Properties

Label 2800.2.a.br.1.1
Level $2800$
Weight $2$
Character 2800.1
Self dual yes
Analytic conductor $22.358$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
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 1.1
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 2800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.76156 q^{3} -1.00000 q^{7} +0.103084 q^{9} +O(q^{10})\) \(q-1.76156 q^{3} -1.00000 q^{7} +0.103084 q^{9} +0.626198 q^{11} +5.49084 q^{13} +0.896916 q^{17} -6.38776 q^{19} +1.76156 q^{21} +3.72928 q^{23} +5.10308 q^{27} -7.87859 q^{29} -7.52311 q^{31} -1.10308 q^{33} +6.00000 q^{37} -9.67243 q^{39} +7.72928 q^{41} -1.72928 q^{43} -5.87859 q^{47} +1.00000 q^{49} -1.57997 q^{51} -6.77551 q^{53} +11.2524 q^{57} +0.593923 q^{59} +7.13536 q^{61} -0.103084 q^{63} +5.79383 q^{67} -6.56934 q^{69} -5.52311 q^{71} -3.72928 q^{73} -0.626198 q^{77} +5.67243 q^{79} -9.29862 q^{81} +17.4340 q^{83} +13.8786 q^{87} +14.2986 q^{89} -5.49084 q^{91} +13.2524 q^{93} -10.1493 q^{97} +0.0645508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{3} - 3q^{7} + 4q^{9} + O(q^{10}) \) \( 3q + q^{3} - 3q^{7} + 4q^{9} - 7q^{11} + 5q^{13} - q^{17} - 4q^{19} - q^{21} + 6q^{23} + 19q^{27} + 3q^{29} - 10q^{31} - 7q^{33} + 18q^{37} + 5q^{39} + 18q^{41} + 9q^{47} + 3q^{49} - 21q^{51} + 10q^{53} + 16q^{57} - 6q^{59} + 24q^{61} - 4q^{63} + 10q^{67} + 18q^{69} - 4q^{71} - 6q^{73} + 7q^{77} - 17q^{79} + 15q^{81} + 12q^{83} + 15q^{87} - 5q^{91} + 22q^{93} - 9q^{97} - 2q^{99} + O(q^{100}) \)

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
\(3\) −1.76156 −1.01704 −0.508518 0.861052i \(-0.669806\pi\)
−0.508518 + 0.861052i \(0.669806\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0.103084 0.0343612
\(10\) 0 0
\(11\) 0.626198 0.188806 0.0944029 0.995534i \(-0.469906\pi\)
0.0944029 + 0.995534i \(0.469906\pi\)
\(12\) 0 0
\(13\) 5.49084 1.52288 0.761442 0.648233i \(-0.224492\pi\)
0.761442 + 0.648233i \(0.224492\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.896916 0.217534 0.108767 0.994067i \(-0.465310\pi\)
0.108767 + 0.994067i \(0.465310\pi\)
\(18\) 0 0
\(19\) −6.38776 −1.46545 −0.732726 0.680524i \(-0.761752\pi\)
−0.732726 + 0.680524i \(0.761752\pi\)
\(20\) 0 0
\(21\) 1.76156 0.384403
\(22\) 0 0
\(23\) 3.72928 0.777609 0.388805 0.921320i \(-0.372888\pi\)
0.388805 + 0.921320i \(0.372888\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.10308 0.982089
\(28\) 0 0
\(29\) −7.87859 −1.46302 −0.731509 0.681832i \(-0.761184\pi\)
−0.731509 + 0.681832i \(0.761184\pi\)
\(30\) 0 0
\(31\) −7.52311 −1.35119 −0.675596 0.737272i \(-0.736113\pi\)
−0.675596 + 0.737272i \(0.736113\pi\)
\(32\) 0 0
\(33\) −1.10308 −0.192022
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) −9.67243 −1.54883
\(40\) 0 0
\(41\) 7.72928 1.20711 0.603556 0.797321i \(-0.293750\pi\)
0.603556 + 0.797321i \(0.293750\pi\)
\(42\) 0 0
\(43\) −1.72928 −0.263713 −0.131856 0.991269i \(-0.542094\pi\)
−0.131856 + 0.991269i \(0.542094\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.87859 −0.857481 −0.428741 0.903428i \(-0.641043\pi\)
−0.428741 + 0.903428i \(0.641043\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.57997 −0.221240
\(52\) 0 0
\(53\) −6.77551 −0.930688 −0.465344 0.885130i \(-0.654069\pi\)
−0.465344 + 0.885130i \(0.654069\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 11.2524 1.49042
\(58\) 0 0
\(59\) 0.593923 0.0773221 0.0386611 0.999252i \(-0.487691\pi\)
0.0386611 + 0.999252i \(0.487691\pi\)
\(60\) 0 0
\(61\) 7.13536 0.913589 0.456795 0.889572i \(-0.348997\pi\)
0.456795 + 0.889572i \(0.348997\pi\)
\(62\) 0 0
\(63\) −0.103084 −0.0129873
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.79383 0.707829 0.353915 0.935278i \(-0.384850\pi\)
0.353915 + 0.935278i \(0.384850\pi\)
\(68\) 0 0
\(69\) −6.56934 −0.790856
\(70\) 0 0
\(71\) −5.52311 −0.655473 −0.327737 0.944769i \(-0.606286\pi\)
−0.327737 + 0.944769i \(0.606286\pi\)
\(72\) 0 0
\(73\) −3.72928 −0.436479 −0.218240 0.975895i \(-0.570031\pi\)
−0.218240 + 0.975895i \(0.570031\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.626198 −0.0713619
\(78\) 0 0
\(79\) 5.67243 0.638198 0.319099 0.947721i \(-0.396620\pi\)
