Properties

Label 608.2.a.j.1.3
Level $608$
Weight $2$
Character 608.1
Self dual yes
Analytic conductor $4.855$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.85490444289\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.69353\) of defining polynomial
Character \(\chi\) \(=\) 608.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{3} -4.20608 q^{5} +1.74252 q^{7} +3.56155 q^{9} +O(q^{10})\) \(q+2.56155 q^{3} -4.20608 q^{5} +1.74252 q^{7} +3.56155 q^{9} +6.20608 q^{11} +4.82550 q^{13} -10.7741 q^{15} -5.12957 q^{17} +1.00000 q^{19} +4.46356 q^{21} +0.463560 q^{23} +12.6911 q^{25} +1.43845 q^{27} +4.82550 q^{29} -1.63806 q^{31} +15.8972 q^{33} -7.32919 q^{35} +3.63806 q^{37} +12.3608 q^{39} -5.28906 q^{41} +1.08298 q^{43} -14.9802 q^{45} -0.555087 q^{47} -3.96362 q^{49} -13.1397 q^{51} +2.65955 q^{53} -26.1033 q^{55} +2.56155 q^{57} -1.85061 q^{59} -9.32919 q^{61} +6.20608 q^{63} -20.2964 q^{65} +4.72751 q^{67} +1.18743 q^{69} -11.4850 q^{71} -0.00646614 q^{73} +32.5090 q^{75} +10.8142 q^{77} +6.76117 q^{79} -7.00000 q^{81} -11.8972 q^{83} +21.5754 q^{85} +12.3608 q^{87} -7.31909 q^{89} +8.40853 q^{91} -4.19598 q^{93} -4.20608 q^{95} -10.4122 q^{97} +22.1033 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + q^{5} - q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + q^{5} - q^{7} + 6 q^{9} + 7 q^{11} + 10 q^{13} - 8 q^{15} + 5 q^{17} + 4 q^{19} + 8 q^{21} - 8 q^{23} + 17 q^{25} + 14 q^{27} + 10 q^{29} - 6 q^{31} + 12 q^{33} + 5 q^{35} + 14 q^{37} - 12 q^{39} - 2 q^{41} + 3 q^{43} - 7 q^{45} - 3 q^{47} + 7 q^{49} - 6 q^{51} + 4 q^{53} - 35 q^{55} + 2 q^{57} + 20 q^{59} - 3 q^{61} + 7 q^{63} - 12 q^{65} + 8 q^{67} - 4 q^{69} - 30 q^{71} + 9 q^{73} + 34 q^{75} - 7 q^{77} + 10 q^{79} - 28 q^{81} + 4 q^{83} + 19 q^{85} - 12 q^{87} - 16 q^{89} + 10 q^{91} - 20 q^{93} + q^{95} - 6 q^{97} + 19 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.56155 1.47891 0.739457 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) 0 0
\(5\) −4.20608 −1.88102 −0.940508 0.339770i \(-0.889651\pi\)
−0.940508 + 0.339770i \(0.889651\pi\)
\(6\) 0 0
\(7\) 1.74252 0.658611 0.329306 0.944223i \(-0.393185\pi\)
0.329306 + 0.944223i \(0.393185\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) 6.20608 1.87120 0.935602 0.353056i \(-0.114858\pi\)
0.935602 + 0.353056i \(0.114858\pi\)
\(12\) 0 0
\(13\) 4.82550 1.33835 0.669176 0.743104i \(-0.266647\pi\)
0.669176 + 0.743104i \(0.266647\pi\)
\(14\) 0 0
\(15\) −10.7741 −2.78186
\(16\) 0 0
\(17\) −5.12957 −1.24410 −0.622052 0.782976i \(-0.713701\pi\)
−0.622052 + 0.782976i \(0.713701\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.46356 0.974029
\(22\) 0 0
\(23\) 0.463560 0.0966590 0.0483295 0.998831i \(-0.484610\pi\)
0.0483295 + 0.998831i \(0.484610\pi\)
\(24\) 0 0
\(25\) 12.6911 2.53822
\(26\) 0 0
\(27\) 1.43845 0.276829
\(28\) 0 0
\(29\) 4.82550 0.896072 0.448036 0.894015i \(-0.352124\pi\)
0.448036 + 0.894015i \(0.352124\pi\)
\(30\) 0 0
\(31\) −1.63806 −0.294205 −0.147102 0.989121i \(-0.546995\pi\)
−0.147102 + 0.989121i \(0.546995\pi\)
\(32\) 0 0
\(33\) 15.8972 2.76735
\(34\) 0 0
\(35\) −7.32919 −1.23886
\(36\) 0 0
\(37\) 3.63806 0.598094 0.299047 0.954238i \(-0.403331\pi\)
0.299047 + 0.954238i \(0.403331\pi\)
\(38\) 0 0
\(39\) 12.3608 1.97931
\(40\) 0 0
\(41\) −5.28906 −0.826012 −0.413006 0.910728i \(-0.635521\pi\)
−0.413006 + 0.910728i \(0.635521\pi\)
\(42\) 0 0
\(43\) 1.08298 0.165152 0.0825762 0.996585i \(-0.473685\pi\)
0.0825762 + 0.996585i \(0.473685\pi\)
\(44\) 0 0
\(45\) −14.9802 −2.23311
\(46\) 0 0
\(47\) −0.555087 −0.0809677 −0.0404839 0.999180i \(-0.512890\pi\)
−0.0404839 + 0.999180i \(0.512890\pi\)
\(48\) 0 0
\(49\) −3.96362 −0.566231
\(50\) 0 0
\(51\) −13.1397 −1.83992
\(52\) 0 0
\(53\) 2.65955 0.365317 0.182658 0.983176i \(-0.441530\pi\)
0.182658 + 0.983176i \(0.441530\pi\)
\(54\) 0 0
\(55\) −26.1033 −3.51977
\(56\) 0 0
\(57\) 2.56155 0.339286
\(58\) 0 0
\(59\) −1.85061 −0.240929 −0.120465 0.992718i \(-0.538438\pi\)
−0.120465 + 0.992718i \(0.538438\pi\)
\(60\) 0 0
\(61\) −9.32919 −1.19448 −0.597240 0.802063i \(-0.703736\pi\)
−0.597240 + 0.802063i \(0.703736\pi\)
\(62\) 0 0
\(63\) 6.20608 0.781893
\(64\) 0 0
\(65\) −20.2964 −2.51746
