Properties

Label 3920.2.a.bz.1.2
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +O(q^{10})\) \(q+2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +4.82843 q^{11} +2.00000 q^{13} +2.41421 q^{15} +3.65685 q^{17} -5.65685 q^{19} +8.41421 q^{23} +1.00000 q^{25} -0.414214 q^{27} -2.17157 q^{29} +4.82843 q^{31} +11.6569 q^{33} -5.65685 q^{37} +4.82843 q^{39} +0.171573 q^{41} -12.8995 q^{43} +2.82843 q^{45} -0.343146 q^{47} +8.82843 q^{51} +5.65685 q^{53} +4.82843 q^{55} -13.6569 q^{57} +4.00000 q^{59} -4.65685 q^{61} +2.00000 q^{65} -6.89949 q^{67} +20.3137 q^{69} +12.0000 q^{71} -7.65685 q^{73} +2.41421 q^{75} +4.00000 q^{79} -9.48528 q^{81} +13.2426 q^{83} +3.65685 q^{85} -5.24264 q^{87} -16.6569 q^{89} +11.6569 q^{93} -5.65685 q^{95} +6.00000 q^{97} +13.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} + 4q^{11} + 4q^{13} + 2q^{15} - 4q^{17} + 14q^{23} + 2q^{25} + 2q^{27} - 10q^{29} + 4q^{31} + 12q^{33} + 4q^{39} + 6q^{41} - 6q^{43} - 12q^{47} + 12q^{51} + 4q^{55} - 16q^{57} + 8q^{59} + 2q^{61} + 4q^{65} + 6q^{67} + 18q^{69} + 24q^{71} - 4q^{73} + 2q^{75} + 8q^{79} - 2q^{81} + 18q^{83} - 4q^{85} - 2q^{87} - 22q^{89} + 12q^{93} + 12q^{97} + 16q^{99} + O(q^{100}) \)

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
\(5\) 1.00000 0.447214
\(6\) 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 \(-0.240491\pi\)
0.727913 + 0.685670i \(0.240491\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 2.41421 0.623347
\(16\) 0 0
\(17\) 3.65685 0.886917 0.443459 0.896295i \(-0.353751\pi\)
0.443459 + 0.896295i \(0.353751\pi\)
\(18\) 0 0
\(19\) −5.65685 −1.29777 −0.648886 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 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
\(25\) 1.00000 0.200000
\(26\) 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 \(-0.357241\pi\)
0.433606 + 0.901103i \(0.357241\pi\)
\(32\) 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 \(-0.653942\pi\)
−0.464991 + 0.885316i \(0.653942\pi\)
\(38\) 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
\(45\) 2.82843 0.421637
\(46\) 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 \(-0.372988\pi\)
0.388514 + 0.921443i \(0.372988\pi\)
\(54\) 0 0
\(55\) 4.82843 0.651065
\(56\) 0 0
\(57\) −13.6569 −1.80889
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 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
\(65\) 2.00000 0.248069
\(66\) 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 \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −7.65685 −0.896167 −0.448084 0.893992i \(-0.647893\pi\)
−0.448084 + 0.893992i \(0.647893\pi\)
\(74\) 0 0
\(75\) 2.41421 0.278769
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 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
\(85\) 3.65685 0.396642
\(86\) 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
\(95\) −5.65685 −0.580381
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 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 \(-0.678617\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) 0 0
\(115\) 8.41421 0.784629
\(116\) 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
\(125\) 1.00000 0.0894427
\(126\) 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 \(-0.532640\pi\)
−0.102361 + 0.994747i \(0.532640\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.414214 −0.0356498
\(136\) 0 0
\(137\) −4.00000 −0.341743 −0.170872 0.985293i \(-0.554658\pi\)
−0.170872 + 0.985293i \(0.554658\pi\)
\(138\) 0 0
\(139\) 14.4853 1.22863 0.614313 0.789063i \(-0.289433\pi\)
0.614313 + 0.789063i \(0.289433\pi\)
\(140\) 0 0
\(141\) −0.828427 −0.0697661
\(142\) 0 0
\(143\) 9.65685 0.807547
\(144\) 0 0
\(145\) −2.17157 −0.180339
\(146\) 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 \(-0.259916\pi\)
0.684739 + 0.728788i \(0.259916\pi\)
\(152\) 0 0
\(153\) 10.3431 0.836194
