Properties

Label 7800.2.a.y.1.1
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.73205 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.73205 q^{7} +1.00000 q^{9} -1.73205 q^{11} +1.00000 q^{13} +4.46410 q^{17} +3.46410 q^{19} +1.73205 q^{21} +2.00000 q^{23} -1.00000 q^{27} -9.92820 q^{29} +1.19615 q^{31} +1.73205 q^{33} -7.46410 q^{37} -1.00000 q^{39} -9.46410 q^{41} -4.92820 q^{43} +10.6603 q^{47} -4.00000 q^{49} -4.46410 q^{51} +1.92820 q^{53} -3.46410 q^{57} -1.73205 q^{59} -5.53590 q^{61} -1.73205 q^{63} +7.73205 q^{67} -2.00000 q^{69} +6.39230 q^{71} +3.46410 q^{73} +3.00000 q^{77} +14.0000 q^{79} +1.00000 q^{81} +10.2679 q^{83} +9.92820 q^{87} +10.9282 q^{89} -1.73205 q^{91} -1.19615 q^{93} -8.92820 q^{97} -1.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} + 2 q^{13} + 2 q^{17} + 4 q^{23} - 2 q^{27} - 6 q^{29} - 8 q^{31} - 8 q^{37} - 2 q^{39} - 12 q^{41} + 4 q^{43} + 4 q^{47} - 8 q^{49} - 2 q^{51} - 10 q^{53} - 18 q^{61} + 12 q^{67} - 4 q^{69} - 8 q^{71} + 6 q^{77} + 28 q^{79} + 2 q^{81} + 24 q^{83} + 6 q^{87} + 8 q^{89} + 8 q^{93} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.73205 −0.654654 −0.327327 0.944911i \(-0.606148\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.73205 −0.522233 −0.261116 0.965307i \(-0.584091\pi\)
−0.261116 + 0.965307i \(0.584091\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.46410 1.08270 0.541352 0.840796i \(-0.317913\pi\)
0.541352 + 0.840796i \(0.317913\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 1.73205 0.377964
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −9.92820 −1.84362 −0.921811 0.387641i \(-0.873290\pi\)
−0.921811 + 0.387641i \(0.873290\pi\)
\(30\) 0 0
\(31\) 1.19615 0.214835 0.107418 0.994214i \(-0.465742\pi\)
0.107418 + 0.994214i \(0.465742\pi\)
\(32\) 0 0
\(33\) 1.73205 0.301511
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.46410 −1.22709 −0.613545 0.789659i \(-0.710257\pi\)
−0.613545 + 0.789659i \(0.710257\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −9.46410 −1.47804 −0.739022 0.673681i \(-0.764712\pi\)
−0.739022 + 0.673681i \(0.764712\pi\)
\(42\) 0 0
\(43\) −4.92820 −0.751544 −0.375772 0.926712i \(-0.622622\pi\)
−0.375772 + 0.926712i \(0.622622\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.6603 1.55496 0.777479 0.628909i \(-0.216498\pi\)
0.777479 + 0.628909i \(0.216498\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) −4.46410 −0.625099
\(52\) 0 0
\(53\) 1.92820 0.264859 0.132430 0.991192i \(-0.457722\pi\)
0.132430 + 0.991192i \(0.457722\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.46410 −0.458831
\(58\) 0 0
\(59\) −1.73205 −0.225494 −0.112747 0.993624i \(-0.535965\pi\)
−0.112747 + 0.993624i \(0.535965\pi\)
\(60\) 0 0
\(61\) −5.53590 −0.708799 −0.354400 0.935094i \(-0.615315\pi\)
−0.354400 + 0.935094i \(0.615315\pi\)
\(62\) 0 0
\(63\) −1.73205 −0.218218
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.73205 0.944620 0.472310 0.881432i \(-0.343420\pi\)
0.472310 + 0.881432i \(0.343420\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) 6.39230 0.758627 0.379314 0.925268i \(-0.376160\pi\)
0.379314 + 0.925268i \(0.376160\pi\)
\(72\) 0 0
\(73\) 3.46410 0.405442 0.202721 0.979236i \(-0.435021\pi\)
0.202721 + 0.979236i \(0.435021\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.2679 1.12705 0.563527 0.826098i \(-0.309444\pi\)
0.563527 + 0.826098i \(0.309444\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.92820 1.06442
\(88\) 0 0
\(89\) 10.9282 1.15839 0.579194 0.815190i \(-0.303368\pi\)
0.579194 + 0.815190i \(0.303368\pi\)
\(90\) 0 0
\(91\) −1.73205 −0.181568
\(92\) 0 0
\(93\) −1.19615 −0.124035
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.92820 −0.906522 −0.453261 0.891378i \(-0.649739\pi\)
−0.453261 + 0.891378i \(0.649739\pi\)
