\(885\) −2.87273e6 −0.123293
\(886\) 2.78595e7 1.19231
\(887\) −1.53139e7 −0.653547 −0.326773 0.945103i \(-0.605961\pi\)
−0.326773 + 0.945103i \(0.605961\pi\)
\(888\) 2.17079e7 0.923814
\(889\) 0 0
\(890\) 1.70777e7 0.722693
\(891\) 1.21005e6 0.0510632
\(892\) −98725.0 −0.00415447
\(893\) −3.79376e7 −1.59199
\(894\) −2.47757e7 −1.03677
\(895\) −7.82466e6 −0.326519
\(896\) 0 0
\(897\) −182311. −0.00756539
\(898\) 3.24377e7 1.34233
\(899\) −1.12972e7 −0.466200
\(900\) 346471. 0.0142581
\(901\) 1.83735e7 0.754016
\(902\) 9.01554e6 0.368957
\(903\) 0 0
\(904\) 4.97172e6 0.202342
\(905\) 6.77194e6 0.274847
\(906\) 9.46512e6 0.383094
\(907\) −2.17757e7 −0.878930 −0.439465 0.898260i \(-0.644832\pi\)
−0.439465 + 0.898260i \(0.644832\pi\)
\(908\) −1.73623e6 −0.0698865
\(909\) 1.80085e6 0.0722883
\(910\) 0 0
\(911\) −5.35112e6 −0.213623 −0.106812 0.994279i \(-0.534064\pi\)
−0.106812 + 0.994279i \(0.534064\pi\)
\(912\) 1.60736e7 0.639919
\(913\) −1.87552e7 −0.744639
\(914\) −1.41511e7 −0.560306
\(915\) 1.33321e7 0.526435
\(916\) −2.20124e6 −0.0866818
\(917\) 0 0
\(918\) −7.81507e6 −0.306074
\(919\) −2.67858e7 −1.04620 −0.523102 0.852270i \(-0.675225\pi\)
−0.523102 + 0.852270i \(0.675225\pi\)
\(920\) −922285. −0.0359249
\(921\) 2.96100e7 1.15024
\(922\) 2.45904e7 0.952661
\(923\) −9.44411e6 −0.364886
\(924\) 0 0
\(925\) −2.35890e7 −0.906474
\(926\) −3.02336e7 −1.15868
\(927\) 1.10104e7 0.420826
\(928\) −1.06428e6 −0.0405683
\(929\) 739974. 0.0281305 0.0140652 0.999901i \(-0.495523\pi\)
0.0140652 + 0.999901i \(0.495523\pi\)
\(930\) −1.58265e7 −0.600037
\(931\) 0 0
\(932\) 1.19817e6 0.0451834
\(933\) 6.29741e6 0.236841
\(934\) −4.33166e7 −1.62475
\(935\) 1.30691e7 0.488895
\(936\) −2.24001e6 −0.0835719
\(937\) −122654. −0.00456385 −0.00228193 0.999997i \(-0.500726\pi\)
−0.00228193 + 0.999997i \(0.500726\pi\)
\(938\) 0 0
\(939\) −5.60897e6 −0.207596
\(940\) 1.68772e6 0.0622991
\(941\) 1.22466e7 0.450861 0.225431 0.974259i \(-0.427621\pi\)
0.225431 + 0.974259i \(0.427621\pi\)
\(942\) −1.29855e7 −0.476794
\(943\) 1.22956e6 0.0450266
\(944\) −8.36722e6 −0.305599
\(945\) 0 0
\(946\) 1.35992e7 0.494065
\(947\) −3.71174e6 −0.134494 −0.0672470 0.997736i \(-0.521422\pi\)
−0.0672470 + 0.997736i \(0.521422\pi\)
\(948\) 357048. 0.0129034
\(949\) 6.10701e6 0.220122
\(950\) −1.88516e7 −0.677702
\(951\) 5.75319e6 0.206280
\(952\) 0 0
\(953\) −2.32857e7 −0.830533 −0.415266 0.909700i \(-0.636312\pi\)
−0.415266 + 0.909700i \(0.636312\pi\)
\(954\) 4.11778e6 0.146485
\(955\) 6.24681e6 0.221641
\(956\) 269702. 0.00954418
\(957\) 2.09075e6 0.0737940
\(958\) −3.57474e7 −1.25843
\(959\) 0 0
\(960\) −1.12752e7 −0.394862
\(961\) 5.18142e7 1.80984
\(962\) 1.03867e7 0.361859
\(963\) 9.52772e6 0.331073
\(964\) 2.12787e6 0.0737483
\(965\) 4.24013e6 0.146575
\(966\) 0 0
\(967\) 2.25257e7 0.774661 0.387330 0.921941i \(-0.373397\pi\)
0.387330 + 0.921941i \(0.373397\pi\)
\(968\) 2.37579e7 0.814927
\(969\) −3.35267e7 −1.14705
\(970\) −2.31565e7 −0.790212
\(971\) 1.99456e7 0.678890 0.339445 0.940626i \(-0.389761\pi\)
0.339445 + 0.940626i \(0.389761\pi\)
\(972\) 138096. 0.00468830
\(973\) 0 0
\(974\) 2.53995e7 0.857884
\(975\) 2.43412e6 0.0820032
\(976\) 3.88314e7 1.30484
\(977\) −959342. −0.0321542 −0.0160771 0.999871i \(-0.505118\pi\)
−0.0160771 + 0.999871i \(0.505118\pi\)
\(978\) 2.64242e7 0.883393
\(979\) 1.60643e7 0.535681
\(980\) 0 0
\(981\) −2.80780e6 −0.0931524
\(982\) −4.99542e6 −0.165308
\(983\) 5.22097e7 1.72333 0.861663 0.507481i \(-0.169423\pi\)
0.861663 + 0.507481i \(0.169423\pi\)
\(984\) 1.51073e7 0.497391
\(985\) 8.09042e6 0.265693
\(986\) −1.35030e7 −0.442323
\(987\) 0 0
\(988\) −654475. −0.0213305
\(989\) 1.85468e6 0.0602945
\(990\) 2.92897e6 0.0949789
\(991\) −1.76305e7 −0.570269 −0.285134 0.958488i \(-0.592038\pi\)
−0.285134 + 0.958488i \(0.592038\pi\)
\(992\) 7.57836e6 0.244510
\(993\) −1.26867e7 −0.408296
\(994\) 0 0
\(995\) 2.66498e7 0.853369
\(996\) −2.14043e6 −0.0683680
\(997\) 4.04875e7 1.28998 0.644990 0.764191i \(-0.276862\pi\)
0.644990 + 0.764191i \(0.276862\pi\)
\(998\) 1.70704e6 0.0542522
\(999\) −9.40207e6 −0.298064
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 147.6.a.h.1.2 2
3.2 odd 2 441.6.a.q.1.1 2
7.2 even 3 147.6.e.n.67.1 4
7.3 odd 6 147.6.e.m.79.1 4
7.4 even 3 147.6.e.n.79.1 4
7.5 odd 6 147.6.e.m.67.1 4
7.6 odd 2 147.6.a.j.1.2 yes 2
21.20 even 2 441.6.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.h.1.2 2 1.1 even 1 trivial
147.6.a.j.1.2 yes 2 7.6 odd 2
147.6.e.m.67.1 4 7.5 odd 6
147.6.e.m.79.1 4 7.3 odd 6
147.6.e.n.67.1 4 7.2 even 3
147.6.e.n.79.1 4 7.4 even 3
441.6.a.q.1.1 2 3.2 odd 2
441.6.a.r.1.1 2 21.20 even 2