Properties

Label 8450.2.a.bf.1.1
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 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} +0.732051 q^{6} +3.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} +0.732051 q^{6} +3.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +3.00000 q^{11} -0.732051 q^{12} -3.00000 q^{14} +1.00000 q^{16} +8.19615 q^{17} +2.46410 q^{18} -0.464102 q^{19} -2.19615 q^{21} -3.00000 q^{22} +9.46410 q^{23} +0.732051 q^{24} +4.00000 q^{27} +3.00000 q^{28} -2.53590 q^{29} +4.73205 q^{31} -1.00000 q^{32} -2.19615 q^{33} -8.19615 q^{34} -2.46410 q^{36} +0.803848 q^{37} +0.464102 q^{38} +10.3923 q^{41} +2.19615 q^{42} +2.00000 q^{43} +3.00000 q^{44} -9.46410 q^{46} +3.00000 q^{47} -0.732051 q^{48} +2.00000 q^{49} -6.00000 q^{51} -0.464102 q^{53} -4.00000 q^{54} -3.00000 q^{56} +0.339746 q^{57} +2.53590 q^{58} -10.3923 q^{59} +6.19615 q^{61} -4.73205 q^{62} -7.39230 q^{63} +1.00000 q^{64} +2.19615 q^{66} +8.19615 q^{68} -6.92820 q^{69} -6.00000 q^{71} +2.46410 q^{72} +11.6603 q^{73} -0.803848 q^{74} -0.464102 q^{76} +9.00000 q^{77} -4.19615 q^{79} +4.46410 q^{81} -10.3923 q^{82} +8.19615 q^{83} -2.19615 q^{84} -2.00000 q^{86} +1.85641 q^{87} -3.00000 q^{88} +6.80385 q^{89} +9.46410 q^{92} -3.46410 q^{93} -3.00000 q^{94} +0.732051 q^{96} -9.12436 q^{97} -2.00000 q^{98} -7.39230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 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
\(3\) −0.732051 −0.422650 −0.211325 0.977416i \(-0.567778\pi\)
−0.211325 + 0.977416i \(0.567778\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0.732051 0.298858
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −0.732051 −0.211325
\(13\) 0 0
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 8.19615 1.98786 0.993929 0.110019i \(-0.0350912\pi\)
0.993929 + 0.110019i \(0.0350912\pi\)
\(18\) 2.46410 0.580794
\(19\) −0.464102 −0.106472 −0.0532361 0.998582i \(-0.516954\pi\)
−0.0532361 + 0.998582i \(0.516954\pi\)
\(20\) 0 0
\(21\) −2.19615 −0.479240
\(22\) −3.00000 −0.639602
\(23\) 9.46410 1.97340 0.986701 0.162547i \(-0.0519709\pi\)
0.986701 + 0.162547i \(0.0519709\pi\)
\(24\) 0.732051 0.149429
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 3.00000 0.566947
\(29\) −2.53590 −0.470905 −0.235452 0.971886i \(-0.575657\pi\)
−0.235452 + 0.971886i \(0.575657\pi\)
\(30\) 0 0
\(31\) 4.73205 0.849901 0.424951 0.905216i \(-0.360291\pi\)
0.424951 + 0.905216i \(0.360291\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.19615 −0.382301
\(34\) −8.19615 −1.40563
\(35\) 0 0
\(36\) −2.46410 −0.410684
\(37\) 0.803848 0.132152 0.0660759 0.997815i \(-0.478952\pi\)
0.0660759 + 0.997815i \(0.478952\pi\)
\(38\) 0.464102 0.0752872
\(39\) 0 0
\(40\) 0 0
\(41\) 10.3923 1.62301 0.811503 0.584349i \(-0.198650\pi\)
0.811503 + 0.584349i \(0.198650\pi\)
\(42\) 2.19615 0.338874
\(43\) 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
\(46\) −9.46410 −1.39541
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −0.732051 −0.105662
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) −0.464102 −0.0637493 −0.0318746 0.999492i \(-0.510148\pi\)
−0.0318746 + 0.999492i \(0.510148\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 0.339746 0.0450005
\(58\) 2.53590 0.332980
\(59\) −10.3923 −1.35296 −0.676481 0.736460i \(-0.736496\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) 6.19615 0.793336 0.396668 0.917962i \(-0.370166\pi\)
0.396668 + 0.917962i \(0.370166\pi\)
\(62\) −4.73205 −0.600971
\(63\) −7.39230 −0.931343
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.19615 0.270328
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 8.19615 0.993929
\(69\) −6.92820 −0.834058
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 2.46410 0.290397
\(73\) 11.6603 1.36473 0.682365 0.731012i \(-0.260952\pi\)
0.682365 + 0.731012i \(0.260952\pi\)
\(74\) −0.803848 −0.0934454
\(75\) 0 0
\(76\) −0.464102 −0.0532361
\(77\) 9.00000 1.02565
\(78\) 0 0
\(79\) −4.19615 −0.472104 −0.236052 0.971740i \(-0.575854\pi\)
−0.236052 + 0.971740i \(0.575854\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) −10.3923 −1.14764
\(83\) 8.19615 0.899645 0.449822 0.893118i \(-0.351487\pi\)
0.449822 + 0.893118i \(0.351487\pi\)
\(84\) −2.19615 −0.239620
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 1.85641 0.199028
\(88\) −3.00000 −0.319801
\(89\) 6.80385 0.721206 0.360603 0.932719i \(-0.382571\pi\)
0.360603 + 0.932719i \(0.382571\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.46410 0.986701
\(93\) −3.46410 −0.359211
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) 0.732051 0.0747146
