Properties

Label 29.12.a.a.1.11
Level $29$
Weight $12$
Character 29.1
Self dual yes
Analytic conductor $22.282$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.2819522362\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} - 388180519304 x^{4} + 193065378004825 x^{3} + 1279291654973975 x^{2} - 65244901875230266 x - 758324542468966858\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(81.6399\) of defining polynomial
Character \(\chi\) \(=\) 29.1

$q$-expansion

\(f(q)\) \(=\) \(q+78.6399 q^{2} -108.951 q^{3} +4136.23 q^{4} -12464.4 q^{5} -8567.92 q^{6} -230.276 q^{7} +164218. q^{8} -165277. q^{9} +O(q^{10})\) \(q+78.6399 q^{2} -108.951 q^{3} +4136.23 q^{4} -12464.4 q^{5} -8567.92 q^{6} -230.276 q^{7} +164218. q^{8} -165277. q^{9} -980202. q^{10} -26731.2 q^{11} -450648. q^{12} -1.35410e6 q^{13} -18108.8 q^{14} +1.35802e6 q^{15} +4.44308e6 q^{16} +3.19725e6 q^{17} -1.29973e7 q^{18} -9.37456e6 q^{19} -5.15558e7 q^{20} +25088.8 q^{21} -2.10214e6 q^{22} -1.17129e7 q^{23} -1.78918e7 q^{24} +1.06534e8 q^{25} -1.06486e8 q^{26} +3.73075e7 q^{27} -952472. q^{28} +2.05111e7 q^{29} +1.06794e8 q^{30} +1.69267e8 q^{31} +1.30850e7 q^{32} +2.91240e6 q^{33} +2.51431e8 q^{34} +2.87026e6 q^{35} -6.83622e8 q^{36} -6.21479e8 q^{37} -7.37214e8 q^{38} +1.47531e8 q^{39} -2.04688e9 q^{40} -9.80521e8 q^{41} +1.97298e6 q^{42} +1.55730e9 q^{43} -1.10566e8 q^{44} +2.06008e9 q^{45} -9.21103e8 q^{46} +7.66462e8 q^{47} -4.84080e8 q^{48} -1.97727e9 q^{49} +8.37783e9 q^{50} -3.48345e8 q^{51} -5.60086e9 q^{52} -1.24890e7 q^{53} +2.93386e9 q^{54} +3.33190e8 q^{55} -3.78154e7 q^{56} +1.02137e9 q^{57} +1.61299e9 q^{58} +3.94786e9 q^{59} +5.61707e9 q^{60} -6.43946e9 q^{61} +1.33112e10 q^{62} +3.80592e7 q^{63} -8.07043e9 q^{64} +1.68781e10 q^{65} +2.29031e8 q^{66} +4.26072e9 q^{67} +1.32246e10 q^{68} +1.27614e9 q^{69} +2.25716e8 q^{70} +2.70019e10 q^{71} -2.71414e10 q^{72} -8.00729e9 q^{73} -4.88730e10 q^{74} -1.16070e10 q^{75} -3.87753e10 q^{76} +6.15555e6 q^{77} +1.16018e10 q^{78} +4.92961e10 q^{79} -5.53805e10 q^{80} +2.52136e10 q^{81} -7.71080e10 q^{82} -7.07907e10 q^{83} +1.03773e8 q^{84} -3.98519e10 q^{85} +1.22466e11 q^{86} -2.23472e9 q^{87} -4.38975e9 q^{88} -8.54214e10 q^{89} +1.62004e11 q^{90} +3.11816e8 q^{91} -4.84473e10 q^{92} -1.84419e10 q^{93} +6.02744e10 q^{94} +1.16849e11 q^{95} -1.42563e9 q^{96} -4.48857e10 q^{97} -1.55493e11 q^{98} +4.41805e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 32q^{2} - 982q^{3} + 9146q^{4} - 2740q^{5} - 28202q^{6} - 49432q^{7} - 150054q^{8} + 330749q^{9} + O(q^{10}) \) \( 11q - 32q^{2} - 982q^{3} + 9146q^{4} - 2740q^{5} - 28202q^{6} - 49432q^{7} - 150054q^{8} + 330749q^{9} - 685834q^{10} - 612246q^{11} + 2578538q^{12} + 1510364q^{13} + 3955400q^{14} - 2462818q^{15} + 3024818q^{16} - 3291098q^{17} - 27885614q^{18} - 44121388q^{19} - 49472662q^{20} - 46916800q^{21} - 43435618q^{22} - 88684076q^{23} - 224700678q^{24} - 44195521q^{25} - 324999762q^{26} - 236304286q^{27} - 391274848q^{28} + 225622639q^{29} - 494910382q^{30} - 292235934q^{31} - 632542514q^{32} - 1079766410q^{33} - 1113307936q^{34} - 1312820120q^{35} - 2236726492q^{36} - 1380429338q^{37} - 1222857284q^{38} - 1186931090q^{39} - 2713154106q^{40} - 1062067494q^{41} + 205598960q^{42} + 74588594q^{43} + 52891466q^{44} + 4527996830q^{45} - 87670324q^{46} - 1821239394q^{47} + 2666035542q^{48} + 4692522003q^{49} + 9494259926q^{50} + 8768158380q^{51} + 3266669866q^{52} + 7818635688q^{53} + 17402728558q^{54} - 191002682q^{55} + 11263587512q^{56} + 15495358340q^{57} - 656356768q^{58} + 1230002712q^{59} + 31834046430q^{60} - 18602654230q^{61} + 22075953162q^{62} - 9964531456q^{63} + 11813658086q^{64} + 32245789334q^{65} + 42677188354q^{66} + 27481284652q^{67} + 29588811820q^{68} - 20565315068q^{69} + 42862666712q^{70} - 20347168516q^{71} + 47061083616q^{72} - 57740010478q^{73} - 2640709564q^{74} - 23544691000q^{75} - 33350650772q^{76} + 871959792q^{77} - 15384525342q^{78} - 120245016462q^{79} - 84319695274q^{80} - 48880047865q^{81} - 111495532412q^{82} - 142463983824q^{83} - 134146226376q^{84} - 181628566552q^{85} + 47870165542q^{86} - 20141948318q^{87} - 180608014462q^{88} - 96700717270q^{89} - 25522461244q^{90} - 355162031176q^{91} - 22429477796q^{92} - 172582115142q^{93} + 172608565078q^{94} - 195922150708q^{95} + 226391047758q^{96} - 303190852014q^{97} - 123776497136q^{98} - 139125462440q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 78.6399 1.73771 0.868856 0.495065i \(-0.164856\pi\)
0.868856 + 0.495065i \(0.164856\pi\)
\(3\) −108.951 −0.258860 −0.129430 0.991589i \(-0.541315\pi\)
−0.129430 + 0.991589i \(0.541315\pi\)
\(4\) 4136.23 2.01964
\(5\) −12464.4 −1.78377 −0.891883 0.452267i \(-0.850615\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(6\) −8567.92 −0.449825
\(7\) −230.276 −0.00517856 −0.00258928 0.999997i \(-0.500824\pi\)
−0.00258928 + 0.999997i \(0.500824\pi\)
\(8\) 164218. 1.77184
\(9\) −165277. −0.932991
\(10\) −980202. −3.09967
\(11\) −26731.2 −0.0500448 −0.0250224 0.999687i \(-0.507966\pi\)
−0.0250224 + 0.999687i \(0.507966\pi\)
\(12\) −450648. −0.522806
\(13\) −1.35410e6 −1.01149 −0.505745 0.862683i \(-0.668782\pi\)
−0.505745 + 0.862683i \(0.668782\pi\)
\(14\) −18108.8 −0.00899884
\(15\) 1.35802e6 0.461746
\(16\) 4.44308e6 1.05931
\(17\) 3.19725e6 0.546144 0.273072 0.961994i \(-0.411960\pi\)
0.273072 + 0.961994i \(0.411960\pi\)
\(18\) −1.29973e7 −1.62127
\(19\) −9.37456e6 −0.868573 −0.434286 0.900775i \(-0.642999\pi\)
−0.434286 + 0.900775i \(0.642999\pi\)
\(20\) −5.15558e7 −3.60257
\(21\) 25088.8 0.00134052
\(22\) −2.10214e6 −0.0869635
\(23\) −1.17129e7 −0.379457 −0.189728 0.981837i \(-0.560761\pi\)
−0.189728 + 0.981837i \(0.560761\pi\)
\(24\) −1.78918e7 −0.458661
\(25\) 1.06534e8 2.18182
\(26\) −1.06486e8 −1.75768
