Properties

Label 8550.2.a.cq.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: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
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 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.07838 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.07838 q^{7} +1.00000 q^{8} -6.34017 q^{11} +3.41855 q^{13} -1.07838 q^{14} +1.00000 q^{16} +5.41855 q^{17} +1.00000 q^{19} -6.34017 q^{22} -6.34017 q^{23} +3.41855 q^{26} -1.07838 q^{28} +0.340173 q^{29} +1.07838 q^{31} +1.00000 q^{32} +5.41855 q^{34} +3.41855 q^{37} +1.00000 q^{38} -7.60197 q^{41} -11.1773 q^{43} -6.34017 q^{44} -6.34017 q^{46} +6.34017 q^{47} -5.83710 q^{49} +3.41855 q^{52} -6.00000 q^{53} -1.07838 q^{56} +0.340173 q^{58} -0.738205 q^{59} -2.68035 q^{61} +1.07838 q^{62} +1.00000 q^{64} +2.83710 q^{67} +5.41855 q^{68} +2.83710 q^{71} +6.83710 q^{73} +3.41855 q^{74} +1.00000 q^{76} +6.83710 q^{77} -1.07838 q^{79} -7.60197 q^{82} -0.894960 q^{83} -11.1773 q^{86} -6.34017 q^{88} -6.92162 q^{89} -3.68649 q^{91} -6.34017 q^{92} +6.34017 q^{94} -3.65983 q^{97} -5.83710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} + 3q^{8} - 8q^{11} - 4q^{13} + 3q^{16} + 2q^{17} + 3q^{19} - 8q^{22} - 8q^{23} - 4q^{26} - 10q^{29} + 3q^{32} + 2q^{34} - 4q^{37} + 3q^{38} - 4q^{41} + 6q^{43} - 8q^{44} - 8q^{46} + 8q^{47} + 11q^{49} - 4q^{52} - 18q^{53} - 10q^{58} - 10q^{59} + 14q^{61} + 3q^{64} - 20q^{67} + 2q^{68} - 20q^{71} - 8q^{73} - 4q^{74} + 3q^{76} - 8q^{77} - 4q^{82} - 4q^{83} + 6q^{86} - 8q^{88} - 24q^{89} - 24q^{91} - 8q^{92} + 8q^{94} - 22q^{97} + 11q^{98} + O(q^{100}) \)

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
\(7\) −1.07838 −0.407588 −0.203794 0.979014i \(-0.565327\pi\)
−0.203794 + 0.979014i \(0.565327\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −6.34017 −1.91163 −0.955817 0.293962i \(-0.905026\pi\)
−0.955817 + 0.293962i \(0.905026\pi\)
\(12\) 0 0
\(13\) 3.41855 0.948135 0.474068 0.880488i \(-0.342785\pi\)
0.474068 + 0.880488i \(0.342785\pi\)
\(14\) −1.07838 −0.288209
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.41855 1.31419 0.657096 0.753807i \(-0.271785\pi\)
0.657096 + 0.753807i \(0.271785\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −6.34017 −1.35173
\(23\) −6.34017 −1.32202 −0.661009 0.750378i \(-0.729871\pi\)
−0.661009 + 0.750378i \(0.729871\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.41855 0.670433
\(27\) 0 0
\(28\) −1.07838 −0.203794
\(29\) 0.340173 0.0631685 0.0315843 0.999501i \(-0.489945\pi\)
0.0315843 + 0.999501i \(0.489945\pi\)
\(30\) 0 0
\(31\) 1.07838 0.193682 0.0968412 0.995300i \(-0.469126\pi\)
0.0968412 + 0.995300i \(0.469126\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 5.41855 0.929274
\(35\) 0 0
\(36\) 0 0
\(37\) 3.41855 0.562006 0.281003 0.959707i \(-0.409333\pi\)
0.281003 + 0.959707i \(0.409333\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −7.60197 −1.18723 −0.593614 0.804750i \(-0.702299\pi\)
−0.593614 + 0.804750i \(0.702299\pi\)
\(42\) 0 0
\(43\) −11.1773 −1.70452 −0.852259 0.523120i \(-0.824768\pi\)
−0.852259 + 0.523120i \(0.824768\pi\)
\(44\) −6.34017 −0.955817
\(45\) 0 0
\(46\) −6.34017 −0.934808
\(47\) 6.34017 0.924809 0.462405 0.886669i \(-0.346987\pi\)
0.462405 + 0.886669i \(0.346987\pi\)
\(48\) 0 0
\(49\) −5.83710 −0.833872
\(50\) 0 0
\(51\) 0 0
\(52\) 3.41855 0.474068
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.07838 −0.144104
\(57\) 0 0
\(58\) 0.340173 0.0446669
\(59\) −0.738205 −0.0961061 −0.0480530 0.998845i \(-0.515302\pi\)
−0.0480530 + 0.998845i \(0.515302\pi\)
\(60\) 0 0
\(61\) −2.68035 −0.343183 −0.171592 0.985168i \(-0.554891\pi\)
−0.171592 + 0.985168i \(0.554891\pi\)
\(62\) 1.07838 0.136954
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.83710 0.346607 0.173304 0.984868i \(-0.444556\pi\)
0.173304 + 0.984868i \(0.444556\pi\)
\(68\) 5.41855 0.657096
\(69\) 0 0
\(70\) 0 0
\(71\) 2.83710 0.336702 0.168351 0.985727i \(-0.446156\pi\)
0.168351 + 0.985727i \(0.446156\pi\)
\(72\) 0 0
\(73\) 6.83710 0.800222 0.400111 0.916467i \(-0.368972\pi\)
0.400111 + 0.916467i \(0.368972\pi\)
\(74\) 3.41855 0.397398
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 6.83710 0.779160
\(78\) 0 0
\(79\) −1.07838 −0.121327 −0.0606635 0.998158i \(-0.519322\pi\)
