Properties

Label 9800.2.a.bz.1.2
Level $9800$
Weight $2$
Character 9800.1
Self dual yes
Analytic conductor $78.253$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.2533939809\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9800.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{3} +2.82843 q^{9} +O(q^{10})\) \(q+2.41421 q^{3} +2.82843 q^{9} -4.82843 q^{11} -2.00000 q^{13} -3.65685 q^{17} +5.65685 q^{19} +8.41421 q^{23} -0.414214 q^{27} -2.17157 q^{29} -4.82843 q^{31} -11.6569 q^{33} +5.65685 q^{37} -4.82843 q^{39} +0.171573 q^{41} -12.8995 q^{43} -0.343146 q^{47} -8.82843 q^{51} -5.65685 q^{53} +13.6569 q^{57} -4.00000 q^{59} -4.65685 q^{61} -6.89949 q^{67} +20.3137 q^{69} -12.0000 q^{71} +7.65685 q^{73} -4.00000 q^{79} -9.48528 q^{81} +13.2426 q^{83} -5.24264 q^{87} -16.6569 q^{89} -11.6569 q^{93} -6.00000 q^{97} -13.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{11} - 4q^{13} + 4q^{17} + 14q^{23} + 2q^{27} - 10q^{29} - 4q^{31} - 12q^{33} - 4q^{39} + 6q^{41} - 6q^{43} - 12q^{47} - 12q^{51} + 16q^{57} - 8q^{59} + 2q^{61} + 6q^{67} + 18q^{69} - 24q^{71} + 4q^{73} - 8q^{79} - 2q^{81} + 18q^{83} - 2q^{87} - 22q^{89} - 12q^{93} - 12q^{97} - 16q^{99} + O(q^{100}) \)

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
\(3\) 2.41421 1.39385 0.696923 0.717146i \(-0.254552\pi\)
0.696923 + 0.717146i \(0.254552\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) −4.82843 −1.45583 −0.727913 0.685670i \(-0.759509\pi\)
−0.727913 + 0.685670i \(0.759509\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.65685 −0.886917 −0.443459 0.896295i \(-0.646249\pi\)
−0.443459 + 0.896295i \(0.646249\pi\)
\(18\) 0 0
\(19\) 5.65685 1.29777 0.648886 0.760886i \(-0.275235\pi\)
0.648886 + 0.760886i \(0.275235\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.41421 1.75448 0.877242 0.480048i \(-0.159381\pi\)
0.877242 + 0.480048i \(0.159381\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.414214 −0.0797154
\(28\) 0 0
\(29\) −2.17157 −0.403251 −0.201625 0.979463i \(-0.564622\pi\)
−0.201625 + 0.979463i \(0.564622\pi\)
\(30\) 0 0
\(31\) −4.82843 −0.867211 −0.433606 0.901103i \(-0.642759\pi\)
−0.433606 + 0.901103i \(0.642759\pi\)
\(32\) 0 0
\(33\) −11.6569 −2.02920
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.65685 0.929981 0.464991 0.885316i \(-0.346058\pi\)
0.464991 + 0.885316i \(0.346058\pi\)
\(38\) 0 0
\(39\) −4.82843 −0.773167
\(40\) 0 0
\(41\) 0.171573 0.0267952 0.0133976 0.999910i \(-0.495735\pi\)
0.0133976 + 0.999910i \(0.495735\pi\)
\(42\) 0 0
\(43\) −12.8995 −1.96715 −0.983577 0.180488i \(-0.942232\pi\)
−0.983577 + 0.180488i \(0.942232\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.343146 −0.0500530 −0.0250265 0.999687i \(-0.507967\pi\)
−0.0250265 + 0.999687i \(0.507967\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −8.82843 −1.23623
\(52\) 0 0
\(53\) −5.65685 −0.777029 −0.388514 0.921443i \(-0.627012\pi\)
−0.388514 + 0.921443i \(0.627012\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 13.6569 1.80889
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −4.65685 −0.596249 −0.298125 0.954527i \(-0.596361\pi\)
−0.298125 + 0.954527i \(0.596361\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.89949 −0.842907 −0.421454 0.906850i \(-0.638480\pi\)
−0.421454 + 0.906850i \(0.638480\pi\)
\(68\) 0 0
\(69\) 20.3137 2.44548
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 7.65685 0.896167 0.448084 0.893992i \(-0.352107\pi\)
0.448084 + 0.893992i \(0.352107\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) 13.2426 1.45357 0.726784 0.686866i \(-0.241014\pi\)
0.726784 + 0.686866i \(0.241014\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −5.24264 −0.562070
\(88\) 0 0
\(89\) −16.6569 −1.76562 −0.882812 0.469727i \(-0.844352\pi\)
