Properties

Label 8450.2.a.bf.1.2
Level $8450$
Weight $2$
Character 8450.1
Self dual yes
Analytic conductor $67.474$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.4735897080\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -2.73205 q^{6} +3.00000 q^{7} -1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -2.73205 q^{6} +3.00000 q^{7} -1.00000 q^{8} +4.46410 q^{9} +3.00000 q^{11} +2.73205 q^{12} -3.00000 q^{14} +1.00000 q^{16} -2.19615 q^{17} -4.46410 q^{18} +6.46410 q^{19} +8.19615 q^{21} -3.00000 q^{22} +2.53590 q^{23} -2.73205 q^{24} +4.00000 q^{27} +3.00000 q^{28} -9.46410 q^{29} +1.26795 q^{31} -1.00000 q^{32} +8.19615 q^{33} +2.19615 q^{34} +4.46410 q^{36} +11.1962 q^{37} -6.46410 q^{38} -10.3923 q^{41} -8.19615 q^{42} +2.00000 q^{43} +3.00000 q^{44} -2.53590 q^{46} +3.00000 q^{47} +2.73205 q^{48} +2.00000 q^{49} -6.00000 q^{51} +6.46410 q^{53} -4.00000 q^{54} -3.00000 q^{56} +17.6603 q^{57} +9.46410 q^{58} +10.3923 q^{59} -4.19615 q^{61} -1.26795 q^{62} +13.3923 q^{63} +1.00000 q^{64} -8.19615 q^{66} -2.19615 q^{68} +6.92820 q^{69} -6.00000 q^{71} -4.46410 q^{72} -5.66025 q^{73} -11.1962 q^{74} +6.46410 q^{76} +9.00000 q^{77} +6.19615 q^{79} -2.46410 q^{81} +10.3923 q^{82} -2.19615 q^{83} +8.19615 q^{84} -2.00000 q^{86} -25.8564 q^{87} -3.00000 q^{88} +17.1962 q^{89} +2.53590 q^{92} +3.46410 q^{93} -3.00000 q^{94} -2.73205 q^{96} +15.1244 q^{97} -2.00000 q^{98} +13.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} + 6 q^{11} + 2 q^{12} - 6 q^{14} + 2 q^{16} + 6 q^{17} - 2 q^{18} + 6 q^{19} + 6 q^{21} - 6 q^{22} + 12 q^{23} - 2 q^{24} + 8 q^{27} + 6 q^{28} - 12 q^{29} + 6 q^{31} - 2 q^{32} + 6 q^{33} - 6 q^{34} + 2 q^{36} + 12 q^{37} - 6 q^{38} - 6 q^{42} + 4 q^{43} + 6 q^{44} - 12 q^{46} + 6 q^{47} + 2 q^{48} + 4 q^{49} - 12 q^{51} + 6 q^{53} - 8 q^{54} - 6 q^{56} + 18 q^{57} + 12 q^{58} + 2 q^{61} - 6 q^{62} + 6 q^{63} + 2 q^{64} - 6 q^{66} + 6 q^{68} - 12 q^{71} - 2 q^{72} + 6 q^{73} - 12 q^{74} + 6 q^{76} + 18 q^{77} + 2 q^{79} + 2 q^{81} + 6 q^{83} + 6 q^{84} - 4 q^{86} - 24 q^{87} - 6 q^{88} + 24 q^{89} + 12 q^{92} - 6 q^{94} - 2 q^{96} + 6 q^{97} - 4 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.73205 1.57735 0.788675 0.614810i \(-0.210767\pi\)
0.788675 + 0.614810i \(0.210767\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.73205 −1.11536
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 2.73205 0.788675
\(13\) 0 0
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.19615 −0.532645 −0.266323 0.963884i \(-0.585809\pi\)
−0.266323 + 0.963884i \(0.585809\pi\)
\(18\) −4.46410 −1.05220
\(19\) 6.46410 1.48297 0.741483 0.670971i \(-0.234123\pi\)
0.741483 + 0.670971i \(0.234123\pi\)
\(20\) 0 0
\(21\) 8.19615 1.78855
\(22\) −3.00000 −0.639602
\(23\) 2.53590 0.528771 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(24\) −2.73205 −0.557678
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 3.00000 0.566947
\(29\) −9.46410 −1.75744 −0.878720 0.477338i \(-0.841602\pi\)
−0.878720 + 0.477338i \(0.841602\pi\)
\(30\) 0 0
\(31\) 1.26795 0.227730 0.113865 0.993496i \(-0.463677\pi\)
0.113865 + 0.993496i \(0.463677\pi\)
\(32\) −1.00000 −0.176777
\(33\) 8.19615 1.42677
\(34\) 2.19615 0.376637
\(35\) 0 0
\(36\) 4.46410 0.744017
\(37\) 11.1962 1.84064 0.920318 0.391171i \(-0.127930\pi\)
0.920318 + 0.391171i \(0.127930\pi\)
\(38\) −6.46410 −1.04862
\(39\) 0 0
\(40\) 0 0
\(41\) −10.3923 −1.62301 −0.811503 0.584349i \(-0.801350\pi\)
−0.811503 + 0.584349i \(0.801350\pi\)
\(42\) −8.19615 −1.26469
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −2.53590 −0.373898
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 2.73205 0.394338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 6.46410 0.887913 0.443956 0.896048i \(-0.353575\pi\)
0.443956 + 0.896048i \(0.353575\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 17.6603 2.33916
\(58\) 9.46410 1.24270
\(59\) 10.3923 1.35296 0.676481 0.736460i \(-0.263504\pi\)
0.676481 + 0.736460i \(0.263504\pi\)
\(60\) 0 0
\(61\) −4.19615 −0.537262 −0.268631 0.963243i \(-0.586571\pi\)
−0.268631 + 0.963243i \(0.586571\pi\)
\(62\) −1.26795 −0.161030
\(63\) 13.3923 1.68727
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −8.19615 −1.00888
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −2.19615 −0.266323
\(69\) 6.92820 0.834058
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −4.46410 −0.526099
\(73\) −5.66025 −0.662483 −0.331241 0.943546i \(-0.607467\pi\)
−0.331241 + 0.943546i \(0.607467\pi\)
\(74\) −11.1962 −1.30153
\(75\) 0 0
\(76\) 6.46410 0.741483
\(77\) 9.00000 1.02565
\(78\) 0 0
\(79\) 6.19615 0.697122 0.348561 0.937286i \(-0.386670\pi\)
0.348561 + 0.937286i \(0.386670\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 10.3923 1.14764
