Properties

Label 8550.2.a.cv.1.1
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1710)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.93185\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.86370 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.86370 q^{7} +1.00000 q^{8} +1.03528 q^{11} +3.86370 q^{13} -3.86370 q^{14} +1.00000 q^{16} -0.535898 q^{17} -1.00000 q^{19} +1.03528 q^{22} -5.46410 q^{23} +3.86370 q^{26} -3.86370 q^{28} +4.62158 q^{29} -10.9282 q^{31} +1.00000 q^{32} -0.535898 q^{34} +1.79315 q^{37} -1.00000 q^{38} +1.79315 q^{41} +6.69213 q^{43} +1.03528 q^{44} -5.46410 q^{46} -2.53590 q^{47} +7.92820 q^{49} +3.86370 q^{52} -6.00000 q^{53} -3.86370 q^{56} +4.62158 q^{58} +6.96953 q^{59} -2.00000 q^{61} -10.9282 q^{62} +1.00000 q^{64} -13.3843 q^{67} -0.535898 q^{68} +4.14110 q^{71} +3.58630 q^{73} +1.79315 q^{74} -1.00000 q^{76} -4.00000 q^{77} -4.00000 q^{79} +1.79315 q^{82} -16.3923 q^{83} +6.69213 q^{86} +1.03528 q^{88} -3.86370 q^{89} -14.9282 q^{91} -5.46410 q^{92} -2.53590 q^{94} -2.55103 q^{97} +7.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{4} + 4q^{8} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{4} + 4q^{8} + 4q^{16} - 16q^{17} - 4q^{19} - 8q^{23} - 16q^{31} + 4q^{32} - 16q^{34} - 4q^{38} - 8q^{46} - 24q^{47} + 4q^{49} - 24q^{53} - 8q^{61} - 16q^{62} + 4q^{64} - 16q^{68} - 4q^{76} - 16q^{77} - 16q^{79} - 24q^{83} - 32q^{91} - 8q^{92} - 24q^{94} + 4q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −3.86370 −1.46034 −0.730171 0.683264i \(-0.760560\pi\)
−0.730171 + 0.683264i \(0.760560\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 1.03528 0.312148 0.156074 0.987745i \(-0.450116\pi\)
0.156074 + 0.987745i \(0.450116\pi\)
\(12\) 0 0
\(13\) 3.86370 1.07160 0.535799 0.844345i \(-0.320010\pi\)
0.535799 + 0.844345i \(0.320010\pi\)
\(14\) −3.86370 −1.03262
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.535898 −0.129974 −0.0649872 0.997886i \(-0.520701\pi\)
−0.0649872 + 0.997886i \(0.520701\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 1.03528 0.220722
\(23\) −5.46410 −1.13934 −0.569672 0.821872i \(-0.692930\pi\)
−0.569672 + 0.821872i \(0.692930\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.86370 0.757735
\(27\) 0 0
\(28\) −3.86370 −0.730171
\(29\) 4.62158 0.858206 0.429103 0.903256i \(-0.358830\pi\)
0.429103 + 0.903256i \(0.358830\pi\)
\(30\) 0 0
\(31\) −10.9282 −1.96276 −0.981382 0.192068i \(-0.938481\pi\)
−0.981382 + 0.192068i \(0.938481\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.535898 −0.0919058
\(35\) 0 0
\(36\) 0 0
\(37\) 1.79315 0.294792 0.147396 0.989078i \(-0.452911\pi\)
0.147396 + 0.989078i \(0.452911\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 1.79315 0.280043 0.140022 0.990148i \(-0.455283\pi\)
0.140022 + 0.990148i \(0.455283\pi\)
\(42\) 0 0
\(43\) 6.69213 1.02054 0.510270 0.860014i \(-0.329545\pi\)
0.510270 + 0.860014i \(0.329545\pi\)
\(44\) 1.03528 0.156074
\(45\) 0 0
\(46\) −5.46410 −0.805638
\(47\) −2.53590 −0.369899 −0.184949 0.982748i \(-0.559212\pi\)
−0.184949 + 0.982748i \(0.559212\pi\)
\(48\) 0 0
\(49\) 7.92820 1.13260
\(50\) 0 0
\(51\) 0 0
\(52\) 3.86370 0.535799
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.86370 −0.516309
\(57\) 0 0
\(58\) 4.62158 0.606843
\(59\) 6.96953 0.907356 0.453678 0.891166i \(-0.350112\pi\)
0.453678 + 0.891166i \(0.350112\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −10.9282 −1.38788
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −13.3843 −1.63515 −0.817574 0.575824i \(-0.804681\pi\)
−0.817574 + 0.575824i \(0.804681\pi\)
\(68\) −0.535898 −0.0649872
\(69\) 0 0
\(70\) 0 0
\(71\) 4.14110 0.491459 0.245729 0.969338i \(-0.420973\pi\)
0.245729 + 0.969338i \(0.420973\pi\)
\(72\) 0 0
\(73\) 3.58630 0.419745 0.209872 0.977729i \(-0.432695\pi\)
0.209872 + 0.977729i \(0.432695\pi\)
\(74\) 1.79315 0.208450
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.79315 0.198020
\(83\) −16.3923 −1.79929 −0.899645 0.436623i \(-0.856174\pi\)
−0.899645 + 0.436623i \(0.856174\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.69213 0.721631
\(87\) 0 0
\(88\) 1.03528 0.110361
\(89\) −3.86370 −0.409552 −0.204776 0.978809i \(-0.565647\pi\)
