Properties

Label 768.4.a.r.1.1
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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: \(0\)
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 1.1
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 \(-0.634268\pi\)
−0.409418 + 0.912347i \(0.634268\pi\)
\(6\) 0 0
\(7\) −27.4175 −1.48040 −0.740202 0.672385i \(-0.765270\pi\)
−0.740202 + 0.672385i \(0.765270\pi\)
\(8\) 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 \(-0.388944\pi\)
0.341857 + 0.939752i \(0.388944\pi\)
\(14\) 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 \(-0.513228\pi\)
−0.0415447 + 0.999137i \(0.513228\pi\)
\(24\) 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 \(-0.458033\pi\)
0.131461 + 0.991321i \(0.458033\pi\)
\(30\) 0 0
\(31\) −187.606 −1.08694 −0.543468 0.839430i \(-0.682889\pi\)
−0.543468 + 0.839430i \(0.682889\pi\)
\(32\) 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 \(-0.581597\pi\)
−0.253546 + 0.967323i \(0.581597\pi\)
\(38\) 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 \(-0.527024\pi\)
−0.0847978 + 0.996398i \(0.527024\pi\)
\(48\) 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 \(-0.109012\pi\)
0.941928 + 0.335815i \(0.109012\pi\)
\(54\) 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 \(-0.209278\pi\)
0.791543 + 0.611114i \(0.209278\pi\)
\(62\) 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 \(-0.581242\pi\)
−0.252467 + 0.967606i \(0.581242\pi\)
\(72\) 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 \(-0.431060\pi\)
0.214892 + 0.976638i \(0.431060\pi\)
\(80\) 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 \(-0.531608\pi\)
−0.0991360 + 0.995074i \(0.531608\pi\)
\(102\) 0 0
\(103\) −682.440 −0.652843 −0.326421 0.945224i \(-0.605843\pi\)
−0.326421 + 0.945224i \(0.605843\pi\)
\(104\) 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 \(-0.411334\pi\)
0.274963 + 0.961455i \(0.411334\pi\)
\(110\) 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 \(-0.591096\pi\)
−0.282296 + 0.959327i \(0.591096\pi\)
\(128\) 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 \(-0.384559\pi\)
0.354770 + 0.934954i \(0.384559\pi\)
\(150\) 0 0
\(151\) −1175.51 −0.633521 −0.316761 0.948505i \(-0.602595\pi\)
−0.316761 + 0.948505i \(0.602595\pi\)
\(152\) 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 \(-0.589530\pi\)
−0.277574 + 0.960704i \(0.589530\pi\)
\(158\) 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 \(-0.503378\pi\)
−0.0106114 + 0.999944i \(0.503378\pi\)
\(168\) 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 \(-0.318630\pi\)
0.539455 + 0.842014i \(0.318630\pi\)
\(174\) 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 \(-0.774569\pi\)
−0.759526 + 0.650477i \(0.774569\pi\)
\(182\) 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 \(-0.0810960\pi\)
0.967721 + 0.252023i \(0.0810960\pi\)
\(192\) 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 \(-0.329891\pi\)
0.509337 + 0.860567i \(0.329891\pi\)
\(198\) 0 0
\(199\) 948.556 0.337896 0.168948 0.985625i \(-0.445963\pi\)
0.168948 + 0.985625i \(0.445963\pi\)
\(200\) 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 \(-0.742088\pi\)
−0.689315 + 0.724462i \(0.742088\pi\)
\(224\) 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 \(-0.498410\pi\)
0.00499644 + 0.999988i \(0.498410\pi\)
\(230\) 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 \(-0.528238\pi\)
−0.0885962 + 0.996068i \(0.528238\pi\)
\(240\) 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 \(-0.168252\pi\)
0.863524 + 0.504308i \(0.168252\pi\)
\(264\) 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 \(-0.851251\pi\)
−0.892784 + 0.450484i \(0.851251\pi\)
\(270\) 0 0
\(271\) −5399.92 −1.21041 −0.605206 0.796069i \(-0.706909\pi\)
