Properties

Label 531.8.a.c.1.9
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
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: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-3.62331\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.62331 q^{2} -114.872 q^{4} -300.817 q^{5} +1353.78 q^{7} -880.000 q^{8} +O(q^{10})\) \(q+3.62331 q^{2} -114.872 q^{4} -300.817 q^{5} +1353.78 q^{7} -880.000 q^{8} -1089.96 q^{10} -8127.65 q^{11} +1822.67 q^{13} +4905.17 q^{14} +11515.0 q^{16} -1656.90 q^{17} +18541.4 q^{19} +34555.4 q^{20} -29449.0 q^{22} -15601.2 q^{23} +12366.0 q^{25} +6604.11 q^{26} -155511. q^{28} -150338. q^{29} -248513. q^{31} +154363. q^{32} -6003.46 q^{34} -407240. q^{35} -170725. q^{37} +67181.2 q^{38} +264719. q^{40} -283414. q^{41} +39428.6 q^{43} +933636. q^{44} -56528.0 q^{46} +77426.6 q^{47} +1.00917e6 q^{49} +44805.9 q^{50} -209373. q^{52} -1.98028e6 q^{53} +2.44494e6 q^{55} -1.19133e6 q^{56} -544721. q^{58} +205379. q^{59} -938894. q^{61} -900440. q^{62} -914622. q^{64} -548291. q^{65} +2.93505e6 q^{67} +190331. q^{68} -1.47556e6 q^{70} +5.87325e6 q^{71} -2.53313e6 q^{73} -618590. q^{74} -2.12988e6 q^{76} -1.10030e7 q^{77} -1.99924e6 q^{79} -3.46392e6 q^{80} -1.02690e6 q^{82} -5.33794e6 q^{83} +498424. q^{85} +142862. q^{86} +7.15233e6 q^{88} -2.14988e6 q^{89} +2.46749e6 q^{91} +1.79213e6 q^{92} +280541. q^{94} -5.57757e6 q^{95} -3.36902e6 q^{97} +3.65656e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + O(q^{10}) \) \( 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + 6521q^{10} + 1764q^{11} + 18192q^{13} + 7827q^{14} + 139226q^{16} + 15507q^{17} + 52083q^{19} - 721q^{20} - 234434q^{22} - 63823q^{23} + 202153q^{25} + 367956q^{26} + 182306q^{28} + 502955q^{29} + 347531q^{31} + 243908q^{32} - 330872q^{34} - 92641q^{35} + 447615q^{37} - 775669q^{38} + 2203270q^{40} - 940335q^{41} + 478562q^{43} + 596924q^{44} - 3078663q^{46} - 703121q^{47} + 1895082q^{49} + 876967q^{50} + 6278296q^{52} + 1005974q^{53} + 5212846q^{55} - 3425294q^{56} + 6710166q^{58} + 3491443q^{59} + 11510749q^{61} - 5996234q^{62} + 29496941q^{64} - 11094180q^{65} + 14007144q^{67} - 19688159q^{68} + 30909708q^{70} - 5229074q^{71} + 5452211q^{73} - 12819662q^{74} + 41929340q^{76} - 9930777q^{77} + 15275654q^{79} - 36576105q^{80} + 32025935q^{82} - 7826609q^{83} + 11836945q^{85} - 51649136q^{86} + 30223741q^{88} + 6436185q^{89} + 11633535q^{91} - 43357972q^{92} - 4494252q^{94} - 23741055q^{95} + 26377540q^{97} - 26517816q^{98} + O(q^{100}) \)

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 \(-0.448809\pi\)
0.160129 + 0.987096i \(0.448809\pi\)
\(3\) 0 0
\(4\) −114.872 −0.897434
\(5\) −300.817 −1.07624 −0.538118 0.842869i \(-0.680865\pi\)
−0.538118 + 0.842869i \(0.680865\pi\)
\(6\) 0 0
\(7\) 1353.78 1.49178 0.745890 0.666070i \(-0.232025\pi\)
0.745890 + 0.666070i \(0.232025\pi\)
\(8\) −880.000 −0.607670
\(9\) 0 0
\(10\) −1089.96 −0.344674
\(11\) −8127.65 −1.84116 −0.920578 0.390558i \(-0.872282\pi\)
−0.920578 + 0.390558i \(0.872282\pi\)
\(12\) 0 0
\(13\) 1822.67 0.230095 0.115047 0.993360i \(-0.463298\pi\)
0.115047 + 0.993360i \(0.463298\pi\)
\(14\) 4905.17 0.477755
\(15\) 0 0
\(16\) 11515.0 0.702823
\(17\) −1656.90 −0.0817946 −0.0408973 0.999163i \(-0.513022\pi\)
−0.0408973 + 0.999163i \(0.513022\pi\)
\(18\) 0 0
\(19\) 18541.4 0.620161 0.310081 0.950710i \(-0.399644\pi\)
0.310081 + 0.950710i \(0.399644\pi\)
\(20\) 34555.4 0.965852
\(21\) 0 0
\(22\) −29449.0 −0.589646
\(23\) −15601.2 −0.267369 −0.133684 0.991024i \(-0.542681\pi\)
−0.133684 + 0.991024i \(0.542681\pi\)
\(24\) 0 0
\(25\) 12366.0 0.158285
\(26\) 6604.11 0.0736899
\(27\) 0 0
\(28\) −155511. −1.33877
\(29\) −150338. −1.14466 −0.572328 0.820025i \(-0.693960\pi\)
−0.572328 + 0.820025i \(0.693960\pi\)
\(30\) 0 0
\(31\) −248513. −1.49825 −0.749123 0.662431i \(-0.769525\pi\)
−0.749123 + 0.662431i \(0.769525\pi\)
\(32\) 154363. 0.832755
\(33\) 0 0
\(34\) −6003.46 −0.0261954
\(35\) −407240. −1.60551
\(36\) 0 0
\(37\) −170725. −0.554104 −0.277052 0.960855i \(-0.589357\pi\)
−0.277052 + 0.960855i \(0.589357\pi\)
\(38\) 67181.2 0.198612
\(39\) 0 0
\(40\) 264719. 0.653997
\(41\) −283414. −0.642210 −0.321105 0.947044i \(-0.604054\pi\)
−0.321105 + 0.947044i \(0.604054\pi\)
\(42\) 0 0
\(43\) 39428.6 0.0756261 0.0378130 0.999285i \(-0.487961\pi\)
0.0378130 + 0.999285i \(0.487961\pi\)
\(44\) 933636. 1.65232
\(45\) 0 0
\(46\) −56528.0 −0.0856271
\(47\) 77426.6 0.108780 0.0543898 0.998520i \(-0.482679\pi\)
0.0543898 + 0.998520i \(0.482679\pi\)
\(48\) 0 0
\(49\) 1.00917e6 1.22541
\(50\) 44805.9 0.0506921
\(51\) 0 0
\(52\) −209373. −0.206495
\(53\) −1.98028e6 −1.82709 −0.913546 0.406735i \(-0.866667\pi\)
−0.913546 + 0.406735i \(0.866667\pi\)
