Properties

Label 7800.2.a.bb.1.2
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.37228 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.37228 q^{7} +1.00000 q^{9} +2.37228 q^{11} +1.00000 q^{13} -4.37228 q^{17} +4.74456 q^{19} +2.37228 q^{21} +6.37228 q^{23} +1.00000 q^{27} -2.00000 q^{29} -4.74456 q^{31} +2.37228 q^{33} +0.372281 q^{37} +1.00000 q^{39} +4.37228 q^{41} +4.00000 q^{43} +12.7446 q^{47} -1.37228 q^{49} -4.37228 q^{51} +3.62772 q^{53} +4.74456 q^{57} -8.00000 q^{59} -9.11684 q^{61} +2.37228 q^{63} +4.00000 q^{67} +6.37228 q^{69} -5.62772 q^{71} -7.48913 q^{73} +5.62772 q^{77} +11.8614 q^{79} +1.00000 q^{81} +12.0000 q^{83} -2.00000 q^{87} +7.62772 q^{89} +2.37228 q^{91} -4.74456 q^{93} -8.37228 q^{97} +2.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 3 q^{17} - 2 q^{19} - q^{21} + 7 q^{23} + 2 q^{27} - 4 q^{29} + 2 q^{31} - q^{33} - 5 q^{37} + 2 q^{39} + 3 q^{41} + 8 q^{43} + 14 q^{47} + 3 q^{49} - 3 q^{51} + 13 q^{53} - 2 q^{57} - 16 q^{59} - q^{61} - q^{63} + 8 q^{67} + 7 q^{69} - 17 q^{71} + 8 q^{73} + 17 q^{77} - 5 q^{79} + 2 q^{81} + 24 q^{83} - 4 q^{87} + 21 q^{89} - q^{91} + 2 q^{93} - 11 q^{97} - q^{99}+O(q^{100}) \) 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
\(7\) 2.37228 0.896638 0.448319 0.893874i \(-0.352023\pi\)
0.448319 + 0.893874i \(0.352023\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.37228 0.715270 0.357635 0.933862i \(-0.383583\pi\)
0.357635 + 0.933862i \(0.383583\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.37228 −1.06043 −0.530217 0.847862i \(-0.677890\pi\)
−0.530217 + 0.847862i \(0.677890\pi\)
\(18\) 0 0
\(19\) 4.74456 1.08848 0.544239 0.838930i \(-0.316819\pi\)
0.544239 + 0.838930i \(0.316819\pi\)
\(20\) 0 0
\(21\) 2.37228 0.517674
\(22\) 0 0
\(23\) 6.37228 1.32871 0.664356 0.747416i \(-0.268706\pi\)
0.664356 + 0.747416i \(0.268706\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.74456 −0.852149 −0.426074 0.904688i \(-0.640104\pi\)
−0.426074 + 0.904688i \(0.640104\pi\)
\(32\) 0 0
\(33\) 2.37228 0.412961
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.372281 0.0612027 0.0306013 0.999532i \(-0.490258\pi\)
0.0306013 + 0.999532i \(0.490258\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 4.37228 0.682836 0.341418 0.939912i \(-0.389093\pi\)
0.341418 + 0.939912i \(0.389093\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.7446 1.85899 0.929493 0.368840i \(-0.120245\pi\)
0.929493 + 0.368840i \(0.120245\pi\)
\(48\) 0 0
\(49\) −1.37228 −0.196040
\(50\) 0 0
\(51\) −4.37228 −0.612242
\(52\) 0 0
\(53\) 3.62772 0.498305 0.249153 0.968464i \(-0.419848\pi\)
0.249153 + 0.968464i \(0.419848\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.74456 0.628433
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −9.11684 −1.16729 −0.583646 0.812008i \(-0.698374\pi\)
−0.583646 + 0.812008i \(0.698374\pi\)
\(62\) 0 0
\(63\) 2.37228 0.298879
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 6.37228 0.767133
\(70\) 0 0
\(71\) −5.62772 −0.667887 −0.333944 0.942593i \(-0.608380\pi\)
−0.333944 + 0.942593i \(0.608380\pi\)
\(72\) 0 0
\(73\) −7.48913 −0.876536 −0.438268 0.898844i \(-0.644408\pi\)
−0.438268 + 0.898844i \(0.644408\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.62772 0.641338
\(78\) 0 0
\(79\) 11.8614 1.33451 0.667256 0.744828i \(-0.267469\pi\)
0.667256 + 0.744828i \(0.267469\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 7.62772 0.808537 0.404268 0.914640i \(-0.367526\pi\)
0.404268 + 0.914640i \(0.367526\pi\)
\(90\) 0 0
\(91\) 2.37228 0.248683
\(92\) 0 0
\(93\) −4.74456 −0.491988
