Properties

Label 8550.2.a.cf.1.1
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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 570)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -4.42864 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -4.42864 q^{7} -1.00000 q^{8} +5.80642 q^{11} +6.42864 q^{13} +4.42864 q^{14} +1.00000 q^{16} -3.37778 q^{17} -1.00000 q^{19} -5.80642 q^{22} -6.42864 q^{23} -6.42864 q^{26} -4.42864 q^{28} -7.80642 q^{29} +9.05086 q^{31} -1.00000 q^{32} +3.37778 q^{34} -3.67307 q^{37} +1.00000 q^{38} -4.42864 q^{41} +1.05086 q^{43} +5.80642 q^{44} +6.42864 q^{46} +5.18421 q^{47} +12.6128 q^{49} +6.42864 q^{52} -4.75557 q^{53} +4.42864 q^{56} +7.80642 q^{58} -4.62222 q^{59} +2.00000 q^{61} -9.05086 q^{62} +1.00000 q^{64} -2.75557 q^{67} -3.37778 q^{68} -7.61285 q^{71} -11.6128 q^{73} +3.67307 q^{74} -1.00000 q^{76} -25.7146 q^{77} +2.94914 q^{79} +4.42864 q^{82} -0.133353 q^{83} -1.05086 q^{86} -5.80642 q^{88} +3.18421 q^{89} -28.4701 q^{91} -6.42864 q^{92} -5.18421 q^{94} +11.4193 q^{97} -12.6128 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + 4 q^{11} + 6 q^{13} + 3 q^{16} - 10 q^{17} - 3 q^{19} - 4 q^{22} - 6 q^{23} - 6 q^{26} - 10 q^{29} + 14 q^{31} - 3 q^{32} + 10 q^{34} + 2 q^{37} + 3 q^{38} - 10 q^{43} + 4 q^{44} + 6 q^{46} + 2 q^{47} + 11 q^{49} + 6 q^{52} - 14 q^{53} + 10 q^{58} - 14 q^{59} + 6 q^{61} - 14 q^{62} + 3 q^{64} - 8 q^{67} - 10 q^{68} + 4 q^{71} - 8 q^{73} - 2 q^{74} - 3 q^{76} - 24 q^{77} + 22 q^{79} + 10 q^{86} - 4 q^{88} - 4 q^{89} - 32 q^{91} - 6 q^{92} - 2 q^{94} - 6 q^{97} - 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −4.42864 −1.67387 −0.836934 0.547304i \(-0.815654\pi\)
−0.836934 + 0.547304i \(0.815654\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 5.80642 1.75070 0.875351 0.483487i \(-0.160630\pi\)
0.875351 + 0.483487i \(0.160630\pi\)
\(12\) 0 0
\(13\) 6.42864 1.78298 0.891492 0.453037i \(-0.149659\pi\)
0.891492 + 0.453037i \(0.149659\pi\)
\(14\) 4.42864 1.18360
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.37778 −0.819233 −0.409617 0.912258i \(-0.634337\pi\)
−0.409617 + 0.912258i \(0.634337\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −5.80642 −1.23793
\(23\) −6.42864 −1.34046 −0.670232 0.742152i \(-0.733805\pi\)
−0.670232 + 0.742152i \(0.733805\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −6.42864 −1.26076
\(27\) 0 0
\(28\) −4.42864 −0.836934
\(29\) −7.80642 −1.44962 −0.724808 0.688951i \(-0.758072\pi\)
−0.724808 + 0.688951i \(0.758072\pi\)
\(30\) 0 0
\(31\) 9.05086 1.62558 0.812791 0.582556i \(-0.197947\pi\)
0.812791 + 0.582556i \(0.197947\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.37778 0.579285
\(35\) 0 0
\(36\) 0 0
\(37\) −3.67307 −0.603849 −0.301925 0.953332i \(-0.597629\pi\)
−0.301925 + 0.953332i \(0.597629\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −4.42864 −0.691637 −0.345819 0.938301i \(-0.612399\pi\)
−0.345819 + 0.938301i \(0.612399\pi\)
\(42\) 0 0
\(43\) 1.05086 0.160254 0.0801270 0.996785i \(-0.474467\pi\)
0.0801270 + 0.996785i \(0.474467\pi\)
\(44\) 5.80642 0.875351
\(45\) 0 0
\(46\) 6.42864 0.947851
\(47\) 5.18421 0.756194 0.378097 0.925766i \(-0.376578\pi\)
0.378097 + 0.925766i \(0.376578\pi\)
\(48\) 0 0
\(49\) 12.6128 1.80184
\(50\) 0 0
\(51\) 0 0
\(52\) 6.42864 0.891492
\(53\) −4.75557 −0.653228 −0.326614 0.945158i \(-0.605908\pi\)
−0.326614 + 0.945158i \(0.605908\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.42864 0.591802
\(57\) 0 0
\(58\) 7.80642 1.02503
\(59\) −4.62222 −0.601761 −0.300881 0.953662i \(-0.597281\pi\)
−0.300881 + 0.953662i \(0.597281\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −9.05086 −1.14946
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.75557 −0.336646 −0.168323 0.985732i \(-0.553835\pi\)
−0.168323 + 0.985732i \(0.553835\pi\)
\(68\) −3.37778 −0.409617
\(69\) 0 0
\(70\) 0 0
\(71\) −7.61285 −0.903479 −0.451739 0.892150i \(-0.649196\pi\)
−0.451739 + 0.892150i \(0.649196\pi\)
\(72\) 0 0
\(73\) −11.6128 −1.35918 −0.679591 0.733592i \(-0.737843\pi\)
−0.679591 + 0.733592i \(0.737843\pi\)
\(74\) 3.67307 0.426986
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −25.7146 −2.93045
\(78\) 0 0
\(79\) 2.94914 0.331805 0.165902 0.986142i \(-0.446946\pi\)
