Properties

Label 177.8.a.c.1.9
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 1.9
Root \(-3.62331\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.62331 q^{2} -27.0000 q^{3} -114.872 q^{4} +300.817 q^{5} +97.8295 q^{6} +1353.78 q^{7} +880.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-3.62331 q^{2} -27.0000 q^{3} -114.872 q^{4} +300.817 q^{5} +97.8295 q^{6} +1353.78 q^{7} +880.000 q^{8} +729.000 q^{9} -1089.96 q^{10} +8127.65 q^{11} +3101.53 q^{12} +1822.67 q^{13} -4905.17 q^{14} -8122.07 q^{15} +11515.0 q^{16} +1656.90 q^{17} -2641.40 q^{18} +18541.4 q^{19} -34555.4 q^{20} -36552.0 q^{21} -29449.0 q^{22} +15601.2 q^{23} -23760.0 q^{24} +12366.0 q^{25} -6604.11 q^{26} -19683.0 q^{27} -155511. q^{28} +150338. q^{29} +29428.8 q^{30} -248513. q^{31} -154363. q^{32} -219447. q^{33} -6003.46 q^{34} +407240. q^{35} -83741.4 q^{36} -170725. q^{37} -67181.2 q^{38} -49212.1 q^{39} +264719. q^{40} +283414. q^{41} +132439. q^{42} +39428.6 q^{43} -933636. q^{44} +219296. q^{45} -56528.0 q^{46} -77426.6 q^{47} -310906. q^{48} +1.00917e6 q^{49} -44805.9 q^{50} -44736.3 q^{51} -209373. q^{52} +1.98028e6 q^{53} +71317.7 q^{54} +2.44494e6 q^{55} +1.19133e6 q^{56} -500617. q^{57} -544721. q^{58} -205379. q^{59} +932995. q^{60} -938894. q^{61} +900440. q^{62} +986905. q^{63} -914622. q^{64} +548291. q^{65} +795124. q^{66} +2.93505e6 q^{67} -190331. q^{68} -421232. q^{69} -1.47556e6 q^{70} -5.87325e6 q^{71} +641520. q^{72} -2.53313e6 q^{73} +618590. q^{74} -333882. q^{75} -2.12988e6 q^{76} +1.10030e7 q^{77} +178311. q^{78} -1.99924e6 q^{79} +3.46392e6 q^{80} +531441. q^{81} -1.02690e6 q^{82} +5.33794e6 q^{83} +4.19879e6 q^{84} +498424. q^{85} -142862. q^{86} -4.05912e6 q^{87} +7.15233e6 q^{88} +2.14988e6 q^{89} -794577. q^{90} +2.46749e6 q^{91} -1.79213e6 q^{92} +6.70985e6 q^{93} +280541. q^{94} +5.57757e6 q^{95} +4.16779e6 q^{96} -3.36902e6 q^{97} -3.65656e6 q^{98} +5.92506e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.62331 −0.320259 −0.160129 0.987096i \(-0.551191\pi\)
−0.160129 + 0.987096i \(0.551191\pi\)
\(3\) −27.0000 −0.577350
\(4\) −114.872 −0.897434
\(5\) 300.817 1.07624 0.538118 0.842869i \(-0.319135\pi\)
0.538118 + 0.842869i \(0.319135\pi\)
\(6\) 97.8295 0.184901
\(7\) 1353.78 1.49178 0.745890 0.666070i \(-0.232025\pi\)
0.745890 + 0.666070i \(0.232025\pi\)
\(8\) 880.000 0.607670
\(9\) 729.000 0.333333
\(10\) −1089.96 −0.344674
\(11\) 8127.65 1.84116 0.920578 0.390558i \(-0.127718\pi\)
0.920578 + 0.390558i \(0.127718\pi\)
\(12\) 3101.53 0.518134
\(13\) 1822.67 0.230095 0.115047 0.993360i \(-0.463298\pi\)
0.115047 + 0.993360i \(0.463298\pi\)
\(14\) −4905.17 −0.477755
\(15\) −8122.07 −0.621365
\(16\) 11515.0 0.702823
\(17\) 1656.90 0.0817946 0.0408973 0.999163i \(-0.486978\pi\)
0.0408973 + 0.999163i \(0.486978\pi\)
\(18\) −2641.40 −0.106753
\(19\) 18541.4 0.620161 0.310081 0.950710i \(-0.399644\pi\)
0.310081 + 0.950710i \(0.399644\pi\)
\(20\) −34555.4 −0.965852
\(21\) −36552.0 −0.861279
\(22\) −29449.0 −0.589646
\(23\) 15601.2 0.267369 0.133684 0.991024i \(-0.457319\pi\)
0.133684 + 0.991024i \(0.457319\pi\)
\(24\) −23760.0 −0.350838
\(25\) 12366.0 0.158285
\(26\) −6604.11 −0.0736899
\(27\) −19683.0 −0.192450
\(28\) −155511. −1.33877
\(29\) 150338. 1.14466 0.572328 0.820025i \(-0.306040\pi\)
0.572328 + 0.820025i \(0.306040\pi\)
\(30\) 29428.8 0.198998
\(31\) −248513. −1.49825 −0.749123 0.662431i \(-0.769525\pi\)
−0.749123 + 0.662431i \(0.769525\pi\)
\(32\) −154363. −0.832755
\(33\) −219447. −1.06299
\(34\) −6003.46 −0.0261954
\(35\) 407240. 1.60551
\(36\) −83741.4 −0.299145
\(37\) −170725. −0.554104 −0.277052 0.960855i \(-0.589357\pi\)
−0.277052 + 0.960855i \(0.589357\pi\)
\(38\) −67181.2 −0.198612
\(39\) −49212.1 −0.132845
\(40\) 264719. 0.653997
\(41\) 283414. 0.642210 0.321105 0.947044i \(-0.395946\pi\)
0.321105 + 0.947044i \(0.395946\pi\)
\(42\) 132439. 0.275832
\(43\) 39428.6 0.0756261 0.0378130 0.999285i \(-0.487961\pi\)
0.0378130 + 0.999285i \(0.487961\pi\)
\(44\) −933636. −1.65232
\(45\) 219296. 0.358745
\(46\) −56528.0 −0.0856271
\(47\) −77426.6 −0.108780 −0.0543898 0.998520i \(-0.517321\pi\)
−0.0543898 + 0.998520i \(0.517321\pi\)
\(48\) −310906. −0.405775
\(49\) 1.00917e6 1.22541
\(50\) −44805.9 −0.0506921
\(51\) −44736.3 −0.0472241
\(52\) −209373. −0.206495
\(53\) 1.98028e6 1.82709 0.913546 0.406735i \(-0.133333\pi\)
0.913546 + 0.406735i \(0.133333\pi\)
\(54\) 71317.7 0.0616338
\(55\) 2.44494e6 1.98152
\(56\) 1.19133e6 0.906509
\(57\) −500617. −0.358050
\(58\) −544721. −0.366586
\(59\) −205379. −0.130189
\(60\) 932995. 0.557635
\(61\) −938894. −0.529617 −0.264808 0.964301i \(-0.585309\pi\)
−0.264808 + 0.964301i \(0.585309\pi\)
\(62\) 900440. 0.479826
\(63\) 986905. 0.497260
\(64\) −914622. −0.436126
\(65\) 548291. 0.247636
\(66\) 795124. 0.340433
\(67\) 2.93505e6 1.19221 0.596107 0.802905i \(-0.296714\pi\)
0.596107 + 0.802905i \(0.296714\pi\)
\(68\) −190331. −0.0734053
\(69\) −421232. −0.154365
\(70\) −1.47556e6 −0.514178
\(71\) −5.87325e6 −1.94749 −0.973743 0.227650i \(-0.926896\pi\)
−0.973743 + 0.227650i \(0.926896\pi\)
\(72\) 641520. 0.202557
\(73\) −2.53313e6 −0.762126 −0.381063 0.924549i \(-0.624442\pi\)
−0.381063 + 0.924549i \(0.624442\pi\)
\(74\) 618590. 0.177457
\(75\) −333882. −0.0913859
\(76\) −2.12988e6 −0.556554
\(77\) 1.10030e7 2.74660
\(78\) 178311. 0.0425449
