Properties

Label 9200.2.a.cv.1.3
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.4623698596\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.521397.1
Defining polynomial: \(x^{5} - 9 x^{3} - 3 x^{2} + 18 x + 12\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4600)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.794805\) of defining polynomial
Character \(\chi\) \(=\) 9200.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.794805 q^{3} +2.47193 q^{7} -2.36829 q^{9} +O(q^{10})\) \(q-0.794805 q^{3} +2.47193 q^{7} -2.36829 q^{9} +2.29993 q^{11} +3.84022 q^{13} +7.74682 q^{17} +2.29993 q^{19} -1.96471 q^{21} -1.00000 q^{23} +4.26674 q^{27} +5.28380 q^{29} +6.40148 q^{31} -1.82800 q^{33} -8.56457 q^{37} -3.05223 q^{39} +4.27699 q^{41} -1.88954 q^{43} +12.3432 q^{47} -0.889540 q^{49} -6.15721 q^{51} +7.57482 q^{53} -1.82800 q^{57} -6.07180 q^{59} -0.635155 q^{61} -5.85425 q^{63} +11.1333 q^{67} +0.794805 q^{69} -8.58163 q^{71} -16.5849 q^{73} +5.68528 q^{77} -0.335225 q^{79} +3.71363 q^{81} +15.1937 q^{83} -4.19959 q^{87} +5.55735 q^{89} +9.49277 q^{91} -5.08792 q^{93} -6.42786 q^{97} -5.44689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{7} + 3 q^{9} + O(q^{10}) \) \( 5 q + 4 q^{7} + 3 q^{9} - 4 q^{13} - 6 q^{17} - 5 q^{23} + 9 q^{27} + 12 q^{29} + 18 q^{31} - 6 q^{33} - 10 q^{37} - 9 q^{39} - 6 q^{41} + 10 q^{43} + 22 q^{47} + 15 q^{49} + 6 q^{51} - 10 q^{53} - 6 q^{57} + q^{59} + 10 q^{61} + 8 q^{67} - 8 q^{71} - 6 q^{73} - 27 q^{81} - 2 q^{83} + 39 q^{87} + 14 q^{89} + 46 q^{91} + 3 q^{93} - 6 q^{97} + 6 q^{99} + O(q^{100}) \)

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
\(3\) −0.794805 −0.458881 −0.229440 0.973323i \(-0.573690\pi\)
−0.229440 + 0.973323i \(0.573690\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.47193 0.934303 0.467152 0.884177i \(-0.345280\pi\)
0.467152 + 0.884177i \(0.345280\pi\)
\(8\) 0 0
\(9\) −2.36829 −0.789428
\(10\) 0 0
\(11\) 2.29993 0.693455 0.346728 0.937966i \(-0.387293\pi\)
0.346728 + 0.937966i \(0.387293\pi\)
\(12\) 0 0
\(13\) 3.84022 1.06509 0.532543 0.846403i \(-0.321237\pi\)
0.532543 + 0.846403i \(0.321237\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.74682 1.87888 0.939440 0.342713i \(-0.111346\pi\)
0.939440 + 0.342713i \(0.111346\pi\)
\(18\) 0 0
\(19\) 2.29993 0.527640 0.263820 0.964572i \(-0.415017\pi\)
0.263820 + 0.964572i \(0.415017\pi\)
\(20\) 0 0
\(21\) −1.96471 −0.428734
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.26674 0.821134
\(28\) 0 0
\(29\) 5.28380 0.981177 0.490589 0.871391i \(-0.336782\pi\)
0.490589 + 0.871391i \(0.336782\pi\)
\(30\) 0 0
\(31\) 6.40148 1.14974 0.574870 0.818245i \(-0.305053\pi\)
0.574870 + 0.818245i \(0.305053\pi\)
\(32\) 0 0
\(33\) −1.82800 −0.318213
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.56457 −1.40801 −0.704003 0.710197i \(-0.748606\pi\)
−0.704003 + 0.710197i \(0.748606\pi\)
\(38\) 0 0
\(39\) −3.05223 −0.488747
\(40\) 0 0
\(41\) 4.27699 0.667954 0.333977 0.942581i \(-0.391609\pi\)
0.333977 + 0.942581i \(0.391609\pi\)
\(42\) 0 0
\(43\) −1.88954 −0.288152 −0.144076 0.989567i \(-0.546021\pi\)
−0.144076 + 0.989567i \(0.546021\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.3432 1.80045 0.900223 0.435428i \(-0.143403\pi\)
0.900223 + 0.435428i \(0.143403\pi\)
\(48\) 0 0
\(49\) −0.889540 −0.127077
\(50\) 0 0
\(51\) −6.15721 −0.862182
\(52\) 0 0
\(53\) 7.57482 1.04048 0.520241 0.854020i \(-0.325842\pi\)
0.520241 + 0.854020i \(0.325842\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.82800 −0.242124
\(58\) 0 0
\(59\) −6.07180 −0.790480 −0.395240 0.918578i \(-0.629339\pi\)
−0.395240 + 0.918578i \(0.629339\pi\)
\(60\) 0 0
\(61\) −0.635155 −0.0813233 −0.0406616 0.999173i \(-0.512947\pi\)
−0.0406616 + 0.999173i \(0.512947\pi\)
\(62\) 0 0
\(63\) −5.85425 −0.737566
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.1333 1.36015 0.680077 0.733141i \(-0.261946\pi\)
0.680077 + 0.733141i \(0.261946\pi\)
\(68\) 0 0
\(69\) 0.794805 0.0956833
\(70\) 0 0
\(71\) −8.58163 −1.01845 −0.509226 0.860633i \(-0.670068\pi\)
−0.509226 + 0.860633i \(0.670068\pi\)
