Properties

Label 1089.3.b.j.485.11
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 48 x^{14} + 921 x^{12} + 8986 x^{10} + 46812 x^{8} + 125072 x^{6} + 152129 x^{4} + 65614 x^{2} + 5041\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 485.11
Root \(1.35141i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.j.485.6

$q$-expansion

\(f(q)\) \(=\) \(q+1.35141i q^{2} +2.17369 q^{4} +5.10469i q^{5} +13.1679 q^{7} +8.34318i q^{8} +O(q^{10})\) \(q+1.35141i q^{2} +2.17369 q^{4} +5.10469i q^{5} +13.1679 q^{7} +8.34318i q^{8} -6.89852 q^{10} +2.67027 q^{13} +17.7952i q^{14} -2.58027 q^{16} +17.2161i q^{17} -30.6226 q^{19} +11.0960i q^{20} +15.5309i q^{23} -1.05786 q^{25} +3.60862i q^{26} +28.6230 q^{28} -10.3828i q^{29} +16.8678 q^{31} +29.8857i q^{32} -23.2660 q^{34} +67.2181i q^{35} +18.7510 q^{37} -41.3837i q^{38} -42.5894 q^{40} -13.3582i q^{41} -10.5356 q^{43} -20.9886 q^{46} -82.2065i q^{47} +124.394 q^{49} -1.42960i q^{50} +5.80435 q^{52} -62.9849i q^{53} +109.862i q^{56} +14.0314 q^{58} +65.9058i q^{59} -36.5562 q^{61} +22.7954i q^{62} -50.7089 q^{64} +13.6309i q^{65} +60.5815 q^{67} +37.4225i q^{68} -90.8392 q^{70} -49.1893i q^{71} -6.37308 q^{73} +25.3402i q^{74} -66.5643 q^{76} -115.053 q^{79} -13.1715i q^{80} +18.0523 q^{82} -40.4929i q^{83} -87.8828 q^{85} -14.2379i q^{86} +71.1308i q^{89} +35.1619 q^{91} +33.7594i q^{92} +111.095 q^{94} -156.319i q^{95} +18.6682 q^{97} +168.107i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 32q^{4} + 8q^{7} + O(q^{10}) \) \( 16q - 32q^{4} + 8q^{7} - 24q^{10} - 4q^{13} + 28q^{16} + 20q^{19} - 44q^{25} - 16q^{28} + 28q^{31} + 148q^{34} - 148q^{37} - 224q^{40} - 272q^{43} - 208q^{46} + 348q^{49} - 520q^{52} - 44q^{58} - 224q^{61} + 436q^{64} + 24q^{67} - 664q^{70} + 4q^{73} - 1052q^{76} - 216q^{79} + 348q^{82} - 416q^{85} - 168q^{91} - 1140q^{94} - 44q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.35141i 0.675704i 0.941199 + 0.337852i \(0.109700\pi\)
−0.941199 + 0.337852i \(0.890300\pi\)
\(3\) 0 0
\(4\) 2.17369 0.543424
\(5\) 5.10469i 1.02094i 0.859896 + 0.510469i \(0.170528\pi\)
−0.859896 + 0.510469i \(0.829472\pi\)
\(6\) 0 0
\(7\) 13.1679 1.88113 0.940565 0.339613i \(-0.110296\pi\)
0.940565 + 0.339613i \(0.110296\pi\)
\(8\) 8.34318i 1.04290i
\(9\) 0 0
\(10\) −6.89852 −0.689852
\(11\) 0 0
\(12\) 0 0
\(13\) 2.67027 0.205405 0.102703 0.994712i \(-0.467251\pi\)
0.102703 + 0.994712i \(0.467251\pi\)
\(14\) 17.7952i 1.27109i
\(15\) 0 0
\(16\) −2.58027 −0.161267
\(17\) 17.2161i 1.01271i 0.862325 + 0.506356i \(0.169008\pi\)
−0.862325 + 0.506356i \(0.830992\pi\)
\(18\) 0 0
\(19\) −30.6226 −1.61172 −0.805859 0.592107i \(-0.798296\pi\)
−0.805859 + 0.592107i \(0.798296\pi\)
\(20\) 11.0960i 0.554802i
\(21\) 0 0
\(22\) 0 0
\(23\) 15.5309i 0.675257i 0.941279 + 0.337628i \(0.109625\pi\)
−0.941279 + 0.337628i \(0.890375\pi\)
\(24\) 0 0
\(25\) −1.05786 −0.0423145
\(26\) 3.60862i 0.138793i
\(27\) 0 0
\(28\) 28.6230 1.02225
\(29\) − 10.3828i − 0.358028i −0.983847 0.179014i \(-0.942709\pi\)
0.983847 0.179014i \(-0.0572908\pi\)
\(30\) 0 0
\(31\) 16.8678 0.544124 0.272062 0.962280i \(-0.412294\pi\)
0.272062 + 0.962280i \(0.412294\pi\)
\(32\) 29.8857i 0.933929i
\(33\) 0 0
\(34\) −23.2660 −0.684294
\(35\) 67.2181i 1.92052i
\(36\) 0 0
\(37\) 18.7510 0.506783 0.253391 0.967364i \(-0.418454\pi\)
0.253391 + 0.967364i \(0.418454\pi\)
\(38\) − 41.3837i − 1.08904i
\(39\) 0 0
\(40\) −42.5894 −1.06473
\(41\) − 13.3582i − 0.325809i −0.986642 0.162904i \(-0.947914\pi\)
0.986642 0.162904i \(-0.0520862\pi\)
\(42\) 0 0
\(43\) −10.5356 −0.245015 −0.122507 0.992468i \(-0.539093\pi\)
−0.122507 + 0.992468i \(0.539093\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −20.9886 −0.456274
\(47\) − 82.2065i − 1.74907i −0.484958 0.874537i \(-0.661165\pi\)
0.484958 0.874537i \(-0.338835\pi\)
\(48\) 0 0
\(49\) 124.394 2.53865
\(50\) − 1.42960i − 0.0285921i
\(51\) 0 0
\(52\) 5.80435 0.111622
\(53\) − 62.9849i − 1.18839i −0.804320 0.594197i \(-0.797470\pi\)
0.804320 0.594197i \(-0.202530\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 109.862i 1.96183i
\(57\) 0 0
\(58\) 14.0314 0.241921
\(59\) 65.9058i 1.11705i 0.829488 + 0.558524i \(0.188632\pi\)
−0.829488 + 0.558524i \(0.811368\pi\)
\(60\) 0 0
\(61\) −36.5562 −0.599282 −0.299641 0.954052i \(-0.596867\pi\)
−0.299641 + 0.954052i \(0.596867\pi\)
\(62\) 22.7954i 0.367667i
\(63\) 0 0
\(64\) −50.7089 −0.792327
\(65\) 13.6309i 0.209706i
\(66\) 0 0
\(67\) 60.5815 0.904202 0.452101 0.891967i \(-0.350675\pi\)
0.452101 + 0.891967i \(0.350675\pi\)
\(68\) 37.4225i 0.550331i
\(69\) 0 0