\(98\) 0 0
\(99\) −1.73205 −0.174078
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) 19.3205 1.90371 0.951853 0.306554i \(-0.0991762\pi\)
0.951853 + 0.306554i \(0.0991762\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.535898 −0.0518073 −0.0259036 0.999664i \(-0.508246\pi\)
−0.0259036 + 0.999664i \(0.508246\pi\)
\(108\) 0 0
\(109\) −9.46410 −0.906497 −0.453248 0.891384i \(-0.649735\pi\)
−0.453248 + 0.891384i \(0.649735\pi\)
\(110\) 0 0
\(111\) 7.46410 0.708461
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −7.73205 −0.708796
\(120\) 0 0
\(121\) −8.00000 −0.727273
\(122\) 0 0
\(123\) 9.46410 0.853349
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −12.9282 −1.14719 −0.573596 0.819138i \(-0.694452\pi\)
−0.573596 + 0.819138i \(0.694452\pi\)
\(128\) 0 0
\(129\) 4.92820 0.433904
\(130\) 0 0
\(131\) −14.5359 −1.27001 −0.635004 0.772509i \(-0.719001\pi\)
−0.635004 + 0.772509i \(0.719001\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.9282 −0.933659 −0.466830 0.884347i \(-0.654604\pi\)
−0.466830 + 0.884347i \(0.654604\pi\)
\(138\) 0 0
\(139\) 6.39230 0.542188 0.271094 0.962553i \(-0.412615\pi\)
0.271094 + 0.962553i \(0.412615\pi\)
\(140\) 0 0
\(141\) −10.6603 −0.897755
\(142\) 0 0
\(143\) −1.73205 −0.144841
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.00000 0.329914
\(148\) 0 0
\(149\) −15.8564 −1.29901 −0.649504 0.760358i \(-0.725023\pi\)
−0.649504 + 0.760358i \(0.725023\pi\)
\(150\) 0 0
\(151\) −16.1244 −1.31218 −0.656091 0.754682i \(-0.727791\pi\)
−0.656091 + 0.754682i \(0.727791\pi\)
\(152\) 0 0
\(153\) 4.46410 0.360901
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.3923 0.909205 0.454602 0.890694i \(-0.349781\pi\)
0.454602 + 0.890694i \(0.349781\pi\)
\(158\) 0 0
\(159\) −1.92820 −0.152916
\(160\) 0 0
\(161\) −3.46410 −0.273009
\(162\) 0 0
\(163\) 14.3923 1.12729 0.563646 0.826016i \(-0.309398\pi\)
0.563646 + 0.826016i \(0.309398\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.3205 −1.34030 −0.670151 0.742225i \(-0.733770\pi\)
−0.670151 + 0.742225i \(0.733770\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 3.46410 0.264906
\(172\) 0 0
\(173\) −9.92820 −0.754827 −0.377414 0.926045i \(-0.623187\pi\)
−0.377414 + 0.926045i \(0.623187\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.73205 0.130189
\(178\) 0 0
\(179\) 20.3923 1.52419 0.762096 0.647464i \(-0.224170\pi\)
0.762096 + 0.647464i \(0.224170\pi\)
\(180\) 0 0
\(181\) −10.4641 −0.777791 −0.388895 0.921282i \(-0.627143\pi\)
−0.388895 + 0.921282i \(0.627143\pi\)
\(182\) 0 0
\(183\) 5.53590 0.409225
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −7.73205 −0.565424
\(188\) 0 0
\(189\) 1.73205 0.125988
\(190\) 0 0
\(191\) −6.39230 −0.462531 −0.231265 0.972891i \(-0.574287\pi\)
−0.231265 + 0.972891i \(0.574287\pi\)
\(192\) 0 0
\(193\) −5.46410 −0.393315 −0.196657 0.980472i \(-0.563009\pi\)
−0.196657 + 0.980472i \(0.563009\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −16.3923 −1.16790 −0.583952 0.811788i \(-0.698494\pi\)
−0.583952 + 0.811788i \(0.698494\pi\)
\(198\) 0 0
\(199\) −13.3205 −0.944266 −0.472133 0.881527i \(-0.656516\pi\)
−0.472133 + 0.881527i \(0.656516\pi\)
\(200\) 0 0
\(201\) −7.73205 −0.545377
\(202\) 0 0
\(203\) 17.1962 1.20693
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −26.0000 −1.78991 −0.894957 0.446153i \(-0.852794\pi\)
−0.894957 + 0.446153i \(0.852794\pi\)
\(212\) 0 0
\(213\) −6.39230 −0.437994
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.07180 −0.140643
\(218\) 0 0
\(219\) −3.46410 −0.234082
\(220\) 0 0
\(221\) 4.46410 0.300288
\(222\) 0 0
\(223\) −20.2487 −1.35595 −0.677977 0.735083i \(-0.737143\pi\)
−0.677977 + 0.735083i \(0.737143\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.73205 −0.380450 −0.190225 0.981741i \(-0.560922\pi\)
