Properties

Label 5520.2.a.cc.1.3
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.20087896.1
Defining polynomial: \( x^{5} - x^{4} - 21x^{3} + 5x^{2} + 84x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.39144\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +0.944775 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +0.944775 q^{7} +1.00000 q^{9} -5.62816 q^{11} +4.42401 q^{13} +1.00000 q^{15} +3.41410 q^{17} +2.35888 q^{19} +0.944775 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.57294 q^{29} +8.99695 q^{31} -5.62816 q^{33} +0.944775 q^{35} +1.05522 q^{37} +4.42401 q^{39} -9.35583 q^{41} -4.78289 q^{43} +1.00000 q^{45} -2.35888 q^{47} -6.10740 q^{49} +3.41410 q^{51} -11.4663 q^{53} -5.62816 q^{55} +2.35888 q^{57} -13.2454 q^{59} +11.6933 q^{61} +0.944775 q^{63} +4.42401 q^{65} +8.93182 q^{67} +1.00000 q^{69} +4.68339 q^{71} +1.53067 q^{73} +1.00000 q^{75} -5.31735 q^{77} -16.7699 q^{79} +1.00000 q^{81} +5.41410 q^{83} +3.41410 q^{85} +8.57294 q^{87} +18.8351 q^{89} +4.17970 q^{91} +8.99695 q^{93} +2.35888 q^{95} +8.47344 q^{97} -5.62816 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} + 5 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{3} + 5 q^{5} + 4 q^{7} + 5 q^{9} + 4 q^{11} + 4 q^{13} + 5 q^{15} + 10 q^{17} + 4 q^{19} + 4 q^{21} + 5 q^{23} + 5 q^{25} + 5 q^{27} + 10 q^{29} - 6 q^{31} + 4 q^{33} + 4 q^{35} + 6 q^{37} + 4 q^{39} + 12 q^{41} + 2 q^{43} + 5 q^{45} - 4 q^{47} + 19 q^{49} + 10 q^{51} + 4 q^{55} + 4 q^{57} - 6 q^{59} + 16 q^{61} + 4 q^{63} + 4 q^{65} + 4 q^{67} + 5 q^{69} - 8 q^{71} + 14 q^{73} + 5 q^{75} + 16 q^{77} - 18 q^{79} + 5 q^{81} + 20 q^{83} + 10 q^{85} + 10 q^{87} + 18 q^{89} - 20 q^{91} - 6 q^{93} + 4 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.944775 0.357091 0.178546 0.983932i \(-0.442861\pi\)
0.178546 + 0.983932i \(0.442861\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.62816 −1.69696 −0.848478 0.529231i \(-0.822480\pi\)
−0.848478 + 0.529231i \(0.822480\pi\)
\(12\) 0 0
\(13\) 4.42401 1.22700 0.613500 0.789695i \(-0.289761\pi\)
0.613500 + 0.789695i \(0.289761\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 3.41410 0.828041 0.414021 0.910267i \(-0.364124\pi\)
0.414021 + 0.910267i \(0.364124\pi\)
\(18\) 0 0
\(19\) 2.35888 0.541164 0.270582 0.962697i \(-0.412784\pi\)
0.270582 + 0.962697i \(0.412784\pi\)
\(20\) 0 0
\(21\) 0.944775 0.206167
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.57294 1.59195 0.795977 0.605326i \(-0.206957\pi\)
0.795977 + 0.605326i \(0.206957\pi\)
\(30\) 0 0
\(31\) 8.99695 1.61590 0.807950 0.589251i \(-0.200577\pi\)
0.807950 + 0.589251i \(0.200577\pi\)
\(32\) 0 0
\(33\) −5.62816 −0.979738
\(34\) 0 0
\(35\) 0.944775 0.159696
\(36\) 0 0
\(37\) 1.05522 0.173478 0.0867389 0.996231i \(-0.472355\pi\)
0.0867389 + 0.996231i \(0.472355\pi\)
\(38\) 0 0
\(39\) 4.42401 0.708409
\(40\) 0 0
\(41\) −9.35583 −1.46113 −0.730567 0.682841i \(-0.760744\pi\)
−0.730567 + 0.682841i \(0.760744\pi\)
\(42\) 0 0
\(43\) −4.78289 −0.729384 −0.364692 0.931128i \(-0.618826\pi\)
−0.364692 + 0.931128i \(0.618826\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −2.35888 −0.344078 −0.172039 0.985090i \(-0.555035\pi\)
−0.172039 + 0.985090i \(0.555035\pi\)
\(48\) 0 0
\(49\) −6.10740 −0.872486
\(50\) 0 0
\(51\) 3.41410 0.478070
\(52\) 0 0
\(53\) −11.4663 −1.57501 −0.787507 0.616306i \(-0.788629\pi\)
−0.787507 + 0.616306i \(0.788629\pi\)
\(54\) 0 0
\(55\) −5.62816 −0.758901
\(56\) 0 0
\(57\) 2.35888 0.312441
\(58\) 0 0
\(59\) −13.2454 −1.72440 −0.862201 0.506567i \(-0.830914\pi\)
−0.862201 + 0.506567i \(0.830914\pi\)
\(60\) 0 0
\(61\) 11.6933 1.49717 0.748587 0.663037i \(-0.230733\pi\)
0.748587 + 0.663037i \(0.230733\pi\)
\(62\) 0 0
\(63\) 0.944775 0.119030
\(64\) 0 0
\(65\) 4.42401 0.548731
\(66\) 0 0
\(67\) 8.93182 1.09119 0.545597 0.838047i \(-0.316303\pi\)
0.545597 + 0.838047i \(0.316303\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 4.68339 0.555816 0.277908 0.960608i \(-0.410359\pi\)
0.277908 + 0.960608i \(0.410359\pi\)
\(72\) 0 0
\(73\) 1.53067 0.179152 0.0895759 0.995980i \(-0.471449\pi\)
