Properties

Label 273.12.a.c.1.10
Level $273$
Weight $12$
Character 273.1
Self dual yes
Analytic conductor $209.758$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 273.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(209.757688293\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{15} - 26640 x^{14} - 38138 x^{13} + 279954048 x^{12} + 940095168 x^{11} - 1473489774048 x^{10} - 6951601887328 x^{9} + 4072481535854976 x^{8} + 21355761647276288 x^{7} - 5703452025696599040 x^{6} - 25041700255043457024 x^{5} + 3737994671267475603456 x^{4} + 6711995200063092490240 x^{3} - 963518272786595267739648 x^{2} + 1287697466669064032092160 x + 45555354651594891689197568\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{12}\cdot 5\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(18.3601\) of defining polynomial
Character \(\chi\) \(=\) 273.1

$q$-expansion

\(f(q)\) \(=\) \(q+14.3601 q^{2} -243.000 q^{3} -1841.79 q^{4} +4424.64 q^{5} -3489.51 q^{6} +16807.0 q^{7} -55857.8 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+14.3601 q^{2} -243.000 q^{3} -1841.79 q^{4} +4424.64 q^{5} -3489.51 q^{6} +16807.0 q^{7} -55857.8 q^{8} +59049.0 q^{9} +63538.3 q^{10} +338272. q^{11} +447554. q^{12} -371293. q^{13} +241350. q^{14} -1.07519e6 q^{15} +2.96986e6 q^{16} -3.75161e6 q^{17} +847950. q^{18} -1.16379e7 q^{19} -8.14925e6 q^{20} -4.08410e6 q^{21} +4.85763e6 q^{22} +5.77416e7 q^{23} +1.35734e7 q^{24} -2.92507e7 q^{25} -5.33181e6 q^{26} -1.43489e7 q^{27} -3.09549e7 q^{28} -1.08561e8 q^{29} -1.54398e7 q^{30} -9.12180e7 q^{31} +1.57044e8 q^{32} -8.22002e7 q^{33} -5.38735e7 q^{34} +7.43650e7 q^{35} -1.08756e8 q^{36} -5.76597e8 q^{37} -1.67122e8 q^{38} +9.02242e7 q^{39} -2.47151e8 q^{40} +1.17326e9 q^{41} -5.86481e7 q^{42} +1.62472e9 q^{43} -6.23026e8 q^{44} +2.61271e8 q^{45} +8.29176e8 q^{46} +7.56156e8 q^{47} -7.21675e8 q^{48} +2.82475e8 q^{49} -4.20043e8 q^{50} +9.11641e8 q^{51} +6.83843e8 q^{52} -2.53658e9 q^{53} -2.06052e8 q^{54} +1.49673e9 q^{55} -9.38802e8 q^{56} +2.82802e9 q^{57} -1.55895e9 q^{58} +1.62567e9 q^{59} +1.98027e9 q^{60} +1.26515e8 q^{61} -1.30990e9 q^{62} +9.92437e8 q^{63} -3.82709e9 q^{64} -1.64284e9 q^{65} -1.18040e9 q^{66} -1.00750e10 q^{67} +6.90967e9 q^{68} -1.40312e10 q^{69} +1.06789e9 q^{70} +1.02472e10 q^{71} -3.29835e9 q^{72} +1.14591e10 q^{73} -8.28000e9 q^{74} +7.10791e9 q^{75} +2.14346e10 q^{76} +5.68534e9 q^{77} +1.29563e9 q^{78} +2.44618e10 q^{79} +1.31406e10 q^{80} +3.48678e9 q^{81} +1.68481e10 q^{82} -1.15405e10 q^{83} +7.52205e9 q^{84} -1.65995e10 q^{85} +2.33311e10 q^{86} +2.63803e10 q^{87} -1.88951e10 q^{88} +2.29828e10 q^{89} +3.75188e9 q^{90} -6.24032e9 q^{91} -1.06348e11 q^{92} +2.21660e10 q^{93} +1.08585e10 q^{94} -5.14937e10 q^{95} -3.81617e10 q^{96} -5.73728e10 q^{97} +4.05638e9 q^{98} +1.99746e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + O(q^{10}) \) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + 1056711q^{10} - 466884q^{11} - 5044923q^{12} - 5940688q^{13} - 1058841q^{14} + 769824q^{15} + 40058265q^{16} + 1753452q^{17} - 3720087q^{18} + 4237800q^{19} - 20710503q^{20} - 65345616q^{21} - 37479661q^{22} - 150481440q^{23} + 45675495q^{24} + 150983176q^{25} + 23391459q^{26} - 229582512q^{27} + 348930127q^{28} - 111411432q^{29} - 256780773q^{30} - 90536236q^{31} - 609941313q^{32} + 113452812q^{33} + 443745577q^{34} - 53244576q^{35} + 1225916289q^{36} - 1756337900q^{37} - 4378868973q^{38} + 1443587184q^{39} + 57535383q^{40} + 649575720q^{41} + 257298363q^{42} - 1889554520q^{43} - 4979587689q^{44} - 187067232q^{45} - 3204000867q^{46} - 4036082940q^{47} - 9734158395q^{48} + 4519603984q^{49} - 15928886862q^{50} - 426088836q^{51} - 7708413973q^{52} + 511144020q^{53} + 903981141q^{54} - 8519365572q^{55} - 3159127755q^{56} - 1029785400q^{57} - 7851149577q^{58} + 3728287296q^{59} + 5032652229q^{60} + 26227205052q^{61} + 2996618490q^{62} + 15878984688q^{63} + 51104408465q^{64} + 1176256224q^{65} + 9107557623q^{66} - 5295680024q^{67} + 7919540325q^{68} + 36566989920q^{69} + 17760141777q^{70} + 17082800928q^{71} - 11099145285q^{72} + 21076301488q^{73} + 52794333675q^{74} - 36688911768q^{75} + 81459179177q^{76} - 7846919388q^{77} - 5684124537q^{78} - 83677977852q^{79} - 163988465427q^{80} + 55788550416q^{81} - 61543728760q^{82} - 72857072340q^{83} - 84790020861q^{84} - 12608565060q^{85} - 20177818011q^{86} + 27072977976q^{87} - 202213679643q^{88} + 32238476676q^{89} + 62397727839q^{90} - 99845143216q^{91} - 487057197033q^{92} + 22000305348q^{93} + 183904776602q^{94} - 143022915504q^{95} + 148215739059q^{96} + 6973535140q^{97} - 17795940687q^{98} - 27569033316q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.3601 0.317317 0.158658 0.987334i \(-0.449283\pi\)
0.158658 + 0.987334i \(0.449283\pi\)
\(3\) −243.000 −0.577350
\(4\) −1841.79 −0.899310
\(5\) 4424.64 0.633203 0.316602 0.948559i \(-0.397458\pi\)
0.316602 + 0.948559i \(0.397458\pi\)
\(6\) −3489.51 −0.183203
\(7\) 16807.0 0.377964
\(8\) −55857.8 −0.602683
\(9\) 59049.0 0.333333
\(10\) 63538.3 0.200926
\(11\) 338272. 0.633296 0.316648 0.948543i \(-0.397443\pi\)
0.316648 + 0.948543i \(0.397443\pi\)
\(12\) 447554. 0.519217
\(13\) −371293. −0.277350
\(14\) 241350. 0.119934
\(15\) −1.07519e6 −0.365580
\(16\) 2.96986e6 0.708069
\(17\) −3.75161e6 −0.640838 −0.320419 0.947276i \(-0.603824\pi\)
−0.320419 + 0.947276i \(0.603824\pi\)
\(18\) 847950. 0.105772
\(19\) −1.16379e7 −1.07828 −0.539139 0.842217i \(-0.681250\pi\)
−0.539139 + 0.842217i \(0.681250\pi\)
\(20\) −8.14925e6 −0.569446
\(21\) −4.08410e6 −0.218218
\(22\) 4.85763e6 0.200955
\(23\) 5.77416e7 1.87062 0.935311 0.353827i \(-0.115120\pi\)
0.935311 + 0.353827i \(0.115120\pi\)
\(24\) 1.35734e7 0.347959
\(25\) −2.92507e7 −0.599054
\(26\) −5.33181e6 −0.0880078
\(27\) −1.43489e7 −0.192450