\(72\) 0 0
\(73\) −16.5849 −1.94112 −0.970560 0.240859i \(-0.922571\pi\)
−0.970560 + 0.240859i \(0.922571\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.68528 0.647897
\(78\) 0 0
\(79\) −0.335225 −0.0377157 −0.0188579 0.999822i \(-0.506003\pi\)
−0.0188579 + 0.999822i \(0.506003\pi\)
\(80\) 0 0
\(81\) 3.71363 0.412626
\(82\) 0 0
\(83\) 15.1937 1.66773 0.833863 0.551971i \(-0.186124\pi\)
0.833863 + 0.551971i \(0.186124\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.19959 −0.450243
\(88\) 0 0
\(89\) 5.55735 0.589078 0.294539 0.955639i \(-0.404834\pi\)
0.294539 + 0.955639i \(0.404834\pi\)
\(90\) 0 0
\(91\) 9.49277 0.995113
\(92\) 0 0
\(93\) −5.08792 −0.527593
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.42786 −0.652650 −0.326325 0.945258i \(-0.605810\pi\)
−0.326325 + 0.945258i \(0.605810\pi\)
\(98\) 0 0
\(99\) −5.44689 −0.547433
\(100\) 0 0
\(101\) −14.6041 −1.45316 −0.726581 0.687081i \(-0.758892\pi\)
−0.726581 + 0.687081i \(0.758892\pi\)
\(102\) 0 0
\(103\) −13.3980 −1.32014 −0.660071 0.751203i \(-0.729474\pi\)
−0.660071 + 0.751203i \(0.729474\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.63549 0.158109 0.0790544 0.996870i \(-0.474810\pi\)
0.0790544 + 0.996870i \(0.474810\pi\)
\(108\) 0 0
\(109\) 1.50182 0.143848 0.0719239 0.997410i \(-0.477086\pi\)
0.0719239 + 0.997410i \(0.477086\pi\)
\(110\) 0 0
\(111\) 6.80716 0.646107
\(112\) 0 0
\(113\) −17.6451 −1.65992 −0.829958 0.557826i \(-0.811636\pi\)
−0.829958 + 0.557826i \(0.811636\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −9.09474 −0.840809
\(118\) 0 0
\(119\) 19.1496 1.75544
\(120\) 0 0
\(121\) −5.71032 −0.519120
\(122\) 0 0
\(123\) −3.39937 −0.306511
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.97362 0.352602 0.176301 0.984336i \(-0.443587\pi\)
0.176301 + 0.984336i \(0.443587\pi\)
\(128\) 0 0
\(129\) 1.50182 0.132228
\(130\) 0 0
\(131\) 14.3032 1.24968 0.624840 0.780753i \(-0.285164\pi\)
0.624840 + 0.780753i \(0.285164\pi\)
\(132\) 0 0
\(133\) 5.68528 0.492976
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.25585 −0.363602 −0.181801 0.983335i \(-0.558193\pi\)
−0.181801 + 0.983335i \(0.558193\pi\)
\(138\) 0 0
\(139\) 9.86950 0.837120 0.418560 0.908189i \(-0.362535\pi\)
0.418560 + 0.908189i \(0.362535\pi\)
\(140\) 0 0
\(141\) −9.81047 −0.826191
\(142\) 0 0
\(143\) 8.83224 0.738589
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.707011 0.0583133
\(148\) 0 0
\(149\) 9.55735 0.782969 0.391484 0.920185i \(-0.371962\pi\)
0.391484 + 0.920185i \(0.371962\pi\)
\(150\) 0 0
\(151\) 24.4557 1.99018 0.995090 0.0989730i \(-0.0315557\pi\)
0.995090 + 0.0989730i \(0.0315557\pi\)
\(152\) 0 0
\(153\) −18.3467 −1.48324
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.1922 −0.973047 −0.486524 0.873667i \(-0.661735\pi\)
−0.486524 + 0.873667i \(0.661735\pi\)
\(158\) 0 0
\(159\) −6.02050 −0.477457
\(160\) 0 0
\(161\) −2.47193 −0.194816
\(162\) 0 0
\(163\) −18.5762 −1.45500 −0.727498 0.686109i \(-0.759317\pi\)
−0.727498 + 0.686109i \(0.759317\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.8639 −1.30497 −0.652484 0.757803i \(-0.726273\pi\)
−0.652484 + 0.757803i \(0.726273\pi\)
\(168\) 0 0
\(169\) 1.74729 0.134407
\(170\) 0 0
\(171\) −5.44689 −0.416534
\(172\) 0 0
\(173\) −24.4626 −1.85985 −0.929927 0.367745i \(-0.880130\pi\)
−0.929927 + 0.367745i \(0.880130\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.82589 0.362736
\(178\) 0 0
\(179\) −2.26173 −0.169050 −0.0845249 0.996421i \(-0.526937\pi\)
−0.0845249 + 0.996421i \(0.526937\pi\)
\(180\) 0 0
\(181\) 14.7187 1.09404 0.547018 0.837121i \(-0.315763\pi\)
0.547018 + 0.837121i \(0.315763\pi\)
\(182\) 0 0
\(183\) 0.504824 0.0373177
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 17.8171 1.30292
\(188\) 0 0
\(189\) 10.5471 0.767189
\(190\) 0 0
\(191\) 23.7055 1.71527 0.857636 0.514258i \(-0.171933\pi\)
0.857636 + 0.514258i \(0.171933\pi\)
\(192\) 0 0
\(193\) 20.4146 1.46947 0.734736 0.678353i \(-0.237306\pi\)
0.734736 + 0.678353i \(0.237306\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −17.7607 −1.26540 −0.632699 0.774398i \(-0.718053\pi\)
