Properties

Label 3.24.a.b.1.2
Level $3$
Weight $24$
Character 3.1
Self dual yes
Analytic conductor $10.056$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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 1.2
Root \(-363.643\) of defining polynomial
Character \(\chi\) \(=\) 3.1

$q$-expansion

\(f(q)\) \(=\) \(q+1563.86 q^{2} -177147. q^{3} -5.94295e6 q^{4} +1.13630e8 q^{5} -2.77033e8 q^{6} +7.85264e9 q^{7} -2.24125e10 q^{8} +3.13811e10 q^{9} +O(q^{10})\) \(q+1563.86 q^{2} -177147. q^{3} -5.94295e6 q^{4} +1.13630e8 q^{5} -2.77033e8 q^{6} +7.85264e9 q^{7} -2.24125e10 q^{8} +3.13811e10 q^{9} +1.77701e11 q^{10} +1.03137e12 q^{11} +1.05278e12 q^{12} +8.08921e12 q^{13} +1.22804e13 q^{14} -2.01292e13 q^{15} +1.48030e13 q^{16} -1.38649e14 q^{17} +4.90756e13 q^{18} -1.41905e14 q^{19} -6.75297e14 q^{20} -1.39107e15 q^{21} +1.61292e15 q^{22} +4.80770e15 q^{23} +3.97031e15 q^{24} +9.90836e14 q^{25} +1.26504e16 q^{26} -5.55906e15 q^{27} -4.66679e16 q^{28} -1.45614e16 q^{29} -3.14792e16 q^{30} -8.10883e16 q^{31} +2.11160e17 q^{32} -1.82704e17 q^{33} -2.16828e17 q^{34} +8.92295e17 q^{35} -1.86496e17 q^{36} +1.19418e18 q^{37} -2.21919e17 q^{38} -1.43298e18 q^{39} -2.54674e18 q^{40} +6.63767e18 q^{41} -2.17544e18 q^{42} -1.06144e19 q^{43} -6.12940e18 q^{44} +3.56583e18 q^{45} +7.51856e18 q^{46} +1.16951e19 q^{47} -2.62232e18 q^{48} +3.42952e19 q^{49} +1.54953e18 q^{50} +2.45613e19 q^{51} -4.80738e19 q^{52} -7.56809e19 q^{53} -8.69359e18 q^{54} +1.17195e20 q^{55} -1.75998e20 q^{56} +2.51380e19 q^{57} -2.27720e19 q^{58} -4.19627e19 q^{59} +1.19627e20 q^{60} +7.45869e19 q^{61} -1.26811e20 q^{62} +2.46424e20 q^{63} +2.06047e20 q^{64} +9.19177e20 q^{65} -2.85724e20 q^{66} -1.29554e21 q^{67} +8.23985e20 q^{68} -8.51670e20 q^{69} +1.39542e21 q^{70} -2.37787e21 q^{71} -7.03329e20 q^{72} -3.03208e21 q^{73} +1.86752e21 q^{74} -1.75524e20 q^{75} +8.43334e20 q^{76} +8.09899e21 q^{77} -2.24098e21 q^{78} +3.75756e20 q^{79} +1.68207e21 q^{80} +9.84771e20 q^{81} +1.03804e22 q^{82} -9.99994e21 q^{83} +8.26707e21 q^{84} -1.57547e22 q^{85} -1.65994e22 q^{86} +2.57951e21 q^{87} -2.31157e22 q^{88} -5.36043e21 q^{89} +5.57645e21 q^{90} +6.35217e22 q^{91} -2.85719e22 q^{92} +1.43645e22 q^{93} +1.82895e22 q^{94} -1.61246e22 q^{95} -3.74063e22 q^{96} +1.11058e22 q^{97} +5.36328e22 q^{98} +3.23655e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1242 q^{2} - 354294 q^{3} - 6458716 q^{4} - 46808820 q^{5} + 220016574 q^{6} - 211963904 q^{7} + 2571869016 q^{8} + 62762119218 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1563.86 0.539949 0.269974 0.962867i \(-0.412985\pi\)
0.269974 + 0.962867i \(0.412985\pi\)
\(3\) −177147. −0.577350
\(4\) −5.94295e6 −0.708455
\(5\) 1.13630e8 1.04073 0.520365 0.853944i \(-0.325796\pi\)
0.520365 + 0.853944i \(0.325796\pi\)
\(6\) −2.77033e8 −0.311740
\(7\) 7.85264e9 1.50103 0.750513 0.660856i \(-0.229807\pi\)
0.750513 + 0.660856i \(0.229807\pi\)
\(8\) −2.24125e10 −0.922478
\(9\) 3.13811e10 0.333333
\(10\) 1.77701e11 0.561941
\(11\) 1.03137e12 1.08993 0.544966 0.838458i \(-0.316543\pi\)
0.544966 + 0.838458i \(0.316543\pi\)
\(12\) 1.05278e12 0.409027
\(13\) 8.08921e12 1.25187 0.625933 0.779877i \(-0.284718\pi\)
0.625933 + 0.779877i \(0.284718\pi\)
\(14\) 1.22804e13 0.810477
\(15\) −2.01292e13 −0.600865
\(16\) 1.48030e13 0.210364
\(17\) −1.38649e14 −0.981193 −0.490597 0.871387i \(-0.663221\pi\)
−0.490597 + 0.871387i \(0.663221\pi\)
\(18\) 4.90756e13 0.179983
\(19\) −1.41905e14 −0.279467 −0.139734 0.990189i \(-0.544625\pi\)
−0.139734 + 0.990189i \(0.544625\pi\)
\(20\) −6.75297e14 −0.737310
\(21\) −1.39107e15 −0.866618
\(22\) 1.61292e15 0.588508
\(23\) 4.80770e15 1.05212 0.526062 0.850446i \(-0.323668\pi\)
0.526062 + 0.850446i \(0.323668\pi\)
\(24\) 3.97031e15 0.532593
\(25\) 9.90836e14 0.0831174
\(26\) 1.26504e16 0.675944
\(27\) −5.55906e15 −0.192450
\(28\) −4.66679e16 −1.06341
\(29\) −1.45614e16 −0.221629 −0.110814 0.993841i \(-0.535346\pi\)
−0.110814 + 0.993841i \(0.535346\pi\)
\(30\) −3.14792e16 −0.324437
\(31\) −8.10883e16 −0.573190 −0.286595 0.958052i \(-0.592523\pi\)
−0.286595 + 0.958052i \(0.592523\pi\)
\(32\) 2.11160e17 1.03606
\(33\) −1.82704e17 −0.629273
\(34\) −2.16828e17 −0.529794
\(35\) 8.92295e17 1.56216
\(36\) −1.86496e17 −0.236152
\(37\) 1.19418e18 1.10344 0.551720 0.834029i \(-0.313972\pi\)
0.551720 + 0.834029i \(0.313972\pi\)
\(38\) −2.21919e17 −0.150898
\(39\) −1.43298e18 −0.722765
\(40\) −2.54674e18 −0.960050
\(41\) 6.63767e18 1.88365 0.941827 0.336098i \(-0.109107\pi\)
0.941827 + 0.336098i \(0.109107\pi\)
\(42\) −2.17544e18 −0.467929
\(43\) −1.06144e19 −1.74184 −0.870920 0.491425i \(-0.836476\pi\)
−0.870920 + 0.491425i \(0.836476\pi\)
\(44\) −6.12940e18 −0.772168
\(45\) 3.56583e18 0.346910
\(46\) 7.51856e18 0.568094
\(47\) 1.16951e19 0.690049 0.345024 0.938594i \(-0.387871\pi\)
0.345024 + 0.938594i \(0.387871\pi\)
\(48\) −2.62232e18 −0.121454
\(49\) 3.42952e19 1.25308
\(50\) 1.54953e18 0.0448791
\(51\) 2.45613e19 0.566492
\(52\) −4.80738e19 −0.886891
\(53\) −7.56809e19 −1.12154 −0.560769 0.827972i \(-0.689494\pi\)
−0.560769 + 0.827972i \(0.689494\pi\)
\(54\) −8.69359e18 −0.103913
\(55\) 1.17195e20 1.13432
\(56\) −1.75998e20 −1.38466
\(57\) 2.51380e19 0.161350
\(58\) −2.27720e19 −0.119668
\(59\) −4.19627e19 −0.181161 −0.0905806 0.995889i \(-0.528872\pi\)
−0.0905806 + 0.995889i \(0.528872\pi\)
\(60\) 1.19627e20 0.425686
\(61\) 7.45869e19 0.219467 0.109734 0.993961i \(-0.465000\pi\)
0.109734 + 0.993961i \(0.465000\pi\)
\(62\) −1.26811e20 −0.309493