\(79\) −1.99924e6 −0.456217 −0.228108 0.973636i \(-0.573254\pi\)
−0.228108 + 0.973636i \(0.573254\pi\)
\(80\) 3.46392e6 0.756403
\(81\) 531441. 0.111111
\(82\) −1.02690e6 −0.205673
\(83\) 5.33794e6 1.02471 0.512354 0.858774i \(-0.328773\pi\)
0.512354 + 0.858774i \(0.328773\pi\)
\(84\) 4.19879e6 0.772942
\(85\) 498424. 0.0880303
\(86\) −142862. −0.0242199
\(87\) −4.05912e6 −0.660867
\(88\) 7.15233e6 1.11882
\(89\) 2.14988e6 0.323258 0.161629 0.986852i \(-0.448325\pi\)
0.161629 + 0.986852i \(0.448325\pi\)
\(90\) −794577. −0.114891
\(91\) 2.46749e6 0.343251
\(92\) −1.79213e6 −0.239946
\(93\) 6.70985e6 0.865012
\(94\) 280541. 0.0348376
\(95\) 5.57757e6 0.667440
\(96\) 4.16779e6 0.480791
\(97\) −3.36902e6 −0.374802 −0.187401 0.982283i \(-0.560006\pi\)
−0.187401 + 0.982283i \(0.560006\pi\)
\(98\) −3.65656e6 −0.392447
\(99\) 5.92506e6 0.613719
\(100\) −1.42050e6 −0.142050
\(101\) 1.18390e6 0.114338 0.0571688 0.998365i \(-0.481793\pi\)
0.0571688 + 0.998365i \(0.481793\pi\)
\(102\) 162094. 0.0151239
\(103\) −1.00671e7 −0.907764 −0.453882 0.891062i \(-0.649961\pi\)
−0.453882 + 0.891062i \(0.649961\pi\)
\(104\) 1.60395e6 0.139822
\(105\) −1.09955e7 −0.926940
\(106\) −7.17517e6 −0.585142
\(107\) −5.24702e6 −0.414066 −0.207033 0.978334i \(-0.566381\pi\)
−0.207033 + 0.978334i \(0.566381\pi\)
\(108\) 2.26102e6 0.172711
\(109\) 1.31376e7 0.971681 0.485841 0.874047i \(-0.338514\pi\)
0.485841 + 0.874047i \(0.338514\pi\)
\(110\) −8.85878e6 −0.634599
\(111\) 4.60958e6 0.319912
\(112\) 1.55888e7 1.04846
\(113\) 9.32454e6 0.607929 0.303965 0.952683i \(-0.401690\pi\)
0.303965 + 0.952683i \(0.401690\pi\)
\(114\) 1.81389e6 0.114669
\(115\) 4.69311e6 0.287752
\(116\) −1.72695e7 −1.02725
\(117\) 1.32873e6 0.0766983
\(118\) 744153. 0.0416941
\(119\) 2.24307e6 0.122020
\(120\) −7.14742e6 −0.377585
\(121\) 4.65715e7 2.38986
\(122\) 3.40191e6 0.169614
\(123\) −7.65217e6 −0.370780
\(124\) 2.85471e7 1.34458
\(125\) −1.97814e7 −0.905884
\(126\) −3.57587e6 −0.159252
\(127\) 1.14096e7 0.494263 0.247131 0.968982i \(-0.420512\pi\)
0.247131 + 0.968982i \(0.420512\pi\)
\(128\) 2.30724e7 0.972428
\(129\) −1.06457e6 −0.0436627
\(130\) −1.98663e6 −0.0793077
\(131\) 1.68449e7 0.654666 0.327333 0.944909i \(-0.393850\pi\)
0.327333 + 0.944909i \(0.393850\pi\)
\(132\) 2.52082e7 0.953966
\(133\) 2.51009e7 0.925143
\(134\) −1.06346e7 −0.381817
\(135\) −5.92099e6 −0.207122
\(136\) 1.45807e6 0.0497041
\(137\) −5.53033e7 −1.83751 −0.918754 0.394830i \(-0.870804\pi\)
−0.918754 + 0.394830i \(0.870804\pi\)
\(138\) 1.52626e6 0.0494368
\(139\) −392568. −0.0123983 −0.00619916 0.999981i \(-0.501973\pi\)
−0.00619916 + 0.999981i \(0.501973\pi\)
\(140\) −4.67803e7 −1.44084
\(141\) 2.09052e6 0.0628040
\(142\) 2.12806e7 0.623699
\(143\) 1.48140e7 0.423640
\(144\) 8.39447e6 0.234274
\(145\) 4.52242e7 1.23192
\(146\) 9.17831e6 0.244078
\(147\) −2.72477e7 −0.707488
\(148\) 1.96115e7 0.497272
\(149\) −3.39826e7 −0.841598 −0.420799 0.907154i \(-0.638250\pi\)
−0.420799 + 0.907154i \(0.638250\pi\)
\(150\) 1.20976e6 0.0292671
\(151\) 1.86602e7 0.441059 0.220530 0.975380i \(-0.429221\pi\)
0.220530 + 0.975380i \(0.429221\pi\)
\(152\) 1.63164e7 0.376853
\(153\) 1.20788e6 0.0272649
\(154\) −3.98675e7 −0.879622
\(155\) −7.47570e7 −1.61247
\(156\) 5.65308e6 0.119220
\(157\) 3.31311e7 0.683262 0.341631 0.939834i \(-0.389021\pi\)
0.341631 + 0.939834i \(0.389021\pi\)
\(158\) 7.24389e6 0.146107
\(159\) −5.34675e7 −1.05487
\(160\) −4.64349e7 −0.896241
\(161\) 2.11206e7 0.398855
\(162\) −1.92558e6 −0.0355843
\(163\) −7.04997e7 −1.27506 −0.637530 0.770426i \(-0.720044\pi\)
−0.637530 + 0.770426i \(0.720044\pi\)
\(164\) −3.25562e7 −0.576342
\(165\) −6.60133e7 −1.14403
\(166\) −1.93410e7 −0.328172
\(167\) −5.02367e7 −0.834667 −0.417333 0.908753i \(-0.637035\pi\)
−0.417333 + 0.908753i \(0.637035\pi\)
\(168\) −3.21658e7 −0.523373
\(169\) −5.94264e7 −0.947056
\(170\) −1.80595e6 −0.0281925
\(171\) 1.35167e7 0.206720
\(172\) −4.52922e6 −0.0678694
\(173\) 3.60070e7 0.528719 0.264360 0.964424i \(-0.414839\pi\)
0.264360 + 0.964424i \(0.414839\pi\)
\(174\) 1.47075e7 0.211649
\(175\) 1.67408e7 0.236126
\(176\) 9.35903e7 1.29401
\(177\) 5.54523e6 0.0751646
\(178\) −7.78968e6 −0.103526
\(179\) −6.95659e7 −0.906589 −0.453295 0.891361i \(-0.649751\pi\)
−0.453295 + 0.891361i \(0.649751\pi\)
\(180\) −2.51909e7 −0.321951
\(181\) 1.45844e8 1.82816 0.914079 0.405536i \(-0.132915\pi\)
0.914079 + 0.405536i \(0.132915\pi\)
\(182\) −8.94051e6 −0.109929
\(183\) 2.53501e7 0.305775
\(184\) 1.37291e7 0.162472
\(185\) −5.13570e7 −0.596347
\(186\) −2.43119e7 −0.277028
\(187\) 1.34667e7 0.150597
\(188\) 8.89412e6 0.0976226
\(189\) −2.66464e7 −0.287093
\(190\) −2.02093e7 −0.213753
\(191\) 3.79780e7 0.394380 0.197190 0.980365i \(-0.436818\pi\)
0.197190 + 0.980365i \(0.436818\pi\)
\(192\) 2.46948e7 0.251797
\(193\) −1.18553e8 −1.18704 −0.593518 0.804821i \(-0.702261\pi\)
−0.593518 + 0.804821i \(0.702261\pi\)
\(194\) 1.22070e7 0.120034
\(195\) −1.48039e7 −0.142973
\(196\) −1.15925e8 −1.09972
\(197\) −1.37665e8 −1.28290 −0.641450 0.767165i \(-0.721667\pi\)
−0.641450 + 0.767165i \(0.721667\pi\)
\(198\) −2.14683e7 −0.196549
\(199\) −1.00525e8 −0.904248 −0.452124 0.891955i \(-0.649334\pi\)
−0.452124 + 0.891955i \(0.649334\pi\)
\(200\) 1.08821e7 0.0961850
\(201\) −7.92464e7 −0.688325
\(202\) −4.28963e6 −0.0366176
