Properties

Label 7800.2.a.br.1.1
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.788.1
Defining polynomial: \(x^{3} - x^{2} - 7 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.87740\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.87740 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.87740 q^{7} +1.00000 q^{9} +4.15686 q^{11} -1.00000 q^{13} -1.00000 q^{17} -3.47536 q^{19} -1.87740 q^{21} +1.47536 q^{23} +1.00000 q^{27} -1.80410 q^{29} -8.96095 q^{31} +4.15686 q^{33} -1.72055 q^{37} -1.00000 q^{39} -8.83836 q^{41} +4.27945 q^{43} +1.07331 q^{47} -3.47536 q^{49} -1.00000 q^{51} +11.5932 q^{53} -3.47536 q^{57} -10.4363 q^{59} -0.195903 q^{61} -1.87740 q^{63} -6.15686 q^{67} +1.47536 q^{69} +4.03426 q^{71} -1.08355 q^{73} -7.80410 q^{77} -8.27945 q^{79} +1.00000 q^{81} +4.43631 q^{83} -1.80410 q^{87} -1.19590 q^{89} +1.87740 q^{91} -8.96095 q^{93} -4.24519 q^{97} +4.15686 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} + q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} - 3 q^{17} - 6 q^{19} + q^{21} + 3 q^{27} - q^{29} - 7 q^{31} - 3 q^{33} - 14 q^{37} - 3 q^{39} + 4 q^{43} + q^{47} - 6 q^{49} - 3 q^{51} - 5 q^{53} - 6 q^{57} - 7 q^{59} - 5 q^{61} + q^{63} - 3 q^{67} - 10 q^{71} + 10 q^{73} - 19 q^{77} - 16 q^{79} + 3 q^{81} - 11 q^{83} - q^{87} - 8 q^{89} - q^{91} - 7 q^{93} - 26 q^{97} - 3 q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.87740 −0.709592 −0.354796 0.934944i \(-0.615450\pi\)
−0.354796 + 0.934944i \(0.615450\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.15686 1.25334 0.626670 0.779285i \(-0.284418\pi\)
0.626670 + 0.779285i \(0.284418\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) −3.47536 −0.797301 −0.398651 0.917103i \(-0.630521\pi\)
−0.398651 + 0.917103i \(0.630521\pi\)
\(20\) 0 0
\(21\) −1.87740 −0.409683
\(22\) 0 0
\(23\) 1.47536 0.307633 0.153816 0.988099i \(-0.450844\pi\)
0.153816 + 0.988099i \(0.450844\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.80410 −0.335012 −0.167506 0.985871i \(-0.553571\pi\)
−0.167506 + 0.985871i \(0.553571\pi\)
\(30\) 0 0
\(31\) −8.96095 −1.60943 −0.804717 0.593658i \(-0.797683\pi\)
−0.804717 + 0.593658i \(0.797683\pi\)
\(32\) 0 0
\(33\) 4.15686 0.723616
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.72055 −0.282856 −0.141428 0.989949i \(-0.545169\pi\)
−0.141428 + 0.989949i \(0.545169\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −8.83836 −1.38032 −0.690160 0.723657i \(-0.742460\pi\)
−0.690160 + 0.723657i \(0.742460\pi\)
\(42\) 0 0
\(43\) 4.27945 0.652610 0.326305 0.945264i \(-0.394196\pi\)
0.326305 + 0.945264i \(0.394196\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.07331 0.156558 0.0782789 0.996931i \(-0.475058\pi\)
0.0782789 + 0.996931i \(0.475058\pi\)
\(48\) 0 0
\(49\) −3.47536 −0.496479
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) 11.5932 1.59244 0.796222 0.605005i \(-0.206829\pi\)
0.796222 + 0.605005i \(0.206829\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.47536 −0.460322
\(58\) 0 0
\(59\) −10.4363 −1.35869 −0.679346 0.733818i \(-0.737736\pi\)
−0.679346 + 0.733818i \(0.737736\pi\)
\(60\) 0 0
\(61\) −0.195903 −0.0250828 −0.0125414 0.999921i \(-0.503992\pi\)
−0.0125414 + 0.999921i \(0.503992\pi\)
\(62\) 0 0
\(63\) −1.87740 −0.236531
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.15686 −0.752180 −0.376090 0.926583i \(-0.622732\pi\)
−0.376090 + 0.926583i \(0.622732\pi\)
\(68\) 0 0
\(69\) 1.47536 0.177612
\(70\) 0 0
\(71\) 4.03426 0.478779 0.239389 0.970924i \(-0.423053\pi\)
0.239389 + 0.970924i \(0.423053\pi\)
\(72\) 0 0
\(73\) −1.08355 −0.126820 −0.0634099 0.997988i \(-0.520198\pi\)
−0.0634099 + 0.997988i \(0.520198\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.80410 −0.889359
\(78\) 0 0
\(79\) −8.27945 −0.931511 −0.465756 0.884913i \(-0.654217\pi\)
−0.465756 + 0.884913i \(0.654217\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.43631 0.486948 0.243474 0.969907i \(-0.421713\pi\)
