Properties

Label 3.22.a.c.1.1
Level $3$
Weight $22$
Character 3.1
Self dual yes
Analytic conductor $8.384$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.38432032861\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{649}) \)
Defining polynomial: \( x^{2} - x - 162 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(13.2377\) of defining polynomial
Character \(\chi\) \(=\) 3.1

$q$-expansion

\(f(q)\) \(=\) \(q-1271.96 q^{2} +59049.0 q^{3} -479282. q^{4} -2.12776e7 q^{5} -7.51077e7 q^{6} +6.32076e8 q^{7} +3.27711e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q-1271.96 q^{2} +59049.0 q^{3} -479282. q^{4} -2.12776e7 q^{5} -7.51077e7 q^{6} +6.32076e8 q^{7} +3.27711e9 q^{8} +3.48678e9 q^{9} +2.70641e10 q^{10} +5.97585e10 q^{11} -2.83011e10 q^{12} +7.38499e11 q^{13} -8.03972e11 q^{14} -1.25642e12 q^{15} -3.16321e12 q^{16} -8.35876e12 q^{17} -4.43503e12 q^{18} +4.19061e13 q^{19} +1.01980e13 q^{20} +3.73234e13 q^{21} -7.60101e13 q^{22} +4.48926e13 q^{23} +1.93510e14 q^{24} -2.41012e13 q^{25} -9.39338e14 q^{26} +2.05891e14 q^{27} -3.02943e14 q^{28} -2.76669e15 q^{29} +1.59811e15 q^{30} +8.36452e15 q^{31} -2.84914e15 q^{32} +3.52868e15 q^{33} +1.06320e16 q^{34} -1.34490e16 q^{35} -1.67115e15 q^{36} -1.77675e16 q^{37} -5.33027e16 q^{38} +4.36077e16 q^{39} -6.97290e16 q^{40} +1.45253e17 q^{41} -4.74737e16 q^{42} +1.24744e17 q^{43} -2.86412e16 q^{44} -7.41904e16 q^{45} -5.71014e16 q^{46} +4.28566e17 q^{47} -1.86784e17 q^{48} -1.59026e17 q^{49} +3.06556e16 q^{50} -4.93577e17 q^{51} -3.53950e17 q^{52} -4.77017e17 q^{53} -2.61884e17 q^{54} -1.27152e18 q^{55} +2.07138e18 q^{56} +2.47452e18 q^{57} +3.51911e18 q^{58} +1.61959e18 q^{59} +6.02180e17 q^{60} -3.76882e18 q^{61} -1.06393e19 q^{62} +2.20391e18 q^{63} +1.02577e19 q^{64} -1.57135e19 q^{65} -4.48832e18 q^{66} -2.81797e18 q^{67} +4.00621e18 q^{68} +2.65086e18 q^{69} +1.71066e19 q^{70} +1.00228e19 q^{71} +1.14266e19 q^{72} -1.72739e19 q^{73} +2.25995e19 q^{74} -1.42315e18 q^{75} -2.00849e19 q^{76} +3.77719e19 q^{77} -5.54670e19 q^{78} -3.28276e19 q^{79} +6.73054e19 q^{80} +1.21577e19 q^{81} -1.84755e20 q^{82} +3.05240e17 q^{83} -1.78885e19 q^{84} +1.77854e20 q^{85} -1.58669e20 q^{86} -1.63370e20 q^{87} +1.95835e20 q^{88} +2.34593e20 q^{89} +9.43668e19 q^{90} +4.66787e20 q^{91} -2.15162e19 q^{92} +4.93917e20 q^{93} -5.45116e20 q^{94} -8.91662e20 q^{95} -1.68239e20 q^{96} -5.92086e20 q^{97} +2.02274e20 q^{98} +2.08365e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 666 q^{2} + 118098 q^{3} + 1179236 q^{4} + 996876 q^{5} + 39326634 q^{6} + 679896112 q^{7} + 2427055848 q^{8} + 6973568802 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 666 q^{2} + 118098 q^{3} + 1179236 q^{4} + 996876 q^{5} + 39326634 q^{6} + 679896112 q^{7} + 2427055848 q^{8} + 6973568802 q^{9} + 70231066524 q^{10} + 219869122968 q^{11} + 69632706564 q^{12} - 48468909956 q^{13} - 711297706896 q^{14} + 58864530924 q^{15} - 8288736440560 q^{16} - 11333529041436 q^{17} + 2322198411066 q^{18} + 11960585011624 q^{19} + 47140581172824 q^{20} + 40147185517488 q^{21} + 234277148563128 q^{22} - 146508390063504 q^{23} + 143315220768552 q^{24} - 4786354247074 q^{25} - 24\!\cdots\!16 q^{26}+ \cdots + 76\!\cdots\!68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1271.96 −0.878328 −0.439164 0.898407i \(-0.644725\pi\)
−0.439164 + 0.898407i \(0.644725\pi\)
\(3\) 59049.0 0.577350
\(4\) −479282. −0.228540
\(5\) −2.12776e7 −0.974400 −0.487200 0.873290i \(-0.661982\pi\)
−0.487200 + 0.873290i \(0.661982\pi\)
\(6\) −7.51077e7 −0.507103
\(7\) 6.32076e8 0.845745 0.422873 0.906189i \(-0.361022\pi\)
0.422873 + 0.906189i \(0.361022\pi\)
\(8\) 3.27711e9 1.07906
\(9\) 3.48678e9 0.333333
\(10\) 2.70641e10 0.855843
\(11\) 5.97585e10 0.694666 0.347333 0.937742i \(-0.387087\pi\)
0.347333 + 0.937742i \(0.387087\pi\)
\(12\) −2.83011e10 −0.131947
\(13\) 7.38499e11 1.48575 0.742874 0.669432i \(-0.233462\pi\)
0.742874 + 0.669432i \(0.233462\pi\)
\(14\) −8.03972e11 −0.742842
\(15\) −1.25642e12 −0.562570
\(16\) −3.16321e12 −0.719230
\(17\) −8.35876e12 −1.00561 −0.502804 0.864401i \(-0.667698\pi\)
−0.502804 + 0.864401i \(0.667698\pi\)
\(18\) −4.43503e12 −0.292776
\(19\) 4.19061e13 1.56807 0.784034 0.620718i \(-0.213159\pi\)
0.784034 + 0.620718i \(0.213159\pi\)
\(20\) 1.01980e13 0.222689
\(21\) 3.73234e13 0.488291
\(22\) −7.60101e13 −0.610145
\(23\) 4.48926e13 0.225961 0.112980 0.993597i \(-0.463960\pi\)
0.112980 + 0.993597i \(0.463960\pi\)
\(24\) 1.93510e14 0.622996
\(25\) −2.41012e13 −0.0505438
\(26\) −9.39338e14 −1.30497
\(27\) 2.05891e14 0.192450
\(28\) −3.02943e14 −0.193286
\(29\) −2.76669e15 −1.22119 −0.610593 0.791945i \(-0.709069\pi\)
−0.610593 + 0.791945i \(0.709069\pi\)
\(30\) 1.59811e15 0.494121
\(31\) 8.36452e15 1.83292 0.916459 0.400128i \(-0.131034\pi\)
0.916459 + 0.400128i \(0.131034\pi\)
\(32\) −2.84914e15 −0.447341
\(33\) 3.52868e15 0.401066
\(34\) 1.06320e16 0.883253
\(35\) −1.34490e16 −0.824095
\(36\) −1.67115e15 −0.0761798
\(37\) −1.77675e16 −0.607447 −0.303724 0.952760i \(-0.598230\pi\)
−0.303724 + 0.952760i \(0.598230\pi\)
\(38\) −5.33027e16 −1.37728
\(39\) 4.36077e16 0.857797
\(40\) −6.97290e16 −1.05144
\(41\) 1.45253e17 1.69003 0.845013 0.534745i \(-0.179592\pi\)
0.845013 + 0.534745i \(0.179592\pi\)
\(42\) −4.74737e16 −0.428880
\(43\) 1.24744e17 0.880239 0.440119 0.897939i \(-0.354936\pi\)
0.440119 + 0.897939i \(0.354936\pi\)
\(44\) −2.86412e16 −0.158759
\(45\) −7.41904e16 −0.324800
\(46\) −5.71014e16 −0.198468
\(47\) 4.28566e17 1.18847 0.594237 0.804290i \(-0.297454\pi\)
0.594237 + 0.804290i \(0.297454\pi\)
\(48\) −1.86784e17 −0.415248
\(49\) −1.59026e17 −0.284715
\(50\) 3.06556e16 0.0443941
