Properties

Label 177.12.a.b.1.11
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 1.11
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-19.9033 q^{2} +243.000 q^{3} -1651.86 q^{4} +11169.4 q^{5} -4836.50 q^{6} -27660.5 q^{7} +73639.4 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-19.9033 q^{2} +243.000 q^{3} -1651.86 q^{4} +11169.4 q^{5} -4836.50 q^{6} -27660.5 q^{7} +73639.4 q^{8} +59049.0 q^{9} -222307. q^{10} -1.04155e6 q^{11} -401402. q^{12} +162421. q^{13} +550535. q^{14} +2.71416e6 q^{15} +1.91734e6 q^{16} +6.41060e6 q^{17} -1.17527e6 q^{18} +1.86314e7 q^{19} -1.84502e7 q^{20} -6.72150e6 q^{21} +2.07302e7 q^{22} -5.10201e7 q^{23} +1.78944e7 q^{24} +7.59268e7 q^{25} -3.23270e6 q^{26} +1.43489e7 q^{27} +4.56912e7 q^{28} -2.37753e6 q^{29} -5.40207e7 q^{30} -2.59764e8 q^{31} -1.88975e8 q^{32} -2.53096e8 q^{33} -1.27592e8 q^{34} -3.08950e8 q^{35} -9.75406e7 q^{36} +1.23520e8 q^{37} -3.70826e8 q^{38} +3.94682e7 q^{39} +8.22506e8 q^{40} +1.74139e8 q^{41} +1.33780e8 q^{42} -1.00184e9 q^{43} +1.72049e9 q^{44} +6.59540e8 q^{45} +1.01547e9 q^{46} +1.25095e9 q^{47} +4.65914e8 q^{48} -1.21222e9 q^{49} -1.51119e9 q^{50} +1.55777e9 q^{51} -2.68296e8 q^{52} +1.52944e9 q^{53} -2.85590e8 q^{54} -1.16334e10 q^{55} -2.03690e9 q^{56} +4.52742e9 q^{57} +4.73207e7 q^{58} -7.14924e8 q^{59} -4.48341e9 q^{60} +6.43509e9 q^{61} +5.17015e9 q^{62} -1.63332e9 q^{63} -1.65494e8 q^{64} +1.81414e9 q^{65} +5.03745e9 q^{66} -1.91932e9 q^{67} -1.05894e10 q^{68} -1.23979e10 q^{69} +6.14913e9 q^{70} -1.22406e10 q^{71} +4.34833e9 q^{72} +2.33155e10 q^{73} -2.45845e9 q^{74} +1.84502e10 q^{75} -3.07764e10 q^{76} +2.88097e10 q^{77} -7.85547e8 q^{78} -1.85989e10 q^{79} +2.14155e10 q^{80} +3.48678e9 q^{81} -3.46593e9 q^{82} +3.10782e10 q^{83} +1.11030e10 q^{84} +7.16024e10 q^{85} +1.99398e10 q^{86} -5.77740e8 q^{87} -7.66989e10 q^{88} -1.78818e10 q^{89} -1.31270e10 q^{90} -4.49263e9 q^{91} +8.42781e10 q^{92} -6.31226e10 q^{93} -2.48981e10 q^{94} +2.08101e11 q^{95} -4.59209e10 q^{96} -1.24517e11 q^{97} +2.41272e10 q^{98} -6.15024e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.9033 −0.439805 −0.219902 0.975522i \(-0.570574\pi\)
−0.219902 + 0.975522i \(0.570574\pi\)
\(3\) 243.000 0.577350
\(4\) −1651.86 −0.806572
\(5\) 11169.4 1.59843 0.799215 0.601045i \(-0.205249\pi\)
0.799215 + 0.601045i \(0.205249\pi\)
\(6\) −4836.50 −0.253921
\(7\) −27660.5 −0.622043 −0.311022 0.950403i \(-0.600671\pi\)
−0.311022 + 0.950403i \(0.600671\pi\)
\(8\) 73639.4 0.794539
\(9\) 59049.0 0.333333
\(10\) −222307. −0.702998
\(11\) −1.04155e6 −1.94993 −0.974966 0.222353i \(-0.928626\pi\)
−0.974966 + 0.222353i \(0.928626\pi\)
\(12\) −401402. −0.465674
\(13\) 162421. 0.121326 0.0606628 0.998158i \(-0.480679\pi\)
0.0606628 + 0.998158i \(0.480679\pi\)
\(14\) 550535. 0.273578
\(15\) 2.71416e6 0.922855
\(16\) 1.91734e6 0.457130
\(17\) 6.41060e6 1.09504 0.547519 0.836793i \(-0.315572\pi\)
0.547519 + 0.836793i \(0.315572\pi\)
\(18\) −1.17527e6 −0.146602
\(19\) 1.86314e7 1.72624 0.863118 0.505003i \(-0.168508\pi\)
0.863118 + 0.505003i \(0.168508\pi\)
\(20\) −1.84502e7 −1.28925
\(21\) −6.72150e6 −0.359137
\(22\) 2.07302e7 0.857590
\(23\) −5.10201e7 −1.65287 −0.826435 0.563032i \(-0.809635\pi\)
−0.826435 + 0.563032i \(0.809635\pi\)
\(24\) 1.78944e7 0.458727
\(25\) 7.59268e7 1.55498
\(26\) −3.23270e6 −0.0533596
\(27\) 1.43489e7 0.192450
\(28\) 4.56912e7 0.501723
\(29\) −2.37753e6 −0.0215247 −0.0107623 0.999942i \(-0.503426\pi\)
−0.0107623 + 0.999942i \(0.503426\pi\)
\(30\) −5.40207e7 −0.405876
\(31\) −2.59764e8 −1.62963 −0.814815 0.579721i \(-0.803162\pi\)
−0.814815 + 0.579721i \(0.803162\pi\)
\(32\) −1.88975e8 −0.995587
\(33\) −2.53096e8 −1.12579
\(34\) −1.27592e8 −0.481603
\(35\) −3.08950e8 −0.994293
\(36\) −9.75406e7 −0.268857
\(37\) 1.23520e8 0.292838 0.146419 0.989223i \(-0.453225\pi\)
0.146419 + 0.989223i \(0.453225\pi\)
\(38\) −3.70826e8 −0.759207
\(39\) 3.94682e7 0.0700474
\(40\) 8.22506e8 1.27002
\(41\) 1.74139e8 0.234738 0.117369 0.993088i \(-0.462554\pi\)
0.117369 + 0.993088i \(0.462554\pi\)
\(42\) 1.33780e8 0.157950
\(43\) −1.00184e9 −1.03925 −0.519625 0.854394i \(-0.673928\pi\)
−0.519625 + 0.854394i \(0.673928\pi\)
\(44\) 1.72049e9 1.57276
\(45\) 6.59540e8 0.532810
\(46\) 1.01547e9 0.726940
\(47\) 1.25095e9 0.795615 0.397807 0.917469i \(-0.369771\pi\)
0.397807 + 0.917469i \(0.369771\pi\)
\(48\) 4.65914e8 0.263924
\(49\) −1.21222e9 −0.613062
\(50\) −1.51119e9 −0.683888
\(51\) 1.55777e9 0.632221
\(52\) −2.68296e8 −0.0978578
\(53\) 1.52944e9 0.502361 0.251180 0.967940i \(-0.419181\pi\)
0.251180 + 0.967940i \(0.419181\pi\)
\(54\) −2.85590e8 −0.0846405
\(55\) −1.16334e10 −3.11683
\(56\) −2.03690e9 −0.494238
\(57\) 4.52742e9 0.996643
\(58\) 4.73207e7 0.00946666
\(59\) −7.14924e8 −0.130189
\(60\) −4.48341e9 −0.744348
\(61\) 6.43509e9 0.975530 0.487765 0.872975i \(-0.337812\pi\)
0.487765 + 0.872975i \(0.337812\pi\)
\(62\) 5.17015e9 0.716719
\(63\) −1.63332e9 −0.207348
\(64\) −1.65494e8 −0.0192661
\(65\) 1.81414e9 0.193931
\(66\) 5.03745e9 0.495130
\(67\) −1.91932e9 −0.173674 −0.0868371 0.996223i \(-0.527676\pi\)
−0.0868371 + 0.996223i \(0.527676\pi\)
\(68\) −1.05894e10 −0.883227
\(69\) −1.23979e10 −0.954285
\(70\) 6.14913e9 0.437295
\(71\) −1.22406e10 −0.805156 −0.402578 0.915386i \(-0.631886\pi\)
−0.402578 + 0.915386i \(0.631886\pi\)
\(72\) 4.34833e9 0.264846
\(73\) 2.33155e10 1.31634 0.658170 0.752869i \(-0.271331\pi\)
0.658170 + 0.752869i \(0.271331\pi\)
