Properties

Label 5850.2.a.cj.1.1
Level $5850$
Weight $2$
Character 5850.1
Self dual yes
Analytic conductor $46.712$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.7124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1950)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 5850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -4.70156 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -4.70156 q^{7} +1.00000 q^{8} -4.70156 q^{11} +1.00000 q^{13} -4.70156 q^{14} +1.00000 q^{16} -0.701562 q^{17} -1.70156 q^{19} -4.70156 q^{22} +1.00000 q^{26} -4.70156 q^{28} +6.40312 q^{29} -10.1047 q^{31} +1.00000 q^{32} -0.701562 q^{34} +1.70156 q^{37} -1.70156 q^{38} +3.70156 q^{41} +11.4031 q^{43} -4.70156 q^{44} +7.00000 q^{47} +15.1047 q^{49} +1.00000 q^{52} +2.40312 q^{53} -4.70156 q^{56} +6.40312 q^{58} -2.70156 q^{59} +14.1047 q^{61} -10.1047 q^{62} +1.00000 q^{64} +6.40312 q^{67} -0.701562 q^{68} +1.70156 q^{71} +12.0000 q^{73} +1.70156 q^{74} -1.70156 q^{76} +22.1047 q^{77} -5.70156 q^{79} +3.70156 q^{82} -10.7016 q^{83} +11.4031 q^{86} -4.70156 q^{88} -11.4031 q^{89} -4.70156 q^{91} +7.00000 q^{94} +2.59688 q^{97} +15.1047 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 3 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 3 q^{7} + 2 q^{8} - 3 q^{11} + 2 q^{13} - 3 q^{14} + 2 q^{16} + 5 q^{17} + 3 q^{19} - 3 q^{22} + 2 q^{26} - 3 q^{28} - q^{31} + 2 q^{32} + 5 q^{34} - 3 q^{37} + 3 q^{38} + q^{41} + 10 q^{43} - 3 q^{44} + 14 q^{47} + 11 q^{49} + 2 q^{52} - 8 q^{53} - 3 q^{56} + q^{59} + 9 q^{61} - q^{62} + 2 q^{64} + 5 q^{68} - 3 q^{71} + 24 q^{73} - 3 q^{74} + 3 q^{76} + 25 q^{77} - 5 q^{79} + q^{82} - 15 q^{83} + 10 q^{86} - 3 q^{88} - 10 q^{89} - 3 q^{91} + 14 q^{94} + 18 q^{97} + 11 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −4.70156 −1.77702 −0.888512 0.458854i \(-0.848260\pi\)
−0.888512 + 0.458854i \(0.848260\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −4.70156 −1.41757 −0.708787 0.705422i \(-0.750757\pi\)
−0.708787 + 0.705422i \(0.750757\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −4.70156 −1.25655
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.701562 −0.170154 −0.0850769 0.996374i \(-0.527114\pi\)
−0.0850769 + 0.996374i \(0.527114\pi\)
\(18\) 0 0
\(19\) −1.70156 −0.390365 −0.195183 0.980767i \(-0.562530\pi\)
−0.195183 + 0.980767i \(0.562530\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.70156 −1.00238
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −4.70156 −0.888512
\(29\) 6.40312 1.18903 0.594515 0.804084i \(-0.297344\pi\)
0.594515 + 0.804084i \(0.297344\pi\)
\(30\) 0 0
\(31\) −10.1047 −1.81486 −0.907428 0.420208i \(-0.861957\pi\)
−0.907428 + 0.420208i \(0.861957\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.701562 −0.120317
\(35\) 0 0
\(36\) 0 0
\(37\) 1.70156 0.279735 0.139868 0.990170i \(-0.455332\pi\)
0.139868 + 0.990170i \(0.455332\pi\)
\(38\) −1.70156 −0.276030
\(39\) 0 0
\(40\) 0 0
\(41\) 3.70156 0.578087 0.289043 0.957316i \(-0.406663\pi\)
0.289043 + 0.957316i \(0.406663\pi\)
\(42\) 0 0
\(43\) 11.4031 1.73896 0.869480 0.493968i \(-0.164454\pi\)
0.869480 + 0.493968i \(0.164454\pi\)
\(44\) −4.70156 −0.708787
\(45\) 0 0
\(46\) 0 0
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 0 0
\(49\) 15.1047 2.15781
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 2.40312 0.330095 0.165047 0.986286i \(-0.447222\pi\)
0.165047 + 0.986286i \(0.447222\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.70156 −0.628273
\(57\) 0 0
\(58\) 6.40312 0.840771
\(59\) −2.70156 −0.351713 −0.175857 0.984416i \(-0.556270\pi\)
−0.175857 + 0.984416i \(0.556270\pi\)
\(60\) 0 0
\(61\) 14.1047 1.80592 0.902960 0.429725i \(-0.141389\pi\)
0.902960 + 0.429725i \(0.141389\pi\)
\(62\) −10.1047 −1.28330
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.40312 0.782266 0.391133 0.920334i \(-0.372083\pi\)
0.391133 + 0.920334i \(0.372083\pi\)
\(68\) −0.701562 −0.0850769
\(69\) 0 0
\(70\) 0 0
\(71\) 1.70156 0.201938 0.100969 0.994890i \(-0.467806\pi\)
0.100969 + 0.994890i \(0.467806\pi\)
\(72\) 0 0
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 1.70156 0.197803
\(75\) 0 0
