Properties

Label 768.4.a.q.1.3
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.3134668844\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1436.1
Defining polynomial: \( x^{3} - 11x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.28282\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +9.15486 q^{5} +27.4175 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +9.15486 q^{5} +27.4175 q^{7} +9.00000 q^{9} -20.5252 q^{11} -32.0471 q^{13} -27.4646 q^{15} -111.764 q^{17} -129.764 q^{19} -82.2524 q^{21} +9.16510 q^{23} -41.1885 q^{25} -27.0000 q^{27} -41.0606 q^{29} +187.606 q^{31} +61.5755 q^{33} +251.003 q^{35} +114.127 q^{37} +96.1414 q^{39} -282.915 q^{41} +89.3870 q^{43} +82.3937 q^{45} +54.6464 q^{47} +408.717 q^{49} +335.292 q^{51} -726.878 q^{53} -187.905 q^{55} +389.292 q^{57} +216.579 q^{59} -754.222 q^{61} +246.757 q^{63} -293.387 q^{65} -379.433 q^{67} -27.4953 q^{69} +302.080 q^{71} +504.396 q^{73} +123.566 q^{75} -562.748 q^{77} -301.780 q^{79} +81.0000 q^{81} -599.003 q^{83} -1023.18 q^{85} +123.182 q^{87} +277.528 q^{89} -878.651 q^{91} -562.818 q^{93} -1187.97 q^{95} -765.905 q^{97} -184.727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 9 q^{3} - 10 q^{5} + 14 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 9 q^{3} - 10 q^{5} + 14 q^{7} + 27 q^{9} - 52 q^{13} + 30 q^{15} + 26 q^{17} - 28 q^{19} - 42 q^{21} + 164 q^{23} + 53 q^{25} - 81 q^{27} - 174 q^{29} + 318 q^{31} + 92 q^{35} - 296 q^{37} + 156 q^{39} - 118 q^{41} + 260 q^{43} - 90 q^{45} + 204 q^{47} + 327 q^{49} - 78 q^{51} - 1086 q^{53} + 512 q^{55} + 84 q^{57} - 196 q^{59} - 1536 q^{61} + 126 q^{63} - 872 q^{65} + 660 q^{67} - 492 q^{69} - 852 q^{71} - 478 q^{73} - 159 q^{75} - 2304 q^{77} + 22 q^{79} + 243 q^{81} - 1136 q^{83} - 2732 q^{85} + 522 q^{87} + 110 q^{89} + 632 q^{91} - 954 q^{93} - 2552 q^{95} - 1222 q^{97}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 9.15486 0.818836 0.409418 0.912347i \(-0.365732\pi\)
0.409418 + 0.912347i \(0.365732\pi\)
\(6\) 0 0
\(7\) 27.4175 1.48040 0.740202 0.672385i \(-0.234730\pi\)
0.740202 + 0.672385i \(0.234730\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −20.5252 −0.562598 −0.281299 0.959620i \(-0.590765\pi\)
−0.281299 + 0.959620i \(0.590765\pi\)
\(12\) 0 0
\(13\) −32.0471 −0.683713 −0.341857 0.939752i \(-0.611056\pi\)
−0.341857 + 0.939752i \(0.611056\pi\)
\(14\) 0 0
\(15\) −27.4646 −0.472755
\(16\) 0 0
\(17\) −111.764 −1.59452 −0.797258 0.603639i \(-0.793717\pi\)
−0.797258 + 0.603639i \(0.793717\pi\)
\(18\) 0 0
\(19\) −129.764 −1.56684 −0.783419 0.621494i \(-0.786526\pi\)
−0.783419 + 0.621494i \(0.786526\pi\)
\(20\) 0 0
\(21\) −82.2524 −0.854711
\(22\) 0 0
\(23\) 9.16510 0.0830893 0.0415447 0.999137i \(-0.486772\pi\)
0.0415447 + 0.999137i \(0.486772\pi\)
\(24\) 0 0
\(25\) −41.1885 −0.329508
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −41.0606 −0.262923 −0.131461 0.991321i \(-0.541967\pi\)
−0.131461 + 0.991321i \(0.541967\pi\)
\(30\) 0 0
\(31\) 187.606 1.08694 0.543468 0.839430i \(-0.317111\pi\)
0.543468 + 0.839430i \(0.317111\pi\)
\(32\) 0 0
\(33\) 61.5755 0.324816
\(34\) 0 0
\(35\) 251.003 1.21221
\(36\) 0 0
\(37\) 114.127 0.507093 0.253546 0.967323i \(-0.418403\pi\)
0.253546 + 0.967323i \(0.418403\pi\)
\(38\) 0 0
\(39\) 96.1414 0.394742
\(40\) 0 0
\(41\) −282.915 −1.07766 −0.538828 0.842416i \(-0.681133\pi\)
−0.538828 + 0.842416i \(0.681133\pi\)
\(42\) 0 0
\(43\) 89.3870 0.317009 0.158505 0.987358i \(-0.449333\pi\)
0.158505 + 0.987358i \(0.449333\pi\)
\(44\) 0 0
\(45\) 82.3937 0.272945
\(46\) 0 0
\(47\) 54.6464 0.169596 0.0847978 0.996398i \(-0.472976\pi\)
0.0847978 + 0.996398i \(0.472976\pi\)
\(48\) 0 0
\(49\) 408.717 1.19159
\(50\) 0 0
\(51\) 335.292 0.920594
\(52\) 0 0
\(53\) −726.878 −1.88386 −0.941928 0.335815i \(-0.890988\pi\)
−0.941928 + 0.335815i \(0.890988\pi\)
\(54\) 0 0
\(55\) −187.905 −0.460675
\(56\) 0 0
\(57\) 389.292 0.904614
\(58\) 0 0
\(59\) 216.579 0.477900 0.238950 0.971032i \(-0.423197\pi\)
0.238950 + 0.971032i \(0.423197\pi\)
\(60\) 0 0
\(61\) −754.222 −1.58309 −0.791543 0.611114i \(-0.790722\pi\)
−0.791543 + 0.611114i \(0.790722\pi\)
\(62\) 0 0
\(63\) 246.757 0.493468
\(64\) 0 0
\(65\) −293.387 −0.559849
\(66\) 0 0
\(67\) −379.433 −0.691868 −0.345934 0.938259i \(-0.612438\pi\)