\(83\) −2.19615 −0.241059 −0.120530 0.992710i \(-0.538459\pi\)
−0.120530 + 0.992710i \(0.538459\pi\)
\(84\) 8.19615 0.894274
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) −25.8564 −2.77210
\(88\) −3.00000 −0.319801
\(89\) 17.1962 1.82279 0.911394 0.411534i \(-0.135007\pi\)
0.911394 + 0.411534i \(0.135007\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.53590 0.264386
\(93\) 3.46410 0.359211
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) −2.73205 −0.278839
\(97\) 15.1244 1.53565 0.767823 0.640662i \(-0.221340\pi\)
0.767823 + 0.640662i \(0.221340\pi\)
\(98\) −2.00000 −0.202031
\(99\) 13.3923 1.34598
\(100\) 0 0
\(101\) −7.26795 −0.723188 −0.361594 0.932336i \(-0.617767\pi\)
−0.361594 + 0.932336i \(0.617767\pi\)
\(102\) 6.00000 0.594089
\(103\) 1.19615 0.117860 0.0589302 0.998262i \(-0.481231\pi\)
0.0589302 + 0.998262i \(0.481231\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −6.46410 −0.627849
\(107\) −0.339746 −0.0328445 −0.0164222 0.999865i \(-0.505228\pi\)
−0.0164222 + 0.999865i \(0.505228\pi\)
\(108\) 4.00000 0.384900
\(109\) −15.4641 −1.48119 −0.740596 0.671950i \(-0.765457\pi\)
−0.740596 + 0.671950i \(0.765457\pi\)
\(110\) 0 0
\(111\) 30.5885 2.90333
\(112\) 3.00000 0.283473
\(113\) 6.92820 0.651751 0.325875 0.945413i \(-0.394341\pi\)
0.325875 + 0.945413i \(0.394341\pi\)
\(114\) −17.6603 −1.65403
\(115\) 0 0
\(116\) −9.46410 −0.878720
\(117\) 0 0
\(118\) −10.3923 −0.956689
\(119\) −6.58846 −0.603963
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 4.19615 0.379902
\(123\) −28.3923 −2.56005
\(124\) 1.26795 0.113865
\(125\) 0 0
\(126\) −13.3923 −1.19308
\(127\) 21.1962 1.88085 0.940427 0.339995i \(-0.110425\pi\)
0.940427 + 0.339995i \(0.110425\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.46410 0.481087
\(130\) 0 0
\(131\) −18.1244 −1.58353 −0.791766 0.610824i \(-0.790838\pi\)
−0.791766 + 0.610824i \(0.790838\pi\)
\(132\) 8.19615 0.713384
\(133\) 19.3923 1.68153
\(134\) 0 0
\(135\) 0 0
\(136\) 2.19615 0.188319
\(137\) −8.19615 −0.700245 −0.350122 0.936704i \(-0.613860\pi\)
−0.350122 + 0.936704i \(0.613860\pi\)
\(138\) −6.92820 −0.589768
\(139\) −9.19615 −0.780007 −0.390004 0.920813i \(-0.627526\pi\)
−0.390004 + 0.920813i \(0.627526\pi\)
\(140\) 0 0
\(141\) 8.19615 0.690241
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 4.46410 0.372008
\(145\) 0 0
\(146\) 5.66025 0.468446
\(147\) 5.46410 0.450672
\(148\) 11.1962 0.920318
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 6.33975 0.515921 0.257961 0.966155i \(-0.416950\pi\)
0.257961 + 0.966155i \(0.416950\pi\)
\(152\) −6.46410 −0.524308
\(153\) −9.80385 −0.792594
\(154\) −9.00000 −0.725241
\(155\) 0 0
\(156\) 0 0
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) −6.19615 −0.492939
\(159\) 17.6603 1.40055
\(160\) 0 0
\(161\) 7.60770 0.599570
\(162\) 2.46410 0.193598
\(163\) −7.26795 −0.569270 −0.284635 0.958636i \(-0.591872\pi\)
−0.284635 + 0.958636i \(0.591872\pi\)
\(164\) −10.3923 −0.811503
\(165\) 0 0
\(166\) 2.19615 0.170454
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) −8.19615 −0.632347
\(169\) 0 0
\(170\) 0 0
\(171\) 28.8564 2.20670
\(172\) 2.00000 0.152499
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) 25.8564 1.96017
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 28.3923 2.13410
\(178\) −17.1962 −1.28891
\(179\) 2.53590 0.189542 0.0947710 0.995499i \(-0.469788\pi\)
0.0947710 + 0.995499i \(0.469788\pi\)
\(180\) 0 0
\(181\) 16.5885 1.23301 0.616505 0.787351i \(-0.288548\pi\)
0.616505 + 0.787351i \(0.288548\pi\)
\(182\) 0 0
\(183\) −11.4641 −0.847451
\(184\) −2.53590 −0.186949
\(185\) 0 0
\(186\) −3.46410 −0.254000
\(187\) −6.58846 −0.481796
\(188\) 3.00000 0.218797
\(189\) 12.0000 0.872872
\(190\) 0 0
\(191\) −19.2679 −1.39418 −0.697090 0.716984i \(-0.745522\pi\)
−0.697090 + 0.716984i \(0.745522\pi\)
\(192\) 2.73205 0.197169
\(193\) −4.39230 −0.316165 −0.158083 0.987426i \(-0.550531\pi\)
−0.158083 + 0.987426i \(0.550531\pi\)
\(194\) −15.1244 −1.08587
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −9.58846 −0.683149 −0.341575 0.939855i \(-0.610960\pi\)
−0.341575 + 0.939855i \(0.610960\pi\)
\(198\) −13.3923 −0.951750
\(199\) −14.3923 −1.02024 −0.510122 0.860102i \(-0.670400\pi\)
−0.510122 + 0.860102i \(0.670400\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 7.26795 0.511371
\(203\) −28.3923 −1.99275
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −1.19615 −0.0833399
\(207\) 11.3205 0.786830
\(208\) 0 0
\(209\) 19.3923 1.34139
\(210\) 0 0
\(211\) 13.5885 0.935468 0.467734 0.883869i \(-0.345071\pi\)
0.467734 + 0.883869i \(0.345071\pi\)
\(212\) 6.46410 0.443956
\(213\) −16.3923 −1.12318
