Properties

Label 4600.2.e.u.4049.4
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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 4049.4
Root \(-1.31091i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.u.4049.7

$q$-expansion

\(f(q)\) \(=\) \(q-1.31091i q^{3} +4.66212i q^{7} +1.28151 q^{9} +O(q^{10})\) \(q-1.31091i q^{3} +4.66212i q^{7} +1.28151 q^{9} +2.23020 q^{11} -2.80072i q^{13} -7.63271i q^{17} +1.36222 q^{19} +6.11163 q^{21} -1.00000i q^{23} -5.61268i q^{27} -8.94362 q^{29} -1.58140 q^{31} -2.92360i q^{33} +1.40251i q^{37} -3.67149 q^{39} +10.7134 q^{41} +7.26391i q^{47} -14.7353 q^{49} -10.0058 q^{51} -8.38212i q^{53} -1.78575i q^{57} +4.88331 q^{59} +4.33282 q^{61} +5.97454i q^{63} -8.54355i q^{67} -1.31091 q^{69} +8.81604 q^{71} -5.26391i q^{73} +10.3974i q^{77} -7.08222 q^{79} -3.51322 q^{81} -4.59749i q^{83} +11.7243i q^{87} +4.70241 q^{89} +13.0573 q^{91} +2.07308i q^{93} -16.5160i q^{97} +2.85802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 26 q^{9} + O(q^{10}) \) \( 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99} + O(q^{100}) \)

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
\(3\) − 1.31091i − 0.756856i −0.925631 0.378428i \(-0.876465\pi\)
0.925631 0.378428i \(-0.123535\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.66212i 1.76211i 0.473010 + 0.881057i \(0.343168\pi\)
−0.473010 + 0.881057i \(0.656832\pi\)
\(8\) 0 0
\(9\) 1.28151 0.427169
\(10\) 0 0
\(11\) 2.23020 0.672430 0.336215 0.941785i \(-0.390853\pi\)
0.336215 + 0.941785i \(0.390853\pi\)
\(12\) 0 0
\(13\) − 2.80072i − 0.776779i −0.921495 0.388389i \(-0.873032\pi\)
0.921495 0.388389i \(-0.126968\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7.63271i − 1.85120i −0.378498 0.925602i \(-0.623559\pi\)
0.378498 0.925602i \(-0.376441\pi\)
\(18\) 0 0
\(19\) 1.36222 0.312515 0.156258 0.987716i \(-0.450057\pi\)
0.156258 + 0.987716i \(0.450057\pi\)
\(20\) 0 0
\(21\) 6.11163 1.33367
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.61268i − 1.08016i
\(28\) 0 0
\(29\) −8.94362 −1.66079 −0.830395 0.557176i \(-0.811885\pi\)
−0.830395 + 0.557176i \(0.811885\pi\)
\(30\) 0 0
\(31\) −1.58140 −0.284028 −0.142014 0.989865i \(-0.545358\pi\)
−0.142014 + 0.989865i \(0.545358\pi\)
\(32\) 0 0
\(33\) − 2.92360i − 0.508933i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.40251i 0.230572i 0.993332 + 0.115286i \(0.0367784\pi\)
−0.993332 + 0.115286i \(0.963222\pi\)
\(38\) 0 0
\(39\) −3.67149 −0.587909
\(40\) 0 0
\(41\) 10.7134 1.67316 0.836578 0.547848i \(-0.184553\pi\)
0.836578 + 0.547848i \(0.184553\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.26391i 1.05955i 0.848138 + 0.529775i \(0.177724\pi\)
−0.848138 + 0.529775i \(0.822276\pi\)
\(48\) 0 0
\(49\) −14.7353 −2.10505
\(50\) 0 0
\(51\) −10.0058 −1.40109
\(52\) 0 0
\(53\) − 8.38212i − 1.15137i −0.817671 0.575686i \(-0.804735\pi\)
0.817671 0.575686i \(-0.195265\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.78575i − 0.236529i
\(58\) 0 0
\(59\) 4.88331 0.635752 0.317876 0.948132i \(-0.397030\pi\)
0.317876 + 0.948132i \(0.397030\pi\)
\(60\) 0 0
\(61\) 4.33282 0.554760 0.277380 0.960760i \(-0.410534\pi\)
0.277380 + 0.960760i \(0.410534\pi\)
\(62\) 0 0
\(63\) 5.97454i 0.752721i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 8.54355i − 1.04376i −0.853019 0.521880i \(-0.825231\pi\)
0.853019 0.521880i \(-0.174769\pi\)
\(68\) 0 0
\(69\) −1.31091 −0.157815
\(70\) 0 0
\(71\) 8.81604 1.04627 0.523136 0.852249i \(-0.324762\pi\)
0.523136 + 0.852249i \(0.324762\pi\)
\(72\) 0 0
\(73\) − 5.26391i − 0.616095i −0.951371 0.308047i \(-0.900325\pi\)
0.951371 0.308047i \(-0.0996755\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3974i 1.18490i
\(78\) 0 0
\(79\) −7.08222 −0.796812 −0.398406 0.917209i \(-0.630437\pi\)
−0.398406 + 0.917209i \(0.630437\pi\)
\(80\) 0 0
\(81\) −3.51322 −0.390357
\(82\) 0 0
\(83\) − 4.59749i − 0.504640i −0.967644 0.252320i \(-0.918806\pi\)
0.967644 0.252320i \(-0.0811935\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 11.7243i 1.25698i
\(88\) 0 0