\(203\) 2.03524e8 1.70757
\(204\) 5.13893e6 0.0423806
\(205\) 8.52557e7 0.691170
\(206\) 3.64762e7 0.290719
\(207\) 1.13733e7 0.0891229
\(208\) 2.09882e7 0.161716
\(209\) 1.50698e8 1.14181
\(210\) 3.98401e7 0.296861
\(211\) 2.62670e8 1.92496 0.962480 0.271354i \(-0.0874714\pi\)
0.962480 + 0.271354i \(0.0874714\pi\)
\(212\) −2.27478e8 −1.63970
\(213\) 1.58578e8 1.12438
\(214\) 1.90116e7 0.132608
\(215\) 1.18608e7 0.0813915
\(216\) −1.73210e7 −0.116946
\(217\) −3.36432e8 −2.23505
\(218\) −4.76017e7 −0.311189
\(219\) 6.83944e7 0.440014
\(220\) −2.80854e8 −1.77828
\(221\) 3.01998e6 0.0188205
\(222\) −1.67019e7 −0.102455
\(223\) 2.67872e8 1.61756 0.808781 0.588110i \(-0.200128\pi\)
0.808781 + 0.588110i \(0.200128\pi\)
\(224\) −2.08973e8 −1.24229
\(225\) 9.01482e6 0.0527616
\(226\) −3.37857e7 −0.194695
\(227\) −9.46663e7 −0.537161 −0.268581 0.963257i \(-0.586555\pi\)
−0.268581 + 0.963257i \(0.586555\pi\)
\(228\) 5.75067e7 0.321326
\(229\) 2.26035e8 1.24380 0.621900 0.783096i \(-0.286361\pi\)
0.621900 + 0.783096i \(0.286361\pi\)
\(230\) −1.70046e7 −0.0921550
\(231\) −2.97082e8 −1.58575
\(232\) 1.32297e8 0.695573
\(233\) −6.28293e7 −0.325399 −0.162700 0.986676i \(-0.552020\pi\)
−0.162700 + 0.986676i \(0.552020\pi\)
\(234\) −4.81440e6 −0.0245633
\(235\) −2.32913e7 −0.117073
\(236\) 2.35922e7 0.116836
\(237\) 5.39796e7 0.263397
\(238\) −8.12736e6 −0.0390778
\(239\) −1.86677e7 −0.0884499 −0.0442250 0.999022i \(-0.514082\pi\)
−0.0442250 + 0.999022i \(0.514082\pi\)
\(240\) −9.35260e7 −0.436710
\(241\) −2.43058e7 −0.111854 −0.0559268 0.998435i \(-0.517811\pi\)
−0.0559268 + 0.998435i \(0.517811\pi\)
\(242\) −1.68743e8 −0.765372
\(243\) −1.43489e7 −0.0641500
\(244\) 1.07852e8 0.475296
\(245\) 3.03577e8 1.31883
\(246\) 2.77262e7 0.118746
\(247\) 3.37949e7 0.142696
\(248\) −2.18691e8 −0.910439
\(249\) −1.44124e8 −0.591616
\(250\) 7.16744e7 0.290117
\(251\) −1.85068e8 −0.738708 −0.369354 0.929289i \(-0.620421\pi\)
−0.369354 + 0.929289i \(0.620421\pi\)
\(252\) −1.13367e8 −0.446258
\(253\) 1.26801e8 0.492267
\(254\) −4.13406e7 −0.158292
\(255\) −1.34574e7 −0.0508243
\(256\) 3.34731e7 0.124697
\(257\) 7.99627e7 0.293847 0.146924 0.989148i \(-0.453063\pi\)
0.146924 + 0.989148i \(0.453063\pi\)
\(258\) 3.85728e6 0.0139834
\(259\) −2.31124e8 −0.826601
\(260\) −6.29831e7 −0.222237
\(261\) 1.09596e8 0.381552
\(262\) −6.10345e7 −0.209663
\(263\) −4.27501e7 −0.144908 −0.0724539 0.997372i \(-0.523083\pi\)
−0.0724539 + 0.997372i \(0.523083\pi\)
\(264\) −1.93113e8 −0.645948
\(265\) 5.95702e8 1.96638
\(266\) −9.09486e7 −0.296285
\(267\) −5.80467e7 −0.186633
\(268\) −3.37154e8 −1.06993
\(269\) −2.49600e8 −0.781830 −0.390915 0.920427i \(-0.627841\pi\)
−0.390915 + 0.920427i \(0.627841\pi\)
\(270\) 2.14536e7 0.0663326
\(271\) −6.59051e7 −0.201153 −0.100577 0.994929i \(-0.532069\pi\)
−0.100577 + 0.994929i \(0.532069\pi\)
\(272\) 1.90793e7 0.0574871
\(273\) −6.66224e7 −0.198176
\(274\) 2.00381e8 0.588478
\(275\) 1.00507e8 0.291427
\(276\) 4.83876e7 0.138533
\(277\) 9.39872e7 0.265699 0.132849 0.991136i \(-0.457587\pi\)
0.132849 + 0.991136i \(0.457587\pi\)
\(278\) 1.42240e6 0.00397067
\(279\) −1.81166e8 −0.499415
\(280\) 3.58371e8 0.975619
\(281\) 1.08507e8 0.291733 0.145867 0.989304i \(-0.453403\pi\)
0.145867 + 0.989304i \(0.453403\pi\)
\(282\) −7.57460e6 −0.0201135
\(283\) 6.20243e8 1.62671 0.813353 0.581770i \(-0.197640\pi\)
0.813353 + 0.581770i \(0.197640\pi\)
\(284\) 6.74670e8 1.74774
\(285\) −1.50594e8 −0.385347
\(286\) −5.36759e7 −0.135675
\(287\) 3.83679e8 0.958036
\(288\) −1.12530e8 −0.277585
\(289\) −4.07593e8 −0.993310
\(290\) −1.63861e8 −0.394533
\(291\) 9.09634e7 0.216392
\(292\) 2.90984e8 0.683958
\(293\) 6.28432e8 1.45956 0.729779 0.683683i \(-0.239623\pi\)
0.729779 + 0.683683i \(0.239623\pi\)
\(294\) 9.87270e7 0.226579
\(295\) −6.17815e7 −0.140114
\(296\) −1.50238e8 −0.336712
\(297\) −1.59977e8 −0.354331
\(298\) 1.23130e8 0.269529
\(299\) 2.84359e7 0.0615201
\(300\) 3.83536e7 0.0820128
\(301\) 5.33776e7 0.112817
\(302\) −6.76117e7 −0.141253
\(303\) −3.19652e7 −0.0660129
\(304\) 2.13505e8 0.435863
\(305\) −2.82435e8 −0.569993
\(306\) −4.37653e6 −0.00873181
\(307\) 7.45356e8 1.47021 0.735105 0.677953i \(-0.237133\pi\)
0.735105 + 0.677953i \(0.237133\pi\)
\(308\) −1.26394e9 −2.46489
\(309\) 2.71811e8 0.524098
\(310\) 2.70868e8 0.516407
\(311\) 7.31974e8 1.37986 0.689929 0.723877i \(-0.257642\pi\)
0.689929 + 0.723877i \(0.257642\pi\)
\(312\) −4.33067e7 −0.0807261
\(313\) 3.03184e8 0.558857 0.279429 0.960166i \(-0.409855\pi\)
0.279429 + 0.960166i \(0.409855\pi\)
\(314\) −1.20044e8 −0.218821
\(315\) 2.96878e8 0.535169
\(316\) 2.29656e8 0.409425
\(317\) 2.79655e8 0.493077 0.246539 0.969133i \(-0.420707\pi\)
0.246539 + 0.969133i \(0.420707\pi\)
\(318\) 1.93730e8 0.337832
\(319\) 1.22189e9 2.10749
\(320\) −2.75134e8 −0.469374
\(321\) 1.41670e8 0.239061
\(322\) −7.65264e7 −0.127737
\(323\) 3.07212e7 0.0507258
\(324\) −6.10475e7 −0.0997149
\(325\) 2.25392e7 0.0364205
\(326\) 2.55443e8 0.408349
\(327\) −3.54716e8 −0.561000
\(328\) 2.49404e8 0.390252
\(329\) −1.04818e8 −0.162275
\(330\) 2.39187e8 0.366386
\(331\) 1.20182e9 1.82156 0.910778 0.412897i \(-0.135483\pi\)
0.910778 + 0.412897i \(0.135483\pi\)
\(332\) −6.13178e8 −0.919609
\(333\) −1.24459e8 −0.184701
\(334\) 1.82023e8 0.267309
\(335\) 8.82914e8 1.28310
\(336\) −4.20898e8 −0.605327