−0.0606635 + 0.998158i \(0.519322\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.60197 −0.839497
\(83\) −0.894960 −0.0982347 −0.0491173 0.998793i \(-0.515641\pi\)
−0.0491173 + 0.998793i \(0.515641\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −11.1773 −1.20528
\(87\) 0 0
\(88\) −6.34017 −0.675865
\(89\) −6.92162 −0.733690 −0.366845 0.930282i \(-0.619562\pi\)
−0.366845 + 0.930282i \(0.619562\pi\)
\(90\) 0 0
\(91\) −3.68649 −0.386449
\(92\) −6.34017 −0.661009
\(93\) 0 0
\(94\) 6.34017 0.653939
\(95\) 0 0
\(96\) 0 0
\(97\) −3.65983 −0.371599 −0.185800 0.982588i \(-0.559488\pi\)
−0.185800 + 0.982588i \(0.559488\pi\)
\(98\) −5.83710 −0.589636
\(99\) 0 0
\(100\) 0 0
\(101\) −19.6020 −1.95047 −0.975234 0.221174i \(-0.929011\pi\)
−0.975234 + 0.221174i \(0.929011\pi\)
\(102\) 0 0
\(103\) −11.7587 −1.15862 −0.579311 0.815107i \(-0.696678\pi\)
−0.579311 + 0.815107i \(0.696678\pi\)
\(104\) 3.41855 0.335216
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −6.15676 −0.595196 −0.297598 0.954691i \(-0.596186\pi\)
−0.297598 + 0.954691i \(0.596186\pi\)
\(108\) 0 0
\(109\) 17.9155 1.71599 0.857996 0.513657i \(-0.171709\pi\)
0.857996 + 0.513657i \(0.171709\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.07838 −0.101897
\(113\) 13.2039 1.24212 0.621061 0.783762i \(-0.286702\pi\)
0.621061 + 0.783762i \(0.286702\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.340173 0.0315843
\(117\) 0 0
\(118\) −0.738205 −0.0679573
\(119\) −5.84324 −0.535649
\(120\) 0 0
\(121\) 29.1978 2.65434
\(122\) −2.68035 −0.242667
\(123\) 0 0
\(124\) 1.07838 0.0968412
\(125\) 0 0
\(126\) 0 0
\(127\) −0.921622 −0.0817808 −0.0408904 0.999164i \(-0.513019\pi\)
−0.0408904 + 0.999164i \(0.513019\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −10.6537 −0.930817 −0.465408 0.885096i \(-0.654093\pi\)
−0.465408 + 0.885096i \(0.654093\pi\)
\(132\) 0 0
\(133\) −1.07838 −0.0935072
\(134\) 2.83710 0.245088
\(135\) 0 0
\(136\) 5.41855 0.464637
\(137\) −20.6225 −1.76190 −0.880949 0.473211i \(-0.843095\pi\)
−0.880949 + 0.473211i \(0.843095\pi\)
\(138\) 0 0
\(139\) 18.8371 1.59774 0.798871 0.601502i \(-0.205431\pi\)
0.798871 + 0.601502i \(0.205431\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.83710 0.238084
\(143\) −21.6742 −1.81249
\(144\) 0 0
\(145\) 0 0
\(146\) 6.83710 0.565843
\(147\) 0 0
\(148\) 3.41855 0.281003
\(149\) 5.44521 0.446089 0.223045 0.974808i \(-0.428400\pi\)
0.223045 + 0.974808i \(0.428400\pi\)
\(150\) 0 0
\(151\) −23.1194 −1.88143 −0.940716 0.339196i \(-0.889845\pi\)
−0.940716 + 0.339196i \(0.889845\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 6.83710 0.550949
\(155\) 0 0
\(156\) 0 0
\(157\) −9.73206 −0.776703 −0.388352 0.921511i \(-0.626955\pi\)
−0.388352 + 0.921511i \(0.626955\pi\)
\(158\) −1.07838 −0.0857911
\(159\) 0 0
\(160\) 0 0
\(161\) 6.83710 0.538839
\(162\) 0 0
\(163\) −6.49693 −0.508879 −0.254439 0.967089i \(-0.581891\pi\)
−0.254439 + 0.967089i \(0.581891\pi\)
\(164\) −7.60197 −0.593614
\(165\) 0 0
\(166\) −0.894960 −0.0694624
\(167\) −20.9939 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(168\) 0 0
\(169\) −1.31351 −0.101039
\(170\) 0 0
\(171\) 0 0
\(172\) −11.1773 −0.852259
\(173\) 6.99386 0.531733 0.265867 0.964010i \(-0.414342\pi\)
0.265867 + 0.964010i \(0.414342\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6.34017 −0.477909
\(177\) 0 0
\(178\) −6.92162 −0.518798
\(179\) 4.73820 0.354150 0.177075 0.984197i \(-0.443336\pi\)
0.177075 + 0.984197i \(0.443336\pi\)
\(180\) 0 0
\(181\) −12.0722 −0.897322 −0.448661 0.893702i \(-0.648099\pi\)
−0.448661 + 0.893702i \(0.648099\pi\)
\(182\) −3.68649 −0.273261
\(183\) 0 0
\(184\) −6.34017 −0.467404
\(185\) 0 0
\(186\) 0 0
\(187\) −34.3545 −2.51225
\(188\) 6.34017 0.462405
\(189\) 0 0
\(190\) 0 0
\(191\) 7.26180 0.525445 0.262723 0.964871i \(-0.415380\pi\)
0.262723 + 0.964871i \(0.415380\pi\)
\(192\) 0 0
\(193\) −18.8638 −1.35784 −0.678922 0.734211i \(-0.737552\pi\)
−0.678922 + 0.734211i \(0.737552\pi\)
\(194\) −3.65983 −0.262760
\(195\) 0 0
\(196\) −5.83710 −0.416936
\(197\) −17.7009 −1.26113 −0.630567 0.776135i \(-0.717178\pi\)
−0.630567 + 0.776135i \(0.717178\pi\)
\(198\) 0 0
\(199\) −0.313511 −0.0222242 −0.0111121 0.999938i \(-0.503537\pi\)
−0.0111121 + 0.999938i \(0.503537\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −19.6020 −1.37919
