Properties

Label 768.4.d.c.385.1
Level $768$
Weight $4$
Character 768.385
Analytic conductor $45.313$
Analytic rank $0$
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.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 385.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 768.385
Dual form 768.4.d.c.385.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} +6.00000i q^{5} -16.0000 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +6.00000i q^{5} -16.0000 q^{7} -9.00000 q^{9} -12.0000i q^{11} -38.0000i q^{13} +18.0000 q^{15} -126.000 q^{17} +20.0000i q^{19} +48.0000i q^{21} +168.000 q^{23} +89.0000 q^{25} +27.0000i q^{27} -30.0000i q^{29} +88.0000 q^{31} -36.0000 q^{33} -96.0000i q^{35} +254.000i q^{37} -114.000 q^{39} -42.0000 q^{41} +52.0000i q^{43} -54.0000i q^{45} +96.0000 q^{47} -87.0000 q^{49} +378.000i q^{51} +198.000i q^{53} +72.0000 q^{55} +60.0000 q^{57} +660.000i q^{59} +538.000i q^{61} +144.000 q^{63} +228.000 q^{65} +884.000i q^{67} -504.000i q^{69} +792.000 q^{71} -218.000 q^{73} -267.000i q^{75} +192.000i q^{77} +520.000 q^{79} +81.0000 q^{81} -492.000i q^{83} -756.000i q^{85} -90.0000 q^{87} -810.000 q^{89} +608.000i q^{91} -264.000i q^{93} -120.000 q^{95} +1154.00 q^{97} +108.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{7} - 18q^{9} + O(q^{10}) \) \( 2q - 32q^{7} - 18q^{9} + 36q^{15} - 252q^{17} + 336q^{23} + 178q^{25} + 176q^{31} - 72q^{33} - 228q^{39} - 84q^{41} + 192q^{47} - 174q^{49} + 144q^{55} + 120q^{57} + 288q^{63} + 456q^{65} + 1584q^{71} - 436q^{73} + 1040q^{79} + 162q^{81} - 180q^{87} - 1620q^{89} - 240q^{95} + 2308q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(5\) 6.00000i 0.536656i 0.963328 + 0.268328i \(0.0864711\pi\)
−0.963328 + 0.268328i \(0.913529\pi\)
\(6\) 0 0
\(7\) −16.0000 −0.863919 −0.431959 0.901893i \(-0.642178\pi\)
−0.431959 + 0.901893i \(0.642178\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) − 12.0000i − 0.328921i −0.986384 0.164461i \(-0.947412\pi\)
0.986384 0.164461i \(-0.0525884\pi\)
\(12\) 0 0
\(13\) − 38.0000i − 0.810716i −0.914158 0.405358i \(-0.867147\pi\)
0.914158 0.405358i \(-0.132853\pi\)
\(14\) 0 0
\(15\) 18.0000 0.309839
\(16\) 0 0
\(17\) −126.000 −1.79762 −0.898808 0.438342i \(-0.855566\pi\)
−0.898808 + 0.438342i \(0.855566\pi\)
\(18\) 0 0
\(19\) 20.0000i 0.241490i 0.992684 + 0.120745i \(0.0385284\pi\)
−0.992684 + 0.120745i \(0.961472\pi\)
\(20\) 0 0
\(21\) 48.0000i 0.498784i
\(22\) 0 0
\(23\) 168.000 1.52306 0.761531 0.648129i \(-0.224448\pi\)
0.761531 + 0.648129i \(0.224448\pi\)
\(24\) 0 0
\(25\) 89.0000 0.712000
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) − 30.0000i − 0.192099i −0.995377 0.0960493i \(-0.969379\pi\)
0.995377 0.0960493i \(-0.0306207\pi\)
\(30\) 0 0
\(31\) 88.0000 0.509847 0.254924 0.966961i \(-0.417950\pi\)
0.254924 + 0.966961i \(0.417950\pi\)
\(32\) 0 0
\(33\) −36.0000 −0.189903
\(34\) 0 0
\(35\) − 96.0000i − 0.463627i
\(36\) 0 0
\(37\) 254.000i 1.12858i 0.825578 + 0.564288i \(0.190849\pi\)
−0.825578 + 0.564288i \(0.809151\pi\)
\(38\) 0 0
\(39\) −114.000 −0.468067
\(40\) 0 0
\(41\) −42.0000 −0.159983 −0.0799914 0.996796i \(-0.525489\pi\)
−0.0799914 + 0.996796i \(0.525489\pi\)
\(42\) 0 0
\(43\) 52.0000i 0.184417i 0.995740 + 0.0922084i \(0.0293926\pi\)
−0.995740 + 0.0922084i \(0.970607\pi\)
\(44\) 0 0
\(45\) − 54.0000i − 0.178885i
\(46\) 0 0
\(47\) 96.0000 0.297937 0.148969 0.988842i \(-0.452405\pi\)
0.148969 + 0.988842i \(0.452405\pi\)
\(48\) 0 0
\(49\) −87.0000 −0.253644
\(50\) 0 0
\(51\) 378.000i 1.03785i
\(52\) 0 0
\(53\) 198.000i 0.513158i 0.966523 + 0.256579i \(0.0825954\pi\)
−0.966523 + 0.256579i \(0.917405\pi\)
\(54\) 0 0
\(55\) 72.0000 0.176518
\(56\) 0 0
\(57\) 60.0000 0.139424
\(58\) 0 0
\(59\) 660.000i 1.45635i 0.685391 + 0.728175i \(0.259631\pi\)
−0.685391 + 0.728175i \(0.740369\pi\)
\(60\) 0 0
\(61\) 538.000i 1.12924i 0.825350 + 0.564622i \(0.190978\pi\)
−0.825350 + 0.564622i \(0.809022\pi\)
\(62\) 0 0
\(63\) 144.000 0.287973
\(64\) 0 0
\(65\) 228.000 0.435076
\(66\) 0 0
\(67\) 884.000i 1.61191i 0.591979 + 0.805954i \(0.298347\pi\)
−0.591979 + 0.805954i \(0.701653\pi\)
\(68\) 0 0
\(69\) − 504.000i − 0.879340i
\(70\) 0 0
\(71\) 792.000 1.32385 0.661923 0.749572i \(-0.269740\pi\)
0.661923 + 0.749572i \(0.269740\pi\)
\(72\) 0 0
\(73\) −218.000 −0.349520 −0.174760 0.984611i \(-0.555915\pi\)
−0.174760 + 0.984611i \(0.555915\pi\)
\(74\) 0 0
\(75\) − 267.000i − 0.411073i
\(76\) 0 0
\(77\) 192.000i 0.284161i
\(78\) 0 0
\(79\) 520.000 0.740564 0.370282 0.928919i \(-0.379261\pi\)