\(337\) −1.16073e9 −1.65206 −0.826031 0.563624i \(-0.809407\pi\)
−0.826031 + 0.563624i \(0.809407\pi\)
\(338\) 2.15320e8 0.303303
\(339\) −2.51763e8 −0.350988
\(340\) −5.72547e7 −0.0790015
\(341\) −2.01983e9 −2.75850
\(342\) −4.89751e7 −0.0662040
\(343\) 2.51304e8 0.336255
\(344\) 3.46971e7 0.0459557
\(345\) −1.26714e8 −0.166134
\(346\) −1.30465e8 −0.169327
\(347\) −1.18359e8 −0.152071 −0.0760356 0.997105i \(-0.524226\pi\)
−0.0760356 + 0.997105i \(0.524226\pi\)
\(348\) 4.66277e8 0.593085
\(349\) 2.87181e8 0.361632 0.180816 0.983517i \(-0.442126\pi\)
0.180816 + 0.983517i \(0.442126\pi\)
\(350\) −6.06573e7 −0.0756215
\(351\) −3.58757e7 −0.0442818
\(352\) −1.25461e9 −1.53323
\(353\) −4.86525e8 −0.588700 −0.294350 0.955698i \(-0.595103\pi\)
−0.294350 + 0.955698i \(0.595103\pi\)
\(354\) −2.00921e7 −0.0240721
\(355\) −1.76678e9 −2.09596
\(356\) −2.46960e8 −0.290102
\(357\) −6.05630e7 −0.0704480
\(358\) 2.52059e8 0.290343
\(359\) 1.52624e9 1.74098 0.870488 0.492189i \(-0.163803\pi\)
0.870488 + 0.492189i \(0.163803\pi\)
\(360\) 1.92980e8 0.217999
\(361\) −5.50089e8 −0.615400
\(362\) −5.28439e8 −0.585484
\(363\) −1.25743e9 −1.37978
\(364\) −2.83445e8 −0.308045
\(365\) −7.62008e8 −0.820228
\(366\) −9.18515e7 −0.0979270
\(367\) −1.48262e8 −0.156566 −0.0782832 0.996931i \(-0.524944\pi\)
−0.0782832 + 0.996931i \(0.524944\pi\)
\(368\) 1.79648e8 0.187913
\(369\) 2.06609e8 0.214070
\(370\) 1.86083e8 0.190985
\(371\) 2.68086e9 2.72562
\(372\) −7.70771e8 −0.776292
\(373\) −1.55988e9 −1.55636 −0.778180 0.628041i \(-0.783857\pi\)
−0.778180 + 0.628041i \(0.783857\pi\)
\(374\) −4.87941e7 −0.0482299
\(375\) 5.34099e8 0.523013
\(376\) −6.81354e7 −0.0661021
\(377\) 2.74016e8 0.263379
\(378\) 9.65484e7 0.0919441
\(379\) 1.77311e9 1.67301 0.836504 0.547961i \(-0.184596\pi\)
0.836504 + 0.547961i \(0.184596\pi\)
\(380\) −6.40704e8 −0.598983
\(381\) −3.08059e8 −0.285363
\(382\) −1.37606e8 −0.126304
\(383\) 1.71331e9 1.55826 0.779129 0.626863i \(-0.215662\pi\)
0.779129 + 0.626863i \(0.215662\pi\)
\(384\) −6.22954e8 −0.561432
\(385\) 3.30990e9 2.95599
\(386\) 4.29557e8 0.380159
\(387\) 2.87434e7 0.0252087
\(388\) 3.87004e8 0.336360
\(389\) −9.34250e8 −0.804711 −0.402355 0.915484i \(-0.631808\pi\)
−0.402355 + 0.915484i \(0.631808\pi\)
\(390\) 5.36390e7 0.0457883
\(391\) 2.58496e7 0.0218693
\(392\) 8.88073e8 0.744642
\(393\) −4.54814e8 −0.377972
\(394\) 4.98805e8 0.410860
\(395\) −6.01407e8 −0.490997
\(396\) −6.80621e8 −0.550772
\(397\) 7.43414e8 0.596299 0.298150 0.954519i \(-0.403631\pi\)
0.298150 + 0.954519i \(0.403631\pi\)
\(398\) 3.64233e8 0.289593
\(399\) −6.77725e8 −0.534132
\(400\) 1.42395e8 0.111246
\(401\) 2.43334e9 1.88450 0.942252 0.334905i \(-0.108704\pi\)
0.942252 + 0.334905i \(0.108704\pi\)
\(402\) 2.87135e8 0.220442
\(403\) −4.52957e8 −0.344739
\(404\) −1.35996e8 −0.102611
\(405\) 1.59867e8 0.119582
\(406\) −7.37432e8 −0.546866
\(407\) −1.38759e9 −1.02019
\(408\) −3.93679e7 −0.0286967
\(409\) −2.52851e9 −1.82740 −0.913701 0.406388i \(-0.866788\pi\)
−0.913701 + 0.406388i \(0.866788\pi\)
\(410\) −3.08908e8 −0.221353
\(411\) 1.49319e9 1.06089
\(412\) 1.15642e9 0.814659
\(413\) −2.78038e8 −0.194213
\(414\) −4.12089e7 −0.0285424
\(415\) 1.60574e9 1.10283
\(416\) −2.81352e8 −0.191613
\(417\) 1.05993e7 0.00715818
\(418\) −5.46026e8 −0.365676
\(419\) −1.34539e9 −0.893507 −0.446753 0.894657i \(-0.647420\pi\)
−0.446753 + 0.894657i \(0.647420\pi\)
\(420\) 1.26307e9 0.831868
\(421\) 1.01782e9 0.664790 0.332395 0.943140i \(-0.392143\pi\)
0.332395 + 0.943140i \(0.392143\pi\)
\(422\) −9.51735e8 −0.616485
\(423\) −5.64440e7 −0.0362599
\(424\) 1.74264e9 1.11027
\(425\) 2.04892e7 0.0129469
\(426\) −5.74577e8 −0.360093
\(427\) −1.27105e9 −0.790072
\(428\) 6.02734e8 0.371597
\(429\) −3.99979e8 −0.244589
\(430\) −4.29754e7 −0.0260663
\(431\) −2.08550e9 −1.25470 −0.627349 0.778738i \(-0.715860\pi\)
−0.627349 + 0.778738i \(0.715860\pi\)
\(432\) −2.26651e8 −0.135258
\(433\) −1.27758e9 −0.756277 −0.378139 0.925749i \(-0.623436\pi\)
−0.378139 + 0.925749i \(0.623436\pi\)
\(434\) 1.21900e9 0.715795
\(435\) −1.22105e9 −0.711250
\(436\) −1.50914e9 −0.872020
\(437\) 2.89268e8 0.165812
\(438\) −2.47814e8 −0.140918
\(439\) −3.72523e8 −0.210149 −0.105074 0.994464i \(-0.533508\pi\)
−0.105074 + 0.994464i \(0.533508\pi\)
\(440\) 2.15155e9 1.20411
\(441\) 7.35688e8 0.408469
\(442\) −1.09423e7 −0.00602743
\(443\) 2.51167e8 0.137262 0.0686309 0.997642i \(-0.478137\pi\)
0.0686309 + 0.997642i \(0.478137\pi\)
\(444\) −5.29509e8 −0.287100
\(445\) 6.46720e8 0.347902
\(446\) −9.70586e8 −0.518038
\(447\) 9.17531e8 0.485897
\(448\) −1.23820e9 −0.650603
\(449\) −7.22359e8 −0.376609 −0.188304 0.982111i \(-0.560299\pi\)
−0.188304 + 0.982111i \(0.560299\pi\)
\(450\) −3.26635e7 −0.0168974
\(451\) 2.30349e9 1.18241
\(452\) −1.07113e9 −0.545577
\(453\) −5.03825e8 −0.254646
\(454\) 3.43006e8 0.172031
\(455\) 7.42265e8 0.369419
\(456\) −4.40543e8 −0.217576
\(457\) −3.81278e9 −1.86868 −0.934341 0.356380i \(-0.884011\pi\)
−0.934341 + 0.356380i \(0.884011\pi\)
\(458\) −8.18995e8 −0.398338
\(459\) −3.26127e7 −0.0157414
\(460\) −5.39105e8 −0.258238
\(461\) −2.43626e9 −1.15817 −0.579083 0.815269i \(-0.696589\pi\)
−0.579083 + 0.815269i \(0.696589\pi\)
\(462\) 1.07642e9 0.507850
\(463\) −2.87487e9 −1.34612 −0.673061 0.739587i \(-0.735021\pi\)
