Properties

Label 4704.2.a.bx.1.4
Level $4704$
Weight $2$
Character 4704.1
Self dual yes
Analytic conductor $37.562$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4704.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.5616291108\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2624.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.77462\) of defining polynomial
Character \(\chi\) \(=\) 4704.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.13503 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +4.13503 q^{5} +1.00000 q^{9} -2.29857 q^{11} -2.43361 q^{13} -4.13503 q^{15} +3.71279 q^{17} -7.84782 q^{19} -1.70143 q^{23} +12.0985 q^{25} -1.00000 q^{27} +3.25067 q^{29} +1.80904 q^{31} +2.29857 q^{33} -5.65685 q^{37} +2.43361 q^{39} -6.81128 q^{41} +5.44164 q^{43} +4.13503 q^{45} +13.2895 q^{47} -3.71279 q^{51} +9.09849 q^{53} -9.50467 q^{55} +7.84782 q^{57} +14.9463 q^{59} +2.16354 q^{61} -10.0630 q^{65} +10.2540 q^{67} +1.70143 q^{69} +3.95543 q^{71} +9.68428 q^{73} -12.0985 q^{75} -10.0388 q^{79} +1.00000 q^{81} +11.6956 q^{83} +15.3525 q^{85} -3.25067 q^{87} +12.7725 q^{89} -1.80904 q^{93} -32.4510 q^{95} +12.0274 q^{97} -2.29857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{5} + 4q^{9} - 4q^{11} + 8q^{13} - 4q^{15} + 4q^{17} - 8q^{19} - 12q^{23} + 12q^{25} - 4q^{27} + 8q^{31} + 4q^{33} - 8q^{39} + 20q^{41} + 8q^{43} + 4q^{45} + 16q^{47} - 4q^{51} + 8q^{55} + 8q^{57} + 16q^{61} - 8q^{65} + 8q^{67} + 12q^{69} - 12q^{71} + 8q^{73} - 12q^{75} - 16q^{79} + 4q^{81} - 8q^{85} + 28q^{89} - 8q^{93} - 24q^{95} + 40q^{97} - 4q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 4.13503 1.84924 0.924621 0.380888i \(-0.124382\pi\)
0.924621 + 0.380888i \(0.124382\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.29857 −0.693046 −0.346523 0.938042i \(-0.612638\pi\)
−0.346523 + 0.938042i \(0.612638\pi\)
\(12\) 0 0
\(13\) −2.43361 −0.674961 −0.337480 0.941333i \(-0.609575\pi\)
−0.337480 + 0.941333i \(0.609575\pi\)
\(14\) 0 0
\(15\) −4.13503 −1.06766
\(16\) 0 0
\(17\) 3.71279 0.900483 0.450241 0.892907i \(-0.351338\pi\)
0.450241 + 0.892907i \(0.351338\pi\)
\(18\) 0 0
\(19\) −7.84782 −1.80041 −0.900207 0.435463i \(-0.856585\pi\)
−0.900207 + 0.435463i \(0.856585\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.70143 −0.354772 −0.177386 0.984141i \(-0.556764\pi\)
−0.177386 + 0.984141i \(0.556764\pi\)
\(24\) 0 0
\(25\) 12.0985 2.41970
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.25067 0.603635 0.301817 0.953366i \(-0.402407\pi\)
0.301817 + 0.953366i \(0.402407\pi\)
\(30\) 0 0
\(31\) 1.80904 0.324912 0.162456 0.986716i \(-0.448058\pi\)
0.162456 + 0.986716i \(0.448058\pi\)
\(32\) 0 0
\(33\) 2.29857 0.400130
\(34\) 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
\(39\) 2.43361 0.389689
\(40\) 0 0
\(41\) −6.81128 −1.06374 −0.531871 0.846825i \(-0.678511\pi\)
−0.531871 + 0.846825i \(0.678511\pi\)
\(42\) 0 0
\(43\) 5.44164 0.829842 0.414921 0.909857i \(-0.363809\pi\)
0.414921 + 0.909857i \(0.363809\pi\)
\(44\) 0 0
\(45\) 4.13503 0.616414
\(46\) 0 0
\(47\) 13.2895 1.93847 0.969233 0.246144i \(-0.0791637\pi\)
0.969233 + 0.246144i \(0.0791637\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −3.71279 −0.519894
\(52\) 0 0
\(53\) 9.09849 1.24977 0.624887 0.780715i \(-0.285145\pi\)
0.624887 + 0.780715i \(0.285145\pi\)
\(54\) 0 0
\(55\) −9.50467 −1.28161
\(56\) 0 0
\(57\) 7.84782 1.03947
\(58\) 0 0
\(59\) 14.9463 1.94584 0.972922 0.231134i \(-0.0742436\pi\)
0.972922 + 0.231134i \(0.0742436\pi\)
\(60\) 0 0
\(61\) 2.16354 0.277013 0.138506 0.990362i \(-0.455770\pi\)
0.138506 + 0.990362i \(0.455770\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10.0630 −1.24817
\(66\) 0 0
\(67\) 10.2540 1.25273 0.626363 0.779532i \(-0.284543\pi\)
0.626363 + 0.779532i \(0.284543\pi\)
\(68\) 0 0
\(69\) 1.70143 0.204828
\(70\) 0 0
\(71\) 3.95543 0.469423 0.234711 0.972065i \(-0.424585\pi\)
0.234711 + 0.972065i \(0.424585\pi\)
\(72\) 0 0
\(73\) 9.68428 1.13346 0.566730 0.823904i \(-0.308208\pi\)
0.566730 + 0.823904i \(0.308208\pi\)
\(74\) 0 0
\(75\) −12.0985 −1.39701