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.37228 −0.850076 −0.425038 0.905175i \(-0.639739\pi\)
−0.425038 + 0.905175i \(0.639739\pi\)
\(98\) 0 0
\(99\) 2.37228 0.238423
\(100\) 0 0
\(101\) −6.74456 −0.671109 −0.335555 0.942021i \(-0.608924\pi\)
−0.335555 + 0.942021i \(0.608924\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.3723 1.38942 0.694710 0.719290i \(-0.255533\pi\)
0.694710 + 0.719290i \(0.255533\pi\)
\(108\) 0 0
\(109\) −2.74456 −0.262881 −0.131441 0.991324i \(-0.541960\pi\)
−0.131441 + 0.991324i \(0.541960\pi\)
\(110\) 0 0
\(111\) 0.372281 0.0353354
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −10.3723 −0.950825
\(120\) 0 0
\(121\) −5.37228 −0.488389
\(122\) 0 0
\(123\) 4.37228 0.394235
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −18.2337 −1.61798 −0.808989 0.587824i \(-0.799985\pi\)
−0.808989 + 0.587824i \(0.799985\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −7.25544 −0.633911 −0.316955 0.948440i \(-0.602661\pi\)
−0.316955 + 0.948440i \(0.602661\pi\)
\(132\) 0 0
\(133\) 11.2554 0.975970
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.4891 0.981582 0.490791 0.871277i \(-0.336708\pi\)
0.490791 + 0.871277i \(0.336708\pi\)
\(138\) 0 0
\(139\) 1.62772 0.138061 0.0690306 0.997615i \(-0.478009\pi\)
0.0690306 + 0.997615i \(0.478009\pi\)
\(140\) 0 0
\(141\) 12.7446 1.07329
\(142\) 0 0
\(143\) 2.37228 0.198380
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.37228 −0.113184
\(148\) 0 0
\(149\) 7.62772 0.624887 0.312444 0.949936i \(-0.398852\pi\)
0.312444 + 0.949936i \(0.398852\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −4.37228 −0.353478
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 3.62772 0.287697
\(160\) 0 0
\(161\) 15.1168 1.19137
\(162\) 0 0
\(163\) 6.37228 0.499116 0.249558 0.968360i \(-0.419715\pi\)
0.249558 + 0.968360i \(0.419715\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 4.74456 0.362826
\(172\) 0 0
\(173\) 15.4891 1.17762 0.588808 0.808273i \(-0.299597\pi\)
0.588808 + 0.808273i \(0.299597\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −8.00000 −0.601317
\(178\) 0 0
\(179\) −10.2337 −0.764902 −0.382451 0.923976i \(-0.624920\pi\)
−0.382451 + 0.923976i \(0.624920\pi\)
\(180\) 0 0
\(181\) 0.372281 0.0276715 0.0138357 0.999904i \(-0.495596\pi\)
0.0138357 + 0.999904i \(0.495596\pi\)
\(182\) 0 0
\(183\) −9.11684 −0.673936
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −10.3723 −0.758496
\(188\) 0 0
\(189\) 2.37228 0.172558
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −3.62772 −0.261129 −0.130564 0.991440i \(-0.541679\pi\)
−0.130564 + 0.991440i \(0.541679\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −24.2337 −1.72658 −0.863289 0.504709i \(-0.831600\pi\)
−0.863289 + 0.504709i \(0.831600\pi\)
\(198\) 0 0
\(199\) −1.48913 −0.105561 −0.0527806 0.998606i \(-0.516808\pi\)
−0.0527806 + 0.998606i \(0.516808\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) −4.74456 −0.333003
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.37228 0.442904
\(208\) 0 0
\(209\) 11.2554 0.778555
\(210\) 0 0
\(211\) −22.9783 −1.58189 −0.790944 0.611889i \(-0.790410\pi\)
−0.790944 + 0.611889i \(0.790410\pi\)
\(212\) 0 0
\(213\) −5.62772 −0.385605
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −11.2554 −0.764069
\(218\) 0 0
\(219\) −7.48913 −0.506068
\(220\) 0 0
\(221\) −4.37228 −0.294111
\(222\) 0 0
\(223\) 26.9783 1.80660 0.903299 0.429012i \(-0.141138\pi\)
0.903299 + 0.429012i \(0.141138\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.25544 0.481560 0.240780 0.970580i \(-0.422597\pi\)
