Properties

Label 128.5.d.c.63.4
Level $128$
Weight $5$
Character 128.63
Analytic conductor $13.231$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 128.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 63.4
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 128.63
Dual form 128.5.d.c.63.3

$q$-expansion

\(f(q)\) \(=\) \(q+9.79796 q^{3} +8.00000i q^{5} +78.3837i q^{7} +15.0000 q^{9} +O(q^{10})\) \(q+9.79796 q^{3} +8.00000i q^{5} +78.3837i q^{7} +15.0000 q^{9} -107.778 q^{11} +216.000i q^{13} +78.3837i q^{15} -162.000 q^{17} +440.908 q^{19} +768.000i q^{21} -705.453i q^{23} +561.000 q^{25} -646.665 q^{27} +1304.00i q^{29} -627.069i q^{31} -1056.00 q^{33} -627.069 q^{35} +1512.00i q^{37} +2116.36i q^{39} +1890.00 q^{41} +2909.99 q^{43} +120.000i q^{45} +1410.91i q^{47} -3743.00 q^{49} -1587.27 q^{51} -1976.00i q^{53} -862.220i q^{55} +4320.00 q^{57} -2263.33 q^{59} -2376.00i q^{61} +1175.76i q^{63} -1728.00 q^{65} -1675.45 q^{67} -6912.00i q^{69} -7759.98i q^{71} +2750.00 q^{73} +5496.65 q^{75} -8448.00i q^{77} -7995.13i q^{79} -7551.00 q^{81} +9337.45 q^{83} -1296.00i q^{85} +12776.5i q^{87} +2430.00 q^{89} -16930.9 q^{91} -6144.00i q^{93} +3527.27i q^{95} +7454.00 q^{97} -1616.66 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 60q^{9} + O(q^{10}) \) \( 4q + 60q^{9} - 648q^{17} + 2244q^{25} - 4224q^{33} + 7560q^{41} - 14972q^{49} + 17280q^{57} - 6912q^{65} + 11000q^{73} - 30204q^{81} + 9720q^{89} + 29816q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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
\(3\) 9.79796 1.08866 0.544331 0.838870i \(-0.316784\pi\)
0.544331 + 0.838870i \(0.316784\pi\)
\(4\) 0 0
\(5\) 8.00000i 0.320000i 0.987117 + 0.160000i \(0.0511494\pi\)
−0.987117 + 0.160000i \(0.948851\pi\)
\(6\) 0 0
\(7\) 78.3837i 1.59967i 0.600222 + 0.799833i \(0.295079\pi\)
−0.600222 + 0.799833i \(0.704921\pi\)
\(8\) 0 0
\(9\) 15.0000 0.185185
\(10\) 0 0
\(11\) −107.778 −0.890724 −0.445362 0.895351i \(-0.646925\pi\)
−0.445362 + 0.895351i \(0.646925\pi\)
\(12\) 0 0
\(13\) 216.000i 1.27811i 0.769162 + 0.639053i \(0.220674\pi\)
−0.769162 + 0.639053i \(0.779326\pi\)
\(14\) 0 0
\(15\) 78.3837i 0.348372i
\(16\) 0 0
\(17\) −162.000 −0.560554 −0.280277 0.959919i \(-0.590426\pi\)
−0.280277 + 0.959919i \(0.590426\pi\)
\(18\) 0 0
\(19\) 440.908 1.22135 0.610676 0.791880i \(-0.290898\pi\)
0.610676 + 0.791880i \(0.290898\pi\)
\(20\) 0 0
\(21\) 768.000i 1.74150i
\(22\) 0 0
\(23\) − 705.453i − 1.33356i −0.745255 0.666780i \(-0.767672\pi\)
0.745255 0.666780i \(-0.232328\pi\)
\(24\) 0 0
\(25\) 561.000 0.897600
\(26\) 0 0
\(27\) −646.665 −0.887058
\(28\) 0 0
\(29\) 1304.00i 1.55054i 0.631633 + 0.775268i \(0.282385\pi\)
−0.631633 + 0.775268i \(0.717615\pi\)
\(30\) 0 0
\(31\) − 627.069i − 0.652518i −0.945280 0.326259i \(-0.894212\pi\)
0.945280 0.326259i \(-0.105788\pi\)
\(32\) 0 0
\(33\) −1056.00 −0.969697
\(34\) 0 0
\(35\) −627.069 −0.511893
\(36\) 0 0
\(37\) 1512.00i 1.10446i 0.833693 + 0.552228i \(0.186222\pi\)
−0.833693 + 0.552228i \(0.813778\pi\)
\(38\) 0 0
\(39\) 2116.36i 1.39143i
\(40\) 0 0
\(41\) 1890.00 1.12433 0.562165 0.827025i \(-0.309968\pi\)
0.562165 + 0.827025i \(0.309968\pi\)
\(42\) 0 0
\(43\) 2909.99 1.57382 0.786910 0.617068i \(-0.211679\pi\)
0.786910 + 0.617068i \(0.211679\pi\)
\(44\) 0 0
\(45\) 120.000i 0.0592593i
\(46\) 0 0
\(47\) 1410.91i 0.638708i 0.947635 + 0.319354i \(0.103466\pi\)
−0.947635 + 0.319354i \(0.896534\pi\)
\(48\) 0 0
\(49\) −3743.00 −1.55893
\(50\) 0 0
\(51\) −1587.27 −0.610253
\(52\) 0 0
\(53\) − 1976.00i − 0.703453i −0.936103 0.351727i \(-0.885595\pi\)
0.936103 0.351727i \(-0.114405\pi\)
\(54\) 0 0
\(55\) − 862.220i − 0.285032i
\(56\) 0 0
\(57\) 4320.00 1.32964
\(58\) 0 0
\(59\) −2263.33 −0.650195 −0.325097 0.945681i \(-0.605397\pi\)
−0.325097 + 0.945681i \(0.605397\pi\)
\(60\) 0 0
\(61\) − 2376.00i − 0.638538i −0.947664 0.319269i \(-0.896563\pi\)
0.947664 0.319269i \(-0.103437\pi\)
\(62\) 0 0
\(63\) 1175.76i 0.296235i
\(64\) 0 0
\(65\) −1728.00 −0.408994
\(66\) 0 0
\(67\) −1675.45 −0.373235 −0.186617 0.982433i \(-0.559752\pi\)
−0.186617 + 0.982433i \(0.559752\pi\)
\(68\) 0 0
\(69\) − 6912.00i − 1.45180i
\(70\) 0 0
\(71\) − 7759.98i − 1.53937i −0.638422 0.769687i \(-0.720412\pi\)
0.638422 0.769687i \(-0.279588\pi\)
\(72\) 0 0
\(73\) 2750.00 0.516044 0.258022 0.966139i \(-0.416929\pi\)
