Properties

Label 7168.2.a.bb.1.1
Level $7168$
Weight $2$
Character 7168.1
Self dual yes
Analytic conductor $57.237$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7168 = 2^{10} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7168.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.2367681689\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.9433055232.1
Defining polynomial: \(x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 9 x^{4} - 76 x^{3} - 4 x^{2} + 48 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1792)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.822080\) of defining polynomial
Character \(\chi\) \(=\) 7168.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.88693 q^{3} -0.992279 q^{5} +1.00000 q^{7} +5.33435 q^{9} +O(q^{10})\) \(q-2.88693 q^{3} -0.992279 q^{5} +1.00000 q^{7} +5.33435 q^{9} +3.42224 q^{11} -2.77325 q^{13} +2.86464 q^{15} +6.93050 q^{17} +1.96503 q^{19} -2.88693 q^{21} -2.05612 q^{23} -4.01538 q^{25} -6.73909 q^{27} -7.55775 q^{29} -5.23708 q^{31} -9.87975 q^{33} -0.992279 q^{35} -9.31119 q^{37} +8.00617 q^{39} +0.949797 q^{41} +8.41942 q^{43} -5.29316 q^{45} +4.64785 q^{47} +1.00000 q^{49} -20.0079 q^{51} -10.2501 q^{53} -3.39582 q^{55} -5.67289 q^{57} +12.1346 q^{59} -3.98105 q^{61} +5.33435 q^{63} +2.75184 q^{65} -12.8297 q^{67} +5.93586 q^{69} +3.60510 q^{71} +6.53179 q^{73} +11.5921 q^{75} +3.42224 q^{77} +13.7819 q^{79} +3.45223 q^{81} -6.35508 q^{83} -6.87699 q^{85} +21.8187 q^{87} -0.428825 q^{89} -2.77325 q^{91} +15.1191 q^{93} -1.94986 q^{95} +14.6339 q^{97} +18.2554 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + 8q^{7} + 12q^{9} + O(q^{10}) \) \( 8q - 8q^{5} + 8q^{7} + 12q^{9} - 12q^{11} - 20q^{13} + 4q^{17} - 4q^{19} + 8q^{23} + 12q^{25} + 12q^{27} - 8q^{29} - 4q^{31} + 8q^{33} - 8q^{35} - 8q^{37} - 16q^{39} - 12q^{41} + 4q^{43} - 52q^{45} + 20q^{47} + 8q^{49} - 32q^{51} - 40q^{53} + 24q^{55} - 4q^{57} - 4q^{59} + 8q^{61} + 12q^{63} + 36q^{65} - 28q^{67} - 4q^{69} - 16q^{71} + 16q^{73} + 28q^{75} - 12q^{77} + 20q^{81} + 8q^{83} - 16q^{85} + 20q^{87} + 16q^{89} - 20q^{91} - 16q^{93} - 40q^{95} - 36q^{97} + 4q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.88693 −1.66677 −0.833384 0.552694i \(-0.813600\pi\)
−0.833384 + 0.552694i \(0.813600\pi\)
\(4\) 0 0
\(5\) −0.992279 −0.443761 −0.221880 0.975074i \(-0.571219\pi\)
−0.221880 + 0.975074i \(0.571219\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 5.33435 1.77812
\(10\) 0 0
\(11\) 3.42224 1.03184 0.515922 0.856636i \(-0.327449\pi\)
0.515922 + 0.856636i \(0.327449\pi\)
\(12\) 0 0
\(13\) −2.77325 −0.769161 −0.384581 0.923091i \(-0.625654\pi\)
−0.384581 + 0.923091i \(0.625654\pi\)
\(14\) 0 0
\(15\) 2.86464 0.739646
\(16\) 0 0
\(17\) 6.93050 1.68089 0.840447 0.541894i \(-0.182293\pi\)
0.840447 + 0.541894i \(0.182293\pi\)
\(18\) 0 0
\(19\) 1.96503 0.450808 0.225404 0.974265i \(-0.427630\pi\)
0.225404 + 0.974265i \(0.427630\pi\)
\(20\) 0 0
\(21\) −2.88693 −0.629979
\(22\) 0 0
\(23\) −2.05612 −0.428730 −0.214365 0.976754i \(-0.568768\pi\)
−0.214365 + 0.976754i \(0.568768\pi\)
\(24\) 0 0
\(25\) −4.01538 −0.803076
\(26\) 0 0
\(27\) −6.73909 −1.29694
\(28\) 0 0
\(29\) −7.55775 −1.40344 −0.701720 0.712453i \(-0.747584\pi\)
−0.701720 + 0.712453i \(0.747584\pi\)
\(30\) 0 0
\(31\) −5.23708 −0.940607 −0.470304 0.882505i \(-0.655856\pi\)
−0.470304 + 0.882505i \(0.655856\pi\)
\(32\) 0 0
\(33\) −9.87975 −1.71984
\(34\) 0 0
\(35\) −0.992279 −0.167726
\(36\) 0 0
\(37\) −9.31119 −1.53075 −0.765375 0.643584i \(-0.777447\pi\)
−0.765375 + 0.643584i \(0.777447\pi\)
\(38\) 0 0
\(39\) 8.00617 1.28201
\(40\) 0 0
\(41\) 0.949797 0.148333 0.0741667 0.997246i \(-0.476370\pi\)
0.0741667 + 0.997246i \(0.476370\pi\)
\(42\) 0 0
\(43\) 8.41942 1.28395 0.641975 0.766726i \(-0.278115\pi\)
0.641975 + 0.766726i \(0.278115\pi\)
\(44\) 0 0
\(45\) −5.29316 −0.789058
\(46\) 0 0
\(47\) 4.64785 0.677959 0.338980 0.940794i \(-0.389918\pi\)
0.338980 + 0.940794i \(0.389918\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −20.0079 −2.80166
\(52\) 0 0
\(53\) −10.2501 −1.40796 −0.703979 0.710221i \(-0.748595\pi\)
−0.703979 + 0.710221i \(0.748595\pi\)
\(54\) 0 0
\(55\) −3.39582 −0.457892
\(56\) 0 0
\(57\) −5.67289 −0.751393
\(58\) 0 0
\(59\) 12.1346 1.57979 0.789897 0.613239i \(-0.210134\pi\)