\(54\) 0 0
\(55\) 2.44494e6 1.98152
\(56\) −1.19133e6 −0.906509
\(57\) 0 0
\(58\) −544721. −0.366586
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) −938894. −0.529617 −0.264808 0.964301i \(-0.585309\pi\)
−0.264808 + 0.964301i \(0.585309\pi\)
\(62\) −900440. −0.479826
\(63\) 0 0
\(64\) −914622. −0.436126
\(65\) −548291. −0.247636
\(66\) 0 0
\(67\) 2.93505e6 1.19221 0.596107 0.802905i \(-0.296714\pi\)
0.596107 + 0.802905i \(0.296714\pi\)
\(68\) 190331. 0.0734053
\(69\) 0 0
\(70\) −1.47556e6 −0.514178
\(71\) 5.87325e6 1.94749 0.973743 0.227650i \(-0.0731043\pi\)
0.973743 + 0.227650i \(0.0731043\pi\)
\(72\) 0 0
\(73\) −2.53313e6 −0.762126 −0.381063 0.924549i \(-0.624442\pi\)
−0.381063 + 0.924549i \(0.624442\pi\)
\(74\) −618590. −0.177457
\(75\) 0 0
\(76\) −2.12988e6 −0.556554
\(77\) −1.10030e7 −2.74660
\(78\) 0 0
\(79\) −1.99924e6 −0.456217 −0.228108 0.973636i \(-0.573254\pi\)
−0.228108 + 0.973636i \(0.573254\pi\)
\(80\) −3.46392e6 −0.756403
\(81\) 0 0
\(82\) −1.02690e6 −0.205673
\(83\) −5.33794e6 −1.02471 −0.512354 0.858774i \(-0.671227\pi\)
−0.512354 + 0.858774i \(0.671227\pi\)
\(84\) 0 0
\(85\) 498424. 0.0880303
\(86\) 142862. 0.0242199
\(87\) 0 0
\(88\) 7.15233e6 1.11882
\(89\) −2.14988e6 −0.323258 −0.161629 0.986852i \(-0.551675\pi\)
−0.161629 + 0.986852i \(0.551675\pi\)
\(90\) 0 0
\(91\) 2.46749e6 0.343251
\(92\) 1.79213e6 0.239946
\(93\) 0 0
\(94\) 280541. 0.0348376
\(95\) −5.57757e6 −0.667440
\(96\) 0 0
\(97\) −3.36902e6 −0.374802 −0.187401 0.982283i \(-0.560006\pi\)
−0.187401 + 0.982283i \(0.560006\pi\)
\(98\) 3.65656e6 0.392447
\(99\) 0 0
\(100\) −1.42050e6 −0.142050
\(101\) −1.18390e6 −0.114338 −0.0571688 0.998365i \(-0.518207\pi\)
−0.0571688 + 0.998365i \(0.518207\pi\)
\(102\) 0 0
\(103\) −1.00671e7 −0.907764 −0.453882 0.891062i \(-0.649961\pi\)
−0.453882 + 0.891062i \(0.649961\pi\)
\(104\) −1.60395e6 −0.139822
\(105\) 0 0
\(106\) −7.17517e6 −0.585142
\(107\) 5.24702e6 0.414066 0.207033 0.978334i \(-0.433619\pi\)
0.207033 + 0.978334i \(0.433619\pi\)
\(108\) 0 0
\(109\) 1.31376e7 0.971681 0.485841 0.874047i \(-0.338514\pi\)
0.485841 + 0.874047i \(0.338514\pi\)
\(110\) 8.85878e6 0.634599
\(111\) 0 0
\(112\) 1.55888e7 1.04846
\(113\) −9.32454e6 −0.607929 −0.303965 0.952683i \(-0.598310\pi\)
−0.303965 + 0.952683i \(0.598310\pi\)
\(114\) 0 0
\(115\) 4.69311e6 0.287752
\(116\) 1.72695e7 1.02725
\(117\) 0 0
\(118\) 744153. 0.0416941
\(119\) −2.24307e6 −0.122020
\(120\) 0 0
\(121\) 4.65715e7 2.38986
\(122\) −3.40191e6 −0.169614
\(123\) 0 0
\(124\) 2.85471e7 1.34458
\(125\) 1.97814e7 0.905884
\(126\) 0 0
\(127\) 1.14096e7 0.494263 0.247131 0.968982i \(-0.420512\pi\)
0.247131 + 0.968982i \(0.420512\pi\)
\(128\) −2.30724e7 −0.972428
\(129\) 0 0
\(130\) −1.98663e6 −0.0793077
\(131\) −1.68449e7 −0.654666 −0.327333 0.944909i \(-0.606150\pi\)
−0.327333 + 0.944909i \(0.606150\pi\)
\(132\) 0 0
\(133\) 2.51009e7 0.925143
\(134\) 1.06346e7 0.381817
\(135\) 0 0
\(136\) 1.45807e6 0.0497041
\(137\) 5.53033e7 1.83751 0.918754 0.394830i \(-0.129196\pi\)
0.918754 + 0.394830i \(0.129196\pi\)
\(138\) 0 0
\(139\) −392568. −0.0123983 −0.00619916 0.999981i \(-0.501973\pi\)
−0.00619916 + 0.999981i \(0.501973\pi\)
\(140\) 4.67803e7 1.44084
\(141\) 0 0
\(142\) 2.12806e7 0.623699
\(143\) −1.48140e7 −0.423640
\(144\) 0 0
\(145\) 4.52242e7 1.23192
\(146\) −9.17831e6 −0.244078
\(147\) 0 0
\(148\) 1.96115e7 0.497272
\(149\) 3.39826e7 0.841598 0.420799 0.907154i \(-0.361750\pi\)
0.420799 + 0.907154i \(0.361750\pi\)
\(150\) 0 0
\(151\) 1.86602e7 0.441059 0.220530 0.975380i \(-0.429221\pi\)
0.220530 + 0.975380i \(0.429221\pi\)
\(152\) −1.63164e7 −0.376853
\(153\) 0 0
\(154\) −3.98675e7 −0.879622
\(155\) 7.47570e7 1.61247
\(156\) 0 0
\(157\) 3.31311e7 0.683262 0.341631 0.939834i \(-0.389021\pi\)
0.341631 + 0.939834i \(0.389021\pi\)
\(158\) −7.24389e6 −0.146107
\(159\) 0 0
\(160\) −4.64349e7 −0.896241
\(161\) −2.11206e7 −0.398855
\(162\) 0 0
\(163\) −7.04997e7 −1.27506 −0.637530 0.770426i \(-0.720044\pi\)
−0.637530 + 0.770426i \(0.720044\pi\)
\(164\) 3.25562e7 0.576342
\(165\) 0 0
\(166\) −1.93410e7 −0.328172
\(167\) 5.02367e7 0.834667 0.417333 0.908753i \(-0.362965\pi\)
0.417333 + 0.908753i \(0.362965\pi\)
\(168\) 0 0
\(169\) −5.94264e7 −0.947056
\(170\) 1.80595e6 0.0281925
\(171\) 0 0
\(172\) −4.52922e6 −0.0678694
\(173\) −3.60070e7 −0.528719 −0.264360 0.964424i \(-0.585161\pi\)
−0.264360 + 0.964424i \(0.585161\pi\)
\(174\) 0 0
\(175\) 1.67408e7 0.236126
\(176\) −9.35903e7 −1.29401
\(177\) 0 0
\(178\) −7.78968e6 −0.103526
\(179\) 6.95659e7 0.906589 0.453295 0.891361i \(-0.350249\pi\)
0.453295 + 0.891361i \(0.350249\pi\)
\(180\) 0 0
\(181\) 1.45844e8 1.82816 0.914079 0.405536i \(-0.132915\pi\)
0.914079 + 0.405536i \(0.132915\pi\)
