Properties

Label 5312.2.a.h.1.1
Level $5312$
Weight $2$
Character 5312.1
Self dual yes
Analytic conductor $42.417$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5312 = 2^{6} \cdot 83 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5312.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.4165335537\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 83)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5312.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.00000 q^{5} +3.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.00000 q^{5} +3.00000 q^{7} -2.00000 q^{9} +3.00000 q^{11} +6.00000 q^{13} -2.00000 q^{15} +5.00000 q^{17} +2.00000 q^{19} -3.00000 q^{21} +4.00000 q^{23} -1.00000 q^{25} +5.00000 q^{27} +7.00000 q^{29} -5.00000 q^{31} -3.00000 q^{33} +6.00000 q^{35} +11.0000 q^{37} -6.00000 q^{39} -2.00000 q^{41} -8.00000 q^{43} -4.00000 q^{45} +2.00000 q^{49} -5.00000 q^{51} -6.00000 q^{53} +6.00000 q^{55} -2.00000 q^{57} +5.00000 q^{59} -5.00000 q^{61} -6.00000 q^{63} +12.0000 q^{65} -2.00000 q^{67} -4.00000 q^{69} -2.00000 q^{71} +1.00000 q^{75} +9.00000 q^{77} -14.0000 q^{79} +1.00000 q^{81} -1.00000 q^{83} +10.0000 q^{85} -7.00000 q^{87} +18.0000 q^{91} +5.00000 q^{93} +4.00000 q^{95} -8.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 7.00000 1.29987 0.649934 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 6.00000 1.01419
\(36\) 0 0
\(37\) 11.0000 1.80839 0.904194 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 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 \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) −4.00000 −0.596285
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 0 0
\(63\) −6.00000 −0.755929
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 9.00000 1.02565
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.00000 −0.109764
\(84\) 0 0
\(85\) 10.0000 1.08465
\(86\) 0 0
\(87\) −7.00000 −0.750479
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 18.0000 1.88691
\(92\) 0 0
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) −6.00000 −0.585540
\(106\) 0 0
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) 17.0000 1.62830 0.814152 0.580651i \(-0.197202\pi\)
0.814152 + 0.580651i \(0.197202\pi\)
\(110\) 0 0
\(111\) −11.0000 −1.04407
\(112\) 0 0
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 0 0
\(117\) −12.0000 −1.10940
\(118\) 0 0
\(119\) 15.0000 1.37505
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) 0 0
\(135\) 10.0000 0.860663
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 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
\(143\) 18.0000 1.50524
\(144\) 0 0
\(145\) 14.0000 1.16264
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) 16.0000 1.31077 0.655386 0.755295i \(-0.272506\pi\)
0.655386 + 0.755295i \(0.272506\pi\)
\(150\) 0 0
\(151\) −9.00000 −0.732410 −0.366205 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(152\) 0 0
\(153\) −10.0000 −0.808452
\(154\) 0 0
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) −22.0000 −1.72317 −0.861586 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) 0 0
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) −21.0000 −1.62503 −0.812514 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) −21.0000 −1.59660 −0.798300 0.602260i \(-0.794267\pi\)
−0.798300 + 0.602260i \(0.794267\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 0 0
\(177\) −5.00000 −0.375823
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) 5.00000 0.369611
\(184\) 0 0
\(185\) 22.0000 1.61747
\(186\) 0 0
\(187\) 15.0000 1.09691
\(188\) 0 0
\(189\) 15.0000 1.09109
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) −12.0000 −0.859338
\(196\) 0 0
\(197\) −11.0000 −0.783718 −0.391859 0.920025i \(-0.628168\pi\)
−0.391859 + 0.920025i \(0.628168\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 0 0
\(203\) 21.0000 1.47391
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 0 0
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) −15.0000 −1.01827
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 30.0000 2.01802
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 0 0
