Properties

Label 4600.2.e.u.4049.6
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 28 x^{8} + 260 x^{6} + 897 x^{4} + 1056 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.6
Root \(0.568386i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.u.4049.5

$q$-expansion

\(f(q)\) \(=\) \(q+0.568386i q^{3} +4.73770i q^{7} +2.67694 q^{9} +O(q^{10})\) \(q+0.568386i q^{3} +4.73770i q^{7} +2.67694 q^{9} -0.360532 q^{11} -5.26123i q^{13} +0.370852i q^{17} +4.60586 q^{19} -2.69284 q^{21} +1.00000i q^{23} +3.22669i q^{27} -0.939238 q^{29} +9.66662 q^{31} -0.204921i q^{33} +3.26862i q^{37} +2.99041 q^{39} +5.29977 q^{41} -1.25491i q^{47} -15.4458 q^{49} -0.210787 q^{51} -10.9278i q^{53} +2.61790i q^{57} +9.66955 q^{59} +9.71441 q^{61} +12.6825i q^{63} -7.07001i q^{67} -0.568386 q^{69} +11.3747 q^{71} -0.745086i q^{73} -1.70809i q^{77} -0.415709 q^{79} +6.19680 q^{81} +9.26862i q^{83} -0.533850i q^{87} -12.6122 q^{89} +24.9261 q^{91} +5.49437i q^{93} +14.0404i q^{97} -0.965121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 26q^{9} + O(q^{10}) \) \( 10q - 26q^{9} - 2q^{11} - 14q^{19} + 12q^{21} - 8q^{29} + 38q^{31} - 38q^{39} + 50q^{41} - 50q^{49} + 38q^{51} + 2q^{59} - 10q^{61} + 2q^{71} + 4q^{79} + 114q^{81} - 12q^{89} + 22q^{91} + 130q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 0.568386i 0.328158i 0.986447 + 0.164079i \(0.0524652\pi\)
−0.986447 + 0.164079i \(0.947535\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.73770i 1.79068i 0.445381 + 0.895341i \(0.353068\pi\)
−0.445381 + 0.895341i \(0.646932\pi\)
\(8\) 0 0
\(9\) 2.67694 0.892312
\(10\) 0 0
\(11\) −0.360532 −0.108704 −0.0543522 0.998522i \(-0.517309\pi\)
−0.0543522 + 0.998522i \(0.517309\pi\)
\(12\) 0 0
\(13\) − 5.26123i − 1.45920i −0.683873 0.729601i \(-0.739706\pi\)
0.683873 0.729601i \(-0.260294\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.370852i 0.0899449i 0.998988 + 0.0449724i \(0.0143200\pi\)
−0.998988 + 0.0449724i \(0.985680\pi\)
\(18\) 0 0
\(19\) 4.60586 1.05666 0.528328 0.849040i \(-0.322819\pi\)
0.528328 + 0.849040i \(0.322819\pi\)
\(20\) 0 0
\(21\) −2.69284 −0.587626
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.22669i 0.620977i
\(28\) 0 0
\(29\) −0.939238 −0.174412 −0.0872061 0.996190i \(-0.527794\pi\)
−0.0872061 + 0.996190i \(0.527794\pi\)
\(30\) 0 0
\(31\) 9.66662 1.73618 0.868088 0.496411i \(-0.165349\pi\)
0.868088 + 0.496411i \(0.165349\pi\)
\(32\) 0 0
\(33\) − 0.204921i − 0.0356722i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.26862i 0.537357i 0.963230 + 0.268679i \(0.0865869\pi\)
−0.963230 + 0.268679i \(0.913413\pi\)
\(38\) 0 0
\(39\) 2.99041 0.478849
\(40\) 0 0
\(41\) 5.29977 0.827685 0.413843 0.910348i \(-0.364186\pi\)
0.413843 + 0.910348i \(0.364186\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.25491i − 0.183048i −0.995803 0.0915240i \(-0.970826\pi\)
0.995803 0.0915240i \(-0.0291738\pi\)
\(48\) 0 0
\(49\) −15.4458 −2.20654
\(50\) 0 0
\(51\) −0.210787 −0.0295161
\(52\) 0 0
\(53\) − 10.9278i − 1.50106i −0.660839 0.750528i \(-0.729799\pi\)
0.660839 0.750528i \(-0.270201\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.61790i 0.346750i
\(58\) 0 0
\(59\) 9.66955 1.25887 0.629434 0.777054i \(-0.283287\pi\)
0.629434 + 0.777054i \(0.283287\pi\)
\(60\) 0 0
\(61\) 9.71441 1.24380 0.621901 0.783096i \(-0.286361\pi\)
0.621901 + 0.783096i \(0.286361\pi\)
\(62\) 0 0
\(63\) 12.6825i 1.59785i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 7.07001i − 0.863739i −0.901936 0.431870i \(-0.857854\pi\)
0.901936 0.431870i \(-0.142146\pi\)
\(68\) 0 0
\(69\) −0.568386 −0.0684257
\(70\) 0 0
\(71\) 11.3747 1.34993 0.674965 0.737850i \(-0.264159\pi\)
0.674965 + 0.737850i \(0.264159\pi\)
\(72\) 0 0
\(73\) − 0.745086i − 0.0872057i −0.999049 0.0436029i \(-0.986116\pi\)
0.999049 0.0436029i \(-0.0138836\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.70809i − 0.194655i
\(78\) 0 0
\(79\) −0.415709 −0.0467709 −0.0233854 0.999727i \(-0.507444\pi\)
−0.0233854 + 0.999727i \(0.507444\pi\)
\(80\) 0 0
\(81\) 6.19680 0.688534
\(82\) 0 0
\(83\) 9.26862i 1.01736i 0.860955 + 0.508681i \(0.169867\pi\)