0.789897 + 0.613239i \(0.210134\pi\)
\(60\) 0 0
\(61\) −3.98105 −0.509721 −0.254860 0.966978i \(-0.582030\pi\)
−0.254860 + 0.966978i \(0.582030\pi\)
\(62\) 0 0
\(63\) 5.33435 0.672065
\(64\) 0 0
\(65\) 2.75184 0.341324
\(66\) 0 0
\(67\) −12.8297 −1.56740 −0.783700 0.621140i \(-0.786670\pi\)
−0.783700 + 0.621140i \(0.786670\pi\)
\(68\) 0 0
\(69\) 5.93586 0.714593
\(70\) 0 0
\(71\) 3.60510 0.427847 0.213923 0.976850i \(-0.431376\pi\)
0.213923 + 0.976850i \(0.431376\pi\)
\(72\) 0 0
\(73\) 6.53179 0.764488 0.382244 0.924061i \(-0.375151\pi\)
0.382244 + 0.924061i \(0.375151\pi\)
\(74\) 0 0
\(75\) 11.5921 1.33854
\(76\) 0 0
\(77\) 3.42224 0.390000
\(78\) 0 0
\(79\) 13.7819 1.55059 0.775293 0.631602i \(-0.217602\pi\)
0.775293 + 0.631602i \(0.217602\pi\)
\(80\) 0 0
\(81\) 3.45223 0.383581
\(82\) 0 0
\(83\) −6.35508 −0.697561 −0.348780 0.937205i \(-0.613404\pi\)
−0.348780 + 0.937205i \(0.613404\pi\)
\(84\) 0 0
\(85\) −6.87699 −0.745915
\(86\) 0 0
\(87\) 21.8187 2.33921
\(88\) 0 0
\(89\) −0.428825 −0.0454554 −0.0227277 0.999742i \(-0.507235\pi\)
−0.0227277 + 0.999742i \(0.507235\pi\)
\(90\) 0 0
\(91\) −2.77325 −0.290716
\(92\) 0 0
\(93\) 15.1191 1.56777
\(94\) 0 0
\(95\) −1.94986 −0.200051
\(96\) 0 0
\(97\) 14.6339 1.48584 0.742922 0.669378i \(-0.233439\pi\)
0.742922 + 0.669378i \(0.233439\pi\)
\(98\) 0 0
\(99\) 18.2554 1.83474
\(100\) 0 0
\(101\) 15.8402 1.57616 0.788078 0.615575i \(-0.211076\pi\)
0.788078 + 0.615575i \(0.211076\pi\)
\(102\) 0 0
\(103\) −18.4673 −1.81964 −0.909819 0.415005i \(-0.863780\pi\)
−0.909819 + 0.415005i \(0.863780\pi\)
\(104\) 0 0
\(105\) 2.86464 0.279560
\(106\) 0 0
\(107\) 0.206717 0.0199841 0.00999205 0.999950i \(-0.496819\pi\)
0.00999205 + 0.999950i \(0.496819\pi\)
\(108\) 0 0
\(109\) −0.437577 −0.0419123 −0.0209561 0.999780i \(-0.506671\pi\)
−0.0209561 + 0.999780i \(0.506671\pi\)
\(110\) 0 0
\(111\) 26.8807 2.55141
\(112\) 0 0
\(113\) −16.7193 −1.57282 −0.786409 0.617706i \(-0.788062\pi\)
−0.786409 + 0.617706i \(0.788062\pi\)
\(114\) 0 0
\(115\) 2.04024 0.190254
\(116\) 0 0
\(117\) −14.7935 −1.36766
\(118\) 0 0
\(119\) 6.93050 0.635318
\(120\) 0 0
\(121\) 0.711717 0.0647016
\(122\) 0 0
\(123\) −2.74200 −0.247237
\(124\) 0 0
\(125\) 8.94578 0.800135
\(126\) 0 0
\(127\) 4.91639 0.436259 0.218130 0.975920i \(-0.430004\pi\)
0.218130 + 0.975920i \(0.430004\pi\)
\(128\) 0 0
\(129\) −24.3063 −2.14005
\(130\) 0 0
\(131\) −1.56133 −0.136414 −0.0682069 0.997671i \(-0.521728\pi\)
−0.0682069 + 0.997671i \(0.521728\pi\)
\(132\) 0 0
\(133\) 1.96503 0.170390
\(134\) 0 0
\(135\) 6.68706 0.575531
\(136\) 0 0
\(137\) −0.184924 −0.0157991 −0.00789957 0.999969i \(-0.502515\pi\)
−0.00789957 + 0.999969i \(0.502515\pi\)
\(138\) 0 0
\(139\) −5.78000 −0.490253 −0.245126 0.969491i \(-0.578829\pi\)
−0.245126 + 0.969491i \(0.578829\pi\)
\(140\) 0 0
\(141\) −13.4180 −1.13000
\(142\) 0 0
\(143\) −9.49073 −0.793654
\(144\) 0 0
\(145\) 7.49940 0.622791
\(146\) 0 0
\(147\) −2.88693 −0.238110
\(148\) 0 0
\(149\) 7.59491 0.622199 0.311100 0.950377i \(-0.399303\pi\)
0.311100 + 0.950377i \(0.399303\pi\)
\(150\) 0 0
\(151\) 11.4266 0.929880 0.464940 0.885342i \(-0.346076\pi\)
0.464940 + 0.885342i \(0.346076\pi\)
\(152\) 0 0
\(153\) 36.9697 2.98882
\(154\) 0 0
\(155\) 5.19665 0.417405
\(156\) 0 0
\(157\) −13.3194 −1.06300 −0.531502 0.847057i \(-0.678372\pi\)
−0.531502 + 0.847057i \(0.678372\pi\)
\(158\) 0 0
\(159\) 29.5913 2.34674
\(160\) 0 0
\(161\) −2.05612 −0.162045
\(162\) 0 0
\(163\) 17.6451 1.38207 0.691035 0.722822i \(-0.257155\pi\)
0.691035 + 0.722822i \(0.257155\pi\)
\(164\) 0 0
\(165\) 9.80348 0.763200
\(166\) 0 0
\(167\) 16.0783 1.24418 0.622088 0.782947i \(-0.286284\pi\)
0.622088 + 0.782947i \(0.286284\pi\)
\(168\) 0 0
\(169\) −5.30908 −0.408391
\(170\) 0 0
\(171\) 10.4821 0.801590
\(172\) 0 0
\(173\) 20.4009 1.55105 0.775526 0.631315i \(-0.217485\pi\)
0.775526 + 0.631315i \(0.217485\pi\)
\(174\) 0 0
\(175\) −4.01538 −0.303534
\(176\) 0 0
\(177\) −35.0318 −2.63315
\(178\) 0 0
\(179\) −4.70472 −0.351648 −0.175824 0.984422i \(-0.556259\pi\)
−0.175824 + 0.984422i \(0.556259\pi\)
\(180\) 0 0
\(181\) −14.5522 −1.08166 −0.540828 0.841134i \(-0.681889\pi\)
−0.540828 + 0.841134i \(0.681889\pi\)
\(182\) 0 0
\(183\) 11.4930 0.849586
\(184\) 0 0
\(185\) 9.23930 0.679287
\(186\) 0 0
\(187\) 23.7178 1.73442
\(188\) 0 0
