Properties

Label 273.12.a.c.1.5
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 1.5
Root \(-37.3711\) of defining polynomial
Character \(\chi\) \(=\) 273.1

$q$-expansion

\(f(q)\) \(=\) \(q-41.3711 q^{2} -243.000 q^{3} -336.429 q^{4} -4784.52 q^{5} +10053.2 q^{6} +16807.0 q^{7} +98646.5 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-41.3711 q^{2} -243.000 q^{3} -336.429 q^{4} -4784.52 q^{5} +10053.2 q^{6} +16807.0 q^{7} +98646.5 q^{8} +59049.0 q^{9} +197941. q^{10} +884509. q^{11} +81752.2 q^{12} -371293. q^{13} -695325. q^{14} +1.16264e6 q^{15} -3.39211e6 q^{16} +6.33691e6 q^{17} -2.44292e6 q^{18} -7.31264e6 q^{19} +1.60965e6 q^{20} -4.08410e6 q^{21} -3.65931e7 q^{22} -1.47316e7 q^{23} -2.39711e7 q^{24} -2.59365e7 q^{25} +1.53608e7 q^{26} -1.43489e7 q^{27} -5.65436e6 q^{28} +1.00927e8 q^{29} -4.80997e7 q^{30} -2.95315e8 q^{31} -6.16925e7 q^{32} -2.14936e8 q^{33} -2.62165e8 q^{34} -8.04135e7 q^{35} -1.98658e7 q^{36} -2.53738e8 q^{37} +3.02532e8 q^{38} +9.02242e7 q^{39} -4.71977e8 q^{40} +1.20683e9 q^{41} +1.68964e8 q^{42} +4.41770e7 q^{43} -2.97574e8 q^{44} -2.82521e8 q^{45} +6.09463e8 q^{46} -1.91405e9 q^{47} +8.24284e8 q^{48} +2.82475e8 q^{49} +1.07302e9 q^{50} -1.53987e9 q^{51} +1.24914e8 q^{52} +6.36600e8 q^{53} +5.93631e8 q^{54} -4.23195e9 q^{55} +1.65795e9 q^{56} +1.77697e9 q^{57} -4.17545e9 q^{58} +4.83453e9 q^{59} -3.91145e8 q^{60} +1.25696e10 q^{61} +1.22175e10 q^{62} +9.92437e8 q^{63} +9.49934e9 q^{64} +1.77646e9 q^{65} +8.89213e9 q^{66} -1.37371e10 q^{67} -2.13192e9 q^{68} +3.57978e9 q^{69} +3.32680e9 q^{70} -4.32356e9 q^{71} +5.82498e9 q^{72} -1.27289e10 q^{73} +1.04974e10 q^{74} +6.30256e9 q^{75} +2.46018e9 q^{76} +1.48659e10 q^{77} -3.73268e9 q^{78} -2.47713e9 q^{79} +1.62296e10 q^{80} +3.48678e9 q^{81} -4.99278e10 q^{82} -6.01852e10 q^{83} +1.37401e9 q^{84} -3.03191e10 q^{85} -1.82765e9 q^{86} -2.45251e10 q^{87} +8.72537e10 q^{88} +6.43004e10 q^{89} +1.16882e10 q^{90} -6.24032e9 q^{91} +4.95613e9 q^{92} +7.17616e10 q^{93} +7.91866e10 q^{94} +3.49875e10 q^{95} +1.49913e10 q^{96} +1.31580e11 q^{97} -1.16863e10 q^{98} +5.22294e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + O(q^{10}) \) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + 1056711q^{10} - 466884q^{11} - 5044923q^{12} - 5940688q^{13} - 1058841q^{14} + 769824q^{15} + 40058265q^{16} + 1753452q^{17} - 3720087q^{18} + 4237800q^{19} - 20710503q^{20} - 65345616q^{21} - 37479661q^{22} - 150481440q^{23} + 45675495q^{24} + 150983176q^{25} + 23391459q^{26} - 229582512q^{27} + 348930127q^{28} - 111411432q^{29} - 256780773q^{30} - 90536236q^{31} - 609941313q^{32} + 113452812q^{33} + 443745577q^{34} - 53244576q^{35} + 1225916289q^{36} - 1756337900q^{37} - 4378868973q^{38} + 1443587184q^{39} + 57535383q^{40} + 649575720q^{41} + 257298363q^{42} - 1889554520q^{43} - 4979587689q^{44} - 187067232q^{45} - 3204000867q^{46} - 4036082940q^{47} - 9734158395q^{48} + 4519603984q^{49} - 15928886862q^{50} - 426088836q^{51} - 7708413973q^{52} + 511144020q^{53} + 903981141q^{54} - 8519365572q^{55} - 3159127755q^{56} - 1029785400q^{57} - 7851149577q^{58} + 3728287296q^{59} + 5032652229q^{60} + 26227205052q^{61} + 2996618490q^{62} + 15878984688q^{63} + 51104408465q^{64} + 1176256224q^{65} + 9107557623q^{66} - 5295680024q^{67} + 7919540325q^{68} + 36566989920q^{69} + 17760141777q^{70} + 17082800928q^{71} - 11099145285q^{72} + 21076301488q^{73} + 52794333675q^{74} - 36688911768q^{75} + 81459179177q^{76} - 7846919388q^{77} - 5684124537q^{78} - 83677977852q^{79} - 163988465427q^{80} + 55788550416q^{81} - 61543728760q^{82} - 72857072340q^{83} - 84790020861q^{84} - 12608565060q^{85} - 20177818011q^{86} + 27072977976q^{87} - 202213679643q^{88} + 32238476676q^{89} + 62397727839q^{90} - 99845143216q^{91} - 487057197033q^{92} + 22000305348q^{93} + 183904776602q^{94} - 143022915504q^{95} + 148215739059q^{96} + 6973535140q^{97} - 17795940687q^{98} - 27569033316q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −41.3711 −0.914182 −0.457091 0.889420i \(-0.651109\pi\)
−0.457091 + 0.889420i \(0.651109\pi\)
\(3\) −243.000 −0.577350
\(4\) −336.429 −0.164272
\(5\) −4784.52 −0.684705 −0.342353 0.939572i \(-0.611224\pi\)
−0.342353 + 0.939572i \(0.611224\pi\)
\(6\) 10053.2 0.527803
\(7\) 16807.0 0.377964
\(8\) 98646.5 1.06436
\(9\) 59049.0 0.333333
\(10\) 197941. 0.625945
\(11\) 884509. 1.65593 0.827966 0.560779i \(-0.189498\pi\)
0.827966 + 0.560779i \(0.189498\pi\)
\(12\) 81752.2 0.0948424
\(13\) −371293. −0.277350
\(14\) −695325. −0.345528
\(15\) 1.16264e6 0.395315
\(16\) −3.39211e6 −0.808743
\(17\) 6.33691e6 1.08245 0.541226 0.840877i \(-0.317961\pi\)
0.541226 + 0.840877i \(0.317961\pi\)
\(18\) −2.44292e6 −0.304727
\(19\) −7.31264e6 −0.677531 −0.338766 0.940871i \(-0.610009\pi\)
−0.338766 + 0.940871i \(0.610009\pi\)
\(20\) 1.60965e6 0.112478
\(21\) −4.08410e6 −0.218218
\(22\) −3.65931e7 −1.51382
\(23\) −1.47316e7 −0.477251 −0.238626 0.971112i \(-0.576697\pi\)
−0.238626 + 0.971112i \(0.576697\pi\)
\(24\) −2.39711e7 −0.614506
\(25\) −2.59365e7 −0.531179
\(26\) 1.53608e7 0.253548
\(27\) −1.43489e7 −0.192450
\(28\) −5.65436e6 −0.0620889
\(29\) 1.00927e8 0.913727 0.456863 0.889537i \(-0.348973\pi\)
0.456863 + 0.889537i \(0.348973\pi\)
\(30\) −4.80997e7 −0.361389
\(31\) −2.95315e8 −1.85266 −0.926331 0.376710i \(-0.877055\pi\)
−0.926331 + 0.376710i \(0.877055\pi\)
\(32\) −6.16925e7 −0.325018
\(33\) −2.14936e8 −0.956052
\(34\) −2.62165e8 −0.989557
\(35\) −8.04135e7 −0.258794
\(36\) −1.98658e7 −0.0547573
\(37\) −2.53738e8 −0.601556 −0.300778 0.953694i \(-0.597246\pi\)
−0.300778 + 0.953694i \(0.597246\pi\)
\(38\) 3.02532e8 0.619387
\(39\) 9.02242e7 0.160128
\(40\) −4.71977e8 −0.728770
