Properties

Label 7623.2.a.dc.1.13
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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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: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(2.08213\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.08213 q^{2} +2.33525 q^{4} +1.33222 q^{5} +1.00000 q^{7} +0.698024 q^{8} +O(q^{10})\) \(q+2.08213 q^{2} +2.33525 q^{4} +1.33222 q^{5} +1.00000 q^{7} +0.698024 q^{8} +2.77386 q^{10} +5.74691 q^{13} +2.08213 q^{14} -3.21712 q^{16} +4.27240 q^{17} +0.162553 q^{19} +3.11107 q^{20} +1.12680 q^{23} -3.22518 q^{25} +11.9658 q^{26} +2.33525 q^{28} +9.83547 q^{29} +9.09480 q^{31} -8.09449 q^{32} +8.89567 q^{34} +1.33222 q^{35} -8.42718 q^{37} +0.338455 q^{38} +0.929924 q^{40} +4.50712 q^{41} -8.02855 q^{43} +2.34613 q^{46} -2.79374 q^{47} +1.00000 q^{49} -6.71523 q^{50} +13.4204 q^{52} -10.4883 q^{53} +0.698024 q^{56} +20.4787 q^{58} -13.6125 q^{59} +8.06786 q^{61} +18.9365 q^{62} -10.4195 q^{64} +7.65617 q^{65} +9.10333 q^{67} +9.97710 q^{68} +2.77386 q^{70} +1.01904 q^{71} -7.64824 q^{73} -17.5464 q^{74} +0.379601 q^{76} -0.00547271 q^{79} -4.28592 q^{80} +9.38438 q^{82} +2.83233 q^{83} +5.69179 q^{85} -16.7165 q^{86} +10.5254 q^{89} +5.74691 q^{91} +2.63135 q^{92} -5.81693 q^{94} +0.216557 q^{95} +12.1613 q^{97} +2.08213 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 20q^{4} + 16q^{7} + O(q^{10}) \) \( 16q + 20q^{4} + 16q^{7} - 6q^{10} + 32q^{16} + 10q^{19} + 44q^{25} + 20q^{28} + 30q^{31} + 12q^{34} + 38q^{37} - 68q^{40} + 16q^{43} + 80q^{46} + 16q^{49} + 2q^{52} + 18q^{58} - 28q^{61} + 34q^{64} + 52q^{67} - 6q^{70} - 14q^{73} - 14q^{76} + 54q^{79} + 64q^{82} + 30q^{85} - 60q^{94} + 108q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.08213 1.47228 0.736142 0.676827i \(-0.236645\pi\)
0.736142 + 0.676827i \(0.236645\pi\)
\(3\) 0 0
\(4\) 2.33525 1.16762
\(5\) 1.33222 0.595789 0.297894 0.954599i \(-0.403716\pi\)
0.297894 + 0.954599i \(0.403716\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0.698024 0.246789
\(9\) 0 0
\(10\) 2.77386 0.877171
\(11\) 0 0
\(12\) 0 0
\(13\) 5.74691 1.59391 0.796953 0.604041i \(-0.206444\pi\)
0.796953 + 0.604041i \(0.206444\pi\)
\(14\) 2.08213 0.556471
\(15\) 0 0
\(16\) −3.21712 −0.804280
\(17\) 4.27240 1.03621 0.518105 0.855317i \(-0.326638\pi\)
0.518105 + 0.855317i \(0.326638\pi\)
\(18\) 0 0
\(19\) 0.162553 0.0372922 0.0186461 0.999826i \(-0.494064\pi\)
0.0186461 + 0.999826i \(0.494064\pi\)
\(20\) 3.11107 0.695657
\(21\) 0 0
\(22\) 0 0
\(23\) 1.12680 0.234954 0.117477 0.993076i \(-0.462519\pi\)
0.117477 + 0.993076i \(0.462519\pi\)
\(24\) 0 0
\(25\) −3.22518 −0.645036
\(26\) 11.9658 2.34668
\(27\) 0 0
\(28\) 2.33525 0.441320
\(29\) 9.83547 1.82640 0.913201 0.407510i \(-0.133603\pi\)
0.913201 + 0.407510i \(0.133603\pi\)
\(30\) 0 0
\(31\) 9.09480 1.63347 0.816737 0.577009i \(-0.195780\pi\)
0.816737 + 0.577009i \(0.195780\pi\)
\(32\) −8.09449 −1.43092
\(33\) 0 0
\(34\) 8.89567 1.52559
\(35\) 1.33222 0.225187
\(36\) 0 0
\(37\) −8.42718 −1.38542 −0.692710 0.721217i \(-0.743583\pi\)
−0.692710 + 0.721217i \(0.743583\pi\)
\(38\) 0.338455 0.0549047
\(39\) 0 0
\(40\) 0.929924 0.147034
\(41\) 4.50712 0.703894 0.351947 0.936020i \(-0.385520\pi\)
0.351947 + 0.936020i \(0.385520\pi\)
\(42\) 0 0
\(43\) −8.02855 −1.22434 −0.612172 0.790725i \(-0.709704\pi\)
−0.612172 + 0.790725i \(0.709704\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 2.34613 0.345919
\(47\) −2.79374 −0.407510 −0.203755 0.979022i \(-0.565315\pi\)
−0.203755 + 0.979022i \(0.565315\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −6.71523 −0.949677
\(51\) 0 0
\(52\) 13.4204 1.86108
\(53\) −10.4883 −1.44067 −0.720336 0.693625i \(-0.756013\pi\)
−0.720336 + 0.693625i \(0.756013\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.698024 0.0932774
\(57\) 0 0
\(58\) 20.4787 2.68898
\(59\) −13.6125 −1.77220 −0.886099 0.463497i \(-0.846595\pi\)
−0.886099 + 0.463497i \(0.846595\pi\)
\(60\) 0 0
\(61\) 8.06786 1.03298 0.516492 0.856292i \(-0.327238\pi\)
0.516492 + 0.856292i \(0.327238\pi\)
\(62\) 18.9365 2.40494
\(63\) 0 0
\(64\) −10.4195 −1.30244
\(65\) 7.65617 0.949631
\(66\) 0 0
\(67\) 9.10333 1.11215 0.556074 0.831133i \(-0.312307\pi\)
0.556074 + 0.831133i \(0.312307\pi\)
\(68\) 9.97710 1.20990
\(69\) 0 0
\(70\) 2.77386 0.331539
\(71\) 1.01904 0.120938 0.0604689 0.998170i \(-0.480740\pi\)
0.0604689 + 0.998170i \(0.480740\pi\)
