Properties

Label 531.8.a.b.1.10
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $16$
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: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(4.55626\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.55626 q^{2} -107.241 q^{4} +540.445 q^{5} -1238.10 q^{7} -1071.82 q^{8} +O(q^{10})\) \(q+4.55626 q^{2} -107.241 q^{4} +540.445 q^{5} -1238.10 q^{7} -1071.82 q^{8} +2462.40 q^{10} -3521.59 q^{11} -4832.75 q^{13} -5641.10 q^{14} +8843.32 q^{16} -2243.59 q^{17} +53136.6 q^{19} -57957.6 q^{20} -16045.3 q^{22} +56390.0 q^{23} +213955. q^{25} -22019.3 q^{26} +132774. q^{28} +137347. q^{29} -121430. q^{31} +177485. q^{32} -10222.4 q^{34} -669124. q^{35} -476697. q^{37} +242104. q^{38} -579257. q^{40} -140564. q^{41} +696919. q^{43} +377657. q^{44} +256927. q^{46} -783363. q^{47} +709348. q^{49} +974835. q^{50} +518267. q^{52} -1.05434e6 q^{53} -1.90322e6 q^{55} +1.32702e6 q^{56} +625790. q^{58} -205379. q^{59} +378219. q^{61} -553265. q^{62} -323278. q^{64} -2.61184e6 q^{65} -1.12190e6 q^{67} +240604. q^{68} -3.04870e6 q^{70} -3.82821e6 q^{71} -1.68355e6 q^{73} -2.17195e6 q^{74} -5.69840e6 q^{76} +4.36008e6 q^{77} +5.56662e6 q^{79} +4.77932e6 q^{80} -640447. q^{82} +1.80553e6 q^{83} -1.21254e6 q^{85} +3.17534e6 q^{86} +3.77449e6 q^{88} -8.26085e6 q^{89} +5.98343e6 q^{91} -6.04729e6 q^{92} -3.56920e6 q^{94} +2.87174e7 q^{95} -9.36799e6 q^{97} +3.23197e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.55626 0.402720 0.201360 0.979517i \(-0.435464\pi\)
0.201360 + 0.979517i \(0.435464\pi\)
\(3\) 0 0
\(4\) −107.241 −0.837817
\(5\) 540.445 1.93355 0.966777 0.255623i \(-0.0822806\pi\)
0.966777 + 0.255623i \(0.0822806\pi\)
\(6\) 0 0
\(7\) −1238.10 −1.36431 −0.682154 0.731208i \(-0.738957\pi\)
−0.682154 + 0.731208i \(0.738957\pi\)
\(8\) −1071.82 −0.740125
\(9\) 0 0
\(10\) 2462.40 0.778680
\(11\) −3521.59 −0.797745 −0.398872 0.917006i \(-0.630598\pi\)
−0.398872 + 0.917006i \(0.630598\pi\)
\(12\) 0 0
\(13\) −4832.75 −0.610089 −0.305044 0.952338i \(-0.598671\pi\)
−0.305044 + 0.952338i \(0.598671\pi\)
\(14\) −5641.10 −0.549434
\(15\) 0 0
\(16\) 8843.32 0.539753
\(17\) −2243.59 −0.110757 −0.0553787 0.998465i \(-0.517637\pi\)
−0.0553787 + 0.998465i \(0.517637\pi\)
\(18\) 0 0
\(19\) 53136.6 1.77728 0.888641 0.458603i \(-0.151650\pi\)
0.888641 + 0.458603i \(0.151650\pi\)
\(20\) −57957.6 −1.61996
\(21\) 0 0
\(22\) −16045.3 −0.321268
\(23\) 56390.0 0.966394 0.483197 0.875512i \(-0.339475\pi\)
0.483197 + 0.875512i \(0.339475\pi\)
\(24\) 0 0
\(25\) 213955. 2.73863
\(26\) −22019.3 −0.245695
\(27\) 0 0
\(28\) 132774. 1.14304
\(29\) 137347. 1.04575 0.522874 0.852410i \(-0.324860\pi\)
0.522874 + 0.852410i \(0.324860\pi\)
\(30\) 0 0
\(31\) −121430. −0.732081 −0.366041 0.930599i \(-0.619287\pi\)
−0.366041 + 0.930599i \(0.619287\pi\)
\(32\) 177485. 0.957495
\(33\) 0 0
\(34\) −10222.4 −0.0446042
\(35\) −669124. −2.63796
\(36\) 0 0
\(37\) −476697. −1.54716 −0.773581 0.633697i \(-0.781537\pi\)
−0.773581 + 0.633697i \(0.781537\pi\)
\(38\) 242104. 0.715747
\(39\) 0 0
\(40\) −579257. −1.43107
\(41\) −140564. −0.318516 −0.159258 0.987237i \(-0.550910\pi\)
−0.159258 + 0.987237i \(0.550910\pi\)
\(42\) 0 0
\(43\) 696919. 1.33673 0.668363 0.743835i \(-0.266995\pi\)
0.668363 + 0.743835i \(0.266995\pi\)
\(44\) 377657. 0.668364
\(45\) 0 0
\(46\) 256927. 0.389186
\(47\) −783363. −1.10058 −0.550289 0.834974i \(-0.685482\pi\)
−0.550289 + 0.834974i \(0.685482\pi\)
\(48\) 0 0
\(49\) 709348. 0.861337
\(50\) 974835. 1.10290
\(51\) 0 0
\(52\) 518267. 0.511142
\(53\) −1.05434e6 −0.972781 −0.486390 0.873742i \(-0.661687\pi\)
−0.486390 + 0.873742i \(0.661687\pi\)
\(54\) 0 0
\(55\) −1.90322e6 −1.54248
\(56\) 1.32702e6 1.00976
\(57\) 0 0
\(58\) 625790. 0.421144
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) 378219. 0.213348 0.106674 0.994294i \(-0.465980\pi\)
0.106674 + 0.994294i \(0.465980\pi\)
\(62\) −553265. −0.294824
\(63\) 0 0
\(64\) −323278. −0.154151
\(65\) −2.61184e6 −1.17964
\(66\) 0 0
\(67\) −1.12190e6 −0.455714 −0.227857 0.973695i \(-0.573172\pi\)
−0.227857 + 0.973695i \(0.573172\pi\)
\(68\) 240604. 0.0927943
\(69\) 0 0
\(70\) −3.04870e6 −1.06236
\(71\) −3.82821e6 −1.26938 −0.634689 0.772767i \(-0.718872\pi\)
−0.634689 + 0.772767i \(0.718872\pi\)
\(72\) 0 0
\(73\) −1.68355e6 −0.506518 −0.253259 0.967398i \(-0.581503\pi\)
−0.253259 + 0.967398i \(0.581503\pi\)
\(74\) −2.17195e6 −0.623073
\(75\) 0 0
\(76\) −5.69840e6 −1.48904
\(77\) 4.36008e6 1.08837
\(78\) 0 0
\(79\) 5.56662e6 1.27027 0.635136 0.772401i \(-0.280944\pi\)
0.635136 + 0.772401i \(0.280944\pi\)