\(203\) −0.366835 −0.0257468
\(204\) 0 0
\(205\) 0 0
\(206\) −11.7587 −0.819269
\(207\) 0 0
\(208\) 3.41855 0.237034
\(209\) −6.34017 −0.438559
\(210\) 0 0
\(211\) 14.8371 1.02143 0.510714 0.859751i \(-0.329381\pi\)
0.510714 + 0.859751i \(0.329381\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −6.15676 −0.420867
\(215\) 0 0
\(216\) 0 0
\(217\) −1.16290 −0.0789427
\(218\) 17.9155 1.21339
\(219\) 0 0
\(220\) 0 0
\(221\) 18.5236 1.24603
\(222\) 0 0
\(223\) 1.28846 0.0862815 0.0431407 0.999069i \(-0.486264\pi\)
0.0431407 + 0.999069i \(0.486264\pi\)
\(224\) −1.07838 −0.0720521
\(225\) 0 0
\(226\) 13.2039 0.878313
\(227\) −20.9939 −1.39341 −0.696706 0.717357i \(-0.745352\pi\)
−0.696706 + 0.717357i \(0.745352\pi\)
\(228\) 0 0
\(229\) −2.36683 −0.156405 −0.0782024 0.996938i \(-0.524918\pi\)
−0.0782024 + 0.996938i \(0.524918\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.340173 0.0223334
\(233\) −5.10504 −0.334442 −0.167221 0.985919i \(-0.553479\pi\)
−0.167221 + 0.985919i \(0.553479\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.738205 −0.0480530
\(237\) 0 0
\(238\) −5.84324 −0.378761
\(239\) −10.4124 −0.673523 −0.336761 0.941590i \(-0.609332\pi\)
−0.336761 + 0.941590i \(0.609332\pi\)
\(240\) 0 0
\(241\) 3.36069 0.216481 0.108241 0.994125i \(-0.465478\pi\)
0.108241 + 0.994125i \(0.465478\pi\)
\(242\) 29.1978 1.87691
\(243\) 0 0
\(244\) −2.68035 −0.171592
\(245\) 0 0
\(246\) 0 0
\(247\) 3.41855 0.217517
\(248\) 1.07838 0.0684771
\(249\) 0 0
\(250\) 0 0
\(251\) −21.8576 −1.37964 −0.689820 0.723981i \(-0.742311\pi\)
−0.689820 + 0.723981i \(0.742311\pi\)
\(252\) 0 0
\(253\) 40.1978 2.52721
\(254\) −0.921622 −0.0578277
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 26.8781 1.67661 0.838306 0.545200i \(-0.183546\pi\)
0.838306 + 0.545200i \(0.183546\pi\)
\(258\) 0 0
\(259\) −3.68649 −0.229067
\(260\) 0 0
\(261\) 0 0
\(262\) −10.6537 −0.658187
\(263\) −22.3402 −1.37755 −0.688777 0.724973i \(-0.741852\pi\)
−0.688777 + 0.724973i \(0.741852\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.07838 −0.0661196
\(267\) 0 0
\(268\) 2.83710 0.173304
\(269\) 17.3340 1.05687 0.528437 0.848972i \(-0.322778\pi\)
0.528437 + 0.848972i \(0.322778\pi\)
\(270\) 0 0
\(271\) −13.3607 −0.811604 −0.405802 0.913961i \(-0.633008\pi\)
−0.405802 + 0.913961i \(0.633008\pi\)
\(272\) 5.41855 0.328548
\(273\) 0 0
\(274\) −20.6225 −1.24585
\(275\) 0 0
\(276\) 0 0
\(277\) 0.738205 0.0443544 0.0221772 0.999754i \(-0.492940\pi\)
0.0221772 + 0.999754i \(0.492940\pi\)
\(278\) 18.8371 1.12977
\(279\) 0 0
\(280\) 0 0
\(281\) 31.7998 1.89701 0.948507 0.316755i \(-0.102593\pi\)
0.948507 + 0.316755i \(0.102593\pi\)
\(282\) 0 0
\(283\) −4.70701 −0.279803 −0.139901 0.990165i \(-0.544679\pi\)
−0.139901 + 0.990165i \(0.544679\pi\)
\(284\) 2.83710 0.168351
\(285\) 0 0
\(286\) −21.6742 −1.28162
\(287\) 8.19779 0.483900
\(288\) 0 0
\(289\) 12.3607 0.727100
\(290\) 0 0
\(291\) 0 0
\(292\) 6.83710 0.400111
\(293\) −26.1978 −1.53049 −0.765246 0.643738i \(-0.777383\pi\)
−0.765246 + 0.643738i \(0.777383\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.41855 0.198699
\(297\) 0 0
\(298\) 5.44521 0.315433
\(299\) −21.6742 −1.25345
\(300\) 0 0
\(301\) 12.0533 0.694742
\(302\) −23.1194 −1.33037
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) 32.1978 1.83763 0.918813 0.394694i \(-0.129149\pi\)
0.918813 + 0.394694i \(0.129149\pi\)
\(308\) 6.83710 0.389580
\(309\) 0 0
\(310\) 0 0
\(311\) −16.7382 −0.949137 −0.474568 0.880219i \(-0.657396\pi\)
−0.474568 + 0.880219i \(0.657396\pi\)
\(312\) 0 0
\(313\) −14.1568 −0.800187 −0.400094 0.916474i \(-0.631022\pi\)
−0.400094 + 0.916474i \(0.631022\pi\)
\(314\) −9.73206 −0.549212
\(315\) 0 0
\(316\) −1.07838 −0.0606635
\(317\) 21.8843 1.22914 0.614572 0.788861i \(-0.289329\pi\)
0.614572 + 0.788861i \(0.289329\pi\)
\(318\) 0 0
\(319\) −2.15676 −0.120755
\(320\) 0 0
\(321\) 0 0
\(322\) 6.83710 0.381017
\(323\) 5.41855 0.301496
\(324\) 0 0
\(325\) 0 0
\(326\) −6.49693 −0.359832
\(327\) 0 0
\(328\) −7.60197 −0.419748
\(329\) −6.83710 −0.376942
\(330\) 0 0
\(331\) −26.1568 −1.43771 −0.718853 0.695162i \(-0.755332\pi\)
−0.718853 + 0.695162i \(0.755332\pi\)
\(332\) −0.894960 −0.0491173
\(333\) 0 0
\(334\) −20.9939 −1.14873
\(335\) 0 0