\(72\) 0 0
\(73\) −7.64824 −0.895159 −0.447579 0.894244i \(-0.647714\pi\)
−0.447579 + 0.894244i \(0.647714\pi\)
\(74\) −17.5464 −2.03973
\(75\) 0 0
\(76\) 0.379601 0.0435432
\(77\) 0 0
\(78\) 0 0
\(79\) −0.00547271 −0.000615728 0 −0.000307864 1.00000i \(-0.500098\pi\)
−0.000307864 1.00000i \(0.500098\pi\)
\(80\) −4.28592 −0.479181
\(81\) 0 0
\(82\) 9.38438 1.03633
\(83\) 2.83233 0.310888 0.155444 0.987845i \(-0.450319\pi\)
0.155444 + 0.987845i \(0.450319\pi\)
\(84\) 0 0
\(85\) 5.69179 0.617362
\(86\) −16.7165 −1.80258
\(87\) 0 0
\(88\) 0 0
\(89\) 10.5254 1.11569 0.557845 0.829945i \(-0.311628\pi\)
0.557845 + 0.829945i \(0.311628\pi\)
\(90\) 0 0
\(91\) 5.74691 0.602440
\(92\) 2.63135 0.274337
\(93\) 0 0
\(94\) −5.81693 −0.599970
\(95\) 0.216557 0.0222182
\(96\) 0 0
\(97\) 12.1613 1.23479 0.617396 0.786652i \(-0.288188\pi\)
0.617396 + 0.786652i \(0.288188\pi\)
\(98\) 2.08213 0.210326
\(99\) 0 0
\(100\) −7.53159 −0.753159
\(101\) 1.52188 0.151433 0.0757164 0.997129i \(-0.475876\pi\)
0.0757164 + 0.997129i \(0.475876\pi\)
\(102\) 0 0
\(103\) 15.4665 1.52396 0.761979 0.647602i \(-0.224228\pi\)
0.761979 + 0.647602i \(0.224228\pi\)
\(104\) 4.01148 0.393358
\(105\) 0 0
\(106\) −21.8379 −2.12108
\(107\) −8.25043 −0.797599 −0.398799 0.917038i \(-0.630573\pi\)
−0.398799 + 0.917038i \(0.630573\pi\)
\(108\) 0 0
\(109\) 0.305943 0.0293040 0.0146520 0.999893i \(-0.495336\pi\)
0.0146520 + 0.999893i \(0.495336\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.21712 −0.303989
\(113\) 10.1503 0.954857 0.477428 0.878671i \(-0.341569\pi\)
0.477428 + 0.878671i \(0.341569\pi\)
\(114\) 0 0
\(115\) 1.50115 0.139983
\(116\) 22.9682 2.13255
\(117\) 0 0
\(118\) −28.3429 −2.60918
\(119\) 4.27240 0.391650
\(120\) 0 0
\(121\) 0 0
\(122\) 16.7983 1.52085
\(123\) 0 0
\(124\) 21.2386 1.90728
\(125\) −10.9578 −0.980094
\(126\) 0 0
\(127\) 3.80774 0.337882 0.168941 0.985626i \(-0.445965\pi\)
0.168941 + 0.985626i \(0.445965\pi\)
\(128\) −5.50574 −0.486643
\(129\) 0 0
\(130\) 15.9411 1.39813
\(131\) −14.6612 −1.28096 −0.640479 0.767976i \(-0.721264\pi\)
−0.640479 + 0.767976i \(0.721264\pi\)
\(132\) 0 0
\(133\) 0.162553 0.0140951
\(134\) 18.9543 1.63740
\(135\) 0 0
\(136\) 2.98224 0.255725
\(137\) 16.8306 1.43794 0.718968 0.695043i \(-0.244615\pi\)
0.718968 + 0.695043i \(0.244615\pi\)
\(138\) 0 0
\(139\) −7.79467 −0.661135 −0.330568 0.943782i \(-0.607240\pi\)
−0.330568 + 0.943782i \(0.607240\pi\)
\(140\) 3.11107 0.262933
\(141\) 0 0
\(142\) 2.12177 0.178055
\(143\) 0 0
\(144\) 0 0
\(145\) 13.1031 1.08815
\(146\) −15.9246 −1.31793
\(147\) 0 0
\(148\) −19.6795 −1.61765
\(149\) 9.71989 0.796284 0.398142 0.917324i \(-0.369655\pi\)
0.398142 + 0.917324i \(0.369655\pi\)
\(150\) 0 0
\(151\) 10.1015 0.822051 0.411026 0.911624i \(-0.365171\pi\)
0.411026 + 0.911624i \(0.365171\pi\)
\(152\) 0.113466 0.00920328
\(153\) 0 0
\(154\) 0 0
\(155\) 12.1163 0.973206
\(156\) 0 0
\(157\) −4.94915 −0.394985 −0.197493 0.980304i \(-0.563280\pi\)
−0.197493 + 0.980304i \(0.563280\pi\)
\(158\) −0.0113949 −0.000906526 0
\(159\) 0 0
\(160\) −10.7837 −0.852524
\(161\) 1.12680 0.0888041
\(162\) 0 0
\(163\) 6.73297 0.527367 0.263683 0.964609i \(-0.415063\pi\)
0.263683 + 0.964609i \(0.415063\pi\)
\(164\) 10.5252 0.821882
\(165\) 0 0
\(166\) 5.89726 0.457716
\(167\) 13.9874 1.08238 0.541188 0.840902i \(-0.317975\pi\)
0.541188 + 0.840902i \(0.317975\pi\)
\(168\) 0 0
\(169\) 20.0270 1.54054
\(170\) 11.8510 0.908932
\(171\) 0 0
\(172\) −18.7486 −1.42957
\(173\) −3.50877 −0.266767 −0.133383 0.991065i \(-0.542584\pi\)
−0.133383 + 0.991065i \(0.542584\pi\)
\(174\) 0 0
\(175\) −3.22518 −0.243801
\(176\) 0 0
\(177\) 0 0
\(178\) 21.9152 1.64261
\(179\) 2.22735 0.166480 0.0832399 0.996530i \(-0.473473\pi\)
0.0832399 + 0.996530i \(0.473473\pi\)
\(180\) 0 0
\(181\) 1.11432 0.0828265 0.0414132 0.999142i \(-0.486814\pi\)
0.0414132 + 0.999142i \(0.486814\pi\)
\(182\) 11.9658 0.886963
\(183\) 0 0
\(184\) 0.786532 0.0579839
\(185\) −11.2269 −0.825417
\(186\) 0 0
\(187\) 0 0
\(188\) −6.52408 −0.475818
\(189\) 0 0
\(190\) 0.450898 0.0327116
\(191\) 10.2314 0.740320 0.370160 0.928968i \(-0.379303\pi\)
0.370160 + 0.928968i \(0.379303\pi\)
\(192\) 0 0
\(193\) −17.9336 −1.29089 −0.645443 0.763808i \(-0.723327\pi\)
−0.645443 + 0.763808i \(0.723327\pi\)
\(194\) 25.3213 1.81797
\(195\) 0 0
\(196\) 2.33525 0.166803
\(197\) 4.30284 0.306564 0.153282 0.988182i \(-0.451016\pi\)
