Properties

Label 3150.2.a.bs.1.2
Level 3150
Weight 2
Character 3150.1
Self dual yes
Analytic conductor 25.153
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\)
Character \(\chi\) = 3150.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +4.89898 q^{11} -4.44949 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +1.55051 q^{19} -4.89898 q^{22} +2.89898 q^{23} +4.44949 q^{26} +1.00000 q^{28} -6.89898 q^{29} +8.89898 q^{31} -1.00000 q^{32} -2.00000 q^{34} -2.00000 q^{37} -1.55051 q^{38} +1.10102 q^{41} +0.898979 q^{43} +4.89898 q^{44} -2.89898 q^{46} +8.89898 q^{47} +1.00000 q^{49} -4.44949 q^{52} -10.8990 q^{53} -1.00000 q^{56} +6.89898 q^{58} +1.55051 q^{59} +3.55051 q^{61} -8.89898 q^{62} +1.00000 q^{64} +8.00000 q^{67} +2.00000 q^{68} +1.10102 q^{71} -2.89898 q^{73} +2.00000 q^{74} +1.55051 q^{76} +4.89898 q^{77} +6.89898 q^{79} -1.10102 q^{82} -2.44949 q^{83} -0.898979 q^{86} -4.89898 q^{88} +10.0000 q^{89} -4.44949 q^{91} +2.89898 q^{92} -8.89898 q^{94} -15.7980 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 2q^{7} - 2q^{8} - 4q^{13} - 2q^{14} + 2q^{16} + 4q^{17} + 8q^{19} - 4q^{23} + 4q^{26} + 2q^{28} - 4q^{29} + 8q^{31} - 2q^{32} - 4q^{34} - 4q^{37} - 8q^{38} + 12q^{41} - 8q^{43} + 4q^{46} + 8q^{47} + 2q^{49} - 4q^{52} - 12q^{53} - 2q^{56} + 4q^{58} + 8q^{59} + 12q^{61} - 8q^{62} + 2q^{64} + 16q^{67} + 4q^{68} + 12q^{71} + 4q^{73} + 4q^{74} + 8q^{76} + 4q^{79} - 12q^{82} + 8q^{86} + 20q^{89} - 4q^{91} - 4q^{92} - 8q^{94} - 12q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 4.89898 1.47710 0.738549 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) 0 0
\(13\) −4.44949 −1.23407 −0.617033 0.786937i \(-0.711666\pi\)
−0.617033 + 0.786937i \(0.711666\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 1.55051 0.355711 0.177856 0.984057i \(-0.443084\pi\)
0.177856 + 0.984057i \(0.443084\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.89898 −1.04447
\(23\) 2.89898 0.604479 0.302240 0.953232i \(-0.402266\pi\)
0.302240 + 0.953232i \(0.402266\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.44949 0.872617
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −6.89898 −1.28111 −0.640554 0.767913i \(-0.721295\pi\)
−0.640554 + 0.767913i \(0.721295\pi\)
\(30\) 0 0
\(31\) 8.89898 1.59830 0.799152 0.601129i \(-0.205282\pi\)
0.799152 + 0.601129i \(0.205282\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −1.55051 −0.251526
\(39\) 0 0
\(40\) 0 0
\(41\) 1.10102 0.171951 0.0859753 0.996297i \(-0.472599\pi\)
0.0859753 + 0.996297i \(0.472599\pi\)
\(42\) 0 0
\(43\) 0.898979 0.137093 0.0685465 0.997648i \(-0.478164\pi\)
0.0685465 + 0.997648i \(0.478164\pi\)
\(44\) 4.89898 0.738549
\(45\) 0 0
\(46\) −2.89898 −0.427431
\(47\) 8.89898 1.29805 0.649025 0.760767i \(-0.275177\pi\)
0.649025 + 0.760767i \(0.275177\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −4.44949 −0.617033
\(53\) −10.8990 −1.49709 −0.748545 0.663084i \(-0.769247\pi\)
−0.748545 + 0.663084i \(0.769247\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 6.89898 0.905880
\(59\) 1.55051 0.201859 0.100930 0.994894i \(-0.467818\pi\)
0.100930 + 0.994894i \(0.467818\pi\)
\(60\) 0 0
\(61\) 3.55051 0.454596 0.227298 0.973825i \(-0.427011\pi\)
0.227298 + 0.973825i \(0.427011\pi\)
\(62\) −8.89898 −1.13017
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 1.10102 0.130667 0.0653335 0.997863i \(-0.479189\pi\)
0.0653335 + 0.997863i \(0.479189\pi\)
\(72\) 0 0
\(73\) −2.89898 −0.339300 −0.169650 0.985504i \(-0.554264\pi\)
−0.169650 + 0.985504i \(0.554264\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 1.55051 0.177856
\(77\) 4.89898 0.558291
\(78\) 0 0
\(79\) 6.89898 0.776196 0.388098 0.921618i \(-0.373132\pi\)
0.388098 + 0.921618i \(0.373132\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.10102 −0.121587
\(83\) −2.44949 −0.268866 −0.134433 0.990923i \(-0.542921\pi\)
−0.134433 + 0.990923i \(0.542921\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.898979 −0.0969395
\(87\) 0 0
\(88\) −4.89898 −0.522233
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −4.44949 −0.466433
\(92\) 2.89898 0.302240
\(93\) 0 0
\(94\) −8.89898 −0.917860
\(95\) 0 0
\(96\) 0 0
\(97\) −15.7980 −1.60404 −0.802020 0.597297i \(-0.796241\pi\)
