Properties

Label 5520.2.a.cc.1.2
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 1.2
Root \(-3.43588\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -2.62057 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} -2.62057 q^{7} +1.00000 q^{9} +6.55627 q^{11} +7.06828 q^{13} +1.00000 q^{15} +6.42405 q^{17} +1.80348 q^{19} -2.62057 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.17684 q^{29} -4.10856 q^{31} +6.55627 q^{33} -2.62057 q^{35} +4.62057 q^{37} +7.06828 q^{39} +4.30508 q^{41} -6.87176 q^{43} +1.00000 q^{45} -1.80348 q^{47} -0.132589 q^{49} +6.42405 q^{51} -4.93606 q^{53} +6.55627 q^{55} +1.80348 q^{57} +7.54623 q^{59} +2.70854 q^{61} -2.62057 q^{63} +7.06828 q^{65} -7.37337 q^{67} +1.00000 q^{69} -3.93569 q^{71} -5.04462 q^{73} +1.00000 q^{75} -17.1812 q^{77} -6.11897 q^{79} +1.00000 q^{81} +8.42405 q^{83} +6.42405 q^{85} -7.17684 q^{87} +11.3838 q^{89} -18.5230 q^{91} -4.10856 q^{93} +1.80348 q^{95} -17.9843 q^{97} +6.55627 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} + 5 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 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
\(7\) −2.62057 −0.990484 −0.495242 0.868755i \(-0.664921\pi\)
−0.495242 + 0.868755i \(0.664921\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.55627 1.97679 0.988395 0.151907i \(-0.0485414\pi\)
0.988395 + 0.151907i \(0.0485414\pi\)
\(12\) 0 0
\(13\) 7.06828 1.96039 0.980195 0.198037i \(-0.0634565\pi\)
0.980195 + 0.198037i \(0.0634565\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 6.42405 1.55806 0.779030 0.626986i \(-0.215712\pi\)
0.779030 + 0.626986i \(0.215712\pi\)
\(18\) 0 0
\(19\) 1.80348 0.413746 0.206873 0.978368i \(-0.433671\pi\)
0.206873 + 0.978368i \(0.433671\pi\)
\(20\) 0 0
\(21\) −2.62057 −0.571856
\(22\) 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
\(29\) −7.17684 −1.33271 −0.666353 0.745636i \(-0.732146\pi\)
−0.666353 + 0.745636i \(0.732146\pi\)
\(30\) 0 0
\(31\) −4.10856 −0.737919 −0.368960 0.929445i \(-0.620286\pi\)
−0.368960 + 0.929445i \(0.620286\pi\)
\(32\) 0 0
\(33\) 6.55627 1.14130
\(34\) 0 0
\(35\) −2.62057 −0.442958
\(36\) 0 0
\(37\) 4.62057 0.759618 0.379809 0.925065i \(-0.375990\pi\)
0.379809 + 0.925065i \(0.375990\pi\)
\(38\) 0 0
\(39\) 7.06828 1.13183
\(40\) 0 0
\(41\) 4.30508 0.672341 0.336171 0.941801i \(-0.390868\pi\)
0.336171 + 0.941801i \(0.390868\pi\)
\(42\) 0 0
\(43\) −6.87176 −1.04793 −0.523967 0.851739i \(-0.675548\pi\)
−0.523967 + 0.851739i \(0.675548\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −1.80348 −0.263064 −0.131532 0.991312i \(-0.541990\pi\)
−0.131532 + 0.991312i \(0.541990\pi\)
\(48\) 0 0
\(49\) −0.132589 −0.0189413
\(50\) 0 0
\(51\) 6.42405 0.899547
\(52\) 0 0
\(53\) −4.93606 −0.678021 −0.339010 0.940783i \(-0.610092\pi\)
−0.339010 + 0.940783i \(0.610092\pi\)
\(54\) 0 0
\(55\) 6.55627 0.884047
\(56\) 0 0
\(57\) 1.80348 0.238876
\(58\) 0 0
\(59\) 7.54623 0.982436 0.491218 0.871037i \(-0.336552\pi\)
0.491218 + 0.871037i \(0.336552\pi\)
\(60\) 0 0
\(61\) 2.70854 0.346793 0.173396 0.984852i \(-0.444526\pi\)
0.173396 + 0.984852i \(0.444526\pi\)
\(62\) 0 0
\(63\) −2.62057 −0.330161
\(64\) 0 0
\(65\) 7.06828 0.876713
\(66\) 0 0
\(67\) −7.37337 −0.900800 −0.450400 0.892827i \(-0.648719\pi\)
−0.450400 + 0.892827i \(0.648719\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −3.93569 −0.467081 −0.233541 0.972347i \(-0.575031\pi\)
−0.233541 + 0.972347i \(0.575031\pi\)
\(72\) 0 0
\(73\) −5.04462 −0.590429 −0.295214 0.955431i \(-0.595391\pi\)
−0.295214 + 0.955431i \(0.595391\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −17.1812 −1.95798
\(78\) 0 0
\(79\) −6.11897 −0.688437 −0.344219 0.938889i \(-0.611856\pi\)
−0.344219 + 0.938889i \(0.611856\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.42405 0.924660 0.462330 0.886708i \(-0.347014\pi\)
0.462330 + 0.886708i \(0.347014\pi\)
\(84\) 0 0
\(85\) 6.42405 0.696786
\(86\) 0 0
\(87\) −7.17684 −0.769438
\(88\) 0 0
\(89\) 11.3838 1.20668 0.603339 0.797485i \(-0.293837\pi\)
0.603339 + 0.797485i \(0.293837\pi\)