0.165902 + 0.986142i \(0.446946\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.42864 0.489061
\(83\) −0.133353 −0.0146374 −0.00731870 0.999973i \(-0.502330\pi\)
−0.00731870 + 0.999973i \(0.502330\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.05086 −0.113317
\(87\) 0 0
\(88\) −5.80642 −0.618967
\(89\) 3.18421 0.337525 0.168763 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(90\) 0 0
\(91\) −28.4701 −2.98448
\(92\) −6.42864 −0.670232
\(93\) 0 0
\(94\) −5.18421 −0.534710
\(95\) 0 0
\(96\) 0 0
\(97\) 11.4193 1.15945 0.579726 0.814812i \(-0.303160\pi\)
0.579726 + 0.814812i \(0.303160\pi\)
\(98\) −12.6128 −1.27409
\(99\) 0 0
\(100\) 0 0
\(101\) 1.86665 0.185738 0.0928692 0.995678i \(-0.470396\pi\)
0.0928692 + 0.995678i \(0.470396\pi\)
\(102\) 0 0
\(103\) −10.6222 −1.04664 −0.523319 0.852137i \(-0.675306\pi\)
−0.523319 + 0.852137i \(0.675306\pi\)
\(104\) −6.42864 −0.630380
\(105\) 0 0
\(106\) 4.75557 0.461902
\(107\) 7.61285 0.735962 0.367981 0.929833i \(-0.380049\pi\)
0.367981 + 0.929833i \(0.380049\pi\)
\(108\) 0 0
\(109\) 5.53972 0.530609 0.265304 0.964165i \(-0.414528\pi\)
0.265304 + 0.964165i \(0.414528\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.42864 −0.418467
\(113\) 12.3684 1.16352 0.581761 0.813360i \(-0.302364\pi\)
0.581761 + 0.813360i \(0.302364\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −7.80642 −0.724808
\(117\) 0 0
\(118\) 4.62222 0.425509
\(119\) 14.9590 1.37129
\(120\) 0 0
\(121\) 22.7146 2.06496
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 9.05086 0.812791
\(125\) 0 0
\(126\) 0 0
\(127\) −1.76494 −0.156613 −0.0783064 0.996929i \(-0.524951\pi\)
−0.0783064 + 0.996929i \(0.524951\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −9.80642 −0.856791 −0.428396 0.903591i \(-0.640921\pi\)
−0.428396 + 0.903591i \(0.640921\pi\)
\(132\) 0 0
\(133\) 4.42864 0.384012
\(134\) 2.75557 0.238045
\(135\) 0 0
\(136\) 3.37778 0.289643
\(137\) 1.47949 0.126402 0.0632009 0.998001i \(-0.479869\pi\)
0.0632009 + 0.998001i \(0.479869\pi\)
\(138\) 0 0
\(139\) −4.85728 −0.411989 −0.205995 0.978553i \(-0.566043\pi\)
−0.205995 + 0.978553i \(0.566043\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.61285 0.638856
\(143\) 37.3274 3.12147
\(144\) 0 0
\(145\) 0 0
\(146\) 11.6128 0.961086
\(147\) 0 0
\(148\) −3.67307 −0.301925
\(149\) −4.62222 −0.378667 −0.189333 0.981913i \(-0.560633\pi\)
−0.189333 + 0.981913i \(0.560633\pi\)
\(150\) 0 0
\(151\) −11.4193 −0.929287 −0.464644 0.885498i \(-0.653818\pi\)
−0.464644 + 0.885498i \(0.653818\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 25.7146 2.07214
\(155\) 0 0
\(156\) 0 0
\(157\) −7.37778 −0.588811 −0.294406 0.955681i \(-0.595122\pi\)
−0.294406 + 0.955681i \(0.595122\pi\)
\(158\) −2.94914 −0.234621
\(159\) 0 0
\(160\) 0 0
\(161\) 28.4701 2.24376
\(162\) 0 0
\(163\) 5.90813 0.462761 0.231380 0.972863i \(-0.425676\pi\)
0.231380 + 0.972863i \(0.425676\pi\)
\(164\) −4.42864 −0.345819
\(165\) 0 0
\(166\) 0.133353 0.0103502
\(167\) 2.75557 0.213232 0.106616 0.994300i \(-0.465998\pi\)
0.106616 + 0.994300i \(0.465998\pi\)
\(168\) 0 0
\(169\) 28.3274 2.17903
\(170\) 0 0
\(171\) 0 0
\(172\) 1.05086 0.0801270
\(173\) −8.10171 −0.615962 −0.307981 0.951393i \(-0.599653\pi\)
−0.307981 + 0.951393i \(0.599653\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.80642 0.437676
\(177\) 0 0
\(178\) −3.18421 −0.238666
\(179\) 0.235063 0.0175695 0.00878473 0.999961i \(-0.497204\pi\)
0.00878473 + 0.999961i \(0.497204\pi\)
\(180\) 0 0
\(181\) 11.3176 0.841228 0.420614 0.907240i \(-0.361815\pi\)
0.420614 + 0.907240i \(0.361815\pi\)
\(182\) 28.4701 2.11035
\(183\) 0 0
\(184\) 6.42864 0.473926
\(185\) 0 0
\(186\) 0 0
\(187\) −19.6128 −1.43423
\(188\) 5.18421 0.378097
\(189\) 0 0
\(190\) 0 0
\(191\) 6.32693 0.457801 0.228900 0.973450i \(-0.426487\pi\)
0.228900 + 0.973450i \(0.426487\pi\)
\(192\) 0 0
\(193\) −11.5397 −0.830647 −0.415324 0.909674i \(-0.636332\pi\)
−0.415324 + 0.909674i \(0.636332\pi\)
\(194\) −11.4193 −0.819856
\(195\) 0 0
\(196\) 12.6128 0.900918
\(197\) 20.0415 1.42790 0.713948 0.700198i \(-0.246905\pi\)
0.713948 + 0.700198i \(0.246905\pi\)
\(198\) 0 0
\(199\) −20.4701 −1.45109 −0.725544 0.688175i \(-0.758412\pi\)
−0.725544 + 0.688175i \(0.758412\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −1.86665 −0.131337