\(41\) 1.20683e9 1.62680 0.813398 0.581707i \(-0.197615\pi\)
0.813398 + 0.581707i \(0.197615\pi\)
\(42\) 1.68964e8 0.199491
\(43\) 4.41770e7 0.0458269 0.0229134 0.999737i \(-0.492706\pi\)
0.0229134 + 0.999737i \(0.492706\pi\)
\(44\) −2.97574e8 −0.272023
\(45\) −2.82521e8 −0.228235
\(46\) 6.09463e8 0.436294
\(47\) −1.91405e9 −1.21735 −0.608676 0.793419i \(-0.708299\pi\)
−0.608676 + 0.793419i \(0.708299\pi\)
\(48\) 8.24284e8 0.466928
\(49\) 2.82475e8 0.142857
\(50\) 1.07302e9 0.485594
\(51\) −1.53987e9 −0.624953
\(52\) 1.24914e8 0.0455608
\(53\) 6.36600e8 0.209098 0.104549 0.994520i \(-0.466660\pi\)
0.104549 + 0.994520i \(0.466660\pi\)
\(54\) 5.93631e8 0.175934
\(55\) −4.23195e9 −1.13382
\(56\) 1.65795e9 0.402289
\(57\) 1.77697e9 0.391173
\(58\) −4.17545e9 −0.835312
\(59\) 4.83453e9 0.880376 0.440188 0.897906i \(-0.354912\pi\)
0.440188 + 0.897906i \(0.354912\pi\)
\(60\) −3.91145e8 −0.0649391
\(61\) 1.25696e10 1.90549 0.952743 0.303777i \(-0.0982478\pi\)
0.952743 + 0.303777i \(0.0982478\pi\)
\(62\) 1.22175e10 1.69367
\(63\) 9.92437e8 0.125988
\(64\) 9.49934e9 1.10587
\(65\) 1.77646e9 0.189903
\(66\) 8.89213e9 0.874006
\(67\) −1.37371e10 −1.24303 −0.621516 0.783402i \(-0.713483\pi\)
−0.621516 + 0.783402i \(0.713483\pi\)
\(68\) −2.13192e9 −0.177816
\(69\) 3.57978e9 0.275541
\(70\) 3.32680e9 0.236585
\(71\) −4.32356e9 −0.284394 −0.142197 0.989838i \(-0.545417\pi\)
−0.142197 + 0.989838i \(0.545417\pi\)
\(72\) 5.82498e9 0.354785
\(73\) −1.27289e10 −0.718648 −0.359324 0.933213i \(-0.616993\pi\)
−0.359324 + 0.933213i \(0.616993\pi\)
\(74\) 1.04974e10 0.549931
\(75\) 6.30256e9 0.306676
\(76\) 2.46018e9 0.111299
\(77\) 1.48659e10 0.625883
\(78\) −3.73268e9 −0.146386
\(79\) −2.47713e9 −0.0905731 −0.0452866 0.998974i \(-0.514420\pi\)
−0.0452866 + 0.998974i \(0.514420\pi\)
\(80\) 1.62296e10 0.553750
\(81\) 3.48678e9 0.111111
\(82\) −4.99278e10 −1.48719
\(83\) −6.01852e10 −1.67711 −0.838553 0.544820i \(-0.816598\pi\)
−0.838553 + 0.544820i \(0.816598\pi\)
\(84\) 1.37401e9 0.0358470
\(85\) −3.03191e10 −0.741160
\(86\) −1.82765e9 −0.0418941
\(87\) −2.45251e10 −0.527540
\(88\) 8.72537e10 1.76250
\(89\) 6.43004e10 1.22059 0.610293 0.792176i \(-0.291052\pi\)
0.610293 + 0.792176i \(0.291052\pi\)
\(90\) 1.16882e10 0.208648
\(91\) −6.24032e9 −0.104828
\(92\) 4.95613e9 0.0783989
\(93\) 7.17616e10 1.06964
\(94\) 7.91866e10 1.11288
\(95\) 3.49875e10 0.463909
\(96\) 1.49913e10 0.187649
\(97\) 1.31580e11 1.55577 0.777885 0.628406i \(-0.216293\pi\)
0.777885 + 0.628406i \(0.216293\pi\)
\(98\) −1.16863e10 −0.130597
\(99\) 5.22294e10 0.551977
\(100\) 8.72577e9 0.0872577
\(101\) −1.89831e11 −1.79721 −0.898605 0.438759i \(-0.855418\pi\)
−0.898605 + 0.438759i \(0.855418\pi\)
\(102\) 6.37061e10 0.571321
\(103\) −1.05355e11 −0.895465 −0.447733 0.894167i \(-0.647768\pi\)
−0.447733 + 0.894167i \(0.647768\pi\)
\(104\) −3.66268e10 −0.295199
\(105\) 1.95405e10 0.149415
\(106\) −2.63369e10 −0.191153
\(107\) 2.04540e10 0.140983 0.0704915 0.997512i \(-0.477543\pi\)
0.0704915 + 0.997512i \(0.477543\pi\)
\(108\) 4.82738e9 0.0316141
\(109\) −1.94751e11 −1.21237 −0.606183 0.795325i \(-0.707300\pi\)
−0.606183 + 0.795325i \(0.707300\pi\)
\(110\) 1.75081e11 1.03652
\(111\) 6.16583e10 0.347308
\(112\) −5.70113e10 −0.305676
\(113\) 3.68331e11 1.88064 0.940322 0.340285i \(-0.110524\pi\)
0.940322 + 0.340285i \(0.110524\pi\)
\(114\) −7.35153e10 −0.357603
\(115\) 7.04837e10 0.326776
\(116\) −3.39546e10 −0.150100
\(117\) −2.19245e10 −0.0924500
\(118\) −2.00010e11 −0.804823
\(119\) 1.06504e11 0.409128
\(120\) 1.14690e11 0.420756
\(121\) 4.97044e11 1.74211
\(122\) −5.20017e11 −1.74196
\(123\) −2.93259e11 −0.939232
\(124\) 9.93524e10 0.304340
\(125\) 3.57713e11 1.04841
\(126\) −4.10582e10 −0.115176
\(127\) −3.23542e11 −0.868980 −0.434490 0.900677i \(-0.643071\pi\)
−0.434490 + 0.900677i \(0.643071\pi\)
\(128\) −2.66652e11 −0.685947
\(129\) −1.07350e10 −0.0264582
\(130\) −7.34942e10 −0.173606
\(131\) −5.51792e10 −0.124964 −0.0624818 0.998046i \(-0.519902\pi\)
−0.0624818 + 0.998046i \(0.519902\pi\)
\(132\) 7.23105e10 0.157052
\(133\) −1.22904e11 −0.256083
\(134\) 5.68318e11 1.13636
\(135\) 6.86527e10 0.131772
\(136\) 6.25114e11 1.15211
\(137\) −5.47782e11 −0.969716 −0.484858 0.874593i \(-0.661129\pi\)
−0.484858 + 0.874593i \(0.661129\pi\)
\(138\) −1.48100e11 −0.251895
\(139\) 3.99759e11 0.653457 0.326729 0.945118i \(-0.394054\pi\)
0.326729 + 0.945118i \(0.394054\pi\)
\(140\) 2.70534e10 0.0425126
\(141\) 4.65115e11 0.702838
\(142\) 1.78871e11 0.259988
\(143\) −3.28412e11 −0.459273
\(144\) −2.00301e11 −0.269581
\(145\) −4.82885e11 −0.625633
\(146\) 5.26611e11 0.656975
\(147\) −6.86415e10 −0.0824786
\(148\) 8.53647e10 0.0988186
\(149\) 3.17758e10 0.0354464 0.0177232 0.999843i \(-0.494358\pi\)
0.0177232 + 0.999843i \(0.494358\pi\)
\(150\) −2.60744e11 −0.280358
\(151\) 7.42401e11 0.769600 0.384800 0.923000i \(-0.374270\pi\)
0.384800 + 0.923000i \(0.374270\pi\)
\(152\) −7.21367e11 −0.721135
\(153\) 3.74188e11 0.360817
\(154\) −6.15021e11 −0.572171
\(155\) 1.41294e12 1.26853
\(156\) −3.03540e10 −0.0263045
\(157\) 4.60184e11 0.385020 0.192510 0.981295i \(-0.438337\pi\)
0.192510 + 0.981295i \(0.438337\pi\)
\(158\) 1.02482e11 0.0828003
\(159\) −1.54694e11 −0.120723
\(160\) 2.95169e11 0.222541
\(161\) −2.47594e11 −0.180384
\(162\) −1.44252e11 −0.101576
\(163\) 4.85673e11 0.330607 0.165303 0.986243i \(-0.447140\pi\)
0.165303 + 0.986243i \(0.447140\pi\)
\(164\) −4.06011e11 −0.267237
\(165\) 1.02836e12 0.654614
\(166\) 2.48993e12 1.53318
\(167\) 1.04722e12 0.623872 0.311936 0.950103i \(-0.399023\pi\)
0.311936 + 0.950103i \(0.399023\pi\)
\(168\) −4.02882e11 −0.232262
\(169\) 1.37858e11 0.0769231
\(170\) 1.25433e12 0.677555