−0.204776 + 0.978809i \(0.565647\pi\)
\(90\) 0 0
\(91\) −14.9282 −1.56490
\(92\) −5.46410 −0.569672
\(93\) 0 0
\(94\) −2.53590 −0.261558
\(95\) 0 0
\(96\) 0 0
\(97\) −2.55103 −0.259017 −0.129509 0.991578i \(-0.541340\pi\)
−0.129509 + 0.991578i \(0.541340\pi\)
\(98\) 7.92820 0.800869
\(99\) 0 0
\(100\) 0 0
\(101\) −10.5558 −1.05034 −0.525172 0.850996i \(-0.675999\pi\)
−0.525172 + 0.850996i \(0.675999\pi\)
\(102\) 0 0
\(103\) −4.89898 −0.482711 −0.241355 0.970437i \(-0.577592\pi\)
−0.241355 + 0.970437i \(0.577592\pi\)
\(104\) 3.86370 0.378867
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 2.92820 0.283080 0.141540 0.989933i \(-0.454795\pi\)
0.141540 + 0.989933i \(0.454795\pi\)
\(108\) 0 0
\(109\) 8.92820 0.855167 0.427583 0.903976i \(-0.359365\pi\)
0.427583 + 0.903976i \(0.359365\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.86370 −0.365086
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4.62158 0.429103
\(117\) 0 0
\(118\) 6.96953 0.641597
\(119\) 2.07055 0.189807
\(120\) 0 0
\(121\) −9.92820 −0.902564
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) −10.9282 −0.981382
\(125\) 0 0
\(126\) 0 0
\(127\) −8.48528 −0.752947 −0.376473 0.926427i \(-0.622863\pi\)
−0.376473 + 0.926427i \(0.622863\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 3.10583 0.271357 0.135679 0.990753i \(-0.456678\pi\)
0.135679 + 0.990753i \(0.456678\pi\)
\(132\) 0 0
\(133\) 3.86370 0.335026
\(134\) −13.3843 −1.15622
\(135\) 0 0
\(136\) −0.535898 −0.0459529
\(137\) −0.535898 −0.0457849 −0.0228924 0.999738i \(-0.507288\pi\)
−0.0228924 + 0.999738i \(0.507288\pi\)
\(138\) 0 0
\(139\) 6.92820 0.587643 0.293821 0.955860i \(-0.405073\pi\)
0.293821 + 0.955860i \(0.405073\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.14110 0.347514
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 3.58630 0.296804
\(147\) 0 0
\(148\) 1.79315 0.147396
\(149\) 20.3538 1.66745 0.833724 0.552182i \(-0.186205\pi\)
0.833724 + 0.552182i \(0.186205\pi\)
\(150\) 0 0
\(151\) 5.85641 0.476588 0.238294 0.971193i \(-0.423412\pi\)
0.238294 + 0.971193i \(0.423412\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) −14.1421 −1.12867 −0.564333 0.825547i \(-0.690866\pi\)
−0.564333 + 0.825547i \(0.690866\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) 0 0
\(161\) 21.1117 1.66383
\(162\) 0 0
\(163\) −23.6627 −1.85341 −0.926703 0.375796i \(-0.877369\pi\)
−0.926703 + 0.375796i \(0.877369\pi\)
\(164\) 1.79315 0.140022
\(165\) 0 0
\(166\) −16.3923 −1.27229
\(167\) −10.9282 −0.845650 −0.422825 0.906211i \(-0.638961\pi\)
−0.422825 + 0.906211i \(0.638961\pi\)
\(168\) 0 0
\(169\) 1.92820 0.148323
\(170\) 0 0
\(171\) 0 0
\(172\) 6.69213 0.510270
\(173\) 22.7846 1.73228 0.866141 0.499800i \(-0.166593\pi\)
0.866141 + 0.499800i \(0.166593\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.03528 0.0780369
\(177\) 0 0
\(178\) −3.86370 −0.289597
\(179\) −16.7675 −1.25326 −0.626631 0.779316i \(-0.715566\pi\)
−0.626631 + 0.779316i \(0.715566\pi\)
\(180\) 0 0
\(181\) 19.8564 1.47592 0.737958 0.674847i \(-0.235790\pi\)
0.737958 + 0.674847i \(0.235790\pi\)
\(182\) −14.9282 −1.10655
\(183\) 0 0
\(184\) −5.46410 −0.402819
\(185\) 0 0
\(186\) 0 0
\(187\) −0.554803 −0.0405712
\(188\) −2.53590 −0.184949
\(189\) 0 0
\(190\) 0 0
\(191\) −17.2480 −1.24802 −0.624009 0.781417i \(-0.714497\pi\)
−0.624009 + 0.781417i \(0.714497\pi\)
\(192\) 0 0
\(193\) −8.76268 −0.630752 −0.315376 0.948967i \(-0.602131\pi\)
−0.315376 + 0.948967i \(0.602131\pi\)
\(194\) −2.55103 −0.183153
\(195\) 0 0
\(196\) 7.92820 0.566300
\(197\) −23.4641 −1.67175 −0.835874 0.548921i \(-0.815039\pi\)
−0.835874 + 0.548921i \(0.815039\pi\)
\(198\) 0 0
\(199\) 13.0718 0.926635 0.463318 0.886192i \(-0.346659\pi\)
0.463318 + 0.886192i \(0.346659\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −10.5558 −0.742706
\(203\) −17.8564 −1.25327
\(204\) 0 0
\(205\) 0 0
\(206\) −4.89898 −0.341328
\(207\) 0 0
\(208\) 3.86370 0.267900
\(209\) −1.03528 −0.0716116
\(210\) 0 0
\(211\) −17.8564 −1.22929 −0.614643 0.788806i \(-0.710700\pi\)
−0.614643 + 0.788806i \(0.710700\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 2.92820 0.200168
\(215\) 0 0
\(216\) 0 0
