Properties

Label 768.4.a.a.1.1
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,4,Mod(1,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 768.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -8.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -8.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} +12.0000 q^{11} +20.0000 q^{13} +24.0000 q^{15} +62.0000 q^{17} -108.000 q^{19} -36.0000 q^{21} -72.0000 q^{23} -61.0000 q^{25} -27.0000 q^{27} -128.000 q^{29} +204.000 q^{31} -36.0000 q^{33} -96.0000 q^{35} -228.000 q^{37} -60.0000 q^{39} +22.0000 q^{41} +204.000 q^{43} -72.0000 q^{45} +600.000 q^{47} -199.000 q^{49} -186.000 q^{51} +256.000 q^{53} -96.0000 q^{55} +324.000 q^{57} +828.000 q^{59} -84.0000 q^{61} +108.000 q^{63} -160.000 q^{65} -348.000 q^{67} +216.000 q^{69} +456.000 q^{71} -822.000 q^{73} +183.000 q^{75} +144.000 q^{77} +1356.00 q^{79} +81.0000 q^{81} -108.000 q^{83} -496.000 q^{85} +384.000 q^{87} +938.000 q^{89} +240.000 q^{91} -612.000 q^{93} +864.000 q^{95} +1278.00 q^{97} +108.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −8.00000 −0.715542 −0.357771 0.933809i \(-0.616463\pi\)
−0.357771 + 0.933809i \(0.616463\pi\)
\(6\) 0 0
\(7\) 12.0000 0.647939 0.323970 0.946068i \(-0.394982\pi\)
0.323970 + 0.946068i \(0.394982\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 12.0000 0.328921 0.164461 0.986384i \(-0.447412\pi\)
0.164461 + 0.986384i \(0.447412\pi\)
\(12\) 0 0
\(13\) 20.0000 0.426692 0.213346 0.976977i \(-0.431564\pi\)
0.213346 + 0.976977i \(0.431564\pi\)
\(14\) 0 0
\(15\) 24.0000 0.413118
\(16\) 0 0
\(17\) 62.0000 0.884542 0.442271 0.896882i \(-0.354173\pi\)
0.442271 + 0.896882i \(0.354173\pi\)
\(18\) 0 0
\(19\) −108.000 −1.30405 −0.652024 0.758199i \(-0.726080\pi\)
−0.652024 + 0.758199i \(0.726080\pi\)
\(20\) 0 0
\(21\) −36.0000 −0.374088
\(22\) 0 0
\(23\) −72.0000 −0.652741 −0.326370 0.945242i \(-0.605826\pi\)
−0.326370 + 0.945242i \(0.605826\pi\)
\(24\) 0 0
\(25\) −61.0000 −0.488000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −128.000 −0.819621 −0.409810 0.912171i \(-0.634405\pi\)
−0.409810 + 0.912171i \(0.634405\pi\)
\(30\) 0 0
\(31\) 204.000 1.18192 0.590959 0.806701i \(-0.298749\pi\)
0.590959 + 0.806701i \(0.298749\pi\)
\(32\) 0 0
\(33\) −36.0000 −0.189903
\(34\) 0 0
\(35\) −96.0000 −0.463627
\(36\) 0 0
\(37\) −228.000 −1.01305 −0.506527 0.862224i \(-0.669071\pi\)
−0.506527 + 0.862224i \(0.669071\pi\)
\(38\) 0 0
\(39\) −60.0000 −0.246351
\(40\) 0 0
\(41\) 22.0000 0.0838006 0.0419003 0.999122i \(-0.486659\pi\)
0.0419003 + 0.999122i \(0.486659\pi\)
\(42\) 0 0
\(43\) 204.000 0.723482 0.361741 0.932279i \(-0.382183\pi\)
0.361741 + 0.932279i \(0.382183\pi\)
\(44\) 0 0
\(45\) −72.0000 −0.238514
\(46\) 0 0
\(47\) 600.000 1.86211 0.931053 0.364884i \(-0.118891\pi\)
0.931053 + 0.364884i \(0.118891\pi\)
\(48\) 0 0
\(49\) −199.000 −0.580175
\(50\) 0 0
\(51\) −186.000 −0.510690
\(52\) 0 0
\(53\) 256.000 0.663477 0.331739 0.943371i \(-0.392365\pi\)
0.331739 + 0.943371i \(0.392365\pi\)
\(54\) 0 0
\(55\) −96.0000 −0.235357
\(56\) 0 0
\(57\) 324.000 0.752892
\(58\) 0 0
\(59\) 828.000 1.82706 0.913529 0.406774i \(-0.133346\pi\)
0.913529 + 0.406774i \(0.133346\pi\)
\(60\) 0 0
\(61\) −84.0000 −0.176313 −0.0881565 0.996107i \(-0.528098\pi\)
−0.0881565 + 0.996107i \(0.528098\pi\)
\(62\) 0 0
\(63\) 108.000 0.215980
\(64\) 0 0
\(65\) −160.000 −0.305316
\(66\) 0 0
\(67\) −348.000 −0.634552 −0.317276 0.948333i \(-0.602768\pi\)
−0.317276 + 0.948333i \(0.602768\pi\)
\(68\) 0 0
\(69\) 216.000 0.376860
\(70\) 0 0
\(71\) 456.000 0.762215 0.381107 0.924531i \(-0.375543\pi\)
0.381107 + 0.924531i \(0.375543\pi\)
\(72\) 0 0
\(73\) −822.000 −1.31792 −0.658958 0.752180i \(-0.729002\pi\)
−0.658958 + 0.752180i \(0.729002\pi\)
\(74\) 0 0
\(75\) 183.000 0.281747
\(76\) 0 0
\(77\) 144.000 0.213121
\(78\) 0 0
\(79\) 1356.00 1.93116 0.965582 0.260100i \(-0.0837554\pi\)
0.965582 + 0.260100i \(0.0837554\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −108.000 −0.142826 −0.0714129 0.997447i \(-0.522751\pi\)
−0.0714129 + 0.997447i \(0.522751\pi\)
\(84\) 0 0
\(85\) −496.000 −0.632927
\(86\) 0 0
\(87\) 384.000 0.473208
\(88\) 0 0
