Properties

Label 7168.2.a.bf.1.2
Level $7168$
Weight $2$
Character 7168.1
Self dual yes
Analytic conductor $57.237$
Analytic rank $0$
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: \(0\)
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 1.2
Root \(-1.38887\) of defining polynomial
Character \(\chi\) \(=\) 7168.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.09974 q^{3} +2.59647 q^{5} +1.00000 q^{7} +1.40890 q^{9} +O(q^{10})\) \(q-2.09974 q^{3} +2.59647 q^{5} +1.00000 q^{7} +1.40890 q^{9} -0.454591 q^{11} +6.53068 q^{13} -5.45192 q^{15} +1.84172 q^{17} -5.50056 q^{19} -2.09974 q^{21} -5.88497 q^{23} +1.74168 q^{25} +3.34090 q^{27} -8.69134 q^{29} +5.69821 q^{31} +0.954522 q^{33} +2.59647 q^{35} +2.35999 q^{37} -13.7127 q^{39} +10.7333 q^{41} +0.753925 q^{43} +3.65817 q^{45} +0.465401 q^{47} +1.00000 q^{49} -3.86712 q^{51} +0.881386 q^{53} -1.18033 q^{55} +11.5497 q^{57} +10.3546 q^{59} +10.7092 q^{61} +1.40890 q^{63} +16.9567 q^{65} +8.71619 q^{67} +12.3569 q^{69} -0.162532 q^{71} -3.49118 q^{73} -3.65707 q^{75} -0.454591 q^{77} -8.28703 q^{79} -11.2417 q^{81} -3.55356 q^{83} +4.78197 q^{85} +18.2495 q^{87} +1.60040 q^{89} +6.53068 q^{91} -11.9647 q^{93} -14.2821 q^{95} -8.88621 q^{97} -0.640472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{5} + 8q^{7} + 12q^{9} + O(q^{10}) \) \( 8q + 8q^{5} + 8q^{7} + 12q^{9} + 12q^{11} + 20q^{13} + 4q^{17} + 4q^{19} + 8q^{23} + 12q^{25} - 12q^{27} + 8q^{29} - 4q^{31} + 8q^{33} + 8q^{35} + 8q^{37} - 16q^{39} - 12q^{41} - 4q^{43} + 52q^{45} + 20q^{47} + 8q^{49} + 32q^{51} + 40q^{53} + 24q^{55} - 4q^{57} + 4q^{59} - 8q^{61} + 12q^{63} + 36q^{65} + 28q^{67} + 4q^{69} - 16q^{71} + 16q^{73} - 28q^{75} + 12q^{77} + 20q^{81} - 8q^{83} + 16q^{85} + 20q^{87} + 16q^{89} + 20q^{91} + 16q^{93} - 40q^{95} - 36q^{97} - 4q^{99} + O(q^{100}) \)

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
\(3\) −2.09974 −1.21228 −0.606142 0.795356i \(-0.707284\pi\)
−0.606142 + 0.795356i \(0.707284\pi\)
\(4\) 0 0
\(5\) 2.59647 1.16118 0.580589 0.814196i \(-0.302822\pi\)
0.580589 + 0.814196i \(0.302822\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.40890 0.469633
\(10\) 0 0
\(11\) −0.454591 −0.137064 −0.0685322 0.997649i \(-0.521832\pi\)
−0.0685322 + 0.997649i \(0.521832\pi\)
\(12\) 0 0
\(13\) 6.53068 1.81128 0.905642 0.424044i \(-0.139390\pi\)
0.905642 + 0.424044i \(0.139390\pi\)
\(14\) 0 0
\(15\) −5.45192 −1.40768
\(16\) 0 0
\(17\) 1.84172 0.446682 0.223341 0.974740i \(-0.428304\pi\)
0.223341 + 0.974740i \(0.428304\pi\)
\(18\) 0 0
\(19\) −5.50056 −1.26191 −0.630957 0.775818i \(-0.717338\pi\)
−0.630957 + 0.775818i \(0.717338\pi\)
\(20\) 0 0
\(21\) −2.09974 −0.458200
\(22\) 0 0
\(23\) −5.88497 −1.22710 −0.613550 0.789656i \(-0.710259\pi\)
−0.613550 + 0.789656i \(0.710259\pi\)
\(24\) 0 0
\(25\) 1.74168 0.348336
\(26\) 0 0
\(27\) 3.34090 0.642956
\(28\) 0 0
\(29\) −8.69134 −1.61394 −0.806970 0.590592i \(-0.798894\pi\)
−0.806970 + 0.590592i \(0.798894\pi\)
\(30\) 0 0
\(31\) 5.69821 1.02343 0.511714 0.859156i \(-0.329011\pi\)
0.511714 + 0.859156i \(0.329011\pi\)
\(32\) 0 0
\(33\) 0.954522 0.166161
\(34\) 0 0
\(35\) 2.59647 0.438884
\(36\) 0 0
\(37\) 2.35999 0.387980 0.193990 0.981003i \(-0.437857\pi\)
0.193990 + 0.981003i \(0.437857\pi\)
\(38\) 0 0
\(39\) −13.7127 −2.19579
\(40\) 0 0
\(41\) 10.7333 1.67626 0.838132 0.545468i \(-0.183648\pi\)
0.838132 + 0.545468i \(0.183648\pi\)
\(42\) 0 0
\(43\) 0.753925 0.114972 0.0574862 0.998346i \(-0.481691\pi\)
0.0574862 + 0.998346i \(0.481691\pi\)
\(44\) 0 0
\(45\) 3.65817 0.545327
\(46\) 0 0
\(47\) 0.465401 0.0678856 0.0339428 0.999424i \(-0.489194\pi\)
0.0339428 + 0.999424i \(0.489194\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.86712 −0.541505
\(52\) 0 0
\(53\) 0.881386 0.121068 0.0605338 0.998166i \(-0.480720\pi\)
0.0605338 + 0.998166i \(0.480720\pi\)
\(54\) 0 0
\(55\) −1.18033 −0.159156
\(56\) 0 0
\(57\) 11.5497 1.52980
\(58\) 0 0
\(59\) 10.3546 1.34806 0.674030 0.738704i \(-0.264562\pi\)
0.674030 + 0.738704i \(0.264562\pi\)
\(60\) 0 0
\(61\) 10.7092 1.37118 0.685589 0.727989i \(-0.259545\pi\)
0.685589 + 0.727989i \(0.259545\pi\)
\(62\) 0 0
\(63\) 1.40890 0.177504
\(64\) 0 0
\(65\) 16.9567 2.10322
\(66\) 0 0
\(67\) 8.71619 1.06485 0.532426 0.846477i \(-0.321281\pi\)