\(66\) 0 0
\(67\) 4.72751 0.577557 0.288778 0.957396i \(-0.406751\pi\)
0.288778 + 0.957396i \(0.406751\pi\)
\(68\) 0 0
\(69\) 1.18743 0.142950
\(70\) 0 0
\(71\) −11.4850 −1.36302 −0.681512 0.731807i \(-0.738677\pi\)
−0.681512 + 0.731807i \(0.738677\pi\)
\(72\) 0 0
\(73\) −0.00646614 −0.000756804 0 −0.000378402 1.00000i \(-0.500120\pi\)
−0.000378402 1.00000i \(0.500120\pi\)
\(74\) 0 0
\(75\) 32.5090 3.75381
\(76\) 0 0
\(77\) 10.8142 1.23240
\(78\) 0 0
\(79\) 6.76117 0.760691 0.380345 0.924844i \(-0.375805\pi\)
0.380345 + 0.924844i \(0.375805\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −11.8972 −1.30589 −0.652944 0.757406i \(-0.726466\pi\)
−0.652944 + 0.757406i \(0.726466\pi\)
\(84\) 0 0
\(85\) 21.5754 2.34018
\(86\) 0 0
\(87\) 12.3608 1.32521
\(88\) 0 0
\(89\) −7.31909 −0.775822 −0.387911 0.921697i \(-0.626803\pi\)
−0.387911 + 0.921697i \(0.626803\pi\)
\(90\) 0 0
\(91\) 8.40853 0.881454
\(92\) 0 0
\(93\) −4.19598 −0.435103
\(94\) 0 0
\(95\) −4.20608 −0.431535
\(96\) 0 0
\(97\) −10.4122 −1.05720 −0.528598 0.848873i \(-0.677282\pi\)
−0.528598 + 0.848873i \(0.677282\pi\)
\(98\) 0 0
\(99\) 22.1033 2.22146
\(100\) 0 0
\(101\) 0.246211 0.0244989 0.0122495 0.999925i \(-0.496101\pi\)
0.0122495 + 0.999925i \(0.496101\pi\)
\(102\) 0 0
\(103\) 0.710942 0.0700512 0.0350256 0.999386i \(-0.488849\pi\)
0.0350256 + 0.999386i \(0.488849\pi\)
\(104\) 0 0
\(105\) −18.7741 −1.83216
\(106\) 0 0
\(107\) −11.6976 −1.13085 −0.565424 0.824800i \(-0.691288\pi\)
−0.565424 + 0.824800i \(0.691288\pi\)
\(108\) 0 0
\(109\) −2.99145 −0.286529 −0.143264 0.989684i \(-0.545760\pi\)
−0.143264 + 0.989684i \(0.545760\pi\)
\(110\) 0 0
\(111\) 9.31909 0.884529
\(112\) 0 0
\(113\) 6.41216 0.603206 0.301603 0.953434i \(-0.402478\pi\)
0.301603 + 0.953434i \(0.402478\pi\)
\(114\) 0 0
\(115\) −1.94977 −0.181817
\(116\) 0 0
\(117\) 17.1863 1.58887
\(118\) 0 0
\(119\) −8.93839 −0.819381
\(120\) 0 0
\(121\) 27.5155 2.50140
\(122\) 0 0
\(123\) −13.5482 −1.22160
\(124\) 0 0
\(125\) −32.3495 −2.89343
\(126\) 0 0
\(127\) −18.1434 −1.60997 −0.804984 0.593297i \(-0.797826\pi\)
−0.804984 + 0.593297i \(0.797826\pi\)
\(128\) 0 0
\(129\) 2.77410 0.244246
\(130\) 0 0
\(131\) 12.4652 1.08909 0.544546 0.838731i \(-0.316702\pi\)
0.544546 + 0.838731i \(0.316702\pi\)
\(132\) 0 0
\(133\) 1.74252 0.151096
\(134\) 0 0
\(135\) −6.05023 −0.520721
\(136\) 0 0
\(137\) 8.41863 0.719252 0.359626 0.933097i \(-0.382904\pi\)
0.359626 + 0.933097i \(0.382904\pi\)
\(138\) 0 0
\(139\) 11.3292 0.960929 0.480465 0.877014i \(-0.340468\pi\)
0.480465 + 0.877014i \(0.340468\pi\)
\(140\) 0 0
\(141\) −1.42188 −0.119744
\(142\) 0 0
\(143\) 29.9474 2.50433
\(144\) 0 0
\(145\) −20.2964 −1.68553
\(146\) 0 0
\(147\) −10.1530 −0.837407
\(148\) 0 0
\(149\) −0.321808 −0.0263635 −0.0131818 0.999913i \(-0.504196\pi\)
−0.0131818 + 0.999913i \(0.504196\pi\)
\(150\) 0 0
\(151\) −2.95715 −0.240650 −0.120325 0.992735i \(-0.538394\pi\)
−0.120325 + 0.992735i \(0.538394\pi\)
\(152\) 0 0
\(153\) −18.2692 −1.47698
\(154\) 0 0
\(155\) 6.88983 0.553404
\(156\) 0 0
\(157\) 6.52789 0.520982 0.260491 0.965476i \(-0.416116\pi\)
0.260491 + 0.965476i \(0.416116\pi\)
\(158\) 0 0
\(159\) 6.81257 0.540272
\(160\) 0 0
\(161\) 0.807764 0.0636607
\(162\) 0 0
\(163\) 2.05023 0.160586 0.0802931 0.996771i \(-0.474414\pi\)
0.0802931 + 0.996771i \(0.474414\pi\)
\(164\) 0 0
\(165\) −66.8650 −5.20543
\(166\) 0 0
\(167\) −10.2462 −0.792876 −0.396438 0.918062i \(-0.629754\pi\)
−0.396438 + 0.918062i \(0.629754\pi\)
\(168\) 0 0
\(169\) 10.2854 0.791187
\(170\) 0 0
\(171\) 3.56155 0.272359
\(172\) 0 0
\(173\) 8.05023 0.612047 0.306024 0.952024i \(-0.401001\pi\)
0.306024 + 0.952024i \(0.401001\pi\)
\(174\) 0 0
\(175\) 22.1146 1.67170
\(176\) 0 0
\(177\) −4.74044 −0.356313
\(178\) 0 0
\(179\) 1.75379 0.131084 0.0655422 0.997850i \(-0.479122\pi\)
0.0655422 + 0.997850i \(0.479122\pi\)
\(180\) 0 0
\(181\) −11.0203 −0.819133 −0.409567 0.912280i \(-0.634320\pi\)
−0.409567 + 0.912280i \(0.634320\pi\)
\(182\) 0 0
\(183\) −23.8972 −1.76653
\(184\) 0 0
\(185\) −15.3020 −1.12502
\(186\) 0 0
\(187\) −31.8345 −2.32797
\(188\) 0 0
\(189\) 2.50652 0.182323
\(190\) 0 0
\(191\) −13.2276 −0.957113 −0.478556 0.878057i \(-0.658840\pi\)