\(80\) 4.77932e6 1.04364
\(81\) 0 0
\(82\) −640447. −0.128273
\(83\) 1.80553e6 0.346602 0.173301 0.984869i \(-0.444557\pi\)
0.173301 + 0.984869i \(0.444557\pi\)
\(84\) 0 0
\(85\) −1.21254e6 −0.214155
\(86\) 3.17534e6 0.538327
\(87\) 0 0
\(88\) 3.77449e6 0.590431
\(89\) −8.26085e6 −1.24211 −0.621055 0.783767i \(-0.713296\pi\)
−0.621055 + 0.783767i \(0.713296\pi\)
\(90\) 0 0
\(91\) 5.98343e6 0.832349
\(92\) −6.04729e6 −0.809661
\(93\) 0 0
\(94\) −3.56920e6 −0.443225
\(95\) 2.87174e7 3.43647
\(96\) 0 0
\(97\) −9.36799e6 −1.04219 −0.521093 0.853500i \(-0.674476\pi\)
−0.521093 + 0.853500i \(0.674476\pi\)
\(98\) 3.23197e6 0.346878
\(99\) 0 0
\(100\) −2.29447e7 −2.29447
\(101\) −5.17821e6 −0.500097 −0.250049 0.968233i \(-0.580447\pi\)
−0.250049 + 0.968233i \(0.580447\pi\)
\(102\) 0 0
\(103\) −1.63595e7 −1.47516 −0.737579 0.675261i \(-0.764031\pi\)
−0.737579 + 0.675261i \(0.764031\pi\)
\(104\) 5.17982e6 0.451542
\(105\) 0 0
\(106\) −4.80384e6 −0.391758
\(107\) 1.15281e7 0.909734 0.454867 0.890559i \(-0.349687\pi\)
0.454867 + 0.890559i \(0.349687\pi\)
\(108\) 0 0
\(109\) −1.66988e6 −0.123507 −0.0617535 0.998091i \(-0.519669\pi\)
−0.0617535 + 0.998091i \(0.519669\pi\)
\(110\) −8.67157e6 −0.621188
\(111\) 0 0
\(112\) −1.09489e7 −0.736390
\(113\) 2.82765e6 0.184353 0.0921767 0.995743i \(-0.470618\pi\)
0.0921767 + 0.995743i \(0.470618\pi\)
\(114\) 0 0
\(115\) 3.04756e7 1.86857
\(116\) −1.47292e7 −0.876145
\(117\) 0 0
\(118\) −935759. −0.0524297
\(119\) 2.77779e6 0.151107
\(120\) 0 0
\(121\) −7.08560e6 −0.363603
\(122\) 1.72326e6 0.0859195
\(123\) 0 0
\(124\) 1.30222e7 0.613350
\(125\) 7.34087e7 3.36173
\(126\) 0 0
\(127\) 4.15154e7 1.79844 0.899221 0.437496i \(-0.144134\pi\)
0.899221 + 0.437496i \(0.144134\pi\)
\(128\) −2.41910e7 −1.01957
\(129\) 0 0
\(130\) −1.19002e7 −0.475064
\(131\) −4.63759e7 −1.80237 −0.901183 0.433438i \(-0.857300\pi\)
−0.901183 + 0.433438i \(0.857300\pi\)
\(132\) 0 0
\(133\) −6.57885e7 −2.42476
\(134\) −5.11167e6 −0.183525
\(135\) 0 0
\(136\) 2.40472e6 0.0819743
\(137\) 3.20418e6 0.106462 0.0532310 0.998582i \(-0.483048\pi\)
0.0532310 + 0.998582i \(0.483048\pi\)
\(138\) 0 0
\(139\) 3.74591e7 1.18306 0.591529 0.806284i \(-0.298525\pi\)
0.591529 + 0.806284i \(0.298525\pi\)
\(140\) 7.17572e7 2.21013
\(141\) 0 0
\(142\) −1.74423e7 −0.511204
\(143\) 1.70190e7 0.486695
\(144\) 0 0
\(145\) 7.42286e7 2.02201
\(146\) −7.67067e6 −0.203985
\(147\) 0 0
\(148\) 5.11212e7 1.29624
\(149\) −2.39531e7 −0.593212 −0.296606 0.955000i \(-0.595855\pi\)
−0.296606 + 0.955000i \(0.595855\pi\)
\(150\) 0 0
\(151\) −6.59208e7 −1.55813 −0.779064 0.626945i \(-0.784305\pi\)
−0.779064 + 0.626945i \(0.784305\pi\)
\(152\) −5.69527e7 −1.31541
\(153\) 0 0
\(154\) 1.98656e7 0.438308
\(155\) −6.56261e7 −1.41552
\(156\) 0 0
\(157\) −6.45045e7 −1.33027 −0.665137 0.746721i \(-0.731627\pi\)
−0.665137 + 0.746721i \(0.731627\pi\)
\(158\) 2.53629e7 0.511564
\(159\) 0 0
\(160\) 9.59207e7 1.85137
\(161\) −6.98164e7 −1.31846
\(162\) 0 0
\(163\) −3.88407e7 −0.702474 −0.351237 0.936287i \(-0.614239\pi\)
−0.351237 + 0.936287i \(0.614239\pi\)
\(164\) 1.50742e7 0.266858
\(165\) 0 0
\(166\) 8.22644e6 0.139583
\(167\) −2.62525e7 −0.436178 −0.218089 0.975929i \(-0.569982\pi\)
−0.218089 + 0.975929i \(0.569982\pi\)
\(168\) 0 0
\(169\) −3.93930e7 −0.627792
\(170\) −5.52463e6 −0.0862446
\(171\) 0 0
\(172\) −7.47380e7 −1.11993
\(173\) 9.81294e6 0.144091 0.0720457 0.997401i \(-0.477047\pi\)
0.0720457 + 0.997401i \(0.477047\pi\)
\(174\) 0 0
\(175\) −2.64898e8 −3.73633
\(176\) −3.11425e7 −0.430585
\(177\) 0 0
\(178\) −3.76386e7 −0.500222
\(179\) 3.88790e7 0.506675 0.253337 0.967378i \(-0.418472\pi\)
0.253337 + 0.967378i \(0.418472\pi\)
\(180\) 0 0
\(181\) −5.75554e7 −0.721458 −0.360729 0.932671i \(-0.617472\pi\)
−0.360729 + 0.932671i \(0.617472\pi\)
\(182\) 2.72621e7 0.335204
\(183\) 0 0
\(184\) −6.04397e7 −0.715253
\(185\) −2.57628e8 −2.99152
\(186\) 0 0
\(187\) 7.90100e6 0.0883561
\(188\) 8.40083e7 0.922082
\(189\) 0 0
\(190\) 1.30844e8 1.38394
\(191\) −1.56708e8 −1.62733 −0.813663 0.581337i \(-0.802530\pi\)
−0.813663 + 0.581337i \(0.802530\pi\)
\(192\) 0 0
\(193\) 5.44941e7 0.545631 0.272816 0.962066i \(-0.412045\pi\)
0.272816 + 0.962066i \(0.412045\pi\)
\(194\) −4.26830e7 −0.419709
\(195\) 0 0
\(196\) −7.60709e7 −0.721642
\(197\) −1.46168e8 −1.36213 −0.681066 0.732222i \(-0.738483\pi\)
−0.681066 + 0.732222i \(0.738483\pi\)
\(198\) 0 0
\(199\) 5.06585e7 0.455687 0.227843 0.973698i \(-0.426833\pi\)
0.227843 + 0.973698i \(0.426833\pi\)
\(200\) −2.29321e8 −2.02693
\(201\) 0 0
\(202\) −2.35932e7 −0.201399