\(74\) −2.45845e9 −0.128792
\(75\) 1.84502e10 0.897769
\(76\) −3.07764e10 −1.39233
\(77\) 2.88097e10 1.21294
\(78\) −7.85547e8 −0.0308072
\(79\) −1.85989e10 −0.680044 −0.340022 0.940417i \(-0.610435\pi\)
−0.340022 + 0.940417i \(0.610435\pi\)
\(80\) 2.14155e10 0.730690
\(81\) 3.48678e9 0.111111
\(82\) −3.46593e9 −0.103239
\(83\) 3.10782e10 0.866016 0.433008 0.901390i \(-0.357452\pi\)
0.433008 + 0.901390i \(0.357452\pi\)
\(84\) 1.11030e10 0.289670
\(85\) 7.16024e10 1.75034
\(86\) 1.99398e10 0.457067
\(87\) −5.77740e8 −0.0124273
\(88\) −7.66989e10 −1.54930
\(89\) −1.78818e10 −0.339443 −0.169721 0.985492i \(-0.554287\pi\)
−0.169721 + 0.985492i \(0.554287\pi\)
\(90\) −1.31270e10 −0.234333
\(91\) −4.49263e9 −0.0754698
\(92\) 8.42781e10 1.33316
\(93\) −6.31226e10 −0.940867
\(94\) −2.48981e10 −0.349915
\(95\) 2.08101e11 2.75927
\(96\) −4.59209e10 −0.574802
\(97\) −1.24517e11 −1.47225 −0.736127 0.676844i \(-0.763347\pi\)
−0.736127 + 0.676844i \(0.763347\pi\)
\(98\) 2.41272e10 0.269628
\(99\) −6.15024e10 −0.649977
\(100\) −1.25420e11 −1.25420
\(101\) −3.84734e10 −0.364245 −0.182122 0.983276i \(-0.558297\pi\)
−0.182122 + 0.983276i \(0.558297\pi\)
\(102\) −3.10048e10 −0.278054
\(103\) −1.97395e11 −1.67777 −0.838884 0.544310i \(-0.816791\pi\)
−0.838884 + 0.544310i \(0.816791\pi\)
\(104\) 1.19605e10 0.0963979
\(105\) −7.50749e10 −0.574055
\(106\) −3.04409e10 −0.220941
\(107\) −3.82896e10 −0.263919 −0.131959 0.991255i \(-0.542127\pi\)
−0.131959 + 0.991255i \(0.542127\pi\)
\(108\) −2.37024e10 −0.155225
\(109\) 1.87575e11 1.16770 0.583849 0.811862i \(-0.301546\pi\)
0.583849 + 0.811862i \(0.301546\pi\)
\(110\) 2.31544e11 1.37080
\(111\) 3.00154e10 0.169070
\(112\) −5.30346e10 −0.284355
\(113\) −2.62880e11 −1.34223 −0.671113 0.741355i \(-0.734183\pi\)
−0.671113 + 0.741355i \(0.734183\pi\)
\(114\) −9.01106e10 −0.438328
\(115\) −5.69863e11 −2.64200
\(116\) 3.92734e9 0.0173612
\(117\) 9.59077e9 0.0404419
\(118\) 1.42293e10 0.0572577
\(119\) −1.77320e11 −0.681161
\(120\) 1.99869e11 0.733244
\(121\) 7.99511e11 2.80224
\(122\) −1.28080e11 −0.429043
\(123\) 4.23157e10 0.135526
\(124\) 4.29093e11 1.31441
\(125\) 3.02676e11 0.887100
\(126\) 3.25085e10 0.0911925
\(127\) −3.83464e11 −1.02992 −0.514961 0.857214i \(-0.672194\pi\)
−0.514961 + 0.857214i \(0.672194\pi\)
\(128\) 3.90314e11 1.00406
\(129\) −2.43446e11 −0.600011
\(130\) −3.61073e10 −0.0852916
\(131\) 2.09931e11 0.475428 0.237714 0.971335i \(-0.423602\pi\)
0.237714 + 0.971335i \(0.423602\pi\)
\(132\) 4.18079e11 0.908034
\(133\) −5.15353e11 −1.07379
\(134\) 3.82007e10 0.0763827
\(135\) 1.60268e11 0.307618
\(136\) 4.72072e11 0.870051
\(137\) 2.00324e11 0.354626 0.177313 0.984155i \(-0.443260\pi\)
0.177313 + 0.984155i \(0.443260\pi\)
\(138\) 2.46759e11 0.419699
\(139\) −2.83228e11 −0.462972 −0.231486 0.972838i \(-0.574359\pi\)
−0.231486 + 0.972838i \(0.574359\pi\)
\(140\) 5.10342e11 0.801969
\(141\) 3.03982e11 0.459348
\(142\) 2.43627e11 0.354112
\(143\) −1.69169e11 −0.236577
\(144\) 1.13217e11 0.152377
\(145\) −2.65555e10 −0.0344057
\(146\) −4.64054e11 −0.578933
\(147\) −2.94570e11 −0.353952
\(148\) −2.04038e11 −0.236195
\(149\) −1.13062e12 −1.26123 −0.630613 0.776097i \(-0.717196\pi\)
−0.630613 + 0.776097i \(0.717196\pi\)
\(150\) −3.67220e11 −0.394843
\(151\) 1.43392e12 1.48645 0.743226 0.669041i \(-0.233295\pi\)
0.743226 + 0.669041i \(0.233295\pi\)
\(152\) 1.37200e12 1.37156
\(153\) 3.78539e11 0.365013
\(154\) −5.73408e11 −0.533458
\(155\) −2.90140e12 −2.60485
\(156\) −6.51959e10 −0.0564982
\(157\) −7.28991e11 −0.609922 −0.304961 0.952365i \(-0.598643\pi\)
−0.304961 + 0.952365i \(0.598643\pi\)
\(158\) 3.70178e11 0.299087
\(159\) 3.71654e11 0.290038
\(160\) −2.11073e12 −1.59138
\(161\) 1.41124e12 1.02816
\(162\) −6.93985e10 −0.0488672
\(163\) −4.56368e11 −0.310658 −0.155329 0.987863i \(-0.549644\pi\)
−0.155329 + 0.987863i \(0.549644\pi\)
\(164\) −2.87652e11 −0.189333
\(165\) −2.82693e12 −1.79950
\(166\) −6.18558e11 −0.380878
\(167\) −3.91774e11 −0.233397 −0.116699 0.993167i \(-0.537231\pi\)
−0.116699 + 0.993167i \(0.537231\pi\)
\(168\) −4.94967e11 −0.285348
\(169\) −1.76578e12 −0.985280
\(170\) −1.42512e12 −0.769809
\(171\) 1.10016e12 0.575412
\(172\) 1.65489e12 0.838230
\(173\) −2.32022e12 −1.13835 −0.569174 0.822217i \(-0.692737\pi\)
−0.569174 + 0.822217i \(0.692737\pi\)
\(174\) 1.14989e10 0.00546558
\(175\) −2.10017e12 −0.967266
\(176\) −1.99700e12 −0.891372
\(177\) −1.73727e11 −0.0751646
\(178\) 3.55907e11 0.149288
\(179\) −1.20055e12 −0.488303 −0.244152 0.969737i \(-0.578509\pi\)
−0.244152 + 0.969737i \(0.578509\pi\)
\(180\) −1.08947e12 −0.429750
\(181\) −3.13673e12 −1.20018 −0.600088 0.799934i \(-0.704868\pi\)
−0.600088 + 0.799934i \(0.704868\pi\)
\(182\) 8.94181e10 0.0331920
\(183\) 1.56373e12 0.563223
\(184\) −3.75709e12 −1.31327
\(185\) 1.37964e12 0.468082
\(186\) 1.25635e12 0.413798
\(187\) −6.67694e12 −2.13525
\(188\) −2.06640e12 −0.641721
\(189\) −3.96898e11 −0.119712
\(190\) −4.14189e12 −1.21354
\(191\) −3.68407e12 −1.04868 −0.524341 0.851508i \(-0.675688\pi\)
−0.524341 + 0.851508i \(0.675688\pi\)
\(192\) −4.02151e10 −0.0111233
\(193\) 1.28396e12 0.345133 0.172566 0.984998i \(-0.444794\pi\)
0.172566 + 0.984998i \(0.444794\pi\)
\(194\) 2.47829e12 0.647504
\(195\) 4.40835e11 0.111966
\(196\) 2.00242e12 0.494479
\(197\) −3.61276e12 −0.867510 −0.433755 0.901031i \(-0.642812\pi\)
−0.433755 + 0.901031i \(0.642812\pi\)
\(198\) 1.22410e12 0.285863
