Properties

Label 177.12.a.c.1.3
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $0$
Dimension $27$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(0\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-79.2286 q^{2} -243.000 q^{3} +4229.17 q^{4} +9672.04 q^{5} +19252.5 q^{6} +84867.6 q^{7} -172811. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-79.2286 q^{2} -243.000 q^{3} +4229.17 q^{4} +9672.04 q^{5} +19252.5 q^{6} +84867.6 q^{7} -172811. q^{8} +59049.0 q^{9} -766302. q^{10} +465513. q^{11} -1.02769e6 q^{12} +1.10167e6 q^{13} -6.72394e6 q^{14} -2.35031e6 q^{15} +5.03021e6 q^{16} -2.02733e6 q^{17} -4.67837e6 q^{18} -5.75353e6 q^{19} +4.09047e7 q^{20} -2.06228e7 q^{21} -3.68820e7 q^{22} +3.68952e6 q^{23} +4.19930e7 q^{24} +4.47203e7 q^{25} -8.72834e7 q^{26} -1.43489e7 q^{27} +3.58919e8 q^{28} +2.53977e7 q^{29} +1.86211e8 q^{30} +1.59360e8 q^{31} -4.46200e7 q^{32} -1.13120e8 q^{33} +1.60622e8 q^{34} +8.20843e8 q^{35} +2.49728e8 q^{36} +6.00742e7 q^{37} +4.55844e8 q^{38} -2.67705e8 q^{39} -1.67143e9 q^{40} +1.17324e8 q^{41} +1.63392e9 q^{42} +1.58206e9 q^{43} +1.96873e9 q^{44} +5.71125e8 q^{45} -2.92316e8 q^{46} -2.01137e9 q^{47} -1.22234e9 q^{48} +5.22519e9 q^{49} -3.54313e9 q^{50} +4.92640e8 q^{51} +4.65913e9 q^{52} +3.31605e8 q^{53} +1.13684e9 q^{54} +4.50247e9 q^{55} -1.46660e10 q^{56} +1.39811e9 q^{57} -2.01222e9 q^{58} -7.14924e8 q^{59} -9.93984e9 q^{60} -8.69345e9 q^{61} -1.26258e10 q^{62} +5.01135e9 q^{63} -6.76669e9 q^{64} +1.06554e10 q^{65} +8.96232e9 q^{66} -3.59954e9 q^{67} -8.57390e9 q^{68} -8.96554e8 q^{69} -6.50343e10 q^{70} +1.77404e10 q^{71} -1.02043e10 q^{72} +9.43622e9 q^{73} -4.75959e9 q^{74} -1.08670e10 q^{75} -2.43326e10 q^{76} +3.95070e10 q^{77} +2.12099e10 q^{78} +3.35436e10 q^{79} +4.86524e10 q^{80} +3.48678e9 q^{81} -9.29540e9 q^{82} -5.21982e10 q^{83} -8.72174e10 q^{84} -1.96084e10 q^{85} -1.25344e11 q^{86} -6.17164e9 q^{87} -8.04457e10 q^{88} +6.06182e10 q^{89} -4.52494e10 q^{90} +9.34957e10 q^{91} +1.56036e10 q^{92} -3.87244e10 q^{93} +1.59358e11 q^{94} -5.56484e10 q^{95} +1.08427e10 q^{96} +3.05431e10 q^{97} -4.13984e11 q^{98} +2.74881e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 46q^{2} - 6561q^{3} + 26142q^{4} - 2442q^{5} + 11178q^{6} + 170093q^{7} - 19341q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 46q^{2} - 6561q^{3} + 26142q^{4} - 2442q^{5} + 11178q^{6} + 170093q^{7} - 19341q^{8} + 1594323q^{9} + 140249q^{10} + 256992q^{11} - 6352506q^{12} + 2436978q^{13} + 5233061q^{14} + 593406q^{15} + 28295194q^{16} - 4565351q^{17} - 2716254q^{18} + 33607699q^{19} - 19208463q^{20} - 41332599q^{21} + 79735622q^{22} + 43966161q^{23} + 4699863q^{24} + 406675819q^{25} + 42605404q^{26} - 387420489q^{27} + 635747682q^{28} - 107217773q^{29} - 34080507q^{30} + 570926627q^{31} + 526569236q^{32} - 62449056q^{33} + 129790240q^{34} + 134356079q^{35} + 1543658958q^{36} - 107121371q^{37} + 208302581q^{38} - 592185654q^{39} - 958762162q^{40} - 1935967559q^{41} - 1271633823q^{42} + 1725943824q^{43} + 196885756q^{44} - 144197658q^{45} - 13265966407q^{46} + 1801256065q^{47} - 6875732142q^{48} + 10484289252q^{49} - 10067682271q^{50} + 1109380293q^{51} - 882697024q^{52} - 6214238922q^{53} + 660049722q^{54} + 4460552366q^{55} + 28328012310q^{56} - 8166670857q^{57} + 12220116750q^{58} - 19302956073q^{59} + 4667656509q^{60} + 13167821039q^{61} - 1162130230q^{62} + 10043821557q^{63} - 5337557395q^{64} - 16849896006q^{65} - 19375756146q^{66} - 16856763152q^{67} - 36171071977q^{68} - 10683777123q^{69} - 120177261588q^{70} - 5198545690q^{71} - 1142066709q^{72} - 25075321857q^{73} - 182979651978q^{74} - 98822224017q^{75} - 3501293988q^{76} - 42787697701q^{77} - 10353113172q^{78} + 6850314702q^{79} - 261464428159q^{80} + 94143178827q^{81} - 148881516273q^{82} + 30908370899q^{83} - 154486686726q^{84} - 49419624969q^{85} - 220725475224q^{86} + 26053918839q^{87} - 53091280787q^{88} + 28988060121q^{89} + 8281563201q^{90} + 97120614047q^{91} + 45374597708q^{92} - 138735170361q^{93} + 208966927220q^{94} - 125253904969q^{95} - 127956324348q^{96} + 367722840268q^{97} - 48265639912q^{98} + 15175120608q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −79.2286 −1.75072 −0.875360 0.483471i \(-0.839376\pi\)
−0.875360 + 0.483471i \(0.839376\pi\)
\(3\) −243.000 −0.577350
\(4\) 4229.17 2.06502
\(5\) 9672.04 1.38415 0.692075 0.721825i \(-0.256697\pi\)
0.692075 + 0.721825i \(0.256697\pi\)
\(6\) 19252.5 1.01078
\(7\) 84867.6 1.90855 0.954273 0.298935i \(-0.0966314\pi\)
0.954273 + 0.298935i \(0.0966314\pi\)
\(8\) −172811. −1.86456
\(9\) 59049.0 0.333333
\(10\) −766302. −2.42326
\(11\) 465513. 0.871510 0.435755 0.900065i \(-0.356481\pi\)
0.435755 + 0.900065i \(0.356481\pi\)
\(12\) −1.02769e6 −1.19224
\(13\) 1.10167e6 0.822927 0.411463 0.911426i \(-0.365018\pi\)
0.411463 + 0.911426i \(0.365018\pi\)
\(14\) −6.72394e6 −3.34133
\(15\) −2.35031e6 −0.799140
\(16\) 5.03021e6 1.19930
\(17\) −2.02733e6 −0.346302 −0.173151 0.984895i \(-0.555395\pi\)
−0.173151 + 0.984895i \(0.555395\pi\)
\(18\) −4.67837e6 −0.583574
\(19\) −5.75353e6 −0.533076 −0.266538 0.963824i \(-0.585880\pi\)
−0.266538 + 0.963824i \(0.585880\pi\)
\(20\) 4.09047e7 2.85830
\(21\) −2.06228e7 −1.10190
\(22\) −3.68820e7 −1.52577
\(23\) 3.68952e6 0.119527 0.0597636 0.998213i \(-0.480965\pi\)
0.0597636 + 0.998213i \(0.480965\pi\)
\(24\) 4.19930e7 1.07650
\(25\) 4.47203e7 0.915872
\(26\) −8.72834e7 −1.44071
\(27\) −1.43489e7 −0.192450
\(28\) 3.58919e8 3.94119
\(29\) 2.53977e7 0.229935 0.114968 0.993369i \(-0.463324\pi\)
0.114968 + 0.993369i \(0.463324\pi\)
\(30\) 1.86211e8 1.39907
\(31\) 1.59360e8 0.999745 0.499873 0.866099i \(-0.333380\pi\)
0.499873 + 0.866099i \(0.333380\pi\)