0.243474 + 0.969907i \(0.421713\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.80410 −0.193420
\(88\) 0 0
\(89\) −1.19590 −0.126765 −0.0633827 0.997989i \(-0.520189\pi\)
−0.0633827 + 0.997989i \(0.520189\pi\)
\(90\) 0 0
\(91\) 1.87740 0.196805
\(92\) 0 0
\(93\) −8.96095 −0.929208
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.24519 −0.431034 −0.215517 0.976500i \(-0.569144\pi\)
−0.215517 + 0.976500i \(0.569144\pi\)
\(98\) 0 0
\(99\) 4.15686 0.417780
\(100\) 0 0
\(101\) −5.67126 −0.564311 −0.282156 0.959369i \(-0.591049\pi\)
−0.282156 + 0.959369i \(0.591049\pi\)
\(102\) 0 0
\(103\) −10.3480 −1.01962 −0.509808 0.860288i \(-0.670284\pi\)
−0.509808 + 0.860288i \(0.670284\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.36300 0.518461 0.259230 0.965815i \(-0.416531\pi\)
0.259230 + 0.965815i \(0.416531\pi\)
\(108\) 0 0
\(109\) −0.916451 −0.0877800 −0.0438900 0.999036i \(-0.513975\pi\)
−0.0438900 + 0.999036i \(0.513975\pi\)
\(110\) 0 0
\(111\) −1.72055 −0.163307
\(112\) 0 0
\(113\) 0.804097 0.0756431 0.0378215 0.999285i \(-0.487958\pi\)
0.0378215 + 0.999285i \(0.487958\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 1.87740 0.172101
\(120\) 0 0
\(121\) 6.27945 0.570859
\(122\) 0 0
\(123\) −8.83836 −0.796928
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.03426 0.535454 0.267727 0.963495i \(-0.413727\pi\)
0.267727 + 0.963495i \(0.413727\pi\)
\(128\) 0 0
\(129\) 4.27945 0.376785
\(130\) 0 0
\(131\) 17.1521 1.49858 0.749292 0.662240i \(-0.230394\pi\)
0.749292 + 0.662240i \(0.230394\pi\)
\(132\) 0 0
\(133\) 6.52464 0.565758
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.636998 −0.0544224 −0.0272112 0.999630i \(-0.508663\pi\)
−0.0272112 + 0.999630i \(0.508663\pi\)
\(138\) 0 0
\(139\) −15.1178 −1.28228 −0.641138 0.767426i \(-0.721537\pi\)
−0.641138 + 0.767426i \(0.721537\pi\)
\(140\) 0 0
\(141\) 1.07331 0.0903887
\(142\) 0 0
\(143\) −4.15686 −0.347614
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.47536 −0.286642
\(148\) 0 0
\(149\) −6.14661 −0.503550 −0.251775 0.967786i \(-0.581014\pi\)
−0.251775 + 0.967786i \(0.581014\pi\)
\(150\) 0 0
\(151\) −10.8281 −0.881179 −0.440590 0.897709i \(-0.645231\pi\)
−0.440590 + 0.897709i \(0.645231\pi\)
\(152\) 0 0
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −9.52464 −0.760149 −0.380075 0.924956i \(-0.624102\pi\)
−0.380075 + 0.924956i \(0.624102\pi\)
\(158\) 0 0
\(159\) 11.5932 0.919398
\(160\) 0 0
\(161\) −2.76984 −0.218294
\(162\) 0 0
\(163\) 22.2014 1.73894 0.869472 0.493982i \(-0.164459\pi\)
0.869472 + 0.493982i \(0.164459\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.27945 −0.485919 −0.242959 0.970036i \(-0.578118\pi\)
−0.242959 + 0.970036i \(0.578118\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −3.47536 −0.265767
\(172\) 0 0
\(173\) −5.83836 −0.443882 −0.221941 0.975060i \(-0.571239\pi\)
−0.221941 + 0.975060i \(0.571239\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −10.4363 −0.784441
\(178\) 0 0
\(179\) 18.3480 1.37139 0.685696 0.727888i \(-0.259498\pi\)
0.685696 + 0.727888i \(0.259498\pi\)
\(180\) 0 0
\(181\) −0.887647 −0.0659783 −0.0329891 0.999456i \(-0.510503\pi\)
−0.0329891 + 0.999456i \(0.510503\pi\)
\(182\) 0 0
\(183\) −0.195903 −0.0144816
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −4.15686 −0.303979
\(188\) 0 0
\(189\) −1.87740 −0.136561
\(190\) 0 0
\(191\) −15.6082 −1.12937 −0.564685 0.825307i \(-0.691002\pi\)
−0.564685 + 0.825307i \(0.691002\pi\)
\(192\) 0 0
\(193\) 5.96574 0.429423 0.214712 0.976677i \(-0.431119\pi\)
0.214712 + 0.976677i \(0.431119\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −23.2302 −1.65508 −0.827540 0.561406i \(-0.810260\pi\)
−0.827540 + 0.561406i \(0.810260\pi\)
\(198\) 0 0
\(199\) 2.55890 0.181396 0.0906980 0.995878i \(-0.471090\pi\)
0.0906980 + 0.995878i \(0.471090\pi\)
\(200\) 0 0
\(201\) −6.15686 −0.434271
\(202\) 0 0
\(203\) 3.38702 0.237722
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.47536 0.102544
\(208\) 0 0
