Properties

Label 273.12.a.c.1.1
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 1.1
Root \(-84.0758\) of defining polynomial
Character \(\chi\) \(=\) 273.1

$q$-expansion

\(f(q)\) \(=\) \(q-88.0758 q^{2} -243.000 q^{3} +5709.35 q^{4} -13401.9 q^{5} +21402.4 q^{6} +16807.0 q^{7} -322476. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-88.0758 q^{2} -243.000 q^{3} +5709.35 q^{4} -13401.9 q^{5} +21402.4 q^{6} +16807.0 q^{7} -322476. q^{8} +59049.0 q^{9} +1.18039e6 q^{10} +966243. q^{11} -1.38737e6 q^{12} -371293. q^{13} -1.48029e6 q^{14} +3.25667e6 q^{15} +1.67096e7 q^{16} -3.26911e6 q^{17} -5.20079e6 q^{18} +7.04693e6 q^{19} -7.65164e7 q^{20} -4.08410e6 q^{21} -8.51026e7 q^{22} -2.58288e7 q^{23} +7.83618e7 q^{24} +1.30784e8 q^{25} +3.27019e7 q^{26} -1.43489e7 q^{27} +9.59571e7 q^{28} -9.40200e7 q^{29} -2.86834e8 q^{30} +1.08890e8 q^{31} -8.11283e8 q^{32} -2.34797e8 q^{33} +2.87929e8 q^{34} -2.25247e8 q^{35} +3.37131e8 q^{36} -5.39687e8 q^{37} -6.20665e8 q^{38} +9.02242e7 q^{39} +4.32181e9 q^{40} +6.21045e8 q^{41} +3.59711e8 q^{42} +2.48281e8 q^{43} +5.51662e9 q^{44} -7.91372e8 q^{45} +2.27489e9 q^{46} +1.09250e8 q^{47} -4.06044e9 q^{48} +2.82475e8 q^{49} -1.15189e10 q^{50} +7.94393e8 q^{51} -2.11984e9 q^{52} +3.34527e8 q^{53} +1.26379e9 q^{54} -1.29495e10 q^{55} -5.41986e9 q^{56} -1.71240e9 q^{57} +8.28089e9 q^{58} -7.67466e9 q^{59} +1.85935e10 q^{60} -1.30287e9 q^{61} -9.59061e9 q^{62} +9.92437e8 q^{63} +3.72331e10 q^{64} +4.97605e9 q^{65} +2.06799e10 q^{66} -1.72145e10 q^{67} -1.86645e10 q^{68} +6.27640e9 q^{69} +1.98388e10 q^{70} -2.59950e10 q^{71} -1.90419e10 q^{72} +9.41694e9 q^{73} +4.75334e10 q^{74} -3.17805e10 q^{75} +4.02334e10 q^{76} +1.62396e10 q^{77} -7.94657e9 q^{78} -1.58309e10 q^{79} -2.23942e11 q^{80} +3.48678e9 q^{81} -5.46991e10 q^{82} +1.74906e10 q^{83} -2.33176e10 q^{84} +4.38124e10 q^{85} -2.18676e10 q^{86} +2.28469e10 q^{87} -3.11590e11 q^{88} +3.74891e10 q^{89} +6.97007e10 q^{90} -6.24032e9 q^{91} -1.47466e11 q^{92} -2.64604e10 q^{93} -9.62226e9 q^{94} -9.44426e10 q^{95} +1.97142e11 q^{96} -3.78023e10 q^{97} -2.48792e10 q^{98} +5.70557e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 63 q^{2} - 3888 q^{3} + 20761 q^{4} - 3168 q^{5} + 15309 q^{6} + 268912 q^{7} - 187965 q^{8} + 944784 q^{9} + O(q^{10}) \) \( 16 q - 63 q^{2} - 3888 q^{3} + 20761 q^{4} - 3168 q^{5} + 15309 q^{6} + 268912 q^{7} - 187965 q^{8} + 944784 q^{9} + 1056711 q^{10} - 466884 q^{11} - 5044923 q^{12} - 5940688 q^{13} - 1058841 q^{14} + 769824 q^{15} + 40058265 q^{16} + 1753452 q^{17} - 3720087 q^{18} + 4237800 q^{19} - 20710503 q^{20} - 65345616 q^{21} - 37479661 q^{22} - 150481440 q^{23} + 45675495 q^{24} + 150983176 q^{25} + 23391459 q^{26} - 229582512 q^{27} + 348930127 q^{28} - 111411432 q^{29} - 256780773 q^{30} - 90536236 q^{31} - 609941313 q^{32} + 113452812 q^{33} + 443745577 q^{34} - 53244576 q^{35} + 1225916289 q^{36} - 1756337900 q^{37} - 4378868973 q^{38} + 1443587184 q^{39} + 57535383 q^{40} + 649575720 q^{41} + 257298363 q^{42} - 1889554520 q^{43} - 4979587689 q^{44} - 187067232 q^{45} - 3204000867 q^{46} - 4036082940 q^{47} - 9734158395 q^{48} + 4519603984 q^{49} - 15928886862 q^{50} - 426088836 q^{51} - 7708413973 q^{52} + 511144020 q^{53} + 903981141 q^{54} - 8519365572 q^{55} - 3159127755 q^{56} - 1029785400 q^{57} - 7851149577 q^{58} + 3728287296 q^{59} + 5032652229 q^{60} + 26227205052 q^{61} + 2996618490 q^{62} + 15878984688 q^{63} + 51104408465 q^{64} + 1176256224 q^{65} + 9107557623 q^{66} - 5295680024 q^{67} + 7919540325 q^{68} + 36566989920 q^{69} + 17760141777 q^{70} + 17082800928 q^{71} - 11099145285 q^{72} + 21076301488 q^{73} + 52794333675 q^{74} - 36688911768 q^{75} + 81459179177 q^{76} - 7846919388 q^{77} - 5684124537 q^{78} - 83677977852 q^{79} - 163988465427 q^{80} + 55788550416 q^{81} - 61543728760 q^{82} - 72857072340 q^{83} - 84790020861 q^{84} - 12608565060 q^{85} - 20177818011 q^{86} + 27072977976 q^{87} - 202213679643 q^{88} + 32238476676 q^{89} + 62397727839 q^{90} - 99845143216 q^{91} - 487057197033 q^{92} + 22000305348 q^{93} + 183904776602 q^{94} - 143022915504 q^{95} + 148215739059 q^{96} + 6973535140 q^{97} - 17795940687 q^{98} - 27569033316 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −88.0758 −1.94622 −0.973110 0.230343i \(-0.926015\pi\)
−0.973110 + 0.230343i \(0.926015\pi\)
\(3\) −243.000 −0.577350
\(4\) 5709.35 2.78777
\(5\) −13401.9 −1.91793 −0.958965 0.283523i \(-0.908497\pi\)
−0.958965 + 0.283523i \(0.908497\pi\)
\(6\) 21402.4 1.12365
\(7\) 16807.0 0.377964
\(8\) −322476. −3.47939
\(9\) 59049.0 0.333333
\(10\) 1.18039e6 3.73271
\(11\) 966243. 1.80895 0.904475 0.426527i \(-0.140263\pi\)
0.904475 + 0.426527i \(0.140263\pi\)
\(12\) −1.38737e6 −1.60952
\(13\) −371293. −0.277350
\(14\) −1.48029e6 −0.735602
\(15\) 3.25667e6 1.10732
\(16\) 1.67096e7 3.98389
\(17\) −3.26911e6 −0.558419 −0.279209 0.960230i \(-0.590072\pi\)
−0.279209 + 0.960230i \(0.590072\pi\)
\(18\) −5.20079e6 −0.648740
\(19\) 7.04693e6 0.652913 0.326457 0.945212i \(-0.394145\pi\)
0.326457 + 0.945212i \(0.394145\pi\)
\(20\) −7.65164e7 −5.34675
\(21\) −4.08410e6 −0.218218
\(22\) −8.51026e7 −3.52061
\(23\) −2.58288e7 −0.836761 −0.418380 0.908272i \(-0.637402\pi\)
−0.418380 + 0.908272i \(0.637402\pi\)
\(24\) 7.83618e7 2.00883
\(25\) 1.30784e8 2.67846
\(26\) 3.27019e7 0.539784
\(27\) −1.43489e7 −0.192450
\(28\) 9.59571e7 1.05368
\(29\) −9.40200e7 −0.851199 −0.425600 0.904912i \(-0.639937\pi\)
−0.425600 + 0.904912i \(0.639937\pi\)
\(30\) −2.86834e8 −2.15508
\(31\) 1.08890e8 0.683125 0.341563 0.939859i \(-0.389044\pi\)
0.341563 + 0.939859i \(0.389044\pi\)
\(32\) −8.11283e8 −4.27413
\(33\) −2.34797e8 −1.04440