−0.802020 + 0.597297i \(0.796241\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 0 0
\(101\) −3.55051 −0.353289 −0.176644 0.984275i \(-0.556524\pi\)
−0.176644 + 0.984275i \(0.556524\pi\)
\(102\) 0 0
\(103\) −12.8990 −1.27097 −0.635487 0.772111i \(-0.719201\pi\)
−0.635487 + 0.772111i \(0.719201\pi\)
\(104\) 4.44949 0.436308
\(105\) 0 0
\(106\) 10.8990 1.05860
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) 6.89898 0.660802 0.330401 0.943841i \(-0.392816\pi\)
0.330401 + 0.943841i \(0.392816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 19.7980 1.86244 0.931218 0.364464i \(-0.118748\pi\)
0.931218 + 0.364464i \(0.118748\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.89898 −0.640554
\(117\) 0 0
\(118\) −1.55051 −0.142736
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) −3.55051 −0.321448
\(123\) 0 0
\(124\) 8.89898 0.799152
\(125\) 0 0
\(126\) 0 0
\(127\) 14.8990 1.32207 0.661035 0.750355i \(-0.270117\pi\)
0.661035 + 0.750355i \(0.270117\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 6.44949 0.563495 0.281747 0.959489i \(-0.409086\pi\)
0.281747 + 0.959489i \(0.409086\pi\)
\(132\) 0 0
\(133\) 1.55051 0.134446
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −1.79796 −0.153610 −0.0768050 0.997046i \(-0.524472\pi\)
−0.0768050 + 0.997046i \(0.524472\pi\)
\(138\) 0 0
\(139\) 1.55051 0.131513 0.0657563 0.997836i \(-0.479054\pi\)
0.0657563 + 0.997836i \(0.479054\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.10102 −0.0923956
\(143\) −21.7980 −1.82284
\(144\) 0 0
\(145\) 0 0
\(146\) 2.89898 0.239921
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 3.79796 0.311141 0.155570 0.987825i \(-0.450278\pi\)
0.155570 + 0.987825i \(0.450278\pi\)
\(150\) 0 0
\(151\) 19.5959 1.59469 0.797347 0.603522i \(-0.206236\pi\)
0.797347 + 0.603522i \(0.206236\pi\)
\(152\) −1.55051 −0.125763
\(153\) 0 0
\(154\) −4.89898 −0.394771
\(155\) 0 0
\(156\) 0 0
\(157\) −3.55051 −0.283362 −0.141681 0.989912i \(-0.545251\pi\)
−0.141681 + 0.989912i \(0.545251\pi\)
\(158\) −6.89898 −0.548853
\(159\) 0 0
\(160\) 0 0
\(161\) 2.89898 0.228472
\(162\) 0 0
\(163\) 7.10102 0.556195 0.278097 0.960553i \(-0.410296\pi\)
0.278097 + 0.960553i \(0.410296\pi\)
\(164\) 1.10102 0.0859753
\(165\) 0 0
\(166\) 2.44949 0.190117
\(167\) −4.89898 −0.379094 −0.189547 0.981872i \(-0.560702\pi\)
−0.189547 + 0.981872i \(0.560702\pi\)
\(168\) 0 0
\(169\) 6.79796 0.522920
\(170\) 0 0
\(171\) 0 0
\(172\) 0.898979 0.0685465
\(173\) −6.24745 −0.474985 −0.237492 0.971389i \(-0.576325\pi\)
−0.237492 + 0.971389i \(0.576325\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.89898 0.369274
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) −13.7980 −1.03131 −0.515654 0.856797i \(-0.672451\pi\)
−0.515654 + 0.856797i \(0.672451\pi\)
\(180\) 0 0
\(181\) −10.2474 −0.761687 −0.380843 0.924640i \(-0.624366\pi\)
−0.380843 + 0.924640i \(0.624366\pi\)
\(182\) 4.44949 0.329818
\(183\) 0 0
\(184\) −2.89898 −0.213716
\(185\) 0 0
\(186\) 0 0
\(187\) 9.79796 0.716498
\(188\) 8.89898 0.649025
\(189\) 0 0
\(190\) 0 0
\(191\) −12.6969 −0.918718 −0.459359 0.888251i \(-0.651921\pi\)
−0.459359 + 0.888251i \(0.651921\pi\)
\(192\) 0 0
\(193\) 21.5959 1.55451 0.777254 0.629187i \(-0.216612\pi\)
0.777254 + 0.629187i \(0.216612\pi\)
\(194\) 15.7980 1.13423
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.8990 1.34650 0.673248 0.739417i \(-0.264899\pi\)
0.673248 + 0.739417i \(0.264899\pi\)
\(198\) 0 0
\(199\) 16.8990 1.19794 0.598968 0.800773i \(-0.295577\pi\)
0.598968 + 0.800773i \(0.295577\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 3.55051 0.249813
\(203\) −6.89898 −0.484213
\(204\) 0 0
\(205\) 0 0
\(206\) 12.8990 0.898714
\(207\) 0 0
\(208\) −4.44949 −0.308517
\(209\) 7.59592 0.525421
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −10.8990 −0.748545
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 0 0
\(217\) 8.89898 0.604102
\(218\) −6.89898 −0.467258
\(219\) 0 0
\(220\) 0 0
\(221\) −8.89898 −0.598610
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −19.7980 −1.31694
\(227\) 7.34847 0.487735 0.243868 0.969809i \(-0.421584\pi\)
0.243868 + 0.969809i \(0.421584\pi\)
\(228\) 0 0
\(229\) −19.1464 −1.26523 −0.632616 0.774466i \(-0.718019\pi\)