\(51\) −4.93577e17 −0.580588
\(52\) −3.53950e17 −0.339552
\(53\) −4.77017e17 −0.374660 −0.187330 0.982297i \(-0.559983\pi\)
−0.187330 + 0.982297i \(0.559983\pi\)
\(54\) −2.61884e17 −0.169034
\(55\) −1.27152e18 −0.676883
\(56\) 2.07138e18 0.912611
\(57\) 2.47452e18 0.905324
\(58\) 3.51911e18 1.07260
\(59\) 1.61959e18 0.412533 0.206267 0.978496i \(-0.433869\pi\)
0.206267 + 0.978496i \(0.433869\pi\)
\(60\) 6.02180e17 0.128570
\(61\) −3.76882e18 −0.676461 −0.338230 0.941063i \(-0.609828\pi\)
−0.338230 + 0.941063i \(0.609828\pi\)
\(62\) −1.06393e19 −1.60990
\(63\) 2.20391e18 0.281915
\(64\) 1.02577e19 1.11214
\(65\) −1.57135e19 −1.44771
\(66\) −4.48832e18 −0.352267
\(67\) −2.81797e18 −0.188865 −0.0944324 0.995531i \(-0.530104\pi\)
−0.0944324 + 0.995531i \(0.530104\pi\)
\(68\) 4.00621e18 0.229821
\(69\) 2.65086e18 0.130458
\(70\) 1.71066e19 0.723826
\(71\) 1.00228e19 0.365407 0.182704 0.983168i \(-0.441515\pi\)
0.182704 + 0.983168i \(0.441515\pi\)
\(72\) 1.14266e19 0.359687
\(73\) −1.72739e19 −0.470435 −0.235218 0.971943i \(-0.575580\pi\)
−0.235218 + 0.971943i \(0.575580\pi\)
\(74\) 2.25995e19 0.533538
\(75\) −1.42315e18 −0.0291815
\(76\) −2.00849e19 −0.358365
\(77\) 3.77719e19 0.587511
\(78\) −5.54670e19 −0.753427
\(79\) −3.28276e19 −0.390080 −0.195040 0.980795i \(-0.562484\pi\)
−0.195040 + 0.980795i \(0.562484\pi\)
\(80\) 6.73054e19 0.700818
\(81\) 1.21577e19 0.111111
\(82\) −1.84755e20 −1.48440
\(83\) 3.05240e17 0.00215934 0.00107967 0.999999i \(-0.499656\pi\)
0.00107967 + 0.999999i \(0.499656\pi\)
\(84\) −1.78885e19 −0.111594
\(85\) 1.77854e20 0.979864
\(86\) −1.58669e20 −0.773139
\(87\) −1.63370e20 −0.705052
\(88\) 1.95835e20 0.749587
\(89\) 2.34593e20 0.797480 0.398740 0.917064i \(-0.369448\pi\)
0.398740 + 0.917064i \(0.369448\pi\)
\(90\) 9.43668e19 0.285281
\(91\) 4.66787e20 1.25656
\(92\) −2.15162e19 −0.0516409
\(93\) 4.93917e20 1.05824
\(94\) −5.45116e20 −1.04387
\(95\) −8.91662e20 −1.52793
\(96\) −1.68239e20 −0.258272
\(97\) −5.92086e20 −0.815233 −0.407616 0.913153i \(-0.633640\pi\)
−0.407616 + 0.913153i \(0.633640\pi\)
\(98\) 2.02274e20 0.250073
\(99\) 2.08365e20 0.231555
\(100\) 1.15513e19 0.0115513
\(101\) −1.66229e21 −1.49739 −0.748693 0.662917i \(-0.769318\pi\)
−0.748693 + 0.662917i \(0.769318\pi\)
\(102\) 6.27807e20 0.509946
\(103\) 1.17919e21 0.864552 0.432276 0.901741i \(-0.357711\pi\)
0.432276 + 0.901741i \(0.357711\pi\)
\(104\) 2.42014e21 1.60321
\(105\) −7.94153e20 −0.475791
\(106\) 6.06745e20 0.329075
\(107\) −8.69160e20 −0.427140 −0.213570 0.976928i \(-0.568509\pi\)
−0.213570 + 0.976928i \(0.568509\pi\)
\(108\) −9.86799e19 −0.0439825
\(109\) 4.12021e20 0.166702 0.0833510 0.996520i \(-0.473438\pi\)
0.0833510 + 0.996520i \(0.473438\pi\)
\(110\) 1.61731e21 0.594526
\(111\) −1.04915e21 −0.350710
\(112\) −1.99939e21 −0.608286
\(113\) 2.45943e21 0.681569 0.340785 0.940141i \(-0.389307\pi\)
0.340785 + 0.940141i \(0.389307\pi\)
\(114\) −3.14747e21 −0.795172
\(115\) −9.55207e20 −0.220176
\(116\) 1.32602e21 0.279089
\(117\) 2.57499e21 0.495249
\(118\) −2.06005e21 −0.362340
\(119\) −5.28337e21 −0.850487
\(120\) −4.11743e21 −0.607048
\(121\) −3.82917e21 −0.517439
\(122\) 4.79378e21 0.594155
\(123\) 8.57702e21 0.975737
\(124\) −4.00896e21 −0.418894
\(125\) 1.06588e22 1.02365
\(126\) −2.80328e21 −0.247614
\(127\) 5.53783e21 0.450195 0.225097 0.974336i \(-0.427730\pi\)
0.225097 + 0.974336i \(0.427730\pi\)
\(128\) −7.07226e21 −0.529485
\(129\) 7.36600e21 0.508206
\(130\) 1.99869e22 1.27157
\(131\) −2.54630e22 −1.49472 −0.747362 0.664417i \(-0.768680\pi\)
−0.747362 + 0.664417i \(0.768680\pi\)
\(132\) −1.69123e21 −0.0916594
\(133\) 2.64878e22 1.32619
\(134\) 3.58433e21 0.165885
\(135\) −4.38087e21 −0.187523
\(136\) −2.73926e22 −1.08511
\(137\) −4.82088e22 −1.76832 −0.884158 0.467188i \(-0.845267\pi\)
−0.884158 + 0.467188i \(0.845267\pi\)
\(138\) −3.37178e21 −0.114585
\(139\) −2.18194e22 −0.687363 −0.343682 0.939086i \(-0.611674\pi\)
−0.343682 + 0.939086i \(0.611674\pi\)
\(140\) 6.44589e21 0.188338
\(141\) 2.53064e22 0.686166
\(142\) −1.27486e22 −0.320948
\(143\) 4.41316e22 1.03210
\(144\) −1.10294e22 −0.239743
\(145\) 5.88685e22 1.18992
\(146\) 2.19716e22 0.413197
\(147\) −9.39035e21 −0.164380
\(148\) 8.51565e21 0.138826
\(149\) 4.84140e22 0.735386 0.367693 0.929947i \(-0.380148\pi\)
0.367693 + 0.929947i \(0.380148\pi\)
\(150\) 1.81018e21 0.0256309
\(151\) −1.22948e23 −1.62354 −0.811770 0.583977i \(-0.801496\pi\)
−0.811770 + 0.583977i \(0.801496\pi\)
\(152\) 1.37331e23 1.69204
\(153\) −2.91452e22 −0.335202
\(154\) −4.80441e22 −0.516027
\(155\) −1.77977e23 −1.78600
\(156\) −2.09004e22 −0.196040
\(157\) 5.89828e22 0.517344 0.258672 0.965965i \(-0.416715\pi\)
0.258672 + 0.965965i \(0.416715\pi\)
\(158\) 4.17552e22 0.342619
\(159\) −2.81674e22 −0.216310
\(160\) 6.06228e22 0.435889
\(161\) 2.83755e22 0.191105
\(162\) −1.54640e22 −0.0975920
\(163\) −3.18244e22 −0.188274 −0.0941369 0.995559i \(-0.530009\pi\)
−0.0941369 + 0.995559i \(0.530009\pi\)
\(164\) −6.96170e22 −0.386238
\(165\) −7.50818e22 −0.390799
\(166\) −3.88252e20 −0.00189661
\(167\) −3.62822e22 −0.166407 −0.0832033 0.996533i \(-0.526515\pi\)
−0.0832033 + 0.996533i \(0.526515\pi\)
\(168\) 1.22313e23 0.526896
\(169\) 2.98317e23 1.20744
\(170\) −2.26223e23 −0.860642
\(171\) 1.46118e23 0.522689
\(172\) −5.97875e22 −0.201169
\(173\) 3.21510e23 1.01791 0.508956 0.860792i \(-0.330031\pi\)
0.508956 + 0.860792i \(0.330031\pi\)
\(174\) 2.07800e23 0.619267
\(175\) −1.52338e22 −0.0427472
\(176\) −1.89028e23 −0.499625
\(177\) 9.56351e22 0.238176
\(178\) −2.98392e23 −0.700449
\(179\) −1.80418e23 −0.399321 −0.199661 0.979865i \(-0.563984\pi\)