\(171\) −4.31804e11 −0.225844
\(172\) −1.48624e10 −0.00752806
\(173\) 1.72683e12 0.847217 0.423609 0.905845i \(-0.360763\pi\)
0.423609 + 0.905845i \(0.360763\pi\)
\(174\) 1.01463e12 0.482268
\(175\) −4.35914e11 −0.200767
\(176\) −3.00035e12 −1.33922
\(177\) −1.17479e12 −0.508285
\(178\) −2.66018e12 −1.11584
\(179\) −3.60323e11 −0.146555 −0.0732775 0.997312i \(-0.523346\pi\)
−0.0732775 + 0.997312i \(0.523346\pi\)
\(180\) 9.50482e10 0.0374926
\(181\) 4.36143e12 1.66877 0.834385 0.551182i \(-0.185823\pi\)
0.834385 + 0.551182i \(0.185823\pi\)
\(182\) 2.58169e11 0.0958323
\(183\) −3.05440e12 −1.10013
\(184\) −1.45322e12 −0.507965
\(185\) 1.21401e12 0.411888
\(186\) −2.96886e12 −0.977841
\(187\) 5.60505e12 1.79247
\(188\) 6.43942e11 0.199976
\(189\) −2.41162e11 −0.0727393
\(190\) −1.44747e12 −0.424097
\(191\) −5.06514e12 −1.44181 −0.720905 0.693034i \(-0.756274\pi\)
−0.720905 + 0.693034i \(0.756274\pi\)
\(192\) −2.30834e12 −0.638473
\(193\) 1.80649e12 0.485589 0.242795 0.970078i \(-0.421936\pi\)
0.242795 + 0.970078i \(0.421936\pi\)
\(194\) −5.44362e12 −1.42226
\(195\) −4.31680e11 −0.109641
\(196\) −9.50328e10 −0.0234674
\(197\) −1.98212e12 −0.475955 −0.237978 0.971271i \(-0.576484\pi\)
−0.237978 + 0.971271i \(0.576484\pi\)
\(198\) −2.16079e12 −0.504607
\(199\) −7.04044e11 −0.159922 −0.0799609 0.996798i \(-0.525480\pi\)
−0.0799609 + 0.996798i \(0.525480\pi\)
\(200\) −2.55854e12 −0.565363
\(201\) 3.33810e12 0.717665
\(202\) 7.85351e12 1.64298
\(203\) 1.69627e12 0.345356
\(204\) 5.18056e11 0.102662
\(205\) −5.77408e12 −1.11388
\(206\) 4.35864e12 0.818618
\(207\) −8.69886e11 −0.159084
\(208\) 1.25947e12 0.224305
\(209\) −6.46809e12 −1.12195
\(210\) −8.08412e11 −0.136592
\(211\) −6.92729e12 −1.14028 −0.570138 0.821549i \(-0.693110\pi\)
−0.570138 + 0.821549i \(0.693110\pi\)
\(212\) −2.14170e11 −0.0343489
\(213\) 1.05062e12 0.164195
\(214\) −8.46204e11 −0.128884
\(215\) −2.11366e11 −0.0313779
\(216\) −1.41547e12 −0.204835
\(217\) −4.96336e12 −0.700241
\(218\) 8.05707e12 1.10832
\(219\) 3.09313e12 0.414912
\(220\) 1.42375e12 0.186255
\(221\) −2.35285e12 −0.300218
\(222\) −2.55087e12 −0.317503
\(223\) −1.05933e13 −1.28634 −0.643171 0.765723i \(-0.722381\pi\)
−0.643171 + 0.765723i \(0.722381\pi\)
\(224\) −1.03687e12 −0.122845
\(225\) −1.53152e12 −0.177060
\(226\) −1.52383e13 −1.71925
\(227\) 9.52631e12 1.04902 0.524508 0.851405i \(-0.324249\pi\)
0.524508 + 0.851405i \(0.324249\pi\)
\(228\) −5.97824e11 −0.0642587
\(229\) 1.45565e12 0.152744 0.0763718 0.997079i \(-0.475666\pi\)
0.0763718 + 0.997079i \(0.475666\pi\)
\(230\) −2.91599e12 −0.298733
\(231\) −3.61242e12 −0.361354
\(232\) 9.95605e12 0.972531
\(233\) 3.13813e12 0.299373 0.149687 0.988734i \(-0.452174\pi\)
0.149687 + 0.988734i \(0.452174\pi\)
\(234\) 9.07041e11 0.0845161
\(235\) 9.15783e12 0.833527
\(236\) −1.62647e12 −0.144621
\(237\) 6.01942e11 0.0522924
\(238\) −4.40621e12 −0.374017
\(239\) −4.36200e12 −0.361824 −0.180912 0.983499i \(-0.557905\pi\)
−0.180912 + 0.983499i \(0.557905\pi\)
\(240\) −3.94380e12 −0.319708
\(241\) 2.27001e12 0.179860 0.0899299 0.995948i \(-0.471336\pi\)
0.0899299 + 0.995948i \(0.471336\pi\)
\(242\) −2.05633e13 −1.59260
\(243\) −8.47289e11 −0.0641500
\(244\) −4.22876e12 −0.313018
\(245\) −1.35151e12 −0.0978150
\(246\) 1.21324e13 0.858628
\(247\) 2.71513e12 0.187913
\(248\) −2.91318e13 −1.97189
\(249\) 1.46250e13 0.968277
\(250\) −1.47990e13 −0.958434
\(251\) 6.55497e11 0.0415303 0.0207652 0.999784i \(-0.493390\pi\)
0.0207652 + 0.999784i \(0.493390\pi\)
\(252\) −3.33884e11 −0.0206963
\(253\) −1.30302e13 −0.790295
\(254\) 1.33853e13 0.794406
\(255\) 7.36754e12 0.427909
\(256\) −8.42293e12 −0.478788
\(257\) 1.53232e13 0.852545 0.426272 0.904595i \(-0.359826\pi\)
0.426272 + 0.904595i \(0.359826\pi\)
\(258\) 4.44120e11 0.0241876
\(259\) −4.26457e12 −0.227367
\(260\) −5.97652e11 −0.0311957
\(261\) 5.95961e12 0.304576
\(262\) 2.28283e12 0.114239
\(263\) −1.98777e13 −0.974112 −0.487056 0.873371i \(-0.661929\pi\)
−0.487056 + 0.873371i \(0.661929\pi\)
\(264\) −2.12027e13 −1.01758
\(265\) −3.04583e12 −0.143170
\(266\) 5.08466e12 0.234106
\(267\) −1.56250e13 −0.704706
\(268\) 4.62154e12 0.204195
\(269\) −2.47772e13 −1.07254 −0.536272 0.844045i \(-0.680168\pi\)
−0.536272 + 0.844045i \(0.680168\pi\)
\(270\) −2.84024e12 −0.120463
\(271\) −3.90044e12 −0.162100 −0.0810500 0.996710i \(-0.525827\pi\)
−0.0810500 + 0.996710i \(0.525827\pi\)
\(272\) −2.14955e13 −0.875425
\(273\) 1.51640e12 0.0605228
\(274\) 2.26624e13 0.886497
\(275\) −2.29410e13 −0.879596
\(276\) −1.20434e12 −0.0452636
\(277\) −2.32051e13 −0.854957 −0.427478 0.904026i \(-0.640598\pi\)
−0.427478 + 0.904026i \(0.640598\pi\)
\(278\) −1.65385e13 −0.597379
\(279\) −1.74381e13 −0.617554
\(280\) −7.93251e12 −0.275449
\(281\) −4.66297e12 −0.158774 −0.0793868 0.996844i \(-0.525296\pi\)
−0.0793868 + 0.996844i \(0.525296\pi\)
\(282\) −1.92423e13 −0.642522
\(283\) −8.46371e12 −0.277163 −0.138582 0.990351i \(-0.544254\pi\)
−0.138582 + 0.990351i \(0.544254\pi\)
\(284\) 1.45457e12 0.0467179
\(285\) −8.50196e12 −0.267838
\(286\) 1.35868e13 0.419859
\(287\) 2.02831e13 0.614871
\(288\) −3.64288e12 −0.108339
\(289\) 5.88450e12 0.171701
\(290\) 1.99775e13 0.571943
\(291\) −3.19739e13 −0.898224
\(292\) 4.28238e12 0.118054
\(293\) 6.79340e13 1.83787 0.918937 0.394405i \(-0.129049\pi\)
0.918937 + 0.394405i \(0.129049\pi\)
\(294\) 2.83978e12 0.0754004
\(295\) −2.31309e13 −0.602798
\(296\) −2.50304e13 −0.640269
\(297\) −1.26917e13 −0.318684
\(298\) −1.31460e12 −0.0324044
\(299\) 5.46974e12 0.132366
\(300\) −2.12036e12 −0.0503783
\(301\) 7.42484e11 0.0173209
\(302\) −3.07140e13 −0.703555
\(303\) 4.61288e13 1.03762
\(304\) 2.48053e13 0.547949