\(90\) 0 0
\(91\) −18.5230 −1.94173
\(92\) 0 0
\(93\) −4.10856 −0.426038
\(94\) 0 0
\(95\) 1.80348 0.185033
\(96\) 0 0
\(97\) −17.9843 −1.82603 −0.913014 0.407927i \(-0.866252\pi\)
−0.913014 + 0.407927i \(0.866252\pi\)
\(98\) 0 0
\(99\) 6.55627 0.658930
\(100\) 0 0
\(101\) 1.67126 0.166296 0.0831481 0.996537i \(-0.473503\pi\)
0.0831481 + 0.996537i \(0.473503\pi\)
\(102\) 0 0
\(103\) −6.72913 −0.663041 −0.331521 0.943448i \(-0.607562\pi\)
−0.331521 + 0.943448i \(0.607562\pi\)
\(104\) 0 0
\(105\) −2.62057 −0.255742
\(106\) 0 0
\(107\) −17.9296 −1.73332 −0.866662 0.498896i \(-0.833739\pi\)
−0.866662 + 0.498896i \(0.833739\pi\)
\(108\) 0 0
\(109\) −12.6692 −1.21349 −0.606744 0.794898i \(-0.707525\pi\)
−0.606744 + 0.794898i \(0.707525\pi\)
\(110\) 0 0
\(111\) 4.62057 0.438566
\(112\) 0 0
\(113\) 21.1708 1.99158 0.995790 0.0916634i \(-0.0292184\pi\)
0.995790 + 0.0916634i \(0.0292184\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 7.06828 0.653463
\(118\) 0 0
\(119\) −16.8347 −1.54323
\(120\) 0 0
\(121\) 31.9847 2.90770
\(122\) 0 0
\(123\) 4.30508 0.388176
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.04425 0.358870 0.179435 0.983770i \(-0.442573\pi\)
0.179435 + 0.983770i \(0.442573\pi\)
\(128\) 0 0
\(129\) −6.87176 −0.605025
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −4.72614 −0.409808
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 5.75316 0.491526 0.245763 0.969330i \(-0.420962\pi\)
0.245763 + 0.969330i \(0.420962\pi\)
\(138\) 0 0
\(139\) −4.50161 −0.381821 −0.190911 0.981607i \(-0.561144\pi\)
−0.190911 + 0.981607i \(0.561144\pi\)
\(140\) 0 0
\(141\) −1.80348 −0.151880
\(142\) 0 0
\(143\) 46.3416 3.87528
\(144\) 0 0
\(145\) −7.17684 −0.596004
\(146\) 0 0
\(147\) −0.132589 −0.0109358
\(148\) 0 0
\(149\) −14.1572 −1.15980 −0.579900 0.814688i \(-0.696908\pi\)
−0.579900 + 0.814688i \(0.696908\pi\)
\(150\) 0 0
\(151\) 18.9847 1.54495 0.772475 0.635045i \(-0.219018\pi\)
0.772475 + 0.635045i \(0.219018\pi\)
\(152\) 0 0
\(153\) 6.42405 0.519354
\(154\) 0 0
\(155\) −4.10856 −0.330007
\(156\) 0 0
\(157\) 8.22753 0.656628 0.328314 0.944569i \(-0.393520\pi\)
0.328314 + 0.944569i \(0.393520\pi\)
\(158\) 0 0
\(159\) −4.93606 −0.391455
\(160\) 0 0
\(161\) −2.62057 −0.206530
\(162\) 0 0
\(163\) −0.264439 −0.0207124 −0.0103562 0.999946i \(-0.503297\pi\)
−0.0103562 + 0.999946i \(0.503297\pi\)
\(164\) 0 0
\(165\) 6.55627 0.510405
\(166\) 0 0
\(167\) −9.45775 −0.731862 −0.365931 0.930642i \(-0.619249\pi\)
−0.365931 + 0.930642i \(0.619249\pi\)
\(168\) 0 0
\(169\) 36.9606 2.84313
\(170\) 0 0
\(171\) 1.80348 0.137915
\(172\) 0 0
\(173\) −0.758851 −0.0576944 −0.0288472 0.999584i \(-0.509184\pi\)
−0.0288472 + 0.999584i \(0.509184\pi\)
\(174\) 0 0
\(175\) −2.62057 −0.198097
\(176\) 0 0
\(177\) 7.54623 0.567210
\(178\) 0 0
\(179\) −21.9606 −1.64142 −0.820708 0.571348i \(-0.806420\pi\)
−0.820708 + 0.571348i \(0.806420\pi\)
\(180\) 0 0
\(181\) −14.4727 −1.07574 −0.537872 0.843027i \(-0.680772\pi\)
−0.537872 + 0.843027i \(0.680772\pi\)
\(182\) 0 0
\(183\) 2.70854 0.200221
\(184\) 0 0
\(185\) 4.62057 0.339711
\(186\) 0 0
\(187\) 42.1178 3.07996
\(188\) 0 0
\(189\) −2.62057 −0.190619
\(190\) 0 0
\(191\) 11.9400 0.863951 0.431976 0.901885i \(-0.357817\pi\)
0.431976 + 0.901885i \(0.357817\pi\)
\(192\) 0 0
\(193\) 11.6342 0.837448 0.418724 0.908114i \(-0.362477\pi\)
0.418724 + 0.908114i \(0.362477\pi\)
\(194\) 0 0
\(195\) 7.06828 0.506170
\(196\) 0 0
\(197\) 24.6181 1.75397 0.876984 0.480519i \(-0.159552\pi\)
0.876984 + 0.480519i \(0.159552\pi\)
\(198\) 0 0
\(199\) −18.7291 −1.32767 −0.663837 0.747878i \(-0.731073\pi\)
−0.663837 + 0.747878i \(0.731073\pi\)
\(200\) 0 0
\(201\) −7.37337 −0.520077
\(202\) 0 0
\(203\) 18.8075 1.32002
\(204\) 0 0
\(205\) 4.30508 0.300680
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 11.8241 0.817888
\(210\) 0 0
\(211\) −3.49839 −0.240839 −0.120420 0.992723i \(-0.538424\pi\)
−0.120420 + 0.992723i \(0.538424\pi\)
\(212\) 0 0
\(213\) −3.93569 −0.269669
\(214\) 0 0
\(215\) −6.87176 −0.468650
