Properties

Label 6840.2.a.bf.1.1
Level $6840$
Weight $2$
Character 6840.1
Self dual yes
Analytic conductor $54.618$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.6176749826\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} -3.30777 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -3.30777 q^{7} -2.24914 q^{11} -3.19051 q^{13} -3.55691 q^{17} -1.00000 q^{19} -1.55691 q^{23} +1.00000 q^{25} -4.24914 q^{29} -7.43965 q^{31} +3.30777 q^{35} -5.30777 q^{37} -2.36641 q^{41} +3.80605 q^{43} +9.55691 q^{47} +3.94137 q^{49} +3.55691 q^{53} +2.24914 q^{55} -0.498281 q^{59} -1.17590 q^{61} +3.19051 q^{65} -12.9966 q^{67} +10.3810 q^{71} -0.117266 q^{73} +7.43965 q^{77} +12.1725 q^{83} +3.55691 q^{85} -10.6302 q^{89} +10.5535 q^{91} +1.00000 q^{95} +7.68879 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 2 q^{7} + O(q^{10}) \) \( 3 q - 3 q^{5} - 2 q^{7} + 2 q^{11} + 6 q^{17} - 3 q^{19} + 12 q^{23} + 3 q^{25} - 4 q^{29} - 4 q^{31} + 2 q^{35} - 8 q^{37} - 14 q^{43} + 12 q^{47} + 11 q^{49} - 6 q^{53} - 2 q^{55} + 16 q^{59} - 6 q^{61} - 4 q^{67} + 12 q^{71} - 2 q^{73} + 4 q^{77} + 4 q^{83} - 6 q^{85} - 4 q^{89} - 20 q^{91} + 3 q^{95} - 4 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.30777 −1.25022 −0.625110 0.780536i \(-0.714946\pi\)
−0.625110 + 0.780536i \(0.714946\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.24914 −0.678141 −0.339071 0.940761i \(-0.610113\pi\)
−0.339071 + 0.940761i \(0.610113\pi\)
\(12\) 0 0
\(13\) −3.19051 −0.884888 −0.442444 0.896796i \(-0.645888\pi\)
−0.442444 + 0.896796i \(0.645888\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.55691 −0.862678 −0.431339 0.902190i \(-0.641959\pi\)
−0.431339 + 0.902190i \(0.641959\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.55691 −0.324639 −0.162320 0.986738i \(-0.551898\pi\)
−0.162320 + 0.986738i \(0.551898\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.24914 −0.789046 −0.394523 0.918886i \(-0.629090\pi\)
−0.394523 + 0.918886i \(0.629090\pi\)
\(30\) 0 0
\(31\) −7.43965 −1.33620 −0.668100 0.744071i \(-0.732892\pi\)
−0.668100 + 0.744071i \(0.732892\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.30777 0.559116
\(36\) 0 0
\(37\) −5.30777 −0.872593 −0.436296 0.899803i \(-0.643710\pi\)
−0.436296 + 0.899803i \(0.643710\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.36641 −0.369571 −0.184785 0.982779i \(-0.559159\pi\)
−0.184785 + 0.982779i \(0.559159\pi\)
\(42\) 0 0
\(43\) 3.80605 0.580418 0.290209 0.956963i \(-0.406275\pi\)
0.290209 + 0.956963i \(0.406275\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.55691 1.39402 0.697010 0.717062i \(-0.254513\pi\)
0.697010 + 0.717062i \(0.254513\pi\)
\(48\) 0 0
\(49\) 3.94137 0.563052
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.55691 0.488580 0.244290 0.969702i \(-0.421445\pi\)
0.244290 + 0.969702i \(0.421445\pi\)
\(54\) 0 0
\(55\) 2.24914 0.303274
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.498281 −0.0648707 −0.0324353 0.999474i \(-0.510326\pi\)
−0.0324353 + 0.999474i \(0.510326\pi\)
\(60\) 0 0
\(61\) −1.17590 −0.150559 −0.0752793 0.997162i \(-0.523985\pi\)
−0.0752793 + 0.997162i \(0.523985\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.19051 0.395734
\(66\) 0 0
\(67\) −12.9966 −1.58778 −0.793891 0.608060i \(-0.791948\pi\)
−0.793891 + 0.608060i \(0.791948\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3810 1.23200 0.616000 0.787746i \(-0.288752\pi\)
0.616000 + 0.787746i \(0.288752\pi\)
\(72\) 0 0
\(73\) −0.117266 −0.0137250 −0.00686249 0.999976i \(-0.502184\pi\)
−0.00686249 + 0.999976i \(0.502184\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.43965 0.847827
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.1725 1.33610 0.668051 0.744116i \(-0.267129\pi\)
0.668051 + 0.744116i \(0.267129\pi\)
\(84\) 0 0
\(85\) 3.55691 0.385802
