Properties

Label 1143.3.b.a.890.6
Level $1143$
Weight $3$
Character 1143.890
Analytic conductor $31.144$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1143.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.1444942164\)
Analytic rank: \(0\)
Dimension: \(84\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 890.6
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.79

$q$-expansion

\(f(q)\) \(=\) \(q-3.62368i q^{2} -9.13107 q^{4} -3.89223i q^{5} -8.96960 q^{7} +18.5934i q^{8} +O(q^{10})\) \(q-3.62368i q^{2} -9.13107 q^{4} -3.89223i q^{5} -8.96960 q^{7} +18.5934i q^{8} -14.1042 q^{10} -3.39451i q^{11} +4.51211 q^{13} +32.5030i q^{14} +30.8521 q^{16} +0.588323i q^{17} -29.7256 q^{19} +35.5402i q^{20} -12.3006 q^{22} +25.8399i q^{23} +9.85055 q^{25} -16.3504i q^{26} +81.9021 q^{28} +30.9303i q^{29} +11.9611 q^{31} -37.4249i q^{32} +2.13190 q^{34} +34.9117i q^{35} +5.42650 q^{37} +107.716i q^{38} +72.3696 q^{40} -40.4950i q^{41} -32.6461 q^{43} +30.9955i q^{44} +93.6356 q^{46} -41.3299i q^{47} +31.4538 q^{49} -35.6953i q^{50} -41.2004 q^{52} -16.8242i q^{53} -13.2122 q^{55} -166.775i q^{56} +112.081 q^{58} +60.6872i q^{59} +75.7718 q^{61} -43.3430i q^{62} -12.2073 q^{64} -17.5622i q^{65} +33.9228 q^{67} -5.37202i q^{68} +126.509 q^{70} +30.0600i q^{71} -0.689938 q^{73} -19.6639i q^{74} +271.427 q^{76} +30.4474i q^{77} -72.2732 q^{79} -120.084i q^{80} -146.741 q^{82} -21.0132i q^{83} +2.28989 q^{85} +118.299i q^{86} +63.1153 q^{88} -139.493i q^{89} -40.4718 q^{91} -235.946i q^{92} -149.766 q^{94} +115.699i q^{95} -32.4602 q^{97} -113.978i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times\).

\(n\) \(128\) \(892\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.62368i − 1.81184i −0.423448 0.905920i \(-0.639180\pi\)
0.423448 0.905920i \(-0.360820\pi\)
\(3\) 0 0
\(4\) −9.13107 −2.28277
\(5\) − 3.89223i − 0.778446i −0.921144 0.389223i \(-0.872744\pi\)
0.921144 0.389223i \(-0.127256\pi\)
\(6\) 0 0
\(7\) −8.96960 −1.28137 −0.640686 0.767803i \(-0.721350\pi\)
−0.640686 + 0.767803i \(0.721350\pi\)
\(8\) 18.5934i 2.32417i
\(9\) 0 0
\(10\) −14.1042 −1.41042
\(11\) − 3.39451i − 0.308592i −0.988025 0.154296i \(-0.950689\pi\)
0.988025 0.154296i \(-0.0493109\pi\)
\(12\) 0 0
\(13\) 4.51211 0.347085 0.173543 0.984826i \(-0.444479\pi\)
0.173543 + 0.984826i \(0.444479\pi\)
\(14\) 32.5030i 2.32164i
\(15\) 0 0
\(16\) 30.8521 1.92826
\(17\) 0.588323i 0.0346073i 0.999850 + 0.0173036i \(0.00550819\pi\)
−0.999850 + 0.0173036i \(0.994492\pi\)
\(18\) 0 0
\(19\) −29.7256 −1.56451 −0.782253 0.622960i \(-0.785930\pi\)
−0.782253 + 0.622960i \(0.785930\pi\)
\(20\) 35.5402i 1.77701i
\(21\) 0 0
\(22\) −12.3006 −0.559119
\(23\) 25.8399i 1.12347i 0.827316 + 0.561737i \(0.189867\pi\)
−0.827316 + 0.561737i \(0.810133\pi\)
\(24\) 0 0
\(25\) 9.85055 0.394022
\(26\) − 16.3504i − 0.628863i
\(27\) 0 0
\(28\) 81.9021 2.92507
\(29\) 30.9303i 1.06656i 0.845938 + 0.533281i \(0.179041\pi\)
−0.845938 + 0.533281i \(0.820959\pi\)
\(30\) 0 0
\(31\) 11.9611 0.385840 0.192920 0.981214i \(-0.438204\pi\)
0.192920 + 0.981214i \(0.438204\pi\)
\(32\) − 37.4249i − 1.16953i
\(33\) 0 0
\(34\) 2.13190 0.0627029
\(35\) 34.9117i 0.997479i
\(36\) 0 0
\(37\) 5.42650 0.146662 0.0733311 0.997308i \(-0.476637\pi\)
0.0733311 + 0.997308i \(0.476637\pi\)
\(38\) 107.716i 2.83464i
\(39\) 0 0
\(40\) 72.3696 1.80924
\(41\) − 40.4950i − 0.987682i −0.869552 0.493841i \(-0.835593\pi\)
0.869552 0.493841i \(-0.164407\pi\)
\(42\) 0 0
\(43\) −32.6461 −0.759211 −0.379606 0.925148i \(-0.623940\pi\)
−0.379606 + 0.925148i \(0.623940\pi\)
\(44\) 30.9955i 0.704443i
\(45\) 0 0
\(46\) 93.6356 2.03556
\(47\) − 41.3299i − 0.879359i −0.898155 0.439680i \(-0.855092\pi\)
0.898155 0.439680i \(-0.144908\pi\)
\(48\) 0 0
\(49\) 31.4538 0.641914
\(50\) − 35.6953i − 0.713905i
\(51\) 0 0
\(52\) −41.2004 −0.792315
\(53\) − 16.8242i − 0.317437i −0.987324 0.158719i \(-0.949264\pi\)
0.987324 0.158719i \(-0.0507363\pi\)
\(54\) 0 0
\(55\) −13.2122 −0.240222
\(56\) − 166.775i − 2.97813i
\(57\) 0 0
\(58\) 112.081 1.93244
\(59\) 60.6872i 1.02860i 0.857611 + 0.514298i \(0.171948\pi\)
−0.857611 + 0.514298i \(0.828052\pi\)
\(60\) 0 0
\(61\) 75.7718 1.24216 0.621080 0.783747i \(-0.286694\pi\)
0.621080 + 0.783747i \(0.286694\pi\)
\(62\) − 43.3430i − 0.699081i
\(63\) 0 0
\(64\) −12.2073 −0.190739
\(65\) − 17.5622i − 0.270187i
\(66\) 0 0
\(67\) 33.9228 0.506311 0.253155 0.967426i \(-0.418532\pi\)
0.253155 + 0.967426i \(0.418532\pi\)
\(68\) − 5.37202i − 0.0790003i
\(69\) 0 0
\(70\) 126.509 1.80727
\(71\) 30.0600i 0.423380i 0.977337 + 0.211690i \(0.0678967\pi\)