0.532426 + 0.846477i \(0.321281\pi\)
\(68\) 0 0
\(69\) 12.3569 1.48759
\(70\) 0 0
\(71\) −0.162532 −0.0192890 −0.00964452 0.999953i \(-0.503070\pi\)
−0.00964452 + 0.999953i \(0.503070\pi\)
\(72\) 0 0
\(73\) −3.49118 −0.408612 −0.204306 0.978907i \(-0.565494\pi\)
−0.204306 + 0.978907i \(0.565494\pi\)
\(74\) 0 0
\(75\) −3.65707 −0.422283
\(76\) 0 0
\(77\) −0.454591 −0.0518054
\(78\) 0 0
\(79\) −8.28703 −0.932363 −0.466182 0.884689i \(-0.654371\pi\)
−0.466182 + 0.884689i \(0.654371\pi\)
\(80\) 0 0
\(81\) −11.2417 −1.24908
\(82\) 0 0
\(83\) −3.55356 −0.390054 −0.195027 0.980798i \(-0.562479\pi\)
−0.195027 + 0.980798i \(0.562479\pi\)
\(84\) 0 0
\(85\) 4.78197 0.518677
\(86\) 0 0
\(87\) 18.2495 1.95655
\(88\) 0 0
\(89\) 1.60040 0.169642 0.0848209 0.996396i \(-0.472968\pi\)
0.0848209 + 0.996396i \(0.472968\pi\)
\(90\) 0 0
\(91\) 6.53068 0.684601
\(92\) 0 0
\(93\) −11.9647 −1.24069
\(94\) 0 0
\(95\) −14.2821 −1.46531
\(96\) 0 0
\(97\) −8.88621 −0.902258 −0.451129 0.892459i \(-0.648979\pi\)
−0.451129 + 0.892459i \(0.648979\pi\)
\(98\) 0 0
\(99\) −0.640472 −0.0643699
\(100\) 0 0
\(101\) 13.8668 1.37980 0.689901 0.723904i \(-0.257654\pi\)
0.689901 + 0.723904i \(0.257654\pi\)
\(102\) 0 0
\(103\) −14.3236 −1.41135 −0.705674 0.708537i \(-0.749356\pi\)
−0.705674 + 0.708537i \(0.749356\pi\)
\(104\) 0 0
\(105\) −5.45192 −0.532052
\(106\) 0 0
\(107\) 15.1314 1.46280 0.731401 0.681947i \(-0.238867\pi\)
0.731401 + 0.681947i \(0.238867\pi\)
\(108\) 0 0
\(109\) −10.4465 −1.00059 −0.500296 0.865854i \(-0.666776\pi\)
−0.500296 + 0.865854i \(0.666776\pi\)
\(110\) 0 0
\(111\) −4.95537 −0.470343
\(112\) 0 0
\(113\) −8.05995 −0.758217 −0.379108 0.925352i \(-0.623769\pi\)
−0.379108 + 0.925352i \(0.623769\pi\)
\(114\) 0 0
\(115\) −15.2802 −1.42488
\(116\) 0 0
\(117\) 9.20105 0.850638
\(118\) 0 0
\(119\) 1.84172 0.168830
\(120\) 0 0
\(121\) −10.7933 −0.981213
\(122\) 0 0
\(123\) −22.5372 −2.03211
\(124\) 0 0
\(125\) −8.46014 −0.756698
\(126\) 0 0
\(127\) 0.367471 0.0326078 0.0163039 0.999867i \(-0.494810\pi\)
0.0163039 + 0.999867i \(0.494810\pi\)
\(128\) 0 0
\(129\) −1.58304 −0.139379
\(130\) 0 0
\(131\) 13.1642 1.15016 0.575081 0.818096i \(-0.304970\pi\)
0.575081 + 0.818096i \(0.304970\pi\)
\(132\) 0 0
\(133\) −5.50056 −0.476959
\(134\) 0 0
\(135\) 8.67456 0.746587
\(136\) 0 0
\(137\) 6.85872 0.585980 0.292990 0.956115i \(-0.405350\pi\)
0.292990 + 0.956115i \(0.405350\pi\)
\(138\) 0 0
\(139\) 22.8721 1.93998 0.969991 0.243141i \(-0.0781777\pi\)
0.969991 + 0.243141i \(0.0781777\pi\)
\(140\) 0 0
\(141\) −0.977219 −0.0822967
\(142\) 0 0
\(143\) −2.96879 −0.248262
\(144\) 0 0
\(145\) −22.5668 −1.87407
\(146\) 0 0
\(147\) −2.09974 −0.173183
\(148\) 0 0
\(149\) 21.4973 1.76112 0.880562 0.473931i \(-0.157165\pi\)
0.880562 + 0.473931i \(0.157165\pi\)
\(150\) 0 0
\(151\) 4.76306 0.387612 0.193806 0.981040i \(-0.437917\pi\)
0.193806 + 0.981040i \(0.437917\pi\)
\(152\) 0 0
\(153\) 2.59479 0.209776
\(154\) 0 0
\(155\) 14.7952 1.18838
\(156\) 0 0
\(157\) 16.6718 1.33055 0.665277 0.746596i \(-0.268313\pi\)
0.665277 + 0.746596i \(0.268313\pi\)
\(158\) 0 0
\(159\) −1.85068 −0.146768
\(160\) 0 0
\(161\) −5.88497 −0.463800
\(162\) 0 0
\(163\) −13.6908 −1.07234 −0.536171 0.844109i \(-0.680130\pi\)
−0.536171 + 0.844109i \(0.680130\pi\)
\(164\) 0 0
\(165\) 2.47839 0.192942
\(166\) 0 0
\(167\) −13.1916 −1.02079 −0.510397 0.859939i \(-0.670501\pi\)
−0.510397 + 0.859939i \(0.670501\pi\)
\(168\) 0 0
\(169\) 29.6497 2.28075
\(170\) 0 0
\(171\) −7.74972 −0.592636
\(172\) 0 0
\(173\) 9.06309 0.689054 0.344527 0.938776i \(-0.388039\pi\)
0.344527 + 0.938776i \(0.388039\pi\)
\(174\) 0 0
\(175\) 1.74168 0.131659
\(176\) 0 0
\(177\) −21.7420 −1.63423
\(178\) 0 0
\(179\) 12.8125 0.957653 0.478826 0.877910i \(-0.341062\pi\)
0.478826 + 0.877910i \(0.341062\pi\)
\(180\) 0 0
\(181\) −19.3980 −1.44184 −0.720920 0.693019i \(-0.756280\pi\)
−0.720920 + 0.693019i \(0.756280\pi\)
\(182\) 0 0
\(183\) −22.4866 −1.66226
\(184\) 0 0
\(185\) 6.12766 0.450515
\(186\) 0 0
\(187\) −0.837227 −0.0612241
\(188\) 0 0
\(189\) 3.34090 0.243015
\(190\) 0 0
\(191\) −19.2622 −1.39376 −0.696882 0.717186i \(-0.745430\pi\)
−0.696882 + 0.717186i \(0.745430\pi\)
\(192\) 0 0
\(193\) −2.38323 −0.171549 −0.0857744 0.996315i \(-0.527336\pi\)