\(199\) −2.63629e12 −0.598827 −0.299413 0.954123i \(-0.596791\pi\)
−0.299413 + 0.954123i \(0.596791\pi\)
\(200\) 5.59120e12 1.23549
\(201\) −4.66394e11 −0.100271
\(202\) 7.65747e11 0.160196
\(203\) 6.57636e10 0.0133893
\(204\) −2.57322e12 −0.509931
\(205\) 1.94502e12 0.375213
\(206\) 3.92881e12 0.737890
\(207\) −3.01269e12 −0.550957
\(208\) 3.11416e11 0.0554615
\(209\) −1.94055e13 −3.36604
\(210\) 1.49424e12 0.252472
\(211\) 7.03889e11 0.115865 0.0579323 0.998321i \(-0.481549\pi\)
0.0579323 + 0.998321i \(0.481549\pi\)
\(212\) −2.52642e12 −0.405190
\(213\) −2.97446e12 −0.464857
\(214\) 7.62090e11 0.116073
\(215\) −1.11899e13 −1.66117
\(216\) 1.05664e12 0.152909
\(217\) 7.18519e12 1.01370
\(218\) −3.73337e12 −0.513559
\(219\) 5.66566e12 0.759989
\(220\) 1.92168e13 2.51395
\(221\) 1.04121e12 0.132856
\(222\) −5.97405e11 −0.0743579
\(223\) −1.44869e13 −1.75913 −0.879565 0.475778i \(-0.842166\pi\)
−0.879565 + 0.475778i \(0.842166\pi\)
\(224\) 5.22714e12 0.619298
\(225\) 4.48340e12 0.518327
\(226\) 5.23217e12 0.590317
\(227\) 9.81667e12 1.08099 0.540496 0.841347i \(-0.318237\pi\)
0.540496 + 0.841347i \(0.318237\pi\)
\(228\) −7.47867e12 −0.803864
\(229\) −7.88896e12 −0.827799 −0.413899 0.910323i \(-0.635833\pi\)
−0.413899 + 0.910323i \(0.635833\pi\)
\(230\) 1.13422e13 1.16196
\(231\) 7.00076e12 0.700293
\(232\) −1.75080e11 −0.0171022
\(233\) −4.80416e12 −0.458311 −0.229155 0.973390i \(-0.573596\pi\)
−0.229155 + 0.973390i \(0.573596\pi\)
\(234\) −1.90888e11 −0.0177865
\(235\) 1.39724e13 1.27174
\(236\) 1.18095e12 0.105007
\(237\) −4.51952e12 −0.392624
\(238\) 3.52926e12 0.299578
\(239\) 8.16712e12 0.677455 0.338728 0.940884i \(-0.390003\pi\)
0.338728 + 0.940884i \(0.390003\pi\)
\(240\) 5.20397e12 0.421864
\(241\) 2.05703e12 0.162984 0.0814922 0.996674i \(-0.474031\pi\)
0.0814922 + 0.996674i \(0.474031\pi\)
\(242\) −1.59129e13 −1.23244
\(243\) 8.47289e11 0.0641500
\(244\) −1.06299e13 −0.786835
\(245\) −1.35398e13 −0.979938
\(246\) −8.42221e11 −0.0596051
\(247\) 3.02612e12 0.209437
\(248\) −1.91288e13 −1.29480
\(249\) 7.55199e12 0.499994
\(250\) −6.02424e12 −0.390151
\(251\) −1.86917e13 −1.18425 −0.592125 0.805846i \(-0.701711\pi\)
−0.592125 + 0.805846i \(0.701711\pi\)
\(252\) 2.69802e12 0.167241
\(253\) 5.31399e13 3.22298
\(254\) 7.63220e12 0.452965
\(255\) 1.73994e13 1.01056
\(256\) −7.42961e12 −0.422324
\(257\) 2.71267e13 1.50926 0.754631 0.656149i \(-0.227816\pi\)
0.754631 + 0.656149i \(0.227816\pi\)
\(258\) 4.84538e12 0.263888
\(259\) −3.41662e12 −0.182158
\(260\) −2.99670e12 −0.156419
\(261\) −1.40391e11 −0.00717490
\(262\) −4.17832e12 −0.209096
\(263\) −2.38754e13 −1.17002 −0.585011 0.811026i \(-0.698910\pi\)
−0.585011 + 0.811026i \(0.698910\pi\)
\(264\) −1.86378e13 −0.894487
\(265\) 1.70829e13 0.802989
\(266\) 1.02572e13 0.472259
\(267\) −4.34528e12 −0.195977
\(268\) 3.17044e12 0.140081
\(269\) −4.08460e13 −1.76812 −0.884061 0.467371i \(-0.845201\pi\)
−0.884061 + 0.467371i \(0.845201\pi\)
\(270\) −3.18987e12 −0.135292
\(271\) 1.27174e13 0.528528 0.264264 0.964450i \(-0.414871\pi\)
0.264264 + 0.964450i \(0.414871\pi\)
\(272\) 1.22913e13 0.500575
\(273\) −1.09171e12 −0.0435725
\(274\) −3.98711e12 −0.155966
\(275\) −7.90815e13 −3.03211
\(276\) 2.04796e13 0.769699
\(277\) −2.49980e13 −0.921014 −0.460507 0.887656i \(-0.652332\pi\)
−0.460507 + 0.887656i \(0.652332\pi\)
\(278\) 5.63717e12 0.203617
\(279\) −1.53388e13 −0.543210
\(280\) −2.27509e13 −0.790005
\(281\) 4.57687e13 1.55842 0.779208 0.626766i \(-0.215622\pi\)
0.779208 + 0.626766i \(0.215622\pi\)
\(282\) −6.05024e12 −0.202024
\(283\) −3.59305e13 −1.17662 −0.588312 0.808634i \(-0.700207\pi\)
−0.588312 + 0.808634i \(0.700207\pi\)
\(284\) 2.02197e13 0.649416
\(285\) 5.05685e13 1.59306
\(286\) 3.36702e12 0.104048
\(287\) −4.81676e12 −0.146017
\(288\) −1.11588e13 −0.331862
\(289\) 6.82384e12 0.199109
\(290\) 5.28542e11 0.0151318
\(291\) −3.02575e13 −0.850006
\(292\) −3.85138e13 −1.06172
\(293\) −1.97133e13 −0.533319 −0.266660 0.963791i \(-0.585920\pi\)
−0.266660 + 0.963791i \(0.585920\pi\)
\(294\) 5.86292e12 0.155670
\(295\) −7.98526e12 −0.208098
\(296\) 9.09594e12 0.232671
\(297\) −1.49451e13 −0.375265
\(298\) 2.25031e13 0.554693
\(299\) −8.28672e12 −0.200535
\(300\) −3.04772e13 −0.724115
\(301\) 2.77113e13 0.646459
\(302\) −2.85397e13 −0.653749
\(303\) −9.34904e12 −0.210297
\(304\) 3.57227e13 0.789114
\(305\) 7.18760e13 1.55932
\(306\) −7.53418e12 −0.160534
\(307\) 8.56709e13 1.79297 0.896483 0.443078i \(-0.146114\pi\)
0.896483 + 0.443078i \(0.146114\pi\)
\(308\) −4.75896e13 −0.978325
\(309\) −4.79670e13 −0.968660
\(310\) 5.77474e13 1.14563
\(311\) −1.78231e13 −0.347376 −0.173688 0.984801i \(-0.555568\pi\)
−0.173688 + 0.984801i \(0.555568\pi\)
\(312\) 2.90641e12 0.0556554
\(313\) −1.53485e13 −0.288784 −0.144392 0.989521i \(-0.546123\pi\)
−0.144392 + 0.989521i \(0.546123\pi\)
\(314\) 1.45093e13 0.268247
\(315\) −1.82432e13 −0.331431
\(316\) 3.07227e13 0.548505
\(317\) 6.02868e13 1.05778 0.528891 0.848690i \(-0.322608\pi\)
0.528891 + 0.848690i \(0.322608\pi\)
\(318\) −7.39714e12 −0.127560
\(319\) 2.47631e12 0.0419717
\(320\) −1.84847e12 −0.0307955
\(321\) −9.30438e12 −0.152374
\(322\) −2.80884e13 −0.452188
\(323\) 1.19438e14 1.89029
\(324\) −5.75968e12 −0.0896191
\(325\) 1.23321e13 0.188659
\(326\) 9.08322e12 0.136629
\(327\) 4.55808e13 0.674170
\(328\) 1.28235e13 0.186509
\(329\) −3.46020e13 −0.494907
\(330\) 5.62651e13 0.791430
\(331\) −9.07651e13 −1.25564 −0.627820 0.778359i \(-0.716053\pi\)