\(214\) 0.339746 0.0232246
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 3.80385 0.258222
\(218\) 15.4641 1.04736
\(219\) −15.4641 −1.04497
\(220\) 0 0
\(221\) 0 0
\(222\) −30.5885 −2.05296
\(223\) 0.464102 0.0310785 0.0155393 0.999879i \(-0.495053\pi\)
0.0155393 + 0.999879i \(0.495053\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) −6.92820 −0.460857
\(227\) 4.39230 0.291528 0.145764 0.989319i \(-0.453436\pi\)
0.145764 + 0.989319i \(0.453436\pi\)
\(228\) 17.6603 1.16958
\(229\) 4.73205 0.312703 0.156351 0.987701i \(-0.450027\pi\)
0.156351 + 0.987701i \(0.450027\pi\)
\(230\) 0 0
\(231\) 24.5885 1.61780
\(232\) 9.46410 0.621349
\(233\) 1.26795 0.0830661 0.0415331 0.999137i \(-0.486776\pi\)
0.0415331 + 0.999137i \(0.486776\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 10.3923 0.676481
\(237\) 16.9282 1.09960
\(238\) 6.58846 0.427066
\(239\) −8.19615 −0.530165 −0.265083 0.964226i \(-0.585399\pi\)
−0.265083 + 0.964226i \(0.585399\pi\)
\(240\) 0 0
\(241\) −8.66025 −0.557856 −0.278928 0.960312i \(-0.589979\pi\)
−0.278928 + 0.960312i \(0.589979\pi\)
\(242\) 2.00000 0.128565
\(243\) −18.7321 −1.20166
\(244\) −4.19615 −0.268631
\(245\) 0 0
\(246\) 28.3923 1.81023
\(247\) 0 0
\(248\) −1.26795 −0.0805149
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −6.80385 −0.429455 −0.214728 0.976674i \(-0.568886\pi\)
−0.214728 + 0.976674i \(0.568886\pi\)
\(252\) 13.3923 0.843636
\(253\) 7.60770 0.478292
\(254\) −21.1962 −1.32996
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 13.8564 0.864339 0.432169 0.901792i \(-0.357748\pi\)
0.432169 + 0.901792i \(0.357748\pi\)
\(258\) −5.46410 −0.340180
\(259\) 33.5885 2.08709
\(260\) 0 0
\(261\) −42.2487 −2.61513
\(262\) 18.1244 1.11973
\(263\) 27.5885 1.70118 0.850589 0.525832i \(-0.176246\pi\)
0.850589 + 0.525832i \(0.176246\pi\)
\(264\) −8.19615 −0.504438
\(265\) 0 0
\(266\) −19.3923 −1.18902
\(267\) 46.9808 2.87518
\(268\) 0 0
\(269\) −2.87564 −0.175331 −0.0876656 0.996150i \(-0.527941\pi\)
−0.0876656 + 0.996150i \(0.527941\pi\)
\(270\) 0 0
\(271\) 2.53590 0.154045 0.0770224 0.997029i \(-0.475459\pi\)
0.0770224 + 0.997029i \(0.475459\pi\)
\(272\) −2.19615 −0.133161
\(273\) 0 0
\(274\) 8.19615 0.495148
\(275\) 0 0
\(276\) 6.92820 0.417029
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) 9.19615 0.551549
\(279\) 5.66025 0.338871
\(280\) 0 0
\(281\) −10.3923 −0.619953 −0.309976 0.950744i \(-0.600321\pi\)
−0.309976 + 0.950744i \(0.600321\pi\)
\(282\) −8.19615 −0.488074
\(283\) −30.3923 −1.80663 −0.903317 0.428973i \(-0.858876\pi\)
−0.903317 + 0.428973i \(0.858876\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) −31.1769 −1.84032
\(288\) −4.46410 −0.263050
\(289\) −12.1769 −0.716289
\(290\) 0 0
\(291\) 41.3205 2.42225
\(292\) −5.66025 −0.331241
\(293\) 0.803848 0.0469613 0.0234806 0.999724i \(-0.492525\pi\)
0.0234806 + 0.999724i \(0.492525\pi\)
\(294\) −5.46410 −0.318673
\(295\) 0 0
\(296\) −11.1962 −0.650763
\(297\) 12.0000 0.696311
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 6.00000 0.345834
\(302\) −6.33975 −0.364811
\(303\) −19.8564 −1.14072
\(304\) 6.46410 0.370742
\(305\) 0 0
\(306\) 9.80385 0.560449
\(307\) −20.5359 −1.17205 −0.586023 0.810295i \(-0.699307\pi\)
−0.586023 + 0.810295i \(0.699307\pi\)
\(308\) 9.00000 0.512823
\(309\) 3.26795 0.185907
\(310\) 0 0
\(311\) −15.1244 −0.857624 −0.428812 0.903394i \(-0.641068\pi\)
−0.428812 + 0.903394i \(0.641068\pi\)
\(312\) 0 0
\(313\) −5.60770 −0.316966 −0.158483 0.987362i \(-0.550660\pi\)
−0.158483 + 0.987362i \(0.550660\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) 6.19615 0.348561
\(317\) 12.8038 0.719136 0.359568 0.933119i \(-0.382924\pi\)
0.359568 + 0.933119i \(0.382924\pi\)
\(318\) −17.6603 −0.990338
\(319\) −28.3923 −1.58966
\(320\) 0 0
\(321\) −0.928203 −0.0518073
\(322\) −7.60770 −0.423960
\(323\) −14.1962 −0.789895
\(324\) −2.46410 −0.136895
\(325\) 0 0
\(326\) 7.26795 0.402534
\(327\) −42.2487 −2.33636
\(328\) 10.3923 0.573819
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −0.928203 −0.0510187 −0.0255093 0.999675i \(-0.508121\pi\)
−0.0255093 + 0.999675i \(0.508121\pi\)
\(332\) −2.19615 −0.120530
\(333\) 49.9808 2.73893
\(334\) 3.00000 0.164153
\(335\) 0 0
\(336\) 8.19615 0.447137
\(337\) −4.19615 −0.228579 −0.114289 0.993447i \(-0.536459\pi\)
−0.114289 + 0.993447i \(0.536459\pi\)
\(338\) 0 0
\(339\) 18.9282 1.02804
\(340\) 0 0
\(341\) 3.80385 0.205990
\(342\) −28.8564 −1.56038
\(343\) −15.0000 −0.809924
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) 15.0000 0.806405
\(347\) 25.2679 1.35645 0.678227 0.734852i \(-0.262748\pi\)
0.678227 + 0.734852i \(0.262748\pi\)
\(348\) −25.8564 −1.38605
\(349\) 34.0526 1.82279 0.911396 0.411531i \(-0.135006\pi\)