0.319099 + 0.947721i \(0.396620\pi\)
\(80\) 0 0
\(81\) −9.29862 −1.03318
\(82\) 0 0
\(83\) 17.4340 1.91363 0.956814 0.290700i \(-0.0938882\pi\)
0.956814 + 0.290700i \(0.0938882\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 13.8786 1.48794
\(88\) 0 0
\(89\) 14.2986 1.51565 0.757826 0.652457i \(-0.226262\pi\)
0.757826 + 0.652457i \(0.226262\pi\)
\(90\) 0 0
\(91\) −5.49084 −0.575596
\(92\) 0 0
\(93\) 13.2524 1.37421
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.1493 −1.03051 −0.515253 0.857038i \(-0.672302\pi\)
−0.515253 + 0.857038i \(0.672302\pi\)
\(98\) 0 0
\(99\) 0.0645508 0.00648760
\(100\) 0 0
\(101\) 9.64015 0.959231 0.479615 0.877479i \(-0.340776\pi\)
0.479615 + 0.877479i \(0.340776\pi\)
\(102\) 0 0
\(103\) 0.626198 0.0617011 0.0308506 0.999524i \(-0.490178\pi\)
0.0308506 + 0.999524i \(0.490178\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 18.9248 1.81267 0.906335 0.422561i \(-0.138869\pi\)
0.906335 + 0.422561i \(0.138869\pi\)
\(110\) 0 0
\(111\) −10.5693 −1.00320
\(112\) 0 0
\(113\) −1.04623 −0.0984209 −0.0492105 0.998788i \(-0.515671\pi\)
−0.0492105 + 0.998788i \(0.515671\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.566016 0.0523282
\(118\) 0 0
\(119\) −0.896916 −0.0822202
\(120\) 0 0
\(121\) −10.6079 −0.964352
\(122\) 0 0
\(123\) −13.6156 −1.22767
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 21.2803 1.88832 0.944161 0.329485i \(-0.106875\pi\)
0.944161 + 0.329485i \(0.106875\pi\)
\(128\) 0 0
\(129\) 3.04623 0.268205
\(130\) 0 0
\(131\) 9.91087 0.865917 0.432958 0.901414i \(-0.357470\pi\)
0.432958 + 0.901414i \(0.357470\pi\)
\(132\) 0 0
\(133\) 6.38776 0.553889
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.45856 0.295485 0.147743 0.989026i \(-0.452799\pi\)
0.147743 + 0.989026i \(0.452799\pi\)
\(138\) 0 0
\(139\) 20.4157 1.73163 0.865817 0.500361i \(-0.166799\pi\)
0.865817 + 0.500361i \(0.166799\pi\)
\(140\) 0 0
\(141\) 10.3555 0.872089
\(142\) 0 0
\(143\) 3.43835 0.287530
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.76156 −0.145291
\(148\) 0 0
\(149\) −17.0462 −1.39648 −0.698241 0.715863i \(-0.746033\pi\)
−0.698241 + 0.715863i \(0.746033\pi\)
\(150\) 0 0
\(151\) −2.89692 −0.235748 −0.117874 0.993029i \(-0.537608\pi\)
−0.117874 + 0.993029i \(0.537608\pi\)
\(152\) 0 0
\(153\) 0.0924575 0.00747474
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.1170 0.807427 0.403714 0.914885i \(-0.367719\pi\)
0.403714 + 0.914885i \(0.367719\pi\)
\(158\) 0 0
\(159\) 11.9354 0.946543
\(160\) 0 0
\(161\) −3.72928 −0.293909
\(162\) 0 0
\(163\) −0.476886 −0.0373526 −0.0186763 0.999826i \(-0.505945\pi\)
−0.0186763 + 0.999826i \(0.505945\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.1676 1.01894 0.509471 0.860488i \(-0.329841\pi\)
0.509471 + 0.860488i \(0.329841\pi\)
\(168\) 0 0
\(169\) 17.1493 1.31918
\(170\) 0 0
\(171\) −0.658473 −0.0503547
\(172\) 0 0
\(173\) 13.9677 1.06195 0.530973 0.847389i \(-0.321826\pi\)
0.530973 + 0.847389i \(0.321826\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.04623 −0.0786394
\(178\) 0 0
\(179\) −7.45856 −0.557479 −0.278740 0.960367i \(-0.589917\pi\)
−0.278740 + 0.960367i \(0.589917\pi\)
\(180\) 0 0
\(181\) −14.1170 −1.04931 −0.524656 0.851315i \(-0.675806\pi\)
−0.524656 + 0.851315i \(0.675806\pi\)
\(182\) 0 0
\(183\) −12.5693 −0.929153
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.561647 0.0410717
\(188\) 0 0
\(189\) −5.10308 −0.371195
\(190\) 0 0
\(191\) 8.42003 0.609252 0.304626 0.952472i \(-0.401469\pi\)
0.304626 + 0.952472i \(0.401469\pi\)
\(192\) 0 0
\(193\) −22.2986 −1.60509 −0.802545 0.596591i \(-0.796521\pi\)
−0.802545 + 0.596591i \(0.796521\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.3632 1.87830 0.939149 0.343509i \(-0.111616\pi\)
0.939149 + 0.343509i \(0.111616\pi\)
\(198\) 0 0
\(199\) −22.4402 −1.59075 −0.795373 0.606120i \(-0.792725\pi\)
−0.795373 + 0.606120i \(0.792725\pi\)
\(200\) 0 0
\(201\) −10.2062 −0.719888
\(202\) 0 0
\(203\) 7.87859 0.552969
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.384428 0.0267196
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 15.1955 1.04610 0.523052 0.852301i \(-0.324793\pi\)
