Properties

Label 768.4.a.g.1.1
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.3134668844\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 192)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -10.3923 q^{5} -3.46410 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -10.3923 q^{5} -3.46410 q^{7} +9.00000 q^{9} +55.4256 q^{13} +31.1769 q^{15} -90.0000 q^{17} +116.000 q^{19} +10.3923 q^{21} -103.923 q^{23} -17.0000 q^{25} -27.0000 q^{27} +259.808 q^{29} +301.377 q^{31} +36.0000 q^{35} -34.6410 q^{37} -166.277 q^{39} +54.0000 q^{41} +20.0000 q^{43} -93.5307 q^{45} -394.908 q^{47} -331.000 q^{49} +270.000 q^{51} -488.438 q^{53} -348.000 q^{57} +324.000 q^{59} -575.041 q^{61} -31.1769 q^{63} -576.000 q^{65} -116.000 q^{67} +311.769 q^{69} +1101.58 q^{71} -1106.00 q^{73} +51.0000 q^{75} +148.956 q^{79} +81.0000 q^{81} -1152.00 q^{83} +935.307 q^{85} -779.423 q^{87} -918.000 q^{89} -192.000 q^{91} -904.131 q^{93} -1205.51 q^{95} +190.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 18 q^{9} - 180 q^{17} + 232 q^{19} - 34 q^{25} - 54 q^{27} + 72 q^{35} + 108 q^{41} + 40 q^{43} - 662 q^{49} + 540 q^{51} - 696 q^{57} + 648 q^{59} - 1152 q^{65} - 232 q^{67} - 2212 q^{73} + 102 q^{75} + 162 q^{81} - 2304 q^{83} - 1836 q^{89} - 384 q^{91} + 380 q^{97}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −10.3923 −0.929516 −0.464758 0.885438i \(-0.653859\pi\)
−0.464758 + 0.885438i \(0.653859\pi\)
\(6\) 0 0
\(7\) −3.46410 −0.187044 −0.0935220 0.995617i \(-0.529813\pi\)
−0.0935220 + 0.995617i \(0.529813\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 55.4256 1.18248 0.591242 0.806494i \(-0.298638\pi\)
0.591242 + 0.806494i \(0.298638\pi\)
\(14\) 0 0
\(15\) 31.1769 0.536656
\(16\) 0 0
\(17\) −90.0000 −1.28401 −0.642006 0.766700i \(-0.721898\pi\)
−0.642006 + 0.766700i \(0.721898\pi\)
\(18\) 0 0
\(19\) 116.000 1.40064 0.700322 0.713827i \(-0.253040\pi\)
0.700322 + 0.713827i \(0.253040\pi\)
\(20\) 0 0
\(21\) 10.3923 0.107990
\(22\) 0 0
\(23\) −103.923 −0.942150 −0.471075 0.882093i \(-0.656134\pi\)
−0.471075 + 0.882093i \(0.656134\pi\)
\(24\) 0 0
\(25\) −17.0000 −0.136000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 259.808 1.66362 0.831811 0.555058i \(-0.187304\pi\)
0.831811 + 0.555058i \(0.187304\pi\)
\(30\) 0 0
\(31\) 301.377 1.74609 0.873046 0.487637i \(-0.162141\pi\)
0.873046 + 0.487637i \(0.162141\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 36.0000 0.173860
\(36\) 0 0
\(37\) −34.6410 −0.153918 −0.0769588 0.997034i \(-0.524521\pi\)
−0.0769588 + 0.997034i \(0.524521\pi\)
\(38\) 0 0
\(39\) −166.277 −0.682708
\(40\) 0 0
\(41\) 54.0000 0.205692 0.102846 0.994697i \(-0.467205\pi\)
0.102846 + 0.994697i \(0.467205\pi\)
\(42\) 0 0
\(43\) 20.0000 0.0709296 0.0354648 0.999371i \(-0.488709\pi\)
0.0354648 + 0.999371i \(0.488709\pi\)
\(44\) 0 0
\(45\) −93.5307 −0.309839
\(46\) 0 0
\(47\) −394.908 −1.22560 −0.612800 0.790238i \(-0.709957\pi\)
−0.612800 + 0.790238i \(0.709957\pi\)
\(48\) 0 0
\(49\) −331.000 −0.965015
\(50\) 0 0
\(51\) 270.000 0.741325
\(52\) 0 0
\(53\) −488.438 −1.26589 −0.632945 0.774197i \(-0.718154\pi\)
−0.632945 + 0.774197i \(0.718154\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −348.000 −0.808662
\(58\) 0 0
\(59\) 324.000 0.714936 0.357468 0.933925i \(-0.383640\pi\)
0.357468 + 0.933925i \(0.383640\pi\)
\(60\) 0 0
\(61\) −575.041 −1.20699 −0.603495 0.797366i \(-0.706226\pi\)
−0.603495 + 0.797366i \(0.706226\pi\)
\(62\) 0 0
\(63\) −31.1769 −0.0623480
\(64\) 0 0
\(65\) −576.000 −1.09914
\(66\) 0 0
\(67\) −116.000 −0.211517 −0.105759 0.994392i \(-0.533727\pi\)
−0.105759 + 0.994392i \(0.533727\pi\)
\(68\) 0 0
\(69\) 311.769 0.543951
\(70\) 0 0
\(71\) 1101.58 1.84132 0.920662 0.390361i \(-0.127650\pi\)
0.920662 + 0.390361i \(0.127650\pi\)
\(72\) 0 0
\(73\) −1106.00 −1.77325 −0.886627 0.462486i \(-0.846958\pi\)
−0.886627 + 0.462486i \(0.846958\pi\)
\(74\) 0 0
\(75\) 51.0000 0.0785196
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 148.956 0.212138 0.106069 0.994359i \(-0.466174\pi\)
0.106069 + 0.994359i \(0.466174\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1152.00 −1.52348 −0.761738 0.647886i \(-0.775653\pi\)