\(34\) 2.87929e8 1.08680
\(35\) −2.25247e8 −0.724910
\(36\) 3.37131e8 0.929256
\(37\) −5.39687e8 −1.27948 −0.639739 0.768592i \(-0.720958\pi\)
−0.639739 + 0.768592i \(0.720958\pi\)
\(38\) −6.20665e8 −1.27071
\(39\) 9.02242e7 0.160128
\(40\) 4.32181e9 6.67323
\(41\) 6.21045e8 0.837167 0.418584 0.908178i \(-0.362527\pi\)
0.418584 + 0.908178i \(0.362527\pi\)
\(42\) 3.59711e8 0.424700
\(43\) 2.48281e8 0.257554 0.128777 0.991674i \(-0.458895\pi\)
0.128777 + 0.991674i \(0.458895\pi\)
\(44\) 5.51662e9 5.04293
\(45\) −7.91372e8 −0.639310
\(46\) 2.27489e9 1.62852
\(47\) 1.09250e8 0.0694835 0.0347418 0.999396i \(-0.488939\pi\)
0.0347418 + 0.999396i \(0.488939\pi\)
\(48\) −4.06044e9 −2.30010
\(49\) 2.82475e8 0.142857
\(50\) −1.15189e10 −5.21287
\(51\) 7.94393e8 0.322403
\(52\) −2.11984e9 −0.773188
\(53\) 3.34527e8 0.109879 0.0549394 0.998490i \(-0.482503\pi\)
0.0549394 + 0.998490i \(0.482503\pi\)
\(54\) 1.26379e9 0.374550
\(55\) −1.29495e10 −3.46944
\(56\) −5.41986e9 −1.31509
\(57\) −1.71240e9 −0.376960
\(58\) 8.28089e9 1.65662
\(59\) −7.67466e9 −1.39757 −0.698784 0.715333i \(-0.746275\pi\)
−0.698784 + 0.715333i \(0.746275\pi\)
\(60\) 1.85935e10 3.08695
\(61\) −1.30287e9 −0.197508 −0.0987542 0.995112i \(-0.531486\pi\)
−0.0987542 + 0.995112i \(0.531486\pi\)
\(62\) −9.59061e9 −1.32951
\(63\) 9.92437e8 0.125988
\(64\) 3.72331e10 4.33450
\(65\) 4.97605e9 0.531938
\(66\) 2.06799e10 2.03263
\(67\) −1.72145e10 −1.55770 −0.778849 0.627211i \(-0.784196\pi\)
−0.778849 + 0.627211i \(0.784196\pi\)
\(68\) −1.86645e10 −1.55674
\(69\) 6.27640e9 0.483104
\(70\) 1.98388e10 1.41083
\(71\) −2.59950e10 −1.70989 −0.854947 0.518715i \(-0.826410\pi\)
−0.854947 + 0.518715i \(0.826410\pi\)
\(72\) −1.90419e10 −1.15980
\(73\) 9.41694e9 0.531660 0.265830 0.964020i \(-0.414354\pi\)
0.265830 + 0.964020i \(0.414354\pi\)
\(74\) 4.75334e10 2.49014
\(75\) −3.17805e10 −1.54641
\(76\) 4.02334e10 1.82017
\(77\) 1.62396e10 0.683719
\(78\) −7.94657e9 −0.311644
\(79\) −1.58309e10 −0.578837 −0.289419 0.957203i \(-0.593462\pi\)
−0.289419 + 0.957203i \(0.593462\pi\)
\(80\) −2.23942e11 −7.64082
\(81\) 3.48678e9 0.111111
\(82\) −5.46991e10 −1.62931
\(83\) 1.74906e10 0.487387 0.243694 0.969852i \(-0.421641\pi\)
0.243694 + 0.969852i \(0.421641\pi\)
\(84\) −2.33176e10 −0.608341
\(85\) 4.38124e10 1.07101
\(86\) −2.18676e10 −0.501256
\(87\) 2.28469e10 0.491440
\(88\) −3.11590e11 −6.29404
\(89\) 3.74891e10 0.711640 0.355820 0.934554i \(-0.384202\pi\)
0.355820 + 0.934554i \(0.384202\pi\)
\(90\) 6.97007e10 1.24424
\(91\) −6.24032e9 −0.104828
\(92\) −1.47466e11 −2.33270
\(93\) −2.64604e10 −0.394402
\(94\) −9.62226e9 −0.135230
\(95\) −9.44426e10 −1.25224
\(96\) 1.97142e11 2.46767
\(97\) −3.78023e10 −0.446965 −0.223483 0.974708i \(-0.571743\pi\)
−0.223483 + 0.974708i \(0.571743\pi\)
\(98\) −2.48792e10 −0.278031
\(99\) 5.70557e10 0.602983
\(100\) 7.46692e11 7.46692
\(101\) 1.36161e11 1.28910 0.644548 0.764564i \(-0.277045\pi\)
0.644548 + 0.764564i \(0.277045\pi\)
\(102\) −6.99668e10 −0.627467
\(103\) 7.29756e10 0.620259 0.310130 0.950694i \(-0.399628\pi\)
0.310130 + 0.950694i \(0.399628\pi\)
\(104\) 1.19733e11 0.965009
\(105\) 5.47349e10 0.418527
\(106\) −2.94637e10 −0.213848
\(107\) 2.71362e11 1.87042 0.935208 0.354098i \(-0.115212\pi\)
0.935208 + 0.354098i \(0.115212\pi\)
\(108\) −8.19229e10 −0.536506
\(109\) 6.06812e9 0.0377753 0.0188877 0.999822i \(-0.493988\pi\)
0.0188877 + 0.999822i \(0.493988\pi\)
\(110\) 1.14054e12 6.75229
\(111\) 1.31144e11 0.738707
\(112\) 2.80839e11 1.50577
\(113\) −3.32091e11 −1.69561 −0.847805 0.530308i \(-0.822076\pi\)
−0.847805 + 0.530308i \(0.822076\pi\)
\(114\) 1.50821e11 0.733646
\(115\) 3.46156e11 1.60485
\(116\) −5.36793e11 −2.37295
\(117\) −2.19245e10 −0.0924500
\(118\) 6.75952e11 2.71997
\(119\) −5.49439e10 −0.211062
\(120\) −1.05020e12 −3.85279
\(121\) 6.48313e11 2.27230
\(122\) 1.14751e11 0.384395
\(123\) −1.50914e11 −0.483339
\(124\) 6.21693e11 1.90439
\(125\) −1.09837e12 −3.21917
\(126\) −8.74097e10 −0.245201
\(127\) 2.44398e11 0.656412 0.328206 0.944606i \(-0.393556\pi\)
0.328206 + 0.944606i \(0.393556\pi\)
\(128\) −1.61783e12 −4.16176
\(129\) −6.03323e10 −0.148699
\(130\) −4.38270e11 −1.03527
\(131\) 6.62940e11 1.50135 0.750675 0.660672i \(-0.229729\pi\)
0.750675 + 0.660672i \(0.229729\pi\)
\(132\) −1.34054e12 −2.91154
\(133\) 1.18438e11 0.246778
\(134\) 1.51618e12 3.03162
\(135\) 1.92303e11 0.369106
\(136\) 1.05421e12 1.94296
\(137\) −7.76673e11 −1.37491 −0.687456 0.726226i \(-0.741273\pi\)
−0.687456 + 0.726226i \(0.741273\pi\)
\(138\) −5.52799e11 −0.940226
\(139\) 7.23049e11 1.18192 0.590958 0.806703i \(-0.298750\pi\)
0.590958 + 0.806703i \(0.298750\pi\)
\(140\) −1.28601e12 −2.02088
\(141\) −2.65477e10 −0.0401163
\(142\) 2.28953e12 3.32783
\(143\) −3.58759e11 −0.501712
\(144\) 9.86687e11 1.32796
\(145\) 1.26005e12 1.63254
\(146\) −8.29404e11 −1.03473
\(147\) −6.86415e10 −0.0824786
\(148\) −3.08126e12 −3.56689
\(149\) 6.93308e11 0.773395 0.386698 0.922207i \(-0.373616\pi\)
0.386698 + 0.922207i \(0.373616\pi\)
\(150\) 2.79910e12 3.00965
\(151\) −7.94031e11 −0.823122 −0.411561 0.911382i \(-0.635016\pi\)
−0.411561 + 0.911382i \(0.635016\pi\)
\(152\) −2.27247e12 −2.27174
\(153\) −1.93037e11 −0.186140
\(154\) −1.43032e12 −1.33067
\(155\) −1.45934e12 −1.31019
\(156\) 5.15122e11 0.446400
\(157\) 1.24474e12 1.04143 0.520716 0.853730i \(-0.325665\pi\)
0.520716 + 0.853730i \(0.325665\pi\)
\(158\) 1.39432e12 1.12654
\(159\) −8.12900e10 −0.0634385
\(160\) 1.08728e13 8.19748
\(161\) −4.34105e11 −0.316266
\(162\) −3.07101e11 −0.216247
\(163\) 1.59796e12 1.08777 0.543883 0.839161i \(-0.316954\pi\)
0.543883 + 0.839161i \(0.316954\pi\)
