Properties

Label 3330.2.a.f.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -5.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -5.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +5.00000 q^{11} -1.00000 q^{13} +5.00000 q^{14} +1.00000 q^{16} +5.00000 q^{17} -3.00000 q^{19} +1.00000 q^{20} -5.00000 q^{22} -3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} -5.00000 q^{28} -6.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} -5.00000 q^{34} -5.00000 q^{35} -1.00000 q^{37} +3.00000 q^{38} -1.00000 q^{40} +4.00000 q^{43} +5.00000 q^{44} +3.00000 q^{46} +18.0000 q^{49} -1.00000 q^{50} -1.00000 q^{52} +3.00000 q^{53} +5.00000 q^{55} +5.00000 q^{56} +6.00000 q^{58} +10.0000 q^{59} +10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -14.0000 q^{67} +5.00000 q^{68} +5.00000 q^{70} -6.00000 q^{71} -9.00000 q^{73} +1.00000 q^{74} -3.00000 q^{76} -25.0000 q^{77} +1.00000 q^{80} -5.00000 q^{83} +5.00000 q^{85} -4.00000 q^{86} -5.00000 q^{88} -13.0000 q^{89} +5.00000 q^{91} -3.00000 q^{92} -3.00000 q^{95} -4.00000 q^{97} -18.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −5.00000 −1.88982 −0.944911 0.327327i \(-0.893852\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 5.00000 1.33631
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −5.00000 −0.944911
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −5.00000 −0.857493
\(35\) −5.00000 −0.845154
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 3.00000 0.486664
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 5.00000 0.668153
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −14.0000 −1.71037 −0.855186 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 5.00000 0.606339
\(69\) 0 0
\(70\) 5.00000 0.597614
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −9.00000 −1.05337 −0.526685 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) −25.0000 −2.84901
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 0 0
\(83\) −5.00000 −0.548821 −0.274411 0.961613i \(-0.588483\pi\)
−0.274411 + 0.961613i \(0.588483\pi\)
\(84\) 0 0
\(85\) 5.00000 0.542326
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −5.00000 −0.533002
\(89\) −13.0000 −1.37800 −0.688999 0.724763i \(-0.741949\pi\)
−0.688999 + 0.724763i \(0.741949\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) −3.00000 −0.312772
\(93\) 0 0
\(94\) 0 0
\(95\) −3.00000 −0.307794
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) −18.0000 −1.81827
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) −15.0000 −1.45010 −0.725052 0.688694i \(-0.758184\pi\)
−0.725052 + 0.688694i \(0.758184\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) −5.00000 −0.476731
\(111\) 0 0
\(112\) −5.00000 −0.472456
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) −25.0000 −2.29175
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −3.00000 −0.266207 −0.133103 0.991102i \(-0.542494\pi\)
−0.133103 + 0.991102i \(0.542494\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 1.00000 0.0877058
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) 15.0000 1.30066
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) −5.00000 −0.428746
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) −5.00000 −0.422577
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) −5.00000 −0.418121
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 9.00000 0.744845
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 15.0000 1.22068 0.610341 0.792139i \(-0.291032\pi\)
0.610341 + 0.792139i \(0.291032\pi\)
\(152\) 3.00000 0.243332
\(153\) 0 0
\(154\) 25.0000 2.01456
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 15.0000 1.18217
\(162\) 0 0
\(163\) −23.0000 −1.80150 −0.900750 0.434339i \(-0.856982\pi\)
−0.900750 + 0.434339i \(0.856982\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 5.00000 0.388075
\(167\) −5.00000 −0.386912 −0.193456 0.981109i \(-0.561970\pi\)
−0.193456 + 0.981109i \(0.561970\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −5.00000 −0.383482
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −11.0000 −0.836315 −0.418157 0.908375i \(-0.637324\pi\)
−0.418157 + 0.908375i \(0.637324\pi\)
\(174\) 0 0
\(175\) −5.00000 −0.377964