−0.882812 + 0.469727i \(0.844352\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −11.6569 −1.20876
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −13.6569 −1.37257
\(100\) 0 0
\(101\) 5.48528 0.545806 0.272903 0.962042i \(-0.412016\pi\)
0.272903 + 0.962042i \(0.412016\pi\)
\(102\) 0 0
\(103\) −10.4142 −1.02614 −0.513071 0.858346i \(-0.671492\pi\)
−0.513071 + 0.858346i \(0.671492\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.41421 0.813433 0.406716 0.913554i \(-0.366674\pi\)
0.406716 + 0.913554i \(0.366674\pi\)
\(108\) 0 0
\(109\) −4.31371 −0.413178 −0.206589 0.978428i \(-0.566236\pi\)
−0.206589 + 0.978428i \(0.566236\pi\)
\(110\) 0 0
\(111\) 13.6569 1.29625
\(112\) 0 0
\(113\) 11.3137 1.06430 0.532152 0.846649i \(-0.321383\pi\)
0.532152 + 0.846649i \(0.321383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −5.65685 −0.522976
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 0 0
\(123\) 0.414214 0.0373484
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 15.6569 1.38932 0.694661 0.719338i \(-0.255555\pi\)
0.694661 + 0.719338i \(0.255555\pi\)
\(128\) 0 0
\(129\) −31.1421 −2.74191
\(130\) 0 0
\(131\) 2.34315 0.204722 0.102361 0.994747i \(-0.467360\pi\)
0.102361 + 0.994747i \(0.467360\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 0 0
\(139\) −14.4853 −1.22863 −0.614313 0.789063i \(-0.710567\pi\)
−0.614313 + 0.789063i \(0.710567\pi\)
\(140\) 0 0
\(141\) −0.828427 −0.0697661
\(142\) 0 0
\(143\) 9.65685 0.807547
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.65685 −0.545351 −0.272675 0.962106i \(-0.587908\pi\)
−0.272675 + 0.962106i \(0.587908\pi\)
\(150\) 0 0
\(151\) −16.8284 −1.36948 −0.684739 0.728788i \(-0.740084\pi\)
−0.684739 + 0.728788i \(0.740084\pi\)
\(152\) 0 0
\(153\) −10.3431 −0.836194
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 21.3137 1.70102 0.850510 0.525960i \(-0.176294\pi\)
0.850510 + 0.525960i \(0.176294\pi\)
\(158\) 0 0
\(159\) −13.6569 −1.08306
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.34315 0.340181 0.170091 0.985428i \(-0.445594\pi\)
0.170091 + 0.985428i \(0.445594\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0711 0.934087 0.467044 0.884234i \(-0.345319\pi\)
0.467044 + 0.884234i \(0.345319\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 16.0000 1.22355
\(172\) 0 0
\(173\) −21.6569 −1.64654 −0.823270 0.567650i \(-0.807853\pi\)
−0.823270 + 0.567650i \(0.807853\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −9.65685 −0.725854
\(178\) 0 0
\(179\) 10.4853 0.783707 0.391853 0.920028i \(-0.371834\pi\)
0.391853 + 0.920028i \(0.371834\pi\)
\(180\) 0 0
\(181\) −9.82843 −0.730541 −0.365271 0.930901i \(-0.619024\pi\)
−0.365271 + 0.930901i \(0.619024\pi\)
\(182\) 0 0
\(183\) −11.2426 −0.831080
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 17.6569 1.29120
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −22.4853 −1.62698 −0.813489 0.581580i \(-0.802435\pi\)
−0.813489 + 0.581580i \(0.802435\pi\)
\(192\) 0 0
\(193\) −17.3137 −1.24627 −0.623134 0.782115i \(-0.714141\pi\)
−0.623134 + 0.782115i \(0.714141\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.6569 −0.830516 −0.415258 0.909704i \(-0.636309\pi\)
−0.415258 + 0.909704i \(0.636309\pi\)
\(198\) 0 0
\(199\) −0.686292 −0.0486499 −0.0243250 0.999704i \(-0.507744\pi\)
−0.0243250 + 0.999704i \(0.507744\pi\)
\(200\) 0 0
\(201\) −16.6569 −1.17488
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 23.7990 1.65414
\(208\) 0 0
\(209\) −27.3137 −1.88933
\(210\) 0 0
\(211\) −18.6274 −1.28236 −0.641182 0.767389i \(-0.721556\pi\)
−0.641182 + 0.767389i \(0.721556\pi\)
\(212\) 0 0
\(213\) −28.9706 −1.98503
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 18.4853 1.24912
\(220\) 0 0
\(221\) 7.31371 0.491973
