Properties

Label 6724.2.a.c.1.4
Level $6724$
Weight $2$
Character 6724.1
Self dual yes
Analytic conductor $53.691$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6724 = 2^{2} \cdot 41^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6724.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.6914103191\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
Defining polynomial: \(x^{4} - 10 x^{2} - 6 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.707500\) of defining polynomial
Character \(\chi\) \(=\) 6724.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.24028 q^{3} +2.56613 q^{5} +0.858626 q^{7} +7.49944 q^{9} +O(q^{10})\) \(q+3.24028 q^{3} +2.56613 q^{5} +0.858626 q^{7} +7.49944 q^{9} -6.20694 q^{11} -1.41500 q^{13} +8.31498 q^{15} +3.93332 q^{17} -3.82529 q^{19} +2.78219 q^{21} +3.06557 q^{23} +1.58500 q^{25} +14.5795 q^{27} +8.48057 q^{29} +1.13225 q^{31} -20.1123 q^{33} +2.20334 q^{35} +8.49944 q^{37} -4.58500 q^{39} +5.34832 q^{43} +19.2445 q^{45} +6.65528 q^{47} -6.26276 q^{49} +12.7451 q^{51} -6.41389 q^{53} -15.9278 q^{55} -12.3950 q^{57} -3.06557 q^{59} +7.41500 q^{61} +6.43922 q^{63} -3.63107 q^{65} -1.79306 q^{67} +9.93332 q^{69} +3.02422 q^{71} +0.632807 q^{73} +5.13585 q^{75} -5.32944 q^{77} +14.3059 q^{79} +24.7433 q^{81} -8.76332 q^{83} +10.0934 q^{85} +27.4795 q^{87} +7.89557 q^{89} -1.21496 q^{91} +3.66882 q^{93} -9.81616 q^{95} -9.86664 q^{97} -46.5486 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 4q^{5} + 12q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 4q^{5} + 12q^{9} - 4q^{11} + 10q^{15} + 4q^{17} - 6q^{19} - 12q^{23} + 12q^{25} + 10q^{27} + 4q^{29} - 8q^{31} - 20q^{33} + 26q^{35} + 16q^{37} - 24q^{39} + 4q^{43} + 4q^{45} + 6q^{47} + 16q^{49} - 4q^{51} + 16q^{53} + 2q^{55} + 4q^{57} + 12q^{59} + 24q^{61} + 10q^{63} - 4q^{65} - 28q^{67} + 28q^{69} + 2q^{71} + 8q^{73} - 30q^{75} + 8q^{77} + 18q^{79} + 28q^{81} - 12q^{83} - 32q^{85} + 44q^{87} - 4q^{89} + 36q^{91} + 28q^{93} - 14q^{95} - 16q^{97} - 58q^{99} + O(q^{100}) \)

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
\(3\) 3.24028 1.87078 0.935390 0.353619i \(-0.115049\pi\)
0.935390 + 0.353619i \(0.115049\pi\)
\(4\) 0 0
\(5\) 2.56613 1.14761 0.573803 0.818993i \(-0.305467\pi\)
0.573803 + 0.818993i \(0.305467\pi\)
\(6\) 0 0
\(7\) 0.858626 0.324530 0.162265 0.986747i \(-0.448120\pi\)
0.162265 + 0.986747i \(0.448120\pi\)
\(8\) 0 0
\(9\) 7.49944 2.49981
\(10\) 0 0
\(11\) −6.20694 −1.87146 −0.935732 0.352712i \(-0.885260\pi\)
−0.935732 + 0.352712i \(0.885260\pi\)
\(12\) 0 0
\(13\) −1.41500 −0.392450 −0.196225 0.980559i \(-0.562868\pi\)
−0.196225 + 0.980559i \(0.562868\pi\)
\(14\) 0 0
\(15\) 8.31498 2.14692
\(16\) 0 0
\(17\) 3.93332 0.953970 0.476985 0.878911i \(-0.341730\pi\)
0.476985 + 0.878911i \(0.341730\pi\)
\(18\) 0 0
\(19\) −3.82529 −0.877581 −0.438790 0.898589i \(-0.644593\pi\)
−0.438790 + 0.898589i \(0.644593\pi\)
\(20\) 0 0
\(21\) 2.78219 0.607124
\(22\) 0 0
\(23\) 3.06557 0.639215 0.319608 0.947550i \(-0.396449\pi\)
0.319608 + 0.947550i \(0.396449\pi\)
\(24\) 0 0
\(25\) 1.58500 0.317000
\(26\) 0 0
\(27\) 14.5795 2.80582
\(28\) 0 0
\(29\) 8.48057 1.57480 0.787401 0.616441i \(-0.211426\pi\)
0.787401 + 0.616441i \(0.211426\pi\)
\(30\) 0 0
\(31\) 1.13225 0.203358 0.101679 0.994817i \(-0.467578\pi\)
0.101679 + 0.994817i \(0.467578\pi\)
\(32\) 0 0
\(33\) −20.1123 −3.50110
\(34\) 0 0
\(35\) 2.20334 0.372433
\(36\) 0 0
\(37\) 8.49944 1.39730 0.698650 0.715464i \(-0.253784\pi\)
0.698650 + 0.715464i \(0.253784\pi\)
\(38\) 0 0
\(39\) −4.58500 −0.734188
\(40\) 0 0
\(41\) 0 0
\(42\) 0 0
\(43\) 5.34832 0.815611 0.407805 0.913069i \(-0.366294\pi\)
0.407805 + 0.913069i \(0.366294\pi\)
\(44\) 0 0
\(45\) 19.2445 2.86880
\(46\) 0 0
\(47\) 6.65528 0.970773 0.485386 0.874300i \(-0.338679\pi\)
0.485386 + 0.874300i \(0.338679\pi\)
\(48\) 0 0
\(49\) −6.26276 −0.894680
\(50\) 0 0
\(51\) 12.7451 1.78467
\(52\) 0 0
\(53\) −6.41389 −0.881015 −0.440508 0.897749i \(-0.645202\pi\)
−0.440508 + 0.897749i \(0.645202\pi\)
\(54\) 0 0
\(55\) −15.9278 −2.14770
\(56\) 0 0
\(57\) −12.3950 −1.64176
\(58\) 0 0
\(59\) −3.06557 −0.399103 −0.199552 0.979887i \(-0.563949\pi\)
−0.199552 + 0.979887i \(0.563949\pi\)
\(60\) 0 0
\(61\) 7.41500 0.949393 0.474697 0.880149i \(-0.342558\pi\)