−0.605206 + 0.796069i \(0.706909\pi\)
\(272\) 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 \(-0.658981\pi\)
−0.478946 + 0.877844i \(0.658981\pi\)
\(278\) 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 \(-0.719133\pi\)
−0.635324 + 0.772246i \(0.719133\pi\)
\(294\) 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 \(-0.235274\pi\)
0.739052 + 0.673648i \(0.235274\pi\)
\(312\) 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 \(-0.327277\pi\)
0.516385 + 0.856357i \(0.327277\pi\)
\(318\) 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 \(-0.760372\pi\)
−0.729768 + 0.683695i \(0.760372\pi\)
\(350\) 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 \(-0.442902\pi\)
0.178419 + 0.983955i \(0.442902\pi\)
\(360\) 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 \(-0.383762\pi\)
0.357109 + 0.934063i \(0.383762\pi\)
\(368\) 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 \(-0.570894\pi\)
−0.220882 + 0.975300i \(0.570894\pi\)
\(374\) 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 \(-0.336884\pi\)
0.490309 + 0.871549i \(0.336884\pi\)
\(384\) 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 \(-0.822073\pi\)
−0.847800 + 0.530317i \(0.822073\pi\)
\(390\) 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 \(-0.599770\pi\)
−0.308330 + 0.951280i \(0.599770\pi\)
\(398\) 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 \(-0.586716\pi\)
−0.269070 + 0.963121i \(0.586716\pi\)
\(422\) 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 \(-0.254085\pi\)
0.697975 + 0.716122i \(0.254085\pi\)
\(432\) 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 \(-0.548526\pi\)
−0.151858 + 0.988402i \(0.548526\pi\)
\(440\) 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 \(-0.515785\pi\)
−0.0495705 + 0.998771i \(0.515785\pi\)
\(462\) 0 0
\(463\) −14082.7 −1.41356 −0.706782 0.707431i \(-0.749854\pi\)
−0.706782 + 0.707431i \(0.749854\pi\)
\(464\) 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 \(-0.876536\pi\)
−0.925715 + 0.378222i \(0.876536\pi\)
\(480\) 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 \(-0.690773\pi\)
−0.564091 + 0.825712i \(0.690773\pi\)
\(488\) 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 \(-0.316684\pi\)
0.544592 + 0.838701i \(0.316684\pi\)
\(504\) 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 \(-0.493757\pi\)
0.0196124 + 0.999808i \(0.493757\pi\)
\(510\) 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 \(-0.338260\pi\)
0.486537 + 0.873660i \(0.338260\pi\)
\(542\) 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 \(-0.283181\pi\)
0.629692 + 0.776845i \(0.283181\pi\)
\(558\) 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 \(-0.750539\pi\)
−0.708302 + 0.705909i \(0.750539\pi\)
\(600\) 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 \(-0.378225\pi\)
0.373304 + 0.927709i \(0.378225\pi\)
\(608\) 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 \(-0.318148\pi\)
0.540731 + 0.841195i \(0.318148\pi\)
\(614\) 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 \(-0.361315\pi\)
0.422039 + 0.906578i \(0.361315\pi\)
\(632\) 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 \(-0.420183\pi\)
0.248132 + 0.968726i \(0.420183\pi\)
\(648\) 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 \(-0.428389\pi\)
0.223080 + 0.974800i \(0.428389\pi\)
\(654\) 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 \(-0.523706\pi\)
−0.0744046 + 0.997228i \(0.523706\pi\)
\(662\) 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 \(-0.259070\pi\)
0.686673 + 0.726966i \(0.259070\pi\)
\(678\) 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 \(-0.746316\pi\)
−0.698876 + 0.715243i \(0.746316\pi\)
\(702\) 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 \(-0.453571\pi\)
