Properties

Label 4560.2.a.bh.1.1
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -0.732051 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -0.732051 q^{7} +1.00000 q^{9} -1.26795 q^{11} -2.73205 q^{13} -1.00000 q^{15} -1.00000 q^{19} +0.732051 q^{21} +3.46410 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.19615 q^{29} +4.92820 q^{31} +1.26795 q^{33} -0.732051 q^{35} +4.19615 q^{37} +2.73205 q^{39} +4.73205 q^{41} +6.19615 q^{43} +1.00000 q^{45} -3.46410 q^{47} -6.46410 q^{49} -2.53590 q^{53} -1.26795 q^{55} +1.00000 q^{57} -9.46410 q^{59} -13.4641 q^{61} -0.732051 q^{63} -2.73205 q^{65} -8.00000 q^{67} -3.46410 q^{69} -16.3923 q^{71} -3.07180 q^{73} -1.00000 q^{75} +0.928203 q^{77} -2.92820 q^{79} +1.00000 q^{81} -0.928203 q^{83} +2.19615 q^{87} +7.26795 q^{89} +2.00000 q^{91} -4.92820 q^{93} -1.00000 q^{95} +4.19615 q^{97} -1.26795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} - 2 q^{19} - 2 q^{21} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 4 q^{31} + 6 q^{33} + 2 q^{35} - 2 q^{37} + 2 q^{39} + 6 q^{41} + 2 q^{43} + 2 q^{45} - 6 q^{49} - 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 20 q^{61} + 2 q^{63} - 2 q^{65} - 16 q^{67} - 12 q^{71} - 20 q^{73} - 2 q^{75} - 12 q^{77} + 8 q^{79} + 2 q^{81} + 12 q^{83} - 6 q^{87} + 18 q^{89} + 4 q^{91} + 4 q^{93} - 2 q^{95} - 2 q^{97} - 6 q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.732051 −0.276689 −0.138345 0.990384i \(-0.544178\pi\)
−0.138345 + 0.990384i \(0.544178\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.26795 −0.382301 −0.191151 0.981561i \(-0.561222\pi\)
−0.191151 + 0.981561i \(0.561222\pi\)
\(12\) 0 0
\(13\) −2.73205 −0.757735 −0.378867 0.925451i \(-0.623686\pi\)
−0.378867 + 0.925451i \(0.623686\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.732051 0.159747
\(22\) 0 0
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.19615 −0.407815 −0.203908 0.978990i \(-0.565364\pi\)
−0.203908 + 0.978990i \(0.565364\pi\)
\(30\) 0 0
\(31\) 4.92820 0.885131 0.442566 0.896736i \(-0.354068\pi\)
0.442566 + 0.896736i \(0.354068\pi\)
\(32\) 0 0
\(33\) 1.26795 0.220722
\(34\) 0 0
\(35\) −0.732051 −0.123739
\(36\) 0 0
\(37\) 4.19615 0.689843 0.344922 0.938631i \(-0.387905\pi\)
0.344922 + 0.938631i \(0.387905\pi\)
\(38\) 0 0
\(39\) 2.73205 0.437478
\(40\) 0 0
\(41\) 4.73205 0.739022 0.369511 0.929226i \(-0.379525\pi\)
0.369511 + 0.929226i \(0.379525\pi\)
\(42\) 0 0
\(43\) 6.19615 0.944904 0.472452 0.881356i \(-0.343369\pi\)
0.472452 + 0.881356i \(0.343369\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −3.46410 −0.505291 −0.252646 0.967559i \(-0.581301\pi\)
−0.252646 + 0.967559i \(0.581301\pi\)
\(48\) 0 0
\(49\) −6.46410 −0.923443
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.53590 −0.348332 −0.174166 0.984716i \(-0.555723\pi\)
−0.174166 + 0.984716i \(0.555723\pi\)
\(54\) 0 0
\(55\) −1.26795 −0.170970
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −9.46410 −1.23212 −0.616061 0.787699i \(-0.711272\pi\)
−0.616061 + 0.787699i \(0.711272\pi\)
\(60\) 0 0
\(61\) −13.4641 −1.72390 −0.861951 0.506992i \(-0.830757\pi\)
−0.861951 + 0.506992i \(0.830757\pi\)
\(62\) 0 0
\(63\) −0.732051 −0.0922297
\(64\) 0 0
\(65\) −2.73205 −0.338869
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −3.46410 −0.417029
\(70\) 0 0
\(71\) −16.3923 −1.94541 −0.972704 0.232048i \(-0.925457\pi\)
−0.972704 + 0.232048i \(0.925457\pi\)
\(72\) 0 0
\(73\) −3.07180 −0.359527 −0.179763 0.983710i \(-0.557533\pi\)
−0.179763 + 0.983710i \(0.557533\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0.928203 0.105779
\(78\) 0 0
\(79\) −2.92820 −0.329449 −0.164724 0.986340i \(-0.552673\pi\)
−0.164724 + 0.986340i \(0.552673\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.928203 −0.101884 −0.0509418 0.998702i \(-0.516222\pi\)
−0.0509418 + 0.998702i \(0.516222\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.19615 0.235452
