Properties

Label 1536.2.a.n.1.4
Level $1536$
Weight $2$
Character 1536.1
Self dual yes
Analytic conductor $12.265$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.2650217505\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \( x^{4} - 6x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.27133\) of defining polynomial
Character \(\chi\) \(=\) 1536.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.95687 q^{5} -1.63899 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.95687 q^{5} -1.63899 q^{7} +1.00000 q^{9} +4.82843 q^{11} +5.59587 q^{13} +3.95687 q^{15} -0.828427 q^{17} -2.82843 q^{19} -1.63899 q^{21} -7.91375 q^{23} +10.6569 q^{25} +1.00000 q^{27} -7.23486 q^{29} +1.63899 q^{31} +4.82843 q^{33} -6.48528 q^{35} -2.31788 q^{37} +5.59587 q^{39} +3.17157 q^{41} +4.48528 q^{43} +3.95687 q^{45} -7.91375 q^{47} -4.31371 q^{49} -0.828427 q^{51} -0.678892 q^{53} +19.1055 q^{55} -2.82843 q^{57} +9.65685 q^{59} -2.31788 q^{61} -1.63899 q^{63} +22.1421 q^{65} -13.6569 q^{67} -7.91375 q^{69} +3.27798 q^{71} +4.00000 q^{73} +10.6569 q^{75} -7.91375 q^{77} -1.63899 q^{79} +1.00000 q^{81} +8.82843 q^{83} -3.27798 q^{85} -7.23486 q^{87} +10.0000 q^{89} -9.17157 q^{91} +1.63899 q^{93} -11.1917 q^{95} +11.6569 q^{97} +4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} + 8 q^{11} + 8 q^{17} + 20 q^{25} + 4 q^{27} + 8 q^{33} + 8 q^{35} + 24 q^{41} - 16 q^{43} + 28 q^{49} + 8 q^{51} + 16 q^{59} + 32 q^{65} - 32 q^{67} + 16 q^{73} + 20 q^{75} + 4 q^{81} + 24 q^{83} + 40 q^{89} - 48 q^{91} + 24 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 3.95687 1.76957 0.884784 0.466001i \(-0.154306\pi\)
0.884784 + 0.466001i \(0.154306\pi\)
\(6\) 0 0
\(7\) −1.63899 −0.619480 −0.309740 0.950821i \(-0.600242\pi\)
−0.309740 + 0.950821i \(0.600242\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.82843 1.45583 0.727913 0.685670i \(-0.240491\pi\)
0.727913 + 0.685670i \(0.240491\pi\)
\(12\) 0 0
\(13\) 5.59587 1.55201 0.776007 0.630724i \(-0.217242\pi\)
0.776007 + 0.630724i \(0.217242\pi\)
\(14\) 0 0
\(15\) 3.95687 1.02166
\(16\) 0 0
\(17\) −0.828427 −0.200923 −0.100462 0.994941i \(-0.532032\pi\)
−0.100462 + 0.994941i \(0.532032\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) −1.63899 −0.357657
\(22\) 0 0
\(23\) −7.91375 −1.65013 −0.825065 0.565037i \(-0.808862\pi\)
−0.825065 + 0.565037i \(0.808862\pi\)
\(24\) 0 0
\(25\) 10.6569 2.13137
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.23486 −1.34348 −0.671740 0.740787i \(-0.734453\pi\)
−0.671740 + 0.740787i \(0.734453\pi\)
\(30\) 0 0
\(31\) 1.63899 0.294371 0.147186 0.989109i \(-0.452978\pi\)
0.147186 + 0.989109i \(0.452978\pi\)
\(32\) 0 0
\(33\) 4.82843 0.840521
\(34\) 0 0
\(35\) −6.48528 −1.09621
\(36\) 0 0
\(37\) −2.31788 −0.381058 −0.190529 0.981682i \(-0.561020\pi\)
−0.190529 + 0.981682i \(0.561020\pi\)
\(38\) 0 0
\(39\) 5.59587 0.896056
\(40\) 0 0
\(41\) 3.17157 0.495316 0.247658 0.968847i \(-0.420339\pi\)
0.247658 + 0.968847i \(0.420339\pi\)
\(42\) 0 0
\(43\) 4.48528 0.683999 0.341999 0.939700i \(-0.388896\pi\)
0.341999 + 0.939700i \(0.388896\pi\)
\(44\) 0 0
\(45\) 3.95687 0.589856
\(46\) 0 0
\(47\) −7.91375 −1.15434 −0.577169 0.816624i \(-0.695843\pi\)
−0.577169 + 0.816624i \(0.695843\pi\)
\(48\) 0 0
\(49\) −4.31371 −0.616244
\(50\) 0 0
\(51\) −0.828427 −0.116003
\(52\) 0 0
\(53\) −0.678892 −0.0932530 −0.0466265 0.998912i \(-0.514847\pi\)
−0.0466265 + 0.998912i \(0.514847\pi\)
\(54\) 0 0
\(55\) 19.1055 2.57618
\(56\) 0 0
\(57\) −2.82843 −0.374634
\(58\) 0 0
\(59\) 9.65685 1.25722 0.628608 0.777723i \(-0.283625\pi\)
0.628608 + 0.777723i \(0.283625\pi\)
\(60\) 0 0
\(61\) −2.31788 −0.296775 −0.148387 0.988929i \(-0.547408\pi\)
−0.148387 + 0.988929i \(0.547408\pi\)
\(62\) 0 0
\(63\) −1.63899 −0.206493
\(64\) 0 0
\(65\) 22.1421 2.74639
\(66\) 0 0
\(67\) −13.6569 −1.66845 −0.834225 0.551424i \(-0.814085\pi\)
−0.834225 + 0.551424i \(0.814085\pi\)
\(68\) 0 0
\(69\) −7.91375 −0.952703
\(70\) 0 0
\(71\) 3.27798 0.389025 0.194512 0.980900i \(-0.437688\pi\)
0.194512 + 0.980900i \(0.437688\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 10.6569 1.23055
\(76\) 0 0
