Properties

Label 3.24.a.b.1.1
Level $3$
Weight $24$
Character 3.1
Self dual yes
Analytic conductor $10.056$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.0561211204\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{530401}) \)
Defining polynomial: \( x^{2} - x - 132600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(364.643\) of defining polynomial
Character \(\chi\) \(=\) 3.1

$q$-expansion

\(f(q)\) \(=\) \(q-2805.86 q^{2} -177147. q^{3} -515763. q^{4} -1.60439e8 q^{5} +4.97050e8 q^{6} -8.06460e9 q^{7} +2.49844e10 q^{8} +3.13811e10 q^{9} +O(q^{10})\) \(q-2805.86 q^{2} -177147. q^{3} -515763. q^{4} -1.60439e8 q^{5} +4.97050e8 q^{6} -8.06460e9 q^{7} +2.49844e10 q^{8} +3.13811e10 q^{9} +4.50169e11 q^{10} +4.37600e11 q^{11} +9.13659e10 q^{12} +2.40244e12 q^{13} +2.26281e13 q^{14} +2.84212e13 q^{15} -6.57762e13 q^{16} -7.22389e13 q^{17} -8.80508e13 q^{18} -7.65478e14 q^{19} +8.27484e13 q^{20} +1.42862e15 q^{21} -1.22785e15 q^{22} +5.30922e15 q^{23} -4.42591e15 q^{24} +1.38197e16 q^{25} -6.74091e15 q^{26} -5.55906e15 q^{27} +4.15942e15 q^{28} +3.30401e16 q^{29} -7.97460e16 q^{30} -1.91705e17 q^{31} -2.50257e16 q^{32} -7.75196e16 q^{33} +2.02692e17 q^{34} +1.29387e18 q^{35} -1.61852e16 q^{36} -1.19466e18 q^{37} +2.14782e18 q^{38} -4.25585e17 q^{39} -4.00847e18 q^{40} -1.08195e18 q^{41} -4.00851e18 q^{42} +9.41608e18 q^{43} -2.25698e17 q^{44} -5.03474e18 q^{45} -1.48969e19 q^{46} +1.46134e19 q^{47} +1.16521e19 q^{48} +3.76691e19 q^{49} -3.87760e19 q^{50} +1.27969e19 q^{51} -1.23909e18 q^{52} +3.45534e19 q^{53} +1.55979e19 q^{54} -7.02081e19 q^{55} -2.01489e20 q^{56} +1.35602e20 q^{57} -9.27060e19 q^{58} -2.65574e20 q^{59} -1.46586e19 q^{60} +5.20374e20 q^{61} +5.37898e20 q^{62} -2.53076e20 q^{63} +6.21989e20 q^{64} -3.85445e20 q^{65} +2.17509e20 q^{66} -3.34691e20 q^{67} +3.72581e19 q^{68} -9.40513e20 q^{69} -3.63043e21 q^{70} +1.99393e21 q^{71} +7.84037e20 q^{72} +2.19877e20 q^{73} +3.35204e21 q^{74} -2.44811e21 q^{75} +3.94805e20 q^{76} -3.52907e21 q^{77} +1.19413e21 q^{78} -3.54687e20 q^{79} +1.05531e22 q^{80} +9.84771e20 q^{81} +3.03580e21 q^{82} -4.81732e21 q^{83} -7.36829e20 q^{84} +1.15899e22 q^{85} -2.64202e22 q^{86} -5.85296e21 q^{87} +1.09332e22 q^{88} -3.80894e22 q^{89} +1.41268e22 q^{90} -1.93747e22 q^{91} -2.73830e21 q^{92} +3.39600e22 q^{93} -4.10032e22 q^{94} +1.22812e23 q^{95} +4.43322e21 q^{96} +9.60308e22 q^{97} -1.05694e23 q^{98} +1.37324e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1242 q^{2} - 354294 q^{3} - 6458716 q^{4} - 46808820 q^{5} + 220016574 q^{6} - 211963904 q^{7} + 2571869016 q^{8} + 62762119218 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1242 q^{2} - 354294 q^{3} - 6458716 q^{4} - 46808820 q^{5} + 220016574 q^{6} - 211963904 q^{7} + 2571869016 q^{8} + 62762119218 q^{9} + 627869790180 q^{10} + 1468972366488 q^{11} + 1144142163252 q^{12} + 10491654264748 q^{13} + 34908555004416 q^{14} + 8292042036540 q^{15} - 50973150676720 q^{16} - 210888011520828 q^{17} - 38975276034378 q^{18} - 907382448537944 q^{19} - 592549041758760 q^{20} + 37548769701888 q^{21} + 385075304370504 q^{22} + 10\!\cdots\!12 q^{23}+ \cdots + 46\!\cdots\!92 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2805.86 −0.968770 −0.484385 0.874855i \(-0.660957\pi\)
−0.484385 + 0.874855i \(0.660957\pi\)
\(3\) −177147. −0.577350
\(4\) −515763. −0.0614838
\(5\) −1.60439e8 −1.46945 −0.734724 0.678366i \(-0.762688\pi\)
−0.734724 + 0.678366i \(0.762688\pi\)
\(6\) 4.97050e8 0.559320
\(7\) −8.06460e9 −1.54154 −0.770771 0.637112i \(-0.780129\pi\)
−0.770771 + 0.637112i \(0.780129\pi\)
\(8\) 2.49844e10 1.02833
\(9\) 3.13811e10 0.333333
\(10\) 4.50169e11 1.42356
\(11\) 4.37600e11 0.462447 0.231223 0.972901i \(-0.425727\pi\)
0.231223 + 0.972901i \(0.425727\pi\)
\(12\) 9.13659e10 0.0354977
\(13\) 2.40244e12 0.371795 0.185898 0.982569i \(-0.440481\pi\)
0.185898 + 0.982569i \(0.440481\pi\)
\(14\) 2.26281e13 1.49340
\(15\) 2.84212e13 0.848386
\(16\) −6.57762e13 −0.934736
\(17\) −7.22389e13 −0.511221 −0.255610 0.966780i \(-0.582276\pi\)
−0.255610 + 0.966780i \(0.582276\pi\)
\(18\) −8.80508e13 −0.322923
\(19\) −7.65478e14 −1.50753 −0.753765 0.657144i \(-0.771764\pi\)
−0.753765 + 0.657144i \(0.771764\pi\)
\(20\) 8.27484e13 0.0903472
\(21\) 1.42862e15 0.890010
\(22\) −1.22785e15 −0.448005
\(23\) 5.30922e15 1.16188 0.580940 0.813947i \(-0.302685\pi\)
0.580940 + 0.813947i \(0.302685\pi\)
\(24\) −4.42591e15 −0.593709
\(25\) 1.38197e16 1.15928
\(26\) −6.74091e15 −0.360184
\(27\) −5.55906e15 −0.192450
\(28\) 4.15942e15 0.0947798
\(29\) 3.30401e16 0.502881 0.251440 0.967873i \(-0.419096\pi\)
0.251440 + 0.967873i \(0.419096\pi\)
\(30\) −7.97460e16 −0.821892
\(31\) −1.91705e17 −1.35511 −0.677555 0.735472i \(-0.736961\pi\)
−0.677555 + 0.735472i \(0.736961\pi\)
\(32\) −2.50257e16 −0.122789
\(33\) −7.75196e16 −0.266994
\(34\) 2.02692e17 0.495255
\(35\) 1.29387e18 2.26522
\(36\) −1.61852e16 −0.0204946
\(37\) −1.19466e18 −1.10388 −0.551941 0.833883i \(-0.686113\pi\)
−0.551941 + 0.833883i \(0.686113\pi\)
\(38\) 2.14782e18 1.46045
\(39\) −4.25585e17 −0.214656
\(40\) −4.00847e18 −1.51108
\(41\) −1.08195e18 −0.307039 −0.153519 0.988146i \(-0.549061\pi\)
−0.153519 + 0.988146i \(0.549061\pi\)
\(42\) −4.00851e18 −0.862215
\(43\) 9.41608e18 1.54519 0.772597 0.634897i \(-0.218957\pi\)
0.772597 + 0.634897i \(0.218957\pi\)
\(44\) −2.25698e17 −0.0284330
\(45\) −5.03474e18 −0.489816
\(46\) −1.48969e19 −1.12559
\(47\) 1.46134e19 0.862237 0.431119 0.902295i \(-0.358119\pi\)
0.431119 + 0.902295i \(0.358119\pi\)
\(48\) 1.16521e19 0.539670
\(49\) 3.76691e19 1.37635
\(50\) −3.87760e19 −1.12307