\(203\) 34.5718 2.42647
\(204\) 0 0
\(205\) 0 0
\(206\) 10.6222 0.740085
\(207\) 0 0
\(208\) 6.42864 0.445746
\(209\) −5.80642 −0.401639
\(210\) 0 0
\(211\) 2.75557 0.189701 0.0948506 0.995492i \(-0.469763\pi\)
0.0948506 + 0.995492i \(0.469763\pi\)
\(212\) −4.75557 −0.326614
\(213\) 0 0
\(214\) −7.61285 −0.520404
\(215\) 0 0
\(216\) 0 0
\(217\) −40.0830 −2.72101
\(218\) −5.53972 −0.375197
\(219\) 0 0
\(220\) 0 0
\(221\) −21.7146 −1.46068
\(222\) 0 0
\(223\) −10.6222 −0.711316 −0.355658 0.934616i \(-0.615743\pi\)
−0.355658 + 0.934616i \(0.615743\pi\)
\(224\) 4.42864 0.295901
\(225\) 0 0
\(226\) −12.3684 −0.822735
\(227\) 15.3461 1.01856 0.509280 0.860601i \(-0.329912\pi\)
0.509280 + 0.860601i \(0.329912\pi\)
\(228\) 0 0
\(229\) −19.7146 −1.30277 −0.651387 0.758745i \(-0.725813\pi\)
−0.651387 + 0.758745i \(0.725813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 7.80642 0.512517
\(233\) −29.5625 −1.93670 −0.968351 0.249593i \(-0.919703\pi\)
−0.968351 + 0.249593i \(0.919703\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.62222 −0.300881
\(237\) 0 0
\(238\) −14.9590 −0.969647
\(239\) −1.67307 −0.108222 −0.0541110 0.998535i \(-0.517232\pi\)
−0.0541110 + 0.998535i \(0.517232\pi\)
\(240\) 0 0
\(241\) −16.9590 −1.09242 −0.546212 0.837647i \(-0.683931\pi\)
−0.546212 + 0.837647i \(0.683931\pi\)
\(242\) −22.7146 −1.46015
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) −6.42864 −0.409045
\(248\) −9.05086 −0.574730
\(249\) 0 0
\(250\) 0 0
\(251\) −4.94914 −0.312387 −0.156194 0.987726i \(-0.549922\pi\)
−0.156194 + 0.987726i \(0.549922\pi\)
\(252\) 0 0
\(253\) −37.3274 −2.34675
\(254\) 1.76494 0.110742
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 5.34614 0.333483 0.166742 0.986001i \(-0.446675\pi\)
0.166742 + 0.986001i \(0.446675\pi\)
\(258\) 0 0
\(259\) 16.2667 1.01076
\(260\) 0 0
\(261\) 0 0
\(262\) 9.80642 0.605843
\(263\) −9.45091 −0.582768 −0.291384 0.956606i \(-0.594116\pi\)
−0.291384 + 0.956606i \(0.594116\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.42864 −0.271537
\(267\) 0 0
\(268\) −2.75557 −0.168323
\(269\) −2.94914 −0.179813 −0.0899063 0.995950i \(-0.528657\pi\)
−0.0899063 + 0.995950i \(0.528657\pi\)
\(270\) 0 0
\(271\) 30.9590 1.88062 0.940312 0.340313i \(-0.110533\pi\)
0.940312 + 0.340313i \(0.110533\pi\)
\(272\) −3.37778 −0.204808
\(273\) 0 0
\(274\) −1.47949 −0.0893795
\(275\) 0 0
\(276\) 0 0
\(277\) 13.0923 0.786643 0.393321 0.919401i \(-0.371326\pi\)
0.393321 + 0.919401i \(0.371326\pi\)
\(278\) 4.85728 0.291320
\(279\) 0 0
\(280\) 0 0
\(281\) −18.5303 −1.10543 −0.552714 0.833371i \(-0.686408\pi\)
−0.552714 + 0.833371i \(0.686408\pi\)
\(282\) 0 0
\(283\) 26.3783 1.56802 0.784012 0.620745i \(-0.213170\pi\)
0.784012 + 0.620745i \(0.213170\pi\)
\(284\) −7.61285 −0.451739
\(285\) 0 0
\(286\) −37.3274 −2.20722
\(287\) 19.6128 1.15771
\(288\) 0 0
\(289\) −5.59057 −0.328857
\(290\) 0 0
\(291\) 0 0
\(292\) −11.6128 −0.679591
\(293\) −28.5718 −1.66918 −0.834592 0.550868i \(-0.814297\pi\)
−0.834592 + 0.550868i \(0.814297\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.67307 0.213493
\(297\) 0 0
\(298\) 4.62222 0.267758
\(299\) −41.3274 −2.39003
\(300\) 0 0
\(301\) −4.65386 −0.268244
\(302\) 11.4193 0.657105
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −22.7556 −1.29873 −0.649364 0.760477i \(-0.724965\pi\)
−0.649364 + 0.760477i \(0.724965\pi\)
\(308\) −25.7146 −1.46522
\(309\) 0 0
\(310\) 0 0
\(311\) 6.32693 0.358767 0.179384 0.983779i \(-0.442590\pi\)
0.179384 + 0.983779i \(0.442590\pi\)
\(312\) 0 0
\(313\) 13.7146 0.775193 0.387596 0.921829i \(-0.373305\pi\)
0.387596 + 0.921829i \(0.373305\pi\)
\(314\) 7.37778 0.416352
\(315\) 0 0
\(316\) 2.94914 0.165902
\(317\) −32.5718 −1.82942 −0.914708 0.404115i \(-0.867580\pi\)
−0.914708 + 0.404115i \(0.867580\pi\)
\(318\) 0 0
\(319\) −45.3274 −2.53785
\(320\) 0 0
\(321\) 0 0
\(322\) −28.4701 −1.58658
\(323\) 3.37778 0.187945
\(324\) 0 0
\(325\) 0 0
\(326\) −5.90813 −0.327221
\(327\) 0 0
\(328\) 4.42864 0.244531
\(329\) −22.9590 −1.26577
\(330\) 0 0
\(331\) 5.63158 0.309540 0.154770 0.987951i \(-0.450536\pi\)
0.154770 + 0.987951i \(0.450536\pi\)
\(332\) −0.133353 −0.00731870
\(333\) 0 0
\(334\) −2.75557 −0.150778
\(335\) 0 0
\(336\) 0 0
\(337\) −5.70471 −0.310756 −0.155378 0.987855i \(-0.549659\pi\)
