Properties

Label 784.4.f.g.783.6
Level $784$
Weight $4$
Character 784.783
Analytic conductor $46.257$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.2574974445\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.12258833328.1
Defining polynomial: \(x^{6} - x^{5} + 29 x^{4} - 20 x^{3} + 808 x^{2} - 672 x + 576\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 783.6
Root \(0.426664 - 0.739004i\) of defining polynomial
Character \(\chi\) \(=\) 784.783
Dual form 784.4.f.g.783.5

$q$-expansion

\(f(q)\) \(=\) \(q+7.06258 q^{3} +1.22397i q^{5} +22.8800 q^{9} +O(q^{10})\) \(q+7.06258 q^{3} +1.22397i q^{5} +22.8800 q^{9} +4.57968i q^{11} -41.0611i q^{13} +8.64436i q^{15} +119.455i q^{17} +60.1826 q^{19} +8.35305i q^{23} +123.502 q^{25} -29.0976 q^{27} +164.382 q^{29} +287.559 q^{31} +32.3443i q^{33} -148.742 q^{37} -289.998i q^{39} +358.778i q^{41} -360.388i q^{43} +28.0044i q^{45} +225.434 q^{47} +843.660i q^{51} +684.724 q^{53} -5.60537 q^{55} +425.044 q^{57} +85.0522 q^{59} -209.127i q^{61} +50.2574 q^{65} +417.225i q^{67} +58.9941i q^{69} +982.974i q^{71} -341.526i q^{73} +872.242 q^{75} -76.2934i q^{79} -823.265 q^{81} -523.643 q^{83} -146.209 q^{85} +1160.96 q^{87} -972.055i q^{89} +2030.91 q^{93} +73.6614i q^{95} -676.730i q^{97} +104.783i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 14q^{3} + 156q^{9} + O(q^{10}) \) \( 6q - 14q^{3} + 156q^{9} + 286q^{19} - 612q^{25} - 362q^{27} - 348q^{29} + 410q^{31} + 498q^{37} + 150q^{47} + 1290q^{53} - 918q^{55} - 6q^{57} + 642q^{59} + 1224q^{65} + 8276q^{75} - 450q^{81} - 24q^{83} + 3786q^{85} + 7284q^{87} + 5982q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 7.06258 1.35919 0.679597 0.733586i \(-0.262155\pi\)
0.679597 + 0.733586i \(0.262155\pi\)
\(4\) 0 0
\(5\) 1.22397i 0.109475i 0.998501 + 0.0547374i \(0.0174322\pi\)
−0.998501 + 0.0547374i \(0.982568\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 22.8800 0.847408
\(10\) 0 0
\(11\) 4.57968i 0.125529i 0.998028 + 0.0627647i \(0.0199918\pi\)
−0.998028 + 0.0627647i \(0.980008\pi\)
\(12\) 0 0
\(13\) − 41.0611i − 0.876024i −0.898969 0.438012i \(-0.855683\pi\)
0.898969 0.438012i \(-0.144317\pi\)
\(14\) 0 0
\(15\) 8.64436i 0.148798i
\(16\) 0 0
\(17\) 119.455i 1.70424i 0.523346 + 0.852120i \(0.324683\pi\)
−0.523346 + 0.852120i \(0.675317\pi\)
\(18\) 0 0
\(19\) 60.1826 0.726675 0.363337 0.931658i \(-0.381637\pi\)
0.363337 + 0.931658i \(0.381637\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.35305i 0.0757275i 0.999283 + 0.0378637i \(0.0120553\pi\)
−0.999283 + 0.0378637i \(0.987945\pi\)
\(24\) 0 0
\(25\) 123.502 0.988015
\(26\) 0 0
\(27\) −29.0976 −0.207401
\(28\) 0 0
\(29\) 164.382 1.05258 0.526292 0.850304i \(-0.323582\pi\)
0.526292 + 0.850304i \(0.323582\pi\)
\(30\) 0 0
\(31\) 287.559 1.66604 0.833019 0.553244i \(-0.186610\pi\)
0.833019 + 0.553244i \(0.186610\pi\)
\(32\) 0 0
\(33\) 32.3443i 0.170619i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −148.742 −0.660892 −0.330446 0.943825i \(-0.607199\pi\)
−0.330446 + 0.943825i \(0.607199\pi\)
\(38\) 0 0
\(39\) − 289.998i − 1.19069i
\(40\) 0 0
\(41\) 358.778i 1.36663i 0.730125 + 0.683314i \(0.239462\pi\)
−0.730125 + 0.683314i \(0.760538\pi\)
\(42\) 0 0
\(43\) − 360.388i − 1.27811i −0.769161 0.639055i \(-0.779326\pi\)
0.769161 0.639055i \(-0.220674\pi\)
\(44\) 0 0
\(45\) 28.0044i 0.0927699i
\(46\) 0 0
\(47\) 225.434 0.699637 0.349819 0.936817i \(-0.386243\pi\)
0.349819 + 0.936817i \(0.386243\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 843.660i 2.31639i
\(52\) 0 0
\(53\) 684.724 1.77460 0.887302 0.461189i \(-0.152577\pi\)
0.887302 + 0.461189i \(0.152577\pi\)
\(54\) 0 0
\(55\) −5.60537 −0.0137423
\(56\) 0 0
\(57\) 425.044 0.987692
\(58\) 0 0
\(59\) 85.0522 0.187675 0.0938377 0.995588i \(-0.470087\pi\)
0.0938377 + 0.995588i \(0.470087\pi\)
\(60\) 0 0
\(61\) − 209.127i − 0.438949i −0.975618 0.219475i \(-0.929566\pi\)
0.975618 0.219475i \(-0.0704344\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 50.2574 0.0959026
\(66\) 0 0
\(67\) 417.225i 0.760778i 0.924827 + 0.380389i \(0.124210\pi\)
−0.924827 + 0.380389i \(0.875790\pi\)
\(68\) 0 0
\(69\) 58.9941i 0.102928i
\(70\) 0 0
\(71\) 982.974i 1.64306i 0.570162 + 0.821532i \(0.306880\pi\)
−0.570162 + 0.821532i \(0.693120\pi\)