\(164\) 3.54577e12 2.33383
\(165\) 3.14674e12 2.00308
\(166\) −1.54050e12 −0.948563
\(167\) 3.20978e12 1.91221 0.956103 0.293031i \(-0.0946640\pi\)
0.956103 + 0.293031i \(0.0946640\pi\)
\(168\) 1.31703e12 0.759265
\(169\) 1.37858e11 0.0769231
\(170\) −3.85881e12 −2.08442
\(171\) 4.16114e11 0.217638
\(172\) 1.41752e12 0.718000
\(173\) −2.68266e12 −1.31617 −0.658084 0.752945i \(-0.728633\pi\)
−0.658084 + 0.752945i \(0.728633\pi\)
\(174\) −2.01226e12 −0.956450
\(175\) 2.19809e12 1.01236
\(176\) 1.61456e13 7.20665
\(177\) 1.86494e12 0.806887
\(178\) −3.30189e12 −1.38501
\(179\) −1.42958e12 −0.581455 −0.290728 0.956806i \(-0.593897\pi\)
−0.290728 + 0.956806i \(0.593897\pi\)
\(180\) −4.51822e12 −1.78225
\(181\) −4.83814e12 −1.85117 −0.925584 0.378542i \(-0.876425\pi\)
−0.925584 + 0.378542i \(0.876425\pi\)
\(182\) 5.49621e11 0.204019
\(183\) 3.16597e11 0.114032
\(184\) 8.32918e12 2.91142
\(185\) 7.23286e12 2.45395
\(186\) 2.33052e12 0.767594
\(187\) −3.15875e12 −1.01015
\(188\) 6.23745e11 0.193704
\(189\) −2.41162e11 −0.0727393
\(190\) 8.31811e12 2.43714
\(191\) 3.48943e11 0.0993279 0.0496639 0.998766i \(-0.484185\pi\)
0.0496639 + 0.998766i \(0.484185\pi\)
\(192\) −9.04763e12 −2.50252
\(193\) −2.28986e12 −0.615522 −0.307761 0.951464i \(-0.599580\pi\)
−0.307761 + 0.951464i \(0.599580\pi\)
\(194\) 3.32947e12 0.869892
\(195\) −1.20918e12 −0.307115
\(196\) 1.61275e12 0.398253
\(197\) 2.35434e11 0.0565333 0.0282667 0.999600i \(-0.491001\pi\)
0.0282667 + 0.999600i \(0.491001\pi\)
\(198\) −5.02522e12 −1.17354
\(199\) 5.99332e12 1.36137 0.680684 0.732578i \(-0.261683\pi\)
0.680684 + 0.732578i \(0.261683\pi\)
\(200\) −4.21748e13 −9.31940
\(201\) 4.18313e12 0.899338
\(202\) −1.19925e13 −2.50886
\(203\) −1.58019e12 −0.321723
\(204\) 4.53547e12 0.898785
\(205\) −8.32322e12 −1.60563
\(206\) −6.42739e12 −1.20716
\(207\) −1.52517e12 −0.278920
\(208\) −6.20417e12 −1.10493
\(209\) 6.80905e12 1.18109
\(210\) −4.82082e12 −0.814545
\(211\) 1.12195e13 1.84680 0.923398 0.383843i \(-0.125400\pi\)
0.923398 + 0.383843i \(0.125400\pi\)
\(212\) 1.90993e12 0.306317
\(213\) 6.31679e12 0.987208
\(214\) −2.39004e13 −3.64024
\(215\) −3.32745e12 −0.493970
\(216\) 4.62718e12 0.669609
\(217\) 1.83012e12 0.258197
\(218\) −5.34455e11 −0.0735191
\(219\) −2.28832e12 −0.306954
\(220\) −7.39334e13 −9.67199
\(221\) 1.21380e12 0.154877
\(222\) −1.15506e13 −1.43769
\(223\) −1.04277e13 −1.26623 −0.633115 0.774058i \(-0.718224\pi\)
−0.633115 + 0.774058i \(0.718224\pi\)
\(224\) −1.36352e13 −1.61547
\(225\) 7.72267e12 0.892819
\(226\) 2.92492e13 3.30003
\(227\) −3.75088e12 −0.413039 −0.206519 0.978443i \(-0.566214\pi\)
−0.206519 + 0.978443i \(0.566214\pi\)
\(228\) −9.77672e12 −1.05088
\(229\) 9.22961e12 0.968474 0.484237 0.874937i \(-0.339097\pi\)
0.484237 + 0.874937i \(0.339097\pi\)
\(230\) −3.04880e13 −3.12339
\(231\) −3.94623e12 −0.394745
\(232\) 3.03192e13 2.96165
\(233\) −8.78175e12 −0.837767 −0.418884 0.908040i \(-0.637579\pi\)
−0.418884 + 0.908040i \(0.637579\pi\)
\(234\) 1.93102e12 0.179928
\(235\) −1.46416e12 −0.133265
\(236\) −4.38173e13 −3.89610
\(237\) 3.84691e12 0.334192
\(238\) 4.83923e12 0.410774
\(239\) 7.03033e12 0.583159 0.291580 0.956547i \(-0.405819\pi\)
0.291580 + 0.956547i \(0.405819\pi\)
\(240\) 5.44178e13 4.41143
\(241\) 1.59569e12 0.126431 0.0632156 0.998000i \(-0.479864\pi\)
0.0632156 + 0.998000i \(0.479864\pi\)
\(242\) −5.71007e13 −4.42239
\(243\) −8.47289e11 −0.0641500
\(244\) −7.43852e12 −0.550608
\(245\) −3.78572e12 −0.273990
\(246\) 1.32919e13 0.940683
\(247\) −2.61648e12 −0.181086
\(248\) −3.51146e13 −2.37686
\(249\) −4.25021e12 −0.281393
\(250\) 9.67398e13 6.26520
\(251\) 3.17410e12 0.201101 0.100551 0.994932i \(-0.467940\pi\)
0.100551 + 0.994932i \(0.467940\pi\)
\(252\) 5.66617e12 0.351226
\(253\) −2.49569e13 −1.51366
\(254\) −2.15255e13 −1.27752
\(255\) −1.06464e13 −0.618347
\(256\) 6.62380e13 3.76520
\(257\) −1.59775e12 −0.0888949 −0.0444474 0.999012i \(-0.514153\pi\)
−0.0444474 + 0.999012i \(0.514153\pi\)
\(258\) 5.31382e12 0.289400
\(259\) −9.07053e12 −0.483597
\(260\) 2.84100e13 1.48292
\(261\) −5.55179e12 −0.283733
\(262\) −5.83889e13 −2.92196
\(263\) 2.51476e13 1.23237 0.616184 0.787602i \(-0.288678\pi\)
0.616184 + 0.787602i \(0.288678\pi\)
\(264\) 7.57165e13 3.63387
\(265\) −4.48331e12 −0.210740
\(266\) −1.04315e13 −0.480284
\(267\) −9.10986e12 −0.410866
\(268\) −9.82837e13 −4.34250
\(269\) 2.66015e13 1.15151 0.575756 0.817621i \(-0.304708\pi\)
0.575756 + 0.817621i \(0.304708\pi\)
\(270\) −1.69373e13 −0.718361
\(271\) −4.43812e12 −0.184446 −0.0922228 0.995738i \(-0.529397\pi\)
−0.0922228 + 0.995738i \(0.529397\pi\)
\(272\) −5.46255e13 −2.22468
\(273\) 1.51640e12 0.0605228
\(274\) 6.84061e13 2.67588
\(275\) 1.26369e14 4.84519
\(276\) 3.58342e13 1.34678
\(277\) 3.22412e13 1.18788 0.593939 0.804510i \(-0.297572\pi\)
0.593939 + 0.804510i \(0.297572\pi\)
\(278\) −6.36831e13 −2.30027
\(279\) 6.42987e12 0.227708
\(280\) 7.26367e13 2.52224
\(281\) −6.18824e12 −0.210709 −0.105354 0.994435i \(-0.533598\pi\)
−0.105354 + 0.994435i \(0.533598\pi\)
\(282\) 2.33821e12 0.0780752
\(283\) −4.63157e13 −1.51671 −0.758356 0.651841i \(-0.773997\pi\)
−0.758356 + 0.651841i \(0.773997\pi\)
\(284\) −1.48415e14 −4.76679
\(285\) 2.29496e13 0.722982
\(286\) 3.15980e13 0.976442
\(287\) 1.04379e13 0.316419
\(288\) −4.79054e13 −1.42471
\(289\) −2.35848e13 −0.688169
\(290\) −1.10980e14 −3.17728
\(291\) 9.18596e12 0.258055
\(292\) 5.37646e13 1.48214
\(293\) −1.38698e13 −0.375231 −0.187615 0.982243i \(-0.560076\pi\)
−0.187615 + 0.982243i \(0.560076\pi\)
\(294\) 6.04566e12 0.160521
\(295\) 1.02855e14 2.68044
\(296\) 1.74036e14 4.45180
\(297\) −1.38645e13 −0.348132