\(63\) 2.46424e20 0.500342
\(64\) 2.06047e20 0.349058
\(65\) 9.19177e20 1.30285
\(66\) −2.85724e20 −0.339775
\(67\) −1.29554e21 −1.29595 −0.647976 0.761661i \(-0.724384\pi\)
−0.647976 + 0.761661i \(0.724384\pi\)
\(68\) 8.23985e20 0.695131
\(69\) −8.51670e20 −0.607444
\(70\) 1.39542e21 0.843487
\(71\) −2.37787e21 −1.22101 −0.610503 0.792014i \(-0.709033\pi\)
−0.610503 + 0.792014i \(0.709033\pi\)
\(72\) −7.03329e20 −0.307493
\(73\) −3.03208e21 −1.13117 −0.565585 0.824690i \(-0.691349\pi\)
−0.565585 + 0.824690i \(0.691349\pi\)
\(74\) 1.86752e21 0.595801
\(75\) −1.75524e20 −0.0479878
\(76\) 8.43334e20 0.197990
\(77\) 8.09899e21 1.63602
\(78\) −2.24098e21 −0.390256
\(79\) 3.75756e20 0.0565190 0.0282595 0.999601i \(-0.491004\pi\)
0.0282595 + 0.999601i \(0.491004\pi\)
\(80\) 1.68207e21 0.218932
\(81\) 9.84771e20 0.111111
\(82\) 1.03804e22 1.01708
\(83\) −9.99994e21 −0.852314 −0.426157 0.904649i \(-0.640133\pi\)
−0.426157 + 0.904649i \(0.640133\pi\)
\(84\) 8.26707e21 0.613960
\(85\) −1.57547e22 −1.02116
\(86\) −1.65994e22 −0.940505
\(87\) 2.57951e21 0.127957
\(88\) −2.31157e22 −1.00544
\(89\) −5.36043e21 −0.204746 −0.102373 0.994746i \(-0.532643\pi\)
−0.102373 + 0.994746i \(0.532643\pi\)
\(90\) 5.57645e21 0.187314
\(91\) 6.35217e22 1.87908
\(92\) −2.85719e22 −0.745383
\(93\) 1.43645e22 0.330931
\(94\) 1.82895e22 0.372591
\(95\) −1.61246e22 −0.290850
\(96\) −3.74063e22 −0.598172
\(97\) 1.11058e22 0.157643 0.0788214 0.996889i \(-0.474884\pi\)
0.0788214 + 0.996889i \(0.474884\pi\)
\(98\) 5.36328e22 0.676598
\(99\) 3.23655e22 0.363311
\(100\) −5.88849e21 −0.0588849
\(101\) 8.99589e22 0.802321 0.401160 0.916008i \(-0.368607\pi\)
0.401160 + 0.916008i \(0.368607\pi\)
\(102\) 3.84104e22 0.305877
\(103\) −8.83460e22 −0.628867 −0.314434 0.949279i \(-0.601815\pi\)
−0.314434 + 0.949279i \(0.601815\pi\)
\(104\) −1.81300e23 −1.15482
\(105\) −1.58067e23 −0.901914
\(106\) −1.18354e23 −0.605573
\(107\) 2.32082e23 1.06593 0.532963 0.846138i \(-0.321078\pi\)
0.532963 + 0.846138i \(0.321078\pi\)
\(108\) 3.30372e22 0.136342
\(109\) −2.76863e23 −1.02768 −0.513842 0.857885i \(-0.671778\pi\)
−0.513842 + 0.857885i \(0.671778\pi\)
\(110\) 1.83276e23 0.612477
\(111\) −2.11545e23 −0.637071
\(112\) 1.16243e23 0.315762
\(113\) 3.71795e23 0.911803 0.455901 0.890030i \(-0.349317\pi\)
0.455901 + 0.890030i \(0.349317\pi\)
\(114\) 3.93123e22 0.0871210
\(115\) 5.46299e23 1.09498
\(116\) 8.65376e22 0.157014
\(117\) 2.53848e23 0.417289
\(118\) −6.56237e22 −0.0978178
\(119\) −1.08876e24 −1.47280
\(120\) 4.51147e23 0.554285
\(121\) 1.68298e23 0.187952
\(122\) 1.16643e23 0.118501
\(123\) −1.17584e24 −1.08753
\(124\) 4.81904e23 0.406080
\(125\) −1.24199e24 −0.954227
\(126\) 3.85373e23 0.270159
\(127\) −1.60553e24 −1.02772 −0.513861 0.857874i \(-0.671785\pi\)
−0.513861 + 0.857874i \(0.671785\pi\)
\(128\) −1.44911e24 −0.847591
\(129\) 1.88031e24 1.00565
\(130\) 1.43746e24 0.703474
\(131\) 2.07814e24 0.931228 0.465614 0.884988i \(-0.345834\pi\)
0.465614 + 0.884988i \(0.345834\pi\)
\(132\) 1.08580e24 0.445811
\(133\) −1.11433e24 −0.419487
\(134\) −2.02603e24 −0.699748
\(135\) −6.31676e23 −0.200288
\(136\) 3.10748e24 0.905130
\(137\) 4.86669e24 1.30301 0.651504 0.758645i \(-0.274138\pi\)
0.651504 + 0.758645i \(0.274138\pi\)
\(138\) −1.33189e24 −0.327989
\(139\) 9.05962e23 0.205324 0.102662 0.994716i \(-0.467264\pi\)
0.102662 + 0.994716i \(0.467264\pi\)
\(140\) −5.30287e24 −1.10672
\(141\) −2.07176e24 −0.398400
\(142\) −3.71866e24 −0.659281
\(143\) 8.34299e24 1.36445
\(144\) 4.64535e23 0.0701213
\(145\) −1.65461e24 −0.230655
\(146\) −4.74175e24 −0.610774
\(147\) −6.07529e24 −0.723465
\(148\) −7.09694e24 −0.781738
\(149\) 9.86509e24 1.00568 0.502839 0.864380i \(-0.332289\pi\)
0.502839 + 0.864380i \(0.332289\pi\)
\(150\) −2.74494e23 −0.0259110
\(151\) −4.82721e24 −0.422145 −0.211072 0.977470i \(-0.567696\pi\)
−0.211072 + 0.977470i \(0.567696\pi\)
\(152\) 3.18045e24 0.257802
\(153\) −4.35096e24 −0.327064
\(154\) 1.26657e25 0.883365
\(155\) −9.21406e24 −0.596536
\(156\) 8.51613e24 0.512047
\(157\) 1.87063e24 0.104506 0.0522529 0.998634i \(-0.483360\pi\)
0.0522529 + 0.998634i \(0.483360\pi\)
\(158\) 5.87629e23 0.0305174
\(159\) 1.34067e25 0.647520
\(160\) 2.39941e25 1.07826
\(161\) 3.77531e25 1.57927
\(162\) 1.54004e24 0.0599943
\(163\) −3.92032e25 −1.42286 −0.711432 0.702755i \(-0.751953\pi\)
−0.711432 + 0.702755i \(0.751953\pi\)
\(164\) −3.94473e25 −1.33448
\(165\) −2.07607e25 −0.654902
\(166\) −1.56385e25 −0.460206
\(167\) −4.64021e25 −1.27438 −0.637188 0.770708i \(-0.719903\pi\)
−0.637188 + 0.770708i \(0.719903\pi\)
\(168\) 3.11774e25 0.799436
\(169\) 2.36815e25 0.567168
\(170\) −2.46381e25 −0.551372
\(171\) −4.45312e24 −0.0931557
\(172\) 6.30809e25 1.23402
\(173\) −4.74788e25 −0.868900 −0.434450 0.900696i \(-0.643057\pi\)
−0.434450 + 0.900696i \(0.643057\pi\)
\(174\) 4.03398e24 0.0690904
\(175\) 7.78068e24 0.124761
\(176\) 1.52674e25 0.229282
\(177\) 7.43356e24 0.104593
\(178\) −8.38295e24 −0.110552
\(179\) 5.97561e25 0.738878 0.369439 0.929255i \(-0.379550\pi\)
0.369439 + 0.929255i \(0.379550\pi\)
\(180\) −2.11915e25 −0.245770
\(181\) 3.92753e25 0.427381 0.213691 0.976901i \(-0.431452\pi\)
0.213691 + 0.976901i \(0.431452\pi\)
\(182\) 9.93389e25 1.01461
\(183\) −1.32128e25 −0.126709
\(184\) −1.07753e26 −0.970562
\(185\) 1.35694e26 1.14838
\(186\) 2.24641e25 0.178686
\(187\) −1.42999e26 −1.06943
\(188\) −6.95036e25 −0.488869
\(189\) −4.36533e25 −0.288873
\(190\) −2.52167e25 −0.157044
\(191\) −9.94591e25 −0.583124 −0.291562 0.956552i \(-0.594175\pi\)
