Properties

Label 177.12.a.c.1.9
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 1.9
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-41.5867 q^{2} -243.000 q^{3} -318.543 q^{4} +9657.75 q^{5} +10105.6 q^{6} -11578.9 q^{7} +98416.8 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-41.5867 q^{2} -243.000 q^{3} -318.543 q^{4} +9657.75 q^{5} +10105.6 q^{6} -11578.9 q^{7} +98416.8 q^{8} +59049.0 q^{9} -401634. q^{10} -521143. q^{11} +77405.9 q^{12} +2.19264e6 q^{13} +481530. q^{14} -2.34683e6 q^{15} -3.44046e6 q^{16} -5.78339e6 q^{17} -2.45566e6 q^{18} +1.37354e7 q^{19} -3.07641e6 q^{20} +2.81368e6 q^{21} +2.16727e7 q^{22} +5.21100e7 q^{23} -2.39153e7 q^{24} +4.44440e7 q^{25} -9.11849e7 q^{26} -1.43489e7 q^{27} +3.68839e6 q^{28} +1.77755e8 q^{29} +9.75971e7 q^{30} -1.22823e8 q^{31} -5.84802e7 q^{32} +1.26638e8 q^{33} +2.40513e8 q^{34} -1.11826e8 q^{35} -1.88096e7 q^{36} +3.62082e8 q^{37} -5.71212e8 q^{38} -5.32812e8 q^{39} +9.50485e8 q^{40} +1.34909e9 q^{41} -1.17012e8 q^{42} +1.17172e9 q^{43} +1.66007e8 q^{44} +5.70280e8 q^{45} -2.16709e9 q^{46} +5.28251e8 q^{47} +8.36031e8 q^{48} -1.84325e9 q^{49} -1.84828e9 q^{50} +1.40536e9 q^{51} -6.98451e8 q^{52} -5.18449e9 q^{53} +5.96724e8 q^{54} -5.03307e9 q^{55} -1.13956e9 q^{56} -3.33771e9 q^{57} -7.39224e9 q^{58} -7.14924e8 q^{59} +7.47567e8 q^{60} -9.00303e9 q^{61} +5.10781e9 q^{62} -6.83725e8 q^{63} +9.47806e9 q^{64} +2.11760e10 q^{65} -5.26646e9 q^{66} +5.11986e9 q^{67} +1.84226e9 q^{68} -1.26627e10 q^{69} +4.65050e9 q^{70} -2.99744e9 q^{71} +5.81141e9 q^{72} -2.74521e10 q^{73} -1.50578e10 q^{74} -1.07999e10 q^{75} -4.37533e9 q^{76} +6.03429e9 q^{77} +2.21579e10 q^{78} +2.06456e10 q^{79} -3.32271e10 q^{80} +3.48678e9 q^{81} -5.61044e10 q^{82} +3.14626e10 q^{83} -8.96278e8 q^{84} -5.58546e10 q^{85} -4.87280e10 q^{86} -4.31944e10 q^{87} -5.12893e10 q^{88} +2.32505e10 q^{89} -2.37161e10 q^{90} -2.53885e10 q^{91} -1.65993e10 q^{92} +2.98460e10 q^{93} -2.19682e10 q^{94} +1.32653e11 q^{95} +1.42107e10 q^{96} -1.39153e11 q^{97} +7.66550e10 q^{98} -3.07730e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 46q^{2} - 6561q^{3} + 26142q^{4} - 2442q^{5} + 11178q^{6} + 170093q^{7} - 19341q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 46q^{2} - 6561q^{3} + 26142q^{4} - 2442q^{5} + 11178q^{6} + 170093q^{7} - 19341q^{8} + 1594323q^{9} + 140249q^{10} + 256992q^{11} - 6352506q^{12} + 2436978q^{13} + 5233061q^{14} + 593406q^{15} + 28295194q^{16} - 4565351q^{17} - 2716254q^{18} + 33607699q^{19} - 19208463q^{20} - 41332599q^{21} + 79735622q^{22} + 43966161q^{23} + 4699863q^{24} + 406675819q^{25} + 42605404q^{26} - 387420489q^{27} + 635747682q^{28} - 107217773q^{29} - 34080507q^{30} + 570926627q^{31} + 526569236q^{32} - 62449056q^{33} + 129790240q^{34} + 134356079q^{35} + 1543658958q^{36} - 107121371q^{37} + 208302581q^{38} - 592185654q^{39} - 958762162q^{40} - 1935967559q^{41} - 1271633823q^{42} + 1725943824q^{43} + 196885756q^{44} - 144197658q^{45} - 13265966407q^{46} + 1801256065q^{47} - 6875732142q^{48} + 10484289252q^{49} - 10067682271q^{50} + 1109380293q^{51} - 882697024q^{52} - 6214238922q^{53} + 660049722q^{54} + 4460552366q^{55} + 28328012310q^{56} - 8166670857q^{57} + 12220116750q^{58} - 19302956073q^{59} + 4667656509q^{60} + 13167821039q^{61} - 1162130230q^{62} + 10043821557q^{63} - 5337557395q^{64} - 16849896006q^{65} - 19375756146q^{66} - 16856763152q^{67} - 36171071977q^{68} - 10683777123q^{69} - 120177261588q^{70} - 5198545690q^{71} - 1142066709q^{72} - 25075321857q^{73} - 182979651978q^{74} - 98822224017q^{75} - 3501293988q^{76} - 42787697701q^{77} - 10353113172q^{78} + 6850314702q^{79} - 261464428159q^{80} + 94143178827q^{81} - 148881516273q^{82} + 30908370899q^{83} - 154486686726q^{84} - 49419624969q^{85} - 220725475224q^{86} + 26053918839q^{87} - 53091280787q^{88} + 28988060121q^{89} + 8281563201q^{90} + 97120614047q^{91} + 45374597708q^{92} - 138735170361q^{93} + 208966927220q^{94} - 125253904969q^{95} - 127956324348q^{96} + 367722840268q^{97} - 48265639912q^{98} + 15175120608q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −41.5867 −0.918946 −0.459473 0.888192i \(-0.651962\pi\)
−0.459473 + 0.888192i \(0.651962\pi\)
\(3\) −243.000 −0.577350
\(4\) −318.543 −0.155539
\(5\) 9657.75 1.38210 0.691052 0.722805i \(-0.257147\pi\)
0.691052 + 0.722805i \(0.257147\pi\)
\(6\) 10105.6 0.530554
\(7\) −11578.9 −0.260393 −0.130197 0.991488i \(-0.541561\pi\)
−0.130197 + 0.991488i \(0.541561\pi\)
\(8\) 98416.8 1.06188
\(9\) 59049.0 0.333333
\(10\) −401634. −1.27008
\(11\) −521143. −0.975658 −0.487829 0.872939i \(-0.662211\pi\)
−0.487829 + 0.872939i \(0.662211\pi\)
\(12\) 77405.9 0.0898002
\(13\) 2.19264e6 1.63787 0.818935 0.573886i \(-0.194565\pi\)
0.818935 + 0.573886i \(0.194565\pi\)
\(14\) 481530. 0.239287
\(15\) −2.34683e6 −0.797958
\(16\) −3.44046e6 −0.820269
\(17\) −5.78339e6 −0.987902 −0.493951 0.869490i \(-0.664448\pi\)
−0.493951 + 0.869490i \(0.664448\pi\)
\(18\) −2.45566e6 −0.306315
\(19\) 1.37354e7 1.27262 0.636309 0.771435i \(-0.280460\pi\)
0.636309 + 0.771435i \(0.280460\pi\)
\(20\) −3.07641e6 −0.214970
\(21\) 2.81368e6 0.150338
\(22\) 2.16727e7 0.896577
\(23\) 5.21100e7 1.68818 0.844089 0.536203i \(-0.180142\pi\)
0.844089 + 0.536203i \(0.180142\pi\)
\(24\) −2.39153e7 −0.613075
\(25\) 4.44440e7 0.910213
\(26\) −9.11849e7 −1.50511
\(27\) −1.43489e7 −0.192450
\(28\) 3.68839e6 0.0405012
\(29\) 1.77755e8 1.60928 0.804641 0.593761i \(-0.202358\pi\)
0.804641 + 0.593761i \(0.202358\pi\)
\(30\) 9.75971e7 0.733281
\(31\) −1.22823e8 −0.770531 −0.385266 0.922806i \(-0.625890\pi\)
−0.385266 + 0.922806i \(0.625890\pi\)
\(32\) −5.84802e7 −0.308094
\(33\) 1.26638e8 0.563296