−0.977337 + 0.211690i \(0.932103\pi\)
\(72\) 0 0
\(73\) −0.689938 −0.00945120 −0.00472560 0.999989i \(-0.501504\pi\)
−0.00472560 + 0.999989i \(0.501504\pi\)
\(74\) − 19.6639i − 0.265728i
\(75\) 0 0
\(76\) 271.427 3.57140
\(77\) 30.4474i 0.395421i
\(78\) 0 0
\(79\) −72.2732 −0.914850 −0.457425 0.889248i \(-0.651228\pi\)
−0.457425 + 0.889248i \(0.651228\pi\)
\(80\) − 120.084i − 1.50104i
\(81\) 0 0
\(82\) −146.741 −1.78952
\(83\) − 21.0132i − 0.253171i −0.991956 0.126585i \(-0.959598\pi\)
0.991956 0.126585i \(-0.0404018\pi\)
\(84\) 0 0
\(85\) 2.28989 0.0269399
\(86\) 118.299i 1.37557i
\(87\) 0 0
\(88\) 63.1153 0.717219
\(89\) − 139.493i − 1.56733i −0.621183 0.783666i \(-0.713347\pi\)
0.621183 0.783666i \(-0.286653\pi\)
\(90\) 0 0
\(91\) −40.4718 −0.444745
\(92\) − 235.946i − 2.56463i
\(93\) 0 0
\(94\) −149.766 −1.59326
\(95\) 115.699i 1.21788i
\(96\) 0 0
\(97\) −32.4602 −0.334641 −0.167321 0.985903i \(-0.553512\pi\)
−0.167321 + 0.985903i \(0.553512\pi\)
\(98\) − 113.978i − 1.16305i
\(99\) 0 0
\(100\) −89.9461 −0.899461
\(101\) 13.2198i 0.130889i 0.997856 + 0.0654447i \(0.0208466\pi\)
−0.997856 + 0.0654447i \(0.979153\pi\)
\(102\) 0 0
\(103\) 106.910 1.03796 0.518981 0.854786i \(-0.326311\pi\)
0.518981 + 0.854786i \(0.326311\pi\)
\(104\) 83.8952i 0.806685i
\(105\) 0 0
\(106\) −60.9655 −0.575146
\(107\) − 9.87832i − 0.0923207i −0.998934 0.0461604i \(-0.985301\pi\)
0.998934 0.0461604i \(-0.0146985\pi\)
\(108\) 0 0
\(109\) 7.38727 0.0677731 0.0338866 0.999426i \(-0.489212\pi\)
0.0338866 + 0.999426i \(0.489212\pi\)
\(110\) 47.8768i 0.435244i
\(111\) 0 0
\(112\) −276.731 −2.47082
\(113\) 145.173i 1.28472i 0.766404 + 0.642359i \(0.222044\pi\)
−0.766404 + 0.642359i \(0.777956\pi\)
\(114\) 0 0
\(115\) 100.575 0.874564
\(116\) − 282.426i − 2.43471i
\(117\) 0 0
\(118\) 219.911 1.86365
\(119\) − 5.27703i − 0.0443448i
\(120\) 0 0
\(121\) 109.477 0.904771
\(122\) − 274.573i − 2.25060i
\(123\) 0 0
\(124\) −109.217 −0.880784
\(125\) − 135.646i − 1.08517i
\(126\) 0 0
\(127\) 11.2694 0.0887357
\(128\) − 105.464i − 0.823938i
\(129\) 0 0
\(130\) −63.6397 −0.489536
\(131\) 127.633i 0.974301i 0.873318 + 0.487151i \(0.161964\pi\)
−0.873318 + 0.487151i \(0.838036\pi\)
\(132\) 0 0
\(133\) 266.627 2.00471
\(134\) − 122.925i − 0.917354i
\(135\) 0 0
\(136\) −10.9389 −0.0804332
\(137\) 172.488i 1.25903i 0.776987 + 0.629517i \(0.216747\pi\)
−0.776987 + 0.629517i \(0.783253\pi\)
\(138\) 0 0
\(139\) 243.875 1.75450 0.877249 0.480035i \(-0.159376\pi\)
0.877249 + 0.480035i \(0.159376\pi\)
\(140\) − 318.782i − 2.27701i
\(141\) 0 0
\(142\) 108.928 0.767097
\(143\) − 15.3164i − 0.107108i
\(144\) 0 0
\(145\) 120.388 0.830260
\(146\) 2.50011i 0.0171241i
\(147\) 0 0
\(148\) −49.5497 −0.334796
\(149\) 100.923i 0.677337i 0.940906 + 0.338668i \(0.109976\pi\)
−0.940906 + 0.338668i \(0.890024\pi\)
\(150\) 0 0
\(151\) −261.415 −1.73122 −0.865611 0.500717i \(-0.833070\pi\)
−0.865611 + 0.500717i \(0.833070\pi\)
\(152\) − 552.699i − 3.63618i
\(153\) 0 0
\(154\) 110.332 0.716439
\(155\) − 46.5551i − 0.300356i
\(156\) 0 0
\(157\) −120.944 −0.770345 −0.385172 0.922845i \(-0.625858\pi\)
−0.385172 + 0.922845i \(0.625858\pi\)
\(158\) 261.895i 1.65756i
\(159\) 0 0
\(160\) −145.666 −0.910414
\(161\) − 231.774i − 1.43959i
\(162\) 0 0
\(163\) −75.6333 −0.464008 −0.232004 0.972715i \(-0.574528\pi\)
−0.232004 + 0.972715i \(0.574528\pi\)
\(164\) 369.762i 2.25465i
\(165\) 0 0
\(166\) −76.1450 −0.458705
\(167\) 141.550i 0.847606i 0.905754 + 0.423803i \(0.139305\pi\)
−0.905754 + 0.423803i \(0.860695\pi\)
\(168\) 0 0
\(169\) −148.641 −0.879532
\(170\) − 8.29783i − 0.0488108i
\(171\) 0 0
\(172\) 298.094 1.73310
\(173\) 131.050i 0.757512i 0.925497 + 0.378756i \(0.123648\pi\)
−0.925497 + 0.378756i \(0.876352\pi\)
\(174\) 0 0
\(175\) −88.3555 −0.504889
\(176\) − 104.728i − 0.595044i
\(177\) 0 0
\(178\) −505.477 −2.83976
\(179\) 179.416i 1.00233i 0.865353 + 0.501163i \(0.167094\pi\)
−0.865353 + 0.501163i \(0.832906\pi\)
\(180\) 0 0
\(181\) 339.063 1.87327 0.936637 0.350301i \(-0.113921\pi\)
0.936637 + 0.350301i \(0.113921\pi\)
\(182\) 146.657i 0.805808i
\(183\) 0 0
\(184\) −480.451 −2.61114
\(185\) − 21.1212i − 0.114169i
\(186\) 0 0
\(187\) 1.99707 0.0106795
\(188\) 377.386i 2.00737i
\(189\) 0 0
\(190\) 419.256 2.20661
\(191\) 314.303i 1.64557i 0.568354 + 0.822784i \(0.307580\pi\)
−0.568354 + 0.822784i \(0.692420\pi\)
\(192\) 0 0
\(193\) −151.480 −0.784872 −0.392436 0.919779i \(-0.628367\pi\)
−0.392436 + 0.919779i \(0.628367\pi\)
\(194\) 117.625i 0.606317i
\(195\) 0 0
\(196\) −287.207 −1.46534
