Properties

Label 192.4.d.a.97.4
Level $192$
Weight $4$
Character 192.97
Analytic conductor $11.328$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 192.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.3283667211\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.4
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 192.97
Dual form 192.4.d.a.97.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000i q^{3} +10.3923i q^{5} -3.46410 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} +10.3923i q^{5} -3.46410 q^{7} -9.00000 q^{9} +55.4256i q^{13} -31.1769 q^{15} -90.0000 q^{17} -116.000i q^{19} -10.3923i q^{21} -103.923 q^{23} +17.0000 q^{25} -27.0000i q^{27} +259.808i q^{29} -301.377 q^{31} -36.0000i q^{35} +34.6410i q^{37} -166.277 q^{39} -54.0000 q^{41} +20.0000i q^{43} -93.5307i q^{45} +394.908 q^{47} -331.000 q^{49} -270.000i q^{51} +488.438i q^{53} +348.000 q^{57} +324.000i q^{59} -575.041i q^{61} +31.1769 q^{63} -576.000 q^{65} +116.000i q^{67} -311.769i q^{69} +1101.58 q^{71} +1106.00 q^{73} +51.0000i q^{75} -148.956 q^{79} +81.0000 q^{81} +1152.00i q^{83} -935.307i q^{85} -779.423 q^{87} +918.000 q^{89} -192.000i q^{91} -904.131i q^{93} +1205.51 q^{95} +190.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{9} - 360 q^{17} + 68 q^{25} - 216 q^{41} - 1324 q^{49} + 1392 q^{57} - 2304 q^{65} + 4424 q^{73} + 324 q^{81} + 3672 q^{89} + 760 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 10.3923i 0.929516i 0.885438 + 0.464758i \(0.153859\pi\)
−0.885438 + 0.464758i \(0.846141\pi\)
\(6\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 55.4256i 1.18248i 0.806494 + 0.591242i \(0.201362\pi\)
−0.806494 + 0.591242i \(0.798638\pi\)
\(14\) 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
\(19\) − 116.000i − 1.40064i −0.713827 0.700322i \(-0.753040\pi\)
0.713827 0.700322i \(-0.246960\pi\)
\(20\) 0 0
\(21\) − 10.3923i − 0.107990i
\(22\) 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
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) 259.808i 1.66362i 0.555058 + 0.831811i \(0.312696\pi\)
−0.555058 + 0.831811i \(0.687304\pi\)
\(30\) 0 0
\(31\) −301.377 −1.74609 −0.873046 0.487637i \(-0.837859\pi\)
−0.873046 + 0.487637i \(0.837859\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 36.0000i − 0.173860i
\(36\) 0 0
\(37\) 34.6410i 0.153918i 0.997034 + 0.0769588i \(0.0245210\pi\)
−0.997034 + 0.0769588i \(0.975479\pi\)
\(38\) 0 0
\(39\) −166.277 −0.682708
\(40\) 0 0
\(41\) −54.0000 −0.205692 −0.102846 0.994697i \(-0.532795\pi\)
−0.102846 + 0.994697i \(0.532795\pi\)
\(42\) 0 0
\(43\) 20.0000i 0.0709296i 0.999371 + 0.0354648i \(0.0112912\pi\)
−0.999371 + 0.0354648i \(0.988709\pi\)
\(44\) 0 0
\(45\) − 93.5307i − 0.309839i
\(46\) 0 0
\(47\) 394.908 1.22560 0.612800 0.790238i \(-0.290043\pi\)
0.612800 + 0.790238i \(0.290043\pi\)
\(48\) 0 0
\(49\) −331.000 −0.965015
\(50\) 0 0
\(51\) − 270.000i − 0.741325i
\(52\) 0 0
\(53\) 488.438i 1.26589i 0.774197 + 0.632945i \(0.218154\pi\)
−0.774197 + 0.632945i \(0.781846\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 348.000 0.808662
\(58\) 0 0
\(59\) 324.000i 0.714936i 0.933925 + 0.357468i \(0.116360\pi\)
−0.933925 + 0.357468i \(0.883640\pi\)
\(60\) 0 0
\(61\) − 575.041i − 1.20699i −0.797366 0.603495i \(-0.793774\pi\)
0.797366 0.603495i \(-0.206226\pi\)
\(62\) 0 0
\(63\) 31.1769 0.0623480
\(64\) 0 0
\(65\) −576.000 −1.09914
\(66\) 0 0
\(67\) 116.000i 0.211517i 0.994392 + 0.105759i \(0.0337271\pi\)
−0.994392 + 0.105759i \(0.966273\pi\)
\(68\) 0 0
\(69\) − 311.769i − 0.543951i
\(70\) 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 \(-0.153042\pi\)
0.886627 + 0.462486i \(0.153042\pi\)
\(74\) 0 0
\(75\) 51.0000i 0.0785196i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −148.956 −0.212138 −0.106069 0.994359i \(-0.533826\pi\)