\(298\) −6.10636e13 −1.50520
\(299\) 9.59005e12 0.232076
\(300\) −1.81446e14 −4.31103
\(301\) 4.17286e12 0.0973461
\(302\) 6.99349e13 1.60198
\(303\) −3.30871e13 −0.744260
\(304\) 1.17752e14 2.60113
\(305\) 1.74609e13 0.378808
\(306\) 1.70019e13 0.362268
\(307\) 6.33986e13 1.32684 0.663420 0.748247i \(-0.269104\pi\)
0.663420 + 0.748247i \(0.269104\pi\)
\(308\) 9.27178e13 1.90605
\(309\) −1.77331e13 −0.358107
\(310\) 1.28533e14 2.54991
\(311\) −4.99398e12 −0.0973341 −0.0486670 0.998815i \(-0.515497\pi\)
−0.0486670 + 0.998815i \(0.515497\pi\)
\(312\) −2.90952e13 −0.557148
\(313\) −3.34290e13 −0.628970 −0.314485 0.949262i \(-0.601832\pi\)
−0.314485 + 0.949262i \(0.601832\pi\)
\(314\) −1.09632e14 −2.02685
\(315\) −1.33006e13 −0.241637
\(316\) −9.03842e13 −1.61367
\(317\) −2.50776e13 −0.440007 −0.220004 0.975499i \(-0.570607\pi\)
−0.220004 + 0.975499i \(0.570607\pi\)
\(318\) 7.15969e12 0.123465
\(319\) −9.08461e13 −1.53978
\(320\) −4.98996e14 −8.31327
\(321\) −6.59410e13 −1.07989
\(322\) 3.82341e13 0.615523
\(323\) −2.30372e13 −0.364599
\(324\) 1.99073e13 0.309752
\(325\) −4.85592e13 −0.742871
\(326\) −1.40742e14 −2.11703
\(327\) −1.47455e12 −0.0218096
\(328\) −2.00273e14 −2.91283
\(329\) 1.83616e12 0.0262623
\(330\) −2.77151e14 −3.89844
\(331\) −1.09568e14 −1.51576 −0.757882 0.652392i \(-0.773766\pi\)
−0.757882 + 0.652392i \(0.773766\pi\)
\(332\) 9.98598e13 1.35872
\(333\) −3.18680e13 −0.426493
\(334\) −2.82704e14 −3.72157
\(335\) 2.30708e14 2.98756
\(336\) −6.82438e13 −0.869355
\(337\) 2.22866e12 0.0279305 0.0139652 0.999902i \(-0.495555\pi\)
0.0139652 + 0.999902i \(0.495555\pi\)
\(338\) −1.21420e13 −0.149709
\(339\) 8.06981e13 0.978961
\(340\) 2.50140e14 2.98572
\(341\) 1.05215e14 1.23574
\(342\) −3.66496e13 −0.423571
\(343\) 4.74756e12 0.0539949
\(344\) −8.00649e13 −0.896129
\(345\) −8.41160e13 −0.926560
\(346\) 2.36277e14 2.56155
\(347\) 1.65759e14 1.76875 0.884374 0.466779i \(-0.154586\pi\)
0.884374 + 0.466779i \(0.154586\pi\)
\(348\) 1.30441e14 1.37002
\(349\) −7.00848e13 −0.724576 −0.362288 0.932066i \(-0.618004\pi\)
−0.362288 + 0.932066i \(0.618004\pi\)
\(350\) −1.93598e14 −1.97028
\(351\) 5.32765e12 0.0533761
\(352\) −7.83896e14 −7.73168
\(353\) 1.61695e14 1.57013 0.785066 0.619412i \(-0.212629\pi\)
0.785066 + 0.619412i \(0.212629\pi\)
\(354\) −1.64256e14 −1.57038
\(355\) 3.48384e14 3.27946
\(356\) 2.14039e14 1.98389
\(357\) 1.33514e13 0.121857
\(358\) 1.25911e14 1.13164
\(359\) −8.69841e13 −0.769875 −0.384937 0.922943i \(-0.625777\pi\)
−0.384937 + 0.922943i \(0.625777\pi\)
\(360\) 2.55199e14 2.22441
\(361\) −6.68310e13 −0.573704
\(362\) 4.26123e14 3.60278
\(363\) −1.57540e14 −1.31191
\(364\) −3.56282e13 −0.292238
\(365\) −1.26205e14 −1.01969
\(366\) −2.78845e13 −0.221930
\(367\) 1.22953e14 0.963998 0.481999 0.876172i \(-0.339911\pi\)
0.481999 + 0.876172i \(0.339911\pi\)
\(368\) −4.31590e14 −3.33356
\(369\) 3.66721e13 0.279056
\(370\) −6.37040e14 −4.77592
\(371\) 5.62239e12 0.0415303
\(372\) −1.51072e14 −1.09950
\(373\) 5.85866e13 0.420146 0.210073 0.977686i \(-0.432630\pi\)
0.210073 + 0.977686i \(0.432630\pi\)
\(374\) 2.78209e14 1.96598
\(375\) 2.66904e14 1.85859
\(376\) −3.52305e13 −0.241760
\(377\) 3.49090e13 0.236080
\(378\) 2.12405e13 0.141567
\(379\) −6.41085e13 −0.421114 −0.210557 0.977582i \(-0.567528\pi\)
−0.210557 + 0.977582i \(0.567528\pi\)
\(380\) −5.39206e14 −3.49096
\(381\) −5.93886e13 −0.378980
\(382\) −3.07335e13 −0.193314
\(383\) −2.80586e12 −0.0173969 −0.00869847 0.999962i \(-0.502769\pi\)
−0.00869847 + 0.999962i \(0.502769\pi\)
\(384\) 3.93132e14 2.40279
\(385\) −2.17643e14 −1.31132
\(386\) 2.01681e14 1.19794
\(387\) 1.46608e13 0.0858512
\(388\) −2.15827e14 −1.24604
\(389\) −2.86526e14 −1.63095 −0.815477 0.578789i \(-0.803525\pi\)
−0.815477 + 0.578789i \(0.803525\pi\)
\(390\) 1.06500e14 0.597712
\(391\) 8.44371e13 0.467263
\(392\) −9.10916e13 −0.497056
\(393\) −1.61094e14 −0.866805
\(394\) −2.07360e13 −0.110026
\(395\) 2.12165e14 1.11017
\(396\) 3.25751e14 1.68098
\(397\) −3.33362e14 −1.69655 −0.848277 0.529552i \(-0.822360\pi\)
−0.848277 + 0.529552i \(0.822360\pi\)
\(398\) −5.27866e14 −2.64952
\(399\) −2.87804e13 −0.142477
\(400\) 2.18535e15 10.6707
\(401\) 1.48196e14 0.713746 0.356873 0.934153i \(-0.383843\pi\)
0.356873 + 0.934153i \(0.383843\pi\)
\(402\) −3.68432e14 −1.75031
\(403\) −4.04302e13 −0.189465
\(404\) 7.77391e14 3.59370
\(405\) −4.67297e13 −0.213103
\(406\) 1.39177e14 0.626144
\(407\) −5.21469e14 −2.31451
\(408\) −2.56173e14 −1.12177
\(409\) −3.25562e14 −1.40655 −0.703277 0.710916i \(-0.748280\pi\)
−0.703277 + 0.710916i \(0.748280\pi\)
\(410\) 7.33074e14 3.12490
\(411\) 1.88731e14 0.793806
\(412\) 4.16644e14 1.72914
\(413\) −1.28988e14 −0.528231
\(414\) 1.34330e14 0.542840
\(415\) −2.34408e14 −0.934775
\(416\) 3.01224e14 1.18543
\(417\) −1.75701e14 −0.682379
\(418\) −5.99712e14 −2.29865
\(419\) −8.95562e13 −0.338781 −0.169390 0.985549i \(-0.554180\pi\)
−0.169390 + 0.985549i \(0.554180\pi\)
\(420\) 3.12501e14 1.16676
\(421\) −2.87090e14 −1.05795 −0.528977 0.848636i \(-0.677424\pi\)
−0.528977 + 0.848636i \(0.677424\pi\)
\(422\) −9.88165e14 −3.59427
\(423\) 6.45108e12 0.0231612
\(424\) −1.07877e14 −0.382311
\(425\) −4.27547e14 −1.49570
\(426\) −5.56356e14 −1.92132
\(427\) −2.18973e13 −0.0746512
\(428\) 1.54930e15 5.21429
\(429\) 8.71785e13 0.289664
\(430\) 2.93068e14 0.961373
\(431\) −4.24606e14 −1.37519 −0.687593 0.726096i \(-0.741333\pi\)
−0.687593 + 0.726096i \(0.741333\pi\)
\(432\) −2.39765e14 −0.766699
\(433\) 4.40990e14 1.39234 0.696170 0.717877i \(-0.254886\pi\)
0.696170 + 0.717877i \(0.254886\pi\)
\(434\) −1.61189e14 −0.502508
\(435\) −3.06192e14 −0.942548
\(436\) 3.46450e13 0.105309