\(97\) −9.12436 −0.926438 −0.463219 0.886244i \(-0.653306\pi\)
−0.463219 + 0.886244i \(0.653306\pi\)
\(98\) −2.00000 −0.202031
\(99\) −7.39230 −0.742955
\(100\) 0 0
\(101\) −10.7321 −1.06788 −0.533939 0.845523i \(-0.679289\pi\)
−0.533939 + 0.845523i \(0.679289\pi\)
\(102\) 6.00000 0.594089
\(103\) −9.19615 −0.906124 −0.453062 0.891479i \(-0.649668\pi\)
−0.453062 + 0.891479i \(0.649668\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.464102 0.0450775
\(107\) −17.6603 −1.70728 −0.853641 0.520862i \(-0.825610\pi\)
−0.853641 + 0.520862i \(0.825610\pi\)
\(108\) 4.00000 0.384900
\(109\) −8.53590 −0.817591 −0.408795 0.912626i \(-0.634051\pi\)
−0.408795 + 0.912626i \(0.634051\pi\)
\(110\) 0 0
\(111\) −0.588457 −0.0558539
\(112\) 3.00000 0.283473
\(113\) −6.92820 −0.651751 −0.325875 0.945413i \(-0.605659\pi\)
−0.325875 + 0.945413i \(0.605659\pi\)
\(114\) −0.339746 −0.0318201
\(115\) 0 0
\(116\) −2.53590 −0.235452
\(117\) 0 0
\(118\) 10.3923 0.956689
\(119\) 24.5885 2.25402
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −6.19615 −0.560973
\(123\) −7.60770 −0.685963
\(124\) 4.73205 0.424951
\(125\) 0 0
\(126\) 7.39230 0.658559
\(127\) 10.8038 0.958686 0.479343 0.877628i \(-0.340875\pi\)
0.479343 + 0.877628i \(0.340875\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.46410 −0.128907
\(130\) 0 0
\(131\) 6.12436 0.535087 0.267544 0.963546i \(-0.413788\pi\)
0.267544 + 0.963546i \(0.413788\pi\)
\(132\) −2.19615 −0.191151
\(133\) −1.39230 −0.120728
\(134\) 0 0
\(135\) 0 0
\(136\) −8.19615 −0.702814
\(137\) 2.19615 0.187630 0.0938150 0.995590i \(-0.470094\pi\)
0.0938150 + 0.995590i \(0.470094\pi\)
\(138\) 6.92820 0.589768
\(139\) 1.19615 0.101456 0.0507282 0.998712i \(-0.483846\pi\)
0.0507282 + 0.998712i \(0.483846\pi\)
\(140\) 0 0
\(141\) −2.19615 −0.184949
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) −2.46410 −0.205342
\(145\) 0 0
\(146\) −11.6603 −0.965009
\(147\) −1.46410 −0.120757
\(148\) 0.803848 0.0660759
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 23.6603 1.92544 0.962722 0.270492i \(-0.0871865\pi\)
0.962722 + 0.270492i \(0.0871865\pi\)
\(152\) 0.464102 0.0376436
\(153\) −20.1962 −1.63276
\(154\) −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 \(0.326395\pi\)
\(158\) 4.19615 0.333828
\(159\) 0.339746 0.0269436
\(160\) 0 0
\(161\) 28.3923 2.23763
\(162\) −4.46410 −0.350733
\(163\) −10.7321 −0.840599 −0.420300 0.907385i \(-0.638075\pi\)
−0.420300 + 0.907385i \(0.638075\pi\)
\(164\) 10.3923 0.811503
\(165\) 0 0
\(166\) −8.19615 −0.636145
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 2.19615 0.169437
\(169\) 0 0
\(170\) 0 0
\(171\) 1.14359 0.0874528
\(172\) 2.00000 0.152499
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) −1.85641 −0.140734
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 7.60770 0.571829
\(178\) −6.80385 −0.509970
\(179\) 9.46410 0.707380 0.353690 0.935363i \(-0.384927\pi\)
0.353690 + 0.935363i \(0.384927\pi\)
\(180\) 0 0
\(181\) −14.5885 −1.08435 −0.542176 0.840265i \(-0.682399\pi\)
−0.542176 + 0.840265i \(0.682399\pi\)
\(182\) 0 0
\(183\) −4.53590 −0.335303
\(184\) −9.46410 −0.697703
\(185\) 0 0
\(186\) 3.46410 0.254000
\(187\) 24.5885 1.79809
\(188\) 3.00000 0.218797
\(189\) 12.0000 0.872872
\(190\) 0 0
\(191\) −22.7321 −1.64483 −0.822417 0.568886i \(-0.807375\pi\)
−0.822417 + 0.568886i \(0.807375\pi\)
\(192\) −0.732051 −0.0528312
\(193\) 16.3923 1.17994 0.589972 0.807424i \(-0.299139\pi\)
0.589972 + 0.807424i \(0.299139\pi\)
\(194\) 9.12436 0.655091
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 21.5885 1.53811 0.769057 0.639180i \(-0.220726\pi\)
0.769057 + 0.639180i \(0.220726\pi\)
\(198\) 7.39230 0.525348
\(199\) 6.39230 0.453138 0.226569 0.973995i \(-0.427249\pi\)
0.226569 + 0.973995i \(0.427249\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 10.7321 0.755104
\(203\) −7.60770 −0.533956
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 9.19615 0.640726
\(207\) −23.3205 −1.62089
\(208\) 0 0
\(209\) −1.39230 −0.0963077
\(210\) 0 0
\(211\) −17.5885 −1.21084 −0.605420 0.795906i \(-0.706995\pi\)
−0.605420 + 0.795906i \(0.706995\pi\)
\(212\) −0.464102 −0.0318746
\(213\) 4.39230 0.300956
\(214\) 17.6603 1.20723
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 14.1962 0.963698
\(218\) 8.53590 0.578124
\(219\) −8.53590 −0.576803
\(220\) 0 0
\(221\) 0 0
\(222\) 0.588457 0.0394947
\(223\) −6.46410 −0.432868 −0.216434 0.976297i \(-0.569443\pi\)
−0.216434 + 0.976297i \(0.569443\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) 6.92820 0.460857
\(227\) −16.3923 −1.08800 −0.543998 0.839087i \(-0.683090\pi\)
−0.543998 + 0.839087i \(0.683090\pi\)