−0.106069 + 0.994359i \(0.533826\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1152.00i 1.52348i 0.647886 + 0.761738i \(0.275653\pi\)
−0.647886 + 0.761738i \(0.724347\pi\)
\(84\) 0 0
\(85\) − 935.307i − 1.19351i
\(86\) 0 0
\(87\) −779.423 −0.960493
\(88\) 0 0
\(89\) 918.000 1.09335 0.546673 0.837346i \(-0.315894\pi\)
0.546673 + 0.837346i \(0.315894\pi\)
\(90\) 0 0
\(91\) − 192.000i − 0.221177i
\(92\) 0 0
\(93\) − 904.131i − 1.00811i
\(94\) 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
\(101\) − 10.3923i − 0.0102383i −0.999987 0.00511917i \(-0.998371\pi\)
0.999987 0.00511917i \(-0.00162949\pi\)
\(102\) 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
\(107\) − 252.000i − 0.227680i −0.993499 0.113840i \(-0.963685\pi\)
0.993499 0.113840i \(-0.0363151\pi\)
\(108\) 0 0
\(109\) − 457.261i − 0.401814i −0.979610 0.200907i \(-0.935611\pi\)
0.979610 0.200907i \(-0.0643889\pi\)
\(110\) 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
\(115\) − 1080.00i − 0.875744i
\(116\) 0 0
\(117\) − 498.831i − 0.394162i
\(118\) 0 0
\(119\) 311.769 0.240167
\(120\) 0 0
\(121\) 1331.00 1.00000
\(122\) 0 0
\(123\) − 162.000i − 0.118756i
\(124\) 0 0
\(125\) 1475.71i 1.05593i
\(126\) 0 0
\(127\) −696.284 −0.486498 −0.243249 0.969964i \(-0.578213\pi\)
−0.243249 + 0.969964i \(0.578213\pi\)
\(128\) 0 0
\(129\) −60.0000 −0.0409512
\(130\) 0 0
\(131\) 2268.00i 1.51264i 0.654201 + 0.756321i \(0.273005\pi\)
−0.654201 + 0.756321i \(0.726995\pi\)
\(132\) 0 0
\(133\) 401.836i 0.261982i
\(134\) 0 0
\(135\) 280.592 0.178885
\(136\) 0 0
\(137\) 522.000 0.325529 0.162764 0.986665i \(-0.447959\pi\)
0.162764 + 0.986665i \(0.447959\pi\)
\(138\) 0 0
\(139\) 676.000i 0.412501i 0.978499 + 0.206250i \(0.0661261\pi\)
−0.978499 + 0.206250i \(0.933874\pi\)
\(140\) 0 0
\(141\) 1184.72i 0.707600i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −2700.00 −1.54636
\(146\) 0 0
\(147\) − 993.000i − 0.557151i
\(148\) 0 0
\(149\) 1465.31i 0.805660i 0.915275 + 0.402830i \(0.131973\pi\)
−0.915275 + 0.402830i \(0.868027\pi\)
\(150\) 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
\(155\) − 3132.00i − 1.62302i
\(156\) 0 0
\(157\) 2016.11i 1.02486i 0.858729 + 0.512430i \(0.171254\pi\)
−0.858729 + 0.512430i \(0.828746\pi\)
\(158\) 0 0
\(159\) −1465.31 −0.730862
\(160\) 0 0
\(161\) 360.000 0.176223
\(162\) 0 0
\(163\) 388.000i 0.186445i 0.995645 + 0.0932224i \(0.0297168\pi\)
−0.995645 + 0.0932224i \(0.970283\pi\)
\(164\) 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
\(171\) 1044.00i 0.466881i
\(172\) 0 0
\(173\) 197.454i 0.0867753i 0.999058 + 0.0433877i \(0.0138151\pi\)
−0.999058 + 0.0433877i \(0.986185\pi\)
\(174\) 0 0
\(175\) −58.8897 −0.0254380
\(176\) 0 0
\(177\) −972.000 −0.412768
\(178\) 0 0
\(179\) − 2844.00i − 1.18754i −0.804633 0.593772i \(-0.797638\pi\)
0.804633 0.593772i \(-0.202362\pi\)
\(180\) 0 0
\(181\) − 96.9948i − 0.0398319i −0.999802 0.0199159i \(-0.993660\pi\)
0.999802 0.0199159i \(-0.00633986\pi\)
\(182\) 0 0
\(183\) 1725.12 0.696856
\(184\) 0 0
\(185\) −360.000 −0.143069
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 93.5307i 0.0359966i
\(190\) 0 0
\(191\) −3200.83 −1.21259 −0.606293 0.795241i \(-0.707344\pi\)
−0.606293 + 0.795241i \(0.707344\pi\)
\(192\) 0 0
\(193\) −1342.00 −0.500514 −0.250257 0.968179i \(-0.580515\pi\)
−0.250257 + 0.968179i \(0.580515\pi\)
\(194\) 0 0
\(195\) − 1728.00i − 0.634588i
\(196\) 0 0
\(197\) − 966.484i − 0.349539i −0.984609 0.174769i \(-0.944082\pi\)
0.984609 0.174769i \(-0.0559180\pi\)
\(198\) 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
\(203\) − 900.000i − 0.311171i
\(204\) 0 0
\(205\) − 561.184i − 0.191194i
\(206\) 0 0
\(207\) 935.307 0.314050