\(27\) 3.73075e7 0.500375
\(28\) −952472. −0.0104588
\(29\) 2.05111e7 0.185695
\(30\) 1.06794e8 0.802382
\(31\) 1.69267e8 1.06190 0.530951 0.847403i \(-0.321835\pi\)
0.530951 + 0.847403i \(0.321835\pi\)
\(32\) 1.30850e7 0.0689364
\(33\) 2.91240e6 0.0129546
\(34\) 2.51431e8 0.949041
\(35\) 2.87026e6 0.00923733
\(36\) −6.83622e8 −1.88431
\(37\) −6.21479e8 −1.47339 −0.736694 0.676227i \(-0.763614\pi\)
−0.736694 + 0.676227i \(0.763614\pi\)
\(38\) −7.37214e8 −1.50933
\(39\) 1.47531e8 0.261835
\(40\) −2.04688e9 −3.16055
\(41\) −9.80521e8 −1.32174 −0.660869 0.750501i \(-0.729812\pi\)
−0.660869 + 0.750501i \(0.729812\pi\)
\(42\) 1.97298e6 0.00232944
\(43\) 1.55730e9 1.61546 0.807729 0.589554i \(-0.200697\pi\)
0.807729 + 0.589554i \(0.200697\pi\)
\(44\) −1.10566e8 −0.101073
\(45\) 2.06008e9 1.66424
\(46\) −9.21103e8 −0.659386
\(47\) 7.66462e8 0.487475 0.243737 0.969841i \(-0.421626\pi\)
0.243737 + 0.969841i \(0.421626\pi\)
\(48\) −4.84080e8 −0.274214
\(49\) −1.97727e9 −0.999973
\(50\) 8.37783e9 3.79137
\(51\) −3.48345e8 −0.141375
\(52\) −5.60086e9 −2.04285
\(53\) −1.24890e7 −0.00410215 −0.00205108 0.999998i \(-0.500653\pi\)
−0.00205108 + 0.999998i \(0.500653\pi\)
\(54\) 2.93386e9 0.869508
\(55\) 3.33190e8 0.0892682
\(56\) −3.78154e7 −0.00917560
\(57\) 1.02137e9 0.224839
\(58\) 1.61299e9 0.322685
\(59\) 3.94786e9 0.718911 0.359456 0.933162i \(-0.382962\pi\)
0.359456 + 0.933162i \(0.382962\pi\)
\(60\) 5.61707e9 0.932562
\(61\) −6.43946e9 −0.976193 −0.488096 0.872790i \(-0.662309\pi\)
−0.488096 + 0.872790i \(0.662309\pi\)
\(62\) 1.33112e10 1.84528
\(63\) 3.80592e7 0.00483155
\(64\) −8.07043e9 −0.939522
\(65\) 1.68781e10 1.80426
\(66\) 2.29031e8 0.0225114
\(67\) 4.26072e9 0.385542 0.192771 0.981244i \(-0.438253\pi\)
0.192771 + 0.981244i \(0.438253\pi\)
\(68\) 1.32246e10 1.10302
\(69\) 1.27614e9 0.0982263
\(70\) 2.25716e8 0.0160518
\(71\) 2.70019e10 1.77612 0.888062 0.459724i \(-0.152052\pi\)
0.888062 + 0.459724i \(0.152052\pi\)
\(72\) −2.71414e10 −1.65312
\(73\) −8.00729e9 −0.452075 −0.226037 0.974119i \(-0.572577\pi\)
−0.226037 + 0.974119i \(0.572577\pi\)
\(74\) −4.88730e10 −2.56032
\(75\) −1.16070e10 −0.564786
\(76\) −3.87753e10 −1.75421
\(77\) 6.15555e6 0.000259160 0
\(78\) 1.16018e10 0.454994
\(79\) 4.92961e10 1.80245 0.901225 0.433351i \(-0.142669\pi\)
0.901225 + 0.433351i \(0.142669\pi\)
\(80\) −5.53805e10 −1.88957
\(81\) 2.52136e10 0.803464
\(82\) −7.71080e10 −2.29680
\(83\) −7.07907e10 −1.97264 −0.986318 0.164856i \(-0.947284\pi\)
−0.986318 + 0.164856i \(0.947284\pi\)
\(84\) 1.03773e8 0.00270738
\(85\) −3.98519e10 −0.974193
\(86\) 1.22466e11 2.80720
\(87\) −2.23472e9 −0.0480692
\(88\) −4.38975e9 −0.0886717
\(89\) −8.54214e10 −1.62152 −0.810758 0.585381i \(-0.800945\pi\)
−0.810758 + 0.585381i \(0.800945\pi\)
\(90\) 1.62004e11 2.89196
\(91\) 3.11816e8 0.00523806
\(92\) −4.84473e10 −0.766367
\(93\) −1.84419e10 −0.274884
\(94\) 6.02744e10 0.847091
\(95\) 1.16849e11 1.54933
\(96\) −1.42563e9 −0.0178449
\(97\) −4.48857e10 −0.530718 −0.265359 0.964150i \(-0.585490\pi\)
−0.265359 + 0.964150i \(0.585490\pi\)
\(98\) −1.55493e11 −1.73767
\(99\) 4.41805e9 0.0466914
\(100\) 4.40649e11 4.40649
\(101\) −6.57247e9 −0.0622245 −0.0311122 0.999516i \(-0.509905\pi\)
−0.0311122 + 0.999516i \(0.509905\pi\)
\(102\) −2.73938e10 −0.245669
\(103\) −1.46925e11 −1.24880 −0.624398 0.781106i \(-0.714656\pi\)
−0.624398 + 0.781106i \(0.714656\pi\)
\(104\) −2.22367e11 −1.79220
\(105\) −3.12718e8 −0.00239118
\(106\) −9.82136e8 −0.00712836
\(107\) −1.16620e11 −0.803825 −0.401912 0.915678i \(-0.631654\pi\)
−0.401912 + 0.915678i \(0.631654\pi\)
\(108\) 1.54312e11 1.01058
\(109\) −2.91271e11 −1.81322 −0.906612 0.421966i \(-0.861340\pi\)
−0.906612 + 0.421966i \(0.861340\pi\)
\(110\) 2.62020e10 0.155122
\(111\) 6.77110e10 0.381402
\(112\) −1.02313e9 −0.00548571
\(113\) −3.60464e10 −0.184048 −0.0920239 0.995757i \(-0.529334\pi\)
−0.0920239 + 0.995757i \(0.529334\pi\)
\(114\) 8.03205e10 0.390706
\(115\) 1.45995e11 0.676862
\(116\) 8.48388e10 0.375038
\(117\) 2.23801e11 0.943712
\(118\) 3.10459e11 1.24926
\(119\) −7.36248e8 −0.00282824
\(120\) 2.23011e11 0.818143
\(121\) −2.84597e11 −0.997496
\(122\) −5.06398e11 −1.69634
\(123\) 1.06829e11 0.342146
\(124\) 7.00129e11 2.14466
\(125\) −7.19272e11 −2.10809
\(126\) 2.99297e9 0.00839584
\(127\) −3.25875e11 −0.875247 −0.437624 0.899158i \(-0.644180\pi\)
−0.437624 + 0.899158i \(0.644180\pi\)
\(128\) −6.61456e11 −1.70155
\(129\) −1.69670e11 −0.418178
\(130\) 1.32729e12 3.13529
\(131\) 1.38770e11 0.314271 0.157136 0.987577i \(-0.449774\pi\)
0.157136 + 0.987577i \(0.449774\pi\)
\(132\) 1.20464e10 0.0261637
\(133\) 2.15873e9 0.00449795
\(134\) 3.35063e11 0.669961
\(135\) −4.65017e11 −0.892551
\(136\) 5.25046e11 0.967683
\(137\) 8.88362e11 1.57263 0.786315 0.617825i \(-0.211986\pi\)
0.786315 + 0.617825i \(0.211986\pi\)
\(138\) 1.00355e11 0.170689
\(139\) −5.26116e11 −0.860003 −0.430001 0.902828i \(-0.641487\pi\)
−0.430001 + 0.902828i \(0.641487\pi\)
\(140\) 1.18720e10 0.0186561
\(141\) −8.35070e10 −0.126188
\(142\) 2.12342e12 3.08639
\(143\) 3.61967e10 0.0506199
\(144\) −7.34337e11 −0.988330
\(145\) −2.55660e11 −0.331237
\(146\) −6.29692e11 −0.785575
\(147\) 2.15427e11 0.258854
\(148\) −2.57058e12 −2.97572
\(149\) −2.24927e11 −0.250910 −0.125455 0.992099i \(-0.540039\pi\)
−0.125455 + 0.992099i \(0.540039\pi\)
\(150\) −9.12775e11 −0.981436
\(151\) 7.38831e11 0.765900 0.382950 0.923769i \(-0.374908\pi\)
0.382950 + 0.923769i \(0.374908\pi\)
\(152\) −1.53947e12 −1.53898
\(153\) −5.28431e11 −0.509548
\(154\) 4.84072e8 0.000450345 0
\(155\) −2.10982e12 −1.89418
\(156\) 6.10221e11 0.528813
\(157\) 1.40898e12 1.17884 0.589422 0.807825i \(-0.299355\pi\)
0.589422 + 0.807825i \(0.299355\pi\)
\(158\) 3.87664e12 3.13214
\(159\) 1.36070e9 0.00106188