−0.478556 + 0.878057i \(0.658840\pi\)
\(192\) 0 0
\(193\) 1.47211 0.105965 0.0529824 0.998595i \(-0.483127\pi\)
0.0529824 + 0.998595i \(0.483127\pi\)
\(194\) 0 0
\(195\) −51.9904 −3.72311
\(196\) 0 0
\(197\) 7.84698 0.559074 0.279537 0.960135i \(-0.409819\pi\)
0.279537 + 0.960135i \(0.409819\pi\)
\(198\) 0 0
\(199\) −11.8057 −0.836882 −0.418441 0.908244i \(-0.637423\pi\)
−0.418441 + 0.908244i \(0.637423\pi\)
\(200\) 0 0
\(201\) 12.1098 0.854156
\(202\) 0 0
\(203\) 8.40853 0.590163
\(204\) 0 0
\(205\) 22.2462 1.55374
\(206\) 0 0
\(207\) 1.65100 0.114752
\(208\) 0 0
\(209\) 6.20608 0.429284
\(210\) 0 0
\(211\) −14.5745 −1.00335 −0.501674 0.865056i \(-0.667282\pi\)
−0.501674 + 0.865056i \(0.667282\pi\)
\(212\) 0 0
\(213\) −29.4195 −2.01579
\(214\) 0 0
\(215\) −4.55509 −0.310654
\(216\) 0 0
\(217\) −2.85436 −0.193767
\(218\) 0 0
\(219\) −0.0165634 −0.00111925
\(220\) 0 0
\(221\) −24.7527 −1.66505
\(222\) 0 0
\(223\) 26.6713 1.78604 0.893021 0.450014i \(-0.148581\pi\)
0.893021 + 0.450014i \(0.148581\pi\)
\(224\) 0 0
\(225\) 45.2001 3.01334
\(226\) 0 0
\(227\) −4.01656 −0.266589 −0.133294 0.991076i \(-0.542556\pi\)
−0.133294 + 0.991076i \(0.542556\pi\)
\(228\) 0 0
\(229\) 23.1834 1.53200 0.766002 0.642838i \(-0.222243\pi\)
0.766002 + 0.642838i \(0.222243\pi\)
\(230\) 0 0
\(231\) 27.7012 1.82261
\(232\) 0 0
\(233\) 22.2692 1.45891 0.729453 0.684031i \(-0.239775\pi\)
0.729453 + 0.684031i \(0.239775\pi\)
\(234\) 0 0
\(235\) 2.33474 0.152302
\(236\) 0 0
\(237\) 17.3191 1.12500
\(238\) 0 0
\(239\) −23.0818 −1.49304 −0.746519 0.665364i \(-0.768276\pi\)
−0.746519 + 0.665364i \(0.768276\pi\)
\(240\) 0 0
\(241\) −0.565184 −0.0364067 −0.0182033 0.999834i \(-0.505795\pi\)
−0.0182033 + 0.999834i \(0.505795\pi\)
\(242\) 0 0
\(243\) −22.2462 −1.42710
\(244\) 0 0
\(245\) 16.6713 1.06509
\(246\) 0 0
\(247\) 4.82550 0.307039
\(248\) 0 0
\(249\) −30.4753 −1.93130
\(250\) 0 0
\(251\) 15.8441 1.00007 0.500037 0.866004i \(-0.333320\pi\)
0.500037 + 0.866004i \(0.333320\pi\)
\(252\) 0 0
\(253\) 2.87689 0.180869
\(254\) 0 0
\(255\) 55.2665 3.46092
\(256\) 0 0
\(257\) 12.0632 0.752479 0.376240 0.926522i \(-0.377217\pi\)
0.376240 + 0.926522i \(0.377217\pi\)
\(258\) 0 0
\(259\) 6.33940 0.393911
\(260\) 0 0
\(261\) 17.1863 1.06380
\(262\) 0 0
\(263\) 23.2135 1.43140 0.715702 0.698406i \(-0.246107\pi\)
0.715702 + 0.698406i \(0.246107\pi\)
\(264\) 0 0
\(265\) −11.1863 −0.687167
\(266\) 0 0
\(267\) −18.7482 −1.14737
\(268\) 0 0
\(269\) 26.2964 1.60332 0.801661 0.597779i \(-0.203950\pi\)
0.801661 + 0.597779i \(0.203950\pi\)
\(270\) 0 0
\(271\) 12.9560 0.787020 0.393510 0.919320i \(-0.371261\pi\)
0.393510 + 0.919320i \(0.371261\pi\)
\(272\) 0 0
\(273\) 21.5389 1.30359
\(274\) 0 0
\(275\) 78.7622 4.74954
\(276\) 0 0
\(277\) −10.8645 −0.652782 −0.326391 0.945235i \(-0.605833\pi\)
−0.326391 + 0.945235i \(0.605833\pi\)
\(278\) 0 0
\(279\) −5.83405 −0.349275
\(280\) 0 0
\(281\) −15.8470 −0.945352 −0.472676 0.881236i \(-0.656712\pi\)
−0.472676 + 0.881236i \(0.656712\pi\)
\(282\) 0 0
\(283\) −27.6740 −1.64505 −0.822525 0.568729i \(-0.807435\pi\)
−0.822525 + 0.568729i \(0.807435\pi\)
\(284\) 0 0
\(285\) −10.7741 −0.638203
\(286\) 0 0
\(287\) −9.21630 −0.544021
\(288\) 0 0
\(289\) 9.31251 0.547795
\(290\) 0 0
\(291\) −26.6713 −1.56350
\(292\) 0 0
\(293\) −21.3179 −1.24541 −0.622703 0.782458i \(-0.713966\pi\)
−0.622703 + 0.782458i \(0.713966\pi\)
\(294\) 0 0
\(295\) 7.78382 0.453192
\(296\) 0 0
\(297\) 8.92712 0.518004
\(298\) 0 0
\(299\) 2.23691 0.129364
\(300\) 0 0
\(301\) 1.88711 0.108771
\(302\) 0 0
\(303\) 0.630683 0.0362318
\(304\) 0 0
\(305\) 39.2393 2.24684
\(306\) 0 0
\(307\) −26.6584 −1.52147 −0.760737 0.649060i \(-0.775162\pi\)
−0.760737 + 0.649060i \(0.775162\pi\)
\(308\) 0 0
\(309\) 1.82112 0.103600
\(310\) 0 0
\(311\) 20.0690 1.13801 0.569004 0.822335i \(-0.307329\pi\)
0.569004 + 0.822335i \(0.307329\pi\)
\(312\) 0 0
\(313\) −11.1397 −0.629651 −0.314826 0.949150i \(-0.601946\pi\)
−0.314826 + 0.949150i \(0.601946\pi\)
\(314\) 0 0
\(315\) −26.1033 −1.47075
\(316\) 0 0
\(317\) 33.1349 1.86104 0.930520 0.366242i \(-0.119356\pi\)
0.930520 + 0.366242i \(0.119356\pi\)
\(318\) 0 0
\(319\) 29.9474 1.67673