−0.190225 + 0.981741i \(0.560922\pi\)
\(228\) 0 0
\(229\) 22.7846 1.50565 0.752825 0.658221i \(-0.228691\pi\)
0.752825 + 0.658221i \(0.228691\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −7.85641 −0.514690 −0.257345 0.966320i \(-0.582848\pi\)
−0.257345 + 0.966320i \(0.582848\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −14.0000 −0.909398
\(238\) 0 0
\(239\) −7.19615 −0.465480 −0.232740 0.972539i \(-0.574769\pi\)
−0.232740 + 0.972539i \(0.574769\pi\)
\(240\) 0 0
\(241\) 0.392305 0.0252706 0.0126353 0.999920i \(-0.495978\pi\)
0.0126353 + 0.999920i \(0.495978\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.46410 0.220416
\(248\) 0 0
\(249\) −10.2679 −0.650705
\(250\) 0 0
\(251\) −16.3923 −1.03467 −0.517337 0.855782i \(-0.673076\pi\)
−0.517337 + 0.855782i \(0.673076\pi\)
\(252\) 0 0
\(253\) −3.46410 −0.217786
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 29.2487 1.82448 0.912242 0.409651i \(-0.134350\pi\)
0.912242 + 0.409651i \(0.134350\pi\)
\(258\) 0 0
\(259\) 12.9282 0.803319
\(260\) 0 0
\(261\) −9.92820 −0.614540
\(262\) 0 0
\(263\) −11.3205 −0.698052 −0.349026 0.937113i \(-0.613488\pi\)
−0.349026 + 0.937113i \(0.613488\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −10.9282 −0.668795
\(268\) 0 0
\(269\) −19.0000 −1.15845 −0.579225 0.815168i \(-0.696645\pi\)
−0.579225 + 0.815168i \(0.696645\pi\)
\(270\) 0 0
\(271\) −18.2679 −1.10970 −0.554849 0.831951i \(-0.687224\pi\)
−0.554849 + 0.831951i \(0.687224\pi\)
\(272\) 0 0
\(273\) 1.73205 0.104828
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7.07180 −0.424903 −0.212452 0.977172i \(-0.568145\pi\)
−0.212452 + 0.977172i \(0.568145\pi\)
\(278\) 0 0
\(279\) 1.19615 0.0716118
\(280\) 0 0
\(281\) −0.928203 −0.0553720 −0.0276860 0.999617i \(-0.508814\pi\)
−0.0276860 + 0.999617i \(0.508814\pi\)
\(282\) 0 0
\(283\) −25.3205 −1.50515 −0.752574 0.658508i \(-0.771188\pi\)
−0.752574 + 0.658508i \(0.771188\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 16.3923 0.967607
\(288\) 0 0
\(289\) 2.92820 0.172247
\(290\) 0 0
\(291\) 8.92820 0.523381
\(292\) 0 0
\(293\) 11.8564 0.692659 0.346329 0.938113i \(-0.387428\pi\)
0.346329 + 0.938113i \(0.387428\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.73205 0.100504
\(298\) 0 0
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 8.53590 0.492001
\(302\) 0 0
\(303\) −3.00000 −0.172345
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 11.4641 0.654291 0.327145 0.944974i \(-0.393913\pi\)
0.327145 + 0.944974i \(0.393913\pi\)
\(308\) 0 0
\(309\) −19.3205 −1.09911
\(310\) 0 0
\(311\) −0.535898 −0.0303880 −0.0151940 0.999885i \(-0.504837\pi\)
−0.0151940 + 0.999885i \(0.504837\pi\)
\(312\) 0 0
\(313\) 11.9282 0.674222 0.337111 0.941465i \(-0.390550\pi\)
0.337111 + 0.941465i \(0.390550\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −19.4641 −1.09321 −0.546606 0.837390i \(-0.684081\pi\)
−0.546606 + 0.837390i \(0.684081\pi\)
\(318\) 0 0
\(319\) 17.1962 0.962800
\(320\) 0 0
\(321\) 0.535898 0.0299109
\(322\) 0 0
\(323\) 15.4641 0.860446
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 9.46410 0.523366
\(328\) 0 0
\(329\) −18.4641 −1.01796
\(330\) 0 0
\(331\) −11.4641 −0.630124 −0.315062 0.949071i \(-0.602025\pi\)
−0.315062 + 0.949071i \(0.602025\pi\)
\(332\) 0 0
\(333\) −7.46410 −0.409030
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −30.8564 −1.68086 −0.840428 0.541924i \(-0.817696\pi\)
−0.840428 + 0.541924i \(0.817696\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) −2.07180 −0.112194
\(342\) 0 0
\(343\) 19.0526 1.02874
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −32.2487 −1.73120 −0.865601 0.500735i \(-0.833063\pi\)
−0.865601 + 0.500735i \(0.833063\pi\)
\(348\) 0 0
\(349\) 6.39230 0.342172 0.171086 0.985256i \(-0.445272\pi\)