−0.155378 + 0.987855i \(0.549659\pi\)
\(338\) −28.3274 −1.54081
\(339\) 0 0
\(340\) 0 0
\(341\) 52.5531 2.84591
\(342\) 0 0
\(343\) −24.8573 −1.34217
\(344\) −1.05086 −0.0566583
\(345\) 0 0
\(346\) 8.10171 0.435551
\(347\) −2.62222 −0.140768 −0.0703840 0.997520i \(-0.522422\pi\)
−0.0703840 + 0.997520i \(0.522422\pi\)
\(348\) 0 0
\(349\) −24.1017 −1.29013 −0.645067 0.764126i \(-0.723171\pi\)
−0.645067 + 0.764126i \(0.723171\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −5.80642 −0.309483
\(353\) 3.64449 0.193977 0.0969883 0.995286i \(-0.469079\pi\)
0.0969883 + 0.995286i \(0.469079\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3.18421 0.168763
\(357\) 0 0
\(358\) −0.235063 −0.0124235
\(359\) −9.08250 −0.479356 −0.239678 0.970852i \(-0.577042\pi\)
−0.239678 + 0.970852i \(0.577042\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −11.3176 −0.594838
\(363\) 0 0
\(364\) −28.4701 −1.49224
\(365\) 0 0
\(366\) 0 0
\(367\) −3.95851 −0.206633 −0.103316 0.994649i \(-0.532945\pi\)
−0.103316 + 0.994649i \(0.532945\pi\)
\(368\) −6.42864 −0.335116
\(369\) 0 0
\(370\) 0 0
\(371\) 21.0607 1.09342
\(372\) 0 0
\(373\) −12.5303 −0.648797 −0.324398 0.945921i \(-0.605162\pi\)
−0.324398 + 0.945921i \(0.605162\pi\)
\(374\) 19.6128 1.01416
\(375\) 0 0
\(376\) −5.18421 −0.267355
\(377\) −50.1847 −2.58464
\(378\) 0 0
\(379\) 26.8385 1.37860 0.689302 0.724474i \(-0.257917\pi\)
0.689302 + 0.724474i \(0.257917\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −6.32693 −0.323714
\(383\) 17.5111 0.894777 0.447389 0.894340i \(-0.352354\pi\)
0.447389 + 0.894340i \(0.352354\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 11.5397 0.587356
\(387\) 0 0
\(388\) 11.4193 0.579726
\(389\) 29.4795 1.49467 0.747335 0.664448i \(-0.231333\pi\)
0.747335 + 0.664448i \(0.231333\pi\)
\(390\) 0 0
\(391\) 21.7146 1.09815
\(392\) −12.6128 −0.637045
\(393\) 0 0
\(394\) −20.0415 −1.00968
\(395\) 0 0
\(396\) 0 0
\(397\) −9.21279 −0.462377 −0.231188 0.972909i \(-0.574261\pi\)
−0.231188 + 0.972909i \(0.574261\pi\)
\(398\) 20.4701 1.02607
\(399\) 0 0
\(400\) 0 0
\(401\) −29.2859 −1.46247 −0.731234 0.682126i \(-0.761055\pi\)
−0.731234 + 0.682126i \(0.761055\pi\)
\(402\) 0 0
\(403\) 58.1847 2.89839
\(404\) 1.86665 0.0928692
\(405\) 0 0
\(406\) −34.5718 −1.71577
\(407\) −21.3274 −1.05716
\(408\) 0 0
\(409\) −17.8796 −0.884087 −0.442044 0.896994i \(-0.645746\pi\)
−0.442044 + 0.896994i \(0.645746\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −10.6222 −0.523319
\(413\) 20.4701 1.00727
\(414\) 0 0
\(415\) 0 0
\(416\) −6.42864 −0.315190
\(417\) 0 0
\(418\) 5.80642 0.284001
\(419\) −30.8671 −1.50796 −0.753979 0.656899i \(-0.771868\pi\)
−0.753979 + 0.656899i \(0.771868\pi\)
\(420\) 0 0
\(421\) −18.3970 −0.896615 −0.448307 0.893879i \(-0.647973\pi\)
−0.448307 + 0.893879i \(0.647973\pi\)
\(422\) −2.75557 −0.134139
\(423\) 0 0
\(424\) 4.75557 0.230951
\(425\) 0 0
\(426\) 0 0
\(427\) −8.85728 −0.428634
\(428\) 7.61285 0.367981
\(429\) 0 0
\(430\) 0 0
\(431\) −29.5941 −1.42550 −0.712749 0.701419i \(-0.752550\pi\)
−0.712749 + 0.701419i \(0.752550\pi\)
\(432\) 0 0
\(433\) −27.2988 −1.31190 −0.655949 0.754805i \(-0.727731\pi\)
−0.655949 + 0.754805i \(0.727731\pi\)
\(434\) 40.0830 1.92404
\(435\) 0 0
\(436\) 5.53972 0.265304
\(437\) 6.42864 0.307524
\(438\) 0 0
\(439\) 9.64143 0.460160 0.230080 0.973172i \(-0.426101\pi\)
0.230080 + 0.973172i \(0.426101\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 21.7146 1.03286
\(443\) 6.70519 0.318573 0.159287 0.987232i \(-0.449081\pi\)
0.159287 + 0.987232i \(0.449081\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 10.6222 0.502976
\(447\) 0 0
\(448\) −4.42864 −0.209234
\(449\) 13.9398 0.657859 0.328929 0.944355i \(-0.393312\pi\)
0.328929 + 0.944355i \(0.393312\pi\)
\(450\) 0 0
\(451\) −25.7146 −1.21085
\(452\) 12.3684 0.581761
\(453\) 0 0
\(454\) −15.3461 −0.720230
\(455\) 0 0
\(456\) 0 0
\(457\) −23.2257 −1.08645 −0.543226 0.839586i \(-0.682797\pi\)
−0.543226 + 0.839586i \(0.682797\pi\)
\(458\) 19.7146 0.921201
\(459\) 0 0
\(460\) 0 0
\(461\) −19.9684 −0.930019 −0.465010 0.885306i \(-0.653949\pi\)
−0.465010 + 0.885306i \(0.653949\pi\)
\(462\) 0 0
\(463\) 2.79706 0.129990 0.0649951 0.997886i \(-0.479297\pi\)
0.0649951 + 0.997886i \(0.479297\pi\)