0.523052 + 0.852301i \(0.324793\pi\)
\(212\) 0 0
\(213\) 9.72928 0.666639
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.52311 0.510702
\(218\) 0 0
\(219\) 6.56934 0.443915
\(220\) 0 0
\(221\) 4.92482 0.331279
\(222\) 0 0
\(223\) 6.89692 0.461852 0.230926 0.972971i \(-0.425825\pi\)
0.230926 + 0.972971i \(0.425825\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.77988 0.184507 0.0922535 0.995736i \(-0.470593\pi\)
0.0922535 + 0.995736i \(0.470593\pi\)
\(228\) 0 0
\(229\) 18.6585 1.23299 0.616493 0.787360i \(-0.288553\pi\)
0.616493 + 0.787360i \(0.288553\pi\)
\(230\) 0 0
\(231\) 1.10308 0.0725776
\(232\) 0 0
\(233\) −11.0462 −0.723663 −0.361831 0.932244i \(-0.617848\pi\)
−0.361831 + 0.932244i \(0.617848\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −9.99230 −0.649070
\(238\) 0 0
\(239\) 20.1493 1.30335 0.651675 0.758498i \(-0.274066\pi\)
0.651675 + 0.758498i \(0.274066\pi\)
\(240\) 0 0
\(241\) 3.72928 0.240224 0.120112 0.992760i \(-0.461675\pi\)
0.120112 + 0.992760i \(0.461675\pi\)
\(242\) 0 0
\(243\) 1.07081 0.0686924
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −35.0741 −2.23171
\(248\) 0 0
\(249\) −30.7110 −1.94623
\(250\) 0 0
\(251\) 21.4985 1.35698 0.678488 0.734612i \(-0.262636\pi\)
0.678488 + 0.734612i \(0.262636\pi\)
\(252\) 0 0
\(253\) 2.33527 0.146817
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.27072 0.266400 0.133200 0.991089i \(-0.457475\pi\)
0.133200 + 0.991089i \(0.457475\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −0.812155 −0.0502711
\(262\) 0 0
\(263\) 11.4586 0.706565 0.353283 0.935517i \(-0.385065\pi\)
0.353283 + 0.935517i \(0.385065\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −25.1878 −1.54147
\(268\) 0 0
\(269\) −7.07081 −0.431115 −0.215557 0.976491i \(-0.569157\pi\)
−0.215557 + 0.976491i \(0.569157\pi\)
\(270\) 0 0
\(271\) −17.0096 −1.03326 −0.516629 0.856209i \(-0.672813\pi\)
−0.516629 + 0.856209i \(0.672813\pi\)
\(272\) 0 0
\(273\) 9.67243 0.585402
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.7755 −1.12811 −0.564056 0.825737i \(-0.690760\pi\)
−0.564056 + 0.825737i \(0.690760\pi\)
\(278\) 0 0
\(279\) −0.775511 −0.0464286
\(280\) 0 0
\(281\) −4.59829 −0.274311 −0.137156 0.990550i \(-0.543796\pi\)
−0.137156 + 0.990550i \(0.543796\pi\)
\(282\) 0 0
\(283\) 13.5833 0.807443 0.403722 0.914882i \(-0.367716\pi\)
0.403722 + 0.914882i \(0.367716\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7.72928 −0.456245
\(288\) 0 0
\(289\) −16.1955 −0.952679
\(290\) 0 0
\(291\) 17.8786 1.04806
\(292\) 0 0
\(293\) 11.8261 0.690889 0.345444 0.938439i \(-0.387728\pi\)
0.345444 + 0.938439i \(0.387728\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.19554 0.185424
\(298\) 0 0
\(299\) 20.4769 1.18421
\(300\) 0 0
\(301\) 1.72928 0.0996741
\(302\) 0 0
\(303\) −16.9817 −0.975572
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.9860 1.19774 0.598868 0.800847i \(-0.295617\pi\)
0.598868 + 0.800847i \(0.295617\pi\)
\(308\) 0 0
\(309\) −1.10308 −0.0627522
\(310\) 0 0
\(311\) 22.5048 1.27613 0.638065 0.769983i \(-0.279735\pi\)
0.638065 + 0.769983i \(0.279735\pi\)
\(312\) 0 0
\(313\) −12.4846 −0.705670 −0.352835 0.935686i \(-0.614782\pi\)
−0.352835 + 0.935686i \(0.614782\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 25.5231 1.43352 0.716760 0.697319i \(-0.245624\pi\)
0.716760 + 0.697319i \(0.245624\pi\)
\(318\) 0 0
\(319\) −4.93356 −0.276226
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.72928 −0.318786
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −33.3372 −1.84355
\(328\) 0 0
\(329\) 5.87859 0.324097
\(330\) 0 0
\(331\) 2.68305 0.147474 0.0737370 0.997278i \(-0.476507\pi\)
0.0737370 + 0.997278i \(0.476507\pi\)
\(332\) 0 0
\(333\) 0.618502 0.0338937
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 12.5048 0.681179 0.340590 0.940212i \(-0.389373\pi\)
0.340590 + 0.940212i \(0.389373\pi\)
\(338\) 0 0
\(339\) 1.84299 0.100098
\(340\) 0 0
\(341\) −4.71096 −0.255113
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.9450 1.55385 0.776925 0.629593i \(-0.216778\pi\)
0.776925 + 0.629593i \(0.216778\pi\)
\(348\) 0 0