\(72\) 0 0
\(73\) − 341.526i − 0.547570i −0.961791 0.273785i \(-0.911724\pi\)
0.961791 0.273785i \(-0.0882757\pi\)
\(74\) 0 0
\(75\) 872.242 1.34290
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 76.2934i − 0.108654i −0.998523 0.0543271i \(-0.982699\pi\)
0.998523 0.0543271i \(-0.0173014\pi\)
\(80\) 0 0
\(81\) −823.265 −1.12931
\(82\) 0 0
\(83\) −523.643 −0.692498 −0.346249 0.938143i \(-0.612545\pi\)
−0.346249 + 0.938143i \(0.612545\pi\)
\(84\) 0 0
\(85\) −146.209 −0.186571
\(86\) 0 0
\(87\) 1160.96 1.43067
\(88\) 0 0
\(89\) − 972.055i − 1.15773i −0.815425 0.578863i \(-0.803497\pi\)
0.815425 0.578863i \(-0.196503\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2030.91 2.26447
\(94\) 0 0
\(95\) 73.6614i 0.0795526i
\(96\) 0 0
\(97\) − 676.730i − 0.708367i −0.935176 0.354183i \(-0.884759\pi\)
0.935176 0.354183i \(-0.115241\pi\)
\(98\) 0 0
\(99\) 104.783i 0.106375i
\(100\) 0 0
\(101\) − 838.786i − 0.826360i −0.910649 0.413180i \(-0.864418\pi\)
0.910649 0.413180i \(-0.135582\pi\)
\(102\) 0 0
\(103\) 1515.20 1.44949 0.724744 0.689018i \(-0.241958\pi\)
0.724744 + 0.689018i \(0.241958\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1687.67i 1.52480i 0.647107 + 0.762399i \(0.275979\pi\)
−0.647107 + 0.762399i \(0.724021\pi\)
\(108\) 0 0
\(109\) −1576.23 −1.38510 −0.692550 0.721370i \(-0.743513\pi\)
−0.692550 + 0.721370i \(0.743513\pi\)
\(110\) 0 0
\(111\) −1050.50 −0.898281
\(112\) 0 0
\(113\) −2192.01 −1.82484 −0.912422 0.409251i \(-0.865790\pi\)
−0.912422 + 0.409251i \(0.865790\pi\)
\(114\) 0 0
\(115\) −10.2239 −0.00829025
\(116\) 0 0
\(117\) − 939.480i − 0.742350i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1310.03 0.984242
\(122\) 0 0
\(123\) 2533.90i 1.85751i
\(124\) 0 0
\(125\) 304.158i 0.217638i
\(126\) 0 0
\(127\) 1085.60i 0.758515i 0.925291 + 0.379258i \(0.123821\pi\)
−0.925291 + 0.379258i \(0.876179\pi\)
\(128\) 0 0
\(129\) − 2545.27i − 1.73720i
\(130\) 0 0
\(131\) −1533.35 −1.02267 −0.511334 0.859382i \(-0.670849\pi\)
−0.511334 + 0.859382i \(0.670849\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) − 35.6145i − 0.0227052i
\(136\) 0 0
\(137\) 920.175 0.573838 0.286919 0.957955i \(-0.407369\pi\)
0.286919 + 0.957955i \(0.407369\pi\)
\(138\) 0 0
\(139\) −1424.66 −0.869339 −0.434669 0.900590i \(-0.643135\pi\)
−0.434669 + 0.900590i \(0.643135\pi\)
\(140\) 0 0
\(141\) 1592.15 0.950943
\(142\) 0 0
\(143\) 188.047 0.109967
\(144\) 0 0
\(145\) 201.198i 0.115232i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1378.25 −0.757791 −0.378896 0.925439i \(-0.623696\pi\)
−0.378896 + 0.925439i \(0.623696\pi\)
\(150\) 0 0
\(151\) − 1333.78i − 0.718817i −0.933180 0.359409i \(-0.882978\pi\)
0.933180 0.359409i \(-0.117022\pi\)
\(152\) 0 0
\(153\) 2733.13i 1.44419i
\(154\) 0 0
\(155\) 351.963i 0.182389i
\(156\) 0 0
\(157\) − 1255.89i − 0.638416i −0.947685 0.319208i \(-0.896583\pi\)
0.947685 0.319208i \(-0.103417\pi\)
\(158\) 0 0
\(159\) 4835.92 2.41203
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3254.58i 1.56392i 0.623330 + 0.781959i \(0.285779\pi\)
−0.623330 + 0.781959i \(0.714221\pi\)
\(164\) 0 0
\(165\) −39.5884 −0.0186785
\(166\) 0 0
\(167\) 2736.24 1.26788 0.633941 0.773381i \(-0.281436\pi\)
0.633941 + 0.773381i \(0.281436\pi\)
\(168\) 0 0
\(169\) 510.983 0.232582
\(170\) 0 0
\(171\) 1376.98 0.615791
\(172\) 0 0
\(173\) − 3380.13i − 1.48547i −0.669584 0.742737i \(-0.733527\pi\)
0.669584 0.742737i \(-0.266473\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 600.688 0.255087
\(178\) 0 0
\(179\) − 1217.98i − 0.508584i −0.967128 0.254292i \(-0.918158\pi\)
0.967128 0.254292i \(-0.0818423\pi\)
\(180\) 0 0
\(181\) − 1757.61i − 0.721781i −0.932608 0.360890i \(-0.882473\pi\)
0.932608 0.360890i \(-0.117527\pi\)
\(182\) 0 0
\(183\) − 1476.97i − 0.596618i
\(184\) 0 0
\(185\) − 182.055i − 0.0723511i
\(186\) 0 0
\(187\) −547.065 −0.213932
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 956.433i 0.362330i 0.983453 + 0.181165i \(0.0579868\pi\)
−0.983453 + 0.181165i \(0.942013\pi\)
\(192\) 0 0
\(193\) 1333.55 0.497362 0.248681 0.968585i \(-0.420003\pi\)
0.248681 + 0.968585i \(0.420003\pi\)
\(194\) 0 0
\(195\) 354.947 0.130350
\(196\) 0 0
\(197\) 1098.75 0.397373 0.198686 0.980063i \(-0.436332\pi\)
