Properties

Label 4600.2.a.be.1.4
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.13955077.1
Defining polynomial: \(x^{5} - 14 x^{3} - x^{2} + 32 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.31091\) of defining polynomial
Character \(\chi\) \(=\) 4600.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.31091 q^{3} +4.66212 q^{7} -1.28151 q^{9} +O(q^{10})\) \(q+1.31091 q^{3} +4.66212 q^{7} -1.28151 q^{9} +2.23020 q^{11} +2.80072 q^{13} -7.63271 q^{17} -1.36222 q^{19} +6.11163 q^{21} +1.00000 q^{23} -5.61268 q^{27} +8.94362 q^{29} -1.58140 q^{31} +2.92360 q^{33} +1.40251 q^{37} +3.67149 q^{39} +10.7134 q^{41} +7.26391 q^{47} +14.7353 q^{49} -10.0058 q^{51} +8.38212 q^{53} -1.78575 q^{57} -4.88331 q^{59} +4.33282 q^{61} -5.97454 q^{63} -8.54355 q^{67} +1.31091 q^{69} +8.81604 q^{71} +5.26391 q^{73} +10.3974 q^{77} +7.08222 q^{79} -3.51322 q^{81} +4.59749 q^{83} +11.7243 q^{87} -4.70241 q^{89} +13.0573 q^{91} -2.07308 q^{93} -16.5160 q^{97} -2.85802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{7} + 13 q^{9} + O(q^{10}) \) \( 5 q + 2 q^{7} + 13 q^{9} - q^{11} - 4 q^{13} - 4 q^{17} + 7 q^{19} + 6 q^{21} + 5 q^{23} - 3 q^{27} + 4 q^{29} + 19 q^{31} - 17 q^{33} - 15 q^{37} + 19 q^{39} + 25 q^{41} + 11 q^{47} + 25 q^{49} + 19 q^{51} - 3 q^{53} - 48 q^{57} - q^{59} - 5 q^{61} + 41 q^{63} - 9 q^{67} + q^{71} + q^{73} - 18 q^{77} - 2 q^{79} + 57 q^{81} + 45 q^{83} + 9 q^{87} + 6 q^{89} + 11 q^{91} + 39 q^{93} - 25 q^{97} - 65 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.31091 0.756856 0.378428 0.925631i \(-0.376465\pi\)
0.378428 + 0.925631i \(0.376465\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.66212 1.76211 0.881057 0.473010i \(-0.156832\pi\)
0.881057 + 0.473010i \(0.156832\pi\)
\(8\) 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
\(13\) 2.80072 0.776779 0.388389 0.921495i \(-0.373032\pi\)
0.388389 + 0.921495i \(0.373032\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.63271 −1.85120 −0.925602 0.378498i \(-0.876441\pi\)
−0.925602 + 0.378498i \(0.876441\pi\)
\(18\) 0 0
\(19\) −1.36222 −0.312515 −0.156258 0.987716i \(-0.549943\pi\)
−0.156258 + 0.987716i \(0.549943\pi\)
\(20\) 0 0
\(21\) 6.11163 1.33367
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.61268 −1.08016
\(28\) 0 0
\(29\) 8.94362 1.66079 0.830395 0.557176i \(-0.188115\pi\)
0.830395 + 0.557176i \(0.188115\pi\)
\(30\) 0 0
\(31\) −1.58140 −0.284028 −0.142014 0.989865i \(-0.545358\pi\)
−0.142014 + 0.989865i \(0.545358\pi\)
\(32\) 0 0
\(33\) 2.92360 0.508933
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.40251 0.230572 0.115286 0.993332i \(-0.463222\pi\)
0.115286 + 0.993332i \(0.463222\pi\)
\(38\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.26391 1.05955 0.529775 0.848138i \(-0.322276\pi\)
0.529775 + 0.848138i \(0.322276\pi\)
\(48\) 0 0
\(49\) 14.7353 2.10505
\(50\) 0 0
\(51\) −10.0058 −1.40109
\(52\) 0 0
\(53\) 8.38212 1.15137 0.575686 0.817671i \(-0.304735\pi\)
0.575686 + 0.817671i \(0.304735\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.78575 −0.236529
\(58\) 0 0
\(59\) −4.88331 −0.635752 −0.317876 0.948132i \(-0.602970\pi\)
−0.317876 + 0.948132i \(0.602970\pi\)
\(60\) 0 0
\(61\) 4.33282 0.554760 0.277380 0.960760i \(-0.410534\pi\)
0.277380 + 0.960760i \(0.410534\pi\)
\(62\) 0 0
\(63\) −5.97454 −0.752721
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.54355 −1.04376 −0.521880 0.853019i \(-0.674769\pi\)
−0.521880 + 0.853019i \(0.674769\pi\)
\(68\) 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
\(73\) 5.26391 0.616095 0.308047 0.951371i \(-0.400325\pi\)
0.308047 + 0.951371i \(0.400325\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3974 1.18490