\(76\) −1.70156 −0.195183
\(77\) 22.1047 2.51906
\(78\) 0 0
\(79\) −5.70156 −0.641476 −0.320738 0.947168i \(-0.603931\pi\)
−0.320738 + 0.947168i \(0.603931\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.70156 0.408769
\(83\) −10.7016 −1.17465 −0.587325 0.809352i \(-0.699819\pi\)
−0.587325 + 0.809352i \(0.699819\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 11.4031 1.22963
\(87\) 0 0
\(88\) −4.70156 −0.501188
\(89\) −11.4031 −1.20873 −0.604364 0.796708i \(-0.706573\pi\)
−0.604364 + 0.796708i \(0.706573\pi\)
\(90\) 0 0
\(91\) −4.70156 −0.492858
\(92\) 0 0
\(93\) 0 0
\(94\) 7.00000 0.721995
\(95\) 0 0
\(96\) 0 0
\(97\) 2.59688 0.263673 0.131836 0.991271i \(-0.457913\pi\)
0.131836 + 0.991271i \(0.457913\pi\)
\(98\) 15.1047 1.52580
\(99\) 0 0
\(100\) 0 0
\(101\) −6.70156 −0.666830 −0.333415 0.942780i \(-0.608201\pi\)
−0.333415 + 0.942780i \(0.608201\pi\)
\(102\) 0 0
\(103\) −1.40312 −0.138254 −0.0691270 0.997608i \(-0.522021\pi\)
−0.0691270 + 0.997608i \(0.522021\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 2.40312 0.233412
\(107\) 19.1047 1.84692 0.923460 0.383695i \(-0.125349\pi\)
0.923460 + 0.383695i \(0.125349\pi\)
\(108\) 0 0
\(109\) 4.29844 0.411716 0.205858 0.978582i \(-0.434002\pi\)
0.205858 + 0.978582i \(0.434002\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.70156 −0.444256
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.40312 0.594515
\(117\) 0 0
\(118\) −2.70156 −0.248699
\(119\) 3.29844 0.302367
\(120\) 0 0
\(121\) 11.1047 1.00952
\(122\) 14.1047 1.27698
\(123\) 0 0
\(124\) −10.1047 −0.907428
\(125\) 0 0
\(126\) 0 0
\(127\) 6.29844 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −22.5078 −1.96652 −0.983258 0.182217i \(-0.941673\pi\)
−0.983258 + 0.182217i \(0.941673\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 6.40312 0.553146
\(135\) 0 0
\(136\) −0.701562 −0.0601585
\(137\) 13.7016 1.17060 0.585302 0.810816i \(-0.300976\pi\)
0.585302 + 0.810816i \(0.300976\pi\)
\(138\) 0 0
\(139\) 9.40312 0.797563 0.398781 0.917046i \(-0.369433\pi\)
0.398781 + 0.917046i \(0.369433\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.70156 0.142792
\(143\) −4.70156 −0.393164
\(144\) 0 0
\(145\) 0 0
\(146\) 12.0000 0.993127
\(147\) 0 0
\(148\) 1.70156 0.139868
\(149\) −6.59688 −0.540437 −0.270219 0.962799i \(-0.587096\pi\)
−0.270219 + 0.962799i \(0.587096\pi\)
\(150\) 0 0
\(151\) −14.1047 −1.14782 −0.573912 0.818917i \(-0.694575\pi\)
−0.573912 + 0.818917i \(0.694575\pi\)
\(152\) −1.70156 −0.138015
\(153\) 0 0
\(154\) 22.1047 1.78125
\(155\) 0 0
\(156\) 0 0
\(157\) 22.7016 1.81178 0.905891 0.423511i \(-0.139203\pi\)
0.905891 + 0.423511i \(0.139203\pi\)
\(158\) −5.70156 −0.453592
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 18.8062 1.47302 0.736510 0.676427i \(-0.236473\pi\)
0.736510 + 0.676427i \(0.236473\pi\)
\(164\) 3.70156 0.289043
\(165\) 0 0
\(166\) −10.7016 −0.830602
\(167\) −1.10469 −0.0854832 −0.0427416 0.999086i \(-0.513609\pi\)
−0.0427416 + 0.999086i \(0.513609\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 11.4031 0.869480
\(173\) −0.193752 −0.0147307 −0.00736533 0.999973i \(-0.502344\pi\)
−0.00736533 + 0.999973i \(0.502344\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.70156 −0.354394
\(177\) 0 0
\(178\) −11.4031 −0.854700
\(179\) 24.2094 1.80949 0.904747 0.425950i \(-0.140060\pi\)
0.904747 + 0.425950i \(0.140060\pi\)
\(180\) 0 0
\(181\) 11.2984 0.839806 0.419903 0.907569i \(-0.362064\pi\)
0.419903 + 0.907569i \(0.362064\pi\)
\(182\) −4.70156 −0.348503
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.29844 0.241206
\(188\) 7.00000 0.510527
\(189\) 0 0
\(190\) 0 0
\(191\) 12.8062 0.926628 0.463314 0.886194i \(-0.346660\pi\)
0.463314 + 0.886194i \(0.346660\pi\)
\(192\) 0 0
\(193\) −16.2094 −1.16678 −0.583388 0.812194i \(-0.698273\pi\)
−0.583388 + 0.812194i \(0.698273\pi\)
\(194\) 2.59688 0.186445
\(195\) 0 0
\(196\) 15.1047 1.07891
\(197\) −5.40312 −0.384957 −0.192478 0.981301i \(-0.561653\pi\)
−0.192478 + 0.981301i \(0.561653\pi\)
\(198\) 0 0
\(199\) 8.29844 0.588261 0.294130 0.955765i \(-0.404970\pi\)
0.294130 + 0.955765i \(0.404970\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −6.70156 −0.471520