\(28\) −3.09549e7 −0.339907
\(29\) −1.08561e8 −0.982843 −0.491422 0.870922i \(-0.663523\pi\)
−0.491422 + 0.870922i \(0.663523\pi\)
\(30\) −1.54398e7 −0.116005
\(31\) −9.12180e7 −0.572257 −0.286128 0.958191i \(-0.592368\pi\)
−0.286128 + 0.958191i \(0.592368\pi\)
\(32\) 1.57044e8 0.827365
\(33\) −8.22002e7 −0.365634
\(34\) −5.38735e7 −0.203349
\(35\) 7.43650e7 0.239328
\(36\) −1.08756e8 −0.299770
\(37\) −5.76597e8 −1.36698 −0.683492 0.729958i \(-0.739539\pi\)
−0.683492 + 0.729958i \(0.739539\pi\)
\(38\) −1.67122e8 −0.342156
\(39\) 9.02242e7 0.160128
\(40\) −2.47151e8 −0.381621
\(41\) 1.17326e9 1.58154 0.790772 0.612111i \(-0.209679\pi\)
0.790772 + 0.612111i \(0.209679\pi\)
\(42\) −5.86481e7 −0.0692442
\(43\) 1.62472e9 1.68539 0.842697 0.538387i \(-0.180966\pi\)
0.842697 + 0.538387i \(0.180966\pi\)
\(44\) −6.23026e8 −0.569530
\(45\) 2.61271e8 0.211068
\(46\) 8.29176e8 0.593579
\(47\) 7.56156e8 0.480920 0.240460 0.970659i \(-0.422702\pi\)
0.240460 + 0.970659i \(0.422702\pi\)
\(48\) −7.21675e8 −0.408804
\(49\) 2.82475e8 0.142857
\(50\) −4.20043e8 −0.190090
\(51\) 9.11641e8 0.369988
\(52\) 6.83843e8 0.249424
\(53\) −2.53658e9 −0.833164 −0.416582 0.909098i \(-0.636772\pi\)
−0.416582 + 0.909098i \(0.636772\pi\)
\(54\) −2.06052e8 −0.0610676
\(55\) 1.49673e9 0.401005
\(56\) −9.38802e8 −0.227793
\(57\) 2.82802e9 0.622544
\(58\) −1.55895e9 −0.311873
\(59\) 1.62567e9 0.296038 0.148019 0.988985i \(-0.452710\pi\)
0.148019 + 0.988985i \(0.452710\pi\)
\(60\) 1.98027e9 0.328770
\(61\) 1.26515e8 0.0191790 0.00958951 0.999954i \(-0.496948\pi\)
0.00958951 + 0.999954i \(0.496948\pi\)
\(62\) −1.30990e9 −0.181587
\(63\) 9.92437e8 0.125988
\(64\) −3.82709e9 −0.445532
\(65\) −1.64284e9 −0.175619
\(66\) −1.18040e9 −0.116022
\(67\) −1.00750e10 −0.911664 −0.455832 0.890066i \(-0.650658\pi\)
−0.455832 + 0.890066i \(0.650658\pi\)
\(68\) 6.90967e9 0.576313
\(69\) −1.40312e10 −1.08000
\(70\) 1.06789e9 0.0759428
\(71\) 1.02472e10 0.674036 0.337018 0.941498i \(-0.390582\pi\)
0.337018 + 0.941498i \(0.390582\pi\)
\(72\) −3.29835e9 −0.200894
\(73\) 1.14591e10 0.646956 0.323478 0.946236i \(-0.395148\pi\)
0.323478 + 0.946236i \(0.395148\pi\)
\(74\) −8.28000e9 −0.433767
\(75\) 7.10791e9 0.345864
\(76\) 2.14346e10 0.969707
\(77\) 5.68534e9 0.239363
\(78\) 1.29563e9 0.0508113
\(79\) 2.44618e10 0.894417 0.447208 0.894430i \(-0.352418\pi\)
0.447208 + 0.894430i \(0.352418\pi\)
\(80\) 1.31406e10 0.448352
\(81\) 3.48678e9 0.111111
\(82\) 1.68481e10 0.501850
\(83\) −1.15405e10 −0.321585 −0.160793 0.986988i \(-0.551405\pi\)
−0.160793 + 0.986988i \(0.551405\pi\)
\(84\) 7.52205e9 0.196246
\(85\) −1.65995e10 −0.405781
\(86\) 2.33311e10 0.534804
\(87\) 2.63803e10 0.567445
\(88\) −1.88951e10 −0.381677
\(89\) 2.29828e10 0.436273 0.218136 0.975918i \(-0.430002\pi\)
0.218136 + 0.975918i \(0.430002\pi\)
\(90\) 3.75188e9 0.0669753
\(91\) −6.24032e9 −0.104828
\(92\) −1.06348e11 −1.68227
\(93\) 2.21660e10 0.330393
\(94\) 1.08585e10 0.152604
\(95\) −5.14937e10 −0.682769
\(96\) −3.81617e10 −0.477679
\(97\) −5.73728e10 −0.678362 −0.339181 0.940721i \(-0.610150\pi\)
−0.339181 + 0.940721i \(0.610150\pi\)
\(98\) 4.05638e9 0.0453309
\(99\) 1.99746e10 0.211099
\(100\) 5.38735e10 0.538735
\(101\) 1.65759e11 1.56931 0.784655 0.619933i \(-0.212840\pi\)
0.784655 + 0.619933i \(0.212840\pi\)
\(102\) 1.30913e10 0.117403
\(103\) 1.25356e11 1.06547 0.532736 0.846282i \(-0.321164\pi\)
0.532736 + 0.846282i \(0.321164\pi\)
\(104\) 2.07396e10 0.167154
\(105\) −1.80707e10 −0.138176
\(106\) −3.64255e10 −0.264377
\(107\) 2.37292e11 1.63558 0.817791 0.575516i \(-0.195199\pi\)
0.817791 + 0.575516i \(0.195199\pi\)
\(108\) 2.64276e10 0.173072
\(109\) −2.15796e11 −1.34338 −0.671688 0.740834i \(-0.734431\pi\)
−0.671688 + 0.740834i \(0.734431\pi\)
\(110\) 2.14933e10 0.127246
\(111\) 1.40113e11 0.789228
\(112\) 4.99144e10 0.267625
\(113\) −2.51250e10 −0.128285 −0.0641423 0.997941i \(-0.520431\pi\)
−0.0641423 + 0.997941i \(0.520431\pi\)
\(114\) 4.06106e10 0.197544
\(115\) 2.55486e11 1.18448
\(116\) 1.99946e11 0.883881
\(117\) −2.19245e10 −0.0924500
\(118\) 2.33449e10 0.0939378
\(119\) −6.30533e10 −0.242214
\(120\) 6.00576e10 0.220329
\(121\) −1.70883e11 −0.598936
\(122\) 1.81676e9 0.00608582
\(123\) −2.85101e11 −0.913105
\(124\) 1.68004e11 0.514636
\(125\) −3.45471e11 −1.01253
\(126\) 1.42515e10 0.0399781
\(127\) −4.50271e11 −1.20936 −0.604678 0.796470i \(-0.706698\pi\)
−0.604678 + 0.796470i \(0.706698\pi\)
\(128\) −3.76584e11 −0.968739
\(129\) −3.94807e11 −0.973063
\(130\) −2.35913e10 −0.0557268
\(131\) −2.39336e11 −0.542022 −0.271011 0.962576i \(-0.587358\pi\)
−0.271011 + 0.962576i \(0.587358\pi\)
\(132\) 1.51395e11 0.328818
\(133\) −1.95599e11 −0.407551
\(134\) −1.44678e11 −0.289286
\(135\) −6.34888e10 −0.121860
\(136\) 2.09557e11 0.386222
\(137\) 1.02628e11 0.181678 0.0908388 0.995866i \(-0.471045\pi\)
0.0908388 + 0.995866i \(0.471045\pi\)
\(138\) −2.01490e11 −0.342703
\(139\) −5.39795e10 −0.0882363 −0.0441181 0.999026i \(-0.514048\pi\)
−0.0441181 + 0.999026i \(0.514048\pi\)
\(140\) −1.36964e11 −0.215230
\(141\) −1.83746e11 −0.277659
\(142\) 1.47151e11 0.213883
\(143\) −1.25598e11 −0.175645
\(144\) 1.75367e11 0.236023
\(145\) −4.80343e11 −0.622340
\(146\) 1.64554e11 0.205290
\(147\) −6.86415e10 −0.0824786
\(148\) 1.06197e12 1.22934
\(149\) −1.61055e12 −1.79659 −0.898294 0.439394i \(-0.855193\pi\)
−0.898294 + 0.439394i \(0.855193\pi\)
\(150\) 1.02070e11 0.109748
\(151\) −6.54417e11 −0.678393 −0.339197 0.940715i \(-0.610155\pi\)
−0.339197 + 0.940715i \(0.610155\pi\)
\(152\) 6.50069e11 0.649860
\(153\) −2.21529e11 −0.213613
\(154\) 8.16422e10 0.0759540
\(155\) −4.03607e11 −0.362355
\(156\) −1.66174e11 −0.144005
\(157\) −9.44292e11 −0.790056 −0.395028 0.918669i \(-0.629265\pi\)
−0.395028 + 0.918669i \(0.629265\pi\)