0.198686 + 0.980063i \(0.436332\pi\)
\(198\) 0 0
\(199\) −248.579 −0.0885491 −0.0442745 0.999019i \(-0.514098\pi\)
−0.0442745 + 0.999019i \(0.514098\pi\)
\(200\) 0 0
\(201\) 2946.68i 1.03405i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −439.132 −0.149611
\(206\) 0 0
\(207\) 191.118i 0.0641721i
\(208\) 0 0
\(209\) 275.617i 0.0912191i
\(210\) 0 0
\(211\) 2029.24i 0.662078i 0.943617 + 0.331039i \(0.107399\pi\)
−0.943617 + 0.331039i \(0.892601\pi\)
\(212\) 0 0
\(213\) 6942.33i 2.23324i
\(214\) 0 0
\(215\) 441.103 0.139921
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 2412.06i − 0.744254i
\(220\) 0 0
\(221\) 4904.96 1.49295
\(222\) 0 0
\(223\) −3904.24 −1.17241 −0.586204 0.810164i \(-0.699378\pi\)
−0.586204 + 0.810164i \(0.699378\pi\)
\(224\) 0 0
\(225\) 2825.73 0.837253
\(226\) 0 0
\(227\) 4255.75 1.24434 0.622168 0.782884i \(-0.286252\pi\)
0.622168 + 0.782884i \(0.286252\pi\)
\(228\) 0 0
\(229\) 1732.25i 0.499871i 0.968262 + 0.249936i \(0.0804095\pi\)
−0.968262 + 0.249936i \(0.919591\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −874.910 −0.245997 −0.122998 0.992407i \(-0.539251\pi\)
−0.122998 + 0.992407i \(0.539251\pi\)
\(234\) 0 0
\(235\) 275.924i 0.0765927i
\(236\) 0 0
\(237\) − 538.828i − 0.147682i
\(238\) 0 0
\(239\) − 1842.84i − 0.498760i −0.968406 0.249380i \(-0.919773\pi\)
0.968406 0.249380i \(-0.0802269\pi\)
\(240\) 0 0
\(241\) 4683.60i 1.25185i 0.779881 + 0.625927i \(0.215279\pi\)
−0.779881 + 0.625927i \(0.784721\pi\)
\(242\) 0 0
\(243\) −5028.74 −1.32755
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 2471.16i − 0.636585i
\(248\) 0 0
\(249\) −3698.27 −0.941239
\(250\) 0 0
\(251\) −1015.59 −0.255393 −0.127696 0.991813i \(-0.540758\pi\)
−0.127696 + 0.991813i \(0.540758\pi\)
\(252\) 0 0
\(253\) −38.2543 −0.00950603
\(254\) 0 0
\(255\) −1032.61 −0.253587
\(256\) 0 0
\(257\) 3016.56i 0.732171i 0.930581 + 0.366085i \(0.119302\pi\)
−0.930581 + 0.366085i \(0.880698\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3761.06 0.891969
\(262\) 0 0
\(263\) − 2651.05i − 0.621561i −0.950482 0.310780i \(-0.899410\pi\)
0.950482 0.310780i \(-0.100590\pi\)
\(264\) 0 0
\(265\) 838.078i 0.194275i
\(266\) 0 0
\(267\) − 6865.21i − 1.57357i
\(268\) 0 0
\(269\) − 452.582i − 0.102581i −0.998684 0.0512907i \(-0.983666\pi\)
0.998684 0.0512907i \(-0.0163335\pi\)
\(270\) 0 0
\(271\) −164.726 −0.0369239 −0.0184620 0.999830i \(-0.505877\pi\)
−0.0184620 + 0.999830i \(0.505877\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 565.599i 0.124025i
\(276\) 0 0
\(277\) 1939.61 0.420721 0.210360 0.977624i \(-0.432536\pi\)
0.210360 + 0.977624i \(0.432536\pi\)
\(278\) 0 0
\(279\) 6579.37 1.41181
\(280\) 0 0
\(281\) −7152.79 −1.51850 −0.759252 0.650796i \(-0.774435\pi\)
−0.759252 + 0.650796i \(0.774435\pi\)
\(282\) 0 0
\(283\) −3295.55 −0.692226 −0.346113 0.938193i \(-0.612499\pi\)
−0.346113 + 0.938193i \(0.612499\pi\)
\(284\) 0 0
\(285\) 520.239i 0.108127i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9356.49 −1.90443
\(290\) 0 0
\(291\) − 4779.46i − 0.962808i
\(292\) 0 0
\(293\) − 7368.65i − 1.46922i −0.678491 0.734609i \(-0.737366\pi\)
0.678491 0.734609i \(-0.262634\pi\)
\(294\) 0 0
\(295\) 104.101i 0.0205457i
\(296\) 0 0
\(297\) − 133.258i − 0.0260350i
\(298\) 0 0
\(299\) 342.986 0.0663391
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 5923.99i − 1.12318i
\(304\) 0 0
\(305\) 255.964 0.0480539
\(306\) 0 0
\(307\) 2692.96 0.500636 0.250318 0.968164i \(-0.419465\pi\)
0.250318 + 0.968164i \(0.419465\pi\)
\(308\) 0 0
\(309\) 10701.2 1.97014
\(310\) 0 0
\(311\) −2971.99 −0.541885 −0.270943 0.962596i \(-0.587335\pi\)
−0.270943 + 0.962596i \(0.587335\pi\)
\(312\) 0 0
\(313\) − 1140.00i − 0.205868i −0.994688 0.102934i \(-0.967177\pi\)
0.994688 0.102934i \(-0.0328230\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5645.11 −1.00019 −0.500097 0.865970i \(-0.666702\pi\)
−0.500097 + 0.865970i \(0.666702\pi\)
\(318\) 0 0
\(319\) 752.816i 0.132130i
\(320\) 0 0
\(321\) 11919.3i 2.07250i
\(322\) 0 0
\(323\) 7189.10i 1.23843i
\(324\) 0 0
\(325\) − 5071.13i − 0.865525i
\(326\) 0 0
\(327\) −11132.3 −1.88262
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 7359.98i − 1.22218i −0.791562 0.611089i \(-0.790732\pi\)
0.791562 0.611089i \(-0.209268\pi\)