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) − 2764.00i − 0.901809i −0.892572 0.450904i \(-0.851102\pi\)
0.892572 0.450904i \(-0.148898\pi\)
\(212\) 0 0
\(213\) 3304.75i 1.06309i
\(214\) 0 0
\(215\) −207.846 −0.0659302
\(216\) 0 0
\(217\) 1044.00 0.326596
\(218\) 0 0
\(219\) 3318.00i 1.02379i
\(220\) 0 0
\(221\) − 4988.31i − 1.51832i
\(222\) 0 0
\(223\) 4292.02 1.28886 0.644428 0.764665i \(-0.277095\pi\)
0.644428 + 0.764665i \(0.277095\pi\)
\(224\) 0 0
\(225\) −153.000 −0.0453333
\(226\) 0 0
\(227\) 5688.00i 1.66311i 0.555443 + 0.831555i \(0.312549\pi\)
−0.555443 + 0.831555i \(0.687451\pi\)
\(228\) 0 0
\(229\) 5570.28i 1.60740i 0.595036 + 0.803699i \(0.297138\pi\)
−0.595036 + 0.803699i \(0.702862\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2718.00 0.764215 0.382108 0.924118i \(-0.375198\pi\)
0.382108 + 0.924118i \(0.375198\pi\)
\(234\) 0 0
\(235\) 4104.00i 1.13921i
\(236\) 0 0
\(237\) − 446.869i − 0.122478i
\(238\) 0 0
\(239\) −3574.95 −0.967550 −0.483775 0.875192i \(-0.660735\pi\)
−0.483775 + 0.875192i \(0.660735\pi\)
\(240\) 0 0
\(241\) 4490.00 1.20011 0.600055 0.799959i \(-0.295146\pi\)
0.600055 + 0.799959i \(0.295146\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) − 3439.85i − 0.896996i
\(246\) 0 0
\(247\) 6429.37 1.65624
\(248\) 0 0
\(249\) −3456.00 −0.879579
\(250\) 0 0
\(251\) − 4608.00i − 1.15878i −0.815050 0.579391i \(-0.803290\pi\)
0.815050 0.579391i \(-0.196710\pi\)
\(252\) 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
\(259\) − 120.000i − 0.0287893i
\(260\) 0 0
\(261\) − 2338.27i − 0.554541i
\(262\) 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
\(267\) 2754.00i 0.631244i
\(268\) 0 0
\(269\) − 3148.87i − 0.713717i −0.934158 0.356859i \(-0.883848\pi\)
0.934158 0.356859i \(-0.116152\pi\)
\(270\) 0 0
\(271\) 5345.11 1.19813 0.599063 0.800702i \(-0.295540\pi\)
0.599063 + 0.800702i \(0.295540\pi\)
\(272\) 0 0
\(273\) 576.000 0.127696
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 6526.37i − 1.41564i −0.706394 0.707818i \(-0.749679\pi\)
0.706394 0.707818i \(-0.250321\pi\)
\(278\) 0 0
\(279\) 2712.39 0.582031
\(280\) 0 0
\(281\) −1170.00 −0.248386 −0.124193 0.992258i \(-0.539634\pi\)
−0.124193 + 0.992258i \(0.539634\pi\)
\(282\) 0 0
\(283\) 5740.00i 1.20568i 0.797862 + 0.602840i \(0.205964\pi\)
−0.797862 + 0.602840i \(0.794036\pi\)
\(284\) 0 0
\(285\) 3616.52i 0.751664i
\(286\) 0 0
\(287\) 187.061 0.0384735
\(288\) 0 0
\(289\) 3187.00 0.648687
\(290\) 0 0
\(291\) 570.000i 0.114825i
\(292\) 0 0
\(293\) − 7991.68i − 1.59344i −0.604346 0.796722i \(-0.706566\pi\)
0.604346 0.796722i \(-0.293434\pi\)
\(294\) 0 0
\(295\) −3367.11 −0.664544
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 5760.00i − 1.11408i
\(300\) 0 0
\(301\) − 69.2820i − 0.0132669i
\(302\) 0 0
\(303\) 31.1769 0.00591111
\(304\) 0 0
\(305\) 5976.00 1.12192
\(306\) 0 0
\(307\) − 5452.00i − 1.01356i −0.862076 0.506779i \(-0.830836\pi\)
0.862076 0.506779i \(-0.169164\pi\)
\(308\) 0 0
\(309\) 2379.84i 0.438137i
\(310\) 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 \(-0.529762\pi\)
−0.0933628 + 0.995632i \(0.529762\pi\)
\(314\) 0 0
\(315\) 324.000i 0.0579534i
\(316\) 0 0
\(317\) − 2650.04i − 0.469530i −0.972052 0.234765i \(-0.924568\pi\)
0.972052 0.234765i \(-0.0754320\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 756.000 0.131451
\(322\) 0 0
\(323\) 10440.0i 1.79844i
\(324\) 0 0
\(325\) 942.236i 0.160818i
\(326\) 0 0
\(327\) 1371.78 0.231987
\(328\) 0 0
\(329\) −1368.00 −0.229241
\(330\) 0 0
\(331\) 4132.00i 0.686149i 0.939308 + 0.343074i \(0.111468\pi\)
−0.939308 + 0.343074i \(0.888532\pi\)
\(332\) 0 0
\(333\) − 311.769i − 0.0513058i
\(334\) 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