0.171086 + 0.985256i \(0.445272\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 27.3205 1.45412 0.727062 0.686572i \(-0.240885\pi\)
0.727062 + 0.686572i \(0.240885\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 7.73205 0.409224
\(358\) 0 0
\(359\) 0.267949 0.0141418 0.00707091 0.999975i \(-0.497749\pi\)
0.00707091 + 0.999975i \(0.497749\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) 0 0
\(363\) 8.00000 0.419891
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −6.92820 −0.361649 −0.180825 0.983515i \(-0.557877\pi\)
−0.180825 + 0.983515i \(0.557877\pi\)
\(368\) 0 0
\(369\) −9.46410 −0.492681
\(370\) 0 0
\(371\) −3.33975 −0.173391
\(372\) 0 0
\(373\) −0.607695 −0.0314653 −0.0157326 0.999876i \(-0.505008\pi\)
−0.0157326 + 0.999876i \(0.505008\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9.92820 −0.511328
\(378\) 0 0
\(379\) 15.1962 0.780574 0.390287 0.920693i \(-0.372376\pi\)
0.390287 + 0.920693i \(0.372376\pi\)
\(380\) 0 0
\(381\) 12.9282 0.662332
\(382\) 0 0
\(383\) 15.4641 0.790179 0.395089 0.918643i \(-0.370714\pi\)
0.395089 + 0.918643i \(0.370714\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.92820 −0.250515
\(388\) 0 0
\(389\) 19.8564 1.00676 0.503380 0.864065i \(-0.332090\pi\)
0.503380 + 0.864065i \(0.332090\pi\)
\(390\) 0 0
\(391\) 8.92820 0.451519
\(392\) 0 0
\(393\) 14.5359 0.733239
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) 38.1051 1.90288 0.951439 0.307836i \(-0.0996049\pi\)
0.951439 + 0.307836i \(0.0996049\pi\)
\(402\) 0 0
\(403\) 1.19615 0.0595846
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.9282 0.640827
\(408\) 0 0
\(409\) −23.3205 −1.15312 −0.576562 0.817053i \(-0.695606\pi\)
−0.576562 + 0.817053i \(0.695606\pi\)
\(410\) 0 0
\(411\) 10.9282 0.539049
\(412\) 0 0
\(413\) 3.00000 0.147620
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6.39230 −0.313033
\(418\) 0 0
\(419\) 11.3205 0.553043 0.276522 0.961008i \(-0.410818\pi\)
0.276522 + 0.961008i \(0.410818\pi\)
\(420\) 0 0
\(421\) −18.9282 −0.922504 −0.461252 0.887269i \(-0.652600\pi\)
−0.461252 + 0.887269i \(0.652600\pi\)
\(422\) 0 0
\(423\) 10.6603 0.518319
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 9.58846 0.464018
\(428\) 0 0
\(429\) 1.73205 0.0836242
\(430\) 0 0
\(431\) −18.3923 −0.885926 −0.442963 0.896540i \(-0.646073\pi\)
−0.442963 + 0.896540i \(0.646073\pi\)
\(432\) 0 0
\(433\) −12.9282 −0.621290 −0.310645 0.950526i \(-0.600545\pi\)
−0.310645 + 0.950526i \(0.600545\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.92820 0.331421
\(438\) 0 0
\(439\) −30.7846 −1.46927 −0.734635 0.678463i \(-0.762646\pi\)
−0.734635 + 0.678463i \(0.762646\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) −4.92820 −0.234146 −0.117073 0.993123i \(-0.537351\pi\)
−0.117073 + 0.993123i \(0.537351\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 15.8564 0.749982
\(448\) 0 0
\(449\) −21.0718 −0.994440 −0.497220 0.867625i \(-0.665646\pi\)
−0.497220 + 0.867625i \(0.665646\pi\)
\(450\) 0 0
\(451\) 16.3923 0.771883
\(452\) 0 0
\(453\) 16.1244 0.757588
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −23.3205 −1.09089 −0.545444 0.838147i \(-0.683639\pi\)
−0.545444 + 0.838147i \(0.683639\pi\)
\(458\) 0 0
\(459\) −4.46410 −0.208366
\(460\) 0 0
\(461\) −20.9282 −0.974724 −0.487362 0.873200i \(-0.662041\pi\)
−0.487362 + 0.873200i \(0.662041\pi\)
\(462\) 0 0
\(463\) −9.73205 −0.452287 −0.226143 0.974094i \(-0.572612\pi\)
−0.226143 + 0.974094i \(0.572612\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.85641 0.0859042 0.0429521 0.999077i \(-0.486324\pi\)
0.0429521 + 0.999077i \(0.486324\pi\)
\(468\) 0 0
\(469\) −13.3923 −0.618399
\(470\) 0 0
\(471\) −11.3923 −0.524930
\(472\) 0 0
\(473\) 8.53590 0.392481
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.92820 0.0882864
