Properties

Label 177.12.a.b.1.5
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
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: \(1\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-68.7593 q^{2} +243.000 q^{3} +2679.84 q^{4} +7324.59 q^{5} -16708.5 q^{6} -63928.8 q^{7} -43445.2 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-68.7593 q^{2} +243.000 q^{3} +2679.84 q^{4} +7324.59 q^{5} -16708.5 q^{6} -63928.8 q^{7} -43445.2 q^{8} +59049.0 q^{9} -503634. q^{10} -307718. q^{11} +651202. q^{12} -1.14204e6 q^{13} +4.39570e6 q^{14} +1.77988e6 q^{15} -2.50106e6 q^{16} +1.41880e6 q^{17} -4.06017e6 q^{18} +4.00245e6 q^{19} +1.96288e7 q^{20} -1.55347e7 q^{21} +2.11585e7 q^{22} +4.77119e7 q^{23} -1.05572e7 q^{24} +4.82151e6 q^{25} +7.85260e7 q^{26} +1.43489e7 q^{27} -1.71319e8 q^{28} +1.63918e8 q^{29} -1.22383e8 q^{30} +1.17737e8 q^{31} +2.60947e8 q^{32} -7.47754e7 q^{33} -9.75560e7 q^{34} -4.68252e8 q^{35} +1.58242e8 q^{36} -5.45888e8 q^{37} -2.75206e8 q^{38} -2.77516e8 q^{39} -3.18218e8 q^{40} -5.77402e8 q^{41} +1.06815e9 q^{42} -1.16727e9 q^{43} -8.24635e8 q^{44} +4.32510e8 q^{45} -3.28064e9 q^{46} -3.29117e8 q^{47} -6.07758e8 q^{48} +2.10956e9 q^{49} -3.31524e8 q^{50} +3.44769e8 q^{51} -3.06049e9 q^{52} +5.23739e9 q^{53} -9.86621e8 q^{54} -2.25391e9 q^{55} +2.77740e9 q^{56} +9.72596e8 q^{57} -1.12709e10 q^{58} -7.14924e8 q^{59} +4.76979e9 q^{60} +6.69141e9 q^{61} -8.09555e9 q^{62} -3.77493e9 q^{63} -1.28204e10 q^{64} -8.36499e9 q^{65} +5.14151e9 q^{66} -1.09510e10 q^{67} +3.80217e9 q^{68} +1.15940e10 q^{69} +3.21967e10 q^{70} -1.13733e10 q^{71} -2.56539e9 q^{72} -1.17767e10 q^{73} +3.75349e10 q^{74} +1.17163e9 q^{75} +1.07259e10 q^{76} +1.96720e10 q^{77} +1.90818e10 q^{78} +4.67627e10 q^{79} -1.83192e10 q^{80} +3.48678e9 q^{81} +3.97018e10 q^{82} -4.41617e9 q^{83} -4.16305e10 q^{84} +1.03922e10 q^{85} +8.02607e10 q^{86} +3.98320e10 q^{87} +1.33688e10 q^{88} +2.79794e9 q^{89} -2.97391e10 q^{90} +7.30093e10 q^{91} +1.27861e11 q^{92} +2.86102e10 q^{93} +2.26299e10 q^{94} +2.93163e10 q^{95} +6.34101e10 q^{96} -1.07713e11 q^{97} -1.45052e11 q^{98} -1.81704e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −68.7593 −1.51938 −0.759690 0.650285i \(-0.774649\pi\)
−0.759690 + 0.650285i \(0.774649\pi\)
\(3\) 243.000 0.577350
\(4\) 2679.84 1.30852
\(5\) 7324.59 1.04821 0.524105 0.851654i \(-0.324400\pi\)
0.524105 + 0.851654i \(0.324400\pi\)
\(6\) −16708.5 −0.877215
\(7\) −63928.8 −1.43766 −0.718832 0.695184i \(-0.755323\pi\)
−0.718832 + 0.695184i \(0.755323\pi\)
\(8\) −43445.2 −0.468756
\(9\) 59049.0 0.333333
\(10\) −503634. −1.59263
\(11\) −307718. −0.576093 −0.288047 0.957616i \(-0.593006\pi\)
−0.288047 + 0.957616i \(0.593006\pi\)
\(12\) 651202. 0.755473
\(13\) −1.14204e6 −0.853088 −0.426544 0.904467i \(-0.640269\pi\)
−0.426544 + 0.904467i \(0.640269\pi\)
\(14\) 4.39570e6 2.18436
\(15\) 1.77988e6 0.605184
\(16\) −2.50106e6 −0.596299
\(17\) 1.41880e6 0.242356 0.121178 0.992631i \(-0.461333\pi\)
0.121178 + 0.992631i \(0.461333\pi\)
\(18\) −4.06017e6 −0.506460
\(19\) 4.00245e6 0.370836 0.185418 0.982660i \(-0.440636\pi\)
0.185418 + 0.982660i \(0.440636\pi\)
\(20\) 1.96288e7 1.37160
\(21\) −1.55347e7 −0.830035
\(22\) 2.11585e7 0.875305
\(23\) 4.77119e7 1.54570 0.772848 0.634591i \(-0.218832\pi\)
0.772848 + 0.634591i \(0.218832\pi\)
\(24\) −1.05572e7 −0.270636
\(25\) 4.82151e6 0.0987445
\(26\) 7.85260e7 1.29616
\(27\) 1.43489e7 0.192450
\(28\) −1.71319e8 −1.88121
\(29\) 1.63918e8 1.48401 0.742006 0.670393i \(-0.233875\pi\)
0.742006 + 0.670393i \(0.233875\pi\)
\(30\) −1.22383e8 −0.919505
\(31\) 1.17737e8 0.738627 0.369314 0.929305i \(-0.379593\pi\)
0.369314 + 0.929305i \(0.379593\pi\)
\(32\) 2.60947e8 1.37476
\(33\) −7.47754e7 −0.332608
\(34\) −9.75560e7 −0.368231
\(35\) −4.68252e8 −1.50697
\(36\) 1.58242e8 0.436173
\(37\) −5.45888e8 −1.29418 −0.647089 0.762414i \(-0.724014\pi\)
−0.647089 + 0.762414i \(0.724014\pi\)
\(38\) −2.75206e8 −0.563440
\(39\) −2.77516e8 −0.492530
\(40\) −3.18218e8 −0.491354
\(41\) −5.77402e8 −0.778336 −0.389168 0.921167i \(-0.627237\pi\)
−0.389168 + 0.921167i \(0.627237\pi\)
\(42\) 1.06815e9 1.26114
\(43\) −1.16727e9 −1.21086 −0.605431 0.795897i \(-0.706999\pi\)
−0.605431 + 0.795897i \(0.706999\pi\)
\(44\) −8.24635e8 −0.753828
\(45\) 4.32510e8 0.349403
\(46\) −3.28064e9 −2.34850
\(47\) −3.29117e8 −0.209321 −0.104660 0.994508i \(-0.533376\pi\)
−0.104660 + 0.994508i \(0.533376\pi\)
\(48\) −6.07758e8 −0.344274
\(49\) 2.10956e9 1.06688
\(50\) −3.31524e8 −0.150030
\(51\) 3.44769e8 0.139924
\(52\) −3.06049e9 −1.11628
\(53\) 5.23739e9 1.72027 0.860137 0.510063i \(-0.170378\pi\)
0.860137 + 0.510063i \(0.170378\pi\)
\(54\) −9.86621e8 −0.292405
\(55\) −2.25391e9 −0.603867
\(56\) 2.77740e9 0.673913
\(57\) 9.72596e8 0.214102
\(58\) −1.12709e10 −2.25478
\(59\) −7.14924e8 −0.130189
\(60\) 4.76979e9 0.791894
\(61\) 6.69141e9 1.01439 0.507193 0.861832i \(-0.330683\pi\)
0.507193 + 0.861832i \(0.330683\pi\)
\(62\) −8.09555e9 −1.12226
\(63\) −3.77493e9 −0.479221
\(64\) −1.28204e10 −1.49249
\(65\) −8.36499e9 −0.894215
\(66\) 5.14151e9 0.505357
\(67\) −1.09510e10 −0.990930 −0.495465 0.868628i \(-0.665002\pi\)
−0.495465 + 0.868628i \(0.665002\pi\)
\(68\) 3.80217e9 0.317127
\(69\) 1.15940e10 0.892408
\(70\) 3.21967e10 2.28967
\(71\) −1.13733e10 −0.748109 −0.374055 0.927407i \(-0.622033\pi\)
−0.374055 + 0.927407i \(0.622033\pi\)
\(72\) −2.56539e9 −0.156252