0.258022 + 0.966139i \(0.416929\pi\)
\(74\) 0 0
\(75\) 5496.65 0.977183
\(76\) 0 0
\(77\) − 8448.00i − 1.42486i
\(78\) 0 0
\(79\) − 7995.13i − 1.28107i −0.767931 0.640533i \(-0.778713\pi\)
0.767931 0.640533i \(-0.221287\pi\)
\(80\) 0 0
\(81\) −7551.00 −1.15089
\(82\) 0 0
\(83\) 9337.45 1.35542 0.677708 0.735332i \(-0.262974\pi\)
0.677708 + 0.735332i \(0.262974\pi\)
\(84\) 0 0
\(85\) − 1296.00i − 0.179377i
\(86\) 0 0
\(87\) 12776.5i 1.68801i
\(88\) 0 0
\(89\) 2430.00 0.306779 0.153390 0.988166i \(-0.450981\pi\)
0.153390 + 0.988166i \(0.450981\pi\)
\(90\) 0 0
\(91\) −16930.9 −2.04454
\(92\) 0 0
\(93\) − 6144.00i − 0.710371i
\(94\) 0 0
\(95\) 3527.27i 0.390833i
\(96\) 0 0
\(97\) 7454.00 0.792220 0.396110 0.918203i \(-0.370360\pi\)
0.396110 + 0.918203i \(0.370360\pi\)
\(98\) 0 0
\(99\) −1616.66 −0.164949
\(100\) 0 0
\(101\) − 1496.00i − 0.146652i −0.997308 0.0733261i \(-0.976639\pi\)
0.997308 0.0733261i \(-0.0233614\pi\)
\(102\) 0 0
\(103\) 2586.66i 0.243818i 0.992541 + 0.121909i \(0.0389015\pi\)
−0.992541 + 0.121909i \(0.961098\pi\)
\(104\) 0 0
\(105\) −6144.00 −0.557279
\(106\) 0 0
\(107\) 1067.98 0.0932813 0.0466406 0.998912i \(-0.485148\pi\)
0.0466406 + 0.998912i \(0.485148\pi\)
\(108\) 0 0
\(109\) 14904.0i 1.25444i 0.778842 + 0.627220i \(0.215807\pi\)
−0.778842 + 0.627220i \(0.784193\pi\)
\(110\) 0 0
\(111\) 14814.5i 1.20238i
\(112\) 0 0
\(113\) −702.000 −0.0549769 −0.0274884 0.999622i \(-0.508751\pi\)
−0.0274884 + 0.999622i \(0.508751\pi\)
\(114\) 0 0
\(115\) 5643.62 0.426739
\(116\) 0 0
\(117\) 3240.00i 0.236686i
\(118\) 0 0
\(119\) − 12698.2i − 0.896699i
\(120\) 0 0
\(121\) −3025.00 −0.206612
\(122\) 0 0
\(123\) 18518.1 1.22402
\(124\) 0 0
\(125\) 9488.00i 0.607232i
\(126\) 0 0
\(127\) − 3448.88i − 0.213831i −0.994268 0.106916i \(-0.965903\pi\)
0.994268 0.106916i \(-0.0340974\pi\)
\(128\) 0 0
\(129\) 28512.0 1.71336
\(130\) 0 0
\(131\) −9592.20 −0.558954 −0.279477 0.960152i \(-0.590161\pi\)
−0.279477 + 0.960152i \(0.590161\pi\)
\(132\) 0 0
\(133\) 34560.0i 1.95376i
\(134\) 0 0
\(135\) − 5173.32i − 0.283859i
\(136\) 0 0
\(137\) −32670.0 −1.74064 −0.870318 0.492490i \(-0.836087\pi\)
−0.870318 + 0.492490i \(0.836087\pi\)
\(138\) 0 0
\(139\) −1322.72 −0.0684605 −0.0342302 0.999414i \(-0.510898\pi\)
−0.0342302 + 0.999414i \(0.510898\pi\)
\(140\) 0 0
\(141\) 13824.0i 0.695337i
\(142\) 0 0
\(143\) − 23280.0i − 1.13844i
\(144\) 0 0
\(145\) −10432.0 −0.496171
\(146\) 0 0
\(147\) −36673.8 −1.69715
\(148\) 0 0
\(149\) 872.000i 0.0392775i 0.999807 + 0.0196388i \(0.00625161\pi\)
−0.999807 + 0.0196388i \(0.993748\pi\)
\(150\) 0 0
\(151\) − 9484.42i − 0.415965i −0.978133 0.207983i \(-0.933310\pi\)
0.978133 0.207983i \(-0.0666898\pi\)
\(152\) 0 0
\(153\) −2430.00 −0.103806
\(154\) 0 0
\(155\) 5016.55 0.208806
\(156\) 0 0
\(157\) 28728.0i 1.16548i 0.812657 + 0.582742i \(0.198020\pi\)
−0.812657 + 0.582742i \(0.801980\pi\)
\(158\) 0 0
\(159\) − 19360.8i − 0.765823i
\(160\) 0 0
\(161\) 55296.0 2.13325
\(162\) 0 0
\(163\) 31128.1 1.17160 0.585798 0.810457i \(-0.300781\pi\)
0.585798 + 0.810457i \(0.300781\pi\)
\(164\) 0 0
\(165\) − 8448.00i − 0.310303i
\(166\) 0 0
\(167\) 54319.9i 1.94772i 0.227155 + 0.973859i \(0.427058\pi\)
−0.227155 + 0.973859i \(0.572942\pi\)
\(168\) 0 0
\(169\) −18095.0 −0.633556
\(170\) 0 0
\(171\) 6613.62 0.226176
\(172\) 0 0
\(173\) − 25256.0i − 0.843864i −0.906628 0.421932i \(-0.861352\pi\)
0.906628 0.421932i \(-0.138648\pi\)
\(174\) 0 0
\(175\) 43973.2i 1.43586i
\(176\) 0 0
\(177\) −22176.0 −0.707843
\(178\) 0 0
\(179\) 8945.54 0.279190 0.139595 0.990209i \(-0.455420\pi\)
0.139595 + 0.990209i \(0.455420\pi\)
\(180\) 0 0
\(181\) 36072.0i 1.10107i 0.834814 + 0.550533i \(0.185575\pi\)
−0.834814 + 0.550533i \(0.814425\pi\)
\(182\) 0 0
\(183\) − 23280.0i − 0.695152i
\(184\) 0 0
\(185\) −12096.0 −0.353426
\(186\) 0 0
\(187\) 17460.0 0.499298
\(188\) 0 0
\(189\) − 50688.0i − 1.41900i
\(190\) 0 0
\(191\) 31039.9i 0.850852i 0.904993 + 0.425426i \(0.139876\pi\)
−0.904993 + 0.425426i \(0.860124\pi\)
\(192\) 0 0
\(193\) 41374.0 1.11074 0.555371 0.831603i \(-0.312576\pi\)
0.555371 + 0.831603i \(0.312576\pi\)
\(194\) 0 0
\(195\) −16930.9 −0.445256
\(196\) 0 0
\(197\) − 67640.0i − 1.74289i −0.490489 0.871447i \(-0.663182\pi\)
0.490489 0.871447i \(-0.336818\pi\)
\(198\) 0 0