\(77\) −7.91375 −0.901855
\(78\) 0 0
\(79\) −1.63899 −0.184401 −0.0922004 0.995740i \(-0.529390\pi\)
−0.0922004 + 0.995740i \(0.529390\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.82843 0.969046 0.484523 0.874779i \(-0.338993\pi\)
0.484523 + 0.874779i \(0.338993\pi\)
\(84\) 0 0
\(85\) −3.27798 −0.355547
\(86\) 0 0
\(87\) −7.23486 −0.775658
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −9.17157 −0.961442
\(92\) 0 0
\(93\) 1.63899 0.169955
\(94\) 0 0
\(95\) −11.1917 −1.14825
\(96\) 0 0
\(97\) 11.6569 1.18357 0.591787 0.806094i \(-0.298423\pi\)
0.591787 + 0.806094i \(0.298423\pi\)
\(98\) 0 0
\(99\) 4.82843 0.485275
\(100\) 0 0
\(101\) 0.678892 0.0675523 0.0337762 0.999429i \(-0.489247\pi\)
0.0337762 + 0.999429i \(0.489247\pi\)
\(102\) 0 0
\(103\) −17.4665 −1.72102 −0.860512 0.509430i \(-0.829856\pi\)
−0.860512 + 0.509430i \(0.829856\pi\)
\(104\) 0 0
\(105\) −6.48528 −0.632899
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −16.7876 −1.60796 −0.803980 0.594656i \(-0.797288\pi\)
−0.803980 + 0.594656i \(0.797288\pi\)
\(110\) 0 0
\(111\) −2.31788 −0.220004
\(112\) 0 0
\(113\) 0.343146 0.0322804 0.0161402 0.999870i \(-0.494862\pi\)
0.0161402 + 0.999870i \(0.494862\pi\)
\(114\) 0 0
\(115\) −31.3137 −2.92002
\(116\) 0 0
\(117\) 5.59587 0.517338
\(118\) 0 0
\(119\) 1.35778 0.124468
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 0 0
\(123\) 3.17157 0.285971
\(124\) 0 0
\(125\) 22.3835 2.00204
\(126\) 0 0
\(127\) 14.1885 1.25903 0.629513 0.776990i \(-0.283254\pi\)
0.629513 + 0.776990i \(0.283254\pi\)
\(128\) 0 0
\(129\) 4.48528 0.394907
\(130\) 0 0
\(131\) −7.31371 −0.639002 −0.319501 0.947586i \(-0.603515\pi\)
−0.319501 + 0.947586i \(0.603515\pi\)
\(132\) 0 0
\(133\) 4.63577 0.401972
\(134\) 0 0
\(135\) 3.95687 0.340554
\(136\) 0 0
\(137\) −16.1421 −1.37912 −0.689558 0.724231i \(-0.742195\pi\)
−0.689558 + 0.724231i \(0.742195\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −7.91375 −0.666458
\(142\) 0 0
\(143\) 27.0192 2.25946
\(144\) 0 0
\(145\) −28.6274 −2.37738
\(146\) 0 0
\(147\) −4.31371 −0.355789
\(148\) 0 0
\(149\) 3.95687 0.324160 0.162080 0.986778i \(-0.448180\pi\)
0.162080 + 0.986778i \(0.448180\pi\)
\(150\) 0 0
\(151\) −14.1885 −1.15464 −0.577322 0.816516i \(-0.695902\pi\)
−0.577322 + 0.816516i \(0.695902\pi\)
\(152\) 0 0
\(153\) −0.828427 −0.0669744
\(154\) 0 0
\(155\) 6.48528 0.520910
\(156\) 0 0
\(157\) 18.1454 1.44816 0.724080 0.689717i \(-0.242265\pi\)
0.724080 + 0.689717i \(0.242265\pi\)
\(158\) 0 0
\(159\) −0.678892 −0.0538397
\(160\) 0 0
\(161\) 12.9706 1.02222
\(162\) 0 0
\(163\) 10.1421 0.794393 0.397197 0.917734i \(-0.369983\pi\)
0.397197 + 0.917734i \(0.369983\pi\)
\(164\) 0 0
\(165\) 19.1055 1.48736
\(166\) 0 0
\(167\) 4.63577 0.358726 0.179363 0.983783i \(-0.442596\pi\)
0.179363 + 0.983783i \(0.442596\pi\)
\(168\) 0 0
\(169\) 18.3137 1.40875
\(170\) 0 0
\(171\) −2.82843 −0.216295
\(172\) 0 0
\(173\) 11.8706 0.902507 0.451253 0.892396i \(-0.350977\pi\)
0.451253 + 0.892396i \(0.350977\pi\)
\(174\) 0 0
\(175\) −17.4665 −1.32034
\(176\) 0 0
\(177\) 9.65685 0.725854
\(178\) 0 0
\(179\) 12.9706 0.969465 0.484733 0.874662i \(-0.338917\pi\)
0.484733 + 0.874662i \(0.338917\pi\)
\(180\) 0 0
\(181\) 16.7876 1.24781 0.623906 0.781499i \(-0.285545\pi\)
0.623906 + 0.781499i \(0.285545\pi\)
\(182\) 0 0
\(183\) −2.31788 −0.171343
\(184\) 0 0
\(185\) −9.17157 −0.674307
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) −1.63899 −0.119219
\(190\) 0 0
\(191\) 17.7477 1.28418 0.642089 0.766630i \(-0.278068\pi\)
0.642089 + 0.766630i \(0.278068\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 22.1421 1.58563
\(196\) 0 0
\(197\) −16.5064 −1.17603 −0.588016 0.808849i \(-0.700091\pi\)
−0.588016 + 0.808849i \(0.700091\pi\)
\(198\) 0 0
\(199\) −1.63899 −0.116185 −0.0580925 0.998311i \(-0.518502\pi\)
−0.0580925 + 0.998311i \(0.518502\pi\)
\(200\) 0 0
\(201\) −13.6569 −0.963280
\(202\) 0 0
\(203\) 11.8579 0.832259
\(204\) 0 0
\(205\) 12.5495 0.876496
\(206\) 0 0
\(207\) −7.91375 −0.550044
\(208\) 0 0
\(209\) −13.6569 −0.944664
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 3.27798 0.224604
