Properties

Label 177.8.a.c.1.8
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-3.88629\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.88629 q^{2} -27.0000 q^{3} -112.897 q^{4} -26.2639 q^{5} +104.930 q^{6} -644.460 q^{7} +936.194 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-3.88629 q^{2} -27.0000 q^{3} -112.897 q^{4} -26.2639 q^{5} +104.930 q^{6} -644.460 q^{7} +936.194 q^{8} +729.000 q^{9} +102.069 q^{10} -5692.26 q^{11} +3048.21 q^{12} -606.275 q^{13} +2504.56 q^{14} +709.124 q^{15} +10812.5 q^{16} -39283.3 q^{17} -2833.10 q^{18} -34062.8 q^{19} +2965.10 q^{20} +17400.4 q^{21} +22121.8 q^{22} -21244.4 q^{23} -25277.2 q^{24} -77435.2 q^{25} +2356.16 q^{26} -19683.0 q^{27} +72757.5 q^{28} +26193.7 q^{29} -2755.86 q^{30} -310358. q^{31} -161853. q^{32} +153691. q^{33} +152666. q^{34} +16926.0 q^{35} -82301.7 q^{36} -330264. q^{37} +132378. q^{38} +16369.4 q^{39} -24588.1 q^{40} +318533. q^{41} -67623.0 q^{42} +16755.2 q^{43} +642638. q^{44} -19146.3 q^{45} +82562.0 q^{46} +764733. q^{47} -291937. q^{48} -408214. q^{49} +300935. q^{50} +1.06065e6 q^{51} +68446.5 q^{52} -1.38308e6 q^{53} +76493.8 q^{54} +149501. q^{55} -603340. q^{56} +919695. q^{57} -101796. q^{58} -205379. q^{59} -80057.8 q^{60} -354332. q^{61} +1.20614e6 q^{62} -469811. q^{63} -754988. q^{64} +15923.1 q^{65} -597287. q^{66} +4.54645e6 q^{67} +4.43496e6 q^{68} +573600. q^{69} -65779.3 q^{70} -367019. q^{71} +682485. q^{72} -1.13809e6 q^{73} +1.28350e6 q^{74} +2.09075e6 q^{75} +3.84558e6 q^{76} +3.66843e6 q^{77} -63616.3 q^{78} +3.04988e6 q^{79} -283977. q^{80} +531441. q^{81} -1.23791e6 q^{82} -8.39695e6 q^{83} -1.96445e6 q^{84} +1.03173e6 q^{85} -65115.7 q^{86} -707229. q^{87} -5.32906e6 q^{88} -3.28661e6 q^{89} +74408.2 q^{90} +390720. q^{91} +2.39843e6 q^{92} +8.37967e6 q^{93} -2.97197e6 q^{94} +894620. q^{95} +4.37004e6 q^{96} +8.46055e6 q^{97} +1.58644e6 q^{98} -4.14966e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.88629 −0.343503 −0.171751 0.985140i \(-0.554943\pi\)
−0.171751 + 0.985140i \(0.554943\pi\)
\(3\) −27.0000 −0.577350
\(4\) −112.897 −0.882006
\(5\) −26.2639 −0.0939644 −0.0469822 0.998896i \(-0.514960\pi\)
−0.0469822 + 0.998896i \(0.514960\pi\)
\(6\) 104.930 0.198321
\(7\) −644.460 −0.710155 −0.355077 0.934837i \(-0.615545\pi\)
−0.355077 + 0.934837i \(0.615545\pi\)
\(8\) 936.194 0.646474
\(9\) 729.000 0.333333
\(10\) 102.069 0.0322770
\(11\) −5692.26 −1.28947 −0.644734 0.764407i \(-0.723032\pi\)
−0.644734 + 0.764407i \(0.723032\pi\)
\(12\) 3048.21 0.509226
\(13\) −606.275 −0.0765364 −0.0382682 0.999268i \(-0.512184\pi\)
−0.0382682 + 0.999268i \(0.512184\pi\)
\(14\) 2504.56 0.243940
\(15\) 709.124 0.0542504
\(16\) 10812.5 0.659941
\(17\) −39283.3 −1.93926 −0.969632 0.244570i \(-0.921353\pi\)
−0.969632 + 0.244570i \(0.921353\pi\)
\(18\) −2833.10 −0.114501
\(19\) −34062.8 −1.13931 −0.569656 0.821883i \(-0.692923\pi\)
−0.569656 + 0.821883i \(0.692923\pi\)
\(20\) 2965.10 0.0828772
\(21\) 17400.4 0.410008
\(22\) 22121.8 0.442935
\(23\) −21244.4 −0.364081 −0.182040 0.983291i \(-0.558270\pi\)
−0.182040 + 0.983291i \(0.558270\pi\)
\(24\) −25277.2 −0.373242
\(25\) −77435.2 −0.991171
\(26\) 2356.16 0.0262905
\(27\) −19683.0 −0.192450
\(28\) 72757.5 0.626361
\(29\) 26193.7 0.199436 0.0997179 0.995016i \(-0.468206\pi\)
0.0997179 + 0.995016i \(0.468206\pi\)
\(30\) −2755.86 −0.0186351
\(31\) −310358. −1.87110 −0.935551 0.353193i \(-0.885096\pi\)
−0.935551 + 0.353193i \(0.885096\pi\)
\(32\) −161853. −0.873165
\(33\) 153691. 0.744474
\(34\) 152666. 0.666142
\(35\) 16926.0 0.0667292
\(36\) −82301.7 −0.294002
\(37\) −330264. −1.07190 −0.535951 0.844249i \(-0.680047\pi\)
−0.535951 + 0.844249i \(0.680047\pi\)
\(38\) 132378. 0.391356
\(39\) 16369.4 0.0441883
\(40\) −24588.1 −0.0607455
\(41\) 318533. 0.721790 0.360895 0.932607i \(-0.382471\pi\)
0.360895 + 0.932607i \(0.382471\pi\)
\(42\) −67623.0 −0.140839
\(43\) 16755.2 0.0321374 0.0160687 0.999871i \(-0.494885\pi\)
0.0160687 + 0.999871i \(0.494885\pi\)
\(44\) 642638. 1.13732
\(45\) −19146.3 −0.0313215
\(46\) 82562.0 0.125063
\(47\) 764733. 1.07440 0.537202 0.843454i \(-0.319481\pi\)
0.537202 + 0.843454i \(0.319481\pi\)
\(48\) −291937. −0.381017
\(49\) −408214. −0.495681
\(50\) 300935. 0.340470
\(51\) 1.06065e6 1.11963
\(52\) 68446.5 0.0675056
\(53\) −1.38308e6 −1.27609 −0.638046 0.769998i \(-0.720257\pi\)
−0.638046 + 0.769998i \(0.720257\pi\)
\(54\) 76493.8 0.0661071
\(55\) 149501. 0.121164
\(56\) −603340. −0.459096
\(57\) 919695. 0.657782
\(58\) −101796. −0.0685067
\(59\) −205379. −0.130189
\(60\) −80057.8 −0.0478492
\(61\) −354332. −0.199874 −0.0999371 0.994994i \(-0.531864\pi\)
−0.0999371 + 0.994994i \(0.531864\pi\)
\(62\) 1.20614e6 0.642728
\(63\) −469811. −0.236718
\(64\) −754988. −0.360006
\(65\) 15923.1 0.00719170
\(66\) −597287. −0.255729
\(67\) 4.54645e6 1.84676 0.923380 0.383887i \(-0.125415\pi\)
0.923380 + 0.383887i \(0.125415\pi\)
\(68\) 4.43496e6 1.71044
\(69\) 573600. 0.210202
\(70\) −65779.3 −0.0229217
\(71\) −367019. −0.121698 −0.0608491 0.998147i \(-0.519381\pi\)
−0.0608491 + 0.998147i \(0.519381\pi\)
\(72\) 682485. 0.215491
\(73\) −1.13809e6 −0.342411 −0.171205 0.985235i \(-0.554766\pi\)
−0.171205 + 0.985235i \(0.554766\pi\)
\(74\) 1.28350e6 0.368201
\(75\) 2.09075e6 0.572253
\(76\) 3.84558e6 1.00488
\(77\) 3.66843e6 0.915721
\(78\) −63616.3 −0.0151788