\(339\) − 6642.00i − 1.06414i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 2334.80 0.367544
\(344\) 0 0
\(345\) 3240.00 0.505611
\(346\) 0 0
\(347\) − 11016.0i − 1.70424i −0.523350 0.852118i \(-0.675318\pi\)
0.523350 0.852118i \(-0.324682\pi\)
\(348\) 0 0
\(349\) 2528.79i 0.387860i 0.981015 + 0.193930i \(0.0621235\pi\)
−0.981015 + 0.193930i \(0.937876\pi\)
\(350\) 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
\(355\) 11448.0i 1.71154i
\(356\) 0 0
\(357\) 935.307i 0.138660i
\(358\) 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
\(363\) 3993.00i 0.577350i
\(364\) 0 0
\(365\) 11493.9i 1.64827i
\(366\) 0 0
\(367\) −12799.9 −1.82056 −0.910282 0.413989i \(-0.864135\pi\)
−0.910282 + 0.413989i \(0.864135\pi\)
\(368\) 0 0
\(369\) 486.000 0.0685641
\(370\) 0 0
\(371\) − 1692.00i − 0.236777i
\(372\) 0 0
\(373\) 4981.38i 0.691491i 0.938328 + 0.345745i \(0.112374\pi\)
−0.938328 + 0.345745i \(0.887626\pi\)
\(374\) 0 0
\(375\) −4427.12 −0.609642
\(376\) 0 0
\(377\) −14400.0 −1.96721
\(378\) 0 0
\(379\) − 9892.00i − 1.34068i −0.742054 0.670340i \(-0.766148\pi\)
0.742054 0.670340i \(-0.233852\pi\)
\(380\) 0 0
\(381\) − 2088.85i − 0.280880i
\(382\) 0 0
\(383\) 8771.11 1.17019 0.585095 0.810965i \(-0.301057\pi\)
0.585095 + 0.810965i \(0.301057\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 180.000i − 0.0236432i
\(388\) 0 0
\(389\) 9903.87i 1.29086i 0.763818 + 0.645432i \(0.223323\pi\)
−0.763818 + 0.645432i \(0.776677\pi\)
\(390\) 0 0
\(391\) 9353.07 1.20973
\(392\) 0 0
\(393\) −6804.00 −0.873324
\(394\) 0 0
\(395\) − 1548.00i − 0.197186i
\(396\) 0 0
\(397\) − 103.923i − 0.0131379i −0.999978 0.00656895i \(-0.997909\pi\)
0.999978 0.00656895i \(-0.00209098\pi\)
\(398\) 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
\(403\) − 16704.0i − 2.06473i
\(404\) 0 0
\(405\) 841.777i 0.103280i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −8614.00 −1.04141 −0.520703 0.853738i \(-0.674330\pi\)
−0.520703 + 0.853738i \(0.674330\pi\)
\(410\) 0 0
\(411\) 1566.00i 0.187944i
\(412\) 0 0
\(413\) − 1122.37i − 0.133724i
\(414\) 0 0
\(415\) −11971.9 −1.41609
\(416\) 0 0
\(417\) −2028.00 −0.238157
\(418\) 0 0
\(419\) 10440.0i 1.21725i 0.793458 + 0.608625i \(0.208278\pi\)
−0.793458 + 0.608625i \(0.791722\pi\)
\(420\) 0 0
\(421\) − 900.666i − 0.104266i −0.998640 0.0521328i \(-0.983398\pi\)
0.998640 0.0521328i \(-0.0166019\pi\)
\(422\) 0 0
\(423\) −3554.17 −0.408533
\(424\) 0 0
\(425\) −1530.00 −0.174626
\(426\) 0 0
\(427\) 1992.00i 0.225760i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −394.908 −0.0441346 −0.0220673 0.999756i \(-0.507025\pi\)
−0.0220673 + 0.999756i \(0.507025\pi\)
\(432\) 0 0
\(433\) 12958.0 1.43816 0.719078 0.694929i \(-0.244564\pi\)
0.719078 + 0.694929i \(0.244564\pi\)
\(434\) 0 0
\(435\) − 8100.00i − 0.892794i
\(436\) 0 0
\(437\) 12055.1i 1.31962i
\(438\) 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
\(443\) − 1800.00i − 0.193049i −0.995331 0.0965244i \(-0.969227\pi\)
0.995331 0.0965244i \(-0.0307726\pi\)
\(444\) 0 0
\(445\) 9540.14i 1.01628i
\(446\) 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
\(453\) − 7160.30i − 0.742649i
\(454\) 0 0
\(455\) 1995.32 0.205587
\(456\) 0 0
\(457\) 12602.0 1.28993 0.644964 0.764213i \(-0.276873\pi\)
0.644964 + 0.764213i \(0.276873\pi\)
\(458\) 0 0
\(459\) 2430.00i 0.247108i
\(460\) 0 0
\(461\) − 1839.44i − 0.185838i −0.995674 0.0929188i \(-0.970380\pi\)
0.995674 0.0929188i \(-0.0296197\pi\)
\(462\) 0 0
\(463\) 11012.4 1.10538 0.552688 0.833389i \(-0.313602\pi\)
0.552688 + 0.833389i \(0.313602\pi\)
\(464\) 0 0
\(465\) 9396.00 0.937052
\(466\) 0 0
\(467\) 9144.00i 0.906068i 0.891493 + 0.453034i \(0.149658\pi\)
−0.891493 + 0.453034i \(0.850342\pi\)
\(468\) 0 0
\(469\) − 401.836i − 0.0395630i