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.6302 −1.12679 −0.563397 0.826186i \(-0.690506\pi\)
−0.563397 + 0.826186i \(0.690506\pi\)
\(90\) 0 0
\(91\) 10.5535 1.10630
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 7.68879 0.780678 0.390339 0.920671i \(-0.372358\pi\)
0.390339 + 0.920671i \(0.372358\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.38101 −0.833942 −0.416971 0.908920i \(-0.636908\pi\)
−0.416971 + 0.908920i \(0.636908\pi\)
\(102\) 0 0
\(103\) −3.76547 −0.371023 −0.185511 0.982642i \(-0.559394\pi\)
−0.185511 + 0.982642i \(0.559394\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.11727 0.591378 0.295689 0.955284i \(-0.404451\pi\)
0.295689 + 0.955284i \(0.404451\pi\)
\(108\) 0 0
\(109\) −15.4948 −1.48414 −0.742068 0.670324i \(-0.766155\pi\)
−0.742068 + 0.670324i \(0.766155\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.94137 0.841133 0.420567 0.907262i \(-0.361831\pi\)
0.420567 + 0.907262i \(0.361831\pi\)
\(114\) 0 0
\(115\) 1.55691 0.145183
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 11.7655 1.07854
\(120\) 0 0
\(121\) −5.94137 −0.540124
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −10.1173 −0.897762 −0.448881 0.893591i \(-0.648177\pi\)
−0.448881 + 0.893591i \(0.648177\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.51633 0.132482 0.0662410 0.997804i \(-0.478899\pi\)
0.0662410 + 0.997804i \(0.478899\pi\)
\(132\) 0 0
\(133\) 3.30777 0.286820
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.8827 1.01521 0.507605 0.861590i \(-0.330531\pi\)
0.507605 + 0.861590i \(0.330531\pi\)
\(138\) 0 0
\(139\) 16.4983 1.39937 0.699683 0.714453i \(-0.253325\pi\)
0.699683 + 0.714453i \(0.253325\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 7.17590 0.600079
\(144\) 0 0
\(145\) 4.24914 0.352872
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.23109 0.592394 0.296197 0.955127i \(-0.404281\pi\)
0.296197 + 0.955127i \(0.404281\pi\)
\(150\) 0 0
\(151\) −3.67418 −0.299001 −0.149500 0.988762i \(-0.547766\pi\)
−0.149500 + 0.988762i \(0.547766\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.43965 0.597567
\(156\) 0 0
\(157\) 0.381015 0.0304083 0.0152041 0.999884i \(-0.495160\pi\)
0.0152041 + 0.999884i \(0.495160\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.14992 0.405871
\(162\) 0 0
\(163\) 2.80949 0.220056 0.110028 0.993928i \(-0.464906\pi\)
0.110028 + 0.993928i \(0.464906\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.17590 0.555288 0.277644 0.960684i \(-0.410446\pi\)
0.277644 + 0.960684i \(0.410446\pi\)
\(168\) 0 0
\(169\) −2.82066 −0.216974
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.94137 0.375685 0.187843 0.982199i \(-0.439851\pi\)
0.187843 + 0.982199i \(0.439851\pi\)
\(174\) 0 0
\(175\) −3.30777 −0.250044
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.61555 −0.195495 −0.0977476 0.995211i \(-0.531164\pi\)
−0.0977476 + 0.995211i \(0.531164\pi\)
\(180\) 0 0
\(181\) −17.6121 −1.30910 −0.654549 0.756020i \(-0.727141\pi\)
−0.654549 + 0.756020i \(0.727141\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.30777 0.390235
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.7440 −0.849765 −0.424882 0.905249i \(-0.639685\pi\)
−0.424882 + 0.905249i \(0.639685\pi\)
\(192\) 0 0
\(193\) −0.574960 −0.0413865 −0.0206933 0.999786i \(-0.506587\pi\)
−0.0206933 + 0.999786i \(0.506587\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.5569 0.823396 0.411698 0.911320i \(-0.364936\pi\)
0.411698 + 0.911320i \(0.364936\pi\)
\(198\) 0 0
\(199\) 3.50172 0.248230 0.124115 0.992268i \(-0.460391\pi\)
0.124115 + 0.992268i \(0.460391\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 14.0552 0.986481
\(204\) 0 0
\(205\) 2.36641 0.165277
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.24914 0.155576
\(210\) 0 0
\(211\) 9.23109 0.635495 0.317747 0.948175i \(-0.397074\pi\)