\(34\) 2.40513e8 0.907828
\(35\) −1.11826e8 −0.359890
\(36\) −1.88096e7 −0.0518462
\(37\) 3.62082e8 0.858414 0.429207 0.903206i \(-0.358793\pi\)
0.429207 + 0.903206i \(0.358793\pi\)
\(38\) −5.71212e8 −1.16947
\(39\) −5.32812e8 −0.945625
\(40\) 9.50485e8 1.46763
\(41\) 1.34909e9 1.81857 0.909286 0.416172i \(-0.136628\pi\)
0.909286 + 0.416172i \(0.136628\pi\)
\(42\) −1.17012e8 −0.138152
\(43\) 1.17172e9 1.21548 0.607739 0.794136i \(-0.292076\pi\)
0.607739 + 0.794136i \(0.292076\pi\)
\(44\) 1.66007e8 0.151752
\(45\) 5.70280e8 0.460701
\(46\) −2.16709e9 −1.55134
\(47\) 5.28251e8 0.335971 0.167986 0.985789i \(-0.446274\pi\)
0.167986 + 0.985789i \(0.446274\pi\)
\(48\) 8.36031e8 0.473583
\(49\) −1.84325e9 −0.932195
\(50\) −1.84828e9 −0.836436
\(51\) 1.40536e9 0.570365
\(52\) −6.98451e8 −0.254752
\(53\) −5.18449e9 −1.70290 −0.851450 0.524436i \(-0.824276\pi\)
−0.851450 + 0.524436i \(0.824276\pi\)
\(54\) 5.96724e8 0.176851
\(55\) −5.03307e9 −1.34846
\(56\) −1.13956e9 −0.276506
\(57\) −3.33771e9 −0.734746
\(58\) −7.39224e9 −1.47884
\(59\) −7.14924e8 −0.130189
\(60\) 7.47567e8 0.124113
\(61\) −9.00303e9 −1.36482 −0.682409 0.730970i \(-0.739068\pi\)
−0.682409 + 0.730970i \(0.739068\pi\)
\(62\) 5.10781e9 0.708076
\(63\) −6.83725e8 −0.0867977
\(64\) 9.47806e9 1.10339
\(65\) 2.11760e10 2.26371
\(66\) −5.26646e9 −0.517639
\(67\) 5.11986e9 0.463284 0.231642 0.972801i \(-0.425590\pi\)
0.231642 + 0.972801i \(0.425590\pi\)
\(68\) 1.84226e9 0.153657
\(69\) −1.26627e10 −0.974670
\(70\) 4.65050e9 0.330720
\(71\) −2.99744e9 −0.197165 −0.0985823 0.995129i \(-0.531431\pi\)
−0.0985823 + 0.995129i \(0.531431\pi\)
\(72\) 5.81141e9 0.353959
\(73\) −2.74521e10 −1.54989 −0.774943 0.632032i \(-0.782221\pi\)
−0.774943 + 0.632032i \(0.782221\pi\)
\(74\) −1.50578e10 −0.788836
\(75\) −1.07999e10 −0.525512
\(76\) −4.37533e9 −0.197941
\(77\) 6.03429e9 0.254055
\(78\) 2.21579e10 0.868978
\(79\) 2.06456e10 0.754882 0.377441 0.926034i \(-0.376804\pi\)
0.377441 + 0.926034i \(0.376804\pi\)
\(80\) −3.32271e10 −1.13370
\(81\) 3.48678e9 0.111111
\(82\) −5.61044e10 −1.67117
\(83\) 3.14626e10 0.876729 0.438365 0.898797i \(-0.355558\pi\)
0.438365 + 0.898797i \(0.355558\pi\)
\(84\) −8.96278e8 −0.0233834
\(85\) −5.58546e10 −1.36538
\(86\) −4.87280e10 −1.11696
\(87\) −4.31944e10 −0.929120
\(88\) −5.12893e10 −1.03603
\(89\) 2.32505e10 0.441355 0.220678 0.975347i \(-0.429173\pi\)
0.220678 + 0.975347i \(0.429173\pi\)
\(90\) −2.37161e10 −0.423360
\(91\) −2.53885e10 −0.426490
\(92\) −1.65993e10 −0.262577
\(93\) 2.98460e10 0.444866
\(94\) −2.19682e10 −0.308739
\(95\) 1.32653e11 1.75889
\(96\) 1.42107e10 0.177878
\(97\) −1.39153e11 −1.64531 −0.822656 0.568539i \(-0.807509\pi\)
−0.822656 + 0.568539i \(0.807509\pi\)
\(98\) 7.66550e10 0.856637
\(99\) −3.07730e10 −0.325219
\(100\) −1.41573e10 −0.141573
\(101\) −4.73242e10 −0.448039 −0.224020 0.974585i \(-0.571918\pi\)
−0.224020 + 0.974585i \(0.571918\pi\)
\(102\) −5.84446e10 −0.524135
\(103\) 1.63494e11 1.38962 0.694812 0.719192i \(-0.255488\pi\)
0.694812 + 0.719192i \(0.255488\pi\)
\(104\) 2.15793e11 1.73922
\(105\) 2.71738e10 0.207783
\(106\) 2.15606e11 1.56487
\(107\) −4.94181e10 −0.340624 −0.170312 0.985390i \(-0.554478\pi\)
−0.170312 + 0.985390i \(0.554478\pi\)
\(108\) 4.57074e9 0.0299334
\(109\) −6.78251e10 −0.422226 −0.211113 0.977462i \(-0.567709\pi\)
−0.211113 + 0.977462i \(0.567709\pi\)
\(110\) 2.09309e11 1.23916
\(111\) −8.79858e10 −0.495606
\(112\) 3.98369e10 0.213592
\(113\) −1.69239e11 −0.864110 −0.432055 0.901847i \(-0.642211\pi\)
−0.432055 + 0.901847i \(0.642211\pi\)
\(114\) 1.38805e11 0.675192
\(115\) 5.03266e11 2.33324
\(116\) −5.66225e10 −0.250305
\(117\) 1.29473e11 0.545957
\(118\) 2.97314e10 0.119637
\(119\) 6.69656e10 0.257243
\(120\) −2.30968e11 −0.847334
\(121\) −1.37211e10 −0.0480917
\(122\) 3.74407e11 1.25419
\(123\) −3.27829e11 −1.04995
\(124\) 3.91244e10 0.119847
\(125\) −4.23410e10 −0.124095
\(126\) 2.84339e10 0.0797624
\(127\) 8.54400e10 0.229478 0.114739 0.993396i \(-0.463397\pi\)
0.114739 + 0.993396i \(0.463397\pi\)
\(128\) −2.74394e11 −0.705863
\(129\) −2.84728e11 −0.701757
\(130\) −8.80641e11 −2.08023
\(131\) −2.76029e11 −0.625118 −0.312559 0.949898i \(-0.601186\pi\)
−0.312559 + 0.949898i \(0.601186\pi\)
\(132\) −4.03396e10 −0.0876143
\(133\) −1.59042e11 −0.331381
\(134\) −2.12918e11 −0.425733
\(135\) −1.38578e11 −0.265986
\(136\) −5.69183e11 −1.04903
\(137\) 4.55733e11 0.806765 0.403383 0.915031i \(-0.367834\pi\)
0.403383 + 0.915031i \(0.367834\pi\)
\(138\) 5.26602e11 0.895669
\(139\) −2.37619e11 −0.388418 −0.194209 0.980960i \(-0.562214\pi\)
−0.194209 + 0.980960i \(0.562214\pi\)
\(140\) 3.56215e10 0.0559768
\(141\) −1.28365e11 −0.193973
\(142\) 1.24654e11 0.181184
\(143\) −1.14268e12 −1.59800
\(144\) −2.03156e11 −0.273423
\(145\) 1.71671e12 2.22420
\(146\) 1.14164e12 1.42426
\(147\) 4.47911e11 0.538203
\(148\) −1.15339e11 −0.133516
\(149\) 4.93394e11 0.550388 0.275194 0.961389i \(-0.411258\pi\)
0.275194 + 0.961389i \(0.411258\pi\)
\(150\) 4.49132e11 0.482917
\(151\) 1.58009e12 1.63798 0.818991 0.573806i \(-0.194534\pi\)
0.818991 + 0.573806i \(0.194534\pi\)
\(152\) 1.35180e12 1.35136
\(153\) −3.41504e11 −0.329301
\(154\) −2.50946e11 −0.233462
\(155\) −1.18619e12 −1.06495
\(156\) 1.69724e11 0.147081
\(157\) 5.92635e11 0.495837 0.247919 0.968781i \(-0.420253\pi\)
0.247919 + 0.968781i \(0.420253\pi\)
\(158\) −8.58585e11 −0.693696
\(159\) 1.25983e12 0.983170
\(160\) −5.64787e11 −0.425818
\(161\) −6.03379e11 −0.439590
\(162\) −1.45004e11 −0.102105
\(163\) −5.95948e11 −0.405674 −0.202837 0.979213i \(-0.565016\pi\)
−0.202837 + 0.979213i \(0.565016\pi\)
\(164\) −4.29744e11 −0.282858
\(165\) 1.22304e12 0.778534
\(166\) −1.30843e12 −0.805667