0.474697 + 0.880149i \(0.342558\pi\)
\(62\) 0 0
\(63\) 6.43922 0.811265
\(64\) 0 0
\(65\) −3.63107 −0.450378
\(66\) 0 0
\(67\) −1.79306 −0.219057 −0.109528 0.993984i \(-0.534934\pi\)
−0.109528 + 0.993984i \(0.534934\pi\)
\(68\) 0 0
\(69\) 9.93332 1.19583
\(70\) 0 0
\(71\) 3.02422 0.358909 0.179454 0.983766i \(-0.442567\pi\)
0.179454 + 0.983766i \(0.442567\pi\)
\(72\) 0 0
\(73\) 0.632807 0.0740645 0.0370322 0.999314i \(-0.488210\pi\)
0.0370322 + 0.999314i \(0.488210\pi\)
\(74\) 0 0
\(75\) 5.13585 0.593037
\(76\) 0 0
\(77\) −5.32944 −0.607346
\(78\) 0 0
\(79\) 14.3059 1.60953 0.804767 0.593591i \(-0.202290\pi\)
0.804767 + 0.593591i \(0.202290\pi\)
\(80\) 0 0
\(81\) 24.7433 2.74926
\(82\) 0 0
\(83\) −8.76332 −0.961899 −0.480950 0.876748i \(-0.659708\pi\)
−0.480950 + 0.876748i \(0.659708\pi\)
\(84\) 0 0
\(85\) 10.0934 1.09478
\(86\) 0 0
\(87\) 27.4795 2.94611
\(88\) 0 0
\(89\) 7.89557 0.836929 0.418464 0.908233i \(-0.362569\pi\)
0.418464 + 0.908233i \(0.362569\pi\)
\(90\) 0 0
\(91\) −1.21496 −0.127362
\(92\) 0 0
\(93\) 3.66882 0.380439
\(94\) 0 0
\(95\) −9.81616 −1.00712
\(96\) 0 0
\(97\) −9.86664 −1.00181 −0.500903 0.865504i \(-0.666999\pi\)
−0.500903 + 0.865504i \(0.666999\pi\)
\(98\) 0 0
\(99\) −46.5486 −4.67831
\(100\) 0 0
\(101\) 4.51832 0.449590 0.224795 0.974406i \(-0.427829\pi\)
0.224795 + 0.974406i \(0.427829\pi\)
\(102\) 0 0
\(103\) −16.4139 −1.61731 −0.808654 0.588284i \(-0.799804\pi\)
−0.808654 + 0.588284i \(0.799804\pi\)
\(104\) 0 0
\(105\) 7.13946 0.696739
\(106\) 0 0
\(107\) −16.0645 −1.55301 −0.776505 0.630111i \(-0.783009\pi\)
−0.776505 + 0.630111i \(0.783009\pi\)
\(108\) 0 0
\(109\) 5.61282 0.537611 0.268805 0.963195i \(-0.413371\pi\)
0.268805 + 0.963195i \(0.413371\pi\)
\(110\) 0 0
\(111\) 27.5406 2.61404
\(112\) 0 0
\(113\) 5.63170 0.529785 0.264893 0.964278i \(-0.414663\pi\)
0.264893 + 0.964278i \(0.414663\pi\)
\(114\) 0 0
\(115\) 7.86664 0.733568
\(116\) 0 0
\(117\) −10.6117 −0.981053
\(118\) 0 0
\(119\) 3.37725 0.309592
\(120\) 0 0
\(121\) 27.5262 2.50238
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −8.76332 −0.783815
\(126\) 0 0
\(127\) 1.69775 0.150651 0.0753254 0.997159i \(-0.476000\pi\)
0.0753254 + 0.997159i \(0.476000\pi\)
\(128\) 0 0
\(129\) 17.3301 1.52583
\(130\) 0 0
\(131\) −2.48057 −0.216728 −0.108364 0.994111i \(-0.534561\pi\)
−0.108364 + 0.994111i \(0.534561\pi\)
\(132\) 0 0
\(133\) −3.28449 −0.284801
\(134\) 0 0
\(135\) 37.4128 3.21998
\(136\) 0 0
\(137\) −2.76332 −0.236086 −0.118043 0.993008i \(-0.537662\pi\)
−0.118043 + 0.993008i \(0.537662\pi\)
\(138\) 0 0
\(139\) 8.99889 0.763276 0.381638 0.924312i \(-0.375360\pi\)
0.381638 + 0.924312i \(0.375360\pi\)
\(140\) 0 0
\(141\) 21.5650 1.81610
\(142\) 0 0
\(143\) 8.78282 0.734456
\(144\) 0 0
\(145\) 21.7622 1.80725
\(146\) 0 0
\(147\) −20.2931 −1.67375
\(148\) 0 0
\(149\) 8.48057 0.694755 0.347378 0.937725i \(-0.387072\pi\)
0.347378 + 0.937725i \(0.387072\pi\)
\(150\) 0 0
\(151\) 7.58971 0.617642 0.308821 0.951120i \(-0.400066\pi\)
0.308821 + 0.951120i \(0.400066\pi\)
\(152\) 0 0
\(153\) 29.4977 2.38475
\(154\) 0 0
\(155\) 2.90550 0.233375
\(156\) 0 0
\(157\) 6.78282 0.541328 0.270664 0.962674i \(-0.412757\pi\)
0.270664 + 0.962674i \(0.412757\pi\)
\(158\) 0 0
\(159\) −20.7828 −1.64818
\(160\) 0 0
\(161\) 2.63218 0.207445
\(162\) 0 0
\(163\) 15.0278 1.17707 0.588535 0.808472i \(-0.299705\pi\)
0.588535 + 0.808472i \(0.299705\pi\)
\(164\) 0 0
\(165\) −51.6106 −4.01788
\(166\) 0 0
\(167\) −5.79306 −0.448280 −0.224140 0.974557i \(-0.571957\pi\)
−0.224140 + 0.974557i \(0.571957\pi\)
\(168\) 0 0
\(169\) −10.9978 −0.845983
\(170\) 0 0
\(171\) −28.6875 −2.19379
\(172\) 0 0
\(173\) −5.43450 −0.413178 −0.206589 0.978428i \(-0.566236\pi\)
−0.206589 + 0.978428i \(0.566236\pi\)
\(174\) 0 0
\(175\) 1.36092 0.102876
\(176\) 0 0
\(177\) −9.93332 −0.746634
\(178\) 0 0
\(179\) −8.10251 −0.605610 −0.302805 0.953052i \(-0.597923\pi\)
−0.302805 + 0.953052i \(0.597923\pi\)
\(180\) 0 0
\(181\) 9.24389 0.687093 0.343546 0.939136i \(-0.388372\pi\)
0.343546 + 0.939136i \(0.388372\pi\)
\(182\) 0 0
\(183\) 24.0267 1.77611
\(184\) 0 0
\(185\) 21.8106 1.60355
\(186\) 0 0
\(187\) −24.4139 −1.78532
\(188\) 0 0
\(189\) 12.5183 0.910574
\(190\) 0 0
