Properties

Label 177.12.a.b.1.19
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 1.19
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+33.0193 q^{2} +243.000 q^{3} -957.729 q^{4} -9842.24 q^{5} +8023.68 q^{6} +66020.7 q^{7} -99246.9 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+33.0193 q^{2} +243.000 q^{3} -957.729 q^{4} -9842.24 q^{5} +8023.68 q^{6} +66020.7 q^{7} -99246.9 q^{8} +59049.0 q^{9} -324984. q^{10} -949141. q^{11} -232728. q^{12} +2.41136e6 q^{13} +2.17995e6 q^{14} -2.39167e6 q^{15} -1.31563e6 q^{16} +3.48871e6 q^{17} +1.94975e6 q^{18} +4.12185e6 q^{19} +9.42620e6 q^{20} +1.60430e7 q^{21} -3.13399e7 q^{22} +6.06828e6 q^{23} -2.41170e7 q^{24} +4.80417e7 q^{25} +7.96214e7 q^{26} +1.43489e7 q^{27} -6.32299e7 q^{28} +2.69799e7 q^{29} -7.89710e7 q^{30} -5.69028e6 q^{31} +1.59817e8 q^{32} -2.30641e8 q^{33} +1.15195e8 q^{34} -6.49792e8 q^{35} -5.65529e7 q^{36} -5.82949e8 q^{37} +1.36101e8 q^{38} +5.85961e8 q^{39} +9.76813e8 q^{40} -8.77528e8 q^{41} +5.29729e8 q^{42} +1.13932e9 q^{43} +9.09020e8 q^{44} -5.81175e8 q^{45} +2.00370e8 q^{46} -2.51964e9 q^{47} -3.19698e8 q^{48} +2.38140e9 q^{49} +1.58630e9 q^{50} +8.47756e8 q^{51} -2.30943e9 q^{52} +4.99466e9 q^{53} +4.73790e8 q^{54} +9.34168e9 q^{55} -6.55235e9 q^{56} +1.00161e9 q^{57} +8.90855e8 q^{58} -7.14924e8 q^{59} +2.29057e9 q^{60} -2.72632e9 q^{61} -1.87889e8 q^{62} +3.89846e9 q^{63} +7.97143e9 q^{64} -2.37332e10 q^{65} -7.61560e9 q^{66} -1.65369e10 q^{67} -3.34124e9 q^{68} +1.47459e9 q^{69} -2.14556e10 q^{70} -2.39838e10 q^{71} -5.86043e9 q^{72} +2.31878e10 q^{73} -1.92485e10 q^{74} +1.16741e10 q^{75} -3.94762e9 q^{76} -6.26629e10 q^{77} +1.93480e10 q^{78} -4.76466e10 q^{79} +1.29488e10 q^{80} +3.48678e9 q^{81} -2.89753e10 q^{82} -3.98712e10 q^{83} -1.53649e10 q^{84} -3.43367e10 q^{85} +3.76194e10 q^{86} +6.55611e9 q^{87} +9.41993e10 q^{88} +3.08001e10 q^{89} -1.91900e10 q^{90} +1.59200e11 q^{91} -5.81176e9 q^{92} -1.38274e9 q^{93} -8.31965e10 q^{94} -4.05683e10 q^{95} +3.88354e10 q^{96} -8.23911e10 q^{97} +7.86322e10 q^{98} -5.60458e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 33.0193 0.729629 0.364815 0.931080i \(-0.381132\pi\)
0.364815 + 0.931080i \(0.381132\pi\)
\(3\) 243.000 0.577350
\(4\) −957.729 −0.467641
\(5\) −9842.24 −1.40851 −0.704254 0.709948i \(-0.748718\pi\)
−0.704254 + 0.709948i \(0.748718\pi\)
\(6\) 8023.68 0.421252
\(7\) 66020.7 1.48471 0.742354 0.670008i \(-0.233709\pi\)
0.742354 + 0.670008i \(0.233709\pi\)
\(8\) −99246.9 −1.07083
\(9\) 59049.0 0.333333
\(10\) −324984. −1.02769
\(11\) −949141. −1.77693 −0.888466 0.458942i \(-0.848228\pi\)
−0.888466 + 0.458942i \(0.848228\pi\)
\(12\) −232728. −0.269993
\(13\) 2.41136e6 1.80125 0.900626 0.434595i \(-0.143109\pi\)
0.900626 + 0.434595i \(0.143109\pi\)
\(14\) 2.17995e6 1.08329
\(15\) −2.39167e6 −0.813202
\(16\) −1.31563e6 −0.313671
\(17\) 3.48871e6 0.595931 0.297965 0.954577i \(-0.403692\pi\)
0.297965 + 0.954577i \(0.403692\pi\)
\(18\) 1.94975e6 0.243210
\(19\) 4.12185e6 0.381898 0.190949 0.981600i \(-0.438843\pi\)
0.190949 + 0.981600i \(0.438843\pi\)
\(20\) 9.42620e6 0.658676
\(21\) 1.60430e7 0.857196
\(22\) −3.13399e7 −1.29650
\(23\) 6.06828e6 0.196590 0.0982952 0.995157i \(-0.468661\pi\)
0.0982952 + 0.995157i \(0.468661\pi\)
\(24\) −2.41170e7 −0.618246
\(25\) 4.80417e7 0.983893
\(26\) 7.96214e7 1.31425
\(27\) 1.43489e7 0.192450
\(28\) −6.32299e7 −0.694310
\(29\) 2.69799e7 0.244259 0.122130 0.992514i \(-0.461028\pi\)
0.122130 + 0.992514i \(0.461028\pi\)
\(30\) −7.89710e7 −0.593336
\(31\) −5.69028e6 −0.0356980 −0.0178490 0.999841i \(-0.505682\pi\)
−0.0178490 + 0.999841i \(0.505682\pi\)
\(32\) 1.59817e8 0.841971
\(33\) −2.30641e8 −1.02591
\(34\) 1.15195e8 0.434809
\(35\) −6.49792e8 −2.09122
\(36\) −5.65529e7 −0.155880
\(37\) −5.82949e8 −1.38204 −0.691021 0.722835i \(-0.742839\pi\)
−0.691021 + 0.722835i \(0.742839\pi\)
\(38\) 1.36101e8 0.278644
\(39\) 5.85961e8 1.03995
\(40\) 9.76813e8 1.50828
\(41\) −8.77528e8 −1.18290 −0.591452 0.806340i \(-0.701445\pi\)
−0.591452 + 0.806340i \(0.701445\pi\)
\(42\) 5.29729e8 0.625435
\(43\) 1.13932e9 1.18187 0.590933 0.806720i \(-0.298760\pi\)
0.590933 + 0.806720i \(0.298760\pi\)
\(44\) 9.09020e8 0.830966
\(45\) −5.81175e8 −0.469502
\(46\) 2.00370e8 0.143438
\(47\) −2.51964e9 −1.60251 −0.801253 0.598326i \(-0.795833\pi\)
−0.801253 + 0.598326i \(0.795833\pi\)
\(48\) −3.19698e8 −0.181098
\(49\) 2.38140e9 1.20436
\(50\) 1.58630e9 0.717877
\(51\) 8.47756e8 0.344061
\(52\) −2.30943e9 −0.842339
\(53\) 4.99466e9 1.64055 0.820273 0.571972i \(-0.193822\pi\)
0.820273 + 0.571972i \(0.193822\pi\)
\(54\) 4.73790e8 0.140417
\(55\) 9.34168e9 2.50282
\(56\) −6.55235e9 −1.58987
\(57\) 1.00161e9 0.220489
\(58\) 8.90855e8 0.178219
\(59\) −7.14924e8 −0.130189
\(60\) 2.29057e9 0.380287
\(61\) −2.72632e9 −0.413298 −0.206649 0.978415i \(-0.566256\pi\)
−0.206649 + 0.978415i \(0.566256\pi\)
\(62\) −1.87889e8 −0.0260463
\(63\) 3.89846e9 0.494902
\(64\) 7.97143e9 0.927997
\(65\) −2.37332e10 −2.53708
\(66\) −7.61560e9 −0.748536
\(67\) −1.65369e10 −1.49639 −0.748193 0.663481i \(-0.769079\pi\)
−0.748193 + 0.663481i \(0.769079\pi\)
\(68\) −3.34124e9 −0.278682
\(69\) 1.47459e9 0.113502
\(70\) −2.14556e10 −1.52582
\(71\) −2.39838e10 −1.57760 −0.788801 0.614649i \(-0.789298\pi\)
−0.788801 + 0.614649i \(0.789298\pi\)
\(72\) −5.86043e9 −0.356945
\(73\) 2.31878e10 1.30913 0.654566 0.756005i \(-0.272851\pi\)
0.654566 + 0.756005i \(0.272851\pi\)
