Properties

Label 177.10.a.a.1.8
Level $177$
Weight $10$
Character 177.1
Self dual yes
Analytic conductor $91.161$
Analytic rank $1$
Dimension $21$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(91.1613430010\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-18.1702 q^{2} +81.0000 q^{3} -181.842 q^{4} +1096.53 q^{5} -1471.79 q^{6} -2885.78 q^{7} +12607.3 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-18.1702 q^{2} +81.0000 q^{3} -181.842 q^{4} +1096.53 q^{5} -1471.79 q^{6} -2885.78 q^{7} +12607.3 q^{8} +6561.00 q^{9} -19924.2 q^{10} +1677.58 q^{11} -14729.2 q^{12} -12863.8 q^{13} +52435.4 q^{14} +88818.7 q^{15} -135974. q^{16} -100811. q^{17} -119215. q^{18} -509381. q^{19} -199395. q^{20} -233749. q^{21} -30482.0 q^{22} +2.00745e6 q^{23} +1.02119e6 q^{24} -750753. q^{25} +233738. q^{26} +531441. q^{27} +524758. q^{28} +1.00906e6 q^{29} -1.61386e6 q^{30} -9.60546e6 q^{31} -3.98425e6 q^{32} +135884. q^{33} +1.83177e6 q^{34} -3.16434e6 q^{35} -1.19307e6 q^{36} +1.57704e7 q^{37} +9.25558e6 q^{38} -1.04197e6 q^{39} +1.38242e7 q^{40} +1.77391e7 q^{41} +4.24727e6 q^{42} -3.57341e7 q^{43} -305054. q^{44} +7.19432e6 q^{45} -3.64759e7 q^{46} -142504. q^{47} -1.10139e7 q^{48} -3.20259e7 q^{49} +1.36414e7 q^{50} -8.16572e6 q^{51} +2.33918e6 q^{52} +6.97093e7 q^{53} -9.65641e6 q^{54} +1.83951e6 q^{55} -3.63819e7 q^{56} -4.12599e7 q^{57} -1.83349e7 q^{58} +1.21174e7 q^{59} -1.61510e7 q^{60} +7.35248e6 q^{61} +1.74534e8 q^{62} -1.89336e7 q^{63} +1.42013e8 q^{64} -1.41055e7 q^{65} -2.46904e6 q^{66} -1.33128e8 q^{67} +1.83318e7 q^{68} +1.62604e8 q^{69} +5.74969e7 q^{70} +2.69166e8 q^{71} +8.27164e7 q^{72} +2.14588e8 q^{73} -2.86551e8 q^{74} -6.08110e7 q^{75} +9.26270e7 q^{76} -4.84112e6 q^{77} +1.89328e7 q^{78} -6.11948e8 q^{79} -1.49099e8 q^{80} +4.30467e7 q^{81} -3.22323e8 q^{82} +4.77356e8 q^{83} +4.25054e7 q^{84} -1.10542e8 q^{85} +6.49297e8 q^{86} +8.17342e7 q^{87} +2.11497e7 q^{88} -7.71476e8 q^{89} -1.30722e8 q^{90} +3.71221e7 q^{91} -3.65040e8 q^{92} -7.78043e8 q^{93} +2.58934e6 q^{94} -5.58550e8 q^{95} -3.22724e8 q^{96} +1.38667e9 q^{97} +5.81917e8 q^{98} +1.10066e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} - 54663q^{10} - 151769q^{11} + 421686q^{12} - 153611q^{13} - 286771q^{14} - 240084q^{15} + 805530q^{16} - 723621q^{17} - 433026q^{18} - 549388q^{19} - 527311q^{20} - 2492775q^{21} + 2973158q^{22} + 169962q^{23} - 1994301q^{24} + 8035779q^{25} - 2337392q^{26} + 11160261q^{27} - 22659054q^{28} - 16845442q^{29} - 4427703q^{30} - 19307976q^{31} - 44923568q^{32} - 12293289q^{33} - 35547496q^{34} - 34882596q^{35} + 34156566q^{36} - 41561129q^{37} - 52335371q^{38} - 12442491q^{39} - 125735038q^{40} - 68169291q^{41} - 23228451q^{42} - 25719587q^{43} - 126277032q^{44} - 19446804q^{45} - 292814271q^{46} - 174095332q^{47} + 65247930q^{48} + 7479350q^{49} - 227877439q^{50} - 58613301q^{51} - 232397708q^{52} - 228390500q^{53} - 35075106q^{54} - 29426208q^{55} + 326778474q^{56} - 44500428q^{57} + 480343762q^{58} + 254464581q^{59} - 42712191q^{60} - 183928964q^{61} - 21753862q^{62} - 201914775q^{63} + 310571245q^{64} + 5308466q^{65} + 240825798q^{66} - 82724114q^{67} - 138336205q^{68} + 13766922q^{69} + 1030274876q^{70} - 404721965q^{71} - 161538381q^{72} + 154162574q^{73} + 36352054q^{74} + 650898099q^{75} + 1068940636q^{76} - 448535481q^{77} - 189328752q^{78} + 272529635q^{79} - 345587859q^{80} + 903981141q^{81} - 38412637q^{82} + 432518643q^{83} - 1835383374q^{84} - 126211490q^{85} - 3699273072q^{86} - 1364480802q^{87} + 170111045q^{88} - 1255621070q^{89} - 358643943q^{90} + 1448885849q^{91} + 1568933320q^{92} - 1563946056q^{93} - 1908445164q^{94} - 2896546490q^{95} - 3638809008q^{96} + 1007235486q^{97} - 9506868248q^{98} - 995756409q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −18.1702 −0.803019 −0.401509 0.915855i \(-0.631514\pi\)
−0.401509 + 0.915855i \(0.631514\pi\)
\(3\) 81.0000 0.577350
\(4\) −181.842 −0.355161
\(5\) 1096.53 0.784611 0.392306 0.919835i \(-0.371678\pi\)
0.392306 + 0.919835i \(0.371678\pi\)
\(6\) −1471.79 −0.463623
\(7\) −2885.78 −0.454279 −0.227140 0.973862i \(-0.572937\pi\)
−0.227140 + 0.973862i \(0.572937\pi\)
\(8\) 12607.3 1.08822
\(9\) 6561.00 0.333333
\(10\) −19924.2 −0.630057
\(11\) 1677.58 0.0345474 0.0172737 0.999851i \(-0.494501\pi\)
0.0172737 + 0.999851i \(0.494501\pi\)
\(12\) −14729.2 −0.205052
\(13\) −12863.8 −0.124918 −0.0624588 0.998048i \(-0.519894\pi\)
−0.0624588 + 0.998048i \(0.519894\pi\)
\(14\) 52435.4 0.364795
\(15\) 88818.7 0.452995
\(16\) −135974. −0.518700
\(17\) −100811. −0.292745 −0.146372 0.989230i \(-0.546760\pi\)
−0.146372 + 0.989230i \(0.546760\pi\)
\(18\) −119215. −0.267673
\(19\) −509381. −0.896709 −0.448355 0.893856i \(-0.647990\pi\)
−0.448355 + 0.893856i \(0.647990\pi\)
\(20\) −199395. −0.278663
\(21\) −233749. −0.262278
\(22\) −30482.0 −0.0277422
\(23\) 2.00745e6 1.49579 0.747894 0.663818i \(-0.231065\pi\)
0.747894 + 0.663818i \(0.231065\pi\)
\(24\) 1.02119e6 0.628284
\(25\) −750753. −0.384385
\(26\) 233738. 0.100311
\(27\) 531441. 0.192450
\(28\) 524758. 0.161342
\(29\) 1.00906e6 0.264928 0.132464 0.991188i \(-0.457711\pi\)
0.132464 + 0.991188i \(0.457711\pi\)
\(30\) −1.61386e6 −0.363764
\(31\) −9.60546e6 −1.86806 −0.934030 0.357195i \(-0.883733\pi\)
−0.934030 + 0.357195i \(0.883733\pi\)
\(32\) −3.98425e6 −0.671694
\(33\) 135884. 0.0199459
\(34\) 1.83177e6 0.235080
\(35\) −3.16434e6 −0.356432
\(36\) −1.19307e6 −0.118387
\(37\) 1.57704e7 1.38336 0.691678 0.722206i \(-0.256872\pi\)
0.691678 + 0.722206i \(0.256872\pi\)
\(38\) 9.25558e6 0.720074
\(39\) −1.04197e6 −0.0721212
\(40\) 1.38242e7 0.853829
\(41\) 1.77391e7 0.980401 0.490200 0.871610i \(-0.336924\pi\)