\(158\) 3.51275e11 0.283813
\(159\) 6.16388e11 0.481028
\(160\) 6.94864e11 0.523890
\(161\) 9.70464e11 0.707029
\(162\) 5.00706e10 0.0352574
\(163\) 6.94267e11 0.472601 0.236301 0.971680i \(-0.424065\pi\)
0.236301 + 0.971680i \(0.424065\pi\)
\(164\) −2.16089e12 −1.42230
\(165\) −3.63706e11 −0.231520
\(166\) −1.65723e11 −0.102044
\(167\) −2.56300e12 −1.52689 −0.763444 0.645874i \(-0.776493\pi\)
−0.763444 + 0.645874i \(0.776493\pi\)
\(168\) 2.28129e11 0.131516
\(169\) 1.37858e11 0.0769231
\(170\) −2.38371e11 −0.128761
\(171\) −6.87208e11 −0.359426
\(172\) −2.99239e12 −1.51569
\(173\) 3.50023e12 1.71729 0.858645 0.512571i \(-0.171307\pi\)
0.858645 + 0.512571i \(0.171307\pi\)
\(174\) 3.78824e11 0.180060
\(175\) −4.91616e11 −0.226421
\(176\) 1.00462e12 0.448417
\(177\) −3.95039e11 −0.170918
\(178\) 3.30036e11 0.138437
\(179\) 3.94665e12 1.60523 0.802615 0.596497i \(-0.203441\pi\)
0.802615 + 0.596497i \(0.203441\pi\)
\(180\) −4.81205e11 −0.189815
\(181\) −4.53603e12 −1.73558 −0.867788 0.496934i \(-0.834459\pi\)
−0.867788 + 0.496934i \(0.834459\pi\)
\(182\) −8.96117e10 −0.0332638
\(183\) −3.07430e10 −0.0110730
\(184\) −3.22532e12 −1.12739
\(185\) −2.55124e12 −0.865578
\(186\) 3.18306e11 0.104839
\(187\) −1.26907e12 −0.405840
\(188\) −1.39268e12 −0.432496
\(189\) −2.41162e11 −0.0727393
\(190\) −7.39455e11 −0.216654
\(191\) 4.14824e12 1.18081 0.590405 0.807107i \(-0.298968\pi\)
0.590405 + 0.807107i \(0.298968\pi\)
\(192\) 9.29984e11 0.257228
\(193\) −7.06168e12 −1.89820 −0.949102 0.314970i \(-0.898005\pi\)
−0.949102 + 0.314970i \(0.898005\pi\)
\(194\) −8.23880e11 −0.215256
\(195\) 3.99210e11 0.101394
\(196\) −5.20259e11 −0.128473
\(197\) 3.37833e11 0.0811219 0.0405610 0.999177i \(-0.487085\pi\)
0.0405610 + 0.999177i \(0.487085\pi\)
\(198\) 2.86838e11 0.0669851
\(199\) −3.00148e12 −0.681780 −0.340890 0.940103i \(-0.610728\pi\)
−0.340890 + 0.940103i \(0.610728\pi\)
\(200\) 1.63388e12 0.361039
\(201\) 2.44823e12 0.526349
\(202\) 2.38031e12 0.497968
\(203\) −1.82458e12 −0.371480
\(204\) −1.67905e12 −0.332734
\(205\) 5.19123e12 1.00144
\(206\) 1.80013e12 0.338092
\(207\) 3.40959e12 0.623541
\(208\) −1.10269e12 −0.196383
\(209\) −3.93679e12 −0.682870
\(210\) −2.59497e11 −0.0438456
\(211\) −3.07714e12 −0.506516 −0.253258 0.967399i \(-0.581502\pi\)
−0.253258 + 0.967399i \(0.581502\pi\)
\(212\) 4.67183e12 0.749273
\(213\) −2.49006e12 −0.389155
\(214\) 3.40754e12 0.518997
\(215\) 7.18880e12 1.06720
\(216\) 8.01498e11 0.115986
\(217\) −1.53310e12 −0.216293
\(218\) −3.09885e12 −0.426276
\(219\) −2.78456e12 −0.373520
\(220\) −2.75667e12 −0.360628
\(221\) 1.39295e12 0.177737
\(222\) 2.01204e12 0.250435
\(223\) −8.21247e12 −0.997234 −0.498617 0.866823i \(-0.666159\pi\)
−0.498617 + 0.866823i \(0.666159\pi\)
\(224\) 2.63944e12 0.312714
\(225\) −1.72722e12 −0.199685
\(226\) −3.60798e11 −0.0407068
\(227\) 2.91907e12 0.321442 0.160721 0.987000i \(-0.448618\pi\)
0.160721 + 0.987000i \(0.448618\pi\)
\(228\) −5.20861e12 −0.559861
\(229\) −9.73078e12 −1.02106 −0.510531 0.859859i \(-0.670551\pi\)
−0.510531 + 0.859859i \(0.670551\pi\)
\(230\) 3.66881e12 0.375856
\(231\) −1.38154e12 −0.138197
\(232\) 6.06397e12 0.592343
\(233\) 7.20907e12 0.687735 0.343868 0.939018i \(-0.388263\pi\)
0.343868 + 0.939018i \(0.388263\pi\)
\(234\) −3.14838e11 −0.0293359
\(235\) 3.34572e12 0.304520
\(236\) −2.99415e12 −0.266230
\(237\) −5.94422e12 −0.516392
\(238\) −9.05452e11 −0.0768586
\(239\) −1.17393e13 −0.973762 −0.486881 0.873468i \(-0.661865\pi\)
−0.486881 + 0.873468i \(0.661865\pi\)
\(240\) −3.19315e12 −0.258856
\(241\) −1.26058e13 −0.998798 −0.499399 0.866372i \(-0.666446\pi\)
−0.499399 + 0.866372i \(0.666446\pi\)
\(242\) −2.45391e12 −0.190052
\(243\) −8.47289e11 −0.0641500
\(244\) −2.33013e11 −0.0172479
\(245\) 1.24985e12 0.0904576
\(246\) −4.09408e12 −0.289743
\(247\) 4.32108e12 0.299061
\(248\) 5.09523e12 0.344889
\(249\) 2.80435e12 0.185667
\(250\) −4.96100e12 −0.321291
\(251\) 2.08449e13 1.32067 0.660334 0.750972i \(-0.270415\pi\)
0.660334 + 0.750972i \(0.270415\pi\)
\(252\) −1.82786e12 −0.113302
\(253\) 1.95324e13 1.18466
\(254\) −6.46595e12 −0.383749
\(255\) 4.03369e12 0.234278
\(256\) 2.43010e12 0.138135
\(257\) −2.39536e12 −0.133272 −0.0666361 0.997777i \(-0.521227\pi\)
−0.0666361 + 0.997777i \(0.521227\pi\)
\(258\) −5.66947e12 −0.308769
\(259\) −9.69087e12 −0.516671
\(260\) 3.02576e12 0.157936
\(261\) −6.41041e12 −0.327614
\(262\) −3.43690e12 −0.171993
\(263\) −1.40462e13 −0.688340 −0.344170 0.938907i \(-0.611840\pi\)
−0.344170 + 0.938907i \(0.611840\pi\)
\(264\) 4.59152e12 0.220361
\(265\) −1.12234e13 −0.527562
\(266\) −2.80882e12 −0.129323
\(267\) −5.58482e12 −0.251882
\(268\) 1.85561e13 0.819869
\(269\) 1.56369e13 0.676884 0.338442 0.940987i \(-0.390100\pi\)
0.338442 + 0.940987i \(0.390100\pi\)
\(270\) −9.11706e11 −0.0386682
\(271\) −2.91728e13 −1.21240 −0.606202 0.795311i \(-0.707308\pi\)
−0.606202 + 0.795311i \(0.707308\pi\)
\(272\) −1.11417e13 −0.453758
\(273\) 1.51640e12 0.0605228
\(274\) 1.47375e12 0.0576493
\(275\) −9.89469e12 −0.379378
\(276\) 2.58425e13 0.971259
\(277\) −1.45533e13 −0.536193 −0.268097 0.963392i \(-0.586395\pi\)
−0.268097 + 0.963392i \(0.586395\pi\)
\(278\) −7.75151e11 −0.0279988
\(279\) −5.38633e12 −0.190752
\(280\) −4.15386e12 −0.144239
\(281\) −4.74697e13 −1.61634 −0.808168 0.588952i \(-0.799541\pi\)
−0.808168 + 0.588952i \(0.799541\pi\)
\(282\) −2.63861e12 −0.0881059
\(283\) −2.66240e13 −0.871864 −0.435932 0.899980i \(-0.643581\pi\)
−0.435932 + 0.899980i \(0.643581\pi\)
\(284\) −1.88731e13 −0.606168
\(285\) 1.25130e13 0.394197
\(286\) −1.80360e12 −0.0557350
\(287\) 1.97189e13 0.597768
\(288\) 9.27330e12 0.275788
\(289\) −2.01973e13 −0.589326
\(290\) −6.89778e12 −0.197479
\(291\) 1.39416e13 0.391653
\(292\) −2.11052e13 −0.581814