0.0895759 + 0.995980i \(0.471449\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −5.31735 −0.605968
\(78\) 0 0
\(79\) −16.7699 −1.88676 −0.943382 0.331708i \(-0.892375\pi\)
−0.943382 + 0.331708i \(0.892375\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.41410 0.594275 0.297137 0.954835i \(-0.403968\pi\)
0.297137 + 0.954835i \(0.403968\pi\)
\(84\) 0 0
\(85\) 3.41410 0.370311
\(86\) 0 0
\(87\) 8.57294 0.919116
\(88\) 0 0
\(89\) 18.8351 1.99651 0.998256 0.0590273i \(-0.0187999\pi\)
0.998256 + 0.0590273i \(0.0187999\pi\)
\(90\) 0 0
\(91\) 4.17970 0.438151
\(92\) 0 0
\(93\) 8.99695 0.932940
\(94\) 0 0
\(95\) 2.35888 0.242016
\(96\) 0 0
\(97\) 8.47344 0.860347 0.430174 0.902746i \(-0.358452\pi\)
0.430174 + 0.902746i \(0.358452\pi\)
\(98\) 0 0
\(99\) −5.62816 −0.565652
\(100\) 0 0
\(101\) 11.4011 1.13446 0.567228 0.823561i \(-0.308016\pi\)
0.567228 + 0.823561i \(0.308016\pi\)
\(102\) 0 0
\(103\) 9.94173 0.979587 0.489794 0.871838i \(-0.337072\pi\)
0.489794 + 0.871838i \(0.337072\pi\)
\(104\) 0 0
\(105\) 0.944775 0.0922006
\(106\) 0 0
\(107\) 10.5600 1.02087 0.510436 0.859916i \(-0.329484\pi\)
0.510436 + 0.859916i \(0.329484\pi\)
\(108\) 0 0
\(109\) 8.73483 0.836645 0.418322 0.908299i \(-0.362618\pi\)
0.418322 + 0.908299i \(0.362618\pi\)
\(110\) 0 0
\(111\) 1.05522 0.100158
\(112\) 0 0
\(113\) −14.4495 −1.35930 −0.679649 0.733538i \(-0.737868\pi\)
−0.679649 + 0.733538i \(0.737868\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 4.42401 0.409000
\(118\) 0 0
\(119\) 3.22556 0.295686
\(120\) 0 0
\(121\) 20.6762 1.87966
\(122\) 0 0
\(123\) −9.35583 −0.843587
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −17.6803 −1.56888 −0.784438 0.620207i \(-0.787049\pi\)
−0.784438 + 0.620207i \(0.787049\pi\)
\(128\) 0 0
\(129\) −4.78289 −0.421110
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 2.22861 0.193245
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 8.16262 0.697380 0.348690 0.937238i \(-0.386627\pi\)
0.348690 + 0.937238i \(0.386627\pi\)
\(138\) 0 0
\(139\) 9.71470 0.823990 0.411995 0.911186i \(-0.364832\pi\)
0.411995 + 0.911186i \(0.364832\pi\)
\(140\) 0 0
\(141\) −2.35888 −0.198653
\(142\) 0 0
\(143\) −24.8991 −2.08216
\(144\) 0 0
\(145\) 8.57294 0.711944
\(146\) 0 0
\(147\) −6.10740 −0.503730
\(148\) 0 0
\(149\) 16.7870 1.37524 0.687622 0.726069i \(-0.258655\pi\)
0.687622 + 0.726069i \(0.258655\pi\)
\(150\) 0 0
\(151\) 7.67623 0.624682 0.312341 0.949970i \(-0.398887\pi\)
0.312341 + 0.949970i \(0.398887\pi\)
\(152\) 0 0
\(153\) 3.41410 0.276014
\(154\) 0 0
\(155\) 8.99695 0.722652
\(156\) 0 0
\(157\) 5.77298 0.460734 0.230367 0.973104i \(-0.426007\pi\)
0.230367 + 0.973104i \(0.426007\pi\)
\(158\) 0 0
\(159\) −11.4663 −0.909335
\(160\) 0 0
\(161\) 0.944775 0.0744587
\(162\) 0 0
\(163\) 18.0845 1.41649 0.708245 0.705967i \(-0.249487\pi\)
0.708245 + 0.705967i \(0.249487\pi\)
\(164\) 0 0
\(165\) −5.62816 −0.438152
\(166\) 0 0
\(167\) −18.9860 −1.46918 −0.734590 0.678511i \(-0.762626\pi\)
−0.734590 + 0.678511i \(0.762626\pi\)
\(168\) 0 0
\(169\) 6.57188 0.505529
\(170\) 0 0
\(171\) 2.35888 0.180388
\(172\) 0 0
\(173\) −7.88955 −0.599831 −0.299916 0.953966i \(-0.596959\pi\)
−0.299916 + 0.953966i \(0.596959\pi\)
\(174\) 0 0
\(175\) 0.944775 0.0714183
\(176\) 0 0
\(177\) −13.2454 −0.995584
\(178\) 0 0
\(179\) 8.42812 0.629948 0.314974 0.949100i \(-0.398004\pi\)
0.314974 + 0.949100i \(0.398004\pi\)
\(180\) 0 0
\(181\) 6.37595 0.473921 0.236960 0.971519i \(-0.423849\pi\)
0.236960 + 0.971519i \(0.423849\pi\)
\(182\) 0 0
\(183\) 11.6933 0.864393
\(184\) 0 0
\(185\) 1.05522 0.0775817
\(186\) 0 0
\(187\) −19.2151 −1.40515
\(188\) 0 0
\(189\) 0.944775 0.0687223
\(190\) 0 0
\(191\) 7.20690 0.521473 0.260736 0.965410i \(-0.416035\pi\)
0.260736 + 0.965410i \(0.416035\pi\)
\(192\) 0 0
\(193\) 3.39270 0.244212 0.122106 0.992517i \(-0.461035\pi\)
0.122106 + 0.992517i \(0.461035\pi\)
\(194\) 0 0
\(195\) 4.42401 0.316810
\(196\) 0 0
\(197\) −25.2304 −1.79759 −0.898796 0.438367i \(-0.855557\pi\)
−0.898796 + 0.438367i \(0.855557\pi\)
\(198\) 0 0
\(199\) −2.05827 −0.145907 −0.0729536 0.997335i \(-0.523243\pi\)
