Properties

Label 1536.2.d.g.769.8
Level $1536$
Weight $2$
Character 1536.769
Analytic conductor $12.265$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2650217505\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.8
Root \(0.500000 - 0.691860i\) of defining polynomial
Character \(\chi\) \(=\) 1536.769
Dual form 1536.2.d.g.769.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +3.95687i q^{5} -1.63899 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +3.95687i q^{5} -1.63899 q^{7} -1.00000 q^{9} -4.82843i q^{11} -5.59587i q^{13} -3.95687 q^{15} -0.828427 q^{17} -2.82843i q^{19} -1.63899i q^{21} -7.91375 q^{23} -10.6569 q^{25} -1.00000i q^{27} +7.23486i q^{29} -1.63899 q^{31} +4.82843 q^{33} -6.48528i q^{35} -2.31788i q^{37} +5.59587 q^{39} -3.17157 q^{41} -4.48528i q^{43} -3.95687i q^{45} +7.91375 q^{47} -4.31371 q^{49} -0.828427i q^{51} -0.678892i q^{53} +19.1055 q^{55} +2.82843 q^{57} -9.65685i q^{59} +2.31788i q^{61} +1.63899 q^{63} +22.1421 q^{65} -13.6569i q^{67} -7.91375i q^{69} +3.27798 q^{71} -4.00000 q^{73} -10.6569i q^{75} +7.91375i q^{77} +1.63899 q^{79} +1.00000 q^{81} +8.82843i q^{83} -3.27798i q^{85} -7.23486 q^{87} -10.0000 q^{89} +9.17157i q^{91} -1.63899i q^{93} +11.1917 q^{95} +11.6569 q^{97} +4.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} + 16 q^{17} - 40 q^{25} + 16 q^{33} - 48 q^{41} + 56 q^{49} + 64 q^{65} - 32 q^{73} + 8 q^{81} - 80 q^{89} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(517\) \(1025\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 3.95687i 1.76957i 0.466001 + 0.884784i \(0.345694\pi\)
−0.466001 + 0.884784i \(0.654306\pi\)
\(6\) 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
\(11\) − 4.82843i − 1.45583i −0.685670 0.727913i \(-0.740491\pi\)
0.685670 0.727913i \(-0.259509\pi\)
\(12\) 0 0
\(13\) − 5.59587i − 1.55201i −0.630724 0.776007i \(-0.717242\pi\)
0.630724 0.776007i \(-0.282758\pi\)
\(14\) 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
\(19\) − 2.82843i − 0.648886i −0.945905 0.324443i \(-0.894823\pi\)
0.945905 0.324443i \(-0.105177\pi\)
\(20\) 0 0
\(21\) − 1.63899i − 0.357657i
\(22\) 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
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 7.23486i 1.34348i 0.740787 + 0.671740i \(0.234453\pi\)
−0.740787 + 0.671740i \(0.765547\pi\)
\(30\) 0 0
\(31\) −1.63899 −0.294371 −0.147186 0.989109i \(-0.547022\pi\)
−0.147186 + 0.989109i \(0.547022\pi\)
\(32\) 0 0
\(33\) 4.82843 0.840521
\(34\) 0 0
\(35\) − 6.48528i − 1.09621i
\(36\) 0 0
\(37\) − 2.31788i − 0.381058i −0.981682 0.190529i \(-0.938980\pi\)
0.981682 0.190529i \(-0.0610203\pi\)
\(38\) 0 0
\(39\) 5.59587 0.896056
\(40\) 0 0
\(41\) −3.17157 −0.495316 −0.247658 0.968847i \(-0.579661\pi\)
−0.247658 + 0.968847i \(0.579661\pi\)
\(42\) 0 0
\(43\) − 4.48528i − 0.683999i −0.939700 0.341999i \(-0.888896\pi\)
0.939700 0.341999i \(-0.111104\pi\)
\(44\) 0 0
\(45\) − 3.95687i − 0.589856i
\(46\) 0 0
\(47\) 7.91375 1.15434 0.577169 0.816624i \(-0.304157\pi\)
0.577169 + 0.816624i \(0.304157\pi\)
\(48\) 0 0
\(49\) −4.31371 −0.616244
\(50\) 0 0
\(51\) − 0.828427i − 0.116003i
\(52\) 0 0
\(53\) − 0.678892i − 0.0932530i −0.998912 0.0466265i \(-0.985153\pi\)
0.998912 0.0466265i \(-0.0148471\pi\)
\(54\) 0 0
\(55\) 19.1055 2.57618
\(56\) 0 0
\(57\) 2.82843 0.374634
\(58\) 0 0
\(59\) − 9.65685i − 1.25722i −0.777723 0.628608i \(-0.783625\pi\)
0.777723 0.628608i \(-0.216375\pi\)
\(60\) 0 0
\(61\) 2.31788i 0.296775i 0.988929 + 0.148387i \(0.0474082\pi\)
−0.988929 + 0.148387i \(0.952592\pi\)
\(62\) 0 0
\(63\) 1.63899 0.206493
\(64\) 0 0
\(65\) 22.1421 2.74639
\(66\) 0 0
\(67\) − 13.6569i − 1.66845i −0.551424 0.834225i \(-0.685915\pi\)
0.551424 0.834225i \(-0.314085\pi\)
\(68\) 0 0
\(69\) − 7.91375i − 0.952703i