\(437\) −1.82014e14 −0.546332
\(438\) 2.01545e14 0.597400
\(439\) −3.86345e14 −1.13089 −0.565445 0.824786i \(-0.691295\pi\)
−0.565445 + 0.824786i \(0.691295\pi\)
\(440\) 4.17592e15 12.0715
\(441\) 1.66799e13 0.0476190
\(442\) −1.06906e14 −0.301425
\(443\) 5.59438e14 1.55787 0.778935 0.627105i \(-0.215760\pi\)
0.778935 + 0.627105i \(0.215760\pi\)
\(444\) 7.48747e14 2.05934
\(445\) −5.02428e14 −1.36488
\(446\) 9.18430e14 2.46436
\(447\) −1.68474e14 −0.446520
\(448\) 6.25776e14 1.63829
\(449\) −1.87991e13 −0.0486163 −0.0243081 0.999705i \(-0.507738\pi\)
−0.0243081 + 0.999705i \(0.507738\pi\)
\(450\) −6.80180e14 −1.73762
\(451\) 6.00081e14 1.51439
\(452\) −1.89602e15 −4.72697
\(453\) 1.92950e14 0.475230
\(454\) 3.30362e14 0.803864
\(455\) 8.36325e13 0.201054
\(456\) 5.52210e14 1.31159
\(457\) 2.40929e14 0.565393 0.282696 0.959209i \(-0.408771\pi\)
0.282696 + 0.959209i \(0.408771\pi\)
\(458\) −8.12905e14 −1.88486
\(459\) 4.69081e13 0.107468
\(460\) 1.97633e15 4.47395
\(461\) 2.18518e12 0.00488800 0.00244400 0.999997i \(-0.499222\pi\)
0.00244400 + 0.999997i \(0.499222\pi\)
\(462\) 3.47568e14 0.768260
\(463\) −2.69300e14 −0.588221 −0.294110 0.955771i \(-0.595023\pi\)
−0.294110 + 0.955771i \(0.595023\pi\)
\(464\) −1.57104e15 −3.39108
\(465\) 3.54620e14 0.756437
\(466\) 7.73460e14 1.63048
\(467\) 2.73339e14 0.569453 0.284727 0.958609i \(-0.408097\pi\)
0.284727 + 0.958609i \(0.408097\pi\)
\(468\) −1.25175e14 −0.257729
\(469\) −2.89324e14 −0.588755
\(470\) 1.28957e14 0.259362
\(471\) −3.02472e14 −0.601271
\(472\) 2.47490e15 4.86269
\(473\) 2.39900e14 0.465901
\(474\) −3.38820e14 −0.650411
\(475\) 9.21627e14 1.74880
\(476\) −3.13694e14 −0.588393
\(477\) 1.97535e13 0.0366263
\(478\) −6.19202e14 −1.13496
\(479\) 1.44477e14 0.261790 0.130895 0.991396i \(-0.458215\pi\)
0.130895 + 0.991396i \(0.458215\pi\)
\(480\) −2.64208e15 −4.73282
\(481\) 2.00382e14 0.354863
\(482\) −1.40542e14 −0.246063
\(483\) 1.05487e14 0.182596
\(484\) 3.70145e15 6.33464
\(485\) 5.06625e14 0.857248
\(486\) 7.46256e13 0.124850
\(487\) −1.54874e14 −0.256193 −0.128097 0.991762i \(-0.540887\pi\)
−0.128097 + 0.991762i \(0.540887\pi\)
\(488\) 4.20144e14 0.687209
\(489\) −3.88305e14 −0.628022
\(490\) 3.33430e14 0.533245
\(491\) −1.00549e15 −1.59012 −0.795060 0.606530i \(-0.792561\pi\)
−0.795060 + 0.606530i \(0.792561\pi\)
\(492\) −8.61621e14 −1.34744
\(493\) 3.07361e14 0.475326
\(494\) 2.30448e14 0.352432
\(495\) −7.64657e14 −1.15648
\(496\) 1.81952e15 2.72149
\(497\) −4.36898e14 −0.646279
\(498\) 3.74341e14 0.547653
\(499\) −5.41005e14 −0.782796 −0.391398 0.920222i \(-0.628008\pi\)
−0.391398 + 0.920222i \(0.628008\pi\)
\(500\) −6.27098e15 −8.97429
\(501\) −7.79976e14 −1.10401
\(502\) −2.79561e14 −0.391387
\(503\) 2.17448e14 0.301115 0.150557 0.988601i \(-0.451893\pi\)
0.150557 + 0.988601i \(0.451893\pi\)
\(504\) −3.20037e14 −0.438362
\(505\) −1.82482e15 −2.47240
\(506\) 2.19810e15 2.94591
\(507\) −3.34996e13 −0.0444116
\(508\) 1.39535e15 1.82992
\(509\) −4.82712e14 −0.626239 −0.313120 0.949714i \(-0.601374\pi\)
−0.313120 + 0.949714i \(0.601374\pi\)
\(510\) 9.37691e14 1.20344
\(511\) 1.58270e14 0.200949
\(512\) −2.52066e15 −3.16614
\(513\) −1.01116e14 −0.125653
\(514\) 1.40723e14 0.173009
\(515\) −9.78016e14 −1.18961
\(516\) −3.44459e14 −0.414537
\(517\) 1.05562e14 0.125692
\(518\) 7.98894e14 0.941186
\(519\) 6.51885e14 0.759890
\(520\) −1.60466e15 −1.85082
\(521\) −9.80262e13 −0.111875 −0.0559377 0.998434i \(-0.517815\pi\)
−0.0559377 + 0.998434i \(0.517815\pi\)
\(522\) 4.88978e14 0.552207
\(523\) 4.41445e14 0.493307 0.246654 0.969104i \(-0.420669\pi\)
0.246654 + 0.969104i \(0.420669\pi\)
\(524\) 3.78495e15 4.18542
\(525\) −5.34135e14 −0.584487
\(526\) −2.21490e15 −2.39846
\(527\) −3.55974e14 −0.381470
\(528\) −3.92337e15 −4.16076
\(529\) −2.85683e14 −0.299832
\(530\) 3.94871e14 0.410146
\(531\) −4.53181e14 −0.465856
\(532\) 6.76203e14 0.687960
\(533\) −2.30590e14 −0.232188
\(534\) 8.02359e14 0.799635
\(535\) −3.63678e15 −3.58733
\(536\) 5.55128e15 5.41984
\(537\) 3.47387e14 0.335703
\(538\) −2.34295e15 −2.24110
\(539\) 2.72940e14 0.258421
\(540\) 1.09793e15 1.02898
\(541\) 1.26578e15 1.17429 0.587143 0.809483i \(-0.300253\pi\)
0.587143 + 0.809483i \(0.300253\pi\)
\(542\) 3.90891e14 0.358971
\(543\) 1.17567e15 1.06877
\(544\) 2.65217e15 2.38675
\(545\) −8.13246e13 −0.0724505
\(546\) −1.33558e14 −0.117791
\(547\) −1.51359e14 −0.132153 −0.0660767 0.997815i \(-0.521048\pi\)
−0.0660767 + 0.997815i \(0.521048\pi\)
\(548\) −4.43430e15 −3.83294
\(549\) −7.69330e13 −0.0658362
\(550\) −1.11301e16 −9.42981
\(551\) −6.62553e14 −0.555759
\(552\) −2.02399e15 −1.68091
\(553\) −2.66070e14 −0.218780
\(554\) −2.83967e15 −2.31187
\(555\) −1.75759e15 −1.41679
\(556\) 4.12814e15 3.29491
\(557\) 6.00776e14 0.474798 0.237399 0.971412i \(-0.423705\pi\)
0.237399 + 0.971412i \(0.423705\pi\)
\(558\) −5.66316e14 −0.443170
\(559\) −9.21851e13 −0.0714325
\(560\) −3.76379e15 −2.88796
\(561\) 7.67576e14 0.583211
\(562\) 5.45034e14 0.410085
\(563\) 1.79235e15 1.33544 0.667722 0.744411i \(-0.267269\pi\)
0.667722 + 0.744411i \(0.267269\pi\)
\(564\) −1.51570e14 −0.111835
\(565\) 4.45067e15 3.25206
\(566\) 4.07930e15 2.95185
\(567\) 5.86024e13 0.0419961
\(568\) 8.38278e15 5.94939
\(569\) −7.18981e14 −0.505359 −0.252679 0.967550i \(-0.581312\pi\)
−0.252679 + 0.967550i \(0.581312\pi\)
\(570\) −2.02130e15 −1.40708
\(571\) 1.32240e15 0.911724 0.455862 0.890050i \(-0.349331\pi\)
0.455862 + 0.890050i \(0.349331\pi\)
\(572\) −2.04828e15 −1.39866
\(573\) −8.47932e13 −0.0573470
\(574\) −9.19328e14 −0.615822
\(575\) −3.37800e15 −2.24123
\(576\) 2.19858e15 1.44483
\(577\) 1.29152e15 0.840685 0.420343 0.907365i \(-0.361910\pi\)