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0388 −1.12945 −0.564726 0.825279i \(-0.691018\pi\)
−0.564726 + 0.825279i \(0.691018\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.6956 1.28376 0.641881 0.766804i \(-0.278154\pi\)
0.641881 + 0.766804i \(0.278154\pi\)
\(84\) 0 0
\(85\) 15.3525 1.66521
\(86\) 0 0
\(87\) −3.25067 −0.348509
\(88\) 0 0
\(89\) 12.7725 1.35388 0.676941 0.736037i \(-0.263305\pi\)
0.676941 + 0.736037i \(0.263305\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.80904 −0.187588
\(94\) 0 0
\(95\) −32.4510 −3.32940
\(96\) 0 0
\(97\) 12.0274 1.22120 0.610600 0.791939i \(-0.290928\pi\)
0.610600 + 0.791939i \(0.290928\pi\)
\(98\) 0 0
\(99\) −2.29857 −0.231015
\(100\) 0 0
\(101\) 1.30661 0.130012 0.0650060 0.997885i \(-0.479293\pi\)
0.0650060 + 0.997885i \(0.479293\pi\)
\(102\) 0 0
\(103\) 11.0033 1.08419 0.542095 0.840317i \(-0.317631\pi\)
0.542095 + 0.840317i \(0.317631\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.8387 1.04782 0.523908 0.851775i \(-0.324473\pi\)
0.523908 + 0.851775i \(0.324473\pi\)
\(108\) 0 0
\(109\) −9.44164 −0.904345 −0.452172 0.891931i \(-0.649351\pi\)
−0.452172 + 0.891931i \(0.649351\pi\)
\(110\) 0 0
\(111\) 5.65685 0.536925
\(112\) 0 0
\(113\) 9.31371 0.876160 0.438080 0.898936i \(-0.355659\pi\)
0.438080 + 0.898936i \(0.355659\pi\)
\(114\) 0 0
\(115\) −7.03546 −0.656060
\(116\) 0 0
\(117\) −2.43361 −0.224987
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.71656 −0.519688
\(122\) 0 0
\(123\) 6.81128 0.614152
\(124\) 0 0
\(125\) 29.3525 2.62537
\(126\) 0 0
\(127\) −13.1373 −1.16574 −0.582872 0.812564i \(-0.698071\pi\)
−0.582872 + 0.812564i \(0.698071\pi\)
\(128\) 0 0
\(129\) −5.44164 −0.479109
\(130\) 0 0
\(131\) −8.81236 −0.769940 −0.384970 0.922929i \(-0.625788\pi\)
−0.384970 + 0.922929i \(0.625788\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −4.13503 −0.355887
\(136\) 0 0
\(137\) 0.367398 0.0313889 0.0156945 0.999877i \(-0.495004\pi\)
0.0156945 + 0.999877i \(0.495004\pi\)
\(138\) 0 0
\(139\) −15.3137 −1.29889 −0.649446 0.760408i \(-0.724999\pi\)
−0.649446 + 0.760408i \(0.724999\pi\)
\(140\) 0 0
\(141\) −13.2895 −1.11917
\(142\) 0 0
\(143\) 5.59382 0.467779
\(144\) 0 0
\(145\) 13.4416 1.11627
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −16.4122 −1.34454 −0.672270 0.740306i \(-0.734681\pi\)
−0.672270 + 0.740306i \(0.734681\pi\)
\(150\) 0 0
\(151\) 7.78478 0.633517 0.316758 0.948506i \(-0.397406\pi\)
0.316758 + 0.948506i \(0.397406\pi\)
\(152\) 0 0
\(153\) 3.71279 0.300161
\(154\) 0 0
\(155\) 7.48042 0.600842
\(156\) 0 0
\(157\) −17.2050 −1.37311 −0.686555 0.727078i \(-0.740878\pi\)
−0.686555 + 0.727078i \(0.740878\pi\)
\(158\) 0 0
\(159\) −9.09849 −0.721557
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −6.71656 −0.526082 −0.263041 0.964785i \(-0.584725\pi\)
−0.263041 + 0.964785i \(0.584725\pi\)
\(164\) 0 0
\(165\) 9.50467 0.739938
\(166\) 0 0
\(167\) 1.21189 0.0937789 0.0468894 0.998900i \(-0.485069\pi\)
0.0468894 + 0.998900i \(0.485069\pi\)
\(168\) 0 0
\(169\) −7.07757 −0.544428
\(170\) 0 0
\(171\) −7.84782 −0.600138
\(172\) 0 0
\(173\) −10.2772 −0.781359 −0.390679 0.920527i \(-0.627760\pi\)
−0.390679 + 0.920527i \(0.627760\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −14.9463 −1.12343
\(178\) 0 0
\(179\) −12.1221 −0.906051 −0.453026 0.891498i \(-0.649655\pi\)
−0.453026 + 0.891498i \(0.649655\pi\)
\(180\) 0 0
\(181\) 23.2460 1.72786 0.863930 0.503613i \(-0.167996\pi\)
0.863930 + 0.503613i \(0.167996\pi\)
\(182\) 0 0
\(183\) −2.16354 −0.159934
\(184\) 0 0
\(185\) −23.3913 −1.71976
\(186\) 0 0
\(187\) −8.53411 −0.624076
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 23.2691 1.68370 0.841848 0.539715i \(-0.181468\pi\)
0.841848 + 0.539715i \(0.181468\pi\)
\(192\) 0 0
\(193\) −4.28613 −0.308522 −0.154261 0.988030i \(-0.549300\pi\)
−0.154261 + 0.988030i \(0.549300\pi\)
\(194\) 0 0
\(195\) 10.0630 0.720629
\(196\) 0 0
\(197\) −11.9818 −0.853666 −0.426833 0.904331i \(-0.640371\pi\)
−0.426833 + 0.904331i \(0.640371\pi\)
\(198\) 0 0
\(199\) 22.1970 1.57350 0.786751 0.617270i \(-0.211761\pi\)