\(154\) 0 0
\(155\) 4.82843 0.387829
\(156\) 0 0
\(157\) −21.3137 −1.70102 −0.850510 0.525960i \(-0.823706\pi\)
−0.850510 + 0.525960i \(0.823706\pi\)
\(158\) 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
\(165\) 11.6569 0.907485
\(166\) 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 \(-0.192147\pi\)
0.823270 + 0.567650i \(0.192147\pi\)
\(174\) 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 \(-0.628166\pi\)
−0.391853 + 0.920028i \(0.628166\pi\)
\(180\) 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
\(185\) −5.65685 −0.415900
\(186\) 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 \(-0.197565\pi\)
0.813489 + 0.581580i \(0.197565\pi\)
\(192\) 0 0
\(193\) 17.3137 1.24627 0.623134 0.782115i \(-0.285859\pi\)
0.623134 + 0.782115i \(0.285859\pi\)
\(194\) 0 0
\(195\) 4.82843 0.345771
\(196\) 0 0
\(197\) 11.6569 0.830516 0.415258 0.909704i \(-0.363691\pi\)
0.415258 + 0.909704i \(0.363691\pi\)
\(198\) 0 0
\(199\) 0.686292 0.0486499 0.0243250 0.999704i \(-0.492256\pi\)
0.0243250 + 0.999704i \(0.492256\pi\)
\(200\) 0 0
\(201\) −16.6569 −1.17488
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0.171573 0.0119832
\(206\) 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 \(-0.278444\pi\)
0.641182 + 0.767389i \(0.278444\pi\)
\(212\) 0 0
\(213\) 28.9706 1.98503
\(214\) 0 0
\(215\) −12.8995 −0.879738
\(216\) 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
\(225\) 2.82843 0.188562
\(226\) 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 \(-0.496422\pi\)
0.0112401 + 0.999937i \(0.496422\pi\)
\(234\) 0 0
\(235\) −0.343146 −0.0223844
\(236\) 0 0
\(237\) 9.65685 0.627280
\(238\) 0 0
\(239\) 13.5147 0.874194 0.437097 0.899414i \(-0.356007\pi\)
0.437097 + 0.899414i \(0.356007\pi\)
\(240\) 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 \(-0.765298\pi\)
−0.740260 + 0.672321i \(0.765298\pi\)
\(252\) 0 0
\(253\) 40.6274 2.55422
\(254\) 0 0
\(255\) 8.82843 0.552858
\(256\) 0 0
\(257\) −7.31371 −0.456217 −0.228108 0.973636i \(-0.573254\pi\)
−0.228108 + 0.973636i \(0.573254\pi\)
\(258\) 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
\(265\) 5.65685 0.347498
\(266\) 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 \(-0.699539\pi\)
−0.586612 + 0.809868i \(0.699539\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.82843 0.291165
\(276\) 0 0
\(277\) −16.6274 −0.999045 −0.499522 0.866301i \(-0.666491\pi\)
−0.499522 + 0.866301i \(0.666491\pi\)
\(278\) 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
\(285\) −13.6569 −0.808962
\(286\) 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 \(-0.665094\pi\)
−0.495715 + 0.868485i \(0.665094\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 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
\(305\) −4.65685 −0.266651
\(306\) 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 \(-0.594738\pi\)
−0.293253 + 0.956035i \(0.594738\pi\)
\(312\) 0 0
\(313\) −12.9706 −0.733140 −0.366570 0.930391i \(-0.619468\pi\)
−0.366570 + 0.930391i \(0.619468\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 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
\(325\) 2.00000 0.110940
\(326\) 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 \(-0.584213\pi\)
−0.261488 + 0.965207i \(0.584213\pi\)
\(332\) 0 0
\(333\) −16.0000 −0.876795
\(334\) 0 0
\(335\) −6.89949 −0.376960
\(336\) 0 0
\(337\) −8.97056 −0.488658 −0.244329 0.969692i \(-0.578568\pi\)
−0.244329 + 0.969692i \(0.578568\pi\)
\(338\) 0 0
\(339\) −27.3137 −1.48348
\(340\) 0 0
\(341\) 23.3137 1.26251
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 20.3137 1.09365
\(346\) 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 \(-0.297309\pi\)
0.594603 + 0.804020i \(0.297309\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.4853 −0.553392 −0.276696 0.960958i \(-0.589239\pi\)