\(51\) 1.27969e19 0.295153
\(52\) −1.23909e18 −0.0228594
\(53\) 3.45534e19 0.512058 0.256029 0.966669i \(-0.417586\pi\)
0.256029 + 0.966669i \(0.417586\pi\)
\(54\) 1.55979e19 0.186440
\(55\) −7.02081e19 −0.679542
\(56\) −2.01489e20 −1.58522
\(57\) 1.35602e20 0.870373
\(58\) −9.27060e19 −0.487176
\(59\) −2.65574e20 −1.14654 −0.573268 0.819368i \(-0.694325\pi\)
−0.573268 + 0.819368i \(0.694325\pi\)
\(60\) −1.46586e19 −0.0521620
\(61\) 5.20374e20 1.53117 0.765584 0.643336i \(-0.222450\pi\)
0.765584 + 0.643336i \(0.222450\pi\)
\(62\) 5.37898e20 1.31279
\(63\) −2.53076e20 −0.513847
\(64\) 6.21989e20 1.05369
\(65\) −3.85445e20 −0.546334
\(66\) 2.17509e20 0.258656
\(67\) −3.34691e20 −0.334799 −0.167399 0.985889i \(-0.553537\pi\)
−0.167399 + 0.985889i \(0.553537\pi\)
\(68\) 3.72581e19 0.0314318
\(69\) −9.40513e20 −0.670811
\(70\) −3.63043e21 −2.19448
\(71\) 1.99393e21 1.02386 0.511928 0.859028i \(-0.328931\pi\)
0.511928 + 0.859028i \(0.328931\pi\)
\(72\) 7.84037e20 0.342778
\(73\) 2.19877e20 0.0820290 0.0410145 0.999159i \(-0.486941\pi\)
0.0410145 + 0.999159i \(0.486941\pi\)
\(74\) 3.35204e21 1.06941
\(75\) −2.44811e21 −0.669309
\(76\) 3.94805e20 0.0926886
\(77\) −3.52907e21 −0.712881
\(78\) 1.19413e21 0.207953
\(79\) −3.54687e20 −0.0533498 −0.0266749 0.999644i \(-0.508492\pi\)
−0.0266749 + 0.999644i \(0.508492\pi\)
\(80\) 1.05531e22 1.37355
\(81\) 9.84771e20 0.111111
\(82\) 3.03580e21 0.297450
\(83\) −4.81732e21 −0.410589 −0.205295 0.978700i \(-0.565815\pi\)
−0.205295 + 0.978700i \(0.565815\pi\)
\(84\) −7.36829e20 −0.0547211
\(85\) 1.15899e22 0.751212
\(86\) −2.64202e22 −1.49694
\(87\) −5.85296e21 −0.290338
\(88\) 1.09332e22 0.475550
\(89\) −3.80894e22 −1.45485 −0.727426 0.686186i \(-0.759284\pi\)
−0.727426 + 0.686186i \(0.759284\pi\)
\(90\) 1.41268e22 0.474519
\(91\) −1.93747e22 −0.573138
\(92\) −2.73830e21 −0.0714367
\(93\) 3.39600e22 0.782373
\(94\) −4.10032e22 −0.835310
\(95\) 1.22812e23 2.21524
\(96\) 4.43322e21 0.0708925
\(97\) 9.60308e22 1.36312 0.681562 0.731760i \(-0.261301\pi\)
0.681562 + 0.731760i \(0.261301\pi\)
\(98\) −1.05694e23 −1.33337
\(99\) 1.37324e22 0.154149
\(100\) −7.12768e21 −0.0712768
\(101\) 1.42212e22 0.126835 0.0634176 0.997987i \(-0.479800\pi\)
0.0634176 + 0.997987i \(0.479800\pi\)
\(102\) −3.59063e22 −0.285936
\(103\) −8.52949e22 −0.607149 −0.303574 0.952808i \(-0.598180\pi\)
−0.303574 + 0.952808i \(0.598180\pi\)
\(104\) 6.00236e22 0.382330
\(105\) −2.29206e23 −1.30782
\(106\) −9.69521e22 −0.496066
\(107\) 6.63340e21 0.0304665 0.0152333 0.999884i \(-0.495151\pi\)
0.0152333 + 0.999884i \(0.495151\pi\)
\(108\) 2.86716e21 0.0118326
\(109\) −6.22821e22 −0.231184 −0.115592 0.993297i \(-0.536877\pi\)
−0.115592 + 0.993297i \(0.536877\pi\)
\(110\) 1.96994e23 0.658320
\(111\) 2.11630e23 0.637327
\(112\) 5.30459e23 1.44094
\(113\) 2.16525e23 0.531015 0.265508 0.964109i \(-0.414460\pi\)
0.265508 + 0.964109i \(0.414460\pi\)
\(114\) −3.80480e23 −0.843192
\(115\) −8.51805e23 −1.70732
\(116\) −1.70409e22 −0.0309190
\(117\) 7.53911e22 0.123932
\(118\) 7.45164e23 1.11073
\(119\) 5.82578e23 0.788068
\(120\) 7.10088e23 0.872425
\(121\) −7.03936e23 −0.786143
\(122\) −1.46010e24 −1.48335
\(123\) 1.91664e23 0.177269
\(124\) 9.88745e22 0.0833173
\(125\) −3.04632e23 −0.234051
\(126\) 7.10095e23 0.497800
\(127\) −3.88241e23 −0.248518 −0.124259 0.992250i \(-0.539655\pi\)
−0.124259 + 0.992250i \(0.539655\pi\)
\(128\) −1.53528e24 −0.897995
\(129\) −1.66803e24 −0.892118
\(130\) 1.08150e24 0.529272
\(131\) −2.29586e24 −1.02879 −0.514394 0.857554i \(-0.671983\pi\)
−0.514394 + 0.857554i \(0.671983\pi\)
\(132\) 3.99817e22 0.0164158
\(133\) 6.17327e24 2.32392
\(134\) 9.39095e23 0.324343
\(135\) 8.91889e23 0.282795
\(136\) −1.80485e24 −0.525706
\(137\) 5.74414e24 1.53794 0.768968 0.639287i \(-0.220770\pi\)
0.768968 + 0.639287i \(0.220770\pi\)
\(138\) 2.63895e24 0.649862
\(139\) −3.37226e24 −0.764278 −0.382139 0.924105i \(-0.624812\pi\)
−0.382139 + 0.924105i \(0.624812\pi\)
\(140\) −6.67333e23 −0.139274
\(141\) −2.58873e24 −0.497813
\(142\) −5.59469e24 −0.991882
\(143\) 1.05131e24 0.171936
\(144\) −2.06413e24 −0.311579
\(145\) −5.30092e24 −0.738957
\(146\) −6.16945e23 −0.0794673
\(147\) −6.67296e24 −0.794638
\(148\) 6.16159e23 0.0678708
\(149\) 1.40559e25 1.43290 0.716451 0.697637i \(-0.245765\pi\)
0.716451 + 0.697637i \(0.245765\pi\)
\(150\) 6.86906e24 0.648407
\(151\) −5.05826e24 −0.442350 −0.221175 0.975234i \(-0.570989\pi\)
−0.221175 + 0.975234i \(0.570989\pi\)
\(152\) −1.91250e25 −1.55024
\(153\) −2.26693e24 −0.170407
\(154\) 9.90208e24 0.690619
\(155\) 3.07570e25 1.99126
\(156\) 2.19501e23 0.0131979
\(157\) 9.65397e24 0.539337 0.269668 0.962953i \(-0.413086\pi\)
0.269668 + 0.962953i \(0.413086\pi\)
\(158\) 9.95200e23 0.0516838
\(159\) −6.12104e24 −0.295637
\(160\) 4.01509e24 0.180433
\(161\) −4.28168e25 −1.79109
\(162\) −2.76313e24 −0.107641
\(163\) 1.70611e25 0.619225 0.309613 0.950863i \(-0.399801\pi\)
0.309613 + 0.950863i \(0.399801\pi\)
\(164\) 5.58031e23 0.0188779
\(165\) 1.24371e25 0.392334
\(166\) 1.35167e25 0.397767
\(167\) −4.07224e25 −1.11839 −0.559195 0.829036i \(-0.688890\pi\)
−0.559195 + 0.829036i \(0.688890\pi\)
\(168\) 3.56932e25 0.915228
\(169\) −3.59822e25 −0.861768
\(170\) −3.25197e25 −0.727752
\(171\) −2.40215e25 −0.502510
\(172\) −4.85647e24 −0.0950043
\(173\) 8.85089e25 1.61978 0.809891 0.586580i \(-0.199526\pi\)
0.809891 + 0.586580i \(0.199526\pi\)
\(174\) 1.64226e25 0.281271
\(175\) −1.11450e26 −1.78708
\(176\) −2.87837e25 −0.432266
\(177\) 4.70457e25 0.661953
\(178\) 1.06873e26 1.40942
\(179\) −5.75965e25 −0.712174 −0.356087 0.934453i \(-0.615889\pi\)
