Properties

Label 3.20.a.b.1.1
Level $3$
Weight $20$
Character 3.1
Self dual yes
Analytic conductor $6.865$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.86450089669\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{87481}) \)
Defining polynomial: \( x^{2} - x - 21870 \) 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 \(148.386\) of defining polynomial
Character \(\chi\) \(=\) 3.1

$q$-expansion

\(f(q)\) \(=\) \(q-536.316 q^{2} -19683.0 q^{3} -236654. q^{4} -683163. q^{5} +1.05563e7 q^{6} +1.13677e8 q^{7} +4.08105e8 q^{8} +3.87420e8 q^{9} +O(q^{10})\) \(q-536.316 q^{2} -19683.0 q^{3} -236654. q^{4} -683163. q^{5} +1.05563e7 q^{6} +1.13677e8 q^{7} +4.08105e8 q^{8} +3.87420e8 q^{9} +3.66391e8 q^{10} +6.43152e9 q^{11} +4.65805e9 q^{12} -5.75132e10 q^{13} -6.09670e10 q^{14} +1.34467e10 q^{15} -9.47984e10 q^{16} +6.85083e11 q^{17} -2.07780e11 q^{18} +1.22275e12 q^{19} +1.61673e11 q^{20} -2.23751e12 q^{21} -3.44932e12 q^{22} +5.02230e12 q^{23} -8.03273e12 q^{24} -1.86068e13 q^{25} +3.08453e13 q^{26} -7.62560e12 q^{27} -2.69022e13 q^{28} +1.53748e14 q^{29} -7.21167e12 q^{30} +3.92872e13 q^{31} -1.63123e14 q^{32} -1.26592e14 q^{33} -3.67421e14 q^{34} -7.76602e13 q^{35} -9.16844e13 q^{36} +1.35709e15 q^{37} -6.55779e14 q^{38} +1.13203e15 q^{39} -2.78802e14 q^{40} -5.75410e14 q^{41} +1.20001e15 q^{42} +3.36667e14 q^{43} -1.52204e15 q^{44} -2.64671e14 q^{45} -2.69354e15 q^{46} -6.99998e15 q^{47} +1.86592e15 q^{48} +1.52367e15 q^{49} +9.97910e15 q^{50} -1.34845e16 q^{51} +1.36107e16 q^{52} +1.69895e16 q^{53} +4.08973e15 q^{54} -4.39377e15 q^{55} +4.63923e16 q^{56} -2.40674e16 q^{57} -8.24573e16 q^{58} -2.70689e16 q^{59} -3.18221e15 q^{60} -5.11533e16 q^{61} -2.10704e16 q^{62} +4.40410e16 q^{63} +1.37187e17 q^{64} +3.92909e16 q^{65} +6.78930e16 q^{66} +2.07437e17 q^{67} -1.62127e17 q^{68} -9.88540e16 q^{69} +4.16504e16 q^{70} -2.35123e17 q^{71} +1.58108e17 q^{72} +7.43858e16 q^{73} -7.27828e17 q^{74} +3.66237e17 q^{75} -2.89368e17 q^{76} +7.31118e17 q^{77} -6.07127e17 q^{78} -6.98697e17 q^{79} +6.47627e16 q^{80} +1.50095e17 q^{81} +3.08601e17 q^{82} +3.25532e18 q^{83} +5.29515e17 q^{84} -4.68023e17 q^{85} -1.80560e17 q^{86} -3.02622e18 q^{87} +2.62473e18 q^{88} -1.43575e18 q^{89} +1.41947e17 q^{90} -6.53796e18 q^{91} -1.18855e18 q^{92} -7.73291e17 q^{93} +3.75420e18 q^{94} -8.35337e17 q^{95} +3.21074e18 q^{96} +1.66911e18 q^{97} -8.17166e17 q^{98} +2.49170e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 702 q^{2} - 39366 q^{3} + 772484 q^{4} + 6016140 q^{5} - 13817466 q^{6} + 113892064 q^{7} + 1008501624 q^{8} + 774840978 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 702 q^{2} - 39366 q^{3} + 772484 q^{4} + 6016140 q^{5} - 13817466 q^{6} + 113892064 q^{7} + 1008501624 q^{8} + 774840978 q^{9} + 8662242420 q^{10} - 6650071272 q^{11} - 15204802572 q^{12} - 44072356148 q^{13} - 60701219424 q^{14} - 118415683620 q^{15} + 119603620880 q^{16} + 336281471748 q^{17} + 271969183278 q^{18} + 602118925096 q^{19} + 6922191196440 q^{20} - 2241737495712 q^{21} - 19648459326744 q^{22} + 2368252165968 q^{23} - 19850337465192 q^{24} + 7200399078350 q^{25} + 47489309789364 q^{26} - 15251194969974 q^{27} - 26685589805888 q^{28} + 280977251970492 q^{29} - 170498917552860 q^{30} + 41610149253712 q^{31} - 212406109003296 q^{32} + 130893352846776 q^{33} - 799347349171332 q^{34} - 76222408017600 q^{35} + 299276129024676 q^{36} + 637994163989884 q^{37} - 14\!\cdots\!32 q^{38}+ \cdots - 25\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −536.316 −0.740688 −0.370344 0.928895i \(-0.620760\pi\)
−0.370344 + 0.928895i \(0.620760\pi\)
\(3\) −19683.0 −0.577350
\(4\) −236654. −0.451381
\(5\) −683163. −0.156426 −0.0782131 0.996937i \(-0.524921\pi\)
−0.0782131 + 0.996937i \(0.524921\pi\)
\(6\) 1.05563e7 0.427637
\(7\) 1.13677e8 1.06474 0.532369 0.846512i \(-0.321302\pi\)
0.532369 + 0.846512i \(0.321302\pi\)
\(8\) 4.08105e8 1.07502
\(9\) 3.87420e8 0.333333
\(10\) 3.66391e8 0.115863
\(11\) 6.43152e9 0.822400 0.411200 0.911545i \(-0.365110\pi\)
0.411200 + 0.911545i \(0.365110\pi\)
\(12\) 4.65805e9 0.260605
\(13\) −5.75132e10 −1.50420 −0.752101 0.659048i \(-0.770959\pi\)
−0.752101 + 0.659048i \(0.770959\pi\)
\(14\) −6.09670e10 −0.788639
\(15\) 1.34467e10 0.0903127
\(16\) −9.47984e10 −0.344874
\(17\) 6.85083e11 1.40113 0.700565 0.713588i \(-0.252931\pi\)
0.700565 + 0.713588i \(0.252931\pi\)
\(18\) −2.07780e11 −0.246896
\(19\) 1.22275e12 0.869317 0.434658 0.900595i \(-0.356869\pi\)
0.434658 + 0.900595i \(0.356869\pi\)
\(20\) 1.61673e11 0.0706078
\(21\) −2.23751e12 −0.614727
\(22\) −3.44932e12 −0.609142
\(23\) 5.02230e12 0.581418 0.290709 0.956811i \(-0.406109\pi\)
0.290709 + 0.956811i \(0.406109\pi\)
\(24\) −8.03273e12 −0.620664
\(25\) −1.86068e13 −0.975531
\(26\) 3.08453e13 1.11414
\(27\) −7.62560e12 −0.192450
\(28\) −2.69022e13 −0.480603
\(29\) 1.53748e14 1.96801 0.984006 0.178134i \(-0.0570060\pi\)
0.984006 + 0.178134i \(0.0570060\pi\)
\(30\) −7.21167e12 −0.0668935
\(31\) 3.92872e13 0.266880 0.133440 0.991057i \(-0.457398\pi\)
0.133440 + 0.991057i \(0.457398\pi\)
\(32\) −1.63123e14 −0.819576
\(33\) −1.26592e14 −0.474813
\(34\) −3.67421e14 −1.03780
\(35\) −7.76602e13 −0.166553
\(36\) −9.16844e13 −0.150460
\(37\) 1.35709e15 1.71669 0.858346 0.513072i \(-0.171493\pi\)
0.858346 + 0.513072i \(0.171493\pi\)
\(38\) −6.55779e14 −0.643893
\(39\) 1.13203e15 0.868451
\(40\) −2.78802e14 −0.168161
\(41\) −5.75410e14 −0.274493 −0.137246 0.990537i \(-0.543825\pi\)
−0.137246 + 0.990537i \(0.543825\pi\)
\(42\) 1.20001e15 0.455321
\(43\) 3.36667e14 0.102153 0.0510763 0.998695i \(-0.483735\pi\)
0.0510763 + 0.998695i \(0.483735\pi\)
\(44\) −1.52204e15 −0.371215