−0.0729536 + 0.997335i \(0.523243\pi\)
\(200\) 0 0
\(201\) 8.93182 0.630002
\(202\) 0 0
\(203\) 8.09950 0.568473
\(204\) 0 0
\(205\) −9.35583 −0.653439
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −13.2761 −0.918330
\(210\) 0 0
\(211\) −17.7147 −1.21953 −0.609765 0.792582i \(-0.708736\pi\)
−0.609765 + 0.792582i \(0.708736\pi\)
\(212\) 0 0
\(213\) 4.68339 0.320900
\(214\) 0 0
\(215\) −4.78289 −0.326190
\(216\) 0 0
\(217\) 8.50010 0.577024
\(218\) 0 0
\(219\) 1.53067 0.103433
\(220\) 0 0
\(221\) 15.1040 1.01601
\(222\) 0 0
\(223\) −7.32146 −0.490281 −0.245141 0.969488i \(-0.578834\pi\)
−0.245141 + 0.969488i \(0.578834\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 3.11352 0.206652 0.103326 0.994648i \(-0.467052\pi\)
0.103326 + 0.994648i \(0.467052\pi\)
\(228\) 0 0
\(229\) −6.57093 −0.434219 −0.217110 0.976147i \(-0.569663\pi\)
−0.217110 + 0.976147i \(0.569663\pi\)
\(230\) 0 0
\(231\) −5.31735 −0.349856
\(232\) 0 0
\(233\) 5.67623 0.371862 0.185931 0.982563i \(-0.440470\pi\)
0.185931 + 0.982563i \(0.440470\pi\)
\(234\) 0 0
\(235\) −2.35888 −0.153876
\(236\) 0 0
\(237\) −16.7699 −1.08932
\(238\) 0 0
\(239\) 19.3950 1.25456 0.627281 0.778793i \(-0.284168\pi\)
0.627281 + 0.778793i \(0.284168\pi\)
\(240\) 0 0
\(241\) 20.7870 1.33901 0.669504 0.742808i \(-0.266507\pi\)
0.669504 + 0.742808i \(0.266507\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −6.10740 −0.390187
\(246\) 0 0
\(247\) 10.4357 0.664008
\(248\) 0 0
\(249\) 5.41410 0.343105
\(250\) 0 0
\(251\) −4.20794 −0.265603 −0.132801 0.991143i \(-0.542397\pi\)
−0.132801 + 0.991143i \(0.542397\pi\)
\(252\) 0 0
\(253\) −5.62816 −0.353840
\(254\) 0 0
\(255\) 3.41410 0.213799
\(256\) 0 0
\(257\) 1.30977 0.0817015 0.0408507 0.999165i \(-0.486993\pi\)
0.0408507 + 0.999165i \(0.486993\pi\)
\(258\) 0 0
\(259\) 0.996950 0.0619475
\(260\) 0 0
\(261\) 8.57294 0.530652
\(262\) 0 0
\(263\) 30.3143 1.86926 0.934630 0.355622i \(-0.115731\pi\)
0.934630 + 0.355622i \(0.115731\pi\)
\(264\) 0 0
\(265\) −11.4663 −0.704368
\(266\) 0 0
\(267\) 18.8351 1.15269
\(268\) 0 0
\(269\) 16.2492 0.990729 0.495365 0.868685i \(-0.335034\pi\)
0.495365 + 0.868685i \(0.335034\pi\)
\(270\) 0 0
\(271\) −15.1074 −0.917709 −0.458855 0.888511i \(-0.651740\pi\)
−0.458855 + 0.888511i \(0.651740\pi\)
\(272\) 0 0
\(273\) 4.17970 0.252967
\(274\) 0 0
\(275\) −5.62816 −0.339391
\(276\) 0 0
\(277\) 14.4591 0.868764 0.434382 0.900729i \(-0.356967\pi\)
0.434382 + 0.900729i \(0.356967\pi\)
\(278\) 0 0
\(279\) 8.99695 0.538633
\(280\) 0 0
\(281\) −27.0835 −1.61567 −0.807833 0.589412i \(-0.799360\pi\)
−0.807833 + 0.589412i \(0.799360\pi\)
\(282\) 0 0
\(283\) −7.53912 −0.448154 −0.224077 0.974571i \(-0.571937\pi\)
−0.224077 + 0.974571i \(0.571937\pi\)
\(284\) 0 0
\(285\) 2.35888 0.139728
\(286\) 0 0
\(287\) −8.83915 −0.521759
\(288\) 0 0
\(289\) −5.34391 −0.314348
\(290\) 0 0
\(291\) 8.47344 0.496722
\(292\) 0 0
\(293\) 20.4819 1.19656 0.598282 0.801285i \(-0.295850\pi\)
0.598282 + 0.801285i \(0.295850\pi\)
\(294\) 0 0
\(295\) −13.2454 −0.771176
\(296\) 0 0
\(297\) −5.62816 −0.326579
\(298\) 0 0
\(299\) 4.42401 0.255847
\(300\) 0 0
\(301\) −4.51875 −0.260457
\(302\) 0 0
\(303\) 11.4011 0.654979
\(304\) 0 0
\(305\) 11.6933 0.669556
\(306\) 0 0
\(307\) −11.2069 −0.639612 −0.319806 0.947483i \(-0.603618\pi\)
−0.319806 + 0.947483i \(0.603618\pi\)
\(308\) 0 0
\(309\) 9.94173 0.565565
\(310\) 0 0
\(311\) −6.80271 −0.385746 −0.192873 0.981224i \(-0.561781\pi\)
−0.192873 + 0.981224i \(0.561781\pi\)
\(312\) 0 0
\(313\) −7.06208 −0.399173 −0.199586 0.979880i \(-0.563960\pi\)
−0.199586 + 0.979880i \(0.563960\pi\)
\(314\) 0 0
\(315\) 0.944775 0.0532321
\(316\) 0 0
\(317\) −2.69023 −0.151098 −0.0755491 0.997142i \(-0.524071\pi\)
−0.0755491 + 0.997142i \(0.524071\pi\)
\(318\) 0 0
\(319\) −48.2499 −2.70148
\(320\) 0 0
\(321\) 10.5600 0.589401
\(322\) 0 0
\(323\) 8.05345 0.448106
\(324\) 0 0
\(325\) 4.42401 0.245400
\(326\) 0 0
\(327\) 8.73483 0.483037
\(328\) 0 0
\(329\) −2.22861 −0.122867
\(330\) 0 0
\(331\) −26.2731 −1.44410 −0.722050 0.691841i \(-0.756800\pi\)