\(70\) 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 \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) − 10.6569i − 1.23055i
\(76\) 0 0
\(77\) 7.91375i 0.901855i
\(78\) 0 0
\(79\) 1.63899 0.184401 0.0922004 0.995740i \(-0.470610\pi\)
0.0922004 + 0.995740i \(0.470610\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.82843i 0.969046i 0.874779 + 0.484523i \(0.161007\pi\)
−0.874779 + 0.484523i \(0.838993\pi\)
\(84\) 0 0
\(85\) − 3.27798i − 0.355547i
\(86\) 0 0
\(87\) −7.23486 −0.775658
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 9.17157i 0.961442i
\(92\) 0 0
\(93\) − 1.63899i − 0.169955i
\(94\) 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
\(99\) 4.82843i 0.485275i
\(100\) 0 0
\(101\) 0.678892i 0.0675523i 0.999429 + 0.0337762i \(0.0107533\pi\)
−0.999429 + 0.0337762i \(0.989247\pi\)
\(102\) 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
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) 16.7876i 1.60796i 0.594656 + 0.803980i \(0.297288\pi\)
−0.594656 + 0.803980i \(0.702712\pi\)
\(110\) 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
\(115\) − 31.3137i − 2.92002i
\(116\) 0 0
\(117\) 5.59587i 0.517338i
\(118\) 0 0
\(119\) 1.35778 0.124468
\(120\) 0 0
\(121\) −12.3137 −1.11943
\(122\) 0 0
\(123\) − 3.17157i − 0.285971i
\(124\) 0 0
\(125\) − 22.3835i − 2.00204i
\(126\) 0 0
\(127\) −14.1885 −1.25903 −0.629513 0.776990i \(-0.716746\pi\)
−0.629513 + 0.776990i \(0.716746\pi\)
\(128\) 0 0
\(129\) 4.48528 0.394907
\(130\) 0 0
\(131\) − 7.31371i − 0.639002i −0.947586 0.319501i \(-0.896485\pi\)
0.947586 0.319501i \(-0.103515\pi\)
\(132\) 0 0
\(133\) 4.63577i 0.401972i
\(134\) 0 0
\(135\) 3.95687 0.340554
\(136\) 0 0
\(137\) 16.1421 1.37912 0.689558 0.724231i \(-0.257805\pi\)
0.689558 + 0.724231i \(0.257805\pi\)
\(138\) 0 0
\(139\) 8.00000i 0.678551i 0.940687 + 0.339276i \(0.110182\pi\)
−0.940687 + 0.339276i \(0.889818\pi\)
\(140\) 0 0
\(141\) 7.91375i 0.666458i
\(142\) 0 0
\(143\) −27.0192 −2.25946
\(144\) 0 0
\(145\) −28.6274 −2.37738
\(146\) 0 0
\(147\) − 4.31371i − 0.355789i
\(148\) 0 0
\(149\) 3.95687i 0.324160i 0.986778 + 0.162080i \(0.0518202\pi\)
−0.986778 + 0.162080i \(0.948180\pi\)
\(150\) 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
\(155\) − 6.48528i − 0.520910i
\(156\) 0 0
\(157\) − 18.1454i − 1.44816i −0.689717 0.724080i \(-0.742265\pi\)
0.689717 0.724080i \(-0.257735\pi\)
\(158\) 0 0
\(159\) 0.678892 0.0538397
\(160\) 0 0
\(161\) 12.9706 1.02222
\(162\) 0 0
\(163\) 10.1421i 0.794393i 0.917734 + 0.397197i \(0.130017\pi\)
−0.917734 + 0.397197i \(0.869983\pi\)
\(164\) 0 0
\(165\) 19.1055i 1.48736i
\(166\) 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
\(171\) 2.82843i 0.216295i
\(172\) 0 0
\(173\) − 11.8706i − 0.902507i −0.892396 0.451253i \(-0.850977\pi\)
0.892396 0.451253i \(-0.149023\pi\)
\(174\) 0 0
\(175\) 17.4665 1.32034
\(176\) 0 0
\(177\) 9.65685 0.725854
\(178\) 0 0
\(179\) 12.9706i 0.969465i 0.874662 + 0.484733i \(0.161083\pi\)
−0.874662 + 0.484733i \(0.838917\pi\)
\(180\) 0 0
\(181\) 16.7876i 1.24781i 0.781499 + 0.623906i \(0.214455\pi\)
−0.781499 + 0.623906i \(0.785545\pi\)
\(182\) 0 0
\(183\) −2.31788 −0.171343
\(184\) 0 0
\(185\) 9.17157 0.674307
\(186\) 0 0
\(187\) 4.00000i 0.292509i
\(188\) 0 0
\(189\) 1.63899i 0.119219i
\(190\) 0 0
\(191\) −17.7477 −1.28418 −0.642089 0.766630i \(-0.721932\pi\)
−0.642089 + 0.766630i \(0.721932\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 22.1421i 1.58563i
\(196\) 0 0
\(197\) − 16.5064i − 1.17603i −0.808849 0.588016i \(-0.799909\pi\)
0.808849 0.588016i \(-0.200091\pi\)
\(198\) 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
\(203\) − 11.8579i − 0.832259i
\(204\) 0 0
\(205\) − 12.5495i − 0.876496i
\(206\) 0 0
\(207\) 7.91375 0.550044
\(208\) 0 0
\(209\) −13.6569 −0.944664
\(210\) 0 0