−0.632699 + 0.774398i \(0.718053\pi\)
\(198\) 0 0
\(199\) −11.1379 −0.789546 −0.394773 0.918779i \(-0.629177\pi\)
−0.394773 + 0.918779i \(0.629177\pi\)
\(200\) 0 0
\(201\) −8.84883 −0.624149
\(202\) 0 0
\(203\) 13.0612 0.916717
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.36829 0.164607
\(208\) 0 0
\(209\) 5.28968 0.365895
\(210\) 0 0
\(211\) −4.17347 −0.287314 −0.143657 0.989628i \(-0.545886\pi\)
−0.143657 + 0.989628i \(0.545886\pi\)
\(212\) 0 0
\(213\) 6.82072 0.467348
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 15.8240 1.07421
\(218\) 0 0
\(219\) 13.1818 0.890743
\(220\) 0 0
\(221\) 29.7495 2.00117
\(222\) 0 0
\(223\) 11.4218 0.764863 0.382432 0.923984i \(-0.375087\pi\)
0.382432 + 0.923984i \(0.375087\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.40014 0.358420 0.179210 0.983811i \(-0.442646\pi\)
0.179210 + 0.983811i \(0.442646\pi\)
\(228\) 0 0
\(229\) −18.4333 −1.21811 −0.609054 0.793129i \(-0.708451\pi\)
−0.609054 + 0.793129i \(0.708451\pi\)
\(230\) 0 0
\(231\) −4.51869 −0.297308
\(232\) 0 0
\(233\) 7.82122 0.512385 0.256193 0.966626i \(-0.417532\pi\)
0.256193 + 0.966626i \(0.417532\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.266438 0.0173070
\(238\) 0 0
\(239\) 2.26133 0.146273 0.0731365 0.997322i \(-0.476699\pi\)
0.0731365 + 0.997322i \(0.476699\pi\)
\(240\) 0 0
\(241\) 13.4906 0.869006 0.434503 0.900670i \(-0.356924\pi\)
0.434503 + 0.900670i \(0.356924\pi\)
\(242\) 0 0
\(243\) −15.7518 −1.01048
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.83224 0.561982
\(248\) 0 0
\(249\) −12.0760 −0.765288
\(250\) 0 0
\(251\) −15.5642 −0.982406 −0.491203 0.871045i \(-0.663443\pi\)
−0.491203 + 0.871045i \(0.663443\pi\)
\(252\) 0 0
\(253\) −2.29993 −0.144595
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −25.2354 −1.57414 −0.787069 0.616865i \(-0.788402\pi\)
−0.787069 + 0.616865i \(0.788402\pi\)
\(258\) 0 0
\(259\) −21.1710 −1.31550
\(260\) 0 0
\(261\) −12.5135 −0.774569
\(262\) 0 0
\(263\) −12.6243 −0.778448 −0.389224 0.921143i \(-0.627257\pi\)
−0.389224 + 0.921143i \(0.627257\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −4.41701 −0.270317
\(268\) 0 0
\(269\) 25.0644 1.52820 0.764101 0.645096i \(-0.223183\pi\)
0.764101 + 0.645096i \(0.223183\pi\)
\(270\) 0 0
\(271\) 17.6777 1.07384 0.536922 0.843632i \(-0.319587\pi\)
0.536922 + 0.843632i \(0.319587\pi\)
\(272\) 0 0
\(273\) −7.54490 −0.456638
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.22724 0.434243 0.217121 0.976145i \(-0.430333\pi\)
0.217121 + 0.976145i \(0.430333\pi\)
\(278\) 0 0
\(279\) −15.1605 −0.907637
\(280\) 0 0
\(281\) −5.61315 −0.334852 −0.167426 0.985885i \(-0.553546\pi\)
−0.167426 + 0.985885i \(0.553546\pi\)
\(282\) 0 0
\(283\) −0.873541 −0.0519266 −0.0259633 0.999663i \(-0.508265\pi\)
−0.0259633 + 0.999663i \(0.508265\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.5724 0.624071
\(288\) 0 0
\(289\) 43.0132 2.53019
\(290\) 0 0
\(291\) 5.10889 0.299489
\(292\) 0 0
\(293\) −10.2737 −0.600195 −0.300097 0.953909i \(-0.597019\pi\)
−0.300097 + 0.953909i \(0.597019\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 9.81320 0.569420
\(298\) 0 0
\(299\) −3.84022 −0.222086
\(300\) 0 0
\(301\) −4.67082 −0.269222
\(302\) 0 0
\(303\) 11.6074 0.666828
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 26.3994 1.50670 0.753348 0.657622i \(-0.228438\pi\)
0.753348 + 0.657622i \(0.228438\pi\)
\(308\) 0 0
\(309\) 10.6488 0.605788
\(310\) 0 0
\(311\) −31.6429 −1.79430 −0.897151 0.441725i \(-0.854367\pi\)
−0.897151 + 0.441725i \(0.854367\pi\)
\(312\) 0 0
\(313\) 24.1327 1.36406 0.682032 0.731323i \(-0.261097\pi\)
0.682032 + 0.731323i \(0.261097\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.973154 0.0546578 0.0273289 0.999626i \(-0.491300\pi\)
0.0273289 + 0.999626i \(0.491300\pi\)
\(318\) 0 0
\(319\) 12.1524 0.680402
\(320\) 0 0
\(321\) −1.29990 −0.0725531
\(322\) 0 0
\(323\) 17.8171 0.991373
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −1.19365 −0.0660090
\(328\) 0 0
\(329\) 30.5117 1.68216