\(73\) −1.17767e10 −0.664886 −0.332443 0.943123i \(-0.607873\pi\)
−0.332443 + 0.943123i \(0.607873\pi\)
\(74\) 3.75349e10 1.96635
\(75\) 1.17163e9 0.0570102
\(76\) 1.07259e10 0.485245
\(77\) 1.96720e10 0.828228
\(78\) 1.90818e10 0.748341
\(79\) 4.67627e10 1.70982 0.854910 0.518776i \(-0.173612\pi\)
0.854910 + 0.518776i \(0.173612\pi\)
\(80\) −1.83192e10 −0.625047
\(81\) 3.48678e9 0.111111
\(82\) 3.97018e10 1.18259
\(83\) −4.41617e9 −0.123060 −0.0615299 0.998105i \(-0.519598\pi\)
−0.0615299 + 0.998105i \(0.519598\pi\)
\(84\) −4.16305e10 −1.08612
\(85\) 1.03922e10 0.254040
\(86\) 8.02607e10 1.83976
\(87\) 3.98320e10 0.856795
\(88\) 1.33688e10 0.270047
\(89\) 2.79794e9 0.0531121 0.0265561 0.999647i \(-0.491546\pi\)
0.0265561 + 0.999647i \(0.491546\pi\)
\(90\) −2.97391e10 −0.530877
\(91\) 7.30093e10 1.22645
\(92\) 1.27861e11 2.02257
\(93\) 2.86102e10 0.426447
\(94\) 2.26299e10 0.318038
\(95\) 2.93163e10 0.388714
\(96\) 6.34101e10 0.793719
\(97\) −1.07713e11 −1.27357 −0.636783 0.771043i \(-0.719735\pi\)
−0.636783 + 0.771043i \(0.719735\pi\)
\(98\) −1.45052e11 −1.62099
\(99\) −1.81704e10 −0.192031
\(100\) 1.29209e10 0.129209
\(101\) 1.21405e11 1.14939 0.574695 0.818368i \(-0.305121\pi\)
0.574695 + 0.818368i \(0.305121\pi\)
\(102\) −2.37061e10 −0.212598
\(103\) 1.13157e11 0.961784 0.480892 0.876780i \(-0.340313\pi\)
0.480892 + 0.876780i \(0.340313\pi\)
\(104\) 4.96162e10 0.399890
\(105\) −1.13785e11 −0.870051
\(106\) −3.60119e11 −2.61375
\(107\) −9.86871e10 −0.680220 −0.340110 0.940386i \(-0.610464\pi\)
−0.340110 + 0.940386i \(0.610464\pi\)
\(108\) 3.84528e10 0.251824
\(109\) −1.67870e11 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(110\) 1.54977e11 0.917503
\(111\) −1.32651e11 −0.747194
\(112\) 1.59890e11 0.857278
\(113\) 8.34253e10 0.425958 0.212979 0.977057i \(-0.431683\pi\)
0.212979 + 0.977057i \(0.431683\pi\)
\(114\) −6.68750e10 −0.325302
\(115\) 3.49470e11 1.62021
\(116\) 4.39274e11 1.94186
\(117\) −6.74364e10 −0.284363
\(118\) 4.91577e10 0.197807
\(119\) −9.07024e10 −0.348426
\(120\) −7.73270e10 −0.283684
\(121\) −1.90621e11 −0.668117
\(122\) −4.60097e11 −1.54124
\(123\) −1.40309e11 −0.449372
\(124\) 3.15518e11 0.966506
\(125\) −3.22330e11 −0.944705
\(126\) 2.59562e11 0.728119
\(127\) 3.84746e11 1.03336 0.516682 0.856177i \(-0.327167\pi\)
0.516682 + 0.856177i \(0.327167\pi\)
\(128\) 3.47100e11 0.892894
\(129\) −2.83647e11 −0.699092
\(130\) 5.75171e11 1.35865
\(131\) −4.26964e11 −0.966939 −0.483470 0.875361i \(-0.660624\pi\)
−0.483470 + 0.875361i \(0.660624\pi\)
\(132\) −2.00386e11 −0.435223
\(133\) −2.55872e11 −0.533137
\(134\) 7.52985e11 1.50560
\(135\) 1.05100e11 0.201728
\(136\) −6.16402e10 −0.113606
\(137\) 3.05982e11 0.541668 0.270834 0.962626i \(-0.412701\pi\)
0.270834 + 0.962626i \(0.412701\pi\)
\(138\) −7.97196e11 −1.35591
\(139\) −9.62226e11 −1.57288 −0.786440 0.617666i \(-0.788078\pi\)
−0.786440 + 0.617666i \(0.788078\pi\)
\(140\) −1.25484e12 −1.97190
\(141\) −7.99755e10 −0.120851
\(142\) 7.82019e11 1.13666
\(143\) 3.51427e11 0.491458
\(144\) −1.47685e11 −0.198766
\(145\) 1.20063e12 1.55556
\(146\) 8.09756e11 1.01021
\(147\) 5.12623e11 0.615961
\(148\) −1.46289e12 −1.69346
\(149\) −1.17212e12 −1.30752 −0.653761 0.756701i \(-0.726810\pi\)
−0.653761 + 0.756701i \(0.726810\pi\)
\(150\) −8.05602e10 −0.0866201
\(151\) 9.13900e11 0.947382 0.473691 0.880691i \(-0.342921\pi\)
0.473691 + 0.880691i \(0.342921\pi\)
\(152\) −1.73887e11 −0.173831
\(153\) 8.37790e10 0.0807853
\(154\) −1.35263e12 −1.25839
\(155\) 8.62378e11 0.774236
\(156\) −7.43700e11 −0.644485
\(157\) 2.00126e12 1.67439 0.837193 0.546908i \(-0.184195\pi\)
0.837193 + 0.546908i \(0.184195\pi\)
\(158\) −3.21537e12 −2.59787
\(159\) 1.27269e12 0.993201
\(160\) 1.91133e12 1.44104
\(161\) −3.05017e12 −2.22219
\(162\) −2.39749e11 −0.168820
\(163\) −2.90005e12 −1.97412 −0.987059 0.160355i \(-0.948736\pi\)
−0.987059 + 0.160355i \(0.948736\pi\)
\(164\) −1.54735e12 −1.01847
\(165\) −5.47699e11 −0.348643
\(166\) 3.03653e11 0.186975
\(167\) −1.12393e11 −0.0669577 −0.0334788 0.999439i \(-0.510659\pi\)
−0.0334788 + 0.999439i \(0.510659\pi\)
\(168\) 6.74907e11 0.389084
\(169\) −4.87900e11 −0.272242
\(170\) −7.14558e11 −0.385983
\(171\) 2.36341e11 0.123612
\(172\) −3.12810e12 −1.58444
\(173\) −1.78076e12 −0.873677 −0.436839 0.899540i \(-0.643902\pi\)
−0.436839 + 0.899540i \(0.643902\pi\)
\(174\) −2.73882e12 −1.30180
\(175\) −3.08233e11 −0.141961
\(176\) 7.69621e11 0.343524
\(177\) −1.73727e11 −0.0751646
\(178\) −1.92385e11 −0.0806976
\(179\) −2.97323e12 −1.20931 −0.604654 0.796488i \(-0.706689\pi\)
−0.604654 + 0.796488i \(0.706689\pi\)
\(180\) 1.15906e12 0.457200
\(181\) 2.21255e12 0.846568 0.423284 0.905997i \(-0.360877\pi\)
0.423284 + 0.905997i \(0.360877\pi\)
\(182\) −5.02007e12 −1.86345
\(183\) 1.62601e12 0.585657
\(184\) −2.07285e12 −0.724553
\(185\) −3.99841e12 −1.35657
\(186\) −1.96722e12 −0.647935
\(187\) −4.36591e11 −0.139620
\(188\) −8.81983e11 −0.273900
\(189\) −9.17308e11 −0.276678
\(190\) −2.01577e12 −0.590604
\(191\) 9.71854e11 0.276642 0.138321 0.990387i \(-0.455829\pi\)
0.138321 + 0.990387i \(0.455829\pi\)
\(192\) −3.11535e12 −0.861687
\(193\) −1.87888e12 −0.505050 −0.252525 0.967590i \(-0.581261\pi\)
−0.252525 + 0.967590i \(0.581261\pi\)
\(194\) 7.40624e12 1.93503
\(195\) −2.03269e12 −0.516275
\(196\) 5.65329e12 1.39602
\(197\) −2.04544e12 −0.491159 −0.245579 0.969376i \(-0.578978\pi\)
−0.245579 + 0.969376i \(0.578978\pi\)
\(198\) 1.24939e12 0.291768