\(211\) 8.00000i 0.550743i 0.961338 + 0.275371i \(0.0888008\pi\)
−0.961338 + 0.275371i \(0.911199\pi\)
\(212\) 0 0
\(213\) 3.27798i 0.224604i
\(214\) 0 0
\(215\) 17.7477 1.21038
\(216\) 0 0
\(217\) 2.68629 0.182357
\(218\) 0 0
\(219\) − 4.00000i − 0.270295i
\(220\) 0 0
\(221\) 4.63577i 0.311835i
\(222\) 0 0
\(223\) 14.1885 0.950133 0.475066 0.879950i \(-0.342424\pi\)
0.475066 + 0.879950i \(0.342424\pi\)
\(224\) 0 0
\(225\) 10.6569 0.710457
\(226\) 0 0
\(227\) − 12.1421i − 0.805902i −0.915222 0.402951i \(-0.867985\pi\)
0.915222 0.402951i \(-0.132015\pi\)
\(228\) 0 0
\(229\) − 21.4234i − 1.41570i −0.706365 0.707848i \(-0.749666\pi\)
0.706365 0.707848i \(-0.250334\pi\)
\(230\) 0 0
\(231\) −7.91375 −0.520686
\(232\) 0 0
\(233\) 21.3137 1.39631 0.698154 0.715948i \(-0.254005\pi\)
0.698154 + 0.715948i \(0.254005\pi\)
\(234\) 0 0
\(235\) 31.3137i 2.04268i
\(236\) 0 0
\(237\) 1.63899i 0.106464i
\(238\) 0 0
\(239\) −4.63577 −0.299863 −0.149931 0.988696i \(-0.547905\pi\)
−0.149931 + 0.988696i \(0.547905\pi\)
\(240\) 0 0
\(241\) −16.9706 −1.09317 −0.546585 0.837404i \(-0.684072\pi\)
−0.546585 + 0.837404i \(0.684072\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) − 17.0688i − 1.09049i
\(246\) 0 0
\(247\) −15.8275 −1.00708
\(248\) 0 0
\(249\) −8.82843 −0.559479
\(250\) 0 0
\(251\) − 20.8284i − 1.31468i −0.753595 0.657339i \(-0.771682\pi\)
0.753595 0.657339i \(-0.228318\pi\)
\(252\) 0 0
\(253\) 38.2110i 2.40230i
\(254\) 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
\(259\) 3.79899i 0.236058i
\(260\) 0 0
\(261\) − 7.23486i − 0.447826i
\(262\) 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
\(267\) − 10.0000i − 0.611990i
\(268\) 0 0
\(269\) − 7.23486i − 0.441117i −0.975374 0.220558i \(-0.929212\pi\)
0.975374 0.220558i \(-0.0707880\pi\)
\(270\) 0 0
\(271\) −17.4665 −1.06101 −0.530507 0.847681i \(-0.677998\pi\)
−0.530507 + 0.847681i \(0.677998\pi\)
\(272\) 0 0
\(273\) −9.17157 −0.555089
\(274\) 0 0
\(275\) 51.4558i 3.10290i
\(276\) 0 0
\(277\) − 14.8674i − 0.893295i −0.894710 0.446648i \(-0.852618\pi\)
0.894710 0.446648i \(-0.147382\pi\)
\(278\) 0 0
\(279\) 1.63899 0.0981238
\(280\) 0 0
\(281\) −0.343146 −0.0204704 −0.0102352 0.999948i \(-0.503258\pi\)
−0.0102352 + 0.999948i \(0.503258\pi\)
\(282\) 0 0
\(283\) 28.9706i 1.72212i 0.508502 + 0.861061i \(0.330199\pi\)
−0.508502 + 0.861061i \(0.669801\pi\)
\(284\) 0 0
\(285\) 11.1917i 0.662941i
\(286\) 0 0
\(287\) 5.19818 0.306839
\(288\) 0 0
\(289\) −16.3137 −0.959630
\(290\) 0 0
\(291\) 11.6569i 0.683337i
\(292\) 0 0
\(293\) − 0.678892i − 0.0396613i −0.999803 0.0198307i \(-0.993687\pi\)
0.999803 0.0198307i \(-0.00631271\pi\)
\(294\) 0 0
\(295\) 38.2110 2.22473
\(296\) 0 0
\(297\) −4.82843 −0.280174
\(298\) 0 0
\(299\) 44.2843i 2.56103i
\(300\) 0 0
\(301\) 7.35134i 0.423724i
\(302\) 0 0
\(303\) −0.678892 −0.0390013
\(304\) 0 0
\(305\) −9.17157 −0.525163
\(306\) 0 0
\(307\) − 21.6569i − 1.23602i −0.786169 0.618011i \(-0.787939\pi\)
0.786169 0.618011i \(-0.212061\pi\)
\(308\) 0 0
\(309\) − 17.4665i − 0.993634i
\(310\) 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 \(-0.566272\pi\)
−0.206698 + 0.978405i \(0.566272\pi\)
\(314\) 0 0
\(315\) 6.48528i 0.365404i
\(316\) 0 0
\(317\) 21.7046i 1.21905i 0.792767 + 0.609525i \(0.208640\pi\)
−0.792767 + 0.609525i \(0.791360\pi\)
\(318\) 0 0
\(319\) 34.9330 1.95587
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 2.34315i 0.130376i
\(324\) 0 0
\(325\) 59.6343i 3.30792i
\(326\) 0 0
\(327\) −16.7876 −0.928356
\(328\) 0 0
\(329\) −12.9706 −0.715090
\(330\) 0 0
\(331\) − 10.3431i − 0.568511i −0.958749 0.284255i \(-0.908254\pi\)
0.958749 0.284255i \(-0.0917463\pi\)
\(332\) 0 0
\(333\) 2.31788i 0.127019i
\(334\) 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
\(339\) 0.343146i 0.0186371i
\(340\) 0 0
\(341\) 7.91375i 0.428554i