\(197\) − 13.8619i − 0.0703650i −0.999381 0.0351825i \(-0.988799\pi\)
0.999381 0.0351825i \(-0.0112013\pi\)
\(198\) 0 0
\(199\) −351.838 −1.76803 −0.884015 0.467459i \(-0.845169\pi\)
−0.884015 + 0.467459i \(0.845169\pi\)
\(200\) 183.155i 0.915774i
\(201\) 0 0
\(202\) 47.9045 0.237151
\(203\) − 277.432i − 1.36666i
\(204\) 0 0
\(205\) −157.616 −0.768857
\(206\) − 387.408i − 1.88062i
\(207\) 0 0
\(208\) 139.208 0.669270
\(209\) 100.904i 0.482794i
\(210\) 0 0
\(211\) −193.186 −0.915574 −0.457787 0.889062i \(-0.651358\pi\)
−0.457787 + 0.889062i \(0.651358\pi\)
\(212\) 153.623i 0.724635i
\(213\) 0 0
\(214\) −35.7959 −0.167270
\(215\) 127.066i 0.591005i
\(216\) 0 0
\(217\) −107.286 −0.494405
\(218\) − 26.7691i − 0.122794i
\(219\) 0 0
\(220\) 120.642 0.548371
\(221\) 2.65458i 0.0120117i
\(222\) 0 0
\(223\) 312.551 1.40158 0.700788 0.713370i \(-0.252832\pi\)
0.700788 + 0.713370i \(0.252832\pi\)
\(224\) 335.686i 1.49860i
\(225\) 0 0
\(226\) 526.061 2.32770
\(227\) 111.603i 0.491644i 0.969315 + 0.245822i \(0.0790579\pi\)
−0.969315 + 0.245822i \(0.920942\pi\)
\(228\) 0 0
\(229\) 123.668 0.540034 0.270017 0.962855i \(-0.412971\pi\)
0.270017 + 0.962855i \(0.412971\pi\)
\(230\) − 364.451i − 1.58457i
\(231\) 0 0
\(232\) −575.098 −2.47887
\(233\) − 327.222i − 1.40439i −0.711986 0.702194i \(-0.752204\pi\)
0.711986 0.702194i \(-0.247796\pi\)
\(234\) 0 0
\(235\) −160.865 −0.684534
\(236\) − 554.139i − 2.34805i
\(237\) 0 0
\(238\) −19.1223 −0.0803457
\(239\) 16.7075i 0.0699059i 0.999389 + 0.0349529i \(0.0111281\pi\)
−0.999389 + 0.0349529i \(0.988872\pi\)
\(240\) 0 0
\(241\) 84.7568 0.351688 0.175844 0.984418i \(-0.443735\pi\)
0.175844 + 0.984418i \(0.443735\pi\)
\(242\) − 396.711i − 1.63930i
\(243\) 0 0
\(244\) −691.877 −2.83556
\(245\) − 122.425i − 0.499695i
\(246\) 0 0
\(247\) −134.125 −0.543017
\(248\) 222.396i 0.896758i
\(249\) 0 0
\(250\) −491.539 −1.96616
\(251\) 132.843i 0.529255i 0.964351 + 0.264628i \(0.0852491\pi\)
−0.964351 + 0.264628i \(0.914751\pi\)
\(252\) 0 0
\(253\) 87.7138 0.346695
\(254\) − 40.8368i − 0.160775i
\(255\) 0 0
\(256\) −430.998 −1.68358
\(257\) 285.101i 1.10934i 0.832070 + 0.554671i \(0.187156\pi\)
−0.832070 + 0.554671i \(0.812844\pi\)
\(258\) 0 0
\(259\) −48.6735 −0.187929
\(260\) 160.361i 0.616774i
\(261\) 0 0
\(262\) 462.503 1.76528
\(263\) 299.719i 1.13962i 0.821777 + 0.569809i \(0.192983\pi\)
−0.821777 + 0.569809i \(0.807017\pi\)
\(264\) 0 0
\(265\) −65.4836 −0.247108
\(266\) − 966.172i − 3.63222i
\(267\) 0 0
\(268\) −309.751 −1.15579
\(269\) 263.792i 0.980641i 0.871542 + 0.490320i \(0.163120\pi\)
−0.871542 + 0.490320i \(0.836880\pi\)
\(270\) 0 0
\(271\) −61.5576 −0.227150 −0.113575 0.993529i \(-0.536230\pi\)
−0.113575 + 0.993529i \(0.536230\pi\)
\(272\) 18.1510i 0.0667317i
\(273\) 0 0
\(274\) 625.041 2.28117
\(275\) − 33.4378i − 0.121592i
\(276\) 0 0
\(277\) −494.241 −1.78426 −0.892132 0.451774i \(-0.850791\pi\)
−0.892132 + 0.451774i \(0.850791\pi\)
\(278\) − 883.726i − 3.17887i
\(279\) 0 0
\(280\) −649.127 −2.31831
\(281\) 329.072i 1.17107i 0.810646 + 0.585537i \(0.199116\pi\)
−0.810646 + 0.585537i \(0.800884\pi\)
\(282\) 0 0
\(283\) −66.0556 −0.233412 −0.116706 0.993167i \(-0.537234\pi\)
−0.116706 + 0.993167i \(0.537234\pi\)
\(284\) − 274.480i − 0.966478i
\(285\) 0 0
\(286\) −55.5017 −0.194062
\(287\) 363.224i 1.26559i
\(288\) 0 0
\(289\) 288.654 0.998802
\(290\) − 436.247i − 1.50430i
\(291\) 0 0
\(292\) 6.29987 0.0215749
\(293\) 369.480i 1.26102i 0.776180 + 0.630512i \(0.217155\pi\)
−0.776180 + 0.630512i \(0.782845\pi\)
\(294\) 0 0
\(295\) 236.209 0.800707
\(296\) 100.897i 0.340868i
\(297\) 0 0
\(298\) 365.714 1.22723
\(299\) 116.592i 0.389941i
\(300\) 0 0
\(301\) 292.822 0.972832
\(302\) 947.283i 3.13670i
\(303\) 0 0
\(304\) −917.099 −3.01677
\(305\) − 294.921i − 0.966955i
\(306\) 0 0
\(307\) 162.715 0.530016 0.265008 0.964246i \(-0.414625\pi\)
0.265008 + 0.964246i \(0.414625\pi\)
\(308\) − 278.017i − 0.902653i
\(309\) 0 0
\(310\) −168.701 −0.544197
\(311\) − 308.574i − 0.992201i −0.868265 0.496100i \(-0.834765\pi\)
0.868265 0.496100i \(-0.165235\pi\)
\(312\) 0 0
\(313\) 136.494 0.436083 0.218041 0.975940i \(-0.430033\pi\)
0.218041 + 0.975940i \(0.430033\pi\)
\(314\) 438.263i 1.39574i
\(315\) 0 0
\(316\) 659.931 2.08839
\(317\) − 145.648i − 0.459457i −0.973255 0.229729i \(-0.926216\pi\)
0.973255 0.229729i \(-0.0737839\pi\)
\(318\) 0 0
\(319\) 104.993 0.329132
\(320\) 47.5137i 0.148480i
\(321\) 0 0
\(322\) −839.874 −2.60830
\(323\) − 17.4883i − 0.0541433i
\(324\) 0 0
\(325\) 44.4468 0.136759
\(326\) 274.071i 0.840708i
\(327\) 0 0
