Properties

Label 7623.2.a.ct.1.5
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.226211\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.226211 q^{2} -1.94883 q^{4} +2.49552 q^{5} +1.00000 q^{7} +0.893270 q^{8} +O(q^{10})\) \(q-0.226211 q^{2} -1.94883 q^{4} +2.49552 q^{5} +1.00000 q^{7} +0.893270 q^{8} -0.564516 q^{10} +5.13499 q^{13} -0.226211 q^{14} +3.69559 q^{16} +1.43752 q^{17} -6.06848 q^{19} -4.86335 q^{20} -7.08292 q^{23} +1.22764 q^{25} -1.16159 q^{26} -1.94883 q^{28} -6.51769 q^{29} +7.68895 q^{31} -2.62252 q^{32} -0.325184 q^{34} +2.49552 q^{35} -3.98432 q^{37} +1.37276 q^{38} +2.22918 q^{40} -6.74900 q^{41} -0.802299 q^{43} +1.60224 q^{46} -6.75222 q^{47} +1.00000 q^{49} -0.277706 q^{50} -10.0072 q^{52} -6.58167 q^{53} +0.893270 q^{56} +1.47437 q^{58} -2.87625 q^{59} -0.855342 q^{61} -1.73933 q^{62} -6.79793 q^{64} +12.8145 q^{65} -1.64668 q^{67} -2.80149 q^{68} -0.564516 q^{70} -4.52077 q^{71} -14.8479 q^{73} +0.901299 q^{74} +11.8264 q^{76} +2.45291 q^{79} +9.22243 q^{80} +1.52670 q^{82} -2.24780 q^{83} +3.58738 q^{85} +0.181489 q^{86} -1.73566 q^{89} +5.13499 q^{91} +13.8034 q^{92} +1.52743 q^{94} -15.1440 q^{95} -12.0776 q^{97} -0.226211 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + O(q^{10}) \) \( 8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + 6q^{10} - 6q^{13} - q^{14} + q^{16} + 5q^{17} - 13q^{19} - 23q^{20} - 16q^{23} + 16q^{25} + 6q^{26} + 7q^{28} - 9q^{29} + 9q^{31} - 16q^{32} - 12q^{34} - 10q^{35} + 7q^{37} + 10q^{38} + 5q^{40} + 10q^{41} - 4q^{43} + 4q^{46} - 16q^{47} + 8q^{49} - 6q^{50} - 41q^{52} - 37q^{53} - 15q^{58} - q^{59} + 19q^{61} + 18q^{62} - 4q^{64} + 4q^{65} - 19q^{67} - 9q^{68} + 6q^{70} - 13q^{71} - 25q^{73} - 33q^{74} + 26q^{76} - 4q^{80} - 13q^{82} + 25q^{83} + 3q^{85} - 4q^{86} - 37q^{89} - 6q^{91} - 35q^{92} - 42q^{94} - 21q^{95} + 15q^{97} - q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.226211 −0.159956 −0.0799778 0.996797i \(-0.525485\pi\)
−0.0799778 + 0.996797i \(0.525485\pi\)
\(3\) 0 0
\(4\) −1.94883 −0.974414
\(5\) 2.49552 1.11603 0.558016 0.829830i \(-0.311563\pi\)
0.558016 + 0.829830i \(0.311563\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0.893270 0.315819
\(9\) 0 0
\(10\) −0.564516 −0.178516
\(11\) 0 0
\(12\) 0 0
\(13\) 5.13499 1.42419 0.712094 0.702084i \(-0.247747\pi\)
0.712094 + 0.702084i \(0.247747\pi\)
\(14\) −0.226211 −0.0604575
\(15\) 0 0
\(16\) 3.69559 0.923897
\(17\) 1.43752 0.348651 0.174325 0.984688i \(-0.444226\pi\)
0.174325 + 0.984688i \(0.444226\pi\)
\(18\) 0 0
\(19\) −6.06848 −1.39220 −0.696102 0.717943i \(-0.745084\pi\)
−0.696102 + 0.717943i \(0.745084\pi\)
\(20\) −4.86335 −1.08748
\(21\) 0 0
\(22\) 0 0
\(23\) −7.08292 −1.47689 −0.738446 0.674313i \(-0.764440\pi\)
−0.738446 + 0.674313i \(0.764440\pi\)
\(24\) 0 0
\(25\) 1.22764 0.245528
\(26\) −1.16159 −0.227807
\(27\) 0 0
\(28\) −1.94883 −0.368294
\(29\) −6.51769 −1.21030 −0.605152 0.796110i \(-0.706888\pi\)
−0.605152 + 0.796110i \(0.706888\pi\)
\(30\) 0 0
\(31\) 7.68895 1.38098 0.690488 0.723344i \(-0.257396\pi\)
0.690488 + 0.723344i \(0.257396\pi\)
\(32\) −2.62252 −0.463601
\(33\) 0 0
\(34\) −0.325184 −0.0557687
\(35\) 2.49552 0.421820
\(36\) 0 0
\(37\) −3.98432 −0.655019 −0.327509 0.944848i \(-0.606209\pi\)
−0.327509 + 0.944848i \(0.606209\pi\)
\(38\) 1.37276 0.222691
\(39\) 0 0
\(40\) 2.22918 0.352464
\(41\) −6.74900 −1.05402 −0.527008 0.849860i \(-0.676686\pi\)
−0.527008 + 0.849860i \(0.676686\pi\)
\(42\) 0 0
\(43\) −0.802299 −0.122349 −0.0611747 0.998127i \(-0.519485\pi\)
−0.0611747 + 0.998127i \(0.519485\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.60224 0.236237
\(47\) −6.75222 −0.984912 −0.492456 0.870337i \(-0.663901\pi\)
−0.492456 + 0.870337i \(0.663901\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −0.277706 −0.0392735
\(51\) 0 0
\(52\) −10.0072 −1.38775
\(53\) −6.58167 −0.904062 −0.452031 0.892002i \(-0.649300\pi\)
−0.452031 + 0.892002i \(0.649300\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.893270 0.119368
\(57\) 0 0
\(58\) 1.47437 0.193595
\(59\) −2.87625 −0.374456 −0.187228 0.982316i \(-0.559950\pi\)
−0.187228 + 0.982316i \(0.559950\pi\)
\(60\) 0 0
\(61\) −0.855342 −0.109515 −0.0547576 0.998500i \(-0.517439\pi\)
−0.0547576 + 0.998500i \(0.517439\pi\)
\(62\) −1.73933 −0.220895
\(63\) 0 0
\(64\) −6.79793 −0.849742
\(65\) 12.8145 1.58944
\(66\) 0 0
\(67\) −1.64668 −0.201174 −0.100587 0.994928i \(-0.532072\pi\)
−0.100587 + 0.994928i \(0.532072\pi\)
\(68\) −2.80149 −0.339730
\(69\) 0 0
\(70\) −0.564516 −0.0674725
\(71\) −4.52077 −0.536517 −0.268258 0.963347i \(-0.586448\pi\)
−0.268258 + 0.963347i \(0.586448\pi\)