\(89\) 938.000 1.11717 0.558583 0.829449i \(-0.311345\pi\)
0.558583 + 0.829449i \(0.311345\pi\)
\(90\) 0 0
\(91\) 240.000 0.276471
\(92\) 0 0
\(93\) −612.000 −0.682381
\(94\) 0 0
\(95\) 864.000 0.933100
\(96\) 0 0
\(97\) 1278.00 1.33774 0.668872 0.743377i \(-0.266777\pi\)
0.668872 + 0.743377i \(0.266777\pi\)
\(98\) 0 0
\(99\) 108.000 0.109640
\(100\) 0 0
\(101\) 608.000 0.598993 0.299496 0.954097i \(-0.403181\pi\)
0.299496 + 0.954097i \(0.403181\pi\)
\(102\) 0 0
\(103\) 948.000 0.906886 0.453443 0.891285i \(-0.350196\pi\)
0.453443 + 0.891285i \(0.350196\pi\)
\(104\) 0 0
\(105\) 288.000 0.267675
\(106\) 0 0
\(107\) −1044.00 −0.943246 −0.471623 0.881800i \(-0.656332\pi\)
−0.471623 + 0.881800i \(0.656332\pi\)
\(108\) 0 0
\(109\) 1780.00 1.56416 0.782078 0.623180i \(-0.214160\pi\)
0.782078 + 0.623180i \(0.214160\pi\)
\(110\) 0 0
\(111\) 684.000 0.584887
\(112\) 0 0
\(113\) −622.000 −0.517813 −0.258906 0.965902i \(-0.583362\pi\)
−0.258906 + 0.965902i \(0.583362\pi\)
\(114\) 0 0
\(115\) 576.000 0.467063
\(116\) 0 0
\(117\) 180.000 0.142231
\(118\) 0 0
\(119\) 744.000 0.573129
\(120\) 0 0
\(121\) −1187.00 −0.891811
\(122\) 0 0
\(123\) −66.0000 −0.0483823
\(124\) 0 0
\(125\) 1488.00 1.06473
\(126\) 0 0
\(127\) −204.000 −0.142536 −0.0712680 0.997457i \(-0.522705\pi\)
−0.0712680 + 0.997457i \(0.522705\pi\)
\(128\) 0 0
\(129\) −612.000 −0.417702
\(130\) 0 0
\(131\) −348.000 −0.232098 −0.116049 0.993243i \(-0.537023\pi\)
−0.116049 + 0.993243i \(0.537023\pi\)
\(132\) 0 0
\(133\) −1296.00 −0.844943
\(134\) 0 0
\(135\) 216.000 0.137706
\(136\) 0 0
\(137\) −106.000 −0.0661036 −0.0330518 0.999454i \(-0.510523\pi\)
−0.0330518 + 0.999454i \(0.510523\pi\)
\(138\) 0 0
\(139\) −1188.00 −0.724927 −0.362463 0.931998i \(-0.618064\pi\)
−0.362463 + 0.931998i \(0.618064\pi\)
\(140\) 0 0
\(141\) −1800.00 −1.07509
\(142\) 0 0
\(143\) 240.000 0.140348
\(144\) 0 0
\(145\) 1024.00 0.586473
\(146\) 0 0
\(147\) 597.000 0.334964
\(148\) 0 0
\(149\) 2648.00 1.45592 0.727962 0.685618i \(-0.240468\pi\)
0.727962 + 0.685618i \(0.240468\pi\)
\(150\) 0 0
\(151\) 3420.00 1.84315 0.921575 0.388200i \(-0.126903\pi\)
0.921575 + 0.388200i \(0.126903\pi\)
\(152\) 0 0
\(153\) 558.000 0.294847
\(154\) 0 0
\(155\) −1632.00 −0.845712
\(156\) 0 0
\(157\) 60.0000 0.0305001 0.0152501 0.999884i \(-0.495146\pi\)
0.0152501 + 0.999884i \(0.495146\pi\)
\(158\) 0 0
\(159\) −768.000 −0.383059
\(160\) 0 0
\(161\) −864.000 −0.422936
\(162\) 0 0
\(163\) 228.000 0.109560 0.0547802 0.998498i \(-0.482554\pi\)
0.0547802 + 0.998498i \(0.482554\pi\)
\(164\) 0 0
\(165\) 288.000 0.135883
\(166\) 0 0
\(167\) 4128.00 1.91278 0.956390 0.292093i \(-0.0943517\pi\)
0.956390 + 0.292093i \(0.0943517\pi\)
\(168\) 0 0
\(169\) −1797.00 −0.817934
\(170\) 0 0
\(171\) −972.000 −0.434682
\(172\) 0 0
\(173\) −1352.00 −0.594166 −0.297083 0.954852i \(-0.596014\pi\)
−0.297083 + 0.954852i \(0.596014\pi\)
\(174\) 0 0
\(175\) −732.000 −0.316194
\(176\) 0 0
\(177\) −2484.00 −1.05485
\(178\) 0 0
\(179\) 1716.00 0.716536 0.358268 0.933619i \(-0.383367\pi\)
0.358268 + 0.933619i \(0.383367\pi\)
\(180\) 0 0
\(181\) −3692.00 −1.51616 −0.758078 0.652164i \(-0.773861\pi\)
−0.758078 + 0.652164i \(0.773861\pi\)
\(182\) 0 0
\(183\) 252.000 0.101794
\(184\) 0 0
\(185\) 1824.00 0.724882
\(186\) 0 0
\(187\) 744.000 0.290945
\(188\) 0 0
\(189\) −324.000 −0.124696
\(190\) 0 0
\(191\) −48.0000 −0.0181841 −0.00909204 0.999959i \(-0.502894\pi\)
−0.00909204 + 0.999959i \(0.502894\pi\)
\(192\) 0 0
\(193\) 2414.00 0.900329 0.450165 0.892946i \(-0.351365\pi\)
0.450165 + 0.892946i \(0.351365\pi\)
\(194\) 0 0
\(195\) 480.000 0.176274
\(196\) 0 0
\(197\) 4304.00 1.55659 0.778293 0.627902i \(-0.216086\pi\)
0.778293 + 0.627902i \(0.216086\pi\)
\(198\) 0 0
\(199\) −204.000 −0.0726692 −0.0363346 0.999340i \(-0.511568\pi\)
−0.0363346 + 0.999340i \(0.511568\pi\)
\(200\) 0 0
\(201\) 1044.00 0.366359
\(202\) 0 0
\(203\) −1536.00 −0.531064
\(204\) 0 0
\(205\) −176.000 −0.0599628
\(206\) 0 0
\(207\) −648.000 −0.217580
\(208\) 0 0
\(209\) −1296.00 −0.428929
\(210\) 0 0
\(211\) 4020.00 1.31160 0.655801 0.754933i \(-0.272331\pi\)
0.655801 + 0.754933i \(0.272331\pi\)
\(212\) 0 0
\(213\) −1368.00 −0.440065
\(214\) 0 0
