Properties

Label 8880.2.a.i.1.1
Level $8880$
Weight $2$
Character 8880.1
Self dual yes
Analytic conductor $70.907$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8880.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.9071569949\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 8880.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +5.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +5.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} -1.00000 q^{13} +1.00000 q^{15} -5.00000 q^{17} +3.00000 q^{19} -5.00000 q^{21} -3.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.00000 q^{29} +6.00000 q^{31} -5.00000 q^{33} -5.00000 q^{35} -1.00000 q^{37} +1.00000 q^{39} -4.00000 q^{43} -1.00000 q^{45} +18.0000 q^{49} +5.00000 q^{51} -3.00000 q^{53} -5.00000 q^{55} -3.00000 q^{57} +10.0000 q^{59} +10.0000 q^{61} +5.00000 q^{63} +1.00000 q^{65} +14.0000 q^{67} +3.00000 q^{69} -6.00000 q^{71} -9.00000 q^{73} -1.00000 q^{75} +25.0000 q^{77} +1.00000 q^{81} -5.00000 q^{83} +5.00000 q^{85} -6.00000 q^{87} +13.0000 q^{89} -5.00000 q^{91} -6.00000 q^{93} -3.00000 q^{95} -4.00000 q^{97} +5.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 5.00000 1.88982 0.944911 0.327327i \(-0.106148\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 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 \(0.544284\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) −5.00000 −1.09109
\(22\) 0 0
\(23\) −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
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) −5.00000 −0.845154
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) 5.00000 0.700140
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) 0 0
\(59\) 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 \(0.278858\pi\)
\(62\) 0 0
\(63\) 5.00000 0.629941
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −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 \(0.676565\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 25.0000 2.84901
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 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
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 13.0000 1.37800 0.688999 0.724763i \(-0.258051\pi\)
0.688999 + 0.724763i \(0.258051\pi\)
\(90\) 0 0
\(91\) −5.00000 −0.524142
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) 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 \(0.565092\pi\)
\(98\) 0 0
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 5.00000 0.487950
\(106\) 0 0
\(107\) −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 \(0.576969\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −25.0000 −2.29175
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 3.00000 0.266207 0.133103 0.991102i \(-0.457506\pi\)
0.133103 + 0.991102i \(0.457506\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 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
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.00000 −0.418121
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) −18.0000 −1.48461
\(148\) 0 0
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) −15.0000 −1.22068 −0.610341 0.792139i \(-0.708968\pi\)
−0.610341 + 0.792139i \(0.708968\pi\)
\(152\) 0 0
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) −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
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −15.0000 −1.18217
\(162\) 0 0
\(163\) 23.0000 1.80150 0.900750 0.434339i \(-0.143018\pi\)
0.900750 + 0.434339i \(0.143018\pi\)
\(164\) 0 0
\(165\) 5.00000 0.389249
\(166\) 0 0
\(167\) −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
\(170\) 0 0
\(171\) 3.00000 0.229416
\(172\) 0 0
\(173\) 11.0000 0.836315 0.418157 0.908375i \(-0.362676\pi\)
0.418157 + 0.908375i \(0.362676\pi\)
\(174\) 0 0
\(175\) 5.00000 0.377964
\(176\) 0 0
\(177\) −10.0000 −0.751646
\(178\) 0 0
\(179\) −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 \(0.621207\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −25.0000 −1.82818
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 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 \(0.522932\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) 23.0000 1.63868 0.819341 0.573306i \(-0.194340\pi\)