\(320\) 0 0
\(321\) −29.9640 −1.67243
\(322\) 0 0
\(323\) −5.12957 −0.285417
\(324\) 0 0
\(325\) 61.2410 3.39704
\(326\) 0 0
\(327\) −7.66276 −0.423751
\(328\) 0 0
\(329\) −0.967250 −0.0533262
\(330\) 0 0
\(331\) 4.30241 0.236482 0.118241 0.992985i \(-0.462274\pi\)
0.118241 + 0.992985i \(0.462274\pi\)
\(332\) 0 0
\(333\) 12.9572 0.710048
\(334\) 0 0
\(335\) −19.8843 −1.08639
\(336\) 0 0
\(337\) 31.3393 1.70716 0.853580 0.520962i \(-0.174427\pi\)
0.853580 + 0.520962i \(0.174427\pi\)
\(338\) 0 0
\(339\) 16.4251 0.892089
\(340\) 0 0
\(341\) −10.1660 −0.550517
\(342\) 0 0
\(343\) −19.1043 −1.03154
\(344\) 0 0
\(345\) −4.99445 −0.268892
\(346\) 0 0
\(347\) −21.6782 −1.16375 −0.581873 0.813280i \(-0.697680\pi\)
−0.581873 + 0.813280i \(0.697680\pi\)
\(348\) 0 0
\(349\) 4.50486 0.241140 0.120570 0.992705i \(-0.461528\pi\)
0.120570 + 0.992705i \(0.461528\pi\)
\(350\) 0 0
\(351\) 6.94122 0.370495
\(352\) 0 0
\(353\) −22.4158 −1.19307 −0.596536 0.802586i \(-0.703457\pi\)
−0.596536 + 0.802586i \(0.703457\pi\)
\(354\) 0 0
\(355\) 48.3070 2.56387
\(356\) 0 0
\(357\) −22.8962 −1.21179
\(358\) 0 0
\(359\) 11.4940 0.606629 0.303314 0.952891i \(-0.401907\pi\)
0.303314 + 0.952891i \(0.401907\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 70.4823 3.69936
\(364\) 0 0
\(365\) 0.0271971 0.00142356
\(366\) 0 0
\(367\) −19.7944 −1.03326 −0.516630 0.856209i \(-0.672814\pi\)
−0.516630 + 0.856209i \(0.672814\pi\)
\(368\) 0 0
\(369\) −18.8373 −0.980629
\(370\) 0 0
\(371\) 4.63431 0.240602
\(372\) 0 0
\(373\) −16.6199 −0.860546 −0.430273 0.902699i \(-0.641583\pi\)
−0.430273 + 0.902699i \(0.641583\pi\)
\(374\) 0 0
\(375\) −82.8650 −4.27913
\(376\) 0 0
\(377\) 23.2854 1.19926
\(378\) 0 0
\(379\) 16.3956 0.842185 0.421093 0.907018i \(-0.361647\pi\)
0.421093 + 0.907018i \(0.361647\pi\)
\(380\) 0 0
\(381\) −46.4753 −2.38100
\(382\) 0 0
\(383\) −30.6915 −1.56826 −0.784131 0.620595i \(-0.786891\pi\)
−0.784131 + 0.620595i \(0.786891\pi\)
\(384\) 0 0
\(385\) −45.4855 −2.31816
\(386\) 0 0
\(387\) 3.85708 0.196066
\(388\) 0 0
\(389\) −2.24905 −0.114031 −0.0570156 0.998373i \(-0.518158\pi\)
−0.0570156 + 0.998373i \(0.518158\pi\)
\(390\) 0 0
\(391\) −2.37787 −0.120254
\(392\) 0 0
\(393\) 31.9303 1.61067
\(394\) 0 0
\(395\) −28.4380 −1.43087
\(396\) 0 0
\(397\) −4.12582 −0.207069 −0.103535 0.994626i \(-0.533015\pi\)
−0.103535 + 0.994626i \(0.533015\pi\)
\(398\) 0 0
\(399\) 4.46356 0.223458
\(400\) 0 0
\(401\) −22.3992 −1.11856 −0.559282 0.828977i \(-0.688923\pi\)
−0.559282 + 0.828977i \(0.688923\pi\)
\(402\) 0 0
\(403\) −7.90447 −0.393750
\(404\) 0 0
\(405\) 29.4426 1.46301
\(406\) 0 0
\(407\) 22.5781 1.11916
\(408\) 0 0
\(409\) 13.1992 0.652658 0.326329 0.945256i \(-0.394188\pi\)
0.326329 + 0.945256i \(0.394188\pi\)
\(410\) 0 0
\(411\) 21.5648 1.06371
\(412\) 0 0
\(413\) −3.22473 −0.158679
\(414\) 0 0
\(415\) 50.0406 2.45640
\(416\) 0 0
\(417\) 29.0203 1.42113
\(418\) 0 0
\(419\) −31.5984 −1.54368 −0.771842 0.635814i \(-0.780664\pi\)
−0.771842 + 0.635814i \(0.780664\pi\)
\(420\) 0 0
\(421\) −25.8587 −1.26028 −0.630139 0.776482i \(-0.717002\pi\)
−0.630139 + 0.776482i \(0.717002\pi\)
\(422\) 0 0
\(423\) −1.97697 −0.0961236
\(424\) 0 0
\(425\) −65.1000 −3.15782
\(426\) 0 0
\(427\) −16.2563 −0.786698
\(428\) 0 0
\(429\) 76.7119 3.70369
\(430\) 0 0
\(431\) −31.8972 −1.53643 −0.768217 0.640189i \(-0.778856\pi\)
−0.768217 + 0.640189i \(0.778856\pi\)
\(432\) 0 0
\(433\) 23.1006 1.11014 0.555071 0.831803i \(-0.312691\pi\)
0.555071 + 0.831803i \(0.312691\pi\)
\(434\) 0 0
\(435\) −51.9904 −2.49275
\(436\) 0 0
\(437\) 0.463560 0.0221751
\(438\) 0 0
\(439\) 15.7944 0.753826 0.376913 0.926249i \(-0.376986\pi\)
0.376913 + 0.926249i \(0.376986\pi\)
\(440\) 0 0
\(441\) −14.1166 −0.672221
\(442\) 0 0
\(443\) 40.5584 1.92699 0.963494 0.267729i \(-0.0862732\pi\)
0.963494 + 0.267729i \(0.0862732\pi\)
\(444\) 0 0
\(445\) 30.7847 1.45933
\(446\) 0 0
\(447\) −0.824327 −0.0389893
\(448\) 0 0
\(449\) −30.4026 −1.43479 −0.717393 0.696669i \(-0.754665\pi\)
−0.717393 + 0.696669i \(0.754665\pi\)
\(450\) 0 0
\(451\) −32.8243 −1.54564
\(452\) 0 0
\(453\) −7.57490 −0.355900
\(454\) 0 0
\(455\) −35.3670 −1.65803
\(456\) 0 0