−0.673061 + 0.739587i \(0.735021\pi\)
\(464\) 1.73115e9 0.804490
\(465\) 2.01844e9 0.930958
\(466\) 2.27650e8 0.104212
\(467\) −1.80431e9 −0.819787 −0.409894 0.912133i \(-0.634434\pi\)
−0.409894 + 0.912133i \(0.634434\pi\)
\(468\) −1.52633e8 −0.0688317
\(469\) 3.97341e9 1.77852
\(470\) 8.43915e7 0.0374935
\(471\) −8.94541e8 −0.394482
\(472\) −1.80734e8 −0.0791119
\(473\) 3.20462e8 0.139239
\(474\) −1.95585e8 −0.0843551
\(475\) 2.29283e8 0.0981621
\(476\) −2.57666e8 −0.109504
\(477\) 1.44362e9 0.609031
\(478\) 6.76388e7 0.0283269
\(479\) −2.36338e9 −0.982558 −0.491279 0.871002i \(-0.663471\pi\)
−0.491279 + 0.871002i \(0.663471\pi\)
\(480\) 1.25374e9 0.517445
\(481\) −3.11176e8 −0.127496
\(482\) 8.80674e7 0.0358221
\(483\) −5.70255e8 −0.230279
\(484\) −5.34975e9 −2.14474
\(485\) −1.01346e9 −0.403376
\(486\) 5.19906e7 0.0205446
\(487\) 3.50576e9 1.37541 0.687703 0.725993i \(-0.258619\pi\)
0.687703 + 0.725993i \(0.258619\pi\)
\(488\) −8.26226e8 −0.321832
\(489\) 1.90349e9 0.736156
\(490\) −1.09995e9 −0.422366
\(491\) −3.92494e9 −1.49640 −0.748199 0.663474i \(-0.769081\pi\)
−0.748199 + 0.663474i \(0.769081\pi\)
\(492\) 8.79017e8 0.332751
\(493\) 2.49094e8 0.0936267
\(494\) −1.22449e8 −0.0456996
\(495\) 1.78236e9 0.660506
\(496\) −2.86164e9 −1.05300
\(497\) −7.95108e9 −2.90522
\(498\) 5.22208e8 0.189470
\(499\) −3.58721e9 −1.29242 −0.646212 0.763158i \(-0.723648\pi\)
−0.646212 + 0.763158i \(0.723648\pi\)
\(500\) 2.27233e9 0.812972
\(501\) 1.35639e9 0.481895
\(502\) 6.70559e8 0.236578
\(503\) 2.48168e9 0.869475 0.434737 0.900557i \(-0.356841\pi\)
0.434737 + 0.900557i \(0.356841\pi\)
\(504\) 8.68476e8 0.302170
\(505\) 3.56137e8 0.123054
\(506\) −4.59440e8 −0.157653
\(507\) 1.60451e9 0.546783
\(508\) −1.31064e9 −0.443568
\(509\) 3.73752e9 1.25624 0.628118 0.778118i \(-0.283826\pi\)
0.628118 + 0.778118i \(0.283826\pi\)
\(510\) 4.87605e7 0.0162769
\(511\) −3.42929e9 −1.13692
\(512\) −3.07455e9 −1.01236
\(513\) −3.64950e8 −0.119350
\(514\) −2.89730e8 −0.0941071
\(515\) −3.02835e9 −0.976969
\(516\) 1.22289e8 0.0391844
\(517\) −6.29296e8 −0.200280
\(518\) 8.37435e8 0.264726
\(519\) −9.72188e8 −0.305256
\(520\) 4.82496e8 0.150481
\(521\) −3.12643e9 −0.968539 −0.484270 0.874919i \(-0.660915\pi\)
−0.484270 + 0.874919i \(0.660915\pi\)
\(522\) −3.97101e8 −0.122195
\(523\) 3.47385e9 1.06183 0.530915 0.847425i \(-0.321848\pi\)
0.530915 + 0.847425i \(0.321848\pi\)
\(524\) −1.93501e9 −0.587520
\(525\) −4.52003e8 −0.136328
\(526\) 1.54897e8 0.0464080
\(527\) −4.11761e8 −0.122548
\(528\) −2.52694e9 −0.747095
\(529\) −3.16143e9 −0.928514
\(530\) −2.15841e9 −0.629752
\(531\) −1.49721e8 −0.0433963
\(532\) −2.88338e9 −0.830255
\(533\) 5.16570e8 0.147769
\(534\) 2.10321e8 0.0597708
\(535\) −1.57839e9 −0.445633
\(536\) 2.58285e9 0.724472
\(537\) 1.87828e9 0.523419
\(538\) 9.04381e8 0.250388
\(539\) 8.20222e9 2.25616
\(540\) 6.80153e8 0.185878
\(541\) 6.14788e8 0.166930 0.0834651 0.996511i \(-0.473401\pi\)
0.0834651 + 0.996511i \(0.473401\pi\)
\(542\) 2.38795e8 0.0644210
\(543\) −3.93779e9 −1.05549
\(544\) −2.55763e8 −0.0681149
\(545\) 3.95202e9 1.04576
\(546\) 2.41394e8 0.0634676
\(547\) −3.36033e9 −0.877863 −0.438931 0.898521i \(-0.644643\pi\)
−0.438931 + 0.898521i \(0.644643\pi\)
\(548\) 6.35278e9 1.64904
\(549\) −6.84453e8 −0.176539
\(550\) −3.64167e8 −0.0933321
\(551\) 2.78747e9 0.709871
\(552\) −3.70684e8 −0.0938032
\(553\) −2.70654e9 −0.680575
\(554\) −3.40545e8 −0.0850923
\(555\) 1.38664e9 0.344301
\(556\) 4.50949e7 0.0111267
\(557\) −2.92483e8 −0.0717145 −0.0358572 0.999357i \(-0.511416\pi\)
−0.0358572 + 0.999357i \(0.511416\pi\)
\(558\) 6.56421e8 0.159942
\(559\) 7.18654e7 0.0174012
\(560\) 4.68939e9 1.12839
\(561\) −3.63601e8 −0.0869470
\(562\) −3.93156e8 −0.0934302
\(563\) 5.83190e9 1.37731 0.688653 0.725091i \(-0.258202\pi\)
0.688653 + 0.725091i \(0.258202\pi\)
\(564\) −2.40141e8 −0.0563624
\(565\) 2.80498e9 0.654276
\(566\) −2.24733e9 −0.520967
\(567\) 7.19454e8 0.165753
\(568\) −5.16846e9 −1.18343
\(569\) −1.28209e9 −0.291760 −0.145880 0.989302i \(-0.546601\pi\)
−0.145880 + 0.989302i \(0.546601\pi\)
\(570\) 5.45650e8 0.123411
\(571\) 2.52700e9 0.568041 0.284020 0.958818i \(-0.408332\pi\)
0.284020 + 0.958818i \(0.408332\pi\)
\(572\) −1.70171e9 −0.380189
\(573\) −1.02541e9 −0.227696
\(574\) −1.39019e9 −0.306819
\(575\) 1.92925e8 0.0423204
\(576\) −6.66759e8 −0.145375
\(577\) −3.17996e9 −0.689139 −0.344570 0.938761i \(-0.611975\pi\)
−0.344570 + 0.938761i \(0.611975\pi\)
\(578\) 1.47684e9 0.318116
\(579\) 3.20094e9 0.685335
\(580\) −5.19497e9 −1.10557
\(581\) 7.22639e9 1.52864
\(582\) −3.29589e8 −0.0693015
\(583\) 1.60950e10 3.36396
\(584\) −2.22915e9 −0.463121
\(585\) 3.99704e8 0.0825455
\(586\) −2.27701e9 −0.467437
\(587\) 3.74135e9 0.763474 0.381737 0.924271i \(-0.375326\pi\)
0.381737 + 0.924271i \(0.375326\pi\)
\(588\) 3.12999e9 0.634924
\(589\) −4.60777e9 −0.929154
\(590\) 2.23854e8 0.0448728
\(591\) 3.71696e9 0.740683
\(592\) −1.96591e9 −0.389437
\(593\) −1.49511e8 −0.0294430 −0.0147215 0.999892i \(-0.504686\pi\)
−0.0147215 + 0.999892i \(0.504686\pi\)
\(594\) 5.79645e8 0.113478
\(595\) 6.74755e8 0.131322
\(596\) 3.90364e9 0.755279
\(597\) 2.71417e9 0.522068
\(598\) −1.03032e8 −0.0197024
\(599\) −4.90943e9 −0.933335 −0.466668 0.884433i \(-0.654546\pi\)
−0.466668 + 0.884433i \(0.654546\pi\)
\(600\) −2.93816e8 −0.0555324