\(199\) − 25004.4i − 0.631408i −0.948858 0.315704i \(-0.897759\pi\)
0.948858 0.315704i \(-0.102241\pi\)
\(200\) 0 0
\(201\) −16416.0 −0.406327
\(202\) 0 0
\(203\) −102212. −2.48034
\(204\) 0 0
\(205\) 15120.0i 0.359786i
\(206\) 0 0
\(207\) − 10581.8i − 0.246955i
\(208\) 0 0
\(209\) −47520.0 −1.08789
\(210\) 0 0
\(211\) 26189.9 0.588260 0.294130 0.955765i \(-0.404970\pi\)
0.294130 + 0.955765i \(0.404970\pi\)
\(212\) 0 0
\(213\) − 76032.0i − 1.67586i
\(214\) 0 0
\(215\) 23280.0i 0.503623i
\(216\) 0 0
\(217\) 49152.0 1.04381
\(218\) 0 0
\(219\) 26944.4 0.561798
\(220\) 0 0
\(221\) − 34992.0i − 0.716447i
\(222\) 0 0
\(223\) − 20693.3i − 0.416121i −0.978116 0.208061i \(-0.933285\pi\)
0.978116 0.208061i \(-0.0667151\pi\)
\(224\) 0 0
\(225\) 8415.00 0.166222
\(226\) 0 0
\(227\) 77923.2 1.51222 0.756110 0.654445i \(-0.227098\pi\)
0.756110 + 0.654445i \(0.227098\pi\)
\(228\) 0 0
\(229\) 14472.0i 0.275967i 0.990435 + 0.137984i \(0.0440621\pi\)
−0.990435 + 0.137984i \(0.955938\pi\)
\(230\) 0 0
\(231\) − 82773.2i − 1.55119i
\(232\) 0 0
\(233\) −2754.00 −0.0507285 −0.0253643 0.999678i \(-0.508075\pi\)
−0.0253643 + 0.999678i \(0.508075\pi\)
\(234\) 0 0
\(235\) −11287.2 −0.204387
\(236\) 0 0
\(237\) − 78336.0i − 1.39465i
\(238\) 0 0
\(239\) − 91708.9i − 1.60552i −0.596302 0.802760i \(-0.703364\pi\)
0.596302 0.802760i \(-0.296636\pi\)
\(240\) 0 0
\(241\) −97570.0 −1.67990 −0.839948 0.542667i \(-0.817414\pi\)
−0.839948 + 0.542667i \(0.817414\pi\)
\(242\) 0 0
\(243\) −21604.5 −0.365874
\(244\) 0 0
\(245\) − 29944.0i − 0.498859i
\(246\) 0 0
\(247\) 95236.2i 1.56102i
\(248\) 0 0
\(249\) 91488.0 1.47559
\(250\) 0 0
\(251\) 67811.7 1.07636 0.538179 0.842830i \(-0.319112\pi\)
0.538179 + 0.842830i \(0.319112\pi\)
\(252\) 0 0
\(253\) 76032.0i 1.18783i
\(254\) 0 0
\(255\) − 12698.2i − 0.195281i
\(256\) 0 0
\(257\) 80514.0 1.21900 0.609502 0.792785i \(-0.291369\pi\)
0.609502 + 0.792785i \(0.291369\pi\)
\(258\) 0 0
\(259\) −118516. −1.76676
\(260\) 0 0
\(261\) 19560.0i 0.287136i
\(262\) 0 0
\(263\) 14814.5i 0.214179i 0.994249 + 0.107089i \(0.0341531\pi\)
−0.994249 + 0.107089i \(0.965847\pi\)
\(264\) 0 0
\(265\) 15808.0 0.225105
\(266\) 0 0
\(267\) 23809.0 0.333979
\(268\) 0 0
\(269\) − 44872.0i − 0.620113i −0.950718 0.310057i \(-0.899652\pi\)
0.950718 0.310057i \(-0.100348\pi\)
\(270\) 0 0
\(271\) 91395.4i 1.24447i 0.782829 + 0.622237i \(0.213776\pi\)
−0.782829 + 0.622237i \(0.786224\pi\)
\(272\) 0 0
\(273\) −165888. −2.22582
\(274\) 0 0
\(275\) −60463.2 −0.799513
\(276\) 0 0
\(277\) 90504.0i 1.17953i 0.807576 + 0.589764i \(0.200779\pi\)
−0.807576 + 0.589764i \(0.799221\pi\)
\(278\) 0 0
\(279\) − 9406.04i − 0.120837i
\(280\) 0 0
\(281\) 23166.0 0.293385 0.146693 0.989182i \(-0.453137\pi\)
0.146693 + 0.989182i \(0.453137\pi\)
\(282\) 0 0
\(283\) −70809.8 −0.884140 −0.442070 0.896981i \(-0.645756\pi\)
−0.442070 + 0.896981i \(0.645756\pi\)
\(284\) 0 0
\(285\) 34560.0i 0.425485i
\(286\) 0 0
\(287\) 148145.i 1.79855i
\(288\) 0 0
\(289\) −57277.0 −0.685780
\(290\) 0 0
\(291\) 73034.0 0.862460
\(292\) 0 0
\(293\) 37768.0i 0.439935i 0.975507 + 0.219968i \(0.0705952\pi\)
−0.975507 + 0.219968i \(0.929405\pi\)
\(294\) 0 0
\(295\) − 18106.6i − 0.208062i
\(296\) 0 0
\(297\) 69696.0 0.790123
\(298\) 0 0
\(299\) 152378. 1.70443
\(300\) 0 0
\(301\) 228096.i 2.51759i
\(302\) 0 0
\(303\) − 14657.7i − 0.159655i
\(304\) 0 0
\(305\) 19008.0 0.204332
\(306\) 0 0
\(307\) 38535.4 0.408868 0.204434 0.978880i \(-0.434465\pi\)
0.204434 + 0.978880i \(0.434465\pi\)
\(308\) 0 0
\(309\) 25344.0i 0.265435i
\(310\) 0 0
\(311\) − 85359.8i − 0.882537i −0.897375 0.441268i \(-0.854529\pi\)
0.897375 0.441268i \(-0.145471\pi\)
\(312\) 0 0
\(313\) 93602.0 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(314\) 0 0
\(315\) −9406.04 −0.0947951
\(316\) 0 0
\(317\) − 49192.0i − 0.489526i −0.969583 0.244763i \(-0.921290\pi\)
0.969583 0.244763i \(-0.0787102\pi\)
\(318\) 0 0
\(319\) − 140542.i − 1.38110i
\(320\) 0 0
\(321\) 10464.0 0.101552
\(322\) 0 0
\(323\) −71427.1 −0.684633
\(324\) 0 0
\(325\) 121176.i 1.14723i
\(326\) 0 0
\(327\) 146029.i 1.36566i
\(328\) 0 0
\(329\) −110592. −1.02172
\(330\) 0 0
\(331\) −16490.0 −0.150509 −0.0752547 0.997164i \(-0.523977\pi\)
−0.0752547 + 0.997164i \(0.523977\pi\)
\(332\) 0 0
\(333\) 22680.0i 0.204529i
\(334\) 0 0
\(335\) − 13403.6i − 0.119435i
\(336\) 0 0