0.420343 + 0.907365i \(0.361910\pi\)
\(578\) 2.07725e15 1.33933
\(579\) 5.56436e14 0.355372
\(580\) 7.19407e15 4.55115
\(581\) 2.93964e14 0.184215
\(582\) −8.09061e14 −0.502233
\(583\) 3.23234e14 0.198765
\(584\) −3.03674e15 −1.84985
\(585\) 2.93831e14 0.177313
\(586\) 1.22159e15 0.730281
\(587\) −1.83623e15 −1.08747 −0.543737 0.839256i \(-0.682991\pi\)
−0.543737 + 0.839256i \(0.682991\pi\)
\(588\) −3.91898e14 −0.229931
\(589\) 7.67343e14 0.446021
\(590\) −9.05907e15 −5.21672
\(591\) −5.72104e13 −0.0326395
\(592\) −9.01798e15 −5.09729
\(593\) −1.32174e15 −0.740194 −0.370097 0.928993i \(-0.620676\pi\)
−0.370097 + 0.928993i \(0.620676\pi\)
\(594\) 1.22113e15 0.677542
\(595\) 7.36355e14 0.404803
\(596\) 3.95834e15 2.15605
\(597\) −1.45638e15 −0.785986
\(598\) −8.44652e14 −0.451670
\(599\) 3.75903e15 1.99172 0.995859 0.0909106i \(-0.0289777\pi\)
0.995859 + 0.0909106i \(0.0289777\pi\)
\(600\) 1.02485e16 5.38056
\(601\) −4.00228e14 −0.208208 −0.104104 0.994566i \(-0.533198\pi\)
−0.104104 + 0.994566i \(0.533198\pi\)
\(602\) −3.67528e14 −0.189457
\(603\) −1.01650e15 −0.519233
\(604\) −4.53340e15 −2.29467
\(605\) −8.68866e15 −4.35811
\(606\) 2.91418e15 1.44849
\(607\) 1.50336e15 0.740498 0.370249 0.928933i \(-0.379272\pi\)
0.370249 + 0.928933i \(0.379272\pi\)
\(608\) −5.71705e15 −2.79063
\(609\) 3.83987e14 0.185747
\(610\) −1.53789e15 −0.737243
\(611\) −4.05636e13 −0.0192713
\(612\) −1.10212e15 −0.518914
\(613\) −1.77000e15 −0.825924 −0.412962 0.910748i \(-0.635506\pi\)
−0.412962 + 0.910748i \(0.635506\pi\)
\(614\) −5.58388e15 −2.58232
\(615\) 2.02254e15 0.927010
\(616\) −5.23690e15 −2.37892
\(617\) −6.28931e14 −0.283162 −0.141581 0.989927i \(-0.545219\pi\)
−0.141581 + 0.989927i \(0.545219\pi\)
\(618\) 1.56186e15 0.696954
\(619\) 1.27006e15 0.561728 0.280864 0.959748i \(-0.409379\pi\)
0.280864 + 0.959748i \(0.409379\pi\)
\(620\) −8.33190e15 −3.65250
\(621\) 3.70615e14 0.161035
\(622\) 4.39849e14 0.189433
\(623\) 6.30080e14 0.268975
\(624\) 1.50761e15 0.637932
\(625\) 8.33435e15 3.49568
\(626\) 2.94429e15 1.22411
\(627\) −1.65460e15 −0.681901
\(628\) 7.10666e15 2.90327
\(629\) 1.76429e15 0.714484
\(630\) 1.17146e15 0.470278
\(631\) 2.62244e15 1.04362 0.521812 0.853060i \(-0.325256\pi\)
0.521812 + 0.853060i \(0.325256\pi\)
\(632\) 5.10509e15 2.01400
\(633\) −2.72633e15 −1.06625
\(634\) 2.20873e15 0.856351
\(635\) −3.27540e15 −1.25895
\(636\) −4.64113e14 −0.176852
\(637\) −1.04881e14 −0.0396214
\(638\) 8.00135e15 2.99674
\(639\) −1.53498e15 −0.569965
\(640\) 2.16820e16 7.98196
\(641\) −1.80604e15 −0.659186 −0.329593 0.944123i \(-0.606912\pi\)
−0.329593 + 0.944123i \(0.606912\pi\)
\(642\) 5.80781e15 2.10169
\(643\) 2.19272e14 0.0786723 0.0393361 0.999226i \(-0.487476\pi\)
0.0393361 + 0.999226i \(0.487476\pi\)
\(644\) −2.47846e15 −0.881676
\(645\) 8.08571e14 0.285194
\(646\) 2.02902e15 0.709589
\(647\) 1.55330e15 0.538621 0.269310 0.963053i \(-0.413204\pi\)
0.269310 + 0.963053i \(0.413204\pi\)
\(648\) −1.12441e15 −0.386599
\(649\) −7.41558e15 −2.52813
\(650\) 4.27689e15 1.44579
\(651\) −4.44719e14 −0.149070
\(652\) 9.12334e15 3.03244
\(653\) 1.58385e15 0.522025 0.261012 0.965335i \(-0.415944\pi\)
0.261012 + 0.965335i \(0.415944\pi\)
\(654\) 1.29872e14 0.0424463
\(655\) −8.88468e15 −2.87948
\(656\) 1.03774e16 3.33518
\(657\) 5.56061e14 0.177220
\(658\) −1.61721e14 −0.0511122
\(659\) 2.46153e15 0.771498 0.385749 0.922604i \(-0.373943\pi\)
0.385749 + 0.922604i \(0.373943\pi\)
\(660\) 1.79658e16 5.58413
\(661\) 1.14409e15 0.352655 0.176328 0.984332i \(-0.443578\pi\)
0.176328 + 0.984332i \(0.443578\pi\)
\(662\) 9.65033e15 2.95001
\(663\) −2.94952e14 −0.0894185
\(664\) −5.64030e15 −1.69581
\(665\) −1.58730e15 −0.473303
\(666\) 2.80680e15 0.830048
\(667\) 2.42842e15 0.712250
\(668\) 1.83258e16 5.33079
\(669\) 2.53394e15 0.731058
\(670\) −2.03198e16 −5.81444
\(671\) −1.25888e15 −0.357283
\(672\) 3.31336e15 0.932691
\(673\) −2.59131e15 −0.723496 −0.361748 0.932276i \(-0.617820\pi\)
−0.361748 + 0.932276i \(0.617820\pi\)
\(674\) −1.96291e14 −0.0543589
\(675\) −1.87661e15 −0.515469
\(676\) 7.87082e14 0.214444
\(677\) 2.25648e15 0.609809 0.304905 0.952383i \(-0.401375\pi\)
0.304905 + 0.952383i \(0.401375\pi\)
\(678\) −7.10756e15 −1.90527
\(679\) −6.35343e14 −0.168937
\(680\) −1.41285e16 −3.72645
\(681\) 9.11463e14 0.238468
\(682\) −9.26686e15 −2.40502
\(683\) 5.97129e14 0.153728 0.0768642 0.997042i \(-0.475509\pi\)
0.0768642 + 0.997042i \(0.475509\pi\)
\(684\) 2.37574e15 0.606724
\(685\) 1.04089e16 2.63699
\(686\) −4.18145e14 −0.105086
\(687\) −2.24279e15 −0.559149
\(688\) 4.14869e15 1.02606
\(689\) −1.24207e14 −0.0304749
\(690\) 7.40858e15 1.80329
\(691\) 7.80419e15 1.88451 0.942255 0.334897i \(-0.108702\pi\)
0.942255 + 0.334897i \(0.108702\pi\)
\(692\) −1.53162e16 −3.66917
\(693\) 9.58934e14 0.227906
\(694\) −1.45994e16 −3.44237
\(695\) −9.69026e15 −2.26683
\(696\) −7.36757e15 −1.70991
\(697\) −2.03026e15 −0.467490
\(698\) 6.17278e15 1.41018
\(699\) 2.13396e15 0.483685
\(700\) 1.25497e16 2.82223
\(701\) −6.06863e15 −1.35407 −0.677036 0.735950i \(-0.736736\pi\)
−0.677036 + 0.735950i \(0.736736\pi\)
\(702\) −4.69237e14 −0.103881
\(703\) −3.80314e15 −0.835388
\(704\) 3.59762e16 7.84089
\(705\) 3.55791e14 0.0769404
\(706\) −1.42414e16 −3.05582
\(707\) 2.28846e15 0.487233
\(708\) 1.06476e16 2.24941
\(709\) −6.42694e15 −1.34726 −0.673628 0.739071i \(-0.735265\pi\)
−0.673628 + 0.739071i \(0.735265\pi\)
\(710\) −3.06842e16 −6.38254
\(711\) −9.34799e14 −0.192946
\(712\) −1.20894e16 −2.47607
\(713\) −2.81251e15 −0.571612
\(714\) −1.17593e15 −0.237160
\(715\) 4.80807e15 0.962249
\(716\) −8.16196e15 −1.62096
\(717\) −1.70837e15 −0.336687