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −9.00000 −0.592157
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 14.0000 0.909398
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 4.00000 0.255551
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) 0 0
\(249\) 1.00000 0.0633724
\(250\) 0 0
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) −10.0000 −0.626224
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 33.0000 2.05052
\(260\) 0 0
\(261\) −14.0000 −0.866578
\(262\) 0 0
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 0 0
\(273\) −18.0000 −1.08941
\(274\) 0 0
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 26.0000 1.54554 0.772770 0.634686i \(-0.218871\pi\)
0.772770 + 0.634686i \(0.218871\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 10.0000 0.582223
\(296\) 0 0
\(297\) 15.0000 0.870388
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 11.0000 0.621757 0.310878 0.950450i \(-0.399377\pi\)
0.310878 + 0.950450i \(0.399377\pi\)
\(314\) 0 0
\(315\) −12.0000 −0.676123
\(316\) 0 0
\(317\) −3.00000 −0.168497 −0.0842484 0.996445i \(-0.526849\pi\)
−0.0842484 + 0.996445i \(0.526849\pi\)
\(318\) 0 0
\(319\) 21.0000 1.17577
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) 10.0000 0.556415
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) 0 0
\(327\) −17.0000 −0.940102
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −22.0000 −1.20923 −0.604615 0.796518i \(-0.706673\pi\)
−0.604615 + 0.796518i \(0.706673\pi\)
\(332\) 0 0
\(333\) −22.0000 −1.20559
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −15.0000 −0.814688
\(340\) 0 0
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) −8.00000 −0.430706
\(346\) 0 0
\(347\) 14.0000 0.751559 0.375780 0.926709i \(-0.377375\pi\)
0.375780 + 0.926709i \(0.377375\pi\)
\(348\) 0 0
\(349\) −23.0000 −1.23116 −0.615581 0.788074i \(-0.711079\pi\)
−0.615581 + 0.788074i \(0.711079\pi\)
\(350\) 0 0
\(351\) 30.0000 1.60128
\(352\) 0 0
\(353\) −1.00000 −0.0532246 −0.0266123 0.999646i \(-0.508472\pi\)
−0.0266123 + 0.999646i \(0.508472\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 0 0
\(357\) −15.0000 −0.793884
\(358\) 0 0
\(359\) −33.0000 −1.74167 −0.870837 0.491572i \(-0.836422\pi\)
−0.870837 + 0.491572i \(0.836422\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) 0 0
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) −18.0000 −0.934513
\(372\) 0 0
\(373\) 11.0000 0.569558 0.284779 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 42.0000 2.16311
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 13.0000 0.666010
\(382\) 0 0
\(383\) 29.0000 1.48183 0.740915 0.671598i \(-0.234392\pi\)
0.740915 + 0.671598i \(0.234392\pi\)
\(384\) 0 0
\(385\) 18.0000 0.917365
\(386\) 0 0
\(387\) 16.0000 0.813326
\(388\) 0 0
\(389\) −4.00000 −0.202808 −0.101404 0.994845i \(-0.532333\pi\)
−0.101404 + 0.994845i \(0.532333\pi\)
\(390\) 0 0
\(391\) 20.0000 1.01144
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) −28.0000 −1.40883
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −30.0000 −1.49441
\(404\) 0 0
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 33.0000 1.63575
\(408\) 0 0
\(409\) −21.0000 −1.03838 −0.519192 0.854658i \(-0.673767\pi\)
−0.519192 + 0.854658i \(0.673767\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) 0 0
\(413\) 15.0000 0.738102
\(414\) 0 0
\(415\) −2.00000 −0.0981761
\(416\) 0 0
\(417\) 2.00000 0.0979404
\(418\) 0 0
\(419\) 27.0000 1.31904 0.659518 0.751689i \(-0.270760\pi\)
0.659518 + 0.751689i \(0.270760\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.00000 −0.242536
\(426\) 0 0
\(427\) −15.0000 −0.725901
\(428\) 0 0
\(429\) −18.0000 −0.869048
\(430\) 0 0
\(431\) −21.0000 −1.01153 −0.505767 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) −14.0000 −0.671249
\(436\) 0 0
\(437\) 8.00000 0.382692
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) 13.0000 0.617649 0.308824 0.951119i \(-0.400064\pi\)
0.308824 + 0.951119i \(0.400064\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −16.0000 −0.756774
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 0 0
