Properties

Label 1232.2.a.l.1.1
Level $1232$
Weight $2$
Character 1232.1
Self dual yes
Analytic conductor $9.838$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1232.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} +1.00000 q^{11} -4.00000 q^{13} -3.00000 q^{15} +2.00000 q^{17} +6.00000 q^{19} +3.00000 q^{21} +5.00000 q^{23} -4.00000 q^{25} +9.00000 q^{27} +10.0000 q^{29} -1.00000 q^{31} +3.00000 q^{33} -1.00000 q^{35} -5.00000 q^{37} -12.0000 q^{39} -2.00000 q^{41} +8.00000 q^{43} -6.00000 q^{45} -8.00000 q^{47} +1.00000 q^{49} +6.00000 q^{51} -6.00000 q^{53} -1.00000 q^{55} +18.0000 q^{57} -3.00000 q^{59} -2.00000 q^{61} +6.00000 q^{63} +4.00000 q^{65} +3.00000 q^{67} +15.0000 q^{69} -1.00000 q^{71} +10.0000 q^{73} -12.0000 q^{75} +1.00000 q^{77} -6.00000 q^{79} +9.00000 q^{81} -12.0000 q^{83} -2.00000 q^{85} +30.0000 q^{87} -15.0000 q^{89} -4.00000 q^{91} -3.00000 q^{93} -6.00000 q^{95} -5.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 5.00000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −12.0000 −1.92154
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) −6.00000 −0.894427
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 18.0000 2.38416
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 3.00000 0.366508 0.183254 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) 0 0
\(69\) 15.0000 1.80579
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) −12.0000 −1.38564
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 30.0000 3.21634
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) −3.00000 −0.311086
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −5.00000 −0.507673 −0.253837 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) −15.0000 −1.42374
\(112\) 0 0
\(113\) −19.0000 −1.78737 −0.893685 0.448695i \(-0.851889\pi\)
−0.893685 + 0.448695i \(0.851889\pi\)
\(114\) 0 0
\(115\) −5.00000 −0.466252
\(116\) 0 0
\(117\) −24.0000 −2.21880
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 24.0000 2.11308
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) 0 0
\(135\) −9.00000 −0.774597
\(136\) 0 0
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −10.0000 −0.830455
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 0 0
\(159\) −18.0000 −1.42749
\(160\) 0 0
\(161\) 5.00000 0.394055
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) −3.00000 −0.233550
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 36.0000 2.75299
\(172\) 0 0
\(173\) 16.0000 1.21646 0.608229 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −9.00000 −0.676481
\(178\) 0 0
\(179\) −1.00000 −0.0747435 −0.0373718 0.999301i \(-0.511899\pi\)
−0.0373718 + 0.999301i \(0.511899\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) 5.00000 0.367607
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 0 0
\(189\) 9.00000 0.654654
\(190\) 0 0
\(191\) −5.00000 −0.361787 −0.180894 0.983503i \(-0.557899\pi\)
−0.180894 + 0.983503i \(0.557899\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) 12.0000 0.859338
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 9.00000 0.634811
\(202\) 0 0
\(203\) 10.0000 0.701862
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) 30.0000 2.08514
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 0 0
\(213\) −3.00000 −0.205557
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −1.00000 −0.0678844
\(218\) 0 0
\(219\) 30.0000 2.02721
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) −1.00000 −0.0669650 −0.0334825 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) −24.0000 −1.60000