\(228\) 0.339746 0.0225002
\(229\) 1.26795 0.0837884 0.0418942 0.999122i \(-0.486661\pi\)
0.0418942 + 0.999122i \(0.486661\pi\)
\(230\) 0 0
\(231\) −6.58846 −0.433489
\(232\) 2.53590 0.166490
\(233\) 4.73205 0.310007 0.155003 0.987914i \(-0.450461\pi\)
0.155003 + 0.987914i \(0.450461\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −10.3923 −0.676481
\(237\) 3.07180 0.199535
\(238\) −24.5885 −1.59383
\(239\) 2.19615 0.142057 0.0710286 0.997474i \(-0.477372\pi\)
0.0710286 + 0.997474i \(0.477372\pi\)
\(240\) 0 0
\(241\) 8.66025 0.557856 0.278928 0.960312i \(-0.410021\pi\)
0.278928 + 0.960312i \(0.410021\pi\)
\(242\) 2.00000 0.128565
\(243\) −15.2679 −0.979439
\(244\) 6.19615 0.396668
\(245\) 0 0
\(246\) 7.60770 0.485049
\(247\) 0 0
\(248\) −4.73205 −0.300486
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −17.1962 −1.08541 −0.542706 0.839923i \(-0.682600\pi\)
−0.542706 + 0.839923i \(0.682600\pi\)
\(252\) −7.39230 −0.465671
\(253\) 28.3923 1.78501
\(254\) −10.8038 −0.677894
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −13.8564 −0.864339 −0.432169 0.901792i \(-0.642252\pi\)
−0.432169 + 0.901792i \(0.642252\pi\)
\(258\) 1.46410 0.0911510
\(259\) 2.41154 0.149846
\(260\) 0 0
\(261\) 6.24871 0.386786
\(262\) −6.12436 −0.378364
\(263\) −3.58846 −0.221274 −0.110637 0.993861i \(-0.535289\pi\)
−0.110637 + 0.993861i \(0.535289\pi\)
\(264\) 2.19615 0.135164
\(265\) 0 0
\(266\) 1.39230 0.0853677
\(267\) −4.98076 −0.304818
\(268\) 0 0
\(269\) −27.1244 −1.65380 −0.826901 0.562348i \(-0.809898\pi\)
−0.826901 + 0.562348i \(0.809898\pi\)
\(270\) 0 0
\(271\) 9.46410 0.574903 0.287452 0.957795i \(-0.407192\pi\)
0.287452 + 0.957795i \(0.407192\pi\)
\(272\) 8.19615 0.496965
\(273\) 0 0
\(274\) −2.19615 −0.132674
\(275\) 0 0
\(276\) −6.92820 −0.417029
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) −1.19615 −0.0717405
\(279\) −11.6603 −0.698081
\(280\) 0 0
\(281\) 10.3923 0.619953 0.309976 0.950744i \(-0.399679\pi\)
0.309976 + 0.950744i \(0.399679\pi\)
\(282\) 2.19615 0.130779
\(283\) −9.60770 −0.571118 −0.285559 0.958361i \(-0.592179\pi\)
−0.285559 + 0.958361i \(0.592179\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) 31.1769 1.84032
\(288\) 2.46410 0.145199
\(289\) 50.1769 2.95158
\(290\) 0 0
\(291\) 6.67949 0.391559
\(292\) 11.6603 0.682365
\(293\) 11.1962 0.654086 0.327043 0.945009i \(-0.393948\pi\)
0.327043 + 0.945009i \(0.393948\pi\)
\(294\) 1.46410 0.0853881
\(295\) 0 0
\(296\) −0.803848 −0.0467227
\(297\) 12.0000 0.696311
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 6.00000 0.345834
\(302\) −23.6603 −1.36149
\(303\) 7.85641 0.451339
\(304\) −0.464102 −0.0266181
\(305\) 0 0
\(306\) 20.1962 1.15454
\(307\) −27.4641 −1.56746 −0.783730 0.621102i \(-0.786685\pi\)
−0.783730 + 0.621102i \(0.786685\pi\)
\(308\) 9.00000 0.512823
\(309\) 6.73205 0.382973
\(310\) 0 0
\(311\) 9.12436 0.517395 0.258697 0.965958i \(-0.416707\pi\)
0.258697 + 0.965958i \(0.416707\pi\)
\(312\) 0 0
\(313\) −26.3923 −1.49178 −0.745891 0.666068i \(-0.767976\pi\)
−0.745891 + 0.666068i \(0.767976\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) −4.19615 −0.236052
\(317\) 23.1962 1.30283 0.651413 0.758723i \(-0.274177\pi\)
0.651413 + 0.758723i \(0.274177\pi\)
\(318\) −0.339746 −0.0190520
\(319\) −7.60770 −0.425949
\(320\) 0 0
\(321\) 12.9282 0.721582
\(322\) −28.3923 −1.58224
\(323\) −3.80385 −0.211652
\(324\) 4.46410 0.248006
\(325\) 0 0
\(326\) 10.7321 0.594393
\(327\) 6.24871 0.345555
\(328\) −10.3923 −0.573819
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 12.9282 0.710598 0.355299 0.934753i \(-0.384379\pi\)
0.355299 + 0.934753i \(0.384379\pi\)
\(332\) 8.19615 0.449822
\(333\) −1.98076 −0.108545
\(334\) 3.00000 0.164153
\(335\) 0 0
\(336\) −2.19615 −0.119810
\(337\) 6.19615 0.337526 0.168763 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(338\) 0 0
\(339\) 5.07180 0.275462
\(340\) 0 0
\(341\) 14.1962 0.768765
\(342\) −1.14359 −0.0618385
\(343\) −15.0000 −0.809924
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) 15.0000 0.806405
\(347\) 28.7321 1.54242 0.771209 0.636582i \(-0.219653\pi\)
0.771209 + 0.636582i \(0.219653\pi\)
\(348\) 1.85641 0.0995138
\(349\) −4.05256 −0.216929 −0.108464 0.994100i \(-0.534593\pi\)
−0.108464 + 0.994100i \(0.534593\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) 9.80385 0.521806 0.260903 0.965365i \(-0.415980\pi\)
0.260903 + 0.965365i \(0.415980\pi\)
\(354\) −7.60770 −0.404344
\(355\) 0 0
\(356\) 6.80385 0.360603
\(357\) −18.0000 −0.952661
\(358\) −9.46410 −0.500193
\(359\) −1.60770 −0.0848509 −0.0424255 0.999100i \(-0.513508\pi\)
−0.0424255 + 0.999100i \(0.513508\pi\)