−0.356087 + 0.934453i \(0.615889\pi\)
\(180\) 2.59673e24 0.0301157
\(181\) 5.86096e25 0.637771 0.318886 0.947793i \(-0.396691\pi\)
0.318886 + 0.947793i \(0.396691\pi\)
\(182\) 5.43627e25 0.555240
\(183\) −9.21827e25 −0.884020
\(184\) 1.32648e26 1.19480
\(185\) 1.91669e26 1.62210
\(186\) −9.52870e25 −0.757940
\(187\) −3.16118e25 −0.236412
\(188\) −7.53707e24 −0.0530136
\(189\) 4.48316e25 0.296670
\(190\) −3.44594e26 −2.14606
\(191\) −4.89385e25 −0.286924 −0.143462 0.989656i \(-0.545823\pi\)
−0.143462 + 0.989656i \(0.545823\pi\)
\(192\) −1.10184e26 −0.608349
\(193\) −4.30761e25 −0.224041 −0.112021 0.993706i \(-0.535732\pi\)
−0.112021 + 0.993706i \(0.535732\pi\)
\(194\) −2.69449e26 −1.32055
\(195\) 6.82803e25 0.315426
\(196\) −1.94283e25 −0.0846233
\(197\) 2.77750e26 1.14102 0.570509 0.821291i \(-0.306746\pi\)
0.570509 + 0.821291i \(0.306746\pi\)
\(198\) −3.85311e25 −0.149335
\(199\) 2.40263e26 0.878774 0.439387 0.898298i \(-0.355196\pi\)
0.439387 + 0.898298i \(0.355196\pi\)
\(200\) 3.45276e26 1.19213
\(201\) 5.92895e25 0.193296
\(202\) −3.99026e25 −0.122874
\(203\) −2.66456e26 −0.775212
\(204\) −6.60017e24 −0.0181471
\(205\) 1.73587e26 0.451178
\(206\) 2.39326e26 0.588188
\(207\) 1.66609e26 0.387293
\(208\) −1.58023e26 −0.347531
\(209\) −3.34973e26 −0.697153
\(210\) 6.43120e26 1.26698
\(211\) −3.16141e26 −0.589703 −0.294851 0.955543i \(-0.595270\pi\)
−0.294851 + 0.955543i \(0.595270\pi\)
\(212\) −1.78214e25 −0.0314832
\(213\) −3.53219e26 −0.591124
\(214\) −1.86124e25 −0.0295151
\(215\) −1.51070e27 −2.27058
\(216\) −1.38890e26 −0.197903
\(217\) 1.54603e27 2.08896
\(218\) 1.74755e26 0.223965
\(219\) −3.89506e25 −0.0473595
\(220\) 3.62107e25 0.0417808
\(221\) −1.73550e26 −0.190069
\(222\) −5.93803e26 −0.617423
\(223\) −4.77095e26 −0.471084 −0.235542 0.971864i \(-0.575687\pi\)
−0.235542 + 0.971864i \(0.575687\pi\)
\(224\) 2.01822e26 0.189285
\(225\) 4.33676e26 0.386426
\(226\) −6.07540e26 −0.514432
\(227\) −4.56159e26 −0.367129 −0.183565 0.983008i \(-0.558764\pi\)
−0.183565 + 0.983008i \(0.558764\pi\)
\(228\) −6.99385e25 −0.0535138
\(229\) 1.36743e26 0.0994942 0.0497471 0.998762i \(-0.484158\pi\)
0.0497471 + 0.998762i \(0.484158\pi\)
\(230\) 2.39005e27 1.65400
\(231\) 6.25165e26 0.411582
\(232\) 8.25488e26 0.517129
\(233\) 1.01159e27 0.603130 0.301565 0.953446i \(-0.402491\pi\)
0.301565 + 0.953446i \(0.402491\pi\)
\(234\) −2.11537e26 −0.120061
\(235\) −2.34456e27 −1.26701
\(236\) 1.36973e26 0.0704934
\(237\) 6.28317e25 0.0308015
\(238\) −1.63463e27 −0.763457
\(239\) 2.91639e27 1.29799 0.648993 0.760794i \(-0.275190\pi\)
0.648993 + 0.760794i \(0.275190\pi\)
\(240\) −1.86944e27 −0.793017
\(241\) 1.34542e27 0.544079 0.272039 0.962286i \(-0.412302\pi\)
0.272039 + 0.962286i \(0.412302\pi\)
\(242\) 1.97515e27 0.761592
\(243\) −1.74449e26 −0.0641500
\(244\) −2.68390e26 −0.0941419
\(245\) −6.04358e27 −2.02248
\(246\) −5.37783e26 −0.171733
\(247\) −1.83901e27 −0.560493
\(248\) −4.78964e27 −1.39351
\(249\) 8.53374e26 0.237054
\(250\) 8.54753e26 0.226741
\(251\) 3.42333e27 0.867363 0.433681 0.901066i \(-0.357214\pi\)
0.433681 + 0.901066i \(0.357214\pi\)
\(252\) 1.30527e26 0.0315933
\(253\) 2.32332e27 0.537307
\(254\) 1.08935e27 0.240757
\(255\) −2.05312e27 −0.433713
\(256\) −9.09835e26 −0.183740
\(257\) 3.47997e27 0.671962 0.335981 0.941869i \(-0.390932\pi\)
0.335981 + 0.941869i \(0.390932\pi\)
\(258\) 4.68026e27 0.864258
\(259\) 9.63442e27 1.70168
\(260\) 1.98798e26 0.0335907
\(261\) 1.03683e27 0.167627
\(262\) 6.44187e27 0.996660
\(263\) −7.94565e27 −1.17663 −0.588313 0.808634i \(-0.700208\pi\)
−0.588313 + 0.808634i \(0.700208\pi\)
\(264\) −1.93678e27 −0.274559
\(265\) −5.54371e27 −0.752442
\(266\) −1.73213e28 −2.25135
\(267\) 6.74742e27 0.839959
\(268\) 1.72621e26 0.0205847
\(269\) 8.25439e27 0.943048 0.471524 0.881853i \(-0.343704\pi\)
0.471524 + 0.881853i \(0.343704\pi\)
\(270\) −2.50251e27 −0.273964
\(271\) −1.26880e28 −1.33120 −0.665602 0.746307i \(-0.731825\pi\)
−0.665602 + 0.746307i \(0.731825\pi\)
\(272\) 4.75160e27 0.477856
\(273\) 3.43217e27 0.330902
\(274\) −1.61172e28 −1.48991
\(275\) 6.04749e27 0.536104
\(276\) 4.85082e26 0.0412440
\(277\) −9.00378e27 −0.734358 −0.367179 0.930150i \(-0.619676\pi\)
−0.367179 + 0.930150i \(0.619676\pi\)
\(278\) 9.46209e27 0.740410
\(279\) −6.01592e27 −0.451703
\(280\) 3.23267e28 2.32940
\(281\) −1.44072e28 −0.996454 −0.498227 0.867047i \(-0.666015\pi\)
−0.498227 + 0.867047i \(0.666015\pi\)
\(282\) 7.26360e27 0.482267
\(283\) 1.15566e28 0.736692 0.368346 0.929689i \(-0.379924\pi\)
0.368346 + 0.929689i \(0.379924\pi\)
\(284\) −1.02840e27 −0.0629505
\(285\) −2.17558e28 −1.27897
\(286\) −2.94982e27 −0.166566
\(287\) 8.72551e27 0.473313
\(288\) −7.85332e26 −0.0409298
\(289\) −1.47491e28 −0.738653
\(290\) 1.48736e28 0.715880
\(291\) −1.70116e28 −0.787000
\(292\) −1.13405e26 −0.00504345
\(293\) 4.02839e28 1.72248 0.861238 0.508201i \(-0.169689\pi\)
0.861238 + 0.508201i \(0.169689\pi\)
\(294\) 1.87234e28 0.769822
\(295\) 4.26084e28 1.68478
\(296\) −2.98478e28 −1.13516
\(297\) −2.43265e27 −0.0889979
\(298\) −3.94389e28 −1.38815
\(299\) 1.27551e28 0.431981
\(300\) 1.26265e27 0.0411517
\(301\) −7.59369e28 −2.38198
\(302\) 1.41928e28 0.428536
\(303\) −2.51924e27 −0.0732283
\(304\) 5.03502e28 1.40914
\(305\) −8.34882e28 −2.24997
\(306\) 6.36069e27 0.165085
\(307\) 3.90618e28 0.976475 0.488237 0.872711i \(-0.337640\pi\)
0.488237 + 0.872711i \(0.337640\pi\)
\(308\) 1.82017e27 0.0438306
\(309\) 1.51097e28 0.350537
\(310\) −8.62997e28 −1.92908
\(311\) 9.65010e27 0.207868 0.103934 0.994584i \(-0.466857\pi\)
0.103934 + 0.994584i \(0.466857\pi\)
\(312\) −1.06330e28 −0.220738