\(203\) −30.1047 −2.11293
\(204\) 0 0
\(205\) 0 0
\(206\) −1.40312 −0.0977603
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 6.80625 0.468561 0.234281 0.972169i \(-0.424727\pi\)
0.234281 + 0.972169i \(0.424727\pi\)
\(212\) 2.40312 0.165047
\(213\) 0 0
\(214\) 19.1047 1.30597
\(215\) 0 0
\(216\) 0 0
\(217\) 47.5078 3.22504
\(218\) 4.29844 0.291127
\(219\) 0 0
\(220\) 0 0
\(221\) −0.701562 −0.0471922
\(222\) 0 0
\(223\) −11.4031 −0.763610 −0.381805 0.924243i \(-0.624697\pi\)
−0.381805 + 0.924243i \(0.624697\pi\)
\(224\) −4.70156 −0.314136
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) −16.1047 −1.06891 −0.534453 0.845198i \(-0.679482\pi\)
−0.534453 + 0.845198i \(0.679482\pi\)
\(228\) 0 0
\(229\) 15.7016 1.03759 0.518794 0.854899i \(-0.326381\pi\)
0.518794 + 0.854899i \(0.326381\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.40312 0.420386
\(233\) −20.2094 −1.32396 −0.661980 0.749521i \(-0.730284\pi\)
−0.661980 + 0.749521i \(0.730284\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.70156 −0.175857
\(237\) 0 0
\(238\) 3.29844 0.213806
\(239\) −6.10469 −0.394879 −0.197440 0.980315i \(-0.563263\pi\)
−0.197440 + 0.980315i \(0.563263\pi\)
\(240\) 0 0
\(241\) −2.59688 −0.167279 −0.0836397 0.996496i \(-0.526654\pi\)
−0.0836397 + 0.996496i \(0.526654\pi\)
\(242\) 11.1047 0.713836
\(243\) 0 0
\(244\) 14.1047 0.902960
\(245\) 0 0
\(246\) 0 0
\(247\) −1.70156 −0.108268
\(248\) −10.1047 −0.641648
\(249\) 0 0
\(250\) 0 0
\(251\) −0.298438 −0.0188372 −0.00941862 0.999956i \(-0.502998\pi\)
−0.00941862 + 0.999956i \(0.502998\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 6.29844 0.395199
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −15.2984 −0.954290 −0.477145 0.878824i \(-0.658328\pi\)
−0.477145 + 0.878824i \(0.658328\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) −22.5078 −1.39054
\(263\) 2.00000 0.123325 0.0616626 0.998097i \(-0.480360\pi\)
0.0616626 + 0.998097i \(0.480360\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) 0 0
\(268\) 6.40312 0.391133
\(269\) −31.2094 −1.90287 −0.951435 0.307851i \(-0.900390\pi\)
−0.951435 + 0.307851i \(0.900390\pi\)
\(270\) 0 0
\(271\) 19.5078 1.18502 0.592508 0.805565i \(-0.298138\pi\)
0.592508 + 0.805565i \(0.298138\pi\)
\(272\) −0.701562 −0.0425385
\(273\) 0 0
\(274\) 13.7016 0.827742
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 9.40312 0.563962
\(279\) 0 0
\(280\) 0 0
\(281\) 21.9109 1.30710 0.653548 0.756885i \(-0.273280\pi\)
0.653548 + 0.756885i \(0.273280\pi\)
\(282\) 0 0
\(283\) −21.4031 −1.27228 −0.636142 0.771572i \(-0.719471\pi\)
−0.636142 + 0.771572i \(0.719471\pi\)
\(284\) 1.70156 0.100969
\(285\) 0 0
\(286\) −4.70156 −0.278009
\(287\) −17.4031 −1.02727
\(288\) 0 0
\(289\) −16.5078 −0.971048
\(290\) 0 0
\(291\) 0 0
\(292\) 12.0000 0.702247
\(293\) 21.4031 1.25038 0.625192 0.780471i \(-0.285021\pi\)
0.625192 + 0.780471i \(0.285021\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.70156 0.0989013
\(297\) 0 0
\(298\) −6.59688 −0.382147
\(299\) 0 0
\(300\) 0 0
\(301\) −53.6125 −3.09017
\(302\) −14.1047 −0.811633
\(303\) 0 0
\(304\) −1.70156 −0.0975913
\(305\) 0 0
\(306\) 0 0
\(307\) 5.70156 0.325405 0.162703 0.986675i \(-0.447979\pi\)
0.162703 + 0.986675i \(0.447979\pi\)
\(308\) 22.1047 1.25953
\(309\) 0 0
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) 21.2094 1.19882 0.599412 0.800440i \(-0.295401\pi\)
0.599412 + 0.800440i \(0.295401\pi\)
\(314\) 22.7016 1.28112
\(315\) 0 0
\(316\) −5.70156 −0.320738
\(317\) 1.19375 0.0670478 0.0335239 0.999438i \(-0.489327\pi\)
0.0335239 + 0.999438i \(0.489327\pi\)
\(318\) 0 0
\(319\) −30.1047 −1.68554
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.19375 0.0664221
\(324\) 0 0
\(325\) 0 0
\(326\) 18.8062 1.04158
\(327\) 0 0
\(328\) 3.70156 0.204385
\(329\) −32.9109 −1.81444
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −10.7016 −0.587325
\(333\) 0 0
\(334\) −1.10469 −0.0604457
\(335\) 0 0
\(336\) 0 0
\(337\) −8.10469 −0.441490 −0.220745 0.975332i \(-0.570849\pi\)
−0.220745 + 0.975332i \(0.570849\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) 0 0
\(341\) 47.5078 2.57269
