Properties

Label 8550.2.a.cl.1.2
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.363328\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -0.636672 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -0.636672 q^{7} +1.00000 q^{8} -3.50466 q^{11} +0.141336 q^{13} -0.636672 q^{14} +1.00000 q^{16} +2.14134 q^{17} +1.00000 q^{19} -3.50466 q^{22} +4.91934 q^{23} +0.141336 q^{26} -0.636672 q^{28} -7.15066 q^{29} -7.78734 q^{31} +1.00000 q^{32} +2.14134 q^{34} -3.27334 q^{37} +1.00000 q^{38} +4.23132 q^{41} -2.49534 q^{43} -3.50466 q^{44} +4.91934 q^{46} -10.2827 q^{47} -6.59465 q^{49} +0.141336 q^{52} +8.14134 q^{53} -0.636672 q^{56} -7.15066 q^{58} +5.64600 q^{59} -6.49534 q^{61} -7.78734 q^{62} +1.00000 q^{64} -8.37266 q^{67} +2.14134 q^{68} +8.95798 q^{71} +3.69735 q^{73} -3.27334 q^{74} +1.00000 q^{76} +2.23132 q^{77} -4.17997 q^{79} +4.23132 q^{82} -9.00933 q^{83} -2.49534 q^{86} -3.50466 q^{88} +6.77801 q^{89} -0.0899847 q^{91} +4.91934 q^{92} -10.2827 q^{94} -14.5653 q^{97} -6.59465 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} - 4q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} - 4q^{7} + 3q^{8} - 8q^{13} - 4q^{14} + 3q^{16} - 2q^{17} + 3q^{19} - 8q^{26} - 4q^{28} + 8q^{29} + 4q^{31} + 3q^{32} - 2q^{34} - 14q^{37} + 3q^{38} - 2q^{41} - 18q^{43} - 14q^{47} - 3q^{49} - 8q^{52} + 16q^{53} - 4q^{56} + 8q^{58} - 2q^{59} - 30q^{61} + 4q^{62} + 3q^{64} - 2q^{67} - 2q^{68} + 8q^{71} - 10q^{73} - 14q^{74} + 3q^{76} - 8q^{77} - 2q^{82} - 6q^{83} - 18q^{86} + 14q^{89} + 6q^{91} - 14q^{94} - 10q^{97} - 3q^{98} + O(q^{100}) \)

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
\(7\) −0.636672 −0.240639 −0.120320 0.992735i \(-0.538392\pi\)
−0.120320 + 0.992735i \(0.538392\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −3.50466 −1.05670 −0.528348 0.849028i \(-0.677188\pi\)
−0.528348 + 0.849028i \(0.677188\pi\)
\(12\) 0 0
\(13\) 0.141336 0.0391996 0.0195998 0.999808i \(-0.493761\pi\)
0.0195998 + 0.999808i \(0.493761\pi\)
\(14\) −0.636672 −0.170158
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.14134 0.519350 0.259675 0.965696i \(-0.416385\pi\)
0.259675 + 0.965696i \(0.416385\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −3.50466 −0.747197
\(23\) 4.91934 1.02575 0.512877 0.858462i \(-0.328580\pi\)
0.512877 + 0.858462i \(0.328580\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.141336 0.0277183
\(27\) 0 0
\(28\) −0.636672 −0.120320
\(29\) −7.15066 −1.32785 −0.663923 0.747801i \(-0.731110\pi\)
−0.663923 + 0.747801i \(0.731110\pi\)
\(30\) 0 0
\(31\) −7.78734 −1.39865 −0.699323 0.714805i \(-0.746515\pi\)
−0.699323 + 0.714805i \(0.746515\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.14134 0.367236
\(35\) 0 0
\(36\) 0 0
\(37\) −3.27334 −0.538134 −0.269067 0.963121i \(-0.586715\pi\)
−0.269067 + 0.963121i \(0.586715\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 4.23132 0.660821 0.330411 0.943837i \(-0.392813\pi\)
0.330411 + 0.943837i \(0.392813\pi\)
\(42\) 0 0
\(43\) −2.49534 −0.380535 −0.190268 0.981732i \(-0.560936\pi\)
−0.190268 + 0.981732i \(0.560936\pi\)
\(44\) −3.50466 −0.528348
\(45\) 0 0
\(46\) 4.91934 0.725318
\(47\) −10.2827 −1.49988 −0.749941 0.661505i \(-0.769918\pi\)
−0.749941 + 0.661505i \(0.769918\pi\)
\(48\) 0 0
\(49\) −6.59465 −0.942093
\(50\) 0 0
\(51\) 0 0
\(52\) 0.141336 0.0195998
\(53\) 8.14134 1.11830 0.559149 0.829067i \(-0.311128\pi\)
0.559149 + 0.829067i \(0.311128\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.636672 −0.0850788
\(57\) 0 0
\(58\) −7.15066 −0.938928
\(59\) 5.64600 0.735047 0.367523 0.930014i \(-0.380206\pi\)
0.367523 + 0.930014i \(0.380206\pi\)
\(60\) 0 0
\(61\) −6.49534 −0.831643 −0.415821 0.909446i \(-0.636506\pi\)
−0.415821 + 0.909446i \(0.636506\pi\)
\(62\) −7.78734 −0.988993
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.37266 −1.02288 −0.511441 0.859318i \(-0.670888\pi\)
−0.511441 + 0.859318i \(0.670888\pi\)
\(68\) 2.14134 0.259675
\(69\) 0 0
\(70\) 0 0
\(71\) 8.95798 1.06312 0.531558 0.847022i \(-0.321607\pi\)
0.531558 + 0.847022i \(0.321607\pi\)
\(72\) 0 0
\(73\) 3.69735 0.432742 0.216371 0.976311i \(-0.430578\pi\)
0.216371 + 0.976311i \(0.430578\pi\)
\(74\) −3.27334 −0.380518
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 2.23132 0.254283
\(78\) 0 0
\(79\) −4.17997 −0.470283 −0.235142 0.971961i \(-0.575555\pi\)