0.153282 + 0.988182i \(0.451016\pi\)
\(198\) 0 0
\(199\) 22.0560 1.56351 0.781753 0.623588i \(-0.214326\pi\)
0.781753 + 0.623588i \(0.214326\pi\)
\(200\) −2.25125 −0.159188
\(201\) 0 0
\(202\) 3.16875 0.222952
\(203\) 9.83547 0.690315
\(204\) 0 0
\(205\) 6.00449 0.419372
\(206\) 32.2031 2.24370
\(207\) 0 0
\(208\) −18.4885 −1.28195
\(209\) 0 0
\(210\) 0 0
\(211\) −14.8525 −1.02249 −0.511244 0.859435i \(-0.670815\pi\)
−0.511244 + 0.859435i \(0.670815\pi\)
\(212\) −24.4927 −1.68216
\(213\) 0 0
\(214\) −17.1784 −1.17429
\(215\) −10.6958 −0.729450
\(216\) 0 0
\(217\) 9.09480 0.617395
\(218\) 0.637011 0.0431438
\(219\) 0 0
\(220\) 0 0
\(221\) 24.5531 1.65162
\(222\) 0 0
\(223\) 5.59782 0.374857 0.187429 0.982278i \(-0.439985\pi\)
0.187429 + 0.982278i \(0.439985\pi\)
\(224\) −8.09449 −0.540836
\(225\) 0 0
\(226\) 21.1341 1.40582
\(227\) −7.52880 −0.499704 −0.249852 0.968284i \(-0.580382\pi\)
−0.249852 + 0.968284i \(0.580382\pi\)
\(228\) 0 0
\(229\) −27.2330 −1.79961 −0.899805 0.436292i \(-0.856292\pi\)
−0.899805 + 0.436292i \(0.856292\pi\)
\(230\) 3.12558 0.206094
\(231\) 0 0
\(232\) 6.86539 0.450735
\(233\) −5.48986 −0.359653 −0.179826 0.983698i \(-0.557554\pi\)
−0.179826 + 0.983698i \(0.557554\pi\)
\(234\) 0 0
\(235\) −3.72189 −0.242790
\(236\) −31.7885 −2.06926
\(237\) 0 0
\(238\) 8.89567 0.576621
\(239\) −24.8413 −1.60685 −0.803426 0.595405i \(-0.796992\pi\)
−0.803426 + 0.595405i \(0.796992\pi\)
\(240\) 0 0
\(241\) 17.3238 1.11592 0.557961 0.829867i \(-0.311584\pi\)
0.557961 + 0.829867i \(0.311584\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 18.8404 1.20613
\(245\) 1.33222 0.0851127
\(246\) 0 0
\(247\) 0.934176 0.0594402
\(248\) 6.34839 0.403123
\(249\) 0 0
\(250\) −22.8155 −1.44298
\(251\) −24.6655 −1.55687 −0.778436 0.627724i \(-0.783987\pi\)
−0.778436 + 0.627724i \(0.783987\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 7.92819 0.497459
\(255\) 0 0
\(256\) 9.37538 0.585961
\(257\) 3.87897 0.241964 0.120982 0.992655i \(-0.461396\pi\)
0.120982 + 0.992655i \(0.461396\pi\)
\(258\) 0 0
\(259\) −8.42718 −0.523639
\(260\) 17.8790 1.10881
\(261\) 0 0
\(262\) −30.5265 −1.88593
\(263\) −5.19075 −0.320075 −0.160038 0.987111i \(-0.551162\pi\)
−0.160038 + 0.987111i \(0.551162\pi\)
\(264\) 0 0
\(265\) −13.9727 −0.858336
\(266\) 0.338455 0.0207520
\(267\) 0 0
\(268\) 21.2585 1.29857
\(269\) −16.8397 −1.02673 −0.513366 0.858170i \(-0.671602\pi\)
−0.513366 + 0.858170i \(0.671602\pi\)
\(270\) 0 0
\(271\) −14.0214 −0.851739 −0.425870 0.904785i \(-0.640032\pi\)
−0.425870 + 0.904785i \(0.640032\pi\)
\(272\) −13.7448 −0.833402
\(273\) 0 0
\(274\) 35.0434 2.11705
\(275\) 0 0
\(276\) 0 0
\(277\) 0.945328 0.0567992 0.0283996 0.999597i \(-0.490959\pi\)
0.0283996 + 0.999597i \(0.490959\pi\)
\(278\) −16.2295 −0.973380
\(279\) 0 0
\(280\) 0.929924 0.0555736
\(281\) 4.03539 0.240731 0.120366 0.992730i \(-0.461593\pi\)
0.120366 + 0.992730i \(0.461593\pi\)
\(282\) 0 0
\(283\) 4.80601 0.285688 0.142844 0.989745i \(-0.454375\pi\)
0.142844 + 0.989745i \(0.454375\pi\)
\(284\) 2.37971 0.141210
\(285\) 0 0
\(286\) 0 0
\(287\) 4.50712 0.266047
\(288\) 0 0
\(289\) 1.25340 0.0737292
\(290\) 27.2822 1.60207
\(291\) 0 0
\(292\) −17.8605 −1.04521
\(293\) −1.04668 −0.0611474 −0.0305737 0.999533i \(-0.509733\pi\)
−0.0305737 + 0.999533i \(0.509733\pi\)
\(294\) 0 0
\(295\) −18.1349 −1.05585
\(296\) −5.88237 −0.341906
\(297\) 0 0
\(298\) 20.2380 1.17236
\(299\) 6.47561 0.374494
\(300\) 0 0
\(301\) −8.02855 −0.462758
\(302\) 21.0327 1.21029
\(303\) 0 0
\(304\) −0.522952 −0.0299933
\(305\) 10.7482 0.615440
\(306\) 0 0
\(307\) −26.1929 −1.49491 −0.747455 0.664313i \(-0.768724\pi\)
−0.747455 + 0.664313i \(0.768724\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 25.2277 1.43284
\(311\) −22.0318 −1.24931 −0.624655 0.780901i \(-0.714760\pi\)
−0.624655 + 0.780901i \(0.714760\pi\)
\(312\) 0 0
\(313\) 30.4119 1.71898 0.859491 0.511151i \(-0.170781\pi\)
0.859491 + 0.511151i \(0.170781\pi\)
\(314\) −10.3047 −0.581531
\(315\) 0 0
\(316\) −0.0127801 −0.000718938 0
\(317\) −19.0776 −1.07151 −0.535753 0.844374i \(-0.679972\pi\)
−0.535753 + 0.844374i \(0.679972\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −13.8811 −0.775978
\(321\) 0 0
\(322\) 2.34613 0.130745
\(323\) 0.694490 0.0386425
\(324\) 0 0
\(325\) −18.5348 −1.02813
\(326\) 14.0189 0.776434
\(327\) 0 0
\(328\) 3.14608 0.173713
\(329\) −2.79374 −0.154024
\(330\) 0 0
\(331\) 3.61239 0.198555 0.0992774 0.995060i \(-0.468347\pi\)
