Properties

Label 6400.2.a.cp.1.3
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.1042572936\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 3200)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.28825\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.23607 q^{3} -2.82843 q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+2.23607 q^{3} -2.82843 q^{7} +2.00000 q^{9} -2.23607 q^{11} +6.32456 q^{13} -5.00000 q^{17} -2.23607 q^{19} -6.32456 q^{21} +5.65685 q^{23} -2.23607 q^{27} -6.32456 q^{29} -5.00000 q^{33} -6.32456 q^{37} +14.1421 q^{39} -3.00000 q^{41} +8.94427 q^{43} +2.82843 q^{47} +1.00000 q^{49} -11.1803 q^{51} -12.6491 q^{53} -5.00000 q^{57} -8.94427 q^{59} +6.32456 q^{61} -5.65685 q^{63} -11.1803 q^{67} +12.6491 q^{69} +14.1421 q^{71} -15.0000 q^{73} +6.32456 q^{77} -14.1421 q^{79} -11.0000 q^{81} +6.70820 q^{83} -14.1421 q^{87} -1.00000 q^{89} -17.8885 q^{91} +10.0000 q^{97} -4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{9} - 20 q^{17} - 20 q^{33} - 12 q^{41} + 4 q^{49} - 20 q^{57} - 60 q^{73} - 44 q^{81} - 4 q^{89} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.23607 1.29099 0.645497 0.763763i \(-0.276650\pi\)
0.645497 + 0.763763i \(0.276650\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −2.23607 −0.674200 −0.337100 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(12\) 0 0
\(13\) 6.32456 1.75412 0.877058 0.480384i \(-0.159503\pi\)
0.877058 + 0.480384i \(0.159503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) −2.23607 −0.512989 −0.256495 0.966546i \(-0.582568\pi\)
−0.256495 + 0.966546i \(0.582568\pi\)
\(20\) 0 0
\(21\) −6.32456 −1.38013
\(22\) 0 0
\(23\) 5.65685 1.17954 0.589768 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.23607 −0.430331
\(28\) 0 0
\(29\) −6.32456 −1.17444 −0.587220 0.809427i \(-0.699778\pi\)
−0.587220 + 0.809427i \(0.699778\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.32456 −1.03975 −0.519875 0.854242i \(-0.674022\pi\)
−0.519875 + 0.854242i \(0.674022\pi\)
\(38\) 0 0
\(39\) 14.1421 2.26455
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) 8.94427 1.36399 0.681994 0.731357i \(-0.261113\pi\)
0.681994 + 0.731357i \(0.261113\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −11.1803 −1.56556
\(52\) 0 0
\(53\) −12.6491 −1.73749 −0.868744 0.495261i \(-0.835073\pi\)
−0.868744 + 0.495261i \(0.835073\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.00000 −0.662266
\(58\) 0 0
\(59\) −8.94427 −1.16445 −0.582223 0.813029i \(-0.697817\pi\)
−0.582223 + 0.813029i \(0.697817\pi\)
\(60\) 0 0
\(61\) 6.32456 0.809776 0.404888 0.914366i \(-0.367310\pi\)
0.404888 + 0.914366i \(0.367310\pi\)
\(62\) 0 0
\(63\) −5.65685 −0.712697
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −11.1803 −1.36590 −0.682948 0.730467i \(-0.739302\pi\)
−0.682948 + 0.730467i \(0.739302\pi\)
\(68\) 0 0
\(69\) 12.6491 1.52277
\(70\) 0 0
\(71\) 14.1421 1.67836 0.839181 0.543852i \(-0.183035\pi\)
0.839181 + 0.543852i \(0.183035\pi\)
\(72\) 0 0
\(73\) −15.0000 −1.75562 −0.877809 0.479012i \(-0.840995\pi\)
−0.877809 + 0.479012i \(0.840995\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.32456 0.720750
\(78\) 0 0
\(79\) −14.1421 −1.59111 −0.795557 0.605878i \(-0.792822\pi\)
−0.795557 + 0.605878i \(0.792822\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 6.70820 0.736321 0.368161 0.929762i \(-0.379988\pi\)
0.368161 + 0.929762i \(0.379988\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −14.1421 −1.51620
\(88\) 0 0
\(89\) −1.00000 −0.106000 −0.0529999 0.998595i \(-0.516878\pi\)
−0.0529999 + 0.998595i \(0.516878\pi\)
\(90\) 0 0
\(91\) −17.8885 −1.87523
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) −4.47214 −0.449467
\(100\) 0 0
\(101\) 6.32456 0.629317 0.314658 0.949205i \(-0.398110\pi\)