\(216\) 0 0
\(217\) 10.7668 0.730897
\(218\) 0 0
\(219\) −5.04462 −0.340884
\(220\) 0 0
\(221\) 45.4070 3.05441
\(222\) 0 0
\(223\) 13.8477 0.927313 0.463656 0.886015i \(-0.346537\pi\)
0.463656 + 0.886015i \(0.346537\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −19.5772 −1.29939 −0.649693 0.760196i \(-0.725103\pi\)
−0.649693 + 0.760196i \(0.725103\pi\)
\(228\) 0 0
\(229\) 25.4959 1.68482 0.842410 0.538838i \(-0.181136\pi\)
0.842410 + 0.538838i \(0.181136\pi\)
\(230\) 0 0
\(231\) −17.1812 −1.13044
\(232\) 0 0
\(233\) 16.9847 1.11270 0.556351 0.830947i \(-0.312201\pi\)
0.556351 + 0.830947i \(0.312201\pi\)
\(234\) 0 0
\(235\) −1.80348 −0.117646
\(236\) 0 0
\(237\) −6.11897 −0.397470
\(238\) 0 0
\(239\) −16.5459 −1.07026 −0.535131 0.844769i \(-0.679738\pi\)
−0.535131 + 0.844769i \(0.679738\pi\)
\(240\) 0 0
\(241\) −10.1572 −0.654280 −0.327140 0.944976i \(-0.606085\pi\)
−0.327140 + 0.944976i \(0.606085\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.132589 −0.00847081
\(246\) 0 0
\(247\) 12.7475 0.811103
\(248\) 0 0
\(249\) 8.42405 0.533852
\(250\) 0 0
\(251\) −5.72950 −0.361643 −0.180822 0.983516i \(-0.557876\pi\)
−0.180822 + 0.983516i \(0.557876\pi\)
\(252\) 0 0
\(253\) 6.55627 0.412189
\(254\) 0 0
\(255\) 6.42405 0.402290
\(256\) 0 0
\(257\) −19.5269 −1.21806 −0.609028 0.793149i \(-0.708440\pi\)
−0.609028 + 0.793149i \(0.708440\pi\)
\(258\) 0 0
\(259\) −12.1086 −0.752389
\(260\) 0 0
\(261\) −7.17684 −0.444235
\(262\) 0 0
\(263\) 29.0726 1.79270 0.896348 0.443352i \(-0.146211\pi\)
0.896348 + 0.443352i \(0.146211\pi\)
\(264\) 0 0
\(265\) −4.93606 −0.303220
\(266\) 0 0
\(267\) 11.3838 0.696676
\(268\) 0 0
\(269\) 11.8078 0.719936 0.359968 0.932965i \(-0.382788\pi\)
0.359968 + 0.932965i \(0.382788\pi\)
\(270\) 0 0
\(271\) −9.13259 −0.554765 −0.277383 0.960760i \(-0.589467\pi\)
−0.277383 + 0.960760i \(0.589467\pi\)
\(272\) 0 0
\(273\) −18.5230 −1.12106
\(274\) 0 0
\(275\) 6.55627 0.395358
\(276\) 0 0
\(277\) 27.8564 1.67373 0.836865 0.547409i \(-0.184386\pi\)
0.836865 + 0.547409i \(0.184386\pi\)
\(278\) 0 0
\(279\) −4.10856 −0.245973
\(280\) 0 0
\(281\) −11.9461 −0.712645 −0.356322 0.934363i \(-0.615969\pi\)
−0.356322 + 0.934363i \(0.615969\pi\)
\(282\) 0 0
\(283\) 17.0076 1.01099 0.505497 0.862828i \(-0.331309\pi\)
0.505497 + 0.862828i \(0.331309\pi\)
\(284\) 0 0
\(285\) 1.80348 0.106829
\(286\) 0 0
\(287\) −11.2818 −0.665943
\(288\) 0 0
\(289\) 24.2684 1.42755
\(290\) 0 0
\(291\) −17.9843 −1.05426
\(292\) 0 0
\(293\) −23.9472 −1.39901 −0.699506 0.714626i \(-0.746597\pi\)
−0.699506 + 0.714626i \(0.746597\pi\)
\(294\) 0 0
\(295\) 7.54623 0.439359
\(296\) 0 0
\(297\) 6.55627 0.380433
\(298\) 0 0
\(299\) 7.06828 0.408769
\(300\) 0 0
\(301\) 18.0080 1.03796
\(302\) 0 0
\(303\) 1.67126 0.0960111
\(304\) 0 0
\(305\) 2.70854 0.155091
\(306\) 0 0
\(307\) −15.9400 −0.909746 −0.454873 0.890556i \(-0.650315\pi\)
−0.454873 + 0.890556i \(0.650315\pi\)
\(308\) 0 0
\(309\) −6.72913 −0.382807
\(310\) 0 0
\(311\) −8.16023 −0.462724 −0.231362 0.972868i \(-0.574318\pi\)
−0.231362 + 0.972868i \(0.574318\pi\)
\(312\) 0 0
\(313\) 2.84375 0.160738 0.0803692 0.996765i \(-0.474390\pi\)
0.0803692 + 0.996765i \(0.474390\pi\)
\(314\) 0 0
\(315\) −2.62057 −0.147653
\(316\) 0 0
\(317\) −23.5269 −1.32140 −0.660702 0.750648i \(-0.729741\pi\)
−0.660702 + 0.750648i \(0.729741\pi\)
\(318\) 0 0
\(319\) −47.0533 −2.63448
\(320\) 0 0
\(321\) −17.9296 −1.00073
\(322\) 0 0
\(323\) 11.5856 0.644641
\(324\) 0 0
\(325\) 7.06828 0.392078
\(326\) 0 0
\(327\) −12.6692 −0.700607
\(328\) 0 0
\(329\) 4.72614 0.260561
\(330\) 0 0
\(331\) 11.9326 0.655877 0.327938 0.944699i \(-0.393646\pi\)
0.327938 + 0.944699i \(0.393646\pi\)
\(332\) 0 0
\(333\) 4.62057 0.253206
\(334\) 0 0
\(335\) −7.37337 −0.402850
\(336\) 0 0
\(337\) 12.6073 0.686765 0.343382 0.939196i \(-0.388427\pi\)
0.343382 + 0.939196i \(0.388427\pi\)
\(338\) 0 0
\(339\) 21.1708 1.14984
\(340\) 0 0
\(341\) −26.9368 −1.45871
\(342\) 0 0
\(343\) 18.6915 1.00925