0.786751 + 0.617270i \(0.211761\pi\)
\(200\) 0 0
\(201\) −10.2540 −0.723261
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −28.1649 −1.96712
\(206\) 0 0
\(207\) −1.70143 −0.118257
\(208\) 0 0
\(209\) 18.0388 1.24777
\(210\) 0 0
\(211\) −0.540129 −0.0371840 −0.0185920 0.999827i \(-0.505918\pi\)
−0.0185920 + 0.999827i \(0.505918\pi\)
\(212\) 0 0
\(213\) −3.95543 −0.271021
\(214\) 0 0
\(215\) 22.5013 1.53458
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −9.68428 −0.654403
\(220\) 0 0
\(221\) −9.03546 −0.607791
\(222\) 0 0
\(223\) −10.1194 −0.677646 −0.338823 0.940850i \(-0.610029\pi\)
−0.338823 + 0.940850i \(0.610029\pi\)
\(224\) 0 0
\(225\) 12.0985 0.806566
\(226\) 0 0
\(227\) −27.7587 −1.84241 −0.921204 0.389080i \(-0.872793\pi\)
−0.921204 + 0.389080i \(0.872793\pi\)
\(228\) 0 0
\(229\) 14.8962 0.984366 0.492183 0.870492i \(-0.336199\pi\)
0.492183 + 0.870492i \(0.336199\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.5644 −0.954144 −0.477072 0.878864i \(-0.658302\pi\)
−0.477072 + 0.878864i \(0.658302\pi\)
\(234\) 0 0
\(235\) 54.9523 3.58469
\(236\) 0 0
\(237\) 10.0388 0.652089
\(238\) 0 0
\(239\) −1.70143 −0.110056 −0.0550281 0.998485i \(-0.517525\pi\)
−0.0550281 + 0.998485i \(0.517525\pi\)
\(240\) 0 0
\(241\) 27.0141 1.74013 0.870064 0.492939i \(-0.164077\pi\)
0.870064 + 0.492939i \(0.164077\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 19.0985 1.21521
\(248\) 0 0
\(249\) −11.6956 −0.741181
\(250\) 0 0
\(251\) −12.8269 −0.809626 −0.404813 0.914399i \(-0.632663\pi\)
−0.404813 + 0.914399i \(0.632663\pi\)
\(252\) 0 0
\(253\) 3.91085 0.245873
\(254\) 0 0
\(255\) −15.3525 −0.961410
\(256\) 0 0
\(257\) −1.99892 −0.124689 −0.0623445 0.998055i \(-0.519858\pi\)
−0.0623445 + 0.998055i \(0.519858\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3.25067 0.201212
\(262\) 0 0
\(263\) −1.07215 −0.0661118 −0.0330559 0.999454i \(-0.510524\pi\)
−0.0330559 + 0.999454i \(0.510524\pi\)
\(264\) 0 0
\(265\) 37.6226 2.31114
\(266\) 0 0
\(267\) −12.7725 −0.781664
\(268\) 0 0
\(269\) 31.7188 1.93393 0.966965 0.254910i \(-0.0820458\pi\)
0.966965 + 0.254910i \(0.0820458\pi\)
\(270\) 0 0
\(271\) 0.614744 0.0373430 0.0186715 0.999826i \(-0.494056\pi\)
0.0186715 + 0.999826i \(0.494056\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −27.8093 −1.67696
\(276\) 0 0
\(277\) −14.6747 −0.881718 −0.440859 0.897576i \(-0.645326\pi\)
−0.440859 + 0.897576i \(0.645326\pi\)
\(278\) 0 0
\(279\) 1.80904 0.108304
\(280\) 0 0
\(281\) 12.1406 0.724248 0.362124 0.932130i \(-0.382052\pi\)
0.362124 + 0.932130i \(0.382052\pi\)
\(282\) 0 0
\(283\) 7.54346 0.448412 0.224206 0.974542i \(-0.428021\pi\)
0.224206 + 0.974542i \(0.428021\pi\)
\(284\) 0 0
\(285\) 32.4510 1.92223
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −3.21522 −0.189130
\(290\) 0 0
\(291\) −12.0274 −0.705060
\(292\) 0 0
\(293\) 21.4648 1.25399 0.626994 0.779024i \(-0.284285\pi\)
0.626994 + 0.779024i \(0.284285\pi\)
\(294\) 0 0
\(295\) 61.8035 3.59834
\(296\) 0 0
\(297\) 2.29857 0.133377
\(298\) 0 0
\(299\) 4.14060 0.239457
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1.30661 −0.0750625
\(304\) 0 0
\(305\) 8.94631 0.512264
\(306\) 0 0
\(307\) 7.41738 0.423333 0.211666 0.977342i \(-0.432111\pi\)
0.211666 + 0.977342i \(0.432111\pi\)
\(308\) 0 0
\(309\) −11.0033 −0.625957
\(310\) 0 0
\(311\) 14.7881 0.838557 0.419278 0.907858i \(-0.362283\pi\)
0.419278 + 0.907858i \(0.362283\pi\)
\(312\) 0 0
\(313\) −9.95434 −0.562653 −0.281326 0.959612i \(-0.590774\pi\)
−0.281326 + 0.959612i \(0.590774\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.97907 0.167321 0.0836607 0.996494i \(-0.473339\pi\)
0.0836607 + 0.996494i \(0.473339\pi\)
\(318\) 0 0
\(319\) −7.47191 −0.418347
\(320\) 0 0
\(321\) −10.8387 −0.604957
\(322\) 0 0
\(323\) −29.1373 −1.62124
\(324\) 0 0
\(325\) −29.4430 −1.63320
\(326\) 0 0
\(327\) 9.44164 0.522124
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6.80571 0.374076 0.187038 0.982353i \(-0.440111\pi\)
0.187038 + 0.982353i \(0.440111\pi\)
\(332\) 0 0
\(333\) −5.65685 −0.309994
\(334\) 0 0
\(335\) 42.4006 2.31659