\(330\) 0 0
\(331\) −1.10352 −0.0606549 −0.0303274 0.999540i \(-0.509655\pi\)
−0.0303274 + 0.999540i \(0.509655\pi\)
\(332\) 0 0
\(333\) 20.2833 1.11152
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.33526 0.508524 0.254262 0.967135i \(-0.418167\pi\)
0.254262 + 0.967135i \(0.418167\pi\)
\(338\) 0 0
\(339\) 14.0244 0.761703
\(340\) 0 0
\(341\) 14.7229 0.797292
\(342\) 0 0
\(343\) −19.5024 −1.05303
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.4052 1.04173 0.520863 0.853641i \(-0.325610\pi\)
0.520863 + 0.853641i \(0.325610\pi\)
\(348\) 0 0
\(349\) 26.0825 1.39616 0.698081 0.716019i \(-0.254038\pi\)
0.698081 + 0.716019i \(0.254038\pi\)
\(350\) 0 0
\(351\) 16.3852 0.874578
\(352\) 0 0
\(353\) −24.5504 −1.30669 −0.653343 0.757062i \(-0.726634\pi\)
−0.653343 + 0.757062i \(0.726634\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −15.2202 −0.805540
\(358\) 0 0
\(359\) 10.2809 0.542607 0.271303 0.962494i \(-0.412545\pi\)
0.271303 + 0.962494i \(0.412545\pi\)
\(360\) 0 0
\(361\) −13.7103 −0.721596
\(362\) 0 0
\(363\) 4.53859 0.238214
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −13.2290 −0.690549 −0.345274 0.938502i \(-0.612214\pi\)
−0.345274 + 0.938502i \(0.612214\pi\)
\(368\) 0 0
\(369\) −10.1291 −0.527302
\(370\) 0 0
\(371\) 18.7245 0.972125
\(372\) 0 0
\(373\) −34.2573 −1.77378 −0.886889 0.461983i \(-0.847138\pi\)
−0.886889 + 0.461983i \(0.847138\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.2910 1.04504
\(378\) 0 0
\(379\) 10.2939 0.528763 0.264382 0.964418i \(-0.414832\pi\)
0.264382 + 0.964418i \(0.414832\pi\)
\(380\) 0 0
\(381\) −3.15825 −0.161802
\(382\) 0 0
\(383\) −26.2028 −1.33890 −0.669450 0.742857i \(-0.733470\pi\)
−0.669450 + 0.742857i \(0.733470\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.47497 0.227476
\(388\) 0 0
\(389\) 5.59078 0.283464 0.141732 0.989905i \(-0.454733\pi\)
0.141732 + 0.989905i \(0.454733\pi\)
\(390\) 0 0
\(391\) −7.74682 −0.391774
\(392\) 0 0
\(393\) −11.3683 −0.573454
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.98084 0.149604 0.0748019 0.997198i \(-0.476168\pi\)
0.0748019 + 0.997198i \(0.476168\pi\)
\(398\) 0 0
\(399\) −4.51869 −0.226217
\(400\) 0 0
\(401\) −19.8229 −0.989906 −0.494953 0.868920i \(-0.664815\pi\)
−0.494953 + 0.868920i \(0.664815\pi\)
\(402\) 0 0
\(403\) 24.5831 1.22457
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −19.6979 −0.976389
\(408\) 0 0
\(409\) 12.8521 0.635498 0.317749 0.948175i \(-0.397073\pi\)
0.317749 + 0.948175i \(0.397073\pi\)
\(410\) 0 0
\(411\) 3.38257 0.166850
\(412\) 0 0
\(413\) −15.0091 −0.738549
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7.84433 −0.384139
\(418\) 0 0
\(419\) 5.42665 0.265109 0.132555 0.991176i \(-0.457682\pi\)
0.132555 + 0.991176i \(0.457682\pi\)
\(420\) 0 0
\(421\) 19.2456 0.937973 0.468987 0.883205i \(-0.344619\pi\)
0.468987 + 0.883205i \(0.344619\pi\)
\(422\) 0 0
\(423\) −29.2323 −1.42132
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.57006 −0.0759806
\(428\) 0 0
\(429\) −7.01991 −0.338924
\(430\) 0 0
\(431\) −8.53348 −0.411043 −0.205522 0.978653i \(-0.565889\pi\)
−0.205522 + 0.978653i \(0.565889\pi\)
\(432\) 0 0
\(433\) −22.7745 −1.09447 −0.547237 0.836978i \(-0.684320\pi\)
−0.547237 + 0.836978i \(0.684320\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.29993 −0.110021
\(438\) 0 0
\(439\) −4.67135 −0.222952 −0.111476 0.993767i \(-0.535558\pi\)
−0.111476 + 0.993767i \(0.535558\pi\)
\(440\) 0 0
\(441\) 2.10668 0.100318
\(442\) 0 0
\(443\) 3.00378 0.142714 0.0713568 0.997451i \(-0.477267\pi\)
0.0713568 + 0.997451i \(0.477267\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −7.59623 −0.359289
\(448\) 0 0
\(449\) 5.56914 0.262824 0.131412 0.991328i \(-0.458049\pi\)
0.131412 + 0.991328i \(0.458049\pi\)
\(450\) 0 0
\(451\) 9.83678 0.463196
\(452\) 0 0
\(453\) −19.4375 −0.913256
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 31.0180 1.45096 0.725481 0.688242i \(-0.241617\pi\)
0.725481 + 0.688242i \(0.241617\pi\)
\(458\) 0 0
\(459\) 33.0537 1.54281
\(460\) 0 0
\(461\) −9.92290 −0.462155 −0.231078 0.972935i \(-0.574225\pi\)