−0.0857744 + 0.996315i \(0.527336\pi\)
\(194\) 0 0
\(195\) −35.6047 −2.54971
\(196\) 0 0
\(197\) 8.71532 0.620941 0.310470 0.950583i \(-0.399513\pi\)
0.310470 + 0.950583i \(0.399513\pi\)
\(198\) 0 0
\(199\) 9.33136 0.661483 0.330741 0.943721i \(-0.392701\pi\)
0.330741 + 0.943721i \(0.392701\pi\)
\(200\) 0 0
\(201\) −18.3017 −1.29090
\(202\) 0 0
\(203\) −8.69134 −0.610012
\(204\) 0 0
\(205\) 27.8688 1.94644
\(206\) 0 0
\(207\) −8.29132 −0.576286
\(208\) 0 0
\(209\) 2.50050 0.172963
\(210\) 0 0
\(211\) −27.3470 −1.88264 −0.941321 0.337513i \(-0.890414\pi\)
−0.941321 + 0.337513i \(0.890414\pi\)
\(212\) 0 0
\(213\) 0.341275 0.0233838
\(214\) 0 0
\(215\) 1.95755 0.133504
\(216\) 0 0
\(217\) 5.69821 0.386819
\(218\) 0 0
\(219\) 7.33057 0.495354
\(220\) 0 0
\(221\) 12.0277 0.809067
\(222\) 0 0
\(223\) 19.2604 1.28977 0.644886 0.764279i \(-0.276905\pi\)
0.644886 + 0.764279i \(0.276905\pi\)
\(224\) 0 0
\(225\) 2.45385 0.163590
\(226\) 0 0
\(227\) 0.530703 0.0352240 0.0176120 0.999845i \(-0.494394\pi\)
0.0176120 + 0.999845i \(0.494394\pi\)
\(228\) 0 0
\(229\) −13.2167 −0.873382 −0.436691 0.899612i \(-0.643850\pi\)
−0.436691 + 0.899612i \(0.643850\pi\)
\(230\) 0 0
\(231\) 0.954522 0.0628029
\(232\) 0 0
\(233\) 13.8761 0.909055 0.454527 0.890733i \(-0.349808\pi\)
0.454527 + 0.890733i \(0.349808\pi\)
\(234\) 0 0
\(235\) 1.20840 0.0788274
\(236\) 0 0
\(237\) 17.4006 1.13029
\(238\) 0 0
\(239\) 15.2148 0.984165 0.492082 0.870549i \(-0.336236\pi\)
0.492082 + 0.870549i \(0.336236\pi\)
\(240\) 0 0
\(241\) 28.2964 1.82273 0.911364 0.411601i \(-0.135030\pi\)
0.911364 + 0.411601i \(0.135030\pi\)
\(242\) 0 0
\(243\) 13.5819 0.871281
\(244\) 0 0
\(245\) 2.59647 0.165883
\(246\) 0 0
\(247\) −35.9223 −2.28568
\(248\) 0 0
\(249\) 7.46154 0.472856
\(250\) 0 0
\(251\) −12.6157 −0.796295 −0.398147 0.917321i \(-0.630347\pi\)
−0.398147 + 0.917321i \(0.630347\pi\)
\(252\) 0 0
\(253\) 2.67525 0.168192
\(254\) 0 0
\(255\) −10.0409 −0.628784
\(256\) 0 0
\(257\) −27.4810 −1.71422 −0.857110 0.515133i \(-0.827742\pi\)
−0.857110 + 0.515133i \(0.827742\pi\)
\(258\) 0 0
\(259\) 2.35999 0.146643
\(260\) 0 0
\(261\) −12.2452 −0.757959
\(262\) 0 0
\(263\) −14.9308 −0.920674 −0.460337 0.887744i \(-0.652271\pi\)
−0.460337 + 0.887744i \(0.652271\pi\)
\(264\) 0 0
\(265\) 2.28850 0.140581
\(266\) 0 0
\(267\) −3.36042 −0.205654
\(268\) 0 0
\(269\) −0.00645316 −0.000393456 0 −0.000196728 1.00000i \(-0.500063\pi\)
−0.000196728 1.00000i \(0.500063\pi\)
\(270\) 0 0
\(271\) 5.74567 0.349025 0.174512 0.984655i \(-0.444165\pi\)
0.174512 + 0.984655i \(0.444165\pi\)
\(272\) 0 0
\(273\) −13.7127 −0.829931
\(274\) 0 0
\(275\) −0.791752 −0.0477445
\(276\) 0 0
\(277\) −15.9643 −0.959203 −0.479602 0.877486i \(-0.659219\pi\)
−0.479602 + 0.877486i \(0.659219\pi\)
\(278\) 0 0
\(279\) 8.02819 0.480635
\(280\) 0 0
\(281\) 9.71094 0.579306 0.289653 0.957132i \(-0.406460\pi\)
0.289653 + 0.957132i \(0.406460\pi\)
\(282\) 0 0
\(283\) −17.8953 −1.06377 −0.531884 0.846817i \(-0.678516\pi\)
−0.531884 + 0.846817i \(0.678516\pi\)
\(284\) 0 0
\(285\) 29.9886 1.77637
\(286\) 0 0
\(287\) 10.7333 0.633568
\(288\) 0 0
\(289\) −13.6081 −0.800475
\(290\) 0 0
\(291\) 18.6587 1.09379
\(292\) 0 0
\(293\) 1.28175 0.0748807 0.0374403 0.999299i \(-0.488080\pi\)
0.0374403 + 0.999299i \(0.488080\pi\)
\(294\) 0 0
\(295\) 26.8856 1.56534
\(296\) 0 0
\(297\) −1.51874 −0.0881263
\(298\) 0 0
\(299\) −38.4328 −2.22263
\(300\) 0 0
\(301\) 0.753925 0.0434555
\(302\) 0 0
\(303\) −29.1167 −1.67271
\(304\) 0 0
\(305\) 27.8063 1.59218
\(306\) 0 0
\(307\) 20.9201 1.19398 0.596988 0.802251i \(-0.296364\pi\)
0.596988 + 0.802251i \(0.296364\pi\)
\(308\) 0 0
\(309\) 30.0758 1.71095
\(310\) 0 0
\(311\) 11.6043 0.658022 0.329011 0.944326i \(-0.393285\pi\)
0.329011 + 0.944326i \(0.393285\pi\)
\(312\) 0 0
\(313\) 18.5621 1.04919 0.524596 0.851351i \(-0.324216\pi\)
0.524596 + 0.851351i \(0.324216\pi\)
\(314\) 0 0
\(315\) 3.65817 0.206114
\(316\) 0 0
\(317\) 22.4666 1.26185 0.630924 0.775845i \(-0.282676\pi\)
0.630924 + 0.775845i \(0.282676\pi\)
\(318\) 0 0
\(319\) 3.95100 0.221214
\(320\) 0 0
\(321\) −31.7719 −1.77333
\(322\) 0 0
\(323\) −10.1305 −0.563674
\(324\) 0 0
\(325\) 11.3744 0.630936
\(326\) 0 0
\(327\) 21.9349 1.21300
\(328\) 0 0
