Properties

Label 4600.2.e.w.4049.10
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: \(x^{10} + 18 x^{8} + 117 x^{6} + 333 x^{4} + 396 x^{2} + 144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.10
Root \(2.61696i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.w.4049.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.61696i q^{3} -3.83744i q^{7} -3.84849 q^{9} +O(q^{10})\) \(q+2.61696i q^{3} -3.83744i q^{7} -3.84849 q^{9} -0.508005 q^{11} +1.01106i q^{13} -1.44705i q^{17} +0.508005 q^{19} +10.0424 q^{21} -1.00000i q^{23} -2.22047i q^{27} -7.51040 q^{29} -0.439038 q^{31} -1.32943i q^{33} +7.02642i q^{37} -2.64590 q^{39} +5.47041 q^{41} +6.72592i q^{43} +2.64098i q^{47} -7.72592 q^{49} +3.78688 q^{51} +4.77648i q^{53} +1.32943i q^{57} -3.85345 q^{59} -9.05844 q^{61} +14.7683i q^{63} -3.45696i q^{67} +2.61696 q^{69} -2.73649 q^{71} -9.21300i q^{73} +1.94944i q^{77} -10.5504 q^{79} -5.73458 q^{81} -1.40211i q^{83} -19.6544i q^{87} -6.77086 q^{89} +3.87986 q^{91} -1.14895i q^{93} +0.313420i q^{97} +1.95506 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 6q^{9} + O(q^{10}) \) \( 10q - 6q^{9} - 24q^{29} - 36q^{31} - 18q^{39} - 12q^{41} - 30q^{49} - 12q^{51} + 2q^{59} + 20q^{61} + 16q^{71} - 54q^{81} - 28q^{89} - 92q^{91} + 12q^{99} + O(q^{100}) \)

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
\(3\) 2.61696i 1.51090i 0.655204 + 0.755452i \(0.272583\pi\)
−0.655204 + 0.755452i \(0.727417\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.83744i − 1.45041i −0.688531 0.725207i \(-0.741744\pi\)
0.688531 0.725207i \(-0.258256\pi\)
\(8\) 0 0
\(9\) −3.84849 −1.28283
\(10\) 0 0
\(11\) −0.508005 −0.153169 −0.0765847 0.997063i \(-0.524402\pi\)
−0.0765847 + 0.997063i \(0.524402\pi\)
\(12\) 0 0
\(13\) 1.01106i 0.280417i 0.990122 + 0.140208i \(0.0447772\pi\)
−0.990122 + 0.140208i \(0.955223\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.44705i − 0.350961i −0.984483 0.175481i \(-0.943852\pi\)
0.984483 0.175481i \(-0.0561479\pi\)
\(18\) 0 0
\(19\) 0.508005 0.116544 0.0582722 0.998301i \(-0.481441\pi\)
0.0582722 + 0.998301i \(0.481441\pi\)
\(20\) 0 0
\(21\) 10.0424 2.19144
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 2.22047i − 0.427330i
\(28\) 0 0
\(29\) −7.51040 −1.39465 −0.697323 0.716757i \(-0.745626\pi\)
−0.697323 + 0.716757i \(0.745626\pi\)
\(30\) 0 0
\(31\) −0.439038 −0.0788536 −0.0394268 0.999222i \(-0.512553\pi\)
−0.0394268 + 0.999222i \(0.512553\pi\)
\(32\) 0 0
\(33\) − 1.32943i − 0.231424i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.02642i 1.15514i 0.816343 + 0.577568i \(0.195998\pi\)
−0.816343 + 0.577568i \(0.804002\pi\)
\(38\) 0 0
\(39\) −2.64590 −0.423683
\(40\) 0 0
\(41\) 5.47041 0.854335 0.427167 0.904173i \(-0.359512\pi\)
0.427167 + 0.904173i \(0.359512\pi\)
\(42\) 0 0
\(43\) 6.72592i 1.02569i 0.858480 + 0.512847i \(0.171409\pi\)
−0.858480 + 0.512847i \(0.828591\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.64098i 0.385227i 0.981275 + 0.192614i \(0.0616964\pi\)
−0.981275 + 0.192614i \(0.938304\pi\)
\(48\) 0 0
\(49\) −7.72592 −1.10370
\(50\) 0 0
\(51\) 3.78688 0.530269
\(52\) 0 0
\(53\) 4.77648i 0.656100i 0.944660 + 0.328050i \(0.106391\pi\)
−0.944660 + 0.328050i \(0.893609\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.32943i 0.176087i
\(58\) 0 0
\(59\) −3.85345 −0.501676 −0.250838 0.968029i \(-0.580706\pi\)
−0.250838 + 0.968029i \(0.580706\pi\)
\(60\) 0 0
\(61\) −9.05844 −1.15981 −0.579907 0.814683i \(-0.696911\pi\)
−0.579907 + 0.814683i \(0.696911\pi\)
\(62\) 0 0
\(63\) 14.7683i 1.86064i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.45696i − 0.422335i −0.977450 0.211167i \(-0.932273\pi\)
0.977450 0.211167i \(-0.0677265\pi\)
\(68\) 0 0
\(69\) 2.61696 0.315045
\(70\) 0 0
\(71\) −2.73649 −0.324762 −0.162381 0.986728i \(-0.551917\pi\)
−0.162381 + 0.986728i \(0.551917\pi\)
\(72\) 0 0
\(73\) − 9.21300i − 1.07830i −0.842210 0.539150i \(-0.818746\pi\)
0.842210 0.539150i \(-0.181254\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.94944i 0.222159i
\(78\) 0 0
\(79\) −10.5504 −1.18702 −0.593508 0.804828i \(-0.702258\pi\)