\(74\) −1.92485e10 −1.00838
\(75\) 1.16741e10 0.568051
\(76\) −3.94762e9 −0.178591
\(77\) −6.26629e10 −2.63822
\(78\) 1.93480e10 0.758780
\(79\) −4.76466e10 −1.74214 −0.871069 0.491160i \(-0.836573\pi\)
−0.871069 + 0.491160i \(0.836573\pi\)
\(80\) 1.29488e10 0.441808
\(81\) 3.48678e9 0.111111
\(82\) −2.89753e10 −0.863082
\(83\) −3.98712e10 −1.11104 −0.555520 0.831503i \(-0.687481\pi\)
−0.555520 + 0.831503i \(0.687481\pi\)
\(84\) −1.53649e10 −0.400860
\(85\) −3.43367e10 −0.839373
\(86\) 3.76194e10 0.862325
\(87\) 6.55611e9 0.141023
\(88\) 9.41993e10 1.90280
\(89\) 3.08001e10 0.584664 0.292332 0.956317i \(-0.405569\pi\)
0.292332 + 0.956317i \(0.405569\pi\)
\(90\) −1.91900e10 −0.342563
\(91\) 1.59200e11 2.67433
\(92\) −5.81176e9 −0.0919337
\(93\) −1.38274e9 −0.0206103
\(94\) −8.31965e10 −1.16924
\(95\) −4.05683e10 −0.537907
\(96\) 3.88354e10 0.486112
\(97\) −8.23911e10 −0.974173 −0.487086 0.873354i \(-0.661940\pi\)
−0.487086 + 0.873354i \(0.661940\pi\)
\(98\) 7.86322e10 0.878733
\(99\) −5.60458e10 −0.592311
\(100\) −4.60109e10 −0.460109
\(101\) −6.85656e10 −0.649140 −0.324570 0.945862i \(-0.605220\pi\)
−0.324570 + 0.945862i \(0.605220\pi\)
\(102\) 2.79923e10 0.251037
\(103\) −3.56213e10 −0.302764 −0.151382 0.988475i \(-0.548372\pi\)
−0.151382 + 0.988475i \(0.548372\pi\)
\(104\) −2.39320e11 −1.92884
\(105\) −1.57899e11 −1.20737
\(106\) 1.64920e11 1.19699
\(107\) −1.83661e11 −1.26592 −0.632961 0.774184i \(-0.718161\pi\)
−0.632961 + 0.774184i \(0.718161\pi\)
\(108\) −1.37424e10 −0.0899976
\(109\) 1.51726e11 0.944526 0.472263 0.881458i \(-0.343437\pi\)
0.472263 + 0.881458i \(0.343437\pi\)
\(110\) 3.08455e11 1.82613
\(111\) −1.41657e11 −0.797922
\(112\) −8.68588e10 −0.465709
\(113\) −1.36916e10 −0.0699073 −0.0349536 0.999389i \(-0.511128\pi\)
−0.0349536 + 0.999389i \(0.511128\pi\)
\(114\) 3.30724e10 0.160875
\(115\) −5.97255e10 −0.276899
\(116\) −2.58394e10 −0.114226
\(117\) 1.42389e11 0.600417
\(118\) −2.36063e10 −0.0949896
\(119\) 2.30327e11 0.884783
\(120\) 2.37365e11 0.870804
\(121\) 6.15556e11 2.15749
\(122\) −9.00212e10 −0.301554
\(123\) −2.13239e11 −0.682950
\(124\) 5.44975e9 0.0166939
\(125\) 7.74057e9 0.0226865
\(126\) 1.28724e11 0.361095
\(127\) 5.93794e10 0.159483 0.0797417 0.996816i \(-0.474590\pi\)
0.0797417 + 0.996816i \(0.474590\pi\)
\(128\) −6.40935e10 −0.164877
\(129\) 2.76854e11 0.682351
\(130\) −7.83654e11 −1.85113
\(131\) −2.61661e11 −0.592581 −0.296290 0.955098i \(-0.595750\pi\)
−0.296290 + 0.955098i \(0.595750\pi\)
\(132\) 2.20892e11 0.479759
\(133\) 2.72128e11 0.567007
\(134\) −5.46038e11 −1.09181
\(135\) −1.41225e11 −0.271067
\(136\) −3.46244e11 −0.638143
\(137\) −2.57483e11 −0.455812 −0.227906 0.973683i \(-0.573188\pi\)
−0.227906 + 0.973683i \(0.573188\pi\)
\(138\) 4.86899e10 0.0828140
\(139\) 4.25003e11 0.694721 0.347360 0.937732i \(-0.387078\pi\)
0.347360 + 0.937732i \(0.387078\pi\)
\(140\) 6.22324e11 0.977941
\(141\) −6.12272e11 −0.925207
\(142\) −7.91928e11 −1.15106
\(143\) −2.28872e12 −3.20070
\(144\) −7.76867e10 −0.104557
\(145\) −2.65542e11 −0.344041
\(146\) 7.65643e11 0.955181
\(147\) 5.78681e11 0.695335
\(148\) 5.58307e11 0.646299
\(149\) 4.39403e11 0.490161 0.245080 0.969503i \(-0.421186\pi\)
0.245080 + 0.969503i \(0.421186\pi\)
\(150\) 3.85471e11 0.414467
\(151\) −1.33918e12 −1.38824 −0.694122 0.719857i \(-0.744207\pi\)
−0.694122 + 0.719857i \(0.744207\pi\)
\(152\) −4.09081e11 −0.408950
\(153\) 2.06005e11 0.198644
\(154\) −2.06908e12 −1.92493
\(155\) 5.60052e10 0.0502810
\(156\) −5.61192e11 −0.486325
\(157\) −2.14367e12 −1.79354 −0.896768 0.442500i \(-0.854092\pi\)
−0.896768 + 0.442500i \(0.854092\pi\)
\(158\) −1.57325e12 −1.27112
\(159\) 1.21370e12 0.947170
\(160\) −1.57295e12 −1.18592
\(161\) 4.00632e11 0.291879
\(162\) 1.15131e11 0.0810699
\(163\) −1.32179e12 −0.899768 −0.449884 0.893087i \(-0.648535\pi\)
−0.449884 + 0.893087i \(0.648535\pi\)
\(164\) 8.40434e11 0.553175
\(165\) 2.27003e12 1.44501
\(166\) −1.31652e12 −0.810648
\(167\) 6.76182e11 0.402831 0.201415 0.979506i \(-0.435446\pi\)
0.201415 + 0.979506i \(0.435446\pi\)
\(168\) −1.59222e12 −0.917915
\(169\) 4.02252e12 2.24451
\(170\) −1.13377e12 −0.612431
\(171\) 2.43391e11 0.127299
\(172\) −1.09116e12 −0.552690
\(173\) 1.22961e12 0.603273 0.301637 0.953423i \(-0.402467\pi\)
0.301637 + 0.953423i \(0.402467\pi\)
\(174\) 2.16478e11 0.102895
\(175\) 3.17174e12 1.46079
\(176\) 1.24872e12 0.557372
\(177\) −1.73727e11 −0.0751646
\(178\) 1.01699e12 0.426588
\(179\) −2.71937e12 −1.10605 −0.553027 0.833164i \(-0.686527\pi\)
−0.553027 + 0.833164i \(0.686527\pi\)
\(180\) 5.56608e11 0.219559
\(181\) 9.07548e11 0.347246 0.173623 0.984812i \(-0.444453\pi\)
0.173623 + 0.984812i \(0.444453\pi\)
\(182\) 5.25666e12 1.95127
\(183\) −6.62497e11 −0.238618
\(184\) −6.02258e11 −0.210516
\(185\) 5.73753e12 1.94662
\(186\) −4.56570e10 −0.0150379
\(187\) −3.31128e12 −1.05893
\(188\) 2.41313e12 0.749398
\(189\) 9.47325e11 0.285732
\(190\) −1.33953e12 −0.392472
\(191\) −2.31320e12 −0.658462 −0.329231 0.944249i \(-0.606789\pi\)
−0.329231 + 0.944249i \(0.606789\pi\)
\(192\) 1.93706e12 0.535779
\(193\) −6.12626e12 −1.64676 −0.823380 0.567491i \(-0.807914\pi\)
−0.823380 + 0.567491i \(0.807914\pi\)
\(194\) −2.72049e12 −0.710785
\(195\) −5.76718e12 −1.46478
\(196\) −2.28074e12 −0.563206
\(197\) 2.53885e12 0.609638 0.304819 0.952410i \(-0.401404\pi\)
0.304819 + 0.952410i \(0.401404\pi\)
\(198\) −1.85059e12 −0.432167