−0.722050 + 0.691841i \(0.756800\pi\)
\(332\) 0 0
\(333\) 1.05522 0.0578260
\(334\) 0 0
\(335\) 8.93182 0.487997
\(336\) 0 0
\(337\) 28.8674 1.57251 0.786254 0.617903i \(-0.212018\pi\)
0.786254 + 0.617903i \(0.212018\pi\)
\(338\) 0 0
\(339\) −14.4495 −0.784791
\(340\) 0 0
\(341\) −50.6363 −2.74211
\(342\) 0 0
\(343\) −12.3835 −0.668649
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) −28.6012 −1.53539 −0.767697 0.640814i \(-0.778597\pi\)
−0.767697 + 0.640814i \(0.778597\pi\)
\(348\) 0 0
\(349\) 22.0324 1.17937 0.589683 0.807635i \(-0.299253\pi\)
0.589683 + 0.807635i \(0.299253\pi\)
\(350\) 0 0
\(351\) 4.42401 0.236136
\(352\) 0 0
\(353\) −30.4830 −1.62245 −0.811224 0.584735i \(-0.801199\pi\)
−0.811224 + 0.584735i \(0.801199\pi\)
\(354\) 0 0
\(355\) 4.68339 0.248568
\(356\) 0 0
\(357\) 3.22556 0.170715
\(358\) 0 0
\(359\) −10.9860 −0.579819 −0.289909 0.957054i \(-0.593625\pi\)
−0.289909 + 0.957054i \(0.593625\pi\)
\(360\) 0 0
\(361\) −13.4357 −0.707142
\(362\) 0 0
\(363\) 20.6762 1.08522
\(364\) 0 0
\(365\) 1.53067 0.0801191
\(366\) 0 0
\(367\) 21.7669 1.13622 0.568111 0.822952i \(-0.307674\pi\)
0.568111 + 0.822952i \(0.307674\pi\)
\(368\) 0 0
\(369\) −9.35583 −0.487045
\(370\) 0 0
\(371\) −10.8331 −0.562424
\(372\) 0 0
\(373\) −12.6595 −0.655483 −0.327742 0.944767i \(-0.606288\pi\)
−0.327742 + 0.944767i \(0.606288\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 37.9268 1.95333
\(378\) 0 0
\(379\) 13.4751 0.692172 0.346086 0.938203i \(-0.387511\pi\)
0.346086 + 0.938203i \(0.387511\pi\)
\(380\) 0 0
\(381\) −17.6803 −0.905791
\(382\) 0 0
\(383\) 22.6122 1.15543 0.577714 0.816240i \(-0.303945\pi\)
0.577714 + 0.816240i \(0.303945\pi\)
\(384\) 0 0
\(385\) −5.31735 −0.270997
\(386\) 0 0
\(387\) −4.78289 −0.243128
\(388\) 0 0
\(389\) 15.6646 0.794228 0.397114 0.917769i \(-0.370012\pi\)
0.397114 + 0.917769i \(0.370012\pi\)
\(390\) 0 0
\(391\) 3.41410 0.172659
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.7699 −0.843787
\(396\) 0 0
\(397\) −29.5164 −1.48139 −0.740694 0.671843i \(-0.765503\pi\)
−0.740694 + 0.671843i \(0.765503\pi\)
\(398\) 0 0
\(399\) 2.22861 0.111570
\(400\) 0 0
\(401\) 24.4778 1.22236 0.611181 0.791491i \(-0.290695\pi\)
0.611181 + 0.791491i \(0.290695\pi\)
\(402\) 0 0
\(403\) 39.8026 1.98271
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −5.93898 −0.294384
\(408\) 0 0
\(409\) 2.82387 0.139631 0.0698157 0.997560i \(-0.477759\pi\)
0.0698157 + 0.997560i \(0.477759\pi\)
\(410\) 0 0
\(411\) 8.16262 0.402633
\(412\) 0 0
\(413\) −12.5139 −0.615769
\(414\) 0 0
\(415\) 5.41410 0.265768
\(416\) 0 0
\(417\) 9.71470 0.475731
\(418\) 0 0
\(419\) 27.4148 1.33930 0.669651 0.742676i \(-0.266444\pi\)
0.669651 + 0.742676i \(0.266444\pi\)
\(420\) 0 0
\(421\) −13.3619 −0.651222 −0.325611 0.945504i \(-0.605570\pi\)
−0.325611 + 0.945504i \(0.605570\pi\)
\(422\) 0 0
\(423\) −2.35888 −0.114693
\(424\) 0 0
\(425\) 3.41410 0.165608
\(426\) 0 0
\(427\) 11.0475 0.534628
\(428\) 0 0
\(429\) −24.8991 −1.20214
\(430\) 0 0
\(431\) 11.1261 0.535923 0.267962 0.963430i \(-0.413650\pi\)
0.267962 + 0.963430i \(0.413650\pi\)
\(432\) 0 0
\(433\) 28.2986 1.35994 0.679972 0.733238i \(-0.261992\pi\)
0.679972 + 0.733238i \(0.261992\pi\)
\(434\) 0 0
\(435\) 8.57294 0.411041
\(436\) 0 0
\(437\) 2.35888 0.112840
\(438\) 0 0
\(439\) −38.8419 −1.85382 −0.926912 0.375279i \(-0.877547\pi\)
−0.926912 + 0.375279i \(0.877547\pi\)
\(440\) 0 0
\(441\) −6.10740 −0.290829
\(442\) 0 0
\(443\) 2.44951 0.116380 0.0581898 0.998306i \(-0.481467\pi\)
0.0581898 + 0.998306i \(0.481467\pi\)
\(444\) 0 0
\(445\) 18.8351 0.892868
\(446\) 0 0
\(447\) 16.7870 0.793998
\(448\) 0 0
\(449\) −32.8590 −1.55071 −0.775355 0.631525i \(-0.782429\pi\)
−0.775355 + 0.631525i \(0.782429\pi\)
\(450\) 0 0
\(451\) 52.6561 2.47948
\(452\) 0 0
\(453\) 7.67623 0.360661
\(454\) 0 0
\(455\) 4.17970 0.195947
\(456\) 0 0
\(457\) −17.3798 −0.812991 −0.406495 0.913653i \(-0.633249\pi\)
−0.406495 + 0.913653i \(0.633249\pi\)
\(458\) 0 0
\(459\) 3.41410 0.159357
\(460\) 0 0
\(461\) −29.2996 −1.36462 −0.682308 0.731065i \(-0.739024\pi\)
−0.682308 + 0.731065i \(0.739024\pi\)