\(199\) 3.90343e12 0.886654 0.443327 0.896360i \(-0.353798\pi\)
0.443327 + 0.896360i \(0.353798\pi\)
\(200\) −2.09471e11 −0.0462870
\(201\) −2.66110e12 −0.572114
\(202\) −8.34769e12 −1.74636
\(203\) −1.04791e13 −2.13351
\(204\) 9.23928e11 0.183093
\(205\) −4.22923e12 −0.815860
\(206\) −7.78061e12 −1.46132
\(207\) 2.81734e12 0.515232
\(208\) 2.85632e12 0.508696
\(209\) −1.23163e12 −0.213636
\(210\) 7.82380e12 1.32194
\(211\) 4.79581e12 0.789421 0.394711 0.918805i \(-0.370845\pi\)
0.394711 + 0.918805i \(0.370845\pi\)
\(212\) 1.40354e13 2.25101
\(213\) −2.76371e12 −0.431921
\(214\) 6.78566e12 1.03351
\(215\) −8.54978e12 −1.26924
\(216\) −6.23391e11 −0.0902121
\(217\) −7.52681e12 −1.06190
\(218\) 1.15426e13 1.58780
\(219\) −2.86173e12 −0.383872
\(220\) −6.04012e12 −0.790170
\(221\) −1.62033e12 −0.206751
\(222\) 9.12098e12 1.13527
\(223\) −5.10096e12 −0.619405 −0.309703 0.950833i \(-0.600230\pi\)
−0.309703 + 0.950833i \(0.600230\pi\)
\(224\) −1.66820e13 −1.97644
\(225\) 2.84705e11 0.0329148
\(226\) −5.73627e12 −0.647192
\(227\) −1.38386e13 −1.52387 −0.761937 0.647651i \(-0.775752\pi\)
−0.761937 + 0.647651i \(0.775752\pi\)
\(228\) 2.60641e12 0.280156
\(229\) −3.00157e12 −0.314959 −0.157479 0.987522i \(-0.550337\pi\)
−0.157479 + 0.987522i \(0.550337\pi\)
\(230\) −2.40293e13 −2.46172
\(231\) 4.78030e12 0.478178
\(232\) −7.12144e12 −0.695639
\(233\) −2.94172e12 −0.280636 −0.140318 0.990106i \(-0.544812\pi\)
−0.140318 + 0.990106i \(0.544812\pi\)
\(234\) 4.63688e12 0.432055
\(235\) −2.41065e12 −0.219412
\(236\) −1.91589e12 −0.170354
\(237\) 1.13633e13 0.987165
\(238\) 6.23664e12 0.529392
\(239\) −1.24328e13 −1.03129 −0.515646 0.856802i \(-0.672448\pi\)
−0.515646 + 0.856802i \(0.672448\pi\)
\(240\) −4.45158e12 −0.360871
\(241\) 4.39171e11 0.0347969 0.0173984 0.999849i \(-0.494462\pi\)
0.0173984 + 0.999849i \(0.494462\pi\)
\(242\) 1.31070e13 1.01512
\(243\) 8.47289e11 0.0641500
\(244\) 1.79319e13 1.32734
\(245\) 1.54517e13 1.11831
\(246\) 9.64753e12 0.682768
\(247\) −4.57097e12 −0.316355
\(248\) −5.11512e12 −0.346236
\(249\) −1.07313e12 −0.0710486
\(250\) 2.21632e13 1.43537
\(251\) 2.10997e13 1.33681 0.668405 0.743797i \(-0.266977\pi\)
0.668405 + 0.743797i \(0.266977\pi\)
\(252\) −1.01162e13 −0.627069
\(253\) −1.46818e13 −0.890465
\(254\) −2.64548e13 −1.57007
\(255\) 2.52530e12 0.146670
\(256\) 2.38974e12 0.135841
\(257\) 9.06650e11 0.0504438 0.0252219 0.999682i \(-0.491971\pi\)
0.0252219 + 0.999682i \(0.491971\pi\)
\(258\) 1.95034e13 1.06219
\(259\) 3.48980e13 1.86059
\(260\) −2.24169e13 −1.17010
\(261\) 9.67919e12 0.494671
\(262\) 2.93578e13 1.46915
\(263\) −5.28789e12 −0.259135 −0.129567 0.991571i \(-0.541359\pi\)
−0.129567 + 0.991571i \(0.541359\pi\)
\(264\) 3.24863e12 0.155912
\(265\) 3.83617e13 1.80321
\(266\) 1.75936e13 0.810037
\(267\) 6.79900e11 0.0306643
\(268\) −2.93470e13 −1.29665
\(269\) −6.36340e12 −0.275455 −0.137728 0.990470i \(-0.543980\pi\)
−0.137728 + 0.990470i \(0.543980\pi\)
\(270\) −7.22660e12 −0.306502
\(271\) −2.73829e13 −1.13802 −0.569008 0.822332i \(-0.692673\pi\)
−0.569008 + 0.822332i \(0.692673\pi\)
\(272\) −3.54852e12 −0.144517
\(273\) 1.77413e13 0.708093
\(274\) −2.10391e13 −0.823000
\(275\) −1.48366e12 −0.0568860
\(276\) 3.10701e13 1.16773
\(277\) −1.41836e13 −0.522573 −0.261287 0.965261i \(-0.584147\pi\)
−0.261287 + 0.965261i \(0.584147\pi\)
\(278\) 6.61620e13 2.38980
\(279\) 6.95228e12 0.246209
\(280\) 2.03433e13 0.706402
\(281\) 2.12548e13 0.723722 0.361861 0.932232i \(-0.382142\pi\)
0.361861 + 0.932232i \(0.382142\pi\)
\(282\) 5.49906e12 0.183619
\(283\) −1.55797e13 −0.510192 −0.255096 0.966916i \(-0.582107\pi\)
−0.255096 + 0.966916i \(0.582107\pi\)
\(284\) −3.04786e13 −0.978914
\(285\) 7.12387e12 0.224424
\(286\) −2.41638e13 −0.746712
\(287\) 3.69126e13 1.11898
\(288\) 1.54087e13 0.458254
\(289\) −3.22589e13 −0.941264
\(290\) −8.25546e13 −2.36348
\(291\) −2.61741e13 −0.735294
\(292\) −3.15596e13 −0.870014
\(293\) 6.97384e13 1.88669 0.943345 0.331815i \(-0.107661\pi\)
0.943345 + 0.331815i \(0.107661\pi\)
\(294\) −3.52476e13 −0.935879
\(295\) −5.23653e12 −0.136465
\(296\) 2.37162e13 0.606653
\(297\) −4.41541e12 −0.110869
\(298\) 8.05945e13 1.98662
\(299\) −5.44890e13 −1.31861
\(300\) 3.13978e12 0.0745988
\(301\) 7.46222e13 1.74081
\(302\) −6.28391e13 −1.43943
\(303\) 2.95013e13 0.663600
\(304\) −1.00104e13 −0.221129
\(305\) 4.90118e13 1.06329
\(306\) −5.76059e12 −0.122744
\(307\) −8.59060e12 −0.179789 −0.0898944 0.995951i \(-0.528653\pi\)
−0.0898944 + 0.995951i \(0.528653\pi\)
\(308\) 5.27179e13 1.08375
\(309\) 2.74972e13 0.555286
\(310\) −5.92966e13 −1.17636
\(311\) −1.43460e13 −0.279608 −0.139804 0.990179i \(-0.544647\pi\)
−0.139804 + 0.990179i \(0.544647\pi\)
\(312\) 1.20567e13 0.230876
\(313\) −4.78260e13 −0.899850 −0.449925 0.893066i \(-0.648549\pi\)
−0.449925 + 0.893066i \(0.648549\pi\)
\(314\) −1.37605e14 −2.54403
\(315\) −2.76498e13 −0.502324
\(316\) 1.25317e14 2.23733
\(317\) 3.44196e13 0.603920 0.301960 0.953321i \(-0.402359\pi\)
0.301960 + 0.953321i \(0.402359\pi\)
\(318\) −8.75090e13 −1.50905
\(319\) −5.04404e13 −0.854929
\(320\) −9.39039e13 −1.56444
\(321\) −2.39810e13 −0.392725
\(322\) 2.09727e14 3.37635
\(323\) 5.67870e12 0.0898742
\(324\) 9.34404e12 0.145391
\(325\) −5.50636e12 −0.0842377
\(326\) 1.99405e14 2.99944
\(327\) −4.07925e13 −0.603348
\(328\) 2.50853e13 0.364849
\(329\) 2.10401e13 0.300933
\(330\) 3.76594e13 0.529721