0.490200 + 0.871610i \(0.336924\pi\)
\(42\) 4.24727e6 0.210614
\(43\) −3.57341e7 −1.59395 −0.796975 0.604012i \(-0.793568\pi\)
−0.796975 + 0.604012i \(0.793568\pi\)
\(44\) −305054. −0.0122699
\(45\) 7.19432e6 0.261537
\(46\) −3.64759e7 −1.20115
\(47\) −142504. −0.00425978 −0.00212989 0.999998i \(-0.500678\pi\)
−0.00212989 + 0.999998i \(0.500678\pi\)
\(48\) −1.10139e7 −0.299472
\(49\) −3.20259e7 −0.793631
\(50\) 1.36414e7 0.308669
\(51\) −8.16572e6 −0.169016
\(52\) 2.33918e6 0.0443658
\(53\) 6.97093e7 1.21353 0.606764 0.794882i \(-0.292467\pi\)
0.606764 + 0.794882i \(0.292467\pi\)
\(54\) −9.65641e6 −0.154541
\(55\) 1.83951e6 0.0271063
\(56\) −3.63819e7 −0.494355
\(57\) −4.12599e7 −0.517715
\(58\) −1.83349e7 −0.212742
\(59\) 1.21174e7 0.130189
\(60\) −1.61510e7 −0.160886
\(61\) 7.35248e6 0.0679907 0.0339953 0.999422i \(-0.489177\pi\)
0.0339953 + 0.999422i \(0.489177\pi\)
\(62\) 1.74534e8 1.50009
\(63\) −1.89336e7 −0.151426
\(64\) 1.42013e8 1.05808
\(65\) −1.41055e7 −0.0980117
\(66\) −2.46904e6 −0.0160170
\(67\) −1.33128e8 −0.807113 −0.403556 0.914955i \(-0.632226\pi\)
−0.403556 + 0.914955i \(0.632226\pi\)
\(68\) 1.83318e7 0.103971
\(69\) 1.62604e8 0.863594
\(70\) 5.74969e7 0.286222
\(71\) 2.69166e8 1.25707 0.628533 0.777783i \(-0.283656\pi\)
0.628533 + 0.777783i \(0.283656\pi\)
\(72\) 8.27164e7 0.362740
\(73\) 2.14588e8 0.884409 0.442205 0.896914i \(-0.354196\pi\)
0.442205 + 0.896914i \(0.354196\pi\)
\(74\) −2.86551e8 −1.11086
\(75\) −6.08110e7 −0.221925
\(76\) 9.26270e7 0.318476
\(77\) −4.84112e6 −0.0156942
\(78\) 1.89328e7 0.0579147
\(79\) −6.11948e8 −1.76763 −0.883817 0.467832i \(-0.845035\pi\)
−0.883817 + 0.467832i \(0.845035\pi\)
\(80\) −1.49099e8 −0.406978
\(81\) 4.30467e7 0.111111
\(82\) −3.22323e8 −0.787280
\(83\) 4.77356e8 1.10405 0.552027 0.833826i \(-0.313854\pi\)
0.552027 + 0.833826i \(0.313854\pi\)
\(84\) 4.25054e7 0.0931509
\(85\) −1.10542e8 −0.229691
\(86\) 6.49297e8 1.27997
\(87\) 8.17342e7 0.152956
\(88\) 2.11497e7 0.0375951
\(89\) −7.71476e8 −1.30337 −0.651684 0.758490i \(-0.725937\pi\)
−0.651684 + 0.758490i \(0.725937\pi\)
\(90\) −1.30722e8 −0.210019
\(91\) 3.71221e7 0.0567474
\(92\) −3.65040e8 −0.531245
\(93\) −7.78043e8 −1.07852
\(94\) 2.58934e6 0.00342069
\(95\) −5.58550e8 −0.703568
\(96\) −3.22724e8 −0.387803
\(97\) 1.38667e9 1.59038 0.795190 0.606361i \(-0.207371\pi\)
0.795190 + 0.606361i \(0.207371\pi\)
\(98\) 5.81917e8 0.637300
\(99\) 1.10066e7 0.0115158
\(100\) 1.36519e8 0.136519
\(101\) −5.68775e8 −0.543869 −0.271934 0.962316i \(-0.587663\pi\)
−0.271934 + 0.962316i \(0.587663\pi\)
\(102\) 1.48373e8 0.135723
\(103\) 5.90682e7 0.0517114 0.0258557 0.999666i \(-0.491769\pi\)
0.0258557 + 0.999666i \(0.491769\pi\)
\(104\) −1.62177e8 −0.135938
\(105\) −2.56312e8 −0.205786
\(106\) −1.26664e9 −0.974485
\(107\) −1.58696e8 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(108\) −9.66385e7 −0.0683507
\(109\) −2.56370e9 −1.73959 −0.869797 0.493410i \(-0.835750\pi\)
−0.869797 + 0.493410i \(0.835750\pi\)
\(110\) −3.34243e7 −0.0217668
\(111\) 1.27740e9 0.798681
\(112\) 3.92392e8 0.235635
\(113\) −1.92831e8 −0.111256 −0.0556280 0.998452i \(-0.517716\pi\)
−0.0556280 + 0.998452i \(0.517716\pi\)
\(114\) 7.49702e8 0.415735
\(115\) 2.20123e9 1.17361
\(116\) −1.83491e8 −0.0940920
\(117\) −8.43992e7 −0.0416392
\(118\) −2.20175e8 −0.104544
\(119\) 2.90920e8 0.132988
\(120\) 1.11976e9 0.492958
\(121\) −2.35513e9 −0.998806
\(122\) −1.33596e8 −0.0545978
\(123\) 1.43687e9 0.566035
\(124\) 1.74668e9 0.663461
\(125\) −2.96488e9 −1.08620
\(126\) 3.44029e8 0.121598
\(127\) −3.38046e9 −1.15308 −0.576539 0.817070i \(-0.695597\pi\)
−0.576539 + 0.817070i \(0.695597\pi\)
\(128\) −5.40485e8 −0.177967
\(129\) −2.89446e9 −0.920268
\(130\) 2.56300e8 0.0787052
\(131\) −5.38013e9 −1.59614 −0.798071 0.602563i \(-0.794146\pi\)
−0.798071 + 0.602563i \(0.794146\pi\)
\(132\) −2.47094e7 −0.00708402
\(133\) 1.46996e9 0.407356
\(134\) 2.41898e9 0.648127
\(135\) 5.82740e8 0.150998
\(136\) −1.27096e9 −0.318571
\(137\) −7.34840e9 −1.78218 −0.891088 0.453831i \(-0.850057\pi\)
−0.891088 + 0.453831i \(0.850057\pi\)
\(138\) −2.95455e9 −0.693482
\(139\) −6.11127e9 −1.38856 −0.694280 0.719705i \(-0.744277\pi\)
−0.694280 + 0.719705i \(0.744277\pi\)
\(140\) 5.75411e8 0.126591
\(141\) −1.15428e7 −0.00245939
\(142\) −4.89082e9 −1.00945
\(143\) −2.15800e7 −0.00431557
\(144\) −8.92126e8 −0.172900
\(145\) 1.10647e9 0.207865
\(146\) −3.89912e9 −0.710197
\(147\) −2.59409e9 −0.458203
\(148\) −2.86772e9 −0.491314
\(149\) −6.73461e9 −1.11937 −0.559686 0.828705i \(-0.689078\pi\)
−0.559686 + 0.828705i \(0.689078\pi\)
\(150\) 1.10495e9 0.178210
\(151\) 1.63533e9 0.255981 0.127991 0.991775i \(-0.459147\pi\)
0.127991 + 0.991775i \(0.459147\pi\)
\(152\) −6.42191e9 −0.975816
\(153\) −6.61423e8 −0.0975816
\(154\) 8.79644e7 0.0126027
\(155\) −1.05327e10 −1.46570
\(156\) 1.89473e8 0.0256146
\(157\) −6.72353e8 −0.0883179 −0.0441589 0.999025i \(-0.514061\pi\)
−0.0441589 + 0.999025i \(0.514061\pi\)
\(158\) 1.11192e10 1.41944
\(159\) 5.64646e9 0.700630
\(160\) −4.36883e9 −0.527018
\(161\) −5.79308e9 −0.679505
\(162\) −7.82169e8 −0.0892243
\(163\) −1.02090e10 −1.13276 −0.566379 0.824145i \(-0.691656\pi\)
−0.566379 + 0.824145i \(0.691656\pi\)
\(164\) −3.22572e9 −0.348200
\(165\) 1.49000e8 0.0156498
\(166\) −8.67367e9 −0.886577
\(167\) −8.98649e9 −0.894059 −0.447029 0.894519i \(-0.647518\pi\)
−0.447029 + 0.894519i \(0.647518\pi\)
\(168\) −2.94693e9 −0.285416
\(169\) −1.04390e10 −0.984396
\(170\) 2.00858e9 0.184446
\(171\) −3.34205e9 −0.298903
\(172\) 6.49797e9 0.566109
\(173\) 1.26827e10 1.07648 0.538238 0.842793i \(-0.319090\pi\)