\(215\) −1632.00 −0.517681
\(216\) 0 0
\(217\) 2448.00 0.765811
\(218\) 0 0
\(219\) 2466.00 0.760899
\(220\) 0 0
\(221\) 1240.00 0.377427
\(222\) 0 0
\(223\) 516.000 0.154950 0.0774751 0.996994i \(-0.475314\pi\)
0.0774751 + 0.996994i \(0.475314\pi\)
\(224\) 0 0
\(225\) −549.000 −0.162667
\(226\) 0 0
\(227\) 1428.00 0.417532 0.208766 0.977966i \(-0.433055\pi\)
0.208766 + 0.977966i \(0.433055\pi\)
\(228\) 0 0
\(229\) −6028.00 −1.73948 −0.869741 0.493508i \(-0.835714\pi\)
−0.869741 + 0.493508i \(0.835714\pi\)
\(230\) 0 0
\(231\) −432.000 −0.123046
\(232\) 0 0
\(233\) −2630.00 −0.739472 −0.369736 0.929137i \(-0.620552\pi\)
−0.369736 + 0.929137i \(0.620552\pi\)
\(234\) 0 0
\(235\) −4800.00 −1.33241
\(236\) 0 0
\(237\) −4068.00 −1.11496
\(238\) 0 0
\(239\) −4416.00 −1.19518 −0.597588 0.801803i \(-0.703874\pi\)
−0.597588 + 0.801803i \(0.703874\pi\)
\(240\) 0 0
\(241\) 4830.00 1.29099 0.645493 0.763766i \(-0.276652\pi\)
0.645493 + 0.763766i \(0.276652\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 1592.00 0.415139
\(246\) 0 0
\(247\) −2160.00 −0.556427
\(248\) 0 0
\(249\) 324.000 0.0824605
\(250\) 0 0
\(251\) 5532.00 1.39114 0.695571 0.718457i \(-0.255151\pi\)
0.695571 + 0.718457i \(0.255151\pi\)
\(252\) 0 0
\(253\) −864.000 −0.214700
\(254\) 0 0
\(255\) 1488.00 0.365420
\(256\) 0 0
\(257\) −254.000 −0.0616501 −0.0308251 0.999525i \(-0.509813\pi\)
−0.0308251 + 0.999525i \(0.509813\pi\)
\(258\) 0 0
\(259\) −2736.00 −0.656397
\(260\) 0 0
\(261\) −1152.00 −0.273207
\(262\) 0 0
\(263\) −4272.00 −1.00161 −0.500804 0.865561i \(-0.666962\pi\)
−0.500804 + 0.865561i \(0.666962\pi\)
\(264\) 0 0
\(265\) −2048.00 −0.474746
\(266\) 0 0
\(267\) −2814.00 −0.644996
\(268\) 0 0
\(269\) 4544.00 1.02994 0.514968 0.857210i \(-0.327804\pi\)
0.514968 + 0.857210i \(0.327804\pi\)
\(270\) 0 0
\(271\) −2076.00 −0.465343 −0.232672 0.972555i \(-0.574747\pi\)
−0.232672 + 0.972555i \(0.574747\pi\)
\(272\) 0 0
\(273\) −720.000 −0.159620
\(274\) 0 0
\(275\) −732.000 −0.160514
\(276\) 0 0
\(277\) 484.000 0.104985 0.0524923 0.998621i \(-0.483283\pi\)
0.0524923 + 0.998621i \(0.483283\pi\)
\(278\) 0 0
\(279\) 1836.00 0.393973
\(280\) 0 0
\(281\) −406.000 −0.0861919 −0.0430960 0.999071i \(-0.513722\pi\)
−0.0430960 + 0.999071i \(0.513722\pi\)
\(282\) 0 0
\(283\) 8172.00 1.71652 0.858260 0.513216i \(-0.171546\pi\)
0.858260 + 0.513216i \(0.171546\pi\)
\(284\) 0 0
\(285\) −2592.00 −0.538726
\(286\) 0 0
\(287\) 264.000 0.0542977
\(288\) 0 0
\(289\) −1069.00 −0.217586
\(290\) 0 0
\(291\) −3834.00 −0.772347
\(292\) 0 0
\(293\) 4960.00 0.988963 0.494482 0.869188i \(-0.335358\pi\)
0.494482 + 0.869188i \(0.335358\pi\)
\(294\) 0 0
\(295\) −6624.00 −1.30734
\(296\) 0 0
\(297\) −324.000 −0.0633010
\(298\) 0 0
\(299\) −1440.00 −0.278520
\(300\) 0 0
\(301\) 2448.00 0.468772
\(302\) 0 0
\(303\) −1824.00 −0.345829
\(304\) 0 0
\(305\) 672.000 0.126159
\(306\) 0 0
\(307\) −6684.00 −1.24259 −0.621296 0.783576i \(-0.713394\pi\)
−0.621296 + 0.783576i \(0.713394\pi\)
\(308\) 0 0
\(309\) −2844.00 −0.523591
\(310\) 0 0
\(311\) −4992.00 −0.910194 −0.455097 0.890442i \(-0.650395\pi\)
−0.455097 + 0.890442i \(0.650395\pi\)
\(312\) 0 0
\(313\) −5402.00 −0.975524 −0.487762 0.872977i \(-0.662187\pi\)
−0.487762 + 0.872977i \(0.662187\pi\)
\(314\) 0 0
\(315\) −864.000 −0.154542
\(316\) 0 0
\(317\) 7856.00 1.39191 0.695957 0.718083i \(-0.254980\pi\)
0.695957 + 0.718083i \(0.254980\pi\)
\(318\) 0 0
\(319\) −1536.00 −0.269591
\(320\) 0 0
\(321\) 3132.00 0.544583
\(322\) 0 0
\(323\) −6696.00 −1.15348
\(324\) 0 0
\(325\) −1220.00 −0.208226
\(326\) 0 0
\(327\) −5340.00 −0.903066
\(328\) 0 0
\(329\) 7200.00 1.20653
\(330\) 0 0
\(331\) −3732.00 −0.619726 −0.309863 0.950781i \(-0.600283\pi\)
−0.309863 + 0.950781i \(0.600283\pi\)
\(332\) 0 0
\(333\) −2052.00 −0.337684
\(334\) 0 0
\(335\) 2784.00 0.454048
\(336\) 0 0
\(337\) −5598.00 −0.904874 −0.452437 0.891796i \(-0.649445\pi\)
−0.452437 + 0.891796i \(0.649445\pi\)
\(338\) 0 0
\(339\) 1866.00 0.298959
\(340\) 0 0
\(341\) 2448.00 0.388758
\(342\) 0 0
\(343\) −6504.00 −1.02386
\(344\) 0 0
\(345\) −1728.00 −0.269659
\(346\) 0 0
\(347\) 8220.00 1.27168 0.635840 0.771821i \(-0.280654\pi\)
0.635840 + 0.771821i \(0.280654\pi\)