\(72\) 0 0
\(73\) −14.8479 −1.73782 −0.868910 0.494970i \(-0.835179\pi\)
−0.868910 + 0.494970i \(0.835179\pi\)
\(74\) 0.901299 0.104774
\(75\) 0 0
\(76\) 11.8264 1.35658
\(77\) 0 0
\(78\) 0 0
\(79\) 2.45291 0.275973 0.137987 0.990434i \(-0.455937\pi\)
0.137987 + 0.990434i \(0.455937\pi\)
\(80\) 9.22243 1.03110
\(81\) 0 0
\(82\) 1.52670 0.168596
\(83\) −2.24780 −0.246728 −0.123364 0.992361i \(-0.539368\pi\)
−0.123364 + 0.992361i \(0.539368\pi\)
\(84\) 0 0
\(85\) 3.58738 0.389106
\(86\) 0.181489 0.0195705
\(87\) 0 0
\(88\) 0 0
\(89\) −1.73566 −0.183980 −0.0919898 0.995760i \(-0.529323\pi\)
−0.0919898 + 0.995760i \(0.529323\pi\)
\(90\) 0 0
\(91\) 5.13499 0.538293
\(92\) 13.8034 1.43910
\(93\) 0 0
\(94\) 1.52743 0.157542
\(95\) −15.1440 −1.55374
\(96\) 0 0
\(97\) −12.0776 −1.22629 −0.613145 0.789970i \(-0.710096\pi\)
−0.613145 + 0.789970i \(0.710096\pi\)
\(98\) −0.226211 −0.0228508
\(99\) 0 0
\(100\) −2.39246 −0.239246
\(101\) 3.69338 0.367505 0.183753 0.982973i \(-0.441175\pi\)
0.183753 + 0.982973i \(0.441175\pi\)
\(102\) 0 0
\(103\) −1.15156 −0.113467 −0.0567334 0.998389i \(-0.518069\pi\)
−0.0567334 + 0.998389i \(0.518069\pi\)
\(104\) 4.58693 0.449785
\(105\) 0 0
\(106\) 1.48885 0.144610
\(107\) −1.16714 −0.112831 −0.0564157 0.998407i \(-0.517967\pi\)
−0.0564157 + 0.998407i \(0.517967\pi\)
\(108\) 0 0
\(109\) 9.30234 0.891003 0.445501 0.895281i \(-0.353025\pi\)
0.445501 + 0.895281i \(0.353025\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.69559 0.349200
\(113\) −3.29733 −0.310187 −0.155093 0.987900i \(-0.549568\pi\)
−0.155093 + 0.987900i \(0.549568\pi\)
\(114\) 0 0
\(115\) −17.6756 −1.64826
\(116\) 12.7019 1.17934
\(117\) 0 0
\(118\) 0.650640 0.0598963
\(119\) 1.43752 0.131778
\(120\) 0 0
\(121\) 0 0
\(122\) 0.193488 0.0175176
\(123\) 0 0
\(124\) −14.9844 −1.34564
\(125\) −9.41402 −0.842015
\(126\) 0 0
\(127\) −0.289205 −0.0256628 −0.0128314 0.999918i \(-0.504084\pi\)
−0.0128314 + 0.999918i \(0.504084\pi\)
\(128\) 6.78282 0.599522
\(129\) 0 0
\(130\) −2.89878 −0.254240
\(131\) 16.5059 1.44212 0.721062 0.692871i \(-0.243654\pi\)
0.721062 + 0.692871i \(0.243654\pi\)
\(132\) 0 0
\(133\) −6.06848 −0.526204
\(134\) 0.372498 0.0321789
\(135\) 0 0
\(136\) 1.28410 0.110110
\(137\) −9.32588 −0.796763 −0.398382 0.917220i \(-0.630428\pi\)
−0.398382 + 0.917220i \(0.630428\pi\)
\(138\) 0 0
\(139\) 4.82649 0.409377 0.204689 0.978827i \(-0.434382\pi\)
0.204689 + 0.978827i \(0.434382\pi\)
\(140\) −4.86335 −0.411028
\(141\) 0 0
\(142\) 1.02265 0.0858189
\(143\) 0 0
\(144\) 0 0
\(145\) −16.2650 −1.35074
\(146\) 3.35877 0.277974
\(147\) 0 0
\(148\) 7.76476 0.638260
\(149\) −0.921915 −0.0755262 −0.0377631 0.999287i \(-0.512023\pi\)
−0.0377631 + 0.999287i \(0.512023\pi\)
\(150\) 0 0
\(151\) 18.0964 1.47267 0.736333 0.676620i \(-0.236556\pi\)
0.736333 + 0.676620i \(0.236556\pi\)
\(152\) −5.42079 −0.439684
\(153\) 0 0
\(154\) 0 0
\(155\) 19.1880 1.54121
\(156\) 0 0
\(157\) −12.2733 −0.979518 −0.489759 0.871858i \(-0.662915\pi\)
−0.489759 + 0.871858i \(0.662915\pi\)
\(158\) −0.554875 −0.0441435
\(159\) 0 0
\(160\) −6.54457 −0.517394
\(161\) −7.08292 −0.558212
\(162\) 0 0
\(163\) 8.09913 0.634373 0.317186 0.948363i \(-0.397262\pi\)
0.317186 + 0.948363i \(0.397262\pi\)
\(164\) 13.1526 1.02705
\(165\) 0 0
\(166\) 0.508478 0.0394655
\(167\) 13.0516 1.00996 0.504982 0.863130i \(-0.331499\pi\)
0.504982 + 0.863130i \(0.331499\pi\)
\(168\) 0 0
\(169\) 13.3681 1.02831
\(170\) −0.811505 −0.0622396
\(171\) 0 0
\(172\) 1.56354 0.119219
\(173\) −5.91219 −0.449495 −0.224748 0.974417i \(-0.572156\pi\)
−0.224748 + 0.974417i \(0.572156\pi\)
\(174\) 0 0
\(175\) 1.22764 0.0928007
\(176\) 0 0
\(177\) 0 0
\(178\) 0.392626 0.0294286
\(179\) −4.33508 −0.324019 −0.162009 0.986789i \(-0.551798\pi\)
−0.162009 + 0.986789i \(0.551798\pi\)
\(180\) 0 0
\(181\) 10.8307 0.805040 0.402520 0.915411i \(-0.368134\pi\)
0.402520 + 0.915411i \(0.368134\pi\)
\(182\) −1.16159 −0.0861029
\(183\) 0 0
\(184\) −6.32696 −0.466430
\(185\) −9.94297 −0.731022
\(186\) 0 0
\(187\) 0 0
\(188\) 13.1589 0.959713
\(189\) 0 0
\(190\) 3.42575 0.248530
\(191\) −11.6556 −0.843370 −0.421685 0.906742i \(-0.638561\pi\)
−0.421685 + 0.906742i \(0.638561\pi\)
\(192\) 0 0
\(193\) 22.4454 1.61566 0.807829 0.589418i \(-0.200643\pi\)
0.807829 + 0.589418i \(0.200643\pi\)
\(194\) 2.73208 0.196152
\(195\) 0 0
\(196\) −1.94883 −0.139202
\(197\) −24.1022 −1.71721 −0.858604 0.512639i \(-0.828668\pi\)
−0.858604 + 0.512639i \(0.828668\pi\)
\(198\) 0 0
\(199\) 18.7205 1.32706 0.663531 0.748148i \(-0.269057\pi\)
0.663531 + 0.748148i \(0.269057\pi\)
\(200\) 1.09661 0.0775422