\(453\) 9.00000 0.422857
\(454\) 0 0
\(455\) 36.0000 1.68771
\(456\) 0 0
\(457\) 30.0000 1.40334 0.701670 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) 0 0
\(459\) 25.0000 1.16690
\(460\) 0 0
\(461\) −20.0000 −0.931493 −0.465746 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) 0 0
\(463\) 5.00000 0.232370 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) 0 0
\(465\) 10.0000 0.463739
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −6.00000 −0.277054
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) −24.0000 −1.10352
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) −21.0000 −0.959514 −0.479757 0.877401i \(-0.659275\pi\)
−0.479757 + 0.877401i \(0.659275\pi\)
\(480\) 0 0
\(481\) 66.0000 3.00934
\(482\) 0 0
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) −16.0000 −0.726523
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 22.0000 0.994874
\(490\) 0 0
\(491\) −10.0000 −0.451294 −0.225647 0.974209i \(-0.572450\pi\)
−0.225647 + 0.974209i \(0.572450\pi\)
\(492\) 0 0
\(493\) 35.0000 1.57632
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 0 0
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) −9.00000 −0.402895 −0.201448 0.979499i \(-0.564565\pi\)
−0.201448 + 0.979499i \(0.564565\pi\)
\(500\) 0 0
\(501\) 21.0000 0.938211
\(502\) 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
\(507\) −23.0000 −1.02147
\(508\) 0 0
\(509\) 33.0000 1.46270 0.731350 0.682003i \(-0.238891\pi\)
0.731350 + 0.682003i \(0.238891\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 10.0000 0.441511
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 21.0000 0.921798
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) 29.0000 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 0 0
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) −25.0000 −1.08902
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −36.0000 −1.55642
\(536\) 0 0
\(537\) 6.00000 0.258919
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 0 0
\(543\) −26.0000 −1.11577
\(544\) 0 0
\(545\) 34.0000 1.45640
\(546\) 0 0
\(547\) 13.0000 0.555840 0.277920 0.960604i \(-0.410355\pi\)
0.277920 + 0.960604i \(0.410355\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 14.0000 0.596420
\(552\) 0 0
\(553\) −42.0000 −1.78602
\(554\) 0 0
\(555\) −22.0000 −0.933848
\(556\) 0 0
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 0 0
\(559\) −48.0000 −2.03018
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) 21.0000 0.885044 0.442522 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) 0 0
\(565\) 30.0000 1.26211
\(566\) 0 0
\(567\) 3.00000 0.125988
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) −3.00000 −0.125327
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) −3.00000 −0.124461
\(582\) 0 0
\(583\) −18.0000 −0.745484
\(584\) 0 0
\(585\) −24.0000 −0.992278
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) 11.0000 0.452480
\(592\) 0 0
\(593\) −15.0000 −0.615976 −0.307988 0.951390i \(-0.599656\pi\)
−0.307988 + 0.951390i \(0.599656\pi\)
\(594\) 0 0
\(595\) 30.0000 1.22988
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −4.00000 −0.162623
\(606\) 0 0
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 0 0
\(609\) −21.0000 −0.850963
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) 0 0
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) 11.0000 0.442843 0.221422 0.975178i \(-0.428930\pi\)
0.221422 + 0.975178i \(0.428930\pi\)
\(618\) 0 0
\(619\) 23.0000 0.924448 0.462224 0.886763i \(-0.347052\pi\)
0.462224 + 0.886763i \(0.347052\pi\)
\(620\) 0 0
\(621\) 20.0000 0.802572
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −6.00000 −0.239617
\(628\) 0 0
\(629\) 55.0000 2.19299
\(630\) 0 0
\(631\) 12.0000 0.477712 0.238856 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) 0 0
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) −26.0000 −1.03178
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) 8.00000 0.315981 0.157991 0.987441i \(-0.449498\pi\)
0.157991 + 0.987441i \(0.449498\pi\)
\(642\) 0 0
\(643\) 8.00000 0.315489 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 0 0