\(32\) −4.46200e7 −0.235074
\(33\) −1.13120e8 −0.503167
\(34\) 1.60622e8 0.606278
\(35\) 8.20843e8 2.64172
\(36\) 2.49728e8 0.688341
\(37\) 6.00742e7 0.142422 0.0712112 0.997461i \(-0.477314\pi\)
0.0712112 + 0.997461i \(0.477314\pi\)
\(38\) 4.55844e8 0.933268
\(39\) −2.67705e8 −0.475117
\(40\) −1.67143e9 −2.58083
\(41\) 1.17324e8 0.158152 0.0790761 0.996869i \(-0.474803\pi\)
0.0790761 + 0.996869i \(0.474803\pi\)
\(42\) 1.63392e9 1.92912
\(43\) 1.58206e9 1.64114 0.820569 0.571547i \(-0.193657\pi\)
0.820569 + 0.571547i \(0.193657\pi\)
\(44\) 1.96873e9 1.79969
\(45\) 5.71125e8 0.461383
\(46\) −2.92316e8 −0.209259
\(47\) −2.01137e9 −1.27925 −0.639624 0.768688i \(-0.720910\pi\)
−0.639624 + 0.768688i \(0.720910\pi\)
\(48\) −1.22234e9 −0.692413
\(49\) 5.22519e9 2.64255
\(50\) −3.54313e9 −1.60344
\(51\) 4.92640e8 0.199937
\(52\) 4.65913e9 1.69936
\(53\) 3.31605e8 0.108919 0.0544595 0.998516i \(-0.482656\pi\)
0.0544595 + 0.998516i \(0.482656\pi\)
\(54\) 1.13684e9 0.336926
\(55\) 4.50247e9 1.20630
\(56\) −1.46660e10 −3.55859
\(57\) 1.39811e9 0.307772
\(58\) −2.01222e9 −0.402552
\(59\) −7.14924e8 −0.130189
\(60\) −9.93984e9 −1.65024
\(61\) −8.69345e9 −1.31789 −0.658944 0.752192i \(-0.728996\pi\)
−0.658944 + 0.752192i \(0.728996\pi\)
\(62\) −1.26258e10 −1.75027
\(63\) 5.01135e9 0.636182
\(64\) −6.76669e9 −0.787746
\(65\) 1.06554e10 1.13905
\(66\) 8.96232e9 0.880904
\(67\) −3.59954e9 −0.325714 −0.162857 0.986650i \(-0.552071\pi\)
−0.162857 + 0.986650i \(0.552071\pi\)
\(68\) −8.57390e9 −0.715121
\(69\) −8.96554e8 −0.0690091
\(70\) −6.50343e10 −4.62491
\(71\) 1.77404e10 1.16693 0.583463 0.812140i \(-0.301697\pi\)
0.583463 + 0.812140i \(0.301697\pi\)
\(72\) −1.02043e10 −0.621519
\(73\) 9.43622e9 0.532749 0.266374 0.963870i \(-0.414174\pi\)
0.266374 + 0.963870i \(0.414174\pi\)
\(74\) −4.75959e9 −0.249342
\(75\) −1.08670e10 −0.528779
\(76\) −2.43326e10 −1.10081
\(77\) 3.95070e10 1.66332
\(78\) 2.12099e10 0.831797
\(79\) 3.35436e10 1.22648 0.613240 0.789897i \(-0.289866\pi\)
0.613240 + 0.789897i \(0.289866\pi\)
\(80\) 4.86524e10 1.66000
\(81\) 3.48678e9 0.111111
\(82\) −9.29540e9 −0.276880
\(83\) −5.21982e10 −1.45454 −0.727270 0.686351i \(-0.759211\pi\)
−0.727270 + 0.686351i \(0.759211\pi\)
\(84\) −8.72174e10 −2.27545
\(85\) −1.96084e10 −0.479334
\(86\) −1.25344e11 −2.87317
\(87\) −6.17164e9 −0.132753
\(88\) −8.04457e10 −1.62498
\(89\) 6.06182e10 1.15069 0.575345 0.817911i \(-0.304868\pi\)
0.575345 + 0.817911i \(0.304868\pi\)
\(90\) −4.52494e10 −0.807753
\(91\) 9.34957e10 1.57059
\(92\) 1.56036e10 0.246826
\(93\) −3.87244e10 −0.577203
\(94\) 1.59358e11 2.23960
\(95\) −5.56484e10 −0.737858
\(96\) 1.08427e10 0.135720
\(97\) 3.05431e10 0.361134 0.180567 0.983563i \(-0.442207\pi\)
0.180567 + 0.983563i \(0.442207\pi\)
\(98\) −4.13984e11 −4.62637
\(99\) 2.74881e10 0.290503
\(100\) 1.89130e11 1.89130
\(101\) −4.43366e10 −0.419754 −0.209877 0.977728i \(-0.567306\pi\)
−0.209877 + 0.977728i \(0.567306\pi\)
\(102\) −3.90312e10 −0.350034
\(103\) −1.25879e11 −1.06991 −0.534956 0.844880i \(-0.679672\pi\)
−0.534956 + 0.844880i \(0.679672\pi\)
\(104\) −1.90379e11 −1.53439
\(105\) −1.99465e11 −1.52520
\(106\) −2.62726e10 −0.190687
\(107\) −2.47750e11 −1.70766 −0.853832 0.520549i \(-0.825727\pi\)
−0.853832 + 0.520549i \(0.825727\pi\)
\(108\) −6.06839e10 −0.397414
\(109\) 1.58261e11 0.985210 0.492605 0.870253i \(-0.336045\pi\)
0.492605 + 0.870253i \(0.336045\pi\)
\(110\) −3.56724e11 −2.11190
\(111\) −1.45980e10 −0.0822276
\(112\) 4.26902e11 2.28891
\(113\) 3.15802e11 1.61244 0.806220 0.591616i \(-0.201510\pi\)
0.806220 + 0.591616i \(0.201510\pi\)
\(114\) −1.10770e11 −0.538822
\(115\) 3.56852e10 0.165444
\(116\) 1.07411e11 0.474821
\(117\) 6.50522e10 0.274309
\(118\) 5.66424e10 0.227924
\(119\) −1.72054e11 −0.660933
\(120\) 4.06158e11 1.49004
\(121\) −6.86090e10 −0.240470
\(122\) 6.88770e11 2.30725
\(123\) −2.85097e10 −0.0913092
\(124\) 6.73959e11 2.06450
\(125\) −3.97308e10 −0.116445
\(126\) −3.97042e11 −1.11378
\(127\) 1.92351e11 0.516623 0.258312 0.966062i \(-0.416834\pi\)
0.258312 + 0.966062i \(0.416834\pi\)
\(128\) 6.27497e11 1.61420
\(129\) −3.84439e11 −0.947512
\(130\) −8.44209e11 −1.99417
\(131\) −2.14235e11 −0.485174 −0.242587 0.970130i \(-0.577996\pi\)
−0.242587 + 0.970130i \(0.577996\pi\)
\(132\) −4.78402e11 −1.03905
\(133\) −4.88288e11 −1.01740
\(134\) 2.85187e11 0.570234
\(135\) −1.38783e11 −0.266380
\(136\) 3.50344e11 0.645699
\(137\) 2.55783e11 0.452801 0.226401 0.974034i \(-0.427304\pi\)
0.226401 + 0.974034i \(0.427304\pi\)
\(138\) 7.10327e10 0.120816
\(139\) 7.46108e11 1.21961 0.609804 0.792552i \(-0.291248\pi\)
0.609804 + 0.792552i \(0.291248\pi\)
\(140\) 3.47148e12 5.45520
\(141\) 4.88764e11 0.738574
\(142\) −1.40555e12 −2.04296
\(143\) 5.12840e11 0.717189
\(144\) 2.97029e11 0.399765
\(145\) 2.45648e11 0.318265
\(146\) −7.47618e11 −0.932694
\(147\) −1.26972e12 −1.52568
\(148\) 2.54064e11 0.294106
\(149\) −1.12183e11 −0.125142 −0.0625710 0.998041i \(-0.519930\pi\)
−0.0625710 + 0.998041i \(0.519930\pi\)
\(150\) 8.60980e11 0.925744
\(151\) 1.23881e12 1.28419 0.642097 0.766623i \(-0.278065\pi\)
0.642097 + 0.766623i \(0.278065\pi\)
\(152\) 9.94271e11 0.993951
\(153\) −1.19712e11 −0.115434
\(154\) −3.13008e12 −2.91200
\(155\) 1.54133e12 1.38380
\(156\) −1.13217e12 −0.981127
\(157\) 2.33390e12 1.95270 0.976348 0.216206i \(-0.0693684\pi\)
0.976348 + 0.216206i \(0.0693684\pi\)
\(158\) −2.65761e12 −2.14722
\(159\) −8.05799e10 −0.0628844
\(160\) −4.31567e11 −0.325378
\(161\) 3.13121e11 0.228123
\(162\) −2.76253e11 −0.194525
\(163\) −1.13885e12 −0.775238 −0.387619 0.921820i \(-0.626702\pi\)
−0.387619 + 0.921820i \(0.626702\pi\)
\(164\) 4.96182e11 0.326588