0.538238 + 0.842793i \(0.319090\pi\)
\(174\) −1.48513e9 −0.122827
\(175\) 2.16651e9 0.174618
\(176\) −2.28107e8 −0.0179197
\(177\) 9.81506e8 0.0751646
\(178\) 1.40179e10 1.04663
\(179\) −1.31061e10 −0.954190 −0.477095 0.878852i \(-0.658310\pi\)
−0.477095 + 0.878852i \(0.658310\pi\)
\(180\) −1.30823e9 −0.0928877
\(181\) 2.68008e10 1.85607 0.928034 0.372495i \(-0.121498\pi\)
0.928034 + 0.372495i \(0.121498\pi\)
\(182\) −6.74517e8 −0.0455692
\(183\) 5.95551e8 0.0392544
\(184\) 2.53085e10 1.62775
\(185\) 1.72926e10 1.08540
\(186\) 1.41372e10 0.866076
\(187\) −1.69119e8 −0.0101136
\(188\) 2.59133e7 0.00151291
\(189\) −1.53362e9 −0.0874261
\(190\) 1.01490e10 0.564978
\(191\) 2.93131e9 0.159372 0.0796861 0.996820i \(-0.474608\pi\)
0.0796861 + 0.996820i \(0.474608\pi\)
\(192\) 1.15031e10 0.610884
\(193\) −6.49197e9 −0.336797 −0.168399 0.985719i \(-0.553860\pi\)
−0.168399 + 0.985719i \(0.553860\pi\)
\(194\) −2.51962e10 −1.27710
\(195\) −1.14254e9 −0.0565871
\(196\) 5.82366e9 0.281866
\(197\) −9.40339e9 −0.444822 −0.222411 0.974953i \(-0.571393\pi\)
−0.222411 + 0.974953i \(0.571393\pi\)
\(198\) −1.99992e8 −0.00924740
\(199\) 2.14054e10 0.967575 0.483787 0.875186i \(-0.339261\pi\)
0.483787 + 0.875186i \(0.339261\pi\)
\(200\) −9.46495e9 −0.418296
\(201\) −1.07834e10 −0.465987
\(202\) 1.03348e10 0.436737
\(203\) −2.91194e9 −0.120351
\(204\) 1.48487e9 0.0600280
\(205\) 1.94514e10 0.769233
\(206\) −1.07328e9 −0.0415252
\(207\) 1.31709e10 0.498596
\(208\) 1.74914e9 0.0647947
\(209\) −8.54526e8 −0.0309790
\(210\) 4.65725e9 0.165250
\(211\) −4.98106e10 −1.73002 −0.865009 0.501756i \(-0.832687\pi\)
−0.865009 + 0.501756i \(0.832687\pi\)
\(212\) −1.26761e10 −0.430997
\(213\) 2.18025e10 0.725768
\(214\) 2.88354e9 0.0939862
\(215\) −3.91834e10 −1.25063
\(216\) 6.70003e9 0.209428
\(217\) 2.77193e10 0.848620
\(218\) 4.65830e10 1.39693
\(219\) 1.73817e10 0.510614
\(220\) −3.34500e8 −0.00962708
\(221\) 1.29681e9 0.0365690
\(222\) −2.32107e10 −0.641356
\(223\) 8.74921e9 0.236917 0.118459 0.992959i \(-0.462205\pi\)
0.118459 + 0.992959i \(0.462205\pi\)
\(224\) 1.14977e10 0.305136
\(225\) −4.92569e9 −0.128128
\(226\) 3.50378e9 0.0893406
\(227\) 3.86038e10 0.964969 0.482484 0.875905i \(-0.339735\pi\)
0.482484 + 0.875905i \(0.339735\pi\)
\(228\) 7.50279e9 0.183872
\(229\) 3.12351e10 0.750555 0.375278 0.926912i \(-0.377547\pi\)
0.375278 + 0.926912i \(0.377547\pi\)
\(230\) −3.99968e10 −0.942432
\(231\) −3.92131e8 −0.00906102
\(232\) 1.27216e10 0.288300
\(233\) −5.00071e10 −1.11155 −0.555777 0.831332i \(-0.687579\pi\)
−0.555777 + 0.831332i \(0.687579\pi\)
\(234\) 1.53355e9 0.0334370
\(235\) −1.56260e8 −0.00334227
\(236\) −2.20345e9 −0.0462380
\(237\) −4.95678e10 −1.02054
\(238\) −5.28608e9 −0.106792
\(239\) 6.27521e10 1.24405 0.622025 0.782997i \(-0.286310\pi\)
0.622025 + 0.782997i \(0.286310\pi\)
\(240\) −1.20770e10 −0.234969
\(241\) 5.35838e10 1.02319 0.511596 0.859226i \(-0.329055\pi\)
0.511596 + 0.859226i \(0.329055\pi\)
\(242\) 4.27933e10 0.802060
\(243\) 3.48678e9 0.0641500
\(244\) −1.33699e9 −0.0241476
\(245\) −3.51172e10 −0.622691
\(246\) −2.61082e10 −0.454537
\(247\) 6.55256e9 0.112015
\(248\) −1.21099e11 −2.03286
\(249\) 3.86658e10 0.637426
\(250\) 5.38725e10 0.872242
\(251\) 7.31890e10 1.16390 0.581948 0.813226i \(-0.302291\pi\)
0.581948 + 0.813226i \(0.302291\pi\)
\(252\) 3.44294e9 0.0537807
\(253\) 3.36766e9 0.0516756
\(254\) 6.14237e10 0.925943
\(255\) −8.95393e9 −0.132612
\(256\) −6.28902e10 −0.915172
\(257\) −7.91834e10 −1.13223 −0.566115 0.824326i \(-0.691554\pi\)
−0.566115 + 0.824326i \(0.691554\pi\)
\(258\) 5.25931e10 0.738993
\(259\) −4.55099e10 −0.628430
\(260\) 2.56497e9 0.0348099
\(261\) 6.62047e9 0.0883093
\(262\) 9.77582e10 1.28173
\(263\) 1.99558e10 0.257198 0.128599 0.991697i \(-0.458952\pi\)
0.128599 + 0.991697i \(0.458952\pi\)
\(264\) 1.71312e9 0.0217056
\(265\) 7.64382e10 0.952147
\(266\) −2.67096e10 −0.327115
\(267\) −6.24895e10 −0.752500
\(268\) 2.42084e10 0.286655
\(269\) −1.24707e11 −1.45213 −0.726066 0.687625i \(-0.758653\pi\)
−0.726066 + 0.687625i \(0.758653\pi\)
\(270\) −1.05885e10 −0.121255
\(271\) −1.08239e11 −1.21905 −0.609525 0.792767i \(-0.708640\pi\)
−0.609525 + 0.792767i \(0.708640\pi\)
\(272\) 1.37077e10 0.151847
\(273\) 3.00689e9 0.0327631
\(274\) 1.33522e11 1.43112
\(275\) −1.25945e9 −0.0132795
\(276\) −2.95682e10 −0.306715
\(277\) 1.24654e11 1.27218 0.636090 0.771615i \(-0.280551\pi\)
0.636090 + 0.771615i \(0.280551\pi\)
\(278\) 1.11043e11 1.11504
\(279\) −6.30214e10 −0.622686
\(280\) −3.98938e10 −0.387877
\(281\) 1.22215e11 1.16936 0.584679 0.811265i \(-0.301220\pi\)
0.584679 + 0.811265i \(0.301220\pi\)
\(282\) 2.09736e8 0.00197493
\(283\) −1.35737e11 −1.25794 −0.628969 0.777431i \(-0.716523\pi\)
−0.628969 + 0.777431i \(0.716523\pi\)
\(284\) −4.89458e10 −0.446461
\(285\) −4.52426e10 −0.406205
\(286\) 3.92113e8 0.00346549
\(287\) −5.11912e10 −0.445376
\(288\) −2.61406e10 −0.223898
\(289\) −1.08425e11 −0.914300
\(290\) −2.01048e10 −0.166920
\(291\) 1.12320e11 0.918206
\(292\) −3.90212e10 −0.314108
\(293\) 7.79389e10 0.617803 0.308902 0.951094i \(-0.400039\pi\)
0.308902 + 0.951094i \(0.400039\pi\)
\(294\) 4.71353e10 0.367945
\(295\) 1.32870e10 0.102148
\(296\) 1.98821e11 1.50540
\(297\) 8.91533e8 0.00664865
\(298\) 1.22370e11 0.898877
\(299\) −2.58234e10 −0.186850
\(300\) 1.10580e10 0.0788191
\(301\) 1.03121e11 0.724099
\(302\) −2.97143e10 −0.205558
\(303\) −4.60707e10 −0.314003
\(304\) 6.92626e10 0.465123
\(305\) 8.06219e9 0.0533462
\(306\) 1.20182e10 0.0783599
\(307\) 1.99796e11 1.28370 0.641850 0.766830i \(-0.278167\pi\)
0.641850 + 0.766830i \(0.278167\pi\)
\(308\) 8.80321e8 0.00557395
\(309\) 4.78453e9 0.0298556