\(336\) 0 0
\(337\) −7.69564 −0.419208 −0.209604 0.977786i \(-0.567217\pi\)
−0.209604 + 0.977786i \(0.567217\pi\)
\(338\) 0 0
\(339\) −9.31371 −0.505851
\(340\) 0 0
\(341\) −4.15820 −0.225179
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 7.03546 0.378776
\(346\) 0 0
\(347\) 10.2094 0.548071 0.274035 0.961720i \(-0.411641\pi\)
0.274035 + 0.961720i \(0.411641\pi\)
\(348\) 0 0
\(349\) −27.6742 −1.48137 −0.740684 0.671854i \(-0.765498\pi\)
−0.740684 + 0.671854i \(0.765498\pi\)
\(350\) 0 0
\(351\) 2.43361 0.129896
\(352\) 0 0
\(353\) −2.51849 −0.134046 −0.0670230 0.997751i \(-0.521350\pi\)
−0.0670230 + 0.997751i \(0.521350\pi\)
\(354\) 0 0
\(355\) 16.3558 0.868077
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.58470 0.241971 0.120986 0.992654i \(-0.461394\pi\)
0.120986 + 0.992654i \(0.461394\pi\)
\(360\) 0 0
\(361\) 42.5883 2.24149
\(362\) 0 0
\(363\) 5.71656 0.300042
\(364\) 0 0
\(365\) 40.0448 2.09604
\(366\) 0 0
\(367\) −9.38462 −0.489873 −0.244937 0.969539i \(-0.578767\pi\)
−0.244937 + 0.969539i \(0.578767\pi\)
\(368\) 0 0
\(369\) −6.81128 −0.354581
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −11.0027 −0.569698 −0.284849 0.958572i \(-0.591943\pi\)
−0.284849 + 0.958572i \(0.591943\pi\)
\(374\) 0 0
\(375\) −29.3525 −1.51576
\(376\) 0 0
\(377\) −7.91085 −0.407430
\(378\) 0 0
\(379\) −26.7050 −1.37174 −0.685871 0.727723i \(-0.740579\pi\)
−0.685871 + 0.727723i \(0.740579\pi\)
\(380\) 0 0
\(381\) 13.1373 0.673043
\(382\) 0 0
\(383\) −28.3940 −1.45086 −0.725432 0.688294i \(-0.758360\pi\)
−0.725432 + 0.688294i \(0.758360\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.44164 0.276614
\(388\) 0 0
\(389\) 20.4450 1.03660 0.518300 0.855199i \(-0.326565\pi\)
0.518300 + 0.855199i \(0.326565\pi\)
\(390\) 0 0
\(391\) −6.31704 −0.319466
\(392\) 0 0
\(393\) 8.81236 0.444525
\(394\) 0 0
\(395\) −41.5107 −2.08863
\(396\) 0 0
\(397\) −15.8752 −0.796756 −0.398378 0.917221i \(-0.630427\pi\)
−0.398378 + 0.917221i \(0.630427\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −19.2507 −0.961333 −0.480666 0.876904i \(-0.659605\pi\)
−0.480666 + 0.876904i \(0.659605\pi\)
\(402\) 0 0
\(403\) −4.40248 −0.219303
\(404\) 0 0
\(405\) 4.13503 0.205471
\(406\) 0 0
\(407\) 13.0027 0.644519
\(408\) 0 0
\(409\) −1.27963 −0.0632737 −0.0316368 0.999499i \(-0.510072\pi\)
−0.0316368 + 0.999499i \(0.510072\pi\)
\(410\) 0 0
\(411\) −0.367398 −0.0181224
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 48.3618 2.37399
\(416\) 0 0
\(417\) 15.3137 0.749916
\(418\) 0 0
\(419\) 16.8754 0.824417 0.412209 0.911090i \(-0.364758\pi\)
0.412209 + 0.911090i \(0.364758\pi\)
\(420\) 0 0
\(421\) 23.6774 1.15397 0.576983 0.816756i \(-0.304230\pi\)
0.576983 + 0.816756i \(0.304230\pi\)
\(422\) 0 0
\(423\) 13.2895 0.646155
\(424\) 0 0
\(425\) 44.9191 2.17890
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −5.59382 −0.270072
\(430\) 0 0
\(431\) −11.8208 −0.569390 −0.284695 0.958618i \(-0.591892\pi\)
−0.284695 + 0.958618i \(0.591892\pi\)
\(432\) 0 0
\(433\) −21.9771 −1.05615 −0.528075 0.849198i \(-0.677086\pi\)
−0.528075 + 0.849198i \(0.677086\pi\)
\(434\) 0 0
\(435\) −13.4416 −0.644477
\(436\) 0 0
\(437\) 13.3525 0.638736
\(438\) 0 0
\(439\) 35.3137 1.68543 0.842716 0.538359i \(-0.180956\pi\)
0.842716 + 0.538359i \(0.180956\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −17.1248 −0.813625 −0.406813 0.913512i \(-0.633360\pi\)
−0.406813 + 0.913512i \(0.633360\pi\)
\(444\) 0 0
\(445\) 52.8147 2.50366
\(446\) 0 0
\(447\) 16.4122 0.776270
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 15.6562 0.737223
\(452\) 0 0
\(453\) −7.78478 −0.365761
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.80571 0.224802 0.112401 0.993663i \(-0.464146\pi\)
0.112401 + 0.993663i \(0.464146\pi\)
\(458\) 0 0
\(459\) −3.71279 −0.173298
\(460\) 0 0
\(461\) −21.0798 −0.981785 −0.490892 0.871220i \(-0.663329\pi\)
−0.490892 + 0.871220i \(0.663329\pi\)
\(462\) 0 0
\(463\) 14.2861 0.663933 0.331966 0.943291i \(-0.392288\pi\)
0.331966 + 0.943291i \(0.392288\pi\)
\(464\) 0 0
\(465\) −7.48042 −0.346896
\(466\) 0 0