−0.860955 + 0.508681i \(0.830133\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 0.533850i − 0.0572347i
\(88\) 0 0
\(89\) −12.6122 −1.33689 −0.668444 0.743763i \(-0.733039\pi\)
−0.668444 + 0.743763i \(0.733039\pi\)
\(90\) 0 0
\(91\) 24.9261 2.61297
\(92\) 0 0
\(93\) 5.49437i 0.569740i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0404i 1.42559i 0.701374 + 0.712793i \(0.252570\pi\)
−0.701374 + 0.712793i \(0.747430\pi\)
\(98\) 0 0
\(99\) −0.965121 −0.0969983
\(100\) 0 0
\(101\) −12.7440 −1.26808 −0.634038 0.773302i \(-0.718604\pi\)
−0.634038 + 0.773302i \(0.718604\pi\)
\(102\) 0 0
\(103\) 3.31347i 0.326486i 0.986586 + 0.163243i \(0.0521955\pi\)
−0.986586 + 0.163243i \(0.947805\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 8.13184i − 0.786135i −0.919510 0.393067i \(-0.871414\pi\)
0.919510 0.393067i \(-0.128586\pi\)
\(108\) 0 0
\(109\) 5.79508 0.555068 0.277534 0.960716i \(-0.410483\pi\)
0.277534 + 0.960716i \(0.410483\pi\)
\(110\) 0 0
\(111\) −1.85784 −0.176338
\(112\) 0 0
\(113\) 7.60724i 0.715629i 0.933793 + 0.357815i \(0.116478\pi\)
−0.933793 + 0.357815i \(0.883522\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 14.0840i − 1.30206i
\(118\) 0 0
\(119\) −1.75699 −0.161063
\(120\) 0 0
\(121\) −10.8700 −0.988183
\(122\) 0 0
\(123\) 3.01232i 0.271611i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 4.13077i − 0.366547i −0.983062 0.183273i \(-0.941331\pi\)
0.983062 0.183273i \(-0.0586693\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −17.7582 −1.55154 −0.775770 0.631016i \(-0.782638\pi\)
−0.775770 + 0.631016i \(0.782638\pi\)
\(132\) 0 0
\(133\) 21.8212i 1.89213i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.86954i 0.586905i 0.955974 + 0.293452i \(0.0948042\pi\)
−0.955974 + 0.293452i \(0.905196\pi\)
\(138\) 0 0
\(139\) −20.6635 −1.75266 −0.876330 0.481712i \(-0.840015\pi\)
−0.876330 + 0.481712i \(0.840015\pi\)
\(140\) 0 0
\(141\) 0.713276 0.0600686
\(142\) 0 0
\(143\) 1.89684i 0.158622i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 8.77917i − 0.724094i
\(148\) 0 0
\(149\) 16.4763 1.34979 0.674896 0.737912i \(-0.264188\pi\)
0.674896 + 0.737912i \(0.264188\pi\)
\(150\) 0 0
\(151\) −8.89224 −0.723640 −0.361820 0.932248i \(-0.617844\pi\)
−0.361820 + 0.932248i \(0.617844\pi\)
\(152\) 0 0
\(153\) 0.992748i 0.0802589i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 6.62249i − 0.528532i −0.964450 0.264266i \(-0.914870\pi\)
0.964450 0.264266i \(-0.0851297\pi\)
\(158\) 0 0
\(159\) 6.21124 0.492583
\(160\) 0 0
\(161\) −4.73770 −0.373383
\(162\) 0 0
\(163\) 16.0779i 1.25932i 0.776872 + 0.629658i \(0.216805\pi\)
−0.776872 + 0.629658i \(0.783195\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.8763i 1.07378i 0.843651 + 0.536891i \(0.180402\pi\)
−0.843651 + 0.536891i \(0.819598\pi\)
\(168\) 0 0
\(169\) −14.6805 −1.12927
\(170\) 0 0
\(171\) 12.3296 0.942867
\(172\) 0 0
\(173\) 2.05518i 0.156252i 0.996943 + 0.0781261i \(0.0248937\pi\)
−0.996943 + 0.0781261i \(0.975106\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.49604i 0.413108i
\(178\) 0 0
\(179\) −17.6478 −1.31906 −0.659531 0.751678i \(-0.729245\pi\)
−0.659531 + 0.751678i \(0.729245\pi\)
\(180\) 0 0
\(181\) 2.85477 0.212193 0.106097 0.994356i \(-0.466165\pi\)
0.106097 + 0.994356i \(0.466165\pi\)
\(182\) 0 0
\(183\) 5.52153i 0.408164i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 0.133704i − 0.00977741i
\(188\) 0 0
\(189\) −15.2871 −1.11197
\(190\) 0 0
\(191\) 1.13677 0.0822540 0.0411270 0.999154i \(-0.486905\pi\)
0.0411270 + 0.999154i \(0.486905\pi\)
\(192\) 0 0
\(193\) 26.4694i 1.90531i 0.304055 + 0.952654i \(0.401659\pi\)
−0.304055 + 0.952654i \(0.598341\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.44424i 0.387886i 0.981013 + 0.193943i \(0.0621277\pi\)
−0.981013 + 0.193943i \(0.937872\pi\)
\(198\) 0 0
\(199\) 18.6122 1.31938 0.659691 0.751537i \(-0.270687\pi\)
0.659691 + 0.751537i \(0.270687\pi\)
\(200\) 0 0
\(201\) 4.01850 0.283443
\(202\) 0 0
\(203\) − 4.44983i − 0.312317i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.67694i 0.186060i
\(208\) 0 0