0.370282 + 0.928919i \(0.379261\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 492.000i − 0.650651i −0.945602 0.325325i \(-0.894526\pi\)
0.945602 0.325325i \(-0.105474\pi\)
\(84\) 0 0
\(85\) − 756.000i − 0.964703i
\(86\) 0 0
\(87\) −90.0000 −0.110908
\(88\) 0 0
\(89\) −810.000 −0.964717 −0.482359 0.875974i \(-0.660220\pi\)
−0.482359 + 0.875974i \(0.660220\pi\)
\(90\) 0 0
\(91\) 608.000i 0.700393i
\(92\) 0 0
\(93\) − 264.000i − 0.294360i
\(94\) 0 0
\(95\) −120.000 −0.129597
\(96\) 0 0
\(97\) 1154.00 1.20795 0.603974 0.797004i \(-0.293583\pi\)
0.603974 + 0.797004i \(0.293583\pi\)
\(98\) 0 0
\(99\) 108.000i 0.109640i
\(100\) 0 0
\(101\) − 618.000i − 0.608845i −0.952537 0.304422i \(-0.901537\pi\)
0.952537 0.304422i \(-0.0984634\pi\)
\(102\) 0 0
\(103\) 128.000 0.122449 0.0612243 0.998124i \(-0.480499\pi\)
0.0612243 + 0.998124i \(0.480499\pi\)
\(104\) 0 0
\(105\) −288.000 −0.267675
\(106\) 0 0
\(107\) 1476.00i 1.33355i 0.745257 + 0.666777i \(0.232327\pi\)
−0.745257 + 0.666777i \(0.767673\pi\)
\(108\) 0 0
\(109\) − 1190.00i − 1.04570i −0.852425 0.522850i \(-0.824869\pi\)
0.852425 0.522850i \(-0.175131\pi\)
\(110\) 0 0
\(111\) 762.000 0.651584
\(112\) 0 0
\(113\) −462.000 −0.384613 −0.192307 0.981335i \(-0.561597\pi\)
−0.192307 + 0.981335i \(0.561597\pi\)
\(114\) 0 0
\(115\) 1008.00i 0.817361i
\(116\) 0 0
\(117\) 342.000i 0.270239i
\(118\) 0 0
\(119\) 2016.00 1.55300
\(120\) 0 0
\(121\) 1187.00 0.891811
\(122\) 0 0
\(123\) 126.000i 0.0923662i
\(124\) 0 0
\(125\) 1284.00i 0.918756i
\(126\) 0 0
\(127\) 2536.00 1.77192 0.885959 0.463763i \(-0.153501\pi\)
0.885959 + 0.463763i \(0.153501\pi\)
\(128\) 0 0
\(129\) 156.000 0.106473
\(130\) 0 0
\(131\) 2292.00i 1.52865i 0.644832 + 0.764324i \(0.276927\pi\)
−0.644832 + 0.764324i \(0.723073\pi\)
\(132\) 0 0
\(133\) − 320.000i − 0.208628i
\(134\) 0 0
\(135\) −162.000 −0.103280
\(136\) 0 0
\(137\) 726.000 0.452747 0.226374 0.974041i \(-0.427313\pi\)
0.226374 + 0.974041i \(0.427313\pi\)
\(138\) 0 0
\(139\) − 380.000i − 0.231879i −0.993256 0.115939i \(-0.963012\pi\)
0.993256 0.115939i \(-0.0369879\pi\)
\(140\) 0 0
\(141\) − 288.000i − 0.172014i
\(142\) 0 0
\(143\) −456.000 −0.266662
\(144\) 0 0
\(145\) 180.000 0.103091
\(146\) 0 0
\(147\) 261.000i 0.146442i
\(148\) 0 0
\(149\) 1590.00i 0.874214i 0.899410 + 0.437107i \(0.143997\pi\)
−0.899410 + 0.437107i \(0.856003\pi\)
\(150\) 0 0
\(151\) 2432.00 1.31068 0.655342 0.755332i \(-0.272524\pi\)
0.655342 + 0.755332i \(0.272524\pi\)
\(152\) 0 0
\(153\) 1134.00 0.599206
\(154\) 0 0
\(155\) 528.000i 0.273613i
\(156\) 0 0
\(157\) − 614.000i − 0.312118i −0.987748 0.156059i \(-0.950121\pi\)
0.987748 0.156059i \(-0.0498790\pi\)
\(158\) 0 0
\(159\) 594.000 0.296272
\(160\) 0 0
\(161\) −2688.00 −1.31580
\(162\) 0 0
\(163\) − 1852.00i − 0.889938i −0.895546 0.444969i \(-0.853215\pi\)
0.895546 0.444969i \(-0.146785\pi\)
\(164\) 0 0
\(165\) − 216.000i − 0.101913i
\(166\) 0 0
\(167\) −2136.00 −0.989752 −0.494876 0.868964i \(-0.664787\pi\)
−0.494876 + 0.868964i \(0.664787\pi\)
\(168\) 0 0
\(169\) 753.000 0.342740
\(170\) 0 0
\(171\) − 180.000i − 0.0804967i
\(172\) 0 0
\(173\) − 1758.00i − 0.772591i −0.922375 0.386296i \(-0.873754\pi\)
0.922375 0.386296i \(-0.126246\pi\)
\(174\) 0 0
\(175\) −1424.00 −0.615110
\(176\) 0 0
\(177\) 1980.00 0.840824
\(178\) 0 0
\(179\) − 540.000i − 0.225483i −0.993624 0.112742i \(-0.964037\pi\)
0.993624 0.112742i \(-0.0359632\pi\)
\(180\) 0 0
\(181\) 1982.00i 0.813928i 0.913444 + 0.406964i \(0.133412\pi\)
−0.913444 + 0.406964i \(0.866588\pi\)
\(182\) 0 0
\(183\) 1614.00 0.651969
\(184\) 0 0
\(185\) −1524.00 −0.605658
\(186\) 0 0
\(187\) 1512.00i 0.591275i
\(188\) 0 0
\(189\) − 432.000i − 0.166261i
\(190\) 0 0
\(191\) 2688.00 1.01831 0.509154 0.860675i \(-0.329958\pi\)
0.509154 + 0.860675i \(0.329958\pi\)
\(192\) 0 0
\(193\) −2302.00 −0.858557 −0.429279 0.903172i \(-0.641232\pi\)
−0.429279 + 0.903172i \(0.641232\pi\)
\(194\) 0 0
\(195\) − 684.000i − 0.251191i
\(196\) 0 0
\(197\) 4374.00i 1.58190i 0.611880 + 0.790951i \(0.290414\pi\)
−0.611880 + 0.790951i \(0.709586\pi\)
\(198\) 0 0
\(199\) −1600.00 −0.569955 −0.284977 0.958534i \(-0.591986\pi\)
−0.284977 + 0.958534i \(0.591986\pi\)
\(200\) 0 0
\(201\) 2652.00 0.930635
\(202\) 0 0
\(203\) 480.000i 0.165958i
\(204\) 0 0
\(205\) − 252.000i − 0.0858558i
\(206\) 0 0
\(207\) −1512.00 −0.507687
\(208\) 0 0
\(209\) 240.000 0.0794313
\(210\) 0 0
\(211\) 3332.00i 1.08713i 0.839367 + 0.543565i \(0.182926\pi\)
−0.839367 + 0.543565i \(0.817074\pi\)
\(212\) 0 0
\(213\) − 2376.00i − 0.764323i
\(214\) 0 0
\(215\) −312.000 −0.0989685
\(216\) 0 0