\(293\) −3.08487e13 −0.834574 −0.417287 0.908775i \(-0.637019\pi\)
−0.417287 + 0.908775i \(0.637019\pi\)
\(294\) −9.85699e11 −0.0261718
\(295\) 7.19303e12 0.187452
\(296\) 3.22075e13 0.823857
\(297\) −4.85384e12 −0.121878
\(298\) −2.31276e13 −0.570087
\(299\) −2.14391e13 −0.518817
\(300\) −1.30913e13 −0.311039
\(301\) 2.73066e13 0.637019
\(302\) −9.39750e12 −0.215265
\(303\) −4.02793e13 −0.906041
\(304\) −3.45630e13 −0.763496
\(305\) 5.59782e11 0.0121442
\(306\) −3.18118e12 −0.0677829
\(307\) 7.57475e13 1.58528 0.792642 0.609687i \(-0.208705\pi\)
0.792642 + 0.609687i \(0.208705\pi\)
\(308\) −1.04712e13 −0.215262
\(309\) −3.04616e13 −0.615150
\(310\) −5.79584e12 −0.114981
\(311\) −5.43810e13 −1.05990 −0.529950 0.848029i \(-0.677789\pi\)
−0.529950 + 0.848029i \(0.677789\pi\)
\(312\) −5.03972e12 −0.0965065
\(313\) 7.63285e13 1.43613 0.718064 0.695977i \(-0.245029\pi\)
0.718064 + 0.695977i \(0.245029\pi\)
\(314\) −1.35601e13 −0.250698
\(315\) 4.39118e12 0.0797761
\(316\) −4.50535e13 −0.804358
\(317\) 1.24347e13 0.218178 0.109089 0.994032i \(-0.465207\pi\)
0.109089 + 0.994032i \(0.465207\pi\)
\(318\) 8.85140e12 0.152638
\(319\) −3.67231e13 −0.622431
\(320\) −1.69335e13 −0.282113
\(321\) −5.76619e13 −0.944303
\(322\) 1.39360e13 0.224352
\(323\) 4.36610e13 0.691002
\(324\) −6.42192e12 −0.0999234
\(325\) 1.08606e13 0.166148
\(326\) 9.96975e12 0.149964
\(327\) 5.24384e13 0.775599
\(328\) −6.55354e13 −0.953169
\(329\) 1.27087e13 0.181771
\(330\) −5.22286e12 −0.0734653
\(331\) 2.35824e12 0.0326238 0.0163119 0.999867i \(-0.494808\pi\)
0.0163119 + 0.999867i \(0.494808\pi\)
\(332\) 2.12552e13 0.289205
\(333\) −3.40475e13 −0.455661
\(334\) −3.68049e13 −0.484507
\(335\) −4.45784e13 −0.577268
\(336\) −1.21292e13 −0.154513
\(337\) −1.18171e13 −0.148097 −0.0740487 0.997255i \(-0.523592\pi\)
−0.0740487 + 0.997255i \(0.523592\pi\)
\(338\) 1.97966e12 0.0244090
\(339\) 6.10537e12 0.0740652
\(340\) 3.05728e13 0.364923
\(341\) −3.08565e13 −0.362408
\(342\) −9.86838e12 −0.114052
\(343\) 4.74756e12 0.0539949
\(344\) −9.07532e13 −1.01576
\(345\) −6.20831e13 −0.683862
\(346\) 5.02638e13 0.544925
\(347\) −6.85979e11 −0.00731980 −0.00365990 0.999993i \(-0.501165\pi\)
−0.00365990 + 0.999993i \(0.501165\pi\)
\(348\) −4.85869e13 −0.510309
\(349\) −3.85641e13 −0.398697 −0.199349 0.979929i \(-0.563883\pi\)
−0.199349 + 0.979929i \(0.563883\pi\)
\(350\) −7.05966e12 −0.0718471
\(351\) 5.32765e12 0.0533761
\(352\) 5.31237e13 0.523967
\(353\) 2.41846e13 0.234843 0.117422 0.993082i \(-0.462537\pi\)
0.117422 + 0.993082i \(0.462537\pi\)
\(354\) −5.67280e12 −0.0542350
\(355\) 4.53401e13 0.426802
\(356\) −4.23295e13 −0.392345
\(357\) 1.53220e13 0.139842
\(358\) 5.66744e13 0.509366
\(359\) −1.30749e14 −1.15723 −0.578615 0.815601i \(-0.696407\pi\)
−0.578615 + 0.815601i \(0.696407\pi\)
\(360\) −1.45940e13 −0.127207
\(361\) 1.89512e13 0.162685
\(362\) −6.51379e13 −0.550727
\(363\) 4.15247e13 0.345796
\(364\) 1.14933e13 0.0942733
\(365\) 5.07024e13 0.409655
\(366\) −4.41473e11 −0.00351365
\(367\) −1.88991e14 −1.48176 −0.740882 0.671636i \(-0.765592\pi\)
−0.740882 + 0.671636i \(0.765592\pi\)
\(368\) 1.71484e14 1.32453
\(369\) 6.92796e13 0.527181
\(370\) −3.66360e13 −0.274662
\(371\) −4.26322e13 −0.314906
\(372\) −4.08250e13 −0.297125
\(373\) −6.86251e13 −0.492135 −0.246068 0.969253i \(-0.579139\pi\)
−0.246068 + 0.969253i \(0.579139\pi\)
\(374\) −1.82239e13 −0.128780
\(375\) 8.39494e13 0.584582
\(376\) −4.22372e13 −0.289842
\(377\) 4.03079e13 0.272592
\(378\) −3.46311e12 −0.0230814
\(379\) 8.54281e13 0.561158 0.280579 0.959831i \(-0.409474\pi\)
0.280579 + 0.959831i \(0.409474\pi\)
\(380\) 9.48404e13 0.614021
\(381\) 1.09416e14 0.698222
\(382\) 5.95692e13 0.374691
\(383\) −2.73679e14 −1.69687 −0.848433 0.529303i \(-0.822454\pi\)
−0.848433 + 0.529303i \(0.822454\pi\)
\(384\) 9.15099e13 0.559302
\(385\) 2.51556e13 0.151566
\(386\) −1.01406e14 −0.602331
\(387\) 9.59380e13 0.561798
\(388\) 1.05669e14 0.610058
\(389\) 9.67709e13 0.550835 0.275418 0.961325i \(-0.411184\pi\)
0.275418 + 0.961325i \(0.411184\pi\)
\(390\) 5.73270e12 0.0321739
\(391\) −2.16624e14 −1.19877
\(392\) −1.57784e13 −0.0860975
\(393\) 5.81588e13 0.312936
\(394\) 4.85132e12 0.0257413
\(395\) 1.08235e14 0.566347
\(396\) −3.67890e13 −0.189843
\(397\) −4.26925e13 −0.217272 −0.108636 0.994082i \(-0.534648\pi\)
−0.108636 + 0.994082i \(0.534648\pi\)
\(398\) −4.31016e13 −0.216340
\(399\) 4.75305e13 0.235300
\(400\) −8.68703e13 −0.424171
\(401\) 2.31618e14 1.11552 0.557762 0.830001i \(-0.311660\pi\)
0.557762 + 0.830001i \(0.311660\pi\)
\(402\) 3.51569e13 0.167019
\(403\) 3.38686e13 0.158716
\(404\) −3.05292e14 −1.41130
\(405\) 1.54278e13 0.0703559
\(406\) −2.62012e13 −0.117877
\(407\) −1.95047e14 −0.865705
\(408\) −5.09222e13 −0.222985
\(409\) −1.08333e14 −0.468039 −0.234019 0.972232i \(-0.575188\pi\)
−0.234019 + 0.972232i \(0.575188\pi\)
\(410\) 7.45467e13 0.317773
\(411\) −2.49385e13 −0.104892
\(412\) −2.30880e14 −0.958189
\(413\) 2.73227e13 0.111892
\(414\) 4.89620e13 0.197860
\(415\) −5.10627e13 −0.203629
\(416\) −5.83094e13 −0.229470
\(417\) 1.31170e13 0.0509433
\(418\) −5.65327e13 −0.216686
\(419\) −2.76667e14 −1.04660 −0.523299 0.852149i \(-0.675299\pi\)
−0.523299 + 0.852149i \(0.675299\pi\)
\(420\) 3.32824e13 0.124263
\(421\) −4.58583e14 −1.68992 −0.844960 0.534830i \(-0.820376\pi\)
−0.844960 + 0.534830i \(0.820376\pi\)
\(422\) −4.41880e13 −0.160726
\(423\) 4.46502e13 0.160307
\(424\) 1.41687e14 0.502134
\(425\) 1.09737e14 0.383897
\(426\) −3.57576e13 −0.123485
\(427\) 2.12633e12 0.00724899
\(428\) −4.37041e14 −1.47089
\(429\) 3.05204e13 0.101409
\(430\) 1.03232e14 0.338639
\(431\) 1.04507e14 0.338470 0.169235 0.985576i \(-0.445870\pi\)
0.169235 + 0.985576i \(0.445870\pi\)
\(432\) −4.26142e13 −0.136268