\(467\) −22.5159 −1.04191 −0.520955 0.853584i \(-0.674424\pi\)
−0.520955 + 0.853584i \(0.674424\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 17.2050 0.792765
\(472\) 0 0
\(473\) −12.5080 −0.575118
\(474\) 0 0
\(475\) −94.9468 −4.35646
\(476\) 0 0
\(477\) 9.09849 0.416591
\(478\) 0 0
\(479\) −8.90753 −0.406995 −0.203498 0.979075i \(-0.565231\pi\)
−0.203498 + 0.979075i \(0.565231\pi\)
\(480\) 0 0
\(481\) 13.7665 0.627701
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 49.7338 2.25829
\(486\) 0 0
\(487\) 23.7848 1.07779 0.538896 0.842372i \(-0.318842\pi\)
0.538896 + 0.842372i \(0.318842\pi\)
\(488\) 0 0
\(489\) 6.71656 0.303733
\(490\) 0 0
\(491\) −14.1524 −0.638689 −0.319345 0.947639i \(-0.603463\pi\)
−0.319345 + 0.947639i \(0.603463\pi\)
\(492\) 0 0
\(493\) 12.0691 0.543563
\(494\) 0 0
\(495\) −9.50467 −0.427203
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −22.4122 −1.00331 −0.501654 0.865068i \(-0.667275\pi\)
−0.501654 + 0.865068i \(0.667275\pi\)
\(500\) 0 0
\(501\) −1.21189 −0.0541432
\(502\) 0 0
\(503\) 6.61538 0.294965 0.147483 0.989065i \(-0.452883\pi\)
0.147483 + 0.989065i \(0.452883\pi\)
\(504\) 0 0
\(505\) 5.40285 0.240424
\(506\) 0 0
\(507\) 7.07757 0.314326
\(508\) 0 0
\(509\) 1.93588 0.0858064 0.0429032 0.999079i \(-0.486339\pi\)
0.0429032 + 0.999079i \(0.486339\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 7.84782 0.346490
\(514\) 0 0
\(515\) 45.4991 2.00493
\(516\) 0 0
\(517\) −30.5468 −1.34345
\(518\) 0 0
\(519\) 10.2772 0.451118
\(520\) 0 0
\(521\) 10.7079 0.469123 0.234561 0.972101i \(-0.424635\pi\)
0.234561 + 0.972101i \(0.424635\pi\)
\(522\) 0 0
\(523\) 32.0896 1.40318 0.701590 0.712581i \(-0.252474\pi\)
0.701590 + 0.712581i \(0.252474\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.71656 0.292578
\(528\) 0 0
\(529\) −20.1051 −0.874137
\(530\) 0 0
\(531\) 14.9463 0.648615
\(532\) 0 0
\(533\) 16.5760 0.717985
\(534\) 0 0
\(535\) 44.8184 1.93767
\(536\) 0 0
\(537\) 12.1221 0.523109
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −17.4804 −0.751542 −0.375771 0.926713i \(-0.622622\pi\)
−0.375771 + 0.926713i \(0.622622\pi\)
\(542\) 0 0
\(543\) −23.2460 −0.997580
\(544\) 0 0
\(545\) −39.0415 −1.67235
\(546\) 0 0
\(547\) 34.6243 1.48043 0.740215 0.672370i \(-0.234724\pi\)
0.740215 + 0.672370i \(0.234724\pi\)
\(548\) 0 0
\(549\) 2.16354 0.0923377
\(550\) 0 0
\(551\) −25.5107 −1.08679
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 23.3913 0.992904
\(556\) 0 0
\(557\) 11.9042 0.504397 0.252199 0.967676i \(-0.418846\pi\)
0.252199 + 0.967676i \(0.418846\pi\)
\(558\) 0 0
\(559\) −13.2428 −0.560111
\(560\) 0 0
\(561\) 8.53411 0.360310
\(562\) 0 0
\(563\) −18.1339 −0.764255 −0.382127 0.924110i \(-0.624808\pi\)
−0.382127 + 0.924110i \(0.624808\pi\)
\(564\) 0 0
\(565\) 38.5125 1.62023
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −19.7102 −0.826293 −0.413147 0.910665i \(-0.635570\pi\)
−0.413147 + 0.910665i \(0.635570\pi\)
\(570\) 0 0
\(571\) 5.96122 0.249469 0.124735 0.992190i \(-0.460192\pi\)
0.124735 + 0.992190i \(0.460192\pi\)
\(572\) 0 0
\(573\) −23.2691 −0.972082
\(574\) 0 0
\(575\) −20.5847 −0.858441
\(576\) 0 0
\(577\) 34.9249 1.45394 0.726971 0.686668i \(-0.240927\pi\)
0.726971 + 0.686668i \(0.240927\pi\)
\(578\) 0 0
\(579\) 4.28613 0.178125
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −20.9135 −0.866151
\(584\) 0 0
\(585\) −10.0630 −0.416055
\(586\) 0 0
\(587\) −9.05369 −0.373686 −0.186843 0.982390i \(-0.559826\pi\)
−0.186843 + 0.982390i \(0.559826\pi\)
\(588\) 0 0
\(589\) −14.1970 −0.584977
\(590\) 0 0
\(591\) 11.9818 0.492864
\(592\) 0 0
\(593\) −0.129013 −0.00529795 −0.00264897 0.999996i \(-0.500843\pi\)
−0.00264897 + 0.999996i \(0.500843\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −22.1970 −0.908462
\(598\) 0 0
\(599\) 21.0357 0.859495 0.429747 0.902949i \(-0.358603\pi\)
0.429747 + 0.902949i \(0.358603\pi\)
\(600\) 0 0
\(601\) 4.67307 0.190619 0.0953093 0.995448i \(-0.469616\pi\)
0.0953093 + 0.995448i \(0.469616\pi\)
\(602\) 0 0
\(603\) 10.2540 0.417575
\(604\) 0 0
\(605\) −23.6382 −0.961028