\(336\) 0 0
\(337\) −29.7009 −1.61791 −0.808955 0.587871i \(-0.799966\pi\)
−0.808955 + 0.587871i \(0.799966\pi\)
\(338\) −1.31351 −0.0714456
\(339\) 0 0
\(340\) 0 0
\(341\) −6.83710 −0.370250
\(342\) 0 0
\(343\) 13.8432 0.747465
\(344\) −11.1773 −0.602638
\(345\) 0 0
\(346\) 6.99386 0.375992
\(347\) −23.4186 −1.25717 −0.628587 0.777739i \(-0.716366\pi\)
−0.628587 + 0.777739i \(0.716366\pi\)
\(348\) 0 0
\(349\) 6.48255 0.347003 0.173502 0.984834i \(-0.444492\pi\)
0.173502 + 0.984834i \(0.444492\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6.34017 −0.337932
\(353\) −2.58145 −0.137397 −0.0686983 0.997637i \(-0.521885\pi\)
−0.0686983 + 0.997637i \(0.521885\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.92162 −0.366845
\(357\) 0 0
\(358\) 4.73820 0.250422
\(359\) −28.1399 −1.48517 −0.742584 0.669752i \(-0.766400\pi\)
−0.742584 + 0.669752i \(0.766400\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −12.0722 −0.634503
\(363\) 0 0
\(364\) −3.68649 −0.193225
\(365\) 0 0
\(366\) 0 0
\(367\) 10.5548 0.550955 0.275478 0.961307i \(-0.411164\pi\)
0.275478 + 0.961307i \(0.411164\pi\)
\(368\) −6.34017 −0.330504
\(369\) 0 0
\(370\) 0 0
\(371\) 6.47027 0.335919
\(372\) 0 0
\(373\) 17.8888 0.926248 0.463124 0.886294i \(-0.346728\pi\)
0.463124 + 0.886294i \(0.346728\pi\)
\(374\) −34.3545 −1.77643
\(375\) 0 0
\(376\) 6.34017 0.326969
\(377\) 1.16290 0.0598923
\(378\) 0 0
\(379\) 4.36683 0.224309 0.112155 0.993691i \(-0.464225\pi\)
0.112155 + 0.993691i \(0.464225\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 7.26180 0.371546
\(383\) −22.5236 −1.15090 −0.575451 0.817836i \(-0.695173\pi\)
−0.575451 + 0.817836i \(0.695173\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −18.8638 −0.960140
\(387\) 0 0
\(388\) −3.65983 −0.185800
\(389\) 9.07838 0.460292 0.230146 0.973156i \(-0.426080\pi\)
0.230146 + 0.973156i \(0.426080\pi\)
\(390\) 0 0
\(391\) −34.3545 −1.73738
\(392\) −5.83710 −0.294818
\(393\) 0 0
\(394\) −17.7009 −0.891757
\(395\) 0 0
\(396\) 0 0
\(397\) −3.74435 −0.187923 −0.0939617 0.995576i \(-0.529953\pi\)
−0.0939617 + 0.995576i \(0.529953\pi\)
\(398\) −0.313511 −0.0157149
\(399\) 0 0
\(400\) 0 0
\(401\) 11.9155 0.595031 0.297515 0.954717i \(-0.403842\pi\)
0.297515 + 0.954717i \(0.403842\pi\)
\(402\) 0 0
\(403\) 3.68649 0.183637
\(404\) −19.6020 −0.975234
\(405\) 0 0
\(406\) −0.366835 −0.0182057
\(407\) −21.6742 −1.07435
\(408\) 0 0
\(409\) −2.99386 −0.148037 −0.0740183 0.997257i \(-0.523582\pi\)
−0.0740183 + 0.997257i \(0.523582\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −11.7587 −0.579311
\(413\) 0.796064 0.0391717
\(414\) 0 0
\(415\) 0 0
\(416\) 3.41855 0.167608
\(417\) 0 0
\(418\) −6.34017 −0.310108
\(419\) 33.5441 1.63874 0.819368 0.573267i \(-0.194324\pi\)
0.819368 + 0.573267i \(0.194324\pi\)
\(420\) 0 0
\(421\) 31.5897 1.53959 0.769793 0.638293i \(-0.220359\pi\)
0.769793 + 0.638293i \(0.220359\pi\)
\(422\) 14.8371 0.722259
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 2.89043 0.139877
\(428\) −6.15676 −0.297598
\(429\) 0 0
\(430\) 0 0
\(431\) 40.5113 1.95136 0.975680 0.219198i \(-0.0703440\pi\)
0.975680 + 0.219198i \(0.0703440\pi\)
\(432\) 0 0
\(433\) 5.13624 0.246832 0.123416 0.992355i \(-0.460615\pi\)
0.123416 + 0.992355i \(0.460615\pi\)
\(434\) −1.16290 −0.0558209
\(435\) 0 0
\(436\) 17.9155 0.857996
\(437\) −6.34017 −0.303292
\(438\) 0 0
\(439\) −17.7054 −0.845033 −0.422516 0.906355i \(-0.638853\pi\)
−0.422516 + 0.906355i \(0.638853\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 18.5236 0.881077
\(443\) −6.73820 −0.320142 −0.160071 0.987106i \(-0.551172\pi\)
−0.160071 + 0.987106i \(0.551172\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.28846 0.0610102
\(447\) 0 0
\(448\) −1.07838 −0.0509486
\(449\) 36.2290 1.70975 0.854876 0.518833i \(-0.173633\pi\)
0.854876 + 0.518833i \(0.173633\pi\)
\(450\) 0 0
\(451\) 48.1978 2.26955
\(452\) 13.2039 0.621061
\(453\) 0 0
\(454\) −20.9939 −0.985291
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −2.36683 −0.110595
\(459\) 0 0
\(460\) 0 0
\(461\) −35.7464 −1.66488 −0.832439 0.554117i \(-0.813056\pi\)
−0.832439 + 0.554117i \(0.813056\pi\)
\(462\) 0 0
\(463\) −37.3295 −1.73485 −0.867424 0.497569i \(-0.834226\pi\)
−0.867424 + 0.497569i \(0.834226\pi\)
\(464\) 0.340173 0.0157921