\(433\) −3.84367e14 −1.21357 −0.606783 0.794868i \(-0.707540\pi\)
−0.606783 + 0.794868i \(0.707540\pi\)
\(434\) −2.20155e13 −0.0686333
\(435\) 1.16723e14 0.359308
\(436\) 3.97450e14 1.20811
\(437\) −6.71993e14 −2.01705
\(438\) −3.99866e13 −0.118524
\(439\) 5.48221e14 1.60473 0.802363 0.596837i \(-0.203576\pi\)
0.802363 + 0.596837i \(0.203576\pi\)
\(440\) −8.36042e13 −0.241679
\(441\) 1.66799e13 0.0476190
\(442\) 2.00029e13 0.0563988
\(443\) 1.57696e14 0.439138 0.219569 0.975597i \(-0.429535\pi\)
0.219569 + 0.975597i \(0.429535\pi\)
\(444\) −2.58059e14 −0.709761
\(445\) 1.01691e14 0.276249
\(446\) −1.17932e14 −0.316439
\(447\) 3.91363e14 1.03726
\(448\) −6.43220e13 −0.168395
\(449\) −3.11807e14 −0.806365 −0.403182 0.915120i \(-0.632096\pi\)
−0.403182 + 0.915120i \(0.632096\pi\)
\(450\) −2.48031e13 −0.0633632
\(451\) 3.96880e14 1.00159
\(452\) 4.62749e13 0.115368
\(453\) 1.59023e14 0.391671
\(454\) 4.19182e13 0.101999
\(455\) −2.76112e13 −0.0663777
\(456\) −1.57967e14 −0.375197
\(457\) −1.77079e14 −0.415554 −0.207777 0.978176i \(-0.566623\pi\)
−0.207777 + 0.978176i \(0.566623\pi\)
\(458\) −1.39735e14 −0.324000
\(459\) 5.38315e13 0.123329
\(460\) −4.70551e14 −1.06522
\(461\) 5.44865e14 1.21880 0.609401 0.792862i \(-0.291410\pi\)
0.609401 + 0.792862i \(0.291410\pi\)
\(462\) −1.98390e13 −0.0438521
\(463\) −4.79572e14 −1.04751 −0.523755 0.851869i \(-0.675469\pi\)
−0.523755 + 0.851869i \(0.675469\pi\)
\(464\) −3.22410e14 −0.695921
\(465\) 9.80764e13 0.209206
\(466\) 1.03523e14 0.218230
\(467\) 3.66938e14 0.764452 0.382226 0.924069i \(-0.375158\pi\)
0.382226 + 0.924069i \(0.375158\pi\)
\(468\) 4.03802e13 0.0831413
\(469\) −1.69331e14 −0.344577
\(470\) 4.80449e13 0.0966293
\(471\) 2.29463e14 0.456139
\(472\) −9.08066e13 −0.178417
\(473\) 5.49597e14 1.06735
\(474\) −8.53597e13 −0.163860
\(475\) 3.40417e14 0.645947
\(476\) 1.16131e14 0.217826
\(477\) −1.49782e14 −0.277721
\(478\) −1.68577e14 −0.308991
\(479\) −6.21358e13 −0.112589 −0.0562945 0.998414i \(-0.517929\pi\)
−0.0562945 + 0.998414i \(0.517929\pi\)
\(480\) −1.68852e14 −0.302468
\(481\) 2.14087e14 0.379133
\(482\) −1.81021e14 −0.316935
\(483\) −2.35823e14 −0.408203
\(484\) 3.14731e14 0.538629
\(485\) −2.53854e14 −0.429541
\(486\) −1.21672e13 −0.0203559
\(487\) −5.76505e14 −0.953661 −0.476831 0.878995i \(-0.658215\pi\)
−0.476831 + 0.878995i \(0.658215\pi\)
\(488\) −7.06682e12 −0.0115589
\(489\) −1.68707e14 −0.272856
\(490\) 1.79480e13 0.0287037
\(491\) 7.52984e14 1.19080 0.595398 0.803431i \(-0.296994\pi\)
0.595398 + 0.803431i \(0.296994\pi\)
\(492\) 5.25095e14 0.821165
\(493\) 4.07278e14 0.629844
\(494\) 6.20512e13 0.0948969
\(495\) 8.83807e13 0.133668
\(496\) −2.70904e14 −0.405197
\(497\) 1.72224e14 0.254762
\(498\) 4.02707e13 0.0589153
\(499\) −1.17700e15 −1.70303 −0.851516 0.524329i \(-0.824316\pi\)
−0.851516 + 0.524329i \(0.824316\pi\)
\(500\) 6.36284e14 0.910575
\(501\) 6.22808e14 0.881549
\(502\) 2.99335e14 0.419070
\(503\) 1.40677e15 1.94805 0.974023 0.226451i \(-0.0727122\pi\)
0.974023 + 0.226451i \(0.0727122\pi\)
\(504\) −5.54353e13 −0.0759309
\(505\) 7.33423e14 0.993692
\(506\) 2.80487e14 0.375911
\(507\) −3.34996e13 −0.0444116
\(508\) 8.29304e14 1.08759
\(509\) −2.38636e14 −0.309590 −0.154795 0.987947i \(-0.549472\pi\)
−0.154795 + 0.987947i \(0.549472\pi\)
\(510\) 5.79242e13 0.0743402
\(511\) 1.92593e14 0.244527
\(512\) 8.06141e14 1.01257
\(513\) 1.66992e14 0.207515
\(514\) −3.43977e13 −0.0422895
\(515\) 5.54657e14 0.674660
\(516\) 7.27150e14 0.875086
\(517\) 2.55787e14 0.304565
\(518\) −1.39162e14 −0.163948
\(519\) −8.50557e14 −0.991478
\(520\) 9.17653e13 0.105842
\(521\) 7.26317e14 0.828932 0.414466 0.910065i \(-0.363968\pi\)
0.414466 + 0.910065i \(0.363968\pi\)
\(522\) −9.20542e13 −0.103958
\(523\) −3.93852e14 −0.440123 −0.220061 0.975486i \(-0.570626\pi\)
−0.220061 + 0.975486i \(0.570626\pi\)
\(524\) 4.40807e14 0.487446
\(525\) 1.19463e14 0.130724
\(526\) −2.01705e14 −0.218422
\(527\) 3.42214e14 0.366724
\(528\) −2.44123e14 −0.258894
\(529\) 2.38129e15 2.49923
\(530\) −1.61170e14 −0.167404
\(531\) 9.59945e13 0.0986794
\(532\) 3.60251e14 0.366515
\(533\) −4.35622e14 −0.438641
\(534\) −8.01987e13 −0.0799264
\(535\) 1.04993e15 1.03566
\(536\) 5.62768e14 0.549444
\(537\) −9.59037e14 −0.926780
\(538\) 2.24548e14 0.214786
\(539\) 9.55536e13 0.0904709
\(540\) 1.16933e14 0.109590
\(541\) 6.71611e14 0.623064 0.311532 0.950236i \(-0.399158\pi\)
0.311532 + 0.950236i \(0.399158\pi\)
\(542\) −4.18925e14 −0.384716
\(543\) 1.10226e15 1.00204
\(544\) −5.89168e14 −0.530207
\(545\) −9.54820e14 −0.850630
\(546\) 2.17756e13 0.0192049
\(547\) 1.41217e15 1.23298 0.616491 0.787362i \(-0.288554\pi\)
0.616491 + 0.787362i \(0.288554\pi\)
\(548\) −1.89018e14 −0.163385
\(549\) 7.47056e12 0.00639301
\(550\) −1.42089e14 −0.120383
\(551\) 1.26342e15 1.05978
\(552\) 7.83752e14 0.650900
\(553\) 4.11130e14 0.338058
\(554\) −2.08986e14 −0.170143
\(555\) 6.19951e14 0.499742
\(556\) 9.94187e13 0.0793518
\(557\) −2.05467e15 −1.62382 −0.811909 0.583783i \(-0.801572\pi\)
−0.811909 + 0.583783i \(0.801572\pi\)
\(558\) −7.73483e13 −0.0605289
\(559\) −6.03247e14 −0.467444
\(560\) 2.20853e14 0.169461
\(561\) 3.08383e14 0.234312
\(562\) −6.81670e14 −0.512890
\(563\) 2.82561e14 0.210531 0.105266 0.994444i \(-0.466431\pi\)
0.105266 + 0.994444i \(0.466431\pi\)
\(564\) 3.38421e14 0.249702
\(565\) −1.11169e14 −0.0812302
\(566\) −3.82324e14 −0.276657
\(567\) 5.86024e13 0.0419961
\(568\) −5.72385e14 −0.406230
\(569\) 4.92020e14 0.345832 0.172916 0.984937i \(-0.444681\pi\)
0.172916 + 0.984937i \(0.444681\pi\)
\(570\) 1.79688e14 0.125085
\(571\) −3.14202e14 −0.216626 −0.108313 0.994117i \(-0.534545\pi\)
−0.108313 + 0.994117i \(0.534545\pi\)
\(572\) 2.31325e14 0.157959
\(573\) −1.00802e15 −0.681741
\(574\) 2.83166e14 0.189682