0.317747 + 0.948175i \(0.397074\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.80605 −0.259571
\(216\) 0 0
\(217\) 24.6087 1.67055
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 11.3484 0.763373
\(222\) 0 0
\(223\) 17.9931 1.20491 0.602454 0.798153i \(-0.294190\pi\)
0.602454 + 0.798153i \(0.294190\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.94480 0.394571 0.197285 0.980346i \(-0.436787\pi\)
0.197285 + 0.980346i \(0.436787\pi\)
\(228\) 0 0
\(229\) 14.1725 0.936543 0.468271 0.883585i \(-0.344877\pi\)
0.468271 + 0.883585i \(0.344877\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.3484 0.874480 0.437240 0.899345i \(-0.355956\pi\)
0.437240 + 0.899345i \(0.355956\pi\)
\(234\) 0 0
\(235\) −9.55691 −0.623424
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.2423 1.56810 0.784051 0.620697i \(-0.213150\pi\)
0.784051 + 0.620697i \(0.213150\pi\)
\(240\) 0 0
\(241\) −11.2311 −0.723458 −0.361729 0.932283i \(-0.617814\pi\)
−0.361729 + 0.932283i \(0.617814\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.94137 −0.251805
\(246\) 0 0
\(247\) 3.19051 0.203007
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.48024 0.472148 0.236074 0.971735i \(-0.424139\pi\)
0.236074 + 0.971735i \(0.424139\pi\)
\(252\) 0 0
\(253\) 3.50172 0.220151
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −0.0551953 −0.00344299 −0.00172149 0.999999i \(-0.500548\pi\)
−0.00172149 + 0.999999i \(0.500548\pi\)
\(258\) 0 0
\(259\) 17.5569 1.09093
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.93793 −0.242823 −0.121412 0.992602i \(-0.538742\pi\)
−0.121412 + 0.992602i \(0.538742\pi\)
\(264\) 0 0
\(265\) −3.55691 −0.218500
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −17.8681 −1.08944 −0.544719 0.838618i \(-0.683364\pi\)
−0.544719 + 0.838618i \(0.683364\pi\)
\(270\) 0 0
\(271\) −27.8466 −1.69156 −0.845782 0.533529i \(-0.820865\pi\)
−0.845782 + 0.533529i \(0.820865\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.24914 −0.135628
\(276\) 0 0
\(277\) −11.9931 −0.720597 −0.360299 0.932837i \(-0.617325\pi\)
−0.360299 + 0.932837i \(0.617325\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.8647 0.648133 0.324066 0.946034i \(-0.394950\pi\)
0.324066 + 0.946034i \(0.394950\pi\)
\(282\) 0 0
\(283\) 16.6922 0.992250 0.496125 0.868251i \(-0.334756\pi\)
0.496125 + 0.868251i \(0.334756\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.82754 0.462045
\(288\) 0 0
\(289\) −4.34836 −0.255786
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −31.1690 −1.82091 −0.910457 0.413604i \(-0.864270\pi\)
−0.910457 + 0.413604i \(0.864270\pi\)
\(294\) 0 0
\(295\) 0.498281 0.0290110
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.96735 0.287269
\(300\) 0 0
\(301\) −12.5896 −0.725651
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.17590 0.0673318
\(306\) 0 0
\(307\) −0.498281 −0.0284384 −0.0142192 0.999899i \(-0.504526\pi\)
−0.0142192 + 0.999899i \(0.504526\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.01805 −0.0577281 −0.0288640 0.999583i \(-0.509189\pi\)
−0.0288640 + 0.999583i \(0.509189\pi\)
\(312\) 0 0
\(313\) 12.3810 0.699816 0.349908 0.936784i \(-0.386213\pi\)
0.349908 + 0.936784i \(0.386213\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.78801 −0.268921 −0.134461 0.990919i \(-0.542930\pi\)
−0.134461 + 0.990919i \(0.542930\pi\)
\(318\) 0 0
\(319\) 9.55691 0.535084
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.55691 0.197912
\(324\) 0 0
\(325\) −3.19051 −0.176978
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −31.6121 −1.74283
\(330\) 0 0
\(331\) −12.4362 −0.683556 −0.341778 0.939781i \(-0.611029\pi\)
−0.341778 + 0.939781i \(0.611029\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.9966 0.710078
\(336\) 0 0
\(337\) −15.4543 −0.841847 −0.420923 0.907096i \(-0.638294\pi\)