\(329\) 0.465401 0.0256584
\(330\) 0 0
\(331\) −7.77317 −0.427252 −0.213626 0.976915i \(-0.568527\pi\)
−0.213626 + 0.976915i \(0.568527\pi\)
\(332\) 0 0
\(333\) 3.32499 0.182208
\(334\) 0 0
\(335\) 22.6314 1.23648
\(336\) 0 0
\(337\) 2.73515 0.148993 0.0744966 0.997221i \(-0.476265\pi\)
0.0744966 + 0.997221i \(0.476265\pi\)
\(338\) 0 0
\(339\) 16.9238 0.919174
\(340\) 0 0
\(341\) −2.59035 −0.140275
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 32.0843 1.72736
\(346\) 0 0
\(347\) 35.3848 1.89956 0.949779 0.312922i \(-0.101308\pi\)
0.949779 + 0.312922i \(0.101308\pi\)
\(348\) 0 0
\(349\) 12.3409 0.660596 0.330298 0.943877i \(-0.392851\pi\)
0.330298 + 0.943877i \(0.392851\pi\)
\(350\) 0 0
\(351\) 21.8183 1.16458
\(352\) 0 0
\(353\) −22.2643 −1.18501 −0.592504 0.805567i \(-0.701861\pi\)
−0.592504 + 0.805567i \(0.701861\pi\)
\(354\) 0 0
\(355\) −0.422011 −0.0223980
\(356\) 0 0
\(357\) −3.86712 −0.204670
\(358\) 0 0
\(359\) 18.3142 0.966588 0.483294 0.875458i \(-0.339440\pi\)
0.483294 + 0.875458i \(0.339440\pi\)
\(360\) 0 0
\(361\) 11.2561 0.592427
\(362\) 0 0
\(363\) 22.6632 1.18951
\(364\) 0 0
\(365\) −9.06477 −0.474472
\(366\) 0 0
\(367\) 19.3398 1.00953 0.504764 0.863257i \(-0.331580\pi\)
0.504764 + 0.863257i \(0.331580\pi\)
\(368\) 0 0
\(369\) 15.1222 0.787228
\(370\) 0 0
\(371\) 0.881386 0.0457593
\(372\) 0 0
\(373\) 7.97146 0.412747 0.206373 0.978473i \(-0.433834\pi\)
0.206373 + 0.978473i \(0.433834\pi\)
\(374\) 0 0
\(375\) 17.7641 0.917333
\(376\) 0 0
\(377\) −56.7603 −2.92330
\(378\) 0 0
\(379\) 4.58476 0.235503 0.117752 0.993043i \(-0.462431\pi\)
0.117752 + 0.993043i \(0.462431\pi\)
\(380\) 0 0
\(381\) −0.771592 −0.0395299
\(382\) 0 0
\(383\) 5.39804 0.275827 0.137913 0.990444i \(-0.455960\pi\)
0.137913 + 0.990444i \(0.455960\pi\)
\(384\) 0 0
\(385\) −1.18033 −0.0601554
\(386\) 0 0
\(387\) 1.06220 0.0539948
\(388\) 0 0
\(389\) 22.1427 1.12268 0.561340 0.827586i \(-0.310286\pi\)
0.561340 + 0.827586i \(0.310286\pi\)
\(390\) 0 0
\(391\) −10.8384 −0.548123
\(392\) 0 0
\(393\) −27.6414 −1.39432
\(394\) 0 0
\(395\) −21.5171 −1.08264
\(396\) 0 0
\(397\) 3.17580 0.159389 0.0796945 0.996819i \(-0.474606\pi\)
0.0796945 + 0.996819i \(0.474606\pi\)
\(398\) 0 0
\(399\) 11.5497 0.578209
\(400\) 0 0
\(401\) −19.3307 −0.965331 −0.482665 0.875805i \(-0.660331\pi\)
−0.482665 + 0.875805i \(0.660331\pi\)
\(402\) 0 0
\(403\) 37.2131 1.85372
\(404\) 0 0
\(405\) −29.1888 −1.45040
\(406\) 0 0
\(407\) −1.07283 −0.0531783
\(408\) 0 0
\(409\) 0.497591 0.0246043 0.0123021 0.999924i \(-0.496084\pi\)
0.0123021 + 0.999924i \(0.496084\pi\)
\(410\) 0 0
\(411\) −14.4015 −0.710375
\(412\) 0 0
\(413\) 10.3546 0.509519
\(414\) 0 0
\(415\) −9.22673 −0.452922
\(416\) 0 0
\(417\) −48.0253 −2.35181
\(418\) 0 0
\(419\) 4.96496 0.242554 0.121277 0.992619i \(-0.461301\pi\)
0.121277 + 0.992619i \(0.461301\pi\)
\(420\) 0 0
\(421\) 1.91040 0.0931073 0.0465537 0.998916i \(-0.485176\pi\)
0.0465537 + 0.998916i \(0.485176\pi\)
\(422\) 0 0
\(423\) 0.655702 0.0318813
\(424\) 0 0
\(425\) 3.20768 0.155595
\(426\) 0 0
\(427\) 10.7092 0.518256
\(428\) 0 0
\(429\) 6.23367 0.300964
\(430\) 0 0
\(431\) 19.6166 0.944896 0.472448 0.881358i \(-0.343370\pi\)
0.472448 + 0.881358i \(0.343370\pi\)
\(432\) 0 0
\(433\) 2.76217 0.132741 0.0663706 0.997795i \(-0.478858\pi\)
0.0663706 + 0.997795i \(0.478858\pi\)
\(434\) 0 0
\(435\) 47.3844 2.27191
\(436\) 0 0
\(437\) 32.3706 1.54850
\(438\) 0 0
\(439\) −3.05296 −0.145710 −0.0728550 0.997343i \(-0.523211\pi\)
−0.0728550 + 0.997343i \(0.523211\pi\)
\(440\) 0 0
\(441\) 1.40890 0.0670904
\(442\) 0 0
\(443\) 1.93814 0.0920836 0.0460418 0.998940i \(-0.485339\pi\)
0.0460418 + 0.998940i \(0.485339\pi\)
\(444\) 0 0
\(445\) 4.15539 0.196985
\(446\) 0 0
\(447\) −45.1386 −2.13498
\(448\) 0 0
\(449\) 38.6173 1.82246 0.911232 0.411894i \(-0.135133\pi\)
0.911232 + 0.411894i \(0.135133\pi\)
\(450\) 0 0
\(451\) −4.87927 −0.229756
\(452\) 0 0
\(453\) −10.0012 −0.469896
\(454\) 0 0
\(455\) 16.9567 0.794944
\(456\) 0 0
\(457\) −36.0959 −1.68849 −0.844247 0.535955i \(-0.819952\pi\)
−0.844247 + 0.535955i \(0.819952\pi\)
\(458\) 0 0
\(459\) 6.15298 0.287197
\(460\) 0 0
\(461\) −6.07115 −0.282762 −0.141381 0.989955i \(-0.545154\pi\)
−0.141381 + 0.989955i \(0.545154\pi\)
\(462\) 0 0