\(89\) 4.70241 0.498454 0.249227 0.968445i \(-0.419823\pi\)
0.249227 + 0.968445i \(0.419823\pi\)
\(90\) 0 0
\(91\) 13.0573 1.36877
\(92\) 0 0
\(93\) 2.07308i 0.214968i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 16.5160i − 1.67695i −0.544942 0.838474i \(-0.683448\pi\)
0.544942 0.838474i \(-0.316552\pi\)
\(98\) 0 0
\(99\) 2.85802 0.287241
\(100\) 0 0
\(101\) 10.7267 1.06735 0.533676 0.845689i \(-0.320810\pi\)
0.533676 + 0.845689i \(0.320810\pi\)
\(102\) 0 0
\(103\) 1.95300i 0.192435i 0.995360 + 0.0962175i \(0.0306744\pi\)
−0.995360 + 0.0962175i \(0.969326\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.97566i 0.190994i 0.995430 + 0.0954972i \(0.0304441\pi\)
−0.995430 + 0.0954972i \(0.969556\pi\)
\(108\) 0 0
\(109\) 8.92360 0.854725 0.427363 0.904080i \(-0.359443\pi\)
0.427363 + 0.904080i \(0.359443\pi\)
\(110\) 0 0
\(111\) 1.83857 0.174510
\(112\) 0 0
\(113\) 17.3486i 1.63202i 0.578040 + 0.816008i \(0.303818\pi\)
−0.578040 + 0.816008i \(0.696182\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 3.58914i − 0.331816i
\(118\) 0 0
\(119\) 35.5846 3.26203
\(120\) 0 0
\(121\) −6.02621 −0.547838
\(122\) 0 0
\(123\) − 14.0444i − 1.26634i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 19.4872i − 1.72921i −0.502455 0.864603i \(-0.667570\pi\)
0.502455 0.864603i \(-0.332430\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.68050 0.496308 0.248154 0.968721i \(-0.420176\pi\)
0.248154 + 0.968721i \(0.420176\pi\)
\(132\) 0 0
\(133\) 6.35084i 0.550687i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.68645i 0.742134i 0.928606 + 0.371067i \(0.121008\pi\)
−0.928606 + 0.371067i \(0.878992\pi\)
\(138\) 0 0
\(139\) 9.22569 0.782513 0.391256 0.920282i \(-0.372041\pi\)
0.391256 + 0.920282i \(0.372041\pi\)
\(140\) 0 0
\(141\) 9.52236 0.801927
\(142\) 0 0
\(143\) − 6.24615i − 0.522329i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 19.3167i 1.59322i
\(148\) 0 0
\(149\) −18.6056 −1.52423 −0.762115 0.647441i \(-0.775839\pi\)
−0.762115 + 0.647441i \(0.775839\pi\)
\(150\) 0 0
\(151\) 17.4316 1.41856 0.709280 0.704927i \(-0.249020\pi\)
0.709280 + 0.704927i \(0.249020\pi\)
\(152\) 0 0
\(153\) − 9.78138i − 0.790778i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 0.839497i − 0.0669992i −0.999439 0.0334996i \(-0.989335\pi\)
0.999439 0.0334996i \(-0.0106652\pi\)
\(158\) 0 0
\(159\) −10.9882 −0.871423
\(160\) 0 0
\(161\) 4.66212 0.367426
\(162\) 0 0
\(163\) − 14.5673i − 1.14100i −0.821297 0.570500i \(-0.806749\pi\)
0.821297 0.570500i \(-0.193251\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.03842i 0.389884i 0.980815 + 0.194942i \(0.0624519\pi\)
−0.980815 + 0.194942i \(0.937548\pi\)
\(168\) 0 0
\(169\) 5.15599 0.396615
\(170\) 0 0
\(171\) 1.74570 0.133497
\(172\) 0 0
\(173\) − 11.3124i − 0.860067i −0.902813 0.430034i \(-0.858502\pi\)
0.902813 0.430034i \(-0.141498\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 6.40159i − 0.481173i
\(178\) 0 0
\(179\) 24.3053 1.81667 0.908333 0.418247i \(-0.137355\pi\)
0.908333 + 0.418247i \(0.137355\pi\)
\(180\) 0 0
\(181\) −19.4829 −1.44815 −0.724075 0.689721i \(-0.757733\pi\)
−0.724075 + 0.689721i \(0.757733\pi\)
\(182\) 0 0
\(183\) − 5.67994i − 0.419874i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 17.0225i − 1.24481i
\(188\) 0 0
\(189\) 26.1670 1.90337
\(190\) 0 0
\(191\) 2.62183 0.189709 0.0948543 0.995491i \(-0.469761\pi\)
0.0948543 + 0.995491i \(0.469761\pi\)
\(192\) 0 0
\(193\) 17.4332i 1.25487i 0.778668 + 0.627436i \(0.215895\pi\)
−0.778668 + 0.627436i \(0.784105\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.4402i 1.67004i 0.550217 + 0.835022i \(0.314545\pi\)
−0.550217 + 0.835022i \(0.685455\pi\)
\(198\) 0 0
\(199\) 1.29759 0.0919839 0.0459920 0.998942i \(-0.485355\pi\)
0.0459920 + 0.998942i \(0.485355\pi\)
\(200\) 0 0
\(201\) −11.1998 −0.789976
\(202\) 0 0
\(203\) − 41.6962i − 2.92650i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.28151i − 0.0890709i
\(208\) 0 0
\(209\) 3.03803 0.210145
\(210\) 0 0
\(211\) 14.7619 1.01625 0.508127 0.861282i \(-0.330338\pi\)
0.508127 + 0.861282i \(0.330338\pi\)