\(601\) 9.51517e9 1.78795 0.893976 0.448114i \(-0.147904\pi\)
0.893976 + 0.448114i \(0.147904\pi\)
\(602\) −1.93404e8 −0.0361308
\(603\) 2.13965e9 0.397404
\(604\) −2.14353e9 −0.395822
\(605\) 1.40095e10 2.57205
\(606\) 1.15820e8 0.0211412
\(607\) 6.23277e9 1.13115 0.565576 0.824696i \(-0.308654\pi\)
0.565576 + 0.824696i \(0.308654\pi\)
\(608\) −2.86210e9 −0.516442
\(609\) −5.49515e9 −0.985868
\(610\) 1.02335e9 0.182545
\(611\) −1.41123e8 −0.0250296
\(612\) −1.38751e8 −0.0244684
\(613\) −2.97929e9 −0.522397 −0.261198 0.965285i \(-0.584118\pi\)
−0.261198 + 0.965285i \(0.584118\pi\)
\(614\) −2.70066e9 −0.470848
\(615\) −2.30190e9 −0.399047
\(616\) 9.68268e9 1.66903
\(617\) 3.18056e9 0.545137 0.272569 0.962136i \(-0.412127\pi\)
0.272569 + 0.962136i \(0.412127\pi\)
\(618\) −9.84857e8 −0.167847
\(619\) 1.21436e8 0.0205793 0.0102896 0.999947i \(-0.496725\pi\)
0.0102896 + 0.999947i \(0.496725\pi\)
\(620\) 8.58745e9 1.44708
\(621\) −3.07078e8 −0.0514551
\(622\) −2.65217e9 −0.441912
\(623\) 2.91046e9 0.482229
\(624\) −5.66680e8 −0.0933667
\(625\) −6.91669e9 −1.13323
\(626\) −1.09853e9 −0.178979
\(627\) −4.06884e9 −0.659226
\(628\) −3.80583e9 −0.613183
\(629\) −2.82874e8 −0.0453227
\(630\) −1.07568e9 −0.171393
\(631\) −5.92643e9 −0.939053 −0.469527 0.882918i \(-0.655575\pi\)
−0.469527 + 0.882918i \(0.655575\pi\)
\(632\) −1.75934e9 −0.277229
\(633\) −7.09208e9 −1.11138
\(634\) −1.01328e9 −0.157912
\(635\) 3.43221e9 0.531943
\(636\) 6.14190e9 0.946679
\(637\) 1.83939e9 0.281959
\(638\) −4.42730e9 −0.674942
\(639\) −4.28160e9 −0.649162
\(640\) 6.94057e9 1.04656
\(641\) −7.39788e8 −0.110944 −0.0554721 0.998460i \(-0.517666\pi\)
−0.0554721 + 0.998460i \(0.517666\pi\)
\(642\) −5.13313e8 −0.0765614
\(643\) 5.85977e9 0.869245 0.434622 0.900613i \(-0.356882\pi\)
0.434622 + 0.900613i \(0.356882\pi\)
\(644\) −2.42615e9 −0.357946
\(645\) −3.20241e8 −0.0469914
\(646\) −1.11313e8 −0.0162454
\(647\) −7.09749e9 −1.03024 −0.515122 0.857117i \(-0.672253\pi\)
−0.515122 + 0.857117i \(0.672253\pi\)
\(648\) 4.67668e8 0.0675189
\(649\) −1.66925e9 −0.239698
\(650\) −8.16665e7 −0.0116640
\(651\) 9.08365e9 1.29041
\(652\) 8.09841e9 1.14428
\(653\) 3.83880e9 0.539510 0.269755 0.962929i \(-0.413057\pi\)
0.269755 + 0.962929i \(0.413057\pi\)
\(654\) 1.28525e9 0.179665
\(655\) 5.06725e9 0.704576
\(656\) 3.26352e9 0.451360
\(657\) −1.84665e9 −0.254042
\(658\) 3.79790e8 0.0519701
\(659\) −4.88094e9 −0.664361 −0.332181 0.943216i \(-0.607784\pi\)
−0.332181 + 0.943216i \(0.607784\pi\)
\(660\) 7.58305e9 1.02669
\(661\) −5.63851e9 −0.759381 −0.379690 0.925114i \(-0.623969\pi\)
−0.379690 + 0.925114i \(0.623969\pi\)
\(662\) −4.35458e9 −0.583369
\(663\) −8.15395e7 −0.0108660
\(664\) 4.69739e9 0.622685
\(665\) 7.55079e9 0.995673
\(666\) 4.50952e8 0.0591522
\(667\) 2.34545e9 0.306045
\(668\) 5.77076e9 0.749059
\(669\) −7.23255e9 −0.933900
\(670\) −3.19907e9 −0.410925
\(671\) −7.63100e9 −0.975108
\(672\) 5.64227e9 0.717235
\(673\) 1.34178e10 1.69680 0.848399 0.529358i \(-0.177567\pi\)
0.848399 + 0.529358i \(0.177567\pi\)
\(674\) 4.20569e9 0.529088
\(675\) −2.43400e8 −0.0304620
\(676\) 6.82640e9 0.849921
\(677\) −1.49956e10 −1.85740 −0.928699 0.370835i \(-0.879072\pi\)
−0.928699 + 0.370835i \(0.879072\pi\)
\(678\) 9.12215e8 0.112407
\(679\) −4.56090e9 −0.559122
\(680\) 4.38613e8 0.0534934
\(681\) 2.55599e9 0.310130
\(682\) 7.31846e9 0.883435
\(683\) −6.32321e9 −0.759391 −0.379695 0.925112i \(-0.623971\pi\)
−0.379695 + 0.925112i \(0.623971\pi\)
\(684\) −1.55268e9 −0.185518
\(685\) −1.66362e10 −1.97759
\(686\) −9.10552e8 −0.107689
\(687\) −6.10294e9 −0.718109
\(688\) 4.54022e8 0.0531517
\(689\) 3.60940e9 0.420404
\(690\) 4.59124e8 0.0532057
\(691\) −1.51402e10 −1.74565 −0.872826 0.488031i \(-0.837715\pi\)
−0.872826 + 0.488031i \(0.837715\pi\)
\(692\) −4.13618e9 −0.474491
\(693\) 8.02122e9 0.915533
\(694\) 4.28850e8 0.0487021
\(695\) −1.18091e8 −0.0133435
\(696\) −3.57202e9 −0.401589
\(697\) 4.69588e8 0.0525293
\(698\) −1.04055e9 −0.115816
\(699\) 1.69639e9 0.187869
\(700\) −1.92305e9 −0.211908
\(701\) 1.32383e10 1.45151 0.725755 0.687953i \(-0.241491\pi\)
0.725755 + 0.687953i \(0.241491\pi\)
\(702\) 1.29989e8 0.0141816
\(703\) −3.16548e9 −0.343634
\(704\) −7.43373e9 −0.802975
\(705\) 6.28864e8 0.0675919
\(706\) 1.76283e9 0.188536
\(707\) 1.60274e9 0.170567
\(708\) −6.36990e8 −0.0674553
\(709\) −6.66730e9 −0.702568 −0.351284 0.936269i \(-0.614255\pi\)
−0.351284 + 0.936269i \(0.614255\pi\)
\(710\) 6.40158e9 0.671248
\(711\) −1.45745e9 −0.152072
\(712\) 1.89189e9 0.196434
\(713\) −3.87710e9 −0.400584
\(714\) 2.19439e8 0.0225616
\(715\) 4.45632e9 0.455937
\(716\) 7.99114e9 0.813604
\(717\) 5.04027e8 0.0510666
\(718\) −5.53005e9 −0.557563
\(719\) −5.24566e9 −0.526319 −0.263159 0.964752i \(-0.584765\pi\)
−0.263159 + 0.964752i \(0.584765\pi\)
\(720\) 2.52520e9 0.252134
\(721\) −1.36286e10 −1.35418
\(722\) 1.99315e9 0.197087
\(723\) 6.56256e8 0.0645787
\(724\) −1.67533e10 −1.64065
\(725\) 1.85908e9 0.181182
\(726\) 4.55607e9 0.441888
\(727\) 4.33721e9 0.418639 0.209320 0.977847i \(-0.432875\pi\)
0.209320 + 0.977847i \(0.432875\pi\)
\(728\) 2.17140e9 0.208583
\(729\) 3.87420e8 0.0370370
\(730\) 2.76099e9 0.262685
\(731\) 6.53291e7 0.00618580
\(732\) −2.91201e9 −0.274413
\(733\) −1.14178e10 −1.07083 −0.535413 0.844590i \(-0.679844\pi\)
−0.535413 + 0.844590i \(0.679844\pi\)