−0.593508 + 0.804828i \(0.702258\pi\)
\(80\) 0 0
\(81\) −5.73458 −0.637176
\(82\) 0 0
\(83\) − 1.40211i − 0.153901i −0.997035 0.0769505i \(-0.975482\pi\)
0.997035 0.0769505i \(-0.0245183\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 19.6544i − 2.10718i
\(88\) 0 0
\(89\) −6.77086 −0.717710 −0.358855 0.933393i \(-0.616833\pi\)
−0.358855 + 0.933393i \(0.616833\pi\)
\(90\) 0 0
\(91\) 3.87986 0.406720
\(92\) 0 0
\(93\) − 1.14895i − 0.119140i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.313420i 0.0318230i 0.999873 + 0.0159115i \(0.00506500\pi\)
−0.999873 + 0.0159115i \(0.994935\pi\)
\(98\) 0 0
\(99\) 1.95506 0.196490
\(100\) 0 0
\(101\) −4.83182 −0.480784 −0.240392 0.970676i \(-0.577276\pi\)
−0.240392 + 0.970676i \(0.577276\pi\)
\(102\) 0 0
\(103\) 8.07746i 0.795896i 0.917408 + 0.397948i \(0.130278\pi\)
−0.917408 + 0.397948i \(0.869722\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.1587i 1.07876i 0.842064 + 0.539378i \(0.181341\pi\)
−0.842064 + 0.539378i \(0.818659\pi\)
\(108\) 0 0
\(109\) −17.6015 −1.68592 −0.842958 0.537979i \(-0.819188\pi\)
−0.842958 + 0.537979i \(0.819188\pi\)
\(110\) 0 0
\(111\) −18.3879 −1.74530
\(112\) 0 0
\(113\) − 4.06454i − 0.382360i −0.981555 0.191180i \(-0.938769\pi\)
0.981555 0.191180i \(-0.0612314\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 3.89104i − 0.359727i
\(118\) 0 0
\(119\) −5.55296 −0.509039
\(120\) 0 0
\(121\) −10.7419 −0.976539
\(122\) 0 0
\(123\) 14.3159i 1.29082i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 4.75246i − 0.421713i −0.977517 0.210856i \(-0.932375\pi\)
0.977517 0.210856i \(-0.0676252\pi\)
\(128\) 0 0
\(129\) −17.6015 −1.54972
\(130\) 0 0
\(131\) 1.96851 0.171989 0.0859946 0.996296i \(-0.472593\pi\)
0.0859946 + 0.996296i \(0.472593\pi\)
\(132\) 0 0
\(133\) − 1.94944i − 0.169038i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.64285i 0.482101i 0.970513 + 0.241051i \(0.0774920\pi\)
−0.970513 + 0.241051i \(0.922508\pi\)
\(138\) 0 0
\(139\) 7.59724 0.644390 0.322195 0.946673i \(-0.395579\pi\)
0.322195 + 0.946673i \(0.395579\pi\)
\(140\) 0 0
\(141\) −6.91136 −0.582041
\(142\) 0 0
\(143\) − 0.513622i − 0.0429512i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 20.2184i − 1.66759i
\(148\) 0 0
\(149\) −10.7709 −0.882384 −0.441192 0.897413i \(-0.645444\pi\)
−0.441192 + 0.897413i \(0.645444\pi\)
\(150\) 0 0
\(151\) −16.5333 −1.34546 −0.672729 0.739889i \(-0.734878\pi\)
−0.672729 + 0.739889i \(0.734878\pi\)
\(152\) 0 0
\(153\) 5.56896i 0.450224i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.5436i 0.921281i 0.887587 + 0.460640i \(0.152380\pi\)
−0.887587 + 0.460640i \(0.847620\pi\)
\(158\) 0 0
\(159\) −12.4999 −0.991304
\(160\) 0 0
\(161\) −3.83744 −0.302432
\(162\) 0 0
\(163\) 3.32143i 0.260155i 0.991504 + 0.130077i \(0.0415226\pi\)
−0.991504 + 0.130077i \(0.958477\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 12.6742i − 0.980755i −0.871510 0.490378i \(-0.836859\pi\)
0.871510 0.490378i \(-0.163141\pi\)
\(168\) 0 0
\(169\) 11.9778 0.921367
\(170\) 0 0
\(171\) −1.95506 −0.149507
\(172\) 0 0
\(173\) 17.5733i 1.33607i 0.744130 + 0.668035i \(0.232864\pi\)
−0.744130 + 0.668035i \(0.767136\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 10.0843i − 0.757984i
\(178\) 0 0
\(179\) 14.8952 1.11332 0.556659 0.830741i \(-0.312083\pi\)
0.556659 + 0.830741i \(0.312083\pi\)
\(180\) 0 0
\(181\) −23.2868 −1.73089 −0.865446 0.501003i \(-0.832965\pi\)
−0.865446 + 0.501003i \(0.832965\pi\)
\(182\) 0 0
\(183\) − 23.7056i − 1.75237i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.735109i 0.0537565i
\(188\) 0 0
\(189\) −8.52093 −0.619806
\(190\) 0 0
\(191\) 6.92360 0.500974 0.250487 0.968120i \(-0.419409\pi\)
0.250487 + 0.968120i \(0.419409\pi\)
\(192\) 0 0
\(193\) − 7.74318i − 0.557366i −0.960383 0.278683i \(-0.910102\pi\)
0.960383 0.278683i \(-0.0898979\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.8908i 1.70215i 0.525045 + 0.851074i \(0.324048\pi\)
−0.525045 + 0.851074i \(0.675952\pi\)
\(198\) 0 0
\(199\) 23.9444 1.69737 0.848687 0.528896i \(-0.177394\pi\)
0.848687 + 0.528896i \(0.177394\pi\)
\(200\) 0 0
\(201\) 9.04673 0.638107