\(222\) 0 0
\(223\) −18.9706 −1.27036 −0.635181 0.772363i \(-0.719075\pi\)
−0.635181 + 0.772363i \(0.719075\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.343146 −0.0224802 −0.0112401 0.999937i \(-0.503578\pi\)
−0.0112401 + 0.999937i \(0.503578\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −9.65685 −0.627280
\(238\) 0 0
\(239\) −13.5147 −0.874194 −0.437097 0.899414i \(-0.643993\pi\)
−0.437097 + 0.899414i \(0.643993\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) −21.6569 −1.38929
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −11.3137 −0.719874
\(248\) 0 0
\(249\) 31.9706 2.02605
\(250\) 0 0
\(251\) 23.4558 1.48052 0.740260 0.672321i \(-0.234702\pi\)
0.740260 + 0.672321i \(0.234702\pi\)
\(252\) 0 0
\(253\) −40.6274 −2.55422
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.31371 0.456217 0.228108 0.973636i \(-0.426746\pi\)
0.228108 + 0.973636i \(0.426746\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.14214 −0.380189
\(262\) 0 0
\(263\) −9.72792 −0.599849 −0.299925 0.953963i \(-0.596962\pi\)
−0.299925 + 0.953963i \(0.596962\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −40.2132 −2.46101
\(268\) 0 0
\(269\) 4.65685 0.283933 0.141967 0.989871i \(-0.454657\pi\)
0.141967 + 0.989871i \(0.454657\pi\)
\(270\) 0 0
\(271\) 19.3137 1.17322 0.586612 0.809868i \(-0.300461\pi\)
0.586612 + 0.809868i \(0.300461\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 16.6274 0.999045 0.499522 0.866301i \(-0.333509\pi\)
0.499522 + 0.866301i \(0.333509\pi\)
\(278\) 0 0
\(279\) −13.6569 −0.817614
\(280\) 0 0
\(281\) 25.3137 1.51009 0.755045 0.655673i \(-0.227615\pi\)
0.755045 + 0.655673i \(0.227615\pi\)
\(282\) 0 0
\(283\) −18.0000 −1.06999 −0.534994 0.844856i \(-0.679686\pi\)
−0.534994 + 0.844856i \(0.679686\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −3.62742 −0.213377
\(290\) 0 0
\(291\) −14.4853 −0.849142
\(292\) 0 0
\(293\) 16.9706 0.991431 0.495715 0.868485i \(-0.334906\pi\)
0.495715 + 0.868485i \(0.334906\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) −16.8284 −0.973213
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 13.2426 0.760770
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −13.2426 −0.755797 −0.377899 0.925847i \(-0.623353\pi\)
−0.377899 + 0.925847i \(0.623353\pi\)
\(308\) 0 0
\(309\) −25.1421 −1.43029
\(310\) 0 0
\(311\) 10.3431 0.586506 0.293253 0.956035i \(-0.405262\pi\)
0.293253 + 0.956035i \(0.405262\pi\)
\(312\) 0 0
\(313\) 12.9706 0.733140 0.366570 0.930391i \(-0.380532\pi\)
0.366570 + 0.930391i \(0.380532\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) 10.4853 0.587063
\(320\) 0 0
\(321\) 20.3137 1.13380
\(322\) 0 0
\(323\) −20.6863 −1.15102
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −10.4142 −0.575907
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.51472 0.522976 0.261488 0.965207i \(-0.415787\pi\)
0.261488 + 0.965207i \(0.415787\pi\)
\(332\) 0 0
\(333\) 16.0000 0.876795
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.97056 0.488658 0.244329 0.969692i \(-0.421432\pi\)
0.244329 + 0.969692i \(0.421432\pi\)
\(338\) 0 0
\(339\) 27.3137 1.48348
\(340\) 0 0
\(341\) 23.3137 1.26251
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 25.3848 1.36273 0.681363 0.731946i \(-0.261387\pi\)
0.681363 + 0.731946i \(0.261387\pi\)
\(348\) 0 0
\(349\) 4.17157 0.223299 0.111650 0.993748i \(-0.464387\pi\)
0.111650 + 0.993748i \(0.464387\pi\)
\(350\) 0 0
\(351\) 0.828427 0.0442182
\(352\) 0 0
\(353\) −22.3431 −1.18921 −0.594603 0.804020i \(-0.702691\pi\)
−0.594603 + 0.804020i \(0.702691\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.4853 0.553392 0.276696 0.960958i \(-0.410761\pi\)
0.276696 + 0.960958i \(0.410761\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 29.7279 1.56031