\(209\) −14.4466 −0.999289
\(210\) 0 0
\(211\) −1.96574 −0.135327 −0.0676636 0.997708i \(-0.521554\pi\)
−0.0676636 + 0.997708i \(0.521554\pi\)
\(212\) 0 0
\(213\) 4.03426 0.276423
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.8233 1.14204
\(218\) 0 0
\(219\) −1.08355 −0.0732195
\(220\) 0 0
\(221\) 1.00000 0.0672673
\(222\) 0 0
\(223\) −16.6617 −1.11575 −0.557874 0.829925i \(-0.688383\pi\)
−0.557874 + 0.829925i \(0.688383\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.0102429 0.000679842 0 0.000339921 1.00000i \(-0.499892\pi\)
0.000339921 1.00000i \(0.499892\pi\)
\(228\) 0 0
\(229\) −9.21639 −0.609036 −0.304518 0.952507i \(-0.598495\pi\)
−0.304518 + 0.952507i \(0.598495\pi\)
\(230\) 0 0
\(231\) −7.80410 −0.513472
\(232\) 0 0
\(233\) 7.19590 0.471419 0.235710 0.971824i \(-0.424259\pi\)
0.235710 + 0.971824i \(0.424259\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −8.27945 −0.537808
\(238\) 0 0
\(239\) 6.82811 0.441674 0.220837 0.975311i \(-0.429121\pi\)
0.220837 + 0.975311i \(0.429121\pi\)
\(240\) 0 0
\(241\) 16.2014 1.04362 0.521811 0.853061i \(-0.325257\pi\)
0.521811 + 0.853061i \(0.325257\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.47536 0.221132
\(248\) 0 0
\(249\) 4.43631 0.281140
\(250\) 0 0
\(251\) −23.9562 −1.51210 −0.756050 0.654514i \(-0.772873\pi\)
−0.756050 + 0.654514i \(0.772873\pi\)
\(252\) 0 0
\(253\) 6.13284 0.385568
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −29.6617 −1.85025 −0.925123 0.379669i \(-0.876038\pi\)
−0.925123 + 0.379669i \(0.876038\pi\)
\(258\) 0 0
\(259\) 3.23016 0.200713
\(260\) 0 0
\(261\) −1.80410 −0.111671
\(262\) 0 0
\(263\) 6.21093 0.382983 0.191491 0.981494i \(-0.438668\pi\)
0.191491 + 0.981494i \(0.438668\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1.19590 −0.0731881
\(268\) 0 0
\(269\) 18.1178 1.10466 0.552331 0.833625i \(-0.313738\pi\)
0.552331 + 0.833625i \(0.313738\pi\)
\(270\) 0 0
\(271\) −30.8966 −1.87684 −0.938418 0.345501i \(-0.887709\pi\)
−0.938418 + 0.345501i \(0.887709\pi\)
\(272\) 0 0
\(273\) 1.87740 0.113626
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −26.7740 −1.60870 −0.804348 0.594158i \(-0.797485\pi\)
−0.804348 + 0.594158i \(0.797485\pi\)
\(278\) 0 0
\(279\) −8.96095 −0.536478
\(280\) 0 0
\(281\) 4.16710 0.248588 0.124294 0.992245i \(-0.460333\pi\)
0.124294 + 0.992245i \(0.460333\pi\)
\(282\) 0 0
\(283\) 20.4603 1.21624 0.608120 0.793845i \(-0.291924\pi\)
0.608120 + 0.793845i \(0.291924\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 16.5932 0.979464
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −4.24519 −0.248858
\(292\) 0 0
\(293\) −18.6274 −1.08823 −0.544113 0.839012i \(-0.683134\pi\)
−0.544113 + 0.839012i \(0.683134\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.15686 0.241205
\(298\) 0 0
\(299\) −1.47536 −0.0853220
\(300\) 0 0
\(301\) −8.03426 −0.463087
\(302\) 0 0
\(303\) −5.67126 −0.325805
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 25.7891 1.47186 0.735930 0.677058i \(-0.236745\pi\)
0.735930 + 0.677058i \(0.236745\pi\)
\(308\) 0 0
\(309\) −10.3480 −0.588676
\(310\) 0 0
\(311\) −16.4603 −0.933379 −0.466690 0.884421i \(-0.654554\pi\)
−0.466690 + 0.884421i \(0.654554\pi\)
\(312\) 0 0
\(313\) 33.1028 1.87108 0.935540 0.353221i \(-0.114914\pi\)
0.935540 + 0.353221i \(0.114914\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −21.0835 −1.18417 −0.592085 0.805875i \(-0.701695\pi\)
−0.592085 + 0.805875i \(0.701695\pi\)
\(318\) 0 0
\(319\) −7.49937 −0.419884
\(320\) 0 0
\(321\) 5.36300 0.299334
\(322\) 0 0
\(323\) 3.47536 0.193374
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −0.916451 −0.0506798
\(328\) 0 0
\(329\) −2.01503 −0.111092
\(330\) 0 0
\(331\) −5.08355 −0.279417 −0.139709 0.990193i \(-0.544617\pi\)
−0.139709 + 0.990193i \(0.544617\pi\)
\(332\) 0 0
\(333\) −1.72055 −0.0942854
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.67126 0.199986 0.0999931 0.994988i \(-0.468118\pi\)
0.0999931 + 0.994988i \(0.468118\pi\)