0.240780 + 0.970580i \(0.422597\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 5.62772 0.370277
\(232\) 0 0
\(233\) 24.3723 1.59668 0.798341 0.602206i \(-0.205711\pi\)
0.798341 + 0.602206i \(0.205711\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 11.8614 0.770481
\(238\) 0 0
\(239\) 29.3505 1.89853 0.949264 0.314480i \(-0.101830\pi\)
0.949264 + 0.314480i \(0.101830\pi\)
\(240\) 0 0
\(241\) 20.9783 1.35133 0.675664 0.737210i \(-0.263857\pi\)
0.675664 + 0.737210i \(0.263857\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.74456 0.301889
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −5.48913 −0.346471 −0.173235 0.984880i \(-0.555422\pi\)
−0.173235 + 0.984880i \(0.555422\pi\)
\(252\) 0 0
\(253\) 15.1168 0.950388
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.0000 −0.623783 −0.311891 0.950118i \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) 0 0
\(259\) 0.883156 0.0548766
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −5.48913 −0.338474 −0.169237 0.985575i \(-0.554130\pi\)
−0.169237 + 0.985575i \(0.554130\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 7.62772 0.466809
\(268\) 0 0
\(269\) 1.25544 0.0765454 0.0382727 0.999267i \(-0.487814\pi\)
0.0382727 + 0.999267i \(0.487814\pi\)
\(270\) 0 0
\(271\) 28.7446 1.74611 0.873054 0.487624i \(-0.162136\pi\)
0.873054 + 0.487624i \(0.162136\pi\)
\(272\) 0 0
\(273\) 2.37228 0.143577
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −20.9783 −1.26046 −0.630230 0.776408i \(-0.717040\pi\)
−0.630230 + 0.776408i \(0.717040\pi\)
\(278\) 0 0
\(279\) −4.74456 −0.284050
\(280\) 0 0
\(281\) −32.9783 −1.96732 −0.983659 0.180043i \(-0.942376\pi\)
−0.983659 + 0.180043i \(0.942376\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.3723 0.612256
\(288\) 0 0
\(289\) 2.11684 0.124520
\(290\) 0 0
\(291\) −8.37228 −0.490792
\(292\) 0 0
\(293\) 26.7446 1.56243 0.781217 0.624260i \(-0.214599\pi\)
0.781217 + 0.624260i \(0.214599\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.37228 0.137654
\(298\) 0 0
\(299\) 6.37228 0.368519
\(300\) 0 0
\(301\) 9.48913 0.546944
\(302\) 0 0
\(303\) −6.74456 −0.387465
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −9.62772 −0.549483 −0.274741 0.961518i \(-0.588592\pi\)
−0.274741 + 0.961518i \(0.588592\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −16.2337 −0.911775 −0.455887 0.890037i \(-0.650678\pi\)
−0.455887 + 0.890037i \(0.650678\pi\)
\(318\) 0 0
\(319\) −4.74456 −0.265645
\(320\) 0 0
\(321\) 14.3723 0.802183
\(322\) 0 0
\(323\) −20.7446 −1.15426
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.74456 −0.151775
\(328\) 0 0
\(329\) 30.2337 1.66684
\(330\) 0 0
\(331\) 1.48913 0.0818497 0.0409249 0.999162i \(-0.486970\pi\)
0.0409249 + 0.999162i \(0.486970\pi\)
\(332\) 0 0
\(333\) 0.372281 0.0204009
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −16.2337 −0.884305 −0.442153 0.896940i \(-0.645785\pi\)
−0.442153 + 0.896940i \(0.645785\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −11.2554 −0.609516
\(342\) 0 0
\(343\) −19.8614 −1.07242
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.86141 0.422022 0.211011 0.977484i \(-0.432324\pi\)
0.211011 + 0.977484i \(0.432324\pi\)
\(348\) 0 0
\(349\) 32.2337 1.72543 0.862715 0.505691i \(-0.168762\pi\)
0.862715 + 0.505691i \(0.168762\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −10.3723 −0.548959
\(358\) 0 0
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 3.51087 0.184783
\(362\) 0 0
\(363\) −5.37228 −0.281972
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −13.4891 −0.704127 −0.352063 0.935976i \(-0.614520\pi\)