\(214\) 0 0
\(215\) 17.7477 1.21038
\(216\) 0 0
\(217\) −2.68629 −0.182357
\(218\) 0 0
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −4.63577 −0.311835
\(222\) 0 0
\(223\) −14.1885 −0.950133 −0.475066 0.879950i \(-0.657576\pi\)
−0.475066 + 0.879950i \(0.657576\pi\)
\(224\) 0 0
\(225\) 10.6569 0.710457
\(226\) 0 0
\(227\) −12.1421 −0.805902 −0.402951 0.915222i \(-0.632015\pi\)
−0.402951 + 0.915222i \(0.632015\pi\)
\(228\) 0 0
\(229\) −21.4234 −1.41570 −0.707848 0.706365i \(-0.750334\pi\)
−0.707848 + 0.706365i \(0.750334\pi\)
\(230\) 0 0
\(231\) −7.91375 −0.520686
\(232\) 0 0
\(233\) −21.3137 −1.39631 −0.698154 0.715948i \(-0.745995\pi\)
−0.698154 + 0.715948i \(0.745995\pi\)
\(234\) 0 0
\(235\) −31.3137 −2.04268
\(236\) 0 0
\(237\) −1.63899 −0.106464
\(238\) 0 0
\(239\) 4.63577 0.299863 0.149931 0.988696i \(-0.452095\pi\)
0.149931 + 0.988696i \(0.452095\pi\)
\(240\) 0 0
\(241\) −16.9706 −1.09317 −0.546585 0.837404i \(-0.684072\pi\)
−0.546585 + 0.837404i \(0.684072\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −17.0688 −1.09049
\(246\) 0 0
\(247\) −15.8275 −1.00708
\(248\) 0 0
\(249\) 8.82843 0.559479
\(250\) 0 0
\(251\) 20.8284 1.31468 0.657339 0.753595i \(-0.271682\pi\)
0.657339 + 0.753595i \(0.271682\pi\)
\(252\) 0 0
\(253\) −38.2110 −2.40230
\(254\) 0 0
\(255\) −3.27798 −0.205275
\(256\) 0 0
\(257\) −19.6569 −1.22616 −0.613080 0.790020i \(-0.710070\pi\)
−0.613080 + 0.790020i \(0.710070\pi\)
\(258\) 0 0
\(259\) 3.79899 0.236058
\(260\) 0 0
\(261\) −7.23486 −0.447826
\(262\) 0 0
\(263\) 15.8275 0.975965 0.487983 0.872853i \(-0.337733\pi\)
0.487983 + 0.872853i \(0.337733\pi\)
\(264\) 0 0
\(265\) −2.68629 −0.165018
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 0 0
\(269\) 7.23486 0.441117 0.220558 0.975374i \(-0.429212\pi\)
0.220558 + 0.975374i \(0.429212\pi\)
\(270\) 0 0
\(271\) 17.4665 1.06101 0.530507 0.847681i \(-0.322002\pi\)
0.530507 + 0.847681i \(0.322002\pi\)
\(272\) 0 0
\(273\) −9.17157 −0.555089
\(274\) 0 0
\(275\) 51.4558 3.10290
\(276\) 0 0
\(277\) −14.8674 −0.893295 −0.446648 0.894710i \(-0.647382\pi\)
−0.446648 + 0.894710i \(0.647382\pi\)
\(278\) 0 0
\(279\) 1.63899 0.0981238
\(280\) 0 0
\(281\) 0.343146 0.0204704 0.0102352 0.999948i \(-0.496742\pi\)
0.0102352 + 0.999948i \(0.496742\pi\)
\(282\) 0 0
\(283\) −28.9706 −1.72212 −0.861061 0.508502i \(-0.830199\pi\)
−0.861061 + 0.508502i \(0.830199\pi\)
\(284\) 0 0
\(285\) −11.1917 −0.662941
\(286\) 0 0
\(287\) −5.19818 −0.306839
\(288\) 0 0
\(289\) −16.3137 −0.959630
\(290\) 0 0
\(291\) 11.6569 0.683337
\(292\) 0 0
\(293\) −0.678892 −0.0396613 −0.0198307 0.999803i \(-0.506313\pi\)
−0.0198307 + 0.999803i \(0.506313\pi\)
\(294\) 0 0
\(295\) 38.2110 2.22473
\(296\) 0 0
\(297\) 4.82843 0.280174
\(298\) 0 0
\(299\) −44.2843 −2.56103
\(300\) 0 0
\(301\) −7.35134 −0.423724
\(302\) 0 0
\(303\) 0.678892 0.0390013
\(304\) 0 0
\(305\) −9.17157 −0.525163
\(306\) 0 0
\(307\) −21.6569 −1.23602 −0.618011 0.786169i \(-0.712061\pi\)
−0.618011 + 0.786169i \(0.712061\pi\)
\(308\) 0 0
\(309\) −17.4665 −0.993634
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 7.31371 0.413395 0.206698 0.978405i \(-0.433728\pi\)
0.206698 + 0.978405i \(0.433728\pi\)
\(314\) 0 0
\(315\) −6.48528 −0.365404
\(316\) 0 0
\(317\) −21.7046 −1.21905 −0.609525 0.792767i \(-0.708640\pi\)
−0.609525 + 0.792767i \(0.708640\pi\)
\(318\) 0 0
\(319\) −34.9330 −1.95587
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 2.34315 0.130376
\(324\) 0 0
\(325\) 59.6343 3.30792
\(326\) 0 0
\(327\) −16.7876 −0.928356
\(328\) 0 0
\(329\) 12.9706 0.715090
\(330\) 0 0
\(331\) 10.3431 0.568511 0.284255 0.958749i \(-0.408254\pi\)
0.284255 + 0.958749i \(0.408254\pi\)
\(332\) 0 0
\(333\) −2.31788 −0.127019
\(334\) 0 0
\(335\) −54.0385 −2.95244
\(336\) 0 0
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 0 0
\(339\) 0.343146 0.0186371
\(340\) 0 0
\(341\) 7.91375 0.428554
\(342\) 0 0
\(343\) 18.5431 1.00123
\(344\) 0 0
\(345\) −31.3137 −1.68587
\(346\) 0 0
\(347\) 0.828427 0.0444723 0.0222361 0.999753i \(-0.492921\pi\)
0.0222361 + 0.999753i \(0.492921\pi\)
\(348\) 0 0