\(165\) −1.09410e12 −0.696458
\(166\) 4.13559e12 2.54649
\(167\) 2.48575e12 1.48087 0.740436 0.672127i \(-0.234619\pi\)
0.740436 + 0.672127i \(0.234619\pi\)
\(168\) 3.56385e12 2.05456
\(169\) −5.78494e11 −0.322791
\(170\) 1.55354e12 0.839179
\(171\) −3.39740e11 −0.177692
\(172\) 6.69077e12 3.38899
\(173\) −2.18667e12 −1.07283 −0.536414 0.843955i \(-0.680221\pi\)
−0.536414 + 0.843955i \(0.680221\pi\)
\(174\) 4.88970e11 0.232414
\(175\) 3.79531e12 1.74799
\(176\) 2.34163e12 1.04520
\(177\) 1.73727e11 0.0751646
\(178\) −4.80269e12 −2.01454
\(179\) −1.47087e12 −0.598249 −0.299125 0.954214i \(-0.596695\pi\)
−0.299125 + 0.954214i \(0.596695\pi\)
\(180\) 2.41538e12 0.952767
\(181\) −4.92496e12 −1.88439 −0.942195 0.335064i \(-0.891242\pi\)
−0.942195 + 0.335064i \(0.891242\pi\)
\(182\) −7.40753e12 −2.74967
\(183\) 2.11251e12 0.760883
\(184\) −6.37589e11 −0.222865
\(185\) 5.81040e11 0.197134
\(186\) 3.06808e12 1.01052
\(187\) −9.43748e11 −0.301805
\(188\) −8.50643e12 −2.64167
\(189\) −1.21776e12 −0.367300
\(190\) 4.40894e12 1.29178
\(191\) −3.46630e12 −0.986695 −0.493347 0.869832i \(-0.664227\pi\)
−0.493347 + 0.869832i \(0.664227\pi\)
\(192\) 1.64431e12 0.454805
\(193\) −9.90934e11 −0.266366 −0.133183 0.991091i \(-0.542520\pi\)
−0.133183 + 0.991091i \(0.542520\pi\)
\(194\) −2.41989e12 −0.632245
\(195\) −2.58925e12 −0.657633
\(196\) 2.20982e13 5.45693
\(197\) −1.24232e12 −0.298311 −0.149155 0.988814i \(-0.547655\pi\)
−0.149155 + 0.988814i \(0.547655\pi\)
\(198\) −2.17784e12 −0.508590
\(199\) −2.76684e12 −0.628481 −0.314241 0.949343i \(-0.601750\pi\)
−0.314241 + 0.949343i \(0.601750\pi\)
\(200\) −7.72815e12 −1.70770
\(201\) 8.74689e11 0.188051
\(202\) 3.51273e12 0.734872
\(203\) 2.15544e12 0.438842
\(204\) 2.08346e12 0.412875
\(205\) 1.13476e12 0.218906
\(206\) 9.97321e12 1.87312
\(207\) 2.17863e11 0.0398424
\(208\) 5.54161e12 0.986932
\(209\) −2.67834e12 −0.464581
\(210\) 1.58033e13 2.67019
\(211\) 6.86381e12 1.12983 0.564914 0.825150i \(-0.308910\pi\)
0.564914 + 0.825150i \(0.308910\pi\)
\(212\) 1.40241e12 0.224920
\(213\) −4.31092e12 −0.673725
\(214\) 1.96288e13 2.98964
\(215\) 1.53017e13 2.27158
\(216\) 2.47964e12 0.358834
\(217\) 1.35245e13 1.90806
\(218\) −1.25388e13 −1.72483
\(219\) −2.29300e12 −0.307583
\(220\) 1.90417e13 2.49104
\(221\) −2.23344e12 −0.284981
\(222\) 1.15658e12 0.143958
\(223\) 1.20175e13 1.45928 0.729640 0.683831i \(-0.239688\pi\)
0.729640 + 0.683831i \(0.239688\pi\)
\(224\) −3.78679e12 −0.448650
\(225\) 2.64069e12 0.305291
\(226\) −2.50205e13 −2.82293
\(227\) −6.98874e11 −0.0769585 −0.0384792 0.999259i \(-0.512251\pi\)
−0.0384792 + 0.999259i \(0.512251\pi\)
\(228\) 5.91283e12 0.635556
\(229\) 1.89982e13 1.99351 0.996753 0.0805184i \(-0.0256576\pi\)
0.996753 + 0.0805184i \(0.0256576\pi\)
\(230\) −2.82729e12 −0.289646
\(231\) −9.60020e12 −0.960317
\(232\) −4.38899e12 −0.428727
\(233\) −1.53061e13 −1.46018 −0.730090 0.683351i \(-0.760522\pi\)
−0.730090 + 0.683351i \(0.760522\pi\)
\(234\) −5.15400e12 −0.480238
\(235\) −1.94541e13 −1.77067
\(236\) −3.02353e12 −0.268843
\(237\) −8.15109e12 −0.708109
\(238\) 1.36316e13 1.15711
\(239\) −2.05455e13 −1.70423 −0.852114 0.523357i \(-0.824680\pi\)
−0.852114 + 0.523357i \(0.824680\pi\)
\(240\) −1.18225e13 −0.958404
\(241\) 8.72286e12 0.691139 0.345570 0.938393i \(-0.387686\pi\)
0.345570 + 0.938393i \(0.387686\pi\)
\(242\) 5.43579e12 0.420996
\(243\) −8.47289e11 −0.0641500
\(244\) −3.67661e13 −2.72147
\(245\) 5.05382e13 3.65769
\(246\) 2.25878e12 0.159857
\(247\) −6.33846e12 −0.438683
\(248\) −2.75391e13 −1.86408
\(249\) 1.26842e13 0.839779
\(250\) 3.14782e12 0.203863
\(251\) −2.73350e13 −1.73187 −0.865933 0.500160i \(-0.833275\pi\)
−0.865933 + 0.500160i \(0.833275\pi\)
\(252\) 2.11938e13 1.31373
\(253\) 1.71752e12 0.104169
\(254\) −1.52397e13 −0.904463
\(255\) 4.76484e12 0.276743
\(256\) −3.58575e13 −2.03826
\(257\) −2.42947e12 −0.135169 −0.0675847 0.997714i \(-0.521529\pi\)
−0.0675847 + 0.997714i \(0.521529\pi\)
\(258\) 3.04586e13 1.65883
\(259\) 5.09835e12 0.271820
\(260\) 4.50633e13 2.35217
\(261\) 1.49971e12 0.0766451
\(262\) 1.69735e13 0.849405
\(263\) 4.85645e12 0.237992 0.118996 0.992895i \(-0.462032\pi\)
0.118996 + 0.992895i \(0.462032\pi\)
\(264\) 1.95483e13 0.938182
\(265\) 3.20730e12 0.150760
\(266\) 3.86864e13 1.78119
\(267\) −1.47302e13 −0.664351
\(268\) −1.52231e13 −0.672606
\(269\) −3.64117e13 −1.57617 −0.788085 0.615566i \(-0.788927\pi\)
−0.788085 + 0.615566i \(0.788927\pi\)
\(270\) 1.09956e13 0.466357
\(271\) −1.70716e12 −0.0709486 −0.0354743 0.999371i \(-0.511294\pi\)
−0.0354743 + 0.999371i \(0.511294\pi\)
\(272\) −1.01979e13 −0.415318
\(273\) −2.27195e13 −0.906783
\(274\) −2.02653e13 −0.792729
\(275\) 2.08179e13 0.798192
\(276\) −3.79167e12 −0.142505
\(277\) −3.21307e13 −1.18381 −0.591905 0.806008i \(-0.701624\pi\)
−0.591905 + 0.806008i \(0.701624\pi\)
\(278\) −5.91131e13 −2.13519
\(279\) 9.41003e12 0.333248
\(280\) −1.41850e14 −4.92563
\(281\) −3.49986e13 −1.19170 −0.595849 0.803096i \(-0.703184\pi\)
−0.595849 + 0.803096i \(0.703184\pi\)
\(282\) −3.87241e13 −1.29304
\(283\) 3.29943e13 1.08047 0.540235 0.841514i \(-0.318335\pi\)
0.540235 + 0.841514i \(0.318335\pi\)
\(284\) 7.50272e13 2.40973
\(285\) 1.35226e13 0.426002
\(286\) −4.06316e13 −1.25560
\(287\) 9.95700e12 0.301841
\(288\) −2.63477e12 −0.0783580
\(289\) −3.01618e13 −0.880075
\(290\) −1.94623e13 −0.557193
\(291\) −7.42198e12 −0.208501
\(292\) 3.99073e13 1.10014
\(293\) 1.27340e13 0.344503 0.172252 0.985053i \(-0.444896\pi\)
0.172252 + 0.985053i \(0.444896\pi\)
\(294\) 1.00598e14 2.67104
\(295\) −6.91478e12 −0.180201
\(296\) −1.03815e13 −0.265555
\(297\) −6.67961e12 −0.167722
\(298\) 8.88811e12 0.219089