\(217\) −1408.00 −0.440467
\(218\) 0 0
\(219\) 654.000i 0.201796i
\(220\) 0 0
\(221\) 4788.00i 1.45736i
\(222\) 0 0
\(223\) −2648.00 −0.795171 −0.397586 0.917565i \(-0.630152\pi\)
−0.397586 + 0.917565i \(0.630152\pi\)
\(224\) 0 0
\(225\) −801.000 −0.237333
\(226\) 0 0
\(227\) 2244.00i 0.656121i 0.944657 + 0.328061i \(0.106395\pi\)
−0.944657 + 0.328061i \(0.893605\pi\)
\(228\) 0 0
\(229\) − 5650.00i − 1.63040i −0.579177 0.815202i \(-0.696626\pi\)
0.579177 0.815202i \(-0.303374\pi\)
\(230\) 0 0
\(231\) 576.000 0.164061
\(232\) 0 0
\(233\) −4698.00 −1.32093 −0.660464 0.750858i \(-0.729640\pi\)
−0.660464 + 0.750858i \(0.729640\pi\)
\(234\) 0 0
\(235\) 576.000i 0.159890i
\(236\) 0 0
\(237\) − 1560.00i − 0.427565i
\(238\) 0 0
\(239\) 1200.00 0.324776 0.162388 0.986727i \(-0.448080\pi\)
0.162388 + 0.986727i \(0.448080\pi\)
\(240\) 0 0
\(241\) −718.000 −0.191911 −0.0959553 0.995386i \(-0.530591\pi\)
−0.0959553 + 0.995386i \(0.530591\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) − 522.000i − 0.136120i
\(246\) 0 0
\(247\) 760.000 0.195780
\(248\) 0 0
\(249\) −1476.00 −0.375653
\(250\) 0 0
\(251\) − 6012.00i − 1.51185i −0.654659 0.755924i \(-0.727188\pi\)
0.654659 0.755924i \(-0.272812\pi\)
\(252\) 0 0
\(253\) − 2016.00i − 0.500968i
\(254\) 0 0
\(255\) −2268.00 −0.556971
\(256\) 0 0
\(257\) −2046.00 −0.496599 −0.248300 0.968683i \(-0.579872\pi\)
−0.248300 + 0.968683i \(0.579872\pi\)
\(258\) 0 0
\(259\) − 4064.00i − 0.974999i
\(260\) 0 0
\(261\) 270.000i 0.0640329i
\(262\) 0 0
\(263\) −6072.00 −1.42363 −0.711817 0.702365i \(-0.752127\pi\)
−0.711817 + 0.702365i \(0.752127\pi\)
\(264\) 0 0
\(265\) −1188.00 −0.275390
\(266\) 0 0
\(267\) 2430.00i 0.556980i
\(268\) 0 0
\(269\) 6930.00i 1.57074i 0.619025 + 0.785371i \(0.287528\pi\)
−0.619025 + 0.785371i \(0.712472\pi\)
\(270\) 0 0
\(271\) −1352.00 −0.303056 −0.151528 0.988453i \(-0.548419\pi\)
−0.151528 + 0.988453i \(0.548419\pi\)
\(272\) 0 0
\(273\) 1824.00 0.404372
\(274\) 0 0
\(275\) − 1068.00i − 0.234192i
\(276\) 0 0
\(277\) − 1186.00i − 0.257256i −0.991693 0.128628i \(-0.958943\pi\)
0.991693 0.128628i \(-0.0410573\pi\)
\(278\) 0 0
\(279\) −792.000 −0.169949
\(280\) 0 0
\(281\) −2442.00 −0.518425 −0.259213 0.965820i \(-0.583463\pi\)
−0.259213 + 0.965820i \(0.583463\pi\)
\(282\) 0 0
\(283\) − 2828.00i − 0.594018i −0.954875 0.297009i \(-0.904011\pi\)
0.954875 0.297009i \(-0.0959892\pi\)
\(284\) 0 0
\(285\) 360.000i 0.0748230i
\(286\) 0 0
\(287\) 672.000 0.138212
\(288\) 0 0
\(289\) 10963.0 2.23143
\(290\) 0 0
\(291\) − 3462.00i − 0.697409i
\(292\) 0 0
\(293\) 4758.00i 0.948687i 0.880340 + 0.474344i \(0.157315\pi\)
−0.880340 + 0.474344i \(0.842685\pi\)
\(294\) 0 0
\(295\) −3960.00 −0.781560
\(296\) 0 0
\(297\) 324.000 0.0633010
\(298\) 0 0
\(299\) − 6384.00i − 1.23477i
\(300\) 0 0
\(301\) − 832.000i − 0.159321i
\(302\) 0 0
\(303\) −1854.00 −0.351517
\(304\) 0 0
\(305\) −3228.00 −0.606016
\(306\) 0 0
\(307\) − 8476.00i − 1.57574i −0.615844 0.787868i \(-0.711185\pi\)
0.615844 0.787868i \(-0.288815\pi\)
\(308\) 0 0
\(309\) − 384.000i − 0.0706958i
\(310\) 0 0
\(311\) 4632.00 0.844555 0.422278 0.906467i \(-0.361231\pi\)
0.422278 + 0.906467i \(0.361231\pi\)
\(312\) 0 0
\(313\) 4822.00 0.870785 0.435392 0.900241i \(-0.356610\pi\)
0.435392 + 0.900241i \(0.356610\pi\)
\(314\) 0 0
\(315\) 864.000i 0.154542i
\(316\) 0 0
\(317\) 3426.00i 0.607014i 0.952829 + 0.303507i \(0.0981575\pi\)
−0.952829 + 0.303507i \(0.901842\pi\)
\(318\) 0 0
\(319\) −360.000 −0.0631854
\(320\) 0 0
\(321\) 4428.00 0.769928
\(322\) 0 0
\(323\) − 2520.00i − 0.434107i
\(324\) 0 0
\(325\) − 3382.00i − 0.577230i
\(326\) 0 0
\(327\) −3570.00 −0.603735
\(328\) 0 0
\(329\) −1536.00 −0.257393
\(330\) 0 0
\(331\) 2788.00i 0.462968i 0.972839 + 0.231484i \(0.0743581\pi\)
−0.972839 + 0.231484i \(0.925642\pi\)
\(332\) 0 0
\(333\) − 2286.00i − 0.376192i
\(334\) 0 0
\(335\) −5304.00 −0.865040
\(336\) 0 0
\(337\) 434.000 0.0701528 0.0350764 0.999385i \(-0.488833\pi\)
0.0350764 + 0.999385i \(0.488833\pi\)
\(338\) 0 0
\(339\) 1386.00i 0.222057i
\(340\) 0 0
\(341\) − 1056.00i − 0.167700i
\(342\) 0 0
\(343\) 6880.00 1.08305
\(344\) 0 0
\(345\) 3024.00 0.471903
\(346\) 0 0
\(347\) − 6684.00i − 1.03405i −0.855970 0.517026i \(-0.827039\pi\)
0.855970 0.517026i \(-0.172961\pi\)
\(348\) 0 0
\(349\) − 2630.00i − 0.403383i −0.979449 0.201692i \(-0.935356\pi\)
0.979449 0.201692i \(-0.0646438\pi\)
\(350\) 0 0
\(351\) 1026.00 0.156022
\(352\) 0 0
\(353\) −7422.00 −1.11907 −0.559537 0.828805i \(-0.689021\pi\)
−0.559537 + 0.828805i \(0.689021\pi\)
\(354\) 0 0