−0.352063 + 0.935976i \(0.614520\pi\)
\(368\) 0 0
\(369\) 4.37228 0.227612
\(370\) 0 0
\(371\) 8.60597 0.446800
\(372\) 0 0
\(373\) 26.7446 1.38478 0.692390 0.721523i \(-0.256558\pi\)
0.692390 + 0.721523i \(0.256558\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 3.25544 0.167221 0.0836103 0.996499i \(-0.473355\pi\)
0.0836103 + 0.996499i \(0.473355\pi\)
\(380\) 0 0
\(381\) −18.2337 −0.934140
\(382\) 0 0
\(383\) −20.7446 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) −8.23369 −0.417465 −0.208732 0.977973i \(-0.566934\pi\)
−0.208732 + 0.977973i \(0.566934\pi\)
\(390\) 0 0
\(391\) −27.8614 −1.40901
\(392\) 0 0
\(393\) −7.25544 −0.365988
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 27.3505 1.37268 0.686342 0.727279i \(-0.259215\pi\)
0.686342 + 0.727279i \(0.259215\pi\)
\(398\) 0 0
\(399\) 11.2554 0.563477
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) −4.74456 −0.236343
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.883156 0.0437764
\(408\) 0 0
\(409\) −32.9783 −1.63067 −0.815335 0.578990i \(-0.803447\pi\)
−0.815335 + 0.578990i \(0.803447\pi\)
\(410\) 0 0
\(411\) 11.4891 0.566717
\(412\) 0 0
\(413\) −18.9783 −0.933859
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.62772 0.0797097
\(418\) 0 0
\(419\) 2.23369 0.109123 0.0545614 0.998510i \(-0.482624\pi\)
0.0545614 + 0.998510i \(0.482624\pi\)
\(420\) 0 0
\(421\) −29.7228 −1.44860 −0.724301 0.689484i \(-0.757837\pi\)
−0.724301 + 0.689484i \(0.757837\pi\)
\(422\) 0 0
\(423\) 12.7446 0.619662
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −21.6277 −1.04664
\(428\) 0 0
\(429\) 2.37228 0.114535
\(430\) 0 0
\(431\) −26.9783 −1.29950 −0.649748 0.760149i \(-0.725126\pi\)
−0.649748 + 0.760149i \(0.725126\pi\)
\(432\) 0 0
\(433\) 28.2337 1.35682 0.678412 0.734681i \(-0.262668\pi\)
0.678412 + 0.734681i \(0.262668\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 30.2337 1.44627
\(438\) 0 0
\(439\) 40.6060 1.93802 0.969009 0.247027i \(-0.0794537\pi\)
0.969009 + 0.247027i \(0.0794537\pi\)
\(440\) 0 0
\(441\) −1.37228 −0.0653467
\(442\) 0 0
\(443\) 33.3505 1.58453 0.792266 0.610176i \(-0.208901\pi\)
0.792266 + 0.610176i \(0.208901\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 7.62772 0.360779
\(448\) 0 0
\(449\) 4.37228 0.206341 0.103170 0.994664i \(-0.467101\pi\)
0.103170 + 0.994664i \(0.467101\pi\)
\(450\) 0 0
\(451\) 10.3723 0.488412
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.8832 −1.07043 −0.535214 0.844716i \(-0.679769\pi\)
−0.535214 + 0.844716i \(0.679769\pi\)
\(458\) 0 0
\(459\) −4.37228 −0.204081
\(460\) 0 0
\(461\) 31.3505 1.46014 0.730070 0.683372i \(-0.239487\pi\)
0.730070 + 0.683372i \(0.239487\pi\)
\(462\) 0 0
\(463\) −8.60597 −0.399953 −0.199977 0.979801i \(-0.564087\pi\)
−0.199977 + 0.979801i \(0.564087\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 41.3505 1.91347 0.956737 0.290953i \(-0.0939725\pi\)
0.956737 + 0.290953i \(0.0939725\pi\)
\(468\) 0 0
\(469\) 9.48913 0.438167
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 9.48913 0.436310
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.62772 0.166102
\(478\) 0 0
\(479\) 31.1168 1.42176 0.710882 0.703311i \(-0.248296\pi\)
0.710882 + 0.703311i \(0.248296\pi\)
\(480\) 0 0
\(481\) 0.372281 0.0169746
\(482\) 0 0
\(483\) 15.1168 0.687840
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 15.1168 0.685010 0.342505 0.939516i \(-0.388725\pi\)
0.342505 + 0.939516i \(0.388725\pi\)
\(488\) 0 0
\(489\) 6.37228 0.288165
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 8.74456 0.393835