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −24.4142 −1.27441 −0.637206 0.770694i \(-0.719910\pi\)
−0.637206 + 0.770694i \(0.719910\pi\)
\(368\) 0 0
\(369\) 0.485281 0.0252627
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −12.0000 −0.621336 −0.310668 0.950518i \(-0.600553\pi\)
−0.310668 + 0.950518i \(0.600553\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.34315 0.223683
\(378\) 0 0
\(379\) 27.3137 1.40301 0.701505 0.712664i \(-0.252512\pi\)
0.701505 + 0.712664i \(0.252512\pi\)
\(380\) 0 0
\(381\) 37.7990 1.93650
\(382\) 0 0
\(383\) −18.8995 −0.965719 −0.482860 0.875698i \(-0.660402\pi\)
−0.482860 + 0.875698i \(0.660402\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −36.4853 −1.85465
\(388\) 0 0
\(389\) −17.3137 −0.877840 −0.438920 0.898526i \(-0.644639\pi\)
−0.438920 + 0.898526i \(0.644639\pi\)
\(390\) 0 0
\(391\) −30.7696 −1.55608
\(392\) 0 0
\(393\) 5.65685 0.285351
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −20.6274 −1.03526 −0.517630 0.855604i \(-0.673186\pi\)
−0.517630 + 0.855604i \(0.673186\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.68629 −0.483710 −0.241855 0.970312i \(-0.577756\pi\)
−0.241855 + 0.970312i \(0.577756\pi\)
\(402\) 0 0
\(403\) 9.65685 0.481042
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −27.3137 −1.35389
\(408\) 0 0
\(409\) 3.14214 0.155369 0.0776843 0.996978i \(-0.475247\pi\)
0.0776843 + 0.996978i \(0.475247\pi\)
\(410\) 0 0
\(411\) 9.65685 0.476337
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −34.9706 −1.71252
\(418\) 0 0
\(419\) 7.31371 0.357298 0.178649 0.983913i \(-0.442827\pi\)
0.178649 + 0.983913i \(0.442827\pi\)
\(420\) 0 0
\(421\) 38.6569 1.88402 0.942010 0.335585i \(-0.108934\pi\)
0.942010 + 0.335585i \(0.108934\pi\)
\(422\) 0 0
\(423\) −0.970563 −0.0471904
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 23.3137 1.12560
\(430\) 0 0
\(431\) 4.82843 0.232577 0.116289 0.993215i \(-0.462900\pi\)
0.116289 + 0.993215i \(0.462900\pi\)
\(432\) 0 0
\(433\) −3.31371 −0.159247 −0.0796233 0.996825i \(-0.525372\pi\)
−0.0796233 + 0.996825i \(0.525372\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 47.5980 2.27692
\(438\) 0 0
\(439\) −29.6569 −1.41544 −0.707722 0.706491i \(-0.750277\pi\)
−0.707722 + 0.706491i \(0.750277\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.4142 0.874886 0.437443 0.899246i \(-0.355884\pi\)
0.437443 + 0.899246i \(0.355884\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −16.0711 −0.760135
\(448\) 0 0
\(449\) 9.48528 0.447638 0.223819 0.974631i \(-0.428148\pi\)
0.223819 + 0.974631i \(0.428148\pi\)
\(450\) 0 0
\(451\) −0.828427 −0.0390091
\(452\) 0 0
\(453\) −40.6274 −1.90884
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.97056 0.232513 0.116257 0.993219i \(-0.462911\pi\)
0.116257 + 0.993219i \(0.462911\pi\)
\(458\) 0 0
\(459\) 1.51472 0.0707010
\(460\) 0 0
\(461\) −21.3137 −0.992678 −0.496339 0.868129i \(-0.665323\pi\)
−0.496339 + 0.868129i \(0.665323\pi\)
\(462\) 0 0
\(463\) 4.89949 0.227699 0.113849 0.993498i \(-0.463682\pi\)
0.113849 + 0.993498i \(0.463682\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −15.8701 −0.734379 −0.367189 0.930146i \(-0.619680\pi\)
−0.367189 + 0.930146i \(0.619680\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 51.4558 2.37096
\(472\) 0 0
\(473\) 62.2843 2.86383
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −16.0000 −0.732590
\(478\) 0 0
\(479\) −18.4853 −0.844614 −0.422307 0.906453i \(-0.638780\pi\)
−0.422307 + 0.906453i \(0.638780\pi\)
\(480\) 0 0
\(481\) −11.3137 −0.515861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −18.2843 −0.828539 −0.414270 0.910154i \(-0.635963\pi\)
−0.414270 + 0.910154i \(0.635963\pi\)
\(488\) 0 0
\(489\) 10.4853 0.474161
\(490\) 0 0