\(191\) −24.3381 −1.76104 −0.880521 0.474007i \(-0.842807\pi\)
−0.880521 + 0.474007i \(0.842807\pi\)
\(192\) 0 0
\(193\) −13.2634 −0.954720 −0.477360 0.878708i \(-0.658406\pi\)
−0.477360 + 0.878708i \(0.658406\pi\)
\(194\) 0 0
\(195\) −11.7657 −0.842558
\(196\) 0 0
\(197\) −18.1116 −1.29040 −0.645200 0.764013i \(-0.723226\pi\)
−0.645200 + 0.764013i \(0.723226\pi\)
\(198\) 0 0
\(199\) −19.7726 −1.40164 −0.700821 0.713337i \(-0.747183\pi\)
−0.700821 + 0.713337i \(0.747183\pi\)
\(200\) 0 0
\(201\) −5.81001 −0.409807
\(202\) 0 0
\(203\) 7.28164 0.511071
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 22.9901 1.59792
\(208\) 0 0
\(209\) 23.7433 1.64236
\(210\) 0 0
\(211\) −1.53924 −0.105966 −0.0529828 0.998595i \(-0.516873\pi\)
−0.0529828 + 0.998595i \(0.516873\pi\)
\(212\) 0 0
\(213\) 9.79933 0.671439
\(214\) 0 0
\(215\) 13.7245 0.936000
\(216\) 0 0
\(217\) 0.972180 0.0659959
\(218\) 0 0
\(219\) 2.05047 0.138558
\(220\) 0 0
\(221\) −5.56564 −0.374386
\(222\) 0 0
\(223\) −27.6578 −1.85210 −0.926051 0.377399i \(-0.876819\pi\)
−0.926051 + 0.377399i \(0.876819\pi\)
\(224\) 0 0
\(225\) 11.8866 0.792442
\(226\) 0 0
\(227\) 1.79635 0.119228 0.0596141 0.998221i \(-0.481013\pi\)
0.0596141 + 0.998221i \(0.481013\pi\)
\(228\) 0 0
\(229\) −21.6128 −1.42822 −0.714108 0.700036i \(-0.753167\pi\)
−0.714108 + 0.700036i \(0.753167\pi\)
\(230\) 0 0
\(231\) −17.2689 −1.13621
\(232\) 0 0
\(233\) 18.3312 1.20092 0.600458 0.799656i \(-0.294985\pi\)
0.600458 + 0.799656i \(0.294985\pi\)
\(234\) 0 0
\(235\) 17.0783 1.11407
\(236\) 0 0
\(237\) 46.3550 3.01108
\(238\) 0 0
\(239\) −11.9586 −0.773541 −0.386770 0.922176i \(-0.626409\pi\)
−0.386770 + 0.922176i \(0.626409\pi\)
\(240\) 0 0
\(241\) 7.41500 0.477642 0.238821 0.971064i \(-0.423239\pi\)
0.238821 + 0.971064i \(0.423239\pi\)
\(242\) 0 0
\(243\) 36.4370 2.33743
\(244\) 0 0
\(245\) −16.0710 −1.02674
\(246\) 0 0
\(247\) 5.41278 0.344407
\(248\) 0 0
\(249\) −28.3956 −1.79950
\(250\) 0 0
\(251\) −1.91507 −0.120878 −0.0604392 0.998172i \(-0.519250\pi\)
−0.0604392 + 0.998172i \(0.519250\pi\)
\(252\) 0 0
\(253\) −19.0278 −1.19627
\(254\) 0 0
\(255\) 32.7055 2.04810
\(256\) 0 0
\(257\) −2.93443 −0.183045 −0.0915224 0.995803i \(-0.529173\pi\)
−0.0915224 + 0.995803i \(0.529173\pi\)
\(258\) 0 0
\(259\) 7.29784 0.453466
\(260\) 0 0
\(261\) 63.5996 3.93671
\(262\) 0 0
\(263\) 3.10692 0.191581 0.0957905 0.995402i \(-0.469462\pi\)
0.0957905 + 0.995402i \(0.469462\pi\)
\(264\) 0 0
\(265\) −16.4588 −1.01106
\(266\) 0 0
\(267\) 25.5839 1.56571
\(268\) 0 0
\(269\) −10.2450 −0.624649 −0.312324 0.949976i \(-0.601108\pi\)
−0.312324 + 0.949976i \(0.601108\pi\)
\(270\) 0 0
\(271\) −11.1989 −0.680287 −0.340143 0.940374i \(-0.610476\pi\)
−0.340143 + 0.940374i \(0.610476\pi\)
\(272\) 0 0
\(273\) −3.93680 −0.238266
\(274\) 0 0
\(275\) −9.83801 −0.593254
\(276\) 0 0
\(277\) 2.52838 0.151915 0.0759577 0.997111i \(-0.475799\pi\)
0.0759577 + 0.997111i \(0.475799\pi\)
\(278\) 0 0
\(279\) 8.49125 0.508358
\(280\) 0 0
\(281\) 23.5656 1.40581 0.702904 0.711285i \(-0.251886\pi\)
0.702904 + 0.711285i \(0.251886\pi\)
\(282\) 0 0
\(283\) 8.99889 0.534928 0.267464 0.963568i \(-0.413814\pi\)
0.267464 + 0.963568i \(0.413814\pi\)
\(284\) 0 0
\(285\) −31.8072 −1.88409
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.52900 −0.0899415
\(290\) 0 0
\(291\) −31.9707 −1.87416
\(292\) 0 0
\(293\) −3.17721 −0.185614 −0.0928072 0.995684i \(-0.529584\pi\)
−0.0928072 + 0.995684i \(0.529584\pi\)
\(294\) 0 0
\(295\) −7.86664 −0.458013
\(296\) 0 0
\(297\) −90.4940 −5.25100
\(298\) 0 0
\(299\) −4.33778 −0.250860
\(300\) 0 0
\(301\) 4.59220 0.264690
\(302\) 0 0
\(303\) 14.6406 0.841083
\(304\) 0 0
\(305\) 19.0278 1.08953
\(306\) 0 0
\(307\) −17.5839 −1.00357 −0.501783 0.864994i \(-0.667322\pi\)
−0.501783 + 0.864994i \(0.667322\pi\)
\(308\) 0 0
\(309\) −53.1857 −3.02563
\(310\) 0 0
\(311\) −1.55749 −0.0883169 −0.0441584 0.999025i \(-0.514061\pi\)
−0.0441584 + 0.999025i \(0.514061\pi\)
\(312\) 0 0
\(313\) −20.5255 −1.16017 −0.580086 0.814556i \(-0.696981\pi\)
−0.580086 + 0.814556i \(0.696981\pi\)
\(314\) 0 0
\(315\) 16.5238 0.931013
\(316\) 0 0
\(317\) −26.1094 −1.46645 −0.733225 0.679986i \(-0.761986\pi\)
−0.733225 + 0.679986i \(0.761986\pi\)
\(318\) 0 0
\(319\) −52.6384 −2.94719
\(320\) 0 0