−0.345934 + 0.938259i \(0.612438\pi\)
\(68\) 0 0
\(69\) −27.4953 −0.0479717
\(70\) 0 0
\(71\) 302.080 0.504933 0.252467 0.967606i \(-0.418758\pi\)
0.252467 + 0.967606i \(0.418758\pi\)
\(72\) 0 0
\(73\) 504.396 0.808700 0.404350 0.914604i \(-0.367498\pi\)
0.404350 + 0.914604i \(0.367498\pi\)
\(74\) 0 0
\(75\) 123.566 0.190242
\(76\) 0 0
\(77\) −562.748 −0.832872
\(78\) 0 0
\(79\) −301.780 −0.429784 −0.214892 0.976638i \(-0.568940\pi\)
−0.214892 + 0.976638i \(0.568940\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −599.003 −0.792158 −0.396079 0.918216i \(-0.629629\pi\)
−0.396079 + 0.918216i \(0.629629\pi\)
\(84\) 0 0
\(85\) −1023.18 −1.30565
\(86\) 0 0
\(87\) 123.182 0.151799
\(88\) 0 0
\(89\) 277.528 0.330538 0.165269 0.986248i \(-0.447151\pi\)
0.165269 + 0.986248i \(0.447151\pi\)
\(90\) 0 0
\(91\) −878.651 −1.01217
\(92\) 0 0
\(93\) −562.818 −0.627543
\(94\) 0 0
\(95\) −1187.97 −1.28298
\(96\) 0 0
\(97\) −765.905 −0.801710 −0.400855 0.916141i \(-0.631287\pi\)
−0.400855 + 0.916141i \(0.631287\pi\)
\(98\) 0 0
\(99\) −184.727 −0.187533
\(100\) 0 0
\(101\) 201.253 0.198272 0.0991360 0.995074i \(-0.468392\pi\)
0.0991360 + 0.995074i \(0.468392\pi\)
\(102\) 0 0
\(103\) 682.440 0.652843 0.326421 0.945224i \(-0.394157\pi\)
0.326421 + 0.945224i \(0.394157\pi\)
\(104\) 0 0
\(105\) −753.009 −0.699868
\(106\) 0 0
\(107\) −457.252 −0.413123 −0.206562 0.978434i \(-0.566227\pi\)
−0.206562 + 0.978434i \(0.566227\pi\)
\(108\) 0 0
\(109\) −625.812 −0.549926 −0.274963 0.961455i \(-0.588666\pi\)
−0.274963 + 0.961455i \(0.588666\pi\)
\(110\) 0 0
\(111\) −342.382 −0.292770
\(112\) 0 0
\(113\) 981.151 0.816805 0.408402 0.912802i \(-0.366086\pi\)
0.408402 + 0.912802i \(0.366086\pi\)
\(114\) 0 0
\(115\) 83.9052 0.0680365
\(116\) 0 0
\(117\) −288.424 −0.227904
\(118\) 0 0
\(119\) −3064.29 −2.36053
\(120\) 0 0
\(121\) −909.717 −0.683484
\(122\) 0 0
\(123\) 848.745 0.622185
\(124\) 0 0
\(125\) −1521.43 −1.08865
\(126\) 0 0
\(127\) 808.055 0.564593 0.282296 0.959327i \(-0.408904\pi\)
0.282296 + 0.959327i \(0.408904\pi\)
\(128\) 0 0
\(129\) −268.161 −0.183025
\(130\) 0 0
\(131\) 1110.85 0.740884 0.370442 0.928856i \(-0.379206\pi\)
0.370442 + 0.928856i \(0.379206\pi\)
\(132\) 0 0
\(133\) −3557.80 −2.31955
\(134\) 0 0
\(135\) −247.181 −0.157585
\(136\) 0 0
\(137\) 466.765 0.291084 0.145542 0.989352i \(-0.453507\pi\)
0.145542 + 0.989352i \(0.453507\pi\)
\(138\) 0 0
\(139\) −351.773 −0.214654 −0.107327 0.994224i \(-0.534229\pi\)
−0.107327 + 0.994224i \(0.534229\pi\)
\(140\) 0 0
\(141\) −163.939 −0.0979161
\(142\) 0 0
\(143\) 657.773 0.384656
\(144\) 0 0
\(145\) −375.904 −0.215291
\(146\) 0 0
\(147\) −1226.15 −0.687967
\(148\) 0 0
\(149\) −1290.49 −0.709540 −0.354770 0.934954i \(-0.615441\pi\)
−0.354770 + 0.934954i \(0.615441\pi\)
\(150\) 0 0
\(151\) 1175.51 0.633521 0.316761 0.948505i \(-0.397405\pi\)
0.316761 + 0.948505i \(0.397405\pi\)
\(152\) 0 0
\(153\) −1005.88 −0.531505
\(154\) 0 0
\(155\) 1717.51 0.890022
\(156\) 0 0
\(157\) 1092.09 0.555148 0.277574 0.960704i \(-0.410470\pi\)
0.277574 + 0.960704i \(0.410470\pi\)
\(158\) 0 0
\(159\) 2180.63 1.08764
\(160\) 0 0
\(161\) 251.284 0.123006
\(162\) 0 0
\(163\) 3626.97 1.74286 0.871430 0.490519i \(-0.163193\pi\)
0.871430 + 0.490519i \(0.163193\pi\)
\(164\) 0 0
\(165\) 563.716 0.265971
\(166\) 0 0
\(167\) 45.8012 0.0212228 0.0106114 0.999944i \(-0.496622\pi\)
0.0106114 + 0.999944i \(0.496622\pi\)
\(168\) 0 0
\(169\) −1169.98 −0.532536
\(170\) 0 0
\(171\) −1167.88 −0.522279
\(172\) 0 0
\(173\) −2455.02 −1.07891 −0.539455 0.842014i \(-0.681370\pi\)
−0.539455 + 0.842014i \(0.681370\pi\)
\(174\) 0 0
\(175\) −1129.28 −0.487805
\(176\) 0 0
\(177\) −649.736 −0.275916
\(178\) 0 0
\(179\) −1026.28 −0.428533 −0.214267 0.976775i \(-0.568736\pi\)
−0.214267 + 0.976775i \(0.568736\pi\)
\(180\) 0 0
\(181\) 3699.05 1.51905 0.759526 0.650477i \(-0.225431\pi\)
0.759526 + 0.650477i \(0.225431\pi\)
\(182\) 0 0
\(183\) 2262.66 0.913995
\(184\) 0 0
\(185\) 1044.82 0.415226
\(186\) 0 0
\(187\) 2293.98 0.897071
\(188\) 0 0
\(189\) −740.271 −0.284904
\(190\) 0 0
\(191\) −5108.93 −1.93544 −0.967721 0.252023i \(-0.918904\pi\)
−0.967721 + 0.252023i \(0.918904\pi\)
\(192\) 0 0
\(193\) −1414.13 −0.527417 −0.263709 0.964602i \(-0.584946\pi\)
−0.263709 + 0.964602i \(0.584946\pi\)