\(457\) 1.33516 0.0624561 0.0312280 0.999512i \(-0.490058\pi\)
0.0312280 + 0.999512i \(0.490058\pi\)
\(458\) 0 0
\(459\) −7.37862 −0.344404
\(460\) 0 0
\(461\) −23.0604 −1.07403 −0.537016 0.843572i \(-0.680448\pi\)
−0.537016 + 0.843572i \(0.680448\pi\)
\(462\) 0 0
\(463\) 9.29916 0.432168 0.216084 0.976375i \(-0.430671\pi\)
0.216084 + 0.976375i \(0.430671\pi\)
\(464\) 0 0
\(465\) 17.6487 0.818437
\(466\) 0 0
\(467\) −18.8344 −0.871553 −0.435777 0.900055i \(-0.643526\pi\)
−0.435777 + 0.900055i \(0.643526\pi\)
\(468\) 0 0
\(469\) 8.23778 0.380385
\(470\) 0 0
\(471\) 16.7215 0.770488
\(472\) 0 0
\(473\) 6.72104 0.309034
\(474\) 0 0
\(475\) 12.6911 0.582309
\(476\) 0 0
\(477\) 9.47211 0.433698
\(478\) 0 0
\(479\) 10.9701 0.501236 0.250618 0.968086i \(-0.419366\pi\)
0.250618 + 0.968086i \(0.419366\pi\)
\(480\) 0 0
\(481\) 17.5555 0.800460
\(482\) 0 0
\(483\) 2.06913 0.0941487
\(484\) 0 0
\(485\) 43.7944 1.98860
\(486\) 0 0
\(487\) 19.3895 0.878623 0.439311 0.898335i \(-0.355223\pi\)
0.439311 + 0.898335i \(0.355223\pi\)
\(488\) 0 0
\(489\) 5.25176 0.237493
\(490\) 0 0
\(491\) 33.1562 1.49632 0.748160 0.663518i \(-0.230938\pi\)
0.748160 + 0.663518i \(0.230938\pi\)
\(492\) 0 0
\(493\) −24.7527 −1.11481
\(494\) 0 0
\(495\) −92.9682 −4.17861
\(496\) 0 0
\(497\) −20.0129 −0.897703
\(498\) 0 0
\(499\) −25.8814 −1.15861 −0.579306 0.815110i \(-0.696676\pi\)
−0.579306 + 0.815110i \(0.696676\pi\)
\(500\) 0 0
\(501\) −26.2462 −1.17259
\(502\) 0 0
\(503\) 10.3178 0.460048 0.230024 0.973185i \(-0.426120\pi\)
0.230024 + 0.973185i \(0.426120\pi\)
\(504\) 0 0
\(505\) −1.03558 −0.0460829
\(506\) 0 0
\(507\) 26.3467 1.17010
\(508\) 0 0
\(509\) −6.21618 −0.275527 −0.137764 0.990465i \(-0.543991\pi\)
−0.137764 + 0.990465i \(0.543991\pi\)
\(510\) 0 0
\(511\) −0.0112674 −0.000498440 0
\(512\) 0 0
\(513\) 1.43845 0.0635090
\(514\) 0 0
\(515\) −2.99028 −0.131767
\(516\) 0 0
\(517\) −3.44491 −0.151507
\(518\) 0 0
\(519\) 20.6211 0.905165
\(520\) 0 0
\(521\) 22.8917 1.00290 0.501451 0.865186i \(-0.332800\pi\)
0.501451 + 0.865186i \(0.332800\pi\)
\(522\) 0 0
\(523\) 9.02628 0.394692 0.197346 0.980334i \(-0.436768\pi\)
0.197346 + 0.980334i \(0.436768\pi\)
\(524\) 0 0
\(525\) 56.6476 2.47230
\(526\) 0 0
\(527\) 8.40256 0.366021
\(528\) 0 0
\(529\) −22.7851 −0.990657
\(530\) 0 0
\(531\) −6.59105 −0.286027
\(532\) 0 0
\(533\) −25.5223 −1.10550
\(534\) 0 0
\(535\) 49.2010 2.12715
\(536\) 0 0
\(537\) 4.49242 0.193862
\(538\) 0 0
\(539\) −24.5985 −1.05953
\(540\) 0 0
\(541\) −21.2937 −0.915489 −0.457744 0.889084i \(-0.651342\pi\)
−0.457744 + 0.889084i \(0.651342\pi\)
\(542\) 0 0
\(543\) −28.2291 −1.21143
\(544\) 0 0
\(545\) 12.5823 0.538966
\(546\) 0 0
\(547\) 3.60803 0.154268 0.0771341 0.997021i \(-0.475423\pi\)
0.0771341 + 0.997021i \(0.475423\pi\)
\(548\) 0 0
\(549\) −33.2264 −1.41807
\(550\) 0 0
\(551\) 4.82550 0.205573
\(552\) 0 0
\(553\) 11.7815 0.501000
\(554\) 0 0
\(555\) −39.1969 −1.66381
\(556\) 0 0
\(557\) −34.8645 −1.47725 −0.738627 0.674114i \(-0.764526\pi\)
−0.738627 + 0.674114i \(0.764526\pi\)
\(558\) 0 0
\(559\) 5.22590 0.221032
\(560\) 0 0
\(561\) −81.5459 −3.44287
\(562\) 0 0
\(563\) 41.3167 1.74129 0.870647 0.491909i \(-0.163701\pi\)
0.870647 + 0.491909i \(0.163701\pi\)
\(564\) 0 0
\(565\) −26.9701 −1.13464
\(566\) 0 0
\(567\) −12.1976 −0.512253
\(568\) 0 0
\(569\) 41.0835 1.72231 0.861154 0.508344i \(-0.169742\pi\)
0.861154 + 0.508344i \(0.169742\pi\)
\(570\) 0 0
\(571\) −36.8243 −1.54105 −0.770525 0.637410i \(-0.780006\pi\)
−0.770525 + 0.637410i \(0.780006\pi\)
\(572\) 0 0
\(573\) −33.8831 −1.41549
\(574\) 0 0
\(575\) 5.88310 0.245342
\(576\) 0 0
\(577\) 15.7004 0.653617 0.326809 0.945091i \(-0.394027\pi\)
0.326809 + 0.945091i \(0.394027\pi\)
\(578\) 0 0
\(579\) 3.77089 0.156713
\(580\) 0 0
\(581\) −20.7311 −0.860072
\(582\) 0 0
\(583\) 16.5054 0.683582
\(584\) 0 0
\(585\) −72.2868 −2.98869
\(586\) 0 0
\(587\) 34.3099 1.41612 0.708060 0.706152i \(-0.249571\pi\)
0.708060 + 0.706152i \(0.249571\pi\)
\(588\) 0 0
\(589\) −1.63806 −0.0674952
\(590\) 0 0
\(591\) 20.1005 0.826822
\(592\) 0 0
\(593\) 7.11584 0.292213 0.146106 0.989269i \(-0.453326\pi\)
0.146106 + 0.989269i \(0.453326\pi\)
\(594\) 0 0
\(595\) 37.5956 1.54127
\(596\) 0 0