\(88\) 0 0
\(89\) 7.26795 0.770401 0.385201 0.922833i \(-0.374132\pi\)
0.385201 + 0.922833i \(0.374132\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −4.92820 −0.511031
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 4.19615 0.426055 0.213027 0.977046i \(-0.431668\pi\)
0.213027 + 0.977046i \(0.431668\pi\)
\(98\) 0 0
\(99\) −1.26795 −0.127434
\(100\) 0 0
\(101\) 10.3923 1.03407 0.517036 0.855963i \(-0.327035\pi\)
0.517036 + 0.855963i \(0.327035\pi\)
\(102\) 0 0
\(103\) 17.8564 1.75944 0.879722 0.475488i \(-0.157729\pi\)
0.879722 + 0.475488i \(0.157729\pi\)
\(104\) 0 0
\(105\) 0.732051 0.0714408
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 6.39230 0.612272 0.306136 0.951988i \(-0.400964\pi\)
0.306136 + 0.951988i \(0.400964\pi\)
\(110\) 0 0
\(111\) −4.19615 −0.398281
\(112\) 0 0
\(113\) 5.07180 0.477115 0.238557 0.971128i \(-0.423326\pi\)
0.238557 + 0.971128i \(0.423326\pi\)
\(114\) 0 0
\(115\) 3.46410 0.323029
\(116\) 0 0
\(117\) −2.73205 −0.252578
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −9.39230 −0.853846
\(122\) 0 0
\(123\) −4.73205 −0.426675
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) −6.19615 −0.545541
\(130\) 0 0
\(131\) −15.1244 −1.32142 −0.660711 0.750641i \(-0.729745\pi\)
−0.660711 + 0.750641i \(0.729745\pi\)
\(132\) 0 0
\(133\) 0.732051 0.0634769
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −7.85641 −0.671218 −0.335609 0.942001i \(-0.608942\pi\)
−0.335609 + 0.942001i \(0.608942\pi\)
\(138\) 0 0
\(139\) −12.3923 −1.05110 −0.525551 0.850762i \(-0.676141\pi\)
−0.525551 + 0.850762i \(0.676141\pi\)
\(140\) 0 0
\(141\) 3.46410 0.291730
\(142\) 0 0
\(143\) 3.46410 0.289683
\(144\) 0 0
\(145\) −2.19615 −0.182381
\(146\) 0 0
\(147\) 6.46410 0.533150
\(148\) 0 0
\(149\) 7.85641 0.643622 0.321811 0.946804i \(-0.395708\pi\)
0.321811 + 0.946804i \(0.395708\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.92820 0.395843
\(156\) 0 0
\(157\) −14.3923 −1.14863 −0.574315 0.818634i \(-0.694732\pi\)
−0.574315 + 0.818634i \(0.694732\pi\)
\(158\) 0 0
\(159\) 2.53590 0.201110
\(160\) 0 0
\(161\) −2.53590 −0.199857
\(162\) 0 0
\(163\) −12.7321 −0.997251 −0.498626 0.866817i \(-0.666162\pi\)
−0.498626 + 0.866817i \(0.666162\pi\)
\(164\) 0 0
\(165\) 1.26795 0.0987097
\(166\) 0 0
\(167\) 3.46410 0.268060 0.134030 0.990977i \(-0.457208\pi\)
0.134030 + 0.990977i \(0.457208\pi\)
\(168\) 0 0
\(169\) −5.53590 −0.425838
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −6.92820 −0.526742 −0.263371 0.964695i \(-0.584834\pi\)
−0.263371 + 0.964695i \(0.584834\pi\)
\(174\) 0 0
\(175\) −0.732051 −0.0553378
\(176\) 0 0
\(177\) 9.46410 0.711365
\(178\) 0 0
\(179\) 11.3205 0.846135 0.423067 0.906098i \(-0.360953\pi\)
0.423067 + 0.906098i \(0.360953\pi\)
\(180\) 0 0
\(181\) 18.3923 1.36709 0.683545 0.729909i \(-0.260437\pi\)
0.683545 + 0.729909i \(0.260437\pi\)
\(182\) 0 0
\(183\) 13.4641 0.995295
\(184\) 0 0
\(185\) 4.19615 0.308507
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.732051 0.0532489
\(190\) 0 0
\(191\) −17.6603 −1.27785 −0.638926 0.769269i \(-0.720621\pi\)
−0.638926 + 0.769269i \(0.720621\pi\)
\(192\) 0 0
\(193\) −7.12436 −0.512822 −0.256411 0.966568i \(-0.582540\pi\)
−0.256411 + 0.966568i \(0.582540\pi\)
\(194\) 0 0
\(195\) 2.73205 0.195646
\(196\) 0 0
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 0 0
\(199\) −19.3205 −1.36959 −0.684797 0.728734i \(-0.740109\pi\)
−0.684797 + 0.728734i \(0.740109\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 1.60770 0.112838
\(204\) 0 0
\(205\) 4.73205 0.330501
\(206\) 0 0
\(207\) 3.46410 0.240772
\(208\) 0 0
\(209\) 1.26795 0.0877059
\(210\) 0 0
\(211\) −14.9282 −1.02770 −0.513850 0.857880i \(-0.671781\pi\)
−0.513850 + 0.857880i \(0.671781\pi\)
\(212\) 0 0
\(213\) 16.3923 1.12318
\(214\) 0 0
\(215\) 6.19615 0.422574
\(216\) 0 0
\(217\) −3.60770 −0.244906
\(218\) 0 0