\(160\) −1.63097e11 −0.122966
\(161\) 2.69720e9 0.00196504
\(162\) 1.98279e12 1.39619
\(163\) −9.72492e11 −0.661994 −0.330997 0.943632i \(-0.607385\pi\)
−0.330997 + 0.943632i \(0.607385\pi\)
\(164\) −4.05566e12 −2.66944
\(165\) −3.63015e10 −0.0231080
\(166\) −5.56697e12 −3.42787
\(167\) −3.91628e10 −0.0233310 −0.0116655 0.999932i \(-0.503713\pi\)
−0.0116655 + 0.999932i \(0.503713\pi\)
\(168\) 4.12004e9 0.00237520
\(169\) 4.14236e10 0.0231138
\(170\) −3.13395e12 −1.69287
\(171\) 1.54940e12 0.810371
\(172\) 6.44134e12 3.26265
\(173\) −2.21018e12 −1.08436 −0.542181 0.840262i \(-0.682401\pi\)
−0.542181 + 0.840262i \(0.682401\pi\)
\(174\) −1.75738e11 −0.0835304
\(175\) −2.45322e10 −0.0112987
\(176\) −1.18769e11 −0.0530132
\(177\) −4.30124e11 −0.186098
\(178\) −6.71752e12 −2.81773
\(179\) 2.23234e12 0.907965 0.453983 0.891010i \(-0.350003\pi\)
0.453983 + 0.891010i \(0.350003\pi\)
\(180\) 8.52096e12 3.36116
\(181\) −7.82120e10 −0.0299255 −0.0149627 0.999888i \(-0.504763\pi\)
−0.0149627 + 0.999888i \(0.504763\pi\)
\(182\) 2.45212e10 0.00910224
\(183\) 7.01588e11 0.252698
\(184\) −1.92347e12 −0.672338
\(185\) 7.74638e12 2.62818
\(186\) −1.45027e12 −0.477670
\(187\) −8.54664e10 −0.0273317
\(188\) 3.17026e12 0.984525
\(189\) −8.59101e9 −0.00259122
\(190\) 9.18896e12 2.69229
\(191\) 2.06678e12 0.588317 0.294158 0.955757i \(-0.404961\pi\)
0.294158 + 0.955757i \(0.404961\pi\)
\(192\) 8.79284e11 0.243205
\(193\) 2.83320e12 0.761574 0.380787 0.924663i \(-0.375653\pi\)
0.380787 + 0.924663i \(0.375653\pi\)
\(194\) −3.52981e12 −0.922235
\(195\) −1.83889e12 −0.467052
\(196\) −8.17845e12 −2.01959
\(197\) 4.94999e12 1.18861 0.594305 0.804239i \(-0.297427\pi\)
0.594305 + 0.804239i \(0.297427\pi\)
\(198\) 3.47435e11 0.0811362
\(199\) −3.92808e12 −0.892253 −0.446127 0.894970i \(-0.647197\pi\)
−0.446127 + 0.894970i \(0.647197\pi\)
\(200\) 1.74948e13 3.86584
\(201\) −4.64211e11 −0.0998016
\(202\) −5.16858e11 −0.108128
\(203\) −4.72322e9 −0.000961634 0
\(204\) −1.44083e12 −0.285527
\(205\) 1.22216e13 2.35767
\(206\) −1.15542e13 −2.17005
\(207\) 1.93587e12 0.354030
\(208\) −6.01637e12 −1.07149
\(209\) 2.50594e11 0.0434676
\(210\) −2.45921e10 −0.00415518
\(211\) −6.85663e12 −1.12865 −0.564323 0.825554i \(-0.690863\pi\)
−0.564323 + 0.825554i \(0.690863\pi\)
\(212\) −5.16575e10 −0.00828488
\(213\) −2.94189e12 −0.459768
\(214\) −9.17096e12 −1.39682
\(215\) −1.94109e13 −2.88160
\(216\) 6.12656e12 0.886587
\(217\) −3.89782e10 −0.00549911
\(218\) −2.29055e13 −3.15086
\(219\) 8.72405e11 0.117024
\(220\) 1.37815e12 0.180290
\(221\) −4.32939e12 −0.552420
\(222\) 5.32478e12 0.662766
\(223\) −4.05692e12 −0.492629 −0.246314 0.969190i \(-0.579220\pi\)
−0.246314 + 0.969190i \(0.579220\pi\)
\(224\) −3.01315e9 −0.000356991 0
\(225\) −1.76076e13 −2.03562
\(226\) −2.83469e12 −0.319822
\(227\) 1.11423e13 1.22696 0.613481 0.789709i \(-0.289769\pi\)
0.613481 + 0.789709i \(0.289769\pi\)
\(228\) 4.22462e12 0.454095
\(229\) 5.75727e12 0.604118 0.302059 0.953289i \(-0.402326\pi\)
0.302059 + 0.953289i \(0.402326\pi\)
\(230\) 1.14810e13 1.17619
\(231\) −6.70656e8 −6.70863e−5 0
\(232\) 3.36830e12 0.329023
\(233\) 3.46144e12 0.330217 0.165108 0.986275i \(-0.447203\pi\)
0.165108 + 0.986275i \(0.447203\pi\)
\(234\) 1.75997e13 1.63990
\(235\) −9.55351e12 −0.869541
\(236\) 1.63292e13 1.45194
\(237\) −5.37087e12 −0.466583
\(238\) −5.78985e10 −0.00491466
\(239\) 1.86431e13 1.54643 0.773215 0.634144i \(-0.218647\pi\)
0.773215 + 0.634144i \(0.218647\pi\)
\(240\) 6.03378e12 0.489134
\(241\) −3.22030e12 −0.255154 −0.127577 0.991829i \(-0.540720\pi\)
−0.127577 + 0.991829i \(0.540720\pi\)
\(242\) −2.23807e13 −1.73336
\(243\) −9.35596e12 −0.708360
\(244\) −2.66351e13 −1.97156
\(245\) 2.46456e13 1.78372
\(246\) 8.40102e12 0.594551
\(247\) 1.26941e13 0.878553
\(248\) 2.77967e13 1.88152
\(249\) 7.71274e12 0.510637
\(250\) −5.65635e13 −3.66325
\(251\) 1.10256e13 0.698547 0.349273 0.937021i \(-0.386428\pi\)
0.349273 + 0.937021i \(0.386428\pi\)
\(252\) 1.57421e11 0.00975800
\(253\) 3.13101e11 0.0189899
\(254\) −2.56268e13 −1.52093
\(255\) 4.34192e12 0.252180
\(256\) −3.54885e13 −2.01729
\(257\) −1.84306e13 −1.02543 −0.512716 0.858558i \(-0.671361\pi\)
−0.512716 + 0.858558i \(0.671361\pi\)
\(258\) −1.33428e13 −0.726673
\(259\) 1.43111e11 0.00763002
\(260\) 6.98116e13 3.64396
\(261\) −3.39001e12 −0.173252
\(262\) 1.09129e13 0.546113
\(263\) 1.30599e13 0.640005 0.320003 0.947417i \(-0.396316\pi\)
0.320003 + 0.947417i \(0.396316\pi\)
\(264\) 4.78269e11 0.0229536
\(265\) 1.55669e11 0.00731728
\(266\) 1.69762e11 0.00781614
\(267\) 9.30677e12 0.419747
\(268\) 1.76233e13 0.778657
\(269\) −3.11337e13 −1.34770 −0.673850 0.738868i \(-0.735361\pi\)
−0.673850 + 0.738868i \(0.735361\pi\)
\(270\) −3.65689e13 −1.55100
\(271\) 2.84357e13 1.18177 0.590885 0.806756i \(-0.298779\pi\)
0.590885 + 0.806756i \(0.298779\pi\)
\(272\) 1.42056e13 0.578538
\(273\) −3.39728e10 −0.00135593
\(274\) 6.98607e13 2.73278
\(275\) −2.84779e12 −0.109189
\(276\) 5.27840e12 0.198382
\(277\) −2.26144e13 −0.833193 −0.416597 0.909091i \(-0.636777\pi\)
−0.416597 + 0.909091i \(0.636777\pi\)
\(278\) −4.13737e13 −1.49444
\(279\) −2.79759e13 −0.990744
\(280\) 4.71347e11 0.0163671
\(281\) 3.67511e13 1.25137 0.625684 0.780076i \(-0.284820\pi\)
0.625684 + 0.780076i \(0.284820\pi\)
\(282\) −6.56698e12 −0.219278
\(283\) 1.41159e13 0.462255 0.231128 0.972923i \(-0.425759\pi\)
0.231128 + 0.972923i \(0.425759\pi\)
\(284\) 1.11686e14 3.58713
\(285\) −1.27308e13 −0.401060
\(286\) 2.84651e12 0.0879628
\(287\) 2.25790e11 0.00684470
\(288\) −2.16264e12 −0.0643170
\(289\) −2.40495e13 −0.701726
\(290\) −2.01051e13 −0.575594
\(291\) 4.89036e12 0.137382
\(292\) −3.31200e13 −0.913029
\(293\) −3.80422e12 −0.102918 −0.0514592 0.998675i \(-0.516387\pi\)