\(209\) −1.66056 −0.114863
\(210\) 0 0
\(211\) 6.10003 0.419944 0.209972 0.977707i \(-0.432663\pi\)
0.209972 + 0.977707i \(0.432663\pi\)
\(212\) 0 0
\(213\) 6.46523i 0.442990i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 45.7975i 3.10894i
\(218\) 0 0
\(219\) 0.423497 0.0286173
\(220\) 0 0
\(221\) 1.95114 0.131248
\(222\) 0 0
\(223\) 16.4136i 1.09913i 0.835450 + 0.549567i \(0.185207\pi\)
−0.835450 + 0.549567i \(0.814793\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.21171i 0.0804240i 0.999191 + 0.0402120i \(0.0128033\pi\)
−0.999191 + 0.0402120i \(0.987197\pi\)
\(228\) 0 0
\(229\) 22.8293 1.50860 0.754300 0.656529i \(-0.227976\pi\)
0.754300 + 0.656529i \(0.227976\pi\)
\(230\) 0 0
\(231\) 0.970856 0.0638776
\(232\) 0 0
\(233\) − 21.9420i − 1.43747i −0.695284 0.718735i \(-0.744722\pi\)
0.695284 0.718735i \(-0.255278\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 0.236283i − 0.0153482i
\(238\) 0 0
\(239\) 8.58274 0.555172 0.277586 0.960701i \(-0.410466\pi\)
0.277586 + 0.960701i \(0.410466\pi\)
\(240\) 0 0
\(241\) 15.3857 0.991079 0.495540 0.868585i \(-0.334970\pi\)
0.495540 + 0.868585i \(0.334970\pi\)
\(242\) 0 0
\(243\) 13.2023i 0.846925i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 24.2325i − 1.54187i
\(248\) 0 0
\(249\) −5.26815 −0.333856
\(250\) 0 0
\(251\) −8.85477 −0.558908 −0.279454 0.960159i \(-0.590153\pi\)
−0.279454 + 0.960159i \(0.590153\pi\)
\(252\) 0 0
\(253\) − 0.360532i − 0.0226664i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 22.6008i − 1.40980i −0.709308 0.704899i \(-0.750992\pi\)
0.709308 0.704899i \(-0.249008\pi\)
\(258\) 0 0
\(259\) −15.4857 −0.962236
\(260\) 0 0
\(261\) −2.51428 −0.155630
\(262\) 0 0
\(263\) 7.73590i 0.477016i 0.971141 + 0.238508i \(0.0766583\pi\)
−0.971141 + 0.238508i \(0.923342\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 7.16858i − 0.438710i
\(268\) 0 0
\(269\) 3.87154 0.236052 0.118026 0.993011i \(-0.462343\pi\)
0.118026 + 0.993011i \(0.462343\pi\)
\(270\) 0 0
\(271\) 4.24705 0.257990 0.128995 0.991645i \(-0.458825\pi\)
0.128995 + 0.991645i \(0.458825\pi\)
\(272\) 0 0
\(273\) 14.1677i 0.857466i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 7.26754i − 0.436664i −0.975875 0.218332i \(-0.929938\pi\)
0.975875 0.218332i \(-0.0700616\pi\)
\(278\) 0 0
\(279\) 25.8769 1.54921
\(280\) 0 0
\(281\) −13.0131 −0.776297 −0.388148 0.921597i \(-0.626885\pi\)
−0.388148 + 0.921597i \(0.626885\pi\)
\(282\) 0 0
\(283\) − 17.8342i − 1.06013i −0.847956 0.530067i \(-0.822167\pi\)
0.847956 0.530067i \(-0.177833\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 25.1087i 1.48212i
\(288\) 0 0
\(289\) 16.8625 0.991910
\(290\) 0 0
\(291\) −7.98037 −0.467818
\(292\) 0 0
\(293\) − 7.32291i − 0.427809i −0.976855 0.213905i \(-0.931382\pi\)
0.976855 0.213905i \(-0.0686182\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1.16333i − 0.0675030i
\(298\) 0 0
\(299\) 5.26123 0.304265
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 7.24352i − 0.416129i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 29.1127i − 1.66155i −0.556608 0.830775i \(-0.687897\pi\)
0.556608 0.830775i \(-0.312103\pi\)
\(308\) 0 0
\(309\) −1.88333 −0.107139
\(310\) 0 0
\(311\) 0.458912 0.0260225 0.0130113 0.999915i \(-0.495858\pi\)
0.0130113 + 0.999915i \(0.495858\pi\)
\(312\) 0 0
\(313\) 17.8279i 1.00769i 0.863794 + 0.503846i \(0.168082\pi\)
−0.863794 + 0.503846i \(0.831918\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.8799i 1.34123i 0.741807 + 0.670614i \(0.233969\pi\)
−0.741807 + 0.670614i \(0.766031\pi\)
\(318\) 0 0
\(319\) 0.338625 0.0189594
\(320\) 0 0
\(321\) 4.62203 0.257976
\(322\) 0 0
\(323\) 1.70809i 0.0950407i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.29384i 0.182150i
\(328\) 0 0
\(329\) 5.94540 0.327781
\(330\) 0 0
\(331\) 30.0214 1.65013 0.825063 0.565041i \(-0.191140\pi\)
0.825063 + 0.565041i \(0.191140\pi\)
\(332\) 0 0
\(333\) 8.74988i 0.479490i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 11.9440i 0.650631i 0.945605 + 0.325316i \(0.105471\pi\)
−0.945605 + 0.325316i \(0.894529\pi\)
\(338\) 0 0
\(339\) −4.32385 −0.234839