−0.235142 + 0.971961i \(0.575555\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.23132 0.467271
\(83\) −9.00933 −0.988902 −0.494451 0.869205i \(-0.664631\pi\)
−0.494451 + 0.869205i \(0.664631\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.49534 −0.269079
\(87\) 0 0
\(88\) −3.50466 −0.373598
\(89\) 6.77801 0.718467 0.359234 0.933248i \(-0.383038\pi\)
0.359234 + 0.933248i \(0.383038\pi\)
\(90\) 0 0
\(91\) −0.0899847 −0.00943296
\(92\) 4.91934 0.512877
\(93\) 0 0
\(94\) −10.2827 −1.06058
\(95\) 0 0
\(96\) 0 0
\(97\) −14.5653 −1.47889 −0.739443 0.673219i \(-0.764911\pi\)
−0.739443 + 0.673219i \(0.764911\pi\)
\(98\) −6.59465 −0.666160
\(99\) 0 0
\(100\) 0 0
\(101\) 16.6167 1.65342 0.826712 0.562626i \(-0.190209\pi\)
0.826712 + 0.562626i \(0.190209\pi\)
\(102\) 0 0
\(103\) −9.06068 −0.892775 −0.446388 0.894840i \(-0.647290\pi\)
−0.446388 + 0.894840i \(0.647290\pi\)
\(104\) 0.141336 0.0138591
\(105\) 0 0
\(106\) 8.14134 0.790756
\(107\) 0.0899847 0.00869915 0.00434958 0.999991i \(-0.498615\pi\)
0.00434958 + 0.999991i \(0.498615\pi\)
\(108\) 0 0
\(109\) −13.5946 −1.30213 −0.651066 0.759021i \(-0.725678\pi\)
−0.651066 + 0.759021i \(0.725678\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.636672 −0.0601598
\(113\) 11.5233 1.08402 0.542011 0.840371i \(-0.317663\pi\)
0.542011 + 0.840371i \(0.317663\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −7.15066 −0.663923
\(117\) 0 0
\(118\) 5.64600 0.519756
\(119\) −1.36333 −0.124976
\(120\) 0 0
\(121\) 1.28267 0.116607
\(122\) −6.49534 −0.588060
\(123\) 0 0
\(124\) −7.78734 −0.699323
\(125\) 0 0
\(126\) 0 0
\(127\) 3.29200 0.292118 0.146059 0.989276i \(-0.453341\pi\)
0.146059 + 0.989276i \(0.453341\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −18.0187 −1.57430 −0.787149 0.616763i \(-0.788444\pi\)
−0.787149 + 0.616763i \(0.788444\pi\)
\(132\) 0 0
\(133\) −0.636672 −0.0552064
\(134\) −8.37266 −0.723287
\(135\) 0 0
\(136\) 2.14134 0.183618
\(137\) −14.4240 −1.23233 −0.616163 0.787619i \(-0.711314\pi\)
−0.616163 + 0.787619i \(0.711314\pi\)
\(138\) 0 0
\(139\) 15.4720 1.31232 0.656158 0.754624i \(-0.272181\pi\)
0.656158 + 0.754624i \(0.272181\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.95798 0.751737
\(143\) −0.495336 −0.0414220
\(144\) 0 0
\(145\) 0 0
\(146\) 3.69735 0.305995
\(147\) 0 0
\(148\) −3.27334 −0.269067
\(149\) 17.1893 1.40820 0.704101 0.710100i \(-0.251350\pi\)
0.704101 + 0.710100i \(0.251350\pi\)
\(150\) 0 0
\(151\) 3.29200 0.267899 0.133950 0.990988i \(-0.457234\pi\)
0.133950 + 0.990988i \(0.457234\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 2.23132 0.179805
\(155\) 0 0
\(156\) 0 0
\(157\) −15.1893 −1.21224 −0.606119 0.795374i \(-0.707274\pi\)
−0.606119 + 0.795374i \(0.707274\pi\)
\(158\) −4.17997 −0.332541
\(159\) 0 0
\(160\) 0 0
\(161\) −3.13201 −0.246837
\(162\) 0 0
\(163\) 14.0700 1.10205 0.551024 0.834489i \(-0.314237\pi\)
0.551024 + 0.834489i \(0.314237\pi\)
\(164\) 4.23132 0.330411
\(165\) 0 0
\(166\) −9.00933 −0.699260
\(167\) −14.7967 −1.14500 −0.572500 0.819905i \(-0.694026\pi\)
−0.572500 + 0.819905i \(0.694026\pi\)
\(168\) 0 0
\(169\) −12.9800 −0.998463
\(170\) 0 0
\(171\) 0 0
\(172\) −2.49534 −0.190268
\(173\) −17.2920 −1.31469 −0.657343 0.753591i \(-0.728320\pi\)
−0.657343 + 0.753591i \(0.728320\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.50466 −0.264174
\(177\) 0 0
\(178\) 6.77801 0.508033
\(179\) −17.7360 −1.32565 −0.662825 0.748774i \(-0.730643\pi\)
−0.662825 + 0.748774i \(0.730643\pi\)
\(180\) 0 0
\(181\) 6.17997 0.459354 0.229677 0.973267i \(-0.426233\pi\)
0.229677 + 0.973267i \(0.426233\pi\)
\(182\) −0.0899847 −0.00667011
\(183\) 0 0
\(184\) 4.91934 0.362659
\(185\) 0 0
\(186\) 0 0
\(187\) −7.50466 −0.548795
\(188\) −10.2827 −0.749941
\(189\) 0 0
\(190\) 0 0
\(191\) −14.6367 −1.05907 −0.529536 0.848287i \(-0.677634\pi\)
−0.529536 + 0.848287i \(0.677634\pi\)
\(192\) 0 0
\(193\) −20.0187 −1.44097 −0.720487 0.693468i \(-0.756082\pi\)
−0.720487 + 0.693468i \(0.756082\pi\)
\(194\) −14.5653 −1.04573
\(195\) 0 0
\(196\) −6.59465 −0.471046
\(197\) −9.94865 −0.708812 −0.354406 0.935092i \(-0.615317\pi\)
−0.354406 + 0.935092i \(0.615317\pi\)
\(198\) 0 0
\(199\) 9.74870 0.691067 0.345534 0.938406i \(-0.387698\pi\)
0.345534 + 0.938406i \(0.387698\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 16.6167 1.16915