\(78\) 0 0
\(79\) 7.08222 0.796812 0.398406 0.917209i \(-0.369563\pi\)
0.398406 + 0.917209i \(0.369563\pi\)
\(80\) 0 0
\(81\) −3.51322 −0.390357
\(82\) 0 0
\(83\) 4.59749 0.504640 0.252320 0.967644i \(-0.418806\pi\)
0.252320 + 0.967644i \(0.418806\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 11.7243 1.25698
\(88\) 0 0
\(89\) −4.70241 −0.498454 −0.249227 0.968445i \(-0.580177\pi\)
−0.249227 + 0.968445i \(0.580177\pi\)
\(90\) 0 0
\(91\) 13.0573 1.36877
\(92\) 0 0
\(93\) −2.07308 −0.214968
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −16.5160 −1.67695 −0.838474 0.544942i \(-0.816552\pi\)
−0.838474 + 0.544942i \(0.816552\pi\)
\(98\) 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
\(103\) −1.95300 −0.192435 −0.0962175 0.995360i \(-0.530674\pi\)
−0.0962175 + 0.995360i \(0.530674\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.97566 0.190994 0.0954972 0.995430i \(-0.469556\pi\)
0.0954972 + 0.995430i \(0.469556\pi\)
\(108\) 0 0
\(109\) −8.92360 −0.854725 −0.427363 0.904080i \(-0.640557\pi\)
−0.427363 + 0.904080i \(0.640557\pi\)
\(110\) 0 0
\(111\) 1.83857 0.174510
\(112\) 0 0
\(113\) −17.3486 −1.63202 −0.816008 0.578040i \(-0.803818\pi\)
−0.816008 + 0.578040i \(0.803818\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.58914 −0.331816
\(118\) 0 0
\(119\) −35.5846 −3.26203
\(120\) 0 0
\(121\) −6.02621 −0.547838
\(122\) 0 0
\(123\) 14.0444 1.26634
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −19.4872 −1.72921 −0.864603 0.502455i \(-0.832430\pi\)
−0.864603 + 0.502455i \(0.832430\pi\)
\(128\) 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
\(133\) −6.35084 −0.550687
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.68645 0.742134 0.371067 0.928606i \(-0.378992\pi\)
0.371067 + 0.928606i \(0.378992\pi\)
\(138\) 0 0
\(139\) −9.22569 −0.782513 −0.391256 0.920282i \(-0.627959\pi\)
−0.391256 + 0.920282i \(0.627959\pi\)
\(140\) 0 0
\(141\) 9.52236 0.801927
\(142\) 0 0
\(143\) 6.24615 0.522329
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 19.3167 1.59322
\(148\) 0 0
\(149\) 18.6056 1.52423 0.762115 0.647441i \(-0.224161\pi\)
0.762115 + 0.647441i \(0.224161\pi\)
\(150\) 0 0
\(151\) 17.4316 1.41856 0.709280 0.704927i \(-0.249020\pi\)
0.709280 + 0.704927i \(0.249020\pi\)
\(152\) 0 0
\(153\) 9.78138 0.790778
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.839497 −0.0669992 −0.0334996 0.999439i \(-0.510665\pi\)
−0.0334996 + 0.999439i \(0.510665\pi\)
\(158\) 0 0
\(159\) 10.9882 0.871423
\(160\) 0 0
\(161\) 4.66212 0.367426
\(162\) 0 0
\(163\) 14.5673 1.14100 0.570500 0.821297i \(-0.306749\pi\)
0.570500 + 0.821297i \(0.306749\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.03842 0.389884 0.194942 0.980815i \(-0.437548\pi\)
0.194942 + 0.980815i \(0.437548\pi\)
\(168\) 0 0
\(169\) −5.15599 −0.396615
\(170\) 0 0
\(171\) 1.74570 0.133497
\(172\) 0 0
\(173\) 11.3124 0.860067 0.430034 0.902813i \(-0.358502\pi\)
0.430034 + 0.902813i \(0.358502\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.40159 −0.481173
\(178\) 0 0
\(179\) −24.3053 −1.81667 −0.908333 0.418247i \(-0.862645\pi\)
−0.908333 + 0.418247i \(0.862645\pi\)
\(180\) 0 0
\(181\) −19.4829 −1.44815 −0.724075 0.689721i \(-0.757733\pi\)
−0.724075 + 0.689721i \(0.757733\pi\)
\(182\) 0 0
\(183\) 5.67994 0.419874
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −17.0225 −1.24481
\(188\) 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
\(193\) −17.4332 −1.25487 −0.627436 0.778668i \(-0.715895\pi\)
−0.627436 + 0.778668i \(0.715895\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.4402 1.67004 0.835022 0.550217i \(-0.185455\pi\)