\(299\) 4.06462e12 0.0983622
\(300\) −4.59585e13 −1.09194
\(301\) 1.34265e14 3.13219
\(302\) −9.81490e13 −2.24826
\(303\) 1.07738e13 0.242345
\(304\) −2.89414e13 −0.639316
\(305\) −8.40835e13 −1.82415
\(306\) 9.48458e12 0.202093
\(307\) −4.39040e13 −0.918847 −0.459423 0.888217i \(-0.651944\pi\)
−0.459423 + 0.888217i \(0.651944\pi\)
\(308\) 1.67082e14 3.43479
\(309\) 3.05886e13 0.617714
\(310\) −1.22118e14 −2.42264
\(311\) 3.76975e13 0.734735 0.367368 0.930076i \(-0.380259\pi\)
0.367368 + 0.930076i \(0.380259\pi\)
\(312\) 4.62622e13 0.885883
\(313\) −7.29934e13 −1.37338 −0.686688 0.726952i \(-0.740936\pi\)
−0.686688 + 0.726952i \(0.740936\pi\)
\(314\) −1.84912e14 −3.41862
\(315\) 4.84700e13 0.880572
\(316\) 1.41861e14 2.53271
\(317\) 1.79108e13 0.314260 0.157130 0.987578i \(-0.449776\pi\)
0.157130 + 0.987578i \(0.449776\pi\)
\(318\) 6.38423e12 0.110093
\(319\) 1.18230e13 0.200391
\(320\) −6.54477e13 −1.09036
\(321\) 6.02032e13 0.985920
\(322\) −2.48081e13 −0.399380
\(323\) 1.16643e13 0.184605
\(324\) 1.47462e13 0.229447
\(325\) 4.92668e13 0.753696
\(326\) 9.02296e13 1.35723
\(327\) −3.84575e13 −0.568811
\(328\) −2.02748e13 −0.294884
\(329\) −1.70701e14 −2.44150
\(330\) 8.66839e13 1.21930
\(331\) −2.70114e13 −0.373674 −0.186837 0.982391i \(-0.559824\pi\)
−0.186837 + 0.982391i \(0.559824\pi\)
\(332\) −2.20755e14 −3.00366
\(333\) 3.54732e12 0.0474741
\(334\) −1.96943e14 −2.59259
\(335\) −3.48149e13 −0.450837
\(336\) −1.03737e14 −1.32150
\(337\) 9.45958e13 1.18552 0.592758 0.805381i \(-0.298039\pi\)
0.592758 + 0.805381i \(0.298039\pi\)
\(338\) 4.58333e13 0.565118
\(339\) −7.67399e13 −0.930942
\(340\) −8.29271e13 −0.989835
\(341\) 7.41841e13 0.871288
\(342\) 2.69171e13 0.311089
\(343\) 2.75638e14 3.13489
\(344\) −2.73396e14 −3.06000
\(345\) −8.67151e12 −0.0955190
\(346\) 1.73247e14 1.87822
\(347\) −2.02122e13 −0.215676 −0.107838 0.994168i \(-0.534393\pi\)
−0.107838 + 0.994168i \(0.534393\pi\)
\(348\) −2.61009e13 −0.274138
\(349\) −5.64261e13 −0.583365 −0.291682 0.956515i \(-0.594215\pi\)
−0.291682 + 0.956515i \(0.594215\pi\)
\(350\) −3.00697e14 −3.06023
\(351\) −1.58077e13 −0.158372
\(352\) −2.07712e13 −0.204869
\(353\) 7.43872e13 0.722333 0.361166 0.932501i \(-0.382379\pi\)
0.361166 + 0.932501i \(0.382379\pi\)
\(354\) −1.37641e13 −0.131592
\(355\) 1.71586e14 1.61520
\(356\) 2.56364e14 2.37620
\(357\) 4.18092e13 0.381590
\(358\) 1.16535e14 1.04737
\(359\) −1.13256e14 −1.00240 −0.501200 0.865331i \(-0.667108\pi\)
−0.501200 + 0.865331i \(0.667108\pi\)
\(360\) −9.86964e13 −0.860276
\(361\) −8.33872e13 −0.715830
\(362\) 3.90198e14 3.29904
\(363\) 1.66720e13 0.138836
\(364\) 3.95409e14 3.24331
\(365\) 9.12675e13 0.737404
\(366\) −1.67371e14 −1.33209
\(367\) −1.91005e13 −0.149755 −0.0748774 0.997193i \(-0.523857\pi\)
−0.0748774 + 0.997193i \(0.523857\pi\)
\(368\) 1.85591e13 0.143348
\(369\) 6.92786e12 0.0527174
\(370\) −4.60350e13 −0.345127
\(371\) 2.81425e13 0.207877
\(372\) −1.63772e14 −1.19194
\(373\) 2.11166e14 1.51435 0.757174 0.653213i \(-0.226579\pi\)
0.757174 + 0.653213i \(0.226579\pi\)
\(374\) 7.47718e13 0.528377
\(375\) 9.65459e12 0.0672298
\(376\) 3.47587e14 2.38523
\(377\) 2.79798e13 0.189220
\(378\) 9.64812e13 0.643040
\(379\) −2.39226e14 −1.57142 −0.785710 0.618594i \(-0.787702\pi\)
−0.785710 + 0.618594i \(0.787702\pi\)
\(380\) −2.35346e14 −1.52369
\(381\) −4.67413e13 −0.298273
\(382\) 2.74630e14 1.72743
\(383\) −1.68715e14 −1.04607 −0.523034 0.852312i \(-0.675200\pi\)
−0.523034 + 0.852312i \(0.675200\pi\)
\(384\) −1.52482e14 −0.931957
\(385\) 3.82114e14 2.30228
\(386\) 7.85102e13 0.466333
\(387\) 9.34188e13 0.547046
\(388\) 1.29172e14 0.745750
\(389\) 2.35191e14 1.33874 0.669372 0.742927i \(-0.266563\pi\)
0.669372 + 0.742927i \(0.266563\pi\)
\(390\) 2.05143e14 1.15133
\(391\) −7.47986e12 −0.0413925
\(392\) −9.02968e14 −4.92719
\(393\) 5.20591e13 0.280116
\(394\) 9.84272e13 0.522259
\(395\) 3.24435e14 1.69763
\(396\) 1.16252e14 0.599896
\(397\) 2.65418e14 1.35077 0.675387 0.737464i \(-0.263977\pi\)
0.675387 + 0.737464i \(0.263977\pi\)
\(398\) 2.19213e14 1.10029
\(399\) 1.18654e14 0.587397
\(400\) 2.24953e14 1.09840
\(401\) 2.13150e14 1.02658 0.513289 0.858216i \(-0.328427\pi\)
0.513289 + 0.858216i \(0.328427\pi\)
\(402\) −6.93004e13 −0.329224
\(403\) 1.75561e14 0.822717
\(404\) −1.87507e14 −0.866802
\(405\) 3.37243e13 0.153794
\(406\) −1.70773e14 −0.768290
\(407\) 2.79653e13 0.124123
\(408\) −8.51335e13 −0.372795
\(409\) −1.94878e14 −0.841947 −0.420974 0.907073i \(-0.638312\pi\)
−0.420974 + 0.907073i \(0.638312\pi\)
\(410\) −8.99056e13 −0.383244
\(411\) −6.21552e13 −0.261425
\(412\) −5.32363e14 −2.20939
\(413\) −6.06739e13 −0.248472
\(414\) −1.72609e13 −0.0697529
\(415\) −5.04863e14 −2.01330
\(416\) −4.91563e13 −0.193449
\(417\) −1.81304e14 −0.704141
\(418\) 2.12201e14 0.813352
\(419\) 4.34434e14 1.64341 0.821707 0.569910i \(-0.193022\pi\)
0.821707 + 0.569910i \(0.193022\pi\)
\(420\) −8.43570e14 −3.14956
\(421\) 1.94537e14 0.716886 0.358443 0.933552i \(-0.383308\pi\)
0.358443 + 0.933552i \(0.383308\pi\)
\(422\) −5.43810e14 −1.97801
\(423\) −1.18770e14 −0.426416
\(424\) −5.73048e13 −0.203086
\(425\) −9.06627e13 −0.317168
\(426\) 3.41548e14 1.17950
\(427\) −7.37793e14 −2.51525
\(428\) −1.04777e15 −3.52636
\(429\) −1.24620e14 −0.414069
\(430\) −1.21233e15 −3.97691
\(431\) −2.48713e14 −0.805515 −0.402758 0.915307i \(-0.631948\pi\)
−0.402758 + 0.915307i \(0.631948\pi\)
\(432\) −7.21780e13 −0.230804
\(433\) 3.64798e14 1.15178 0.575889 0.817528i \(-0.304656\pi\)
0.575889 + 0.817528i \(0.304656\pi\)
\(434\) −1.07153e15 −3.34048
\(435\) −5.96924e13 −0.183750
\(436\) 6.69313e14 2.03448
\(437\) −2.12278e13 −0.0637172