\(45\) −2.64671e14 −0.0521420
\(46\) −2.69354e15 −0.430650
\(47\) −6.99998e15 −0.912362 −0.456181 0.889887i \(-0.650783\pi\)
−0.456181 + 0.889887i \(0.650783\pi\)
\(48\) 1.86592e15 0.199113
\(49\) 1.52367e15 0.133668
\(50\) 9.97910e15 0.722564
\(51\) −1.34845e16 −0.808943
\(52\) 1.36107e16 0.678968
\(53\) 1.69895e16 0.707231 0.353615 0.935391i \(-0.384952\pi\)
0.353615 + 0.935391i \(0.384952\pi\)
\(54\) 4.08973e15 0.142546
\(55\) −4.39377e15 −0.128645
\(56\) 4.63923e16 1.14462
\(57\) −2.40674e16 −0.501900
\(58\) −8.24573e16 −1.45768
\(59\) −2.70689e16 −0.406796 −0.203398 0.979096i \(-0.565198\pi\)
−0.203398 + 0.979096i \(0.565198\pi\)
\(60\) −3.18221e15 −0.0407654
\(61\) −5.11533e16 −0.560068 −0.280034 0.959990i \(-0.590346\pi\)
−0.280034 + 0.959990i \(0.590346\pi\)
\(62\) −2.10704e16 −0.197675
\(63\) 4.40410e16 0.354913
\(64\) 1.37187e17 0.951925
\(65\) 3.92909e16 0.235296
\(66\) 6.78930e16 0.351688
\(67\) 2.07437e17 0.931487 0.465743 0.884920i \(-0.345787\pi\)
0.465743 + 0.884920i \(0.345787\pi\)
\(68\) −1.62127e17 −0.632443
\(69\) −9.88540e16 −0.335682
\(70\) 4.16504e16 0.123364
\(71\) −2.35123e17 −0.608613 −0.304306 0.952574i \(-0.598425\pi\)
−0.304306 + 0.952574i \(0.598425\pi\)
\(72\) 1.58108e17 0.358340
\(73\) 7.43858e16 0.147885 0.0739423 0.997263i \(-0.476442\pi\)
0.0739423 + 0.997263i \(0.476442\pi\)
\(74\) −7.27828e17 −1.27153
\(75\) 3.66237e17 0.563223
\(76\) −2.89368e17 −0.392393
\(77\) 7.31118e17 0.875641
\(78\) −6.07127e17 −0.643252
\(79\) −6.98697e17 −0.655890 −0.327945 0.944697i \(-0.606356\pi\)
−0.327945 + 0.944697i \(0.606356\pi\)
\(80\) 6.47627e16 0.0539474
\(81\) 1.50095e17 0.111111
\(82\) 3.08601e17 0.203314
\(83\) 3.25532e18 1.91140 0.955701 0.294338i \(-0.0950992\pi\)
0.955701 + 0.294338i \(0.0950992\pi\)
\(84\) 5.29515e17 0.277476
\(85\) −4.68023e17 −0.219173
\(86\) −1.80560e17 −0.0756633
\(87\) −3.02622e18 −1.13623
\(88\) 2.62473e18 0.884097
\(89\) −1.43575e18 −0.434385 −0.217193 0.976129i \(-0.569690\pi\)
−0.217193 + 0.976129i \(0.569690\pi\)
\(90\) 1.41947e17 0.0386210
\(91\) −6.53796e18 −1.60158
\(92\) −1.18855e18 −0.262441
\(93\) −7.73291e17 −0.154083
\(94\) 3.75420e18 0.675776
\(95\) −8.35337e17 −0.135984
\(96\) 3.21074e18 0.473183
\(97\) 1.66911e18 0.222922 0.111461 0.993769i \(-0.464447\pi\)
0.111461 + 0.993769i \(0.464447\pi\)
\(98\) −8.17166e17 −0.0990062
\(99\) 2.49170e18 0.274133
\(100\) 4.40336e18 0.440336
\(101\) 9.99068e18 0.908955 0.454477 0.890758i \(-0.349826\pi\)
0.454477 + 0.890758i \(0.349826\pi\)
\(102\) 7.23194e18 0.599175
\(103\) 1.17501e18 0.0887333 0.0443666 0.999015i \(-0.485873\pi\)
0.0443666 + 0.999015i \(0.485873\pi\)
\(104\) −2.34714e19 −1.61705
\(105\) 1.52859e18 0.0961594
\(106\) −9.11176e18 −0.523837
\(107\) −1.73164e19 −0.910568 −0.455284 0.890346i \(-0.650462\pi\)
−0.455284 + 0.890346i \(0.650462\pi\)
\(108\) 1.80462e18 0.0868683
\(109\) 1.57748e19 0.695685 0.347842 0.937553i \(-0.386914\pi\)
0.347842 + 0.937553i \(0.386914\pi\)
\(110\) 2.35645e18 0.0952857
\(111\) −2.67116e19 −0.991132
\(112\) −1.07764e19 −0.367201
\(113\) 4.78817e19 1.49942 0.749712 0.661764i \(-0.230192\pi\)
0.749712 + 0.661764i \(0.230192\pi\)
\(114\) 1.29077e19 0.371752
\(115\) −3.43105e18 −0.0909490
\(116\) −3.63850e19 −0.888323
\(117\) −2.22818e19 −0.501400
\(118\) 1.45175e19 0.301309
\(119\) 7.78785e19 1.49184
\(120\) 5.48766e18 0.0970880
\(121\) −1.97947e19 −0.323659
\(122\) 2.74343e19 0.414836
\(123\) 1.13258e19 0.158478
\(124\) −9.29747e18 −0.120464
\(125\) 2.57418e19 0.309025
\(126\) −2.36199e19 −0.262880
\(127\) −1.25232e20 −1.29294 −0.646472 0.762938i \(-0.723756\pi\)
−0.646472 + 0.762938i \(0.723756\pi\)
\(128\) 1.19478e19 0.114497
\(129\) −6.62661e18 −0.0589779
\(130\) −2.10723e19 −0.174281
\(131\) −2.09329e20 −1.60972 −0.804862 0.593462i \(-0.797761\pi\)
−0.804862 + 0.593462i \(0.797761\pi\)
\(132\) 2.99583e19 0.214321
\(133\) 1.38999e20 0.925595
\(134\) −1.11252e20 −0.689941
\(135\) 5.20953e18 0.0301042
\(136\) 2.79586e20 1.50624
\(137\) 1.53205e20 0.769888 0.384944 0.922940i \(-0.374221\pi\)
0.384944 + 0.922940i \(0.374221\pi\)
\(138\) 5.30169e19 0.248636
\(139\) −2.54374e19 −0.111386 −0.0556932 0.998448i \(-0.517737\pi\)
−0.0556932 + 0.998448i \(0.517737\pi\)
\(140\) 1.83786e19 0.0751788
\(141\) 1.37781e20 0.526752
\(142\) 1.26100e20 0.450792
\(143\) −3.69897e20 −1.23705
\(144\) −3.67268e19 −0.114958
\(145\) −1.05035e20 −0.307849
\(146\) −3.98942e19 −0.109536
\(147\) −2.99903e19 −0.0771732
\(148\) −3.21160e20 −0.774882
\(149\) 1.67659e20 0.379453 0.189727 0.981837i \(-0.439240\pi\)
0.189727 + 0.981837i \(0.439240\pi\)
\(150\) −1.96419e20 −0.417173
\(151\) −4.29868e20 −0.857144 −0.428572 0.903508i \(-0.640983\pi\)
−0.428572 + 0.903508i \(0.640983\pi\)
\(152\) 4.99010e20 0.934533
\(153\) 2.65415e20 0.467043
\(154\) −3.92110e20 −0.648577
\(155\) −2.68396e19 −0.0417470
\(156\) −2.67900e20 −0.392002
\(157\) 9.16306e20 1.26181 0.630905 0.775860i \(-0.282684\pi\)
0.630905 + 0.775860i \(0.282684\pi\)
\(158\) 3.74722e20 0.485810
\(159\) −3.34405e20 −0.408320
\(160\) 1.11439e20 0.128203
\(161\) 5.70923e20 0.619058
\(162\) −8.04981e19 −0.0822987
\(163\) 4.50070e20 0.434008 0.217004 0.976171i \(-0.430372\pi\)
0.217004 + 0.976171i \(0.430372\pi\)
\(164\) 1.36173e20 0.123901
\(165\) 8.64827e19 0.0742731
\(166\) −1.74588e21 −1.41575
\(167\) −1.62414e20 −0.124399 −0.0621993 0.998064i \(-0.519811\pi\)
−0.0621993 + 0.998064i \(0.519811\pi\)
\(168\) −9.13140e20 −0.660844
\(169\) 1.84585e21 1.26262
\(170\) 2.51008e20 0.162339
\(171\) 4.73718e20 0.289772
\(172\) −7.96734e19 −0.0461098
\(173\) −3.29538e21 −1.80496 −0.902481 0.430731i \(-0.858256\pi\)
−0.902481 + 0.430731i \(0.858256\pi\)
\(174\) 1.62301e21 0.841594
\(175\) −2.11517e21 −1.03869
