Properties

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

$q$-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 \(-0.727417\pi\)
0.655204 0.755452i \(-0.272583\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.83744i 1.45041i 0.688531 + 0.725207i \(0.258256\pi\)
−0.688531 + 0.725207i \(0.741744\pi\)
\(8\) 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 \(-0.955223\pi\)
0.990122 0.140208i \(-0.0447772\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.44705i 0.350961i 0.984483 + 0.175481i \(0.0561479\pi\)
−0.984483 + 0.175481i \(0.943852\pi\)
\(18\) 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 \(-0.804002\pi\)
0.816343 0.577568i \(-0.195998\pi\)
\(38\) 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 \(-0.828591\pi\)
0.858480 0.512847i \(-0.171409\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.64098i − 0.385227i −0.981275 0.192614i \(-0.938304\pi\)
0.981275 0.192614i \(-0.0616964\pi\)
\(48\) 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 \(-0.893609\pi\)
0.944660 0.328050i \(-0.106391\pi\)
\(54\) 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 \(0.0677265\pi\)
−0.977450 + 0.211167i \(0.932273\pi\)
\(68\) 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 \(0.181254\pi\)
−0.842210 + 0.539150i \(0.818746\pi\)
\(74\) 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 \(0.0245183\pi\)
−0.997035 + 0.0769505i \(0.975482\pi\)
\(84\) 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 \(-0.994935\pi\)
0.999873 0.0159115i \(-0.00506500\pi\)
\(98\) 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 \(-0.869722\pi\)
0.917408 0.397948i \(-0.130278\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 11.1587i − 1.07876i −0.842064 0.539378i \(-0.818659\pi\)
0.842064 0.539378i \(-0.181341\pi\)
\(108\) 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 \(0.0612314\pi\)
−0.981555 + 0.191180i \(0.938769\pi\)
\(114\) 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 \(0.0676252\pi\)
−0.977517 + 0.210856i \(0.932375\pi\)
\(128\) 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 \(-0.922508\pi\)
0.970513 0.241051i \(-0.0774920\pi\)
\(138\) 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 \(-0.847620\pi\)
0.887587 0.460640i \(-0.152380\pi\)
\(158\) 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 \(-0.958477\pi\)
0.991504 0.130077i \(-0.0415226\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.6742i 0.980755i 0.871510 + 0.490378i \(0.163141\pi\)
−0.871510 + 0.490378i \(0.836859\pi\)
\(168\) 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 \(-0.767136\pi\)
0.744130 0.668035i \(-0.232864\pi\)
\(174\) 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 \(0.0898979\pi\)
−0.960383 + 0.278683i \(0.910102\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 23.8908i − 1.70215i −0.525045 0.851074i \(-0.675952\pi\)
0.525045 0.851074i \(-0.324048\pi\)
\(198\) 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 \(0.0508140\pi\)
−0.987285 + 0.158960i \(0.949186\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.98399i 0.596288i 0.954521 + 0.298144i \(0.0963676\pi\)
−0.954521 + 0.298144i \(0.903632\pi\)
\(228\) 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 \(-0.674332\pi\)
0.520707 0.853735i \(-0.325668\pi\)
\(234\) 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 \(0.304640\pi\)
−0.575929 + 0.817500i \(0.695360\pi\)
\(258\) 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 \(0.0557605\pi\)
−0.984696 + 0.174282i \(0.944240\pi\)
\(264\) 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 \(0.364865\pi\)
−0.411900 + 0.911229i \(0.635135\pi\)
\(278\) 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 \(0.125896\pi\)
−0.922798 + 0.385283i \(0.874104\pi\)
\(284\) 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 \(-0.726217\pi\)
0.652350 0.757918i \(-0.273783\pi\)
\(294\) 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 \(0.111872\pi\)
−0.938872 + 0.344266i \(0.888128\pi\)
\(308\) 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 \(0.0387895\pi\)
−0.992584 + 0.121560i \(0.961210\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 6.28317i − 0.352898i −0.984310 0.176449i \(-0.943539\pi\)
0.984310 0.176449i \(-0.0564611\pi\)
\(318\) 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 \(0.0891008\pi\)
−0.961078 + 0.276277i \(0.910899\pi\)
\(338\) 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 \(-0.830893\pi\)
0.862167 0.506625i \(-0.169107\pi\)
\(348\) 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 \(-0.931324\pi\)
0.976816 0.214081i \(-0.0686755\pi\)
\(354\) 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 \(-0.927595\pi\)
0.974241 0.225511i \(-0.0724050\pi\)
\(368\) 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 \(0.164113\pi\)
−0.870009 + 0.493037i \(0.835887\pi\)
\(374\) 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 \(0.0770760\pi\)
−0.970827 + 0.239782i \(0.922924\pi\)
\(384\) 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 \(0.106156\pi\)
−0.944903 + 0.327350i \(0.893844\pi\)
\(398\) 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 \(-0.975075\pi\)
0.996936 0.0782257i \(-0.0249255\pi\)
\(434\) 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 \(0.124167\pi\)
−0.924878 + 0.380263i \(0.875833\pi\)
\(444\) 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 \(-0.934321\pi\)
0.978788 0.204876i \(-0.0656790\pi\)
\(458\) 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 \(-0.938706\pi\)