\(70\) −90.8392 −1.29770
\(71\) − 49.1893i − 0.692807i −0.938086 0.346404i \(-0.887403\pi\)
0.938086 0.346404i \(-0.112597\pi\)
\(72\) 0 0
\(73\) −6.37308 −0.0873024 −0.0436512 0.999047i \(-0.513899\pi\)
−0.0436512 + 0.999047i \(0.513899\pi\)
\(74\) 25.3402i 0.342435i
\(75\) 0 0
\(76\) −66.5643 −0.875846
\(77\) 0 0
\(78\) 0 0
\(79\) −115.053 −1.45637 −0.728183 0.685383i \(-0.759635\pi\)
−0.728183 + 0.685383i \(0.759635\pi\)
\(80\) − 13.1715i − 0.164644i
\(81\) 0 0
\(82\) 18.0523 0.220150
\(83\) − 40.4929i − 0.487867i −0.969792 0.243933i \(-0.921562\pi\)
0.969792 0.243933i \(-0.0784378\pi\)
\(84\) 0 0
\(85\) −87.8828 −1.03392
\(86\) − 14.2379i − 0.165557i
\(87\) 0 0
\(88\) 0 0
\(89\) 71.1308i 0.799222i 0.916685 + 0.399611i \(0.130855\pi\)
−0.916685 + 0.399611i \(0.869145\pi\)
\(90\) 0 0
\(91\) 35.1619 0.386394
\(92\) 33.7594i 0.366950i
\(93\) 0 0
\(94\) 111.095 1.18186
\(95\) − 156.319i − 1.64546i
\(96\) 0 0
\(97\) 18.6682 0.192456 0.0962279 0.995359i \(-0.469322\pi\)
0.0962279 + 0.995359i \(0.469322\pi\)
\(98\) 168.107i 1.71538i
\(99\) 0 0
\(100\) −2.29947 −0.0229947
\(101\) 47.0627i 0.465967i 0.972481 + 0.232983i \(0.0748488\pi\)
−0.972481 + 0.232983i \(0.925151\pi\)
\(102\) 0 0
\(103\) 111.438 1.08192 0.540960 0.841048i \(-0.318061\pi\)
0.540960 + 0.841048i \(0.318061\pi\)
\(104\) 22.2785i 0.214217i
\(105\) 0 0
\(106\) 85.1183 0.803003
\(107\) 110.880i 1.03626i 0.855301 + 0.518131i \(0.173372\pi\)
−0.855301 + 0.518131i \(0.826628\pi\)
\(108\) 0 0
\(109\) −118.337 −1.08566 −0.542832 0.839841i \(-0.682648\pi\)
−0.542832 + 0.839841i \(0.682648\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −33.9768 −0.303364
\(113\) 69.3440i 0.613664i 0.951764 + 0.306832i \(0.0992689\pi\)
−0.951764 + 0.306832i \(0.900731\pi\)
\(114\) 0 0
\(115\) −79.2805 −0.689395
\(116\) − 22.5691i − 0.194561i
\(117\) 0 0
\(118\) −89.0657 −0.754794
\(119\) 226.700i 1.90504i
\(120\) 0 0
\(121\) 0 0
\(122\) − 49.4023i − 0.404937i
\(123\) 0 0
\(124\) 36.6655 0.295690
\(125\) 122.217i 0.977738i
\(126\) 0 0
\(127\) −10.7748 −0.0848412 −0.0424206 0.999100i \(-0.513507\pi\)
−0.0424206 + 0.999100i \(0.513507\pi\)
\(128\) 51.0145i 0.398550i
\(129\) 0 0
\(130\) −18.4209 −0.141699
\(131\) 92.0724i 0.702843i 0.936217 + 0.351421i \(0.114302\pi\)
−0.936217 + 0.351421i \(0.885698\pi\)
\(132\) 0 0
\(133\) −403.236 −3.03185
\(134\) 81.8704i 0.610973i
\(135\) 0 0
\(136\) −143.637 −1.05615
\(137\) − 198.651i − 1.45001i −0.688746 0.725003i \(-0.741838\pi\)
0.688746 0.725003i \(-0.258162\pi\)
\(138\) 0 0
\(139\) 59.5350 0.428310 0.214155 0.976800i \(-0.431300\pi\)
0.214155 + 0.976800i \(0.431300\pi\)
\(140\) 146.112i 1.04365i
\(141\) 0 0
\(142\) 66.4748 0.468133
\(143\) 0 0
\(144\) 0 0
\(145\) 53.0011 0.365525
\(146\) − 8.61263i − 0.0589906i
\(147\) 0 0
\(148\) 40.7588 0.275398
\(149\) − 14.5644i − 0.0977477i −0.998805 0.0488738i \(-0.984437\pi\)
0.998805 0.0488738i \(-0.0155632\pi\)
\(150\) 0 0
\(151\) 243.598 1.61323 0.806616 0.591076i \(-0.201297\pi\)
0.806616 + 0.591076i \(0.201297\pi\)
\(152\) − 255.490i − 1.68086i
\(153\) 0 0
\(154\) 0 0
\(155\) 86.1051i 0.555517i
\(156\) 0 0
\(157\) −111.033 −0.707218 −0.353609 0.935393i \(-0.615046\pi\)
−0.353609 + 0.935393i \(0.615046\pi\)
\(158\) − 155.483i − 0.984072i
\(159\) 0 0
\(160\) −152.557 −0.953484
\(161\) 204.510i 1.27025i
\(162\) 0 0
\(163\) 123.154 0.755546 0.377773 0.925898i \(-0.376690\pi\)
0.377773 + 0.925898i \(0.376690\pi\)
\(164\) − 29.0365i − 0.177052i
\(165\) 0 0
\(166\) 54.7225 0.329654
\(167\) 229.010i 1.37132i 0.727924 + 0.685658i \(0.240485\pi\)
−0.727924 + 0.685658i \(0.759515\pi\)
\(168\) 0 0
\(169\) −161.870 −0.957809
\(170\) − 118.766i − 0.698621i
\(171\) 0 0
\(172\) −22.9012 −0.133147
\(173\) − 172.678i − 0.998138i −0.866562 0.499069i \(-0.833675\pi\)
0.866562 0.499069i \(-0.166325\pi\)
\(174\) 0 0
\(175\) −13.9298 −0.0795991
\(176\) 0 0
\(177\) 0 0
\(178\) −96.1267 −0.540038
\(179\) − 182.492i − 1.01951i −0.860320 0.509755i \(-0.829736\pi\)
0.860320 0.509755i \(-0.170264\pi\)
\(180\) 0 0
\(181\) 121.741 0.672600 0.336300 0.941755i \(-0.390824\pi\)
0.336300 + 0.941755i \(0.390824\pi\)
\(182\) 47.5181i 0.261088i
\(183\) 0 0
\(184\) −129.577 −0.704224
\(185\) 95.7178i 0.517394i
\(186\) 0 0
\(187\) 0 0
\(188\) − 178.692i − 0.950488i
\(189\) 0 0
\(190\) 211.251 1.11185
\(191\) − 314.416i − 1.64616i −0.567926 0.823080i \(-0.692254\pi\)
0.567926 0.823080i \(-0.307746\pi\)
\(192\) 0 0
\(193\) −124.391 −0.644511 −0.322255 0.946653i \(-0.604441\pi\)
−0.322255 + 0.946653i \(0.604441\pi\)
\(194\) 25.2284i 0.130043i
\(195\) 0 0
\(196\) 270.395 1.37956