0.819341 + 0.573306i \(0.194340\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) −14.0000 −0.987484
\(202\) 0 0
\(203\) 30.0000 2.10559
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.00000 −0.208514
\(208\) 0 0
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 0 0
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 30.0000 2.03653
\(218\) 0 0
\(219\) 9.00000 0.608164
\(220\) 0 0
\(221\) 5.00000 0.336336
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −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 \(0.436476\pi\)
\(230\) 0 0
\(231\) −25.0000 −1.64488
\(232\) 0 0
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −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 \(0.696846\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −18.0000 −1.14998
\(246\) 0 0
\(247\) −3.00000 −0.190885
\(248\) 0 0
\(249\) 5.00000 0.316862
\(250\) 0 0
\(251\) −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
\(254\) 0 0
\(255\) −5.00000 −0.313112
\(256\) 0 0
\(257\) 15.0000 0.935674 0.467837 0.883815i \(-0.345033\pi\)
0.467837 + 0.883815i \(0.345033\pi\)
\(258\) 0 0
\(259\) −5.00000 −0.310685
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) −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
\(266\) 0 0
\(267\) −13.0000 −0.795587
\(268\) 0 0
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) 5.00000 0.302614
\(274\) 0 0
\(275\) 5.00000 0.301511
\(276\) 0 0
\(277\) 25.0000 1.50210 0.751052 0.660243i \(-0.229547\pi\)
0.751052 + 0.660243i \(0.229547\pi\)
\(278\) 0 0
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 1.00000 0.0596550 0.0298275 0.999555i \(-0.490504\pi\)
0.0298275 + 0.999555i \(0.490504\pi\)
\(282\) 0 0
\(283\) −5.00000 −0.297219 −0.148610 0.988896i \(-0.547480\pi\)
−0.148610 + 0.988896i \(0.547480\pi\)
\(284\) 0 0
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) 0 0
\(293\) −11.0000 −0.642627 −0.321313 0.946973i \(-0.604124\pi\)
−0.321313 + 0.946973i \(0.604124\pi\)
\(294\) 0 0
\(295\) −10.0000 −0.582223
\(296\) 0 0
\(297\) −5.00000 −0.290129
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) 30.0000 1.71219 0.856095 0.516818i \(-0.172884\pi\)
0.856095 + 0.516818i \(0.172884\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −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 \(0.262726\pi\)
\(314\) 0 0
\(315\) −5.00000 −0.281718
\(316\) 0 0
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) 30.0000 1.67968
\(320\) 0 0
\(321\) 15.0000 0.837218
\(322\) 0 0
\(323\) −15.0000 −0.834622
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) 5.00000 0.276501
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) −1.00000 −0.0547997
\(334\) 0 0
\(335\) −14.0000 −0.764902
\(336\) 0 0
\(337\) −9.00000 −0.490261 −0.245131 0.969490i \(-0.578831\pi\)
−0.245131 + 0.969490i \(0.578831\pi\)
\(338\) 0 0
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) 30.0000 1.62459
\(342\) 0 0
\(343\) 55.0000 2.96972
\(344\) 0 0
\(345\) −3.00000 −0.161515
\(346\) 0 0
\(347\) 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 \(0.277962\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 0 0
\(357\) 25.0000 1.32314
\(358\) 0 0
\(359\) −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
\(362\) 0 0
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) 9.00000 0.471082
\(366\) 0 0
\(367\) 5.00000 0.260998 0.130499 0.991448i \(-0.458342\pi\)
0.130499 + 0.991448i \(0.458342\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −15.0000 −0.778761
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) −3.00000 −0.153695
\(382\) 0 0
\(383\) −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
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) 0 0
\(391\) 15.0000 0.758583
\(392\) 0 0
\(393\) −20.0000 −1.00887
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −36.0000 −1.80679 −0.903394 0.428811i \(-0.858933\pi\)
−0.903394 + 0.428811i \(0.858933\pi\)
\(398\) 0 0
\(399\) −15.0000 −0.750939
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) −6.00000 −0.298881
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −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