\(217\) 42.2233 2.86631
\(218\) 8.92820 0.604694
\(219\) 0 0
\(220\) 0 0
\(221\) −2.07055 −0.139280
\(222\) 0 0
\(223\) 10.5558 0.706871 0.353435 0.935459i \(-0.385013\pi\)
0.353435 + 0.935459i \(0.385013\pi\)
\(224\) −3.86370 −0.258155
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 1.85641 0.123214 0.0616070 0.998100i \(-0.480377\pi\)
0.0616070 + 0.998100i \(0.480377\pi\)
\(228\) 0 0
\(229\) −3.85641 −0.254839 −0.127419 0.991849i \(-0.540669\pi\)
−0.127419 + 0.991849i \(0.540669\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.62158 0.303421
\(233\) −12.5359 −0.821254 −0.410627 0.911803i \(-0.634690\pi\)
−0.410627 + 0.911803i \(0.634690\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.96953 0.453678
\(237\) 0 0
\(238\) 2.07055 0.134214
\(239\) 13.1069 0.847812 0.423906 0.905706i \(-0.360659\pi\)
0.423906 + 0.905706i \(0.360659\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −9.92820 −0.638209
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) −3.86370 −0.245842
\(248\) −10.9282 −0.693942
\(249\) 0 0
\(250\) 0 0
\(251\) −28.3586 −1.78998 −0.894990 0.446087i \(-0.852817\pi\)
−0.894990 + 0.446087i \(0.852817\pi\)
\(252\) 0 0
\(253\) −5.65685 −0.355643
\(254\) −8.48528 −0.532414
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.7846 −1.42126 −0.710632 0.703563i \(-0.751591\pi\)
−0.710632 + 0.703563i \(0.751591\pi\)
\(258\) 0 0
\(259\) −6.92820 −0.430498
\(260\) 0 0
\(261\) 0 0
\(262\) 3.10583 0.191879
\(263\) −19.3205 −1.19135 −0.595677 0.803224i \(-0.703116\pi\)
−0.595677 + 0.803224i \(0.703116\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3.86370 0.236899
\(267\) 0 0
\(268\) −13.3843 −0.817574
\(269\) −31.9449 −1.94772 −0.973858 0.227160i \(-0.927056\pi\)
−0.973858 + 0.227160i \(0.927056\pi\)
\(270\) 0 0
\(271\) 10.9282 0.663841 0.331921 0.943307i \(-0.392303\pi\)
0.331921 + 0.943307i \(0.392303\pi\)
\(272\) −0.535898 −0.0324936
\(273\) 0 0
\(274\) −0.535898 −0.0323748
\(275\) 0 0
\(276\) 0 0
\(277\) 18.2832 1.09853 0.549267 0.835647i \(-0.314907\pi\)
0.549267 + 0.835647i \(0.314907\pi\)
\(278\) 6.92820 0.415526
\(279\) 0 0
\(280\) 0 0
\(281\) −31.1870 −1.86046 −0.930231 0.366974i \(-0.880394\pi\)
−0.930231 + 0.366974i \(0.880394\pi\)
\(282\) 0 0
\(283\) −5.17638 −0.307704 −0.153852 0.988094i \(-0.549168\pi\)
−0.153852 + 0.988094i \(0.549168\pi\)
\(284\) 4.14110 0.245729
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) −6.92820 −0.408959
\(288\) 0 0
\(289\) −16.7128 −0.983107
\(290\) 0 0
\(291\) 0 0
\(292\) 3.58630 0.209872
\(293\) 19.8564 1.16002 0.580012 0.814608i \(-0.303048\pi\)
0.580012 + 0.814608i \(0.303048\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.79315 0.104225
\(297\) 0 0
\(298\) 20.3538 1.17906
\(299\) −21.1117 −1.22092
\(300\) 0 0
\(301\) −25.8564 −1.49034
\(302\) 5.85641 0.336998
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −17.5254 −1.00023 −0.500113 0.865960i \(-0.666708\pi\)
−0.500113 + 0.865960i \(0.666708\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) 0 0
\(311\) −1.79315 −0.101680 −0.0508401 0.998707i \(-0.516190\pi\)
−0.0508401 + 0.998707i \(0.516190\pi\)
\(312\) 0 0
\(313\) 19.5959 1.10763 0.553813 0.832641i \(-0.313172\pi\)
0.553813 + 0.832641i \(0.313172\pi\)
\(314\) −14.1421 −0.798087
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −14.0000 −0.786318 −0.393159 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) 0 0
\(319\) 4.78461 0.267887
\(320\) 0 0
\(321\) 0 0
\(322\) 21.1117 1.17651
\(323\) 0.535898 0.0298182
\(324\) 0 0
\(325\) 0 0
\(326\) −23.6627 −1.31056
\(327\) 0 0
\(328\) 1.79315 0.0990102
\(329\) 9.79796 0.540179
\(330\) 0 0
\(331\) 13.8564 0.761617 0.380808 0.924654i \(-0.375646\pi\)
0.380808 + 0.924654i \(0.375646\pi\)
\(332\) −16.3923 −0.899645
\(333\) 0 0
\(334\) −10.9282 −0.597965
\(335\) 0 0
\(336\) 0 0
\(337\) −1.03528 −0.0563951 −0.0281975 0.999602i \(-0.508977\pi\)
−0.0281975 + 0.999602i \(0.508977\pi\)
\(338\) 1.92820 0.104880
\(339\) 0 0
\(340\) 0 0
\(341\) −11.3137 −0.612672
\(342\) 0 0
\(343\) −3.58630 −0.193642
\(344\) 6.69213 0.360815
\(345\) 0 0
\(346\) 22.7846 1.22491
\(347\) −18.5359 −0.995059 −0.497530 0.867447i \(-0.665759\pi\)