−0.199661 + 0.979865i \(0.563984\pi\)
\(180\) 3.55581e22 0.0742297
\(181\) −6.36646e21 −0.0125393 −0.00626965 0.999980i \(-0.501996\pi\)
−0.00626965 + 0.999980i \(0.501996\pi\)
\(182\) −5.93733e23 −1.10368
\(183\) −2.22545e23 −0.390555
\(184\) 1.47118e23 0.243825
\(185\) 3.78050e23 0.591897
\(186\) −6.28240e23 −0.929479
\(187\) −4.99507e23 −0.698561
\(188\) −2.05404e23 −0.271613
\(189\) 1.30139e23 0.162764
\(190\) 1.13415e24 1.34202
\(191\) 8.17882e23 0.915883 0.457942 0.888982i \(-0.348587\pi\)
0.457942 + 0.888982i \(0.348587\pi\)
\(192\) 6.05707e23 0.642096
\(193\) −1.80064e23 −0.180748 −0.0903742 0.995908i \(-0.528806\pi\)
−0.0903742 + 0.995908i \(0.528806\pi\)
\(194\) 7.53107e23 0.716042
\(195\) −9.27866e23 −0.835837
\(196\) 7.62185e22 0.0650686
\(197\) 1.65216e24 1.33708 0.668541 0.743675i \(-0.266919\pi\)
0.668541 + 0.743675i \(0.266919\pi\)
\(198\) −2.65031e23 −0.203382
\(199\) −1.52537e24 −1.11024 −0.555120 0.831770i \(-0.687328\pi\)
−0.555120 + 0.831770i \(0.687328\pi\)
\(200\) −7.89822e22 −0.0545399
\(201\) −1.66398e23 −0.109041
\(202\) 2.11436e24 1.31520
\(203\) −1.74876e24 −1.03281
\(204\) 2.36562e23 0.132687
\(205\) −3.09063e24 −1.64676
\(206\) −1.49987e24 −0.759361
\(207\) 1.56531e23 0.0753202
\(208\) −2.33603e24 −1.06859
\(209\) 2.50425e24 1.08928
\(210\) 1.01013e24 0.417901
\(211\) 1.35748e24 0.534277 0.267138 0.963658i \(-0.413922\pi\)
0.267138 + 0.963658i \(0.413922\pi\)
\(212\) 2.28626e23 0.0856247
\(213\) 5.91838e23 0.210968
\(214\) 1.10553e24 0.375169
\(215\) −2.65425e24 −0.857705
\(216\) 6.74728e23 0.207665
\(217\) 5.28701e24 1.55018
\(218\) −5.24072e23 −0.146419
\(219\) −1.02001e24 −0.271606
\(220\) 6.09415e23 0.154695
\(221\) −6.17294e24 −1.49408
\(222\) 1.33448e24 0.308038
\(223\) 4.73192e24 1.04192 0.520962 0.853580i \(-0.325573\pi\)
0.520962 + 0.853580i \(0.325573\pi\)
\(224\) −1.80087e24 −0.378336
\(225\) −8.40356e22 −0.0168479
\(226\) −3.12828e24 −0.598642
\(227\) 8.00226e24 1.46198 0.730989 0.682389i \(-0.239059\pi\)
0.730989 + 0.682389i \(0.239059\pi\)
\(228\) −1.18599e24 −0.206902
\(229\) 2.72111e24 0.453391 0.226696 0.973966i \(-0.427208\pi\)
0.226696 + 0.973966i \(0.427208\pi\)
\(230\) 1.21498e24 0.193387
\(231\) 2.23039e24 0.339200
\(232\) −9.06674e24 −1.31773
\(233\) −6.60115e24 −0.917028 −0.458514 0.888687i \(-0.651618\pi\)
−0.458514 + 0.888687i \(0.651618\pi\)
\(234\) −3.27527e24 −0.434991
\(235\) −9.11885e24 −1.15805
\(236\) −7.76240e23 −0.0942801
\(237\) −1.93843e24 −0.225213
\(238\) 6.72021e24 0.747007
\(239\) 9.92224e24 1.05544 0.527718 0.849420i \(-0.323048\pi\)
0.527718 + 0.849420i \(0.323048\pi\)
\(240\) 3.97432e24 0.404618
\(241\) −1.54177e25 −1.50259 −0.751295 0.659966i \(-0.770571\pi\)
−0.751295 + 0.659966i \(0.770571\pi\)
\(242\) 4.87054e24 0.454481
\(243\) 7.17898e23 0.0641500
\(244\) 1.80633e24 0.154598
\(245\) 3.38370e24 0.277426
\(246\) −1.09096e25 −0.857018
\(247\) 3.09477e25 2.32975
\(248\) 2.74114e25 1.97783
\(249\) 1.80241e22 0.00124670
\(250\) −1.35575e25 −0.899101
\(251\) −1.00009e25 −0.636014 −0.318007 0.948088i \(-0.603014\pi\)
−0.318007 + 0.948088i \(0.603014\pi\)
\(252\) −1.05630e24 −0.0644287
\(253\) 2.68271e24 0.156967
\(254\) −7.04387e24 −0.395419
\(255\) 1.05021e25 0.565725
\(256\) −1.25164e25 −0.647080
\(257\) −3.06596e25 −1.52149 −0.760744 0.649052i \(-0.775166\pi\)
−0.760744 + 0.649052i \(0.775166\pi\)
\(258\) −9.36923e24 −0.446372
\(259\) −1.12304e25 −0.513746
\(260\) 7.53119e24 0.330860
\(261\) −9.64685e24 −0.407062
\(262\) 3.23878e25 1.31286
\(263\) −2.33455e25 −0.909216 −0.454608 0.890692i \(-0.650221\pi\)
−0.454608 + 0.890692i \(0.650221\pi\)
\(264\) 1.15639e25 0.432775
\(265\) 1.01498e25 0.365069
\(266\) −3.36914e25 −1.16483
\(267\) 1.38525e25 0.460425
\(268\) 1.35060e24 0.0431631
\(269\) 2.29737e25 0.706043 0.353022 0.935615i \(-0.385154\pi\)
0.353022 + 0.935615i \(0.385154\pi\)
\(270\) 5.57227e24 0.164707
\(271\) −4.02085e25 −1.14325 −0.571624 0.820516i \(-0.693686\pi\)
−0.571624 + 0.820516i \(0.693686\pi\)
\(272\) 2.64405e25 0.723263
\(273\) 2.75633e25 0.725477
\(274\) 6.13194e25 1.55316
\(275\) −1.44025e24 −0.0351111
\(276\) −1.27051e24 −0.0298149
\(277\) −4.48722e25 −1.01377 −0.506886 0.862013i \(-0.669203\pi\)
−0.506886 + 0.862013i \(0.669203\pi\)
\(278\) 2.77532e25 0.603731
\(279\) 2.91653e25 0.610973
\(280\) −4.40740e25 −0.889248
\(281\) −6.06489e25 −1.17871 −0.589355 0.807874i \(-0.700618\pi\)
−0.589355 + 0.807874i \(0.700618\pi\)
\(282\) −3.21886e25 −0.602679
\(283\) 1.04229e26 1.88032 0.940158 0.340739i \(-0.110677\pi\)
0.940158 + 0.340739i \(0.110677\pi\)
\(284\) −4.80376e24 −0.0835100
\(285\) −5.26517e25 −0.882148
\(286\) −5.61334e25 −0.906521
\(287\) 9.18107e25 1.42933
\(288\) −9.93433e24 −0.149114
\(289\) 7.76980e23 0.0112456
\(290\) −7.48781e25 −1.04514
\(291\) −3.49621e25 −0.470675
\(292\) 8.27907e24 0.107513
\(293\) 6.24634e25 0.782556 0.391278 0.920273i \(-0.372033\pi\)
0.391278 + 0.920273i \(0.372033\pi\)
\(294\) 1.19441e25 0.144380
\(295\) −3.44610e25 −0.401972
\(296\) −5.82261e25 −0.655472
\(297\) 1.23037e25 0.133689
\(298\) −6.15805e25 −0.645911
\(299\) 3.31532e25 0.335720
\(300\) 6.82091e23 0.00666913
\(301\) 7.88476e25 0.744458
\(302\) 1.56384e26 1.42600
\(303\) −9.81568e25 −0.864516
\(304\) −1.32558e26 −1.12780
\(305\) 8.01915e25 0.659144
\(306\) 3.70714e25 0.294418
\(307\) 7.14411e25 0.548271 0.274135 0.961691i \(-0.411608\pi\)
0.274135 + 0.961691i \(0.411608\pi\)
\(308\) −1.81034e25 −0.134269
\(309\) 6.96297e25 0.499150
\(310\) 2.26379e26 1.56869
\(311\) −6.01915e25 −0.403228 −0.201614 0.979465i \(-0.564619\pi\)
−0.201614 + 0.979465i \(0.564619\pi\)
\(312\) 1.42907e26 0.925615