\(194\) 0 0
\(195\) 880.161 0.323229
\(196\) 0 0
\(197\) −2816.66 −1.01867 −0.509337 0.860567i \(-0.670109\pi\)
−0.509337 + 0.860567i \(0.670109\pi\)
\(198\) 0 0
\(199\) −948.556 −0.337896 −0.168948 0.985625i \(-0.554037\pi\)
−0.168948 + 0.985625i \(0.554037\pi\)
\(200\) 0 0
\(201\) 1138.30 0.399450
\(202\) 0 0
\(203\) −1125.78 −0.389232
\(204\) 0 0
\(205\) −2590.05 −0.882424
\(206\) 0 0
\(207\) 82.4859 0.0276964
\(208\) 0 0
\(209\) 2663.43 0.881499
\(210\) 0 0
\(211\) −4487.28 −1.46406 −0.732032 0.681271i \(-0.761428\pi\)
−0.732032 + 0.681271i \(0.761428\pi\)
\(212\) 0 0
\(213\) −906.239 −0.291523
\(214\) 0 0
\(215\) 818.326 0.259578
\(216\) 0 0
\(217\) 5143.68 1.60910
\(218\) 0 0
\(219\) −1513.19 −0.466903
\(220\) 0 0
\(221\) 3581.72 1.09019
\(222\) 0 0
\(223\) 4590.98 1.37863 0.689315 0.724462i \(-0.257912\pi\)
0.689315 + 0.724462i \(0.257912\pi\)
\(224\) 0 0
\(225\) −370.697 −0.109836
\(226\) 0 0
\(227\) −2897.47 −0.847189 −0.423594 0.905852i \(-0.639232\pi\)
−0.423594 + 0.905852i \(0.639232\pi\)
\(228\) 0 0
\(229\) −34.6293 −0.00999288 −0.00499644 0.999988i \(-0.501590\pi\)
−0.00499644 + 0.999988i \(0.501590\pi\)
\(230\) 0 0
\(231\) 1688.24 0.480859
\(232\) 0 0
\(233\) 1054.02 0.296355 0.148178 0.988961i \(-0.452659\pi\)
0.148178 + 0.988961i \(0.452659\pi\)
\(234\) 0 0
\(235\) 500.280 0.138871
\(236\) 0 0
\(237\) 905.341 0.248136
\(238\) 0 0
\(239\) 654.700 0.177192 0.0885962 0.996068i \(-0.471762\pi\)
0.0885962 + 0.996068i \(0.471762\pi\)
\(240\) 0 0
\(241\) 3194.00 0.853707 0.426854 0.904321i \(-0.359622\pi\)
0.426854 + 0.904321i \(0.359622\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 3741.74 0.975719
\(246\) 0 0
\(247\) 4158.57 1.07127
\(248\) 0 0
\(249\) 1797.01 0.457353
\(250\) 0 0
\(251\) −5042.90 −1.26815 −0.634074 0.773273i \(-0.718618\pi\)
−0.634074 + 0.773273i \(0.718618\pi\)
\(252\) 0 0
\(253\) −188.115 −0.0467459
\(254\) 0 0
\(255\) 3069.55 0.753815
\(256\) 0 0
\(257\) −5166.64 −1.25403 −0.627016 0.779007i \(-0.715724\pi\)
−0.627016 + 0.779007i \(0.715724\pi\)
\(258\) 0 0
\(259\) 3129.08 0.750702
\(260\) 0 0
\(261\) −369.545 −0.0876409
\(262\) 0 0
\(263\) −7366.11 −1.72705 −0.863524 0.504308i \(-0.831748\pi\)
−0.863524 + 0.504308i \(0.831748\pi\)
\(264\) 0 0
\(265\) −6654.47 −1.54257
\(266\) 0 0
\(267\) −832.584 −0.190836
\(268\) 0 0
\(269\) 7877.80 1.78557 0.892784 0.450484i \(-0.148749\pi\)
0.892784 + 0.450484i \(0.148749\pi\)
\(270\) 0 0
\(271\) 5399.92 1.21041 0.605206 0.796069i \(-0.293091\pi\)
0.605206 + 0.796069i \(0.293091\pi\)
\(272\) 0 0
\(273\) 2635.95 0.584378
\(274\) 0 0
\(275\) 845.402 0.185381
\(276\) 0 0
\(277\) 4416.07 0.957892 0.478946 0.877844i \(-0.341019\pi\)
0.478946 + 0.877844i \(0.341019\pi\)
\(278\) 0 0
\(279\) 1688.45 0.362312
\(280\) 0 0
\(281\) −8068.94 −1.71300 −0.856499 0.516148i \(-0.827365\pi\)
−0.856499 + 0.516148i \(0.827365\pi\)
\(282\) 0 0
\(283\) 5241.13 1.10089 0.550447 0.834870i \(-0.314457\pi\)
0.550447 + 0.834870i \(0.314457\pi\)
\(284\) 0 0
\(285\) 3563.92 0.740730
\(286\) 0 0
\(287\) −7756.81 −1.59537
\(288\) 0 0
\(289\) 7578.21 1.54248
\(290\) 0 0
\(291\) 2297.72 0.462868
\(292\) 0 0
\(293\) 6372.75 1.27065 0.635324 0.772246i \(-0.280867\pi\)
0.635324 + 0.772246i \(0.280867\pi\)
\(294\) 0 0
\(295\) 1982.75 0.391322
\(296\) 0 0
\(297\) 554.180 0.108272
\(298\) 0 0
\(299\) −293.715 −0.0568093
\(300\) 0 0
\(301\) 2450.76 0.469301
\(302\) 0 0
\(303\) −603.760 −0.114472
\(304\) 0 0
\(305\) −6904.79 −1.29629
\(306\) 0 0
\(307\) −3810.22 −0.708342 −0.354171 0.935181i \(-0.615237\pi\)
−0.354171 + 0.935181i \(0.615237\pi\)
\(308\) 0 0
\(309\) −2047.32 −0.376919
\(310\) 0 0
\(311\) −8106.73 −1.47810 −0.739052 0.673648i \(-0.764726\pi\)
−0.739052 + 0.673648i \(0.764726\pi\)
\(312\) 0 0
\(313\) 559.983 0.101125 0.0505625 0.998721i \(-0.483899\pi\)
0.0505625 + 0.998721i \(0.483899\pi\)
\(314\) 0 0
\(315\) 2259.03 0.404069
\(316\) 0 0
\(317\) −5828.98 −1.03277 −0.516385 0.856357i \(-0.672723\pi\)
−0.516385 + 0.856357i \(0.672723\pi\)
\(318\) 0 0
\(319\) 842.776 0.147920
\(320\) 0 0
\(321\) 1371.76 0.238517
\(322\) 0 0
\(323\) 14503.0 2.49835
\(324\) 0 0
\(325\) 1319.97 0.225289
\(326\) 0 0
\(327\) 1877.44 0.317500
\(328\) 0 0
\(329\) 1498.26 0.251070
\(330\) 0 0
\(331\) −2847.98 −0.472928 −0.236464 0.971640i \(-0.575989\pi\)