−0.632616 + 0.774466i \(0.718019\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.89898 0.452940
\(233\) 29.7980 1.95213 0.976065 0.217481i \(-0.0697840\pi\)
0.976065 + 0.217481i \(0.0697840\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.55051 0.100930
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) 6.20204 0.401177 0.200588 0.979676i \(-0.435715\pi\)
0.200588 + 0.979676i \(0.435715\pi\)
\(240\) 0 0
\(241\) −8.69694 −0.560219 −0.280110 0.959968i \(-0.590371\pi\)
−0.280110 + 0.959968i \(0.590371\pi\)
\(242\) −13.0000 −0.835672
\(243\) 0 0
\(244\) 3.55051 0.227298
\(245\) 0 0
\(246\) 0 0
\(247\) −6.89898 −0.438972
\(248\) −8.89898 −0.565086
\(249\) 0 0
\(250\) 0 0
\(251\) 6.44949 0.407088 0.203544 0.979066i \(-0.434754\pi\)
0.203544 + 0.979066i \(0.434754\pi\)
\(252\) 0 0
\(253\) 14.2020 0.892875
\(254\) −14.8990 −0.934845
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −8.69694 −0.542500 −0.271250 0.962509i \(-0.587437\pi\)
−0.271250 + 0.962509i \(0.587437\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) −6.44949 −0.398451
\(263\) 9.79796 0.604168 0.302084 0.953281i \(-0.402318\pi\)
0.302084 + 0.953281i \(0.402318\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.55051 −0.0950679
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) −19.1464 −1.16738 −0.583689 0.811977i \(-0.698391\pi\)
−0.583689 + 0.811977i \(0.698391\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 1.79796 0.108619
\(275\) 0 0
\(276\) 0 0
\(277\) 14.8990 0.895193 0.447596 0.894236i \(-0.352280\pi\)
0.447596 + 0.894236i \(0.352280\pi\)
\(278\) −1.55051 −0.0929934
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −3.75255 −0.223066 −0.111533 0.993761i \(-0.535576\pi\)
−0.111533 + 0.993761i \(0.535576\pi\)
\(284\) 1.10102 0.0653335
\(285\) 0 0
\(286\) 21.7980 1.28894
\(287\) 1.10102 0.0649912
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) −2.89898 −0.169650
\(293\) 18.2474 1.06603 0.533014 0.846107i \(-0.321059\pi\)
0.533014 + 0.846107i \(0.321059\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) −3.79796 −0.220010
\(299\) −12.8990 −0.745967
\(300\) 0 0
\(301\) 0.898979 0.0518163
\(302\) −19.5959 −1.12762
\(303\) 0 0
\(304\) 1.55051 0.0889279
\(305\) 0 0
\(306\) 0 0
\(307\) 20.2474 1.15558 0.577791 0.816184i \(-0.303915\pi\)
0.577791 + 0.816184i \(0.303915\pi\)
\(308\) 4.89898 0.279145
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 21.5959 1.22067 0.610337 0.792142i \(-0.291034\pi\)
0.610337 + 0.792142i \(0.291034\pi\)
\(314\) 3.55051 0.200367
\(315\) 0 0
\(316\) 6.89898 0.388098
\(317\) −22.4949 −1.26344 −0.631720 0.775197i \(-0.717651\pi\)
−0.631720 + 0.775197i \(0.717651\pi\)
\(318\) 0 0
\(319\) −33.7980 −1.89232
\(320\) 0 0
\(321\) 0 0
\(322\) −2.89898 −0.161554
\(323\) 3.10102 0.172545
\(324\) 0 0
\(325\) 0 0
\(326\) −7.10102 −0.393289
\(327\) 0 0
\(328\) −1.10102 −0.0607937
\(329\) 8.89898 0.490617
\(330\) 0 0
\(331\) −18.6969 −1.02768 −0.513838 0.857887i \(-0.671777\pi\)
−0.513838 + 0.857887i \(0.671777\pi\)
\(332\) −2.44949 −0.134433
\(333\) 0 0
\(334\) 4.89898 0.268060
\(335\) 0 0
\(336\) 0 0
\(337\) −9.59592 −0.522723 −0.261361 0.965241i \(-0.584171\pi\)
−0.261361 + 0.965241i \(0.584171\pi\)
\(338\) −6.79796 −0.369760
\(339\) 0 0
\(340\) 0 0
\(341\) 43.5959 2.36085
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −0.898979 −0.0484697
\(345\) 0 0
\(346\) 6.24745 0.335865
\(347\) 28.8990 1.55138 0.775689 0.631115i \(-0.217402\pi\)
0.775689 + 0.631115i \(0.217402\pi\)
\(348\) 0 0
\(349\) −8.44949 −0.452291 −0.226145 0.974094i \(-0.572612\pi\)
−0.226145 + 0.974094i \(0.572612\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.89898 −0.261116
\(353\) 22.8990 1.21879 0.609395 0.792867i \(-0.291412\pi\)
0.609395 + 0.792867i \(0.291412\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 13.7980 0.729245
\(359\) 27.5959 1.45646 0.728228 0.685334i \(-0.240344\pi\)
0.728228 + 0.685334i \(0.240344\pi\)
\(360\) 0 0
\(361\) −16.5959 −0.873469
\(362\) 10.2474 0.538594
\(363\) 0 0
\(364\) −4.44949 −0.233217
\(365\) 0 0
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 2.89898 0.151120
\(369\) 0 0
\(370\) 0 0
\(371\) −10.8990 −0.565847
\(372\) 0 0
\(373\) 4.69694 0.243198 0.121599 0.992579i \(-0.461198\pi\)