\(219\) 3.07180 0.207573
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −9.85641 −0.660034 −0.330017 0.943975i \(-0.607054\pi\)
−0.330017 + 0.943975i \(0.607054\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 10.3923 0.689761 0.344881 0.938647i \(-0.387919\pi\)
0.344881 + 0.938647i \(0.387919\pi\)
\(228\) 0 0
\(229\) −25.4641 −1.68272 −0.841358 0.540479i \(-0.818243\pi\)
−0.841358 + 0.540479i \(0.818243\pi\)
\(230\) 0 0
\(231\) −0.928203 −0.0610713
\(232\) 0 0
\(233\) 19.8564 1.30084 0.650418 0.759576i \(-0.274594\pi\)
0.650418 + 0.759576i \(0.274594\pi\)
\(234\) 0 0
\(235\) −3.46410 −0.225973
\(236\) 0 0
\(237\) 2.92820 0.190207
\(238\) 0 0
\(239\) 20.1962 1.30638 0.653190 0.757194i \(-0.273430\pi\)
0.653190 + 0.757194i \(0.273430\pi\)
\(240\) 0 0
\(241\) −16.9282 −1.09044 −0.545221 0.838293i \(-0.683554\pi\)
−0.545221 + 0.838293i \(0.683554\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −6.46410 −0.412976
\(246\) 0 0
\(247\) 2.73205 0.173836
\(248\) 0 0
\(249\) 0.928203 0.0588225
\(250\) 0 0
\(251\) 10.0526 0.634512 0.317256 0.948340i \(-0.397239\pi\)
0.317256 + 0.948340i \(0.397239\pi\)
\(252\) 0 0
\(253\) −4.39230 −0.276142
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 0 0
\(259\) −3.07180 −0.190872
\(260\) 0 0
\(261\) −2.19615 −0.135938
\(262\) 0 0
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) −2.53590 −0.155779
\(266\) 0 0
\(267\) −7.26795 −0.444791
\(268\) 0 0
\(269\) −30.5885 −1.86501 −0.932506 0.361156i \(-0.882382\pi\)
−0.932506 + 0.361156i \(0.882382\pi\)
\(270\) 0 0
\(271\) 20.3923 1.23874 0.619372 0.785098i \(-0.287387\pi\)
0.619372 + 0.785098i \(0.287387\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) −1.26795 −0.0764602
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 4.92820 0.295044
\(280\) 0 0
\(281\) 4.73205 0.282290 0.141145 0.989989i \(-0.454922\pi\)
0.141145 + 0.989989i \(0.454922\pi\)
\(282\) 0 0
\(283\) 26.9808 1.60384 0.801920 0.597432i \(-0.203812\pi\)
0.801920 + 0.597432i \(0.203812\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) −3.46410 −0.204479
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −4.19615 −0.245983
\(292\) 0 0
\(293\) −27.7128 −1.61900 −0.809500 0.587120i \(-0.800262\pi\)
−0.809500 + 0.587120i \(0.800262\pi\)
\(294\) 0 0
\(295\) −9.46410 −0.551021
\(296\) 0 0
\(297\) 1.26795 0.0735739
\(298\) 0 0
\(299\) −9.46410 −0.547323
\(300\) 0 0
\(301\) −4.53590 −0.261445
\(302\) 0 0
\(303\) −10.3923 −0.597022
\(304\) 0 0
\(305\) −13.4641 −0.770952
\(306\) 0 0
\(307\) 11.6077 0.662486 0.331243 0.943545i \(-0.392532\pi\)
0.331243 + 0.943545i \(0.392532\pi\)
\(308\) 0 0
\(309\) −17.8564 −1.01582
\(310\) 0 0
\(311\) 26.4449 1.49955 0.749775 0.661692i \(-0.230162\pi\)
0.749775 + 0.661692i \(0.230162\pi\)
\(312\) 0 0
\(313\) −14.3923 −0.813501 −0.406751 0.913539i \(-0.633338\pi\)
−0.406751 + 0.913539i \(0.633338\pi\)
\(314\) 0 0
\(315\) −0.732051 −0.0412464
\(316\) 0 0
\(317\) 23.3205 1.30981 0.654905 0.755711i \(-0.272709\pi\)
0.654905 + 0.755711i \(0.272709\pi\)
\(318\) 0 0
\(319\) 2.78461 0.155908
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −2.73205 −0.151547
\(326\) 0 0
\(327\) −6.39230 −0.353495
\(328\) 0 0
\(329\) 2.53590 0.139809
\(330\) 0 0
\(331\) −29.7128 −1.63316 −0.816582 0.577230i \(-0.804134\pi\)
−0.816582 + 0.577230i \(0.804134\pi\)
\(332\) 0 0
\(333\) 4.19615 0.229948
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −19.1244 −1.04177 −0.520885 0.853627i \(-0.674398\pi\)
−0.520885 + 0.853627i \(0.674398\pi\)
\(338\) 0 0
\(339\) −5.07180 −0.275462
\(340\) 0 0
\(341\) −6.24871 −0.338387
\(342\) 0 0
\(343\) 9.85641 0.532196
\(344\) 0 0
\(345\) −3.46410 −0.186501
\(346\) 0 0
\(347\) −12.9282 −0.694022 −0.347011 0.937861i \(-0.612803\pi\)
−0.347011 + 0.937861i \(0.612803\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 2.73205 0.145826
\(352\) 0 0