−0.0514592 + 0.998675i \(0.516387\pi\)
\(294\) 1.69411e13 0.449813
\(295\) −4.92078e13 −1.28237
\(296\) −1.02058e14 −2.61061
\(297\) −9.97276e11 −0.0250412
\(298\) −1.76882e13 −0.436009
\(299\) 1.58605e13 0.383817
\(300\) −4.80093e13 −1.14067
\(301\) −3.58608e11 −0.00836574
\(302\) 5.81016e13 1.33091
\(303\) 7.16080e11 0.0161075
\(304\) −4.16519e13 −0.920090
\(305\) 8.02643e13 1.74130
\(306\) −4.15557e13 −0.885447
\(307\) 7.16897e13 1.50036 0.750180 0.661233i \(-0.229967\pi\)
0.750180 + 0.661233i \(0.229967\pi\)
\(308\) 2.54608e10 0.000523411 0
\(309\) 1.60077e13 0.323264
\(310\) −1.65916e14 −3.29154
\(311\) 5.89462e13 1.14888 0.574439 0.818547i \(-0.305220\pi\)
0.574439 + 0.818547i \(0.305220\pi\)
\(312\) 2.42272e13 0.463931
\(313\) −3.15570e13 −0.593747 −0.296874 0.954917i \(-0.595944\pi\)
−0.296874 + 0.954917i \(0.595944\pi\)
\(314\) 1.10802e14 2.04849
\(315\) −4.74386e11 −0.00861835
\(316\) 2.03900e14 3.64030
\(317\) 4.00150e13 0.702096 0.351048 0.936357i \(-0.385825\pi\)
0.351048 + 0.936357i \(0.385825\pi\)
\(318\) 1.07005e11 0.00184525
\(319\) −5.48288e11 −0.00929309
\(320\) 1.00593e14 1.67589
\(321\) 1.27059e13 0.208078
\(322\) 2.12107e11 0.00341467
\(323\) −2.99728e13 −0.474366
\(324\) 1.04289e14 1.62271
\(325\) −1.44258e14 −2.20689
\(326\) −7.64766e13 −1.15035
\(327\) 3.17344e13 0.469372
\(328\) −1.61019e14 −2.34192
\(329\) −1.76497e11 −0.00252442
\(330\) −2.85474e12 −0.0401551
\(331\) −7.96381e13 −1.10171 −0.550855 0.834601i \(-0.685698\pi\)
−0.550855 + 0.834601i \(0.685698\pi\)
\(332\) −2.92807e14 −3.98402
\(333\) 1.02716e14 1.37466
\(334\) −3.07976e12 −0.0405425
\(335\) −5.31075e13 −0.687717
\(336\) 1.11472e11 0.00142003
\(337\) 1.59262e13 0.199594 0.0997971 0.995008i \(-0.468181\pi\)
0.0997971 + 0.995008i \(0.468181\pi\)
\(338\) 3.25755e12 0.0401651
\(339\) 3.92731e12 0.0476427
\(340\) −1.64837e14 −1.96752
\(341\) −4.52473e12 −0.0531427
\(342\) 1.21844e14 1.40819
\(343\) 9.10648e11 0.0103570
\(344\) 2.55736e14 2.86234
\(345\) −1.59064e13 −0.175213
\(346\) −1.73808e14 −1.88431
\(347\) −1.48839e14 −1.58820 −0.794100 0.607787i \(-0.792057\pi\)
−0.794100 + 0.607787i \(0.792057\pi\)
\(348\) −9.24330e12 −0.0970826
\(349\) 1.26082e14 1.30351 0.651755 0.758430i \(-0.274033\pi\)
0.651755 + 0.758430i \(0.274033\pi\)
\(350\) −1.92921e12 −0.0196338
\(351\) −5.05181e13 −0.506125
\(352\) −3.49778e11 −0.00344991
\(353\) −7.45266e12 −0.0723687 −0.0361843 0.999345i \(-0.511520\pi\)
−0.0361843 + 0.999345i \(0.511520\pi\)
\(354\) −3.38249e13 −0.323384
\(355\) −3.36563e14 −3.16819
\(356\) −3.53322e14 −3.27488
\(357\) 8.02153e10 0.000732119 0
\(358\) 1.75551e14 1.57778
\(359\) −3.60657e13 −0.319209 −0.159604 0.987181i \(-0.551022\pi\)
−0.159604 + 0.987181i \(0.551022\pi\)
\(360\) 3.38302e14 2.94877
\(361\) −2.86079e13 −0.245582
\(362\) −6.15058e12 −0.0520019
\(363\) 3.10072e13 0.258212
\(364\) 1.28974e12 0.0105790
\(365\) 9.98064e13 0.806395
\(366\) 5.51728e13 0.439116
\(367\) 4.67133e13 0.366250 0.183125 0.983090i \(-0.441379\pi\)
0.183125 + 0.983090i \(0.441379\pi\)
\(368\) −5.20415e13 −0.401964
\(369\) 1.62057e14 1.23317
\(370\) 6.09175e14 4.56701
\(371\) 2.87592e9 2.12432e−5 0
\(372\) −7.62800e13 −0.555168
\(373\) −1.54301e14 −1.10655 −0.553274 0.832999i \(-0.686622\pi\)
−0.553274 + 0.832999i \(0.686622\pi\)
\(374\) −6.72107e12 −0.0474946
\(375\) 7.83657e13 0.545700
\(376\) 1.25867e14 0.863730
\(377\) −2.77741e13 −0.187829
\(378\) −6.75596e11 −0.00450279
\(379\) 8.46962e13 0.556350 0.278175 0.960530i \(-0.410270\pi\)
0.278175 + 0.960530i \(0.410270\pi\)
\(380\) 4.83313e14 3.12909
\(381\) 3.55045e13 0.226567
\(382\) 1.62531e14 1.02233
\(383\) 3.82978e13 0.237455 0.118727 0.992927i \(-0.462119\pi\)
0.118727 + 0.992927i \(0.462119\pi\)
\(384\) 7.20665e13 0.440465
\(385\) −7.67255e10 −0.000462281 0
\(386\) 2.22802e14 1.32340
\(387\) −2.57385e14 −1.50721
\(388\) −1.85658e14 −1.07186
\(389\) 2.06383e14 1.17476 0.587381 0.809310i \(-0.300159\pi\)
0.587381 + 0.809310i \(0.300159\pi\)
\(390\) −1.44610e14 −0.811602
\(391\) −3.74491e13 −0.207238
\(392\) −3.24704e14 −1.77180
\(393\) −1.51192e13 −0.0813525
\(394\) 3.89266e14 2.06546
\(395\) −6.14448e14 −3.21515
\(396\) 1.82741e13 0.0942999
\(397\) −2.05709e12 −0.0104690 −0.00523452 0.999986i \(-0.501666\pi\)
−0.00523452 + 0.999986i \(0.501666\pi\)
\(398\) −3.08903e14 −1.55048
\(399\) −2.35197e11 −0.00116434
\(400\) 4.73340e14 2.31123
\(401\) −3.68472e14 −1.77464 −0.887320 0.461154i \(-0.847436\pi\)
−0.887320 + 0.461154i \(0.847436\pi\)
\(402\) −3.65055e13 −0.173426
\(403\) −2.29205e14 −1.07410
\(404\) −2.71852e13 −0.125671
\(405\) −3.14273e14 −1.43319
\(406\) −3.71433e11 −0.00167104
\(407\) 1.66129e13 0.0737354
\(408\) −5.72044e13 −0.250495
\(409\) 1.82553e14 0.788700 0.394350 0.918960i \(-0.370970\pi\)
0.394350 + 0.918960i \(0.370970\pi\)
\(410\) 9.61108e14 4.09695
\(411\) −9.67882e13 −0.407092
\(412\) −6.07716e14 −2.52212
\(413\) −9.09095e11 −0.00372292
\(414\) 1.52237e14 0.615202
\(415\) 8.82366e14 3.51872
\(416\) −1.77184e13 −0.0697285
\(417\) 5.73210e13 0.222621
\(418\) 1.97066e13 0.0755341
\(419\) −3.84878e14 −1.45595 −0.727974 0.685604i \(-0.759538\pi\)
−0.727974 + 0.685604i \(0.759538\pi\)
\(420\) −1.29347e12 −0.00482933
\(421\) 1.47090e14 0.542042 0.271021 0.962573i \(-0.412639\pi\)
0.271021 + 0.962573i \(0.412639\pi\)
\(422\) −5.39205e14 −1.96126
\(423\) −1.26678e14 −0.454810
\(424\) −2.05092e12 −0.00726838
\(425\) 3.40616e14 1.19159
\(426\) −2.31350e14 −0.798945
\(427\) 1.48285e12 0.00505527
\(428\) −4.82366e14 −1.62344
\(429\) −3.94368e12 −0.0131035
\(430\) −1.52647e15 −5.00739
\(431\) 2.42526e14 0.785477 0.392738 0.919650i \(-0.371528\pi\)
0.392738 + 0.919650i \(0.371528\pi\)
\(432\) 1.65760e14 0.530054
\(433\) −7.22429e13 −0.228093 −0.114046 0.993475i \(-0.536381\pi\)
−0.114046 + 0.993475i \(0.536381\pi\)