\(349\) 32.1449 1.72068 0.860340 0.509721i \(-0.170251\pi\)
0.860340 + 0.509721i \(0.170251\pi\)
\(350\) 0 0
\(351\) 28.0202 1.49561
\(352\) 0 0
\(353\) 8.17722 0.435229 0.217615 0.976035i \(-0.430172\pi\)
0.217615 + 0.976035i \(0.430172\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.57997 0.0836208
\(358\) 0 0
\(359\) −18.5048 −0.976646 −0.488323 0.872663i \(-0.662391\pi\)
−0.488323 + 0.872663i \(0.662391\pi\)
\(360\) 0 0
\(361\) 21.8034 1.14755
\(362\) 0 0
\(363\) 18.6864 0.980781
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 27.4942 1.43518 0.717592 0.696464i \(-0.245244\pi\)
0.717592 + 0.696464i \(0.245244\pi\)
\(368\) 0 0
\(369\) 0.796763 0.0414778
\(370\) 0 0
\(371\) 6.77551 0.351767
\(372\) 0 0
\(373\) 6.06455 0.314011 0.157005 0.987598i \(-0.449816\pi\)
0.157005 + 0.987598i \(0.449816\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −43.2601 −2.22801
\(378\) 0 0
\(379\) 5.72928 0.294293 0.147147 0.989115i \(-0.452991\pi\)
0.147147 + 0.989115i \(0.452991\pi\)
\(380\) 0 0
\(381\) −37.4865 −1.92049
\(382\) 0 0
\(383\) 1.72928 0.0883622 0.0441811 0.999024i \(-0.485932\pi\)
0.0441811 + 0.999024i \(0.485932\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.178261 −0.00906150
\(388\) 0 0
\(389\) −36.7187 −1.86171 −0.930855 0.365389i \(-0.880936\pi\)
−0.930855 + 0.365389i \(0.880936\pi\)
\(390\) 0 0
\(391\) 3.34485 0.169157
\(392\) 0 0
\(393\) −17.4586 −0.880668
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −2.03228 −0.101997 −0.0509985 0.998699i \(-0.516240\pi\)
−0.0509985 + 0.998699i \(0.516240\pi\)
\(398\) 0 0
\(399\) −11.2524 −0.563324
\(400\) 0 0
\(401\) −1.64452 −0.0821234 −0.0410617 0.999157i \(-0.513074\pi\)
−0.0410617 + 0.999157i \(0.513074\pi\)
\(402\) 0 0
\(403\) −41.3082 −2.05771
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.75719 0.186237
\(408\) 0 0
\(409\) −4.98168 −0.246328 −0.123164 0.992386i \(-0.539304\pi\)
−0.123164 + 0.992386i \(0.539304\pi\)
\(410\) 0 0
\(411\) −6.09246 −0.300519
\(412\) 0 0
\(413\) −0.593923 −0.0292250
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −35.9634 −1.76113
\(418\) 0 0
\(419\) −22.9205 −1.11974 −0.559869 0.828581i \(-0.689148\pi\)
−0.559869 + 0.828581i \(0.689148\pi\)
\(420\) 0 0
\(421\) −20.9527 −1.02117 −0.510587 0.859826i \(-0.670572\pi\)
−0.510587 + 0.859826i \(0.670572\pi\)
\(422\) 0 0
\(423\) −0.605987 −0.0294641
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −7.13536 −0.345304
\(428\) 0 0
\(429\) −6.05685 −0.292428
\(430\) 0 0
\(431\) −33.1589 −1.59721 −0.798604 0.601857i \(-0.794428\pi\)
−0.798604 + 0.601857i \(0.794428\pi\)
\(432\) 0 0
\(433\) −18.5414 −0.891045 −0.445522 0.895271i \(-0.646982\pi\)
−0.445522 + 0.895271i \(0.646982\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −23.8217 −1.13955
\(438\) 0 0
\(439\) 20.8401 0.994642 0.497321 0.867567i \(-0.334317\pi\)
0.497321 + 0.867567i \(0.334317\pi\)
\(440\) 0 0
\(441\) 0.103084 0.00490875
\(442\) 0 0
\(443\) −17.5510 −0.833874 −0.416937 0.908935i \(-0.636896\pi\)
−0.416937 + 0.908935i \(0.636896\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 30.0279 1.42027
\(448\) 0 0
\(449\) −13.1955 −0.622736 −0.311368 0.950289i \(-0.600787\pi\)
−0.311368 + 0.950289i \(0.600787\pi\)
\(450\) 0 0
\(451\) 4.84006 0.227910
\(452\) 0 0
\(453\) 5.10308 0.239764
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −29.1020 −1.36134 −0.680668 0.732592i \(-0.738310\pi\)
−0.680668 + 0.732592i \(0.738310\pi\)
\(458\) 0 0
\(459\) 4.57704 0.213638
\(460\) 0 0
\(461\) −18.8280 −0.876907 −0.438454 0.898754i \(-0.644474\pi\)
−0.438454 + 0.898754i \(0.644474\pi\)
\(462\) 0 0
\(463\) 14.0925 0.654932 0.327466 0.944863i \(-0.393805\pi\)
0.327466 + 0.944863i \(0.393805\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.2663 0.567619 0.283809 0.958881i \(-0.408402\pi\)
0.283809 + 0.958881i \(0.408402\pi\)
\(468\) 0 0
\(469\) −5.79383 −0.267534
\(470\) 0 0
\(471\) −17.8217 −0.821182
\(472\) 0 0
\(473\) −1.08287 −0.0497905
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.698445 −0.0319796
\(478\) 0 0
\(479\) 14.1570 0.646850 0.323425 0.946254i \(-0.395166\pi\)
0.323425 + 0.946254i \(0.395166\pi\)