\(201\) 0 0
\(202\) −0.835485 −0.0587845
\(203\) −6.51769 −0.457452
\(204\) 0 0
\(205\) −16.8423 −1.17632
\(206\) 0.260497 0.0181497
\(207\) 0 0
\(208\) 18.9768 1.31580
\(209\) 0 0
\(210\) 0 0
\(211\) −7.56636 −0.520890 −0.260445 0.965489i \(-0.583869\pi\)
−0.260445 + 0.965489i \(0.583869\pi\)
\(212\) 12.8266 0.880931
\(213\) 0 0
\(214\) 0.264019 0.0180480
\(215\) −2.00216 −0.136546
\(216\) 0 0
\(217\) 7.68895 0.521960
\(218\) −2.10430 −0.142521
\(219\) 0 0
\(220\) 0 0
\(221\) 7.38167 0.496545
\(222\) 0 0
\(223\) 17.5244 1.17352 0.586760 0.809761i \(-0.300403\pi\)
0.586760 + 0.809761i \(0.300403\pi\)
\(224\) −2.62252 −0.175225
\(225\) 0 0
\(226\) 0.745893 0.0496161
\(227\) 25.6773 1.70426 0.852130 0.523330i \(-0.175311\pi\)
0.852130 + 0.523330i \(0.175311\pi\)
\(228\) 0 0
\(229\) −19.8369 −1.31086 −0.655429 0.755257i \(-0.727512\pi\)
−0.655429 + 0.755257i \(0.727512\pi\)
\(230\) 3.99842 0.263648
\(231\) 0 0
\(232\) −5.82205 −0.382236
\(233\) 20.2146 1.32430 0.662151 0.749371i \(-0.269644\pi\)
0.662151 + 0.749371i \(0.269644\pi\)
\(234\) 0 0
\(235\) −16.8503 −1.09919
\(236\) 5.60532 0.364875
\(237\) 0 0
\(238\) −0.325184 −0.0210786
\(239\) −17.1004 −1.10613 −0.553066 0.833137i \(-0.686542\pi\)
−0.553066 + 0.833137i \(0.686542\pi\)
\(240\) 0 0
\(241\) −24.1529 −1.55582 −0.777912 0.628373i \(-0.783721\pi\)
−0.777912 + 0.628373i \(0.783721\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 1.66691 0.106713
\(245\) 2.49552 0.159433
\(246\) 0 0
\(247\) −31.1615 −1.98276
\(248\) 6.86831 0.436138
\(249\) 0 0
\(250\) 2.12956 0.134685
\(251\) −11.8947 −0.750790 −0.375395 0.926865i \(-0.622493\pi\)
−0.375395 + 0.926865i \(0.622493\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0.0654215 0.00410491
\(255\) 0 0
\(256\) 12.0615 0.753845
\(257\) −22.9609 −1.43226 −0.716131 0.697966i \(-0.754089\pi\)
−0.716131 + 0.697966i \(0.754089\pi\)
\(258\) 0 0
\(259\) −3.98432 −0.247574
\(260\) −24.9732 −1.54877
\(261\) 0 0
\(262\) −3.73381 −0.230676
\(263\) −1.93774 −0.119486 −0.0597432 0.998214i \(-0.519028\pi\)
−0.0597432 + 0.998214i \(0.519028\pi\)
\(264\) 0 0
\(265\) −16.4247 −1.00896
\(266\) 1.37276 0.0841692
\(267\) 0 0
\(268\) 3.20910 0.196027
\(269\) −7.18676 −0.438184 −0.219092 0.975704i \(-0.570310\pi\)
−0.219092 + 0.975704i \(0.570310\pi\)
\(270\) 0 0
\(271\) 1.19110 0.0723543 0.0361772 0.999345i \(-0.488482\pi\)
0.0361772 + 0.999345i \(0.488482\pi\)
\(272\) 5.31250 0.322118
\(273\) 0 0
\(274\) 2.10962 0.127447
\(275\) 0 0
\(276\) 0 0
\(277\) 10.3402 0.621285 0.310642 0.950527i \(-0.399456\pi\)
0.310642 + 0.950527i \(0.399456\pi\)
\(278\) −1.09181 −0.0654822
\(279\) 0 0
\(280\) 2.22918 0.133219
\(281\) 13.1513 0.784541 0.392271 0.919850i \(-0.371690\pi\)
0.392271 + 0.919850i \(0.371690\pi\)
\(282\) 0 0
\(283\) 0.300031 0.0178350 0.00891748 0.999960i \(-0.497161\pi\)
0.00891748 + 0.999960i \(0.497161\pi\)
\(284\) 8.81021 0.522790
\(285\) 0 0
\(286\) 0 0
\(287\) −6.74900 −0.398381
\(288\) 0 0
\(289\) −14.9335 −0.878443
\(290\) 3.67934 0.216058
\(291\) 0 0
\(292\) 28.9361 1.69336
\(293\) −16.1441 −0.943148 −0.471574 0.881826i \(-0.656314\pi\)
−0.471574 + 0.881826i \(0.656314\pi\)
\(294\) 0 0
\(295\) −7.17775 −0.417905
\(296\) −3.55908 −0.206867
\(297\) 0 0
\(298\) 0.208548 0.0120808
\(299\) −36.3707 −2.10337
\(300\) 0 0
\(301\) −0.802299 −0.0462437
\(302\) −4.09361 −0.235561
\(303\) 0 0
\(304\) −22.4266 −1.28625
\(305\) −2.13453 −0.122223
\(306\) 0 0
\(307\) −28.6376 −1.63443 −0.817217 0.576330i \(-0.804484\pi\)
−0.817217 + 0.576330i \(0.804484\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.34053 −0.246526
\(311\) 31.8228 1.80450 0.902252 0.431210i \(-0.141913\pi\)
0.902252 + 0.431210i \(0.141913\pi\)
\(312\) 0 0
\(313\) 0.0342232 0.00193441 0.000967206 1.00000i \(-0.499692\pi\)
0.000967206 1.00000i \(0.499692\pi\)
\(314\) 2.77636 0.156679
\(315\) 0 0
\(316\) −4.78029 −0.268912
\(317\) −22.3894 −1.25752 −0.628758 0.777601i \(-0.716436\pi\)
−0.628758 + 0.777601i \(0.716436\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −16.9644 −0.948339
\(321\) 0 0
\(322\) 1.60224 0.0892892
\(323\) −8.72359 −0.485393
\(324\) 0 0
\(325\) 6.30391 0.349678
\(326\) −1.83211 −0.101471
\(327\) 0 0
\(328\) −6.02868 −0.332878
\(329\) −6.75222 −0.372262
\(330\) 0 0
\(331\) 10.7577 0.591297 0.295648 0.955297i \(-0.404464\pi\)
0.295648 + 0.955297i \(0.404464\pi\)
\(332\) 4.38057 0.240415
\(333\) 0 0
\(334\) −2.95242 −0.161549
\(335\) −4.10933 −0.224517
\(336\) 0 0
\(337\) 7.50492 0.408819 0.204410 0.978885i \(-0.434473\pi\)
0.204410 + 0.978885i \(0.434473\pi\)
\(338\) −3.02401 −0.164485
\(339\) 0 0
\(340\) −6.99118 −0.379150