0.835022 + 0.550217i \(0.185455\pi\)
\(198\) 0 0
\(199\) −1.29759 −0.0919839 −0.0459920 0.998942i \(-0.514645\pi\)
−0.0459920 + 0.998942i \(0.514645\pi\)
\(200\) 0 0
\(201\) −11.1998 −0.789976
\(202\) 0 0
\(203\) 41.6962 2.92650
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.28151 −0.0890709
\(208\) 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
\(213\) 11.5571 0.791877
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −7.37268 −0.500490
\(218\) 0 0
\(219\) 6.90053 0.466295
\(220\) 0 0
\(221\) −21.3771 −1.43798
\(222\) 0 0
\(223\) −11.8434 −0.793095 −0.396548 0.918014i \(-0.629792\pi\)
−0.396548 + 0.918014i \(0.629792\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.27556 0.350151 0.175075 0.984555i \(-0.443983\pi\)
0.175075 + 0.984555i \(0.443983\pi\)
\(228\) 0 0
\(229\) −1.23878 −0.0818611 −0.0409305 0.999162i \(-0.513032\pi\)
−0.0409305 + 0.999162i \(0.513032\pi\)
\(230\) 0 0
\(231\) 13.6301 0.896798
\(232\) 0 0
\(233\) −2.66412 −0.174533 −0.0872663 0.996185i \(-0.527813\pi\)
−0.0872663 + 0.996185i \(0.527813\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 9.28418 0.603072
\(238\) 0 0
\(239\) 26.2577 1.69847 0.849235 0.528014i \(-0.177063\pi\)
0.849235 + 0.528014i \(0.177063\pi\)
\(240\) 0 0
\(241\) −2.22326 −0.143213 −0.0716063 0.997433i \(-0.522813\pi\)
−0.0716063 + 0.997433i \(0.522813\pi\)
\(242\) 0 0
\(243\) 12.2325 0.784717
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3.81520 −0.242755
\(248\) 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
\(253\) 2.23020 0.140211
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −22.0281 −1.37408 −0.687039 0.726620i \(-0.741090\pi\)
−0.687039 + 0.726620i \(0.741090\pi\)
\(258\) 0 0
\(259\) 6.53868 0.406294
\(260\) 0 0
\(261\) −11.4613 −0.709438
\(262\) 0 0
\(263\) −22.3164 −1.37609 −0.688043 0.725670i \(-0.741530\pi\)
−0.688043 + 0.725670i \(0.741530\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.16445 −0.377258
\(268\) 0 0
\(269\) 10.5050 0.640501 0.320251 0.947333i \(-0.396233\pi\)
0.320251 + 0.947333i \(0.396233\pi\)
\(270\) 0 0
\(271\) −3.84696 −0.233686 −0.116843 0.993150i \(-0.537277\pi\)
−0.116843 + 0.993150i \(0.537277\pi\)
\(272\) 0 0
\(273\) 17.1169 1.03596
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −14.8653 −0.893172 −0.446586 0.894741i \(-0.647360\pi\)
−0.446586 + 0.894741i \(0.647360\pi\)
\(278\) 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
\(283\) −3.88495 −0.230936 −0.115468 0.993311i \(-0.536837\pi\)
−0.115468 + 0.993311i \(0.536837\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 49.9472 2.94829
\(288\) 0 0
\(289\) 41.2583 2.42696
\(290\) 0 0
\(291\) −21.6511 −1.26921
\(292\) 0 0
\(293\) 25.0257 1.46202 0.731009 0.682368i \(-0.239050\pi\)
0.731009 + 0.682368i \(0.239050\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −12.5174 −0.726333
\(298\) 0 0
\(299\) 2.80072 0.161970
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 14.0618 0.807831
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −22.0197 −1.25673 −0.628365 0.777918i \(-0.716276\pi\)
−0.628365 + 0.777918i \(0.716276\pi\)
\(308\) 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
\(313\) 17.9496 1.01457 0.507285 0.861778i \(-0.330649\pi\)
0.507285 + 0.861778i \(0.330649\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −34.3185 −1.92752 −0.963761 0.266768i \(-0.914044\pi\)
−0.963761 + 0.266768i \(0.914044\pi\)
\(318\) 0 0
\(319\) 19.9461 1.11676
\(320\) 0 0
\(321\) 2.58992 0.144555
\(322\) 0 0
\(323\) 10.3974 0.578529
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −11.6981 −0.646904
\(328\) 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