\(349\) 4.23808 0.226859 0.113430 0.993546i \(-0.463816\pi\)
0.113430 + 0.993546i \(0.463816\pi\)
\(350\) 0 0
\(351\) 5.59587 0.298685
\(352\) 0 0
\(353\) 1.31371 0.0699216 0.0349608 0.999389i \(-0.488869\pi\)
0.0349608 + 0.999389i \(0.488869\pi\)
\(354\) 0 0
\(355\) 12.9706 0.688406
\(356\) 0 0
\(357\) 1.35778 0.0718616
\(358\) 0 0
\(359\) −34.9330 −1.84369 −0.921846 0.387556i \(-0.873319\pi\)
−0.921846 + 0.387556i \(0.873319\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 12.3137 0.646302
\(364\) 0 0
\(365\) 15.8275 0.828449
\(366\) 0 0
\(367\) 24.0225 1.25396 0.626981 0.779035i \(-0.284290\pi\)
0.626981 + 0.779035i \(0.284290\pi\)
\(368\) 0 0
\(369\) 3.17157 0.165105
\(370\) 0 0
\(371\) 1.11270 0.0577684
\(372\) 0 0
\(373\) −20.0656 −1.03896 −0.519478 0.854484i \(-0.673874\pi\)
−0.519478 + 0.854484i \(0.673874\pi\)
\(374\) 0 0
\(375\) 22.3835 1.15588
\(376\) 0 0
\(377\) −40.4853 −2.08510
\(378\) 0 0
\(379\) −0.485281 −0.0249272 −0.0124636 0.999922i \(-0.503967\pi\)
−0.0124636 + 0.999922i \(0.503967\pi\)
\(380\) 0 0
\(381\) 14.1885 0.726899
\(382\) 0 0
\(383\) 11.1917 0.571871 0.285935 0.958249i \(-0.407696\pi\)
0.285935 + 0.958249i \(0.407696\pi\)
\(384\) 0 0
\(385\) −31.3137 −1.59589
\(386\) 0 0
\(387\) 4.48528 0.228000
\(388\) 0 0
\(389\) −3.95687 −0.200621 −0.100311 0.994956i \(-0.531984\pi\)
−0.100311 + 0.994956i \(0.531984\pi\)
\(390\) 0 0
\(391\) 6.55596 0.331549
\(392\) 0 0
\(393\) −7.31371 −0.368928
\(394\) 0 0
\(395\) −6.48528 −0.326310
\(396\) 0 0
\(397\) 18.1454 0.910691 0.455345 0.890315i \(-0.349516\pi\)
0.455345 + 0.890315i \(0.349516\pi\)
\(398\) 0 0
\(399\) 4.63577 0.232079
\(400\) 0 0
\(401\) 24.1421 1.20560 0.602800 0.797892i \(-0.294052\pi\)
0.602800 + 0.797892i \(0.294052\pi\)
\(402\) 0 0
\(403\) 9.17157 0.456869
\(404\) 0 0
\(405\) 3.95687 0.196619
\(406\) 0 0
\(407\) −11.1917 −0.554753
\(408\) 0 0
\(409\) −3.65685 −0.180820 −0.0904099 0.995905i \(-0.528818\pi\)
−0.0904099 + 0.995905i \(0.528818\pi\)
\(410\) 0 0
\(411\) −16.1421 −0.796233
\(412\) 0 0
\(413\) −15.8275 −0.778820
\(414\) 0 0
\(415\) 34.9330 1.71479
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 15.1716 0.741180 0.370590 0.928797i \(-0.379156\pi\)
0.370590 + 0.928797i \(0.379156\pi\)
\(420\) 0 0
\(421\) −21.4234 −1.04411 −0.522055 0.852912i \(-0.674835\pi\)
−0.522055 + 0.852912i \(0.674835\pi\)
\(422\) 0 0
\(423\) −7.91375 −0.384780
\(424\) 0 0
\(425\) −8.82843 −0.428242
\(426\) 0 0
\(427\) 3.79899 0.183846
\(428\) 0 0
\(429\) 27.0192 1.30450
\(430\) 0 0
\(431\) 14.4697 0.696982 0.348491 0.937312i \(-0.386694\pi\)
0.348491 + 0.937312i \(0.386694\pi\)
\(432\) 0 0
\(433\) 24.6274 1.18352 0.591759 0.806115i \(-0.298434\pi\)
0.591759 + 0.806115i \(0.298434\pi\)
\(434\) 0 0
\(435\) −28.6274 −1.37258
\(436\) 0 0
\(437\) 22.3835 1.07075
\(438\) 0 0
\(439\) −39.8499 −1.90193 −0.950967 0.309292i \(-0.899908\pi\)
−0.950967 + 0.309292i \(0.899908\pi\)
\(440\) 0 0
\(441\) −4.31371 −0.205415
\(442\) 0 0
\(443\) −18.4853 −0.878262 −0.439131 0.898423i \(-0.644714\pi\)
−0.439131 + 0.898423i \(0.644714\pi\)
\(444\) 0 0
\(445\) 39.5687 1.87574
\(446\) 0 0
\(447\) 3.95687 0.187154
\(448\) 0 0
\(449\) 17.5147 0.826571 0.413285 0.910602i \(-0.364381\pi\)
0.413285 + 0.910602i \(0.364381\pi\)
\(450\) 0 0
\(451\) 15.3137 0.721094
\(452\) 0 0
\(453\) −14.1885 −0.666634
\(454\) 0 0
\(455\) −36.2908 −1.70134
\(456\) 0 0
\(457\) −23.6569 −1.10662 −0.553310 0.832975i \(-0.686636\pi\)
−0.553310 + 0.832975i \(0.686636\pi\)
\(458\) 0 0
\(459\) −0.828427 −0.0386677
\(460\) 0 0
\(461\) −32.8963 −1.53213 −0.766067 0.642761i \(-0.777789\pi\)
−0.766067 + 0.642761i \(0.777789\pi\)
\(462\) 0 0
\(463\) −10.9105 −0.507055 −0.253528 0.967328i \(-0.581591\pi\)
−0.253528 + 0.967328i \(0.581591\pi\)
\(464\) 0 0
\(465\) 6.48528 0.300748
\(466\) 0 0
\(467\) 37.7990 1.74913 0.874564 0.484910i \(-0.161147\pi\)
0.874564 + 0.484910i \(0.161147\pi\)
\(468\) 0 0
\(469\) 22.3835 1.03357
\(470\) 0 0
\(471\) 18.1454 0.836095
\(472\) 0 0
\(473\) 21.6569 0.995783
\(474\) 0 0
\(475\) −30.1421 −1.38302
\(476\) 0 0
\(477\) −0.678892 −0.0310843
\(478\) 0 0
\(479\) 32.2174 1.47205 0.736025 0.676954i \(-0.236700\pi\)