\(199\) −4.70949e12 −1.06975 −0.534875 0.844931i \(-0.679641\pi\)
−0.534875 + 0.844931i \(0.679641\pi\)
\(200\) −4.76799e12 −1.05359
\(201\) −4.01848e12 −0.863939
\(202\) −2.26398e12 −0.473632
\(203\) 1.78123e12 0.362653
\(204\) −8.11921e11 −0.160897
\(205\) 8.63685e12 1.66613
\(206\) −1.17619e12 −0.220906
\(207\) 3.58326e11 0.0655301
\(208\) −3.17246e12 −0.565000
\(209\) −3.91222e12 −0.678607
\(210\) −5.21372e12 −0.880930
\(211\) 7.88349e12 1.29767 0.648836 0.760928i \(-0.275256\pi\)
0.648836 + 0.760928i \(0.275256\pi\)
\(212\) −4.78353e12 −0.767187
\(213\) −5.82807e12 −0.910829
\(214\) −6.06436e12 −0.923654
\(215\) −1.12135e13 −1.66467
\(216\) −1.42408e12 −0.206082
\(217\) −3.75676e11 −0.0530011
\(218\) 5.00987e12 0.689154
\(219\) 5.63463e12 0.755828
\(220\) −8.94679e12 −1.17042
\(221\) 8.41255e12 1.07342
\(222\) −4.67740e12 −0.582187
\(223\) 3.02311e12 0.367094 0.183547 0.983011i \(-0.441242\pi\)
0.183547 + 0.983011i \(0.441242\pi\)
\(224\) 1.05512e13 1.25008
\(225\) 2.83681e12 0.327964
\(226\) −4.52086e11 −0.0510064
\(227\) −1.20851e13 −1.33078 −0.665391 0.746495i \(-0.731735\pi\)
−0.665391 + 0.746495i \(0.731735\pi\)
\(228\) −9.59271e11 −0.103110
\(229\) 7.73469e12 0.811611 0.405805 0.913960i \(-0.366991\pi\)
0.405805 + 0.913960i \(0.366991\pi\)
\(230\) −1.97209e12 −0.202034
\(231\) −1.52271e13 −1.52318
\(232\) −2.67767e12 −0.261561
\(233\) −6.85784e12 −0.654229 −0.327115 0.944985i \(-0.606076\pi\)
−0.327115 + 0.944985i \(0.606076\pi\)
\(234\) 4.70157e12 0.438082
\(235\) 2.47989e13 2.25714
\(236\) 6.84704e11 0.0608817
\(237\) −1.15781e13 −1.00582
\(238\) 7.60523e12 0.645563
\(239\) −2.25392e12 −0.186960 −0.0934802 0.995621i \(-0.529799\pi\)
−0.0934802 + 0.995621i \(0.529799\pi\)
\(240\) 3.14655e12 0.255078
\(241\) 6.16812e12 0.488719 0.244359 0.969685i \(-0.421422\pi\)
0.244359 + 0.969685i \(0.421422\pi\)
\(242\) 2.03252e13 1.57417
\(243\) 8.47289e11 0.0641500
\(244\) 2.61108e12 0.193275
\(245\) −2.34384e13 −1.69634
\(246\) −7.04100e12 −0.498301
\(247\) 9.93929e12 0.687895
\(248\) 5.64743e11 0.0382267
\(249\) −9.68871e12 −0.641460
\(250\) 2.55588e11 0.0165527
\(251\) −2.70928e13 −1.71652 −0.858260 0.513215i \(-0.828454\pi\)
−0.858260 + 0.513215i \(0.828454\pi\)
\(252\) −3.73366e12 −0.231437
\(253\) −5.75965e12 −0.349328
\(254\) 1.96066e12 0.116364
\(255\) −8.34383e12 −0.484612
\(256\) −1.84418e13 −1.04830
\(257\) 2.86679e13 1.59501 0.797507 0.603310i \(-0.206152\pi\)
0.797507 + 0.603310i \(0.206152\pi\)
\(258\) 9.14152e12 0.497863
\(259\) −3.84867e13 −2.05193
\(260\) 2.27300e13 1.18644
\(261\) 1.59313e12 0.0814197
\(262\) −8.63986e12 −0.432364
\(263\) −5.44223e12 −0.266698 −0.133349 0.991069i \(-0.542573\pi\)
−0.133349 + 0.991069i \(0.542573\pi\)
\(264\) 2.28904e13 1.09858
\(265\) −4.91586e13 −2.31072
\(266\) 8.98545e12 0.413705
\(267\) 7.48441e12 0.337556
\(268\) 1.58379e13 0.699772
\(269\) 3.06659e13 1.32745 0.663726 0.747976i \(-0.268974\pi\)
0.663726 + 0.747976i \(0.268974\pi\)
\(270\) −4.66316e12 −0.197779
\(271\) 2.95139e13 1.22658 0.613290 0.789858i \(-0.289846\pi\)
0.613290 + 0.789858i \(0.289846\pi\)
\(272\) −4.58985e12 −0.186926
\(273\) 3.86856e13 1.54403
\(274\) −8.50191e12 −0.332574
\(275\) −4.55983e13 −1.74831
\(276\) −1.41226e12 −0.0530780
\(277\) −2.11178e13 −0.778053 −0.389027 0.921227i \(-0.627189\pi\)
−0.389027 + 0.921227i \(0.627189\pi\)
\(278\) 1.40333e13 0.506889
\(279\) −3.36005e11 −0.0118993
\(280\) 6.44898e13 2.23935
\(281\) −2.82525e12 −0.0961994 −0.0480997 0.998843i \(-0.515317\pi\)
−0.0480997 + 0.998843i \(0.515317\pi\)
\(282\) −2.02168e13 −0.675058
\(283\) −5.38064e13 −1.76201 −0.881005 0.473107i \(-0.843132\pi\)
−0.881005 + 0.473107i \(0.843132\pi\)
\(284\) 2.29700e13 0.737751
\(285\) −9.85810e12 −0.310561
\(286\) −7.55720e13 −2.33533
\(287\) −5.79350e13 −1.75627
\(288\) 9.43701e12 0.280657
\(289\) −2.21008e13 −0.644867
\(290\) −8.76801e12 −0.251022
\(291\) −2.00210e13 −0.562439
\(292\) −2.22076e13 −0.612204
\(293\) 4.23289e13 1.14516 0.572578 0.819850i \(-0.305943\pi\)
0.572578 + 0.819850i \(0.305943\pi\)
\(294\) 1.91076e13 0.507337
\(295\) 7.03646e12 0.183372
\(296\) 5.78559e13 1.47994
\(297\) −1.36191e13 −0.341971
\(298\) 1.45088e13 0.357636
\(299\) 1.46328e13 0.354109
\(300\) −1.11806e13 −0.265644
\(301\) 7.52186e13 1.75473
\(302\) −4.42187e13 −1.01290
\(303\) −1.66614e13 −0.374781
\(304\) −5.42284e12 −0.119790
\(305\) 2.68331e13 0.582133
\(306\) 6.80213e12 0.144936
\(307\) −6.19623e13 −1.29678 −0.648390 0.761308i \(-0.724557\pi\)
−0.648390 + 0.761308i \(0.724557\pi\)
\(308\) 6.00141e13 1.23374
\(309\) −8.65597e12 −0.174801
\(310\) 1.84925e12 0.0366865
\(311\) 2.65568e13 0.517600 0.258800 0.965931i \(-0.416673\pi\)
0.258800 + 0.965931i \(0.416673\pi\)
\(312\) −5.81549e13 −1.11362
\(313\) 2.26996e13 0.427095 0.213548 0.976933i \(-0.431498\pi\)
0.213548 + 0.976933i \(0.431498\pi\)
\(314\) −7.07825e13 −1.30862
\(315\) −3.83696e13 −0.697074
\(316\) 4.56325e13 0.814696
\(317\) 7.46373e13 1.30957 0.654787 0.755813i \(-0.272758\pi\)
0.654787 + 0.755813i \(0.272758\pi\)
\(318\) 4.00755e13 0.691083
\(319\) −2.56077e13 −0.434032
\(320\) −7.84568e13 −1.30709
\(321\) −4.46297e13 −0.730880
\(322\) 1.32286e13 0.212964
\(323\) 1.43800e13 0.227585
\(324\) −3.33939e12 −0.0519601
\(325\) 1.15846e14 1.77224
\(326\) −4.36445e13 −0.656497
\(327\) 3.68694e13 0.545322
\(328\) 8.70920e13 1.26669
\(329\) −1.66348e14 −2.37925
\(330\) 7.49546e13 1.05432
\(331\) 3.58231e13 0.495574 0.247787 0.968814i \(-0.420297\pi\)