\(342\) 0 0
\(343\) 18.5431 1.00123
\(344\) 0 0
\(345\) 31.3137 1.68587
\(346\) 0 0
\(347\) − 0.828427i − 0.0444723i −0.999753 0.0222361i \(-0.992921\pi\)
0.999753 0.0222361i \(-0.00707857\pi\)
\(348\) 0 0
\(349\) − 4.23808i − 0.226859i −0.993546 0.113430i \(-0.963816\pi\)
0.993546 0.113430i \(-0.0361836\pi\)
\(350\) 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
\(355\) 12.9706i 0.688406i
\(356\) 0 0
\(357\) 1.35778i 0.0718616i
\(358\) 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
\(363\) − 12.3137i − 0.646302i
\(364\) 0 0
\(365\) − 15.8275i − 0.828449i
\(366\) 0 0
\(367\) −24.0225 −1.25396 −0.626981 0.779035i \(-0.715710\pi\)
−0.626981 + 0.779035i \(0.715710\pi\)
\(368\) 0 0
\(369\) 3.17157 0.165105
\(370\) 0 0
\(371\) 1.11270i 0.0577684i
\(372\) 0 0
\(373\) − 20.0656i − 1.03896i −0.854484 0.519478i \(-0.826126\pi\)
0.854484 0.519478i \(-0.173874\pi\)
\(374\) 0 0
\(375\) 22.3835 1.15588
\(376\) 0 0
\(377\) 40.4853 2.08510
\(378\) 0 0
\(379\) 0.485281i 0.0249272i 0.999922 + 0.0124636i \(0.00396739\pi\)
−0.999922 + 0.0124636i \(0.996033\pi\)
\(380\) 0 0
\(381\) − 14.1885i − 0.726899i
\(382\) 0 0
\(383\) −11.1917 −0.571871 −0.285935 0.958249i \(-0.592304\pi\)
−0.285935 + 0.958249i \(0.592304\pi\)
\(384\) 0 0
\(385\) −31.3137 −1.59589
\(386\) 0 0
\(387\) 4.48528i 0.228000i
\(388\) 0 0
\(389\) − 3.95687i − 0.200621i −0.994956 0.100311i \(-0.968016\pi\)
0.994956 0.100311i \(-0.0319837\pi\)
\(390\) 0 0
\(391\) 6.55596 0.331549
\(392\) 0 0
\(393\) 7.31371 0.368928
\(394\) 0 0
\(395\) 6.48528i 0.326310i
\(396\) 0 0
\(397\) − 18.1454i − 0.910691i −0.890315 0.455345i \(-0.849516\pi\)
0.890315 0.455345i \(-0.150484\pi\)
\(398\) 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
\(403\) 9.17157i 0.456869i
\(404\) 0 0
\(405\) 3.95687i 0.196619i
\(406\) 0 0
\(407\) −11.1917 −0.554753
\(408\) 0 0
\(409\) 3.65685 0.180820 0.0904099 0.995905i \(-0.471182\pi\)
0.0904099 + 0.995905i \(0.471182\pi\)
\(410\) 0 0
\(411\) 16.1421i 0.796233i
\(412\) 0 0
\(413\) 15.8275i 0.778820i
\(414\) 0 0
\(415\) −34.9330 −1.71479
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 15.1716i 0.741180i 0.928797 + 0.370590i \(0.120844\pi\)
−0.928797 + 0.370590i \(0.879156\pi\)
\(420\) 0 0
\(421\) − 21.4234i − 1.04411i −0.852912 0.522055i \(-0.825165\pi\)
0.852912 0.522055i \(-0.174835\pi\)
\(422\) 0 0
\(423\) −7.91375 −0.384780
\(424\) 0 0
\(425\) 8.82843 0.428242
\(426\) 0 0
\(427\) − 3.79899i − 0.183846i
\(428\) 0 0
\(429\) − 27.0192i − 1.30450i
\(430\) 0 0
\(431\) −14.4697 −0.696982 −0.348491 0.937312i \(-0.613306\pi\)
−0.348491 + 0.937312i \(0.613306\pi\)
\(432\) 0 0
\(433\) 24.6274 1.18352 0.591759 0.806115i \(-0.298434\pi\)
0.591759 + 0.806115i \(0.298434\pi\)
\(434\) 0 0
\(435\) − 28.6274i − 1.37258i
\(436\) 0 0
\(437\) 22.3835i 1.07075i
\(438\) 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
\(443\) 18.4853i 0.878262i 0.898423 + 0.439131i \(0.144714\pi\)
−0.898423 + 0.439131i \(0.855286\pi\)
\(444\) 0 0
\(445\) − 39.5687i − 1.87574i
\(446\) 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
\(451\) 15.3137i 0.721094i
\(452\) 0 0
\(453\) − 14.1885i − 0.666634i
\(454\) 0 0
\(455\) −36.2908 −1.70134
\(456\) 0 0
\(457\) 23.6569 1.10662 0.553310 0.832975i \(-0.313364\pi\)
0.553310 + 0.832975i \(0.313364\pi\)
\(458\) 0 0
\(459\) 0.828427i 0.0386677i
\(460\) 0 0
\(461\) 32.8963i 1.53213i 0.642761 + 0.766067i \(0.277789\pi\)
−0.642761 + 0.766067i \(0.722211\pi\)
\(462\) 0 0
\(463\) 10.9105 0.507055 0.253528 0.967328i \(-0.418409\pi\)
0.253528 + 0.967328i \(0.418409\pi\)
\(464\) 0 0
\(465\) 6.48528 0.300748
\(466\) 0 0
\(467\) 37.7990i 1.74913i 0.484910 + 0.874564i \(0.338853\pi\)
−0.484910 + 0.874564i \(0.661147\pi\)
\(468\) 0 0
\(469\) 22.3835i 1.03357i
\(470\) 0 0
\(471\) 18.1454 0.836095
\(472\) 0 0