−0.276696 + 0.960958i \(0.589239\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 29.7279 1.56031
\(364\) 0 0
\(365\) −7.65685 −0.400778
\(366\) 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 \(-0.399447\pi\)
0.310668 + 0.950518i \(0.399447\pi\)
\(374\) 0 0
\(375\) 2.41421 0.124669
\(376\) 0 0
\(377\) −4.34315 −0.223683
\(378\) 0 0
\(379\) −27.3137 −1.40301 −0.701505 0.712664i \(-0.747488\pi\)
−0.701505 + 0.712664i \(0.747488\pi\)
\(380\) 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
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) 20.6274 1.03526 0.517630 0.855604i \(-0.326814\pi\)
0.517630 + 0.855604i \(0.326814\pi\)
\(398\) 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
\(405\) −9.48528 −0.471327
\(406\) 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
\(415\) 13.2426 0.650056
\(416\) 0 0
\(417\) 34.9706 1.71252
\(418\) 0 0
\(419\) −7.31371 −0.357298 −0.178649 0.983913i \(-0.557173\pi\)
−0.178649 + 0.983913i \(0.557173\pi\)
\(420\) 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
\(425\) 3.65685 0.177383
\(426\) 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 \(-0.537100\pi\)
−0.116289 + 0.993215i \(0.537100\pi\)
\(432\) 0 0
\(433\) 3.31371 0.159247 0.0796233 0.996825i \(-0.474628\pi\)
0.0796233 + 0.996825i \(0.474628\pi\)
\(434\) 0 0
\(435\) −5.24264 −0.251365
\(436\) 0 0
\(437\) −47.5980 −2.27692
\(438\) 0 0
\(439\) 29.6569 1.41544 0.707722 0.706491i \(-0.249723\pi\)
0.707722 + 0.706491i \(0.249723\pi\)
\(440\) 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
\(445\) −16.6569 −0.789611
\(446\) 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 \(-0.537089\pi\)
−0.116257 + 0.993219i \(0.537089\pi\)
\(458\) 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
\(465\) 11.6569 0.540574
\(466\) 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
\(475\) −5.65685 −0.259554
\(476\) 0 0
\(477\) 16.0000 0.732590
\(478\) 0 0
\(479\) 18.4853 0.844614 0.422307 0.906453i \(-0.361220\pi\)
0.422307 + 0.906453i \(0.361220\pi\)
\(480\) 0 0
\(481\) −11.3137 −0.515861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.00000 0.272446
\(486\) 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 \(-0.636959\pi\)
−0.417115 + 0.908854i \(0.636959\pi\)
\(492\) 0 0
\(493\) −7.94113 −0.357650
\(494\) 0 0
\(495\) 13.6569 0.613830
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −15.8579 −0.709896 −0.354948 0.934886i \(-0.615501\pi\)
−0.354948 + 0.934886i \(0.615501\pi\)
\(500\) 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
\(505\) 5.48528 0.244092
\(506\) 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
\(515\) −10.4142 −0.458905
\(516\) 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
\(535\) 8.41421 0.363778
\(536\) 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
\(545\) −4.31371 −0.184779
\(546\) 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
\(555\) −13.6569 −0.579701
\(556\) 0 0
\(557\) 5.31371 0.225149 0.112575 0.993643i \(-0.464090\pi\)
0.112575 + 0.993643i \(0.464090\pi\)
\(558\) 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
\(565\) −11.3137 −0.475971
\(566\) 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 \(-0.587487\pi\)
−0.271401 + 0.962466i \(0.587487\pi\)
\(572\) 0 0
\(573\) 54.2843 2.26776
\(574\) 0 0
\(575\) 8.41421 0.350897
\(576\) 0 0
\(577\) 34.2843 1.42727 0.713636 0.700516i \(-0.247047\pi\)
0.713636 + 0.700516i \(0.247047\pi\)
\(578\) 0 0
\(579\) 41.7990 1.73711
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 27.3137 1.13122
\(584\) 0 0
\(585\) 5.65685 0.233882
\(586\) 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 \(-0.784287\pi\)
−0.779028 + 0.626990i \(0.784287\pi\)
\(594\) 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 \(-0.482906\pi\)
0.0536767 + 0.998558i \(0.482906\pi\)
\(600\) 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