\(463\) −3.38348 −0.157243 −0.0786217 0.996905i \(-0.525052\pi\)
−0.0786217 + 0.996905i \(0.525052\pi\)
\(464\) 0 0
\(465\) −31.0661 −1.44066
\(466\) 0 0
\(467\) −30.7940 −1.42498 −0.712489 0.701683i \(-0.752432\pi\)
−0.712489 + 0.701683i \(0.752432\pi\)
\(468\) 0 0
\(469\) 8.71619 0.402476
\(470\) 0 0
\(471\) −35.0064 −1.61301
\(472\) 0 0
\(473\) −0.342727 −0.0157586
\(474\) 0 0
\(475\) −9.58022 −0.439570
\(476\) 0 0
\(477\) 1.24178 0.0568573
\(478\) 0 0
\(479\) −6.74054 −0.307983 −0.153992 0.988072i \(-0.549213\pi\)
−0.153992 + 0.988072i \(0.549213\pi\)
\(480\) 0 0
\(481\) 15.4123 0.702743
\(482\) 0 0
\(483\) 12.3569 0.562258
\(484\) 0 0
\(485\) −23.0728 −1.04768
\(486\) 0 0
\(487\) 30.4472 1.37970 0.689848 0.723954i \(-0.257677\pi\)
0.689848 + 0.723954i \(0.257677\pi\)
\(488\) 0 0
\(489\) 28.7470 1.29998
\(490\) 0 0
\(491\) 15.7806 0.712166 0.356083 0.934454i \(-0.384112\pi\)
0.356083 + 0.934454i \(0.384112\pi\)
\(492\) 0 0
\(493\) −16.0070 −0.720918
\(494\) 0 0
\(495\) −1.66297 −0.0747449
\(496\) 0 0
\(497\) −0.162532 −0.00729057
\(498\) 0 0
\(499\) −24.6859 −1.10509 −0.552547 0.833482i \(-0.686344\pi\)
−0.552547 + 0.833482i \(0.686344\pi\)
\(500\) 0 0
\(501\) 27.6988 1.23749
\(502\) 0 0
\(503\) 26.7354 1.19207 0.596035 0.802958i \(-0.296742\pi\)
0.596035 + 0.802958i \(0.296742\pi\)
\(504\) 0 0
\(505\) 36.0049 1.60220
\(506\) 0 0
\(507\) −62.2566 −2.76491
\(508\) 0 0
\(509\) −19.4609 −0.862588 −0.431294 0.902212i \(-0.641943\pi\)
−0.431294 + 0.902212i \(0.641943\pi\)
\(510\) 0 0
\(511\) −3.49118 −0.154441
\(512\) 0 0
\(513\) −18.3768 −0.811355
\(514\) 0 0
\(515\) −37.1909 −1.63883
\(516\) 0 0
\(517\) −0.211567 −0.00930470
\(518\) 0 0
\(519\) −19.0301 −0.835330
\(520\) 0 0
\(521\) 36.5859 1.60286 0.801428 0.598091i \(-0.204074\pi\)
0.801428 + 0.598091i \(0.204074\pi\)
\(522\) 0 0
\(523\) −5.68929 −0.248775 −0.124388 0.992234i \(-0.539697\pi\)
−0.124388 + 0.992234i \(0.539697\pi\)
\(524\) 0 0
\(525\) −3.65707 −0.159608
\(526\) 0 0
\(527\) 10.4945 0.457147
\(528\) 0 0
\(529\) 11.6328 0.505776
\(530\) 0 0
\(531\) 14.5886 0.633093
\(532\) 0 0
\(533\) 70.0958 3.03619
\(534\) 0 0
\(535\) 39.2882 1.69858
\(536\) 0 0
\(537\) −26.9029 −1.16095
\(538\) 0 0
\(539\) −0.454591 −0.0195806
\(540\) 0 0
\(541\) 25.3799 1.09117 0.545584 0.838056i \(-0.316308\pi\)
0.545584 + 0.838056i \(0.316308\pi\)
\(542\) 0 0
\(543\) 40.7306 1.74792
\(544\) 0 0
\(545\) −27.1241 −1.16187
\(546\) 0 0
\(547\) 26.7752 1.14482 0.572412 0.819966i \(-0.306008\pi\)
0.572412 + 0.819966i \(0.306008\pi\)
\(548\) 0 0
\(549\) 15.0882 0.643950
\(550\) 0 0
\(551\) 47.8072 2.03665
\(552\) 0 0
\(553\) −8.28703 −0.352400
\(554\) 0 0
\(555\) −12.8665 −0.546152
\(556\) 0 0
\(557\) 42.9236 1.81873 0.909366 0.415998i \(-0.136568\pi\)
0.909366 + 0.415998i \(0.136568\pi\)
\(558\) 0 0
\(559\) 4.92364 0.208248
\(560\) 0 0
\(561\) 1.75796 0.0742210
\(562\) 0 0
\(563\) 25.7662 1.08592 0.542958 0.839760i \(-0.317304\pi\)
0.542958 + 0.839760i \(0.317304\pi\)
\(564\) 0 0
\(565\) −20.9275 −0.880425
\(566\) 0 0
\(567\) −11.2417 −0.472107
\(568\) 0 0
\(569\) −40.3261 −1.69056 −0.845279 0.534325i \(-0.820566\pi\)
−0.845279 + 0.534325i \(0.820566\pi\)
\(570\) 0 0
\(571\) −25.6720 −1.07434 −0.537169 0.843475i \(-0.680506\pi\)
−0.537169 + 0.843475i \(0.680506\pi\)
\(572\) 0 0
\(573\) 40.4456 1.68964
\(574\) 0 0
\(575\) −10.2497 −0.427444
\(576\) 0 0
\(577\) 28.8535 1.20119 0.600594 0.799554i \(-0.294931\pi\)
0.600594 + 0.799554i \(0.294931\pi\)
\(578\) 0 0
\(579\) 5.00416 0.207966
\(580\) 0 0
\(581\) −3.55356 −0.147427
\(582\) 0 0
\(583\) −0.400670 −0.0165941
\(584\) 0 0
\(585\) 23.8903 0.987743
\(586\) 0 0
\(587\) 31.1230 1.28458 0.642292 0.766460i \(-0.277984\pi\)
0.642292 + 0.766460i \(0.277984\pi\)
\(588\) 0 0
\(589\) −31.3433 −1.29148
\(590\) 0 0
\(591\) −18.2999 −0.752757
\(592\) 0 0
\(593\) −24.6606 −1.01269 −0.506344 0.862331i \(-0.669004\pi\)
−0.506344 + 0.862331i \(0.669004\pi\)
\(594\) 0 0
\(595\) 4.78197 0.196042
\(596\) 0 0
\(597\) −19.5934 −0.801905
\(598\) 0 0
\(599\) −10.8236 −0.442239 −0.221120 0.975247i \(-0.570971\pi\)
−0.221120 + 0.975247i \(0.570971\pi\)
\(600\) 0 0
\(601\) 16.6417 0.678828 0.339414 0.940637i \(-0.389771\pi\)
0.339414 + 0.940637i \(0.389771\pi\)
\(602\) 0 0
\(603\) 12.2802 0.500089