\(203\) −1.70050e8 −1.42672
\(204\) 0 0
\(205\) −7.59672e7 −0.615868
\(206\) −7.45379e7 −0.594076
\(207\) 0 0
\(208\) −4.27376e7 −0.329297
\(209\) −1.87125e8 −1.41782
\(210\) 0 0
\(211\) 3.36247e7 0.246416 0.123208 0.992381i \(-0.460682\pi\)
0.123208 + 0.992381i \(0.460682\pi\)
\(212\) 1.13068e8 0.815012
\(213\) 0 0
\(214\) 5.25250e7 0.366368
\(215\) 3.76646e8 2.58463
\(216\) 0 0
\(217\) 1.50342e8 0.998784
\(218\) −7.60839e6 −0.0497388
\(219\) 0 0
\(220\) 2.04103e8 1.29232
\(221\) 1.08427e7 0.0675718
\(222\) 0 0
\(223\) 8.68957e7 0.524725 0.262362 0.964969i \(-0.415498\pi\)
0.262362 + 0.964969i \(0.415498\pi\)
\(224\) −2.19744e8 −1.30632
\(225\) 0 0
\(226\) 1.28835e7 0.0742428
\(227\) −3.64696e7 −0.206938 −0.103469 0.994633i \(-0.532994\pi\)
−0.103469 + 0.994633i \(0.532994\pi\)
\(228\) 0 0
\(229\) 2.19546e8 1.20809 0.604047 0.796949i \(-0.293554\pi\)
0.604047 + 0.796949i \(0.293554\pi\)
\(230\) 1.38855e8 0.752512
\(231\) 0 0
\(232\) −1.47211e8 −0.773985
\(233\) −1.19989e8 −0.621435 −0.310718 0.950502i \(-0.600569\pi\)
−0.310718 + 0.950502i \(0.600569\pi\)
\(234\) 0 0
\(235\) −4.23364e8 −2.12803
\(236\) 2.20250e7 0.109074
\(237\) 0 0
\(238\) 1.26563e7 0.0608539
\(239\) 2.23797e8 1.06038 0.530189 0.847879i \(-0.322121\pi\)
0.530189 + 0.847879i \(0.322121\pi\)
\(240\) 0 0
\(241\) −1.78627e8 −0.822030 −0.411015 0.911629i \(-0.634826\pi\)
−0.411015 + 0.911629i \(0.634826\pi\)
\(242\) −3.22838e7 −0.146430
\(243\) 0 0
\(244\) −4.05604e7 −0.178746
\(245\) 3.83363e8 1.66544
\(246\) 0 0
\(247\) −2.56796e8 −1.08430
\(248\) 1.30150e8 0.541832
\(249\) 0 0
\(250\) 3.34469e8 1.35384
\(251\) 3.44628e8 1.37560 0.687800 0.725900i \(-0.258577\pi\)
0.687800 + 0.725900i \(0.258577\pi\)
\(252\) 0 0
\(253\) −1.98582e8 −0.770936
\(254\) 1.89155e8 0.724268
\(255\) 0 0
\(256\) −6.88409e7 −0.256452
\(257\) 1.65611e7 0.0608588 0.0304294 0.999537i \(-0.490313\pi\)
0.0304294 + 0.999537i \(0.490313\pi\)
\(258\) 0 0
\(259\) 5.90198e8 2.11081
\(260\) 2.80095e8 0.988321
\(261\) 0 0
\(262\) −2.11301e8 −0.725849
\(263\) −3.89298e8 −1.31958 −0.659792 0.751448i \(-0.729356\pi\)
−0.659792 + 0.751448i \(0.729356\pi\)
\(264\) 0 0
\(265\) −5.69812e8 −1.88092
\(266\) −2.99749e8 −0.976500
\(267\) 0 0
\(268\) 1.20313e8 0.381805
\(269\) 2.69260e8 0.843410 0.421705 0.906733i \(-0.361432\pi\)
0.421705 + 0.906733i \(0.361432\pi\)
\(270\) 0 0
\(271\) −4.95264e8 −1.51163 −0.755813 0.654787i \(-0.772758\pi\)
−0.755813 + 0.654787i \(0.772758\pi\)
\(272\) −1.98408e7 −0.0597816
\(273\) 0 0
\(274\) 1.45990e7 0.0428743
\(275\) −7.53462e8 −2.18473
\(276\) 0 0
\(277\) 3.16965e8 0.896051 0.448025 0.894021i \(-0.352127\pi\)
0.448025 + 0.894021i \(0.352127\pi\)
\(278\) 1.70673e8 0.476441
\(279\) 0 0
\(280\) 7.17178e8 1.95242
\(281\) 4.50536e8 1.21132 0.605658 0.795725i \(-0.292910\pi\)
0.605658 + 0.795725i \(0.292910\pi\)
\(282\) 0 0
\(283\) 4.67571e8 1.22629 0.613147 0.789969i \(-0.289903\pi\)
0.613147 + 0.789969i \(0.289903\pi\)
\(284\) 4.10539e8 1.06351
\(285\) 0 0
\(286\) 7.75428e7 0.196002
\(287\) 1.74033e8 0.434554
\(288\) 0 0
\(289\) −4.05305e8 −0.987733
\(290\) 3.38205e8 0.814304
\(291\) 0 0
\(292\) 1.80544e8 0.424369
\(293\) 1.60508e7 0.0372786 0.0186393 0.999826i \(-0.494067\pi\)
0.0186393 + 0.999826i \(0.494067\pi\)
\(294\) 0 0
\(295\) −1.10996e8 −0.251727
\(296\) 5.10931e8 1.14509
\(297\) 0 0
\(298\) −1.09136e8 −0.238898
\(299\) −2.72519e8 −0.589586
\(300\) 0 0
\(301\) −8.62855e8 −1.82371
\(302\) −3.00352e8 −0.627489
\(303\) 0 0
\(304\) 4.69904e8 0.959294
\(305\) 2.04406e8 0.412520
\(306\) 0 0
\(307\) −6.56164e7 −0.129428 −0.0647140 0.997904i \(-0.520614\pi\)
−0.0647140 + 0.997904i \(0.520614\pi\)
\(308\) −4.67577e8 −0.911854
\(309\) 0 0
\(310\) −2.99009e8 −0.570057
\(311\) −4.44680e8 −0.838275 −0.419138 0.907923i \(-0.637668\pi\)
−0.419138 + 0.907923i \(0.637668\pi\)
\(312\) 0 0
\(313\) 2.94757e8 0.543324 0.271662 0.962393i \(-0.412427\pi\)
0.271662 + 0.962393i \(0.412427\pi\)
\(314\) −2.93899e8 −0.535728
\(315\) 0 0
\(316\) −5.96967e8 −1.06425
\(317\) 3.97400e8 0.700682 0.350341 0.936622i \(-0.386066\pi\)
0.350341 + 0.936622i \(0.386066\pi\)
\(318\) 0 0
\(319\) −4.83680e8 −0.834240
\(320\) −1.74714e8 −0.298059
\(321\) 0 0
\(322\) −3.18101e8 −0.530970
\(323\) −1.19217e8 −0.196847
\(324\) 0 0
\(325\) −1.03399e9 −1.67081
\(326\) −1.76968e8 −0.282900
\(327\) 0 0
\(328\) 1.50659e8 0.235742
\(329\) 9.69882e8 1.50153
\(330\) 0 0
\(331\) −1.15349e9 −1.74830 −0.874150 0.485656i \(-0.838581\pi\)
−0.874150 + 0.485656i \(0.838581\pi\)
\(332\) −1.93626e8 −0.290389
\(333\) 0 0
\(334\) −1.19613e8 −0.175658
\(335\) −6.06325e8 −0.881148