\(202\) 0 0
\(203\) 28.8207i 2.02282i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.84849i 0.267489i
\(208\) 0 0
\(209\) −0.258070 −0.0178510
\(210\) 0 0
\(211\) 14.4709 0.996220 0.498110 0.867114i \(-0.334027\pi\)
0.498110 + 0.867114i \(0.334027\pi\)
\(212\) 0 0
\(213\) − 7.16129i − 0.490684i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.68478i 0.114370i
\(218\) 0 0
\(219\) 24.1101 1.62921
\(220\) 0 0
\(221\) 1.46305 0.0984153
\(222\) 0 0
\(223\) − 4.74755i − 0.317919i −0.987285 0.158960i \(-0.949186\pi\)
0.987285 0.158960i \(-0.0508140\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 8.98399i − 0.596288i −0.954521 0.298144i \(-0.903632\pi\)
0.954521 0.298144i \(-0.0963676\pi\)
\(228\) 0 0
\(229\) −20.7290 −1.36981 −0.684906 0.728631i \(-0.740157\pi\)
−0.684906 + 0.728631i \(0.740157\pi\)
\(230\) 0 0
\(231\) −5.10161 −0.335661
\(232\) 0 0
\(233\) 26.0634i 1.70747i 0.520707 + 0.853735i \(0.325668\pi\)
−0.520707 + 0.853735i \(0.674332\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 27.6101i − 1.79347i
\(238\) 0 0
\(239\) 2.94209 0.190308 0.0951540 0.995463i \(-0.469666\pi\)
0.0951540 + 0.995463i \(0.469666\pi\)
\(240\) 0 0
\(241\) −24.9413 −1.60661 −0.803306 0.595567i \(-0.796927\pi\)
−0.803306 + 0.595567i \(0.796927\pi\)
\(242\) 0 0
\(243\) − 21.6686i − 1.39004i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.513622i 0.0326810i
\(248\) 0 0
\(249\) 3.66926 0.232530
\(250\) 0 0
\(251\) 21.1908 1.33755 0.668774 0.743465i \(-0.266819\pi\)
0.668774 + 0.743465i \(0.266819\pi\)
\(252\) 0 0
\(253\) 0.508005i 0.0319380i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 26.2110i − 1.63500i −0.575929 0.817500i \(-0.695360\pi\)
0.575929 0.817500i \(-0.304640\pi\)
\(258\) 0 0
\(259\) 26.9634 1.67543
\(260\) 0 0
\(261\) 28.9037 1.78910
\(262\) 0 0
\(263\) − 5.65276i − 0.348564i −0.984696 0.174282i \(-0.944240\pi\)
0.984696 0.174282i \(-0.0557605\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 17.7191i − 1.08439i
\(268\) 0 0
\(269\) −9.65772 −0.588841 −0.294421 0.955676i \(-0.595127\pi\)
−0.294421 + 0.955676i \(0.595127\pi\)
\(270\) 0 0
\(271\) 8.39040 0.509680 0.254840 0.966983i \(-0.417977\pi\)
0.254840 + 0.966983i \(0.417977\pi\)
\(272\) 0 0
\(273\) 10.1535i 0.614515i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 30.3317i − 1.82246i −0.411900 0.911229i \(-0.635135\pi\)
0.411900 0.911229i \(-0.364865\pi\)
\(278\) 0 0
\(279\) 1.68964 0.101156
\(280\) 0 0
\(281\) −25.3132 −1.51006 −0.755028 0.655692i \(-0.772377\pi\)
−0.755028 + 0.655692i \(0.772377\pi\)
\(282\) 0 0
\(283\) − 12.9629i − 0.770566i −0.922798 0.385283i \(-0.874104\pi\)
0.922798 0.385283i \(-0.125896\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 20.9924i − 1.23914i
\(288\) 0 0
\(289\) 14.9060 0.876826
\(290\) 0 0
\(291\) −0.820209 −0.0480815
\(292\) 0 0
\(293\) 25.9469i 1.51584i 0.652350 + 0.757918i \(0.273783\pi\)
−0.652350 + 0.757918i \(0.726217\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.12801i 0.0654539i
\(298\) 0 0
\(299\) 1.01106 0.0584709
\(300\) 0 0
\(301\) 25.8103 1.48768
\(302\) 0 0
\(303\) − 12.6447i − 0.726419i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 12.0640i − 0.688531i −0.938872 0.344266i \(-0.888128\pi\)
0.938872 0.344266i \(-0.111872\pi\)
\(308\) 0 0
\(309\) −21.1384 −1.20252
\(310\) 0 0
\(311\) −31.8125 −1.80392 −0.901959 0.431821i \(-0.857871\pi\)
−0.901959 + 0.431821i \(0.857871\pi\)
\(312\) 0 0
\(313\) − 4.30122i − 0.243119i −0.992584 0.121560i \(-0.961210\pi\)
0.992584 0.121560i \(-0.0387895\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.28317i 0.352898i 0.984310 + 0.176449i \(0.0564611\pi\)
−0.984310 + 0.176449i \(0.943539\pi\)
\(318\) 0 0
\(319\) 3.81532 0.213617
\(320\) 0 0
\(321\) −29.2020 −1.62990
\(322\) 0 0
\(323\) − 0.735109i − 0.0409026i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 46.0624i − 2.54726i
\(328\) 0 0
\(329\) 10.1346 0.558739
\(330\) 0 0
\(331\) −8.00053 −0.439749 −0.219874 0.975528i \(-0.570565\pi\)
−0.219874 + 0.975528i \(0.570565\pi\)
\(332\) 0 0