\(605\) 12.3137 0.500623
\(606\) 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 \(-0.467614\pi\)
0.101569 + 0.994829i \(0.467614\pi\)
\(614\) 0 0
\(615\) 0.414214 0.0167027
\(616\) 0 0
\(617\) −35.3137 −1.42168 −0.710838 0.703356i \(-0.751684\pi\)
−0.710838 + 0.703356i \(0.751684\pi\)
\(618\) 0 0
\(619\) 11.4558 0.460449 0.230225 0.973138i \(-0.426054\pi\)
0.230225 + 0.973138i \(0.426054\pi\)
\(620\) 0 0
\(621\) −3.48528 −0.139860
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 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 \(-0.380062\pi\)
0.367944 + 0.929848i \(0.380062\pi\)
\(632\) 0 0
\(633\) 44.9706 1.78742
\(634\) 0 0
\(635\) 15.6569 0.621323
\(636\) 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
\(645\) −31.1421 −1.22622
\(646\) 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 \(-0.527083\pi\)
−0.0849802 + 0.996383i \(0.527083\pi\)
\(654\) 0 0
\(655\) −2.34315 −0.0915543
\(656\) 0 0
\(657\) −21.6569 −0.844914
\(658\) 0 0
\(659\) −35.3137 −1.37563 −0.687813 0.725888i \(-0.741429\pi\)
−0.687813 + 0.725888i \(0.741429\pi\)
\(660\) 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 \(-0.465226\pi\)
0.109028 + 0.994039i \(0.465226\pi\)
\(674\) 0 0
\(675\) −0.414214 −0.0159431
\(676\) 0 0
\(677\) −10.9706 −0.421633 −0.210816 0.977526i \(-0.567612\pi\)
−0.210816 + 0.977526i \(0.567612\pi\)
\(678\) 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
\(685\) −4.00000 −0.152832
\(686\) 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 \(-0.699009\pi\)
−0.585264 + 0.810842i \(0.699009\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 14.4853 0.549458
\(696\) 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
\(705\) −0.828427 −0.0312004
\(706\) 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
\(715\) 9.65685 0.361146
\(716\) 0 0
\(717\) 32.6274 1.21849
\(718\) 0 0
\(719\) −13.7990 −0.514615 −0.257308 0.966330i \(-0.582835\pi\)
−0.257308 + 0.966330i \(0.582835\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −24.1421 −0.897856
\(724\) 0 0
\(725\) −2.17157 −0.0806502
\(726\) 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 \(-0.521513\pi\)
−0.0675345 + 0.997717i \(0.521513\pi\)
\(734\) 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 \(-0.565874\pi\)
−0.205476 + 0.978662i \(0.565874\pi\)
\(740\) 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
\(745\) −6.65685 −0.243888
\(746\) 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 \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 0 0
\(753\) −56.6274 −2.06362
\(754\) 0 0
\(755\) 16.8284 0.612449
\(756\) 0 0
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 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
\(765\) 10.3431 0.373957
\(766\) 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 \(-0.278419\pi\)
0.641244 + 0.767337i \(0.278419\pi\)
\(774\) 0 0
\(775\) 4.82843 0.173442
\(776\) 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
\(785\) −21.3137 −0.760719
\(786\) 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
\(795\) 13.6569 0.484359
\(796\) 0 0
\(797\) −12.6863 −0.449372 −0.224686 0.974431i \(-0.572136\pi\)
−0.224686 + 0.974431i \(0.572136\pi\)
\(798\) 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 \(-0.637578\pi\)
−0.418881 + 0.908041i \(0.637578\pi\)
\(812\) 0 0
\(813\) −46.6274 −1.63529
\(814\) 0 0
\(815\) 4.34315 0.152134
\(816\) 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
\(825\) 11.6569 0.405840
\(826\) 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
\(835\) 12.0711 0.417737
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) 8.14214 0.281098 0.140549 0.990074i \(-0.455113\pi\)
0.140549 + 0.990074i \(0.455113\pi\)
\(840\) 0 0
\(841\) −24.2843 −0.837389
\(842\) 0 0
\(843\) 61.1127 2.10483