\(212\) 0 0
\(213\) − 11.5571i − 0.791877i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 7.37268i − 0.500490i
\(218\) 0 0
\(219\) −6.90053 −0.466295
\(220\) 0 0
\(221\) −21.3771 −1.43798
\(222\) 0 0
\(223\) 11.8434i 0.793095i 0.918014 + 0.396548i \(0.129792\pi\)
−0.918014 + 0.396548i \(0.870208\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.27556i 0.350151i 0.984555 + 0.175075i \(0.0560170\pi\)
−0.984555 + 0.175075i \(0.943983\pi\)
\(228\) 0 0
\(229\) 1.23878 0.0818611 0.0409305 0.999162i \(-0.486968\pi\)
0.0409305 + 0.999162i \(0.486968\pi\)
\(230\) 0 0
\(231\) 13.6301 0.896798
\(232\) 0 0
\(233\) 2.66412i 0.174533i 0.996185 + 0.0872663i \(0.0278131\pi\)
−0.996185 + 0.0872663i \(0.972187\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 9.28418i 0.603072i
\(238\) 0 0
\(239\) −26.2577 −1.69847 −0.849235 0.528014i \(-0.822937\pi\)
−0.849235 + 0.528014i \(0.822937\pi\)
\(240\) 0 0
\(241\) −2.22326 −0.143213 −0.0716063 0.997433i \(-0.522813\pi\)
−0.0716063 + 0.997433i \(0.522813\pi\)
\(242\) 0 0
\(243\) − 12.2325i − 0.784717i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 3.81520i − 0.242755i
\(248\) 0 0
\(249\) −6.02690 −0.381940
\(250\) 0 0
\(251\) 13.4829 0.851031 0.425515 0.904951i \(-0.360093\pi\)
0.425515 + 0.904951i \(0.360093\pi\)
\(252\) 0 0
\(253\) − 2.23020i − 0.140211i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 22.0281i − 1.37408i −0.726620 0.687039i \(-0.758910\pi\)
0.726620 0.687039i \(-0.241090\pi\)
\(258\) 0 0
\(259\) −6.53868 −0.406294
\(260\) 0 0
\(261\) −11.4613 −0.709438
\(262\) 0 0
\(263\) 22.3164i 1.37609i 0.725670 + 0.688043i \(0.241530\pi\)
−0.725670 + 0.688043i \(0.758470\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 6.16445i − 0.377258i
\(268\) 0 0
\(269\) −10.5050 −0.640501 −0.320251 0.947333i \(-0.603767\pi\)
−0.320251 + 0.947333i \(0.603767\pi\)
\(270\) 0 0
\(271\) −3.84696 −0.233686 −0.116843 0.993150i \(-0.537277\pi\)
−0.116843 + 0.993150i \(0.537277\pi\)
\(272\) 0 0
\(273\) − 17.1169i − 1.03596i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 14.8653i − 0.893172i −0.894741 0.446586i \(-0.852640\pi\)
0.894741 0.446586i \(-0.147360\pi\)
\(278\) 0 0
\(279\) −2.02658 −0.121328
\(280\) 0 0
\(281\) 4.41659 0.263472 0.131736 0.991285i \(-0.457945\pi\)
0.131736 + 0.991285i \(0.457945\pi\)
\(282\) 0 0
\(283\) 3.88495i 0.230936i 0.993311 + 0.115468i \(0.0368368\pi\)
−0.993311 + 0.115468i \(0.963163\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 49.9472i 2.94829i
\(288\) 0 0
\(289\) −41.2583 −2.42696
\(290\) 0 0
\(291\) −21.6511 −1.26921
\(292\) 0 0
\(293\) − 25.0257i − 1.46202i −0.682368 0.731009i \(-0.739050\pi\)
0.682368 0.731009i \(-0.260950\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 12.5174i − 0.726333i
\(298\) 0 0
\(299\) −2.80072 −0.161970
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 14.0618i − 0.807831i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 22.0197i − 1.25673i −0.777918 0.628365i \(-0.783724\pi\)
0.777918 0.628365i \(-0.216276\pi\)
\(308\) 0 0
\(309\) 2.56021 0.145645
\(310\) 0 0
\(311\) 19.6217 1.11264 0.556322 0.830967i \(-0.312212\pi\)
0.556322 + 0.830967i \(0.312212\pi\)
\(312\) 0 0
\(313\) − 17.9496i − 1.01457i −0.861778 0.507285i \(-0.830649\pi\)
0.861778 0.507285i \(-0.169351\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 34.3185i − 1.92752i −0.266768 0.963761i \(-0.585956\pi\)
0.266768 0.963761i \(-0.414044\pi\)
\(318\) 0 0
\(319\) −19.9461 −1.11676
\(320\) 0 0
\(321\) 2.58992 0.144555
\(322\) 0 0
\(323\) − 10.3974i − 0.578529i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 11.6981i − 0.646904i
\(328\) 0 0
\(329\) −33.8652 −1.86705
\(330\) 0 0
\(331\) −0.299762 −0.0164764 −0.00823821 0.999966i \(-0.502622\pi\)
−0.00823821 + 0.999966i \(0.502622\pi\)
\(332\) 0 0
\(333\) 1.79733i 0.0984931i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.2965i 1.21457i 0.794485 + 0.607284i \(0.207741\pi\)
−0.794485 + 0.607284i \(0.792259\pi\)
\(338\) 0 0
\(339\) 22.7425 1.23520
\(340\) 0 0
\(341\) −3.52684 −0.190989
\(342\) 0 0