−0.497530 + 0.867447i \(0.665759\pi\)
\(348\) 0 0
\(349\) 4.14359 0.221801 0.110901 0.993831i \(-0.464626\pi\)
0.110901 + 0.993831i \(0.464626\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.03528 0.0551804
\(353\) −21.3205 −1.13478 −0.567388 0.823451i \(-0.692046\pi\)
−0.567388 + 0.823451i \(0.692046\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3.86370 −0.204776
\(357\) 0 0
\(358\) −16.7675 −0.886189
\(359\) 34.7733 1.83527 0.917633 0.397429i \(-0.130097\pi\)
0.917633 + 0.397429i \(0.130097\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 19.8564 1.04363
\(363\) 0 0
\(364\) −14.9282 −0.782450
\(365\) 0 0
\(366\) 0 0
\(367\) 5.37945 0.280805 0.140403 0.990094i \(-0.455160\pi\)
0.140403 + 0.990094i \(0.455160\pi\)
\(368\) −5.46410 −0.284836
\(369\) 0 0
\(370\) 0 0
\(371\) 23.1822 1.20356
\(372\) 0 0
\(373\) 24.9754 1.29318 0.646588 0.762840i \(-0.276195\pi\)
0.646588 + 0.762840i \(0.276195\pi\)
\(374\) −0.554803 −0.0286882
\(375\) 0 0
\(376\) −2.53590 −0.130779
\(377\) 17.8564 0.919652
\(378\) 0 0
\(379\) −35.7128 −1.83444 −0.917222 0.398376i \(-0.869574\pi\)
−0.917222 + 0.398376i \(0.869574\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −17.2480 −0.882483
\(383\) 22.9282 1.17158 0.585788 0.810464i \(-0.300785\pi\)
0.585788 + 0.810464i \(0.300785\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8.76268 −0.446009
\(387\) 0 0
\(388\) −2.55103 −0.129509
\(389\) −19.7990 −1.00385 −0.501924 0.864912i \(-0.667374\pi\)
−0.501924 + 0.864912i \(0.667374\pi\)
\(390\) 0 0
\(391\) 2.92820 0.148086
\(392\) 7.92820 0.400435
\(393\) 0 0
\(394\) −23.4641 −1.18210
\(395\) 0 0
\(396\) 0 0
\(397\) −18.2832 −0.917610 −0.458805 0.888537i \(-0.651722\pi\)
−0.458805 + 0.888537i \(0.651722\pi\)
\(398\) 13.0718 0.655230
\(399\) 0 0
\(400\) 0 0
\(401\) 9.52056 0.475434 0.237717 0.971334i \(-0.423601\pi\)
0.237717 + 0.971334i \(0.423601\pi\)
\(402\) 0 0
\(403\) −42.2233 −2.10329
\(404\) −10.5558 −0.525172
\(405\) 0 0
\(406\) −17.8564 −0.886199
\(407\) 1.85641 0.0920187
\(408\) 0 0
\(409\) −20.9282 −1.03483 −0.517417 0.855734i \(-0.673106\pi\)
−0.517417 + 0.855734i \(0.673106\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.89898 −0.241355
\(413\) −26.9282 −1.32505
\(414\) 0 0
\(415\) 0 0
\(416\) 3.86370 0.189434
\(417\) 0 0
\(418\) −1.03528 −0.0506370
\(419\) 29.3195 1.43235 0.716177 0.697919i \(-0.245890\pi\)
0.716177 + 0.697919i \(0.245890\pi\)
\(420\) 0 0
\(421\) 29.7128 1.44811 0.724057 0.689740i \(-0.242275\pi\)
0.724057 + 0.689740i \(0.242275\pi\)
\(422\) −17.8564 −0.869236
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 7.72741 0.373955
\(428\) 2.92820 0.141540
\(429\) 0 0
\(430\) 0 0
\(431\) −26.2137 −1.26267 −0.631335 0.775510i \(-0.717493\pi\)
−0.631335 + 0.775510i \(0.717493\pi\)
\(432\) 0 0
\(433\) 10.8332 0.520612 0.260306 0.965526i \(-0.416177\pi\)
0.260306 + 0.965526i \(0.416177\pi\)
\(434\) 42.2233 2.02678
\(435\) 0 0
\(436\) 8.92820 0.427583
\(437\) 5.46410 0.261383
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.07055 −0.0984861
\(443\) −16.3923 −0.778822 −0.389411 0.921064i \(-0.627321\pi\)
−0.389411 + 0.921064i \(0.627321\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 10.5558 0.499833
\(447\) 0 0
\(448\) −3.86370 −0.182543
\(449\) −40.4302 −1.90802 −0.954009 0.299777i \(-0.903088\pi\)
−0.954009 + 0.299777i \(0.903088\pi\)
\(450\) 0 0
\(451\) 1.85641 0.0874148
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 1.85641 0.0871255
\(455\) 0 0
\(456\) 0 0
\(457\) 33.9411 1.58770 0.793849 0.608114i \(-0.208074\pi\)
0.793849 + 0.608114i \(0.208074\pi\)
\(458\) −3.85641 −0.180198
\(459\) 0 0
\(460\) 0 0
\(461\) 27.5264 1.28203 0.641016 0.767527i \(-0.278513\pi\)
0.641016 + 0.767527i \(0.278513\pi\)
\(462\) 0 0
\(463\) −19.8733 −0.923591 −0.461796 0.886986i \(-0.652795\pi\)
−0.461796 + 0.886986i \(0.652795\pi\)
\(464\) 4.62158 0.214551
\(465\) 0 0
\(466\) −12.5359 −0.580714
\(467\) −2.53590 −0.117347 −0.0586737 0.998277i \(-0.518687\pi\)
−0.0586737 + 0.998277i \(0.518687\pi\)
\(468\) 0 0
\(469\) 51.7128 2.38788
\(470\) 0 0
\(471\) 0 0
\(472\) 6.96953 0.320799
\(473\) 6.92820 0.318559
\(474\) 0 0
\(475\) 0 0
\(476\) 2.07055 0.0949036
\(477\) 0 0