\(465\) 0 0
\(466\) −5.10504 −0.236486
\(467\) 12.0989 0.559870 0.279935 0.960019i \(-0.409687\pi\)
0.279935 + 0.960019i \(0.409687\pi\)
\(468\) 0 0
\(469\) −3.05947 −0.141273
\(470\) 0 0
\(471\) 0 0
\(472\) −0.738205 −0.0339786
\(473\) 70.8659 3.25842
\(474\) 0 0
\(475\) 0 0
\(476\) −5.84324 −0.267825
\(477\) 0 0
\(478\) −10.4124 −0.476252
\(479\) 34.6102 1.58138 0.790690 0.612216i \(-0.209722\pi\)
0.790690 + 0.612216i \(0.209722\pi\)
\(480\) 0 0
\(481\) 11.6865 0.532858
\(482\) 3.36069 0.153075
\(483\) 0 0
\(484\) 29.1978 1.32717
\(485\) 0 0
\(486\) 0 0
\(487\) 30.2823 1.37222 0.686111 0.727497i \(-0.259316\pi\)
0.686111 + 0.727497i \(0.259316\pi\)
\(488\) −2.68035 −0.121334
\(489\) 0 0
\(490\) 0 0
\(491\) 27.0205 1.21942 0.609709 0.792625i \(-0.291286\pi\)
0.609709 + 0.792625i \(0.291286\pi\)
\(492\) 0 0
\(493\) 1.84324 0.0830156
\(494\) 3.41855 0.153808
\(495\) 0 0
\(496\) 1.07838 0.0484206
\(497\) −3.05947 −0.137236
\(498\) 0 0
\(499\) −35.0349 −1.56838 −0.784189 0.620522i \(-0.786921\pi\)
−0.784189 + 0.620522i \(0.786921\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −21.8576 −0.975553
\(503\) 29.5441 1.31731 0.658653 0.752447i \(-0.271126\pi\)
0.658653 + 0.752447i \(0.271126\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 40.1978 1.78701
\(507\) 0 0
\(508\) −0.921622 −0.0408904
\(509\) −29.2183 −1.29508 −0.647539 0.762032i \(-0.724202\pi\)
−0.647539 + 0.762032i \(0.724202\pi\)
\(510\) 0 0
\(511\) −7.37298 −0.326161
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 26.8781 1.18554
\(515\) 0 0
\(516\) 0 0
\(517\) −40.1978 −1.76790
\(518\) −3.68649 −0.161975
\(519\) 0 0
\(520\) 0 0
\(521\) 16.3980 0.718411 0.359205 0.933259i \(-0.383048\pi\)
0.359205 + 0.933259i \(0.383048\pi\)
\(522\) 0 0
\(523\) −13.8432 −0.605323 −0.302661 0.953098i \(-0.597875\pi\)
−0.302661 + 0.953098i \(0.597875\pi\)
\(524\) −10.6537 −0.465408
\(525\) 0 0
\(526\) −22.3402 −0.974078
\(527\) 5.84324 0.254536
\(528\) 0 0
\(529\) 17.1978 0.747730
\(530\) 0 0
\(531\) 0 0
\(532\) −1.07838 −0.0467536
\(533\) −25.9877 −1.12565
\(534\) 0 0
\(535\) 0 0
\(536\) 2.83710 0.122544
\(537\) 0 0
\(538\) 17.3340 0.747323
\(539\) 37.0082 1.59406
\(540\) 0 0
\(541\) −5.20394 −0.223735 −0.111867 0.993723i \(-0.535683\pi\)
−0.111867 + 0.993723i \(0.535683\pi\)
\(542\) −13.3607 −0.573891
\(543\) 0 0
\(544\) 5.41855 0.232318
\(545\) 0 0
\(546\) 0 0
\(547\) 19.8843 0.850191 0.425095 0.905149i \(-0.360241\pi\)
0.425095 + 0.905149i \(0.360241\pi\)
\(548\) −20.6225 −0.880949
\(549\) 0 0
\(550\) 0 0
\(551\) 0.340173 0.0144919
\(552\) 0 0
\(553\) 1.16290 0.0494515
\(554\) 0.738205 0.0313633
\(555\) 0 0
\(556\) 18.8371 0.798871
\(557\) −17.0205 −0.721183 −0.360591 0.932724i \(-0.617425\pi\)
−0.360591 + 0.932724i \(0.617425\pi\)
\(558\) 0 0
\(559\) −38.2101 −1.61611
\(560\) 0 0
\(561\) 0 0
\(562\) 31.7998 1.34139
\(563\) −12.6270 −0.532166 −0.266083 0.963950i \(-0.585729\pi\)
−0.266083 + 0.963950i \(0.585729\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.70701 −0.197850
\(567\) 0 0
\(568\) 2.83710 0.119042
\(569\) 31.9155 1.33797 0.668983 0.743277i \(-0.266730\pi\)
0.668983 + 0.743277i \(0.266730\pi\)
\(570\) 0 0
\(571\) 24.5113 1.02577 0.512883 0.858458i \(-0.328577\pi\)
0.512883 + 0.858458i \(0.328577\pi\)
\(572\) −21.6742 −0.906244
\(573\) 0 0
\(574\) 8.19779 0.342169
\(575\) 0 0
\(576\) 0 0
\(577\) −32.0000 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) 12.3607 0.514137
\(579\) 0 0
\(580\) 0 0
\(581\) 0.965105 0.0400393
\(582\) 0 0
\(583\) 38.0410 1.57550
\(584\) 6.83710 0.282921
\(585\) 0 0
\(586\) −26.1978 −1.08222
\(587\) 8.58145 0.354194 0.177097 0.984193i \(-0.443329\pi\)
0.177097 + 0.984193i \(0.443329\pi\)
\(588\) 0 0
\(589\) 1.07838 0.0444338
\(590\) 0 0
\(591\) 0 0
\(592\) 3.41855 0.140502
\(593\) 10.7792 0.442650 0.221325 0.975200i \(-0.428962\pi\)
0.221325 + 0.975200i \(0.428962\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 5.44521 0.223045
\(597\) 0 0
\(598\) −21.6742 −0.886324
\(599\) 18.2101 0.744044 0.372022 0.928224i \(-0.378665\pi\)
0.372022 + 0.928224i \(0.378665\pi\)
\(600\) 0 0
\(601\) −32.0410 −1.30698 −0.653491 0.756935i \(-0.726696\pi\)
−0.653491 + 0.756935i \(0.726696\pi\)
\(602\) 12.0533 0.491257
\(603\) 0 0
\(604\) −23.1194 −0.940716
\(605\) 0 0
\(606\) 0 0