−0.420923 + 0.907096i \(0.638294\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.7328 0.906133
\(342\) 0 0
\(343\) 10.1173 0.546281
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.8241 0.688434 0.344217 0.938890i \(-0.388144\pi\)
0.344217 + 0.938890i \(0.388144\pi\)
\(348\) 0 0
\(349\) 28.8793 1.54587 0.772937 0.634483i \(-0.218787\pi\)
0.772937 + 0.634483i \(0.218787\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 0 0
\(355\) −10.3810 −0.550967
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −19.7440 −1.04205 −0.521024 0.853542i \(-0.674450\pi\)
−0.521024 + 0.853542i \(0.674450\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.117266 0.00613800
\(366\) 0 0
\(367\) 13.6888 0.714549 0.357274 0.933999i \(-0.383706\pi\)
0.357274 + 0.933999i \(0.383706\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −11.7655 −0.610833
\(372\) 0 0
\(373\) 30.1510 1.56116 0.780579 0.625057i \(-0.214924\pi\)
0.780579 + 0.625057i \(0.214924\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 13.5569 0.698217
\(378\) 0 0
\(379\) −25.6673 −1.31844 −0.659220 0.751950i \(-0.729114\pi\)
−0.659220 + 0.751950i \(0.729114\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 23.8759 1.22000 0.610000 0.792402i \(-0.291170\pi\)
0.610000 + 0.792402i \(0.291170\pi\)
\(384\) 0 0
\(385\) −7.43965 −0.379160
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −29.3484 −1.48802 −0.744010 0.668168i \(-0.767079\pi\)
−0.744010 + 0.668168i \(0.767079\pi\)
\(390\) 0 0
\(391\) 5.53781 0.280059
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −12.1173 −0.608148 −0.304074 0.952648i \(-0.598347\pi\)
−0.304074 + 0.952648i \(0.598347\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 27.4734 1.37195 0.685977 0.727623i \(-0.259375\pi\)
0.685977 + 0.727623i \(0.259375\pi\)
\(402\) 0 0
\(403\) 23.7363 1.18239
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.9379 0.591741
\(408\) 0 0
\(409\) 32.9605 1.62979 0.814895 0.579608i \(-0.196794\pi\)
0.814895 + 0.579608i \(0.196794\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.64820 0.0811027
\(414\) 0 0
\(415\) −12.1725 −0.597523
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.7474 0.720459 0.360229 0.932864i \(-0.382698\pi\)
0.360229 + 0.932864i \(0.382698\pi\)
\(420\) 0 0
\(421\) 24.9897 1.21792 0.608961 0.793200i \(-0.291586\pi\)
0.608961 + 0.793200i \(0.291586\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.55691 −0.172536
\(426\) 0 0
\(427\) 3.88961 0.188231
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10.8501 −0.522630 −0.261315 0.965254i \(-0.584156\pi\)
−0.261315 + 0.965254i \(0.584156\pi\)
\(432\) 0 0
\(433\) −11.0371 −0.530412 −0.265206 0.964192i \(-0.585440\pi\)
−0.265206 + 0.964192i \(0.585440\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.55691 0.0744773
\(438\) 0 0
\(439\) 30.2277 1.44269 0.721344 0.692577i \(-0.243525\pi\)
0.721344 + 0.692577i \(0.243525\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −21.9018 −1.04059 −0.520294 0.853987i \(-0.674178\pi\)
−0.520294 + 0.853987i \(0.674178\pi\)
\(444\) 0 0
\(445\) 10.6302 0.503918
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.01805 −0.142430 −0.0712152 0.997461i \(-0.522688\pi\)
−0.0712152 + 0.997461i \(0.522688\pi\)
\(450\) 0 0
\(451\) 5.32238 0.250621
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −10.5535 −0.494755
\(456\) 0 0
\(457\) 13.5017 0.631584 0.315792 0.948828i \(-0.397730\pi\)
0.315792 + 0.948828i \(0.397730\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.64820 0.169914 0.0849568 0.996385i \(-0.472925\pi\)
0.0849568 + 0.996385i \(0.472925\pi\)
\(462\) 0 0
\(463\) −8.53887 −0.396835 −0.198417 0.980118i \(-0.563580\pi\)
−0.198417 + 0.980118i \(0.563580\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −22.5535 −1.04365 −0.521825 0.853052i \(-0.674749\pi\)