\(348\) 0 0
\(349\) −11844.0 −1.81660 −0.908302 0.418315i \(-0.862621\pi\)
−0.908302 + 0.418315i \(0.862621\pi\)
\(350\) 0 0
\(351\) −540.000 −0.0821170
\(352\) 0 0
\(353\) −7006.00 −1.05635 −0.528175 0.849135i \(-0.677124\pi\)
−0.528175 + 0.849135i \(0.677124\pi\)
\(354\) 0 0
\(355\) −3648.00 −0.545396
\(356\) 0 0
\(357\) −2232.00 −0.330896
\(358\) 0 0
\(359\) −7512.00 −1.10437 −0.552184 0.833722i \(-0.686205\pi\)
−0.552184 + 0.833722i \(0.686205\pi\)
\(360\) 0 0
\(361\) 4805.00 0.700539
\(362\) 0 0
\(363\) 3561.00 0.514887
\(364\) 0 0
\(365\) 6576.00 0.943023
\(366\) 0 0
\(367\) −5076.00 −0.721976 −0.360988 0.932571i \(-0.617560\pi\)
−0.360988 + 0.932571i \(0.617560\pi\)
\(368\) 0 0
\(369\) 198.000 0.0279335
\(370\) 0 0
\(371\) 3072.00 0.429893
\(372\) 0 0
\(373\) 4860.00 0.674641 0.337321 0.941390i \(-0.390479\pi\)
0.337321 + 0.941390i \(0.390479\pi\)
\(374\) 0 0
\(375\) −4464.00 −0.614720
\(376\) 0 0
\(377\) −2560.00 −0.349726
\(378\) 0 0
\(379\) 5964.00 0.808312 0.404156 0.914690i \(-0.367565\pi\)
0.404156 + 0.914690i \(0.367565\pi\)
\(380\) 0 0
\(381\) 612.000 0.0822932
\(382\) 0 0
\(383\) 432.000 0.0576349 0.0288175 0.999585i \(-0.490826\pi\)
0.0288175 + 0.999585i \(0.490826\pi\)
\(384\) 0 0
\(385\) −1152.00 −0.152497
\(386\) 0 0
\(387\) 1836.00 0.241161
\(388\) 0 0
\(389\) 8888.00 1.15846 0.579228 0.815165i \(-0.303354\pi\)
0.579228 + 0.815165i \(0.303354\pi\)
\(390\) 0 0
\(391\) −4464.00 −0.577376
\(392\) 0 0
\(393\) 1044.00 0.134002
\(394\) 0 0
\(395\) −10848.0 −1.38183
\(396\) 0 0
\(397\) −2676.00 −0.338299 −0.169149 0.985590i \(-0.554102\pi\)
−0.169149 + 0.985590i \(0.554102\pi\)
\(398\) 0 0
\(399\) 3888.00 0.487828
\(400\) 0 0
\(401\) 13790.0 1.71731 0.858653 0.512557i \(-0.171302\pi\)
0.858653 + 0.512557i \(0.171302\pi\)
\(402\) 0 0
\(403\) 4080.00 0.504316
\(404\) 0 0
\(405\) −648.000 −0.0795046
\(406\) 0 0
\(407\) −2736.00 −0.333215
\(408\) 0 0
\(409\) 1974.00 0.238650 0.119325 0.992855i \(-0.461927\pi\)
0.119325 + 0.992855i \(0.461927\pi\)
\(410\) 0 0
\(411\) 318.000 0.0381649
\(412\) 0 0
\(413\) 9936.00 1.18382
\(414\) 0 0
\(415\) 864.000 0.102198
\(416\) 0 0
\(417\) 3564.00 0.418537
\(418\) 0 0
\(419\) −4764.00 −0.555457 −0.277729 0.960660i \(-0.589582\pi\)
−0.277729 + 0.960660i \(0.589582\pi\)
\(420\) 0 0
\(421\) −92.0000 −0.0106504 −0.00532518 0.999986i \(-0.501695\pi\)
−0.00532518 + 0.999986i \(0.501695\pi\)
\(422\) 0 0
\(423\) 5400.00 0.620702
\(424\) 0 0
\(425\) −3782.00 −0.431656
\(426\) 0 0
\(427\) −1008.00 −0.114240
\(428\) 0 0
\(429\) −720.000 −0.0810301
\(430\) 0 0
\(431\) −10488.0 −1.17213 −0.586066 0.810263i \(-0.699324\pi\)
−0.586066 + 0.810263i \(0.699324\pi\)
\(432\) 0 0
\(433\) −13138.0 −1.45813 −0.729067 0.684442i \(-0.760046\pi\)
−0.729067 + 0.684442i \(0.760046\pi\)
\(434\) 0 0
\(435\) −3072.00 −0.338600
\(436\) 0 0
\(437\) 7776.00 0.851205
\(438\) 0 0
\(439\) −3612.00 −0.392691 −0.196346 0.980535i \(-0.562907\pi\)
−0.196346 + 0.980535i \(0.562907\pi\)
\(440\) 0 0
\(441\) −1791.00 −0.193392
\(442\) 0 0
\(443\) 12972.0 1.39124 0.695619 0.718411i \(-0.255130\pi\)
0.695619 + 0.718411i \(0.255130\pi\)
\(444\) 0 0
\(445\) −7504.00 −0.799379
\(446\) 0 0
\(447\) −7944.00 −0.840578
\(448\) 0 0
\(449\) 5998.00 0.630430 0.315215 0.949020i \(-0.397923\pi\)
0.315215 + 0.949020i \(0.397923\pi\)
\(450\) 0 0
\(451\) 264.000 0.0275638
\(452\) 0 0
\(453\) −10260.0 −1.06414
\(454\) 0 0
\(455\) −1920.00 −0.197826
\(456\) 0 0
\(457\) 8934.00 0.914475 0.457237 0.889345i \(-0.348839\pi\)
0.457237 + 0.889345i \(0.348839\pi\)
\(458\) 0 0
\(459\) −1674.00 −0.170230
\(460\) 0 0
\(461\) 7448.00 0.752468 0.376234 0.926525i \(-0.377219\pi\)
0.376234 + 0.926525i \(0.377219\pi\)
\(462\) 0 0
\(463\) 4356.00 0.437236 0.218618 0.975810i \(-0.429845\pi\)
0.218618 + 0.975810i \(0.429845\pi\)
\(464\) 0 0
\(465\) 4896.00 0.488272
\(466\) 0 0
\(467\) 8580.00 0.850182 0.425091 0.905151i \(-0.360242\pi\)
0.425091 + 0.905151i \(0.360242\pi\)
\(468\) 0 0
\(469\) −4176.00 −0.411151
\(470\) 0 0
\(471\) −180.000 −0.0176093
\(472\) 0 0
\(473\) 2448.00 0.237969
\(474\) 0 0
\(475\) 6588.00 0.636375
\(476\) 0 0
\(477\) 2304.00 0.221159
\(478\) 0 0