\(203\) 4.55263 0.319532
\(204\) 0 0
\(205\) 0 0
\(206\) −9.06068 −0.631287
\(207\) 0 0
\(208\) 0.141336 0.00979990
\(209\) −3.50466 −0.242423
\(210\) 0 0
\(211\) −20.7580 −1.42904 −0.714521 0.699614i \(-0.753355\pi\)
−0.714521 + 0.699614i \(0.753355\pi\)
\(212\) 8.14134 0.559149
\(213\) 0 0
\(214\) 0.0899847 0.00615123
\(215\) 0 0
\(216\) 0 0
\(217\) 4.95798 0.336569
\(218\) −13.5946 −0.920746
\(219\) 0 0
\(220\) 0 0
\(221\) 0.302648 0.0203583
\(222\) 0 0
\(223\) 10.7267 0.718310 0.359155 0.933278i \(-0.383065\pi\)
0.359155 + 0.933278i \(0.383065\pi\)
\(224\) −0.636672 −0.0425394
\(225\) 0 0
\(226\) 11.5233 0.766520
\(227\) 12.5526 0.833147 0.416574 0.909102i \(-0.363231\pi\)
0.416574 + 0.909102i \(0.363231\pi\)
\(228\) 0 0
\(229\) 25.4720 1.68324 0.841618 0.540074i \(-0.181604\pi\)
0.841618 + 0.540074i \(0.181604\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −7.15066 −0.469464
\(233\) −3.11203 −0.203876 −0.101938 0.994791i \(-0.532504\pi\)
−0.101938 + 0.994791i \(0.532504\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 5.64600 0.367523
\(237\) 0 0
\(238\) −1.36333 −0.0883714
\(239\) 1.54330 0.0998276 0.0499138 0.998754i \(-0.484105\pi\)
0.0499138 + 0.998754i \(0.484105\pi\)
\(240\) 0 0
\(241\) 10.2827 0.662365 0.331183 0.943567i \(-0.392552\pi\)
0.331183 + 0.943567i \(0.392552\pi\)
\(242\) 1.28267 0.0824533
\(243\) 0 0
\(244\) −6.49534 −0.415821
\(245\) 0 0
\(246\) 0 0
\(247\) 0.141336 0.00899300
\(248\) −7.78734 −0.494496
\(249\) 0 0
\(250\) 0 0
\(251\) 2.51399 0.158682 0.0793409 0.996848i \(-0.474718\pi\)
0.0793409 + 0.996848i \(0.474718\pi\)
\(252\) 0 0
\(253\) −17.2406 −1.08391
\(254\) 3.29200 0.206559
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −24.7967 −1.54677 −0.773387 0.633934i \(-0.781439\pi\)
−0.773387 + 0.633934i \(0.781439\pi\)
\(258\) 0 0
\(259\) 2.08405 0.129496
\(260\) 0 0
\(261\) 0 0
\(262\) −18.0187 −1.11320
\(263\) −22.5653 −1.39144 −0.695719 0.718314i \(-0.744914\pi\)
−0.695719 + 0.718314i \(0.744914\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.636672 −0.0390369
\(267\) 0 0
\(268\) −8.37266 −0.511441
\(269\) 26.5653 1.61972 0.809859 0.586625i \(-0.199544\pi\)
0.809859 + 0.586625i \(0.199544\pi\)
\(270\) 0 0
\(271\) −24.9380 −1.51488 −0.757438 0.652907i \(-0.773549\pi\)
−0.757438 + 0.652907i \(0.773549\pi\)
\(272\) 2.14134 0.129838
\(273\) 0 0
\(274\) −14.4240 −0.871386
\(275\) 0 0
\(276\) 0 0
\(277\) −18.5467 −1.11436 −0.557181 0.830391i \(-0.688117\pi\)
−0.557181 + 0.830391i \(0.688117\pi\)
\(278\) 15.4720 0.927947
\(279\) 0 0
\(280\) 0 0
\(281\) −24.7967 −1.47925 −0.739623 0.673022i \(-0.764996\pi\)
−0.739623 + 0.673022i \(0.764996\pi\)
\(282\) 0 0
\(283\) −13.5747 −0.806931 −0.403465 0.914995i \(-0.632194\pi\)
−0.403465 + 0.914995i \(0.632194\pi\)
\(284\) 8.95798 0.531558
\(285\) 0 0
\(286\) −0.495336 −0.0292898
\(287\) −2.69396 −0.159020
\(288\) 0 0
\(289\) −12.4147 −0.730275
\(290\) 0 0
\(291\) 0 0
\(292\) 3.69735 0.216371
\(293\) −15.6133 −0.912139 −0.456070 0.889944i \(-0.650743\pi\)
−0.456070 + 0.889944i \(0.650743\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −3.27334 −0.190259
\(297\) 0 0
\(298\) 17.1893 0.995749
\(299\) 0.695281 0.0402091
\(300\) 0 0
\(301\) 1.58871 0.0915717
\(302\) 3.29200 0.189433
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −34.0187 −1.94155 −0.970774 0.239997i \(-0.922854\pi\)
−0.970774 + 0.239997i \(0.922854\pi\)
\(308\) 2.23132 0.127141
\(309\) 0 0
\(310\) 0 0
\(311\) −8.93800 −0.506828 −0.253414 0.967358i \(-0.581553\pi\)
−0.253414 + 0.967358i \(0.581553\pi\)
\(312\) 0 0
\(313\) 18.4240 1.04139 0.520693 0.853744i \(-0.325674\pi\)
0.520693 + 0.853744i \(0.325674\pi\)
\(314\) −15.1893 −0.857182
\(315\) 0 0
\(316\) −4.17997 −0.235142
\(317\) 33.5547 1.88462 0.942310 0.334742i \(-0.108649\pi\)
0.942310 + 0.334742i \(0.108649\pi\)
\(318\) 0 0
\(319\) 25.0607 1.40313
\(320\) 0 0
\(321\) 0 0
\(322\) −3.13201 −0.174540
\(323\) 2.14134 0.119147
\(324\) 0 0
\(325\) 0 0
\(326\) 14.0700 0.779266
\(327\) 0 0
\(328\) 4.23132 0.233636
\(329\) 6.54669 0.360931
\(330\) 0 0
\(331\) 2.25130 0.123742 0.0618712 0.998084i \(-0.480293\pi\)
0.0618712 + 0.998084i \(0.480293\pi\)
\(332\) −9.00933 −0.494451
\(333\) 0 0
\(334\) −14.7967 −0.809638
\(335\) 0 0
\(336\) 0 0
\(337\) 21.3620 1.16366 0.581831 0.813309i \(-0.302336\pi\)
0.581831 + 0.813309i \(0.302336\pi\)