\(360\) 0 0
\(361\) −18.7846 −0.988664
\(362\) 14.5885 0.766752
\(363\) 1.46410 0.0768454
\(364\) 0 0
\(365\) 0 0
\(366\) 4.53590 0.237095
\(367\) 5.60770 0.292719 0.146360 0.989231i \(-0.453244\pi\)
0.146360 + 0.989231i \(0.453244\pi\)
\(368\) 9.46410 0.493350
\(369\) −25.6077 −1.33308
\(370\) 0 0
\(371\) −1.39230 −0.0722849
\(372\) −3.46410 −0.179605
\(373\) 0.392305 0.0203128 0.0101564 0.999948i \(-0.496767\pi\)
0.0101564 + 0.999948i \(0.496767\pi\)
\(374\) −24.5885 −1.27144
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) 0 0
\(378\) −12.0000 −0.617213
\(379\) −29.7846 −1.52993 −0.764966 0.644070i \(-0.777244\pi\)
−0.764966 + 0.644070i \(0.777244\pi\)
\(380\) 0 0
\(381\) −7.90897 −0.405189
\(382\) 22.7321 1.16307
\(383\) −22.3923 −1.14419 −0.572097 0.820186i \(-0.693870\pi\)
−0.572097 + 0.820186i \(0.693870\pi\)
\(384\) 0.732051 0.0373573
\(385\) 0 0
\(386\) −16.3923 −0.834346
\(387\) −4.92820 −0.250515
\(388\) −9.12436 −0.463219
\(389\) −22.7321 −1.15256 −0.576280 0.817252i \(-0.695496\pi\)
−0.576280 + 0.817252i \(0.695496\pi\)
\(390\) 0 0
\(391\) 77.5692 3.92284
\(392\) −2.00000 −0.101015
\(393\) −4.48334 −0.226155
\(394\) −21.5885 −1.08761
\(395\) 0 0
\(396\) −7.39230 −0.371477
\(397\) −11.1962 −0.561919 −0.280959 0.959720i \(-0.590653\pi\)
−0.280959 + 0.959720i \(0.590653\pi\)
\(398\) −6.39230 −0.320417
\(399\) 1.01924 0.0510257
\(400\) 0 0
\(401\) −5.19615 −0.259483 −0.129742 0.991548i \(-0.541415\pi\)
−0.129742 + 0.991548i \(0.541415\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −10.7321 −0.533939
\(405\) 0 0
\(406\) 7.60770 0.377564
\(407\) 2.41154 0.119536
\(408\) 6.00000 0.297044
\(409\) −16.2679 −0.804398 −0.402199 0.915552i \(-0.631754\pi\)
−0.402199 + 0.915552i \(0.631754\pi\)
\(410\) 0 0
\(411\) −1.60770 −0.0793018
\(412\) −9.19615 −0.453062
\(413\) −31.1769 −1.53412
\(414\) 23.3205 1.14614
\(415\) 0 0
\(416\) 0 0
\(417\) −0.875644 −0.0428805
\(418\) 1.39230 0.0680999
\(419\) −17.3205 −0.846162 −0.423081 0.906092i \(-0.639051\pi\)
−0.423081 + 0.906092i \(0.639051\pi\)
\(420\) 0 0
\(421\) 2.87564 0.140150 0.0700752 0.997542i \(-0.477676\pi\)
0.0700752 + 0.997542i \(0.477676\pi\)
\(422\) 17.5885 0.856193
\(423\) −7.39230 −0.359426
\(424\) 0.464102 0.0225388
\(425\) 0 0
\(426\) −4.39230 −0.212808
\(427\) 18.5885 0.899558
\(428\) −17.6603 −0.853641
\(429\) 0 0
\(430\) 0 0
\(431\) 8.19615 0.394795 0.197397 0.980324i \(-0.436751\pi\)
0.197397 + 0.980324i \(0.436751\pi\)
\(432\) 4.00000 0.192450
\(433\) 8.39230 0.403308 0.201654 0.979457i \(-0.435368\pi\)
0.201654 + 0.979457i \(0.435368\pi\)
\(434\) −14.1962 −0.681437
\(435\) 0 0
\(436\) −8.53590 −0.408795
\(437\) −4.39230 −0.210112
\(438\) 8.53590 0.407861
\(439\) 3.41154 0.162824 0.0814120 0.996681i \(-0.474057\pi\)
0.0814120 + 0.996681i \(0.474057\pi\)
\(440\) 0 0
\(441\) −4.92820 −0.234676
\(442\) 0 0
\(443\) 22.3923 1.06389 0.531945 0.846779i \(-0.321461\pi\)
0.531945 + 0.846779i \(0.321461\pi\)
\(444\) −0.588457 −0.0279269
\(445\) 0 0
\(446\) 6.46410 0.306084
\(447\) −4.39230 −0.207749
\(448\) 3.00000 0.141737
\(449\) −15.5885 −0.735665 −0.367832 0.929892i \(-0.619900\pi\)
−0.367832 + 0.929892i \(0.619900\pi\)
\(450\) 0 0
\(451\) 31.1769 1.46806
\(452\) −6.92820 −0.325875
\(453\) −17.3205 −0.813788
\(454\) 16.3923 0.769329
\(455\) 0 0
\(456\) −0.339746 −0.0159101
\(457\) 18.3397 0.857897 0.428949 0.903329i \(-0.358884\pi\)
0.428949 + 0.903329i \(0.358884\pi\)
\(458\) −1.26795 −0.0592474
\(459\) 32.7846 1.53025
\(460\) 0 0
\(461\) −30.5885 −1.42465 −0.712323 0.701852i \(-0.752357\pi\)
−0.712323 + 0.701852i \(0.752357\pi\)
\(462\) 6.58846 0.306523
\(463\) 12.9282 0.600825 0.300412 0.953809i \(-0.402876\pi\)
0.300412 + 0.953809i \(0.402876\pi\)
\(464\) −2.53590 −0.117726
\(465\) 0 0
\(466\) −4.73205 −0.219208
\(467\) −37.8564 −1.75179 −0.875893 0.482506i \(-0.839727\pi\)
−0.875893 + 0.482506i \(0.839727\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −9.51666 −0.438505
\(472\) 10.3923 0.478345
\(473\) 6.00000 0.275880
\(474\) −3.07180 −0.141092
\(475\) 0 0
\(476\) 24.5885 1.12701
\(477\) 1.14359 0.0523616
\(478\) −2.19615 −0.100450
\(479\) −40.9808 −1.87246 −0.936229 0.351389i \(-0.885709\pi\)
−0.936229 + 0.351389i \(0.885709\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −8.66025 −0.394464
\(483\) −20.7846 −0.945732
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 15.2679 0.692568
\(487\) 15.2487 0.690985 0.345493 0.938421i \(-0.387712\pi\)
0.345493 + 0.938421i \(0.387712\pi\)
\(488\) −6.19615 −0.280487
\(489\) 7.85641 0.355279
\(490\) 0 0
\(491\) −20.6603 −0.932384 −0.466192 0.884684i \(-0.654374\pi\)