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) −7.00000 −0.462573 −0.231287 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) −18.0000 −1.16923
\(238\) 0 0
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −24.0000 −1.52708
\(248\) 0 0
\(249\) −36.0000 −2.28141
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) 5.00000 0.314347
\(254\) 0 0
\(255\) −6.00000 −0.375735
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) −5.00000 −0.310685
\(260\) 0 0
\(261\) 60.0000 3.71391
\(262\) 0 0
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −45.0000 −2.75396
\(268\) 0 0
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) −12.0000 −0.726273
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 24.0000 1.44202 0.721010 0.692925i \(-0.243678\pi\)
0.721010 + 0.692925i \(0.243678\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) −18.0000 −1.06623
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −15.0000 −0.879316
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 3.00000 0.174667
\(296\) 0 0
\(297\) 9.00000 0.522233
\(298\) 0 0
\(299\) −20.0000 −1.15663
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) −36.0000 −2.06815
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) 36.0000 2.04797
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −23.0000 −1.30004 −0.650018 0.759918i \(-0.725239\pi\)
−0.650018 + 0.759918i \(0.725239\pi\)
\(314\) 0 0
\(315\) −6.00000 −0.338062
\(316\) 0 0
\(317\) 9.00000 0.505490 0.252745 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) 30.0000 1.67444
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 16.0000 0.887520
\(326\) 0 0
\(327\) 12.0000 0.663602
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 0 0
\(333\) −30.0000 −1.64399
\(334\) 0 0
\(335\) −3.00000 −0.163908
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) −57.0000 −3.09582
\(340\) 0 0
\(341\) −1.00000 −0.0541530
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −15.0000 −0.807573
\(346\) 0 0
\(347\) −14.0000 −0.751559 −0.375780 0.926709i \(-0.622625\pi\)
−0.375780 + 0.926709i \(0.622625\pi\)
\(348\) 0 0
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) −36.0000 −1.92154
\(352\) 0 0
\(353\) 9.00000 0.479022 0.239511 0.970894i \(-0.423013\pi\)
0.239511 + 0.970894i \(0.423013\pi\)
\(354\) 0 0
\(355\) 1.00000 0.0530745
\(356\) 0 0
\(357\) 6.00000 0.317554
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 3.00000 0.157459
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 11.0000 0.574195 0.287098 0.957901i \(-0.407310\pi\)
0.287098 + 0.957901i \(0.407310\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 27.0000 1.39427
\(376\) 0 0
\(377\) −40.0000 −2.06010
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) −17.0000 −0.868659 −0.434330 0.900754i \(-0.643015\pi\)
−0.434330 + 0.900754i \(0.643015\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) 48.0000 2.43998
\(388\) 0 0
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) 10.0000 0.505722
\(392\) 0 0
\(393\) −54.0000 −2.72394
\(394\) 0 0
\(395\) 6.00000 0.301893
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) 18.0000 0.901127
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) 0 0
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) −5.00000 −0.247841
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 0 0
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 30.0000 1.46911
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) −48.0000 −2.33384
\(424\) 0 0
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 0 0