−0.291562 + 0.956552i \(0.594175\pi\)
\(192\) −3.65007e25 −0.201529
\(193\) 2.11752e26 1.10133 0.550667 0.834725i \(-0.314373\pi\)
0.550667 + 0.834725i \(0.314373\pi\)
\(194\) 1.73679e25 0.0851190
\(195\) −1.62829e26 −0.752203
\(196\) −2.03815e26 −0.887750
\(197\) 2.43182e26 0.999009 0.499505 0.866311i \(-0.333515\pi\)
0.499505 + 0.866311i \(0.333515\pi\)
\(198\) 5.06151e25 0.196169
\(199\) −2.49955e26 −0.914220 −0.457110 0.889410i \(-0.651115\pi\)
−0.457110 + 0.889410i \(0.651115\pi\)
\(200\) −2.22072e25 −0.0766740
\(201\) 2.29500e26 0.748218
\(202\) 1.40683e26 0.433212
\(203\) −1.14345e26 −0.332670
\(204\) −1.45967e26 −0.401334
\(205\) 7.54238e26 1.96037
\(206\) −1.38161e26 −0.339556
\(207\) 1.50871e26 0.350708
\(208\) 1.19745e26 0.263347
\(209\) −1.46357e26 −0.304600
\(210\) −2.47195e26 −0.486988
\(211\) −2.68608e26 −0.501038 −0.250519 0.968112i \(-0.580601\pi\)
−0.250519 + 0.968112i \(0.580601\pi\)
\(212\) 4.49768e26 0.794560
\(213\) 4.21233e26 0.704948
\(214\) 3.62943e26 0.575546
\(215\) −1.20611e27 −1.81278
\(216\) 1.24593e26 0.177531
\(217\) −6.36757e26 −0.860373
\(218\) −4.32974e26 −0.554897
\(219\) 5.37124e26 0.653081
\(220\) −6.96483e26 −0.803618
\(221\) −1.12156e27 −1.22832
\(222\) −3.30826e26 −0.343986
\(223\) 4.28148e25 0.0422754 0.0211377 0.999777i \(-0.493271\pi\)
0.0211377 + 0.999777i \(0.493271\pi\)
\(224\) 1.65816e27 1.55516
\(225\) 3.10935e25 0.0277058
\(226\) 5.81434e26 0.492327
\(227\) 1.04141e27 0.838154 0.419077 0.907951i \(-0.362354\pi\)
0.419077 + 0.907951i \(0.362354\pi\)
\(228\) −1.49394e26 −0.114310
\(229\) 3.17314e26 0.230877 0.115439 0.993315i \(-0.463173\pi\)
0.115439 + 0.993315i \(0.463173\pi\)
\(230\) 8.54334e26 0.591232
\(231\) −1.43471e27 −0.944554
\(232\) 3.26358e26 0.204448
\(233\) 1.07463e27 0.640715 0.320357 0.947297i \(-0.396197\pi\)
0.320357 + 0.947297i \(0.396197\pi\)
\(234\) 3.96983e26 0.225315
\(235\) 1.32892e27 0.718154
\(236\) 2.49382e26 0.128345
\(237\) −6.65640e25 −0.0326312
\(238\) −1.70267e27 −0.795235
\(239\) −2.91187e27 −1.29598 −0.647988 0.761651i \(-0.724389\pi\)
−0.647988 + 0.761651i \(0.724389\pi\)
\(240\) −2.97974e26 −0.126400
\(241\) −1.77639e27 −0.718362 −0.359181 0.933268i \(-0.616944\pi\)
−0.359181 + 0.933268i \(0.616944\pi\)
\(242\) 2.63194e26 0.101484
\(243\) −1.74449e26 −0.0641500
\(244\) −4.43266e26 −0.155483
\(245\) 3.89696e27 1.30411
\(246\) −1.83885e27 −0.587210
\(247\) −1.14790e27 −0.349855
\(248\) 1.81739e27 0.528756
\(249\) 1.77146e27 0.492084
\(250\) −1.94229e27 −0.515234
\(251\) −4.39892e27 −1.11455 −0.557274 0.830329i \(-0.688153\pi\)
−0.557274 + 0.830329i \(0.688153\pi\)
\(252\) −1.46449e27 −0.354470
\(253\) 4.95853e27 1.14674
\(254\) −2.51082e27 −0.554917
\(255\) 2.79090e27 0.589565
\(256\) −3.99465e27 −0.806714
\(257\) 3.06305e27 0.591457 0.295729 0.955272i \(-0.404438\pi\)
0.295729 + 0.955272i \(0.404438\pi\)
\(258\) 2.94054e27 0.543001
\(259\) 9.37744e27 1.65629
\(260\) −5.46263e27 −0.923013
\(261\) −4.56952e26 −0.0738762
\(262\) 3.24992e27 0.502815
\(263\) 4.37920e27 0.648491 0.324246 0.945973i \(-0.394890\pi\)
0.324246 + 0.945973i \(0.394890\pi\)
\(264\) 4.09487e27 0.580490
\(265\) −8.59962e27 −1.16722
\(266\) −1.74265e27 −0.226502
\(267\) 9.49584e26 0.118210
\(268\) 7.69930e27 0.918124
\(269\) −1.36648e28 −1.56118 −0.780588 0.625046i \(-0.785080\pi\)
−0.780588 + 0.625046i \(0.785080\pi\)
\(270\) −9.87852e26 −0.108146
\(271\) −7.06980e27 −0.741755 −0.370878 0.928682i \(-0.620943\pi\)
−0.370878 + 0.928682i \(0.620943\pi\)
\(272\) −2.05243e27 −0.206408
\(273\) −1.12527e28 −1.08489
\(274\) 7.61081e27 0.703558
\(275\) 1.02192e27 0.0905923
\(276\) 5.06143e27 0.430347
\(277\) 1.02188e28 0.833460 0.416730 0.909030i \(-0.363176\pi\)
0.416730 + 0.909030i \(0.363176\pi\)
\(278\) 1.41680e27 0.110865
\(279\) −2.54464e27 −0.191063
\(280\) −1.99986e28 −1.44106
\(281\) −8.03841e26 −0.0555965 −0.0277983 0.999614i \(-0.508850\pi\)
−0.0277983 + 0.999614i \(0.508850\pi\)
\(282\) −3.23994e27 −0.215116
\(283\) −2.15414e28 −1.37318 −0.686592 0.727043i \(-0.740894\pi\)
−0.686592 + 0.727043i \(0.740894\pi\)
\(284\) 1.41316e28 0.865028
\(285\) 2.85643e27 0.167922
\(286\) 1.30473e28 0.736733
\(287\) 5.21232e28 2.82741
\(288\) 6.62642e27 0.345355
\(289\) −7.43986e26 −0.0372597
\(290\) −2.58758e27 −0.124542
\(291\) −1.96735e27 −0.0910151
\(292\) 1.80195e28 0.801383
\(293\) −2.33106e28 −0.996725 −0.498363 0.866969i \(-0.666065\pi\)
−0.498363 + 0.866969i \(0.666065\pi\)
\(294\) −9.50089e27 −0.390634
\(295\) −4.76822e27 −0.188540
\(296\) −2.67645e28 −1.01790
\(297\) −5.73346e27 −0.209758
\(298\) 1.54276e28 0.543015
\(299\) 3.88905e28 1.31712
\(300\) 1.04313e27 0.0339972
\(301\) −8.33511e28 −2.61455
\(302\) −7.54907e27 −0.227936
\(303\) −1.59359e28 −0.463220
\(304\) −2.10062e27 −0.0587898
\(305\) 8.47531e27 0.228406
\(306\) −6.80428e27 −0.176598
\(307\) 6.97395e28 1.74336 0.871681 0.490074i \(-0.163030\pi\)
0.871681 + 0.490074i \(0.163030\pi\)
\(308\) −4.81319e28 −1.15904
\(309\) 1.56502e28 0.363077
\(310\) −1.44095e28 −0.322099
\(311\) −4.01460e28 −0.864765 −0.432382 0.901690i \(-0.642327\pi\)
−0.432382 + 0.901690i \(0.642327\pi\)
\(312\) 3.21167e28 0.666735
\(313\) 7.08075e28 1.41683 0.708417 0.705794i \(-0.249409\pi\)
0.708417 + 0.705794i \(0.249409\pi\)
\(314\) 2.92539e27 0.0564278
\(315\) 2.80012e28 0.520720
\(316\) −2.23310e27 −0.0400412
\(317\) −3.21150e28 −0.555298 −0.277649 0.960683i \(-0.589555\pi\)
−0.277649 + 0.960683i \(0.589555\pi\)
\(318\) 2.09661e28 0.349628
\(319\) −1.50182e28 −0.241560
\(320\) 2.34132e28 0.363275