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −0.716669 −0.0386402
\(345\) 0 0
\(346\) 1.33740 0.0718993
\(347\) 27.6818 1.48604 0.743019 0.669271i \(-0.233393\pi\)
0.743019 + 0.669271i \(0.233393\pi\)
\(348\) 0 0
\(349\) −11.0605 −0.592055 −0.296028 0.955179i \(-0.595662\pi\)
−0.296028 + 0.955179i \(0.595662\pi\)
\(350\) −0.277706 −0.0148440
\(351\) 0 0
\(352\) 0 0
\(353\) −31.9202 −1.69894 −0.849469 0.527638i \(-0.823078\pi\)
−0.849469 + 0.527638i \(0.823078\pi\)
\(354\) 0 0
\(355\) −11.2817 −0.598770
\(356\) 3.38251 0.179272
\(357\) 0 0
\(358\) 0.980644 0.0518286
\(359\) −3.57826 −0.188854 −0.0944268 0.995532i \(-0.530102\pi\)
−0.0944268 + 0.995532i \(0.530102\pi\)
\(360\) 0 0
\(361\) 17.8264 0.938233
\(362\) −2.45003 −0.128771
\(363\) 0 0
\(364\) −10.0072 −0.524520
\(365\) −37.0534 −1.93946
\(366\) 0 0
\(367\) 2.29397 0.119744 0.0598720 0.998206i \(-0.480931\pi\)
0.0598720 + 0.998206i \(0.480931\pi\)
\(368\) −26.1756 −1.36450
\(369\) 0 0
\(370\) 2.24921 0.116931
\(371\) −6.58167 −0.341703
\(372\) 0 0
\(373\) 7.96856 0.412596 0.206298 0.978489i \(-0.433858\pi\)
0.206298 + 0.978489i \(0.433858\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.03155 −0.311054
\(377\) −33.4682 −1.72370
\(378\) 0 0
\(379\) −11.6212 −0.596941 −0.298470 0.954419i \(-0.596476\pi\)
−0.298470 + 0.954419i \(0.596476\pi\)
\(380\) 29.5131 1.51399
\(381\) 0 0
\(382\) 2.63663 0.134902
\(383\) −12.5785 −0.642729 −0.321364 0.946956i \(-0.604141\pi\)
−0.321364 + 0.946956i \(0.604141\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.07741 −0.258433
\(387\) 0 0
\(388\) 23.5371 1.19491
\(389\) 0.438294 0.0222224 0.0111112 0.999938i \(-0.496463\pi\)
0.0111112 + 0.999938i \(0.496463\pi\)
\(390\) 0 0
\(391\) −10.1819 −0.514920
\(392\) 0.893270 0.0451169
\(393\) 0 0
\(394\) 5.45218 0.274677
\(395\) 6.12128 0.307995
\(396\) 0 0
\(397\) 16.8147 0.843905 0.421952 0.906618i \(-0.361345\pi\)
0.421952 + 0.906618i \(0.361345\pi\)
\(398\) −4.23480 −0.212271
\(399\) 0 0
\(400\) 4.53685 0.226842
\(401\) −36.8609 −1.84074 −0.920372 0.391043i \(-0.872114\pi\)
−0.920372 + 0.391043i \(0.872114\pi\)
\(402\) 0 0
\(403\) 39.4827 1.96677
\(404\) −7.19776 −0.358102
\(405\) 0 0
\(406\) 1.47437 0.0731720
\(407\) 0 0
\(408\) 0 0
\(409\) 16.2739 0.804692 0.402346 0.915488i \(-0.368195\pi\)
0.402346 + 0.915488i \(0.368195\pi\)
\(410\) 3.80992 0.188158
\(411\) 0 0
\(412\) 2.24420 0.110564
\(413\) −2.87625 −0.141531
\(414\) 0 0
\(415\) −5.60944 −0.275356
\(416\) −13.4666 −0.660255
\(417\) 0 0
\(418\) 0 0
\(419\) −5.56352 −0.271796 −0.135898 0.990723i \(-0.543392\pi\)
−0.135898 + 0.990723i \(0.543392\pi\)
\(420\) 0 0
\(421\) 21.4914 1.04743 0.523713 0.851895i \(-0.324546\pi\)
0.523713 + 0.851895i \(0.324546\pi\)
\(422\) 1.71160 0.0833192
\(423\) 0 0
\(424\) −5.87921 −0.285520
\(425\) 1.76476 0.0856035
\(426\) 0 0
\(427\) −0.855342 −0.0413929
\(428\) 2.27455 0.109944
\(429\) 0 0
\(430\) 0.452910 0.0218413
\(431\) 27.7188 1.33517 0.667583 0.744536i \(-0.267329\pi\)
0.667583 + 0.744536i \(0.267329\pi\)
\(432\) 0 0
\(433\) 9.28812 0.446358 0.223179 0.974777i \(-0.428356\pi\)
0.223179 + 0.974777i \(0.428356\pi\)
\(434\) −1.73933 −0.0834904
\(435\) 0 0
\(436\) −18.1287 −0.868206
\(437\) 42.9826 2.05613
\(438\) 0 0
\(439\) −14.7118 −0.702156 −0.351078 0.936346i \(-0.614185\pi\)
−0.351078 + 0.936346i \(0.614185\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.66982 −0.0794251
\(443\) −2.44345 −0.116092 −0.0580459 0.998314i \(-0.518487\pi\)
−0.0580459 + 0.998314i \(0.518487\pi\)
\(444\) 0 0
\(445\) −4.33138 −0.205327
\(446\) −3.96421 −0.187711
\(447\) 0 0
\(448\) −6.79793 −0.321172
\(449\) −4.76935 −0.225080 −0.112540 0.993647i \(-0.535899\pi\)
−0.112540 + 0.993647i \(0.535899\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.42593 0.302250
\(453\) 0 0
\(454\) −5.80849 −0.272606
\(455\) 12.8145 0.600752
\(456\) 0 0
\(457\) 31.0875 1.45421 0.727106 0.686525i \(-0.240865\pi\)
0.727106 + 0.686525i \(0.240865\pi\)
\(458\) 4.48733 0.209679
\(459\) 0 0
\(460\) 34.4467 1.60609
\(461\) −29.7215 −1.38427 −0.692134 0.721769i \(-0.743329\pi\)
−0.692134 + 0.721769i \(0.743329\pi\)
\(462\) 0 0
\(463\) −25.4553 −1.18301 −0.591505 0.806302i \(-0.701466\pi\)
−0.591505 + 0.806302i \(0.701466\pi\)
\(464\) −24.0867 −1.11820
\(465\) 0 0
\(466\) −4.57277 −0.211829
\(467\) 3.18491 0.147380 0.0736901 0.997281i \(-0.476522\pi\)
0.0736901 + 0.997281i \(0.476522\pi\)
\(468\) 0 0
\(469\) −1.64668 −0.0760366
\(470\) 3.81173 0.175822
\(471\) 0 0
\(472\) −2.56927 −0.118260
\(473\) 0 0
\(474\) 0 0
\(475\) −7.44990 −0.341825
\(476\) −2.80149 −0.128406
\(477\) 0 0