−0.761738 + 0.647886i \(0.775653\pi\)
\(84\) 0 0
\(85\) 935.307 1.19351
\(86\) 0 0
\(87\) −779.423 −0.960493
\(88\) 0 0
\(89\) −918.000 −1.09335 −0.546673 0.837346i \(-0.684106\pi\)
−0.546673 + 0.837346i \(0.684106\pi\)
\(90\) 0 0
\(91\) −192.000 −0.221177
\(92\) 0 0
\(93\) −904.131 −1.00811
\(94\) 0 0
\(95\) −1205.51 −1.30192
\(96\) 0 0
\(97\) 190.000 0.198882 0.0994411 0.995043i \(-0.468295\pi\)
0.0994411 + 0.995043i \(0.468295\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.3923 0.0102383 0.00511917 0.999987i \(-0.498371\pi\)
0.00511917 + 0.999987i \(0.498371\pi\)
\(102\) 0 0
\(103\) 793.279 0.758875 0.379438 0.925217i \(-0.376118\pi\)
0.379438 + 0.925217i \(0.376118\pi\)
\(104\) 0 0
\(105\) −108.000 −0.100378
\(106\) 0 0
\(107\) −252.000 −0.227680 −0.113840 0.993499i \(-0.536315\pi\)
−0.113840 + 0.993499i \(0.536315\pi\)
\(108\) 0 0
\(109\) −457.261 −0.401814 −0.200907 0.979610i \(-0.564389\pi\)
−0.200907 + 0.979610i \(0.564389\pi\)
\(110\) 0 0
\(111\) 103.923 0.0888643
\(112\) 0 0
\(113\) −2214.00 −1.84315 −0.921573 0.388204i \(-0.873096\pi\)
−0.921573 + 0.388204i \(0.873096\pi\)
\(114\) 0 0
\(115\) 1080.00 0.875744
\(116\) 0 0
\(117\) 498.831 0.394162
\(118\) 0 0
\(119\) 311.769 0.240167
\(120\) 0 0
\(121\) −1331.00 −1.00000
\(122\) 0 0
\(123\) −162.000 −0.118756
\(124\) 0 0
\(125\) 1475.71 1.05593
\(126\) 0 0
\(127\) 696.284 0.486498 0.243249 0.969964i \(-0.421787\pi\)
0.243249 + 0.969964i \(0.421787\pi\)
\(128\) 0 0
\(129\) −60.0000 −0.0409512
\(130\) 0 0
\(131\) −2268.00 −1.51264 −0.756321 0.654201i \(-0.773005\pi\)
−0.756321 + 0.654201i \(0.773005\pi\)
\(132\) 0 0
\(133\) −401.836 −0.261982
\(134\) 0 0
\(135\) 280.592 0.178885
\(136\) 0 0
\(137\) −522.000 −0.325529 −0.162764 0.986665i \(-0.552041\pi\)
−0.162764 + 0.986665i \(0.552041\pi\)
\(138\) 0 0
\(139\) 676.000 0.412501 0.206250 0.978499i \(-0.433874\pi\)
0.206250 + 0.978499i \(0.433874\pi\)
\(140\) 0 0
\(141\) 1184.72 0.707600
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −2700.00 −1.54636
\(146\) 0 0
\(147\) 993.000 0.557151
\(148\) 0 0
\(149\) −1465.31 −0.805660 −0.402830 0.915275i \(-0.631973\pi\)
−0.402830 + 0.915275i \(0.631973\pi\)
\(150\) 0 0
\(151\) −2386.77 −1.28631 −0.643153 0.765738i \(-0.722374\pi\)
−0.643153 + 0.765738i \(0.722374\pi\)
\(152\) 0 0
\(153\) −810.000 −0.428004
\(154\) 0 0
\(155\) −3132.00 −1.62302
\(156\) 0 0
\(157\) 2016.11 1.02486 0.512430 0.858729i \(-0.328746\pi\)
0.512430 + 0.858729i \(0.328746\pi\)
\(158\) 0 0
\(159\) 1465.31 0.730862
\(160\) 0 0
\(161\) 360.000 0.176223
\(162\) 0 0
\(163\) −388.000 −0.186445 −0.0932224 0.995645i \(-0.529717\pi\)
−0.0932224 + 0.995645i \(0.529717\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2203.17 −1.02088 −0.510438 0.859915i \(-0.670517\pi\)
−0.510438 + 0.859915i \(0.670517\pi\)
\(168\) 0 0
\(169\) 875.000 0.398270
\(170\) 0 0
\(171\) 1044.00 0.466881
\(172\) 0 0
\(173\) 197.454 0.0867753 0.0433877 0.999058i \(-0.486185\pi\)
0.0433877 + 0.999058i \(0.486185\pi\)
\(174\) 0 0
\(175\) 58.8897 0.0254380
\(176\) 0 0
\(177\) −972.000 −0.412768
\(178\) 0 0
\(179\) 2844.00 1.18754 0.593772 0.804633i \(-0.297638\pi\)
0.593772 + 0.804633i \(0.297638\pi\)
\(180\) 0 0
\(181\) 96.9948 0.0398319 0.0199159 0.999802i \(-0.493660\pi\)
0.0199159 + 0.999802i \(0.493660\pi\)
\(182\) 0 0
\(183\) 1725.12 0.696856
\(184\) 0 0
\(185\) 360.000 0.143069
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 93.5307 0.0359966
\(190\) 0 0
\(191\) 3200.83 1.21259 0.606293 0.795241i \(-0.292656\pi\)
0.606293 + 0.795241i \(0.292656\pi\)
\(192\) 0 0
\(193\) −1342.00 −0.500514 −0.250257 0.968179i \(-0.580515\pi\)
−0.250257 + 0.968179i \(0.580515\pi\)
\(194\) 0 0
\(195\) 1728.00 0.634588
\(196\) 0 0
\(197\) 966.484 0.349539 0.174769 0.984609i \(-0.444082\pi\)
0.174769 + 0.984609i \(0.444082\pi\)
\(198\) 0 0
\(199\) −1784.01 −0.635504 −0.317752 0.948174i \(-0.602928\pi\)
−0.317752 + 0.948174i \(0.602928\pi\)
\(200\) 0 0
\(201\) 348.000 0.122120
\(202\) 0 0
\(203\) −900.000 −0.311171
\(204\) 0 0
\(205\) −561.184 −0.191194
\(206\) 0 0
\(207\) −935.307 −0.314050
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 2764.00 0.901809 0.450904 0.892572i \(-0.351102\pi\)