\(604\) 0 0
\(605\) −28.0247 −1.13936
\(606\) 0 0
\(607\) 22.3716 0.908037 0.454019 0.890992i \(-0.349990\pi\)
0.454019 + 0.890992i \(0.349990\pi\)
\(608\) 0 0
\(609\) 18.2495 0.739508
\(610\) 0 0
\(611\) 3.03938 0.122960
\(612\) 0 0
\(613\) 46.2891 1.86960 0.934800 0.355175i \(-0.115579\pi\)
0.934800 + 0.355175i \(0.115579\pi\)
\(614\) 0 0
\(615\) −58.5172 −2.35964
\(616\) 0 0
\(617\) 3.37531 0.135885 0.0679424 0.997689i \(-0.478357\pi\)
0.0679424 + 0.997689i \(0.478357\pi\)
\(618\) 0 0
\(619\) 5.05532 0.203190 0.101595 0.994826i \(-0.467605\pi\)
0.101595 + 0.994826i \(0.467605\pi\)
\(620\) 0 0
\(621\) −19.6611 −0.788972
\(622\) 0 0
\(623\) 1.60040 0.0641186
\(624\) 0 0
\(625\) −30.6750 −1.22700
\(626\) 0 0
\(627\) −5.25040 −0.209681
\(628\) 0 0
\(629\) 4.34644 0.173304
\(630\) 0 0
\(631\) −31.2089 −1.24241 −0.621204 0.783649i \(-0.713356\pi\)
−0.621204 + 0.783649i \(0.713356\pi\)
\(632\) 0 0
\(633\) 57.4214 2.28230
\(634\) 0 0
\(635\) 0.954129 0.0378634
\(636\) 0 0
\(637\) 6.53068 0.258755
\(638\) 0 0
\(639\) −0.228991 −0.00905876
\(640\) 0 0
\(641\) 25.7829 1.01836 0.509182 0.860659i \(-0.329948\pi\)
0.509182 + 0.860659i \(0.329948\pi\)
\(642\) 0 0
\(643\) −2.57885 −0.101700 −0.0508500 0.998706i \(-0.516193\pi\)
−0.0508500 + 0.998706i \(0.516193\pi\)
\(644\) 0 0
\(645\) −4.11033 −0.161844
\(646\) 0 0
\(647\) 10.0909 0.396713 0.198356 0.980130i \(-0.436440\pi\)
0.198356 + 0.980130i \(0.436440\pi\)
\(648\) 0 0
\(649\) −4.70713 −0.184771
\(650\) 0 0
\(651\) −11.9647 −0.468935
\(652\) 0 0
\(653\) −19.6009 −0.767041 −0.383520 0.923532i \(-0.625288\pi\)
−0.383520 + 0.923532i \(0.625288\pi\)
\(654\) 0 0
\(655\) 34.1805 1.33554
\(656\) 0 0
\(657\) −4.91872 −0.191898
\(658\) 0 0
\(659\) −19.6681 −0.766162 −0.383081 0.923715i \(-0.625137\pi\)
−0.383081 + 0.923715i \(0.625137\pi\)
\(660\) 0 0
\(661\) 1.60575 0.0624564 0.0312282 0.999512i \(-0.490058\pi\)
0.0312282 + 0.999512i \(0.490058\pi\)
\(662\) 0 0
\(663\) −25.2549 −0.980820
\(664\) 0 0
\(665\) −14.2821 −0.553834
\(666\) 0 0
\(667\) 51.1482 1.98047
\(668\) 0 0
\(669\) −40.4418 −1.56357
\(670\) 0 0
\(671\) −4.86832 −0.187939
\(672\) 0 0
\(673\) 9.91726 0.382282 0.191141 0.981563i \(-0.438781\pi\)
0.191141 + 0.981563i \(0.438781\pi\)
\(674\) 0 0
\(675\) 5.81878 0.223965
\(676\) 0 0
\(677\) −23.2883 −0.895042 −0.447521 0.894273i \(-0.647693\pi\)
−0.447521 + 0.894273i \(0.647693\pi\)
\(678\) 0 0
\(679\) −8.88621 −0.341021
\(680\) 0 0
\(681\) −1.11434 −0.0427015
\(682\) 0 0
\(683\) 7.80039 0.298474 0.149237 0.988801i \(-0.452318\pi\)
0.149237 + 0.988801i \(0.452318\pi\)
\(684\) 0 0
\(685\) 17.8085 0.680428
\(686\) 0 0
\(687\) 27.7515 1.05879
\(688\) 0 0
\(689\) 5.75605 0.219288
\(690\) 0 0
\(691\) 44.0204 1.67462 0.837308 0.546732i \(-0.184128\pi\)
0.837308 + 0.546732i \(0.184128\pi\)
\(692\) 0 0
\(693\) −0.640472 −0.0243295
\(694\) 0 0
\(695\) 59.3867 2.25267
\(696\) 0 0
\(697\) 19.7677 0.748756
\(698\) 0 0
\(699\) −29.1362 −1.10203
\(700\) 0 0
\(701\) −30.7777 −1.16246 −0.581229 0.813740i \(-0.697428\pi\)
−0.581229 + 0.813740i \(0.697428\pi\)
\(702\) 0 0
\(703\) −12.9813 −0.489598
\(704\) 0 0
\(705\) −2.53732 −0.0955612
\(706\) 0 0
\(707\) 13.8668 0.521516
\(708\) 0 0
\(709\) −18.3503 −0.689159 −0.344580 0.938757i \(-0.611979\pi\)
−0.344580 + 0.938757i \(0.611979\pi\)
\(710\) 0 0
\(711\) −11.6756 −0.437868
\(712\) 0 0
\(713\) −33.5338 −1.25585
\(714\) 0 0
\(715\) −7.70838 −0.288277
\(716\) 0 0
\(717\) −31.9471 −1.19309
\(718\) 0 0
\(719\) 36.4527 1.35946 0.679728 0.733464i \(-0.262098\pi\)
0.679728 + 0.733464i \(0.262098\pi\)
\(720\) 0 0
\(721\) −14.3236 −0.533439
\(722\) 0 0
\(723\) −59.4149 −2.20966
\(724\) 0 0
\(725\) −15.1375 −0.562194
\(726\) 0 0
\(727\) 37.9281 1.40668 0.703338 0.710855i \(-0.251692\pi\)
0.703338 + 0.710855i \(0.251692\pi\)
\(728\) 0 0
\(729\) 5.20661 0.192838
\(730\) 0 0
\(731\) 1.38852 0.0513561
\(732\) 0 0
\(733\) −18.2291 −0.673306 −0.336653 0.941629i \(-0.609295\pi\)
−0.336653 + 0.941629i \(0.609295\pi\)
\(734\) 0 0
\(735\) −5.45192 −0.201097
\(736\) 0 0
\(737\) −3.96230 −0.145953
\(738\) 0 0
\(739\) 11.2720 0.414645 0.207323 0.978273i \(-0.433525\pi\)
0.207323 + 0.978273i \(0.433525\pi\)
\(740\) 0 0
\(741\) 75.4275 2.77090
\(742\) 0 0
\(743\) −41.0385 −1.50556 −0.752779 0.658273i \(-0.771287\pi\)