−0.231078 + 0.972935i \(0.574225\pi\)
\(462\) 0 0
\(463\) 10.0111 0.465256 0.232628 0.972566i \(-0.425267\pi\)
0.232628 + 0.972566i \(0.425267\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.9345 1.70912 0.854562 0.519350i \(-0.173826\pi\)
0.854562 + 0.519350i \(0.173826\pi\)
\(468\) 0 0
\(469\) 27.5209 1.27080
\(470\) 0 0
\(471\) 9.69046 0.446513
\(472\) 0 0
\(473\) −4.34581 −0.199821
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −17.9393 −0.821385
\(478\) 0 0
\(479\) 16.7734 0.766396 0.383198 0.923666i \(-0.374823\pi\)
0.383198 + 0.923666i \(0.374823\pi\)
\(480\) 0 0
\(481\) −32.8898 −1.49965
\(482\) 0 0
\(483\) 1.96471 0.0893972
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 5.47391 0.248046 0.124023 0.992279i \(-0.460420\pi\)
0.124023 + 0.992279i \(0.460420\pi\)
\(488\) 0 0
\(489\) 14.7644 0.667670
\(490\) 0 0
\(491\) −35.4536 −1.60000 −0.800000 0.600001i \(-0.795167\pi\)
−0.800000 + 0.600001i \(0.795167\pi\)
\(492\) 0 0
\(493\) 40.9327 1.84351
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −21.2132 −0.951543
\(498\) 0 0
\(499\) 33.4110 1.49568 0.747840 0.663879i \(-0.231091\pi\)
0.747840 + 0.663879i \(0.231091\pi\)
\(500\) 0 0
\(501\) 13.4035 0.598825
\(502\) 0 0
\(503\) −1.97733 −0.0881650 −0.0440825 0.999028i \(-0.514036\pi\)
−0.0440825 + 0.999028i \(0.514036\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.38875 −0.0616766
\(508\) 0 0
\(509\) −23.0984 −1.02382 −0.511909 0.859040i \(-0.671061\pi\)
−0.511909 + 0.859040i \(0.671061\pi\)
\(510\) 0 0
\(511\) −40.9969 −1.81360
\(512\) 0 0
\(513\) 9.81320 0.433264
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 28.3886 1.24853
\(518\) 0 0
\(519\) 19.4430 0.853451
\(520\) 0 0
\(521\) 21.2867 0.932588 0.466294 0.884630i \(-0.345589\pi\)
0.466294 + 0.884630i \(0.345589\pi\)
\(522\) 0 0
\(523\) 28.1943 1.23285 0.616425 0.787414i \(-0.288580\pi\)
0.616425 + 0.787414i \(0.288580\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 49.5911 2.16022
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 14.3797 0.624028
\(532\) 0 0
\(533\) 16.4246 0.711428
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.79763 0.0775737
\(538\) 0 0
\(539\) −2.04588 −0.0881223
\(540\) 0 0
\(541\) 14.8271 0.637468 0.318734 0.947844i \(-0.396742\pi\)
0.318734 + 0.947844i \(0.396742\pi\)
\(542\) 0 0
\(543\) −11.6985 −0.502032
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −9.09950 −0.389067 −0.194533 0.980896i \(-0.562319\pi\)
−0.194533 + 0.980896i \(0.562319\pi\)
\(548\) 0 0
\(549\) 1.50423 0.0641989
\(550\) 0 0
\(551\) 12.1524 0.517709
\(552\) 0 0
\(553\) −0.828654 −0.0352379
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 32.9824 1.39751 0.698754 0.715362i \(-0.253738\pi\)
0.698754 + 0.715362i \(0.253738\pi\)
\(558\) 0 0
\(559\) −7.25625 −0.306907
\(560\) 0 0
\(561\) −14.1612 −0.597885
\(562\) 0 0
\(563\) −21.9384 −0.924595 −0.462297 0.886725i \(-0.652975\pi\)
−0.462297 + 0.886725i \(0.652975\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9.17985 0.385517
\(568\) 0 0
\(569\) −28.2664 −1.18499 −0.592495 0.805574i \(-0.701857\pi\)
−0.592495 + 0.805574i \(0.701857\pi\)
\(570\) 0 0
\(571\) 4.13671 0.173116 0.0865580 0.996247i \(-0.472413\pi\)
0.0865580 + 0.996247i \(0.472413\pi\)
\(572\) 0 0
\(573\) −18.8413 −0.787105
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −13.8146 −0.575111 −0.287556 0.957764i \(-0.592843\pi\)
−0.287556 + 0.957764i \(0.592843\pi\)
\(578\) 0 0
\(579\) −16.2256 −0.674313
\(580\) 0 0
\(581\) 37.5579 1.55816
\(582\) 0 0
\(583\) 17.4216 0.721527
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −17.2506 −0.712010 −0.356005 0.934484i \(-0.615861\pi\)
−0.356005 + 0.934484i \(0.615861\pi\)
\(588\) 0 0
\(589\) 14.7229 0.606649
\(590\) 0 0
\(591\) 14.1163 0.580667
\(592\) 0 0
\(593\) −27.0042 −1.10893 −0.554466 0.832207i \(-0.687077\pi\)
−0.554466 + 0.832207i \(0.687077\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.85247 0.362307
\(598\) 0 0
\(599\) 11.5259 0.470934 0.235467 0.971882i \(-0.424338\pi\)
0.235467 + 0.971882i \(0.424338\pi\)