0.450904 + 0.892572i \(0.351102\pi\)
\(212\) 0 0
\(213\) −3304.75 −1.06309
\(214\) 0 0
\(215\) −207.846 −0.0659302
\(216\) 0 0
\(217\) −1044.00 −0.326596
\(218\) 0 0
\(219\) 3318.00 1.02379
\(220\) 0 0
\(221\) −4988.31 −1.51832
\(222\) 0 0
\(223\) −4292.02 −1.28886 −0.644428 0.764665i \(-0.722905\pi\)
−0.644428 + 0.764665i \(0.722905\pi\)
\(224\) 0 0
\(225\) −153.000 −0.0453333
\(226\) 0 0
\(227\) −5688.00 −1.66311 −0.831555 0.555443i \(-0.812549\pi\)
−0.831555 + 0.555443i \(0.812549\pi\)
\(228\) 0 0
\(229\) −5570.28 −1.60740 −0.803699 0.595036i \(-0.797138\pi\)
−0.803699 + 0.595036i \(0.797138\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2718.00 −0.764215 −0.382108 0.924118i \(-0.624802\pi\)
−0.382108 + 0.924118i \(0.624802\pi\)
\(234\) 0 0
\(235\) 4104.00 1.13921
\(236\) 0 0
\(237\) −446.869 −0.122478
\(238\) 0 0
\(239\) 3574.95 0.967550 0.483775 0.875192i \(-0.339265\pi\)
0.483775 + 0.875192i \(0.339265\pi\)
\(240\) 0 0
\(241\) 4490.00 1.20011 0.600055 0.799959i \(-0.295146\pi\)
0.600055 + 0.799959i \(0.295146\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 3439.85 0.896996
\(246\) 0 0
\(247\) 6429.37 1.65624
\(248\) 0 0
\(249\) 3456.00 0.879579
\(250\) 0 0
\(251\) −4608.00 −1.15878 −0.579391 0.815050i \(-0.696710\pi\)
−0.579391 + 0.815050i \(0.696710\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −2805.92 −0.689073
\(256\) 0 0
\(257\) 4626.00 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 120.000 0.0287893
\(260\) 0 0
\(261\) 2338.27 0.554541
\(262\) 0 0
\(263\) 1995.32 0.467821 0.233910 0.972258i \(-0.424848\pi\)
0.233910 + 0.972258i \(0.424848\pi\)
\(264\) 0 0
\(265\) 5076.00 1.17666
\(266\) 0 0
\(267\) 2754.00 0.631244
\(268\) 0 0
\(269\) −3148.87 −0.713717 −0.356859 0.934158i \(-0.616152\pi\)
−0.356859 + 0.934158i \(0.616152\pi\)
\(270\) 0 0
\(271\) −5345.11 −1.19813 −0.599063 0.800702i \(-0.704460\pi\)
−0.599063 + 0.800702i \(0.704460\pi\)
\(272\) 0 0
\(273\) 576.000 0.127696
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6526.37 1.41564 0.707818 0.706394i \(-0.249679\pi\)
0.707818 + 0.706394i \(0.249679\pi\)
\(278\) 0 0
\(279\) 2712.39 0.582031
\(280\) 0 0
\(281\) 1170.00 0.248386 0.124193 0.992258i \(-0.460366\pi\)
0.124193 + 0.992258i \(0.460366\pi\)
\(282\) 0 0
\(283\) 5740.00 1.20568 0.602840 0.797862i \(-0.294036\pi\)
0.602840 + 0.797862i \(0.294036\pi\)
\(284\) 0 0
\(285\) 3616.52 0.751664
\(286\) 0 0
\(287\) −187.061 −0.0384735
\(288\) 0 0
\(289\) 3187.00 0.648687
\(290\) 0 0
\(291\) −570.000 −0.114825
\(292\) 0 0
\(293\) 7991.68 1.59344 0.796722 0.604346i \(-0.206566\pi\)
0.796722 + 0.604346i \(0.206566\pi\)
\(294\) 0 0
\(295\) −3367.11 −0.664544
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5760.00 −1.11408
\(300\) 0 0
\(301\) −69.2820 −0.0132669
\(302\) 0 0
\(303\) −31.1769 −0.00591111
\(304\) 0 0
\(305\) 5976.00 1.12192
\(306\) 0 0
\(307\) 5452.00 1.01356 0.506779 0.862076i \(-0.330836\pi\)
0.506779 + 0.862076i \(0.330836\pi\)
\(308\) 0 0
\(309\) −2379.84 −0.438137
\(310\) 0 0
\(311\) −2203.17 −0.401705 −0.200852 0.979622i \(-0.564371\pi\)
−0.200852 + 0.979622i \(0.564371\pi\)
\(312\) 0 0
\(313\) 1034.00 0.186726 0.0933628 0.995632i \(-0.470238\pi\)
0.0933628 + 0.995632i \(0.470238\pi\)
\(314\) 0 0
\(315\) 324.000 0.0579534
\(316\) 0 0
\(317\) −2650.04 −0.469530 −0.234765 0.972052i \(-0.575432\pi\)
−0.234765 + 0.972052i \(0.575432\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 756.000 0.131451
\(322\) 0 0
\(323\) −10440.0 −1.79844
\(324\) 0 0
\(325\) −942.236 −0.160818
\(326\) 0 0
\(327\) 1371.78 0.231987
\(328\) 0 0
\(329\) 1368.00 0.229241
\(330\) 0 0
\(331\) 4132.00 0.686149 0.343074 0.939308i \(-0.388532\pi\)
0.343074 + 0.939308i \(0.388532\pi\)
\(332\) 0 0
\(333\) −311.769 −0.0513058
\(334\) 0 0
\(335\) 1205.51 0.196609
\(336\) 0 0
\(337\) −458.000 −0.0740322 −0.0370161 0.999315i \(-0.511785\pi\)
−0.0370161 + 0.999315i \(0.511785\pi\)
\(338\) 0 0
\(339\) 6642.00 1.06414
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 2334.80 0.367544
\(344\) 0 0
\(345\) −3240.00 −0.505611
\(346\) 0 0
\(347\) −11016.0 −1.70424 −0.852118 0.523350i \(-0.824682\pi\)