\(167\) −3.42070e11 −0.203786 −0.101893 0.994795i \(-0.532490\pi\)
−0.101893 + 0.994795i \(0.532490\pi\)
\(168\) 2.76914e11 0.159641
\(169\) 3.01552e12 1.68262
\(170\) 2.32281e12 1.25471
\(171\) 8.11064e11 0.424206
\(172\) −3.73243e11 −0.189054
\(173\) 2.76695e12 1.35753 0.678763 0.734357i \(-0.262516\pi\)
0.678763 + 0.734357i \(0.262516\pi\)
\(174\) 1.79631e12 0.853811
\(175\) −5.14614e11 −0.237013
\(176\) 1.79297e12 0.800302
\(177\) 1.73727e11 0.0751646
\(178\) −9.66915e11 −0.405581
\(179\) 8.79091e11 0.357555 0.178777 0.983890i \(-0.442786\pi\)
0.178777 + 0.983890i \(0.442786\pi\)
\(180\) −1.81659e11 −0.0716568
\(181\) 1.33882e12 0.512258 0.256129 0.966643i \(-0.417553\pi\)
0.256129 + 0.966643i \(0.417553\pi\)
\(182\) 1.05582e12 0.391921
\(183\) 2.18774e12 0.787978
\(184\) 5.12850e12 1.79264
\(185\) 3.49689e12 1.18642
\(186\) −1.24120e12 −0.408808
\(187\) 3.01398e12 0.963854
\(188\) −1.68271e11 −0.0522565
\(189\) 1.66145e11 0.0501127
\(190\) −5.51662e12 −1.61632
\(191\) −2.43082e12 −0.691941 −0.345971 0.938245i \(-0.612450\pi\)
−0.345971 + 0.938245i \(0.612450\pi\)
\(192\) −2.30317e12 −0.637043
\(193\) −4.51748e12 −1.21431 −0.607157 0.794582i \(-0.707690\pi\)
−0.607157 + 0.794582i \(0.707690\pi\)
\(194\) 5.78692e12 1.51195
\(195\) −5.14577e12 −1.30695
\(196\) 5.87156e11 0.144992
\(197\) 1.97497e12 0.474238 0.237119 0.971481i \(-0.423797\pi\)
0.237119 + 0.971481i \(0.423797\pi\)
\(198\) 1.27975e12 0.298859
\(199\) 5.67332e12 1.28868 0.644340 0.764739i \(-0.277132\pi\)
0.644340 + 0.764739i \(0.277132\pi\)
\(200\) 4.37403e12 0.966534
\(201\) −1.24413e12 −0.267477
\(202\) 1.96806e12 0.411724
\(203\) −2.05821e12 −0.419046
\(204\) −4.47669e11 −0.0887138
\(205\) 1.30292e13 2.51346
\(206\) −6.79918e12 −1.27699
\(207\) 3.07705e12 0.562726
\(208\) −7.54370e12 −1.34349
\(209\) −7.15814e12 −1.24164
\(210\) −1.13007e12 −0.190941
\(211\) 5.87075e12 0.966363 0.483181 0.875520i \(-0.339481\pi\)
0.483181 + 0.875520i \(0.339481\pi\)
\(212\) 1.65148e12 0.264867
\(213\) 7.28377e11 0.113833
\(214\) 2.05514e12 0.313015
\(215\) 1.13162e13 1.67992
\(216\) −1.41217e12 −0.204358
\(217\) 1.42216e12 0.200641
\(218\) 2.82063e12 0.388003
\(219\) 6.67086e12 0.894827
\(220\) 1.60325e12 0.209738
\(221\) −1.26809e13 −1.61806
\(222\) 3.65904e12 0.455435
\(223\) 5.76894e12 0.700518 0.350259 0.936653i \(-0.386094\pi\)
0.350259 + 0.936653i \(0.386094\pi\)
\(224\) 6.77138e11 0.0802256
\(225\) 2.62437e12 0.303404
\(226\) 7.03810e12 0.794070
\(227\) −1.03297e13 −1.13749 −0.568744 0.822515i \(-0.692570\pi\)
−0.568744 + 0.822515i \(0.692570\pi\)
\(228\) 1.06320e12 0.114281
\(229\) 7.46366e12 0.783171 0.391586 0.920142i \(-0.371927\pi\)
0.391586 + 0.920142i \(0.371927\pi\)
\(230\) −2.09292e13 −2.14412
\(231\) −1.46633e12 −0.146678
\(232\) 1.74941e13 1.70886
\(233\) −1.19773e13 −1.14262 −0.571309 0.820735i \(-0.693564\pi\)
−0.571309 + 0.820735i \(0.693564\pi\)
\(234\) −5.38438e12 −0.501705
\(235\) 5.10171e12 0.464347
\(236\) 2.27734e11 0.0202494
\(237\) −5.01689e12 −0.435832
\(238\) −2.78488e12 −0.236392
\(239\) 1.09656e13 0.909585 0.454792 0.890598i \(-0.349713\pi\)
0.454792 + 0.890598i \(0.349713\pi\)
\(240\) 8.07418e12 0.654541
\(241\) −5.17342e12 −0.409906 −0.204953 0.978772i \(-0.565704\pi\)
−0.204953 + 0.978772i \(0.565704\pi\)
\(242\) 5.70617e11 0.0441937
\(243\) −8.47289e11 −0.0641500
\(244\) 2.86785e12 0.212282
\(245\) −1.78017e13 −1.28839
\(246\) 1.36334e13 0.964850
\(247\) 3.01169e13 2.08438
\(248\) −1.20878e13 −0.818209
\(249\) −7.64542e12 −0.506180
\(250\) 1.76082e12 0.114037
\(251\) 1.53032e13 0.969567 0.484784 0.874634i \(-0.338898\pi\)
0.484784 + 0.874634i \(0.338898\pi\)
\(252\) 2.17796e11 0.0135004
\(253\) −2.71568e13 −1.64708
\(254\) −3.55317e12 −0.210878
\(255\) 1.35727e13 0.788305
\(256\) −7.99990e12 −0.454742
\(257\) −1.06963e13 −0.595116 −0.297558 0.954704i \(-0.596172\pi\)
−0.297558 + 0.954704i \(0.596172\pi\)
\(258\) 1.18409e13 0.644877
\(259\) −4.19252e12 −0.223525
\(260\) −6.74546e12 −0.352094
\(261\) 1.04962e13 0.536427
\(262\) 1.14791e13 0.574450
\(263\) 7.37052e12 0.361195 0.180597 0.983557i \(-0.442197\pi\)
0.180597 + 0.983557i \(0.442197\pi\)
\(264\) 1.24633e13 0.598152
\(265\) −5.00705e13 −2.35359
\(266\) 6.61403e12 0.304521
\(267\) −5.64988e12 −0.254816
\(268\) −1.63090e12 −0.0720585
\(269\) 3.47705e13 1.50513 0.752564 0.658519i \(-0.228817\pi\)
0.752564 + 0.658519i \(0.228817\pi\)
\(270\) 5.76301e12 0.244427
\(271\) −6.38370e12 −0.265303 −0.132651 0.991163i \(-0.542349\pi\)
−0.132651 + 0.991163i \(0.542349\pi\)
\(272\) 1.98975e13 0.810345
\(273\) 6.16940e12 0.246234
\(274\) −1.89524e13 −0.741374
\(275\) −2.31617e13 −0.888056
\(276\) 4.03362e12 0.151599
\(277\) 2.57063e13 0.947112 0.473556 0.880764i \(-0.342970\pi\)
0.473556 + 0.880764i \(0.342970\pi\)
\(278\) 9.88179e12 0.356935
\(279\) −7.25257e12 −0.256844
\(280\) −1.10056e13 −0.382159
\(281\) −3.20177e13 −1.09020 −0.545098 0.838372i \(-0.683508\pi\)
−0.545098 + 0.838372i \(0.683508\pi\)
\(282\) 5.33828e12 0.178251
\(283\) −1.70757e13 −0.559182 −0.279591 0.960119i \(-0.590199\pi\)
−0.279591 + 0.960119i \(0.590199\pi\)
\(284\) 9.54812e11 0.0306667
\(285\) −3.22348e13 −1.01550
\(286\) 4.75204e13 1.46848
\(287\) −1.56211e13 −0.473544
\(288\) −3.45319e12 −0.102698
\(289\) −8.24239e11 −0.0240500
\(290\) −7.13924e13 −2.04392
\(291\) 3.38142e13 0.949922
\(292\) 8.74466e12 0.241067
\(293\) −4.38934e13 −1.18748 −0.593742 0.804656i \(-0.702350\pi\)
−0.593742 + 0.804656i \(0.702350\pi\)
\(294\) −1.86272e13 −0.494580
\(295\) −6.90456e12 −0.179935
\(296\) 3.56349e13 0.911531
\(297\) 7.47784e12 0.187765
\(298\) −2.05186e13 −0.505777
\(299\) 1.14259e14 2.76502
\(300\) 3.44023e12 0.0817373