\(321\) −52.0534 −2.90534
\(322\) 0 0
\(323\) −15.0461 −0.837185
\(324\) 0 0
\(325\) −2.24277 −0.124407
\(326\) 0 0
\(327\) 18.1871 1.00575
\(328\) 0 0
\(329\) 5.71440 0.315045
\(330\) 0 0
\(331\) −3.68862 −0.202745 −0.101373 0.994849i \(-0.532323\pi\)
−0.101373 + 0.994849i \(0.532323\pi\)
\(332\) 0 0
\(333\) 63.7411 3.49299
\(334\) 0 0
\(335\) −4.60121 −0.251391
\(336\) 0 0
\(337\) 30.9240 1.68454 0.842269 0.539057i \(-0.181219\pi\)
0.842269 + 0.539057i \(0.181219\pi\)
\(338\) 0 0
\(339\) 18.2483 0.991111
\(340\) 0 0
\(341\) −7.02782 −0.380578
\(342\) 0 0
\(343\) −11.3878 −0.614881
\(344\) 0 0
\(345\) 25.4901 1.37234
\(346\) 0 0
\(347\) −27.1264 −1.45622 −0.728111 0.685459i \(-0.759602\pi\)
−0.728111 + 0.685459i \(0.759602\pi\)
\(348\) 0 0
\(349\) −9.30051 −0.497845 −0.248922 0.968523i \(-0.580076\pi\)
−0.248922 + 0.968523i \(0.580076\pi\)
\(350\) 0 0
\(351\) −20.6300 −1.10115
\(352\) 0 0
\(353\) 13.3961 0.713004 0.356502 0.934295i \(-0.383969\pi\)
0.356502 + 0.934295i \(0.383969\pi\)
\(354\) 0 0
\(355\) 7.76052 0.411886
\(356\) 0 0
\(357\) 10.9432 0.579178
\(358\) 0 0
\(359\) −28.9611 −1.52851 −0.764255 0.644914i \(-0.776893\pi\)
−0.764255 + 0.644914i \(0.776893\pi\)
\(360\) 0 0
\(361\) −4.36719 −0.229852
\(362\) 0 0
\(363\) 89.1926 4.68140
\(364\) 0 0
\(365\) 1.62386 0.0849968
\(366\) 0 0
\(367\) −31.4795 −1.64321 −0.821607 0.570054i \(-0.806922\pi\)
−0.821607 + 0.570054i \(0.806922\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.50713 −0.285916
\(372\) 0 0
\(373\) −15.5266 −0.803939 −0.401969 0.915653i \(-0.631674\pi\)
−0.401969 + 0.915653i \(0.631674\pi\)
\(374\) 0 0
\(375\) −28.3956 −1.46634
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 1.97107 0.101247 0.0506235 0.998718i \(-0.483879\pi\)
0.0506235 + 0.998718i \(0.483879\pi\)
\(380\) 0 0
\(381\) 5.50119 0.281834
\(382\) 0 0
\(383\) 12.8404 0.656113 0.328056 0.944658i \(-0.393606\pi\)
0.328056 + 0.944658i \(0.393606\pi\)
\(384\) 0 0
\(385\) −13.6760 −0.696995
\(386\) 0 0
\(387\) 40.1094 2.03888
\(388\) 0 0
\(389\) 1.03886 0.0526724 0.0263362 0.999653i \(-0.491616\pi\)
0.0263362 + 0.999653i \(0.491616\pi\)
\(390\) 0 0
\(391\) 12.0579 0.609792
\(392\) 0 0
\(393\) −8.03775 −0.405451
\(394\) 0 0
\(395\) 36.7106 1.84711
\(396\) 0 0
\(397\) 24.6406 1.23668 0.618339 0.785911i \(-0.287806\pi\)
0.618339 + 0.785911i \(0.287806\pi\)
\(398\) 0 0
\(399\) −10.6427 −0.532800
\(400\) 0 0
\(401\) 0.397236 0.0198370 0.00991851 0.999951i \(-0.496843\pi\)
0.00991851 + 0.999951i \(0.496843\pi\)
\(402\) 0 0
\(403\) −1.60213 −0.0798080
\(404\) 0 0
\(405\) 63.4945 3.15507
\(406\) 0 0
\(407\) −52.7556 −2.61500
\(408\) 0 0
\(409\) −2.76395 −0.136668 −0.0683342 0.997662i \(-0.521768\pi\)
−0.0683342 + 0.997662i \(0.521768\pi\)
\(410\) 0 0
\(411\) −8.95393 −0.441665
\(412\) 0 0
\(413\) −2.63218 −0.129521
\(414\) 0 0
\(415\) −22.4878 −1.10388
\(416\) 0 0
\(417\) 29.1590 1.42792
\(418\) 0 0
\(419\) 1.48168 0.0723848 0.0361924 0.999345i \(-0.488477\pi\)
0.0361924 + 0.999345i \(0.488477\pi\)
\(420\) 0 0
\(421\) 18.9039 0.921319 0.460659 0.887577i \(-0.347613\pi\)
0.460659 + 0.887577i \(0.347613\pi\)
\(422\) 0 0
\(423\) 49.9109 2.42675
\(424\) 0 0
\(425\) 6.23431 0.302409
\(426\) 0 0
\(427\) 6.36671 0.308107
\(428\) 0 0
\(429\) 28.4588 1.37401
\(430\) 0 0
\(431\) −7.28164 −0.350744 −0.175372 0.984502i \(-0.556113\pi\)
−0.175372 + 0.984502i \(0.556113\pi\)
\(432\) 0 0
\(433\) −0.397865 −0.0191202 −0.00956009 0.999954i \(-0.503043\pi\)
−0.00956009 + 0.999954i \(0.503043\pi\)
\(434\) 0 0
\(435\) 70.5157 3.38097
\(436\) 0 0
\(437\) −11.7267 −0.560963
\(438\) 0 0
\(439\) −33.9798 −1.62177 −0.810885 0.585206i \(-0.801014\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(440\) 0 0
\(441\) −46.9672 −2.23653
\(442\) 0 0
\(443\) 16.2050 0.769924 0.384962 0.922932i \(-0.374215\pi\)
0.384962 + 0.922932i \(0.374215\pi\)
\(444\) 0 0
\(445\) 20.2610 0.960464
\(446\) 0 0
\(447\) 27.4795 1.29973
\(448\) 0 0
\(449\) −23.1118 −1.09071 −0.545356 0.838204i \(-0.683606\pi\)
−0.545356 + 0.838204i \(0.683606\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 24.5928 1.15547
\(454\) 0 0
\(455\) −3.11773 −0.146161
\(456\) 0 0
\(457\) −35.6301 −1.66671 −0.833353 0.552741i \(-0.813582\pi\)
−0.833353 + 0.552741i \(0.813582\pi\)