\(479\) −12648.0 −1.20648 −0.603238 0.797561i \(-0.706123\pi\)
−0.603238 + 0.797561i \(0.706123\pi\)
\(480\) 0 0
\(481\) −4560.00 −0.432262
\(482\) 0 0
\(483\) 2592.00 0.244182
\(484\) 0 0
\(485\) −10224.0 −0.957212
\(486\) 0 0
\(487\) 15036.0 1.39907 0.699534 0.714599i \(-0.253391\pi\)
0.699534 + 0.714599i \(0.253391\pi\)
\(488\) 0 0
\(489\) −684.000 −0.0632547
\(490\) 0 0
\(491\) −15684.0 −1.44157 −0.720783 0.693161i \(-0.756218\pi\)
−0.720783 + 0.693161i \(0.756218\pi\)
\(492\) 0 0
\(493\) −7936.00 −0.724989
\(494\) 0 0
\(495\) −864.000 −0.0784523
\(496\) 0 0
\(497\) 5472.00 0.493869
\(498\) 0 0
\(499\) 16308.0 1.46302 0.731509 0.681831i \(-0.238816\pi\)
0.731509 + 0.681831i \(0.238816\pi\)
\(500\) 0 0
\(501\) −12384.0 −1.10434
\(502\) 0 0
\(503\) 10344.0 0.916931 0.458465 0.888712i \(-0.348399\pi\)
0.458465 + 0.888712i \(0.348399\pi\)
\(504\) 0 0
\(505\) −4864.00 −0.428604
\(506\) 0 0
\(507\) 5391.00 0.472234
\(508\) 0 0
\(509\) −5648.00 −0.491833 −0.245917 0.969291i \(-0.579089\pi\)
−0.245917 + 0.969291i \(0.579089\pi\)
\(510\) 0 0
\(511\) −9864.00 −0.853929
\(512\) 0 0
\(513\) 2916.00 0.250964
\(514\) 0 0
\(515\) −7584.00 −0.648915
\(516\) 0 0
\(517\) 7200.00 0.612487
\(518\) 0 0
\(519\) 4056.00 0.343042
\(520\) 0 0
\(521\) −19498.0 −1.63958 −0.819792 0.572662i \(-0.805911\pi\)
−0.819792 + 0.572662i \(0.805911\pi\)
\(522\) 0 0
\(523\) −22596.0 −1.88920 −0.944602 0.328217i \(-0.893552\pi\)
−0.944602 + 0.328217i \(0.893552\pi\)
\(524\) 0 0
\(525\) 2196.00 0.182555
\(526\) 0 0
\(527\) 12648.0 1.04546
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 7452.00 0.609019
\(532\) 0 0
\(533\) 440.000 0.0357571
\(534\) 0 0
\(535\) 8352.00 0.674932
\(536\) 0 0
\(537\) −5148.00 −0.413692
\(538\) 0 0
\(539\) −2388.00 −0.190832
\(540\) 0 0
\(541\) −812.000 −0.0645298 −0.0322649 0.999479i \(-0.510272\pi\)
−0.0322649 + 0.999479i \(0.510272\pi\)
\(542\) 0 0
\(543\) 11076.0 0.875353
\(544\) 0 0
\(545\) −14240.0 −1.11922
\(546\) 0 0
\(547\) 6132.00 0.479315 0.239658 0.970857i \(-0.422965\pi\)
0.239658 + 0.970857i \(0.422965\pi\)
\(548\) 0 0
\(549\) −756.000 −0.0587710
\(550\) 0 0
\(551\) 13824.0 1.06882
\(552\) 0 0
\(553\) 16272.0 1.25128
\(554\) 0 0
\(555\) −5472.00 −0.418511
\(556\) 0 0
\(557\) 2792.00 0.212389 0.106195 0.994345i \(-0.466133\pi\)
0.106195 + 0.994345i \(0.466133\pi\)
\(558\) 0 0
\(559\) 4080.00 0.308704
\(560\) 0 0
\(561\) −2232.00 −0.167977
\(562\) 0 0
\(563\) 6468.00 0.484181 0.242090 0.970254i \(-0.422167\pi\)
0.242090 + 0.970254i \(0.422167\pi\)
\(564\) 0 0
\(565\) 4976.00 0.370517
\(566\) 0 0
\(567\) 972.000 0.0719932
\(568\) 0 0
\(569\) −10522.0 −0.775229 −0.387614 0.921822i \(-0.626701\pi\)
−0.387614 + 0.921822i \(0.626701\pi\)
\(570\) 0 0
\(571\) 1068.00 0.0782739 0.0391370 0.999234i \(-0.487539\pi\)
0.0391370 + 0.999234i \(0.487539\pi\)
\(572\) 0 0
\(573\) 144.000 0.0104986
\(574\) 0 0
\(575\) 4392.00 0.318537
\(576\) 0 0
\(577\) 3602.00 0.259884 0.129942 0.991522i \(-0.458521\pi\)
0.129942 + 0.991522i \(0.458521\pi\)
\(578\) 0 0
\(579\) −7242.00 −0.519805
\(580\) 0 0
\(581\) −1296.00 −0.0925424
\(582\) 0 0
\(583\) 3072.00 0.218232
\(584\) 0 0
\(585\) −1440.00 −0.101772
\(586\) 0 0
\(587\) −17940.0 −1.26144 −0.630718 0.776012i \(-0.717240\pi\)
−0.630718 + 0.776012i \(0.717240\pi\)
\(588\) 0 0
\(589\) −22032.0 −1.54128
\(590\) 0 0
\(591\) −12912.0 −0.898695
\(592\) 0 0
\(593\) 18034.0 1.24885 0.624425 0.781085i \(-0.285334\pi\)
0.624425 + 0.781085i \(0.285334\pi\)
\(594\) 0 0
\(595\) −5952.00 −0.410098
\(596\) 0 0
\(597\) 612.000 0.0419556
\(598\) 0 0
\(599\) 12264.0 0.836550 0.418275 0.908320i \(-0.362635\pi\)
0.418275 + 0.908320i \(0.362635\pi\)
\(600\) 0 0
\(601\) 12634.0 0.857490 0.428745 0.903426i \(-0.358956\pi\)
0.428745 + 0.903426i \(0.358956\pi\)
\(602\) 0 0
\(603\) −3132.00 −0.211517
\(604\) 0 0
\(605\) 9496.00 0.638128
\(606\) 0 0
\(607\) −2796.00 −0.186962 −0.0934812 0.995621i \(-0.529799\pi\)
−0.0934812 + 0.995621i \(0.529799\pi\)
\(608\) 0 0
\(609\) 4608.00 0.306610
\(610\) 0 0
\(611\) 12000.0 0.794547
\(612\) 0 0
\(613\) 13788.0 0.908470 0.454235 0.890882i \(-0.349913\pi\)
0.454235 + 0.890882i \(0.349913\pi\)
\(614\) 0 0
\(615\) 528.000 0.0346195
\(616\) 0 0