\(336\) 0 0
\(337\) −1.00153e9 −1.42547 −0.712735 0.701434i \(-0.752544\pi\)
−0.712735 + 0.701434i \(0.752544\pi\)
\(338\) −1.79485e8 −0.252824
\(339\) 0 0
\(340\) 1.30033e8 0.179423
\(341\) 4.27625e8 0.584014
\(342\) 0 0
\(343\) 1.41385e8 0.189179
\(344\) −7.46969e8 −0.989346
\(345\) 0 0
\(346\) 4.47103e7 0.0580285
\(347\) 2.60064e8 0.334139 0.167069 0.985945i \(-0.446570\pi\)
0.167069 + 0.985945i \(0.446570\pi\)
\(348\) 0 0
\(349\) −2.45544e8 −0.309201 −0.154600 0.987977i \(-0.549409\pi\)
−0.154600 + 0.987977i \(0.549409\pi\)
\(350\) −1.20694e9 −1.50470
\(351\) 0 0
\(352\) −6.25028e8 −0.763836
\(353\) 8.28455e8 1.00244 0.501219 0.865320i \(-0.332885\pi\)
0.501219 + 0.865320i \(0.332885\pi\)
\(354\) 0 0
\(355\) −2.06893e9 −2.45441
\(356\) 8.85898e8 1.04066
\(357\) 0 0
\(358\) 1.77143e8 0.204048
\(359\) −8.66890e8 −0.988857 −0.494428 0.869218i \(-0.664623\pi\)
−0.494428 + 0.869218i \(0.664623\pi\)
\(360\) 0 0
\(361\) 1.92963e9 2.15873
\(362\) −2.62237e8 −0.290546
\(363\) 0 0
\(364\) −6.41666e8 −0.697356
\(365\) −9.09863e8 −0.979380
\(366\) 0 0
\(367\) −1.45949e9 −1.54124 −0.770620 0.637295i \(-0.780053\pi\)
−0.770620 + 0.637295i \(0.780053\pi\)
\(368\) 4.98674e8 0.521615
\(369\) 0 0
\(370\) −1.17382e9 −1.20475
\(371\) 1.30538e9 1.32717
\(372\) 0 0
\(373\) −1.35964e9 −1.35657 −0.678287 0.734798i \(-0.737277\pi\)
−0.678287 + 0.734798i \(0.737277\pi\)
\(374\) 3.59990e7 0.0355828
\(375\) 0 0
\(376\) 8.39622e8 0.814566
\(377\) −6.63766e8 −0.637999
\(378\) 0 0
\(379\) 1.81549e9 1.71300 0.856500 0.516147i \(-0.172634\pi\)
0.856500 + 0.516147i \(0.172634\pi\)
\(380\) −3.07967e9 −2.87913
\(381\) 0 0
\(382\) −7.14002e8 −0.655357
\(383\) 5.43039e8 0.493896 0.246948 0.969029i \(-0.420572\pi\)
0.246948 + 0.969029i \(0.420572\pi\)
\(384\) 0 0
\(385\) 2.35638e9 2.10442
\(386\) 2.48289e8 0.219737
\(387\) 0 0
\(388\) 1.00463e9 0.873161
\(389\) −1.90428e9 −1.64024 −0.820121 0.572190i \(-0.806094\pi\)
−0.820121 + 0.572190i \(0.806094\pi\)
\(390\) 0 0
\(391\) −1.26516e8 −0.107035
\(392\) −7.60291e8 −0.637498
\(393\) 0 0
\(394\) −6.65977e8 −0.548558
\(395\) 3.00845e9 2.45614
\(396\) 0 0
\(397\) −2.75416e8 −0.220913 −0.110457 0.993881i \(-0.535231\pi\)
−0.110457 + 0.993881i \(0.535231\pi\)
\(398\) 2.30813e8 0.183514
\(399\) 0 0
\(400\) 1.89207e9 1.47818
\(401\) −1.17696e9 −0.911501 −0.455751 0.890108i \(-0.650629\pi\)
−0.455751 + 0.890108i \(0.650629\pi\)
\(402\) 0 0
\(403\) 5.86840e8 0.446635
\(404\) 5.55314e8 0.418990
\(405\) 0 0
\(406\) −7.74790e8 −0.574570
\(407\) 1.67873e9 1.23424
\(408\) 0 0
\(409\) −1.75837e9 −1.27080 −0.635402 0.772181i \(-0.719166\pi\)
−0.635402 + 0.772181i \(0.719166\pi\)
\(410\) −3.46126e8 −0.248022
\(411\) 0 0
\(412\) 1.75440e9 1.23591
\(413\) 2.54280e8 0.177618
\(414\) 0 0
\(415\) 9.75787e8 0.670173
\(416\) −8.57741e8 −0.584157
\(417\) 0 0
\(418\) −8.52591e8 −0.570984
\(419\) −2.47153e9 −1.64141 −0.820705 0.571353i \(-0.806419\pi\)
−0.820705 + 0.571353i \(0.806419\pi\)
\(420\) 0 0
\(421\) −1.28349e9 −0.838313 −0.419157 0.907914i \(-0.637674\pi\)
−0.419157 + 0.907914i \(0.637674\pi\)
\(422\) 1.53203e8 0.0992367
\(423\) 0 0
\(424\) 1.13006e9 0.719980
\(425\) −4.80028e8 −0.303323
\(426\) 0 0
\(427\) −4.68272e8 −0.291072
\(428\) −1.23628e9 −0.762191
\(429\) 0 0
\(430\) 1.71610e9 1.04088
\(431\) −1.78192e9 −1.07206 −0.536030 0.844199i \(-0.680077\pi\)
−0.536030 + 0.844199i \(0.680077\pi\)
\(432\) 0 0
\(433\) 8.17222e8 0.483763 0.241881 0.970306i \(-0.422236\pi\)
0.241881 + 0.970306i \(0.422236\pi\)
\(434\) 6.84998e8 0.402230
\(435\) 0 0
\(436\) 1.79079e8 0.103476
\(437\) 2.99637e9 1.71756
\(438\) 0 0
\(439\) −2.53136e9 −1.42800 −0.714001 0.700145i \(-0.753119\pi\)
−0.714001 + 0.700145i \(0.753119\pi\)
\(440\) 2.03990e9 1.14163
\(441\) 0 0
\(442\) 4.94022e7 0.0272125
\(443\) 7.10880e8 0.388493 0.194247 0.980953i \(-0.437774\pi\)
0.194247 + 0.980953i \(0.437774\pi\)
\(444\) 0 0
\(445\) −4.46453e9 −2.40169
\(446\) 3.95919e8 0.211317
\(447\) 0 0
\(448\) 4.00250e8 0.210309
\(449\) −1.22783e9 −0.640139 −0.320070 0.947394i \(-0.603706\pi\)
−0.320070 + 0.947394i \(0.603706\pi\)
\(450\) 0 0
\(451\) 4.95009e8 0.254095
\(452\) −3.03239e8 −0.154454
\(453\) 0 0
\(454\) −1.66165e8 −0.0833382
\(455\) 3.23371e9 1.60939
\(456\) 0 0
\(457\) 3.07883e9 1.50897 0.754483 0.656319i \(-0.227888\pi\)
0.754483 + 0.656319i \(0.227888\pi\)
\(458\) 1.00031e9 0.486524
\(459\) 0 0
\(460\) −3.26822e9 −1.56552
\(461\) −1.77409e9 −0.843379 −0.421689 0.906740i \(-0.638563\pi\)
−0.421689 + 0.906740i \(0.638563\pi\)
\(462\) 0 0
\(463\) 3.91361e9 1.83250 0.916249 0.400608i \(-0.131201\pi\)