\(176\) −6.09697e20 −0.283625
\(177\) 5.32797e20 0.234864
\(178\) 7.70017e20 0.321744
\(179\) −9.81720e20 −0.388941 −0.194471 0.980908i \(-0.562299\pi\)
−0.194471 + 0.980908i \(0.562299\pi\)
\(180\) 6.26354e19 0.0235359
\(181\) 1.51701e20 0.0540807 0.0270403 0.999634i \(-0.491392\pi\)
0.0270403 + 0.999634i \(0.491392\pi\)
\(182\) 3.50641e21 1.18627
\(183\) 1.00685e21 0.323355
\(184\) 2.04963e21 0.625037
\(185\) −9.27114e20 −0.268535
\(186\) 4.14728e20 0.114128
\(187\) 4.40612e21 1.15229
\(188\) 1.65657e21 0.411823
\(189\) −8.66858e20 −0.204909
\(190\) 4.48004e20 0.100722
\(191\) −5.13044e21 −1.09733 −0.548666 0.836042i \(-0.684864\pi\)
−0.548666 + 0.836042i \(0.684864\pi\)
\(192\) −2.70025e21 −0.549594
\(193\) −9.72873e21 −1.88478 −0.942392 0.334510i \(-0.891429\pi\)
−0.942392 + 0.334510i \(0.891429\pi\)
\(194\) −8.95169e20 −0.165116
\(195\) −7.73363e20 −0.135848
\(196\) −3.60581e20 −0.0603351
\(197\) 7.68506e21 1.22523 0.612616 0.790381i \(-0.290117\pi\)
0.612616 + 0.790381i \(0.290117\pi\)
\(198\) −1.33634e21 −0.203047
\(199\) 4.55786e21 0.660172 0.330086 0.943951i \(-0.392922\pi\)
0.330086 + 0.943951i \(0.392922\pi\)
\(200\) −7.59351e21 −1.04872
\(201\) −4.08299e21 −0.537794
\(202\) −5.35816e21 −0.673252
\(203\) 1.74777e22 2.09542
\(204\) 3.19115e21 0.365141
\(205\) 3.93099e20 0.0429378
\(206\) −6.30174e20 −0.0657237
\(207\) 1.94574e21 0.193806
\(208\) 5.45216e21 0.518761
\(209\) 7.86413e21 0.714926
\(210\) −8.19805e20 −0.0712241
\(211\) −1.71334e22 −1.42285 −0.711426 0.702761i \(-0.751950\pi\)
−0.711426 + 0.702761i \(0.751950\pi\)
\(212\) −4.02064e21 −0.319230
\(213\) 4.62793e21 0.351383
\(214\) 9.28707e21 0.674447
\(215\) −2.29998e20 −0.0159794
\(216\) −3.11204e21 −0.206888
\(217\) 4.46607e21 0.284157
\(218\) −8.46028e21 −0.515286
\(219\) −1.46414e21 −0.0853812
\(220\) 1.03980e21 0.0580678
\(221\) −3.94014e22 −2.10758
\(222\) 1.43258e22 0.734120
\(223\) 3.17895e22 1.56095 0.780473 0.625189i \(-0.214978\pi\)
0.780473 + 0.625189i \(0.214978\pi\)
\(224\) −1.85434e22 −0.872634
\(225\) −7.20865e21 −0.325177
\(226\) −2.56797e22 −1.11061
\(227\) 2.55774e22 1.06074 0.530372 0.847765i \(-0.322052\pi\)
0.530372 + 0.847765i \(0.322052\pi\)
\(228\) 5.69563e21 0.226548
\(229\) −2.23445e22 −0.852576 −0.426288 0.904588i \(-0.640179\pi\)
−0.426288 + 0.904588i \(0.640179\pi\)
\(230\) 1.84013e21 0.0673648
\(231\) −1.43906e22 −0.505551
\(232\) 6.27452e22 2.11565
\(233\) −2.52369e21 −0.0816874 −0.0408437 0.999166i \(-0.513005\pi\)
−0.0408437 + 0.999166i \(0.513005\pi\)
\(234\) 1.19501e22 0.371381
\(235\) 4.78213e21 0.142717
\(236\) 6.40595e21 0.183620
\(237\) 1.37524e22 0.378678
\(238\) −4.17675e22 −1.10499
\(239\) 3.93020e22 0.999158 0.499579 0.866268i \(-0.333488\pi\)
0.499579 + 0.866268i \(0.333488\pi\)
\(240\) −1.27472e21 −0.0311465
\(241\) −1.99943e22 −0.469618 −0.234809 0.972042i \(-0.575447\pi\)
−0.234809 + 0.972042i \(0.575447\pi\)
\(242\) 1.06162e22 0.239730
\(243\) −2.95431e21 −0.0641500
\(244\) 1.21056e22 0.252804
\(245\) −1.04091e21 −0.0209091
\(246\) −6.07420e21 −0.117383
\(247\) −7.03242e22 −1.30763
\(248\) 1.60333e22 0.286901
\(249\) −6.40745e22 −1.10355
\(250\) −1.38057e22 −0.228891
\(251\) −4.94152e22 −0.788788 −0.394394 0.918942i \(-0.629045\pi\)
−0.394394 + 0.918942i \(0.629045\pi\)
\(252\) −1.04225e22 −0.160201
\(253\) 3.23010e22 0.478158
\(254\) 6.71638e22 0.957668
\(255\) 9.21211e21 0.126540
\(256\) −7.83332e22 −1.03673
\(257\) 1.79354e22 0.228742 0.114371 0.993438i \(-0.463515\pi\)
0.114371 + 0.993438i \(0.463515\pi\)
\(258\) 3.55395e21 0.0436842
\(259\) 1.54271e23 1.82783
\(260\) −9.29834e21 −0.106208
\(261\) 5.95650e22 0.656004
\(262\) 1.12266e23 1.19230
\(263\) 2.64384e22 0.270804 0.135402 0.990791i \(-0.456767\pi\)
0.135402 + 0.990791i \(0.456767\pi\)
\(264\) −5.16626e22 −0.510434
\(265\) −1.16066e22 −0.110629
\(266\) −7.45473e22 −0.685577
\(267\) 2.82599e22 0.250792
\(268\) −4.90908e22 −0.420455
\(269\) −6.26512e22 −0.517944 −0.258972 0.965885i \(-0.583384\pi\)
−0.258972 + 0.965885i \(0.583384\pi\)
\(270\) −2.79395e21 −0.0222978
\(271\) −8.23488e22 −0.634525 −0.317263 0.948338i \(-0.602764\pi\)
−0.317263 + 0.948338i \(0.602764\pi\)
\(272\) −6.49448e22 −0.483214
\(273\) 1.28687e23 0.924673
\(274\) −8.21662e22 −0.570247
\(275\) −1.19670e23 −0.802276
\(276\) 2.33942e22 0.151520
\(277\) 1.42704e23 0.893057 0.446528 0.894769i \(-0.352660\pi\)
0.446528 + 0.894769i \(0.352660\pi\)
\(278\) 1.36425e22 0.0825026
\(279\) 1.52207e22 0.0889599
\(280\) −3.16935e22 −0.179048
\(281\) 4.36986e22 0.238648 0.119324 0.992855i \(-0.461927\pi\)
0.119324 + 0.992855i \(0.461927\pi\)
\(282\) −7.38939e22 −0.390159
\(283\) −2.42461e23 −1.23786 −0.618929 0.785447i \(-0.712433\pi\)
−0.618929 + 0.785447i \(0.712433\pi\)
\(284\) 5.56427e22 0.274716
\(285\) 1.64419e22 0.0785103
\(286\) 1.98382e23 0.916272
\(287\) −6.54112e22 −0.292263
\(288\) −6.31970e22 −0.273192
\(289\) 2.30266e23 0.963166
\(290\) 5.63318e22 0.228020
\(291\) −3.28531e22 −0.128704
\(292\) −1.76037e22 −0.0667522
\(293\) 1.65262e22 0.0606638 0.0303319 0.999540i \(-0.490344\pi\)
0.0303319 + 0.999540i \(0.490344\pi\)
\(294\) 1.60843e22 0.0571613
\(295\) 1.84925e22 0.0636335
\(296\) 5.53835e23 1.84548
\(297\) −4.90442e22 −0.158271
\(298\) −8.99184e22 −0.281057
\(299\) −2.88849e23 −0.874570
\(300\) −8.66713e22 −0.254228
\(301\) 3.82714e22 0.108766
\(302\) 2.30545e23 0.634876
\(303\) −1.96647e23 −0.524785
\(304\) −1.15915e23 −0.299805
\(305\) 3.49461e22 0.0876092
\(306\) −1.42346e23 −0.345934
\(307\) −7.14354e23 −1.68306 −0.841528 0.540213i \(-0.818344\pi\)
−0.841528 + 0.540213i \(0.818344\pi\)
\(308\) −1.73022e23 −0.395247
\(309\) −2.31276e22 −0.0512302
\(310\) 1.43945e22 0.0309215