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 20.0000 0.963366 0.481683 0.876346i \(-0.340026\pi\)
0.481683 + 0.876346i \(0.340026\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 0 0
\(435\) −30.0000 −1.43839
\(436\) 0 0
\(437\) 30.0000 1.43509
\(438\) 0 0
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) 39.0000 1.85295 0.926473 0.376361i \(-0.122825\pi\)
0.926473 + 0.376361i \(0.122825\pi\)
\(444\) 0 0
\(445\) 15.0000 0.711068
\(446\) 0 0
\(447\) −66.0000 −3.12169
\(448\) 0 0
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 0 0
\(453\) −18.0000 −0.845714
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) 18.0000 0.840168
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −13.0000 −0.604161 −0.302081 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(464\) 0 0
\(465\) 3.00000 0.139122
\(466\) 0 0
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 0 0
\(469\) 3.00000 0.138527
\(470\) 0 0
\(471\) 21.0000 0.967629
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) −24.0000 −1.10120
\(476\) 0 0
\(477\) −36.0000 −1.64833
\(478\) 0 0
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) 15.0000 0.682524
\(484\) 0 0
\(485\) 5.00000 0.227038
\(486\) 0 0
\(487\) 13.0000 0.589086 0.294543 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 0 0
\(497\) −1.00000 −0.0448561
\(498\) 0 0
\(499\) −44.0000 −1.96971 −0.984855 0.173379i \(-0.944532\pi\)
−0.984855 + 0.173379i \(0.944532\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) −31.0000 −1.37405 −0.687025 0.726633i \(-0.741084\pi\)
−0.687025 + 0.726633i \(0.741084\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 0 0
\(513\) 54.0000 2.38416
\(514\) 0 0
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) 0 0
\(519\) 48.0000 2.10697
\(520\) 0 0
\(521\) 7.00000 0.306676 0.153338 0.988174i \(-0.450998\pi\)
0.153338 + 0.988174i \(0.450998\pi\)
\(522\) 0 0
\(523\) −32.0000 −1.39926 −0.699631 0.714504i \(-0.746652\pi\)
−0.699631 + 0.714504i \(0.746652\pi\)
\(524\) 0 0
\(525\) −12.0000 −0.523723
\(526\) 0 0
\(527\) −2.00000 −0.0871214
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) −18.0000 −0.781133
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −10.0000 −0.432338
\(536\) 0 0
\(537\) −3.00000 −0.129460
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) 0 0
\(543\) 15.0000 0.643712
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) 24.0000 1.02617 0.513083 0.858339i \(-0.328503\pi\)
0.513083 + 0.858339i \(0.328503\pi\)
\(548\) 0 0
\(549\) −12.0000 −0.512148
\(550\) 0 0
\(551\) 60.0000 2.55609
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) 0 0
\(555\) 15.0000 0.636715
\(556\) 0 0
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 0 0
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) 19.0000 0.799336
\(566\) 0 0
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) −15.0000 −0.626634
\(574\) 0 0
\(575\) −20.0000 −0.834058
\(576\) 0 0
\(577\) −25.0000 −1.04076 −0.520382 0.853934i \(-0.674210\pi\)
−0.520382 + 0.853934i \(0.674210\pi\)
\(578\) 0 0
\(579\) 42.0000 1.74546
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) 0 0
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) 54.0000 2.22126
\(592\) 0 0
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) 0 0
\(597\) 24.0000 0.982255
\(598\) 0 0
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 0 0
\(603\) 18.0000 0.733017
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 10.0000 0.405887 0.202944 0.979190i \(-0.434949\pi\)
0.202944 + 0.979190i \(0.434949\pi\)
\(608\) 0 0
\(609\) 30.0000 1.21566