\(313\) −1.69049e26 −1.05876 −0.529379 0.848385i \(-0.677575\pi\)
−0.529379 + 0.848385i \(0.677575\pi\)
\(314\) −7.50235e25 −0.454398
\(315\) −4.68939e25 −0.274698
\(316\) 1.57337e25 0.0891488
\(317\) −5.71691e25 −0.313357 −0.156678 0.987650i \(-0.550079\pi\)
−0.156678 + 0.987650i \(0.550079\pi\)
\(318\) 3.58277e25 0.189991
\(319\) −1.65333e26 −0.848316
\(320\) −2.18259e26 −1.08367
\(321\) −5.13230e25 −0.246609
\(322\) −3.60924e25 −0.167853
\(323\) −3.50283e26 −1.57686
\(324\) −5.82695e24 −0.0253933
\(325\) −1.77987e25 −0.0750954
\(326\) 4.04792e25 0.165366
\(327\) 2.43294e25 0.0962454
\(328\) 4.76009e26 1.82364
\(329\) 2.70886e26 1.00515
\(330\) 9.55007e25 0.343250
\(331\) 6.67957e25 0.232571 0.116285 0.993216i \(-0.462901\pi\)
0.116285 + 0.993216i \(0.462901\pi\)
\(332\) −1.46296e23 −0.000493495 0
\(333\) −6.19515e25 −0.202482
\(334\) 4.61493e25 0.146160
\(335\) 5.99597e25 0.184030
\(336\) −1.18062e26 −0.351194
\(337\) 3.55365e26 1.02461 0.512307 0.858802i \(-0.328791\pi\)
0.512307 + 0.858802i \(0.328791\pi\)
\(338\) −3.79446e26 −1.06053
\(339\) 1.45227e26 0.393504
\(340\) −8.52424e25 −0.223938
\(341\) 4.99851e26 1.27327
\(342\) −1.85855e26 −0.459093
\(343\) −4.53560e26 −1.08654
\(344\) 4.08799e26 0.949832
\(345\) −5.64040e25 −0.127119
\(346\) −4.08947e26 −0.894062
\(347\) −3.94238e25 −0.0836179 −0.0418090 0.999126i \(-0.513312\pi\)
−0.0418090 + 0.999126i \(0.513312\pi\)
\(348\) 7.83004e25 0.161132
\(349\) −5.84299e26 −1.16672 −0.583362 0.812212i \(-0.698263\pi\)
−0.583362 + 0.812212i \(0.698263\pi\)
\(350\) 1.93767e25 0.0375461
\(351\) 1.52050e26 0.285932
\(352\) −1.70260e26 −0.310753
\(353\) 4.96062e26 0.878823 0.439412 0.898286i \(-0.355187\pi\)
0.439412 + 0.898286i \(0.355187\pi\)
\(354\) −1.21644e26 −0.209197
\(355\) −2.13262e26 −0.356053
\(356\) −1.12436e26 −0.182256
\(357\) −3.11978e26 −0.491029
\(358\) 2.29483e26 0.350735
\(359\) −3.35136e26 −0.497426 −0.248713 0.968577i \(-0.580008\pi\)
−0.248713 + 0.968577i \(0.580008\pi\)
\(360\) −2.43130e26 −0.350479
\(361\) 1.04192e27 1.45884
\(362\) 8.09785e24 0.0110136
\(363\) −2.26109e26 −0.298743
\(364\) −2.23723e26 −0.287174
\(365\) 3.67547e26 0.458392
\(366\) 2.83068e26 0.343035
\(367\) 9.70198e25 0.114253 0.0571264 0.998367i \(-0.481806\pi\)
0.0571264 + 0.998367i \(0.481806\pi\)
\(368\) −1.42005e26 −0.162518
\(369\) 5.06465e26 0.563342
\(370\) −4.80862e26 −0.519880
\(371\) −3.01511e26 −0.316867
\(372\) −2.36725e26 −0.241849
\(373\) 1.03252e27 1.02554 0.512771 0.858525i \(-0.328619\pi\)
0.512771 + 0.858525i \(0.328619\pi\)
\(374\) 6.35350e26 0.613566
\(375\) 6.29389e26 0.591005
\(376\) 1.40446e27 1.28244
\(377\) −2.04320e27 −1.81437
\(378\) −1.65531e26 −0.142960
\(379\) 2.20881e26 0.185544 0.0927721 0.995687i \(-0.470427\pi\)
0.0927721 + 0.995687i \(0.470427\pi\)
\(380\) 4.27358e26 0.349191
\(381\) 3.27003e26 0.259920
\(382\) −1.04031e27 −0.804446
\(383\) −1.59547e27 −1.20033 −0.600165 0.799876i \(-0.704899\pi\)
−0.600165 + 0.799876i \(0.704899\pi\)
\(384\) −4.17610e26 −0.305698
\(385\) −8.03695e26 −0.572471
\(386\) 2.29033e26 0.158756
\(387\) 4.34955e26 0.293413
\(388\) 2.83776e26 0.186313
\(389\) −3.60474e26 −0.230358 −0.115179 0.993345i \(-0.536744\pi\)
−0.115179 + 0.993345i \(0.536744\pi\)
\(390\) 1.18020e27 0.734140
\(391\) −3.75247e26 −0.227228
\(392\) −5.21147e26 −0.307225
\(393\) −1.50357e27 −0.862980
\(394\) −2.10148e27 −1.17440
\(395\) 6.98492e26 0.380095
\(396\) −9.98656e25 −0.0529196
\(397\) −1.47191e27 −0.759595 −0.379798 0.925070i \(-0.624006\pi\)
−0.379798 + 0.925070i \(0.624006\pi\)
\(398\) 1.94020e27 0.975156
\(399\) 1.56408e27 0.765674
\(400\) 7.62370e25 0.0363527
\(401\) 1.00401e27 0.466360 0.233180 0.972434i \(-0.425087\pi\)
0.233180 + 0.972434i \(0.425087\pi\)
\(402\) 2.11651e26 0.0957740
\(403\) 6.17719e27 2.72325
\(404\) 7.96708e26 0.342212
\(405\) −2.58686e26 −0.108267
\(406\) 2.22434e27 0.907148
\(407\) −1.06176e27 −0.421973
\(408\) −1.61750e27 −0.626489
\(409\) −4.41838e27 −1.66789 −0.833947 0.551845i \(-0.813924\pi\)
−0.833947 + 0.551845i \(0.813924\pi\)
\(410\) 3.93114e27 1.44640
\(411\) −2.84668e27 −1.02094
\(412\) −5.65163e26 −0.197584
\(413\) 1.02370e27 0.348898
\(414\) −1.99100e26 −0.0661559
\(415\) −6.49477e24 −0.00210406
\(416\) −2.10409e27 −0.664635
\(417\) −1.28841e27 −0.396849
\(418\) −3.18529e27 −0.956749
\(419\) −2.36639e27 −0.693169 −0.346584 0.938019i \(-0.612659\pi\)
−0.346584 + 0.938019i \(0.612659\pi\)
\(420\) 3.80623e26 0.108737
\(421\) 2.72067e27 0.758078 0.379039 0.925381i \(-0.376255\pi\)
0.379039 + 0.925381i \(0.376255\pi\)
\(422\) −1.72665e27 −0.469271
\(423\) 1.49432e27 0.396158
\(424\) −1.56324e27 −0.404281
\(425\) 2.01456e26 0.0508272
\(426\) −7.52791e26 −0.185299
\(427\) −2.38218e27 −0.572114
\(428\) 4.16573e26 0.0976183
\(429\) 2.60593e27 0.595883
\(430\) 3.37609e27 0.753347
\(431\) 8.87397e27 1.93244 0.966222 0.257713i \(-0.0829688\pi\)
0.966222 + 0.257713i \(0.0829688\pi\)
\(432\) −6.51276e26 −0.138416
\(433\) −3.39994e27 −0.705258 −0.352629 0.935763i \(-0.614712\pi\)
−0.352629 + 0.935763i \(0.614712\pi\)
\(434\) −6.72484e27 −1.36157
\(435\) 3.47613e27 0.687003
\(436\) −1.97474e26 −0.0380980
\(437\) 1.88128e27 0.354322
\(438\) 1.29740e27 0.238559
\(439\) 2.29516e27 0.412037 0.206018 0.978548i \(-0.433949\pi\)
0.206018 + 0.978548i \(0.433949\pi\)
\(440\) −4.16690e27 −0.730398
\(441\) −5.54491e26 −0.0949050
\(442\) 7.85170e27 1.31229
\(443\) −5.65265e27 −0.922599 −0.461299 0.887245i \(-0.652617\pi\)
−0.461299 + 0.887245i \(0.652617\pi\)
\(444\) 5.02841e26 0.0801510
\(445\) −4.99157e27 −0.777065
\(446\) −6.01879e27 −0.915152
\(447\) 2.85880e27 0.424576
\(448\) 6.48364e27 0.940589
\(449\) 1.23324e28 1.74768 0.873839 0.486216i \(-0.161623\pi\)