\(79\) 3.04988e6 0.695967 0.347983 0.937501i \(-0.386867\pi\)
0.347983 + 0.937501i \(0.386867\pi\)
\(80\) −283977. −0.0620109
\(81\) 531441. 0.111111
\(82\) −1.23791e6 −0.247937
\(83\) −8.39695e6 −1.61194 −0.805969 0.591958i \(-0.798355\pi\)
−0.805969 + 0.591958i \(0.798355\pi\)
\(84\) −1.96445e6 −0.361629
\(85\) 1.03173e6 0.182222
\(86\) −65115.7 −0.0110393
\(87\) −707229. −0.115144
\(88\) −5.32906e6 −0.833607
\(89\) −3.28661e6 −0.494177 −0.247089 0.968993i \(-0.579474\pi\)
−0.247089 + 0.968993i \(0.579474\pi\)
\(90\) 74408.2 0.0107590
\(91\) 390720. 0.0543527
\(92\) 2.39843e6 0.321121
\(93\) 8.37967e6 1.08028
\(94\) −2.97197e6 −0.369060
\(95\) 894620. 0.107055
\(96\) 4.37004e6 0.504122
\(97\) 8.46055e6 0.941234 0.470617 0.882338i \(-0.344031\pi\)
0.470617 + 0.882338i \(0.344031\pi\)
\(98\) 1.58644e6 0.170268
\(99\) −4.14966e6 −0.429822
\(100\) 8.74219e6 0.874219
\(101\) 7.61349e6 0.735290 0.367645 0.929966i \(-0.380164\pi\)
0.367645 + 0.929966i \(0.380164\pi\)
\(102\) −4.12199e6 −0.384597
\(103\) 4.15554e6 0.374712 0.187356 0.982292i \(-0.440008\pi\)
0.187356 + 0.982292i \(0.440008\pi\)
\(104\) −567591. −0.0494788
\(105\) −457002. −0.0385262
\(106\) 5.37505e6 0.438341
\(107\) 1.11011e7 0.876038 0.438019 0.898966i \(-0.355680\pi\)
0.438019 + 0.898966i \(0.355680\pi\)
\(108\) 2.22215e6 0.169742
\(109\) 8.22798e6 0.608556 0.304278 0.952583i \(-0.401585\pi\)
0.304278 + 0.952583i \(0.401585\pi\)
\(110\) −581002. −0.0416201
\(111\) 8.91712e6 0.618863
\(112\) −6.96820e6 −0.468660
\(113\) −1.87058e7 −1.21956 −0.609779 0.792571i \(-0.708742\pi\)
−0.609779 + 0.792571i \(0.708742\pi\)
\(114\) −3.57420e6 −0.225950
\(115\) 557961. 0.0342106
\(116\) −2.95718e6 −0.175904
\(117\) −441975. −0.0255121
\(118\) 798162. 0.0447202
\(119\) 2.53165e7 1.37718
\(120\) 663878. 0.0350714
\(121\) 1.29146e7 0.662725
\(122\) 1.37704e6 0.0686573
\(123\) −8.60039e6 −0.416725
\(124\) 3.50384e7 1.65032
\(125\) 4.08561e6 0.187099
\(126\) 1.82582e6 0.0813133
\(127\) 1.39760e7 0.605437 0.302718 0.953080i \(-0.402106\pi\)
0.302718 + 0.953080i \(0.402106\pi\)
\(128\) 2.36513e7 0.996828
\(129\) −452392. −0.0185546
\(130\) −61881.8 −0.00247037
\(131\) 1.69407e7 0.658387 0.329193 0.944263i \(-0.393223\pi\)
0.329193 + 0.944263i \(0.393223\pi\)
\(132\) −1.73512e7 −0.656631
\(133\) 2.19521e7 0.809087
\(134\) −1.76688e7 −0.634367
\(135\) 516951. 0.0180835
\(136\) −3.67768e7 −1.25368
\(137\) −2.48882e7 −0.826935 −0.413467 0.910519i \(-0.635682\pi\)
−0.413467 + 0.910519i \(0.635682\pi\)
\(138\) −2.22917e6 −0.0722049
\(139\) −5.91087e7 −1.86681 −0.933404 0.358826i \(-0.883177\pi\)
−0.933404 + 0.358826i \(0.883177\pi\)
\(140\) −1.91089e6 −0.0588556
\(141\) −2.06478e7 −0.620307
\(142\) 1.42634e6 0.0418036
\(143\) 3.45108e6 0.0986912
\(144\) 7.88229e6 0.219980
\(145\) −687946. −0.0187399
\(146\) 4.42295e6 0.117619
\(147\) 1.10218e7 0.286181
\(148\) 3.72857e7 0.945424
\(149\) −4.77206e7 −1.18183 −0.590914 0.806735i \(-0.701233\pi\)
−0.590914 + 0.806735i \(0.701233\pi\)
\(150\) −8.12526e6 −0.196570
\(151\) 1.71936e7 0.406393 0.203197 0.979138i \(-0.434867\pi\)
0.203197 + 0.979138i \(0.434867\pi\)
\(152\) −3.18894e7 −0.736535
\(153\) −2.86375e7 −0.646421
\(154\) −1.42566e7 −0.314552
\(155\) 8.15120e6 0.175817
\(156\) −1.84806e6 −0.0389744
\(157\) −2.53780e7 −0.523371 −0.261685 0.965153i \(-0.584278\pi\)
−0.261685 + 0.965153i \(0.584278\pi\)
\(158\) −1.18527e7 −0.239066
\(159\) 3.73432e7 0.736752
\(160\) 4.25089e6 0.0820464
\(161\) 1.36912e7 0.258554
\(162\) −2.06533e6 −0.0381669
\(163\) 6.34001e7 1.14666 0.573328 0.819326i \(-0.305652\pi\)
0.573328 + 0.819326i \(0.305652\pi\)
\(164\) −3.59613e7 −0.636623
\(165\) −4.03652e6 −0.0699541
\(166\) 3.26330e7 0.553705
\(167\) −3.68194e7 −0.611743 −0.305872 0.952073i \(-0.598948\pi\)
−0.305872 + 0.952073i \(0.598948\pi\)
\(168\) 1.62902e7 0.265059
\(169\) −6.23809e7 −0.994142
\(170\) −4.00960e6 −0.0625936
\(171\) −2.48318e7 −0.379771
\(172\) −1.89161e6 −0.0283454
\(173\) −6.29918e7 −0.924959 −0.462480 0.886630i \(-0.653040\pi\)
−0.462480 + 0.886630i \(0.653040\pi\)
\(174\) 2.74849e6 0.0395524
\(175\) 4.99039e7 0.703884
\(176\) −6.15474e7 −0.850972
\(177\) 5.54523e6 0.0751646
\(178\) 1.27727e7 0.169751
\(179\) −1.68796e7 −0.219977 −0.109989 0.993933i \(-0.535081\pi\)
−0.109989 + 0.993933i \(0.535081\pi\)
\(180\) 2.16156e6 0.0276257
\(181\) 5.41326e7 0.678554 0.339277 0.940687i \(-0.389818\pi\)
0.339277 + 0.940687i \(0.389818\pi\)
\(182\) −1.51845e6 −0.0186703
\(183\) 9.56698e6 0.115397
\(184\) −1.98889e7 −0.235369
\(185\) 8.67400e6 0.100721
\(186\) −3.25658e7 −0.371079
\(187\) 2.23611e8 2.50062
\(188\) −8.63359e7 −0.947630
\(189\) 1.26849e7 0.136669
\(190\) −3.47675e6 −0.0367736
\(191\) −8.74794e7 −0.908425 −0.454213 0.890893i \(-0.650079\pi\)
−0.454213 + 0.890893i \(0.650079\pi\)
\(192\) 2.03847e7 0.207850
\(193\) −8.18019e7 −0.819054 −0.409527 0.912298i \(-0.634306\pi\)
−0.409527 + 0.912298i \(0.634306\pi\)
\(194\) −3.28801e7 −0.323316
\(195\) −429924. −0.00415213
\(196\) 4.60861e7 0.437193
\(197\) −1.92295e8 −1.79199 −0.895996 0.444061i \(-0.853537\pi\)
−0.895996 + 0.444061i \(0.853537\pi\)
\(198\) 1.61268e7 0.147645
\(199\) −1.44937e6 −0.0130375 −0.00651873 0.999979i \(-0.502075\pi\)
−0.00651873 + 0.999979i \(0.502075\pi\)
\(200\) −7.24944e7 −0.640766
\(201\) −1.22754e8 −1.06623
\(202\) −2.95882e7 −0.252574