0.121599 + 0.992579i \(0.461198\pi\)
\(374\) −9.79796 −0.506640
\(375\) 0 0
\(376\) −8.89898 −0.458930
\(377\) 30.6969 1.58097
\(378\) 0 0
\(379\) 30.6969 1.57680 0.788398 0.615166i \(-0.210911\pi\)
0.788398 + 0.615166i \(0.210911\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 12.6969 0.649632
\(383\) −7.10102 −0.362845 −0.181423 0.983405i \(-0.558070\pi\)
−0.181423 + 0.983405i \(0.558070\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −21.5959 −1.09920
\(387\) 0 0
\(388\) −15.7980 −0.802020
\(389\) −13.1010 −0.664248 −0.332124 0.943236i \(-0.607765\pi\)
−0.332124 + 0.943236i \(0.607765\pi\)
\(390\) 0 0
\(391\) 5.79796 0.293215
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −18.8990 −0.952117
\(395\) 0 0
\(396\) 0 0
\(397\) 2.65153 0.133077 0.0665383 0.997784i \(-0.478805\pi\)
0.0665383 + 0.997784i \(0.478805\pi\)
\(398\) −16.8990 −0.847069
\(399\) 0 0
\(400\) 0 0
\(401\) 29.3939 1.46786 0.733930 0.679225i \(-0.237684\pi\)
0.733930 + 0.679225i \(0.237684\pi\)
\(402\) 0 0
\(403\) −39.5959 −1.97241
\(404\) −3.55051 −0.176644
\(405\) 0 0
\(406\) 6.89898 0.342391
\(407\) −9.79796 −0.485667
\(408\) 0 0
\(409\) 34.4949 1.70566 0.852831 0.522186i \(-0.174883\pi\)
0.852831 + 0.522186i \(0.174883\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −12.8990 −0.635487
\(413\) 1.55051 0.0762956
\(414\) 0 0
\(415\) 0 0
\(416\) 4.44949 0.218154
\(417\) 0 0
\(418\) −7.59592 −0.371528
\(419\) 1.55051 0.0757474 0.0378737 0.999283i \(-0.487942\pi\)
0.0378737 + 0.999283i \(0.487942\pi\)
\(420\) 0 0
\(421\) −4.20204 −0.204795 −0.102397 0.994744i \(-0.532651\pi\)
−0.102397 + 0.994744i \(0.532651\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) 10.8990 0.529301
\(425\) 0 0
\(426\) 0 0
\(427\) 3.55051 0.171821
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) 1.79796 0.0866046 0.0433023 0.999062i \(-0.486212\pi\)
0.0433023 + 0.999062i \(0.486212\pi\)
\(432\) 0 0
\(433\) 0.202041 0.00970947 0.00485474 0.999988i \(-0.498455\pi\)
0.00485474 + 0.999988i \(0.498455\pi\)
\(434\) −8.89898 −0.427165
\(435\) 0 0
\(436\) 6.89898 0.330401
\(437\) 4.49490 0.215020
\(438\) 0 0
\(439\) −21.3939 −1.02107 −0.510537 0.859856i \(-0.670553\pi\)
−0.510537 + 0.859856i \(0.670553\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 8.89898 0.423281
\(443\) 9.79796 0.465515 0.232758 0.972535i \(-0.425225\pi\)
0.232758 + 0.972535i \(0.425225\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 5.39388 0.253988
\(452\) 19.7980 0.931218
\(453\) 0 0
\(454\) −7.34847 −0.344881
\(455\) 0 0
\(456\) 0 0
\(457\) −29.5959 −1.38444 −0.692219 0.721687i \(-0.743367\pi\)
−0.692219 + 0.721687i \(0.743367\pi\)
\(458\) 19.1464 0.894654
\(459\) 0 0
\(460\) 0 0
\(461\) −17.3485 −0.807999 −0.403999 0.914759i \(-0.632380\pi\)
−0.403999 + 0.914759i \(0.632380\pi\)
\(462\) 0 0
\(463\) −3.59592 −0.167116 −0.0835582 0.996503i \(-0.526628\pi\)
−0.0835582 + 0.996503i \(0.526628\pi\)
\(464\) −6.89898 −0.320277
\(465\) 0 0
\(466\) −29.7980 −1.38036
\(467\) 10.4495 0.483545 0.241772 0.970333i \(-0.422271\pi\)
0.241772 + 0.970333i \(0.422271\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) −1.55051 −0.0713680
\(473\) 4.40408 0.202500
\(474\) 0 0
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) −6.20204 −0.283675
\(479\) −9.30306 −0.425068 −0.212534 0.977154i \(-0.568172\pi\)
−0.212534 + 0.977154i \(0.568172\pi\)
\(480\) 0 0
\(481\) 8.89898 0.405759
\(482\) 8.69694 0.396135
\(483\) 0 0
\(484\) 13.0000 0.590909
\(485\) 0 0
\(486\) 0 0
\(487\) 7.30306 0.330933 0.165467 0.986215i \(-0.447087\pi\)
0.165467 + 0.986215i \(0.447087\pi\)
\(488\) −3.55051 −0.160724
\(489\) 0 0
\(490\) 0 0
\(491\) −19.5959 −0.884351 −0.442176 0.896928i \(-0.645793\pi\)
−0.442176 + 0.896928i \(0.645793\pi\)
\(492\) 0 0
\(493\) −13.7980 −0.621429
\(494\) 6.89898 0.310400
\(495\) 0 0
\(496\) 8.89898 0.399576
\(497\) 1.10102 0.0493875
\(498\) 0 0
\(499\) 6.20204 0.277641 0.138821 0.990318i \(-0.455669\pi\)
0.138821 + 0.990318i \(0.455669\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −6.44949 −0.287855
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −14.2020 −0.631358
\(507\) 0 0
\(508\) 14.8990 0.661035
\(509\) 31.5505 1.39845 0.699226 0.714901i \(-0.253528\pi\)
0.699226 + 0.714901i \(0.253528\pi\)