\(478\) 3.86830 0.176932
\(479\) −3.23354 −0.147744 −0.0738720 0.997268i \(-0.523536\pi\)
−0.0738720 + 0.997268i \(0.523536\pi\)
\(480\) 0 0
\(481\) −20.4594 −0.932870
\(482\) 5.46366 0.248863
\(483\) 0 0
\(484\) 0 0
\(485\) −30.1398 −1.36858
\(486\) 0 0
\(487\) 9.87096 0.447296 0.223648 0.974670i \(-0.428203\pi\)
0.223648 + 0.974670i \(0.428203\pi\)
\(488\) −0.764051 −0.0345870
\(489\) 0 0
\(490\) −0.564516 −0.0255022
\(491\) −4.22600 −0.190717 −0.0953584 0.995443i \(-0.530400\pi\)
−0.0953584 + 0.995443i \(0.530400\pi\)
\(492\) 0 0
\(493\) −9.36933 −0.421974
\(494\) 7.04910 0.317154
\(495\) 0 0
\(496\) 28.4152 1.27588
\(497\) −4.52077 −0.202784
\(498\) 0 0
\(499\) −20.3953 −0.913018 −0.456509 0.889719i \(-0.650900\pi\)
−0.456509 + 0.889719i \(0.650900\pi\)
\(500\) 18.3463 0.820472
\(501\) 0 0
\(502\) 2.69073 0.120093
\(503\) 23.9593 1.06829 0.534145 0.845393i \(-0.320634\pi\)
0.534145 + 0.845393i \(0.320634\pi\)
\(504\) 0 0
\(505\) 9.21692 0.410147
\(506\) 0 0
\(507\) 0 0
\(508\) 0.563611 0.0250062
\(509\) −3.69341 −0.163708 −0.0818538 0.996644i \(-0.526084\pi\)
−0.0818538 + 0.996644i \(0.526084\pi\)
\(510\) 0 0
\(511\) −14.8479 −0.656834
\(512\) −16.2941 −0.720104
\(513\) 0 0
\(514\) 5.19402 0.229098
\(515\) −2.87375 −0.126633
\(516\) 0 0
\(517\) 0 0
\(518\) 0.901299 0.0396008
\(519\) 0 0
\(520\) 11.4468 0.501975
\(521\) 0.238270 0.0104388 0.00521939 0.999986i \(-0.498339\pi\)
0.00521939 + 0.999986i \(0.498339\pi\)
\(522\) 0 0
\(523\) −21.9914 −0.961618 −0.480809 0.876825i \(-0.659657\pi\)
−0.480809 + 0.876825i \(0.659657\pi\)
\(524\) −32.1671 −1.40523
\(525\) 0 0
\(526\) 0.438339 0.0191125
\(527\) 11.0531 0.481479
\(528\) 0 0
\(529\) 27.1678 1.18121
\(530\) 3.71546 0.161389
\(531\) 0 0
\(532\) 11.8264 0.512740
\(533\) −34.6560 −1.50112
\(534\) 0 0
\(535\) −2.91262 −0.125923
\(536\) −1.47093 −0.0635345
\(537\) 0 0
\(538\) 1.62573 0.0700900
\(539\) 0 0
\(540\) 0 0
\(541\) 28.6309 1.23094 0.615470 0.788161i \(-0.288966\pi\)
0.615470 + 0.788161i \(0.288966\pi\)
\(542\) −0.269441 −0.0115735
\(543\) 0 0
\(544\) −3.76994 −0.161635
\(545\) 23.2142 0.994388
\(546\) 0 0
\(547\) 6.40847 0.274007 0.137003 0.990571i \(-0.456253\pi\)
0.137003 + 0.990571i \(0.456253\pi\)
\(548\) 18.1745 0.776378
\(549\) 0 0
\(550\) 0 0
\(551\) 39.5524 1.68499
\(552\) 0 0
\(553\) 2.45291 0.104308
\(554\) −2.33908 −0.0993780
\(555\) 0 0
\(556\) −9.40600 −0.398903
\(557\) 22.5193 0.954172 0.477086 0.878857i \(-0.341693\pi\)
0.477086 + 0.878857i \(0.341693\pi\)
\(558\) 0 0
\(559\) −4.11979 −0.174249
\(560\) 9.22243 0.389719
\(561\) 0 0
\(562\) −2.97498 −0.125492
\(563\) 29.8267 1.25705 0.628523 0.777791i \(-0.283660\pi\)
0.628523 + 0.777791i \(0.283660\pi\)
\(564\) 0 0
\(565\) −8.22856 −0.346178
\(566\) −0.0678703 −0.00285280
\(567\) 0 0
\(568\) −4.03827 −0.169442
\(569\) −29.2764 −1.22733 −0.613665 0.789567i \(-0.710305\pi\)
−0.613665 + 0.789567i \(0.710305\pi\)
\(570\) 0 0
\(571\) 37.9252 1.58712 0.793559 0.608493i \(-0.208226\pi\)
0.793559 + 0.608493i \(0.208226\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1.52670 0.0637232
\(575\) −8.69527 −0.362618
\(576\) 0 0
\(577\) 8.77263 0.365209 0.182605 0.983186i \(-0.441547\pi\)
0.182605 + 0.983186i \(0.441547\pi\)
\(578\) 3.37813 0.140512
\(579\) 0 0
\(580\) 31.6978 1.31618
\(581\) −2.24780 −0.0932544
\(582\) 0 0
\(583\) 0 0
\(584\) −13.2632 −0.548836
\(585\) 0 0
\(586\) 3.65198 0.150862
\(587\) −9.17011 −0.378491 −0.189246 0.981930i \(-0.560604\pi\)
−0.189246 + 0.981930i \(0.560604\pi\)
\(588\) 0 0
\(589\) −46.6602 −1.92260
\(590\) 1.62369 0.0668462
\(591\) 0 0
\(592\) −14.7244 −0.605170
\(593\) −7.25596 −0.297967 −0.148983 0.988840i \(-0.547600\pi\)
−0.148983 + 0.988840i \(0.547600\pi\)
\(594\) 0 0
\(595\) 3.58738 0.147068
\(596\) 1.79665 0.0735938
\(597\) 0 0
\(598\) 8.22747 0.336446
\(599\) 20.0181 0.817916 0.408958 0.912553i \(-0.365892\pi\)
0.408958 + 0.912553i \(0.365892\pi\)
\(600\) 0 0
\(601\) −11.2706 −0.459736 −0.229868 0.973222i \(-0.573829\pi\)
−0.229868 + 0.973222i \(0.573829\pi\)
\(602\) 0.181489 0.00739694
\(603\) 0 0
\(604\) −35.2668 −1.43499
\(605\) 0 0
\(606\) 0 0
\(607\) −13.5260 −0.549005 −0.274502 0.961586i \(-0.588513\pi\)
−0.274502 + 0.961586i \(0.588513\pi\)
\(608\) 15.9147 0.645427
\(609\) 0 0
\(610\) 0.482854 0.0195502
\(611\) −34.6725 −1.40270
\(612\) 0 0
\(613\) −30.7968 −1.24387 −0.621935 0.783069i \(-0.713653\pi\)
−0.621935 + 0.783069i \(0.713653\pi\)
\(614\) 6.47815 0.261437
\(615\) 0 0
\(616\) 0 0
\(617\) −23.6896 −0.953707 −0.476853 0.878983i \(-0.658223\pi\)
−0.476853 + 0.878983i \(0.658223\pi\)
\(618\) 0 0
\(619\) −32.4878 −1.30579 −0.652897 0.757446i \(-0.726447\pi\)