\(333\) −1.79733 −0.0984931
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.2965 1.21457 0.607284 0.794485i \(-0.292259\pi\)
0.607284 + 0.794485i \(0.292259\pi\)
\(338\) 0 0
\(339\) −22.7425 −1.23520
\(340\) 0 0
\(341\) −3.52684 −0.190989
\(342\) 0 0
\(343\) 36.0630 1.94722
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −33.1240 −1.77819 −0.889094 0.457725i \(-0.848664\pi\)
−0.889094 + 0.457725i \(0.848664\pi\)
\(348\) 0 0
\(349\) −22.0041 −1.17785 −0.588926 0.808187i \(-0.700449\pi\)
−0.588926 + 0.808187i \(0.700449\pi\)
\(350\) 0 0
\(351\) −15.7195 −0.839046
\(352\) 0 0
\(353\) 15.6085 0.830759 0.415379 0.909648i \(-0.363649\pi\)
0.415379 + 0.909648i \(0.363649\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −46.6483 −2.46889
\(358\) 0 0
\(359\) 0.263781 0.0139218 0.00696092 0.999976i \(-0.497784\pi\)
0.00696092 + 0.999976i \(0.497784\pi\)
\(360\) 0 0
\(361\) −17.1444 −0.902334
\(362\) 0 0
\(363\) −7.89984 −0.414634
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −10.8615 −0.566967 −0.283484 0.958977i \(-0.591490\pi\)
−0.283484 + 0.958977i \(0.591490\pi\)
\(368\) 0 0
\(369\) −13.7293 −0.714721
\(370\) 0 0
\(371\) 39.0784 2.02885
\(372\) 0 0
\(373\) −26.8889 −1.39225 −0.696127 0.717919i \(-0.745095\pi\)
−0.696127 + 0.717919i \(0.745095\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 25.0485 1.29007
\(378\) 0 0
\(379\) −2.15824 −0.110861 −0.0554306 0.998463i \(-0.517653\pi\)
−0.0554306 + 0.998463i \(0.517653\pi\)
\(380\) 0 0
\(381\) −25.5460 −1.30876
\(382\) 0 0
\(383\) 4.62814 0.236487 0.118244 0.992985i \(-0.462274\pi\)
0.118244 + 0.992985i \(0.462274\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.4988 −0.937929 −0.468964 0.883217i \(-0.655373\pi\)
−0.468964 + 0.883217i \(0.655373\pi\)
\(390\) 0 0
\(391\) −7.63271 −0.386003
\(392\) 0 0
\(393\) 7.44664 0.375634
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 10.2685 0.515360 0.257680 0.966230i \(-0.417042\pi\)
0.257680 + 0.966230i \(0.417042\pi\)
\(398\) 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
\(403\) −4.42906 −0.220627
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.12788 0.155043
\(408\) 0 0
\(409\) 24.3497 1.20401 0.602006 0.798491i \(-0.294368\pi\)
0.602006 + 0.798491i \(0.294368\pi\)
\(410\) 0 0
\(411\) 11.3872 0.561688
\(412\) 0 0
\(413\) −22.7665 −1.12027
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −12.0941 −0.592249
\(418\) 0 0
\(419\) 25.4535 1.24348 0.621742 0.783222i \(-0.286425\pi\)
0.621742 + 0.783222i \(0.286425\pi\)
\(420\) 0 0
\(421\) 15.2376 0.742634 0.371317 0.928506i \(-0.378906\pi\)
0.371317 + 0.928506i \(0.378906\pi\)
\(422\) 0 0
\(423\) −9.30876 −0.452607
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.2001 0.977551
\(428\) 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
\(433\) −28.4774 −1.36854 −0.684268 0.729230i \(-0.739878\pi\)
−0.684268 + 0.729230i \(0.739878\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.36222 −0.0651639
\(438\) 0 0
\(439\) 21.6009 1.03095 0.515477 0.856904i \(-0.327615\pi\)
0.515477 + 0.856904i \(0.327615\pi\)
\(440\) 0 0
\(441\) −18.8834 −0.899211
\(442\) 0 0
\(443\) −40.3700 −1.91804 −0.959018 0.283346i \(-0.908555\pi\)
−0.959018 + 0.283346i \(0.908555\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 24.3903 1.15362
\(448\) 0 0
\(449\) 5.95484 0.281026 0.140513 0.990079i \(-0.455125\pi\)
0.140513 + 0.990079i \(0.455125\pi\)
\(450\) 0 0
\(451\) 23.8931 1.12508
\(452\) 0 0
\(453\) 22.8512 1.07365
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.66169 0.264843 0.132421 0.991194i \(-0.457725\pi\)
0.132421 + 0.991194i \(0.457725\pi\)