\(310\) 1.91381e11 1.17698
\(311\) 2.73103e10 0.165541 0.0827703 0.996569i \(-0.473623\pi\)
0.0827703 + 0.996569i \(0.473623\pi\)
\(312\) −1.31364e10 −0.0784837
\(313\) −2.95243e11 −1.73872 −0.869362 0.494177i \(-0.835470\pi\)
−0.869362 + 0.494177i \(0.835470\pi\)
\(314\) 1.22168e10 0.0709209
\(315\) −2.07612e10 −0.118811
\(316\) 1.11278e11 0.627794
\(317\) −3.10967e11 −1.72961 −0.864805 0.502109i \(-0.832558\pi\)
−0.864805 + 0.502109i \(0.832558\pi\)
\(318\) −1.02597e11 −0.562619
\(319\) 1.69278e9 0.00915257
\(320\) 1.55722e11 0.830183
\(321\) −1.28544e10 −0.0675737
\(322\) 1.05262e11 0.545656
\(323\) 5.13514e10 0.262507
\(324\) −7.82772e9 −0.0394623
\(325\) 9.65752e9 0.0480165
\(326\) 1.85499e11 0.909627
\(327\) −2.07660e11 −1.00436
\(328\) 2.23642e11 1.06689
\(329\) 4.11237e8 0.00193513
\(330\) −2.70737e9 −0.0125671
\(331\) −2.95514e11 −1.35317 −0.676586 0.736364i \(-0.736541\pi\)
−0.676586 + 0.736364i \(0.736541\pi\)
\(332\) −8.68035e10 −0.392117
\(333\) 1.03469e11 0.461119
\(334\) 1.63287e11 0.717946
\(335\) −1.45979e11 −0.633270
\(336\) 3.17838e10 0.136044
\(337\) 6.54404e10 0.276383 0.138191 0.990406i \(-0.455871\pi\)
0.138191 + 0.990406i \(0.455871\pi\)
\(338\) 1.89680e11 0.790488
\(339\) −1.56193e10 −0.0642336
\(340\) 2.01013e10 0.0815772
\(341\) −1.61139e10 −0.0645366
\(342\) 6.07258e10 0.240025
\(343\) 2.08872e11 0.814809
\(344\) −4.50510e11 −1.73457
\(345\) 1.78299e11 0.677585
\(346\) −2.30448e11 −0.864431
\(347\) −2.55074e11 −0.944459 −0.472230 0.881476i \(-0.656551\pi\)
−0.472230 + 0.881476i \(0.656551\pi\)
\(348\) −1.48627e10 −0.0543240
\(349\) −2.16864e11 −0.782479 −0.391240 0.920289i \(-0.627954\pi\)
−0.391240 + 0.920289i \(0.627954\pi\)
\(350\) −3.93660e10 −0.140222
\(351\) −6.83634e9 −0.0240404
\(352\) −6.68388e9 −0.0232053
\(353\) 3.04446e11 1.04358 0.521789 0.853075i \(-0.325265\pi\)
0.521789 + 0.853075i \(0.325265\pi\)
\(354\) −1.78342e10 −0.0603586
\(355\) 2.95148e11 0.986308
\(356\) 1.40287e11 0.462905
\(357\) 2.35645e10 0.0767806
\(358\) 2.38141e11 0.766233
\(359\) −2.99907e11 −0.952932 −0.476466 0.879193i \(-0.658082\pi\)
−0.476466 + 0.879193i \(0.658082\pi\)
\(360\) 9.07008e10 0.284610
\(361\) −6.32186e10 −0.195913
\(362\) −4.86977e11 −1.49046
\(363\) −1.90766e11 −0.576661
\(364\) −6.75037e9 −0.0201545
\(365\) 2.35302e11 0.693917
\(366\) −1.08213e10 −0.0315221
\(367\) −1.37612e11 −0.395967 −0.197983 0.980205i \(-0.563439\pi\)
−0.197983 + 0.980205i \(0.563439\pi\)
\(368\) −2.72962e11 −0.775865
\(369\) 1.16386e11 0.326800
\(370\) −3.14211e11 −0.871594
\(371\) −2.01166e11 −0.551280
\(372\) 1.41481e11 0.383050
\(373\) 5.96688e11 1.59609 0.798045 0.602598i \(-0.205868\pi\)
0.798045 + 0.602598i \(0.205868\pi\)
\(374\) 3.07293e9 0.00812139
\(375\) −2.40155e11 −0.627120
\(376\) −1.79659e9 −0.00463558
\(377\) −1.29804e10 −0.0330941
\(378\) 2.78663e10 0.0702048
\(379\) −4.36706e10 −0.108721 −0.0543604 0.998521i \(-0.517312\pi\)
−0.0543604 + 0.998521i \(0.517312\pi\)
\(380\) 1.01568e11 0.249880
\(381\) −2.73817e11 −0.665730
\(382\) −5.32627e10 −0.127979
\(383\) 7.11868e11 1.69046 0.845230 0.534403i \(-0.179463\pi\)
0.845230 + 0.534403i \(0.179463\pi\)
\(384\) −4.37793e10 −0.102749
\(385\) −5.30842e9 −0.0123138
\(386\) 1.17961e11 0.270454
\(387\) −2.34452e11 −0.531317
\(388\) −2.52156e11 −0.564840
\(389\) 6.46995e11 1.43261 0.716305 0.697787i \(-0.245832\pi\)
0.716305 + 0.697787i \(0.245832\pi\)
\(390\) 2.07603e10 0.0454405
\(391\) −2.02374e11 −0.437884
\(392\) −4.03759e11 −0.863644
\(393\) −4.35790e11 −0.921534
\(394\) 1.70862e11 0.357200
\(395\) −6.71017e11 −1.38691
\(396\) −2.00146e9 −0.00408996
\(397\) 1.09495e11 0.221226 0.110613 0.993864i \(-0.464719\pi\)
0.110613 + 0.993864i \(0.464719\pi\)
\(398\) −3.88941e11 −0.776981
\(399\) 1.19067e11 0.235187
\(400\) 1.02083e11 0.199381
\(401\) −2.69349e11 −0.520194 −0.260097 0.965583i \(-0.583754\pi\)
−0.260097 + 0.965583i \(0.583754\pi\)
\(402\) 1.95937e11 0.374196
\(403\) 1.23563e11 0.233353
\(404\) 1.03427e11 0.193161
\(405\) 4.72019e10 0.0871790
\(406\) 5.29107e10 0.0966443
\(407\) 2.64560e10 0.0477913
\(408\) −1.02948e11 −0.183927
\(409\) −4.33340e10 −0.0765727 −0.0382863 0.999267i \(-0.512190\pi\)
−0.0382863 + 0.999267i \(0.512190\pi\)
\(410\) −3.53436e11 −0.617709
\(411\) −5.95221e11 −1.02894
\(412\) −1.07411e10 −0.0183659
\(413\) −3.49681e10 −0.0591421
\(414\) −2.39318e11 −0.400382
\(415\) 5.23434e11 0.866254
\(416\) 5.12525e10 0.0839063
\(417\) −4.95013e11 −0.801685
\(418\) 1.55269e10 0.0248767
\(419\) 3.11457e11 0.493668 0.246834 0.969058i \(-0.420610\pi\)
0.246834 + 0.969058i \(0.420610\pi\)
\(420\) 4.66083e10 0.0730872
\(421\) 2.22481e11 0.345162 0.172581 0.984995i \(-0.444789\pi\)
0.172581 + 0.984995i \(0.444789\pi\)
\(422\) 9.05070e11 1.38924
\(423\) −9.34970e8 −0.00141993
\(424\) 8.78845e11 1.32058
\(425\) 7.56844e10 0.112527
\(426\) −3.96156e11 −0.582805
\(427\) −2.12177e10 −0.0308867
\(428\) 2.88576e10 0.0415684
\(429\) −1.74798e9 −0.00249160
\(430\) 7.11972e11 1.00428
\(431\) −4.24757e11 −0.592916 −0.296458 0.955046i \(-0.595805\pi\)
−0.296458 + 0.955046i \(0.595805\pi\)
\(432\) −7.22622e10 −0.0998239
\(433\) 1.33542e12 1.82567 0.912837 0.408325i \(-0.133887\pi\)
0.912837 + 0.408325i \(0.133887\pi\)
\(434\) −5.03666e11 −0.681458
\(435\) 8.96238e10 0.120011
\(436\) 4.66189e11 0.617836
\(437\) −1.02256e12 −1.34129
\(438\) −3.15829e11 −0.410033
\(439\) −1.39607e12 −1.79397 −0.896987 0.442057i \(-0.854249\pi\)
−0.896987 + 0.442057i \(0.854249\pi\)
\(440\) 2.31912e10 0.0294976
\(441\) −2.10122e11 −0.264544
\(442\) −2.35634e10 −0.0293656
\(443\) 5.81905e11 0.717852 0.358926 0.933366i \(-0.383143\pi\)
0.358926 + 0.933366i \(0.383143\pi\)