\(313\) 5.80452e28 1.16147 0.580733 0.814094i \(-0.302766\pi\)
0.580733 + 0.814094i \(0.302766\pi\)
\(314\) −2.70877e28 −0.522494
\(315\) 4.06032e28 0.755072
\(316\) 1.82934e26 0.00328015
\(317\) 8.81588e28 1.52435 0.762174 0.647373i \(-0.224132\pi\)
0.762174 + 0.647373i \(0.224132\pi\)
\(318\) 1.71748e28 0.286404
\(319\) 1.44584e28 0.232556
\(320\) −9.97912e28 −1.54834
\(321\) −1.17509e27 −0.0175898
\(322\) 1.20138e29 1.73515
\(323\) 5.52972e28 0.770681
\(324\) −5.07908e26 −0.00683153
\(325\) 3.32009e28 0.431014
\(326\) −4.78709e28 −0.599887
\(327\) 1.10331e28 0.133474
\(328\) −2.70319e28 −0.315739
\(329\) −1.17852e29 −1.32918
\(330\) −3.48969e28 −0.380081
\(331\) −1.57171e29 −1.65329 −0.826646 0.562723i \(-0.809754\pi\)
−0.826646 + 0.562723i \(0.809754\pi\)
\(332\) 2.48460e27 0.0252446
\(333\) −3.74896e28 −0.367961
\(334\) 1.14261e29 1.08346
\(335\) 5.36974e28 0.491969
\(336\) −9.39692e28 −0.831924
\(337\) −7.23440e28 −0.618954 −0.309477 0.950907i \(-0.600154\pi\)
−0.309477 + 0.950907i \(0.600154\pi\)
\(338\) 1.00961e29 0.834856
\(339\) −3.83568e28 −0.306582
\(340\) −5.97765e27 −0.0461873
\(341\) −8.38903e28 −0.626667
\(342\) 6.74009e28 0.486817
\(343\) −8.30679e28 −0.580164
\(344\) 2.35255e29 1.58898
\(345\) 1.50895e29 0.985722
\(346\) −2.48343e29 −1.56920
\(347\) 2.40365e28 0.146921 0.0734603 0.997298i \(-0.476596\pi\)
0.0734603 + 0.997298i \(0.476596\pi\)
\(348\) 3.01874e27 0.0178511
\(349\) 2.63663e29 1.50854 0.754269 0.656566i \(-0.227992\pi\)
0.754269 + 0.656566i \(0.227992\pi\)
\(350\) 3.12713e29 1.73127
\(351\) −1.33553e28 −0.0715521
\(352\) −1.09512e28 −0.0567836
\(353\) −5.21932e28 −0.261942 −0.130971 0.991386i \(-0.541809\pi\)
−0.130971 + 0.991386i \(0.541809\pi\)
\(354\) −1.32004e29 −0.641281
\(355\) −3.19904e29 −1.50450
\(356\) 1.96451e28 0.0894498
\(357\) −1.03202e29 −0.454991
\(358\) 1.61608e29 0.689933
\(359\) −1.93870e29 −0.801538 −0.400769 0.916179i \(-0.631257\pi\)
−0.400769 + 0.916179i \(0.631257\pi\)
\(360\) −1.25790e29 −0.503695
\(361\) 3.28126e29 1.27265
\(362\) −1.64450e29 −0.617854
\(363\) 1.24700e29 0.453880
\(364\) 9.99277e27 0.0352387
\(365\) −3.52768e28 −0.120537
\(366\) 2.58652e29 0.856412
\(367\) 5.00067e29 1.60460 0.802302 0.596918i \(-0.203608\pi\)
0.802302 + 0.596918i \(0.203608\pi\)
\(368\) −3.49221e29 −1.08605
\(369\) −3.39528e28 −0.102346
\(370\) −5.37796e29 −1.57144
\(371\) −2.78660e29 −0.789359
\(372\) −1.75153e28 −0.0481032
\(373\) −5.08622e29 −1.35439 −0.677195 0.735804i \(-0.736805\pi\)
−0.677195 + 0.735804i \(0.736805\pi\)
\(374\) 8.86981e28 0.229029
\(375\) 5.39646e28 0.135129
\(376\) 3.65108e29 0.886668
\(377\) 7.93770e28 0.186969
\(378\) −1.25791e29 −0.287405
\(379\) 3.25520e29 0.721483 0.360742 0.932666i \(-0.382524\pi\)
0.360742 + 0.932666i \(0.382524\pi\)
\(380\) −6.33420e28 −0.136201
\(381\) 6.87757e28 0.143482
\(382\) 1.37314e29 0.277963
\(383\) −3.64395e29 −0.715792 −0.357896 0.933761i \(-0.616506\pi\)
−0.357896 + 0.933761i \(0.616506\pi\)
\(384\) 2.71971e29 0.518458
\(385\) 5.66200e29 1.04754
\(386\) 1.20866e29 0.217044
\(387\) 2.95487e29 0.515065
\(388\) −4.95291e28 −0.0838100
\(389\) 3.87947e29 0.637312 0.318656 0.947870i \(-0.396768\pi\)
0.318656 + 0.947870i \(0.396768\pi\)
\(390\) −1.91585e29 −0.305576
\(391\) −3.83532e29 −0.593977
\(392\) 9.41139e29 1.41535
\(393\) 4.06705e29 0.593971
\(394\) −7.79327e29 −1.10538
\(395\) 5.69055e28 0.0783948
\(396\) −7.08265e27 −0.00947766
\(397\) −1.90170e29 −0.247202 −0.123601 0.992332i \(-0.539444\pi\)
−0.123601 + 0.992332i \(0.539444\pi\)
\(398\) −6.74145e29 −0.851330
\(399\) −1.09358e30 −1.34172
\(400\) −9.09005e29 −1.08362
\(401\) 3.60649e29 0.417757 0.208879 0.977942i \(-0.433019\pi\)
0.208879 + 0.977942i \(0.433019\pi\)
\(402\) −1.66358e29 −0.187259
\(403\) −4.60561e29 −0.503824
\(404\) −7.33476e27 −0.00779830
\(405\) −1.57995e29 −0.163272
\(406\) 7.47637e29 0.751002
\(407\) −5.22782e29 −0.510487
\(408\) 3.19723e29 0.303516
\(409\) 1.06335e30 0.981432 0.490716 0.871320i \(-0.336735\pi\)
0.490716 + 0.871320i \(0.336735\pi\)
\(410\) −4.87060e29 −0.437088
\(411\) −1.01756e30 −0.887928
\(412\) 4.39920e28 0.0373298
\(413\) 2.14175e30 1.76744
\(414\) −4.67482e29 −0.375198
\(415\) 7.72885e29 0.603339
\(416\) −6.01227e28 −0.0456526
\(417\) 5.97386e29 0.441256
\(418\) 9.39888e29 0.675381
\(419\) 1.01398e29 0.0708874 0.0354437 0.999372i \(-0.488716\pi\)
0.0354437 + 0.999372i \(0.488716\pi\)
\(420\) 1.18216e29 0.0804099
\(421\) −1.31228e30 −0.868523 −0.434262 0.900787i \(-0.642991\pi\)
−0.434262 + 0.900787i \(0.642991\pi\)
\(422\) 8.87048e29 0.571287
\(423\) 4.58585e29 0.287412
\(424\) 8.63297e29 0.526566
\(425\) −9.98317e29 −0.592647
\(426\) 9.91082e29 0.572663
\(427\) −4.19661e30 −2.36036
\(428\) −3.42126e27 −0.00187320
\(429\) −1.86236e29 −0.0992671
\(430\) 4.23882e30 2.19967
\(431\) 2.07677e30 1.04930 0.524649 0.851319i \(-0.324197\pi\)
0.524649 + 0.851319i \(0.324197\pi\)
\(432\) 3.65654e29 0.179890
\(433\) −3.77508e30 −1.80849 −0.904243 0.427019i \(-0.859564\pi\)
−0.904243 + 0.427019i \(0.859564\pi\)
\(434\) −4.33793e30 −2.02372
\(435\) 9.39042e29 0.426637
\(436\) 3.21228e28 0.0142141
\(437\) −4.06409e30 −1.75157
\(438\) 1.09290e29 0.0458805
\(439\) 1.39779e30 0.571609 0.285805 0.958288i \(-0.407739\pi\)
0.285805 + 0.958288i \(0.407739\pi\)
\(440\) −1.75411e30 −0.698796
\(441\) 1.18209e30 0.458784
\(442\) 4.86956e29 0.184134
\(443\) 4.24438e30 1.56376 0.781882 0.623426i \(-0.214260\pi\)
0.781882 + 0.623426i \(0.214260\pi\)
\(444\) −1.09151e29 −0.0391852
\(445\) 6.11101e30 2.13783
\(446\) 1.33866e30 0.456373
\(447\) −2.48996e30 −0.827287
\(448\) −5.01610e30 −1.62431