−0.236464 + 0.971640i \(0.575989\pi\)
\(332\) 0 0
\(333\) 1027.15 0.169031
\(334\) 0 0
\(335\) −3473.66 −0.566526
\(336\) 0 0
\(337\) −10127.8 −1.63707 −0.818537 0.574454i \(-0.805214\pi\)
−0.818537 + 0.574454i \(0.805214\pi\)
\(338\) 0 0
\(339\) −2943.45 −0.471582
\(340\) 0 0
\(341\) −3850.65 −0.611508
\(342\) 0 0
\(343\) 1801.78 0.283636
\(344\) 0 0
\(345\) −251.716 −0.0392809
\(346\) 0 0
\(347\) 10148.2 1.56999 0.784993 0.619505i \(-0.212667\pi\)
0.784993 + 0.619505i \(0.212667\pi\)
\(348\) 0 0
\(349\) 9515.96 1.45954 0.729768 0.683695i \(-0.239628\pi\)
0.729768 + 0.683695i \(0.239628\pi\)
\(350\) 0 0
\(351\) 865.273 0.131581
\(352\) 0 0
\(353\) 2813.56 0.424223 0.212111 0.977245i \(-0.431966\pi\)
0.212111 + 0.977245i \(0.431966\pi\)
\(354\) 0 0
\(355\) 2765.50 0.413457
\(356\) 0 0
\(357\) 9192.86 1.36285
\(358\) 0 0
\(359\) −2427.25 −0.356839 −0.178419 0.983955i \(-0.557098\pi\)
−0.178419 + 0.983955i \(0.557098\pi\)
\(360\) 0 0
\(361\) 9979.71 1.45498
\(362\) 0 0
\(363\) 2729.15 0.394610
\(364\) 0 0
\(365\) 4617.67 0.662192
\(366\) 0 0
\(367\) −5021.46 −0.714219 −0.357109 0.934063i \(-0.616238\pi\)
−0.357109 + 0.934063i \(0.616238\pi\)
\(368\) 0 0
\(369\) −2546.24 −0.359219
\(370\) 0 0
\(371\) −19929.1 −2.78887
\(372\) 0 0
\(373\) 3182.40 0.441765 0.220882 0.975300i \(-0.429106\pi\)
0.220882 + 0.975300i \(0.429106\pi\)
\(374\) 0 0
\(375\) 4564.30 0.628532
\(376\) 0 0
\(377\) 1315.87 0.179764
\(378\) 0 0
\(379\) −5868.93 −0.795426 −0.397713 0.917510i \(-0.630196\pi\)
−0.397713 + 0.917510i \(0.630196\pi\)
\(380\) 0 0
\(381\) −2424.16 −0.325968
\(382\) 0 0
\(383\) −7350.18 −0.980618 −0.490309 0.871549i \(-0.663116\pi\)
−0.490309 + 0.871549i \(0.663116\pi\)
\(384\) 0 0
\(385\) −5151.88 −0.681985
\(386\) 0 0
\(387\) 804.483 0.105670
\(388\) 0 0
\(389\) 13009.1 1.69560 0.847800 0.530317i \(-0.177927\pi\)
0.847800 + 0.530317i \(0.177927\pi\)
\(390\) 0 0
\(391\) −1024.33 −0.132487
\(392\) 0 0
\(393\) −3332.56 −0.427750
\(394\) 0 0
\(395\) −2762.76 −0.351923
\(396\) 0 0
\(397\) 4877.88 0.616659 0.308330 0.951280i \(-0.400230\pi\)
0.308330 + 0.951280i \(0.400230\pi\)
\(398\) 0 0
\(399\) 10673.4 1.33919
\(400\) 0 0
\(401\) 5552.33 0.691446 0.345723 0.938337i \(-0.387634\pi\)
0.345723 + 0.938337i \(0.387634\pi\)
\(402\) 0 0
\(403\) −6012.23 −0.743153
\(404\) 0 0
\(405\) 741.544 0.0909817
\(406\) 0 0
\(407\) −2342.49 −0.285289
\(408\) 0 0
\(409\) −6989.27 −0.844981 −0.422491 0.906367i \(-0.638844\pi\)
−0.422491 + 0.906367i \(0.638844\pi\)
\(410\) 0 0
\(411\) −1400.30 −0.168057
\(412\) 0 0
\(413\) 5938.03 0.707485
\(414\) 0 0
\(415\) −5483.79 −0.648647
\(416\) 0 0
\(417\) 1055.32 0.123931
\(418\) 0 0
\(419\) 10461.0 1.21970 0.609849 0.792518i \(-0.291230\pi\)
0.609849 + 0.792518i \(0.291230\pi\)
\(420\) 0 0
\(421\) 4648.55 0.538139 0.269070 0.963121i \(-0.413284\pi\)
0.269070 + 0.963121i \(0.413284\pi\)
\(422\) 0 0
\(423\) 491.817 0.0565319
\(424\) 0 0
\(425\) 4603.40 0.525406
\(426\) 0 0
\(427\) −20678.8 −2.34360
\(428\) 0 0
\(429\) −1973.32 −0.222081
\(430\) 0 0
\(431\) −12490.7 −1.39595 −0.697975 0.716122i \(-0.745915\pi\)
−0.697975 + 0.716122i \(0.745915\pi\)
\(432\) 0 0
\(433\) 9446.37 1.04842 0.524208 0.851590i \(-0.324362\pi\)
0.524208 + 0.851590i \(0.324362\pi\)
\(434\) 0 0
\(435\) 1127.71 0.124298
\(436\) 0 0
\(437\) −1189.30 −0.130188
\(438\) 0 0
\(439\) 2793.60 0.303716 0.151858 0.988402i \(-0.451474\pi\)
0.151858 + 0.988402i \(0.451474\pi\)
\(440\) 0 0
\(441\) 3678.45 0.397198
\(442\) 0 0
\(443\) 7601.37 0.815241 0.407621 0.913151i \(-0.366359\pi\)
0.407621 + 0.913151i \(0.366359\pi\)
\(444\) 0 0
\(445\) 2540.73 0.270657
\(446\) 0 0
\(447\) 3871.48 0.409653
\(448\) 0 0
\(449\) 10708.8 1.12557 0.562785 0.826603i \(-0.309730\pi\)
0.562785 + 0.826603i \(0.309730\pi\)
\(450\) 0 0
\(451\) 5806.89 0.606287
\(452\) 0 0
\(453\) −3526.53 −0.365764
\(454\) 0 0
\(455\) −8043.92 −0.828802
\(456\) 0 0
\(457\) −233.840 −0.0239356 −0.0119678 0.999928i \(-0.503810\pi\)
−0.0119678 + 0.999928i \(0.503810\pi\)
\(458\) 0 0
\(459\) 3017.63 0.306865
\(460\) 0 0
\(461\) 981.307 0.0991410 0.0495705 0.998771i \(-0.484215\pi\)
0.0495705 + 0.998771i \(0.484215\pi\)
\(462\) 0 0
\(463\) 14082.7 1.41356 0.706782 0.707431i \(-0.250146\pi\)
0.706782 + 0.707431i \(0.250146\pi\)
\(464\) 0 0