\(197\) − 193.993i − 0.984738i −0.870387 0.492369i \(-0.836131\pi\)
0.870387 0.492369i \(-0.163869\pi\)
\(198\) 0 0
\(199\) −67.6934 −0.340168 −0.170084 0.985430i \(-0.554404\pi\)
−0.170084 + 0.985430i \(0.554404\pi\)
\(200\) − 8.82594i − 0.0441297i
\(201\) 0 0
\(202\) −63.6009 −0.314856
\(203\) − 136.720i − 0.673498i
\(204\) 0 0
\(205\) 68.1892 0.332630
\(206\) 150.598i 0.731058i
\(207\) 0 0
\(208\) −6.89002 −0.0331251
\(209\) 0 0
\(210\) 0 0
\(211\) −75.0484 −0.355679 −0.177840 0.984059i \(-0.556911\pi\)
−0.177840 + 0.984059i \(0.556911\pi\)
\(212\) − 136.910i − 0.645801i
\(213\) 0 0
\(214\) −149.844 −0.700207
\(215\) − 53.7811i − 0.250145i
\(216\) 0 0
\(217\) 222.114 1.02357
\(218\) − 159.922i − 0.733588i
\(219\) 0 0
\(220\) 0 0
\(221\) 45.9716i 0.208016i
\(222\) 0 0
\(223\) −5.97825 −0.0268083 −0.0134042 0.999910i \(-0.504267\pi\)
−0.0134042 + 0.999910i \(0.504267\pi\)
\(224\) 393.533i 1.75684i
\(225\) 0 0
\(226\) −93.7121 −0.414655
\(227\) − 168.380i − 0.741762i −0.928680 0.370881i \(-0.879056\pi\)
0.928680 0.370881i \(-0.120944\pi\)
\(228\) 0 0
\(229\) −352.188 −1.53794 −0.768969 0.639287i \(-0.779230\pi\)
−0.768969 + 0.639287i \(0.779230\pi\)
\(230\) − 107.140i − 0.465827i
\(231\) 0 0
\(232\) 86.6257 0.373387
\(233\) − 356.092i − 1.52829i −0.645043 0.764146i \(-0.723160\pi\)
0.645043 0.764146i \(-0.276840\pi\)
\(234\) 0 0
\(235\) 419.639 1.78570
\(236\) 143.259i 0.607030i
\(237\) 0 0
\(238\) −306.364 −1.28725
\(239\) − 17.4384i − 0.0729639i −0.999334 0.0364820i \(-0.988385\pi\)
0.999334 0.0364820i \(-0.0116151\pi\)
\(240\) 0 0
\(241\) 98.6722 0.409428 0.204714 0.978822i \(-0.434374\pi\)
0.204714 + 0.978822i \(0.434374\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −79.4620 −0.325664
\(245\) 634.993i 2.59181i
\(246\) 0 0
\(247\) −81.7707 −0.331055
\(248\) 140.732i 0.567466i
\(249\) 0 0
\(250\) −165.165 −0.660661
\(251\) 177.305i 0.706394i 0.935549 + 0.353197i \(0.114906\pi\)
−0.935549 + 0.353197i \(0.885094\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 14.5612i − 0.0573275i
\(255\) 0 0
\(256\) −271.777 −1.06163
\(257\) − 115.325i − 0.448735i −0.974505 0.224367i \(-0.927968\pi\)
0.974505 0.224367i \(-0.0720315\pi\)
\(258\) 0 0
\(259\) 246.911 0.953324
\(260\) 29.6294i 0.113959i
\(261\) 0 0
\(262\) −124.427 −0.474914
\(263\) 236.953i 0.900964i 0.892786 + 0.450482i \(0.148748\pi\)
−0.892786 + 0.450482i \(0.851252\pi\)
\(264\) 0 0
\(265\) 321.518 1.21328
\(266\) − 544.937i − 2.04864i
\(267\) 0 0
\(268\) 131.686 0.491365
\(269\) 528.479i 1.96461i 0.187297 + 0.982303i \(0.440027\pi\)
−0.187297 + 0.982303i \(0.559973\pi\)
\(270\) 0 0
\(271\) 178.936 0.660280 0.330140 0.943932i \(-0.392904\pi\)
0.330140 + 0.943932i \(0.392904\pi\)
\(272\) − 44.4222i − 0.163317i
\(273\) 0 0
\(274\) 268.458 0.979775
\(275\) 0 0
\(276\) 0 0
\(277\) 286.215 1.03327 0.516634 0.856206i \(-0.327185\pi\)
0.516634 + 0.856206i \(0.327185\pi\)
\(278\) 80.4562i 0.289411i
\(279\) 0 0
\(280\) −560.813 −2.00290
\(281\) − 253.434i − 0.901902i −0.892549 0.450951i \(-0.851085\pi\)
0.892549 0.450951i \(-0.148915\pi\)
\(282\) 0 0
\(283\) 283.270 1.00095 0.500477 0.865750i \(-0.333158\pi\)
0.500477 + 0.865750i \(0.333158\pi\)
\(284\) − 106.923i − 0.376488i
\(285\) 0 0
\(286\) 0 0
\(287\) − 175.899i − 0.612889i
\(288\) 0 0
\(289\) −7.39394 −0.0255846
\(290\) 71.6261i 0.246986i
\(291\) 0 0
\(292\) −13.8531 −0.0474422
\(293\) − 299.293i − 1.02148i −0.859736 0.510739i \(-0.829372\pi\)
0.859736 0.510739i \(-0.170628\pi\)
\(294\) 0 0
\(295\) −336.429 −1.14044
\(296\) 156.443i 0.528523i
\(297\) 0 0
\(298\) 19.6825 0.0660485
\(299\) 41.4717i 0.138701i
\(300\) 0 0
\(301\) −138.732 −0.460905
\(302\) 329.200i 1.09007i
\(303\) 0 0
\(304\) 79.0148 0.259917
\(305\) − 186.608i − 0.611829i
\(306\) 0 0
\(307\) −111.839 −0.364296 −0.182148 0.983271i \(-0.558305\pi\)
−0.182148 + 0.983271i \(0.558305\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −116.363 −0.375365
\(311\) − 356.284i − 1.14561i −0.819692 0.572805i \(-0.805855\pi\)
0.819692 0.572805i \(-0.194145\pi\)
\(312\) 0 0
\(313\) 338.742 1.08224 0.541121 0.840945i \(-0.318000\pi\)
0.541121 + 0.840945i \(0.318000\pi\)
\(314\) − 150.051i − 0.477870i
\(315\) 0 0
\(316\) −250.090 −0.791423
\(317\) 404.642i 1.27647i 0.769840 + 0.638237i \(0.220336\pi\)
−0.769840 + 0.638237i \(0.779664\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 258.853i − 0.808917i
\(321\) 0 0
\(322\) −276.376 −0.858311
\(323\) − 527.202i − 1.63221i
\(324\) 0 0
\(325\) −2.82478 −0.00869162
\(326\) 166.431i 0.510526i
\(327\) 0 0
\(328\) 111.450 0.339785