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) 50.0000 2.46034
\(414\) 0 0
\(415\) 5.00000 0.245440
\(416\) 0 0
\(417\) 2.00000 0.0979404
\(418\) 0 0
\(419\) 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 \(0.546708\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.00000 −0.242536
\(426\) 0 0
\(427\) 50.0000 2.41967
\(428\) 0 0
\(429\) 5.00000 0.241402
\(430\) 0 0
\(431\) 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 \(0.331641\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 0 0
\(437\) −9.00000 −0.430528
\(438\) 0 0
\(439\) −36.0000 −1.71819 −0.859093 0.511819i \(-0.828972\pi\)
−0.859093 + 0.511819i \(0.828972\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) 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
\(446\) 0 0
\(447\) −2.00000 −0.0945968
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 15.0000 0.704761
\(454\) 0 0
\(455\) 5.00000 0.234404
\(456\) 0 0
\(457\) 4.00000 0.187112 0.0935561 0.995614i \(-0.470177\pi\)
0.0935561 + 0.995614i \(0.470177\pi\)
\(458\) 0 0
\(459\) 5.00000 0.233380
\(460\) 0 0
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) 28.0000 1.30127 0.650635 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) 0 0
\(465\) 6.00000 0.278243
\(466\) 0 0
\(467\) 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
\(471\) 20.0000 0.921551
\(472\) 0 0
\(473\) −20.0000 −0.919601
\(474\) 0 0
\(475\) 3.00000 0.137649
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) 0 0
\(479\) 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
\(482\) 0 0
\(483\) 15.0000 0.682524
\(484\) 0 0
\(485\) 4.00000 0.181631
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 0 0
\(489\) −23.0000 −1.04010
\(490\) 0 0
\(491\) 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
\(494\) 0 0
\(495\) −5.00000 −0.224733
\(496\) 0 0
\(497\) −30.0000 −1.34568
\(498\) 0 0
\(499\) 5.00000 0.223831 0.111915 0.993718i \(-0.464301\pi\)
0.111915 + 0.993718i \(0.464301\pi\)
\(500\) 0 0
\(501\) 5.00000 0.223384
\(502\) 0 0
\(503\) 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
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) −45.0000 −1.99068
\(512\) 0 0
\(513\) −3.00000 −0.132453
\(514\) 0 0
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −11.0000 −0.482846
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 0 0
\(525\) −5.00000 −0.218218
\(526\) 0 0
\(527\) −30.0000 −1.30682
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 15.0000 0.648507
\(536\) 0 0
\(537\) 8.00000 0.345225
\(538\) 0 0
\(539\) 90.0000 3.87657
\(540\) 0 0
\(541\) 5.00000 0.214967 0.107483 0.994207i \(-0.465721\pi\)
0.107483 + 0.994207i \(0.465721\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) 5.00000 0.214176
\(546\) 0 0
\(547\) 43.0000 1.83855 0.919274 0.393619i \(-0.128777\pi\)
0.919274 + 0.393619i \(0.128777\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 18.0000 0.766826
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.00000 −0.0424476
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 25.0000 1.05550
\(562\) 0 0
\(563\) −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
\(566\) 0 0
\(567\) 5.00000 0.209980
\(568\) 0 0
\(569\) −13.0000 −0.544988 −0.272494 0.962157i \(-0.587849\pi\)
−0.272494 + 0.962157i \(0.587849\pi\)
\(570\) 0 0
\(571\) 10.0000 0.418487 0.209243 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) 0 0
\(573\) −7.00000 −0.292429
\(574\) 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 \(0.317971\pi\)
\(578\) 0 0
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) −25.0000 −1.03717
\(582\) 0 0
\(583\) −15.0000 −0.621237
\(584\) 0 0
\(585\) 1.00000 0.0413449
\(586\) 0 0
\(587\) 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
\(590\) 0 0
\(591\) −23.0000 −0.946094
\(592\) 0 0
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 0 0
\(595\) 25.0000 1.02490
\(596\) 0 0
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) −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 \(0.585418\pi\)
\(602\) 0 0
\(603\) 14.0000 0.570124
\(604\) 0 0
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) 18.0000 0.730597 0.365299 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) 0 0
\(609\) −30.0000 −1.21566