\(575\) −1.68898e15 −1.12060
\(576\) −2.25986e14 −0.148511
\(577\) 1.24691e15 0.811649 0.405824 0.913951i \(-0.366984\pi\)
0.405824 + 0.913951i \(0.366984\pi\)
\(578\) −2.90036e14 −0.187003
\(579\) 1.71599e15 1.09593
\(580\) 8.84690e14 0.559676
\(581\) −1.93962e14 −0.121548
\(582\) 2.00203e14 0.124278
\(583\) −8.58054e14 −0.527640
\(584\) −6.40080e14 −0.389909
\(585\) −9.70080e13 −0.0585397
\(586\) −4.42990e14 −0.264824
\(587\) 1.13347e14 0.0671276 0.0335638 0.999437i \(-0.489314\pi\)
0.0335638 + 0.999437i \(0.489314\pi\)
\(588\) 1.26423e14 0.0741739
\(589\) 1.06159e15 0.617052
\(590\) 1.03293e14 0.0594817
\(591\) −8.20935e13 −0.0468358
\(592\) −1.71241e15 −0.967919
\(593\) −1.19853e15 −0.671196 −0.335598 0.942005i \(-0.608938\pi\)
−0.335598 + 0.942005i \(0.608938\pi\)
\(594\) −6.97017e13 −0.0386739
\(595\) −2.78988e14 −0.153371
\(596\) 2.96628e15 1.61569
\(597\) 7.29361e14 0.393626
\(598\) −3.07867e14 −0.164629
\(599\) 2.49233e15 1.32056 0.660280 0.751020i \(-0.270438\pi\)
0.660280 + 0.751020i \(0.270438\pi\)
\(600\) −3.97032e14 −0.208446
\(601\) −1.73260e15 −0.901341 −0.450670 0.892690i \(-0.648815\pi\)
−0.450670 + 0.892690i \(0.648815\pi\)
\(602\) 3.92126e14 0.202137
\(603\) −5.94920e14 −0.303888
\(604\) 1.20530e15 0.610086
\(605\) −7.56098e14 −0.379248
\(606\) −5.78416e14 −0.287502
\(607\) −2.42676e15 −1.19533 −0.597666 0.801746i \(-0.703905\pi\)
−0.597666 + 0.801746i \(0.703905\pi\)
\(608\) −1.82767e15 −0.892130
\(609\) 4.43374e14 0.214474
\(610\) 8.03853e12 0.00385356
\(611\) −2.80755e14 −0.133383
\(612\) 4.08009e14 0.192104
\(613\) 2.24780e15 1.04888 0.524439 0.851448i \(-0.324275\pi\)
0.524439 + 0.851448i \(0.324275\pi\)
\(614\) 1.08774e15 0.503037
\(615\) −1.26147e15 −0.578181
\(616\) −3.17571e14 −0.144260
\(617\) −2.85833e15 −1.28690 −0.643450 0.765488i \(-0.722497\pi\)
−0.643450 + 0.765488i \(0.722497\pi\)
\(618\) −4.37432e14 −0.195197
\(619\) 4.48909e15 1.98545 0.992727 0.120384i \(-0.0384126\pi\)
0.992727 + 0.120384i \(0.0384126\pi\)
\(620\) 7.43358e14 0.325869
\(621\) −8.28529e14 −0.360001
\(622\) −7.80917e14 −0.336324
\(623\) 3.86272e14 0.164896
\(624\) 2.67953e14 0.113382
\(625\) −1.00329e14 −0.0420809
\(626\) 1.09609e15 0.455707
\(627\) 9.56640e14 0.394255
\(628\) 1.73918e15 0.710506
\(629\) 2.16317e15 0.876016
\(630\) 6.30578e13 0.0253143
\(631\) −3.15742e15 −1.25652 −0.628261 0.778002i \(-0.716233\pi\)
−0.628261 + 0.778002i \(0.716233\pi\)
\(632\) −1.36638e15 −0.539049
\(633\) 7.47745e14 0.292437
\(634\) 1.78564e14 0.0692314
\(635\) −1.99229e15 −0.765768
\(636\) −1.13526e15 −0.432593
\(637\) −1.04881e14 −0.0396214
\(638\) −5.27348e14 −0.197508
\(639\) 6.05086e14 0.224679
\(640\) −1.66625e15 −0.613409
\(641\) −3.46542e15 −1.26484 −0.632422 0.774624i \(-0.717939\pi\)
−0.632422 + 0.774624i \(0.717939\pi\)
\(642\) −8.28032e14 −0.299643
\(643\) 2.12153e15 0.761183 0.380592 0.924743i \(-0.375720\pi\)
0.380592 + 0.924743i \(0.375720\pi\)
\(644\) −1.78739e15 −0.635838
\(645\) −1.74688e15 −0.616147
\(646\) 6.26976e14 0.219266
\(647\) −7.97510e14 −0.276543 −0.138271 0.990394i \(-0.544155\pi\)
−0.138271 + 0.990394i \(0.544155\pi\)
\(648\) −1.94764e14 −0.0669647
\(649\) 5.49921e14 0.187480
\(650\) 1.55959e14 0.0527214
\(651\) 3.72543e14 0.124877
\(652\) −1.27869e15 −0.425015
\(653\) 3.52532e15 1.16192 0.580960 0.813932i \(-0.302677\pi\)
0.580960 + 0.813932i \(0.302677\pi\)
\(654\) 7.53022e14 0.246110
\(655\) −1.05898e15 −0.343210
\(656\) 3.48440e15 1.11984
\(657\) 6.76649e14 0.215652
\(658\) 1.82498e14 0.0576789
\(659\) −5.45502e13 −0.0170973 −0.00854863 0.999963i \(-0.502721\pi\)
−0.00854863 + 0.999963i \(0.502721\pi\)
\(660\) 6.69870e14 0.208209
\(661\) 5.37751e15 1.65758 0.828788 0.559563i \(-0.189031\pi\)
0.828788 + 0.559563i \(0.189031\pi\)
\(662\) 3.38646e13 0.0103521
\(663\) −3.38486e14 −0.102616
\(664\) 6.44628e14 0.193814
\(665\) −8.65454e14 −0.258063
\(666\) −4.88926e14 −0.144589
\(667\) −6.26848e15 −1.83853
\(668\) 4.72049e15 1.37315
\(669\) 1.99563e15 0.575753
\(670\) −6.40150e14 −0.183177
\(671\) 4.27964e13 0.0121460
\(672\) −6.41384e14 −0.180546
\(673\) 2.35400e15 0.657240 0.328620 0.944462i \(-0.393416\pi\)
0.328620 + 0.944462i \(0.393416\pi\)
\(674\) −1.69695e14 −0.0469938
\(675\) 4.19715e14 0.115288
\(676\) −2.53906e14 −0.0691777
\(677\) −1.68449e15 −0.455230 −0.227615 0.973751i \(-0.573093\pi\)
−0.227615 + 0.973751i \(0.573093\pi\)
\(678\) 8.76738e13 0.0235021
\(679\) −9.64265e14 −0.256397
\(680\) 9.27213e14 0.244557
\(681\) −7.09334e14 −0.185585
\(682\) −4.43103e14 −0.114998
\(683\) 2.86844e15 0.738467 0.369234 0.929337i \(-0.379620\pi\)
0.369234 + 0.929337i \(0.379620\pi\)
\(684\) 1.26569e15 0.323236
\(685\) 4.54091e14 0.115039
\(686\) 6.81755e13 0.0171335
\(687\) 2.36458e15 0.589511
\(688\) 4.82518e15 1.19338
\(689\) 9.41813e14 0.231078
\(690\) −8.91520e14 −0.217001
\(691\) −5.57355e15 −1.34587 −0.672934 0.739703i \(-0.734966\pi\)
−0.672934 + 0.739703i \(0.734966\pi\)
\(692\) −6.44669e15 −1.54438
\(693\) 3.35714e14 0.0797878
\(694\) −9.85074e12 −0.00232269
\(695\) −2.38840e14 −0.0558715
\(696\) −1.47354e15 −0.341989
\(697\) −4.40160e15 −1.01351
\(698\) −5.53785e14 −0.126513
\(699\) −1.75180e15 −0.397064
\(700\) 9.05452e14 0.203623
\(701\) −7.00824e15 −1.56372 −0.781861 0.623452i \(-0.785729\pi\)
−0.781861 + 0.623452i \(0.785729\pi\)
\(702\) 7.65056e13 0.0169371
\(703\) 6.71040e15 1.47399
\(704\) −1.29460e15 −0.282154
\(705\) −8.13009e14 −0.175815
\(706\) 3.47294e14 0.0745197
\(707\) 2.78591e15 0.593143
\(708\) 7.27578e14 0.153708
\(709\) 4.11385e15 0.862371 0.431185 0.902263i \(-0.358095\pi\)
0.431185 + 0.902263i \(0.358095\pi\)
\(710\) 6.51089e14 0.135431
\(711\) 1.44445e15 0.298139
\(712\) −1.28377e15 −0.262934
\(713\) −5.26707e15 −1.07048
\(714\) 2.20025e14 0.0443743
\(715\) −5.55727e14 −0.111219