−0.466192 + 0.884684i \(0.654374\pi\)
\(492\) −7.60770 −0.342981
\(493\) −20.7846 −0.936092
\(494\) 0 0
\(495\) 0 0
\(496\) 4.73205 0.212475
\(497\) −18.0000 −0.807410
\(498\) 6.00000 0.268866
\(499\) 25.8564 1.15749 0.578746 0.815508i \(-0.303542\pi\)
0.578746 + 0.815508i \(0.303542\pi\)
\(500\) 0 0
\(501\) 2.19615 0.0981169
\(502\) 17.1962 0.767502
\(503\) 26.6603 1.18872 0.594361 0.804198i \(-0.297405\pi\)
0.594361 + 0.804198i \(0.297405\pi\)
\(504\) 7.39230 0.329279
\(505\) 0 0
\(506\) −28.3923 −1.26219
\(507\) 0 0
\(508\) 10.8038 0.479343
\(509\) 22.3923 0.992521 0.496261 0.868174i \(-0.334706\pi\)
0.496261 + 0.868174i \(0.334706\pi\)
\(510\) 0 0
\(511\) 34.9808 1.54746
\(512\) −1.00000 −0.0441942
\(513\) −1.85641 −0.0819623
\(514\) 13.8564 0.611180
\(515\) 0 0
\(516\) −1.46410 −0.0644535
\(517\) 9.00000 0.395820
\(518\) −2.41154 −0.105957
\(519\) 10.9808 0.482002
\(520\) 0 0
\(521\) 6.46410 0.283197 0.141599 0.989924i \(-0.454776\pi\)
0.141599 + 0.989924i \(0.454776\pi\)
\(522\) −6.24871 −0.273499
\(523\) 2.39230 0.104608 0.0523041 0.998631i \(-0.483343\pi\)
0.0523041 + 0.998631i \(0.483343\pi\)
\(524\) 6.12436 0.267544
\(525\) 0 0
\(526\) 3.58846 0.156464
\(527\) 38.7846 1.68948
\(528\) −2.19615 −0.0955753
\(529\) 66.5692 2.89431
\(530\) 0 0
\(531\) 25.6077 1.11128
\(532\) −1.39230 −0.0603641
\(533\) 0 0
\(534\) 4.98076 0.215539
\(535\) 0 0
\(536\) 0 0
\(537\) −6.92820 −0.298974
\(538\) 27.1244 1.16941
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −22.0526 −0.948114 −0.474057 0.880494i \(-0.657211\pi\)
−0.474057 + 0.880494i \(0.657211\pi\)
\(542\) −9.46410 −0.406518
\(543\) 10.6795 0.458301
\(544\) −8.19615 −0.351407
\(545\) 0 0
\(546\) 0 0
\(547\) 6.78461 0.290089 0.145044 0.989425i \(-0.453667\pi\)
0.145044 + 0.989425i \(0.453667\pi\)
\(548\) 2.19615 0.0938150
\(549\) −15.2679 −0.651620
\(550\) 0 0
\(551\) 1.17691 0.0501382
\(552\) 6.92820 0.294884
\(553\) −12.5885 −0.535316
\(554\) 1.00000 0.0424859
\(555\) 0 0
\(556\) 1.19615 0.0507282
\(557\) 6.80385 0.288288 0.144144 0.989557i \(-0.453957\pi\)
0.144144 + 0.989557i \(0.453957\pi\)
\(558\) 11.6603 0.493618
\(559\) 0 0
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) −10.3923 −0.438373
\(563\) 14.5359 0.612615 0.306308 0.951933i \(-0.400906\pi\)
0.306308 + 0.951933i \(0.400906\pi\)
\(564\) −2.19615 −0.0924747
\(565\) 0 0
\(566\) 9.60770 0.403842
\(567\) 13.3923 0.562424
\(568\) 6.00000 0.251754
\(569\) 27.2487 1.14233 0.571163 0.820837i \(-0.306493\pi\)
0.571163 + 0.820837i \(0.306493\pi\)
\(570\) 0 0
\(571\) 38.3731 1.60586 0.802931 0.596071i \(-0.203272\pi\)
0.802931 + 0.596071i \(0.203272\pi\)
\(572\) 0 0
\(573\) 16.6410 0.695188
\(574\) −31.1769 −1.30130
\(575\) 0 0
\(576\) −2.46410 −0.102671
\(577\) 26.4449 1.10091 0.550457 0.834863i \(-0.314453\pi\)
0.550457 + 0.834863i \(0.314453\pi\)
\(578\) −50.1769 −2.08708
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) 24.5885 1.02010
\(582\) −6.67949 −0.276874
\(583\) −1.39230 −0.0576634
\(584\) −11.6603 −0.482505
\(585\) 0 0
\(586\) −11.1962 −0.462509
\(587\) 30.0000 1.23823 0.619116 0.785299i \(-0.287491\pi\)
0.619116 + 0.785299i \(0.287491\pi\)
\(588\) −1.46410 −0.0603785
\(589\) −2.19615 −0.0904909
\(590\) 0 0
\(591\) −15.8038 −0.650083
\(592\) 0.803848 0.0330379
\(593\) −20.7846 −0.853522 −0.426761 0.904365i \(-0.640345\pi\)
−0.426761 + 0.904365i \(0.640345\pi\)
\(594\) −12.0000 −0.492366
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −4.67949 −0.191519
\(598\) 0 0
\(599\) −19.8564 −0.811311 −0.405655 0.914026i \(-0.632957\pi\)
−0.405655 + 0.914026i \(0.632957\pi\)
\(600\) 0 0
\(601\) −37.7846 −1.54127 −0.770633 0.637279i \(-0.780060\pi\)
−0.770633 + 0.637279i \(0.780060\pi\)
\(602\) −6.00000 −0.244542
\(603\) 0 0
\(604\) 23.6603 0.962722
\(605\) 0 0
\(606\) −7.85641 −0.319145
\(607\) 37.5885 1.52567 0.762834 0.646594i \(-0.223807\pi\)
0.762834 + 0.646594i \(0.223807\pi\)
\(608\) 0.464102 0.0188218
\(609\) 5.56922 0.225676
\(610\) 0 0
\(611\) 0 0
\(612\) −20.1962 −0.816381
\(613\) 38.9090 1.57152 0.785759 0.618533i \(-0.212273\pi\)
0.785759 + 0.618533i \(0.212273\pi\)
\(614\) 27.4641 1.10836
\(615\) 0 0
\(616\) −9.00000 −0.362620
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −6.73205 −0.270803
\(619\) 31.3923 1.26176 0.630882 0.775879i \(-0.282693\pi\)
0.630882 + 0.775879i \(0.282693\pi\)
\(620\) 0 0
\(621\) 37.8564 1.51913
\(622\) −9.12436 −0.365853
\(623\) 20.4115 0.817771
\(624\) 0 0
\(625\) 0 0
\(626\) 26.3923 1.05485
\(627\) 1.01924 0.0407044
\(628\) 13.0000 0.518756
\(629\) 6.58846 0.262699
\(630\) 0 0
\(631\) 20.7846 0.827422 0.413711 0.910408i \(-0.364232\pi\)