\(464\) −7.80642 −0.362404
\(465\) 0 0
\(466\) 29.5625 1.36945
\(467\) −30.9719 −1.43321 −0.716604 0.697480i \(-0.754305\pi\)
−0.716604 + 0.697480i \(0.754305\pi\)
\(468\) 0 0
\(469\) 12.2034 0.563502
\(470\) 0 0
\(471\) 0 0
\(472\) 4.62222 0.212755
\(473\) 6.10171 0.280557
\(474\) 0 0
\(475\) 0 0
\(476\) 14.9590 0.685644
\(477\) 0 0
\(478\) 1.67307 0.0765245
\(479\) −29.8765 −1.36509 −0.682546 0.730843i \(-0.739127\pi\)
−0.682546 + 0.730843i \(0.739127\pi\)
\(480\) 0 0
\(481\) −23.6128 −1.07665
\(482\) 16.9590 0.772461
\(483\) 0 0
\(484\) 22.7146 1.03248
\(485\) 0 0
\(486\) 0 0
\(487\) −17.8479 −0.808766 −0.404383 0.914590i \(-0.632514\pi\)
−0.404383 + 0.914590i \(0.632514\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) 0 0
\(491\) 8.68244 0.391833 0.195916 0.980621i \(-0.437232\pi\)
0.195916 + 0.980621i \(0.437232\pi\)
\(492\) 0 0
\(493\) 26.3684 1.18757
\(494\) 6.42864 0.289238
\(495\) 0 0
\(496\) 9.05086 0.406395
\(497\) 33.7146 1.51230
\(498\) 0 0
\(499\) 11.2257 0.502531 0.251266 0.967918i \(-0.419153\pi\)
0.251266 + 0.967918i \(0.419153\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.94914 0.220891
\(503\) −17.6543 −0.787168 −0.393584 0.919289i \(-0.628765\pi\)
−0.393584 + 0.919289i \(0.628765\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 37.3274 1.65941
\(507\) 0 0
\(508\) −1.76494 −0.0783064
\(509\) −14.1748 −0.628289 −0.314144 0.949375i \(-0.601718\pi\)
−0.314144 + 0.949375i \(0.601718\pi\)
\(510\) 0 0
\(511\) 51.4291 2.27509
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −5.34614 −0.235808
\(515\) 0 0
\(516\) 0 0
\(517\) 30.1017 1.32387
\(518\) −16.2667 −0.714718
\(519\) 0 0
\(520\) 0 0
\(521\) 15.0005 0.657183 0.328591 0.944472i \(-0.393426\pi\)
0.328591 + 0.944472i \(0.393426\pi\)
\(522\) 0 0
\(523\) −40.2864 −1.76160 −0.880801 0.473488i \(-0.842995\pi\)
−0.880801 + 0.473488i \(0.842995\pi\)
\(524\) −9.80642 −0.428396
\(525\) 0 0
\(526\) 9.45091 0.412079
\(527\) −30.5718 −1.33173
\(528\) 0 0
\(529\) 18.3274 0.796844
\(530\) 0 0
\(531\) 0 0
\(532\) 4.42864 0.192006
\(533\) −28.4701 −1.23318
\(534\) 0 0
\(535\) 0 0
\(536\) 2.75557 0.119022
\(537\) 0 0
\(538\) 2.94914 0.127147
\(539\) 73.2355 3.15448
\(540\) 0 0
\(541\) −22.3872 −0.962499 −0.481249 0.876584i \(-0.659817\pi\)
−0.481249 + 0.876584i \(0.659817\pi\)
\(542\) −30.9590 −1.32980
\(543\) 0 0
\(544\) 3.37778 0.144821
\(545\) 0 0
\(546\) 0 0
\(547\) −19.7333 −0.843735 −0.421867 0.906658i \(-0.638625\pi\)
−0.421867 + 0.906658i \(0.638625\pi\)
\(548\) 1.47949 0.0632009
\(549\) 0 0
\(550\) 0 0
\(551\) 7.80642 0.332565
\(552\) 0 0
\(553\) −13.0607 −0.555397
\(554\) −13.0923 −0.556240
\(555\) 0 0
\(556\) −4.85728 −0.205995
\(557\) 15.6543 0.663295 0.331648 0.943403i \(-0.392395\pi\)
0.331648 + 0.943403i \(0.392395\pi\)
\(558\) 0 0
\(559\) 6.75557 0.285730
\(560\) 0 0
\(561\) 0 0
\(562\) 18.5303 0.781656
\(563\) 36.9403 1.55685 0.778423 0.627740i \(-0.216020\pi\)
0.778423 + 0.627740i \(0.216020\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −26.3783 −1.10876
\(567\) 0 0
\(568\) 7.61285 0.319428
\(569\) 1.20294 0.0504300 0.0252150 0.999682i \(-0.491973\pi\)
0.0252150 + 0.999682i \(0.491973\pi\)
\(570\) 0 0
\(571\) 37.7975 1.58178 0.790889 0.611960i \(-0.209619\pi\)
0.790889 + 0.611960i \(0.209619\pi\)
\(572\) 37.3274 1.56074
\(573\) 0 0
\(574\) −19.6128 −0.818624
\(575\) 0 0
\(576\) 0 0
\(577\) −23.2257 −0.966898 −0.483449 0.875372i \(-0.660616\pi\)
−0.483449 + 0.875372i \(0.660616\pi\)
\(578\) 5.59057 0.232537
\(579\) 0 0
\(580\) 0 0
\(581\) 0.590573 0.0245011
\(582\) 0 0
\(583\) −27.6128 −1.14361
\(584\) 11.6128 0.480543
\(585\) 0 0
\(586\) 28.5718 1.18029
\(587\) 16.8069 0.693695 0.346848 0.937922i \(-0.387252\pi\)
0.346848 + 0.937922i \(0.387252\pi\)
\(588\) 0 0
\(589\) −9.05086 −0.372934
\(590\) 0 0
\(591\) 0 0
\(592\) −3.67307 −0.150962
\(593\) −16.3555 −0.671640 −0.335820 0.941926i \(-0.609013\pi\)
−0.335820 + 0.941926i \(0.609013\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.62222 −0.189333
\(597\) 0 0
\(598\) 41.3274 1.69000
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 7.89829 0.322178 0.161089 0.986940i \(-0.448499\pi\)
0.161089 + 0.986940i \(0.448499\pi\)
\(602\) 4.65386 0.189677
\(603\) 0 0
\(604\) −11.4193 −0.464644
\(605\) 0 0