\(449\) −3.75377e30 −1.18477 −0.592386 0.805654i \(-0.701814\pi\)
−0.592386 + 0.805654i \(0.701814\pi\)
\(450\) −1.21683e30 −0.374358
\(451\) −4.73462e29 −0.141989
\(452\) −1.11676e29 −0.0326488
\(453\) 8.96055e29 0.255391
\(454\) 1.27992e30 0.355664
\(455\) 3.10846e30 0.842197
\(456\) 3.38794e30 0.895034
\(457\) −4.50794e29 −0.116129 −0.0580646 0.998313i \(-0.518493\pi\)
−0.0580646 + 0.998313i \(0.518493\pi\)
\(458\) −3.83682e29 −0.0963870
\(459\) 4.01580e29 0.0983845
\(460\) 4.39330e29 0.104972
\(461\) 3.64054e30 0.848410 0.424205 0.905566i \(-0.360554\pi\)
0.424205 + 0.905566i \(0.360554\pi\)
\(462\) −1.75412e30 −0.398729
\(463\) 7.58411e30 1.68160 0.840801 0.541345i \(-0.182085\pi\)
0.840801 + 0.541345i \(0.182085\pi\)
\(464\) −2.17325e30 −0.470061
\(465\) −5.44850e30 −1.14966
\(466\) −2.83837e30 −0.584294
\(467\) −4.97036e29 −0.0998259 −0.0499130 0.998754i \(-0.515894\pi\)
−0.0499130 + 0.998754i \(0.515894\pi\)
\(468\) −3.88840e28 −0.00761979
\(469\) 2.69915e30 0.516106
\(470\) 6.57851e30 1.22744
\(471\) −1.71017e30 −0.311386
\(472\) −6.63522e30 −1.17902
\(473\) 4.12048e30 0.714570
\(474\) −1.76297e29 −0.0298396
\(475\) −1.05786e31 −1.74765
\(476\) −3.00472e29 −0.0484534
\(477\) 1.08432e30 0.170686
\(478\) −8.18299e30 −1.25745
\(479\) 6.87924e29 0.103201 0.0516003 0.998668i \(-0.483568\pi\)
0.0516003 + 0.998668i \(0.483568\pi\)
\(480\) −7.11261e29 −0.104173
\(481\) −2.87009e30 −0.410418
\(482\) −3.77506e30 −0.527087
\(483\) 7.58486e30 1.03408
\(484\) 3.63064e29 0.0483350
\(485\) −1.54071e31 −2.00304
\(486\) 4.89480e29 0.0621467
\(487\) 1.42477e30 0.176670 0.0883349 0.996091i \(-0.471845\pi\)
0.0883349 + 0.996091i \(0.471845\pi\)
\(488\) 1.30012e31 1.57455
\(489\) −3.02232e30 −0.357510
\(490\) 1.69574e31 1.95932
\(491\) 1.42373e31 1.60690 0.803452 0.595369i \(-0.202994\pi\)
0.803452 + 0.595369i \(0.202994\pi\)
\(492\) −9.88534e28 −0.0108992
\(493\) −2.38678e30 −0.257083
\(494\) 5.16001e30 0.542989
\(495\) −2.20320e30 −0.226514
\(496\) 1.26096e31 1.26667
\(497\) −1.60803e31 −1.57832
\(498\) −2.39445e30 −0.229651
\(499\) 1.65690e31 1.55289 0.776445 0.630185i \(-0.217021\pi\)
0.776445 + 0.630185i \(0.217021\pi\)
\(500\) 1.57118e29 0.0143903
\(501\) 7.21385e30 0.645703
\(502\) −9.60538e30 −0.840276
\(503\) −2.06539e31 −1.76592 −0.882960 0.469448i \(-0.844453\pi\)
−0.882960 + 0.469448i \(0.844453\pi\)
\(504\) −6.32295e30 −0.528407
\(505\) −2.28163e30 −0.186378
\(506\) −6.51890e30 −0.520527
\(507\) 6.37414e30 0.497542
\(508\) 2.00240e29 0.0152798
\(509\) −2.07213e31 −1.54583 −0.772916 0.634508i \(-0.781203\pi\)
−0.772916 + 0.634508i \(0.781203\pi\)
\(510\) 5.76076e30 0.420168
\(511\) −1.77322e30 −0.126451
\(512\) 1.54318e31 1.07600
\(513\) 4.25534e30 0.290124
\(514\) −9.76431e30 −0.650977
\(515\) 1.36846e31 0.892174
\(516\) 8.60308e29 0.0548508
\(517\) 6.39484e30 0.398739
\(518\) −2.70328e31 −1.64854
\(519\) −1.56791e31 −0.935182
\(520\) −9.63010e30 −0.561814
\(521\) 2.06129e31 1.17626 0.588131 0.808765i \(-0.299864\pi\)
0.588131 + 0.808765i \(0.299864\pi\)
\(522\) −2.90921e30 −0.162392
\(523\) −2.36613e31 −1.29202 −0.646009 0.763329i \(-0.723563\pi\)
−0.646009 + 0.763329i \(0.723563\pi\)
\(524\) 1.18412e30 0.0632538
\(525\) 1.97431e31 1.03177
\(526\) 2.22944e31 1.13988
\(527\) 1.38486e31 0.692760
\(528\) 5.09894e30 0.249569
\(529\) 7.30739e30 0.349963
\(530\) 1.55549e31 0.728944
\(531\) −8.33401e30 −0.382179
\(532\) −3.18395e30 −0.142883
\(533\) −2.59932e30 −0.114156
\(534\) −1.89323e31 −0.813728
\(535\) −1.06425e30 −0.0447690
\(536\) −8.36205e30 −0.344285
\(537\) 1.02030e31 0.411174
\(538\) −2.31607e31 −0.913597
\(539\) 1.64840e31 0.636490
\(540\) −4.60003e29 −0.0173873
\(541\) 1.35546e31 0.501553 0.250776 0.968045i \(-0.419314\pi\)
0.250776 + 0.968045i \(0.419314\pi\)
\(542\) 3.56006e31 1.28963
\(543\) −1.03825e31 −0.368218
\(544\) 1.80783e30 0.0627725
\(545\) 9.99246e30 0.339713
\(546\) −9.63020e30 −0.320568
\(547\) −5.86214e30 −0.191074 −0.0955371 0.995426i \(-0.530457\pi\)
−0.0955371 + 0.995426i \(0.530457\pi\)
\(548\) −2.96262e30 −0.0945581
\(549\) 1.63299e31 0.510389
\(550\) −1.69684e31 −0.519362
\(551\) −2.52915e31 −0.758108
\(552\) −2.34982e31 −0.689818
\(553\) 2.86041e30 0.0822410
\(554\) 2.52633e31 0.711424
\(555\) −3.39536e31 −0.936519
\(556\) 1.73929e30 0.0469907
\(557\) −3.63450e31 −0.961857 −0.480928 0.876760i \(-0.659700\pi\)
−0.480928 + 0.876760i \(0.659700\pi\)
\(558\) 1.68798e31 0.437597
\(559\) 2.26216e31 0.574496
\(560\) −8.51062e31 −2.11738
\(561\) 5.59993e30 0.136493
\(562\) 4.04246e31 0.965336
\(563\) 5.87315e31 1.37412 0.687060 0.726601i \(-0.258901\pi\)
0.687060 + 0.726601i \(0.258901\pi\)
\(564\) 1.33517e30 0.0306074
\(565\) −3.47391e31 −0.780299
\(566\) −3.24262e31 −0.713686
\(567\) −7.94179e30 −0.171282
\(568\) 4.98172e31 1.05287
\(569\) −3.38209e31 −0.700477 −0.350238 0.936661i \(-0.613899\pi\)
−0.350238 + 0.936661i \(0.613899\pi\)
\(570\) 6.10438e31 1.23903
\(571\) −7.49909e31 −1.49174 −0.745871 0.666091i \(-0.767966\pi\)
−0.745871 + 0.666091i \(0.767966\pi\)
\(572\) −5.42226e29 −0.0105712
\(573\) 8.66930e30 0.165656
\(574\) −2.44825e31 −0.458532
\(575\) 7.33717e31 1.34694
\(576\) 1.95187e31 0.351230
\(577\) 6.33342e31 1.11716 0.558581 0.829450i \(-0.311346\pi\)
0.558581 + 0.829450i \(0.311346\pi\)
\(578\) 4.13839e31 0.715586
\(579\) 7.63081e30 0.129350
\(580\) 2.73402e30 0.0454338
\(581\) 3.88498e31 0.632940
\(582\) 4.77321e31 0.762423
\(583\) 1.51206e31 0.236799
\(584\) 5.49350e30 0.0843532
\(585\) −1.20957e31 −0.182111
\(586\) −1.13031e32 −1.66868
\(587\) −8.61217e31 −1.24674 −0.623368 0.781929i \(-0.714236\pi\)