0.911396 + 0.411531i \(0.135006\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) 20.1962 1.07493 0.537466 0.843285i \(-0.319382\pi\)
0.537466 + 0.843285i \(0.319382\pi\)
\(354\) −28.3923 −1.50903
\(355\) 0 0
\(356\) 17.1962 0.911394
\(357\) −18.0000 −0.952661
\(358\) −2.53590 −0.134026
\(359\) −22.3923 −1.18182 −0.590910 0.806737i \(-0.701231\pi\)
−0.590910 + 0.806737i \(0.701231\pi\)
\(360\) 0 0
\(361\) 22.7846 1.19919
\(362\) −16.5885 −0.871870
\(363\) −5.46410 −0.286791
\(364\) 0 0
\(365\) 0 0
\(366\) 11.4641 0.599238
\(367\) 26.3923 1.37767 0.688834 0.724920i \(-0.258123\pi\)
0.688834 + 0.724920i \(0.258123\pi\)
\(368\) 2.53590 0.132193
\(369\) −46.3923 −2.41509
\(370\) 0 0
\(371\) 19.3923 1.00680
\(372\) 3.46410 0.179605
\(373\) −20.3923 −1.05587 −0.527937 0.849284i \(-0.677034\pi\)
−0.527937 + 0.849284i \(0.677034\pi\)
\(374\) 6.58846 0.340681
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) 0 0
\(378\) −12.0000 −0.617213
\(379\) 11.7846 0.605335 0.302667 0.953096i \(-0.402123\pi\)
0.302667 + 0.953096i \(0.402123\pi\)
\(380\) 0 0
\(381\) 57.9090 2.96677
\(382\) 19.2679 0.985834
\(383\) −1.60770 −0.0821494 −0.0410747 0.999156i \(-0.513078\pi\)
−0.0410747 + 0.999156i \(0.513078\pi\)
\(384\) −2.73205 −0.139419
\(385\) 0 0
\(386\) 4.39230 0.223562
\(387\) 8.92820 0.453846
\(388\) 15.1244 0.767823
\(389\) −19.2679 −0.976924 −0.488462 0.872585i \(-0.662442\pi\)
−0.488462 + 0.872585i \(0.662442\pi\)
\(390\) 0 0
\(391\) −5.56922 −0.281648
\(392\) −2.00000 −0.101015
\(393\) −49.5167 −2.49779
\(394\) 9.58846 0.483059
\(395\) 0 0
\(396\) 13.3923 0.672989
\(397\) −0.803848 −0.0403440 −0.0201720 0.999797i \(-0.506421\pi\)
−0.0201720 + 0.999797i \(0.506421\pi\)
\(398\) 14.3923 0.721421
\(399\) 52.9808 2.65236
\(400\) 0 0
\(401\) 5.19615 0.259483 0.129742 0.991548i \(-0.458585\pi\)
0.129742 + 0.991548i \(0.458585\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −7.26795 −0.361594
\(405\) 0 0
\(406\) 28.3923 1.40909
\(407\) 33.5885 1.66492
\(408\) 6.00000 0.297044
\(409\) −19.7321 −0.975687 −0.487844 0.872931i \(-0.662216\pi\)
−0.487844 + 0.872931i \(0.662216\pi\)
\(410\) 0 0
\(411\) −22.3923 −1.10453
\(412\) 1.19615 0.0589302
\(413\) 31.1769 1.53412
\(414\) −11.3205 −0.556373
\(415\) 0 0
\(416\) 0 0
\(417\) −25.1244 −1.23034
\(418\) −19.3923 −0.948509
\(419\) 17.3205 0.846162 0.423081 0.906092i \(-0.360949\pi\)
0.423081 + 0.906092i \(0.360949\pi\)
\(420\) 0 0
\(421\) 27.1244 1.32196 0.660980 0.750403i \(-0.270141\pi\)
0.660980 + 0.750403i \(0.270141\pi\)
\(422\) −13.5885 −0.661476
\(423\) 13.3923 0.651156
\(424\) −6.46410 −0.313925
\(425\) 0 0
\(426\) 16.3923 0.794210
\(427\) −12.5885 −0.609198
\(428\) −0.339746 −0.0164222
\(429\) 0 0
\(430\) 0 0
\(431\) −2.19615 −0.105785 −0.0528925 0.998600i \(-0.516844\pi\)
−0.0528925 + 0.998600i \(0.516844\pi\)
\(432\) 4.00000 0.192450
\(433\) −12.3923 −0.595536 −0.297768 0.954638i \(-0.596242\pi\)
−0.297768 + 0.954638i \(0.596242\pi\)
\(434\) −3.80385 −0.182591
\(435\) 0 0
\(436\) −15.4641 −0.740596
\(437\) 16.3923 0.784150
\(438\) 15.4641 0.738903
\(439\) 34.5885 1.65082 0.825408 0.564536i \(-0.190945\pi\)
0.825408 + 0.564536i \(0.190945\pi\)
\(440\) 0 0
\(441\) 8.92820 0.425153
\(442\) 0 0
\(443\) 1.60770 0.0763839 0.0381920 0.999270i \(-0.487840\pi\)
0.0381920 + 0.999270i \(0.487840\pi\)
\(444\) 30.5885 1.45166
\(445\) 0 0
\(446\) −0.464102 −0.0219758
\(447\) 16.3923 0.775329
\(448\) 3.00000 0.141737
\(449\) 15.5885 0.735665 0.367832 0.929892i \(-0.380100\pi\)
0.367832 + 0.929892i \(0.380100\pi\)
\(450\) 0 0
\(451\) −31.1769 −1.46806
\(452\) 6.92820 0.325875
\(453\) 17.3205 0.813788
\(454\) −4.39230 −0.206141
\(455\) 0 0
\(456\) −17.6603 −0.827017
\(457\) 35.6603 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(458\) −4.73205 −0.221114
\(459\) −8.78461 −0.410030
\(460\) 0 0
\(461\) 0.588457 0.0274072 0.0137036 0.999906i \(-0.495638\pi\)
0.0137036 + 0.999906i \(0.495638\pi\)
\(462\) −24.5885 −1.14396
\(463\) −0.928203 −0.0431373 −0.0215686 0.999767i \(-0.506866\pi\)
−0.0215686 + 0.999767i \(0.506866\pi\)
\(464\) −9.46410 −0.439360
\(465\) 0 0
\(466\) −1.26795 −0.0587366
\(467\) −10.1436 −0.469390 −0.234695 0.972069i \(-0.575409\pi\)
−0.234695 + 0.972069i \(0.575409\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 35.5167 1.63652
\(472\) −10.3923 −0.478345
\(473\) 6.00000 0.275880
\(474\) −16.9282 −0.777538
\(475\) 0 0
\(476\) −6.58846 −0.301981
\(477\) 28.8564 1.32124
\(478\) 8.19615 0.374883
\(479\) 10.9808 0.501724 0.250862 0.968023i \(-0.419286\pi\)
0.250862 + 0.968023i \(0.419286\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 8.66025 0.394464
\(483\) 20.7846 0.945732