\(465\) −5152.52 −0.513855
\(466\) 0 0
\(467\) −9286.49 −0.920188 −0.460094 0.887870i \(-0.652184\pi\)
−0.460094 + 0.887870i \(0.652184\pi\)
\(468\) 0 0
\(469\) −10403.1 −1.02424
\(470\) 0 0
\(471\) −3276.27 −0.320515
\(472\) 0 0
\(473\) −1834.68 −0.178349
\(474\) 0 0
\(475\) 5344.79 0.516286
\(476\) 0 0
\(477\) −6541.90 −0.627952
\(478\) 0 0
\(479\) 19409.3 1.85143 0.925715 0.378222i \(-0.123464\pi\)
0.925715 + 0.378222i \(0.123464\pi\)
\(480\) 0 0
\(481\) −3657.46 −0.346706
\(482\) 0 0
\(483\) −753.851 −0.0710174
\(484\) 0 0
\(485\) −7011.76 −0.656469
\(486\) 0 0
\(487\) 12124.8 1.12818 0.564091 0.825712i \(-0.309227\pi\)
0.564091 + 0.825712i \(0.309227\pi\)
\(488\) 0 0
\(489\) −10880.9 −1.00624
\(490\) 0 0
\(491\) −5100.69 −0.468820 −0.234410 0.972138i \(-0.575316\pi\)
−0.234410 + 0.972138i \(0.575316\pi\)
\(492\) 0 0
\(493\) 4589.10 0.419235
\(494\) 0 0
\(495\) −1691.15 −0.153558
\(496\) 0 0
\(497\) 8282.26 0.747505
\(498\) 0 0
\(499\) 85.2797 0.00765058 0.00382529 0.999993i \(-0.498782\pi\)
0.00382529 + 0.999993i \(0.498782\pi\)
\(500\) 0 0
\(501\) −137.404 −0.0122530
\(502\) 0 0
\(503\) −12287.2 −1.08918 −0.544592 0.838701i \(-0.683316\pi\)
−0.544592 + 0.838701i \(0.683316\pi\)
\(504\) 0 0
\(505\) 1842.45 0.162352
\(506\) 0 0
\(507\) 3509.94 0.307460
\(508\) 0 0
\(509\) −450.441 −0.0392248 −0.0196124 0.999808i \(-0.506243\pi\)
−0.0196124 + 0.999808i \(0.506243\pi\)
\(510\) 0 0
\(511\) 13829.2 1.19720
\(512\) 0 0
\(513\) 3503.63 0.301538
\(514\) 0 0
\(515\) 6247.64 0.534571
\(516\) 0 0
\(517\) −1121.63 −0.0954141
\(518\) 0 0
\(519\) 7365.05 0.622909
\(520\) 0 0
\(521\) 15088.1 1.26876 0.634378 0.773023i \(-0.281256\pi\)
0.634378 + 0.773023i \(0.281256\pi\)
\(522\) 0 0
\(523\) 17719.4 1.48149 0.740743 0.671789i \(-0.234474\pi\)
0.740743 + 0.671789i \(0.234474\pi\)
\(524\) 0 0
\(525\) 3387.85 0.281634
\(526\) 0 0
\(527\) −20967.6 −1.73314
\(528\) 0 0
\(529\) −12083.0 −0.993096
\(530\) 0 0
\(531\) 1949.21 0.159300
\(532\) 0 0
\(533\) 9066.62 0.736808
\(534\) 0 0
\(535\) −4186.08 −0.338280
\(536\) 0 0
\(537\) 3078.83 0.247414
\(538\) 0 0
\(539\) −8388.98 −0.670388
\(540\) 0 0
\(541\) −12244.5 −0.973074 −0.486537 0.873660i \(-0.661740\pi\)
−0.486537 + 0.873660i \(0.661740\pi\)
\(542\) 0 0
\(543\) −11097.2 −0.877025
\(544\) 0 0
\(545\) −5729.22 −0.450299
\(546\) 0 0
\(547\) 7822.46 0.611452 0.305726 0.952119i \(-0.401101\pi\)
0.305726 + 0.952119i \(0.401101\pi\)
\(548\) 0 0
\(549\) −6787.99 −0.527695
\(550\) 0 0
\(551\) 5328.19 0.411957
\(552\) 0 0
\(553\) −8274.05 −0.636254
\(554\) 0 0
\(555\) −3134.46 −0.239731
\(556\) 0 0
\(557\) −16555.5 −1.25938 −0.629692 0.776845i \(-0.716819\pi\)
−0.629692 + 0.776845i \(0.716819\pi\)
\(558\) 0 0
\(559\) −2864.60 −0.216743
\(560\) 0 0
\(561\) −6881.93 −0.517924
\(562\) 0 0
\(563\) 12580.7 0.941766 0.470883 0.882196i \(-0.343935\pi\)
0.470883 + 0.882196i \(0.343935\pi\)
\(564\) 0 0
\(565\) 8982.30 0.668829
\(566\) 0 0
\(567\) 2220.81 0.164489
\(568\) 0 0
\(569\) −2657.93 −0.195828 −0.0979141 0.995195i \(-0.531217\pi\)
−0.0979141 + 0.995195i \(0.531217\pi\)
\(570\) 0 0
\(571\) 17669.0 1.29496 0.647481 0.762081i \(-0.275822\pi\)
0.647481 + 0.762081i \(0.275822\pi\)
\(572\) 0 0
\(573\) 15326.8 1.11743
\(574\) 0 0
\(575\) −377.497 −0.0273786
\(576\) 0 0
\(577\) 14617.7 1.05467 0.527334 0.849658i \(-0.323192\pi\)
0.527334 + 0.849658i \(0.323192\pi\)
\(578\) 0 0
\(579\) 4242.40 0.304505
\(580\) 0 0
\(581\) −16423.1 −1.17271
\(582\) 0 0
\(583\) 14919.3 1.05985
\(584\) 0 0
\(585\) −2640.48 −0.186616
\(586\) 0 0
\(587\) −4096.53 −0.288044 −0.144022 0.989574i \(-0.546004\pi\)
−0.144022 + 0.989574i \(0.546004\pi\)
\(588\) 0 0
\(589\) −24344.5 −1.70305
\(590\) 0 0
\(591\) 8449.99 0.588132
\(592\) 0 0
\(593\) −21988.3 −1.52269 −0.761343 0.648349i \(-0.775460\pi\)
−0.761343 + 0.648349i \(0.775460\pi\)
\(594\) 0 0
\(595\) −28053.1 −1.93288
\(596\) 0 0
\(597\) 2845.67 0.195084
\(598\) 0 0
\(599\) 20767.7 1.41660 0.708302 0.705909i \(-0.249461\pi\)
0.708302 + 0.705909i \(0.249461\pi\)
\(600\) 0 0
\(601\) −5382.61 −0.365326 −0.182663 0.983176i \(-0.558472\pi\)
−0.182663 + 0.983176i \(0.558472\pi\)
\(602\) 0 0
\(603\) −3414.90 −0.230623
\(604\) 0 0
\(605\) −8328.33 −0.559661
\(606\) 0 0
\(607\) −11165.4 −0.746607 −0.373304 0.927709i \(-0.621775\pi\)