\(182\) 8.94051e6 0.109929
\(183\) 0 0
\(184\) 1.37291e7 0.162472
\(185\) 5.13570e7 0.596347
\(186\) 0 0
\(187\) 1.34667e7 0.150597
\(188\) −8.89412e6 −0.0976226
\(189\) 0 0
\(190\) −2.02093e7 −0.213753
\(191\) −3.79780e7 −0.394380 −0.197190 0.980365i \(-0.563182\pi\)
−0.197190 + 0.980365i \(0.563182\pi\)
\(192\) 0 0
\(193\) −1.18553e8 −1.18704 −0.593518 0.804821i \(-0.702261\pi\)
−0.593518 + 0.804821i \(0.702261\pi\)
\(194\) −1.22070e7 −0.120034
\(195\) 0 0
\(196\) −1.15925e8 −1.09972
\(197\) 1.37665e8 1.28290 0.641450 0.767165i \(-0.278333\pi\)
0.641450 + 0.767165i \(0.278333\pi\)
\(198\) 0 0
\(199\) −1.00525e8 −0.904248 −0.452124 0.891955i \(-0.649334\pi\)
−0.452124 + 0.891955i \(0.649334\pi\)
\(200\) −1.08821e7 −0.0961850
\(201\) 0 0
\(202\) −4.28963e6 −0.0366176
\(203\) −2.03524e8 −1.70757
\(204\) 0 0
\(205\) 8.52557e7 0.691170
\(206\) −3.64762e7 −0.290719
\(207\) 0 0
\(208\) 2.09882e7 0.161716
\(209\) −1.50698e8 −1.14181
\(210\) 0 0
\(211\) 2.62670e8 1.92496 0.962480 0.271354i \(-0.0874714\pi\)
0.962480 + 0.271354i \(0.0874714\pi\)
\(212\) 2.27478e8 1.63970
\(213\) 0 0
\(214\) 1.90116e7 0.132608
\(215\) −1.18608e7 −0.0813915
\(216\) 0 0
\(217\) −3.36432e8 −2.23505
\(218\) 4.76017e7 0.311189
\(219\) 0 0
\(220\) −2.80854e8 −1.77828
\(221\) −3.01998e6 −0.0188205
\(222\) 0 0
\(223\) 2.67872e8 1.61756 0.808781 0.588110i \(-0.200128\pi\)
0.808781 + 0.588110i \(0.200128\pi\)
\(224\) 2.08973e8 1.24229
\(225\) 0 0
\(226\) −3.37857e7 −0.194695
\(227\) 9.46663e7 0.537161 0.268581 0.963257i \(-0.413445\pi\)
0.268581 + 0.963257i \(0.413445\pi\)
\(228\) 0 0
\(229\) 2.26035e8 1.24380 0.621900 0.783096i \(-0.286361\pi\)
0.621900 + 0.783096i \(0.286361\pi\)
\(230\) 1.70046e7 0.0921550
\(231\) 0 0
\(232\) 1.32297e8 0.695573
\(233\) 6.28293e7 0.325399 0.162700 0.986676i \(-0.447980\pi\)
0.162700 + 0.986676i \(0.447980\pi\)
\(234\) 0 0
\(235\) −2.32913e7 −0.117073
\(236\) −2.35922e7 −0.116836
\(237\) 0 0
\(238\) −8.12736e6 −0.0390778
\(239\) 1.86677e7 0.0884499 0.0442250 0.999022i \(-0.485918\pi\)
0.0442250 + 0.999022i \(0.485918\pi\)
\(240\) 0 0
\(241\) −2.43058e7 −0.111854 −0.0559268 0.998435i \(-0.517811\pi\)
−0.0559268 + 0.998435i \(0.517811\pi\)
\(242\) 1.68743e8 0.765372
\(243\) 0 0
\(244\) 1.07852e8 0.475296
\(245\) −3.03577e8 −1.31883
\(246\) 0 0
\(247\) 3.37949e7 0.142696
\(248\) 2.18691e8 0.910439
\(249\) 0 0
\(250\) 7.16744e7 0.290117
\(251\) 1.85068e8 0.738708 0.369354 0.929289i \(-0.379579\pi\)
0.369354 + 0.929289i \(0.379579\pi\)
\(252\) 0 0
\(253\) 1.26801e8 0.492267
\(254\) 4.13406e7 0.158292
\(255\) 0 0
\(256\) 3.34731e7 0.124697
\(257\) −7.99627e7 −0.293847 −0.146924 0.989148i \(-0.546937\pi\)
−0.146924 + 0.989148i \(0.546937\pi\)
\(258\) 0 0
\(259\) −2.31124e8 −0.826601
\(260\) 6.29831e7 0.222237
\(261\) 0 0
\(262\) −6.10345e7 −0.209663
\(263\) 4.27501e7 0.144908 0.0724539 0.997372i \(-0.476917\pi\)
0.0724539 + 0.997372i \(0.476917\pi\)
\(264\) 0 0
\(265\) 5.95702e8 1.96638
\(266\) 9.09486e7 0.296285
\(267\) 0 0
\(268\) −3.37154e8 −1.06993
\(269\) 2.49600e8 0.781830 0.390915 0.920427i \(-0.372159\pi\)
0.390915 + 0.920427i \(0.372159\pi\)
\(270\) 0 0
\(271\) −6.59051e7 −0.201153 −0.100577 0.994929i \(-0.532069\pi\)
−0.100577 + 0.994929i \(0.532069\pi\)
\(272\) −1.90793e7 −0.0574871
\(273\) 0 0
\(274\) 2.00381e8 0.588478
\(275\) −1.00507e8 −0.291427
\(276\) 0 0
\(277\) 9.39872e7 0.265699 0.132849 0.991136i \(-0.457587\pi\)
0.132849 + 0.991136i \(0.457587\pi\)
\(278\) −1.42240e6 −0.00397067
\(279\) 0 0
\(280\) 3.58371e8 0.975619
\(281\) −1.08507e8 −0.291733 −0.145867 0.989304i \(-0.546597\pi\)
−0.145867 + 0.989304i \(0.546597\pi\)
\(282\) 0 0
\(283\) 6.20243e8 1.62671 0.813353 0.581770i \(-0.197640\pi\)
0.813353 + 0.581770i \(0.197640\pi\)
\(284\) −6.74670e8 −1.74774
\(285\) 0 0
\(286\) −5.36759e7 −0.135675
\(287\) −3.83679e8 −0.958036
\(288\) 0 0
\(289\) −4.07593e8 −0.993310
\(290\) 1.63861e8 0.394533
\(291\) 0 0
\(292\) 2.90984e8 0.683958
\(293\) −6.28432e8 −1.45956 −0.729779 0.683683i \(-0.760377\pi\)
−0.729779 + 0.683683i \(0.760377\pi\)
\(294\) 0 0
\(295\) −6.17815e7 −0.140114
\(296\) 1.50238e8 0.336712
\(297\) 0 0
\(298\) 1.23130e8 0.269529
\(299\) −2.84359e7 −0.0615201
\(300\) 0 0
\(301\) 5.33776e7 0.112817
\(302\) 6.76117e7 0.141253
\(303\) 0 0
\(304\) 2.13505e8 0.435863
\(305\) 2.82435e8 0.569993
\(306\) 0 0
\(307\) 7.45356e8 1.47021 0.735105 0.677953i \(-0.237133\pi\)
0.735105 + 0.677953i \(0.237133\pi\)
\(308\) 1.26394e9 2.46489
\(309\) 0 0
\(310\) 2.70868e8 0.516407
\(311\) −7.31974e8 −1.37986 −0.689929 0.723877i \(-0.742358\pi\)
−0.689929 + 0.723877i \(0.742358\pi\)
\(312\) 0 0
\(313\) 3.03184e8 0.558857 0.279429 0.960166i \(-0.409855\pi\)
0.279429 + 0.960166i \(0.409855\pi\)