\(189\) −6.73909 −0.490197
\(190\) 0 0
\(191\) −26.6094 −1.92539 −0.962693 0.270595i \(-0.912780\pi\)
−0.962693 + 0.270595i \(0.912780\pi\)
\(192\) 0 0
\(193\) −25.2624 −1.81843 −0.909215 0.416327i \(-0.863317\pi\)
−0.909215 + 0.416327i \(0.863317\pi\)
\(194\) 0 0
\(195\) −7.94436 −0.568908
\(196\) 0 0
\(197\) −9.85072 −0.701835 −0.350917 0.936406i \(-0.614130\pi\)
−0.350917 + 0.936406i \(0.614130\pi\)
\(198\) 0 0
\(199\) 15.2541 1.08133 0.540666 0.841238i \(-0.318172\pi\)
0.540666 + 0.841238i \(0.318172\pi\)
\(200\) 0 0
\(201\) 37.0385 2.61249
\(202\) 0 0
\(203\) −7.55775 −0.530450
\(204\) 0 0
\(205\) −0.942464 −0.0658246
\(206\) 0 0
\(207\) −10.9680 −0.762332
\(208\) 0 0
\(209\) 6.72480 0.465164
\(210\) 0 0
\(211\) −3.49273 −0.240449 −0.120225 0.992747i \(-0.538361\pi\)
−0.120225 + 0.992747i \(0.538361\pi\)
\(212\) 0 0
\(213\) −10.4077 −0.713121
\(214\) 0 0
\(215\) −8.35442 −0.569767
\(216\) 0 0
\(217\) −5.23708 −0.355516
\(218\) 0 0
\(219\) −18.8568 −1.27422
\(220\) 0 0
\(221\) −19.2200 −1.29288
\(222\) 0 0
\(223\) −12.2245 −0.818610 −0.409305 0.912398i \(-0.634229\pi\)
−0.409305 + 0.912398i \(0.634229\pi\)
\(224\) 0 0
\(225\) −21.4194 −1.42796
\(226\) 0 0
\(227\) −15.0878 −1.00141 −0.500707 0.865617i \(-0.666927\pi\)
−0.500707 + 0.865617i \(0.666927\pi\)
\(228\) 0 0
\(229\) −9.55003 −0.631084 −0.315542 0.948912i \(-0.602186\pi\)
−0.315542 + 0.948912i \(0.602186\pi\)
\(230\) 0 0
\(231\) −9.87975 −0.650040
\(232\) 0 0
\(233\) 7.42241 0.486258 0.243129 0.969994i \(-0.421826\pi\)
0.243129 + 0.969994i \(0.421826\pi\)
\(234\) 0 0
\(235\) −4.61197 −0.300852
\(236\) 0 0
\(237\) −39.7874 −2.58447
\(238\) 0 0
\(239\) 3.44262 0.222685 0.111342 0.993782i \(-0.464485\pi\)
0.111342 + 0.993782i \(0.464485\pi\)
\(240\) 0 0
\(241\) −23.0386 −1.48404 −0.742022 0.670376i \(-0.766133\pi\)
−0.742022 + 0.670376i \(0.766133\pi\)
\(242\) 0 0
\(243\) 10.2510 0.657599
\(244\) 0 0
\(245\) −0.992279 −0.0633944
\(246\) 0 0
\(247\) −5.44952 −0.346744
\(248\) 0 0
\(249\) 18.3466 1.16267
\(250\) 0 0
\(251\) 17.0883 1.07861 0.539303 0.842112i \(-0.318688\pi\)
0.539303 + 0.842112i \(0.318688\pi\)
\(252\) 0 0
\(253\) −7.03652 −0.442382
\(254\) 0 0
\(255\) 19.8534 1.24327
\(256\) 0 0
\(257\) 2.15842 0.134638 0.0673191 0.997731i \(-0.478555\pi\)
0.0673191 + 0.997731i \(0.478555\pi\)
\(258\) 0 0
\(259\) −9.31119 −0.578569
\(260\) 0 0
\(261\) −40.3157 −2.49548
\(262\) 0 0
\(263\) −0.124319 −0.00766581 −0.00383290 0.999993i \(-0.501220\pi\)
−0.00383290 + 0.999993i \(0.501220\pi\)
\(264\) 0 0
\(265\) 10.1710 0.624797
\(266\) 0 0
\(267\) 1.23799 0.0757636
\(268\) 0 0
\(269\) 11.2239 0.684334 0.342167 0.939639i \(-0.388839\pi\)
0.342167 + 0.939639i \(0.388839\pi\)
\(270\) 0 0
\(271\) 0.326600 0.0198395 0.00991976 0.999951i \(-0.496842\pi\)
0.00991976 + 0.999951i \(0.496842\pi\)
\(272\) 0 0
\(273\) 8.00617 0.484556
\(274\) 0 0
\(275\) −13.7416 −0.828649
\(276\) 0 0
\(277\) 21.2121 1.27451 0.637255 0.770653i \(-0.280070\pi\)
0.637255 + 0.770653i \(0.280070\pi\)
\(278\) 0 0
\(279\) −27.9364 −1.67251
\(280\) 0 0
\(281\) −27.8495 −1.66136 −0.830680 0.556750i \(-0.812048\pi\)
−0.830680 + 0.556750i \(0.812048\pi\)
\(282\) 0 0
\(283\) −23.2706 −1.38329 −0.691645 0.722237i \(-0.743114\pi\)
−0.691645 + 0.722237i \(0.743114\pi\)
\(284\) 0 0
\(285\) 5.62910 0.333439
\(286\) 0 0
\(287\) 0.949797 0.0560648
\(288\) 0 0
\(289\) 31.0318 1.82540
\(290\) 0 0
\(291\) −42.2469 −2.47656
\(292\) 0 0
\(293\) −28.3586 −1.65672 −0.828362 0.560193i \(-0.810727\pi\)
−0.828362 + 0.560193i \(0.810727\pi\)
\(294\) 0 0
\(295\) −12.0409 −0.701051
\(296\) 0 0
\(297\) −23.0628 −1.33824
\(298\) 0 0
\(299\) 5.70213 0.329763
\(300\) 0 0
\(301\) 8.41942 0.485287
\(302\) 0 0
\(303\) −45.7294 −2.62709
\(304\) 0 0
\(305\) 3.95031 0.226194
\(306\) 0 0
\(307\) 26.5949 1.51785 0.758926 0.651177i \(-0.225724\pi\)
0.758926 + 0.651177i \(0.225724\pi\)
\(308\) 0 0
\(309\) 53.3138 3.03292
\(310\) 0 0
\(311\) −5.25843 −0.298178 −0.149089 0.988824i \(-0.547634\pi\)
−0.149089 + 0.988824i \(0.547634\pi\)
\(312\) 0 0
\(313\) 7.82442 0.442262 0.221131 0.975244i \(-0.429025\pi\)
0.221131 + 0.975244i \(0.429025\pi\)
\(314\) 0 0
\(315\) −5.29316 −0.298236
\(316\) 0 0
\(317\) −25.5817 −1.43681 −0.718407 0.695623i \(-0.755128\pi\)
−0.718407 + 0.695623i \(0.755128\pi\)
\(318\) 0 0