\(321\) −4.11126e28 −0.615413
\(322\) 5.90406e28 0.852723
\(323\) 1.96750e28 0.274211
\(324\) −5.85245e27 −0.0787172
\(325\) 8.01509e27 0.104052
\(326\) −6.13082e28 −0.768274
\(327\) 4.90454e28 0.593333
\(328\) −1.48767e29 −1.73763
\(329\) 9.18377e28 1.03578
\(330\) −3.24668e28 −0.353614
\(331\) 8.40986e28 0.884640 0.442320 0.896857i \(-0.354155\pi\)
0.442320 + 0.896857i \(0.354155\pi\)
\(332\) 5.94292e28 0.603826
\(333\) 3.74745e28 0.367813
\(334\) −7.25663e28 −0.688098
\(335\) −1.47212e29 −1.34874
\(336\) −2.05921e28 −0.182305
\(337\) −1.63996e29 −1.40311 −0.701553 0.712618i \(-0.747510\pi\)
−0.701553 + 0.712618i \(0.747510\pi\)
\(338\) 3.70345e28 0.306242
\(339\) −6.58623e28 −0.526430
\(340\) 9.36294e28 0.723444
\(341\) −8.36322e28 −0.624738
\(342\) −6.96406e27 −0.0502993
\(343\) 5.43907e28 0.379877
\(344\) 2.37896e29 1.60681
\(345\) −9.67752e28 −0.632185
\(346\) −7.42502e28 −0.469162
\(347\) 6.78465e28 0.414704 0.207352 0.978266i \(-0.433515\pi\)
0.207352 + 0.978266i \(0.433515\pi\)
\(348\) −1.53299e28 −0.0906520
\(349\) 1.82821e29 1.04601 0.523003 0.852331i \(-0.324812\pi\)
0.523003 + 0.852331i \(0.324812\pi\)
\(350\) 1.21679e28 0.0673647
\(351\) −4.49684e28 −0.240922
\(352\) 2.17784e29 1.12924
\(353\) 1.44105e29 0.723218 0.361609 0.932330i \(-0.382228\pi\)
0.361609 + 0.932330i \(0.382228\pi\)
\(354\) 1.16250e28 0.0564751
\(355\) −2.70198e29 −1.27074
\(356\) 3.18568e28 0.145053
\(357\) 1.92871e29 0.850319
\(358\) 9.34501e28 0.398956
\(359\) −3.54388e28 −0.146519 −0.0732593 0.997313i \(-0.523340\pi\)
−0.0732593 + 0.997313i \(0.523340\pi\)
\(360\) −7.99193e28 −0.320017
\(361\) −2.37693e29 −0.921898
\(362\) 6.14210e28 0.230764
\(363\) −2.98135e28 −0.108514
\(364\) −3.77506e29 −1.33125
\(365\) −3.44535e29 −1.17724
\(366\) −2.06630e28 −0.0684166
\(367\) 1.57063e29 0.503981 0.251990 0.967730i \(-0.418915\pi\)
0.251990 + 0.967730i \(0.418915\pi\)
\(368\) 7.11686e28 0.221329
\(369\) 2.08297e29 0.627885
\(370\) 2.12207e29 0.620068
\(371\) −5.94295e29 −1.68346
\(372\) −8.53678e28 −0.234450
\(373\) 4.68656e29 1.24796 0.623982 0.781438i \(-0.285514\pi\)
0.623982 + 0.781438i \(0.285514\pi\)
\(374\) −2.23630e29 −0.577440
\(375\) 2.20014e29 0.550923
\(376\) −2.62118e29 −0.636555
\(377\) −1.17790e29 −0.277449
\(378\) −6.82676e28 −0.155976
\(379\) 4.57537e28 0.101409 0.0507044 0.998714i \(-0.483853\pi\)
0.0507044 + 0.998714i \(0.483853\pi\)
\(380\) 9.58280e28 0.206054
\(381\) 2.84415e29 0.593355
\(382\) −1.55540e29 −0.314857
\(383\) 4.53749e29 0.891312 0.445656 0.895204i \(-0.352970\pi\)
0.445656 + 0.895204i \(0.352970\pi\)
\(384\) 2.56705e29 0.489357
\(385\) 9.20288e29 1.70265
\(386\) 3.31151e29 0.594664
\(387\) −3.33091e29 −0.580613
\(388\) −6.60011e28 −0.111683
\(389\) 3.59874e29 0.591195 0.295597 0.955313i \(-0.404481\pi\)
0.295597 + 0.955313i \(0.404481\pi\)
\(390\) −2.54642e29 −0.406151
\(391\) −6.66583e29 −1.03234
\(392\) −7.68642e29 −1.15594
\(393\) −3.68137e29 −0.537644
\(394\) 3.80302e29 0.539414
\(395\) 4.26971e28 0.0588210
\(396\) −1.92347e29 −0.257389
\(397\) −9.62669e28 −0.125137 −0.0625686 0.998041i \(-0.519929\pi\)
−0.0625686 + 0.998041i \(0.519929\pi\)
\(398\) −3.90894e29 −0.493632
\(399\) 1.97400e29 0.242191
\(400\) 1.46674e28 0.0174849
\(401\) 4.97920e29 0.576766 0.288383 0.957515i \(-0.406882\pi\)
0.288383 + 0.957515i \(0.406882\pi\)
\(402\) 3.58906e29 0.404000
\(403\) −6.55941e29 −0.717557
\(404\) −5.34621e29 −0.568408
\(405\) 1.11899e29 0.115637
\(406\) −1.78820e29 −0.179625
\(407\) 1.23164e30 1.20267
\(408\) −5.50481e29 −0.522577
\(409\) −1.48649e30 −1.37197 −0.685985 0.727616i \(-0.740628\pi\)
−0.685985 + 0.727616i \(0.740628\pi\)
\(410\) 1.17952e30 1.05850
\(411\) −8.62119e29 −0.752292
\(412\) 5.25036e29 0.445524
\(413\) −3.29518e29 −0.271928
\(414\) 2.35941e29 0.189365
\(415\) −1.13629e30 −0.887028
\(416\) 1.70812e30 1.29701
\(417\) −1.60489e29 −0.118544
\(418\) −2.28881e29 −0.164469
\(419\) −5.07194e29 −0.354579 −0.177289 0.984159i \(-0.556733\pi\)
−0.177289 + 0.984159i \(0.556733\pi\)
\(420\) 9.39387e29 0.638966
\(421\) 2.56153e30 1.69533 0.847666 0.530531i \(-0.178007\pi\)
0.847666 + 0.530531i \(0.178007\pi\)
\(422\) −4.20065e29 −0.270535
\(423\) 3.67006e29 0.230016
\(424\) 1.69620e30 1.03459
\(425\) −1.37379e29 −0.0815542
\(426\) 6.58749e29 0.380636
\(427\) 5.85704e29 0.329426
\(428\) −1.37925e30 −0.755161
\(429\) −1.47794e30 −0.787765
\(430\) −1.88619e30 −0.978811
\(431\) −1.21646e30 −0.614625 −0.307312 0.951609i \(-0.599430\pi\)
−0.307312 + 0.951609i \(0.599430\pi\)
\(432\) −8.22910e28 −0.0404846
\(433\) 1.79430e30 0.859577 0.429789 0.902930i \(-0.358588\pi\)
0.429789 + 0.902930i \(0.358588\pi\)
\(434\) −9.95798e29 −0.464558
\(435\) 2.93109e29 0.133169
\(436\) 1.64538e30 0.728068
\(437\) −6.82236e29 −0.294034
\(438\) 8.39986e29 0.352630
\(439\) −3.97063e30 −1.62374 −0.811871 0.583837i \(-0.801551\pi\)
−0.811871 + 0.583837i \(0.801551\pi\)
\(440\) −2.62663e30 −1.04639
\(441\) 1.07622e30 0.417693
\(442\) −1.75397e30 −0.663231
\(443\) −1.28735e30 −0.474301 −0.237151 0.971473i \(-0.576213\pi\)
−0.237151 + 0.971473i \(0.576213\pi\)
\(444\) 1.25720e30 0.451337
\(445\) −6.09105e29 −0.213085
\(446\) 6.69562e28 0.0228265
\(447\) −1.74757e30 −0.580628
\(448\) 1.61802e30 0.523945
\(449\) 3.17012e30 1.00056 0.500279 0.865864i \(-0.333231\pi\)
0.500279 + 0.865864i \(0.333231\pi\)
\(450\) 4.86258e28 0.0149597
\(451\) 6.84590e30 2.05305
\(452\) −2.20956e30 −0.645971
\(453\) 8.55126e29 0.243725
\(454\) 1.62862e30 0.452560
\(455\) 7.21796e30 1.95562
\(456\) −5.63407e29 −0.148842