0.873839 + 0.486216i \(0.161623\pi\)
\(450\) 1.06890e26 0.0147980
\(451\) 8.68008e27 1.17400
\(452\) −1.17876e27 −0.155766
\(453\) −7.25995e27 −0.937351
\(454\) −1.01785e28 −1.28410
\(455\) −9.93211e27 −1.22440
\(456\) 8.10926e27 0.976900
\(457\) −7.62646e27 −0.897848 −0.448924 0.893570i \(-0.648193\pi\)
−0.448924 + 0.893570i \(0.648193\pi\)
\(458\) −3.46113e27 −0.398226
\(459\) −1.72100e27 −0.193529
\(460\) 4.57814e26 0.0503189
\(461\) −9.04851e26 −0.0972114 −0.0486057 0.998818i \(-0.515478\pi\)
−0.0486057 + 0.998818i \(0.515478\pi\)
\(462\) −2.83696e27 −0.297929
\(463\) −1.19705e27 −0.122889 −0.0614443 0.998111i \(-0.519571\pi\)
−0.0614443 + 0.998111i \(0.519571\pi\)
\(464\) 8.75161e27 0.878313
\(465\) −1.05094e28 −1.03115
\(466\) 8.39637e27 0.805452
\(467\) −1.16951e28 −1.09693 −0.548464 0.836174i \(-0.684787\pi\)
−0.548464 + 0.836174i \(0.684787\pi\)
\(468\) −1.23415e27 −0.113184
\(469\) −1.78117e27 −0.159732
\(470\) 1.15988e28 1.01715
\(471\) 3.48288e27 0.298689
\(472\) 5.30757e27 0.445148
\(473\) 7.45451e27 0.611472
\(474\) 2.46560e27 0.197811
\(475\) −1.00999e27 −0.0792562
\(476\) 2.53222e27 0.194370
\(477\) −1.66326e27 −0.124887
\(478\) −1.26206e28 −0.927019
\(479\) 2.04856e28 1.47206 0.736029 0.676950i \(-0.236699\pi\)
0.736029 + 0.676950i \(0.236699\pi\)
\(480\) 3.57971e27 0.251661
\(481\) −1.31213e28 −0.902513
\(482\) 1.96106e28 1.31977
\(483\) 1.67555e27 0.110335
\(484\) 1.83525e27 0.118255
\(485\) 1.25982e28 0.794363
\(486\) −9.13134e26 −0.0563448
\(487\) −3.75693e27 −0.226871 −0.113436 0.993545i \(-0.536186\pi\)
−0.113436 + 0.993545i \(0.536186\pi\)
\(488\) −1.23508e28 −0.729942
\(489\) −1.87920e27 −0.108700
\(490\) −4.30391e27 −0.243671
\(491\) −1.53173e28 −0.848841 −0.424420 0.905465i \(-0.639522\pi\)
−0.424420 + 0.905465i \(0.639522\pi\)
\(492\) −4.11081e27 −0.222995
\(493\) 2.31261e28 1.22803
\(494\) −3.93640e28 −2.04629
\(495\) −4.43350e27 −0.225628
\(496\) −2.64587e28 −1.31829
\(497\) 6.33518e27 0.309042
\(498\) −2.29259e25 −0.00109501
\(499\) 3.84836e27 0.179978 0.0899890 0.995943i \(-0.471317\pi\)
0.0899890 + 0.995943i \(0.471317\pi\)
\(500\) −5.10855e27 −0.233945
\(501\) −2.14243e27 −0.0960749
\(502\) 1.27208e28 0.558629
\(503\) −2.25177e28 −0.968411 −0.484205 0.874954i \(-0.660891\pi\)
−0.484205 + 0.874954i \(0.660891\pi\)
\(504\) 7.22246e27 0.304204
\(505\) 3.53696e28 1.45905
\(506\) −3.41229e27 −0.137869
\(507\) 1.76153e28 0.697119
\(508\) −2.65418e27 −0.102887
\(509\) 2.09122e27 0.0794078 0.0397039 0.999211i \(-0.487359\pi\)
0.0397039 + 0.999211i \(0.487359\pi\)
\(510\) −1.33582e28 −0.496892
\(511\) −1.09184e28 −0.397869
\(512\) 3.07519e28 1.09783
\(513\) 8.62810e27 0.301775
\(514\) 3.89976e28 1.33637
\(515\) −2.50902e28 −0.842420
\(516\) −3.53039e27 −0.116145
\(517\) 2.56104e28 0.825594
\(518\) 1.42846e28 0.451237
\(519\) 1.89849e28 0.587692
\(520\) −5.14948e28 −1.56217
\(521\) −3.76486e27 −0.111932 −0.0559659 0.998433i \(-0.517824\pi\)
−0.0559659 + 0.998433i \(0.517824\pi\)
\(522\) 1.22704e28 0.357534
\(523\) −1.74944e28 −0.499611 −0.249806 0.968296i \(-0.580367\pi\)
−0.249806 + 0.968296i \(0.580367\pi\)
\(524\) 1.22040e28 0.341604
\(525\) −8.99539e26 −0.0246801
\(526\) 2.96944e28 0.798590
\(527\) −6.99170e28 −1.84320
\(528\) −1.11619e28 −0.288459
\(529\) −3.74562e28 −0.948942
\(530\) −1.29101e28 −0.320650
\(531\) 5.64716e27 0.137511
\(532\) −1.26952e28 −0.303086
\(533\) 1.07269e29 2.51095
\(534\) −1.76197e28 −0.404404
\(535\) 1.84936e28 0.416205
\(536\) −9.23480e27 −0.203797
\(537\) −1.06535e28 −0.230548
\(538\) −2.92215e28 −0.620138
\(539\) −9.50317e27 −0.197782
\(540\) 2.09967e27 0.0428565
\(541\) −6.99960e28 −1.40121 −0.700603 0.713552i \(-0.747085\pi\)
−0.700603 + 0.713552i \(0.747085\pi\)
\(542\) 5.11434e28 1.00415
\(543\) −3.75933e26 −0.00723956
\(544\) 2.38153e28 0.449849
\(545\) −8.76680e27 −0.162434
\(546\) −3.50593e28 −0.637207
\(547\) 3.00896e28 0.536476 0.268238 0.963353i \(-0.413559\pi\)
0.268238 + 0.963353i \(0.413559\pi\)
\(548\) 2.31056e28 0.404130
\(549\) −1.31411e28 −0.225487
\(550\) 1.83193e27 0.0308391
\(551\) −1.15941e29 −1.91490
\(552\) 8.68717e27 0.140773
\(553\) −2.07495e28 −0.329909
\(554\) 5.70754e28 0.890424
\(555\) 2.23235e28 0.341732
\(556\) 1.04576e28 0.157090
\(557\) 1.17263e29 1.72855 0.864277 0.503017i \(-0.167777\pi\)
0.864277 + 0.503017i \(0.167777\pi\)
\(558\) −3.70969e28 −0.536635
\(559\) 9.21233e28 1.30781
\(560\) 4.25421e28 0.592714
\(561\) −2.94954e28 −0.403315
\(562\) 7.71427e28 1.03529
\(563\) 6.46986e28 0.852230 0.426115 0.904669i \(-0.359882\pi\)
0.426115 + 0.904669i \(0.359882\pi\)
\(564\) −1.21289e28 −0.156816
\(565\) −5.23307e28 −0.664121
\(566\) −1.32575e29 −1.65153
\(567\) 7.68456e27 0.0939717
\(568\) 3.28459e28 0.394297
\(569\) −1.35009e29 −1.59105 −0.795527 0.605918i \(-0.792806\pi\)
−0.795527 + 0.605918i \(0.792806\pi\)
\(570\) 6.69707e28 0.774816
\(571\) 2.55750e28 0.290494 0.145247 0.989395i \(-0.453602\pi\)
0.145247 + 0.989395i \(0.453602\pi\)
\(572\) −2.11515e28 −0.235875
\(573\) 4.82951e28 0.528786
\(574\) −1.16779e29 −1.25542
\(575\) −1.08197e27 −0.0114209
\(576\) 3.57664e28 0.370714
\(577\) −1.52580e29 −1.55293 −0.776465 0.630160i \(-0.782989\pi\)
−0.776465 + 0.630160i \(0.782989\pi\)
\(578\) −9.88284e26 −0.00987733
\(579\) −1.06326e28 −0.104355
\(580\) −2.82146e28 −0.271944
\(581\) 1.92935e26 0.00182625
\(582\) 4.44702e28 0.413407
\(583\) −2.85058e28 −0.260264
\(584\) −5.66084e28 −0.507629
\(585\) −5.47896e28 −0.482571
\(586\) −7.94506e28 −0.687341
\(587\) −1.08485e29 −0.921868 −0.460934 0.887434i \(-0.652486\pi\)
−0.460934 + 0.887434i \(0.652486\pi\)
\(588\) 4.50063e27 0.0375674
\(589\) 3.50525e29 2.87414