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −13.3505 −0.598853
\(498\) 0 0
\(499\) −3.25544 −0.145733 −0.0728667 0.997342i \(-0.523215\pi\)
−0.0728667 + 0.997342i \(0.523215\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −22.9783 −1.02455 −0.512275 0.858822i \(-0.671197\pi\)
−0.512275 + 0.858822i \(0.671197\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 6.13859 0.272088 0.136044 0.990703i \(-0.456561\pi\)
0.136044 + 0.990703i \(0.456561\pi\)
\(510\) 0 0
\(511\) −17.7663 −0.785935
\(512\) 0 0
\(513\) 4.74456 0.209478
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 30.2337 1.32968
\(518\) 0 0
\(519\) 15.4891 0.679897
\(520\) 0 0
\(521\) 0.510875 0.0223818 0.0111909 0.999937i \(-0.496438\pi\)
0.0111909 + 0.999937i \(0.496438\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.7446 0.903647
\(528\) 0 0
\(529\) 17.6060 0.765477
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 4.37228 0.189385
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −10.2337 −0.441616
\(538\) 0 0
\(539\) −3.25544 −0.140222
\(540\) 0 0
\(541\) 0.510875 0.0219642 0.0109821 0.999940i \(-0.496504\pi\)
0.0109821 + 0.999940i \(0.496504\pi\)
\(542\) 0 0
\(543\) 0.372281 0.0159761
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 38.9783 1.66659 0.833295 0.552829i \(-0.186452\pi\)
0.833295 + 0.552829i \(0.186452\pi\)
\(548\) 0 0
\(549\) −9.11684 −0.389097
\(550\) 0 0
\(551\) −9.48913 −0.404250
\(552\) 0 0
\(553\) 28.1386 1.19657
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 32.9783 1.39733 0.698667 0.715447i \(-0.253777\pi\)
0.698667 + 0.715447i \(0.253777\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −10.3723 −0.437918
\(562\) 0 0
\(563\) −19.1168 −0.805679 −0.402839 0.915271i \(-0.631977\pi\)
−0.402839 + 0.915271i \(0.631977\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.37228 0.0996265
\(568\) 0 0
\(569\) 33.7228 1.41373 0.706867 0.707347i \(-0.250108\pi\)
0.706867 + 0.707347i \(0.250108\pi\)
\(570\) 0 0
\(571\) 17.3505 0.726097 0.363049 0.931770i \(-0.381736\pi\)
0.363049 + 0.931770i \(0.381736\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −22.8832 −0.952638 −0.476319 0.879272i \(-0.658029\pi\)
−0.476319 + 0.879272i \(0.658029\pi\)
\(578\) 0 0
\(579\) −3.62772 −0.150763
\(580\) 0 0
\(581\) 28.4674 1.18103
\(582\) 0 0
\(583\) 8.60597 0.356423
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.74456 −0.360927 −0.180463 0.983582i \(-0.557760\pi\)
−0.180463 + 0.983582i \(0.557760\pi\)
\(588\) 0 0
\(589\) −22.5109 −0.927544
\(590\) 0 0
\(591\) −24.2337 −0.996841
\(592\) 0 0
\(593\) 22.7446 0.934007 0.467004 0.884255i \(-0.345333\pi\)
0.467004 + 0.884255i \(0.345333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.48913 −0.0609458
\(598\) 0 0
\(599\) −17.4891 −0.714586 −0.357293 0.933992i \(-0.616300\pi\)
−0.357293 + 0.933992i \(0.616300\pi\)
\(600\) 0 0
\(601\) 2.60597 0.106300 0.0531499 0.998587i \(-0.483074\pi\)
0.0531499 + 0.998587i \(0.483074\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 0 0
\(609\) −4.74456 −0.192259
\(610\) 0 0
\(611\) 12.7446 0.515590
\(612\) 0 0
\(613\) −13.8614 −0.559857 −0.279928 0.960021i \(-0.590311\pi\)
−0.279928 + 0.960021i \(0.590311\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.4674 0.904502 0.452251 0.891891i \(-0.350621\pi\)
0.452251 + 0.891891i \(0.350621\pi\)
\(618\) 0 0
\(619\) −15.7228 −0.631953 −0.315977 0.948767i \(-0.602332\pi\)
−0.315977 + 0.948767i \(0.602332\pi\)
\(620\) 0 0
\(621\) 6.37228 0.255711
\(622\) 0 0
\(623\) 18.0951 0.724965
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 11.2554 0.449499