\(329\) − 1082.49i − 3.29024i
\(330\) 0 0
\(331\) 634.065 1.91561 0.957803 0.287426i \(-0.0927995\pi\)
0.957803 + 0.287426i \(0.0927995\pi\)
\(332\) − 88.0193i − 0.265118i
\(333\) 0 0
\(334\) −309.486 −0.926604
\(335\) 309.250i 0.923134i
\(336\) 0 0
\(337\) −237.412 −0.704488 −0.352244 0.935908i \(-0.614581\pi\)
−0.352244 + 0.935908i \(0.614581\pi\)
\(338\) − 218.752i − 0.647195i
\(339\) 0 0
\(340\) −191.030 −0.561854
\(341\) 0 0
\(342\) 0 0
\(343\) 992.782 2.89441
\(344\) − 87.9007i − 0.255525i
\(345\) 0 0
\(346\) 233.358 0.674446
\(347\) 487.484i 1.40485i 0.711757 + 0.702426i \(0.247900\pi\)
−0.711757 + 0.702426i \(0.752100\pi\)
\(348\) 0 0
\(349\) −38.0498 −0.109025 −0.0545126 0.998513i \(-0.517361\pi\)
−0.0545126 + 0.998513i \(0.517361\pi\)
\(350\) − 18.8249i − 0.0537855i
\(351\) 0 0
\(352\) 0 0
\(353\) − 90.5379i − 0.256481i −0.991743 0.128241i \(-0.959067\pi\)
0.991743 0.128241i \(-0.0409330\pi\)
\(354\) 0 0
\(355\) 251.096 0.707313
\(356\) 154.617i 0.434316i
\(357\) 0 0
\(358\) 246.621 0.688887
\(359\) − 111.987i − 0.311941i −0.987762 0.155971i \(-0.950150\pi\)
0.987762 0.155971i \(-0.0498505\pi\)
\(360\) 0 0
\(361\) 576.747 1.59764
\(362\) 164.521i 0.454478i
\(363\) 0 0
\(364\) 76.4312 0.209976
\(365\) − 32.5326i − 0.0891304i
\(366\) 0 0
\(367\) 338.791 0.923137 0.461568 0.887105i \(-0.347287\pi\)
0.461568 + 0.887105i \(0.347287\pi\)
\(368\) − 40.0740i − 0.108897i
\(369\) 0 0
\(370\) −129.354 −0.349605
\(371\) − 829.380i − 2.23552i
\(372\) 0 0
\(373\) 554.596 1.48685 0.743427 0.668818i \(-0.233199\pi\)
0.743427 + 0.668818i \(0.233199\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 685.864 1.82411
\(377\) − 27.7249i − 0.0735409i
\(378\) 0 0
\(379\) −251.309 −0.663084 −0.331542 0.943440i \(-0.607569\pi\)
−0.331542 + 0.943440i \(0.607569\pi\)
\(380\) − 339.790i − 0.894184i
\(381\) 0 0
\(382\) 424.905 1.11232
\(383\) − 514.214i − 1.34259i −0.741188 0.671297i \(-0.765737\pi\)
0.741188 0.671297i \(-0.234263\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 168.102i − 0.435499i
\(387\) 0 0
\(388\) 40.5790 0.104585
\(389\) 194.151i 0.499102i 0.968362 + 0.249551i \(0.0802830\pi\)
−0.968362 + 0.249551i \(0.919717\pi\)
\(390\) 0 0
\(391\) −267.382 −0.683840
\(392\) 1037.84i 2.64756i
\(393\) 0 0
\(394\) 262.164 0.665392
\(395\) − 587.309i − 1.48686i
\(396\) 0 0
\(397\) −260.248 −0.655537 −0.327769 0.944758i \(-0.606297\pi\)
−0.327769 + 0.944758i \(0.606297\pi\)
\(398\) − 91.4815i − 0.229853i
\(399\) 0 0
\(400\) 2.72957 0.00682393
\(401\) − 379.370i − 0.946059i −0.881047 0.473029i \(-0.843160\pi\)
0.881047 0.473029i \(-0.156840\pi\)
\(402\) 0 0
\(403\) 45.0417 0.111766
\(404\) 102.300i 0.253217i
\(405\) 0 0
\(406\) 184.765 0.455085
\(407\) 0 0
\(408\) 0 0
\(409\) −682.061 −1.66763 −0.833815 0.552044i \(-0.813848\pi\)
−0.833815 + 0.552044i \(0.813848\pi\)
\(410\) 92.1515i 0.224760i
\(411\) 0 0
\(412\) 242.232 0.587941
\(413\) 867.842i 2.10131i
\(414\) 0 0
\(415\) 206.704 0.498082
\(416\) 79.8030i 0.191834i
\(417\) 0 0
\(418\) 0 0
\(419\) − 275.319i − 0.657085i −0.944489 0.328543i \(-0.893443\pi\)
0.944489 0.328543i \(-0.106557\pi\)
\(420\) 0 0
\(421\) 121.470 0.288527 0.144263 0.989539i \(-0.453919\pi\)
0.144263 + 0.989539i \(0.453919\pi\)
\(422\) − 101.421i − 0.240334i
\(423\) 0 0
\(424\) 525.494 1.23937
\(425\) − 18.2123i − 0.0428524i
\(426\) 0 0
\(427\) −481.369 −1.12733
\(428\) 241.019i 0.563130i
\(429\) 0 0
\(430\) 72.6803 0.169024
\(431\) − 98.7461i − 0.229109i −0.993417 0.114555i \(-0.963456\pi\)
0.993417 0.114555i \(-0.0365441\pi\)
\(432\) 0 0
\(433\) 559.992 1.29328 0.646642 0.762794i \(-0.276173\pi\)
0.646642 + 0.762794i \(0.276173\pi\)
\(434\) 300.167i 0.691630i
\(435\) 0 0
\(436\) −257.230 −0.589976
\(437\) − 475.597i − 1.08832i
\(438\) 0 0
\(439\) 577.439 1.31535 0.657676 0.753301i \(-0.271540\pi\)
0.657676 + 0.753301i \(0.271540\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −62.1264 −0.140558
\(443\) 87.3800i 0.197246i 0.995125 + 0.0986230i \(0.0314438\pi\)
−0.995125 + 0.0986230i \(0.968556\pi\)
\(444\) 0 0
\(445\) −363.101 −0.815956
\(446\) − 8.07906i − 0.0181145i
\(447\) 0 0
\(448\) −667.731 −1.49047
\(449\) 96.2092i 0.214274i 0.994244 + 0.107137i \(0.0341684\pi\)
−0.994244 + 0.107137i \(0.965832\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 150.733i 0.333479i
\(453\) 0 0
\(454\) 227.550 0.501212
\(455\) 179.490i 0.394485i
\(456\) 0 0
\(457\) 691.788 1.51376 0.756879 0.653555i \(-0.226723\pi\)
0.756879 + 0.653555i \(0.226723\pi\)
\(458\) − 475.949i − 1.03919i
\(459\) 0 0
\(460\) −172.332 −0.374634