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −17.0000 −0.683288 −0.341644 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) 0 0
\(621\) 45.0000 1.80579
\(622\) 0 0
\(623\) −15.0000 −0.600962
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 18.0000 0.718851
\(628\) 0 0
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) −27.0000 −1.07485 −0.537427 0.843311i \(-0.680603\pi\)
−0.537427 + 0.843311i \(0.680603\pi\)
\(632\) 0 0
\(633\) 6.00000 0.238479
\(634\) 0 0
\(635\) 2.00000 0.0793676
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 0 0
\(643\) 29.0000 1.14365 0.571824 0.820376i \(-0.306236\pi\)
0.571824 + 0.820376i \(0.306236\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) 21.0000 0.825595 0.412798 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) 0 0
\(649\) −3.00000 −0.117760
\(650\) 0 0
\(651\) −3.00000 −0.117579
\(652\) 0 0
\(653\) −17.0000 −0.665261 −0.332631 0.943057i \(-0.607936\pi\)
−0.332631 + 0.943057i \(0.607936\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) 0 0
\(657\) 60.0000 2.34082
\(658\) 0 0
\(659\) 2.00000 0.0779089 0.0389545 0.999241i \(-0.487597\pi\)
0.0389545 + 0.999241i \(0.487597\pi\)
\(660\) 0 0
\(661\) 35.0000 1.36134 0.680671 0.732589i \(-0.261688\pi\)
0.680671 + 0.732589i \(0.261688\pi\)
\(662\) 0 0
\(663\) −24.0000 −0.932083
\(664\) 0 0
\(665\) −6.00000 −0.232670
\(666\) 0 0
\(667\) 50.0000 1.93601
\(668\) 0 0
\(669\) −3.00000 −0.115987
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) 0 0
\(675\) −36.0000 −1.38564
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) −5.00000 −0.191882
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 3.00000 0.114624
\(686\) 0 0
\(687\) −21.0000 −0.801200
\(688\) 0 0
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −15.0000 −0.570627 −0.285313 0.958434i \(-0.592098\pi\)
−0.285313 + 0.958434i \(0.592098\pi\)
\(692\) 0 0
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) −10.0000 −0.379322
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −30.0000 −1.13147
\(704\) 0 0
\(705\) 24.0000 0.903892
\(706\) 0 0
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 39.0000 1.46468 0.732338 0.680941i \(-0.238429\pi\)
0.732338 + 0.680941i \(0.238429\pi\)
\(710\) 0 0
\(711\) −36.0000 −1.35011
\(712\) 0 0
\(713\) −5.00000 −0.187251
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) 11.0000 0.410231 0.205115 0.978738i \(-0.434243\pi\)
0.205115 + 0.978738i \(0.434243\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) −36.0000 −1.33885
\(724\) 0 0
\(725\) −40.0000 −1.48556
\(726\) 0 0
\(727\) 19.0000 0.704671 0.352335 0.935874i \(-0.385388\pi\)
0.352335 + 0.935874i \(0.385388\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 0 0
\(735\) −3.00000 −0.110657
\(736\) 0 0
\(737\) 3.00000 0.110506
\(738\) 0 0
\(739\) 18.0000 0.662141 0.331070 0.943606i \(-0.392590\pi\)
0.331070 + 0.943606i \(0.392590\pi\)
\(740\) 0 0
\(741\) −72.0000 −2.64499
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 22.0000 0.806018
\(746\) 0 0
\(747\) −72.0000 −2.63434
\(748\) 0 0
\(749\) 10.0000 0.365392
\(750\) 0 0
\(751\) 23.0000 0.839282 0.419641 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) 0 0
\(753\) 63.0000 2.29585
\(754\) 0 0
\(755\) 6.00000 0.218362
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) 15.0000 0.544466
\(760\) 0 0
\(761\) −48.0000 −1.74000 −0.869999 0.493053i \(-0.835881\pi\)
−0.869999 + 0.493053i \(0.835881\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 0 0
\(765\) −12.0000 −0.433861
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 0 0