\(331\) 6.96318e13 0.963282 0.481641 0.876369i \(-0.340041\pi\)
0.481641 + 0.876369i \(0.340041\pi\)
\(332\) −1.18346e13 −0.161026
\(333\) −3.22341e13 −0.431393
\(334\) 7.72810e12 0.101734
\(335\) −8.02117e13 −1.03870
\(336\) 3.88532e13 0.494949
\(337\) −5.62053e13 −0.704389 −0.352195 0.935927i \(-0.614564\pi\)
−0.352195 + 0.935927i \(0.614564\pi\)
\(338\) 3.35477e13 0.413639
\(339\) 2.02723e13 0.245927
\(340\) 2.78494e13 0.332416
\(341\) −3.62299e13 −0.425518
\(342\) −1.62506e13 −0.187813
\(343\) −8.45354e12 −0.0961437
\(344\) 5.07123e13 0.567599
\(345\) 8.49213e13 0.935431
\(346\) 1.22444e14 1.32745
\(347\) −4.96603e13 −0.529904 −0.264952 0.964262i \(-0.585356\pi\)
−0.264952 + 0.964262i \(0.585356\pi\)
\(348\) 1.06744e14 1.12113
\(349\) −3.32863e13 −0.344133 −0.172066 0.985085i \(-0.555044\pi\)
−0.172066 + 0.985085i \(0.555044\pi\)
\(350\) 2.11939e13 0.215693
\(351\) −1.63871e13 −0.164177
\(352\) −8.02980e13 −0.791991
\(353\) −1.82488e14 −1.77204 −0.886021 0.463646i \(-0.846541\pi\)
−0.886021 + 0.463646i \(0.846541\pi\)
\(354\) 1.19453e13 0.114204
\(355\) −8.33047e13 −0.784176
\(356\) 7.49805e12 0.0694982
\(357\) −2.20407e13 −0.201164
\(358\) 2.04437e14 1.83740
\(359\) −2.06373e14 −1.82655 −0.913277 0.407339i \(-0.866457\pi\)
−0.913277 + 0.407339i \(0.866457\pi\)
\(360\) −1.87905e13 −0.163785
\(361\) −1.00471e14 −0.862481
\(362\) −1.52134e14 −1.28626
\(363\) −4.63210e13 −0.385737
\(364\) 1.95654e14 1.60483
\(365\) −8.62593e13 −0.696940
\(366\) −1.11804e14 −0.889835
\(367\) 8.31297e13 0.651768 0.325884 0.945410i \(-0.394338\pi\)
0.325884 + 0.945410i \(0.394338\pi\)
\(368\) −1.19330e14 −0.921697
\(369\) −3.40950e13 −0.259445
\(370\) 2.74928e14 2.06115
\(371\) −3.34820e14 −2.47317
\(372\) 7.66709e13 0.558013
\(373\) −2.44630e14 −1.75433 −0.877164 0.480192i \(-0.840567\pi\)
−0.877164 + 0.480192i \(0.840567\pi\)
\(374\) 3.00197e13 0.212135
\(375\) −7.83263e13 −0.545426
\(376\) 1.42986e13 0.0981203
\(377\) −1.87201e14 −1.26599
\(378\) 6.30735e13 0.420380
\(379\) −9.15400e13 −0.601306 −0.300653 0.953734i \(-0.597205\pi\)
−0.300653 + 0.953734i \(0.597205\pi\)
\(380\) 7.85632e13 0.508639
\(381\) 9.34932e13 0.596613
\(382\) −6.68240e13 −0.420324
\(383\) −3.80408e13 −0.235861 −0.117931 0.993022i \(-0.537626\pi\)
−0.117931 + 0.993022i \(0.537626\pi\)
\(384\) 8.43453e13 0.515512
\(385\) 1.44089e14 0.868157
\(386\) 1.29191e14 0.767363
\(387\) −6.89262e13 −0.403621
\(388\) −2.88653e14 −1.66648
\(389\) 1.69152e14 0.962841 0.481420 0.876490i \(-0.340121\pi\)
0.481420 + 0.876490i \(0.340121\pi\)
\(390\) 1.39767e14 0.784419
\(391\) 6.76939e13 0.374608
\(392\) −9.16502e13 −0.500104
\(393\) −1.03752e14 −0.558263
\(394\) 1.40643e14 0.746257
\(395\) 3.42518e14 1.79225
\(396\) −4.86939e13 −0.251276
\(397\) 2.92592e14 1.48907 0.744535 0.667583i \(-0.232671\pi\)
0.744535 + 0.667583i \(0.232671\pi\)
\(398\) −2.68397e14 −1.34717
\(399\) −6.21769e13 −0.307807
\(400\) −1.20589e13 −0.0588813
\(401\) 1.76860e14 0.851798 0.425899 0.904771i \(-0.359958\pi\)
0.425899 + 0.904771i \(0.359958\pi\)
\(402\) 1.82975e14 0.869259
\(403\) −1.34461e14 −0.630114
\(404\) 3.25345e14 1.50400
\(405\) 2.55393e13 0.116468
\(406\) 7.20534e14 3.24161
\(407\) 1.67979e14 0.745567
\(408\) −1.49786e13 −0.0655903
\(409\) −9.26772e13 −0.400401 −0.200200 0.979755i \(-0.564159\pi\)
−0.200200 + 0.979755i \(0.564159\pi\)
\(410\) 2.90799e14 1.23960
\(411\) 7.43537e13 0.312732
\(412\) 3.03244e14 1.25851
\(413\) 4.57042e13 0.187168
\(414\) −1.93719e14 −0.782833
\(415\) −3.23466e13 −0.128993
\(416\) −2.98012e14 −1.17279
\(417\) −2.33821e14 −0.908103
\(418\) 8.46857e13 0.324594
\(419\) 3.72528e12 0.0140923 0.00704614 0.999975i \(-0.497757\pi\)
0.00704614 + 0.999975i \(0.497757\pi\)
\(420\) −3.04927e14 −1.13848
\(421\) 4.18454e14 1.54204 0.771022 0.636809i \(-0.219746\pi\)
0.771022 + 0.636809i \(0.219746\pi\)
\(422\) −3.29757e14 −1.19943
\(423\) −1.94340e13 −0.0697736
\(424\) −2.27539e14 −0.806388
\(425\) 6.84078e12 0.0239313
\(426\) 1.90031e14 0.656252
\(427\) −4.27774e14 −1.45835
\(428\) −2.64466e14 −0.890080
\(429\) 8.53966e13 0.283743
\(430\) 5.87877e14 1.92846
\(431\) −3.15215e14 −1.02090 −0.510448 0.859909i \(-0.670520\pi\)
−0.510448 + 0.859909i \(0.670520\pi\)
\(432\) −3.58875e13 −0.114758
\(433\) −6.15459e14 −1.94319 −0.971596 0.236644i \(-0.923953\pi\)
−0.971596 + 0.236644i \(0.923953\pi\)
\(434\) 5.17538e14 1.61343
\(435\) 2.91753e14 0.898101
\(436\) −4.49866e14 −1.36744
\(437\) 1.90965e14 0.573199
\(438\) 1.96771e14 0.583248
\(439\) −5.68329e14 −1.66358 −0.831792 0.555087i \(-0.812685\pi\)
−0.831792 + 0.555087i \(0.812685\pi\)
\(440\) 9.79213e13 0.283066
\(441\) 1.24567e14 0.355625
\(442\) 1.11413e14 0.314133
\(443\) −1.07083e14 −0.298195 −0.149097 0.988823i \(-0.547637\pi\)
−0.149097 + 0.988823i \(0.547637\pi\)
\(444\) −3.55483e14 −0.977717
\(445\) 2.04938e13 0.0556727
\(446\) 3.50738e14 0.941113
\(447\) −2.84826e14 −0.754899
\(448\) 8.19590e14 2.14569
\(449\) −1.43883e14 −0.372097 −0.186048 0.982541i \(-0.559568\pi\)
−0.186048 + 0.982541i \(0.559568\pi\)
\(450\) −1.95761e13 −0.0500102
\(451\) 1.77677e14 0.448394
\(452\) 2.23567e14 0.557373
\(453\) 2.22078e14 0.546971
\(454\) 9.51531e14 2.31535
\(455\) 5.34764e14 1.28558
\(456\) −4.22546e13 −0.100362
\(457\) −5.29269e14 −1.24204 −0.621022 0.783793i \(-0.713282\pi\)
−0.621022 + 0.783793i \(0.713282\pi\)
\(458\) 2.06386e14 0.478542
\(459\) 2.03583e13 0.0466414
\(460\) 9.36526e14 2.12008
\(461\) 3.99873e14 0.894473 0.447236 0.894416i \(-0.352408\pi\)