\(734\) 5.37200e8 0.0501417
\(735\) −8.19658e9 −0.761425
\(736\) −2.40824e9 −0.222653
\(737\) 2.38551e10 2.19505
\(738\) −7.48608e8 −0.0685578
\(739\) −1.87923e9 −0.171287 −0.0856433 0.996326i \(-0.527295\pi\)
−0.0856433 + 0.996326i \(0.527295\pi\)
\(740\) 5.89946e9 0.535182
\(741\) −9.12461e8 −0.0823855
\(742\) −9.71359e9 −0.872903
\(743\) −9.66780e9 −0.864703 −0.432351 0.901705i \(-0.642316\pi\)
−0.432351 + 0.901705i \(0.642316\pi\)
\(744\) 5.90467e9 0.525642
\(745\) −1.02226e10 −0.905759
\(746\) 5.65193e9 0.498438
\(747\) 3.89136e9 0.341570
\(748\) −1.54694e9 −0.135151
\(749\) −7.10331e9 −0.617695
\(750\) −1.93521e9 −0.167499
\(751\) −1.07300e10 −0.924403 −0.462202 0.886775i \(-0.652940\pi\)
−0.462202 + 0.886775i \(0.652940\pi\)
\(752\) −8.91571e8 −0.0764528
\(753\) 4.99683e9 0.426493
\(754\) −9.92847e8 −0.0843496
\(755\) 5.61331e9 0.474684
\(756\) 3.06092e9 0.257647
\(757\) 1.90038e10 1.59223 0.796115 0.605145i \(-0.206885\pi\)
0.796115 + 0.605145i \(0.206885\pi\)
\(758\) −6.42453e9 −0.535796
\(759\) −3.42363e9 −0.284211
\(760\) 4.90826e9 0.405583
\(761\) 8.07062e9 0.663836 0.331918 0.943308i \(-0.392304\pi\)
0.331918 + 0.943308i \(0.392304\pi\)
\(762\) 1.11620e9 0.0913899
\(763\) 1.77854e10 1.44953
\(764\) −4.36259e9 −0.353930
\(765\) 3.63351e8 0.0293434
\(766\) −6.20785e9 −0.499046
\(767\) −3.74339e8 −0.0299558
\(768\) −9.03773e8 −0.0719938
\(769\) −4.19821e9 −0.332906 −0.166453 0.986049i \(-0.553231\pi\)
−0.166453 + 0.986049i \(0.553231\pi\)
\(770\) −1.19928e10 −0.946682
\(771\) −2.15899e9 −0.169653
\(772\) 1.36184e10 1.06529
\(773\) −1.26866e10 −0.987910 −0.493955 0.869487i \(-0.664449\pi\)
−0.493955 + 0.869487i \(0.664449\pi\)
\(774\) −1.04146e8 −0.00807330
\(775\) −3.07311e9 −0.237150
\(776\) −2.96473e9 −0.227756
\(777\) 6.24035e9 0.477238
\(778\) 3.38508e9 0.257716
\(779\) 5.25488e9 0.398274
\(780\) 1.70054e9 0.128309
\(781\) −4.77357e10 −3.58563
\(782\) −9.36612e7 −0.00700384
\(783\) −2.95910e9 −0.220289
\(784\) 1.16207e10 0.861243
\(785\) 9.96642e9 0.735352
\(786\) 1.64793e9 0.121049
\(787\) 1.93383e10 1.41419 0.707093 0.707121i \(-0.250006\pi\)
0.707093 + 0.707121i \(0.250006\pi\)
\(788\) 1.58138e10 1.15132
\(789\) 1.15425e9 0.0836625
\(790\) 2.17909e9 0.157246
\(791\) 1.26234e10 0.906896
\(792\) 5.21405e9 0.372938
\(793\) −1.71129e9 −0.121862
\(794\) −2.69362e9 −0.190970
\(795\) −1.60839e10 −1.13529
\(796\) 1.15474e10 0.811503
\(797\) −2.47116e10 −1.72901 −0.864505 0.502624i \(-0.832368\pi\)
−0.864505 + 0.502624i \(0.832368\pi\)
\(798\) 2.45561e9 0.171060
\(799\) −1.28288e8 −0.00889759
\(800\) −1.90885e9 −0.131813
\(801\) 1.56726e9 0.107753
\(802\) −8.81675e9 −0.603529
\(803\) −2.05884e10 −1.40319
\(804\) 9.10316e9 0.617726
\(805\) 6.35343e9 0.429262
\(806\) 1.64121e9 0.110406
\(807\) 6.73921e9 0.451390
\(808\) 1.04183e9 0.0694796
\(809\) 2.28215e10 1.51539 0.757695 0.652609i \(-0.226326\pi\)
0.757695 + 0.652609i \(0.226326\pi\)
\(810\) −5.79247e8 −0.0382971
\(811\) −2.67364e10 −1.76007 −0.880037 0.474906i \(-0.842482\pi\)
−0.880037 + 0.474906i \(0.842482\pi\)
\(812\) −2.33791e10 −1.53244
\(813\) 1.77944e9 0.116136
\(814\) 5.02769e9 0.326725
\(815\) −2.12075e10 −1.37227
\(816\) −5.15140e8 −0.0331902
\(817\) 7.31060e8 0.0469003
\(818\) 9.16160e9 0.585241
\(819\) 1.79880e9 0.114417
\(820\) −9.79346e9 −0.620280
\(821\) 1.71127e10 1.07924 0.539618 0.841910i \(-0.318569\pi\)
0.539618 + 0.841910i \(0.318569\pi\)
\(822\) −5.41030e9 −0.339758
\(823\) −2.37659e10 −1.48612 −0.743061 0.669224i \(-0.766627\pi\)
−0.743061 + 0.669224i \(0.766627\pi\)
\(824\) −8.85903e9 −0.551621
\(825\) −2.71368e9 −0.168256
\(826\) 1.00742e9 0.0621985
\(827\) −6.09517e9 −0.374728 −0.187364 0.982291i \(-0.559994\pi\)
−0.187364 + 0.982291i \(0.559994\pi\)
\(828\) −1.30647e9 −0.0799819
\(829\) −5.59971e9 −0.341370 −0.170685 0.985326i \(-0.554598\pi\)
−0.170685 + 0.985326i \(0.554598\pi\)
\(830\) −5.81812e9 −0.353191
\(831\) −2.53765e9 −0.153401
\(832\) −1.66706e9 −0.100350
\(833\) 1.67210e9 0.100232
\(834\) −3.84047e7 −0.00229247
\(835\) −1.51121e10 −0.898299
\(836\) −1.73109e10 −1.02470
\(837\) 4.89148e9 0.288337
\(838\) 4.87476e9 0.286153
\(839\) −1.48782e10 −0.869730 −0.434865 0.900496i \(-0.643204\pi\)
−0.434865 + 0.900496i \(0.643204\pi\)
\(840\) −9.67602e9 −0.563274
\(841\) 5.35156e9 0.310237
\(842\) −3.68789e9 −0.212905
\(843\) −2.92969e9 −0.168432
\(844\) −3.01733e10 −1.72752
\(845\) −1.78765e10 −1.01926
\(846\) 2.04514e8 0.0116125
\(847\) 6.30476e10 3.56514
\(848\) 2.28030e10 1.28412
\(849\) −1.67466e10 −0.939179
\(850\) −7.42389e7 −0.00414634
\(851\) −2.66351e9 −0.148150
\(852\) −1.82161e10 −1.00906
\(853\) 3.87811e9 0.213943 0.106972 0.994262i \(-0.465885\pi\)
0.106972 + 0.994262i \(0.465885\pi\)
\(854\) 4.60543e9 0.253027
\(855\) 4.06605e9 0.222480
\(856\) −4.61738e9 −0.251616
\(857\) −1.92983e10 −1.04733 −0.523667 0.851923i \(-0.675436\pi\)
−0.523667 + 0.851923i \(0.675436\pi\)
\(858\) 1.44925e9 0.0783317
\(859\) 2.34053e10 1.25990 0.629952 0.776634i \(-0.283074\pi\)
0.629952 + 0.776634i \(0.283074\pi\)
\(860\) −1.36247e9 −0.0730435
\(861\) −1.03593e10 −0.553122
\(862\) 7.55642e9 0.401828
\(863\) 2.15437e10 1.14099 0.570495 0.821301i \(-0.306751\pi\)
0.570495 + 0.821301i \(0.306751\pi\)
\(864\) 3.03832e9 0.160264
\(865\) 1.08315e10 0.569027
\(866\) 4.62908e9 0.242204
\(867\) 1.10050e10 0.573488