\(328\) 752.937 2.29554
\(329\) 370.713i 1.12679i
\(330\) 0 0
\(331\) −133.660 −0.403806 −0.201903 0.979406i \(-0.564713\pi\)
−0.201903 + 0.979406i \(0.564713\pi\)
\(332\) 191.873i 0.577930i
\(333\) 0 0
\(334\) 512.933 1.53573
\(335\) − 132.035i − 0.394135i
\(336\) 0 0
\(337\) 43.6985 0.129669 0.0648346 0.997896i \(-0.479348\pi\)
0.0648346 + 0.997896i \(0.479348\pi\)
\(338\) 538.627i 1.59357i
\(339\) 0 0
\(340\) −20.9091 −0.0614975
\(341\) − 40.6019i − 0.119067i
\(342\) 0 0
\(343\) 157.383 0.458842
\(344\) − 607.000i − 1.76454i
\(345\) 0 0
\(346\) 474.882 1.37249
\(347\) 142.865i 0.411715i 0.978582 + 0.205858i \(0.0659984\pi\)
−0.978582 + 0.205858i \(0.934002\pi\)
\(348\) 0 0
\(349\) 404.034 1.15769 0.578845 0.815438i \(-0.303504\pi\)
0.578845 + 0.815438i \(0.303504\pi\)
\(350\) 320.172i 0.914778i
\(351\) 0 0
\(352\) −127.039 −0.360906
\(353\) 212.275i 0.601345i 0.953727 + 0.300672i \(0.0972111\pi\)
−0.953727 + 0.300672i \(0.902789\pi\)
\(354\) 0 0
\(355\) 117.000 0.329579
\(356\) 1273.72i 3.57785i
\(357\) 0 0
\(358\) 650.147 1.81605
\(359\) − 107.708i − 0.300024i −0.988684 0.150012i \(-0.952069\pi\)
0.988684 0.150012i \(-0.0479312\pi\)
\(360\) 0 0
\(361\) 522.613 1.44768
\(362\) − 1228.66i − 3.39408i
\(363\) 0 0
\(364\) 369.551 1.01525
\(365\) 2.68540i 0.00735725i
\(366\) 0 0
\(367\) 60.3564 0.164459 0.0822295 0.996613i \(-0.473796\pi\)
0.0822295 + 0.996613i \(0.473796\pi\)
\(368\) 797.216i 2.16635i
\(369\) 0 0
\(370\) −76.5364 −0.206855
\(371\) 150.906i 0.406755i
\(372\) 0 0
\(373\) −288.400 −0.773191 −0.386596 0.922249i \(-0.626349\pi\)
−0.386596 + 0.922249i \(0.626349\pi\)
\(374\) − 7.23674i − 0.0193496i
\(375\) 0 0
\(376\) 768.462 2.04378
\(377\) 139.561i 0.370188i
\(378\) 0 0
\(379\) 369.425 0.974735 0.487368 0.873197i \(-0.337957\pi\)
0.487368 + 0.873197i \(0.337957\pi\)
\(380\) − 1056.45i − 2.78014i
\(381\) 0 0
\(382\) 1138.94 2.98151
\(383\) 209.087i 0.545918i 0.962026 + 0.272959i \(0.0880023\pi\)
−0.962026 + 0.272959i \(0.911998\pi\)
\(384\) 0 0
\(385\) 118.508 0.307814
\(386\) 548.916i 1.42206i
\(387\) 0 0
\(388\) 296.396 0.763908
\(389\) 686.324i 1.76433i 0.470942 + 0.882164i \(0.343914\pi\)
−0.470942 + 0.882164i \(0.656086\pi\)
\(390\) 0 0
\(391\) −15.2022 −0.0388804
\(392\) 584.831i 1.49192i
\(393\) 0 0
\(394\) −50.2312 −0.127490
\(395\) 281.304i 0.712161i
\(396\) 0 0
\(397\) 178.613 0.449906 0.224953 0.974370i \(-0.427777\pi\)
0.224953 + 0.974370i \(0.427777\pi\)
\(398\) 1274.95i 3.20339i
\(399\) 0 0
\(400\) 303.911 0.759776
\(401\) 26.1163i 0.0651279i 0.999470 + 0.0325640i \(0.0103673\pi\)
−0.999470 + 0.0325640i \(0.989633\pi\)
\(402\) 0 0
\(403\) 53.9696 0.133919
\(404\) − 120.711i − 0.298790i
\(405\) 0 0
\(406\) −1005.33 −2.47617
\(407\) − 18.4203i − 0.0452587i
\(408\) 0 0
\(409\) 194.587 0.475763 0.237882 0.971294i \(-0.423547\pi\)
0.237882 + 0.971294i \(0.423547\pi\)
\(410\) 571.149i 1.39305i
\(411\) 0 0
\(412\) −976.203 −2.36943
\(413\) − 544.340i − 1.31801i
\(414\) 0 0
\(415\) −81.7880 −0.197080
\(416\) − 168.865i − 0.405926i
\(417\) 0 0
\(418\) 365.644 0.874745
\(419\) − 138.890i − 0.331481i −0.986169 0.165740i \(-0.946999\pi\)
0.986169 0.165740i \(-0.0530014\pi\)
\(420\) 0 0
\(421\) 582.377 1.38332 0.691659 0.722224i \(-0.256880\pi\)
0.691659 + 0.722224i \(0.256880\pi\)
\(422\) 700.045i 1.65887i
\(423\) 0 0
\(424\) 312.818 0.737778
\(425\) 5.79531i 0.0136360i
\(426\) 0 0
\(427\) −679.643 −1.59167
\(428\) 90.1996i 0.210747i
\(429\) 0 0
\(430\) 460.447 1.07081
\(431\) − 489.076i − 1.13475i −0.823461 0.567373i \(-0.807960\pi\)
0.823461 0.567373i \(-0.192040\pi\)
\(432\) 0 0
\(433\) −684.180 −1.58009 −0.790046 0.613047i \(-0.789944\pi\)
−0.790046 + 0.613047i \(0.789944\pi\)
\(434\) 388.770i 0.895783i
\(435\) 0 0
\(436\) −67.4537 −0.154710
\(437\) − 768.107i − 1.75768i
\(438\) 0 0
\(439\) −350.343 −0.798048 −0.399024 0.916941i \(-0.630651\pi\)
−0.399024 + 0.916941i \(0.630651\pi\)
\(440\) − 245.659i − 0.558316i
\(441\) 0 0
\(442\) 9.61935 0.0217632
\(443\) − 419.851i − 0.947745i −0.880593 0.473873i \(-0.842856\pi\)
0.880593 0.473873i \(-0.157144\pi\)
\(444\) 0 0
\(445\) −542.937 −1.22008
\(446\) − 1132.59i − 2.53943i
\(447\) 0 0
\(448\) 109.495 0.244408
\(449\) 338.737i 0.754426i 0.926126 + 0.377213i \(0.123118\pi\)
−0.926126 + 0.377213i \(0.876882\pi\)
\(450\) 0 0
\(451\) −137.460 −0.304790
\(452\) − 1325.58i − 2.93271i
\(453\) 0 0
\(454\) 404.415 0.890781
\(455\) 157.526i 0.346210i
\(456\) 0 0
\(457\) 401.316 0.878153 0.439076 0.898450i \(-0.355306\pi\)
0.439076 + 0.898450i \(0.355306\pi\)
\(458\) − 448.133i − 0.978456i
\(459\) 0 0