\(606\) 0 0
\(607\) 32.3164 1.31168 0.655841 0.754899i \(-0.272314\pi\)
0.655841 + 0.754899i \(0.272314\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −32.3413 −1.30839
\(612\) 0 0
\(613\) −42.6480 −1.72254 −0.861268 0.508152i \(-0.830329\pi\)
−0.861268 + 0.508152i \(0.830329\pi\)
\(614\) 0 0
\(615\) 28.1649 1.13572
\(616\) 0 0
\(617\) −22.3741 −0.900745 −0.450373 0.892841i \(-0.648709\pi\)
−0.450373 + 0.892841i \(0.648709\pi\)
\(618\) 0 0
\(619\) −31.3428 −1.25977 −0.629886 0.776687i \(-0.716898\pi\)
−0.629886 + 0.776687i \(0.716898\pi\)
\(620\) 0 0
\(621\) 1.70143 0.0682759
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 60.8810 2.43524
\(626\) 0 0
\(627\) −18.0388 −0.720400
\(628\) 0 0
\(629\) −21.0027 −0.837432
\(630\) 0 0
\(631\) −2.79413 −0.111233 −0.0556163 0.998452i \(-0.517712\pi\)
−0.0556163 + 0.998452i \(0.517712\pi\)
\(632\) 0 0
\(633\) 0.540129 0.0214682
\(634\) 0 0
\(635\) −54.3231 −2.15574
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3.95543 0.156474
\(640\) 0 0
\(641\) −2.17042 −0.0857262 −0.0428631 0.999081i \(-0.513648\pi\)
−0.0428631 + 0.999081i \(0.513648\pi\)
\(642\) 0 0
\(643\) 8.91604 0.351614 0.175807 0.984425i \(-0.443746\pi\)
0.175807 + 0.984425i \(0.443746\pi\)
\(644\) 0 0
\(645\) −22.5013 −0.885990
\(646\) 0 0
\(647\) −18.1018 −0.711656 −0.355828 0.934551i \(-0.615801\pi\)
−0.355828 + 0.934551i \(0.615801\pi\)
\(648\) 0 0
\(649\) −34.3552 −1.34856
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.826893 −0.0323588 −0.0161794 0.999869i \(-0.505150\pi\)
−0.0161794 + 0.999869i \(0.505150\pi\)
\(654\) 0 0
\(655\) −36.4394 −1.42381
\(656\) 0 0
\(657\) 9.68428 0.377820
\(658\) 0 0
\(659\) −32.4956 −1.26585 −0.632924 0.774214i \(-0.718145\pi\)
−0.632924 + 0.774214i \(0.718145\pi\)
\(660\) 0 0
\(661\) 27.4127 1.06623 0.533115 0.846043i \(-0.321021\pi\)
0.533115 + 0.846043i \(0.321021\pi\)
\(662\) 0 0
\(663\) 9.03546 0.350908
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.53078 −0.214153
\(668\) 0 0
\(669\) 10.1194 0.391239
\(670\) 0 0
\(671\) −4.97306 −0.191983
\(672\) 0 0
\(673\) 31.9884 1.23306 0.616531 0.787330i \(-0.288537\pi\)
0.616531 + 0.787330i \(0.288537\pi\)
\(674\) 0 0
\(675\) −12.0985 −0.465671
\(676\) 0 0
\(677\) −40.6979 −1.56415 −0.782073 0.623186i \(-0.785838\pi\)
−0.782073 + 0.623186i \(0.785838\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 27.7587 1.06371
\(682\) 0 0
\(683\) 17.0357 0.651852 0.325926 0.945395i \(-0.394324\pi\)
0.325926 + 0.945395i \(0.394324\pi\)
\(684\) 0 0
\(685\) 1.51920 0.0580458
\(686\) 0 0
\(687\) −14.8962 −0.568324
\(688\) 0 0
\(689\) −22.1421 −0.843548
\(690\) 0 0
\(691\) −25.8926 −0.985002 −0.492501 0.870312i \(-0.663917\pi\)
−0.492501 + 0.870312i \(0.663917\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −63.3227 −2.40197
\(696\) 0 0
\(697\) −25.2888 −0.957882
\(698\) 0 0
\(699\) 14.5644 0.550875
\(700\) 0 0
\(701\) −25.8781 −0.977402 −0.488701 0.872451i \(-0.662529\pi\)
−0.488701 + 0.872451i \(0.662529\pi\)
\(702\) 0 0
\(703\) 44.3940 1.67435
\(704\) 0 0
\(705\) −54.9523 −2.06962
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 30.2660 1.13666 0.568332 0.822799i \(-0.307589\pi\)
0.568332 + 0.822799i \(0.307589\pi\)
\(710\) 0 0
\(711\) −10.0388 −0.376484
\(712\) 0 0
\(713\) −3.07794 −0.115270
\(714\) 0 0
\(715\) 23.1306 0.865036
\(716\) 0 0
\(717\) 1.70143 0.0635410
\(718\) 0 0
\(719\) −15.2652 −0.569296 −0.284648 0.958632i \(-0.591877\pi\)
−0.284648 + 0.958632i \(0.591877\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −27.0141 −1.00466
\(724\) 0 0
\(725\) 39.3282 1.46061
\(726\) 0 0
\(727\) −19.0154 −0.705241 −0.352620 0.935766i \(-0.614709\pi\)
−0.352620 + 0.935766i \(0.614709\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 20.2036 0.747259
\(732\) 0 0
\(733\) −9.21353 −0.340309 −0.170155 0.985417i \(-0.554427\pi\)
−0.170155 + 0.985417i \(0.554427\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −23.5696 −0.868196
\(738\) 0 0
\(739\) 38.1078 1.40182 0.700910 0.713250i \(-0.252778\pi\)
0.700910 + 0.713250i \(0.252778\pi\)
\(740\) 0 0
\(741\) −19.0985 −0.701601