\(301\) −1.35673e13 −0.316502
\(302\) −6.57109e13 −1.50522
\(303\) 1.14998e13 0.258676
\(304\) −4.72562e13 −1.04389
\(305\) −8.69490e13 −1.88632
\(306\) 1.42020e13 0.302609
\(307\) 6.67684e13 1.39737 0.698683 0.715432i \(-0.253770\pi\)
0.698683 + 0.715432i \(0.253770\pi\)
\(308\) −1.92218e12 −0.0395153
\(309\) −3.97291e13 −0.802300
\(310\) 4.93299e13 0.978635
\(311\) −3.26129e13 −0.635634 −0.317817 0.948152i \(-0.602950\pi\)
−0.317817 + 0.948152i \(0.602950\pi\)
\(312\) −5.24377e13 −1.00414
\(313\) 2.01566e13 0.379247 0.189624 0.981857i \(-0.439273\pi\)
0.189624 + 0.981857i \(0.439273\pi\)
\(314\) −2.46458e13 −0.455648
\(315\) −6.60324e12 −0.119963
\(316\) −6.57652e12 −0.117413
\(317\) 8.99054e12 0.157746 0.0788732 0.996885i \(-0.474868\pi\)
0.0788732 + 0.996885i \(0.474868\pi\)
\(318\) −5.23923e13 −0.903480
\(319\) −9.26357e13 −1.57011
\(320\) 9.15367e13 1.52500
\(321\) 1.20086e13 0.196660
\(322\) 2.50926e13 0.403959
\(323\) −7.94375e13 −1.25722
\(324\) −1.11069e12 −0.0172821
\(325\) 9.74498e13 1.49081
\(326\) 2.47836e13 0.372792
\(327\) 1.64815e13 0.243772
\(328\) 1.32773e14 1.93110
\(329\) −6.11658e12 −0.0874846
\(330\) −5.08621e13 −0.715431
\(331\) 1.91983e13 0.265588 0.132794 0.991144i \(-0.457605\pi\)
0.132794 + 0.991144i \(0.457605\pi\)
\(332\) −1.00222e13 −0.136365
\(333\) 2.13806e13 0.286138
\(334\) 1.42256e13 0.187268
\(335\) 4.94464e13 0.640306
\(336\) −9.68035e12 −0.123318
\(337\) −1.52067e14 −1.90577 −0.952884 0.303334i \(-0.901900\pi\)
−0.952884 + 0.303334i \(0.901900\pi\)
\(338\) −1.25406e14 −1.54624
\(339\) 4.11251e13 0.498894
\(340\) 1.77921e13 0.212370
\(341\) 6.40084e13 0.751775
\(342\) −3.37295e13 −0.389822
\(343\) 4.42383e13 0.503130
\(344\) 1.15317e14 1.29069
\(345\) −1.22294e14 −1.34710
\(346\) −1.15069e14 −1.24749
\(347\) −2.53920e13 −0.270948 −0.135474 0.990781i \(-0.543256\pi\)
−0.135474 + 0.990781i \(0.543256\pi\)
\(348\) 1.37593e13 0.144514
\(349\) −1.71683e14 −1.77495 −0.887477 0.460851i \(-0.847544\pi\)
−0.887477 + 0.460851i \(0.847544\pi\)
\(350\) 2.14011e13 0.217802
\(351\) −3.14620e13 −0.315208
\(352\) 3.04766e13 0.300595
\(353\) 1.59851e13 0.155223 0.0776114 0.996984i \(-0.475271\pi\)
0.0776114 + 0.996984i \(0.475271\pi\)
\(354\) −7.22472e12 −0.0690722
\(355\) −2.89485e13 −0.272502
\(356\) −7.40630e12 −0.0686477
\(357\) −1.62726e13 −0.148519
\(358\) −3.65586e13 −0.328573
\(359\) −9.11753e13 −0.806970 −0.403485 0.914986i \(-0.632201\pi\)
−0.403485 + 0.914986i \(0.632201\pi\)
\(360\) 5.61252e13 0.489208
\(361\) 7.21721e13 0.619555
\(362\) −5.56770e13 −0.470737
\(363\) 3.33423e12 0.0277658
\(364\) 8.08732e12 0.0663356
\(365\) −2.65125e14 −2.14210
\(366\) −9.09809e13 −0.724109
\(367\) 2.08132e14 1.63184 0.815918 0.578168i \(-0.196232\pi\)
0.815918 + 0.578168i \(0.196232\pi\)
\(368\) −1.79282e14 −1.38476
\(369\) 7.96626e13 0.606191
\(370\) −1.45424e14 −1.09025
\(371\) 6.00309e13 0.443423
\(372\) −9.50722e12 −0.0691938
\(373\) −7.64019e13 −0.547905 −0.273953 0.961743i \(-0.588331\pi\)
−0.273953 + 0.961743i \(0.588331\pi\)
\(374\) −1.25342e14 −0.885730
\(375\) 1.02889e13 0.0716466
\(376\) 5.19888e13 0.356760
\(377\) 3.89753e14 2.63580
\(378\) −6.90943e12 −0.0460508
\(379\) 2.62872e14 1.72675 0.863373 0.504566i \(-0.168348\pi\)
0.863373 + 0.504566i \(0.168348\pi\)
\(380\) −4.22558e13 −0.273575
\(381\) −2.07619e13 −0.132489
\(382\) 1.01090e14 0.635857
\(383\) 8.49676e12 0.0526817 0.0263409 0.999653i \(-0.491614\pi\)
0.0263409 + 0.999653i \(0.491614\pi\)
\(384\) 6.66778e13 0.407530
\(385\) 5.82776e13 0.351130
\(386\) 1.87867e14 1.11589
\(387\) 6.91889e13 0.405160
\(388\) 4.43262e13 0.255909
\(389\) 1.44073e14 0.820085 0.410042 0.912066i \(-0.365514\pi\)
0.410042 + 0.912066i \(0.365514\pi\)
\(390\) 2.13996e14 1.20102
\(391\) −3.01373e14 −1.66775
\(392\) −1.81407e14 −0.989877
\(393\) 6.70750e13 0.360912
\(394\) −8.21326e13 −0.435799
\(395\) 1.99390e14 1.04333
\(396\) 9.80252e12 0.0505841
\(397\) 2.01462e14 1.02529 0.512643 0.858602i \(-0.328666\pi\)
0.512643 + 0.858602i \(0.328666\pi\)
\(398\) −2.35935e14 −1.18423
\(399\) 3.86472e13 0.191323
\(400\) −1.52908e14 −0.746619
\(401\) −3.32883e14 −1.60323 −0.801617 0.597838i \(-0.796027\pi\)
−0.801617 + 0.597838i \(0.796027\pi\)
\(402\) 5.17392e13 0.245797
\(403\) −2.69307e14 −1.26203
\(404\) 1.50748e13 0.0696874
\(405\) 3.36745e13 0.153567
\(406\) 8.55943e13 0.385081
\(407\) −1.88696e14 −0.837519
\(408\) 1.38312e14 0.605658
\(409\) 4.12130e14 1.78056 0.890280 0.455414i \(-0.150509\pi\)
0.890280 + 0.455414i \(0.150509\pi\)
\(410\) −5.41842e14 −2.30973
\(411\) −1.10743e14 −0.465786
\(412\) −5.20799e13 −0.216140
\(413\) 8.27806e12 0.0339003
\(414\) −1.27964e14 −0.517115
\(415\) 3.03858e14 1.21173
\(416\) −1.28226e14 −0.504619
\(417\) 5.77413e13 0.224253
\(418\) 2.97684e14 1.14100
\(419\) 4.57021e14 1.72886 0.864429 0.502755i \(-0.167680\pi\)
0.864429 + 0.502755i \(0.167680\pi\)
\(420\) −8.65603e12 −0.0323182
\(421\) 2.48837e13 0.0916987 0.0458494 0.998948i \(-0.485401\pi\)
0.0458494 + 0.998948i \(0.485401\pi\)
\(422\) −2.44145e14 −0.888035
\(423\) 3.11927e13 0.111990
\(424\) −5.10241e14 −1.80827
\(425\) −2.57037e14 −0.899201
\(426\) −3.02908e13 −0.104606
\(427\) 1.04246e14 0.355389
\(428\) 1.57418e13 0.0529802
\(429\) 2.77672e14 0.922606
\(430\) −4.70603e14 −1.54375
\(431\) 4.08017e14 1.32146 0.660729 0.750625i \(-0.270247\pi\)
0.660729 + 0.750625i \(0.270247\pi\)
\(432\) 4.93668e13 0.157861
\(433\) 5.62181e14 1.77498 0.887489 0.460829i \(-0.152448\pi\)
0.887489 + 0.460829i \(0.152448\pi\)
\(434\) −5.91430e13 −0.184378
\(435\) −4.17161e14 −1.28414
\(436\) 2.16052e13 0.0656724
\(437\) 7.15754e14 2.14840
\(438\) −2.77419e14 −0.822297
\(439\) −1.79597e14 −0.525708 −0.262854 0.964836i \(-0.584664\pi\)