\(311\) 3.74882e23 0.781037 0.390518 0.920595i \(-0.372296\pi\)
0.390518 + 0.920595i \(0.372296\pi\)
\(312\) 4.61988e23 0.933603
\(313\) −2.04132e23 −0.400166 −0.200083 0.979779i \(-0.564121\pi\)
−0.200083 + 0.979779i \(0.564121\pi\)
\(314\) −4.91429e23 −0.934607
\(315\) −3.00872e22 −0.0555176
\(316\) 1.65349e23 0.296056
\(317\) −2.74437e23 −0.476848 −0.238424 0.971161i \(-0.576631\pi\)
−0.238424 + 0.971161i \(0.576631\pi\)
\(318\) 1.79347e23 0.302438
\(319\) 9.88832e23 1.61849
\(320\) −9.37210e22 −0.148906
\(321\) 3.40839e23 0.525717
\(322\) −3.06195e23 −0.458529
\(323\) 8.37684e23 1.21803
\(324\) −3.55204e22 −0.0501534
\(325\) 1.07014e24 1.46739
\(326\) −2.41379e23 −0.321464
\(327\) −3.10496e23 −0.401654
\(328\) −2.34828e23 −0.295085
\(329\) −7.95740e23 −0.971427
\(330\) −4.63820e22 −0.0550132
\(331\) −2.31592e23 −0.266906 −0.133453 0.991055i \(-0.542607\pi\)
−0.133453 + 0.991055i \(0.542607\pi\)
\(332\) −7.70384e23 −0.862771
\(333\) 5.25764e23 0.572230
\(334\) 8.71049e22 0.0921406
\(335\) −1.41714e23 −0.145709
\(336\) 2.12113e23 0.212004
\(337\) −1.11237e23 −0.108085 −0.0540425 0.998539i \(-0.517211\pi\)
−0.0540425 + 0.998539i \(0.517211\pi\)
\(338\) −9.89960e23 −0.935209
\(339\) −9.42456e23 −0.865693
\(340\) 1.10759e23 0.0989307
\(341\) 2.52677e23 0.219482
\(342\) −2.54062e23 −0.214631
\(343\) −1.12259e24 −0.922417
\(344\) 1.37395e23 0.109816
\(345\) 6.75334e22 0.0525094
\(346\) 1.76737e24 1.33691
\(347\) 2.05609e24 1.51326 0.756629 0.653844i \(-0.226845\pi\)
0.756629 + 0.653844i \(0.226845\pi\)
\(348\) 7.16165e23 0.512874
\(349\) −1.75492e24 −1.22297 −0.611487 0.791255i \(-0.709428\pi\)
−0.611487 + 0.791255i \(0.709428\pi\)
\(350\) 1.13440e24 0.769342
\(351\) 4.38573e23 0.289484
\(352\) −1.04913e24 −0.674019
\(353\) 8.21430e23 0.513702 0.256851 0.966451i \(-0.417315\pi\)
0.256851 + 0.966451i \(0.417315\pi\)
\(354\) −2.85747e23 −0.173961
\(355\) 1.60627e23 0.0952029
\(356\) 3.39776e23 0.196073
\(357\) −1.53288e24 −0.861313
\(358\) 5.26512e23 0.288084
\(359\) 5.29683e23 0.282240 0.141120 0.989992i \(-0.454930\pi\)
0.141120 + 0.989992i \(0.454930\pi\)
\(360\) −1.08014e23 −0.0560538
\(361\) −4.83305e23 −0.244289
\(362\) −8.13596e22 −0.0400569
\(363\) 3.89618e23 0.186864
\(364\) 1.54723e24 0.722923
\(365\) −5.08176e22 −0.0231330
\(366\) −5.39990e23 −0.239505
\(367\) −2.30180e24 −0.994809 −0.497404 0.867519i \(-0.665713\pi\)
−0.497404 + 0.867519i \(0.665713\pi\)
\(368\) −4.76106e23 −0.200516
\(369\) −2.22926e23 −0.0914976
\(370\) 4.97225e23 0.198901
\(371\) 1.93133e24 0.753016
\(372\) 1.83002e23 0.0695501
\(373\) 1.64953e24 0.611120 0.305560 0.952173i \(-0.401156\pi\)
0.305560 + 0.952173i \(0.401156\pi\)
\(374\) −2.36307e24 −0.853487
\(375\) −5.06675e23 −0.178415
\(376\) −2.85673e24 −0.980808
\(377\) −8.84253e24 −2.96029
\(378\) 4.64910e23 0.151774
\(379\) 3.36454e24 1.07116 0.535579 0.844485i \(-0.320094\pi\)
0.535579 + 0.844485i \(0.320094\pi\)
\(380\) 1.97685e23 0.0613805
\(381\) 2.46494e24 0.746482
\(382\) 2.75154e24 0.812780
\(383\) 1.55219e24 0.447257 0.223628 0.974674i \(-0.428210\pi\)
0.223628 + 0.974674i \(0.428210\pi\)
\(384\) −2.35168e23 −0.0661046
\(385\) −4.99473e23 −0.136973
\(386\) 5.21767e24 1.39604
\(387\) 1.30432e23 0.0340509
\(388\) −3.95000e23 −0.100623
\(389\) −1.19996e23 −0.0298295 −0.0149148 0.999889i \(-0.504748\pi\)
−0.0149148 + 0.999889i \(0.504748\pi\)
\(390\) 4.14767e23 0.100621
\(391\) 3.44070e24 0.814643
\(392\) 6.21815e23 0.143696
\(393\) 4.12022e24 0.929375
\(394\) −4.12162e24 −0.907515
\(395\) 4.77324e23 0.102598
\(396\) −5.89670e23 −0.123738
\(397\) −1.23812e24 −0.253661 −0.126831 0.991924i \(-0.540480\pi\)
−0.126831 + 0.991924i \(0.540480\pi\)
\(398\) −2.44445e24 −0.488981
\(399\) −2.73592e24 −0.534392
\(400\) 1.76389e24 0.336436
\(401\) −6.55287e24 −1.22056 −0.610281 0.792185i \(-0.708943\pi\)
−0.610281 + 0.792185i \(0.708943\pi\)
\(402\) 2.18977e24 0.398338
\(403\) −2.25954e24 −0.401441
\(404\) −2.36433e24 −0.410285
\(405\) −1.02539e23 −0.0173807
\(406\) −9.37354e24 −1.55205
\(407\) 8.72815e24 1.41181
\(408\) −5.50309e24 −0.869631
\(409\) 8.56848e24 1.32292 0.661458 0.749982i \(-0.269938\pi\)
0.661458 + 0.749982i \(0.269938\pi\)
\(410\) −2.10825e23 −0.0318036
\(411\) −3.01553e24 −0.444495
\(412\) −2.78069e23 −0.0400525
\(413\) −3.07712e24 −0.433131
\(414\) −1.04353e24 −0.143550
\(415\) −2.22392e24 −0.298993
\(416\) 9.38171e24 1.23281
\(417\) 5.00685e23 0.0643090
\(418\) −4.21766e24 −0.529537
\(419\) 8.09987e24 0.994134 0.497067 0.867712i \(-0.334410\pi\)
0.497067 + 0.867712i \(0.334410\pi\)
\(420\) −3.61745e23 −0.0434045
\(421\) 9.99748e24 1.17276 0.586382 0.810035i \(-0.300552\pi\)
0.586382 + 0.810035i \(0.300552\pi\)
\(422\) 9.18890e24 1.05389
\(423\) −2.71194e24 −0.304121
\(424\) 6.93352e24 0.760288
\(425\) −1.27472e25 −1.36685
\(426\) −2.48203e24 −0.260265
\(427\) −5.81498e24 −0.596326
\(428\) 4.09799e24 0.411013
\(429\) 7.28069e24 0.714214
\(430\) 1.23352e23 0.0118357
\(431\) −4.59733e24 −0.431491 −0.215745 0.976450i \(-0.569218\pi\)
−0.215745 + 0.976450i \(0.569218\pi\)
\(432\) 7.22894e23 0.0663711
\(433\) −7.51468e24 −0.674956 −0.337478 0.941333i \(-0.609574\pi\)
−0.337478 + 0.941333i \(0.609574\pi\)
\(434\) −2.39523e24 −0.210472
\(435\) 2.06740e24 0.177736
\(436\) −3.73317e24 −0.314019
\(437\) 6.14102e24 0.505436
\(438\) 7.85238e23 0.0632408
\(439\) −2.04197e25 −1.60930 −0.804649 0.593751i \(-0.797647\pi\)
−0.804649 + 0.593751i \(0.797647\pi\)
\(440\) −1.79312e24 −0.138296
\(441\) 5.90299e23 0.0445560
\(442\) 2.11316e25 1.56106
\(443\) −1.84340e25 −1.33286 −0.666428 0.745569i \(-0.732178\pi\)
−0.666428 + 0.745569i \(0.732178\pi\)
\(444\) 6.32140e24 0.447378
\(445\) 9.80854e23 0.0679492