\(510\) 0 0
\(511\) −2.89898 −0.128243
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 8.69694 0.383606
\(515\) 0 0
\(516\) 0 0
\(517\) 43.5959 1.91735
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) 0 0
\(521\) −32.6969 −1.43248 −0.716239 0.697855i \(-0.754138\pi\)
−0.716239 + 0.697855i \(0.754138\pi\)
\(522\) 0 0
\(523\) 33.1464 1.44939 0.724696 0.689069i \(-0.241980\pi\)
0.724696 + 0.689069i \(0.241980\pi\)
\(524\) 6.44949 0.281747
\(525\) 0 0
\(526\) −9.79796 −0.427211
\(527\) 17.7980 0.775291
\(528\) 0 0
\(529\) −14.5959 −0.634605
\(530\) 0 0
\(531\) 0 0
\(532\) 1.55051 0.0672231
\(533\) −4.89898 −0.212198
\(534\) 0 0
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 19.1464 0.825461
\(539\) 4.89898 0.211014
\(540\) 0 0
\(541\) 9.59592 0.412561 0.206280 0.978493i \(-0.433864\pi\)
0.206280 + 0.978493i \(0.433864\pi\)
\(542\) −12.0000 −0.515444
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) 18.6969 0.799423 0.399712 0.916641i \(-0.369110\pi\)
0.399712 + 0.916641i \(0.369110\pi\)
\(548\) −1.79796 −0.0768050
\(549\) 0 0
\(550\) 0 0
\(551\) −10.6969 −0.455705
\(552\) 0 0
\(553\) 6.89898 0.293374
\(554\) −14.8990 −0.632997
\(555\) 0 0
\(556\) 1.55051 0.0657563
\(557\) 12.6969 0.537987 0.268993 0.963142i \(-0.413309\pi\)
0.268993 + 0.963142i \(0.413309\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) −30.0454 −1.26626 −0.633131 0.774044i \(-0.718231\pi\)
−0.633131 + 0.774044i \(0.718231\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 3.75255 0.157731
\(567\) 0 0
\(568\) −1.10102 −0.0461978
\(569\) −33.7980 −1.41688 −0.708442 0.705769i \(-0.750602\pi\)
−0.708442 + 0.705769i \(0.750602\pi\)
\(570\) 0 0
\(571\) −11.1010 −0.464563 −0.232282 0.972649i \(-0.574619\pi\)
−0.232282 + 0.972649i \(0.574619\pi\)
\(572\) −21.7980 −0.911418
\(573\) 0 0
\(574\) −1.10102 −0.0459557
\(575\) 0 0
\(576\) 0 0
\(577\) 2.49490 0.103864 0.0519320 0.998651i \(-0.483462\pi\)
0.0519320 + 0.998651i \(0.483462\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 0 0
\(581\) −2.44949 −0.101622
\(582\) 0 0
\(583\) −53.3939 −2.21135
\(584\) 2.89898 0.119961
\(585\) 0 0
\(586\) −18.2474 −0.753795
\(587\) 1.14643 0.0473182 0.0236591 0.999720i \(-0.492468\pi\)
0.0236591 + 0.999720i \(0.492468\pi\)
\(588\) 0 0
\(589\) 13.7980 0.568535
\(590\) 0 0
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) −10.8990 −0.447567 −0.223784 0.974639i \(-0.571841\pi\)
−0.223784 + 0.974639i \(0.571841\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.79796 0.155570
\(597\) 0 0
\(598\) 12.8990 0.527478
\(599\) 13.1010 0.535293 0.267647 0.963517i \(-0.413754\pi\)
0.267647 + 0.963517i \(0.413754\pi\)
\(600\) 0 0
\(601\) −39.3939 −1.60691 −0.803455 0.595366i \(-0.797007\pi\)
−0.803455 + 0.595366i \(0.797007\pi\)
\(602\) −0.898979 −0.0366397
\(603\) 0 0
\(604\) 19.5959 0.797347
\(605\) 0 0
\(606\) 0 0
\(607\) −33.3939 −1.35542 −0.677708 0.735331i \(-0.737027\pi\)
−0.677708 + 0.735331i \(0.737027\pi\)
\(608\) −1.55051 −0.0628815
\(609\) 0 0
\(610\) 0 0
\(611\) −39.5959 −1.60188
\(612\) 0 0
\(613\) 27.7980 1.12275 0.561374 0.827562i \(-0.310273\pi\)
0.561374 + 0.827562i \(0.310273\pi\)
\(614\) −20.2474 −0.817121
\(615\) 0 0
\(616\) −4.89898 −0.197386
\(617\) 29.5959 1.19149 0.595743 0.803175i \(-0.296858\pi\)
0.595743 + 0.803175i \(0.296858\pi\)
\(618\) 0 0
\(619\) −41.5505 −1.67006 −0.835028 0.550207i \(-0.814549\pi\)
−0.835028 + 0.550207i \(0.814549\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) 0 0
\(626\) −21.5959 −0.863146
\(627\) 0 0
\(628\) −3.55051 −0.141681
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −42.4949 −1.69170 −0.845848 0.533425i \(-0.820905\pi\)
−0.845848 + 0.533425i \(0.820905\pi\)
\(632\) −6.89898 −0.274427
\(633\) 0 0
\(634\) 22.4949 0.893387
\(635\) 0 0
\(636\) 0 0
\(637\) −4.44949 −0.176295
\(638\) 33.7980 1.33807
\(639\) 0 0
\(640\) 0 0
\(641\) −25.7980 −1.01896 −0.509479 0.860483i \(-0.670162\pi\)
−0.509479 + 0.860483i \(0.670162\pi\)
\(642\) 0 0
\(643\) −25.1464 −0.991678 −0.495839 0.868414i \(-0.665139\pi\)
−0.495839 + 0.868414i \(0.665139\pi\)
\(644\) 2.89898 0.114236
\(645\) 0 0
\(646\) −3.10102 −0.122008
\(647\) −46.2929 −1.81996 −0.909980 0.414652i \(-0.863903\pi\)
−0.909980 + 0.414652i \(0.863903\pi\)