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 18.7321 0.849703
\(487\) −33.2487 −1.50664 −0.753321 0.657652i \(-0.771550\pi\)
−0.753321 + 0.657652i \(0.771550\pi\)
\(488\) 4.19615 0.189951
\(489\) −19.8564 −0.897938
\(490\) 0 0
\(491\) −3.33975 −0.150721 −0.0753603 0.997156i \(-0.524011\pi\)
−0.0753603 + 0.997156i \(0.524011\pi\)
\(492\) −28.3923 −1.28002
\(493\) 20.7846 0.936092
\(494\) 0 0
\(495\) 0 0
\(496\) 1.26795 0.0569326
\(497\) −18.0000 −0.807410
\(498\) 6.00000 0.268866
\(499\) −1.85641 −0.0831042 −0.0415521 0.999136i \(-0.513230\pi\)
−0.0415521 + 0.999136i \(0.513230\pi\)
\(500\) 0 0
\(501\) −8.19615 −0.366177
\(502\) 6.80385 0.303671
\(503\) 9.33975 0.416439 0.208219 0.978082i \(-0.433233\pi\)
0.208219 + 0.978082i \(0.433233\pi\)
\(504\) −13.3923 −0.596541
\(505\) 0 0
\(506\) −7.60770 −0.338203
\(507\) 0 0
\(508\) 21.1962 0.940427
\(509\) 1.60770 0.0712598 0.0356299 0.999365i \(-0.488656\pi\)
0.0356299 + 0.999365i \(0.488656\pi\)
\(510\) 0 0
\(511\) −16.9808 −0.751185
\(512\) −1.00000 −0.0441942
\(513\) 25.8564 1.14159
\(514\) −13.8564 −0.611180
\(515\) 0 0
\(516\) 5.46410 0.240544
\(517\) 9.00000 0.395820
\(518\) −33.5885 −1.47579
\(519\) −40.9808 −1.79886
\(520\) 0 0
\(521\) −0.464102 −0.0203327 −0.0101663 0.999948i \(-0.503236\pi\)
−0.0101663 + 0.999948i \(0.503236\pi\)
\(522\) 42.2487 1.84918
\(523\) −18.3923 −0.804239 −0.402120 0.915587i \(-0.631726\pi\)
−0.402120 + 0.915587i \(0.631726\pi\)
\(524\) −18.1244 −0.791766
\(525\) 0 0
\(526\) −27.5885 −1.20291
\(527\) −2.78461 −0.121300
\(528\) 8.19615 0.356692
\(529\) −16.5692 −0.720401
\(530\) 0 0
\(531\) 46.3923 2.01325
\(532\) 19.3923 0.840763
\(533\) 0 0
\(534\) −46.9808 −2.03306
\(535\) 0 0
\(536\) 0 0
\(537\) 6.92820 0.298974
\(538\) 2.87564 0.123978
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 16.0526 0.690153 0.345077 0.938574i \(-0.387853\pi\)
0.345077 + 0.938574i \(0.387853\pi\)
\(542\) −2.53590 −0.108926
\(543\) 45.3205 1.94489
\(544\) 2.19615 0.0941593
\(545\) 0 0
\(546\) 0 0
\(547\) −34.7846 −1.48728 −0.743641 0.668579i \(-0.766903\pi\)
−0.743641 + 0.668579i \(0.766903\pi\)
\(548\) −8.19615 −0.350122
\(549\) −18.7321 −0.799464
\(550\) 0 0
\(551\) −61.1769 −2.60622
\(552\) −6.92820 −0.294884
\(553\) 18.5885 0.790462
\(554\) 1.00000 0.0424859
\(555\) 0 0
\(556\) −9.19615 −0.390004
\(557\) 17.1962 0.728624 0.364312 0.931277i \(-0.381304\pi\)
0.364312 + 0.931277i \(0.381304\pi\)
\(558\) −5.66025 −0.239618
\(559\) 0 0
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) 10.3923 0.438373
\(563\) 21.4641 0.904604 0.452302 0.891865i \(-0.350603\pi\)
0.452302 + 0.891865i \(0.350603\pi\)
\(564\) 8.19615 0.345120
\(565\) 0 0
\(566\) 30.3923 1.27748
\(567\) −7.39230 −0.310448
\(568\) 6.00000 0.251754
\(569\) −21.2487 −0.890792 −0.445396 0.895334i \(-0.646937\pi\)
−0.445396 + 0.895334i \(0.646937\pi\)
\(570\) 0 0
\(571\) −34.3731 −1.43847 −0.719234 0.694768i \(-0.755507\pi\)
−0.719234 + 0.694768i \(0.755507\pi\)
\(572\) 0 0
\(573\) −52.6410 −2.19911
\(574\) 31.1769 1.30130
\(575\) 0 0
\(576\) 4.46410 0.186004
\(577\) −32.4449 −1.35070 −0.675349 0.737499i \(-0.736007\pi\)
−0.675349 + 0.737499i \(0.736007\pi\)
\(578\) 12.1769 0.506493
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) −6.58846 −0.273335
\(582\) −41.3205 −1.71279
\(583\) 19.3923 0.803147
\(584\) 5.66025 0.234223
\(585\) 0 0
\(586\) −0.803848 −0.0332066
\(587\) 30.0000 1.23823 0.619116 0.785299i \(-0.287491\pi\)
0.619116 + 0.785299i \(0.287491\pi\)
\(588\) 5.46410 0.225336
\(589\) 8.19615 0.337717
\(590\) 0 0
\(591\) −26.1962 −1.07757
\(592\) 11.1962 0.460159
\(593\) 20.7846 0.853522 0.426761 0.904365i \(-0.359655\pi\)
0.426761 + 0.904365i \(0.359655\pi\)
\(594\) −12.0000 −0.492366
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −39.3205 −1.60928
\(598\) 0 0
\(599\) 7.85641 0.321004 0.160502 0.987036i \(-0.448689\pi\)
0.160502 + 0.987036i \(0.448689\pi\)
\(600\) 0 0
\(601\) 3.78461 0.154377 0.0771887 0.997016i \(-0.475406\pi\)
0.0771887 + 0.997016i \(0.475406\pi\)
\(602\) −6.00000 −0.244542
\(603\) 0 0
\(604\) 6.33975 0.257961
\(605\) 0 0
\(606\) 19.8564 0.806611
\(607\) 6.41154 0.260236 0.130118 0.991498i \(-0.458464\pi\)
0.130118 + 0.991498i \(0.458464\pi\)
\(608\) −6.46410 −0.262154
\(609\) −77.5692 −3.14326
\(610\) 0 0
\(611\) 0 0
\(612\) −9.80385 −0.396297
\(613\) −26.9090 −1.08684 −0.543421 0.839460i \(-0.682871\pi\)
−0.543421 + 0.839460i \(0.682871\pi\)
\(614\) 20.5359 0.828761
\(615\) 0 0
\(616\) −9.00000 −0.362620
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −3.26795 −0.131456
\(619\) 10.6077 0.426359 0.213180 0.977013i \(-0.431618\pi\)
0.213180 + 0.977013i \(0.431618\pi\)
\(620\) 0 0
\(621\) 10.1436 0.407048
\(622\) 15.1244 0.606431