\(473\) −21.6569 −0.995783
\(474\) 0 0
\(475\) 30.1421i 1.38302i
\(476\) 0 0
\(477\) 0.678892i 0.0310843i
\(478\) 0 0
\(479\) −32.2174 −1.47205 −0.736025 0.676954i \(-0.763300\pi\)
−0.736025 + 0.676954i \(0.763300\pi\)
\(480\) 0 0
\(481\) −12.9706 −0.591407
\(482\) 0 0
\(483\) 12.9706i 0.590181i
\(484\) 0 0
\(485\) 46.1247i 2.09442i
\(486\) 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
\(491\) − 1.65685i − 0.0747728i −0.999301 0.0373864i \(-0.988097\pi\)
0.999301 0.0373864i \(-0.0119032\pi\)
\(492\) 0 0
\(493\) − 5.99355i − 0.269936i
\(494\) 0 0
\(495\) −19.1055 −0.858727
\(496\) 0 0
\(497\) −5.37258 −0.240993
\(498\) 0 0
\(499\) − 30.3431i − 1.35835i −0.733978 0.679173i \(-0.762339\pi\)
0.733978 0.679173i \(-0.237661\pi\)
\(500\) 0 0
\(501\) 4.63577i 0.207111i
\(502\) 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
\(507\) − 18.3137i − 0.813340i
\(508\) 0 0
\(509\) − 8.59264i − 0.380862i −0.981701 0.190431i \(-0.939011\pi\)
0.981701 0.190431i \(-0.0609886\pi\)
\(510\) 0 0
\(511\) 6.55596 0.290019
\(512\) 0 0
\(513\) −2.82843 −0.124878
\(514\) 0 0
\(515\) − 69.1127i − 3.04547i
\(516\) 0 0
\(517\) − 38.2110i − 1.68052i
\(518\) 0 0
\(519\) 11.8706 0.521063
\(520\) 0 0
\(521\) −15.1716 −0.664679 −0.332339 0.943160i \(-0.607838\pi\)
−0.332339 + 0.943160i \(0.607838\pi\)
\(522\) 0 0
\(523\) − 13.1716i − 0.575953i −0.957638 0.287976i \(-0.907018\pi\)
0.957638 0.287976i \(-0.0929824\pi\)
\(524\) 0 0
\(525\) 17.4665i 0.762300i
\(526\) 0 0
\(527\) 1.35778 0.0591460
\(528\) 0 0
\(529\) 39.6274 1.72293
\(530\) 0 0
\(531\) 9.65685i 0.419072i
\(532\) 0 0
\(533\) 17.7477i 0.768738i
\(534\) 0 0
\(535\) −15.8275 −0.684282
\(536\) 0 0
\(537\) −12.9706 −0.559721
\(538\) 0 0
\(539\) 20.8284i 0.897144i
\(540\) 0 0
\(541\) − 10.2316i − 0.439892i −0.975512 0.219946i \(-0.929412\pi\)
0.975512 0.219946i \(-0.0705882\pi\)
\(542\) 0 0
\(543\) −16.7876 −0.720425
\(544\) 0 0
\(545\) −66.4264 −2.84539
\(546\) 0 0
\(547\) − 7.79899i − 0.333461i −0.986003 0.166730i \(-0.946679\pi\)
0.986003 0.166730i \(-0.0533209\pi\)
\(548\) 0 0
\(549\) − 2.31788i − 0.0989248i
\(550\) 0 0
\(551\) 20.4633 0.871764
\(552\) 0 0
\(553\) −2.68629 −0.114233
\(554\) 0 0
\(555\) 9.17157i 0.389312i
\(556\) 0 0
\(557\) 27.6981i 1.17361i 0.809729 + 0.586804i \(0.199614\pi\)
−0.809729 + 0.586804i \(0.800386\pi\)
\(558\) 0 0
\(559\) −25.0990 −1.06158
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 4.82843i 0.203494i 0.994810 + 0.101747i \(0.0324432\pi\)
−0.994810 + 0.101747i \(0.967557\pi\)
\(564\) 0 0
\(565\) 1.35778i 0.0571224i
\(566\) 0 0
\(567\) −1.63899 −0.0688312
\(568\) 0 0
\(569\) 27.4558 1.15101 0.575504 0.817799i \(-0.304806\pi\)
0.575504 + 0.817799i \(0.304806\pi\)
\(570\) 0 0
\(571\) − 0.686292i − 0.0287204i −0.999897 0.0143602i \(-0.995429\pi\)
0.999897 0.0143602i \(-0.00457115\pi\)
\(572\) 0 0
\(573\) − 17.7477i − 0.741421i
\(574\) 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
\(579\) − 4.00000i − 0.166234i
\(580\) 0 0
\(581\) − 14.4697i − 0.600305i
\(582\) 0 0
\(583\) −3.27798 −0.135760
\(584\) 0 0
\(585\) −22.1421 −0.915465
\(586\) 0 0
\(587\) − 28.9706i − 1.19574i −0.801592 0.597872i \(-0.796013\pi\)
0.801592 0.597872i \(-0.203987\pi\)
\(588\) 0 0
\(589\) 4.63577i 0.191013i
\(590\) 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
\(595\) 5.37258i 0.220254i
\(596\) 0 0
\(597\) − 1.63899i − 0.0670794i
\(598\) 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 \(-0.311924\pi\)
0.557075 + 0.830462i \(0.311924\pi\)
\(602\) 0 0
\(603\) 13.6569i 0.556150i
\(604\) 0 0
\(605\) − 48.7238i − 1.98090i
\(606\) 0 0
\(607\) 8.19496 0.332623 0.166311 0.986073i \(-0.446814\pi\)
0.166311 + 0.986073i \(0.446814\pi\)
\(608\) 0 0
\(609\) 11.8579 0.480505
\(610\) 0 0
\(611\) − 44.2843i − 1.79155i
\(612\) 0 0