\(314\) 1.20044e8 0.218821
\(315\) 0 0
\(316\) 2.29656e8 0.409425
\(317\) −2.79655e8 −0.493077 −0.246539 0.969133i \(-0.579293\pi\)
−0.246539 + 0.969133i \(0.579293\pi\)
\(318\) 0 0
\(319\) 1.22189e9 2.10749
\(320\) 2.75134e8 0.469374
\(321\) 0 0
\(322\) −7.65264e7 −0.127737
\(323\) −3.07212e7 −0.0507258
\(324\) 0 0
\(325\) 2.25392e7 0.0364205
\(326\) −2.55443e8 −0.408349
\(327\) 0 0
\(328\) 2.49404e8 0.390252
\(329\) 1.04818e8 0.162275
\(330\) 0 0
\(331\) 1.20182e9 1.82156 0.910778 0.412897i \(-0.135483\pi\)
0.910778 + 0.412897i \(0.135483\pi\)
\(332\) 6.13178e8 0.919609
\(333\) 0 0
\(334\) 1.82023e8 0.267309
\(335\) −8.82914e8 −1.28310
\(336\) 0 0
\(337\) −1.16073e9 −1.65206 −0.826031 0.563624i \(-0.809407\pi\)
−0.826031 + 0.563624i \(0.809407\pi\)
\(338\) −2.15320e8 −0.303303
\(339\) 0 0
\(340\) −5.72547e7 −0.0790015
\(341\) 2.01983e9 2.75850
\(342\) 0 0
\(343\) 2.51304e8 0.336255
\(344\) −3.46971e7 −0.0459557
\(345\) 0 0
\(346\) −1.30465e8 −0.169327
\(347\) 1.18359e8 0.152071 0.0760356 0.997105i \(-0.475774\pi\)
0.0760356 + 0.997105i \(0.475774\pi\)
\(348\) 0 0
\(349\) 2.87181e8 0.361632 0.180816 0.983517i \(-0.442126\pi\)
0.180816 + 0.983517i \(0.442126\pi\)
\(350\) 6.06573e7 0.0756215
\(351\) 0 0
\(352\) −1.25461e9 −1.53323
\(353\) 4.86525e8 0.588700 0.294350 0.955698i \(-0.404897\pi\)
0.294350 + 0.955698i \(0.404897\pi\)
\(354\) 0 0
\(355\) −1.76678e9 −2.09596
\(356\) 2.46960e8 0.290102
\(357\) 0 0
\(358\) 2.52059e8 0.290343
\(359\) −1.52624e9 −1.74098 −0.870488 0.492189i \(-0.836197\pi\)
−0.870488 + 0.492189i \(0.836197\pi\)
\(360\) 0 0
\(361\) −5.50089e8 −0.615400
\(362\) 5.28439e8 0.585484
\(363\) 0 0
\(364\) −2.83445e8 −0.308045
\(365\) 7.62008e8 0.820228
\(366\) 0 0
\(367\) −1.48262e8 −0.156566 −0.0782832 0.996931i \(-0.524944\pi\)
−0.0782832 + 0.996931i \(0.524944\pi\)
\(368\) −1.79648e8 −0.187913
\(369\) 0 0
\(370\) 1.86083e8 0.190985
\(371\) −2.68086e9 −2.72562
\(372\) 0 0
\(373\) −1.55988e9 −1.55636 −0.778180 0.628041i \(-0.783857\pi\)
−0.778180 + 0.628041i \(0.783857\pi\)
\(374\) 4.87941e7 0.0482299
\(375\) 0 0
\(376\) −6.81354e7 −0.0661021
\(377\) −2.74016e8 −0.263379
\(378\) 0 0
\(379\) 1.77311e9 1.67301 0.836504 0.547961i \(-0.184596\pi\)
0.836504 + 0.547961i \(0.184596\pi\)
\(380\) 6.40704e8 0.598983
\(381\) 0 0
\(382\) −1.37606e8 −0.126304
\(383\) −1.71331e9 −1.55826 −0.779129 0.626863i \(-0.784338\pi\)
−0.779129 + 0.626863i \(0.784338\pi\)
\(384\) 0 0
\(385\) 3.30990e9 2.95599
\(386\) −4.29557e8 −0.380159
\(387\) 0 0
\(388\) 3.87004e8 0.336360
\(389\) 9.34250e8 0.804711 0.402355 0.915484i \(-0.368192\pi\)
0.402355 + 0.915484i \(0.368192\pi\)
\(390\) 0 0
\(391\) 2.58496e7 0.0218693
\(392\) −8.88073e8 −0.744642
\(393\) 0 0
\(394\) 4.98805e8 0.410860
\(395\) 6.01407e8 0.490997
\(396\) 0 0
\(397\) 7.43414e8 0.596299 0.298150 0.954519i \(-0.403631\pi\)
0.298150 + 0.954519i \(0.403631\pi\)
\(398\) −3.64233e8 −0.289593
\(399\) 0 0
\(400\) 1.42395e8 0.111246
\(401\) −2.43334e9 −1.88450 −0.942252 0.334905i \(-0.891296\pi\)
−0.942252 + 0.334905i \(0.891296\pi\)
\(402\) 0 0
\(403\) −4.52957e8 −0.344739
\(404\) 1.35996e8 0.102611
\(405\) 0 0
\(406\) −7.37432e8 −0.546866
\(407\) 1.38759e9 1.02019
\(408\) 0 0
\(409\) −2.52851e9 −1.82740 −0.913701 0.406388i \(-0.866788\pi\)
−0.913701 + 0.406388i \(0.866788\pi\)
\(410\) 3.08908e8 0.221353
\(411\) 0 0
\(412\) 1.15642e9 0.814659
\(413\) 2.78038e8 0.194213
\(414\) 0 0
\(415\) 1.60574e9 1.10283
\(416\) 2.81352e8 0.191613
\(417\) 0 0
\(418\) −5.46026e8 −0.365676
\(419\) 1.34539e9 0.893507 0.446753 0.894657i \(-0.352580\pi\)
0.446753 + 0.894657i \(0.352580\pi\)
\(420\) 0 0
\(421\) 1.01782e9 0.664790 0.332395 0.943140i \(-0.392143\pi\)
0.332395 + 0.943140i \(0.392143\pi\)
\(422\) 9.51735e8 0.616485
\(423\) 0 0
\(424\) 1.74264e9 1.11027
\(425\) −2.04892e7 −0.0129469
\(426\) 0 0
\(427\) −1.27105e9 −0.790072
\(428\) −6.02734e8 −0.371597
\(429\) 0 0
\(430\) −4.29754e7 −0.0260663
\(431\) 2.08550e9 1.25470 0.627349 0.778738i \(-0.284140\pi\)
0.627349 + 0.778738i \(0.284140\pi\)
\(432\) 0 0
\(433\) −1.27758e9 −0.756277 −0.378139 0.925749i \(-0.623436\pi\)
−0.378139 + 0.925749i \(0.623436\pi\)
\(434\) −1.21900e9 −0.715795
\(435\) 0 0
\(436\) −1.50914e9 −0.872020
\(437\) −2.89268e8 −0.165812
\(438\) 0 0
\(439\) −3.72523e8 −0.210149 −0.105074 0.994464i \(-0.533508\pi\)
−0.105074 + 0.994464i \(0.533508\pi\)
\(440\) −2.15155e9 −1.20411
\(441\) 0 0
\(442\) −1.09423e7 −0.00602743
\(443\) −2.51167e8 −0.137262 −0.0686309 0.997642i \(-0.521863\pi\)
−0.0686309 + 0.997642i \(0.521863\pi\)
\(444\) 0 0
\(445\) 6.46720e8 0.347902
\(446\) 9.70586e8 0.518038
\(447\) 0 0
\(448\) −1.23820e9 −0.650603
\(449\) 7.22359e8 0.376609 0.188304 0.982111i \(-0.439701\pi\)