\(600\) 0 0
\(601\) −10.8541 −0.442750 −0.221375 0.975189i \(-0.571054\pi\)
−0.221375 + 0.975189i \(0.571054\pi\)
\(602\) 0 0
\(603\) −26.3669 −1.07374
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −5.16930 −0.209816 −0.104908 0.994482i \(-0.533455\pi\)
−0.104908 + 0.994482i \(0.533455\pi\)
\(608\) 0 0
\(609\) −10.3811 −0.420664
\(610\) 0 0
\(611\) 47.4008 1.91763
\(612\) 0 0
\(613\) 27.3190 1.10340 0.551701 0.834042i \(-0.313979\pi\)
0.551701 + 0.834042i \(0.313979\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.08325 −0.204644 −0.102322 0.994751i \(-0.532627\pi\)
−0.102322 + 0.994751i \(0.532627\pi\)
\(618\) 0 0
\(619\) 0.207299 0.00833203 0.00416602 0.999991i \(-0.498674\pi\)
0.00416602 + 0.999991i \(0.498674\pi\)
\(620\) 0 0
\(621\) −4.26674 −0.171218
\(622\) 0 0
\(623\) 13.7374 0.550378
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.20426 −0.167902
\(628\) 0 0
\(629\) −66.3482 −2.64547
\(630\) 0 0
\(631\) −40.7410 −1.62187 −0.810937 0.585133i \(-0.801042\pi\)
−0.810937 + 0.585133i \(0.801042\pi\)
\(632\) 0 0
\(633\) 3.31710 0.131843
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.41603 −0.135348
\(638\) 0 0
\(639\) 20.3237 0.803995
\(640\) 0 0
\(641\) −29.7426 −1.17476 −0.587381 0.809310i \(-0.699841\pi\)
−0.587381 + 0.809310i \(0.699841\pi\)
\(642\) 0 0
\(643\) −34.3329 −1.35396 −0.676978 0.736003i \(-0.736711\pi\)
−0.676978 + 0.736003i \(0.736711\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.0002 −0.589717 −0.294859 0.955541i \(-0.595273\pi\)
−0.294859 + 0.955541i \(0.595273\pi\)
\(648\) 0 0
\(649\) −13.9647 −0.548163
\(650\) 0 0
\(651\) −12.5770 −0.492932
\(652\) 0 0
\(653\) 14.2038 0.555837 0.277918 0.960605i \(-0.410356\pi\)
0.277918 + 0.960605i \(0.410356\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 39.2779 1.53238
\(658\) 0 0
\(659\) −47.9718 −1.86872 −0.934359 0.356334i \(-0.884027\pi\)
−0.934359 + 0.356334i \(0.884027\pi\)
\(660\) 0 0
\(661\) 43.6518 1.69786 0.848928 0.528508i \(-0.177248\pi\)
0.848928 + 0.528508i \(0.177248\pi\)
\(662\) 0 0
\(663\) −23.6450 −0.918297
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.28380 −0.204590
\(668\) 0 0
\(669\) −9.07814 −0.350981
\(670\) 0 0
\(671\) −1.46081 −0.0563940
\(672\) 0 0
\(673\) 11.9330 0.459983 0.229991 0.973193i \(-0.426130\pi\)
0.229991 + 0.973193i \(0.426130\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.7598 0.720999 0.360500 0.932759i \(-0.382606\pi\)
0.360500 + 0.932759i \(0.382606\pi\)
\(678\) 0 0
\(679\) −15.8892 −0.609773
\(680\) 0 0
\(681\) −4.29206 −0.164472
\(682\) 0 0
\(683\) 10.4542 0.400020 0.200010 0.979794i \(-0.435902\pi\)
0.200010 + 0.979794i \(0.435902\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 14.6509 0.558966
\(688\) 0 0
\(689\) 29.0890 1.10820
\(690\) 0 0
\(691\) 20.8518 0.793240 0.396620 0.917983i \(-0.370183\pi\)
0.396620 + 0.917983i \(0.370183\pi\)
\(692\) 0 0
\(693\) −13.4644 −0.511469
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 33.1331 1.25500
\(698\) 0 0
\(699\) −6.21634 −0.235124
\(700\) 0 0
\(701\) 22.1231 0.835578 0.417789 0.908544i \(-0.362805\pi\)
0.417789 + 0.908544i \(0.362805\pi\)
\(702\) 0 0
\(703\) −19.6979 −0.742921
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −36.1004 −1.35769
\(708\) 0 0
\(709\) 14.1669 0.532049 0.266024 0.963966i \(-0.414290\pi\)
0.266024 + 0.963966i \(0.414290\pi\)
\(710\) 0 0
\(711\) 0.793908 0.0297739
\(712\) 0 0
\(713\) −6.40148 −0.239737
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.79731 −0.0671219
\(718\) 0 0
\(719\) 6.86389 0.255980 0.127990 0.991775i \(-0.459147\pi\)
0.127990 + 0.991775i \(0.459147\pi\)
\(720\) 0 0
\(721\) −33.1189 −1.23341
\(722\) 0 0
\(723\) −10.7224 −0.398770
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.72298 −0.0639016 −0.0319508 0.999489i \(-0.510172\pi\)
−0.0319508 + 0.999489i \(0.510172\pi\)
\(728\) 0 0
\(729\) 1.37874 0.0510645
\(730\) 0 0
\(731\) −14.6379 −0.541403
\(732\) 0 0
\(733\) −13.2855 −0.490712 −0.245356 0.969433i \(-0.578905\pi\)
−0.245356 + 0.969433i \(0.578905\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 25.6059 0.943206