−0.627820 + 0.778359i \(0.716053\pi\)
\(332\) −5.13367e13 −0.698504
\(333\) 7.29373e12 0.0976128
\(334\) 7.79760e12 0.102649
\(335\) −2.14376e13 −0.277606
\(336\) −1.28874e13 −0.164172
\(337\) −7.77690e13 −0.974635 −0.487318 0.873225i \(-0.662025\pi\)
−0.487318 + 0.873225i \(0.662025\pi\)
\(338\) 3.51448e13 0.433331
\(339\) −6.38797e13 −0.774934
\(340\) −1.18277e14 −1.41178
\(341\) 2.70556e14 3.17767
\(342\) −2.18969e13 −0.253069
\(343\) 8.82245e13 1.00339
\(344\) −7.37746e13 −0.825725
\(345\) −1.38477e14 −1.52536
\(346\) 4.61799e13 0.500651
\(347\) −3.53719e13 −0.377439 −0.188719 0.982031i \(-0.560434\pi\)
−0.188719 + 0.982031i \(0.560434\pi\)
\(348\) 9.54345e11 0.0100235
\(349\) 1.56494e14 1.61792 0.808960 0.587863i \(-0.200031\pi\)
0.808960 + 0.587863i \(0.200031\pi\)
\(350\) 4.18004e13 0.425408
\(351\) 2.33056e12 0.0233491
\(352\) 1.96826e14 1.94133
\(353\) 7.88223e13 0.765400 0.382700 0.923873i \(-0.374994\pi\)
0.382700 + 0.923873i \(0.374994\pi\)
\(354\) 3.45773e12 0.0330577
\(355\) −1.36719e14 −1.28699
\(356\) 2.95382e13 0.273785
\(357\) −4.30888e13 −0.393269
\(358\) 2.38950e13 0.214758
\(359\) −8.06824e12 −0.0714101 −0.0357050 0.999362i \(-0.511368\pi\)
−0.0357050 + 0.999362i \(0.511368\pi\)
\(360\) 4.85681e13 0.423338
\(361\) 2.30638e14 1.97989
\(362\) 6.24313e13 0.527843
\(363\) 1.94281e14 1.61787
\(364\) 7.42119e12 0.0608718
\(365\) 2.60419e14 2.10408
\(366\) −3.11233e13 −0.247708
\(367\) −6.53952e13 −0.512723 −0.256361 0.966581i \(-0.582524\pi\)
−0.256361 + 0.966581i \(0.582524\pi\)
\(368\) −9.78230e13 −0.755576
\(369\) 1.02827e13 0.0782461
\(370\) −2.74594e13 −0.205865
\(371\) −4.23051e13 −0.312490
\(372\) 1.04270e14 0.758877
\(373\) −8.32583e13 −0.597075 −0.298538 0.954398i \(-0.596499\pi\)
−0.298538 + 0.954398i \(0.596499\pi\)
\(374\) 1.32893e14 0.939093
\(375\) 7.35502e13 0.512167
\(376\) 9.21194e13 0.632147
\(377\) −3.86160e11 −0.00261150
\(378\) 7.89957e12 0.0526500
\(379\) −1.64786e14 −1.08244 −0.541220 0.840881i \(-0.682037\pi\)
−0.541220 + 0.840881i \(0.682037\pi\)
\(380\) −3.43753e14 −2.22555
\(381\) −9.31818e13 −0.594626
\(382\) 7.33251e13 0.461216
\(383\) −1.93688e14 −1.20090 −0.600452 0.799661i \(-0.705013\pi\)
−0.600452 + 0.799661i \(0.705013\pi\)
\(384\) 9.48464e13 0.579694
\(385\) 3.21787e14 1.93880
\(386\) −2.55550e13 −0.151791
\(387\) −5.91574e13 −0.346417
\(388\) 2.05684e14 1.18748
\(389\) 1.12626e14 0.641086 0.320543 0.947234i \(-0.396135\pi\)
0.320543 + 0.947234i \(0.396135\pi\)
\(390\) −8.77407e12 −0.0492431
\(391\) −3.27070e14 −1.80996
\(392\) −8.92674e13 −0.487102
\(393\) 5.10133e13 0.274489
\(394\) 7.19058e13 0.381535
\(395\) −2.07738e14 −1.08700
\(396\) 1.01593e14 0.524253
\(397\) −1.50136e14 −0.764077 −0.382038 0.924146i \(-0.624778\pi\)
−0.382038 + 0.924146i \(0.624778\pi\)
\(398\) 5.24708e13 0.263367
\(399\) −1.25231e14 −0.619955
\(400\) 1.45578e14 0.710828
\(401\) −1.55254e14 −0.747738 −0.373869 0.927481i \(-0.621969\pi\)
−0.373869 + 0.927481i \(0.621969\pi\)
\(402\) 9.28277e12 0.0440996
\(403\) −4.21909e13 −0.197716
\(404\) 6.35526e13 0.293789
\(405\) 3.89452e13 0.177603
\(406\) −1.30891e12 −0.00588867
\(407\) −1.28652e14 −0.571015
\(408\) 1.14714e14 0.502324
\(409\) −3.98266e14 −1.72066 −0.860330 0.509737i \(-0.829743\pi\)
−0.860330 + 0.509737i \(0.829743\pi\)
\(410\) −3.87123e13 −0.165020
\(411\) 4.86788e13 0.204743
\(412\) 3.26069e14 1.35324
\(413\) 1.97752e13 0.0809831
\(414\) 5.99624e13 0.242313
\(415\) 3.47124e14 1.38427
\(416\) −3.06934e13 −0.120790
\(417\) −6.88244e13 −0.267297
\(418\) 3.86233e14 1.48040
\(419\) −1.89952e14 −0.718567 −0.359284 0.933228i \(-0.616979\pi\)
−0.359284 + 0.933228i \(0.616979\pi\)
\(420\) 1.24013e14 0.463017
\(421\) −3.90610e14 −1.43943 −0.719717 0.694268i \(-0.755728\pi\)
−0.719717 + 0.694268i \(0.755728\pi\)
\(422\) −1.40097e13 −0.0509578
\(423\) 7.38675e13 0.265205
\(424\) 1.12627e14 0.399145
\(425\) 4.86736e14 1.70276
\(426\) 5.92014e13 0.204446
\(427\) −1.77998e14 −0.606822
\(428\) 6.32491e13 0.212870
\(429\) −4.11080e13 −0.136588
\(430\) 2.22716e14 0.730590
\(431\) −3.74617e14 −1.21328 −0.606642 0.794975i \(-0.707484\pi\)
−0.606642 + 0.794975i \(0.707484\pi\)
\(432\) 2.75118e13 0.0879747
\(433\) 2.01352e14 0.635729 0.317864 0.948136i \(-0.397034\pi\)
0.317864 + 0.948136i \(0.397034\pi\)
\(434\) −1.43009e14 −0.445830
\(435\) −6.45299e12 −0.0198642
\(436\) −3.09848e14 −0.941832
\(437\) −9.50575e14 −2.85324
\(438\) −1.12765e14 −0.334247
\(439\) −2.08787e14 −0.611150 −0.305575 0.952168i \(-0.598849\pi\)
−0.305575 + 0.952168i \(0.598849\pi\)
\(440\) −8.56679e14 −2.47644
\(441\) −7.15806e13 −0.204354
\(442\) −2.07235e13 −0.0584308
\(443\) 6.34887e14 1.76797 0.883987 0.467511i \(-0.154849\pi\)
0.883987 + 0.467511i \(0.154849\pi\)
\(444\) −4.95812e13 −0.136367
\(445\) −1.99729e14 −0.542575
\(446\) 2.88337e14 0.773674
\(447\) −2.74741e14 −0.728169
\(448\) 4.57765e12 0.0119843
\(449\) 4.50994e14 1.16631 0.583157 0.812359i \(-0.301817\pi\)
0.583157 + 0.812359i \(0.301817\pi\)
\(450\) −8.92345e13 −0.227963
\(451\) −1.81374e14 −0.457724
\(452\) 4.34240e14 1.08260
\(453\) 3.48442e14 0.858203
\(454\) −1.95384e14 −0.475425
\(455\) −5.01799e13 −0.120633
\(456\) 3.33397e14 0.791871
\(457\) −4.41598e13 −0.103631 −0.0518153 0.998657i \(-0.516501\pi\)
−0.0518153 + 0.998657i \(0.516501\pi\)
\(458\) 1.57016e14 0.364070
\(459\) 9.19850e13 0.210740
\(460\) 9.41334e14 2.13096
\(461\) −7.43702e13 −0.166358 −0.0831790 0.996535i \(-0.526507\pi\)