\(355\) 4752.00i 0.710451i
\(356\) 0 0
\(357\) − 6048.00i − 0.896622i
\(358\) 0 0
\(359\) −10440.0 −1.53482 −0.767412 0.641154i \(-0.778456\pi\)
−0.767412 + 0.641154i \(0.778456\pi\)
\(360\) 0 0
\(361\) 6459.00 0.941682
\(362\) 0 0
\(363\) − 3561.00i − 0.514887i
\(364\) 0 0
\(365\) − 1308.00i − 0.187572i
\(366\) 0 0
\(367\) −10424.0 −1.48264 −0.741319 0.671153i \(-0.765800\pi\)
−0.741319 + 0.671153i \(0.765800\pi\)
\(368\) 0 0
\(369\) 378.000 0.0533276
\(370\) 0 0
\(371\) − 3168.00i − 0.443327i
\(372\) 0 0
\(373\) 3278.00i 0.455036i 0.973774 + 0.227518i \(0.0730610\pi\)
−0.973774 + 0.227518i \(0.926939\pi\)
\(374\) 0 0
\(375\) 3852.00 0.530444
\(376\) 0 0
\(377\) −1140.00 −0.155737
\(378\) 0 0
\(379\) − 6140.00i − 0.832165i −0.909327 0.416083i \(-0.863403\pi\)
0.909327 0.416083i \(-0.136597\pi\)
\(380\) 0 0
\(381\) − 7608.00i − 1.02302i
\(382\) 0 0
\(383\) 3072.00 0.409848 0.204924 0.978778i \(-0.434305\pi\)
0.204924 + 0.978778i \(0.434305\pi\)
\(384\) 0 0
\(385\) −1152.00 −0.152497
\(386\) 0 0
\(387\) − 468.000i − 0.0614723i
\(388\) 0 0
\(389\) 6150.00i 0.801587i 0.916168 + 0.400794i \(0.131266\pi\)
−0.916168 + 0.400794i \(0.868734\pi\)
\(390\) 0 0
\(391\) −21168.0 −2.73788
\(392\) 0 0
\(393\) 6876.00 0.882566
\(394\) 0 0
\(395\) 3120.00i 0.397428i
\(396\) 0 0
\(397\) 106.000i 0.0134005i 0.999978 + 0.00670024i \(0.00213277\pi\)
−0.999978 + 0.00670024i \(0.997867\pi\)
\(398\) 0 0
\(399\) −960.000 −0.120451
\(400\) 0 0
\(401\) −1758.00 −0.218929 −0.109464 0.993991i \(-0.534914\pi\)
−0.109464 + 0.993991i \(0.534914\pi\)
\(402\) 0 0
\(403\) − 3344.00i − 0.413341i
\(404\) 0 0
\(405\) 486.000i 0.0596285i
\(406\) 0 0
\(407\) 3048.00 0.371213
\(408\) 0 0
\(409\) 3670.00 0.443691 0.221846 0.975082i \(-0.428792\pi\)
0.221846 + 0.975082i \(0.428792\pi\)
\(410\) 0 0
\(411\) − 2178.00i − 0.261394i
\(412\) 0 0
\(413\) − 10560.0i − 1.25817i
\(414\) 0 0
\(415\) 2952.00 0.349176
\(416\) 0 0
\(417\) −1140.00 −0.133875
\(418\) 0 0
\(419\) − 9660.00i − 1.12631i −0.826353 0.563153i \(-0.809588\pi\)
0.826353 0.563153i \(-0.190412\pi\)
\(420\) 0 0
\(421\) 8462.00i 0.979602i 0.871834 + 0.489801i \(0.162931\pi\)
−0.871834 + 0.489801i \(0.837069\pi\)
\(422\) 0 0
\(423\) −864.000 −0.0993123
\(424\) 0 0
\(425\) −11214.0 −1.27990
\(426\) 0 0
\(427\) − 8608.00i − 0.975575i
\(428\) 0 0
\(429\) 1368.00i 0.153957i
\(430\) 0 0
\(431\) −9792.00 −1.09435 −0.547174 0.837019i \(-0.684296\pi\)
−0.547174 + 0.837019i \(0.684296\pi\)
\(432\) 0 0
\(433\) −7342.00 −0.814859 −0.407430 0.913237i \(-0.633575\pi\)
−0.407430 + 0.913237i \(0.633575\pi\)
\(434\) 0 0
\(435\) − 540.000i − 0.0595196i
\(436\) 0 0
\(437\) 3360.00i 0.367805i
\(438\) 0 0
\(439\) 10640.0 1.15676 0.578382 0.815766i \(-0.303684\pi\)
0.578382 + 0.815766i \(0.303684\pi\)
\(440\) 0 0
\(441\) 783.000 0.0845481
\(442\) 0 0
\(443\) 17412.0i 1.86742i 0.358024 + 0.933712i \(0.383451\pi\)
−0.358024 + 0.933712i \(0.616549\pi\)
\(444\) 0 0
\(445\) − 4860.00i − 0.517722i
\(446\) 0 0
\(447\) 4770.00 0.504728
\(448\) 0 0
\(449\) −1710.00 −0.179732 −0.0898662 0.995954i \(-0.528644\pi\)
−0.0898662 + 0.995954i \(0.528644\pi\)
\(450\) 0 0
\(451\) 504.000i 0.0526218i
\(452\) 0 0
\(453\) − 7296.00i − 0.756724i
\(454\) 0 0
\(455\) −3648.00 −0.375870
\(456\) 0 0
\(457\) 646.000 0.0661239 0.0330619 0.999453i \(-0.489474\pi\)
0.0330619 + 0.999453i \(0.489474\pi\)
\(458\) 0 0
\(459\) − 3402.00i − 0.345952i
\(460\) 0 0
\(461\) 6018.00i 0.607996i 0.952673 + 0.303998i \(0.0983216\pi\)
−0.952673 + 0.303998i \(0.901678\pi\)
\(462\) 0 0
\(463\) 6712.00 0.673722 0.336861 0.941554i \(-0.390635\pi\)
0.336861 + 0.941554i \(0.390635\pi\)
\(464\) 0 0
\(465\) 1584.00 0.157970
\(466\) 0 0
\(467\) 5364.00i 0.531512i 0.964040 + 0.265756i \(0.0856216\pi\)
−0.964040 + 0.265756i \(0.914378\pi\)
\(468\) 0 0
\(469\) − 14144.0i − 1.39256i
\(470\) 0 0
\(471\) −1842.00 −0.180201
\(472\) 0 0
\(473\) 624.000 0.0606587
\(474\) 0 0
\(475\) 1780.00i 0.171941i
\(476\) 0 0
\(477\) − 1782.00i − 0.171053i
\(478\) 0 0
\(479\) −9840.00 −0.938624 −0.469312 0.883032i \(-0.655498\pi\)
−0.469312 + 0.883032i \(0.655498\pi\)
\(480\) 0 0
\(481\) 9652.00 0.914955
\(482\) 0 0
\(483\) 8064.00i 0.759678i
\(484\) 0 0
\(485\) 6924.00i 0.648253i
\(486\) 0 0
\(487\) 1424.00 0.132500 0.0662501 0.997803i \(-0.478896\pi\)
0.0662501 + 0.997803i \(0.478896\pi\)
\(488\) 0 0
\(489\) −5556.00 −0.513806
\(490\) 0 0
\(491\) 4548.00i 0.418021i 0.977913 + 0.209011i \(0.0670243\pi\)
−0.977913 + 0.209011i \(0.932976\pi\)
\(492\) 0 0
\(493\) 3780.00i 0.345320i
\(494\) 0 0
\(495\) −648.000 −0.0588393
\(496\) 0 0