\(460\) −918.355 −1.99642
\(461\) − 187.767i − 0.407303i −0.979043 0.203651i \(-0.934719\pi\)
0.979043 0.203651i \(-0.0652809\pi\)
\(462\) 0 0
\(463\) 489.118 1.05641 0.528205 0.849117i \(-0.322865\pi\)
0.528205 + 0.849117i \(0.322865\pi\)
\(464\) 954.265i 2.05661i
\(465\) 0 0
\(466\) −1185.75 −2.54453
\(467\) − 371.837i − 0.796225i −0.917337 0.398112i \(-0.869665\pi\)
0.917337 0.398112i \(-0.130335\pi\)
\(468\) 0 0
\(469\) −304.274 −0.648772
\(470\) 582.925i 1.24027i
\(471\) 0 0
\(472\) −1128.38 −2.39063
\(473\) 110.817i 0.234286i
\(474\) 0 0
\(475\) −292.814 −0.616450
\(476\) 48.1849i 0.101229i
\(477\) 0 0
\(478\) 60.5427 0.126658
\(479\) 733.351i 1.53100i 0.643433 + 0.765502i \(0.277509\pi\)
−0.643433 + 0.765502i \(0.722491\pi\)
\(480\) 0 0
\(481\) 24.4850 0.0509043
\(482\) − 307.132i − 0.637203i
\(483\) 0 0
\(484\) −999.645 −2.06538
\(485\) 126.343i 0.260500i
\(486\) 0 0
\(487\) 248.594 0.510459 0.255229 0.966881i \(-0.417849\pi\)
0.255229 + 0.966881i \(0.417849\pi\)
\(488\) 1408.85i 2.88699i
\(489\) 0 0
\(490\) −443.630 −0.905368
\(491\) 324.038i 0.659955i 0.943989 + 0.329977i \(0.107041\pi\)
−0.943989 + 0.329977i \(0.892959\pi\)
\(492\) 0 0
\(493\) −18.1970 −0.0369108
\(494\) 486.027i 0.983861i
\(495\) 0 0
\(496\) 369.024 0.744000
\(497\) − 269.626i − 0.542507i
\(498\) 0 0
\(499\) −682.389 −1.36751 −0.683756 0.729710i \(-0.739655\pi\)
−0.683756 + 0.729710i \(0.739655\pi\)
\(500\) 1238.60i 2.47719i
\(501\) 0 0
\(502\) 481.381 0.958927
\(503\) 778.156i 1.54703i 0.633779 + 0.773515i \(0.281503\pi\)
−0.633779 + 0.773515i \(0.718497\pi\)
\(504\) 0 0
\(505\) 51.4546 0.101890
\(506\) − 317.847i − 0.628156i
\(507\) 0 0
\(508\) −102.902 −0.202563
\(509\) − 147.442i − 0.289669i −0.989456 0.144835i \(-0.953735\pi\)
0.989456 0.144835i \(-0.0462650\pi\)
\(510\) 0 0
\(511\) 6.18847 0.0121105
\(512\) 1139.94i 2.22645i
\(513\) 0 0
\(514\) 1033.11 2.00995
\(515\) − 416.119i − 0.807997i
\(516\) 0 0
\(517\) −140.295 −0.271363
\(518\) 176.377i 0.340497i
\(519\) 0 0
\(520\) 326.540 0.627961
\(521\) 361.413i 0.693690i 0.937922 + 0.346845i \(0.112747\pi\)
−0.937922 + 0.346845i \(0.887253\pi\)
\(522\) 0 0
\(523\) −19.7777 −0.0378159 −0.0189079 0.999821i \(-0.506019\pi\)
−0.0189079 + 0.999821i \(0.506019\pi\)
\(524\) − 1165.43i − 2.22410i
\(525\) 0 0
\(526\) 1086.09 2.06480
\(527\) 7.03697i 0.0133529i
\(528\) 0 0
\(529\) −138.700 −0.262194
\(530\) 237.292i 0.447720i
\(531\) 0 0
\(532\) −2434.59 −4.57630
\(533\) − 182.718i − 0.342810i
\(534\) 0 0
\(535\) −38.4487 −0.0718667
\(536\) 630.739i 1.17675i
\(537\) 0 0
\(538\) 955.899 1.77676
\(539\) − 106.770i − 0.198089i
\(540\) 0 0
\(541\) −663.746 −1.22689 −0.613444 0.789739i \(-0.710216\pi\)
−0.613444 + 0.789739i \(0.710216\pi\)
\(542\) 223.065i 0.411559i
\(543\) 0 0
\(544\) 22.0179 0.0404741
\(545\) − 28.7529i − 0.0527577i
\(546\) 0 0
\(547\) 171.872 0.314209 0.157105 0.987582i \(-0.449784\pi\)
0.157105 + 0.987582i \(0.449784\pi\)
\(548\) − 1575.00i − 2.87408i
\(549\) 0 0
\(550\) −121.168 −0.220305
\(551\) − 919.422i − 1.66864i
\(552\) 0 0
\(553\) 648.262 1.17226
\(554\) 1790.97i 3.23280i
\(555\) 0 0
\(556\) −2226.84 −4.00511
\(557\) − 200.437i − 0.359852i −0.983680 0.179926i \(-0.942414\pi\)
0.983680 0.179926i \(-0.0575858\pi\)
\(558\) 0 0
\(559\) −147.303 −0.263511
\(560\) 1077.10i 1.92340i
\(561\) 0 0
\(562\) 1192.45 2.12180
\(563\) − 753.744i − 1.33880i −0.742902 0.669400i \(-0.766551\pi\)
0.742902 0.669400i \(-0.233449\pi\)
\(564\) 0 0
\(565\) 565.047 1.00008
\(566\) 239.364i 0.422905i
\(567\) 0 0
\(568\) −558.916 −0.984007
\(569\) − 85.1626i − 0.149671i −0.997196 0.0748354i \(-0.976157\pi\)
0.997196 0.0748354i \(-0.0238431\pi\)
\(570\) 0 0
\(571\) −826.091 −1.44674 −0.723372 0.690458i \(-0.757409\pi\)
−0.723372 + 0.690458i \(0.757409\pi\)
\(572\) 139.855i 0.244502i
\(573\) 0 0
\(574\) 1316.21 2.29304
\(575\) 254.537i 0.442674i
\(576\) 0 0
\(577\) 253.424 0.439209 0.219605 0.975589i \(-0.429523\pi\)
0.219605 + 0.975589i \(0.429523\pi\)
\(578\) − 1045.99i − 1.80967i
\(579\) 0 0
\(580\) −1099.27 −1.89529
\(581\) 188.480i 0.324406i
\(582\) 0 0
\(583\) −57.1098 −0.0979585
\(584\) − 12.8283i − 0.0219662i
\(585\) 0 0
\(586\) 1338.88 2.28477
\(587\) 385.659i 0.657000i 0.944504 + 0.328500i \(0.106543\pi\)
−0.944504 + 0.328500i \(0.893457\pi\)
\(588\) 0 0
\(589\) −355.550 −0.603650
\(590\) − 855.945i − 1.45075i
\(591\) 0 0
\(592\) 167.419 0.282803
\(593\) − 809.736i − 1.36549i −0.730656 0.682745i \(-0.760786\pi\)
0.730656 0.682745i \(-0.239214\pi\)
\(594\) 0 0
\(595\) −20.5394 −0.0345200
\(596\) − 921.537i − 1.54620i
\(597\) 0 0