\(176\) 5.00000 0.376889
\(177\) 0 0
\(178\) 13.0000 0.974391
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −5.00000 −0.370625
\(183\) 0 0
\(184\) 3.00000 0.221163
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) 25.0000 1.82818
\(188\) 0 0
\(189\) 0 0
\(190\) 3.00000 0.217643
\(191\) 7.00000 0.506502 0.253251 0.967401i \(-0.418500\pi\)
0.253251 + 0.967401i \(0.418500\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 4.00000 0.287183
\(195\) 0 0
\(196\) 18.0000 1.28571
\(197\) −23.0000 −1.63868 −0.819341 0.573306i \(-0.805660\pi\)
−0.819341 + 0.573306i \(0.805660\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 30.0000 2.10559
\(204\) 0 0
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 3.00000 0.206041
\(213\) 0 0
\(214\) 15.0000 1.02538
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 30.0000 2.03653
\(218\) 5.00000 0.338643
\(219\) 0 0
\(220\) 5.00000 0.337100
\(221\) −5.00000 −0.336336
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 5.00000 0.334077
\(225\) 0 0
\(226\) −10.0000 −0.665190
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 3.00000 0.197814
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) 25.0000 1.62051
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −14.0000 −0.899954
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 18.0000 1.14998
\(246\) 0 0
\(247\) 3.00000 0.190885
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −30.0000 −1.89358 −0.946792 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) 0 0
\(253\) −15.0000 −0.943042
\(254\) 3.00000 0.188237
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −15.0000 −0.935674 −0.467837 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) 0 0
\(259\) 5.00000 0.310685
\(260\) −1.00000 −0.0620174
\(261\) 0 0
\(262\) −20.0000 −1.23560
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 3.00000 0.184289
\(266\) −15.0000 −0.919709
\(267\) 0 0
\(268\) −14.0000 −0.855186
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 5.00000 0.303170
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 5.00000 0.301511
\(276\) 0 0
\(277\) 25.0000 1.50210 0.751052 0.660243i \(-0.229547\pi\)
0.751052 + 0.660243i \(0.229547\pi\)
\(278\) −2.00000 −0.119952
\(279\) 0 0
\(280\) 5.00000 0.298807
\(281\) −1.00000 −0.0596550 −0.0298275 0.999555i \(-0.509496\pi\)
−0.0298275 + 0.999555i \(0.509496\pi\)
\(282\) 0 0
\(283\) 5.00000 0.297219 0.148610 0.988896i \(-0.452520\pi\)
0.148610 + 0.988896i \(0.452520\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 5.00000 0.295656
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −9.00000 −0.526685
\(293\) 11.0000 0.642627 0.321313 0.946973i \(-0.395876\pi\)
0.321313 + 0.946973i \(0.395876\pi\)
\(294\) 0 0
\(295\) 10.0000 0.582223
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) −15.0000 −0.863153
\(303\) 0 0
\(304\) −3.00000 −0.172062
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) −30.0000 −1.71219 −0.856095 0.516818i \(-0.827116\pi\)
−0.856095 + 0.516818i \(0.827116\pi\)
\(308\) −25.0000 −1.42451
\(309\) 0 0
\(310\) 6.00000 0.340777
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 24.0000 1.35656 0.678280 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) 20.0000 1.12867
\(315\) 0 0
\(316\) 0 0
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) −30.0000 −1.67968
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −15.0000 −0.835917
\(323\) −15.0000 −0.834622
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 23.0000 1.27385
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −5.00000 −0.274411
\(333\) 0 0
\(334\) 5.00000 0.273588
\(335\) −14.0000 −0.764902
\(336\) 0 0
\(337\) −9.00000 −0.490261 −0.245131 0.969490i \(-0.578831\pi\)
−0.245131 + 0.969490i \(0.578831\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) 5.00000 0.271163
\(341\) −30.0000 −1.62459
\(342\) 0 0
\(343\) −55.0000 −2.96972
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 11.0000 0.591364
\(347\) 36.0000 1.93258 0.966291 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) 0 0
\(349\) 24.0000 1.28469 0.642345 0.766415i \(-0.277962\pi\)
0.642345 + 0.766415i \(0.277962\pi\)
\(350\) 5.00000 0.267261