\(337\) −93758.0 −0.825560 −0.412780 0.910831i \(-0.635442\pi\)
−0.412780 + 0.910831i \(0.635442\pi\)
\(338\) 0 0
\(339\) −6878.17 −0.0598513
\(340\) 0 0
\(341\) 67584.0i 0.581213i
\(342\) 0 0
\(343\) − 105191.i − 0.894108i
\(344\) 0 0
\(345\) 55296.0 0.464575
\(346\) 0 0
\(347\) −220405. −1.83047 −0.915235 0.402920i \(-0.867995\pi\)
−0.915235 + 0.402920i \(0.867995\pi\)
\(348\) 0 0
\(349\) 48600.0i 0.399012i 0.979897 + 0.199506i \(0.0639337\pi\)
−0.979897 + 0.199506i \(0.936066\pi\)
\(350\) 0 0
\(351\) − 139680.i − 1.13375i
\(352\) 0 0
\(353\) 108162. 0.868011 0.434006 0.900910i \(-0.357100\pi\)
0.434006 + 0.900910i \(0.357100\pi\)
\(354\) 0 0
\(355\) 62079.9 0.492600
\(356\) 0 0
\(357\) − 124416.i − 0.976202i
\(358\) 0 0
\(359\) − 50087.2i − 0.388631i −0.980939 0.194316i \(-0.937751\pi\)
0.980939 0.194316i \(-0.0622486\pi\)
\(360\) 0 0
\(361\) 64079.0 0.491701
\(362\) 0 0
\(363\) −29638.8 −0.224930
\(364\) 0 0
\(365\) 22000.0i 0.165134i
\(366\) 0 0
\(367\) − 61296.0i − 0.455093i −0.973767 0.227547i \(-0.926930\pi\)
0.973767 0.227547i \(-0.0730704\pi\)
\(368\) 0 0
\(369\) 28350.0 0.208209
\(370\) 0 0
\(371\) 154886. 1.12529
\(372\) 0 0
\(373\) − 54648.0i − 0.392787i −0.980525 0.196393i \(-0.937077\pi\)
0.980525 0.196393i \(-0.0629229\pi\)
\(374\) 0 0
\(375\) 92963.0i 0.661070i
\(376\) 0 0
\(377\) −281664. −1.98175
\(378\) 0 0
\(379\) 103790. 0.722564 0.361282 0.932457i \(-0.382339\pi\)
0.361282 + 0.932457i \(0.382339\pi\)
\(380\) 0 0
\(381\) − 33792.0i − 0.232790i
\(382\) 0 0
\(383\) 124160.i 0.846415i 0.906033 + 0.423207i \(0.139096\pi\)
−0.906033 + 0.423207i \(0.860904\pi\)
\(384\) 0 0
\(385\) 67584.0 0.455955
\(386\) 0 0
\(387\) 43649.9 0.291448
\(388\) 0 0
\(389\) − 129112.i − 0.853233i −0.904433 0.426616i \(-0.859705\pi\)
0.904433 0.426616i \(-0.140295\pi\)
\(390\) 0 0
\(391\) 114283.i 0.747532i
\(392\) 0 0
\(393\) −93984.0 −0.608512
\(394\) 0 0
\(395\) 63961.1 0.409941
\(396\) 0 0
\(397\) − 232200.i − 1.47327i −0.676293 0.736633i \(-0.736415\pi\)
0.676293 0.736633i \(-0.263585\pi\)
\(398\) 0 0
\(399\) 338617.i 2.12698i
\(400\) 0 0
\(401\) 151902. 0.944658 0.472329 0.881422i \(-0.343413\pi\)
0.472329 + 0.881422i \(0.343413\pi\)
\(402\) 0 0
\(403\) 135447. 0.833987
\(404\) 0 0
\(405\) − 60408.0i − 0.368285i
\(406\) 0 0
\(407\) − 162960.i − 0.983765i
\(408\) 0 0
\(409\) 28450.0 0.170073 0.0850366 0.996378i \(-0.472899\pi\)
0.0850366 + 0.996378i \(0.472899\pi\)
\(410\) 0 0
\(411\) −320099. −1.89496
\(412\) 0 0
\(413\) − 177408.i − 1.04010i
\(414\) 0 0
\(415\) 74699.6i 0.433733i
\(416\) 0 0
\(417\) −12960.0 −0.0745303
\(418\) 0 0
\(419\) −291382. −1.65972 −0.829858 0.557974i \(-0.811579\pi\)
−0.829858 + 0.557974i \(0.811579\pi\)
\(420\) 0 0
\(421\) 119880.i 0.676367i 0.941080 + 0.338184i \(0.109812\pi\)
−0.941080 + 0.338184i \(0.890188\pi\)
\(422\) 0 0
\(423\) 21163.6i 0.118279i
\(424\) 0 0
\(425\) −90882.0 −0.503153
\(426\) 0 0
\(427\) 186240. 1.02145
\(428\) 0 0
\(429\) − 228096.i − 1.23938i
\(430\) 0 0
\(431\) − 170720.i − 0.919028i −0.888170 0.459514i \(-0.848024\pi\)
0.888170 0.459514i \(-0.151976\pi\)
\(432\) 0 0
\(433\) 215518. 1.14950 0.574748 0.818330i \(-0.305100\pi\)
0.574748 + 0.818330i \(0.305100\pi\)
\(434\) 0 0
\(435\) −102212. −0.540163
\(436\) 0 0
\(437\) − 311040.i − 1.62875i
\(438\) 0 0
\(439\) 75169.9i 0.390045i 0.980799 + 0.195023i \(0.0624781\pi\)
−0.980799 + 0.195023i \(0.937522\pi\)
\(440\) 0 0
\(441\) −56145.0 −0.288691
\(442\) 0 0
\(443\) 185270. 0.944054 0.472027 0.881584i \(-0.343523\pi\)
0.472027 + 0.881584i \(0.343523\pi\)
\(444\) 0 0
\(445\) 19440.0i 0.0981694i
\(446\) 0 0
\(447\) 8543.82i 0.0427599i
\(448\) 0 0
\(449\) 177822. 0.882049 0.441025 0.897495i \(-0.354615\pi\)
0.441025 + 0.897495i \(0.354615\pi\)
\(450\) 0 0
\(451\) −203700. −1.00147
\(452\) 0 0
\(453\) − 92928.0i − 0.452846i
\(454\) 0 0
\(455\) − 135447.i − 0.654254i
\(456\) 0 0
\(457\) −203294. −0.973402 −0.486701 0.873569i \(-0.661800\pi\)
−0.486701 + 0.873569i \(0.661800\pi\)
\(458\) 0 0
\(459\) 104760. 0.497244
\(460\) 0 0
\(461\) 286648.i 1.34880i 0.738367 + 0.674399i \(0.235597\pi\)
−0.738367 + 0.674399i \(0.764403\pi\)
\(462\) 0 0
\(463\) − 332817.i − 1.55254i −0.630399 0.776271i \(-0.717109\pi\)
0.630399 0.776271i \(-0.282891\pi\)
\(464\) 0 0
\(465\) 49152.0 0.227319
\(466\) 0 0
\(467\) −58738.8 −0.269334 −0.134667 0.990891i \(-0.542996\pi\)