\(353\) 26.7846 1.42560 0.712800 0.701367i \(-0.247427\pi\)
0.712800 + 0.701367i \(0.247427\pi\)
\(354\) 0 0
\(355\) −16.3923 −0.870013
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −17.6603 −0.932073 −0.466036 0.884766i \(-0.654318\pi\)
−0.466036 + 0.884766i \(0.654318\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 9.39230 0.492968
\(364\) 0 0
\(365\) −3.07180 −0.160785
\(366\) 0 0
\(367\) −5.80385 −0.302958 −0.151479 0.988460i \(-0.548404\pi\)
−0.151479 + 0.988460i \(0.548404\pi\)
\(368\) 0 0
\(369\) 4.73205 0.246341
\(370\) 0 0
\(371\) 1.85641 0.0963798
\(372\) 0 0
\(373\) 4.19615 0.217269 0.108634 0.994082i \(-0.465352\pi\)
0.108634 + 0.994082i \(0.465352\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) −7.07180 −0.363254 −0.181627 0.983368i \(-0.558136\pi\)
−0.181627 + 0.983368i \(0.558136\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 30.9282 1.58036 0.790179 0.612877i \(-0.209988\pi\)
0.790179 + 0.612877i \(0.209988\pi\)
\(384\) 0 0
\(385\) 0.928203 0.0473056
\(386\) 0 0
\(387\) 6.19615 0.314968
\(388\) 0 0
\(389\) −19.8564 −1.00676 −0.503380 0.864065i \(-0.667910\pi\)
−0.503380 + 0.864065i \(0.667910\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 15.1244 0.762923
\(394\) 0 0
\(395\) −2.92820 −0.147334
\(396\) 0 0
\(397\) −4.92820 −0.247339 −0.123670 0.992323i \(-0.539466\pi\)
−0.123670 + 0.992323i \(0.539466\pi\)
\(398\) 0 0
\(399\) −0.732051 −0.0366484
\(400\) 0 0
\(401\) 4.05256 0.202375 0.101188 0.994867i \(-0.467736\pi\)
0.101188 + 0.994867i \(0.467736\pi\)
\(402\) 0 0
\(403\) −13.4641 −0.670695
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −5.32051 −0.263728
\(408\) 0 0
\(409\) −5.60770 −0.277283 −0.138641 0.990343i \(-0.544274\pi\)
−0.138641 + 0.990343i \(0.544274\pi\)
\(410\) 0 0
\(411\) 7.85641 0.387528
\(412\) 0 0
\(413\) 6.92820 0.340915
\(414\) 0 0
\(415\) −0.928203 −0.0455637
\(416\) 0 0
\(417\) 12.3923 0.606854
\(418\) 0 0
\(419\) −10.0526 −0.491100 −0.245550 0.969384i \(-0.578968\pi\)
−0.245550 + 0.969384i \(0.578968\pi\)
\(420\) 0 0
\(421\) 22.7846 1.11045 0.555227 0.831699i \(-0.312631\pi\)
0.555227 + 0.831699i \(0.312631\pi\)
\(422\) 0 0
\(423\) −3.46410 −0.168430
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 9.85641 0.476985
\(428\) 0 0
\(429\) −3.46410 −0.167248
\(430\) 0 0
\(431\) −23.3205 −1.12331 −0.561655 0.827372i \(-0.689835\pi\)
−0.561655 + 0.827372i \(0.689835\pi\)
\(432\) 0 0
\(433\) 20.5885 0.989418 0.494709 0.869059i \(-0.335275\pi\)
0.494709 + 0.869059i \(0.335275\pi\)
\(434\) 0 0
\(435\) 2.19615 0.105297
\(436\) 0 0
\(437\) −3.46410 −0.165710
\(438\) 0 0
\(439\) −13.0718 −0.623883 −0.311941 0.950101i \(-0.600979\pi\)
−0.311941 + 0.950101i \(0.600979\pi\)
\(440\) 0 0
\(441\) −6.46410 −0.307814
\(442\) 0 0
\(443\) −29.3205 −1.39306 −0.696530 0.717528i \(-0.745274\pi\)
−0.696530 + 0.717528i \(0.745274\pi\)
\(444\) 0 0
\(445\) 7.26795 0.344534
\(446\) 0 0
\(447\) −7.85641 −0.371595
\(448\) 0 0
\(449\) 11.6603 0.550281 0.275141 0.961404i \(-0.411276\pi\)
0.275141 + 0.961404i \(0.411276\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 0 0
\(453\) 14.0000 0.657777
\(454\) 0 0
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 11.4641 0.536268 0.268134 0.963382i \(-0.413593\pi\)
0.268134 + 0.963382i \(0.413593\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −9.51666 −0.442277 −0.221138 0.975242i \(-0.570977\pi\)
−0.221138 + 0.975242i \(0.570977\pi\)
\(464\) 0 0
\(465\) −4.92820 −0.228540
\(466\) 0 0
\(467\) −27.4641 −1.27089 −0.635444 0.772147i \(-0.719183\pi\)
−0.635444 + 0.772147i \(0.719183\pi\)
\(468\) 0 0
\(469\) 5.85641 0.270424
\(470\) 0 0
\(471\) 14.3923 0.663162
\(472\) 0 0
\(473\) −7.85641 −0.361238
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −2.53590 −0.116111
\(478\) 0 0
\(479\) 19.5167 0.891739 0.445869 0.895098i \(-0.352895\pi\)
0.445869 + 0.895098i \(0.352895\pi\)
\(480\) 0 0