\(351\) 0 0
\(352\) −5.00000 −0.266501
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) −13.0000 −0.688999
\(357\) 0 0
\(358\) 8.00000 0.422813
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) 5.00000 0.262071
\(365\) −9.00000 −0.471082
\(366\) 0 0
\(367\) −5.00000 −0.260998 −0.130499 0.991448i \(-0.541658\pi\)
−0.130499 + 0.991448i \(0.541658\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) −15.0000 −0.778761
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −25.0000 −1.29272
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) −3.00000 −0.153897
\(381\) 0 0
\(382\) −7.00000 −0.358151
\(383\) −9.00000 −0.459879 −0.229939 0.973205i \(-0.573853\pi\)
−0.229939 + 0.973205i \(0.573853\pi\)
\(384\) 0 0
\(385\) −25.0000 −1.27412
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) −4.00000 −0.203069
\(389\) −34.0000 −1.72387 −0.861934 0.507020i \(-0.830747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) 0 0
\(391\) −15.0000 −0.758583
\(392\) −18.0000 −0.909137
\(393\) 0 0
\(394\) 23.0000 1.15872
\(395\) 0 0
\(396\) 0 0
\(397\) −36.0000 −1.80679 −0.903394 0.428811i \(-0.858933\pi\)
−0.903394 + 0.428811i \(0.858933\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) −30.0000 −1.48888
\(407\) −5.00000 −0.247841
\(408\) 0 0
\(409\) 24.0000 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) −50.0000 −2.46034
\(414\) 0 0
\(415\) −5.00000 −0.245440
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 15.0000 0.733674
\(419\) 19.0000 0.928211 0.464105 0.885780i \(-0.346376\pi\)
0.464105 + 0.885780i \(0.346376\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 2.00000 0.0973585
\(423\) 0 0
\(424\) −3.00000 −0.145693
\(425\) 5.00000 0.242536
\(426\) 0 0
\(427\) −50.0000 −2.41967
\(428\) −15.0000 −0.725052
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 33.0000 1.58955 0.794777 0.606902i \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) 0 0
\(433\) 21.0000 1.00920 0.504598 0.863355i \(-0.331641\pi\)
0.504598 + 0.863355i \(0.331641\pi\)
\(434\) −30.0000 −1.44005
\(435\) 0 0
\(436\) −5.00000 −0.239457
\(437\) 9.00000 0.430528
\(438\) 0 0
\(439\) 36.0000 1.71819 0.859093 0.511819i \(-0.171028\pi\)
0.859093 + 0.511819i \(0.171028\pi\)
\(440\) −5.00000 −0.238366
\(441\) 0 0
\(442\) 5.00000 0.237826
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 0 0
\(445\) −13.0000 −0.616259
\(446\) 8.00000 0.378811
\(447\) 0 0
\(448\) −5.00000 −0.236228
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 10.0000 0.470360
\(453\) 0 0
\(454\) 8.00000 0.375459
\(455\) 5.00000 0.234404
\(456\) 0 0
\(457\) 4.00000 0.187112 0.0935561 0.995614i \(-0.470177\pi\)
0.0935561 + 0.995614i \(0.470177\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) −3.00000 −0.139876
\(461\) −26.0000 −1.21094 −0.605470 0.795868i \(-0.707015\pi\)
−0.605470 + 0.795868i \(0.707015\pi\)
\(462\) 0 0
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 70.0000 3.23230
\(470\) 0 0
\(471\) 0 0
\(472\) −10.0000 −0.460287
\(473\) 20.0000 0.919601
\(474\) 0 0
\(475\) −3.00000 −0.137649
\(476\) −25.0000 −1.14587
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) 21.0000 0.959514 0.479757 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(480\) 0 0
\(481\) 1.00000 0.0455961
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) −18.0000 −0.813157
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 0 0
\(493\) −30.0000 −1.35113
\(494\) −3.00000 −0.134976
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 30.0000 1.34568
\(498\) 0 0
\(499\) −5.00000 −0.223831 −0.111915 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 30.0000 1.33897
\(503\) 20.0000 0.891756 0.445878 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 15.0000 0.666831
\(507\) 0 0
\(508\) −3.00000 −0.133103
\(509\) −15.0000 −0.664863 −0.332432 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 0 0
\(511\) 45.0000 1.99068
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 15.0000 0.661622
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) −5.00000 −0.219687
\(519\) 0 0
\(520\) 1.00000 0.0438529
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −30.0000 −1.30682