\(338\) 0 0
\(339\) 0.804097 0.0436726
\(340\) 0 0
\(341\) −37.2494 −2.01717
\(342\) 0 0
\(343\) 19.6665 1.06189
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.21639 −0.0652992 −0.0326496 0.999467i \(-0.510395\pi\)
−0.0326496 + 0.999467i \(0.510395\pi\)
\(348\) 0 0
\(349\) −17.2302 −0.922309 −0.461155 0.887320i \(-0.652565\pi\)
−0.461155 + 0.887320i \(0.652565\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −30.2494 −1.61001 −0.805006 0.593266i \(-0.797838\pi\)
−0.805006 + 0.593266i \(0.797838\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.87740 0.0993627
\(358\) 0 0
\(359\) −6.33773 −0.334493 −0.167246 0.985915i \(-0.553488\pi\)
−0.167246 + 0.985915i \(0.553488\pi\)
\(360\) 0 0
\(361\) −6.92191 −0.364311
\(362\) 0 0
\(363\) 6.27945 0.329586
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.88219 0.150449 0.0752246 0.997167i \(-0.476033\pi\)
0.0752246 + 0.997167i \(0.476033\pi\)
\(368\) 0 0
\(369\) −8.83836 −0.460106
\(370\) 0 0
\(371\) −21.7651 −1.12999
\(372\) 0 0
\(373\) −22.0055 −1.13940 −0.569700 0.821853i \(-0.692940\pi\)
−0.569700 + 0.821853i \(0.692940\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.80410 0.0929157
\(378\) 0 0
\(379\) 29.4418 1.51232 0.756161 0.654386i \(-0.227073\pi\)
0.756161 + 0.654386i \(0.227073\pi\)
\(380\) 0 0
\(381\) 6.03426 0.309144
\(382\) 0 0
\(383\) 24.1028 1.23159 0.615797 0.787905i \(-0.288834\pi\)
0.615797 + 0.787905i \(0.288834\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.27945 0.217537
\(388\) 0 0
\(389\) 19.8233 1.00508 0.502541 0.864553i \(-0.332398\pi\)
0.502541 + 0.864553i \(0.332398\pi\)
\(390\) 0 0
\(391\) −1.47536 −0.0746119
\(392\) 0 0
\(393\) 17.1521 0.865207
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.70552 0.135786 0.0678930 0.997693i \(-0.478372\pi\)
0.0678930 + 0.997693i \(0.478372\pi\)
\(398\) 0 0
\(399\) 6.52464 0.326641
\(400\) 0 0
\(401\) −3.88765 −0.194140 −0.0970699 0.995278i \(-0.530947\pi\)
−0.0970699 + 0.995278i \(0.530947\pi\)
\(402\) 0 0
\(403\) 8.96095 0.446377
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.15207 −0.354515
\(408\) 0 0
\(409\) −19.7891 −0.978506 −0.489253 0.872142i \(-0.662731\pi\)
−0.489253 + 0.872142i \(0.662731\pi\)
\(410\) 0 0
\(411\) −0.636998 −0.0314208
\(412\) 0 0
\(413\) 19.5932 0.964117
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −15.1178 −0.740322
\(418\) 0 0
\(419\) −9.78907 −0.478227 −0.239114 0.970992i \(-0.576857\pi\)
−0.239114 + 0.970992i \(0.576857\pi\)
\(420\) 0 0
\(421\) 20.6959 1.00866 0.504329 0.863511i \(-0.331740\pi\)
0.504329 + 0.863511i \(0.331740\pi\)
\(422\) 0 0
\(423\) 1.07331 0.0521860
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.367789 0.0177985
\(428\) 0 0
\(429\) −4.15686 −0.200695
\(430\) 0 0
\(431\) −27.5439 −1.32674 −0.663371 0.748291i \(-0.730875\pi\)
−0.663371 + 0.748291i \(0.730875\pi\)
\(432\) 0 0
\(433\) −0.872617 −0.0419353 −0.0209676 0.999780i \(-0.506675\pi\)
−0.0209676 + 0.999780i \(0.506675\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.12738 −0.245276
\(438\) 0 0
\(439\) −32.2014 −1.53689 −0.768443 0.639918i \(-0.778968\pi\)
−0.768443 + 0.639918i \(0.778968\pi\)
\(440\) 0 0
\(441\) −3.47536 −0.165493
\(442\) 0 0
\(443\) −23.2302 −1.10370 −0.551849 0.833944i \(-0.686078\pi\)
−0.551849 + 0.833944i \(0.686078\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −6.14661 −0.290725
\(448\) 0 0
\(449\) −8.06852 −0.380777 −0.190388 0.981709i \(-0.560975\pi\)
−0.190388 + 0.981709i \(0.560975\pi\)
\(450\) 0 0
\(451\) −36.7398 −1.73001
\(452\) 0 0
\(453\) −10.8281 −0.508749
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −23.8576 −1.11601 −0.558005 0.829837i \(-0.688433\pi\)
−0.558005 + 0.829837i \(0.688433\pi\)
\(458\) 0 0
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −19.0493 −0.887214 −0.443607 0.896221i \(-0.646301\pi\)
−0.443607 + 0.896221i \(0.646301\pi\)
\(462\) 0 0
\(463\) 29.8336 1.38648 0.693242 0.720705i \(-0.256182\pi\)