\(462\) 0 0
\(463\) 22.8521 1.06203 0.531014 0.847363i \(-0.321811\pi\)
0.531014 + 0.847363i \(0.321811\pi\)
\(464\) 0 0
\(465\) 8.99695 0.417224
\(466\) 0 0
\(467\) −22.2703 −1.03055 −0.515274 0.857025i \(-0.672310\pi\)
−0.515274 + 0.857025i \(0.672310\pi\)
\(468\) 0 0
\(469\) 8.43856 0.389656
\(470\) 0 0
\(471\) 5.77298 0.266005
\(472\) 0 0
\(473\) 26.9189 1.23773
\(474\) 0 0
\(475\) 2.35888 0.108233
\(476\) 0 0
\(477\) −11.4663 −0.525005
\(478\) 0 0
\(479\) −38.6582 −1.76634 −0.883169 0.469054i \(-0.844595\pi\)
−0.883169 + 0.469054i \(0.844595\pi\)
\(480\) 0 0
\(481\) 4.66833 0.212857
\(482\) 0 0
\(483\) 0.944775 0.0429888
\(484\) 0 0
\(485\) 8.47344 0.384759
\(486\) 0 0
\(487\) −25.6544 −1.16251 −0.581256 0.813720i \(-0.697439\pi\)
−0.581256 + 0.813720i \(0.697439\pi\)
\(488\) 0 0
\(489\) 18.0845 0.817811
\(490\) 0 0
\(491\) 24.2038 1.09230 0.546152 0.837686i \(-0.316092\pi\)
0.546152 + 0.837686i \(0.316092\pi\)
\(492\) 0 0
\(493\) 29.2689 1.31820
\(494\) 0 0
\(495\) −5.62816 −0.252967
\(496\) 0 0
\(497\) 4.42475 0.198477
\(498\) 0 0
\(499\) 31.9295 1.42936 0.714680 0.699451i \(-0.246572\pi\)
0.714680 + 0.699451i \(0.246572\pi\)
\(500\) 0 0
\(501\) −18.9860 −0.848232
\(502\) 0 0
\(503\) 8.09950 0.361139 0.180569 0.983562i \(-0.442206\pi\)
0.180569 + 0.983562i \(0.442206\pi\)
\(504\) 0 0
\(505\) 11.4011 0.507344
\(506\) 0 0
\(507\) 6.57188 0.291867
\(508\) 0 0
\(509\) −38.1436 −1.69068 −0.845342 0.534226i \(-0.820603\pi\)
−0.845342 + 0.534226i \(0.820603\pi\)
\(510\) 0 0
\(511\) 1.44614 0.0639736
\(512\) 0 0
\(513\) 2.35888 0.104147
\(514\) 0 0
\(515\) 9.94173 0.438085
\(516\) 0 0
\(517\) 13.2761 0.583884
\(518\) 0 0
\(519\) −7.88955 −0.346313
\(520\) 0 0
\(521\) −28.7740 −1.26061 −0.630307 0.776346i \(-0.717071\pi\)
−0.630307 + 0.776346i \(0.717071\pi\)
\(522\) 0 0
\(523\) −32.3426 −1.41424 −0.707121 0.707093i \(-0.750006\pi\)
−0.707121 + 0.707093i \(0.750006\pi\)
\(524\) 0 0
\(525\) 0.944775 0.0412334
\(526\) 0 0
\(527\) 30.7165 1.33803
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −13.2454 −0.574800
\(532\) 0 0
\(533\) −41.3903 −1.79281
\(534\) 0 0
\(535\) 10.5600 0.456548
\(536\) 0 0
\(537\) 8.42812 0.363700
\(538\) 0 0
\(539\) 34.3734 1.48057
\(540\) 0 0
\(541\) −32.7376 −1.40750 −0.703749 0.710449i \(-0.748492\pi\)
−0.703749 + 0.710449i \(0.748492\pi\)
\(542\) 0 0
\(543\) 6.37595 0.273618
\(544\) 0 0
\(545\) 8.73483 0.374159
\(546\) 0 0
\(547\) 34.5127 1.47565 0.737827 0.674990i \(-0.235852\pi\)
0.737827 + 0.674990i \(0.235852\pi\)
\(548\) 0 0
\(549\) 11.6933 0.499058
\(550\) 0 0
\(551\) 20.2225 0.861508
\(552\) 0 0
\(553\) −15.8438 −0.673747
\(554\) 0 0
\(555\) 1.05522 0.0447918
\(556\) 0 0
\(557\) −30.2257 −1.28071 −0.640353 0.768081i \(-0.721212\pi\)
−0.640353 + 0.768081i \(0.721212\pi\)
\(558\) 0 0
\(559\) −21.1596 −0.894954
\(560\) 0 0
\(561\) −19.2151 −0.811263
\(562\) 0 0
\(563\) 38.2703 1.61290 0.806451 0.591300i \(-0.201385\pi\)
0.806451 + 0.591300i \(0.201385\pi\)
\(564\) 0 0
\(565\) −14.4495 −0.607896
\(566\) 0 0
\(567\) 0.944775 0.0396768
\(568\) 0 0
\(569\) 2.00894 0.0842192 0.0421096 0.999113i \(-0.486592\pi\)
0.0421096 + 0.999113i \(0.486592\pi\)
\(570\) 0 0
\(571\) 7.84220 0.328186 0.164093 0.986445i \(-0.447530\pi\)
0.164093 + 0.986445i \(0.447530\pi\)
\(572\) 0 0
\(573\) 7.20690 0.301073
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −19.0470 −0.792938 −0.396469 0.918048i \(-0.629765\pi\)
−0.396469 + 0.918048i \(0.629765\pi\)
\(578\) 0 0
\(579\) 3.39270 0.140996
\(580\) 0 0
\(581\) 5.11511 0.212210
\(582\) 0 0
\(583\) 64.5341 2.67273
\(584\) 0 0
\(585\) 4.42401 0.182910
\(586\) 0 0
\(587\) 16.6231 0.686109 0.343054 0.939316i \(-0.388538\pi\)
0.343054 + 0.939316i \(0.388538\pi\)
\(588\) 0 0
\(589\) 21.2227 0.874466
\(590\) 0 0
\(591\) −25.2304 −1.03784
\(592\) 0 0
\(593\) 8.16537 0.335312 0.167656 0.985846i \(-0.446380\pi\)
0.167656 + 0.985846i \(0.446380\pi\)
\(594\) 0 0
\(595\) 3.22556 0.132235
\(596\) 0 0
\(597\) −2.05827 −0.0842396
\(598\) 0 0
\(599\) −11.3902 −0.465391 −0.232695 0.972550i \(-0.574754\pi\)
−0.232695 + 0.972550i \(0.574754\pi\)