\(340\) 0 0
\(341\) −3.48512 −0.188730
\(342\) 0 0
\(343\) − 40.0136i − 2.16053i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 0.873132i − 0.0468722i −0.999725 0.0234361i \(-0.992539\pi\)
0.999725 0.0234361i \(-0.00746062\pi\)
\(348\) 0 0
\(349\) 18.8868 1.01099 0.505493 0.862831i \(-0.331311\pi\)
0.505493 + 0.862831i \(0.331311\pi\)
\(350\) 0 0
\(351\) 16.9764 0.906131
\(352\) 0 0
\(353\) 5.43875i 0.289475i 0.989470 + 0.144738i \(0.0462338\pi\)
−0.989470 + 0.144738i \(0.953766\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 0.998646i − 0.0528540i
\(358\) 0 0
\(359\) 23.4229 1.23622 0.618108 0.786093i \(-0.287899\pi\)
0.618108 + 0.786093i \(0.287899\pi\)
\(360\) 0 0
\(361\) 2.21390 0.116521
\(362\) 0 0
\(363\) − 6.17837i − 0.324280i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 27.2014i 1.41990i 0.704252 + 0.709951i \(0.251283\pi\)
−0.704252 + 0.709951i \(0.748717\pi\)
\(368\) 0 0
\(369\) 14.1872 0.738554
\(370\) 0 0
\(371\) 51.7728 2.68791
\(372\) 0 0
\(373\) − 20.0431i − 1.03779i −0.854837 0.518897i \(-0.826343\pi\)
0.854837 0.518897i \(-0.173657\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.94155i 0.254503i
\(378\) 0 0
\(379\) 12.8364 0.659362 0.329681 0.944092i \(-0.393059\pi\)
0.329681 + 0.944092i \(0.393059\pi\)
\(380\) 0 0
\(381\) 2.34787 0.120285
\(382\) 0 0
\(383\) 30.5405i 1.56055i 0.625439 + 0.780273i \(0.284920\pi\)
−0.625439 + 0.780273i \(0.715080\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −32.3392 −1.63966 −0.819831 0.572605i \(-0.805933\pi\)
−0.819831 + 0.572605i \(0.805933\pi\)
\(390\) 0 0
\(391\) −0.370852 −0.0187548
\(392\) 0 0
\(393\) − 10.0935i − 0.509150i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 28.9147i 1.45119i 0.688123 + 0.725594i \(0.258435\pi\)
−0.688123 + 0.725594i \(0.741565\pi\)
\(398\) 0 0
\(399\) −12.4028 −0.620919
\(400\) 0 0
\(401\) 24.2862 1.21279 0.606397 0.795162i \(-0.292614\pi\)
0.606397 + 0.795162i \(0.292614\pi\)
\(402\) 0 0
\(403\) − 50.8583i − 2.53343i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 1.17844i − 0.0584131i
\(408\) 0 0
\(409\) −31.8371 −1.57424 −0.787122 0.616797i \(-0.788430\pi\)
−0.787122 + 0.616797i \(0.788430\pi\)
\(410\) 0 0
\(411\) −3.90455 −0.192597
\(412\) 0 0
\(413\) 45.8114i 2.25423i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 11.7449i − 0.575149i
\(418\) 0 0
\(419\) 21.4880 1.04976 0.524879 0.851177i \(-0.324110\pi\)
0.524879 + 0.851177i \(0.324110\pi\)
\(420\) 0 0
\(421\) −13.2930 −0.647859 −0.323930 0.946081i \(-0.605004\pi\)
−0.323930 + 0.946081i \(0.605004\pi\)
\(422\) 0 0
\(423\) − 3.35933i − 0.163336i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 46.0239i 2.22725i
\(428\) 0 0
\(429\) −1.07814 −0.0520530
\(430\) 0 0
\(431\) −0.534460 −0.0257440 −0.0128720 0.999917i \(-0.504097\pi\)
−0.0128720 + 0.999917i \(0.504097\pi\)
\(432\) 0 0
\(433\) − 16.3377i − 0.785140i −0.919722 0.392570i \(-0.871586\pi\)
0.919722 0.392570i \(-0.128414\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.60586i 0.220328i
\(438\) 0 0
\(439\) −28.7573 −1.37251 −0.686255 0.727361i \(-0.740746\pi\)
−0.686255 + 0.727361i \(0.740746\pi\)
\(440\) 0 0
\(441\) −41.3474 −1.96892
\(442\) 0 0
\(443\) 15.6542i 0.743751i 0.928283 + 0.371876i \(0.121285\pi\)
−0.928283 + 0.371876i \(0.878715\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 9.36491i 0.442945i
\(448\) 0 0
\(449\) −38.4302 −1.81363 −0.906817 0.421525i \(-0.861495\pi\)
−0.906817 + 0.421525i \(0.861495\pi\)
\(450\) 0 0
\(451\) −1.91074 −0.0899730
\(452\) 0 0
\(453\) − 5.05422i − 0.237468i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 37.2199i − 1.74107i −0.492104 0.870536i \(-0.663772\pi\)
0.492104 0.870536i \(-0.336228\pi\)
\(458\) 0 0
\(459\) −1.19663 −0.0558537
\(460\) 0 0
\(461\) −3.25491 −0.151596 −0.0757982 0.997123i \(-0.524150\pi\)
−0.0757982 + 0.997123i \(0.524150\pi\)
\(462\) 0 0
\(463\) − 29.9534i − 1.39205i −0.718016 0.696027i \(-0.754950\pi\)
0.718016 0.696027i \(-0.245050\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 39.3302i − 1.81999i −0.414623 0.909993i \(-0.636087\pi\)