0.413711 + 0.910408i \(0.364232\pi\)
\(632\) 4.19615 0.166914
\(633\) 12.8756 0.511761
\(634\) −23.1962 −0.921237
\(635\) 0 0
\(636\) 0.339746 0.0134718
\(637\) 0 0
\(638\) 7.60770 0.301192
\(639\) 14.7846 0.584870
\(640\) 0 0
\(641\) −32.0718 −1.26676 −0.633380 0.773841i \(-0.718333\pi\)
−0.633380 + 0.773841i \(0.718333\pi\)
\(642\) −12.9282 −0.510235
\(643\) 10.7321 0.423231 0.211615 0.977353i \(-0.432128\pi\)
0.211615 + 0.977353i \(0.432128\pi\)
\(644\) 28.3923 1.11881
\(645\) 0 0
\(646\) 3.80385 0.149660
\(647\) −0.803848 −0.0316025 −0.0158013 0.999875i \(-0.505030\pi\)
−0.0158013 + 0.999875i \(0.505030\pi\)
\(648\) −4.46410 −0.175366
\(649\) −31.1769 −1.22380
\(650\) 0 0
\(651\) −10.3923 −0.407307
\(652\) −10.7321 −0.420300
\(653\) −1.39230 −0.0544851 −0.0272425 0.999629i \(-0.508673\pi\)
−0.0272425 + 0.999629i \(0.508673\pi\)
\(654\) −6.24871 −0.244344
\(655\) 0 0
\(656\) 10.3923 0.405751
\(657\) −28.7321 −1.12094
\(658\) −9.00000 −0.350857
\(659\) −5.32051 −0.207258 −0.103629 0.994616i \(-0.533045\pi\)
−0.103629 + 0.994616i \(0.533045\pi\)
\(660\) 0 0
\(661\) 35.9090 1.39670 0.698348 0.715758i \(-0.253919\pi\)
0.698348 + 0.715758i \(0.253919\pi\)
\(662\) −12.9282 −0.502469
\(663\) 0 0
\(664\) −8.19615 −0.318072
\(665\) 0 0
\(666\) 1.98076 0.0767530
\(667\) −24.0000 −0.929284
\(668\) −3.00000 −0.116073
\(669\) 4.73205 0.182952
\(670\) 0 0
\(671\) 18.5885 0.717599
\(672\) 2.19615 0.0847184
\(673\) 24.3923 0.940254 0.470127 0.882599i \(-0.344208\pi\)
0.470127 + 0.882599i \(0.344208\pi\)
\(674\) −6.19615 −0.238667
\(675\) 0 0
\(676\) 0 0
\(677\) 25.8564 0.993742 0.496871 0.867824i \(-0.334482\pi\)
0.496871 + 0.867824i \(0.334482\pi\)
\(678\) −5.07180 −0.194781
\(679\) −27.3731 −1.05048
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) −14.1962 −0.543599
\(683\) 31.1769 1.19295 0.596476 0.802631i \(-0.296567\pi\)
0.596476 + 0.802631i \(0.296567\pi\)
\(684\) 1.14359 0.0437264
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) −0.928203 −0.0354132
\(688\) 2.00000 0.0762493
\(689\) 0 0
\(690\) 0 0
\(691\) 36.7128 1.39662 0.698311 0.715795i \(-0.253935\pi\)
0.698311 + 0.715795i \(0.253935\pi\)
\(692\) −15.0000 −0.570214
\(693\) −22.1769 −0.842431
\(694\) −28.7321 −1.09065
\(695\) 0 0
\(696\) −1.85641 −0.0703669
\(697\) 85.1769 3.22631
\(698\) 4.05256 0.153392
\(699\) −3.46410 −0.131024
\(700\) 0 0
\(701\) −35.9090 −1.35626 −0.678131 0.734941i \(-0.737210\pi\)
−0.678131 + 0.734941i \(0.737210\pi\)
\(702\) 0 0
\(703\) −0.373067 −0.0140705
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −9.80385 −0.368973
\(707\) −32.1962 −1.21086
\(708\) 7.60770 0.285915
\(709\) −10.9808 −0.412391 −0.206196 0.978511i \(-0.566108\pi\)
−0.206196 + 0.978511i \(0.566108\pi\)
\(710\) 0 0
\(711\) 10.3397 0.387771
\(712\) −6.80385 −0.254985
\(713\) 44.7846 1.67720
\(714\) 18.0000 0.673633
\(715\) 0 0
\(716\) 9.46410 0.353690
\(717\) −1.60770 −0.0600405
\(718\) 1.60770 0.0599987
\(719\) −25.8564 −0.964281 −0.482141 0.876094i \(-0.660141\pi\)
−0.482141 + 0.876094i \(0.660141\pi\)
\(720\) 0 0
\(721\) −27.5885 −1.02745
\(722\) 18.7846 0.699091
\(723\) −6.33975 −0.235778
\(724\) −14.5885 −0.542176
\(725\) 0 0
\(726\) −1.46410 −0.0543379
\(727\) 34.3731 1.27483 0.637413 0.770522i \(-0.280004\pi\)
0.637413 + 0.770522i \(0.280004\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) 16.3923 0.606291
\(732\) −4.53590 −0.167652
\(733\) −14.9090 −0.550675 −0.275338 0.961348i \(-0.588790\pi\)
−0.275338 + 0.961348i \(0.588790\pi\)
\(734\) −5.60770 −0.206984
\(735\) 0 0
\(736\) −9.46410 −0.348851
\(737\) 0 0
\(738\) 25.6077 0.942632
\(739\) 32.5692 1.19808 0.599039 0.800720i \(-0.295549\pi\)
0.599039 + 0.800720i \(0.295549\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.39230 0.0511131
\(743\) −13.6077 −0.499218 −0.249609 0.968347i \(-0.580302\pi\)
−0.249609 + 0.968347i \(0.580302\pi\)
\(744\) 3.46410 0.127000
\(745\) 0 0
\(746\) −0.392305 −0.0143633
\(747\) −20.1962 −0.738939
\(748\) 24.5885 0.899043
\(749\) −52.9808 −1.93587
\(750\) 0 0
\(751\) 20.3923 0.744126 0.372063 0.928208i \(-0.378651\pi\)
0.372063 + 0.928208i \(0.378651\pi\)
\(752\) 3.00000 0.109399
\(753\) 12.5885 0.458749
\(754\) 0 0
\(755\) 0 0
\(756\) 12.0000 0.436436
\(757\) 37.7846 1.37330 0.686652 0.726986i \(-0.259079\pi\)
0.686652 + 0.726986i \(0.259079\pi\)
\(758\) 29.7846 1.08183
\(759\) −20.7846 −0.754434
\(760\) 0 0
\(761\) −18.8038 −0.681639 −0.340819 0.940129i \(-0.610704\pi\)
−0.340819 + 0.940129i \(0.610704\pi\)
\(762\) 7.90897 0.286512
\(763\) −25.6077 −0.927061
\(764\) −22.7321 −0.822417
\(765\) 0 0
\(766\) 22.3923 0.809067
\(767\) 0 0
\(768\) −0.732051 −0.0264156