\(606\) 0 0
\(607\) −17.8479 −0.724424 −0.362212 0.932096i \(-0.617978\pi\)
−0.362212 + 0.932096i \(0.617978\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 33.3274 1.34828
\(612\) 0 0
\(613\) −0.622216 −0.0251311 −0.0125655 0.999921i \(-0.504000\pi\)
−0.0125655 + 0.999921i \(0.504000\pi\)
\(614\) 22.7556 0.918340
\(615\) 0 0
\(616\) 25.7146 1.03607
\(617\) −37.4795 −1.50887 −0.754434 0.656376i \(-0.772088\pi\)
−0.754434 + 0.656376i \(0.772088\pi\)
\(618\) 0 0
\(619\) 28.6735 1.15249 0.576244 0.817278i \(-0.304518\pi\)
0.576244 + 0.817278i \(0.304518\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.32693 −0.253687
\(623\) −14.1017 −0.564973
\(624\) 0 0
\(625\) 0 0
\(626\) −13.7146 −0.548144
\(627\) 0 0
\(628\) −7.37778 −0.294406
\(629\) 12.4068 0.494693
\(630\) 0 0
\(631\) 1.24443 0.0495400 0.0247700 0.999693i \(-0.492115\pi\)
0.0247700 + 0.999693i \(0.492115\pi\)
\(632\) −2.94914 −0.117311
\(633\) 0 0
\(634\) 32.5718 1.29359
\(635\) 0 0
\(636\) 0 0
\(637\) 81.0835 3.21264
\(638\) 45.3274 1.79453
\(639\) 0 0
\(640\) 0 0
\(641\) 17.4064 0.687510 0.343755 0.939059i \(-0.388301\pi\)
0.343755 + 0.939059i \(0.388301\pi\)
\(642\) 0 0
\(643\) 35.1526 1.38628 0.693141 0.720802i \(-0.256226\pi\)
0.693141 + 0.720802i \(0.256226\pi\)
\(644\) 28.4701 1.12188
\(645\) 0 0
\(646\) −3.37778 −0.132897
\(647\) 34.4286 1.35353 0.676765 0.736199i \(-0.263381\pi\)
0.676765 + 0.736199i \(0.263381\pi\)
\(648\) 0 0
\(649\) −26.8385 −1.05350
\(650\) 0 0
\(651\) 0 0
\(652\) 5.90813 0.231380
\(653\) −4.30819 −0.168593 −0.0842963 0.996441i \(-0.526864\pi\)
−0.0842963 + 0.996441i \(0.526864\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −4.42864 −0.172909
\(657\) 0 0
\(658\) 22.9590 0.895035
\(659\) 27.0736 1.05464 0.527319 0.849667i \(-0.323197\pi\)
0.527319 + 0.849667i \(0.323197\pi\)
\(660\) 0 0
\(661\) −43.2543 −1.68240 −0.841198 0.540727i \(-0.818149\pi\)
−0.841198 + 0.540727i \(0.818149\pi\)
\(662\) −5.63158 −0.218878
\(663\) 0 0
\(664\) 0.133353 0.00517510
\(665\) 0 0
\(666\) 0 0
\(667\) 50.1847 1.94316
\(668\) 2.75557 0.106616
\(669\) 0 0
\(670\) 0 0
\(671\) 11.6128 0.448309
\(672\) 0 0
\(673\) −15.0321 −0.579446 −0.289723 0.957111i \(-0.593563\pi\)
−0.289723 + 0.957111i \(0.593563\pi\)
\(674\) 5.70471 0.219737
\(675\) 0 0
\(676\) 28.3274 1.08952
\(677\) 22.9403 0.881666 0.440833 0.897589i \(-0.354683\pi\)
0.440833 + 0.897589i \(0.354683\pi\)
\(678\) 0 0
\(679\) −50.5718 −1.94077
\(680\) 0 0
\(681\) 0 0
\(682\) −52.5531 −2.01236
\(683\) 9.77784 0.374139 0.187069 0.982347i \(-0.440101\pi\)
0.187069 + 0.982347i \(0.440101\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 24.8573 0.949055
\(687\) 0 0
\(688\) 1.05086 0.0400635
\(689\) −30.5718 −1.16469
\(690\) 0 0
\(691\) −44.0830 −1.67700 −0.838498 0.544905i \(-0.816566\pi\)
−0.838498 + 0.544905i \(0.816566\pi\)
\(692\) −8.10171 −0.307981
\(693\) 0 0
\(694\) 2.62222 0.0995379
\(695\) 0 0
\(696\) 0 0
\(697\) 14.9590 0.566612
\(698\) 24.1017 0.912263
\(699\) 0 0
\(700\) 0 0
\(701\) 12.7685 0.482259 0.241129 0.970493i \(-0.422482\pi\)
0.241129 + 0.970493i \(0.422482\pi\)
\(702\) 0 0
\(703\) 3.67307 0.138532
\(704\) 5.80642 0.218838
\(705\) 0 0
\(706\) −3.64449 −0.137162
\(707\) −8.26671 −0.310901
\(708\) 0 0
\(709\) −47.5941 −1.78743 −0.893717 0.448631i \(-0.851912\pi\)
−0.893717 + 0.448631i \(0.851912\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.18421 −0.119333
\(713\) −58.1847 −2.17903
\(714\) 0 0
\(715\) 0 0
\(716\) 0.235063 0.00878473
\(717\) 0 0
\(718\) 9.08250 0.338956
\(719\) 47.0005 1.75282 0.876411 0.481564i \(-0.159931\pi\)
0.876411 + 0.481564i \(0.159931\pi\)
\(720\) 0 0
\(721\) 47.0420 1.75193
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 11.3176 0.420614
\(725\) 0 0
\(726\) 0 0
\(727\) −52.6321 −1.95202 −0.976008 0.217737i \(-0.930133\pi\)
−0.976008 + 0.217737i \(0.930133\pi\)
\(728\) 28.4701 1.05517
\(729\) 0 0
\(730\) 0 0
\(731\) −3.54956 −0.131285
\(732\) 0 0
\(733\) −27.3145 −1.00888 −0.504442 0.863446i \(-0.668302\pi\)
−0.504442 + 0.863446i \(0.668302\pi\)
\(734\) 3.95851 0.146111
\(735\) 0 0
\(736\) 6.42864 0.236963
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 25.3274 0.931684 0.465842 0.884868i \(-0.345752\pi\)
0.465842 + 0.884868i \(0.345752\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −21.0607 −0.773163