\(333\) − 27.0411i − 1.48184i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 10.1436i − 0.552554i −0.961078 0.276277i \(-0.910899\pi\)
0.961078 0.276277i \(-0.0891008\pi\)
\(338\) 0 0
\(339\) 10.6367 0.577709
\(340\) 0 0
\(341\) 0.223034 0.0120780
\(342\) 0 0
\(343\) 2.78567i 0.150412i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.8747i 1.01325i 0.862167 + 0.506625i \(0.169107\pi\)
−0.862167 + 0.506625i \(0.830893\pi\)
\(348\) 0 0
\(349\) −9.26611 −0.496004 −0.248002 0.968760i \(-0.579774\pi\)
−0.248002 + 0.968760i \(0.579774\pi\)
\(350\) 0 0
\(351\) 2.24502 0.119831
\(352\) 0 0
\(353\) 8.04442i 0.428161i 0.976816 + 0.214081i \(0.0686755\pi\)
−0.976816 + 0.214081i \(0.931324\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 14.5319i − 0.769109i
\(358\) 0 0
\(359\) −20.5443 −1.08429 −0.542144 0.840285i \(-0.682387\pi\)
−0.542144 + 0.840285i \(0.682387\pi\)
\(360\) 0 0
\(361\) −18.7419 −0.986417
\(362\) 0 0
\(363\) − 28.1112i − 1.47546i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.64032i 0.451021i 0.974241 + 0.225511i \(0.0724050\pi\)
−0.974241 + 0.225511i \(0.927595\pi\)
\(368\) 0 0
\(369\) −21.0528 −1.09597
\(370\) 0 0
\(371\) 18.3294 0.951617
\(372\) 0 0
\(373\) − 19.0442i − 0.986073i −0.870009 0.493037i \(-0.835887\pi\)
0.870009 0.493037i \(-0.164113\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 7.59344i − 0.391082i
\(378\) 0 0
\(379\) −0.926121 −0.0475717 −0.0237858 0.999717i \(-0.507572\pi\)
−0.0237858 + 0.999717i \(0.507572\pi\)
\(380\) 0 0
\(381\) 12.4370 0.637167
\(382\) 0 0
\(383\) − 9.38526i − 0.479564i −0.970827 0.239782i \(-0.922924\pi\)
0.970827 0.239782i \(-0.0770760\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 25.8847i − 1.31579i
\(388\) 0 0
\(389\) −2.22865 −0.112997 −0.0564985 0.998403i \(-0.517994\pi\)
−0.0564985 + 0.998403i \(0.517994\pi\)
\(390\) 0 0
\(391\) −1.44705 −0.0731805
\(392\) 0 0
\(393\) 5.15151i 0.259859i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 13.0448i − 0.654701i −0.944903 0.327350i \(-0.893844\pi\)
0.944903 0.327350i \(-0.106156\pi\)
\(398\) 0 0
\(399\) 5.10161 0.255400
\(400\) 0 0
\(401\) −2.22221 −0.110972 −0.0554859 0.998459i \(-0.517671\pi\)
−0.0554859 + 0.998459i \(0.517671\pi\)
\(402\) 0 0
\(403\) − 0.443893i − 0.0221119i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 3.56946i − 0.176931i
\(408\) 0 0
\(409\) 19.5232 0.965362 0.482681 0.875796i \(-0.339663\pi\)
0.482681 + 0.875796i \(0.339663\pi\)
\(410\) 0 0
\(411\) −14.7671 −0.728409
\(412\) 0 0
\(413\) 14.7874i 0.727638i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 19.8817i 0.973611i
\(418\) 0 0
\(419\) 24.9180 1.21732 0.608662 0.793430i \(-0.291707\pi\)
0.608662 + 0.793430i \(0.291707\pi\)
\(420\) 0 0
\(421\) 6.10721 0.297647 0.148824 0.988864i \(-0.452451\pi\)
0.148824 + 0.988864i \(0.452451\pi\)
\(422\) 0 0
\(423\) − 10.1638i − 0.494181i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 34.7612i 1.68221i
\(428\) 0 0
\(429\) 1.34413 0.0648952
\(430\) 0 0
\(431\) 4.44095 0.213913 0.106956 0.994264i \(-0.465889\pi\)
0.106956 + 0.994264i \(0.465889\pi\)
\(432\) 0 0
\(433\) 3.25554i 0.156451i 0.996936 + 0.0782257i \(0.0249255\pi\)
−0.996936 + 0.0782257i \(0.975075\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 0.508005i − 0.0243012i
\(438\) 0 0
\(439\) 26.9611 1.28678 0.643392 0.765537i \(-0.277527\pi\)
0.643392 + 0.765537i \(0.277527\pi\)
\(440\) 0 0
\(441\) 29.7331 1.41586
\(442\) 0 0
\(443\) − 16.0072i − 0.760526i −0.924878 0.380263i \(-0.875833\pi\)
0.924878 0.380263i \(-0.124167\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 28.1869i − 1.33320i
\(448\) 0 0
\(449\) 37.4278 1.76633 0.883163 0.469066i \(-0.155409\pi\)
0.883163 + 0.469066i \(0.155409\pi\)
\(450\) 0 0
\(451\) −2.77900 −0.130858
\(452\) 0 0
\(453\) − 43.2670i − 2.03286i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.75949i 0.409752i 0.978788 + 0.204876i \(0.0656790\pi\)
−0.978788 + 0.204876i \(0.934321\pi\)
\(458\) 0 0
\(459\) −3.21314 −0.149976
\(460\) 0 0
\(461\) 11.4459 0.533089 0.266545 0.963823i \(-0.414118\pi\)
0.266545 + 0.963823i \(0.414118\pi\)
\(462\) 0 0
\(463\) 8.23570i 0.382745i 0.981517 + 0.191373i \(0.0612939\pi\)