0.916249 + 0.400608i \(0.131201\pi\)
\(464\) 1.21461e9 0.564446
\(465\) 0 0
\(466\) −5.46701e8 −0.250264
\(467\) 3.15577e9 1.43383 0.716914 0.697162i \(-0.245554\pi\)
0.716914 + 0.697162i \(0.245554\pi\)
\(468\) 0 0
\(469\) 1.38902e9 0.621735
\(470\) −1.92896e9 −0.856998
\(471\) 0 0
\(472\) 2.20129e8 0.0963561
\(473\) −2.45426e9 −1.06637
\(474\) 0 0
\(475\) 1.13689e10 4.86732
\(476\) −2.97892e8 −0.126600
\(477\) 0 0
\(478\) 1.01967e9 0.427036
\(479\) 3.08907e9 1.28426 0.642130 0.766596i \(-0.278051\pi\)
0.642130 + 0.766596i \(0.278051\pi\)
\(480\) 0 0
\(481\) 2.30376e9 0.943907
\(482\) −8.13870e8 −0.331048
\(483\) 0 0
\(484\) 7.59864e8 0.304633
\(485\) −5.06288e9 −2.01512
\(486\) 0 0
\(487\) −8.99853e8 −0.353037 −0.176518 0.984297i \(-0.556484\pi\)
−0.176518 + 0.984297i \(0.556484\pi\)
\(488\) −4.05381e8 −0.157904
\(489\) 0 0
\(490\) 1.74670e9 0.670706
\(491\) 8.98595e8 0.342593 0.171297 0.985220i \(-0.445204\pi\)
0.171297 + 0.985220i \(0.445204\pi\)
\(492\) 0 0
\(493\) −3.08151e8 −0.115824
\(494\) −1.17003e9 −0.436669
\(495\) 0 0
\(496\) −1.07384e9 −0.395143
\(497\) 4.73970e9 1.73182
\(498\) 0 0
\(499\) −2.94063e8 −0.105947 −0.0529734 0.998596i \(-0.516870\pi\)
−0.0529734 + 0.998596i \(0.516870\pi\)
\(500\) −7.87239e9 −2.81651
\(501\) 0 0
\(502\) 1.57021e9 0.553982
\(503\) 3.01250e9 1.05545 0.527727 0.849414i \(-0.323044\pi\)
0.527727 + 0.849414i \(0.323044\pi\)
\(504\) 0 0
\(505\) −2.79853e9 −0.966965
\(506\) −9.04791e8 −0.310471
\(507\) 0 0
\(508\) −4.45213e9 −1.50676
\(509\) −7.20984e8 −0.242333 −0.121167 0.992632i \(-0.538664\pi\)
−0.121167 + 0.992632i \(0.538664\pi\)
\(510\) 0 0
\(511\) 2.08440e9 0.691047
\(512\) 2.78279e9 0.916296
\(513\) 0 0
\(514\) 7.54567e7 0.0245091
\(515\) −8.84138e9 −2.85230
\(516\) 0 0
\(517\) 2.75868e9 0.877980
\(518\) 2.68909e9 0.850064
\(519\) 0 0
\(520\) 2.79941e9 0.873081
\(521\) 3.99688e9 1.23820 0.619098 0.785314i \(-0.287498\pi\)
0.619098 + 0.785314i \(0.287498\pi\)
\(522\) 0 0
\(523\) 2.17294e9 0.664190 0.332095 0.943246i \(-0.392245\pi\)
0.332095 + 0.943246i \(0.392245\pi\)
\(524\) 4.97338e9 1.51005
\(525\) 0 0
\(526\) −1.77374e9 −0.531423
\(527\) 2.72439e8 0.0810834
\(528\) 0 0
\(529\) −2.24997e8 −0.0660819
\(530\) −2.59621e9 −0.757485
\(531\) 0 0
\(532\) 7.05519e9 2.03151
\(533\) 6.79313e8 0.194323
\(534\) 0 0
\(535\) 6.23030e9 1.75902
\(536\) 1.20247e9 0.337286
\(537\) 0 0
\(538\) 1.22682e9 0.339658
\(539\) −2.49803e9 −0.687127
\(540\) 0 0
\(541\) −4.30836e9 −1.16983 −0.584914 0.811096i \(-0.698872\pi\)
−0.584914 + 0.811096i \(0.698872\pi\)
\(542\) −2.25655e9 −0.608762
\(543\) 0 0
\(544\) −3.98204e8 −0.106050
\(545\) −9.02476e8 −0.238808
\(546\) 0 0
\(547\) 7.88729e8 0.206050 0.103025 0.994679i \(-0.467148\pi\)
0.103025 + 0.994679i \(0.467148\pi\)
\(548\) −3.43617e8 −0.0891956
\(549\) 0 0
\(550\) −3.43297e9 −0.879833
\(551\) 7.29818e9 1.85859
\(552\) 0 0
\(553\) −6.89203e9 −1.73304
\(554\) 1.44418e9 0.360858
\(555\) 0 0
\(556\) −4.01714e9 −0.991186
\(557\) −5.29917e9 −1.29932 −0.649658 0.760226i \(-0.725088\pi\)
−0.649658 + 0.760226i \(0.725088\pi\)
\(558\) 0 0
\(559\) −3.36804e9 −0.815522
\(560\) −5.91728e9 −1.42385
\(561\) 0 0
\(562\) 2.05276e9 0.487821
\(563\) 5.56156e9 1.31346 0.656731 0.754125i \(-0.271939\pi\)
0.656731 + 0.754125i \(0.271939\pi\)
\(564\) 0 0
\(565\) 1.52819e9 0.356457
\(566\) 2.13037e9 0.493853
\(567\) 0 0
\(568\) 4.10314e9 0.939500
\(569\) −3.31033e9 −0.753319 −0.376659 0.926352i \(-0.622927\pi\)
−0.376659 + 0.926352i \(0.622927\pi\)
\(570\) 0 0
\(571\) −3.86639e8 −0.0869118 −0.0434559 0.999055i \(-0.513837\pi\)
−0.0434559 + 0.999055i \(0.513837\pi\)
\(572\) −1.82512e9 −0.407761
\(573\) 0 0
\(574\) 7.92938e8 0.175004
\(575\) 1.20649e10 2.64659
\(576\) 0 0
\(577\) 3.94872e9 0.855739 0.427870 0.903840i \(-0.359264\pi\)
0.427870 + 0.903840i \(0.359264\pi\)
\(578\) −1.84667e9 −0.397780
\(579\) 0 0
\(580\) −7.96031e9 −1.69407
\(581\) −2.23542e9 −0.472871
\(582\) 0 0
\(583\) 3.71295e9 0.776031
\(584\) 1.80445e9 0.374887
\(585\) 0 0
\(586\) 7.31315e7 0.0150128
\(587\) 1.13309e9 0.231223 0.115612 0.993294i \(-0.463117\pi\)
0.115612 + 0.993294i \(0.463117\pi\)
\(588\) 0 0
\(589\) −6.45237e9 −1.30112
\(590\) −5.05726e8 −0.101376
\(591\) 0 0
\(592\) −4.21558e9 −0.835086
\(593\) −6.61359e9 −1.30240 −0.651202 0.758904i \(-0.725735\pi\)
−0.651202 + 0.758904i \(0.725735\pi\)
\(594\) 0 0
\(595\) 1.50124e9 0.292174
\(596\) 2.56874e9 0.497003
\(597\) 0 0
\(598\) −1.24167e9 −0.237438
\(599\) 2.36330e9 0.449289 0.224644 0.974441i \(-0.427878\pi\)
0.224644 + 0.974441i \(0.427878\pi\)
\(600\) 0 0