\(203\) −1.68808e7 −0.141630
\(204\) −1.19744e8 −0.987524
\(205\) −8.36590e6 −0.0678225
\(206\) −1.61496e7 −0.128715
\(207\) −1.54872e7 −0.121360
\(208\) −6.55533e6 −0.0505095
\(209\) 1.93894e8 1.46910
\(210\) 1.77604e6 0.0132338
\(211\) −2.14625e8 −1.57286 −0.786432 0.617676i \(-0.788074\pi\)
−0.786432 + 0.617676i \(0.788074\pi\)
\(212\) 1.56145e8 1.12552
\(213\) 9.90950e6 0.0702625
\(214\) −4.31421e7 −0.300921
\(215\) −440057. −0.00301977
\(216\) −1.84271e7 −0.124414
\(217\) 2.00013e8 1.32877
\(218\) −3.19763e7 −0.209041
\(219\) 3.07285e7 0.197691
\(220\) −1.68781e7 −0.106867
\(221\) 2.38165e7 0.148424
\(222\) −3.46545e7 −0.212581
\(223\) −2.05595e8 −1.24150 −0.620748 0.784011i \(-0.713171\pi\)
−0.620748 + 0.784011i \(0.713171\pi\)
\(224\) 1.04308e8 0.620082
\(225\) −5.64503e7 −0.330390
\(226\) 7.26962e7 0.418921
\(227\) 1.87007e8 1.06113 0.530563 0.847646i \(-0.321981\pi\)
0.530563 + 0.847646i \(0.321981\pi\)
\(228\) −1.03831e8 −0.580168
\(229\) 1.01441e8 0.558201 0.279100 0.960262i \(-0.409964\pi\)
0.279100 + 0.960262i \(0.409964\pi\)
\(230\) −2.16839e6 −0.0117514
\(231\) −9.90477e7 −0.528692
\(232\) 2.45223e7 0.128930
\(233\) −1.98406e8 −1.02757 −0.513783 0.857920i \(-0.671756\pi\)
−0.513783 + 0.857920i \(0.671756\pi\)
\(234\) 1.71764e6 0.00876348
\(235\) −2.00848e7 −0.100956
\(236\) 2.31866e7 0.114827
\(237\) −8.23468e7 −0.401816
\(238\) −9.83873e7 −0.473064
\(239\) 9.06262e7 0.429399 0.214700 0.976680i \(-0.431123\pi\)
0.214700 + 0.976680i \(0.431123\pi\)
\(240\) 7.66738e6 0.0358020
\(241\) −2.22377e8 −1.02336 −0.511681 0.859175i \(-0.670977\pi\)
−0.511681 + 0.859175i \(0.670977\pi\)
\(242\) −5.01900e7 −0.227648
\(243\) −1.43489e7 −0.0641500
\(244\) 4.00030e7 0.176290
\(245\) 1.07213e7 0.0465763
\(246\) 3.34236e7 0.143146
\(247\) 2.06514e7 0.0871988
\(248\) −2.90555e8 −1.20962
\(249\) 2.26718e8 0.930653
\(250\) −1.58779e7 −0.0642690
\(251\) 4.71405e8 1.88164 0.940819 0.338909i \(-0.110058\pi\)
0.940819 + 0.338909i \(0.110058\pi\)
\(252\) 5.30402e7 0.208787
\(253\) 1.20929e8 0.469470
\(254\) −5.43146e7 −0.207969
\(255\) −2.78567e7 −0.105206
\(256\) 4.72267e6 0.0175933
\(257\) −6.85596e6 −0.0251943 −0.0125972 0.999921i \(-0.504010\pi\)
−0.0125972 + 0.999921i \(0.504010\pi\)
\(258\) 1.75812e6 0.00637354
\(259\) 2.12842e8 0.761216
\(260\) −1.79767e6 −0.00634312
\(261\) 1.90952e7 0.0664786
\(262\) −6.58363e7 −0.226158
\(263\) −4.52181e8 −1.53273 −0.766367 0.642403i \(-0.777938\pi\)
−0.766367 + 0.642403i \(0.777938\pi\)
\(264\) 1.43885e8 0.481283
\(265\) 3.63250e7 0.119907
\(266\) −8.53122e7 −0.277924
\(267\) 8.87384e7 0.285313
\(268\) −5.13279e8 −1.62885
\(269\) 6.31747e7 0.197884 0.0989419 0.995093i \(-0.468454\pi\)
0.0989419 + 0.995093i \(0.468454\pi\)
\(270\) −2.00902e6 −0.00621171
\(271\) 5.80213e8 1.77091 0.885453 0.464730i \(-0.153849\pi\)
0.885453 + 0.464730i \(0.153849\pi\)
\(272\) −4.24749e8 −1.27980
\(273\) −1.05494e7 −0.0313805
\(274\) 9.67226e7 0.284054
\(275\) 4.40781e8 1.27808
\(276\) −6.47575e7 −0.185400
\(277\) 2.08756e8 0.590146 0.295073 0.955475i \(-0.404656\pi\)
0.295073 + 0.955475i \(0.404656\pi\)
\(278\) 2.29714e8 0.641253
\(279\) −2.26251e8 −0.623700
\(280\) 1.58460e7 0.0431387
\(281\) −6.75157e7 −0.181523 −0.0907617 0.995873i \(-0.528930\pi\)
−0.0907617 + 0.995873i \(0.528930\pi\)
\(282\) 8.02433e7 0.213077
\(283\) 5.48132e8 1.43758 0.718791 0.695226i \(-0.244696\pi\)
0.718791 + 0.695226i \(0.244696\pi\)
\(284\) 4.14352e7 0.107338
\(285\) −2.41547e7 −0.0618081
\(286\) −1.34119e7 −0.0339007
\(287\) −2.05282e8 −0.512582
\(288\) −1.17991e8 −0.291055
\(289\) 1.13284e9 2.76074
\(290\) 2.67356e6 0.00643719
\(291\) −2.28435e8 −0.543421
\(292\) 1.28487e8 0.302008
\(293\) 2.36630e8 0.549582 0.274791 0.961504i \(-0.411391\pi\)
0.274791 + 0.961504i \(0.411391\pi\)
\(294\) −4.28338e7 −0.0983040
\(295\) 5.39404e6 0.0122331
\(296\) −3.09191e8 −0.692956
\(297\) 1.12041e8 0.248158
\(298\) 1.85456e8 0.405961
\(299\) 1.28800e7 0.0278654
\(300\) −2.36039e8 −0.504730
\(301\) −1.07981e7 −0.0228225
\(302\) −6.68191e7 −0.139597
\(303\) −2.05564e8 −0.424520
\(304\) −3.68303e8 −0.751878
\(305\) 9.30614e6 0.0187811
\(306\) 1.11294e8 0.222047
\(307\) −2.84146e7 −0.0560477 −0.0280238 0.999607i \(-0.508921\pi\)
−0.0280238 + 0.999607i \(0.508921\pi\)
\(308\) −4.14154e8 −0.807671
\(309\) −1.12200e8 −0.216340
\(310\) −3.16779e7 −0.0603936
\(311\) 3.86506e8 0.728609 0.364305 0.931280i \(-0.381307\pi\)
0.364305 + 0.931280i \(0.381307\pi\)
\(312\) 1.53250e7 0.0285666
\(313\) 7.79214e8 1.43632 0.718160 0.695878i \(-0.244984\pi\)
0.718160 + 0.695878i \(0.244984\pi\)
\(314\) 9.86264e7 0.179779
\(315\) 1.23391e7 0.0222431
\(316\) −3.44322e8 −0.613847
\(317\) −4.02260e8 −0.709251 −0.354625 0.935008i \(-0.615392\pi\)
−0.354625 + 0.935008i \(0.615392\pi\)
\(318\) −1.45126e8 −0.253076
\(319\) −1.49101e8 −0.257166
\(320\) 1.98289e7 0.0338278
\(321\) −2.99730e8 −0.505781
\(322\) −5.32079e7 −0.0888138
\(323\) 1.33810e9 2.20943
\(324\) −5.99980e7 −0.0980007
\(325\) 4.69471e7 0.0758607
\(326\) −2.46391e8 −0.393879
\(327\) −2.22155e8 −0.351350
\(328\) 2.98209e8 0.466618
\(329\) −4.92840e8 −0.762993
\(330\) 1.56871e7 0.0240294
\(331\) −6.06425e8 −0.919135 −0.459568 0.888143i \(-0.651996\pi\)
−0.459568 + 0.888143i \(0.651996\pi\)
\(332\) 9.47989e8 1.42174
\(333\) −2.40762e8 −0.357301
\(334\) 1.43091e8 0.210135
\(335\) −1.19407e8 −0.173530