\(458\) 0 0
\(459\) 57.3457 2.67667
\(460\) 0 0
\(461\) 6.63058 0.308817 0.154409 0.988007i \(-0.450653\pi\)
0.154409 + 0.988007i \(0.450653\pi\)
\(462\) 0 0
\(463\) 21.0220 0.976975 0.488487 0.872571i \(-0.337549\pi\)
0.488487 + 0.872571i \(0.337549\pi\)
\(464\) 0 0
\(465\) 9.41464 0.436594
\(466\) 0 0
\(467\) 15.8933 0.735456 0.367728 0.929933i \(-0.380136\pi\)
0.367728 + 0.929933i \(0.380136\pi\)
\(468\) 0 0
\(469\) −1.53956 −0.0710905
\(470\) 0 0
\(471\) 21.9783 1.01271
\(472\) 0 0
\(473\) −33.1967 −1.52639
\(474\) 0 0
\(475\) −6.06308 −0.278193
\(476\) 0 0
\(477\) −48.1006 −2.20237
\(478\) 0 0
\(479\) 34.8564 1.59263 0.796315 0.604882i \(-0.206780\pi\)
0.796315 + 0.604882i \(0.206780\pi\)
\(480\) 0 0
\(481\) −12.0267 −0.548371
\(482\) 0 0
\(483\) 8.52900 0.388083
\(484\) 0 0
\(485\) −25.3190 −1.14968
\(486\) 0 0
\(487\) −9.44393 −0.427945 −0.213973 0.976840i \(-0.568640\pi\)
−0.213973 + 0.976840i \(0.568640\pi\)
\(488\) 0 0
\(489\) 48.6944 2.20204
\(490\) 0 0
\(491\) 33.1577 1.49639 0.748193 0.663481i \(-0.230922\pi\)
0.748193 + 0.663481i \(0.230922\pi\)
\(492\) 0 0
\(493\) 33.3568 1.50231
\(494\) 0 0
\(495\) −119.450 −5.36886
\(496\) 0 0
\(497\) 2.59667 0.116477
\(498\) 0 0
\(499\) −12.7597 −0.571203 −0.285602 0.958348i \(-0.592193\pi\)
−0.285602 + 0.958348i \(0.592193\pi\)
\(500\) 0 0
\(501\) −18.7712 −0.838633
\(502\) 0 0
\(503\) 17.0025 0.758104 0.379052 0.925375i \(-0.376250\pi\)
0.379052 + 0.925375i \(0.376250\pi\)
\(504\) 0 0
\(505\) 11.5946 0.515952
\(506\) 0 0
\(507\) −35.6359 −1.58265
\(508\) 0 0
\(509\) 2.50950 0.111232 0.0556158 0.998452i \(-0.482288\pi\)
0.0556158 + 0.998452i \(0.482288\pi\)
\(510\) 0 0
\(511\) 0.543345 0.0240361
\(512\) 0 0
\(513\) −55.7707 −2.46234
\(514\) 0 0
\(515\) −42.1201 −1.85603
\(516\) 0 0
\(517\) −41.3090 −1.81677
\(518\) 0 0
\(519\) −17.6093 −0.772964
\(520\) 0 0
\(521\) −13.1300 −0.575237 −0.287618 0.957745i \(-0.592864\pi\)
−0.287618 + 0.957745i \(0.592864\pi\)
\(522\) 0 0
\(523\) 23.8301 1.04202 0.521010 0.853551i \(-0.325556\pi\)
0.521010 + 0.853551i \(0.325556\pi\)
\(524\) 0 0
\(525\) 4.40978 0.192458
\(526\) 0 0
\(527\) 4.45350 0.193998
\(528\) 0 0
\(529\) −13.6023 −0.591404
\(530\) 0 0
\(531\) −22.9901 −0.997684
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −41.2234 −1.78224
\(536\) 0 0
\(537\) −26.2544 −1.13296
\(538\) 0 0
\(539\) 38.8726 1.67436
\(540\) 0 0
\(541\) 14.6995 0.631980 0.315990 0.948762i \(-0.397663\pi\)
0.315990 + 0.948762i \(0.397663\pi\)
\(542\) 0 0
\(543\) 29.9528 1.28540
\(544\) 0 0
\(545\) 14.4032 0.616965
\(546\) 0 0
\(547\) −20.2209 −0.864584 −0.432292 0.901734i \(-0.642295\pi\)
−0.432292 + 0.901734i \(0.642295\pi\)
\(548\) 0 0
\(549\) 55.6084 2.37331
\(550\) 0 0
\(551\) −32.4406 −1.38202
\(552\) 0 0
\(553\) 12.2834 0.522342
\(554\) 0 0
\(555\) 70.6727 2.99989
\(556\) 0 0
\(557\) 40.6784 1.72360 0.861799 0.507249i \(-0.169338\pi\)
0.861799 + 0.507249i \(0.169338\pi\)
\(558\) 0 0
\(559\) −7.56787 −0.320087
\(560\) 0 0
\(561\) −79.1079 −3.33994
\(562\) 0 0
\(563\) 19.4486 0.819661 0.409831 0.912162i \(-0.365588\pi\)
0.409831 + 0.912162i \(0.365588\pi\)
\(564\) 0 0
\(565\) 14.4516 0.607985
\(566\) 0 0
\(567\) 21.2453 0.892217
\(568\) 0 0
\(569\) −31.1850 −1.30734 −0.653672 0.756778i \(-0.726773\pi\)
−0.653672 + 0.756778i \(0.726773\pi\)
\(570\) 0 0
\(571\) 0.190921 0.00798981 0.00399491 0.999992i \(-0.498728\pi\)
0.00399491 + 0.999992i \(0.498728\pi\)
\(572\) 0 0
\(573\) −78.8623 −3.29452
\(574\) 0 0
\(575\) 4.85893 0.202631
\(576\) 0 0
\(577\) 29.0278 1.20844 0.604222 0.796816i \(-0.293484\pi\)
0.604222 + 0.796816i \(0.293484\pi\)
\(578\) 0 0
\(579\) −42.9772 −1.78607
\(580\) 0 0
\(581\) −7.52441 −0.312165
\(582\) 0 0
\(583\) 39.8106 1.64879
\(584\) 0 0
\(585\) −27.2310 −1.12586
\(586\) 0 0
\(587\) 11.5735 0.477690 0.238845 0.971058i \(-0.423231\pi\)
0.238845 + 0.971058i \(0.423231\pi\)
\(588\) 0 0
\(589\) −4.33118 −0.178463
\(590\) 0 0
\(591\) −58.6869 −2.41405
\(592\) 0 0
\(593\) −34.5557 −1.41903 −0.709517 0.704689i \(-0.751087\pi\)
−0.709517 + 0.704689i \(0.751087\pi\)
\(594\) 0 0
\(595\) 8.66645 0.355290
\(596\) 0 0
\(597\) −64.0688 −2.62216
\(598\) 0 0
\(599\) 26.9967 1.10305 0.551527 0.834157i \(-0.314045\pi\)
0.551527 + 0.834157i \(0.314045\pi\)
\(600\) 0 0