0.247787 + 0.968814i \(0.420297\pi\)
\(332\) 3.81858e13 0.519568
\(333\) −3.44226e13 −0.460681
\(334\) 2.23270e13 0.293917
\(335\) 1.62761e14 2.10767
\(336\) −2.11067e13 −0.268877
\(337\) 2.67577e13 0.335340 0.167670 0.985843i \(-0.446376\pi\)
0.167670 + 0.985843i \(0.446376\pi\)
\(338\) 1.32820e14 1.63766
\(339\) −3.32706e12 −0.0403610
\(340\) 3.28853e13 0.392525
\(341\) 5.40088e12 0.0634330
\(342\) 8.03660e12 0.0928814
\(343\) 2.66775e13 0.303408
\(344\) −1.13074e14 −1.26558
\(345\) −1.45133e13 −0.159868
\(346\) 4.06008e13 0.440166
\(347\) 1.06220e13 0.113343 0.0566714 0.998393i \(-0.481951\pi\)
0.0566714 + 0.998393i \(0.481951\pi\)
\(348\) −6.27897e12 −0.0659482
\(349\) 7.47328e13 0.772629 0.386315 0.922367i \(-0.373748\pi\)
0.386315 + 0.922367i \(0.373748\pi\)
\(350\) 1.04729e14 1.06584
\(351\) 3.46004e13 0.346651
\(352\) −1.51688e14 −1.49612
\(353\) 1.13873e14 1.10576 0.552881 0.833260i \(-0.313528\pi\)
0.552881 + 0.833260i \(0.313528\pi\)
\(354\) −5.73632e12 −0.0548423
\(355\) 2.36055e14 2.22206
\(356\) −2.94981e13 −0.273413
\(357\) 5.59695e13 0.510829
\(358\) −8.97914e13 −0.807009
\(359\) −1.86736e14 −1.65275 −0.826376 0.563119i \(-0.809601\pi\)
−0.826376 + 0.563119i \(0.809601\pi\)
\(360\) 5.76798e13 0.502759
\(361\) −9.95006e13 −0.854154
\(362\) 2.99666e13 0.253361
\(363\) 1.49580e14 1.24563
\(364\) −1.52470e14 −1.25063
\(365\) −2.28220e14 −1.84392
\(366\) −2.18751e13 −0.174102
\(367\) 1.29786e12 0.0101757 0.00508785 0.999987i \(-0.498380\pi\)
0.00508785 + 0.999987i \(0.498380\pi\)
\(368\) −7.98361e12 −0.0616646
\(369\) −5.18171e13 −0.394302
\(370\) 1.89449e14 1.42031
\(371\) 3.29751e14 2.43573
\(372\) 1.32429e12 0.00963821
\(373\) −1.29408e14 −0.928033 −0.464017 0.885826i \(-0.653592\pi\)
−0.464017 + 0.885826i \(0.653592\pi\)
\(374\) −1.09336e14 −0.772625
\(375\) 1.88096e12 0.0130981
\(376\) 2.50066e14 1.71602
\(377\) 6.50583e13 0.439972
\(378\) 3.12800e13 0.208478
\(379\) −7.19913e13 −0.472895 −0.236447 0.971644i \(-0.575983\pi\)
−0.236447 + 0.971644i \(0.575983\pi\)
\(380\) 3.88534e13 0.251547
\(381\) 1.44292e13 0.0920778
\(382\) −7.63803e13 −0.480433
\(383\) 1.96384e14 1.21762 0.608811 0.793315i \(-0.291647\pi\)
0.608811 + 0.793315i \(0.291647\pi\)
\(384\) −1.55747e13 −0.0951916
\(385\) 6.16744e14 3.71596
\(386\) −2.02285e14 −1.20152
\(387\) 6.72756e13 0.393956
\(388\) 7.89084e13 0.455563
\(389\) −2.50996e14 −1.42871 −0.714355 0.699783i \(-0.753280\pi\)
−0.714355 + 0.699783i \(0.753280\pi\)
\(390\) −1.90428e14 −1.06875
\(391\) 2.11705e13 0.117154
\(392\) −2.36347e14 −1.28966
\(393\) −6.35837e13 −0.342127
\(394\) 8.38308e13 0.444810
\(395\) 4.68949e14 2.45382
\(396\) 5.36767e13 0.276989
\(397\) 9.23990e13 0.470240 0.235120 0.971966i \(-0.424452\pi\)
0.235120 + 0.971966i \(0.424452\pi\)
\(398\) −1.55504e14 −0.780521
\(399\) 6.61270e13 0.327362
\(400\) −6.32051e13 −0.308619
\(401\) −1.25590e14 −0.604871 −0.302435 0.953170i \(-0.597800\pi\)
−0.302435 + 0.953170i \(0.597800\pi\)
\(402\) −1.32687e14 −0.630355
\(403\) −1.37213e13 −0.0643012
\(404\) 6.56672e13 0.303565
\(405\) −3.43178e13 −0.156501
\(406\) 5.88149e13 0.264602
\(407\) 5.53301e14 2.45579
\(408\) −8.41372e13 −0.368432
\(409\) −1.29788e14 −0.560732 −0.280366 0.959893i \(-0.590456\pi\)
−0.280366 + 0.959893i \(0.590456\pi\)
\(410\) 2.85182e14 1.21566
\(411\) −6.25685e13 −0.263163
\(412\) 3.41155e13 0.141585
\(413\) −4.71998e13 −0.193292
\(414\) 1.18316e13 0.0478127
\(415\) 3.92422e14 1.56491
\(416\) 3.85376e14 1.51660
\(417\) 1.03276e14 0.401097
\(418\) −1.29179e14 −0.495132
\(419\) 3.12673e14 1.18280 0.591402 0.806377i \(-0.298575\pi\)
0.591402 + 0.806377i \(0.298575\pi\)
\(420\) 1.51225e14 0.564614
\(421\) 1.73027e14 0.637620 0.318810 0.947819i \(-0.396717\pi\)
0.318810 + 0.947819i \(0.396717\pi\)
\(422\) 2.60307e14 0.946820
\(423\) −1.48782e14 −0.534169
\(424\) −4.95704e14 −1.75675
\(425\) 1.67603e14 0.586332
\(426\) −1.92438e14 −0.664567
\(427\) −1.79994e14 −0.613627
\(428\) 1.75898e14 0.591997
\(429\) −5.56160e14 −1.84793
\(430\) −3.70260e14 −1.21459
\(431\) −2.13247e14 −0.690649 −0.345325 0.938483i \(-0.612231\pi\)
−0.345325 + 0.938483i \(0.612231\pi\)
\(432\) −1.88779e13 −0.0603660
\(433\) 1.15307e14 0.364060 0.182030 0.983293i \(-0.441733\pi\)
0.182030 + 0.983293i \(0.441733\pi\)
\(434\) −1.24046e13 −0.0386712
\(435\) −6.45268e13 −0.198632
\(436\) −1.45312e14 −0.441699
\(437\) 2.50125e13 0.0750775
\(438\) 1.86051e14 0.551474
\(439\) 1.67741e14 0.491002 0.245501 0.969396i \(-0.421047\pi\)
0.245501 + 0.969396i \(0.421047\pi\)
\(440\) −9.27133e14 −2.68011
\(441\) 1.40620e14 0.401452
\(442\) 2.77776e14 0.783199
\(443\) −1.91042e14 −0.531996 −0.265998 0.963974i \(-0.585702\pi\)
−0.265998 + 0.963974i \(0.585702\pi\)
\(444\) 1.35669e14 0.373141
\(445\) −3.03142e14 −0.823504
\(446\) 9.98209e13 0.267843
\(447\) 1.06775e14 0.282994
\(448\) 5.26280e14 1.37780
\(449\) 2.10346e14 0.543976 0.271988 0.962301i \(-0.412319\pi\)
0.271988 + 0.962301i \(0.412319\pi\)
\(450\) 9.36694e13 0.239292
\(451\) 8.32898e14 2.10194
\(452\) 1.31128e13 0.0326915
\(453\) −3.25421e14 −0.801503
\(454\) −3.99040e14 −0.970977
\(455\) −1.56688e15 −3.76682
\(456\) −9.94068e13 −0.236107
\(457\) −3.40224e14 −0.798410 −0.399205 0.916862i \(-0.630714\pi\)
−0.399205 + 0.916862i \(0.630714\pi\)
\(458\) 2.55394e14 0.592175
\(459\) 5.00592e13 0.114687
\(460\) 5.72008e13 0.129489
\(461\) −1.24588e13 −0.0278690 −0.0139345 0.999903i \(-0.504436\pi\)
−0.0139345 + 0.999903i \(0.504436\pi\)