−0.981517 + 0.191373i \(0.938706\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.9436i 1.33935i 0.742654 + 0.669675i \(0.233567\pi\)
−0.742654 + 0.669675i \(0.766433\pi\)
\(468\) 0 0
\(469\) −13.2659 −0.612561
\(470\) 0 0
\(471\) −30.2092 −1.39197
\(472\) 0 0
\(473\) − 3.41680i − 0.157105i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 18.3823i − 0.841666i
\(478\) 0 0
\(479\) −31.1031 −1.42114 −0.710569 0.703627i \(-0.751563\pi\)
−0.710569 + 0.703627i \(0.751563\pi\)
\(480\) 0 0
\(481\) −7.10410 −0.323919
\(482\) 0 0
\(483\) − 10.0424i − 0.456946i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 1.90588i − 0.0863635i −0.999067 0.0431817i \(-0.986251\pi\)
0.999067 0.0431817i \(-0.0137494\pi\)
\(488\) 0 0
\(489\) −8.69206 −0.393069
\(490\) 0 0
\(491\) −27.2222 −1.22852 −0.614259 0.789104i \(-0.710545\pi\)
−0.614259 + 0.789104i \(0.710545\pi\)
\(492\) 0 0
\(493\) 10.8679i 0.489467i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10.5011i 0.471039i
\(498\) 0 0
\(499\) 9.38913 0.420315 0.210158 0.977668i \(-0.432602\pi\)
0.210158 + 0.977668i \(0.432602\pi\)
\(500\) 0 0
\(501\) 33.1678 1.48183
\(502\) 0 0
\(503\) 29.5613i 1.31807i 0.752110 + 0.659037i \(0.229036\pi\)
−0.752110 + 0.659037i \(0.770964\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 31.3454i 1.39210i
\(508\) 0 0
\(509\) −0.448890 −0.0198967 −0.00994835 0.999951i \(-0.503167\pi\)
−0.00994835 + 0.999951i \(0.503167\pi\)
\(510\) 0 0
\(511\) −35.3543 −1.56398
\(512\) 0 0
\(513\) − 1.12801i − 0.0498030i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 1.34163i − 0.0590050i
\(518\) 0 0
\(519\) −45.9886 −2.01867
\(520\) 0 0
\(521\) 26.8712 1.17725 0.588623 0.808407i \(-0.299670\pi\)
0.588623 + 0.808407i \(0.299670\pi\)
\(522\) 0 0
\(523\) 2.90473i 0.127015i 0.997981 + 0.0635075i \(0.0202287\pi\)
−0.997981 + 0.0635075i \(0.979771\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.635311i 0.0276746i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 14.8300 0.643566
\(532\) 0 0
\(533\) 5.53089i 0.239570i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 38.9801i 1.68212i
\(538\) 0 0
\(539\) 3.92481 0.169053
\(540\) 0 0
\(541\) 34.0386 1.46343 0.731716 0.681610i \(-0.238720\pi\)
0.731716 + 0.681610i \(0.238720\pi\)
\(542\) 0 0
\(543\) − 60.9406i − 2.61521i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 23.8364i 1.01917i 0.860420 + 0.509586i \(0.170202\pi\)
−0.860420 + 0.509586i \(0.829798\pi\)
\(548\) 0 0
\(549\) 34.8613 1.48785
\(550\) 0 0
\(551\) −3.81532 −0.162538
\(552\) 0 0
\(553\) 40.4866i 1.72167i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 40.4201i − 1.71265i −0.516435 0.856326i \(-0.672741\pi\)
0.516435 0.856326i \(-0.327259\pi\)
\(558\) 0 0
\(559\) −6.80028 −0.287621
\(560\) 0 0
\(561\) −1.92375 −0.0812209
\(562\) 0 0
\(563\) − 6.54758i − 0.275948i −0.990436 0.137974i \(-0.955941\pi\)
0.990436 0.137974i \(-0.0440590\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 22.0061i 0.924169i
\(568\) 0 0
\(569\) −20.4590 −0.857686 −0.428843 0.903379i \(-0.641079\pi\)
−0.428843 + 0.903379i \(0.641079\pi\)
\(570\) 0 0
\(571\) 4.71300 0.197233 0.0986164 0.995126i \(-0.468558\pi\)
0.0986164 + 0.995126i \(0.468558\pi\)
\(572\) 0 0
\(573\) 18.1188i 0.756924i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 25.9331i 1.07961i 0.841790 + 0.539805i \(0.181502\pi\)
−0.841790 + 0.539805i \(0.818498\pi\)
\(578\) 0 0
\(579\) 20.2636 0.842127
\(580\) 0 0
\(581\) −5.38049 −0.223220
\(582\) 0 0
\(583\) − 2.42648i − 0.100494i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 10.4711i − 0.432189i −0.976372 0.216095i \(-0.930668\pi\)
0.976372 0.216095i \(-0.0693320\pi\)
\(588\) 0 0
\(589\) −0.223034 −0.00918995
\(590\) 0 0
\(591\) −62.5213 −2.57178
\(592\) 0 0
\(593\) 20.8158i 0.854803i 0.904062 + 0.427401i \(0.140571\pi\)
−0.904062 + 0.427401i \(0.859429\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 62.6616i 2.56457i
\(598\) 0 0
\(599\) 14.7951 0.604511 0.302256 0.953227i \(-0.402260\pi\)
0.302256 + 0.953227i \(0.402260\pi\)
\(600\) 0 0
\(601\) −33.8995 −1.38279 −0.691394 0.722477i \(-0.743003\pi\)