\(742\) 0 0
\(743\) −2.46528 −0.0904425 −0.0452213 0.998977i \(-0.514399\pi\)
−0.0452213 + 0.998977i \(0.514399\pi\)
\(744\) 0 0
\(745\) −67.8650 −2.48638
\(746\) 0 0
\(747\) 11.6956 0.427921
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 28.2843 1.03211 0.516054 0.856556i \(-0.327400\pi\)
0.516054 + 0.856556i \(0.327400\pi\)
\(752\) 0 0
\(753\) 12.8269 0.467438
\(754\) 0 0
\(755\) 32.1903 1.17153
\(756\) 0 0
\(757\) −32.0691 −1.16557 −0.582785 0.812627i \(-0.698037\pi\)
−0.582785 + 0.812627i \(0.698037\pi\)
\(758\) 0 0
\(759\) −3.91085 −0.141955
\(760\) 0 0
\(761\) −3.99136 −0.144687 −0.0723434 0.997380i \(-0.523048\pi\)
−0.0723434 + 0.997380i \(0.523048\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 15.3525 0.555071
\(766\) 0 0
\(767\) −36.3734 −1.31337
\(768\) 0 0
\(769\) 21.8082 0.786423 0.393212 0.919448i \(-0.371364\pi\)
0.393212 + 0.919448i \(0.371364\pi\)
\(770\) 0 0
\(771\) 1.99892 0.0719892
\(772\) 0 0
\(773\) −18.9956 −0.683224 −0.341612 0.939841i \(-0.610973\pi\)
−0.341612 + 0.939841i \(0.610973\pi\)
\(774\) 0 0
\(775\) 21.8866 0.786190
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 53.4537 1.91518
\(780\) 0 0
\(781\) −9.09184 −0.325332
\(782\) 0 0
\(783\) −3.25067 −0.116170
\(784\) 0 0
\(785\) −71.1433 −2.53921
\(786\) 0 0
\(787\) 45.2063 1.61143 0.805716 0.592302i \(-0.201781\pi\)
0.805716 + 0.592302i \(0.201781\pi\)
\(788\) 0 0
\(789\) 1.07215 0.0381696
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −5.26520 −0.186973
\(794\) 0 0
\(795\) −37.6226 −1.33433
\(796\) 0 0
\(797\) 36.5517 1.29473 0.647364 0.762181i \(-0.275871\pi\)
0.647364 + 0.762181i \(0.275871\pi\)
\(798\) 0 0
\(799\) 49.3409 1.74556
\(800\) 0 0
\(801\) 12.7725 0.451294
\(802\) 0 0
\(803\) −22.2600 −0.785539
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −31.7188 −1.11655
\(808\) 0 0
\(809\) −36.0829 −1.26861 −0.634304 0.773083i \(-0.718713\pi\)
−0.634304 + 0.773083i \(0.718713\pi\)
\(810\) 0 0
\(811\) 26.6565 0.936036 0.468018 0.883719i \(-0.344968\pi\)
0.468018 + 0.883719i \(0.344968\pi\)
\(812\) 0 0
\(813\) −0.614744 −0.0215600
\(814\) 0 0
\(815\) −27.7732 −0.972853
\(816\) 0 0
\(817\) −42.7050 −1.49406
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 42.0369 1.46710 0.733549 0.679636i \(-0.237862\pi\)
0.733549 + 0.679636i \(0.237862\pi\)
\(822\) 0 0
\(823\) 26.4692 0.922659 0.461329 0.887229i \(-0.347373\pi\)
0.461329 + 0.887229i \(0.347373\pi\)
\(824\) 0 0
\(825\) 27.8093 0.968194
\(826\) 0 0
\(827\) −55.8663 −1.94266 −0.971330 0.237733i \(-0.923596\pi\)
−0.971330 + 0.237733i \(0.923596\pi\)
\(828\) 0 0
\(829\) −0.394822 −0.0137127 −0.00685637 0.999976i \(-0.502182\pi\)
−0.00685637 + 0.999976i \(0.502182\pi\)
\(830\) 0 0
\(831\) 14.6747 0.509060
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 5.01120 0.173420
\(836\) 0 0
\(837\) −1.80904 −0.0625294
\(838\) 0 0
\(839\) −20.9851 −0.724486 −0.362243 0.932084i \(-0.617989\pi\)
−0.362243 + 0.932084i \(0.617989\pi\)
\(840\) 0 0
\(841\) −18.4331 −0.635625
\(842\) 0 0
\(843\) −12.1406 −0.418145
\(844\) 0 0
\(845\) −29.2660 −1.00678
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −7.54346 −0.258891
\(850\) 0 0
\(851\) 9.62473 0.329931
\(852\) 0 0
\(853\) 8.76517 0.300114 0.150057 0.988677i \(-0.452054\pi\)
0.150057 + 0.988677i \(0.452054\pi\)
\(854\) 0 0
\(855\) −32.4510 −1.10980
\(856\) 0 0
\(857\) 9.45879 0.323106 0.161553 0.986864i \(-0.448350\pi\)
0.161553 + 0.986864i \(0.448350\pi\)
\(858\) 0 0
\(859\) −32.7377 −1.11700 −0.558499 0.829505i \(-0.688622\pi\)
−0.558499 + 0.829505i \(0.688622\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.73086 −0.297202 −0.148601 0.988897i \(-0.547477\pi\)
−0.148601 + 0.988897i \(0.547477\pi\)
\(864\) 0 0
\(865\) −42.4964 −1.44492
\(866\) 0 0
\(867\) 3.21522 0.109194
\(868\) 0 0
\(869\) 23.0749 0.782762
\(870\) 0 0
\(871\) −24.9542 −0.845540
\(872\) 0 0
\(873\) 12.0274 0.407067
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 36.8814 1.24540 0.622698 0.782462i \(-0.286036\pi\)
0.622698 + 0.782462i \(0.286036\pi\)
\(878\) 0 0
\(879\) −21.4648 −0.723990