\(336\) 1.88141e8 0.270581
\(337\) 7.34722e8 1.04573 0.522864 0.852416i \(-0.324864\pi\)
0.522864 + 0.852416i \(0.324864\pi\)
\(338\) 2.42430e8 0.341490
\(339\) 5.05057e8 0.704112
\(340\) −1.16479e8 −0.160721
\(341\) 1.76664e9 2.41272
\(342\) 9.65034e7 0.130452
\(343\) 7.93818e8 1.06216
\(344\) 1.56862e7 0.0207760
\(345\) −1.50649e7 −0.0197515
\(346\) 2.44804e8 0.317726
\(347\) −1.10182e9 −1.41566 −0.707830 0.706382i \(-0.750326\pi\)
−0.707830 + 0.706382i \(0.750326\pi\)
\(348\) 7.98438e7 0.101558
\(349\) −1.38141e8 −0.173953 −0.0869766 0.996210i \(-0.527721\pi\)
−0.0869766 + 0.996210i \(0.527721\pi\)
\(350\) −1.93941e8 −0.241786
\(351\) 1.19333e7 0.0147294
\(352\) 9.21310e8 1.12592
\(353\) −2.28995e8 −0.277086 −0.138543 0.990356i \(-0.544242\pi\)
−0.138543 + 0.990356i \(0.544242\pi\)
\(354\) −2.15504e7 −0.0258192
\(355\) 9.63932e6 0.0114353
\(356\) 3.71047e8 0.435867
\(357\) −6.83546e8 −0.795113
\(358\) 6.55991e7 0.0755627
\(359\) −1.59391e9 −1.81817 −0.909083 0.416615i \(-0.863216\pi\)
−0.909083 + 0.416615i \(0.863216\pi\)
\(360\) −1.79247e7 −0.0202485
\(361\) 2.66402e8 0.298031
\(362\) −2.10375e8 −0.233085
\(363\) −3.48695e8 −0.382625
\(364\) −4.41111e7 −0.0479394
\(365\) 2.98907e7 0.0321744
\(366\) −3.71800e7 −0.0396393
\(367\) −3.97853e8 −0.420137 −0.210069 0.977687i \(-0.567369\pi\)
−0.210069 + 0.977687i \(0.567369\pi\)
\(368\) −2.29705e8 −0.240272
\(369\) 2.32210e8 0.240597
\(370\) −3.37097e7 −0.0345978
\(371\) 8.91340e8 0.906222
\(372\) −9.46038e8 −0.952814
\(373\) 1.35221e9 1.34916 0.674582 0.738200i \(-0.264324\pi\)
0.674582 + 0.738200i \(0.264324\pi\)
\(374\) −8.69016e8 −0.858968
\(375\) −1.10311e8 −0.108022
\(376\) 7.15939e8 0.694574
\(377\) −1.58806e7 −0.0152641
\(378\) −4.92972e7 −0.0469462
\(379\) 1.69493e9 1.59924 0.799621 0.600506i \(-0.205034\pi\)
0.799621 + 0.600506i \(0.205034\pi\)
\(380\) −1.01000e8 −0.0944229
\(381\) −3.77351e8 −0.349549
\(382\) 3.39970e8 0.312046
\(383\) −1.90311e9 −1.73088 −0.865442 0.501010i \(-0.832962\pi\)
−0.865442 + 0.501010i \(0.832962\pi\)
\(384\) −6.38585e8 −0.575519
\(385\) −9.63472e7 −0.0860452
\(386\) 3.17906e8 0.281347
\(387\) 1.22146e7 0.0107125
\(388\) −9.55169e8 −0.830174
\(389\) −1.65443e9 −1.42503 −0.712516 0.701655i \(-0.752445\pi\)
−0.712516 + 0.701655i \(0.752445\pi\)
\(390\) 1.67081e6 0.00142627
\(391\) 8.34551e8 0.706048
\(392\) −3.82168e8 −0.320445
\(393\) −4.57398e8 −0.380120
\(394\) 7.47314e8 0.615554
\(395\) −8.01017e7 −0.0653961
\(396\) 4.68483e8 0.379106
\(397\) −1.76432e9 −1.41518 −0.707590 0.706623i \(-0.750218\pi\)
−0.707590 + 0.706623i \(0.750218\pi\)
\(398\) 5.63266e6 0.00447840
\(399\) −5.92707e8 −0.467127
\(400\) −8.37266e8 −0.654114
\(401\) 9.40134e8 0.728089 0.364045 0.931381i \(-0.381396\pi\)
0.364045 + 0.931381i \(0.381396\pi\)
\(402\) 4.77058e8 0.366252
\(403\) 1.88163e8 0.143207
\(404\) −8.59538e8 −0.648530
\(405\) −1.39577e7 −0.0104405
\(406\) 6.56035e7 0.0486503
\(407\) 1.87995e9 1.38218
\(408\) 9.92974e8 0.723814
\(409\) −9.46509e8 −0.684059 −0.342029 0.939689i \(-0.611114\pi\)
−0.342029 + 0.939689i \(0.611114\pi\)
\(410\) 3.25123e7 0.0232972
\(411\) 6.71981e8 0.477431
\(412\) −4.69148e8 −0.330498
\(413\) 1.32359e8 0.0924542
\(414\) 6.01877e7 0.0416875
\(415\) 2.20536e8 0.151465
\(416\) 9.81276e7 0.0668289
\(417\) 1.59594e9 1.07780
\(418\) −7.53528e8 −0.504641
\(419\) −7.99568e8 −0.531014 −0.265507 0.964109i \(-0.585539\pi\)
−0.265507 + 0.964109i \(0.585539\pi\)
\(420\) 5.15941e7 0.0339803
\(421\) −1.69924e9 −1.10986 −0.554930 0.831897i \(-0.687255\pi\)
−0.554930 + 0.831897i \(0.687255\pi\)
\(422\) 8.34094e8 0.540283
\(423\) 5.57491e8 0.358135
\(424\) −1.29483e9 −0.824960
\(425\) 3.04191e9 1.92214
\(426\) −3.85112e7 −0.0241353
\(427\) 2.28353e8 0.141941
\(428\) −1.25328e9 −0.772671
\(429\) −9.31791e7 −0.0569794
\(430\) 1.71019e6 0.00103730
\(431\) 1.67268e9 1.00634 0.503168 0.864188i \(-0.332168\pi\)
0.503168 + 0.864188i \(0.332168\pi\)
\(432\) −2.12822e8 −0.127006
\(433\) 1.83377e9 1.08552 0.542759 0.839889i \(-0.317380\pi\)
0.542759 + 0.839889i \(0.317380\pi\)
\(434\) −7.77310e8 −0.456436
\(435\) 1.85745e7 0.0108195
\(436\) −9.28912e8 −0.536750
\(437\) 7.23644e8 0.414801
\(438\) −1.19420e8 −0.0679073
\(439\) −8.26984e8 −0.466521 −0.233261 0.972414i \(-0.574940\pi\)
−0.233261 + 0.972414i \(0.574940\pi\)
\(440\) 1.39962e8 0.0783294
\(441\) −2.97588e8 −0.165227
\(442\) −9.25578e7 −0.0509841
\(443\) 4.49548e8 0.245676 0.122838 0.992427i \(-0.460800\pi\)
0.122838 + 0.992427i \(0.460800\pi\)
\(444\) −1.00671e9 −0.545841
\(445\) 8.63190e7 0.0464351
\(446\) 7.99000e8 0.426457
\(447\) 1.28846e9 0.682328
\(448\) 4.86559e8 0.255660
\(449\) −1.26664e9 −0.660377 −0.330188 0.943915i \(-0.607112\pi\)
−0.330188 + 0.943915i \(0.607112\pi\)
\(450\) 2.19382e8 0.113490
\(451\) −1.81317e9 −0.930724
\(452\) 2.11183e9 1.07566
\(453\) −4.64226e8 −0.234631
\(454\) −7.26762e8 −0.364499
\(455\) −1.02618e7 −0.00510722
\(456\) 8.61013e8 0.425239
\(457\) −2.61328e8 −0.128079 −0.0640397 0.997947i \(-0.520398\pi\)
−0.0640397 + 0.997947i \(0.520398\pi\)
\(458\) −3.94230e8 −0.191743
\(459\) 7.73213e8 0.373211
\(460\) −6.29919e7 −0.0301740
\(461\) −2.60747e9 −1.23956 −0.619779 0.784777i \(-0.712778\pi\)
−0.619779 + 0.784777i \(0.712778\pi\)
\(462\) 3.84928e8 0.181607
\(463\) 3.07005e9 1.43751 0.718757 0.695261i \(-0.244711\pi\)
0.718757 + 0.695261i \(0.244711\pi\)