0.447236 + 0.894416i \(0.352408\pi\)
\(462\) −3.28690e14 −0.726534
\(463\) −7.10906e14 −1.55280 −0.776402 0.630237i \(-0.782958\pi\)
−0.776402 + 0.630237i \(0.782958\pi\)
\(464\) −4.09969e14 −0.884915
\(465\) 2.09558e14 0.447006
\(466\) 2.02270e14 0.426393
\(467\) 8.69930e13 0.181235 0.0906174 0.995886i \(-0.471116\pi\)
0.0906174 + 0.995886i \(0.471116\pi\)
\(468\) −1.80719e14 −0.372093
\(469\) 7.00085e14 1.42462
\(470\) 1.65755e14 0.333371
\(471\) 4.86306e14 0.966707
\(472\) 3.10600e13 0.0610268
\(473\) 3.59190e14 0.697570
\(474\) −7.81335e14 −1.49988
\(475\) 1.92979e13 0.0366180
\(476\) −2.43068e14 −0.455922
\(477\) 3.09263e14 0.573425
\(478\) 8.54873e14 1.56692
\(479\) −7.89655e14 −1.43084 −0.715422 0.698693i \(-0.753765\pi\)
−0.715422 + 0.698693i \(0.753765\pi\)
\(480\) 4.64453e14 0.831984
\(481\) 6.23427e14 1.10405
\(482\) −3.01971e13 −0.0528697
\(483\) −7.41190e14 −1.28298
\(484\) −5.10836e14 −0.874242
\(485\) −7.88950e14 −1.33497
\(486\) −5.82590e13 −0.0974683
\(487\) −6.74215e14 −1.11529 −0.557647 0.830078i \(-0.688296\pi\)
−0.557647 + 0.830078i \(0.688296\pi\)
\(488\) −2.90709e14 −0.475500
\(489\) −7.04712e14 −1.13976
\(490\) −1.06245e15 −1.69914
\(491\) −5.64669e14 −0.892988 −0.446494 0.894787i \(-0.647328\pi\)
−0.446494 + 0.894787i \(0.647328\pi\)
\(492\) −3.76005e14 −0.588012
\(493\) 2.32567e14 0.359659
\(494\) 3.14297e14 0.480664
\(495\) −1.33091e14 −0.201289
\(496\) −2.94468e14 −0.440443
\(497\) 7.27080e14 1.07553
\(498\) 7.37876e13 0.107950
\(499\) −4.45620e14 −0.644780 −0.322390 0.946607i \(-0.604486\pi\)
−0.322390 + 0.946607i \(0.604486\pi\)
\(500\) −8.63795e14 −1.23616
\(501\) −2.73116e13 −0.0386580
\(502\) −1.45080e15 −2.03112
\(503\) 3.31592e14 0.459177 0.229588 0.973288i \(-0.426262\pi\)
0.229588 + 0.973288i \(0.426262\pi\)
\(504\) 1.64002e14 0.224638
\(505\) 8.89238e14 1.20480
\(506\) 1.00951e15 1.35295
\(507\) −1.18560e14 −0.157179
\(508\) 1.03106e15 1.35217
\(509\) −4.37985e14 −0.568214 −0.284107 0.958793i \(-0.591697\pi\)
−0.284107 + 0.958793i \(0.591697\pi\)
\(510\) −1.73638e14 −0.222848
\(511\) 7.52868e14 0.955881
\(512\) −8.75178e14 −1.09929
\(513\) 5.74308e13 0.0713673
\(514\) −6.23407e13 −0.0766433
\(515\) 8.28830e14 1.00815
\(516\) −7.60129e14 −0.914774
\(517\) 1.01275e14 0.120588
\(518\) −2.39956e15 −2.82695
\(519\) −4.32724e14 −0.504418
\(520\) 3.63418e14 0.419168
\(521\) 1.25952e15 1.43746 0.718732 0.695287i \(-0.244723\pi\)
0.718732 + 0.695287i \(0.244723\pi\)
\(522\) −6.65534e14 −0.751593
\(523\) −7.52673e14 −0.841099 −0.420550 0.907270i \(-0.638163\pi\)
−0.420550 + 0.907270i \(0.638163\pi\)
\(524\) −1.14420e15 −1.26526
\(525\) −7.49006e13 −0.0819614
\(526\) 3.63592e14 0.393724
\(527\) 1.67046e14 0.179011
\(528\) 1.87018e14 0.198334
\(529\) 1.32362e15 1.38917
\(530\) −2.63773e15 −2.73976
\(531\) −4.22156e13 −0.0433963
\(532\) −6.85697e14 −0.697619
\(533\) 6.59417e14 0.663989
\(534\) −4.67495e13 −0.0465908
\(535\) −7.22843e14 −0.713014
\(536\) 4.75769e14 0.464504
\(537\) −7.22495e14 −0.698194
\(538\) 4.37543e14 0.418522
\(539\) −6.49149e14 −0.614619
\(540\) 2.81651e14 0.263965
\(541\) −1.77185e15 −1.64378 −0.821888 0.569649i \(-0.807079\pi\)
−0.821888 + 0.569649i \(0.807079\pi\)
\(542\) 1.88283e15 1.72908
\(543\) 5.37651e14 0.488766
\(544\) 3.70233e14 0.333181
\(545\) −1.22958e15 −1.09541
\(546\) −1.21988e15 −1.07586
\(547\) 5.08668e14 0.444124 0.222062 0.975033i \(-0.428721\pi\)
0.222062 + 0.975033i \(0.428721\pi\)
\(548\) 8.19985e14 0.708782
\(549\) 3.95121e14 0.338129
\(550\) 1.02016e14 0.0864315
\(551\) 6.56074e14 0.550324
\(552\) −5.03703e14 −0.418321
\(553\) −2.98948e15 −2.45815
\(554\) 9.75254e14 0.793988
\(555\) −9.71613e14 −0.783217
\(556\) −2.57862e15 −2.05814
\(557\) 5.44481e14 0.430307 0.215154 0.976580i \(-0.430975\pi\)
0.215154 + 0.976580i \(0.430975\pi\)
\(558\) −4.78034e14 −0.374085
\(559\) 1.33307e15 1.03297
\(560\) 1.17113e15 0.898607
\(561\) −1.06092e14 −0.0806094
\(562\) −1.46146e15 −1.09961
\(563\) −1.32189e15 −0.984919 −0.492459 0.870335i \(-0.663902\pi\)
−0.492459 + 0.870335i \(0.663902\pi\)
\(564\) −2.14322e14 −0.158136
\(565\) 6.11056e14 0.446493
\(566\) 1.07125e15 0.775177
\(567\) −2.22906e14 −0.159740
\(568\) 4.94114e14 0.350680
\(569\) 1.85007e15 1.30038 0.650191 0.759771i \(-0.274689\pi\)
0.650191 + 0.759771i \(0.274689\pi\)
\(570\) −4.89832e14 −0.340985
\(571\) −1.66054e15 −1.14486 −0.572429 0.819954i \(-0.693999\pi\)
−0.572429 + 0.819954i \(0.693999\pi\)
\(572\) 9.41768e14 0.643081
\(573\) 2.36161e14 0.159719
\(574\) −2.53809e15 −1.70016
\(575\) 2.30043e14 0.152629
\(576\) −7.57029e14 −0.497495
\(577\) 2.32812e15 1.51544 0.757721 0.652579i \(-0.226313\pi\)
0.757721 + 0.652579i \(0.226313\pi\)
\(578\) 2.21810e15 1.43014
\(579\) −4.56568e14 −0.291591
\(580\) 3.21751e15 2.03547
\(581\) 2.82320e14 0.176919
\(582\) 1.79972e15 1.11719
\(583\) −1.61164e15 −0.991038
\(584\) 5.11640e14 0.311669
\(585\) −4.93944e14 −0.298072
\(586\) −4.79517e15 −2.86660
\(587\) 1.56510e15 0.926898 0.463449 0.886124i \(-0.346612\pi\)
0.463449 + 0.886124i \(0.346612\pi\)
\(588\) 1.37375e15 0.805995
\(589\) 4.71238e14 0.273909
\(590\) 3.60060e14 0.207343
\(591\) −4.97041e14 −0.283571
\(592\) 1.36530e15 0.771718
\(593\) −5.65084e14 −0.316455 −0.158227 0.987403i \(-0.550578\pi\)
−0.158227 + 0.987403i \(0.550578\pi\)
\(594\) 3.03601e14 0.168452
\(595\) −6.64358e14 −0.365224
\(596\) −3.14111e15 −1.71092
\(597\) 9.48533e14 0.511910