−0.521825 + 0.853052i \(0.674749\pi\)
\(468\) 0 0
\(469\) 42.9897 1.98508
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.56035 −0.393605
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 33.6267 1.53644 0.768222 0.640184i \(-0.221142\pi\)
0.768222 + 0.640184i \(0.221142\pi\)
\(480\) 0 0
\(481\) 16.9345 0.772146
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.68879 −0.349130
\(486\) 0 0
\(487\) 11.8466 0.536823 0.268411 0.963304i \(-0.413501\pi\)
0.268411 + 0.963304i \(0.413501\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −35.3269 −1.59428 −0.797140 0.603795i \(-0.793655\pi\)
−0.797140 + 0.603795i \(0.793655\pi\)
\(492\) 0 0
\(493\) 15.1138 0.680693
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −34.3380 −1.54027
\(498\) 0 0
\(499\) 30.7259 1.37548 0.687741 0.725956i \(-0.258602\pi\)
0.687741 + 0.725956i \(0.258602\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −42.1656 −1.88007 −0.940035 0.341077i \(-0.889208\pi\)
−0.940035 + 0.341077i \(0.889208\pi\)
\(504\) 0 0
\(505\) 8.38101 0.372950
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −41.7440 −1.85027 −0.925135 0.379639i \(-0.876048\pi\)
−0.925135 + 0.379639i \(0.876048\pi\)
\(510\) 0 0
\(511\) 0.387890 0.0171593
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.76547 0.165926
\(516\) 0 0
\(517\) −21.4948 −0.945342
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18.7405 0.821038 0.410519 0.911852i \(-0.365348\pi\)
0.410519 + 0.911852i \(0.365348\pi\)
\(522\) 0 0
\(523\) −14.7259 −0.643920 −0.321960 0.946753i \(-0.604342\pi\)
−0.321960 + 0.946753i \(0.604342\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 26.4622 1.15271
\(528\) 0 0
\(529\) −20.5760 −0.894609
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.55004 0.327028
\(534\) 0 0
\(535\) −6.11727 −0.264472
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −8.86469 −0.381829
\(540\) 0 0
\(541\) −40.2277 −1.72952 −0.864761 0.502184i \(-0.832530\pi\)
−0.864761 + 0.502184i \(0.832530\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 15.4948 0.663726
\(546\) 0 0
\(547\) 37.7294 1.61319 0.806596 0.591103i \(-0.201307\pi\)
0.806596 + 0.591103i \(0.201307\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.24914 0.181019
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19.9931 0.847136 0.423568 0.905864i \(-0.360777\pi\)
0.423568 + 0.905864i \(0.360777\pi\)
\(558\) 0 0
\(559\) −12.1432 −0.513605
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −44.0483 −1.85642 −0.928208 0.372063i \(-0.878651\pi\)
−0.928208 + 0.372063i \(0.878651\pi\)
\(564\) 0 0
\(565\) −8.94137 −0.376166
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 16.7474 0.702088 0.351044 0.936359i \(-0.385827\pi\)
0.351044 + 0.936359i \(0.385827\pi\)
\(570\) 0 0
\(571\) 36.1396 1.51240 0.756198 0.654343i \(-0.227055\pi\)
0.756198 + 0.654343i \(0.227055\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.55691 −0.0649278
\(576\) 0 0
\(577\) −16.3810 −0.681951 −0.340975 0.940072i \(-0.610757\pi\)
−0.340975 + 0.940072i \(0.610757\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −40.2637 −1.67042
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.5604 −0.518421 −0.259211 0.965821i \(-0.583462\pi\)
−0.259211 + 0.965821i \(0.583462\pi\)
\(588\) 0 0
\(589\) 7.43965 0.306545
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 38.9966 1.60140 0.800698 0.599068i \(-0.204462\pi\)
0.800698 + 0.599068i \(0.204462\pi\)
\(594\) 0 0
\(595\) −11.7655 −0.482337
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −17.3415 −0.708554 −0.354277 0.935141i \(-0.615273\pi\)
−0.354277 + 0.935141i \(0.615273\pi\)
\(600\) 0 0
\(601\) −15.9119 −0.649062 −0.324531 0.945875i \(-0.605206\pi\)
−0.324531 + 0.945875i \(0.605206\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.94137 0.241551
\(606\) 0 0