\(600\) 0 0
\(601\) −20.7836 −0.847782 −0.423891 0.905713i \(-0.639336\pi\)
−0.423891 + 0.905713i \(0.639336\pi\)
\(602\) 0 0
\(603\) 8.93182 0.363732
\(604\) 0 0
\(605\) 20.6762 0.840608
\(606\) 0 0
\(607\) −16.6114 −0.674237 −0.337118 0.941462i \(-0.609452\pi\)
−0.337118 + 0.941462i \(0.609452\pi\)
\(608\) 0 0
\(609\) 8.09950 0.328208
\(610\) 0 0
\(611\) −10.4357 −0.422183
\(612\) 0 0
\(613\) 29.3085 1.18376 0.591880 0.806026i \(-0.298386\pi\)
0.591880 + 0.806026i \(0.298386\pi\)
\(614\) 0 0
\(615\) −9.35583 −0.377263
\(616\) 0 0
\(617\) −20.8838 −0.840748 −0.420374 0.907351i \(-0.638101\pi\)
−0.420374 + 0.907351i \(0.638101\pi\)
\(618\) 0 0
\(619\) −4.94688 −0.198832 −0.0994159 0.995046i \(-0.531697\pi\)
−0.0994159 + 0.995046i \(0.531697\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 17.7949 0.712938
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −13.2761 −0.530198
\(628\) 0 0
\(629\) 3.60265 0.143647
\(630\) 0 0
\(631\) −7.23168 −0.287889 −0.143944 0.989586i \(-0.545979\pi\)
−0.143944 + 0.989586i \(0.545979\pi\)
\(632\) 0 0
\(633\) −17.7147 −0.704096
\(634\) 0 0
\(635\) −17.6803 −0.701623
\(636\) 0 0
\(637\) −27.0192 −1.07054
\(638\) 0 0
\(639\) 4.68339 0.185272
\(640\) 0 0
\(641\) 9.40116 0.371324 0.185662 0.982614i \(-0.440557\pi\)
0.185662 + 0.982614i \(0.440557\pi\)
\(642\) 0 0
\(643\) 34.0007 1.34086 0.670429 0.741974i \(-0.266110\pi\)
0.670429 + 0.741974i \(0.266110\pi\)
\(644\) 0 0
\(645\) −4.78289 −0.188326
\(646\) 0 0
\(647\) −6.77878 −0.266501 −0.133251 0.991082i \(-0.542542\pi\)
−0.133251 + 0.991082i \(0.542542\pi\)
\(648\) 0 0
\(649\) 74.5472 2.92623
\(650\) 0 0
\(651\) 8.50010 0.333145
\(652\) 0 0
\(653\) 25.3372 0.991520 0.495760 0.868460i \(-0.334890\pi\)
0.495760 + 0.868460i \(0.334890\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.53067 0.0597173
\(658\) 0 0
\(659\) −10.8428 −0.422374 −0.211187 0.977446i \(-0.567733\pi\)
−0.211187 + 0.977446i \(0.567733\pi\)
\(660\) 0 0
\(661\) 4.61667 0.179568 0.0897838 0.995961i \(-0.471382\pi\)
0.0897838 + 0.995961i \(0.471382\pi\)
\(662\) 0 0
\(663\) 15.1040 0.586592
\(664\) 0 0
\(665\) 2.22861 0.0864217
\(666\) 0 0
\(667\) 8.57294 0.331946
\(668\) 0 0
\(669\) −7.32146 −0.283064
\(670\) 0 0
\(671\) −65.8118 −2.54064
\(672\) 0 0
\(673\) 7.64112 0.294544 0.147272 0.989096i \(-0.452951\pi\)
0.147272 + 0.989096i \(0.452951\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −44.4956 −1.71011 −0.855053 0.518541i \(-0.826475\pi\)
−0.855053 + 0.518541i \(0.826475\pi\)
\(678\) 0 0
\(679\) 8.00549 0.307223
\(680\) 0 0
\(681\) 3.11352 0.119310
\(682\) 0 0
\(683\) −10.3726 −0.396896 −0.198448 0.980111i \(-0.563590\pi\)
−0.198448 + 0.980111i \(0.563590\pi\)
\(684\) 0 0
\(685\) 8.16262 0.311878
\(686\) 0 0
\(687\) −6.57093 −0.250697
\(688\) 0 0
\(689\) −50.7269 −1.93254
\(690\) 0 0
\(691\) 18.8623 0.717557 0.358779 0.933423i \(-0.383193\pi\)
0.358779 + 0.933423i \(0.383193\pi\)
\(692\) 0 0
\(693\) −5.31735 −0.201989
\(694\) 0 0
\(695\) 9.71470 0.368500
\(696\) 0 0
\(697\) −31.9417 −1.20988
\(698\) 0 0
\(699\) 5.67623 0.214695
\(700\) 0 0
\(701\) −48.9820 −1.85002 −0.925012 0.379938i \(-0.875945\pi\)
−0.925012 + 0.379938i \(0.875945\pi\)
\(702\) 0 0
\(703\) 2.48915 0.0938799
\(704\) 0 0
\(705\) −2.35888 −0.0888405
\(706\) 0 0
\(707\) 10.7715 0.405105
\(708\) 0 0
\(709\) −28.2039 −1.05922 −0.529609 0.848242i \(-0.677661\pi\)
−0.529609 + 0.848242i \(0.677661\pi\)
\(710\) 0 0
\(711\) −16.7699 −0.628921
\(712\) 0 0
\(713\) 8.99695 0.336938
\(714\) 0 0
\(715\) −24.8991 −0.931172
\(716\) 0 0
\(717\) 19.3950 0.724321
\(718\) 0 0
\(719\) −36.0426 −1.34416 −0.672081 0.740478i \(-0.734599\pi\)
−0.672081 + 0.740478i \(0.734599\pi\)
\(720\) 0 0
\(721\) 9.39270 0.349802
\(722\) 0 0
\(723\) 20.7870 0.773077
\(724\) 0 0
\(725\) 8.57294 0.318391
\(726\) 0 0
\(727\) −13.8971 −0.515417 −0.257708 0.966223i \(-0.582967\pi\)
−0.257708 + 0.966223i \(0.582967\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.3293 −0.603960
\(732\) 0 0
\(733\) −29.4629 −1.08824 −0.544119 0.839008i \(-0.683136\pi\)
−0.544119 + 0.839008i \(0.683136\pi\)