\(649\) 15.0000 0.588802
\(650\) 0 0
\(651\) 15.0000 0.587896
\(652\) 0 0
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) −30.0000 −1.16510
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) 28.0000 1.08416
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) −15.0000 −0.579069
\(672\) 0 0
\(673\) −23.0000 −0.886585 −0.443292 0.896377i \(-0.646190\pi\)
−0.443292 + 0.896377i \(0.646190\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) 8.00000 0.307465 0.153732 0.988113i \(-0.450871\pi\)
0.153732 + 0.988113i \(0.450871\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) 0 0
\(683\) 28.0000 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) 0 0
\(685\) −28.0000 −1.06983
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 0 0
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) −35.0000 −1.33146 −0.665731 0.746191i \(-0.731880\pi\)
−0.665731 + 0.746191i \(0.731880\pi\)
\(692\) 0 0
\(693\) −18.0000 −0.683763
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −10.0000 −0.378777
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 22.0000 0.829746
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.0000 0.676960
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) 28.0000 1.05008
\(712\) 0 0
\(713\) −20.0000 −0.749006
\(714\) 0 0
\(715\) 36.0000 1.34632
\(716\) 0 0
\(717\) −8.00000 −0.298765
\(718\) 0 0
\(719\) −44.0000 −1.64092 −0.820462 0.571702i \(-0.806283\pi\)
−0.820462 + 0.571702i \(0.806283\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 1.00000 0.0371904
\(724\) 0 0
\(725\) −7.00000 −0.259973
\(726\) 0 0
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −40.0000 −1.47945
\(732\) 0 0
\(733\) 1.00000 0.0369358 0.0184679 0.999829i \(-0.494121\pi\)
0.0184679 + 0.999829i \(0.494121\pi\)
\(734\) 0 0
\(735\) −4.00000 −0.147542
\(736\) 0 0
\(737\) −6.00000 −0.221013
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) 32.0000 1.17239
\(746\) 0 0
\(747\) 2.00000 0.0731762
\(748\) 0 0
\(749\) −54.0000 −1.97312
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) −8.00000 −0.291536
\(754\) 0 0
\(755\) −18.0000 −0.655087
\(756\) 0 0
\(757\) −23.0000 −0.835949 −0.417975 0.908459i \(-0.637260\pi\)
−0.417975 + 0.908459i \(0.637260\pi\)
\(758\) 0 0
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) 51.0000 1.84632
\(764\) 0 0
\(765\) −20.0000 −0.723102
\(766\) 0 0
\(767\) 30.0000 1.08324
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 5.00000 0.179605
\(776\) 0 0
\(777\) −33.0000 −1.18387
\(778\) 0 0
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 0 0
\(783\) 35.0000 1.25080
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 0 0
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) 45.0000 1.60002
\(792\) 0 0
\(793\) −30.0000 −1.06533
\(794\) 0 0
\(795\) 12.0000 0.425596
\(796\) 0 0
\(797\) 24.0000 0.850124 0.425062 0.905164i \(-0.360252\pi\)
0.425062 + 0.905164i \(0.360252\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 24.0000 0.845889
\(806\) 0 0
\(807\) 10.0000 0.352017
\(808\) 0 0
\(809\) −28.0000 −0.984428 −0.492214 0.870474i \(-0.663812\pi\)
−0.492214 + 0.870474i \(0.663812\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) −32.0000 −1.12229
\(814\) 0 0
\(815\) −44.0000 −1.54125
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) −36.0000 −1.25794
\(820\) 0 0
\(821\) 32.0000 1.11681 0.558404 0.829569i \(-0.311414\pi\)
0.558404 + 0.829569i \(0.311414\pi\)
\(822\) 0 0
\(823\) −36.0000 −1.25488 −0.627441 0.778664i \(-0.715897\pi\)
−0.627441 + 0.778664i \(0.715897\pi\)
\(824\) 0 0
\(825\) 3.00000 0.104447
\(826\) 0 0
\(827\) −32.0000 −1.11275 −0.556375 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 10.0000 0.346479
\(834\) 0 0
\(835\) −42.0000 −1.45347
\(836\) 0 0
\(837\) −25.0000 −0.864126
\(838\) 0 0
\(839\) 45.0000 1.55357 0.776786 0.629764i \(-0.216849\pi\)
0.776786 + 0.629764i \(0.216849\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 46.0000 1.58245
\(846\) 0 0
\(847\) −6.00000 −0.206162
\(848\) 0 0
\(849\) −26.0000 −0.892318
\(850\) 0 0
\(851\) 44.0000 1.50830
\(852\) 0 0