\(305\) −6.01393e13 −1.30470
\(306\) −1.54806e13 −0.329852
\(307\) −6.82850e13 −1.42910 −0.714552 0.699582i \(-0.753370\pi\)
−0.714552 + 0.699582i \(0.753370\pi\)
\(308\) −5.00133e12 −0.102815
\(309\) 2.56012e13 0.516997
\(310\) −5.84550e13 −1.15966
\(311\) 9.04928e13 1.76373 0.881865 0.471502i \(-0.156288\pi\)
0.881865 + 0.471502i \(0.156288\pi\)
\(312\) 8.90030e12 0.170433
\(313\) −9.62908e13 −1.81172 −0.905860 0.423577i \(-0.860774\pi\)
−0.905860 + 0.423577i \(0.860774\pi\)
\(314\) −1.90383e13 −0.351978
\(315\) −4.74834e12 −0.0862647
\(316\) 8.33377e11 0.0148786
\(317\) −5.05202e13 −0.886419 −0.443210 0.896418i \(-0.646160\pi\)
−0.443210 + 0.896418i \(0.646160\pi\)
\(318\) 6.39986e12 0.110362
\(319\) 8.92704e13 1.51307
\(320\) −4.54498e13 −0.757194
\(321\) −4.97031e12 −0.0813965
\(322\) 1.02432e13 0.164904
\(323\) −4.63395e13 −0.733395
\(324\) −1.17305e12 −0.0182524
\(325\) 9.63003e12 0.147323
\(326\) −2.00928e13 −0.302235
\(327\) 4.73245e13 0.699960
\(328\) 1.19049e14 1.73149
\(329\) −3.21695e13 −0.460116
\(330\) −4.25446e13 −0.598436
\(331\) 7.18309e12 0.0993705 0.0496853 0.998765i \(-0.484178\pi\)
0.0496853 + 0.998765i \(0.484178\pi\)
\(332\) 2.02480e13 0.275501
\(333\) −1.49830e13 −0.200519
\(334\) −4.33245e13 −0.570333
\(335\) 6.57252e13 0.851110
\(336\) 1.38537e13 0.176482
\(337\) 4.46834e12 0.0559992 0.0279996 0.999608i \(-0.491086\pi\)
0.0279996 + 0.999608i \(0.491086\pi\)
\(338\) −5.70336e12 −0.0703217
\(339\) −8.95044e13 −1.08579
\(340\) 1.02002e13 0.121752
\(341\) −2.61209e14 −3.06788
\(342\) 1.78642e13 0.206462
\(343\) 4.74756e12 0.0539949
\(344\) 4.35791e12 0.0487761
\(345\) −1.71275e13 −0.188664
\(346\) −7.14407e13 −0.774511
\(347\) 6.77393e13 0.722817 0.361409 0.932408i \(-0.382296\pi\)
0.361409 + 0.932408i \(0.382296\pi\)
\(348\) 8.25096e12 0.0866600
\(349\) −8.30163e12 −0.0858270 −0.0429135 0.999079i \(-0.513664\pi\)
−0.0429135 + 0.999079i \(0.513664\pi\)
\(350\) 1.80343e13 0.183537
\(351\) 5.32765e12 0.0533761
\(352\) −5.45675e13 −0.538207
\(353\) 1.95520e14 1.89859 0.949295 0.314386i \(-0.101799\pi\)
0.949295 + 0.314386i \(0.101799\pi\)
\(354\) 4.86024e13 0.464665
\(355\) 2.06862e13 0.194726
\(356\) −2.16325e13 −0.200508
\(357\) −2.58806e13 −0.236210
\(358\) 1.49070e13 0.133978
\(359\) −1.70243e14 −1.50678 −0.753388 0.657576i \(-0.771582\pi\)
−0.753388 + 0.657576i \(0.771582\pi\)
\(360\) −2.78697e13 −0.242923
\(361\) −6.30156e13 −0.540951
\(362\) −1.80437e14 −1.52556
\(363\) −1.20782e14 −1.00581
\(364\) 2.09942e12 0.0172204
\(365\) 6.09019e13 0.492062
\(366\) 1.26364e14 1.00572
\(367\) 1.06393e14 0.834161 0.417081 0.908869i \(-0.363053\pi\)
0.417081 + 0.908869i \(0.363053\pi\)
\(368\) 4.99713e13 0.385973
\(369\) 7.12618e13 0.542266
\(370\) −5.02252e13 −0.376541
\(371\) 1.06993e13 0.0790315
\(372\) −2.41426e13 −0.175711
\(373\) −2.33613e14 −1.67532 −0.837661 0.546191i \(-0.816077\pi\)
−0.837661 + 0.546191i \(0.816077\pi\)
\(374\) −2.31887e14 −1.63864
\(375\) −8.69242e13 −0.605298
\(376\) −1.88815e14 −1.29570
\(377\) −3.74733e13 −0.253422
\(378\) 9.97715e12 0.0664969
\(379\) −4.56431e13 −0.299819 −0.149910 0.988700i \(-0.547898\pi\)
−0.149910 + 0.988700i \(0.547898\pi\)
\(380\) −1.17708e13 −0.0762072
\(381\) 7.86207e13 0.501706
\(382\) 2.09551e14 1.31808
\(383\) 1.47555e13 0.0914872 0.0457436 0.998953i \(-0.485434\pi\)
0.0457436 + 0.998953i \(0.485434\pi\)
\(384\) 6.47965e13 0.396032
\(385\) −7.11264e13 −0.428545
\(386\) −7.47364e13 −0.443917
\(387\) 2.60861e12 0.0152756
\(388\) −4.42673e13 −0.255569
\(389\) −1.10147e14 −0.626977 −0.313489 0.949592i \(-0.601498\pi\)
−0.313489 + 0.949592i \(0.601498\pi\)
\(390\) 1.78591e13 0.100231
\(391\) −9.33528e13 −0.516601
\(392\) 2.78652e13 0.152051
\(393\) 1.34085e13 0.0721477
\(394\) 8.20026e13 0.435109
\(395\) 1.18519e13 0.0620159
\(396\) −1.75715e13 −0.0906743
\(397\) −2.07100e14 −1.05398 −0.526989 0.849872i \(-0.676679\pi\)
−0.526989 + 0.849872i \(0.676679\pi\)
\(398\) 2.91271e13 0.146198
\(399\) 2.98656e13 0.147849
\(400\) 8.79795e13 0.429587
\(401\) 1.31143e14 0.631615 0.315808 0.948823i \(-0.397725\pi\)
0.315808 + 0.948823i \(0.397725\pi\)
\(402\) −1.38101e14 −0.656076
\(403\) 1.09648e14 0.513836
\(404\) 6.38645e13 0.295231
\(405\) −1.66826e13 −0.0760783
\(406\) −7.01767e13 −0.315718
\(407\) −2.24433e14 −0.996135
\(408\) −1.51903e14 −0.665173
\(409\) 1.18023e14 0.509902 0.254951 0.966954i \(-0.417941\pi\)
0.254951 + 0.966954i \(0.417941\pi\)
\(410\) 2.38880e14 1.01829
\(411\) 1.33111e14 0.559866
\(412\) 3.54443e13 0.147100
\(413\) 8.12539e13 0.332751
\(414\) 3.59882e13 0.145431
\(415\) 2.87958e14 1.14832
\(416\) 2.29060e13 0.0901438
\(417\) −9.71415e13 −0.377274
\(418\) 2.67592e14 1.02566
\(419\) 1.67909e14 0.635179 0.317590 0.948228i \(-0.397127\pi\)
0.317590 + 0.948228i \(0.397127\pi\)
\(420\) −6.57397e12 −0.0245447
\(421\) 4.39224e14 1.61858 0.809290 0.587409i \(-0.199852\pi\)
0.809290 + 0.587409i \(0.199852\pi\)
\(422\) 2.86590e14 1.04242
\(423\) −1.13023e14 −0.405784
\(424\) 6.27984e13 0.222554
\(425\) −1.64357e14 −0.574975
\(426\) −4.34655e13 −0.150104
\(427\) 2.11257e14 0.720206
\(428\) −6.88130e12 −0.0231595
\(429\) 7.98041e13 0.265161
\(430\) 8.74445e12 0.0286851
\(431\) 2.26409e13 0.0733277 0.0366639 0.999328i \(-0.488327\pi\)
0.0366639 + 0.999328i \(0.488327\pi\)
\(432\) 4.86731e13 0.155643
\(433\) 3.58830e14 1.13294 0.566468 0.824084i \(-0.308310\pi\)
0.566468 + 0.824084i \(0.308310\pi\)
\(434\) 2.05340e14 0.640147
\(435\) 1.17341e14 0.361210
\(436\) 6.55198e13 0.199157
\(437\) 1.07727e14 0.323353
\(438\) −1.27966e14 −0.379305
\(439\) 2.85693e14 0.836267 0.418134 0.908386i \(-0.362684\pi\)
0.418134 + 0.908386i \(0.362684\pi\)
\(440\) −4.17467e14 −1.20679