−0.852118 + 0.523350i \(0.824682\pi\)
\(348\) 0 0
\(349\) 2528.79 0.387860 0.193930 0.981015i \(-0.437876\pi\)
0.193930 + 0.981015i \(0.437876\pi\)
\(350\) 0 0
\(351\) −1496.49 −0.227569
\(352\) 0 0
\(353\) 5562.00 0.838627 0.419314 0.907841i \(-0.362271\pi\)
0.419314 + 0.907841i \(0.362271\pi\)
\(354\) 0 0
\(355\) −11448.0 −1.71154
\(356\) 0 0
\(357\) −935.307 −0.138660
\(358\) 0 0
\(359\) −8875.03 −1.30475 −0.652376 0.757895i \(-0.726228\pi\)
−0.652376 + 0.757895i \(0.726228\pi\)
\(360\) 0 0
\(361\) 6597.00 0.961802
\(362\) 0 0
\(363\) 3993.00 0.577350
\(364\) 0 0
\(365\) 11493.9 1.64827
\(366\) 0 0
\(367\) 12799.9 1.82056 0.910282 0.413989i \(-0.135865\pi\)
0.910282 + 0.413989i \(0.135865\pi\)
\(368\) 0 0
\(369\) 486.000 0.0685641
\(370\) 0 0
\(371\) 1692.00 0.236777
\(372\) 0 0
\(373\) −4981.38 −0.691491 −0.345745 0.938328i \(-0.612374\pi\)
−0.345745 + 0.938328i \(0.612374\pi\)
\(374\) 0 0
\(375\) −4427.12 −0.609642
\(376\) 0 0
\(377\) 14400.0 1.96721
\(378\) 0 0
\(379\) −9892.00 −1.34068 −0.670340 0.742054i \(-0.733852\pi\)
−0.670340 + 0.742054i \(0.733852\pi\)
\(380\) 0 0
\(381\) −2088.85 −0.280880
\(382\) 0 0
\(383\) −8771.11 −1.17019 −0.585095 0.810965i \(-0.698943\pi\)
−0.585095 + 0.810965i \(0.698943\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 180.000 0.0236432
\(388\) 0 0
\(389\) −9903.87 −1.29086 −0.645432 0.763818i \(-0.723323\pi\)
−0.645432 + 0.763818i \(0.723323\pi\)
\(390\) 0 0
\(391\) 9353.07 1.20973
\(392\) 0 0
\(393\) 6804.00 0.873324
\(394\) 0 0
\(395\) −1548.00 −0.197186
\(396\) 0 0
\(397\) −103.923 −0.0131379 −0.00656895 0.999978i \(-0.502091\pi\)
−0.00656895 + 0.999978i \(0.502091\pi\)
\(398\) 0 0
\(399\) 1205.51 0.151255
\(400\) 0 0
\(401\) 1062.00 0.132254 0.0661269 0.997811i \(-0.478936\pi\)
0.0661269 + 0.997811i \(0.478936\pi\)
\(402\) 0 0
\(403\) 16704.0 2.06473
\(404\) 0 0
\(405\) −841.777 −0.103280
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 8614.00 1.04141 0.520703 0.853738i \(-0.325670\pi\)
0.520703 + 0.853738i \(0.325670\pi\)
\(410\) 0 0
\(411\) 1566.00 0.187944
\(412\) 0 0
\(413\) −1122.37 −0.133724
\(414\) 0 0
\(415\) 11971.9 1.41609
\(416\) 0 0
\(417\) −2028.00 −0.238157
\(418\) 0 0
\(419\) −10440.0 −1.21725 −0.608625 0.793458i \(-0.708278\pi\)
−0.608625 + 0.793458i \(0.708278\pi\)
\(420\) 0 0
\(421\) 900.666 0.104266 0.0521328 0.998640i \(-0.483398\pi\)
0.0521328 + 0.998640i \(0.483398\pi\)
\(422\) 0 0
\(423\) −3554.17 −0.408533
\(424\) 0 0
\(425\) 1530.00 0.174626
\(426\) 0 0
\(427\) 1992.00 0.225760
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 394.908 0.0441346 0.0220673 0.999756i \(-0.492975\pi\)
0.0220673 + 0.999756i \(0.492975\pi\)
\(432\) 0 0
\(433\) 12958.0 1.43816 0.719078 0.694929i \(-0.244564\pi\)
0.719078 + 0.694929i \(0.244564\pi\)
\(434\) 0 0
\(435\) 8100.00 0.892794
\(436\) 0 0
\(437\) −12055.1 −1.31962
\(438\) 0 0
\(439\) −11441.9 −1.24395 −0.621974 0.783038i \(-0.713669\pi\)
−0.621974 + 0.783038i \(0.713669\pi\)
\(440\) 0 0
\(441\) −2979.00 −0.321672
\(442\) 0 0
\(443\) −1800.00 −0.193049 −0.0965244 0.995331i \(-0.530773\pi\)
−0.0965244 + 0.995331i \(0.530773\pi\)
\(444\) 0 0
\(445\) 9540.14 1.01628
\(446\) 0 0
\(447\) 4395.94 0.465148
\(448\) 0 0
\(449\) −13626.0 −1.43218 −0.716092 0.698006i \(-0.754071\pi\)
−0.716092 + 0.698006i \(0.754071\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 7160.30 0.742649
\(454\) 0 0
\(455\) 1995.32 0.205587
\(456\) 0 0
\(457\) −12602.0 −1.28993 −0.644964 0.764213i \(-0.723127\pi\)
−0.644964 + 0.764213i \(0.723127\pi\)
\(458\) 0 0
\(459\) 2430.00 0.247108
\(460\) 0 0
\(461\) −1839.44 −0.185838 −0.0929188 0.995674i \(-0.529620\pi\)
−0.0929188 + 0.995674i \(0.529620\pi\)
\(462\) 0 0
\(463\) −11012.4 −1.10538 −0.552688 0.833389i \(-0.686398\pi\)
−0.552688 + 0.833389i \(0.686398\pi\)
\(464\) 0 0
\(465\) 9396.00 0.937052
\(466\) 0 0
\(467\) −9144.00 −0.906068 −0.453034 0.891493i \(-0.649658\pi\)
−0.453034 + 0.891493i \(0.649658\pi\)
\(468\) 0 0
\(469\) 401.836 0.0395630
\(470\) 0 0
\(471\) −6048.32 −0.591703
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1972.00 −0.190488
\(476\) 0 0
\(477\) −4395.94 −0.421963
\(478\) 0 0