\(319\) −25.8644 −1.44813
\(320\) 0 0
\(321\) −0.596777 −0.0333089
\(322\) 0 0
\(323\) 13.6186 0.757761
\(324\) 0 0
\(325\) 11.1357 0.617695
\(326\) 0 0
\(327\) 1.26325 0.0698581
\(328\) 0 0
\(329\) 4.64785 0.256244
\(330\) 0 0
\(331\) 0.994122 0.0546419 0.0273210 0.999627i \(-0.491302\pi\)
0.0273210 + 0.999627i \(0.491302\pi\)
\(332\) 0 0
\(333\) −49.6691 −2.72185
\(334\) 0 0
\(335\) 12.7307 0.695551
\(336\) 0 0
\(337\) −13.4691 −0.733710 −0.366855 0.930278i \(-0.619566\pi\)
−0.366855 + 0.930278i \(0.619566\pi\)
\(338\) 0 0
\(339\) 48.2674 2.62152
\(340\) 0 0
\(341\) −17.9225 −0.970560
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −5.89003 −0.317109
\(346\) 0 0
\(347\) 3.59625 0.193057 0.0965283 0.995330i \(-0.469226\pi\)
0.0965283 + 0.995330i \(0.469226\pi\)
\(348\) 0 0
\(349\) 2.56209 0.137146 0.0685728 0.997646i \(-0.478155\pi\)
0.0685728 + 0.997646i \(0.478155\pi\)
\(350\) 0 0
\(351\) 18.6892 0.997556
\(352\) 0 0
\(353\) −4.72060 −0.251252 −0.125626 0.992078i \(-0.540094\pi\)
−0.125626 + 0.992078i \(0.540094\pi\)
\(354\) 0 0
\(355\) −3.57727 −0.189862
\(356\) 0 0
\(357\) −20.0079 −1.05893
\(358\) 0 0
\(359\) 16.9233 0.893177 0.446589 0.894739i \(-0.352639\pi\)
0.446589 + 0.894739i \(0.352639\pi\)
\(360\) 0 0
\(361\) −15.1387 −0.796772
\(362\) 0 0
\(363\) −2.05468 −0.107842
\(364\) 0 0
\(365\) −6.48136 −0.339250
\(366\) 0 0
\(367\) −19.8872 −1.03810 −0.519051 0.854743i \(-0.673714\pi\)
−0.519051 + 0.854743i \(0.673714\pi\)
\(368\) 0 0
\(369\) 5.06655 0.263754
\(370\) 0 0
\(371\) −10.2501 −0.532158
\(372\) 0 0
\(373\) 9.68661 0.501554 0.250777 0.968045i \(-0.419314\pi\)
0.250777 + 0.968045i \(0.419314\pi\)
\(374\) 0 0
\(375\) −25.8258 −1.33364
\(376\) 0 0
\(377\) 20.9595 1.07947
\(378\) 0 0
\(379\) 16.8921 0.867688 0.433844 0.900988i \(-0.357157\pi\)
0.433844 + 0.900988i \(0.357157\pi\)
\(380\) 0 0
\(381\) −14.1933 −0.727143
\(382\) 0 0
\(383\) 24.7947 1.26695 0.633476 0.773763i \(-0.281628\pi\)
0.633476 + 0.773763i \(0.281628\pi\)
\(384\) 0 0
\(385\) −3.39582 −0.173067
\(386\) 0 0
\(387\) 44.9121 2.28301
\(388\) 0 0
\(389\) −3.09208 −0.156775 −0.0783874 0.996923i \(-0.524977\pi\)
−0.0783874 + 0.996923i \(0.524977\pi\)
\(390\) 0 0
\(391\) −14.2499 −0.720649
\(392\) 0 0
\(393\) 4.50744 0.227370
\(394\) 0 0
\(395\) −13.6755 −0.688089
\(396\) 0 0
\(397\) 0.384971 0.0193211 0.00966057 0.999953i \(-0.496925\pi\)
0.00966057 + 0.999953i \(0.496925\pi\)
\(398\) 0 0
\(399\) −5.67289 −0.284000
\(400\) 0 0
\(401\) 14.6370 0.730937 0.365468 0.930824i \(-0.380909\pi\)
0.365468 + 0.930824i \(0.380909\pi\)
\(402\) 0 0
\(403\) 14.5237 0.723479
\(404\) 0 0
\(405\) −3.42557 −0.170218
\(406\) 0 0
\(407\) −31.8651 −1.57950
\(408\) 0 0
\(409\) −5.20342 −0.257292 −0.128646 0.991691i \(-0.541063\pi\)
−0.128646 + 0.991691i \(0.541063\pi\)
\(410\) 0 0
\(411\) 0.533863 0.0263335
\(412\) 0 0
\(413\) 12.1346 0.597106
\(414\) 0 0
\(415\) 6.30601 0.309550
\(416\) 0 0
\(417\) 16.6864 0.817138
\(418\) 0 0
\(419\) −14.0184 −0.684843 −0.342421 0.939546i \(-0.611247\pi\)
−0.342421 + 0.939546i \(0.611247\pi\)
\(420\) 0 0
\(421\) 8.30911 0.404961 0.202481 0.979286i \(-0.435100\pi\)
0.202481 + 0.979286i \(0.435100\pi\)
\(422\) 0 0
\(423\) 24.7933 1.20549
\(424\) 0 0
\(425\) −27.8286 −1.34989
\(426\) 0 0
\(427\) −3.98105 −0.192656
\(428\) 0 0
\(429\) 27.3990 1.32284
\(430\) 0 0
\(431\) 18.4777 0.890039 0.445019 0.895521i \(-0.353197\pi\)
0.445019 + 0.895521i \(0.353197\pi\)
\(432\) 0 0
\(433\) −8.69984 −0.418088 −0.209044 0.977906i \(-0.567035\pi\)
−0.209044 + 0.977906i \(0.567035\pi\)
\(434\) 0 0
\(435\) −21.6502 −1.03805
\(436\) 0 0
\(437\) −4.04033 −0.193275
\(438\) 0 0
\(439\) 2.08886 0.0996959 0.0498479 0.998757i \(-0.484126\pi\)
0.0498479 + 0.998757i \(0.484126\pi\)
\(440\) 0 0
\(441\) 5.33435 0.254017
\(442\) 0 0
\(443\) 2.22591 0.105756 0.0528780 0.998601i \(-0.483161\pi\)
0.0528780 + 0.998601i \(0.483161\pi\)
\(444\) 0 0
\(445\) 0.425515 0.0201713
\(446\) 0 0
\(447\) −21.9260 −1.03706
\(448\) 0 0
\(449\) −22.6235 −1.06767 −0.533835 0.845589i \(-0.679250\pi\)
−0.533835 + 0.845589i \(0.679250\pi\)
\(450\) 0 0
\(451\) 3.25043 0.153057
\(452\) 0 0
\(453\) −32.9876 −1.54989
\(454\) 0 0
\(455\) 2.75184 0.129008
\(456\) 0 0
\(457\) 22.9357 1.07289 0.536443 0.843937i \(-0.319768\pi\)
0.536443 + 0.843937i \(0.319768\pi\)