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) 5.85132 0.314115 0.157058 0.987589i \(-0.449799\pi\)
0.157058 + 0.987589i \(0.449799\pi\)
\(348\) 0 0
\(349\) −29.7034 −1.58999 −0.794993 0.606618i \(-0.792526\pi\)
−0.794993 + 0.606618i \(0.792526\pi\)
\(350\) 0 0
\(351\) 7.06828 0.377277
\(352\) 0 0
\(353\) −10.1160 −0.538419 −0.269209 0.963082i \(-0.586762\pi\)
−0.269209 + 0.963082i \(0.586762\pi\)
\(354\) 0 0
\(355\) −3.93569 −0.208885
\(356\) 0 0
\(357\) −16.8347 −0.890987
\(358\) 0 0
\(359\) −1.45775 −0.0769369 −0.0384684 0.999260i \(-0.512248\pi\)
−0.0384684 + 0.999260i \(0.512248\pi\)
\(360\) 0 0
\(361\) −15.7475 −0.828815
\(362\) 0 0
\(363\) 31.9847 1.67876
\(364\) 0 0
\(365\) −5.04462 −0.264048
\(366\) 0 0
\(367\) −1.98959 −0.103856 −0.0519280 0.998651i \(-0.516537\pi\)
−0.0519280 + 0.998651i \(0.516537\pi\)
\(368\) 0 0
\(369\) 4.30508 0.224114
\(370\) 0 0
\(371\) 12.9353 0.671569
\(372\) 0 0
\(373\) 5.12218 0.265217 0.132608 0.991169i \(-0.457665\pi\)
0.132608 + 0.991169i \(0.457665\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −50.7280 −2.61262
\(378\) 0 0
\(379\) 9.79084 0.502922 0.251461 0.967867i \(-0.419089\pi\)
0.251461 + 0.967867i \(0.419089\pi\)
\(380\) 0 0
\(381\) 4.04425 0.207193
\(382\) 0 0
\(383\) −15.4176 −0.787804 −0.393902 0.919152i \(-0.628875\pi\)
−0.393902 + 0.919152i \(0.628875\pi\)
\(384\) 0 0
\(385\) −17.1812 −0.875635
\(386\) 0 0
\(387\) −6.87176 −0.349311
\(388\) 0 0
\(389\) −38.3616 −1.94501 −0.972506 0.232877i \(-0.925186\pi\)
−0.972506 + 0.232877i \(0.925186\pi\)
\(390\) 0 0
\(391\) 6.42405 0.324878
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6.11897 −0.307879
\(396\) 0 0
\(397\) 2.87102 0.144092 0.0720462 0.997401i \(-0.477047\pi\)
0.0720462 + 0.997401i \(0.477047\pi\)
\(398\) 0 0
\(399\) −4.72614 −0.236603
\(400\) 0 0
\(401\) −29.8453 −1.49040 −0.745201 0.666840i \(-0.767646\pi\)
−0.745201 + 0.666840i \(0.767646\pi\)
\(402\) 0 0
\(403\) −29.0405 −1.44661
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 30.2937 1.50160
\(408\) 0 0
\(409\) −6.21788 −0.307454 −0.153727 0.988113i \(-0.549128\pi\)
−0.153727 + 0.988113i \(0.549128\pi\)
\(410\) 0 0
\(411\) 5.75316 0.283783
\(412\) 0 0
\(413\) −19.7755 −0.973087
\(414\) 0 0
\(415\) 8.42405 0.413520
\(416\) 0 0
\(417\) −4.50161 −0.220445
\(418\) 0 0
\(419\) 33.6695 1.64487 0.822433 0.568863i \(-0.192616\pi\)
0.822433 + 0.568863i \(0.192616\pi\)
\(420\) 0 0
\(421\) 17.0149 0.829256 0.414628 0.909991i \(-0.363912\pi\)
0.414628 + 0.909991i \(0.363912\pi\)
\(422\) 0 0
\(423\) −1.80348 −0.0876880
\(424\) 0 0
\(425\) 6.42405 0.311612
\(426\) 0 0
\(427\) −7.09793 −0.343493
\(428\) 0 0
\(429\) 46.3416 2.23739
\(430\) 0 0
\(431\) −19.6422 −0.946129 −0.473065 0.881028i \(-0.656852\pi\)
−0.473065 + 0.881028i \(0.656852\pi\)
\(432\) 0 0
\(433\) −5.24476 −0.252047 −0.126023 0.992027i \(-0.540221\pi\)
−0.126023 + 0.992027i \(0.540221\pi\)
\(434\) 0 0
\(435\) −7.17684 −0.344103
\(436\) 0 0
\(437\) 1.80348 0.0862719
\(438\) 0 0
\(439\) −17.9194 −0.855249 −0.427624 0.903957i \(-0.640649\pi\)
−0.427624 + 0.903957i \(0.640649\pi\)
\(440\) 0 0
\(441\) −0.132589 −0.00631377
\(442\) 0 0
\(443\) 9.75616 0.463529 0.231764 0.972772i \(-0.425550\pi\)
0.231764 + 0.972772i \(0.425550\pi\)
\(444\) 0 0
\(445\) 11.3838 0.539643
\(446\) 0 0
\(447\) −14.1572 −0.669611
\(448\) 0 0
\(449\) −34.5703 −1.63147 −0.815736 0.578425i \(-0.803668\pi\)
−0.815736 + 0.578425i \(0.803668\pi\)
\(450\) 0 0
\(451\) 28.2253 1.32908
\(452\) 0 0
\(453\) 18.9847 0.891978
\(454\) 0 0
\(455\) −18.5230 −0.868370
\(456\) 0 0
\(457\) 30.0455 1.40547 0.702736 0.711451i \(-0.251962\pi\)
0.702736 + 0.711451i \(0.251962\pi\)
\(458\) 0 0
\(459\) 6.42405 0.299849
\(460\) 0 0
\(461\) −15.2849 −0.711888 −0.355944 0.934507i \(-0.615841\pi\)
−0.355944 + 0.934507i \(0.615841\pi\)
\(462\) 0 0
\(463\) −4.89235 −0.227367 −0.113684 0.993517i \(-0.536265\pi\)
−0.113684 + 0.993517i \(0.536265\pi\)
\(464\) 0 0
\(465\) −4.10856 −0.190530
\(466\) 0 0
\(467\) 35.4972 1.64262 0.821308 0.570485i \(-0.193245\pi\)
0.821308 + 0.570485i \(0.193245\pi\)