\(461\) 70.5491i 0.153035i 0.997068 + 0.0765175i \(0.0243801\pi\)
−0.997068 + 0.0765175i \(0.975620\pi\)
\(462\) 0 0
\(463\) 248.292 0.536267 0.268134 0.963382i \(-0.413593\pi\)
0.268134 + 0.963382i \(0.413593\pi\)
\(464\) 26.7905i 0.0577381i
\(465\) 0 0
\(466\) 481.226 1.03267
\(467\) 254.158i 0.544235i 0.962264 + 0.272118i \(0.0877240\pi\)
−0.962264 + 0.272118i \(0.912276\pi\)
\(468\) 0 0
\(469\) 797.732 1.70092
\(470\) 567.103i 1.20660i
\(471\) 0 0
\(472\) −549.864 −1.16497
\(473\) 0 0
\(474\) 0 0
\(475\) 32.3946 0.0681991
\(476\) 492.777i 1.03525i
\(477\) 0 0
\(478\) 23.5664 0.0493020
\(479\) − 18.2786i − 0.0381599i −0.999818 0.0190800i \(-0.993926\pi\)
0.999818 0.0190800i \(-0.00607371\pi\)
\(480\) 0 0
\(481\) 50.0701 0.104096
\(482\) 133.346i 0.276652i
\(483\) 0 0
\(484\) 0 0
\(485\) 95.2954i 0.196485i
\(486\) 0 0
\(487\) −243.375 −0.499744 −0.249872 0.968279i \(-0.580389\pi\)
−0.249872 + 0.968279i \(0.580389\pi\)
\(488\) − 304.995i − 0.624990i
\(489\) 0 0
\(490\) −858.135 −1.75130
\(491\) − 316.374i − 0.644347i −0.946681 0.322173i \(-0.895587\pi\)
0.946681 0.322173i \(-0.104413\pi\)
\(492\) 0 0
\(493\) 178.752 0.362579
\(494\) − 110.506i − 0.223696i
\(495\) 0 0
\(496\) −43.5236 −0.0877493
\(497\) − 647.721i − 1.30326i
\(498\) 0 0
\(499\) 26.8038 0.0537149 0.0268575 0.999639i \(-0.491450\pi\)
0.0268575 + 0.999639i \(0.491450\pi\)
\(500\) 265.663i 0.531326i
\(501\) 0 0
\(502\) −239.611 −0.477314
\(503\) 182.625i 0.363071i 0.983384 + 0.181535i \(0.0581067\pi\)
−0.983384 + 0.181535i \(0.941893\pi\)
\(504\) 0 0
\(505\) −240.240 −0.475723
\(506\) 0 0
\(507\) 0 0
\(508\) −23.4212 −0.0461047
\(509\) 216.345i 0.425040i 0.977157 + 0.212520i \(0.0681670\pi\)
−0.977157 + 0.212520i \(0.931833\pi\)
\(510\) 0 0
\(511\) −83.9201 −0.164227
\(512\) − 163.224i − 0.318797i
\(513\) 0 0
\(514\) 155.851 0.303212
\(515\) 568.855i 1.10457i
\(516\) 0 0
\(517\) 0 0
\(518\) 333.678i 0.644165i
\(519\) 0 0
\(520\) −113.725 −0.218702
\(521\) − 571.189i − 1.09633i −0.836370 0.548166i \(-0.815326\pi\)
0.836370 0.548166i \(-0.184674\pi\)
\(522\) 0 0
\(523\) −767.452 −1.46740 −0.733702 0.679472i \(-0.762209\pi\)
−0.733702 + 0.679472i \(0.762209\pi\)
\(524\) 200.137i 0.381941i
\(525\) 0 0
\(526\) −320.221 −0.608785
\(527\) 290.398i 0.551041i
\(528\) 0 0
\(529\) 287.791 0.544028
\(530\) 434.503i 0.819816i
\(531\) 0 0
\(532\) −876.513 −1.64758
\(533\) − 35.6699i − 0.0669228i
\(534\) 0 0
\(535\) −566.008 −1.05796
\(536\) 505.443i 0.942990i
\(537\) 0 0
\(538\) −714.191 −1.32749
\(539\) 0 0
\(540\) 0 0
\(541\) −205.807 −0.380420 −0.190210 0.981743i \(-0.560917\pi\)
−0.190210 + 0.981743i \(0.560917\pi\)
\(542\) 241.815i 0.446154i
\(543\) 0 0
\(544\) −514.516 −0.945801
\(545\) − 604.076i − 1.10840i
\(546\) 0 0
\(547\) −435.348 −0.795883 −0.397942 0.917411i \(-0.630275\pi\)
−0.397942 + 0.917411i \(0.630275\pi\)
\(548\) − 431.806i − 0.787968i
\(549\) 0 0
\(550\) 0 0
\(551\) 317.949i 0.577040i
\(552\) 0 0
\(553\) −1515.01 −2.73961
\(554\) 386.794i 0.698184i
\(555\) 0 0
\(556\) 129.411 0.232754
\(557\) − 738.780i − 1.32636i −0.748462 0.663178i \(-0.769207\pi\)
0.748462 0.663178i \(-0.230793\pi\)
\(558\) 0 0
\(559\) −28.1330 −0.0503273
\(560\) − 173.441i − 0.309716i
\(561\) 0 0
\(562\) 342.493 0.609419
\(563\) 65.8590i 0.116979i 0.998288 + 0.0584893i \(0.0186284\pi\)
−0.998288 + 0.0584893i \(0.981372\pi\)
\(564\) 0 0
\(565\) −353.980 −0.626513
\(566\) 382.813i 0.676348i
\(567\) 0 0
\(568\) 410.395 0.722527
\(569\) 697.641i 1.22608i 0.790051 + 0.613041i \(0.210054\pi\)
−0.790051 + 0.613041i \(0.789946\pi\)
\(570\) 0 0
\(571\) 256.388 0.449017 0.224508 0.974472i \(-0.427922\pi\)
0.224508 + 0.974472i \(0.427922\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 237.711 0.414132
\(575\) − 16.4296i − 0.0285732i
\(576\) 0 0
\(577\) 454.911 0.788407 0.394203 0.919023i \(-0.371021\pi\)
0.394203 + 0.919023i \(0.371021\pi\)
\(578\) − 9.99223i − 0.0172876i
\(579\) 0 0
\(580\) 115.208 0.198635
\(581\) − 533.207i − 0.917741i
\(582\) 0 0
\(583\) 0 0
\(584\) − 53.1718i − 0.0910475i
\(585\) 0 0
\(586\) 404.467 0.690217
\(587\) − 110.294i − 0.187894i −0.995577 0.0939470i \(-0.970052\pi\)
0.995577 0.0939470i \(-0.0299484\pi\)
\(588\) 0 0
\(589\) −516.538 −0.876975
\(590\) − 454.653i − 0.770598i
\(591\) 0 0
\(592\) −48.3826 −0.0817273
\(593\) 250.416i 0.422287i 0.977455 + 0.211144i \(0.0677188\pi\)
−0.977455 + 0.211144i \(0.932281\pi\)
\(594\) 0 0
\(595\) −1157.23 −1.94493
\(596\) − 31.6586i − 0.0531184i
\(597\) 0 0
\(598\) −56.0452 −0.0937211
\(599\) − 285.443i − 0.476533i −0.971200 0.238267i \(-0.923421\pi\)
0.971200 0.238267i \(-0.0765792\pi\)