0.414623 0.909993i \(-0.363913\pi\)
\(468\) 0 0
\(469\) 33.4956 1.54668
\(470\) 0 0
\(471\) 3.76413 0.173442
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 29.2532i − 1.33941i
\(478\) 0 0
\(479\) −8.18660 −0.374055 −0.187028 0.982355i \(-0.559885\pi\)
−0.187028 + 0.982355i \(0.559885\pi\)
\(480\) 0 0
\(481\) 17.1969 0.784113
\(482\) 0 0
\(483\) − 2.69284i − 0.122529i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 31.9963i − 1.44989i −0.688806 0.724946i \(-0.741865\pi\)
0.688806 0.724946i \(-0.258135\pi\)
\(488\) 0 0
\(489\) −9.13844 −0.413255
\(490\) 0 0
\(491\) −7.12584 −0.321585 −0.160792 0.986988i \(-0.551405\pi\)
−0.160792 + 0.986988i \(0.551405\pi\)
\(492\) 0 0
\(493\) − 0.348319i − 0.0156875i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 53.8899i 2.41729i
\(498\) 0 0
\(499\) −23.4219 −1.04851 −0.524254 0.851562i \(-0.675656\pi\)
−0.524254 + 0.851562i \(0.675656\pi\)
\(500\) 0 0
\(501\) −7.88711 −0.352370
\(502\) 0 0
\(503\) 3.94170i 0.175752i 0.996131 + 0.0878758i \(0.0280079\pi\)
−0.996131 + 0.0878758i \(0.971992\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 8.34421i − 0.370579i
\(508\) 0 0
\(509\) −38.5237 −1.70753 −0.853766 0.520656i \(-0.825687\pi\)
−0.853766 + 0.520656i \(0.825687\pi\)
\(510\) 0 0
\(511\) 3.52999 0.156158
\(512\) 0 0
\(513\) 14.8617i 0.656159i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0.452436i 0.0198981i
\(518\) 0 0
\(519\) −1.16813 −0.0512754
\(520\) 0 0
\(521\) −39.2428 −1.71926 −0.859630 0.510917i \(-0.829306\pi\)
−0.859630 + 0.510917i \(0.829306\pi\)
\(522\) 0 0
\(523\) 39.0190i 1.70618i 0.521763 + 0.853090i \(0.325274\pi\)
−0.521763 + 0.853090i \(0.674726\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.58489i 0.156160i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 25.8848 1.12330
\(532\) 0 0
\(533\) − 27.8833i − 1.20776i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 10.0308i − 0.432860i
\(538\) 0 0
\(539\) 5.56870 0.239861
\(540\) 0 0
\(541\) 2.54061 0.109230 0.0546148 0.998508i \(-0.482607\pi\)
0.0546148 + 0.998508i \(0.482607\pi\)
\(542\) 0 0
\(543\) 1.62261i 0.0696329i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6.78776i 0.290224i 0.989415 + 0.145112i \(0.0463542\pi\)
−0.989415 + 0.145112i \(0.953646\pi\)
\(548\) 0 0
\(549\) 26.0049 1.10986
\(550\) 0 0
\(551\) −4.32600 −0.184294
\(552\) 0 0
\(553\) − 1.96950i − 0.0837517i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 24.7086i − 1.04694i −0.852045 0.523468i \(-0.824638\pi\)
0.852045 0.523468i \(-0.175362\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0.0759955 0.00320853
\(562\) 0 0
\(563\) 28.1423i 1.18606i 0.805181 + 0.593029i \(0.202068\pi\)
−0.805181 + 0.593029i \(0.797932\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 29.3586i 1.23294i
\(568\) 0 0
\(569\) −41.1938 −1.72694 −0.863468 0.504404i \(-0.831712\pi\)
−0.863468 + 0.504404i \(0.831712\pi\)
\(570\) 0 0
\(571\) 32.4687 1.35877 0.679387 0.733780i \(-0.262246\pi\)
0.679387 + 0.733780i \(0.262246\pi\)
\(572\) 0 0
\(573\) 0.646126i 0.0269923i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 6.61770i 0.275498i 0.990467 + 0.137749i \(0.0439868\pi\)
−0.990467 + 0.137749i \(0.956013\pi\)
\(578\) 0 0
\(579\) −15.0448 −0.625242
\(580\) 0 0
\(581\) −43.9119 −1.82177
\(582\) 0 0
\(583\) 3.93984i 0.163171i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 8.18008i − 0.337628i −0.985648 0.168814i \(-0.946006\pi\)
0.985648 0.168814i \(-0.0539937\pi\)
\(588\) 0 0
\(589\) 44.5230 1.83454
\(590\) 0 0
\(591\) −3.09443 −0.127288
\(592\) 0 0
\(593\) − 29.7489i − 1.22164i −0.791768 0.610821i \(-0.790839\pi\)
0.791768 0.610821i \(-0.209161\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10.5789i 0.432966i
\(598\) 0 0
\(599\) −39.4069 −1.61012 −0.805061 0.593192i \(-0.797868\pi\)
−0.805061 + 0.593192i \(0.797868\pi\)
\(600\) 0 0
\(601\) −34.3183 −1.39987 −0.699937 0.714205i \(-0.746788\pi\)
−0.699937 + 0.714205i \(0.746788\pi\)
\(602\) 0 0
\(603\) − 18.9260i − 0.770725i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 11.7570i − 0.477200i −0.971118 0.238600i \(-0.923312\pi\)