\(470\) 0 0
\(471\) −6048.32 −0.591703
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) − 1972.00i − 0.190488i
\(476\) 0 0
\(477\) − 4395.94i − 0.421963i
\(478\) 0 0
\(479\) −6173.03 −0.588837 −0.294418 0.955677i \(-0.595126\pi\)
−0.294418 + 0.955677i \(0.595126\pi\)
\(480\) 0 0
\(481\) −1920.00 −0.182005
\(482\) 0 0
\(483\) 1080.00i 0.101743i
\(484\) 0 0
\(485\) 1974.54i 0.184864i
\(486\) 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
\(491\) 396.000i 0.0363976i 0.999834 + 0.0181988i \(0.00579318\pi\)
−0.999834 + 0.0181988i \(0.994207\pi\)
\(492\) 0 0
\(493\) − 23382.7i − 2.13611i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3816.00 −0.344408
\(498\) 0 0
\(499\) − 12436.0i − 1.11565i −0.829957 0.557827i \(-0.811635\pi\)
0.829957 0.557827i \(-0.188365\pi\)
\(500\) 0 0
\(501\) − 6609.51i − 0.589403i
\(502\) 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
\(507\) − 2625.00i − 0.229942i
\(508\) 0 0
\(509\) 9155.62i 0.797280i 0.917107 + 0.398640i \(0.130518\pi\)
−0.917107 + 0.398640i \(0.869482\pi\)
\(510\) 0 0
\(511\) −3831.30 −0.331676
\(512\) 0 0
\(513\) −3132.00 −0.269554
\(514\) 0 0
\(515\) 8244.00i 0.705386i
\(516\) 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 \(-0.395765\pi\)
0.321644 + 0.946861i \(0.395765\pi\)
\(522\) 0 0
\(523\) 18332.0i 1.53270i 0.642423 + 0.766350i \(0.277929\pi\)
−0.642423 + 0.766350i \(0.722071\pi\)
\(524\) 0 0
\(525\) − 176.669i − 0.0146866i
\(526\) 0 0
\(527\) 27123.9 2.24200
\(528\) 0 0
\(529\) −1367.00 −0.112353
\(530\) 0 0
\(531\) − 2916.00i − 0.238312i
\(532\) 0 0
\(533\) − 2992.98i − 0.243228i
\(534\) 0 0
\(535\) 2618.86 0.211632
\(536\) 0 0
\(537\) 8532.00 0.685629
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 16863.2i − 1.34012i −0.742305 0.670062i \(-0.766267\pi\)
0.742305 0.670062i \(-0.233733\pi\)
\(542\) 0 0
\(543\) 290.985 0.0229969
\(544\) 0 0
\(545\) 4752.00 0.373492
\(546\) 0 0
\(547\) 1684.00i 0.131632i 0.997832 + 0.0658159i \(0.0209650\pi\)
−0.997832 + 0.0658159i \(0.979035\pi\)
\(548\) 0 0
\(549\) 5175.37i 0.402330i
\(550\) 0 0
\(551\) 30137.7 2.33014
\(552\) 0 0
\(553\) 516.000 0.0396791
\(554\) 0 0
\(555\) − 1080.00i − 0.0826008i
\(556\) 0 0
\(557\) 2275.91i 0.173130i 0.996246 + 0.0865652i \(0.0275891\pi\)
−0.996246 + 0.0865652i \(0.972411\pi\)
\(558\) 0 0
\(559\) −1108.51 −0.0838731
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 7992.00i − 0.598264i −0.954212 0.299132i \(-0.903303\pi\)
0.954212 0.299132i \(-0.0966971\pi\)
\(564\) 0 0
\(565\) − 23008.6i − 1.71323i
\(566\) 0 0
\(567\) −280.592 −0.0207827
\(568\) 0 0
\(569\) −5526.00 −0.407139 −0.203569 0.979061i \(-0.565254\pi\)
−0.203569 + 0.979061i \(0.565254\pi\)
\(570\) 0 0
\(571\) 13420.0i 0.983554i 0.870721 + 0.491777i \(0.163653\pi\)
−0.870721 + 0.491777i \(0.836347\pi\)
\(572\) 0 0
\(573\) − 9602.49i − 0.700087i
\(574\) 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
\(579\) − 4026.00i − 0.288972i
\(580\) 0 0
\(581\) − 3990.65i − 0.284957i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 5184.00 0.366380
\(586\) 0 0
\(587\) 18684.0i 1.31375i 0.754000 + 0.656875i \(0.228122\pi\)
−0.754000 + 0.656875i \(0.771878\pi\)
\(588\) 0 0
\(589\) 34959.7i 2.44565i
\(590\) 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
\(595\) 3240.00i 0.223239i
\(596\) 0 0
\(597\) − 5352.04i − 0.366908i
\(598\) 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 \(-0.110105\pi\)
0.940769 + 0.339049i \(0.110105\pi\)
\(602\) 0 0
\(603\) − 1044.00i − 0.0705057i
\(604\) 0 0
\(605\) 13832.2i 0.929516i
\(606\) 0 0
\(607\) 26684.0 1.78430 0.892149 0.451741i \(-0.149197\pi\)
0.892149 + 0.451741i \(0.149197\pi\)
\(608\) 0 0
\(609\) 2700.00 0.179654
\(610\) 0 0
\(611\) 21888.0i 1.44925i
\(612\) 0 0