\(480\) 0 0
\(481\) 32.9450 1.50216
\(482\) 0 0
\(483\) 6.56934 0.298915
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −30.2341 −1.37004 −0.685018 0.728526i \(-0.740206\pi\)
−0.685018 + 0.728526i \(0.740206\pi\)
\(488\) 0 0
\(489\) 0.840061 0.0379889
\(490\) 0 0
\(491\) −39.9065 −1.80096 −0.900478 0.434902i \(-0.856783\pi\)
−0.900478 + 0.434902i \(0.856783\pi\)
\(492\) 0 0
\(493\) −7.06644 −0.318256
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.52311 0.247746
\(498\) 0 0
\(499\) −27.3246 −1.22322 −0.611610 0.791160i \(-0.709478\pi\)
−0.611610 + 0.791160i \(0.709478\pi\)
\(500\) 0 0
\(501\) −23.1955 −1.03630
\(502\) 0 0
\(503\) −1.40171 −0.0624991 −0.0312495 0.999512i \(-0.509949\pi\)
−0.0312495 + 0.999512i \(0.509949\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −30.2095 −1.34165
\(508\) 0 0
\(509\) 18.6585 0.827022 0.413511 0.910499i \(-0.364302\pi\)
0.413511 + 0.910499i \(0.364302\pi\)
\(510\) 0 0
\(511\) 3.72928 0.164974
\(512\) 0 0
\(513\) −32.5972 −1.43920
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −3.68116 −0.161897
\(518\) 0 0
\(519\) −24.6049 −1.08004
\(520\) 0 0
\(521\) −4.02791 −0.176466 −0.0882329 0.996100i \(-0.528122\pi\)
−0.0882329 + 0.996100i \(0.528122\pi\)
\(522\) 0 0
\(523\) −38.4436 −1.68102 −0.840510 0.541796i \(-0.817744\pi\)
−0.840510 + 0.541796i \(0.817744\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.74760 −0.293930
\(528\) 0 0
\(529\) −9.09246 −0.395324
\(530\) 0 0
\(531\) 0.0612237 0.00265688
\(532\) 0 0
\(533\) 42.4402 1.83829
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 13.1387 0.566976
\(538\) 0 0
\(539\) 0.626198 0.0269723
\(540\) 0 0
\(541\) 12.4846 0.536754 0.268377 0.963314i \(-0.413513\pi\)
0.268377 + 0.963314i \(0.413513\pi\)
\(542\) 0 0
\(543\) 24.8680 1.06719
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −37.7851 −1.61557 −0.807787 0.589474i \(-0.799335\pi\)
−0.807787 + 0.589474i \(0.799335\pi\)
\(548\) 0 0
\(549\) 0.735539 0.0313921
\(550\) 0 0
\(551\) 50.3265 2.14398
\(552\) 0 0
\(553\) −5.67243 −0.241216
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.93545 0.0820076 0.0410038 0.999159i \(-0.486944\pi\)
0.0410038 + 0.999159i \(0.486944\pi\)
\(558\) 0 0
\(559\) −9.49521 −0.401604
\(560\) 0 0
\(561\) −0.989374 −0.0417714
\(562\) 0 0
\(563\) 29.8463 1.25787 0.628936 0.777457i \(-0.283491\pi\)
0.628936 + 0.777457i \(0.283491\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9.29862 0.390506
\(568\) 0 0
\(569\) 15.3449 0.643290 0.321645 0.946860i \(-0.395764\pi\)
0.321645 + 0.946860i \(0.395764\pi\)
\(570\) 0 0
\(571\) −35.8217 −1.49909 −0.749547 0.661952i \(-0.769728\pi\)
−0.749547 + 0.661952i \(0.769728\pi\)
\(572\) 0 0
\(573\) −14.8324 −0.619631
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.92482 0.121762 0.0608810 0.998145i \(-0.480609\pi\)
0.0608810 + 0.998145i \(0.480609\pi\)
\(578\) 0 0
\(579\) 39.2803 1.63243
\(580\) 0 0
\(581\) −17.4340 −0.723284
\(582\) 0 0
\(583\) −4.24281 −0.175719
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.5756 −0.807972 −0.403986 0.914765i \(-0.632375\pi\)
−0.403986 + 0.914765i \(0.632375\pi\)
\(588\) 0 0
\(589\) 48.0558 1.98011
\(590\) 0 0
\(591\) −46.4402 −1.91030
\(592\) 0 0
\(593\) 8.00770 0.328837 0.164418 0.986391i \(-0.447425\pi\)
0.164418 + 0.986391i \(0.447425\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 39.5298 1.61785
\(598\) 0 0
\(599\) 29.3005 1.19719 0.598593 0.801053i \(-0.295727\pi\)
0.598593 + 0.801053i \(0.295727\pi\)
\(600\) 0 0
\(601\) 21.3449 0.870675 0.435337 0.900267i \(-0.356629\pi\)
0.435337 + 0.900267i \(0.356629\pi\)
\(602\) 0 0
\(603\) 0.597250 0.0243219
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −9.53081 −0.386844 −0.193422 0.981116i \(-0.561959\pi\)
−0.193422 + 0.981116i \(0.561959\pi\)
\(608\) 0 0
\(609\) −13.8786 −0.562389
\(610\) 0 0
\(611\) −32.2784 −1.30584
\(612\) 0 0
\(613\) −9.75719 −0.394089 −0.197045 0.980395i \(-0.563134\pi\)
−0.197045 + 0.980395i \(0.563134\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −37.9267 −1.52687 −0.763436 0.645884i \(-0.776489\pi\)
−0.763436 + 0.645884i \(0.776489\pi\)
\(618\) 0 0