\(479\) 6173.03 0.588837 0.294418 0.955677i \(-0.404874\pi\)
0.294418 + 0.955677i \(0.404874\pi\)
\(480\) 0 0
\(481\) −1920.00 −0.182005
\(482\) 0 0
\(483\) −1080.00 −0.101743
\(484\) 0 0
\(485\) −1974.54 −0.184864
\(486\) 0 0
\(487\) −3204.29 −0.298153 −0.149076 0.988826i \(-0.547630\pi\)
−0.149076 + 0.988826i \(0.547630\pi\)
\(488\) 0 0
\(489\) 1164.00 0.107644
\(490\) 0 0
\(491\) 396.000 0.0363976 0.0181988 0.999834i \(-0.494207\pi\)
0.0181988 + 0.999834i \(0.494207\pi\)
\(492\) 0 0
\(493\) −23382.7 −2.13611
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3816.00 −0.344408
\(498\) 0 0
\(499\) 12436.0 1.11565 0.557827 0.829957i \(-0.311635\pi\)
0.557827 + 0.829957i \(0.311635\pi\)
\(500\) 0 0
\(501\) 6609.51 0.589403
\(502\) 0 0
\(503\) 16482.2 1.46104 0.730522 0.682890i \(-0.239277\pi\)
0.730522 + 0.682890i \(0.239277\pi\)
\(504\) 0 0
\(505\) −108.000 −0.00951671
\(506\) 0 0
\(507\) −2625.00 −0.229942
\(508\) 0 0
\(509\) 9155.62 0.797280 0.398640 0.917107i \(-0.369482\pi\)
0.398640 + 0.917107i \(0.369482\pi\)
\(510\) 0 0
\(511\) 3831.30 0.331676
\(512\) 0 0
\(513\) −3132.00 −0.269554
\(514\) 0 0
\(515\) −8244.00 −0.705386
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −592.361 −0.0500998
\(520\) 0 0
\(521\) −7650.00 −0.643287 −0.321644 0.946861i \(-0.604235\pi\)
−0.321644 + 0.946861i \(0.604235\pi\)
\(522\) 0 0
\(523\) 18332.0 1.53270 0.766350 0.642423i \(-0.222071\pi\)
0.766350 + 0.642423i \(0.222071\pi\)
\(524\) 0 0
\(525\) −176.669 −0.0146866
\(526\) 0 0
\(527\) −27123.9 −2.24200
\(528\) 0 0
\(529\) −1367.00 −0.112353
\(530\) 0 0
\(531\) 2916.00 0.238312
\(532\) 0 0
\(533\) 2992.98 0.243228
\(534\) 0 0
\(535\) 2618.86 0.211632
\(536\) 0 0
\(537\) −8532.00 −0.685629
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −16863.2 −1.34012 −0.670062 0.742305i \(-0.733733\pi\)
−0.670062 + 0.742305i \(0.733733\pi\)
\(542\) 0 0
\(543\) −290.985 −0.0229969
\(544\) 0 0
\(545\) 4752.00 0.373492
\(546\) 0 0
\(547\) −1684.00 −0.131632 −0.0658159 0.997832i \(-0.520965\pi\)
−0.0658159 + 0.997832i \(0.520965\pi\)
\(548\) 0 0
\(549\) −5175.37 −0.402330
\(550\) 0 0
\(551\) 30137.7 2.33014
\(552\) 0 0
\(553\) −516.000 −0.0396791
\(554\) 0 0
\(555\) −1080.00 −0.0826008
\(556\) 0 0
\(557\) 2275.91 0.173130 0.0865652 0.996246i \(-0.472411\pi\)
0.0865652 + 0.996246i \(0.472411\pi\)
\(558\) 0 0
\(559\) 1108.51 0.0838731
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7992.00 0.598264 0.299132 0.954212i \(-0.403303\pi\)
0.299132 + 0.954212i \(0.403303\pi\)
\(564\) 0 0
\(565\) 23008.6 1.71323
\(566\) 0 0
\(567\) −280.592 −0.0207827
\(568\) 0 0
\(569\) 5526.00 0.407139 0.203569 0.979061i \(-0.434746\pi\)
0.203569 + 0.979061i \(0.434746\pi\)
\(570\) 0 0
\(571\) 13420.0 0.983554 0.491777 0.870721i \(-0.336347\pi\)
0.491777 + 0.870721i \(0.336347\pi\)
\(572\) 0 0
\(573\) −9602.49 −0.700087
\(574\) 0 0
\(575\) 1766.69 0.128132
\(576\) 0 0
\(577\) −10178.0 −0.734343 −0.367171 0.930153i \(-0.619674\pi\)
−0.367171 + 0.930153i \(0.619674\pi\)
\(578\) 0 0
\(579\) 4026.00 0.288972
\(580\) 0 0
\(581\) 3990.65 0.284957
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −5184.00 −0.366380
\(586\) 0 0
\(587\) 18684.0 1.31375 0.656875 0.754000i \(-0.271878\pi\)
0.656875 + 0.754000i \(0.271878\pi\)
\(588\) 0 0
\(589\) 34959.7 2.44565
\(590\) 0 0
\(591\) −2899.45 −0.201806
\(592\) 0 0
\(593\) −5094.00 −0.352758 −0.176379 0.984322i \(-0.556438\pi\)
−0.176379 + 0.984322i \(0.556438\pi\)
\(594\) 0 0
\(595\) −3240.00 −0.223239
\(596\) 0 0
\(597\) 5352.04 0.366908
\(598\) 0 0
\(599\) −19433.6 −1.32560 −0.662801 0.748795i \(-0.730633\pi\)
−0.662801 + 0.748795i \(0.730633\pi\)
\(600\) 0 0
\(601\) −27722.0 −1.88154 −0.940769 0.339049i \(-0.889895\pi\)
−0.940769 + 0.339049i \(0.889895\pi\)
\(602\) 0 0
\(603\) −1044.00 −0.0705057
\(604\) 0 0
\(605\) 13832.2 0.929516
\(606\) 0 0
\(607\) −26684.0 −1.78430 −0.892149 0.451741i \(-0.850803\pi\)
−0.892149 + 0.451741i \(0.850803\pi\)
\(608\) 0 0
\(609\) 2700.00 0.179654
\(610\) 0 0
\(611\) −21888.0 −1.44925
\(612\) 0 0
\(613\) 16911.7 1.11429 0.557144 0.830416i \(-0.311897\pi\)
0.557144 + 0.830416i \(0.311897\pi\)
\(614\) 0 0
\(615\) 1683.55 0.110386