\(738\) 0 0
\(739\) 28.8043 1.05958 0.529791 0.848128i \(-0.322270\pi\)
0.529791 + 0.848128i \(0.322270\pi\)
\(740\) 0 0
\(741\) −7.01991 −0.257883
\(742\) 0 0
\(743\) 15.7966 0.579521 0.289761 0.957099i \(-0.406424\pi\)
0.289761 + 0.957099i \(0.406424\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −35.9830 −1.31655
\(748\) 0 0
\(749\) 4.04282 0.147722
\(750\) 0 0
\(751\) 12.7384 0.464829 0.232415 0.972617i \(-0.425337\pi\)
0.232415 + 0.972617i \(0.425337\pi\)
\(752\) 0 0
\(753\) 12.3705 0.450807
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.07420 0.111734 0.0558668 0.998438i \(-0.482208\pi\)
0.0558668 + 0.998438i \(0.482208\pi\)
\(758\) 0 0
\(759\) 1.82800 0.0663520
\(760\) 0 0
\(761\) 30.6818 1.11222 0.556108 0.831110i \(-0.312294\pi\)
0.556108 + 0.831110i \(0.312294\pi\)
\(762\) 0 0
\(763\) 3.71239 0.134398
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −23.3170 −0.841929
\(768\) 0 0
\(769\) −34.5153 −1.24465 −0.622327 0.782758i \(-0.713812\pi\)
−0.622327 + 0.782758i \(0.713812\pi\)
\(770\) 0 0
\(771\) 20.0572 0.722342
\(772\) 0 0
\(773\) 36.6662 1.31879 0.659396 0.751796i \(-0.270812\pi\)
0.659396 + 0.751796i \(0.270812\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 16.8269 0.603660
\(778\) 0 0
\(779\) 9.83678 0.352439
\(780\) 0 0
\(781\) −19.7371 −0.706251
\(782\) 0 0
\(783\) 22.5446 0.805678
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 17.6991 0.630906 0.315453 0.948941i \(-0.397844\pi\)
0.315453 + 0.948941i \(0.397844\pi\)
\(788\) 0 0
\(789\) 10.0339 0.357215
\(790\) 0 0
\(791\) −43.6176 −1.55086
\(792\) 0 0
\(793\) −2.43913 −0.0866162
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −26.6535 −0.944117 −0.472058 0.881567i \(-0.656489\pi\)
−0.472058 + 0.881567i \(0.656489\pi\)
\(798\) 0 0
\(799\) 95.6209 3.38282
\(800\) 0 0
\(801\) −13.1614 −0.465035
\(802\) 0 0
\(803\) −38.1442 −1.34608
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −19.9213 −0.701263
\(808\) 0 0
\(809\) −24.1080 −0.847591 −0.423795 0.905758i \(-0.639302\pi\)
−0.423795 + 0.905758i \(0.639302\pi\)
\(810\) 0 0
\(811\) 11.3363 0.398071 0.199035 0.979992i \(-0.436219\pi\)
0.199035 + 0.979992i \(0.436219\pi\)
\(812\) 0 0
\(813\) −14.0503 −0.492766
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4.34581 −0.152041
\(818\) 0 0
\(819\) −22.4816 −0.785570
\(820\) 0 0
\(821\) −3.95739 −0.138114 −0.0690569 0.997613i \(-0.521999\pi\)
−0.0690569 + 0.997613i \(0.521999\pi\)
\(822\) 0 0
\(823\) 46.5023 1.62097 0.810484 0.585761i \(-0.199204\pi\)
0.810484 + 0.585761i \(0.199204\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.7619 0.965375 0.482687 0.875793i \(-0.339661\pi\)
0.482687 + 0.875793i \(0.339661\pi\)
\(828\) 0 0
\(829\) 37.8234 1.31366 0.656831 0.754038i \(-0.271896\pi\)
0.656831 + 0.754038i \(0.271896\pi\)
\(830\) 0 0
\(831\) −5.74424 −0.199266
\(832\) 0 0
\(833\) −6.89111 −0.238763
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 27.3134 0.944090
\(838\) 0 0
\(839\) −20.0661 −0.692758 −0.346379 0.938095i \(-0.612589\pi\)
−0.346379 + 0.938095i \(0.612589\pi\)
\(840\) 0 0
\(841\) −1.08145 −0.0372913
\(842\) 0 0
\(843\) 4.46136 0.153657
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −14.1155 −0.485016
\(848\) 0 0
\(849\) 0.694295 0.0238281
\(850\) 0 0
\(851\) 8.56457 0.293590
\(852\) 0 0
\(853\) 8.35933 0.286218 0.143109 0.989707i \(-0.454290\pi\)
0.143109 + 0.989707i \(0.454290\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 17.0931 0.583889 0.291945 0.956435i \(-0.405698\pi\)
0.291945 + 0.956435i \(0.405698\pi\)
\(858\) 0 0
\(859\) −23.0254 −0.785617 −0.392809 0.919620i \(-0.628497\pi\)
−0.392809 + 0.919620i \(0.628497\pi\)
\(860\) 0 0
\(861\) −8.40303 −0.286374
\(862\) 0 0
\(863\) 5.49197 0.186949 0.0934745 0.995622i \(-0.470203\pi\)
0.0934745 + 0.995622i \(0.470203\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −34.1871 −1.16106
\(868\) 0 0
\(869\) −0.770994 −0.0261542
\(870\) 0 0
\(871\) 42.7545 1.44868
\(872\) 0 0
\(873\) 15.2230 0.515220
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −7.98307 −0.269569 −0.134785 0.990875i \(-0.543034\pi\)