−0.134667 + 0.990891i \(0.542996\pi\)
\(468\) 0 0
\(469\) − 131328.i − 0.597051i
\(470\) 0 0
\(471\) 281476.i 1.26882i
\(472\) 0 0
\(473\) −313632. −1.40184
\(474\) 0 0
\(475\) 247349. 1.09629
\(476\) 0 0
\(477\) − 29640.0i − 0.130269i
\(478\) 0 0
\(479\) − 186240.i − 0.811710i −0.913937 0.405855i \(-0.866974\pi\)
0.913937 0.405855i \(-0.133026\pi\)
\(480\) 0 0
\(481\) −326592. −1.41161
\(482\) 0 0
\(483\) 541788. 2.32239
\(484\) 0 0
\(485\) 59632.0i 0.253510i
\(486\) 0 0
\(487\) − 363308.i − 1.53185i −0.642927 0.765927i \(-0.722280\pi\)
0.642927 0.765927i \(-0.277720\pi\)
\(488\) 0 0
\(489\) 304992. 1.27547
\(490\) 0 0
\(491\) −107533. −0.446043 −0.223022 0.974813i \(-0.571592\pi\)
−0.223022 + 0.974813i \(0.571592\pi\)
\(492\) 0 0
\(493\) − 211248.i − 0.869158i
\(494\) 0 0
\(495\) − 12933.3i − 0.0527836i
\(496\) 0 0
\(497\) 608256. 2.46249
\(498\) 0 0
\(499\) −385089. −1.54654 −0.773268 0.634079i \(-0.781379\pi\)
−0.773268 + 0.634079i \(0.781379\pi\)
\(500\) 0 0
\(501\) 532224.i 2.12041i
\(502\) 0 0
\(503\) 210930.i 0.833688i 0.908978 + 0.416844i \(0.136864\pi\)
−0.908978 + 0.416844i \(0.863136\pi\)
\(504\) 0 0
\(505\) 11968.0 0.0469287
\(506\) 0 0
\(507\) −177294. −0.689729
\(508\) 0 0
\(509\) 171512.i 0.662001i 0.943630 + 0.331001i \(0.107386\pi\)
−0.943630 + 0.331001i \(0.892614\pi\)
\(510\) 0 0
\(511\) 215555.i 0.825499i
\(512\) 0 0
\(513\) −285120. −1.08341
\(514\) 0 0
\(515\) −20693.3 −0.0780216
\(516\) 0 0
\(517\) − 152064.i − 0.568912i
\(518\) 0 0
\(519\) − 247457.i − 0.918683i
\(520\) 0 0
\(521\) −184734. −0.680568 −0.340284 0.940323i \(-0.610523\pi\)
−0.340284 + 0.940323i \(0.610523\pi\)
\(522\) 0 0
\(523\) −420009. −1.53552 −0.767760 0.640738i \(-0.778629\pi\)
−0.767760 + 0.640738i \(0.778629\pi\)
\(524\) 0 0
\(525\) 430848.i 1.56317i
\(526\) 0 0
\(527\) 101585.i 0.365771i
\(528\) 0 0
\(529\) −217823. −0.778381
\(530\) 0 0
\(531\) −33949.9 −0.120406
\(532\) 0 0
\(533\) 408240.i 1.43701i
\(534\) 0 0
\(535\) 8543.82i 0.0298500i
\(536\) 0 0
\(537\) 87648.0 0.303944
\(538\) 0 0
\(539\) 403411. 1.38858
\(540\) 0 0
\(541\) 230904.i 0.788927i 0.918912 + 0.394464i \(0.129070\pi\)
−0.918912 + 0.394464i \(0.870930\pi\)
\(542\) 0 0
\(543\) 353432.i 1.19869i
\(544\) 0 0
\(545\) −119232. −0.401421
\(546\) 0 0
\(547\) −39769.9 −0.132917 −0.0664584 0.997789i \(-0.521170\pi\)
−0.0664584 + 0.997789i \(0.521170\pi\)
\(548\) 0 0
\(549\) − 35640.0i − 0.118248i
\(550\) 0 0
\(551\) 574944.i 1.89375i
\(552\) 0 0
\(553\) 626688. 2.04928
\(554\) 0 0
\(555\) −118516. −0.384761
\(556\) 0 0
\(557\) − 23144.0i − 0.0745981i −0.999304 0.0372991i \(-0.988125\pi\)
0.999304 0.0372991i \(-0.0118754\pi\)
\(558\) 0 0
\(559\) 628559.i 2.01151i
\(560\) 0 0
\(561\) 171072. 0.543567
\(562\) 0 0
\(563\) 491123. 1.54943 0.774717 0.632308i \(-0.217892\pi\)
0.774717 + 0.632308i \(0.217892\pi\)
\(564\) 0 0
\(565\) − 5616.00i − 0.0175926i
\(566\) 0 0
\(567\) − 591875.i − 1.84104i
\(568\) 0 0
\(569\) −127710. −0.394458 −0.197229 0.980357i \(-0.563194\pi\)
−0.197229 + 0.980357i \(0.563194\pi\)
\(570\) 0 0
\(571\) −39064.5 −0.119815 −0.0599073 0.998204i \(-0.519081\pi\)
−0.0599073 + 0.998204i \(0.519081\pi\)
\(572\) 0 0
\(573\) 304128.i 0.926290i
\(574\) 0 0
\(575\) − 395759.i − 1.19700i
\(576\) 0 0
\(577\) −270718. −0.813140 −0.406570 0.913620i \(-0.633275\pi\)
−0.406570 + 0.913620i \(0.633275\pi\)
\(578\) 0 0
\(579\) 405381. 1.20922
\(580\) 0 0
\(581\) 731904.i 2.16821i
\(582\) 0 0
\(583\) 212968.i 0.626582i
\(584\) 0 0
\(585\) −25920.0 −0.0757396
\(586\) 0 0
\(587\) −273353. −0.793319 −0.396660 0.917966i \(-0.629831\pi\)
−0.396660 + 0.917966i \(0.629831\pi\)
\(588\) 0 0
\(589\) − 276480.i − 0.796954i
\(590\) 0 0
\(591\) − 662734.i − 1.89742i
\(592\) 0 0
\(593\) −62910.0 −0.178900 −0.0894500 0.995991i \(-0.528511\pi\)
−0.0894500 + 0.995991i \(0.528511\pi\)
\(594\) 0 0
\(595\) 101585. 0.286944
\(596\) 0 0
\(597\) − 244992.i − 0.687390i
\(598\) 0 0
\(599\) 261723.i 0.729438i 0.931118 + 0.364719i \(0.118835\pi\)
−0.931118 + 0.364719i \(0.881165\pi\)
\(600\) 0 0
\(601\) 203902. 0.564511 0.282256 0.959339i \(-0.408917\pi\)
0.282256 + 0.959339i \(0.408917\pi\)
\(602\) 0 0
\(603\) −25131.8 −0.0691176
\(604\) 0 0
\(605\) − 24200.0i − 0.0661157i
\(606\) 0 0
\(607\) − 369657.i − 1.00328i −0.865077 0.501640i \(-0.832730\pi\)