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −34.0000 −1.36879 −0.684394 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) 0 0
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) 65.0000 2.60417
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −15.0000 −0.599042
\(628\) 0 0
\(629\) 5.00000 0.199363
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) −2.00000 −0.0794929
\(634\) 0 0
\(635\) −3.00000 −0.119051
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −23.0000 −0.907031 −0.453516 0.891248i \(-0.649830\pi\)
−0.453516 + 0.891248i \(0.649830\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) −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
\(650\) 0 0
\(651\) −30.0000 −1.17579
\(652\) 0 0
\(653\) −48.0000 −1.87839 −0.939193 0.343391i \(-0.888424\pi\)
−0.939193 + 0.343391i \(0.888424\pi\)
\(654\) 0 0
\(655\) −20.0000 −0.781465
\(656\) 0 0
\(657\) −9.00000 −0.351123
\(658\) 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 \(0.867057\pi\)
\(662\) 0 0
\(663\) −5.00000 −0.194184
\(664\) 0 0
\(665\) −15.0000 −0.581675
\(666\) 0 0
\(667\) −18.0000 −0.696963
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 50.0000 1.93023
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −9.00000 −0.345898 −0.172949 0.984931i \(-0.555330\pi\)
−0.172949 + 0.984931i \(0.555330\pi\)
\(678\) 0 0
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) −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
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) 0 0
\(689\) 3.00000 0.114291
\(690\) 0 0
\(691\) 22.0000 0.836919 0.418460 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) 0 0
\(693\) 25.0000 0.949671
\(694\) 0 0
\(695\) 2.00000 0.0758643
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −3.00000 −0.113147
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −30.0000 −1.12827
\(708\) 0 0
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −18.0000 −0.674105
\(714\) 0 0
\(715\) 5.00000 0.186989
\(716\) 0 0
\(717\) 12.0000 0.448148
\(718\) 0 0
\(719\) 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
\(722\) 0 0
\(723\) 18.0000 0.669427
\(724\) 0 0
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 30.0000 1.11264 0.556319 0.830969i \(-0.312213\pi\)
0.556319 + 0.830969i \(0.312213\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 20.0000 0.739727
\(732\) 0 0
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) 0 0
\(735\) 18.0000 0.663940
\(736\) 0 0
\(737\) 70.0000 2.57848
\(738\) 0 0
\(739\) −26.0000 −0.956425 −0.478213 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) 0 0
\(741\) 3.00000 0.110208
\(742\) 0 0
\(743\) −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
\(746\) 0 0
\(747\) −5.00000 −0.182940
\(748\) 0 0
\(749\) −75.0000 −2.74044
\(750\) 0 0
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 0 0
\(753\) 30.0000 1.09326
\(754\) 0 0
\(755\) 15.0000 0.545906
\(756\) 0 0
\(757\) 37.0000 1.34479 0.672394 0.740193i \(-0.265266\pi\)
0.672394 + 0.740193i \(0.265266\pi\)
\(758\) 0 0
\(759\) 15.0000 0.544466
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) −25.0000 −0.905061
\(764\) 0 0
\(765\) 5.00000 0.180775
\(766\) 0 0
\(767\) −10.0000 −0.361079
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −15.0000 −0.540212
\(772\) 0 0
\(773\) −21.0000 −0.755318 −0.377659 0.925945i \(-0.623271\pi\)
−0.377659 + 0.925945i \(0.623271\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) 0 0
\(777\) 5.00000 0.179374
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −30.0000 −1.07348
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) −42.0000 −1.49714 −0.748569 0.663057i \(-0.769259\pi\)
−0.748569 + 0.663057i \(0.769259\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −50.0000 −1.77780
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) −3.00000 −0.106399
\(796\) 0 0
\(797\) −14.0000 −0.495905 −0.247953 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 13.0000 0.459332
\(802\) 0 0
\(803\) −45.0000 −1.58802
\(804\) 0 0
\(805\) 15.0000 0.528681
\(806\) 0 0
\(807\) −9.00000 −0.316815
\(808\) 0 0
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) −23.0000 −0.805655
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) 0 0