\(607\) 11.5609 0.469244 0.234622 0.972087i \(-0.424615\pi\)
0.234622 + 0.972087i \(0.424615\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 21.6742 0.876844
\(612\) 0 0
\(613\) −2.58145 −0.104264 −0.0521319 0.998640i \(-0.516602\pi\)
−0.0521319 + 0.998640i \(0.516602\pi\)
\(614\) 32.1978 1.29940
\(615\) 0 0
\(616\) 6.83710 0.275475
\(617\) 9.90110 0.398603 0.199302 0.979938i \(-0.436133\pi\)
0.199302 + 0.979938i \(0.436133\pi\)
\(618\) 0 0
\(619\) 11.2039 0.450324 0.225162 0.974321i \(-0.427709\pi\)
0.225162 + 0.974321i \(0.427709\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −16.7382 −0.671141
\(623\) 7.46412 0.299044
\(624\) 0 0
\(625\) 0 0
\(626\) −14.1568 −0.565818
\(627\) 0 0
\(628\) −9.73206 −0.388352
\(629\) 18.5236 0.738584
\(630\) 0 0
\(631\) −23.3197 −0.928341 −0.464170 0.885746i \(-0.653647\pi\)
−0.464170 + 0.885746i \(0.653647\pi\)
\(632\) −1.07838 −0.0428956
\(633\) 0 0
\(634\) 21.8843 0.869136
\(635\) 0 0
\(636\) 0 0
\(637\) −19.9544 −0.790623
\(638\) −2.15676 −0.0853868
\(639\) 0 0
\(640\) 0 0
\(641\) 19.2885 0.761848 0.380924 0.924606i \(-0.375606\pi\)
0.380924 + 0.924606i \(0.375606\pi\)
\(642\) 0 0
\(643\) 31.3751 1.23731 0.618656 0.785662i \(-0.287677\pi\)
0.618656 + 0.785662i \(0.287677\pi\)
\(644\) 6.83710 0.269420
\(645\) 0 0
\(646\) 5.41855 0.213190
\(647\) −33.2306 −1.30643 −0.653215 0.757173i \(-0.726580\pi\)
−0.653215 + 0.757173i \(0.726580\pi\)
\(648\) 0 0
\(649\) 4.68035 0.183720
\(650\) 0 0
\(651\) 0 0
\(652\) −6.49693 −0.254439
\(653\) −7.85762 −0.307492 −0.153746 0.988110i \(-0.549134\pi\)
−0.153746 + 0.988110i \(0.549134\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −7.60197 −0.296807
\(657\) 0 0
\(658\) −6.83710 −0.266538
\(659\) 25.4186 0.990166 0.495083 0.868846i \(-0.335138\pi\)
0.495083 + 0.868846i \(0.335138\pi\)
\(660\) 0 0
\(661\) 30.9093 1.20223 0.601117 0.799161i \(-0.294723\pi\)
0.601117 + 0.799161i \(0.294723\pi\)
\(662\) −26.1568 −1.01661
\(663\) 0 0
\(664\) −0.894960 −0.0347312
\(665\) 0 0
\(666\) 0 0
\(667\) −2.15676 −0.0835099
\(668\) −20.9939 −0.812277
\(669\) 0 0
\(670\) 0 0
\(671\) 16.9939 0.656041
\(672\) 0 0
\(673\) −27.9733 −1.07829 −0.539146 0.842212i \(-0.681253\pi\)
−0.539146 + 0.842212i \(0.681253\pi\)
\(674\) −29.7009 −1.14403
\(675\) 0 0
\(676\) −1.31351 −0.0505197
\(677\) 4.15676 0.159757 0.0798785 0.996805i \(-0.474547\pi\)
0.0798785 + 0.996805i \(0.474547\pi\)
\(678\) 0 0
\(679\) 3.94668 0.151460
\(680\) 0 0
\(681\) 0 0
\(682\) −6.83710 −0.261806
\(683\) 19.5708 0.748855 0.374427 0.927256i \(-0.377839\pi\)
0.374427 + 0.927256i \(0.377839\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13.8432 0.528538
\(687\) 0 0
\(688\) −11.1773 −0.426130
\(689\) −20.5113 −0.781418
\(690\) 0 0
\(691\) −10.8371 −0.412263 −0.206131 0.978524i \(-0.566087\pi\)
−0.206131 + 0.978524i \(0.566087\pi\)
\(692\) 6.99386 0.265867
\(693\) 0 0
\(694\) −23.4186 −0.888956
\(695\) 0 0
\(696\) 0 0
\(697\) −41.1917 −1.56025
\(698\) 6.48255 0.245368
\(699\) 0 0
\(700\) 0 0
\(701\) 42.0098 1.58669 0.793345 0.608772i \(-0.208338\pi\)
0.793345 + 0.608772i \(0.208338\pi\)
\(702\) 0 0
\(703\) 3.41855 0.128933
\(704\) −6.34017 −0.238954
\(705\) 0 0
\(706\) −2.58145 −0.0971541
\(707\) 21.1383 0.794989
\(708\) 0 0
\(709\) 23.6209 0.887101 0.443550 0.896249i \(-0.353719\pi\)
0.443550 + 0.896249i \(0.353719\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −6.92162 −0.259399
\(713\) −6.83710 −0.256051
\(714\) 0 0
\(715\) 0 0
\(716\) 4.73820 0.177075
\(717\) 0 0
\(718\) −28.1399 −1.05017
\(719\) −30.9770 −1.15525 −0.577624 0.816303i \(-0.696020\pi\)
−0.577624 + 0.816303i \(0.696020\pi\)
\(720\) 0 0
\(721\) 12.6803 0.472241
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −12.0722 −0.448661
\(725\) 0 0
\(726\) 0 0
\(727\) 23.7464 0.880707 0.440353 0.897825i \(-0.354853\pi\)
0.440353 + 0.897825i \(0.354853\pi\)
\(728\) −3.68649 −0.136630
\(729\) 0 0
\(730\) 0 0
\(731\) −60.5646 −2.24006
\(732\) 0 0
\(733\) 41.9832 1.55068 0.775342 0.631542i \(-0.217577\pi\)
0.775342 + 0.631542i \(0.217577\pi\)
\(734\) 10.5548 0.389584
\(735\) 0 0
\(736\) −6.34017 −0.233702
\(737\) −17.9877 −0.662586
\(738\) 0 0
\(739\) −41.3074 −1.51952 −0.759758 0.650206i \(-0.774683\pi\)
−0.759758 + 0.650206i \(0.774683\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 6.47027 0.237531