\(600\) 0 0
\(601\) 223.484 0.371854 0.185927 0.982564i \(-0.440471\pi\)
0.185927 + 0.982564i \(0.440471\pi\)
\(602\) − 187.484i − 0.311435i
\(603\) 0 0
\(604\) 529.508 0.876668
\(605\) 0 0
\(606\) 0 0
\(607\) −954.863 −1.57309 −0.786543 0.617536i \(-0.788131\pi\)
−0.786543 + 0.617536i \(0.788131\pi\)
\(608\) − 915.180i − 1.50523i
\(609\) 0 0
\(610\) 252.184 0.413416
\(611\) − 219.513i − 0.359269i
\(612\) 0 0
\(613\) 340.137 0.554872 0.277436 0.960744i \(-0.410515\pi\)
0.277436 + 0.960744i \(0.410515\pi\)
\(614\) − 151.140i − 0.246156i
\(615\) 0 0
\(616\) 0 0
\(617\) − 812.456i − 1.31678i −0.752675 0.658392i \(-0.771237\pi\)
0.752675 0.658392i \(-0.228763\pi\)
\(618\) 0 0
\(619\) 913.830 1.47630 0.738150 0.674636i \(-0.235699\pi\)
0.738150 + 0.674636i \(0.235699\pi\)
\(620\) 187.166i 0.301881i
\(621\) 0 0
\(622\) 481.486 0.774093
\(623\) 936.644i 1.50344i
\(624\) 0 0
\(625\) −650.327 −1.04052
\(626\) 457.779i 0.731276i
\(627\) 0 0
\(628\) −241.352 −0.384319
\(629\) 322.818i 0.513224i
\(630\) 0 0
\(631\) −603.870 −0.957005 −0.478502 0.878086i \(-0.658820\pi\)
−0.478502 + 0.878086i \(0.658820\pi\)
\(632\) − 959.907i − 1.51884i
\(633\) 0 0
\(634\) −546.837 −0.862518
\(635\) − 55.0021i − 0.0866176i
\(636\) 0 0
\(637\) 332.165 0.521453
\(638\) 0 0
\(639\) 0 0
\(640\) −260.413 −0.406895
\(641\) − 1077.99i − 1.68174i −0.541241 0.840868i \(-0.682045\pi\)
0.541241 0.840868i \(-0.317955\pi\)
\(642\) 0 0
\(643\) 384.547 0.598052 0.299026 0.954245i \(-0.403338\pi\)
0.299026 + 0.954245i \(0.403338\pi\)
\(644\) 444.542i 0.690282i
\(645\) 0 0
\(646\) 712.466 1.10289
\(647\) 595.205i 0.919946i 0.887933 + 0.459973i \(0.152141\pi\)
−0.887933 + 0.459973i \(0.847859\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) − 3.81743i − 0.00587297i
\(651\) 0 0
\(652\) 267.699 0.410582
\(653\) − 418.619i − 0.641071i −0.947237 0.320535i \(-0.896137\pi\)
0.947237 0.320535i \(-0.103863\pi\)
\(654\) 0 0
\(655\) −470.001 −0.717559
\(656\) 34.4677i 0.0525422i
\(657\) 0 0
\(658\) 1462.88 2.22323
\(659\) 1112.39i 1.68800i 0.536347 + 0.843998i \(0.319804\pi\)
−0.536347 + 0.843998i \(0.680196\pi\)
\(660\) 0 0
\(661\) −736.025 −1.11350 −0.556751 0.830679i \(-0.687952\pi\)
−0.556751 + 0.830679i \(0.687952\pi\)
\(662\) 856.882i 1.29438i
\(663\) 0 0
\(664\) 337.840 0.508795
\(665\) − 2058.40i − 3.09533i
\(666\) 0 0
\(667\) 161.255 0.241761
\(668\) 497.797i 0.745206i
\(669\) 0 0
\(670\) −417.923 −0.623765
\(671\) 0 0
\(672\) 0 0
\(673\) −545.629 −0.810742 −0.405371 0.914152i \(-0.632858\pi\)
−0.405371 + 0.914152i \(0.632858\pi\)
\(674\) − 320.841i − 0.476025i
\(675\) 0 0
\(676\) −351.855 −0.520496
\(677\) − 984.875i − 1.45476i −0.686233 0.727382i \(-0.740737\pi\)
0.686233 0.727382i \(-0.259263\pi\)
\(678\) 0 0
\(679\) 245.821 0.362034
\(680\) − 733.223i − 1.07827i
\(681\) 0 0
\(682\) 0 0
\(683\) 924.817i 1.35405i 0.735959 + 0.677026i \(0.236731\pi\)
−0.735959 + 0.677026i \(0.763269\pi\)
\(684\) 0 0
\(685\) 1014.05 1.48037
\(686\) 1341.65i 1.95576i
\(687\) 0 0
\(688\) 27.1848 0.0395128
\(689\) − 168.187i − 0.244102i
\(690\) 0 0
\(691\) −79.1299 −0.114515 −0.0572575 0.998359i \(-0.518236\pi\)
−0.0572575 + 0.998359i \(0.518236\pi\)
\(692\) − 375.349i − 0.542412i
\(693\) 0 0
\(694\) −658.790 −0.949265
\(695\) 303.908i 0.437278i
\(696\) 0 0
\(697\) 229.975 0.329950
\(698\) − 51.4208i − 0.0736688i
\(699\) 0 0
\(700\) −30.2792 −0.0432560
\(701\) − 1232.29i − 1.75790i −0.476912 0.878951i \(-0.658244\pi\)
0.476912 0.878951i \(-0.341756\pi\)
\(702\) 0 0
\(703\) −574.204 −0.816791
\(704\) 0 0
\(705\) 0 0
\(706\) 122.354 0.173306
\(707\) 619.717i 0.876545i
\(708\) 0 0
\(709\) −267.568 −0.377388 −0.188694 0.982036i \(-0.560425\pi\)
−0.188694 + 0.982036i \(0.560425\pi\)
\(710\) 339.333i 0.477934i
\(711\) 0 0
\(712\) −593.457 −0.833507
\(713\) 261.973i 0.367423i
\(714\) 0 0
\(715\) 0 0
\(716\) − 396.682i − 0.554025i
\(717\) 0 0
\(718\) 151.340 0.210780
\(719\) 686.620i 0.954965i 0.878641 + 0.477482i \(0.158451\pi\)
−0.878641 + 0.477482i \(0.841549\pi\)
\(720\) 0 0
\(721\) 1467.40 2.03523
\(722\) 779.420i 1.07953i
\(723\) 0 0
\(724\) 264.627 0.365507
\(725\) 10.9836i 0.0151498i
\(726\) 0 0
\(727\) −1265.58 −1.74082 −0.870410 0.492327i \(-0.836146\pi\)
−0.870410 + 0.492327i \(0.836146\pi\)
\(728\) 293.362i 0.402970i
\(729\) 0 0
\(730\) 43.9648 0.0602258
\(731\) − 181.382i − 0.248129i
\(732\) 0 0
\(733\) −99.5964 −0.135875 −0.0679375 0.997690i \(-0.521642\pi\)
−0.0679375 + 0.997690i \(0.521642\pi\)
\(734\) 457.845i 0.623767i
\(735\) 0 0
\(736\) −464.152 −0.630642
\(737\) 0 0
\(738\) 0 0