\(623\) 51.5885 2.06685
\(624\) 0 0
\(625\) 0 0
\(626\) 5.60770 0.224129
\(627\) 52.9808 2.11585
\(628\) 13.0000 0.518756
\(629\) −24.5885 −0.980406
\(630\) 0 0
\(631\) −20.7846 −0.827422 −0.413711 0.910408i \(-0.635768\pi\)
−0.413711 + 0.910408i \(0.635768\pi\)
\(632\) −6.19615 −0.246470
\(633\) 37.1244 1.47556
\(634\) −12.8038 −0.508506
\(635\) 0 0
\(636\) 17.6603 0.700275
\(637\) 0 0
\(638\) 28.3923 1.12406
\(639\) −26.7846 −1.05958
\(640\) 0 0
\(641\) −45.9282 −1.81405 −0.907027 0.421071i \(-0.861654\pi\)
−0.907027 + 0.421071i \(0.861654\pi\)
\(642\) 0.928203 0.0366333
\(643\) 7.26795 0.286620 0.143310 0.989678i \(-0.454225\pi\)
0.143310 + 0.989678i \(0.454225\pi\)
\(644\) 7.60770 0.299785
\(645\) 0 0
\(646\) 14.1962 0.558540
\(647\) −11.1962 −0.440166 −0.220083 0.975481i \(-0.570633\pi\)
−0.220083 + 0.975481i \(0.570633\pi\)
\(648\) 2.46410 0.0967991
\(649\) 31.1769 1.22380
\(650\) 0 0
\(651\) 10.3923 0.407307
\(652\) −7.26795 −0.284635
\(653\) 19.3923 0.758880 0.379440 0.925216i \(-0.376117\pi\)
0.379440 + 0.925216i \(0.376117\pi\)
\(654\) 42.2487 1.65206
\(655\) 0 0
\(656\) −10.3923 −0.405751
\(657\) −25.2679 −0.985797
\(658\) −9.00000 −0.350857
\(659\) 29.3205 1.14216 0.571082 0.820893i \(-0.306524\pi\)
0.571082 + 0.820893i \(0.306524\pi\)
\(660\) 0 0
\(661\) −29.9090 −1.16332 −0.581662 0.813431i \(-0.697597\pi\)
−0.581662 + 0.813431i \(0.697597\pi\)
\(662\) 0.928203 0.0360756
\(663\) 0 0
\(664\) 2.19615 0.0852272
\(665\) 0 0
\(666\) −49.9808 −1.93672
\(667\) −24.0000 −0.929284
\(668\) −3.00000 −0.116073
\(669\) 1.26795 0.0490217
\(670\) 0 0
\(671\) −12.5885 −0.485972
\(672\) −8.19615 −0.316173
\(673\) 3.60770 0.139066 0.0695332 0.997580i \(-0.477849\pi\)
0.0695332 + 0.997580i \(0.477849\pi\)
\(674\) 4.19615 0.161630
\(675\) 0 0
\(676\) 0 0
\(677\) −1.85641 −0.0713475 −0.0356737 0.999363i \(-0.511358\pi\)
−0.0356737 + 0.999363i \(0.511358\pi\)
\(678\) −18.9282 −0.726933
\(679\) 45.3731 1.74126
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) −3.80385 −0.145657
\(683\) −31.1769 −1.19295 −0.596476 0.802631i \(-0.703433\pi\)
−0.596476 + 0.802631i \(0.703433\pi\)
\(684\) 28.8564 1.10335
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) 12.9282 0.493242
\(688\) 2.00000 0.0762493
\(689\) 0 0
\(690\) 0 0
\(691\) −18.7128 −0.711869 −0.355934 0.934511i \(-0.615837\pi\)
−0.355934 + 0.934511i \(0.615837\pi\)
\(692\) −15.0000 −0.570214
\(693\) 40.1769 1.52619
\(694\) −25.2679 −0.959158
\(695\) 0 0
\(696\) 25.8564 0.980085
\(697\) 22.8231 0.864486
\(698\) −34.0526 −1.28891
\(699\) 3.46410 0.131024
\(700\) 0 0
\(701\) 29.9090 1.12965 0.564823 0.825212i \(-0.308944\pi\)
0.564823 + 0.825212i \(0.308944\pi\)
\(702\) 0 0
\(703\) 72.3731 2.72960
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −20.1962 −0.760092
\(707\) −21.8038 −0.820018
\(708\) 28.3923 1.06705
\(709\) 40.9808 1.53906 0.769532 0.638608i \(-0.220489\pi\)
0.769532 + 0.638608i \(0.220489\pi\)
\(710\) 0 0
\(711\) 27.6603 1.03734
\(712\) −17.1962 −0.644453
\(713\) 3.21539 0.120417
\(714\) 18.0000 0.673633
\(715\) 0 0
\(716\) 2.53590 0.0947710
\(717\) −22.3923 −0.836256
\(718\) 22.3923 0.835673
\(719\) 1.85641 0.0692323 0.0346161 0.999401i \(-0.488979\pi\)
0.0346161 + 0.999401i \(0.488979\pi\)
\(720\) 0 0
\(721\) 3.58846 0.133641
\(722\) −22.7846 −0.847955
\(723\) −23.6603 −0.879934
\(724\) 16.5885 0.616505
\(725\) 0 0
\(726\) 5.46410 0.202792
\(727\) −38.3731 −1.42318 −0.711589 0.702596i \(-0.752024\pi\)
−0.711589 + 0.702596i \(0.752024\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) −4.39230 −0.162455
\(732\) −11.4641 −0.423725
\(733\) 50.9090 1.88037 0.940183 0.340670i \(-0.110654\pi\)
0.940183 + 0.340670i \(0.110654\pi\)
\(734\) −26.3923 −0.974158
\(735\) 0 0
\(736\) −2.53590 −0.0934745
\(737\) 0 0
\(738\) 46.3923 1.70772
\(739\) −50.5692 −1.86022 −0.930109 0.367283i \(-0.880288\pi\)
−0.930109 + 0.367283i \(0.880288\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −19.3923 −0.711914
\(743\) −34.3923 −1.26173 −0.630866 0.775892i \(-0.717300\pi\)
−0.630866 + 0.775892i \(0.717300\pi\)
\(744\) −3.46410 −0.127000
\(745\) 0 0
\(746\) 20.3923 0.746615
\(747\) −9.80385 −0.358704
\(748\) −6.58846 −0.240898
\(749\) −1.01924 −0.0372421
\(750\) 0 0
\(751\) −0.392305 −0.0143154 −0.00715770 0.999974i \(-0.502278\pi\)
−0.00715770 + 0.999974i \(0.502278\pi\)
\(752\) 3.00000 0.109399
\(753\) −18.5885 −0.677401
\(754\) 0 0
\(755\) 0 0
\(756\) 12.0000 0.436436
\(757\) −3.78461 −0.137554 −0.0687770 0.997632i \(-0.521910\pi\)
−0.0687770 + 0.997632i \(0.521910\pi\)
\(758\) −11.7846 −0.428036
\(759\) 20.7846 0.754434
\(760\) 0 0
\(761\) −29.1962 −1.05836 −0.529180 0.848510i \(-0.677500\pi\)
−0.529180 + 0.848510i \(0.677500\pi\)