0.314658 + 0.949205i \(0.398110\pi\)
\(102\) 0 0
\(103\) 8.48528 0.836080 0.418040 0.908429i \(-0.362717\pi\)
0.418040 + 0.908429i \(0.362717\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.6525 −1.51318 −0.756591 0.653888i \(-0.773137\pi\)
−0.756591 + 0.653888i \(0.773137\pi\)
\(108\) 0 0
\(109\) 6.32456 0.605783 0.302891 0.953025i \(-0.402048\pi\)
0.302891 + 0.953025i \(0.402048\pi\)
\(110\) 0 0
\(111\) −14.1421 −1.34231
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 12.6491 1.16941
\(118\) 0 0
\(119\) 14.1421 1.29641
\(120\) 0 0
\(121\) −6.00000 −0.545455
\(122\) 0 0
\(123\) −6.70820 −0.604858
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.9706 1.50589 0.752947 0.658081i \(-0.228632\pi\)
0.752947 + 0.658081i \(0.228632\pi\)
\(128\) 0 0
\(129\) 20.0000 1.76090
\(130\) 0 0
\(131\) −8.94427 −0.781465 −0.390732 0.920504i \(-0.627778\pi\)
−0.390732 + 0.920504i \(0.627778\pi\)
\(132\) 0 0
\(133\) 6.32456 0.548408
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.00000 −0.427179 −0.213589 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(138\) 0 0
\(139\) 15.6525 1.32763 0.663813 0.747899i \(-0.268937\pi\)
0.663813 + 0.747899i \(0.268937\pi\)
\(140\) 0 0
\(141\) 6.32456 0.532624
\(142\) 0 0
\(143\) −14.1421 −1.18262
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.23607 0.184428
\(148\) 0 0
\(149\) −18.9737 −1.55438 −0.777192 0.629264i \(-0.783356\pi\)
−0.777192 + 0.629264i \(0.783356\pi\)
\(150\) 0 0
\(151\) −14.1421 −1.15087 −0.575435 0.817847i \(-0.695167\pi\)
−0.575435 + 0.817847i \(0.695167\pi\)
\(152\) 0 0
\(153\) −10.0000 −0.808452
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) −28.2843 −2.24309
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 6.70820 0.525427 0.262714 0.964874i \(-0.415383\pi\)
0.262714 + 0.964874i \(0.415383\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.82843 0.218870 0.109435 0.993994i \(-0.465096\pi\)
0.109435 + 0.993994i \(0.465096\pi\)
\(168\) 0 0
\(169\) 27.0000 2.07692
\(170\) 0 0
\(171\) −4.47214 −0.341993
\(172\) 0 0
\(173\) −12.6491 −0.961694 −0.480847 0.876804i \(-0.659671\pi\)
−0.480847 + 0.876804i \(0.659671\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −20.0000 −1.50329
\(178\) 0 0
\(179\) −2.23607 −0.167132 −0.0835658 0.996502i \(-0.526631\pi\)
−0.0835658 + 0.996502i \(0.526631\pi\)
\(180\) 0 0
\(181\) 12.6491 0.940201 0.470100 0.882613i \(-0.344218\pi\)
0.470100 + 0.882613i \(0.344218\pi\)
\(182\) 0 0
\(183\) 14.1421 1.04542
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 11.1803 0.817587
\(188\) 0 0
\(189\) 6.32456 0.460044
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −25.0000 −1.76336
\(202\) 0 0
\(203\) 17.8885 1.25553
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 11.3137 0.786357
\(208\) 0 0
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) −6.70820 −0.461812 −0.230906 0.972976i \(-0.574169\pi\)
−0.230906 + 0.972976i \(0.574169\pi\)
\(212\) 0 0
\(213\) 31.6228 2.16676
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −33.5410 −2.26649
\(220\) 0 0
\(221\) −31.6228 −2.12718
\(222\) 0 0
\(223\) −22.6274 −1.51524 −0.757622 0.652694i \(-0.773639\pi\)
−0.757622 + 0.652694i \(0.773639\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.94427 −0.593652 −0.296826 0.954932i \(-0.595928\pi\)
−0.296826 + 0.954932i \(0.595928\pi\)
\(228\) 0 0
\(229\) −12.6491 −0.835877 −0.417938 0.908475i \(-0.637247\pi\)
−0.417938 + 0.908475i \(0.637247\pi\)
\(230\) 0 0
\(231\) 14.1421 0.930484
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −31.6228 −2.05412
\(238\) 0 0
\(239\) −14.1421 −0.914779 −0.457389 0.889267i \(-0.651215\pi\)
−0.457389 + 0.889267i \(0.651215\pi\)
\(240\) 0 0
\(241\) 13.0000 0.837404 0.418702 0.908124i \(-0.362485\pi\)