\(648\) 0 0
\(649\) 7.59592 0.298166
\(650\) 0 0
\(651\) 0 0
\(652\) 7.10102 0.278097
\(653\) −20.2020 −0.790567 −0.395283 0.918559i \(-0.629354\pi\)
−0.395283 + 0.918559i \(0.629354\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.10102 0.0429876
\(657\) 0 0
\(658\) −8.89898 −0.346918
\(659\) −16.8990 −0.658291 −0.329145 0.944279i \(-0.606761\pi\)
−0.329145 + 0.944279i \(0.606761\pi\)
\(660\) 0 0
\(661\) −40.9444 −1.59255 −0.796276 0.604933i \(-0.793200\pi\)
−0.796276 + 0.604933i \(0.793200\pi\)
\(662\) 18.6969 0.726677
\(663\) 0 0
\(664\) 2.44949 0.0950586
\(665\) 0 0
\(666\) 0 0
\(667\) −20.0000 −0.774403
\(668\) −4.89898 −0.189547
\(669\) 0 0
\(670\) 0 0
\(671\) 17.3939 0.671483
\(672\) 0 0
\(673\) 17.7980 0.686061 0.343030 0.939324i \(-0.388547\pi\)
0.343030 + 0.939324i \(0.388547\pi\)
\(674\) 9.59592 0.369621
\(675\) 0 0
\(676\) 6.79796 0.261460
\(677\) −36.4495 −1.40087 −0.700434 0.713717i \(-0.747010\pi\)
−0.700434 + 0.713717i \(0.747010\pi\)
\(678\) 0 0
\(679\) −15.7980 −0.606270
\(680\) 0 0
\(681\) 0 0
\(682\) −43.5959 −1.66937
\(683\) 3.59592 0.137594 0.0687970 0.997631i \(-0.478084\pi\)
0.0687970 + 0.997631i \(0.478084\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 0.898979 0.0342733
\(689\) 48.4949 1.84751
\(690\) 0 0
\(691\) 21.1464 0.804448 0.402224 0.915541i \(-0.368237\pi\)
0.402224 + 0.915541i \(0.368237\pi\)
\(692\) −6.24745 −0.237492
\(693\) 0 0
\(694\) −28.8990 −1.09699
\(695\) 0 0
\(696\) 0 0
\(697\) 2.20204 0.0834083
\(698\) 8.44949 0.319818
\(699\) 0 0
\(700\) 0 0
\(701\) −11.3031 −0.426911 −0.213455 0.976953i \(-0.568472\pi\)
−0.213455 + 0.976953i \(0.568472\pi\)
\(702\) 0 0
\(703\) −3.10102 −0.116957
\(704\) 4.89898 0.184637
\(705\) 0 0
\(706\) −22.8990 −0.861814
\(707\) −3.55051 −0.133531
\(708\) 0 0
\(709\) −28.2929 −1.06256 −0.531280 0.847196i \(-0.678289\pi\)
−0.531280 + 0.847196i \(0.678289\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) 25.7980 0.966141
\(714\) 0 0
\(715\) 0 0
\(716\) −13.7980 −0.515654
\(717\) 0 0
\(718\) −27.5959 −1.02987
\(719\) 4.49490 0.167631 0.0838157 0.996481i \(-0.473289\pi\)
0.0838157 + 0.996481i \(0.473289\pi\)
\(720\) 0 0
\(721\) −12.8990 −0.480383
\(722\) 16.5959 0.617636
\(723\) 0 0
\(724\) −10.2474 −0.380843
\(725\) 0 0
\(726\) 0 0
\(727\) −22.6969 −0.841783 −0.420891 0.907111i \(-0.638283\pi\)
−0.420891 + 0.907111i \(0.638283\pi\)
\(728\) 4.44949 0.164909
\(729\) 0 0
\(730\) 0 0
\(731\) 1.79796 0.0664999
\(732\) 0 0
\(733\) −39.6413 −1.46419 −0.732093 0.681205i \(-0.761456\pi\)
−0.732093 + 0.681205i \(0.761456\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) −2.89898 −0.106858
\(737\) 39.1918 1.44365
\(738\) 0 0
\(739\) 4.49490 0.165347 0.0826737 0.996577i \(-0.473654\pi\)
0.0826737 + 0.996577i \(0.473654\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 10.8990 0.400114
\(743\) −44.6969 −1.63977 −0.819886 0.572527i \(-0.805963\pi\)
−0.819886 + 0.572527i \(0.805963\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −4.69694 −0.171967
\(747\) 0 0
\(748\) 9.79796 0.358249
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) −41.7980 −1.52523 −0.762615 0.646853i \(-0.776085\pi\)
−0.762615 + 0.646853i \(0.776085\pi\)
\(752\) 8.89898 0.324512
\(753\) 0 0
\(754\) −30.6969 −1.11792
\(755\) 0 0
\(756\) 0 0
\(757\) 51.7980 1.88263 0.941314 0.337531i \(-0.109592\pi\)
0.941314 + 0.337531i \(0.109592\pi\)
\(758\) −30.6969 −1.11496
\(759\) 0 0
\(760\) 0 0
\(761\) 21.1010 0.764911 0.382456 0.923974i \(-0.375078\pi\)
0.382456 + 0.923974i \(0.375078\pi\)
\(762\) 0 0
\(763\) 6.89898 0.249760
\(764\) −12.6969 −0.459359
\(765\) 0 0
\(766\) 7.10102 0.256570
\(767\) −6.89898 −0.249108
\(768\) 0 0
\(769\) −40.6969 −1.46757 −0.733785 0.679382i \(-0.762248\pi\)
−0.733785 + 0.679382i \(0.762248\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 21.5959 0.777254
\(773\) 1.34847 0.0485011 0.0242505 0.999706i \(-0.492280\pi\)
0.0242505 + 0.999706i \(0.492280\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 15.7980 0.567114
\(777\) 0 0
\(778\) 13.1010 0.469694
\(779\) 1.70714 0.0611648
\(780\) 0 0
\(781\) 5.39388 0.193008
\(782\) −5.79796 −0.207335
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) −50.4495 −1.79833 −0.899165 0.437610i \(-0.855825\pi\)