\(446\) −1.70492e25 −1.15617
\(447\) −3.30004e24 −0.219077
\(448\) 1.55951e25 1.01355
\(449\) 7.82616e24 0.497976 0.248988 0.968507i \(-0.419902\pi\)
0.248988 + 0.968507i \(0.419902\pi\)
\(450\) 3.86611e24 0.240855
\(451\) −3.70076e24 −0.225743
\(452\) −1.13314e25 −0.676811
\(453\) 8.46108e24 0.494872
\(454\) −1.37176e25 −0.785681
\(455\) 4.46649e24 0.250529
\(456\) −9.82201e24 −0.539553
\(457\) 2.15612e25 1.16003 0.580015 0.814606i \(-0.303047\pi\)
0.580015 + 0.814606i \(0.303047\pi\)
\(458\) 1.19837e25 0.631493
\(459\) −5.22417e24 −0.269648
\(460\) 8.11971e23 0.0410526
\(461\) 4.52754e24 0.224235 0.112118 0.993695i \(-0.464237\pi\)
0.112118 + 0.993695i \(0.464237\pi\)
\(462\) 7.71791e24 0.374456
\(463\) 1.12552e25 0.534974 0.267487 0.963562i \(-0.413807\pi\)
0.267487 + 0.963562i \(0.413807\pi\)
\(464\) −1.45750e25 −0.678717
\(465\) 5.28284e23 0.0241026
\(466\) 1.35350e24 0.0605049
\(467\) 1.73792e25 0.761237 0.380618 0.924732i \(-0.375711\pi\)
0.380618 + 0.924732i \(0.375711\pi\)
\(468\) 5.27307e24 0.226323
\(469\) 2.35810e25 0.991790
\(470\) −2.56473e24 −0.105709
\(471\) −1.80356e25 −0.728506
\(472\) −1.10469e25 −0.437314
\(473\) 2.16528e24 0.0840104
\(474\) −7.37565e24 −0.280483
\(475\) −2.27514e25 −0.848045
\(476\) −1.84302e25 −0.673387
\(477\) 6.58210e24 0.235744
\(478\) −2.10783e25 −0.740065
\(479\) 4.26798e25 1.46904 0.734522 0.678585i \(-0.237407\pi\)
0.734522 + 0.678585i \(0.237407\pi\)
\(480\) −2.19346e24 −0.0740181
\(481\) −7.80506e25 −2.58225
\(482\) 1.07233e25 0.347841
\(483\) −1.12375e25 −0.357413
\(484\) 4.68448e24 0.146093
\(485\) −1.14027e24 −0.0348708
\(486\) 1.58444e24 0.0475152
\(487\) −6.57071e24 −0.193236 −0.0966178 0.995322i \(-0.530802\pi\)
−0.0966178 + 0.995322i \(0.530802\pi\)
\(488\) −2.08759e25 −0.602085
\(489\) −8.85872e24 −0.250574
\(490\) 5.58257e23 0.0154872
\(491\) 5.24168e25 1.42625 0.713126 0.701035i \(-0.247279\pi\)
0.713126 + 0.701035i \(0.247279\pi\)
\(492\) −2.68029e24 −0.0715342
\(493\) 1.05330e26 2.75744
\(494\) 3.77160e25 0.968544
\(495\) −1.70224e24 −0.0428816
\(496\) −3.72437e24 −0.0920400
\(497\) −2.67282e25 −0.648013
\(498\) 3.43642e25 0.817386
\(499\) −2.55146e25 −0.595434 −0.297717 0.954654i \(-0.596225\pi\)
−0.297717 + 0.954654i \(0.596225\pi\)
\(500\) −6.09188e24 −0.139488
\(501\) 3.19679e24 0.0718216
\(502\) 2.65021e25 0.584246
\(503\) −5.69466e25 −1.23189 −0.615945 0.787789i \(-0.711225\pi\)
−0.615945 + 0.787789i \(0.711225\pi\)
\(504\) 1.79733e25 0.381539
\(505\) −6.82527e24 −0.142184
\(506\) −1.73236e25 −0.354166
\(507\) −3.63319e25 −0.728975
\(508\) 2.96366e25 0.583610
\(509\) −3.26878e25 −0.631781 −0.315891 0.948796i \(-0.602303\pi\)
−0.315891 + 0.948796i \(0.602303\pi\)
\(510\) −4.94060e24 −0.0937266
\(511\) 8.45598e24 0.157458
\(512\) 3.57472e25 0.653398
\(513\) −9.32419e24 −0.167300
\(514\) −9.61903e24 −0.169427
\(515\) −8.02721e23 −0.0138802
\(516\) 1.56821e24 0.0266215
\(517\) −4.50205e25 −0.750326
\(518\) −8.27377e25 −1.35385
\(519\) 6.48630e25 1.04209
\(520\) 1.60348e25 0.252949
\(521\) −9.30404e25 −1.44116 −0.720581 0.693370i \(-0.756125\pi\)
−0.720581 + 0.693370i \(0.756125\pi\)
\(522\) −3.19457e25 −0.485895
\(523\) 4.88002e25 0.728879 0.364440 0.931227i \(-0.381261\pi\)
0.364440 + 0.931227i \(0.381261\pi\)
\(524\) 4.95384e25 0.726599
\(525\) 4.16329e25 0.599685
\(526\) −1.41793e25 −0.200582
\(527\) 2.69150e25 0.373933
\(528\) 1.20007e25 0.163751
\(529\) −4.93919e25 −0.661953
\(530\) 6.22482e24 0.0819419
\(531\) −1.04870e25 −0.135599
\(532\) −3.28946e25 −0.417796
\(533\) 3.30937e25 0.412892
\(534\) −1.51562e25 −0.185759
\(535\) 1.18299e25 0.142437
\(536\) 8.46562e25 1.00137
\(537\) 1.93232e25 0.224555
\(538\) 3.36008e25 0.383635
\(539\) 9.79948e24 0.109928
\(540\) −1.23285e24 −0.0135885
\(541\) 8.45858e25 0.916059 0.458030 0.888937i \(-0.348555\pi\)
0.458030 + 0.888937i \(0.348555\pi\)
\(542\) 4.41649e25 0.469985
\(543\) −2.98593e24 −0.0312235
\(544\) −1.11753e26 −1.14833
\(545\) −1.07768e25 −0.108823
\(546\) −6.90167e25 −0.684895
\(547\) −2.08591e25 −0.203430 −0.101715 0.994814i \(-0.532433\pi\)
−0.101715 + 0.994814i \(0.532433\pi\)
\(548\) −3.62565e25 −0.347513
\(549\) −1.98179e25 −0.186689
\(550\) 6.41808e25 0.594237
\(551\) 1.87995e26 1.71083
\(552\) −4.03428e25 −0.360865
\(553\) −7.94260e25 −0.698352
\(554\) −7.65345e25 −0.661477
\(555\) 1.82484e25 0.155039
\(556\) 6.01985e24 0.0502777
\(557\) −1.19573e25 −0.0981768 −0.0490884 0.998794i \(-0.515632\pi\)
−0.0490884 + 0.998794i \(0.515632\pi\)
\(558\) −8.16309e24 −0.0658916
\(559\) −1.93628e25 −0.153658
\(560\) 7.36206e24 0.0574398
\(561\) −8.67257e25 −0.665274
\(562\) −2.34362e25 −0.176764
\(563\) −1.51109e26 −1.12063 −0.560313 0.828281i \(-0.689319\pi\)
−0.560313 + 0.828281i \(0.689319\pi\)
\(564\) −3.26063e25 −0.237766
\(565\) −3.27110e25 −0.234549
\(566\) 1.30036e26 0.916866
\(567\) 1.70624e25 0.118304
\(568\) −9.59549e25 −0.654271
\(569\) −1.31930e26 −0.884661 −0.442331 0.896852i \(-0.645848\pi\)
−0.442331 + 0.896852i \(0.645848\pi\)
\(570\) −8.81806e24 −0.0581517
\(571\) 2.65239e26 1.72026 0.860130 0.510076i \(-0.170383\pi\)
0.860130 + 0.510076i \(0.170383\pi\)
\(572\) 8.75375e25 0.558383
\(573\) 1.00982e26 0.633544
\(574\) 3.50810e25 0.216476
\(575\) −9.34489e25 −0.567191
\(576\) 5.31490e25 0.317308
\(577\) −9.99355e25 −0.586880 −0.293440 0.955977i \(-0.594800\pi\)
−0.293440 + 0.955977i \(0.594800\pi\)
\(578\) −1.23496e26 −0.713406
\(579\) 1.91491e26 1.08818
\(580\) 2.48569e25 0.138957
\(581\) 3.70057e26 2.03514
\(582\) 1.76196e25 0.0953296
\(583\) 1.09269e26 0.581626
\(584\) 3.03572e25 0.158979
\(585\) 1.52221e25 0.0784321
\(586\) −8.86324e24 −0.0449330
\(587\) −1.90494e26 −0.950209 −0.475104 0.879930i \(-0.657590\pi\)