0.971118 0.238600i \(-0.0766885\pi\)
\(608\) 0 0
\(609\) 2.52922 0.102489
\(610\) 0 0
\(611\) −6.60239 −0.267104
\(612\) 0 0
\(613\) − 5.86862i − 0.237031i −0.992952 0.118516i \(-0.962186\pi\)
0.992952 0.118516i \(-0.0378136\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 8.45842i − 0.340523i −0.985399 0.170262i \(-0.945539\pi\)
0.985399 0.170262i \(-0.0544613\pi\)
\(618\) 0 0
\(619\) −0.851047 −0.0342065 −0.0171032 0.999854i \(-0.505444\pi\)
−0.0171032 + 0.999854i \(0.505444\pi\)
\(620\) 0 0
\(621\) −3.22669 −0.129483
\(622\) 0 0
\(623\) − 59.7527i − 2.39394i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 0.943838i − 0.0376933i
\(628\) 0 0
\(629\) −1.21217 −0.0483325
\(630\) 0 0
\(631\) 19.7131 0.784768 0.392384 0.919802i \(-0.371650\pi\)
0.392384 + 0.919802i \(0.371650\pi\)
\(632\) 0 0
\(633\) 3.46717i 0.137808i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 81.2638i 3.21979i
\(638\) 0 0
\(639\) 30.4494 1.20456
\(640\) 0 0
\(641\) −34.3593 −1.35711 −0.678555 0.734550i \(-0.737393\pi\)
−0.678555 + 0.734550i \(0.737393\pi\)
\(642\) 0 0
\(643\) 24.2817i 0.957578i 0.877930 + 0.478789i \(0.158924\pi\)
−0.877930 + 0.478789i \(0.841076\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 30.1465i − 1.18518i −0.805504 0.592590i \(-0.798105\pi\)
0.805504 0.592590i \(-0.201895\pi\)
\(648\) 0 0
\(649\) −3.48618 −0.136845
\(650\) 0 0
\(651\) −26.0307 −1.02022
\(652\) 0 0
\(653\) 7.17298i 0.280700i 0.990102 + 0.140350i \(0.0448228\pi\)
−0.990102 + 0.140350i \(0.955177\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 1.99455i − 0.0778148i
\(658\) 0 0
\(659\) 12.8521 0.500645 0.250323 0.968162i \(-0.419463\pi\)
0.250323 + 0.968162i \(0.419463\pi\)
\(660\) 0 0
\(661\) 3.72426 0.144857 0.0724285 0.997374i \(-0.476925\pi\)
0.0724285 + 0.997374i \(0.476925\pi\)
\(662\) 0 0
\(663\) 1.10900i 0.0430700i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 0.939238i − 0.0363675i
\(668\) 0 0
\(669\) −9.32924 −0.360689
\(670\) 0 0
\(671\) −3.50235 −0.135207
\(672\) 0 0
\(673\) − 13.5204i − 0.521175i −0.965450 0.260587i \(-0.916084\pi\)
0.965450 0.260587i \(-0.0839162\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 17.8534i − 0.686161i −0.939306 0.343081i \(-0.888530\pi\)
0.939306 0.343081i \(-0.111470\pi\)
\(678\) 0 0
\(679\) −66.5192 −2.55277
\(680\) 0 0
\(681\) −0.688719 −0.0263918
\(682\) 0 0
\(683\) − 15.8200i − 0.605337i −0.953096 0.302669i \(-0.902122\pi\)
0.953096 0.302669i \(-0.0978776\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 12.9758i 0.495059i
\(688\) 0 0
\(689\) −57.4939 −2.19034
\(690\) 0 0
\(691\) 9.45476 0.359676 0.179838 0.983696i \(-0.442443\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(692\) 0 0
\(693\) − 4.57245i − 0.173693i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.96543i 0.0744460i
\(698\) 0 0
\(699\) 12.4715 0.471717
\(700\) 0 0
\(701\) 29.0096 1.09568 0.547838 0.836584i \(-0.315451\pi\)
0.547838 + 0.836584i \(0.315451\pi\)
\(702\) 0 0
\(703\) 15.0548i 0.567801i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 60.3773i − 2.27072i
\(708\) 0 0
\(709\) 8.22423 0.308868 0.154434 0.988003i \(-0.450645\pi\)
0.154434 + 0.988003i \(0.450645\pi\)
\(710\) 0 0
\(711\) −1.11283 −0.0417342
\(712\) 0 0
\(713\) 9.66662i 0.362018i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.87831i 0.182184i
\(718\) 0 0
\(719\) 12.5468 0.467917 0.233958 0.972247i \(-0.424832\pi\)
0.233958 + 0.972247i \(0.424832\pi\)
\(720\) 0 0
\(721\) −15.6982 −0.584633
\(722\) 0 0
\(723\) 8.74501i 0.325230i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 29.1952i − 1.08279i −0.840768 0.541395i \(-0.817896\pi\)
0.840768 0.541395i \(-0.182104\pi\)
\(728\) 0 0
\(729\) 11.0864 0.410609
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 36.0414i 1.33122i 0.746299 + 0.665611i \(0.231829\pi\)
−0.746299 + 0.665611i \(0.768171\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.54896i 0.0938923i
\(738\) 0 0
\(739\) −0.728002 −0.0267800 −0.0133900 0.999910i \(-0.504262\pi\)
−0.0133900 + 0.999910i \(0.504262\pi\)
\(740\) 0 0
\(741\) 13.7734 0.505978
\(742\) 0 0