\(444\) −2.32285e11 −0.283660
\(445\) −8.45944e11 −1.02264
\(446\) −1.58975e11 −0.190249
\(447\) −5.45504e11 −0.646270
\(448\) −4.09820e11 −0.480665
\(449\) −5.29072e11 −0.614336 −0.307168 0.951655i \(-0.599381\pi\)
−0.307168 + 0.951655i \(0.599381\pi\)
\(450\) 8.95010e10 0.102890
\(451\) 2.97587e10 0.0338703
\(452\) 3.50648e10 0.0395137
\(453\) 1.32462e11 0.147791
\(454\) −7.01439e11 −0.774888
\(455\) 4.07054e10 0.0445247
\(456\) −5.20175e11 −0.563388
\(457\) −1.28899e12 −1.38237 −0.691187 0.722676i \(-0.742912\pi\)
−0.691187 + 0.722676i \(0.742912\pi\)
\(458\) −5.67549e11 −0.602710
\(459\) −5.35753e10 −0.0563388
\(460\) −4.00276e11 −0.416821
\(461\) 4.40656e11 0.454407 0.227204 0.973847i \(-0.427042\pi\)
0.227204 + 0.973847i \(0.427042\pi\)
\(462\) 7.12512e9 0.00727617
\(463\) −4.52942e10 −0.0458066 −0.0229033 0.999738i \(-0.507291\pi\)
−0.0229033 + 0.999738i \(0.507291\pi\)
\(464\) −1.37207e11 −0.137418
\(465\) −8.53145e11 −0.846222
\(466\) 9.08642e11 0.892598
\(467\) 9.95270e11 0.968311 0.484156 0.874982i \(-0.339127\pi\)
0.484156 + 0.874982i \(0.339127\pi\)
\(468\) 1.53474e10 0.0147886
\(469\) 3.84180e11 0.366654
\(470\) 2.83928e9 0.00268391
\(471\) −5.44606e10 −0.0509904
\(472\) 1.52767e11 0.141674
\(473\) −5.99467e10 −0.0550668
\(474\) 9.00658e11 0.819516
\(475\) 3.82419e11 0.344682
\(476\) −5.29015e10 −0.0472321
\(477\) 4.57363e11 0.404509
\(478\) −1.14022e12 −0.998996
\(479\) 2.36169e11 0.204981 0.102490 0.994734i \(-0.467319\pi\)
0.102490 + 0.994734i \(0.467319\pi\)
\(480\) −3.53876e11 −0.304274
\(481\) −2.02866e11 −0.172805
\(482\) −9.73631e11 −0.821642
\(483\) −4.69239e11 −0.392313
\(484\) 4.28263e11 0.354737
\(485\) 1.52052e12 1.24783
\(486\) −6.33557e10 −0.0515137
\(487\) 8.97688e11 0.723178 0.361589 0.932338i \(-0.382234\pi\)
0.361589 + 0.932338i \(0.382234\pi\)
\(488\) 9.26948e10 0.0739888
\(489\) −8.26926e11 −0.653999
\(490\) 6.38088e11 0.500033
\(491\) −1.38110e12 −1.07240 −0.536202 0.844090i \(-0.680141\pi\)
−0.536202 + 0.844090i \(0.680141\pi\)
\(492\) −2.61283e11 −0.201033
\(493\) −1.01725e11 −0.0775563
\(494\) −1.19062e11 −0.0899499
\(495\) 1.20690e10 0.00903542
\(496\) 1.30609e12 0.968963
\(497\) −7.76756e11 −0.571059
\(498\) −7.02567e11 −0.511865
\(499\) −2.23948e12 −1.61694 −0.808472 0.588535i \(-0.799705\pi\)
−0.808472 + 0.588535i \(0.799705\pi\)
\(500\) 5.39140e11 0.385777
\(501\) −7.27906e11 −0.516185
\(502\) −1.32986e12 −0.934630
\(503\) −1.10093e12 −0.766840 −0.383420 0.923574i \(-0.625254\pi\)
−0.383420 + 0.923574i \(0.625254\pi\)
\(504\) −2.38702e11 −0.164785
\(505\) −6.23677e11 −0.426725
\(506\) −6.11911e10 −0.0414965
\(507\) −8.45561e11 −0.568341
\(508\) 6.14710e11 0.409528
\(509\) 1.30158e12 0.859492 0.429746 0.902950i \(-0.358603\pi\)
0.429746 + 0.902950i \(0.358603\pi\)
\(510\) 1.62695e11 0.106490
\(511\) −6.19256e11 −0.401769
\(512\) 1.41946e12 0.912867
\(513\) −2.70706e11 −0.172572
\(514\) 1.43878e12 0.909203
\(515\) 6.47699e10 0.0405733
\(516\) 5.26336e11 0.326843
\(517\) −2.39062e8 −0.000147164 0
\(518\) 8.26925e11 0.504641
\(519\) 1.02730e12 0.621504
\(520\) −1.77832e11 −0.106658
\(521\) −2.03385e12 −1.20934 −0.604670 0.796476i \(-0.706695\pi\)
−0.604670 + 0.796476i \(0.706695\pi\)
\(522\) −1.20296e11 −0.0709140
\(523\) 2.47262e12 1.44511 0.722554 0.691314i \(-0.242968\pi\)
0.722554 + 0.691314i \(0.242968\pi\)
\(524\) 9.78335e11 0.566887
\(525\) 1.75487e11 0.100816
\(526\) −3.62602e11 −0.206535
\(527\) 9.68340e11 0.546865
\(528\) −1.84767e10 −0.0103460
\(529\) 2.22871e12 1.23738
\(530\) −1.38890e12 −0.764592
\(531\) 7.95020e10 0.0433963
\(532\) −2.67302e11 −0.144677
\(533\) −2.28192e11 −0.122469
\(534\) 1.13545e12 0.604272
\(535\) −1.74014e11 −0.0918318
\(536\) −1.67839e12 −0.878316
\(537\) −1.06159e12 −0.550902
\(538\) 2.26596e12 1.16609
\(539\) −5.37258e10 −0.0274179
\(540\) −1.05967e11 −0.0536287
\(541\) −3.02073e12 −1.51609 −0.758043 0.652205i \(-0.773844\pi\)
−0.758043 + 0.652205i \(0.773844\pi\)
\(542\) 1.96673e12 0.978921
\(543\) 2.17086e12 1.07160
\(544\) 4.01657e11 0.196635
\(545\) −2.81117e12 −1.36490
\(546\) −5.46359e10 −0.0263094
\(547\) 3.81968e12 1.82425 0.912124 0.409914i \(-0.134441\pi\)
0.912124 + 0.409914i \(0.134441\pi\)
\(548\) 1.33625e12 0.632959
\(549\) 4.82396e10 0.0226636
\(550\) 2.28844e10 0.0106637
\(551\) −5.13998e11 −0.237563
\(552\) 2.04999e12 0.939780
\(553\) 1.76595e12 0.802999
\(554\) −2.26500e12 −1.02158
\(555\) 1.40070e12 0.626654
\(556\) 1.11129e12 0.493162
\(557\) 8.89967e11 0.391765 0.195882 0.980627i \(-0.437243\pi\)
0.195882 + 0.980627i \(0.437243\pi\)
\(558\) 1.14511e12 0.500029
\(559\) 4.59676e11 0.199112
\(560\) 4.30269e11 0.184882
\(561\) −1.36986e10 −0.00583907
\(562\) −2.22068e12 −0.939017
\(563\) 7.76854e11 0.325875 0.162938 0.986636i \(-0.447903\pi\)
0.162938 + 0.986636i \(0.447903\pi\)
\(564\) 2.09898e9 0.000873478 0
\(565\) −2.11444e11 −0.0872926
\(566\) 2.46637e12 1.01015
\(567\) −1.24224e11 −0.0504755
\(568\) 3.39346e12 1.36796
\(569\) −3.91130e12 −1.56429 −0.782143 0.623099i \(-0.785874\pi\)
−0.782143 + 0.623099i \(0.785874\pi\)
\(570\) 8.22068e11 0.326190
\(571\) −1.32146e12 −0.520225 −0.260113 0.965578i \(-0.583760\pi\)
−0.260113 + 0.965578i \(0.583760\pi\)
\(572\) 3.92415e9 0.00153272
\(573\) 2.37437e11 0.0920135
\(574\) 9.30156e11 0.357645
\(575\) −1.50710e12 −0.574959
\(576\) 9.31750e11 0.352694
\(577\) 9.16698e11 0.344298 0.172149 0.985071i \(-0.444929\pi\)
0.172149 + 0.985071i \(0.444929\pi\)
\(578\) 1.97011e12 0.734200
\(579\) −5.25849e11 −0.194450
\(580\) −2.01202e11 −0.0738256
\(581\) −1.37755e12 −0.501549
\(582\) −2.04089e12 −0.737337
\(583\) 1.16943e11 0.0419242
\(584\) 2.70538e12 0.962432