\(458\) 0 0
\(459\) −46.7053 −2.18002
\(460\) 0 0
\(461\) 13.8235 0.643827 0.321913 0.946769i \(-0.395674\pi\)
0.321913 + 0.946769i \(0.395674\pi\)
\(462\) 0 0
\(463\) −22.4440 −1.04306 −0.521531 0.853233i \(-0.674639\pi\)
−0.521531 + 0.853233i \(0.674639\pi\)
\(464\) 0 0
\(465\) −15.0023 −0.695717
\(466\) 0 0
\(467\) 2.20341 0.101961 0.0509807 0.998700i \(-0.483765\pi\)
0.0509807 + 0.998700i \(0.483765\pi\)
\(468\) 0 0
\(469\) −12.8297 −0.592421
\(470\) 0 0
\(471\) 38.4522 1.77178
\(472\) 0 0
\(473\) 28.8133 1.32484
\(474\) 0 0
\(475\) −7.89034 −0.362034
\(476\) 0 0
\(477\) −54.6775 −2.50351
\(478\) 0 0
\(479\) −36.0952 −1.64923 −0.824615 0.565694i \(-0.808608\pi\)
−0.824615 + 0.565694i \(0.808608\pi\)
\(480\) 0 0
\(481\) 25.8223 1.17739
\(482\) 0 0
\(483\) 5.93586 0.270091
\(484\) 0 0
\(485\) −14.5209 −0.659359
\(486\) 0 0
\(487\) −40.4748 −1.83409 −0.917044 0.398785i \(-0.869432\pi\)
−0.917044 + 0.398785i \(0.869432\pi\)
\(488\) 0 0
\(489\) −50.9400 −2.30359
\(490\) 0 0
\(491\) −38.3653 −1.73140 −0.865701 0.500562i \(-0.833127\pi\)
−0.865701 + 0.500562i \(0.833127\pi\)
\(492\) 0 0
\(493\) −52.3790 −2.35903
\(494\) 0 0
\(495\) −18.1145 −0.814185
\(496\) 0 0
\(497\) 3.60510 0.161711
\(498\) 0 0
\(499\) 1.96133 0.0878010 0.0439005 0.999036i \(-0.486022\pi\)
0.0439005 + 0.999036i \(0.486022\pi\)
\(500\) 0 0
\(501\) −46.4169 −2.07375
\(502\) 0 0
\(503\) −16.6445 −0.742140 −0.371070 0.928605i \(-0.621009\pi\)
−0.371070 + 0.928605i \(0.621009\pi\)
\(504\) 0 0
\(505\) −15.7179 −0.699436
\(506\) 0 0
\(507\) 15.3269 0.680692
\(508\) 0 0
\(509\) −24.6463 −1.09243 −0.546214 0.837646i \(-0.683931\pi\)
−0.546214 + 0.837646i \(0.683931\pi\)
\(510\) 0 0
\(511\) 6.53179 0.288949
\(512\) 0 0
\(513\) −13.2425 −0.584671
\(514\) 0 0
\(515\) 18.3247 0.807484
\(516\) 0 0
\(517\) 15.9061 0.699548
\(518\) 0 0
\(519\) −58.8960 −2.58525
\(520\) 0 0
\(521\) 2.69192 0.117935 0.0589676 0.998260i \(-0.481219\pi\)
0.0589676 + 0.998260i \(0.481219\pi\)
\(522\) 0 0
\(523\) 15.5510 0.679997 0.339998 0.940426i \(-0.389573\pi\)
0.339998 + 0.940426i \(0.389573\pi\)
\(524\) 0 0
\(525\) 11.5921 0.505921
\(526\) 0 0
\(527\) −36.2956 −1.58106
\(528\) 0 0
\(529\) −18.7724 −0.816191
\(530\) 0 0
\(531\) 64.7303 2.80906
\(532\) 0 0
\(533\) −2.63403 −0.114092
\(534\) 0 0
\(535\) −0.205121 −0.00886816
\(536\) 0 0
\(537\) 13.5822 0.586115
\(538\) 0 0
\(539\) 3.42224 0.147406
\(540\) 0 0
\(541\) 9.14601 0.393218 0.196609 0.980482i \(-0.437007\pi\)
0.196609 + 0.980482i \(0.437007\pi\)
\(542\) 0 0
\(543\) 42.0111 1.80287
\(544\) 0 0
\(545\) 0.434199 0.0185990
\(546\) 0 0
\(547\) −29.0039 −1.24012 −0.620059 0.784555i \(-0.712891\pi\)
−0.620059 + 0.784555i \(0.712891\pi\)
\(548\) 0 0
\(549\) −21.2363 −0.906343
\(550\) 0 0
\(551\) −14.8512 −0.632682
\(552\) 0 0
\(553\) 13.7819 0.586066
\(554\) 0 0
\(555\) −26.6732 −1.13221
\(556\) 0 0
\(557\) 15.6343 0.662448 0.331224 0.943552i \(-0.392538\pi\)
0.331224 + 0.943552i \(0.392538\pi\)
\(558\) 0 0
\(559\) −23.3492 −0.987565
\(560\) 0 0
\(561\) −68.4716 −2.89087
\(562\) 0 0
\(563\) −21.7492 −0.916619 −0.458309 0.888793i \(-0.651545\pi\)
−0.458309 + 0.888793i \(0.651545\pi\)
\(564\) 0 0
\(565\) 16.5902 0.697955
\(566\) 0 0
\(567\) 3.45223 0.144980
\(568\) 0 0
\(569\) 9.21322 0.386238 0.193119 0.981175i \(-0.438140\pi\)
0.193119 + 0.981175i \(0.438140\pi\)
\(570\) 0 0
\(571\) −14.6331 −0.612375 −0.306187 0.951971i \(-0.599053\pi\)
−0.306187 + 0.951971i \(0.599053\pi\)
\(572\) 0 0
\(573\) 76.8193 3.20917
\(574\) 0 0
\(575\) 8.25609 0.344303
\(576\) 0 0
\(577\) 2.75852 0.114839 0.0574193 0.998350i \(-0.481713\pi\)
0.0574193 + 0.998350i \(0.481713\pi\)
\(578\) 0 0
\(579\) 72.9308 3.03090
\(580\) 0 0
\(581\) −6.35508 −0.263653
\(582\) 0 0
\(583\) −35.0783 −1.45279
\(584\) 0 0
\(585\) 14.6793 0.606913
\(586\) 0 0
\(587\) −5.92116 −0.244392 −0.122196 0.992506i \(-0.538994\pi\)
−0.122196 + 0.992506i \(0.538994\pi\)
\(588\) 0 0
\(589\) −10.2910 −0.424034
\(590\) 0 0
\(591\) 28.4383 1.16980
\(592\) 0 0
\(593\) 3.11656 0.127982 0.0639908 0.997950i \(-0.479617\pi\)
0.0639908 + 0.997950i \(0.479617\pi\)
\(594\) 0 0
\(595\) −6.87699 −0.281929
\(596\) 0 0
\(597\) −44.0373 −1.80233
\(598\) 0 0
\(599\) −5.12382 −0.209354 −0.104677 0.994506i \(-0.533381\pi\)
−0.104677 + 0.994506i \(0.533381\pi\)
\(600\) 0 0