\(743\) −11.8796 −0.435819 −0.217909 0.975969i \(-0.569924\pi\)
−0.217909 + 0.975969i \(0.569924\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 12.5303 0.458769
\(747\) 0 0
\(748\) −19.6128 −0.717117
\(749\) −33.7146 −1.23190
\(750\) 0 0
\(751\) −45.5210 −1.66108 −0.830542 0.556956i \(-0.811969\pi\)
−0.830542 + 0.556956i \(0.811969\pi\)
\(752\) 5.18421 0.189049
\(753\) 0 0
\(754\) 50.1847 1.82762
\(755\) 0 0
\(756\) 0 0
\(757\) 51.7275 1.88007 0.940033 0.341083i \(-0.110794\pi\)
0.940033 + 0.341083i \(0.110794\pi\)
\(758\) −26.8385 −0.974820
\(759\) 0 0
\(760\) 0 0
\(761\) −28.3684 −1.02835 −0.514177 0.857684i \(-0.671903\pi\)
−0.514177 + 0.857684i \(0.671903\pi\)
\(762\) 0 0
\(763\) −24.5334 −0.888169
\(764\) 6.32693 0.228900
\(765\) 0 0
\(766\) −17.5111 −0.632703
\(767\) −29.7146 −1.07293
\(768\) 0 0
\(769\) −0.285442 −0.0102933 −0.00514665 0.999987i \(-0.501638\pi\)
−0.00514665 + 0.999987i \(0.501638\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −11.5397 −0.415324
\(773\) 49.8163 1.79177 0.895883 0.444289i \(-0.146544\pi\)
0.895883 + 0.444289i \(0.146544\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −11.4193 −0.409928
\(777\) 0 0
\(778\) −29.4795 −1.05689
\(779\) 4.42864 0.158672
\(780\) 0 0
\(781\) −44.2034 −1.58172
\(782\) −21.7146 −0.776511
\(783\) 0 0
\(784\) 12.6128 0.450459
\(785\) 0 0
\(786\) 0 0
\(787\) 20.7368 0.739188 0.369594 0.929193i \(-0.379497\pi\)
0.369594 + 0.929193i \(0.379497\pi\)
\(788\) 20.0415 0.713948
\(789\) 0 0
\(790\) 0 0
\(791\) −54.7753 −1.94758
\(792\) 0 0
\(793\) 12.8573 0.456575
\(794\) 9.21279 0.326950
\(795\) 0 0
\(796\) −20.4701 −0.725544
\(797\) −21.3461 −0.756119 −0.378060 0.925781i \(-0.623409\pi\)
−0.378060 + 0.925781i \(0.623409\pi\)
\(798\) 0 0
\(799\) −17.5111 −0.619500
\(800\) 0 0
\(801\) 0 0
\(802\) 29.2859 1.03412
\(803\) −67.4291 −2.37952
\(804\) 0 0
\(805\) 0 0
\(806\) −58.1847 −2.04947
\(807\) 0 0
\(808\) −1.86665 −0.0656684
\(809\) 18.6735 0.656527 0.328263 0.944586i \(-0.393537\pi\)
0.328263 + 0.944586i \(0.393537\pi\)
\(810\) 0 0
\(811\) −14.8385 −0.521052 −0.260526 0.965467i \(-0.583896\pi\)
−0.260526 + 0.965467i \(0.583896\pi\)
\(812\) 34.5718 1.21323
\(813\) 0 0
\(814\) 21.3274 0.747525
\(815\) 0 0
\(816\) 0 0
\(817\) −1.05086 −0.0367648
\(818\) 17.8796 0.625144
\(819\) 0 0
\(820\) 0 0
\(821\) −16.1521 −0.563712 −0.281856 0.959457i \(-0.590950\pi\)
−0.281856 + 0.959457i \(0.590950\pi\)
\(822\) 0 0
\(823\) 47.6543 1.66113 0.830563 0.556925i \(-0.188019\pi\)
0.830563 + 0.556925i \(0.188019\pi\)
\(824\) 10.6222 0.370042
\(825\) 0 0
\(826\) −20.4701 −0.712247
\(827\) 26.1017 0.907645 0.453823 0.891092i \(-0.350060\pi\)
0.453823 + 0.891092i \(0.350060\pi\)
\(828\) 0 0
\(829\) −13.8894 −0.482399 −0.241199 0.970476i \(-0.577541\pi\)
−0.241199 + 0.970476i \(0.577541\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 6.42864 0.222873
\(833\) −42.6035 −1.47612
\(834\) 0 0
\(835\) 0 0
\(836\) −5.80642 −0.200819
\(837\) 0 0
\(838\) 30.8671 1.06629
\(839\) −31.6958 −1.09426 −0.547131 0.837047i \(-0.684280\pi\)
−0.547131 + 0.837047i \(0.684280\pi\)
\(840\) 0 0
\(841\) 31.9403 1.10139
\(842\) 18.3970 0.634002
\(843\) 0 0
\(844\) 2.75557 0.0948506
\(845\) 0 0
\(846\) 0 0
\(847\) −100.595 −3.45647
\(848\) −4.75557 −0.163307
\(849\) 0 0
\(850\) 0 0
\(851\) 23.6128 0.809438
\(852\) 0 0
\(853\) −34.8702 −1.19393 −0.596966 0.802266i \(-0.703627\pi\)
−0.596966 + 0.802266i \(0.703627\pi\)
\(854\) 8.85728 0.303090
\(855\) 0 0
\(856\) −7.61285 −0.260202
\(857\) −33.9367 −1.15926 −0.579628 0.814881i \(-0.696802\pi\)
−0.579628 + 0.814881i \(0.696802\pi\)
\(858\) 0 0
\(859\) −37.4479 −1.27770 −0.638852 0.769330i \(-0.720590\pi\)
−0.638852 + 0.769330i \(0.720590\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 29.5941 1.00798
\(863\) 15.5299 0.528643 0.264322 0.964435i \(-0.414852\pi\)
0.264322 + 0.964435i \(0.414852\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 27.2988 0.927652
\(867\) 0 0
\(868\) −40.0830 −1.36050
\(869\) 17.1240 0.580891
\(870\) 0 0
\(871\) −17.7146 −0.600235
\(872\) −5.53972 −0.187599
\(873\) 0 0
\(874\) −6.42864 −0.217452
\(875\) 0 0
\(876\) 0 0
\(877\) 7.87649 0.265970 0.132985 0.991118i \(-0.457544\pi\)
0.132985 + 0.991118i \(0.457544\pi\)