\(598\) 3.74663e15 2.00348
\(599\) 1.21451e15 0.643507 0.321753 0.946824i \(-0.395728\pi\)
0.321753 + 0.946824i \(0.395728\pi\)
\(600\) −5.09015e13 −0.0267238
\(601\) −1.08552e15 −0.564716 −0.282358 0.959309i \(-0.591117\pi\)
−0.282358 + 0.959309i \(0.591117\pi\)
\(602\) −5.13097e15 −2.64496
\(603\) −6.46647e14 −0.330310
\(604\) 2.44911e15 1.23967
\(605\) −1.39622e15 −0.700327
\(606\) −2.02849e15 −1.00826
\(607\) −1.50458e15 −0.741102 −0.370551 0.928812i \(-0.620831\pi\)
−0.370551 + 0.928812i \(0.620831\pi\)
\(608\) 1.04443e15 0.509810
\(609\) −2.54641e15 −1.23178
\(610\) −3.37002e15 −1.61554
\(611\) 3.75866e14 0.178569
\(612\) 2.24515e14 0.105709
\(613\) −7.91295e14 −0.369237 −0.184619 0.982810i \(-0.559105\pi\)
−0.184619 + 0.982810i \(0.559105\pi\)
\(614\) 5.90684e14 0.273168
\(615\) −1.02770e15 −0.471037
\(616\) −8.54654e14 −0.388236
\(617\) 1.66604e15 0.750096 0.375048 0.927005i \(-0.377626\pi\)
0.375048 + 0.927005i \(0.377626\pi\)
\(618\) −1.89069e15 −0.843691
\(619\) 3.78034e15 1.67198 0.835992 0.548741i \(-0.184893\pi\)
0.835992 + 0.548741i \(0.184893\pi\)
\(620\) 2.31104e15 1.01310
\(621\) 6.84614e14 0.297469
\(622\) 9.86422e14 0.424830
\(623\) −1.78869e14 −0.0763574
\(624\) 6.94085e14 0.293696
\(625\) −2.59636e15 −1.08899
\(626\) 3.28848e15 1.36721
\(627\) −2.99285e14 −0.123343
\(628\) 5.36306e15 2.19096
\(629\) −7.74508e14 −0.313652
\(630\) 1.90118e15 0.763222
\(631\) −9.43582e14 −0.375507 −0.187753 0.982216i \(-0.560121\pi\)
−0.187753 + 0.982216i \(0.560121\pi\)
\(632\) −2.03161e15 −0.801488
\(633\) 1.16538e15 0.455773
\(634\) −2.36667e15 −0.917585
\(635\) 2.81810e15 1.08318
\(636\) 3.41060e15 1.29962
\(637\) −2.40921e15 −0.910138
\(638\) 3.46825e15 1.29896
\(639\) −6.71581e14 −0.249370
\(640\) 2.54237e15 0.935940
\(641\) 1.91334e15 0.698350 0.349175 0.937058i \(-0.386462\pi\)
0.349175 + 0.937058i \(0.386462\pi\)
\(642\) 1.64891e15 0.596699
\(643\) 1.55238e15 0.556977 0.278489 0.960440i \(-0.410167\pi\)
0.278489 + 0.960440i \(0.410167\pi\)
\(644\) −8.17397e15 −2.90777
\(645\) −2.07760e15 −0.732795
\(646\) −3.90463e14 −0.136553
\(647\) −1.34109e15 −0.465034 −0.232517 0.972592i \(-0.574696\pi\)
−0.232517 + 0.972592i \(0.574696\pi\)
\(648\) −1.51484e14 −0.0520840
\(649\) 2.19995e14 0.0750009
\(650\) 3.78614e14 0.127989
\(651\) −1.82901e15 −0.613086
\(652\) −7.77168e15 −2.58317
\(653\) −1.19942e15 −0.395319 −0.197659 0.980271i \(-0.563334\pi\)
−0.197659 + 0.980271i \(0.563334\pi\)
\(654\) 2.80486e15 0.916715
\(655\) −3.12734e15 −1.01356
\(656\) 1.44412e15 0.464121
\(657\) −6.95401e14 −0.221629
\(658\) −1.44670e15 −0.457231
\(659\) −4.29128e15 −1.34498 −0.672492 0.740104i \(-0.734776\pi\)
−0.672492 + 0.740104i \(0.734776\pi\)
\(660\) −1.46775e15 −0.456205
\(661\) 2.71996e15 0.838406 0.419203 0.907893i \(-0.362310\pi\)
0.419203 + 0.907893i \(0.362310\pi\)
\(662\) −4.78783e15 −1.46359
\(663\) −3.93741e14 −0.119368
\(664\) 1.91861e14 0.0576850
\(665\) −1.87416e15 −0.558839
\(666\) 2.21640e15 0.655450
\(667\) 7.82084e15 2.29383
\(668\) −3.01197e14 −0.0876153
\(669\) −1.23953e15 −0.357614
\(670\) 5.51530e15 1.57819
\(671\) −2.05907e15 −0.584381
\(672\) −4.05373e15 −1.14110
\(673\) 4.33839e15 1.21128 0.605641 0.795738i \(-0.292917\pi\)
0.605641 + 0.795738i \(0.292917\pi\)
\(674\) 3.86464e15 1.07024
\(675\) 6.91834e13 0.0190034
\(676\) −1.30750e15 −0.356233
\(677\) −4.29274e15 −1.16010 −0.580052 0.814580i \(-0.696968\pi\)
−0.580052 + 0.814580i \(0.696968\pi\)
\(678\) −1.39391e15 −0.373656
\(679\) 6.88593e15 1.83096
\(680\) −4.51489e14 −0.119083
\(681\) −3.36277e15 −0.879809
\(682\) 2.49114e15 0.646524
\(683\) 1.80140e15 0.463763 0.231881 0.972744i \(-0.425512\pi\)
0.231881 + 0.972744i \(0.425512\pi\)
\(684\) 6.33356e14 0.161748
\(685\) 2.24119e15 0.567782
\(686\) 5.81260e14 0.146079
\(687\) −7.29382e14 −0.181841
\(688\) 2.91941e15 0.722037
\(689\) −5.98132e15 −1.46754
\(690\) −5.83913e15 −1.42128
\(691\) 5.96668e15 1.44080 0.720399 0.693560i \(-0.243959\pi\)
0.720399 + 0.693560i \(0.243959\pi\)
\(692\) −4.77215e15 −1.14322
\(693\) 1.16161e15 0.276076
\(694\) 3.41461e15 0.805126
\(695\) −7.04791e15 −1.64871
\(696\) −1.73051e15 −0.401627
\(697\) −8.19221e14 −0.188634
\(698\) 2.28875e15 0.522869
\(699\) −7.14837e14 −0.162025
\(700\) −8.26016e14 −0.185759
\(701\) −4.27057e15 −0.952876 −0.476438 0.879208i \(-0.658072\pi\)
−0.476438 + 0.879208i \(0.658072\pi\)
\(702\) 1.12676e15 0.249447
\(703\) −2.18489e15 −0.479927
\(704\) 3.94505e15 0.859811
\(705\) −5.85788e14 −0.126678
\(706\) 1.25478e16 2.69241
\(707\) −7.76124e15 −1.65244
\(708\) −4.65560e14 −0.0983542
\(709\) 6.74401e15 1.41372 0.706861 0.707353i \(-0.250111\pi\)
0.706861 + 0.707353i \(0.250111\pi\)
\(710\) 5.72797e15 1.19146
\(711\) 2.76129e15 0.569940
\(712\) −1.21557e14 −0.0248966
\(713\) 5.61748e15 1.14169
\(714\) 1.51550e15 0.305644
\(715\) 2.57406e15 0.515151
\(716\) −7.96780e15 −1.58240
\(717\) −3.02118e15 −0.595416
\(718\) 1.41900e16 2.77523
\(719\) 6.94235e15 1.34740 0.673702 0.739003i \(-0.264703\pi\)
0.673702 + 0.739003i \(0.264703\pi\)
\(720\) −1.08173e15 −0.208349
\(721\) −7.23400e15 −1.38272
\(722\) 6.90829e15 1.31044
\(723\) 1.06719e14 0.0200900
\(724\) 5.92930e15 1.10775
\(725\) 7.90331e14 0.146538
\(726\) 3.18500e15 0.586082
\(727\) −1.32393e15 −0.241783 −0.120892 0.992666i \(-0.538575\pi\)
−0.120892 + 0.992666i \(0.538575\pi\)
\(728\) −3.17190e15 −0.574906
\(729\) 2.05891e14 0.0370370
\(730\) 5.93113e15 1.05892