\(462\) −5.02787e14 −1.11136
\(463\) 2.39213e14 0.522503 0.261252 0.965271i \(-0.415865\pi\)
0.261252 + 0.965271i \(0.415865\pi\)
\(464\) −3.54955e13 −0.0766170
\(465\) 1.36093e13 0.0290297
\(466\) −2.26441e14 −0.477345
\(467\) −6.56649e14 −1.36801 −0.684007 0.729476i \(-0.739764\pi\)
−0.684007 + 0.729476i \(0.739764\pi\)
\(468\) −1.36370e14 −0.280780
\(469\) −1.09178e15 −2.22170
\(470\) 8.18841e14 1.64688
\(471\) −5.20912e14 −1.03550
\(472\) 7.09540e13 0.139411
\(473\) −1.08137e15 −2.10010
\(474\) −3.82301e14 −0.733879
\(475\) 1.98021e14 0.375747
\(476\) −2.20591e14 −0.413761
\(477\) 2.94929e14 0.546849
\(478\) −7.44227e13 −0.136412
\(479\) 7.96048e14 1.44243 0.721213 0.692713i \(-0.243585\pi\)
0.721213 + 0.692713i \(0.243585\pi\)
\(480\) −3.82228e14 −0.684692
\(481\) −1.40570e15 −2.48940
\(482\) 2.03667e14 0.356584
\(483\) 9.73535e13 0.168516
\(484\) −5.89536e14 −1.00893
\(485\) 8.10914e14 1.37213
\(486\) 2.79768e13 0.0468057
\(487\) 6.45751e13 0.106821 0.0534104 0.998573i \(-0.482991\pi\)
0.0534104 + 0.998573i \(0.482991\pi\)
\(488\) 2.70579e14 0.442574
\(489\) −3.21195e14 −0.519481
\(490\) −7.73917e14 −1.23770
\(491\) −4.12415e14 −0.652208 −0.326104 0.945334i \(-0.605736\pi\)
−0.326104 + 0.945334i \(0.605736\pi\)
\(492\) 2.04225e14 0.319376
\(493\) 9.41249e13 0.145562
\(494\) 3.28188e14 0.501908
\(495\) 5.51617e14 0.834274
\(496\) 7.48631e12 0.0111974
\(497\) −1.58343e15 −2.34228
\(498\) −3.19914e14 −0.468028
\(499\) −5.97089e13 −0.0863944 −0.0431972 0.999067i \(-0.513754\pi\)
−0.0431972 + 0.999067i \(0.513754\pi\)
\(500\) −7.41336e12 −0.0106091
\(501\) 1.64312e14 0.232575
\(502\) −8.94585e14 −1.25242
\(503\) −3.73346e14 −0.516996 −0.258498 0.966012i \(-0.583228\pi\)
−0.258498 + 0.966012i \(0.583228\pi\)
\(504\) −3.86910e14 −0.529958
\(505\) 6.74839e14 0.914319
\(506\) −1.90179e14 −0.254880
\(507\) 9.77472e14 1.29587
\(508\) −5.68694e13 −0.0745810
\(509\) −4.57037e14 −0.592930 −0.296465 0.955044i \(-0.595808\pi\)
−0.296465 + 0.955044i \(0.595808\pi\)
\(510\) −2.75507e14 −0.353587
\(511\) 1.53087e15 1.94368
\(512\) −4.77672e14 −0.599991
\(513\) 5.91441e13 0.0734964
\(514\) 9.46594e14 1.16377
\(515\) 3.50593e14 0.426446
\(516\) −2.65151e14 −0.319095
\(517\) 2.39149e15 2.84754
\(518\) −1.27080e15 −1.49715
\(519\) 2.98795e14 0.348300
\(520\) 2.35545e15 2.71679
\(521\) −1.14282e15 −1.30428 −0.652139 0.758099i \(-0.726128\pi\)
−0.652139 + 0.758099i \(0.726128\pi\)
\(522\) 5.26041e13 0.0594062
\(523\) 1.56677e15 1.75084 0.875418 0.483366i \(-0.160586\pi\)
0.875418 + 0.483366i \(0.160586\pi\)
\(524\) 2.50601e14 0.277115
\(525\) 7.70734e14 0.843389
\(526\) −1.79698e14 −0.194591
\(527\) −1.98517e13 −0.0212736
\(528\) 3.03439e14 0.321799
\(529\) −9.15986e14 −0.961352
\(530\) −1.62318e15 −1.68597
\(531\) −4.22156e13 −0.0433963
\(532\) −2.60625e14 −0.265156
\(533\) −2.11604e15 −2.13071
\(534\) 2.47130e14 0.246291
\(535\) 1.80764e15 1.78306
\(536\) 1.64124e15 1.60238
\(537\) −6.60806e14 −0.638580
\(538\) 1.01257e15 0.968547
\(539\) −2.26029e15 −2.14006
\(540\) 1.35256e14 0.126762
\(541\) 7.31287e14 0.678426 0.339213 0.940710i \(-0.389839\pi\)
0.339213 + 0.940710i \(0.389839\pi\)
\(542\) 9.74528e14 0.894949
\(543\) 2.20534e14 0.200483
\(544\) 5.57554e14 0.501756
\(545\) −1.49332e15 −1.33037
\(546\) 1.27737e15 1.12657
\(547\) −2.95927e14 −0.258377 −0.129189 0.991620i \(-0.541237\pi\)
−0.129189 + 0.991620i \(0.541237\pi\)
\(548\) 2.46599e14 0.213157
\(549\) −1.60987e14 −0.137766
\(550\) −1.50562e15 −1.27562
\(551\) 1.11207e14 0.0932822
\(552\) −1.46349e14 −0.121541
\(553\) −3.14566e15 −2.58657
\(554\) −6.97293e14 −0.567690
\(555\) 1.39422e15 1.12388
\(556\) −4.07037e14 −0.324880
\(557\) −1.63315e15 −1.29069 −0.645344 0.763892i \(-0.723286\pi\)
−0.645344 + 0.763892i \(0.723286\pi\)
\(558\) −1.10946e13 −0.00868211
\(559\) 2.74731e15 2.12884
\(560\) 8.54886e14 0.655955
\(561\) −8.04640e14 −0.611373
\(562\) −9.32877e13 −0.0701899
\(563\) 1.53617e15 1.14457 0.572287 0.820053i \(-0.306056\pi\)
0.572287 + 0.820053i \(0.306056\pi\)
\(564\) 5.86390e14 0.432665
\(565\) 1.34756e14 0.0984649
\(566\) −1.77665e15 −1.28561
\(567\) 2.30200e14 0.164967
\(568\) 2.38032e15 1.68935
\(569\) 2.07875e15 1.46112 0.730559 0.682849i \(-0.239259\pi\)
0.730559 + 0.682849i \(0.239259\pi\)
\(570\) −3.25507e14 −0.226594
\(571\) 9.23391e14 0.636630 0.318315 0.947985i \(-0.396883\pi\)
0.318315 + 0.947985i \(0.396883\pi\)
\(572\) 2.19198e15 1.49678
\(573\) −5.62109e14 −0.380163
\(574\) −1.91297e15 −1.28142
\(575\) 2.91530e14 0.193424
\(576\) 4.70705e14 0.309332
\(577\) 1.92165e15 1.25086 0.625429 0.780281i \(-0.284924\pi\)
0.625429 + 0.780281i \(0.284924\pi\)
\(578\) −7.29752e14 −0.470514
\(579\) −1.48868e15 −0.950757
\(580\) 2.54318e14 0.160888
\(581\) −2.63233e15 −1.64957
\(582\) −6.61080e14 −0.410372
\(583\) −4.74063e15 −2.91514
\(584\) −2.30132e15 −1.40186
\(585\) −1.40142e15 −0.845692
\(586\) 1.39767e15 0.835540
\(587\) −1.85374e15 −1.09784 −0.548921 0.835874i \(-0.684961\pi\)
−0.548921 + 0.835874i \(0.684961\pi\)
\(588\) −5.54220e14 −0.325167
\(589\) −2.34545e13 −0.0136330
\(590\) 2.32339e14 0.133794
\(591\) 6.16939e14 0.351975
\(592\) 7.66946e14 0.433506
\(593\) −5.93724e14 −0.332494 −0.166247 0.986084i \(-0.553165\pi\)
−0.166247 + 0.986084i \(0.553165\pi\)
\(594\) −4.49694e14 −0.249512
\(595\) −2.26693e15 −1.24622
\(596\) −4.20829e14 −0.229219
\(597\) −1.14441e15 −0.617620
\(598\) 4.83165e14 0.258368