\(458\) 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
\(463\) −37.9899 −1.76554 −0.882769 0.469807i \(-0.844324\pi\)
−0.882769 + 0.469807i \(0.844324\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.2865 1.21640 0.608198 0.793785i \(-0.291893\pi\)
0.608198 + 0.793785i \(0.291893\pi\)
\(468\) 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
\(477\) −10.7417 −0.491831
\(478\) 0 0
\(479\) −31.0770 −1.41994 −0.709971 0.704231i \(-0.751292\pi\)
−0.709971 + 0.704231i \(0.751292\pi\)
\(480\) 0 0
\(481\) 3.92804 0.179103
\(482\) 0 0
\(483\) 6.11163 0.278089
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −14.9591 −0.677860 −0.338930 0.940812i \(-0.610065\pi\)
−0.338930 + 0.940812i \(0.610065\pi\)
\(488\) 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
\(493\) −68.2641 −3.07446
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 41.1014 1.84365
\(498\) 0 0
\(499\) −17.7266 −0.793552 −0.396776 0.917915i \(-0.629871\pi\)
−0.396776 + 0.917915i \(0.629871\pi\)
\(500\) 0 0
\(501\) 6.60492 0.295086
\(502\) 0 0
\(503\) 7.69566 0.343133 0.171566 0.985173i \(-0.445117\pi\)
0.171566 + 0.985173i \(0.445117\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.75906 −0.300180
\(508\) 0 0
\(509\) −33.0564 −1.46520 −0.732600 0.680659i \(-0.761693\pi\)
−0.732600 + 0.680659i \(0.761693\pi\)
\(510\) 0 0
\(511\) 24.5410 1.08563
\(512\) 0 0
\(513\) 7.64572 0.337567
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.2000 0.712474
\(518\) 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
\(523\) 34.5128 1.50914 0.754570 0.656219i \(-0.227845\pi\)
0.754570 + 0.656219i \(0.227845\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0704 0.525794
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 6.25799 0.271574
\(532\) 0 0
\(533\) 30.0053 1.29967
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −31.8622 −1.37495
\(538\) 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
\(543\) −25.5403 −1.09604
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −34.6190 −1.48020 −0.740101 0.672496i \(-0.765222\pi\)
−0.740101 + 0.672496i \(0.765222\pi\)
\(548\) 0 0
\(549\) −5.55254 −0.236977
\(550\) 0 0
\(551\) −12.1832 −0.519022
\(552\) 0 0
\(553\) 33.0181 1.40407
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −25.2490 −1.06983 −0.534917 0.844905i \(-0.679657\pi\)
−0.534917 + 0.844905i \(0.679657\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −22.3150 −0.942139
\(562\) 0 0
\(563\) −41.0793 −1.73128 −0.865642 0.500663i \(-0.833090\pi\)
−0.865642 + 0.500663i \(0.833090\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −16.3790 −0.687854
\(568\) 0 0
\(569\) −28.4230 −1.19155 −0.595776 0.803150i \(-0.703156\pi\)
−0.595776 + 0.803150i \(0.703156\pi\)
\(570\) 0 0
\(571\) 13.4690 0.563659 0.281830 0.959464i \(-0.409059\pi\)
0.281830 + 0.959464i \(0.409059\pi\)
\(572\) 0 0
\(573\) 3.43698 0.143582
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −3.69392 −0.153780 −0.0768899 0.997040i \(-0.524499\pi\)
−0.0768899 + 0.997040i \(0.524499\pi\)
\(578\) 0 0
\(579\) −22.8534 −0.949757
\(580\) 0 0
\(581\) 21.4340 0.889233
\(582\) 0 0
\(583\) 18.6938 0.774218
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.1042 −0.788513 −0.394257 0.919000i \(-0.628998\pi\)
−0.394257 + 0.919000i \(0.628998\pi\)
\(588\) 0 0
\(589\) 2.15422 0.0887631
\(590\) 0 0
\(591\) 30.7280 1.26398
\(592\) 0 0
\(593\) −13.9194 −0.571602 −0.285801 0.958289i \(-0.592260\pi\)
−0.285801 + 0.958289i \(0.592260\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.70103 −0.0696186
\(598\) 0 0
\(599\) 10.9498 0.447396 0.223698 0.974659i \(-0.428187\pi\)
0.223698 + 0.974659i \(0.428187\pi\)
\(600\) 0 0