\(338\) −12.9800 −0.706020
\(339\) 0 0
\(340\) 0 0
\(341\) 27.2920 1.47794
\(342\) 0 0
\(343\) 8.65533 0.467344
\(344\) −2.49534 −0.134539
\(345\) 0 0
\(346\) −17.2920 −0.929624
\(347\) 11.5560 0.620359 0.310180 0.950678i \(-0.399611\pi\)
0.310180 + 0.950678i \(0.399611\pi\)
\(348\) 0 0
\(349\) −17.1120 −0.915986 −0.457993 0.888956i \(-0.651432\pi\)
−0.457993 + 0.888956i \(0.651432\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.50466 −0.186799
\(353\) 11.6974 0.622587 0.311294 0.950314i \(-0.399238\pi\)
0.311294 + 0.950314i \(0.399238\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.77801 0.359234
\(357\) 0 0
\(358\) −17.7360 −0.937376
\(359\) 4.47536 0.236200 0.118100 0.993002i \(-0.462320\pi\)
0.118100 + 0.993002i \(0.462320\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 6.17997 0.324812
\(363\) 0 0
\(364\) −0.0899847 −0.00471648
\(365\) 0 0
\(366\) 0 0
\(367\) −18.7453 −0.978497 −0.489249 0.872144i \(-0.662729\pi\)
−0.489249 + 0.872144i \(0.662729\pi\)
\(368\) 4.91934 0.256439
\(369\) 0 0
\(370\) 0 0
\(371\) −5.18336 −0.269107
\(372\) 0 0
\(373\) −1.69735 −0.0878855 −0.0439428 0.999034i \(-0.513992\pi\)
−0.0439428 + 0.999034i \(0.513992\pi\)
\(374\) −7.50466 −0.388057
\(375\) 0 0
\(376\) −10.2827 −0.530288
\(377\) −1.01065 −0.0520510
\(378\) 0 0
\(379\) 2.63667 0.135437 0.0677184 0.997704i \(-0.478428\pi\)
0.0677184 + 0.997704i \(0.478428\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −14.6367 −0.748877
\(383\) 12.4953 0.638482 0.319241 0.947674i \(-0.396572\pi\)
0.319241 + 0.947674i \(0.396572\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −20.0187 −1.01892
\(387\) 0 0
\(388\) −14.5653 −0.739443
\(389\) 4.51399 0.228869 0.114434 0.993431i \(-0.463494\pi\)
0.114434 + 0.993431i \(0.463494\pi\)
\(390\) 0 0
\(391\) 10.5340 0.532726
\(392\) −6.59465 −0.333080
\(393\) 0 0
\(394\) −9.94865 −0.501206
\(395\) 0 0
\(396\) 0 0
\(397\) 35.6774 1.79060 0.895298 0.445468i \(-0.146963\pi\)
0.895298 + 0.445468i \(0.146963\pi\)
\(398\) 9.74870 0.488658
\(399\) 0 0
\(400\) 0 0
\(401\) −15.3434 −0.766210 −0.383105 0.923705i \(-0.625145\pi\)
−0.383105 + 0.923705i \(0.625145\pi\)
\(402\) 0 0
\(403\) −1.10063 −0.0548264
\(404\) 16.6167 0.826712
\(405\) 0 0
\(406\) 4.55263 0.225943
\(407\) 11.4720 0.568644
\(408\) 0 0
\(409\) 29.3620 1.45186 0.725929 0.687770i \(-0.241410\pi\)
0.725929 + 0.687770i \(0.241410\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −9.06068 −0.446388
\(413\) −3.59465 −0.176881
\(414\) 0 0
\(415\) 0 0
\(416\) 0.141336 0.00692957
\(417\) 0 0
\(418\) −3.50466 −0.171419
\(419\) −25.1379 −1.22807 −0.614035 0.789279i \(-0.710454\pi\)
−0.614035 + 0.789279i \(0.710454\pi\)
\(420\) 0 0
\(421\) 14.5454 0.708898 0.354449 0.935075i \(-0.384668\pi\)
0.354449 + 0.935075i \(0.384668\pi\)
\(422\) −20.7580 −1.01049
\(423\) 0 0
\(424\) 8.14134 0.395378
\(425\) 0 0
\(426\) 0 0
\(427\) 4.13540 0.200126
\(428\) 0.0899847 0.00434958
\(429\) 0 0
\(430\) 0 0
\(431\) −19.4020 −0.934560 −0.467280 0.884109i \(-0.654766\pi\)
−0.467280 + 0.884109i \(0.654766\pi\)
\(432\) 0 0
\(433\) −5.50466 −0.264537 −0.132269 0.991214i \(-0.542226\pi\)
−0.132269 + 0.991214i \(0.542226\pi\)
\(434\) 4.95798 0.237991
\(435\) 0 0
\(436\) −13.5946 −0.651066
\(437\) 4.91934 0.235324
\(438\) 0 0
\(439\) 12.2500 0.584660 0.292330 0.956318i \(-0.405570\pi\)
0.292330 + 0.956318i \(0.405570\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.302648 0.0143955
\(443\) 31.6006 1.50139 0.750695 0.660649i \(-0.229719\pi\)
0.750695 + 0.660649i \(0.229719\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 10.7267 0.507922
\(447\) 0 0
\(448\) −0.636672 −0.0300799
\(449\) 36.0187 1.69983 0.849913 0.526923i \(-0.176655\pi\)
0.849913 + 0.526923i \(0.176655\pi\)
\(450\) 0 0
\(451\) −14.8294 −0.698287
\(452\) 11.5233 0.542011
\(453\) 0 0
\(454\) 12.5526 0.589124
\(455\) 0 0
\(456\) 0 0
\(457\) −22.1413 −1.03573 −0.517864 0.855463i \(-0.673273\pi\)
−0.517864 + 0.855463i \(0.673273\pi\)
\(458\) 25.4720 1.19023
\(459\) 0 0
\(460\) 0 0
\(461\) 2.31537 0.107837 0.0539187 0.998545i \(-0.482829\pi\)
0.0539187 + 0.998545i \(0.482829\pi\)
\(462\) 0 0
\(463\) −15.8387 −0.736086 −0.368043 0.929809i \(-0.619972\pi\)
−0.368043 + 0.929809i \(0.619972\pi\)
\(464\) −7.15066 −0.331961
\(465\) 0 0
\(466\) −3.11203 −0.144162
\(467\) 23.1379 1.07070 0.535348 0.844631i \(-0.320180\pi\)