\(868\) 3.86464e10 2.00581
\(869\) −1.62492e10 −0.839966
\(870\) 4.42426e9 0.227784
\(871\) 5.34964e9 0.274322
\(872\) 1.15611e10 0.590462
\(873\) −2.45601e9 −0.124934
\(874\) −1.04811e9 −0.0531026
\(875\) −2.67797e10 −1.35138
\(876\) −7.85657e9 −0.394883
\(877\) −1.77005e10 −0.886108 −0.443054 0.896495i \(-0.646105\pi\)
−0.443054 + 0.896495i \(0.646105\pi\)
\(878\) 1.34977e9 0.0673021
\(879\) −1.69677e10 −0.842677
\(880\) 2.81536e10 1.39266
\(881\) −2.35041e10 −1.15805 −0.579027 0.815308i \(-0.696567\pi\)
−0.579027 + 0.815308i \(0.696567\pi\)
\(882\) −2.66563e9 −0.130816
\(883\) 6.99377e9 0.341860 0.170930 0.985283i \(-0.445323\pi\)
0.170930 + 0.985283i \(0.445323\pi\)
\(884\) −3.46910e8 −0.0168902
\(885\) 1.66810e9 0.0808949
\(886\) −9.10057e8 −0.0439593
\(887\) −1.04556e10 −0.503056 −0.251528 0.967850i \(-0.580933\pi\)
−0.251528 + 0.967850i \(0.580933\pi\)
\(888\) 4.05643e9 0.194401
\(889\) 1.54461e10 0.737331
\(890\) −2.34327e9 −0.111419
\(891\) 4.31937e9 0.204573
\(892\) −3.07709e10 −1.45166
\(893\) −1.43560e9 −0.0674609
\(894\) −3.32450e9 −0.155613
\(895\) −2.09266e10 −0.975704
\(896\) 3.12349e10 1.45065
\(897\) −7.67768e8 −0.0355187
\(898\) 2.61733e9 0.120612
\(899\) −3.73609e10 −1.71498
\(900\) −1.03555e9 −0.0473501
\(901\) 3.28112e9 0.149446
\(902\) −8.34626e9 −0.378677
\(903\) −1.44119e9 −0.0651352
\(904\) 8.20560e9 0.369420
\(905\) 4.38724e10 1.96753
\(906\) 1.82552e9 0.0815525
\(907\) 2.84631e10 1.26665 0.633325 0.773886i \(-0.281690\pi\)
0.633325 + 0.773886i \(0.281690\pi\)
\(908\) 1.08745e10 0.482067
\(909\) 8.63061e8 0.0381126
\(910\) −2.68946e9 −0.118310
\(911\) −8.69725e9 −0.381125 −0.190562 0.981675i \(-0.561031\pi\)
−0.190562 + 0.981675i \(0.561031\pi\)
\(912\) −5.76463e9 −0.251646
\(913\) 4.33849e10 1.88665
\(914\) 1.38149e10 0.598462
\(915\) 7.62575e9 0.329086
\(916\) −2.59650e10 −1.11623
\(917\) 2.28043e10 0.976618
\(918\) 1.18166e8 0.00504131
\(919\) 1.81679e10 0.772148 0.386074 0.922468i \(-0.373831\pi\)
0.386074 + 0.922468i \(0.373831\pi\)
\(920\) 4.12993e9 0.174858
\(921\) −2.01246e10 −0.848826
\(922\) 8.82735e9 0.370913
\(923\) −1.07050e10 −0.448106
\(924\) 3.41263e10 1.42311
\(925\) −2.11119e9 −0.0877063
\(926\) 1.04166e10 0.431107
\(927\) −7.33890e9 −0.302588
\(928\) −2.32065e10 −0.953218
\(929\) 2.23626e10 0.915098 0.457549 0.889184i \(-0.348727\pi\)
0.457549 + 0.889184i \(0.348727\pi\)
\(930\) −7.31344e9 −0.298147
\(931\) 1.87115e10 0.759949
\(932\) 7.21731e9 0.292025
\(933\) −1.97633e10 −0.796661
\(934\) 6.53757e9 0.262544
\(935\) 4.05101e9 0.162078
\(936\) 1.16928e9 0.0466072
\(937\) −4.79626e8 −0.0190465 −0.00952323 0.999955i \(-0.503031\pi\)
−0.00952323 + 0.999955i \(0.503031\pi\)
\(938\) −1.43969e10 −0.569586
\(939\) −8.18596e9 −0.322656
\(940\) 2.67550e9 0.105065
\(941\) −3.41566e10 −1.33632 −0.668161 0.744017i \(-0.732918\pi\)
−0.668161 + 0.744017i \(0.732918\pi\)
\(942\) 3.24120e9 0.126336
\(943\) 4.42159e9 0.171707
\(944\) −2.36495e9 −0.0914997
\(945\) −8.01571e9 −0.308980
\(946\) −1.16113e9 −0.0445926
\(947\) −1.00641e10 −0.385079 −0.192539 0.981289i \(-0.561672\pi\)
−0.192539 + 0.981289i \(0.561672\pi\)
\(948\) −6.20072e9 −0.236381
\(949\) −4.61706e9 −0.175361
\(950\) −8.30764e8 −0.0314373
\(951\) −7.55068e9 −0.284678
\(952\) 1.97391e9 0.0741476
\(953\) −2.94200e10 −1.10108 −0.550539 0.834810i \(-0.685578\pi\)
−0.550539 + 0.834810i \(0.685578\pi\)
\(954\) −5.23070e9 −0.195047
\(955\) 1.14244e10 0.424446
\(956\) 2.14438e9 0.0793780
\(957\) −3.29911e10 −1.21676
\(958\) 8.56325e9 0.314673
\(959\) −7.48685e10 −2.74116
\(960\) 7.42862e9 0.270993
\(961\) 3.42460e10 1.24474
\(962\) 1.12749e9 0.0408318
\(963\) −3.82508e9 −0.138022
\(964\) 2.79204e9 0.100381
\(965\) −3.56629e10 −1.27753
\(966\) 2.06621e9 0.0737489
\(967\) 9.18664e9 0.326711 0.163356 0.986567i \(-0.447768\pi\)
0.163356 + 0.986567i \(0.447768\pi\)
\(968\) 4.09830e10 1.45224
\(969\) −8.29472e8 −0.0292866
\(970\) 3.67208e9 0.129185
\(971\) 2.46294e10 0.863348 0.431674 0.902030i \(-0.357923\pi\)
0.431674 + 0.902030i \(0.357923\pi\)
\(972\) 1.64828e9 0.0575704
\(973\) −5.31450e8 −0.0184956
\(974\) −1.27025e10 −0.440486
\(975\) −6.08558e8 −0.0210274
\(976\) −1.08114e10 −0.372227
\(977\) 3.26473e10 1.12000 0.559998 0.828494i \(-0.310802\pi\)
0.559998 + 0.828494i \(0.310802\pi\)
\(978\) −6.89695e9 −0.235760
\(979\) 1.74735e10 0.595168
\(980\) −3.48724e10 −1.18356
\(981\) 9.57732e9 0.323894
\(982\) 1.42213e10 0.479235
\(983\) 4.98914e9 0.167528 0.0837641 0.996486i \(-0.473306\pi\)
0.0837641 + 0.996486i \(0.473306\pi\)
\(984\) −6.73391e9 −0.225312
\(985\) −4.14121e10 −1.38070
\(986\) −9.02547e8 −0.0299848
\(987\) 2.83010e9 0.0936897
\(988\) −3.88207e9 −0.128060
\(989\) 6.15133e8 0.0202200
\(990\) −6.45805e9 −0.211533
\(991\) −1.15183e10 −0.375950 −0.187975 0.982174i \(-0.560192\pi\)
−0.187975 + 0.982174i \(0.560192\pi\)
\(992\) 3.83611e10 1.24767
\(993\) −3.24492e10 −1.05168
\(994\) 2.88093e10 0.930422
\(995\) −3.02396e10 −0.973185
\(996\) 1.65558e10 0.530936
\(997\) 2.00277e10 0.640026 0.320013 0.947413i \(-0.396313\pi\)
0.320013 + 0.947413i \(0.396313\pi\)
\(998\) 1.29976e10 0.413910
\(999\) 3.36038e9 0.106637
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.1.9 17
3.2 odd 2 531.8.a.c.1.9 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.9 17 1.1 even 1 trivial
531.8.a.c.1.9 17 3.2 odd 2