\(601\) 28.7034 1.17084 0.585418 0.810731i \(-0.300930\pi\)
0.585418 + 0.810731i \(0.300930\pi\)
\(602\) 0 0
\(603\) 10.9486 0.445862
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 43.7745 1.77675 0.888376 0.459117i \(-0.151834\pi\)
0.888376 + 0.459117i \(0.151834\pi\)
\(608\) 0 0
\(609\) 54.6601 2.21494
\(610\) 0 0
\(611\) 20.3442 0.823036
\(612\) 0 0
\(613\) −6.59492 −0.266366 −0.133183 0.991091i \(-0.542520\pi\)
−0.133183 + 0.991091i \(0.542520\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −20.3939 −0.821029 −0.410514 0.911854i \(-0.634651\pi\)
−0.410514 + 0.911854i \(0.634651\pi\)
\(618\) 0 0
\(619\) 0.513389 0.0206348 0.0103174 0.999947i \(-0.496716\pi\)
0.0103174 + 0.999947i \(0.496716\pi\)
\(620\) 0 0
\(621\) −5.61268 −0.225229
\(622\) 0 0
\(623\) −21.9232 −0.878333
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −3.98259 −0.159049
\(628\) 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
\(633\) 19.3516 0.769157
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 41.2695 1.63516
\(638\) 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
\(643\) 2.18090 0.0860062 0.0430031 0.999075i \(-0.486307\pi\)
0.0430031 + 0.999075i \(0.486307\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 34.7346 1.36556 0.682778 0.730625i \(-0.260771\pi\)
0.682778 + 0.730625i \(0.260771\pi\)
\(648\) 0 0
\(649\) −10.8907 −0.427499
\(650\) 0 0
\(651\) −9.66494 −0.378799
\(652\) 0 0
\(653\) 6.68309 0.261530 0.130765 0.991413i \(-0.458257\pi\)
0.130765 + 0.991413i \(0.458257\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −6.74575 −0.263177
\(658\) 0 0
\(659\) −45.8903 −1.78763 −0.893815 0.448435i \(-0.851982\pi\)
−0.893815 + 0.448435i \(0.851982\pi\)
\(660\) 0 0
\(661\) 13.6251 0.529957 0.264978 0.964254i \(-0.414635\pi\)
0.264978 + 0.964254i \(0.414635\pi\)
\(662\) 0 0
\(663\) −28.0234 −1.08834
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.94362 0.346299
\(668\) 0 0
\(669\) −15.5257 −0.600259
\(670\) 0 0
\(671\) 9.66304 0.373038
\(672\) 0 0
\(673\) 25.3475 0.977075 0.488537 0.872543i \(-0.337531\pi\)
0.488537 + 0.872543i \(0.337531\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.2279 0.662123 0.331062 0.943609i \(-0.392593\pi\)
0.331062 + 0.943609i \(0.392593\pi\)
\(678\) 0 0
\(679\) −76.9996 −2.95497
\(680\) 0 0
\(681\) 6.91579 0.265014
\(682\) 0 0
\(683\) −39.5454 −1.51316 −0.756582 0.653899i \(-0.773132\pi\)
−0.756582 + 0.653899i \(0.773132\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.62394 −0.0619570
\(688\) 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
\(693\) −13.3244 −0.506152
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −81.7725 −3.09735
\(698\) 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
\(703\) −1.91053 −0.0720571
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 50.0093 1.88079
\(708\) 0 0
\(709\) −14.8606 −0.558103 −0.279052 0.960276i \(-0.590020\pi\)
−0.279052 + 0.960276i \(0.590020\pi\)
\(710\) 0 0
\(711\) −9.07592 −0.340374
\(712\) 0 0
\(713\) −1.58140 −0.0592240
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 34.4216 1.28550
\(718\) 0 0
\(719\) 27.4365 1.02321 0.511604 0.859221i \(-0.329051\pi\)
0.511604 + 0.859221i \(0.329051\pi\)
\(720\) 0 0
\(721\) −9.10512 −0.339092
\(722\) 0 0
\(723\) −2.91449 −0.108391
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 23.3674 0.866649 0.433325 0.901238i \(-0.357340\pi\)
0.433325 + 0.901238i \(0.357340\pi\)
\(728\) 0 0
\(729\) 26.5754 0.984275
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 2.53381 0.0935884 0.0467942 0.998905i \(-0.485100\pi\)
0.0467942 + 0.998905i \(0.485100\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −19.0538 −0.701856