\(343\) − 36.0630i − 1.94722i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 33.1240i − 1.77819i −0.457725 0.889094i \(-0.651336\pi\)
0.457725 0.889094i \(-0.348664\pi\)
\(348\) 0 0
\(349\) 22.0041 1.17785 0.588926 0.808187i \(-0.299551\pi\)
0.588926 + 0.808187i \(0.299551\pi\)
\(350\) 0 0
\(351\) −15.7195 −0.839046
\(352\) 0 0
\(353\) − 15.6085i − 0.830759i −0.909648 0.415379i \(-0.863649\pi\)
0.909648 0.415379i \(-0.136351\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 46.6483i − 2.46889i
\(358\) 0 0
\(359\) −0.263781 −0.0139218 −0.00696092 0.999976i \(-0.502216\pi\)
−0.00696092 + 0.999976i \(0.502216\pi\)
\(360\) 0 0
\(361\) −17.1444 −0.902334
\(362\) 0 0
\(363\) 7.89984i 0.414634i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 10.8615i − 0.566967i −0.958977 0.283484i \(-0.908510\pi\)
0.958977 0.283484i \(-0.0914902\pi\)
\(368\) 0 0
\(369\) 13.7293 0.714721
\(370\) 0 0
\(371\) 39.0784 2.02885
\(372\) 0 0
\(373\) 26.8889i 1.39225i 0.717919 + 0.696127i \(0.245095\pi\)
−0.717919 + 0.696127i \(0.754905\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 25.0485i 1.29007i
\(378\) 0 0
\(379\) 2.15824 0.110861 0.0554306 0.998463i \(-0.482347\pi\)
0.0554306 + 0.998463i \(0.482347\pi\)
\(380\) 0 0
\(381\) −25.5460 −1.30876
\(382\) 0 0
\(383\) − 4.62814i − 0.236487i −0.992985 0.118244i \(-0.962274\pi\)
0.992985 0.118244i \(-0.0377264\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 18.4988 0.937929 0.468964 0.883217i \(-0.344627\pi\)
0.468964 + 0.883217i \(0.344627\pi\)
\(390\) 0 0
\(391\) −7.63271 −0.386003
\(392\) 0 0
\(393\) − 7.44664i − 0.375634i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 10.2685i 0.515360i 0.966230 + 0.257680i \(0.0829581\pi\)
−0.966230 + 0.257680i \(0.917042\pi\)
\(398\) 0 0
\(399\) 8.32539 0.416791
\(400\) 0 0
\(401\) −0.885607 −0.0442251 −0.0221125 0.999755i \(-0.507039\pi\)
−0.0221125 + 0.999755i \(0.507039\pi\)
\(402\) 0 0
\(403\) 4.42906i 0.220627i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.12788i 0.155043i
\(408\) 0 0
\(409\) −24.3497 −1.20401 −0.602006 0.798491i \(-0.705632\pi\)
−0.602006 + 0.798491i \(0.705632\pi\)
\(410\) 0 0
\(411\) 11.3872 0.561688
\(412\) 0 0
\(413\) 22.7665i 1.12027i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 12.0941i − 0.592249i
\(418\) 0 0
\(419\) −25.4535 −1.24348 −0.621742 0.783222i \(-0.713575\pi\)
−0.621742 + 0.783222i \(0.713575\pi\)
\(420\) 0 0
\(421\) 15.2376 0.742634 0.371317 0.928506i \(-0.378906\pi\)
0.371317 + 0.928506i \(0.378906\pi\)
\(422\) 0 0
\(423\) 9.30876i 0.452607i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.2001i 0.977551i
\(428\) 0 0
\(429\) −8.18816 −0.395328
\(430\) 0 0
\(431\) −34.6166 −1.66742 −0.833711 0.552202i \(-0.813788\pi\)
−0.833711 + 0.552202i \(0.813788\pi\)
\(432\) 0 0
\(433\) 28.4774i 1.36854i 0.729230 + 0.684268i \(0.239878\pi\)
−0.729230 + 0.684268i \(0.760122\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 1.36222i − 0.0651639i
\(438\) 0 0
\(439\) −21.6009 −1.03095 −0.515477 0.856904i \(-0.672385\pi\)
−0.515477 + 0.856904i \(0.672385\pi\)
\(440\) 0 0
\(441\) −18.8834 −0.899211
\(442\) 0 0
\(443\) 40.3700i 1.91804i 0.283346 + 0.959018i \(0.408555\pi\)
−0.283346 + 0.959018i \(0.591445\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 24.3903i 1.15362i
\(448\) 0 0
\(449\) −5.95484 −0.281026 −0.140513 0.990079i \(-0.544875\pi\)
−0.140513 + 0.990079i \(0.544875\pi\)
\(450\) 0 0
\(451\) 23.8931 1.12508
\(452\) 0 0
\(453\) − 22.8512i − 1.07365i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.66169i 0.264843i 0.991194 + 0.132421i \(0.0422752\pi\)
−0.991194 + 0.132421i \(0.957725\pi\)
\(458\) 0 0
\(459\) −42.8400 −1.99960
\(460\) 0 0
\(461\) −9.26391 −0.431463 −0.215732 0.976453i \(-0.569214\pi\)
−0.215732 + 0.976453i \(0.569214\pi\)
\(462\) 0 0
\(463\) 37.9899i 1.76554i 0.469807 + 0.882769i \(0.344324\pi\)
−0.469807 + 0.882769i \(0.655676\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.2865i 1.21640i 0.793785 + 0.608198i \(0.208107\pi\)
−0.793785 + 0.608198i \(0.791893\pi\)
\(468\) 0 0
\(469\) 39.8310 1.83922