\(762\) −57.9090 −2.09782
\(763\) −46.3923 −1.67951
\(764\) −19.2679 −0.697090
\(765\) 0 0
\(766\) 1.60770 0.0580884
\(767\) 0 0
\(768\) 2.73205 0.0985844
\(769\) −38.1051 −1.37411 −0.687053 0.726607i \(-0.741096\pi\)
−0.687053 + 0.726607i \(0.741096\pi\)
\(770\) 0 0
\(771\) 37.8564 1.36337
\(772\) −4.39230 −0.158083
\(773\) −33.5885 −1.20809 −0.604046 0.796949i \(-0.706446\pi\)
−0.604046 + 0.796949i \(0.706446\pi\)
\(774\) −8.92820 −0.320918
\(775\) 0 0
\(776\) −15.1244 −0.542933
\(777\) 91.7654 3.29206
\(778\) 19.2679 0.690789
\(779\) −67.1769 −2.40686
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 5.56922 0.199155
\(783\) −37.8564 −1.35288
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 49.5167 1.76620
\(787\) 36.8372 1.31310 0.656552 0.754281i \(-0.272014\pi\)
0.656552 + 0.754281i \(0.272014\pi\)
\(788\) −9.58846 −0.341575
\(789\) 75.3731 2.68335
\(790\) 0 0
\(791\) 20.7846 0.739016
\(792\) −13.3923 −0.475875
\(793\) 0 0
\(794\) 0.803848 0.0285275
\(795\) 0 0
\(796\) −14.3923 −0.510122
\(797\) 0.928203 0.0328786 0.0164393 0.999865i \(-0.494767\pi\)
0.0164393 + 0.999865i \(0.494767\pi\)
\(798\) −52.9808 −1.87550
\(799\) −6.58846 −0.233083
\(800\) 0 0
\(801\) 76.7654 2.71237
\(802\) −5.19615 −0.183483
\(803\) −16.9808 −0.599238
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −7.85641 −0.276559
\(808\) 7.26795 0.255686
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) 22.6077 0.793864 0.396932 0.917848i \(-0.370075\pi\)
0.396932 + 0.917848i \(0.370075\pi\)
\(812\) −28.3923 −0.996375
\(813\) 6.92820 0.242983
\(814\) −33.5885 −1.17727
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 12.9282 0.452301
\(818\) 19.7321 0.689915
\(819\) 0 0
\(820\) 0 0
\(821\) −9.80385 −0.342157 −0.171078 0.985257i \(-0.554725\pi\)
−0.171078 + 0.985257i \(0.554725\pi\)
\(822\) 22.3923 0.781021
\(823\) −16.8038 −0.585745 −0.292873 0.956151i \(-0.594611\pi\)
−0.292873 + 0.956151i \(0.594611\pi\)
\(824\) −1.19615 −0.0416699
\(825\) 0 0
\(826\) −31.1769 −1.08478
\(827\) −11.4115 −0.396818 −0.198409 0.980119i \(-0.563577\pi\)
−0.198409 + 0.980119i \(0.563577\pi\)
\(828\) 11.3205 0.393415
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 0 0
\(831\) −2.73205 −0.0947738
\(832\) 0 0
\(833\) −4.39230 −0.152184
\(834\) 25.1244 0.869985
\(835\) 0 0
\(836\) 19.3923 0.670697
\(837\) 5.07180 0.175307
\(838\) −17.3205 −0.598327
\(839\) −32.1962 −1.11153 −0.555767 0.831338i \(-0.687575\pi\)
−0.555767 + 0.831338i \(0.687575\pi\)
\(840\) 0 0
\(841\) 60.5692 2.08859
\(842\) −27.1244 −0.934767
\(843\) −28.3923 −0.977883
\(844\) 13.5885 0.467734
\(845\) 0 0
\(846\) −13.3923 −0.460437
\(847\) −6.00000 −0.206162
\(848\) 6.46410 0.221978
\(849\) −83.0333 −2.84970
\(850\) 0 0
\(851\) 28.3923 0.973276
\(852\) −16.3923 −0.561591
\(853\) −13.8564 −0.474434 −0.237217 0.971457i \(-0.576235\pi\)
−0.237217 + 0.971457i \(0.576235\pi\)
\(854\) 12.5885 0.430768
\(855\) 0 0
\(856\) 0.339746 0.0116123
\(857\) −13.2679 −0.453225 −0.226612 0.973985i \(-0.572765\pi\)
−0.226612 + 0.973985i \(0.572765\pi\)
\(858\) 0 0
\(859\) 52.3731 1.78695 0.893473 0.449117i \(-0.148261\pi\)
0.893473 + 0.449117i \(0.148261\pi\)
\(860\) 0 0
\(861\) −85.1769 −2.90282
\(862\) 2.19615 0.0748012
\(863\) 19.1769 0.652790 0.326395 0.945234i \(-0.394166\pi\)
0.326395 + 0.945234i \(0.394166\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 12.3923 0.421108
\(867\) −33.2679 −1.12984
\(868\) 3.80385 0.129111
\(869\) 18.5885 0.630570
\(870\) 0 0
\(871\) 0 0
\(872\) 15.4641 0.523681
\(873\) 67.5167 2.28509
\(874\) −16.3923 −0.554478
\(875\) 0 0
\(876\) −15.4641 −0.522484
\(877\) 20.5359 0.693448 0.346724 0.937967i \(-0.387294\pi\)
0.346724 + 0.937967i \(0.387294\pi\)
\(878\) −34.5885 −1.16730
\(879\) 2.19615 0.0740744
\(880\) 0 0
\(881\) 8.32051 0.280325 0.140163 0.990129i \(-0.455237\pi\)
0.140163 + 0.990129i \(0.455237\pi\)
\(882\) −8.92820 −0.300628
\(883\) −26.5885 −0.894773 −0.447386 0.894341i \(-0.647645\pi\)
−0.447386 + 0.894341i \(0.647645\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1.60770 −0.0540116
\(887\) −52.7654 −1.77169 −0.885844 0.463983i \(-0.846420\pi\)
−0.885844 + 0.463983i \(0.846420\pi\)
\(888\) −30.5885 −1.02648
\(889\) 63.5885 2.13269
\(890\) 0 0
\(891\) −7.39230 −0.247652
\(892\) 0.464102 0.0155393
\(893\) 19.3923 0.648939
\(894\) −16.3923 −0.548241
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) −15.5885 −0.520194
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −14.1962 −0.472942
\(902\) 31.1769 1.03808
\(903\) 16.3923 0.545502
\(904\) −6.92820 −0.230429
\(905\) 0 0
\(906\) −17.3205 −0.575435
\(907\) −26.5885 −0.882855 −0.441428 0.897297i \(-0.645528\pi\)
−0.441428 + 0.897297i \(0.645528\pi\)