\(880\) 0 0
\(881\) 10.7543 0.362320 0.181160 0.983454i \(-0.442015\pi\)
0.181160 + 0.983454i \(0.442015\pi\)
\(882\) 0 0
\(883\) 13.8266 0.465303 0.232652 0.972560i \(-0.425260\pi\)
0.232652 + 0.972560i \(0.425260\pi\)
\(884\) 0 0
\(885\) −61.8035 −2.07750
\(886\) 0 0
\(887\) 34.8657 1.17067 0.585337 0.810790i \(-0.300962\pi\)
0.585337 + 0.810790i \(0.300962\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −2.29857 −0.0770051
\(892\) 0 0
\(893\) −104.293 −3.49004
\(894\) 0 0
\(895\) −50.1254 −1.67551
\(896\) 0 0
\(897\) −4.14060 −0.138251
\(898\) 0 0
\(899\) 5.88058 0.196128
\(900\) 0 0
\(901\) 33.7808 1.12540
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 96.1228 3.19523
\(906\) 0 0
\(907\) −26.9202 −0.893871 −0.446935 0.894566i \(-0.647485\pi\)
−0.446935 + 0.894566i \(0.647485\pi\)
\(908\) 0 0
\(909\) 1.30661 0.0433374
\(910\) 0 0
\(911\) −16.3288 −0.540999 −0.270499 0.962720i \(-0.587189\pi\)
−0.270499 + 0.962720i \(0.587189\pi\)
\(912\) 0 0
\(913\) −26.8833 −0.889707
\(914\) 0 0
\(915\) −8.94631 −0.295756
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.51920 0.0501139 0.0250570 0.999686i \(-0.492023\pi\)
0.0250570 + 0.999686i \(0.492023\pi\)
\(920\) 0 0
\(921\) −7.41738 −0.244411
\(922\) 0 0
\(923\) −9.62595 −0.316842
\(924\) 0 0
\(925\) −68.4394 −2.25027
\(926\) 0 0
\(927\) 11.0033 0.361397
\(928\) 0 0
\(929\) 8.39057 0.275286 0.137643 0.990482i \(-0.456047\pi\)
0.137643 + 0.990482i \(0.456047\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −14.7881 −0.484141
\(934\) 0 0
\(935\) −35.2888 −1.15407
\(936\) 0 0
\(937\) −28.6019 −0.934382 −0.467191 0.884156i \(-0.654734\pi\)
−0.467191 + 0.884156i \(0.654734\pi\)
\(938\) 0 0
\(939\) 9.95434 0.324848
\(940\) 0 0
\(941\) 30.8743 1.00647 0.503237 0.864148i \(-0.332142\pi\)
0.503237 + 0.864148i \(0.332142\pi\)
\(942\) 0 0
\(943\) 11.5889 0.377386
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −25.0151 −0.812883 −0.406441 0.913677i \(-0.633230\pi\)
−0.406441 + 0.913677i \(0.633230\pi\)
\(948\) 0 0
\(949\) −23.5677 −0.765040
\(950\) 0 0
\(951\) −2.97907 −0.0966031
\(952\) 0 0
\(953\) 4.11942 0.133441 0.0667205 0.997772i \(-0.478746\pi\)
0.0667205 + 0.997772i \(0.478746\pi\)
\(954\) 0 0
\(955\) 96.2186 3.11356
\(956\) 0 0
\(957\) 7.47191 0.241533
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27.7274 −0.894432
\(962\) 0 0
\(963\) 10.8387 0.349272
\(964\) 0 0
\(965\) −17.7233 −0.570533
\(966\) 0 0
\(967\) −23.5901 −0.758607 −0.379303 0.925272i \(-0.623836\pi\)
−0.379303 + 0.925272i \(0.623836\pi\)
\(968\) 0 0
\(969\) 29.1373 0.936024
\(970\) 0 0
\(971\) −14.8833 −0.477627 −0.238814 0.971065i \(-0.576758\pi\)
−0.238814 + 0.971065i \(0.576758\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 29.4430 0.942929
\(976\) 0 0
\(977\) −44.6353 −1.42801 −0.714005 0.700141i \(-0.753121\pi\)
−0.714005 + 0.700141i \(0.753121\pi\)
\(978\) 0 0
\(979\) −29.3585 −0.938302
\(980\) 0 0
\(981\) −9.44164 −0.301448
\(982\) 0 0
\(983\) 13.1167 0.418359 0.209179 0.977877i \(-0.432921\pi\)
0.209179 + 0.977877i \(0.432921\pi\)
\(984\) 0 0
\(985\) −49.5450 −1.57863
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9.25855 −0.294405
\(990\) 0 0
\(991\) 37.5968 1.19430 0.597150 0.802129i \(-0.296300\pi\)
0.597150 + 0.802129i \(0.296300\pi\)
\(992\) 0 0
\(993\) −6.80571 −0.215973
\(994\) 0 0
\(995\) 91.7852 2.90979
\(996\) 0 0
\(997\) 26.1044 0.826733 0.413367 0.910565i \(-0.364353\pi\)
0.413367 + 0.910565i \(0.364353\pi\)
\(998\) 0 0
\(999\) 5.65685 0.178975
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 4704.2.a.bx.1.4 yes 4
4.3 odd 2 4704.2.a.bz.1.4 yes 4
7.6 odd 2 4704.2.a.by.1.1 yes 4
8.3 odd 2 9408.2.a.ek.1.1 4
8.5 even 2 9408.2.a.em.1.1 4
28.27 even 2 4704.2.a.bw.1.1 4
56.13 odd 2 9408.2.a.el.1.4 4
56.27 even 2 9408.2.a.en.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bw.1.1 4 28.27 even 2
4704.2.a.bx.1.4 yes 4 1.1 even 1 trivial
4704.2.a.by.1.1 yes 4 7.6 odd 2
4704.2.a.bz.1.4 yes 4 4.3 odd 2
9408.2.a.ek.1.1 4 8.3 odd 2
9408.2.a.el.1.4 4 56.13 odd 2
9408.2.a.em.1.1 4 8.5 even 2
9408.2.a.en.1.4 4 56.27 even 2