\(468\) 0 0
\(469\) 19.3225 0.892228
\(470\) 0 0
\(471\) 8.22753 0.379104
\(472\) 0 0
\(473\) −45.0531 −2.07154
\(474\) 0 0
\(475\) 1.80348 0.0827491
\(476\) 0 0
\(477\) −4.93606 −0.226007
\(478\) 0 0
\(479\) −7.80421 −0.356584 −0.178292 0.983978i \(-0.557057\pi\)
−0.178292 + 0.983978i \(0.557057\pi\)
\(480\) 0 0
\(481\) 32.6595 1.48915
\(482\) 0 0
\(483\) −2.62057 −0.119240
\(484\) 0 0
\(485\) −17.9843 −0.816625
\(486\) 0 0
\(487\) 21.5498 0.976517 0.488258 0.872699i \(-0.337632\pi\)
0.488258 + 0.872699i \(0.337632\pi\)
\(488\) 0 0
\(489\) −0.264439 −0.0119583
\(490\) 0 0
\(491\) 15.8315 0.714465 0.357232 0.934016i \(-0.383720\pi\)
0.357232 + 0.934016i \(0.383720\pi\)
\(492\) 0 0
\(493\) −46.1044 −2.07644
\(494\) 0 0
\(495\) 6.55627 0.294682
\(496\) 0 0
\(497\) 10.3138 0.462636
\(498\) 0 0
\(499\) 5.76357 0.258013 0.129006 0.991644i \(-0.458821\pi\)
0.129006 + 0.991644i \(0.458821\pi\)
\(500\) 0 0
\(501\) −9.45775 −0.422541
\(502\) 0 0
\(503\) 18.8075 0.838583 0.419291 0.907852i \(-0.362279\pi\)
0.419291 + 0.907852i \(0.362279\pi\)
\(504\) 0 0
\(505\) 1.67126 0.0743699
\(506\) 0 0
\(507\) 36.9606 1.64148
\(508\) 0 0
\(509\) 3.21675 0.142580 0.0712900 0.997456i \(-0.477288\pi\)
0.0712900 + 0.997456i \(0.477288\pi\)
\(510\) 0 0
\(511\) 13.2198 0.584810
\(512\) 0 0
\(513\) 1.80348 0.0796254
\(514\) 0 0
\(515\) −6.72913 −0.296521
\(516\) 0 0
\(517\) −11.8241 −0.520022
\(518\) 0 0
\(519\) −0.758851 −0.0333099
\(520\) 0 0
\(521\) 14.9100 0.653217 0.326609 0.945160i \(-0.394094\pi\)
0.326609 + 0.945160i \(0.394094\pi\)
\(522\) 0 0
\(523\) −12.3982 −0.542134 −0.271067 0.962561i \(-0.587376\pi\)
−0.271067 + 0.962561i \(0.587376\pi\)
\(524\) 0 0
\(525\) −2.62057 −0.114371
\(526\) 0 0
\(527\) −26.3936 −1.14972
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 7.54623 0.327479
\(532\) 0 0
\(533\) 30.4296 1.31805
\(534\) 0 0
\(535\) −17.9296 −0.775166
\(536\) 0 0
\(537\) −21.9606 −0.947672
\(538\) 0 0
\(539\) −0.869290 −0.0374430
\(540\) 0 0
\(541\) −30.8954 −1.32830 −0.664149 0.747600i \(-0.731206\pi\)
−0.664149 + 0.747600i \(0.731206\pi\)
\(542\) 0 0
\(543\) −14.4727 −0.621081
\(544\) 0 0
\(545\) −12.6692 −0.542688
\(546\) 0 0
\(547\) −14.2251 −0.608220 −0.304110 0.952637i \(-0.598359\pi\)
−0.304110 + 0.952637i \(0.598359\pi\)
\(548\) 0 0
\(549\) 2.70854 0.115598
\(550\) 0 0
\(551\) −12.9433 −0.551401
\(552\) 0 0
\(553\) 16.0352 0.681886
\(554\) 0 0
\(555\) 4.62057 0.196132
\(556\) 0 0
\(557\) −14.6989 −0.622811 −0.311406 0.950277i \(-0.600800\pi\)
−0.311406 + 0.950277i \(0.600800\pi\)
\(558\) 0 0
\(559\) −48.5715 −2.05436
\(560\) 0 0
\(561\) 42.1178 1.77821
\(562\) 0 0
\(563\) −19.4972 −0.821710 −0.410855 0.911701i \(-0.634770\pi\)
−0.410855 + 0.911701i \(0.634770\pi\)
\(564\) 0 0
\(565\) 21.1708 0.890662
\(566\) 0 0
\(567\) −2.62057 −0.110054
\(568\) 0 0
\(569\) −17.8854 −0.749795 −0.374898 0.927066i \(-0.622322\pi\)
−0.374898 + 0.927066i \(0.622322\pi\)
\(570\) 0 0
\(571\) 23.3904 0.978856 0.489428 0.872044i \(-0.337206\pi\)
0.489428 + 0.872044i \(0.337206\pi\)
\(572\) 0 0
\(573\) 11.9400 0.498802
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −45.7515 −1.90466 −0.952329 0.305072i \(-0.901320\pi\)
−0.952329 + 0.305072i \(0.901320\pi\)
\(578\) 0 0
\(579\) 11.6342 0.483501
\(580\) 0 0
\(581\) −22.0759 −0.915861
\(582\) 0 0
\(583\) −32.3622 −1.34030
\(584\) 0 0
\(585\) 7.06828 0.292238
\(586\) 0 0
\(587\) −24.9839 −1.03120 −0.515599 0.856830i \(-0.672430\pi\)
−0.515599 + 0.856830i \(0.672430\pi\)
\(588\) 0 0
\(589\) −7.40969 −0.305311
\(590\) 0 0
\(591\) 24.6181 1.01265
\(592\) 0 0
\(593\) 25.3178 1.03968 0.519838 0.854265i \(-0.325992\pi\)
0.519838 + 0.854265i \(0.325992\pi\)
\(594\) 0 0
\(595\) −16.8347 −0.690155
\(596\) 0 0
\(597\) −18.7291 −0.766532
\(598\) 0 0
\(599\) −5.23756 −0.214001 −0.107000 0.994259i \(-0.534125\pi\)
−0.107000 + 0.994259i \(0.534125\pi\)
\(600\) 0 0
\(601\) −26.1173 −1.06535 −0.532673 0.846321i \(-0.678812\pi\)
−0.532673 + 0.846321i \(0.678812\pi\)
\(602\) 0 0
\(603\) −7.37337 −0.300267
\(604\) 0 0