\(617\) 18074.0 1.17931 0.589653 0.807657i \(-0.299264\pi\)
0.589653 + 0.807657i \(0.299264\pi\)
\(618\) 0 0
\(619\) −5940.00 −0.385701 −0.192850 0.981228i \(-0.561773\pi\)
−0.192850 + 0.981228i \(0.561773\pi\)
\(620\) 0 0
\(621\) 1944.00 0.125620
\(622\) 0 0
\(623\) 11256.0 0.723856
\(624\) 0 0
\(625\) −4279.00 −0.273856
\(626\) 0 0
\(627\) 3888.00 0.247642
\(628\) 0 0
\(629\) −14136.0 −0.896088
\(630\) 0 0
\(631\) 8700.00 0.548877 0.274439 0.961605i \(-0.411508\pi\)
0.274439 + 0.961605i \(0.411508\pi\)
\(632\) 0 0
\(633\) −12060.0 −0.757254
\(634\) 0 0
\(635\) 1632.00 0.101990
\(636\) 0 0
\(637\) −3980.00 −0.247556
\(638\) 0 0
\(639\) 4104.00 0.254072
\(640\) 0 0
\(641\) −16306.0 −1.00476 −0.502378 0.864648i \(-0.667541\pi\)
−0.502378 + 0.864648i \(0.667541\pi\)
\(642\) 0 0
\(643\) −22668.0 −1.39026 −0.695131 0.718883i \(-0.744654\pi\)
−0.695131 + 0.718883i \(0.744654\pi\)
\(644\) 0 0
\(645\) 4896.00 0.298883
\(646\) 0 0
\(647\) 11928.0 0.724788 0.362394 0.932025i \(-0.381959\pi\)
0.362394 + 0.932025i \(0.381959\pi\)
\(648\) 0 0
\(649\) 9936.00 0.600959
\(650\) 0 0
\(651\) −7344.00 −0.442141
\(652\) 0 0
\(653\) −2552.00 −0.152936 −0.0764682 0.997072i \(-0.524364\pi\)
−0.0764682 + 0.997072i \(0.524364\pi\)
\(654\) 0 0
\(655\) 2784.00 0.166076
\(656\) 0 0
\(657\) −7398.00 −0.439305
\(658\) 0 0
\(659\) 2196.00 0.129809 0.0649044 0.997891i \(-0.479326\pi\)
0.0649044 + 0.997891i \(0.479326\pi\)
\(660\) 0 0
\(661\) −4260.00 −0.250673 −0.125336 0.992114i \(-0.540001\pi\)
−0.125336 + 0.992114i \(0.540001\pi\)
\(662\) 0 0
\(663\) −3720.00 −0.217908
\(664\) 0 0
\(665\) 10368.0 0.604592
\(666\) 0 0
\(667\) 9216.00 0.535000
\(668\) 0 0
\(669\) −1548.00 −0.0894606
\(670\) 0 0
\(671\) −1008.00 −0.0579932
\(672\) 0 0
\(673\) −2018.00 −0.115584 −0.0577921 0.998329i \(-0.518406\pi\)
−0.0577921 + 0.998329i \(0.518406\pi\)
\(674\) 0 0
\(675\) 1647.00 0.0939156
\(676\) 0 0
\(677\) 9256.00 0.525461 0.262730 0.964869i \(-0.415377\pi\)
0.262730 + 0.964869i \(0.415377\pi\)
\(678\) 0 0
\(679\) 15336.0 0.866777
\(680\) 0 0
\(681\) −4284.00 −0.241062
\(682\) 0 0
\(683\) 29244.0 1.63835 0.819173 0.573546i \(-0.194433\pi\)
0.819173 + 0.573546i \(0.194433\pi\)
\(684\) 0 0
\(685\) 848.000 0.0472999
\(686\) 0 0
\(687\) 18084.0 1.00429
\(688\) 0 0
\(689\) 5120.00 0.283101
\(690\) 0 0
\(691\) 3684.00 0.202816 0.101408 0.994845i \(-0.467665\pi\)
0.101408 + 0.994845i \(0.467665\pi\)
\(692\) 0 0
\(693\) 1296.00 0.0710404
\(694\) 0 0
\(695\) 9504.00 0.518715
\(696\) 0 0
\(697\) 1364.00 0.0741251
\(698\) 0 0
\(699\) 7890.00 0.426934
\(700\) 0 0
\(701\) 13456.0 0.725002 0.362501 0.931983i \(-0.381923\pi\)
0.362501 + 0.931983i \(0.381923\pi\)
\(702\) 0 0
\(703\) 24624.0 1.32107
\(704\) 0 0
\(705\) 14400.0 0.769270
\(706\) 0 0
\(707\) 7296.00 0.388111
\(708\) 0 0
\(709\) −6460.00 −0.342187 −0.171093 0.985255i \(-0.554730\pi\)
−0.171093 + 0.985255i \(0.554730\pi\)
\(710\) 0 0
\(711\) 12204.0 0.643721
\(712\) 0 0
\(713\) −14688.0 −0.771487
\(714\) 0 0
\(715\) −1920.00 −0.100425
\(716\) 0 0
\(717\) 13248.0 0.690036
\(718\) 0 0
\(719\) 17160.0 0.890070 0.445035 0.895513i \(-0.353191\pi\)
0.445035 + 0.895513i \(0.353191\pi\)
\(720\) 0 0
\(721\) 11376.0 0.587607
\(722\) 0 0
\(723\) −14490.0 −0.745351
\(724\) 0 0
\(725\) 7808.00 0.399975
\(726\) 0 0
\(727\) 11820.0 0.602998 0.301499 0.953466i \(-0.402513\pi\)
0.301499 + 0.953466i \(0.402513\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 12648.0 0.639950
\(732\) 0 0
\(733\) 23924.0 1.20553 0.602765 0.797919i \(-0.294066\pi\)
0.602765 + 0.797919i \(0.294066\pi\)
\(734\) 0 0
\(735\) −4776.00 −0.239681
\(736\) 0 0
\(737\) −4176.00 −0.208718
\(738\) 0 0
\(739\) 11796.0 0.587176 0.293588 0.955932i \(-0.405151\pi\)
0.293588 + 0.955932i \(0.405151\pi\)
\(740\) 0 0
\(741\) 6480.00 0.321253
\(742\) 0 0
\(743\) −27024.0 −1.33434 −0.667170 0.744906i \(-0.732494\pi\)
−0.667170 + 0.744906i \(0.732494\pi\)
\(744\) 0 0
\(745\) −21184.0 −1.04177
\(746\) 0 0
\(747\) −972.000 −0.0476086
\(748\) 0 0
\(749\) −12528.0 −0.611166
\(750\) 0 0
\(751\) 17340.0 0.842537 0.421269 0.906936i \(-0.361585\pi\)
0.421269 + 0.906936i \(0.361585\pi\)
\(752\) 0 0
\(753\) −16596.0 −0.803176
\(754\) 0 0