−0.752779 + 0.658273i \(0.771287\pi\)
\(744\) 0 0
\(745\) 55.8171 2.04498
\(746\) 0 0
\(747\) −5.00660 −0.183182
\(748\) 0 0
\(749\) 15.1314 0.552888
\(750\) 0 0
\(751\) −16.8700 −0.615594 −0.307797 0.951452i \(-0.599592\pi\)
−0.307797 + 0.951452i \(0.599592\pi\)
\(752\) 0 0
\(753\) 26.4896 0.965336
\(754\) 0 0
\(755\) 12.3672 0.450087
\(756\) 0 0
\(757\) −30.6915 −1.11550 −0.557750 0.830009i \(-0.688335\pi\)
−0.557750 + 0.830009i \(0.688335\pi\)
\(758\) 0 0
\(759\) −5.61733 −0.203896
\(760\) 0 0
\(761\) −8.35285 −0.302791 −0.151395 0.988473i \(-0.548377\pi\)
−0.151395 + 0.988473i \(0.548377\pi\)
\(762\) 0 0
\(763\) −10.4465 −0.378188
\(764\) 0 0
\(765\) 6.73731 0.243588
\(766\) 0 0
\(767\) 67.6229 2.44172
\(768\) 0 0
\(769\) −21.8388 −0.787527 −0.393764 0.919212i \(-0.628827\pi\)
−0.393764 + 0.919212i \(0.628827\pi\)
\(770\) 0 0
\(771\) 57.7030 2.07812
\(772\) 0 0
\(773\) −9.26946 −0.333399 −0.166700 0.986008i \(-0.553311\pi\)
−0.166700 + 0.986008i \(0.553311\pi\)
\(774\) 0 0
\(775\) 9.92446 0.356497
\(776\) 0 0
\(777\) −4.95537 −0.177773
\(778\) 0 0
\(779\) −59.0392 −2.11530
\(780\) 0 0
\(781\) 0.0738857 0.00264384
\(782\) 0 0
\(783\) −29.0369 −1.03769
\(784\) 0 0
\(785\) 43.2879 1.54501
\(786\) 0 0
\(787\) −25.2810 −0.901170 −0.450585 0.892733i \(-0.648785\pi\)
−0.450585 + 0.892733i \(0.648785\pi\)
\(788\) 0 0
\(789\) 31.3508 1.11612
\(790\) 0 0
\(791\) −8.05995 −0.286579
\(792\) 0 0
\(793\) 69.9386 2.48359
\(794\) 0 0
\(795\) −4.80524 −0.170424
\(796\) 0 0
\(797\) 22.6248 0.801410 0.400705 0.916207i \(-0.368765\pi\)
0.400705 + 0.916207i \(0.368765\pi\)
\(798\) 0 0
\(799\) 0.857136 0.0303233
\(800\) 0 0
\(801\) 2.25480 0.0796694
\(802\) 0 0
\(803\) 1.58706 0.0560062
\(804\) 0 0
\(805\) −15.2802 −0.538555
\(806\) 0 0
\(807\) 0.0135499 0.000476981 0
\(808\) 0 0
\(809\) 13.5401 0.476046 0.238023 0.971260i \(-0.423501\pi\)
0.238023 + 0.971260i \(0.423501\pi\)
\(810\) 0 0
\(811\) 2.11986 0.0744384 0.0372192 0.999307i \(-0.488150\pi\)
0.0372192 + 0.999307i \(0.488150\pi\)
\(812\) 0 0
\(813\) −12.0644 −0.423117
\(814\) 0 0
\(815\) −35.5477 −1.24518
\(816\) 0 0
\(817\) −4.14701 −0.145085
\(818\) 0 0
\(819\) 9.20105 0.321511
\(820\) 0 0
\(821\) −0.815513 −0.0284616 −0.0142308 0.999899i \(-0.504530\pi\)
−0.0142308 + 0.999899i \(0.504530\pi\)
\(822\) 0 0
\(823\) 14.8651 0.518164 0.259082 0.965855i \(-0.416580\pi\)
0.259082 + 0.965855i \(0.416580\pi\)
\(824\) 0 0
\(825\) 1.66247 0.0578799
\(826\) 0 0
\(827\) 31.9223 1.11005 0.555023 0.831835i \(-0.312709\pi\)
0.555023 + 0.831835i \(0.312709\pi\)
\(828\) 0 0
\(829\) −21.8241 −0.757983 −0.378992 0.925400i \(-0.623729\pi\)
−0.378992 + 0.925400i \(0.623729\pi\)
\(830\) 0 0
\(831\) 33.5209 1.16283
\(832\) 0 0
\(833\) 1.84172 0.0638117
\(834\) 0 0
\(835\) −34.2516 −1.18532
\(836\) 0 0
\(837\) 19.0371 0.658019
\(838\) 0 0
\(839\) −37.1968 −1.28418 −0.642089 0.766630i \(-0.721932\pi\)
−0.642089 + 0.766630i \(0.721932\pi\)
\(840\) 0 0
\(841\) 46.5393 1.60480
\(842\) 0 0
\(843\) −20.3904 −0.702284
\(844\) 0 0
\(845\) 76.9848 2.64836
\(846\) 0 0
\(847\) −10.7933 −0.370864
\(848\) 0 0
\(849\) 37.5755 1.28959
\(850\) 0 0
\(851\) −13.8885 −0.476091
\(852\) 0 0
\(853\) −31.7608 −1.08747 −0.543735 0.839257i \(-0.682990\pi\)
−0.543735 + 0.839257i \(0.682990\pi\)
\(854\) 0 0
\(855\) −20.1220 −0.688156
\(856\) 0 0
\(857\) −47.6664 −1.62825 −0.814126 0.580688i \(-0.802784\pi\)
−0.814126 + 0.580688i \(0.802784\pi\)
\(858\) 0 0
\(859\) 31.6231 1.07897 0.539484 0.841996i \(-0.318620\pi\)
0.539484 + 0.841996i \(0.318620\pi\)
\(860\) 0 0
\(861\) −22.5372 −0.768064
\(862\) 0 0
\(863\) −6.01295 −0.204683 −0.102342 0.994749i \(-0.532633\pi\)
−0.102342 + 0.994749i \(0.532633\pi\)
\(864\) 0 0
\(865\) 23.5321 0.800115
\(866\) 0 0
\(867\) 28.5734 0.970404
\(868\) 0 0
\(869\) 3.76721 0.127794
\(870\) 0 0
\(871\) 56.9226 1.92875
\(872\) 0 0
\(873\) −12.5198 −0.423730
\(874\) 0 0
\(875\) −8.46014 −0.286005
\(876\) 0 0
\(877\) −10.6592 −0.359937 −0.179968 0.983672i \(-0.557600\pi\)
−0.179968 + 0.983672i \(0.557600\pi\)
\(878\) 0 0
\(879\) −2.69134 −0.0907766
\(880\) 0 0
\(881\) 18.6023 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(882\) 0 0
\(883\) −21.3704 −0.719170 −0.359585 0.933112i \(-0.617082\pi\)