\(464\) 2.83218e8 0.131616
\(465\) −2.20082e8 −0.101508
\(466\) 7.71064e8 0.352971
\(467\) −2.89908e9 −1.31720 −0.658599 0.752494i \(-0.728851\pi\)
−0.658599 + 0.752494i \(0.728851\pi\)
\(468\) 4.98975e7 0.0225019
\(469\) −2.93000e9 −1.31149
\(470\) 7.80555e7 0.0346785
\(471\) 6.85207e8 0.302168
\(472\) −1.92275e8 −0.0841637
\(473\) −9.53752e7 −0.0414402
\(474\) 3.20023e8 0.138025
\(475\) 2.63766e9 1.12925
\(476\) −2.85815e9 −1.21468
\(477\) −1.00827e9 −0.425364
\(478\) −3.52199e8 −0.147500
\(479\) −2.72155e9 −1.13147 −0.565733 0.824588i \(-0.691407\pi\)
−0.565733 + 0.824588i \(0.691407\pi\)
\(480\) −1.14774e8 −0.0473695
\(481\) 2.00231e8 0.0820395
\(482\) 8.64220e8 0.351528
\(483\) −3.69662e8 −0.149276
\(484\) −1.45802e9 −0.584528
\(485\) −2.22207e8 −0.0884425
\(486\) 5.57640e7 0.0220357
\(487\) −4.18049e9 −1.64012 −0.820060 0.572278i \(-0.806060\pi\)
−0.820060 + 0.572278i \(0.806060\pi\)
\(488\) −3.31724e8 −0.129213
\(489\) −1.71180e9 −0.662023
\(490\) −4.16660e7 −0.0159991
\(491\) 1.16233e9 0.443144 0.221572 0.975144i \(-0.428881\pi\)
0.221572 + 0.975144i \(0.428881\pi\)
\(492\) 9.70956e8 0.367554
\(493\) −1.02897e9 −0.386758
\(494\) −8.02574e7 −0.0299530
\(495\) 1.08986e8 0.0403880
\(496\) −3.35574e9 −1.23482
\(497\) 2.36529e8 0.0864245
\(498\) −8.81090e8 −0.319682
\(499\) −2.46041e9 −0.886453 −0.443227 0.896410i \(-0.646166\pi\)
−0.443227 + 0.896410i \(0.646166\pi\)
\(500\) −4.61252e8 −0.165023
\(501\) 9.94124e8 0.353190
\(502\) −1.83202e9 −0.646347
\(503\) −3.44444e9 −1.20679 −0.603394 0.797443i \(-0.706185\pi\)
−0.603394 + 0.797443i \(0.706185\pi\)
\(504\) −4.39835e8 −0.153032
\(505\) −1.99960e8 −0.0690911
\(506\) −4.69964e8 −0.161264
\(507\) 1.68429e9 0.573968
\(508\) −1.57784e9 −0.533999
\(509\) −1.85883e9 −0.624781 −0.312391 0.949954i \(-0.601130\pi\)
−0.312391 + 0.949954i \(0.601130\pi\)
\(510\) 1.08259e8 0.0361384
\(511\) 7.33455e8 0.243164
\(512\) −3.04572e9 −1.00287
\(513\) 6.70458e8 0.219261
\(514\) 2.66442e7 0.00865431
\(515\) −1.09141e8 −0.0352096
\(516\) 5.10735e7 0.0163652
\(517\) −4.35306e9 −1.38541
\(518\) −8.27164e8 −0.261480
\(519\) 1.70078e9 0.534025
\(520\) 1.49071e7 0.00464925
\(521\) 1.66724e9 0.516496 0.258248 0.966079i \(-0.416855\pi\)
0.258248 + 0.966079i \(0.416855\pi\)
\(522\) −7.42093e7 −0.0228356
\(523\) −5.43856e8 −0.166237 −0.0831186 0.996540i \(-0.526488\pi\)
−0.0831186 + 0.996540i \(0.526488\pi\)
\(524\) −1.91255e9 −0.580701
\(525\) −1.34741e9 −0.406388
\(526\) 1.75730e9 0.526498
\(527\) 1.21919e10 3.62856
\(528\) 1.66178e9 0.491309
\(529\) −2.95350e9 −0.867445
\(530\) −1.41169e8 −0.0411884
\(531\) −1.49721e8 −0.0433963
\(532\) −2.47832e9 −0.713620
\(533\) −1.93119e8 −0.0552432
\(534\) −3.44863e8 −0.0980059
\(535\) −2.91558e8 −0.0823164
\(536\) 4.25636e9 1.19388
\(537\) 4.55750e8 0.127004
\(538\) −2.45515e8 −0.0679736
\(539\) 2.32366e9 0.639164
\(540\) −5.83621e7 −0.0159497
\(541\) −2.30286e8 −0.0625283 −0.0312642 0.999511i \(-0.509953\pi\)
−0.0312642 + 0.999511i \(0.509953\pi\)
\(542\) −2.25488e9 −0.608310
\(543\) −1.46158e9 −0.391763
\(544\) 6.35813e9 1.69330
\(545\) −2.16098e8 −0.0571826
\(546\) 4.09982e7 0.0107793
\(547\) 4.00304e9 1.04576 0.522882 0.852405i \(-0.324857\pi\)
0.522882 + 0.852405i \(0.324857\pi\)
\(548\) 2.80979e9 0.729361
\(549\) −2.58308e8 −0.0666247
\(550\) −1.71300e9 −0.439024
\(551\) −8.92229e8 −0.227219
\(552\) 5.37001e8 0.135890
\(553\) −1.96553e9 −0.494244
\(554\) −8.11285e8 −0.202717
\(555\) −2.34198e8 −0.0581511
\(556\) 6.67318e9 1.64654
\(557\) −3.49935e9 −0.858014 −0.429007 0.903301i \(-0.641136\pi\)
−0.429007 + 0.903301i \(0.641136\pi\)
\(558\) 8.79277e8 0.214243
\(559\) −1.01583e7 −0.00245968
\(560\) 1.83012e8 0.0440373
\(561\) −6.03749e9 −1.44373
\(562\) 2.62385e8 0.0623538
\(563\) 3.07572e9 0.726385 0.363193 0.931714i \(-0.381687\pi\)
0.363193 + 0.931714i \(0.381687\pi\)
\(564\) 2.33107e9 0.547115
\(565\) 4.91287e8 0.114595
\(566\) −2.13020e9 −0.493813
\(567\) −3.42492e8 −0.0789061
\(568\) −3.43601e8 −0.0786747
\(569\) −2.50567e9 −0.570206 −0.285103 0.958497i \(-0.592028\pi\)
−0.285103 + 0.958497i \(0.592028\pi\)
\(570\) 9.38722e7 0.0212312
\(571\) −6.38840e9 −1.43604 −0.718019 0.696023i \(-0.754951\pi\)
−0.718019 + 0.696023i \(0.754951\pi\)
\(572\) −3.89615e8 −0.0870462
\(573\) 2.36194e9 0.524479
\(574\) 7.97784e8 0.176073
\(575\) 1.64507e9 0.360866
\(576\) −5.50386e8 −0.120002
\(577\) −3.86927e9 −0.838520 −0.419260 0.907866i \(-0.637710\pi\)
−0.419260 + 0.907866i \(0.637710\pi\)
\(578\) −4.40254e9 −0.948322
\(579\) 2.20865e9 0.472881
\(580\) 7.76669e7 0.0165287
\(581\) 5.41150e9 1.14473
\(582\) 8.87763e8 0.186667
\(583\) 7.87285e9 1.64548
\(584\) −1.06547e9 −0.221360
\(585\) 1.16080e7 0.00239723
\(586\) −9.19611e8 −0.188783
\(587\) −3.35794e9 −0.685236 −0.342618 0.939475i \(-0.611314\pi\)
−0.342618 + 0.939475i \(0.611314\pi\)
\(588\) −1.24432e9 −0.252414
\(589\) 1.05717e10 2.13177
\(590\) −2.09628e7 −0.00420211
\(591\) 5.19197e9 1.03461
\(592\) −3.57097e9 −0.707391
\(593\) 9.52200e9 1.87515 0.937576 0.347780i \(-0.113064\pi\)
0.937576 + 0.347780i \(0.113064\pi\)
\(594\) −4.35422e8 −0.0852429
\(595\) −6.64909e8 −0.129406
\(596\) 5.38750e9 1.04238
\(597\) 3.91329e7 0.00752718
\(598\) −5.00553e7 −0.00957185
\(599\) 4.99052e8 0.0948750 0.0474375 0.998874i \(-0.484895\pi\)
0.0474375 + 0.998874i \(0.484895\pi\)
\(600\) 1.95735e9 0.369946