\(590\) 4.38328e28 0.353064
\(591\) 9.75587e28 0.771965
\(592\) 5.62023e28 0.436894
\(593\) 8.82363e28 0.673865 0.336933 0.941529i \(-0.390611\pi\)
0.336933 + 0.941529i \(0.390611\pi\)
\(594\) −1.56498e28 −0.117422
\(595\) 1.12417e29 0.828715
\(596\) −2.32040e28 −0.168065
\(597\) −9.00714e28 −0.640998
\(598\) −4.21694e28 −0.294873
\(599\) 9.23342e28 0.634426 0.317213 0.948354i \(-0.397253\pi\)
0.317213 + 0.948354i \(0.397253\pi\)
\(600\) −4.66382e27 −0.0314886
\(601\) −7.30699e28 −0.484793 −0.242397 0.970177i \(-0.577934\pi\)
−0.242397 + 0.970177i \(0.577934\pi\)
\(602\) −1.00291e29 −0.653878
\(603\) −9.82566e27 −0.0629550
\(604\) 5.89267e28 0.371043
\(605\) 8.14756e28 0.504192
\(606\) 1.24851e29 0.759329
\(607\) 8.07544e28 0.482709 0.241354 0.970437i \(-0.422408\pi\)
0.241354 + 0.970437i \(0.422408\pi\)
\(608\) −1.19396e29 −0.701461
\(609\) −1.03262e29 −0.596294
\(610\) −1.02000e29 −0.578945
\(611\) 3.16496e29 1.76577
\(612\) 1.39688e28 0.0766070
\(613\) −2.21326e29 −1.19316 −0.596578 0.802555i \(-0.703473\pi\)
−0.596578 + 0.802555i \(0.703473\pi\)
\(614\) −9.08699e28 −0.481562
\(615\) −1.82498e29 −0.950759
\(616\) 1.23783e29 0.633960
\(617\) −2.16026e29 −1.08771 −0.543854 0.839180i \(-0.683035\pi\)
−0.543854 + 0.839180i \(0.683035\pi\)
\(618\) −8.85659e28 −0.438417
\(619\) −2.09908e29 −1.02159 −0.510795 0.859702i \(-0.670649\pi\)
−0.510795 + 0.859702i \(0.670649\pi\)
\(620\) 8.53011e28 0.408171
\(621\) 9.24299e27 0.0434861
\(622\) 7.65609e28 0.354167
\(623\) 1.48280e29 0.674465
\(624\) −1.37940e29 −0.616953
\(625\) −2.15300e29 −0.946901
\(626\) 2.15023e29 0.929938
\(627\) 1.47873e29 0.628899
\(628\) −2.82694e28 −0.118234
\(629\) 1.48514e29 0.610853
\(630\) 5.96470e28 0.241275
\(631\) 4.81094e28 0.191391 0.0956955 0.995411i \(-0.469492\pi\)
0.0956955 + 0.995411i \(0.469492\pi\)
\(632\) −1.07580e29 −0.420921
\(633\) 8.01576e28 0.308465
\(634\) 7.27166e28 0.275230
\(635\) −1.17832e29 −0.438670
\(636\) 1.35001e28 0.0494354
\(637\) −1.17441e29 −0.423014
\(638\) 2.10296e29 0.745100
\(639\) 3.49474e28 0.121802
\(640\) 1.50481e29 0.515930
\(641\) −5.04867e28 −0.170282 −0.0851408 0.996369i \(-0.527134\pi\)
−0.0851408 + 0.996369i \(0.527134\pi\)
\(642\) 6.52806e28 0.216604
\(643\) −3.93930e29 −1.28589 −0.642945 0.765912i \(-0.722288\pi\)
−0.642945 + 0.765912i \(0.722288\pi\)
\(644\) −1.35999e28 −0.0436751
\(645\) −1.56731e29 −0.495196
\(646\) 4.45545e29 1.38500
\(647\) −1.38475e29 −0.423523 −0.211762 0.977321i \(-0.567920\pi\)
−0.211762 + 0.977321i \(0.567920\pi\)
\(648\) 3.98420e28 0.119896
\(649\) 9.67842e28 0.286573
\(650\) 2.26392e28 0.0659584
\(651\) 3.12193e29 0.894998
\(652\) 1.52528e28 0.0430280
\(653\) 1.16655e29 0.323830 0.161915 0.986805i \(-0.448233\pi\)
0.161915 + 0.986805i \(0.448233\pi\)
\(654\) −3.09459e28 −0.0845350
\(655\) 5.41792e29 1.45646
\(656\) −4.59464e29 −1.21552
\(657\) −6.02303e28 −0.156812
\(658\) −3.44555e29 −0.882849
\(659\) −3.36819e29 −0.849377 −0.424688 0.905340i \(-0.639616\pi\)
−0.424688 + 0.905340i \(0.639616\pi\)
\(660\) 3.59854e28 0.0893130
\(661\) 4.31449e29 1.05394 0.526968 0.849885i \(-0.323329\pi\)
0.526968 + 0.849885i \(0.323329\pi\)
\(662\) −8.49612e28 −0.204273
\(663\) −3.64506e29 −0.862606
\(664\) 1.00030e27 0.00233006
\(665\) −5.63598e29 −1.29224
\(666\) 7.87995e28 0.177846
\(667\) −1.24204e29 −0.275940
\(668\) 1.73894e28 0.0380305
\(669\) 2.79415e29 0.601555
\(670\) −7.62660e28 −0.161639
\(671\) −2.25219e29 −0.469915
\(672\) −1.06340e29 −0.218433
\(673\) −1.19710e29 −0.242088 −0.121044 0.992647i \(-0.538624\pi\)
−0.121044 + 0.992647i \(0.538624\pi\)
\(674\) −4.52008e29 −0.899948
\(675\) −4.96222e27 −0.00972717
\(676\) −1.42978e29 −0.275949
\(677\) 7.71503e29 1.46608 0.733038 0.680188i \(-0.238102\pi\)
0.733038 + 0.680188i \(0.238102\pi\)
\(678\) −1.84722e29 −0.345626
\(679\) −3.74243e29 −0.689479
\(680\) 5.82848e29 1.05733
\(681\) 4.72525e29 0.844074
\(682\) −6.35788e29 −1.11835
\(683\) 6.40488e29 1.10941 0.554707 0.832046i \(-0.312830\pi\)
0.554707 + 0.832046i \(0.312830\pi\)
\(684\) −7.00316e28 −0.119455
\(685\) 1.02577e30 1.72305
\(686\) 5.76908e29 0.954340
\(687\) 1.60679e29 0.261766
\(688\) −3.94591e29 −0.633094
\(689\) −3.52277e29 −0.556650
\(690\) 7.17434e28 0.111652
\(691\) 1.07309e30 1.64482 0.822408 0.568898i \(-0.192630\pi\)
0.822408 + 0.568898i \(0.192630\pi\)
\(692\) −1.54094e29 −0.232633
\(693\) 1.31702e29 0.195837
\(694\) 5.01454e28 0.0734440
\(695\) 4.64263e29 0.669767
\(696\) −5.35382e29 −0.760794
\(697\) −1.21413e30 −1.69950
\(698\) 7.43202e29 1.02477
\(699\) −3.89791e29 −0.529446
\(700\) 7.30127e27 0.00976943
\(701\) −7.18115e29 −0.946574 −0.473287 0.880908i \(-0.656933\pi\)
−0.473287 + 0.880908i \(0.656933\pi\)
\(702\) −1.93401e29 −0.251142
\(703\) −7.44568e29 −0.952518
\(704\) 6.12985e29 0.772568
\(705\) −5.38459e29 −0.668601
\(706\) −6.30969e29 −0.771895
\(707\) −1.05070e30 −1.26641
\(708\) −4.58362e28 −0.0544327
\(709\) 4.24360e29 0.496534 0.248267 0.968692i \(-0.420139\pi\)
0.248267 + 0.968692i \(0.420139\pi\)
\(710\) 2.71259e29 0.312732
\(711\) −1.14463e29 −0.130027
\(712\) 7.68786e29 0.860529
\(713\) 3.75505e29 0.414167
\(714\) 3.96822e29 0.431285
\(715\) −9.39014e29 −1.00568
\(716\) 8.64710e28 0.0912607
\(717\) 5.85898e29 0.609356
\(718\) 4.26277e29 0.436904
\(719\) −5.73272e29 −0.579038 −0.289519 0.957172i \(-0.593495\pi\)
−0.289519 + 0.957172i \(0.593495\pi\)
\(720\) 2.34680e29 0.233606
\(721\) 7.45334e29 0.731191
\(722\) −1.32527e30 −1.28134
\(723\) −9.10400e29 −0.867521
\(724\) 3.05133e27 0.00286572
\(725\) 6.66805e28 0.0617234
\(726\) 2.87600e29 0.262395
\(727\) 1.42987e30 1.28583 0.642916 0.765937i \(-0.277725\pi\)