\(470\) 0 0
\(471\) −1.10051 −0.0507087
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 10.7417i − 0.491831i
\(478\) 0 0
\(479\) 31.0770 1.41994 0.709971 0.704231i \(-0.248708\pi\)
0.709971 + 0.704231i \(0.248708\pi\)
\(480\) 0 0
\(481\) 3.92804 0.179103
\(482\) 0 0
\(483\) − 6.11163i − 0.278089i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 14.9591i − 0.677860i −0.940812 0.338930i \(-0.889935\pi\)
0.940812 0.338930i \(-0.110065\pi\)
\(488\) 0 0
\(489\) −19.0965 −0.863573
\(490\) 0 0
\(491\) 24.1333 1.08912 0.544561 0.838721i \(-0.316696\pi\)
0.544561 + 0.838721i \(0.316696\pi\)
\(492\) 0 0
\(493\) 68.2641i 3.07446i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 41.1014i 1.84365i
\(498\) 0 0
\(499\) 17.7266 0.793552 0.396776 0.917915i \(-0.370129\pi\)
0.396776 + 0.917915i \(0.370129\pi\)
\(500\) 0 0
\(501\) 6.60492 0.295086
\(502\) 0 0
\(503\) − 7.69566i − 0.343133i −0.985173 0.171566i \(-0.945117\pi\)
0.985173 0.171566i \(-0.0548828\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 6.75906i − 0.300180i
\(508\) 0 0
\(509\) 33.0564 1.46520 0.732600 0.680659i \(-0.238307\pi\)
0.732600 + 0.680659i \(0.238307\pi\)
\(510\) 0 0
\(511\) 24.5410 1.08563
\(512\) 0 0
\(513\) − 7.64572i − 0.337567i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.2000i 0.712474i
\(518\) 0 0
\(519\) −14.8296 −0.650947
\(520\) 0 0
\(521\) 10.6047 0.464598 0.232299 0.972644i \(-0.425375\pi\)
0.232299 + 0.972644i \(0.425375\pi\)
\(522\) 0 0
\(523\) − 34.5128i − 1.50914i −0.656219 0.754570i \(-0.727845\pi\)
0.656219 0.754570i \(-0.272155\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0704i 0.525794i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 6.25799 0.271574
\(532\) 0 0
\(533\) − 30.0053i − 1.29967i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 31.8622i − 1.37495i
\(538\) 0 0
\(539\) −32.8627 −1.41550
\(540\) 0 0
\(541\) −27.3344 −1.17520 −0.587598 0.809153i \(-0.699926\pi\)
−0.587598 + 0.809153i \(0.699926\pi\)
\(542\) 0 0
\(543\) 25.5403i 1.09604i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 34.6190i − 1.48020i −0.672496 0.740101i \(-0.734778\pi\)
0.672496 0.740101i \(-0.265222\pi\)
\(548\) 0 0
\(549\) 5.55254 0.236977
\(550\) 0 0
\(551\) −12.1832 −0.519022
\(552\) 0 0
\(553\) − 33.0181i − 1.40407i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 25.2490i − 1.06983i −0.844905 0.534917i \(-0.820343\pi\)
0.844905 0.534917i \(-0.179657\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −22.3150 −0.942139
\(562\) 0 0
\(563\) 41.0793i 1.73128i 0.500663 + 0.865642i \(0.333090\pi\)
−0.500663 + 0.865642i \(0.666910\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 16.3790i − 0.687854i
\(568\) 0 0
\(569\) 28.4230 1.19155 0.595776 0.803150i \(-0.296844\pi\)
0.595776 + 0.803150i \(0.296844\pi\)
\(570\) 0 0
\(571\) 13.4690 0.563659 0.281830 0.959464i \(-0.409059\pi\)
0.281830 + 0.959464i \(0.409059\pi\)
\(572\) 0 0
\(573\) − 3.43698i − 0.143582i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 3.69392i − 0.153780i −0.997040 0.0768899i \(-0.975501\pi\)
0.997040 0.0768899i \(-0.0244990\pi\)
\(578\) 0 0
\(579\) 22.8534 0.949757
\(580\) 0 0
\(581\) 21.4340 0.889233
\(582\) 0 0
\(583\) − 18.6938i − 0.774218i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 19.1042i − 0.788513i −0.919000 0.394257i \(-0.871002\pi\)
0.919000 0.394257i \(-0.128998\pi\)
\(588\) 0 0
\(589\) −2.15422 −0.0887631
\(590\) 0 0
\(591\) 30.7280 1.26398
\(592\) 0 0
\(593\) 13.9194i 0.571602i 0.958289 + 0.285801i \(0.0922596\pi\)
−0.958289 + 0.285801i \(0.907740\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 1.70103i − 0.0696186i
\(598\) 0 0
\(599\) −10.9498 −0.447396 −0.223698 0.974659i \(-0.571813\pi\)
−0.223698 + 0.974659i \(0.571813\pi\)
\(600\) 0 0
\(601\) 28.7034 1.17084 0.585418 0.810731i \(-0.300930\pi\)
0.585418 + 0.810731i \(0.300930\pi\)
\(602\) 0 0
\(603\) − 10.9486i − 0.445862i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 43.7745i 1.77675i 0.459117 + 0.888376i \(0.348166\pi\)
−0.459117 + 0.888376i \(0.651834\pi\)
\(608\) 0 0