\(853\) −1.00000 −0.0342393 −0.0171197 0.999853i \(-0.505450\pi\)
−0.0171197 + 0.999853i \(0.505450\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) −49.0000 −1.67381 −0.836904 0.547350i \(-0.815637\pi\)
−0.836904 + 0.547350i \(0.815637\pi\)
\(858\) 0 0
\(859\) 41.0000 1.39890 0.699451 0.714681i \(-0.253428\pi\)
0.699451 + 0.714681i \(0.253428\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 0 0
\(863\) −37.0000 −1.25949 −0.629747 0.776800i \(-0.716842\pi\)
−0.629747 + 0.776800i \(0.716842\pi\)
\(864\) 0 0
\(865\) −42.0000 −1.42804
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −42.0000 −1.42475
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) 16.0000 0.541518
\(874\) 0 0
\(875\) −36.0000 −1.21702
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 0 0
\(883\) −46.0000 −1.54802 −0.774012 0.633171i \(-0.781753\pi\)
−0.774012 + 0.633171i \(0.781753\pi\)
\(884\) 0 0
\(885\) −10.0000 −0.336146
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) −39.0000 −1.30802
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 0 0
\(899\) −35.0000 −1.16732
\(900\) 0 0
\(901\) −30.0000 −0.999445
\(902\) 0 0
\(903\) 24.0000 0.798670
\(904\) 0 0
\(905\) 52.0000 1.72854
\(906\) 0 0
\(907\) −47.0000 −1.56061 −0.780305 0.625400i \(-0.784936\pi\)
−0.780305 + 0.625400i \(0.784936\pi\)
\(908\) 0 0
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −3.00000 −0.0993944 −0.0496972 0.998764i \(-0.515826\pi\)
−0.0496972 + 0.998764i \(0.515826\pi\)
\(912\) 0 0
\(913\) −3.00000 −0.0992855
\(914\) 0 0
\(915\) 10.0000 0.330590
\(916\) 0 0
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −11.0000 −0.361678
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −35.0000 −1.14831 −0.574156 0.818746i \(-0.694670\pi\)
−0.574156 + 0.818746i \(0.694670\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 30.0000 0.981105
\(936\) 0 0
\(937\) 12.0000 0.392023 0.196011 0.980602i \(-0.437201\pi\)
0.196011 + 0.980602i \(0.437201\pi\)
\(938\) 0 0
\(939\) −11.0000 −0.358971
\(940\) 0 0
\(941\) 25.0000 0.814977 0.407488 0.913210i \(-0.366405\pi\)
0.407488 + 0.913210i \(0.366405\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 30.0000 0.975900
\(946\) 0 0
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 3.00000 0.0972817
\(952\) 0 0
\(953\) −27.0000 −0.874616 −0.437308 0.899312i \(-0.644068\pi\)
−0.437308 + 0.899312i \(0.644068\pi\)
\(954\) 0 0
\(955\) 6.00000 0.194155
\(956\) 0 0
\(957\) −21.0000 −0.678834
\(958\) 0 0
\(959\) −42.0000 −1.35625
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 36.0000 1.16008
\(964\) 0 0
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 0 0
\(969\) −10.0000 −0.321246
\(970\) 0 0
\(971\) −22.0000 −0.706014 −0.353007 0.935621i \(-0.614841\pi\)
−0.353007 + 0.935621i \(0.614841\pi\)
\(972\) 0 0
\(973\) −6.00000 −0.192351
\(974\) 0 0
\(975\) 6.00000 0.192154
\(976\) 0 0
\(977\) −19.0000 −0.607864 −0.303932 0.952694i \(-0.598300\pi\)
−0.303932 + 0.952694i \(0.598300\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −34.0000 −1.08554
\(982\) 0 0
\(983\) 44.0000 1.40338 0.701691 0.712481i \(-0.252429\pi\)
0.701691 + 0.712481i \(0.252429\pi\)
\(984\) 0 0
\(985\) −22.0000 −0.700978
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) −1.00000 −0.0317660 −0.0158830 0.999874i \(-0.505056\pi\)
−0.0158830 + 0.999874i \(0.505056\pi\)
\(992\) 0 0
\(993\) 22.0000 0.698149
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 9.00000 0.285033 0.142516 0.989792i \(-0.454481\pi\)
0.142516 + 0.989792i \(0.454481\pi\)
\(998\) 0 0
\(999\) 55.0000 1.74012
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 5312.2.a.h.1.1 1
4.3 odd 2 5312.2.a.l.1.1 1
8.3 odd 2 83.2.a.a.1.1 1
8.5 even 2 1328.2.a.c.1.1 1
24.11 even 2 747.2.a.d.1.1 1
40.19 odd 2 2075.2.a.d.1.1 1
56.27 even 2 4067.2.a.a.1.1 1
664.331 even 2 6889.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
83.2.a.a.1.1 1 8.3 odd 2
747.2.a.d.1.1 1 24.11 even 2
1328.2.a.c.1.1 1 8.5 even 2
2075.2.a.d.1.1 1 40.19 odd 2
4067.2.a.a.1.1 1 56.27 even 2
5312.2.a.h.1.1 1 1.1 even 1 trivial
5312.2.a.l.1.1 1 4.3 odd 2
6889.2.a.a.1.1 1 664.331 even 2