0.693242 + 0.720705i \(0.256182\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.82333 −0.0843735 −0.0421868 0.999110i \(-0.513432\pi\)
−0.0421868 + 0.999110i \(0.513432\pi\)
\(468\) 0 0
\(469\) 11.5589 0.533741
\(470\) 0 0
\(471\) −9.52464 −0.438872
\(472\) 0 0
\(473\) 17.7891 0.817942
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 11.5932 0.530815
\(478\) 0 0
\(479\) −4.30347 −0.196631 −0.0983153 0.995155i \(-0.531345\pi\)
−0.0983153 + 0.995155i \(0.531345\pi\)
\(480\) 0 0
\(481\) 1.72055 0.0784502
\(482\) 0 0
\(483\) −2.76984 −0.126032
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.33353 −0.105742 −0.0528711 0.998601i \(-0.516837\pi\)
−0.0528711 + 0.998601i \(0.516837\pi\)
\(488\) 0 0
\(489\) 22.2014 1.00398
\(490\) 0 0
\(491\) 18.6274 0.840644 0.420322 0.907375i \(-0.361917\pi\)
0.420322 + 0.907375i \(0.361917\pi\)
\(492\) 0 0
\(493\) 1.80410 0.0812524
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.57393 −0.339737
\(498\) 0 0
\(499\) 15.9254 0.712921 0.356460 0.934310i \(-0.383984\pi\)
0.356460 + 0.934310i \(0.383984\pi\)
\(500\) 0 0
\(501\) −6.27945 −0.280545
\(502\) 0 0
\(503\) −18.4946 −0.824633 −0.412316 0.911041i \(-0.635280\pi\)
−0.412316 + 0.911041i \(0.635280\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −23.6425 −1.04793 −0.523967 0.851739i \(-0.675548\pi\)
−0.523967 + 0.851739i \(0.675548\pi\)
\(510\) 0 0
\(511\) 2.03426 0.0899904
\(512\) 0 0
\(513\) −3.47536 −0.153441
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 4.46158 0.196220
\(518\) 0 0
\(519\) −5.83836 −0.256275
\(520\) 0 0
\(521\) 0.391806 0.0171653 0.00858266 0.999963i \(-0.497268\pi\)
0.00858266 + 0.999963i \(0.497268\pi\)
\(522\) 0 0
\(523\) 18.6370 0.814939 0.407470 0.913219i \(-0.366411\pi\)
0.407470 + 0.913219i \(0.366411\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.96095 0.390345
\(528\) 0 0
\(529\) −20.8233 −0.905362
\(530\) 0 0
\(531\) −10.4363 −0.452897
\(532\) 0 0
\(533\) 8.83836 0.382832
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 18.3480 0.791773
\(538\) 0 0
\(539\) −14.4466 −0.622257
\(540\) 0 0
\(541\) 32.8864 1.41390 0.706948 0.707265i \(-0.250071\pi\)
0.706948 + 0.707265i \(0.250071\pi\)
\(542\) 0 0
\(543\) −0.887647 −0.0380926
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.23016 −0.0525980 −0.0262990 0.999654i \(-0.508372\pi\)
−0.0262990 + 0.999654i \(0.508372\pi\)
\(548\) 0 0
\(549\) −0.195903 −0.00836093
\(550\) 0 0
\(551\) 6.26988 0.267106
\(552\) 0 0
\(553\) 15.5439 0.660993
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.1028 0.428069 0.214034 0.976826i \(-0.431340\pi\)
0.214034 + 0.976826i \(0.431340\pi\)
\(558\) 0 0
\(559\) −4.27945 −0.181002
\(560\) 0 0
\(561\) −4.15686 −0.175503
\(562\) 0 0
\(563\) −13.7410 −0.579116 −0.289558 0.957161i \(-0.593508\pi\)
−0.289558 + 0.957161i \(0.593508\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.87740 −0.0788435
\(568\) 0 0
\(569\) 35.4850 1.48761 0.743805 0.668397i \(-0.233019\pi\)
0.743805 + 0.668397i \(0.233019\pi\)
\(570\) 0 0
\(571\) 15.8672 0.664020 0.332010 0.943276i \(-0.392273\pi\)
0.332010 + 0.943276i \(0.392273\pi\)
\(572\) 0 0
\(573\) −15.6082 −0.652042
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 17.5000 0.728536 0.364268 0.931294i \(-0.381319\pi\)
0.364268 + 0.931294i \(0.381319\pi\)
\(578\) 0 0
\(579\) 5.96574 0.247928
\(580\) 0 0
\(581\) −8.32874 −0.345534
\(582\) 0 0
\(583\) 48.1911 1.99587
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 35.3227 1.45792 0.728962 0.684554i \(-0.240003\pi\)
0.728962 + 0.684554i \(0.240003\pi\)
\(588\) 0 0
\(589\) 31.1425 1.28320
\(590\) 0 0
\(591\) −23.2302 −0.955561
\(592\) 0 0
\(593\) 7.04929 0.289480 0.144740 0.989470i \(-0.453765\pi\)
0.144740 + 0.989470i \(0.453765\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.55890 0.104729
\(598\) 0 0
\(599\) 1.57393 0.0643092 0.0321546 0.999483i \(-0.489763\pi\)
0.0321546 + 0.999483i \(0.489763\pi\)
\(600\) 0 0
\(601\) −15.7603 −0.642875 −0.321437 0.946931i \(-0.604166\pi\)