\(716\) −7.26890e15 −1.44360
\(717\) 2.85264e15 0.562202
\(718\) −1.87757e15 −0.367208
\(719\) −9.76609e14 −0.189545 −0.0947724 0.995499i \(-0.530212\pi\)
−0.0947724 + 0.995499i \(0.530212\pi\)
\(720\) 7.75936e14 0.149451
\(721\) 2.10686e15 0.402710
\(722\) 2.72141e14 0.0516225
\(723\) 3.06322e15 0.576656
\(724\) 8.35440e15 1.56082
\(725\) 3.17548e15 0.588776
\(726\) 5.96299e14 0.109727
\(727\) −8.85628e15 −1.61738 −0.808690 0.588235i \(-0.799823\pi\)
−0.808690 + 0.588235i \(0.799823\pi\)
\(728\) 3.48570e14 0.0631783
\(729\) 2.05891e14 0.0370370
\(730\) 7.28092e14 0.129990
\(731\) −6.09531e15 −1.08007
\(732\) 5.66221e13 0.00995807
\(733\) 4.63449e15 0.808965 0.404483 0.914546i \(-0.367452\pi\)
0.404483 + 0.914546i \(0.367452\pi\)
\(734\) −2.71394e15 −0.470188
\(735\) −3.03714e14 −0.0522257
\(736\) 9.06799e15 1.54769
\(737\) −3.40810e15 −0.577353
\(738\) 9.94862e14 0.167283
\(739\) −7.36735e15 −1.22961 −0.614804 0.788680i \(-0.710765\pi\)
−0.614804 + 0.788680i \(0.710765\pi\)
\(740\) 4.69884e15 0.778423
\(741\) −1.05002e15 −0.172663
\(742\) −6.12204e14 −0.0999250
\(743\) −2.70268e15 −0.437882 −0.218941 0.975738i \(-0.570260\pi\)
−0.218941 + 0.975738i \(0.570260\pi\)
\(744\) −1.23814e15 −0.199122
\(745\) −7.12609e15 −1.13761
\(746\) −9.85464e14 −0.156163
\(747\) −6.81456e14 −0.107195
\(748\) 2.33735e15 0.364976
\(749\) 3.98816e15 0.618192
\(750\) 1.20552e15 0.185498
\(751\) −3.02136e15 −0.461512 −0.230756 0.973012i \(-0.574120\pi\)
−0.230756 + 0.973012i \(0.574120\pi\)
\(752\) 2.24567e15 0.340525
\(753\) −5.06530e15 −0.762488
\(754\) 5.78826e14 0.0864979
\(755\) −2.89556e15 −0.429561
\(756\) 4.44169e14 0.0654152
\(757\) 8.01585e15 1.17199 0.585993 0.810316i \(-0.300705\pi\)
0.585993 + 0.810316i \(0.300705\pi\)
\(758\) 1.22676e15 0.178065
\(759\) −4.74637e15 −0.683962
\(760\) 2.87632e15 0.411493
\(761\) −1.34585e15 −0.191153 −0.0955763 0.995422i \(-0.530469\pi\)
−0.0955763 + 0.995422i \(0.530469\pi\)
\(762\) 1.57123e15 0.221557
\(763\) −3.62688e15 −0.507749
\(764\) −7.64017e15 −1.06191
\(765\) −9.80186e14 −0.135260
\(766\) −3.93005e15 −0.538444
\(767\) −6.03602e14 −0.0821062
\(768\) −5.90515e14 −0.0797525
\(769\) 9.79480e15 1.31341 0.656706 0.754147i \(-0.271949\pi\)
0.656706 + 0.754147i \(0.271949\pi\)
\(770\) 3.61237e14 0.0480943
\(771\) 5.82073e14 0.0769447
\(772\) 1.30061e16 1.70707
\(773\) −1.30070e16 −1.69508 −0.847542 0.530729i \(-0.821918\pi\)
−0.847542 + 0.530729i \(0.821918\pi\)
\(774\) 1.37768e15 0.178268
\(775\) 2.66819e15 0.342813
\(776\) 3.20472e15 0.408837
\(777\) 2.35488e15 0.298300
\(778\) 1.38964e15 0.174789
\(779\) −1.36543e16 −1.70535
\(780\) −7.35259e14 −0.0911843
\(781\) 3.46634e15 0.426865
\(782\) −3.11075e15 −0.380388
\(783\) 1.55773e15 0.189148
\(784\) 8.38911e14 0.101153
\(785\) −4.17815e15 −0.500266
\(786\) 8.35166e14 0.0992999
\(787\) −4.70397e15 −0.555397 −0.277699 0.960668i \(-0.589572\pi\)
−0.277699 + 0.960668i \(0.589572\pi\)
\(788\) −6.22217e14 −0.0729538
\(789\) 3.41323e15 0.397413
\(790\) 1.55426e15 0.179711
\(791\) −4.22276e14 −0.0484870
\(792\) −1.11574e15 −0.127226
\(793\) −4.69740e13 −0.00531930
\(794\) −6.13069e14 −0.0689440
\(795\) 2.72730e15 0.304588
\(796\) 5.52809e15 0.613131
\(797\) 4.79116e15 0.527740 0.263870 0.964558i \(-0.415001\pi\)
0.263870 + 0.964558i \(0.415001\pi\)
\(798\) 6.82543e14 0.0746645
\(799\) −2.83680e15 −0.308192
\(800\) −4.59365e15 −0.495636
\(801\) 1.35711e15 0.145424
\(802\) 3.32606e15 0.353974
\(803\) 3.87630e15 0.409715
\(804\) −4.50912e15 −0.473351
\(805\) 4.29395e15 0.447693
\(806\) 4.86357e14 0.0503631
\(807\) −3.79977e15 −0.390799
\(808\) −9.25891e15 −0.945796
\(809\) −4.14348e15 −0.420386 −0.210193 0.977660i \(-0.567409\pi\)
−0.210193 + 0.977660i \(0.567409\pi\)
\(810\) 2.21544e14 0.0223251
\(811\) 1.70586e16 1.70738 0.853689 0.520783i \(-0.174360\pi\)
0.853689 + 0.520783i \(0.174360\pi\)
\(812\) 3.36049e15 0.334076
\(813\) 7.08899e15 0.699981
\(814\) −2.80090e15 −0.274703
\(815\) 3.07188e15 0.299253
\(816\) 2.70744e15 0.261977
\(817\) −1.89084e16 −1.81732
\(818\) −1.55567e15 −0.148516
\(819\) −3.68485e14 −0.0349428
\(820\) −9.56115e15 −0.900604
\(821\) −9.45209e15 −0.884383 −0.442191 0.896921i \(-0.645799\pi\)
−0.442191 + 0.896921i \(0.645799\pi\)
\(822\) −3.58120e14 −0.0332839
\(823\) −1.45167e16 −1.34019 −0.670097 0.742274i \(-0.733747\pi\)
−0.670097 + 0.742274i \(0.733747\pi\)
\(824\) −7.00213e15 −0.642141
\(825\) 2.40441e15 0.219034
\(826\) 3.92357e14 0.0355051
\(827\) −1.13804e16 −1.02301 −0.511504 0.859281i \(-0.670911\pi\)
−0.511504 + 0.859281i \(0.670911\pi\)
\(828\) −6.27973e15 −0.560756
\(829\) −1.11977e16 −0.993297 −0.496649 0.867952i \(-0.665436\pi\)
−0.496649 + 0.867952i \(0.665436\pi\)
\(830\) −7.33266e14 −0.0646148
\(831\) 3.53644e15 0.309571
\(832\) 1.42097e15 0.123568
\(833\) −1.05974e15 −0.0915484
\(834\) 1.88362e14 0.0161651
\(835\) −1.13403e16 −0.966831
\(836\) 7.25073e15 0.614112
\(837\) 1.30888e15 0.110131
\(838\) −3.97297e15 −0.332103
\(839\) −4.70211e15 −0.390483 −0.195242 0.980755i \(-0.562549\pi\)
−0.195242 + 0.980755i \(0.562549\pi\)
\(840\) 1.00939e15 0.0832764
\(841\) −4.15045e14 −0.0340187
\(842\) −6.58530e15 −0.536240
\(843\) 1.15351e16 0.933192
\(844\) 5.66743e15 0.455515
\(845\) 6.09974e14 0.0487079
\(846\) 6.41182e14 0.0508680
\(847\) −2.87204e15 −0.226377
\(848\) −7.53327e15 −0.589938
\(849\) 6.46964e15 0.503371
\(850\) 1.57584e15 0.121817
\(851\) −3.32937e16 −2.55711
\(852\) 4.58617e15 0.349971
\(853\) −7.88595e15 −0.597908 −0.298954 0.954268i \(-0.596638\pi\)
−0.298954 + 0.954268i \(0.596638\pi\)
\(854\) 3.05343e13 0.00230022
\(855\) −3.04065e15 −0.227590
\(856\) −1.32546e16 −0.985736
\(857\) 1.22466e16 0.904941 0.452471 0.891779i \(-0.350543\pi\)
0.452471 + 0.891779i \(0.350543\pi\)
\(858\) 4.38276e14 0.0321786