\(605\) 31.9847 1.30036
\(606\) 0 0
\(607\) 24.6631 1.00105 0.500523 0.865723i \(-0.333141\pi\)
0.500523 + 0.865723i \(0.333141\pi\)
\(608\) 0 0
\(609\) 18.8075 0.762117
\(610\) 0 0
\(611\) −12.7475 −0.515708
\(612\) 0 0
\(613\) −4.60052 −0.185813 −0.0929067 0.995675i \(-0.529616\pi\)
−0.0929067 + 0.995675i \(0.529616\pi\)
\(614\) 0 0
\(615\) 4.30508 0.173598
\(616\) 0 0
\(617\) 18.9143 0.761461 0.380731 0.924686i \(-0.375673\pi\)
0.380731 + 0.924686i \(0.375673\pi\)
\(618\) 0 0
\(619\) 47.9686 1.92802 0.964010 0.265865i \(-0.0856575\pi\)
0.964010 + 0.265865i \(0.0856575\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −29.8320 −1.19519
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 11.8241 0.472208
\(628\) 0 0
\(629\) 29.6828 1.18353
\(630\) 0 0
\(631\) 29.5445 1.17615 0.588074 0.808807i \(-0.299886\pi\)
0.588074 + 0.808807i \(0.299886\pi\)
\(632\) 0 0
\(633\) −3.49839 −0.139049
\(634\) 0 0
\(635\) 4.04425 0.160491
\(636\) 0 0
\(637\) −0.937178 −0.0371323
\(638\) 0 0
\(639\) −3.93569 −0.155694
\(640\) 0 0
\(641\) −43.2557 −1.70850 −0.854248 0.519865i \(-0.825982\pi\)
−0.854248 + 0.519865i \(0.825982\pi\)
\(642\) 0 0
\(643\) 37.2455 1.46882 0.734410 0.678707i \(-0.237459\pi\)
0.734410 + 0.678707i \(0.237459\pi\)
\(644\) 0 0
\(645\) −6.87176 −0.270575
\(646\) 0 0
\(647\) −41.9007 −1.64729 −0.823643 0.567109i \(-0.808062\pi\)
−0.823643 + 0.567109i \(0.808062\pi\)
\(648\) 0 0
\(649\) 49.4751 1.94207
\(650\) 0 0
\(651\) 10.7668 0.421984
\(652\) 0 0
\(653\) 36.4697 1.42717 0.713584 0.700570i \(-0.247071\pi\)
0.713584 + 0.700570i \(0.247071\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −5.04462 −0.196810
\(658\) 0 0
\(659\) 17.8247 0.694350 0.347175 0.937800i \(-0.387141\pi\)
0.347175 + 0.937800i \(0.387141\pi\)
\(660\) 0 0
\(661\) −2.70189 −0.105091 −0.0525456 0.998619i \(-0.516733\pi\)
−0.0525456 + 0.998619i \(0.516733\pi\)
\(662\) 0 0
\(663\) 45.4070 1.76346
\(664\) 0 0
\(665\) −4.72614 −0.183272
\(666\) 0 0
\(667\) −7.17684 −0.277889
\(668\) 0 0
\(669\) 13.8477 0.535384
\(670\) 0 0
\(671\) 17.7579 0.685537
\(672\) 0 0
\(673\) 8.19652 0.315953 0.157976 0.987443i \(-0.449503\pi\)
0.157976 + 0.987443i \(0.449503\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 26.8759 1.03292 0.516462 0.856310i \(-0.327249\pi\)
0.516462 + 0.856310i \(0.327249\pi\)
\(678\) 0 0
\(679\) 47.1292 1.80865
\(680\) 0 0
\(681\) −19.5772 −0.750201
\(682\) 0 0
\(683\) 17.1252 0.655277 0.327638 0.944803i \(-0.393747\pi\)
0.327638 + 0.944803i \(0.393747\pi\)
\(684\) 0 0
\(685\) 5.75316 0.219817
\(686\) 0 0
\(687\) 25.4959 0.972731
\(688\) 0 0
\(689\) −34.8895 −1.32918
\(690\) 0 0
\(691\) −15.7042 −0.597414 −0.298707 0.954345i \(-0.596555\pi\)
−0.298707 + 0.954345i \(0.596555\pi\)
\(692\) 0 0
\(693\) −17.1812 −0.652660
\(694\) 0 0
\(695\) −4.50161 −0.170756
\(696\) 0 0
\(697\) 27.6561 1.04755
\(698\) 0 0
\(699\) 16.9847 0.642419
\(700\) 0 0
\(701\) −6.81955 −0.257571 −0.128785 0.991672i \(-0.541108\pi\)
−0.128785 + 0.991672i \(0.541108\pi\)
\(702\) 0 0
\(703\) 8.33309 0.314289
\(704\) 0 0
\(705\) −1.80348 −0.0679228
\(706\) 0 0
\(707\) −4.37965 −0.164714
\(708\) 0 0
\(709\) 23.0955 0.867368 0.433684 0.901065i \(-0.357213\pi\)
0.433684 + 0.901065i \(0.357213\pi\)
\(710\) 0 0
\(711\) −6.11897 −0.229479
\(712\) 0 0
\(713\) −4.10856 −0.153867
\(714\) 0 0
\(715\) 46.3416 1.73308
\(716\) 0 0
\(717\) −16.5459 −0.617917
\(718\) 0 0
\(719\) 22.5152 0.839675 0.419838 0.907599i \(-0.362087\pi\)
0.419838 + 0.907599i \(0.362087\pi\)
\(720\) 0 0
\(721\) 17.6342 0.656732
\(722\) 0 0
\(723\) −10.1572 −0.377749
\(724\) 0 0
\(725\) −7.17684 −0.266541
\(726\) 0 0
\(727\) 3.45998 0.128323 0.0641617 0.997940i \(-0.479563\pi\)
0.0641617 + 0.997940i \(0.479563\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −44.1445 −1.63274
\(732\) 0 0
\(733\) −16.2835 −0.601446 −0.300723 0.953711i \(-0.597228\pi\)
−0.300723 + 0.953711i \(0.597228\pi\)
\(734\) 0 0
\(735\) −0.132589 −0.00489062
\(736\) 0 0
\(737\) −48.3418 −1.78069
\(738\) 0 0