\(478\) 0 0
\(479\) 13.0526 0.596387 0.298193 0.954505i \(-0.403616\pi\)
0.298193 + 0.954505i \(0.403616\pi\)
\(480\) 0 0
\(481\) −7.46410 −0.340334
\(482\) 0 0
\(483\) 3.46410 0.157622
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −17.1962 −0.779232 −0.389616 0.920977i \(-0.627392\pi\)
−0.389616 + 0.920977i \(0.627392\pi\)
\(488\) 0 0
\(489\) −14.3923 −0.650843
\(490\) 0 0
\(491\) −37.3205 −1.68425 −0.842125 0.539282i \(-0.818696\pi\)
−0.842125 + 0.539282i \(0.818696\pi\)
\(492\) 0 0
\(493\) −44.3205 −1.99610
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −11.0718 −0.496638
\(498\) 0 0
\(499\) −21.5885 −0.966432 −0.483216 0.875501i \(-0.660531\pi\)
−0.483216 + 0.875501i \(0.660531\pi\)
\(500\) 0 0
\(501\) 17.3205 0.773823
\(502\) 0 0
\(503\) 21.4641 0.957037 0.478518 0.878077i \(-0.341174\pi\)
0.478518 + 0.878077i \(0.341174\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 34.2487 1.51805 0.759024 0.651063i \(-0.225677\pi\)
0.759024 + 0.651063i \(0.225677\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 0 0
\(513\) −3.46410 −0.152944
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −18.4641 −0.812050
\(518\) 0 0
\(519\) 9.92820 0.435800
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 13.3205 0.582465 0.291233 0.956652i \(-0.405935\pi\)
0.291233 + 0.956652i \(0.405935\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.33975 0.232603
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) −1.73205 −0.0751646
\(532\) 0 0
\(533\) −9.46410 −0.409936
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −20.3923 −0.879993
\(538\) 0 0
\(539\) 6.92820 0.298419
\(540\) 0 0
\(541\) 2.39230 0.102853 0.0514266 0.998677i \(-0.483623\pi\)
0.0514266 + 0.998677i \(0.483623\pi\)
\(542\) 0 0
\(543\) 10.4641 0.449058
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.53590 −0.108427 −0.0542136 0.998529i \(-0.517265\pi\)
−0.0542136 + 0.998529i \(0.517265\pi\)
\(548\) 0 0
\(549\) −5.53590 −0.236266
\(550\) 0 0
\(551\) −34.3923 −1.46516
\(552\) 0 0
\(553\) −24.2487 −1.03116
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −11.6077 −0.491834 −0.245917 0.969291i \(-0.579089\pi\)
−0.245917 + 0.969291i \(0.579089\pi\)
\(558\) 0 0
\(559\) −4.92820 −0.208441
\(560\) 0 0
\(561\) 7.73205 0.326447
\(562\) 0 0
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.73205 −0.0727393
\(568\) 0 0
\(569\) −29.2487 −1.22617 −0.613085 0.790017i \(-0.710072\pi\)
−0.613085 + 0.790017i \(0.710072\pi\)
\(570\) 0 0
\(571\) −0.143594 −0.00600920 −0.00300460 0.999995i \(-0.500956\pi\)
−0.00300460 + 0.999995i \(0.500956\pi\)
\(572\) 0 0
\(573\) 6.39230 0.267042
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 36.6410 1.52539 0.762693 0.646761i \(-0.223877\pi\)
0.762693 + 0.646761i \(0.223877\pi\)
\(578\) 0 0
\(579\) 5.46410 0.227080
\(580\) 0 0
\(581\) −17.7846 −0.737830
\(582\) 0 0
\(583\) −3.33975 −0.138318
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.58846 −0.313209 −0.156605 0.987661i \(-0.550055\pi\)
−0.156605 + 0.987661i \(0.550055\pi\)
\(588\) 0 0
\(589\) 4.14359 0.170734
\(590\) 0 0
\(591\) 16.3923 0.674289
\(592\) 0 0
\(593\) 0.928203 0.0381167 0.0190584 0.999818i \(-0.493933\pi\)
0.0190584 + 0.999818i \(0.493933\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13.3205 0.545172
\(598\) 0 0
\(599\) −33.1769 −1.35557 −0.677786 0.735259i \(-0.737060\pi\)
−0.677786 + 0.735259i \(0.737060\pi\)
\(600\) 0 0
\(601\) −32.7128 −1.33438 −0.667192 0.744886i \(-0.732504\pi\)
−0.667192 + 0.744886i \(0.732504\pi\)
\(602\) 0 0
\(603\) 7.73205 0.314873
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 45.3205 1.83950 0.919751 0.392502i \(-0.128390\pi\)
0.919751 + 0.392502i \(0.128390\pi\)
\(608\) 0 0
\(609\) −17.1962 −0.696823
\(610\) 0 0
\(611\) 10.6603 0.431268
\(612\) 0 0