\(601\) −47.8534 −1.95198 −0.975990 0.217816i \(-0.930107\pi\)
−0.975990 + 0.217816i \(0.930107\pi\)
\(602\) 0 0
\(603\) −13.4469 −0.547601
\(604\) 0 0
\(605\) 70.6356 2.87174
\(606\) 0 0
\(607\) 12.3761 0.502332 0.251166 0.967944i \(-0.419186\pi\)
0.251166 + 0.967944i \(0.419186\pi\)
\(608\) 0 0
\(609\) 23.5946 0.956100
\(610\) 0 0
\(611\) −9.41722 −0.380980
\(612\) 0 0
\(613\) 12.9061 0.521274 0.260637 0.965437i \(-0.416067\pi\)
0.260637 + 0.965437i \(0.416067\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −40.5268 −1.63155 −0.815773 0.578372i \(-0.803688\pi\)
−0.815773 + 0.578372i \(0.803688\pi\)
\(618\) 0 0
\(619\) −2.65889 −0.106870 −0.0534348 0.998571i \(-0.517017\pi\)
−0.0534348 + 0.998571i \(0.517017\pi\)
\(620\) 0 0
\(621\) 44.6944 1.79353
\(622\) 0 0
\(623\) 6.77934 0.271609
\(624\) 0 0
\(625\) −30.4128 −1.21651
\(626\) 0 0
\(627\) 76.9351 3.07249
\(628\) 0 0
\(629\) 33.4310 1.33298
\(630\) 0 0
\(631\) −0.139958 −0.00557165 −0.00278583 0.999996i \(-0.500887\pi\)
−0.00278583 + 0.999996i \(0.500887\pi\)
\(632\) 0 0
\(633\) −4.98757 −0.198238
\(634\) 0 0
\(635\) 4.35663 0.172888
\(636\) 0 0
\(637\) 8.86180 0.351117
\(638\) 0 0
\(639\) 22.6800 0.897205
\(640\) 0 0
\(641\) −9.39675 −0.371149 −0.185575 0.982630i \(-0.559415\pi\)
−0.185575 + 0.982630i \(0.559415\pi\)
\(642\) 0 0
\(643\) −6.02863 −0.237746 −0.118873 0.992909i \(-0.537928\pi\)
−0.118873 + 0.992909i \(0.537928\pi\)
\(644\) 0 0
\(645\) 44.4711 1.75105
\(646\) 0 0
\(647\) 17.3861 0.683517 0.341758 0.939788i \(-0.388978\pi\)
0.341758 + 0.939788i \(0.388978\pi\)
\(648\) 0 0
\(649\) 19.0278 0.746907
\(650\) 0 0
\(651\) 3.15014 0.123464
\(652\) 0 0
\(653\) 8.02832 0.314173 0.157086 0.987585i \(-0.449790\pi\)
0.157086 + 0.987585i \(0.449790\pi\)
\(654\) 0 0
\(655\) −6.36545 −0.248719
\(656\) 0 0
\(657\) 4.74570 0.185147
\(658\) 0 0
\(659\) 33.8809 1.31981 0.659907 0.751348i \(-0.270596\pi\)
0.659907 + 0.751348i \(0.270596\pi\)
\(660\) 0 0
\(661\) 34.3840 1.33738 0.668691 0.743541i \(-0.266855\pi\)
0.668691 + 0.743541i \(0.266855\pi\)
\(662\) 0 0
\(663\) −18.0343 −0.700393
\(664\) 0 0
\(665\) −8.42841 −0.326840
\(666\) 0 0
\(667\) 25.9978 1.00664
\(668\) 0 0
\(669\) −89.6191 −3.46487
\(670\) 0 0
\(671\) −46.0245 −1.77676
\(672\) 0 0
\(673\) −10.3689 −0.399693 −0.199847 0.979827i \(-0.564044\pi\)
−0.199847 + 0.979827i \(0.564044\pi\)
\(674\) 0 0
\(675\) 23.1085 0.889446
\(676\) 0 0
\(677\) 28.0616 1.07850 0.539248 0.842147i \(-0.318709\pi\)
0.539248 + 0.842147i \(0.318709\pi\)
\(678\) 0 0
\(679\) −8.47175 −0.325116
\(680\) 0 0
\(681\) 5.82070 0.223050
\(682\) 0 0
\(683\) −12.2209 −0.467621 −0.233810 0.972282i \(-0.575119\pi\)
−0.233810 + 0.972282i \(0.575119\pi\)
\(684\) 0 0
\(685\) −7.09102 −0.270934
\(686\) 0 0
\(687\) −70.0317 −2.67188
\(688\) 0 0
\(689\) 9.07565 0.345755
\(690\) 0 0
\(691\) −46.1797 −1.75676 −0.878379 0.477964i \(-0.841375\pi\)
−0.878379 + 0.477964i \(0.841375\pi\)
\(692\) 0 0
\(693\) −39.9679 −1.51825
\(694\) 0 0
\(695\) 23.0923 0.875940
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 59.3983 2.24665
\(700\) 0 0
\(701\) −46.8161 −1.76822 −0.884110 0.467279i \(-0.845234\pi\)
−0.884110 + 0.467279i \(0.845234\pi\)
\(702\) 0 0
\(703\) −32.5128 −1.22624
\(704\) 0 0
\(705\) 55.3385 2.08417
\(706\) 0 0
\(707\) 3.87955 0.145905
\(708\) 0 0
\(709\) −42.0840 −1.58050 −0.790248 0.612787i \(-0.790048\pi\)
−0.790248 + 0.612787i \(0.790048\pi\)
\(710\) 0 0
\(711\) 107.286 4.02354
\(712\) 0 0
\(713\) 3.47100 0.129990
\(714\) 0 0
\(715\) 22.5378 0.842867
\(716\) 0 0
\(717\) −38.7494 −1.44712
\(718\) 0 0
\(719\) −1.77685 −0.0662653 −0.0331327 0.999451i \(-0.510548\pi\)
−0.0331327 + 0.999451i \(0.510548\pi\)
\(720\) 0 0
\(721\) −14.0934 −0.524865
\(722\) 0 0
\(723\) 24.0267 0.893563
\(724\) 0 0
\(725\) 13.4417 0.499212
\(726\) 0 0
\(727\) 17.4346 0.646614 0.323307 0.946294i \(-0.395205\pi\)
0.323307 + 0.946294i \(0.395205\pi\)
\(728\) 0 0
\(729\) 43.8362 1.62356
\(730\) 0 0
\(731\) 21.0366 0.778068
\(732\) 0 0
\(733\) −8.31827 −0.307242 −0.153621 0.988130i \(-0.549094\pi\)
−0.153621 + 0.988130i \(0.549094\pi\)
\(734\) 0 0
\(735\) −52.0747 −1.92080
\(736\) 0 0
\(737\) 11.1294 0.409957
\(738\) 0 0
\(739\) −0.139958 −0.00514845 −0.00257422 0.999997i \(-0.500819\pi\)