\(434\) −3.06524e12 −0.00955588
\(435\) 2.78545e13 0.0857441
\(436\) −1.20476e15 −3.66206
\(437\) 1.09804e14 0.329586
\(438\) 6.86058e13 0.203354
\(439\) 4.25961e14 1.24685 0.623426 0.781882i \(-0.285740\pi\)
0.623426 + 0.781882i \(0.285740\pi\)
\(440\) 5.47158e13 0.158169
\(441\) 3.26797e14 0.932966
\(442\) −3.40463e14 −0.959947
\(443\) −2.61970e14 −0.729509 −0.364754 0.931104i \(-0.618847\pi\)
−0.364754 + 0.931104i \(0.618847\pi\)
\(444\) 2.80068e14 0.770295
\(445\) 1.06473e15 2.89240
\(446\) −3.19036e14 −0.856047
\(447\) 2.45061e13 0.0649506
\(448\) 1.85842e12 0.00486537
\(449\) 5.02452e14 1.29939 0.649695 0.760195i \(-0.274897\pi\)
0.649695 + 0.760195i \(0.274897\pi\)
\(450\) −1.38466e15 −3.53732
\(451\) 2.62105e13 0.0661462
\(452\) −1.49096e14 −0.371711
\(453\) −8.04966e13 −0.198261
\(454\) 8.76226e14 2.13211
\(455\) −3.88661e12 −0.00934347
\(456\) 1.67727e14 0.398380
\(457\) −3.12924e14 −0.734344 −0.367172 0.930153i \(-0.619674\pi\)
−0.367172 + 0.930153i \(0.619674\pi\)
\(458\) 4.52751e14 1.04978
\(459\) 1.19281e14 0.273277
\(460\) 6.03869e14 1.36702
\(461\) −4.17648e14 −0.934234 −0.467117 0.884196i \(-0.654707\pi\)
−0.467117 + 0.884196i \(0.654707\pi\)
\(462\) −5.27403e10 −0.000116577 0
\(463\) −2.26782e14 −0.495351 −0.247676 0.968843i \(-0.579667\pi\)
−0.247676 + 0.968843i \(0.579667\pi\)
\(464\) 9.11327e13 0.196710
\(465\) 2.29868e14 0.490329
\(466\) 2.72207e14 0.573821
\(467\) 9.21047e14 1.91884 0.959421 0.281979i \(-0.0909909\pi\)
0.959421 + 0.281979i \(0.0909909\pi\)
\(468\) 9.25691e14 1.90596
\(469\) −9.81140e11 −0.00199655
\(470\) −7.51287e14 −1.51101
\(471\) −1.53510e14 −0.305156
\(472\) 6.48309e14 1.27380
\(473\) −4.16285e13 −0.0808453
\(474\) −4.22365e14 −0.810787
\(475\) −9.98710e14 −1.89507
\(476\) −3.04529e12 −0.00571203
\(477\) 2.06415e12 0.00382727
\(478\) 1.46609e15 2.68725
\(479\) 2.79890e14 0.507156 0.253578 0.967315i \(-0.418393\pi\)
0.253578 + 0.967315i \(0.418393\pi\)
\(480\) 1.77696e13 0.0318311
\(481\) 8.41544e14 1.49032
\(482\) −2.53244e14 −0.443384
\(483\) −2.93864e11 −0.000508671 0
\(484\) −1.17716e15 −2.01458
\(485\) 5.59476e14 0.946676
\(486\) −7.35752e14 −1.23093
\(487\) −3.65476e14 −0.604574 −0.302287 0.953217i \(-0.597750\pi\)
−0.302287 + 0.953217i \(0.597750\pi\)
\(488\) −1.05748e15 −1.72966
\(489\) 1.05954e14 0.171364
\(490\) 1.93813e15 3.09959
\(491\) 8.51205e14 1.34613 0.673063 0.739585i \(-0.264978\pi\)
0.673063 + 0.739585i \(0.264978\pi\)
\(492\) 4.41869e14 0.691012
\(493\) 6.55793e13 0.101416
\(494\) 9.98261e14 1.52667
\(495\) −5.50685e13 −0.0832865
\(496\) 7.52069e14 1.12489
\(497\) −6.21787e12 −0.00919776
\(498\) 6.06529e14 0.887340
\(499\) −1.13855e15 −1.64740 −0.823702 0.567023i \(-0.808095\pi\)
−0.823702 + 0.567023i \(0.808095\pi\)
\(500\) −2.97507e15 −4.25758
\(501\) 4.26684e12 0.00603947
\(502\) 8.67049e14 1.21387
\(503\) −1.06306e15 −1.47208 −0.736041 0.676936i \(-0.763307\pi\)
−0.736041 + 0.676936i \(0.763307\pi\)
\(504\) 6.25000e12 0.00856075
\(505\) 8.19222e13 0.110994
\(506\) 2.46222e13 0.0329989
\(507\) −4.51316e12 −0.00598325
\(508\) −1.34789e15 −1.76769
\(509\) −5.69558e14 −0.738908 −0.369454 0.929249i \(-0.620455\pi\)
−0.369454 + 0.929249i \(0.620455\pi\)
\(510\) 3.41448e14 0.438216
\(511\) 1.84388e12 0.00234109
\(512\) −1.43615e15 −1.80391
\(513\) −3.49742e14 −0.434612
\(514\) −1.44938e15 −1.78190
\(515\) 1.83134e15 2.22756
\(516\) −7.01793e14 −0.844570
\(517\) −2.04885e13 −0.0243956
\(518\) 1.12543e13 0.0132588
\(519\) 2.40802e14 0.280699
\(520\) 2.77168e15 3.19687
\(521\) −5.33971e14 −0.609411 −0.304706 0.952447i \(-0.598558\pi\)
−0.304706 + 0.952447i \(0.598558\pi\)
\(522\) −2.66590e14 −0.301062
\(523\) −1.01418e15 −1.13333 −0.566663 0.823949i \(-0.691766\pi\)
−0.566663 + 0.823949i \(0.691766\pi\)
\(524\) 5.73986e14 0.634716
\(525\) 2.67282e12 0.00292478
\(526\) 1.02703e15 1.11214
\(527\) 5.41190e14 0.579951
\(528\) 1.29401e13 0.0137230
\(529\) −8.15617e14 −0.856013
\(530\) 1.22418e13 0.0127153
\(531\) −6.52488e14 −0.670738
\(532\) 8.92901e12 0.00908425
\(533\) 1.32772e15 1.33693
\(534\) 7.31883e14 0.729399
\(535\) 1.45360e15 1.43383
\(536\) 6.99687e14 0.683121
\(537\) −2.43217e14 −0.235036
\(538\) −2.44835e15 −2.34191
\(539\) 5.28550e13 0.0500435
\(540\) −1.92342e15 −1.80263
\(541\) −2.04525e15 −1.89741 −0.948707 0.316158i \(-0.897607\pi\)
−0.948707 + 0.316158i \(0.897607\pi\)
\(542\) 2.23618e15 2.05358
\(543\) 8.52130e12 0.00774652
\(544\) 4.18360e13 0.0376492
\(545\) 3.63053e15 3.23436
\(546\) −2.67161e12 −0.00235621
\(547\) −2.04513e15 −1.78562 −0.892811 0.450431i \(-0.851270\pi\)
−0.892811 + 0.450431i \(0.851270\pi\)
\(548\) 3.67447e15 3.17615
\(549\) 1.06429e15 0.910779
\(550\) −2.23950e14 −0.189739
\(551\) −1.92283e14 −0.161290
\(552\) 2.09565e14 0.174042
\(553\) −1.13517e13 −0.00933409
\(554\) −1.77839e15 −1.44785
\(555\) −8.43979e14 −0.680331
\(556\) −2.17613e15 −1.73690
\(557\) −3.44808e14 −0.272504 −0.136252 0.990674i \(-0.543506\pi\)
−0.136252 + 0.990674i \(0.543506\pi\)
\(558\) −2.20002e15 −1.72163
\(559\) −2.10874e15 −1.63402
\(560\) 1.27528e13 0.00978522
\(561\) 9.31168e12 0.00707510
\(562\) 2.89010e15 2.17452
\(563\) −4.52289e14 −0.336992 −0.168496 0.985702i \(-0.553891\pi\)
−0.168496 + 0.985702i \(0.553891\pi\)
\(564\) −3.45404e14 −0.254855
\(565\) 4.49298e14 0.328298
\(566\) 1.11007e15 0.803266
\(567\) −5.80606e12 −0.00416078
\(568\) 4.43419e15 3.14702
\(569\) 3.63369e14 0.255405 0.127703 0.991812i \(-0.459240\pi\)
0.127703 + 0.991812i \(0.459240\pi\)
\(570\) −1.00115e15 −0.696927
\(571\) −2.17931e15 −1.50252 −0.751261 0.660005i \(-0.770554\pi\)
−0.751261 + 0.660005i \(0.770554\pi\)
\(572\) 1.49718e14 0.102234
\(573\) −2.25179e14 −0.152292
\(574\) 1.77561e13 0.0118941
\(575\) −1.24783e15 −0.827906
\(576\) 1.33385e15 0.876566