−0.262854 + 0.964836i \(0.584664\pi\)
\(440\) −4.95339e14 −1.43190
\(441\) −1.08842e14 −0.310732
\(442\) 5.27358e14 1.48691
\(443\) 2.18111e14 0.607376 0.303688 0.952771i \(-0.401782\pi\)
0.303688 + 0.952771i \(0.401782\pi\)
\(444\) 2.80273e13 0.0770858
\(445\) 2.24548e14 0.609999
\(446\) −2.39911e14 −0.643738
\(447\) −1.19895e14 −0.317767
\(448\) −1.09746e14 −0.287315
\(449\) −6.26011e12 −0.0161893 −0.00809463 0.999967i \(-0.502577\pi\)
−0.00809463 + 0.999967i \(0.502577\pi\)
\(450\) −1.09139e14 −0.278812
\(451\) −7.03071e14 −1.77430
\(452\) 5.39099e13 0.134402
\(453\) −3.83962e14 −0.945689
\(454\) 4.29580e14 1.04529
\(455\) −2.45196e14 −0.589454
\(456\) −3.28487e14 −0.780210
\(457\) −1.30469e13 −0.0306173 −0.0153087 0.999883i \(-0.504873\pi\)
−0.0153087 + 0.999883i \(0.504873\pi\)
\(458\) −3.10389e14 −0.719692
\(459\) 8.29854e13 0.190122
\(460\) −1.60312e14 −0.362908
\(461\) 5.92125e14 1.32452 0.662259 0.749275i \(-0.269598\pi\)
0.662259 + 0.749275i \(0.269598\pi\)
\(462\) 6.09800e13 0.134790
\(463\) −6.22058e14 −1.35874 −0.679368 0.733798i \(-0.737746\pi\)
−0.679368 + 0.733798i \(0.737746\pi\)
\(464\) −6.11558e14 −1.32004
\(465\) 2.88245e14 0.614852
\(466\) 4.98097e14 1.05000
\(467\) −4.68824e14 −0.976712 −0.488356 0.872644i \(-0.662403\pi\)
−0.488356 + 0.872644i \(0.662403\pi\)
\(468\) −4.12428e13 −0.0849173
\(469\) −5.92826e13 −0.120636
\(470\) −2.12164e14 −0.426710
\(471\) −1.44010e14 −0.286272
\(472\) −7.03606e13 −0.138245
\(473\) −6.10634e14 −1.18589
\(474\) 2.08636e14 0.400506
\(475\) 6.10458e14 1.15835
\(476\) −2.13314e13 −0.0400112
\(477\) −3.06139e14 −0.567633
\(478\) −4.56023e14 −0.835859
\(479\) 6.74900e14 1.22291 0.611454 0.791280i \(-0.290585\pi\)
0.611454 + 0.791280i \(0.290585\pi\)
\(480\) 1.37243e14 0.245846
\(481\) 7.93916e14 1.40597
\(482\) 2.15146e14 0.376681
\(483\) 1.46621e14 0.253797
\(484\) 4.37077e12 0.00748012
\(485\) −1.34391e15 −2.27399
\(486\) 3.52360e13 0.0589504
\(487\) 4.12963e14 0.683127 0.341564 0.939859i \(-0.389044\pi\)
0.341564 + 0.939859i \(0.389044\pi\)
\(488\) −8.86050e14 −1.44927
\(489\) 1.44815e14 0.234216
\(490\) 7.40314e14 1.18396
\(491\) 9.20913e14 1.45636 0.728182 0.685383i \(-0.240365\pi\)
0.728182 + 0.685383i \(0.240365\pi\)
\(492\) 1.04428e14 0.163308
\(493\) −1.02803e15 −1.58981
\(494\) −1.25246e15 −1.91543
\(495\) −2.97198e14 −0.449487
\(496\) 4.22567e14 0.632043
\(497\) 3.47071e13 0.0513403
\(498\) 3.17948e14 0.465152
\(499\) 6.50653e14 0.941449 0.470724 0.882280i \(-0.343993\pi\)
0.470724 + 0.882280i \(0.343993\pi\)
\(500\) 1.34874e13 0.0193016
\(501\) 8.31230e13 0.117656
\(502\) −6.36412e14 −0.890980
\(503\) −7.61481e14 −1.05447 −0.527236 0.849719i \(-0.676772\pi\)
−0.527236 + 0.849719i \(0.676772\pi\)
\(504\) −6.72900e13 −0.0921685
\(505\) −4.57046e14 −0.619237
\(506\) 1.12936e15 1.51358
\(507\) −7.32772e14 −0.971461
\(508\) −2.72163e13 −0.0356926
\(509\) 3.56627e14 0.462665 0.231332 0.972875i \(-0.425692\pi\)
0.231332 + 0.972875i \(0.425692\pi\)
\(510\) −5.64443e14 −0.724409
\(511\) 3.17866e14 0.403579
\(512\) 8.94649e14 1.12375
\(513\) −1.97089e14 −0.244915
\(514\) 4.44825e14 0.546879
\(515\) 1.57898e15 1.92060
\(516\) 9.06981e13 0.109150
\(517\) −2.75295e14 −0.327793
\(518\) 1.74353e14 0.205408
\(519\) −6.72370e14 −0.783768
\(520\) 2.08407e15 2.40378
\(521\) −8.99603e14 −1.02670 −0.513350 0.858179i \(-0.671596\pi\)
−0.513350 + 0.858179i \(0.671596\pi\)
\(522\) −4.36504e14 −0.492948
\(523\) −5.90284e14 −0.659632 −0.329816 0.944045i \(-0.606987\pi\)
−0.329816 + 0.944045i \(0.606987\pi\)
\(524\) 8.79270e13 0.0972300
\(525\) 1.25051e14 0.136840
\(526\) −3.06516e14 −0.331919
\(527\) 7.10334e14 0.761209
\(528\) −4.35692e14 −0.462055
\(529\) 1.76265e15 1.84995
\(530\) 2.08227e15 2.16282
\(531\) −4.22156e13 −0.0433963
\(532\) 5.06616e13 0.0515425
\(533\) 2.95808e15 2.97859
\(534\) 2.34960e14 0.234163
\(535\) −4.77268e14 −0.470778
\(536\) 5.03881e14 0.491950
\(537\) −2.13619e14 −0.206434
\(538\) −1.44599e15 −1.38313
\(539\) 9.60600e14 0.909504
\(540\) 4.41431e13 0.0413711
\(541\) 1.22095e15 1.13270 0.566350 0.824165i \(-0.308355\pi\)
0.566350 + 0.824165i \(0.308355\pi\)
\(542\) 2.65477e14 0.243799
\(543\) −3.25332e14 −0.295752
\(544\) 3.38214e14 0.304367
\(545\) −6.55038e14 −0.583560
\(546\) −2.56565e14 −0.226276
\(547\) 6.79223e12 0.00593037 0.00296519 0.999996i \(-0.499056\pi\)
0.00296519 + 0.999996i \(0.499056\pi\)
\(548\) −1.45170e14 −0.125483
\(549\) −5.31620e14 −0.454939
\(550\) 9.63219e14 0.816076
\(551\) 2.44154e15 2.04800
\(552\) −1.24623e15 −1.03498
\(553\) −2.39054e14 −0.196566
\(554\) −1.06904e15 −0.870345
\(555\) −8.49745e14 −0.684979
\(556\) 7.56917e13 0.0604139
\(557\) −1.01178e15 −0.799619 −0.399810 0.916598i \(-0.630924\pi\)
−0.399810 + 0.916598i \(0.630924\pi\)
\(558\) 3.01611e14 0.236025
\(559\) 2.56916e15 1.99080
\(560\) 3.84734e14 0.295207
\(561\) −7.32397e14 −0.556481
\(562\) 1.33151e15 1.00183
\(563\) 2.02865e15 1.51151 0.755755 0.654854i \(-0.227270\pi\)
0.755755 + 0.654854i \(0.227270\pi\)
\(564\) 4.08897e13 0.0301703
\(565\) −1.63447e15 −1.19429
\(566\) 7.10123e14 0.513858
\(567\) −4.03733e13 −0.0289326
\(568\) −2.94998e14 −0.209365
\(569\) 1.08182e15 0.760395 0.380198 0.924905i \(-0.375856\pi\)
0.380198 + 0.924905i \(0.375856\pi\)
\(570\) 1.34054e15 0.933186
\(571\) 7.81982e14 0.539136 0.269568 0.962981i \(-0.413119\pi\)
0.269568 + 0.962981i \(0.413119\pi\)
\(572\) 3.63993e14 0.248551
\(573\) 5.90689e14 0.399493
\(574\) 6.49629e14 0.435161
\(575\) 2.31598e15 1.53660
\(576\) 5.59670e14 0.367797
\(577\) 2.29077e15 1.49113 0.745564 0.666434i \(-0.232180\pi\)
0.745564 + 0.666434i \(0.232180\pi\)
\(578\) 3.42774e13 0.0221006
\(579\) 1.09775e15 0.701084