\(441\) 1.66799e13 0.0476190
\(442\) 9.73401e13 0.274454
\(443\) −6.68680e14 −1.86208 −0.931039 0.364920i \(-0.881096\pi\)
−0.931039 + 0.364920i \(0.881096\pi\)
\(444\) −2.07436e13 −0.0570530
\(445\) −3.07647e14 −0.835742
\(446\) 4.38259e14 1.17595
\(447\) −7.72152e12 −0.0204650
\(448\) 1.59655e14 0.417979
\(449\) −6.68043e13 −0.172762 −0.0863812 0.996262i \(-0.527530\pi\)
−0.0863812 + 0.996262i \(0.527530\pi\)
\(450\) 6.33608e13 0.161865
\(451\) 1.06745e15 2.69386
\(452\) −1.23917e14 −0.308937
\(453\) −1.80403e14 −0.444329
\(454\) −3.94114e14 −0.958992
\(455\) 2.98570e13 0.0717766
\(456\) 1.75292e14 0.416347
\(457\) 1.13591e14 0.266567 0.133283 0.991078i \(-0.457448\pi\)
0.133283 + 0.991078i \(0.457448\pi\)
\(458\) −6.02221e13 −0.139635
\(459\) −9.09277e13 −0.208318
\(460\) −2.37127e13 −0.0536801
\(461\) −4.77652e14 −1.06846 −0.534228 0.845341i \(-0.679398\pi\)
−0.534228 + 0.845341i \(0.679398\pi\)
\(462\) 1.49450e14 0.330343
\(463\) 7.81818e14 1.70770 0.853848 0.520523i \(-0.174263\pi\)
0.853848 + 0.520523i \(0.174263\pi\)
\(464\) −3.42354e14 −0.738970
\(465\) −3.43345e14 −0.732385
\(466\) −1.29828e14 −0.273681
\(467\) 2.44010e14 0.508353 0.254177 0.967158i \(-0.418196\pi\)
0.254177 + 0.967158i \(0.418196\pi\)
\(468\) 7.37602e12 0.0151869
\(469\) −2.30879e14 −0.469822
\(470\) −3.78870e14 −0.761995
\(471\) −1.11825e14 −0.222291
\(472\) 4.76909e14 0.937033
\(473\) 3.90750e13 0.0758861
\(474\) −2.49030e13 −0.0478048
\(475\) 1.89664e14 0.359890
\(476\) −3.58311e13 −0.0672082
\(477\) 3.75906e13 0.0696992
\(478\) 1.80461e14 0.330773
\(479\) −4.45108e14 −0.806530 −0.403265 0.915083i \(-0.632125\pi\)
−0.403265 + 0.915083i \(0.632125\pi\)
\(480\) −7.17261e13 −0.128484
\(481\) 9.42111e13 0.166842
\(482\) −9.39129e13 −0.164425
\(483\) 6.01654e13 0.104145
\(484\) −1.67220e14 −0.286179
\(485\) −6.29548e14 −1.06524
\(486\) 3.50533e13 0.0586448
\(487\) 2.15894e13 0.0357134 0.0178567 0.999841i \(-0.494316\pi\)
0.0178567 + 0.999841i \(0.494316\pi\)
\(488\) 1.23994e15 2.02812
\(489\) −1.18018e14 −0.190876
\(490\) 5.59135e13 0.0894207
\(491\) 5.13313e14 0.811772 0.405886 0.913924i \(-0.366963\pi\)
0.405886 + 0.913924i \(0.366963\pi\)
\(492\) 9.86606e13 0.154289
\(493\) 6.39562e14 0.989065
\(494\) −1.12328e14 −0.171787
\(495\) −2.49893e14 −0.377942
\(496\) 1.00174e15 1.49833
\(497\) −7.26661e13 −0.107491
\(498\) −6.05053e14 −0.885182
\(499\) −6.42067e13 −0.0929025 −0.0464512 0.998921i \(-0.514791\pi\)
−0.0464512 + 0.998921i \(0.514791\pi\)
\(500\) −1.20345e14 −0.172224
\(501\) −2.54473e14 −0.360193
\(502\) −2.71187e13 −0.0379662
\(503\) −3.93209e14 −0.544502 −0.272251 0.962226i \(-0.587768\pi\)
−0.272251 + 0.962226i \(0.587768\pi\)
\(504\) 9.79004e13 0.134096
\(505\) 9.08249e14 1.23056
\(506\) 5.39076e14 0.722473
\(507\) −3.34996e13 −0.0444116
\(508\) 1.08849e14 0.142749
\(509\) −1.94250e14 −0.252008 −0.126004 0.992030i \(-0.540215\pi\)
−0.126004 + 0.992030i \(0.540215\pi\)
\(510\) −3.04803e14 −0.391186
\(511\) −2.13935e14 −0.271624
\(512\) 8.94570e14 1.12365
\(513\) 1.04928e14 0.130391
\(514\) −6.33938e14 −0.779381
\(515\) 5.04071e14 0.613130
\(516\) 3.61157e12 0.00434633
\(517\) −1.69300e15 −2.01585
\(518\) 1.76430e14 0.207854
\(519\) −4.19619e14 −0.489141
\(520\) 1.75242e14 0.202124
\(521\) −2.73245e14 −0.311850 −0.155925 0.987769i \(-0.549836\pi\)
−0.155925 + 0.987769i \(0.549836\pi\)
\(522\) −2.46556e14 −0.278437
\(523\) 1.52484e15 1.70398 0.851992 0.523555i \(-0.175394\pi\)
0.851992 + 0.523555i \(0.175394\pi\)
\(524\) 1.85639e13 0.0205280
\(525\) 1.05927e14 0.115913
\(526\) 8.22362e14 0.890516
\(527\) −1.87138e15 −2.00542
\(528\) 7.29086e14 0.773201
\(529\) −7.35790e14 −0.772231
\(530\) 1.26009e14 0.130884
\(531\) 2.85474e14 0.293459
\(532\) 4.13483e13 0.0420672
\(533\) −4.48086e14 −0.451192
\(534\) 6.46424e14 0.644229
\(535\) −9.78624e13 −0.0965317
\(536\) −1.35511e15 −1.32303
\(537\) 8.75585e13 0.0846135
\(538\) 1.02506e15 0.980500
\(539\) 2.49852e14 0.236562
\(540\) −2.30967e13 −0.0216464
\(541\) −4.85484e14 −0.450391 −0.225196 0.974314i \(-0.572302\pi\)
−0.225196 + 0.974314i \(0.572302\pi\)
\(542\) 1.61366e14 0.148189
\(543\) −1.05983e15 −0.963465
\(544\) −3.90940e14 −0.351816
\(545\) 9.31790e14 0.830113
\(546\) −6.27351e13 −0.0553288
\(547\) −8.96399e14 −0.782656 −0.391328 0.920251i \(-0.627984\pi\)
−0.391328 + 0.920251i \(0.627984\pi\)
\(548\) 1.84290e14 0.159297
\(549\) 7.42220e14 0.635162
\(550\) 9.49097e14 0.804110
\(551\) −7.38039e14 −0.619079
\(552\) 3.53133e14 0.293274
\(553\) −4.16331e13 −0.0342334
\(554\) 9.60020e14 0.781586
\(555\) −2.95006e14 −0.237804
\(556\) −1.34491e14 −0.107345
\(557\) −4.11768e14 −0.325424 −0.162712 0.986674i \(-0.552024\pi\)
−0.162712 + 0.986674i \(0.552024\pi\)
\(558\) 7.21432e14 0.564557
\(559\) −1.64026e13 −0.0127101
\(560\) 2.72772e14 0.209298
\(561\) −1.36203e15 −1.03488
\(562\) 1.92913e14 0.145148
\(563\) −1.33565e14 −0.0995166 −0.0497583 0.998761i \(-0.515845\pi\)
−0.0497583 + 0.998761i \(0.515845\pi\)
\(564\) −1.56478e14 −0.115456
\(565\) −1.76229e15 −1.28769
\(566\) 3.50154e14 0.253378
\(567\) 5.86024e13 0.0419961
\(568\) −4.26504e14 −0.302696
\(569\) 1.71605e15 1.20618 0.603092 0.797672i \(-0.293935\pi\)
0.603092 + 0.797672i \(0.293935\pi\)
\(570\) 3.51736e14 0.244853
\(571\) 1.39135e15 0.959266 0.479633 0.877469i \(-0.340770\pi\)
0.479633 + 0.877469i \(0.340770\pi\)
\(572\) 1.10487e14 0.0754456
\(573\) 1.23083e15 0.832429
\(574\) −8.39136e14 −0.562104
\(575\) 3.82086e14 0.253506
\(576\) 5.60926e14 0.368623
\(577\) −1.34680e15 −0.876669 −0.438335 0.898812i \(-0.644431\pi\)
−0.438335 + 0.898812i \(0.644431\pi\)
\(578\) −2.43449e14 −0.156966
\(579\) −4.38976e14 −0.280355
\(580\) 1.62456e14 0.102774