\(478\) 13.1069 0.599494
\(479\) 16.6932 0.762730 0.381365 0.924425i \(-0.375454\pi\)
0.381365 + 0.924425i \(0.375454\pi\)
\(480\) 0 0
\(481\) 6.92820 0.315899
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −9.92820 −0.451282
\(485\) 0 0
\(486\) 0 0
\(487\) 3.38323 0.153309 0.0766544 0.997058i \(-0.475576\pi\)
0.0766544 + 0.997058i \(0.475576\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) 0 0
\(491\) 18.5606 0.837630 0.418815 0.908072i \(-0.362446\pi\)
0.418815 + 0.908072i \(0.362446\pi\)
\(492\) 0 0
\(493\) −2.47670 −0.111545
\(494\) −3.86370 −0.173836
\(495\) 0 0
\(496\) −10.9282 −0.490691
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) 25.8564 1.15749 0.578746 0.815508i \(-0.303542\pi\)
0.578746 + 0.815508i \(0.303542\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −28.3586 −1.26571
\(503\) 11.3205 0.504757 0.252378 0.967629i \(-0.418787\pi\)
0.252378 + 0.967629i \(0.418787\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −5.65685 −0.251478
\(507\) 0 0
\(508\) −8.48528 −0.376473
\(509\) −37.0470 −1.64208 −0.821039 0.570873i \(-0.806605\pi\)
−0.821039 + 0.570873i \(0.806605\pi\)
\(510\) 0 0
\(511\) −13.8564 −0.612971
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −22.7846 −1.00499
\(515\) 0 0
\(516\) 0 0
\(517\) −2.62536 −0.115463
\(518\) −6.92820 −0.304408
\(519\) 0 0
\(520\) 0 0
\(521\) 9.52056 0.417103 0.208552 0.978011i \(-0.433125\pi\)
0.208552 + 0.978011i \(0.433125\pi\)
\(522\) 0 0
\(523\) −26.7685 −1.17051 −0.585253 0.810851i \(-0.699005\pi\)
−0.585253 + 0.810851i \(0.699005\pi\)
\(524\) 3.10583 0.135679
\(525\) 0 0
\(526\) −19.3205 −0.842414
\(527\) 5.85641 0.255109
\(528\) 0 0
\(529\) 6.85641 0.298105
\(530\) 0 0
\(531\) 0 0
\(532\) 3.86370 0.167513
\(533\) 6.92820 0.300094
\(534\) 0 0
\(535\) 0 0
\(536\) −13.3843 −0.578112
\(537\) 0 0
\(538\) −31.9449 −1.37724
\(539\) 8.20788 0.353538
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 10.9282 0.469407
\(543\) 0 0
\(544\) −0.535898 −0.0229765
\(545\) 0 0
\(546\) 0 0
\(547\) 10.3528 0.442652 0.221326 0.975200i \(-0.428961\pi\)
0.221326 + 0.975200i \(0.428961\pi\)
\(548\) −0.535898 −0.0228924
\(549\) 0 0
\(550\) 0 0
\(551\) −4.62158 −0.196886
\(552\) 0 0
\(553\) 15.4548 0.657206
\(554\) 18.2832 0.776780
\(555\) 0 0
\(556\) 6.92820 0.293821
\(557\) −31.4641 −1.33318 −0.666588 0.745426i \(-0.732246\pi\)
−0.666588 + 0.745426i \(0.732246\pi\)
\(558\) 0 0
\(559\) 25.8564 1.09361
\(560\) 0 0
\(561\) 0 0
\(562\) −31.1870 −1.31555
\(563\) −25.8564 −1.08972 −0.544859 0.838528i \(-0.683417\pi\)
−0.544859 + 0.838528i \(0.683417\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −5.17638 −0.217580
\(567\) 0 0
\(568\) 4.14110 0.173757
\(569\) 44.0165 1.84527 0.922634 0.385678i \(-0.126032\pi\)
0.922634 + 0.385678i \(0.126032\pi\)
\(570\) 0 0
\(571\) 14.9282 0.624726 0.312363 0.949963i \(-0.398880\pi\)
0.312363 + 0.949963i \(0.398880\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) −6.92820 −0.289178
\(575\) 0 0
\(576\) 0 0
\(577\) −5.65685 −0.235498 −0.117749 0.993043i \(-0.537568\pi\)
−0.117749 + 0.993043i \(0.537568\pi\)
\(578\) −16.7128 −0.695161
\(579\) 0 0
\(580\) 0 0
\(581\) 63.3350 2.62758
\(582\) 0 0
\(583\) −6.21166 −0.257261
\(584\) 3.58630 0.148402
\(585\) 0 0
\(586\) 19.8564 0.820261
\(587\) −24.3923 −1.00678 −0.503389 0.864060i \(-0.667914\pi\)
−0.503389 + 0.864060i \(0.667914\pi\)
\(588\) 0 0
\(589\) 10.9282 0.450289
\(590\) 0 0
\(591\) 0 0
\(592\) 1.79315 0.0736980
\(593\) 4.53590 0.186267 0.0931335 0.995654i \(-0.470312\pi\)
0.0931335 + 0.995654i \(0.470312\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 20.3538 0.833724
\(597\) 0 0
\(598\) −21.1117 −0.863320
\(599\) −20.5569 −0.839931 −0.419965 0.907540i \(-0.637958\pi\)
−0.419965 + 0.907540i \(0.637958\pi\)
\(600\) 0 0
\(601\) −18.7846 −0.766240 −0.383120 0.923699i \(-0.625150\pi\)
−0.383120 + 0.923699i \(0.625150\pi\)
\(602\) −25.8564 −1.05383
\(603\) 0 0
\(604\) 5.85641 0.238294
\(605\) 0 0
\(606\) 0 0
\(607\) 45.6066 1.85111 0.925557 0.378609i \(-0.123598\pi\)
0.925557 + 0.378609i \(0.123598\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −9.79796 −0.396383
\(612\) 0 0
\(613\) 5.85993 0.236680 0.118340 0.992973i \(-0.462243\pi\)