\(601\) 28.1920 1.14997 0.574987 0.818162i \(-0.305007\pi\)
0.574987 + 0.818162i \(0.305007\pi\)
\(602\) 0 0
\(603\) −68.4382 −2.78702
\(604\) 0 0
\(605\) −0.706222 −0.0287120
\(606\) 0 0
\(607\) −40.0403 −1.62518 −0.812592 0.582832i \(-0.801944\pi\)
−0.812592 + 0.582832i \(0.801944\pi\)
\(608\) 0 0
\(609\) 21.8187 0.884137
\(610\) 0 0
\(611\) −12.8897 −0.521460
\(612\) 0 0
\(613\) 17.9351 0.724391 0.362196 0.932102i \(-0.382027\pi\)
0.362196 + 0.932102i \(0.382027\pi\)
\(614\) 0 0
\(615\) 2.72083 0.109714
\(616\) 0 0
\(617\) −35.2965 −1.42098 −0.710491 0.703706i \(-0.751527\pi\)
−0.710491 + 0.703706i \(0.751527\pi\)
\(618\) 0 0
\(619\) −9.06462 −0.364338 −0.182169 0.983267i \(-0.558312\pi\)
−0.182169 + 0.983267i \(0.558312\pi\)
\(620\) 0 0
\(621\) 13.8564 0.556037
\(622\) 0 0
\(623\) −0.428825 −0.0171805
\(624\) 0 0
\(625\) 11.2002 0.448008
\(626\) 0 0
\(627\) −19.4140 −0.775320
\(628\) 0 0
\(629\) −64.5312 −2.57303
\(630\) 0 0
\(631\) −8.85344 −0.352450 −0.176225 0.984350i \(-0.556389\pi\)
−0.176225 + 0.984350i \(0.556389\pi\)
\(632\) 0 0
\(633\) 10.0832 0.400773
\(634\) 0 0
\(635\) −4.87844 −0.193595
\(636\) 0 0
\(637\) −2.77325 −0.109880
\(638\) 0 0
\(639\) 19.2309 0.760761
\(640\) 0 0
\(641\) −0.523959 −0.0206951 −0.0103476 0.999946i \(-0.503294\pi\)
−0.0103476 + 0.999946i \(0.503294\pi\)
\(642\) 0 0
\(643\) −10.6478 −0.419909 −0.209954 0.977711i \(-0.567332\pi\)
−0.209954 + 0.977711i \(0.567332\pi\)
\(644\) 0 0
\(645\) 24.1186 0.949669
\(646\) 0 0
\(647\) −17.9059 −0.703955 −0.351977 0.936009i \(-0.614491\pi\)
−0.351977 + 0.936009i \(0.614491\pi\)
\(648\) 0 0
\(649\) 41.5276 1.63010
\(650\) 0 0
\(651\) 15.1191 0.592563
\(652\) 0 0
\(653\) −10.7527 −0.420784 −0.210392 0.977617i \(-0.567474\pi\)
−0.210392 + 0.977617i \(0.567474\pi\)
\(654\) 0 0
\(655\) 1.54927 0.0605351
\(656\) 0 0
\(657\) 34.8428 1.35935
\(658\) 0 0
\(659\) −17.9988 −0.701133 −0.350567 0.936538i \(-0.614011\pi\)
−0.350567 + 0.936538i \(0.614011\pi\)
\(660\) 0 0
\(661\) 31.5748 1.22812 0.614058 0.789261i \(-0.289536\pi\)
0.614058 + 0.789261i \(0.289536\pi\)
\(662\) 0 0
\(663\) 55.4868 2.15493
\(664\) 0 0
\(665\) −1.94986 −0.0756122
\(666\) 0 0
\(667\) 15.5396 0.601696
\(668\) 0 0
\(669\) 35.2911 1.36443
\(670\) 0 0
\(671\) −13.6241 −0.525952
\(672\) 0 0
\(673\) −29.7030 −1.14497 −0.572483 0.819917i \(-0.694020\pi\)
−0.572483 + 0.819917i \(0.694020\pi\)
\(674\) 0 0
\(675\) 27.0600 1.04154
\(676\) 0 0
\(677\) −1.46765 −0.0564063 −0.0282031 0.999602i \(-0.508979\pi\)
−0.0282031 + 0.999602i \(0.508979\pi\)
\(678\) 0 0
\(679\) 14.6339 0.561596
\(680\) 0 0
\(681\) 43.5575 1.66913
\(682\) 0 0
\(683\) 8.35850 0.319829 0.159915 0.987131i \(-0.448878\pi\)
0.159915 + 0.987131i \(0.448878\pi\)
\(684\) 0 0
\(685\) 0.183497 0.00701104
\(686\) 0 0
\(687\) 27.5702 1.05187
\(688\) 0 0
\(689\) 28.4261 1.08295
\(690\) 0 0
\(691\) −8.93094 −0.339749 −0.169874 0.985466i \(-0.554336\pi\)
−0.169874 + 0.985466i \(0.554336\pi\)
\(692\) 0 0
\(693\) 18.2554 0.693466
\(694\) 0 0
\(695\) 5.73537 0.217555
\(696\) 0 0
\(697\) 6.58257 0.249333
\(698\) 0 0
\(699\) −21.4280 −0.810480
\(700\) 0 0
\(701\) −49.9961 −1.88833 −0.944163 0.329479i \(-0.893127\pi\)
−0.944163 + 0.329479i \(0.893127\pi\)
\(702\) 0 0
\(703\) −18.2968 −0.690075
\(704\) 0 0
\(705\) 13.3144 0.501450
\(706\) 0 0
\(707\) 15.8402 0.595731
\(708\) 0 0
\(709\) 36.2179 1.36019 0.680096 0.733123i \(-0.261938\pi\)
0.680096 + 0.733123i \(0.261938\pi\)
\(710\) 0 0
\(711\) 73.5175 2.75712
\(712\) 0 0
\(713\) 10.7680 0.403267
\(714\) 0 0
\(715\) 9.41745 0.352193
\(716\) 0 0
\(717\) −9.93859 −0.371163
\(718\) 0 0
\(719\) −12.8390 −0.478816 −0.239408 0.970919i \(-0.576953\pi\)
−0.239408 + 0.970919i \(0.576953\pi\)
\(720\) 0 0
\(721\) −18.4673 −0.687759
\(722\) 0 0
\(723\) 66.5106 2.47356
\(724\) 0 0
\(725\) 30.3473 1.12707
\(726\) 0 0
\(727\) −50.1537 −1.86010 −0.930049 0.367436i \(-0.880236\pi\)
−0.930049 + 0.367436i \(0.880236\pi\)
\(728\) 0 0
\(729\) −39.9504 −1.47965
\(730\) 0 0
\(731\) 58.3508 2.15818
\(732\) 0 0
\(733\) 30.3047 1.11933 0.559665 0.828719i \(-0.310930\pi\)
0.559665 + 0.828719i \(0.310930\pi\)
\(734\) 0 0
\(735\) 2.86464 0.105664
\(736\) 0 0
\(737\) −43.9064 −1.61731
\(738\) 0 0
\(739\) 3.34736 0.123135 0.0615673 0.998103i \(-0.480390\pi\)
0.0615673 + 0.998103i \(0.480390\pi\)
\(740\) 0 0