0.0992774 + 0.995060i \(0.468347\pi\)
\(332\) 6.61418 0.363000
\(333\) 0 0
\(334\) 29.1235 1.59357
\(335\) 12.1277 0.662605
\(336\) 0 0
\(337\) −2.67839 −0.145901 −0.0729507 0.997336i \(-0.523242\pi\)
−0.0729507 + 0.997336i \(0.523242\pi\)
\(338\) 41.6987 2.26811
\(339\) 0 0
\(340\) 13.2917 0.720846
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −5.60412 −0.302154
\(345\) 0 0
\(346\) −7.30570 −0.392757
\(347\) −18.4546 −0.990696 −0.495348 0.868695i \(-0.664959\pi\)
−0.495348 + 0.868695i \(0.664959\pi\)
\(348\) 0 0
\(349\) 20.3015 1.08672 0.543358 0.839501i \(-0.317153\pi\)
0.543358 + 0.839501i \(0.317153\pi\)
\(350\) −6.71523 −0.358944
\(351\) 0 0
\(352\) 0 0
\(353\) −12.7771 −0.680054 −0.340027 0.940416i \(-0.610436\pi\)
−0.340027 + 0.940416i \(0.610436\pi\)
\(354\) 0 0
\(355\) 1.35759 0.0720533
\(356\) 24.5794 1.30270
\(357\) 0 0
\(358\) 4.63762 0.245106
\(359\) 10.2976 0.543486 0.271743 0.962370i \(-0.412400\pi\)
0.271743 + 0.962370i \(0.412400\pi\)
\(360\) 0 0
\(361\) −18.9736 −0.998609
\(362\) 2.32015 0.121944
\(363\) 0 0
\(364\) 13.4204 0.703423
\(365\) −10.1892 −0.533325
\(366\) 0 0
\(367\) 2.65025 0.138342 0.0691710 0.997605i \(-0.477965\pi\)
0.0691710 + 0.997605i \(0.477965\pi\)
\(368\) −3.62504 −0.188968
\(369\) 0 0
\(370\) −23.3758 −1.21525
\(371\) −10.4883 −0.544523
\(372\) 0 0
\(373\) 32.9030 1.70365 0.851827 0.523824i \(-0.175495\pi\)
0.851827 + 0.523824i \(0.175495\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −1.95010 −0.100569
\(377\) 56.5236 2.91111
\(378\) 0 0
\(379\) −22.2324 −1.14200 −0.571000 0.820950i \(-0.693444\pi\)
−0.571000 + 0.820950i \(0.693444\pi\)
\(380\) 0.505713 0.0259425
\(381\) 0 0
\(382\) 21.3031 1.08996
\(383\) 25.1289 1.28402 0.642012 0.766694i \(-0.278100\pi\)
0.642012 + 0.766694i \(0.278100\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −37.3399 −1.90055
\(387\) 0 0
\(388\) 28.3996 1.44177
\(389\) −23.0628 −1.16933 −0.584665 0.811275i \(-0.698774\pi\)
−0.584665 + 0.811275i \(0.698774\pi\)
\(390\) 0 0
\(391\) 4.81413 0.243461
\(392\) 0.698024 0.0352555
\(393\) 0 0
\(394\) 8.95905 0.451350
\(395\) −0.00729087 −0.000366844 0
\(396\) 0 0
\(397\) −3.78504 −0.189966 −0.0949828 0.995479i \(-0.530280\pi\)
−0.0949828 + 0.995479i \(0.530280\pi\)
\(398\) 45.9233 2.30193
\(399\) 0 0
\(400\) 10.3758 0.518789
\(401\) −39.0596 −1.95054 −0.975271 0.221011i \(-0.929064\pi\)
−0.975271 + 0.221011i \(0.929064\pi\)
\(402\) 0 0
\(403\) 52.2670 2.60361
\(404\) 3.55396 0.176816
\(405\) 0 0
\(406\) 20.4787 1.01634
\(407\) 0 0
\(408\) 0 0
\(409\) −29.4034 −1.45390 −0.726952 0.686689i \(-0.759064\pi\)
−0.726952 + 0.686689i \(0.759064\pi\)
\(410\) 12.5021 0.617435
\(411\) 0 0
\(412\) 36.1180 1.77941
\(413\) −13.6125 −0.669828
\(414\) 0 0
\(415\) 3.77329 0.185224
\(416\) −46.5183 −2.28075
\(417\) 0 0
\(418\) 0 0
\(419\) 14.5510 0.710863 0.355431 0.934702i \(-0.384334\pi\)
0.355431 + 0.934702i \(0.384334\pi\)
\(420\) 0 0
\(421\) 17.7574 0.865443 0.432722 0.901528i \(-0.357553\pi\)
0.432722 + 0.901528i \(0.357553\pi\)
\(422\) −30.9248 −1.50539
\(423\) 0 0
\(424\) −7.32105 −0.355542
\(425\) −13.7793 −0.668392
\(426\) 0 0
\(427\) 8.06786 0.390431
\(428\) −19.2668 −0.931295
\(429\) 0 0
\(430\) −22.2701 −1.07396
\(431\) −17.8382 −0.859234 −0.429617 0.903011i \(-0.641351\pi\)
−0.429617 + 0.903011i \(0.641351\pi\)
\(432\) 0 0
\(433\) 12.4405 0.597851 0.298925 0.954276i \(-0.403372\pi\)
0.298925 + 0.954276i \(0.403372\pi\)
\(434\) 18.9365 0.908982
\(435\) 0 0
\(436\) 0.714451 0.0342160
\(437\) 0.183164 0.00876193
\(438\) 0 0
\(439\) 16.3740 0.781489 0.390744 0.920499i \(-0.372218\pi\)
0.390744 + 0.920499i \(0.372218\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 51.1226 2.43165
\(443\) 2.22725 0.105820 0.0529100 0.998599i \(-0.483150\pi\)
0.0529100 + 0.998599i \(0.483150\pi\)
\(444\) 0 0
\(445\) 14.0222 0.664715
\(446\) 11.6554 0.551897
\(447\) 0 0
\(448\) −10.4195 −0.492276
\(449\) 31.9494 1.50779 0.753893 0.656997i \(-0.228174\pi\)
0.753893 + 0.656997i \(0.228174\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 23.7034 1.11491
\(453\) 0 0
\(454\) −15.6759 −0.735707
\(455\) 7.65617 0.358927
\(456\) 0 0
\(457\) −22.1342 −1.03539 −0.517697 0.855564i \(-0.673210\pi\)
−0.517697 + 0.855564i \(0.673210\pi\)
\(458\) −56.7026 −2.64954
\(459\) 0 0
\(460\) 3.50555 0.163447
\(461\) −14.9408 −0.695865 −0.347932 0.937520i \(-0.613116\pi\)
−0.347932 + 0.937520i \(0.613116\pi\)