−0.134785 + 0.990875i \(0.543034\pi\)
\(878\) 0 0
\(879\) 8.16557 0.275418
\(880\) 0 0
\(881\) 44.5925 1.50236 0.751180 0.660098i \(-0.229485\pi\)
0.751180 + 0.660098i \(0.229485\pi\)
\(882\) 0 0
\(883\) 45.9134 1.54511 0.772554 0.634949i \(-0.218979\pi\)
0.772554 + 0.634949i \(0.218979\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 53.6109 1.80008 0.900039 0.435810i \(-0.143538\pi\)
0.900039 + 0.435810i \(0.143538\pi\)
\(888\) 0 0
\(889\) 9.82253 0.329437
\(890\) 0 0
\(891\) 8.54109 0.286137
\(892\) 0 0
\(893\) 28.3886 0.949988
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.05223 0.101911
\(898\) 0 0
\(899\) 33.8241 1.12810
\(900\) 0 0
\(901\) 58.6808 1.95494
\(902\) 0 0
\(903\) 3.71239 0.123541
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 45.1277 1.49844 0.749220 0.662322i \(-0.230429\pi\)
0.749220 + 0.662322i \(0.230429\pi\)
\(908\) 0 0
\(909\) 34.5867 1.14717
\(910\) 0 0
\(911\) −6.06248 −0.200859 −0.100429 0.994944i \(-0.532022\pi\)
−0.100429 + 0.994944i \(0.532022\pi\)
\(912\) 0 0
\(913\) 34.9445 1.15649
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 35.3567 1.16758
\(918\) 0 0
\(919\) −38.8692 −1.28218 −0.641089 0.767467i \(-0.721517\pi\)
−0.641089 + 0.767467i \(0.721517\pi\)
\(920\) 0 0
\(921\) −20.9824 −0.691394
\(922\) 0 0
\(923\) −32.9553 −1.08474
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 31.7302 1.04216
\(928\) 0 0
\(929\) −43.4632 −1.42598 −0.712991 0.701173i \(-0.752660\pi\)
−0.712991 + 0.701173i \(0.752660\pi\)
\(930\) 0 0
\(931\) −2.04588 −0.0670510
\(932\) 0 0
\(933\) 25.1499 0.823371
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −19.7812 −0.646223 −0.323112 0.946361i \(-0.604729\pi\)
−0.323112 + 0.946361i \(0.604729\pi\)
\(938\) 0 0
\(939\) −19.1808 −0.625942
\(940\) 0 0
\(941\) 31.1805 1.01645 0.508227 0.861223i \(-0.330301\pi\)
0.508227 + 0.861223i \(0.330301\pi\)
\(942\) 0 0
\(943\) −4.27699 −0.139278
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.65195 0.118673 0.0593363 0.998238i \(-0.481102\pi\)
0.0593363 + 0.998238i \(0.481102\pi\)
\(948\) 0 0
\(949\) −63.6898 −2.06746
\(950\) 0 0
\(951\) −0.773468 −0.0250814
\(952\) 0 0
\(953\) −4.81987 −0.156131 −0.0780655 0.996948i \(-0.524874\pi\)
−0.0780655 + 0.996948i \(0.524874\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −9.65877 −0.312224
\(958\) 0 0
\(959\) −10.5202 −0.339715
\(960\) 0 0
\(961\) 9.97890 0.321900
\(962\) 0 0
\(963\) −3.87331 −0.124816
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −28.9718 −0.931670 −0.465835 0.884872i \(-0.654246\pi\)
−0.465835 + 0.884872i \(0.654246\pi\)
\(968\) 0 0
\(969\) −14.1612 −0.454922
\(970\) 0 0
\(971\) −32.8376 −1.05381 −0.526905 0.849924i \(-0.676648\pi\)
−0.526905 + 0.849924i \(0.676648\pi\)
\(972\) 0 0
\(973\) 24.3968 0.782125
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −58.0091 −1.85587 −0.927937 0.372737i \(-0.878419\pi\)
−0.927937 + 0.372737i \(0.878419\pi\)
\(978\) 0 0
\(979\) 12.7815 0.408499
\(980\) 0 0
\(981\) −3.55673 −0.113558
\(982\) 0 0
\(983\) 4.41757 0.140899 0.0704493 0.997515i \(-0.477557\pi\)
0.0704493 + 0.997515i \(0.477557\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −24.2508 −0.771913
\(988\) 0 0
\(989\) 1.88954 0.0600839
\(990\) 0 0
\(991\) −36.1617 −1.14871 −0.574357 0.818605i \(-0.694748\pi\)
−0.574357 + 0.818605i \(0.694748\pi\)
\(992\) 0 0
\(993\) 0.877082 0.0278334
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −11.6357 −0.368506 −0.184253 0.982879i \(-0.558987\pi\)
−0.184253 + 0.982879i \(0.558987\pi\)
\(998\) 0 0
\(999\) −36.5428 −1.15616
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 9200.2.a.cv.1.3 5
4.3 odd 2 4600.2.a.bd.1.3 5
5.4 even 2 9200.2.a.ct.1.3 5
20.3 even 4 4600.2.e.w.4049.6 10
20.7 even 4 4600.2.e.w.4049.5 10
20.19 odd 2 4600.2.a.bf.1.3 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.3 5 4.3 odd 2
4600.2.a.bf.1.3 yes 5 20.19 odd 2
4600.2.e.w.4049.5 10 20.7 even 4
4600.2.e.w.4049.6 10 20.3 even 4
9200.2.a.ct.1.3 5 5.4 even 2
9200.2.a.cv.1.3 5 1.1 even 1 trivial