\(481\) −11.4641 −0.522718
\(482\) 0 0
\(483\) 2.53590 0.115387
\(484\) 0 0
\(485\) 4.19615 0.190537
\(486\) 0 0
\(487\) 11.6077 0.525995 0.262997 0.964797i \(-0.415289\pi\)
0.262997 + 0.964797i \(0.415289\pi\)
\(488\) 0 0
\(489\) 12.7321 0.575763
\(490\) 0 0
\(491\) 22.0526 0.995218 0.497609 0.867401i \(-0.334211\pi\)
0.497609 + 0.867401i \(0.334211\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −1.26795 −0.0569901
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −17.4641 −0.781801 −0.390900 0.920433i \(-0.627836\pi\)
−0.390900 + 0.920433i \(0.627836\pi\)
\(500\) 0 0
\(501\) −3.46410 −0.154765
\(502\) 0 0
\(503\) 36.9282 1.64655 0.823274 0.567645i \(-0.192145\pi\)
0.823274 + 0.567645i \(0.192145\pi\)
\(504\) 0 0
\(505\) 10.3923 0.462451
\(506\) 0 0
\(507\) 5.53590 0.245858
\(508\) 0 0
\(509\) 28.0526 1.24341 0.621704 0.783252i \(-0.286441\pi\)
0.621704 + 0.783252i \(0.286441\pi\)
\(510\) 0 0
\(511\) 2.24871 0.0994771
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 17.8564 0.786847
\(516\) 0 0
\(517\) 4.39230 0.193173
\(518\) 0 0
\(519\) 6.92820 0.304114
\(520\) 0 0
\(521\) 40.7321 1.78450 0.892252 0.451538i \(-0.149125\pi\)
0.892252 + 0.451538i \(0.149125\pi\)
\(522\) 0 0
\(523\) −43.3205 −1.89427 −0.947137 0.320830i \(-0.896038\pi\)
−0.947137 + 0.320830i \(0.896038\pi\)
\(524\) 0 0
\(525\) 0.732051 0.0319493
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) −9.46410 −0.410707
\(532\) 0 0
\(533\) −12.9282 −0.559983
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −11.3205 −0.488516
\(538\) 0 0
\(539\) 8.19615 0.353033
\(540\) 0 0
\(541\) −13.7128 −0.589560 −0.294780 0.955565i \(-0.595246\pi\)
−0.294780 + 0.955565i \(0.595246\pi\)
\(542\) 0 0
\(543\) −18.3923 −0.789289
\(544\) 0 0
\(545\) 6.39230 0.273816
\(546\) 0 0
\(547\) −8.67949 −0.371108 −0.185554 0.982634i \(-0.559408\pi\)
−0.185554 + 0.982634i \(0.559408\pi\)
\(548\) 0 0
\(549\) −13.4641 −0.574634
\(550\) 0 0
\(551\) 2.19615 0.0935592
\(552\) 0 0
\(553\) 2.14359 0.0911549
\(554\) 0 0
\(555\) −4.19615 −0.178117
\(556\) 0 0
\(557\) −12.9282 −0.547786 −0.273893 0.961760i \(-0.588311\pi\)
−0.273893 + 0.961760i \(0.588311\pi\)
\(558\) 0 0
\(559\) −16.9282 −0.715987
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.5359 0.865485 0.432742 0.901518i \(-0.357546\pi\)
0.432742 + 0.901518i \(0.357546\pi\)
\(564\) 0 0
\(565\) 5.07180 0.213372
\(566\) 0 0
\(567\) −0.732051 −0.0307432
\(568\) 0 0
\(569\) −16.0526 −0.672958 −0.336479 0.941691i \(-0.609236\pi\)
−0.336479 + 0.941691i \(0.609236\pi\)
\(570\) 0 0
\(571\) −14.2487 −0.596290 −0.298145 0.954521i \(-0.596368\pi\)
−0.298145 + 0.954521i \(0.596368\pi\)
\(572\) 0 0
\(573\) 17.6603 0.737768
\(574\) 0 0
\(575\) 3.46410 0.144463
\(576\) 0 0
\(577\) −47.1769 −1.96400 −0.982000 0.188879i \(-0.939515\pi\)
−0.982000 + 0.188879i \(0.939515\pi\)
\(578\) 0 0
\(579\) 7.12436 0.296078
\(580\) 0 0
\(581\) 0.679492 0.0281901
\(582\) 0 0
\(583\) 3.21539 0.133168
\(584\) 0 0
\(585\) −2.73205 −0.112956
\(586\) 0 0
\(587\) −3.46410 −0.142979 −0.0714894 0.997441i \(-0.522775\pi\)
−0.0714894 + 0.997441i \(0.522775\pi\)
\(588\) 0 0
\(589\) −4.92820 −0.203063
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 0 0
\(593\) 2.78461 0.114350 0.0571751 0.998364i \(-0.481791\pi\)
0.0571751 + 0.998364i \(0.481791\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 19.3205 0.790736
\(598\) 0 0
\(599\) −13.8564 −0.566157 −0.283079 0.959097i \(-0.591356\pi\)
−0.283079 + 0.959097i \(0.591356\pi\)
\(600\) 0 0
\(601\) 15.1769 0.619079 0.309540 0.950887i \(-0.399825\pi\)
0.309540 + 0.950887i \(0.399825\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) −9.39230 −0.381851
\(606\) 0 0
\(607\) 32.3923 1.31476 0.657382 0.753558i \(-0.271664\pi\)
0.657382 + 0.753558i \(0.271664\pi\)
\(608\) 0 0
\(609\) −1.60770 −0.0651471
\(610\) 0 0
\(611\) 9.46410 0.382877
\(612\) 0 0