0.535348 + 0.844631i \(0.320180\pi\)
\(468\) 0 0
\(469\) 5.33063 0.246146
\(470\) 0 0
\(471\) 0 0
\(472\) 5.64600 0.259878
\(473\) 8.74531 0.402110
\(474\) 0 0
\(475\) 0 0
\(476\) −1.36333 −0.0624880
\(477\) 0 0
\(478\) 1.54330 0.0705888
\(479\) 10.1214 0.462457 0.231228 0.972900i \(-0.425726\pi\)
0.231228 + 0.972900i \(0.425726\pi\)
\(480\) 0 0
\(481\) −0.462642 −0.0210946
\(482\) 10.2827 0.468363
\(483\) 0 0
\(484\) 1.28267 0.0583033
\(485\) 0 0
\(486\) 0 0
\(487\) −20.3854 −0.923750 −0.461875 0.886945i \(-0.652823\pi\)
−0.461875 + 0.886945i \(0.652823\pi\)
\(488\) −6.49534 −0.294030
\(489\) 0 0
\(490\) 0 0
\(491\) 7.78734 0.351438 0.175719 0.984440i \(-0.443775\pi\)
0.175719 + 0.984440i \(0.443775\pi\)
\(492\) 0 0
\(493\) −15.3120 −0.689617
\(494\) 0.141336 0.00635901
\(495\) 0 0
\(496\) −7.78734 −0.349662
\(497\) −5.70329 −0.255828
\(498\) 0 0
\(499\) 4.31537 0.193182 0.0965912 0.995324i \(-0.469206\pi\)
0.0965912 + 0.995324i \(0.469206\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2.51399 0.112205
\(503\) 18.5526 0.827221 0.413610 0.910454i \(-0.364268\pi\)
0.413610 + 0.910454i \(0.364268\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −17.2406 −0.766440
\(507\) 0 0
\(508\) 3.29200 0.146059
\(509\) 7.73599 0.342892 0.171446 0.985194i \(-0.445156\pi\)
0.171446 + 0.985194i \(0.445156\pi\)
\(510\) 0 0
\(511\) −2.35400 −0.104135
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −24.7967 −1.09373
\(515\) 0 0
\(516\) 0 0
\(517\) 36.0373 1.58492
\(518\) 2.08405 0.0915677
\(519\) 0 0
\(520\) 0 0
\(521\) −15.2080 −0.666273 −0.333136 0.942879i \(-0.608107\pi\)
−0.333136 + 0.942879i \(0.608107\pi\)
\(522\) 0 0
\(523\) 18.2113 0.796327 0.398163 0.917315i \(-0.369648\pi\)
0.398163 + 0.917315i \(0.369648\pi\)
\(524\) −18.0187 −0.787149
\(525\) 0 0
\(526\) −22.5653 −0.983896
\(527\) −16.6753 −0.726388
\(528\) 0 0
\(529\) 1.19995 0.0521715
\(530\) 0 0
\(531\) 0 0
\(532\) −0.636672 −0.0276032
\(533\) 0.598038 0.0259039
\(534\) 0 0
\(535\) 0 0
\(536\) −8.37266 −0.361644
\(537\) 0 0
\(538\) 26.5653 1.14531
\(539\) 23.1120 0.995506
\(540\) 0 0
\(541\) 16.5140 0.709992 0.354996 0.934868i \(-0.384482\pi\)
0.354996 + 0.934868i \(0.384482\pi\)
\(542\) −24.9380 −1.07118
\(543\) 0 0
\(544\) 2.14134 0.0918090
\(545\) 0 0
\(546\) 0 0
\(547\) −16.2827 −0.696197 −0.348098 0.937458i \(-0.613172\pi\)
−0.348098 + 0.937458i \(0.613172\pi\)
\(548\) −14.4240 −0.616163
\(549\) 0 0
\(550\) 0 0
\(551\) −7.15066 −0.304629
\(552\) 0 0
\(553\) 2.66127 0.113169
\(554\) −18.5467 −0.787973
\(555\) 0 0
\(556\) 15.4720 0.656158
\(557\) 37.4533 1.58695 0.793474 0.608604i \(-0.208270\pi\)
0.793474 + 0.608604i \(0.208270\pi\)
\(558\) 0 0
\(559\) −0.352681 −0.0149168
\(560\) 0 0
\(561\) 0 0
\(562\) −24.7967 −1.04598
\(563\) −29.1307 −1.22771 −0.613856 0.789418i \(-0.710382\pi\)
−0.613856 + 0.789418i \(0.710382\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −13.5747 −0.570586
\(567\) 0 0
\(568\) 8.95798 0.375868
\(569\) −14.8480 −0.622461 −0.311231 0.950334i \(-0.600741\pi\)
−0.311231 + 0.950334i \(0.600741\pi\)
\(570\) 0 0
\(571\) 41.9087 1.75382 0.876912 0.480651i \(-0.159599\pi\)
0.876912 + 0.480651i \(0.159599\pi\)
\(572\) −0.495336 −0.0207110
\(573\) 0 0
\(574\) −2.69396 −0.112444
\(575\) 0 0
\(576\) 0 0
\(577\) −16.4427 −0.684517 −0.342259 0.939606i \(-0.611192\pi\)
−0.342259 + 0.939606i \(0.611192\pi\)
\(578\) −12.4147 −0.516383
\(579\) 0 0
\(580\) 0 0
\(581\) 5.73599 0.237969
\(582\) 0 0
\(583\) −28.5327 −1.18170
\(584\) 3.69735 0.152998
\(585\) 0 0
\(586\) −15.6133 −0.644980
\(587\) −42.5327 −1.75551 −0.877755 0.479109i \(-0.840960\pi\)
−0.877755 + 0.479109i \(0.840960\pi\)
\(588\) 0 0
\(589\) −7.78734 −0.320872
\(590\) 0 0
\(591\) 0 0
\(592\) −3.27334 −0.134534
\(593\) 3.92273 0.161087 0.0805437 0.996751i \(-0.474334\pi\)
0.0805437 + 0.996751i \(0.474334\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17.1893 0.704101
\(597\) 0 0
\(598\) 0.695281 0.0284322
\(599\) 10.7594 0.439615 0.219808 0.975543i \(-0.429457\pi\)
0.219808 + 0.975543i \(0.429457\pi\)
\(600\) 0 0
\(601\) 25.2220 1.02883 0.514413 0.857542i \(-0.328010\pi\)
0.514413 + 0.857542i \(0.328010\pi\)
\(602\) 1.58871 0.0647510
\(603\) 0 0
\(604\) 3.29200 0.133950
\(605\) 0 0
\(606\) 0 0
\(607\) −39.2920 −1.59481 −0.797407 0.603442i \(-0.793795\pi\)