0.736025 + 0.676954i \(0.236700\pi\)
\(480\) 0 0
\(481\) −12.9706 −0.591407
\(482\) 0 0
\(483\) 12.9706 0.590181
\(484\) 0 0
\(485\) 46.1247 2.09442
\(486\) 0 0
\(487\) 20.7445 0.940022 0.470011 0.882661i \(-0.344250\pi\)
0.470011 + 0.882661i \(0.344250\pi\)
\(488\) 0 0
\(489\) 10.1421 0.458643
\(490\) 0 0
\(491\) 1.65685 0.0747728 0.0373864 0.999301i \(-0.488097\pi\)
0.0373864 + 0.999301i \(0.488097\pi\)
\(492\) 0 0
\(493\) 5.99355 0.269936
\(494\) 0 0
\(495\) 19.1055 0.858727
\(496\) 0 0
\(497\) −5.37258 −0.240993
\(498\) 0 0
\(499\) −30.3431 −1.35835 −0.679173 0.733978i \(-0.737661\pi\)
−0.679173 + 0.733978i \(0.737661\pi\)
\(500\) 0 0
\(501\) 4.63577 0.207111
\(502\) 0 0
\(503\) −12.5495 −0.559555 −0.279778 0.960065i \(-0.590261\pi\)
−0.279778 + 0.960065i \(0.590261\pi\)
\(504\) 0 0
\(505\) 2.68629 0.119538
\(506\) 0 0
\(507\) 18.3137 0.813340
\(508\) 0 0
\(509\) 8.59264 0.380862 0.190431 0.981701i \(-0.439011\pi\)
0.190431 + 0.981701i \(0.439011\pi\)
\(510\) 0 0
\(511\) −6.55596 −0.290019
\(512\) 0 0
\(513\) −2.82843 −0.124878
\(514\) 0 0
\(515\) −69.1127 −3.04547
\(516\) 0 0
\(517\) −38.2110 −1.68052
\(518\) 0 0
\(519\) 11.8706 0.521063
\(520\) 0 0
\(521\) 15.1716 0.664679 0.332339 0.943160i \(-0.392162\pi\)
0.332339 + 0.943160i \(0.392162\pi\)
\(522\) 0 0
\(523\) 13.1716 0.575953 0.287976 0.957638i \(-0.407018\pi\)
0.287976 + 0.957638i \(0.407018\pi\)
\(524\) 0 0
\(525\) −17.4665 −0.762300
\(526\) 0 0
\(527\) −1.35778 −0.0591460
\(528\) 0 0
\(529\) 39.6274 1.72293
\(530\) 0 0
\(531\) 9.65685 0.419072
\(532\) 0 0
\(533\) 17.7477 0.768738
\(534\) 0 0
\(535\) −15.8275 −0.684282
\(536\) 0 0
\(537\) 12.9706 0.559721
\(538\) 0 0
\(539\) −20.8284 −0.897144
\(540\) 0 0
\(541\) 10.2316 0.439892 0.219946 0.975512i \(-0.429412\pi\)
0.219946 + 0.975512i \(0.429412\pi\)
\(542\) 0 0
\(543\) 16.7876 0.720425
\(544\) 0 0
\(545\) −66.4264 −2.84539
\(546\) 0 0
\(547\) −7.79899 −0.333461 −0.166730 0.986003i \(-0.553321\pi\)
−0.166730 + 0.986003i \(0.553321\pi\)
\(548\) 0 0
\(549\) −2.31788 −0.0989248
\(550\) 0 0
\(551\) 20.4633 0.871764
\(552\) 0 0
\(553\) 2.68629 0.114233
\(554\) 0 0
\(555\) −9.17157 −0.389312
\(556\) 0 0
\(557\) −27.6981 −1.17361 −0.586804 0.809729i \(-0.699614\pi\)
−0.586804 + 0.809729i \(0.699614\pi\)
\(558\) 0 0
\(559\) 25.0990 1.06158
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 4.82843 0.203494 0.101747 0.994810i \(-0.467557\pi\)
0.101747 + 0.994810i \(0.467557\pi\)
\(564\) 0 0
\(565\) 1.35778 0.0571224
\(566\) 0 0
\(567\) −1.63899 −0.0688312
\(568\) 0 0
\(569\) −27.4558 −1.15101 −0.575504 0.817799i \(-0.695194\pi\)
−0.575504 + 0.817799i \(0.695194\pi\)
\(570\) 0 0
\(571\) 0.686292 0.0287204 0.0143602 0.999897i \(-0.495429\pi\)
0.0143602 + 0.999897i \(0.495429\pi\)
\(572\) 0 0
\(573\) 17.7477 0.741421
\(574\) 0 0
\(575\) −84.3357 −3.51704
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) −14.4697 −0.600305
\(582\) 0 0
\(583\) −3.27798 −0.135760
\(584\) 0 0
\(585\) 22.1421 0.915465
\(586\) 0 0
\(587\) 28.9706 1.19574 0.597872 0.801592i \(-0.296013\pi\)
0.597872 + 0.801592i \(0.296013\pi\)
\(588\) 0 0
\(589\) −4.63577 −0.191013
\(590\) 0 0
\(591\) −16.5064 −0.678982
\(592\) 0 0
\(593\) 30.9706 1.27181 0.635904 0.771768i \(-0.280627\pi\)
0.635904 + 0.771768i \(0.280627\pi\)
\(594\) 0 0
\(595\) 5.37258 0.220254
\(596\) 0 0
\(597\) −1.63899 −0.0670794
\(598\) 0 0
\(599\) 46.1247 1.88460 0.942302 0.334763i \(-0.108656\pi\)
0.942302 + 0.334763i \(0.108656\pi\)
\(600\) 0 0
\(601\) −27.3137 −1.11415 −0.557075 0.830462i \(-0.688076\pi\)
−0.557075 + 0.830462i \(0.688076\pi\)
\(602\) 0 0
\(603\) −13.6569 −0.556150
\(604\) 0 0
\(605\) 48.7238 1.98090
\(606\) 0 0
\(607\) −8.19496 −0.332623 −0.166311 0.986073i \(-0.553186\pi\)
−0.166311 + 0.986073i \(0.553186\pi\)
\(608\) 0 0
\(609\) 11.8579 0.480505
\(610\) 0 0
\(611\) −44.2843 −1.79155
\(612\) 0 0
\(613\) −35.8931 −1.44971 −0.724854 0.688903i \(-0.758093\pi\)
−0.724854 + 0.688903i \(0.758093\pi\)
\(614\) 0 0
\(615\) 12.5495 0.506045
\(616\) 0 0
\(617\) 23.9411 0.963833 0.481917 0.876217i \(-0.339941\pi\)