0.418702 + 0.908124i \(0.362485\pi\)
\(242\) 0 0
\(243\) −17.8885 −1.14755
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −14.1421 −0.899843
\(248\) 0 0
\(249\) 15.0000 0.950586
\(250\) 0 0
\(251\) 11.1803 0.705697 0.352848 0.935681i \(-0.385213\pi\)
0.352848 + 0.935681i \(0.385213\pi\)
\(252\) 0 0
\(253\) −12.6491 −0.795243
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.0000 −0.623783 −0.311891 0.950118i \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) 0 0
\(259\) 17.8885 1.11154
\(260\) 0 0
\(261\) −12.6491 −0.782960
\(262\) 0 0
\(263\) −22.6274 −1.39527 −0.697633 0.716455i \(-0.745763\pi\)
−0.697633 + 0.716455i \(0.745763\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −2.23607 −0.136845
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) −40.0000 −2.42091
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.9737 −1.14002 −0.570009 0.821639i \(-0.693060\pi\)
−0.570009 + 0.821639i \(0.693060\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) −11.1803 −0.664602 −0.332301 0.943173i \(-0.607825\pi\)
−0.332301 + 0.943173i \(0.607825\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.48528 0.500870
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 22.3607 1.31081
\(292\) 0 0
\(293\) 6.32456 0.369484 0.184742 0.982787i \(-0.440855\pi\)
0.184742 + 0.982787i \(0.440855\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) 35.7771 2.06904
\(300\) 0 0
\(301\) −25.2982 −1.45817
\(302\) 0 0
\(303\) 14.1421 0.812444
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.1246 1.14857 0.574286 0.818655i \(-0.305280\pi\)
0.574286 + 0.818655i \(0.305280\pi\)
\(308\) 0 0
\(309\) 18.9737 1.07937
\(310\) 0 0
\(311\) 14.1421 0.801927 0.400963 0.916094i \(-0.368675\pi\)
0.400963 + 0.916094i \(0.368675\pi\)
\(312\) 0 0
\(313\) −30.0000 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.6491 0.710445 0.355222 0.934782i \(-0.384405\pi\)
0.355222 + 0.934782i \(0.384405\pi\)
\(318\) 0 0
\(319\) 14.1421 0.791808
\(320\) 0 0
\(321\) −35.0000 −1.95351
\(322\) 0 0
\(323\) 11.1803 0.622091
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 14.1421 0.782062
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 33.5410 1.84358 0.921791 0.387688i \(-0.126726\pi\)
0.921791 + 0.387688i \(0.126726\pi\)
\(332\) 0 0
\(333\) −12.6491 −0.693167
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −15.0000 −0.817102 −0.408551 0.912735i \(-0.633966\pi\)
−0.408551 + 0.912735i \(0.633966\pi\)
\(338\) 0 0
\(339\) −33.5410 −1.82170
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.1246 1.08035 0.540173 0.841554i \(-0.318359\pi\)
0.540173 + 0.841554i \(0.318359\pi\)
\(348\) 0 0
\(349\) 18.9737 1.01564 0.507819 0.861464i \(-0.330452\pi\)
0.507819 + 0.861464i \(0.330452\pi\)
\(350\) 0 0
\(351\) −14.1421 −0.754851
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 31.6228 1.67365
\(358\) 0 0
\(359\) 14.1421 0.746393 0.373197 0.927752i \(-0.378262\pi\)
0.373197 + 0.927752i \(0.378262\pi\)
\(360\) 0 0
\(361\) −14.0000 −0.736842
\(362\) 0 0
\(363\) −13.4164 −0.704179
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −16.9706 −0.885856 −0.442928 0.896557i \(-0.646060\pi\)
−0.442928 + 0.896557i \(0.646060\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 35.7771 1.85745
\(372\) 0 0
\(373\) −12.6491 −0.654946 −0.327473 0.944861i \(-0.606197\pi\)
−0.327473 + 0.944861i \(0.606197\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −40.0000 −2.06010
\(378\) 0 0
\(379\) −6.70820 −0.344577 −0.172289 0.985047i \(-0.555116\pi\)
−0.172289 + 0.985047i \(0.555116\pi\)
\(380\) 0 0
\(381\) 37.9473 1.94410
\(382\) 0 0
\(383\) 19.7990 1.01168 0.505841 0.862627i \(-0.331182\pi\)
0.505841 + 0.862627i \(0.331182\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 17.8885 0.909326