−0.691394 + 0.722477i \(0.743003\pi\)
\(602\) 0 0
\(603\) 13.3041i 0.541784i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 32.9350i − 1.33679i −0.743807 0.668394i \(-0.766982\pi\)
0.743807 0.668394i \(-0.233018\pi\)
\(608\) 0 0
\(609\) −75.4227 −3.05628
\(610\) 0 0
\(611\) −2.67018 −0.108024
\(612\) 0 0
\(613\) − 24.8664i − 1.00434i −0.864768 0.502172i \(-0.832535\pi\)
0.864768 0.502172i \(-0.167465\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 20.1280i 0.810324i 0.914245 + 0.405162i \(0.132785\pi\)
−0.914245 + 0.405162i \(0.867215\pi\)
\(618\) 0 0
\(619\) 15.3719 0.617847 0.308924 0.951087i \(-0.400031\pi\)
0.308924 + 0.951087i \(0.400031\pi\)
\(620\) 0 0
\(621\) −2.22047 −0.0891046
\(622\) 0 0
\(623\) 25.9828i 1.04098i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 0.675358i − 0.0269712i
\(628\) 0 0
\(629\) 10.1676 0.405408
\(630\) 0 0
\(631\) −25.7115 −1.02356 −0.511779 0.859117i \(-0.671013\pi\)
−0.511779 + 0.859117i \(0.671013\pi\)
\(632\) 0 0
\(633\) 37.8699i 1.50519i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 7.81134i − 0.309497i
\(638\) 0 0
\(639\) 10.5314 0.416614
\(640\) 0 0
\(641\) 2.95684 0.116788 0.0583941 0.998294i \(-0.481402\pi\)
0.0583941 + 0.998294i \(0.481402\pi\)
\(642\) 0 0
\(643\) − 16.7764i − 0.661596i −0.943702 0.330798i \(-0.892682\pi\)
0.943702 0.330798i \(-0.107318\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.99247i 0.0783322i 0.999233 + 0.0391661i \(0.0124702\pi\)
−0.999233 + 0.0391661i \(0.987530\pi\)
\(648\) 0 0
\(649\) 1.95757 0.0768414
\(650\) 0 0
\(651\) −4.40901 −0.172803
\(652\) 0 0
\(653\) 46.4989i 1.81964i 0.414999 + 0.909822i \(0.363782\pi\)
−0.414999 + 0.909822i \(0.636218\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 35.4562i 1.38328i
\(658\) 0 0
\(659\) 37.1468 1.44703 0.723517 0.690307i \(-0.242524\pi\)
0.723517 + 0.690307i \(0.242524\pi\)
\(660\) 0 0
\(661\) −7.01756 −0.272952 −0.136476 0.990643i \(-0.543578\pi\)
−0.136476 + 0.990643i \(0.543578\pi\)
\(662\) 0 0
\(663\) 3.82874i 0.148696i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.51040i 0.290804i
\(668\) 0 0
\(669\) 12.4242 0.480346
\(670\) 0 0
\(671\) 4.60174 0.177648
\(672\) 0 0
\(673\) − 6.81155i − 0.262566i −0.991345 0.131283i \(-0.958090\pi\)
0.991345 0.131283i \(-0.0419096\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 0.522812i − 0.0200933i −0.999950 0.0100466i \(-0.996802\pi\)
0.999950 0.0100466i \(-0.00319800\pi\)
\(678\) 0 0
\(679\) 1.20273 0.0461566
\(680\) 0 0
\(681\) 23.5108 0.900934
\(682\) 0 0
\(683\) 2.93412i 0.112271i 0.998423 + 0.0561355i \(0.0178779\pi\)
−0.998423 + 0.0561355i \(0.982122\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 54.2471i − 2.06965i
\(688\) 0 0
\(689\) −4.82929 −0.183981
\(690\) 0 0
\(691\) −31.8456 −1.21146 −0.605731 0.795669i \(-0.707119\pi\)
−0.605731 + 0.795669i \(0.707119\pi\)
\(692\) 0 0
\(693\) − 7.50240i − 0.284993i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 7.91596i − 0.299838i
\(698\) 0 0
\(699\) −68.2070 −2.57982
\(700\) 0 0
\(701\) 11.2070 0.423283 0.211642 0.977347i \(-0.432119\pi\)
0.211642 + 0.977347i \(0.432119\pi\)
\(702\) 0 0
\(703\) 3.56946i 0.134625i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.5418i 0.697336i
\(708\) 0 0
\(709\) −23.7300 −0.891198 −0.445599 0.895233i \(-0.647009\pi\)
−0.445599 + 0.895233i \(0.647009\pi\)
\(710\) 0 0
\(711\) 40.6033 1.52274
\(712\) 0 0
\(713\) 0.439038i 0.0164421i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7.69934i 0.287537i
\(718\) 0 0
\(719\) 27.8687 1.03933 0.519664 0.854371i \(-0.326057\pi\)
0.519664 + 0.854371i \(0.326057\pi\)
\(720\) 0 0
\(721\) 30.9968 1.15438
\(722\) 0 0
\(723\) − 65.2705i − 2.42744i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 47.5773i 1.76455i 0.470739 + 0.882273i \(0.343987\pi\)
−0.470739 + 0.882273i \(0.656013\pi\)
\(728\) 0 0
\(729\) 39.5022 1.46304
\(730\) 0 0
\(731\) 9.73274 0.359978
\(732\) 0 0
\(733\) 15.4086i 0.569129i 0.958657 + 0.284565i \(0.0918490\pi\)
−0.958657 + 0.284565i \(0.908151\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.75615i 0.0646888i
\(738\) 0 0