\(580\) −5.46846e14 −0.345948
\(581\) −3.64304e14 −0.228294
\(582\) −1.40622e15 −0.872926
\(583\) 2.70187e15 1.66145
\(584\) −2.70175e15 −1.64579
\(585\) 1.25042e15 0.754569
\(586\) 1.82538e15 1.09123
\(587\) −1.00599e15 −0.595780 −0.297890 0.954600i \(-0.596283\pi\)
−0.297890 + 0.954600i \(0.596283\pi\)
\(588\) −1.42679e14 −0.0837113
\(589\) −1.68703e15 −0.980591
\(590\) 2.87138e14 0.165350
\(591\) −4.79918e14 −0.273801
\(592\) −1.24573e15 −0.704131
\(593\) −1.68886e15 −0.945787 −0.472894 0.881119i \(-0.656791\pi\)
−0.472894 + 0.881119i \(0.656791\pi\)
\(594\) −3.10979e14 −0.172546
\(595\) 6.46737e14 0.355536
\(596\) −1.57167e14 −0.0856065
\(597\) −1.37862e15 −0.744020
\(598\) −4.75165e15 −2.54090
\(599\) −2.30225e15 −1.21985 −0.609924 0.792460i \(-0.708800\pi\)
−0.609924 + 0.792460i \(0.708800\pi\)
\(600\) −1.06289e15 −0.558029
\(601\) −2.09855e15 −1.09172 −0.545859 0.837877i \(-0.683796\pi\)
−0.545859 + 0.837877i \(0.683796\pi\)
\(602\) 5.64219e14 0.290848
\(603\) 3.02323e14 0.154428
\(604\) −5.03327e14 −0.254769
\(605\) −1.32515e14 −0.0664678
\(606\) −4.78239e14 −0.237709
\(607\) 1.77589e15 0.874739 0.437370 0.899282i \(-0.355910\pi\)
0.437370 + 0.899282i \(0.355910\pi\)
\(608\) −8.03251e14 −0.392086
\(609\) 5.00145e14 0.241936
\(610\) 3.61593e15 1.73343
\(611\) 1.15827e15 0.550277
\(612\) 1.08784e14 0.0512189
\(613\) 2.69677e15 1.25838 0.629190 0.777251i \(-0.283387\pi\)
0.629190 + 0.777251i \(0.283387\pi\)
\(614\) −2.77668e15 −1.28410
\(615\) −3.16609e15 −1.45114
\(616\) 5.93875e14 0.269775
\(617\) 5.08744e14 0.229050 0.114525 0.993420i \(-0.463465\pi\)
0.114525 + 0.993420i \(0.463465\pi\)
\(618\) 1.65220e15 0.737270
\(619\) −3.61935e14 −0.160078 −0.0800391 0.996792i \(-0.525505\pi\)
−0.0800391 + 0.996792i \(0.525505\pi\)
\(620\) 3.77853e14 0.165641
\(621\) −7.47722e14 −0.324890
\(622\) 1.35626e15 0.584113
\(623\) −2.69217e14 −0.114926
\(624\) 1.83312e15 0.775667
\(625\) −2.57903e15 −1.08173
\(626\) −8.38246e14 −0.348508
\(627\) 1.73943e15 0.716861
\(628\) −1.88780e14 −0.0771218
\(629\) −2.09406e15 −0.848029
\(630\) 2.74607e14 0.110240
\(631\) −1.96859e15 −0.783418 −0.391709 0.920089i \(-0.628116\pi\)
−0.391709 + 0.920089i \(0.628116\pi\)
\(632\) 2.03188e15 0.801592
\(633\) −1.42659e15 −0.557930
\(634\) −3.73887e14 −0.144960
\(635\) 8.25158e14 0.317162
\(636\) −4.01311e14 −0.152921
\(637\) −4.04160e15 −1.52682
\(638\) 3.85242e15 1.44285
\(639\) −1.76996e14 −0.0657216
\(640\) −2.65003e15 −0.975576
\(641\) 5.42899e15 1.98153 0.990764 0.135600i \(-0.0432962\pi\)
0.990764 + 0.135600i \(0.0432962\pi\)
\(642\) −4.99399e14 −0.180719
\(643\) 5.23242e15 1.87734 0.938669 0.344820i \(-0.112060\pi\)
0.938669 + 0.344820i \(0.112060\pi\)
\(644\) 1.92202e14 0.0683732
\(645\) −2.74983e15 −0.969902
\(646\) 3.30355e15 1.15532
\(647\) −4.57841e15 −1.58760 −0.793799 0.608180i \(-0.791900\pi\)
−0.793799 + 0.608180i \(0.791900\pi\)
\(648\) 3.43158e14 0.117986
\(649\) 3.72578e14 0.127020
\(650\) −4.05262e15 −1.36997
\(651\) −3.45585e14 −0.115840
\(652\) 1.89835e14 0.0630979
\(653\) −4.77225e14 −0.157290 −0.0786449 0.996903i \(-0.525059\pi\)
−0.0786449 + 0.996903i \(0.525059\pi\)
\(654\) −6.85412e14 −0.224014
\(655\) −2.66582e15 −0.863979
\(656\) −4.64150e15 −1.49172
\(657\) −1.62102e15 −0.516628
\(658\) 2.54369e14 0.0803936
\(659\) −6.04420e15 −1.89439 −0.947195 0.320659i \(-0.896096\pi\)
−0.947195 + 0.320659i \(0.896096\pi\)
\(660\) −3.89590e14 −0.121092
\(661\) −1.66802e15 −0.514153 −0.257077 0.966391i \(-0.582759\pi\)
−0.257077 + 0.966391i \(0.582759\pi\)
\(662\) −7.98394e14 −0.244061
\(663\) 3.08146e15 0.934185
\(664\) 3.09645e15 0.930979
\(665\) −1.53599e15 −0.458003
\(666\) −8.89148e14 −0.262945
\(667\) 9.26281e15 2.71676
\(668\) 1.08964e14 0.0316966
\(669\) −1.40185e15 −0.404444
\(670\) −2.05631e15 −0.588407
\(671\) 4.69187e15 1.33160
\(672\) −1.64545e14 −0.0463183
\(673\) 3.51506e15 0.981409 0.490704 0.871326i \(-0.336739\pi\)
0.490704 + 0.871326i \(0.336739\pi\)
\(674\) 6.32397e15 1.75130
\(675\) −6.37722e14 −0.175171
\(676\) −9.60574e14 −0.261712
\(677\) 6.05494e15 1.63633 0.818167 0.574980i \(-0.194990\pi\)
0.818167 + 0.574980i \(0.194990\pi\)
\(678\) −1.71026e15 −0.458457
\(679\) 1.61124e15 0.428428
\(680\) −5.49703e15 −1.44987
\(681\) 2.51012e15 0.656729
\(682\) −2.66190e15 −0.690840
\(683\) −3.36699e15 −0.866817 −0.433409 0.901198i \(-0.642689\pi\)
−0.433409 + 0.901198i \(0.642689\pi\)
\(684\) −2.58359e14 −0.0659803
\(685\) 4.40135e15 1.11503
\(686\) −1.83973e15 −0.462350
\(687\) −1.81367e15 −0.452164
\(688\) −4.03126e15 −0.997020
\(689\) −1.13677e16 −2.78913
\(690\) 5.08579e15 1.23791
\(691\) 3.69145e14 0.0891389 0.0445695 0.999006i \(-0.485808\pi\)
0.0445695 + 0.999006i \(0.485808\pi\)
\(692\) −8.81393e14 −0.211148
\(693\) 3.56319e14 0.0846849
\(694\) 1.05597e15 0.248986
\(695\) −2.29486e15 −0.536834
\(696\) −4.25106e15 −0.986611
\(697\) −7.80234e15 −1.79657
\(698\) 7.13973e15 1.63109
\(699\) 2.91048e15 0.659691
\(700\) 1.63927e14 0.0368647
\(701\) −7.45701e15 −1.66386 −0.831928 0.554884i \(-0.812763\pi\)
−0.831928 + 0.554884i \(0.812763\pi\)
\(702\) 1.30840e15 0.289659
\(703\) 4.97335e15 1.09243
\(704\) −4.93943e15 −1.07653
\(705\) −1.23972e15 −0.268091
\(706\) −6.64770e14 −0.142641
\(707\) 5.47964e14 0.116666
\(708\) −5.53394e13 −0.0116910
\(709\) 1.02668e15 0.215220 0.107610 0.994193i \(-0.465680\pi\)
0.107610 + 0.994193i \(0.465680\pi\)
\(710\) 1.20387e15 0.250415
\(711\) 1.21910e15 0.251627
\(712\) 2.28824e15 0.468665
\(713\) −6.40031e15 −1.30079
\(714\) 6.76726e14 0.136481
\(715\) −1.10357e16 −2.20860
\(716\) −2.80028e14 −0.0556135
\(717\) −2.66464e15 −0.525149
\(718\) 3.79168e15 0.741562