\(332\) 0 0
\(333\) −3403.22 −0.560046
\(334\) 0 0
\(335\) −510.669 −0.0832861
\(336\) 0 0
\(337\) −10163.9 −1.64292 −0.821459 0.570268i \(-0.806839\pi\)
−0.821459 + 0.570268i \(0.806839\pi\)
\(338\) 0 0
\(339\) −15481.3 −2.48032
\(340\) 0 0
\(341\) 1316.93i 0.209137i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −72.2068 −0.0112681
\(346\) 0 0
\(347\) 10544.8i 1.63133i 0.578522 + 0.815666i \(0.303630\pi\)
−0.578522 + 0.815666i \(0.696370\pi\)
\(348\) 0 0
\(349\) − 6048.02i − 0.927631i −0.885932 0.463816i \(-0.846480\pi\)
0.885932 0.463816i \(-0.153520\pi\)
\(350\) 0 0
\(351\) 1194.78i 0.181689i
\(352\) 0 0
\(353\) − 6289.12i − 0.948261i −0.880454 0.474131i \(-0.842762\pi\)
0.880454 0.474131i \(-0.157238\pi\)
\(354\) 0 0
\(355\) −1203.13 −0.179874
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 154.662i 0.0227375i 0.999935 + 0.0113687i \(0.00361886\pi\)
−0.999935 + 0.0113687i \(0.996381\pi\)
\(360\) 0 0
\(361\) −3237.06 −0.471944
\(362\) 0 0
\(363\) 9252.17 1.33778
\(364\) 0 0
\(365\) 418.016 0.0599451
\(366\) 0 0
\(367\) −5890.42 −0.837813 −0.418907 0.908029i \(-0.637587\pi\)
−0.418907 + 0.908029i \(0.637587\pi\)
\(368\) 0 0
\(369\) 8208.86i 1.15809i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 350.300 0.0486270 0.0243135 0.999704i \(-0.492260\pi\)
0.0243135 + 0.999704i \(0.492260\pi\)
\(374\) 0 0
\(375\) 2148.14i 0.295812i
\(376\) 0 0
\(377\) − 6749.71i − 0.922089i
\(378\) 0 0
\(379\) − 8247.17i − 1.11775i −0.829251 0.558877i \(-0.811233\pi\)
0.829251 0.558877i \(-0.188767\pi\)
\(380\) 0 0
\(381\) 7667.14i 1.03097i
\(382\) 0 0
\(383\) −12680.2 −1.69171 −0.845856 0.533411i \(-0.820910\pi\)
−0.845856 + 0.533411i \(0.820910\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 8245.69i − 1.08308i
\(388\) 0 0
\(389\) −7727.53 −1.00720 −0.503601 0.863937i \(-0.667992\pi\)
−0.503601 + 0.863937i \(0.667992\pi\)
\(390\) 0 0
\(391\) −997.814 −0.129058
\(392\) 0 0
\(393\) −10829.4 −1.39000
\(394\) 0 0
\(395\) 93.3806 0.0118949
\(396\) 0 0
\(397\) − 10096.2i − 1.27636i −0.769889 0.638178i \(-0.779688\pi\)
0.769889 0.638178i \(-0.220312\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7678.50 0.956224 0.478112 0.878299i \(-0.341321\pi\)
0.478112 + 0.878299i \(0.341321\pi\)
\(402\) 0 0
\(403\) − 11807.5i − 1.45949i
\(404\) 0 0
\(405\) − 1007.65i − 0.123631i
\(406\) 0 0
\(407\) − 681.190i − 0.0829615i
\(408\) 0 0
\(409\) − 3711.55i − 0.448715i −0.974507 0.224358i \(-0.927972\pi\)
0.974507 0.224358i \(-0.0720284\pi\)
\(410\) 0 0
\(411\) 6498.81 0.779957
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 640.921i − 0.0758111i
\(416\) 0 0
\(417\) −10061.8 −1.18160
\(418\) 0 0
\(419\) −8868.09 −1.03397 −0.516986 0.855994i \(-0.672946\pi\)
−0.516986 + 0.855994i \(0.672946\pi\)
\(420\) 0 0
\(421\) −9571.99 −1.10810 −0.554050 0.832483i \(-0.686919\pi\)
−0.554050 + 0.832483i \(0.686919\pi\)
\(422\) 0 0
\(423\) 5157.94 0.592879
\(424\) 0 0
\(425\) 14752.9i 1.68382i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 1328.09 0.149466
\(430\) 0 0
\(431\) 7712.13i 0.861903i 0.902375 + 0.430952i \(0.141822\pi\)
−0.902375 + 0.430952i \(0.858178\pi\)
\(432\) 0 0
\(433\) − 2222.71i − 0.246690i −0.992364 0.123345i \(-0.960638\pi\)
0.992364 0.123345i \(-0.0393622\pi\)
\(434\) 0 0
\(435\) 1420.98i 0.156622i
\(436\) 0 0
\(437\) 502.708i 0.0550293i
\(438\) 0 0
\(439\) −10612.6 −1.15378 −0.576890 0.816822i \(-0.695734\pi\)
−0.576890 + 0.816822i \(0.695734\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 15591.1i − 1.67213i −0.548630 0.836065i \(-0.684851\pi\)
0.548630 0.836065i \(-0.315149\pi\)
\(444\) 0 0
\(445\) 1189.76 0.126742
\(446\) 0 0
\(447\) −9734.03 −1.02999
\(448\) 0 0
\(449\) 2759.74 0.290067 0.145033 0.989427i \(-0.453671\pi\)
0.145033 + 0.989427i \(0.453671\pi\)
\(450\) 0 0
\(451\) −1643.09 −0.171552
\(452\) 0 0
\(453\) − 9419.92i − 0.977012i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7672.29 −0.785327 −0.392664 0.919682i \(-0.628446\pi\)
−0.392664 + 0.919682i \(0.628446\pi\)
\(458\) 0 0
\(459\) − 3475.86i − 0.353462i
\(460\) 0 0
\(461\) − 13864.8i − 1.40076i −0.713771 0.700379i \(-0.753014\pi\)
0.713771 0.700379i \(-0.246986\pi\)
\(462\) 0 0
\(463\) 11518.3i 1.15616i 0.815981 + 0.578079i \(0.196197\pi\)