\(598\) 422.494 0.706511
\(599\) 29.2794i 0.0488805i 0.999701 + 0.0244402i \(0.00778034\pi\)
−0.999701 + 0.0244402i \(0.992220\pi\)
\(600\) 0 0
\(601\) −934.241 −1.55448 −0.777239 0.629206i \(-0.783380\pi\)
−0.777239 + 0.629206i \(0.783380\pi\)
\(602\) − 1061.09i − 1.76262i
\(603\) 0 0
\(604\) 2386.99 3.95198
\(605\) − 426.111i − 0.704315i
\(606\) 0 0
\(607\) −531.512 −0.875638 −0.437819 0.899063i \(-0.644249\pi\)
−0.437819 + 0.899063i \(0.644249\pi\)
\(608\) 1112.48i 1.82973i
\(609\) 0 0
\(610\) −1068.70 −1.75197
\(611\) − 186.485i − 0.305213i
\(612\) 0 0
\(613\) −259.694 −0.423645 −0.211822 0.977308i \(-0.567940\pi\)
−0.211822 + 0.977308i \(0.567940\pi\)
\(614\) − 589.627i − 0.960305i
\(615\) 0 0
\(616\) −566.119 −0.919025
\(617\) 702.234i 1.13814i 0.822288 + 0.569071i \(0.192697\pi\)
−0.822288 + 0.569071i \(0.807303\pi\)
\(618\) 0 0
\(619\) −1151.24 −1.85985 −0.929923 0.367755i \(-0.880127\pi\)
−0.929923 + 0.367755i \(0.880127\pi\)
\(620\) 425.098i 0.685642i
\(621\) 0 0
\(622\) −1118.18 −1.79771
\(623\) 1251.19i 2.00833i
\(624\) 0 0
\(625\) −281.703 −0.450725
\(626\) − 494.611i − 0.790113i
\(627\) 0 0
\(628\) 1104.35 1.75852
\(629\) 3.19254i 0.00507558i
\(630\) 0 0
\(631\) −873.923 −1.38498 −0.692490 0.721427i \(-0.743486\pi\)
−0.692490 + 0.721427i \(0.743486\pi\)
\(632\) − 1343.80i − 2.12627i
\(633\) 0 0
\(634\) −527.782 −0.832464
\(635\) − 43.8632i − 0.0690759i
\(636\) 0 0
\(637\) 141.923 0.222799
\(638\) − 380.461i − 0.596335i
\(639\) 0 0
\(640\) −410.490 −0.641391
\(641\) − 614.060i − 0.957973i −0.877822 0.478986i \(-0.841004\pi\)
0.877822 0.478986i \(-0.158996\pi\)
\(642\) 0 0
\(643\) 299.770 0.466205 0.233103 0.972452i \(-0.425112\pi\)
0.233103 + 0.972452i \(0.425112\pi\)
\(644\) 2116.34i 3.28624i
\(645\) 0 0
\(646\) −63.3720 −0.0980990
\(647\) − 976.042i − 1.50857i −0.656549 0.754283i \(-0.727985\pi\)
0.656549 0.754283i \(-0.272015\pi\)
\(648\) 0 0
\(649\) 206.003 0.317416
\(650\) − 161.061i − 0.247786i
\(651\) 0 0
\(652\) 690.613 1.05922
\(653\) 1196.90i 1.83293i 0.400116 + 0.916465i \(0.368970\pi\)
−0.400116 + 0.916465i \(0.631030\pi\)
\(654\) 0 0
\(655\) 496.779 0.758441
\(656\) − 1249.36i − 1.90451i
\(657\) 0 0
\(658\) 1343.34 2.04156
\(659\) 225.322i 0.341916i 0.985278 + 0.170958i \(0.0546862\pi\)
−0.985278 + 0.170958i \(0.945314\pi\)
\(660\) 0 0
\(661\) −1119.76 −1.69405 −0.847023 0.531556i \(-0.821608\pi\)
−0.847023 + 0.531556i \(0.821608\pi\)
\(662\) 484.340i 0.731632i
\(663\) 0 0
\(664\) 390.705 0.588412
\(665\) − 1037.77i − 1.56056i
\(666\) 0 0
\(667\) −799.235 −1.19825
\(668\) − 1292.50i − 1.93489i
\(669\) 0 0
\(670\) −478.454 −0.714110
\(671\) − 257.208i − 0.383320i
\(672\) 0 0
\(673\) 526.066 0.781673 0.390836 0.920460i \(-0.372186\pi\)
0.390836 + 0.920460i \(0.372186\pi\)
\(674\) − 158.349i − 0.234940i
\(675\) 0 0
\(676\) 1357.25 2.00777
\(677\) 277.685i 0.410170i 0.978744 + 0.205085i \(0.0657471\pi\)
−0.978744 + 0.205085i \(0.934253\pi\)
\(678\) 0 0
\(679\) 291.155 0.428800
\(680\) 42.5767i 0.0626129i
\(681\) 0 0
\(682\) −147.128 −0.215731
\(683\) 1275.87i 1.86804i 0.357227 + 0.934018i \(0.383722\pi\)
−0.357227 + 0.934018i \(0.616278\pi\)
\(684\) 0 0
\(685\) 671.362 0.980090
\(686\) − 570.305i − 0.831348i
\(687\) 0 0
\(688\) −1007.20 −1.46396
\(689\) − 75.9125i − 0.110178i
\(690\) 0 0
\(691\) −1181.70 −1.71013 −0.855067 0.518518i \(-0.826484\pi\)
−0.855067 + 0.518518i \(0.826484\pi\)
\(692\) − 1196.62i − 1.72922i
\(693\) 0 0
\(694\) 517.698 0.745963
\(695\) − 949.218i − 1.36578i
\(696\) 0 0
\(697\) 23.8241 0.0341810
\(698\) − 1464.09i − 2.09755i
\(699\) 0 0
\(700\) 806.780 1.15254
\(701\) − 1082.89i − 1.54478i −0.635148 0.772390i \(-0.719061\pi\)
0.635148 0.772390i \(-0.280939\pi\)
\(702\) 0 0
\(703\) −161.306 −0.229454
\(704\) 41.4378i 0.0588606i
\(705\) 0 0
\(706\) 769.216 1.08954
\(707\) − 118.577i − 0.167718i
\(708\) 0 0
\(709\) −208.077 −0.293480 −0.146740 0.989175i \(-0.546878\pi\)
−0.146740 + 0.989175i \(0.546878\pi\)
\(710\) − 423.972i − 0.597144i
\(711\) 0 0
\(712\) 2593.63 3.64275
\(713\) 309.072i 0.433482i
\(714\) 0 0
\(715\) −59.6149 −0.0833775
\(716\) − 1638.26i − 2.28808i
\(717\) 0 0
\(718\) −390.301 −0.543595
\(719\) 1011.36i 1.40662i 0.710884 + 0.703309i \(0.248295\pi\)
−0.710884 + 0.703309i \(0.751705\pi\)
\(720\) 0 0
\(721\) −958.941 −1.33002
\(722\) − 1893.78i − 2.62297i
\(723\) 0 0
\(724\) −3096.00 −4.27625
\(725\) 304.680i 0.420249i
\(726\) 0 0
\(727\) 774.643 1.06553 0.532767 0.846262i \(-0.321152\pi\)
0.532767 + 0.846262i \(0.321152\pi\)
\(728\) − 752.507i − 1.03366i
\(729\) 0 0
\(730\) 9.73102 0.0133302