\(613\) − 16911.7i − 1.11429i −0.830416 0.557144i \(-0.811897\pi\)
0.830416 0.557144i \(-0.188103\pi\)
\(614\) 0 0
\(615\) 1683.55 0.110386
\(616\) 0 0
\(617\) 17694.0 1.15451 0.577256 0.816563i \(-0.304124\pi\)
0.577256 + 0.816563i \(0.304124\pi\)
\(618\) 0 0
\(619\) − 13652.0i − 0.886462i −0.896407 0.443231i \(-0.853832\pi\)
0.896407 0.443231i \(-0.146168\pi\)
\(620\) 0 0
\(621\) 2805.92i 0.181317i
\(622\) 0 0
\(623\) −3180.05 −0.204504
\(624\) 0 0
\(625\) −13211.0 −0.845504
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 3117.69i − 0.197632i
\(630\) 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
\(635\) − 7236.00i − 0.452208i
\(636\) 0 0
\(637\) − 18345.9i − 1.14112i
\(638\) 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
\(643\) 15892.0i 0.974680i 0.873212 + 0.487340i \(0.162033\pi\)
−0.873212 + 0.487340i \(0.837967\pi\)
\(644\) 0 0
\(645\) − 623.538i − 0.0380648i
\(646\) 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
\(651\) 3132.00i 0.188560i
\(652\) 0 0
\(653\) − 24660.9i − 1.47788i −0.673770 0.738941i \(-0.735326\pi\)
0.673770 0.738941i \(-0.264674\pi\)
\(654\) 0 0
\(655\) −23569.7 −1.40602
\(656\) 0 0
\(657\) −9954.00 −0.591085
\(658\) 0 0
\(659\) − 28260.0i − 1.67049i −0.549878 0.835245i \(-0.685326\pi\)
0.549878 0.835245i \(-0.314674\pi\)
\(660\) 0 0
\(661\) 25863.0i 1.52187i 0.648830 + 0.760933i \(0.275258\pi\)
−0.648830 + 0.760933i \(0.724742\pi\)
\(662\) 0 0
\(663\) 14964.9 0.876605
\(664\) 0 0
\(665\) −4176.00 −0.243516
\(666\) 0 0
\(667\) − 27000.0i − 1.56738i
\(668\) 0 0
\(669\) 12876.1i 0.744122i
\(670\) 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
\(675\) − 459.000i − 0.0261732i
\(676\) 0 0
\(677\) 4998.70i 0.283775i 0.989883 + 0.141887i \(0.0453171\pi\)
−0.989883 + 0.141887i \(0.954683\pi\)
\(678\) 0 0
\(679\) −658.179 −0.0371997
\(680\) 0 0
\(681\) −17064.0 −0.960197
\(682\) 0 0
\(683\) − 8064.00i − 0.451772i −0.974154 0.225886i \(-0.927472\pi\)
0.974154 0.225886i \(-0.0725277\pi\)
\(684\) 0 0
\(685\) 5424.78i 0.302584i
\(686\) 0 0
\(687\) −16710.8 −0.928032
\(688\) 0 0
\(689\) −27072.0 −1.49690
\(690\) 0 0
\(691\) − 19244.0i − 1.05944i −0.848171 0.529722i \(-0.822296\pi\)
0.848171 0.529722i \(-0.177704\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7025.20 −0.383426
\(696\) 0 0
\(697\) 4860.00 0.264111
\(698\) 0 0
\(699\) 8154.00i 0.441220i
\(700\) 0 0
\(701\) 6204.21i 0.334279i 0.985933 + 0.167140i \(0.0534530\pi\)
−0.985933 + 0.167140i \(0.946547\pi\)
\(702\) 0 0
\(703\) 4018.36 0.215584
\(704\) 0 0
\(705\) −12312.0 −0.657726
\(706\) 0 0
\(707\) 36.0000i 0.00191502i
\(708\) 0 0
\(709\) 15020.3i 0.795629i 0.917466 + 0.397814i \(0.130231\pi\)
−0.917466 + 0.397814i \(0.869769\pi\)
\(710\) 0 0
\(711\) 1340.61 0.0707127
\(712\) 0 0
\(713\) 31320.0 1.64508
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 10724.9i − 0.558615i
\(718\) 0 0
\(719\) −30740.4 −1.59447 −0.797236 0.603668i \(-0.793705\pi\)
−0.797236 + 0.603668i \(0.793705\pi\)
\(720\) 0 0
\(721\) −2748.00 −0.141943
\(722\) 0 0
\(723\) 13470.0i 0.692883i
\(724\) 0 0
\(725\) 4416.73i 0.226253i
\(726\) 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
\(731\) − 1800.00i − 0.0910744i
\(732\) 0 0
\(733\) 12387.6i 0.624212i 0.950047 + 0.312106i \(0.101034\pi\)
−0.950047 + 0.312106i \(0.898966\pi\)
\(734\) 0 0
\(735\) 10319.6 0.517881
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 11180.0i 0.556513i 0.960507 + 0.278256i \(0.0897565\pi\)
−0.960507 + 0.278256i \(0.910244\pi\)
\(740\) 0 0
\(741\) 19288.1i 0.956230i
\(742\) 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
\(747\) − 10368.0i − 0.507825i
\(748\) 0 0
\(749\) 872.954i 0.0425862i
\(750\) 0 0
\(751\) 37970.0 1.84493 0.922467 0.386076i \(-0.126170\pi\)
0.922467 + 0.386076i \(0.126170\pi\)