\(619\) −41.9109 −1.68454 −0.842270 0.539056i \(-0.818781\pi\)
−0.842270 + 0.539056i \(0.818781\pi\)
\(620\) 0 0
\(621\) 19.0308 0.763681
\(622\) 0 0
\(623\) −14.2986 −0.572862
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 7.04623 0.281399
\(628\) 0 0
\(629\) 5.38150 0.214574
\(630\) 0 0
\(631\) 8.42003 0.335196 0.167598 0.985855i \(-0.446399\pi\)
0.167598 + 0.985855i \(0.446399\pi\)
\(632\) 0 0
\(633\) −26.7678 −1.06393
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.49084 0.217555
\(638\) 0 0
\(639\) −0.569343 −0.0225229
\(640\) 0 0
\(641\) −2.07707 −0.0820392 −0.0410196 0.999158i \(-0.513061\pi\)
−0.0410196 + 0.999158i \(0.513061\pi\)
\(642\) 0 0
\(643\) 27.2480 1.07456 0.537279 0.843405i \(-0.319452\pi\)
0.537279 + 0.843405i \(0.319452\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.3632 −0.486047 −0.243023 0.970020i \(-0.578139\pi\)
−0.243023 + 0.970020i \(0.578139\pi\)
\(648\) 0 0
\(649\) 0.371913 0.0145989
\(650\) 0 0
\(651\) −13.2524 −0.519402
\(652\) 0 0
\(653\) 37.3449 1.46142 0.730709 0.682690i \(-0.239190\pi\)
0.730709 + 0.682690i \(0.239190\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −0.384428 −0.0149980
\(658\) 0 0
\(659\) 14.1772 0.552266 0.276133 0.961119i \(-0.410947\pi\)
0.276133 + 0.961119i \(0.410947\pi\)
\(660\) 0 0
\(661\) −1.76925 −0.0688160 −0.0344080 0.999408i \(-0.510955\pi\)
−0.0344080 + 0.999408i \(0.510955\pi\)
\(662\) 0 0
\(663\) −8.67536 −0.336923
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −29.3815 −1.13766
\(668\) 0 0
\(669\) −12.1493 −0.469720
\(670\) 0 0
\(671\) 4.46815 0.172491
\(672\) 0 0
\(673\) 4.05581 0.156340 0.0781701 0.996940i \(-0.475092\pi\)
0.0781701 + 0.996940i \(0.475092\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −9.37713 −0.360392 −0.180196 0.983631i \(-0.557673\pi\)
−0.180196 + 0.983631i \(0.557673\pi\)
\(678\) 0 0
\(679\) 10.1493 0.389495
\(680\) 0 0
\(681\) −4.89692 −0.187650
\(682\) 0 0
\(683\) 5.49521 0.210268 0.105134 0.994458i \(-0.466473\pi\)
0.105134 + 0.994458i \(0.466473\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −32.8680 −1.25399
\(688\) 0 0
\(689\) −37.2032 −1.41733
\(690\) 0 0
\(691\) 3.03416 0.115425 0.0577125 0.998333i \(-0.481619\pi\)
0.0577125 + 0.998333i \(0.481619\pi\)
\(692\) 0 0
\(693\) −0.0645508 −0.00245208
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 6.93252 0.262588
\(698\) 0 0
\(699\) 19.4586 0.735990
\(700\) 0 0
\(701\) 35.2234 1.33037 0.665186 0.746678i \(-0.268352\pi\)
0.665186 + 0.746678i \(0.268352\pi\)
\(702\) 0 0
\(703\) −38.3265 −1.44551
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −9.64015 −0.362555
\(708\) 0 0
\(709\) 28.7187 1.07855 0.539276 0.842129i \(-0.318698\pi\)
0.539276 + 0.842129i \(0.318698\pi\)
\(710\) 0 0
\(711\) 0.584735 0.0219293
\(712\) 0 0
\(713\) −28.0558 −1.05070
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −35.4942 −1.32555
\(718\) 0 0
\(719\) 8.60599 0.320949 0.160475 0.987040i \(-0.448698\pi\)
0.160475 + 0.987040i \(0.448698\pi\)
\(720\) 0 0
\(721\) −0.626198 −0.0233208
\(722\) 0 0
\(723\) −6.56934 −0.244316
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 15.2803 0.566715 0.283358 0.959014i \(-0.408552\pi\)
0.283358 + 0.959014i \(0.408552\pi\)
\(728\) 0 0
\(729\) 26.0096 0.963318
\(730\) 0 0
\(731\) −1.55102 −0.0573666
\(732\) 0 0
\(733\) −46.0235 −1.69992 −0.849959 0.526849i \(-0.823373\pi\)
−0.849959 + 0.526849i \(0.823373\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.62809 0.133642
\(738\) 0 0
\(739\) −46.3467 −1.70489 −0.852446 0.522815i \(-0.824882\pi\)
−0.852446 + 0.522815i \(0.824882\pi\)
\(740\) 0 0
\(741\) 61.7851 2.26973
\(742\) 0 0
\(743\) −10.7755 −0.395315 −0.197658 0.980271i \(-0.563333\pi\)
−0.197658 + 0.980271i \(0.563333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.79716 0.0657546
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.76781 −0.100999 −0.0504995 0.998724i \(-0.516081\pi\)
−0.0504995 + 0.998724i \(0.516081\pi\)
\(752\) 0 0
\(753\) −37.8709 −1.38009
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 14.8401 0.539371 0.269686 0.962948i \(-0.413080\pi\)
0.269686 + 0.962948i \(0.413080\pi\)
\(758\) 0 0
\(759\) −4.11371 −0.149318