−0.815981 + 0.578079i \(0.803803\pi\)
\(464\) 0 0
\(465\) 2485.77i 0.247902i
\(466\) 0 0
\(467\) 2815.92 0.279027 0.139513 0.990220i \(-0.455446\pi\)
0.139513 + 0.990220i \(0.455446\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) − 8869.86i − 0.867731i
\(472\) 0 0
\(473\) 1650.46 0.160440
\(474\) 0 0
\(475\) 7432.66 0.717966
\(476\) 0 0
\(477\) 15666.5 1.50381
\(478\) 0 0
\(479\) −3120.15 −0.297627 −0.148813 0.988865i \(-0.547545\pi\)
−0.148813 + 0.988865i \(0.547545\pi\)
\(480\) 0 0
\(481\) 6107.51i 0.578957i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 828.295 0.0775483
\(486\) 0 0
\(487\) 1988.38i 0.185015i 0.995712 + 0.0925074i \(0.0294882\pi\)
−0.995712 + 0.0925074i \(0.970512\pi\)
\(488\) 0 0
\(489\) 22985.8i 2.12567i
\(490\) 0 0
\(491\) − 3035.51i − 0.279003i −0.990222 0.139502i \(-0.955450\pi\)
0.990222 0.139502i \(-0.0445501\pi\)
\(492\) 0 0
\(493\) 19636.2i 1.79386i
\(494\) 0 0
\(495\) −128.251 −0.0116454
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 16519.7i 1.48201i 0.671498 + 0.741007i \(0.265651\pi\)
−0.671498 + 0.741007i \(0.734349\pi\)
\(500\) 0 0
\(501\) 19324.9 1.72330
\(502\) 0 0
\(503\) 16446.0 1.45783 0.728917 0.684602i \(-0.240024\pi\)
0.728917 + 0.684602i \(0.240024\pi\)
\(504\) 0 0
\(505\) 1026.65 0.0904656
\(506\) 0 0
\(507\) 3608.86 0.316124
\(508\) 0 0
\(509\) 4685.12i 0.407985i 0.978972 + 0.203992i \(0.0653917\pi\)
−0.978972 + 0.203992i \(0.934608\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1751.17 −0.150713
\(514\) 0 0
\(515\) 1854.56i 0.158682i
\(516\) 0 0
\(517\) 1032.42i 0.0878251i
\(518\) 0 0
\(519\) − 23872.5i − 2.01905i
\(520\) 0 0
\(521\) − 2199.68i − 0.184971i −0.995714 0.0924853i \(-0.970519\pi\)
0.995714 0.0924853i \(-0.0294811\pi\)
\(522\) 0 0
\(523\) 10465.4 0.874988 0.437494 0.899221i \(-0.355866\pi\)
0.437494 + 0.899221i \(0.355866\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 34350.4i 2.83933i
\(528\) 0 0
\(529\) 12097.2 0.994265
\(530\) 0 0
\(531\) 1946.00 0.159038
\(532\) 0 0
\(533\) 14731.8 1.19720
\(534\) 0 0
\(535\) −2065.65 −0.166927
\(536\) 0 0
\(537\) − 8602.11i − 0.691264i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2934.94 −0.233240 −0.116620 0.993177i \(-0.537206\pi\)
−0.116620 + 0.993177i \(0.537206\pi\)
\(542\) 0 0
\(543\) − 12413.3i − 0.981040i
\(544\) 0 0
\(545\) − 1929.26i − 0.151634i
\(546\) 0 0
\(547\) 9168.13i 0.716638i 0.933599 + 0.358319i \(0.116650\pi\)
−0.933599 + 0.358319i \(0.883350\pi\)
\(548\) 0 0
\(549\) − 4784.82i − 0.371970i
\(550\) 0 0
\(551\) 9892.92 0.764887
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 1285.78i − 0.0983391i
\(556\) 0 0
\(557\) 4618.14 0.351305 0.175652 0.984452i \(-0.443797\pi\)
0.175652 + 0.984452i \(0.443797\pi\)
\(558\) 0 0
\(559\) −14797.9 −1.11965
\(560\) 0 0
\(561\) −3863.69 −0.290776
\(562\) 0 0
\(563\) 23789.6 1.78084 0.890418 0.455144i \(-0.150412\pi\)
0.890418 + 0.455144i \(0.150412\pi\)
\(564\) 0 0
\(565\) − 2682.95i − 0.199775i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −14772.6 −1.08840 −0.544200 0.838955i \(-0.683167\pi\)
−0.544200 + 0.838955i \(0.683167\pi\)
\(570\) 0 0
\(571\) − 24295.7i − 1.78064i −0.455339 0.890318i \(-0.650482\pi\)
0.455339 0.890318i \(-0.349518\pi\)
\(572\) 0 0
\(573\) 6754.88i 0.492477i
\(574\) 0 0
\(575\) 1031.62i 0.0748199i
\(576\) 0 0
\(577\) − 10024.6i − 0.723275i −0.932319 0.361638i \(-0.882218\pi\)
0.932319 0.361638i \(-0.117782\pi\)
\(578\) 0 0
\(579\) 9418.30 0.676012
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3135.81i 0.222765i
\(584\) 0 0
\(585\) 1149.89 0.0812686
\(586\) 0 0
\(587\) 10134.7 0.712613 0.356306 0.934369i \(-0.384036\pi\)
0.356306 + 0.934369i \(0.384036\pi\)
\(588\) 0 0
\(589\) 17306.1 1.21067
\(590\) 0 0
\(591\) 7759.98 0.540107
\(592\) 0 0
\(593\) − 2197.51i − 0.152177i −0.997101 0.0760885i \(-0.975757\pi\)
0.997101 0.0760885i \(-0.0242431\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1755.61 −0.120355
\(598\) 0 0
\(599\) 24663.6i 1.68235i 0.540765 + 0.841174i \(0.318135\pi\)
−0.540765 + 0.841174i \(0.681865\pi\)
\(600\) 0 0
\(601\) 21532.0i 1.46142i 0.682691 + 0.730708i \(0.260810\pi\)
−0.682691 + 0.730708i \(0.739190\pi\)
\(602\) 0 0
\(603\) 9546.12i 0.644690i
\(604\) 0 0
\(605\) 1603.43i 0.107750i
\(606\) 0 0