\(388\) 0 0
\(389\) −25.2982 −1.28267 −0.641335 0.767261i \(-0.721619\pi\)
−0.641335 + 0.767261i \(0.721619\pi\)
\(390\) 0 0
\(391\) −28.2843 −1.43040
\(392\) 0 0
\(393\) −20.0000 −1.00887
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 25.2982 1.26968 0.634841 0.772643i \(-0.281066\pi\)
0.634841 + 0.772643i \(0.281066\pi\)
\(398\) 0 0
\(399\) 14.1421 0.707992
\(400\) 0 0
\(401\) 13.0000 0.649189 0.324595 0.945853i \(-0.394772\pi\)
0.324595 + 0.945853i \(0.394772\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 14.1421 0.701000
\(408\) 0 0
\(409\) −11.0000 −0.543915 −0.271957 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) 0 0
\(411\) −11.1803 −0.551485
\(412\) 0 0
\(413\) 25.2982 1.24484
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 35.0000 1.71396
\(418\) 0 0
\(419\) 33.5410 1.63859 0.819293 0.573375i \(-0.194366\pi\)
0.819293 + 0.573375i \(0.194366\pi\)
\(420\) 0 0
\(421\) 12.6491 0.616480 0.308240 0.951309i \(-0.400260\pi\)
0.308240 + 0.951309i \(0.400260\pi\)
\(422\) 0 0
\(423\) 5.65685 0.275046
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −17.8885 −0.865687
\(428\) 0 0
\(429\) −31.6228 −1.52676
\(430\) 0 0
\(431\) 14.1421 0.681203 0.340601 0.940208i \(-0.389369\pi\)
0.340601 + 0.940208i \(0.389369\pi\)
\(432\) 0 0
\(433\) −5.00000 −0.240285 −0.120142 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.6491 −0.605089
\(438\) 0 0
\(439\) −14.1421 −0.674967 −0.337484 0.941331i \(-0.609576\pi\)
−0.337484 + 0.941331i \(0.609576\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −29.0689 −1.38110 −0.690552 0.723283i \(-0.742632\pi\)
−0.690552 + 0.723283i \(0.742632\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −42.4264 −2.00670
\(448\) 0 0
\(449\) 21.0000 0.991051 0.495526 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) 6.70820 0.315877
\(452\) 0 0
\(453\) −31.6228 −1.48577
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5.00000 −0.233890 −0.116945 0.993138i \(-0.537310\pi\)
−0.116945 + 0.993138i \(0.537310\pi\)
\(458\) 0 0
\(459\) 11.1803 0.521854
\(460\) 0 0
\(461\) 37.9473 1.76738 0.883692 0.468069i \(-0.155050\pi\)
0.883692 + 0.468069i \(0.155050\pi\)
\(462\) 0 0
\(463\) 5.65685 0.262896 0.131448 0.991323i \(-0.458037\pi\)
0.131448 + 0.991323i \(0.458037\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8.94427 −0.413892 −0.206946 0.978352i \(-0.566352\pi\)
−0.206946 + 0.978352i \(0.566352\pi\)
\(468\) 0 0
\(469\) 31.6228 1.46020
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −20.0000 −0.919601
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −25.2982 −1.15833
\(478\) 0 0
\(479\) 28.2843 1.29234 0.646171 0.763193i \(-0.276369\pi\)
0.646171 + 0.763193i \(0.276369\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) 0 0
\(483\) −35.7771 −1.62791
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −39.5980 −1.79436 −0.897178 0.441669i \(-0.854386\pi\)
−0.897178 + 0.441669i \(0.854386\pi\)
\(488\) 0 0
\(489\) 15.0000 0.678323
\(490\) 0 0
\(491\) 26.8328 1.21095 0.605474 0.795865i \(-0.292984\pi\)
0.605474 + 0.795865i \(0.292984\pi\)
\(492\) 0 0
\(493\) 31.6228 1.42422
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −40.0000 −1.79425
\(498\) 0 0
\(499\) 26.8328 1.20120 0.600601 0.799549i \(-0.294928\pi\)
0.600601 + 0.799549i \(0.294928\pi\)
\(500\) 0 0
\(501\) 6.32456 0.282560
\(502\) 0 0
\(503\) −33.9411 −1.51336 −0.756680 0.653785i \(-0.773180\pi\)
−0.756680 + 0.653785i \(0.773180\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 60.3738 2.68130
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 42.4264 1.87683
\(512\) 0 0
\(513\) 5.00000 0.220755
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −6.32456 −0.278154