\(609\) −54.6601 −2.21494
\(610\) 0 0
\(611\) 20.3442 0.823036
\(612\) 0 0
\(613\) 6.59492i 0.266366i 0.991091 + 0.133183i \(0.0425199\pi\)
−0.991091 + 0.133183i \(0.957480\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 20.3939i − 0.821029i −0.911854 0.410514i \(-0.865349\pi\)
0.911854 0.410514i \(-0.134651\pi\)
\(618\) 0 0
\(619\) −0.513389 −0.0206348 −0.0103174 0.999947i \(-0.503284\pi\)
−0.0103174 + 0.999947i \(0.503284\pi\)
\(620\) 0 0
\(621\) −5.61268 −0.225229
\(622\) 0 0
\(623\) 21.9232i 0.878333i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 3.98259i − 0.159049i
\(628\) 0 0
\(629\) 10.7050 0.426835
\(630\) 0 0
\(631\) 16.3428 0.650596 0.325298 0.945612i \(-0.394535\pi\)
0.325298 + 0.945612i \(0.394535\pi\)
\(632\) 0 0
\(633\) − 19.3516i − 0.769157i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 41.2695i 1.63516i
\(638\) 0 0
\(639\) 11.2978 0.446935
\(640\) 0 0
\(641\) −33.7798 −1.33422 −0.667110 0.744959i \(-0.732469\pi\)
−0.667110 + 0.744959i \(0.732469\pi\)
\(642\) 0 0
\(643\) − 2.18090i − 0.0860062i −0.999075 0.0430031i \(-0.986307\pi\)
0.999075 0.0430031i \(-0.0136925\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 34.7346i 1.36556i 0.730625 + 0.682778i \(0.239229\pi\)
−0.730625 + 0.682778i \(0.760771\pi\)
\(648\) 0 0
\(649\) 10.8907 0.427499
\(650\) 0 0
\(651\) −9.66494 −0.378799
\(652\) 0 0
\(653\) − 6.68309i − 0.261530i −0.991413 0.130765i \(-0.958257\pi\)
0.991413 0.130765i \(-0.0417433\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 6.74575i − 0.263177i
\(658\) 0 0
\(659\) 45.8903 1.78763 0.893815 0.448435i \(-0.148018\pi\)
0.893815 + 0.448435i \(0.148018\pi\)
\(660\) 0 0
\(661\) 13.6251 0.529957 0.264978 0.964254i \(-0.414635\pi\)
0.264978 + 0.964254i \(0.414635\pi\)
\(662\) 0 0
\(663\) 28.0234i 1.08834i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.94362i 0.346299i
\(668\) 0 0
\(669\) 15.5257 0.600259
\(670\) 0 0
\(671\) 9.66304 0.373038
\(672\) 0 0
\(673\) − 25.3475i − 0.977075i −0.872543 0.488537i \(-0.837531\pi\)
0.872543 0.488537i \(-0.162469\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.2279i 0.662123i 0.943609 + 0.331062i \(0.107407\pi\)
−0.943609 + 0.331062i \(0.892593\pi\)
\(678\) 0 0
\(679\) 76.9996 2.95497
\(680\) 0 0
\(681\) 6.91579 0.265014
\(682\) 0 0
\(683\) 39.5454i 1.51316i 0.653899 + 0.756582i \(0.273132\pi\)
−0.653899 + 0.756582i \(0.726868\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 1.62394i − 0.0619570i
\(688\) 0 0
\(689\) −23.4759 −0.894361
\(690\) 0 0
\(691\) −29.0501 −1.10512 −0.552558 0.833474i \(-0.686348\pi\)
−0.552558 + 0.833474i \(0.686348\pi\)
\(692\) 0 0
\(693\) 13.3244i 0.506152i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 81.7725i − 3.09735i
\(698\) 0 0
\(699\) 3.49244 0.132096
\(700\) 0 0
\(701\) −17.7735 −0.671298 −0.335649 0.941987i \(-0.608956\pi\)
−0.335649 + 0.941987i \(0.608956\pi\)
\(702\) 0 0
\(703\) 1.91053i 0.0720571i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 50.0093i 1.88079i
\(708\) 0 0
\(709\) 14.8606 0.558103 0.279052 0.960276i \(-0.409980\pi\)
0.279052 + 0.960276i \(0.409980\pi\)
\(710\) 0 0
\(711\) −9.07592 −0.340374
\(712\) 0 0
\(713\) 1.58140i 0.0592240i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 34.4216i 1.28550i
\(718\) 0 0
\(719\) −27.4365 −1.02321 −0.511604 0.859221i \(-0.670949\pi\)
−0.511604 + 0.859221i \(0.670949\pi\)
\(720\) 0 0
\(721\) −9.10512 −0.339092
\(722\) 0 0
\(723\) 2.91449i 0.108391i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 23.3674i 0.866649i 0.901238 + 0.433325i \(0.142660\pi\)
−0.901238 + 0.433325i \(0.857340\pi\)
\(728\) 0 0
\(729\) −26.5754 −0.984275
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) − 2.53381i − 0.0935884i −0.998905 0.0467942i \(-0.985100\pi\)
0.998905 0.0467942i \(-0.0149005\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 19.0538i − 0.701856i
\(738\) 0 0
\(739\) −25.9318 −0.953918 −0.476959 0.878925i \(-0.658261\pi\)
−0.476959 + 0.878925i \(0.658261\pi\)
\(740\) 0 0
\(741\) −5.00139 −0.183731
\(742\) 0 0
\(743\) − 2.11370i − 0.0775442i −0.999248 0.0387721i \(-0.987655\pi\)