−0.0831790 + 0.996535i \(0.526507\pi\)
\(462\) −1.39338e14 −0.307992
\(463\) 5.73448e14 1.25256 0.626280 0.779598i \(-0.284576\pi\)
0.626280 + 0.779598i \(0.284576\pi\)
\(464\) −4.55854e12 −0.00983958
\(465\) −7.05040e14 −1.50391
\(466\) 9.56186e13 0.201567
\(467\) 1.50443e14 0.313423 0.156711 0.987644i \(-0.449911\pi\)
0.156711 + 0.987644i \(0.449911\pi\)
\(468\) −1.58426e13 −0.0326193
\(469\) 5.30892e13 0.108033
\(470\) −2.78096e14 −0.559315
\(471\) −1.77145e14 −0.352139
\(472\) −5.26466e13 −0.103440
\(473\) 1.04346e15 2.02647
\(474\) 8.99534e13 0.172678
\(475\) 1.41462e15 2.68426
\(476\) 2.92908e14 0.549405
\(477\) 9.03120e13 0.167454
\(478\) −1.62553e14 −0.297948
\(479\) −1.92548e14 −0.348895 −0.174447 0.984666i \(-0.555814\pi\)
−0.174447 + 0.984666i \(0.555814\pi\)
\(480\) −5.12908e14 −0.918782
\(481\) 2.00622e13 0.0355288
\(482\) −4.09416e13 −0.0716813
\(483\) 3.42932e14 0.593606
\(484\) −1.32068e15 −2.26020
\(485\) −1.39077e15 −2.35330
\(486\) −1.68638e13 −0.0282135
\(487\) 1.08217e15 1.79014 0.895072 0.445922i \(-0.147124\pi\)
0.895072 + 0.445922i \(0.147124\pi\)
\(488\) 4.73876e14 0.775097
\(489\) −1.10897e14 −0.179359
\(490\) 2.69486e14 0.430981
\(491\) 1.12347e15 1.77670 0.888349 0.459168i \(-0.151852\pi\)
0.888349 + 0.459168i \(0.151852\pi\)
\(492\) −6.98995e13 −0.109312
\(493\) −1.52414e13 −0.0235704
\(494\) −6.02297e13 −0.0921112
\(495\) −6.86943e14 −1.03894
\(496\) −4.98056e14 −0.744953
\(497\) 3.38580e14 0.500842
\(498\) −1.50310e14 −0.219900
\(499\) 4.84241e14 0.700662 0.350331 0.936626i \(-0.386069\pi\)
0.350331 + 0.936626i \(0.386069\pi\)
\(500\) −4.99978e14 −0.715510
\(501\) −9.52012e13 −0.134752
\(502\) 3.72026e14 0.520838
\(503\) 4.09842e13 0.0567534 0.0283767 0.999597i \(-0.490966\pi\)
0.0283767 + 0.999597i \(0.490966\pi\)
\(504\) −1.20277e14 −0.164746
\(505\) −4.29724e14 −0.582220
\(506\) −1.05766e15 −1.41748
\(507\) −4.29085e14 −0.568852
\(508\) 6.33429e14 0.830706
\(509\) 4.26043e14 0.552720 0.276360 0.961054i \(-0.410872\pi\)
0.276360 + 0.961054i \(0.410872\pi\)
\(510\) −3.46305e14 −0.444450
\(511\) −6.44917e14 −0.818821
\(512\) −6.51490e14 −0.818320
\(513\) 2.67340e14 0.332214
\(514\) −5.39910e14 −0.663781
\(515\) −2.20478e15 −2.68180
\(516\) 4.02139e14 0.483952
\(517\) −1.30293e15 −1.55140
\(518\) 6.80021e13 0.0801140
\(519\) −5.63813e14 −0.657225
\(520\) 1.33592e14 0.154085
\(521\) −7.57646e14 −0.864687 −0.432343 0.901709i \(-0.642313\pi\)
−0.432343 + 0.901709i \(0.642313\pi\)
\(522\) 2.79424e12 0.00315555
\(523\) −6.73820e14 −0.752982 −0.376491 0.926420i \(-0.622869\pi\)
−0.376491 + 0.926420i \(0.622869\pi\)
\(524\) −3.46777e14 −0.383467
\(525\) −5.10342e14 −0.558451
\(526\) 4.75199e14 0.514581
\(527\) −1.66524e15 −1.78451
\(528\) −4.85272e14 −0.514634
\(529\) 1.65025e15 1.73198
\(530\) −3.40006e14 −0.353158
\(531\) −4.22156e13 −0.0433963
\(532\) 8.51290e14 0.866091
\(533\) 2.82837e13 0.0284798
\(534\) 8.64853e13 0.0861917
\(535\) −4.27671e14 −0.421856
\(536\) −1.41337e14 −0.137991
\(537\) −2.91734e14 −0.281922
\(538\) 8.12970e14 0.777628
\(539\) 1.26259e15 1.19543
\(540\) −2.64741e14 −0.248116
\(541\) 1.40310e15 1.30168 0.650838 0.759216i \(-0.274418\pi\)
0.650838 + 0.759216i \(0.274418\pi\)
\(542\) −2.53119e14 −0.232449
\(543\) −7.62225e14 −0.692922
\(544\) −1.21144e15 −1.09021
\(545\) 2.09510e15 1.86648
\(546\) 2.17286e13 0.0191634
\(547\) −2.11560e14 −0.184715 −0.0923575 0.995726i \(-0.529440\pi\)
−0.0923575 + 0.995726i \(0.529440\pi\)
\(548\) −3.30907e14 −0.286031
\(549\) 3.79986e14 0.325177
\(550\) 1.57398e15 1.33354
\(551\) −4.42966e13 −0.0371567
\(552\) −9.12973e14 −0.758216
\(553\) 5.14453e14 0.423017
\(554\) 4.97542e14 0.405067
\(555\) 3.35253e14 0.270247
\(556\) 4.67853e14 0.373420
\(557\) −7.06813e14 −0.558600 −0.279300 0.960204i \(-0.590102\pi\)
−0.279300 + 0.960204i \(0.590102\pi\)
\(558\) 3.05292e14 0.238906
\(559\) −1.62719e14 −0.126088
\(560\) −5.92363e14 −0.454521
\(561\) −1.62250e15 −1.23279
\(562\) −9.10947e14 −0.685399
\(563\) −2.31313e15 −1.72347 −0.861736 0.507358i \(-0.830622\pi\)
−0.861736 + 0.507358i \(0.830622\pi\)
\(564\) −5.02135e14 −0.370498
\(565\) −2.93620e15 −2.14545
\(566\) 7.15135e14 0.517485
\(567\) −9.64462e13 −0.0691159
\(568\) −9.01387e14 −0.639728
\(569\) 5.76143e14 0.404961 0.202480 0.979286i \(-0.435100\pi\)
0.202480 + 0.979286i \(0.435100\pi\)
\(570\) −1.00648e15 −0.700637
\(571\) 2.30158e15 1.58682 0.793408 0.608690i \(-0.208305\pi\)
0.793408 + 0.608690i \(0.208305\pi\)
\(572\) 2.79443e14 0.190816
\(573\) −8.95229e14 −0.605457
\(574\) 9.58693e13 0.0642191
\(575\) −3.87380e15 −2.57018
\(576\) −9.77227e12 −0.00642202
\(577\) 2.46899e15 1.60714 0.803569 0.595212i \(-0.202932\pi\)
0.803569 + 0.595212i \(0.202932\pi\)
\(578\) −1.35817e14 −0.0875691
\(579\) 3.12002e14 0.199263
\(580\) 4.38660e13 0.0277507
\(581\) −8.59637e14 −0.538699
\(582\) 6.02224e14 0.373837
\(583\) −1.59299e15 −0.979569
\(584\) 1.71694e15 1.04588
\(585\) 1.07123e14 0.0646435
\(586\) 3.92359e14 0.234556
\(587\) 2.89295e15 1.71329 0.856645 0.515906i \(-0.172545\pi\)
0.856645 + 0.515906i \(0.172545\pi\)
\(588\) 4.86589e14 0.285487
\(589\) −4.83975e15 −2.81313
\(590\) 1.58933e14 0.0915225
\(591\) −8.77900e14 −0.500857
\(592\) 2.36830e14 0.133865
\(593\) 1.90299e15 1.06570 0.532850 0.846210i \(-0.321121\pi\)
0.532850 + 0.846210i \(0.321121\pi\)
\(594\) 2.97456e14 0.165043
\(595\) −1.98056e15 −1.08879
\(596\) 1.86763e15 1.01727
\(597\) −6.40618e14 −0.345733