\(438\) 1.81671e14 0.538491
\(439\) −2.03663e14 −0.596153 −0.298076 0.954542i \(-0.596345\pi\)
−0.298076 + 0.954542i \(0.596345\pi\)
\(440\) −7.78074e14 −2.24922
\(441\) 3.08542e14 0.880850
\(442\) 1.76952e14 0.498922
\(443\) −1.65338e14 −0.460418 −0.230209 0.973141i \(-0.573941\pi\)
−0.230209 + 0.973141i \(0.573941\pi\)
\(444\) −6.17375e13 −0.169802
\(445\) 5.86302e14 1.59273
\(446\) −9.52132e14 −2.55479
\(447\) 2.72605e13 0.0722508
\(448\) −5.74273e14 −1.50345
\(449\) 5.98301e14 1.54727 0.773633 0.633634i \(-0.218437\pi\)
0.773633 + 0.633634i \(0.218437\pi\)
\(450\) −2.09218e14 −0.534479
\(451\) 5.46158e13 0.137831
\(452\) 1.33558e15 3.32972
\(453\) −3.01030e14 −0.741430
\(454\) 5.53708e13 0.134733
\(455\) 9.04295e14 2.17394
\(456\) −2.41608e14 −0.573858
\(457\) −4.00303e14 −0.939397 −0.469699 0.882827i \(-0.655637\pi\)
−0.469699 + 0.882827i \(0.655637\pi\)
\(458\) −1.50520e15 −3.49007
\(459\) 2.90899e13 0.0666458
\(460\) 1.50919e14 0.341645
\(461\) −1.68156e14 −0.376146 −0.188073 0.982155i \(-0.560224\pi\)
−0.188073 + 0.982155i \(0.560224\pi\)
\(462\) 7.60610e14 1.68125
\(463\) 4.82686e14 1.05431 0.527156 0.849769i \(-0.323258\pi\)
0.527156 + 0.849769i \(0.323258\pi\)
\(464\) 1.27756e14 0.275760
\(465\) −3.74544e14 −0.798936
\(466\) 1.21268e15 2.55637
\(467\) 3.98412e14 0.830021 0.415011 0.909817i \(-0.363778\pi\)
0.415011 + 0.909817i \(0.363778\pi\)
\(468\) 2.75117e14 0.566454
\(469\) −3.05485e14 −0.621640
\(470\) 1.54132e15 3.09995
\(471\) −5.67138e14 −1.12739
\(472\) 1.23547e14 0.242745
\(473\) 7.36468e14 1.43027
\(474\) 6.45799e14 1.23970
\(475\) −2.57300e14 −0.488230
\(476\) −7.27647e14 −1.36484
\(477\) 1.95809e13 0.0363063
\(478\) 1.62779e15 2.98363
\(479\) 3.18807e14 0.577674 0.288837 0.957378i \(-0.406731\pi\)
0.288837 + 0.957378i \(0.406731\pi\)
\(480\) 1.04871e14 0.187857
\(481\) 6.61816e13 0.117203
\(482\) −6.91100e14 −1.20999
\(483\) −7.60884e13 −0.131707
\(484\) −2.90159e14 −0.496577
\(485\) 2.95414e14 0.499864
\(486\) 6.71295e13 0.112309
\(487\) −5.27807e13 −0.0873103 −0.0436552 0.999047i \(-0.513900\pi\)
−0.0436552 + 0.999047i \(0.513900\pi\)
\(488\) 1.50232e15 2.45728
\(489\) 2.76741e14 0.447584
\(490\) −4.00407e15 −6.40359
\(491\) 1.54389e13 0.0244157 0.0122078 0.999925i \(-0.496114\pi\)
0.0122078 + 0.999925i \(0.496114\pi\)
\(492\) −1.20572e14 −0.188556
\(493\) −5.14894e13 −0.0796270
\(494\) 5.02187e14 0.768011
\(495\) 2.65866e14 0.402100
\(496\) 8.01613e14 1.19899
\(497\) 1.50559e15 2.22713
\(498\) −1.00495e15 −1.47022
\(499\) 9.35812e14 1.35405 0.677026 0.735959i \(-0.263268\pi\)
0.677026 + 0.735959i \(0.263268\pi\)
\(500\) −1.68028e14 −0.240462
\(501\) −6.04038e14 −0.854982
\(502\) 2.16572e15 3.03201
\(503\) 7.11605e14 0.985406 0.492703 0.870198i \(-0.336009\pi\)
0.492703 + 0.870198i \(0.336009\pi\)
\(504\) −8.66014e14 −1.18620
\(505\) −4.28826e14 −0.581003
\(506\) −1.36077e14 −0.182371
\(507\) 1.40574e14 0.186364
\(508\) 8.13484e14 1.06684
\(509\) −5.49290e14 −0.712613 −0.356306 0.934369i \(-0.615964\pi\)
−0.356306 + 0.934369i \(0.615964\pi\)
\(510\) −3.77511e14 −0.484500
\(511\) 8.00830e14 1.01678
\(512\) 1.55582e15 1.95423
\(513\) 8.25568e13 0.102591
\(514\) 1.92483e14 0.236644
\(515\) −1.21751e15 −1.48092
\(516\) −1.62586e15 −1.95663
\(517\) −9.36321e14 −1.11488
\(518\) −4.03935e14 −0.475881
\(519\) 5.31361e14 0.619398
\(520\) −1.84136e15 −2.12383
\(521\) 1.52469e14 0.174010 0.0870050 0.996208i \(-0.472270\pi\)
0.0870050 + 0.996208i \(0.472270\pi\)
\(522\) −1.18820e14 −0.134184
\(523\) −5.46877e13 −0.0611126 −0.0305563 0.999533i \(-0.509728\pi\)
−0.0305563 + 0.999533i \(0.509728\pi\)
\(524\) −9.06034e14 −1.00190
\(525\) −9.22260e14 −1.00920
\(526\) −3.84770e14 −0.416658
\(527\) −3.23074e14 −0.346214
\(528\) −5.69016e14 −0.603445
\(529\) −9.39197e14 −0.985713
\(530\) −2.54109e14 −0.263939
\(531\) −4.22156e13 −0.0433963
\(532\) −2.06505e15 −2.10096
\(533\) 1.29252e14 0.130148
\(534\) 1.16705e15 1.16309
\(535\) −2.39625e15 −2.36366
\(536\) 6.22039e14 0.607312
\(537\) 3.57421e14 0.345399
\(538\) 2.88485e15 2.75943
\(539\) 2.43239e15 2.30301
\(540\) −5.86937e14 −0.550080
\(541\) −1.22752e15 −1.13879 −0.569396 0.822063i \(-0.692823\pi\)
−0.569396 + 0.822063i \(0.692823\pi\)
\(542\) 1.35256e14 0.124211
\(543\) 1.19677e15 1.08795
\(544\) 9.04593e13 0.0814066
\(545\) 1.53071e15 1.36368
\(546\) 1.80003e15 1.58752
\(547\) −6.27744e13 −0.0548090 −0.0274045 0.999624i \(-0.508724\pi\)
−0.0274045 + 0.999624i \(0.508724\pi\)
\(548\) 1.08175e15 0.935045
\(549\) −5.13340e14 −0.439296
\(550\) −1.64937e15 −1.39741
\(551\) −1.46126e14 −0.122573
\(552\) 1.54934e14 0.128671
\(553\) 2.84676e15 2.34079
\(554\) 2.54567e15 2.07252
\(555\) −1.41193e14 −0.113815
\(556\) 3.15541e15 2.51852
\(557\) 1.41405e15 1.11754 0.558768 0.829324i \(-0.311274\pi\)
0.558768 + 0.829324i \(0.311274\pi\)
\(558\) −7.45544e14 −0.583425
\(559\) 1.74290e15 1.35054
\(560\) 4.12901e15 3.16820
\(561\) 2.29331e14 0.174247
\(562\) 2.77289e15 2.08633
\(563\) −7.52424e13 −0.0560617 −0.0280309 0.999607i \(-0.508924\pi\)
−0.0280309 + 0.999607i \(0.508924\pi\)
\(564\) 2.06706e15 1.52517
\(565\) 3.05445e15 2.23186
\(566\) −2.61409e15 −1.89160
\(567\) 2.95915e14 0.212061
\(568\) −3.06574e15 −2.17580
\(569\) −3.78781e14 −0.266238 −0.133119 0.991100i \(-0.542499\pi\)
−0.133119 + 0.991100i \(0.542499\pi\)
\(570\) −1.07137e15 −0.745811
\(571\) 2.32982e15 1.60629 0.803145 0.595783i \(-0.203158\pi\)
0.803145 + 0.595783i \(0.203158\pi\)
\(572\) 2.16888e15 1.48101
\(573\) 8.42311e14 0.569668
\(574\) −7.88879e14 −0.528439
\(575\) 1.64997e14 0.109472
\(576\) −3.99566e14 −0.262582
\(577\) −1.28910e15 −0.839110 −0.419555 0.907730i \(-0.637814\pi\)
−0.419555 + 0.907730i \(0.637814\pi\)