\(462\) 0 0
\(463\) 5.02953 0.233742 0.116871 0.993147i \(-0.462714\pi\)
0.116871 + 0.993147i \(0.462714\pi\)
\(464\) −31.6419 −1.46894
\(465\) 0 0
\(466\) −11.4306 −0.529512
\(467\) −32.1377 −1.48716 −0.743579 0.668649i \(-0.766873\pi\)
−0.743579 + 0.668649i \(0.766873\pi\)
\(468\) 0 0
\(469\) 9.10333 0.420353
\(470\) −7.74945 −0.357455
\(471\) 0 0
\(472\) −9.50185 −0.437358
\(473\) 0 0
\(474\) 0 0
\(475\) −0.524262 −0.0240548
\(476\) 9.97710 0.457300
\(477\) 0 0
\(478\) −51.7228 −2.36574
\(479\) −13.1491 −0.600798 −0.300399 0.953814i \(-0.597120\pi\)
−0.300399 + 0.953814i \(0.597120\pi\)
\(480\) 0 0
\(481\) −48.4302 −2.20823
\(482\) 36.0703 1.64296
\(483\) 0 0
\(484\) 0 0
\(485\) 16.2016 0.735675
\(486\) 0 0
\(487\) 1.05225 0.0476821 0.0238411 0.999716i \(-0.492410\pi\)
0.0238411 + 0.999716i \(0.492410\pi\)
\(488\) 5.63156 0.254929
\(489\) 0 0
\(490\) 2.77386 0.125310
\(491\) 27.3208 1.23297 0.616485 0.787367i \(-0.288556\pi\)
0.616485 + 0.787367i \(0.288556\pi\)
\(492\) 0 0
\(493\) 42.0211 1.89253
\(494\) 1.94507 0.0875129
\(495\) 0 0
\(496\) −29.2591 −1.31377
\(497\) 1.01904 0.0457102
\(498\) 0 0
\(499\) −23.4585 −1.05015 −0.525073 0.851057i \(-0.675962\pi\)
−0.525073 + 0.851057i \(0.675962\pi\)
\(500\) −25.5891 −1.14438
\(501\) 0 0
\(502\) −51.3567 −2.29216
\(503\) −11.4795 −0.511846 −0.255923 0.966697i \(-0.582379\pi\)
−0.255923 + 0.966697i \(0.582379\pi\)
\(504\) 0 0
\(505\) 2.02749 0.0902219
\(506\) 0 0
\(507\) 0 0
\(508\) 8.89201 0.394519
\(509\) 1.40656 0.0623446 0.0311723 0.999514i \(-0.490076\pi\)
0.0311723 + 0.999514i \(0.490076\pi\)
\(510\) 0 0
\(511\) −7.64824 −0.338338
\(512\) 30.5322 1.34934
\(513\) 0 0
\(514\) 8.07651 0.356239
\(515\) 20.6048 0.907956
\(516\) 0 0
\(517\) 0 0
\(518\) −17.5464 −0.770946
\(519\) 0 0
\(520\) 5.34419 0.234358
\(521\) −18.5352 −0.812041 −0.406020 0.913864i \(-0.633084\pi\)
−0.406020 + 0.913864i \(0.633084\pi\)
\(522\) 0 0
\(523\) −36.4670 −1.59459 −0.797295 0.603589i \(-0.793737\pi\)
−0.797295 + 0.603589i \(0.793737\pi\)
\(524\) −34.2376 −1.49568
\(525\) 0 0
\(526\) −10.8078 −0.471242
\(527\) 38.8566 1.69262
\(528\) 0 0
\(529\) −21.7303 −0.944797
\(530\) −29.0929 −1.26372
\(531\) 0 0
\(532\) 0.379601 0.0164578
\(533\) 25.9020 1.12194
\(534\) 0 0
\(535\) −10.9914 −0.475200
\(536\) 6.35434 0.274466
\(537\) 0 0
\(538\) −35.0623 −1.51164
\(539\) 0 0
\(540\) 0 0
\(541\) 22.1641 0.952910 0.476455 0.879199i \(-0.341921\pi\)
0.476455 + 0.879199i \(0.341921\pi\)
\(542\) −29.1943 −1.25400
\(543\) 0 0
\(544\) −34.5829 −1.48273
\(545\) 0.407584 0.0174590
\(546\) 0 0
\(547\) −40.5974 −1.73582 −0.867909 0.496723i \(-0.834537\pi\)
−0.867909 + 0.496723i \(0.834537\pi\)
\(548\) 39.3036 1.67897
\(549\) 0 0
\(550\) 0 0
\(551\) 1.59878 0.0681105
\(552\) 0 0
\(553\) −0.00547271 −0.000232723 0
\(554\) 1.96829 0.0836246
\(555\) 0 0
\(556\) −18.2025 −0.771957
\(557\) 39.9780 1.69392 0.846961 0.531655i \(-0.178430\pi\)
0.846961 + 0.531655i \(0.178430\pi\)
\(558\) 0 0
\(559\) −46.1394 −1.95149
\(560\) −4.28592 −0.181113
\(561\) 0 0
\(562\) 8.40219 0.354425
\(563\) −28.8313 −1.21509 −0.607546 0.794284i \(-0.707846\pi\)
−0.607546 + 0.794284i \(0.707846\pi\)
\(564\) 0 0
\(565\) 13.5224 0.568893
\(566\) 10.0067 0.420613
\(567\) 0 0
\(568\) 0.711314 0.0298461
\(569\) −18.0183 −0.755365 −0.377683 0.925935i \(-0.623279\pi\)
−0.377683 + 0.925935i \(0.623279\pi\)
\(570\) 0 0
\(571\) 18.2208 0.762515 0.381258 0.924469i \(-0.375491\pi\)
0.381258 + 0.924469i \(0.375491\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 9.38438 0.391697
\(575\) −3.63413 −0.151554
\(576\) 0 0
\(577\) 10.5299 0.438364 0.219182 0.975684i \(-0.429661\pi\)
0.219182 + 0.975684i \(0.429661\pi\)
\(578\) 2.60973 0.108550
\(579\) 0 0
\(580\) 30.5988 1.27055
\(581\) 2.83233 0.117505
\(582\) 0 0
\(583\) 0 0
\(584\) −5.33865 −0.220915
\(585\) 0 0
\(586\) −2.17931 −0.0900264
\(587\) −2.45140 −0.101180 −0.0505900 0.998720i \(-0.516110\pi\)
−0.0505900 + 0.998720i \(0.516110\pi\)
\(588\) 0 0
\(589\) 1.47839 0.0609158
\(590\) −37.7591 −1.55452
\(591\) 0 0
\(592\) 27.1112 1.11426
\(593\) 27.7941 1.14137 0.570684 0.821170i \(-0.306678\pi\)
0.570684 + 0.821170i \(0.306678\pi\)
\(594\) 0 0
\(595\) 5.69179 0.233341
\(596\) 22.6983 0.929760
\(597\) 0 0
\(598\) 13.4830 0.551362
\(599\) 0.747037 0.0305231 0.0152616 0.999884i \(-0.495142\pi\)
0.0152616 + 0.999884i \(0.495142\pi\)
\(600\) 0 0
\(601\) −15.0794 −0.615103 −0.307551 0.951531i \(-0.599510\pi\)