\(616\) 0 0
\(617\) −17694.0 −1.15451 −0.577256 0.816563i \(-0.695876\pi\)
−0.577256 + 0.816563i \(0.695876\pi\)
\(618\) 0 0
\(619\) −13652.0 −0.886462 −0.443231 0.896407i \(-0.646168\pi\)
−0.443231 + 0.896407i \(0.646168\pi\)
\(620\) 0 0
\(621\) 2805.92 0.181317
\(622\) 0 0
\(623\) 3180.05 0.204504
\(624\) 0 0
\(625\) −13211.0 −0.845504
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3117.69 0.197632
\(630\) 0 0
\(631\) −9162.55 −0.578059 −0.289030 0.957320i \(-0.593333\pi\)
−0.289030 + 0.957320i \(0.593333\pi\)
\(632\) 0 0
\(633\) −8292.00 −0.520659
\(634\) 0 0
\(635\) −7236.00 −0.452208
\(636\) 0 0
\(637\) −18345.9 −1.14112
\(638\) 0 0
\(639\) 9914.26 0.613775
\(640\) 0 0
\(641\) −5202.00 −0.320541 −0.160270 0.987073i \(-0.551237\pi\)
−0.160270 + 0.987073i \(0.551237\pi\)
\(642\) 0 0
\(643\) −15892.0 −0.974680 −0.487340 0.873212i \(-0.662033\pi\)
−0.487340 + 0.873212i \(0.662033\pi\)
\(644\) 0 0
\(645\) 623.538 0.0380648
\(646\) 0 0
\(647\) 478.046 0.0290478 0.0145239 0.999895i \(-0.495377\pi\)
0.0145239 + 0.999895i \(0.495377\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 3132.00 0.188560
\(652\) 0 0
\(653\) −24660.9 −1.47788 −0.738941 0.673770i \(-0.764674\pi\)
−0.738941 + 0.673770i \(0.764674\pi\)
\(654\) 0 0
\(655\) 23569.7 1.40602
\(656\) 0 0
\(657\) −9954.00 −0.591085
\(658\) 0 0
\(659\) 28260.0 1.67049 0.835245 0.549878i \(-0.185326\pi\)
0.835245 + 0.549878i \(0.185326\pi\)
\(660\) 0 0
\(661\) −25863.0 −1.52187 −0.760933 0.648830i \(-0.775258\pi\)
−0.760933 + 0.648830i \(0.775258\pi\)
\(662\) 0 0
\(663\) 14964.9 0.876605
\(664\) 0 0
\(665\) 4176.00 0.243516
\(666\) 0 0
\(667\) −27000.0 −1.56738
\(668\) 0 0
\(669\) 12876.1 0.744122
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 190.000 0.0108826 0.00544128 0.999985i \(-0.498268\pi\)
0.00544128 + 0.999985i \(0.498268\pi\)
\(674\) 0 0
\(675\) 459.000 0.0261732
\(676\) 0 0
\(677\) −4998.70 −0.283775 −0.141887 0.989883i \(-0.545317\pi\)
−0.141887 + 0.989883i \(0.545317\pi\)
\(678\) 0 0
\(679\) −658.179 −0.0371997
\(680\) 0 0
\(681\) 17064.0 0.960197
\(682\) 0 0
\(683\) −8064.00 −0.451772 −0.225886 0.974154i \(-0.572528\pi\)
−0.225886 + 0.974154i \(0.572528\pi\)
\(684\) 0 0
\(685\) 5424.78 0.302584
\(686\) 0 0
\(687\) 16710.8 0.928032
\(688\) 0 0
\(689\) −27072.0 −1.49690
\(690\) 0 0
\(691\) 19244.0 1.05944 0.529722 0.848171i \(-0.322296\pi\)
0.529722 + 0.848171i \(0.322296\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7025.20 −0.383426
\(696\) 0 0
\(697\) −4860.00 −0.264111
\(698\) 0 0
\(699\) 8154.00 0.441220
\(700\) 0 0
\(701\) 6204.21 0.334279 0.167140 0.985933i \(-0.446547\pi\)
0.167140 + 0.985933i \(0.446547\pi\)
\(702\) 0 0
\(703\) −4018.36 −0.215584
\(704\) 0 0
\(705\) −12312.0 −0.657726
\(706\) 0 0
\(707\) −36.0000 −0.00191502
\(708\) 0 0
\(709\) −15020.3 −0.795629 −0.397814 0.917466i \(-0.630231\pi\)
−0.397814 + 0.917466i \(0.630231\pi\)
\(710\) 0 0
\(711\) 1340.61 0.0707127
\(712\) 0 0
\(713\) −31320.0 −1.64508
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −10724.9 −0.558615
\(718\) 0 0
\(719\) 30740.4 1.59447 0.797236 0.603668i \(-0.206295\pi\)
0.797236 + 0.603668i \(0.206295\pi\)
\(720\) 0 0
\(721\) −2748.00 −0.141943
\(722\) 0 0
\(723\) −13470.0 −0.692883
\(724\) 0 0
\(725\) −4416.73 −0.226253
\(726\) 0 0
\(727\) −12127.8 −0.618701 −0.309351 0.950948i \(-0.600112\pi\)
−0.309351 + 0.950948i \(0.600112\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −1800.00 −0.0910744
\(732\) 0 0
\(733\) 12387.6 0.624212 0.312106 0.950047i \(-0.398966\pi\)
0.312106 + 0.950047i \(0.398966\pi\)
\(734\) 0 0
\(735\) −10319.6 −0.517881
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −11180.0 −0.556513 −0.278256 0.960507i \(-0.589756\pi\)
−0.278256 + 0.960507i \(0.589756\pi\)
\(740\) 0 0
\(741\) −19288.1 −0.956230
\(742\) 0 0
\(743\) −35500.1 −1.75286 −0.876429 0.481532i \(-0.840081\pi\)
−0.876429 + 0.481532i \(0.840081\pi\)
\(744\) 0 0
\(745\) 15228.0 0.748873
\(746\) 0 0
\(747\) −10368.0 −0.507825
\(748\) 0 0
\(749\) 872.954 0.0425862
\(750\) 0 0
\(751\) −37970.0 −1.84493 −0.922467 0.386076i \(-0.873830\pi\)
−0.922467 + 0.386076i \(0.873830\pi\)
\(752\) 0 0