\(613\) − 35.8931i − 1.44971i −0.688903 0.724854i \(-0.741907\pi\)
0.688903 0.724854i \(-0.258093\pi\)
\(614\) 0 0
\(615\) 12.5495 0.506045
\(616\) 0 0
\(617\) −23.9411 −0.963833 −0.481917 0.876217i \(-0.660059\pi\)
−0.481917 + 0.876217i \(0.660059\pi\)
\(618\) 0 0
\(619\) 28.0000i 1.12542i 0.826656 + 0.562708i \(0.190240\pi\)
−0.826656 + 0.562708i \(0.809760\pi\)
\(620\) 0 0
\(621\) 7.91375i 0.317568i
\(622\) 0 0
\(623\) 16.3899 0.656648
\(624\) 0 0
\(625\) 35.2843 1.41137
\(626\) 0 0
\(627\) − 13.6569i − 0.545402i
\(628\) 0 0
\(629\) 1.92020i 0.0765633i
\(630\) 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
\(635\) − 56.1421i − 2.22793i
\(636\) 0 0
\(637\) 24.1389i 0.956419i
\(638\) 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
\(643\) 20.4853i 0.807861i 0.914790 + 0.403930i \(0.132356\pi\)
−0.914790 + 0.403930i \(0.867644\pi\)
\(644\) 0 0
\(645\) 17.7477i 0.698815i
\(646\) 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
\(651\) 2.68629i 0.105284i
\(652\) 0 0
\(653\) 13.2284i 0.517668i 0.965922 + 0.258834i \(0.0833382\pi\)
−0.965922 + 0.258834i \(0.916662\pi\)
\(654\) 0 0
\(655\) 28.9394 1.13076
\(656\) 0 0
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) 1.65685i 0.0645419i 0.999479 + 0.0322709i \(0.0102739\pi\)
−0.999479 + 0.0322709i \(0.989726\pi\)
\(660\) 0 0
\(661\) − 4.23808i − 0.164842i −0.996598 0.0824211i \(-0.973735\pi\)
0.996598 0.0824211i \(-0.0262653\pi\)
\(662\) 0 0
\(663\) −4.63577 −0.180038
\(664\) 0 0
\(665\) −18.3431 −0.711317
\(666\) 0 0
\(667\) − 57.2548i − 2.21692i
\(668\) 0 0
\(669\) 14.1885i 0.548559i
\(670\) 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
\(675\) 10.6569i 0.410183i
\(676\) 0 0
\(677\) − 2.59909i − 0.0998911i −0.998752 0.0499456i \(-0.984095\pi\)
0.998752 0.0499456i \(-0.0159048\pi\)
\(678\) 0 0
\(679\) −19.1055 −0.733201
\(680\) 0 0
\(681\) 12.1421 0.465288
\(682\) 0 0
\(683\) 5.51472i 0.211015i 0.994419 + 0.105507i \(0.0336467\pi\)
−0.994419 + 0.105507i \(0.966353\pi\)
\(684\) 0 0
\(685\) 63.8724i 2.44044i
\(686\) 0 0
\(687\) 21.4234 0.817352
\(688\) 0 0
\(689\) −3.79899 −0.144730
\(690\) 0 0
\(691\) 38.8284i 1.47710i 0.674197 + 0.738551i \(0.264490\pi\)
−0.674197 + 0.738551i \(0.735510\pi\)
\(692\) 0 0
\(693\) − 7.91375i − 0.300618i
\(694\) 0 0
\(695\) −31.6550 −1.20074
\(696\) 0 0
\(697\) 2.62742 0.0995205
\(698\) 0 0
\(699\) 21.3137i 0.806158i
\(700\) 0 0
\(701\) 30.9761i 1.16995i 0.811051 + 0.584976i \(0.198896\pi\)
−0.811051 + 0.584976i \(0.801104\pi\)
\(702\) 0 0
\(703\) −6.55596 −0.247263
\(704\) 0 0
\(705\) −31.3137 −1.17934
\(706\) 0 0
\(707\) − 1.11270i − 0.0418473i
\(708\) 0 0
\(709\) 26.0591i 0.978671i 0.872096 + 0.489336i \(0.162761\pi\)
−0.872096 + 0.489336i \(0.837239\pi\)
\(710\) 0 0
\(711\) −1.63899 −0.0614670
\(712\) 0 0
\(713\) 12.9706 0.485751
\(714\) 0 0
\(715\) − 106.912i − 3.99827i
\(716\) 0 0
\(717\) − 4.63577i − 0.173126i
\(718\) 0 0
\(719\) −34.9330 −1.30278 −0.651390 0.758743i \(-0.725814\pi\)
−0.651390 + 0.758743i \(0.725814\pi\)
\(720\) 0 0
\(721\) 28.6274 1.06614
\(722\) 0 0
\(723\) − 16.9706i − 0.631142i
\(724\) 0 0
\(725\) − 77.1008i − 2.86345i
\(726\) 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
\(731\) 3.71573i 0.137431i
\(732\) 0 0
\(733\) 18.7078i 0.690988i 0.938421 + 0.345494i \(0.112289\pi\)
−0.938421 + 0.345494i \(0.887711\pi\)
\(734\) 0 0
\(735\) 17.0688 0.629592
\(736\) 0 0
\(737\) −65.9411 −2.42897
\(738\) 0 0
\(739\) − 4.00000i − 0.147142i −0.997290 0.0735712i \(-0.976560\pi\)
0.997290 0.0735712i \(-0.0234396\pi\)
\(740\) 0 0
\(741\) − 15.8275i − 0.581438i
\(742\) 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
\(747\) − 8.82843i − 0.323015i
\(748\) 0 0
\(749\) − 6.55596i − 0.239550i
\(750\) 0 0
\(751\) 1.63899 0.0598076 0.0299038 0.999553i \(-0.490480\pi\)
0.0299038 + 0.999553i \(0.490480\pi\)