\(743\) 32.8880i 1.20655i 0.797535 + 0.603273i \(0.206137\pi\)
−0.797535 + 0.603273i \(0.793863\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 24.8115i 0.907805i
\(748\) 0 0
\(749\) 38.5262 1.40772
\(750\) 0 0
\(751\) 14.3337 0.523044 0.261522 0.965197i \(-0.415776\pi\)
0.261522 + 0.965197i \(0.415776\pi\)
\(752\) 0 0
\(753\) − 5.03293i − 0.183410i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 0.0587519i − 0.00213537i −0.999999 0.00106769i \(-0.999660\pi\)
0.999999 0.00106769i \(-0.000339855\pi\)
\(758\) 0 0
\(759\) 0.204921 0.00743817
\(760\) 0 0
\(761\) 45.7135 1.65711 0.828556 0.559906i \(-0.189163\pi\)
0.828556 + 0.559906i \(0.189163\pi\)
\(762\) 0 0
\(763\) 27.4553i 0.993950i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 50.8737i − 1.83694i
\(768\) 0 0
\(769\) 39.3949 1.42061 0.710307 0.703892i \(-0.248556\pi\)
0.710307 + 0.703892i \(0.248556\pi\)
\(770\) 0 0
\(771\) 12.8460 0.462636
\(772\) 0 0
\(773\) − 21.1799i − 0.761788i −0.924619 0.380894i \(-0.875616\pi\)
0.924619 0.380894i \(-0.124384\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 8.80187i − 0.315765i
\(778\) 0 0
\(779\) 24.4100 0.874578
\(780\) 0 0
\(781\) −4.10094 −0.146743
\(782\) 0 0
\(783\) − 3.03063i − 0.108306i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 21.5154i 0.766941i 0.923553 + 0.383470i \(0.125271\pi\)
−0.923553 + 0.383470i \(0.874729\pi\)
\(788\) 0 0
\(789\) −4.39698 −0.156537
\(790\) 0 0
\(791\) −36.0408 −1.28146
\(792\) 0 0
\(793\) − 51.1097i − 1.81496i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 54.7227i − 1.93838i −0.246320 0.969189i \(-0.579221\pi\)
0.246320 0.969189i \(-0.420779\pi\)
\(798\) 0 0
\(799\) 0.465388 0.0164642
\(800\) 0 0
\(801\) −33.7620 −1.19292
\(802\) 0 0
\(803\) 0.268627i 0.00947965i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2.20053i 0.0774623i
\(808\) 0 0
\(809\) 43.5554 1.53133 0.765663 0.643242i \(-0.222411\pi\)
0.765663 + 0.643242i \(0.222411\pi\)
\(810\) 0 0
\(811\) −13.1837 −0.462944 −0.231472 0.972842i \(-0.574354\pi\)
−0.231472 + 0.972842i \(0.574354\pi\)
\(812\) 0 0
\(813\) 2.41397i 0.0846615i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 66.7256 2.33158
\(820\) 0 0
\(821\) −3.63328 −0.126803 −0.0634013 0.997988i \(-0.520195\pi\)
−0.0634013 + 0.997988i \(0.520195\pi\)
\(822\) 0 0
\(823\) 39.5044i 1.37704i 0.725220 + 0.688518i \(0.241738\pi\)
−0.725220 + 0.688518i \(0.758262\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 48.3819i 1.68240i 0.540721 + 0.841202i \(0.318152\pi\)
−0.540721 + 0.841202i \(0.681848\pi\)
\(828\) 0 0
\(829\) 44.5864 1.54855 0.774275 0.632850i \(-0.218115\pi\)
0.774275 + 0.632850i \(0.218115\pi\)
\(830\) 0 0
\(831\) 4.13077 0.143295
\(832\) 0 0
\(833\) − 5.72810i − 0.198467i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 31.1912i 1.07813i
\(838\) 0 0
\(839\) 29.2324 1.00921 0.504606 0.863350i \(-0.331638\pi\)
0.504606 + 0.863350i \(0.331638\pi\)
\(840\) 0 0
\(841\) −28.1178 −0.969580
\(842\) 0 0
\(843\) − 7.39647i − 0.254748i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 51.4989i − 1.76952i
\(848\) 0 0
\(849\) 10.1367 0.347891
\(850\) 0 0
\(851\) −3.26862 −0.112047
\(852\) 0 0
\(853\) − 50.5599i − 1.73114i −0.500790 0.865569i \(-0.666957\pi\)
0.500790 0.865569i \(-0.333043\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 21.9568i − 0.750030i −0.927019 0.375015i \(-0.877638\pi\)
0.927019 0.375015i \(-0.122362\pi\)
\(858\) 0 0
\(859\) −13.7671 −0.469726 −0.234863 0.972029i \(-0.575464\pi\)
−0.234863 + 0.972029i \(0.575464\pi\)
\(860\) 0 0
\(861\) −14.2714 −0.486370
\(862\) 0 0
\(863\) 26.5748i 0.904618i 0.891861 + 0.452309i \(0.149400\pi\)
−0.891861 + 0.452309i \(0.850600\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 9.58439i 0.325503i
\(868\) 0 0
\(869\) 0.149876 0.00508420
\(870\) 0 0
\(871\) −37.1969 −1.26037
\(872\) 0 0
\(873\) 37.5853i 1.27207i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 45.9414i 1.55133i 0.631145 + 0.775665i \(0.282585\pi\)
−0.631145 + 0.775665i \(0.717415\pi\)
\(878\) 0 0
\(879\) 4.16224 0.140389
\(880\) 0 0
\(881\) −23.9054 −0.805392 −0.402696 0.915334i \(-0.631927\pi\)