\(878\) −9.64143 −0.325382
\(879\) 0 0
\(880\) 0 0
\(881\) −33.5496 −1.13031 −0.565157 0.824984i \(-0.691184\pi\)
−0.565157 + 0.824984i \(0.691184\pi\)
\(882\) 0 0
\(883\) −50.9688 −1.71524 −0.857619 0.514286i \(-0.828057\pi\)
−0.857619 + 0.514286i \(0.828057\pi\)
\(884\) −21.7146 −0.730340
\(885\) 0 0
\(886\) −6.70519 −0.225265
\(887\) 17.0035 0.570923 0.285461 0.958390i \(-0.407853\pi\)
0.285461 + 0.958390i \(0.407853\pi\)
\(888\) 0 0
\(889\) 7.81627 0.262149
\(890\) 0 0
\(891\) 0 0
\(892\) −10.6222 −0.355658
\(893\) −5.18421 −0.173483
\(894\) 0 0
\(895\) 0 0
\(896\) 4.42864 0.147950
\(897\) 0 0
\(898\) −13.9398 −0.465176
\(899\) −70.6548 −2.35647
\(900\) 0 0
\(901\) 16.0633 0.535146
\(902\) 25.7146 0.856201
\(903\) 0 0
\(904\) −12.3684 −0.411367
\(905\) 0 0
\(906\) 0 0
\(907\) 21.7778 0.723121 0.361561 0.932349i \(-0.382244\pi\)
0.361561 + 0.932349i \(0.382244\pi\)
\(908\) 15.3461 0.509280
\(909\) 0 0
\(910\) 0 0
\(911\) −10.6351 −0.352357 −0.176179 0.984358i \(-0.556374\pi\)
−0.176179 + 0.984358i \(0.556374\pi\)
\(912\) 0 0
\(913\) −0.774305 −0.0256257
\(914\) 23.2257 0.768238
\(915\) 0 0
\(916\) −19.7146 −0.651387
\(917\) 43.4291 1.43416
\(918\) 0 0
\(919\) 12.2667 0.404641 0.202321 0.979319i \(-0.435152\pi\)
0.202321 + 0.979319i \(0.435152\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 19.9684 0.657623
\(923\) −48.9403 −1.61089
\(924\) 0 0
\(925\) 0 0
\(926\) −2.79706 −0.0919170
\(927\) 0 0
\(928\) 7.80642 0.256258
\(929\) 27.9180 0.915959 0.457980 0.888963i \(-0.348573\pi\)
0.457980 + 0.888963i \(0.348573\pi\)
\(930\) 0 0
\(931\) −12.6128 −0.413369
\(932\) −29.5625 −0.968351
\(933\) 0 0
\(934\) 30.9719 1.01343
\(935\) 0 0
\(936\) 0 0
\(937\) 3.14272 0.102668 0.0513341 0.998682i \(-0.483653\pi\)
0.0513341 + 0.998682i \(0.483653\pi\)
\(938\) −12.2034 −0.398456
\(939\) 0 0
\(940\) 0 0
\(941\) 19.6227 0.639681 0.319841 0.947471i \(-0.396371\pi\)
0.319841 + 0.947471i \(0.396371\pi\)
\(942\) 0 0
\(943\) 28.4701 0.927115
\(944\) −4.62222 −0.150440
\(945\) 0 0
\(946\) −6.10171 −0.198384
\(947\) 34.5018 1.12116 0.560578 0.828101i \(-0.310579\pi\)
0.560578 + 0.828101i \(0.310579\pi\)
\(948\) 0 0
\(949\) −74.6548 −2.42340
\(950\) 0 0
\(951\) 0 0
\(952\) −14.9590 −0.484824
\(953\) −19.0035 −0.615585 −0.307793 0.951454i \(-0.599590\pi\)
−0.307793 + 0.951454i \(0.599590\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.67307 −0.0541110
\(957\) 0 0
\(958\) 29.8765 0.965266
\(959\) −6.55215 −0.211580
\(960\) 0 0
\(961\) 50.9180 1.64252
\(962\) 23.6128 0.761309
\(963\) 0 0
\(964\) −16.9590 −0.546212
\(965\) 0 0
\(966\) 0 0
\(967\) 17.8765 0.574869 0.287435 0.957800i \(-0.407198\pi\)
0.287435 + 0.957800i \(0.407198\pi\)
\(968\) −22.7146 −0.730074
\(969\) 0 0
\(970\) 0 0
\(971\) −34.0701 −1.09336 −0.546680 0.837341i \(-0.684109\pi\)
−0.546680 + 0.837341i \(0.684109\pi\)
\(972\) 0 0
\(973\) 21.5111 0.689615
\(974\) 17.8479 0.571884
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −54.0830 −1.73027 −0.865134 0.501541i \(-0.832767\pi\)
−0.865134 + 0.501541i \(0.832767\pi\)
\(978\) 0 0
\(979\) 18.4889 0.590907
\(980\) 0 0
\(981\) 0 0
\(982\) −8.68244 −0.277068
\(983\) 56.9403 1.81611 0.908056 0.418849i \(-0.137566\pi\)
0.908056 + 0.418849i \(0.137566\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −26.3684 −0.839741
\(987\) 0 0
\(988\) −6.42864 −0.204522
\(989\) −6.75557 −0.214815
\(990\) 0 0
\(991\) −47.0321 −1.49402 −0.747012 0.664810i \(-0.768512\pi\)
−0.747012 + 0.664810i \(0.768512\pi\)
\(992\) −9.05086 −0.287365
\(993\) 0 0
\(994\) −33.7146 −1.06936
\(995\) 0 0
\(996\) 0 0
\(997\) 20.8256 0.659555 0.329777 0.944059i \(-0.393026\pi\)
0.329777 + 0.944059i \(0.393026\pi\)
\(998\) −11.2257 −0.355343
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8550.2.a.cf.1.1 3
3.2 odd 2 2850.2.a.bn.1.1 3
5.2 odd 4 1710.2.d.e.1369.2 6
5.3 odd 4 1710.2.d.e.1369.5 6
5.4 even 2 8550.2.a.cr.1.3 3
15.2 even 4 570.2.d.d.229.5 yes 6
15.8 even 4 570.2.d.d.229.2 6
15.14 odd 2 2850.2.a.bk.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.d.d.229.2 6 15.8 even 4
570.2.d.d.229.5 yes 6 15.2 even 4
1710.2.d.e.1369.2 6 5.2 odd 4
1710.2.d.e.1369.5 6 5.3 odd 4
2850.2.a.bk.1.3 3 15.14 odd 2
2850.2.a.bn.1.1 3 3.2 odd 2
8550.2.a.cf.1.1 3 1.1 even 1 trivial
8550.2.a.cr.1.3 3 5.4 even 2