\(577\) 1.34691e15 0.876740 0.438370 0.898795i \(-0.355556\pi\)
0.438370 + 0.898795i \(0.355556\pi\)
\(578\) −1.89125e15 −1.21940
\(579\) −3.08681e14 −0.197141
\(580\) −1.05747e15 −0.668980
\(581\) 1.63014e13 0.0102154
\(582\) 3.84577e14 0.238730
\(583\) 3.33847e11 0.000205292 0
\(584\) −1.31494e15 −0.801006
\(585\) −2.78955e15 −1.68336
\(586\) −2.99163e14 −0.178843
\(587\) 2.46936e15 1.46243 0.731214 0.682148i \(-0.238954\pi\)
0.731214 + 0.682148i \(0.238954\pi\)
\(588\) 8.91054e14 0.522792
\(589\) −1.58681e15 −0.922338
\(590\) −3.86970e15 −2.22839
\(591\) −5.39308e14 −0.307684
\(592\) −2.76128e15 −1.56078
\(593\) −8.09156e14 −0.453139 −0.226569 0.973995i \(-0.572751\pi\)
−0.226569 + 0.973995i \(0.572751\pi\)
\(594\) −7.84257e13 −0.0435144
\(595\) 9.17692e12 0.00504491
\(596\) −9.30350e14 −0.506748
\(597\) 4.27969e14 0.230969
\(598\) 1.24726e15 0.666963
\(599\) 6.45581e14 0.342061 0.171030 0.985266i \(-0.445290\pi\)
0.171030 + 0.985266i \(0.445290\pi\)
\(600\) −1.90608e15 −1.00071
\(601\) 1.25215e14 0.0651397 0.0325698 0.999469i \(-0.489631\pi\)
0.0325698 + 0.999469i \(0.489631\pi\)
\(602\) −2.82009e13 −0.0145372
\(603\) −7.04198e14 −0.359707
\(604\) 3.05597e15 1.54684
\(605\) 3.54734e15 1.77930
\(606\) 5.63124e13 0.0279901
\(607\) 6.50089e14 0.320210 0.160105 0.987100i \(-0.448817\pi\)
0.160105 + 0.987100i \(0.448817\pi\)
\(608\) −1.22666e14 −0.0598762
\(609\) 5.14601e11 0.000248929 0
\(610\) 6.31197e15 3.02587
\(611\) −1.03786e15 −0.493076
\(612\) −2.18571e15 −1.02910
\(613\) 2.17950e14 0.101701 0.0508504 0.998706i \(-0.483807\pi\)
0.0508504 + 0.998706i \(0.483807\pi\)
\(614\) 5.63767e15 2.60720
\(615\) −1.33156e15 −0.610308
\(616\) 1.01085e12 0.000459191 0
\(617\) −1.34231e15 −0.604343 −0.302172 0.953254i \(-0.597712\pi\)
−0.302172 + 0.953254i \(0.597712\pi\)
\(618\) 1.25884e15 0.561739
\(619\) −1.52758e15 −0.675624 −0.337812 0.941214i \(-0.609687\pi\)
−0.337812 + 0.941214i \(0.609687\pi\)
\(620\) −8.72671e15 −3.82557
\(621\) −4.36980e14 −0.189871
\(622\) 4.63552e15 1.99642
\(623\) 1.96705e13 0.00839711
\(624\) 6.55492e14 0.277365
\(625\) 3.76347e15 1.57851
\(626\) −2.48164e15 −1.03176
\(627\) −2.73025e13 −0.0112520
\(628\) 5.82786e15 2.38084
\(629\) −1.98702e15 −0.804682
\(630\) −3.73057e13 −0.0149762
\(631\) −3.33450e15 −1.32699 −0.663497 0.748179i \(-0.730928\pi\)
−0.663497 + 0.748179i \(0.730928\pi\)
\(632\) 8.09530e15 3.19366
\(633\) 7.47039e14 0.292162
\(634\) 3.14677e15 1.22004
\(635\) 4.06185e15 1.56124
\(636\) 5.62815e12 0.00214463
\(637\) 2.67742e15 1.01146
\(638\) −4.31173e13 −0.0161487
\(639\) −4.46278e15 −1.65711
\(640\) 8.24467e15 3.03517
\(641\) 5.75395e14 0.210013 0.105007 0.994472i \(-0.466514\pi\)
0.105007 + 0.994472i \(0.466514\pi\)
\(642\) 9.99189e14 0.361580
\(643\) −5.23490e14 −0.187823 −0.0939113 0.995581i \(-0.529937\pi\)
−0.0939113 + 0.995581i \(0.529937\pi\)
\(644\) 1.11562e13 0.00396867
\(645\) 2.11484e15 0.745932
\(646\) −2.35706e15 −0.824311
\(647\) −4.35727e15 −1.51092 −0.755458 0.655197i \(-0.772586\pi\)
−0.755458 + 0.655197i \(0.772586\pi\)
\(648\) 4.14052e15 1.42361
\(649\) −1.05531e14 −0.0359778
\(650\) −1.13444e16 −3.83494
\(651\) 4.24672e12 0.00142350
\(652\) −4.02245e15 −1.33699
\(653\) −3.06751e14 −0.101103 −0.0505514 0.998721i \(-0.516098\pi\)
−0.0505514 + 0.998721i \(0.516098\pi\)
\(654\) 2.49559e15 0.815633
\(655\) −1.72970e15 −0.560586
\(656\) −4.35653e15 −1.40014
\(657\) 1.32342e15 0.421782
\(658\) −1.38797e13 −0.00438671
\(659\) 5.67369e14 0.177826 0.0889132 0.996039i \(-0.471661\pi\)
0.0889132 + 0.996039i \(0.471661\pi\)
\(660\) −1.50151e14 −0.0466699
\(661\) 5.48457e15 1.69057 0.845287 0.534312i \(-0.179429\pi\)
0.845287 + 0.534312i \(0.179429\pi\)
\(662\) −6.26273e15 −1.91445
\(663\) 4.71693e14 0.143000
\(664\) −1.16251e16 −3.49520
\(665\) −2.69074e13 −0.00802329
\(666\) 8.07756e15 2.38876
\(667\) −2.40246e14 −0.0704633
\(668\) −1.61986e14 −0.0471202
\(669\) 4.42007e14 0.127522
\(670\) −4.17637e15 −1.19505
\(671\) 1.72135e14 0.0488534
\(672\) 3.28287e11 9.24108e−5 0
\(673\) 6.70532e15 1.87213 0.936066 0.351825i \(-0.114439\pi\)
0.936066 + 0.351825i \(0.114439\pi\)
\(674\) 1.25243e15 0.346837
\(675\) 3.97452e15 1.09173
\(676\) 1.71338e14 0.0466816
\(677\) 6.29656e14 0.170163 0.0850816 0.996374i \(-0.472885\pi\)
0.0850816 + 0.996374i \(0.472885\pi\)
\(678\) 3.08843e14 0.0827893
\(679\) 1.03361e13 0.00274835
\(680\) −6.54440e15 −1.72612
\(681\) −1.21396e15 −0.317612
\(682\) −3.55824e14 −0.0923467
\(683\) 3.10297e15 0.798846 0.399423 0.916767i \(-0.369210\pi\)
0.399423 + 0.916767i \(0.369210\pi\)
\(684\) 6.40865e15 1.63666
\(685\) −1.10729e16 −2.80520
\(686\) 7.16132e13 0.0179974
\(687\) −6.27263e14 −0.156382
\(688\) 6.91921e15 1.71128
\(689\) 1.69114e13 0.00414929
\(690\) −1.25087e15 −0.304469
\(691\) 4.40450e15 1.06357 0.531786 0.846879i \(-0.321521\pi\)
0.531786 + 0.846879i \(0.321521\pi\)
\(692\) −9.14182e15 −2.19002
\(693\) −1.01737e12 −0.000241794 0
\(694\) −1.17047e16 −2.75983
\(695\) 6.55774e15 1.53404
\(696\) −3.66981e14 −0.0851711
\(697\) −3.13497e15 −0.721860
\(698\) 9.91509e15 2.26512
\(699\) −3.77128e14 −0.0854800
\(700\) −1.01471e14 −0.0228193
\(701\) 2.42636e15 0.541385 0.270693 0.962666i \(-0.412747\pi\)
0.270693 + 0.962666i \(0.412747\pi\)
\(702\) −3.97273e15 −0.879499
\(703\) 5.82609e15 1.27974
\(704\) 2.15733e14 0.0470182
\(705\) 1.04087e15 0.225090
\(706\) −5.86076e14 −0.125756
\(707\) 1.51348e12 0.000322233 0
\(708\) −1.77909e15 −0.375851
\(709\) −6.40975e15 −1.34365 −0.671826 0.740709i \(-0.734490\pi\)
−0.671826 + 0.740709i \(0.734490\pi\)
\(710\) −2.64673e16 −5.50540
\(711\) −8.14749e15 −1.68167
\(712\) −1.40277e16 −2.87308
\(713\) −1.98262e15 −0.402946
\(714\) 6.30812e12 0.00127221
\(715\) −4.51172e14 −0.0902940
\(716\) 9.23348e15 1.83377
\(717\) −2.03119e15 −0.400309