\(760\) 0 0
\(761\) 31.3169 1.13524 0.567619 0.823291i \(-0.307865\pi\)
0.567619 + 0.823291i \(0.307865\pi\)
\(762\) 0 0
\(763\) −18.9248 −0.685125
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.26113 0.117753
\(768\) 0 0
\(769\) 8.92586 0.321875 0.160937 0.986965i \(-0.448548\pi\)
0.160937 + 0.986965i \(0.448548\pi\)
\(770\) 0 0
\(771\) −7.52311 −0.270938
\(772\) 0 0
\(773\) 36.7711 1.32257 0.661283 0.750136i \(-0.270012\pi\)
0.661283 + 0.750136i \(0.270012\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 10.5693 0.379173
\(778\) 0 0
\(779\) −49.3728 −1.76896
\(780\) 0 0
\(781\) −3.45856 −0.123757
\(782\) 0 0
\(783\) −40.2051 −1.43681
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −6.53707 −0.233021 −0.116511 0.993189i \(-0.537171\pi\)
−0.116511 + 0.993189i \(0.537171\pi\)
\(788\) 0 0
\(789\) −20.1849 −0.718602
\(790\) 0 0
\(791\) 1.04623 0.0371996
\(792\) 0 0
\(793\) 39.1791 1.39129
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.82611 0.277215 0.138607 0.990347i \(-0.455737\pi\)
0.138607 + 0.990347i \(0.455737\pi\)
\(798\) 0 0
\(799\) −5.27261 −0.186531
\(800\) 0 0
\(801\) 1.47396 0.0520796
\(802\) 0 0
\(803\) −2.33527 −0.0824099
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 12.4556 0.438459
\(808\) 0 0
\(809\) 38.4113 1.35047 0.675235 0.737603i \(-0.264042\pi\)
0.675235 + 0.737603i \(0.264042\pi\)
\(810\) 0 0
\(811\) −7.64015 −0.268282 −0.134141 0.990962i \(-0.542828\pi\)
−0.134141 + 0.990962i \(0.542828\pi\)
\(812\) 0 0
\(813\) 29.9634 1.05086
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 11.0462 0.386459
\(818\) 0 0
\(819\) −0.566016 −0.0197782
\(820\) 0 0
\(821\) 2.56165 0.0894021 0.0447011 0.999000i \(-0.485766\pi\)
0.0447011 + 0.999000i \(0.485766\pi\)
\(822\) 0 0
\(823\) −0.784248 −0.0273372 −0.0136686 0.999907i \(-0.504351\pi\)
−0.0136686 + 0.999907i \(0.504351\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −7.28904 −0.253465 −0.126732 0.991937i \(-0.540449\pi\)
−0.126732 + 0.991937i \(0.540449\pi\)
\(828\) 0 0
\(829\) −11.0708 −0.384505 −0.192253 0.981345i \(-0.561579\pi\)
−0.192253 + 0.981345i \(0.561579\pi\)
\(830\) 0 0
\(831\) 33.0741 1.14733
\(832\) 0 0
\(833\) 0.896916 0.0310763
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −38.3911 −1.32699
\(838\) 0 0
\(839\) 5.55976 0.191944 0.0959721 0.995384i \(-0.469404\pi\)
0.0959721 + 0.995384i \(0.469404\pi\)
\(840\) 0 0
\(841\) 33.0722 1.14042
\(842\) 0 0
\(843\) 8.10015 0.278984
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 10.6079 0.364491
\(848\) 0 0
\(849\) −23.9278 −0.821198
\(850\) 0 0
\(851\) 22.3757 0.767029
\(852\) 0 0
\(853\) 16.0804 0.550582 0.275291 0.961361i \(-0.411226\pi\)
0.275291 + 0.961361i \(0.411226\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.58767 0.0542336 0.0271168 0.999632i \(-0.491367\pi\)
0.0271168 + 0.999632i \(0.491367\pi\)
\(858\) 0 0
\(859\) −4.94171 −0.168609 −0.0843044 0.996440i \(-0.526867\pi\)
−0.0843044 + 0.996440i \(0.526867\pi\)
\(860\) 0 0
\(861\) 13.6156 0.464017
\(862\) 0 0
\(863\) 37.6647 1.28212 0.641061 0.767490i \(-0.278494\pi\)
0.641061 + 0.767490i \(0.278494\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 28.5294 0.968908
\(868\) 0 0
\(869\) 3.55206 0.120495
\(870\) 0 0
\(871\) 31.8130 1.07794
\(872\) 0 0
\(873\) −1.04623 −0.0354095
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −3.66473 −0.123749 −0.0618746 0.998084i \(-0.519708\pi\)
−0.0618746 + 0.998084i \(0.519708\pi\)
\(878\) 0 0
\(879\) −20.8324 −0.702658
\(880\) 0 0
\(881\) −43.4373 −1.46344 −0.731720 0.681605i \(-0.761282\pi\)
−0.731720 + 0.681605i \(0.761282\pi\)
\(882\) 0 0
\(883\) 47.3853 1.59464 0.797321 0.603556i \(-0.206250\pi\)
0.797321 + 0.603556i \(0.206250\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −21.3169 −0.715753 −0.357877 0.933769i \(-0.616499\pi\)
−0.357877 + 0.933769i \(0.616499\pi\)
\(888\) 0 0
\(889\) −21.2803 −0.713718
\(890\) 0 0
\(891\) −5.82278 −0.195071
\(892\) 0 0
\(893\) 37.5510 1.25660
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −36.0712 −1.20438
\(898\) 0 0
\(899\) 59.2716 1.97682
\(900\) 0 0
\(901\) −6.07707 −0.202456
\(902\) 0 0