\(613\) 24.9282 1.00684 0.503420 0.864042i \(-0.332075\pi\)
0.503420 + 0.864042i \(0.332075\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.8564 0.799389 0.399694 0.916648i \(-0.369116\pi\)
0.399694 + 0.916648i \(0.369116\pi\)
\(618\) 0 0
\(619\) −9.32051 −0.374623 −0.187311 0.982301i \(-0.559977\pi\)
−0.187311 + 0.982301i \(0.559977\pi\)
\(620\) 0 0
\(621\) −2.00000 −0.0802572
\(622\) 0 0
\(623\) −18.9282 −0.758342
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 6.00000 0.239617
\(628\) 0 0
\(629\) −33.3205 −1.32858
\(630\) 0 0
\(631\) −8.24871 −0.328376 −0.164188 0.986429i \(-0.552500\pi\)
−0.164188 + 0.986429i \(0.552500\pi\)
\(632\) 0 0
\(633\) 26.0000 1.03341
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) 6.39230 0.252876
\(640\) 0 0
\(641\) 28.6077 1.12994 0.564968 0.825113i \(-0.308888\pi\)
0.564968 + 0.825113i \(0.308888\pi\)
\(642\) 0 0
\(643\) 13.3205 0.525310 0.262655 0.964890i \(-0.415402\pi\)
0.262655 + 0.964890i \(0.415402\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.71281 0.303222 0.151611 0.988440i \(-0.451554\pi\)
0.151611 + 0.988440i \(0.451554\pi\)
\(648\) 0 0
\(649\) 3.00000 0.117760
\(650\) 0 0
\(651\) 2.07180 0.0812001
\(652\) 0 0
\(653\) −22.0718 −0.863736 −0.431868 0.901937i \(-0.642145\pi\)
−0.431868 + 0.901937i \(0.642145\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.46410 0.135147
\(658\) 0 0
\(659\) −25.7128 −1.00163 −0.500814 0.865555i \(-0.666966\pi\)
−0.500814 + 0.865555i \(0.666966\pi\)
\(660\) 0 0
\(661\) −15.6077 −0.607069 −0.303534 0.952820i \(-0.598167\pi\)
−0.303534 + 0.952820i \(0.598167\pi\)
\(662\) 0 0
\(663\) −4.46410 −0.173371
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −19.8564 −0.768843
\(668\) 0 0
\(669\) 20.2487 0.782860
\(670\) 0 0
\(671\) 9.58846 0.370158
\(672\) 0 0
\(673\) −11.1436 −0.429554 −0.214777 0.976663i \(-0.568902\pi\)
−0.214777 + 0.976663i \(0.568902\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −28.6410 −1.10076 −0.550382 0.834913i \(-0.685518\pi\)
−0.550382 + 0.834913i \(0.685518\pi\)
\(678\) 0 0
\(679\) 15.4641 0.593458
\(680\) 0 0
\(681\) 5.73205 0.219653
\(682\) 0 0
\(683\) 4.94744 0.189309 0.0946543 0.995510i \(-0.469825\pi\)
0.0946543 + 0.995510i \(0.469825\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −22.7846 −0.869287
\(688\) 0 0
\(689\) 1.92820 0.0734587
\(690\) 0 0
\(691\) −13.8756 −0.527854 −0.263927 0.964543i \(-0.585018\pi\)
−0.263927 + 0.964543i \(0.585018\pi\)
\(692\) 0 0
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −42.2487 −1.60028
\(698\) 0 0
\(699\) 7.85641 0.297157
\(700\) 0 0
\(701\) 28.8564 1.08989 0.544946 0.838471i \(-0.316550\pi\)
0.544946 + 0.838471i \(0.316550\pi\)
\(702\) 0 0
\(703\) −25.8564 −0.975193
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.19615 −0.195421
\(708\) 0 0
\(709\) −10.5359 −0.395684 −0.197842 0.980234i \(-0.563393\pi\)
−0.197842 + 0.980234i \(0.563393\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 0 0
\(713\) 2.39230 0.0895925
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7.19615 0.268745
\(718\) 0 0
\(719\) −6.53590 −0.243748 −0.121874 0.992546i \(-0.538890\pi\)
−0.121874 + 0.992546i \(0.538890\pi\)
\(720\) 0 0
\(721\) −33.4641 −1.24627
\(722\) 0 0
\(723\) −0.392305 −0.0145900
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −26.2487 −0.973511 −0.486755 0.873538i \(-0.661820\pi\)
−0.486755 + 0.873538i \(0.661820\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −22.0000 −0.813699
\(732\) 0 0
\(733\) −24.7846 −0.915440 −0.457720 0.889096i \(-0.651334\pi\)
−0.457720 + 0.889096i \(0.651334\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.3923 −0.493312
\(738\) 0 0
\(739\) 39.7321 1.46157 0.730784 0.682609i \(-0.239155\pi\)
0.730784 + 0.682609i \(0.239155\pi\)
\(740\) 0 0
\(741\) −3.46410 −0.127257