\(599\) 3.51207e15 1.86087 0.930435 0.366458i \(-0.119430\pi\)
0.930435 + 0.366458i \(0.119430\pi\)
\(600\) −1.15862e15 −0.608288
\(601\) 2.72349e15 1.41683 0.708413 0.705798i \(-0.249411\pi\)
0.708413 + 0.705798i \(0.249411\pi\)
\(602\) 2.48366e15 1.28030
\(603\) −9.76490e14 −0.498796
\(604\) 1.28257e15 0.649200
\(605\) −6.05846e15 −3.03884
\(606\) −5.50148e14 −0.273451
\(607\) −1.27778e15 −0.629386 −0.314693 0.949193i \(-0.601902\pi\)
−0.314693 + 0.949193i \(0.601902\pi\)
\(608\) 6.58741e14 0.321547
\(609\) 4.32839e14 0.209378
\(610\) 8.86010e14 0.424742
\(611\) −6.07576e15 −2.88652
\(612\) −1.97297e14 −0.0928939
\(613\) −1.38355e15 −0.645597 −0.322798 0.946468i \(-0.604624\pi\)
−0.322798 + 0.946468i \(0.604624\pi\)
\(614\) −2.04595e15 −0.946169
\(615\) 2.09875e15 0.961941
\(616\) 6.21910e15 2.82510
\(617\) 1.21360e14 0.0546394 0.0273197 0.999627i \(-0.491303\pi\)
0.0273197 + 0.999627i \(0.491303\pi\)
\(618\) −2.85814e14 −0.127540
\(619\) 1.20408e14 0.0532544 0.0266272 0.999645i \(-0.491523\pi\)
0.0266272 + 0.999645i \(0.491523\pi\)
\(620\) −5.36378e13 −0.0235134
\(621\) 8.70731e13 0.0378338
\(622\) 8.76887e14 0.377656
\(623\) 2.03344e15 0.868055
\(624\) −7.70909e14 −0.326203
\(625\) −2.42197e15 −1.01585
\(626\) 7.49524e14 0.311621
\(627\) −9.50669e14 −0.391794
\(628\) 2.05306e15 0.838731
\(629\) −2.03374e15 −0.823601
\(630\) −1.26693e15 −0.508605
\(631\) −3.62981e15 −1.44452 −0.722259 0.691623i \(-0.756896\pi\)
−0.722259 + 0.691623i \(0.756896\pi\)
\(632\) 4.72878e15 1.86554
\(633\) 1.91569e15 0.749212
\(634\) 2.46447e15 0.955504
\(635\) −5.84427e14 −0.224634
\(636\) −1.16240e15 −0.442935
\(637\) 5.74243e15 2.16935
\(638\) −8.45547e14 −0.316682
\(639\) −1.41622e15 −0.525867
\(640\) 6.30824e14 0.232230
\(641\) 4.57809e15 1.67096 0.835479 0.549522i \(-0.185190\pi\)
0.835479 + 0.549522i \(0.185190\pi\)
\(642\) −1.47364e15 −0.533272
\(643\) 2.98723e15 1.07179 0.535893 0.844286i \(-0.319975\pi\)
0.535893 + 0.844286i \(0.319975\pi\)
\(644\) −3.83697e14 −0.136495
\(645\) −2.72487e15 −0.961097
\(646\) 4.74815e14 0.166053
\(647\) −1.14203e15 −0.396008 −0.198004 0.980201i \(-0.563446\pi\)
−0.198004 + 0.980201i \(0.563446\pi\)
\(648\) −3.46053e14 −0.118982
\(649\) 6.78564e14 0.231337
\(650\) 3.82515e15 1.29308
\(651\) −9.12893e13 −0.0306002
\(652\) 1.26592e15 0.420769
\(653\) 2.65382e15 0.874679 0.437340 0.899296i \(-0.355921\pi\)
0.437340 + 0.899296i \(0.355921\pi\)
\(654\) 1.21740e15 0.397883
\(655\) 2.57534e15 0.834654
\(656\) 1.15450e15 0.371043
\(657\) 1.36922e15 0.436377
\(658\) −5.49269e15 −1.73597
\(659\) −5.23584e13 −0.0164103 −0.00820515 0.999966i \(-0.502612\pi\)
−0.00820515 + 0.999966i \(0.502612\pi\)
\(660\) −2.17407e15 −0.675744
\(661\) 1.59301e15 0.491031 0.245516 0.969393i \(-0.421043\pi\)
0.245516 + 0.969393i \(0.421043\pi\)
\(662\) 1.18285e15 0.361586
\(663\) 2.04425e15 0.619740
\(664\) 3.95710e15 1.18974
\(665\) −2.67835e15 −0.798634
\(666\) −1.13661e15 −0.336126
\(667\) 1.63721e14 0.0480190
\(668\) −6.47599e14 −0.188380
\(669\) 7.34616e14 0.211942
\(670\) 5.37424e15 1.53782
\(671\) 2.58766e15 0.734403
\(672\) 2.56394e15 0.721734
\(673\) −5.57867e15 −1.55757 −0.778785 0.627291i \(-0.784164\pi\)
−0.778785 + 0.627291i \(0.784164\pi\)
\(674\) 8.83520e14 0.244674
\(675\) 6.89345e14 0.189350
\(676\) −3.85248e15 −1.04962
\(677\) 1.41246e15 0.381713 0.190857 0.981618i \(-0.438873\pi\)
0.190857 + 0.981618i \(0.438873\pi\)
\(678\) −1.09857e14 −0.0294486
\(679\) −5.43952e15 −1.44636
\(680\) 3.40782e15 0.898829
\(681\) −2.93667e15 −0.768327
\(682\) 1.78333e14 0.0462826
\(683\) −2.32947e15 −0.599714 −0.299857 0.953984i \(-0.596939\pi\)
−0.299857 + 0.953984i \(0.596939\pi\)
\(684\) −2.33103e14 −0.0595305
\(685\) 2.53421e15 0.642015
\(686\) 8.80870e14 0.221375
\(687\) 1.87953e15 0.468584
\(688\) −1.49892e15 −0.370717
\(689\) 1.20439e16 2.95504
\(690\) −4.79218e14 −0.116644
\(691\) −1.32084e15 −0.318948 −0.159474 0.987202i \(-0.550980\pi\)
−0.159474 + 0.987202i \(0.550980\pi\)
\(692\) −1.17763e15 −0.282115
\(693\) −3.70018e15 −0.879408
\(694\) 3.50730e14 0.0826982
\(695\) −4.18298e15 −0.978519
\(696\) −6.50674e14 −0.151012
\(697\) −3.06144e15 −0.704929
\(698\) 2.46762e15 0.563733
\(699\) −1.66646e15 −0.377719
\(700\) −3.03767e15 −0.683127
\(701\) −2.69385e15 −0.601069 −0.300534 0.953771i \(-0.597165\pi\)
−0.300534 + 0.953771i \(0.597165\pi\)
\(702\) 1.14248e15 0.252927
\(703\) −2.40283e15 −0.527800
\(704\) −7.56601e15 −1.64899
\(705\) 6.02613e15 1.30316
\(706\) 3.76002e15 0.806796
\(707\) −4.52675e15 −0.963783
\(708\) 1.66383e14 0.0351501
\(709\) −2.06989e15 −0.433904 −0.216952 0.976182i \(-0.569611\pi\)
−0.216952 + 0.976182i \(0.569611\pi\)
\(710\) 7.79435e15 1.62128
\(711\) −2.81348e15 −0.580713
\(712\) −3.05681e15 −0.626078
\(713\) −3.45302e13 −0.00701789
\(714\) 1.84807e15 0.372716
\(715\) 2.25262e16 4.50821
\(716\) 2.60442e15 0.517236
\(717\) −5.47702e14 −0.107942
\(718\) −6.16587e15 −1.20590
\(719\) 4.66314e15 0.905044 0.452522 0.891753i \(-0.350524\pi\)
0.452522 + 0.891753i \(0.350524\pi\)
\(720\) 7.64611e14 0.147269
\(721\) −2.35174e15 −0.449517
\(722\) −3.28543e15 −0.623216
\(723\) 1.49885e15 0.282162
\(724\) −8.69185e14 −0.162387
\(725\) 1.29616e15 0.240325
\(726\) 4.93903e15 0.908845
\(727\) 3.21654e15 0.587421 0.293711 0.955894i \(-0.405110\pi\)
0.293711 + 0.955894i \(0.405110\pi\)
\(728\) −1.58001e16 −2.86376
\(729\) 2.05891e14 0.0370370
\(730\) −7.53565e15 −1.34538
\(731\) 3.97475e15 0.704311