0.865077 0.501640i \(-0.167270\pi\)
\(608\) 0 0
\(609\) −1.00147e6 −2.70025
\(610\) 0 0
\(611\) −304756. −0.816337
\(612\) 0 0
\(613\) − 222264.i − 0.591491i −0.955267 0.295746i \(-0.904432\pi\)
0.955267 0.295746i \(-0.0955680\pi\)
\(614\) 0 0
\(615\) 148145.i 0.391685i
\(616\) 0 0
\(617\) 363582. 0.955063 0.477532 0.878615i \(-0.341532\pi\)
0.477532 + 0.878615i \(0.341532\pi\)
\(618\) 0 0
\(619\) −317718. −0.829203 −0.414602 0.910003i \(-0.636079\pi\)
−0.414602 + 0.910003i \(0.636079\pi\)
\(620\) 0 0
\(621\) 456192.i 1.18294i
\(622\) 0 0
\(623\) 190472.i 0.490745i
\(624\) 0 0
\(625\) 274721. 0.703286
\(626\) 0 0
\(627\) −465599. −1.18434
\(628\) 0 0
\(629\) − 244944.i − 0.619107i
\(630\) 0 0
\(631\) − 348415.i − 0.875062i −0.899203 0.437531i \(-0.855853\pi\)
0.899203 0.437531i \(-0.144147\pi\)
\(632\) 0 0
\(633\) 256608. 0.640417
\(634\) 0 0
\(635\) 27591.1 0.0684259
\(636\) 0 0
\(637\) − 808488.i − 1.99248i
\(638\) 0 0
\(639\) − 116400.i − 0.285069i
\(640\) 0 0
\(641\) 67230.0 0.163624 0.0818120 0.996648i \(-0.473929\pi\)
0.0818120 + 0.996648i \(0.473929\pi\)
\(642\) 0 0
\(643\) 74513.5 0.180224 0.0901121 0.995932i \(-0.471277\pi\)
0.0901121 + 0.995932i \(0.471277\pi\)
\(644\) 0 0
\(645\) 228096.i 0.548275i
\(646\) 0 0
\(647\) − 380239.i − 0.908340i −0.890915 0.454170i \(-0.849936\pi\)
0.890915 0.454170i \(-0.150064\pi\)
\(648\) 0 0
\(649\) 243936. 0.579144
\(650\) 0 0
\(651\) 481589. 1.13636
\(652\) 0 0
\(653\) − 89288.0i − 0.209395i −0.994504 0.104698i \(-0.966613\pi\)
0.994504 0.104698i \(-0.0333875\pi\)
\(654\) 0 0
\(655\) − 76737.6i − 0.178865i
\(656\) 0 0
\(657\) 41250.0 0.0955638
\(658\) 0 0
\(659\) −223149. −0.513834 −0.256917 0.966433i \(-0.582707\pi\)
−0.256917 + 0.966433i \(0.582707\pi\)
\(660\) 0 0
\(661\) − 560088.i − 1.28190i −0.767584 0.640949i \(-0.778541\pi\)
0.767584 0.640949i \(-0.221459\pi\)
\(662\) 0 0
\(663\) − 342850.i − 0.779969i
\(664\) 0 0
\(665\) −276480. −0.625202
\(666\) 0 0
\(667\) 919911. 2.06773
\(668\) 0 0
\(669\) − 202752.i − 0.453015i
\(670\) 0 0
\(671\) 256079.i 0.568761i
\(672\) 0 0
\(673\) −810722. −1.78995 −0.894977 0.446112i \(-0.852808\pi\)
−0.894977 + 0.446112i \(0.852808\pi\)
\(674\) 0 0
\(675\) −362779. −0.796223
\(676\) 0 0
\(677\) − 655768.i − 1.43078i −0.698725 0.715390i \(-0.746249\pi\)
0.698725 0.715390i \(-0.253751\pi\)
\(678\) 0 0
\(679\) 584272.i 1.26729i
\(680\) 0 0
\(681\) 763488. 1.64630
\(682\) 0 0
\(683\) −189208. −0.405601 −0.202800 0.979220i \(-0.565004\pi\)
−0.202800 + 0.979220i \(0.565004\pi\)
\(684\) 0 0
\(685\) − 261360.i − 0.557004i
\(686\) 0 0
\(687\) 141796.i 0.300435i
\(688\) 0 0
\(689\) 426816. 0.899088
\(690\) 0 0
\(691\) 569389. 1.19248 0.596242 0.802804i \(-0.296660\pi\)
0.596242 + 0.802804i \(0.296660\pi\)
\(692\) 0 0
\(693\) − 126720.i − 0.263863i
\(694\) 0 0
\(695\) − 10581.8i − 0.0219073i
\(696\) 0 0
\(697\) −306180. −0.630248
\(698\) 0 0
\(699\) −26983.6 −0.0552262
\(700\) 0 0
\(701\) 519224.i 1.05662i 0.849052 + 0.528310i \(0.177174\pi\)
−0.849052 + 0.528310i \(0.822826\pi\)
\(702\) 0 0
\(703\) 666653.i 1.34893i
\(704\) 0 0
\(705\) −110592. −0.222508
\(706\) 0 0
\(707\) 117262. 0.234595
\(708\) 0 0
\(709\) − 919512.i − 1.82922i −0.404342 0.914608i \(-0.632499\pi\)
0.404342 0.914608i \(-0.367501\pi\)
\(710\) 0 0
\(711\) − 119927.i − 0.237234i
\(712\) 0 0
\(713\) −442368. −0.870171
\(714\) 0 0
\(715\) 186240. 0.364301
\(716\) 0 0
\(717\) − 898560.i − 1.74787i
\(718\) 0 0
\(719\) 656071.i 1.26909i 0.772885 + 0.634546i \(0.218813\pi\)
−0.772885 + 0.634546i \(0.781187\pi\)
\(720\) 0 0
\(721\) −202752. −0.390027
\(722\) 0 0
\(723\) −955987. −1.82884
\(724\) 0 0
\(725\) 731544.i 1.39176i
\(726\) 0 0
\(727\) 488722.i 0.924684i 0.886702 + 0.462342i \(0.152991\pi\)
−0.886702 + 0.462342i \(0.847009\pi\)
\(728\) 0 0
\(729\) 399951. 0.752578
\(730\) 0 0
\(731\) −471419. −0.882211
\(732\) 0 0
\(733\) − 706536.i − 1.31500i −0.753454 0.657501i \(-0.771614\pi\)
0.753454 0.657501i \(-0.228386\pi\)
\(734\) 0 0
\(735\) − 293390.i − 0.543089i
\(736\) 0 0
\(737\) 180576. 0.332449
\(738\) 0 0
\(739\) −172748. −0.316318 −0.158159 0.987414i \(-0.550556\pi\)
−0.158159 + 0.987414i \(0.550556\pi\)
\(740\) 0 0
\(741\) 933120.i 1.69942i
\(742\) 0 0
\(743\) − 250436.i − 0.453648i −0.973936 0.226824i \(-0.927166\pi\)
0.973936 0.226824i \(-0.0728342\pi\)
\(744\) 0 0