\(769\) 38.1051 1.37411 0.687053 0.726607i \(-0.258904\pi\)
0.687053 + 0.726607i \(0.258904\pi\)
\(770\) 0 0
\(771\) 10.1436 0.365313
\(772\) 16.3923 0.589972
\(773\) −2.41154 −0.0867372 −0.0433686 0.999059i \(-0.513809\pi\)
−0.0433686 + 0.999059i \(0.513809\pi\)
\(774\) 4.92820 0.177141
\(775\) 0 0
\(776\) 9.12436 0.327545
\(777\) −1.76537 −0.0633324
\(778\) 22.7321 0.814984
\(779\) −4.82309 −0.172805
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) −77.5692 −2.77387
\(783\) −10.1436 −0.362502
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 4.48334 0.159915
\(787\) −42.8372 −1.52698 −0.763490 0.645820i \(-0.776516\pi\)
−0.763490 + 0.645820i \(0.776516\pi\)
\(788\) 21.5885 0.769057
\(789\) 2.62693 0.0935213
\(790\) 0 0
\(791\) −20.7846 −0.739016
\(792\) 7.39230 0.262674
\(793\) 0 0
\(794\) 11.1962 0.397337
\(795\) 0 0
\(796\) 6.39230 0.226569
\(797\) −12.9282 −0.457940 −0.228970 0.973433i \(-0.573536\pi\)
−0.228970 + 0.973433i \(0.573536\pi\)
\(798\) −1.01924 −0.0360806
\(799\) 24.5885 0.869877
\(800\) 0 0
\(801\) −16.7654 −0.592375
\(802\) 5.19615 0.183483
\(803\) 34.9808 1.23444
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19.8564 0.698979
\(808\) 10.7321 0.377552
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) 43.3923 1.52371 0.761855 0.647748i \(-0.224289\pi\)
0.761855 + 0.647748i \(0.224289\pi\)
\(812\) −7.60770 −0.266978
\(813\) −6.92820 −0.242983
\(814\) −2.41154 −0.0845245
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) −0.928203 −0.0324737
\(818\) 16.2679 0.568796
\(819\) 0 0
\(820\) 0 0
\(821\) −20.1962 −0.704851 −0.352425 0.935840i \(-0.614643\pi\)
−0.352425 + 0.935840i \(0.614643\pi\)
\(822\) 1.60770 0.0560748
\(823\) −27.1962 −0.947998 −0.473999 0.880525i \(-0.657190\pi\)
−0.473999 + 0.880525i \(0.657190\pi\)
\(824\) 9.19615 0.320363
\(825\) 0 0
\(826\) 31.1769 1.08478
\(827\) −42.5885 −1.48095 −0.740473 0.672086i \(-0.765398\pi\)
−0.740473 + 0.672086i \(0.765398\pi\)
\(828\) −23.3205 −0.810444
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 0 0
\(831\) 0.732051 0.0253946
\(832\) 0 0
\(833\) 16.3923 0.567960
\(834\) 0.875644 0.0303211
\(835\) 0 0
\(836\) −1.39230 −0.0481539
\(837\) 18.9282 0.654254
\(838\) 17.3205 0.598327
\(839\) −21.8038 −0.752752 −0.376376 0.926467i \(-0.622830\pi\)
−0.376376 + 0.926467i \(0.622830\pi\)
\(840\) 0 0
\(841\) −22.5692 −0.778249
\(842\) −2.87564 −0.0991012
\(843\) −7.60770 −0.262023
\(844\) −17.5885 −0.605420
\(845\) 0 0
\(846\) 7.39230 0.254153
\(847\) −6.00000 −0.206162
\(848\) −0.464102 −0.0159373
\(849\) 7.03332 0.241383
\(850\) 0 0
\(851\) 7.60770 0.260788
\(852\) 4.39230 0.150478
\(853\) 13.8564 0.474434 0.237217 0.971457i \(-0.423765\pi\)
0.237217 + 0.971457i \(0.423765\pi\)
\(854\) −18.5885 −0.636084
\(855\) 0 0
\(856\) 17.6603 0.603615
\(857\) −16.7321 −0.571556 −0.285778 0.958296i \(-0.592252\pi\)
−0.285778 + 0.958296i \(0.592252\pi\)
\(858\) 0 0
\(859\) −20.3731 −0.695120 −0.347560 0.937658i \(-0.612990\pi\)
−0.347560 + 0.937658i \(0.612990\pi\)
\(860\) 0 0
\(861\) −22.8231 −0.777809
\(862\) −8.19615 −0.279162
\(863\) −43.1769 −1.46976 −0.734880 0.678198i \(-0.762761\pi\)
−0.734880 + 0.678198i \(0.762761\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) −8.39230 −0.285182
\(867\) −36.7321 −1.24749
\(868\) 14.1962 0.481849
\(869\) −12.5885 −0.427034
\(870\) 0 0
\(871\) 0 0
\(872\) 8.53590 0.289062
\(873\) 22.4833 0.760946
\(874\) 4.39230 0.148572
\(875\) 0 0
\(876\) −8.53590 −0.288401
\(877\) 27.4641 0.927397 0.463698 0.885993i \(-0.346522\pi\)
0.463698 + 0.885993i \(0.346522\pi\)
\(878\) −3.41154 −0.115134
\(879\) −8.19615 −0.276449
\(880\) 0 0
\(881\) −26.3205 −0.886760 −0.443380 0.896334i \(-0.646221\pi\)
−0.443380 + 0.896334i \(0.646221\pi\)
\(882\) 4.92820 0.165941
\(883\) 4.58846 0.154414 0.0772069 0.997015i \(-0.475400\pi\)
0.0772069 + 0.997015i \(0.475400\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −22.3923 −0.752284
\(887\) 40.7654 1.36877 0.684384 0.729122i \(-0.260071\pi\)
0.684384 + 0.729122i \(0.260071\pi\)
\(888\) 0.588457 0.0197473
\(889\) 32.4115 1.08705
\(890\) 0 0
\(891\) 13.3923 0.448659
\(892\) −6.46410 −0.216434
\(893\) −1.39230 −0.0465917
\(894\) 4.39230 0.146901
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) 15.5885 0.520194
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −3.80385 −0.126725
\(902\) −31.1769 −1.03808
\(903\) −4.39230 −0.146167
\(904\) 6.92820 0.230429
\(905\) 0 0
\(906\) 17.3205 0.575435
\(907\) 4.58846 0.152357 0.0761786 0.997094i \(-0.475728\pi\)
0.0761786 + 0.997094i \(0.475728\pi\)
\(908\) −16.3923 −0.543998
\(909\) 26.4449 0.877121