\(741\) 15.7324 0.577943
\(742\) 0 0
\(743\) 4.56306 0.167403 0.0837013 0.996491i \(-0.473326\pi\)
0.0837013 + 0.996491i \(0.473326\pi\)
\(744\) 0 0
\(745\) −7.53627 −0.276108
\(746\) 0 0
\(747\) −33.9002 −1.24034
\(748\) 0 0
\(749\) 0.206717 0.00755328
\(750\) 0 0
\(751\) 6.04590 0.220618 0.110309 0.993897i \(-0.464816\pi\)
0.110309 + 0.993897i \(0.464816\pi\)
\(752\) 0 0
\(753\) −49.3328 −1.79779
\(754\) 0 0
\(755\) −11.3383 −0.412644
\(756\) 0 0
\(757\) 35.4956 1.29011 0.645054 0.764137i \(-0.276835\pi\)
0.645054 + 0.764137i \(0.276835\pi\)
\(758\) 0 0
\(759\) 20.3139 0.737349
\(760\) 0 0
\(761\) 16.2508 0.589089 0.294545 0.955638i \(-0.404832\pi\)
0.294545 + 0.955638i \(0.404832\pi\)
\(762\) 0 0
\(763\) −0.437577 −0.0158414
\(764\) 0 0
\(765\) −36.6843 −1.32632
\(766\) 0 0
\(767\) −33.6524 −1.21512
\(768\) 0 0
\(769\) −18.6812 −0.673660 −0.336830 0.941565i \(-0.609355\pi\)
−0.336830 + 0.941565i \(0.609355\pi\)
\(770\) 0 0
\(771\) −6.23119 −0.224411
\(772\) 0 0
\(773\) −15.0314 −0.540643 −0.270322 0.962770i \(-0.587130\pi\)
−0.270322 + 0.962770i \(0.587130\pi\)
\(774\) 0 0
\(775\) 21.0289 0.755380
\(776\) 0 0
\(777\) 26.8807 0.964341
\(778\) 0 0
\(779\) 1.86638 0.0668700
\(780\) 0 0
\(781\) 12.3375 0.441471
\(782\) 0 0
\(783\) 50.9324 1.82018
\(784\) 0 0
\(785\) 13.2166 0.471720
\(786\) 0 0
\(787\) −24.3102 −0.866566 −0.433283 0.901258i \(-0.642645\pi\)
−0.433283 + 0.901258i \(0.642645\pi\)
\(788\) 0 0
\(789\) 0.358898 0.0127771
\(790\) 0 0
\(791\) −16.7193 −0.594470
\(792\) 0 0
\(793\) 11.0404 0.392058
\(794\) 0 0
\(795\) −29.3628 −1.04139
\(796\) 0 0
\(797\) −27.6904 −0.980844 −0.490422 0.871485i \(-0.663157\pi\)
−0.490422 + 0.871485i \(0.663157\pi\)
\(798\) 0 0
\(799\) 32.2120 1.13958
\(800\) 0 0
\(801\) −2.28750 −0.0808250
\(802\) 0 0
\(803\) 22.3533 0.788832
\(804\) 0 0
\(805\) 2.04024 0.0719091
\(806\) 0 0
\(807\) −32.4026 −1.14063
\(808\) 0 0
\(809\) −52.7688 −1.85525 −0.927626 0.373511i \(-0.878154\pi\)
−0.927626 + 0.373511i \(0.878154\pi\)
\(810\) 0 0
\(811\) −14.2177 −0.499251 −0.249625 0.968342i \(-0.580307\pi\)
−0.249625 + 0.968342i \(0.580307\pi\)
\(812\) 0 0
\(813\) −0.942869 −0.0330679
\(814\) 0 0
\(815\) −17.5088 −0.613308
\(816\) 0 0
\(817\) 16.5444 0.578815
\(818\) 0 0
\(819\) −14.7935 −0.516926
\(820\) 0 0
\(821\) 5.80931 0.202746 0.101373 0.994848i \(-0.467676\pi\)
0.101373 + 0.994848i \(0.467676\pi\)
\(822\) 0 0
\(823\) −23.4496 −0.817403 −0.408702 0.912668i \(-0.634018\pi\)
−0.408702 + 0.912668i \(0.634018\pi\)
\(824\) 0 0
\(825\) 39.6710 1.38117
\(826\) 0 0
\(827\) −47.1666 −1.64014 −0.820071 0.572262i \(-0.806066\pi\)
−0.820071 + 0.572262i \(0.806066\pi\)
\(828\) 0 0
\(829\) 30.4350 1.05705 0.528526 0.848917i \(-0.322745\pi\)
0.528526 + 0.848917i \(0.322745\pi\)
\(830\) 0 0
\(831\) −61.2377 −2.12431
\(832\) 0 0
\(833\) 6.93050 0.240128
\(834\) 0 0
\(835\) −15.9542 −0.552117
\(836\) 0 0
\(837\) 35.2932 1.21991
\(838\) 0 0
\(839\) 36.6874 1.26659 0.633295 0.773911i \(-0.281702\pi\)
0.633295 + 0.773911i \(0.281702\pi\)
\(840\) 0 0
\(841\) 28.1196 0.969642
\(842\) 0 0
\(843\) 80.3994 2.76910
\(844\) 0 0
\(845\) 5.26809 0.181228
\(846\) 0 0
\(847\) 0.711717 0.0244549
\(848\) 0 0
\(849\) 67.1804 2.30562
\(850\) 0 0
\(851\) 19.1449 0.656279
\(852\) 0 0
\(853\) −8.51186 −0.291441 −0.145720 0.989326i \(-0.546550\pi\)
−0.145720 + 0.989326i \(0.546550\pi\)
\(854\) 0 0
\(855\) −10.4012 −0.355714
\(856\) 0 0
\(857\) −54.0539 −1.84645 −0.923223 0.384266i \(-0.874455\pi\)
−0.923223 + 0.384266i \(0.874455\pi\)
\(858\) 0 0
\(859\) −28.4333 −0.970131 −0.485066 0.874478i \(-0.661204\pi\)
−0.485066 + 0.874478i \(0.661204\pi\)
\(860\) 0 0
\(861\) −2.74200 −0.0934470
\(862\) 0 0
\(863\) 11.1553 0.379732 0.189866 0.981810i \(-0.439195\pi\)
0.189866 + 0.981810i \(0.439195\pi\)
\(864\) 0 0
\(865\) −20.2434 −0.688296
\(866\) 0 0
\(867\) −89.5867 −3.04252
\(868\) 0 0
\(869\) 47.1650 1.59996
\(870\) 0 0
\(871\) 35.5800 1.20558
\(872\) 0 0
\(873\) 78.0621 2.64200
\(874\) 0 0
\(875\) 8.94578 0.302422
\(876\) 0 0
\(877\) 32.7330 1.10531 0.552657 0.833409i \(-0.313614\pi\)
0.552657 + 0.833409i \(0.313614\pi\)
\(878\) 0 0
\(879\) 81.8691 2.76138
\(880\) 0 0
\(881\) 22.2981 0.751244 0.375622 0.926773i \(-0.377429\pi\)
0.375622 + 0.926773i \(0.377429\pi\)
\(882\) 0 0
\(883\) −28.7948 −0.969023 −0.484512 0.874785i \(-0.661003\pi\)