−0.307551 + 0.951531i \(0.599510\pi\)
\(602\) −16.7165 −0.681312
\(603\) 0 0
\(604\) 23.5896 0.959846
\(605\) 0 0
\(606\) 0 0
\(607\) −4.16609 −0.169096 −0.0845481 0.996419i \(-0.526945\pi\)
−0.0845481 + 0.996419i \(0.526945\pi\)
\(608\) −1.31578 −0.0533620
\(609\) 0 0
\(610\) 22.3791 0.906102
\(611\) −16.0554 −0.649532
\(612\) 0 0
\(613\) 36.7033 1.48243 0.741216 0.671266i \(-0.234249\pi\)
0.741216 + 0.671266i \(0.234249\pi\)
\(614\) −54.5370 −2.20093
\(615\) 0 0
\(616\) 0 0
\(617\) −22.8780 −0.921035 −0.460518 0.887651i \(-0.652336\pi\)
−0.460518 + 0.887651i \(0.652336\pi\)
\(618\) 0 0
\(619\) −2.59442 −0.104279 −0.0521393 0.998640i \(-0.516604\pi\)
−0.0521393 + 0.998640i \(0.516604\pi\)
\(620\) 28.2946 1.13634
\(621\) 0 0
\(622\) −45.8730 −1.83934
\(623\) 10.5254 0.421691
\(624\) 0 0
\(625\) 1.52768 0.0611071
\(626\) 63.3214 2.53083
\(627\) 0 0
\(628\) −11.5575 −0.461194
\(629\) −36.0043 −1.43558
\(630\) 0 0
\(631\) 29.7938 1.18607 0.593036 0.805176i \(-0.297929\pi\)
0.593036 + 0.805176i \(0.297929\pi\)
\(632\) −0.00382008 −0.000151955 0
\(633\) 0 0
\(634\) −39.7220 −1.57756
\(635\) 5.07276 0.201306
\(636\) 0 0
\(637\) 5.74691 0.227701
\(638\) 0 0
\(639\) 0 0
\(640\) −7.33488 −0.289937
\(641\) 32.9695 1.30222 0.651108 0.758985i \(-0.274305\pi\)
0.651108 + 0.758985i \(0.274305\pi\)
\(642\) 0 0
\(643\) 20.9257 0.825229 0.412614 0.910906i \(-0.364616\pi\)
0.412614 + 0.910906i \(0.364616\pi\)
\(644\) 2.63135 0.103690
\(645\) 0 0
\(646\) 1.44602 0.0568927
\(647\) −12.9654 −0.509722 −0.254861 0.966978i \(-0.582030\pi\)
−0.254861 + 0.966978i \(0.582030\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −38.5918 −1.51370
\(651\) 0 0
\(652\) 15.7231 0.615765
\(653\) 4.59805 0.179936 0.0899678 0.995945i \(-0.471324\pi\)
0.0899678 + 0.995945i \(0.471324\pi\)
\(654\) 0 0
\(655\) −19.5320 −0.763180
\(656\) −14.4999 −0.566127
\(657\) 0 0
\(658\) −5.81693 −0.226767
\(659\) −27.8639 −1.08542 −0.542712 0.839919i \(-0.682602\pi\)
−0.542712 + 0.839919i \(0.682602\pi\)
\(660\) 0 0
\(661\) 36.2151 1.40861 0.704303 0.709900i \(-0.251260\pi\)
0.704303 + 0.709900i \(0.251260\pi\)
\(662\) 7.52144 0.292329
\(663\) 0 0
\(664\) 1.97703 0.0767237
\(665\) 0.216557 0.00839771
\(666\) 0 0
\(667\) 11.0826 0.429120
\(668\) 32.6640 1.26381
\(669\) 0 0
\(670\) 25.2513 0.975544
\(671\) 0 0
\(672\) 0 0
\(673\) −6.97277 −0.268781 −0.134390 0.990928i \(-0.542908\pi\)
−0.134390 + 0.990928i \(0.542908\pi\)
\(674\) −5.57675 −0.214808
\(675\) 0 0
\(676\) 46.7679 1.79876
\(677\) −21.9071 −0.841958 −0.420979 0.907070i \(-0.638313\pi\)
−0.420979 + 0.907070i \(0.638313\pi\)
\(678\) 0 0
\(679\) 12.1613 0.466708
\(680\) 3.97301 0.152358
\(681\) 0 0
\(682\) 0 0
\(683\) −48.9585 −1.87334 −0.936672 0.350208i \(-0.886111\pi\)
−0.936672 + 0.350208i \(0.886111\pi\)
\(684\) 0 0
\(685\) 22.4221 0.856706
\(686\) 2.08213 0.0794959
\(687\) 0 0
\(688\) 25.8288 0.984714
\(689\) −60.2751 −2.29630
\(690\) 0 0
\(691\) −47.8751 −1.82126 −0.910628 0.413227i \(-0.864402\pi\)
−0.910628 + 0.413227i \(0.864402\pi\)
\(692\) −8.19384 −0.311483
\(693\) 0 0
\(694\) −38.4248 −1.45859
\(695\) −10.3842 −0.393897
\(696\) 0 0
\(697\) 19.2562 0.729381
\(698\) 42.2703 1.59995
\(699\) 0 0
\(700\) −7.53159 −0.284667
\(701\) 6.95807 0.262803 0.131401 0.991329i \(-0.458052\pi\)
0.131401 + 0.991329i \(0.458052\pi\)
\(702\) 0 0
\(703\) −1.36986 −0.0516653
\(704\) 0 0
\(705\) 0 0
\(706\) −26.6034 −1.00123
\(707\) 1.52188 0.0572362
\(708\) 0 0
\(709\) −20.3838 −0.765531 −0.382766 0.923845i \(-0.625028\pi\)
−0.382766 + 0.923845i \(0.625028\pi\)
\(710\) 2.82667 0.106083
\(711\) 0 0
\(712\) 7.34698 0.275340
\(713\) 10.2480 0.383791
\(714\) 0 0
\(715\) 0 0
\(716\) 5.20141 0.194386
\(717\) 0 0
\(718\) 21.4409 0.800166
\(719\) −36.9536 −1.37814 −0.689068 0.724697i \(-0.741980\pi\)
−0.689068 + 0.724697i \(0.741980\pi\)
\(720\) 0 0
\(721\) 15.4665 0.576002
\(722\) −39.5054 −1.47024
\(723\) 0 0
\(724\) 2.60220 0.0967101
\(725\) −31.7212 −1.17809
\(726\) 0 0
\(727\) 40.8565 1.51528 0.757641 0.652671i \(-0.226352\pi\)
0.757641 + 0.652671i \(0.226352\pi\)
\(728\) 4.01148 0.148675
\(729\) 0 0
\(730\) −21.2151 −0.785207
\(731\) −34.3012 −1.26868
\(732\) 0 0
\(733\) −17.9728 −0.663839 −0.331920 0.943308i \(-0.607696\pi\)
−0.331920 + 0.943308i \(0.607696\pi\)
\(734\) 5.51815 0.203679
\(735\) 0 0
\(736\) −9.12086 −0.336199
\(737\) 0 0
\(738\) 0 0
\(739\) −2.34892 −0.0864064 −0.0432032 0.999066i \(-0.513756\pi\)