−0.475104 + 0.879930i \(0.657590\pi\)
\(588\) 7.09732e24 0.0348345
\(589\) 4.80384e25 0.232003
\(590\) −9.91779e24 −0.0471326
\(591\) −1.51265e26 −0.707388
\(592\) −1.28650e26 −0.592043
\(593\) −2.37669e26 −1.07635 −0.538173 0.842834i \(-0.680885\pi\)
−0.538173 + 0.842834i \(0.680885\pi\)
\(594\) 2.63032e25 0.117229
\(595\) −5.32037e25 −0.233362
\(596\) −3.96772e25 −0.171278
\(597\) −8.97124e25 −0.381150
\(598\) 1.54914e26 0.647784
\(599\) −3.08975e26 −1.27165 −0.635826 0.771832i \(-0.719341\pi\)
−0.635826 + 0.771832i \(0.719341\pi\)
\(600\) 1.49463e26 0.605476
\(601\) −2.61533e26 −1.04284 −0.521421 0.853299i \(-0.674598\pi\)
−0.521421 + 0.853299i \(0.674598\pi\)
\(602\) −2.05256e25 −0.0805616
\(603\) 8.03655e25 0.310496
\(604\) 1.01730e26 0.386898
\(605\) 1.35230e25 0.0506287
\(606\) 1.05465e26 0.388702
\(607\) 5.37620e25 0.195067 0.0975334 0.995232i \(-0.468905\pi\)
0.0975334 + 0.995232i \(0.468905\pi\)
\(608\) −1.99458e26 −0.712471
\(609\) −3.44013e26 −1.20979
\(610\) −1.87421e25 −0.0648911
\(611\) 4.02591e26 1.37238
\(612\) −6.28115e25 −0.210814
\(613\) 3.16087e26 1.04456 0.522278 0.852775i \(-0.325082\pi\)
0.522278 + 0.852775i \(0.325082\pi\)
\(614\) 3.83119e26 1.24662
\(615\) −7.73737e24 −0.0247902
\(616\) 2.98373e26 0.941332
\(617\) 1.98562e26 0.616862 0.308431 0.951247i \(-0.400196\pi\)
0.308431 + 0.951247i \(0.400196\pi\)
\(618\) 1.24037e25 0.0379456
\(619\) −4.82992e26 −1.45505 −0.727527 0.686080i \(-0.759330\pi\)
−0.727527 + 0.686080i \(0.759330\pi\)
\(620\) 6.35169e24 0.0188438
\(621\) −3.82981e25 −0.111894
\(622\) −2.01055e26 −0.578505
\(623\) −1.63213e26 −0.462507
\(624\) −1.07315e26 −0.299507
\(625\) 3.37310e26 0.927191
\(626\) 1.09479e26 0.296398
\(627\) −1.54790e26 −0.412763
\(628\) −2.16847e26 −0.569556
\(629\) 9.29719e26 2.40531
\(630\) 1.61362e25 0.0411213
\(631\) −6.27693e26 −1.57568 −0.787841 0.615879i \(-0.788801\pi\)
−0.787841 + 0.615879i \(0.788801\pi\)
\(632\) −2.85141e26 −0.705096
\(633\) 3.37236e26 0.821484
\(634\) 1.47185e26 0.353196
\(635\) 8.55538e25 0.202250
\(636\) 7.91382e25 0.184308
\(637\) −8.76310e25 −0.201063
\(638\) −5.30326e26 −1.19880
\(639\) −9.10915e25 −0.202871
\(640\) −8.16228e24 −0.0179103
\(641\) 2.15915e26 0.466800 0.233400 0.972381i \(-0.425015\pi\)
0.233400 + 0.972381i \(0.425015\pi\)
\(642\) −1.82797e26 −0.389392
\(643\) 4.68779e26 0.983929 0.491964 0.870615i \(-0.336279\pi\)
0.491964 + 0.870615i \(0.336279\pi\)
\(644\) −1.35111e26 −0.279431
\(645\) 4.52706e24 0.00922568
\(646\) −4.49263e26 −0.902177
\(647\) −2.10829e26 −0.417196 −0.208598 0.978001i \(-0.566890\pi\)
−0.208598 + 0.978001i \(0.566890\pi\)
\(648\) 6.12543e25 0.119447
\(649\) −1.74094e26 −0.334549
\(650\) −5.73931e26 −1.08688
\(651\) −8.79057e25 −0.164058
\(652\) −1.06511e26 −0.195903
\(653\) 2.88194e26 0.522408 0.261204 0.965284i \(-0.415880\pi\)
0.261204 + 0.965284i \(0.415880\pi\)
\(654\) 1.66524e26 0.297500
\(655\) 1.43006e26 0.251803
\(656\) 5.45480e25 0.0946656
\(657\) 2.88186e25 0.0492948
\(658\) 4.26768e26 0.719524
\(659\) −7.36091e26 −1.22326 −0.611632 0.791143i \(-0.709486\pi\)
−0.611632 + 0.791143i \(0.709486\pi\)
\(660\) −2.04664e25 −0.0335255
\(661\) 3.05440e25 0.0493187 0.0246594 0.999696i \(-0.492150\pi\)
0.0246594 + 0.999696i \(0.492150\pi\)
\(662\) 1.24207e26 0.197694
\(663\) 7.75537e26 1.21681
\(664\) 1.32851e27 2.05480
\(665\) −9.49589e25 −0.144787
\(666\) −2.81976e26 −0.423844
\(667\) 7.72168e26 1.14424
\(668\) 3.84357e25 0.0561512
\(669\) −6.25713e26 −0.901213
\(670\) 7.60032e25 0.107925
\(671\) −3.28994e26 −0.460600
\(672\) 3.64989e26 0.503816
\(673\) −9.62645e26 −1.31016 −0.655078 0.755561i \(-0.727364\pi\)
−0.655078 + 0.755561i \(0.727364\pi\)
\(674\) 5.96582e25 0.0800573
\(675\) 1.41888e26 0.187741
\(676\) −4.36828e26 −0.569923
\(677\) 1.18974e27 1.53059 0.765295 0.643680i \(-0.222593\pi\)
0.765295 + 0.643680i \(0.222593\pi\)
\(678\) 5.05454e26 0.641208
\(679\) 1.89740e26 0.237354
\(680\) −1.91003e26 −0.235616
\(681\) −5.03440e26 −0.612421
\(682\) −1.35514e26 −0.162568
\(683\) −1.51240e27 −1.78925 −0.894624 0.446819i \(-0.852557\pi\)
−0.894624 + 0.446819i \(0.852557\pi\)
\(684\) −1.12107e26 −0.130798
\(685\) −1.04664e26 −0.120431
\(686\) 6.02063e26 0.683224
\(687\) 4.39806e26 0.492235
\(688\) −3.19155e25 −0.0352299
\(689\) −9.77124e26 −1.06382
\(690\) −3.62192e25 −0.0388931
\(691\) −1.40247e27 −1.48543 −0.742716 0.669606i \(-0.766463\pi\)
−0.742716 + 0.669606i \(0.766463\pi\)
\(692\) 7.79864e26 0.814725
\(693\) 2.83250e26 0.291880
\(694\) −1.10272e27 −1.12085
\(695\) 1.73779e25 0.0174238
\(696\) −1.23501e27 −1.22147
\(697\) −3.94204e26 −0.384600
\(698\) 9.41194e26 0.905842
\(699\) 4.96738e25 0.0471622
\(700\) 5.00563e26 0.468843
\(701\) 6.64167e26 0.613700 0.306850 0.951758i \(-0.400725\pi\)
0.306850 + 0.951758i \(0.400725\pi\)
\(702\) −2.35213e26 −0.214417
\(703\) 1.65938e27 1.49235
\(704\) 8.82320e26 0.782863
\(705\) −9.41266e25 −0.0823978
\(706\) −4.40546e26 −0.380493
\(707\) 1.13572e27 0.967799
\(708\) −1.26088e26 −0.106013
\(709\) 1.03445e27 0.858163 0.429081 0.903266i \(-0.358837\pi\)
0.429081 + 0.903266i \(0.358837\pi\)
\(710\) −8.61470e25 −0.0705157
\(711\) −2.70689e26 −0.218630
\(712\) −5.85938e26 −0.466973
\(713\) 1.97313e26 0.155169
\(714\) 8.22109e26 0.637964
\(715\) 2.52700e26 0.193508
\(716\) 2.32328e26 0.175561
\(717\) −7.73581e26 −0.576864
\(718\) −2.84077e26 −0.209052
\(719\) −1.59383e26 −0.115749 −0.0578744 0.998324i \(-0.518432\pi\)
−0.0578744 + 0.998324i \(0.518432\pi\)
\(720\) 2.50904e25 0.0179825
\(721\) 1.33572e26 0.0944777
\(722\) 2.59204e26 0.180942
\(723\) 3.93548e26 0.271134
\(724\) −3.59006e25 −0.0244110
\(725\) −2.86075e27 −1.91986
\(726\) −2.08958e26 −0.138408