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −3.00000 −0.130312
\(531\) 0 0
\(532\) 15.0000 0.650332
\(533\) 0 0
\(534\) 0 0
\(535\) −15.0000 −0.648507
\(536\) 14.0000 0.604708
\(537\) 0 0
\(538\) 9.00000 0.388018
\(539\) 90.0000 3.87657
\(540\) 0 0
\(541\) 5.00000 0.214967 0.107483 0.994207i \(-0.465721\pi\)
0.107483 + 0.994207i \(0.465721\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) −5.00000 −0.214373
\(545\) −5.00000 −0.214176
\(546\) 0 0
\(547\) −43.0000 −1.83855 −0.919274 0.393619i \(-0.871223\pi\)
−0.919274 + 0.393619i \(0.871223\pi\)
\(548\) 2.00000 0.0854358
\(549\) 0 0
\(550\) −5.00000 −0.213201
\(551\) 18.0000 0.766826
\(552\) 0 0
\(553\) 0 0
\(554\) −25.0000 −1.06215
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) −5.00000 −0.211289
\(561\) 0 0
\(562\) 1.00000 0.0421825
\(563\) −22.0000 −0.927189 −0.463595 0.886047i \(-0.653441\pi\)
−0.463595 + 0.886047i \(0.653441\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) −5.00000 −0.210166
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 13.0000 0.544988 0.272494 0.962157i \(-0.412151\pi\)
0.272494 + 0.962157i \(0.412151\pi\)
\(570\) 0 0
\(571\) −10.0000 −0.418487 −0.209243 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) −5.00000 −0.209061
\(573\) 0 0
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) 26.0000 1.08239 0.541197 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 25.0000 1.03717
\(582\) 0 0
\(583\) 15.0000 0.621237
\(584\) 9.00000 0.372423
\(585\) 0 0
\(586\) −11.0000 −0.454406
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 0 0
\(589\) 18.0000 0.741677
\(590\) −10.0000 −0.411693
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) 24.0000 0.985562 0.492781 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) −25.0000 −1.02490
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) −3.00000 −0.122679
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 20.0000 0.815139
\(603\) 0 0
\(604\) 15.0000 0.610341
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) −18.0000 −0.730597 −0.365299 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 3.00000 0.121666
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 30.0000 1.21070
\(615\) 0 0
\(616\) 25.0000 1.00728
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 0 0
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) −6.00000 −0.240966
\(621\) 0 0
\(622\) 8.00000 0.320771
\(623\) 65.0000 2.60417
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −24.0000 −0.959233
\(627\) 0 0
\(628\) −20.0000 −0.798087
\(629\) −5.00000 −0.199363
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −10.0000 −0.397151
\(635\) −3.00000 −0.119051
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) 30.0000 1.18771
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 23.0000 0.907031 0.453516 0.891248i \(-0.350170\pi\)
0.453516 + 0.891248i \(0.350170\pi\)
\(644\) 15.0000 0.591083
\(645\) 0 0
\(646\) 15.0000 0.590167
\(647\) −3.00000 −0.117942 −0.0589711 0.998260i \(-0.518782\pi\)
−0.0589711 + 0.998260i \(0.518782\pi\)
\(648\) 0 0
\(649\) 50.0000 1.96267
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) −23.0000 −0.900750
\(653\) 48.0000 1.87839 0.939193 0.343391i \(-0.111576\pi\)
0.939193 + 0.343391i \(0.111576\pi\)
\(654\) 0 0
\(655\) 20.0000 0.781465
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) −47.0000 −1.82809 −0.914044 0.405615i \(-0.867057\pi\)
−0.914044 + 0.405615i \(0.867057\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 5.00000 0.194038
\(665\) 15.0000 0.581675
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) −5.00000 −0.193456
\(669\) 0 0
\(670\) 14.0000 0.540867
\(671\) 50.0000 1.93023
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 9.00000 0.346667
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 9.00000 0.345898 0.172949 0.984931i \(-0.444670\pi\)
0.172949 + 0.984931i \(0.444670\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) −5.00000 −0.191741
\(681\) 0 0
\(682\) 30.0000 1.14876
\(683\) −48.0000 −1.83667 −0.918334 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 55.0000 2.09991
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −3.00000 −0.114291
\(690\) 0 0