\(753\) 13824.0 0.669023
\(754\) 0 0
\(755\) 24804.0 1.19564
\(756\) 0 0
\(757\) −39047.4 −1.87477 −0.937385 0.348296i \(-0.886760\pi\)
−0.937385 + 0.348296i \(0.886760\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 12222.0 0.582191 0.291095 0.956694i \(-0.405980\pi\)
0.291095 + 0.956694i \(0.405980\pi\)
\(762\) 0 0
\(763\) 1584.00 0.0751568
\(764\) 0 0
\(765\) 8417.77 0.397837
\(766\) 0 0
\(767\) 17957.9 0.845401
\(768\) 0 0
\(769\) −34030.0 −1.59578 −0.797889 0.602804i \(-0.794050\pi\)
−0.797889 + 0.602804i \(0.794050\pi\)
\(770\) 0 0
\(771\) −13878.0 −0.648254
\(772\) 0 0
\(773\) 4873.99 0.226786 0.113393 0.993550i \(-0.463828\pi\)
0.113393 + 0.993550i \(0.463828\pi\)
\(774\) 0 0
\(775\) −5123.41 −0.237469
\(776\) 0 0
\(777\) −360.000 −0.0166215
\(778\) 0 0
\(779\) 6264.00 0.288102
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −7014.81 −0.320164
\(784\) 0 0
\(785\) −20952.0 −0.952623
\(786\) 0 0
\(787\) −30988.0 −1.40356 −0.701781 0.712393i \(-0.747611\pi\)
−0.701781 + 0.712393i \(0.747611\pi\)
\(788\) 0 0
\(789\) −5985.97 −0.270096
\(790\) 0 0
\(791\) 7669.52 0.344749
\(792\) 0 0
\(793\) −31872.0 −1.42725
\(794\) 0 0
\(795\) −15228.0 −0.679348
\(796\) 0 0
\(797\) −7160.30 −0.318232 −0.159116 0.987260i \(-0.550864\pi\)
−0.159116 + 0.987260i \(0.550864\pi\)
\(798\) 0 0
\(799\) 35541.7 1.57369
\(800\) 0 0
\(801\) −8262.00 −0.364449
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −3741.23 −0.163803
\(806\) 0 0
\(807\) 9446.61 0.412065
\(808\) 0 0
\(809\) −37530.0 −1.63101 −0.815503 0.578752i \(-0.803540\pi\)
−0.815503 + 0.578752i \(0.803540\pi\)
\(810\) 0 0
\(811\) −10852.0 −0.469871 −0.234935 0.972011i \(-0.575488\pi\)
−0.234935 + 0.972011i \(0.575488\pi\)
\(812\) 0 0
\(813\) 16035.3 0.691739
\(814\) 0 0
\(815\) 4032.21 0.173303
\(816\) 0 0
\(817\) 2320.00 0.0993470
\(818\) 0 0
\(819\) −1728.00 −0.0737255
\(820\) 0 0
\(821\) 31353.6 1.33282 0.666411 0.745584i \(-0.267829\pi\)
0.666411 + 0.745584i \(0.267829\pi\)
\(822\) 0 0
\(823\) 32947.1 1.39546 0.697729 0.716361i \(-0.254194\pi\)
0.697729 + 0.716361i \(0.254194\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −10044.0 −0.422327 −0.211163 0.977451i \(-0.567725\pi\)
−0.211163 + 0.977451i \(0.567725\pi\)
\(828\) 0 0
\(829\) 9796.48 0.410429 0.205215 0.978717i \(-0.434211\pi\)
0.205215 + 0.978717i \(0.434211\pi\)
\(830\) 0 0
\(831\) −19579.1 −0.817318
\(832\) 0 0
\(833\) 29790.0 1.23909
\(834\) 0 0
\(835\) 22896.0 0.948921
\(836\) 0 0
\(837\) −8137.17 −0.336036
\(838\) 0 0
\(839\) −21054.8 −0.866380 −0.433190 0.901303i \(-0.642612\pi\)
−0.433190 + 0.901303i \(0.642612\pi\)
\(840\) 0 0
\(841\) 43111.0 1.76764
\(842\) 0 0
\(843\) −3510.00 −0.143405
\(844\) 0 0
\(845\) −9093.27 −0.370199
\(846\) 0 0
\(847\) 4610.72 0.187044
\(848\) 0 0
\(849\) −17220.0 −0.696100
\(850\) 0 0
\(851\) 3600.00 0.145013
\(852\) 0 0
\(853\) 40703.2 1.63382 0.816911 0.576763i \(-0.195684\pi\)
0.816911 + 0.576763i \(0.195684\pi\)
\(854\) 0 0
\(855\) −10849.6 −0.433973
\(856\) 0 0
\(857\) 18342.0 0.731098 0.365549 0.930792i \(-0.380881\pi\)
0.365549 + 0.930792i \(0.380881\pi\)
\(858\) 0 0
\(859\) 26324.0 1.04559 0.522796 0.852458i \(-0.324889\pi\)
0.522796 + 0.852458i \(0.324889\pi\)
\(860\) 0 0
\(861\) 561.184 0.0222127
\(862\) 0 0
\(863\) −12761.8 −0.503378 −0.251689 0.967808i \(-0.580986\pi\)
−0.251689 + 0.967808i \(0.580986\pi\)
\(864\) 0 0
\(865\) −2052.00 −0.0806591
\(866\) 0 0
\(867\) −9561.00 −0.374520
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −6429.37 −0.250116
\(872\) 0 0
\(873\) 1710.00 0.0662941
\(874\) 0 0
\(875\) −5112.00 −0.197505
\(876\) 0 0
\(877\) −2459.51 −0.0946999 −0.0473500 0.998878i \(-0.515078\pi\)
−0.0473500 + 0.998878i \(0.515078\pi\)
\(878\) 0 0
\(879\) −23975.0 −0.919975
\(880\) 0 0
\(881\) 37314.0 1.42695 0.713474 0.700682i \(-0.247121\pi\)
0.713474 + 0.700682i \(0.247121\pi\)
\(882\) 0 0
\(883\) −18244.0 −0.695311 −0.347655 0.937622i \(-0.613022\pi\)
−0.347655 + 0.937622i \(0.613022\pi\)
\(884\) 0 0
\(885\) 10101.3 0.383675
\(886\) 0 0
\(887\) 17957.9 0.679783 0.339891 0.940465i \(-0.389610\pi\)
0.339891 + 0.940465i \(0.389610\pi\)
\(888\) 0 0
\(889\) −2412.00 −0.0909965