\(739\) 320.536 0.433743 0.216872 0.976200i \(-0.430415\pi\)
0.216872 + 0.976200i \(0.430415\pi\)
\(740\) 208.061i 0.281164i
\(741\) 0 0
\(742\) 1120.83 1.51055
\(743\) − 54.6611i − 0.0735681i −0.999323 0.0367840i \(-0.988289\pi\)
0.999323 0.0367840i \(-0.0117114\pi\)
\(744\) 0 0
\(745\) 74.3468 0.0997943
\(746\) 749.486i 1.00467i
\(747\) 0 0
\(748\) 0 0
\(749\) 1460.06i 1.94935i
\(750\) 0 0
\(751\) 145.039 0.193128 0.0965641 0.995327i \(-0.469215\pi\)
0.0965641 + 0.995327i \(0.469215\pi\)
\(752\) 212.115i 0.282068i
\(753\) 0 0
\(754\) 37.4677 0.0496919
\(755\) 1243.49i 1.64701i
\(756\) 0 0
\(757\) 261.648 0.345638 0.172819 0.984954i \(-0.444712\pi\)
0.172819 + 0.984954i \(0.444712\pi\)
\(758\) − 339.621i − 0.448049i
\(759\) 0 0
\(760\) 1304.20 1.71605
\(761\) 1004.63i 1.32014i 0.751204 + 0.660070i \(0.229473\pi\)
−0.751204 + 0.660070i \(0.770527\pi\)
\(762\) 0 0
\(763\) −1558.26 −2.04228
\(764\) − 683.445i − 0.894562i
\(765\) 0 0
\(766\) 694.913 0.907197
\(767\) 175.986i 0.229448i
\(768\) 0 0
\(769\) −1067.91 −1.38869 −0.694347 0.719641i \(-0.744307\pi\)
−0.694347 + 0.719641i \(0.744307\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −270.387 −0.350242
\(773\) 177.483i 0.229603i 0.993388 + 0.114801i \(0.0366232\pi\)
−0.993388 + 0.114801i \(0.963377\pi\)
\(774\) 0 0
\(775\) −17.8439 −0.0230243
\(776\) 155.752i 0.200712i
\(777\) 0 0
\(778\) −262.377 −0.337245
\(779\) 409.062i 0.525112i
\(780\) 0 0
\(781\) 0 0
\(782\) − 361.342i − 0.462074i
\(783\) 0 0
\(784\) −320.970 −0.409401
\(785\) − 566.790i − 0.722026i
\(786\) 0 0
\(787\) −660.507 −0.839271 −0.419636 0.907693i \(-0.637842\pi\)
−0.419636 + 0.907693i \(0.637842\pi\)
\(788\) − 421.682i − 0.535130i
\(789\) 0 0
\(790\) 793.695 1.00468
\(791\) 913.116i 1.15438i
\(792\) 0 0
\(793\) −97.6148 −0.123096
\(794\) − 351.702i − 0.442949i
\(795\) 0 0
\(796\) −147.145 −0.184855
\(797\) 398.568i 0.500086i 0.968235 + 0.250043i \(0.0804447\pi\)
−0.968235 + 0.250043i \(0.919555\pi\)
\(798\) 0 0
\(799\) 1415.27 1.77131
\(800\) − 31.6150i − 0.0395188i
\(801\) 0 0
\(802\) 512.683 0.639256
\(803\) 0 0
\(804\) 0 0
\(805\) −1043.96 −1.29684
\(806\) 60.8697i 0.0755207i
\(807\) 0 0
\(808\) −392.652 −0.485956
\(809\) − 752.975i − 0.930748i −0.885114 0.465374i \(-0.845920\pi\)
0.885114 0.465374i \(-0.154080\pi\)
\(810\) 0 0
\(811\) −1002.83 −1.23654 −0.618270 0.785966i \(-0.712166\pi\)
−0.618270 + 0.785966i \(0.712166\pi\)
\(812\) − 297.188i − 0.365995i
\(813\) 0 0
\(814\) 0 0
\(815\) 628.663i 0.771366i
\(816\) 0 0
\(817\) 322.629 0.394895
\(818\) − 921.743i − 1.12683i
\(819\) 0 0
\(820\) 148.223 0.180759
\(821\) − 1368.74i − 1.66717i −0.552394 0.833583i \(-0.686286\pi\)
0.552394 0.833583i \(-0.313714\pi\)
\(822\) 0 0
\(823\) −1395.78 −1.69597 −0.847985 0.530019i \(-0.822185\pi\)
−0.847985 + 0.530019i \(0.822185\pi\)
\(824\) 929.746i 1.12833i
\(825\) 0 0
\(826\) −1172.81 −1.41987
\(827\) − 202.834i − 0.245265i −0.992452 0.122632i \(-0.960866\pi\)
0.992452 0.122632i \(-0.0391336\pi\)
\(828\) 0 0
\(829\) 569.961 0.687528 0.343764 0.939056i \(-0.388298\pi\)
0.343764 + 0.939056i \(0.388298\pi\)
\(830\) 279.341i 0.336556i
\(831\) 0 0
\(832\) −135.406 −0.162748
\(833\) 2141.58i 2.57092i
\(834\) 0 0
\(835\) −1169.02 −1.40003
\(836\) 0 0
\(837\) 0 0
\(838\) 372.068 0.443995
\(839\) 129.965i 0.154904i 0.996996 + 0.0774522i \(0.0246785\pi\)
−0.996996 + 0.0774522i \(0.975321\pi\)
\(840\) 0 0
\(841\) 733.197 0.871816
\(842\) 164.155i 0.194959i
\(843\) 0 0
\(844\) −163.132 −0.193285
\(845\) − 826.294i − 0.977863i
\(846\) 0 0
\(847\) 0 0
\(848\) 162.518i 0.191649i
\(849\) 0 0
\(850\) 24.6122 0.0289555
\(851\) 291.219i 0.342208i
\(852\) 0 0
\(853\) −322.301 −0.377844 −0.188922 0.981992i \(-0.560499\pi\)
−0.188922 + 0.981992i \(0.560499\pi\)
\(854\) − 650.526i − 0.761740i
\(855\) 0 0
\(856\) −925.093 −1.08072
\(857\) 634.845i 0.740776i 0.928877 + 0.370388i \(0.120775\pi\)
−0.928877 + 0.370388i \(0.879225\pi\)
\(858\) 0 0
\(859\) 1658.25 1.93044 0.965219 0.261443i \(-0.0841984\pi\)
0.965219 + 0.261443i \(0.0841984\pi\)
\(860\) − 116.904i − 0.135935i
\(861\) 0 0
\(862\) 133.446 0.154810
\(863\) 862.333i 0.999227i 0.866249 + 0.499613i \(0.166525\pi\)
−0.866249 + 0.499613i \(0.833475\pi\)
\(864\) 0 0
\(865\) 881.467 1.01904
\(866\) 756.778i 0.873877i
\(867\) 0 0
\(868\) 482.809 0.556231
\(869\) 0 0
\(870\) 0 0
\(871\) 161.769 0.185728
\(872\) − 987.311i − 1.13224i
\(873\) 0 0
\(874\) 642.726 0.735385
\(875\) 1609.35i 1.83925i
\(876\) 0 0
\(877\) −685.376 −0.781500 −0.390750 0.920497i \(-0.627784\pi\)
−0.390750 + 0.920497i \(0.627784\pi\)
\(878\) 780.357i 0.888789i