\(718\) 7.66119e15 1.49835
\(719\) 7.57171e15 1.46955 0.734776 0.678310i \(-0.237287\pi\)
0.734776 + 0.678310i \(0.237287\pi\)
\(720\) −1.32235e16 −2.54694
\(721\) 1.22650e15 0.234436
\(722\) 5.88619e15 1.11655
\(723\) −3.87752e14 −0.0729951
\(724\) −2.76226e16 −5.16063
\(725\) −1.22963e16 −2.27990
\(726\) 1.38755e16 2.55327
\(727\) −4.48914e15 −0.819830 −0.409915 0.912124i \(-0.634442\pi\)
−0.409915 + 0.912124i \(0.634442\pi\)
\(728\) 2.01236e15 0.364739
\(729\) 2.05891e14 0.0370370
\(730\) 1.11156e16 1.98453
\(731\) −8.11658e14 −0.143823
\(732\) 1.80756e15 0.317894
\(733\) −6.39691e15 −1.11660 −0.558301 0.829638i \(-0.688547\pi\)
−0.558301 + 0.829638i \(0.688547\pi\)
\(734\) −1.08292e16 −1.87615
\(735\) 9.19930e14 0.158188
\(736\) 2.09545e16 3.57642
\(737\) −1.66334e16 −2.81780
\(738\) −3.22993e15 −0.543104
\(739\) 2.63169e15 0.439228 0.219614 0.975587i \(-0.429520\pi\)
0.219614 + 0.975587i \(0.429520\pi\)
\(740\) 4.12949e16 6.84104
\(741\) 6.35804e14 0.104550
\(742\) −4.95197e14 −0.0808270
\(743\) −3.59566e15 −0.582560 −0.291280 0.956638i \(-0.594081\pi\)
−0.291280 + 0.956638i \(0.594081\pi\)
\(744\) 8.53285e15 1.37228
\(745\) −9.29167e15 −1.48332
\(746\) −5.16006e15 −0.817696
\(747\) 1.03280e15 0.162462
\(748\) −1.80344e16 −2.81607
\(749\) 4.56078e15 0.706951
\(750\) −2.35078e16 −3.61722
\(751\) −8.29732e15 −1.26741 −0.633706 0.773574i \(-0.718467\pi\)
−0.633706 + 0.773574i \(0.718467\pi\)
\(752\) 1.82552e15 0.276815
\(753\) −7.71306e14 −0.116106
\(754\) −3.07464e15 −0.459464
\(755\) 1.06416e16 1.57869
\(756\) −1.37688e15 −0.202780
\(757\) −2.03120e15 −0.296979 −0.148489 0.988914i \(-0.547441\pi\)
−0.148489 + 0.988914i \(0.547441\pi\)
\(758\) 5.64641e15 0.819581
\(759\) 6.06452e15 0.873911
\(760\) 3.04555e16 4.35704
\(761\) −5.23714e15 −0.743838 −0.371919 0.928265i \(-0.621300\pi\)
−0.371919 + 0.928265i \(0.621300\pi\)
\(762\) 5.23070e15 0.737577
\(763\) 1.01987e14 0.0142777
\(764\) 1.99224e15 0.276903
\(765\) 2.58708e15 0.357003
\(766\) 2.47128e14 0.0338583
\(767\) 2.84955e15 0.387616
\(768\) −1.60958e16 −2.17384
\(769\) 2.37472e15 0.318432 0.159216 0.987244i \(-0.449103\pi\)
0.159216 + 0.987244i \(0.449103\pi\)
\(770\) 1.91691e16 2.55213
\(771\) 3.88253e14 0.0513235
\(772\) −1.30736e16 −1.71593
\(773\) −1.06877e16 −1.39283 −0.696416 0.717639i \(-0.745223\pi\)
−0.696416 + 0.717639i \(0.745223\pi\)
\(774\) −1.29126e15 −0.167085
\(775\) 1.42411e16 1.82972
\(776\) 1.21904e16 1.55517
\(777\) 2.20414e15 0.279205
\(778\) 2.52360e16 3.17419
\(779\) 4.37647e15 0.546597
\(780\) −6.90363e15 −0.856165
\(781\) −2.51175e16 −3.09311
\(782\) −7.43687e15 −0.909396
\(783\) 1.34908e15 0.163813
\(784\) 4.72006e15 0.569127
\(785\) −1.66819e16 −1.99739
\(786\) 1.41885e16 1.68699
\(787\) 5.76104e15 0.680205 0.340102 0.940388i \(-0.389538\pi\)
0.340102 + 0.940388i \(0.389538\pi\)
\(788\) 1.34417e15 0.157602
\(789\) −6.11087e15 −0.711508
\(790\) −1.86866e16 −2.16063
\(791\) −5.58146e15 −0.640880
\(792\) −1.83991e16 −2.09801
\(793\) 4.83745e14 0.0547790
\(794\) 2.93611e16 3.30187
\(795\) 1.08944e15 0.121671
\(796\) 3.42179e16 3.79518
\(797\) 1.19610e16 1.31749 0.658746 0.752365i \(-0.271087\pi\)
0.658746 + 0.752365i \(0.271087\pi\)
\(798\) 2.53486e15 0.277292
\(799\) −3.57149e14 −0.0388009
\(800\) −1.06103e17 −11.4481
\(801\) 2.21370e15 0.237213
\(802\) −1.30525e16 −1.38911
\(803\) 9.09904e15 0.961746
\(804\) 2.38829e16 2.50715
\(805\) 5.81785e15 0.606576
\(806\) 3.56093e15 0.368740
\(807\) −6.46417e15 −0.664826
\(808\) −4.39088e16 −4.48527
\(809\) −5.67680e15 −0.575952 −0.287976 0.957638i \(-0.592982\pi\)
−0.287976 + 0.957638i \(0.592982\pi\)
\(810\) 4.11576e15 0.414746
\(811\) 2.09739e15 0.209925 0.104962 0.994476i \(-0.466528\pi\)
0.104962 + 0.994476i \(0.466528\pi\)
\(812\) −9.02188e15 −0.896890
\(813\) 1.07846e15 0.106490
\(814\) 4.59288e16 4.50454
\(815\) −2.14158e16 −2.08626
\(816\) 1.32740e16 1.28442
\(817\) 1.74962e15 0.168160
\(818\) 2.86742e16 2.73746
\(819\) −3.68485e14 −0.0349428
\(820\) −4.75202e16 −4.47612
\(821\) 9.60765e15 0.898938 0.449469 0.893296i \(-0.351613\pi\)
0.449469 + 0.893296i \(0.351613\pi\)
\(822\) −1.66227e16 −1.54492
\(823\) −9.40279e15 −0.868076 −0.434038 0.900895i \(-0.642912\pi\)
−0.434038 + 0.900895i \(0.642912\pi\)
\(824\) −2.35329e16 −2.15812
\(825\) −3.07077e16 −2.79737
\(826\) 1.13607e16 1.02805
\(827\) 6.48063e15 0.582555 0.291278 0.956639i \(-0.405920\pi\)
0.291278 + 0.956639i \(0.405920\pi\)
\(828\) −8.70770e15 −0.777565
\(829\) 1.52038e16 1.34866 0.674330 0.738430i \(-0.264432\pi\)
0.674330 + 0.738430i \(0.264432\pi\)
\(830\) 2.06456e16 1.81928
\(831\) −7.83461e15 −0.685822
\(832\) −1.38244e16 −1.20217
\(833\) −9.23441e14 −0.0797741
\(834\) 1.54750e16 1.32806
\(835\) −4.30173e16 −3.66748
\(836\) 3.88752e16 3.29260
\(837\) −1.56246e15 −0.131467
\(838\) 7.88773e15 0.659341
\(839\) 4.66734e15 0.387595 0.193798 0.981042i \(-0.437919\pi\)
0.193798 + 0.981042i \(0.437919\pi\)
\(840\) −1.76507e16 −1.45622
\(841\) −3.36075e15 −0.275460
\(842\) 2.52857e16 2.05901
\(843\) 1.50374e15 0.121653
\(844\) 6.40559e16 5.14844
\(845\) −1.84757e15 −0.147533
\(846\) −5.68185e14 −0.0450767
\(847\) 1.08962e16 0.858848
\(848\) 5.58982e15 0.437744
\(849\) 1.12547e16 0.875674
\(850\) 3.76566e16 2.91096
\(851\) 1.39395e16 1.07062
\(852\) 3.60648e16 2.75211
\(853\) 2.11779e16 1.60569 0.802847 0.596185i \(-0.203318\pi\)
0.802847 + 0.596185i \(0.203318\pi\)
\(854\) 1.92862e15 0.145288
\(855\) −5.57674e15 −0.417414
\(856\) −8.75079e16 −6.50791
\(857\) 6.44287e15 0.476085 0.238043 0.971255i \(-0.423494\pi\)
0.238043 + 0.971255i \(0.423494\pi\)
\(858\) −7.67831e15 −0.563749
\(859\) −2.98846e15 −0.218014 −0.109007 0.994041i \(-0.534767\pi\)