\(607\) −24.2637 −0.984835 −0.492418 0.870359i \(-0.663887\pi\)
−0.492418 + 0.870359i \(0.663887\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −30.4914 −1.23355
\(612\) 0 0
\(613\) −44.7259 −1.80646 −0.903232 0.429153i \(-0.858812\pi\)
−0.903232 + 0.429153i \(0.858812\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10.6707 0.429588 0.214794 0.976659i \(-0.431092\pi\)
0.214794 + 0.976659i \(0.431092\pi\)
\(618\) 0 0
\(619\) 47.3776 1.90427 0.952133 0.305685i \(-0.0988853\pi\)
0.952133 + 0.305685i \(0.0988853\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 35.1621 1.40874
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 18.8793 0.752767
\(630\) 0 0
\(631\) 10.4622 0.416493 0.208247 0.978076i \(-0.433224\pi\)
0.208247 + 0.978076i \(0.433224\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 10.1173 0.401491
\(636\) 0 0
\(637\) −12.5750 −0.498238
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −20.6233 −0.814571 −0.407285 0.913301i \(-0.633525\pi\)
−0.407285 + 0.913301i \(0.633525\pi\)
\(642\) 0 0
\(643\) 34.4216 1.35746 0.678728 0.734390i \(-0.262532\pi\)
0.678728 + 0.734390i \(0.262532\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.8207 1.01511 0.507557 0.861618i \(-0.330548\pi\)
0.507557 + 0.861618i \(0.330548\pi\)
\(648\) 0 0
\(649\) 1.12070 0.0439915
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −45.3155 −1.77333 −0.886666 0.462410i \(-0.846985\pi\)
−0.886666 + 0.462410i \(0.846985\pi\)
\(654\) 0 0
\(655\) −1.51633 −0.0592478
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 45.3415 1.76625 0.883127 0.469134i \(-0.155434\pi\)
0.883127 + 0.469134i \(0.155434\pi\)
\(660\) 0 0
\(661\) −17.6121 −0.685032 −0.342516 0.939512i \(-0.611279\pi\)
−0.342516 + 0.939512i \(0.611279\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.30777 −0.128270
\(666\) 0 0
\(667\) 6.61555 0.256155
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.64476 0.102100
\(672\) 0 0
\(673\) −13.8061 −0.532184 −0.266092 0.963948i \(-0.585733\pi\)
−0.266092 + 0.963948i \(0.585733\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 23.7914 0.914380 0.457190 0.889369i \(-0.348856\pi\)
0.457190 + 0.889369i \(0.348856\pi\)
\(678\) 0 0
\(679\) −25.4328 −0.976020
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −7.11383 −0.272203 −0.136102 0.990695i \(-0.543457\pi\)
−0.136102 + 0.990695i \(0.543457\pi\)
\(684\) 0 0
\(685\) −11.8827 −0.454016
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −11.3484 −0.432338
\(690\) 0 0
\(691\) −18.1465 −0.690325 −0.345162 0.938543i \(-0.612176\pi\)
−0.345162 + 0.938543i \(0.612176\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.4983 −0.625815
\(696\) 0 0
\(697\) 8.41711 0.318821
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −24.9897 −0.943847 −0.471924 0.881639i \(-0.656440\pi\)
−0.471924 + 0.881639i \(0.656440\pi\)
\(702\) 0 0
\(703\) 5.30777 0.200186
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 27.7225 1.04261
\(708\) 0 0
\(709\) −31.9311 −1.19920 −0.599598 0.800301i \(-0.704673\pi\)
−0.599598 + 0.800301i \(0.704673\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 11.5829 0.433783
\(714\) 0 0
\(715\) −7.17590 −0.268363
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 38.5941 1.43932 0.719658 0.694329i \(-0.244299\pi\)
0.719658 + 0.694329i \(0.244299\pi\)
\(720\) 0 0
\(721\) 12.4553 0.463860
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.24914 −0.157809
\(726\) 0 0
\(727\) −38.0337 −1.41059 −0.705296 0.708913i \(-0.749186\pi\)
−0.705296 + 0.708913i \(0.749186\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −13.5378 −0.500714
\(732\) 0 0
\(733\) 14.5275 0.536585 0.268293 0.963337i \(-0.413541\pi\)
0.268293 + 0.963337i \(0.413541\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 29.2311 1.07674
\(738\) 0 0
\(739\) −5.46563 −0.201056 −0.100528 0.994934i \(-0.532053\pi\)