0.145343 + 0.989381i \(0.453571\pi\)
\(710\) 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 \(-0.353365\pi\)
0.444547 + 0.895756i \(0.353365\pi\)
\(720\) 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 \(-0.366661\pi\)
0.406754 + 0.913538i \(0.366661\pi\)
\(728\) 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 \(-0.368677\pi\)
0.400958 + 0.916096i \(0.368677\pi\)
\(734\) 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 \(-0.486065\pi\)
0.0437652 + 0.999042i \(0.486065\pi\)
\(744\) 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 \(-0.507785\pi\)
−0.0244562 + 0.999701i \(0.507785\pi\)
\(752\) 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 \(-0.257277\pi\)
0.690758 + 0.723086i \(0.257277\pi\)
\(758\) 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 \(-0.623458\pi\)
−0.378202 + 0.925723i \(0.623458\pi\)
\(774\) 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 \(-0.238312\pi\)
0.732589 + 0.680672i \(0.238312\pi\)
\(798\) 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 \(-0.432890\pi\)
0.209275 + 0.977857i \(0.432890\pi\)
\(822\) 0 0
\(823\) −47001.9 −1.99074 −0.995372 0.0960935i \(-0.969365\pi\)
−0.995372 + 0.0960935i \(0.969365\pi\)
\(824\) 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 \(-0.658286\pi\)
−0.477029 + 0.878888i \(0.658286\pi\)
\(830\) 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 \(-0.572739\pi\)
−0.226532 + 0.974004i \(0.572739\pi\)
\(840\) 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 \(-0.222136\pi\)
0.766219 + 0.642579i \(0.222136\pi\)
\(854\) 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 \(-0.731161\pi\)
−0.664042 + 0.747695i \(0.731161\pi\)
\(864\) 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 \(-0.914994\pi\)
−0.964552 + 0.263892i \(0.914994\pi\)
\(878\) 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 \(-0.404942\pi\)
0.294215 + 0.955739i \(0.404942\pi\)
\(888\) 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 \(-0.488560\pi\)
0.0359313 + 0.999354i \(0.488560\pi\)
\(912\) 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 \(-0.392689\pi\)
0.330776 + 0.943709i \(0.392689\pi\)
\(920\) 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 \(-0.634817\pi\)
−0.410991 + 0.911640i \(0.634817\pi\)
\(942\) 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 \(-0.195741\pi\)
0.816808 + 0.576909i \(0.195741\pi\)
\(968\) 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 \(-0.804959\pi\)
−0.818075 + 0.575112i \(0.804959\pi\)
\(984\) 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 \(-0.543559\pi\)
−0.136418 + 0.990651i \(0.543559\pi\)
\(992\) 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 \(-0.368361\pi\)
0.401869 + 0.915697i \(0.368361\pi\)
\(998\) 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.r.1.1 3
3.2 odd 2 2304.4.a.bt.1.3 3
4.3 odd 2 768.4.a.t.1.1 3
8.3 odd 2 768.4.a.q.1.3 3
8.5 even 2 768.4.a.s.1.3 3
12.11 even 2 2304.4.a.bu.1.3 3
16.3 odd 4 96.4.d.a.49.6 6
16.5 even 4 24.4.d.a.13.4 yes 6
16.11 odd 4 96.4.d.a.49.1 6
16.13 even 4 24.4.d.a.13.3 6
24.5 odd 2 2304.4.a.bv.1.1 3
24.11 even 2 2304.4.a.bw.1.1 3
48.5 odd 4 72.4.d.d.37.3 6
48.11 even 4 288.4.d.d.145.5 6
48.29 odd 4 72.4.d.d.37.4 6
48.35 even 4 288.4.d.d.145.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.d.a.13.3 6 16.13 even 4
24.4.d.a.13.4 yes 6 16.5 even 4
72.4.d.d.37.3 6 48.5 odd 4
72.4.d.d.37.4 6 48.29 odd 4
96.4.d.a.49.1 6 16.11 odd 4
96.4.d.a.49.6 6 16.3 odd 4
288.4.d.d.145.2 6 48.35 even 4
288.4.d.d.145.5 6 48.11 even 4
768.4.a.q.1.3 3 8.3 odd 2
768.4.a.r.1.1 3 1.1 even 1 trivial
768.4.a.s.1.3 3 8.5 even 2
768.4.a.t.1.1 3 4.3 odd 2
2304.4.a.bt.1.3 3 3.2 odd 2
2304.4.a.bu.1.3 3 12.11 even 2
2304.4.a.bv.1.1 3 24.5 odd 2
2304.4.a.bw.1.1 3 24.11 even 2