0.118340 + 0.992973i \(0.462243\pi\)
\(614\) −17.5254 −0.707266
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) −28.2487 −1.13725 −0.568625 0.822597i \(-0.692524\pi\)
−0.568625 + 0.822597i \(0.692524\pi\)
\(618\) 0 0
\(619\) −34.6410 −1.39234 −0.696170 0.717877i \(-0.745114\pi\)
−0.696170 + 0.717877i \(0.745114\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −1.79315 −0.0718988
\(623\) 14.9282 0.598086
\(624\) 0 0
\(625\) 0 0
\(626\) 19.5959 0.783210
\(627\) 0 0
\(628\) −14.1421 −0.564333
\(629\) −0.960947 −0.0383155
\(630\) 0 0
\(631\) 21.8564 0.870090 0.435045 0.900409i \(-0.356732\pi\)
0.435045 + 0.900409i \(0.356732\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) −14.0000 −0.556011
\(635\) 0 0
\(636\) 0 0
\(637\) 30.6322 1.21369
\(638\) 4.78461 0.189425
\(639\) 0 0
\(640\) 0 0
\(641\) 17.2480 0.681254 0.340627 0.940199i \(-0.389361\pi\)
0.340627 + 0.940199i \(0.389361\pi\)
\(642\) 0 0
\(643\) 37.0470 1.46099 0.730495 0.682918i \(-0.239290\pi\)
0.730495 + 0.682918i \(0.239290\pi\)
\(644\) 21.1117 0.831916
\(645\) 0 0
\(646\) 0.535898 0.0210846
\(647\) −13.4641 −0.529328 −0.264664 0.964341i \(-0.585261\pi\)
−0.264664 + 0.964341i \(0.585261\pi\)
\(648\) 0 0
\(649\) 7.21539 0.283229
\(650\) 0 0
\(651\) 0 0
\(652\) −23.6627 −0.926703
\(653\) 24.2487 0.948925 0.474463 0.880276i \(-0.342642\pi\)
0.474463 + 0.880276i \(0.342642\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.79315 0.0700108
\(657\) 0 0
\(658\) 9.79796 0.381964
\(659\) 20.3538 0.792871 0.396436 0.918063i \(-0.370247\pi\)
0.396436 + 0.918063i \(0.370247\pi\)
\(660\) 0 0
\(661\) −10.7846 −0.419473 −0.209736 0.977758i \(-0.567261\pi\)
−0.209736 + 0.977758i \(0.567261\pi\)
\(662\) 13.8564 0.538545
\(663\) 0 0
\(664\) −16.3923 −0.636145
\(665\) 0 0
\(666\) 0 0
\(667\) −25.2528 −0.977791
\(668\) −10.9282 −0.422825
\(669\) 0 0
\(670\) 0 0
\(671\) −2.07055 −0.0799328
\(672\) 0 0
\(673\) 27.2490 1.05037 0.525186 0.850988i \(-0.323996\pi\)
0.525186 + 0.850988i \(0.323996\pi\)
\(674\) −1.03528 −0.0398773
\(675\) 0 0
\(676\) 1.92820 0.0741617
\(677\) −36.6410 −1.40823 −0.704114 0.710087i \(-0.748656\pi\)
−0.704114 + 0.710087i \(0.748656\pi\)
\(678\) 0 0
\(679\) 9.85641 0.378254
\(680\) 0 0
\(681\) 0 0
\(682\) −11.3137 −0.433224
\(683\) −18.9282 −0.724268 −0.362134 0.932126i \(-0.617952\pi\)
−0.362134 + 0.932126i \(0.617952\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −3.58630 −0.136926
\(687\) 0 0
\(688\) 6.69213 0.255135
\(689\) −23.1822 −0.883172
\(690\) 0 0
\(691\) 38.9282 1.48090 0.740449 0.672112i \(-0.234613\pi\)
0.740449 + 0.672112i \(0.234613\pi\)
\(692\) 22.7846 0.866141
\(693\) 0 0
\(694\) −18.5359 −0.703613
\(695\) 0 0
\(696\) 0 0
\(697\) −0.960947 −0.0363985
\(698\) 4.14359 0.156837
\(699\) 0 0
\(700\) 0 0
\(701\) −7.52433 −0.284190 −0.142095 0.989853i \(-0.545384\pi\)
−0.142095 + 0.989853i \(0.545384\pi\)
\(702\) 0 0
\(703\) −1.79315 −0.0676300
\(704\) 1.03528 0.0390184
\(705\) 0 0
\(706\) −21.3205 −0.802408
\(707\) 40.7846 1.53386
\(708\) 0 0
\(709\) −11.8564 −0.445277 −0.222638 0.974901i \(-0.571467\pi\)
−0.222638 + 0.974901i \(0.571467\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.86370 −0.144798
\(713\) 59.7128 2.23626
\(714\) 0 0
\(715\) 0 0
\(716\) −16.7675 −0.626631
\(717\) 0 0
\(718\) 34.7733 1.29773
\(719\) 46.6418 1.73945 0.869724 0.493539i \(-0.164297\pi\)
0.869724 + 0.493539i \(0.164297\pi\)
\(720\) 0 0
\(721\) 18.9282 0.704923
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 19.8564 0.737958
\(725\) 0 0
\(726\) 0 0
\(727\) 15.1774 0.562899 0.281450 0.959576i \(-0.409185\pi\)
0.281450 + 0.959576i \(0.409185\pi\)
\(728\) −14.9282 −0.553276
\(729\) 0 0
\(730\) 0 0
\(731\) −3.58630 −0.132644
\(732\) 0 0
\(733\) 21.8695 0.807770 0.403885 0.914810i \(-0.367660\pi\)
0.403885 + 0.914810i \(0.367660\pi\)
\(734\) 5.37945 0.198559
\(735\) 0 0
\(736\) −5.46410 −0.201409
\(737\) −13.8564 −0.510407
\(738\) 0 0
\(739\) 23.7128 0.872290 0.436145 0.899876i \(-0.356343\pi\)
0.436145 + 0.899876i \(0.356343\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 23.1822 0.851046
\(743\) 34.6410 1.27086 0.635428 0.772160i \(-0.280824\pi\)