−0.0432032 + 0.999066i \(0.513756\pi\)
\(740\) −26.2175 −0.963776
\(741\) 0 0
\(742\) −21.8379 −0.801693
\(743\) −38.2722 −1.40407 −0.702036 0.712141i \(-0.747726\pi\)
−0.702036 + 0.712141i \(0.747726\pi\)
\(744\) 0 0
\(745\) 12.9491 0.474417
\(746\) 68.5082 2.50826
\(747\) 0 0
\(748\) 0 0
\(749\) −8.25043 −0.301464
\(750\) 0 0
\(751\) −2.89080 −0.105487 −0.0527434 0.998608i \(-0.516797\pi\)
−0.0527434 + 0.998608i \(0.516797\pi\)
\(752\) 8.98781 0.327752
\(753\) 0 0
\(754\) 117.689 4.28599
\(755\) 13.4575 0.489769
\(756\) 0 0
\(757\) 24.1721 0.878551 0.439275 0.898353i \(-0.355235\pi\)
0.439275 + 0.898353i \(0.355235\pi\)
\(758\) −46.2906 −1.68135
\(759\) 0 0
\(760\) 0.151162 0.00548321
\(761\) −24.7044 −0.895533 −0.447766 0.894151i \(-0.647780\pi\)
−0.447766 + 0.894151i \(0.647780\pi\)
\(762\) 0 0
\(763\) 0.305943 0.0110759
\(764\) 23.8929 0.864415
\(765\) 0 0
\(766\) 52.3214 1.89045
\(767\) −78.2298 −2.82472
\(768\) 0 0
\(769\) −12.8939 −0.464967 −0.232483 0.972600i \(-0.574685\pi\)
−0.232483 + 0.972600i \(0.574685\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −41.8793 −1.50727
\(773\) −32.4616 −1.16756 −0.583780 0.811912i \(-0.698427\pi\)
−0.583780 + 0.811912i \(0.698427\pi\)
\(774\) 0 0
\(775\) −29.3324 −1.05365
\(776\) 8.48887 0.304733
\(777\) 0 0
\(778\) −48.0196 −1.72159
\(779\) 0.732645 0.0262497
\(780\) 0 0
\(781\) 0 0
\(782\) 10.0236 0.358444
\(783\) 0 0
\(784\) −3.21712 −0.114897
\(785\) −6.59338 −0.235328
\(786\) 0 0
\(787\) 28.1909 1.00490 0.502448 0.864607i \(-0.332433\pi\)
0.502448 + 0.864607i \(0.332433\pi\)
\(788\) 10.0482 0.357952
\(789\) 0 0
\(790\) −0.0151805 −0.000540098 0
\(791\) 10.1503 0.360902
\(792\) 0 0
\(793\) 46.3652 1.64648
\(794\) −7.88093 −0.279684
\(795\) 0 0
\(796\) 51.5061 1.82559
\(797\) 42.5723 1.50799 0.753995 0.656881i \(-0.228124\pi\)
0.753995 + 0.656881i \(0.228124\pi\)
\(798\) 0 0
\(799\) −11.9360 −0.422265
\(800\) 26.1062 0.922993
\(801\) 0 0
\(802\) −81.3270 −2.87175
\(803\) 0 0
\(804\) 0 0
\(805\) 1.50115 0.0529085
\(806\) 108.826 3.83325
\(807\) 0 0
\(808\) 1.06231 0.0373719
\(809\) −14.1894 −0.498873 −0.249437 0.968391i \(-0.580245\pi\)
−0.249437 + 0.968391i \(0.580245\pi\)
\(810\) 0 0
\(811\) 2.58292 0.0906987 0.0453494 0.998971i \(-0.485560\pi\)
0.0453494 + 0.998971i \(0.485560\pi\)
\(812\) 22.9682 0.806027
\(813\) 0 0
\(814\) 0 0
\(815\) 8.96982 0.314199
\(816\) 0 0
\(817\) −1.30506 −0.0456584
\(818\) −61.2215 −2.14056
\(819\) 0 0
\(820\) 14.0220 0.489668
\(821\) −52.5523 −1.83409 −0.917043 0.398787i \(-0.869431\pi\)
−0.917043 + 0.398787i \(0.869431\pi\)
\(822\) 0 0
\(823\) 14.6509 0.510699 0.255349 0.966849i \(-0.417809\pi\)
0.255349 + 0.966849i \(0.417809\pi\)
\(824\) 10.7960 0.376095
\(825\) 0 0
\(826\) −28.3429 −0.986177
\(827\) 15.1928 0.528304 0.264152 0.964481i \(-0.414908\pi\)
0.264152 + 0.964481i \(0.414908\pi\)
\(828\) 0 0
\(829\) −41.6162 −1.44539 −0.722695 0.691167i \(-0.757097\pi\)
−0.722695 + 0.691167i \(0.757097\pi\)
\(830\) 7.85647 0.272702
\(831\) 0 0
\(832\) −59.8800 −2.07596
\(833\) 4.27240 0.148030
\(834\) 0 0
\(835\) 18.6343 0.644867
\(836\) 0 0
\(837\) 0 0
\(838\) 30.2970 1.04659
\(839\) 21.8835 0.755503 0.377752 0.925907i \(-0.376697\pi\)
0.377752 + 0.925907i \(0.376697\pi\)
\(840\) 0 0
\(841\) 67.7365 2.33574
\(842\) 36.9732 1.27418
\(843\) 0 0
\(844\) −34.6842 −1.19388
\(845\) 26.6804 0.917834
\(846\) 0 0
\(847\) 0 0
\(848\) 33.7420 1.15870
\(849\) 0 0
\(850\) −28.6901 −0.984063
\(851\) −9.49572 −0.325509
\(852\) 0 0
\(853\) −22.2054 −0.760299 −0.380149 0.924925i \(-0.624127\pi\)
−0.380149 + 0.924925i \(0.624127\pi\)
\(854\) 16.7983 0.574826
\(855\) 0 0
\(856\) −5.75899 −0.196838
\(857\) 24.6876 0.843311 0.421656 0.906756i \(-0.361449\pi\)
0.421656 + 0.906756i \(0.361449\pi\)
\(858\) 0 0
\(859\) 10.3940 0.354639 0.177320 0.984153i \(-0.443257\pi\)
0.177320 + 0.984153i \(0.443257\pi\)
\(860\) −24.9774 −0.851722
\(861\) 0 0
\(862\) −37.1413 −1.26504
\(863\) −45.9179 −1.56306 −0.781532 0.623865i \(-0.785561\pi\)
−0.781532 + 0.623865i \(0.785561\pi\)
\(864\) 0 0
\(865\) −4.67447 −0.158937
\(866\) 25.9026 0.880207
\(867\) 0 0
\(868\) 21.2386 0.720885
\(869\) 0 0
\(870\) 0 0
\(871\) 52.3160 1.77266
\(872\) 0.213555 0.00723189
\(873\) 0 0
\(874\) 0.381371 0.0129001
\(875\) −10.9578 −0.370441
\(876\) 0 0
\(877\) 34.8872 1.17806 0.589029 0.808112i \(-0.299510\pi\)
0.589029 + 0.808112i \(0.299510\pi\)