\(601\) −4.14166e9 −0.778241 −0.389120 0.921187i \(-0.627221\pi\)
−0.389120 + 0.921187i \(0.627221\pi\)
\(602\) 4.19645e7 0.00783960
\(603\) 3.31436e9 0.615587
\(604\) −1.94110e9 −0.358441
\(605\) −3.39188e8 −0.0622726
\(606\) 7.98881e8 0.145824
\(607\) −4.83852e9 −0.878117 −0.439059 0.898458i \(-0.644688\pi\)
−0.439059 + 0.898458i \(0.644688\pi\)
\(608\) 5.51317e9 0.994807
\(609\) 4.55781e8 0.0817702
\(610\) −3.61663e7 −0.00645134
\(611\) −4.63639e8 −0.0822310
\(612\) 3.23308e9 0.570147
\(613\) 2.78195e9 0.487795 0.243898 0.969801i \(-0.421574\pi\)
0.243898 + 0.969801i \(0.421574\pi\)
\(614\) 1.10427e8 0.0192525
\(615\) 2.25879e8 0.0391574
\(616\) 3.43437e9 0.591990
\(617\) 6.18863e9 1.06071 0.530354 0.847776i \(-0.322059\pi\)
0.530354 + 0.847776i \(0.322059\pi\)
\(618\) 4.36040e8 0.0743134
\(619\) 3.57626e9 0.606055 0.303027 0.952982i \(-0.402003\pi\)
0.303027 + 0.952982i \(0.402003\pi\)
\(620\) −9.20244e8 −0.155072
\(621\) 4.18154e8 0.0700674
\(622\) −1.50207e9 −0.250279
\(623\) 2.11809e9 0.350942
\(624\) 1.76994e8 0.0291617
\(625\) 5.94232e9 0.973590
\(626\) −3.02825e9 −0.493380
\(627\) −5.23514e9 −0.848188
\(628\) 2.86510e9 0.461616
\(629\) 1.29739e10 2.07870
\(630\) −4.79531e7 −0.00764055
\(631\) 1.06672e10 1.69024 0.845121 0.534574i \(-0.179528\pi\)
0.845121 + 0.534574i \(0.179528\pi\)
\(632\) 2.85528e9 0.449924
\(633\) 5.79487e9 0.908094
\(634\) 1.56330e9 0.243629
\(635\) −3.67063e8 −0.0568895
\(636\) −4.21592e9 −0.649820
\(637\) 2.47490e8 0.0379376
\(638\) 5.79449e8 0.0883371
\(639\) −2.67557e8 −0.0405660
\(640\) −6.21174e8 −0.0936664
\(641\) −2.94407e9 −0.441515 −0.220757 0.975329i \(-0.570853\pi\)
−0.220757 + 0.975329i \(0.570853\pi\)
\(642\) 1.16484e9 0.173737
\(643\) −1.24967e10 −1.85378 −0.926889 0.375335i \(-0.877528\pi\)
−0.926889 + 0.375335i \(0.877528\pi\)
\(644\) −1.54569e9 −0.228046
\(645\) 1.18815e7 0.00174347
\(646\) −5.20024e9 −0.758943
\(647\) −1.89232e9 −0.274682 −0.137341 0.990524i \(-0.543856\pi\)
−0.137341 + 0.990524i \(0.543856\pi\)
\(648\) 4.97532e8 0.0718304
\(649\) 1.16907e9 0.167874
\(650\) −1.82450e8 −0.0260583
\(651\) −5.40036e9 −0.767166
\(652\) −7.15767e9 −1.01136
\(653\) −1.18782e9 −0.166937 −0.0834687 0.996510i \(-0.526600\pi\)
−0.0834687 + 0.996510i \(0.526600\pi\)
\(654\) 8.63360e8 0.120690
\(655\) −4.44927e8 −0.0618649
\(656\) 3.44413e9 0.476338
\(657\) −8.29669e8 −0.114137
\(658\) 1.91532e9 0.262090
\(659\) −1.27625e10 −1.73715 −0.868574 0.495559i \(-0.834963\pi\)
−0.868574 + 0.495559i \(0.834963\pi\)
\(660\) 4.55710e8 0.0616999
\(661\) 5.94189e9 0.800239 0.400120 0.916463i \(-0.368969\pi\)
0.400120 + 0.916463i \(0.368969\pi\)
\(662\) 2.35674e9 0.315725
\(663\) −6.43046e8 −0.0856928
\(664\) −7.86118e9 −1.04208
\(665\) −5.76547e8 −0.0760254
\(666\) 9.35671e8 0.122734
\(667\) −5.56469e8 −0.0726107
\(668\) 4.15679e9 0.539561
\(669\) 5.55106e9 0.716778
\(670\) 4.64051e8 0.0596079
\(671\) 2.01695e9 0.257731
\(672\) −2.81631e9 −0.358005
\(673\) −4.95302e9 −0.626350 −0.313175 0.949695i \(-0.601393\pi\)
−0.313175 + 0.949695i \(0.601393\pi\)
\(674\) −2.85534e9 −0.359210
\(675\) 1.52416e9 0.190751
\(676\) 7.04261e9 0.876839
\(677\) 1.35131e9 0.167377 0.0836886 0.996492i \(-0.473330\pi\)
0.0836886 + 0.996492i \(0.473330\pi\)
\(678\) −1.96280e9 −0.241864
\(679\) −5.45249e9 −0.668421
\(680\) 9.65900e8 0.117802
\(681\) −5.04918e9 −0.612641
\(682\) −6.86567e9 −0.828777
\(683\) 1.43057e10 1.71806 0.859029 0.511927i \(-0.171068\pi\)
0.859029 + 0.511927i \(0.171068\pi\)
\(684\) 2.80343e9 0.334960
\(685\) 6.53659e8 0.0777024
\(686\) −3.08501e9 −0.364856
\(687\) −2.73891e9 −0.322277
\(688\) 1.81166e8 0.0212088
\(689\) 8.38527e8 0.0976675
\(690\) 5.85467e7 0.00678470
\(691\) −1.07877e10 −1.24382 −0.621909 0.783089i \(-0.713643\pi\)
−0.621909 + 0.783089i \(0.713643\pi\)
\(692\) 7.11157e9 0.815820
\(693\) 2.67429e9 0.305240
\(694\) 4.28201e9 0.486283
\(695\) 1.55242e9 0.175414
\(696\) −6.62103e8 −0.0744378
\(697\) −1.25130e10 −1.39974
\(698\) 5.36854e8 0.0597533
\(699\) 5.35697e9 0.593265
\(700\) −5.63399e9 −0.620830
\(701\) −3.29304e9 −0.361064 −0.180532 0.983569i \(-0.557782\pi\)
−0.180532 + 0.983569i \(0.557782\pi\)
\(702\) −4.63763e7 −0.00505960
\(703\) 1.12497e10 1.22123
\(704\) 4.29759e9 0.464216
\(705\) 5.42291e8 0.0582868
\(706\) 8.89939e8 0.0951796
\(707\) −4.90659e9 −0.522170
\(708\) −6.26039e8 −0.0662956
\(709\) 4.66010e9 0.491059 0.245530 0.969389i \(-0.421038\pi\)
0.245530 + 0.969389i \(0.421038\pi\)
\(710\) −3.74612e7 −0.00392805
\(711\) 2.22336e9 0.231989
\(712\) −3.07690e9 −0.319473
\(713\) 6.59338e9 0.681232
\(714\) 2.65646e9 0.273123
\(715\) −9.06386e7 −0.00927346
\(716\) 1.90566e9 0.194021
\(717\) −2.44691e9 −0.247914
\(718\) 6.19440e9 0.624544
\(719\) −2.36164e9 −0.236954 −0.118477 0.992957i \(-0.537801\pi\)
−0.118477 + 0.992957i \(0.537801\pi\)
\(720\) −2.07019e8 −0.0206703
\(721\) −2.67808e9 −0.266103
\(722\) −1.03531e9 −0.102374
\(723\) 6.00417e9 0.590839
\(724\) −6.11140e9 −0.598488
\(725\) −2.02831e9 −0.197675
\(726\) 1.35513e9 0.131432
\(727\) 2.97525e9 0.287180 0.143590 0.989637i \(-0.454135\pi\)
0.143590 + 0.989637i \(0.454135\pi\)
\(728\) 3.65790e8 0.0351376
\(729\) 3.87420e8 0.0370370
\(730\) −1.16164e8 −0.0110520
\(731\) −6.58201e8 −0.0623229
\(732\) −1.08008e9 −0.101781
\(733\) −1.37632e10 −1.29079 −0.645397 0.763848i \(-0.723308\pi\)
−0.645397 + 0.763848i \(0.723308\pi\)