\(601\) −4.40437e9 −0.827605 −0.413803 0.910367i \(-0.635800\pi\)
−0.413803 + 0.910367i \(0.635800\pi\)
\(602\) −3.93139e9 −0.734443
\(603\) 0 0
\(604\) 7.06938e9 1.30543
\(605\) −3.82937e9 −0.703047
\(606\) 0 0
\(607\) −2.71725e9 −0.493138 −0.246569 0.969125i \(-0.579303\pi\)
−0.246569 + 0.969125i \(0.579303\pi\)
\(608\) 9.43095e9 1.70174
\(609\) 0 0
\(610\) 9.31327e8 0.166130
\(611\) 3.78580e9 0.671450
\(612\) 0 0
\(613\) −1.02118e10 −1.79056 −0.895282 0.445500i \(-0.853026\pi\)
−0.895282 + 0.445500i \(0.853026\pi\)
\(614\) −2.98965e8 −0.0521233
\(615\) 0 0
\(616\) −4.67320e9 −0.805530
\(617\) −3.23817e9 −0.555011 −0.277506 0.960724i \(-0.589508\pi\)
−0.277506 + 0.960724i \(0.589508\pi\)
\(618\) 0 0
\(619\) −1.72870e9 −0.292956 −0.146478 0.989214i \(-0.546794\pi\)
−0.146478 + 0.989214i \(0.546794\pi\)
\(620\) 7.03777e9 1.18594
\(621\) 0 0
\(622\) −2.02608e9 −0.337590
\(623\) 1.02278e10 1.69462
\(624\) 0 0
\(625\) 2.29581e10 3.76145
\(626\) 1.34299e9 0.218807
\(627\) 0 0
\(628\) 6.91750e9 1.11453
\(629\) 1.06951e9 0.171360
\(630\) 0 0
\(631\) 7.57564e9 1.20037 0.600187 0.799859i \(-0.295093\pi\)
0.600187 + 0.799859i \(0.295093\pi\)
\(632\) −5.96639e9 −0.940160
\(633\) 0 0
\(634\) 1.81066e9 0.282179
\(635\) 2.24368e10 3.47738
\(636\) 0 0
\(637\) −3.42811e9 −0.525492
\(638\) −2.20377e9 −0.335965
\(639\) 0 0
\(640\) −1.30739e10 −1.97140
\(641\) 5.56603e9 0.834723 0.417362 0.908740i \(-0.362955\pi\)
0.417362 + 0.908740i \(0.362955\pi\)
\(642\) 0 0
\(643\) −7.76877e9 −1.15243 −0.576214 0.817299i \(-0.695471\pi\)
−0.576214 + 0.817299i \(0.695471\pi\)
\(644\) 7.48715e9 1.10463
\(645\) 0 0
\(646\) −5.43183e8 −0.0792743
\(647\) −6.50327e9 −0.943989 −0.471994 0.881602i \(-0.656466\pi\)
−0.471994 + 0.881602i \(0.656466\pi\)
\(648\) 0 0
\(649\) 7.23260e8 0.103858
\(650\) −4.71114e9 −0.672867
\(651\) 0 0
\(652\) 4.16530e9 0.588545
\(653\) 9.96747e9 1.40084 0.700420 0.713731i \(-0.252996\pi\)
0.700420 + 0.713731i \(0.252996\pi\)
\(654\) 0 0
\(655\) −2.50636e10 −3.48497
\(656\) −1.24306e9 −0.171920
\(657\) 0 0
\(658\) 4.41903e9 0.604695
\(659\) −1.28708e10 −1.75189 −0.875946 0.482410i \(-0.839762\pi\)
−0.875946 + 0.482410i \(0.839762\pi\)
\(660\) 0 0
\(661\) −4.36910e9 −0.588420 −0.294210 0.955741i \(-0.595056\pi\)
−0.294210 + 0.955741i \(0.595056\pi\)
\(662\) −5.25560e9 −0.704075
\(663\) 0 0
\(664\) −1.93519e9 −0.256529
\(665\) −3.55550e10 −4.68841
\(666\) 0 0
\(667\) 7.74501e9 1.01061
\(668\) 2.81534e9 0.365437
\(669\) 0 0
\(670\) −2.76257e9 −0.354856
\(671\) −1.33193e9 −0.170197
\(672\) 0 0
\(673\) −2.50718e9 −0.317053 −0.158527 0.987355i \(-0.550674\pi\)
−0.158527 + 0.987355i \(0.550674\pi\)
\(674\) −4.56321e9 −0.574065
\(675\) 0 0
\(676\) 4.22453e9 0.525974
\(677\) −4.46850e9 −0.553479 −0.276739 0.960945i \(-0.589254\pi\)
−0.276739 + 0.960945i \(0.589254\pi\)
\(678\) 0 0
\(679\) 1.15985e10 1.42186
\(680\) 1.29962e9 0.158502
\(681\) 0 0
\(682\) 1.94837e9 0.235194
\(683\) 3.36244e9 0.403815 0.201908 0.979405i \(-0.435286\pi\)
0.201908 + 0.979405i \(0.435286\pi\)
\(684\) 0 0
\(685\) 1.73168e9 0.205850
\(686\) 6.44185e8 0.0761862
\(687\) 0 0
\(688\) 6.16308e9 0.721503
\(689\) 5.09536e9 0.593483
\(690\) 0 0
\(691\) 1.73704e9 0.200279 0.100140 0.994973i \(-0.468071\pi\)
0.100140 + 0.994973i \(0.468071\pi\)
\(692\) −1.05235e9 −0.120722
\(693\) 0 0
\(694\) 1.18492e9 0.134564
\(695\) 2.02446e10 2.28751
\(696\) 0 0
\(697\) 3.15369e8 0.0352780
\(698\) −1.11876e9 −0.124521
\(699\) 0 0
\(700\) 2.84078e10 3.13036
\(701\) −7.43794e9 −0.815529 −0.407764 0.913087i \(-0.633692\pi\)
−0.407764 + 0.913087i \(0.633692\pi\)
\(702\) 0 0
\(703\) −2.53301e10 −2.74975
\(704\) 1.13845e9 0.122973
\(705\) 0 0
\(706\) 3.77465e9 0.403702
\(707\) 6.41114e9 0.682287
\(708\) 0 0
\(709\) −3.13976e9 −0.330853 −0.165426 0.986222i \(-0.552900\pi\)
−0.165426 + 0.986222i \(0.552900\pi\)
\(710\) −9.42659e9 −0.988441
\(711\) 0 0
\(712\) 8.85412e9 0.919317
\(713\) −6.84742e9 −0.707479
\(714\) 0 0
\(715\) 9.19780e9 0.941051
\(716\) −4.16940e9 −0.424501
\(717\) 0 0
\(718\) −3.94977e9 −0.398232
\(719\) 1.62139e10 1.62680 0.813402 0.581701i \(-0.197613\pi\)
0.813402 + 0.581701i \(0.197613\pi\)
\(720\) 0 0
\(721\) 2.02546e10 2.01257
\(722\) 8.79190e9 0.869365
\(723\) 0 0
\(724\) 6.17227e9 0.604449
\(725\) 2.93862e10 2.86392
\(726\) 0 0
\(727\) 1.44572e10 1.39545 0.697725 0.716366i \(-0.254196\pi\)
0.697725 + 0.716366i \(0.254196\pi\)
\(728\) −6.41314e9 −0.616043
\(729\) 0 0
\(730\) −4.14557e9 −0.394416
\(731\) −1.56360e9 −0.148052
\(732\) 0 0
\(733\) 8.79960e9 0.825275 0.412638 0.910895i \(-0.364608\pi\)
0.412638 + 0.910895i \(0.364608\pi\)