\(491\) 18.4853 0.834229 0.417115 0.908854i \(-0.363041\pi\)
0.417115 + 0.908854i \(0.363041\pi\)
\(492\) 0 0
\(493\) 7.94113 0.357650
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 15.8579 0.709896 0.354948 0.934886i \(-0.384499\pi\)
0.354948 + 0.934886i \(0.384499\pi\)
\(500\) 0 0
\(501\) 29.1421 1.30197
\(502\) 0 0
\(503\) 18.0711 0.805749 0.402875 0.915255i \(-0.368011\pi\)
0.402875 + 0.915255i \(0.368011\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −21.7279 −0.964971
\(508\) 0 0
\(509\) −16.5147 −0.732002 −0.366001 0.930614i \(-0.619273\pi\)
−0.366001 + 0.930614i \(0.619273\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2.34315 −0.103452
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1.65685 0.0728684
\(518\) 0 0
\(519\) −52.2843 −2.29502
\(520\) 0 0
\(521\) −8.62742 −0.377974 −0.188987 0.981980i \(-0.560520\pi\)
−0.188987 + 0.981980i \(0.560520\pi\)
\(522\) 0 0
\(523\) 39.9411 1.74650 0.873252 0.487269i \(-0.162007\pi\)
0.873252 + 0.487269i \(0.162007\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.6569 0.769145
\(528\) 0 0
\(529\) 47.7990 2.07822
\(530\) 0 0
\(531\) −11.3137 −0.490973
\(532\) 0 0
\(533\) −0.343146 −0.0148633
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 25.3137 1.09237
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −6.51472 −0.280090 −0.140045 0.990145i \(-0.544725\pi\)
−0.140045 + 0.990145i \(0.544725\pi\)
\(542\) 0 0
\(543\) −23.7279 −1.01826
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 27.7279 1.18556 0.592780 0.805364i \(-0.298030\pi\)
0.592780 + 0.805364i \(0.298030\pi\)
\(548\) 0 0
\(549\) −13.1716 −0.562149
\(550\) 0 0
\(551\) −12.2843 −0.523328
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.31371 −0.225149 −0.112575 0.993643i \(-0.535910\pi\)
−0.112575 + 0.993643i \(0.535910\pi\)
\(558\) 0 0
\(559\) 25.7990 1.09118
\(560\) 0 0
\(561\) 42.6274 1.79973
\(562\) 0 0
\(563\) −10.0711 −0.424445 −0.212222 0.977221i \(-0.568070\pi\)
−0.212222 + 0.977221i \(0.568070\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.62742 0.361680 0.180840 0.983513i \(-0.442118\pi\)
0.180840 + 0.983513i \(0.442118\pi\)
\(570\) 0 0
\(571\) 12.9706 0.542801 0.271401 0.962466i \(-0.412513\pi\)
0.271401 + 0.962466i \(0.412513\pi\)
\(572\) 0 0
\(573\) −54.2843 −2.26776
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −34.2843 −1.42727 −0.713636 0.700516i \(-0.752953\pi\)
−0.713636 + 0.700516i \(0.752953\pi\)
\(578\) 0 0
\(579\) −41.7990 −1.73711
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 27.3137 1.13122
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.3137 1.87030 0.935148 0.354256i \(-0.115266\pi\)
0.935148 + 0.354256i \(0.115266\pi\)
\(588\) 0 0
\(589\) −27.3137 −1.12544
\(590\) 0 0
\(591\) −28.1421 −1.15761
\(592\) 0 0
\(593\) 37.9411 1.55806 0.779028 0.626990i \(-0.215713\pi\)
0.779028 + 0.626990i \(0.215713\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.65685 −0.0678105
\(598\) 0 0
\(599\) −2.62742 −0.107353 −0.0536767 0.998558i \(-0.517094\pi\)
−0.0536767 + 0.998558i \(0.517094\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) −19.5147 −0.794701
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −18.7574 −0.761338 −0.380669 0.924711i \(-0.624306\pi\)
−0.380669 + 0.924711i \(0.624306\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.686292 0.0277644
\(612\) 0 0
\(613\) −5.02944 −0.203137 −0.101569 0.994829i \(-0.532386\pi\)
−0.101569 + 0.994829i \(0.532386\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 35.3137 1.42168 0.710838 0.703356i \(-0.248316\pi\)
0.710838 + 0.703356i \(0.248316\pi\)
\(618\) 0 0
\(619\) −11.4558 −0.460449 −0.230225 0.973138i \(-0.573946\pi\)
−0.230225 + 0.973138i \(0.573946\pi\)
\(620\) 0 0
\(621\) −3.48528 −0.139860