−0.373304 + 0.927709i \(0.621775\pi\)
\(608\) 0 0
\(609\) 3377.33 0.224723
\(610\) 0 0
\(611\) −1751.26 −0.115955
\(612\) 0 0
\(613\) −16413.5 −1.08146 −0.540731 0.841195i \(-0.681852\pi\)
−0.540731 + 0.841195i \(0.681852\pi\)
\(614\) 0 0
\(615\) 7770.15 0.509468
\(616\) 0 0
\(617\) 51.5882 0.00336607 0.00168303 0.999999i \(-0.499464\pi\)
0.00168303 + 0.999999i \(0.499464\pi\)
\(618\) 0 0
\(619\) −6349.55 −0.412294 −0.206147 0.978521i \(-0.566093\pi\)
−0.206147 + 0.978521i \(0.566093\pi\)
\(620\) 0 0
\(621\) −247.458 −0.0159906
\(622\) 0 0
\(623\) 7609.11 0.489330
\(624\) 0 0
\(625\) −8779.94 −0.561916
\(626\) 0 0
\(627\) −7990.29 −0.508934
\(628\) 0 0
\(629\) −12755.3 −0.808567
\(630\) 0 0
\(631\) −13379.1 −0.844078 −0.422039 0.906578i \(-0.638685\pi\)
−0.422039 + 0.906578i \(0.638685\pi\)
\(632\) 0 0
\(633\) 13461.9 0.845277
\(634\) 0 0
\(635\) 7397.63 0.462309
\(636\) 0 0
\(637\) −13098.2 −0.814709
\(638\) 0 0
\(639\) 2718.72 0.168311
\(640\) 0 0
\(641\) −20406.3 −1.25741 −0.628705 0.777644i \(-0.716415\pi\)
−0.628705 + 0.777644i \(0.716415\pi\)
\(642\) 0 0
\(643\) 19415.1 1.19076 0.595378 0.803446i \(-0.297002\pi\)
0.595378 + 0.803446i \(0.297002\pi\)
\(644\) 0 0
\(645\) −2454.98 −0.149868
\(646\) 0 0
\(647\) −8167.12 −0.496264 −0.248132 0.968726i \(-0.579817\pi\)
−0.248132 + 0.968726i \(0.579817\pi\)
\(648\) 0 0
\(649\) −4445.31 −0.268866
\(650\) 0 0
\(651\) −15431.0 −0.929017
\(652\) 0 0
\(653\) −7444.93 −0.446160 −0.223080 0.974800i \(-0.571611\pi\)
−0.223080 + 0.974800i \(0.571611\pi\)
\(654\) 0 0
\(655\) 10169.7 0.606662
\(656\) 0 0
\(657\) 4539.56 0.269567
\(658\) 0 0
\(659\) −23780.4 −1.40569 −0.702846 0.711342i \(-0.748088\pi\)
−0.702846 + 0.711342i \(0.748088\pi\)
\(660\) 0 0
\(661\) 2528.90 0.148809 0.0744046 0.997228i \(-0.476294\pi\)
0.0744046 + 0.997228i \(0.476294\pi\)
\(662\) 0 0
\(663\) −10745.2 −0.629423
\(664\) 0 0
\(665\) −32571.2 −1.89933
\(666\) 0 0
\(667\) −376.324 −0.0218461
\(668\) 0 0
\(669\) −13772.9 −0.795953
\(670\) 0 0
\(671\) 15480.5 0.890640
\(672\) 0 0
\(673\) 16733.7 0.958447 0.479224 0.877693i \(-0.340918\pi\)
0.479224 + 0.877693i \(0.340918\pi\)
\(674\) 0 0
\(675\) 1112.09 0.0634139
\(676\) 0 0
\(677\) −24191.5 −1.37335 −0.686673 0.726966i \(-0.740930\pi\)
−0.686673 + 0.726966i \(0.740930\pi\)
\(678\) 0 0
\(679\) −20999.2 −1.18685
\(680\) 0 0
\(681\) 8692.41 0.489125
\(682\) 0 0
\(683\) −13965.2 −0.782376 −0.391188 0.920311i \(-0.627936\pi\)
−0.391188 + 0.920311i \(0.627936\pi\)
\(684\) 0 0
\(685\) 4273.17 0.238350
\(686\) 0 0
\(687\) 103.888 0.00576939
\(688\) 0 0
\(689\) 23294.4 1.28802
\(690\) 0 0
\(691\) −8685.63 −0.478172 −0.239086 0.970998i \(-0.576848\pi\)
−0.239086 + 0.970998i \(0.576848\pi\)
\(692\) 0 0
\(693\) −5064.73 −0.277624
\(694\) 0 0
\(695\) −3220.43 −0.175767
\(696\) 0 0
\(697\) 31619.7 1.71834
\(698\) 0 0
\(699\) −3162.05 −0.171101
\(700\) 0 0
\(701\) 25942.2 1.39775 0.698876 0.715243i \(-0.253684\pi\)
0.698876 + 0.715243i \(0.253684\pi\)
\(702\) 0 0
\(703\) −14809.6 −0.794532
\(704\) 0 0
\(705\) −1500.84 −0.0801772
\(706\) 0 0
\(707\) 5517.86 0.293522
\(708\) 0 0
\(709\) −5487.75 −0.290687 −0.145343 0.989381i \(-0.546429\pi\)
−0.145343 + 0.989381i \(0.546429\pi\)
\(710\) 0 0
\(711\) −2716.02 −0.143261
\(712\) 0 0
\(713\) 1719.43 0.0903128
\(714\) 0 0
\(715\) 6021.82 0.314970
\(716\) 0 0
\(717\) −1964.10 −0.102302
\(718\) 0 0
\(719\) −17141.2 −0.889094 −0.444547 0.895756i \(-0.646635\pi\)
−0.444547 + 0.895756i \(0.646635\pi\)
\(720\) 0 0
\(721\) 18710.8 0.966470
\(722\) 0 0
\(723\) −9581.99 −0.492888
\(724\) 0 0
\(725\) 1691.23 0.0866352
\(726\) 0 0
\(727\) −15946.4 −0.813508 −0.406754 0.913538i \(-0.633339\pi\)
−0.406754 + 0.913538i \(0.633339\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −9990.26 −0.505476
\(732\) 0 0
\(733\) −15914.2 −0.801917 −0.400958 0.916096i \(-0.631323\pi\)
−0.400958 + 0.916096i \(0.631323\pi\)
\(734\) 0 0
\(735\) −11225.2 −0.563332
\(736\) 0 0
\(737\) 7787.94 0.389243
\(738\) 0 0
\(739\) 13555.4 0.674755 0.337377 0.941369i \(-0.390460\pi\)
0.337377 + 0.941369i \(0.390460\pi\)
\(740\) 0 0
\(741\) −12475.7 −0.618497
\(742\) 0 0
\(743\) −1772.73 −0.0875303 −0.0437652 0.999042i \(-0.513935\pi\)
−0.0437652 + 0.999042i \(0.513935\pi\)
\(744\) 0 0
\(745\) −11814.3 −0.580997
\(746\) 0 0