\(613\) 21.6077 0.872727 0.436363 0.899771i \(-0.356266\pi\)
0.436363 + 0.899771i \(0.356266\pi\)
\(614\) 0 0
\(615\) −4.73205 −0.190815
\(616\) 0 0
\(617\) −27.7128 −1.11568 −0.557838 0.829950i \(-0.688369\pi\)
−0.557838 + 0.829950i \(0.688369\pi\)
\(618\) 0 0
\(619\) −19.3205 −0.776557 −0.388278 0.921542i \(-0.626930\pi\)
−0.388278 + 0.921542i \(0.626930\pi\)
\(620\) 0 0
\(621\) −3.46410 −0.139010
\(622\) 0 0
\(623\) −5.32051 −0.213162
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.26795 −0.0506370
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 21.0718 0.838855 0.419427 0.907789i \(-0.362231\pi\)
0.419427 + 0.907789i \(0.362231\pi\)
\(632\) 0 0
\(633\) 14.9282 0.593343
\(634\) 0 0
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 17.6603 0.699725
\(638\) 0 0
\(639\) −16.3923 −0.648470
\(640\) 0 0
\(641\) 17.4115 0.687715 0.343857 0.939022i \(-0.388266\pi\)
0.343857 + 0.939022i \(0.388266\pi\)
\(642\) 0 0
\(643\) 1.80385 0.0711368 0.0355684 0.999367i \(-0.488676\pi\)
0.0355684 + 0.999367i \(0.488676\pi\)
\(644\) 0 0
\(645\) −6.19615 −0.243973
\(646\) 0 0
\(647\) 31.8564 1.25240 0.626202 0.779661i \(-0.284608\pi\)
0.626202 + 0.779661i \(0.284608\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 3.60770 0.141397
\(652\) 0 0
\(653\) 30.9282 1.21031 0.605157 0.796106i \(-0.293110\pi\)
0.605157 + 0.796106i \(0.293110\pi\)
\(654\) 0 0
\(655\) −15.1244 −0.590957
\(656\) 0 0
\(657\) −3.07180 −0.119842
\(658\) 0 0
\(659\) −18.9282 −0.737338 −0.368669 0.929561i \(-0.620186\pi\)
−0.368669 + 0.929561i \(0.620186\pi\)
\(660\) 0 0
\(661\) −23.1769 −0.901477 −0.450739 0.892656i \(-0.648839\pi\)
−0.450739 + 0.892656i \(0.648839\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.732051 0.0283877
\(666\) 0 0
\(667\) −7.60770 −0.294571
\(668\) 0 0
\(669\) 9.85641 0.381071
\(670\) 0 0
\(671\) 17.0718 0.659049
\(672\) 0 0
\(673\) −7.12436 −0.274624 −0.137312 0.990528i \(-0.543846\pi\)
−0.137312 + 0.990528i \(0.543846\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 35.3205 1.35748 0.678739 0.734380i \(-0.262527\pi\)
0.678739 + 0.734380i \(0.262527\pi\)
\(678\) 0 0
\(679\) −3.07180 −0.117885
\(680\) 0 0
\(681\) −10.3923 −0.398234
\(682\) 0 0
\(683\) 18.9282 0.724268 0.362134 0.932126i \(-0.382048\pi\)
0.362134 + 0.932126i \(0.382048\pi\)
\(684\) 0 0
\(685\) −7.85641 −0.300178
\(686\) 0 0
\(687\) 25.4641 0.971516
\(688\) 0 0
\(689\) 6.92820 0.263944
\(690\) 0 0
\(691\) 8.39230 0.319258 0.159629 0.987177i \(-0.448970\pi\)
0.159629 + 0.987177i \(0.448970\pi\)
\(692\) 0 0
\(693\) 0.928203 0.0352595
\(694\) 0 0
\(695\) −12.3923 −0.470067
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −19.8564 −0.751038
\(700\) 0 0
\(701\) 21.7128 0.820082 0.410041 0.912067i \(-0.365514\pi\)
0.410041 + 0.912067i \(0.365514\pi\)
\(702\) 0 0
\(703\) −4.19615 −0.158261
\(704\) 0 0
\(705\) 3.46410 0.130466
\(706\) 0 0
\(707\) −7.60770 −0.286117
\(708\) 0 0
\(709\) 33.1769 1.24599 0.622993 0.782228i \(-0.285917\pi\)
0.622993 + 0.782228i \(0.285917\pi\)
\(710\) 0 0
\(711\) −2.92820 −0.109816
\(712\) 0 0
\(713\) 17.0718 0.639344
\(714\) 0 0
\(715\) 3.46410 0.129550
\(716\) 0 0
\(717\) −20.1962 −0.754239
\(718\) 0 0
\(719\) −5.66025 −0.211092 −0.105546 0.994414i \(-0.533659\pi\)
−0.105546 + 0.994414i \(0.533659\pi\)
\(720\) 0 0
\(721\) −13.0718 −0.486819
\(722\) 0 0
\(723\) 16.9282 0.629567
\(724\) 0 0
\(725\) −2.19615 −0.0815631
\(726\) 0 0
\(727\) −8.33975 −0.309304 −0.154652 0.987969i \(-0.549426\pi\)
−0.154652 + 0.987969i \(0.549426\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 22.7846 0.841569 0.420784 0.907161i \(-0.361755\pi\)
0.420784 + 0.907161i \(0.361755\pi\)
\(734\) 0 0
\(735\) 6.46410 0.238432
\(736\) 0 0
\(737\) 10.1436 0.373644
\(738\) 0 0
\(739\) −33.8564 −1.24543 −0.622714 0.782450i \(-0.713970\pi\)
−0.622714 + 0.782450i \(0.713970\pi\)
\(740\) 0 0
\(741\) −2.73205 −0.100364
\(742\) 0 0