\(457\) −6.89795e30 −1.77699 −0.888493 0.458891i \(-0.848247\pi\)
−0.888493 + 0.458891i \(0.848247\pi\)
\(458\) 4.96234e29 0.124662
\(459\) 7.70759e29 0.188831
\(460\) −3.24663e30 −0.775742
\(461\) −1.25314e30 −0.292038 −0.146019 0.989282i \(-0.546646\pi\)
−0.146019 + 0.989282i \(0.546646\pi\)
\(462\) −2.24369e30 −0.510011
\(463\) 4.87399e30 1.08070 0.540348 0.841442i \(-0.318293\pi\)
0.540348 + 0.841442i \(0.318293\pi\)
\(464\) −2.15553e29 −0.0466227
\(465\) 1.63224e30 0.344410
\(466\) 1.68057e30 0.345953
\(467\) 4.35835e30 0.875343 0.437671 0.899135i \(-0.355803\pi\)
0.437671 + 0.899135i \(0.355803\pi\)
\(468\) −1.50861e30 −0.295630
\(469\) −1.01734e31 −1.94526
\(470\) 2.07824e30 0.387766
\(471\) −3.31376e29 −0.0603365
\(472\) 9.40490e29 0.167117
\(473\) −1.09474e31 −1.89849
\(474\) −1.04097e29 −0.0176192
\(475\) −1.40604e29 −0.0232286
\(476\) 6.47046e30 1.04341
\(477\) −2.37495e30 −0.373846
\(478\) −4.55376e30 −0.699760
\(479\) 4.80333e30 0.720582 0.360291 0.932840i \(-0.382677\pi\)
0.360291 + 0.932840i \(0.382677\pi\)
\(480\) −4.25048e30 −0.622535
\(481\) 9.65995e30 1.38136
\(482\) −2.77803e30 −0.387879
\(483\) −6.68785e30 −0.911790
\(484\) −1.00019e30 −0.133156
\(485\) 1.26195e30 0.164063
\(486\) −2.72814e29 −0.0346377
\(487\) −3.71714e30 −0.460920 −0.230460 0.973082i \(-0.574023\pi\)
−0.230460 + 0.973082i \(0.574023\pi\)
\(488\) −1.67168e30 −0.202454
\(489\) 6.94472e30 0.821491
\(490\) 6.09429e30 0.704155
\(491\) 7.53933e30 0.850933 0.425466 0.904974i \(-0.360110\pi\)
0.425466 + 0.904974i \(0.360110\pi\)
\(492\) 6.98798e30 0.770465
\(493\) 2.01892e30 0.217460
\(494\) −1.79515e30 −0.188904
\(495\) 3.67770e30 0.378108
\(496\) −1.20035e30 −0.120579
\(497\) −1.86726e31 −1.83276
\(498\) 2.77031e30 0.265700
\(499\) 3.09263e30 0.289849 0.144925 0.989443i \(-0.453706\pi\)
0.144925 + 0.989443i \(0.453706\pi\)
\(500\) 7.38106e30 0.676027
\(501\) 8.21999e30 0.735762
\(502\) −6.87930e30 −0.601799
\(503\) −4.43628e30 −0.379304 −0.189652 0.981851i \(-0.560736\pi\)
−0.189652 + 0.981851i \(0.560736\pi\)
\(504\) −5.52299e30 −0.461555
\(505\) 1.02220e31 0.834999
\(506\) 7.75444e30 0.619183
\(507\) −4.19510e30 −0.327455
\(508\) 9.54158e30 0.728094
\(509\) −2.05052e31 −1.52971 −0.764855 0.644203i \(-0.777189\pi\)
−0.764855 + 0.644203i \(0.777189\pi\)
\(510\) 4.36457e30 0.318335
\(511\) −2.38098e31 −1.69791
\(512\) 5.90893e30 0.412007
\(513\) 7.88858e29 0.0537835
\(514\) 4.79018e30 0.319357
\(515\) −1.00388e31 −0.654481
\(516\) −1.11746e31 −0.712459
\(517\) 1.20620e31 0.752106
\(518\) 1.46650e31 0.894313
\(519\) 8.41073e30 0.501660
\(520\) −2.06011e31 −1.20185
\(521\) −2.13126e31 −1.21620 −0.608098 0.793862i \(-0.708067\pi\)
−0.608098 + 0.793862i \(0.708067\pi\)
\(522\) −7.14608e29 −0.0398894
\(523\) 5.37551e30 0.293528 0.146764 0.989172i \(-0.453114\pi\)
0.146764 + 0.989172i \(0.453114\pi\)
\(524\) −1.23503e31 −0.659733
\(525\) −1.37832e30 −0.0720310
\(526\) 6.84846e30 0.350152
\(527\) 1.12428e31 0.562410
\(528\) −2.70458e30 −0.132376
\(529\) 2.23351e30 0.106966
\(530\) −1.34486e31 −0.630238
\(531\) −1.31683e30 −0.0603870
\(532\) 6.62240e30 0.297188
\(533\) 5.36935e31 2.35808
\(534\) 1.48502e30 0.0638273
\(535\) 2.63714e31 1.10934
\(536\) 2.90362e31 1.19549
\(537\) −1.05856e31 −0.426591
\(538\) −2.13698e31 −0.842955
\(539\) 3.53711e31 1.36577
\(540\) 3.75402e30 0.141895
\(541\) −3.74692e30 −0.138646 −0.0693228 0.997594i \(-0.522084\pi\)
−0.0693228 + 0.997594i \(0.522084\pi\)
\(542\) −1.10562e31 −0.400510
\(543\) −6.95750e30 −0.246749
\(544\) −2.92771e31 −1.01658
\(545\) −3.14599e31 −1.06954
\(546\) −1.75976e31 −0.585785
\(547\) −3.43148e30 −0.111848 −0.0559239 0.998435i \(-0.517810\pi\)
−0.0559239 + 0.998435i \(0.517810\pi\)
\(548\) −2.89225e31 −0.923123
\(549\) 2.34062e30 0.0731557
\(550\) 1.59814e30 0.0489152
\(551\) 2.06633e30 0.0619379
\(552\) 1.90881e31 0.560354
\(553\) 2.95068e30 0.0848364
\(554\) 1.59808e31 0.450026
\(555\) −2.40378e31 −0.663019
\(556\) −5.38409e30 −0.145463
\(557\) 3.40780e31 0.901860 0.450930 0.892559i \(-0.351092\pi\)
0.450930 + 0.892559i \(0.351092\pi\)
\(558\) −3.97945e30 −0.103164
\(559\) −8.58622e31 −2.18055
\(560\) 1.32087e31 0.328622
\(561\) 2.53318e31 0.617438
\(562\) −1.25709e30 −0.0300193
\(563\) −2.39242e31 −0.559747 −0.279873 0.960037i \(-0.590292\pi\)
−0.279873 + 0.960037i \(0.590292\pi\)
\(564\) 1.23124e31 0.282248
\(565\) 4.22470e31 0.948940
\(566\) −3.36876e31 −0.741450
\(567\) 7.73305e30 0.166781
\(568\) 5.32942e31 1.12635
\(569\) −1.92696e31 −0.399101 −0.199550 0.979888i \(-0.563948\pi\)
−0.199550 + 0.979888i \(0.563948\pi\)
\(570\) 4.46706e30 0.0906694
\(571\) 1.76715e31 0.351526 0.175763 0.984433i \(-0.443761\pi\)
0.175763 + 0.984433i \(0.443761\pi\)
\(572\) −4.95820e31 −0.966651
\(573\) 1.76189e31 0.336667
\(574\) 8.15133e31 1.52666
\(575\) 4.76364e30 0.0874498
\(576\) 6.46598e30 0.116353
\(577\) 5.86874e31 1.03520 0.517599 0.855624i \(-0.326826\pi\)
0.517599 + 0.855624i \(0.326826\pi\)
\(578\) −1.16349e30 −0.0201183
\(579\) −3.75113e31 −0.635856
\(580\) 9.83327e30 0.163409
\(581\) −7.85259e31 −1.27934
\(582\) −3.07667e30 −0.0491435
\(583\) −7.80552e31 −1.22240
\(584\) 6.79566e31 1.04348
\(585\) 2.88447e31 0.434284
\(586\) −3.64545e31 −0.538181
\(587\) 6.89187e31 0.997697 0.498849 0.866689i \(-0.333756\pi\)
0.498849 + 0.866689i \(0.333756\pi\)
\(588\) 3.61051e31 0.512542
\(589\) 1.15068e31 0.160188
\(590\) −7.45682e30 −0.101802
\(591\) −4.30789e31 −0.576778
\(592\) 1.76775e31 0.232124
\(593\) −7.35782e31 −0.947589 −0.473795 0.880635i \(-0.657116\pi\)