\(734\) 0 0
\(735\) −6.10740 −0.225275
\(736\) 0 0
\(737\) −50.2697 −1.85171
\(738\) 0 0
\(739\) −36.1504 −1.32981 −0.664907 0.746926i \(-0.731529\pi\)
−0.664907 + 0.746926i \(0.731529\pi\)
\(740\) 0 0
\(741\) 10.4357 0.383365
\(742\) 0 0
\(743\) −37.7608 −1.38531 −0.692654 0.721270i \(-0.743559\pi\)
−0.692654 + 0.721270i \(0.743559\pi\)
\(744\) 0 0
\(745\) 16.7870 0.615028
\(746\) 0 0
\(747\) 5.41410 0.198092
\(748\) 0 0
\(749\) 9.97681 0.364545
\(750\) 0 0
\(751\) −37.7015 −1.37575 −0.687874 0.725830i \(-0.741456\pi\)
−0.687874 + 0.725830i \(0.741456\pi\)
\(752\) 0 0
\(753\) −4.20794 −0.153346
\(754\) 0 0
\(755\) 7.67623 0.279366
\(756\) 0 0
\(757\) −25.5481 −0.928560 −0.464280 0.885688i \(-0.653687\pi\)
−0.464280 + 0.885688i \(0.653687\pi\)
\(758\) 0 0
\(759\) −5.62816 −0.204289
\(760\) 0 0
\(761\) −24.5352 −0.889400 −0.444700 0.895680i \(-0.646690\pi\)
−0.444700 + 0.895680i \(0.646690\pi\)
\(762\) 0 0
\(763\) 8.25245 0.298759
\(764\) 0 0
\(765\) 3.41410 0.123437
\(766\) 0 0
\(767\) −58.5977 −2.11584
\(768\) 0 0
\(769\) 46.7550 1.68603 0.843014 0.537892i \(-0.180779\pi\)
0.843014 + 0.537892i \(0.180779\pi\)
\(770\) 0 0
\(771\) 1.30977 0.0471704
\(772\) 0 0
\(773\) −48.3879 −1.74039 −0.870196 0.492706i \(-0.836008\pi\)
−0.870196 + 0.492706i \(0.836008\pi\)
\(774\) 0 0
\(775\) 8.99695 0.323180
\(776\) 0 0
\(777\) 0.996950 0.0357654
\(778\) 0 0
\(779\) −22.0692 −0.790713
\(780\) 0 0
\(781\) −26.3589 −0.943195
\(782\) 0 0
\(783\) 8.57294 0.306372
\(784\) 0 0
\(785\) 5.77298 0.206046
\(786\) 0 0
\(787\) 30.2216 1.07729 0.538643 0.842534i \(-0.318937\pi\)
0.538643 + 0.842534i \(0.318937\pi\)
\(788\) 0 0
\(789\) 30.3143 1.07922
\(790\) 0 0
\(791\) −13.6516 −0.485394
\(792\) 0 0
\(793\) 51.7313 1.83703
\(794\) 0 0
\(795\) −11.4663 −0.406667
\(796\) 0 0
\(797\) 29.7443 1.05360 0.526799 0.849990i \(-0.323392\pi\)
0.526799 + 0.849990i \(0.323392\pi\)
\(798\) 0 0
\(799\) −8.05345 −0.284910
\(800\) 0 0
\(801\) 18.8351 0.665504
\(802\) 0 0
\(803\) −8.61488 −0.304013
\(804\) 0 0
\(805\) 0.944775 0.0332989
\(806\) 0 0
\(807\) 16.2492 0.571998
\(808\) 0 0
\(809\) −53.1705 −1.86938 −0.934688 0.355468i \(-0.884321\pi\)
−0.934688 + 0.355468i \(0.884321\pi\)
\(810\) 0 0
\(811\) 22.0522 0.774357 0.387179 0.922005i \(-0.373450\pi\)
0.387179 + 0.922005i \(0.373450\pi\)
\(812\) 0 0
\(813\) −15.1074 −0.529840
\(814\) 0 0
\(815\) 18.0845 0.633473
\(816\) 0 0
\(817\) −11.2822 −0.394716
\(818\) 0 0
\(819\) 4.17970 0.146050
\(820\) 0 0
\(821\) 45.1708 1.57647 0.788235 0.615374i \(-0.210995\pi\)
0.788235 + 0.615374i \(0.210995\pi\)
\(822\) 0 0
\(823\) 21.8299 0.760943 0.380471 0.924793i \(-0.375762\pi\)
0.380471 + 0.924793i \(0.375762\pi\)
\(824\) 0 0
\(825\) −5.62816 −0.195948
\(826\) 0 0
\(827\) −45.7904 −1.59229 −0.796144 0.605107i \(-0.793130\pi\)
−0.796144 + 0.605107i \(0.793130\pi\)
\(828\) 0 0
\(829\) 4.77813 0.165951 0.0829757 0.996552i \(-0.473558\pi\)
0.0829757 + 0.996552i \(0.473558\pi\)
\(830\) 0 0
\(831\) 14.4591 0.501581
\(832\) 0 0
\(833\) −20.8513 −0.722454
\(834\) 0 0
\(835\) −18.9860 −0.657038
\(836\) 0 0
\(837\) 8.99695 0.310980
\(838\) 0 0
\(839\) −15.2344 −0.525951 −0.262975 0.964803i \(-0.584704\pi\)
−0.262975 + 0.964803i \(0.584704\pi\)
\(840\) 0 0
\(841\) 44.4953 1.53432
\(842\) 0 0
\(843\) −27.0835 −0.932805
\(844\) 0 0
\(845\) 6.57188 0.226079
\(846\) 0 0
\(847\) 19.5344 0.671209
\(848\) 0 0
\(849\) −7.53912 −0.258742
\(850\) 0 0
\(851\) 1.05522 0.0361726
\(852\) 0 0
\(853\) 30.2724 1.03651 0.518253 0.855227i \(-0.326583\pi\)
0.518253 + 0.855227i \(0.326583\pi\)
\(854\) 0 0
\(855\) 2.35888 0.0806719
\(856\) 0 0
\(857\) −0.408305 −0.0139474 −0.00697371 0.999976i \(-0.502220\pi\)
−0.00697371 + 0.999976i \(0.502220\pi\)
\(858\) 0 0
\(859\) −16.6772 −0.569019 −0.284509 0.958673i \(-0.591831\pi\)
−0.284509 + 0.958673i \(0.591831\pi\)
\(860\) 0 0
\(861\) −8.83915 −0.301238
\(862\) 0 0
\(863\) −18.6210 −0.633867 −0.316934 0.948448i \(-0.602653\pi\)
−0.316934 + 0.948448i \(0.602653\pi\)
\(864\) 0 0
\(865\) −7.88955 −0.268253
\(866\) 0 0
\(867\) −5.34391 −0.181489
\(868\) 0 0