\(745\) −6976.00 −0.0125688
\(746\) 0 0
\(747\) 140062. 0.251003
\(748\) 0 0
\(749\) 83712.0i 0.149219i
\(750\) 0 0
\(751\) − 536301.i − 0.950887i −0.879746 0.475443i \(-0.842288\pi\)
0.879746 0.475443i \(-0.157712\pi\)
\(752\) 0 0
\(753\) 664416. 1.17179
\(754\) 0 0
\(755\) 75875.4 0.133109
\(756\) 0 0
\(757\) − 470232.i − 0.820579i −0.911955 0.410290i \(-0.865428\pi\)
0.911955 0.410290i \(-0.134572\pi\)
\(758\) 0 0
\(759\) 744958.i 1.29315i
\(760\) 0 0
\(761\) −691038. −1.19325 −0.596627 0.802519i \(-0.703493\pi\)
−0.596627 + 0.802519i \(0.703493\pi\)
\(762\) 0 0
\(763\) −1.16823e6 −2.00669
\(764\) 0 0
\(765\) − 19440.0i − 0.0332180i
\(766\) 0 0
\(767\) − 488879.i − 0.831018i
\(768\) 0 0
\(769\) 304030. 0.514119 0.257060 0.966396i \(-0.417246\pi\)
0.257060 + 0.966396i \(0.417246\pi\)
\(770\) 0 0
\(771\) 788873. 1.32708
\(772\) 0 0
\(773\) − 533720.i − 0.893212i −0.894731 0.446606i \(-0.852633\pi\)
0.894731 0.446606i \(-0.147367\pi\)
\(774\) 0 0
\(775\) − 351786.i − 0.585700i
\(776\) 0 0
\(777\) −1.16122e6 −1.92341
\(778\) 0 0
\(779\) 833316. 1.37320
\(780\) 0 0
\(781\) 836352.i 1.37116i
\(782\) 0 0
\(783\) − 843252.i − 1.37541i
\(784\) 0 0
\(785\) −229824. −0.372955
\(786\) 0 0
\(787\) 1.17467e6 1.89656 0.948278 0.317442i \(-0.102824\pi\)
0.948278 + 0.317442i \(0.102824\pi\)
\(788\) 0 0
\(789\) 145152.i 0.233168i
\(790\) 0 0
\(791\) − 55025.3i − 0.0879447i
\(792\) 0 0
\(793\) 513216. 0.816120
\(794\) 0 0
\(795\) 154886. 0.245063
\(796\) 0 0
\(797\) 552376.i 0.869597i 0.900528 + 0.434799i \(0.143180\pi\)
−0.900528 + 0.434799i \(0.856820\pi\)
\(798\) 0 0
\(799\) − 228567.i − 0.358030i
\(800\) 0 0
\(801\) 36450.0 0.0568110
\(802\) 0 0
\(803\) −296388. −0.459653
\(804\) 0 0
\(805\) 442368.i 0.682640i
\(806\) 0 0
\(807\) − 439654.i − 0.675094i
\(808\) 0 0
\(809\) 803682. 1.22797 0.613984 0.789318i \(-0.289566\pi\)
0.613984 + 0.789318i \(0.289566\pi\)
\(810\) 0 0
\(811\) 68869.9 0.104710 0.0523549 0.998629i \(-0.483327\pi\)
0.0523549 + 0.998629i \(0.483327\pi\)
\(812\) 0 0
\(813\) 895488.i 1.35481i
\(814\) 0 0
\(815\) 249025.i 0.374910i
\(816\) 0 0
\(817\) 1.28304e6 1.92219
\(818\) 0 0
\(819\) −253963. −0.378619
\(820\) 0 0
\(821\) 867592.i 1.28715i 0.765383 + 0.643575i \(0.222550\pi\)
−0.765383 + 0.643575i \(0.777450\pi\)
\(822\) 0 0
\(823\) − 30177.7i − 0.0445540i −0.999752 0.0222770i \(-0.992908\pi\)
0.999752 0.0222770i \(-0.00709158\pi\)
\(824\) 0 0
\(825\) −592416. −0.870400
\(826\) 0 0
\(827\) −864680. −1.26428 −0.632141 0.774853i \(-0.717824\pi\)
−0.632141 + 0.774853i \(0.717824\pi\)
\(828\) 0 0
\(829\) − 449928.i − 0.654687i −0.944905 0.327344i \(-0.893847\pi\)
0.944905 0.327344i \(-0.106153\pi\)
\(830\) 0 0
\(831\) 886754.i 1.28411i
\(832\) 0 0
\(833\) 606366. 0.873866
\(834\) 0 0
\(835\) −434559. −0.623269
\(836\) 0 0
\(837\) 405504.i 0.578821i
\(838\) 0 0
\(839\) − 800689.i − 1.13747i −0.822521 0.568735i \(-0.807433\pi\)
0.822521 0.568735i \(-0.192567\pi\)
\(840\) 0 0
\(841\) −993135. −1.40416
\(842\) 0 0
\(843\) 226980. 0.319398
\(844\) 0 0
\(845\) − 144760.i − 0.202738i
\(846\) 0 0
\(847\) − 237111.i − 0.330510i
\(848\) 0 0
\(849\) −693792. −0.962529
\(850\) 0 0
\(851\) 1.06665e6 1.47286
\(852\) 0 0
\(853\) 121608.i 0.167134i 0.996502 + 0.0835669i \(0.0266312\pi\)
−0.996502 + 0.0835669i \(0.973369\pi\)
\(854\) 0 0
\(855\) 52909.0i 0.0723764i
\(856\) 0 0
\(857\) −1.26473e6 −1.72202 −0.861009 0.508590i \(-0.830167\pi\)
−0.861009 + 0.508590i \(0.830167\pi\)
\(858\) 0 0
\(859\) −967088. −1.31063 −0.655314 0.755356i \(-0.727464\pi\)
−0.655314 + 0.755356i \(0.727464\pi\)
\(860\) 0 0
\(861\) 1.45152e6i 1.95802i
\(862\) 0 0
\(863\) 428915.i 0.575904i 0.957645 + 0.287952i \(0.0929744\pi\)
−0.957645 + 0.287952i \(0.907026\pi\)
\(864\) 0 0
\(865\) 202048. 0.270036
\(866\) 0 0
\(867\) −561198. −0.746582
\(868\) 0 0
\(869\) 861696.i 1.14108i
\(870\) 0 0
\(871\) − 361897.i − 0.477034i
\(872\) 0 0
\(873\) 111810. 0.146707
\(874\) 0 0
\(875\) −743704. −0.971369
\(876\) 0 0
\(877\) − 519048.i − 0.674852i −0.941352 0.337426i \(-0.890444\pi\)
0.941352 0.337426i \(-0.109556\pi\)
\(878\) 0 0
\(879\) 370049.i 0.478941i
\(880\) 0 0
\(881\) 434754. 0.560134 0.280067 0.959980i \(-0.409643\pi\)
0.280067 + 0.959980i \(0.409643\pi\)
\(882\) 0 0
\(883\) −474329. −0.608357 −0.304178 0.952615i \(-0.598382\pi\)
−0.304178 + 0.952615i \(0.598382\pi\)