−0.473795 + 0.880635i \(0.657116\pi\)
\(594\) −8.96632e30 −0.113258
\(595\) −1.23716e32 −1.53278
\(596\) −5.86278e31 −0.712478
\(597\) 4.42787e31 0.527825
\(598\) 6.08193e31 0.711177
\(599\) 1.35998e32 1.56000 0.779999 0.625780i \(-0.215219\pi\)
0.779999 + 0.625780i \(0.215219\pi\)
\(600\) 3.93393e30 0.0442677
\(601\) −7.89659e31 −0.871732 −0.435866 0.900011i \(-0.643558\pi\)
−0.435866 + 0.900011i \(0.643558\pi\)
\(602\) −1.30349e32 −1.41172
\(603\) −4.06553e31 −0.431984
\(604\) 2.86879e31 0.299070
\(605\) 1.91237e31 0.195607
\(606\) −2.49216e31 −0.250115
\(607\) −4.65196e31 −0.458106 −0.229053 0.973414i \(-0.573563\pi\)
−0.229053 + 0.973414i \(0.573563\pi\)
\(608\) −2.99646e31 −0.289546
\(609\) 2.02559e31 0.192067
\(610\) 1.32542e31 0.123327
\(611\) 9.46044e31 0.863849
\(612\) 2.58575e31 0.231710
\(613\) 1.41819e32 1.24721 0.623606 0.781739i \(-0.285667\pi\)
0.623606 + 0.781739i \(0.285667\pi\)
\(614\) 1.09063e32 0.941326
\(615\) −1.33611e32 −1.13182
\(616\) −1.81519e32 −1.50919
\(617\) 2.38519e32 1.94645 0.973227 0.229848i \(-0.0738228\pi\)
0.973227 + 0.229848i \(0.0738228\pi\)
\(618\) 2.44748e31 0.196043
\(619\) 1.46333e32 1.15054 0.575268 0.817965i \(-0.304898\pi\)
0.575268 + 0.817965i \(0.304898\pi\)
\(620\) 5.47587e31 0.422619
\(621\) −2.67263e31 −0.202481
\(622\) −6.27827e31 −0.466929
\(623\) −4.20935e31 −0.307328
\(624\) −2.12125e31 −0.152044
\(625\) −1.52938e32 −1.07621
\(626\) 1.10733e32 0.765018
\(627\) 2.59266e31 0.175861
\(628\) −1.11170e31 −0.0740377
\(629\) −1.65572e32 −1.08269
\(630\) 4.37899e31 0.281162
\(631\) 2.57610e32 1.62414 0.812072 0.583557i \(-0.198339\pi\)
0.812072 + 0.583557i \(0.198339\pi\)
\(632\) −8.42164e30 −0.0521375
\(633\) 4.75831e31 0.289274
\(634\) −5.02233e31 −0.299833
\(635\) −1.82436e32 −1.06958
\(636\) −7.96751e31 −0.458739
\(637\) 2.77421e32 1.56869
\(638\) −2.34864e31 −0.130430
\(639\) −7.46202e31 −0.407002
\(640\) −1.64662e32 −0.882113
\(641\) −2.91189e32 −1.53217 −0.766086 0.642739i \(-0.777798\pi\)
−0.766086 + 0.642739i \(0.777798\pi\)
\(642\) −6.42943e31 −0.332292
\(643\) −3.58958e32 −1.82229 −0.911144 0.412088i \(-0.864800\pi\)
−0.911144 + 0.412088i \(0.864800\pi\)
\(644\) −2.24365e32 −1.11884
\(645\) 2.13659e32 1.04661
\(646\) 3.07689e31 0.148060
\(647\) 3.52744e32 1.66748 0.833740 0.552157i \(-0.186195\pi\)
0.833740 + 0.552157i \(0.186195\pi\)
\(648\) −2.20712e31 −0.102498
\(649\) −4.32791e31 −0.197453
\(650\) 1.25345e31 0.0561827
\(651\) 1.12800e32 0.496737
\(652\) 2.32983e32 1.00804
\(653\) −2.22542e32 −0.946041 −0.473021 0.881051i \(-0.656836\pi\)
−0.473021 + 0.881051i \(0.656836\pi\)
\(654\) 7.67000e31 0.320370
\(655\) 2.36139e32 0.969156
\(656\) 9.82577e31 0.396253
\(657\) −9.51499e31 −0.377057
\(658\) 1.43621e32 0.559269
\(659\) 1.21818e31 0.0466152 0.0233076 0.999728i \(-0.492580\pi\)
0.0233076 + 0.999728i \(0.492580\pi\)
\(660\) 1.23380e32 0.463969
\(661\) −3.10145e32 −1.14617 −0.573084 0.819496i \(-0.694253\pi\)
−0.573084 + 0.819496i \(0.694253\pi\)
\(662\) 1.31518e32 0.477661
\(663\) 1.98681e32 0.709172
\(664\) 2.24124e32 0.786241
\(665\) −1.26621e32 −0.436573
\(666\) 5.86049e31 0.198600
\(667\) −7.00068e31 −0.233181
\(668\) 2.75765e32 0.902839
\(669\) −7.58451e30 −0.0244077
\(670\) −2.30218e32 −0.728248
\(671\) 7.69268e31 0.239204
\(672\) −2.93738e32 −0.897872
\(673\) 1.82996e32 0.549880 0.274940 0.961461i \(-0.411342\pi\)
0.274940 + 0.961461i \(0.411342\pi\)
\(674\) −2.56467e32 −0.757605
\(675\) −5.50812e30 −0.0159959
\(676\) −1.40738e32 −0.401813
\(677\) −5.90638e32 −1.65788 −0.828938 0.559341i \(-0.811054\pi\)
−0.828938 + 0.559341i \(0.811054\pi\)
\(678\) −1.02999e32 −0.284245
\(679\) 8.72096e31 0.236626
\(680\) 3.53103e32 0.941995
\(681\) −1.84482e32 −0.483908
\(682\) −1.30789e32 −0.337327
\(683\) 3.47709e32 0.881815 0.440907 0.897553i \(-0.354657\pi\)
0.440907 + 0.897553i \(0.354657\pi\)
\(684\) 2.64647e31 0.0659967
\(685\) 5.53001e32 1.35608
\(686\) 8.50594e31 0.205114
\(687\) −5.62112e31 −0.133297
\(688\) −1.57125e32 −0.366420
\(689\) −6.12199e32 −1.40402
\(690\) −1.51343e32 −0.341348
\(691\) −4.28793e32 −0.951151 −0.475576 0.879675i \(-0.657760\pi\)
−0.475576 + 0.879675i \(0.657760\pi\)
\(692\) 2.82164e32 0.615577
\(693\) 2.54155e32 0.545339
\(694\) 1.06102e32 0.223919
\(695\) 1.02944e32 0.213687
\(696\) −5.78133e31 −0.118038
\(697\) −9.20307e32 −1.84823
\(698\) 2.85907e32 0.564790
\(699\) −1.90367e32 −0.369917
\(700\) −4.62402e31 −0.0883878
\(701\) 8.63465e32 1.62363 0.811816 0.583913i \(-0.198479\pi\)
0.811816 + 0.583913i \(0.198479\pi\)
\(702\) −7.03243e31 −0.130085
\(703\) −1.69459e32 −0.308375
\(704\) 2.12511e32 0.380449
\(705\) −2.35414e32 −0.414626
\(706\) 2.25359e32 0.390501
\(707\) 7.06414e32 1.20430
\(708\) −4.41773e31 −0.0740998
\(709\) 7.56772e32 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(710\) −4.22551e32 −0.686133
\(711\) 1.17916e31 0.0188397
\(712\) 1.20141e32 0.188873
\(713\) −3.89848e32 −0.603068
\(714\) 3.01623e32 0.459129
\(715\) 9.48013e32 1.42002
\(716\) −3.55128e32 −0.523462
\(717\) 5.15830e32 0.748232
\(718\) −5.54213e31 −0.0791126
\(719\) −3.98801e32 −0.560240 −0.280120 0.959965i \(-0.590374\pi\)
−0.280120 + 0.959965i \(0.590374\pi\)
\(720\) 5.27851e31 0.0729773
\(721\) −6.93749e32 −0.943946
\(722\) −3.71718e32 −0.497778
\(723\) 3.14683e32 0.414746
\(724\) −2.33411e32 −0.302781
\(725\) −1.44279e31 −0.0184212
\(726\) −4.66241e31 −0.0585921
\(727\) −7.76270e32 −0.960211 −0.480106 0.877211i \(-0.659402\pi\)
−0.480106 + 0.877211i \(0.659402\pi\)
\(728\) −1.42368e33 −1.73341