\(719\) −8.42355e15 −1.63488 −0.817441 0.576013i \(-0.804608\pi\)
−0.817441 + 0.576013i \(0.804608\pi\)
\(720\) −1.96203e15 −0.377899
\(721\) −1.89309e15 −0.361848
\(722\) −3.00140e15 −0.569337
\(723\) 1.25714e15 0.236659
\(724\) −4.26470e14 −0.0796758
\(725\) 7.90013e15 1.46479
\(726\) −1.38660e14 −0.0255152
\(727\) −2.08220e15 −0.380262 −0.190131 0.981759i \(-0.560891\pi\)
−0.190131 + 0.981759i \(0.560891\pi\)
\(728\) −2.49865e15 −0.452880
\(729\) 2.05891e14 0.0370370
\(730\) 1.10257e16 1.96848
\(731\) −6.77652e15 −1.20077
\(732\) −6.96888e14 −0.122561
\(733\) 4.41378e15 0.770439 0.385220 0.922825i \(-0.374126\pi\)
0.385220 + 0.922825i \(0.374126\pi\)
\(734\) −8.65555e15 −1.49957
\(735\) 4.32581e15 0.743853
\(736\) −3.04740e15 −0.520118
\(737\) −2.66818e15 −0.452006
\(738\) −3.31291e15 −0.557056
\(739\) −3.08125e15 −0.514259 −0.257130 0.966377i \(-0.582777\pi\)
−0.257130 + 0.966377i \(0.582777\pi\)
\(740\) −1.11391e15 −0.184534
\(741\) −7.31841e15 −1.20342
\(742\) −2.49649e15 −0.407482
\(743\) −4.93087e15 −0.798887 −0.399443 0.916758i \(-0.630797\pi\)
−0.399443 + 0.916758i \(0.630797\pi\)
\(744\) 2.93735e15 0.472393
\(745\) 4.76507e15 0.760694
\(746\) 3.17730e15 0.503495
\(747\) 1.85784e15 0.292243
\(748\) −9.60081e14 −0.149916
\(749\) 5.72210e14 0.0886962
\(750\) −4.27880e14 −0.0658393
\(751\) 8.61170e12 0.00131543 0.000657717 1.00000i \(-0.499791\pi\)
0.000657717 1.00000i \(0.499791\pi\)
\(752\) −1.81743e15 −0.275587
\(753\) −3.71869e15 −0.559780
\(754\) −1.62085e16 −2.42215
\(755\) 1.52601e16 2.26386
\(756\) −5.29243e13 −0.00779445
\(757\) 2.83321e15 0.414240 0.207120 0.978316i \(-0.433591\pi\)
0.207120 + 0.978316i \(0.433591\pi\)
\(758\) −1.09320e16 −1.58679
\(759\) 6.59910e15 0.950944
\(760\) 1.30553e16 1.86773
\(761\) −1.10169e16 −1.56475 −0.782373 0.622810i \(-0.785991\pi\)
−0.782373 + 0.622810i \(0.785991\pi\)
\(762\) 8.63421e14 0.121750
\(763\) 7.85343e14 0.109945
\(764\) 7.74321e14 0.107624
\(765\) −3.29816e15 −0.455128
\(766\) −3.53352e14 −0.0484116
\(767\) −1.56757e15 −0.213233
\(768\) 1.94398e15 0.262545
\(769\) 7.47818e15 1.00277 0.501385 0.865224i \(-0.332824\pi\)
0.501385 + 0.865224i \(0.332824\pi\)
\(770\) −2.42358e15 −0.322669
\(771\) 2.59920e15 0.343590
\(772\) 1.43901e15 0.188873
\(773\) −2.78492e15 −0.362932 −0.181466 0.983397i \(-0.558084\pi\)
−0.181466 + 0.983397i \(0.558084\pi\)
\(774\) −2.87734e15 −0.372320
\(775\) −5.45874e15 −0.701347
\(776\) −1.36950e16 −1.74712
\(777\) 1.01878e15 0.129052
\(778\) −5.99151e15 −0.753614
\(779\) 1.85304e16 2.31435
\(780\) 1.63915e15 0.203281
\(781\) 1.56209e15 0.192365
\(782\) 1.25331e16 1.53258
\(783\) −2.55059e15 −0.309707
\(784\) 6.34164e15 0.764651
\(785\) 5.72352e15 0.685299
\(786\) −2.78943e15 −0.331659
\(787\) 1.00576e16 1.18750 0.593752 0.804648i \(-0.297646\pi\)
0.593752 + 0.804648i \(0.297646\pi\)
\(788\) −6.29113e14 −0.0737623
\(789\) −1.79104e15 −0.208536
\(790\) −8.29199e15 −0.958760
\(791\) 1.95961e15 0.225008
\(792\) −3.02858e15 −0.345343
\(793\) −1.97404e16 −2.23540
\(794\) −8.37815e15 −0.942183
\(795\) 1.21671e16 1.35884
\(796\) −1.80720e15 −0.200439
\(797\) 6.39013e15 0.703864 0.351932 0.936026i \(-0.385525\pi\)
0.351932 + 0.936026i \(0.385525\pi\)
\(798\) −1.60721e15 −0.175815
\(799\) −3.05508e15 −0.331907
\(800\) −2.59909e15 −0.280431
\(801\) 1.37292e15 0.147118
\(802\) 1.38435e16 1.47329
\(803\) 1.43065e16 1.51216
\(804\) 3.96308e14 0.0416030
\(805\) −5.82728e15 −0.607559
\(806\) 1.11996e16 1.15974
\(807\) −8.44923e15 −0.868986
\(808\) −4.65750e15 −0.475763
\(809\) −7.95064e15 −0.806650 −0.403325 0.915057i \(-0.632146\pi\)
−0.403325 + 0.915057i \(0.632146\pi\)
\(810\) −1.40041e15 −0.141120
\(811\) −1.46618e16 −1.46748 −0.733742 0.679428i \(-0.762228\pi\)
−0.733742 + 0.679428i \(0.762228\pi\)
\(812\) 6.55628e14 0.0651778
\(813\) 1.55124e15 0.153173
\(814\) 7.84727e15 0.769634
\(815\) −5.75552e15 −0.560683
\(816\) −4.83510e15 −0.467853
\(817\) 1.60941e16 1.54684
\(818\) −1.71392e16 −1.63624
\(819\) −1.49916e15 −0.142163
\(820\) −4.15036e15 −0.390939
\(821\) 4.75644e15 0.445036 0.222518 0.974929i \(-0.428572\pi\)
0.222518 + 0.974929i \(0.428572\pi\)
\(822\) 4.60544e15 0.428032
\(823\) 2.04699e16 1.88980 0.944901 0.327357i \(-0.106158\pi\)
0.944901 + 0.327357i \(0.106158\pi\)
\(824\) 1.60906e16 1.47561
\(825\) 5.62829e15 0.512719
\(826\) −3.44258e14 −0.0311525
\(827\) 1.10892e16 0.996832 0.498416 0.866938i \(-0.333915\pi\)
0.498416 + 0.866938i \(0.333915\pi\)
\(828\) −9.80171e14 −0.0875256
\(829\) −1.63458e16 −1.44996 −0.724980 0.688770i \(-0.758151\pi\)
−0.724980 + 0.688770i \(0.758151\pi\)
\(830\) −1.26365e16 −1.11352
\(831\) −6.24664e15 −0.546816
\(832\) 2.07820e16 1.80721
\(833\) 1.06603e16 0.920918
\(834\) −2.40127e15 −0.206077
\(835\) −3.30362e15 −0.281653
\(836\) 2.28017e15 0.193123
\(837\) 1.76237e15 0.148289
\(838\) −1.90060e16 −1.58873
\(839\) −1.76209e16 −1.46331 −0.731656 0.681674i \(-0.761252\pi\)
−0.731656 + 0.681674i \(0.761252\pi\)
\(840\) 2.67436e15 0.220640
\(841\) 1.93962e16 1.58979
\(842\) −1.03483e15 −0.0842662
\(843\) 7.78029e15 0.629425
\(844\) −1.87009e15 −0.150307
\(845\) 2.91232e16 2.32556
\(846\) −1.29720e15 −0.102913
\(847\) 1.58876e14 0.0125228
\(848\) 1.78370e16 1.39684
\(849\) 4.14940e15 0.322844
\(850\) 1.06893e16 0.826317
\(851\) 1.88681e16 1.44916
\(852\) −2.32019e14 −0.0177054
\(853\) 1.75870e16 1.33344 0.666719 0.745309i \(-0.267698\pi\)
0.666719 + 0.745309i \(0.267698\pi\)
\(854\) −4.33523e15 −0.326583
\(855\) 7.83305e15 0.586297
\(856\) −4.86358e15 −0.361701
\(857\) −2.59257e16 −1.91574 −0.957868 0.287209i \(-0.907272\pi\)
−0.957868 + 0.287209i \(0.907272\pi\)
\(858\) −1.15475e16 −0.847825
\(859\) −6.63970e15 −0.484380 −0.242190 0.970229i \(-0.577866\pi\)