\(903\) −3.04623 −0.101372
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 18.9325 0.628644 0.314322 0.949316i \(-0.398223\pi\)
0.314322 + 0.949316i \(0.398223\pi\)
\(908\) 0 0
\(909\) 0.993743 0.0329604
\(910\) 0 0
\(911\) 12.6705 0.419794 0.209897 0.977724i \(-0.432687\pi\)
0.209897 + 0.977724i \(0.432687\pi\)
\(912\) 0 0
\(913\) 10.9171 0.361304
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.91087 −0.327286
\(918\) 0 0
\(919\) −22.8690 −0.754379 −0.377190 0.926136i \(-0.623109\pi\)
−0.377190 + 0.926136i \(0.623109\pi\)
\(920\) 0 0
\(921\) −36.9681 −1.21814
\(922\) 0 0
\(923\) −30.3265 −0.998210
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.0645508 0.00212013
\(928\) 0 0
\(929\) 50.4190 1.65419 0.827097 0.562060i \(-0.189991\pi\)
0.827097 + 0.562060i \(0.189991\pi\)
\(930\) 0 0
\(931\) −6.38776 −0.209350
\(932\) 0 0
\(933\) −39.6435 −1.29787
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −47.9344 −1.56595 −0.782974 0.622054i \(-0.786298\pi\)
−0.782974 + 0.622054i \(0.786298\pi\)
\(938\) 0 0
\(939\) 21.9923 0.717692
\(940\) 0 0
\(941\) 59.9667 1.95486 0.977429 0.211264i \(-0.0677580\pi\)
0.977429 + 0.211264i \(0.0677580\pi\)
\(942\) 0 0
\(943\) 28.8247 0.938660
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.5231 0.894381 0.447191 0.894439i \(-0.352425\pi\)
0.447191 + 0.894439i \(0.352425\pi\)
\(948\) 0 0
\(949\) −20.4769 −0.664708
\(950\) 0 0
\(951\) −44.9604 −1.45794
\(952\) 0 0
\(953\) 0.840061 0.0272123 0.0136061 0.999907i \(-0.495669\pi\)
0.0136061 + 0.999907i \(0.495669\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 8.69075 0.280932
\(958\) 0 0
\(959\) −3.45856 −0.111683
\(960\) 0 0
\(961\) 25.5972 0.825718
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −31.8988 −1.02580 −0.512898 0.858449i \(-0.671428\pi\)
−0.512898 + 0.858449i \(0.671428\pi\)
\(968\) 0 0
\(969\) 10.0925 0.324216
\(970\) 0 0
\(971\) −28.4157 −0.911902 −0.455951 0.890005i \(-0.650701\pi\)
−0.455951 + 0.890005i \(0.650701\pi\)
\(972\) 0 0
\(973\) −20.4157 −0.654496
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.55102 0.305564 0.152782 0.988260i \(-0.451177\pi\)
0.152782 + 0.988260i \(0.451177\pi\)
\(978\) 0 0
\(979\) 8.95377 0.286164
\(980\) 0 0
\(981\) 1.95084 0.0622856
\(982\) 0 0
\(983\) 0.0443400 0.00141423 0.000707113 1.00000i \(-0.499775\pi\)
0.000707113 1.00000i \(0.499775\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −10.3555 −0.329619
\(988\) 0 0
\(989\) −6.44898 −0.205066
\(990\) 0 0
\(991\) 13.9913 0.444447 0.222224 0.974996i \(-0.428669\pi\)
0.222224 + 0.974996i \(0.428669\pi\)
\(992\) 0 0
\(993\) −4.72635 −0.149986
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −19.0419 −0.603062 −0.301531 0.953456i \(-0.597498\pi\)
−0.301531 + 0.953456i \(0.597498\pi\)
\(998\) 0 0
\(999\) 30.6185 0.968727
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 2800.2.a.br.1.1 3
4.3 odd 2 1400.2.a.s.1.3 3
5.2 odd 4 560.2.g.f.449.5 6
5.3 odd 4 560.2.g.f.449.2 6
5.4 even 2 2800.2.a.bq.1.3 3
15.2 even 4 5040.2.t.y.1009.4 6
15.8 even 4 5040.2.t.y.1009.3 6
20.3 even 4 280.2.g.b.169.5 yes 6
20.7 even 4 280.2.g.b.169.2 6
20.19 odd 2 1400.2.a.t.1.1 3
28.27 even 2 9800.2.a.cg.1.1 3
40.3 even 4 2240.2.g.l.449.2 6
40.13 odd 4 2240.2.g.m.449.5 6
40.27 even 4 2240.2.g.l.449.5 6
40.37 odd 4 2240.2.g.m.449.2 6
60.23 odd 4 2520.2.t.g.1009.3 6
60.47 odd 4 2520.2.t.g.1009.4 6
140.27 odd 4 1960.2.g.c.1569.5 6
140.83 odd 4 1960.2.g.c.1569.2 6
140.139 even 2 9800.2.a.cd.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.g.b.169.2 6 20.7 even 4
280.2.g.b.169.5 yes 6 20.3 even 4
560.2.g.f.449.2 6 5.3 odd 4
560.2.g.f.449.5 6 5.2 odd 4
1400.2.a.s.1.3 3 4.3 odd 2
1400.2.a.t.1.1 3 20.19 odd 2
1960.2.g.c.1569.2 6 140.83 odd 4
1960.2.g.c.1569.5 6 140.27 odd 4
2240.2.g.l.449.2 6 40.3 even 4
2240.2.g.l.449.5 6 40.27 even 4
2240.2.g.m.449.2 6 40.37 odd 4
2240.2.g.m.449.5 6 40.13 odd 4
2520.2.t.g.1009.3 6 60.23 odd 4
2520.2.t.g.1009.4 6 60.47 odd 4
2800.2.a.bq.1.3 3 5.4 even 2
2800.2.a.br.1.1 3 1.1 even 1 trivial
5040.2.t.y.1009.3 6 15.8 even 4
5040.2.t.y.1009.4 6 15.2 even 4
9800.2.a.cd.1.3 3 140.139 even 2
9800.2.a.cg.1.1 3 28.27 even 2