\(578\) 2.38968e15 1.54077
\(579\) 2.40797e14 0.153787
\(580\) 1.03888e15 0.657224
\(581\) −4.42993e15 −2.77606
\(582\) 5.88033e14 0.365027
\(583\) 1.54366e14 0.0949239
\(584\) −1.63068e15 −0.993340
\(585\) 6.29188e14 0.379685
\(586\) −1.00890e15 −0.603129
\(587\) 1.71101e15 1.01331 0.506655 0.862149i \(-0.330882\pi\)
0.506655 + 0.862149i \(0.330882\pi\)
\(588\) −5.36986e15 −3.15056
\(589\) −9.16881e14 −0.532941
\(590\) 5.47848e14 0.315482
\(591\) 3.01884e14 0.172230
\(592\) 3.02186e14 0.170807
\(593\) −4.88833e14 −0.273754 −0.136877 0.990588i \(-0.543706\pi\)
−0.136877 + 0.990588i \(0.543706\pi\)
\(594\) 5.29216e14 0.293635
\(595\) −1.66412e15 −0.914831
\(596\) −4.74441e14 −0.258421
\(597\) 6.72342e14 0.362854
\(598\) −3.22034e14 −0.172205
\(599\) 3.48930e15 1.84881 0.924403 0.381418i \(-0.124564\pi\)
0.924403 + 0.381418i \(0.124564\pi\)
\(600\) 1.87794e15 0.985939
\(601\) −1.28836e15 −0.670236 −0.335118 0.942176i \(-0.608776\pi\)
−0.335118 + 0.942176i \(0.608776\pi\)
\(602\) −1.06376e16 −5.48359
\(603\) −2.12549e14 −0.108571
\(604\) 5.23912e15 2.65189
\(605\) −6.63589e14 −0.332847
\(606\) −8.53593e14 −0.424279
\(607\) 1.47555e15 0.726803 0.363402 0.931633i \(-0.381615\pi\)
0.363402 + 0.931633i \(0.381615\pi\)
\(608\) 2.56722e14 0.125312
\(609\) −5.23773e14 −0.253366
\(610\) 6.66181e15 3.19358
\(611\) −2.21586e15 −1.05273
\(612\) −5.06280e14 −0.238374
\(613\) −3.36346e15 −1.56947 −0.784736 0.619830i \(-0.787202\pi\)
−0.784736 + 0.619830i \(0.787202\pi\)
\(614\) 3.47845e15 1.60864
\(615\) −2.75747e14 −0.126386
\(616\) −6.82723e15 −3.10135
\(617\) −6.14389e14 −0.276615 −0.138307 0.990389i \(-0.544166\pi\)
−0.138307 + 0.990389i \(0.544166\pi\)
\(618\) −2.42349e15 −1.08145
\(619\) −4.30442e15 −1.90378 −0.951888 0.306448i \(-0.900860\pi\)
−0.951888 + 0.306448i \(0.900860\pi\)
\(620\) 6.51856e15 2.85757
\(621\) −5.29406e13 −0.0230030
\(622\) −2.98672e15 −1.28632
\(623\) 5.14452e15 2.19614
\(624\) −1.34661e15 −0.569806
\(625\) −2.56789e15 −1.07705
\(626\) 5.78316e15 2.40440
\(627\) 6.50838e14 0.268226
\(628\) 9.87046e15 4.03236
\(629\) −1.21790e14 −0.0493211
\(630\) −3.84021e15 −1.54164
\(631\) −3.85187e15 −1.53289 −0.766444 0.642311i \(-0.777976\pi\)
−0.766444 + 0.642311i \(0.777976\pi\)
\(632\) −5.79669e15 −2.28684
\(633\) −1.66791e15 −0.652306
\(634\) −1.41905e15 −0.550181
\(635\) 1.86043e15 0.715084
\(636\) −3.40786e14 −0.129858
\(637\) 5.75641e15 2.17463
\(638\) −9.36717e14 −0.350828
\(639\) 1.04755e15 0.388975
\(640\) 6.06918e15 2.23429
\(641\) −1.86345e15 −0.680140 −0.340070 0.940400i \(-0.610451\pi\)
−0.340070 + 0.940400i \(0.610451\pi\)
\(642\) −4.76981e15 −1.72607
\(643\) −8.75431e14 −0.314095 −0.157048 0.987591i \(-0.550198\pi\)
−0.157048 + 0.987591i \(0.550198\pi\)
\(644\) 1.32424e15 0.471080
\(645\) −3.71832e15 −1.31150
\(646\) −9.24144e14 −0.323192
\(647\) 3.41312e15 1.18353 0.591764 0.806111i \(-0.298432\pi\)
0.591764 + 0.806111i \(0.298432\pi\)
\(648\) −6.02553e14 −0.207173
\(649\) −3.32807e14 −0.113461
\(650\) −3.90334e15 −1.31951
\(651\) −3.28645e15 −1.10162
\(652\) −4.81639e15 −1.60088
\(653\) 2.79838e14 0.0922325 0.0461163 0.998936i \(-0.485316\pi\)
0.0461163 + 0.998936i \(0.485316\pi\)
\(654\) 3.04693e15 0.995829
\(655\) −2.07209e15 −0.671554
\(656\) 5.90164e14 0.189671
\(657\) 5.57199e14 0.177583
\(658\) 1.35244e16 4.27439
\(659\) −2.21300e15 −0.693605 −0.346803 0.937938i \(-0.612733\pi\)
−0.346803 + 0.937938i \(0.612733\pi\)
\(660\) −4.62713e15 −1.43820
\(661\) −4.73366e15 −1.45911 −0.729557 0.683920i \(-0.760274\pi\)
−0.729557 + 0.683920i \(0.760274\pi\)
\(662\) 2.14007e15 0.654199
\(663\) 5.42725e14 0.164534
\(664\) 9.02040e15 2.71207
\(665\) −4.72275e15 −1.40824
\(666\) −2.81049e14 −0.0831140
\(667\) 9.37054e13 0.0274835
\(668\) 1.05127e16 3.05803
\(669\) −2.92026e15 −0.842516
\(670\) 2.75834e15 0.789289
\(671\) −4.04692e15 −1.14855
\(672\) 9.20191e14 0.259028
\(673\) −2.45606e15 −0.685736 −0.342868 0.939384i \(-0.611398\pi\)
−0.342868 + 0.939384i \(0.611398\pi\)
\(674\) −7.49469e15 −2.07551
\(675\) −6.41688e14 −0.176260
\(676\) −2.44655e15 −0.666572
\(677\) 1.22414e15 0.330821 0.165411 0.986225i \(-0.447105\pi\)
0.165411 + 0.986225i \(0.447105\pi\)
\(678\) 6.07999e15 1.62982
\(679\) 2.59212e15 0.689242
\(680\) 3.38854e15 0.893745
\(681\) 1.69826e14 0.0444320
\(682\) −5.87750e15 −1.52538
\(683\) −6.97032e15 −1.79448 −0.897240 0.441543i \(-0.854431\pi\)
−0.897240 + 0.441543i \(0.854431\pi\)
\(684\) −1.43682e15 −0.366938
\(685\) 2.47394e15 0.626745
\(686\) −2.18384e16 −5.48831
\(687\) −4.61657e15 −1.15095
\(688\) 7.95807e15 1.96821
\(689\) 3.65317e14 0.0896323
\(690\) 6.87031e14 0.167227
\(691\) 7.05580e15 1.70379 0.851897 0.523709i \(-0.175452\pi\)
0.851897 + 0.523709i \(0.175452\pi\)
\(692\) −9.24780e15 −2.21541
\(693\) 2.33285e15 0.554439
\(694\) 1.60139e15 0.377589
\(695\) 7.21639e15 1.68812
\(696\) 1.06653e15 0.247526
\(697\) −2.37854e14 −0.0547684
\(698\) 4.47056e15 1.02131
\(699\) 3.71938e15 0.843035
\(700\) 1.60510e16 3.60963
\(701\) −7.19727e15 −1.60590 −0.802950 0.596046i \(-0.796738\pi\)
−0.802950 + 0.596046i \(0.796738\pi\)
\(702\) 1.25242e15 0.277266
\(703\) −3.45639e14 −0.0759220
\(704\) −3.14998e15 −0.686529
\(705\) 4.72735e15 1.02230
\(706\) −5.89359e15 −1.26460
\(707\) −3.76274e15 −0.801121
\(708\) 7.34719e14 0.155217
\(709\) 2.97413e14 0.0623456 0.0311728 0.999514i \(-0.490076\pi\)
0.0311728 + 0.999514i \(0.490076\pi\)
\(710\) −1.35945e16 −2.82776
\(711\) 1.98072e15 0.408827
\(712\) −1.04755e16 −2.14553
\(713\) 5.87961e14 0.119497
\(714\) −3.31248e15 −0.668057
\(715\) 4.96021e15 0.992697
\(716\) −6.22055e15 −1.23540
\(717\) 4.99255e15 0.983936
\(718\) 8.97310e15 1.75492