\(908\) 4.39230 0.145764
\(909\) −32.4449 −1.07613
\(910\) 0 0
\(911\) −14.5359 −0.481596 −0.240798 0.970575i \(-0.577409\pi\)
−0.240798 + 0.970575i \(0.577409\pi\)
\(912\) 17.6603 0.584789
\(913\) −6.58846 −0.218046
\(914\) −35.6603 −1.17954
\(915\) 0 0
\(916\) 4.73205 0.156351
\(917\) −54.3731 −1.79556
\(918\) 8.78461 0.289935
\(919\) −10.7846 −0.355751 −0.177876 0.984053i \(-0.556923\pi\)
−0.177876 + 0.984053i \(0.556923\pi\)
\(920\) 0 0
\(921\) −56.1051 −1.84873
\(922\) −0.588457 −0.0193798
\(923\) 0 0
\(924\) 24.5885 0.808901
\(925\) 0 0
\(926\) 0.928203 0.0305027
\(927\) 5.33975 0.175380
\(928\) 9.46410 0.310674
\(929\) −44.7846 −1.46934 −0.734668 0.678427i \(-0.762662\pi\)
−0.734668 + 0.678427i \(0.762662\pi\)
\(930\) 0 0
\(931\) 12.9282 0.423705
\(932\) 1.26795 0.0415331
\(933\) −41.3205 −1.35277
\(934\) 10.1436 0.331909
\(935\) 0 0
\(936\) 0 0
\(937\) 30.3923 0.992873 0.496437 0.868073i \(-0.334641\pi\)
0.496437 + 0.868073i \(0.334641\pi\)
\(938\) 0 0
\(939\) −15.3205 −0.499966
\(940\) 0 0
\(941\) 44.7846 1.45994 0.729968 0.683481i \(-0.239535\pi\)
0.729968 + 0.683481i \(0.239535\pi\)
\(942\) −35.5167 −1.15720
\(943\) −26.3538 −0.858199
\(944\) 10.3923 0.338241
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) 57.3731 1.86437 0.932187 0.361977i \(-0.117898\pi\)
0.932187 + 0.361977i \(0.117898\pi\)
\(948\) 16.9282 0.549802
\(949\) 0 0
\(950\) 0 0
\(951\) 34.9808 1.13433
\(952\) 6.58846 0.213533
\(953\) 24.5885 0.796498 0.398249 0.917277i \(-0.369618\pi\)
0.398249 + 0.917277i \(0.369618\pi\)
\(954\) −28.8564 −0.934261
\(955\) 0 0
\(956\) −8.19615 −0.265083
\(957\) −77.5692 −2.50746
\(958\) −10.9808 −0.354772
\(959\) −24.5885 −0.794003
\(960\) 0 0
\(961\) −29.3923 −0.948139
\(962\) 0 0
\(963\) −1.51666 −0.0488737
\(964\) −8.66025 −0.278928
\(965\) 0 0
\(966\) −20.7846 −0.668734
\(967\) −38.5692 −1.24030 −0.620151 0.784482i \(-0.712929\pi\)
−0.620151 + 0.784482i \(0.712929\pi\)
\(968\) 2.00000 0.0642824
\(969\) −38.7846 −1.24594
\(970\) 0 0
\(971\) −46.7654 −1.50077 −0.750386 0.661000i \(-0.770132\pi\)
−0.750386 + 0.661000i \(0.770132\pi\)
\(972\) −18.7321 −0.600831
\(973\) −27.5885 −0.884445
\(974\) 33.2487 1.06536
\(975\) 0 0
\(976\) −4.19615 −0.134316
\(977\) −4.39230 −0.140522 −0.0702611 0.997529i \(-0.522383\pi\)
−0.0702611 + 0.997529i \(0.522383\pi\)
\(978\) 19.8564 0.634938
\(979\) 51.5885 1.64877
\(980\) 0 0
\(981\) −69.0333 −2.20406
\(982\) 3.33975 0.106576
\(983\) −38.5692 −1.23017 −0.615084 0.788462i \(-0.710878\pi\)
−0.615084 + 0.788462i \(0.710878\pi\)
\(984\) 28.3923 0.905114
\(985\) 0 0
\(986\) −20.7846 −0.661917
\(987\) 24.5885 0.782659
\(988\) 0 0
\(989\) 5.07180 0.161274
\(990\) 0 0
\(991\) 51.1769 1.62569 0.812844 0.582481i \(-0.197918\pi\)
0.812844 + 0.582481i \(0.197918\pi\)
\(992\) −1.26795 −0.0402574
\(993\) −2.53590 −0.0804743
\(994\) 18.0000 0.570925
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) 42.5692 1.34818 0.674090 0.738649i \(-0.264536\pi\)
0.674090 + 0.738649i \(0.264536\pi\)
\(998\) 1.85641 0.0587635
\(999\) 44.7846 1.41692
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 8450.2.a.bf.1.2 2
5.4 even 2 1690.2.a.m.1.1 2
13.2 odd 12 650.2.m.a.251.1 4
13.7 odd 12 650.2.m.a.101.1 4
13.12 even 2 8450.2.a.bm.1.2 2
65.2 even 12 650.2.n.b.199.2 4
65.4 even 6 1690.2.e.n.991.2 4
65.7 even 12 650.2.n.a.49.1 4
65.9 even 6 1690.2.e.l.991.2 4
65.19 odd 12 1690.2.l.g.361.1 4
65.24 odd 12 1690.2.l.g.1161.1 4
65.28 even 12 650.2.n.a.199.1 4
65.29 even 6 1690.2.e.l.191.2 4
65.33 even 12 650.2.n.b.49.2 4
65.34 odd 4 1690.2.d.f.1351.3 4
65.44 odd 4 1690.2.d.f.1351.1 4
65.49 even 6 1690.2.e.n.191.2 4
65.54 odd 12 130.2.l.a.121.2 yes 4
65.59 odd 12 130.2.l.a.101.2 4
65.64 even 2 1690.2.a.j.1.1 2
195.59 even 12 1170.2.bs.c.361.1 4
195.119 even 12 1170.2.bs.c.901.1 4
260.59 even 12 1040.2.da.a.881.1 4
260.119 even 12 1040.2.da.a.641.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
130.2.l.a.101.2 4 65.59 odd 12
130.2.l.a.121.2 yes 4 65.54 odd 12
650.2.m.a.101.1 4 13.7 odd 12
650.2.m.a.251.1 4 13.2 odd 12
650.2.n.a.49.1 4 65.7 even 12
650.2.n.a.199.1 4 65.28 even 12
650.2.n.b.49.2 4 65.33 even 12
650.2.n.b.199.2 4 65.2 even 12
1040.2.da.a.641.1 4 260.119 even 12
1040.2.da.a.881.1 4 260.59 even 12
1170.2.bs.c.361.1 4 195.59 even 12
1170.2.bs.c.901.1 4 195.119 even 12
1690.2.a.j.1.1 2 65.64 even 2
1690.2.a.m.1.1 2 5.4 even 2
1690.2.d.f.1351.1 4 65.44 odd 4
1690.2.d.f.1351.3 4 65.34 odd 4
1690.2.e.l.191.2 4 65.29 even 6
1690.2.e.l.991.2 4 65.9 even 6
1690.2.e.n.191.2 4 65.49 even 6
1690.2.e.n.991.2 4 65.4 even 6
1690.2.l.g.361.1 4 65.19 odd 12
1690.2.l.g.1161.1 4 65.24 odd 12
8450.2.a.bf.1.2 2 1.1 even 1 trivial
8450.2.a.bm.1.2 2 13.12 even 2