−0.321437 + 0.946931i \(0.604166\pi\)
\(602\) 0 0
\(603\) −6.15686 −0.250727
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 12.0205 0.487896 0.243948 0.969788i \(-0.421557\pi\)
0.243948 + 0.969788i \(0.421557\pi\)
\(608\) 0 0
\(609\) 3.38702 0.137249
\(610\) 0 0
\(611\) −1.07331 −0.0434213
\(612\) 0 0
\(613\) 14.6274 0.590796 0.295398 0.955374i \(-0.404548\pi\)
0.295398 + 0.955374i \(0.404548\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −21.9219 −0.882543 −0.441271 0.897374i \(-0.645472\pi\)
−0.441271 + 0.897374i \(0.645472\pi\)
\(618\) 0 0
\(619\) −22.9069 −0.920705 −0.460353 0.887736i \(-0.652277\pi\)
−0.460353 + 0.887736i \(0.652277\pi\)
\(620\) 0 0
\(621\) 1.47536 0.0592040
\(622\) 0 0
\(623\) 2.24519 0.0899517
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −14.4466 −0.576940
\(628\) 0 0
\(629\) 1.72055 0.0686027
\(630\) 0 0
\(631\) −15.3287 −0.610228 −0.305114 0.952316i \(-0.598694\pi\)
−0.305114 + 0.952316i \(0.598694\pi\)
\(632\) 0 0
\(633\) −1.96574 −0.0781312
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.47536 0.137699
\(638\) 0 0
\(639\) 4.03426 0.159593
\(640\) 0 0
\(641\) 17.1370 0.676872 0.338436 0.940989i \(-0.390102\pi\)
0.338436 + 0.940989i \(0.390102\pi\)
\(642\) 0 0
\(643\) 15.4068 0.607586 0.303793 0.952738i \(-0.401747\pi\)
0.303793 + 0.952738i \(0.401747\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.8041 0.424753 0.212376 0.977188i \(-0.431880\pi\)
0.212376 + 0.977188i \(0.431880\pi\)
\(648\) 0 0
\(649\) −43.3822 −1.70290
\(650\) 0 0
\(651\) 16.8233 0.659358
\(652\) 0 0
\(653\) −28.8438 −1.12875 −0.564373 0.825520i \(-0.690882\pi\)
−0.564373 + 0.825520i \(0.690882\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.08355 −0.0422733
\(658\) 0 0
\(659\) 39.7891 1.54996 0.774981 0.631985i \(-0.217759\pi\)
0.774981 + 0.631985i \(0.217759\pi\)
\(660\) 0 0
\(661\) 40.6822 1.58235 0.791177 0.611588i \(-0.209469\pi\)
0.791177 + 0.611588i \(0.209469\pi\)
\(662\) 0 0
\(663\) 1.00000 0.0388368
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.66168 −0.103061
\(668\) 0 0
\(669\) −16.6617 −0.644178
\(670\) 0 0
\(671\) −0.814340 −0.0314372
\(672\) 0 0
\(673\) 21.7698 0.839166 0.419583 0.907717i \(-0.362176\pi\)
0.419583 + 0.907717i \(0.362176\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −12.9507 −0.497736 −0.248868 0.968537i \(-0.580059\pi\)
−0.248868 + 0.968537i \(0.580059\pi\)
\(678\) 0 0
\(679\) 7.96994 0.305858
\(680\) 0 0
\(681\) 0.0102429 0.000392507 0
\(682\) 0 0
\(683\) −6.89243 −0.263732 −0.131866 0.991268i \(-0.542097\pi\)
−0.131866 + 0.991268i \(0.542097\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −9.21639 −0.351627
\(688\) 0 0
\(689\) −11.5932 −0.441664
\(690\) 0 0
\(691\) 34.0830 1.29658 0.648289 0.761395i \(-0.275485\pi\)
0.648289 + 0.761395i \(0.275485\pi\)
\(692\) 0 0
\(693\) −7.80410 −0.296453
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8.83836 0.334777
\(698\) 0 0
\(699\) 7.19590 0.272174
\(700\) 0 0
\(701\) −4.58771 −0.173275 −0.0866377 0.996240i \(-0.527612\pi\)
−0.0866377 + 0.996240i \(0.527612\pi\)
\(702\) 0 0
\(703\) 5.97951 0.225522
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.6472 0.400431
\(708\) 0 0
\(709\) 15.7891 0.592971 0.296485 0.955037i \(-0.404185\pi\)
0.296485 + 0.955037i \(0.404185\pi\)
\(710\) 0 0
\(711\) −8.27945 −0.310504
\(712\) 0 0
\(713\) −13.2206 −0.495115
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.82811 0.255001
\(718\) 0 0
\(719\) 41.7891 1.55847 0.779235 0.626732i \(-0.215608\pi\)
0.779235 + 0.626732i \(0.215608\pi\)
\(720\) 0 0
\(721\) 19.4273 0.723511
\(722\) 0 0
\(723\) 16.2014 0.602535
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0.378032 0.0140204 0.00701021 0.999975i \(-0.497769\pi\)
0.00701021 + 0.999975i \(0.497769\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.27945 −0.158281
\(732\) 0 0
\(733\) 41.8138 1.54443 0.772213 0.635364i \(-0.219150\pi\)