\(819\) −5.00000 −0.174714
\(820\) 0 0
\(821\) 39.0000 1.36111 0.680555 0.732697i \(-0.261739\pi\)
0.680555 + 0.732697i \(0.261739\pi\)
\(822\) 0 0
\(823\) 35.0000 1.22002 0.610012 0.792392i \(-0.291165\pi\)
0.610012 + 0.792392i \(0.291165\pi\)
\(824\) 0 0
\(825\) −5.00000 −0.174078
\(826\) 0 0
\(827\) −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 \(0.461210\pi\)
\(830\) 0 0
\(831\) −25.0000 −0.867240
\(832\) 0 0
\(833\) −90.0000 −3.11832
\(834\) 0 0
\(835\) 5.00000 0.173032
\(836\) 0 0
\(837\) −6.00000 −0.207390
\(838\) 0 0
\(839\) 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
\(842\) 0 0
\(843\) −1.00000 −0.0344418
\(844\) 0 0
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) 70.0000 2.40523
\(848\) 0 0
\(849\) 5.00000 0.171600
\(850\) 0 0
\(851\) 3.00000 0.102839
\(852\) 0 0
\(853\) 19.0000 0.650548 0.325274 0.945620i \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) 0 0
\(855\) −3.00000 −0.102598
\(856\) 0 0
\(857\) 9.00000 0.307434 0.153717 0.988115i \(-0.450876\pi\)
0.153717 + 0.988115i \(0.450876\pi\)
\(858\) 0 0
\(859\) −7.00000 −0.238837 −0.119418 0.992844i \(-0.538103\pi\)
−0.119418 + 0.992844i \(0.538103\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 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
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −14.0000 −0.474372
\(872\) 0 0
\(873\) −4.00000 −0.135379
\(874\) 0 0
\(875\) −5.00000 −0.169031
\(876\) 0 0
\(877\) −52.0000 −1.75592 −0.877958 0.478738i \(-0.841094\pi\)
−0.877958 + 0.478738i \(0.841094\pi\)
\(878\) 0 0
\(879\) 11.0000 0.371021
\(880\) 0 0
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 0 0
\(883\) −21.0000 −0.706706 −0.353353 0.935490i \(-0.614959\pi\)
−0.353353 + 0.935490i \(0.614959\pi\)
\(884\) 0 0
\(885\) 10.0000 0.336146
\(886\) 0 0
\(887\) 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
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) 0 0
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) 15.0000 0.499722
\(902\) 0 0
\(903\) 20.0000 0.665558
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −37.0000 −1.22856 −0.614282 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 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
\(914\) 0 0
\(915\) 10.0000 0.330590
\(916\) 0 0
\(917\) 100.000 3.30229
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) −30.0000 −0.988534
\(922\) 0 0
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) 2.00000 0.0656179 0.0328089 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) 0 0
\(931\) 54.0000 1.76978
\(932\) 0 0
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) 25.0000 0.817587
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) −24.0000 −0.783210
\(940\) 0 0
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 5.00000 0.162650
\(946\) 0 0
\(947\) −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
\(950\) 0 0
\(951\) 10.0000 0.324272
\(952\) 0 0
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 0 0
\(955\) −7.00000 −0.226515
\(956\) 0 0
\(957\) −30.0000 −0.969762
\(958\) 0 0
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −15.0000 −0.483368
\(964\) 0 0
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) 18.0000 0.578841 0.289420 0.957202i \(-0.406537\pi\)
0.289420 + 0.957202i \(0.406537\pi\)
\(968\) 0 0
\(969\) 15.0000 0.481869
\(970\) 0 0
\(971\) 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
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) 39.0000 1.24772 0.623860 0.781536i \(-0.285563\pi\)
0.623860 + 0.781536i \(0.285563\pi\)
\(978\) 0 0
\(979\) 65.0000 2.07741
\(980\) 0 0
\(981\) −5.00000 −0.159638
\(982\) 0 0
\(983\) 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
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) −39.0000 −1.23514 −0.617571 0.786515i \(-0.711883\pi\)
−0.617571 + 0.786515i \(0.711883\pi\)
\(998\) 0 0
\(999\) 1.00000 0.0316386
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 8880.2.a.i.1.1 1
4.3 odd 2 1110.2.a.m.1.1 1
12.11 even 2 3330.2.a.f.1.1 1
20.19 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 4.3 odd 2
3330.2.a.f.1.1 1 12.11 even 2
5550.2.a.k.1.1 1 20.19 odd 2
8880.2.a.i.1.1 1 1.1 even 1 trivial