\(607\) 987.430 0.0660273 0.0330136 0.999455i \(-0.489490\pi\)
0.0330136 + 0.999455i \(0.489490\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 9256.58i − 0.612899i
\(612\) 0 0
\(613\) 7278.51 0.479570 0.239785 0.970826i \(-0.422923\pi\)
0.239785 + 0.970826i \(0.422923\pi\)
\(614\) 0 0
\(615\) −3101.41 −0.203351
\(616\) 0 0
\(617\) −2543.97 −0.165991 −0.0829953 0.996550i \(-0.526449\pi\)
−0.0829953 + 0.996550i \(0.526449\pi\)
\(618\) 0 0
\(619\) 7853.15 0.509927 0.254963 0.966951i \(-0.417937\pi\)
0.254963 + 0.966951i \(0.417937\pi\)
\(620\) 0 0
\(621\) − 243.054i − 0.0157060i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15065.5 0.964189
\(626\) 0 0
\(627\) 1946.56i 0.123985i
\(628\) 0 0
\(629\) − 17768.0i − 1.12632i
\(630\) 0 0
\(631\) − 16369.0i − 1.03271i −0.856375 0.516355i \(-0.827289\pi\)
0.856375 0.516355i \(-0.172711\pi\)
\(632\) 0 0
\(633\) 14331.7i 0.899893i
\(634\) 0 0
\(635\) −1328.74 −0.0830383
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 22490.5i 1.39235i
\(640\) 0 0
\(641\) 14497.2 0.893302 0.446651 0.894708i \(-0.352617\pi\)
0.446651 + 0.894708i \(0.352617\pi\)
\(642\) 0 0
\(643\) −29771.6 −1.82594 −0.912969 0.408029i \(-0.866216\pi\)
−0.912969 + 0.408029i \(0.866216\pi\)
\(644\) 0 0
\(645\) 3115.32 0.190179
\(646\) 0 0
\(647\) 11215.5 0.681496 0.340748 0.940155i \(-0.389320\pi\)
0.340748 + 0.940155i \(0.389320\pi\)
\(648\) 0 0
\(649\) 389.512i 0.0235588i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20401.0 1.22259 0.611295 0.791403i \(-0.290649\pi\)
0.611295 + 0.791403i \(0.290649\pi\)
\(654\) 0 0
\(655\) − 1876.77i − 0.111956i
\(656\) 0 0
\(657\) − 7814.13i − 0.464016i
\(658\) 0 0
\(659\) 13427.1i 0.793693i 0.917885 + 0.396846i \(0.129895\pi\)
−0.917885 + 0.396846i \(0.870105\pi\)
\(660\) 0 0
\(661\) 2313.84i 0.136154i 0.997680 + 0.0680771i \(0.0216864\pi\)
−0.997680 + 0.0680771i \(0.978314\pi\)
\(662\) 0 0
\(663\) 34641.6 2.02922
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1373.09i 0.0797096i
\(668\) 0 0
\(669\) −27574.0 −1.59353
\(670\) 0 0
\(671\) 957.732 0.0551011
\(672\) 0 0
\(673\) −16395.5 −0.939081 −0.469540 0.882911i \(-0.655580\pi\)
−0.469540 + 0.882911i \(0.655580\pi\)
\(674\) 0 0
\(675\) −3593.61 −0.204916
\(676\) 0 0
\(677\) − 24962.8i − 1.41713i −0.705645 0.708566i \(-0.749342\pi\)
0.705645 0.708566i \(-0.250658\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 30056.6 1.69129
\(682\) 0 0
\(683\) 20374.5i 1.14145i 0.821142 + 0.570724i \(0.193337\pi\)
−0.821142 + 0.570724i \(0.806663\pi\)
\(684\) 0 0
\(685\) 1126.26i 0.0628208i
\(686\) 0 0
\(687\) 12234.2i 0.679422i
\(688\) 0 0
\(689\) − 28115.5i − 1.55460i
\(690\) 0 0
\(691\) 6393.32 0.351973 0.175986 0.984393i \(-0.443689\pi\)
0.175986 + 0.984393i \(0.443689\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 1743.74i − 0.0951707i
\(696\) 0 0
\(697\) −42857.8 −2.32906
\(698\) 0 0
\(699\) −6179.12 −0.334357
\(700\) 0 0
\(701\) −19401.6 −1.04535 −0.522674 0.852533i \(-0.675066\pi\)
−0.522674 + 0.852533i \(0.675066\pi\)
\(702\) 0 0
\(703\) −8951.66 −0.480254
\(704\) 0 0
\(705\) 1948.73i 0.104104i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 14777.7 0.782776 0.391388 0.920226i \(-0.371995\pi\)
0.391388 + 0.920226i \(0.371995\pi\)
\(710\) 0 0
\(711\) − 1745.60i − 0.0920745i
\(712\) 0 0
\(713\) 2402.00i 0.126165i
\(714\) 0 0
\(715\) 230.163i 0.0120386i
\(716\) 0 0
\(717\) − 13015.2i − 0.677912i
\(718\) 0 0
\(719\) −6437.04 −0.333882 −0.166941 0.985967i \(-0.553389\pi\)
−0.166941 + 0.985967i \(0.553389\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 33078.3i 1.70151i
\(724\) 0 0
\(725\) 20301.5 1.03997
\(726\) 0 0
\(727\) −13490.8 −0.688235 −0.344118 0.938927i \(-0.611822\pi\)
−0.344118 + 0.938927i \(0.611822\pi\)
\(728\) 0 0
\(729\) −13287.7 −0.675086
\(730\) 0 0
\(731\) 43050.2 2.17820
\(732\) 0 0
\(733\) − 20806.6i − 1.04844i −0.851582 0.524221i \(-0.824356\pi\)
0.851582 0.524221i \(-0.175644\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1910.75 −0.0955001
\(738\) 0 0
\(739\) 17283.7i 0.860342i 0.902747 + 0.430171i \(0.141547\pi\)
−0.902747 + 0.430171i \(0.858453\pi\)
\(740\) 0 0
\(741\) − 17452.8i − 0.865242i
\(742\) 0 0
\(743\) 11763.6i 0.580839i 0.956899 + 0.290420i \(0.0937948\pi\)