\(497\) −12672.0 −1.14370
\(498\) 0 0
\(499\) 6500.00i 0.583126i 0.956552 + 0.291563i \(0.0941753\pi\)
−0.956552 + 0.291563i \(0.905825\pi\)
\(500\) 0 0
\(501\) 6408.00i 0.571434i
\(502\) 0 0
\(503\) 12168.0 1.07862 0.539308 0.842108i \(-0.318686\pi\)
0.539308 + 0.842108i \(0.318686\pi\)
\(504\) 0 0
\(505\) 3708.00 0.326740
\(506\) 0 0
\(507\) − 2259.00i − 0.197881i
\(508\) 0 0
\(509\) 21090.0i 1.83654i 0.395957 + 0.918269i \(0.370413\pi\)
−0.395957 + 0.918269i \(0.629587\pi\)
\(510\) 0 0
\(511\) 3488.00 0.301957
\(512\) 0 0
\(513\) −540.000 −0.0464748
\(514\) 0 0
\(515\) 768.000i 0.0657129i
\(516\) 0 0
\(517\) − 1152.00i − 0.0979979i
\(518\) 0 0
\(519\) −5274.00 −0.446056
\(520\) 0 0
\(521\) 5238.00 0.440462 0.220231 0.975448i \(-0.429319\pi\)
0.220231 + 0.975448i \(0.429319\pi\)
\(522\) 0 0
\(523\) − 8588.00i − 0.718025i −0.933333 0.359012i \(-0.883114\pi\)
0.933333 0.359012i \(-0.116886\pi\)
\(524\) 0 0
\(525\) 4272.00i 0.355134i
\(526\) 0 0
\(527\) −11088.0 −0.916510
\(528\) 0 0
\(529\) 16057.0 1.31972
\(530\) 0 0
\(531\) − 5940.00i − 0.485450i
\(532\) 0 0
\(533\) 1596.00i 0.129701i
\(534\) 0 0
\(535\) −8856.00 −0.715660
\(536\) 0 0
\(537\) −1620.00 −0.130183
\(538\) 0 0
\(539\) 1044.00i 0.0834291i
\(540\) 0 0
\(541\) − 3062.00i − 0.243338i −0.992571 0.121669i \(-0.961175\pi\)
0.992571 0.121669i \(-0.0388246\pi\)
\(542\) 0 0
\(543\) 5946.00 0.469921
\(544\) 0 0
\(545\) 7140.00 0.561182
\(546\) 0 0
\(547\) − 8476.00i − 0.662537i −0.943537 0.331268i \(-0.892523\pi\)
0.943537 0.331268i \(-0.107477\pi\)
\(548\) 0 0
\(549\) − 4842.00i − 0.376414i
\(550\) 0 0
\(551\) 600.000 0.0463899
\(552\) 0 0
\(553\) −8320.00 −0.639787
\(554\) 0 0
\(555\) 4572.00i 0.349677i
\(556\) 0 0
\(557\) 12546.0i 0.954383i 0.878799 + 0.477191i \(0.158345\pi\)
−0.878799 + 0.477191i \(0.841655\pi\)
\(558\) 0 0
\(559\) 1976.00 0.149510
\(560\) 0 0
\(561\) 4536.00 0.341373
\(562\) 0 0
\(563\) − 12.0000i 0 0.000898294i −1.00000 0.000449147i \(-0.999857\pi\)
1.00000 0.000449147i \(-0.000142968\pi\)
\(564\) 0 0
\(565\) − 2772.00i − 0.206405i
\(566\) 0 0
\(567\) −1296.00 −0.0959910
\(568\) 0 0
\(569\) −19290.0 −1.42123 −0.710614 0.703582i \(-0.751583\pi\)
−0.710614 + 0.703582i \(0.751583\pi\)
\(570\) 0 0
\(571\) 12148.0i 0.890329i 0.895449 + 0.445165i \(0.146855\pi\)
−0.895449 + 0.445165i \(0.853145\pi\)
\(572\) 0 0
\(573\) − 8064.00i − 0.587920i
\(574\) 0 0
\(575\) 14952.0 1.08442
\(576\) 0 0
\(577\) −10366.0 −0.747907 −0.373953 0.927447i \(-0.621998\pi\)
−0.373953 + 0.927447i \(0.621998\pi\)
\(578\) 0 0
\(579\) 6906.00i 0.495688i
\(580\) 0 0
\(581\) 7872.00i 0.562109i
\(582\) 0 0
\(583\) 2376.00 0.168789
\(584\) 0 0
\(585\) −2052.00 −0.145025
\(586\) 0 0
\(587\) − 7644.00i − 0.537482i −0.963213 0.268741i \(-0.913393\pi\)
0.963213 0.268741i \(-0.0866075\pi\)
\(588\) 0 0
\(589\) 1760.00i 0.123123i
\(590\) 0 0
\(591\) 13122.0 0.913311
\(592\) 0 0
\(593\) 8658.00 0.599564 0.299782 0.954008i \(-0.403086\pi\)
0.299782 + 0.954008i \(0.403086\pi\)
\(594\) 0 0
\(595\) 12096.0i 0.833425i
\(596\) 0 0
\(597\) 4800.00i 0.329064i
\(598\) 0 0
\(599\) 25800.0 1.75987 0.879933 0.475098i \(-0.157587\pi\)
0.879933 + 0.475098i \(0.157587\pi\)
\(600\) 0 0
\(601\) −16202.0 −1.09966 −0.549828 0.835278i \(-0.685307\pi\)
−0.549828 + 0.835278i \(0.685307\pi\)
\(602\) 0 0
\(603\) − 7956.00i − 0.537302i
\(604\) 0 0
\(605\) 7122.00i 0.478596i
\(606\) 0 0
\(607\) 24136.0 1.61392 0.806960 0.590605i \(-0.201111\pi\)
0.806960 + 0.590605i \(0.201111\pi\)
\(608\) 0 0
\(609\) 1440.00 0.0958157
\(610\) 0 0
\(611\) − 3648.00i − 0.241542i
\(612\) 0 0
\(613\) − 4642.00i − 0.305854i −0.988237 0.152927i \(-0.951130\pi\)
0.988237 0.152927i \(-0.0488700\pi\)
\(614\) 0 0
\(615\) −756.000 −0.0495689
\(616\) 0 0
\(617\) 6726.00 0.438863 0.219432 0.975628i \(-0.429580\pi\)
0.219432 + 0.975628i \(0.429580\pi\)
\(618\) 0 0
\(619\) 21220.0i 1.37787i 0.724821 + 0.688937i \(0.241922\pi\)
−0.724821 + 0.688937i \(0.758078\pi\)
\(620\) 0 0
\(621\) 4536.00i 0.293113i
\(622\) 0 0
\(623\) 12960.0 0.833437
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 0 0
\(627\) − 720.000i − 0.0458597i
\(628\) 0 0
\(629\) − 32004.0i − 2.02875i
\(630\) 0 0
\(631\) 29792.0 1.87956 0.939779 0.341783i \(-0.111031\pi\)
0.939779 + 0.341783i \(0.111031\pi\)
\(632\) 0 0
\(633\) 9996.00 0.627655
\(634\) 0 0
\(635\) 15216.0i 0.950911i
\(636\) 0 0
\(637\) 3306.00i 0.205633i
\(638\) 0 0
\(639\) −7128.00 −0.441282
\(640\) 0 0
\(641\) −10158.0 −0.625923 −0.312962 0.949766i \(-0.601321\pi\)
−0.312962 + 0.949766i \(0.601321\pi\)
\(642\) 0 0