\(342\) 0 0
\(343\) −38.1047 −2.05746
\(344\) 11.4031 0.614815
\(345\) 0 0
\(346\) −0.193752 −0.0104161
\(347\) 10.5078 0.564089 0.282044 0.959401i \(-0.408987\pi\)
0.282044 + 0.959401i \(0.408987\pi\)
\(348\) 0 0
\(349\) −11.4031 −0.610395 −0.305198 0.952289i \(-0.598723\pi\)
−0.305198 + 0.952289i \(0.598723\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.70156 −0.250594
\(353\) 14.5078 0.772173 0.386086 0.922463i \(-0.373827\pi\)
0.386086 + 0.922463i \(0.373827\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −11.4031 −0.604364
\(357\) 0 0
\(358\) 24.2094 1.27951
\(359\) 34.6125 1.82678 0.913389 0.407088i \(-0.133456\pi\)
0.913389 + 0.407088i \(0.133456\pi\)
\(360\) 0 0
\(361\) −16.1047 −0.847615
\(362\) 11.2984 0.593833
\(363\) 0 0
\(364\) −4.70156 −0.246429
\(365\) 0 0
\(366\) 0 0
\(367\) 2.29844 0.119977 0.0599887 0.998199i \(-0.480894\pi\)
0.0599887 + 0.998199i \(0.480894\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −11.2984 −0.586586
\(372\) 0 0
\(373\) −5.29844 −0.274343 −0.137171 0.990547i \(-0.543801\pi\)
−0.137171 + 0.990547i \(0.543801\pi\)
\(374\) 3.29844 0.170558
\(375\) 0 0
\(376\) 7.00000 0.360997
\(377\) 6.40312 0.329778
\(378\) 0 0
\(379\) −13.8953 −0.713754 −0.356877 0.934151i \(-0.616159\pi\)
−0.356877 + 0.934151i \(0.616159\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 12.8062 0.655225
\(383\) 32.5078 1.66107 0.830536 0.556965i \(-0.188034\pi\)
0.830536 + 0.556965i \(0.188034\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −16.2094 −0.825035
\(387\) 0 0
\(388\) 2.59688 0.131836
\(389\) 27.1047 1.37426 0.687131 0.726533i \(-0.258870\pi\)
0.687131 + 0.726533i \(0.258870\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 15.1047 0.762902
\(393\) 0 0
\(394\) −5.40312 −0.272205
\(395\) 0 0
\(396\) 0 0
\(397\) 21.9109 1.09968 0.549839 0.835271i \(-0.314689\pi\)
0.549839 + 0.835271i \(0.314689\pi\)
\(398\) 8.29844 0.415963
\(399\) 0 0
\(400\) 0 0
\(401\) −18.2094 −0.909333 −0.454666 0.890662i \(-0.650241\pi\)
−0.454666 + 0.890662i \(0.650241\pi\)
\(402\) 0 0
\(403\) −10.1047 −0.503350
\(404\) −6.70156 −0.333415
\(405\) 0 0
\(406\) −30.1047 −1.49407
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 29.4031 1.45389 0.726945 0.686695i \(-0.240939\pi\)
0.726945 + 0.686695i \(0.240939\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.40312 −0.0691270
\(413\) 12.7016 0.625003
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 8.00000 0.391293
\(419\) −9.91093 −0.484181 −0.242090 0.970254i \(-0.577833\pi\)
−0.242090 + 0.970254i \(0.577833\pi\)
\(420\) 0 0
\(421\) −0.596876 −0.0290899 −0.0145450 0.999894i \(-0.504630\pi\)
−0.0145450 + 0.999894i \(0.504630\pi\)
\(422\) 6.80625 0.331323
\(423\) 0 0
\(424\) 2.40312 0.116706
\(425\) 0 0
\(426\) 0 0
\(427\) −66.3141 −3.20916
\(428\) 19.1047 0.923460
\(429\) 0 0
\(430\) 0 0
\(431\) −35.3141 −1.70102 −0.850509 0.525960i \(-0.823706\pi\)
−0.850509 + 0.525960i \(0.823706\pi\)
\(432\) 0 0
\(433\) 11.1047 0.533657 0.266829 0.963744i \(-0.414024\pi\)
0.266829 + 0.963744i \(0.414024\pi\)
\(434\) 47.5078 2.28045
\(435\) 0 0
\(436\) 4.29844 0.205858
\(437\) 0 0
\(438\) 0 0
\(439\) −10.2984 −0.491518 −0.245759 0.969331i \(-0.579037\pi\)
−0.245759 + 0.969331i \(0.579037\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −0.701562 −0.0333699
\(443\) 32.5078 1.54449 0.772246 0.635323i \(-0.219133\pi\)
0.772246 + 0.635323i \(0.219133\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −11.4031 −0.539954
\(447\) 0 0
\(448\) −4.70156 −0.222128
\(449\) −2.29844 −0.108470 −0.0542350 0.998528i \(-0.517272\pi\)
−0.0542350 + 0.998528i \(0.517272\pi\)
\(450\) 0 0
\(451\) −17.4031 −0.819481
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) −16.1047 −0.755830
\(455\) 0 0
\(456\) 0 0
\(457\) 2.59688 0.121477 0.0607384 0.998154i \(-0.480654\pi\)
0.0607384 + 0.998154i \(0.480654\pi\)
\(458\) 15.7016 0.733686
\(459\) 0 0
\(460\) 0 0
\(461\) −36.2094 −1.68644 −0.843219 0.537570i \(-0.819342\pi\)
−0.843219 + 0.537570i \(0.819342\pi\)
\(462\) 0 0
\(463\) 21.2984 0.989822 0.494911 0.868944i \(-0.335201\pi\)
0.494911 + 0.868944i \(0.335201\pi\)
\(464\) 6.40312 0.297258
\(465\) 0 0