−0.100528 + 0.994934i \(0.532053\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 44.5795 1.63546 0.817731 0.575601i \(-0.195232\pi\)
0.817731 + 0.575601i \(0.195232\pi\)
\(744\) 0 0
\(745\) −7.23109 −0.264927
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −20.2345 −0.739354
\(750\) 0 0
\(751\) 7.43965 0.271477 0.135738 0.990745i \(-0.456659\pi\)
0.135738 + 0.990745i \(0.456659\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.67418 0.133717
\(756\) 0 0
\(757\) −23.6190 −0.858447 −0.429223 0.903198i \(-0.641213\pi\)
−0.429223 + 0.903198i \(0.641213\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.9966 −0.688625 −0.344312 0.938855i \(-0.611888\pi\)
−0.344312 + 0.938855i \(0.611888\pi\)
\(762\) 0 0
\(763\) 51.2534 1.85550
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.58977 0.0574032
\(768\) 0 0
\(769\) −11.2311 −0.405004 −0.202502 0.979282i \(-0.564907\pi\)
−0.202502 + 0.979282i \(0.564907\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −26.0191 −0.935842 −0.467921 0.883770i \(-0.654997\pi\)
−0.467921 + 0.883770i \(0.654997\pi\)
\(774\) 0 0
\(775\) −7.43965 −0.267240
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.36641 0.0847853
\(780\) 0 0
\(781\) −23.3484 −0.835470
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.381015 −0.0135990
\(786\) 0 0
\(787\) 14.1984 0.506120 0.253060 0.967451i \(-0.418563\pi\)
0.253060 + 0.967451i \(0.418563\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −29.5760 −1.05160
\(792\) 0 0
\(793\) 3.75172 0.133227
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 34.1725 1.21045 0.605225 0.796054i \(-0.293083\pi\)
0.605225 + 0.796054i \(0.293083\pi\)
\(798\) 0 0
\(799\) −33.9931 −1.20259
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0.263748 0.00930748
\(804\) 0 0
\(805\) −5.14992 −0.181511
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −20.1173 −0.707285 −0.353643 0.935381i \(-0.615057\pi\)
−0.353643 + 0.935381i \(0.615057\pi\)
\(810\) 0 0
\(811\) −0.762030 −0.0267585 −0.0133792 0.999910i \(-0.504259\pi\)
−0.0133792 + 0.999910i \(0.504259\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.80949 −0.0984122
\(816\) 0 0
\(817\) −3.80605 −0.133157
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.50516 0.157231 0.0786155 0.996905i \(-0.474950\pi\)
0.0786155 + 0.996905i \(0.474950\pi\)
\(822\) 0 0
\(823\) −11.1836 −0.389837 −0.194918 0.980819i \(-0.562444\pi\)
−0.194918 + 0.980819i \(0.562444\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.2829 0.566210 0.283105 0.959089i \(-0.408635\pi\)
0.283105 + 0.959089i \(0.408635\pi\)
\(828\) 0 0
\(829\) 15.3845 0.534324 0.267162 0.963652i \(-0.413914\pi\)
0.267162 + 0.963652i \(0.413914\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −14.0191 −0.485733
\(834\) 0 0
\(835\) −7.17590 −0.248332
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 16.0812 0.555184 0.277592 0.960699i \(-0.410464\pi\)
0.277592 + 0.960699i \(0.410464\pi\)
\(840\) 0 0
\(841\) −10.9448 −0.377407
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.82066 0.0970337
\(846\) 0 0
\(847\) 19.6527 0.675275
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.26375 0.283278
\(852\) 0 0
\(853\) −19.9119 −0.681772 −0.340886 0.940105i \(-0.610727\pi\)
−0.340886 + 0.940105i \(0.610727\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.1364 0.961120 0.480560 0.876962i \(-0.340433\pi\)
0.480560 + 0.876962i \(0.340433\pi\)
\(858\) 0 0
\(859\) 33.5760 1.14560 0.572799 0.819696i \(-0.305857\pi\)
0.572799 + 0.819696i \(0.305857\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −32.8724 −1.11899 −0.559495 0.828834i \(-0.689005\pi\)
−0.559495 + 0.828834i \(0.689005\pi\)
\(864\) 0 0
\(865\) −4.94137 −0.168012
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 41.4656 1.40501
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.30777 0.111823