0.999248 0.0387721i \(-0.0123446\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 5.89172i − 0.215567i
\(748\) 0 0
\(749\) −9.21077 −0.336554
\(750\) 0 0
\(751\) 2.54986 0.0930459 0.0465229 0.998917i \(-0.485186\pi\)
0.0465229 + 0.998917i \(0.485186\pi\)
\(752\) 0 0
\(753\) − 17.6749i − 0.644107i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 30.6897i − 1.11544i −0.830030 0.557718i \(-0.811677\pi\)
0.830030 0.557718i \(-0.188323\pi\)
\(758\) 0 0
\(759\) −2.92360 −0.106120
\(760\) 0 0
\(761\) 19.3112 0.700032 0.350016 0.936744i \(-0.386176\pi\)
0.350016 + 0.936744i \(0.386176\pi\)
\(762\) 0 0
\(763\) 41.6028i 1.50612i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 13.6767i − 0.493839i
\(768\) 0 0
\(769\) 8.52624 0.307464 0.153732 0.988113i \(-0.450871\pi\)
0.153732 + 0.988113i \(0.450871\pi\)
\(770\) 0 0
\(771\) −28.8770 −1.03998
\(772\) 0 0
\(773\) 29.5107i 1.06143i 0.847551 + 0.530713i \(0.178076\pi\)
−0.847551 + 0.530713i \(0.821924\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 8.57164i 0.307506i
\(778\) 0 0
\(779\) 14.5941 0.522887
\(780\) 0 0
\(781\) 19.6615 0.703545
\(782\) 0 0
\(783\) 50.1977i 1.79392i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 37.1733i 1.32508i 0.749025 + 0.662542i \(0.230522\pi\)
−0.749025 + 0.662542i \(0.769478\pi\)
\(788\) 0 0
\(789\) 29.2548 1.04150
\(790\) 0 0
\(791\) −80.8811 −2.87580
\(792\) 0 0
\(793\) − 12.1350i − 0.430926i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 16.9954i − 0.602008i −0.953623 0.301004i \(-0.902678\pi\)
0.953623 0.301004i \(-0.0973217\pi\)
\(798\) 0 0
\(799\) 55.4434 1.96144
\(800\) 0 0
\(801\) 6.02617 0.212924
\(802\) 0 0
\(803\) − 11.7396i − 0.414281i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 13.7711i 0.484767i
\(808\) 0 0
\(809\) −16.8409 −0.592095 −0.296047 0.955173i \(-0.595669\pi\)
−0.296047 + 0.955173i \(0.595669\pi\)
\(810\) 0 0
\(811\) 4.44804 0.156192 0.0780959 0.996946i \(-0.475116\pi\)
0.0780959 + 0.996946i \(0.475116\pi\)
\(812\) 0 0
\(813\) 5.04303i 0.176867i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 16.7330 0.584698
\(820\) 0 0
\(821\) −16.7361 −0.584093 −0.292047 0.956404i \(-0.594336\pi\)
−0.292047 + 0.956404i \(0.594336\pi\)
\(822\) 0 0
\(823\) − 43.6605i − 1.52191i −0.648805 0.760955i \(-0.724731\pi\)
0.648805 0.760955i \(-0.275269\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.9750i 1.32052i 0.751037 + 0.660260i \(0.229554\pi\)
−0.751037 + 0.660260i \(0.770446\pi\)
\(828\) 0 0
\(829\) −17.9046 −0.621851 −0.310925 0.950434i \(-0.600639\pi\)
−0.310925 + 0.950434i \(0.600639\pi\)
\(830\) 0 0
\(831\) −19.4872 −0.676002
\(832\) 0 0
\(833\) 112.471i 3.89687i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.87591i 0.306796i
\(838\) 0 0
\(839\) 42.4503 1.46555 0.732773 0.680473i \(-0.238226\pi\)
0.732773 + 0.680473i \(0.238226\pi\)
\(840\) 0 0
\(841\) 50.9884 1.75822
\(842\) 0 0
\(843\) − 5.78976i − 0.199410i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 28.0949i − 0.965353i
\(848\) 0 0
\(849\) 5.09283 0.174785
\(850\) 0 0
\(851\) 1.40251 0.0480775
\(852\) 0 0
\(853\) − 22.9952i − 0.787339i −0.919252 0.393670i \(-0.871205\pi\)
0.919252 0.393670i \(-0.128795\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.46053i 0.323166i 0.986859 + 0.161583i \(0.0516599\pi\)
−0.986859 + 0.161583i \(0.948340\pi\)
\(858\) 0 0
\(859\) 6.20043 0.211556 0.105778 0.994390i \(-0.466267\pi\)
0.105778 + 0.994390i \(0.466267\pi\)
\(860\) 0 0
\(861\) 65.4765 2.23143
\(862\) 0 0
\(863\) − 9.68754i − 0.329768i −0.986313 0.164884i \(-0.947275\pi\)
0.986313 0.164884i \(-0.0527249\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 54.0860i 1.83686i
\(868\) 0 0
\(869\) −15.7948 −0.535801
\(870\) 0 0
\(871\) −23.9280 −0.810771
\(872\) 0 0
\(873\) − 21.1654i − 0.716340i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 51.5663i − 1.74127i −0.491928 0.870636i \(-0.663708\pi\)
0.491928 0.870636i \(-0.336292\pi\)
\(878\) 0 0
\(879\) −32.8065 −1.10654
\(880\) 0 0
\(881\) −33.5969 −1.13191 −0.565954 0.824437i \(-0.691492\pi\)