\(752\) 0 0
\(753\) 13824.0 0.669023
\(754\) 0 0
\(755\) − 24804.0i − 1.19564i
\(756\) 0 0
\(757\) 39047.4i 1.87477i 0.348296 + 0.937385i \(0.386760\pi\)
−0.348296 + 0.937385i \(0.613240\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −12222.0 −0.582191 −0.291095 0.956694i \(-0.594020\pi\)
−0.291095 + 0.956694i \(0.594020\pi\)
\(762\) 0 0
\(763\) 1584.00i 0.0751568i
\(764\) 0 0
\(765\) 8417.77i 0.397837i
\(766\) 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
\(771\) 13878.0i 0.648254i
\(772\) 0 0
\(773\) − 4873.99i − 0.226786i −0.993550 0.113393i \(-0.963828\pi\)
0.993550 0.113393i \(-0.0361718\pi\)
\(774\) 0 0
\(775\) −5123.41 −0.237469
\(776\) 0 0
\(777\) 360.000 0.0166215
\(778\) 0 0
\(779\) 6264.00i 0.288102i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 7014.81 0.320164
\(784\) 0 0
\(785\) −20952.0 −0.952623
\(786\) 0 0
\(787\) 30988.0i 1.40356i 0.712393 + 0.701781i \(0.247611\pi\)
−0.712393 + 0.701781i \(0.752389\pi\)
\(788\) 0 0
\(789\) 5985.97i 0.270096i
\(790\) 0 0
\(791\) 7669.52 0.344749
\(792\) 0 0
\(793\) 31872.0 1.42725
\(794\) 0 0
\(795\) − 15228.0i − 0.679348i
\(796\) 0 0
\(797\) − 7160.30i − 0.318232i −0.987260 0.159116i \(-0.949136\pi\)
0.987260 0.159116i \(-0.0508644\pi\)
\(798\) 0 0
\(799\) −35541.7 −1.57369
\(800\) 0 0
\(801\) −8262.00 −0.364449
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 3741.23i 0.163803i
\(806\) 0 0
\(807\) 9446.61 0.412065
\(808\) 0 0
\(809\) 37530.0 1.63101 0.815503 0.578752i \(-0.196460\pi\)
0.815503 + 0.578752i \(0.196460\pi\)
\(810\) 0 0
\(811\) − 10852.0i − 0.469871i −0.972011 0.234935i \(-0.924512\pi\)
0.972011 0.234935i \(-0.0754879\pi\)
\(812\) 0 0
\(813\) 16035.3i 0.691739i
\(814\) 0 0
\(815\) −4032.21 −0.173303
\(816\) 0 0
\(817\) 2320.00 0.0993470
\(818\) 0 0
\(819\) 1728.00i 0.0737255i
\(820\) 0 0
\(821\) − 31353.6i − 1.33282i −0.745584 0.666411i \(-0.767829\pi\)
0.745584 0.666411i \(-0.232171\pi\)
\(822\) 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
\(827\) − 10044.0i − 0.422327i −0.977451 0.211163i \(-0.932275\pi\)
0.977451 0.211163i \(-0.0677252\pi\)
\(828\) 0 0
\(829\) 9796.48i 0.410429i 0.978717 + 0.205215i \(0.0657892\pi\)
−0.978717 + 0.205215i \(0.934211\pi\)
\(830\) 0 0
\(831\) 19579.1 0.817318
\(832\) 0 0
\(833\) 29790.0 1.23909
\(834\) 0 0
\(835\) − 22896.0i − 0.948921i
\(836\) 0 0
\(837\) 8137.17i 0.336036i
\(838\) 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
\(843\) − 3510.00i − 0.143405i
\(844\) 0 0
\(845\) − 9093.27i − 0.370199i
\(846\) 0 0
\(847\) −4610.72 −0.187044
\(848\) 0 0
\(849\) −17220.0 −0.696100
\(850\) 0 0
\(851\) − 3600.00i − 0.145013i
\(852\) 0 0
\(853\) − 40703.2i − 1.63382i −0.576763 0.816911i \(-0.695684\pi\)
0.576763 0.816911i \(-0.304316\pi\)
\(854\) 0 0
\(855\) −10849.6 −0.433973
\(856\) 0 0
\(857\) −18342.0 −0.731098 −0.365549 0.930792i \(-0.619119\pi\)
−0.365549 + 0.930792i \(0.619119\pi\)
\(858\) 0 0
\(859\) 26324.0i 1.04559i 0.852458 + 0.522796i \(0.175111\pi\)
−0.852458 + 0.522796i \(0.824889\pi\)
\(860\) 0 0
\(861\) 561.184i 0.0222127i
\(862\) 0 0
\(863\) 12761.8 0.503378 0.251689 0.967808i \(-0.419014\pi\)
0.251689 + 0.967808i \(0.419014\pi\)
\(864\) 0 0
\(865\) −2052.00 −0.0806591
\(866\) 0 0
\(867\) 9561.00i 0.374520i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −6429.37 −0.250116
\(872\) 0 0
\(873\) −1710.00 −0.0662941
\(874\) 0 0
\(875\) − 5112.00i − 0.197505i
\(876\) 0 0
\(877\) − 2459.51i − 0.0946999i −0.998878 0.0473500i \(-0.984922\pi\)
0.998878 0.0473500i \(-0.0150776\pi\)
\(878\) 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
\(883\) 18244.0i 0.695311i 0.937622 + 0.347655i \(0.113022\pi\)
−0.937622 + 0.347655i \(0.886978\pi\)
\(884\) 0 0
\(885\) − 10101.3i − 0.383675i