\(738\) 0 0
\(739\) 25.9318 0.953918 0.476959 0.878925i \(-0.341739\pi\)
0.476959 + 0.878925i \(0.341739\pi\)
\(740\) 0 0
\(741\) −5.00139 −0.183731
\(742\) 0 0
\(743\) 2.11370 0.0775442 0.0387721 0.999248i \(-0.487655\pi\)
0.0387721 + 0.999248i \(0.487655\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −5.89172 −0.215567
\(748\) 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
\(753\) 17.6749 0.644107
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −30.6897 −1.11544 −0.557718 0.830030i \(-0.688323\pi\)
−0.557718 + 0.830030i \(0.688323\pi\)
\(758\) 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
\(763\) −41.6028 −1.50612
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −13.6767 −0.493839
\(768\) 0 0
\(769\) −8.52624 −0.307464 −0.153732 0.988113i \(-0.549129\pi\)
−0.153732 + 0.988113i \(0.549129\pi\)
\(770\) 0 0
\(771\) −28.8770 −1.03998
\(772\) 0 0
\(773\) −29.5107 −1.06143 −0.530713 0.847551i \(-0.678076\pi\)
−0.530713 + 0.847551i \(0.678076\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 8.57164 0.307506
\(778\) 0 0
\(779\) −14.5941 −0.522887
\(780\) 0 0
\(781\) 19.6615 0.703545
\(782\) 0 0
\(783\) −50.1977 −1.79392
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 37.1733 1.32508 0.662542 0.749025i \(-0.269478\pi\)
0.662542 + 0.749025i \(0.269478\pi\)
\(788\) 0 0
\(789\) −29.2548 −1.04150
\(790\) 0 0
\(791\) −80.8811 −2.87580
\(792\) 0 0
\(793\) 12.1350 0.430926
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.9954 −0.602008 −0.301004 0.953623i \(-0.597322\pi\)
−0.301004 + 0.953623i \(0.597322\pi\)
\(798\) 0 0
\(799\) −55.4434 −1.96144
\(800\) 0 0
\(801\) 6.02617 0.212924
\(802\) 0 0
\(803\) 11.7396 0.414281
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 13.7711 0.484767
\(808\) 0 0
\(809\) 16.8409 0.592095 0.296047 0.955173i \(-0.404331\pi\)
0.296047 + 0.955173i \(0.404331\pi\)
\(810\) 0 0
\(811\) 4.44804 0.156192 0.0780959 0.996946i \(-0.475116\pi\)
0.0780959 + 0.996946i \(0.475116\pi\)
\(812\) 0 0
\(813\) −5.04303 −0.176867
\(814\) 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
\(823\) 43.6605 1.52191 0.760955 0.648805i \(-0.224731\pi\)
0.760955 + 0.648805i \(0.224731\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.9750 1.32052 0.660260 0.751037i \(-0.270446\pi\)
0.660260 + 0.751037i \(0.270446\pi\)
\(828\) 0 0
\(829\) 17.9046 0.621851 0.310925 0.950434i \(-0.399361\pi\)
0.310925 + 0.950434i \(0.399361\pi\)
\(830\) 0 0
\(831\) −19.4872 −0.676002
\(832\) 0 0
\(833\) −112.471 −3.89687
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.87591 0.306796
\(838\) 0 0
\(839\) −42.4503 −1.46555 −0.732773 0.680473i \(-0.761774\pi\)
−0.732773 + 0.680473i \(0.761774\pi\)
\(840\) 0 0
\(841\) 50.9884 1.75822
\(842\) 0 0
\(843\) 5.78976 0.199410
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −28.0949 −0.965353
\(848\) 0 0
\(849\) −5.09283 −0.174785
\(850\) 0 0
\(851\) 1.40251 0.0480775
\(852\) 0 0
\(853\) 22.9952 0.787339 0.393670 0.919252i \(-0.371205\pi\)
0.393670 + 0.919252i \(0.371205\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.46053 0.323166 0.161583 0.986859i \(-0.448340\pi\)
0.161583 + 0.986859i \(0.448340\pi\)
\(858\) 0 0
\(859\) −6.20043 −0.211556 −0.105778 0.994390i \(-0.533733\pi\)
−0.105778 + 0.994390i \(0.533733\pi\)
\(860\) 0 0
\(861\) 65.4765 2.23143
\(862\) 0 0
\(863\) 9.68754 0.329768 0.164884 0.986313i \(-0.447275\pi\)
0.164884 + 0.986313i \(0.447275\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 54.0860 1.83686
\(868\) 0 0
\(869\) 15.7948 0.535801
\(870\) 0 0
\(871\) −23.9280 −0.810771
\(872\) 0 0
\(873\) 21.1654 0.716340