\(466\) −20.2094 −0.936181
\(467\) 30.2984 1.40204 0.701022 0.713139i \(-0.252727\pi\)
0.701022 + 0.713139i \(0.252727\pi\)
\(468\) 0 0
\(469\) −30.1047 −1.39011
\(470\) 0 0
\(471\) 0 0
\(472\) −2.70156 −0.124349
\(473\) −53.6125 −2.46511
\(474\) 0 0
\(475\) 0 0
\(476\) 3.29844 0.151184
\(477\) 0 0
\(478\) −6.10469 −0.279222
\(479\) 40.6125 1.85563 0.927816 0.373038i \(-0.121684\pi\)
0.927816 + 0.373038i \(0.121684\pi\)
\(480\) 0 0
\(481\) 1.70156 0.0775846
\(482\) −2.59688 −0.118284
\(483\) 0 0
\(484\) 11.1047 0.504758
\(485\) 0 0
\(486\) 0 0
\(487\) 7.89531 0.357771 0.178885 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(488\) 14.1047 0.638489
\(489\) 0 0
\(490\) 0 0
\(491\) −33.4031 −1.50746 −0.753731 0.657183i \(-0.771748\pi\)
−0.753731 + 0.657183i \(0.771748\pi\)
\(492\) 0 0
\(493\) −4.49219 −0.202318
\(494\) −1.70156 −0.0765569
\(495\) 0 0
\(496\) −10.1047 −0.453714
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 17.2094 0.770397 0.385199 0.922834i \(-0.374133\pi\)
0.385199 + 0.922834i \(0.374133\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −0.298438 −0.0133199
\(503\) −12.2094 −0.544389 −0.272195 0.962242i \(-0.587749\pi\)
−0.272195 + 0.962242i \(0.587749\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 6.29844 0.279448
\(509\) 5.40312 0.239489 0.119745 0.992805i \(-0.461792\pi\)
0.119745 + 0.992805i \(0.461792\pi\)
\(510\) 0 0
\(511\) −56.4187 −2.49582
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −15.2984 −0.674785
\(515\) 0 0
\(516\) 0 0
\(517\) −32.9109 −1.44742
\(518\) −8.00000 −0.351500
\(519\) 0 0
\(520\) 0 0
\(521\) 36.2094 1.58636 0.793181 0.608986i \(-0.208424\pi\)
0.793181 + 0.608986i \(0.208424\pi\)
\(522\) 0 0
\(523\) 24.8062 1.08470 0.542351 0.840152i \(-0.317534\pi\)
0.542351 + 0.840152i \(0.317534\pi\)
\(524\) −22.5078 −0.983258
\(525\) 0 0
\(526\) 2.00000 0.0872041
\(527\) 7.08907 0.308805
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) 3.70156 0.160332
\(534\) 0 0
\(535\) 0 0
\(536\) 6.40312 0.276573
\(537\) 0 0
\(538\) −31.2094 −1.34553
\(539\) −71.0156 −3.05886
\(540\) 0 0
\(541\) −1.79063 −0.0769851 −0.0384925 0.999259i \(-0.512256\pi\)
−0.0384925 + 0.999259i \(0.512256\pi\)
\(542\) 19.5078 0.837932
\(543\) 0 0
\(544\) −0.701562 −0.0300792
\(545\) 0 0
\(546\) 0 0
\(547\) 31.6125 1.35165 0.675826 0.737061i \(-0.263787\pi\)
0.675826 + 0.737061i \(0.263787\pi\)
\(548\) 13.7016 0.585302
\(549\) 0 0
\(550\) 0 0
\(551\) −10.8953 −0.464156
\(552\) 0 0
\(553\) 26.8062 1.13992
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) 9.40312 0.398781
\(557\) 16.8062 0.712104 0.356052 0.934466i \(-0.384123\pi\)
0.356052 + 0.934466i \(0.384123\pi\)
\(558\) 0 0
\(559\) 11.4031 0.482301
\(560\) 0 0
\(561\) 0 0
\(562\) 21.9109 0.924257
\(563\) −32.5078 −1.37004 −0.685020 0.728524i \(-0.740207\pi\)
−0.685020 + 0.728524i \(0.740207\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −21.4031 −0.899640
\(567\) 0 0
\(568\) 1.70156 0.0713960
\(569\) 11.2984 0.473655 0.236828 0.971552i \(-0.423892\pi\)
0.236828 + 0.971552i \(0.423892\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) −4.70156 −0.196582
\(573\) 0 0
\(574\) −17.4031 −0.726392
\(575\) 0 0
\(576\) 0 0
\(577\) −24.2094 −1.00785 −0.503925 0.863748i \(-0.668111\pi\)
−0.503925 + 0.863748i \(0.668111\pi\)
\(578\) −16.5078 −0.686634
\(579\) 0 0
\(580\) 0 0
\(581\) 50.3141 2.08738
\(582\) 0 0
\(583\) −11.2984 −0.467933
\(584\) 12.0000 0.496564
\(585\) 0 0
\(586\) 21.4031 0.884155
\(587\) −7.50781 −0.309881 −0.154940 0.987924i \(-0.549519\pi\)
−0.154940 + 0.987924i \(0.549519\pi\)
\(588\) 0 0
\(589\) 17.1938 0.708456
\(590\) 0 0
\(591\) 0 0
\(592\) 1.70156 0.0699338
\(593\) 17.9109 0.735514 0.367757 0.929922i \(-0.380126\pi\)
0.367757 + 0.929922i \(0.380126\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.59688 −0.270219
\(597\) 0 0
\(598\) 0 0
\(599\) −8.20937 −0.335426 −0.167713 0.985836i \(-0.553638\pi\)
−0.167713 + 0.985836i \(0.553638\pi\)
\(600\) 0 0
\(601\) 10.6125 0.432893 0.216446 0.976295i \(-0.430553\pi\)
0.216446 + 0.976295i \(0.430553\pi\)
\(602\) −53.6125 −2.18508
\(603\) 0 0
\(604\) −14.1047 −0.573912
\(605\) 0 0
\(606\) 0 0