\(742\) 0 0
\(743\) 43.1962 1.58471 0.792357 0.610058i \(-0.208854\pi\)
0.792357 + 0.610058i \(0.208854\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 10.2679 0.375685
\(748\) 0 0
\(749\) 0.928203 0.0339158
\(750\) 0 0
\(751\) 11.8564 0.432646 0.216323 0.976322i \(-0.430594\pi\)
0.216323 + 0.976322i \(0.430594\pi\)
\(752\) 0 0
\(753\) 16.3923 0.597369
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −27.5359 −1.00081 −0.500405 0.865792i \(-0.666815\pi\)
−0.500405 + 0.865792i \(0.666815\pi\)
\(758\) 0 0
\(759\) 3.46410 0.125739
\(760\) 0 0
\(761\) 17.6077 0.638278 0.319139 0.947708i \(-0.396606\pi\)
0.319139 + 0.947708i \(0.396606\pi\)
\(762\) 0 0
\(763\) 16.3923 0.593441
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.73205 −0.0625407
\(768\) 0 0
\(769\) −20.7846 −0.749512 −0.374756 0.927123i \(-0.622274\pi\)
−0.374756 + 0.927123i \(0.622274\pi\)
\(770\) 0 0
\(771\) −29.2487 −1.05337
\(772\) 0 0
\(773\) −15.8564 −0.570315 −0.285158 0.958481i \(-0.592046\pi\)
−0.285158 + 0.958481i \(0.592046\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −12.9282 −0.463797
\(778\) 0 0
\(779\) −32.7846 −1.17463
\(780\) 0 0
\(781\) −11.0718 −0.396180
\(782\) 0 0
\(783\) 9.92820 0.354805
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 23.1962 0.826854 0.413427 0.910537i \(-0.364332\pi\)
0.413427 + 0.910537i \(0.364332\pi\)
\(788\) 0 0
\(789\) 11.3205 0.403021
\(790\) 0 0
\(791\) −17.3205 −0.615846
\(792\) 0 0
\(793\) −5.53590 −0.196586
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.85641 −0.172023 −0.0860114 0.996294i \(-0.527412\pi\)
−0.0860114 + 0.996294i \(0.527412\pi\)
\(798\) 0 0
\(799\) 47.5885 1.68356
\(800\) 0 0
\(801\) 10.9282 0.386129
\(802\) 0 0
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19.0000 0.668832
\(808\) 0 0
\(809\) 24.9282 0.876429 0.438214 0.898870i \(-0.355611\pi\)
0.438214 + 0.898870i \(0.355611\pi\)
\(810\) 0 0
\(811\) −29.5885 −1.03899 −0.519496 0.854473i \(-0.673880\pi\)
−0.519496 + 0.854473i \(0.673880\pi\)
\(812\) 0 0
\(813\) 18.2679 0.640685
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −17.0718 −0.597267
\(818\) 0 0
\(819\) −1.73205 −0.0605228
\(820\) 0 0
\(821\) 20.5359 0.716708 0.358354 0.933586i \(-0.383338\pi\)
0.358354 + 0.933586i \(0.383338\pi\)
\(822\) 0 0
\(823\) −37.8564 −1.31959 −0.659796 0.751445i \(-0.729357\pi\)
−0.659796 + 0.751445i \(0.729357\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −4.12436 −0.143418 −0.0717089 0.997426i \(-0.522845\pi\)
−0.0717089 + 0.997426i \(0.522845\pi\)
\(828\) 0 0
\(829\) 35.3923 1.22923 0.614613 0.788829i \(-0.289312\pi\)
0.614613 + 0.788829i \(0.289312\pi\)
\(830\) 0 0
\(831\) 7.07180 0.245318
\(832\) 0 0
\(833\) −17.8564 −0.618688
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.19615 −0.0413451
\(838\) 0 0
\(839\) 5.32051 0.183684 0.0918422 0.995774i \(-0.470724\pi\)
0.0918422 + 0.995774i \(0.470724\pi\)
\(840\) 0 0
\(841\) 69.5692 2.39894
\(842\) 0 0
\(843\) 0.928203 0.0319690
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 13.8564 0.476112
\(848\) 0 0
\(849\) 25.3205 0.868998
\(850\) 0 0
\(851\) −14.9282 −0.511732
\(852\) 0 0
\(853\) −7.21539 −0.247050 −0.123525 0.992341i \(-0.539420\pi\)
−0.123525 + 0.992341i \(0.539420\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −33.7128 −1.15161 −0.575804 0.817588i \(-0.695311\pi\)
−0.575804 + 0.817588i \(0.695311\pi\)
\(858\) 0 0
\(859\) 45.7128 1.55970 0.779851 0.625966i \(-0.215295\pi\)
0.779851 + 0.625966i \(0.215295\pi\)
\(860\) 0 0
\(861\) −16.3923 −0.558648
\(862\) 0 0
\(863\) 49.0526 1.66977 0.834884 0.550426i \(-0.185535\pi\)
0.834884 + 0.550426i \(0.185535\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −2.92820 −0.0994470
\(868\) 0 0
\(869\) −24.2487 −0.822581
\(870\) 0 0
\(871\) 7.73205 0.261991