−0.797407 + 0.603442i \(0.793795\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −1.45331 −0.0587947
\(612\) 0 0
\(613\) −9.80599 −0.396060 −0.198030 0.980196i \(-0.563454\pi\)
−0.198030 + 0.980196i \(0.563454\pi\)
\(614\) −34.0187 −1.37288
\(615\) 0 0
\(616\) 2.23132 0.0899025
\(617\) −35.0093 −1.40942 −0.704711 0.709494i \(-0.748923\pi\)
−0.704711 + 0.709494i \(0.748923\pi\)
\(618\) 0 0
\(619\) 13.4206 0.539420 0.269710 0.962942i \(-0.413072\pi\)
0.269710 + 0.962942i \(0.413072\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −8.93800 −0.358381
\(623\) −4.31537 −0.172891
\(624\) 0 0
\(625\) 0 0
\(626\) 18.4240 0.736371
\(627\) 0 0
\(628\) −15.1893 −0.606119
\(629\) −7.00933 −0.279480
\(630\) 0 0
\(631\) 40.5254 1.61329 0.806645 0.591036i \(-0.201281\pi\)
0.806645 + 0.591036i \(0.201281\pi\)
\(632\) −4.17997 −0.166270
\(633\) 0 0
\(634\) 33.5547 1.33263
\(635\) 0 0
\(636\) 0 0
\(637\) −0.932062 −0.0369296
\(638\) 25.0607 0.992162
\(639\) 0 0
\(640\) 0 0
\(641\) 26.0700 1.02970 0.514852 0.857279i \(-0.327847\pi\)
0.514852 + 0.857279i \(0.327847\pi\)
\(642\) 0 0
\(643\) −30.1400 −1.18861 −0.594303 0.804241i \(-0.702572\pi\)
−0.594303 + 0.804241i \(0.702572\pi\)
\(644\) −3.13201 −0.123418
\(645\) 0 0
\(646\) 2.14134 0.0842497
\(647\) −20.1086 −0.790552 −0.395276 0.918562i \(-0.629351\pi\)
−0.395276 + 0.918562i \(0.629351\pi\)
\(648\) 0 0
\(649\) −19.7873 −0.776721
\(650\) 0 0
\(651\) 0 0
\(652\) 14.0700 0.551024
\(653\) −28.0373 −1.09718 −0.548592 0.836090i \(-0.684836\pi\)
−0.548592 + 0.836090i \(0.684836\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.23132 0.165205
\(657\) 0 0
\(658\) 6.54669 0.255216
\(659\) −4.90069 −0.190904 −0.0954518 0.995434i \(-0.530430\pi\)
−0.0954518 + 0.995434i \(0.530430\pi\)
\(660\) 0 0
\(661\) −8.03863 −0.312667 −0.156333 0.987704i \(-0.549967\pi\)
−0.156333 + 0.987704i \(0.549967\pi\)
\(662\) 2.25130 0.0874991
\(663\) 0 0
\(664\) −9.00933 −0.349630
\(665\) 0 0
\(666\) 0 0
\(667\) −35.1766 −1.36204
\(668\) −14.7967 −0.572500
\(669\) 0 0
\(670\) 0 0
\(671\) 22.7640 0.878793
\(672\) 0 0
\(673\) 4.82936 0.186158 0.0930791 0.995659i \(-0.470329\pi\)
0.0930791 + 0.995659i \(0.470329\pi\)
\(674\) 21.3620 0.822834
\(675\) 0 0
\(676\) −12.9800 −0.499232
\(677\) 12.8094 0.492305 0.246152 0.969231i \(-0.420834\pi\)
0.246152 + 0.969231i \(0.420834\pi\)
\(678\) 0 0
\(679\) 9.27334 0.355878
\(680\) 0 0
\(681\) 0 0
\(682\) 27.2920 1.04506
\(683\) 37.1307 1.42077 0.710383 0.703815i \(-0.248522\pi\)
0.710383 + 0.703815i \(0.248522\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 8.65533 0.330462
\(687\) 0 0
\(688\) −2.49534 −0.0951338
\(689\) 1.15066 0.0438368
\(690\) 0 0
\(691\) −18.1986 −0.692308 −0.346154 0.938178i \(-0.612513\pi\)
−0.346154 + 0.938178i \(0.612513\pi\)
\(692\) −17.2920 −0.657343
\(693\) 0 0
\(694\) 11.5560 0.438660
\(695\) 0 0
\(696\) 0 0
\(697\) 9.06068 0.343198
\(698\) −17.1120 −0.647700
\(699\) 0 0
\(700\) 0 0
\(701\) 26.2827 0.992683 0.496341 0.868127i \(-0.334676\pi\)
0.496341 + 0.868127i \(0.334676\pi\)
\(702\) 0 0
\(703\) −3.27334 −0.123456
\(704\) −3.50466 −0.132087
\(705\) 0 0
\(706\) 11.6974 0.440236
\(707\) −10.5794 −0.397879
\(708\) 0 0
\(709\) −14.9253 −0.560531 −0.280265 0.959923i \(-0.590422\pi\)
−0.280265 + 0.959923i \(0.590422\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.77801 0.254017
\(713\) −38.3086 −1.43467
\(714\) 0 0
\(715\) 0 0
\(716\) −17.7360 −0.662825
\(717\) 0 0
\(718\) 4.47536 0.167019
\(719\) −32.3327 −1.20581 −0.602903 0.797814i \(-0.705989\pi\)
−0.602903 + 0.797814i \(0.705989\pi\)
\(720\) 0 0
\(721\) 5.76868 0.214837
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 6.17997 0.229677
\(725\) 0 0
\(726\) 0 0
\(727\) 42.0246 1.55861 0.779303 0.626647i \(-0.215573\pi\)
0.779303 + 0.626647i \(0.215573\pi\)
\(728\) −0.0899847 −0.00333506
\(729\) 0 0
\(730\) 0 0
\(731\) −5.34335 −0.197631
\(732\) 0 0
\(733\) −26.5840 −0.981903 −0.490951 0.871187i \(-0.663351\pi\)
−0.490951 + 0.871187i \(0.663351\pi\)
\(734\) −18.7453 −0.691902
\(735\) 0 0
\(736\) 4.91934 0.181329
\(737\) 29.3434 1.08088
\(738\) 0 0
\(739\) 8.14728 0.299702 0.149851 0.988709i \(-0.452121\pi\)
0.149851 + 0.988709i \(0.452121\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −5.18336 −0.190287
\(743\) 35.8247 1.31428 0.657139 0.753769i \(-0.271766\pi\)