\(597\) −30.2409 −1.23768
\(598\) 0 0
\(599\) −13.8340 −0.565244 −0.282622 0.959231i \(-0.591204\pi\)
−0.282622 + 0.959231i \(0.591204\pi\)
\(600\) 0 0
\(601\) 39.2140 1.59957 0.799785 0.600286i \(-0.204947\pi\)
0.799785 + 0.600286i \(0.204947\pi\)
\(602\) 0 0
\(603\) 16.8373 0.685666
\(604\) 0 0
\(605\) −115.732 −4.70518
\(606\) 0 0
\(607\) 35.1733 1.42764 0.713821 0.700328i \(-0.246963\pi\)
0.713821 + 0.700328i \(0.246963\pi\)
\(608\) 0 0
\(609\) 21.5389 0.872800
\(610\) 0 0
\(611\) −2.67857 −0.108363
\(612\) 0 0
\(613\) 6.24905 0.252397 0.126198 0.992005i \(-0.459722\pi\)
0.126198 + 0.992005i \(0.459722\pi\)
\(614\) 0 0
\(615\) 56.9848 2.29785
\(616\) 0 0
\(617\) −42.0378 −1.69238 −0.846189 0.532883i \(-0.821109\pi\)
−0.846189 + 0.532883i \(0.821109\pi\)
\(618\) 0 0
\(619\) 31.0633 1.24854 0.624269 0.781209i \(-0.285397\pi\)
0.624269 + 0.781209i \(0.285397\pi\)
\(620\) 0 0
\(621\) 0.666807 0.0267581
\(622\) 0 0
\(623\) −12.7537 −0.510965
\(624\) 0 0
\(625\) 72.6090 2.90436
\(626\) 0 0
\(627\) 15.8972 0.634873
\(628\) 0 0
\(629\) −18.6617 −0.744091
\(630\) 0 0
\(631\) −40.0378 −1.59388 −0.796940 0.604059i \(-0.793549\pi\)
−0.796940 + 0.604059i \(0.793549\pi\)
\(632\) 0 0
\(633\) −37.3333 −1.48387
\(634\) 0 0
\(635\) 76.3127 3.02838
\(636\) 0 0
\(637\) −19.1264 −0.757817
\(638\) 0 0
\(639\) −40.9046 −1.61816
\(640\) 0 0
\(641\) −20.3619 −0.804248 −0.402124 0.915585i \(-0.631728\pi\)
−0.402124 + 0.915585i \(0.631728\pi\)
\(642\) 0 0
\(643\) 13.7041 0.540435 0.270218 0.962799i \(-0.412904\pi\)
0.270218 + 0.962799i \(0.412904\pi\)
\(644\) 0 0
\(645\) −11.6681 −0.459431
\(646\) 0 0
\(647\) 26.9588 1.05986 0.529930 0.848041i \(-0.322218\pi\)
0.529930 + 0.848041i \(0.322218\pi\)
\(648\) 0 0
\(649\) −11.4850 −0.450827
\(650\) 0 0
\(651\) −7.31159 −0.286564
\(652\) 0 0
\(653\) 30.7842 1.20468 0.602339 0.798240i \(-0.294235\pi\)
0.602339 + 0.798240i \(0.294235\pi\)
\(654\) 0 0
\(655\) −52.4298 −2.04860
\(656\) 0 0
\(657\) −0.0230295 −0.000898466 0
\(658\) 0 0
\(659\) 7.57854 0.295218 0.147609 0.989046i \(-0.452842\pi\)
0.147609 + 0.989046i \(0.452842\pi\)
\(660\) 0 0
\(661\) 47.8191 1.85995 0.929974 0.367626i \(-0.119829\pi\)
0.929974 + 0.367626i \(0.119829\pi\)
\(662\) 0 0
\(663\) −63.4054 −2.46246
\(664\) 0 0
\(665\) −7.32919 −0.284214
\(666\) 0 0
\(667\) 2.23691 0.0866135
\(668\) 0 0
\(669\) 68.3200 2.64140
\(670\) 0 0
\(671\) −57.8977 −2.23512
\(672\) 0 0
\(673\) 12.2389 0.471777 0.235888 0.971780i \(-0.424200\pi\)
0.235888 + 0.971780i \(0.424200\pi\)
\(674\) 0 0
\(675\) 18.2555 0.702655
\(676\) 0 0
\(677\) 23.3609 0.897832 0.448916 0.893574i \(-0.351810\pi\)
0.448916 + 0.893574i \(0.351810\pi\)
\(678\) 0 0
\(679\) −18.1434 −0.696280
\(680\) 0 0
\(681\) −10.2886 −0.394262
\(682\) 0 0
\(683\) 24.7239 0.946033 0.473016 0.881054i \(-0.343165\pi\)
0.473016 + 0.881054i \(0.343165\pi\)
\(684\) 0 0
\(685\) −35.4094 −1.35293
\(686\) 0 0
\(687\) 59.3856 2.26570
\(688\) 0 0
\(689\) 12.8336 0.488922
\(690\) 0 0
\(691\) 6.78837 0.258242 0.129121 0.991629i \(-0.458785\pi\)
0.129121 + 0.991629i \(0.458785\pi\)
\(692\) 0 0
\(693\) 38.5155 1.46308
\(694\) 0 0
\(695\) −47.6515 −1.80752
\(696\) 0 0
\(697\) 27.1306 1.02764
\(698\) 0 0
\(699\) 57.0438 2.15760
\(700\) 0 0
\(701\) 26.9175 1.01666 0.508330 0.861162i \(-0.330263\pi\)
0.508330 + 0.861162i \(0.330263\pi\)
\(702\) 0 0
\(703\) 3.63806 0.137212
\(704\) 0 0
\(705\) 5.98056 0.225241
\(706\) 0 0
\(707\) 0.429028 0.0161353
\(708\) 0 0
\(709\) −21.5984 −0.811146 −0.405573 0.914063i \(-0.632928\pi\)
−0.405573 + 0.914063i \(0.632928\pi\)
\(710\) 0 0
\(711\) 24.0803 0.903080
\(712\) 0 0
\(713\) −0.759341 −0.0284376
\(714\) 0 0
\(715\) −125.961 −4.71069
\(716\) 0 0
\(717\) −59.1253 −2.20807
\(718\) 0 0
\(719\) −22.2575 −0.830064 −0.415032 0.909807i \(-0.636230\pi\)
−0.415032 + 0.909807i \(0.636230\pi\)
\(720\) 0 0
\(721\) 1.23883 0.0461365
\(722\) 0 0
\(723\) −1.44775 −0.0538423
\(724\) 0 0
\(725\) 61.2410 2.27443
\(726\) 0 0
\(727\) −51.9118 −1.92530 −0.962651 0.270745i \(-0.912730\pi\)
−0.962651 + 0.270745i \(0.912730\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) −5.55520 −0.205467
\(732\) 0 0
\(733\) 20.0575 0.740840 0.370420 0.928864i \(-0.379214\pi\)