\(910\) 0 0
\(911\) −21.4641 −0.711137 −0.355569 0.934650i \(-0.615713\pi\)
−0.355569 + 0.934650i \(0.615713\pi\)
\(912\) 0.339746 0.0112501
\(913\) 24.5885 0.813759
\(914\) −18.3397 −0.606625
\(915\) 0 0
\(916\) 1.26795 0.0418942
\(917\) 18.3731 0.606732
\(918\) −32.7846 −1.08205
\(919\) 30.7846 1.01549 0.507745 0.861507i \(-0.330479\pi\)
0.507745 + 0.861507i \(0.330479\pi\)
\(920\) 0 0
\(921\) 20.1051 0.662486
\(922\) 30.5885 1.00738
\(923\) 0 0
\(924\) −6.58846 −0.216744
\(925\) 0 0
\(926\) −12.9282 −0.424847
\(927\) 22.6603 0.744260
\(928\) 2.53590 0.0832449
\(929\) −3.21539 −0.105494 −0.0527468 0.998608i \(-0.516798\pi\)
−0.0527468 + 0.998608i \(0.516798\pi\)
\(930\) 0 0
\(931\) −0.928203 −0.0304206
\(932\) 4.73205 0.155003
\(933\) −6.67949 −0.218677
\(934\) 37.8564 1.23870
\(935\) 0 0
\(936\) 0 0
\(937\) 9.60770 0.313870 0.156935 0.987609i \(-0.449839\pi\)
0.156935 + 0.987609i \(0.449839\pi\)
\(938\) 0 0
\(939\) 19.3205 0.630501
\(940\) 0 0
\(941\) 3.21539 0.104819 0.0524094 0.998626i \(-0.483310\pi\)
0.0524094 + 0.998626i \(0.483310\pi\)
\(942\) 9.51666 0.310070
\(943\) 98.3538 3.20284
\(944\) −10.3923 −0.338241
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) −15.3731 −0.499558 −0.249779 0.968303i \(-0.580358\pi\)
−0.249779 + 0.968303i \(0.580358\pi\)
\(948\) 3.07180 0.0997673
\(949\) 0 0
\(950\) 0 0
\(951\) −16.9808 −0.550639
\(952\) −24.5885 −0.796916
\(953\) −6.58846 −0.213421 −0.106711 0.994290i \(-0.534032\pi\)
−0.106711 + 0.994290i \(0.534032\pi\)
\(954\) −1.14359 −0.0370252
\(955\) 0 0
\(956\) 2.19615 0.0710286
\(957\) 5.56922 0.180027
\(958\) 40.9808 1.32403
\(959\) 6.58846 0.212752
\(960\) 0 0
\(961\) −8.60770 −0.277668
\(962\) 0 0
\(963\) 43.5167 1.40230
\(964\) 8.66025 0.278928
\(965\) 0 0
\(966\) 20.7846 0.668734
\(967\) 44.5692 1.43325 0.716625 0.697459i \(-0.245686\pi\)
0.716625 + 0.697459i \(0.245686\pi\)
\(968\) 2.00000 0.0642824
\(969\) 2.78461 0.0894546
\(970\) 0 0
\(971\) 46.7654 1.50077 0.750386 0.661000i \(-0.229868\pi\)
0.750386 + 0.661000i \(0.229868\pi\)
\(972\) −15.2679 −0.489720
\(973\) 3.58846 0.115041
\(974\) −15.2487 −0.488600
\(975\) 0 0
\(976\) 6.19615 0.198334
\(977\) 16.3923 0.524436 0.262218 0.965009i \(-0.415546\pi\)
0.262218 + 0.965009i \(0.415546\pi\)
\(978\) −7.85641 −0.251220
\(979\) 20.4115 0.652356
\(980\) 0 0
\(981\) 21.0333 0.671542
\(982\) 20.6603 0.659295
\(983\) 44.5692 1.42154 0.710769 0.703426i \(-0.248347\pi\)
0.710769 + 0.703426i \(0.248347\pi\)
\(984\) 7.60770 0.242524
\(985\) 0 0
\(986\) 20.7846 0.661917
\(987\) −6.58846 −0.209713
\(988\) 0 0
\(989\) 18.9282 0.601882
\(990\) 0 0
\(991\) −11.1769 −0.355046 −0.177523 0.984117i \(-0.556808\pi\)
−0.177523 + 0.984117i \(0.556808\pi\)
\(992\) −4.73205 −0.150243
\(993\) −9.46410 −0.300334
\(994\) 18.0000 0.570925
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) −40.5692 −1.28484 −0.642420 0.766353i \(-0.722070\pi\)
−0.642420 + 0.766353i \(0.722070\pi\)
\(998\) −25.8564 −0.818470
\(999\) 3.21539 0.101730
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.1.1 2
5.4 even 2 1690.2.a.m.1.2 2
13.6 odd 12 650.2.m.a.101.2 4
13.11 odd 12 650.2.m.a.251.2 4
13.12 even 2 8450.2.a.bm.1.1 2
65.4 even 6 1690.2.e.n.991.1 4
65.9 even 6 1690.2.e.l.991.1 4
65.19 odd 12 130.2.l.a.101.1 4
65.24 odd 12 130.2.l.a.121.1 yes 4
65.29 even 6 1690.2.e.l.191.1 4
65.32 even 12 650.2.n.b.49.1 4
65.34 odd 4 1690.2.d.f.1351.4 4
65.37 even 12 650.2.n.a.199.2 4
65.44 odd 4 1690.2.d.f.1351.2 4
65.49 even 6 1690.2.e.n.191.1 4
65.54 odd 12 1690.2.l.g.1161.2 4
65.58 even 12 650.2.n.a.49.2 4
65.59 odd 12 1690.2.l.g.361.2 4
65.63 even 12 650.2.n.b.199.1 4
65.64 even 2 1690.2.a.j.1.2 2
195.89 even 12 1170.2.bs.c.901.2 4
195.149 even 12 1170.2.bs.c.361.2 4
260.19 even 12 1040.2.da.a.881.2 4
260.219 even 12 1040.2.da.a.641.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
130.2.l.a.101.1 4 65.19 odd 12
130.2.l.a.121.1 yes 4 65.24 odd 12
650.2.m.a.101.2 4 13.6 odd 12
650.2.m.a.251.2 4 13.11 odd 12
650.2.n.a.49.2 4 65.58 even 12
650.2.n.a.199.2 4 65.37 even 12
650.2.n.b.49.1 4 65.32 even 12
650.2.n.b.199.1 4 65.63 even 12
1040.2.da.a.641.2 4 260.219 even 12
1040.2.da.a.881.2 4 260.19 even 12
1170.2.bs.c.361.2 4 195.149 even 12
1170.2.bs.c.901.2 4 195.89 even 12
1690.2.a.j.1.2 2 65.64 even 2
1690.2.a.m.1.2 2 5.4 even 2
1690.2.d.f.1351.2 4 65.44 odd 4
1690.2.d.f.1351.4 4 65.34 odd 4
1690.2.e.l.191.1 4 65.29 even 6
1690.2.e.l.991.1 4 65.9 even 6
1690.2.e.n.191.1 4 65.49 even 6
1690.2.e.n.991.1 4 65.4 even 6
1690.2.l.g.361.2 4 65.59 odd 12
1690.2.l.g.1161.2 4 65.54 odd 12
8450.2.a.bf.1.1 2 1.1 even 1 trivial
8450.2.a.bm.1.1 2 13.12 even 2