−0.956899 + 0.290420i \(0.906205\pi\)
\(744\) 0 0
\(745\) − 1686.94i − 0.0829591i
\(746\) 0 0
\(747\) −11981.0 −0.586828
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 12377.8i − 0.601427i −0.953715 0.300713i \(-0.902775\pi\)
0.953715 0.300713i \(-0.0972247\pi\)
\(752\) 0 0
\(753\) −7172.70 −0.347129
\(754\) 0 0
\(755\) 1632.50 0.0786924
\(756\) 0 0
\(757\) 18350.8 0.881071 0.440536 0.897735i \(-0.354789\pi\)
0.440536 + 0.897735i \(0.354789\pi\)
\(758\) 0 0
\(759\) −270.174 −0.0129205
\(760\) 0 0
\(761\) 16558.4i 0.788754i 0.918949 + 0.394377i \(0.129040\pi\)
−0.918949 + 0.394377i \(0.870960\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −3345.26 −0.158102
\(766\) 0 0
\(767\) − 3492.34i − 0.164408i
\(768\) 0 0
\(769\) 3663.72i 0.171804i 0.996304 + 0.0859019i \(0.0273772\pi\)
−0.996304 + 0.0859019i \(0.972623\pi\)
\(770\) 0 0
\(771\) 21304.7i 0.995162i
\(772\) 0 0
\(773\) − 39653.9i − 1.84509i −0.385893 0.922544i \(-0.626106\pi\)
0.385893 0.922544i \(-0.373894\pi\)
\(774\) 0 0
\(775\) 35514.1 1.64607
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 21592.2i 0.993094i
\(780\) 0 0
\(781\) −4501.70 −0.206253
\(782\) 0 0
\(783\) −4783.12 −0.218308
\(784\) 0 0
\(785\) 1537.17 0.0698905
\(786\) 0 0
\(787\) −35468.7 −1.60651 −0.803254 0.595637i \(-0.796900\pi\)
−0.803254 + 0.595637i \(0.796900\pi\)
\(788\) 0 0
\(789\) − 18723.2i − 0.844821i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −8586.98 −0.384530
\(794\) 0 0
\(795\) 5919.00i 0.264057i
\(796\) 0 0
\(797\) 32026.0i 1.42336i 0.702503 + 0.711681i \(0.252066\pi\)
−0.702503 + 0.711681i \(0.747934\pi\)
\(798\) 0 0
\(799\) 26929.2i 1.19235i
\(800\) 0 0
\(801\) − 22240.6i − 0.981067i
\(802\) 0 0
\(803\) 1564.08 0.0687362
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 3196.39i − 0.139428i
\(808\) 0 0
\(809\) 3417.20 0.148507 0.0742537 0.997239i \(-0.476343\pi\)
0.0742537 + 0.997239i \(0.476343\pi\)
\(810\) 0 0
\(811\) 27667.8 1.19796 0.598982 0.800763i \(-0.295572\pi\)
0.598982 + 0.800763i \(0.295572\pi\)
\(812\) 0 0
\(813\) −1163.39 −0.0501868
\(814\) 0 0
\(815\) −3983.50 −0.171210
\(816\) 0 0
\(817\) − 21689.1i − 0.928770i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 28684.9 1.21938 0.609690 0.792640i \(-0.291294\pi\)
0.609690 + 0.792640i \(0.291294\pi\)
\(822\) 0 0
\(823\) 21516.6i 0.911326i 0.890152 + 0.455663i \(0.150598\pi\)
−0.890152 + 0.455663i \(0.849402\pi\)
\(824\) 0 0
\(825\) 3994.59i 0.168574i
\(826\) 0 0
\(827\) − 28539.8i − 1.20003i −0.799988 0.600016i \(-0.795161\pi\)
0.799988 0.600016i \(-0.204839\pi\)
\(828\) 0 0
\(829\) 4362.57i 0.182772i 0.995816 + 0.0913862i \(0.0291298\pi\)
−0.995816 + 0.0913862i \(0.970870\pi\)
\(830\) 0 0
\(831\) 13698.6 0.571841
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 3349.06i 0.138801i
\(836\) 0 0
\(837\) −8367.29 −0.345539
\(838\) 0 0
\(839\) 34373.2 1.41442 0.707208 0.707006i \(-0.249955\pi\)
0.707208 + 0.707006i \(0.249955\pi\)
\(840\) 0 0
\(841\) 2632.42 0.107935
\(842\) 0 0
\(843\) −50517.2 −2.06394
\(844\) 0 0
\(845\) 625.426i 0.0254619i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −23275.1 −0.940869
\(850\) 0 0
\(851\) − 1242.45i − 0.0500477i
\(852\) 0 0
\(853\) − 9468.45i − 0.380063i −0.981778 0.190031i \(-0.939141\pi\)
0.981778 0.190031i \(-0.0608590\pi\)
\(854\) 0 0
\(855\) 1685.37i 0.0674136i
\(856\) 0 0
\(857\) 11389.0i 0.453957i 0.973900 + 0.226978i \(0.0728847\pi\)
−0.973900 + 0.226978i \(0.927115\pi\)
\(858\) 0 0
\(859\) −8455.90 −0.335869 −0.167935 0.985798i \(-0.553710\pi\)
−0.167935 + 0.985798i \(0.553710\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7474.49i 0.294826i 0.989075 + 0.147413i \(0.0470946\pi\)
−0.989075 + 0.147413i \(0.952905\pi\)
\(864\) 0 0
\(865\) 4137.17 0.162622
\(866\) 0 0
\(867\) −66080.9 −2.58850
\(868\) 0 0
\(869\) 349.399 0.0136393
\(870\) 0 0
\(871\) 17131.7 0.666460
\(872\) 0 0
\(873\) − 15483.6i − 0.600276i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −32056.6 −1.23429 −0.617147 0.786848i \(-0.711712\pi\)
−0.617147 + 0.786848i \(0.711712\pi\)
\(878\) 0 0
\(879\) − 52041.7i − 1.99695i
\(880\) 0 0
\(881\) 27139.7i 1.03787i 0.854815 + 0.518933i \(0.173671\pi\)
−0.854815 + 0.518933i \(0.826329\pi\)
\(882\) 0 0
\(883\) − 43490.8i − 1.65751i −0.559611 0.828755i \(-0.689049\pi\)