\(518\) 0 0
\(519\) −28.2843 −1.24154
\(520\) 0 0
\(521\) 13.0000 0.569540 0.284770 0.958596i \(-0.408083\pi\)
0.284770 + 0.958596i \(0.408083\pi\)
\(522\) 0 0
\(523\) 20.1246 0.879988 0.439994 0.898001i \(-0.354981\pi\)
0.439994 + 0.898001i \(0.354981\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) −17.8885 −0.776297
\(532\) 0 0
\(533\) −18.9737 −0.821841
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5.00000 −0.215766
\(538\) 0 0
\(539\) −2.23607 −0.0963143
\(540\) 0 0
\(541\) 12.6491 0.543828 0.271914 0.962322i \(-0.412343\pi\)
0.271914 + 0.962322i \(0.412343\pi\)
\(542\) 0 0
\(543\) 28.2843 1.21379
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.1246 0.860466 0.430233 0.902718i \(-0.358431\pi\)
0.430233 + 0.902718i \(0.358431\pi\)
\(548\) 0 0
\(549\) 12.6491 0.539851
\(550\) 0 0
\(551\) 14.1421 0.602475
\(552\) 0 0
\(553\) 40.0000 1.70097
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 44.2719 1.87586 0.937930 0.346825i \(-0.112740\pi\)
0.937930 + 0.346825i \(0.112740\pi\)
\(558\) 0 0
\(559\) 56.5685 2.39259
\(560\) 0 0
\(561\) 25.0000 1.05550
\(562\) 0 0
\(563\) −44.7214 −1.88478 −0.942390 0.334515i \(-0.891427\pi\)
−0.942390 + 0.334515i \(0.891427\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 31.1127 1.30661
\(568\) 0 0
\(569\) 21.0000 0.880366 0.440183 0.897908i \(-0.354914\pi\)
0.440183 + 0.897908i \(0.354914\pi\)
\(570\) 0 0
\(571\) 26.8328 1.12292 0.561459 0.827504i \(-0.310240\pi\)
0.561459 + 0.827504i \(0.310240\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 25.0000 1.04076 0.520382 0.853934i \(-0.325790\pi\)
0.520382 + 0.853934i \(0.325790\pi\)
\(578\) 0 0
\(579\) −11.1803 −0.464639
\(580\) 0 0
\(581\) −18.9737 −0.787160
\(582\) 0 0
\(583\) 28.2843 1.17141
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −11.1803 −0.461462 −0.230731 0.973018i \(-0.574112\pi\)
−0.230731 + 0.973018i \(0.574112\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −15.0000 −0.615976 −0.307988 0.951390i \(-0.599656\pi\)
−0.307988 + 0.951390i \(0.599656\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −14.1421 −0.577832 −0.288916 0.957354i \(-0.593295\pi\)
−0.288916 + 0.957354i \(0.593295\pi\)
\(600\) 0 0
\(601\) −17.0000 −0.693444 −0.346722 0.937968i \(-0.612705\pi\)
−0.346722 + 0.937968i \(0.612705\pi\)
\(602\) 0 0
\(603\) −22.3607 −0.910597
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.9706 0.688814 0.344407 0.938820i \(-0.388080\pi\)
0.344407 + 0.938820i \(0.388080\pi\)
\(608\) 0 0
\(609\) 40.0000 1.62088
\(610\) 0 0
\(611\) 17.8885 0.723693
\(612\) 0 0
\(613\) −25.2982 −1.02179 −0.510893 0.859644i \(-0.670685\pi\)
−0.510893 + 0.859644i \(0.670685\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 26.8328 1.07850 0.539251 0.842145i \(-0.318707\pi\)
0.539251 + 0.842145i \(0.318707\pi\)
\(620\) 0 0
\(621\) −12.6491 −0.507591
\(622\) 0 0
\(623\) 2.82843 0.113319
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 11.1803 0.446500
\(628\) 0 0
\(629\) 31.6228 1.26088
\(630\) 0 0
\(631\) −28.2843 −1.12598 −0.562990 0.826464i \(-0.690349\pi\)
−0.562990 + 0.826464i \(0.690349\pi\)
\(632\) 0 0
\(633\) −15.0000 −0.596196
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.32456 0.250588
\(638\) 0 0
\(639\) 28.2843 1.11891
\(640\) 0 0
\(641\) 22.0000 0.868948 0.434474 0.900684i \(-0.356934\pi\)
0.434474 + 0.900684i \(0.356934\pi\)
\(642\) 0 0
\(643\) 26.8328 1.05818 0.529091 0.848565i \(-0.322533\pi\)
0.529091 + 0.848565i \(0.322533\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 45.2548 1.77915 0.889576 0.456788i \(-0.151000\pi\)
0.889576 + 0.456788i \(0.151000\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.32456 0.247499 0.123749 0.992313i \(-0.460508\pi\)