\(585\) −9.25461e10 −0.0326706
\(586\) −1.41617e12 −0.496108
\(587\) 7.25192e11 0.252105 0.126052 0.992024i \(-0.459769\pi\)
0.126052 + 0.992024i \(0.459769\pi\)
\(588\) 4.71716e11 0.162736
\(589\) 4.89284e12 1.67511
\(590\) −2.41428e11 −0.0820265
\(591\) −7.61674e11 −0.256818
\(592\) −2.14436e12 −0.717547
\(593\) −1.00706e12 −0.334434 −0.167217 0.985920i \(-0.553478\pi\)
−0.167217 + 0.985920i \(0.553478\pi\)
\(594\) −1.61994e10 −0.00533899
\(595\) 3.19002e11 0.104344
\(596\) 1.22464e12 0.397557
\(597\) 1.73384e12 0.558630
\(598\) 4.69218e11 0.150044
\(599\) −5.05825e12 −1.60539 −0.802694 0.596392i \(-0.796601\pi\)
−0.802694 + 0.596392i \(0.796601\pi\)
\(600\) −7.66661e11 −0.241503
\(601\) −4.20153e11 −0.131363 −0.0656814 0.997841i \(-0.520922\pi\)
−0.0656814 + 0.997841i \(0.520922\pi\)
\(602\) −1.87373e12 −0.581465
\(603\) −8.73456e11 −0.269038
\(604\) −2.97372e11 −0.0909146
\(605\) −2.58247e12 −0.783675
\(606\) 8.37116e11 0.252150
\(607\) 2.83924e12 0.848893 0.424446 0.905453i \(-0.360469\pi\)
0.424446 + 0.905453i \(0.360469\pi\)
\(608\) 2.02950e12 0.602314
\(609\) −2.35867e11 −0.0694848
\(610\) −1.46492e11 −0.0428380
\(611\) 1.83314e9 0.000532122 0
\(612\) 1.20275e11 0.0346572
\(613\) 4.68595e12 1.34037 0.670187 0.742193i \(-0.266214\pi\)
0.670187 + 0.742193i \(0.266214\pi\)
\(614\) −3.63034e12 −1.03083
\(615\) 1.57556e12 0.444117
\(616\) −6.10334e10 −0.0170787
\(617\) 6.26248e12 1.73966 0.869828 0.493355i \(-0.164230\pi\)
0.869828 + 0.493355i \(0.164230\pi\)
\(618\) −8.69360e10 −0.0239746
\(619\) −4.44051e12 −1.21569 −0.607847 0.794054i \(-0.707967\pi\)
−0.607847 + 0.794054i \(0.707967\pi\)
\(620\) 1.91528e12 0.520559
\(621\) 1.06684e12 0.287865
\(622\) −4.96235e11 −0.132932
\(623\) 2.22631e12 0.592093
\(624\) 1.41680e11 0.0374093
\(625\) −1.78475e12 −0.467862
\(626\) 5.36464e12 1.39623
\(627\) −6.92166e10 −0.0178857
\(628\) 1.22262e11 0.0313670
\(629\) −1.58983e12 −0.404970
\(630\) 3.77237e11 0.0954073
\(631\) 7.46084e12 1.87351 0.936755 0.349987i \(-0.113814\pi\)
0.936755 + 0.349987i \(0.113814\pi\)
\(632\) −7.71500e12 −1.92357
\(633\) −4.03466e12 −0.998826
\(634\) 5.65035e12 1.38891
\(635\) −3.70676e12 −0.904718
\(636\) −1.02676e12 −0.248836
\(637\) 4.11973e11 0.0991384
\(638\) −3.07583e10 −0.00734968
\(639\) 1.76600e12 0.419022
\(640\) −5.92656e11 −0.139635
\(641\) 6.49252e12 1.51898 0.759490 0.650519i \(-0.225449\pi\)
0.759490 + 0.650519i \(0.225449\pi\)
\(642\) 2.33567e11 0.0542630
\(643\) −4.82725e12 −1.11365 −0.556827 0.830628i \(-0.687981\pi\)
−0.556827 + 0.830628i \(0.687981\pi\)
\(644\) 1.05343e12 0.241334
\(645\) −3.17386e12 −0.722052
\(646\) −9.33067e11 −0.210798
\(647\) −1.71606e12 −0.385002 −0.192501 0.981297i \(-0.561660\pi\)
−0.192501 + 0.981297i \(0.561660\pi\)
\(648\) 5.42702e11 0.120913
\(649\) 2.03278e10 0.00449769
\(650\) −1.75479e11 −0.0385581
\(651\) 2.24526e12 0.489951
\(652\) 1.85642e12 0.402311
\(653\) 5.54650e12 1.19374 0.596869 0.802338i \(-0.296411\pi\)
0.596869 + 0.802338i \(0.296411\pi\)
\(654\) 3.77323e12 0.806516
\(655\) −5.89946e12 −1.25235
\(656\) −2.41206e12 −0.508534
\(657\) 1.40791e12 0.294803
\(658\) −7.47227e9 −0.00155395
\(659\) 5.59305e12 1.15522 0.577610 0.816313i \(-0.303985\pi\)
0.577610 + 0.816313i \(0.303985\pi\)
\(660\) −2.70945e10 −0.00555820
\(661\) −2.39528e12 −0.488033 −0.244017 0.969771i \(-0.578465\pi\)
−0.244017 + 0.969771i \(0.578465\pi\)
\(662\) 5.36957e12 1.08662
\(663\) 1.05042e11 0.0211131
\(664\) 6.01816e12 1.20145
\(665\) 1.61186e12 0.319616
\(666\) −1.88006e12 −0.370287
\(667\) 2.02565e12 0.396276
\(668\) 1.63412e12 0.317535
\(669\) 7.08686e11 0.136784
\(670\) 2.65247e12 0.508527
\(671\) 1.23343e10 0.00234890
\(672\) 9.31312e11 0.176171
\(673\) 1.09679e12 0.206089 0.103045 0.994677i \(-0.467142\pi\)
0.103045 + 0.994677i \(0.467142\pi\)
\(674\) −1.18907e12 −0.221941
\(675\) −3.98981e11 −0.0739750
\(676\) 1.89826e12 0.349619
\(677\) −2.91299e11 −0.0532954 −0.0266477 0.999645i \(-0.508483\pi\)
−0.0266477 + 0.999645i \(0.508483\pi\)
\(678\) 2.83806e11 0.0515808
\(679\) −4.00164e12 −0.722476
\(680\) −1.39364e12 −0.249954
\(681\) 3.12690e12 0.557125
\(682\) 2.92793e11 0.0518241
\(683\) −4.83319e12 −0.849847 −0.424924 0.905229i \(-0.639699\pi\)
−0.424924 + 0.905229i \(0.639699\pi\)
\(684\) 6.07726e11 0.106159
\(685\) −8.05772e12 −1.39831
\(686\) −3.79525e12 −0.654307
\(687\) 2.53004e12 0.433333
\(688\) 4.85891e12 0.826782
\(689\) −8.96725e11 −0.151591
\(690\) −3.23974e12 −0.544114
\(691\) 7.26763e12 1.21267 0.606333 0.795211i \(-0.292640\pi\)
0.606333 + 0.795211i \(0.292640\pi\)
\(692\) −2.30625e12 −0.382322
\(693\) −3.17626e10 −0.00523138
\(694\) 4.63475e12 0.758418
\(695\) −6.70117e12 −1.08948
\(696\) 1.03045e12 0.166450
\(697\) −1.78830e12 −0.287007
\(698\) 3.94047e12 0.628345
\(699\) −4.05058e12 −0.641756
\(700\) −3.93963e11 −0.0620176
\(701\) 1.24764e13 1.95146 0.975728 0.218985i \(-0.0702745\pi\)
0.975728 + 0.218985i \(0.0702745\pi\)
\(702\) 1.24218e11 0.0193049
\(703\) −8.03312e12 −1.24047
\(704\) 2.38238e11 0.0365540
\(705\) −1.26570e10 −0.00192966
\(706\) −5.53187e12 −0.838012
\(707\) 1.64136e12 0.247068
\(708\) −1.78479e11 −0.0266955
\(709\) −9.20508e12 −1.36811 −0.684053 0.729433i \(-0.739784\pi\)
−0.684053 + 0.729433i \(0.739784\pi\)
\(710\) −5.36292e12 −0.792024
\(711\) −4.01499e12 −0.589211
\(712\) −9.72621e12 −1.41835
\(713\) −1.92825e13 −2.79422
\(714\) −4.28173e11 −0.0616562
\(715\) −2.36630e10 −0.00338605
\(716\) 2.38324e12 0.338891
\(717\) 5.08292e12 0.718253
\(718\) 5.44939e12 0.765222
\(719\) 5.25770e12 0.733696 0.366848 0.930281i \(-0.380437\pi\)
0.366848 + 0.930281i \(0.380437\pi\)
\(720\) −9.78241e11 −0.135659
\(721\) −1.70458e11 −0.0234914