0.188304 + 0.982111i \(0.439701\pi\)
\(450\) 0 0
\(451\) 2.30349e9 1.18241
\(452\) 1.07113e9 0.545577
\(453\) 0 0
\(454\) 3.43006e8 0.172031
\(455\) −7.42265e8 −0.369419
\(456\) 0 0
\(457\) −3.81278e9 −1.86868 −0.934341 0.356380i \(-0.884011\pi\)
−0.934341 + 0.356380i \(0.884011\pi\)
\(458\) 8.18995e8 0.398338
\(459\) 0 0
\(460\) −5.39105e8 −0.258238
\(461\) 2.43626e9 1.15817 0.579083 0.815269i \(-0.303411\pi\)
0.579083 + 0.815269i \(0.303411\pi\)
\(462\) 0 0
\(463\) −2.87487e9 −1.34612 −0.673061 0.739587i \(-0.735021\pi\)
−0.673061 + 0.739587i \(0.735021\pi\)
\(464\) −1.73115e9 −0.804490
\(465\) 0 0
\(466\) 2.27650e8 0.104212
\(467\) 1.80431e9 0.819787 0.409894 0.912133i \(-0.365566\pi\)
0.409894 + 0.912133i \(0.365566\pi\)
\(468\) 0 0
\(469\) 3.97341e9 1.77852
\(470\) −8.43915e7 −0.0374935
\(471\) 0 0
\(472\) −1.80734e8 −0.0791119
\(473\) −3.20462e8 −0.139239
\(474\) 0 0
\(475\) 2.29283e8 0.0981621
\(476\) 2.57666e8 0.109504
\(477\) 0 0
\(478\) 6.76388e7 0.0283269
\(479\) 2.36338e9 0.982558 0.491279 0.871002i \(-0.336529\pi\)
0.491279 + 0.871002i \(0.336529\pi\)
\(480\) 0 0
\(481\) −3.11176e8 −0.127496
\(482\) −8.80674e7 −0.0358221
\(483\) 0 0
\(484\) −5.34975e9 −2.14474
\(485\) 1.01346e9 0.403376
\(486\) 0 0
\(487\) 3.50576e9 1.37541 0.687703 0.725993i \(-0.258619\pi\)
0.687703 + 0.725993i \(0.258619\pi\)
\(488\) 8.26226e8 0.321832
\(489\) 0 0
\(490\) −1.09995e9 −0.422366
\(491\) 3.92494e9 1.49640 0.748199 0.663474i \(-0.230919\pi\)
0.748199 + 0.663474i \(0.230919\pi\)
\(492\) 0 0
\(493\) 2.49094e8 0.0936267
\(494\) 1.22449e8 0.0456996
\(495\) 0 0
\(496\) −2.86164e9 −1.05300
\(497\) 7.95108e9 2.90522
\(498\) 0 0
\(499\) −3.58721e9 −1.29242 −0.646212 0.763158i \(-0.723648\pi\)
−0.646212 + 0.763158i \(0.723648\pi\)
\(500\) −2.27233e9 −0.812972
\(501\) 0 0
\(502\) 6.70559e8 0.236578
\(503\) −2.48168e9 −0.869475 −0.434737 0.900557i \(-0.643159\pi\)
−0.434737 + 0.900557i \(0.643159\pi\)
\(504\) 0 0
\(505\) 3.56137e8 0.123054
\(506\) 4.59440e8 0.157653
\(507\) 0 0
\(508\) −1.31064e9 −0.443568
\(509\) −3.73752e9 −1.25624 −0.628118 0.778118i \(-0.716174\pi\)
−0.628118 + 0.778118i \(0.716174\pi\)
\(510\) 0 0
\(511\) −3.42929e9 −1.13692
\(512\) 3.07455e9 1.01236
\(513\) 0 0
\(514\) −2.89730e8 −0.0941071
\(515\) 3.02835e9 0.976969
\(516\) 0 0
\(517\) −6.29296e8 −0.200280
\(518\) −8.37435e8 −0.264726
\(519\) 0 0
\(520\) 4.82496e8 0.150481
\(521\) 3.12643e9 0.968539 0.484270 0.874919i \(-0.339085\pi\)
0.484270 + 0.874919i \(0.339085\pi\)
\(522\) 0 0
\(523\) 3.47385e9 1.06183 0.530915 0.847425i \(-0.321848\pi\)
0.530915 + 0.847425i \(0.321848\pi\)
\(524\) 1.93501e9 0.587520
\(525\) 0 0
\(526\) 1.54897e8 0.0464080
\(527\) 4.11761e8 0.122548
\(528\) 0 0
\(529\) −3.16143e9 −0.928514
\(530\) 2.15841e9 0.629752
\(531\) 0 0
\(532\) −2.88338e9 −0.830255
\(533\) −5.16570e8 −0.147769
\(534\) 0 0
\(535\) −1.57839e9 −0.445633
\(536\) −2.58285e9 −0.724472
\(537\) 0 0
\(538\) 9.04381e8 0.250388
\(539\) −8.20222e9 −2.25616
\(540\) 0 0
\(541\) 6.14788e8 0.166930 0.0834651 0.996511i \(-0.473401\pi\)
0.0834651 + 0.996511i \(0.473401\pi\)
\(542\) −2.38795e8 −0.0644210
\(543\) 0 0
\(544\) −2.55763e8 −0.0681149
\(545\) −3.95202e9 −1.04576
\(546\) 0 0
\(547\) −3.36033e9 −0.877863 −0.438931 0.898521i \(-0.644643\pi\)
−0.438931 + 0.898521i \(0.644643\pi\)
\(548\) −6.35278e9 −1.64904
\(549\) 0 0
\(550\) −3.64167e8 −0.0933321
\(551\) −2.78747e9 −0.709871
\(552\) 0 0
\(553\) −2.70654e9 −0.680575
\(554\) 3.40545e8 0.0850923
\(555\) 0 0
\(556\) 4.50949e7 0.0111267
\(557\) 2.92483e8 0.0717145 0.0358572 0.999357i \(-0.488584\pi\)
0.0358572 + 0.999357i \(0.488584\pi\)
\(558\) 0 0
\(559\) 7.18654e7 0.0174012
\(560\) −4.68939e9 −1.12839
\(561\) 0 0
\(562\) −3.93156e8 −0.0934302
\(563\) −5.83190e9 −1.37731 −0.688653 0.725091i \(-0.741798\pi\)
−0.688653 + 0.725091i \(0.741798\pi\)
\(564\) 0 0
\(565\) 2.80498e9 0.654276
\(566\) 2.24733e9 0.520967
\(567\) 0 0
\(568\) −5.16846e9 −1.18343
\(569\) 1.28209e9 0.291760 0.145880 0.989302i \(-0.453399\pi\)
0.145880 + 0.989302i \(0.453399\pi\)
\(570\) 0 0
\(571\) 2.52700e9 0.568041 0.284020 0.958818i \(-0.408332\pi\)
0.284020 + 0.958818i \(0.408332\pi\)
\(572\) 1.70171e9 0.380189
\(573\) 0 0
\(574\) −1.39019e9 −0.306819
\(575\) −1.92925e8 −0.0423204
\(576\) 0 0
\(577\) −3.17996e9 −0.689139 −0.344570 0.938761i \(-0.611975\pi\)
−0.344570 + 0.938761i \(0.611975\pi\)
\(578\) −1.47684e9 −0.318116
\(579\) 0 0
\(580\) −5.19497e9 −1.10557
\(581\) −7.22639e9 −1.52864
\(582\) 0 0
\(583\) 1.60950e10 3.36396
\(584\) 2.22915e9 0.463121
\(585\) 0 0
\(586\) −2.27701e9 −0.467437
\(587\) −3.74135e9 −0.763474 −0.381737 0.924271i \(-0.624674\pi\)
−0.381737 + 0.924271i \(0.624674\pi\)
\(588\) 0 0
\(589\) −4.60777e9 −0.929154