−0.00257422 + 0.999997i \(0.500819\pi\)
\(740\) 0 0
\(741\) 17.5389 0.644309
\(742\) 0 0
\(743\) −3.92389 −0.143954 −0.0719768 0.997406i \(-0.522931\pi\)
−0.0719768 + 0.997406i \(0.522931\pi\)
\(744\) 0 0
\(745\) 21.7622 0.797306
\(746\) 0 0
\(747\) −65.7200 −2.40457
\(748\) 0 0
\(749\) −13.7934 −0.503998
\(750\) 0 0
\(751\) −38.2786 −1.39681 −0.698403 0.715705i \(-0.746106\pi\)
−0.698403 + 0.715705i \(0.746106\pi\)
\(752\) 0 0
\(753\) −6.20538 −0.226137
\(754\) 0 0
\(755\) 19.4762 0.708810
\(756\) 0 0
\(757\) 31.0640 1.12904 0.564519 0.825420i \(-0.309062\pi\)
0.564519 + 0.825420i \(0.309062\pi\)
\(758\) 0 0
\(759\) −61.6556 −2.23795
\(760\) 0 0
\(761\) −2.83945 −0.102930 −0.0514649 0.998675i \(-0.516389\pi\)
−0.0514649 + 0.998675i \(0.516389\pi\)
\(762\) 0 0
\(763\) 4.81931 0.174471
\(764\) 0 0
\(765\) 75.6948 2.73675
\(766\) 0 0
\(767\) 4.33778 0.156628
\(768\) 0 0
\(769\) 38.7888 1.39876 0.699379 0.714751i \(-0.253460\pi\)
0.699379 + 0.714751i \(0.253460\pi\)
\(770\) 0 0
\(771\) −9.50839 −0.342436
\(772\) 0 0
\(773\) −25.7150 −0.924905 −0.462453 0.886644i \(-0.653030\pi\)
−0.462453 + 0.886644i \(0.653030\pi\)
\(774\) 0 0
\(775\) 1.79462 0.0644646
\(776\) 0 0
\(777\) 23.6471 0.848335
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −18.7712 −0.671685
\(782\) 0 0
\(783\) 123.642 4.41861
\(784\) 0 0
\(785\) 17.4056 0.621232
\(786\) 0 0
\(787\) 41.7817 1.48936 0.744679 0.667423i \(-0.232603\pi\)
0.744679 + 0.667423i \(0.232603\pi\)
\(788\) 0 0
\(789\) 10.0673 0.358406
\(790\) 0 0
\(791\) 4.83552 0.171931
\(792\) 0 0
\(793\) −10.4922 −0.372590
\(794\) 0 0
\(795\) −53.3313 −1.89147
\(796\) 0 0
\(797\) 36.2818 1.28517 0.642583 0.766216i \(-0.277863\pi\)
0.642583 + 0.766216i \(0.277863\pi\)
\(798\) 0 0
\(799\) 26.1774 0.926088
\(800\) 0 0
\(801\) 59.2124 2.09217
\(802\) 0 0
\(803\) −3.92780 −0.138609
\(804\) 0 0
\(805\) 6.75450 0.238065
\(806\) 0 0
\(807\) −33.1967 −1.16858
\(808\) 0 0
\(809\) 37.8556 1.33093 0.665466 0.746428i \(-0.268233\pi\)
0.665466 + 0.746428i \(0.268233\pi\)
\(810\) 0 0
\(811\) −15.5751 −0.546915 −0.273457 0.961884i \(-0.588167\pi\)
−0.273457 + 0.961884i \(0.588167\pi\)
\(812\) 0 0
\(813\) −36.2877 −1.27267
\(814\) 0 0
\(815\) 38.5633 1.35081
\(816\) 0 0
\(817\) −20.4588 −0.715764
\(818\) 0 0
\(819\) −9.11149 −0.318381
\(820\) 0 0
\(821\) −9.42283 −0.328859 −0.164430 0.986389i \(-0.552578\pi\)
−0.164430 + 0.986389i \(0.552578\pi\)
\(822\) 0 0
\(823\) 13.9636 0.486741 0.243371 0.969933i \(-0.421747\pi\)
0.243371 + 0.969933i \(0.421747\pi\)
\(824\) 0 0
\(825\) −31.8780 −1.10985
\(826\) 0 0
\(827\) −19.6886 −0.684641 −0.342320 0.939583i \(-0.611213\pi\)
−0.342320 + 0.939583i \(0.611213\pi\)
\(828\) 0 0
\(829\) 52.9595 1.83936 0.919681 0.392668i \(-0.128448\pi\)
0.919681 + 0.392668i \(0.128448\pi\)
\(830\) 0 0
\(831\) 8.19266 0.284200
\(832\) 0 0
\(833\) −24.6334 −0.853498
\(834\) 0 0
\(835\) −14.8657 −0.514449
\(836\) 0 0
\(837\) 16.5076 0.570587
\(838\) 0 0
\(839\) 17.1392 0.591709 0.295855 0.955233i \(-0.404396\pi\)
0.295855 + 0.955233i \(0.404396\pi\)
\(840\) 0 0
\(841\) 42.9201 1.48000
\(842\) 0 0
\(843\) 76.3594 2.62996
\(844\) 0 0
\(845\) −28.2217 −0.970855
\(846\) 0 0
\(847\) 23.6347 0.812097
\(848\) 0 0
\(849\) 29.1590 1.00073
\(850\) 0 0
\(851\) 26.0556 0.893176
\(852\) 0 0
\(853\) −24.8256 −0.850011 −0.425005 0.905191i \(-0.639728\pi\)
−0.425005 + 0.905191i \(0.639728\pi\)
\(854\) 0 0
\(855\) −73.6158 −2.51761
\(856\) 0 0
\(857\) 25.6584 0.876474 0.438237 0.898859i \(-0.355603\pi\)
0.438237 + 0.898859i \(0.355603\pi\)
\(858\) 0 0
\(859\) 11.9818 0.408812 0.204406 0.978886i \(-0.434474\pi\)
0.204406 + 0.978886i \(0.434474\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −33.9138 −1.15444 −0.577220 0.816589i \(-0.695862\pi\)
−0.577220 + 0.816589i \(0.695862\pi\)
\(864\) 0 0
\(865\) −13.9456 −0.474165
\(866\) 0 0
\(867\) −4.95441 −0.168261
\(868\) 0 0
\(869\) −88.7956 −3.01219
\(870\) 0 0
\(871\) 2.53717 0.0859688
\(872\) 0 0
\(873\) −73.9943 −2.50433
\(874\) 0 0
\(875\) −7.52441 −0.254372
\(876\) 0 0
\(877\) −14.1871 −0.479066 −0.239533 0.970888i \(-0.576994\pi\)
−0.239533 + 0.970888i \(0.576994\pi\)
\(878\) 0 0
\(879\) −10.2950 −0.347243
\(880\) 0 0
\(881\) 42.3201 1.42580 0.712901 0.701265i \(-0.247381\pi\)