\(752\) 0 0
\(753\) 20.8284 0.759030
\(754\) 0 0
\(755\) − 56.1421i − 2.04322i
\(756\) 0 0
\(757\) 0.960099i 0.0348954i 0.999848 + 0.0174477i \(0.00555405\pi\)
−0.999848 + 0.0174477i \(0.994446\pi\)
\(758\) 0 0
\(759\) −38.2110 −1.38697
\(760\) 0 0
\(761\) −30.7696 −1.11540 −0.557698 0.830044i \(-0.688315\pi\)
−0.557698 + 0.830044i \(0.688315\pi\)
\(762\) 0 0
\(763\) − 27.5147i − 0.996100i
\(764\) 0 0
\(765\) 3.27798i 0.118516i
\(766\) 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
\(771\) − 19.6569i − 0.707924i
\(772\) 0 0
\(773\) − 5.87707i − 0.211384i −0.994399 0.105692i \(-0.966294\pi\)
0.994399 0.105692i \(-0.0337057\pi\)
\(774\) 0 0
\(775\) 17.4665 0.627415
\(776\) 0 0
\(777\) −3.79899 −0.136288
\(778\) 0 0
\(779\) 8.97056i 0.321404i
\(780\) 0 0
\(781\) − 15.8275i − 0.566352i
\(782\) 0 0
\(783\) 7.23486 0.258553
\(784\) 0 0
\(785\) 71.7990 2.56262
\(786\) 0 0
\(787\) − 20.7696i − 0.740355i −0.928961 0.370177i \(-0.879297\pi\)
0.928961 0.370177i \(-0.120703\pi\)
\(788\) 0 0
\(789\) 15.8275i 0.563474i
\(790\) 0 0
\(791\) −0.562413 −0.0199971
\(792\) 0 0
\(793\) 12.9706 0.460598
\(794\) 0 0
\(795\) 2.68629i 0.0952729i
\(796\) 0 0
\(797\) − 13.2284i − 0.468574i −0.972167 0.234287i \(-0.924724\pi\)
0.972167 0.234287i \(-0.0752756\pi\)
\(798\) 0 0
\(799\) −6.55596 −0.231933
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) 19.3137i 0.681566i
\(804\) 0 0
\(805\) 51.3229i 1.80889i
\(806\) 0 0
\(807\) 7.23486 0.254679
\(808\) 0 0
\(809\) −47.4558 −1.66846 −0.834229 0.551418i \(-0.814087\pi\)
−0.834229 + 0.551418i \(0.814087\pi\)
\(810\) 0 0
\(811\) 1.45584i 0.0511216i 0.999673 + 0.0255608i \(0.00813714\pi\)
−0.999673 + 0.0255608i \(0.991863\pi\)
\(812\) 0 0
\(813\) − 17.4665i − 0.612576i
\(814\) 0 0
\(815\) −40.1312 −1.40573
\(816\) 0 0
\(817\) −12.6863 −0.443837
\(818\) 0 0
\(819\) − 9.17157i − 0.320481i
\(820\) 0 0
\(821\) − 2.03668i − 0.0710805i −0.999368 0.0355403i \(-0.988685\pi\)
0.999368 0.0355403i \(-0.0113152\pi\)
\(822\) 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
\(827\) − 13.9411i − 0.484780i −0.970179 0.242390i \(-0.922069\pi\)
0.970179 0.242390i \(-0.0779314\pi\)
\(828\) 0 0
\(829\) − 5.59587i − 0.194352i −0.995267 0.0971762i \(-0.969019\pi\)
0.995267 0.0971762i \(-0.0309810\pi\)
\(830\) 0 0
\(831\) 14.8674 0.515744
\(832\) 0 0
\(833\) 3.57359 0.123818
\(834\) 0 0
\(835\) 18.3431i 0.634791i
\(836\) 0 0
\(837\) 1.63899i 0.0566518i
\(838\) 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
\(843\) − 0.343146i − 0.0118186i
\(844\) 0 0
\(845\) − 72.4650i − 2.49287i
\(846\) 0 0
\(847\) 20.1821 0.693464
\(848\) 0 0
\(849\) −28.9706 −0.994267
\(850\) 0 0
\(851\) 18.3431i 0.628795i
\(852\) 0 0
\(853\) 18.1454i 0.621286i 0.950527 + 0.310643i \(0.100544\pi\)
−0.950527 + 0.310643i \(0.899456\pi\)
\(854\) 0 0
\(855\) −11.1917 −0.382749
\(856\) 0 0
\(857\) −13.5147 −0.461654 −0.230827 0.972995i \(-0.574143\pi\)
−0.230827 + 0.972995i \(0.574143\pi\)
\(858\) 0 0
\(859\) 29.4558i 1.00502i 0.864571 + 0.502510i \(0.167590\pi\)
−0.864571 + 0.502510i \(0.832410\pi\)
\(860\) 0 0
\(861\) 5.19818i 0.177153i
\(862\) 0 0
\(863\) 4.63577 0.157803 0.0789017 0.996882i \(-0.474859\pi\)
0.0789017 + 0.996882i \(0.474859\pi\)
\(864\) 0 0
\(865\) 46.9706 1.59705
\(866\) 0 0
\(867\) − 16.3137i − 0.554043i
\(868\) 0 0
\(869\) − 7.91375i − 0.268456i
\(870\) 0 0
\(871\) −76.4219 −2.58946
\(872\) 0 0
\(873\) −11.6569 −0.394525
\(874\) 0 0
\(875\) 36.6863i 1.24022i
\(876\) 0 0
\(877\) − 58.2765i − 1.96786i −0.178558 0.983929i \(-0.557143\pi\)
0.178558 0.983929i \(-0.442857\pi\)
\(878\) 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
\(883\) − 48.7696i − 1.64123i −0.571484 0.820613i \(-0.693632\pi\)
0.571484 0.820613i \(-0.306368\pi\)
\(884\) 0 0
\(885\) 38.2110i 1.28445i