\(718\) −2.83620e15 −0.554693
\(719\) 8.38597e14 0.162759 0.0813794 0.996683i \(-0.474067\pi\)
0.0813794 + 0.996683i \(0.474067\pi\)
\(720\) 9.15310e15 1.76295
\(721\) 3.38333e13 0.00646696
\(722\) −2.24972e15 −0.426750
\(723\) 3.50856e14 0.0660493
\(724\) −3.23503e14 −0.0604388
\(725\) 2.18514e15 0.405153
\(726\) 2.43840e15 0.448698
\(727\) −8.88989e15 −1.62352 −0.811759 0.583993i \(-0.801490\pi\)
−0.811759 + 0.583993i \(0.801490\pi\)
\(728\) 5.12058e13 0.00928103
\(729\) −3.44716e15 −0.620098
\(730\) 7.84876e15 1.40128
\(731\) 4.97907e15 0.882273
\(732\) 2.90193e15 0.510359
\(733\) 3.14574e15 0.549099 0.274550 0.961573i \(-0.411471\pi\)
0.274550 + 0.961573i \(0.411471\pi\)
\(734\) 3.67353e15 0.636437
\(735\) −2.68517e15 −0.461734
\(736\) −1.53263e14 −0.0261584
\(737\) −1.13894e14 −0.0192944
\(738\) 1.27442e16 2.14290
\(739\) 2.14266e15 0.357609 0.178804 0.983885i \(-0.442777\pi\)
0.178804 + 0.983885i \(0.442777\pi\)
\(740\) 3.20408e16 5.30798
\(741\) −1.38304e15 −0.227423
\(742\) 2.26162e11 3.69146e−5 0
\(743\) 3.63029e15 0.588170 0.294085 0.955779i \(-0.404985\pi\)
0.294085 + 0.955779i \(0.404985\pi\)
\(744\) −3.02849e15 −0.487052
\(745\) 2.80359e15 0.447564
\(746\) −1.21342e16 −1.92286
\(747\) 1.17000e16 1.84045
\(748\) −3.53509e14 −0.0552003
\(749\) 2.68547e13 0.00416265
\(750\) 6.16267e15 0.948269
\(751\) −4.09016e15 −0.624770 −0.312385 0.949956i \(-0.601128\pi\)
−0.312385 + 0.949956i \(0.601128\pi\)
\(752\) 3.40545e15 0.516389
\(753\) −1.20125e15 −0.180826
\(754\) −2.18415e15 −0.326393
\(755\) −9.20911e15 −1.36619
\(756\) −3.55344e13 −0.00523334
\(757\) −8.56928e15 −1.25290 −0.626451 0.779461i \(-0.715493\pi\)
−0.626451 + 0.779461i \(0.715493\pi\)
\(758\) 6.66050e15 0.966777
\(759\) −3.41128e13 −0.00491572
\(760\) 1.91886e16 2.74517
\(761\) 3.52288e15 0.500360 0.250180 0.968199i \(-0.419510\pi\)
0.250180 + 0.968199i \(0.419510\pi\)
\(762\) 2.79207e15 0.393708
\(763\) 6.70726e13 0.00938988
\(764\) 8.54868e15 1.18819
\(765\) 6.58659e15 0.908914
\(766\) 3.01174e15 0.412628
\(767\) −5.34579e15 −0.727172
\(768\) 3.86652e15 0.522196
\(769\) −3.44010e15 −0.461292 −0.230646 0.973038i \(-0.574084\pi\)
−0.230646 + 0.973038i \(0.574084\pi\)
\(770\) −6.03368e12 −0.000803310 0
\(771\) 2.00804e15 0.265444
\(772\) 1.17188e16 1.53811
\(773\) −1.09067e16 −1.42137 −0.710686 0.703510i \(-0.751615\pi\)
−0.710686 + 0.703510i \(0.751615\pi\)
\(774\) −2.02407e16 −2.61909
\(775\) 1.80328e16 2.31688
\(776\) −7.37104e15 −0.940350
\(777\) −1.55922e13 −0.00197511
\(778\) 1.62299e16 2.04140
\(779\) 9.19195e15 1.14803
\(780\) −7.60607e15 −0.943278
\(781\) −7.21794e14 −0.0888858
\(782\) −2.94500e15 −0.360120
\(783\) 7.65220e14 0.0929173
\(784\) −8.78519e15 −1.05928
\(785\) −1.75621e16 −2.10278
\(786\) −1.18897e15 −0.141367
\(787\) −8.17365e15 −0.965061 −0.482531 0.875879i \(-0.660282\pi\)
−0.482531 + 0.875879i \(0.660282\pi\)
\(788\) 2.04743e16 2.40057
\(789\) −1.42289e15 −0.165672
\(790\) −4.83201e16 −5.58700
\(791\) 8.30061e12 0.000953102 0
\(792\) 7.25523e14 0.0827299
\(793\) 8.71967e15 0.987410
\(794\) −1.61770e14 −0.0181922
\(795\) −1.69603e13 −0.00189415
\(796\) −1.62474e16 −1.80203
\(797\) 1.09849e16 1.20997 0.604984 0.796238i \(-0.293180\pi\)
0.604984 + 0.796238i \(0.293180\pi\)
\(798\) −1.84958e13 −0.00202329
\(799\) 2.45057e15 0.266232
\(800\) 1.39400e15 0.150407
\(801\) 1.41182e16 1.51286
\(802\) −2.89766e16 −3.08381
\(803\) 2.14045e14 0.0226240
\(804\) −1.92008e15 −0.201564
\(805\) −3.36191e13 −0.00350517
\(806\) −1.80246e16 −1.86648
\(807\) 3.39206e15 0.348866
\(808\) −1.07932e15 −0.110252
\(809\) 7.37110e15 0.747851 0.373926 0.927459i \(-0.378011\pi\)
0.373926 + 0.927459i \(0.378011\pi\)
\(810\) −2.47144e16 −2.49047
\(811\) −1.58534e16 −1.58674 −0.793372 0.608737i \(-0.791676\pi\)
−0.793372 + 0.608737i \(0.791676\pi\)
\(812\) −1.95363e13 −0.00194216
\(813\) −3.09811e15 −0.305914
\(814\) 1.30644e15 0.128131
\(815\) 1.21216e16 1.18084
\(816\) −1.54772e15 −0.149761
\(817\) −1.45990e16 −1.40314
\(818\) 1.43560e16 1.37053
\(819\) −5.15359e13 −0.00488707
\(820\) 5.05515e16 4.76165
\(821\) −5.60490e15 −0.524422 −0.262211 0.965011i \(-0.584452\pi\)
−0.262211 + 0.965011i \(0.584452\pi\)
\(822\) −7.61141e15 −0.707409
\(823\) −8.99004e15 −0.829971 −0.414985 0.909828i \(-0.636213\pi\)
−0.414985 + 0.909828i \(0.636213\pi\)
\(824\) −2.41278e16 −2.21267
\(825\) 3.10270e14 0.0282646
\(826\) −7.14911e13 −0.00646937
\(827\) 1.65577e16 1.48840 0.744199 0.667958i \(-0.232832\pi\)
0.744199 + 0.667958i \(0.232832\pi\)
\(828\) 8.00721e15 0.715014
\(829\) −5.71365e14 −0.0506832 −0.0253416 0.999679i \(-0.508067\pi\)
−0.0253416 + 0.999679i \(0.508067\pi\)
\(830\) 6.93892e16 6.11452
\(831\) 2.46387e15 0.215681
\(832\) 1.09282e16 0.950318
\(833\) −6.32184e15 −0.546130
\(834\) 4.50772e15 0.386851
\(835\) 4.88142e14 0.0416170
\(836\) 1.03651e15 0.0877890
\(837\) 6.31495e15 0.531349
\(838\) −3.02668e16 −2.53002
\(839\) 1.42940e16 1.18703 0.593517 0.804821i \(-0.297739\pi\)
0.593517 + 0.804821i \(0.297739\pi\)
\(840\) −5.13539e13 −0.00423680
\(841\) 4.20707e14 0.0344828
\(842\) 1.15672e16 0.941912
\(843\) −4.00408e15 −0.323930
\(844\) −2.83606e16 −2.27946
\(845\) −5.16322e14 −0.0412296
\(846\) −9.96195e15 −0.790328
\(847\) 6.55358e13 0.00516559
\(848\) −5.54898e13 −0.00434546
\(849\) −1.53794e15 −0.119660
\(850\) 2.67860e16 2.07064
\(851\) 7.27933e15 0.559087
\(852\) −1.21683e16 −0.928567
\(853\) 1.62880e16 1.23495 0.617473 0.786592i \(-0.288156\pi\)
0.617473 + 0.786592i \(0.288156\pi\)
\(854\) 1.16611e14 0.00878460
\(855\) −1.93123e16 −1.44551
\(856\) −1.91511e16 −1.42425
\(857\) −1.41268e16 −1.04388 −0.521940 0.852982i \(-0.674791\pi\)
−0.521940 + 0.852982i \(0.674791\pi\)
\(858\) −3.10131e14 −0.0227701
\(859\) −1.09143e16 −0.796224 −0.398112 0.917337i \(-0.630335\pi\)