0.123749 + 0.992313i \(0.460508\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −30.0000 −1.17041
\(658\) 0 0
\(659\) −6.70820 −0.261315 −0.130657 0.991428i \(-0.541709\pi\)
−0.130657 + 0.991428i \(0.541709\pi\)
\(660\) 0 0
\(661\) −44.2719 −1.72198 −0.860988 0.508625i \(-0.830154\pi\)
−0.860988 + 0.508625i \(0.830154\pi\)
\(662\) 0 0
\(663\) −70.7107 −2.74618
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −35.7771 −1.38529
\(668\) 0 0
\(669\) −50.5964 −1.95617
\(670\) 0 0
\(671\) −14.1421 −0.545951
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −37.9473 −1.45843 −0.729217 0.684282i \(-0.760116\pi\)
−0.729217 + 0.684282i \(0.760116\pi\)
\(678\) 0 0
\(679\) −28.2843 −1.08545
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 2.23607 0.0855608 0.0427804 0.999085i \(-0.486378\pi\)
0.0427804 + 0.999085i \(0.486378\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −28.2843 −1.07911
\(688\) 0 0
\(689\) −80.0000 −3.04776
\(690\) 0 0
\(691\) 33.5410 1.27596 0.637980 0.770053i \(-0.279770\pi\)
0.637980 + 0.770053i \(0.279770\pi\)
\(692\) 0 0
\(693\) 12.6491 0.480500
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 15.0000 0.568166
\(698\) 0 0
\(699\) −22.3607 −0.845759
\(700\) 0 0
\(701\) −31.6228 −1.19438 −0.597188 0.802101i \(-0.703715\pi\)
−0.597188 + 0.802101i \(0.703715\pi\)
\(702\) 0 0
\(703\) 14.1421 0.533381
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −17.8885 −0.672768
\(708\) 0 0
\(709\) −12.6491 −0.475047 −0.237524 0.971382i \(-0.576336\pi\)
−0.237524 + 0.971382i \(0.576336\pi\)
\(710\) 0 0
\(711\) −28.2843 −1.06074
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −31.6228 −1.18097
\(718\) 0 0
\(719\) −14.1421 −0.527413 −0.263706 0.964603i \(-0.584945\pi\)
−0.263706 + 0.964603i \(0.584945\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) 29.0689 1.08108
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 25.4558 0.944105 0.472052 0.881570i \(-0.343513\pi\)
0.472052 + 0.881570i \(0.343513\pi\)
\(728\) 0 0
\(729\) −7.00000 −0.259259
\(730\) 0 0
\(731\) −44.7214 −1.65408
\(732\) 0 0
\(733\) 6.32456 0.233603 0.116801 0.993155i \(-0.462736\pi\)
0.116801 + 0.993155i \(0.462736\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 25.0000 0.920887
\(738\) 0 0
\(739\) 8.94427 0.329020 0.164510 0.986375i \(-0.447396\pi\)
0.164510 + 0.986375i \(0.447396\pi\)
\(740\) 0 0
\(741\) −31.6228 −1.16169
\(742\) 0 0
\(743\) −5.65685 −0.207530 −0.103765 0.994602i \(-0.533089\pi\)
−0.103765 + 0.994602i \(0.533089\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 13.4164 0.490881
\(748\) 0 0
\(749\) 44.2719 1.61766
\(750\) 0 0
\(751\) 42.4264 1.54816 0.774081 0.633087i \(-0.218212\pi\)
0.774081 + 0.633087i \(0.218212\pi\)
\(752\) 0 0
\(753\) 25.0000 0.911051
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 12.6491 0.459740 0.229870 0.973221i \(-0.426170\pi\)
0.229870 + 0.973221i \(0.426170\pi\)
\(758\) 0 0
\(759\) −28.2843 −1.02665
\(760\) 0 0
\(761\) −3.00000 −0.108750 −0.0543750 0.998521i \(-0.517317\pi\)
−0.0543750 + 0.998521i \(0.517317\pi\)
\(762\) 0 0
\(763\) −17.8885 −0.647609
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −56.5685 −2.04257
\(768\) 0 0
\(769\) 21.0000 0.757279 0.378640 0.925544i \(-0.376392\pi\)
0.378640 + 0.925544i \(0.376392\pi\)
\(770\) 0 0
\(771\) −22.3607 −0.805300
\(772\) 0 0
\(773\) 31.6228 1.13739 0.568696 0.822548i \(-0.307448\pi\)
0.568696 + 0.822548i \(0.307448\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 40.0000 1.43499
\(778\) 0 0
\(779\) 6.70820 0.240346
\(780\) 0 0
\(781\) −31.6228 −1.13155
\(782\) 0 0
\(783\) 14.1421 0.505399
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −26.8328 −0.956487 −0.478243 0.878227i \(-0.658726\pi\)
−0.478243 + 0.878227i \(0.658726\pi\)
\(788\) 0 0