\(879\) 0 0
\(880\) 0 0
\(881\) − 551.003i − 0.625429i −0.949847 0.312715i \(-0.898762\pi\)
0.949847 0.312715i \(-0.101238\pi\)
\(882\) 0 0
\(883\) −1592.69 −1.80373 −0.901863 0.432022i \(-0.857800\pi\)
−0.901863 + 0.432022i \(0.857800\pi\)
\(884\) 99.9282i 0.113041i
\(885\) 0 0
\(886\) −118.086 −0.133280
\(887\) − 384.029i − 0.432952i −0.976288 0.216476i \(-0.930544\pi\)
0.976288 0.216476i \(-0.0694564\pi\)
\(888\) 0 0
\(889\) −141.882 −0.159597
\(890\) − 490.697i − 0.551345i
\(891\) 0 0
\(892\) −12.9949 −0.0145683
\(893\) 2517.38i 2.81902i
\(894\) 0 0
\(895\) 931.566 1.04086
\(896\) 671.754i 0.749725i
\(897\) 0 0
\(898\) −130.018 −0.144786
\(899\) − 175.136i − 0.194812i
\(900\) 0 0
\(901\) 1084.35 1.20350
\(902\) 0 0
\(903\) 0 0
\(904\) −578.550 −0.639989
\(905\) 621.448i 0.686682i
\(906\) 0 0
\(907\) −1417.57 −1.56292 −0.781462 0.623953i \(-0.785526\pi\)
−0.781462 + 0.623953i \(0.785526\pi\)
\(908\) − 366.007i − 0.403091i
\(909\) 0 0
\(910\) −242.565 −0.266555
\(911\) − 874.814i − 0.960278i −0.877192 0.480139i \(-0.840586\pi\)
0.877192 0.480139i \(-0.159414\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 934.888i 1.02285i
\(915\) 0 0
\(916\) −765.548 −0.835751
\(917\) 1212.40i 1.32214i
\(918\) 0 0
\(919\) 1162.91 1.26541 0.632705 0.774393i \(-0.281945\pi\)
0.632705 + 0.774393i \(0.281945\pi\)
\(920\) − 661.451i − 0.718969i
\(921\) 0 0
\(922\) −95.3407 −0.103406
\(923\) − 131.349i − 0.142306i
\(924\) 0 0
\(925\) −19.8359 −0.0214443
\(926\) 335.544i 0.362358i
\(927\) 0 0
\(928\) 310.298 0.334373
\(929\) − 554.284i − 0.596646i −0.954465 0.298323i \(-0.903573\pi\)
0.954465 0.298323i \(-0.0964273\pi\)
\(930\) 0 0
\(931\) −3809.27 −4.09159
\(932\) − 774.036i − 0.830511i
\(933\) 0 0
\(934\) −343.471 −0.367742
\(935\) 0 0
\(936\) 0 0
\(937\) 857.873 0.915553 0.457776 0.889067i \(-0.348646\pi\)
0.457776 + 0.889067i \(0.348646\pi\)
\(938\) 1078.06i 1.14932i
\(939\) 0 0
\(940\) 912.166 0.970390
\(941\) − 337.080i − 0.358214i −0.983830 0.179107i \(-0.942679\pi\)
0.983830 0.179107i \(-0.0573209\pi\)
\(942\) 0 0
\(943\) 207.464 0.220004
\(944\) − 170.055i − 0.180143i
\(945\) 0 0
\(946\) 0 0
\(947\) − 405.388i − 0.428076i −0.976825 0.214038i \(-0.931338\pi\)
0.976825 0.214038i \(-0.0686616\pi\)
\(948\) 0 0
\(949\) −17.0178 −0.0179324
\(950\) 43.7783i 0.0460824i
\(951\) 0 0
\(952\) −1891.40 −1.98677
\(953\) − 747.947i − 0.784834i −0.919787 0.392417i \(-0.871639\pi\)
0.919787 0.392417i \(-0.128361\pi\)
\(954\) 0 0
\(955\) 1605.00 1.68063
\(956\) − 37.9057i − 0.0396503i
\(957\) 0 0
\(958\) 24.7019 0.0257848
\(959\) − 2615.82i − 2.72765i
\(960\) 0 0
\(961\) −676.476 −0.703929
\(962\) 67.6652i 0.0703380i
\(963\) 0 0
\(964\) 214.483 0.222493
\(965\) − 634.975i − 0.658005i
\(966\) 0 0
\(967\) 284.479 0.294187 0.147094 0.989123i \(-0.453008\pi\)
0.147094 + 0.989123i \(0.453008\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −128.783 −0.132766
\(971\) − 1179.42i − 1.21464i −0.794456 0.607322i \(-0.792244\pi\)
0.794456 0.607322i \(-0.207756\pi\)
\(972\) 0 0
\(973\) 783.952 0.805707
\(974\) − 328.900i − 0.337679i
\(975\) 0 0
\(976\) 94.3249 0.0966443
\(977\) 13.6463i 0.0139676i 0.999976 + 0.00698379i \(0.00222303\pi\)
−0.999976 + 0.00698379i \(0.997777\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1380.28i 1.40845i
\(981\) 0 0
\(982\) 427.551 0.435388
\(983\) − 501.274i − 0.509943i −0.966949 0.254971i \(-0.917934\pi\)
0.966949 0.254971i \(-0.0820660\pi\)
\(984\) 0 0
\(985\) 990.276 1.00536
\(986\) 241.566i 0.244996i
\(987\) 0 0
\(988\) −177.745 −0.179903
\(989\) − 163.628i − 0.165448i
\(990\) 0 0
\(991\) 118.253 0.119327 0.0596635 0.998219i \(-0.480997\pi\)
0.0596635 + 0.998219i \(0.480997\pi\)
\(992\) 504.108i 0.508173i
\(993\) 0 0
\(994\) 875.335 0.880619
\(995\) − 345.554i − 0.347290i
\(996\) 0 0
\(997\) −495.885 −0.497377 −0.248689 0.968583i \(-0.580000\pi\)
−0.248689 + 0.968583i \(0.580000\pi\)
\(998\) 36.2228i 0.0362954i
\(999\) 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 1089.3.b.j.485.11 16
3.2 odd 2 inner 1089.3.b.j.485.6 16
11.7 odd 10 99.3.l.a.71.6 yes 32
11.8 odd 10 99.3.l.a.53.3 32
11.10 odd 2 1089.3.b.i.485.6 16
33.8 even 10 99.3.l.a.53.6 yes 32
33.29 even 10 99.3.l.a.71.3 yes 32
33.32 even 2 1089.3.b.i.485.11 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.3.l.a.53.3 32 11.8 odd 10
99.3.l.a.53.6 yes 32 33.8 even 10
99.3.l.a.71.3 yes 32 33.29 even 10
99.3.l.a.71.6 yes 32 11.7 odd 10
1089.3.b.i.485.6 16 11.10 odd 2
1089.3.b.i.485.11 16 33.32 even 2
1089.3.b.j.485.6 16 3.2 odd 2 inner
1089.3.b.j.485.11 16 1.1 even 1 trivial