0.635428 + 0.772160i \(0.280824\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 24.9754 0.914413
\(747\) 0 0
\(748\) −0.554803 −0.0202856
\(749\) −11.3137 −0.413394
\(750\) 0 0
\(751\) 5.85641 0.213703 0.106852 0.994275i \(-0.465923\pi\)
0.106852 + 0.994275i \(0.465923\pi\)
\(752\) −2.53590 −0.0924747
\(753\) 0 0
\(754\) 17.8564 0.650292
\(755\) 0 0
\(756\) 0 0
\(757\) −40.3559 −1.46676 −0.733379 0.679820i \(-0.762058\pi\)
−0.733379 + 0.679820i \(0.762058\pi\)
\(758\) −35.7128 −1.29715
\(759\) 0 0
\(760\) 0 0
\(761\) 23.1822 0.840355 0.420177 0.907442i \(-0.361968\pi\)
0.420177 + 0.907442i \(0.361968\pi\)
\(762\) 0 0
\(763\) −34.4959 −1.24884
\(764\) −17.2480 −0.624009
\(765\) 0 0
\(766\) 22.9282 0.828430
\(767\) 26.9282 0.972321
\(768\) 0 0
\(769\) 35.8564 1.29302 0.646508 0.762908i \(-0.276229\pi\)
0.646508 + 0.762908i \(0.276229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.76268 −0.315376
\(773\) 7.85641 0.282575 0.141288 0.989969i \(-0.454876\pi\)
0.141288 + 0.989969i \(0.454876\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −2.55103 −0.0915765
\(777\) 0 0
\(778\) −19.7990 −0.709828
\(779\) −1.79315 −0.0642463
\(780\) 0 0
\(781\) 4.28719 0.153408
\(782\) 2.92820 0.104712
\(783\) 0 0
\(784\) 7.92820 0.283150
\(785\) 0 0
\(786\) 0 0
\(787\) −7.17260 −0.255676 −0.127838 0.991795i \(-0.540804\pi\)
−0.127838 + 0.991795i \(0.540804\pi\)
\(788\) −23.4641 −0.835874
\(789\) 0 0
\(790\) 0 0
\(791\) 23.1822 0.824265
\(792\) 0 0
\(793\) −7.72741 −0.274408
\(794\) −18.2832 −0.648848
\(795\) 0 0
\(796\) 13.0718 0.463318
\(797\) −40.6410 −1.43958 −0.719789 0.694193i \(-0.755762\pi\)
−0.719789 + 0.694193i \(0.755762\pi\)
\(798\) 0 0
\(799\) 1.35898 0.0480774
\(800\) 0 0
\(801\) 0 0
\(802\) 9.52056 0.336183
\(803\) 3.71281 0.131022
\(804\) 0 0
\(805\) 0 0
\(806\) −42.2233 −1.48725
\(807\) 0 0
\(808\) −10.5558 −0.371353
\(809\) 44.8487 1.57680 0.788398 0.615166i \(-0.210911\pi\)
0.788398 + 0.615166i \(0.210911\pi\)
\(810\) 0 0
\(811\) −40.4974 −1.42206 −0.711028 0.703163i \(-0.751770\pi\)
−0.711028 + 0.703163i \(0.751770\pi\)
\(812\) −17.8564 −0.626637
\(813\) 0 0
\(814\) 1.85641 0.0650670
\(815\) 0 0
\(816\) 0 0
\(817\) −6.69213 −0.234128
\(818\) −20.9282 −0.731737
\(819\) 0 0
\(820\) 0 0
\(821\) −7.93048 −0.276776 −0.138388 0.990378i \(-0.544192\pi\)
−0.138388 + 0.990378i \(0.544192\pi\)
\(822\) 0 0
\(823\) 47.1966 1.64517 0.822586 0.568641i \(-0.192531\pi\)
0.822586 + 0.568641i \(0.192531\pi\)
\(824\) −4.89898 −0.170664
\(825\) 0 0
\(826\) −26.9282 −0.936952
\(827\) −6.92820 −0.240917 −0.120459 0.992718i \(-0.538437\pi\)
−0.120459 + 0.992718i \(0.538437\pi\)
\(828\) 0 0
\(829\) 14.7846 0.513491 0.256745 0.966479i \(-0.417350\pi\)
0.256745 + 0.966479i \(0.417350\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.86370 0.133950
\(833\) −4.24871 −0.147209
\(834\) 0 0
\(835\) 0 0
\(836\) −1.03528 −0.0358058
\(837\) 0 0
\(838\) 29.3195 1.01283
\(839\) −28.2843 −0.976481 −0.488241 0.872709i \(-0.662361\pi\)
−0.488241 + 0.872709i \(0.662361\pi\)
\(840\) 0 0
\(841\) −7.64102 −0.263483
\(842\) 29.7128 1.02397
\(843\) 0 0
\(844\) −17.8564 −0.614643
\(845\) 0 0
\(846\) 0 0
\(847\) 38.3596 1.31805
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 0 0
\(851\) −9.79796 −0.335870
\(852\) 0 0
\(853\) 14.6969 0.503214 0.251607 0.967830i \(-0.419041\pi\)
0.251607 + 0.967830i \(0.419041\pi\)
\(854\) 7.72741 0.264426
\(855\) 0 0
\(856\) 2.92820 0.100084
\(857\) −16.6410 −0.568446 −0.284223 0.958758i \(-0.591736\pi\)
−0.284223 + 0.958758i \(0.591736\pi\)
\(858\) 0 0
\(859\) −4.78461 −0.163249 −0.0816244 0.996663i \(-0.526011\pi\)
−0.0816244 + 0.996663i \(0.526011\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −26.2137 −0.892843
\(863\) −20.7846 −0.707516 −0.353758 0.935337i \(-0.615096\pi\)
−0.353758 + 0.935337i \(0.615096\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 10.8332 0.368128
\(867\) 0 0
\(868\) 42.2233 1.43315
\(869\) −4.14110 −0.140477
\(870\) 0 0
\(871\) −51.7128 −1.75222
\(872\) 8.92820 0.302347
\(873\) 0 0
\(874\) 5.46410 0.184826
\(875\) 0 0
\(876\) 0 0
\(877\) −29.6713 −1.00193 −0.500964 0.865468i \(-0.667021\pi\)
−0.500964 + 0.865468i \(0.667021\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) 0 0