\(789\) −50.5964 −1.80128
\(790\) 0 0
\(791\) 42.4264 1.50851
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −25.2982 −0.896109 −0.448054 0.894006i \(-0.647883\pi\)
−0.448054 + 0.894006i \(0.647883\pi\)
\(798\) 0 0
\(799\) −14.1421 −0.500313
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 0 0
\(803\) 33.5410 1.18364
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −34.0000 −1.19538 −0.597688 0.801729i \(-0.703914\pi\)
−0.597688 + 0.801729i \(0.703914\pi\)
\(810\) 0 0
\(811\) −8.94427 −0.314076 −0.157038 0.987593i \(-0.550194\pi\)
−0.157038 + 0.987593i \(0.550194\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −20.0000 −0.699711
\(818\) 0 0
\(819\) −35.7771 −1.25015
\(820\) 0 0
\(821\) 37.9473 1.32437 0.662186 0.749340i \(-0.269629\pi\)
0.662186 + 0.749340i \(0.269629\pi\)
\(822\) 0 0
\(823\) 5.65685 0.197186 0.0985928 0.995128i \(-0.468566\pi\)
0.0985928 + 0.995128i \(0.468566\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.1803 −0.388779 −0.194389 0.980924i \(-0.562272\pi\)
−0.194389 + 0.980924i \(0.562272\pi\)
\(828\) 0 0
\(829\) −6.32456 −0.219661 −0.109830 0.993950i \(-0.535031\pi\)
−0.109830 + 0.993950i \(0.535031\pi\)
\(830\) 0 0
\(831\) −42.4264 −1.47176
\(832\) 0 0
\(833\) −5.00000 −0.173240
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −56.5685 −1.95296 −0.976481 0.215601i \(-0.930829\pi\)
−0.976481 + 0.215601i \(0.930829\pi\)
\(840\) 0 0
\(841\) 11.0000 0.379310
\(842\) 0 0
\(843\) 4.47214 0.154029
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 16.9706 0.583115
\(848\) 0 0
\(849\) −25.0000 −0.857998
\(850\) 0 0
\(851\) −35.7771 −1.22642
\(852\) 0 0
\(853\) 12.6491 0.433097 0.216549 0.976272i \(-0.430520\pi\)
0.216549 + 0.976272i \(0.430520\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −45.0000 −1.53717 −0.768585 0.639747i \(-0.779039\pi\)
−0.768585 + 0.639747i \(0.779039\pi\)
\(858\) 0 0
\(859\) −2.23607 −0.0762937 −0.0381468 0.999272i \(-0.512145\pi\)
−0.0381468 + 0.999272i \(0.512145\pi\)
\(860\) 0 0
\(861\) 18.9737 0.646621
\(862\) 0 0
\(863\) 8.48528 0.288842 0.144421 0.989516i \(-0.453868\pi\)
0.144421 + 0.989516i \(0.453868\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 17.8885 0.607527
\(868\) 0 0
\(869\) 31.6228 1.07273
\(870\) 0 0
\(871\) −70.7107 −2.39594
\(872\) 0 0
\(873\) 20.0000 0.676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −12.6491 −0.427130 −0.213565 0.976929i \(-0.568508\pi\)
−0.213565 + 0.976929i \(0.568508\pi\)
\(878\) 0 0
\(879\) 14.1421 0.477002
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) −15.6525 −0.526748 −0.263374 0.964694i \(-0.584835\pi\)
−0.263374 + 0.964694i \(0.584835\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 11.3137 0.379877 0.189939 0.981796i \(-0.439171\pi\)
0.189939 + 0.981796i \(0.439171\pi\)
\(888\) 0 0
\(889\) −48.0000 −1.60987
\(890\) 0 0
\(891\) 24.5967 0.824022
\(892\) 0 0
\(893\) −6.32456 −0.211643
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 80.0000 2.67112
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 63.2456 2.10701
\(902\) 0 0
\(903\) −56.5685 −1.88248
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 8.94427 0.296990 0.148495 0.988913i \(-0.452557\pi\)
0.148495 + 0.988913i \(0.452557\pi\)
\(908\) 0 0
\(909\) 12.6491 0.419545
\(910\) 0 0
\(911\) −28.2843 −0.937100 −0.468550 0.883437i \(-0.655223\pi\)
−0.468550 + 0.883437i \(0.655223\pi\)
\(912\) 0 0
\(913\) −15.0000 −0.496428
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 25.2982 0.835421
\(918\) 0 0
\(919\) −28.2843 −0.933012 −0.466506 0.884518i \(-0.654487\pi\)
−0.466506 + 0.884518i \(0.654487\pi\)
\(920\) 0 0
\(921\) 45.0000 1.48280
\(922\) 0 0
\(923\) 89.4427 2.94404
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 16.9706 0.557386