\(598\) 1.64933e14 0.0881964
\(599\) −5.98455e13 −0.0317091 −0.0158545 0.999874i \(-0.505047\pi\)
−0.0158545 + 0.999874i \(0.505047\pi\)
\(600\) 1.35866e15 0.713312
\(601\) −2.02557e14 −0.105375 −0.0526875 0.998611i \(-0.516779\pi\)
−0.0526875 + 0.998611i \(0.516779\pi\)
\(602\) −5.51546e14 −0.284316
\(603\) −1.13334e14 −0.0578914
\(604\) −2.36863e15 −1.19893
\(605\) 8.93004e15 4.47918
\(606\) 1.86077e14 0.0924895
\(607\) 2.40107e15 1.18268 0.591339 0.806423i \(-0.298600\pi\)
0.591339 + 0.806423i \(0.298600\pi\)
\(608\) −3.52086e15 −1.71862
\(609\) 1.59806e13 0.00773031
\(610\) −1.43057e15 −0.685795
\(611\) 2.03180e14 0.0965285
\(612\) −6.25294e14 −0.294409
\(613\) 3.48502e15 1.62619 0.813097 0.582128i \(-0.197780\pi\)
0.813097 + 0.582128i \(0.197780\pi\)
\(614\) −1.70513e15 −0.788555
\(615\) 4.72640e14 0.216629
\(616\) 2.12153e15 0.963730
\(617\) −1.10199e15 −0.496144 −0.248072 0.968742i \(-0.579797\pi\)
−0.248072 + 0.968742i \(0.579797\pi\)
\(618\) 9.54702e14 0.426021
\(619\) 1.99223e15 0.881131 0.440566 0.897720i \(-0.354778\pi\)
0.440566 + 0.897720i \(0.354778\pi\)
\(620\) 4.79270e15 2.10100
\(621\) −7.32083e14 −0.318095
\(622\) 3.54738e14 0.152778
\(623\) 4.94619e14 0.211148
\(624\) 7.56740e13 0.0320207
\(625\) −3.26667e14 −0.137014
\(626\) 3.05486e14 0.127009
\(627\) −4.71553e15 −1.94339
\(628\) 1.20419e15 0.491946
\(629\) 7.91837e14 0.320669
\(630\) 3.63100e14 0.145765
\(631\) 1.84059e15 0.732481 0.366240 0.930520i \(-0.380645\pi\)
0.366240 + 0.930520i \(0.380645\pi\)
\(632\) −1.36961e15 −0.540322
\(633\) 1.71045e14 0.0668945
\(634\) −1.19991e15 −0.465218
\(635\) −4.28306e15 −1.64626
\(636\) −6.13920e14 −0.233937
\(637\) −1.96890e14 −0.0743801
\(638\) −4.92868e13 −0.0184594
\(639\) −7.22793e14 −0.268385
\(640\) 4.35957e15 1.60492
\(641\) −3.24375e15 −1.18394 −0.591968 0.805961i \(-0.701649\pi\)
−0.591968 + 0.805961i \(0.701649\pi\)
\(642\) 1.85188e14 0.0670147
\(643\) 2.63199e14 0.0944329 0.0472164 0.998885i \(-0.484965\pi\)
0.0472164 + 0.998885i \(0.484965\pi\)
\(644\) −2.33117e15 −0.829282
\(645\) −2.71914e15 −0.959077
\(646\) −2.37721e15 −0.831360
\(647\) −3.99215e15 −1.38431 −0.692156 0.721748i \(-0.743339\pi\)
−0.692156 + 0.721748i \(0.743339\pi\)
\(648\) 2.56765e14 0.0882821
\(649\) 7.44628e14 0.253860
\(650\) −2.45449e14 −0.0829732
\(651\) 1.74600e15 0.585260
\(652\) 7.53855e14 0.250568
\(653\) −3.51274e14 −0.115777 −0.0578887 0.998323i \(-0.518437\pi\)
−0.0578887 + 0.998323i \(0.518437\pi\)
\(654\) −9.07209e14 −0.296503
\(655\) 2.34480e15 0.759939
\(656\) 3.33883e14 0.107306
\(657\) 1.37675e15 0.438780
\(658\) 6.88693e14 0.217662
\(659\) −1.44034e15 −0.451435 −0.225718 0.974193i \(-0.572473\pi\)
−0.225718 + 0.974193i \(0.572473\pi\)
\(660\) 4.66968e15 1.45143
\(661\) −2.84597e15 −0.877249 −0.438624 0.898671i \(-0.644534\pi\)
−0.438624 + 0.898671i \(0.644534\pi\)
\(662\) 1.80652e15 0.552236
\(663\) 2.53015e14 0.0767046
\(664\) 2.28858e15 0.688083
\(665\) −5.75617e15 −1.71638
\(666\) −1.45169e14 −0.0429306
\(667\) 1.21302e14 0.0355775
\(668\) 6.47156e14 0.188252
\(669\) −3.52031e15 −1.01563
\(670\) 4.26678e14 0.122092
\(671\) −6.70246e15 −1.90222
\(672\) 1.27019e15 0.357552
\(673\) 1.63336e15 0.456037 0.228019 0.973657i \(-0.426775\pi\)
0.228019 + 0.973657i \(0.426775\pi\)
\(674\) 1.54786e15 0.428649
\(675\) 1.08947e15 0.299256
\(676\) 2.91682e15 0.794699
\(677\) −5.78179e14 −0.156252 −0.0781259 0.996944i \(-0.524894\pi\)
−0.0781259 + 0.996944i \(0.524894\pi\)
\(678\) 1.27142e15 0.340820
\(679\) 3.44419e15 0.915806
\(680\) 5.27275e15 1.39072
\(681\) 2.38545e15 0.624110
\(682\) −5.38496e15 −1.39755
\(683\) 9.19834e14 0.236808 0.118404 0.992966i \(-0.462222\pi\)
0.118404 + 0.992966i \(0.462222\pi\)
\(684\) −1.81732e15 −0.464111
\(685\) 2.23750e15 0.566845
\(686\) −1.75596e15 −0.441298
\(687\) −1.91702e15 −0.477930
\(688\) −1.92086e15 −0.475072
\(689\) 2.48413e14 0.0609492
\(690\) 2.75614e15 0.670860
\(691\) 4.43480e15 1.07089 0.535445 0.844570i \(-0.320144\pi\)
0.535445 + 0.844570i \(0.320144\pi\)
\(692\) 3.83267e15 0.918159
\(693\) 1.70119e15 0.404314
\(694\) 7.04018e14 0.165999
\(695\) −3.16348e15 −0.740029
\(696\) −4.25444e13 −0.00987396
\(697\) 1.11633e15 0.257047
\(698\) −3.11474e15 −0.711569
\(699\) −1.16741e15 −0.264606
\(700\) 3.46919e15 0.780169
\(701\) −5.48622e15 −1.22412 −0.612060 0.790811i \(-0.709659\pi\)
−0.612060 + 0.790811i \(0.709659\pi\)
\(702\) −4.63857e13 −0.0102691
\(703\) 2.30135e15 0.505508
\(704\) 1.72370e14 0.0375675
\(705\) 3.39529e15 0.734237
\(706\) −1.56882e15 −0.336626
\(707\) 1.06419e15 0.226576
\(708\) 2.86972e14 0.0606256
\(709\) 5.79662e15 1.21512 0.607562 0.794272i \(-0.292148\pi\)
0.607562 + 0.794272i \(0.292148\pi\)
\(710\) 2.72117e15 0.566023
\(711\) −1.09824e15 −0.226681
\(712\) −1.31680e15 −0.269700
\(713\) 1.32532e16 2.69357
\(714\) 8.57609e14 0.172961
\(715\) −1.88951e15 −0.378152
\(716\) 1.98314e15 0.393852
\(717\) 1.98461e15 0.391129
\(718\) 1.60585e14 0.0314065
\(719\) −6.74615e15 −1.30932 −0.654662 0.755922i \(-0.727189\pi\)
−0.654662 + 0.755922i \(0.727189\pi\)
\(720\) 1.26456e15 0.243563
\(721\) 5.46005e15 1.04364
\(722\) −4.59045e15 −0.870765
\(723\) 4.99857e14 0.0940990
\(724\) 5.18144e15 0.968028
\(725\) −1.80518e14 −0.0334705
\(726\) −3.86683e15 −0.711548
\(727\) 6.39372e15 1.16766 0.583828 0.811878i \(-0.301554\pi\)
0.583828 + 0.811878i \(0.301554\pi\)
\(728\) −3.30834e14 −0.0599637
\(729\) 2.05891e14 0.0370370
\(730\) −5.18320e15 −0.925384