\(581\) −1.01153e15 −0.633886
\(582\) 1.32280e15 0.821140
\(583\) 5.63078e14 0.346251
\(584\) −1.25567e15 −0.764898
\(585\) 1.04898e14 0.0633010
\(586\) −2.81051e15 −1.68015
\(587\) 1.33148e15 0.788543 0.394272 0.918994i \(-0.370997\pi\)
0.394272 + 0.918994i \(0.370997\pi\)
\(588\) 2.30930e13 0.0135489
\(589\) 2.15953e15 1.25524
\(590\) 9.56952e14 0.551067
\(591\) 4.81655e14 0.274793
\(592\) 8.60708e14 0.486504
\(593\) −1.37881e15 −0.772151 −0.386075 0.922467i \(-0.626170\pi\)
−0.386075 + 0.922467i \(0.626170\pi\)
\(594\) 5.25072e14 0.291335
\(595\) −5.09573e14 −0.280132
\(596\) −1.06903e13 −0.00582284
\(597\) 1.71083e14 0.0923309
\(598\) −2.26289e14 −0.121006
\(599\) 1.75769e15 0.931310 0.465655 0.884966i \(-0.345819\pi\)
0.465655 + 0.884966i \(0.345819\pi\)
\(600\) 6.21726e14 0.326413
\(601\) 1.07241e15 0.557891 0.278946 0.960307i \(-0.410015\pi\)
0.278946 + 0.960307i \(0.410015\pi\)
\(602\) −3.07174e13 −0.0158345
\(603\) −8.11159e14 −0.414344
\(604\) −2.49765e14 −0.126424
\(605\) −2.37812e15 −1.19283
\(606\) −1.90840e15 −0.948573
\(607\) 2.49072e15 1.22684 0.613419 0.789757i \(-0.289794\pi\)
0.613419 + 0.789757i \(0.289794\pi\)
\(608\) 4.51135e14 0.220210
\(609\) −4.12194e14 −0.199392
\(610\) 2.48803e15 1.19273
\(611\) 7.10675e14 0.337632
\(612\) −1.25888e14 −0.0592721
\(613\) 3.02850e15 1.41317 0.706586 0.707627i \(-0.250234\pi\)
0.706586 + 0.707627i \(0.250234\pi\)
\(614\) 2.82503e15 1.30646
\(615\) 1.40310e15 0.643097
\(616\) 1.46647e15 0.666163
\(617\) −3.64507e15 −1.64111 −0.820555 0.571568i \(-0.806335\pi\)
−0.820555 + 0.571568i \(0.806335\pi\)
\(618\) −1.05915e15 −0.472629
\(619\) −2.76976e15 −1.22502 −0.612511 0.790462i \(-0.709840\pi\)
−0.612511 + 0.790462i \(0.709840\pi\)
\(620\) −4.75354e14 −0.208383
\(621\) 2.11382e14 0.0918470
\(622\) −3.74379e15 −1.61237
\(623\) 1.08070e15 0.461338
\(624\) −3.06051e14 −0.129503
\(625\) −4.45056e14 −0.186670
\(626\) 3.98366e15 1.65624
\(627\) 1.57175e15 0.647756
\(628\) −1.54819e14 −0.0632479
\(629\) −1.60791e15 −0.651155
\(630\) 1.96444e14 0.0788616
\(631\) 9.80104e14 0.390041 0.195021 0.980799i \(-0.437523\pi\)
0.195021 + 0.980799i \(0.437523\pi\)
\(632\) −2.44360e14 −0.0964021
\(633\) 1.68333e15 0.658338
\(634\) 2.09008e15 0.810348
\(635\) 1.54799e15 0.594995
\(636\) 5.20434e13 0.0198313
\(637\) −1.04881e14 −0.0396214
\(638\) −3.69322e15 −1.38322
\(639\) −2.55302e14 −0.0947980
\(640\) 1.27580e15 0.469671
\(641\) −3.08449e15 −1.12581 −0.562904 0.826522i \(-0.690316\pi\)
−0.562904 + 0.826522i \(0.690316\pi\)
\(642\) 2.05627e14 0.0744112
\(643\) −1.29084e15 −0.463138 −0.231569 0.972818i \(-0.574386\pi\)
−0.231569 + 0.972818i \(0.574386\pi\)
\(644\) 8.32977e13 0.0296320
\(645\) 5.13619e13 0.0181160
\(646\) 1.91712e15 0.670456
\(647\) −2.69945e15 −0.936055 −0.468027 0.883714i \(-0.655035\pi\)
−0.468027 + 0.883714i \(0.655035\pi\)
\(648\) 3.43959e14 0.118262
\(649\) 4.27618e15 1.45784
\(650\) −3.98405e14 −0.134680
\(651\) 1.20610e15 0.404284
\(652\) −1.63394e14 −0.0543094
\(653\) −3.57974e15 −1.17986 −0.589928 0.807456i \(-0.700844\pi\)
−0.589928 + 0.807456i \(0.700844\pi\)
\(654\) −1.95787e15 −0.639890
\(655\) 2.64006e14 0.0855632
\(656\) −4.09369e15 −1.31566
\(657\) −7.51631e14 −0.239549
\(658\) 1.33089e15 0.420629
\(659\) −4.16183e15 −1.30441 −0.652205 0.758043i \(-0.726156\pi\)
−0.652205 + 0.758043i \(0.726156\pi\)
\(660\) −3.45971e14 −0.107535
\(661\) −1.39894e15 −0.431211 −0.215606 0.976481i \(-0.569173\pi\)
−0.215606 + 0.976481i \(0.569173\pi\)
\(662\) −2.97173e14 −0.0908427
\(663\) 5.71742e14 0.173331
\(664\) −5.93706e15 −1.78504
\(665\) 5.88035e14 0.175341
\(666\) 6.19863e14 0.183310
\(667\) −1.48681e15 −0.436077
\(668\) −3.52313e14 −0.102485
\(669\) 2.57418e15 0.742669
\(670\) −2.71913e15 −0.778069
\(671\) 1.11179e16 3.15535
\(672\) 2.51958e14 0.0709247
\(673\) 1.50152e15 0.419226 0.209613 0.977784i \(-0.432780\pi\)
0.209613 + 0.977784i \(0.432780\pi\)
\(674\) −1.84861e14 −0.0511935
\(675\) 3.72160e14 0.102225
\(676\) −4.63795e13 −0.0126363
\(677\) 1.21220e15 0.327595 0.163797 0.986494i \(-0.447626\pi\)
0.163797 + 0.986494i \(0.447626\pi\)
\(678\) 3.70290e15 0.992610
\(679\) 2.21147e15 0.588026
\(680\) −2.99087e15 −0.788858
\(681\) −2.31489e15 −0.605650
\(682\) 1.08065e16 2.80460
\(683\) −2.40913e15 −0.620221 −0.310110 0.950701i \(-0.600366\pi\)
−0.310110 + 0.950701i \(0.600366\pi\)
\(684\) 1.45271e14 0.0370998
\(685\) 2.62088e15 0.663970
\(686\) −1.96412e14 −0.0493612
\(687\) −3.53724e14 −0.0881865
\(688\) −1.49854e14 −0.0370622
\(689\) −2.36365e14 −0.0579933
\(690\) 7.08586e14 0.172474
\(691\) −5.51584e15 −1.33193 −0.665966 0.745982i \(-0.731980\pi\)
−0.665966 + 0.745982i \(0.731980\pi\)
\(692\) −5.80953e14 −0.139174
\(693\) 8.77819e14 0.208628
\(694\) −2.80245e15 −0.660786
\(695\) −1.91266e15 −0.447426
\(696\) −2.41932e15 −0.561491
\(697\) 7.64754e15 1.76093
\(698\) 3.43448e14 0.0784614
\(699\) −7.62564e14 −0.172843
\(700\) 1.46654e14 0.0329803
\(701\) 4.04414e15 0.902355 0.451178 0.892434i \(-0.351004\pi\)
0.451178 + 0.892434i \(0.351004\pi\)
\(702\) −2.20411e14 −0.0487954
\(703\) 1.85549e15 0.407573
\(704\) 8.40225e15 1.83124
\(705\) −2.22535e15 −0.481237
\(706\) −8.08890e15 −1.73566
\(707\) −3.19048e15 −0.679281
\(708\) 3.95233e14 0.0834969
\(709\) 3.08515e15 0.646728 0.323364 0.946275i \(-0.395186\pi\)
0.323364 + 0.946275i \(0.395186\pi\)
\(710\) −8.55810e14 −0.178015
\(711\) −1.46272e14 −0.0301910
\(712\) 6.34301e15 1.29914
\(713\) 4.35046e15 0.884185
\(714\) 1.07071e15 0.215939
\(715\) 1.57129e15 0.314466
\(716\) 1.21223e14 0.0240748
\(717\) 1.05997e15 0.208899
\(718\) 7.04313e15 1.37747
\(719\) −6.93720e15 −1.34640 −0.673201 0.739459i \(-0.735081\pi\)
−0.673201 + 0.739459i \(0.735081\pi\)