\(739\) −24.2459 −0.891899 −0.445949 0.895058i \(-0.647134\pi\)
−0.445949 + 0.895058i \(0.647134\pi\)
\(740\) 0 0
\(741\) 12.7475 0.468290
\(742\) 0 0
\(743\) −30.7202 −1.12702 −0.563508 0.826111i \(-0.690549\pi\)
−0.563508 + 0.826111i \(0.690549\pi\)
\(744\) 0 0
\(745\) −14.1572 −0.518678
\(746\) 0 0
\(747\) 8.42405 0.308220
\(748\) 0 0
\(749\) 46.9860 1.71683
\(750\) 0 0
\(751\) 37.3493 1.36290 0.681448 0.731867i \(-0.261351\pi\)
0.681448 + 0.731867i \(0.261351\pi\)
\(752\) 0 0
\(753\) −5.72950 −0.208795
\(754\) 0 0
\(755\) 18.9847 0.690923
\(756\) 0 0
\(757\) 18.8930 0.686677 0.343338 0.939212i \(-0.388442\pi\)
0.343338 + 0.939212i \(0.388442\pi\)
\(758\) 0 0
\(759\) 6.55627 0.237978
\(760\) 0 0
\(761\) −37.5549 −1.36137 −0.680683 0.732578i \(-0.738317\pi\)
−0.680683 + 0.732578i \(0.738317\pi\)
\(762\) 0 0
\(763\) 33.2005 1.20194
\(764\) 0 0
\(765\) 6.42405 0.232262
\(766\) 0 0
\(767\) 53.3389 1.92596
\(768\) 0 0
\(769\) −35.8799 −1.29386 −0.646931 0.762549i \(-0.723948\pi\)
−0.646931 + 0.762549i \(0.723948\pi\)
\(770\) 0 0
\(771\) −19.5269 −0.703245
\(772\) 0 0
\(773\) −32.3745 −1.16443 −0.582215 0.813035i \(-0.697814\pi\)
−0.582215 + 0.813035i \(0.697814\pi\)
\(774\) 0 0
\(775\) −4.10856 −0.147584
\(776\) 0 0
\(777\) −12.1086 −0.434392
\(778\) 0 0
\(779\) 7.76411 0.278178
\(780\) 0 0
\(781\) −25.8035 −0.923321
\(782\) 0 0
\(783\) −7.17684 −0.256479
\(784\) 0 0
\(785\) 8.22753 0.293653
\(786\) 0 0
\(787\) 47.7278 1.70131 0.850656 0.525723i \(-0.176205\pi\)
0.850656 + 0.525723i \(0.176205\pi\)
\(788\) 0 0
\(789\) 29.0726 1.03501
\(790\) 0 0
\(791\) −55.4796 −1.97263
\(792\) 0 0
\(793\) 19.1447 0.679849
\(794\) 0 0
\(795\) −4.93606 −0.175064
\(796\) 0 0
\(797\) −12.8427 −0.454910 −0.227455 0.973789i \(-0.573041\pi\)
−0.227455 + 0.973789i \(0.573041\pi\)
\(798\) 0 0
\(799\) −11.5856 −0.409870
\(800\) 0 0
\(801\) 11.3838 0.402226
\(802\) 0 0
\(803\) −33.0739 −1.16715
\(804\) 0 0
\(805\) −2.62057 −0.0923631
\(806\) 0 0
\(807\) 11.8078 0.415655
\(808\) 0 0
\(809\) 8.84864 0.311102 0.155551 0.987828i \(-0.450285\pi\)
0.155551 + 0.987828i \(0.450285\pi\)
\(810\) 0 0
\(811\) −30.4149 −1.06801 −0.534006 0.845480i \(-0.679314\pi\)
−0.534006 + 0.845480i \(0.679314\pi\)
\(812\) 0 0
\(813\) −9.13259 −0.320294
\(814\) 0 0
\(815\) −0.264439 −0.00926289
\(816\) 0 0
\(817\) −12.3930 −0.433578
\(818\) 0 0
\(819\) −18.5230 −0.647245
\(820\) 0 0
\(821\) 31.2463 1.09050 0.545251 0.838273i \(-0.316434\pi\)
0.545251 + 0.838273i \(0.316434\pi\)
\(822\) 0 0
\(823\) 50.6232 1.76462 0.882308 0.470673i \(-0.155989\pi\)
0.882308 + 0.470673i \(0.155989\pi\)
\(824\) 0 0
\(825\) 6.55627 0.228260
\(826\) 0 0
\(827\) 32.5478 1.13180 0.565898 0.824475i \(-0.308529\pi\)
0.565898 + 0.824475i \(0.308529\pi\)
\(828\) 0 0
\(829\) −29.0119 −1.00763 −0.503813 0.863813i \(-0.668070\pi\)
−0.503813 + 0.863813i \(0.668070\pi\)
\(830\) 0 0
\(831\) 27.8564 0.966329
\(832\) 0 0
\(833\) −0.851759 −0.0295117
\(834\) 0 0
\(835\) −9.45775 −0.327299
\(836\) 0 0
\(837\) −4.10856 −0.142013
\(838\) 0 0
\(839\) 1.97993 0.0683547 0.0341774 0.999416i \(-0.489119\pi\)
0.0341774 + 0.999416i \(0.489119\pi\)
\(840\) 0 0
\(841\) 22.5071 0.776106
\(842\) 0 0
\(843\) −11.9461 −0.411446
\(844\) 0 0
\(845\) 36.9606 1.27148
\(846\) 0 0
\(847\) −83.8182 −2.88003
\(848\) 0 0
\(849\) 17.0076 0.583698
\(850\) 0 0
\(851\) 4.62057 0.158391
\(852\) 0 0
\(853\) −11.1781 −0.382732 −0.191366 0.981519i \(-0.561292\pi\)
−0.191366 + 0.981519i \(0.561292\pi\)
\(854\) 0 0
\(855\) 1.80348 0.0616776
\(856\) 0 0
\(857\) 29.2491 0.999130 0.499565 0.866276i \(-0.333493\pi\)
0.499565 + 0.866276i \(0.333493\pi\)
\(858\) 0 0
\(859\) −47.5143 −1.62117 −0.810584 0.585623i \(-0.800850\pi\)
−0.810584 + 0.585623i \(0.800850\pi\)
\(860\) 0 0
\(861\) −11.2818 −0.384483
\(862\) 0 0
\(863\) 16.5628 0.563806 0.281903 0.959443i \(-0.409034\pi\)
0.281903 + 0.959443i \(0.409034\pi\)
\(864\) 0 0
\(865\) −0.758851 −0.0258017
\(866\) 0 0
\(867\) 24.2684 0.824199
\(868\) 0 0
\(869\) −40.1176 −1.36090
\(870\) 0 0
\(871\) −52.1171 −1.76592
\(872\) 0 0