\(607\) 35.1047 1.42486 0.712428 0.701746i \(-0.247596\pi\)
0.712428 + 0.701746i \(0.247596\pi\)
\(608\) −1.70156 −0.0690075
\(609\) 0 0
\(610\) 0 0
\(611\) 7.00000 0.283190
\(612\) 0 0
\(613\) 12.8062 0.517240 0.258620 0.965979i \(-0.416732\pi\)
0.258620 + 0.965979i \(0.416732\pi\)
\(614\) 5.70156 0.230096
\(615\) 0 0
\(616\) 22.1047 0.890623
\(617\) −27.1047 −1.09119 −0.545597 0.838048i \(-0.683697\pi\)
−0.545597 + 0.838048i \(0.683697\pi\)
\(618\) 0 0
\(619\) −13.1938 −0.530302 −0.265151 0.964207i \(-0.585422\pi\)
−0.265151 + 0.964207i \(0.585422\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −30.0000 −1.20289
\(623\) 53.6125 2.14794
\(624\) 0 0
\(625\) 0 0
\(626\) 21.2094 0.847697
\(627\) 0 0
\(628\) 22.7016 0.905891
\(629\) −1.19375 −0.0475980
\(630\) 0 0
\(631\) 33.6125 1.33809 0.669046 0.743221i \(-0.266703\pi\)
0.669046 + 0.743221i \(0.266703\pi\)
\(632\) −5.70156 −0.226796
\(633\) 0 0
\(634\) 1.19375 0.0474099
\(635\) 0 0
\(636\) 0 0
\(637\) 15.1047 0.598469
\(638\) −30.1047 −1.19186
\(639\) 0 0
\(640\) 0 0
\(641\) −5.50781 −0.217545 −0.108773 0.994067i \(-0.534692\pi\)
−0.108773 + 0.994067i \(0.534692\pi\)
\(642\) 0 0
\(643\) 10.2984 0.406131 0.203065 0.979165i \(-0.434910\pi\)
0.203065 + 0.979165i \(0.434910\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1.19375 0.0469675
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 12.7016 0.498580
\(650\) 0 0
\(651\) 0 0
\(652\) 18.8062 0.736510
\(653\) −21.2984 −0.833472 −0.416736 0.909027i \(-0.636826\pi\)
−0.416736 + 0.909027i \(0.636826\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.70156 0.144522
\(657\) 0 0
\(658\) −32.9109 −1.28300
\(659\) 19.3141 0.752369 0.376184 0.926545i \(-0.377236\pi\)
0.376184 + 0.926545i \(0.377236\pi\)
\(660\) 0 0
\(661\) −40.1203 −1.56050 −0.780250 0.625468i \(-0.784908\pi\)
−0.780250 + 0.625468i \(0.784908\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) −10.7016 −0.415301
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −1.10469 −0.0427416
\(669\) 0 0
\(670\) 0 0
\(671\) −66.3141 −2.56003
\(672\) 0 0
\(673\) −36.0156 −1.38830 −0.694150 0.719830i \(-0.744220\pi\)
−0.694150 + 0.719830i \(0.744220\pi\)
\(674\) −8.10469 −0.312181
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 31.6125 1.21497 0.607483 0.794332i \(-0.292179\pi\)
0.607483 + 0.794332i \(0.292179\pi\)
\(678\) 0 0
\(679\) −12.2094 −0.468553
\(680\) 0 0
\(681\) 0 0
\(682\) 47.5078 1.81917
\(683\) 44.7016 1.71046 0.855229 0.518251i \(-0.173417\pi\)
0.855229 + 0.518251i \(0.173417\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −38.1047 −1.45484
\(687\) 0 0
\(688\) 11.4031 0.434740
\(689\) 2.40312 0.0915517
\(690\) 0 0
\(691\) −22.1938 −0.844290 −0.422145 0.906528i \(-0.638723\pi\)
−0.422145 + 0.906528i \(0.638723\pi\)
\(692\) −0.193752 −0.00736533
\(693\) 0 0
\(694\) 10.5078 0.398871
\(695\) 0 0
\(696\) 0 0
\(697\) −2.59688 −0.0983637
\(698\) −11.4031 −0.431615
\(699\) 0 0
\(700\) 0 0
\(701\) −30.9109 −1.16749 −0.583745 0.811937i \(-0.698413\pi\)
−0.583745 + 0.811937i \(0.698413\pi\)
\(702\) 0 0
\(703\) −2.89531 −0.109199
\(704\) −4.70156 −0.177197
\(705\) 0 0
\(706\) 14.5078 0.546009
\(707\) 31.5078 1.18497
\(708\) 0 0
\(709\) 15.6125 0.586340 0.293170 0.956060i \(-0.405290\pi\)
0.293170 + 0.956060i \(0.405290\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −11.4031 −0.427350
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 24.2094 0.904747
\(717\) 0 0
\(718\) 34.6125 1.29173
\(719\) 11.0156 0.410813 0.205407 0.978677i \(-0.434148\pi\)
0.205407 + 0.978677i \(0.434148\pi\)
\(720\) 0 0
\(721\) 6.59688 0.245680
\(722\) −16.1047 −0.599354
\(723\) 0 0
\(724\) 11.2984 0.419903
\(725\) 0 0
\(726\) 0 0
\(727\) −7.79063 −0.288938 −0.144469 0.989509i \(-0.546147\pi\)
−0.144469 + 0.989509i \(0.546147\pi\)
\(728\) −4.70156 −0.174251
\(729\) 0 0
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −47.9109 −1.76963 −0.884815 0.465942i \(-0.845716\pi\)
−0.884815 + 0.465942i \(0.845716\pi\)
\(734\) 2.29844 0.0848369
\(735\) 0 0
\(736\) 0 0
\(737\) −30.1047 −1.10892
\(738\) 0 0
\(739\) 4.61250 0.169673 0.0848367 0.996395i \(-0.472963\pi\)