0.481917 + 0.876217i \(0.339941\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) −7.91375 −0.317568
\(622\) 0 0
\(623\) −16.3899 −0.656648
\(624\) 0 0
\(625\) 35.2843 1.41137
\(626\) 0 0
\(627\) −13.6569 −0.545402
\(628\) 0 0
\(629\) 1.92020 0.0765633
\(630\) 0 0
\(631\) 11.4729 0.456730 0.228365 0.973576i \(-0.426662\pi\)
0.228365 + 0.973576i \(0.426662\pi\)
\(632\) 0 0
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) 56.1421 2.22793
\(636\) 0 0
\(637\) −24.1389 −0.956419
\(638\) 0 0
\(639\) 3.27798 0.129675
\(640\) 0 0
\(641\) 27.1716 1.07321 0.536606 0.843833i \(-0.319706\pi\)
0.536606 + 0.843833i \(0.319706\pi\)
\(642\) 0 0
\(643\) 20.4853 0.807861 0.403930 0.914790i \(-0.367644\pi\)
0.403930 + 0.914790i \(0.367644\pi\)
\(644\) 0 0
\(645\) 17.7477 0.698815
\(646\) 0 0
\(647\) −5.19818 −0.204362 −0.102181 0.994766i \(-0.532582\pi\)
−0.102181 + 0.994766i \(0.532582\pi\)
\(648\) 0 0
\(649\) 46.6274 1.83029
\(650\) 0 0
\(651\) −2.68629 −0.105284
\(652\) 0 0
\(653\) −13.2284 −0.517668 −0.258834 0.965922i \(-0.583338\pi\)
−0.258834 + 0.965922i \(0.583338\pi\)
\(654\) 0 0
\(655\) −28.9394 −1.13076
\(656\) 0 0
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) 1.65685 0.0645419 0.0322709 0.999479i \(-0.489726\pi\)
0.0322709 + 0.999479i \(0.489726\pi\)
\(660\) 0 0
\(661\) −4.23808 −0.164842 −0.0824211 0.996598i \(-0.526265\pi\)
−0.0824211 + 0.996598i \(0.526265\pi\)
\(662\) 0 0
\(663\) −4.63577 −0.180038
\(664\) 0 0
\(665\) 18.3431 0.711317
\(666\) 0 0
\(667\) 57.2548 2.21692
\(668\) 0 0
\(669\) −14.1885 −0.548559
\(670\) 0 0
\(671\) −11.1917 −0.432052
\(672\) 0 0
\(673\) 14.6863 0.566115 0.283057 0.959103i \(-0.408651\pi\)
0.283057 + 0.959103i \(0.408651\pi\)
\(674\) 0 0
\(675\) 10.6569 0.410183
\(676\) 0 0
\(677\) −2.59909 −0.0998911 −0.0499456 0.998752i \(-0.515905\pi\)
−0.0499456 + 0.998752i \(0.515905\pi\)
\(678\) 0 0
\(679\) −19.1055 −0.733201
\(680\) 0 0
\(681\) −12.1421 −0.465288
\(682\) 0 0
\(683\) −5.51472 −0.211015 −0.105507 0.994419i \(-0.533647\pi\)
−0.105507 + 0.994419i \(0.533647\pi\)
\(684\) 0 0
\(685\) −63.8724 −2.44044
\(686\) 0 0
\(687\) −21.4234 −0.817352
\(688\) 0 0
\(689\) −3.79899 −0.144730
\(690\) 0 0
\(691\) 38.8284 1.47710 0.738551 0.674197i \(-0.235510\pi\)
0.738551 + 0.674197i \(0.235510\pi\)
\(692\) 0 0
\(693\) −7.91375 −0.300618
\(694\) 0 0
\(695\) −31.6550 −1.20074
\(696\) 0 0
\(697\) −2.62742 −0.0995205
\(698\) 0 0
\(699\) −21.3137 −0.806158
\(700\) 0 0
\(701\) −30.9761 −1.16995 −0.584976 0.811051i \(-0.698896\pi\)
−0.584976 + 0.811051i \(0.698896\pi\)
\(702\) 0 0
\(703\) 6.55596 0.247263
\(704\) 0 0
\(705\) −31.3137 −1.17934
\(706\) 0 0
\(707\) −1.11270 −0.0418473
\(708\) 0 0
\(709\) 26.0591 0.978671 0.489336 0.872096i \(-0.337239\pi\)
0.489336 + 0.872096i \(0.337239\pi\)
\(710\) 0 0
\(711\) −1.63899 −0.0614670
\(712\) 0 0
\(713\) −12.9706 −0.485751
\(714\) 0 0
\(715\) 106.912 3.99827
\(716\) 0 0
\(717\) 4.63577 0.173126
\(718\) 0 0
\(719\) 34.9330 1.30278 0.651390 0.758743i \(-0.274186\pi\)
0.651390 + 0.758743i \(0.274186\pi\)
\(720\) 0 0
\(721\) 28.6274 1.06614
\(722\) 0 0
\(723\) −16.9706 −0.631142
\(724\) 0 0
\(725\) −77.1008 −2.86345
\(726\) 0 0
\(727\) 30.5784 1.13409 0.567045 0.823687i \(-0.308086\pi\)
0.567045 + 0.823687i \(0.308086\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.71573 −0.137431
\(732\) 0 0
\(733\) −18.7078 −0.690988 −0.345494 0.938421i \(-0.612289\pi\)
−0.345494 + 0.938421i \(0.612289\pi\)
\(734\) 0 0
\(735\) −17.0688 −0.629592
\(736\) 0 0
\(737\) −65.9411 −2.42897
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) −15.8275 −0.581438
\(742\) 0 0
\(743\) 33.5752 1.23175 0.615877 0.787842i \(-0.288802\pi\)
0.615877 + 0.787842i \(0.288802\pi\)
\(744\) 0 0
\(745\) 15.6569 0.573623
\(746\) 0 0
\(747\) 8.82843 0.323015
\(748\) 0 0
\(749\) 6.55596 0.239550
\(750\) 0 0
\(751\) −1.63899 −0.0598076 −0.0299038 0.999553i \(-0.509520\pi\)
−0.0299038 + 0.999553i \(0.509520\pi\)
\(752\) 0 0
\(753\) 20.8284 0.759030
\(754\) 0 0
\(755\) −56.1421 −2.04322
\(756\) 0 0
\(757\) 0.960099 0.0348954 0.0174477 0.999848i \(-0.494446\pi\)
0.0174477 + 0.999848i \(0.494446\pi\)
\(758\) 0 0