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −65.9411 −2.63343
\(628\) 0 0
\(629\) −20.6863 −0.824816
\(630\) 0 0
\(631\) −18.4853 −0.735887 −0.367944 0.929848i \(-0.619938\pi\)
−0.367944 + 0.929848i \(0.619938\pi\)
\(632\) 0 0
\(633\) −44.9706 −1.78742
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −33.9411 −1.34269
\(640\) 0 0
\(641\) −48.1127 −1.90034 −0.950169 0.311736i \(-0.899089\pi\)
−0.950169 + 0.311736i \(0.899089\pi\)
\(642\) 0 0
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 29.2426 1.14965 0.574823 0.818277i \(-0.305071\pi\)
0.574823 + 0.818277i \(0.305071\pi\)
\(648\) 0 0
\(649\) 19.3137 0.758129
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.34315 0.169960 0.0849802 0.996383i \(-0.472917\pi\)
0.0849802 + 0.996383i \(0.472917\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 21.6569 0.844914
\(658\) 0 0
\(659\) 35.3137 1.37563 0.687813 0.725888i \(-0.258571\pi\)
0.687813 + 0.725888i \(0.258571\pi\)
\(660\) 0 0
\(661\) −27.6863 −1.07687 −0.538436 0.842666i \(-0.680985\pi\)
−0.538436 + 0.842666i \(0.680985\pi\)
\(662\) 0 0
\(663\) 17.6569 0.685735
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −18.2721 −0.707498
\(668\) 0 0
\(669\) −45.7990 −1.77069
\(670\) 0 0
\(671\) 22.4853 0.868035
\(672\) 0 0
\(673\) −5.65685 −0.218056 −0.109028 0.994039i \(-0.534774\pi\)
−0.109028 + 0.994039i \(0.534774\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.9706 0.421633 0.210816 0.977526i \(-0.432388\pi\)
0.210816 + 0.977526i \(0.432388\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −33.7990 −1.29518
\(682\) 0 0
\(683\) 16.0711 0.614942 0.307471 0.951557i \(-0.400517\pi\)
0.307471 + 0.951557i \(0.400517\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −33.7990 −1.28951
\(688\) 0 0
\(689\) 11.3137 0.431018
\(690\) 0 0
\(691\) 30.7696 1.17053 0.585264 0.810842i \(-0.300991\pi\)
0.585264 + 0.810842i \(0.300991\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −0.627417 −0.0237651
\(698\) 0 0
\(699\) −0.828427 −0.0313340
\(700\) 0 0
\(701\) −11.0000 −0.415464 −0.207732 0.978186i \(-0.566608\pi\)
−0.207732 + 0.978186i \(0.566608\pi\)
\(702\) 0 0
\(703\) 32.0000 1.20690
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 49.4264 1.85625 0.928124 0.372272i \(-0.121421\pi\)
0.928124 + 0.372272i \(0.121421\pi\)
\(710\) 0 0
\(711\) −11.3137 −0.424297
\(712\) 0 0
\(713\) −40.6274 −1.52151
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −32.6274 −1.21849
\(718\) 0 0
\(719\) 13.7990 0.514615 0.257308 0.966330i \(-0.417165\pi\)
0.257308 + 0.966330i \(0.417165\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −24.1421 −0.897856
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −23.9289 −0.887475 −0.443737 0.896157i \(-0.646348\pi\)
−0.443737 + 0.896157i \(0.646348\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) 47.1716 1.74470
\(732\) 0 0
\(733\) 3.65685 0.135069 0.0675345 0.997717i \(-0.478487\pi\)
0.0675345 + 0.997717i \(0.478487\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 33.3137 1.22713
\(738\) 0 0
\(739\) 11.1716 0.410953 0.205476 0.978662i \(-0.434126\pi\)
0.205476 + 0.978662i \(0.434126\pi\)
\(740\) 0 0
\(741\) −27.3137 −1.00339
\(742\) 0 0
\(743\) 19.2426 0.705944 0.352972 0.935634i \(-0.385171\pi\)
0.352972 + 0.935634i \(0.385171\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 37.4558 1.37044
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) 56.6274 2.06362
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 0 0
\(759\) −98.0833 −3.56020
\(760\) 0 0
\(761\) 23.9411 0.867865 0.433933 0.900945i \(-0.357126\pi\)
0.433933 + 0.900945i \(0.357126\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −47.2548 −1.70405 −0.852026 0.523499i \(-0.824626\pi\)
−0.852026 + 0.523499i \(0.824626\pi\)