\(734\) −6.64982e9 −0.620688
\(735\) 0 0
\(736\) 1.00084e10 0.925318
\(737\) 3.95087e9 0.363544
\(738\) 0 0
\(739\) 1.06478e10 0.970523 0.485262 0.874369i \(-0.338724\pi\)
0.485262 + 0.874369i \(0.338724\pi\)
\(740\) 2.76282e10 2.50635
\(741\) 0 0
\(742\) 5.94764e9 0.534479
\(743\) 6.58260e9 0.588758 0.294379 0.955689i \(-0.404887\pi\)
0.294379 + 0.955689i \(0.404887\pi\)
\(744\) 0 0
\(745\) −1.29453e10 −1.14701
\(746\) −6.19487e9 −0.546319
\(747\) 0 0
\(748\) −8.47307e8 −0.0740262
\(749\) −1.42729e10 −1.24116
\(750\) 0 0
\(751\) −8.17849e9 −0.704585 −0.352293 0.935890i \(-0.614598\pi\)
−0.352293 + 0.935890i \(0.614598\pi\)
\(752\) −6.92753e9 −0.594040
\(753\) 0 0
\(754\) −3.02429e9 −0.256935
\(755\) −3.56265e10 −3.01272
\(756\) 0 0
\(757\) 1.76129e10 1.47569 0.737844 0.674972i \(-0.235844\pi\)
0.737844 + 0.674972i \(0.235844\pi\)
\(758\) 8.27185e9 0.689860
\(759\) 0 0
\(760\) −3.07798e10 −2.54342
\(761\) 1.93708e10 1.59331 0.796657 0.604431i \(-0.206600\pi\)
0.796657 + 0.604431i \(0.206600\pi\)
\(762\) 0 0
\(763\) 2.06748e9 0.168502
\(764\) 1.68055e10 1.36340
\(765\) 0 0
\(766\) 2.47423e9 0.198902
\(767\) 9.92546e8 0.0794268
\(768\) 0 0
\(769\) 9.07869e9 0.719915 0.359957 0.932969i \(-0.382791\pi\)
0.359957 + 0.932969i \(0.382791\pi\)
\(770\) 1.07363e10 0.847492
\(771\) 0 0
\(772\) −5.84398e9 −0.457139
\(773\) 1.79639e10 1.39886 0.699428 0.714703i \(-0.253438\pi\)
0.699428 + 0.714703i \(0.253438\pi\)
\(774\) 0 0
\(775\) −2.59805e10 −2.00490
\(776\) 1.00408e10 0.771348
\(777\) 0 0
\(778\) −8.67641e9 −0.660558
\(779\) −7.46912e9 −0.566093
\(780\) 0 0
\(781\) 1.34814e10 1.01264
\(782\) −5.76440e8 −0.0431052
\(783\) 0 0
\(784\) 6.27299e9 0.464909
\(785\) −3.48611e10 −2.57216
\(786\) 0 0
\(787\) −9.95896e9 −0.728287 −0.364143 0.931343i \(-0.618638\pi\)
−0.364143 + 0.931343i \(0.618638\pi\)
\(788\) 1.56751e10 1.14122
\(789\) 0 0
\(790\) 1.37073e10 0.989135
\(791\) −3.50092e9 −0.251515
\(792\) 0 0
\(793\) −1.82784e9 −0.130161
\(794\) −1.25486e9 −0.0889662
\(795\) 0 0
\(796\) −5.43264e9 −0.381782
\(797\) −2.78445e10 −1.94821 −0.974105 0.226095i \(-0.927404\pi\)
−0.974105 + 0.226095i \(0.927404\pi\)
\(798\) 0 0
\(799\) 1.75755e9 0.121897
\(800\) 3.79738e10 2.62222
\(801\) 0 0
\(802\) −5.36254e9 −0.367080
\(803\) 5.92875e9 0.404072
\(804\) 0 0
\(805\) −3.77319e10 −2.54931
\(806\) 2.67380e9 0.179869
\(807\) 0 0
\(808\) 5.55009e9 0.370135
\(809\) −6.81852e9 −0.452762 −0.226381 0.974039i \(-0.572689\pi\)
−0.226381 + 0.974039i \(0.572689\pi\)
\(810\) 0 0
\(811\) 3.94141e9 0.259465 0.129732 0.991549i \(-0.458588\pi\)
0.129732 + 0.991549i \(0.458588\pi\)
\(812\) 1.82362e10 1.19533
\(813\) 0 0
\(814\) 7.64872e9 0.497053
\(815\) −2.09912e10 −1.35827
\(816\) 0 0
\(817\) 3.70319e10 2.37574
\(818\) −8.01159e9 −0.511778
\(819\) 0 0
\(820\) 8.14676e9 0.515985
\(821\) −2.49119e10 −1.57111 −0.785553 0.618795i \(-0.787621\pi\)
−0.785553 + 0.618795i \(0.787621\pi\)
\(822\) 0 0
\(823\) −2.21681e9 −0.138621 −0.0693105 0.997595i \(-0.522080\pi\)
−0.0693105 + 0.997595i \(0.522080\pi\)
\(824\) 1.75343e10 1.09180
\(825\) 0 0
\(826\) 1.15856e9 0.0715302
\(827\) 1.70597e10 1.04882 0.524410 0.851466i \(-0.324286\pi\)
0.524410 + 0.851466i \(0.324286\pi\)
\(828\) 0 0
\(829\) −5.39778e9 −0.329060 −0.164530 0.986372i \(-0.552611\pi\)
−0.164530 + 0.986372i \(0.552611\pi\)
\(830\) 4.44594e9 0.269892
\(831\) 0 0
\(832\) 1.56232e9 0.0940457
\(833\) −1.59149e9 −0.0953994
\(834\) 0 0
\(835\) −1.41880e10 −0.843373
\(836\) 2.00674e10 1.18787
\(837\) 0 0
\(838\) −1.12609e10 −0.661028
\(839\) −1.48286e10 −0.866830 −0.433415 0.901194i \(-0.642692\pi\)
−0.433415 + 0.901194i \(0.642692\pi\)
\(840\) 0 0
\(841\) 1.61441e9 0.0935895
\(842\) −5.84793e9 −0.337606
\(843\) 0 0
\(844\) −3.60593e9 −0.206452
\(845\) −2.12897e10 −1.21387
\(846\) 0 0
\(847\) 8.77268e9 0.496067
\(848\) −9.32386e9 −0.525062
\(849\) 0 0
\(850\) −2.18713e9 −0.122154
\(851\) −2.68809e10 −1.49517
\(852\) 0 0
\(853\) 1.65358e10 0.912227 0.456114 0.889922i \(-0.349241\pi\)
0.456114 + 0.889922i \(0.349241\pi\)
\(854\) −2.13357e9 −0.117221
\(855\) 0 0
\(856\) −1.23560e10 −0.673318
\(857\) −1.32332e10 −0.718176 −0.359088 0.933304i \(-0.616912\pi\)
−0.359088 + 0.933304i \(0.616912\pi\)
\(858\) 0 0
\(859\) −4.23318e9 −0.227872 −0.113936 0.993488i \(-0.536346\pi\)
−0.113936 + 0.993488i \(0.536346\pi\)
\(860\) −4.03917e10 −2.16545
\(861\) 0 0
\(862\) −8.11891e9 −0.431740
\(863\) −3.99379e8 −0.0211518 −0.0105759 0.999944i \(-0.503366\pi\)
−0.0105759 + 0.999944i \(0.503366\pi\)
\(864\) 0 0
\(865\) 5.30335e9 0.278608
\(866\) 3.72347e9 0.194821
\(867\) 0 0