\(590\) −2.23854e8 −0.0448728
\(591\) 0 0
\(592\) −1.96591e9 −0.389437
\(593\) 1.49511e8 0.0294430 0.0147215 0.999892i \(-0.495314\pi\)
0.0147215 + 0.999892i \(0.495314\pi\)
\(594\) 0 0
\(595\) 6.74755e8 0.131322
\(596\) −3.90364e9 −0.755279
\(597\) 0 0
\(598\) −1.03032e8 −0.0197024
\(599\) 4.90943e9 0.933335 0.466668 0.884433i \(-0.345454\pi\)
0.466668 + 0.884433i \(0.345454\pi\)
\(600\) 0 0
\(601\) 9.51517e9 1.78795 0.893976 0.448114i \(-0.147904\pi\)
0.893976 + 0.448114i \(0.147904\pi\)
\(602\) 1.93404e8 0.0361308
\(603\) 0 0
\(604\) −2.14353e9 −0.395822
\(605\) −1.40095e10 −2.57205
\(606\) 0 0
\(607\) 6.23277e9 1.13115 0.565576 0.824696i \(-0.308654\pi\)
0.565576 + 0.824696i \(0.308654\pi\)
\(608\) 2.86210e9 0.516442
\(609\) 0 0
\(610\) 1.02335e9 0.182545
\(611\) 1.41123e8 0.0250296
\(612\) 0 0
\(613\) −2.97929e9 −0.522397 −0.261198 0.965285i \(-0.584118\pi\)
−0.261198 + 0.965285i \(0.584118\pi\)
\(614\) 2.70066e9 0.470848
\(615\) 0 0
\(616\) 9.68268e9 1.66903
\(617\) −3.18056e9 −0.545137 −0.272569 0.962136i \(-0.587873\pi\)
−0.272569 + 0.962136i \(0.587873\pi\)
\(618\) 0 0
\(619\) 1.21436e8 0.0205793 0.0102896 0.999947i \(-0.496725\pi\)
0.0102896 + 0.999947i \(0.496725\pi\)
\(620\) −8.58745e9 −1.44708
\(621\) 0 0
\(622\) −2.65217e9 −0.441912
\(623\) −2.91046e9 −0.482229
\(624\) 0 0
\(625\) −6.91669e9 −1.13323
\(626\) 1.09853e9 0.178979
\(627\) 0 0
\(628\) −3.80583e9 −0.613183
\(629\) 2.82874e8 0.0453227
\(630\) 0 0
\(631\) −5.92643e9 −0.939053 −0.469527 0.882918i \(-0.655575\pi\)
−0.469527 + 0.882918i \(0.655575\pi\)
\(632\) 1.75934e9 0.277229
\(633\) 0 0
\(634\) −1.01328e9 −0.157912
\(635\) −3.43221e9 −0.531943
\(636\) 0 0
\(637\) 1.83939e9 0.281959
\(638\) 4.42730e9 0.674942
\(639\) 0 0
\(640\) 6.94057e9 1.04656
\(641\) 7.39788e8 0.110944 0.0554721 0.998460i \(-0.482334\pi\)
0.0554721 + 0.998460i \(0.482334\pi\)
\(642\) 0 0
\(643\) 5.85977e9 0.869245 0.434622 0.900613i \(-0.356882\pi\)
0.434622 + 0.900613i \(0.356882\pi\)
\(644\) 2.42615e9 0.357946
\(645\) 0 0
\(646\) −1.11313e8 −0.0162454
\(647\) 7.09749e9 1.03024 0.515122 0.857117i \(-0.327747\pi\)
0.515122 + 0.857117i \(0.327747\pi\)
\(648\) 0 0
\(649\) −1.66925e9 −0.239698
\(650\) 8.16665e7 0.0116640
\(651\) 0 0
\(652\) 8.09841e9 1.14428
\(653\) −3.83880e9 −0.539510 −0.269755 0.962929i \(-0.586943\pi\)
−0.269755 + 0.962929i \(0.586943\pi\)
\(654\) 0 0
\(655\) 5.06725e9 0.704576
\(656\) −3.26352e9 −0.451360
\(657\) 0 0
\(658\) 3.79790e8 0.0519701
\(659\) 4.88094e9 0.664361 0.332181 0.943216i \(-0.392216\pi\)
0.332181 + 0.943216i \(0.392216\pi\)
\(660\) 0 0
\(661\) −5.63851e9 −0.759381 −0.379690 0.925114i \(-0.623969\pi\)
−0.379690 + 0.925114i \(0.623969\pi\)
\(662\) 4.35458e9 0.583369
\(663\) 0 0
\(664\) 4.69739e9 0.622685
\(665\) −7.55079e9 −0.995673
\(666\) 0 0
\(667\) 2.34545e9 0.306045
\(668\) −5.77076e9 −0.749059
\(669\) 0 0
\(670\) −3.19907e9 −0.410925
\(671\) 7.63100e9 0.975108
\(672\) 0 0
\(673\) 1.34178e10 1.69680 0.848399 0.529358i \(-0.177567\pi\)
0.848399 + 0.529358i \(0.177567\pi\)
\(674\) −4.20569e9 −0.529088
\(675\) 0 0
\(676\) 6.82640e9 0.849921
\(677\) 1.49956e10 1.85740 0.928699 0.370835i \(-0.120928\pi\)
0.928699 + 0.370835i \(0.120928\pi\)
\(678\) 0 0
\(679\) −4.56090e9 −0.559122
\(680\) −4.38613e8 −0.0534934
\(681\) 0 0
\(682\) 7.31846e9 0.883435
\(683\) 6.32321e9 0.759391 0.379695 0.925112i \(-0.376029\pi\)
0.379695 + 0.925112i \(0.376029\pi\)
\(684\) 0 0
\(685\) −1.66362e10 −1.97759
\(686\) 9.10552e8 0.107689
\(687\) 0 0
\(688\) 4.54022e8 0.0531517
\(689\) −3.60940e9 −0.420404
\(690\) 0 0
\(691\) −1.51402e10 −1.74565 −0.872826 0.488031i \(-0.837715\pi\)
−0.872826 + 0.488031i \(0.837715\pi\)
\(692\) 4.13618e9 0.474491
\(693\) 0 0
\(694\) 4.28850e8 0.0487021
\(695\) 1.18091e8 0.0133435
\(696\) 0 0
\(697\) 4.69588e8 0.0525293
\(698\) 1.04055e9 0.115816
\(699\) 0 0
\(700\) −1.92305e9 −0.211908
\(701\) −1.32383e10 −1.45151 −0.725755 0.687953i \(-0.758509\pi\)
−0.725755 + 0.687953i \(0.758509\pi\)
\(702\) 0 0
\(703\) −3.16548e9 −0.343634
\(704\) 7.43373e9 0.802975
\(705\) 0 0
\(706\) 1.76283e9 0.188536
\(707\) −1.60274e9 −0.170567
\(708\) 0 0
\(709\) −6.66730e9 −0.702568 −0.351284 0.936269i \(-0.614255\pi\)
−0.351284 + 0.936269i \(0.614255\pi\)
\(710\) −6.40158e9 −0.671248
\(711\) 0 0
\(712\) 1.89189e9 0.196434
\(713\) 3.87710e9 0.400584
\(714\) 0 0
\(715\) 4.45632e9 0.455937
\(716\) −7.99114e9 −0.813604
\(717\) 0 0
\(718\) −5.53005e9 −0.557563
\(719\) 5.24566e9 0.526319 0.263159 0.964752i \(-0.415235\pi\)
0.263159 + 0.964752i \(0.415235\pi\)
\(720\) 0 0
\(721\) −1.36286e10 −1.35418
\(722\) −1.99315e9 −0.197087
\(723\) 0 0
\(724\) −1.67533e10 −1.64065
\(725\) −1.85908e9 −0.181182
\(726\) 0 0
\(727\) 4.33721e9 0.418639 0.209320 0.977847i \(-0.432875\pi\)