\(759\) −38.2110 −1.38697
\(760\) 0 0
\(761\) 30.7696 1.11540 0.557698 0.830044i \(-0.311685\pi\)
0.557698 + 0.830044i \(0.311685\pi\)
\(762\) 0 0
\(763\) 27.5147 0.996100
\(764\) 0 0
\(765\) −3.27798 −0.118516
\(766\) 0 0
\(767\) 54.0385 1.95122
\(768\) 0 0
\(769\) 12.0000 0.432731 0.216366 0.976312i \(-0.430580\pi\)
0.216366 + 0.976312i \(0.430580\pi\)
\(770\) 0 0
\(771\) −19.6569 −0.707924
\(772\) 0 0
\(773\) −5.87707 −0.211384 −0.105692 0.994399i \(-0.533706\pi\)
−0.105692 + 0.994399i \(0.533706\pi\)
\(774\) 0 0
\(775\) 17.4665 0.627415
\(776\) 0 0
\(777\) 3.79899 0.136288
\(778\) 0 0
\(779\) −8.97056 −0.321404
\(780\) 0 0
\(781\) 15.8275 0.566352
\(782\) 0 0
\(783\) −7.23486 −0.258553
\(784\) 0 0
\(785\) 71.7990 2.56262
\(786\) 0 0
\(787\) −20.7696 −0.740355 −0.370177 0.928961i \(-0.620703\pi\)
−0.370177 + 0.928961i \(0.620703\pi\)
\(788\) 0 0
\(789\) 15.8275 0.563474
\(790\) 0 0
\(791\) −0.562413 −0.0199971
\(792\) 0 0
\(793\) −12.9706 −0.460598
\(794\) 0 0
\(795\) −2.68629 −0.0952729
\(796\) 0 0
\(797\) 13.2284 0.468574 0.234287 0.972167i \(-0.424724\pi\)
0.234287 + 0.972167i \(0.424724\pi\)
\(798\) 0 0
\(799\) 6.55596 0.231933
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) 19.3137 0.681566
\(804\) 0 0
\(805\) 51.3229 1.80889
\(806\) 0 0
\(807\) 7.23486 0.254679
\(808\) 0 0
\(809\) 47.4558 1.66846 0.834229 0.551418i \(-0.185913\pi\)
0.834229 + 0.551418i \(0.185913\pi\)
\(810\) 0 0
\(811\) −1.45584 −0.0511216 −0.0255608 0.999673i \(-0.508137\pi\)
−0.0255608 + 0.999673i \(0.508137\pi\)
\(812\) 0 0
\(813\) 17.4665 0.612576
\(814\) 0 0
\(815\) 40.1312 1.40573
\(816\) 0 0
\(817\) −12.6863 −0.443837
\(818\) 0 0
\(819\) −9.17157 −0.320481
\(820\) 0 0
\(821\) −2.03668 −0.0710805 −0.0355403 0.999368i \(-0.511315\pi\)
−0.0355403 + 0.999368i \(0.511315\pi\)
\(822\) 0 0
\(823\) 14.1885 0.494580 0.247290 0.968941i \(-0.420460\pi\)
0.247290 + 0.968941i \(0.420460\pi\)
\(824\) 0 0
\(825\) 51.4558 1.79146
\(826\) 0 0
\(827\) 13.9411 0.484780 0.242390 0.970179i \(-0.422069\pi\)
0.242390 + 0.970179i \(0.422069\pi\)
\(828\) 0 0
\(829\) 5.59587 0.194352 0.0971762 0.995267i \(-0.469019\pi\)
0.0971762 + 0.995267i \(0.469019\pi\)
\(830\) 0 0
\(831\) −14.8674 −0.515744
\(832\) 0 0
\(833\) 3.57359 0.123818
\(834\) 0 0
\(835\) 18.3431 0.634791
\(836\) 0 0
\(837\) 1.63899 0.0566518
\(838\) 0 0
\(839\) 32.2174 1.11227 0.556134 0.831092i \(-0.312284\pi\)
0.556134 + 0.831092i \(0.312284\pi\)
\(840\) 0 0
\(841\) 23.3431 0.804936
\(842\) 0 0
\(843\) 0.343146 0.0118186
\(844\) 0 0
\(845\) 72.4650 2.49287
\(846\) 0 0
\(847\) −20.1821 −0.693464
\(848\) 0 0
\(849\) −28.9706 −0.994267
\(850\) 0 0
\(851\) 18.3431 0.628795
\(852\) 0 0
\(853\) 18.1454 0.621286 0.310643 0.950527i \(-0.399456\pi\)
0.310643 + 0.950527i \(0.399456\pi\)
\(854\) 0 0
\(855\) −11.1917 −0.382749
\(856\) 0 0
\(857\) 13.5147 0.461654 0.230827 0.972995i \(-0.425857\pi\)
0.230827 + 0.972995i \(0.425857\pi\)
\(858\) 0 0
\(859\) −29.4558 −1.00502 −0.502510 0.864571i \(-0.667590\pi\)
−0.502510 + 0.864571i \(0.667590\pi\)
\(860\) 0 0
\(861\) −5.19818 −0.177153
\(862\) 0 0
\(863\) −4.63577 −0.157803 −0.0789017 0.996882i \(-0.525141\pi\)
−0.0789017 + 0.996882i \(0.525141\pi\)
\(864\) 0 0
\(865\) 46.9706 1.59705
\(866\) 0 0
\(867\) −16.3137 −0.554043
\(868\) 0 0
\(869\) −7.91375 −0.268456
\(870\) 0 0
\(871\) −76.4219 −2.58946
\(872\) 0 0
\(873\) 11.6569 0.394525
\(874\) 0 0
\(875\) −36.6863 −1.24022
\(876\) 0 0
\(877\) 58.2765 1.96786 0.983929 0.178558i \(-0.0571432\pi\)
0.983929 + 0.178558i \(0.0571432\pi\)
\(878\) 0 0
\(879\) −0.678892 −0.0228985
\(880\) 0 0
\(881\) −5.02944 −0.169446 −0.0847230 0.996405i \(-0.527001\pi\)
−0.0847230 + 0.996405i \(0.527001\pi\)
\(882\) 0 0
\(883\) −48.7696 −1.64123 −0.820613 0.571484i \(-0.806368\pi\)
−0.820613 + 0.571484i \(0.806368\pi\)
\(884\) 0 0
\(885\) 38.2110 1.28445
\(886\) 0 0
\(887\) 15.8275 0.531435 0.265718 0.964051i \(-0.414391\pi\)
0.265718 + 0.964051i \(0.414391\pi\)
\(888\) 0 0
\(889\) −23.2548 −0.779942
\(890\) 0 0
\(891\) 4.82843 0.161758
\(892\) 0 0
\(893\) 22.3835 0.749034
\(894\) 0 0
\(895\) 51.3229 1.71553
\(896\) 0 0