0.370420 + 0.928864i \(0.379214\pi\)
\(734\) 0 0
\(735\) 42.7044 1.57518
\(736\) 0 0
\(737\) 29.3393 1.08073
\(738\) 0 0
\(739\) 7.36465 0.270913 0.135457 0.990783i \(-0.456750\pi\)
0.135457 + 0.990783i \(0.456750\pi\)
\(740\) 0 0
\(741\) 12.3608 0.454084
\(742\) 0 0
\(743\) −2.85436 −0.104716 −0.0523581 0.998628i \(-0.516674\pi\)
−0.0523581 + 0.998628i \(0.516674\pi\)
\(744\) 0 0
\(745\) 1.35355 0.0495902
\(746\) 0 0
\(747\) −42.3725 −1.55033
\(748\) 0 0
\(749\) −20.3833 −0.744790
\(750\) 0 0
\(751\) −22.3725 −0.816385 −0.408193 0.912896i \(-0.633841\pi\)
−0.408193 + 0.912896i \(0.633841\pi\)
\(752\) 0 0
\(753\) 40.5856 1.47902
\(754\) 0 0
\(755\) 12.4380 0.452666
\(756\) 0 0
\(757\) 6.07326 0.220736 0.110368 0.993891i \(-0.464797\pi\)
0.110368 + 0.993891i \(0.464797\pi\)
\(758\) 0 0
\(759\) 7.36932 0.267489
\(760\) 0 0
\(761\) 13.1166 0.475478 0.237739 0.971329i \(-0.423594\pi\)
0.237739 + 0.971329i \(0.423594\pi\)
\(762\) 0 0
\(763\) −5.21267 −0.188711
\(764\) 0 0
\(765\) 76.8419 2.77823
\(766\) 0 0
\(767\) −8.93012 −0.322448
\(768\) 0 0
\(769\) −44.5791 −1.60757 −0.803783 0.594923i \(-0.797182\pi\)
−0.803783 + 0.594923i \(0.797182\pi\)
\(770\) 0 0
\(771\) 30.9004 1.11285
\(772\) 0 0
\(773\) −4.43920 −0.159667 −0.0798334 0.996808i \(-0.525439\pi\)
−0.0798334 + 0.996808i \(0.525439\pi\)
\(774\) 0 0
\(775\) −20.7889 −0.746758
\(776\) 0 0
\(777\) 16.2387 0.582561
\(778\) 0 0
\(779\) −5.28906 −0.189500
\(780\) 0 0
\(781\) −71.2771 −2.55050
\(782\) 0 0
\(783\) 6.94122 0.248059
\(784\) 0 0
\(785\) −27.4568 −0.979977
\(786\) 0 0
\(787\) 0.315342 0.0112407 0.00562036 0.999984i \(-0.498211\pi\)
0.00562036 + 0.999984i \(0.498211\pi\)
\(788\) 0 0
\(789\) 59.4625 2.11692
\(790\) 0 0
\(791\) 11.1733 0.397278
\(792\) 0 0
\(793\) −45.0180 −1.59864
\(794\) 0 0
\(795\) −28.6542 −1.01626
\(796\) 0 0
\(797\) 6.16712 0.218451 0.109225 0.994017i \(-0.465163\pi\)
0.109225 + 0.994017i \(0.465163\pi\)
\(798\) 0 0
\(799\) 2.84736 0.100732
\(800\) 0 0
\(801\) −26.0673 −0.921044
\(802\) 0 0
\(803\) −0.0401294 −0.00141614
\(804\) 0 0
\(805\) −3.39752 −0.119747
\(806\) 0 0
\(807\) 67.3597 2.37117
\(808\) 0 0
\(809\) 20.4057 0.717426 0.358713 0.933448i \(-0.383216\pi\)
0.358713 + 0.933448i \(0.383216\pi\)
\(810\) 0 0
\(811\) −36.5763 −1.28437 −0.642184 0.766551i \(-0.721971\pi\)
−0.642184 + 0.766551i \(0.721971\pi\)
\(812\) 0 0
\(813\) 33.1874 1.16393
\(814\) 0 0
\(815\) −8.62342 −0.302065
\(816\) 0 0
\(817\) 1.08298 0.0378885
\(818\) 0 0
\(819\) 29.9474 1.04645
\(820\) 0 0
\(821\) 19.6686 0.686439 0.343219 0.939255i \(-0.388483\pi\)
0.343219 + 0.939255i \(0.388483\pi\)
\(822\) 0 0
\(823\) −36.7328 −1.28042 −0.640212 0.768198i \(-0.721154\pi\)
−0.640212 + 0.768198i \(0.721154\pi\)
\(824\) 0 0
\(825\) 201.753 7.02415
\(826\) 0 0
\(827\) 27.4920 0.955991 0.477995 0.878362i \(-0.341364\pi\)
0.477995 + 0.878362i \(0.341364\pi\)
\(828\) 0 0
\(829\) −12.1016 −0.420307 −0.210153 0.977668i \(-0.567396\pi\)
−0.210153 + 0.977668i \(0.567396\pi\)
\(830\) 0 0
\(831\) −27.8299 −0.965408
\(832\) 0 0
\(833\) 20.3317 0.704451
\(834\) 0 0
\(835\) 43.0964 1.49141
\(836\) 0 0
\(837\) −2.35627 −0.0814445
\(838\) 0 0
\(839\) −51.4731 −1.77705 −0.888524 0.458829i \(-0.848269\pi\)
−0.888524 + 0.458829i \(0.848269\pi\)
\(840\) 0 0
\(841\) −5.71457 −0.197054
\(842\) 0 0
\(843\) −40.5929 −1.39809
\(844\) 0 0
\(845\) −43.2613 −1.48824
\(846\) 0 0
\(847\) 47.9463 1.64745
\(848\) 0 0
\(849\) −70.8885 −2.43289
\(850\) 0 0
\(851\) 1.68646 0.0578112
\(852\) 0 0
\(853\) −42.6187 −1.45924 −0.729619 0.683854i \(-0.760303\pi\)
−0.729619 + 0.683854i \(0.760303\pi\)
\(854\) 0 0
\(855\) −14.9802 −0.512311
\(856\) 0 0
\(857\) 3.45501 0.118021 0.0590105 0.998257i \(-0.481205\pi\)
0.0590105 + 0.998257i \(0.481205\pi\)
\(858\) 0 0
\(859\) 17.1161 0.583994 0.291997 0.956419i \(-0.405680\pi\)
0.291997 + 0.956419i \(0.405680\pi\)
\(860\) 0 0
\(861\) −23.6080 −0.804560
\(862\) 0 0
\(863\) −13.0130 −0.442969 −0.221485 0.975164i \(-0.571090\pi\)
−0.221485 + 0.975164i \(0.571090\pi\)
\(864\) 0 0
\(865\) −33.8599 −1.15127
\(866\) 0 0
\(867\) 23.8545 0.810141
\(868\) 0 0
\(869\) 41.9604 1.42341
\(870\) 0 0
\(871\) 22.8126 0.772974
\(872\) 0 0