0.657139 + 0.753769i \(0.271766\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.69735 −0.0621445
\(747\) 0 0
\(748\) −7.50466 −0.274398
\(749\) −0.0572907 −0.00209336
\(750\) 0 0
\(751\) −14.8994 −0.543686 −0.271843 0.962342i \(-0.587633\pi\)
−0.271843 + 0.962342i \(0.587633\pi\)
\(752\) −10.2827 −0.374970
\(753\) 0 0
\(754\) −1.01065 −0.0368056
\(755\) 0 0
\(756\) 0 0
\(757\) 47.7920 1.73703 0.868514 0.495664i \(-0.165075\pi\)
0.868514 + 0.495664i \(0.165075\pi\)
\(758\) 2.63667 0.0957682
\(759\) 0 0
\(760\) 0 0
\(761\) 38.9053 1.41032 0.705158 0.709050i \(-0.250876\pi\)
0.705158 + 0.709050i \(0.250876\pi\)
\(762\) 0 0
\(763\) 8.65533 0.313344
\(764\) −14.6367 −0.529536
\(765\) 0 0
\(766\) 12.4953 0.451475
\(767\) 0.797984 0.0288135
\(768\) 0 0
\(769\) 8.74663 0.315412 0.157706 0.987486i \(-0.449590\pi\)
0.157706 + 0.987486i \(0.449590\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −20.0187 −0.720487
\(773\) −23.4707 −0.844181 −0.422090 0.906554i \(-0.638703\pi\)
−0.422090 + 0.906554i \(0.638703\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −14.5653 −0.522865
\(777\) 0 0
\(778\) 4.51399 0.161834
\(779\) 4.23132 0.151603
\(780\) 0 0
\(781\) −31.3947 −1.12339
\(782\) 10.5340 0.376694
\(783\) 0 0
\(784\) −6.59465 −0.235523
\(785\) 0 0
\(786\) 0 0
\(787\) −1.26063 −0.0449364 −0.0224682 0.999748i \(-0.507152\pi\)
−0.0224682 + 0.999748i \(0.507152\pi\)
\(788\) −9.94865 −0.354406
\(789\) 0 0
\(790\) 0 0
\(791\) −7.33657 −0.260859
\(792\) 0 0
\(793\) −0.918026 −0.0326000
\(794\) 35.6774 1.26614
\(795\) 0 0
\(796\) 9.74870 0.345534
\(797\) −38.6481 −1.36898 −0.684492 0.729020i \(-0.739976\pi\)
−0.684492 + 0.729020i \(0.739976\pi\)
\(798\) 0 0
\(799\) −22.0187 −0.778964
\(800\) 0 0
\(801\) 0 0
\(802\) −15.3434 −0.541793
\(803\) −12.9580 −0.457277
\(804\) 0 0
\(805\) 0 0
\(806\) −1.10063 −0.0387681
\(807\) 0 0
\(808\) 16.6167 0.584573
\(809\) −51.6506 −1.81594 −0.907970 0.419036i \(-0.862368\pi\)
−0.907970 + 0.419036i \(0.862368\pi\)
\(810\) 0 0
\(811\) 8.19269 0.287684 0.143842 0.989601i \(-0.454054\pi\)
0.143842 + 0.989601i \(0.454054\pi\)
\(812\) 4.55263 0.159766
\(813\) 0 0
\(814\) 11.4720 0.402092
\(815\) 0 0
\(816\) 0 0
\(817\) −2.49534 −0.0873007
\(818\) 29.3620 1.02662
\(819\) 0 0
\(820\) 0 0
\(821\) 49.9600 1.74362 0.871809 0.489846i \(-0.162947\pi\)
0.871809 + 0.489846i \(0.162947\pi\)
\(822\) 0 0
\(823\) 16.8421 0.587078 0.293539 0.955947i \(-0.405167\pi\)
0.293539 + 0.955947i \(0.405167\pi\)
\(824\) −9.06068 −0.315644
\(825\) 0 0
\(826\) −3.59465 −0.125074
\(827\) −41.7487 −1.45174 −0.725872 0.687829i \(-0.758564\pi\)
−0.725872 + 0.687829i \(0.758564\pi\)
\(828\) 0 0
\(829\) −5.98002 −0.207695 −0.103847 0.994593i \(-0.533115\pi\)
−0.103847 + 0.994593i \(0.533115\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.141336 0.00489995
\(833\) −14.1214 −0.489276
\(834\) 0 0
\(835\) 0 0
\(836\) −3.50466 −0.121211
\(837\) 0 0
\(838\) −25.1379 −0.868376
\(839\) −32.0373 −1.10605 −0.553025 0.833164i \(-0.686527\pi\)
−0.553025 + 0.833164i \(0.686527\pi\)
\(840\) 0 0
\(841\) 22.1320 0.763173
\(842\) 14.5454 0.501267
\(843\) 0 0
\(844\) −20.7580 −0.714521
\(845\) 0 0
\(846\) 0 0
\(847\) −0.816641 −0.0280601
\(848\) 8.14134 0.279575
\(849\) 0 0
\(850\) 0 0
\(851\) −16.1027 −0.551994
\(852\) 0 0
\(853\) 40.8480 1.39861 0.699305 0.714824i \(-0.253493\pi\)
0.699305 + 0.714824i \(0.253493\pi\)
\(854\) 4.13540 0.141510
\(855\) 0 0
\(856\) 0.0899847 0.00307561
\(857\) 28.8667 0.986067 0.493033 0.870010i \(-0.335888\pi\)
0.493033 + 0.870010i \(0.335888\pi\)
\(858\) 0 0
\(859\) −11.5047 −0.392534 −0.196267 0.980550i \(-0.562882\pi\)
−0.196267 + 0.980550i \(0.562882\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −19.4020 −0.660833
\(863\) −31.6074 −1.07593 −0.537964 0.842968i \(-0.680806\pi\)
−0.537964 + 0.842968i \(0.680806\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −5.50466 −0.187056
\(867\) 0 0
\(868\) 4.95798 0.168285
\(869\) 14.6494 0.496947
\(870\) 0 0
\(871\) −1.18336 −0.0400966
\(872\) −13.5946 −0.460373
\(873\) 0 0
\(874\) 4.91934 0.166399
\(875\) 0 0
\(876\) 0 0
\(877\) −12.0641 −0.407375 −0.203687 0.979036i \(-0.565293\pi\)
−0.203687 + 0.979036i \(0.565293\pi\)
\(878\) 12.2500 0.413417
\(879\) 0 0
\(880\) 0 0
\(881\) 22.8480 0.769769 0.384885 0.922965i \(-0.374241\pi\)
0.384885 + 0.922965i \(0.374241\pi\)
\(882\) 0 0