\(739\) −53.0212 −1.95042 −0.975208 0.221292i \(-0.928973\pi\)
−0.975208 + 0.221292i \(0.928973\pi\)
\(740\) 0 0
\(741\) −1.34413 −0.0493778
\(742\) 0 0
\(743\) 15.6611i 0.574551i 0.957848 + 0.287275i \(0.0927495\pi\)
−0.957848 + 0.287275i \(0.907250\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.39599i 0.197429i
\(748\) 0 0
\(749\) 42.8209 1.56464
\(750\) 0 0
\(751\) 34.6216 1.26336 0.631679 0.775230i \(-0.282366\pi\)
0.631679 + 0.775230i \(0.282366\pi\)
\(752\) 0 0
\(753\) 55.4554i 2.02091i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 38.4389i 1.39709i 0.715568 + 0.698543i \(0.246168\pi\)
−0.715568 + 0.698543i \(0.753832\pi\)
\(758\) 0 0
\(759\) −1.32943 −0.0482553
\(760\) 0 0
\(761\) 42.0932 1.52588 0.762939 0.646470i \(-0.223755\pi\)
0.762939 + 0.646470i \(0.223755\pi\)
\(762\) 0 0
\(763\) 67.5446i 2.44528i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 3.89605i − 0.140678i
\(768\) 0 0
\(769\) −34.2859 −1.23638 −0.618191 0.786028i \(-0.712134\pi\)
−0.618191 + 0.786028i \(0.712134\pi\)
\(770\) 0 0
\(771\) 68.5933 2.47033
\(772\) 0 0
\(773\) − 12.7422i − 0.458304i −0.973391 0.229152i \(-0.926405\pi\)
0.973391 0.229152i \(-0.0735953\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 70.5623i 2.53141i
\(778\) 0 0
\(779\) 2.77900 0.0995679
\(780\) 0 0
\(781\) 1.39015 0.0497436
\(782\) 0 0
\(783\) 16.6766i 0.595975i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 28.5534i − 1.01782i −0.860820 0.508910i \(-0.830049\pi\)
0.860820 0.508910i \(-0.169951\pi\)
\(788\) 0 0
\(789\) 14.7931 0.526647
\(790\) 0 0
\(791\) −15.5974 −0.554580
\(792\) 0 0
\(793\) − 9.15859i − 0.325231i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.39925i 0.297517i 0.988874 + 0.148758i \(0.0475276\pi\)
−0.988874 + 0.148758i \(0.952472\pi\)
\(798\) 0 0
\(799\) 3.82164 0.135200
\(800\) 0 0
\(801\) 26.0576 0.920701
\(802\) 0 0
\(803\) 4.68026i 0.165163i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 25.2739i − 0.889683i
\(808\) 0 0
\(809\) 28.8816 1.01542 0.507712 0.861527i \(-0.330491\pi\)
0.507712 + 0.861527i \(0.330491\pi\)
\(810\) 0 0
\(811\) −46.5292 −1.63386 −0.816931 0.576736i \(-0.804326\pi\)
−0.816931 + 0.576736i \(0.804326\pi\)
\(812\) 0 0
\(813\) 21.9574i 0.770078i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3.41680i 0.119539i
\(818\) 0 0
\(819\) −14.9316 −0.521753
\(820\) 0 0
\(821\) −2.99065 −0.104374 −0.0521872 0.998637i \(-0.516619\pi\)
−0.0521872 + 0.998637i \(0.516619\pi\)
\(822\) 0 0
\(823\) 5.14357i 0.179294i 0.995974 + 0.0896468i \(0.0285738\pi\)
−0.995974 + 0.0896468i \(0.971426\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 53.4288i 1.85790i 0.370205 + 0.928950i \(0.379287\pi\)
−0.370205 + 0.928950i \(0.620713\pi\)
\(828\) 0 0
\(829\) −11.5290 −0.400420 −0.200210 0.979753i \(-0.564162\pi\)
−0.200210 + 0.979753i \(0.564162\pi\)
\(830\) 0 0
\(831\) 79.3770 2.75356
\(832\) 0 0
\(833\) 11.1798i 0.387357i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.974874i 0.0336966i
\(838\) 0 0
\(839\) −12.0983 −0.417678 −0.208839 0.977950i \(-0.566968\pi\)
−0.208839 + 0.977950i \(0.566968\pi\)
\(840\) 0 0
\(841\) 27.4061 0.945038
\(842\) 0 0
\(843\) − 66.2436i − 2.28155i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 41.2215i 1.41639i
\(848\) 0 0
\(849\) 33.9235 1.16425
\(850\) 0 0
\(851\) 7.02642 0.240862
\(852\) 0 0
\(853\) 3.10103i 0.106177i 0.998590 + 0.0530886i \(0.0169066\pi\)
−0.998590 + 0.0530886i \(0.983093\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 4.39457i − 0.150116i −0.997179 0.0750578i \(-0.976086\pi\)
0.997179 0.0750578i \(-0.0239141\pi\)
\(858\) 0 0
\(859\) 29.0453 0.991014 0.495507 0.868604i \(-0.334982\pi\)
0.495507 + 0.868604i \(0.334982\pi\)
\(860\) 0 0
\(861\) 54.9362 1.87222
\(862\) 0 0
\(863\) − 40.1020i − 1.36509i −0.730845 0.682543i \(-0.760874\pi\)
0.730845 0.682543i \(-0.239126\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 39.0086i 1.32480i
\(868\) 0 0
\(869\) 5.35968 0.181815
\(870\) 0 0
\(871\) 3.49518 0.118430
\(872\) 0 0
\(873\) − 1.20620i − 0.0408235i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 37.7503i − 1.27474i −0.770559 0.637368i \(-0.780023\pi\)