\(731\) −6.42237e15 −1.13802
\(732\) −2.58306e15 −0.454279
\(733\) −8.08612e15 −1.41146 −0.705730 0.708481i \(-0.749381\pi\)
−0.705730 + 0.708481i \(0.749381\pi\)
\(734\) 1.30158e15 0.225498
\(735\) −3.29017e15 −0.565767
\(736\) 9.64152e15 1.64558
\(737\) 1.99906e15 0.338653
\(738\) −2.04660e14 −0.0344130
\(739\) 7.98503e15 1.33270 0.666349 0.745640i \(-0.267856\pi\)
0.666349 + 0.745640i \(0.267856\pi\)
\(740\) −2.27897e15 −0.377542
\(741\) 7.35346e14 0.120918
\(742\) 8.42011e14 0.137435
\(743\) 6.96575e14 0.112857 0.0564286 0.998407i \(-0.482029\pi\)
0.0564286 + 0.998407i \(0.482029\pi\)
\(744\) −4.64831e15 −0.747556
\(745\) −1.26283e16 −2.01598
\(746\) 1.65711e15 0.262596
\(747\) 1.83513e15 0.288672
\(748\) 1.10294e16 1.72223
\(749\) 1.05911e15 0.164169
\(750\) −1.46389e15 −0.225254
\(751\) −7.26736e15 −1.11009 −0.555043 0.831822i \(-0.687298\pi\)
−0.555043 + 0.831822i \(0.687298\pi\)
\(752\) 2.39850e15 0.363699
\(753\) −4.54208e15 −0.683727
\(754\) 7.68585e12 0.00114855
\(755\) 1.60160e16 2.37599
\(756\) 6.55619e14 0.0965566
\(757\) 6.47637e15 0.946900 0.473450 0.880821i \(-0.343008\pi\)
0.473450 + 0.880821i \(0.343008\pi\)
\(758\) 3.27978e15 0.476062
\(759\) 1.29130e16 1.86079
\(760\) 1.53244e16 2.19235
\(761\) 5.52150e14 0.0784227 0.0392113 0.999231i \(-0.487515\pi\)
0.0392113 + 0.999231i \(0.487515\pi\)
\(762\) 1.85462e15 0.261519
\(763\) −5.18843e15 −0.726358
\(764\) 6.08556e15 0.845838
\(765\) 4.22805e15 0.583448
\(766\) 3.85502e15 0.528163
\(767\) −1.16118e14 −0.0157952
\(768\) −1.80539e15 −0.243829
\(769\) 6.34106e15 0.850290 0.425145 0.905125i \(-0.360223\pi\)
0.425145 + 0.905125i \(0.360223\pi\)
\(770\) −6.40461e15 −0.852696
\(771\) 6.59179e15 0.871373
\(772\) −2.12092e15 −0.278374
\(773\) −2.11511e15 −0.275642 −0.137821 0.990457i \(-0.544010\pi\)
−0.137821 + 0.990457i \(0.544010\pi\)
\(774\) 1.17743e15 0.152356
\(775\) −1.97230e16 −2.53404
\(776\) −9.16932e15 −1.16976
\(777\) −8.30240e14 −0.105169
\(778\) −2.24163e15 −0.281953
\(779\) 3.24444e15 0.405214
\(780\) −7.28197e14 −0.0903085
\(781\) 1.27491e16 1.57000
\(782\) 6.50976e15 0.796027
\(783\) −3.41150e13 −0.00414243
\(784\) −2.32425e15 −0.280249
\(785\) −8.14238e15 −0.974918
\(786\) −1.01533e15 −0.120721
\(787\) −1.46829e16 −1.73361 −0.866805 0.498647i \(-0.833831\pi\)
−0.866805 + 0.498647i \(0.833831\pi\)
\(788\) 5.96777e15 0.699709
\(789\) −5.80172e15 −0.675512
\(790\) 4.13466e15 0.478069
\(791\) 7.27138e15 0.834922
\(792\) −4.52900e15 −0.516432
\(793\) 1.04519e15 0.118357
\(794\) 2.98820e15 0.336045
\(795\) 4.15115e15 0.463606
\(796\) 4.35478e15 0.482997
\(797\) 1.64346e16 1.81025 0.905123 0.425150i \(-0.139779\pi\)
0.905123 + 0.425150i \(0.139779\pi\)
\(798\) 2.49250e15 0.272659
\(799\) 8.01936e15 0.871229
\(800\) −1.43483e16 −1.54812
\(801\) −1.05590e15 −0.113148
\(802\) 3.09007e15 0.328859
\(803\) −2.42842e16 −2.56677
\(804\) 7.70417e14 0.0808756
\(805\) 1.57627e16 1.64344
\(806\) 8.39739e14 0.0869564
\(807\) −9.92558e15 −1.02083
\(808\) −2.83316e15 −0.289406
\(809\) 9.93747e15 1.00823 0.504114 0.863637i \(-0.331819\pi\)
0.504114 + 0.863637i \(0.331819\pi\)
\(810\) −7.75138e14 −0.0781108
\(811\) −1.40232e16 −1.40357 −0.701784 0.712390i \(-0.747613\pi\)
−0.701784 + 0.712390i \(0.747613\pi\)
\(812\) −1.08632e14 −0.0107994
\(813\) 3.09033e15 0.305146
\(814\) 2.56060e15 0.251135
\(815\) −5.09734e15 −0.496566
\(816\) 2.98679e15 0.289007
\(817\) −1.86656e16 −1.79399
\(818\) 7.92680e15 0.756754
\(819\) −2.65285e14 −0.0251566
\(820\) −3.21290e15 −0.302636
\(821\) −1.78706e16 −1.67206 −0.836032 0.548681i \(-0.815130\pi\)
−0.836032 + 0.548681i \(0.815130\pi\)
\(822\) −9.68868e14 −0.0900471
\(823\) 1.28825e16 1.18933 0.594665 0.803974i \(-0.297285\pi\)
0.594665 + 0.803974i \(0.297285\pi\)
\(824\) −1.45361e16 −1.33305
\(825\) −1.92168e16 −1.75059
\(826\) −3.93591e14 −0.0356168
\(827\) 5.58717e15 0.502240 0.251120 0.967956i \(-0.419201\pi\)
0.251120 + 0.967956i \(0.419201\pi\)
\(828\) 4.97654e15 0.444386
\(829\) −3.13657e14 −0.0278231 −0.0139115 0.999903i \(-0.504428\pi\)
−0.0139115 + 0.999903i \(0.504428\pi\)
\(830\) −6.90890e15 −0.608807
\(831\) −6.07451e15 −0.531748
\(832\) −2.68797e13 −0.00233747
\(833\) −7.77108e15 −0.671327
\(834\) 1.36983e15 0.117559
\(835\) −4.37588e15 −0.373069
\(836\) 3.20551e16 2.71496
\(837\) −3.72732e15 −0.313622
\(838\) 3.78068e15 0.316029
\(839\) −6.53730e15 −0.542885 −0.271442 0.962455i \(-0.587501\pi\)
−0.271442 + 0.962455i \(0.587501\pi\)
\(840\) −5.52847e15 −0.456109
\(841\) −1.21949e16 −0.999537
\(842\) 7.77442e15 0.633070
\(843\) 1.11218e16 0.899752
\(844\) −1.16273e15 −0.0934532
\(845\) −1.97227e16 −1.57490
\(846\) −1.47021e15 −0.116638
\(847\) −2.21149e16 −1.74311
\(848\) 2.93246e15 0.229644
\(849\) −8.73110e15 −0.679324
\(850\) −9.68765e15 −0.748884
\(851\) −6.30201e15 −0.484024
\(852\) 4.91338e15 0.374941
\(853\) 7.44956e15 0.564821 0.282411 0.959294i \(-0.408866\pi\)
0.282411 + 0.959294i \(0.408866\pi\)
\(854\) 3.54274e15 0.266883
\(855\) 1.22881e16 0.919756
\(856\) −2.81963e15 −0.209694
\(857\) −6.19950e15 −0.458102 −0.229051 0.973414i \(-0.573562\pi\)
−0.229051 + 0.973414i \(0.573562\pi\)
\(858\) 8.18185e14 0.0600719
\(859\) 1.44810e16 1.05642 0.528211 0.849113i \(-0.322863\pi\)
0.528211 + 0.849113i \(0.322863\pi\)
\(860\) 1.84841e16 1.33985
\(861\) −1.17047e15 −0.0843032
\(862\) 7.45611e15 0.533608
\(863\) 1.29197e16 0.918738 0.459369 0.888246i \(-0.348076\pi\)
0.459369 + 0.888246i \(0.348076\pi\)