\(732\) 6.34492e14 0.111587
\(733\) −1.00800e16 −1.75949 −0.879747 0.475442i \(-0.842288\pi\)
−0.879747 + 0.475442i \(0.842288\pi\)
\(734\) 4.28544e13 0.00742449
\(735\) −5.69552e15 −0.979384
\(736\) 9.69811e14 0.165523
\(737\) 1.56959e16 2.65898
\(738\) −1.71096e15 −0.287694
\(739\) −2.70475e15 −0.451422 −0.225711 0.974194i \(-0.572471\pi\)
−0.225711 + 0.974194i \(0.572471\pi\)
\(740\) −5.49500e15 −0.910318
\(741\) 2.41525e15 0.397156
\(742\) 1.08881e16 1.77718
\(743\) −8.03675e15 −1.30209 −0.651047 0.759038i \(-0.725670\pi\)
−0.651047 + 0.759038i \(0.725670\pi\)
\(744\) 1.37233e14 0.0220702
\(745\) −4.32471e15 −0.690395
\(746\) −4.27297e15 −0.677120
\(747\) −2.35436e15 −0.370347
\(748\) 3.17131e15 0.495198
\(749\) −1.21254e16 −1.87952
\(750\) 6.21078e13 0.00955673
\(751\) −3.59310e15 −0.548844 −0.274422 0.961609i \(-0.588487\pi\)
−0.274422 + 0.961609i \(0.588487\pi\)
\(752\) 3.31491e15 0.502659
\(753\) −6.58356e15 −0.991033
\(754\) 2.14818e15 0.321017
\(755\) 1.31805e16 1.95535
\(756\) −9.07280e14 −0.133620
\(757\) −5.51967e15 −0.807023 −0.403512 0.914975i \(-0.632210\pi\)
−0.403512 + 0.914975i \(0.632210\pi\)
\(758\) −2.37710e15 −0.345038
\(759\) −1.39959e15 −0.201684
\(760\) 4.02628e15 0.576009
\(761\) −5.82429e15 −0.827231 −0.413616 0.910452i \(-0.635734\pi\)
−0.413616 + 0.910452i \(0.635734\pi\)
\(762\) 4.76442e14 0.0671827
\(763\) 1.00170e16 1.40234
\(764\) 2.21542e15 0.307924
\(765\) −2.02755e15 −0.279791
\(766\) 6.48445e15 0.888412
\(767\) −1.72394e15 −0.234503
\(768\) −4.48136e15 −0.605234
\(769\) −6.46214e15 −0.866526 −0.433263 0.901268i \(-0.642638\pi\)
−0.433263 + 0.901268i \(0.642638\pi\)
\(770\) 2.03644e16 2.71127
\(771\) 6.96631e15 0.920881
\(772\) 5.86730e15 0.770092
\(773\) −8.02264e15 −1.04551 −0.522757 0.852482i \(-0.675097\pi\)
−0.522757 + 0.852482i \(0.675097\pi\)
\(774\) 2.22139e15 0.287442
\(775\) −2.73371e14 −0.0351231
\(776\) 8.17707e15 1.04318
\(777\) −9.35227e15 −1.18468
\(778\) −8.28771e15 −1.04243
\(779\) −3.61704e15 −0.451749
\(780\) 5.52339e15 0.684992
\(781\) 2.27640e16 2.80329
\(782\) 6.99033e14 0.0854792
\(783\) 3.87132e14 0.0470077
\(784\) −3.13305e15 −0.377771
\(785\) 2.10985e16 2.52621
\(786\) −2.09949e15 −0.249626
\(787\) −7.74464e15 −0.914408 −0.457204 0.889362i \(-0.651149\pi\)
−0.457204 + 0.889362i \(0.651149\pi\)
\(788\) −2.43153e15 −0.285092
\(789\) −1.32246e15 −0.153978
\(790\) 1.54844e16 1.79038
\(791\) −9.03928e14 −0.103792
\(792\) 5.56237e15 0.634266
\(793\) −6.57416e15 −0.744454
\(794\) 3.05094e15 0.343101
\(795\) −1.19455e16 −1.33410
\(796\) 4.51042e15 0.500259
\(797\) 1.41802e16 1.56193 0.780964 0.624576i \(-0.214728\pi\)
0.780964 + 0.624576i \(0.214728\pi\)
\(798\) 2.18346e15 0.238853
\(799\) −8.79028e15 −0.954983
\(800\) 7.67785e15 0.828409
\(801\) 1.81871e15 0.194888
\(802\) −4.14690e15 −0.441331
\(803\) −2.20085e16 −2.32624
\(804\) 3.84861e15 0.404014
\(805\) −3.94312e15 −0.411114
\(806\) −4.53068e14 −0.0469160
\(807\) 7.45182e15 0.766404
\(808\) 6.80492e15 0.695122
\(809\) −8.31515e15 −0.843633 −0.421816 0.906681i \(-0.638607\pi\)
−0.421816 + 0.906681i \(0.638607\pi\)
\(810\) −1.13315e15 −0.114188
\(811\) −1.73836e16 −1.73990 −0.869952 0.493136i \(-0.835851\pi\)
−0.869952 + 0.493136i \(0.835851\pi\)
\(812\) −1.70593e15 −0.169592
\(813\) 7.17188e15 0.708166
\(814\) 1.82696e16 1.79182
\(815\) 1.30094e16 1.26733
\(816\) −1.11533e15 −0.107922
\(817\) 4.69610e15 0.451353
\(818\) −4.28549e15 −0.409126
\(819\) 9.40060e15 0.891444
\(820\) −8.27176e15 −0.779151
\(821\) −1.41285e16 −1.32193 −0.660967 0.750415i \(-0.729854\pi\)
−0.660967 + 0.750415i \(0.729854\pi\)
\(822\) −2.06596e15 −0.192012
\(823\) 1.17775e16 1.08731 0.543657 0.839308i \(-0.317039\pi\)
0.543657 + 0.839308i \(0.317039\pi\)
\(824\) 3.53530e15 0.324210
\(825\) −1.10804e16 −1.00939
\(826\) −1.55850e15 −0.141032
\(827\) 4.82466e15 0.433697 0.216849 0.976205i \(-0.430422\pi\)
0.216849 + 0.976205i \(0.430422\pi\)
\(828\) −3.43179e14 −0.0306446
\(829\) −6.61916e15 −0.587156 −0.293578 0.955935i \(-0.594846\pi\)
−0.293578 + 0.955935i \(0.594846\pi\)
\(830\) 1.29575e16 1.14180
\(831\) −5.13162e15 −0.449209
\(832\) 1.92220e16 1.67156
\(833\) 8.30803e15 0.717712
\(834\) 3.41009e15 0.292652
\(835\) −6.65514e15 −0.567390
\(836\) 3.74685e15 0.317345
\(837\) −8.16493e13 −0.00687009
\(838\) 1.03242e16 0.863009
\(839\) −1.88761e16 −1.56755 −0.783774 0.621046i \(-0.786708\pi\)
−0.783774 + 0.621046i \(0.786708\pi\)
\(840\) 1.56710e16 1.29289
\(841\) −1.14726e16 −0.940337
\(842\) 5.71322e15 0.465227
\(843\) −6.86536e14 −0.0555408
\(844\) −7.55025e15 −0.606845
\(845\) −3.95906e16 −3.16140
\(846\) −4.91267e15 −0.389745
\(847\) 4.06395e16 3.20324
\(848\) −6.57112e15 −0.514591
\(849\) −1.30750e16 −1.01730
\(850\) 5.53414e15 0.427805
\(851\) −3.53750e15 −0.271696
\(852\) 5.58171e15 0.425941
\(853\) −3.62057e15 −0.274510 −0.137255 0.990536i \(-0.543828\pi\)
−0.137255 + 0.990536i \(0.543828\pi\)
\(854\) −5.94326e15 −0.447720
\(855\) −2.39552e15 −0.179302
\(856\) 1.82278e16 1.35559
\(857\) 2.25387e16 1.66546 0.832731 0.553678i \(-0.186776\pi\)
0.832731 + 0.553678i \(0.186776\pi\)
\(858\) −1.83640e16 −1.34830
\(859\) −6.88093e15 −0.501978 −0.250989 0.967990i \(-0.580756\pi\)
−0.250989 + 0.967990i \(0.580756\pi\)
\(860\) 1.07394e16 0.778467
\(861\) −1.40782e16 −1.01398
\(862\) −7.04125e15 −0.503918
\(863\) 1.85297e15 0.131767 0.0658837 0.997827i \(-0.479013\pi\)
0.0658837 + 0.997827i \(0.479013\pi\)
\(864\) 2.29319e15 0.162037