\(731\) − 19.2065i − 0.0262742i
\(732\) 0 0
\(733\) −1018.11 −1.38896 −0.694480 0.719512i \(-0.744366\pi\)
−0.694480 + 0.719512i \(0.744366\pi\)
\(734\) − 218.713i − 0.297973i
\(735\) 0 0
\(736\) 967.055 1.31393
\(737\) − 115.151i − 0.156243i
\(738\) 0 0
\(739\) −857.914 −1.16091 −0.580456 0.814292i \(-0.697126\pi\)
−0.580456 + 0.814292i \(0.697126\pi\)
\(740\) 192.859i 0.260620i
\(741\) 0 0
\(742\) 546.836 0.736976
\(743\) − 1216.92i − 1.63784i −0.573906 0.818921i \(-0.694572\pi\)
0.573906 0.818921i \(-0.305428\pi\)
\(744\) 0 0
\(745\) 392.816 0.527270
\(746\) 1045.07i 1.40090i
\(747\) 0 0
\(748\) −18.2354 −0.0243788
\(749\) 88.6046i 0.118297i
\(750\) 0 0
\(751\) −50.4386 −0.0671619 −0.0335809 0.999436i \(-0.510691\pi\)
−0.0335809 + 0.999436i \(0.510691\pi\)
\(752\) − 1275.12i − 1.69563i
\(753\) 0 0
\(754\) 505.724 0.670721
\(755\) 1017.49i 1.34766i
\(756\) 0 0
\(757\) 1139.84 1.50573 0.752864 0.658176i \(-0.228672\pi\)
0.752864 + 0.658176i \(0.228672\pi\)
\(758\) − 1338.68i − 1.76607i
\(759\) 0 0
\(760\) −2151.23 −2.83057
\(761\) − 672.607i − 0.883847i −0.897053 0.441923i \(-0.854296\pi\)
0.897053 0.441923i \(-0.145704\pi\)
\(762\) 0 0
\(763\) −66.2609 −0.0868426
\(764\) − 2869.93i − 3.75645i
\(765\) 0 0
\(766\) 757.663 0.989116
\(767\) 273.827i 0.357011i
\(768\) 0 0
\(769\) −742.811 −0.965944 −0.482972 0.875636i \(-0.660443\pi\)
−0.482972 + 0.875636i \(0.660443\pi\)
\(770\) − 429.436i − 0.557709i
\(771\) 0 0
\(772\) 1383.18 1.79168
\(773\) − 1222.52i − 1.58152i −0.612124 0.790762i \(-0.709685\pi\)
0.612124 0.790762i \(-0.290315\pi\)
\(774\) 0 0
\(775\) 117.823 0.152030
\(776\) − 603.544i − 0.777763i
\(777\) 0 0
\(778\) 2487.02 3.19668
\(779\) 1203.74i 1.54523i
\(780\) 0 0
\(781\) 102.039 0.130652
\(782\) 55.0880i 0.0704450i
\(783\) 0 0
\(784\) 970.416 1.23778
\(785\) 470.742i 0.599672i
\(786\) 0 0
\(787\) −337.342 −0.428643 −0.214321 0.976763i \(-0.568754\pi\)
−0.214321 + 0.976763i \(0.568754\pi\)
\(788\) 126.574i 0.160627i
\(789\) 0 0
\(790\) 1019.36 1.29032
\(791\) − 1302.14i − 1.64620i
\(792\) 0 0
\(793\) 341.891 0.431136
\(794\) − 647.235i − 0.815158i
\(795\) 0 0
\(796\) 3212.66 4.03600
\(797\) − 582.790i − 0.731230i −0.930766 0.365615i \(-0.880859\pi\)
0.930766 0.365615i \(-0.119141\pi\)
\(798\) 0 0
\(799\) 24.3153 0.0304322
\(800\) − 368.656i − 0.460820i
\(801\) 0 0
\(802\) 94.6371 0.118001
\(803\) 2.34200i 0.00291656i
\(804\) 0 0
\(805\) −902.116 −1.12064
\(806\) − 195.568i − 0.242641i
\(807\) 0 0
\(808\) −245.801 −0.304209
\(809\) − 850.348i − 1.05111i −0.850759 0.525555i \(-0.823858\pi\)
0.850759 0.525555i \(-0.176142\pi\)
\(810\) 0 0
\(811\) 513.085 0.632658 0.316329 0.948650i \(-0.397550\pi\)
0.316329 + 0.948650i \(0.397550\pi\)
\(812\) 2533.25i 3.11977i
\(813\) 0 0
\(814\) −66.7493 −0.0820016
\(815\) 294.382i 0.361205i
\(816\) 0 0
\(817\) 970.425 1.18779
\(818\) − 705.122i − 0.862007i
\(819\) 0 0
\(820\) 1439.20 1.75512
\(821\) − 507.426i − 0.618058i −0.951053 0.309029i \(-0.899996\pi\)
0.951053 0.309029i \(-0.100004\pi\)
\(822\) 0 0
\(823\) −715.645 −0.869557 −0.434779 0.900537i \(-0.643173\pi\)
−0.434779 + 0.900537i \(0.643173\pi\)
\(824\) 1987.82i 2.41240i
\(825\) 0 0
\(826\) −1972.52 −2.38803
\(827\) − 1010.84i − 1.22230i −0.791516 0.611149i \(-0.790708\pi\)
0.791516 0.611149i \(-0.209292\pi\)
\(828\) 0 0
\(829\) −497.757 −0.600431 −0.300215 0.953871i \(-0.597059\pi\)
−0.300215 + 0.953871i \(0.597059\pi\)
\(830\) 296.374i 0.357077i
\(831\) 0 0
\(832\) −55.0808 −0.0662028
\(833\) 18.5050i 0.0222149i
\(834\) 0 0
\(835\) 550.946 0.659815
\(836\) − 921.360i − 1.10211i
\(837\) 0 0
\(838\) −503.295 −0.600591
\(839\) 918.153i 1.09434i 0.837021 + 0.547171i \(0.184295\pi\)
−0.837021 + 0.547171i \(0.815705\pi\)
\(840\) 0 0
\(841\) −115.682 −0.137553
\(842\) − 2110.35i − 2.50635i
\(843\) 0 0
\(844\) 1764.00 2.09004
\(845\) 578.544i 0.684668i
\(846\) 0 0
\(847\) −981.968 −1.15935
\(848\) − 519.062i − 0.612101i
\(849\) 0 0
\(850\) 21.0004 0.0247063
\(851\) 140.220i 0.164771i
\(852\) 0 0
\(853\) 1103.31 1.29344 0.646720 0.762727i \(-0.276140\pi\)
0.646720 + 0.762727i \(0.276140\pi\)
\(854\) 2462.81i 2.88385i
\(855\) 0 0
\(856\) 183.671 0.214569
\(857\) 785.899i 0.917036i 0.888685 + 0.458518i \(0.151619\pi\)
−0.888685 + 0.458518i \(0.848381\pi\)
\(858\) 0 0
\(859\) −395.687 −0.460637 −0.230319 0.973115i \(-0.573977\pi\)
−0.230319 + 0.973115i \(0.573977\pi\)
\(860\) − 1160.25i − 1.34913i
\(861\) 0 0
\(862\) −1772.26 −2.05598
\(863\) − 896.554i − 1.03888i −0.854507 0.519440i \(-0.826140\pi\)
0.854507 0.519440i \(-0.173860\pi\)
\(864\) 0 0