\(770\) 0 0
\(771\) 17.6569 0.635896
\(772\) 0 0
\(773\) −35.6569 −1.28249 −0.641244 0.767337i \(-0.721581\pi\)
−0.641244 + 0.767337i \(0.721581\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.970563 0.0347740
\(780\) 0 0
\(781\) 57.9411 2.07330
\(782\) 0 0
\(783\) 0.899495 0.0321453
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 4.41421 0.157350 0.0786749 0.996900i \(-0.474931\pi\)
0.0786749 + 0.996900i \(0.474931\pi\)
\(788\) 0 0
\(789\) −23.4853 −0.836098
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 9.31371 0.330739
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.6863 0.449372 0.224686 0.974431i \(-0.427864\pi\)
0.224686 + 0.974431i \(0.427864\pi\)
\(798\) 0 0
\(799\) 1.25483 0.0443928
\(800\) 0 0
\(801\) −47.1127 −1.66465
\(802\) 0 0
\(803\) −36.9706 −1.30466
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 11.2426 0.395760
\(808\) 0 0
\(809\) −2.02944 −0.0713512 −0.0356756 0.999363i \(-0.511358\pi\)
−0.0356756 + 0.999363i \(0.511358\pi\)
\(810\) 0 0
\(811\) 23.8579 0.837763 0.418881 0.908041i \(-0.362422\pi\)
0.418881 + 0.908041i \(0.362422\pi\)
\(812\) 0 0
\(813\) 46.6274 1.63529
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −72.9706 −2.55292
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.6274 0.440700 0.220350 0.975421i \(-0.429280\pi\)
0.220350 + 0.975421i \(0.429280\pi\)
\(822\) 0 0
\(823\) 52.0122 1.81303 0.906516 0.422172i \(-0.138732\pi\)
0.906516 + 0.422172i \(0.138732\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 51.5269 1.79177 0.895883 0.444290i \(-0.146544\pi\)
0.895883 + 0.444290i \(0.146544\pi\)
\(828\) 0 0
\(829\) −17.3137 −0.601330 −0.300665 0.953730i \(-0.597209\pi\)
−0.300665 + 0.953730i \(0.597209\pi\)
\(830\) 0 0
\(831\) 40.1421 1.39252
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) 0 0
\(839\) −8.14214 −0.281098 −0.140549 0.990074i \(-0.544887\pi\)
−0.140549 + 0.990074i \(0.544887\pi\)
\(840\) 0 0
\(841\) −24.2843 −0.837389
\(842\) 0 0
\(843\) 61.1127 2.10483
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −43.4558 −1.49140
\(850\) 0 0
\(851\) 47.5980 1.63164
\(852\) 0 0
\(853\) −35.3137 −1.20912 −0.604559 0.796560i \(-0.706651\pi\)
−0.604559 + 0.796560i \(0.706651\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −14.9706 −0.511385 −0.255692 0.966758i \(-0.582303\pi\)
−0.255692 + 0.966758i \(0.582303\pi\)
\(858\) 0 0
\(859\) 14.6274 0.499081 0.249541 0.968364i \(-0.419720\pi\)
0.249541 + 0.968364i \(0.419720\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −26.0122 −0.885465 −0.442733 0.896654i \(-0.645991\pi\)
−0.442733 + 0.896654i \(0.645991\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −8.75736 −0.297416
\(868\) 0 0
\(869\) 19.3137 0.655173
\(870\) 0 0
\(871\) 13.7990 0.467561
\(872\) 0 0
\(873\) −16.9706 −0.574367
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −28.2843 −0.955092 −0.477546 0.878607i \(-0.658474\pi\)
−0.477546 + 0.878607i \(0.658474\pi\)
\(878\) 0 0
\(879\) 40.9706 1.38190
\(880\) 0 0
\(881\) 24.4558 0.823938 0.411969 0.911198i \(-0.364841\pi\)
0.411969 + 0.911198i \(0.364841\pi\)
\(882\) 0 0
\(883\) −29.3137 −0.986485 −0.493242 0.869892i \(-0.664188\pi\)
−0.493242 + 0.869892i \(0.664188\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −20.4142 −0.685442 −0.342721 0.939437i \(-0.611349\pi\)
−0.342721 + 0.939437i \(0.611349\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 45.7990 1.53432
\(892\) 0 0
\(893\) −1.94113 −0.0649573
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −40.6274 −1.35651
\(898\) 0 0
\(899\) 10.4853 0.349704
\(900\) 0 0
\(901\) 20.6863 0.689160
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −3.78680 −0.125739 −0.0628693 0.998022i \(-0.520025\pi\)
−0.0628693 + 0.998022i \(0.520025\pi\)