−0.484512 + 0.874785i \(0.661003\pi\)
\(884\) 0 0
\(885\) 34.7613 1.16849
\(886\) 0 0
\(887\) −40.1793 −1.34909 −0.674544 0.738234i \(-0.735660\pi\)
−0.674544 + 0.738234i \(0.735660\pi\)
\(888\) 0 0
\(889\) 4.91639 0.164891
\(890\) 0 0
\(891\) 11.8143 0.395795
\(892\) 0 0
\(893\) 9.13316 0.305630
\(894\) 0 0
\(895\) 4.66840 0.156047
\(896\) 0 0
\(897\) −16.4616 −0.549638
\(898\) 0 0
\(899\) 39.5806 1.32009
\(900\) 0 0
\(901\) −71.0382 −2.36663
\(902\) 0 0
\(903\) −24.3063 −0.808861
\(904\) 0 0
\(905\) 14.4398 0.479996
\(906\) 0 0
\(907\) 54.3734 1.80544 0.902720 0.430228i \(-0.141567\pi\)
0.902720 + 0.430228i \(0.141567\pi\)
\(908\) 0 0
\(909\) 84.4970 2.80259
\(910\) 0 0
\(911\) −28.5940 −0.947362 −0.473681 0.880697i \(-0.657075\pi\)
−0.473681 + 0.880697i \(0.657075\pi\)
\(912\) 0 0
\(913\) −21.7486 −0.719773
\(914\) 0 0
\(915\) −11.4043 −0.377013
\(916\) 0 0
\(917\) −1.56133 −0.0515596
\(918\) 0 0
\(919\) 14.6386 0.482881 0.241441 0.970416i \(-0.422380\pi\)
0.241441 + 0.970416i \(0.422380\pi\)
\(920\) 0 0
\(921\) −76.7776 −2.52991
\(922\) 0 0
\(923\) −9.99785 −0.329083
\(924\) 0 0
\(925\) 37.3880 1.22931
\(926\) 0 0
\(927\) −98.5111 −3.23553
\(928\) 0 0
\(929\) 48.8608 1.60307 0.801535 0.597948i \(-0.204017\pi\)
0.801535 + 0.597948i \(0.204017\pi\)
\(930\) 0 0
\(931\) 1.96503 0.0644012
\(932\) 0 0
\(933\) 15.1807 0.496994
\(934\) 0 0
\(935\) −23.5347 −0.769667
\(936\) 0 0
\(937\) 51.4306 1.68016 0.840081 0.542460i \(-0.182507\pi\)
0.840081 + 0.542460i \(0.182507\pi\)
\(938\) 0 0
\(939\) −22.5885 −0.737149
\(940\) 0 0
\(941\) −37.0854 −1.20895 −0.604474 0.796624i \(-0.706617\pi\)
−0.604474 + 0.796624i \(0.706617\pi\)
\(942\) 0 0
\(943\) −1.95289 −0.0635950
\(944\) 0 0
\(945\) 6.68706 0.217530
\(946\) 0 0
\(947\) −12.1662 −0.395347 −0.197674 0.980268i \(-0.563339\pi\)
−0.197674 + 0.980268i \(0.563339\pi\)
\(948\) 0 0
\(949\) −18.1143 −0.588015
\(950\) 0 0
\(951\) 73.8526 2.39484
\(952\) 0 0
\(953\) −26.1027 −0.845549 −0.422775 0.906235i \(-0.638944\pi\)
−0.422775 + 0.906235i \(0.638944\pi\)
\(954\) 0 0
\(955\) 26.4039 0.854411
\(956\) 0 0
\(957\) 74.6687 2.41370
\(958\) 0 0
\(959\) −0.184924 −0.00597152
\(960\) 0 0
\(961\) −3.57299 −0.115258
\(962\) 0 0
\(963\) 1.10270 0.0355340
\(964\) 0 0
\(965\) 25.0674 0.806948
\(966\) 0 0
\(967\) −8.74115 −0.281097 −0.140548 0.990074i \(-0.544887\pi\)
−0.140548 + 0.990074i \(0.544887\pi\)
\(968\) 0 0
\(969\) −39.3160 −1.26301
\(970\) 0 0
\(971\) 42.2285 1.35518 0.677589 0.735441i \(-0.263025\pi\)
0.677589 + 0.735441i \(0.263025\pi\)
\(972\) 0 0
\(973\) −5.78000 −0.185298
\(974\) 0 0
\(975\) −32.1478 −1.02955
\(976\) 0 0
\(977\) 24.3299 0.778383 0.389191 0.921157i \(-0.372755\pi\)
0.389191 + 0.921157i \(0.372755\pi\)
\(978\) 0 0
\(979\) −1.46754 −0.0469029
\(980\) 0 0
\(981\) −2.33419 −0.0745249
\(982\) 0 0
\(983\) −36.0892 −1.15107 −0.575534 0.817778i \(-0.695206\pi\)
−0.575534 + 0.817778i \(0.695206\pi\)
\(984\) 0 0
\(985\) 9.77466 0.311447
\(986\) 0 0
\(987\) −13.4180 −0.427100
\(988\) 0 0
\(989\) −17.3113 −0.550468
\(990\) 0 0
\(991\) 44.3921 1.41016 0.705080 0.709127i \(-0.250911\pi\)
0.705080 + 0.709127i \(0.250911\pi\)
\(992\) 0 0
\(993\) −2.86996 −0.0910754
\(994\) 0 0
\(995\) −15.1363 −0.479852
\(996\) 0 0
\(997\) 2.03994 0.0646055 0.0323027 0.999478i \(-0.489716\pi\)
0.0323027 + 0.999478i \(0.489716\pi\)
\(998\) 0 0
\(999\) 62.7490 1.98529
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 7168.2.a.bb.1.1 8
4.3 odd 2 7168.2.a.ba.1.8 8
8.3 odd 2 7168.2.a.be.1.1 8
8.5 even 2 7168.2.a.bf.1.8 8
32.3 odd 8 1792.2.m.g.1345.7 yes 16
32.5 even 8 1792.2.m.h.449.7 yes 16
32.11 odd 8 1792.2.m.g.449.7 yes 16
32.13 even 8 1792.2.m.h.1345.7 yes 16
32.19 odd 8 1792.2.m.f.1345.2 yes 16
32.21 even 8 1792.2.m.e.449.2 16
32.27 odd 8 1792.2.m.f.449.2 yes 16
32.29 even 8 1792.2.m.e.1345.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.e.449.2 16 32.21 even 8
1792.2.m.e.1345.2 yes 16 32.29 even 8
1792.2.m.f.449.2 yes 16 32.27 odd 8
1792.2.m.f.1345.2 yes 16 32.19 odd 8
1792.2.m.g.449.7 yes 16 32.11 odd 8
1792.2.m.g.1345.7 yes 16 32.3 odd 8
1792.2.m.h.449.7 yes 16 32.5 even 8
1792.2.m.h.1345.7 yes 16 32.13 even 8
7168.2.a.ba.1.8 8 4.3 odd 2
7168.2.a.bb.1.1 8 1.1 even 1 trivial
7168.2.a.be.1.1 8 8.3 odd 2
7168.2.a.bf.1.8 8 8.5 even 2