\(729\) 3.09032e31 0.0370370
\(730\) −5.38804e32 −0.635650
\(731\) 1.47168e33 1.70908
\(732\) 7.85233e31 0.0897679
\(733\) 8.56762e32 0.964195 0.482097 0.876118i \(-0.339875\pi\)
0.482097 + 0.876118i \(0.339875\pi\)
\(734\) 2.45624e32 0.272124
\(735\) −6.90335e32 −0.752931
\(736\) 1.01519e33 1.09007
\(737\) −1.33618e33 −1.41250
\(738\) 3.25747e32 0.339026
\(739\) −5.55833e32 −0.569552 −0.284776 0.958594i \(-0.591919\pi\)
−0.284776 + 0.958594i \(0.591919\pi\)
\(740\) −8.06424e32 −0.813577
\(741\) 2.03347e32 0.201989
\(742\) −9.29394e32 −0.908981
\(743\) −1.55682e33 −1.49923 −0.749613 0.661876i \(-0.769760\pi\)
−0.749613 + 0.661876i \(0.769760\pi\)
\(744\) −3.21946e32 −0.305277
\(745\) 1.12097e33 1.04664
\(746\) 7.32911e32 0.673837
\(747\) −3.13809e32 −0.284105
\(748\) 8.49835e32 0.757646
\(749\) 1.82245e33 1.59998
\(750\) 3.44071e32 0.297470
\(751\) 1.66899e33 1.42100 0.710501 0.703696i \(-0.248468\pi\)
0.710501 + 0.703696i \(0.248468\pi\)
\(752\) 1.73124e32 0.145161
\(753\) 7.79256e32 0.643484
\(754\) −1.84207e32 −0.149808
\(755\) −5.48515e32 −0.439338
\(756\) 2.59429e32 0.204653
\(757\) −7.88666e31 −0.0612760 −0.0306380 0.999531i \(-0.509754\pi\)
−0.0306380 + 0.999531i \(0.509754\pi\)
\(758\) 7.15524e31 0.0547556
\(759\) −8.78388e32 −0.662073
\(760\) 3.61394e32 0.268303
\(761\) 7.62229e32 0.557394 0.278697 0.960379i \(-0.410098\pi\)
0.278697 + 0.960379i \(0.410098\pi\)
\(762\) 4.44784e32 0.320381
\(763\) −2.17410e33 −1.54258
\(764\) 5.91081e32 0.413117
\(765\) −4.94399e32 −0.340386
\(766\) 7.09600e32 0.481263
\(767\) −3.39445e32 −0.226789
\(768\) 7.07641e32 0.465756
\(769\) −2.24557e33 −1.45604 −0.728020 0.685556i \(-0.759559\pi\)
−0.728020 + 0.685556i \(0.759559\pi\)
\(770\) 1.43920e33 0.919344
\(771\) −5.42610e32 −0.341478
\(772\) −1.25843e33 −0.780246
\(773\) 1.77063e33 1.08159 0.540795 0.841154i \(-0.318123\pi\)
0.540795 + 0.841154i \(0.318123\pi\)
\(774\) −5.20908e32 −0.313502
\(775\) −8.03452e31 −0.0476421
\(776\) −2.48909e32 −0.145422
\(777\) −1.66118e33 −0.956260
\(778\) 5.62793e32 0.319215
\(779\) −9.41917e32 −0.526419
\(780\) 9.67688e32 0.532902
\(781\) −2.45247e33 −1.33081
\(782\) −1.04244e33 −0.557410
\(783\) 8.09476e31 0.0426524
\(784\) 5.07673e32 0.263602
\(785\) 2.12559e32 0.108762
\(786\) −5.75714e32 −0.290301
\(787\) 1.41877e33 0.705024 0.352512 0.935807i \(-0.385328\pi\)
0.352512 + 0.935807i \(0.385328\pi\)
\(788\) −1.44522e33 −0.707753
\(789\) −7.75763e32 −0.374406
\(790\) 6.67723e31 0.0317603
\(791\) 2.91957e33 1.36864
\(792\) −7.25394e32 −0.335146
\(793\) 6.03349e32 0.274743
\(794\) −1.50548e32 −0.0675676
\(795\) 1.52340e33 0.673893
\(796\) 1.48547e33 0.647684
\(797\) 2.58083e33 1.10915 0.554573 0.832135i \(-0.312882\pi\)
0.554573 + 0.832135i \(0.312882\pi\)
\(798\) 3.08705e32 0.130771
\(799\) −1.62152e33 −0.677071
\(800\) 2.09225e32 0.0861149
\(801\) −1.68216e32 −0.0682485
\(802\) 7.78677e32 0.311424
\(803\) −3.12720e33 −1.23290
\(804\) −1.36391e33 −0.530079
\(805\) 4.28989e33 1.64359
\(806\) −1.02580e33 −0.387444
\(807\) 2.42068e33 0.901345
\(808\) −2.01621e33 −0.740124
\(809\) −1.24432e33 −0.450323 −0.225161 0.974321i \(-0.572291\pi\)
−0.225161 + 0.974321i \(0.572291\pi\)
\(810\) 1.74995e32 0.0624378
\(811\) 1.55059e32 0.0545453 0.0272727 0.999628i \(-0.491318\pi\)
0.0272727 + 0.999628i \(0.491318\pi\)
\(812\) 6.79549e32 0.235682
\(813\) 1.25239e33 0.428253
\(814\) 1.92611e33 0.649383
\(815\) −4.45465e33 −1.48082
\(816\) 3.63582e32 0.119170
\(817\) 1.50624e33 0.486787
\(818\) −2.32466e33 −0.740793
\(819\) 1.99338e33 0.626361
\(820\) −4.48240e33 −1.38884
\(821\) 5.00186e33 1.52822 0.764108 0.645088i \(-0.223179\pi\)
0.764108 + 0.645088i \(0.223179\pi\)
\(822\) −1.34823e33 −0.406199
\(823\) 2.77022e33 0.823030 0.411515 0.911403i \(-0.365000\pi\)
0.411515 + 0.911403i \(0.365000\pi\)
\(824\) 1.98006e33 0.580116
\(825\) −1.81030e32 −0.0523035
\(826\) −5.15319e32 −0.146827
\(827\) −2.81604e33 −0.791270 −0.395635 0.918408i \(-0.629475\pi\)
−0.395635 + 0.918408i \(0.629475\pi\)
\(828\) −8.96618e32 −0.248461
\(829\) −6.05152e32 −0.165382 −0.0826908 0.996575i \(-0.526351\pi\)
−0.0826908 + 0.996575i \(0.526351\pi\)
\(830\) −1.77700e33 −0.478950
\(831\) −1.81024e33 −0.481198
\(832\) 1.66676e33 0.436974
\(833\) −4.75500e33 −1.22951
\(834\) −2.50981e32 −0.0640077
\(835\) −5.27266e33 −1.32628
\(836\) 8.69791e32 0.215796
\(837\) 4.50775e32 0.110310
\(838\) −7.93180e32 −0.191454
\(839\) −2.21831e33 −0.528152 −0.264076 0.964502i \(-0.585067\pi\)
−0.264076 + 0.964502i \(0.585067\pi\)
\(840\) 3.54269e33 0.831996
\(841\) −4.10469e33 −0.950881
\(842\) 4.00587e33 0.915392
\(843\) 1.42398e32 0.0320987
\(844\) 1.59632e33 0.354963
\(845\) 2.69093e33 0.590268
\(846\) 5.73945e32 0.124197
\(847\) 1.32158e33 0.282121
\(848\) −1.12031e33 −0.235931
\(849\) 3.81599e33 0.792809
\(850\) −2.14841e32 −0.0440351
\(851\) 5.74124e33 1.16096
\(852\) −2.50337e33 −0.499424
\(853\) 9.69232e33 1.90772 0.953858 0.300257i \(-0.0970723\pi\)
0.953858 + 0.300257i \(0.0970723\pi\)
\(854\) 9.15958e32 0.177873
\(855\) −5.06008e32 −0.0969499
\(856\) −5.20154e33 −0.983295
\(857\) −3.54610e33 −0.661411 −0.330705 0.943734i \(-0.607287\pi\)
−0.330705 + 0.943734i \(0.607287\pi\)
\(858\) −2.31128e33 −0.425353
\(859\) 9.58348e33 1.74021 0.870105 0.492866i \(-0.164051\pi\)
0.870105 + 0.492866i \(0.164051\pi\)
\(860\) 7.16788e33 1.28428
\(861\) −9.23347e33 −1.63241
\(862\) −1.90238e33 −0.331866
\(863\) 3.43280e33 0.590915 0.295457 0.955356i \(-0.404528\pi\)
0.295457 + 0.955356i \(0.404528\pi\)
\(864\) −1.17385e33 −0.199391