\(865\) −1.21021e16 −0.849715
\(866\) 3.80735e15 0.265629
\(867\) −5.37049e15 −0.372314
\(868\) 3.59796e14 0.0247855
\(869\) 4.52233e16 3.09566
\(870\) −2.13063e15 −0.144928
\(871\) −3.98766e16 −2.69537
\(872\) −1.50583e16 −1.01143
\(873\) −4.86511e15 −0.324724
\(874\) 8.25896e14 0.0547788
\(875\) 5.11037e14 0.0336828
\(876\) −5.39645e15 −0.353456
\(877\) 1.10203e16 0.717293 0.358646 0.933474i \(-0.383238\pi\)
0.358646 + 0.933474i \(0.383238\pi\)
\(878\) 5.53867e15 0.358250
\(879\) 1.02859e16 0.661156
\(880\) −1.22902e16 −0.785062
\(881\) 7.18031e14 0.0455801 0.0227901 0.999740i \(-0.492745\pi\)
0.0227901 + 0.999740i \(0.492745\pi\)
\(882\) 4.64315e15 0.292911
\(883\) −1.29681e15 −0.0813006 −0.0406503 0.999173i \(-0.512943\pi\)
−0.0406503 + 0.999173i \(0.512943\pi\)
\(884\) −8.05694e15 −0.501976
\(885\) 1.70986e15 0.105870
\(886\) −6.30807e15 −0.388160
\(887\) −3.21768e16 −1.96772 −0.983860 0.178938i \(-0.942734\pi\)
−0.983860 + 0.178938i \(0.942734\pi\)
\(888\) 1.40590e16 0.854442
\(889\) 3.92027e15 0.236786
\(890\) −1.00095e16 −0.600853
\(891\) −3.30945e15 −0.197437
\(892\) −2.89532e15 −0.171668
\(893\) −1.03856e16 −0.611994
\(894\) 3.52563e15 0.206481
\(895\) 2.67647e16 1.55788
\(896\) −4.23150e15 −0.244794
\(897\) 3.55578e15 0.204445
\(898\) 6.94548e15 0.396901
\(899\) −1.53523e14 −0.00871958
\(900\) −2.71690e15 −0.153370
\(901\) 1.74249e16 0.977652
\(902\) 2.75017e16 1.53364
\(903\) 1.82781e16 1.01309
\(904\) 1.35885e15 0.0748591
\(905\) −8.93231e15 −0.489099
\(906\) −1.07452e16 −0.584800
\(907\) −3.41230e16 −1.84590 −0.922949 0.384923i \(-0.874228\pi\)
−0.922949 + 0.384923i \(0.874228\pi\)
\(908\) 1.15742e16 0.622328
\(909\) −4.04873e15 −0.216380
\(910\) −5.17374e16 −2.74838
\(911\) 1.37264e16 0.724778 0.362389 0.932027i \(-0.381961\pi\)
0.362389 + 0.932027i \(0.381961\pi\)
\(912\) −1.31775e15 −0.0691610
\(913\) 3.78434e16 1.97424
\(914\) −1.12339e16 −0.582543
\(915\) 6.52045e15 0.336095
\(916\) −7.40774e15 −0.379543
\(917\) −1.72751e16 −0.879809
\(918\) 1.65292e15 0.0836789
\(919\) −2.55210e16 −1.28429 −0.642144 0.766584i \(-0.721955\pi\)
−0.642144 + 0.766584i \(0.721955\pi\)
\(920\) 5.92757e15 0.296513
\(921\) −1.50568e16 −0.748696
\(922\) −4.11381e14 −0.0203340
\(923\) −5.78337e16 −2.84166
\(924\) 1.45834e16 0.712301
\(925\) −2.80058e16 −1.35978
\(926\) 7.89863e15 0.381234
\(927\) −2.10340e15 −0.100921
\(928\) 4.31183e15 0.205659
\(929\) −3.36070e16 −1.59347 −0.796733 0.604332i \(-0.793440\pi\)
−0.796733 + 0.604332i \(0.793440\pi\)
\(930\) 4.49367e14 0.0211809
\(931\) 9.81580e15 0.459941
\(932\) 6.56795e15 0.305944
\(933\) 6.45331e15 0.298837
\(934\) −2.16821e16 −0.998143
\(935\) 3.25904e16 1.49151
\(936\) −1.41316e16 −0.642947
\(937\) −6.27603e15 −0.283869 −0.141934 0.989876i \(-0.545332\pi\)
−0.141934 + 0.989876i \(0.545332\pi\)
\(938\) −3.60498e16 −1.62101
\(939\) 5.51601e15 0.246583
\(940\) −2.37506e16 −1.05553
\(941\) 3.11211e14 0.0137503 0.00687514 0.999976i \(-0.497812\pi\)
0.00687514 + 0.999976i \(0.497812\pi\)
\(942\) −1.72001e16 −0.755530
\(943\) −5.32508e15 −0.232548
\(944\) 9.40576e14 0.0408365
\(945\) −9.32380e15 −0.402456
\(946\) −3.57061e16 −1.53229
\(947\) −6.02494e15 −0.257056 −0.128528 0.991706i \(-0.541025\pi\)
−0.128528 + 0.991706i \(0.541025\pi\)
\(948\) 1.10887e16 0.470365
\(949\) 5.59142e16 2.35808
\(950\) 6.53850e15 0.274156
\(951\) 1.81369e16 0.756083
\(952\) −2.28592e16 −0.947455
\(953\) 2.55137e16 1.05139 0.525694 0.850674i \(-0.323806\pi\)
0.525694 + 0.850674i \(0.323806\pi\)
\(954\) 9.73835e15 0.398997
\(955\) 2.27671e16 0.927448
\(956\) 2.15864e15 0.0874303
\(957\) −6.22267e15 −0.250588
\(958\) 2.62849e16 1.05244
\(959\) −1.69992e16 −0.676748
\(960\) −1.90650e16 −0.754649
\(961\) −2.53761e16 −0.998726
\(962\) −4.64153e16 −1.81634
\(963\) −1.08450e16 −0.421974
\(964\) −5.90739e15 −0.228545
\(965\) 6.02961e16 2.31947
\(966\) 3.21454e15 0.122955
\(967\) −9.90809e14 −0.0376829 −0.0188415 0.999822i \(-0.505998\pi\)
−0.0188415 + 0.999822i \(0.505998\pi\)
\(968\) −6.10921e16 −2.31031
\(969\) 3.49433e15 0.131396
\(970\) 2.67758e16 1.00115
\(971\) −2.21443e16 −0.823297 −0.411649 0.911343i \(-0.635047\pi\)
−0.411649 + 0.911343i \(0.635047\pi\)
\(972\) −8.11473e14 −0.0299992
\(973\) 2.80590e16 1.03146
\(974\) 2.13222e15 0.0779396
\(975\) 2.81506e16 1.02320
\(976\) 3.58683e15 0.129640
\(977\) −5.98566e15 −0.215126 −0.107563 0.994198i \(-0.534305\pi\)
−0.107563 + 0.994198i \(0.534305\pi\)
\(978\) −1.06056e16 −0.379029
\(979\) −2.92336e16 −1.03891
\(980\) 2.24476e16 0.793280
\(981\) 8.95926e15 0.314842
\(982\) −1.36176e16 −0.475870
\(983\) 4.16155e16 1.44614 0.723071 0.690774i \(-0.242730\pi\)
0.723071 + 0.690774i \(0.242730\pi\)
\(984\) 2.11633e16 0.731326
\(985\) −2.49879e16 −0.858680
\(986\) 3.10793e15 0.106206
\(987\) −4.04226e16 −1.37366
\(988\) −9.51915e15 −0.321688
\(989\) 6.91370e15 0.232344
\(990\) 1.82140e16 0.608711
\(991\) −9.40593e15 −0.312606 −0.156303 0.987709i \(-0.549958\pi\)
−0.156303 + 0.987709i \(0.549958\pi\)
\(992\) −9.09401e14 −0.0300567
\(993\) 8.70501e15 0.286120
\(994\) −5.22836e16 −1.70899
\(995\) 4.63520e16 1.50675
\(996\) 9.27915e15 0.299973
\(997\) 3.14043e15 0.100964 0.0504820 0.998725i \(-0.483924\pi\)
0.0504820 + 0.998725i \(0.483924\pi\)
\(998\) −1.97154e15 −0.0630359
\(999\) −8.36468e15 −0.265974
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.1.19 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.19 27 1.1 even 1 trivial