\(628\) 0 0
\(629\) −1.62772 −0.0649014
\(630\) 0 0
\(631\) −41.4891 −1.65166 −0.825828 0.563922i \(-0.809292\pi\)
−0.825828 + 0.563922i \(0.809292\pi\)
\(632\) 0 0
\(633\) −22.9783 −0.913303
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.37228 −0.0543718
\(638\) 0 0
\(639\) −5.62772 −0.222629
\(640\) 0 0
\(641\) −12.2337 −0.483202 −0.241601 0.970376i \(-0.577672\pi\)
−0.241601 + 0.970376i \(0.577672\pi\)
\(642\) 0 0
\(643\) −11.1168 −0.438406 −0.219203 0.975679i \(-0.570346\pi\)
−0.219203 + 0.975679i \(0.570346\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −23.8614 −0.938089 −0.469044 0.883175i \(-0.655402\pi\)
−0.469044 + 0.883175i \(0.655402\pi\)
\(648\) 0 0
\(649\) −18.9783 −0.744961
\(650\) 0 0
\(651\) −11.2554 −0.441135
\(652\) 0 0
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −7.48913 −0.292179
\(658\) 0 0
\(659\) −37.4891 −1.46037 −0.730184 0.683250i \(-0.760566\pi\)
−0.730184 + 0.683250i \(0.760566\pi\)
\(660\) 0 0
\(661\) 0.233688 0.00908941 0.00454470 0.999990i \(-0.498553\pi\)
0.00454470 + 0.999990i \(0.498553\pi\)
\(662\) 0 0
\(663\) −4.37228 −0.169805
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.7446 −0.493471
\(668\) 0 0
\(669\) 26.9783 1.04304
\(670\) 0 0
\(671\) −21.6277 −0.834929
\(672\) 0 0
\(673\) 31.4891 1.21382 0.606908 0.794772i \(-0.292410\pi\)
0.606908 + 0.794772i \(0.292410\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11.6277 0.446890 0.223445 0.974717i \(-0.428270\pi\)
0.223445 + 0.974717i \(0.428270\pi\)
\(678\) 0 0
\(679\) −19.8614 −0.762211
\(680\) 0 0
\(681\) 7.25544 0.278029
\(682\) 0 0
\(683\) −46.9783 −1.79757 −0.898786 0.438387i \(-0.855550\pi\)
−0.898786 + 0.438387i \(0.855550\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.00000 0.0763048
\(688\) 0 0
\(689\) 3.62772 0.138205
\(690\) 0 0
\(691\) 23.7228 0.902458 0.451229 0.892408i \(-0.350986\pi\)
0.451229 + 0.892408i \(0.350986\pi\)
\(692\) 0 0
\(693\) 5.62772 0.213779
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −19.1168 −0.724102
\(698\) 0 0
\(699\) 24.3723 0.921844
\(700\) 0 0
\(701\) −5.25544 −0.198495 −0.0992476 0.995063i \(-0.531644\pi\)
−0.0992476 + 0.995063i \(0.531644\pi\)
\(702\) 0 0
\(703\) 1.76631 0.0666177
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −16.0000 −0.601742
\(708\) 0 0
\(709\) −50.7446 −1.90575 −0.952876 0.303360i \(-0.901892\pi\)
−0.952876 + 0.303360i \(0.901892\pi\)
\(710\) 0 0
\(711\) 11.8614 0.444838
\(712\) 0 0
\(713\) −30.2337 −1.13226
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 29.3505 1.09612
\(718\) 0 0
\(719\) 5.02175 0.187280 0.0936398 0.995606i \(-0.470150\pi\)
0.0936398 + 0.995606i \(0.470150\pi\)
\(720\) 0 0
\(721\) 9.48913 0.353393
\(722\) 0 0
\(723\) 20.9783 0.780190
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −0.744563 −0.0276143 −0.0138072 0.999905i \(-0.504395\pi\)
−0.0138072 + 0.999905i \(0.504395\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −17.4891 −0.646859
\(732\) 0 0
\(733\) −20.0951 −0.742229 −0.371115 0.928587i \(-0.621024\pi\)
−0.371115 + 0.928587i \(0.621024\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.48913 0.349536
\(738\) 0 0
\(739\) −6.23369 −0.229310 −0.114655 0.993405i \(-0.536576\pi\)
−0.114655 + 0.993405i \(0.536576\pi\)
\(740\) 0 0
\(741\) 4.74456 0.174296
\(742\) 0 0
\(743\) −22.2337 −0.815675 −0.407837 0.913055i \(-0.633717\pi\)
−0.407837 + 0.913055i \(0.633717\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 34.0951 1.24581
\(750\) 0 0
\(751\) −19.8614 −0.724753 −0.362377 0.932032i \(-0.618035\pi\)