\(886\) 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
\(891\) − 4.82843i − 0.161758i
\(892\) 0 0
\(893\) − 22.3835i − 0.749034i
\(894\) 0 0
\(895\) −51.3229 −1.71553
\(896\) 0 0
\(897\) −44.2843 −1.47861
\(898\) 0 0
\(899\) − 11.8579i − 0.395482i
\(900\) 0 0
\(901\) 0.562413i 0.0187367i
\(902\) 0 0
\(903\) −7.35134 −0.244637
\(904\) 0 0
\(905\) −66.4264 −2.20809
\(906\) 0 0
\(907\) − 15.7990i − 0.524597i −0.964987 0.262298i \(-0.915520\pi\)
0.964987 0.262298i \(-0.0844805\pi\)
\(908\) 0 0
\(909\) − 0.678892i − 0.0225174i
\(910\) 0 0
\(911\) 1.92020 0.0636190 0.0318095 0.999494i \(-0.489873\pi\)
0.0318095 + 0.999494i \(0.489873\pi\)
\(912\) 0 0
\(913\) 42.6274 1.41076
\(914\) 0 0
\(915\) − 9.17157i − 0.303203i
\(916\) 0 0
\(917\) 11.9871i 0.395849i
\(918\) 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
\(923\) − 18.3431i − 0.603772i
\(924\) 0 0
\(925\) 24.7013i 0.812175i
\(926\) 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
\(931\) 12.2010i 0.399872i
\(932\) 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 \(-0.269019\pi\)
0.663620 + 0.748070i \(0.269019\pi\)
\(938\) 0 0
\(939\) − 7.31371i − 0.238674i
\(940\) 0 0
\(941\) − 52.7972i − 1.72114i −0.509334 0.860569i \(-0.670108\pi\)
0.509334 0.860569i \(-0.329892\pi\)
\(942\) 0 0
\(943\) 25.0990 0.817337
\(944\) 0 0
\(945\) −6.48528 −0.210966
\(946\) 0 0
\(947\) − 28.9706i − 0.941417i −0.882289 0.470708i \(-0.843998\pi\)
0.882289 0.470708i \(-0.156002\pi\)
\(948\) 0 0
\(949\) 22.3835i 0.726598i
\(950\) 0 0
\(951\) −21.7046 −0.703819
\(952\) 0 0
\(953\) −29.7990 −0.965284 −0.482642 0.875818i \(-0.660323\pi\)
−0.482642 + 0.875818i \(0.660323\pi\)
\(954\) 0 0
\(955\) − 70.2254i − 2.27244i
\(956\) 0 0
\(957\) 34.9330i 1.12922i
\(958\) 0 0
\(959\) −26.4568 −0.854335
\(960\) 0 0
\(961\) −28.3137 −0.913345
\(962\) 0 0
\(963\) − 4.00000i − 0.128898i
\(964\) 0 0
\(965\) − 15.8275i − 0.509505i
\(966\) 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
\(971\) 15.4558i 0.496002i 0.968760 + 0.248001i \(0.0797736\pi\)
−0.968760 + 0.248001i \(0.920226\pi\)
\(972\) 0 0
\(973\) − 13.1119i − 0.420349i
\(974\) 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
\(979\) 48.2843i 1.54317i
\(980\) 0 0
\(981\) − 16.7876i − 0.535987i
\(982\) 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
\(987\) − 12.9706i − 0.412858i
\(988\) 0 0
\(989\) 35.4954i 1.12869i
\(990\) 0 0
\(991\) 46.4059 1.47413 0.737066 0.675821i \(-0.236211\pi\)
0.737066 + 0.675821i \(0.236211\pi\)
\(992\) 0 0
\(993\) 10.3431 0.328230
\(994\) 0 0
\(995\) − 6.48528i − 0.205597i
\(996\) 0 0
\(997\) 29.3371i 0.929116i 0.885543 + 0.464558i \(0.153787\pi\)
−0.885543 + 0.464558i \(0.846213\pi\)
\(998\) 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.d.g.769.8 8
3.2 odd 2 4608.2.d.p.2305.1 8
4.3 odd 2 inner 1536.2.d.g.769.4 8
8.3 odd 2 inner 1536.2.d.g.769.5 8
8.5 even 2 inner 1536.2.d.g.769.1 8
12.11 even 2 4608.2.d.p.2305.2 8
16.3 odd 4 1536.2.a.n.1.4 yes 4
16.5 even 4 1536.2.a.n.1.1 yes 4
16.11 odd 4 1536.2.a.m.1.1 4
16.13 even 4 1536.2.a.m.1.4 yes 4
24.5 odd 2 4608.2.d.p.2305.7 8
24.11 even 2 4608.2.d.p.2305.8 8
48.5 odd 4 4608.2.a.t.1.4 4
48.11 even 4 4608.2.a.ba.1.4 4
48.29 odd 4 4608.2.a.ba.1.1 4
48.35 even 4 4608.2.a.t.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.m.1.1 4 16.11 odd 4
1536.2.a.m.1.4 yes 4 16.13 even 4
1536.2.a.n.1.1 yes 4 16.5 even 4
1536.2.a.n.1.4 yes 4 16.3 odd 4
1536.2.d.g.769.1 8 8.5 even 2 inner
1536.2.d.g.769.4 8 4.3 odd 2 inner
1536.2.d.g.769.5 8 8.3 odd 2 inner
1536.2.d.g.769.8 8 1.1 even 1 trivial
4608.2.a.t.1.1 4 48.35 even 4
4608.2.a.t.1.4 4 48.5 odd 4
4608.2.a.ba.1.1 4 48.29 odd 4
4608.2.a.ba.1.4 4 48.11 even 4
4608.2.d.p.2305.1 8 3.2 odd 2
4608.2.d.p.2305.2 8 12.11 even 2
4608.2.d.p.2305.7 8 24.5 odd 2
4608.2.d.p.2305.8 8 24.11 even 2