\(643\) 29828.0i 1.82940i 0.404138 + 0.914698i \(0.367571\pi\)
−0.404138 + 0.914698i \(0.632429\pi\)
\(644\) 0 0
\(645\) 936.000i 0.0571395i
\(646\) 0 0
\(647\) 1944.00 0.118124 0.0590622 0.998254i \(-0.481189\pi\)
0.0590622 + 0.998254i \(0.481189\pi\)
\(648\) 0 0
\(649\) 7920.00 0.479025
\(650\) 0 0
\(651\) 4224.00i 0.254304i
\(652\) 0 0
\(653\) − 26718.0i − 1.60116i −0.599227 0.800579i \(-0.704525\pi\)
0.599227 0.800579i \(-0.295475\pi\)
\(654\) 0 0
\(655\) −13752.0 −0.820359
\(656\) 0 0
\(657\) 1962.00 0.116507
\(658\) 0 0
\(659\) 4260.00i 0.251815i 0.992042 + 0.125907i \(0.0401842\pi\)
−0.992042 + 0.125907i \(0.959816\pi\)
\(660\) 0 0
\(661\) 22862.0i 1.34528i 0.739971 + 0.672639i \(0.234839\pi\)
−0.739971 + 0.672639i \(0.765161\pi\)
\(662\) 0 0
\(663\) 14364.0 0.841405
\(664\) 0 0
\(665\) 1920.00 0.111962
\(666\) 0 0
\(667\) − 5040.00i − 0.292578i
\(668\) 0 0
\(669\) 7944.00i 0.459092i
\(670\) 0 0
\(671\) 6456.00 0.371432
\(672\) 0 0
\(673\) −32542.0 −1.86390 −0.931948 0.362592i \(-0.881892\pi\)
−0.931948 + 0.362592i \(0.881892\pi\)
\(674\) 0 0
\(675\) 2403.00i 0.137024i
\(676\) 0 0
\(677\) 14214.0i 0.806925i 0.914996 + 0.403463i \(0.132193\pi\)
−0.914996 + 0.403463i \(0.867807\pi\)
\(678\) 0 0
\(679\) −18464.0 −1.04357
\(680\) 0 0
\(681\) 6732.00 0.378812
\(682\) 0 0
\(683\) 7092.00i 0.397317i 0.980069 + 0.198659i \(0.0636585\pi\)
−0.980069 + 0.198659i \(0.936341\pi\)
\(684\) 0 0
\(685\) 4356.00i 0.242970i
\(686\) 0 0
\(687\) −16950.0 −0.941314
\(688\) 0 0
\(689\) 7524.00 0.416026
\(690\) 0 0
\(691\) − 13228.0i − 0.728244i −0.931351 0.364122i \(-0.881369\pi\)
0.931351 0.364122i \(-0.118631\pi\)
\(692\) 0 0
\(693\) − 1728.00i − 0.0947205i
\(694\) 0 0
\(695\) 2280.00 0.124439
\(696\) 0 0
\(697\) 5292.00 0.287588
\(698\) 0 0
\(699\) 14094.0i 0.762638i
\(700\) 0 0
\(701\) − 28062.0i − 1.51196i −0.654592 0.755982i \(-0.727160\pi\)
0.654592 0.755982i \(-0.272840\pi\)
\(702\) 0 0
\(703\) −5080.00 −0.272540
\(704\) 0 0
\(705\) 1728.00 0.0923124
\(706\) 0 0
\(707\) 9888.00i 0.525992i
\(708\) 0 0
\(709\) − 27250.0i − 1.44343i −0.692188 0.721717i \(-0.743353\pi\)
0.692188 0.721717i \(-0.256647\pi\)
\(710\) 0 0
\(711\) −4680.00 −0.246855
\(712\) 0 0
\(713\) 14784.0 0.776529
\(714\) 0 0
\(715\) − 2736.00i − 0.143106i
\(716\) 0 0
\(717\) − 3600.00i − 0.187510i
\(718\) 0 0
\(719\) 14400.0 0.746912 0.373456 0.927648i \(-0.378173\pi\)
0.373456 + 0.927648i \(0.378173\pi\)
\(720\) 0 0
\(721\) −2048.00 −0.105786
\(722\) 0 0
\(723\) 2154.00i 0.110800i
\(724\) 0 0
\(725\) − 2670.00i − 0.136774i
\(726\) 0 0
\(727\) 17984.0 0.917455 0.458727 0.888577i \(-0.348305\pi\)
0.458727 + 0.888577i \(0.348305\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) − 6552.00i − 0.331511i
\(732\) 0 0
\(733\) − 16598.0i − 0.836373i −0.908361 0.418186i \(-0.862666\pi\)
0.908361 0.418186i \(-0.137334\pi\)
\(734\) 0 0
\(735\) −1566.00 −0.0785888
\(736\) 0 0
\(737\) 10608.0 0.530191
\(738\) 0 0
\(739\) 1460.00i 0.0726752i 0.999340 + 0.0363376i \(0.0115692\pi\)
−0.999340 + 0.0363376i \(0.988431\pi\)
\(740\) 0 0
\(741\) − 2280.00i − 0.113034i
\(742\) 0 0
\(743\) −30072.0 −1.48484 −0.742419 0.669936i \(-0.766322\pi\)
−0.742419 + 0.669936i \(0.766322\pi\)
\(744\) 0 0
\(745\) −9540.00 −0.469152
\(746\) 0 0
\(747\) 4428.00i 0.216884i
\(748\) 0 0
\(749\) − 23616.0i − 1.15208i
\(750\) 0 0
\(751\) 18088.0 0.878882 0.439441 0.898271i \(-0.355177\pi\)
0.439441 + 0.898271i \(0.355177\pi\)
\(752\) 0 0
\(753\) −18036.0 −0.872866
\(754\) 0 0
\(755\) 14592.0i 0.703387i
\(756\) 0 0
\(757\) 24734.0i 1.18755i 0.804633 + 0.593773i \(0.202362\pi\)
−0.804633 + 0.593773i \(0.797638\pi\)
\(758\) 0 0
\(759\) −6048.00 −0.289234
\(760\) 0 0
\(761\) 22278.0 1.06120 0.530602 0.847621i \(-0.321966\pi\)
0.530602 + 0.847621i \(0.321966\pi\)
\(762\) 0 0
\(763\) 19040.0i 0.903400i
\(764\) 0 0
\(765\) 6804.00i 0.321568i
\(766\) 0 0
\(767\) 25080.0 1.18069
\(768\) 0 0
\(769\) 16130.0 0.756388 0.378194 0.925726i \(-0.376545\pi\)
0.378194 + 0.925726i \(0.376545\pi\)
\(770\) 0 0
\(771\) 6138.00i 0.286712i
\(772\) 0 0
\(773\) 29718.0i 1.38277i 0.722486 + 0.691386i \(0.242999\pi\)
−0.722486 + 0.691386i \(0.757001\pi\)
\(774\) 0 0
\(775\) 7832.00 0.363011
\(776\) 0 0
\(777\) −12192.0 −0.562916
\(778\) 0 0
\(779\) − 840.000i − 0.0386343i
\(780\) 0 0
\(781\) − 9504.00i − 0.435442i
\(782\) 0 0
\(783\) 810.000 0.0369694
\(784\) 0 0
\(785\) 3684.00 0.167500
\(786\) 0 0
\(787\) 9524.00i 0.431377i 0.976462 + 0.215689i \(0.0691996\pi\)
−0.976462 + 0.215689i \(0.930800\pi\)
\(788\) 0 0
\(789\) 18216.0i 0.821935i
\(790\) 0 0
\(791\) 7392.00 0.332275