\(868\) −1.61228e10 −0.836798
\(869\) −1.96033e10 −1.01335
\(870\) 0 0
\(871\) 5.42187e9 0.278026
\(872\) 1.78980e9 0.0914108
\(873\) 0 0
\(874\) 1.36522e10 0.691694
\(875\) −9.08873e10 −4.58643
\(876\) 0 0
\(877\) 9.41245e9 0.471199 0.235599 0.971850i \(-0.424295\pi\)
0.235599 + 0.971850i \(0.424295\pi\)
\(878\) −1.15335e10 −0.575085
\(879\) 0 0
\(880\) −1.68308e10 −0.832560
\(881\) 6.45929e8 0.0318250 0.0159125 0.999873i \(-0.494935\pi\)
0.0159125 + 0.999873i \(0.494935\pi\)
\(882\) 0 0
\(883\) 3.76342e10 1.83959 0.919793 0.392403i \(-0.128356\pi\)
0.919793 + 0.392403i \(0.128356\pi\)
\(884\) −1.16278e9 −0.0566128
\(885\) 0 0
\(886\) 3.23895e9 0.156454
\(887\) 1.95731e10 0.941732 0.470866 0.882205i \(-0.343941\pi\)
0.470866 + 0.882205i \(0.343941\pi\)
\(888\) 0 0
\(889\) −5.14002e10 −2.45363
\(890\) −2.03416e10 −0.967207
\(891\) 0 0
\(892\) −9.31874e9 −0.439623
\(893\) −4.16253e10 −1.95604
\(894\) 0 0
\(895\) 2.10119e10 0.979683
\(896\) 2.99509e10 1.39101
\(897\) 0 0
\(898\) −5.59429e9 −0.257797
\(899\) −1.66781e10 −0.765573
\(900\) 0 0
\(901\) 2.36551e9 0.107743
\(902\) 2.25539e9 0.102329
\(903\) 0 0
\(904\) −3.03072e9 −0.136445
\(905\) −3.11055e10 −1.39498
\(906\) 0 0
\(907\) −2.68155e10 −1.19333 −0.596665 0.802491i \(-0.703508\pi\)
−0.596665 + 0.802491i \(0.703508\pi\)
\(908\) 3.91102e9 0.173376
\(909\) 0 0
\(910\) 1.47336e10 0.648134
\(911\) 3.43001e10 1.50307 0.751537 0.659690i \(-0.229313\pi\)
0.751537 + 0.659690i \(0.229313\pi\)
\(912\) 0 0
\(913\) −6.35832e9 −0.276500
\(914\) 1.40280e10 0.607691
\(915\) 0 0
\(916\) −2.35442e10 −1.01216
\(917\) 5.74180e10 2.45898
\(918\) 0 0
\(919\) 1.59456e10 0.677698 0.338849 0.940841i \(-0.389962\pi\)
0.338849 + 0.940841i \(0.389962\pi\)
\(920\) −3.26643e10 −1.38298
\(921\) 0 0
\(922\) −8.08321e9 −0.339645
\(923\) 1.85008e10 0.774434
\(924\) 0 0
\(925\) −1.01992e11 −4.23710
\(926\) 1.78314e10 0.737984
\(927\) 0 0
\(928\) 2.43771e10 1.00130
\(929\) −1.42285e9 −0.0582242 −0.0291121 0.999576i \(-0.509268\pi\)
−0.0291121 + 0.999576i \(0.509268\pi\)
\(930\) 0 0
\(931\) 3.76924e10 1.53084
\(932\) 1.28677e10 0.520649
\(933\) 0 0
\(934\) 1.43785e10 0.577431
\(935\) 4.27005e9 0.170841
\(936\) 0 0
\(937\) −2.31807e10 −0.920532 −0.460266 0.887781i \(-0.652246\pi\)
−0.460266 + 0.887781i \(0.652246\pi\)
\(938\) 6.32875e9 0.250385
\(939\) 0 0
\(940\) 4.54018e10 1.78290
\(941\) −3.37421e10 −1.32010 −0.660052 0.751220i \(-0.729466\pi\)
−0.660052 + 0.751220i \(0.729466\pi\)
\(942\) 0 0
\(943\) −7.92642e9 −0.307812
\(944\) −1.81623e9 −0.0702699
\(945\) 0 0
\(946\) −1.11822e10 −0.429447
\(947\) −2.26076e10 −0.865029 −0.432514 0.901627i \(-0.642374\pi\)
−0.432514 + 0.901627i \(0.642374\pi\)
\(948\) 0 0
\(949\) 8.13617e9 0.309021
\(950\) 5.17995e10 1.96017
\(951\) 0 0
\(952\) −2.97728e9 −0.111838
\(953\) −1.78597e10 −0.668420 −0.334210 0.942499i \(-0.608469\pi\)
−0.334210 + 0.942499i \(0.608469\pi\)
\(954\) 0 0
\(955\) −8.46920e10 −3.14652
\(956\) −2.40001e10 −0.888403
\(957\) 0 0
\(958\) 1.40746e10 0.517197
\(959\) −3.96709e9 −0.145247
\(960\) 0 0
\(961\) −1.27674e10 −0.464057
\(962\) 1.04965e10 0.380130
\(963\) 0 0
\(964\) 1.91561e10 0.688710
\(965\) 2.94511e10 1.05501
\(966\) 0 0
\(967\) −1.61338e8 −0.00573778 −0.00286889 0.999996i \(-0.500913\pi\)
−0.00286889 + 0.999996i \(0.500913\pi\)
\(968\) 7.59446e9 0.269112
\(969\) 0 0
\(970\) −2.30678e10 −0.811530
\(971\) 4.94599e10 1.73375 0.866874 0.498527i \(-0.166125\pi\)
0.866874 + 0.498527i \(0.166125\pi\)
\(972\) 0 0
\(973\) −4.63782e10 −1.61406
\(974\) −4.09996e9 −0.142175
\(975\) 0 0
\(976\) 3.34471e9 0.115155
\(977\) 1.43208e10 0.491290 0.245645 0.969360i \(-0.421000\pi\)
0.245645 + 0.969360i \(0.421000\pi\)
\(978\) 0 0
\(979\) 2.90913e10 0.990886
\(980\) −4.11121e10 −1.39533
\(981\) 0 0
\(982\) 4.09423e9 0.137969
\(983\) 5.16305e10 1.73368 0.866840 0.498587i \(-0.166148\pi\)
0.866840 + 0.498587i \(0.166148\pi\)
\(984\) 0 0
\(985\) −7.89955e10 −2.63375
\(986\) −1.40402e9 −0.0466448
\(987\) 0 0
\(988\) 2.75390e10 0.908445
\(989\) 3.92992e10 1.29181
\(990\) 0 0
\(991\) −4.70999e10 −1.53731 −0.768656 0.639662i \(-0.779074\pi\)
−0.768656 + 0.639662i \(0.779074\pi\)
\(992\) −2.15520e10 −0.700964
\(993\) 0 0
\(994\) 2.15953e10 0.697440
\(995\) 2.73781e10 0.881094
\(996\) 0 0
\(997\) −5.20068e10 −1.66198 −0.830992 0.556284i \(-0.812227\pi\)
−0.830992 + 0.556284i \(0.812227\pi\)
\(998\) −1.33982e9 −0.0426669
\(999\) 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.b.1.10 16
3.2 odd 2 177.8.a.a.1.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.7 16 3.2 odd 2
531.8.a.b.1.10 16 1.1 even 1 trivial