\(876\) 0 0
\(877\) −24.1579 −0.815753 −0.407876 0.913037i \(-0.633731\pi\)
−0.407876 + 0.913037i \(0.633731\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −32.4914 −1.09466 −0.547332 0.836916i \(-0.684356\pi\)
−0.547332 + 0.836916i \(0.684356\pi\)
\(882\) 0 0
\(883\) 26.3404 0.886426 0.443213 0.896416i \(-0.353839\pi\)
0.443213 + 0.896416i \(0.353839\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −0.996562 −0.0334613 −0.0167306 0.999860i \(-0.505326\pi\)
−0.0167306 + 0.999860i \(0.505326\pi\)
\(888\) 0 0
\(889\) 33.4656 1.12240
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −9.55691 −0.319810
\(894\) 0 0
\(895\) 2.61555 0.0874281
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 31.6121 1.05432
\(900\) 0 0
\(901\) −12.6516 −0.421487
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 17.6121 0.585446
\(906\) 0 0
\(907\) 43.1430 1.43254 0.716271 0.697823i \(-0.245848\pi\)
0.716271 + 0.697823i \(0.245848\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −19.6413 −0.650746 −0.325373 0.945586i \(-0.605490\pi\)
−0.325373 + 0.945586i \(0.605490\pi\)
\(912\) 0 0
\(913\) −27.3776 −0.906066
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5.01567 −0.165632
\(918\) 0 0
\(919\) −37.8827 −1.24964 −0.624818 0.780770i \(-0.714827\pi\)
−0.624818 + 0.780770i \(0.714827\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −33.1207 −1.09018
\(924\) 0 0
\(925\) −5.30777 −0.174519
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 5.39133 0.176884 0.0884419 0.996081i \(-0.471811\pi\)
0.0884419 + 0.996081i \(0.471811\pi\)
\(930\) 0 0
\(931\) −3.94137 −0.129173
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 26.7328 0.873323 0.436661 0.899626i \(-0.356161\pi\)
0.436661 + 0.899626i \(0.356161\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 4.35953 0.142117 0.0710583 0.997472i \(-0.477362\pi\)
0.0710583 + 0.997472i \(0.477362\pi\)
\(942\) 0 0
\(943\) 3.68429 0.119977
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −31.9379 −1.03784 −0.518922 0.854822i \(-0.673666\pi\)
−0.518922 + 0.854822i \(0.673666\pi\)
\(948\) 0 0
\(949\) 0.374139 0.0121451
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.82754 −0.0591998 −0.0295999 0.999562i \(-0.509423\pi\)
−0.0295999 + 0.999562i \(0.509423\pi\)
\(954\) 0 0
\(955\) 11.7440 0.380026
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −39.3054 −1.26924
\(960\) 0 0
\(961\) 24.3484 0.785431
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0.574960 0.0185086
\(966\) 0 0
\(967\) −0.926759 −0.0298026 −0.0149013 0.999889i \(-0.504743\pi\)
−0.0149013 + 0.999889i \(0.504743\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 58.1035 1.86463 0.932315 0.361647i \(-0.117785\pi\)
0.932315 + 0.361647i \(0.117785\pi\)
\(972\) 0 0
\(973\) −54.5726 −1.74952
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.02598 0.256774 0.128387 0.991724i \(-0.459020\pi\)
0.128387 + 0.991724i \(0.459020\pi\)
\(978\) 0 0
\(979\) 23.9087 0.764126
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 44.8103 1.42923 0.714614 0.699519i \(-0.246602\pi\)
0.714614 + 0.699519i \(0.246602\pi\)
\(984\) 0 0
\(985\) −11.5569 −0.368234
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −5.92570 −0.188426
\(990\) 0 0
\(991\) −36.7620 −1.16778 −0.583892 0.811831i \(-0.698471\pi\)
−0.583892 + 0.811831i \(0.698471\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3.50172 −0.111012
\(996\) 0 0
\(997\) 26.3741 0.835277 0.417639 0.908613i \(-0.362858\pi\)
0.417639 + 0.908613i \(0.362858\pi\)
\(998\) 0 0
\(999\) 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 6840.2.a.bf.1.1 3
3.2 odd 2 2280.2.a.s.1.1 3
12.11 even 2 4560.2.a.bv.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.s.1.1 3 3.2 odd 2
4560.2.a.bv.1.3 3 12.11 even 2
6840.2.a.bf.1.1 3 1.1 even 1 trivial