\(864\) −2.71158e15 −0.191601
\(865\) −2.59154e16 −1.81957
\(866\) −4.00756e15 −0.279597
\(867\) 1.65819e15 0.114956
\(868\) −1.18689e16 −0.817622
\(869\) 1.93716e16 1.32604
\(870\) 1.28436e14 0.00873635
\(871\) −3.11736e14 −0.0210711
\(872\) 1.38129e16 0.927781
\(873\) −7.35258e15 −0.490751
\(874\) 1.89196e16 1.25487
\(875\) −8.37216e15 −0.551814
\(876\) −9.35887e15 −0.612986
\(877\) 1.30252e16 0.847786 0.423893 0.905712i \(-0.360663\pi\)
0.423893 + 0.905712i \(0.360663\pi\)
\(878\) 4.15554e15 0.268787
\(879\) −4.79033e15 −0.307912
\(880\) −2.23053e16 −1.42480
\(881\) −2.73110e16 −1.73368 −0.866842 0.498583i \(-0.833854\pi\)
−0.866842 + 0.498583i \(0.833854\pi\)
\(882\) 1.42469e15 0.0898759
\(883\) 6.04765e15 0.379143 0.189571 0.981867i \(-0.439290\pi\)
0.189571 + 0.981867i \(0.439290\pi\)
\(884\) −1.71994e15 −0.107158
\(885\) −1.94042e15 −0.120145
\(886\) −1.26363e16 −0.777563
\(887\) 1.56651e16 0.957972 0.478986 0.877822i \(-0.341004\pi\)
0.478986 + 0.877822i \(0.341004\pi\)
\(888\) 2.21031e15 0.134333
\(889\) 1.06068e16 0.640656
\(890\) 3.97526e15 0.238627
\(891\) −3.63165e15 −0.216659
\(892\) 2.39303e16 1.41887
\(893\) 2.33070e16 1.37342
\(894\) 5.46825e15 0.320252
\(895\) −1.34094e16 −0.780519
\(896\) −1.07963e16 −0.624569
\(897\) −2.01367e15 −0.115779
\(898\) −8.97626e15 −0.512951
\(899\) 6.17596e14 0.0350773
\(900\) −7.40595e15 −0.418068
\(901\) 9.80463e15 0.550104
\(902\) 3.60993e15 0.201309
\(903\) 6.73384e15 0.373233
\(904\) −1.93583e16 −1.06645
\(905\) −3.50353e16 −1.91840
\(906\) −6.93514e15 −0.377442
\(907\) −5.54810e15 −0.300127 −0.150063 0.988676i \(-0.547948\pi\)
−0.150063 + 0.988676i \(0.547948\pi\)
\(908\) −1.62158e16 −0.871897
\(909\) −2.27182e15 −0.121415
\(910\) 9.98745e14 0.0530551
\(911\) −1.81180e15 −0.0956662 −0.0478331 0.998855i \(-0.515232\pi\)
−0.0478331 + 0.998855i \(0.515232\pi\)
\(912\) 8.68062e15 0.455595
\(913\) −3.23694e16 −1.68867
\(914\) 8.78926e14 0.0455772
\(915\) 1.74659e16 0.900272
\(916\) 1.30315e16 0.667679
\(917\) −5.80680e15 −0.295737
\(918\) −1.83081e15 −0.0926846
\(919\) 2.10479e16 1.05919 0.529595 0.848251i \(-0.322344\pi\)
0.529595 + 0.848251i \(0.322344\pi\)
\(920\) −4.19644e16 −2.09917
\(921\) 2.08180e16 1.03517
\(922\) 1.48021e15 0.0731651
\(923\) −1.98812e15 −0.0976861
\(924\) −1.15643e16 −0.564836
\(925\) 9.37848e15 0.455358
\(926\) −1.14135e16 −0.550882
\(927\) −1.16560e16 −0.559256
\(928\) 4.49293e14 0.0214297
\(929\) 2.42748e16 1.15098 0.575492 0.817808i \(-0.304811\pi\)
0.575492 + 0.817808i \(0.304811\pi\)
\(930\) 1.40326e16 0.661427
\(931\) −2.25854e16 −1.05829
\(932\) 7.93579e15 0.369660
\(933\) −4.33100e15 −0.200558
\(934\) −2.99432e15 −0.137845
\(935\) −7.45773e16 −3.41305
\(936\) 7.06258e14 0.0321326
\(937\) 1.07613e15 0.0486739 0.0243370 0.999704i \(-0.492253\pi\)
0.0243370 + 0.999704i \(0.492253\pi\)
\(938\) −1.05665e15 −0.0475134
\(939\) −3.72969e15 −0.166730
\(940\) −2.30804e16 −1.02575
\(941\) 1.33032e16 0.587780 0.293890 0.955839i \(-0.405050\pi\)
0.293890 + 0.955839i \(0.405050\pi\)
\(942\) 3.52577e15 0.154872
\(943\) −8.88458e15 −0.387992
\(944\) −1.37075e15 −0.0595132
\(945\) −4.43310e15 −0.191352
\(946\) −2.07683e16 −0.891250
\(947\) 3.09201e16 1.31922 0.659608 0.751610i \(-0.270722\pi\)
0.659608 + 0.751610i \(0.270722\pi\)
\(948\) 7.46561e15 0.316679
\(949\) 3.78691e15 0.159706
\(950\) −2.81556e16 −1.18055
\(951\) 1.46497e16 0.610711
\(952\) −1.30577e16 −0.541209
\(953\) −2.13055e16 −0.877974 −0.438987 0.898493i \(-0.644663\pi\)
−0.438987 + 0.898493i \(0.644663\pi\)
\(954\) −1.79751e15 −0.0736469
\(955\) −4.11487e16 −1.67625
\(956\) −1.34909e16 −0.546416
\(957\) 6.01744e14 0.0242324
\(958\) 3.83235e15 0.153446
\(959\) −5.54107e15 −0.220593
\(960\) −4.49178e14 −0.0177798
\(961\) 4.20687e16 1.65569
\(962\) −3.99303e14 −0.0156257
\(963\) −2.26097e15 −0.0879730
\(964\) −3.39792e15 −0.131459
\(965\) 1.43410e16 0.551671
\(966\) −6.82547e15 −0.261071
\(967\) 3.12915e16 1.19009 0.595046 0.803692i \(-0.297134\pi\)
0.595046 + 0.803692i \(0.297134\pi\)
\(968\) 5.88755e16 2.22649
\(969\) 2.90235e16 1.09136
\(970\) 2.76810e16 1.03499
\(971\) −1.49974e16 −0.557584 −0.278792 0.960352i \(-0.589934\pi\)
−0.278792 + 0.960352i \(0.589934\pi\)
\(972\) −1.39960e15 −0.0517416
\(973\) 7.83423e15 0.287989
\(974\) −2.15388e16 −0.787314
\(975\) 2.99669e15 0.108922
\(976\) 1.23383e16 0.445944
\(977\) −1.78330e16 −0.640920 −0.320460 0.947262i \(-0.603837\pi\)
−0.320460 + 0.947262i \(0.603837\pi\)
\(978\) 2.20722e15 0.0788828
\(979\) 1.86248e16 0.661890
\(980\) 2.23658e16 0.790390
\(981\) 1.10761e16 0.389233
\(982\) −2.23608e16 −0.781401
\(983\) −1.53203e16 −0.532380 −0.266190 0.963921i \(-0.585765\pi\)
−0.266190 + 0.963921i \(0.585765\pi\)
\(984\) 3.11610e15 0.107681
\(985\) −4.03523e16 −1.38666
\(986\) 3.03354e14 0.0103664
\(987\) −8.40828e15 −0.285735
\(988\) −4.99872e15 −0.168926
\(989\) 5.11138e16 1.71775
\(990\) 1.36724e16 0.456933
\(991\) 1.56797e15 0.0521113 0.0260557 0.999660i \(-0.491705\pi\)
0.0260557 + 0.999660i \(0.491705\pi\)
\(992\) 4.90888e16 1.62244
\(993\) −2.20559e16 −0.724944
\(994\) −6.73885e15 −0.220273
\(995\) −2.94457e16 −0.957183
\(996\) −1.24748e16 −0.403281
\(997\) 4.31229e16 1.38639 0.693193 0.720752i \(-0.256203\pi\)
0.693193 + 0.720752i \(0.256203\pi\)
\(998\) −9.63799e15 −0.308155
\(999\) 1.77238e15 0.0563568
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.1.11 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.11 27 1.1 even 1 trivial