\(743\) 44.7846 1.64299 0.821494 0.570217i \(-0.193141\pi\)
0.821494 + 0.570217i \(0.193141\pi\)
\(744\) 0 0
\(745\) 7.85641 0.287836
\(746\) 0 0
\(747\) −0.928203 −0.0339612
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 0 0
\(753\) −10.0526 −0.366336
\(754\) 0 0
\(755\) −14.0000 −0.509512
\(756\) 0 0
\(757\) −16.2487 −0.590569 −0.295285 0.955409i \(-0.595415\pi\)
−0.295285 + 0.955409i \(0.595415\pi\)
\(758\) 0 0
\(759\) 4.39230 0.159431
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) −4.67949 −0.169409
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 25.8564 0.933621
\(768\) 0 0
\(769\) 48.6410 1.75404 0.877020 0.480454i \(-0.159528\pi\)
0.877020 + 0.480454i \(0.159528\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) 0 0
\(773\) 37.1769 1.33716 0.668580 0.743640i \(-0.266902\pi\)
0.668580 + 0.743640i \(0.266902\pi\)
\(774\) 0 0
\(775\) 4.92820 0.177026
\(776\) 0 0
\(777\) 3.07180 0.110200
\(778\) 0 0
\(779\) −4.73205 −0.169543
\(780\) 0 0
\(781\) 20.7846 0.743732
\(782\) 0 0
\(783\) 2.19615 0.0784841
\(784\) 0 0
\(785\) −14.3923 −0.513683
\(786\) 0 0
\(787\) −43.3205 −1.54421 −0.772105 0.635495i \(-0.780796\pi\)
−0.772105 + 0.635495i \(0.780796\pi\)
\(788\) 0 0
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) −3.71281 −0.132012
\(792\) 0 0
\(793\) 36.7846 1.30626
\(794\) 0 0
\(795\) 2.53590 0.0899390
\(796\) 0 0
\(797\) 3.21539 0.113895 0.0569475 0.998377i \(-0.481863\pi\)
0.0569475 + 0.998377i \(0.481863\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 7.26795 0.256800
\(802\) 0 0
\(803\) 3.89488 0.137447
\(804\) 0 0
\(805\) −2.53590 −0.0893787
\(806\) 0 0
\(807\) 30.5885 1.07676
\(808\) 0 0
\(809\) −26.7846 −0.941697 −0.470848 0.882214i \(-0.656052\pi\)
−0.470848 + 0.882214i \(0.656052\pi\)
\(810\) 0 0
\(811\) 45.5692 1.60015 0.800076 0.599899i \(-0.204793\pi\)
0.800076 + 0.599899i \(0.204793\pi\)
\(812\) 0 0
\(813\) −20.3923 −0.715189
\(814\) 0 0
\(815\) −12.7321 −0.445984
\(816\) 0 0
\(817\) −6.19615 −0.216776
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −39.4641 −1.37731 −0.688653 0.725091i \(-0.741798\pi\)
−0.688653 + 0.725091i \(0.741798\pi\)
\(822\) 0 0
\(823\) 38.9808 1.35878 0.679392 0.733776i \(-0.262244\pi\)
0.679392 + 0.733776i \(0.262244\pi\)
\(824\) 0 0
\(825\) 1.26795 0.0441443
\(826\) 0 0
\(827\) −29.3205 −1.01957 −0.509787 0.860301i \(-0.670276\pi\)
−0.509787 + 0.860301i \(0.670276\pi\)
\(828\) 0 0
\(829\) −42.1051 −1.46237 −0.731186 0.682179i \(-0.761033\pi\)
−0.731186 + 0.682179i \(0.761033\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 3.46410 0.119880
\(836\) 0 0
\(837\) −4.92820 −0.170344
\(838\) 0 0
\(839\) 40.3923 1.39450 0.697249 0.716829i \(-0.254407\pi\)
0.697249 + 0.716829i \(0.254407\pi\)
\(840\) 0 0
\(841\) −24.1769 −0.833687
\(842\) 0 0
\(843\) −4.73205 −0.162980
\(844\) 0 0
\(845\) −5.53590 −0.190441
\(846\) 0 0
\(847\) 6.87564 0.236250
\(848\) 0 0
\(849\) −26.9808 −0.925977
\(850\) 0 0
\(851\) 14.5359 0.498284
\(852\) 0 0
\(853\) −35.1769 −1.20443 −0.602217 0.798332i \(-0.705716\pi\)
−0.602217 + 0.798332i \(0.705716\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 6.24871 0.213452 0.106726 0.994288i \(-0.465963\pi\)
0.106726 + 0.994288i \(0.465963\pi\)
\(858\) 0 0
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) 3.46410 0.118056
\(862\) 0 0
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 0 0
\(865\) −6.92820 −0.235566
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 3.71281 0.125949
\(870\) 0 0
\(871\) 21.8564 0.740576
\(872\) 0 0
\(873\) 4.19615 0.142018
\(874\) 0 0
\(875\) −0.732051 −0.0247478
\(876\) 0 0
\(877\) 28.8756 0.975061 0.487531 0.873106i \(-0.337898\pi\)
0.487531 + 0.873106i \(0.337898\pi\)
\(878\) 0 0
\(879\) 27.7128 0.934730
\(880\) 0 0
\(881\) 15.4641 0.520999 0.260499 0.965474i \(-0.416113\pi\)
0.260499 + 0.965474i \(0.416113\pi\)
\(882\) 0 0