−0.623368 + 0.781929i \(0.714236\pi\)
\(588\) 3.44167e30 0.0488573
\(589\) 1.46746e32 2.04287
\(590\) −1.19553e32 −1.63216
\(591\) −4.92026e31 −0.658767
\(592\) 7.85799e31 1.03184
\(593\) −1.31209e32 −1.68980 −0.844901 0.534923i \(-0.820341\pi\)
−0.844901 + 0.534923i \(0.820341\pi\)
\(594\) 6.82567e30 0.0862186
\(595\) −9.34681e31 −1.15803
\(596\) −7.24952e30 −0.0881002
\(597\) −4.25619e31 −0.507361
\(598\) −3.57890e31 −0.418491
\(599\) −2.38949e30 −0.0274092 −0.0137046 0.999906i \(-0.504362\pi\)
−0.0137046 + 0.999906i \(0.504362\pi\)
\(600\) −6.11647e31 −0.688274
\(601\) −4.19607e31 −0.463219 −0.231610 0.972809i \(-0.574399\pi\)
−0.231610 + 0.972809i \(0.574399\pi\)
\(602\) 2.13068e32 2.30759
\(603\) −1.05030e31 −0.111600
\(604\) 2.60886e30 0.0271973
\(605\) 1.12939e32 1.15520
\(606\) 7.06863e30 0.0709414
\(607\) −1.86893e32 −1.84045 −0.920225 0.391390i \(-0.871994\pi\)
−0.920225 + 0.391390i \(0.871994\pi\)
\(608\) 1.91566e31 0.185109
\(609\) 4.72018e31 0.447569
\(610\) 2.34256e32 2.17971
\(611\) 3.51079e31 0.320576
\(612\) 1.16920e30 0.0104773
\(613\) 1.22268e32 1.07527 0.537637 0.843176i \(-0.319317\pi\)
0.537637 + 0.843176i \(0.319317\pi\)
\(614\) −1.09602e32 −0.945980
\(615\) −3.07504e31 −0.260488
\(616\) −8.81718e31 −0.733080
\(617\) 4.63661e31 0.378374 0.189187 0.981941i \(-0.439415\pi\)
0.189187 + 0.981941i \(0.439415\pi\)
\(618\) −4.23958e31 −0.339590
\(619\) 2.85976e31 0.224847 0.112424 0.993660i \(-0.464139\pi\)
0.112424 + 0.993660i \(0.464139\pi\)
\(620\) −1.58633e31 −0.122430
\(621\) −2.95143e31 −0.223604
\(622\) −2.70768e31 −0.201376
\(623\) 3.07176e32 2.24272
\(624\) 2.79934e31 0.200647
\(625\) −1.15869e32 −0.815353
\(626\) −1.62867e32 −1.12519
\(627\) 5.93395e31 0.402501
\(628\) −4.97916e30 −0.0331605
\(629\) 8.63006e31 0.564328
\(630\) −1.13927e32 −0.731492
\(631\) −8.23803e31 −0.519381 −0.259690 0.965692i \(-0.583620\pi\)
−0.259690 + 0.965692i \(0.583620\pi\)
\(632\) −8.86163e30 −0.0548615
\(633\) 5.60035e31 0.340465
\(634\) −2.47361e32 −1.47674
\(635\) 6.22889e31 0.365185
\(636\) 3.15701e30 0.0181768
\(637\) 9.04977e31 0.511722
\(638\) −4.05682e31 −0.225293
\(639\) 6.25716e31 0.341285
\(640\) 2.46319e32 1.31956
\(641\) −2.94822e32 −1.55129 −0.775645 0.631169i \(-0.782575\pi\)
−0.775645 + 0.631169i \(0.782575\pi\)
\(642\) 3.29713e30 0.0170405
\(643\) 3.36821e31 0.170991 0.0854953 0.996339i \(-0.472753\pi\)
0.0854953 + 0.996339i \(0.472753\pi\)
\(644\) 2.20833e31 0.110123
\(645\) 2.67617e32 1.31092
\(646\) −1.55156e32 −0.746613
\(647\) −1.54155e32 −0.728715 −0.364357 0.931259i \(-0.618711\pi\)
−0.364357 + 0.931259i \(0.618711\pi\)
\(648\) 2.46039e31 0.114259
\(649\) −1.16215e32 −0.530212
\(650\) −9.31571e31 −0.417554
\(651\) −2.73874e32 −1.20606
\(652\) −8.79946e30 −0.0380723
\(653\) −4.51453e32 −1.91916 −0.959580 0.281434i \(-0.909190\pi\)
−0.959580 + 0.281434i \(0.909190\pi\)
\(654\) −3.09573e31 −0.129306
\(655\) 3.68345e32 1.51175
\(656\) 7.11666e31 0.287000
\(657\) 6.89998e30 0.0273430
\(658\) 3.30675e32 1.28767
\(659\) 5.01392e32 1.91865 0.959323 0.282311i \(-0.0911010\pi\)
0.959323 + 0.282311i \(0.0911010\pi\)
\(660\) −6.41462e30 −0.0241221
\(661\) 2.97857e32 1.10076 0.550378 0.834915i \(-0.314483\pi\)
0.550378 + 0.834915i \(0.314483\pi\)
\(662\) 4.40999e32 1.60166
\(663\) 3.07438e31 0.109737
\(664\) −1.20358e32 −0.422223
\(665\) −9.90432e32 −3.41488
\(666\) 1.05190e32 0.356470
\(667\) 1.75417e32 0.584286
\(668\) 2.10031e31 0.0687629
\(669\) 8.45159e31 0.271981
\(670\) −1.50667e32 −0.476605
\(671\) 2.27716e32 0.708084
\(672\) −3.57522e31 −0.109284
\(673\) −1.11603e31 −0.0335354 −0.0167677 0.999859i \(-0.505338\pi\)
−0.0167677 + 0.999859i \(0.505338\pi\)
\(674\) 2.02987e32 0.599625
\(675\) −7.68244e31 −0.223103
\(676\) 1.85583e31 0.0529847
\(677\) 1.78822e32 0.501939 0.250970 0.967995i \(-0.419251\pi\)
0.250970 + 0.967995i \(0.419251\pi\)
\(678\) 1.07624e32 0.297007
\(679\) −7.74450e32 −2.10131
\(680\) 2.89567e32 0.772497
\(681\) 8.08072e31 0.211962
\(682\) 2.35384e32 0.607096
\(683\) −4.63491e32 −1.17545 −0.587724 0.809062i \(-0.699976\pi\)
−0.587724 + 0.809062i \(0.699976\pi\)
\(684\) 1.23894e31 0.0308962
\(685\) −9.21583e32 −2.25992
\(686\) 2.33077e32 0.562046
\(687\) −2.42236e31 −0.0574430
\(688\) −6.19354e32 −1.44435
\(689\) 8.30126e31 0.190381
\(690\) −4.23389e32 −0.954939
\(691\) 7.28816e32 1.61667 0.808333 0.588726i \(-0.200370\pi\)
0.808333 + 0.588726i \(0.200370\pi\)
\(692\) −4.56496e31 −0.0995903
\(693\) −1.10746e32 −0.237627
\(694\) −6.74431e31 −0.142332
\(695\) 5.41042e32 1.12307
\(696\) −1.46233e32 −0.298565
\(697\) 7.81589e31 0.156965
\(698\) −7.39800e32 −1.46143
\(699\) −1.79200e32 −0.348217
\(700\) 5.74819e31 0.109876
\(701\) 8.88871e32 1.67140 0.835702 0.549183i \(-0.185061\pi\)
0.835702 + 0.549183i \(0.185061\pi\)
\(702\) 3.74731e31 0.0693175
\(703\) 9.14482e32 1.66414
\(704\) 2.72183e32 0.487276
\(705\) 4.15332e32 0.731510
\(706\) 1.46447e32 0.253762
\(707\) −1.14688e32 −0.195522
\(708\) −2.42644e31 −0.0406994
\(709\) 2.94549e32 0.486101 0.243050 0.970014i \(-0.421852\pi\)
0.243050 + 0.970014i \(0.421852\pi\)
\(710\) 8.97605e32 1.45752
\(711\) −1.11304e31 −0.0177833
\(712\) −9.51641e32 −1.49607
\(713\) −1.01781e33 −1.57447
\(714\) 2.89570e32 0.440782
\(715\) −1.68671e32 −0.252650
\(716\) 2.97061e31 0.0437871
\(717\) −5.16630e32 −0.749393
\(718\) 5.43972e32 0.776507
\(719\) −1.42330e32 −0.199947 −0.0999734 0.994990i \(-0.531876\pi\)
−0.0999734 + 0.994990i \(0.531876\pi\)
\(720\) 3.31166e32 0.457849
\(721\) 6.87870e32 0.935945
\(722\) −9.20676e32 −1.23290
\(723\) −2.38337e32 −0.314124
\(724\) −3.02287e31 −0.0392126
\(725\) 4.56604e32 0.582978