−0.109007 + 0.994041i \(0.534767\pi\)
\(860\) −1.89976e16 −1.37707
\(861\) −2.53641e15 −0.182685
\(862\) 3.73975e16 2.67641
\(863\) 6.52500e15 0.464003 0.232002 0.972715i \(-0.425473\pi\)
0.232002 + 0.972715i \(0.425473\pi\)
\(864\) 1.16410e16 0.822556
\(865\) 3.59528e16 2.52432
\(866\) −3.88405e16 −2.70980
\(867\) 5.73112e15 0.397314
\(868\) 1.04488e16 0.719794
\(869\) −1.52965e16 −1.04709
\(870\) 2.69682e16 1.83441
\(871\) 6.39163e15 0.432028
\(872\) −1.95683e15 −0.131435
\(873\) −2.23219e15 −0.148988
\(874\) 1.60310e16 1.06328
\(875\) −1.84603e16 −1.21673
\(876\) −1.30648e16 −0.855717
\(877\) −1.10842e16 −0.721450 −0.360725 0.932672i \(-0.617471\pi\)
−0.360725 + 0.932672i \(0.617471\pi\)
\(878\) 3.40276e16 2.20096
\(879\) 3.37036e15 0.216640
\(880\) −2.16382e17 −13.8219
\(881\) −8.89292e14 −0.0564517 −0.0282258 0.999602i \(-0.508986\pi\)
−0.0282258 + 0.999602i \(0.508986\pi\)
\(882\) −1.46909e15 −0.0926771
\(883\) −2.11288e16 −1.32462 −0.662309 0.749231i \(-0.730423\pi\)
−0.662309 + 0.749231i \(0.730423\pi\)
\(884\) 6.92999e15 0.431763
\(885\) −2.49939e16 −1.54755
\(886\) −4.92729e16 −3.03196
\(887\) 6.69059e14 0.0409152 0.0204576 0.999791i \(-0.493488\pi\)
0.0204576 + 0.999791i \(0.493488\pi\)
\(888\) −4.22909e16 −2.57025
\(889\) 4.10759e15 0.248100
\(890\) 4.42517e16 2.65635
\(891\) 3.36908e15 0.200994
\(892\) −5.95355e16 −3.52996
\(893\) 7.69875e14 0.0453667
\(894\) 1.48385e16 0.869026
\(895\) 1.91591e16 1.11519
\(896\) −2.71908e16 −1.57300
\(897\) −2.33038e15 −0.133989
\(898\) 1.65574e15 0.0946179
\(899\) −1.02379e16 −0.581476
\(900\) 4.40914e16 2.48897
\(901\) −1.09360e15 −0.0613583
\(902\) −5.28526e16 −2.94734
\(903\) −1.01401e15 −0.0562028
\(904\) 1.07092e17 5.89969
\(905\) 6.48404e16 3.55041
\(906\) −1.69942e16 −0.924901
\(907\) −3.32321e16 −1.79770 −0.898851 0.438254i \(-0.855597\pi\)
−0.898851 + 0.438254i \(0.855597\pi\)
\(908\) −2.14151e16 −1.15146
\(909\) 8.04018e15 0.429699
\(910\) −7.36600e15 −0.391295
\(911\) 1.78276e16 0.941331 0.470666 0.882312i \(-0.344014\pi\)
0.470666 + 0.882312i \(0.344014\pi\)
\(912\) −2.86137e16 −1.50176
\(913\) 1.69001e16 0.881659
\(914\) −2.12200e16 −1.10038
\(915\) −4.24301e15 −0.218705
\(916\) 5.26951e16 2.69988
\(917\) 1.11420e16 0.567457
\(918\) −4.13147e15 −0.209156
\(919\) −1.37212e16 −0.690488 −0.345244 0.938513i \(-0.612204\pi\)
−0.345244 + 0.938513i \(0.612204\pi\)
\(920\) −1.11627e17 −5.58389
\(921\) −1.54059e16 −0.766051
\(922\) −1.92461e14 −0.00951312
\(923\) 9.65177e15 0.474239
\(924\) −2.25304e16 −1.10046
\(925\) −7.05825e16 −3.42703
\(926\) 2.37188e16 1.14481
\(927\) 4.30914e15 0.206753
\(928\) 7.62768e16 3.63813
\(929\) 2.01407e16 0.954969 0.477484 0.878640i \(-0.341549\pi\)
0.477484 + 0.878640i \(0.341549\pi\)
\(930\) −3.12335e16 −1.47219
\(931\) 1.99058e15 0.0932733
\(932\) −5.01381e16 −2.33550
\(933\) 1.21354e15 0.0561959
\(934\) −2.40745e16 −1.10828
\(935\) 4.23334e16 1.93740
\(936\) 7.07013e15 0.321670
\(937\) −1.07677e16 −0.487030 −0.243515 0.969897i \(-0.578301\pi\)
−0.243515 + 0.969897i \(0.578301\pi\)
\(938\) 2.54825e16 1.14585
\(939\) 8.12325e15 0.363136
\(940\) −8.35939e15 −0.371511
\(941\) −3.92545e15 −0.173439 −0.0867194 0.996233i \(-0.527638\pi\)
−0.0867194 + 0.996233i \(0.527638\pi\)
\(942\) 2.66405e16 1.17020
\(943\) −1.60409e16 −0.700508
\(944\) −1.28241e17 −5.56775
\(945\) 3.23204e15 0.139509
\(946\) −2.11294e16 −0.906746
\(947\) 2.33959e16 0.998195 0.499098 0.866546i \(-0.333665\pi\)
0.499098 + 0.866546i \(0.333665\pi\)
\(948\) 2.19634e16 0.931650
\(949\) −3.49644e15 −0.147456
\(950\) −8.11730e16 −3.40355
\(951\) 6.09386e15 0.254038
\(952\) 1.77181e16 0.734368
\(953\) 3.62494e16 1.49379 0.746896 0.664940i \(-0.231543\pi\)
0.746896 + 0.664940i \(0.231543\pi\)
\(954\) −1.73980e15 −0.0712827
\(955\) −4.67652e15 −0.190504
\(956\) 4.01386e16 1.62571
\(957\) 2.20756e16 0.888990
\(958\) −1.27249e16 −0.509500
\(959\) −1.30535e16 −0.519668
\(960\) 1.21256e17 4.79967
\(961\) −1.35514e16 −0.533340
\(962\) −1.76488e16 −0.690642
\(963\) 1.60237e16 0.623472
\(964\) 9.11034e15 0.352461
\(965\) 3.06886e16 1.18053
\(966\) −9.29089e15 −0.355372
\(967\) −4.05463e15 −0.154208 −0.0771038 0.997023i \(-0.524567\pi\)
−0.0771038 + 0.997023i \(0.524567\pi\)
\(968\) −2.09066e17 −7.90621
\(969\) 5.59803e15 0.210501
\(970\) −4.46214e16 −1.66839
\(971\) 2.07744e16 0.772366 0.386183 0.922422i \(-0.373793\pi\)
0.386183 + 0.922422i \(0.373793\pi\)
\(972\) −4.83747e15 −0.178835
\(973\) 1.21523e16 0.446722
\(974\) 1.36406e16 0.498608
\(975\) 1.17999e16 0.428897
\(976\) −2.17704e16 −0.786851
\(977\) 1.01946e16 0.366394 0.183197 0.983076i \(-0.441355\pi\)
0.183197 + 0.983076i \(0.441355\pi\)
\(978\) 3.42003e16 1.22227
\(979\) 3.62236e16 1.28732
\(980\) −2.16140e16 −0.763821
\(981\) 3.58316e14 0.0125918
\(982\) 8.85595e16 3.09472
\(983\) −1.06518e16 −0.370151 −0.185075 0.982724i \(-0.559253\pi\)
−0.185075 + 0.982724i \(0.559253\pi\)
\(984\) 4.86662e16 1.68172
\(985\) −3.15527e15 −0.108427
\(986\) −2.70711e16 −0.925088
\(987\) −4.46187e14 −0.0151626
\(988\) −1.49384e16 −0.504825
\(989\) −6.41281e15 −0.215511
\(990\) 6.73478e16 2.25076
\(991\) 7.22558e15 0.240142 0.120071 0.992765i \(-0.461688\pi\)
0.120071 + 0.992765i \(0.461688\pi\)
\(992\) −8.83409e16 −2.91976
\(993\) 2.66251e16 0.875127
\(994\) 3.84802e16 1.25780
\(995\) −8.03221e16 −2.61101
\(996\) −2.42659e16 −0.784459
\(997\) −2.31554e16 −0.744439 −0.372219 0.928145i \(-0.621403\pi\)
−0.372219 + 0.928145i \(0.621403\pi\)
\(998\) 4.76495e16 1.52349
\(999\) 7.74392e15 0.246236
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.1.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.12.a.c.1.1 16 1.1 even 1 trivial