0.772213 + 0.635364i \(0.219150\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −25.5932 −0.942736
\(738\) 0 0
\(739\) −15.9117 −0.585320 −0.292660 0.956217i \(-0.594540\pi\)
−0.292660 + 0.956217i \(0.594540\pi\)
\(740\) 0 0
\(741\) 3.47536 0.127670
\(742\) 0 0
\(743\) −25.4213 −0.932616 −0.466308 0.884622i \(-0.654416\pi\)
−0.466308 + 0.884622i \(0.654416\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4.43631 0.162316
\(748\) 0 0
\(749\) −10.0685 −0.367896
\(750\) 0 0
\(751\) 31.7795 1.15965 0.579825 0.814741i \(-0.303121\pi\)
0.579825 + 0.814741i \(0.303121\pi\)
\(752\) 0 0
\(753\) −23.9562 −0.873011
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −23.2356 −0.844513 −0.422256 0.906476i \(-0.638762\pi\)
−0.422256 + 0.906476i \(0.638762\pi\)
\(758\) 0 0
\(759\) 6.13284 0.222608
\(760\) 0 0
\(761\) −5.08355 −0.184279 −0.0921393 0.995746i \(-0.529370\pi\)
−0.0921393 + 0.995746i \(0.529370\pi\)
\(762\) 0 0
\(763\) 1.72055 0.0622880
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.4363 0.376833
\(768\) 0 0
\(769\) −5.75481 −0.207524 −0.103762 0.994602i \(-0.533088\pi\)
−0.103762 + 0.994602i \(0.533088\pi\)
\(770\) 0 0
\(771\) −29.6617 −1.06824
\(772\) 0 0
\(773\) −8.24519 −0.296559 −0.148279 0.988945i \(-0.547374\pi\)
−0.148279 + 0.988945i \(0.547374\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.23016 0.115881
\(778\) 0 0
\(779\) 30.7164 1.10053
\(780\) 0 0
\(781\) 16.7698 0.600072
\(782\) 0 0
\(783\) −1.80410 −0.0644732
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18.8624 0.672371 0.336186 0.941796i \(-0.390863\pi\)
0.336186 + 0.941796i \(0.390863\pi\)
\(788\) 0 0
\(789\) 6.21093 0.221115
\(790\) 0 0
\(791\) −1.50961 −0.0536757
\(792\) 0 0
\(793\) 0.195903 0.00695671
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.0097 0.850466 0.425233 0.905084i \(-0.360192\pi\)
0.425233 + 0.905084i \(0.360192\pi\)
\(798\) 0 0
\(799\) −1.07331 −0.0379709
\(800\) 0 0
\(801\) −1.19590 −0.0422551
\(802\) 0 0
\(803\) −4.50416 −0.158948
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 18.1178 0.637777
\(808\) 0 0
\(809\) −34.1466 −1.20053 −0.600265 0.799801i \(-0.704938\pi\)
−0.600265 + 0.799801i \(0.704938\pi\)
\(810\) 0 0
\(811\) −5.77882 −0.202922 −0.101461 0.994840i \(-0.532352\pi\)
−0.101461 + 0.994840i \(0.532352\pi\)
\(812\) 0 0
\(813\) −30.8966 −1.08359
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −14.8726 −0.520327
\(818\) 0 0
\(819\) 1.87740 0.0656018
\(820\) 0 0
\(821\) −34.9069 −1.21826 −0.609129 0.793071i \(-0.708481\pi\)
−0.609129 + 0.793071i \(0.708481\pi\)
\(822\) 0 0
\(823\) 27.1383 0.945981 0.472991 0.881067i \(-0.343174\pi\)
0.472991 + 0.881067i \(0.343174\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.13763 0.109106 0.0545530 0.998511i \(-0.482627\pi\)
0.0545530 + 0.998511i \(0.482627\pi\)
\(828\) 0 0
\(829\) −9.34377 −0.324523 −0.162261 0.986748i \(-0.551879\pi\)
−0.162261 + 0.986748i \(0.551879\pi\)
\(830\) 0 0
\(831\) −26.7740 −0.928781
\(832\) 0 0
\(833\) 3.47536 0.120414
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −8.96095 −0.309736
\(838\) 0 0
\(839\) −0.661684 −0.0228439 −0.0114219 0.999935i \(-0.503636\pi\)
−0.0114219 + 0.999935i \(0.503636\pi\)
\(840\) 0 0
\(841\) −25.7452 −0.887767
\(842\) 0 0
\(843\) 4.16710 0.143523
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −11.7891 −0.405077
\(848\) 0 0
\(849\) 20.4603 0.702197
\(850\) 0 0
\(851\) −2.53842 −0.0870159
\(852\) 0 0
\(853\) −37.4110 −1.28093 −0.640465 0.767988i \(-0.721258\pi\)
−0.640465 + 0.767988i \(0.721258\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 25.2164 0.861375 0.430688 0.902501i \(-0.358271\pi\)
0.430688 + 0.902501i \(0.358271\pi\)
\(858\) 0 0
\(859\) 38.0247 1.29739 0.648693 0.761050i \(-0.275316\pi\)
0.648693 + 0.761050i \(0.275316\pi\)
\(860\) 0 0
\(861\) 16.5932 0.565494
\(862\) 0 0
\(863\) 15.9802 0.543972 0.271986 0.962301i \(-0.412320\pi\)
0.271986 + 0.962301i \(0.412320\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) −34.4165 −1.16750