\(897\) −44.2843 −1.47861
\(898\) 0 0
\(899\) −11.8579 −0.395482
\(900\) 0 0
\(901\) 0.562413 0.0187367
\(902\) 0 0
\(903\) −7.35134 −0.244637
\(904\) 0 0
\(905\) 66.4264 2.20809
\(906\) 0 0
\(907\) 15.7990 0.524597 0.262298 0.964987i \(-0.415520\pi\)
0.262298 + 0.964987i \(0.415520\pi\)
\(908\) 0 0
\(909\) 0.678892 0.0225174
\(910\) 0 0
\(911\) −1.92020 −0.0636190 −0.0318095 0.999494i \(-0.510127\pi\)
−0.0318095 + 0.999494i \(0.510127\pi\)
\(912\) 0 0
\(913\) 42.6274 1.41076
\(914\) 0 0
\(915\) −9.17157 −0.303203
\(916\) 0 0
\(917\) 11.9871 0.395849
\(918\) 0 0
\(919\) 4.91697 0.162196 0.0810980 0.996706i \(-0.474157\pi\)
0.0810980 + 0.996706i \(0.474157\pi\)
\(920\) 0 0
\(921\) −21.6569 −0.713618
\(922\) 0 0
\(923\) 18.3431 0.603772
\(924\) 0 0
\(925\) −24.7013 −0.812175
\(926\) 0 0
\(927\) −17.4665 −0.573675
\(928\) 0 0
\(929\) 21.7990 0.715202 0.357601 0.933875i \(-0.383595\pi\)
0.357601 + 0.933875i \(0.383595\pi\)
\(930\) 0 0
\(931\) 12.2010 0.399872
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −15.8275 −0.517615
\(936\) 0 0
\(937\) −40.6274 −1.32724 −0.663620 0.748070i \(-0.730981\pi\)
−0.663620 + 0.748070i \(0.730981\pi\)
\(938\) 0 0
\(939\) 7.31371 0.238674
\(940\) 0 0
\(941\) 52.7972 1.72114 0.860569 0.509334i \(-0.170108\pi\)
0.860569 + 0.509334i \(0.170108\pi\)
\(942\) 0 0
\(943\) −25.0990 −0.817337
\(944\) 0 0
\(945\) −6.48528 −0.210966
\(946\) 0 0
\(947\) −28.9706 −0.941417 −0.470708 0.882289i \(-0.656002\pi\)
−0.470708 + 0.882289i \(0.656002\pi\)
\(948\) 0 0
\(949\) 22.3835 0.726598
\(950\) 0 0
\(951\) −21.7046 −0.703819
\(952\) 0 0
\(953\) 29.7990 0.965284 0.482642 0.875818i \(-0.339677\pi\)
0.482642 + 0.875818i \(0.339677\pi\)
\(954\) 0 0
\(955\) 70.2254 2.27244
\(956\) 0 0
\(957\) −34.9330 −1.12922
\(958\) 0 0
\(959\) 26.4568 0.854335
\(960\) 0 0
\(961\) −28.3137 −0.913345
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 0 0
\(965\) −15.8275 −0.509505
\(966\) 0 0
\(967\) −43.1279 −1.38690 −0.693450 0.720504i \(-0.743910\pi\)
−0.693450 + 0.720504i \(0.743910\pi\)
\(968\) 0 0
\(969\) 2.34315 0.0752727
\(970\) 0 0
\(971\) −15.4558 −0.496002 −0.248001 0.968760i \(-0.579774\pi\)
−0.248001 + 0.968760i \(0.579774\pi\)
\(972\) 0 0
\(973\) 13.1119 0.420349
\(974\) 0 0
\(975\) 59.6343 1.90983
\(976\) 0 0
\(977\) −42.7696 −1.36832 −0.684160 0.729332i \(-0.739831\pi\)
−0.684160 + 0.729332i \(0.739831\pi\)
\(978\) 0 0
\(979\) 48.2843 1.54317
\(980\) 0 0
\(981\) −16.7876 −0.535987
\(982\) 0 0
\(983\) −38.2110 −1.21874 −0.609370 0.792886i \(-0.708578\pi\)
−0.609370 + 0.792886i \(0.708578\pi\)
\(984\) 0 0
\(985\) −65.3137 −2.08107
\(986\) 0 0
\(987\) 12.9706 0.412858
\(988\) 0 0
\(989\) −35.4954 −1.12869
\(990\) 0 0
\(991\) −46.4059 −1.47413 −0.737066 0.675821i \(-0.763789\pi\)
−0.737066 + 0.675821i \(0.763789\pi\)
\(992\) 0 0
\(993\) 10.3431 0.328230
\(994\) 0 0
\(995\) −6.48528 −0.205597
\(996\) 0 0
\(997\) 29.3371 0.929116 0.464558 0.885543i \(-0.346213\pi\)
0.464558 + 0.885543i \(0.346213\pi\)
\(998\) 0 0
\(999\) −2.31788 −0.0733346
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 1536.2.a.n.1.4 yes 4
3.2 odd 2 4608.2.a.t.1.1 4
4.3 odd 2 1536.2.a.m.1.4 yes 4
8.3 odd 2 inner 1536.2.a.n.1.1 yes 4
8.5 even 2 1536.2.a.m.1.1 4
12.11 even 2 4608.2.a.ba.1.1 4
16.3 odd 4 1536.2.d.g.769.1 8
16.5 even 4 1536.2.d.g.769.4 8
16.11 odd 4 1536.2.d.g.769.8 8
16.13 even 4 1536.2.d.g.769.5 8
24.5 odd 2 4608.2.a.ba.1.4 4
24.11 even 2 4608.2.a.t.1.4 4
48.5 odd 4 4608.2.d.p.2305.2 8
48.11 even 4 4608.2.d.p.2305.1 8
48.29 odd 4 4608.2.d.p.2305.8 8
48.35 even 4 4608.2.d.p.2305.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.m.1.1 4 8.5 even 2
1536.2.a.m.1.4 yes 4 4.3 odd 2
1536.2.a.n.1.1 yes 4 8.3 odd 2 inner
1536.2.a.n.1.4 yes 4 1.1 even 1 trivial
1536.2.d.g.769.1 8 16.3 odd 4
1536.2.d.g.769.4 8 16.5 even 4
1536.2.d.g.769.5 8 16.13 even 4
1536.2.d.g.769.8 8 16.11 odd 4
4608.2.a.t.1.1 4 3.2 odd 2
4608.2.a.t.1.4 4 24.11 even 2
4608.2.a.ba.1.1 4 12.11 even 2
4608.2.a.ba.1.4 4 24.5 odd 2
4608.2.d.p.2305.1 8 48.11 even 4
4608.2.d.p.2305.2 8 48.5 odd 4
4608.2.d.p.2305.7 8 48.35 even 4
4608.2.d.p.2305.8 8 48.29 odd 4