−0.652897 + 0.757446i \(0.726447\pi\)
\(620\) −37.3940 −1.50178
\(621\) 0 0
\(622\) −7.19867 −0.288640
\(623\) −1.73566 −0.0695378
\(624\) 0 0
\(625\) −29.6311 −1.18524
\(626\) −0.00774169 −0.000309420 0
\(627\) 0 0
\(628\) 23.9186 0.954456
\(629\) −5.72756 −0.228373
\(630\) 0 0
\(631\) −15.1333 −0.602448 −0.301224 0.953553i \(-0.597395\pi\)
−0.301224 + 0.953553i \(0.597395\pi\)
\(632\) 2.19111 0.0871575
\(633\) 0 0
\(634\) 5.06474 0.201147
\(635\) −0.721718 −0.0286405
\(636\) 0 0
\(637\) 5.13499 0.203456
\(638\) 0 0
\(639\) 0 0
\(640\) 16.9267 0.669086
\(641\) 16.5502 0.653695 0.326848 0.945077i \(-0.394014\pi\)
0.326848 + 0.945077i \(0.394014\pi\)
\(642\) 0 0
\(643\) 1.99506 0.0786773 0.0393387 0.999226i \(-0.487475\pi\)
0.0393387 + 0.999226i \(0.487475\pi\)
\(644\) 13.8034 0.543930
\(645\) 0 0
\(646\) 1.97337 0.0776414
\(647\) −40.4517 −1.59032 −0.795160 0.606399i \(-0.792613\pi\)
−0.795160 + 0.606399i \(0.792613\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −1.42602 −0.0559329
\(651\) 0 0
\(652\) −15.7838 −0.618142
\(653\) 41.4856 1.62346 0.811729 0.584034i \(-0.198527\pi\)
0.811729 + 0.584034i \(0.198527\pi\)
\(654\) 0 0
\(655\) 41.1908 1.60946
\(656\) −24.9415 −0.973803
\(657\) 0 0
\(658\) 1.52743 0.0595454
\(659\) 51.1359 1.99197 0.995985 0.0895158i \(-0.0285320\pi\)
0.995985 + 0.0895158i \(0.0285320\pi\)
\(660\) 0 0
\(661\) −42.8840 −1.66800 −0.833998 0.551768i \(-0.813954\pi\)
−0.833998 + 0.551768i \(0.813954\pi\)
\(662\) −2.43351 −0.0945812
\(663\) 0 0
\(664\) −2.00789 −0.0779213
\(665\) −15.1440 −0.587260
\(666\) 0 0
\(667\) 46.1643 1.78749
\(668\) −25.4353 −0.984122
\(669\) 0 0
\(670\) 0.929577 0.0359127
\(671\) 0 0
\(672\) 0 0
\(673\) −25.0072 −0.963958 −0.481979 0.876183i \(-0.660082\pi\)
−0.481979 + 0.876183i \(0.660082\pi\)
\(674\) −1.69770 −0.0653929
\(675\) 0 0
\(676\) −26.0521 −1.00200
\(677\) 9.91890 0.381214 0.190607 0.981666i \(-0.438954\pi\)
0.190607 + 0.981666i \(0.438954\pi\)
\(678\) 0 0
\(679\) −12.0776 −0.463494
\(680\) 3.20450 0.122887
\(681\) 0 0
\(682\) 0 0
\(683\) 39.8980 1.52666 0.763328 0.646011i \(-0.223564\pi\)
0.763328 + 0.646011i \(0.223564\pi\)
\(684\) 0 0
\(685\) −23.2729 −0.889214
\(686\) −0.226211 −0.00863679
\(687\) 0 0
\(688\) −2.96497 −0.113038
\(689\) −33.7968 −1.28756
\(690\) 0 0
\(691\) −6.61401 −0.251609 −0.125804 0.992055i \(-0.540151\pi\)
−0.125804 + 0.992055i \(0.540151\pi\)
\(692\) 11.5218 0.437995
\(693\) 0 0
\(694\) −6.26194 −0.237700
\(695\) 12.0446 0.456878
\(696\) 0 0
\(697\) −9.70185 −0.367484
\(698\) 2.50201 0.0947025
\(699\) 0 0
\(700\) −2.39246 −0.0904264
\(701\) −14.6016 −0.551495 −0.275748 0.961230i \(-0.588925\pi\)
−0.275748 + 0.961230i \(0.588925\pi\)
\(702\) 0 0
\(703\) 24.1788 0.911920
\(704\) 0 0
\(705\) 0 0
\(706\) 7.22070 0.271755
\(707\) 3.69338 0.138904
\(708\) 0 0
\(709\) −4.14406 −0.155633 −0.0778167 0.996968i \(-0.524795\pi\)
−0.0778167 + 0.996968i \(0.524795\pi\)
\(710\) 2.55205 0.0957766
\(711\) 0 0
\(712\) −1.55041 −0.0581042
\(713\) −54.4602 −2.03955
\(714\) 0 0
\(715\) 0 0
\(716\) 8.44832 0.315729
\(717\) 0 0
\(718\) 0.809444 0.0302082
\(719\) −17.0223 −0.634823 −0.317412 0.948288i \(-0.602814\pi\)
−0.317412 + 0.948288i \(0.602814\pi\)
\(720\) 0 0
\(721\) −1.15156 −0.0428864
\(722\) −4.03254 −0.150076
\(723\) 0 0
\(724\) −21.1072 −0.784442
\(725\) −8.00136 −0.297163
\(726\) 0 0
\(727\) −21.6199 −0.801837 −0.400918 0.916114i \(-0.631309\pi\)
−0.400918 + 0.916114i \(0.631309\pi\)
\(728\) 4.58693 0.170003
\(729\) 0 0
\(730\) 8.38190 0.310228
\(731\) −1.15332 −0.0426572
\(732\) 0 0
\(733\) −48.2326 −1.78151 −0.890755 0.454484i \(-0.849824\pi\)
−0.890755 + 0.454484i \(0.849824\pi\)
\(734\) −0.518921 −0.0191537
\(735\) 0 0
\(736\) 18.5751 0.684688
\(737\) 0 0
\(738\) 0 0
\(739\) −8.16347 −0.300298 −0.150149 0.988663i \(-0.547975\pi\)
−0.150149 + 0.988663i \(0.547975\pi\)
\(740\) 19.3772 0.712318
\(741\) 0 0
\(742\) 1.48885 0.0546574
\(743\) −19.5612 −0.717633 −0.358816 0.933408i \(-0.616820\pi\)
−0.358816 + 0.933408i \(0.616820\pi\)
\(744\) 0 0
\(745\) −2.30066 −0.0842897
\(746\) −1.80258 −0.0659971
\(747\) 0 0
\(748\) 0 0
\(749\) −1.16714 −0.0426462
\(750\) 0 0
\(751\) 1.11630 0.0407343 0.0203671 0.999793i \(-0.493516\pi\)
0.0203671 + 0.999793i \(0.493516\pi\)
\(752\) −24.9534 −0.909958
\(753\) 0 0
\(754\) 7.57089 0.275716
\(755\) 45.1600 1.64354
\(756\) 0 0
\(757\) −26.5773 −0.965969 −0.482984 0.875629i \(-0.660447\pi\)
−0.482984 + 0.875629i \(0.660447\pi\)
\(758\) 2.62885 0.0954840
\(759\) 0 0
\(760\) −13.5277 −0.490701
\(761\) −6.30003 −0.228376 −0.114188 0.993459i \(-0.536427\pi\)