\(873\) −37.0835 −1.25509
\(874\) 0 0
\(875\) −56.3697 −1.90564
\(876\) 0 0
\(877\) 0.908097 0.0306642 0.0153321 0.999882i \(-0.495119\pi\)
0.0153321 + 0.999882i \(0.495119\pi\)
\(878\) 0 0
\(879\) −54.6070 −1.84185
\(880\) 0 0
\(881\) 29.1332 0.981523 0.490761 0.871294i \(-0.336719\pi\)
0.490761 + 0.871294i \(0.336719\pi\)
\(882\) 0 0
\(883\) −31.4780 −1.05932 −0.529660 0.848210i \(-0.677681\pi\)
−0.529660 + 0.848210i \(0.677681\pi\)
\(884\) 0 0
\(885\) 19.9387 0.670231
\(886\) 0 0
\(887\) 37.0933 1.24547 0.622736 0.782432i \(-0.286021\pi\)
0.622736 + 0.782432i \(0.286021\pi\)
\(888\) 0 0
\(889\) −31.6153 −1.06034
\(890\) 0 0
\(891\) −43.4426 −1.45538
\(892\) 0 0
\(893\) −0.555087 −0.0185753
\(894\) 0 0
\(895\) −7.37658 −0.246572
\(896\) 0 0
\(897\) 5.72996 0.191318
\(898\) 0 0
\(899\) −7.90447 −0.263629
\(900\) 0 0
\(901\) −13.6423 −0.454492
\(902\) 0 0
\(903\) 4.83393 0.160863
\(904\) 0 0
\(905\) 46.3523 1.54080
\(906\) 0 0
\(907\) −24.0295 −0.797886 −0.398943 0.916976i \(-0.630623\pi\)
−0.398943 + 0.916976i \(0.630623\pi\)
\(908\) 0 0
\(909\) 0.876894 0.0290848
\(910\) 0 0
\(911\) −45.7644 −1.51624 −0.758121 0.652114i \(-0.773882\pi\)
−0.758121 + 0.652114i \(0.773882\pi\)
\(912\) 0 0
\(913\) −73.8350 −2.44358
\(914\) 0 0
\(915\) 100.514 3.32288
\(916\) 0 0
\(917\) 21.7209 0.717288
\(918\) 0 0
\(919\) −2.91536 −0.0961688 −0.0480844 0.998843i \(-0.515312\pi\)
−0.0480844 + 0.998843i \(0.515312\pi\)
\(920\) 0 0
\(921\) −68.2868 −2.25013
\(922\) 0 0
\(923\) −55.4210 −1.82421
\(924\) 0 0
\(925\) 46.1711 1.51810
\(926\) 0 0
\(927\) 2.53206 0.0831637
\(928\) 0 0
\(929\) −14.4417 −0.473815 −0.236908 0.971532i \(-0.576134\pi\)
−0.236908 + 0.971532i \(0.576134\pi\)
\(930\) 0 0
\(931\) −3.96362 −0.129902
\(932\) 0 0
\(933\) 51.4078 1.68302
\(934\) 0 0
\(935\) 133.899 4.37896
\(936\) 0 0
\(937\) 39.3960 1.28701 0.643505 0.765442i \(-0.277479\pi\)
0.643505 + 0.765442i \(0.277479\pi\)
\(938\) 0 0
\(939\) −28.5349 −0.931200
\(940\) 0 0
\(941\) 37.5954 1.22558 0.612788 0.790247i \(-0.290048\pi\)
0.612788 + 0.790247i \(0.290048\pi\)
\(942\) 0 0
\(943\) −2.45180 −0.0798415
\(944\) 0 0
\(945\) −10.5426 −0.342952
\(946\) 0 0
\(947\) −21.3522 −0.693854 −0.346927 0.937892i \(-0.612775\pi\)
−0.346927 + 0.937892i \(0.612775\pi\)
\(948\) 0 0
\(949\) −0.0312023 −0.00101287
\(950\) 0 0
\(951\) 84.8767 2.75232
\(952\) 0 0
\(953\) 37.5726 1.21709 0.608547 0.793518i \(-0.291753\pi\)
0.608547 + 0.793518i \(0.291753\pi\)
\(954\) 0 0
\(955\) 55.6362 1.80035
\(956\) 0 0
\(957\) 76.7119 2.47974
\(958\) 0 0
\(959\) 14.6696 0.473707
\(960\) 0 0
\(961\) −28.3167 −0.913444
\(962\) 0 0
\(963\) −41.6616 −1.34253
\(964\) 0 0
\(965\) −6.19182 −0.199322
\(966\) 0 0
\(967\) 11.5676 0.371990 0.185995 0.982551i \(-0.440449\pi\)
0.185995 + 0.982551i \(0.440449\pi\)
\(968\) 0 0
\(969\) −13.1397 −0.422107
\(970\) 0 0
\(971\) −60.2009 −1.93194 −0.965970 0.258656i \(-0.916720\pi\)
−0.965970 + 0.258656i \(0.916720\pi\)
\(972\) 0 0
\(973\) 19.7414 0.632879
\(974\) 0 0
\(975\) 156.872 5.02393
\(976\) 0 0
\(977\) 41.2738 1.32047 0.660233 0.751061i \(-0.270458\pi\)
0.660233 + 0.751061i \(0.270458\pi\)
\(978\) 0 0
\(979\) −45.4229 −1.45172
\(980\) 0 0
\(981\) −10.6542 −0.340163
\(982\) 0 0
\(983\) 16.5610 0.528214 0.264107 0.964493i \(-0.414923\pi\)
0.264107 + 0.964493i \(0.414923\pi\)
\(984\) 0 0
\(985\) −33.0050 −1.05163
\(986\) 0 0
\(987\) −2.47766 −0.0788649
\(988\) 0 0
\(989\) 0.502025 0.0159635
\(990\) 0 0
\(991\) 35.6243 1.13164 0.565821 0.824528i \(-0.308559\pi\)
0.565821 + 0.824528i \(0.308559\pi\)
\(992\) 0 0
\(993\) 11.0208 0.349736
\(994\) 0 0
\(995\) 49.6557 1.57419
\(996\) 0 0
\(997\) 41.0563 1.30027 0.650133 0.759821i \(-0.274713\pi\)
0.650133 + 0.759821i \(0.274713\pi\)
\(998\) 0 0
\(999\) 5.23316 0.165570
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 608.2.a.j.1.3 yes 4
3.2 odd 2 5472.2.a.bs.1.4 4
4.3 odd 2 608.2.a.i.1.1 4
8.3 odd 2 1216.2.a.x.1.4 4
8.5 even 2 1216.2.a.w.1.2 4
12.11 even 2 5472.2.a.bt.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.i.1.1 4 4.3 odd 2
608.2.a.j.1.3 yes 4 1.1 even 1 trivial
1216.2.a.w.1.2 4 8.5 even 2
1216.2.a.x.1.4 4 8.3 odd 2
5472.2.a.bs.1.4 4 3.2 odd 2
5472.2.a.bt.1.4 4 12.11 even 2