0.981517 0.191373i \(-0.0612939\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 28.9436i − 1.33935i −0.742654 0.669675i \(-0.766433\pi\)
0.742654 0.669675i \(-0.233567\pi\)
\(468\) 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 \(0.0137494\pi\)
−0.999067 + 0.0431817i \(0.986251\pi\)
\(488\) 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 \(-0.770964\pi\)
0.752110 0.659037i \(-0.229036\pi\)
\(504\) 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 \(-0.979771\pi\)
0.997981 0.0635075i \(-0.0202287\pi\)
\(524\) 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 \(-0.829798\pi\)
0.860420 0.509586i \(-0.170202\pi\)
\(548\) 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 \(0.327259\pi\)
−0.516435 + 0.856326i \(0.672741\pi\)
\(558\) 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 \(0.0440590\pi\)
−0.990436 + 0.137974i \(0.955941\pi\)
\(564\) 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 \(-0.818498\pi\)
0.841790 0.539805i \(-0.181502\pi\)
\(578\) 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 \(0.0693320\pi\)
−0.976372 + 0.216095i \(0.930668\pi\)
\(588\) 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 \(-0.859429\pi\)
0.904062 0.427401i \(-0.140571\pi\)
\(594\) 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 \(0.233018\pi\)
−0.743807 + 0.668394i \(0.766982\pi\)
\(608\) 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 \(0.167465\pi\)
−0.864768 + 0.502172i \(0.832535\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 20.1280i − 0.810324i −0.914245 0.405162i \(-0.867215\pi\)
0.914245 0.405162i \(-0.132785\pi\)
\(618\) 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 \(0.107318\pi\)
−0.943702 + 0.330798i \(0.892682\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 1.99247i − 0.0783322i −0.999233 0.0391661i \(-0.987530\pi\)
0.999233 0.0391661i \(-0.0124702\pi\)
\(648\) 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 \(-0.636218\pi\)
0.414999 0.909822i \(-0.363782\pi\)
\(654\) 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 \(0.0419096\pi\)
−0.991345 + 0.131283i \(0.958090\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0.522812i 0.0200933i 0.999950 + 0.0100466i \(0.00319800\pi\)
−0.999950 + 0.0100466i \(0.996802\pi\)
\(678\) 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 \(-0.982122\pi\)
0.998423 0.0561355i \(-0.0178779\pi\)
\(684\) 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 \(-0.656013\pi\)
0.470739 0.882273i \(-0.343987\pi\)
\(728\) 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 \(-0.908151\pi\)
0.958657 0.284565i \(-0.0918490\pi\)
\(734\) 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 \(-0.907250\pi\)
0.957848 0.287275i \(-0.0927495\pi\)
\(744\) 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 \(-0.753832\pi\)
0.715568 0.698543i \(-0.246168\pi\)
\(758\) 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 \(0.0735953\pi\)
−0.973391 + 0.229152i \(0.926405\pi\)
\(774\) 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 \(0.169951\pi\)
−0.860820 + 0.508910i \(0.830049\pi\)
\(788\) 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 \(-0.952472\pi\)
0.988874 0.148758i \(-0.0475276\pi\)
\(798\) 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 \(-0.971426\pi\)
0.995974 0.0896468i \(-0.0285738\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 53.4288i − 1.85790i −0.370205 0.928950i \(-0.620713\pi\)
0.370205 0.928950i \(-0.379287\pi\)
\(828\) 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 \(-0.983093\pi\)
0.998590 0.0530886i \(-0.0169066\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.39457i 0.150116i 0.997179 + 0.0750578i \(0.0239141\pi\)
−0.997179 + 0.0750578i \(0.976086\pi\)
\(858\) 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 \(0.239126\pi\)
−0.730845 + 0.682543i \(0.760874\pi\)
\(864\) 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 \(0.219977\pi\)
−0.770559 + 0.637368i \(0.780023\pi\)
\(878\) 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 \(0.260580\pi\)
−0.683218 + 0.730214i \(0.739420\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 25.4091i − 0.853155i −0.904451 0.426578i \(-0.859719\pi\)
0.904451 0.426578i \(-0.140281\pi\)
\(888\) 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 \(0.0106124\pi\)
−0.999444 + 0.0333336i \(0.989388\pi\)
\(908\) 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 \(0.172985\pi\)
−0.855930 + 0.517091i \(0.827015\pi\)
\(938\) 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 \(-0.811349\pi\)
0.829456 0.558572i \(-0.188651\pi\)
\(948\) 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 \(-0.713401\pi\)
0.621314 0.783562i \(-0.286599\pi\)
\(954\) 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 \(-0.690621\pi\)
0.563697 0.825982i \(-0.309379\pi\)
\(968\) 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 \(0.171887\pi\)
−0.857710 + 0.514134i \(0.828113\pi\)
\(978\) 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 \(-0.874262\pi\)
0.922990 0.384824i \(-0.125738\pi\)
\(984\) 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 \(-0.717372\pi\)
0.631042 0.775749i \(-0.282628\pi\)
\(998\) 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.4049.1 10
5.2 odd 4 4600.2.a.bd.1.1 5
5.3 odd 4 4600.2.a.bf.1.5 yes 5
5.4 even 2 inner 4600.2.e.w.4049.10 10
20.3 even 4 9200.2.a.ct.1.1 5
20.7 even 4 9200.2.a.cv.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.1 5 5.2 odd 4
4600.2.a.bf.1.5 yes 5 5.3 odd 4
4600.2.e.w.4049.1 10 1.1 even 1 trivial
4600.2.e.w.4049.10 10 5.4 even 2 inner
9200.2.a.ct.1.1 5 20.3 even 4
9200.2.a.cv.1.5 5 20.7 even 4