−0.398112 + 0.917337i \(0.630335\pi\)
\(860\) −8.02877e16 −5.81980
\(861\) −2.46001e13 −0.00177182
\(862\) 1.90722e16 1.36493
\(863\) 6.50838e15 0.462822 0.231411 0.972856i \(-0.425666\pi\)
0.231411 + 0.972856i \(0.425666\pi\)
\(864\) 4.88168e14 0.0344940
\(865\) 2.75487e16 1.93425
\(866\) −5.68117e15 −0.396359
\(867\) 2.62022e15 0.181649
\(868\) −1.61223e14 −0.0111062
\(869\) −1.31774e15 −0.0902033
\(870\) 2.19047e15 0.148999
\(871\) −5.76944e15 −0.389972
\(872\) −4.78319e16 −3.21275
\(873\) 7.41856e15 0.495155
\(874\) 8.63493e15 0.572725
\(875\) 1.65631e14 0.0109168
\(876\) 3.60847e15 0.236347
\(877\) 5.86025e15 0.381433 0.190716 0.981645i \(-0.438919\pi\)
0.190716 + 0.981645i \(0.438919\pi\)
\(878\) 3.34975e16 2.16667
\(879\) 4.14475e14 0.0266415
\(880\) 1.48039e15 0.0945630
\(881\) −1.13320e16 −0.719346 −0.359673 0.933078i \(-0.617112\pi\)
−0.359673 + 0.933078i \(0.617112\pi\)
\(882\) 2.56993e16 1.62123
\(883\) 4.59592e15 0.288130 0.144065 0.989568i \(-0.453983\pi\)
0.144065 + 0.989568i \(0.453983\pi\)
\(884\) −1.79074e16 −1.11569
\(885\) 5.36126e15 0.331955
\(886\) −2.06013e16 −1.26768
\(887\) 1.85397e16 1.13376 0.566881 0.823800i \(-0.308150\pi\)
0.566881 + 0.823800i \(0.308150\pi\)
\(888\) 1.11194e16 0.675784
\(889\) 7.50411e13 0.00453252
\(890\) 8.37302e16 5.02617
\(891\) −6.73989e14 −0.0402092
\(892\) −1.67804e16 −0.994934
\(893\) −7.18524e15 −0.423407
\(894\) 1.92716e15 0.112865
\(895\) −2.78249e16 −1.61960
\(896\) 1.52317e14 0.00881159
\(897\) −1.72802e15 −0.0993550
\(898\) 3.95127e16 2.25796
\(899\) 3.47187e15 0.197190
\(900\) −7.28290e16 −4.11122
\(901\) −3.99306e13 −0.00224037
\(902\) 2.06119e15 0.114943
\(903\) 3.90708e13 0.00216556
\(904\) −5.91947e15 −0.326104
\(905\) 9.74868e14 0.0533800
\(906\) −6.33024e15 −0.344521
\(907\) 1.52365e16 0.824226 0.412113 0.911133i \(-0.364791\pi\)
0.412113 + 0.911133i \(0.364791\pi\)
\(908\) 4.60869e16 2.47802
\(909\) 1.08628e15 0.0580549
\(910\) −3.05642e14 −0.0162363
\(911\) −2.85826e16 −1.50921 −0.754607 0.656177i \(-0.772172\pi\)
−0.754607 + 0.656177i \(0.772172\pi\)
\(912\) 4.53804e15 0.238175
\(913\) 1.89232e15 0.0987202
\(914\) −2.46083e16 −1.27608
\(915\) −8.74490e15 −0.450753
\(916\) 2.38134e16 1.22010
\(917\) −3.19554e13 −0.00162747
\(918\) 9.38027e15 0.474877
\(919\) 3.83197e15 0.192836 0.0964178 0.995341i \(-0.469262\pi\)
0.0964178 + 0.995341i \(0.469262\pi\)
\(920\) 2.39750e16 1.19929
\(921\) −7.81069e15 −0.388384
\(922\) −3.28438e16 −1.62343
\(923\) −3.65632e16 −1.79653
\(924\) −2.77398e12 −0.000135490 0
\(925\) −6.62087e16 −3.21466
\(926\) −1.78341e16 −0.860778
\(927\) 2.42833e16 1.16512
\(928\) 2.68388e14 0.0128012
\(929\) 1.12285e15 0.0532396 0.0266198 0.999646i \(-0.491526\pi\)
0.0266198 + 0.999646i \(0.491526\pi\)
\(930\) 1.80768e16 0.852050
\(931\) 1.85361e16 0.868549
\(932\) 1.43173e16 0.666920
\(933\) −6.42227e15 −0.297399
\(934\) 7.24310e16 3.33439
\(935\) 1.06529e15 0.0487533
\(936\) 3.67521e16 1.67211
\(937\) 2.64634e16 1.19696 0.598478 0.801139i \(-0.295772\pi\)
0.598478 + 0.801139i \(0.295772\pi\)
\(938\) −7.71567e13 −0.00346943
\(939\) 3.43818e15 0.153698
\(940\) −3.95155e16 −1.75616
\(941\) −5.56710e15 −0.245972 −0.122986 0.992408i \(-0.539247\pi\)
−0.122986 + 0.992408i \(0.539247\pi\)
\(942\) −1.20720e16 −0.530274
\(943\) 1.14848e16 0.501543
\(944\) 1.75407e16 0.761552
\(945\) 1.07082e14 0.00462213
\(946\) −3.27366e15 −0.140486
\(947\) −4.76013e15 −0.203093 −0.101546 0.994831i \(-0.532379\pi\)
−0.101546 + 0.994831i \(0.532379\pi\)
\(948\) −2.22152e16 −0.942331
\(949\) 1.08427e16 0.457269
\(950\) −7.85384e16 −3.29308
\(951\) −4.35969e15 −0.181745
\(952\) −1.20905e14 −0.00501120
\(953\) −2.30580e16 −0.950189 −0.475095 0.879935i \(-0.657586\pi\)
−0.475095 + 0.879935i \(0.657586\pi\)
\(954\) 1.62324e14 0.00665070
\(955\) −2.57613e16 −1.04942
\(956\) 7.71122e16 3.12323
\(957\) 5.97368e13 0.00240561
\(958\) 2.20105e16 0.881291
\(959\) −2.04568e14 −0.00814396
\(960\) −1.09598e16 −0.433821
\(961\) 3.24299e15 0.127634
\(962\) 6.61789e16 2.58974
\(963\) 1.92745e16 0.749962
\(964\) −1.33199e16 −0.515320
\(965\) −3.53142e16 −1.35847
\(966\) −2.31094e13 −0.000883923 0
\(967\) 4.58017e16 1.74195 0.870975 0.491327i \(-0.163488\pi\)
0.870975 + 0.491327i \(0.163488\pi\)
\(968\) −4.67360e16 −1.76741
\(969\) 3.26558e15 0.122795
\(970\) 4.39971e16 1.64505
\(971\) 3.23841e16 1.20400 0.601999 0.798497i \(-0.294371\pi\)
0.601999 + 0.798497i \(0.294371\pi\)
\(972\) −3.86984e16 −1.43063
\(973\) 1.21152e14 0.00445357
\(974\) −2.87410e16 −1.05058
\(975\) 1.57171e16 0.571276
\(976\) −2.86111e16 −1.03409
\(977\) −1.53051e16 −0.550069 −0.275035 0.961434i \(-0.588689\pi\)
−0.275035 + 0.961434i \(0.588689\pi\)
\(978\) 8.33223e15 0.297781
\(979\) 2.28342e15 0.0811485
\(980\) 1.01940e17 3.60247
\(981\) 4.81403e16 1.69172
\(982\) 6.69386e16 2.33918
\(983\) −3.08928e16 −1.07353 −0.536764 0.843732i \(-0.680354\pi\)
−0.536764 + 0.843732i \(0.680354\pi\)
\(984\) 1.75433e16 0.606229
\(985\) −6.16988e16 −2.12020
\(986\) 5.15714e15 0.176233
\(987\) 1.92296e13 0.000653471 0
\(988\) 5.25056e16 1.77436
\(989\) −1.82405e16 −0.612996
\(990\) −4.33058e15 −0.144728
\(991\) −1.59520e16 −0.530163 −0.265082 0.964226i \(-0.585399\pi\)
−0.265082 + 0.964226i \(0.585399\pi\)
\(992\) 2.21486e15 0.0732036
\(993\) 8.67668e15 0.285189
\(994\) −4.88973e14 −0.0159830
\(995\) 4.89613e16 1.59157
\(996\) 3.19017e16 1.03130
\(997\) −1.48585e16 −0.477697 −0.238848 0.971057i \(-0.576770\pi\)
−0.238848 + 0.971057i \(0.576770\pi\)
\(998\) −8.95356e16 −2.86271
\(999\) −2.31858e16 −0.737246
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 29.12.a.a.1.11 11
3.2 odd 2 261.12.a.a.1.1 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.12.a.a.1.11 11 1.1 even 1 trivial
261.12.a.a.1.1 11 3.2 odd 2