\(722\) 1.14870e12 0.157322
\(723\) 4.34029e12 0.590740
\(724\) −4.87352e12 −0.659203
\(725\) −7.57558e11 −0.101834
\(726\) 3.46626e12 0.463070
\(727\) −3.84442e12 −0.510418 −0.255209 0.966886i \(-0.582144\pi\)
−0.255209 + 0.966886i \(0.582144\pi\)
\(728\) 4.68009e11 0.0617537
\(729\) 2.82430e11 0.0370370
\(730\) −4.27549e12 −0.557229
\(731\) 3.60240e12 0.466621
\(732\) −1.08296e11 −0.0139416
\(733\) −1.01488e13 −1.29852 −0.649259 0.760567i \(-0.724921\pi\)
−0.649259 + 0.760567i \(0.724921\pi\)
\(734\) 2.50044e12 0.317969
\(735\) −2.84450e12 −0.359511
\(736\) −7.99819e12 −1.00471
\(737\) −2.23333e11 −0.0278836
\(738\) −2.11476e12 −0.262427
\(739\) 8.01395e11 0.0988432 0.0494216 0.998778i \(-0.484262\pi\)
0.0494216 + 0.998778i \(0.484262\pi\)
\(740\) −3.14453e12 −0.385490
\(741\) 5.30758e11 0.0646717
\(742\) 3.65524e12 0.442688
\(743\) −3.34243e12 −0.402358 −0.201179 0.979555i \(-0.564477\pi\)
−0.201179 + 0.979555i \(0.564477\pi\)
\(744\) −9.80900e12 −1.17367
\(745\) −7.38469e12 −0.878272
\(746\) −1.08420e13 −1.28169
\(747\) 3.13193e12 0.368018
\(748\) 3.07529e10 0.00359194
\(749\) 4.57962e11 0.0531693
\(750\) 4.36367e12 0.503589
\(751\) −1.86106e12 −0.213491 −0.106746 0.994286i \(-0.534043\pi\)
−0.106746 + 0.994286i \(0.534043\pi\)
\(752\) 1.93769e10 0.00220955
\(753\) 5.92831e12 0.671975
\(754\) 2.35857e11 0.0265752
\(755\) 1.79318e12 0.200846
\(756\) 2.78878e11 0.0310503
\(757\) −1.12176e13 −1.24157 −0.620784 0.783982i \(-0.713185\pi\)
−0.620784 + 0.783982i \(0.713185\pi\)
\(758\) 7.93505e11 0.0873048
\(759\) 2.72780e11 0.0298349
\(760\) −7.04180e12 −0.765636
\(761\) −1.53092e13 −1.65471 −0.827354 0.561681i \(-0.810155\pi\)
−0.827354 + 0.561681i \(0.810155\pi\)
\(762\) 4.97532e12 0.534594
\(763\) 7.39828e12 0.790261
\(764\) −5.33037e11 −0.0566027
\(765\) −7.25269e11 −0.0765636
\(766\) −1.29348e13 −1.35747
\(767\) −1.55875e11 −0.0162629
\(768\) −5.09410e12 −0.528375
\(769\) −5.51350e12 −0.568537 −0.284269 0.958745i \(-0.591751\pi\)
−0.284269 + 0.958745i \(0.591751\pi\)
\(770\) 9.64554e10 0.00988822
\(771\) −6.41385e12 −0.653694
\(772\) 1.18051e12 0.119617
\(773\) 1.48992e13 1.50091 0.750456 0.660921i \(-0.229834\pi\)
0.750456 + 0.660921i \(0.229834\pi\)
\(774\) 4.26004e12 0.426658
\(775\) 7.21133e12 0.718055
\(776\) 1.74822e13 1.73068
\(777\) −3.68630e12 −0.362824
\(778\) −1.17561e13 −1.15041
\(779\) −9.03595e12 −0.879134
\(780\) 2.07763e11 0.0200975
\(781\) 4.51547e11 0.0434284
\(782\) 3.67718e12 0.351629
\(783\) 5.36258e11 0.0509854
\(784\) 4.35469e12 0.411656
\(785\) −7.37253e11 −0.0692952
\(786\) 7.91842e12 0.740009
\(787\) −6.16580e12 −0.572933 −0.286466 0.958090i \(-0.592481\pi\)
−0.286466 + 0.958090i \(0.592481\pi\)
\(788\) 1.70993e12 0.157983
\(789\) 1.61642e12 0.148494
\(790\) 1.21925e13 1.11371
\(791\) 5.56468e11 0.0505412
\(792\) 1.38763e11 0.0125317
\(793\) −9.45806e10 −0.00849323
\(794\) −1.98955e12 −0.177649
\(795\) 6.19149e12 0.549722
\(796\) −3.89241e12 −0.343645
\(797\) −2.00164e13 −1.75721 −0.878607 0.477546i \(-0.841526\pi\)
−0.878607 + 0.477546i \(0.841526\pi\)
\(798\) −2.16348e12 −0.188860
\(799\) 1.43660e10 0.00124703
\(800\) 2.99118e12 0.258189
\(801\) −5.06165e12 −0.434456
\(802\) 4.89413e12 0.417725
\(803\) 3.59988e11 0.0305540
\(804\) 1.96088e12 0.165500
\(805\) −6.35227e12 −0.533147
\(806\) −2.24516e12 −0.187387
\(807\) −1.01013e13 −0.838389
\(808\) −7.17070e12 −0.591848
\(809\) 1.58868e13 1.30398 0.651988 0.758230i \(-0.273935\pi\)
0.651988 + 0.758230i \(0.273935\pi\)
\(810\) −8.57670e11 −0.0700064
\(811\) −4.96216e12 −0.402789 −0.201394 0.979510i \(-0.564547\pi\)
−0.201394 + 0.979510i \(0.564547\pi\)
\(812\) 5.29514e11 0.0427440
\(813\) −8.76735e12 −0.703819
\(814\) −4.80712e11 −0.0383773
\(815\) −1.11944e13 −0.888775
\(816\) 1.11033e12 0.0876688
\(817\) 1.82023e13 1.42931
\(818\) 7.87389e11 0.0614893
\(819\) 2.43558e11 0.0189158
\(820\) −3.53708e12 −0.273202
\(821\) −2.00520e13 −1.54033 −0.770164 0.637846i \(-0.779826\pi\)
−0.770164 + 0.637846i \(0.779826\pi\)
\(822\) 1.08153e13 0.826258
\(823\) 1.39355e13 1.05882 0.529410 0.848366i \(-0.322413\pi\)
0.529410 + 0.848366i \(0.322413\pi\)
\(824\) 7.44690e11 0.0562734
\(825\) −1.02015e11 −0.00766693
\(826\) 6.35379e11 0.0474922
\(827\) 1.67616e13 1.24606 0.623031 0.782197i \(-0.285901\pi\)
0.623031 + 0.782197i \(0.285901\pi\)
\(828\) −2.39503e12 −0.177082
\(829\) −1.06490e13 −0.783095 −0.391548 0.920158i \(-0.628060\pi\)
−0.391548 + 0.920158i \(0.628060\pi\)
\(830\) −9.51091e12 −0.695618
\(831\) 1.00970e13 0.734493
\(832\) −1.82683e12 −0.132173
\(833\) 3.22857e12 0.232331
\(834\) 8.99450e12 0.643768
\(835\) −9.85393e12 −0.701488
\(836\) 1.55389e11 0.0110025
\(837\) −5.10474e12 −0.359508
\(838\) −5.65925e12 −0.396425
\(839\) 2.13820e13 1.48977 0.744886 0.667192i \(-0.232504\pi\)
0.744886 + 0.667192i \(0.232504\pi\)
\(840\) −3.23139e12 −0.223941
\(841\) −1.34889e13 −0.929813
\(842\) −4.04252e12 −0.277171
\(843\) 9.89945e12 0.675129
\(844\) 9.05767e12 0.614435
\(845\) −1.14467e13 −0.772368
\(846\) 1.69886e10 0.00114023
\(847\) 6.79641e12 0.453737
\(848\) −9.47867e12 −0.629457
\(849\) −1.09947e13 −0.726271
\(850\) −1.37520e12 −0.0903612
\(851\) 3.16583e13 2.06921
\(852\) −3.96461e12 −0.257764
\(853\) −1.06774e13 −0.690548 −0.345274 0.938502i \(-0.612214\pi\)
−0.345274 + 0.938502i \(0.612214\pi\)
\(854\) 3.85530e11 0.0248026
\(855\) −3.66465e12 −0.234523
\(856\) −2.00072e12 −0.127366
\(857\) 1.26338e13 0.800054 0.400027 0.916503i \(-0.369001\pi\)
0.400027 + 0.916503i \(0.369001\pi\)
\(858\) 3.17612e10 0.00200080
\(859\) −2.64136e13 −1.65523 −0.827615 0.561296i \(-0.810303\pi\)
−0.827615 + 0.561296i \(0.810303\pi\)
\(860\) 7.12521e12 0.444175