\(691\) −22.0000 −0.836919 −0.418460 0.908235i \(-0.637430\pi\)
−0.418460 + 0.908235i \(0.637430\pi\)
\(692\) −11.0000 −0.418157
\(693\) 0 0
\(694\) −36.0000 −1.36654
\(695\) 2.00000 0.0758643
\(696\) 0 0
\(697\) 0 0
\(698\) −24.0000 −0.908413
\(699\) 0 0
\(700\) −5.00000 −0.188982
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 3.00000 0.113147
\(704\) 5.00000 0.188445
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) −30.0000 −1.12827
\(708\) 0 0
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 6.00000 0.225176
\(711\) 0 0
\(712\) 13.0000 0.487196
\(713\) 18.0000 0.674105
\(714\) 0 0
\(715\) −5.00000 −0.186989
\(716\) −8.00000 −0.298974
\(717\) 0 0
\(718\) 36.0000 1.34351
\(719\) 26.0000 0.969636 0.484818 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) 0 0
\(721\) 80.0000 2.97936
\(722\) 10.0000 0.372161
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −30.0000 −1.11264 −0.556319 0.830969i \(-0.687787\pi\)
−0.556319 + 0.830969i \(0.687787\pi\)
\(728\) −5.00000 −0.185312
\(729\) 0 0
\(730\) 9.00000 0.333105
\(731\) 20.0000 0.739727
\(732\) 0 0
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) 5.00000 0.184553
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) −70.0000 −2.57848
\(738\) 0 0
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) 15.0000 0.550667
\(743\) −10.0000 −0.366864 −0.183432 0.983032i \(-0.558721\pi\)
−0.183432 + 0.983032i \(0.558721\pi\)
\(744\) 0 0
\(745\) −2.00000 −0.0732743
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) 25.0000 0.914091
\(749\) 75.0000 2.74044
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −6.00000 −0.218507
\(755\) 15.0000 0.545906
\(756\) 0 0
\(757\) 37.0000 1.34479 0.672394 0.740193i \(-0.265266\pi\)
0.672394 + 0.740193i \(0.265266\pi\)
\(758\) 26.0000 0.944363
\(759\) 0 0
\(760\) 3.00000 0.108821
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 25.0000 0.905061
\(764\) 7.00000 0.253251
\(765\) 0 0
\(766\) 9.00000 0.325183
\(767\) −10.0000 −0.361079
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 25.0000 0.900937
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) 21.0000 0.755318 0.377659 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 4.00000 0.143592
\(777\) 0 0
\(778\) 34.0000 1.21896
\(779\) 0 0
\(780\) 0 0
\(781\) −30.0000 −1.07348
\(782\) 15.0000 0.536399
\(783\) 0 0
\(784\) 18.0000 0.642857
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 42.0000 1.49714 0.748569 0.663057i \(-0.230741\pi\)
0.748569 + 0.663057i \(0.230741\pi\)
\(788\) −23.0000 −0.819341
\(789\) 0 0
\(790\) 0 0
\(791\) −50.0000 −1.77780
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 36.0000 1.27759
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) 14.0000 0.495905 0.247953 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −27.0000 −0.953403
\(803\) −45.0000 −1.58802
\(804\) 0 0
\(805\) 15.0000 0.528681
\(806\) −6.00000 −0.211341
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) 40.0000 1.40459 0.702295 0.711886i \(-0.252159\pi\)
0.702295 + 0.711886i \(0.252159\pi\)
\(812\) 30.0000 1.05279
\(813\) 0 0
\(814\) 5.00000 0.175250
\(815\) −23.0000 −0.805655
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) −24.0000 −0.839140
\(819\) 0 0
\(820\) 0 0
\(821\) −39.0000 −1.36111 −0.680555 0.732697i \(-0.738261\pi\)
−0.680555 + 0.732697i \(0.738261\pi\)
\(822\) 0 0
\(823\) −35.0000 −1.22002 −0.610012 0.792392i \(-0.708835\pi\)
−0.610012 + 0.792392i \(0.708835\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 50.0000 1.73972
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 7.00000 0.243120 0.121560 0.992584i \(-0.461210\pi\)
0.121560 + 0.992584i \(0.461210\pi\)
\(830\) 5.00000 0.173553
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 90.0000 3.11832
\(834\) 0 0
\(835\) −5.00000 −0.173032
\(836\) −15.0000 −0.518786
\(837\) 0 0
\(838\) −19.0000 −0.656344
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 6.00000 0.206774
\(843\) 0 0
\(844\) −2.00000 −0.0688428
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) −70.0000 −2.40523
\(848\) 3.00000 0.103020
\(849\) 0 0
\(850\) −5.00000 −0.171499
\(851\) 3.00000 0.102839
\(852\) 0 0