\(719\) 4.12277e15 0.800166 0.400083 0.916479i \(-0.368981\pi\)
0.400083 + 0.916479i \(0.368981\pi\)
\(720\) 2.87288e15 0.553335
\(721\) −1.06830e16 −2.04198
\(722\) 6.60665e15 1.25322
\(723\) −2.11966e15 −0.399029
\(724\) −2.08285e16 −3.89131
\(725\) 1.13579e15 0.210591
\(726\) −1.32090e15 −0.243062
\(727\) −2.20204e15 −0.402148 −0.201074 0.979576i \(-0.564443\pi\)
−0.201074 + 0.979576i \(0.564443\pi\)
\(728\) −1.61571e16 −2.92846
\(729\) 2.05891e14 0.0370370
\(730\) −7.23100e15 −1.29099
\(731\) −3.20734e15 −0.568329
\(732\) 8.93415e15 1.57124
\(733\) 7.67019e15 1.33886 0.669428 0.742877i \(-0.266539\pi\)
0.669428 + 0.742877i \(0.266539\pi\)
\(734\) 1.51330e15 0.262179
\(735\) −1.22808e16 −2.11177
\(736\) −1.64626e14 −0.0280978
\(737\) −1.67564e15 −0.283863
\(738\) −5.48884e14 −0.0922934
\(739\) −4.60255e15 −0.768164 −0.384082 0.923299i \(-0.625482\pi\)
−0.384082 + 0.923299i \(0.625482\pi\)
\(740\) 2.45732e15 0.407086
\(741\) 1.54025e15 0.253274
\(742\) −2.22969e15 −0.363934
\(743\) −7.62174e15 −1.23485 −0.617427 0.786628i \(-0.711825\pi\)
−0.617427 + 0.786628i \(0.711825\pi\)
\(744\) 6.69199e15 1.07623
\(745\) −1.08504e15 −0.173215
\(746\) −1.67304e16 −2.65120
\(747\) −3.08225e15 −0.484847
\(748\) −3.99127e15 −0.623235
\(749\) −2.10259e16 −3.25916
\(750\) −7.64919e14 −0.117701
\(751\) 3.56762e15 0.544953 0.272476 0.962162i \(-0.412157\pi\)
0.272476 + 0.962162i \(0.412157\pi\)
\(752\) −1.01176e16 −1.53420
\(753\) 6.64242e15 0.999893
\(754\) −2.21680e15 −0.331271
\(755\) 1.19818e16 1.77752
\(756\) −5.15010e15 −0.758483
\(757\) 2.86324e15 0.418631 0.209315 0.977848i \(-0.432876\pi\)
0.209315 + 0.977848i \(0.432876\pi\)
\(758\) 1.89535e16 2.75112
\(759\) −4.17358e14 −0.0601421
\(760\) 9.61663e15 1.37578
\(761\) 1.63278e14 0.0231906 0.0115953 0.999933i \(-0.496309\pi\)
0.0115953 + 0.999933i \(0.496309\pi\)
\(762\) 3.70325e15 0.522192
\(763\) 1.34313e16 1.88032
\(764\) −1.46596e16 −2.03755
\(765\) −1.15786e15 −0.159778
\(766\) 1.33670e16 1.83137
\(767\) −7.87607e14 −0.107136
\(768\) 8.71337e15 1.17679
\(769\) 8.05214e15 1.07973 0.539867 0.841750i \(-0.318475\pi\)
0.539867 + 0.841750i \(0.318475\pi\)
\(770\) −3.02743e16 −4.03065
\(771\) 5.90360e14 0.0780401
\(772\) −4.19082e15 −0.550052
\(773\) 7.43296e15 0.968667 0.484334 0.874883i \(-0.339062\pi\)
0.484334 + 0.874883i \(0.339062\pi\)
\(774\) −7.40144e15 −0.957725
\(775\) 7.12662e15 0.915639
\(776\) −5.27817e15 −0.673355
\(777\) −1.23890e15 −0.156935
\(778\) −1.86338e16 −2.34377
\(779\) −6.75026e14 −0.0843072
\(780\) −1.09504e16 −1.35803
\(781\) 8.25841e15 1.01699
\(782\) 5.92619e14 0.0724667
\(783\) −3.64429e14 −0.0442510
\(784\) 2.62838e16 3.16920
\(785\) 2.25736e16 2.70282
\(786\) −4.12456e15 −0.490404
\(787\) −7.39938e14 −0.0873643 −0.0436822 0.999045i \(-0.513909\pi\)
−0.0436822 + 0.999045i \(0.513909\pi\)
\(788\) −5.25398e15 −0.616019
\(789\) −1.18012e15 −0.137405
\(790\) −2.57045e16 −2.97208
\(791\) 2.68014e16 3.07742
\(792\) −4.75024e15 −0.541660
\(793\) −9.57728e15 −1.08452
\(794\) −2.10287e16 −2.36483
\(795\) −7.79373e14 −0.0870414
\(796\) −1.17014e16 −1.29783
\(797\) −1.21756e15 −0.134112 −0.0670560 0.997749i \(-0.521361\pi\)
−0.0670560 + 0.997749i \(0.521361\pi\)
\(798\) −9.40079e15 −1.02837
\(799\) 4.07771e15 0.443006
\(800\) −1.99542e15 −0.215298
\(801\) 3.57944e15 0.383563
\(802\) −1.68876e16 −1.79725
\(803\) 4.39269e15 0.464296
\(804\) 3.69920e15 0.388329
\(805\) 3.02852e15 0.315757
\(806\) −1.39095e16 −1.44035
\(807\) 8.84804e15 0.910002
\(808\) 7.66184e15 0.782656
\(809\) −1.28451e16 −1.30322 −0.651612 0.758552i \(-0.725907\pi\)
−0.651612 + 0.758552i \(0.725907\pi\)
\(810\) −2.67193e15 −0.269251
\(811\) 5.68684e15 0.569189 0.284594 0.958648i \(-0.408141\pi\)
0.284594 + 0.958648i \(0.408141\pi\)
\(812\) 9.11573e15 0.906219
\(813\) 4.14841e14 0.0409622
\(814\) −2.21565e15 −0.217304
\(815\) −1.10150e16 −1.07305
\(816\) 2.47808e15 0.239784
\(817\) −9.10240e15 −0.874852
\(818\) 1.54399e16 1.47401
\(819\) 5.52083e15 0.523531
\(820\) 4.79910e15 0.452047
\(821\) 1.88909e16 1.76753 0.883763 0.467934i \(-0.155002\pi\)
0.883763 + 0.467934i \(0.155002\pi\)
\(822\) 4.92446e15 0.457682
\(823\) −1.30076e16 −1.20087 −0.600437 0.799672i \(-0.705007\pi\)
−0.600437 + 0.799672i \(0.705007\pi\)
\(824\) 2.17532e16 1.99491
\(825\) −5.05875e15 −0.460836
\(826\) 4.80711e15 0.435004
\(827\) 1.61858e16 1.45497 0.727485 0.686124i \(-0.240689\pi\)
0.727485 + 0.686124i \(0.240689\pi\)
\(828\) 9.21377e14 0.0822755
\(829\) −1.82837e15 −0.162186 −0.0810932 0.996707i \(-0.525841\pi\)
−0.0810932 + 0.996707i \(0.525841\pi\)
\(830\) 3.99996e16 3.52473
\(831\) 7.80777e15 0.683473
\(832\) −7.45462e15 −0.648257
\(833\) −1.05932e16 −0.915120
\(834\) 1.43645e16 1.23275
\(835\) 2.40423e16 2.04975
\(836\) −1.13272e16 −0.959371
\(837\) −2.28664e15 −0.192401
\(838\) −3.44196e16 −2.87716
\(839\) 6.27284e15 0.520923 0.260462 0.965484i \(-0.416125\pi\)
0.260462 + 0.965484i \(0.416125\pi\)
\(840\) 3.44697e16 2.84381
\(841\) −1.15555e16 −0.947130
\(842\) −1.54129e16 −1.25507
\(843\) 8.50467e15 0.688027
\(844\) 2.90282e16 2.33312
\(845\) −5.59522e15 −0.446792
\(846\) 9.40995e15 0.746535
\(847\) −5.82268e15 −0.458949
\(848\) 1.66804e15 0.130626
\(849\) −8.01761e15 −0.623810
\(850\) 7.18308e15 0.555273
\(851\) 2.21645e14 0.0170234
\(852\) −1.82316e16 −1.39126
\(853\) −1.56569e16 −1.18709 −0.593546 0.804800i \(-0.702273\pi\)
−0.593546 + 0.804800i \(0.702273\pi\)
\(854\) 5.84543e16 4.40350
\(855\) −3.28598e15 −0.245953
\(856\) 4.28138e16 3.18403
\(857\) −8.16164e15 −0.603091 −0.301546 0.953452i \(-0.597503\pi\)
−0.301546 + 0.953452i \(0.597503\pi\)
\(858\) 9.87347e15 0.724919
\(859\) 9.60173e15 0.700466 0.350233 0.936663i \(-0.386102\pi\)