−0.114188 + 0.993459i \(0.536427\pi\)
\(762\) 0 0
\(763\) 9.30234 0.336767
\(764\) 22.7148 0.821792
\(765\) 0 0
\(766\) 2.84539 0.102808
\(767\) −14.7695 −0.533296
\(768\) 0 0
\(769\) 13.1916 0.475700 0.237850 0.971302i \(-0.423557\pi\)
0.237850 + 0.971302i \(0.423557\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −43.7423 −1.57432
\(773\) −45.8178 −1.64795 −0.823976 0.566625i \(-0.808249\pi\)
−0.823976 + 0.566625i \(0.808249\pi\)
\(774\) 0 0
\(775\) 9.43925 0.339068
\(776\) −10.7885 −0.387285
\(777\) 0 0
\(778\) −0.0991470 −0.00355459
\(779\) 40.9562 1.46741
\(780\) 0 0
\(781\) 0 0
\(782\) 2.30326 0.0823643
\(783\) 0 0
\(784\) 3.69559 0.131985
\(785\) −30.6284 −1.09317
\(786\) 0 0
\(787\) −18.8479 −0.671856 −0.335928 0.941888i \(-0.609050\pi\)
−0.335928 + 0.941888i \(0.609050\pi\)
\(788\) 46.9710 1.67327
\(789\) 0 0
\(790\) −1.38470 −0.0492656
\(791\) −3.29733 −0.117240
\(792\) 0 0
\(793\) −4.39217 −0.155970
\(794\) −3.80367 −0.134987
\(795\) 0 0
\(796\) −36.4831 −1.29311
\(797\) 28.1613 0.997526 0.498763 0.866738i \(-0.333788\pi\)
0.498763 + 0.866738i \(0.333788\pi\)
\(798\) 0 0
\(799\) −9.70648 −0.343391
\(800\) −3.21951 −0.113827
\(801\) 0 0
\(802\) 8.33835 0.294437
\(803\) 0 0
\(804\) 0 0
\(805\) −17.6756 −0.622983
\(806\) −8.93143 −0.314596
\(807\) 0 0
\(808\) 3.29919 0.116065
\(809\) 6.54552 0.230128 0.115064 0.993358i \(-0.463293\pi\)
0.115064 + 0.993358i \(0.463293\pi\)
\(810\) 0 0
\(811\) 18.8046 0.660320 0.330160 0.943925i \(-0.392897\pi\)
0.330160 + 0.943925i \(0.392897\pi\)
\(812\) 12.7019 0.445748
\(813\) 0 0
\(814\) 0 0
\(815\) 20.2116 0.707980
\(816\) 0 0
\(817\) 4.86873 0.170335
\(818\) −3.68134 −0.128715
\(819\) 0 0
\(820\) 32.8227 1.14622
\(821\) 11.5880 0.404424 0.202212 0.979342i \(-0.435187\pi\)
0.202212 + 0.979342i \(0.435187\pi\)
\(822\) 0 0
\(823\) 12.3464 0.430368 0.215184 0.976574i \(-0.430965\pi\)
0.215184 + 0.976574i \(0.430965\pi\)
\(824\) −1.02866 −0.0358349
\(825\) 0 0
\(826\) 0.650640 0.0226387
\(827\) 4.49579 0.156334 0.0781669 0.996940i \(-0.475093\pi\)
0.0781669 + 0.996940i \(0.475093\pi\)
\(828\) 0 0
\(829\) −19.8629 −0.689866 −0.344933 0.938627i \(-0.612098\pi\)
−0.344933 + 0.938627i \(0.612098\pi\)
\(830\) 1.26892 0.0440448
\(831\) 0 0
\(832\) −34.9073 −1.21019
\(833\) 1.43752 0.0498073
\(834\) 0 0
\(835\) 32.5706 1.12715
\(836\) 0 0
\(837\) 0 0
\(838\) 1.25853 0.0434753
\(839\) 43.8528 1.51397 0.756984 0.653434i \(-0.226672\pi\)
0.756984 + 0.653434i \(0.226672\pi\)
\(840\) 0 0
\(841\) 13.4802 0.464836
\(842\) −4.86160 −0.167542
\(843\) 0 0
\(844\) 14.7455 0.507562
\(845\) 33.3604 1.14763
\(846\) 0 0
\(847\) 0 0
\(848\) −24.3232 −0.835261
\(849\) 0 0
\(850\) −0.399209 −0.0136928
\(851\) 28.2207 0.967392
\(852\) 0 0
\(853\) 39.1407 1.34015 0.670076 0.742293i \(-0.266262\pi\)
0.670076 + 0.742293i \(0.266262\pi\)
\(854\) 0.193488 0.00662102
\(855\) 0 0
\(856\) −1.04257 −0.0356342
\(857\) −35.0524 −1.19737 −0.598684 0.800986i \(-0.704309\pi\)
−0.598684 + 0.800986i \(0.704309\pi\)
\(858\) 0 0
\(859\) −32.5206 −1.10959 −0.554794 0.831988i \(-0.687203\pi\)
−0.554794 + 0.831988i \(0.687203\pi\)
\(860\) 3.90186 0.133052
\(861\) 0 0
\(862\) −6.27030 −0.213567
\(863\) −12.6940 −0.432107 −0.216054 0.976381i \(-0.569319\pi\)
−0.216054 + 0.976381i \(0.569319\pi\)
\(864\) 0 0
\(865\) −14.7540 −0.501651
\(866\) −2.10108 −0.0713975
\(867\) 0 0
\(868\) −14.9844 −0.508605
\(869\) 0 0
\(870\) 0 0
\(871\) −8.45568 −0.286510
\(872\) 8.30950 0.281395
\(873\) 0 0
\(874\) −9.72314 −0.328890
\(875\) −9.41402 −0.318252
\(876\) 0 0
\(877\) −28.1340 −0.950017 −0.475008 0.879981i \(-0.657555\pi\)
−0.475008 + 0.879981i \(0.657555\pi\)
\(878\) 3.32798 0.112314
\(879\) 0 0
\(880\) 0 0
\(881\) 36.7964 1.23970 0.619850 0.784720i \(-0.287193\pi\)
0.619850 + 0.784720i \(0.287193\pi\)
\(882\) 0 0
\(883\) −2.28419 −0.0768692 −0.0384346 0.999261i \(-0.512237\pi\)
−0.0384346 + 0.999261i \(0.512237\pi\)
\(884\) −14.3856 −0.483840
\(885\) 0 0
\(886\) 0.552736 0.0185695
\(887\) −42.2676 −1.41921 −0.709604 0.704600i \(-0.751126\pi\)
−0.709604 + 0.704600i \(0.751126\pi\)
\(888\) 0 0
\(889\) −0.289205 −0.00969963
\(890\) 0.979808 0.0328432
\(891\) 0 0
\(892\) −34.1520 −1.14349
\(893\) 40.9757 1.37120
\(894\) 0 0
\(895\) −10.8183 −0.361616
\(896\) 6.78282 0.226598
\(897\) 0 0
\(898\) 1.07888 0.0360027
\(899\) −50.1142 −1.67140
\(900\) 0 0
\(901\) −9.46132 −0.315202
\(902\) 0 0
\(903\) 0 0
\(904\) −2.94540 −0.0979627
\(905\) 27.0283 0.898451
\(906\) 0 0
\(907\) 39.5286 1.31252 0.656262 0.754533i \(-0.272136\pi\)
0.656262 + 0.754533i \(0.272136\pi\)
\(908\) −50.0406 −1.66065
\(909\) 0 0
\(910\) −2.89878 −0.0960936