\(884\) 0 0
\(885\) − 177408.i − 0.226510i
\(886\) 0 0
\(887\) 298407.i 0.379281i 0.981854 + 0.189641i \(0.0607323\pi\)
−0.981854 + 0.189641i \(0.939268\pi\)
\(888\) 0 0
\(889\) 270336. 0.342058
\(890\) 0 0
\(891\) 813828. 1.02513
\(892\) 0 0
\(893\) 622080.i 0.780088i
\(894\) 0 0
\(895\) 71564.3i 0.0893409i
\(896\) 0 0
\(897\) 1.49299e6 1.85555
\(898\) 0 0
\(899\) 817698. 1.01175
\(900\) 0 0
\(901\) 320112.i 0.394323i
\(902\) 0 0
\(903\) 2.23488e6i 2.74080i
\(904\) 0 0
\(905\) −288576. −0.352341
\(906\) 0 0
\(907\) 693549. 0.843067 0.421534 0.906813i \(-0.361492\pi\)
0.421534 + 0.906813i \(0.361492\pi\)
\(908\) 0 0
\(909\) − 22440.0i − 0.0271578i
\(910\) 0 0
\(911\) 1.13296e6i 1.36514i 0.730821 + 0.682570i \(0.239138\pi\)
−0.730821 + 0.682570i \(0.760862\pi\)
\(912\) 0 0
\(913\) −1.00637e6 −1.20730
\(914\) 0 0
\(915\) 186240. 0.222449
\(916\) 0 0
\(917\) − 751872.i − 0.894139i
\(918\) 0 0
\(919\) − 526817.i − 0.623776i −0.950119 0.311888i \(-0.899039\pi\)
0.950119 0.311888i \(-0.100961\pi\)
\(920\) 0 0
\(921\) 377568. 0.445119
\(922\) 0 0
\(923\) 1.67616e6 1.96748
\(924\) 0 0
\(925\) 848232.i 0.991360i
\(926\) 0 0
\(927\) 38799.9i 0.0451514i
\(928\) 0 0
\(929\) −371682. −0.430666 −0.215333 0.976541i \(-0.569084\pi\)
−0.215333 + 0.976541i \(0.569084\pi\)
\(930\) 0 0
\(931\) −1.65032e6 −1.90401
\(932\) 0 0
\(933\) − 836352.i − 0.960784i
\(934\) 0 0
\(935\) 139680.i 0.159775i
\(936\) 0 0
\(937\) −532418. −0.606420 −0.303210 0.952924i \(-0.598058\pi\)
−0.303210 + 0.952924i \(0.598058\pi\)
\(938\) 0 0
\(939\) 917109. 1.04013
\(940\) 0 0
\(941\) − 614600.i − 0.694086i −0.937849 0.347043i \(-0.887186\pi\)
0.937849 0.347043i \(-0.112814\pi\)
\(942\) 0 0
\(943\) − 1.33331e6i − 1.49936i
\(944\) 0 0
\(945\) 405504. 0.454079
\(946\) 0 0
\(947\) −583988. −0.651184 −0.325592 0.945510i \(-0.605564\pi\)
−0.325592 + 0.945510i \(0.605564\pi\)
\(948\) 0 0
\(949\) 594000.i 0.659560i
\(950\) 0 0
\(951\) − 481981.i − 0.532929i
\(952\) 0 0
\(953\) 604962. 0.666104 0.333052 0.942908i \(-0.391921\pi\)
0.333052 + 0.942908i \(0.391921\pi\)
\(954\) 0 0
\(955\) −248319. −0.272273
\(956\) 0 0
\(957\) − 1.37702e6i − 1.50355i
\(958\) 0 0
\(959\) − 2.56079e6i − 2.78444i
\(960\) 0 0
\(961\) 530305. 0.574221
\(962\) 0 0
\(963\) 16019.7 0.0172743
\(964\) 0 0
\(965\) 330992.i 0.355437i
\(966\) 0 0
\(967\) 1.39405e6i 1.49082i 0.666604 + 0.745412i \(0.267747\pi\)
−0.666604 + 0.745412i \(0.732253\pi\)
\(968\) 0 0
\(969\) −699840. −0.745334
\(970\) 0 0
\(971\) −662175. −0.702319 −0.351160 0.936316i \(-0.614213\pi\)
−0.351160 + 0.936316i \(0.614213\pi\)
\(972\) 0 0
\(973\) − 103680.i − 0.109514i
\(974\) 0 0
\(975\) 1.18728e6i 1.24894i
\(976\) 0 0
\(977\) −815778. −0.854639 −0.427320 0.904101i \(-0.640542\pi\)
−0.427320 + 0.904101i \(0.640542\pi\)
\(978\) 0 0
\(979\) −261899. −0.273256
\(980\) 0 0
\(981\) 223560.i 0.232304i
\(982\) 0 0
\(983\) − 1.23384e6i − 1.27688i −0.769671 0.638441i \(-0.779580\pi\)
0.769671 0.638441i \(-0.220420\pi\)
\(984\) 0 0
\(985\) 541120. 0.557726
\(986\) 0 0
\(987\) −1.08358e6 −1.11231
\(988\) 0 0
\(989\) − 2.05286e6i − 2.09878i
\(990\) 0 0
\(991\) 417315.i 0.424929i 0.977169 + 0.212464i \(0.0681490\pi\)
−0.977169 + 0.212464i \(0.931851\pi\)
\(992\) 0 0
\(993\) −161568. −0.163854
\(994\) 0 0
\(995\) 200035. 0.202051
\(996\) 0 0
\(997\) − 1.80252e6i − 1.81338i −0.421793 0.906692i \(-0.638599\pi\)
0.421793 0.906692i \(-0.361401\pi\)
\(998\) 0 0
\(999\) − 977758.i − 0.979716i
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 128.5.d.c.63.4 yes 4
3.2 odd 2 1152.5.b.i.703.2 4
4.3 odd 2 inner 128.5.d.c.63.2 yes 4
8.3 odd 2 inner 128.5.d.c.63.3 yes 4
8.5 even 2 inner 128.5.d.c.63.1 4
12.11 even 2 1152.5.b.i.703.1 4
16.3 odd 4 256.5.c.f.255.1 2
16.5 even 4 256.5.c.c.255.1 2
16.11 odd 4 256.5.c.c.255.2 2
16.13 even 4 256.5.c.f.255.2 2
24.5 odd 2 1152.5.b.i.703.4 4
24.11 even 2 1152.5.b.i.703.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.5.d.c.63.1 4 8.5 even 2 inner
128.5.d.c.63.2 yes 4 4.3 odd 2 inner
128.5.d.c.63.3 yes 4 8.3 odd 2 inner
128.5.d.c.63.4 yes 4 1.1 even 1 trivial
256.5.c.c.255.1 2 16.5 even 4
256.5.c.c.255.2 2 16.11 odd 4
256.5.c.f.255.1 2 16.3 odd 4
256.5.c.f.255.2 2 16.13 even 4
1152.5.b.i.703.1 4 12.11 even 2
1152.5.b.i.703.2 4 3.2 odd 2
1152.5.b.i.703.3 4 24.11 even 2
1152.5.b.i.703.4 4 24.5 odd 2