\(734\) 1.54617e9 0.144318
\(735\) −2.89475e8 −0.0268909
\(736\) 3.43848e9 0.317903
\(737\) −2.58796e10 −2.38134
\(738\) −9.02436e8 −0.0826455
\(739\) 1.14115e10 1.04013 0.520064 0.854127i \(-0.325908\pi\)
0.520064 + 0.854127i \(0.325908\pi\)
\(740\) −9.79266e8 −0.0888362
\(741\) −5.57589e8 −0.0503443
\(742\) −3.46400e9 −0.311290
\(743\) 5.94931e9 0.532115 0.266057 0.963957i \(-0.414279\pi\)
0.266057 + 0.963957i \(0.414279\pi\)
\(744\) 7.84500e9 0.698373
\(745\) 1.25333e9 0.111050
\(746\) −5.25509e9 −0.463441
\(747\) −6.12138e9 −0.537313
\(748\) −2.52449e10 −2.20556
\(749\) −7.15422e9 −0.622123
\(750\) 4.28702e8 0.0371057
\(751\) −7.58856e9 −0.653762 −0.326881 0.945066i \(-0.605998\pi\)
−0.326881 + 0.945066i \(0.605998\pi\)
\(752\) 8.26865e9 0.709043
\(753\) −1.27279e10 −1.08636
\(754\) 6.17164e7 0.00524326
\(755\) −4.51569e8 −0.0381865
\(756\) −1.43209e9 −0.120543
\(757\) −1.43522e10 −1.20250 −0.601248 0.799063i \(-0.705329\pi\)
−0.601248 + 0.799063i \(0.705329\pi\)
\(758\) −6.58698e9 −0.549343
\(759\) −3.26508e9 −0.271049
\(760\) 8.37538e8 0.0692081
\(761\) −5.32785e9 −0.438233 −0.219117 0.975699i \(-0.570318\pi\)
−0.219117 + 0.975699i \(0.570318\pi\)
\(762\) 1.46650e9 0.120071
\(763\) −5.30260e9 −0.432169
\(764\) 9.87614e9 0.801236
\(765\) 7.52132e8 0.0607406
\(766\) 7.39602e9 0.594563
\(767\) 1.24516e8 0.00996419
\(768\) −1.27512e8 −0.0101575
\(769\) 8.89088e9 0.705022 0.352511 0.935808i \(-0.385328\pi\)
0.352511 + 0.935808i \(0.385328\pi\)
\(770\) 3.74433e8 0.0295567
\(771\) 1.85111e8 0.0145459
\(772\) 9.23517e9 0.722411
\(773\) −2.40093e10 −1.86961 −0.934807 0.355155i \(-0.884428\pi\)
−0.934807 + 0.355155i \(0.884428\pi\)
\(774\) −4.74693e7 −0.00367976
\(775\) 2.40327e10 1.85458
\(776\) 7.92071e9 0.608483
\(777\) −5.74673e9 −0.439488
\(778\) 6.42959e9 0.489502
\(779\) −1.08501e10 −0.822343
\(780\) 4.85371e7 0.00366220
\(781\) 2.08917e9 0.156926
\(782\) −3.24331e9 −0.242529
\(783\) −5.15570e8 −0.0383814
\(784\) −4.41380e9 −0.327120
\(785\) 6.66525e8 0.0491782
\(786\) 1.77758e9 0.130572
\(787\) 1.74554e10 1.27649 0.638246 0.769833i \(-0.279660\pi\)
0.638246 + 0.769833i \(0.279660\pi\)
\(788\) 2.17095e10 1.58055
\(789\) 1.22089e10 0.884925
\(790\) 3.11298e8 0.0224637
\(791\) 1.20552e10 0.866075
\(792\) −3.88488e9 −0.277869
\(793\) 2.14823e8 0.0152976
\(794\) 6.85667e9 0.486118
\(795\) −9.80775e8 −0.0692284
\(796\) 1.63629e8 0.0114991
\(797\) −2.13622e10 −1.49466 −0.747329 0.664454i \(-0.768664\pi\)
−0.747329 + 0.664454i \(0.768664\pi\)
\(798\) 2.30343e9 0.160459
\(799\) −3.00413e10 −2.08355
\(800\) 1.25331e10 0.865456
\(801\) −2.39594e9 −0.164726
\(802\) −3.65363e9 −0.250101
\(803\) 6.47831e9 0.441527
\(804\) 1.38585e10 0.940419
\(805\) −3.59583e8 −0.0242948
\(806\) −7.31254e8 −0.0491921
\(807\) −1.70572e9 −0.114248
\(808\) 7.12770e9 0.475346
\(809\) −8.12776e9 −0.539698 −0.269849 0.962903i \(-0.586974\pi\)
−0.269849 + 0.962903i \(0.586974\pi\)
\(810\) 5.42436e7 0.00358633
\(811\) −4.71482e9 −0.310379 −0.155190 0.987885i \(-0.549599\pi\)
−0.155190 + 0.987885i \(0.549599\pi\)
\(812\) 1.90578e9 0.124919
\(813\) −1.56658e10 −1.02243
\(814\) −7.30601e9 −0.474783
\(815\) −1.66513e9 −0.107745
\(816\) 1.14682e10 0.738892
\(817\) −5.70730e8 −0.0366145
\(818\) 3.67841e9 0.234976
\(819\) 2.84835e8 0.0181176
\(820\) 9.44483e8 0.0598199
\(821\) −1.97226e10 −1.24383 −0.621917 0.783083i \(-0.713646\pi\)
−0.621917 + 0.783083i \(0.713646\pi\)
\(822\) −2.61151e9 −0.163999
\(823\) 1.20133e9 0.0751215 0.0375608 0.999294i \(-0.488041\pi\)
0.0375608 + 0.999294i \(0.488041\pi\)
\(824\) 3.89040e9 0.242242
\(825\) −1.19011e10 −0.737901
\(826\) −5.14383e8 −0.0317583
\(827\) 8.23556e9 0.506319 0.253159 0.967425i \(-0.418530\pi\)
0.253159 + 0.967425i \(0.418530\pi\)
\(828\) 1.74845e9 0.107040
\(829\) 3.37482e9 0.205736 0.102868 0.994695i \(-0.467198\pi\)
0.102868 + 0.994695i \(0.467198\pi\)
\(830\) −8.57067e8 −0.0520285
\(831\) −5.63641e9 −0.340721
\(832\) 4.57731e8 0.0275536
\(833\) 1.60360e10 0.961255
\(834\) −6.20226e9 −0.370228
\(835\) 9.67020e8 0.0574821
\(836\) −2.18900e10 −1.29576
\(837\) 6.10878e9 0.360094
\(838\) 3.10735e9 0.182405
\(839\) 2.19151e10 1.28108 0.640540 0.767925i \(-0.278711\pi\)
0.640540 + 0.767925i \(0.278711\pi\)
\(840\) −4.27843e8 −0.0249061
\(841\) −1.65638e10 −0.960225
\(842\) 6.60375e9 0.381240
\(843\) 1.82292e9 0.104803
\(844\) 2.42305e10 1.38728
\(845\) 1.63836e9 0.0934140
\(846\) −2.16657e9 −0.123020
\(847\) −8.32297e9 −0.470637
\(848\) −1.49545e10 −0.842145
\(849\) −1.47996e10 −0.829988
\(850\) −1.18217e10 −0.660260
\(851\) 7.01627e9 0.390259
\(852\) −1.11875e9 −0.0619719
\(853\) −2.68331e10 −1.48030 −0.740148 0.672444i \(-0.765245\pi\)
−0.740148 + 0.672444i \(0.765245\pi\)
\(854\) −8.87446e8 −0.0487573
\(855\) 6.52178e8 0.0356849
\(856\) 1.03928e10 0.566336
\(857\) −3.79081e9 −0.205731 −0.102865 0.994695i \(-0.532801\pi\)
−0.102865 + 0.994695i \(0.532801\pi\)
\(858\) 3.62121e8 0.0195726
\(859\) −1.08182e10 −0.582342 −0.291171 0.956671i \(-0.594045\pi\)
−0.291171 + 0.956671i \(0.594045\pi\)
\(860\) 4.96810e7 0.00266346
\(861\) 5.54260e9 0.295939
\(862\) −6.50053e9 −0.345679
\(863\) −2.34587e10 −1.24241 −0.621206 0.783647i \(-0.713357\pi\)
−0.621206 + 0.783647i \(0.713357\pi\)
\(864\) 3.18576e9 0.168041
\(865\) 1.65441e9 0.0869133
\(866\) −7.12655e9 −0.372878
\(867\) −3.05867e10 −1.59392