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −2.23607 −0.0732842
\(932\) 0 0
\(933\) 31.6228 1.03528
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 25.0000 0.816714 0.408357 0.912822i \(-0.366102\pi\)
0.408357 + 0.912822i \(0.366102\pi\)
\(938\) 0 0
\(939\) −67.0820 −2.18914
\(940\) 0 0
\(941\) 31.6228 1.03087 0.515437 0.856928i \(-0.327630\pi\)
0.515437 + 0.856928i \(0.327630\pi\)
\(942\) 0 0
\(943\) −16.9706 −0.552638
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.8328 0.871949 0.435975 0.899959i \(-0.356404\pi\)
0.435975 + 0.899959i \(0.356404\pi\)
\(948\) 0 0
\(949\) −94.8683 −3.07956
\(950\) 0 0
\(951\) 28.2843 0.917180
\(952\) 0 0
\(953\) −45.0000 −1.45769 −0.728846 0.684677i \(-0.759943\pi\)
−0.728846 + 0.684677i \(0.759943\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 31.6228 1.02222
\(958\) 0 0
\(959\) 14.1421 0.456673
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −31.3050 −1.00879
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −16.9706 −0.545737 −0.272868 0.962051i \(-0.587972\pi\)
−0.272868 + 0.962051i \(0.587972\pi\)
\(968\) 0 0
\(969\) 25.0000 0.803116
\(970\) 0 0
\(971\) −2.23607 −0.0717588 −0.0358794 0.999356i \(-0.511423\pi\)
−0.0358794 + 0.999356i \(0.511423\pi\)
\(972\) 0 0
\(973\) −44.2719 −1.41929
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −45.0000 −1.43968 −0.719839 0.694141i \(-0.755784\pi\)
−0.719839 + 0.694141i \(0.755784\pi\)
\(978\) 0 0
\(979\) 2.23607 0.0714650
\(980\) 0 0
\(981\) 12.6491 0.403855
\(982\) 0 0
\(983\) 19.7990 0.631490 0.315745 0.948844i \(-0.397746\pi\)
0.315745 + 0.948844i \(0.397746\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −17.8885 −0.569399
\(988\) 0 0
\(989\) 50.5964 1.60887
\(990\) 0 0
\(991\) 42.4264 1.34772 0.673860 0.738859i \(-0.264635\pi\)
0.673860 + 0.738859i \(0.264635\pi\)
\(992\) 0 0
\(993\) 75.0000 2.38005
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25.2982 0.801203 0.400601 0.916252i \(-0.368801\pi\)
0.400601 + 0.916252i \(0.368801\pi\)
\(998\) 0 0
\(999\) 14.1421 0.447437
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 6400.2.a.cp.1.3 4
4.3 odd 2 inner 6400.2.a.cp.1.2 4
5.4 even 2 6400.2.a.cs.1.2 4
8.3 odd 2 inner 6400.2.a.cp.1.4 4
8.5 even 2 inner 6400.2.a.cp.1.1 4
16.3 odd 4 3200.2.d.m.1601.1 4
16.5 even 4 3200.2.d.m.1601.2 yes 4
16.11 odd 4 3200.2.d.m.1601.3 yes 4
16.13 even 4 3200.2.d.m.1601.4 yes 4
20.19 odd 2 6400.2.a.cs.1.3 4
40.19 odd 2 6400.2.a.cs.1.1 4
40.29 even 2 6400.2.a.cs.1.4 4
80.3 even 4 3200.2.f.r.449.7 8
80.13 odd 4 3200.2.f.r.449.2 8
80.19 odd 4 3200.2.d.r.1601.4 yes 4
80.27 even 4 3200.2.f.r.449.6 8
80.29 even 4 3200.2.d.r.1601.1 yes 4
80.37 odd 4 3200.2.f.r.449.3 8
80.43 even 4 3200.2.f.r.449.4 8
80.53 odd 4 3200.2.f.r.449.5 8
80.59 odd 4 3200.2.d.r.1601.2 yes 4
80.67 even 4 3200.2.f.r.449.1 8
80.69 even 4 3200.2.d.r.1601.3 yes 4
80.77 odd 4 3200.2.f.r.449.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3200.2.d.m.1601.1 4 16.3 odd 4
3200.2.d.m.1601.2 yes 4 16.5 even 4
3200.2.d.m.1601.3 yes 4 16.11 odd 4
3200.2.d.m.1601.4 yes 4 16.13 even 4
3200.2.d.r.1601.1 yes 4 80.29 even 4
3200.2.d.r.1601.2 yes 4 80.59 odd 4
3200.2.d.r.1601.3 yes 4 80.69 even 4
3200.2.d.r.1601.4 yes 4 80.19 odd 4
3200.2.f.r.449.1 8 80.67 even 4
3200.2.f.r.449.2 8 80.13 odd 4
3200.2.f.r.449.3 8 80.37 odd 4
3200.2.f.r.449.4 8 80.43 even 4
3200.2.f.r.449.5 8 80.53 odd 4
3200.2.f.r.449.6 8 80.27 even 4
3200.2.f.r.449.7 8 80.3 even 4
3200.2.f.r.449.8 8 80.77 odd 4
6400.2.a.cp.1.1 4 8.5 even 2 inner
6400.2.a.cp.1.2 4 4.3 odd 2 inner
6400.2.a.cp.1.3 4 1.1 even 1 trivial
6400.2.a.cp.1.4 4 8.3 odd 2 inner
6400.2.a.cs.1.1 4 40.19 odd 2
6400.2.a.cs.1.2 4 5.4 even 2
6400.2.a.cs.1.3 4 20.19 odd 2
6400.2.a.cs.1.4 4 40.29 even 2