\(727\) 1.66059e27 1.08564 0.542819 0.839850i \(-0.317357\pi\)
0.542819 + 0.839850i \(0.317357\pi\)
\(728\) −2.66817e27 −1.72173
\(729\) 5.81497e25 0.0370370
\(730\) 2.72543e25 0.0171343
\(731\) 2.30645e26 0.143129
\(732\) −2.38275e26 −0.145956
\(733\) −1.36074e27 −0.822786 −0.411393 0.911458i \(-0.634958\pi\)
−0.411393 + 0.911458i \(0.634958\pi\)
\(734\) 1.23449e27 0.736843
\(735\) 2.04883e25 0.0120719
\(736\) −8.19251e26 −0.476517
\(737\) 1.33414e27 0.766054
\(738\) 1.19559e26 0.0677712
\(739\) 9.59784e26 0.537095 0.268548 0.963266i \(-0.413456\pi\)
0.268548 + 0.963266i \(0.413456\pi\)
\(740\) 2.19405e26 0.121212
\(741\) 1.38419e27 0.754959
\(742\) −1.03580e27 −0.557750
\(743\) −1.18287e27 −0.628846 −0.314423 0.949283i \(-0.601811\pi\)
−0.314423 + 0.949283i \(0.601811\pi\)
\(744\) −3.15584e26 −0.165643
\(745\) −1.14539e26 −0.0593564
\(746\) −8.84670e26 −0.452650
\(747\) 1.26118e27 0.637134
\(748\) −1.04273e27 −0.520121
\(749\) −1.96849e27 −0.969516
\(750\) 2.71738e26 0.132150
\(751\) −1.02382e27 −0.491637 −0.245819 0.969316i \(-0.579057\pi\)
−0.245819 + 0.969316i \(0.579057\pi\)
\(752\) 6.63587e26 0.314650
\(753\) 9.72639e26 0.455407
\(754\) 4.74239e27 2.19265
\(755\) 2.93670e26 0.134080
\(756\) 2.05145e26 0.0924920
\(757\) 2.74415e25 0.0122179 0.00610897 0.999981i \(-0.498055\pi\)
0.00610897 + 0.999981i \(0.498055\pi\)
\(758\) −1.80446e27 −0.793394
\(759\) −6.35781e26 −0.276065
\(760\) −3.40905e26 −0.146185
\(761\) 1.89240e27 0.801416 0.400708 0.916206i \(-0.368764\pi\)
0.400708 + 0.916206i \(0.368764\pi\)
\(762\) −1.32199e27 −0.552910
\(763\) 1.79324e27 0.740722
\(764\) 1.21414e27 0.495314
\(765\) −1.81322e26 −0.0730578
\(766\) −8.32465e26 −0.331278
\(767\) 1.55682e27 0.611902
\(768\) 1.54183e27 0.598557
\(769\) −1.93865e27 −0.743359 −0.371679 0.928361i \(-0.621218\pi\)
−0.371679 + 0.928361i \(0.621218\pi\)
\(770\) 2.67875e26 0.101454
\(771\) −3.53022e26 −0.132064
\(772\) 2.30234e27 0.850756
\(773\) −7.63732e24 −0.00278764 −0.00139382 0.999999i \(-0.500444\pi\)
−0.00139382 + 0.999999i \(0.500444\pi\)
\(774\) −6.99525e25 −0.0252211
\(775\) −7.31009e26 −0.260349
\(776\) 6.81171e26 0.239646
\(777\) −3.03651e27 −1.05530
\(778\) 6.43559e25 0.0220944
\(779\) −7.03582e26 −0.238621
\(780\) 1.83019e26 0.0613194
\(781\) −1.51220e27 −0.500523
\(782\) −1.84530e27 −0.603396
\(783\) −1.17242e27 −0.378744
\(784\) −1.44441e26 −0.0460986
\(785\) −6.25986e26 −0.197380
\(786\) −2.20974e27 −0.688377
\(787\) −4.61341e27 −1.41991 −0.709956 0.704246i \(-0.751285\pi\)
−0.709956 + 0.704246i \(0.751285\pi\)
\(788\) −1.81870e27 −0.553046
\(789\) −5.20387e26 −0.156349
\(790\) −2.55996e26 −0.0759934
\(791\) 5.44307e27 1.59649
\(792\) 1.01688e27 0.294699
\(793\) 2.94199e27 0.842455
\(794\) 6.64025e26 0.187884
\(795\) 2.28453e26 0.0638719
\(796\) −1.07863e27 −0.297989
\(797\) 1.01219e27 0.276316 0.138158 0.990410i \(-0.455882\pi\)
0.138158 + 0.990410i \(0.455882\pi\)
\(798\) 1.46731e27 0.395818
\(799\) −4.79557e27 −1.27834
\(800\) 3.03519e27 0.799522
\(801\) −5.56240e26 −0.144795
\(802\) 3.51440e27 0.904056
\(803\) 4.78413e26 0.121620
\(804\) 9.66255e26 0.242750
\(805\) −3.90033e26 −0.0968369
\(806\) 1.21183e27 0.297343
\(807\) 1.23316e27 0.299035
\(808\) 4.07725e27 0.977145
\(809\) 3.00853e27 0.712597 0.356298 0.934372i \(-0.384039\pi\)
0.356298 + 0.934372i \(0.384039\pi\)
\(810\) 5.49933e25 0.0128737
\(811\) 3.54753e27 0.820782 0.410391 0.911910i \(-0.365392\pi\)
0.410391 + 0.911910i \(0.365392\pi\)
\(812\) −4.13615e27 −0.945832
\(813\) 1.62087e27 0.366343
\(814\) −4.68104e27 −1.04571
\(815\) −3.07471e26 −0.0678901
\(816\) 1.27831e27 0.278984
\(817\) 4.11659e26 0.0888030
\(818\) −4.59541e27 −0.979869
\(819\) −2.53294e27 −0.533860
\(820\) −9.30283e25 −0.0193813
\(821\) −4.28797e27 −0.883063 −0.441532 0.897246i \(-0.645565\pi\)
−0.441532 + 0.897246i \(0.645565\pi\)
\(822\) 1.61728e27 0.329232
\(823\) 3.20459e27 0.644874 0.322437 0.946591i \(-0.395498\pi\)
0.322437 + 0.946591i \(0.395498\pi\)
\(824\) 4.79526e26 0.0953901
\(825\) 2.35546e27 0.463194
\(826\) 1.65031e27 0.320815
\(827\) −4.02192e27 −0.772914 −0.386457 0.922308i \(-0.626301\pi\)
−0.386457 + 0.922308i \(0.626301\pi\)
\(828\) −4.60467e26 −0.0874803
\(829\) −1.58908e27 −0.298455 −0.149227 0.988803i \(-0.547679\pi\)
−0.149227 + 0.988803i \(0.547679\pi\)
\(830\) 1.19272e27 0.221461
\(831\) −2.80885e27 −0.515607
\(832\) −7.89006e27 −1.43189
\(833\) 1.04384e27 0.187286
\(834\) −2.68525e26 −0.0476329
\(835\) 1.10955e26 0.0194592
\(836\) −1.86107e27 −0.322704
\(837\) −2.99589e26 −0.0513610
\(838\) −4.34409e27 −0.736343
\(839\) −5.31813e27 −0.891293 −0.445646 0.895209i \(-0.647026\pi\)
−0.445646 + 0.895209i \(0.647026\pi\)
\(840\) 6.23823e26 0.103373
\(841\) 1.75351e28 2.87307
\(842\) −5.36180e27 −0.868652
\(843\) −8.60120e26 −0.137783
\(844\) 4.05467e27 0.642248
\(845\) −1.26102e27 −0.197507
\(846\) 1.45445e27 0.225259
\(847\) −2.25021e27 −0.344612
\(848\) −1.61058e27 −0.243906
\(849\) 4.77236e27 0.714677
\(850\) 6.83652e27 1.01241
\(851\) 6.81572e27 0.998115
\(852\) −1.09522e27 −0.158607
\(853\) −1.60722e27 −0.230176 −0.115088 0.993355i \(-0.536715\pi\)
−0.115088 + 0.993355i \(0.536715\pi\)
\(854\) 3.11867e27 0.441691
\(855\) −3.23627e26 −0.0453279
\(856\) −7.06692e27 −0.978879
\(857\) 8.25782e27 1.13122 0.565611 0.824672i \(-0.308641\pi\)
0.565611 + 0.824672i \(0.308641\pi\)
\(858\) −3.90475e27 −0.529010
\(859\) −9.31408e27 −1.24797 −0.623986 0.781436i \(-0.714488\pi\)
−0.623986 + 0.781436i \(0.714488\pi\)
\(860\) 5.44299e25 0.00721277
\(861\) 1.28749e27 0.168738
\(862\) 2.46562e27 0.319600
\(863\) 6.65457e27 0.853134 0.426567 0.904456i \(-0.359723\pi\)
0.426567 + 0.904456i \(0.359723\pi\)
\(864\) 1.24391e27 0.157728
\(865\) 2.25128e27 0.282343