\(865\) −5.39502e33 −0.904290
\(866\) 2.80604e33 0.464128
\(867\) 1.31795e32 0.0215119
\(868\) 3.78422e33 0.609536
\(869\) 3.87544e32 0.0616018
\(870\) 4.58381e32 0.0719044
\(871\) −1.04799e34 −1.62236
\(872\) 6.20519e33 0.948016
\(873\) 3.48511e32 0.0525476
\(874\) −1.06692e33 −0.158764
\(875\) −9.75287e33 −1.43232
\(876\) −3.19210e33 −0.462679
\(877\) −8.57680e33 −1.22696 −0.613480 0.789711i \(-0.710231\pi\)
−0.613480 + 0.789711i \(0.710231\pi\)
\(878\) −6.20950e33 −0.876738
\(879\) 4.12940e33 0.575460
\(880\) 1.73484e33 0.238621
\(881\) −8.23643e33 −1.11819 −0.559096 0.829103i \(-0.688852\pi\)
−0.559096 + 0.829103i \(0.688852\pi\)
\(882\) 1.68305e33 0.225533
\(883\) 1.05626e34 1.39709 0.698545 0.715566i \(-0.253831\pi\)
0.698545 + 0.715566i \(0.253831\pi\)
\(884\) 6.66539e33 0.870211
\(885\) 8.44675e32 0.108853
\(886\) −2.01324e33 −0.256098
\(887\) 6.30164e33 0.791283 0.395642 0.918405i \(-0.370522\pi\)
0.395642 + 0.918405i \(0.370522\pi\)
\(888\) 4.74126e33 0.587685
\(889\) −1.26076e34 −1.54264
\(890\) −9.52555e32 −0.115055
\(891\) 1.01567e33 0.121104
\(892\) −2.54446e32 −0.0299502
\(893\) −1.65960e33 −0.192846
\(894\) −2.73295e33 −0.313510
\(895\) 6.79008e33 0.768972
\(896\) −1.13793e34 −1.27226
\(897\) −6.88934e33 −0.760439
\(898\) 4.95762e33 0.540250
\(899\) 1.18076e33 0.127035
\(900\) −1.84787e32 −0.0196283
\(901\) 1.04931e34 1.10045
\(902\) 1.07060e34 1.10854
\(903\) 1.47654e34 1.50951
\(904\) −8.33286e33 −0.841118
\(905\) 4.46285e33 0.444788
\(906\) 1.33730e33 0.131599
\(907\) 1.10592e34 1.07458 0.537292 0.843396i \(-0.319447\pi\)
0.537292 + 0.843396i \(0.319447\pi\)
\(908\) −6.18904e33 −0.593795
\(909\) 2.82300e33 0.267440
\(910\) 1.12879e34 1.05593
\(911\) −8.48544e33 −0.783814 −0.391907 0.920005i \(-0.628185\pi\)
−0.391907 + 0.920005i \(0.628185\pi\)
\(912\) 3.72119e32 0.0339423
\(913\) −1.03137e34 −0.928964
\(914\) −1.07874e34 −0.959481
\(915\) −1.50138e33 −0.131870
\(916\) −1.88578e33 −0.163566
\(917\) 1.63189e34 1.39780
\(918\) 1.20536e33 0.101959
\(919\) 2.82637e33 0.236102 0.118051 0.993008i \(-0.462335\pi\)
0.118051 + 0.993008i \(0.462335\pi\)
\(920\) −1.22439e34 −1.01009
\(921\) −1.23541e34 −1.00653
\(922\) −1.95973e33 −0.157685
\(923\) −1.92351e34 −1.52853
\(924\) 8.52643e33 0.669174
\(925\) 1.18323e33 0.0917150
\(926\) 7.62223e33 0.583520
\(927\) −2.77239e33 −0.209622
\(928\) −3.07478e33 −0.229621
\(929\) 2.09040e34 1.54187 0.770936 0.636913i \(-0.219789\pi\)
0.770936 + 0.636913i \(0.219789\pi\)
\(930\) 2.55260e33 0.185964
\(931\) −4.86665e33 −0.350194
\(932\) −6.38646e33 −0.453918
\(933\) 7.11175e33 0.499272
\(934\) 6.81585e33 0.472640
\(935\) −1.62489e34 −1.11299
\(936\) −5.68938e33 −0.384940
\(937\) −9.77881e33 −0.653553 −0.326776 0.945102i \(-0.605962\pi\)
−0.326776 + 0.945102i \(0.605962\pi\)
\(938\) −1.59097e34 −1.05034
\(939\) −1.25433e34 −0.818010
\(940\) −7.89769e33 −0.508780
\(941\) 2.28884e34 1.45658 0.728290 0.685270i \(-0.240316\pi\)
0.728290 + 0.685270i \(0.240316\pi\)
\(942\) −5.18225e32 −0.0325786
\(943\) 3.19119e34 1.98184
\(944\) −6.21175e32 −0.0381098
\(945\) −4.96032e33 −0.300638
\(946\) −1.71202e34 −1.02509
\(947\) 1.58194e34 0.935763 0.467882 0.883791i \(-0.345017\pi\)
0.467882 + 0.883791i \(0.345017\pi\)
\(948\) 3.95587e32 0.0231178
\(949\) −2.45271e34 −1.41607
\(950\) −2.19886e32 −0.0125422
\(951\) 5.68907e33 0.320601
\(952\) 2.44019e34 1.35862
\(953\) −1.70079e34 −0.935582 −0.467791 0.883839i \(-0.654950\pi\)
−0.467791 + 0.883839i \(0.654950\pi\)
\(954\) −3.71408e33 −0.201858
\(955\) −1.13015e34 −0.606874
\(956\) 1.73051e34 0.918140
\(957\) 2.66043e33 0.139465
\(958\) 7.51172e33 0.389078
\(959\) 3.82163e34 1.95585
\(960\) −4.14757e33 −0.209737
\(961\) −1.34380e34 −0.671453
\(962\) 1.51068e34 0.745863
\(963\) 7.28297e33 0.355309
\(964\) 1.05570e34 0.508927
\(965\) 2.40614e34 1.14619
\(966\) −1.04589e34 −0.492320
\(967\) −3.44351e34 −1.60176 −0.800880 0.598825i \(-0.795634\pi\)
−0.800880 + 0.598825i \(0.795634\pi\)
\(968\) −3.77198e33 −0.173382
\(969\) −3.48536e33 −0.158316
\(970\) 1.97351e33 0.0885859
\(971\) 1.73462e33 0.0769456 0.0384728 0.999260i \(-0.487751\pi\)
0.0384728 + 0.999260i \(0.487751\pi\)
\(972\) 1.03674e33 0.0454474
\(973\) 7.11419e33 0.308197
\(974\) −5.81308e33 −0.248873
\(975\) −1.41985e33 −0.0600743
\(976\) 1.10411e33 0.0461680
\(977\) 1.44173e33 0.0595794 0.0297897 0.999556i \(-0.490516\pi\)
0.0297897 + 0.999556i \(0.490516\pi\)
\(978\) 1.08606e34 0.443563
\(979\) −5.52859e33 −0.223159
\(980\) −2.31594e34 −0.923907
\(981\) −8.68824e33 −0.342561
\(982\) 1.17904e34 0.459460
\(983\) −4.62857e34 −1.78271 −0.891357 0.453303i \(-0.850246\pi\)
−0.891357 + 0.453303i \(0.850246\pi\)
\(984\) 2.63536e34 1.00322
\(985\) 2.76327e34 1.03970
\(986\) 3.15731e33 0.117418
\(987\) −1.62688e34 −0.598008
\(988\) 6.82191e33 0.247857
\(989\) −5.10309e34 −1.83263
\(990\) 5.75140e33 0.204159
\(991\) −1.68657e34 −0.591776 −0.295888 0.955223i \(-0.595615\pi\)
−0.295888 + 0.955223i \(0.595615\pi\)
\(992\) −1.71226e34 −0.593862
\(993\) −1.48978e34 −0.510747
\(994\) −2.92013e34 −0.989597
\(995\) −2.84023e34 −0.951455
\(996\) −1.05277e34 −0.348619
\(997\) 2.01747e34 0.660410 0.330205 0.943909i \(-0.392882\pi\)
0.330205 + 0.943909i \(0.392882\pi\)
\(998\) 4.83644e33 0.156504
\(999\) −6.63850e33 −0.212357
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.1.2 2
3.2 odd 2 9.24.a.c.1.1 2
4.3 odd 2 48.24.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.24.a.b.1.2 2 1.1 even 1 trivial
9.24.a.c.1.1 2 3.2 odd 2
48.24.a.g.1.2 2 4.3 odd 2