0.770559 0.637368i \(-0.219977\pi\)
\(878\) 0 0
\(879\) −67.9021 −2.29028
\(880\) 0 0
\(881\) −29.1541 −0.982227 −0.491113 0.871096i \(-0.663410\pi\)
−0.491113 + 0.871096i \(0.663410\pi\)
\(882\) 0 0
\(883\) − 43.3971i − 1.46043i −0.683218 0.730214i \(-0.739420\pi\)
0.683218 0.730214i \(-0.260580\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 25.4091i 0.853155i 0.904451 + 0.426578i \(0.140281\pi\)
−0.904451 + 0.426578i \(0.859719\pi\)
\(888\) 0 0
\(889\) −18.2373 −0.611658
\(890\) 0 0
\(891\) 2.91320 0.0975958
\(892\) 0 0
\(893\) 1.34163i 0.0448961i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.64590i 0.0883439i
\(898\) 0 0
\(899\) 3.29735 0.109973
\(900\) 0 0
\(901\) 6.91181 0.230266
\(902\) 0 0
\(903\) 67.5446i 2.24774i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 2.00778i − 0.0666672i −0.999444 0.0333336i \(-0.989388\pi\)
0.999444 0.0333336i \(-0.0106124\pi\)
\(908\) 0 0
\(909\) 18.5952 0.616765
\(910\) 0 0
\(911\) −8.45792 −0.280223 −0.140112 0.990136i \(-0.544746\pi\)
−0.140112 + 0.990136i \(0.544746\pi\)
\(912\) 0 0
\(913\) 0.712277i 0.0235729i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 7.55402i − 0.249456i
\(918\) 0 0
\(919\) −0.840028 −0.0277100 −0.0138550 0.999904i \(-0.504410\pi\)
−0.0138550 + 0.999904i \(0.504410\pi\)
\(920\) 0 0
\(921\) 31.5711 1.04030
\(922\) 0 0
\(923\) − 2.76675i − 0.0910686i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 31.0861i − 1.02100i
\(928\) 0 0
\(929\) −15.7877 −0.517979 −0.258989 0.965880i \(-0.583389\pi\)
−0.258989 + 0.965880i \(0.583389\pi\)
\(930\) 0 0
\(931\) −3.92481 −0.128630
\(932\) 0 0
\(933\) − 83.2520i − 2.72555i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 31.6568i − 1.03418i −0.855930 0.517091i \(-0.827015\pi\)
0.855930 0.517091i \(-0.172985\pi\)
\(938\) 0 0
\(939\) 11.2561 0.367330
\(940\) 0 0
\(941\) −38.3932 −1.25158 −0.625792 0.779990i \(-0.715224\pi\)
−0.625792 + 0.779990i \(0.715224\pi\)
\(942\) 0 0
\(943\) − 5.47041i − 0.178141i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 34.3783i 1.11714i 0.829456 + 0.558572i \(0.188651\pi\)
−0.829456 + 0.558572i \(0.811349\pi\)
\(948\) 0 0
\(949\) 9.31486 0.302373
\(950\) 0 0
\(951\) −16.4428 −0.533195
\(952\) 0 0
\(953\) 48.3782i 1.56712i 0.621314 + 0.783562i \(0.286599\pi\)
−0.621314 + 0.783562i \(0.713401\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 9.98456i 0.322755i
\(958\) 0 0
\(959\) 21.6541 0.699247
\(960\) 0 0
\(961\) −30.8072 −0.993782
\(962\) 0 0
\(963\) − 42.9443i − 1.38386i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 51.3705i 1.65196i 0.563697 + 0.825982i \(0.309379\pi\)
−0.563697 + 0.825982i \(0.690621\pi\)
\(968\) 0 0
\(969\) 1.92375 0.0617999
\(970\) 0 0
\(971\) 50.4587 1.61930 0.809649 0.586914i \(-0.199657\pi\)
0.809649 + 0.586914i \(0.199657\pi\)
\(972\) 0 0
\(973\) − 29.1539i − 0.934633i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 32.1406i − 1.02827i −0.857710 0.514134i \(-0.828113\pi\)
0.857710 0.514134i \(-0.171887\pi\)
\(978\) 0 0
\(979\) 3.43964 0.109931
\(980\) 0 0
\(981\) 67.7392 2.16275
\(982\) 0 0
\(983\) 24.1306i 0.769647i 0.922990 + 0.384824i \(0.125738\pi\)
−0.922990 + 0.384824i \(0.874262\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 26.5219i 0.844201i
\(988\) 0 0
\(989\) 6.72592 0.213872
\(990\) 0 0
\(991\) 40.1267 1.27467 0.637333 0.770588i \(-0.280038\pi\)
0.637333 + 0.770588i \(0.280038\pi\)
\(992\) 0 0
\(993\) − 20.9371i − 0.664418i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 48.9890i 1.55150i 0.631042 + 0.775749i \(0.282628\pi\)
−0.631042 + 0.775749i \(0.717372\pi\)
\(998\) 0 0
\(999\) 15.6020 0.493625
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.w.4049.10 10
5.2 odd 4 4600.2.a.bf.1.5 yes 5
5.3 odd 4 4600.2.a.bd.1.1 5
5.4 even 2 inner 4600.2.e.w.4049.1 10
20.3 even 4 9200.2.a.cv.1.5 5
20.7 even 4 9200.2.a.ct.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.1 5 5.3 odd 4
4600.2.a.bf.1.5 yes 5 5.2 odd 4
4600.2.e.w.4049.1 10 5.4 even 2 inner
4600.2.e.w.4049.10 10 1.1 even 1 trivial
9200.2.a.ct.1.1 5 20.7 even 4
9200.2.a.cv.1.5 5 20.3 even 4