\(755\) −27360.0 −1.31885
\(756\) 0 0
\(757\) 27236.0 1.30767 0.653837 0.756635i \(-0.273158\pi\)
0.653837 + 0.756635i \(0.273158\pi\)
\(758\) 0 0
\(759\) 2592.00 0.123957
\(760\) 0 0
\(761\) 14758.0 0.702992 0.351496 0.936189i \(-0.385673\pi\)
0.351496 + 0.936189i \(0.385673\pi\)
\(762\) 0 0
\(763\) 21360.0 1.01348
\(764\) 0 0
\(765\) −4464.00 −0.210976
\(766\) 0 0
\(767\) 16560.0 0.779592
\(768\) 0 0
\(769\) 25774.0 1.20863 0.604314 0.796747i \(-0.293447\pi\)
0.604314 + 0.796747i \(0.293447\pi\)
\(770\) 0 0
\(771\) 762.000 0.0355937
\(772\) 0 0
\(773\) −37424.0 −1.74133 −0.870665 0.491877i \(-0.836311\pi\)
−0.870665 + 0.491877i \(0.836311\pi\)
\(774\) 0 0
\(775\) −12444.0 −0.576776
\(776\) 0 0
\(777\) 8208.00 0.378971
\(778\) 0 0
\(779\) −2376.00 −0.109280
\(780\) 0 0
\(781\) 5472.00 0.250709
\(782\) 0 0
\(783\) 3456.00 0.157736
\(784\) 0 0
\(785\) −480.000 −0.0218241
\(786\) 0 0
\(787\) 12804.0 0.579941 0.289970 0.957036i \(-0.406355\pi\)
0.289970 + 0.957036i \(0.406355\pi\)
\(788\) 0 0
\(789\) 12816.0 0.578278
\(790\) 0 0
\(791\) −7464.00 −0.335511
\(792\) 0 0
\(793\) −1680.00 −0.0752315
\(794\) 0 0
\(795\) 6144.00 0.274095
\(796\) 0 0
\(797\) −32024.0 −1.42327 −0.711636 0.702548i \(-0.752046\pi\)
−0.711636 + 0.702548i \(0.752046\pi\)
\(798\) 0 0
\(799\) 37200.0 1.64711
\(800\) 0 0
\(801\) 8442.00 0.372389
\(802\) 0 0
\(803\) −9864.00 −0.433491
\(804\) 0 0
\(805\) 6912.00 0.302629
\(806\) 0 0
\(807\) −13632.0 −0.594633
\(808\) 0 0
\(809\) −38090.0 −1.65534 −0.827672 0.561212i \(-0.810335\pi\)
−0.827672 + 0.561212i \(0.810335\pi\)
\(810\) 0 0
\(811\) 10428.0 0.451512 0.225756 0.974184i \(-0.427515\pi\)
0.225756 + 0.974184i \(0.427515\pi\)
\(812\) 0 0
\(813\) 6228.00 0.268666
\(814\) 0 0
\(815\) −1824.00 −0.0783950
\(816\) 0 0
\(817\) −22032.0 −0.943454
\(818\) 0 0
\(819\) 2160.00 0.0921569
\(820\) 0 0
\(821\) −7984.00 −0.339395 −0.169698 0.985496i \(-0.554279\pi\)
−0.169698 + 0.985496i \(0.554279\pi\)
\(822\) 0 0
\(823\) −28788.0 −1.21930 −0.609652 0.792669i \(-0.708691\pi\)
−0.609652 + 0.792669i \(0.708691\pi\)
\(824\) 0 0
\(825\) 2196.00 0.0926726
\(826\) 0 0
\(827\) −468.000 −0.0196783 −0.00983915 0.999952i \(-0.503132\pi\)
−0.00983915 + 0.999952i \(0.503132\pi\)
\(828\) 0 0
\(829\) 28852.0 1.20877 0.604386 0.796692i \(-0.293419\pi\)
0.604386 + 0.796692i \(0.293419\pi\)
\(830\) 0 0
\(831\) −1452.00 −0.0606129
\(832\) 0 0
\(833\) −12338.0 −0.513189
\(834\) 0 0
\(835\) −33024.0 −1.36867
\(836\) 0 0
\(837\) −5508.00 −0.227460
\(838\) 0 0
\(839\) 1944.00 0.0799932 0.0399966 0.999200i \(-0.487265\pi\)
0.0399966 + 0.999200i \(0.487265\pi\)
\(840\) 0 0
\(841\) −8005.00 −0.328222
\(842\) 0 0
\(843\) 1218.00 0.0497629
\(844\) 0 0
\(845\) 14376.0 0.585266
\(846\) 0 0
\(847\) −14244.0 −0.577839
\(848\) 0 0
\(849\) −24516.0 −0.991033
\(850\) 0 0
\(851\) 16416.0 0.661261
\(852\) 0 0
\(853\) −37044.0 −1.48694 −0.743472 0.668768i \(-0.766822\pi\)
−0.743472 + 0.668768i \(0.766822\pi\)
\(854\) 0 0
\(855\) 7776.00 0.311033
\(856\) 0 0
\(857\) 15046.0 0.599722 0.299861 0.953983i \(-0.403060\pi\)
0.299861 + 0.953983i \(0.403060\pi\)
\(858\) 0 0
\(859\) −12180.0 −0.483791 −0.241895 0.970302i \(-0.577769\pi\)
−0.241895 + 0.970302i \(0.577769\pi\)
\(860\) 0 0
\(861\) −792.000 −0.0313488
\(862\) 0 0
\(863\) 28752.0 1.13410 0.567051 0.823683i \(-0.308084\pi\)
0.567051 + 0.823683i \(0.308084\pi\)
\(864\) 0 0
\(865\) 10816.0 0.425150
\(866\) 0 0
\(867\) 3207.00 0.125623
\(868\) 0 0
\(869\) 16272.0 0.635201
\(870\) 0 0
\(871\) −6960.00 −0.270758
\(872\) 0 0
\(873\) 11502.0 0.445915
\(874\) 0 0
\(875\) 17856.0 0.689878
\(876\) 0 0
\(877\) 31884.0 1.22765 0.613823 0.789443i \(-0.289631\pi\)
0.613823 + 0.789443i \(0.289631\pi\)
\(878\) 0 0
\(879\) −14880.0 −0.570978
\(880\) 0 0
\(881\) 30802.0 1.17792 0.588959 0.808163i \(-0.299538\pi\)
0.588959 + 0.808163i \(0.299538\pi\)
\(882\) 0 0
\(883\) −32460.0 −1.23711 −0.618554 0.785742i \(-0.712281\pi\)
−0.618554 + 0.785742i \(0.712281\pi\)
\(884\) 0 0
\(885\) 19872.0 0.754791
\(886\) 0 0
\(887\) 14832.0 0.561454 0.280727 0.959788i \(-0.409424\pi\)
0.280727 + 0.959788i \(0.409424\pi\)
\(888\) 0 0
\(889\) −2448.00 −0.0923547
\(890\) 0 0
\(891\) 972.000 0.0365468