\(881\) −38.6370 −1.30171 −0.650857 0.759200i \(-0.725590\pi\)
−0.650857 + 0.759200i \(0.725590\pi\)
\(882\) 0 0
\(883\) 49.4703 1.66481 0.832404 0.554170i \(-0.186964\pi\)
0.832404 + 0.554170i \(0.186964\pi\)
\(884\) −2.07055 −0.0696402
\(885\) 0 0
\(886\) −16.3923 −0.550710
\(887\) 15.7128 0.527585 0.263792 0.964580i \(-0.415027\pi\)
0.263792 + 0.964580i \(0.415027\pi\)
\(888\) 0 0
\(889\) 32.7846 1.09956
\(890\) 0 0
\(891\) 0 0
\(892\) 10.5558 0.353435
\(893\) 2.53590 0.0848606
\(894\) 0 0
\(895\) 0 0
\(896\) −3.86370 −0.129077
\(897\) 0 0
\(898\) −40.4302 −1.34917
\(899\) −50.5055 −1.68445
\(900\) 0 0
\(901\) 3.21539 0.107120
\(902\) 1.85641 0.0618116
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) −36.5665 −1.21417 −0.607085 0.794637i \(-0.707661\pi\)
−0.607085 + 0.794637i \(0.707661\pi\)
\(908\) 1.85641 0.0616070
\(909\) 0 0
\(910\) 0 0
\(911\) 45.2548 1.49936 0.749680 0.661801i \(-0.230208\pi\)
0.749680 + 0.661801i \(0.230208\pi\)
\(912\) 0 0
\(913\) −16.9706 −0.561644
\(914\) 33.9411 1.12267
\(915\) 0 0
\(916\) −3.85641 −0.127419
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) 27.7128 0.914161 0.457081 0.889425i \(-0.348895\pi\)
0.457081 + 0.889425i \(0.348895\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 27.5264 0.906534
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) 0 0
\(926\) −19.8733 −0.653078
\(927\) 0 0
\(928\) 4.62158 0.151711
\(929\) 11.3137 0.371191 0.185595 0.982626i \(-0.440579\pi\)
0.185595 + 0.982626i \(0.440579\pi\)
\(930\) 0 0
\(931\) −7.92820 −0.259836
\(932\) −12.5359 −0.410627
\(933\) 0 0
\(934\) −2.53590 −0.0829771
\(935\) 0 0
\(936\) 0 0
\(937\) −20.5569 −0.671563 −0.335782 0.941940i \(-0.609000\pi\)
−0.335782 + 0.941940i \(0.609000\pi\)
\(938\) 51.7128 1.68848
\(939\) 0 0
\(940\) 0 0
\(941\) 15.9353 0.519475 0.259738 0.965679i \(-0.416364\pi\)
0.259738 + 0.965679i \(0.416364\pi\)
\(942\) 0 0
\(943\) −9.79796 −0.319065
\(944\) 6.96953 0.226839
\(945\) 0 0
\(946\) 6.92820 0.225255
\(947\) 25.1769 0.818140 0.409070 0.912503i \(-0.365853\pi\)
0.409070 + 0.912503i \(0.365853\pi\)
\(948\) 0 0
\(949\) 13.8564 0.449798
\(950\) 0 0
\(951\) 0 0
\(952\) 2.07055 0.0671070
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 13.1069 0.423906
\(957\) 0 0
\(958\) 16.6932 0.539332
\(959\) 2.07055 0.0668616
\(960\) 0 0
\(961\) 88.4256 2.85244
\(962\) 6.92820 0.223374
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −17.2480 −0.554657 −0.277329 0.960775i \(-0.589449\pi\)
−0.277329 + 0.960775i \(0.589449\pi\)
\(968\) −9.92820 −0.319105
\(969\) 0 0
\(970\) 0 0
\(971\) 54.2949 1.74241 0.871203 0.490922i \(-0.163340\pi\)
0.871203 + 0.490922i \(0.163340\pi\)
\(972\) 0 0
\(973\) −26.7685 −0.858159
\(974\) 3.38323 0.108406
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −5.71281 −0.182769 −0.0913845 0.995816i \(-0.529129\pi\)
−0.0913845 + 0.995816i \(0.529129\pi\)
\(978\) 0 0
\(979\) −4.00000 −0.127841
\(980\) 0 0
\(981\) 0 0
\(982\) 18.5606 0.592294
\(983\) −7.21539 −0.230135 −0.115068 0.993358i \(-0.536708\pi\)
−0.115068 + 0.993358i \(0.536708\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.47670 −0.0788741
\(987\) 0 0
\(988\) −3.86370 −0.122921
\(989\) −36.5665 −1.16275
\(990\) 0 0
\(991\) −20.7846 −0.660245 −0.330122 0.943938i \(-0.607090\pi\)
−0.330122 + 0.943938i \(0.607090\pi\)
\(992\) −10.9282 −0.346971
\(993\) 0 0
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) −18.2832 −0.579036 −0.289518 0.957173i \(-0.593495\pi\)
−0.289518 + 0.957173i \(0.593495\pi\)
\(998\) 25.8564 0.818470
\(999\) 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.cv.1.1 4
3.2 odd 2 8550.2.a.cu.1.1 4
5.2 odd 4 1710.2.d.g.1369.5 yes 8
5.3 odd 4 1710.2.d.g.1369.2 8
5.4 even 2 8550.2.a.cu.1.4 4
15.2 even 4 1710.2.d.g.1369.4 yes 8
15.8 even 4 1710.2.d.g.1369.7 yes 8
15.14 odd 2 inner 8550.2.a.cv.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1710.2.d.g.1369.2 8 5.3 odd 4
1710.2.d.g.1369.4 yes 8 15.2 even 4
1710.2.d.g.1369.5 yes 8 5.2 odd 4
1710.2.d.g.1369.7 yes 8 15.8 even 4
8550.2.a.cu.1.1 4 3.2 odd 2
8550.2.a.cu.1.4 4 5.4 even 2
8550.2.a.cv.1.1 4 1.1 even 1 trivial
8550.2.a.cv.1.4 4 15.14 odd 2 inner