\(868\) −2.25809e10 −1.17198
\(869\) −1.73607e10 −0.897426
\(870\) −7.21860e7 −0.00371651
\(871\) −2.75640e9 −0.141344
\(872\) 7.70299e9 0.393415
\(873\) 6.16774e9 0.313745
\(874\) −2.81229e9 −0.142485
\(875\) −2.63301e9 −0.132869
\(876\) −3.46915e9 −0.174365
\(877\) 3.42071e10 1.71245 0.856224 0.516605i \(-0.172805\pi\)
0.856224 + 0.516605i \(0.172805\pi\)
\(878\) 3.21390e9 0.160251
\(879\) −6.38900e9 −0.317302
\(880\) 1.61647e9 0.0799611
\(881\) −5.54512e9 −0.273209 −0.136605 0.990626i \(-0.543619\pi\)
−0.136605 + 0.990626i \(0.543619\pi\)
\(882\) 1.15651e9 0.0567558
\(883\) 1.95899e10 0.957568 0.478784 0.877933i \(-0.341078\pi\)
0.478784 + 0.877933i \(0.341078\pi\)
\(884\) −2.68881e9 −0.130911
\(885\) −1.45639e8 −0.00706280
\(886\) −1.74707e9 −0.0843904
\(887\) 2.41962e10 1.16416 0.582081 0.813131i \(-0.302238\pi\)
0.582081 + 0.813131i \(0.302238\pi\)
\(888\) 8.34816e9 0.400078
\(889\) −9.00695e9 −0.429954
\(890\) −3.35460e8 −0.0159506
\(891\) −3.02510e9 −0.143274
\(892\) 2.32110e10 1.09501
\(893\) −2.60489e10 −1.22408
\(894\) −5.00731e9 −0.234382
\(895\) 4.43324e8 0.0206700
\(896\) −1.52423e10 −0.707902
\(897\) −3.47759e8 −0.0160881
\(898\) 4.92254e9 0.226841
\(899\) −8.12941e9 −0.373164
\(900\) 6.37305e9 0.291406
\(901\) 5.43320e10 2.47468
\(902\) 7.04650e9 0.319706
\(903\) 2.91548e8 0.0131766
\(904\) −1.75123e10 −0.788412
\(905\) −1.42173e9 −0.0637599
\(906\) 1.80412e9 0.0805964
\(907\) −1.77861e10 −0.791510 −0.395755 0.918356i \(-0.629517\pi\)
−0.395755 + 0.918356i \(0.629517\pi\)
\(908\) −2.11125e10 −0.935919
\(909\) 5.55023e9 0.245097
\(910\) 3.98804e7 0.00175434
\(911\) −3.79625e10 −1.66357 −0.831783 0.555101i \(-0.812680\pi\)
−0.831783 + 0.555101i \(0.812680\pi\)
\(912\) 9.94417e9 0.434097
\(913\) 4.77976e10 2.07854
\(914\) 1.01560e9 0.0439956
\(915\) −2.51266e8 −0.0108432
\(916\) −1.14524e10 −0.492337
\(917\) −1.09176e10 −0.467556
\(918\) −3.00493e9 −0.128199
\(919\) −4.00064e10 −1.70030 −0.850149 0.526543i \(-0.823488\pi\)
−0.850149 + 0.526543i \(0.823488\pi\)
\(920\) 5.22359e8 0.0221163
\(921\) 7.67195e8 0.0323591
\(922\) 1.01334e10 0.425791
\(923\) 2.22514e8 0.00931434
\(924\) 1.11822e10 0.466309
\(925\) 2.55740e10 1.06244
\(926\) −1.19311e10 −0.493790
\(927\) 3.02939e9 0.124904
\(928\) −4.23953e9 −0.174140
\(929\) −4.55114e10 −1.86237 −0.931183 0.364552i \(-0.881222\pi\)
−0.931183 + 0.364552i \(0.881222\pi\)
\(930\) 8.55304e8 0.0348682
\(931\) 1.39049e10 0.564735
\(932\) 2.23994e10 0.906319
\(933\) −1.04357e10 −0.420663
\(934\) 1.12667e10 0.452461
\(935\) −5.87288e9 −0.234969
\(936\) −4.13774e8 −0.0164929
\(937\) 3.78655e9 0.150368 0.0751839 0.997170i \(-0.476046\pi\)
0.0751839 + 0.997170i \(0.476046\pi\)
\(938\) 1.13868e10 0.450498
\(939\) −2.10388e10 −0.829260
\(940\) 2.26751e9 0.0890435
\(941\) −9.78869e9 −0.382966 −0.191483 0.981496i \(-0.561330\pi\)
−0.191483 + 0.981496i \(0.561330\pi\)
\(942\) −2.66291e9 −0.103796
\(943\) −6.76705e9 −0.262790
\(944\) −2.22065e9 −0.0859170
\(945\) −3.33155e8 −0.0128421
\(946\) 3.70655e8 0.0142348
\(947\) 3.55199e10 1.35909 0.679543 0.733635i \(-0.262178\pi\)
0.679543 + 0.733635i \(0.262178\pi\)
\(948\) 9.29669e9 0.354405
\(949\) 6.89997e8 0.0262069
\(950\) −1.02507e10 −0.387901
\(951\) 1.08610e10 0.409486
\(952\) 2.37012e10 0.890309
\(953\) −1.87406e9 −0.0701389 −0.0350695 0.999385i \(-0.511165\pi\)
−0.0350695 + 0.999385i \(0.511165\pi\)
\(954\) 3.91841e9 0.146114
\(955\) 2.29755e9 0.0853596
\(956\) −1.02314e10 −0.378733
\(957\) 4.02573e9 0.148475
\(958\) 1.05767e10 0.388661
\(959\) 1.60394e10 0.587251
\(960\) −5.35380e8 −0.0195305
\(961\) 6.88096e10 2.50102
\(962\) −7.78154e8 −0.0281808
\(963\) 8.09271e9 0.292013
\(964\) 2.51056e10 0.902612
\(965\) 2.14843e9 0.0769620
\(966\) 1.43661e9 0.0512767
\(967\) 2.52830e10 0.899158 0.449579 0.893241i \(-0.351574\pi\)
0.449579 + 0.893241i \(0.351574\pi\)
\(968\) 1.20906e10 0.428434
\(969\) −3.61287e10 −1.27561
\(970\) 8.63559e8 0.0303802
\(971\) 2.33470e10 0.818396 0.409198 0.912446i \(-0.365808\pi\)
0.409198 + 0.912446i \(0.365808\pi\)
\(972\) 1.61995e9 0.0565807
\(973\) 3.80932e10 1.32572
\(974\) 1.62466e10 0.563385
\(975\) −1.26757e9 −0.0437982
\(976\) −3.83121e9 −0.131905
\(977\) −3.73874e9 −0.128261 −0.0641304 0.997942i \(-0.520427\pi\)
−0.0641304 + 0.997942i \(0.520427\pi\)
\(978\) 6.65256e9 0.227406
\(979\) 1.87082e10 0.637225
\(980\) −1.21040e9 −0.0410806
\(981\) 5.99820e9 0.202852
\(982\) −4.51716e9 −0.152221
\(983\) −4.06366e10 −1.36452 −0.682261 0.731109i \(-0.739003\pi\)
−0.682261 + 0.731109i \(0.739003\pi\)
\(984\) −8.05163e9 −0.269402
\(985\) 5.05041e9 0.168384
\(986\) 3.99889e9 0.132853
\(987\) 1.33067e10 0.440514
\(988\) −2.33148e9 −0.0769099
\(989\) −3.55956e8 −0.0117006
\(990\) −4.23551e8 −0.0138734
\(991\) 2.46369e10 0.804134 0.402067 0.915610i \(-0.368292\pi\)
0.402067 + 0.915610i \(0.368292\pi\)
\(992\) 5.02325e10 1.63378
\(993\) 1.63735e10 0.530663
\(994\) −9.19219e8 −0.0296870
\(995\) 3.80660e7 0.00122506
\(996\) −2.55957e10 −0.820841
\(997\) −1.08445e10 −0.346558 −0.173279 0.984873i \(-0.555436\pi\)
−0.173279 + 0.984873i \(0.555436\pi\)
\(998\) 9.56186e9 0.304499
\(999\) 6.50058e9 0.206288
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 177.8.a.c.1.8 17
3.2 odd 2 531.8.a.c.1.10 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.8 17 1.1 even 1 trivial
531.8.a.c.1.10 17 3.2 odd 2