\(866\) 4.03024e27 0.499932
\(867\) −4.53234e27 −0.556084
\(868\) −1.05691e27 −0.128263
\(869\) −4.49368e27 −0.539404
\(870\) −1.10878e27 −0.131647
\(871\) −1.19304e28 −1.40114
\(872\) 6.43778e27 0.747876
\(873\) 6.46647e26 0.0743074
\(874\) −3.29352e27 −0.374371
\(875\) 2.92626e27 0.329030
\(876\) 3.46493e26 0.0385394
\(877\) −1.08073e27 −0.118911 −0.0594554 0.998231i \(-0.518936\pi\)
−0.0594554 + 0.998231i \(0.518936\pi\)
\(878\) 1.09514e28 1.19199
\(879\) −3.25285e26 −0.0350243
\(880\) 4.16523e26 0.0443663
\(881\) −1.38540e28 −1.45983 −0.729916 0.683536i \(-0.760441\pi\)
−0.729916 + 0.683536i \(0.760441\pi\)
\(882\) −3.16587e26 −0.0330021
\(883\) 6.76336e27 0.697487 0.348743 0.937218i \(-0.386608\pi\)
0.348743 + 0.937218i \(0.386608\pi\)
\(884\) 9.32447e27 0.951322
\(885\) −3.63987e26 −0.0367388
\(886\) 9.88642e27 0.987231
\(887\) −1.60903e28 −1.58960 −0.794802 0.606869i \(-0.792425\pi\)
−0.794802 + 0.606869i \(0.792425\pi\)
\(888\) −1.09011e28 −1.06549
\(889\) −1.42360e28 −1.37665
\(890\) −5.26047e26 −0.0503292
\(891\) 9.65336e26 0.0913777
\(892\) −7.52311e27 −0.704581
\(893\) −8.55921e27 −0.793131
\(894\) 1.76986e27 0.162268
\(895\) 6.70675e26 0.0608406
\(896\) 1.35819e27 0.121909
\(897\) 5.68541e27 0.504933
\(898\) −4.19729e27 −0.368845
\(899\) 6.04033e27 0.525223
\(900\) 1.70595e27 0.146779
\(901\) 1.16393e28 0.990922
\(902\) 1.98478e27 0.167205
\(903\) −7.53296e26 −0.0627960
\(904\) 1.95408e28 1.61191
\(905\) −1.03636e26 −0.00845963
\(906\) −4.53781e27 −0.366546
\(907\) −4.87731e26 −0.0389862 −0.0194931 0.999810i \(-0.506205\pi\)
−0.0194931 + 0.999810i \(0.506205\pi\)
\(908\) −6.05299e27 −0.478800
\(909\) 3.87060e27 0.302985
\(910\) −2.39545e27 −0.185564
\(911\) 2.04975e28 1.57136 0.785680 0.618634i \(-0.212313\pi\)
0.785680 + 0.618634i \(0.212313\pi\)
\(912\) 2.28155e27 0.173093
\(913\) 2.09367e28 1.57194
\(914\) −1.15636e28 −0.859221
\(915\) −6.87844e26 −0.0505812
\(916\) 5.28790e27 0.384836
\(917\) −2.37960e28 −1.71394
\(918\) 2.80180e27 0.199725
\(919\) 7.36209e27 0.519402 0.259701 0.965689i \(-0.416376\pi\)
0.259701 + 0.965689i \(0.416376\pi\)
\(920\) −1.40023e27 −0.0977720
\(921\) 1.40606e28 0.971713
\(922\) −2.42819e27 −0.166088
\(923\) 1.35227e28 0.915476
\(924\) 3.40559e27 0.228196
\(925\) −2.52511e28 −1.67469
\(926\) −6.03632e27 −0.396249
\(927\) 4.55222e26 0.0295778
\(928\) −2.50797e28 −1.61294
\(929\) −9.18844e27 −0.584915 −0.292457 0.956279i \(-0.594473\pi\)
−0.292457 + 0.956279i \(0.594473\pi\)
\(930\) −2.83327e26 −0.0178525
\(931\) 1.86306e27 0.116200
\(932\) 5.97241e26 0.0368721
\(933\) −7.37881e27 −0.450932
\(934\) −9.32074e27 −0.563839
\(935\) −3.01010e27 −0.180248
\(936\) −9.09331e27 −0.539016
\(937\) 1.55615e28 0.913113 0.456557 0.889694i \(-0.349083\pi\)
0.456557 + 0.889694i \(0.349083\pi\)
\(938\) −1.26468e28 −0.734607
\(939\) 4.01793e27 0.231036
\(940\) −1.13171e27 −0.0644198
\(941\) −4.21264e27 −0.237385 −0.118693 0.992931i \(-0.537870\pi\)
−0.118693 + 0.992931i \(0.537870\pi\)
\(942\) 9.67280e27 0.539596
\(943\) −2.88989e27 −0.159595
\(944\) 2.56609e27 0.140293
\(945\) 5.92206e26 0.0320531
\(946\) −1.16127e27 −0.0622255
\(947\) 1.60906e27 0.0853584 0.0426792 0.999089i \(-0.486411\pi\)
0.0426792 + 0.999089i \(0.486411\pi\)
\(948\) −3.25456e27 −0.170928
\(949\) −4.27817e27 −0.222448
\(950\) 1.22019e28 0.628137
\(951\) 5.40175e27 0.275308
\(952\) 3.17826e28 1.60376
\(953\) 2.00721e28 1.00279 0.501396 0.865218i \(-0.332820\pi\)
0.501396 + 0.865218i \(0.332820\pi\)
\(954\) −3.53008e27 −0.174612
\(955\) 3.50493e27 0.171651
\(956\) −9.30095e27 −0.451001
\(957\) −1.94632e28 −0.934437
\(958\) −2.28898e28 −1.08810
\(959\) 1.74160e28 0.819729
\(960\) 1.84471e27 0.0859709
\(961\) −2.01272e28 −0.928775
\(962\) 4.18598e28 1.91264
\(963\) −6.70874e27 −0.303523
\(964\) 4.73173e27 0.211977
\(965\) 6.64631e27 0.294830
\(966\) 6.02683e27 0.264732
\(967\) 3.69606e28 1.60764 0.803818 0.594876i \(-0.202799\pi\)
0.803818 + 0.594876i \(0.202799\pi\)
\(968\) −8.07830e27 −0.347940
\(969\) −1.64881e28 −0.703228
\(970\) 6.11546e26 0.0258284
\(971\) −2.54073e28 −1.06261 −0.531306 0.847180i \(-0.678299\pi\)
−0.531306 + 0.847180i \(0.678299\pi\)
\(972\) 6.99149e26 0.0289561
\(973\) −2.89166e27 −0.118597
\(974\) 3.52397e27 0.143127
\(975\) −2.10635e28 −0.847201
\(976\) 4.84925e27 0.193153
\(977\) 8.52403e27 0.336238 0.168119 0.985767i \(-0.446231\pi\)
0.168119 + 0.985767i \(0.446231\pi\)
\(978\) 4.75107e27 0.185598
\(979\) −9.23408e27 −0.357238
\(980\) 2.46336e26 0.00943799
\(981\) 6.11149e27 0.231895
\(982\) −2.81120e28 −1.05641
\(983\) 3.09592e28 1.15221 0.576105 0.817376i \(-0.304572\pi\)
0.576105 + 0.817376i \(0.304572\pi\)
\(984\) 4.62211e27 0.170368
\(985\) −5.25015e27 −0.191658
\(986\) −5.64901e28 −2.04240
\(987\) 1.56625e28 0.560853
\(988\) 1.66425e28 0.590238
\(989\) 1.69084e27 0.0593934
\(990\) 9.12937e26 0.0317619
\(991\) −2.94856e28 −1.01604 −0.508020 0.861345i \(-0.669622\pi\)
−0.508020 + 0.861345i \(0.669622\pi\)
\(992\) −6.40864e27 −0.218728
\(993\) 4.55843e27 0.154098
\(994\) 1.43347e28 0.479976
\(995\) −3.11376e27 −0.103268
\(996\) 1.51635e28 0.498121
\(997\) −3.29038e28 −1.07064 −0.535318 0.844650i \(-0.679808\pi\)
−0.535318 + 0.844650i \(0.679808\pi\)
\(998\) 1.36839e28 0.441031
\(999\) −1.03486e28 −0.330377
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 3.20.a.b.1.1 2
3.2 odd 2 9.20.a.c.1.2 2
4.3 odd 2 48.20.a.j.1.1 2
5.2 odd 4 75.20.b.b.49.2 4
5.3 odd 4 75.20.b.b.49.3 4
5.4 even 2 75.20.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.20.a.b.1.1 2 1.1 even 1 trivial
9.20.a.c.1.2 2 3.2 odd 2
48.20.a.j.1.1 2 4.3 odd 2
75.20.a.b.1.2 2 5.4 even 2
75.20.b.b.49.2 4 5.2 odd 4
75.20.b.b.49.3 4 5.3 odd 4