−0.242190 + 0.970229i \(0.577866\pi\)
\(860\) −3.60469e15 −0.261292
\(861\) 3.79592e15 0.273401
\(862\) −1.69681e16 −1.21435
\(863\) 1.81587e16 1.29129 0.645646 0.763637i \(-0.276588\pi\)
0.645646 + 0.763637i \(0.276588\pi\)
\(864\) 8.39126e14 0.0592928
\(865\) 2.67225e16 1.87624
\(866\) −2.33793e16 −1.63111
\(867\) 2.00290e14 0.0138853
\(868\) −4.53019e14 −0.0312074
\(869\) −1.07593e16 −0.736507
\(870\) 1.73484e16 1.18006
\(871\) 1.12260e16 0.758799
\(872\) −6.67513e15 −0.448352
\(873\) −8.21685e15 −0.548437
\(874\) −2.97659e16 −1.97427
\(875\) 4.90264e14 0.0323136
\(876\) −2.12495e15 −0.139180
\(877\) −7.62216e14 −0.0496113 −0.0248056 0.999692i \(-0.507897\pi\)
−0.0248056 + 0.999692i \(0.507897\pi\)
\(878\) 7.46886e15 0.483097
\(879\) 1.06661e16 0.685594
\(880\) 1.73161e16 1.10610
\(881\) 9.89653e15 0.628225 0.314113 0.949386i \(-0.398293\pi\)
0.314113 + 0.949386i \(0.398293\pi\)
\(882\) 4.52640e15 0.285546
\(883\) 1.41170e16 0.885034 0.442517 0.896760i \(-0.354086\pi\)
0.442517 + 0.896760i \(0.354086\pi\)
\(884\) 4.03942e15 0.251670
\(885\) 1.67781e15 0.103885
\(886\) −9.07054e15 −0.558146
\(887\) −2.86512e16 −1.75211 −0.876057 0.482207i \(-0.839835\pi\)
−0.876057 + 0.482207i \(0.839835\pi\)
\(888\) −8.65928e15 −0.526273
\(889\) −9.89304e14 −0.0597545
\(890\) −9.33822e15 −0.560556
\(891\) −1.81711e15 −0.108406
\(892\) −1.83765e15 −0.108957
\(893\) 7.25576e15 0.427563
\(894\) 4.98603e15 0.292010
\(895\) 8.49004e15 0.494178
\(896\) 3.17719e15 0.183802
\(897\) −2.77649e16 −1.59638
\(898\) 2.60337e14 0.0148770
\(899\) −2.18324e16 −1.24000
\(900\) −8.35975e14 −0.0471910
\(901\) 2.99840e16 1.68230
\(902\) 2.92384e16 1.63049
\(903\) 3.29685e15 0.182733
\(904\) −1.66560e16 −0.917579
\(905\) 1.29299e16 0.707994
\(906\) 1.59677e16 0.869037
\(907\) −1.59018e16 −0.860213 −0.430106 0.902778i \(-0.641524\pi\)
−0.430106 + 0.902778i \(0.641524\pi\)
\(908\) 3.29046e15 0.176923
\(909\) −2.79445e15 −0.149346
\(910\) 1.01969e16 0.541676
\(911\) −2.43582e16 −1.28616 −0.643080 0.765799i \(-0.722344\pi\)
−0.643080 + 0.765799i \(0.722344\pi\)
\(912\) 1.14833e16 0.602690
\(913\) −1.63965e16 −0.855388
\(914\) 5.42577e14 0.0281357
\(915\) 2.11286e16 1.08907
\(916\) −2.37750e15 −0.121813
\(917\) 3.19612e15 0.162777
\(918\) −3.45109e15 −0.174712
\(919\) −1.12252e16 −0.564885 −0.282442 0.959284i \(-0.591145\pi\)
−0.282442 + 0.959284i \(0.591145\pi\)
\(920\) 4.95298e16 2.47761
\(921\) −1.62247e16 −0.806769
\(922\) −2.46245e16 −1.21716
\(923\) −6.57231e15 −0.322930
\(924\) 4.67090e14 0.0228142
\(925\) 1.60923e16 0.781340
\(926\) 2.58694e16 1.24861
\(927\) 9.65416e15 0.463208
\(928\) −1.03951e16 −0.495811
\(929\) 2.38103e16 1.12896 0.564480 0.825447i \(-0.309077\pi\)
0.564480 + 0.825447i \(0.309077\pi\)
\(930\) −1.19872e16 −0.565015
\(931\) −2.53179e16 −1.18633
\(932\) 3.81528e15 0.177721
\(933\) 7.92493e15 0.366984
\(934\) 1.94968e16 0.897546
\(935\) 2.91082e16 1.33215
\(936\) 1.27424e16 0.579739
\(937\) 8.95352e15 0.404973 0.202486 0.979285i \(-0.435098\pi\)
0.202486 + 0.979285i \(0.435098\pi\)
\(938\) 2.46537e15 0.110858
\(939\) −4.89805e15 −0.218959
\(940\) −1.62511e15 −0.0722239
\(941\) −1.50764e16 −0.666126 −0.333063 0.942905i \(-0.608082\pi\)
−0.333063 + 0.942905i \(0.608082\pi\)
\(942\) 5.98892e15 0.263068
\(943\) 7.03013e16 3.07007
\(944\) 2.45967e15 0.106790
\(945\) 1.60459e15 0.0692609
\(946\) 2.53943e16 1.08977
\(947\) −3.25117e16 −1.38712 −0.693562 0.720397i \(-0.743960\pi\)
−0.693562 + 0.720397i \(0.743960\pi\)
\(948\) 1.59809e15 0.0677886
\(949\) −6.01926e16 −2.53851
\(950\) −2.53869e16 −1.06446
\(951\) −2.18470e15 −0.0910750
\(952\) 6.59054e15 0.273160
\(953\) −3.67755e15 −0.151547 −0.0757735 0.997125i \(-0.524143\pi\)
−0.0757735 + 0.997125i \(0.524143\pi\)
\(954\) 1.27313e16 0.521624
\(955\) −2.34763e16 −0.956335
\(956\) −3.49301e15 −0.141475
\(957\) 2.25105e16 0.906503
\(958\) −2.80669e16 −1.12379
\(959\) −5.27690e15 −0.210076
\(960\) −2.22434e16 −0.880460
\(961\) −1.03230e16 −0.406282
\(962\) −3.30164e16 −1.29201
\(963\) −2.91809e15 −0.113541
\(964\) 1.64796e15 0.0637561
\(965\) −4.36287e16 −1.67831
\(966\) −6.09749e15 −0.233226
\(967\) −4.22043e16 −1.60513 −0.802567 0.596562i \(-0.796533\pi\)
−0.802567 + 0.596562i \(0.796533\pi\)
\(968\) −1.35039e15 −0.0510675
\(969\) 1.93033e16 0.725857
\(970\) 5.58887e16 2.08968
\(971\) 2.20791e16 0.820871 0.410435 0.911890i \(-0.365377\pi\)
0.410435 + 0.911890i \(0.365377\pi\)
\(972\) 2.69898e14 0.00997780
\(973\) 2.75137e15 0.101141
\(974\) −1.71738e16 −0.627757
\(975\) −2.36803e16 −0.860720
\(976\) 3.09746e16 1.11952
\(977\) −3.75496e16 −1.34954 −0.674770 0.738028i \(-0.735757\pi\)
−0.674770 + 0.738028i \(0.735757\pi\)
\(978\) −6.02240e15 −0.215232
\(979\) −1.21169e16 −0.430612
\(980\) 5.67060e15 0.200394
\(981\) −4.00501e15 −0.140742
\(982\) −3.82978e16 −1.33832
\(983\) 4.99505e15 0.173578 0.0867892 0.996227i \(-0.472339\pi\)
0.0867892 + 0.996227i \(0.472339\pi\)
\(984\) −3.22639e16 −1.11492
\(985\) 1.90738e16 0.655446
\(986\) 4.27523e16 1.46095
\(987\) 1.48633e15 0.0505092
\(988\) −9.59353e15 −0.324202
\(989\) 6.10584e16 2.05194
\(990\) 1.23595e16 0.413054
\(991\) −1.30791e16 −0.434684 −0.217342 0.976096i \(-0.569739\pi\)
−0.217342 + 0.976096i \(0.569739\pi\)
\(992\) 7.18270e15 0.237396
\(993\) −4.66518e15 −0.153337
\(994\) −1.44336e15 −0.0471790
\(995\) 5.47915e16 1.78109
\(996\) 2.43539e15 0.0787305
\(997\) 3.07717e16 0.989301 0.494651 0.869092i \(-0.335296\pi\)
0.494651 + 0.869092i \(0.335296\pi\)
\(998\) −2.70586e16 −0.865140
\(999\) −5.19548e15 −0.165202
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.1.9 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.c.1.9 27 1.1 even 1 trivial