\(865\) 510.075 0.589682
\(866\) 2479.25i 2.86288i
\(867\) 0 0
\(868\) 979.635 1.12861
\(869\) 245.332i 0.282315i
\(870\) 0 0
\(871\) 153.063 0.175733
\(872\) 137.354i 0.157516i
\(873\) 0 0
\(874\) −2783.38 −3.18464
\(875\) 1216.69i 1.39051i
\(876\) 0 0
\(877\) −1390.20 −1.58518 −0.792590 0.609755i \(-0.791268\pi\)
−0.792590 + 0.609755i \(0.791268\pi\)
\(878\) 1269.53i 1.44594i
\(879\) 0 0
\(880\) −407.625 −0.463210
\(881\) 1509.07i 1.71290i 0.516228 + 0.856451i \(0.327336\pi\)
−0.516228 + 0.856451i \(0.672664\pi\)
\(882\) 0 0
\(883\) −409.196 −0.463416 −0.231708 0.972785i \(-0.574431\pi\)
−0.231708 + 0.972785i \(0.574431\pi\)
\(884\) − 24.2391i − 0.0274198i
\(885\) 0 0
\(886\) −1521.41 −1.71716
\(887\) 201.476i 0.227143i 0.993530 + 0.113572i \(0.0362292\pi\)
−0.993530 + 0.113572i \(0.963771\pi\)
\(888\) 0 0
\(889\) −101.082 −0.113703
\(890\) 1967.43i 2.21060i
\(891\) 0 0
\(892\) −2853.93 −3.19947
\(893\) 1228.56i 1.37576i
\(894\) 0 0
\(895\) 698.329 0.780256
\(896\) 945.971i 1.05577i
\(897\) 0 0
\(898\) 1227.48 1.36690
\(899\) 369.959i 0.411522i
\(900\) 0 0
\(901\) 9.89806 0.0109856
\(902\) 498.113i 0.552232i
\(903\) 0 0
\(904\) −2699.25 −2.98590
\(905\) − 1319.71i − 1.45824i
\(906\) 0 0
\(907\) 1740.01 1.91843 0.959213 0.282683i \(-0.0912244\pi\)
0.959213 + 0.282683i \(0.0912244\pi\)
\(908\) − 1019.06i − 1.12231i
\(909\) 0 0
\(910\) 570.823 0.627278
\(911\) − 1651.26i − 1.81258i −0.422662 0.906288i \(-0.638904\pi\)
0.422662 0.906288i \(-0.361096\pi\)
\(912\) 0 0
\(913\) −71.3294 −0.0781264
\(914\) − 1454.24i − 1.59107i
\(915\) 0 0
\(916\) −1129.22 −1.23277
\(917\) − 1144.82i − 1.24844i
\(918\) 0 0
\(919\) −157.497 −0.171378 −0.0856892 0.996322i \(-0.527309\pi\)
−0.0856892 + 0.996322i \(0.527309\pi\)
\(920\) 1870.02i 2.03263i
\(921\) 0 0
\(922\) −680.406 −0.737968
\(923\) 135.634i 0.146949i
\(924\) 0 0
\(925\) 53.4540 0.0577881
\(926\) − 1772.41i − 1.91405i
\(927\) 0 0
\(928\) 1157.56 1.24737
\(929\) 262.416i 0.282472i 0.989976 + 0.141236i \(0.0451076\pi\)
−0.989976 + 0.141236i \(0.954892\pi\)
\(930\) 0 0
\(931\) −934.983 −1.00428
\(932\) 2987.89i 3.20589i
\(933\) 0 0
\(934\) −1347.42 −1.44263
\(935\) − 7.77305i − 0.00831342i
\(936\) 0 0
\(937\) −47.0928 −0.0502591 −0.0251295 0.999684i \(-0.508000\pi\)
−0.0251295 + 0.999684i \(0.508000\pi\)
\(938\) 1102.59i 1.17547i
\(939\) 0 0
\(940\) 1468.87 1.56263
\(941\) − 362.486i − 0.385213i −0.981276 0.192607i \(-0.938306\pi\)
0.981276 0.192607i \(-0.0616941\pi\)
\(942\) 0 0
\(943\) 1046.39 1.10963
\(944\) 1872.33i 1.98340i
\(945\) 0 0
\(946\) 401.567 0.424489
\(947\) 775.594i 0.819001i 0.912310 + 0.409500i \(0.134297\pi\)
−0.912310 + 0.409500i \(0.865703\pi\)
\(948\) 0 0
\(949\) −3.11307 −0.00328037
\(950\) 1061.06i 1.11691i
\(951\) 0 0
\(952\) 98.1177 0.103065
\(953\) 1296.53i 1.36047i 0.732994 + 0.680235i \(0.238122\pi\)
−0.732994 + 0.680235i \(0.761878\pi\)
\(954\) 0 0
\(955\) 1223.34 1.28099
\(956\) − 152.557i − 0.159579i
\(957\) 0 0
\(958\) 2657.43 2.77394
\(959\) − 1547.15i − 1.61329i
\(960\) 0 0
\(961\) −817.933 −0.851127
\(962\) − 88.7257i − 0.0922304i
\(963\) 0 0
\(964\) −773.920 −0.802822
\(965\) 589.596i 0.610980i
\(966\) 0 0
\(967\) 1033.76 1.06904 0.534521 0.845155i \(-0.320492\pi\)
0.534521 + 0.845155i \(0.320492\pi\)
\(968\) 2035.55i 2.10284i
\(969\) 0 0
\(970\) 457.825 0.471985
\(971\) 1439.64i 1.48263i 0.671155 + 0.741317i \(0.265799\pi\)
−0.671155 + 0.741317i \(0.734201\pi\)
\(972\) 0 0
\(973\) −2187.46 −2.24816
\(974\) − 900.824i − 0.924870i
\(975\) 0 0
\(976\) 2337.72 2.39521
\(977\) 1126.35i 1.15287i 0.817144 + 0.576434i \(0.195556\pi\)
−0.817144 + 0.576434i \(0.804444\pi\)
\(978\) 0 0
\(979\) −473.509 −0.483666
\(980\) 1117.87i 1.14069i
\(981\) 0 0
\(982\) 1174.21 1.19573
\(983\) − 2.00328i − 0.00203793i −0.999999 0.00101896i \(-0.999676\pi\)
0.999999 0.00101896i \(-0.000324346\pi\)
\(984\) 0 0
\(985\) −53.9537 −0.0547754
\(986\) 65.9402i 0.0668764i
\(987\) 0 0
\(988\) 1224.71 1.23958
\(989\) − 843.571i − 0.852954i
\(990\) 0 0
\(991\) 835.250 0.842836 0.421418 0.906867i \(-0.361533\pi\)
0.421418 + 0.906867i \(0.361533\pi\)
\(992\) − 447.641i − 0.451251i
\(993\) 0 0
\(994\) −977.039 −0.982937
\(995\) 1369.43i 1.37631i
\(996\) 0 0
\(997\) 878.694 0.881338 0.440669 0.897670i \(-0.354741\pi\)
0.440669 + 0.897670i \(0.354741\pi\)
\(998\) 2472.76i 2.47771i
\(999\) 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 1143.3.b.a.890.6 84
3.2 odd 2 inner 1143.3.b.a.890.79 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.6 84 1.1 even 1 trivial
1143.3.b.a.890.79 yes 84 3.2 odd 2 inner