0.350233 + 0.936663i \(0.386102\pi\)
\(860\) 6.47135e16 4.69087
\(861\) −2.41955e15 −0.174268
\(862\) 1.97052e16 1.41023
\(863\) −2.59016e15 −0.184191 −0.0920954 0.995750i \(-0.529356\pi\)
−0.0920954 + 0.995750i \(0.529356\pi\)
\(864\) 6.40248e14 0.0452400
\(865\) −2.11496e16 −1.48496
\(866\) −2.89024e16 −2.01644
\(867\) 7.32933e15 0.508112
\(868\) 5.71973e16 3.94019
\(869\) 1.56150e16 1.06889
\(870\) 4.72934e15 0.321695
\(871\) −3.96549e15 −0.268038
\(872\) −2.73492e16 −1.83698
\(873\) 1.80354e15 0.120378
\(874\) 1.68185e15 0.111551
\(875\) −3.37186e15 −0.222242
\(876\) −9.69748e15 −0.635165
\(877\) −2.83381e15 −0.184448 −0.0922238 0.995738i \(-0.529398\pi\)
−0.0922238 + 0.995738i \(0.529398\pi\)
\(878\) 1.61359e16 1.04370
\(879\) −3.09437e15 −0.198899
\(880\) 2.26483e16 1.44671
\(881\) 1.78252e16 1.13153 0.565765 0.824567i \(-0.308581\pi\)
0.565765 + 0.824567i \(0.308581\pi\)
\(882\) −2.44453e16 −1.54212
\(883\) 2.31060e16 1.44857 0.724286 0.689499i \(-0.242169\pi\)
0.724286 + 0.689499i \(0.242169\pi\)
\(884\) −9.44557e15 −0.588492
\(885\) 1.68029e15 0.104039
\(886\) 1.30995e16 0.806063
\(887\) 2.48053e16 1.51693 0.758465 0.651714i \(-0.225950\pi\)
0.758465 + 0.651714i \(0.225950\pi\)
\(888\) 2.52269e15 0.153318
\(889\) 1.63244e16 0.986000
\(890\) −4.64519e16 −2.78842
\(891\) 1.62314e15 0.0968344
\(892\) 5.08242e16 3.01345
\(893\) 1.15725e16 0.681937
\(894\) −2.15981e15 −0.126491
\(895\) −1.42263e16 −0.828067
\(896\) 5.32542e16 3.08077
\(897\) −9.87702e14 −0.0567894
\(898\) −4.74026e16 −2.70883
\(899\) 4.04737e15 0.229877
\(900\) 1.11679e16 0.630432
\(901\) −6.72271e14 −0.0377188
\(902\) −4.32713e15 −0.241304
\(903\) −3.26265e16 −1.80837
\(904\) −5.45739e16 −3.00648
\(905\) −4.76345e16 −2.60828
\(906\) 2.38502e16 1.29804
\(907\) −1.40277e16 −0.758835 −0.379417 0.925226i \(-0.623876\pi\)
−0.379417 + 0.925226i \(0.623876\pi\)
\(908\) −2.95565e15 −0.158921
\(909\) −2.61803e15 −0.139918
\(910\) −7.16460e16 −3.80596
\(911\) −1.31685e16 −0.695321 −0.347660 0.937621i \(-0.613024\pi\)
−0.347660 + 0.937621i \(0.613024\pi\)
\(912\) 7.03277e15 0.369109
\(913\) −2.42989e16 −1.26765
\(914\) 3.17154e16 1.64462
\(915\) 2.04323e16 1.05318
\(916\) 8.03466e16 4.11664
\(917\) −1.81816e16 −0.925978
\(918\) −2.30475e15 −0.116678
\(919\) 2.07390e16 1.04365 0.521823 0.853054i \(-0.325252\pi\)
0.521823 + 0.853054i \(0.325252\pi\)
\(920\) −6.16679e15 −0.308479
\(921\) 1.06687e16 0.530497
\(922\) 1.33227e16 0.658526
\(923\) 1.95440e16 0.960294
\(924\) −4.06009e16 −1.98308
\(925\) 2.68654e15 0.130441
\(926\) −3.82425e16 −1.84581
\(927\) −7.43303e15 −0.356638
\(928\) −1.13325e15 −0.0540518
\(929\) −3.76027e16 −1.78292 −0.891462 0.453095i \(-0.850320\pi\)
−0.891462 + 0.453095i \(0.850320\pi\)
\(930\) 2.96746e16 1.39871
\(931\) −3.00633e16 −1.40868
\(932\) −6.47319e16 −3.01530
\(933\) −9.16050e15 −0.424200
\(934\) −3.15656e16 −1.45314
\(935\) −9.12797e15 −0.417744
\(936\) −1.12417e16 −0.511465
\(937\) 6.38433e15 0.288767 0.144383 0.989522i \(-0.453880\pi\)
0.144383 + 0.989522i \(0.453880\pi\)
\(938\) 2.42031e16 1.08832
\(939\) 1.77374e16 0.792919
\(940\) −8.22746e16 −3.65648
\(941\) 3.29317e16 1.45503 0.727514 0.686093i \(-0.240676\pi\)
0.727514 + 0.686093i \(0.240676\pi\)
\(942\) 4.49335e16 1.97374
\(943\) 4.32869e14 0.0189035
\(944\) −3.59622e15 −0.156135
\(945\) −1.17782e16 −0.508398
\(946\) −5.83493e16 −2.50400
\(947\) −3.81365e16 −1.62711 −0.813553 0.581491i \(-0.802470\pi\)
−0.813553 + 0.581491i \(0.802470\pi\)
\(948\) −3.44723e16 −1.46226
\(949\) 1.03956e16 0.438413
\(950\) 2.03855e16 0.854754
\(951\) −4.35232e15 −0.181438
\(952\) 2.97328e16 1.23235
\(953\) 3.05216e16 1.25776 0.628878 0.777504i \(-0.283515\pi\)
0.628878 + 0.777504i \(0.283515\pi\)
\(954\) −1.55137e15 −0.0635622
\(955\) −3.35262e16 −1.36573
\(956\) −8.68902e16 −3.51927
\(957\) −2.87298e15 −0.115696
\(958\) −2.52586e16 −1.01135
\(959\) 2.17077e16 0.864193
\(960\) 1.59038e16 0.629519
\(961\) −1.29429e13 −0.000509394 0
\(962\) −5.24348e15 −0.205190
\(963\) −1.46294e16 −0.569221
\(964\) 3.68904e16 1.42722
\(965\) −9.58435e15 −0.368691
\(966\) 6.02837e15 0.230582
\(967\) −2.94657e15 −0.112065 −0.0560327 0.998429i \(-0.517845\pi\)
−0.0560327 + 0.998429i \(0.517845\pi\)
\(968\) 1.18564e16 0.448371
\(969\) −2.83442e15 −0.106582
\(970\) −2.34053e16 −0.875122
\(971\) 1.54015e16 0.572607 0.286304 0.958139i \(-0.407573\pi\)
0.286304 + 0.958139i \(0.407573\pi\)
\(972\) −3.58332e15 −0.132471
\(973\) 6.33204e16 2.32768
\(974\) 4.18174e15 0.152856
\(975\) −1.19718e16 −0.435146
\(976\) −4.37299e16 −1.58054
\(977\) −6.06880e14 −0.0218114 −0.0109057 0.999941i \(-0.503471\pi\)
−0.0109057 + 0.999941i \(0.503471\pi\)
\(978\) −2.19258e16 −0.783595
\(979\) 2.82186e16 1.00284
\(980\) 2.13735e17 7.55321
\(981\) 9.34516e15 0.328403
\(982\) −1.22320e15 −0.0427450
\(983\) 9.73684e15 0.338356 0.169178 0.985586i \(-0.445889\pi\)
0.169178 + 0.985586i \(0.445889\pi\)
\(984\) 4.92678e15 0.170251
\(985\) −1.20158e16 −0.412907
\(986\) 4.07943e15 0.139405
\(987\) 4.14802e16 1.40960
\(988\) −2.68064e16 −0.905890
\(989\) 5.83703e15 0.196161
\(990\) −2.10642e16 −0.703965
\(991\) 2.00885e16 0.667640 0.333820 0.942637i \(-0.391662\pi\)
0.333820 + 0.942637i \(0.391662\pi\)
\(992\) −7.11063e15 −0.235014
\(993\) 6.56377e15 0.215741
\(994\) −1.19286e17 −3.89909
\(995\) −2.67610e16 −0.869912
\(996\) 5.36434e16 1.73416
\(997\) −3.71564e16 −1.19457 −0.597283 0.802030i \(-0.703753\pi\)
−0.597283 + 0.802030i \(0.703753\pi\)
\(998\) −7.41430e16 −2.37057
\(999\) −8.61999e14 −0.0274092
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 177.12.a.c.1.3 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.c.1.3 27 1.1 even 1 trivial