\(908\) 0 0
\(909\) 15.5147 0.514591
\(910\) 0 0
\(911\) −1.51472 −0.0501849 −0.0250924 0.999685i \(-0.507988\pi\)
−0.0250924 + 0.999685i \(0.507988\pi\)
\(912\) 0 0
\(913\) −63.9411 −2.11614
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 57.2548 1.88866 0.944331 0.328996i \(-0.106710\pi\)
0.944331 + 0.328996i \(0.106710\pi\)
\(920\) 0 0
\(921\) −31.9706 −1.05347
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −29.4558 −0.967457
\(928\) 0 0
\(929\) 48.7990 1.60104 0.800521 0.599304i \(-0.204556\pi\)
0.800521 + 0.599304i \(0.204556\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 24.9706 0.817500
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 26.6274 0.869880 0.434940 0.900459i \(-0.356770\pi\)
0.434940 + 0.900459i \(0.356770\pi\)
\(938\) 0 0
\(939\) 31.3137 1.02188
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) 1.44365 0.0470117
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −30.8995 −1.00410 −0.502049 0.864839i \(-0.667420\pi\)
−0.502049 + 0.864839i \(0.667420\pi\)
\(948\) 0 0
\(949\) −15.3137 −0.497104
\(950\) 0 0
\(951\) 53.1127 1.72230
\(952\) 0 0
\(953\) 9.37258 0.303608 0.151804 0.988411i \(-0.451492\pi\)
0.151804 + 0.988411i \(0.451492\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 25.3137 0.818276
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −7.68629 −0.247945
\(962\) 0 0
\(963\) 23.7990 0.766912
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 33.2426 1.06901 0.534506 0.845165i \(-0.320498\pi\)
0.534506 + 0.845165i \(0.320498\pi\)
\(968\) 0 0
\(969\) −49.9411 −1.60434
\(970\) 0 0
\(971\) 8.00000 0.256732 0.128366 0.991727i \(-0.459027\pi\)
0.128366 + 0.991727i \(0.459027\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 26.2843 0.840908 0.420454 0.907314i \(-0.361871\pi\)
0.420454 + 0.907314i \(0.361871\pi\)
\(978\) 0 0
\(979\) 80.4264 2.57044
\(980\) 0 0
\(981\) −12.2010 −0.389548
\(982\) 0 0
\(983\) 4.61522 0.147203 0.0736014 0.997288i \(-0.476551\pi\)
0.0736014 + 0.997288i \(0.476551\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −108.539 −3.45134
\(990\) 0 0
\(991\) −0.828427 −0.0263159 −0.0131579 0.999913i \(-0.504188\pi\)
−0.0131579 + 0.999913i \(0.504188\pi\)
\(992\) 0 0
\(993\) 22.9706 0.728949
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25.6569 0.812561 0.406280 0.913748i \(-0.366826\pi\)
0.406280 + 0.913748i \(0.366826\pi\)
\(998\) 0 0
\(999\) −2.34315 −0.0741339
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 9800.2.a.bz.1.2 2
5.4 even 2 1960.2.a.p.1.1 2
7.2 even 3 1400.2.q.h.1201.1 4
7.4 even 3 1400.2.q.h.401.1 4
7.6 odd 2 9800.2.a.br.1.1 2
20.19 odd 2 3920.2.a.bz.1.2 2
35.2 odd 12 1400.2.bh.g.249.4 8
35.4 even 6 280.2.q.d.121.2 yes 4
35.9 even 6 280.2.q.d.81.2 4
35.18 odd 12 1400.2.bh.g.849.4 8
35.19 odd 6 1960.2.q.q.361.1 4
35.23 odd 12 1400.2.bh.g.249.1 8
35.24 odd 6 1960.2.q.q.961.1 4
35.32 odd 12 1400.2.bh.g.849.1 8
35.34 odd 2 1960.2.a.t.1.2 2
105.44 odd 6 2520.2.bi.k.361.1 4
105.74 odd 6 2520.2.bi.k.1801.1 4
140.39 odd 6 560.2.q.j.401.1 4
140.79 odd 6 560.2.q.j.81.1 4
140.139 even 2 3920.2.a.bp.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.d.81.2 4 35.9 even 6
280.2.q.d.121.2 yes 4 35.4 even 6
560.2.q.j.81.1 4 140.79 odd 6
560.2.q.j.401.1 4 140.39 odd 6
1400.2.q.h.401.1 4 7.4 even 3
1400.2.q.h.1201.1 4 7.2 even 3
1400.2.bh.g.249.1 8 35.23 odd 12
1400.2.bh.g.249.4 8 35.2 odd 12
1400.2.bh.g.849.1 8 35.32 odd 12
1400.2.bh.g.849.4 8 35.18 odd 12
1960.2.a.p.1.1 2 5.4 even 2
1960.2.a.t.1.2 2 35.34 odd 2
1960.2.q.q.361.1 4 35.19 odd 6
1960.2.q.q.961.1 4 35.24 odd 6
2520.2.bi.k.361.1 4 105.44 odd 6
2520.2.bi.k.1801.1 4 105.74 odd 6
3920.2.a.bp.1.1 2 140.139 even 2
3920.2.a.bz.1.2 2 20.19 odd 2
9800.2.a.br.1.1 2 7.6 odd 2
9800.2.a.bz.1.2 2 1.1 even 1 trivial