0.642916 + 0.765937i \(0.277725\pi\)
\(728\) 1.52971e30 1.35591
\(729\) 4.23912e28 0.0370370
\(730\) −4.67503e29 −0.402619
\(731\) −1.04270e30 −0.885175
\(732\) 1.06662e29 0.0892572
\(733\) 1.99143e30 1.64276 0.821380 0.570381i \(-0.193205\pi\)
0.821380 + 0.570381i \(0.193205\pi\)
\(734\) −1.23405e29 −0.100351
\(735\) 1.99804e29 0.160172
\(736\) −1.27905e29 −0.101081
\(737\) −1.68398e29 −0.131198
\(738\) −6.44200e29 −0.494799
\(739\) −2.11124e30 −1.59872 −0.799359 0.600854i \(-0.794827\pi\)
−0.799359 + 0.600854i \(0.794827\pi\)
\(740\) −1.81193e29 −0.135272
\(741\) 1.82743e30 1.34508
\(742\) 3.83509e29 0.278313
\(743\) 1.11538e30 0.798067 0.399033 0.916936i \(-0.369346\pi\)
0.399033 + 0.916936i \(0.369346\pi\)
\(744\) 1.61862e30 1.14190
\(745\) −1.03013e30 −0.716561
\(746\) −1.31331e30 −0.900763
\(747\) 1.06431e27 0.000719781 0
\(748\) 2.39405e29 0.159649
\(749\) −5.49375e29 −0.361251
\(750\) −8.00555e29 −0.519096
\(751\) −2.28371e30 −1.46023 −0.730116 0.683323i \(-0.760534\pi\)
−0.730116 + 0.683323i \(0.760534\pi\)
\(752\) −1.35564e30 −0.854787
\(753\) −5.90546e29 −0.367203
\(754\) 2.59886e30 1.59361
\(755\) 2.61603e30 1.58198
\(756\) −6.23732e28 −0.0371980
\(757\) 9.73411e29 0.572519 0.286259 0.958152i \(-0.407588\pi\)
0.286259 + 0.958152i \(0.407588\pi\)
\(758\) −2.80951e29 −0.162969
\(759\) 1.58412e29 0.0906251
\(760\) −2.92207e30 −1.64873
\(761\) −3.06985e30 −1.70836 −0.854179 0.519979i \(-0.825940\pi\)
−0.854179 + 0.519979i \(0.825940\pi\)
\(762\) −4.15933e29 −0.228295
\(763\) 2.60428e29 0.140987
\(764\) −3.91996e29 −0.209316
\(765\) 6.20140e29 0.326621
\(766\) 2.02936e30 1.05428
\(767\) 1.19607e30 0.612920
\(768\) −7.39078e29 −0.373592
\(769\) −7.51816e29 −0.374874 −0.187437 0.982277i \(-0.560018\pi\)
−0.187437 + 0.982277i \(0.560018\pi\)
\(770\) 1.02226e30 0.502817
\(771\) −1.81042e30 −0.878432
\(772\) 8.63013e28 0.0413081
\(773\) 1.23771e30 0.584434 0.292217 0.956352i \(-0.405607\pi\)
0.292217 + 0.956352i \(0.405607\pi\)
\(774\) −5.53243e29 −0.257713
\(775\) −2.01595e29 −0.0926427
\(776\) −1.94033e30 −0.879686
\(777\) −6.63144e29 −0.296611
\(778\) 4.58507e29 0.202330
\(779\) 6.08698e30 2.65008
\(780\) 4.44710e29 0.191022
\(781\) 5.98949e29 0.253836
\(782\) 4.77297e29 0.199580
\(783\) −5.69637e29 −0.235017
\(784\) 5.03033e29 0.204776
\(785\) −1.25501e30 −0.504100
\(786\) 1.91247e30 0.757979
\(787\) 2.14352e30 0.838288 0.419144 0.907920i \(-0.362330\pi\)
0.419144 + 0.907920i \(0.362330\pi\)
\(788\) −7.91853e29 −0.305576
\(789\) −1.37853e30 −0.524936
\(790\) −8.88450e29 −0.333848
\(791\) 1.55454e30 0.576434
\(792\) 6.82834e29 0.249862
\(793\) −2.78327e30 −1.00505
\(794\) 1.87221e30 0.667174
\(795\) 5.99334e29 0.210773
\(796\) 7.31081e29 0.253734
\(797\) 3.01280e30 1.03195 0.515974 0.856605i \(-0.327430\pi\)
0.515974 + 0.856605i \(0.327430\pi\)
\(798\) −1.98944e30 −0.672513
\(799\) −3.58228e30 −1.19514
\(800\) 6.86676e28 0.0226103
\(801\) 8.17975e29 0.265827
\(802\) −1.27705e30 −0.409617
\(803\) −1.03226e30 −0.326796
\(804\) 7.97518e28 0.0249202
\(805\) −6.03763e29 −0.186213
\(806\) −7.85711e30 −2.39191
\(807\) 1.35657e30 0.407634
\(808\) −5.44752e30 −1.61577
\(809\) −5.76883e30 −1.68900 −0.844498 0.535559i \(-0.820101\pi\)
−0.844498 + 0.535559i \(0.820101\pi\)
\(810\) 3.29037e29 0.0950937
\(811\) 3.98728e30 1.13752 0.568759 0.822504i \(-0.307424\pi\)
0.568759 + 0.822504i \(0.307424\pi\)
\(812\) 8.38148e29 0.236038
\(813\) −2.37427e30 −0.660055
\(814\) 1.35051e30 0.370631
\(815\) 6.77146e29 0.183454
\(816\) 1.56129e30 0.417576
\(817\) 5.22754e30 1.38027
\(818\) 5.61998e30 1.46496
\(819\) 1.62759e30 0.418855
\(820\) 1.48128e30 0.376350
\(821\) −4.09615e30 −1.02748 −0.513739 0.857946i \(-0.671740\pi\)
−0.513739 + 0.857946i \(0.671740\pi\)
\(822\) 3.62085e30 0.896719
\(823\) 1.43384e30 0.350592 0.175296 0.984516i \(-0.443912\pi\)
0.175296 + 0.984516i \(0.443912\pi\)
\(824\) 3.86432e30 0.932905
\(825\) −8.50453e28 −0.0202714
\(826\) −1.30210e30 −0.306447
\(827\) −3.13142e30 −0.727668 −0.363834 0.931464i \(-0.618532\pi\)
−0.363834 + 0.931464i \(0.618532\pi\)
\(828\) −7.50225e28 −0.0172136
\(829\) −4.44329e30 −1.00666 −0.503329 0.864095i \(-0.667892\pi\)
−0.503329 + 0.864095i \(0.667892\pi\)
\(830\) 8.26106e27 0.00184806
\(831\) −2.64966e30 −0.585301
\(832\) 7.57531e30 1.65236
\(833\) 1.32926e30 0.286311
\(834\) 1.63880e30 0.348564
\(835\) 7.71998e29 0.162147
\(836\) −1.20024e30 −0.248944
\(837\) 1.72218e30 0.352745
\(838\) 3.00994e30 0.608829
\(839\) −9.30057e29 −0.185784 −0.0928922 0.995676i \(-0.529611\pi\)
−0.0928922 + 0.995676i \(0.529611\pi\)
\(840\) −2.60252e30 −0.513408
\(841\) 2.52173e30 0.491293
\(842\) −3.46057e30 −0.665842
\(843\) −3.58126e30 −0.680528
\(844\) −6.50614e29 −0.122103
\(845\) −6.34746e30 −1.17653
\(846\) −1.90070e30 −0.347957
\(847\) −2.42033e30 −0.437621
\(848\) 1.50891e30 0.269467
\(849\) 6.15462e30 1.08560
\(850\) −2.56243e29 −0.0446430
\(851\) −7.97630e29 −0.137259
\(852\) −2.83657e29 −0.0482145
\(853\) −2.27471e30 −0.381911 −0.190955 0.981599i \(-0.561159\pi\)
−0.190955 + 0.981599i \(0.561159\pi\)
\(854\) 3.03003e30 0.502503
\(855\) −3.10903e30 −0.509309
\(856\) −2.84833e30 −0.460910
\(857\) 3.05003e30 0.487534 0.243767 0.969834i \(-0.421617\pi\)
0.243767 + 0.969834i \(0.421617\pi\)
\(858\) −3.31462e30 −0.523380
\(859\) 1.34073e30 0.209128 0.104564 0.994518i \(-0.466655\pi\)
0.104564 + 0.994518i \(0.466655\pi\)
\(860\) 1.27213e30 0.196020
\(861\) 5.42133e30 0.825225
\(862\) −1.12873e31 −1.69732
\(863\) 1.14916e31 1.70713 0.853566 0.520985i \(-0.174435\pi\)
0.853566 + 0.520985i \(0.174435\pi\)
\(864\) −5.86612e29 −0.0860908
\(865\) −6.84096e30 −0.991855
\(866\) 4.32457e30 0.619448