−0.565954 + 0.824437i \(0.691492\pi\)
\(882\) 0 0
\(883\) 11.7914i 0.396814i 0.980120 + 0.198407i \(0.0635767\pi\)
−0.980120 + 0.198407i \(0.936423\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 54.9578i 1.84530i 0.385634 + 0.922652i \(0.373983\pi\)
−0.385634 + 0.922652i \(0.626017\pi\)
\(888\) 0 0
\(889\) 90.8515 3.04706
\(890\) 0 0
\(891\) −7.83517 −0.262488
\(892\) 0 0
\(893\) 9.89506i 0.331126i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.67149i 0.122588i
\(898\) 0 0
\(899\) 14.1435 0.471711
\(900\) 0 0
\(901\) −63.9783 −2.13143
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 8.41429i − 0.279392i −0.990194 0.139696i \(-0.955387\pi\)
0.990194 0.139696i \(-0.0446125\pi\)
\(908\) 0 0
\(909\) 13.7464 0.455940
\(910\) 0 0
\(911\) −26.4143 −0.875146 −0.437573 0.899183i \(-0.644162\pi\)
−0.437573 + 0.899183i \(0.644162\pi\)
\(912\) 0 0
\(913\) − 10.2533i − 0.339335i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 26.4832i 0.874551i
\(918\) 0 0
\(919\) −59.2734 −1.95525 −0.977624 0.210359i \(-0.932537\pi\)
−0.977624 + 0.210359i \(0.932537\pi\)
\(920\) 0 0
\(921\) −28.8659 −0.951164
\(922\) 0 0
\(923\) − 24.6912i − 0.812722i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2.50279i 0.0822023i
\(928\) 0 0
\(929\) 17.2475 0.565871 0.282935 0.959139i \(-0.408692\pi\)
0.282935 + 0.959139i \(0.408692\pi\)
\(930\) 0 0
\(931\) −20.0728 −0.657859
\(932\) 0 0
\(933\) − 25.7223i − 0.842111i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 21.6918i 0.708642i 0.935124 + 0.354321i \(0.115288\pi\)
−0.935124 + 0.354321i \(0.884712\pi\)
\(938\) 0 0
\(939\) −23.5303 −0.767883
\(940\) 0 0
\(941\) −11.1158 −0.362365 −0.181182 0.983449i \(-0.557992\pi\)
−0.181182 + 0.983449i \(0.557992\pi\)
\(942\) 0 0
\(943\) − 10.7134i − 0.348877i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40.8093i 1.32613i 0.748564 + 0.663063i \(0.230744\pi\)
−0.748564 + 0.663063i \(0.769256\pi\)
\(948\) 0 0
\(949\) −14.7427 −0.478569
\(950\) 0 0
\(951\) −44.9886 −1.45886
\(952\) 0 0
\(953\) 11.8237i 0.383009i 0.981492 + 0.191504i \(0.0613366\pi\)
−0.981492 + 0.191504i \(0.938663\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 26.1475i 0.845230i
\(958\) 0 0
\(959\) −40.4973 −1.30772
\(960\) 0 0
\(961\) −28.4992 −0.919328
\(962\) 0 0
\(963\) 2.53183i 0.0815869i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 27.8536i − 0.895710i −0.894106 0.447855i \(-0.852188\pi\)
0.894106 0.447855i \(-0.147812\pi\)
\(968\) 0 0
\(969\) −13.6301 −0.437863
\(970\) 0 0
\(971\) −16.5249 −0.530309 −0.265155 0.964206i \(-0.585423\pi\)
−0.265155 + 0.964206i \(0.585423\pi\)
\(972\) 0 0
\(973\) 43.0112i 1.37888i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 50.0059i 1.59983i 0.600114 + 0.799915i \(0.295122\pi\)
−0.600114 + 0.799915i \(0.704878\pi\)
\(978\) 0 0
\(979\) 10.4873 0.335176
\(980\) 0 0
\(981\) 11.4357 0.365112
\(982\) 0 0
\(983\) 22.7894i 0.726869i 0.931620 + 0.363434i \(0.118396\pi\)
−0.931620 + 0.363434i \(0.881604\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 44.3943i 1.41309i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 33.9822 1.07948 0.539740 0.841832i \(-0.318522\pi\)
0.539740 + 0.841832i \(0.318522\pi\)
\(992\) 0 0
\(993\) 0.392962i 0.0124703i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 17.7258i 0.561382i 0.959798 + 0.280691i \(0.0905637\pi\)
−0.959798 + 0.280691i \(0.909436\pi\)
\(998\) 0 0
\(999\) 7.87186 0.249055
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.4049.4 10
5.2 odd 4 920.2.a.j.1.2 5
5.3 odd 4 4600.2.a.be.1.4 5
5.4 even 2 inner 4600.2.e.u.4049.7 10
15.2 even 4 8280.2.a.bs.1.1 5
20.3 even 4 9200.2.a.cu.1.2 5
20.7 even 4 1840.2.a.v.1.4 5
40.27 even 4 7360.2.a.cp.1.2 5
40.37 odd 4 7360.2.a.co.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.2 5 5.2 odd 4
1840.2.a.v.1.4 5 20.7 even 4
4600.2.a.be.1.4 5 5.3 odd 4
4600.2.e.u.4049.4 10 1.1 even 1 trivial
4600.2.e.u.4049.7 10 5.4 even 2 inner
7360.2.a.co.1.4 5 40.37 odd 4
7360.2.a.cp.1.2 5 40.27 even 4
8280.2.a.bs.1.1 5 15.2 even 4
9200.2.a.cu.1.2 5 20.3 even 4