\(870\) 0 0
\(871\) 6.15686 0.208617
\(872\) 0 0
\(873\) −4.24519 −0.143678
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 15.6562 0.528673 0.264337 0.964430i \(-0.414847\pi\)
0.264337 + 0.964430i \(0.414847\pi\)
\(878\) 0 0
\(879\) −18.6274 −0.628287
\(880\) 0 0
\(881\) 2.08355 0.0701966 0.0350983 0.999384i \(-0.488826\pi\)
0.0350983 + 0.999384i \(0.488826\pi\)
\(882\) 0 0
\(883\) −9.89185 −0.332887 −0.166444 0.986051i \(-0.553228\pi\)
−0.166444 + 0.986051i \(0.553228\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.44655 0.216454 0.108227 0.994126i \(-0.465483\pi\)
0.108227 + 0.994126i \(0.465483\pi\)
\(888\) 0 0
\(889\) −11.3287 −0.379954
\(890\) 0 0
\(891\) 4.15686 0.139260
\(892\) 0 0
\(893\) −3.73012 −0.124824
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1.47536 −0.0492607
\(898\) 0 0
\(899\) 16.1664 0.539181
\(900\) 0 0
\(901\) −11.5932 −0.386224
\(902\) 0 0
\(903\) −8.03426 −0.267363
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 29.4110 0.976577 0.488289 0.872682i \(-0.337621\pi\)
0.488289 + 0.872682i \(0.337621\pi\)
\(908\) 0 0
\(909\) −5.67126 −0.188104
\(910\) 0 0
\(911\) −16.7603 −0.555292 −0.277646 0.960683i \(-0.589554\pi\)
−0.277646 + 0.960683i \(0.589554\pi\)
\(912\) 0 0
\(913\) 18.4411 0.610311
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −32.2014 −1.06338
\(918\) 0 0
\(919\) −13.0973 −0.432041 −0.216020 0.976389i \(-0.569308\pi\)
−0.216020 + 0.976389i \(0.569308\pi\)
\(920\) 0 0
\(921\) 25.7891 0.849779
\(922\) 0 0
\(923\) −4.03426 −0.132789
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −10.3480 −0.339872
\(928\) 0 0
\(929\) 12.7945 0.419775 0.209887 0.977726i \(-0.432690\pi\)
0.209887 + 0.977726i \(0.432690\pi\)
\(930\) 0 0
\(931\) 12.0781 0.395844
\(932\) 0 0
\(933\) −16.4603 −0.538887
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.0397 0.720006 0.360003 0.932951i \(-0.382776\pi\)
0.360003 + 0.932951i \(0.382776\pi\)
\(938\) 0 0
\(939\) 33.1028 1.08027
\(940\) 0 0
\(941\) −58.7603 −1.91553 −0.957765 0.287552i \(-0.907158\pi\)
−0.957765 + 0.287552i \(0.907158\pi\)
\(942\) 0 0
\(943\) −13.0397 −0.424632
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 47.2747 1.53622 0.768110 0.640318i \(-0.221197\pi\)
0.768110 + 0.640318i \(0.221197\pi\)
\(948\) 0 0
\(949\) 1.08355 0.0351735
\(950\) 0 0
\(951\) −21.0835 −0.683681
\(952\) 0 0
\(953\) 15.8083 0.512081 0.256040 0.966666i \(-0.417582\pi\)
0.256040 + 0.966666i \(0.417582\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −7.49937 −0.242420
\(958\) 0 0
\(959\) 1.19590 0.0386177
\(960\) 0 0
\(961\) 49.2987 1.59028
\(962\) 0 0
\(963\) 5.36300 0.172820
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −24.6377 −0.792294 −0.396147 0.918187i \(-0.629653\pi\)
−0.396147 + 0.918187i \(0.629653\pi\)
\(968\) 0 0
\(969\) 3.47536 0.111644
\(970\) 0 0
\(971\) −8.60694 −0.276210 −0.138105 0.990418i \(-0.544101\pi\)
−0.138105 + 0.990418i \(0.544101\pi\)
\(972\) 0 0
\(973\) 28.3822 0.909893
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −19.4411 −0.621976 −0.310988 0.950414i \(-0.600660\pi\)
−0.310988 + 0.950414i \(0.600660\pi\)
\(978\) 0 0
\(979\) −4.97120 −0.158880
\(980\) 0 0
\(981\) −0.916451 −0.0292600
\(982\) 0 0
\(983\) −30.5691 −0.975004 −0.487502 0.873122i \(-0.662092\pi\)
−0.487502 + 0.873122i \(0.662092\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2.01503 −0.0641391
\(988\) 0 0
\(989\) 6.31371 0.200764
\(990\) 0 0
\(991\) 18.1713 0.577230 0.288615 0.957445i \(-0.406805\pi\)
0.288615 + 0.957445i \(0.406805\pi\)
\(992\) 0 0
\(993\) −5.08355 −0.161322
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −45.1275 −1.42920 −0.714601 0.699533i \(-0.753392\pi\)
−0.714601 + 0.699533i \(0.753392\pi\)
\(998\) 0 0
\(999\) −1.72055 −0.0544357
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 7800.2.a.br.1.1 yes 3
5.4 even 2 7800.2.a.bg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.bg.1.3 3 5.4 even 2
7800.2.a.br.1.1 yes 3 1.1 even 1 trivial