\(743\) −28.3668 −1.04068 −0.520339 0.853960i \(-0.674194\pi\)
−0.520339 + 0.853960i \(0.674194\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 17.8888 0.654956
\(747\) 0 0
\(748\) −34.3545 −1.25613
\(749\) 6.63931 0.242595
\(750\) 0 0
\(751\) 5.75872 0.210139 0.105069 0.994465i \(-0.466494\pi\)
0.105069 + 0.994465i \(0.466494\pi\)
\(752\) 6.34017 0.231202
\(753\) 0 0
\(754\) 1.16290 0.0423503
\(755\) 0 0
\(756\) 0 0
\(757\) −32.3090 −1.17429 −0.587145 0.809482i \(-0.699748\pi\)
−0.587145 + 0.809482i \(0.699748\pi\)
\(758\) 4.36683 0.158611
\(759\) 0 0
\(760\) 0 0
\(761\) −33.5708 −1.21694 −0.608470 0.793577i \(-0.708216\pi\)
−0.608470 + 0.793577i \(0.708216\pi\)
\(762\) 0 0
\(763\) −19.3197 −0.699418
\(764\) 7.26180 0.262723
\(765\) 0 0
\(766\) −22.5236 −0.813810
\(767\) −2.52359 −0.0911216
\(768\) 0 0
\(769\) −17.8843 −0.644924 −0.322462 0.946582i \(-0.604510\pi\)
−0.322462 + 0.946582i \(0.604510\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −18.8638 −0.678922
\(773\) 46.5113 1.67290 0.836448 0.548047i \(-0.184628\pi\)
0.836448 + 0.548047i \(0.184628\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3.65983 −0.131380
\(777\) 0 0
\(778\) 9.07838 0.325476
\(779\) −7.60197 −0.272369
\(780\) 0 0
\(781\) −17.9877 −0.643651
\(782\) −34.3545 −1.22852
\(783\) 0 0
\(784\) −5.83710 −0.208468
\(785\) 0 0
\(786\) 0 0
\(787\) −41.9877 −1.49670 −0.748350 0.663304i \(-0.769154\pi\)
−0.748350 + 0.663304i \(0.769154\pi\)
\(788\) −17.7009 −0.630567
\(789\) 0 0
\(790\) 0 0
\(791\) −14.2388 −0.506275
\(792\) 0 0
\(793\) −9.16290 −0.325384
\(794\) −3.74435 −0.132882
\(795\) 0 0
\(796\) −0.313511 −0.0111121
\(797\) 11.3074 0.400528 0.200264 0.979742i \(-0.435820\pi\)
0.200264 + 0.979742i \(0.435820\pi\)
\(798\) 0 0
\(799\) 34.3545 1.21538
\(800\) 0 0
\(801\) 0 0
\(802\) 11.9155 0.420750
\(803\) −43.3484 −1.52973
\(804\) 0 0
\(805\) 0 0
\(806\) 3.68649 0.129851
\(807\) 0 0
\(808\) −19.6020 −0.689595
\(809\) 48.5523 1.70701 0.853505 0.521085i \(-0.174473\pi\)
0.853505 + 0.521085i \(0.174473\pi\)
\(810\) 0 0
\(811\) −1.42309 −0.0499713 −0.0249856 0.999688i \(-0.507954\pi\)
−0.0249856 + 0.999688i \(0.507954\pi\)
\(812\) −0.366835 −0.0128734
\(813\) 0 0
\(814\) −21.6742 −0.759680
\(815\) 0 0
\(816\) 0 0
\(817\) −11.1773 −0.391043
\(818\) −2.99386 −0.104678
\(819\) 0 0
\(820\) 0 0
\(821\) 6.55479 0.228764 0.114382 0.993437i \(-0.463511\pi\)
0.114382 + 0.993437i \(0.463511\pi\)
\(822\) 0 0
\(823\) 0.0845208 0.00294621 0.00147311 0.999999i \(-0.499531\pi\)
0.00147311 + 0.999999i \(0.499531\pi\)
\(824\) −11.7587 −0.409635
\(825\) 0 0
\(826\) 0.796064 0.0276986
\(827\) 26.4703 0.920461 0.460231 0.887799i \(-0.347767\pi\)
0.460231 + 0.887799i \(0.347767\pi\)
\(828\) 0 0
\(829\) −12.4924 −0.433879 −0.216939 0.976185i \(-0.569607\pi\)
−0.216939 + 0.976185i \(0.569607\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.41855 0.118517
\(833\) −31.6286 −1.09587
\(834\) 0 0
\(835\) 0 0
\(836\) −6.34017 −0.219279
\(837\) 0 0
\(838\) 33.5441 1.15876
\(839\) 17.5297 0.605194 0.302597 0.953119i \(-0.402146\pi\)
0.302597 + 0.953119i \(0.402146\pi\)
\(840\) 0 0
\(841\) −28.8843 −0.996010
\(842\) 31.5897 1.08865
\(843\) 0 0
\(844\) 14.8371 0.510714
\(845\) 0 0
\(846\) 0 0
\(847\) −31.4863 −1.08188
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 0 0
\(851\) −21.6742 −0.742982
\(852\) 0 0
\(853\) −22.0456 −0.754826 −0.377413 0.926045i \(-0.623186\pi\)
−0.377413 + 0.926045i \(0.623186\pi\)
\(854\) 2.89043 0.0989083
\(855\) 0 0
\(856\) −6.15676 −0.210434
\(857\) −13.1506 −0.449216 −0.224608 0.974449i \(-0.572110\pi\)
−0.224608 + 0.974449i \(0.572110\pi\)
\(858\) 0 0
\(859\) 5.42309 0.185033 0.0925166 0.995711i \(-0.470509\pi\)
0.0925166 + 0.995711i \(0.470509\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 40.5113 1.37982
\(863\) 28.6803 0.976290 0.488145 0.872762i \(-0.337674\pi\)
0.488145 + 0.872762i \(0.337674\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 5.13624 0.174536
\(867\) 0 0
\(868\) −1.16290 −0.0394713
\(869\) 6.83710 0.231933
\(870\) 0 0
\(871\) 9.69878 0.328630
\(872\) 17.9155 0.606695
\(873\) 0 0
\(874\) −6.34017 −0.214460
\(875\) 0 0
\(876\) 0 0
\(877\) −24.4657 −0.826149 −0.413075 0.910697i \(-0.635545\pi\)
−0.413075 + 0.910697i \(0.635545\pi\)