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 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 \(-0.293349\pi\)
0.604559 + 0.796560i \(0.293349\pi\)
\(854\) 0 0
\(855\) −16.0000 −0.547188
\(856\) 0 0
\(857\) 14.9706 0.511385 0.255692 0.966758i \(-0.417697\pi\)
0.255692 + 0.966758i \(0.417697\pi\)
\(858\) 0 0
\(859\) −14.6274 −0.499081 −0.249541 0.968364i \(-0.580280\pi\)
−0.249541 + 0.968364i \(0.580280\pi\)
\(860\) 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
\(865\) 21.6569 0.736355
\(866\) 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 \(-0.341526\pi\)
0.477546 + 0.878607i \(0.341526\pi\)
\(878\) 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
\(885\) 9.65685 0.324612
\(886\) 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
\(895\) −10.4853 −0.350484
\(896\) 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
\(905\) −9.82843 −0.326708
\(906\) 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 \(-0.492012\pi\)
0.0250924 + 0.999685i \(0.492012\pi\)
\(912\) 0 0
\(913\) 63.9411 2.11614
\(914\) 0 0
\(915\) −11.2426 −0.371670
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −57.2548 −1.88866 −0.944331 0.328996i \(-0.893290\pi\)
−0.944331 + 0.328996i \(0.893290\pi\)
\(920\) 0 0
\(921\) −31.9706 −1.05347
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −5.65685 −0.185996
\(926\) 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
\(935\) 17.6569 0.577441
\(936\) 0 0
\(937\) −26.6274 −0.869880 −0.434940 0.900459i \(-0.643230\pi\)
−0.434940 + 0.900459i \(0.643230\pi\)
\(938\) 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 \(-0.548508\pi\)
−0.151804 + 0.988411i \(0.548508\pi\)
\(954\) 0 0
\(955\) 22.4853 0.727607
\(956\) 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
\(965\) 17.3137 0.557348
\(966\) 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 \(-0.540973\pi\)
−0.128366 + 0.991727i \(0.540973\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 4.82843 0.154633
\(976\) 0 0
\(977\) −26.2843 −0.840908 −0.420454 0.907314i \(-0.638129\pi\)
−0.420454 + 0.907314i \(0.638129\pi\)
\(978\) 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
\(985\) 11.6569 0.371418
\(986\) 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 \(-0.495812\pi\)
0.0131579 + 0.999913i \(0.495812\pi\)
\(992\) 0 0
\(993\) −22.9706 −0.728949
\(994\) 0 0
\(995\) 0.686292 0.0217569
\(996\) 0 0
\(997\) −25.6569 −0.812561 −0.406280 0.913748i \(-0.633174\pi\)
−0.406280 + 0.913748i \(0.633174\pi\)
\(998\) 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 3920.2.a.bz.1.2 2
4.3 odd 2 1960.2.a.p.1.1 2
7.2 even 3 560.2.q.j.81.1 4
7.4 even 3 560.2.q.j.401.1 4
7.6 odd 2 3920.2.a.bp.1.1 2
20.19 odd 2 9800.2.a.bz.1.2 2
28.3 even 6 1960.2.q.q.961.1 4
28.11 odd 6 280.2.q.d.121.2 yes 4
28.19 even 6 1960.2.q.q.361.1 4
28.23 odd 6 280.2.q.d.81.2 4
28.27 even 2 1960.2.a.t.1.2 2
84.11 even 6 2520.2.bi.k.1801.1 4
84.23 even 6 2520.2.bi.k.361.1 4
140.23 even 12 1400.2.bh.g.249.4 8
140.39 odd 6 1400.2.q.h.401.1 4
140.67 even 12 1400.2.bh.g.849.4 8
140.79 odd 6 1400.2.q.h.1201.1 4
140.107 even 12 1400.2.bh.g.249.1 8
140.123 even 12 1400.2.bh.g.849.1 8
140.139 even 2 9800.2.a.br.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.d.81.2 4 28.23 odd 6
280.2.q.d.121.2 yes 4 28.11 odd 6
560.2.q.j.81.1 4 7.2 even 3
560.2.q.j.401.1 4 7.4 even 3
1400.2.q.h.401.1 4 140.39 odd 6
1400.2.q.h.1201.1 4 140.79 odd 6
1400.2.bh.g.249.1 8 140.107 even 12
1400.2.bh.g.249.4 8 140.23 even 12
1400.2.bh.g.849.1 8 140.123 even 12
1400.2.bh.g.849.4 8 140.67 even 12
1960.2.a.p.1.1 2 4.3 odd 2
1960.2.a.t.1.2 2 28.27 even 2
1960.2.q.q.361.1 4 28.19 even 6
1960.2.q.q.961.1 4 28.3 even 6
2520.2.bi.k.361.1 4 84.23 even 6
2520.2.bi.k.1801.1 4 84.11 even 6
3920.2.a.bp.1.1 2 7.6 odd 2
3920.2.a.bz.1.2 2 1.1 even 1 trivial
9800.2.a.br.1.1 2 140.139 even 2
9800.2.a.bz.1.2 2 20.19 odd 2