−0.362377 + 0.932032i \(0.618035\pi\)
\(752\) 0 0
\(753\) −5.48913 −0.200035
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 50.7446 1.84434 0.922171 0.386782i \(-0.126413\pi\)
0.922171 + 0.386782i \(0.126413\pi\)
\(758\) 0 0
\(759\) 15.1168 0.548707
\(760\) 0 0
\(761\) −23.4891 −0.851480 −0.425740 0.904846i \(-0.639986\pi\)
−0.425740 + 0.904846i \(0.639986\pi\)
\(762\) 0 0
\(763\) −6.51087 −0.235709
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) −28.2337 −1.01813 −0.509066 0.860727i \(-0.670009\pi\)
−0.509066 + 0.860727i \(0.670009\pi\)
\(770\) 0 0
\(771\) −10.0000 −0.360141
\(772\) 0 0
\(773\) −5.25544 −0.189025 −0.0945125 0.995524i \(-0.530129\pi\)
−0.0945125 + 0.995524i \(0.530129\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0.883156 0.0316830
\(778\) 0 0
\(779\) 20.7446 0.743251
\(780\) 0 0
\(781\) −13.3505 −0.477720
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −38.9783 −1.38942 −0.694712 0.719288i \(-0.744468\pi\)
−0.694712 + 0.719288i \(0.744468\pi\)
\(788\) 0 0
\(789\) −5.48913 −0.195418
\(790\) 0 0
\(791\) 14.2337 0.506092
\(792\) 0 0
\(793\) −9.11684 −0.323749
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 16.3723 0.579936 0.289968 0.957036i \(-0.406355\pi\)
0.289968 + 0.957036i \(0.406355\pi\)
\(798\) 0 0
\(799\) −55.7228 −1.97133
\(800\) 0 0
\(801\) 7.62772 0.269512
\(802\) 0 0
\(803\) −17.7663 −0.626960
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.25544 0.0441935
\(808\) 0 0
\(809\) −34.4674 −1.21181 −0.605904 0.795538i \(-0.707189\pi\)
−0.605904 + 0.795538i \(0.707189\pi\)
\(810\) 0 0
\(811\) 6.51087 0.228628 0.114314 0.993445i \(-0.463533\pi\)
0.114314 + 0.993445i \(0.463533\pi\)
\(812\) 0 0
\(813\) 28.7446 1.00812
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 18.9783 0.663965
\(818\) 0 0
\(819\) 2.37228 0.0828942
\(820\) 0 0
\(821\) −24.0951 −0.840925 −0.420462 0.907310i \(-0.638132\pi\)
−0.420462 + 0.907310i \(0.638132\pi\)
\(822\) 0 0
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.23369 −0.0776729 −0.0388365 0.999246i \(-0.512365\pi\)
−0.0388365 + 0.999246i \(0.512365\pi\)
\(828\) 0 0
\(829\) 24.9783 0.867531 0.433765 0.901026i \(-0.357185\pi\)
0.433765 + 0.901026i \(0.357185\pi\)
\(830\) 0 0
\(831\) −20.9783 −0.727727
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.74456 −0.163996
\(838\) 0 0
\(839\) −5.62772 −0.194290 −0.0971452 0.995270i \(-0.530971\pi\)
−0.0971452 + 0.995270i \(0.530971\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −32.9783 −1.13583
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −12.7446 −0.437908
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 2.37228 0.0813208
\(852\) 0 0
\(853\) 21.1168 0.723027 0.361513 0.932367i \(-0.382260\pi\)
0.361513 + 0.932367i \(0.382260\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18.6060 −0.635568 −0.317784 0.948163i \(-0.602939\pi\)
−0.317784 + 0.948163i \(0.602939\pi\)
\(858\) 0 0
\(859\) −12.8832 −0.439568 −0.219784 0.975549i \(-0.570535\pi\)
−0.219784 + 0.975549i \(0.570535\pi\)
\(860\) 0 0
\(861\) 10.3723 0.353486
\(862\) 0 0
\(863\) −35.2554 −1.20011 −0.600055 0.799959i \(-0.704854\pi\)
−0.600055 + 0.799959i \(0.704854\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.11684 0.0718918
\(868\) 0 0
\(869\) 28.1386 0.954536
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) −8.37228 −0.283359
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.9783 1.11360 0.556798 0.830648i \(-0.312030\pi\)
0.556798 + 0.830648i \(0.312030\pi\)
\(878\) 0 0
\(879\) 26.7446 0.902072