\(726\) −3.49891e32 −0.439705
\(727\) −1.10613e33 −1.36824 −0.684118 0.729371i \(-0.739813\pi\)
−0.684118 + 0.729371i \(0.739813\pi\)
\(728\) −4.84066e32 −0.589378
\(729\) 3.09032e31 0.0370370
\(730\) 9.89819e31 0.116773
\(731\) −6.80207e32 −0.789935
\(732\) 4.75445e31 0.0543529
\(733\) 1.11000e33 1.24918 0.624592 0.780951i \(-0.285265\pi\)
0.624592 + 0.780951i \(0.285265\pi\)
\(734\) −1.40312e33 −1.55449
\(735\) 1.07060e33 1.16768
\(736\) −1.32867e32 −0.142667
\(737\) −1.46461e32 −0.154827
\(738\) 9.52667e31 0.0991501
\(739\) 7.82934e31 0.0802259 0.0401129 0.999195i \(-0.487228\pi\)
0.0401129 + 0.999195i \(0.487228\pi\)
\(740\) −9.88558e31 −0.0997327
\(741\) 3.25776e32 0.323601
\(742\) 7.81880e32 0.764707
\(743\) 1.22041e33 1.17526 0.587631 0.809129i \(-0.300061\pi\)
0.587631 + 0.809129i \(0.300061\pi\)
\(744\) 8.48471e32 0.804541
\(745\) −2.25511e33 −2.10558
\(746\) 1.42712e33 1.31209
\(747\) −1.51173e32 −0.136863
\(748\) 1.63042e31 0.0145355
\(749\) −5.34957e31 −0.0469654
\(750\) −1.51417e32 −0.130909
\(751\) −1.43927e33 −1.22542 −0.612708 0.790309i \(-0.709920\pi\)
−0.612708 + 0.790309i \(0.709920\pi\)
\(752\) −9.61216e32 −0.805964
\(753\) −6.06432e32 −0.500772
\(754\) −2.22721e32 −0.181130
\(755\) 8.11541e32 0.650011
\(756\) −2.31225e31 −0.0182404
\(757\) 7.52652e32 0.584778 0.292389 0.956299i \(-0.405550\pi\)
0.292389 + 0.956299i \(0.405550\pi\)
\(758\) −9.13362e32 −0.698951
\(759\) −4.11569e32 −0.310215
\(760\) 3.06839e33 2.27800
\(761\) 2.01529e33 1.47372 0.736859 0.676046i \(-0.236308\pi\)
0.736859 + 0.676046i \(0.236308\pi\)
\(762\) −1.92975e32 −0.139001
\(763\) 5.02280e32 0.356380
\(764\) 2.52406e31 0.0176412
\(765\) 3.63704e32 0.250404
\(766\) 1.02244e33 0.693438
\(767\) −6.38027e32 −0.426277
\(768\) 1.61175e32 0.106082
\(769\) −2.69734e32 −0.174897 −0.0874487 0.996169i \(-0.527871\pi\)
−0.0874487 + 0.996169i \(0.527871\pi\)
\(770\) −1.58868e33 −1.01483
\(771\) −6.16467e32 −0.387958
\(772\) 2.22171e31 0.0137749
\(773\) −3.48915e31 −0.0213135 −0.0106568 0.999943i \(-0.503392\pi\)
−0.0106568 + 0.999943i \(0.503392\pi\)
\(774\) −8.29094e32 −0.498979
\(775\) −2.64930e33 −1.57095
\(776\) 2.39927e33 1.40175
\(777\) −1.70671e33 −0.982466
\(778\) −1.08853e33 −0.617409
\(779\) 8.28210e32 0.462870
\(780\) −3.52165e31 −0.0193936
\(781\) 8.72545e32 0.473479
\(782\) 1.07614e33 0.575427
\(783\) −1.83672e32 −0.0967794
\(784\) −2.47773e33 −1.28653
\(785\) −1.54887e33 −0.792528
\(786\) −1.14116e33 −0.575422
\(787\) −7.64381e32 −0.379840 −0.189920 0.981800i \(-0.560823\pi\)
−0.189920 + 0.981800i \(0.560823\pi\)
\(788\) −1.43253e32 −0.0701541
\(789\) 1.40755e33 0.679325
\(790\) −1.59669e32 −0.0759466
\(791\) −1.74619e33 −0.818582
\(792\) 3.43095e32 0.158517
\(793\) 1.25017e33 0.569281
\(794\) 5.33591e32 0.239482
\(795\) 9.82052e32 0.434423
\(796\) −1.23919e32 −0.0540303
\(797\) 3.28083e33 1.40998 0.704990 0.709218i \(-0.250952\pi\)
0.704990 + 0.709218i \(0.250952\pi\)
\(798\) 3.06842e33 1.29982
\(799\) −1.05566e33 −0.440794
\(800\) −3.45847e32 −0.142347
\(801\) −1.19529e33 −0.484951
\(802\) −1.01193e33 −0.404711
\(803\) 9.62184e31 0.0379341
\(804\) −3.05793e31 −0.0118846
\(805\) 6.86947e33 2.63191
\(806\) 1.29227e33 0.488090
\(807\) −1.46224e33 −0.544469
\(808\) 3.55308e32 0.130429
\(809\) −1.06623e33 −0.385870 −0.192935 0.981212i \(-0.561801\pi\)
−0.192935 + 0.981212i \(0.561801\pi\)
\(810\) 4.43313e32 0.158173
\(811\) 6.88669e32 0.242254 0.121127 0.992637i \(-0.461349\pi\)
0.121127 + 0.992637i \(0.461349\pi\)
\(812\) 1.37428e32 0.0476629
\(813\) 2.24763e33 0.768571
\(814\) 1.46685e33 0.494545
\(815\) −2.73726e33 −0.909919
\(816\) −8.41731e32 −0.275890
\(817\) −7.20780e33 −2.32943
\(818\) −2.98362e33 −0.950782
\(819\) −6.07999e32 −0.191046
\(820\) −8.95297e31 −0.0277401
\(821\) 1.55564e33 0.475293 0.237647 0.971352i \(-0.423624\pi\)
0.237647 + 0.971352i \(0.423624\pi\)
\(822\) 2.85512e33 0.860199
\(823\) 5.22482e33 1.55229 0.776146 0.630554i \(-0.217172\pi\)
0.776146 + 0.630554i \(0.217172\pi\)
\(824\) −2.13104e33 −0.624352
\(825\) −1.07130e33 −0.309520
\(826\) −6.00945e33 −1.71224
\(827\) −3.58810e33 −1.00821 −0.504105 0.863642i \(-0.668178\pi\)
−0.504105 + 0.863642i \(0.668178\pi\)
\(828\) −8.59308e31 −0.0238122
\(829\) 8.83624e32 0.241485 0.120743 0.992684i \(-0.461472\pi\)
0.120743 + 0.992684i \(0.461472\pi\)
\(830\) −2.16861e33 −0.584497
\(831\) 1.59499e33 0.423982
\(832\) 1.49429e33 0.391757
\(833\) −2.72117e33 −0.703620
\(834\) −1.67618e33 −0.427476
\(835\) 6.53345e33 1.64342
\(836\) 1.72767e32 0.0428636
\(837\) 1.06570e33 0.260791
\(838\) −2.84509e32 −0.0686736
\(839\) 4.70731e32 0.112075 0.0560377 0.998429i \(-0.482153\pi\)
0.0560377 + 0.998429i \(0.482153\pi\)
\(840\) −5.72658e33 −1.34488
\(841\) −3.22507e33 −0.747111
\(842\) 3.68206e33 0.841400
\(843\) 2.55220e33 0.575303
\(844\) 1.63054e32 0.0362571
\(845\) 5.77294e33 1.26632
\(846\) −1.28672e33 −0.278437
\(847\) 5.67696e33 1.21187
\(848\) −2.27279e33 −0.478639
\(849\) −2.04722e33 −0.425329
\(850\) 2.80114e33 0.574139
\(851\) −6.34269e33 −1.28258
\(852\) 1.82177e32 0.0363445
\(853\) −3.75623e33 −0.739329 −0.369665 0.929165i \(-0.620527\pi\)
−0.369665 + 0.929165i \(0.620527\pi\)
\(854\) 1.17751e34 2.28665
\(855\) 3.85398e33 0.738413
\(856\) 1.65732e32 0.0313297
\(857\) 8.04748e33 1.50100 0.750499 0.660871i \(-0.229813\pi\)
0.750499 + 0.660871i \(0.229813\pi\)
\(858\) 5.22553e32 0.0961670
\(859\) −4.64719e33 −0.843858 −0.421929 0.906629i \(-0.638647\pi\)
−0.421929 + 0.906629i \(0.638647\pi\)
\(860\) 7.79165e32 0.139604
\(861\) −1.54570e33 −0.273268
\(862\) −5.82711e33 −1.01653
\(863\) −8.37927e33 −1.44239 −0.721194 0.692733i \(-0.756406\pi\)