0.712901 + 0.701265i \(0.247381\pi\)
\(882\) 0 0
\(883\) −51.7258 −1.74071 −0.870356 0.492422i \(-0.836112\pi\)
−0.870356 + 0.492422i \(0.836112\pi\)
\(884\) 0 0
\(885\) −25.4901 −0.856842
\(886\) 0 0
\(887\) −30.1270 −1.01157 −0.505783 0.862661i \(-0.668796\pi\)
−0.505783 + 0.862661i \(0.668796\pi\)
\(888\) 0 0
\(889\) 1.45773 0.0488907
\(890\) 0 0
\(891\) −153.580 −5.14514
\(892\) 0 0
\(893\) −25.4584 −0.851932
\(894\) 0 0
\(895\) −20.7921 −0.695002
\(896\) 0 0
\(897\) −14.0556 −0.469304
\(898\) 0 0
\(899\) 9.60213 0.320249
\(900\) 0 0
\(901\) −25.2279 −0.840462
\(902\) 0 0
\(903\) 14.8800 0.495177
\(904\) 0 0
\(905\) 23.7210 0.788512
\(906\) 0 0
\(907\) −4.24848 −0.141068 −0.0705342 0.997509i \(-0.522470\pi\)
−0.0705342 + 0.997509i \(0.522470\pi\)
\(908\) 0 0
\(909\) 33.8849 1.12389
\(910\) 0 0
\(911\) −49.1395 −1.62806 −0.814031 0.580821i \(-0.802732\pi\)
−0.814031 + 0.580821i \(0.802732\pi\)
\(912\) 0 0
\(913\) 54.3934 1.80016
\(914\) 0 0
\(915\) 61.6556 2.03827
\(916\) 0 0
\(917\) −2.12988 −0.0703349
\(918\) 0 0
\(919\) −6.59412 −0.217520 −0.108760 0.994068i \(-0.534688\pi\)
−0.108760 + 0.994068i \(0.534688\pi\)
\(920\) 0 0
\(921\) −56.9768 −1.87745
\(922\) 0 0
\(923\) −4.27927 −0.140854
\(924\) 0 0
\(925\) 13.4716 0.442944
\(926\) 0 0
\(927\) −123.095 −4.04297
\(928\) 0 0
\(929\) −54.3833 −1.78426 −0.892130 0.451779i \(-0.850789\pi\)
−0.892130 + 0.451779i \(0.850789\pi\)
\(930\) 0 0
\(931\) 23.9568 0.785154
\(932\) 0 0
\(933\) −5.04669 −0.165221
\(934\) 0 0
\(935\) −62.6491 −2.04884
\(936\) 0 0
\(937\) 43.0888 1.40765 0.703825 0.710374i \(-0.251474\pi\)
0.703825 + 0.710374i \(0.251474\pi\)
\(938\) 0 0
\(939\) −66.5085 −2.17042
\(940\) 0 0
\(941\) 12.6966 0.413899 0.206949 0.978352i \(-0.433646\pi\)
0.206949 + 0.978352i \(0.433646\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 32.1236 1.04498
\(946\) 0 0
\(947\) 29.6867 0.964688 0.482344 0.875982i \(-0.339785\pi\)
0.482344 + 0.875982i \(0.339785\pi\)
\(948\) 0 0
\(949\) −0.895422 −0.0290666
\(950\) 0 0
\(951\) −84.6019 −2.74341
\(952\) 0 0
\(953\) 7.62947 0.247143 0.123571 0.992336i \(-0.460565\pi\)
0.123571 + 0.992336i \(0.460565\pi\)
\(954\) 0 0
\(955\) −62.4546 −2.02098
\(956\) 0 0
\(957\) −170.563 −5.51353
\(958\) 0 0
\(959\) −2.37266 −0.0766171
\(960\) 0 0
\(961\) −29.7180 −0.958645
\(962\) 0 0
\(963\) −120.475 −3.88224
\(964\) 0 0
\(965\) −34.0355 −1.09564
\(966\) 0 0
\(967\) 37.0187 1.19044 0.595221 0.803562i \(-0.297065\pi\)
0.595221 + 0.803562i \(0.297065\pi\)
\(968\) 0 0
\(969\) −48.7535 −1.56619
\(970\) 0 0
\(971\) 17.2425 0.553338 0.276669 0.960965i \(-0.410769\pi\)
0.276669 + 0.960965i \(0.410769\pi\)
\(972\) 0 0
\(973\) 7.72668 0.247706
\(974\) 0 0
\(975\) −7.26723 −0.232738
\(976\) 0 0
\(977\) 31.6112 1.01133 0.505666 0.862729i \(-0.331247\pi\)
0.505666 + 0.862729i \(0.331247\pi\)
\(978\) 0 0
\(979\) −49.0074 −1.56628
\(980\) 0 0
\(981\) 42.0930 1.34393
\(982\) 0 0
\(983\) 51.8543 1.65390 0.826948 0.562278i \(-0.190075\pi\)
0.826948 + 0.562278i \(0.190075\pi\)
\(984\) 0 0
\(985\) −46.4767 −1.48087
\(986\) 0 0
\(987\) 18.5163 0.589380
\(988\) 0 0
\(989\) 16.3956 0.521351
\(990\) 0 0
\(991\) 37.7209 1.19824 0.599121 0.800658i \(-0.295517\pi\)
0.599121 + 0.800658i \(0.295517\pi\)
\(992\) 0 0
\(993\) −11.9522 −0.379291
\(994\) 0 0
\(995\) −50.7389 −1.60853
\(996\) 0 0
\(997\) 19.1882 0.607698 0.303849 0.952720i \(-0.401728\pi\)
0.303849 + 0.952720i \(0.401728\pi\)
\(998\) 0 0
\(999\) 123.917 3.92058
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 6724.2.a.c.1.4 4
41.40 even 2 164.2.a.a.1.1 4
123.122 odd 2 1476.2.a.g.1.2 4
164.163 odd 2 656.2.a.i.1.4 4
205.122 odd 4 4100.2.d.c.1149.8 8
205.163 odd 4 4100.2.d.c.1149.1 8
205.204 even 2 4100.2.a.c.1.4 4
287.286 odd 2 8036.2.a.i.1.4 4
328.163 odd 2 2624.2.a.y.1.1 4
328.245 even 2 2624.2.a.v.1.4 4
492.491 even 2 5904.2.a.bp.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.2.a.a.1.1 4 41.40 even 2
656.2.a.i.1.4 4 164.163 odd 2
1476.2.a.g.1.2 4 123.122 odd 2
2624.2.a.v.1.4 4 328.245 even 2
2624.2.a.y.1.1 4 328.163 odd 2
4100.2.a.c.1.4 4 205.204 even 2
4100.2.d.c.1149.1 8 205.163 odd 4
4100.2.d.c.1149.8 8 205.122 odd 4
5904.2.a.bp.1.2 4 492.491 even 2
6724.2.a.c.1.4 4 1.1 even 1 trivial
8036.2.a.i.1.4 4 287.286 odd 2