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −38.9783 −1.31172 −0.655861 0.754881i \(-0.727694\pi\)
−0.655861 + 0.754881i \(0.727694\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.35053 −0.0453464 −0.0226732 0.999743i \(-0.507218\pi\)
−0.0226732 + 0.999743i \(0.507218\pi\)
\(888\) 0 0
\(889\) −43.2554 −1.45074
\(890\) 0 0
\(891\) 2.37228 0.0794744
\(892\) 0 0
\(893\) 60.4674 2.02346
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 6.37228 0.212764
\(898\) 0 0
\(899\) 9.48913 0.316480
\(900\) 0 0
\(901\) −15.8614 −0.528420
\(902\) 0 0
\(903\) 9.48913 0.315778
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −48.7446 −1.61854 −0.809268 0.587439i \(-0.800136\pi\)
−0.809268 + 0.587439i \(0.800136\pi\)
\(908\) 0 0
\(909\) −6.74456 −0.223703
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 28.4674 0.942133
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −17.2119 −0.568388
\(918\) 0 0
\(919\) −20.1386 −0.664311 −0.332155 0.943225i \(-0.607776\pi\)
−0.332155 + 0.943225i \(0.607776\pi\)
\(920\) 0 0
\(921\) −9.62772 −0.317244
\(922\) 0 0
\(923\) −5.62772 −0.185239
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) 50.6060 1.66033 0.830164 0.557519i \(-0.188247\pi\)
0.830164 + 0.557519i \(0.188247\pi\)
\(930\) 0 0
\(931\) −6.51087 −0.213385
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −19.4891 −0.636682 −0.318341 0.947976i \(-0.603126\pi\)
−0.318341 + 0.947976i \(0.603126\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 31.6277 1.03103 0.515517 0.856879i \(-0.327600\pi\)
0.515517 + 0.856879i \(0.327600\pi\)
\(942\) 0 0
\(943\) 27.8614 0.907292
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.02175 0.0332024 0.0166012 0.999862i \(-0.494715\pi\)
0.0166012 + 0.999862i \(0.494715\pi\)
\(948\) 0 0
\(949\) −7.48913 −0.243107
\(950\) 0 0
\(951\) −16.2337 −0.526413
\(952\) 0 0
\(953\) 8.64947 0.280184 0.140092 0.990139i \(-0.455260\pi\)
0.140092 + 0.990139i \(0.455260\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −4.74456 −0.153370
\(958\) 0 0
\(959\) 27.2554 0.880124
\(960\) 0 0
\(961\) −8.48913 −0.273843
\(962\) 0 0
\(963\) 14.3723 0.463140
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 10.9783 0.353037 0.176518 0.984297i \(-0.443517\pi\)
0.176518 + 0.984297i \(0.443517\pi\)
\(968\) 0 0
\(969\) −20.7446 −0.666411
\(970\) 0 0
\(971\) 29.4891 0.946351 0.473176 0.880968i \(-0.343108\pi\)
0.473176 + 0.880968i \(0.343108\pi\)
\(972\) 0 0
\(973\) 3.86141 0.123791
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −49.2554 −1.57582 −0.787911 0.615790i \(-0.788837\pi\)
−0.787911 + 0.615790i \(0.788837\pi\)
\(978\) 0 0
\(979\) 18.0951 0.578322
\(980\) 0 0
\(981\) −2.74456 −0.0876271
\(982\) 0 0
\(983\) 17.4891 0.557816 0.278908 0.960318i \(-0.410027\pi\)
0.278908 + 0.960318i \(0.410027\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 30.2337 0.962349
\(988\) 0 0
\(989\) 25.4891 0.810507
\(990\) 0 0
\(991\) 24.6060 0.781634 0.390817 0.920468i \(-0.372192\pi\)
0.390817 + 0.920468i \(0.372192\pi\)
\(992\) 0 0
\(993\) 1.48913 0.0472560
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −33.7228 −1.06801 −0.534006 0.845481i \(-0.679314\pi\)
−0.534006 + 0.845481i \(0.679314\pi\)
\(998\) 0 0
\(999\) 0.372281 0.0117785
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.bb.1.2 2
5.4 even 2 1560.2.a.n.1.1 2
15.14 odd 2 4680.2.a.bf.1.1 2
20.19 odd 2 3120.2.a.bd.1.2 2
60.59 even 2 9360.2.a.co.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.n.1.1 2 5.4 even 2
3120.2.a.bd.1.2 2 20.19 odd 2
4680.2.a.bf.1.1 2 15.14 odd 2
7800.2.a.bb.1.2 2 1.1 even 1 trivial
9360.2.a.co.1.2 2 60.59 even 2