0.0848367 + 0.996395i \(0.472963\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −11.2984 −0.414779
\(743\) −30.6125 −1.12306 −0.561532 0.827455i \(-0.689788\pi\)
−0.561532 + 0.827455i \(0.689788\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −5.29844 −0.193990
\(747\) 0 0
\(748\) 3.29844 0.120603
\(749\) −89.8219 −3.28202
\(750\) 0 0
\(751\) 8.50781 0.310454 0.155227 0.987879i \(-0.450389\pi\)
0.155227 + 0.987879i \(0.450389\pi\)
\(752\) 7.00000 0.255264
\(753\) 0 0
\(754\) 6.40312 0.233188
\(755\) 0 0
\(756\) 0 0
\(757\) 17.8953 0.650416 0.325208 0.945642i \(-0.394566\pi\)
0.325208 + 0.945642i \(0.394566\pi\)
\(758\) −13.8953 −0.504701
\(759\) 0 0
\(760\) 0 0
\(761\) 26.7172 0.968497 0.484249 0.874930i \(-0.339093\pi\)
0.484249 + 0.874930i \(0.339093\pi\)
\(762\) 0 0
\(763\) −20.2094 −0.731628
\(764\) 12.8062 0.463314
\(765\) 0 0
\(766\) 32.5078 1.17455
\(767\) −2.70156 −0.0975478
\(768\) 0 0
\(769\) −8.00000 −0.288487 −0.144244 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −16.2094 −0.583388
\(773\) 26.5969 0.956623 0.478312 0.878190i \(-0.341249\pi\)
0.478312 + 0.878190i \(0.341249\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2.59688 0.0932224
\(777\) 0 0
\(778\) 27.1047 0.971750
\(779\) −6.29844 −0.225665
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 15.1047 0.539453
\(785\) 0 0
\(786\) 0 0
\(787\) −33.8953 −1.20824 −0.604119 0.796894i \(-0.706475\pi\)
−0.604119 + 0.796894i \(0.706475\pi\)
\(788\) −5.40312 −0.192478
\(789\) 0 0
\(790\) 0 0
\(791\) 65.8219 2.34036
\(792\) 0 0
\(793\) 14.1047 0.500872
\(794\) 21.9109 0.777590
\(795\) 0 0
\(796\) 8.29844 0.294130
\(797\) 2.91093 0.103111 0.0515553 0.998670i \(-0.483582\pi\)
0.0515553 + 0.998670i \(0.483582\pi\)
\(798\) 0 0
\(799\) −4.91093 −0.173736
\(800\) 0 0
\(801\) 0 0
\(802\) −18.2094 −0.642995
\(803\) −56.4187 −1.99097
\(804\) 0 0
\(805\) 0 0
\(806\) −10.1047 −0.355922
\(807\) 0 0
\(808\) −6.70156 −0.235760
\(809\) −7.19375 −0.252919 −0.126459 0.991972i \(-0.540361\pi\)
−0.126459 + 0.991972i \(0.540361\pi\)
\(810\) 0 0
\(811\) 48.7016 1.71014 0.855072 0.518510i \(-0.173513\pi\)
0.855072 + 0.518510i \(0.173513\pi\)
\(812\) −30.1047 −1.05647
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 0 0
\(817\) −19.4031 −0.678829
\(818\) 29.4031 1.02806
\(819\) 0 0
\(820\) 0 0
\(821\) 20.5969 0.718836 0.359418 0.933177i \(-0.382975\pi\)
0.359418 + 0.933177i \(0.382975\pi\)
\(822\) 0 0
\(823\) −17.1047 −0.596232 −0.298116 0.954530i \(-0.596358\pi\)
−0.298116 + 0.954530i \(0.596358\pi\)
\(824\) −1.40312 −0.0488801
\(825\) 0 0
\(826\) 12.7016 0.441944
\(827\) 20.7016 0.719864 0.359932 0.932979i \(-0.382800\pi\)
0.359932 + 0.932979i \(0.382800\pi\)
\(828\) 0 0
\(829\) −5.50781 −0.191294 −0.0956471 0.995415i \(-0.530492\pi\)
−0.0956471 + 0.995415i \(0.530492\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) −10.5969 −0.367160
\(834\) 0 0
\(835\) 0 0
\(836\) 8.00000 0.276686
\(837\) 0 0
\(838\) −9.91093 −0.342368
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 12.0000 0.413793
\(842\) −0.596876 −0.0205697
\(843\) 0 0
\(844\) 6.80625 0.234281
\(845\) 0 0
\(846\) 0 0
\(847\) −52.2094 −1.79394
\(848\) 2.40312 0.0825236
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −0.507811 −0.0173871 −0.00869355 0.999962i \(-0.502767\pi\)
−0.00869355 + 0.999962i \(0.502767\pi\)
\(854\) −66.3141 −2.26922
\(855\) 0 0
\(856\) 19.1047 0.652985
\(857\) −7.61250 −0.260038 −0.130019 0.991512i \(-0.541504\pi\)
−0.130019 + 0.991512i \(0.541504\pi\)
\(858\) 0 0
\(859\) −6.20937 −0.211861 −0.105931 0.994374i \(-0.533782\pi\)
−0.105931 + 0.994374i \(0.533782\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −35.3141 −1.20280
\(863\) 54.2250 1.84584 0.922920 0.384991i \(-0.125796\pi\)
0.922920 + 0.384991i \(0.125796\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 11.1047 0.377353
\(867\) 0 0
\(868\) 47.5078 1.61252
\(869\) 26.8062 0.909340
\(870\) 0 0
\(871\) 6.40312 0.216962
\(872\) 4.29844 0.145563
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 42.7172 1.44246 0.721228 0.692697i \(-0.243578\pi\)
0.721228 + 0.692697i \(0.243578\pi\)
\(878\) −10.2984 −0.347555
\(879\) 0 0
\(880\) 0 0