\(911\) 35.2296 1.16721 0.583604 0.812039i \(-0.301642\pi\)
0.583604 + 0.812039i \(0.301642\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −7.03234 −0.232609
\(915\) 0 0
\(916\) 38.6587 1.27732
\(917\) 16.5059 0.545072
\(918\) 0 0
\(919\) −39.6105 −1.30663 −0.653314 0.757087i \(-0.726622\pi\)
−0.653314 + 0.757087i \(0.726622\pi\)
\(920\) −15.7891 −0.520551
\(921\) 0 0
\(922\) 6.72334 0.221421
\(923\) −23.2141 −0.764101
\(924\) 0 0
\(925\) −4.89131 −0.160825
\(926\) 5.75828 0.189229
\(927\) 0 0
\(928\) 17.0928 0.561098
\(929\) −2.61657 −0.0858469 −0.0429234 0.999078i \(-0.513667\pi\)
−0.0429234 + 0.999078i \(0.513667\pi\)
\(930\) 0 0
\(931\) −6.06848 −0.198886
\(932\) −39.3947 −1.29042
\(933\) 0 0
\(934\) −0.720463 −0.0235743
\(935\) 0 0
\(936\) 0 0
\(937\) 53.8401 1.75888 0.879439 0.476011i \(-0.157918\pi\)
0.879439 + 0.476011i \(0.157918\pi\)
\(938\) 0.372498 0.0121625
\(939\) 0 0
\(940\) 32.8384 1.07107
\(941\) −29.6476 −0.966484 −0.483242 0.875487i \(-0.660541\pi\)
−0.483242 + 0.875487i \(0.660541\pi\)
\(942\) 0 0
\(943\) 47.8026 1.55667
\(944\) −10.6294 −0.345959
\(945\) 0 0
\(946\) 0 0
\(947\) 15.7861 0.512980 0.256490 0.966547i \(-0.417434\pi\)
0.256490 + 0.966547i \(0.417434\pi\)
\(948\) 0 0
\(949\) −76.2440 −2.47498
\(950\) 1.68525 0.0546768
\(951\) 0 0
\(952\) 1.28410 0.0416178
\(953\) 35.1119 1.13739 0.568693 0.822550i \(-0.307449\pi\)
0.568693 + 0.822550i \(0.307449\pi\)
\(954\) 0 0
\(955\) −29.0869 −0.941229
\(956\) 33.3257 1.07783
\(957\) 0 0
\(958\) 0.731463 0.0236325
\(959\) −9.32588 −0.301148
\(960\) 0 0
\(961\) 28.1200 0.907096
\(962\) 4.62816 0.149218
\(963\) 0 0
\(964\) 47.0698 1.51602
\(965\) 56.0131 1.80313
\(966\) 0 0
\(967\) −49.2820 −1.58480 −0.792401 0.610001i \(-0.791169\pi\)
−0.792401 + 0.610001i \(0.791169\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 6.81797 0.218912
\(971\) −37.3424 −1.19838 −0.599188 0.800608i \(-0.704510\pi\)
−0.599188 + 0.800608i \(0.704510\pi\)
\(972\) 0 0
\(973\) 4.82649 0.154730
\(974\) −2.23292 −0.0715475
\(975\) 0 0
\(976\) −3.16099 −0.101181
\(977\) −40.0485 −1.28126 −0.640632 0.767848i \(-0.721328\pi\)
−0.640632 + 0.767848i \(0.721328\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −4.86335 −0.155354
\(981\) 0 0
\(982\) 0.955970 0.0305062
\(983\) 52.0414 1.65986 0.829932 0.557864i \(-0.188379\pi\)
0.829932 + 0.557864i \(0.188379\pi\)
\(984\) 0 0
\(985\) −60.1475 −1.91646
\(986\) 2.11945 0.0674970
\(987\) 0 0
\(988\) 60.7285 1.93203
\(989\) 5.68262 0.180697
\(990\) 0 0
\(991\) 45.4828 1.44481 0.722404 0.691471i \(-0.243037\pi\)
0.722404 + 0.691471i \(0.243037\pi\)
\(992\) −20.1645 −0.640222
\(993\) 0 0
\(994\) 1.02265 0.0324365
\(995\) 46.7175 1.48104
\(996\) 0 0
\(997\) 27.9594 0.885482 0.442741 0.896650i \(-0.354006\pi\)
0.442741 + 0.896650i \(0.354006\pi\)
\(998\) 4.61364 0.146042
\(999\) 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 7623.2.a.ct.1.5 8
3.2 odd 2 847.2.a.p.1.4 8
11.5 even 5 693.2.m.i.190.2 16
11.9 even 5 693.2.m.i.631.2 16
11.10 odd 2 7623.2.a.cw.1.4 8
21.20 even 2 5929.2.a.bt.1.4 8
33.2 even 10 847.2.f.x.323.2 16
33.5 odd 10 77.2.f.b.36.3 yes 16
33.8 even 10 847.2.f.v.372.3 16
33.14 odd 10 847.2.f.w.372.2 16
33.17 even 10 847.2.f.x.729.2 16
33.20 odd 10 77.2.f.b.15.3 16
33.26 odd 10 847.2.f.w.148.2 16
33.29 even 10 847.2.f.v.148.3 16
33.32 even 2 847.2.a.o.1.5 8
231.5 even 30 539.2.q.f.410.2 32
231.20 even 10 539.2.f.e.246.3 16
231.38 even 30 539.2.q.f.520.3 32
231.53 odd 30 539.2.q.g.422.2 32
231.86 odd 30 539.2.q.g.312.3 32
231.104 even 10 539.2.f.e.344.3 16
231.137 odd 30 539.2.q.g.520.3 32
231.152 even 30 539.2.q.f.312.3 32
231.170 odd 30 539.2.q.g.410.2 32
231.185 even 30 539.2.q.f.422.2 32
231.230 odd 2 5929.2.a.bs.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.b.15.3 16 33.20 odd 10
77.2.f.b.36.3 yes 16 33.5 odd 10
539.2.f.e.246.3 16 231.20 even 10
539.2.f.e.344.3 16 231.104 even 10
539.2.q.f.312.3 32 231.152 even 30
539.2.q.f.410.2 32 231.5 even 30
539.2.q.f.422.2 32 231.185 even 30
539.2.q.f.520.3 32 231.38 even 30
539.2.q.g.312.3 32 231.86 odd 30
539.2.q.g.410.2 32 231.170 odd 30
539.2.q.g.422.2 32 231.53 odd 30
539.2.q.g.520.3 32 231.137 odd 30
693.2.m.i.190.2 16 11.5 even 5
693.2.m.i.631.2 16 11.9 even 5
847.2.a.o.1.5 8 33.32 even 2
847.2.a.p.1.4 8 3.2 odd 2
847.2.f.v.148.3 16 33.29 even 10
847.2.f.v.372.3 16 33.8 even 10
847.2.f.w.148.2 16 33.26 odd 10
847.2.f.w.372.2 16 33.14 odd 10
847.2.f.x.323.2 16 33.2 even 10
847.2.f.x.729.2 16 33.17 even 10
5929.2.a.bs.1.5 8 231.230 odd 2
5929.2.a.bt.1.4 8 21.20 even 2
7623.2.a.ct.1.5 8 1.1 even 1 trivial
7623.2.a.cw.1.4 8 11.10 odd 2