\(747\) −5391.03 −0.264053
\(748\) 0 0
\(749\) −12536.7 −0.611589
\(750\) 0 0
\(751\) 1006.65 0.0489124 0.0244562 0.999701i \(-0.492215\pi\)
0.0244562 + 0.999701i \(0.492215\pi\)
\(752\) 0 0
\(753\) 15128.7 0.732165
\(754\) 0 0
\(755\) 10761.6 0.518750
\(756\) 0 0
\(757\) −28774.0 −1.38152 −0.690758 0.723086i \(-0.742723\pi\)
−0.690758 + 0.723086i \(0.742723\pi\)
\(758\) 0 0
\(759\) 564.346 0.0269887
\(760\) 0 0
\(761\) −18393.5 −0.876166 −0.438083 0.898934i \(-0.644342\pi\)
−0.438083 + 0.898934i \(0.644342\pi\)
\(762\) 0 0
\(763\) −17158.2 −0.814112
\(764\) 0 0
\(765\) −9208.66 −0.435215
\(766\) 0 0
\(767\) −6940.72 −0.326747
\(768\) 0 0
\(769\) −14672.2 −0.688027 −0.344014 0.938965i \(-0.611787\pi\)
−0.344014 + 0.938965i \(0.611787\pi\)
\(770\) 0 0
\(771\) 15499.9 0.724016
\(772\) 0 0
\(773\) 16256.4 0.756405 0.378202 0.925723i \(-0.376542\pi\)
0.378202 + 0.925723i \(0.376542\pi\)
\(774\) 0 0
\(775\) −7727.21 −0.358154
\(776\) 0 0
\(777\) −9387.25 −0.433418
\(778\) 0 0
\(779\) 36712.2 1.68851
\(780\) 0 0
\(781\) −6200.24 −0.284074
\(782\) 0 0
\(783\) 1108.64 0.0505995
\(784\) 0 0
\(785\) 9997.92 0.454575
\(786\) 0 0
\(787\) 23988.3 1.08652 0.543259 0.839565i \(-0.317190\pi\)
0.543259 + 0.839565i \(0.317190\pi\)
\(788\) 0 0
\(789\) 22098.3 0.997112
\(790\) 0 0
\(791\) 26900.7 1.20920
\(792\) 0 0
\(793\) 24170.6 1.08238
\(794\) 0 0
\(795\) 19963.4 0.890602
\(796\) 0 0
\(797\) −32966.9 −1.46518 −0.732589 0.680672i \(-0.761688\pi\)
−0.732589 + 0.680672i \(0.761688\pi\)
\(798\) 0 0
\(799\) −6107.50 −0.270423
\(800\) 0 0
\(801\) 2497.75 0.110179
\(802\) 0 0
\(803\) −10352.8 −0.454973
\(804\) 0 0
\(805\) 2300.47 0.100721
\(806\) 0 0
\(807\) −23633.4 −1.03090
\(808\) 0 0
\(809\) 41700.3 1.81224 0.906122 0.423017i \(-0.139029\pi\)
0.906122 + 0.423017i \(0.139029\pi\)
\(810\) 0 0
\(811\) 5981.80 0.259000 0.129500 0.991579i \(-0.458663\pi\)
0.129500 + 0.991579i \(0.458663\pi\)
\(812\) 0 0
\(813\) −16199.7 −0.698832
\(814\) 0 0
\(815\) 33204.4 1.42712
\(816\) 0 0
\(817\) −11599.2 −0.496702
\(818\) 0 0
\(819\) −7907.86 −0.337391
\(820\) 0 0
\(821\) −9846.06 −0.418550 −0.209275 0.977857i \(-0.567110\pi\)
−0.209275 + 0.977857i \(0.567110\pi\)
\(822\) 0 0
\(823\) 47001.9 1.99074 0.995372 0.0960935i \(-0.0306348\pi\)
0.995372 + 0.0960935i \(0.0306348\pi\)
\(824\) 0 0
\(825\) −2536.21 −0.107030
\(826\) 0 0
\(827\) 21727.4 0.913587 0.456794 0.889573i \(-0.348998\pi\)
0.456794 + 0.889573i \(0.348998\pi\)
\(828\) 0 0
\(829\) 22772.3 0.954058 0.477029 0.878888i \(-0.341714\pi\)
0.477029 + 0.878888i \(0.341714\pi\)
\(830\) 0 0
\(831\) −13248.2 −0.553039
\(832\) 0 0
\(833\) −45679.8 −1.90002
\(834\) 0 0
\(835\) 419.304 0.0173780
\(836\) 0 0
\(837\) −5065.36 −0.209181
\(838\) 0 0
\(839\) 11010.4 0.453064 0.226532 0.974004i \(-0.427261\pi\)
0.226532 + 0.974004i \(0.427261\pi\)
\(840\) 0 0
\(841\) −22703.0 −0.930872
\(842\) 0 0
\(843\) 24206.8 0.989000
\(844\) 0 0
\(845\) −10711.0 −0.436059
\(846\) 0 0
\(847\) −24942.1 −1.01183
\(848\) 0 0
\(849\) −15723.4 −0.635602
\(850\) 0 0
\(851\) 1045.99 0.0421340
\(852\) 0 0
\(853\) −38177.4 −1.53244 −0.766219 0.642579i \(-0.777864\pi\)
−0.766219 + 0.642579i \(0.777864\pi\)
\(854\) 0 0
\(855\) −10691.7 −0.427661
\(856\) 0 0
\(857\) −8848.01 −0.352675 −0.176337 0.984330i \(-0.556425\pi\)
−0.176337 + 0.984330i \(0.556425\pi\)
\(858\) 0 0
\(859\) 4347.66 0.172690 0.0863448 0.996265i \(-0.472481\pi\)
0.0863448 + 0.996265i \(0.472481\pi\)
\(860\) 0 0
\(861\) 23270.4 0.921085
\(862\) 0 0
\(863\) 33669.9 1.32808 0.664042 0.747695i \(-0.268839\pi\)
0.664042 + 0.747695i \(0.268839\pi\)
\(864\) 0 0
\(865\) −22475.3 −0.883450
\(866\) 0 0
\(867\) −22734.6 −0.890552
\(868\) 0 0
\(869\) 6194.10 0.241796
\(870\) 0 0
\(871\) 12159.7 0.473039
\(872\) 0 0
\(873\) −6893.15 −0.267237
\(874\) 0 0
\(875\) −41713.8 −1.61164
\(876\) 0 0
\(877\) 50102.0 1.92910 0.964552 0.263892i \(-0.0850062\pi\)
0.964552 + 0.263892i \(0.0850062\pi\)
\(878\) 0 0
\(879\) −19118.2 −0.733609
\(880\) 0 0
\(881\) 18716.9 0.715766 0.357883 0.933766i \(-0.383499\pi\)
0.357883 + 0.933766i \(0.383499\pi\)
\(882\) 0 0
\(883\) 7514.19 0.286379 0.143189 0.989695i \(-0.454264\pi\)
0.143189 + 0.989695i \(0.454264\pi\)
\(884\) 0 0
\(885\) −5948.24 −0.225930
\(886\) 0 0
\(887\) −15544.6 −0.588429 −0.294215 0.955739i \(-0.595058\pi\)