\(872\) 0 0
\(873\) −8.92820 −0.302174
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −33.0718 −1.11676 −0.558378 0.829587i \(-0.688576\pi\)
−0.558378 + 0.829587i \(0.688576\pi\)
\(878\) 0 0
\(879\) −11.8564 −0.399907
\(880\) 0 0
\(881\) 6.46410 0.217781 0.108891 0.994054i \(-0.465270\pi\)
0.108891 + 0.994054i \(0.465270\pi\)
\(882\) 0 0
\(883\) −34.9282 −1.17543 −0.587714 0.809069i \(-0.699972\pi\)
−0.587714 + 0.809069i \(0.699972\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 38.7846 1.30226 0.651130 0.758966i \(-0.274295\pi\)
0.651130 + 0.758966i \(0.274295\pi\)
\(888\) 0 0
\(889\) 22.3923 0.751014
\(890\) 0 0
\(891\) −1.73205 −0.0580259
\(892\) 0 0
\(893\) 36.9282 1.23576
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) 0 0
\(899\) −11.8756 −0.396075
\(900\) 0 0
\(901\) 8.60770 0.286764
\(902\) 0 0
\(903\) −8.53590 −0.284057
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 3.00000 0.0995037
\(910\) 0 0
\(911\) 12.9282 0.428330 0.214165 0.976797i \(-0.431297\pi\)
0.214165 + 0.976797i \(0.431297\pi\)
\(912\) 0 0
\(913\) −17.7846 −0.588585
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 25.1769 0.831415
\(918\) 0 0
\(919\) 33.0333 1.08967 0.544834 0.838544i \(-0.316593\pi\)
0.544834 + 0.838544i \(0.316593\pi\)
\(920\) 0 0
\(921\) −11.4641 −0.377755
\(922\) 0 0
\(923\) 6.39230 0.210405
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 19.3205 0.634569
\(928\) 0 0
\(929\) 16.6410 0.545974 0.272987 0.962018i \(-0.411988\pi\)
0.272987 + 0.962018i \(0.411988\pi\)
\(930\) 0 0
\(931\) −13.8564 −0.454125
\(932\) 0 0
\(933\) 0.535898 0.0175445
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 23.6410 0.772318 0.386159 0.922432i \(-0.373802\pi\)
0.386159 + 0.922432i \(0.373802\pi\)
\(938\) 0 0
\(939\) −11.9282 −0.389262
\(940\) 0 0
\(941\) 52.2487 1.70326 0.851630 0.524144i \(-0.175615\pi\)
0.851630 + 0.524144i \(0.175615\pi\)
\(942\) 0 0
\(943\) −18.9282 −0.616387
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24.6603 0.801351 0.400675 0.916220i \(-0.368776\pi\)
0.400675 + 0.916220i \(0.368776\pi\)
\(948\) 0 0
\(949\) 3.46410 0.112449
\(950\) 0 0
\(951\) 19.4641 0.631167
\(952\) 0 0
\(953\) −19.2487 −0.623527 −0.311763 0.950160i \(-0.600920\pi\)
−0.311763 + 0.950160i \(0.600920\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −17.1962 −0.555873
\(958\) 0 0
\(959\) 18.9282 0.611224
\(960\) 0 0
\(961\) −29.5692 −0.953846
\(962\) 0 0
\(963\) −0.535898 −0.0172691
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 25.4449 0.818252 0.409126 0.912478i \(-0.365834\pi\)
0.409126 + 0.912478i \(0.365834\pi\)
\(968\) 0 0
\(969\) −15.4641 −0.496779
\(970\) 0 0
\(971\) 35.1769 1.12888 0.564440 0.825474i \(-0.309092\pi\)
0.564440 + 0.825474i \(0.309092\pi\)
\(972\) 0 0
\(973\) −11.0718 −0.354946
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 41.5692 1.32992 0.664959 0.746880i \(-0.268449\pi\)
0.664959 + 0.746880i \(0.268449\pi\)
\(978\) 0 0
\(979\) −18.9282 −0.604948
\(980\) 0 0
\(981\) −9.46410 −0.302166
\(982\) 0 0
\(983\) −12.2679 −0.391287 −0.195643 0.980675i \(-0.562680\pi\)
−0.195643 + 0.980675i \(0.562680\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 18.4641 0.587719
\(988\) 0 0
\(989\) −9.85641 −0.313415
\(990\) 0 0
\(991\) 5.71281 0.181473 0.0907367 0.995875i \(-0.471078\pi\)
0.0907367 + 0.995875i \(0.471078\pi\)
\(992\) 0 0
\(993\) 11.4641 0.363802
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −44.0333 −1.39455 −0.697275 0.716804i \(-0.745604\pi\)
−0.697275 + 0.716804i \(0.745604\pi\)
\(998\) 0 0
\(999\) 7.46410 0.236154
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 7800.2.a.y.1.1 2
5.4 even 2 7800.2.a.bc.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.y.1.1 2 1.1 even 1 trivial
7800.2.a.bc.1.2 yes 2 5.4 even 2