\(867\) 4.58799e28 0.00649265
\(868\) −2.53397e30 −0.354278
\(869\) −1.96173e30 −0.270976
\(870\) −4.42148e30 −0.603414
\(871\) −2.08107e30 −0.280606
\(872\) 1.35024e30 0.179882
\(873\) −2.06448e30 −0.271744
\(874\) −2.39290e30 −0.311211
\(875\) 6.73714e30 0.865747
\(876\) 4.88871e29 0.0620727
\(877\) 9.06320e30 1.13707 0.568533 0.822660i \(-0.307511\pi\)
0.568533 + 0.822660i \(0.307511\pi\)
\(878\) −2.91934e30 −0.361904
\(879\) 3.68840e30 0.451809
\(880\) 4.02207e30 0.486835
\(881\) −6.50767e30 −0.778356 −0.389178 0.921163i \(-0.627241\pi\)
−0.389178 + 0.921163i \(0.627241\pi\)
\(882\) 7.05287e29 0.0833577
\(883\) 5.90742e30 0.689939 0.344969 0.938614i \(-0.387889\pi\)
0.344969 + 0.938614i \(0.387889\pi\)
\(884\) 2.95858e30 0.341456
\(885\) −2.03489e30 −0.232079
\(886\) 7.18992e30 0.810344
\(887\) −3.42312e30 −0.381263 −0.190632 0.981662i \(-0.561054\pi\)
−0.190632 + 0.981662i \(0.561054\pi\)
\(888\) −3.43819e30 −0.378437
\(889\) 3.50032e30 0.380750
\(890\) 6.34905e30 0.682518
\(891\) 7.26524e29 0.0771852
\(892\) −2.26792e30 −0.238121
\(893\) 1.79595e31 1.86361
\(894\) −3.63627e30 −0.372917
\(895\) 3.83886e30 0.389099
\(896\) −4.47020e30 −0.447810
\(897\) 1.95766e30 0.193828
\(898\) −1.56863e31 −1.53503
\(899\) −2.31420e31 −2.23833
\(900\) 4.02768e28 0.00385042
\(901\) 3.98728e30 0.376761
\(902\) −1.10407e31 −1.03116
\(903\) 4.65587e30 0.429813
\(904\) 8.05980e30 0.735455
\(905\) 1.35463e29 0.0122183
\(906\) 9.23433e30 0.823302
\(907\) −1.16251e31 −1.02452 −0.512262 0.858829i \(-0.671192\pi\)
−0.512262 + 0.858829i \(0.671192\pi\)
\(908\) −3.83534e30 −0.334120
\(909\) −5.79606e30 −0.499128
\(910\) 1.26332e31 1.07542
\(911\) −1.34510e31 −1.13191 −0.565957 0.824435i \(-0.691493\pi\)
−0.565957 + 0.824435i \(0.691493\pi\)
\(912\) −7.82741e30 −0.651137
\(913\) 1.82407e28 0.00150002
\(914\) 9.70051e30 0.788605
\(915\) 4.73523e30 0.380557
\(916\) −1.30418e30 −0.103618
\(917\) −1.60945e31 −1.26416
\(918\) 2.18903e30 0.169982
\(919\) −8.55562e30 −0.656808 −0.328404 0.944537i \(-0.606511\pi\)
−0.328404 + 0.944537i \(0.606511\pi\)
\(920\) −3.13032e30 −0.237583
\(921\) 4.21853e30 0.316544
\(922\) 1.15093e30 0.0853835
\(923\) 7.40185e30 0.542903
\(924\) −1.06899e30 −0.0775205
\(925\) 4.28218e29 0.0307027
\(926\) 1.52259e30 0.107937
\(927\) 4.11157e30 0.288184
\(928\) 7.88268e30 0.546286
\(929\) 9.93712e30 0.680919 0.340460 0.940259i \(-0.389417\pi\)
0.340460 + 0.940259i \(0.389417\pi\)
\(930\) 1.33674e31 0.905684
\(931\) −6.66418e30 −0.446452
\(932\) 3.16381e30 0.209577
\(933\) −3.55425e30 −0.232804
\(934\) 1.48757e31 0.963462
\(935\) 1.06283e31 0.680679
\(936\) 8.43852e30 0.534404
\(937\) −2.14272e30 −0.134183 −0.0670917 0.997747i \(-0.521372\pi\)
−0.0670917 + 0.997747i \(0.521372\pi\)
\(938\) 2.26557e30 0.140297
\(939\) −9.98217e30 −0.611275
\(940\) 4.37050e30 0.264660
\(941\) 1.69148e31 1.01292 0.506459 0.862264i \(-0.330954\pi\)
0.506459 + 0.862264i \(0.330954\pi\)
\(942\) −4.43006e30 −0.262347
\(943\) 6.52077e30 0.381879
\(944\) −5.12310e30 −0.296706
\(945\) −2.76904e30 −0.158597
\(946\) −9.48180e30 −0.537074
\(947\) −1.96530e31 −1.10092 −0.550459 0.834862i \(-0.685547\pi\)
−0.550459 + 0.834862i \(0.685547\pi\)
\(948\) 9.29057e29 0.0514701
\(949\) −1.27568e31 −0.698948
\(950\) 1.28466e30 0.0696129
\(951\) −3.37578e30 −0.180917
\(952\) −1.73142e31 −0.917728
\(953\) 2.39001e31 1.25292 0.626462 0.779452i \(-0.284502\pi\)
0.626462 + 0.779452i \(0.284502\pi\)
\(954\) 2.11559e30 0.109692
\(955\) −1.74026e31 −0.892437
\(956\) −4.75555e30 −0.241209
\(957\) −9.76276e30 −0.489776
\(958\) −2.60567e31 −1.29295
\(959\) −3.04716e31 −1.49555
\(960\) −1.28880e31 −0.625658
\(961\) 4.91397e31 2.35959
\(962\) 1.66897e31 0.792703
\(963\) −3.03057e30 −0.142380
\(964\) 7.38943e30 0.343401
\(965\) 3.83132e30 0.176121
\(966\) −2.13122e30 −0.0969100
\(967\) 1.63072e31 0.733504 0.366752 0.930319i \(-0.380470\pi\)
0.366752 + 0.930319i \(0.380470\pi\)
\(968\) −1.25486e31 −0.558348
\(969\) −2.06839e31 −0.910401
\(970\) −1.60243e31 −0.697712
\(971\) 3.09343e31 1.33241 0.666207 0.745767i \(-0.267917\pi\)
0.666207 + 0.745767i \(0.267917\pi\)
\(972\) −3.44076e29 −0.0146608
\(973\) −1.37915e31 −0.581334
\(974\) 4.77865e30 0.199267
\(975\) −1.05100e30 −0.0433563
\(976\) 1.19216e31 0.486531
\(977\) 1.98044e31 0.799592 0.399796 0.916604i \(-0.369081\pi\)
0.399796 + 0.916604i \(0.369081\pi\)
\(978\) 2.39025e30 0.0954742
\(979\) 1.40189e31 0.553982
\(980\) −1.62175e30 −0.0634029
\(981\) 1.43663e30 0.0555673
\(982\) 1.94829e31 0.745561
\(983\) −4.34284e30 −0.164423 −0.0822114 0.996615i \(-0.526198\pi\)
−0.0822114 + 0.996615i \(0.526198\pi\)
\(984\) 2.81078e31 1.05288
\(985\) −3.51541e31 −1.30285
\(986\) −2.94154e31 −1.07862
\(987\) 1.59955e31 0.580322
\(988\) −1.48327e31 −0.532441
\(989\) 5.60008e30 0.198899
\(990\) 5.63922e30 0.198175
\(991\) 4.70795e30 0.163704 0.0818518 0.996645i \(-0.473917\pi\)
0.0818518 + 0.996645i \(0.473917\pi\)
\(992\) −2.38317e31 −0.819939
\(993\) 3.94422e30 0.134275
\(994\) −8.05807e30 −0.271440
\(995\) 3.24561e31 1.08182
\(996\) −8.63864e27 −0.000284920 0
\(997\) −1.58934e31 −0.518700 −0.259350 0.965783i \(-0.583508\pi\)
−0.259350 + 0.965783i \(0.583508\pi\)
\(998\) −4.89494e30 −0.158080
\(999\) −3.65817e30 −0.116903
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 3.22.a.c.1.1 2
3.2 odd 2 9.22.a.e.1.2 2
4.3 odd 2 48.22.a.g.1.1 2
5.2 odd 4 75.22.b.d.49.2 4
5.3 odd 4 75.22.b.d.49.3 4
5.4 even 2 75.22.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.c.1.1 2 1.1 even 1 trivial
9.22.a.e.1.2 2 3.2 odd 2
48.22.a.g.1.1 2 4.3 odd 2
75.22.a.d.1.2 2 5.4 even 2
75.22.b.d.49.2 4 5.2 odd 4
75.22.b.d.49.3 4 5.3 odd 4