0.559611 0.828755i \(-0.310951\pi\)
\(884\) 0 0
\(885\) 735.222i 0.0279257i
\(886\) 0 0
\(887\) −33511.2 −1.26854 −0.634271 0.773111i \(-0.718700\pi\)
−0.634271 + 0.773111i \(0.718700\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 3770.29i − 0.141761i
\(892\) 0 0
\(893\) 13567.2 0.508409
\(894\) 0 0
\(895\) 1490.77 0.0556771
\(896\) 0 0
\(897\) 2422.36 0.0901677
\(898\) 0 0
\(899\) 47269.6 1.75365
\(900\) 0 0
\(901\) 81793.6i 3.02435i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2151.26 0.0790168
\(906\) 0 0
\(907\) − 7530.44i − 0.275683i −0.990454 0.137841i \(-0.955984\pi\)
0.990454 0.137841i \(-0.0440164\pi\)
\(908\) 0 0
\(909\) − 19191.4i − 0.700264i
\(910\) 0 0
\(911\) 19971.6i 0.726332i 0.931724 + 0.363166i \(0.118304\pi\)
−0.931724 + 0.363166i \(0.881696\pi\)
\(912\) 0 0
\(913\) − 2398.12i − 0.0869289i
\(914\) 0 0
\(915\) 1807.76 0.0653146
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 46751.5i − 1.67812i −0.544041 0.839059i \(-0.683106\pi\)
0.544041 0.839059i \(-0.316894\pi\)
\(920\) 0 0
\(921\) 19019.2 0.680461
\(922\) 0 0
\(923\) 40362.0 1.43936
\(924\) 0 0
\(925\) −18369.9 −0.652972
\(926\) 0 0
\(927\) 34667.9 1.22831
\(928\) 0 0
\(929\) − 18772.2i − 0.662968i −0.943461 0.331484i \(-0.892451\pi\)
0.943461 0.331484i \(-0.107549\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −20989.9 −0.736527
\(934\) 0 0
\(935\) − 669.589i − 0.0234202i
\(936\) 0 0
\(937\) − 13717.6i − 0.478263i −0.970987 0.239132i \(-0.923137\pi\)
0.970987 0.239132i \(-0.0768628\pi\)
\(938\) 0 0
\(939\) − 8051.34i − 0.279814i
\(940\) 0 0
\(941\) 18443.8i 0.638949i 0.947595 + 0.319475i \(0.103506\pi\)
−0.947595 + 0.319475i \(0.896494\pi\)
\(942\) 0 0
\(943\) −2996.89 −0.103491
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33239.1i 1.14058i 0.821444 + 0.570289i \(0.193169\pi\)
−0.821444 + 0.570289i \(0.806831\pi\)
\(948\) 0 0
\(949\) −14023.5 −0.479684
\(950\) 0 0
\(951\) −39869.1 −1.35946
\(952\) 0 0
\(953\) 32914.7 1.11880 0.559398 0.828899i \(-0.311033\pi\)
0.559398 + 0.828899i \(0.311033\pi\)
\(954\) 0 0
\(955\) −1170.64 −0.0396660
\(956\) 0 0
\(957\) 5316.82i 0.179591i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 52899.4 1.77568
\(962\) 0 0
\(963\) 38614.0i 1.29213i
\(964\) 0 0
\(965\) 1632.22i 0.0544487i
\(966\) 0 0
\(967\) − 29368.7i − 0.976663i −0.872658 0.488332i \(-0.837606\pi\)
0.872658 0.488332i \(-0.162394\pi\)
\(968\) 0 0
\(969\) 50773.6i 1.68326i
\(970\) 0 0
\(971\) −31567.2 −1.04329 −0.521647 0.853161i \(-0.674682\pi\)
−0.521647 + 0.853161i \(0.674682\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) − 35815.2i − 1.17642i
\(976\) 0 0
\(977\) 23250.0 0.761344 0.380672 0.924710i \(-0.375693\pi\)
0.380672 + 0.924710i \(0.375693\pi\)
\(978\) 0 0
\(979\) 4451.70 0.145329
\(980\) 0 0
\(981\) −36064.3 −1.17374
\(982\) 0 0
\(983\) 4057.54 0.131654 0.0658268 0.997831i \(-0.479032\pi\)
0.0658268 + 0.997831i \(0.479032\pi\)
\(984\) 0 0
\(985\) 1344.83i 0.0435023i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3010.34 0.0967880
\(990\) 0 0
\(991\) − 42406.5i − 1.35932i −0.733526 0.679661i \(-0.762127\pi\)
0.733526 0.679661i \(-0.237873\pi\)
\(992\) 0 0
\(993\) − 51980.4i − 1.66118i
\(994\) 0 0
\(995\) − 304.252i − 0.00969389i
\(996\) 0 0
\(997\) 57689.0i 1.83253i 0.400576 + 0.916263i \(0.368810\pi\)
−0.400576 + 0.916263i \(0.631190\pi\)
\(998\) 0 0
\(999\) 4328.03 0.137070
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 784.4.f.g.783.6 6
4.3 odd 2 784.4.f.h.783.2 6
7.2 even 3 112.4.p.g.31.1 yes 6
7.3 odd 6 112.4.p.f.47.3 yes 6
7.6 odd 2 784.4.f.h.783.1 6
28.3 even 6 112.4.p.g.47.1 yes 6
28.23 odd 6 112.4.p.f.31.3 6
28.27 even 2 inner 784.4.f.g.783.5 6
56.3 even 6 448.4.p.f.383.3 6
56.37 even 6 448.4.p.f.255.3 6
56.45 odd 6 448.4.p.g.383.1 6
56.51 odd 6 448.4.p.g.255.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.4.p.f.31.3 6 28.23 odd 6
112.4.p.f.47.3 yes 6 7.3 odd 6
112.4.p.g.31.1 yes 6 7.2 even 3
112.4.p.g.47.1 yes 6 28.3 even 6
448.4.p.f.255.3 6 56.37 even 6
448.4.p.f.383.3 6 56.3 even 6
448.4.p.g.255.1 6 56.51 odd 6
448.4.p.g.383.1 6 56.45 odd 6
784.4.f.g.783.5 6 28.27 even 2 inner
784.4.f.g.783.6 6 1.1 even 1 trivial
784.4.f.h.783.1 6 7.6 odd 2
784.4.f.h.783.2 6 4.3 odd 2