\(731\) −1.65613e15 −0.293460
\(732\) 4.35746e15 0.766342
\(733\) 6.09764e15 1.06436 0.532182 0.846630i \(-0.321372\pi\)
0.532182 + 0.846630i \(0.321372\pi\)
\(734\) −5.71594e15 −0.990283
\(735\) 3.75476e15 0.645656
\(736\) 1.24503e16 2.12496
\(737\) 3.36982e15 0.570868
\(738\) 2.34435e15 0.394196
\(739\) 3.04358e15 0.507972 0.253986 0.967208i \(-0.418258\pi\)
0.253986 + 0.967208i \(0.418258\pi\)
\(740\) −1.07151e16 −1.77510
\(741\) −1.11075e15 −0.182648
\(742\) 2.30220e16 3.75769
\(743\) −1.38530e15 −0.224442 −0.112221 0.993683i \(-0.535796\pi\)
−0.112221 + 0.993683i \(0.535796\pi\)
\(744\) −1.24297e15 −0.199899
\(745\) −8.58533e15 −1.37056
\(746\) 1.68206e16 2.66549
\(747\) −2.60770e14 −0.0410199
\(748\) −1.17000e15 −0.182695
\(749\) 6.30894e15 0.977927
\(750\) 5.38566e15 0.828709
\(751\) −3.49866e15 −0.534419 −0.267209 0.963638i \(-0.586102\pi\)
−0.267209 + 0.963638i \(0.586102\pi\)
\(752\) 8.23142e14 0.124818
\(753\) 5.12721e15 0.771808
\(754\) 1.28718e16 1.92352
\(755\) 6.69394e15 0.993056
\(756\) −2.45824e15 −0.362039
\(757\) 7.88002e15 1.15213 0.576063 0.817405i \(-0.304588\pi\)
0.576063 + 0.817405i \(0.304588\pi\)
\(758\) 6.29423e15 0.913612
\(759\) −3.56768e15 −0.514110
\(760\) −1.27365e15 −0.182212
\(761\) 8.78969e15 1.24841 0.624206 0.781260i \(-0.285422\pi\)
0.624206 + 0.781260i \(0.285422\pi\)
\(762\) −6.42853e15 −0.906482
\(763\) 1.07317e16 1.50240
\(764\) 2.60442e15 0.361990
\(765\) 6.13647e14 0.0846800
\(766\) 2.61566e15 0.358363
\(767\) 8.16474e14 0.111063
\(768\) 5.80707e14 0.0784279
\(769\) −6.36504e15 −0.853505 −0.426752 0.904369i \(-0.640342\pi\)
−0.426752 + 0.904369i \(0.640342\pi\)
\(770\) −9.90749e15 −1.31906
\(771\) 2.20316e14 0.0291237
\(772\) −5.03511e15 −0.660866
\(773\) −7.57190e15 −0.986774 −0.493387 0.869810i \(-0.664241\pi\)
−0.493387 + 0.869810i \(0.664241\pi\)
\(774\) 4.73932e15 0.613254
\(775\) 5.67672e14 0.0729353
\(776\) 4.67959e15 0.596991
\(777\) 8.48020e15 1.07421
\(778\) −1.16308e16 −1.46292
\(779\) −2.31102e15 −0.288635
\(780\) −5.44730e15 −0.675555
\(781\) 3.49976e15 0.430981
\(782\) −4.65459e15 −0.569173
\(783\) 2.35204e15 0.285598
\(784\) −5.27614e15 −0.636177
\(785\) 1.46584e16 1.75511
\(786\) 7.13393e15 0.848213
\(787\) −1.79619e14 −0.0212075 −0.0106038 0.999944i \(-0.503375\pi\)
−0.0106038 + 0.999944i \(0.503375\pi\)
\(788\) −5.48145e15 −0.642690
\(789\) −1.28496e15 −0.149612
\(790\) −2.35513e16 −2.72311
\(791\) −5.33328e15 −0.612383
\(792\) 7.89417e14 0.0900156
\(793\) −7.64187e15 −0.865361
\(794\) −2.01185e16 −2.26246
\(795\) 9.32190e15 1.04108
\(796\) 1.04606e16 1.16020
\(797\) 1.68156e15 0.185221 0.0926106 0.995702i \(-0.470479\pi\)
0.0926106 + 0.995702i \(0.470479\pi\)
\(798\) 4.27524e15 0.467675
\(799\) −4.66953e14 −0.0507301
\(800\) 1.25816e15 0.135750
\(801\) 1.65216e14 0.0177040
\(802\) −1.21608e16 −1.29420
\(803\) 3.62389e15 0.383036
\(804\) −7.13133e15 −0.748621
\(805\) −2.23412e16 −2.32932
\(806\) 9.24545e15 0.957382
\(807\) −1.54631e15 −0.159034
\(808\) −5.27444e15 −0.538783
\(809\) −6.94771e15 −0.704896 −0.352448 0.935831i \(-0.614651\pi\)
−0.352448 + 0.935831i \(0.614651\pi\)
\(810\) −1.75606e15 −0.176959
\(811\) −5.93624e15 −0.594151 −0.297076 0.954854i \(-0.596011\pi\)
−0.297076 + 0.954854i \(0.596011\pi\)
\(812\) −2.80823e16 −2.79173
\(813\) −6.65405e15 −0.657034
\(814\) −1.15502e16 −1.13280
\(815\) −2.12417e16 −2.06929
\(816\) −8.62289e14 −0.0834367
\(817\) −4.67194e15 −0.449031
\(818\) 6.37242e15 0.608361
\(819\) 4.31113e15 0.408818
\(820\) −1.13337e16 −1.06757
\(821\) 1.57327e16 1.47203 0.736015 0.676965i \(-0.236705\pi\)
0.736015 + 0.676965i \(0.236705\pi\)
\(822\) −5.11251e15 −0.475159
\(823\) −1.40061e16 −1.29306 −0.646528 0.762890i \(-0.723780\pi\)
−0.646528 + 0.762890i \(0.723780\pi\)
\(824\) −4.91613e15 −0.450842
\(825\) −3.60530e14 −0.0328432
\(826\) −3.14259e15 −0.284379
\(827\) 7.21399e15 0.648478 0.324239 0.945975i \(-0.394892\pi\)
0.324239 + 0.945975i \(0.394892\pi\)
\(828\) 7.55004e15 0.674190
\(829\) 1.09885e16 0.974739 0.487369 0.873196i \(-0.337957\pi\)
0.487369 + 0.873196i \(0.337957\pi\)
\(830\) 2.22413e15 0.195989
\(831\) −3.44661e15 −0.301708
\(832\) 1.46414e16 1.27322
\(833\) 2.99305e15 0.258563
\(834\) 1.60774e16 1.37975
\(835\) −8.23236e14 −0.0701857
\(836\) −3.30056e15 −0.279546
\(837\) 1.68940e15 0.142149
\(838\) −2.56148e14 −0.0214116
\(839\) −1.66430e16 −1.38210 −0.691051 0.722806i \(-0.742852\pi\)
−0.691051 + 0.722806i \(0.742852\pi\)
\(840\) 4.94342e15 0.407841
\(841\) 1.46686e16 1.20229
\(842\) −2.87726e16 −2.34295
\(843\) 5.16491e15 0.417841
\(844\) 1.28520e16 1.03297
\(845\) −3.57367e15 −0.285366
\(846\) 1.33627e15 0.106013
\(847\) 1.21862e16 0.960527
\(848\) −1.30990e16 −1.02580
\(849\) −3.78587e15 −0.294560
\(850\) −4.70367e14 −0.0363608
\(851\) −2.60454e16 −2.00041
\(852\) −7.40631e15 −0.565176
\(853\) −2.43050e16 −1.84279 −0.921394 0.388630i \(-0.872948\pi\)
−0.921394 + 0.388630i \(0.872948\pi\)
\(854\) 2.94134e16 2.21578
\(855\) 1.73110e15 0.129571
\(856\) 4.28748e15 0.318857
\(857\) −1.40484e15 −0.103808 −0.0519042 0.998652i \(-0.516529\pi\)
−0.0519042 + 0.998652i \(0.516529\pi\)
\(858\) −5.87182e15 −0.431114
\(859\) −9.01491e15 −0.657656 −0.328828 0.944390i \(-0.606654\pi\)
−0.328828 + 0.944390i \(0.606654\pi\)
\(860\) −2.29121e16 −1.66082
\(861\) 8.96976e15 0.646046
\(862\) 2.16739e16 1.55113
\(863\) −1.45313e16 −1.03334 −0.516672 0.856183i \(-0.672829\pi\)
−0.516672 + 0.856183i \(0.672829\pi\)