\(861\) −4.14648e12 −0.257138
\(862\) 7.71794e12 0.476122
\(863\) 2.48314e13 1.52388 0.761942 0.647645i \(-0.224246\pi\)
0.761942 + 0.647645i \(0.224246\pi\)
\(864\) −2.11739e12 −0.129268
\(865\) 1.39069e13 0.844616
\(866\) −2.42649e13 −1.46605
\(867\) −8.78242e12 −0.527872
\(868\) −5.04054e12 −0.301397
\(869\) −1.02659e12 −0.0610671
\(870\) −1.62849e12 −0.0963712
\(871\) 1.71253e12 0.100823
\(872\) −3.23213e13 −1.89306
\(873\) 9.09795e12 0.530126
\(874\) 1.85801e13 1.07708
\(875\) 8.55599e12 0.493440
\(876\) −3.16072e12 −0.181350
\(877\) −1.05883e13 −0.604405 −0.302203 0.953244i \(-0.597722\pi\)
−0.302203 + 0.953244i \(0.597722\pi\)
\(878\) 2.53669e13 1.44059
\(879\) 6.31305e12 0.356689
\(880\) −2.50125e11 −0.0140600
\(881\) 2.49573e13 1.39575 0.697874 0.716221i \(-0.254130\pi\)
0.697874 + 0.716221i \(0.254130\pi\)
\(882\) 3.81796e12 0.212433
\(883\) −2.63084e13 −1.45637 −0.728184 0.685382i \(-0.759635\pi\)
−0.728184 + 0.685382i \(0.759635\pi\)
\(884\) −2.35816e11 −0.0129879
\(885\) 1.07625e12 0.0589750
\(886\) −1.05733e13 −0.576449
\(887\) 6.05761e12 0.328583 0.164291 0.986412i \(-0.447466\pi\)
0.164291 + 0.986412i \(0.447466\pi\)
\(888\) 1.61045e13 0.869140
\(889\) 9.75527e12 0.523819
\(890\) 1.53710e13 0.821197
\(891\) 7.22142e10 0.00383860
\(892\) −1.59098e12 −0.0841437
\(893\) 7.25890e10 0.00381979
\(894\) 9.91193e12 0.518967
\(895\) −1.43712e13 −0.748668
\(896\) 1.55972e12 0.0808465
\(897\) −2.09170e12 −0.107878
\(898\) 9.61337e12 0.493324
\(899\) −9.69253e12 −0.494901
\(900\) 8.95699e11 0.0455062
\(901\) −7.02749e12 −0.355254
\(902\) −5.40722e11 −0.0271985
\(903\) 8.35280e12 0.418059
\(904\) −2.43107e12 −0.121071
\(905\) 2.93878e13 1.45629
\(906\) −2.40686e12 −0.118679
\(907\) 4.24714e12 0.208384 0.104192 0.994557i \(-0.466774\pi\)
0.104192 + 0.994557i \(0.466774\pi\)
\(908\) −7.01980e12 −0.342719
\(909\) −3.73173e12 −0.181290
\(910\) −7.39627e11 −0.0357541
\(911\) −2.80542e10 −0.00134948 −0.000674739 1.00000i \(-0.500215\pi\)
−0.000674739 1.00000i \(0.500215\pi\)
\(912\) 5.61027e12 0.268539
\(913\) 8.00801e11 0.0381422
\(914\) 2.34212e13 1.11007
\(915\) 6.53038e11 0.0307995
\(916\) −5.67986e12 −0.266568
\(917\) 1.55259e13 0.725094
\(918\) 9.73476e11 0.0452411
\(919\) 9.06347e12 0.419155 0.209578 0.977792i \(-0.432791\pi\)
0.209578 + 0.977792i \(0.432791\pi\)
\(920\) 2.77515e13 1.27715
\(921\) 1.61835e13 0.741144
\(922\) −8.00682e12 −0.364898
\(923\) −3.46250e12 −0.157030
\(924\) 7.13060e10 0.00321812
\(925\) −1.18396e13 −0.531742
\(926\) 8.23007e11 0.0367836
\(927\) 3.87547e11 0.0172371
\(928\) −4.02036e12 −0.177950
\(929\) 1.42413e13 0.627306 0.313653 0.949538i \(-0.398447\pi\)
0.313653 + 0.949538i \(0.398447\pi\)
\(930\) 1.55018e13 0.679532
\(931\) 1.63134e13 0.711656
\(932\) 9.09341e12 0.394780
\(933\) 2.21213e12 0.0955749
\(934\) −1.80843e13 −0.777572
\(935\) −1.85443e11 −0.00793522
\(936\) −1.06405e12 −0.0453126
\(937\) −1.79359e12 −0.0760142 −0.0380071 0.999277i \(-0.512101\pi\)
−0.0380071 + 0.999277i \(0.512101\pi\)
\(938\) −6.98064e12 −0.294430
\(939\) −2.39147e13 −1.00385
\(940\) 2.84146e10 0.00118704
\(941\) 1.93549e13 0.804707 0.402354 0.915484i \(-0.368192\pi\)
0.402354 + 0.915484i \(0.368192\pi\)
\(942\) 9.89562e11 0.0409462
\(943\) 3.56104e13 1.46647
\(944\) −1.64765e12 −0.0675290
\(945\) −1.68166e12 −0.0685954
\(946\) 1.08925e12 0.0442197
\(947\) 8.51355e12 0.343982 0.171991 0.985099i \(-0.444980\pi\)
0.171991 + 0.985099i \(0.444980\pi\)
\(948\) 9.01352e12 0.362457
\(949\) −2.76042e12 −0.110478
\(950\) −6.94865e12 −0.276786
\(951\) −2.51883e13 −0.998590
\(952\) 3.66771e12 0.144720
\(953\) 1.91350e12 0.0751469 0.0375734 0.999294i \(-0.488037\pi\)
0.0375734 + 0.999294i \(0.488037\pi\)
\(954\) −8.31040e12 −0.324828
\(955\) 3.21427e12 0.125045
\(956\) −1.14110e13 −0.441838
\(957\) 1.37115e11 0.00528424
\(958\) −4.29125e12 −0.164603
\(959\) 2.12059e13 0.809605
\(960\) 1.26135e13 0.479307
\(961\) 6.58253e13 2.48965
\(962\) 3.68613e12 0.138766
\(963\) −1.04120e12 −0.0390137
\(964\) −9.74381e12 −0.363398
\(965\) −7.11862e12 −0.264255
\(966\) 8.52619e12 0.315034
\(967\) −2.15966e13 −0.794268 −0.397134 0.917761i \(-0.629995\pi\)
−0.397134 + 0.917761i \(0.629995\pi\)
\(968\) −2.96918e13 −1.08692
\(969\) 4.15946e12 0.151558
\(970\) −2.76283e13 −1.00203
\(971\) −2.99863e13 −1.08252 −0.541261 0.840854i \(-0.682053\pi\)
−0.541261 + 0.840854i \(0.682053\pi\)
\(972\) −6.34045e11 −0.0227836
\(973\) 1.76358e13 0.630794
\(974\) −1.63112e13 −0.580726
\(975\) 7.82259e11 0.0277223
\(976\) −9.99746e11 −0.0352668
\(977\) 1.77038e13 0.621642 0.310821 0.950469i \(-0.399396\pi\)
0.310821 + 0.950469i \(0.399396\pi\)
\(978\) 1.50254e13 0.525173
\(979\) −1.29421e12 −0.0450280
\(980\) 6.38580e12 0.221156
\(981\) −1.68204e13 −0.579865
\(982\) 2.50949e13 0.861161
\(983\) −2.82507e13 −0.965025 −0.482513 0.875889i \(-0.660276\pi\)
−0.482513 + 0.875889i \(0.660276\pi\)
\(984\) 1.81150e13 0.615970
\(985\) −1.03111e13 −0.349012
\(986\) 1.84837e12 0.0622792
\(987\) 3.33102e10 0.00111725
\(988\) −1.19153e12 −0.0397832
\(989\) −7.17345e13 −2.38421
\(990\) −2.19297e11 −0.00725561
\(991\) −2.72770e13 −0.898391 −0.449196 0.893433i \(-0.648289\pi\)
−0.449196 + 0.893433i \(0.648289\pi\)
\(992\) 3.82705e13 1.25476
\(993\) −2.39367e13 −0.781254
\(994\) 1.41138e13 0.458571
\(995\) 2.34716e13 0.759170
\(996\) −7.03108e12 −0.226389
\(997\) 9.74389e12 0.312323 0.156162 0.987732i \(-0.450088\pi\)
0.156162 + 0.987732i \(0.450088\pi\)
\(998\) 4.06919e13 1.29844
\(999\) 8.38102e12 0.266227
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.10.a.a.1.8 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.a.1.8 21 1.1 even 1 trivial