\(720\) 9.58344e14 0.184583
\(721\) −1.77069e15 −0.338454
\(722\) 2.60703e15 0.494528
\(723\) −5.51613e14 −0.103842
\(724\) −1.46731e15 −0.274132
\(725\) −2.61768e15 −0.485352
\(726\) 4.99688e15 0.919491
\(727\) −7.17170e15 −1.30973 −0.654866 0.755745i \(-0.727275\pi\)
−0.654866 + 0.755745i \(0.727275\pi\)
\(728\) −6.15586e14 −0.111575
\(729\) 2.05891e14 0.0370370
\(730\) −2.51958e15 −0.449834
\(731\) 2.79946e14 0.0496053
\(732\) 1.02759e15 0.180721
\(733\) −2.16288e14 −0.0377539 −0.0188769 0.999822i \(-0.506009\pi\)
−0.0188769 + 0.999822i \(0.506009\pi\)
\(734\) −4.40160e15 −0.762575
\(735\) 3.28417e14 0.0564735
\(736\) 9.08829e14 0.155115
\(737\) −1.21505e16 −2.05837
\(738\) −2.94818e15 −0.495729
\(739\) 9.19704e14 0.153498 0.0767492 0.997050i \(-0.475546\pi\)
0.0767492 + 0.997050i \(0.475546\pi\)
\(740\) −4.08429e14 −0.0676616
\(741\) −6.59777e14 −0.108492
\(742\) −4.42644e14 −0.0722492
\(743\) −7.53135e15 −1.22021 −0.610105 0.792321i \(-0.708873\pi\)
−0.610105 + 0.792321i \(0.708873\pi\)
\(744\) 7.07903e15 1.13847
\(745\) −1.52032e14 −0.0242703
\(746\) 9.66483e15 1.53155
\(747\) −3.55388e15 −0.559035
\(748\) −1.88570e15 −0.294451
\(749\) 3.43770e14 0.0532865
\(750\) 3.59615e15 0.553352
\(751\) −1.09968e16 −1.67976 −0.839879 0.542774i \(-0.817374\pi\)
−0.839879 + 0.542774i \(0.817374\pi\)
\(752\) 6.49269e15 0.984524
\(753\) −1.59286e14 −0.0239775
\(754\) 1.55031e15 0.231674
\(755\) −3.55203e15 −0.526949
\(756\) 8.11338e13 0.0119490
\(757\) −8.13398e15 −1.18926 −0.594629 0.804000i \(-0.702701\pi\)
−0.594629 + 0.804000i \(0.702701\pi\)
\(758\) 1.88831e15 0.274089
\(759\) 3.16635e15 0.456277
\(760\) 3.45139e15 0.493765
\(761\) 1.85156e15 0.262980 0.131490 0.991317i \(-0.458024\pi\)
0.131490 + 0.991317i \(0.458024\pi\)
\(762\) −3.25263e15 −0.458651
\(763\) −3.27318e15 −0.458231
\(764\) 1.70406e15 0.236849
\(765\) −1.79031e15 −0.247053
\(766\) −6.10451e14 −0.0836359
\(767\) −1.79503e15 −0.244172
\(768\) 2.04677e15 0.276429
\(769\) 5.75342e15 0.771492 0.385746 0.922605i \(-0.373944\pi\)
0.385746 + 0.922605i \(0.373944\pi\)
\(770\) 2.94258e15 0.391768
\(771\) −3.72354e15 −0.492217
\(772\) −6.07753e14 −0.0797687
\(773\) −1.09432e16 −1.42613 −0.713064 0.701099i \(-0.752693\pi\)
−0.713064 + 0.701099i \(0.752693\pi\)
\(774\) −1.07921e14 −0.0139647
\(775\) 7.65943e15 0.984095
\(776\) 1.29799e16 1.65589
\(777\) 1.03629e15 0.131270
\(778\) 4.55693e15 0.573171
\(779\) −8.82508e15 −1.10221
\(780\) 1.45229e14 0.0180109
\(781\) −3.82423e15 −0.470937
\(782\) 3.86211e15 0.472267
\(783\) −1.44819e15 −0.175847
\(784\) −9.58188e14 −0.115535
\(785\) −2.20176e15 −0.263625
\(786\) −5.54727e14 −0.0659561
\(787\) −6.29593e13 −0.00743360 −0.00371680 0.999993i \(-0.501183\pi\)
−0.00371680 + 0.999993i \(0.501183\pi\)
\(788\) 6.66842e14 0.0781860
\(789\) 4.83028e15 0.562404
\(790\) −4.90326e14 −0.0566938
\(791\) 6.19054e15 0.710817
\(792\) 5.15225e15 0.587500
\(793\) −4.66699e15 −0.528487
\(794\) 8.56795e15 0.963528
\(795\) 7.40136e14 0.0826594
\(796\) 2.36860e14 0.0262706
\(797\) 1.66423e16 1.83312 0.916561 0.399895i \(-0.130953\pi\)
0.916561 + 0.399895i \(0.130953\pi\)
\(798\) −1.23557e15 −0.135161
\(799\) −1.21292e16 −1.31772
\(800\) 1.60008e15 0.172643
\(801\) 3.79687e15 0.406862
\(802\) −5.42555e15 −0.577411
\(803\) −1.12589e16 −1.19003
\(804\) −1.12303e15 −0.117892
\(805\) 1.18462e15 0.123510
\(806\) −4.53628e15 −0.469740
\(807\) 6.02086e15 0.619233
\(808\) −1.87261e16 −1.91287
\(809\) −4.83065e15 −0.490104 −0.245052 0.969510i \(-0.578805\pi\)
−0.245052 + 0.969510i \(0.578805\pi\)
\(810\) 6.90178e14 0.0695494
\(811\) 9.33002e15 0.933830 0.466915 0.884302i \(-0.345365\pi\)
0.466915 + 0.884302i \(0.345365\pi\)
\(812\) −5.70675e14 −0.0567323
\(813\) 9.47808e14 0.0935884
\(814\) 9.28507e15 0.910648
\(815\) −2.32371e15 −0.226368
\(816\) 5.22341e15 0.505427
\(817\) −3.23051e14 −0.0310491
\(818\) −4.88273e15 −0.466143
\(819\) −3.68485e14 −0.0349428
\(820\) 1.94257e15 0.182978
\(821\) 1.15132e16 1.07723 0.538617 0.842551i \(-0.318947\pi\)
0.538617 + 0.842551i \(0.318947\pi\)
\(822\) −5.50696e15 −0.511819
\(823\) 1.06942e16 0.987301 0.493651 0.869660i \(-0.335662\pi\)
0.493651 + 0.869660i \(0.335662\pi\)
\(824\) −1.03929e16 −0.953094
\(825\) 5.57467e15 0.507835
\(826\) −3.36157e15 −0.304195
\(827\) −1.42416e16 −1.28020 −0.640099 0.768292i \(-0.721107\pi\)
−0.640099 + 0.768292i \(0.721107\pi\)
\(828\) 2.92655e14 0.0261330
\(829\) −1.56225e16 −1.38580 −0.692899 0.721034i \(-0.743667\pi\)
−0.692899 + 0.721034i \(0.743667\pi\)
\(830\) −1.19131e16 −1.04978
\(831\) 5.63883e15 0.493610
\(832\) −3.52704e15 −0.306713
\(833\) 1.79002e15 0.154636
\(834\) 4.01886e15 0.344897
\(835\) −5.01043e15 −0.427169
\(836\) 2.17605e15 0.184304
\(837\) 4.23745e15 0.356545
\(838\) −6.94658e15 −0.580669
\(839\) −1.92662e16 −1.59995 −0.799975 0.600033i \(-0.795154\pi\)
−0.799975 + 0.600033i \(0.795154\pi\)
\(840\) 1.92760e15 0.159031
\(841\) −2.01434e15 −0.165103
\(842\) −1.81712e16 −1.47968
\(843\) 1.13310e15 0.0916679
\(844\) 2.33054e15 0.187315
\(845\) −6.59587e14 −0.0526696
\(846\) 4.67589e15 0.370960
\(847\) 8.35382e15 0.658455
\(848\) −2.15942e15 −0.169106
\(849\) 2.05668e15 0.160020
\(850\) 6.79964e15 0.525632
\(851\) 3.73797e15 0.287093
\(852\) −3.53460e14 −0.0269726
\(853\) −2.58227e16 −1.95786 −0.978929 0.204200i \(-0.934541\pi\)
−0.978929 + 0.204200i \(0.934541\pi\)
\(854\) −8.73992e15 −0.658399
\(855\) 2.06598e15 0.154636
\(856\) 2.01771e15 0.150056
\(857\) −1.21194e16 −0.895542 −0.447771 0.894148i \(-0.647782\pi\)
−0.447771 + 0.894148i \(0.647782\pi\)
\(858\) −3.30159e15 −0.242406
\(859\) −1.96206e16 −1.43137 −0.715683 0.698426i \(-0.753884\pi\)
−0.715683 + 0.698426i \(0.753884\pi\)