\(892\) 0 0
\(893\) −64800.0 −2.42827
\(894\) 0 0
\(895\) −13728.0 −0.512711
\(896\) 0 0
\(897\) 4320.00 0.160803
\(898\) 0 0
\(899\) −26112.0 −0.968725
\(900\) 0 0
\(901\) 15872.0 0.586873
\(902\) 0 0
\(903\) −7344.00 −0.270646
\(904\) 0 0
\(905\) 29536.0 1.08487
\(906\) 0 0
\(907\) −6900.00 −0.252603 −0.126301 0.991992i \(-0.540311\pi\)
−0.126301 + 0.991992i \(0.540311\pi\)
\(908\) 0 0
\(909\) 5472.00 0.199664
\(910\) 0 0
\(911\) −32832.0 −1.19404 −0.597021 0.802225i \(-0.703649\pi\)
−0.597021 + 0.802225i \(0.703649\pi\)
\(912\) 0 0
\(913\) −1296.00 −0.0469785
\(914\) 0 0
\(915\) −2016.00 −0.0728381
\(916\) 0 0
\(917\) −4176.00 −0.150386
\(918\) 0 0
\(919\) −8340.00 −0.299359 −0.149680 0.988735i \(-0.547824\pi\)
−0.149680 + 0.988735i \(0.547824\pi\)
\(920\) 0 0
\(921\) 20052.0 0.717411
\(922\) 0 0
\(923\) 9120.00 0.325231
\(924\) 0 0
\(925\) 13908.0 0.494370
\(926\) 0 0
\(927\) 8532.00 0.302295
\(928\) 0 0
\(929\) −39826.0 −1.40651 −0.703255 0.710937i \(-0.748271\pi\)
−0.703255 + 0.710937i \(0.748271\pi\)
\(930\) 0 0
\(931\) 21492.0 0.756576
\(932\) 0 0
\(933\) 14976.0 0.525501
\(934\) 0 0
\(935\) −5952.00 −0.208183
\(936\) 0 0
\(937\) −28550.0 −0.995398 −0.497699 0.867350i \(-0.665822\pi\)
−0.497699 + 0.867350i \(0.665822\pi\)
\(938\) 0 0
\(939\) 16206.0 0.563219
\(940\) 0 0
\(941\) −50632.0 −1.75404 −0.877022 0.480450i \(-0.840473\pi\)
−0.877022 + 0.480450i \(0.840473\pi\)
\(942\) 0 0
\(943\) −1584.00 −0.0547000
\(944\) 0 0
\(945\) 2592.00 0.0892251
\(946\) 0 0
\(947\) −18204.0 −0.624657 −0.312329 0.949974i \(-0.601109\pi\)
−0.312329 + 0.949974i \(0.601109\pi\)
\(948\) 0 0
\(949\) −16440.0 −0.562345
\(950\) 0 0
\(951\) −23568.0 −0.803622
\(952\) 0 0
\(953\) 4934.00 0.167710 0.0838552 0.996478i \(-0.473277\pi\)
0.0838552 + 0.996478i \(0.473277\pi\)
\(954\) 0 0
\(955\) 384.000 0.0130115
\(956\) 0 0
\(957\) 4608.00 0.155648
\(958\) 0 0
\(959\) −1272.00 −0.0428311
\(960\) 0 0
\(961\) 11825.0 0.396932
\(962\) 0 0
\(963\) −9396.00 −0.314415
\(964\) 0 0
\(965\) −19312.0 −0.644223
\(966\) 0 0
\(967\) 13284.0 0.441763 0.220881 0.975301i \(-0.429107\pi\)
0.220881 + 0.975301i \(0.429107\pi\)
\(968\) 0 0
\(969\) 20088.0 0.665964
\(970\) 0 0
\(971\) −50820.0 −1.67960 −0.839800 0.542896i \(-0.817328\pi\)
−0.839800 + 0.542896i \(0.817328\pi\)
\(972\) 0 0
\(973\) −14256.0 −0.469709
\(974\) 0 0
\(975\) 3660.00 0.120219
\(976\) 0 0
\(977\) 11038.0 0.361450 0.180725 0.983534i \(-0.442156\pi\)
0.180725 + 0.983534i \(0.442156\pi\)
\(978\) 0 0
\(979\) 11256.0 0.367460
\(980\) 0 0
\(981\) 16020.0 0.521386
\(982\) 0 0
\(983\) −44112.0 −1.43129 −0.715643 0.698466i \(-0.753866\pi\)
−0.715643 + 0.698466i \(0.753866\pi\)
\(984\) 0 0
\(985\) −34432.0 −1.11380
\(986\) 0 0
\(987\) −21600.0 −0.696591
\(988\) 0 0
\(989\) −14688.0 −0.472246
\(990\) 0 0
\(991\) −56196.0 −1.80134 −0.900668 0.434507i \(-0.856923\pi\)
−0.900668 + 0.434507i \(0.856923\pi\)
\(992\) 0 0
\(993\) 11196.0 0.357799
\(994\) 0 0
\(995\) 1632.00 0.0519979
\(996\) 0 0
\(997\) −45588.0 −1.44813 −0.724065 0.689731i \(-0.757729\pi\)
−0.724065 + 0.689731i \(0.757729\pi\)
\(998\) 0 0
\(999\) 6156.00 0.194962
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 768.4.a.a.1.1 1
3.2 odd 2 2304.4.a.l.1.1 1
4.3 odd 2 768.4.a.c.1.1 1
8.3 odd 2 768.4.a.b.1.1 1
8.5 even 2 768.4.a.d.1.1 1
12.11 even 2 2304.4.a.k.1.1 1
16.3 odd 4 384.4.d.b.193.2 yes 2
16.5 even 4 384.4.d.a.193.2 yes 2
16.11 odd 4 384.4.d.b.193.1 yes 2
16.13 even 4 384.4.d.a.193.1 2
24.5 odd 2 2304.4.a.f.1.1 1
24.11 even 2 2304.4.a.e.1.1 1
48.5 odd 4 1152.4.d.b.577.2 2
48.11 even 4 1152.4.d.g.577.2 2
48.29 odd 4 1152.4.d.b.577.1 2
48.35 even 4 1152.4.d.g.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.a.193.1 2 16.13 even 4
384.4.d.a.193.2 yes 2 16.5 even 4
384.4.d.b.193.1 yes 2 16.11 odd 4
384.4.d.b.193.2 yes 2 16.3 odd 4
768.4.a.a.1.1 1 1.1 even 1 trivial
768.4.a.b.1.1 1 8.3 odd 2
768.4.a.c.1.1 1 4.3 odd 2
768.4.a.d.1.1 1 8.5 even 2
1152.4.d.b.577.1 2 48.29 odd 4
1152.4.d.b.577.2 2 48.5 odd 4
1152.4.d.g.577.1 2 48.35 even 4
1152.4.d.g.577.2 2 48.11 even 4
2304.4.a.e.1.1 1 24.11 even 2
2304.4.a.f.1.1 1 24.5 odd 2
2304.4.a.k.1.1 1 12.11 even 2
2304.4.a.l.1.1 1 3.2 odd 2