\(878\) −17.7054 −0.597528
\(879\) 0 0
\(880\) 0 0
\(881\) −5.51745 −0.185888 −0.0929438 0.995671i \(-0.529628\pi\)
−0.0929438 + 0.995671i \(0.529628\pi\)
\(882\) 0 0
\(883\) −7.23060 −0.243329 −0.121665 0.992571i \(-0.538823\pi\)
−0.121665 + 0.992571i \(0.538823\pi\)
\(884\) 18.5236 0.623016
\(885\) 0 0
\(886\) −6.73820 −0.226374
\(887\) −18.1568 −0.609644 −0.304822 0.952409i \(-0.598597\pi\)
−0.304822 + 0.952409i \(0.598597\pi\)
\(888\) 0 0
\(889\) 0.993857 0.0333329
\(890\) 0 0
\(891\) 0 0
\(892\) 1.28846 0.0431407
\(893\) 6.34017 0.212166
\(894\) 0 0
\(895\) 0 0
\(896\) −1.07838 −0.0360261
\(897\) 0 0
\(898\) 36.2290 1.20898
\(899\) 0.366835 0.0122346
\(900\) 0 0
\(901\) −32.5113 −1.08311
\(902\) 48.1978 1.60481
\(903\) 0 0
\(904\) 13.2039 0.439156
\(905\) 0 0
\(906\) 0 0
\(907\) −19.2039 −0.637656 −0.318828 0.947813i \(-0.603289\pi\)
−0.318828 + 0.947813i \(0.603289\pi\)
\(908\) −20.9939 −0.696706
\(909\) 0 0
\(910\) 0 0
\(911\) −52.8781 −1.75193 −0.875965 0.482374i \(-0.839775\pi\)
−0.875965 + 0.482374i \(0.839775\pi\)
\(912\) 0 0
\(913\) 5.67420 0.187789
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) −2.36683 −0.0782024
\(917\) 11.4887 0.379390
\(918\) 0 0
\(919\) 10.0410 0.331223 0.165612 0.986191i \(-0.447040\pi\)
0.165612 + 0.986191i \(0.447040\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −35.7464 −1.17725
\(923\) 9.69878 0.319239
\(924\) 0 0
\(925\) 0 0
\(926\) −37.3295 −1.22672
\(927\) 0 0
\(928\) 0.340173 0.0111667
\(929\) −11.3074 −0.370983 −0.185491 0.982646i \(-0.559388\pi\)
−0.185491 + 0.982646i \(0.559388\pi\)
\(930\) 0 0
\(931\) −5.83710 −0.191303
\(932\) −5.10504 −0.167221
\(933\) 0 0
\(934\) 12.0989 0.395888
\(935\) 0 0
\(936\) 0 0
\(937\) −27.0349 −0.883192 −0.441596 0.897214i \(-0.645587\pi\)
−0.441596 + 0.897214i \(0.645587\pi\)
\(938\) −3.05947 −0.0998951
\(939\) 0 0
\(940\) 0 0
\(941\) 19.9733 0.651112 0.325556 0.945523i \(-0.394449\pi\)
0.325556 + 0.945523i \(0.394449\pi\)
\(942\) 0 0
\(943\) 48.1978 1.56954
\(944\) −0.738205 −0.0240265
\(945\) 0 0
\(946\) 70.8659 2.30405
\(947\) −12.5281 −0.407109 −0.203555 0.979064i \(-0.565249\pi\)
−0.203555 + 0.979064i \(0.565249\pi\)
\(948\) 0 0
\(949\) 23.3730 0.758719
\(950\) 0 0
\(951\) 0 0
\(952\) −5.84324 −0.189381
\(953\) 46.5113 1.50665 0.753324 0.657649i \(-0.228449\pi\)
0.753324 + 0.657649i \(0.228449\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −10.4124 −0.336761
\(957\) 0 0
\(958\) 34.6102 1.11820
\(959\) 22.2388 0.718129
\(960\) 0 0
\(961\) −29.8371 −0.962487
\(962\) 11.6865 0.376788
\(963\) 0 0
\(964\) 3.36069 0.108241
\(965\) 0 0
\(966\) 0 0
\(967\) 37.2762 1.19872 0.599360 0.800479i \(-0.295422\pi\)
0.599360 + 0.800479i \(0.295422\pi\)
\(968\) 29.1978 0.938453
\(969\) 0 0
\(970\) 0 0
\(971\) 16.4534 0.528016 0.264008 0.964520i \(-0.414955\pi\)
0.264008 + 0.964520i \(0.414955\pi\)
\(972\) 0 0
\(973\) −20.3135 −0.651221
\(974\) 30.2823 0.970308
\(975\) 0 0
\(976\) −2.68035 −0.0857958
\(977\) 34.5646 1.10582 0.552910 0.833241i \(-0.313517\pi\)
0.552910 + 0.833241i \(0.313517\pi\)
\(978\) 0 0
\(979\) 43.8843 1.40255
\(980\) 0 0
\(981\) 0 0
\(982\) 27.0205 0.862259
\(983\) 40.1978 1.28211 0.641055 0.767495i \(-0.278497\pi\)
0.641055 + 0.767495i \(0.278497\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 1.84324 0.0587009
\(987\) 0 0
\(988\) 3.41855 0.108759
\(989\) 70.8659 2.25340
\(990\) 0 0
\(991\) −24.6491 −0.783006 −0.391503 0.920177i \(-0.628045\pi\)
−0.391503 + 0.920177i \(0.628045\pi\)
\(992\) 1.07838 0.0342385
\(993\) 0 0
\(994\) −3.05947 −0.0970404
\(995\) 0 0
\(996\) 0 0
\(997\) 6.89496 0.218366 0.109183 0.994022i \(-0.465177\pi\)
0.109183 + 0.994022i \(0.465177\pi\)
\(998\) −35.0349 −1.10901
\(999\) 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.cq.1.2 3
3.2 odd 2 2850.2.a.bl.1.2 3
5.2 odd 4 1710.2.d.f.1369.5 6
5.3 odd 4 1710.2.d.f.1369.2 6
5.4 even 2 8550.2.a.ce.1.2 3
15.2 even 4 570.2.d.c.229.2 6
15.8 even 4 570.2.d.c.229.5 yes 6
15.14 odd 2 2850.2.a.bm.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.d.c.229.2 6 15.2 even 4
570.2.d.c.229.5 yes 6 15.8 even 4
1710.2.d.f.1369.2 6 5.3 odd 4
1710.2.d.f.1369.5 6 5.2 odd 4
2850.2.a.bl.1.2 3 3.2 odd 2
2850.2.a.bm.1.2 3 15.14 odd 2
8550.2.a.ce.1.2 3 5.4 even 2
8550.2.a.cq.1.2 3 1.1 even 1 trivial