0.209320 + 0.977847i \(0.432875\pi\)
\(728\) −2.17140e9 −0.208583
\(729\) 0 0
\(730\) 2.76099e9 0.262685
\(731\) −6.53291e7 −0.00618580
\(732\) 0 0
\(733\) −1.14178e10 −1.07083 −0.535413 0.844590i \(-0.679844\pi\)
−0.535413 + 0.844590i \(0.679844\pi\)
\(734\) −5.37200e8 −0.0501417
\(735\) 0 0
\(736\) −2.40824e9 −0.222653
\(737\) −2.38551e10 −2.19505
\(738\) 0 0
\(739\) −1.87923e9 −0.171287 −0.0856433 0.996326i \(-0.527295\pi\)
−0.0856433 + 0.996326i \(0.527295\pi\)
\(740\) −5.89946e9 −0.535182
\(741\) 0 0
\(742\) −9.71359e9 −0.872903
\(743\) 9.66780e9 0.864703 0.432351 0.901705i \(-0.357684\pi\)
0.432351 + 0.901705i \(0.357684\pi\)
\(744\) 0 0
\(745\) −1.02226e10 −0.905759
\(746\) −5.65193e9 −0.498438
\(747\) 0 0
\(748\) −1.54694e9 −0.135151
\(749\) 7.10331e9 0.617695
\(750\) 0 0
\(751\) −1.07300e10 −0.924403 −0.462202 0.886775i \(-0.652940\pi\)
−0.462202 + 0.886775i \(0.652940\pi\)
\(752\) 8.91571e8 0.0764528
\(753\) 0 0
\(754\) −9.92847e8 −0.0843496
\(755\) −5.61331e9 −0.474684
\(756\) 0 0
\(757\) 1.90038e10 1.59223 0.796115 0.605145i \(-0.206885\pi\)
0.796115 + 0.605145i \(0.206885\pi\)
\(758\) 6.42453e9 0.535796
\(759\) 0 0
\(760\) 4.90826e9 0.405583
\(761\) −8.07062e9 −0.663836 −0.331918 0.943308i \(-0.607696\pi\)
−0.331918 + 0.943308i \(0.607696\pi\)
\(762\) 0 0
\(763\) 1.77854e10 1.44953
\(764\) 4.36259e9 0.353930
\(765\) 0 0
\(766\) −6.20785e9 −0.499046
\(767\) 3.74339e8 0.0299558
\(768\) 0 0
\(769\) −4.19821e9 −0.332906 −0.166453 0.986049i \(-0.553231\pi\)
−0.166453 + 0.986049i \(0.553231\pi\)
\(770\) 1.19928e10 0.946682
\(771\) 0 0
\(772\) 1.36184e10 1.06529
\(773\) 1.26866e10 0.987910 0.493955 0.869487i \(-0.335551\pi\)
0.493955 + 0.869487i \(0.335551\pi\)
\(774\) 0 0
\(775\) −3.07311e9 −0.237150
\(776\) 2.96473e9 0.227756
\(777\) 0 0
\(778\) 3.38508e9 0.257716
\(779\) −5.25488e9 −0.398274
\(780\) 0 0
\(781\) −4.77357e10 −3.58563
\(782\) 9.36612e7 0.00700384
\(783\) 0 0
\(784\) 1.16207e10 0.861243
\(785\) −9.96642e9 −0.735352
\(786\) 0 0
\(787\) 1.93383e10 1.41419 0.707093 0.707121i \(-0.250006\pi\)
0.707093 + 0.707121i \(0.250006\pi\)
\(788\) −1.58138e10 −1.15132
\(789\) 0 0
\(790\) 2.17909e9 0.157246
\(791\) −1.26234e10 −0.906896
\(792\) 0 0
\(793\) −1.71129e9 −0.121862
\(794\) 2.69362e9 0.190970
\(795\) 0 0
\(796\) 1.15474e10 0.811503
\(797\) 2.47116e10 1.72901 0.864505 0.502624i \(-0.167632\pi\)
0.864505 + 0.502624i \(0.167632\pi\)
\(798\) 0 0
\(799\) −1.28288e8 −0.00889759
\(800\) 1.90885e9 0.131813
\(801\) 0 0
\(802\) −8.81675e9 −0.603529
\(803\) 2.05884e10 1.40319
\(804\) 0 0
\(805\) 6.35343e9 0.429262
\(806\) −1.64121e9 −0.110406
\(807\) 0 0
\(808\) 1.04183e9 0.0694796
\(809\) −2.28215e10 −1.51539 −0.757695 0.652609i \(-0.773674\pi\)
−0.757695 + 0.652609i \(0.773674\pi\)
\(810\) 0 0
\(811\) −2.67364e10 −1.76007 −0.880037 0.474906i \(-0.842482\pi\)
−0.880037 + 0.474906i \(0.842482\pi\)
\(812\) 2.33791e10 1.53244
\(813\) 0 0
\(814\) 5.02769e9 0.326725
\(815\) 2.12075e10 1.37227
\(816\) 0 0
\(817\) 7.31060e8 0.0469003
\(818\) −9.16160e9 −0.585241
\(819\) 0 0
\(820\) −9.79346e9 −0.620280
\(821\) −1.71127e10 −1.07924 −0.539618 0.841910i \(-0.681431\pi\)
−0.539618 + 0.841910i \(0.681431\pi\)
\(822\) 0 0
\(823\) −2.37659e10 −1.48612 −0.743061 0.669224i \(-0.766627\pi\)
−0.743061 + 0.669224i \(0.766627\pi\)
\(824\) 8.85903e9 0.551621
\(825\) 0 0
\(826\) 1.00742e9 0.0621985
\(827\) 6.09517e9 0.374728 0.187364 0.982291i \(-0.440006\pi\)
0.187364 + 0.982291i \(0.440006\pi\)
\(828\) 0 0
\(829\) −5.59971e9 −0.341370 −0.170685 0.985326i \(-0.554598\pi\)
−0.170685 + 0.985326i \(0.554598\pi\)
\(830\) 5.81812e9 0.353191
\(831\) 0 0
\(832\) −1.66706e9 −0.100350
\(833\) −1.67210e9 −0.100232
\(834\) 0 0
\(835\) −1.51121e10 −0.898299
\(836\) 1.73109e10 1.02470
\(837\) 0 0
\(838\) 4.87476e9 0.286153
\(839\) 1.48782e10 0.869730 0.434865 0.900496i \(-0.356796\pi\)
0.434865 + 0.900496i \(0.356796\pi\)
\(840\) 0 0
\(841\) 5.35156e9 0.310237
\(842\) 3.68789e9 0.212905
\(843\) 0 0
\(844\) −3.01733e10 −1.72752
\(845\) 1.78765e10 1.01926
\(846\) 0 0
\(847\) 6.30476e10 3.56514
\(848\) −2.28030e10 −1.28412
\(849\) 0 0
\(850\) −7.42389e7 −0.00414634
\(851\) 2.66351e9 0.148150
\(852\) 0 0
\(853\) 3.87811e9 0.213943 0.106972 0.994262i \(-0.465885\pi\)
0.106972 + 0.994262i \(0.465885\pi\)
\(854\) −4.60543e9 −0.253027
\(855\) 0 0
\(856\) −4.61738e9 −0.251616
\(857\) 1.92983e10 1.04733 0.523667 0.851923i \(-0.324564\pi\)
0.523667 + 0.851923i \(0.324564\pi\)
\(858\) 0 0
\(859\) 2.34053e10 1.25990 0.629952 0.776634i \(-0.283074\pi\)
0.629952 + 0.776634i \(0.283074\pi\)
\(860\) 1.36247e9 0.0730435
\(861\) 0 0
\(862\) 7.55642e9 0.401828
\(863\) −2.15437e10 −1.14099 −0.570495 0.821301i \(-0.693249\pi\)