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −51.5663 −1.74127 −0.870636 0.491928i \(-0.836292\pi\)
−0.870636 + 0.491928i \(0.836292\pi\)
\(878\) 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
\(883\) −11.7914 −0.396814 −0.198407 0.980120i \(-0.563577\pi\)
−0.198407 + 0.980120i \(0.563577\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 54.9578 1.84530 0.922652 0.385634i \(-0.126017\pi\)
0.922652 + 0.385634i \(0.126017\pi\)
\(888\) 0 0
\(889\) −90.8515 −3.04706
\(890\) 0 0
\(891\) −7.83517 −0.262488
\(892\) 0 0
\(893\) −9.89506 −0.331126
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.67149 0.122588
\(898\) 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
\(907\) −8.41429 −0.279392 −0.139696 0.990194i \(-0.544613\pi\)
−0.139696 + 0.990194i \(0.544613\pi\)
\(908\) 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
\(913\) 10.2533 0.339335
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 26.4832 0.874551
\(918\) 0 0
\(919\) 59.2734 1.95525 0.977624 0.210359i \(-0.0674633\pi\)
0.977624 + 0.210359i \(0.0674633\pi\)
\(920\) 0 0
\(921\) −28.8659 −0.951164
\(922\) 0 0
\(923\) 24.6912 0.812722
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2.50279 0.0822023
\(928\) 0 0
\(929\) −17.2475 −0.565871 −0.282935 0.959139i \(-0.591308\pi\)
−0.282935 + 0.959139i \(0.591308\pi\)
\(930\) 0 0
\(931\) −20.0728 −0.657859
\(932\) 0 0
\(933\) 25.7223 0.842111
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 21.6918 0.708642 0.354321 0.935124i \(-0.384712\pi\)
0.354321 + 0.935124i \(0.384712\pi\)
\(938\) 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
\(943\) 10.7134 0.348877
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40.8093 1.32613 0.663063 0.748564i \(-0.269256\pi\)
0.663063 + 0.748564i \(0.269256\pi\)
\(948\) 0 0
\(949\) 14.7427 0.478569
\(950\) 0 0
\(951\) −44.9886 −1.45886
\(952\) 0 0
\(953\) −11.8237 −0.383009 −0.191504 0.981492i \(-0.561337\pi\)
−0.191504 + 0.981492i \(0.561337\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 26.1475 0.845230
\(958\) 0 0
\(959\) 40.4973 1.30772
\(960\) 0 0
\(961\) −28.4992 −0.919328
\(962\) 0 0
\(963\) −2.53183 −0.0815869
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −27.8536 −0.895710 −0.447855 0.894106i \(-0.647812\pi\)
−0.447855 + 0.894106i \(0.647812\pi\)
\(968\) 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
\(973\) −43.0112 −1.37888
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 50.0059 1.59983 0.799915 0.600114i \(-0.204878\pi\)
0.799915 + 0.600114i \(0.204878\pi\)
\(978\) 0 0
\(979\) −10.4873 −0.335176
\(980\) 0 0
\(981\) 11.4357 0.365112
\(982\) 0 0
\(983\) −22.7894 −0.726869 −0.363434 0.931620i \(-0.618396\pi\)
−0.363434 + 0.931620i \(0.618396\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 44.3943 1.41309
\(988\) 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
\(993\) −0.392962 −0.0124703
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 17.7258 0.561382 0.280691 0.959798i \(-0.409436\pi\)
0.280691 + 0.959798i \(0.409436\pi\)
\(998\) 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.a.be.1.4 5
4.3 odd 2 9200.2.a.cu.1.2 5
5.2 odd 4 4600.2.e.u.4049.4 10
5.3 odd 4 4600.2.e.u.4049.7 10
5.4 even 2 920.2.a.j.1.2 5
15.14 odd 2 8280.2.a.bs.1.1 5
20.19 odd 2 1840.2.a.v.1.4 5
40.19 odd 2 7360.2.a.cp.1.2 5
40.29 even 2 7360.2.a.co.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.2 5 5.4 even 2
1840.2.a.v.1.4 5 20.19 odd 2
4600.2.a.be.1.4 5 1.1 even 1 trivial
4600.2.e.u.4049.4 10 5.2 odd 4
4600.2.e.u.4049.7 10 5.3 odd 4
7360.2.a.co.1.4 5 40.29 even 2
7360.2.a.cp.1.2 5 40.19 odd 2
8280.2.a.bs.1.1 5 15.14 odd 2
9200.2.a.cu.1.2 5 4.3 odd 2