\(853\) 19.0000 0.650548 0.325274 0.945620i \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) 50.0000 1.71096
\(855\) 0 0
\(856\) 15.0000 0.512689
\(857\) −9.00000 −0.307434 −0.153717 0.988115i \(-0.549124\pi\)
−0.153717 + 0.988115i \(0.549124\pi\)
\(858\) 0 0
\(859\) 7.00000 0.238837 0.119418 0.992844i \(-0.461897\pi\)
0.119418 + 0.992844i \(0.461897\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) −33.0000 −1.12398
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 0 0
\(865\) −11.0000 −0.374011
\(866\) −21.0000 −0.713609
\(867\) 0 0
\(868\) 30.0000 1.01827
\(869\) 0 0
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 5.00000 0.169321
\(873\) 0 0
\(874\) −9.00000 −0.304430
\(875\) −5.00000 −0.169031
\(876\) 0 0
\(877\) −52.0000 −1.75592 −0.877958 0.478738i \(-0.841094\pi\)
−0.877958 + 0.478738i \(0.841094\pi\)
\(878\) −36.0000 −1.21494
\(879\) 0 0
\(880\) 5.00000 0.168550
\(881\) 40.0000 1.34763 0.673817 0.738898i \(-0.264654\pi\)
0.673817 + 0.738898i \(0.264654\pi\)
\(882\) 0 0
\(883\) 21.0000 0.706706 0.353353 0.935490i \(-0.385041\pi\)
0.353353 + 0.935490i \(0.385041\pi\)
\(884\) −5.00000 −0.168168
\(885\) 0 0
\(886\) −28.0000 −0.940678
\(887\) 38.0000 1.27592 0.637958 0.770072i \(-0.279780\pi\)
0.637958 + 0.770072i \(0.279780\pi\)
\(888\) 0 0
\(889\) 15.0000 0.503084
\(890\) 13.0000 0.435761
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 5.00000 0.167038
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) 15.0000 0.499722
\(902\) 0 0
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) 37.0000 1.22856 0.614282 0.789086i \(-0.289446\pi\)
0.614282 + 0.789086i \(0.289446\pi\)
\(908\) −8.00000 −0.265489
\(909\) 0 0
\(910\) −5.00000 −0.165748
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) −25.0000 −0.827379
\(914\) −4.00000 −0.132308
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) −100.000 −3.30229
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 3.00000 0.0989071
\(921\) 0 0
\(922\) 26.0000 0.856264
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 28.0000 0.920137
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) −2.00000 −0.0656179 −0.0328089 0.999462i \(-0.510445\pi\)
−0.0328089 + 0.999462i \(0.510445\pi\)
\(930\) 0 0
\(931\) −54.0000 −1.76978
\(932\) 26.0000 0.851658
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 25.0000 0.817587
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) −70.0000 −2.28558
\(939\) 0 0
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 9.00000 0.292152
\(950\) 3.00000 0.0973329
\(951\) 0 0
\(952\) 25.0000 0.810255
\(953\) −14.0000 −0.453504 −0.226752 0.973952i \(-0.572811\pi\)
−0.226752 + 0.973952i \(0.572811\pi\)
\(954\) 0 0
\(955\) 7.00000 0.226515
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) −21.0000 −0.678479
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −1.00000 −0.0322413
\(963\) 0 0
\(964\) −18.0000 −0.579741
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −18.0000 −0.578841 −0.289420 0.957202i \(-0.593463\pi\)
−0.289420 + 0.957202i \(0.593463\pi\)
\(968\) −14.0000 −0.449977
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) 0 0
\(973\) −10.0000 −0.320585
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −39.0000 −1.24772 −0.623860 0.781536i \(-0.714437\pi\)
−0.623860 + 0.781536i \(0.714437\pi\)
\(978\) 0 0
\(979\) −65.0000 −2.07741
\(980\) 18.0000 0.574989
\(981\) 0 0
\(982\) −9.00000 −0.287202
\(983\) 58.0000 1.84991 0.924956 0.380073i \(-0.124101\pi\)
0.924956 + 0.380073i \(0.124101\pi\)
\(984\) 0 0
\(985\) −23.0000 −0.732841
\(986\) 30.0000 0.955395
\(987\) 0 0
\(988\) 3.00000 0.0954427
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) −30.0000 −0.951542
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) −39.0000 −1.23514 −0.617571 0.786515i \(-0.711883\pi\)
−0.617571 + 0.786515i \(0.711883\pi\)
\(998\) 5.00000 0.158272
\(999\) 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 3330.2.a.f.1.1 1
3.2 odd 2 1110.2.a.m.1.1 1
12.11 even 2 8880.2.a.i.1.1 1
15.14 odd 2 5550.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.m.1.1 1 3.2 odd 2
3330.2.a.f.1.1 1 1.1 even 1 trivial
5550.2.a.k.1.1 1 15.14 odd 2
8880.2.a.i.1.1 1 12.11 even 2