\(878\) 34.0927 1.15057
\(879\) 0 0
\(880\) 0 0
\(881\) −8.85305 −0.298267 −0.149133 0.988817i \(-0.547648\pi\)
−0.149133 + 0.988817i \(0.547648\pi\)
\(882\) 0 0
\(883\) 17.3511 0.583910 0.291955 0.956432i \(-0.405694\pi\)
0.291955 + 0.956432i \(0.405694\pi\)
\(884\) 57.3375 1.92847
\(885\) 0 0
\(886\) 4.63742 0.155797
\(887\) −38.0505 −1.27761 −0.638806 0.769368i \(-0.720571\pi\)
−0.638806 + 0.769368i \(0.720571\pi\)
\(888\) 0 0
\(889\) 3.80774 0.127707
\(890\) 29.1959 0.978650
\(891\) 0 0
\(892\) 13.0723 0.437692
\(893\) −0.454131 −0.0151969
\(894\) 0 0
\(895\) 2.96733 0.0991868
\(896\) −5.50574 −0.183934
\(897\) 0 0
\(898\) 66.5227 2.21989
\(899\) 89.4517 2.98338
\(900\) 0 0
\(901\) −44.8100 −1.49284
\(902\) 0 0
\(903\) 0 0
\(904\) 7.08513 0.235648
\(905\) 1.48452 0.0493471
\(906\) 0 0
\(907\) −14.1082 −0.468455 −0.234227 0.972182i \(-0.575256\pi\)
−0.234227 + 0.972182i \(0.575256\pi\)
\(908\) −17.5816 −0.583466
\(909\) 0 0
\(910\) 15.9411 0.528443
\(911\) 29.1703 0.966457 0.483228 0.875494i \(-0.339464\pi\)
0.483228 + 0.875494i \(0.339464\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −46.0861 −1.52439
\(915\) 0 0
\(916\) −63.5959 −2.10127
\(917\) −14.6612 −0.484156
\(918\) 0 0
\(919\) 3.84247 0.126751 0.0633757 0.997990i \(-0.479813\pi\)
0.0633757 + 0.997990i \(0.479813\pi\)
\(920\) 1.04784 0.0345461
\(921\) 0 0
\(922\) −31.1087 −1.02451
\(923\) 5.85633 0.192763
\(924\) 0 0
\(925\) 27.1792 0.893645
\(926\) 10.4721 0.344135
\(927\) 0 0
\(928\) −79.6131 −2.61343
\(929\) 11.2254 0.368295 0.184147 0.982899i \(-0.441048\pi\)
0.184147 + 0.982899i \(0.441048\pi\)
\(930\) 0 0
\(931\) 0.162553 0.00532745
\(932\) −12.8202 −0.419939
\(933\) 0 0
\(934\) −66.9148 −2.18952
\(935\) 0 0
\(936\) 0 0
\(937\) 10.9651 0.358213 0.179106 0.983830i \(-0.442679\pi\)
0.179106 + 0.983830i \(0.442679\pi\)
\(938\) 18.9543 0.618879
\(939\) 0 0
\(940\) −8.69154 −0.283487
\(941\) −60.7290 −1.97971 −0.989855 0.142083i \(-0.954620\pi\)
−0.989855 + 0.142083i \(0.954620\pi\)
\(942\) 0 0
\(943\) 5.07861 0.165382
\(944\) 43.7930 1.42534
\(945\) 0 0
\(946\) 0 0
\(947\) −45.2278 −1.46971 −0.734854 0.678226i \(-0.762749\pi\)
−0.734854 + 0.678226i \(0.762749\pi\)
\(948\) 0 0
\(949\) −43.9537 −1.42680
\(950\) −1.09158 −0.0354155
\(951\) 0 0
\(952\) 2.98224 0.0966548
\(953\) −1.48140 −0.0479872 −0.0239936 0.999712i \(-0.507638\pi\)
−0.0239936 + 0.999712i \(0.507638\pi\)
\(954\) 0 0
\(955\) 13.6306 0.441074
\(956\) −58.0106 −1.87620
\(957\) 0 0
\(958\) −27.3781 −0.884545
\(959\) 16.8306 0.543489
\(960\) 0 0
\(961\) 51.7155 1.66824
\(962\) −100.838 −3.25114
\(963\) 0 0
\(964\) 40.4553 1.30298
\(965\) −23.8915 −0.769095
\(966\) 0 0
\(967\) −18.8504 −0.606187 −0.303094 0.952961i \(-0.598019\pi\)
−0.303094 + 0.952961i \(0.598019\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 33.7337 1.08312
\(971\) 58.9217 1.89089 0.945444 0.325785i \(-0.105629\pi\)
0.945444 + 0.325785i \(0.105629\pi\)
\(972\) 0 0
\(973\) −7.79467 −0.249886
\(974\) 2.19092 0.0702017
\(975\) 0 0
\(976\) −25.9552 −0.830807
\(977\) −43.0633 −1.37772 −0.688858 0.724896i \(-0.741888\pi\)
−0.688858 + 0.724896i \(0.741888\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.11107 0.0993795
\(981\) 0 0
\(982\) 56.8853 1.81528
\(983\) 13.2499 0.422605 0.211303 0.977421i \(-0.432229\pi\)
0.211303 + 0.977421i \(0.432229\pi\)
\(984\) 0 0
\(985\) 5.73234 0.182648
\(986\) 87.4931 2.78635
\(987\) 0 0
\(988\) 2.18153 0.0694037
\(989\) −9.04656 −0.287664
\(990\) 0 0
\(991\) 14.1260 0.448728 0.224364 0.974505i \(-0.427970\pi\)
0.224364 + 0.974505i \(0.427970\pi\)
\(992\) −73.6178 −2.33737
\(993\) 0 0
\(994\) 2.12177 0.0672984
\(995\) 29.3835 0.931519
\(996\) 0 0
\(997\) 13.3911 0.424100 0.212050 0.977259i \(-0.431986\pi\)
0.212050 + 0.977259i \(0.431986\pi\)
\(998\) −48.8435 −1.54611
\(999\) 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.dc.1.13 16
3.2 odd 2 inner 7623.2.a.dc.1.4 16
11.3 even 5 693.2.m.k.64.7 yes 32
11.4 even 5 693.2.m.k.379.7 yes 32
11.10 odd 2 7623.2.a.db.1.4 16
33.14 odd 10 693.2.m.k.64.2 32
33.26 odd 10 693.2.m.k.379.2 yes 32
33.32 even 2 7623.2.a.db.1.13 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.k.64.2 32 33.14 odd 10
693.2.m.k.64.7 yes 32 11.3 even 5
693.2.m.k.379.2 yes 32 33.26 odd 10
693.2.m.k.379.7 yes 32 11.4 even 5
7623.2.a.db.1.4 16 11.10 odd 2
7623.2.a.db.1.13 16 33.32 even 2
7623.2.a.dc.1.4 16 3.2 odd 2 inner
7623.2.a.dc.1.13 16 1.1 even 1 trivial