\(792\) 0 0
\(793\) 20444.0 0.915495
\(794\) 0 0
\(795\) 3564.00i 0.158996i
\(796\) 0 0
\(797\) 33906.0i 1.50692i 0.657496 + 0.753458i \(0.271616\pi\)
−0.657496 + 0.753458i \(0.728384\pi\)
\(798\) 0 0
\(799\) −12096.0 −0.535577
\(800\) 0 0
\(801\) 7290.00 0.321572
\(802\) 0 0
\(803\) 2616.00i 0.114965i
\(804\) 0 0
\(805\) − 16128.0i − 0.706133i
\(806\) 0 0
\(807\) 20790.0 0.906868
\(808\) 0 0
\(809\) 630.000 0.0273790 0.0136895 0.999906i \(-0.495642\pi\)
0.0136895 + 0.999906i \(0.495642\pi\)
\(810\) 0 0
\(811\) 20788.0i 0.900081i 0.893008 + 0.450040i \(0.148590\pi\)
−0.893008 + 0.450040i \(0.851410\pi\)
\(812\) 0 0
\(813\) 4056.00i 0.174969i
\(814\) 0 0
\(815\) 11112.0 0.477591
\(816\) 0 0
\(817\) −1040.00 −0.0445349
\(818\) 0 0
\(819\) − 5472.00i − 0.233464i
\(820\) 0 0
\(821\) − 43098.0i − 1.83207i −0.401097 0.916036i \(-0.631371\pi\)
0.401097 0.916036i \(-0.368629\pi\)
\(822\) 0 0
\(823\) −14272.0 −0.604484 −0.302242 0.953231i \(-0.597735\pi\)
−0.302242 + 0.953231i \(0.597735\pi\)
\(824\) 0 0
\(825\) −3204.00 −0.135211
\(826\) 0 0
\(827\) − 13644.0i − 0.573698i −0.957976 0.286849i \(-0.907392\pi\)
0.957976 0.286849i \(-0.0926078\pi\)
\(828\) 0 0
\(829\) 2410.00i 0.100968i 0.998725 + 0.0504842i \(0.0160764\pi\)
−0.998725 + 0.0504842i \(0.983924\pi\)
\(830\) 0 0
\(831\) −3558.00 −0.148527
\(832\) 0 0
\(833\) 10962.0 0.455955
\(834\) 0 0
\(835\) − 12816.0i − 0.531157i
\(836\) 0 0
\(837\) 2376.00i 0.0981202i
\(838\) 0 0
\(839\) 23160.0 0.953006 0.476503 0.879173i \(-0.341904\pi\)
0.476503 + 0.879173i \(0.341904\pi\)
\(840\) 0 0
\(841\) 23489.0 0.963098
\(842\) 0 0
\(843\) 7326.00i 0.299313i
\(844\) 0 0
\(845\) 4518.00i 0.183934i
\(846\) 0 0
\(847\) −18992.0 −0.770452
\(848\) 0 0
\(849\) −8484.00 −0.342957
\(850\) 0 0
\(851\) 42672.0i 1.71889i
\(852\) 0 0
\(853\) 32078.0i 1.28761i 0.765190 + 0.643804i \(0.222645\pi\)
−0.765190 + 0.643804i \(0.777355\pi\)
\(854\) 0 0
\(855\) 1080.00 0.0431991
\(856\) 0 0
\(857\) 14406.0 0.574212 0.287106 0.957899i \(-0.407307\pi\)
0.287106 + 0.957899i \(0.407307\pi\)
\(858\) 0 0
\(859\) − 30620.0i − 1.21623i −0.793849 0.608115i \(-0.791926\pi\)
0.793849 0.608115i \(-0.208074\pi\)
\(860\) 0 0
\(861\) − 2016.00i − 0.0797969i
\(862\) 0 0
\(863\) −17568.0 −0.692957 −0.346478 0.938058i \(-0.612623\pi\)
−0.346478 + 0.938058i \(0.612623\pi\)
\(864\) 0 0
\(865\) 10548.0 0.414616
\(866\) 0 0
\(867\) − 32889.0i − 1.28831i
\(868\) 0 0
\(869\) − 6240.00i − 0.243587i
\(870\) 0 0
\(871\) 33592.0 1.30680
\(872\) 0 0
\(873\) −10386.0 −0.402649
\(874\) 0 0
\(875\) − 20544.0i − 0.793730i
\(876\) 0 0
\(877\) 21706.0i 0.835758i 0.908503 + 0.417879i \(0.137226\pi\)
−0.908503 + 0.417879i \(0.862774\pi\)
\(878\) 0 0
\(879\) 14274.0 0.547725
\(880\) 0 0
\(881\) −14958.0 −0.572018 −0.286009 0.958227i \(-0.592329\pi\)
−0.286009 + 0.958227i \(0.592329\pi\)
\(882\) 0 0
\(883\) − 32812.0i − 1.25052i −0.780415 0.625261i \(-0.784992\pi\)
0.780415 0.625261i \(-0.215008\pi\)
\(884\) 0 0
\(885\) 11880.0i 0.451234i
\(886\) 0 0
\(887\) −38856.0 −1.47086 −0.735432 0.677598i \(-0.763021\pi\)
−0.735432 + 0.677598i \(0.763021\pi\)
\(888\) 0 0
\(889\) −40576.0 −1.53079
\(890\) 0 0
\(891\) − 972.000i − 0.0365468i
\(892\) 0 0
\(893\) 1920.00i 0.0719489i
\(894\) 0 0
\(895\) 3240.00 0.121007
\(896\) 0 0
\(897\) −19152.0 −0.712895
\(898\) 0 0
\(899\) − 2640.00i − 0.0979410i
\(900\) 0 0
\(901\) − 24948.0i − 0.922462i
\(902\) 0 0
\(903\) −2496.00 −0.0919841
\(904\) 0 0
\(905\) −11892.0 −0.436799
\(906\) 0 0
\(907\) 28276.0i 1.03516i 0.855635 + 0.517579i \(0.173167\pi\)
−0.855635 + 0.517579i \(0.826833\pi\)
\(908\) 0 0
\(909\) 5562.00i 0.202948i
\(910\) 0 0
\(911\) −8112.00 −0.295019 −0.147510 0.989061i \(-0.547126\pi\)
−0.147510 + 0.989061i \(0.547126\pi\)
\(912\) 0 0
\(913\) −5904.00 −0.214013
\(914\) 0 0
\(915\) 9684.00i 0.349883i
\(916\) 0 0
\(917\) − 36672.0i − 1.32063i
\(918\) 0 0
\(919\) −26080.0 −0.936126 −0.468063 0.883695i \(-0.655048\pi\)
−0.468063 + 0.883695i \(0.655048\pi\)
\(920\) 0 0
\(921\) −25428.0 −0.909751
\(922\) 0 0
\(923\) − 30096.0i − 1.07326i
\(924\) 0 0
\(925\) 22606.0i 0.803547i
\(926\) 0 0
\(927\) −1152.00 −0.0408162
\(928\) 0 0
\(929\) 49170.0 1.73651 0.868254 0.496120i \(-0.165243\pi\)
0.868254 + 0.496120i \(0.165243\pi\)
\(930\) 0 0
\(931\) − 1740.00i − 0.0612526i
\(932\) 0 0
\(933\) − 13896.0i − 0.487604i
\(934\) 0 0
\(935\) −9072.00 −0.317311
\(936\) 0 0
\(937\) −48314.0 −1.68447 −0.842236 0.539110i \(-0.818761\pi\)
−0.842236 + 0.539110i \(0.818761\pi\)
\(938\) 0 0
\(939\) − 14466.0i − 0.502748i
\(940\) 0 0