\(869\) 94.3839 3.20175
\(870\) 0 0
\(871\) 39.5145 1.33890
\(872\) 0 0
\(873\) 8.47344 0.286782
\(874\) 0 0
\(875\) 0.944775 0.0319392
\(876\) 0 0
\(877\) −6.51918 −0.220137 −0.110068 0.993924i \(-0.535107\pi\)
−0.110068 + 0.993924i \(0.535107\pi\)
\(878\) 0 0
\(879\) 20.4819 0.690837
\(880\) 0 0
\(881\) −42.3358 −1.42633 −0.713165 0.700997i \(-0.752739\pi\)
−0.713165 + 0.700997i \(0.752739\pi\)
\(882\) 0 0
\(883\) 21.5123 0.723948 0.361974 0.932188i \(-0.382103\pi\)
0.361974 + 0.932188i \(0.382103\pi\)
\(884\) 0 0
\(885\) −13.2454 −0.445238
\(886\) 0 0
\(887\) −50.8023 −1.70577 −0.852887 0.522096i \(-0.825150\pi\)
−0.852887 + 0.522096i \(0.825150\pi\)
\(888\) 0 0
\(889\) −16.7039 −0.560232
\(890\) 0 0
\(891\) −5.62816 −0.188551
\(892\) 0 0
\(893\) −5.56430 −0.186202
\(894\) 0 0
\(895\) 8.42812 0.281721
\(896\) 0 0
\(897\) 4.42401 0.147713
\(898\) 0 0
\(899\) 77.1303 2.57244
\(900\) 0 0
\(901\) −39.1470 −1.30418
\(902\) 0 0
\(903\) −4.51875 −0.150375
\(904\) 0 0
\(905\) 6.37595 0.211944
\(906\) 0 0
\(907\) 18.1255 0.601848 0.300924 0.953648i \(-0.402705\pi\)
0.300924 + 0.953648i \(0.402705\pi\)
\(908\) 0 0
\(909\) 11.4011 0.378152
\(910\) 0 0
\(911\) 9.67013 0.320386 0.160193 0.987086i \(-0.448788\pi\)
0.160193 + 0.987086i \(0.448788\pi\)
\(912\) 0 0
\(913\) −30.4715 −1.00846
\(914\) 0 0
\(915\) 11.6933 0.386569
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −23.7324 −0.782860 −0.391430 0.920208i \(-0.628020\pi\)
−0.391430 + 0.920208i \(0.628020\pi\)
\(920\) 0 0
\(921\) −11.2069 −0.369280
\(922\) 0 0
\(923\) 20.7194 0.681986
\(924\) 0 0
\(925\) 1.05522 0.0346956
\(926\) 0 0
\(927\) 9.94173 0.326529
\(928\) 0 0
\(929\) −35.0519 −1.15001 −0.575007 0.818148i \(-0.695001\pi\)
−0.575007 + 0.818148i \(0.695001\pi\)
\(930\) 0 0
\(931\) −14.4066 −0.472157
\(932\) 0 0
\(933\) −6.80271 −0.222711
\(934\) 0 0
\(935\) −19.2151 −0.628402
\(936\) 0 0
\(937\) −23.0767 −0.753884 −0.376942 0.926237i \(-0.623024\pi\)
−0.376942 + 0.926237i \(0.623024\pi\)
\(938\) 0 0
\(939\) −7.06208 −0.230462
\(940\) 0 0
\(941\) 21.9466 0.715437 0.357719 0.933829i \(-0.383555\pi\)
0.357719 + 0.933829i \(0.383555\pi\)
\(942\) 0 0
\(943\) −9.35583 −0.304668
\(944\) 0 0
\(945\) 0.944775 0.0307335
\(946\) 0 0
\(947\) 36.5494 1.18769 0.593847 0.804578i \(-0.297608\pi\)
0.593847 + 0.804578i \(0.297608\pi\)
\(948\) 0 0
\(949\) 6.77172 0.219819
\(950\) 0 0
\(951\) −2.69023 −0.0872366
\(952\) 0 0
\(953\) −18.0180 −0.583661 −0.291831 0.956470i \(-0.594264\pi\)
−0.291831 + 0.956470i \(0.594264\pi\)
\(954\) 0 0
\(955\) 7.20690 0.233210
\(956\) 0 0
\(957\) −48.2499 −1.55970
\(958\) 0 0
\(959\) 7.71185 0.249028
\(960\) 0 0
\(961\) 49.9451 1.61113
\(962\) 0 0
\(963\) 10.5600 0.340291
\(964\) 0 0
\(965\) 3.39270 0.109215
\(966\) 0 0
\(967\) 46.6837 1.50125 0.750623 0.660730i \(-0.229753\pi\)
0.750623 + 0.660730i \(0.229753\pi\)
\(968\) 0 0
\(969\) 8.05345 0.258714
\(970\) 0 0
\(971\) 29.5747 0.949098 0.474549 0.880229i \(-0.342611\pi\)
0.474549 + 0.880229i \(0.342611\pi\)
\(972\) 0 0
\(973\) 9.17821 0.294240
\(974\) 0 0
\(975\) 4.42401 0.141682
\(976\) 0 0
\(977\) 27.3195 0.874027 0.437013 0.899455i \(-0.356036\pi\)
0.437013 + 0.899455i \(0.356036\pi\)
\(978\) 0 0
\(979\) −106.007 −3.38799
\(980\) 0 0
\(981\) 8.73483 0.278882
\(982\) 0 0
\(983\) −30.5133 −0.973223 −0.486612 0.873618i \(-0.661767\pi\)
−0.486612 + 0.873618i \(0.661767\pi\)
\(984\) 0 0
\(985\) −25.2304 −0.803908
\(986\) 0 0
\(987\) −2.22861 −0.0709374
\(988\) 0 0
\(989\) −4.78289 −0.152087
\(990\) 0 0
\(991\) 10.3637 0.329215 0.164607 0.986359i \(-0.447364\pi\)
0.164607 + 0.986359i \(0.447364\pi\)
\(992\) 0 0
\(993\) −26.2731 −0.833752
\(994\) 0 0
\(995\) −2.05827 −0.0652517
\(996\) 0 0
\(997\) 47.0726 1.49080 0.745402 0.666616i \(-0.232258\pi\)
0.745402 + 0.666616i \(0.232258\pi\)
\(998\) 0 0
\(999\) 1.05522 0.0333858
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 5520.2.a.cc.1.3 5
4.3 odd 2 2760.2.a.w.1.3 5
12.11 even 2 8280.2.a.br.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.w.1.3 5 4.3 odd 2
5520.2.a.cc.1.3 5 1.1 even 1 trivial
8280.2.a.br.1.3 5 12.11 even 2