−0.899165 + 0.437610i \(0.855825\pi\)
\(788\) 18.8990 0.673248
\(789\) 0 0
\(790\) 0 0
\(791\) 19.7980 0.703934
\(792\) 0 0
\(793\) −15.7980 −0.561002
\(794\) −2.65153 −0.0940993
\(795\) 0 0
\(796\) 16.8990 0.598968
\(797\) −0.944387 −0.0334519 −0.0167260 0.999860i \(-0.505324\pi\)
−0.0167260 + 0.999860i \(0.505324\pi\)
\(798\) 0 0
\(799\) 17.7980 0.629647
\(800\) 0 0
\(801\) 0 0
\(802\) −29.3939 −1.03793
\(803\) −14.2020 −0.501179
\(804\) 0 0
\(805\) 0 0
\(806\) 39.5959 1.39471
\(807\) 0 0
\(808\) 3.55051 0.124907
\(809\) −47.5959 −1.67338 −0.836692 0.547674i \(-0.815513\pi\)
−0.836692 + 0.547674i \(0.815513\pi\)
\(810\) 0 0
\(811\) 14.9444 0.524768 0.262384 0.964963i \(-0.415491\pi\)
0.262384 + 0.964963i \(0.415491\pi\)
\(812\) −6.89898 −0.242107
\(813\) 0 0
\(814\) 9.79796 0.343418
\(815\) 0 0
\(816\) 0 0
\(817\) 1.39388 0.0487656
\(818\) −34.4949 −1.20609
\(819\) 0 0
\(820\) 0 0
\(821\) −8.20204 −0.286253 −0.143127 0.989704i \(-0.545716\pi\)
−0.143127 + 0.989704i \(0.545716\pi\)
\(822\) 0 0
\(823\) 39.1918 1.36614 0.683071 0.730352i \(-0.260644\pi\)
0.683071 + 0.730352i \(0.260644\pi\)
\(824\) 12.8990 0.449357
\(825\) 0 0
\(826\) −1.55051 −0.0539492
\(827\) −15.5959 −0.542323 −0.271162 0.962534i \(-0.587408\pi\)
−0.271162 + 0.962534i \(0.587408\pi\)
\(828\) 0 0
\(829\) −43.6413 −1.51573 −0.757863 0.652414i \(-0.773756\pi\)
−0.757863 + 0.652414i \(0.773756\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.44949 −0.154258
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) 7.59592 0.262710
\(837\) 0 0
\(838\) −1.55051 −0.0535615
\(839\) −36.8990 −1.27389 −0.636947 0.770907i \(-0.719803\pi\)
−0.636947 + 0.770907i \(0.719803\pi\)
\(840\) 0 0
\(841\) 18.5959 0.641239
\(842\) 4.20204 0.144812
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) 13.0000 0.446685
\(848\) −10.8990 −0.374272
\(849\) 0 0
\(850\) 0 0
\(851\) −5.79796 −0.198751
\(852\) 0 0
\(853\) 33.8434 1.15877 0.579387 0.815052i \(-0.303292\pi\)
0.579387 + 0.815052i \(0.303292\pi\)
\(854\) −3.55051 −0.121496
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) −53.1918 −1.81700 −0.908499 0.417886i \(-0.862771\pi\)
−0.908499 + 0.417886i \(0.862771\pi\)
\(858\) 0 0
\(859\) 53.6413 1.83022 0.915109 0.403206i \(-0.132104\pi\)
0.915109 + 0.403206i \(0.132104\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.79796 −0.0612387
\(863\) −45.3939 −1.54523 −0.772613 0.634878i \(-0.781051\pi\)
−0.772613 + 0.634878i \(0.781051\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −0.202041 −0.00686563
\(867\) 0 0
\(868\) 8.89898 0.302051
\(869\) 33.7980 1.14652
\(870\) 0 0
\(871\) −35.5959 −1.20612
\(872\) −6.89898 −0.233629
\(873\) 0 0
\(874\) −4.49490 −0.152042
\(875\) 0 0
\(876\) 0 0
\(877\) 39.3939 1.33024 0.665118 0.746738i \(-0.268381\pi\)
0.665118 + 0.746738i \(0.268381\pi\)
\(878\) 21.3939 0.722008
\(879\) 0 0
\(880\) 0 0
\(881\) −8.20204 −0.276334 −0.138167 0.990409i \(-0.544121\pi\)
−0.138167 + 0.990409i \(0.544121\pi\)
\(882\) 0 0
\(883\) −22.2020 −0.747158 −0.373579 0.927598i \(-0.621870\pi\)
−0.373579 + 0.927598i \(0.621870\pi\)
\(884\) −8.89898 −0.299305
\(885\) 0 0
\(886\) −9.79796 −0.329169
\(887\) 2.69694 0.0905543 0.0452772 0.998974i \(-0.485583\pi\)
0.0452772 + 0.998974i \(0.485583\pi\)
\(888\) 0 0
\(889\) 14.8990 0.499696
\(890\) 0 0
\(891\) 0 0
\(892\) 4.00000 0.133930
\(893\) 13.7980 0.461731
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −10.0000 −0.333704
\(899\) −61.3939 −2.04760
\(900\) 0 0
\(901\) −21.7980 −0.726195
\(902\) −5.39388 −0.179596
\(903\) 0 0
\(904\) −19.7980 −0.658470
\(905\) 0 0
\(906\) 0 0
\(907\) 41.7980 1.38788 0.693939 0.720034i \(-0.255874\pi\)
0.693939 + 0.720034i \(0.255874\pi\)
\(908\) 7.34847 0.243868
\(909\) 0 0
\(910\) 0 0
\(911\) 35.5959 1.17935 0.589673 0.807642i \(-0.299257\pi\)
0.589673 + 0.807642i \(0.299257\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 29.5959 0.978946
\(915\) 0 0
\(916\) −19.1464 −0.632616
\(917\) 6.44949 0.212981
\(918\) 0 0
\(919\) −26.8990 −0.887315 −0.443658 0.896196i \(-0.646319\pi\)
−0.443658 + 0.896196i \(0.646319\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 17.3485 0.571341
\(923\) −4.89898 −0.161252
\(924\) 0 0
\(925\) 0 0
\(926\) 3.59592 0.118169