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −45809.3 −1.71663
\(894\) 0 0
\(895\) −29555.7 −1.10384
\(896\) 0 0
\(897\) 17280.0 0.643213
\(898\) 0 0
\(899\) 78300.0 2.90484
\(900\) 0 0
\(901\) 43959.4 1.62542
\(902\) 0 0
\(903\) 207.846 0.00765967
\(904\) 0 0
\(905\) −1008.00 −0.0370244
\(906\) 0 0
\(907\) 16388.0 0.599950 0.299975 0.953947i \(-0.403022\pi\)
0.299975 + 0.953947i \(0.403022\pi\)
\(908\) 0 0
\(909\) 93.5307 0.00341278
\(910\) 0 0
\(911\) 25107.8 0.913127 0.456564 0.889691i \(-0.349080\pi\)
0.456564 + 0.889691i \(0.349080\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −17928.0 −0.647739
\(916\) 0 0
\(917\) 7856.58 0.282930
\(918\) 0 0
\(919\) 27155.1 0.974716 0.487358 0.873202i \(-0.337961\pi\)
0.487358 + 0.873202i \(0.337961\pi\)
\(920\) 0 0
\(921\) −16356.0 −0.585178
\(922\) 0 0
\(923\) 61056.0 2.17734
\(924\) 0 0
\(925\) 588.897 0.0209328
\(926\) 0 0
\(927\) 7139.51 0.252958
\(928\) 0 0
\(929\) 48006.0 1.69540 0.847700 0.530477i \(-0.177987\pi\)
0.847700 + 0.530477i \(0.177987\pi\)
\(930\) 0 0
\(931\) −38396.0 −1.35164
\(932\) 0 0
\(933\) 6609.51 0.231924
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −7894.00 −0.275225 −0.137612 0.990486i \(-0.543943\pi\)
−0.137612 + 0.990486i \(0.543943\pi\)
\(938\) 0 0
\(939\) −3102.00 −0.107806
\(940\) 0 0
\(941\) −2670.82 −0.0925253 −0.0462627 0.998929i \(-0.514731\pi\)
−0.0462627 + 0.998929i \(0.514731\pi\)
\(942\) 0 0
\(943\) −5611.84 −0.193793
\(944\) 0 0
\(945\) −972.000 −0.0334594
\(946\) 0 0
\(947\) −22356.0 −0.767130 −0.383565 0.923514i \(-0.625304\pi\)
−0.383565 + 0.923514i \(0.625304\pi\)
\(948\) 0 0
\(949\) −61300.7 −2.09685
\(950\) 0 0
\(951\) 7950.11 0.271083
\(952\) 0 0
\(953\) 14958.0 0.508434 0.254217 0.967147i \(-0.418182\pi\)
0.254217 + 0.967147i \(0.418182\pi\)
\(954\) 0 0
\(955\) −33264.0 −1.12712
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1808.26 0.0608882
\(960\) 0 0
\(961\) 61037.0 2.04884
\(962\) 0 0
\(963\) −2268.00 −0.0758933
\(964\) 0 0
\(965\) 13946.5 0.465236
\(966\) 0 0
\(967\) 17476.4 0.581182 0.290591 0.956847i \(-0.406148\pi\)
0.290591 + 0.956847i \(0.406148\pi\)
\(968\) 0 0
\(969\) 31320.0 1.03833
\(970\) 0 0
\(971\) 48528.0 1.60385 0.801925 0.597425i \(-0.203810\pi\)
0.801925 + 0.597425i \(0.203810\pi\)
\(972\) 0 0
\(973\) −2341.73 −0.0771557
\(974\) 0 0
\(975\) 2826.71 0.0928483
\(976\) 0 0
\(977\) −39978.0 −1.30912 −0.654560 0.756010i \(-0.727146\pi\)
−0.654560 + 0.756010i \(0.727146\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −4115.35 −0.133938
\(982\) 0 0
\(983\) 17957.9 0.582674 0.291337 0.956621i \(-0.405900\pi\)
0.291337 + 0.956621i \(0.405900\pi\)
\(984\) 0 0
\(985\) −10044.0 −0.324902
\(986\) 0 0
\(987\) −4104.00 −0.132352
\(988\) 0 0
\(989\) −2078.46 −0.0668263
\(990\) 0 0
\(991\) −1666.23 −0.0534103 −0.0267052 0.999643i \(-0.508502\pi\)
−0.0267052 + 0.999643i \(0.508502\pi\)
\(992\) 0 0
\(993\) −12396.0 −0.396148
\(994\) 0 0
\(995\) 18540.0 0.590711
\(996\) 0 0
\(997\) 41465.3 1.31717 0.658585 0.752506i \(-0.271155\pi\)
0.658585 + 0.752506i \(0.271155\pi\)
\(998\) 0 0
\(999\) 935.307 0.0296214
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 768.4.a.g.1.1 2
3.2 odd 2 2304.4.a.bg.1.2 2
4.3 odd 2 768.4.a.n.1.1 2
8.3 odd 2 inner 768.4.a.g.1.2 2
8.5 even 2 768.4.a.n.1.2 2
12.11 even 2 2304.4.a.be.1.2 2
16.3 odd 4 192.4.d.a.97.4 yes 4
16.5 even 4 192.4.d.a.97.3 yes 4
16.11 odd 4 192.4.d.a.97.1 4
16.13 even 4 192.4.d.a.97.2 yes 4
24.5 odd 2 2304.4.a.be.1.1 2
24.11 even 2 2304.4.a.bg.1.1 2
48.5 odd 4 576.4.d.g.289.4 4
48.11 even 4 576.4.d.g.289.3 4
48.29 odd 4 576.4.d.g.289.2 4
48.35 even 4 576.4.d.g.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.4.d.a.97.1 4 16.11 odd 4
192.4.d.a.97.2 yes 4 16.13 even 4
192.4.d.a.97.3 yes 4 16.5 even 4
192.4.d.a.97.4 yes 4 16.3 odd 4
576.4.d.g.289.1 4 48.35 even 4
576.4.d.g.289.2 4 48.29 odd 4
576.4.d.g.289.3 4 48.11 even 4
576.4.d.g.289.4 4 48.5 odd 4
768.4.a.g.1.1 2 1.1 even 1 trivial
768.4.a.g.1.2 2 8.3 odd 2 inner
768.4.a.n.1.1 2 4.3 odd 2
768.4.a.n.1.2 2 8.5 even 2
2304.4.a.be.1.1 2 24.5 odd 2
2304.4.a.be.1.2 2 12.11 even 2
2304.4.a.bg.1.1 2 24.11 even 2
2304.4.a.bg.1.2 2 3.2 odd 2