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) −15.0000 −0.538122
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −1.00000 −0.0357828
\(782\) 0 0
\(783\) 90.0000 3.21634
\(784\) 0 0
\(785\) −7.00000 −0.249841
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 0 0
\(789\) −54.0000 −1.92245
\(790\) 0 0
\(791\) −19.0000 −0.675562
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 0 0
\(795\) 18.0000 0.638394
\(796\) 0 0
\(797\) 23.0000 0.814702 0.407351 0.913272i \(-0.366453\pi\)
0.407351 + 0.913272i \(0.366453\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −90.0000 −3.17999
\(802\) 0 0
\(803\) 10.0000 0.352892
\(804\) 0 0
\(805\) −5.00000 −0.176227
\(806\) 0 0
\(807\) −54.0000 −1.90089
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 22.0000 0.772524 0.386262 0.922389i \(-0.373766\pi\)
0.386262 + 0.922389i \(0.373766\pi\)
\(812\) 0 0
\(813\) −48.0000 −1.68343
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 48.0000 1.67931
\(818\) 0 0
\(819\) −24.0000 −0.838628
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 25.0000 0.871445 0.435723 0.900081i \(-0.356493\pi\)
0.435723 + 0.900081i \(0.356493\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) −29.0000 −1.00721 −0.503606 0.863934i \(-0.667994\pi\)
−0.503606 + 0.863934i \(0.667994\pi\)
\(830\) 0 0
\(831\) 72.0000 2.49765
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −2.00000 −0.0692129
\(836\) 0 0
\(837\) −9.00000 −0.311086
\(838\) 0 0
\(839\) −45.0000 −1.55357 −0.776786 0.629764i \(-0.783151\pi\)
−0.776786 + 0.629764i \(0.783151\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) −12.0000 −0.413302
\(844\) 0 0
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −25.0000 −0.856989
\(852\) 0 0
\(853\) −34.0000 −1.16414 −0.582069 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) 0 0
\(855\) −36.0000 −1.23117
\(856\) 0 0
\(857\) −28.0000 −0.956462 −0.478231 0.878234i \(-0.658722\pi\)
−0.478231 + 0.878234i \(0.658722\pi\)
\(858\) 0 0
\(859\) −55.0000 −1.87658 −0.938288 0.345855i \(-0.887589\pi\)
−0.938288 + 0.345855i \(0.887589\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) −52.0000 −1.77010 −0.885050 0.465495i \(-0.845876\pi\)
−0.885050 + 0.465495i \(0.845876\pi\)
\(864\) 0 0
\(865\) −16.0000 −0.544016
\(866\) 0 0
\(867\) −39.0000 −1.32451
\(868\) 0 0
\(869\) −6.00000 −0.203536
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) −30.0000 −1.01535
\(874\) 0 0
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 0 0
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) 27.0000 0.909653 0.454827 0.890580i \(-0.349701\pi\)
0.454827 + 0.890580i \(0.349701\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 9.00000 0.302532
\(886\) 0 0
\(887\) 2.00000 0.0671534 0.0335767 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) 9.00000 0.301511
\(892\) 0 0
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) 1.00000 0.0334263
\(896\) 0 0
\(897\) −60.0000 −2.00334
\(898\) 0 0
\(899\) −10.0000 −0.333519
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 24.0000 0.798670
\(904\) 0 0
\(905\) −5.00000 −0.166206
\(906\) 0 0
\(907\) 40.0000 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) 0 0
\(909\) −72.0000 −2.38809
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) 6.00000 0.198354
\(916\) 0 0
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 0 0
\(921\) 84.0000 2.76789
\(922\) 0 0
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) 20.0000 0.657596
\(926\) 0 0
\(927\) 72.0000 2.36479
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) −24.0000 −0.785725