\(927\) 0 0
\(928\) 6.89898 0.226470
\(929\) 28.2929 0.928259 0.464129 0.885767i \(-0.346367\pi\)
0.464129 + 0.885767i \(0.346367\pi\)
\(930\) 0 0
\(931\) 1.55051 0.0508159
\(932\) 29.7980 0.976065
\(933\) 0 0
\(934\) −10.4495 −0.341918
\(935\) 0 0
\(936\) 0 0
\(937\) 41.1010 1.34271 0.671356 0.741135i \(-0.265712\pi\)
0.671356 + 0.741135i \(0.265712\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) 0 0
\(941\) 19.5505 0.637328 0.318664 0.947868i \(-0.396766\pi\)
0.318664 + 0.947868i \(0.396766\pi\)
\(942\) 0 0
\(943\) 3.19184 0.103940
\(944\) 1.55051 0.0504648
\(945\) 0 0
\(946\) −4.40408 −0.143189
\(947\) 44.0908 1.43276 0.716379 0.697711i \(-0.245798\pi\)
0.716379 + 0.697711i \(0.245798\pi\)
\(948\) 0 0
\(949\) 12.8990 0.418719
\(950\) 0 0
\(951\) 0 0
\(952\) −2.00000 −0.0648204
\(953\) 2.20204 0.0713311 0.0356656 0.999364i \(-0.488645\pi\)
0.0356656 + 0.999364i \(0.488645\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 6.20204 0.200588
\(957\) 0 0
\(958\) 9.30306 0.300568
\(959\) −1.79796 −0.0580591
\(960\) 0 0
\(961\) 48.1918 1.55458
\(962\) −8.89898 −0.286915
\(963\) 0 0
\(964\) −8.69694 −0.280110
\(965\) 0 0
\(966\) 0 0
\(967\) 36.2929 1.16710 0.583550 0.812077i \(-0.301663\pi\)
0.583550 + 0.812077i \(0.301663\pi\)
\(968\) −13.0000 −0.417836
\(969\) 0 0
\(970\) 0 0
\(971\) 9.55051 0.306490 0.153245 0.988188i \(-0.451028\pi\)
0.153245 + 0.988188i \(0.451028\pi\)
\(972\) 0 0
\(973\) 1.55051 0.0497071
\(974\) −7.30306 −0.234005
\(975\) 0 0
\(976\) 3.55051 0.113649
\(977\) −29.3939 −0.940393 −0.470197 0.882562i \(-0.655817\pi\)
−0.470197 + 0.882562i \(0.655817\pi\)
\(978\) 0 0
\(979\) 48.9898 1.56572
\(980\) 0 0
\(981\) 0 0
\(982\) 19.5959 0.625331
\(983\) −13.3031 −0.424302 −0.212151 0.977237i \(-0.568047\pi\)
−0.212151 + 0.977237i \(0.568047\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 13.7980 0.439417
\(987\) 0 0
\(988\) −6.89898 −0.219486
\(989\) 2.60612 0.0828699
\(990\) 0 0
\(991\) 31.3031 0.994375 0.497187 0.867643i \(-0.334366\pi\)
0.497187 + 0.867643i \(0.334366\pi\)
\(992\) −8.89898 −0.282543
\(993\) 0 0
\(994\) −1.10102 −0.0349223
\(995\) 0 0
\(996\) 0 0
\(997\) −57.3485 −1.81624 −0.908122 0.418705i \(-0.862484\pi\)
−0.908122 + 0.418705i \(0.862484\pi\)
\(998\) −6.20204 −0.196322
\(999\) 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 3150.2.a.bs.1.2 2
3.2 odd 2 350.2.a.h.1.2 2
5.2 odd 4 630.2.g.g.379.1 4
5.3 odd 4 630.2.g.g.379.3 4
5.4 even 2 3150.2.a.bt.1.2 2
12.11 even 2 2800.2.a.bl.1.1 2
15.2 even 4 70.2.c.a.29.3 yes 4
15.8 even 4 70.2.c.a.29.2 4
15.14 odd 2 350.2.a.g.1.1 2
20.3 even 4 5040.2.t.t.1009.2 4
20.7 even 4 5040.2.t.t.1009.1 4
21.20 even 2 2450.2.a.bq.1.1 2
60.23 odd 4 560.2.g.e.449.2 4
60.47 odd 4 560.2.g.e.449.4 4
60.59 even 2 2800.2.a.bm.1.2 2
105.2 even 12 490.2.i.c.459.4 8
105.17 odd 12 490.2.i.f.79.2 8
105.23 even 12 490.2.i.c.459.1 8
105.32 even 12 490.2.i.c.79.1 8
105.38 odd 12 490.2.i.f.79.3 8
105.47 odd 12 490.2.i.f.459.3 8
105.53 even 12 490.2.i.c.79.4 8
105.62 odd 4 490.2.c.e.99.4 4
105.68 odd 12 490.2.i.f.459.2 8
105.83 odd 4 490.2.c.e.99.1 4
105.104 even 2 2450.2.a.bl.1.2 2
120.53 even 4 2240.2.g.j.449.1 4
120.77 even 4 2240.2.g.j.449.3 4
120.83 odd 4 2240.2.g.i.449.3 4
120.107 odd 4 2240.2.g.i.449.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.c.a.29.2 4 15.8 even 4
70.2.c.a.29.3 yes 4 15.2 even 4
350.2.a.g.1.1 2 15.14 odd 2
350.2.a.h.1.2 2 3.2 odd 2
490.2.c.e.99.1 4 105.83 odd 4
490.2.c.e.99.4 4 105.62 odd 4
490.2.i.c.79.1 8 105.32 even 12
490.2.i.c.79.4 8 105.53 even 12
490.2.i.c.459.1 8 105.23 even 12
490.2.i.c.459.4 8 105.2 even 12
490.2.i.f.79.2 8 105.17 odd 12
490.2.i.f.79.3 8 105.38 odd 12
490.2.i.f.459.2 8 105.68 odd 12
490.2.i.f.459.3 8 105.47 odd 12
560.2.g.e.449.2 4 60.23 odd 4
560.2.g.e.449.4 4 60.47 odd 4
630.2.g.g.379.1 4 5.2 odd 4
630.2.g.g.379.3 4 5.3 odd 4
2240.2.g.i.449.1 4 120.107 odd 4
2240.2.g.i.449.3 4 120.83 odd 4
2240.2.g.j.449.1 4 120.53 even 4
2240.2.g.j.449.3 4 120.77 even 4
2450.2.a.bl.1.2 2 105.104 even 2
2450.2.a.bq.1.1 2 21.20 even 2
2800.2.a.bl.1.1 2 12.11 even 2
2800.2.a.bm.1.2 2 60.59 even 2
3150.2.a.bs.1.2 2 1.1 even 1 trivial
3150.2.a.bt.1.2 2 5.4 even 2
5040.2.t.t.1009.1 4 20.7 even 4
5040.2.t.t.1009.2 4 20.3 even 4