\(873\) −17.9843 −0.608676
\(874\) 0 0
\(875\) −2.62057 −0.0885916
\(876\) 0 0
\(877\) −4.80976 −0.162414 −0.0812070 0.996697i \(-0.525877\pi\)
−0.0812070 + 0.996697i \(0.525877\pi\)
\(878\) 0 0
\(879\) −23.9472 −0.807720
\(880\) 0 0
\(881\) 29.8046 1.00414 0.502072 0.864826i \(-0.332571\pi\)
0.502072 + 0.864826i \(0.332571\pi\)
\(882\) 0 0
\(883\) 22.1579 0.745673 0.372836 0.927897i \(-0.378385\pi\)
0.372836 + 0.927897i \(0.378385\pi\)
\(884\) 0 0
\(885\) 7.54623 0.253664
\(886\) 0 0
\(887\) −31.3425 −1.05238 −0.526189 0.850367i \(-0.676380\pi\)
−0.526189 + 0.850367i \(0.676380\pi\)
\(888\) 0 0
\(889\) −10.5983 −0.355455
\(890\) 0 0
\(891\) 6.55627 0.219643
\(892\) 0 0
\(893\) −3.25252 −0.108842
\(894\) 0 0
\(895\) −21.9606 −0.734063
\(896\) 0 0
\(897\) 7.06828 0.236003
\(898\) 0 0
\(899\) 29.4865 0.983430
\(900\) 0 0
\(901\) −31.7095 −1.05640
\(902\) 0 0
\(903\) 18.0080 0.599267
\(904\) 0 0
\(905\) −14.4727 −0.481087
\(906\) 0 0
\(907\) −11.3541 −0.377006 −0.188503 0.982073i \(-0.560364\pi\)
−0.188503 + 0.982073i \(0.560364\pi\)
\(908\) 0 0
\(909\) 1.67126 0.0554321
\(910\) 0 0
\(911\) −5.23245 −0.173359 −0.0866794 0.996236i \(-0.527626\pi\)
−0.0866794 + 0.996236i \(0.527626\pi\)
\(912\) 0 0
\(913\) 55.2303 1.82786
\(914\) 0 0
\(915\) 2.70854 0.0895415
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −58.1349 −1.91769 −0.958846 0.283925i \(-0.908363\pi\)
−0.958846 + 0.283925i \(0.908363\pi\)
\(920\) 0 0
\(921\) −15.9400 −0.525242
\(922\) 0 0
\(923\) −27.8186 −0.915661
\(924\) 0 0
\(925\) 4.62057 0.151924
\(926\) 0 0
\(927\) −6.72913 −0.221014
\(928\) 0 0
\(929\) −31.9681 −1.04884 −0.524419 0.851460i \(-0.675717\pi\)
−0.524419 + 0.851460i \(0.675717\pi\)
\(930\) 0 0
\(931\) −0.239121 −0.00783688
\(932\) 0 0
\(933\) −8.16023 −0.267154
\(934\) 0 0
\(935\) 42.1178 1.37740
\(936\) 0 0
\(937\) 44.2567 1.44580 0.722902 0.690951i \(-0.242808\pi\)
0.722902 + 0.690951i \(0.242808\pi\)
\(938\) 0 0
\(939\) 2.84375 0.0928023
\(940\) 0 0
\(941\) 18.4144 0.600292 0.300146 0.953893i \(-0.402965\pi\)
0.300146 + 0.953893i \(0.402965\pi\)
\(942\) 0 0
\(943\) 4.30508 0.140193
\(944\) 0 0
\(945\) −2.62057 −0.0852473
\(946\) 0 0
\(947\) −48.8625 −1.58782 −0.793909 0.608037i \(-0.791957\pi\)
−0.793909 + 0.608037i \(0.791957\pi\)
\(948\) 0 0
\(949\) −35.6568 −1.15747
\(950\) 0 0
\(951\) −23.5269 −0.762913
\(952\) 0 0
\(953\) −49.0643 −1.58935 −0.794674 0.607037i \(-0.792358\pi\)
−0.794674 + 0.607037i \(0.792358\pi\)
\(954\) 0 0
\(955\) 11.9400 0.386371
\(956\) 0 0
\(957\) −47.0533 −1.52102
\(958\) 0 0
\(959\) −15.0766 −0.486849
\(960\) 0 0
\(961\) −14.1197 −0.455475
\(962\) 0 0
\(963\) −17.9296 −0.577775
\(964\) 0 0
\(965\) 11.6342 0.374518
\(966\) 0 0
\(967\) 20.4921 0.658981 0.329491 0.944159i \(-0.393123\pi\)
0.329491 + 0.944159i \(0.393123\pi\)
\(968\) 0 0
\(969\) 11.5856 0.372184
\(970\) 0 0
\(971\) 13.8581 0.444728 0.222364 0.974964i \(-0.428623\pi\)
0.222364 + 0.974964i \(0.428623\pi\)
\(972\) 0 0
\(973\) 11.7968 0.378188
\(974\) 0 0
\(975\) 7.06828 0.226366
\(976\) 0 0
\(977\) −10.1668 −0.325266 −0.162633 0.986687i \(-0.551999\pi\)
−0.162633 + 0.986687i \(0.551999\pi\)
\(978\) 0 0
\(979\) 74.6351 2.38535
\(980\) 0 0
\(981\) −12.6692 −0.404496
\(982\) 0 0
\(983\) −50.6875 −1.61668 −0.808341 0.588715i \(-0.799634\pi\)
−0.808341 + 0.588715i \(0.799634\pi\)
\(984\) 0 0
\(985\) 24.6181 0.784399
\(986\) 0 0
\(987\) 4.72614 0.150435
\(988\) 0 0
\(989\) −6.87176 −0.218509
\(990\) 0 0
\(991\) −19.9799 −0.634684 −0.317342 0.948311i \(-0.602790\pi\)
−0.317342 + 0.948311i \(0.602790\pi\)
\(992\) 0 0
\(993\) 11.9326 0.378671
\(994\) 0 0
\(995\) −18.7291 −0.593753
\(996\) 0 0
\(997\) −7.41454 −0.234821 −0.117410 0.993083i \(-0.537459\pi\)
−0.117410 + 0.993083i \(0.537459\pi\)
\(998\) 0 0
\(999\) 4.62057 0.146189
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.1.2 5
4.3 odd 2 2760.2.a.w.1.4 5
12.11 even 2 8280.2.a.br.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.w.1.4 5 4.3 odd 2
5520.2.a.cc.1.2 5 1.1 even 1 trivial
8280.2.a.br.1.4 5 12.11 even 2