\(883\) 14.9808 0.504143 0.252071 0.967709i \(-0.418888\pi\)
0.252071 + 0.967709i \(0.418888\pi\)
\(884\) 0 0
\(885\) 9.46410 0.318132
\(886\) 0 0
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) −2.92820 −0.0982088
\(890\) 0 0
\(891\) −1.26795 −0.0424779
\(892\) 0 0
\(893\) 3.46410 0.115922
\(894\) 0 0
\(895\) 11.3205 0.378403
\(896\) 0 0
\(897\) 9.46410 0.315997
\(898\) 0 0
\(899\) −10.8231 −0.360970
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 4.53590 0.150945
\(904\) 0 0
\(905\) 18.3923 0.611381
\(906\) 0 0
\(907\) 11.6077 0.385427 0.192714 0.981255i \(-0.438271\pi\)
0.192714 + 0.981255i \(0.438271\pi\)
\(908\) 0 0
\(909\) 10.3923 0.344691
\(910\) 0 0
\(911\) 41.0718 1.36077 0.680385 0.732855i \(-0.261813\pi\)
0.680385 + 0.732855i \(0.261813\pi\)
\(912\) 0 0
\(913\) 1.17691 0.0389502
\(914\) 0 0
\(915\) 13.4641 0.445109
\(916\) 0 0
\(917\) 11.0718 0.365623
\(918\) 0 0
\(919\) 59.4256 1.96027 0.980135 0.198330i \(-0.0635518\pi\)
0.980135 + 0.198330i \(0.0635518\pi\)
\(920\) 0 0
\(921\) −11.6077 −0.382487
\(922\) 0 0
\(923\) 44.7846 1.47410
\(924\) 0 0
\(925\) 4.19615 0.137969
\(926\) 0 0
\(927\) 17.8564 0.586481
\(928\) 0 0
\(929\) −22.3923 −0.734668 −0.367334 0.930089i \(-0.619729\pi\)
−0.367334 + 0.930089i \(0.619729\pi\)
\(930\) 0 0
\(931\) 6.46410 0.211852
\(932\) 0 0
\(933\) −26.4449 −0.865766
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 32.2487 1.05352 0.526760 0.850014i \(-0.323407\pi\)
0.526760 + 0.850014i \(0.323407\pi\)
\(938\) 0 0
\(939\) 14.3923 0.469675
\(940\) 0 0
\(941\) −30.5885 −0.997155 −0.498578 0.866845i \(-0.666144\pi\)
−0.498578 + 0.866845i \(0.666144\pi\)
\(942\) 0 0
\(943\) 16.3923 0.533807
\(944\) 0 0
\(945\) 0.732051 0.0238136
\(946\) 0 0
\(947\) −55.8564 −1.81509 −0.907545 0.419956i \(-0.862046\pi\)
−0.907545 + 0.419956i \(0.862046\pi\)
\(948\) 0 0
\(949\) 8.39230 0.272426
\(950\) 0 0
\(951\) −23.3205 −0.756219
\(952\) 0 0
\(953\) 10.1436 0.328583 0.164292 0.986412i \(-0.447466\pi\)
0.164292 + 0.986412i \(0.447466\pi\)
\(954\) 0 0
\(955\) −17.6603 −0.571472
\(956\) 0 0
\(957\) −2.78461 −0.0900136
\(958\) 0 0
\(959\) 5.75129 0.185719
\(960\) 0 0
\(961\) −6.71281 −0.216542
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −7.12436 −0.229341
\(966\) 0 0
\(967\) −29.1244 −0.936576 −0.468288 0.883576i \(-0.655129\pi\)
−0.468288 + 0.883576i \(0.655129\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −27.7128 −0.889346 −0.444673 0.895693i \(-0.646680\pi\)
−0.444673 + 0.895693i \(0.646680\pi\)
\(972\) 0 0
\(973\) 9.07180 0.290828
\(974\) 0 0
\(975\) 2.73205 0.0874957
\(976\) 0 0
\(977\) 51.0333 1.63270 0.816350 0.577557i \(-0.195994\pi\)
0.816350 + 0.577557i \(0.195994\pi\)
\(978\) 0 0
\(979\) −9.21539 −0.294525
\(980\) 0 0
\(981\) 6.39230 0.204091
\(982\) 0 0
\(983\) 6.67949 0.213043 0.106521 0.994310i \(-0.466029\pi\)
0.106521 + 0.994310i \(0.466029\pi\)
\(984\) 0 0
\(985\) −24.0000 −0.764704
\(986\) 0 0
\(987\) −2.53590 −0.0807185
\(988\) 0 0
\(989\) 21.4641 0.682519
\(990\) 0 0
\(991\) −26.9282 −0.855403 −0.427701 0.903920i \(-0.640676\pi\)
−0.427701 + 0.903920i \(0.640676\pi\)
\(992\) 0 0
\(993\) 29.7128 0.942908
\(994\) 0 0
\(995\) −19.3205 −0.612501
\(996\) 0 0
\(997\) −38.3923 −1.21590 −0.607948 0.793977i \(-0.708007\pi\)
−0.607948 + 0.793977i \(0.708007\pi\)
\(998\) 0 0
\(999\) −4.19615 −0.132760
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 4560.2.a.bh.1.1 2
4.3 odd 2 285.2.a.e.1.2 2
12.11 even 2 855.2.a.f.1.1 2
20.3 even 4 1425.2.c.k.799.2 4
20.7 even 4 1425.2.c.k.799.3 4
20.19 odd 2 1425.2.a.o.1.1 2
60.59 even 2 4275.2.a.t.1.2 2
76.75 even 2 5415.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.e.1.2 2 4.3 odd 2
855.2.a.f.1.1 2 12.11 even 2
1425.2.a.o.1.1 2 20.19 odd 2
1425.2.c.k.799.2 4 20.3 even 4
1425.2.c.k.799.3 4 20.7 even 4
4275.2.a.t.1.2 2 60.59 even 2
4560.2.a.bh.1.1 2 1.1 even 1 trivial
5415.2.a.r.1.1 2 76.75 even 2