−0.359585 + 0.933112i \(0.617082\pi\)
\(884\) 0 0
\(885\) −56.4527 −1.89764
\(886\) 0 0
\(887\) 56.3410 1.89175 0.945873 0.324537i \(-0.105208\pi\)
0.945873 + 0.324537i \(0.105208\pi\)
\(888\) 0 0
\(889\) 0.367471 0.0123246
\(890\) 0 0
\(891\) 5.11037 0.171204
\(892\) 0 0
\(893\) −2.55996 −0.0856658
\(894\) 0 0
\(895\) 33.2674 1.11201
\(896\) 0 0
\(897\) 80.6988 2.69446
\(898\) 0 0
\(899\) −49.5250 −1.65175
\(900\) 0 0
\(901\) 1.62326 0.0540787
\(902\) 0 0
\(903\) −1.58304 −0.0526804
\(904\) 0 0
\(905\) −50.3663 −1.67423
\(906\) 0 0
\(907\) 7.40527 0.245888 0.122944 0.992414i \(-0.460766\pi\)
0.122944 + 0.992414i \(0.460766\pi\)
\(908\) 0 0
\(909\) 19.5370 0.648000
\(910\) 0 0
\(911\) −28.0494 −0.929319 −0.464660 0.885489i \(-0.653823\pi\)
−0.464660 + 0.885489i \(0.653823\pi\)
\(912\) 0 0
\(913\) 1.61542 0.0534625
\(914\) 0 0
\(915\) −58.3859 −1.93018
\(916\) 0 0
\(917\) 13.1642 0.434721
\(918\) 0 0
\(919\) 36.2282 1.19506 0.597529 0.801847i \(-0.296149\pi\)
0.597529 + 0.801847i \(0.296149\pi\)
\(920\) 0 0
\(921\) −43.9268 −1.44744
\(922\) 0 0
\(923\) −1.06145 −0.0349379
\(924\) 0 0
\(925\) 4.11036 0.135148
\(926\) 0 0
\(927\) −20.1805 −0.662815
\(928\) 0 0
\(929\) 16.6482 0.546210 0.273105 0.961984i \(-0.411949\pi\)
0.273105 + 0.961984i \(0.411949\pi\)
\(930\) 0 0
\(931\) −5.50056 −0.180273
\(932\) 0 0
\(933\) −24.3661 −0.797710
\(934\) 0 0
\(935\) −2.17384 −0.0710922
\(936\) 0 0
\(937\) −16.4319 −0.536807 −0.268403 0.963307i \(-0.586496\pi\)
−0.268403 + 0.963307i \(0.586496\pi\)
\(938\) 0 0
\(939\) −38.9755 −1.27192
\(940\) 0 0
\(941\) 16.5679 0.540099 0.270049 0.962846i \(-0.412960\pi\)
0.270049 + 0.962846i \(0.412960\pi\)
\(942\) 0 0
\(943\) −63.1652 −2.05694
\(944\) 0 0
\(945\) 8.67456 0.282183
\(946\) 0 0
\(947\) −17.7431 −0.576574 −0.288287 0.957544i \(-0.593086\pi\)
−0.288287 + 0.957544i \(0.593086\pi\)
\(948\) 0 0
\(949\) −22.7998 −0.740113
\(950\) 0 0
\(951\) −47.1739 −1.52972
\(952\) 0 0
\(953\) 21.9557 0.711213 0.355607 0.934636i \(-0.384274\pi\)
0.355607 + 0.934636i \(0.384274\pi\)
\(954\) 0 0
\(955\) −50.0138 −1.61841
\(956\) 0 0
\(957\) −8.29607 −0.268174
\(958\) 0 0
\(959\) 6.85872 0.221480
\(960\) 0 0
\(961\) 1.46955 0.0474047
\(962\) 0 0
\(963\) 21.3185 0.686980
\(964\) 0 0
\(965\) −6.18800 −0.199199
\(966\) 0 0
\(967\) 22.2420 0.715255 0.357628 0.933864i \(-0.383586\pi\)
0.357628 + 0.933864i \(0.383586\pi\)
\(968\) 0 0
\(969\) 21.2713 0.683333
\(970\) 0 0
\(971\) 26.5681 0.852612 0.426306 0.904579i \(-0.359815\pi\)
0.426306 + 0.904579i \(0.359815\pi\)
\(972\) 0 0
\(973\) 22.8721 0.733244
\(974\) 0 0
\(975\) −23.8832 −0.764873
\(976\) 0 0
\(977\) −9.54985 −0.305527 −0.152763 0.988263i \(-0.548817\pi\)
−0.152763 + 0.988263i \(0.548817\pi\)
\(978\) 0 0
\(979\) −0.727526 −0.0232518
\(980\) 0 0
\(981\) −14.7180 −0.469911
\(982\) 0 0
\(983\) 5.65760 0.180449 0.0902247 0.995921i \(-0.471241\pi\)
0.0902247 + 0.995921i \(0.471241\pi\)
\(984\) 0 0
\(985\) 22.6291 0.721023
\(986\) 0 0
\(987\) −0.977219 −0.0311052
\(988\) 0 0
\(989\) −4.43682 −0.141083
\(990\) 0 0
\(991\) −62.8260 −1.99573 −0.997867 0.0652866i \(-0.979204\pi\)
−0.997867 + 0.0652866i \(0.979204\pi\)
\(992\) 0 0
\(993\) 16.3216 0.517951
\(994\) 0 0
\(995\) 24.2287 0.768100
\(996\) 0 0
\(997\) 8.14444 0.257937 0.128969 0.991649i \(-0.458833\pi\)
0.128969 + 0.991649i \(0.458833\pi\)
\(998\) 0 0
\(999\) 7.88449 0.249454
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.bf.1.2 8
4.3 odd 2 7168.2.a.be.1.7 8
8.3 odd 2 7168.2.a.ba.1.2 8
8.5 even 2 7168.2.a.bb.1.7 8
32.3 odd 8 1792.2.m.f.1345.8 yes 16
32.5 even 8 1792.2.m.e.449.8 16
32.11 odd 8 1792.2.m.f.449.8 yes 16
32.13 even 8 1792.2.m.e.1345.8 yes 16
32.19 odd 8 1792.2.m.g.1345.1 yes 16
32.21 even 8 1792.2.m.h.449.1 yes 16
32.27 odd 8 1792.2.m.g.449.1 yes 16
32.29 even 8 1792.2.m.h.1345.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.e.449.8 16 32.5 even 8
1792.2.m.e.1345.8 yes 16 32.13 even 8
1792.2.m.f.449.8 yes 16 32.11 odd 8
1792.2.m.f.1345.8 yes 16 32.3 odd 8
1792.2.m.g.449.1 yes 16 32.27 odd 8
1792.2.m.g.1345.1 yes 16 32.19 odd 8
1792.2.m.h.449.1 yes 16 32.21 even 8
1792.2.m.h.1345.1 yes 16 32.29 even 8
7168.2.a.ba.1.2 8 8.3 odd 2
7168.2.a.bb.1.7 8 8.5 even 2
7168.2.a.be.1.7 8 4.3 odd 2
7168.2.a.bf.1.2 8 1.1 even 1 trivial