\(883\) 34.1773 1.15016 0.575079 0.818098i \(-0.304971\pi\)
0.575079 + 0.818098i \(0.304971\pi\)
\(884\) 0.302648 0.0101792
\(885\) 0 0
\(886\) 31.6006 1.06164
\(887\) 26.6167 0.893701 0.446851 0.894609i \(-0.352546\pi\)
0.446851 + 0.894609i \(0.352546\pi\)
\(888\) 0 0
\(889\) −2.09592 −0.0702950
\(890\) 0 0
\(891\) 0 0
\(892\) 10.7267 0.359155
\(893\) −10.2827 −0.344097
\(894\) 0 0
\(895\) 0 0
\(896\) −0.636672 −0.0212697
\(897\) 0 0
\(898\) 36.0187 1.20196
\(899\) 55.6846 1.85719
\(900\) 0 0
\(901\) 17.4333 0.580789
\(902\) −14.8294 −0.493764
\(903\) 0 0
\(904\) 11.5233 0.383260
\(905\) 0 0
\(906\) 0 0
\(907\) −13.1434 −0.436420 −0.218210 0.975902i \(-0.570022\pi\)
−0.218210 + 0.975902i \(0.570022\pi\)
\(908\) 12.5526 0.416574
\(909\) 0 0
\(910\) 0 0
\(911\) 32.9253 1.09086 0.545432 0.838155i \(-0.316366\pi\)
0.545432 + 0.838155i \(0.316366\pi\)
\(912\) 0 0
\(913\) 31.5747 1.04497
\(914\) −22.1413 −0.732370
\(915\) 0 0
\(916\) 25.4720 0.841618
\(917\) 11.4720 0.378838
\(918\) 0 0
\(919\) −24.1273 −0.795886 −0.397943 0.917410i \(-0.630276\pi\)
−0.397943 + 0.917410i \(0.630276\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 2.31537 0.0762525
\(923\) 1.26609 0.0416737
\(924\) 0 0
\(925\) 0 0
\(926\) −15.8387 −0.520492
\(927\) 0 0
\(928\) −7.15066 −0.234732
\(929\) 15.9359 0.522841 0.261420 0.965225i \(-0.415809\pi\)
0.261420 + 0.965225i \(0.415809\pi\)
\(930\) 0 0
\(931\) −6.59465 −0.216131
\(932\) −3.11203 −0.101938
\(933\) 0 0
\(934\) 23.1379 0.757097
\(935\) 0 0
\(936\) 0 0
\(937\) −23.5547 −0.769498 −0.384749 0.923021i \(-0.625712\pi\)
−0.384749 + 0.923021i \(0.625712\pi\)
\(938\) 5.33063 0.174051
\(939\) 0 0
\(940\) 0 0
\(941\) 25.6974 0.837710 0.418855 0.908053i \(-0.362432\pi\)
0.418855 + 0.908053i \(0.362432\pi\)
\(942\) 0 0
\(943\) 20.8153 0.677840
\(944\) 5.64600 0.183762
\(945\) 0 0
\(946\) 8.74531 0.284335
\(947\) −16.3013 −0.529722 −0.264861 0.964287i \(-0.585326\pi\)
−0.264861 + 0.964287i \(0.585326\pi\)
\(948\) 0 0
\(949\) 0.522569 0.0169633
\(950\) 0 0
\(951\) 0 0
\(952\) −1.36333 −0.0441857
\(953\) 10.2754 0.332853 0.166427 0.986054i \(-0.446777\pi\)
0.166427 + 0.986054i \(0.446777\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.54330 0.0499138
\(957\) 0 0
\(958\) 10.1214 0.327006
\(959\) 9.18336 0.296546
\(960\) 0 0
\(961\) 29.6426 0.956213
\(962\) −0.462642 −0.0149162
\(963\) 0 0
\(964\) 10.2827 0.331183
\(965\) 0 0
\(966\) 0 0
\(967\) 53.2920 1.71376 0.856878 0.515520i \(-0.172401\pi\)
0.856878 + 0.515520i \(0.172401\pi\)
\(968\) 1.28267 0.0412266
\(969\) 0 0
\(970\) 0 0
\(971\) −2.54669 −0.0817271 −0.0408635 0.999165i \(-0.513011\pi\)
−0.0408635 + 0.999165i \(0.513011\pi\)
\(972\) 0 0
\(973\) −9.85057 −0.315795
\(974\) −20.3854 −0.653190
\(975\) 0 0
\(976\) −6.49534 −0.207911
\(977\) −49.2966 −1.57714 −0.788569 0.614946i \(-0.789178\pi\)
−0.788569 + 0.614946i \(0.789178\pi\)
\(978\) 0 0
\(979\) −23.7546 −0.759202
\(980\) 0 0
\(981\) 0 0
\(982\) 7.78734 0.248504
\(983\) −36.7453 −1.17199 −0.585997 0.810313i \(-0.699297\pi\)
−0.585997 + 0.810313i \(0.699297\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −15.3120 −0.487633
\(987\) 0 0
\(988\) 0.141336 0.00449650
\(989\) −12.2754 −0.390335
\(990\) 0 0
\(991\) −35.5674 −1.12984 −0.564918 0.825147i \(-0.691092\pi\)
−0.564918 + 0.825147i \(0.691092\pi\)
\(992\) −7.78734 −0.247248
\(993\) 0 0
\(994\) −5.70329 −0.180897
\(995\) 0 0
\(996\) 0 0
\(997\) 48.4299 1.53379 0.766896 0.641771i \(-0.221800\pi\)
0.766896 + 0.641771i \(0.221800\pi\)
\(998\) 4.31537 0.136601
\(999\) 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.cl.1.2 3
3.2 odd 2 950.2.a.i.1.2 3
5.2 odd 4 1710.2.d.d.1369.5 6
5.3 odd 4 1710.2.d.d.1369.2 6
5.4 even 2 8550.2.a.ck.1.2 3
12.11 even 2 7600.2.a.cd.1.2 3
15.2 even 4 190.2.b.b.39.2 6
15.8 even 4 190.2.b.b.39.5 yes 6
15.14 odd 2 950.2.a.n.1.2 3
60.23 odd 4 1520.2.d.j.609.4 6
60.47 odd 4 1520.2.d.j.609.3 6
60.59 even 2 7600.2.a.bi.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.b.39.2 6 15.2 even 4
190.2.b.b.39.5 yes 6 15.8 even 4
950.2.a.i.1.2 3 3.2 odd 2
950.2.a.n.1.2 3 15.14 odd 2
1520.2.d.j.609.3 6 60.47 odd 4
1520.2.d.j.609.4 6 60.23 odd 4
1710.2.d.d.1369.2 6 5.3 odd 4
1710.2.d.d.1369.5 6 5.2 odd 4
7600.2.a.bi.1.2 3 60.59 even 2
7600.2.a.cd.1.2 3 12.11 even 2
8550.2.a.ck.1.2 3 5.4 even 2
8550.2.a.cl.1.2 3 1.1 even 1 trivial