−0.402696 + 0.915334i \(0.631927\pi\)
\(882\) 0 0
\(883\) − 52.4002i − 1.76341i −0.471804 0.881703i \(-0.656397\pi\)
0.471804 0.881703i \(-0.343603\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 32.7608i − 1.10000i −0.835165 0.550000i \(-0.814628\pi\)
0.835165 0.550000i \(-0.185372\pi\)
\(888\) 0 0
\(889\) 19.5703 0.656368
\(890\) 0 0
\(891\) −2.23415 −0.0748467
\(892\) 0 0
\(893\) − 5.77995i − 0.193419i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.99041i 0.0998469i
\(898\) 0 0
\(899\) −9.07926 −0.302810
\(900\) 0 0
\(901\) 4.05262 0.135012
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 20.9426i 0.695388i 0.937608 + 0.347694i \(0.113035\pi\)
−0.937608 + 0.347694i \(0.886965\pi\)
\(908\) 0 0
\(909\) −34.1149 −1.13152
\(910\) 0 0
\(911\) 40.6418 1.34652 0.673262 0.739404i \(-0.264893\pi\)
0.673262 + 0.739404i \(0.264893\pi\)
\(912\) 0 0
\(913\) − 3.34163i − 0.110592i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 84.1330i − 2.77831i
\(918\) 0 0
\(919\) 50.3241 1.66004 0.830019 0.557735i \(-0.188330\pi\)
0.830019 + 0.557735i \(0.188330\pi\)
\(920\) 0 0
\(921\) 16.5473 0.545251
\(922\) 0 0
\(923\) − 59.8449i − 1.96982i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 8.86996i 0.291328i
\(928\) 0 0
\(929\) −24.1954 −0.793825 −0.396912 0.917856i \(-0.629918\pi\)
−0.396912 + 0.917856i \(0.629918\pi\)
\(930\) 0 0
\(931\) −71.1411 −2.33155
\(932\) 0 0
\(933\) 0.260839i 0.00853949i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 37.2988i − 1.21850i −0.792978 0.609250i \(-0.791471\pi\)
0.792978 0.609250i \(-0.208529\pi\)
\(938\) 0 0
\(939\) −10.1331 −0.330682
\(940\) 0 0
\(941\) 16.6467 0.542667 0.271334 0.962485i \(-0.412535\pi\)
0.271334 + 0.962485i \(0.412535\pi\)
\(942\) 0 0
\(943\) 5.29977i 0.172584i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 30.1868i − 0.980938i −0.871459 0.490469i \(-0.836825\pi\)
0.871459 0.490469i \(-0.163175\pi\)
\(948\) 0 0
\(949\) −3.92007 −0.127251
\(950\) 0 0
\(951\) −13.5730 −0.440134
\(952\) 0 0
\(953\) − 34.6992i − 1.12402i −0.827132 0.562008i \(-0.810029\pi\)
0.827132 0.562008i \(-0.189971\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0.192470i 0.00622167i
\(958\) 0 0
\(959\) −32.5458 −1.05096
\(960\) 0 0
\(961\) 62.4435 2.01431
\(962\) 0 0
\(963\) − 21.7684i − 0.701478i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 11.4788i − 0.369133i −0.982820 0.184566i \(-0.940912\pi\)
0.982820 0.184566i \(-0.0590881\pi\)
\(968\) 0 0
\(969\) −0.970856 −0.0311884
\(970\) 0 0
\(971\) −46.8441 −1.50330 −0.751650 0.659563i \(-0.770742\pi\)
−0.751650 + 0.659563i \(0.770742\pi\)
\(972\) 0 0
\(973\) − 97.8977i − 3.13846i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 39.3737i 1.25968i 0.776727 + 0.629838i \(0.216879\pi\)
−0.776727 + 0.629838i \(0.783121\pi\)
\(978\) 0 0
\(979\) 4.54709 0.145326
\(980\) 0 0
\(981\) 15.5131 0.495294
\(982\) 0 0
\(983\) − 42.3505i − 1.35077i −0.737465 0.675385i \(-0.763977\pi\)
0.737465 0.675385i \(-0.236023\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 3.37929i 0.107564i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −2.00104 −0.0635652 −0.0317826 0.999495i \(-0.510118\pi\)
−0.0317826 + 0.999495i \(0.510118\pi\)
\(992\) 0 0
\(993\) 17.0638i 0.541502i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.97936i 0.0626869i 0.999509 + 0.0313435i \(0.00997857\pi\)
−0.999509 + 0.0313435i \(0.990021\pi\)
\(998\) 0 0
\(999\) −10.5468 −0.333687
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 4600.2.e.u.4049.6 10
5.2 odd 4 4600.2.a.be.1.3 5
5.3 odd 4 920.2.a.j.1.3 5
5.4 even 2 inner 4600.2.e.u.4049.5 10
15.8 even 4 8280.2.a.bs.1.5 5
20.3 even 4 1840.2.a.v.1.3 5
20.7 even 4 9200.2.a.cu.1.3 5
40.3 even 4 7360.2.a.cp.1.3 5
40.13 odd 4 7360.2.a.co.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.3 5 5.3 odd 4
1840.2.a.v.1.3 5 20.3 even 4
4600.2.a.be.1.3 5 5.2 odd 4
4600.2.e.u.4049.5 10 5.4 even 2 inner
4600.2.e.u.4049.6 10 1.1 even 1 trivial
7360.2.a.co.1.3 5 40.13 odd 4
7360.2.a.cp.1.3 5 40.3 even 4
8280.2.a.bs.1.5 5 15.8 even 4
9200.2.a.cu.1.3 5 20.7 even 4