−0.721194 + 0.692733i \(0.756406\pi\)
\(864\) 1.39119e32 0.0236308
\(865\) −1.42003e34 −2.38019
\(866\) 1.05923e34 1.75201
\(867\) 2.61276e33 0.426462
\(868\) −7.97384e32 −0.128437
\(869\) −1.55211e32 −0.0246715
\(870\) −2.63482e33 −0.413313
\(871\) −8.04075e32 −0.124477
\(872\) −1.55608e33 −0.237735
\(873\) 3.01355e33 0.454375
\(874\) 1.14033e34 1.69687
\(875\) 2.45673e33 0.360799
\(876\) 2.00893e31 0.00291184
\(877\) 7.40167e33 1.05885 0.529425 0.848357i \(-0.322408\pi\)
0.529425 + 0.848357i \(0.322408\pi\)
\(878\) −3.92200e33 −0.553758
\(879\) −7.13616e33 −0.994472
\(880\) 4.61802e33 0.635192
\(881\) 8.68785e32 0.117948 0.0589738 0.998260i \(-0.481217\pi\)
0.0589738 + 0.998260i \(0.481217\pi\)
\(882\) −3.31679e33 −0.444457
\(883\) −7.76396e33 −1.02692 −0.513459 0.858114i \(-0.671636\pi\)
−0.513459 + 0.858114i \(0.671636\pi\)
\(884\) 8.95105e31 0.0116862
\(885\) −7.54796e33 −0.972706
\(886\) −1.19091e34 −1.51493
\(887\) 3.55847e33 0.446829 0.223415 0.974723i \(-0.428280\pi\)
0.223415 + 0.974723i \(0.428280\pi\)
\(888\) 5.28744e33 0.655385
\(889\) 3.13101e33 0.383102
\(890\) −1.71466e34 −2.07107
\(891\) 4.30936e32 0.0513830
\(892\) 2.46068e32 0.0289640
\(893\) −1.11863e34 −1.29985
\(894\) 6.98648e33 0.801451
\(895\) 9.24071e33 1.04650
\(896\) 1.23815e34 1.38430
\(897\) −2.25953e33 −0.249405
\(898\) 1.05326e34 1.14777
\(899\) −6.33397e33 −0.681459
\(900\) −2.23674e32 −0.0237589
\(901\) −2.49610e33 −0.261774
\(902\) 1.32847e33 0.137555
\(903\) 1.34520e34 1.37524
\(904\) 5.40976e33 0.546061
\(905\) −9.40325e33 −0.937172
\(906\) −2.51420e33 −0.247415
\(907\) 1.11723e34 1.08557 0.542786 0.839871i \(-0.317370\pi\)
0.542786 + 0.839871i \(0.317370\pi\)
\(908\) 2.35270e32 0.0225725
\(909\) 4.46275e32 0.0422784
\(910\) −8.72189e33 −0.815896
\(911\) 1.42411e34 1.31548 0.657738 0.753247i \(-0.271513\pi\)
0.657738 + 0.753247i \(0.271513\pi\)
\(912\) −8.91939e33 −0.813569
\(913\) −2.10806e33 −0.189876
\(914\) 1.26486e33 0.112503
\(915\) 1.47897e34 1.29902
\(916\) −7.05271e31 −0.00611727
\(917\) 1.85152e34 1.58592
\(918\) −1.12678e33 −0.0953120
\(919\) 2.28691e34 1.91038 0.955191 0.295991i \(-0.0956499\pi\)
0.955191 + 0.295991i \(0.0956499\pi\)
\(920\) −2.12819e34 −1.75570
\(921\) −6.91968e33 −0.563768
\(922\) −1.02149e34 −0.821914
\(923\) 4.79030e33 0.380665
\(924\) −3.22437e32 −0.0253056
\(925\) −1.65097e34 −1.27971
\(926\) −2.12799e34 −1.62909
\(927\) −2.67664e33 −0.202383
\(928\) −8.26852e32 −0.0617484
\(929\) −1.64699e34 −1.21482 −0.607409 0.794389i \(-0.707791\pi\)
−0.607409 + 0.794389i \(0.707791\pi\)
\(930\) 1.52877e34 1.11375
\(931\) −2.88348e34 −2.07489
\(932\) −5.21740e32 −0.0370827
\(933\) −1.70949e33 −0.120013
\(934\) 1.39461e33 0.0967084
\(935\) 5.07175e33 0.347396
\(936\) 1.88360e33 0.127443
\(937\) 1.12309e34 0.750602 0.375301 0.926903i \(-0.377539\pi\)
0.375301 + 0.926903i \(0.377539\pi\)
\(938\) −7.57343e33 −0.499988
\(939\) −1.02825e34 −0.670573
\(940\) 1.20924e33 0.0779007
\(941\) 1.59102e34 1.01250 0.506251 0.862386i \(-0.331031\pi\)
0.506251 + 0.862386i \(0.331031\pi\)
\(942\) 4.79850e33 0.301662
\(943\) −5.74432e33 −0.356742
\(944\) 1.74685e34 1.07171
\(945\) −7.19273e33 −0.435941
\(946\) −1.15615e34 −0.692254
\(947\) 1.27050e34 0.751537 0.375768 0.926714i \(-0.377379\pi\)
0.375768 + 0.926714i \(0.377379\pi\)
\(948\) −3.24062e31 −0.00189379
\(949\) 5.28242e32 0.0304980
\(950\) 2.96822e34 1.69307
\(951\) −1.56171e34 −0.880082
\(952\) 1.45554e34 0.810398
\(953\) 1.87378e34 1.03074 0.515372 0.856966i \(-0.327654\pi\)
0.515372 + 0.856966i \(0.327654\pi\)
\(954\) −3.04246e33 −0.165355
\(955\) 7.85163e33 0.421620
\(956\) −1.50417e33 −0.0798051
\(957\) −2.56126e33 −0.134266
\(958\) −1.93022e33 −0.0999777
\(959\) −4.63242e34 −2.37079
\(960\) 1.76777e34 0.893937
\(961\) 1.67376e34 0.836324
\(962\) 8.05306e33 0.397601
\(963\) 2.08163e32 0.0101555
\(964\) −6.93918e32 −0.0334520
\(965\) 6.91108e33 0.329217
\(966\) −2.12821e34 −1.00179
\(967\) −1.43720e34 −0.668520 −0.334260 0.942481i \(-0.608486\pi\)
−0.334260 + 0.942481i \(0.608486\pi\)
\(968\) −1.75874e34 −0.808418
\(969\) −9.79574e33 −0.444953
\(970\) 4.32300e34 1.94049
\(971\) 2.53347e34 1.12381 0.561907 0.827201i \(-0.310068\pi\)
0.561907 + 0.827201i \(0.310068\pi\)
\(972\) 8.99745e31 0.00394418
\(973\) 2.71960e34 1.17817
\(974\) −3.99771e33 −0.171153
\(975\) −5.88145e33 −0.248846
\(976\) −3.42282e34 −1.43124
\(977\) −7.40861e32 −0.0306161 −0.0153080 0.999883i \(-0.504873\pi\)
−0.0153080 + 0.999883i \(0.504873\pi\)
\(978\) 8.48019e33 0.346345
\(979\) −1.66679e34 −0.672792
\(980\) 3.11705e33 0.124350
\(981\) −1.95448e33 −0.0770614
\(982\) −3.99478e34 −1.55672
\(983\) −3.84141e34 −1.47953 −0.739767 0.672863i \(-0.765065\pi\)
−0.739767 + 0.672863i \(0.765065\pi\)
\(984\) 4.78862e33 0.182292
\(985\) −4.45619e34 −1.67667
\(986\) 6.69697e33 0.249054
\(987\) 2.08770e34 0.767400
\(988\) 9.48496e32 0.0344612
\(989\) 4.99921e34 1.79533
\(990\) 6.18188e33 0.219440
\(991\) 6.00940e33 0.210855 0.105427 0.994427i \(-0.466379\pi\)
0.105427 + 0.994427i \(0.466379\pi\)
\(992\) 4.79756e33 0.166393
\(993\) 2.78423e34 0.954528
\(994\) 4.51189e34 1.52903
\(995\) −3.85476e34 −1.29131
\(996\) −4.40139e32 −0.0145749
\(997\) −1.73502e34 −0.567950 −0.283975 0.958832i \(-0.591653\pi\)
−0.283975 + 0.958832i \(0.591653\pi\)
\(998\) −4.64904e34 −1.50439
\(999\) 6.64116e33 0.212442
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 3.24.a.b.1.1 2
3.2 odd 2 9.24.a.c.1.2 2
4.3 odd 2 48.24.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.24.a.b.1.1 2 1.1 even 1 trivial
9.24.a.c.1.2 2 3.2 odd 2
48.24.a.g.1.1 2 4.3 odd 2