−0.294215 + 0.955739i \(0.595058\pi\)
\(888\) 0 0
\(889\) 22154.8 0.835825
\(890\) 0 0
\(891\) −1662.54 −0.0625109
\(892\) 0 0
\(893\) −7091.14 −0.265729
\(894\) 0 0
\(895\) −9395.42 −0.350898
\(896\) 0 0
\(897\) 881.145 0.0327989
\(898\) 0 0
\(899\) −7703.21 −0.285780
\(900\) 0 0
\(901\) 81238.8 3.00384
\(902\) 0 0
\(903\) −7352.29 −0.270951
\(904\) 0 0
\(905\) 33864.3 1.24385
\(906\) 0 0
\(907\) −8713.10 −0.318979 −0.159489 0.987200i \(-0.550985\pi\)
−0.159489 + 0.987200i \(0.550985\pi\)
\(908\) 0 0
\(909\) 1811.28 0.0660906
\(910\) 0 0
\(911\) −1975.97 −0.0718627 −0.0359313 0.999354i \(-0.511440\pi\)
−0.0359313 + 0.999354i \(0.511440\pi\)
\(912\) 0 0
\(913\) 12294.6 0.445666
\(914\) 0 0
\(915\) 20714.4 0.748411
\(916\) 0 0
\(917\) 30456.8 1.09681
\(918\) 0 0
\(919\) −18430.5 −0.661552 −0.330776 0.943709i \(-0.607311\pi\)
−0.330776 + 0.943709i \(0.607311\pi\)
\(920\) 0 0
\(921\) 11430.7 0.408961
\(922\) 0 0
\(923\) −9680.79 −0.345230
\(924\) 0 0
\(925\) −4700.74 −0.167091
\(926\) 0 0
\(927\) 6141.96 0.217614
\(928\) 0 0
\(929\) 12506.8 0.441697 0.220848 0.975308i \(-0.429117\pi\)
0.220848 + 0.975308i \(0.429117\pi\)
\(930\) 0 0
\(931\) −53036.7 −1.86703
\(932\) 0 0
\(933\) 24320.2 0.853384
\(934\) 0 0
\(935\) 21001.0 0.734554
\(936\) 0 0
\(937\) −39267.5 −1.36906 −0.684532 0.728982i \(-0.739994\pi\)
−0.684532 + 0.728982i \(0.739994\pi\)
\(938\) 0 0
\(939\) −1679.95 −0.0583846
\(940\) 0 0
\(941\) 23727.2 0.821981 0.410991 0.911640i \(-0.365183\pi\)
0.410991 + 0.911640i \(0.365183\pi\)
\(942\) 0 0
\(943\) −2592.94 −0.0895418
\(944\) 0 0
\(945\) −6777.08 −0.233289
\(946\) 0 0
\(947\) 23399.8 0.802948 0.401474 0.915870i \(-0.368498\pi\)
0.401474 + 0.915870i \(0.368498\pi\)
\(948\) 0 0
\(949\) −16164.4 −0.552919
\(950\) 0 0
\(951\) 17486.9 0.596270
\(952\) 0 0
\(953\) −41497.1 −1.41052 −0.705258 0.708950i \(-0.749169\pi\)
−0.705258 + 0.708950i \(0.749169\pi\)
\(954\) 0 0
\(955\) −46771.6 −1.58481
\(956\) 0 0
\(957\) −2528.33 −0.0854015
\(958\) 0 0
\(959\) 12797.5 0.430921
\(960\) 0 0
\(961\) 5405.00 0.181431
\(962\) 0 0
\(963\) −4115.27 −0.137708
\(964\) 0 0
\(965\) −12946.2 −0.431868
\(966\) 0 0
\(967\) −49123.6 −1.63362 −0.816808 0.576909i \(-0.804259\pi\)
−0.816808 + 0.576909i \(0.804259\pi\)
\(968\) 0 0
\(969\) −43508.9 −1.44242
\(970\) 0 0
\(971\) 20346.8 0.672462 0.336231 0.941780i \(-0.390848\pi\)
0.336231 + 0.941780i \(0.390848\pi\)
\(972\) 0 0
\(973\) −9644.71 −0.317775
\(974\) 0 0
\(975\) −3959.92 −0.130071
\(976\) 0 0
\(977\) 40602.1 1.32955 0.664777 0.747042i \(-0.268526\pi\)
0.664777 + 0.747042i \(0.268526\pi\)
\(978\) 0 0
\(979\) −5696.32 −0.185960
\(980\) 0 0
\(981\) −5632.31 −0.183309
\(982\) 0 0
\(983\) 50425.9 1.63615 0.818075 0.575112i \(-0.195041\pi\)
0.818075 + 0.575112i \(0.195041\pi\)
\(984\) 0 0
\(985\) −25786.2 −0.834127
\(986\) 0 0
\(987\) −4494.79 −0.144955
\(988\) 0 0
\(989\) 819.241 0.0263401
\(990\) 0 0
\(991\) 8511.62 0.272836 0.136418 0.990651i \(-0.456441\pi\)
0.136418 + 0.990651i \(0.456441\pi\)
\(992\) 0 0
\(993\) 8543.94 0.273045
\(994\) 0 0
\(995\) −8683.90 −0.276681
\(996\) 0 0
\(997\) −25302.1 −0.803738 −0.401869 0.915697i \(-0.631639\pi\)
−0.401869 + 0.915697i \(0.631639\pi\)
\(998\) 0 0
\(999\) −3081.44 −0.0975900
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 768.4.a.q.1.3 3
3.2 odd 2 2304.4.a.bw.1.1 3
4.3 odd 2 768.4.a.s.1.3 3
8.3 odd 2 768.4.a.r.1.1 3
8.5 even 2 768.4.a.t.1.1 3
12.11 even 2 2304.4.a.bv.1.1 3
16.3 odd 4 24.4.d.a.13.4 yes 6
16.5 even 4 96.4.d.a.49.6 6
16.11 odd 4 24.4.d.a.13.3 6
16.13 even 4 96.4.d.a.49.1 6
24.5 odd 2 2304.4.a.bu.1.3 3
24.11 even 2 2304.4.a.bt.1.3 3
48.5 odd 4 288.4.d.d.145.2 6
48.11 even 4 72.4.d.d.37.4 6
48.29 odd 4 288.4.d.d.145.5 6
48.35 even 4 72.4.d.d.37.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.d.a.13.3 6 16.11 odd 4
24.4.d.a.13.4 yes 6 16.3 odd 4
72.4.d.d.37.3 6 48.35 even 4
72.4.d.d.37.4 6 48.11 even 4
96.4.d.a.49.1 6 16.13 even 4
96.4.d.a.49.6 6 16.5 even 4
288.4.d.d.145.2 6 48.5 odd 4
288.4.d.d.145.5 6 48.29 odd 4
768.4.a.q.1.3 3 1.1 even 1 trivial
768.4.a.r.1.1 3 8.3 odd 2
768.4.a.s.1.3 3 4.3 odd 2
768.4.a.t.1.1 3 8.5 even 2
2304.4.a.bt.1.3 3 24.11 even 2
2304.4.a.bu.1.3 3 24.5 odd 2
2304.4.a.bv.1.1 3 12.11 even 2
2304.4.a.bw.1.1 3 3.2 odd 2