\(859\) 9.72186e15 0.709230 0.354615 0.935012i \(-0.384612\pi\)
0.354615 + 0.935012i \(0.384612\pi\)
\(860\) −1.32402e16 −0.959741
\(861\) −4.79169e15 −0.345121
\(862\) 1.50073e15 0.107402
\(863\) −1.71694e15 −0.122094 −0.0610471 0.998135i \(-0.519444\pi\)
−0.0610471 + 0.998135i \(0.519444\pi\)
\(864\) −2.25341e15 −0.159226
\(865\) 1.54873e16 1.08739
\(866\) −5.51956e15 −0.385084
\(867\) 4.90795e15 0.340248
\(868\) 2.82364e15 0.194514
\(869\) 8.27476e15 0.566431
\(870\) 1.67616e15 0.114014
\(871\) 3.74079e15 0.252850
\(872\) 1.20539e16 0.809630
\(873\) −3.38781e15 −0.226121
\(874\) −9.64989e15 −0.640044
\(875\) −5.80633e15 −0.382699
\(876\) 5.12857e15 0.335911
\(877\) 9.93119e15 0.646403 0.323202 0.946330i \(-0.395241\pi\)
0.323202 + 0.946330i \(0.395241\pi\)
\(878\) 7.87251e15 0.509206
\(879\) 7.49623e15 0.481841
\(880\) 4.44509e15 0.283939
\(881\) −1.03838e16 −0.659160 −0.329580 0.944128i \(-0.606907\pi\)
−0.329580 + 0.944128i \(0.606907\pi\)
\(882\) 2.39525e14 0.0151103
\(883\) −4.58291e14 −0.0287315 −0.0143657 0.999897i \(-0.504573\pi\)
−0.0143657 + 0.999897i \(0.504573\pi\)
\(884\) −2.56551e15 −0.159840
\(885\) −1.74791e15 −0.108226
\(886\) 2.26454e15 0.139346
\(887\) −1.51623e16 −0.927224 −0.463612 0.886038i \(-0.653447\pi\)
−0.463612 + 0.886038i \(0.653447\pi\)
\(888\) −7.82641e15 −0.475654
\(889\) −7.56771e15 −0.457093
\(890\) 1.46029e15 0.0876585
\(891\) 1.17948e15 0.0703662
\(892\) 1.51256e16 0.896822
\(893\) −8.80009e15 −0.518566
\(894\) 5.62001e15 0.329140
\(895\) 1.74625e16 1.01644
\(896\) −6.32925e15 −0.366149
\(897\) 5.20969e15 0.299539
\(898\) −4.47759e15 −0.255873
\(899\) 9.90270e15 0.562439
\(900\) 3.18118e15 0.179578
\(901\) 9.51624e15 0.533924
\(902\) 5.69924e15 0.317820
\(903\) −6.63551e15 −0.367783
\(904\) 1.40343e15 0.0773149
\(905\) −2.00703e16 −1.09897
\(906\) 2.28359e15 0.124284
\(907\) −3.65610e16 −1.97778 −0.988891 0.148640i \(-0.952510\pi\)
−0.988891 + 0.148640i \(0.952510\pi\)
\(908\) −5.37631e15 −0.289076
\(909\) 9.78788e15 0.523103
\(910\) −3.96500e14 −0.0210628
\(911\) −1.72118e16 −0.908814 −0.454407 0.890794i \(-0.650149\pi\)
−0.454407 + 0.890794i \(0.650149\pi\)
\(912\) 8.39881e15 0.440804
\(913\) −3.90384e15 −0.203659
\(914\) −2.54287e15 −0.131862
\(915\) −1.36027e14 −0.00701147
\(916\) 1.79220e16 0.918252
\(917\) −4.02253e15 −0.204865
\(918\) 7.73026e14 0.0391345
\(919\) −3.23789e16 −1.62940 −0.814699 0.579884i \(-0.803098\pi\)
−0.814699 + 0.579884i \(0.803098\pi\)
\(920\) −1.42709e16 −0.713868
\(921\) −1.84066e16 −0.915264
\(922\) 7.82431e15 0.386746
\(923\) −3.80471e15 −0.186944
\(924\) 2.54450e15 0.124282
\(925\) 1.68659e16 0.818897
\(926\) −6.88671e15 −0.332392
\(927\) 7.40217e15 0.355157
\(928\) −1.70489e16 −0.813170
\(929\) −3.06416e15 −0.145286 −0.0726431 0.997358i \(-0.523143\pi\)
−0.0726431 + 0.997358i \(0.523143\pi\)
\(930\) 1.40839e15 0.0663844
\(931\) −3.28743e15 −0.154040
\(932\) −1.32776e16 −0.618488
\(933\) 1.32146e16 0.611933
\(934\) 5.26927e15 0.242573
\(935\) −5.61516e15 −0.256979
\(936\) 1.22465e15 0.0557180
\(937\) −2.98777e16 −1.35138 −0.675692 0.737184i \(-0.736155\pi\)
−0.675692 + 0.737184i \(0.736155\pi\)
\(938\) −2.43161e15 −0.109340
\(939\) −1.85478e16 −0.829148
\(940\) −6.16210e15 −0.273858
\(941\) −2.65487e16 −1.17300 −0.586502 0.809947i \(-0.699496\pi\)
−0.586502 + 0.809947i \(0.699496\pi\)
\(942\) 3.29511e15 0.144741
\(943\) 6.77457e16 2.95847
\(944\) 4.82802e15 0.209615
\(945\) −1.06706e15 −0.0460588
\(946\) 7.89228e15 0.338689
\(947\) −9.07376e15 −0.387135 −0.193568 0.981087i \(-0.562006\pi\)
−0.193568 + 0.981087i \(0.562006\pi\)
\(948\) 1.09480e16 0.464396
\(949\) −4.25469e15 −0.179433
\(950\) 4.88843e15 0.204970
\(951\) −3.02164e15 −0.125965
\(952\) 3.52202e15 0.145978
\(953\) −3.04447e16 −1.25459 −0.627293 0.778783i \(-0.715837\pi\)
−0.627293 + 0.778783i \(0.715837\pi\)
\(954\) −2.15089e15 −0.0881256
\(955\) 1.83545e16 0.747693
\(956\) 2.16212e16 0.875714
\(957\) 8.92373e15 0.359361
\(958\) −8.92276e14 −0.0357264
\(959\) 1.72486e15 0.0686677
\(960\) 4.11485e15 0.162878
\(961\) −1.70878e16 −0.672522
\(962\) 3.07431e15 0.120305
\(963\) 1.40118e16 0.545194
\(964\) 2.32173e16 0.898229
\(965\) −3.12454e16 −1.20195
\(966\) −3.38644e15 −0.129530
\(967\) 3.12630e16 1.18901 0.594504 0.804093i \(-0.297348\pi\)
0.594504 + 0.804093i \(0.297348\pi\)
\(968\) 9.54517e15 0.360968
\(969\) −1.06096e16 −0.398950
\(970\) −3.64537e15 −0.136301
\(971\) 7.64208e15 0.284123 0.142061 0.989858i \(-0.454627\pi\)
0.142061 + 0.989858i \(0.454627\pi\)
\(972\) 1.56053e15 0.0576908
\(973\) −9.07233e14 −0.0333502
\(974\) −8.27868e15 −0.302613
\(975\) −2.63912e15 −0.0959254
\(976\) 3.75730e14 0.0135801
\(977\) −1.47644e16 −0.530634 −0.265317 0.964161i \(-0.585477\pi\)
−0.265317 + 0.964161i \(0.585477\pi\)
\(978\) −2.42265e15 −0.0865819
\(979\) 7.77445e15 0.276290
\(980\) −2.30196e15 −0.0813494
\(981\) −1.27425e16 −0.447792
\(982\) 1.08129e16 0.377859
\(983\) 3.40812e16 1.18432 0.592162 0.805819i \(-0.298274\pi\)
0.592162 + 0.805819i \(0.298274\pi\)
\(984\) 1.59251e16 0.550313
\(985\) 1.49479e15 0.0513667
\(986\) 5.84856e15 0.199860
\(987\) −3.08822e15 −0.104945
\(988\) −7.95851e15 −0.268948
\(989\) 9.38139e16 3.15274
\(990\) 1.26916e15 0.0424152
\(991\) 5.15004e16 1.71161 0.855806 0.517297i \(-0.173062\pi\)
0.855806 + 0.517297i \(0.173062\pi\)
\(992\) −1.43252e16 −0.473465
\(993\) −5.73052e14 −0.0188353
\(994\) 2.47316e15 0.0808402
\(995\) −1.32805e16 −0.431705
\(996\) −5.16501e15 −0.166972
\(997\) 2.96920e16 0.954588 0.477294 0.878744i \(-0.341618\pi\)
0.477294 + 0.878744i \(0.341618\pi\)
\(998\) −1.69018e16 −0.540400
\(999\) 8.27354e15 0.263076
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 273.12.a.c.1.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.12.a.c.1.10 16 1.1 even 1 trivial