\(881\) 39.7172 1.33811 0.669053 0.743215i \(-0.266700\pi\)
0.669053 + 0.743215i \(0.266700\pi\)
\(882\) 0 0
\(883\) −45.6125 −1.53498 −0.767491 0.641059i \(-0.778495\pi\)
−0.767491 + 0.641059i \(0.778495\pi\)
\(884\) −0.701562 −0.0235961
\(885\) 0 0
\(886\) 32.5078 1.09212
\(887\) −20.0000 −0.671534 −0.335767 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) 0 0
\(889\) −29.6125 −0.993171
\(890\) 0 0
\(891\) 0 0
\(892\) −11.4031 −0.381805
\(893\) −11.9109 −0.398584
\(894\) 0 0
\(895\) 0 0
\(896\) −4.70156 −0.157068
\(897\) 0 0
\(898\) −2.29844 −0.0766999
\(899\) −64.7016 −2.15792
\(900\) 0 0
\(901\) −1.68594 −0.0561668
\(902\) −17.4031 −0.579461
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) 25.0156 0.830630 0.415315 0.909678i \(-0.363671\pi\)
0.415315 + 0.909678i \(0.363671\pi\)
\(908\) −16.1047 −0.534453
\(909\) 0 0
\(910\) 0 0
\(911\) −42.2094 −1.39846 −0.699229 0.714897i \(-0.746473\pi\)
−0.699229 + 0.714897i \(0.746473\pi\)
\(912\) 0 0
\(913\) 50.3141 1.66515
\(914\) 2.59688 0.0858970
\(915\) 0 0
\(916\) 15.7016 0.518794
\(917\) 105.822 3.49455
\(918\) 0 0
\(919\) −4.89531 −0.161481 −0.0807407 0.996735i \(-0.525729\pi\)
−0.0807407 + 0.996735i \(0.525729\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −36.2094 −1.19249
\(923\) 1.70156 0.0560076
\(924\) 0 0
\(925\) 0 0
\(926\) 21.2984 0.699910
\(927\) 0 0
\(928\) 6.40312 0.210193
\(929\) −52.2984 −1.71586 −0.857928 0.513770i \(-0.828249\pi\)
−0.857928 + 0.513770i \(0.828249\pi\)
\(930\) 0 0
\(931\) −25.7016 −0.842335
\(932\) −20.2094 −0.661980
\(933\) 0 0
\(934\) 30.2984 0.991395
\(935\) 0 0
\(936\) 0 0
\(937\) −14.9109 −0.487119 −0.243560 0.969886i \(-0.578315\pi\)
−0.243560 + 0.969886i \(0.578315\pi\)
\(938\) −30.1047 −0.982953
\(939\) 0 0
\(940\) 0 0
\(941\) −28.4187 −0.926425 −0.463212 0.886247i \(-0.653303\pi\)
−0.463212 + 0.886247i \(0.653303\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −2.70156 −0.0879284
\(945\) 0 0
\(946\) −53.6125 −1.74309
\(947\) 47.5078 1.54380 0.771898 0.635746i \(-0.219307\pi\)
0.771898 + 0.635746i \(0.219307\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 3.29844 0.106903
\(953\) −31.2984 −1.01386 −0.506928 0.861988i \(-0.669219\pi\)
−0.506928 + 0.861988i \(0.669219\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −6.10469 −0.197440
\(957\) 0 0
\(958\) 40.6125 1.31213
\(959\) −64.4187 −2.08019
\(960\) 0 0
\(961\) 71.1047 2.29370
\(962\) 1.70156 0.0548606
\(963\) 0 0
\(964\) −2.59688 −0.0836397
\(965\) 0 0
\(966\) 0 0
\(967\) −29.8953 −0.961368 −0.480684 0.876894i \(-0.659612\pi\)
−0.480684 + 0.876894i \(0.659612\pi\)
\(968\) 11.1047 0.356918
\(969\) 0 0
\(970\) 0 0
\(971\) 36.8953 1.18403 0.592013 0.805928i \(-0.298333\pi\)
0.592013 + 0.805928i \(0.298333\pi\)
\(972\) 0 0
\(973\) −44.2094 −1.41729
\(974\) 7.89531 0.252982
\(975\) 0 0
\(976\) 14.1047 0.451480
\(977\) −23.4031 −0.748732 −0.374366 0.927281i \(-0.622140\pi\)
−0.374366 + 0.927281i \(0.622140\pi\)
\(978\) 0 0
\(979\) 53.6125 1.71346
\(980\) 0 0
\(981\) 0 0
\(982\) −33.4031 −1.06594
\(983\) −7.50781 −0.239462 −0.119731 0.992806i \(-0.538203\pi\)
−0.119731 + 0.992806i \(0.538203\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −4.49219 −0.143060
\(987\) 0 0
\(988\) −1.70156 −0.0541339
\(989\) 0 0
\(990\) 0 0
\(991\) −9.10469 −0.289220 −0.144610 0.989489i \(-0.546193\pi\)
−0.144610 + 0.989489i \(0.546193\pi\)
\(992\) −10.1047 −0.320824
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) 0 0
\(997\) 10.4922 0.332291 0.166145 0.986101i \(-0.446868\pi\)
0.166145 + 0.986101i \(0.446868\pi\)
\(998\) 17.2094 0.544753
\(999\) 0 0
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 5850.2.a.cj.1.1 2
3.2 odd 2 1950.2.a.bc.1.1 2
5.2 odd 4 5850.2.e.bi.5149.3 4
5.3 odd 4 5850.2.e.bi.5149.2 4
5.4 even 2 5850.2.a.cg.1.2 2
15.2 even 4 1950.2.e.p.1249.1 4
15.8 even 4 1950.2.e.p.1249.4 4
15.14 odd 2 1950.2.a.bg.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.bc.1.1 2 3.2 odd 2
1950.2.a.bg.1.2 yes 2 15.14 odd 2
1950.2.e.p.1249.1 4 15.2 even 4
1950.2.e.p.1249.4 4 15.8 even 4
5850.2.a.cg.1.2 2 5.4 even 2
5850.2.a.cj.1.1 2 1.1 even 1 trivial
5850.2.e.bi.5149.2 4 5.3 odd 4
5850.2.e.bi.5149.3 4 5.2 odd 4