\(941\) − 34782.0i − 1.20495i −0.798137 0.602477i \(-0.794181\pi\)
0.798137 0.602477i \(-0.205819\pi\)
\(942\) 0 0
\(943\) −7056.00 −0.243664
\(944\) 0 0
\(945\) 2592.00 0.0892251
\(946\) 0 0
\(947\) − 25116.0i − 0.861838i −0.902391 0.430919i \(-0.858190\pi\)
0.902391 0.430919i \(-0.141810\pi\)
\(948\) 0 0
\(949\) 8284.00i 0.283361i
\(950\) 0 0
\(951\) 10278.0 0.350460
\(952\) 0 0
\(953\) 15462.0 0.525565 0.262782 0.964855i \(-0.415360\pi\)
0.262782 + 0.964855i \(0.415360\pi\)
\(954\) 0 0
\(955\) 16128.0i 0.546481i
\(956\) 0 0
\(957\) 1080.00i 0.0364801i
\(958\) 0 0
\(959\) −11616.0 −0.391137
\(960\) 0 0
\(961\) −22047.0 −0.740056
\(962\) 0 0
\(963\) − 13284.0i − 0.444518i
\(964\) 0 0
\(965\) − 13812.0i − 0.460750i
\(966\) 0 0
\(967\) −736.000 −0.0244759 −0.0122379 0.999925i \(-0.503896\pi\)
−0.0122379 + 0.999925i \(0.503896\pi\)
\(968\) 0 0
\(969\) −7560.00 −0.250632
\(970\) 0 0
\(971\) 29268.0i 0.967307i 0.875260 + 0.483653i \(0.160690\pi\)
−0.875260 + 0.483653i \(0.839310\pi\)
\(972\) 0 0
\(973\) 6080.00i 0.200325i
\(974\) 0 0
\(975\) −10146.0 −0.333264
\(976\) 0 0
\(977\) 16674.0 0.546007 0.273003 0.962013i \(-0.411983\pi\)
0.273003 + 0.962013i \(0.411983\pi\)
\(978\) 0 0
\(979\) 9720.00i 0.317316i
\(980\) 0 0
\(981\) 10710.0i 0.348567i
\(982\) 0 0
\(983\) −31272.0 −1.01467 −0.507336 0.861749i \(-0.669370\pi\)
−0.507336 + 0.861749i \(0.669370\pi\)
\(984\) 0 0
\(985\) −26244.0 −0.848937
\(986\) 0 0
\(987\) 4608.00i 0.148606i
\(988\) 0 0
\(989\) 8736.00i 0.280878i
\(990\) 0 0
\(991\) 15928.0 0.510565 0.255282 0.966867i \(-0.417832\pi\)
0.255282 + 0.966867i \(0.417832\pi\)
\(992\) 0 0
\(993\) 8364.00 0.267295
\(994\) 0 0
\(995\) − 9600.00i − 0.305870i
\(996\) 0 0
\(997\) 42014.0i 1.33460i 0.744789 + 0.667300i \(0.232550\pi\)
−0.744789 + 0.667300i \(0.767450\pi\)
\(998\) 0 0
\(999\) −6858.00 −0.217195
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.d.c.385.1 2
4.3 odd 2 768.4.d.n.385.2 2
8.3 odd 2 768.4.d.n.385.1 2
8.5 even 2 inner 768.4.d.c.385.2 2
16.3 odd 4 6.4.a.a.1.1 1
16.5 even 4 192.4.a.c.1.1 1
16.11 odd 4 192.4.a.i.1.1 1
16.13 even 4 48.4.a.c.1.1 1
48.5 odd 4 576.4.a.r.1.1 1
48.11 even 4 576.4.a.q.1.1 1
48.29 odd 4 144.4.a.c.1.1 1
48.35 even 4 18.4.a.a.1.1 1
80.3 even 4 150.4.c.d.49.2 2
80.13 odd 4 1200.4.f.j.49.2 2
80.19 odd 4 150.4.a.i.1.1 1
80.29 even 4 1200.4.a.b.1.1 1
80.67 even 4 150.4.c.d.49.1 2
80.77 odd 4 1200.4.f.j.49.1 2
112.3 even 12 294.4.e.g.79.1 2
112.13 odd 4 2352.4.a.e.1.1 1
112.19 even 12 294.4.e.g.67.1 2
112.51 odd 12 294.4.e.h.67.1 2
112.67 odd 12 294.4.e.h.79.1 2
112.83 even 4 294.4.a.e.1.1 1
144.67 odd 12 162.4.c.f.55.1 2
144.83 even 12 162.4.c.c.109.1 2
144.115 odd 12 162.4.c.f.109.1 2
144.131 even 12 162.4.c.c.55.1 2
176.131 even 4 726.4.a.f.1.1 1
208.51 odd 4 1014.4.a.g.1.1 1
208.83 even 4 1014.4.b.d.337.2 2
208.99 even 4 1014.4.b.d.337.1 2
240.83 odd 4 450.4.c.e.199.1 2
240.179 even 4 450.4.a.h.1.1 1
240.227 odd 4 450.4.c.e.199.2 2
272.67 odd 4 1734.4.a.d.1.1 1
304.227 even 4 2166.4.a.i.1.1 1
336.83 odd 4 882.4.a.n.1.1 1
336.131 odd 12 882.4.g.f.361.1 2
336.179 even 12 882.4.g.i.667.1 2
336.227 odd 12 882.4.g.f.667.1 2
336.275 even 12 882.4.g.i.361.1 2
528.131 odd 4 2178.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.4.a.a.1.1 1 16.3 odd 4
18.4.a.a.1.1 1 48.35 even 4
48.4.a.c.1.1 1 16.13 even 4
144.4.a.c.1.1 1 48.29 odd 4
150.4.a.i.1.1 1 80.19 odd 4
150.4.c.d.49.1 2 80.67 even 4
150.4.c.d.49.2 2 80.3 even 4
162.4.c.c.55.1 2 144.131 even 12
162.4.c.c.109.1 2 144.83 even 12
162.4.c.f.55.1 2 144.67 odd 12
162.4.c.f.109.1 2 144.115 odd 12
192.4.a.c.1.1 1 16.5 even 4
192.4.a.i.1.1 1 16.11 odd 4
294.4.a.e.1.1 1 112.83 even 4
294.4.e.g.67.1 2 112.19 even 12
294.4.e.g.79.1 2 112.3 even 12
294.4.e.h.67.1 2 112.51 odd 12
294.4.e.h.79.1 2 112.67 odd 12
450.4.a.h.1.1 1 240.179 even 4
450.4.c.e.199.1 2 240.83 odd 4
450.4.c.e.199.2 2 240.227 odd 4
576.4.a.q.1.1 1 48.11 even 4
576.4.a.r.1.1 1 48.5 odd 4
726.4.a.f.1.1 1 176.131 even 4
768.4.d.c.385.1 2 1.1 even 1 trivial
768.4.d.c.385.2 2 8.5 even 2 inner
768.4.d.n.385.1 2 8.3 odd 2
768.4.d.n.385.2 2 4.3 odd 2
882.4.a.n.1.1 1 336.83 odd 4
882.4.g.f.361.1 2 336.131 odd 12
882.4.g.f.667.1 2 336.227 odd 12
882.4.g.i.361.1 2 336.275 even 12
882.4.g.i.667.1 2 336.179 even 12
1014.4.a.g.1.1 1 208.51 odd 4
1014.4.b.d.337.1 2 208.99 even 4
1014.4.b.d.337.2 2 208.83 even 4
1200.4.a.b.1.1 1 80.29 even 4
1200.4.f.j.49.1 2 80.77 odd 4
1200.4.f.j.49.2 2 80.13 odd 4
1734.4.a.d.1.1 1 272.67 odd 4
2166.4.a.i.1.1 1 304.227 even 4
2178.4.a.e.1.1 1 528.131 odd 4
2352.4.a.e.1.1 1 112.13 odd 4