\(864\) 3.74430e15 0.264573
\(865\) −1.30433e16 −0.915797
\(866\) 4.23186e16 2.95245
\(867\) −7.83891e15 −0.543439
\(868\) −2.01707e16 −1.38951
\(869\) −1.43897e16 −0.985016
\(870\) −2.00608e16 −1.36456
\(871\) 1.25065e16 0.845350
\(872\) 7.29315e15 0.489863
\(873\) −6.36032e15 −0.424522
\(874\) −1.31306e16 −0.870907
\(875\) 2.06062e16 1.35817
\(876\) −7.66899e15 −0.502303
\(877\) −1.53984e16 −1.00225 −0.501127 0.865374i \(-0.667081\pi\)
−0.501127 + 0.865374i \(0.667081\pi\)
\(878\) 3.90779e16 2.52762
\(879\) 1.69464e16 1.08928
\(880\) 5.63716e15 0.360085
\(881\) −7.14900e15 −0.453814 −0.226907 0.973916i \(-0.572861\pi\)
−0.226907 + 0.973916i \(0.572861\pi\)
\(882\) −8.56517e15 −0.540330
\(883\) −2.60603e16 −1.63379 −0.816893 0.576789i \(-0.804305\pi\)
−0.816893 + 0.576789i \(0.804305\pi\)
\(884\) −4.34224e15 −0.270537
\(885\) −1.27248e15 −0.0787883
\(886\) 7.36296e15 0.453072
\(887\) −1.95473e16 −1.19538 −0.597690 0.801727i \(-0.703915\pi\)
−0.597690 + 0.801727i \(0.703915\pi\)
\(888\) 5.76304e15 0.350251
\(889\) −2.45963e16 −1.48563
\(890\) −1.40914e15 −0.0845880
\(891\) −1.07295e15 −0.0640103
\(892\) −1.36698e16 −0.810503
\(893\) −1.31728e15 −0.0776236
\(894\) 1.95845e16 1.14698
\(895\) −2.17777e16 −1.26761
\(896\) −2.21897e16 −1.28368
\(897\) −1.32408e16 −0.761302
\(898\) 9.89332e15 0.565356
\(899\) 1.92993e16 1.09613
\(900\) 7.62966e14 0.0430696
\(901\) 7.43083e15 0.416918
\(902\) −1.22169e16 −0.681281
\(903\) 1.81332e16 1.00506
\(904\) −3.62443e15 −0.199670
\(905\) 1.62061e16 0.887381
\(906\) −1.52699e16 −0.831058
\(907\) −8.30930e15 −0.449495 −0.224747 0.974417i \(-0.572156\pi\)
−0.224747 + 0.974417i \(0.572156\pi\)
\(908\) −3.70852e16 −1.99402
\(909\) 7.16882e15 0.383130
\(910\) −3.67700e16 −1.95329
\(911\) 1.50691e16 0.795675 0.397837 0.917456i \(-0.369761\pi\)
0.397837 + 0.917456i \(0.369761\pi\)
\(912\) −2.43252e15 −0.127669
\(913\) 1.35893e15 0.0708939
\(914\) 3.63922e16 1.88714
\(915\) 1.19099e16 0.613891
\(916\) −8.04374e15 −0.412129
\(917\) 2.72953e16 1.39013
\(918\) −1.39982e15 −0.0708660
\(919\) 1.38543e16 0.697188 0.348594 0.937274i \(-0.386659\pi\)
0.348594 + 0.937274i \(0.386659\pi\)
\(920\) −1.51828e16 −0.759484
\(921\) −2.08752e15 −0.103801
\(922\) −2.74950e16 −1.35904
\(923\) 1.29888e16 0.638203
\(924\) 1.28105e16 0.625704
\(925\) −2.63200e15 −0.127793
\(926\) 4.88814e16 2.35930
\(927\) 6.68182e15 0.320595
\(928\) 4.27739e16 2.04016
\(929\) −3.12674e16 −1.48254 −0.741269 0.671208i \(-0.765776\pi\)
−0.741269 + 0.671208i \(0.765776\pi\)
\(930\) −1.44091e16 −0.679172
\(931\) 8.44342e15 0.395635
\(932\) −7.88334e15 −0.367217
\(933\) −3.48608e15 −0.161432
\(934\) −5.98158e15 −0.275365
\(935\) −3.19785e15 −0.146351
\(936\) 2.92979e15 0.133297
\(937\) −3.08797e16 −1.39671 −0.698354 0.715753i \(-0.746084\pi\)
−0.698354 + 0.715753i \(0.746084\pi\)
\(938\) −4.81374e16 −2.16455
\(939\) −1.16217e16 −0.519529
\(940\) −6.46016e15 −0.287105
\(941\) 1.21992e16 0.538999 0.269499 0.963001i \(-0.413142\pi\)
0.269499 + 0.963001i \(0.413142\pi\)
\(942\) −3.34381e16 −1.46880
\(943\) −2.75490e16 −1.20307
\(944\) 1.78807e15 0.0776316
\(945\) −6.71891e15 −0.290017
\(946\) −2.46976e16 −1.05987
\(947\) −4.47461e15 −0.190911 −0.0954554 0.995434i \(-0.530431\pi\)
−0.0954554 + 0.995434i \(0.530431\pi\)
\(948\) 3.04520e16 1.29172
\(949\) 1.34495e16 0.567206
\(950\) −1.32691e15 −0.0556366
\(951\) 8.36396e15 0.348674
\(952\) 3.94058e15 0.163327
\(953\) 4.43753e16 1.82865 0.914325 0.404981i \(-0.132722\pi\)
0.914325 + 0.404981i \(0.132722\pi\)
\(954\) −2.12647e16 −0.871250
\(955\) 7.11843e15 0.289979
\(956\) −3.33180e16 −1.34946
\(957\) −1.22570e16 −0.493594
\(958\) 5.42961e16 2.17400
\(959\) −1.95611e16 −0.778736
\(960\) −2.28186e16 −0.903229
\(961\) −1.15464e16 −0.454430
\(962\) −4.28664e16 −1.67747
\(963\) −5.82737e15 −0.226740
\(964\) 1.17691e15 0.0455323
\(965\) −1.37620e16 −0.529398
\(966\) 5.09637e16 1.94934
\(967\) 3.87053e16 1.47206 0.736029 0.676950i \(-0.236699\pi\)
0.736029 + 0.676950i \(0.236699\pi\)
\(968\) 8.28158e15 0.313183
\(969\) 1.37992e15 0.0518889
\(970\) 5.42477e16 2.02832
\(971\) 1.17311e16 0.436146 0.218073 0.975932i \(-0.430023\pi\)
0.218073 + 0.975932i \(0.430023\pi\)
\(972\) 2.27060e15 0.0839414
\(973\) 6.15139e16 2.26127
\(974\) 4.63586e16 1.69456
\(975\) −1.33805e15 −0.0486347
\(976\) −1.67356e16 −0.604878
\(977\) 2.66043e16 0.956161 0.478081 0.878316i \(-0.341333\pi\)
0.478081 + 0.878316i \(0.341333\pi\)
\(978\) 4.84555e16 1.73173
\(979\) −8.60977e14 −0.0305975
\(980\) 4.14081e16 1.46333
\(981\) −9.91257e15 −0.348343
\(982\) 3.88263e16 1.35679
\(983\) 1.81683e16 0.631350 0.315675 0.948867i \(-0.397769\pi\)
0.315675 + 0.948867i \(0.397769\pi\)
\(984\) 6.09573e15 0.210646
\(985\) −1.49820e16 −0.514838
\(986\) −1.59912e16 −0.546459
\(987\) 5.11273e15 0.173744
\(988\) −1.22495e16 −0.413956
\(989\) −5.56927e16 −1.87163
\(990\) 9.15124e15 0.305834
\(991\) 2.95832e16 0.983196 0.491598 0.870822i \(-0.336413\pi\)
0.491598 + 0.870822i \(0.336413\pi\)
\(992\) 3.07232e16 1.01544
\(993\) 1.69205e16 0.556151
\(994\) −4.99935e16 −1.63414
\(995\) 2.85910e16 0.929400
\(996\) −2.87582e15 −0.0929683
\(997\) 2.06404e16 0.663582 0.331791 0.943353i \(-0.392347\pi\)
0.331791 + 0.943353i \(0.392347\pi\)
\(998\) 3.06405e16 0.979667
\(999\) −7.83290e15 −0.249065
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.b.1.5 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.5 27 1.1 even 1 trivial