\(860\) 7.11096e13 0.00515450
\(861\) −4.92880e15 −0.354996
\(862\) −9.36679e14 −0.0670349
\(863\) −1.94564e16 −1.38357 −0.691787 0.722102i \(-0.743176\pi\)
−0.691787 + 0.722102i \(0.743176\pi\)
\(864\) 8.85220e14 0.0625497
\(865\) −8.26204e15 −0.580094
\(866\) −1.48452e16 −1.03571
\(867\) −1.42993e15 −0.0991314
\(868\) 1.66982e15 0.115030
\(869\) −2.19104e15 −0.149983
\(870\) −4.85454e15 −0.330211
\(871\) 5.10047e15 0.344755
\(872\) −1.92115e16 −1.29039
\(873\) 7.76967e15 0.518590
\(874\) −4.45679e15 −0.295603
\(875\) 6.01208e15 0.396260
\(876\) −1.04062e15 −0.0681583
\(877\) −2.63035e16 −1.71205 −0.856024 0.516937i \(-0.827072\pi\)
−0.856024 + 0.516937i \(0.827072\pi\)
\(878\) −1.18195e16 −0.764500
\(879\) −1.65080e16 −1.06110
\(880\) 1.43553e16 0.916973
\(881\) 1.47217e16 0.934525 0.467262 0.884119i \(-0.345240\pi\)
0.467262 + 0.884119i \(0.345240\pi\)
\(882\) −6.90066e14 −0.0435325
\(883\) −2.32000e16 −1.45447 −0.727234 0.686389i \(-0.759195\pi\)
−0.727234 + 0.686389i \(0.759195\pi\)
\(884\) 7.91566e14 0.0493173
\(885\) 5.62081e15 0.348025
\(886\) 2.76641e16 1.70228
\(887\) 2.26901e16 1.38758 0.693789 0.720178i \(-0.255940\pi\)
0.693789 + 0.720178i \(0.255940\pi\)
\(888\) 6.08238e15 0.369660
\(889\) −5.43777e15 −0.328444
\(890\) 1.27277e16 0.764020
\(891\) 3.08409e15 0.183992
\(892\) 3.56390e15 0.211310
\(893\) 1.39968e16 0.824794
\(894\) 3.19448e14 0.0187087
\(895\) 1.72397e15 0.100347
\(896\) −4.48162e15 −0.259264
\(897\) −1.32915e15 −0.0764213
\(898\) 2.76377e15 0.157936
\(899\) −2.98051e16 −1.69283
\(900\) 5.15248e14 0.0290859
\(901\) 4.03407e15 0.226338
\(902\) −4.41615e16 −2.46268
\(903\) −1.80423e14 −0.0100002
\(904\) 3.63346e16 2.00168
\(905\) −2.08673e16 −1.14262
\(906\) 7.46349e15 0.406197
\(907\) 1.59637e16 0.863564 0.431782 0.901978i \(-0.357885\pi\)
0.431782 + 0.901978i \(0.357885\pi\)
\(908\) −3.20492e15 −0.172324
\(909\) −1.12093e16 −0.599070
\(910\) −1.23522e15 −0.0656169
\(911\) 3.71587e16 1.96205 0.981023 0.193891i \(-0.0621107\pi\)
0.981023 + 0.193891i \(0.0621107\pi\)
\(912\) −6.02769e15 −0.316358
\(913\) −5.32344e16 −2.77717
\(914\) −4.69940e15 −0.243690
\(915\) 1.46139e16 0.753267
\(916\) −4.89724e14 −0.0250915
\(917\) −9.27397e14 −0.0472318
\(918\) 3.76178e15 0.190440
\(919\) −2.44138e16 −1.22857 −0.614286 0.789083i \(-0.710556\pi\)
−0.614286 + 0.789083i \(0.710556\pi\)
\(920\) 6.95297e15 0.347806
\(921\) 1.65933e16 0.825094
\(922\) 1.97610e16 0.976763
\(923\) 1.60531e15 0.0788767
\(924\) 1.21532e15 0.0593602
\(925\) 6.58107e15 0.319534
\(926\) −3.23447e16 −1.56114
\(927\) −6.22108e15 −0.298488
\(928\) −6.22641e15 −0.296978
\(929\) −3.17884e16 −1.50724 −0.753621 0.657309i \(-0.771694\pi\)
−0.753621 + 0.657309i \(0.771694\pi\)
\(930\) 1.42046e16 0.669533
\(931\) −2.06564e15 −0.0967902
\(932\) −1.05576e15 −0.0491785
\(933\) −2.19898e16 −1.01829
\(934\) −1.00950e16 −0.464727
\(935\) −2.68175e16 −1.22731
\(936\) −2.16277e15 −0.0983997
\(937\) −2.07294e16 −0.937602 −0.468801 0.883304i \(-0.655314\pi\)
−0.468801 + 0.883304i \(0.655314\pi\)
\(938\) 9.55171e15 0.429502
\(939\) 2.33987e16 1.04600
\(940\) −3.08096e15 −0.136925
\(941\) −1.43488e16 −0.633977 −0.316988 0.948429i \(-0.602672\pi\)
−0.316988 + 0.948429i \(0.602672\pi\)
\(942\) 4.62632e15 0.203215
\(943\) −1.77785e16 −0.776391
\(944\) −1.63993e16 −0.711998
\(945\) 1.15385e15 0.0498050
\(946\) −1.61658e15 −0.0693737
\(947\) 2.54468e15 0.108569 0.0542847 0.998525i \(-0.482712\pi\)
0.0542847 + 0.998525i \(0.482712\pi\)
\(948\) −2.02511e14 −0.00859017
\(949\) 4.72617e15 0.199317
\(950\) −7.84662e15 −0.329005
\(951\) 1.22764e16 0.511774
\(952\) 1.05063e16 0.435458
\(953\) 3.97221e16 1.63690 0.818449 0.574579i \(-0.194834\pi\)
0.818449 + 0.574579i \(0.194834\pi\)
\(954\) −1.55517e15 −0.0637178
\(955\) 2.42343e16 0.987215
\(956\) 1.46750e15 0.0594374
\(957\) −2.16927e16 −0.873571
\(958\) 1.84146e16 0.737315
\(959\) −9.20657e15 −0.366518
\(960\) 1.10443e16 0.437166
\(961\) 6.18025e16 2.43236
\(962\) −3.89762e15 −0.152523
\(963\) 1.20779e15 0.0469943
\(964\) −7.63697e14 −0.0295459
\(965\) −8.64317e15 −0.332486
\(966\) −2.48911e15 −0.0952072
\(967\) 3.01593e16 1.14703 0.573516 0.819194i \(-0.305579\pi\)
0.573516 + 0.819194i \(0.305579\pi\)
\(968\) 4.90317e16 1.85422
\(969\) 1.12605e16 0.423426
\(970\) 2.60451e16 0.973826
\(971\) 1.45389e16 0.540536 0.270268 0.962785i \(-0.412888\pi\)
0.270268 + 0.962785i \(0.412888\pi\)
\(972\) 2.85052e14 0.0105380
\(973\) 6.71876e15 0.246984
\(974\) −8.93178e14 −0.0326485
\(975\) −2.34010e15 −0.0850567
\(976\) −4.26374e16 −1.54105
\(977\) −9.38345e15 −0.337243 −0.168621 0.985681i \(-0.553932\pi\)
−0.168621 + 0.985681i \(0.553932\pi\)
\(978\) 4.88256e15 0.174495
\(979\) 5.68743e16 2.02121
\(980\) 4.54686e14 0.0160682
\(981\) −1.14998e16 −0.404122
\(982\) −2.12364e16 −0.742107
\(983\) −2.35900e16 −0.819753 −0.409876 0.912141i \(-0.634428\pi\)
−0.409876 + 0.912141i \(0.634428\pi\)
\(984\) −2.89289e16 −0.999677
\(985\) 9.48350e15 0.325889
\(986\) −2.64594e16 −0.904185
\(987\) 7.81719e15 0.265648
\(988\) −9.13448e14 −0.0308689
\(989\) −6.50799e14 −0.0218709
\(990\) 1.03383e16 0.345507
\(991\) −1.80350e16 −0.599391 −0.299696 0.954035i \(-0.596885\pi\)
−0.299696 + 0.954035i \(0.596885\pi\)
\(992\) 1.82187e16 0.602149
\(993\) −1.74549e15 −0.0573716
\(994\) 3.00628e15 0.0982662
\(995\) 3.36851e15 0.109499
\(996\) −4.92027e15 −0.159061
\(997\) −5.32014e16 −1.71041 −0.855203 0.518293i \(-0.826568\pi\)
−0.855203 + 0.518293i \(0.826568\pi\)
\(998\) 2.65630e15 0.0849298
\(999\) 3.64086e15 0.115769
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.1.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.12.a.c.1.5 16 1.1 even 1 trivial