\(934\) 0 0
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) 36.0000 1.17607 0.588034 0.808836i \(-0.299902\pi\)
0.588034 + 0.808836i \(0.299902\pi\)
\(938\) 0 0
\(939\) −69.0000 −2.25173
\(940\) 0 0
\(941\) 58.0000 1.89075 0.945373 0.325991i \(-0.105698\pi\)
0.945373 + 0.325991i \(0.105698\pi\)
\(942\) 0 0
\(943\) −10.0000 −0.325645
\(944\) 0 0
\(945\) −9.00000 −0.292770
\(946\) 0 0
\(947\) −5.00000 −0.162478 −0.0812391 0.996695i \(-0.525888\pi\)
−0.0812391 + 0.996695i \(0.525888\pi\)
\(948\) 0 0
\(949\) −40.0000 −1.29845
\(950\) 0 0
\(951\) 27.0000 0.875535
\(952\) 0 0
\(953\) 44.0000 1.42530 0.712650 0.701520i \(-0.247495\pi\)
0.712650 + 0.701520i \(0.247495\pi\)
\(954\) 0 0
\(955\) 5.00000 0.161796
\(956\) 0 0
\(957\) 30.0000 0.969762
\(958\) 0 0
\(959\) −3.00000 −0.0968751
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 60.0000 1.93347
\(964\) 0 0
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 34.0000 1.09337 0.546683 0.837340i \(-0.315890\pi\)
0.546683 + 0.837340i \(0.315890\pi\)
\(968\) 0 0
\(969\) 36.0000 1.15649
\(970\) 0 0
\(971\) −29.0000 −0.930654 −0.465327 0.885139i \(-0.654063\pi\)
−0.465327 + 0.885139i \(0.654063\pi\)
\(972\) 0 0
\(973\) 10.0000 0.320585
\(974\) 0 0
\(975\) 48.0000 1.53723
\(976\) 0 0
\(977\) −31.0000 −0.991778 −0.495889 0.868386i \(-0.665158\pi\)
−0.495889 + 0.868386i \(0.665158\pi\)
\(978\) 0 0
\(979\) −15.0000 −0.479402
\(980\) 0 0
\(981\) 24.0000 0.766261
\(982\) 0 0
\(983\) 27.0000 0.861166 0.430583 0.902551i \(-0.358308\pi\)
0.430583 + 0.902551i \(0.358308\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) 40.0000 1.27193
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) 51.0000 1.61844
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) −12.0000 −0.380044 −0.190022 0.981780i \(-0.560856\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) 0 0
\(999\) −45.0000 −1.42374
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 1232.2.a.l.1.1 1
4.3 odd 2 77.2.a.a.1.1 1
7.6 odd 2 8624.2.a.a.1.1 1
8.3 odd 2 4928.2.a.bj.1.1 1
8.5 even 2 4928.2.a.a.1.1 1
12.11 even 2 693.2.a.c.1.1 1
20.3 even 4 1925.2.b.e.1849.1 2
20.7 even 4 1925.2.b.e.1849.2 2
20.19 odd 2 1925.2.a.h.1.1 1
28.3 even 6 539.2.e.c.177.1 2
28.11 odd 6 539.2.e.f.177.1 2
28.19 even 6 539.2.e.c.67.1 2
28.23 odd 6 539.2.e.f.67.1 2
28.27 even 2 539.2.a.c.1.1 1
44.3 odd 10 847.2.f.i.372.1 4
44.7 even 10 847.2.f.h.148.1 4
44.15 odd 10 847.2.f.i.148.1 4
44.19 even 10 847.2.f.h.372.1 4
44.27 odd 10 847.2.f.i.729.1 4
44.31 odd 10 847.2.f.i.323.1 4
44.35 even 10 847.2.f.h.323.1 4
44.39 even 10 847.2.f.h.729.1 4
44.43 even 2 847.2.a.b.1.1 1
84.83 odd 2 4851.2.a.j.1.1 1
132.131 odd 2 7623.2.a.j.1.1 1
308.307 odd 2 5929.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.a.1.1 1 4.3 odd 2
539.2.a.c.1.1 1 28.27 even 2
539.2.e.c.67.1 2 28.19 even 6
539.2.e.c.177.1 2 28.3 even 6
539.2.e.f.67.1 2 28.23 odd 6
539.2.e.f.177.1 2 28.11 odd 6
693.2.a.c.1.1 1 12.11 even 2
847.2.a.b.1.1 1 44.43 even 2
847.2.f.h.148.1 4 44.7 even 10
847.2.f.h.323.1 4 44.35 even 10
847.2.f.h.372.1 4 44.19 even 10
847.2.f.h.729.1 4 44.39 even 10
847.2.f.i.148.1 4 44.15 odd 10
847.2.f.i.323.1 4 44.31 odd 10
847.2.f.i.372.1 4 44.3 odd 10
847.2.f.i.729.1 4 44.27 odd 10
1232.2.a.l.1.1 1 1.1 even 1 trivial
1925.2.a.h.1.1 1 20.19 odd 2
1925.2.b.e.1849.1 2 20.3 even 4
1925.2.b.e.1849.2 2 20.7 even 4
4851.2.a.j.1.1 1 84.83 odd 2
4928.2.a.a.1.1 1 8.5 even 2
4928.2.a.bj.1.1 1 8.3 odd 2
5929.2.a.f.1.1 1 308.307 odd 2
7623.2.a.j.1.1 1 132.131 odd 2
8624.2.a.a.1.1 1 7.6 odd 2