\(886\) 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
\(893\) − 45809.3i − 1.71663i
\(894\) 0 0
\(895\) 29555.7 1.10384
\(896\) 0 0
\(897\) 17280.0 0.643213
\(898\) 0 0
\(899\) − 78300.0i − 2.90484i
\(900\) 0 0
\(901\) − 43959.4i − 1.62542i
\(902\) 0 0
\(903\) 207.846 0.00765967
\(904\) 0 0
\(905\) 1008.00 0.0370244
\(906\) 0 0
\(907\) 16388.0i 0.599950i 0.953947 + 0.299975i \(0.0969783\pi\)
−0.953947 + 0.299975i \(0.903022\pi\)
\(908\) 0 0
\(909\) 93.5307i 0.00341278i
\(910\) 0 0
\(911\) −25107.8 −0.913127 −0.456564 0.889691i \(-0.650920\pi\)
−0.456564 + 0.889691i \(0.650920\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 17928.0i 0.647739i
\(916\) 0 0
\(917\) − 7856.58i − 0.282930i
\(918\) 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
\(923\) 61056.0i 2.17734i
\(924\) 0 0
\(925\) 588.897i 0.0209328i
\(926\) 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
\(931\) 38396.0i 1.35164i
\(932\) 0 0
\(933\) − 6609.51i − 0.231924i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7894.00 0.275225 0.137612 0.990486i \(-0.456057\pi\)
0.137612 + 0.990486i \(0.456057\pi\)
\(938\) 0 0
\(939\) − 3102.00i − 0.107806i
\(940\) 0 0
\(941\) − 2670.82i − 0.0925253i −0.998929 0.0462627i \(-0.985269\pi\)
0.998929 0.0462627i \(-0.0147311\pi\)
\(942\) 0 0
\(943\) 5611.84 0.193793
\(944\) 0 0
\(945\) −972.000 −0.0334594
\(946\) 0 0
\(947\) 22356.0i 0.767130i 0.923514 + 0.383565i \(0.125304\pi\)
−0.923514 + 0.383565i \(0.874696\pi\)
\(948\) 0 0
\(949\) 61300.7i 2.09685i
\(950\) 0 0
\(951\) 7950.11 0.271083
\(952\) 0 0
\(953\) −14958.0 −0.508434 −0.254217 0.967147i \(-0.581818\pi\)
−0.254217 + 0.967147i \(0.581818\pi\)
\(954\) 0 0
\(955\) − 33264.0i − 1.12712i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1808.26 −0.0608882
\(960\) 0 0
\(961\) 61037.0 2.04884
\(962\) 0 0
\(963\) 2268.00i 0.0758933i
\(964\) 0 0
\(965\) − 13946.5i − 0.465236i
\(966\) 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
\(971\) 48528.0i 1.60385i 0.597425 + 0.801925i \(0.296190\pi\)
−0.597425 + 0.801925i \(0.703810\pi\)
\(972\) 0 0
\(973\) − 2341.73i − 0.0771557i
\(974\) 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
\(981\) 4115.35i 0.133938i
\(982\) 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
\(987\) − 4104.00i − 0.132352i
\(988\) 0 0
\(989\) − 2078.46i − 0.0668263i
\(990\) 0 0
\(991\) 1666.23 0.0534103 0.0267052 0.999643i \(-0.491498\pi\)
0.0267052 + 0.999643i \(0.491498\pi\)
\(992\) 0 0
\(993\) −12396.0 −0.396148
\(994\) 0 0
\(995\) − 18540.0i − 0.590711i
\(996\) 0 0
\(997\) − 41465.3i − 1.31717i −0.752506 0.658585i \(-0.771155\pi\)
0.752506 0.658585i \(-0.228845\pi\)
\(998\) 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 192.4.d.a.97.4 yes 4
3.2 odd 2 576.4.d.g.289.1 4
4.3 odd 2 inner 192.4.d.a.97.2 yes 4
8.3 odd 2 inner 192.4.d.a.97.3 yes 4
8.5 even 2 inner 192.4.d.a.97.1 4
12.11 even 2 576.4.d.g.289.2 4
16.3 odd 4 768.4.a.n.1.2 2
16.5 even 4 768.4.a.n.1.1 2
16.11 odd 4 768.4.a.g.1.1 2
16.13 even 4 768.4.a.g.1.2 2
24.5 odd 2 576.4.d.g.289.3 4
24.11 even 2 576.4.d.g.289.4 4
48.5 odd 4 2304.4.a.be.1.2 2
48.11 even 4 2304.4.a.bg.1.2 2
48.29 odd 4 2304.4.a.bg.1.1 2
48.35 even 4 2304.4.a.be.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.4.d.a.97.1 4 8.5 even 2 inner
192.4.d.a.97.2 yes 4 4.3 odd 2 inner
192.4.d.a.97.3 yes 4 8.3 odd 2 inner
192.4.d.a.97.4 yes 4 1.1 even 1 trivial
576.4.d.g.289.1 4 3.2 odd 2
576.4.d.g.289.2 4 12.11 even 2
576.4.d.g.289.3 4 24.5 odd 2
576.4.d.g.289.4 4 24.11 even 2
768.4.a.g.1.1 2 16.11 odd 4
768.4.a.g.1.2 2 16.13 even 4
768.4.a.n.1.1 2 16.5 even 4
768.4.a.n.1.2 2 16.3 odd 4
2304.4.a.be.1.1 2 48.35 even 4
2304.4.a.be.1.2 2 48.5 odd 4
2304.4.a.bg.1.1 2 48.29 odd 4
2304.4.a.bg.1.2 2 48.11 even 4