−0.570495 + 0.821301i \(0.693249\pi\)
\(864\) 0 0
\(865\) 1.08315e10 0.569027
\(866\) −4.62908e9 −0.242204
\(867\) 0 0
\(868\) 3.86464e10 2.00581
\(869\) 1.62492e10 0.839966
\(870\) 0 0
\(871\) 5.34964e9 0.274322
\(872\) −1.15611e10 −0.590462
\(873\) 0 0
\(874\) −1.04811e9 −0.0531026
\(875\) 2.67797e10 1.35138
\(876\) 0 0
\(877\) −1.77005e10 −0.886108 −0.443054 0.896495i \(-0.646105\pi\)
−0.443054 + 0.896495i \(0.646105\pi\)
\(878\) −1.34977e9 −0.0673021
\(879\) 0 0
\(880\) 2.81536e10 1.39266
\(881\) 2.35041e10 1.15805 0.579027 0.815308i \(-0.303433\pi\)
0.579027 + 0.815308i \(0.303433\pi\)
\(882\) 0 0
\(883\) 6.99377e9 0.341860 0.170930 0.985283i \(-0.445323\pi\)
0.170930 + 0.985283i \(0.445323\pi\)
\(884\) 3.46910e8 0.0168902
\(885\) 0 0
\(886\) −9.10057e8 −0.0439593
\(887\) 1.04556e10 0.503056 0.251528 0.967850i \(-0.419067\pi\)
0.251528 + 0.967850i \(0.419067\pi\)
\(888\) 0 0
\(889\) 1.54461e10 0.737331
\(890\) 2.34327e9 0.111419
\(891\) 0 0
\(892\) −3.07709e10 −1.45166
\(893\) 1.43560e9 0.0674609
\(894\) 0 0
\(895\) −2.09266e10 −0.975704
\(896\) −3.12349e10 −1.45065
\(897\) 0 0
\(898\) 2.61733e9 0.120612
\(899\) 3.73609e10 1.71498
\(900\) 0 0
\(901\) 3.28112e9 0.149446
\(902\) 8.34626e9 0.378677
\(903\) 0 0
\(904\) 8.20560e9 0.369420
\(905\) −4.38724e10 −1.96753
\(906\) 0 0
\(907\) 2.84631e10 1.26665 0.633325 0.773886i \(-0.281690\pi\)
0.633325 + 0.773886i \(0.281690\pi\)
\(908\) −1.08745e10 −0.482067
\(909\) 0 0
\(910\) −2.68946e9 −0.118310
\(911\) 8.69725e9 0.381125 0.190562 0.981675i \(-0.438969\pi\)
0.190562 + 0.981675i \(0.438969\pi\)
\(912\) 0 0
\(913\) 4.33849e10 1.88665
\(914\) −1.38149e10 −0.598462
\(915\) 0 0
\(916\) −2.59650e10 −1.11623
\(917\) −2.28043e10 −0.976618
\(918\) 0 0
\(919\) 1.81679e10 0.772148 0.386074 0.922468i \(-0.373831\pi\)
0.386074 + 0.922468i \(0.373831\pi\)
\(920\) −4.12993e9 −0.174858
\(921\) 0 0
\(922\) 8.82735e9 0.370913
\(923\) 1.07050e10 0.448106
\(924\) 0 0
\(925\) −2.11119e9 −0.0877063
\(926\) −1.04166e10 −0.431107
\(927\) 0 0
\(928\) −2.32065e10 −0.953218
\(929\) −2.23626e10 −0.915098 −0.457549 0.889184i \(-0.651273\pi\)
−0.457549 + 0.889184i \(0.651273\pi\)
\(930\) 0 0
\(931\) 1.87115e10 0.759949
\(932\) −7.21731e9 −0.292025
\(933\) 0 0
\(934\) 6.53757e9 0.262544
\(935\) −4.05101e9 −0.162078
\(936\) 0 0
\(937\) −4.79626e8 −0.0190465 −0.00952323 0.999955i \(-0.503031\pi\)
−0.00952323 + 0.999955i \(0.503031\pi\)
\(938\) 1.43969e10 0.569586
\(939\) 0 0
\(940\) 2.67550e9 0.105065
\(941\) 3.41566e10 1.33632 0.668161 0.744017i \(-0.267082\pi\)
0.668161 + 0.744017i \(0.267082\pi\)
\(942\) 0 0
\(943\) 4.42159e9 0.171707
\(944\) 2.36495e9 0.0914997
\(945\) 0 0
\(946\) −1.16113e9 −0.0445926
\(947\) 1.00641e10 0.385079 0.192539 0.981289i \(-0.438328\pi\)
0.192539 + 0.981289i \(0.438328\pi\)
\(948\) 0 0
\(949\) −4.61706e9 −0.175361
\(950\) 8.30764e8 0.0314373
\(951\) 0 0
\(952\) 1.97391e9 0.0741476
\(953\) 2.94200e10 1.10108 0.550539 0.834810i \(-0.314422\pi\)
0.550539 + 0.834810i \(0.314422\pi\)
\(954\) 0 0
\(955\) 1.14244e10 0.424446
\(956\) −2.14438e9 −0.0793780
\(957\) 0 0
\(958\) 8.56325e9 0.314673
\(959\) 7.48685e10 2.74116
\(960\) 0 0
\(961\) 3.42460e10 1.24474
\(962\) −1.12749e9 −0.0408318
\(963\) 0 0
\(964\) 2.79204e9 0.100381
\(965\) 3.56629e10 1.27753
\(966\) 0 0
\(967\) 9.18664e9 0.326711 0.163356 0.986567i \(-0.447768\pi\)
0.163356 + 0.986567i \(0.447768\pi\)
\(968\) −4.09830e10 −1.45224
\(969\) 0 0
\(970\) 3.67208e9 0.129185
\(971\) −2.46294e10 −0.863348 −0.431674 0.902030i \(-0.642077\pi\)
−0.431674 + 0.902030i \(0.642077\pi\)
\(972\) 0 0
\(973\) −5.31450e8 −0.0184956
\(974\) 1.27025e10 0.440486
\(975\) 0 0
\(976\) −1.08114e10 −0.372227
\(977\) −3.26473e10 −1.12000 −0.559998 0.828494i \(-0.689198\pi\)
−0.559998 + 0.828494i \(0.689198\pi\)
\(978\) 0 0
\(979\) 1.74735e10 0.595168
\(980\) 3.48724e10 1.18356
\(981\) 0 0
\(982\) 1.42213e10 0.479235
\(983\) −4.98914e9 −0.167528 −0.0837641 0.996486i \(-0.526694\pi\)
−0.0837641 + 0.996486i \(0.526694\pi\)
\(984\) 0 0
\(985\) −4.14121e10 −1.38070
\(986\) 9.02547e8 0.0299848
\(987\) 0 0
\(988\) −3.88207e9 −0.128060
\(989\) −6.15133e8 −0.0202200
\(990\) 0 0
\(991\) −1.15183e10 −0.375950 −0.187975 0.982174i \(-0.560192\pi\)
−0.187975 + 0.982174i \(0.560192\pi\)
\(992\) −3.83611e10 −1.24767
\(993\) 0 0
\(994\) 2.88093e10 0.930422
\(995\) 3.02396e10 0.973185
\(996\) 0 0
\(997\) 2.00277e10 0.640026 0.320013 0.947413i \(-0.396313\pi\)
0.320013 + 0.947413i \(0.396313\pi\)
\(998\) −1.29976e10 −0.413910
\(999\) 0 0
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 531.8.a.c.1.9 17
3.2 odd 2 177.8.a.c.1.9 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.9 17 3.2 odd 2
531.8.a.c.1.9 17 1.1 even 1 trivial