Properties

Label 552.2.a.g.1.3
Level $552$
Weight $2$
Character 552.1
Self dual yes
Analytic conductor $4.408$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 552.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.40774219157\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 552.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.35026 q^{5} -2.96239 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.35026 q^{5} -2.96239 q^{7} +1.00000 q^{9} +1.61213 q^{11} +2.00000 q^{13} +3.35026 q^{15} +4.96239 q^{17} -1.35026 q^{19} -2.96239 q^{21} -1.00000 q^{23} +6.22425 q^{25} +1.00000 q^{27} -7.92478 q^{29} +5.92478 q^{31} +1.61213 q^{33} -9.92478 q^{35} -2.31265 q^{37} +2.00000 q^{39} -1.22425 q^{41} -4.57452 q^{43} +3.35026 q^{45} +1.92478 q^{47} +1.77575 q^{49} +4.96239 q^{51} -4.12601 q^{53} +5.40105 q^{55} -1.35026 q^{57} +2.70052 q^{59} +14.3127 q^{61} -2.96239 q^{63} +6.70052 q^{65} +4.57452 q^{67} -1.00000 q^{69} -16.6253 q^{71} +2.00000 q^{73} +6.22425 q^{75} -4.77575 q^{77} -5.03761 q^{79} +1.00000 q^{81} -15.0132 q^{83} +16.6253 q^{85} -7.92478 q^{87} -5.73813 q^{89} -5.92478 q^{91} +5.92478 q^{93} -4.52373 q^{95} +12.7005 q^{97} +1.61213 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 2q^{7} + 3q^{9} + 4q^{11} + 6q^{13} + 4q^{17} + 6q^{19} + 2q^{21} - 3q^{23} + 17q^{25} + 3q^{27} - 2q^{29} - 4q^{31} + 4q^{33} - 8q^{35} + 14q^{37} + 6q^{39} - 2q^{41} - 2q^{43} - 16q^{47} + 7q^{49} + 4q^{51} - 4q^{53} - 24q^{55} + 6q^{57} - 12q^{59} + 22q^{61} + 2q^{63} + 2q^{67} - 3q^{69} - 8q^{71} + 6q^{73} + 17q^{75} - 16q^{77} - 26q^{79} + 3q^{81} - 4q^{83} + 8q^{85} - 2q^{87} - 8q^{89} + 4q^{91} - 4q^{93} - 32q^{95} + 18q^{97} + 4q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 3.35026 1.49828 0.749141 0.662410i \(-0.230466\pi\)
0.749141 + 0.662410i \(0.230466\pi\)
\(6\) 0 0
\(7\) −2.96239 −1.11968 −0.559839 0.828602i \(-0.689137\pi\)
−0.559839 + 0.828602i \(0.689137\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.61213 0.486075 0.243037 0.970017i \(-0.421856\pi\)
0.243037 + 0.970017i \(0.421856\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 3.35026 0.865034
\(16\) 0 0
\(17\) 4.96239 1.20356 0.601778 0.798663i \(-0.294459\pi\)
0.601778 + 0.798663i \(0.294459\pi\)
\(18\) 0 0
\(19\) −1.35026 −0.309771 −0.154886 0.987932i \(-0.549501\pi\)
−0.154886 + 0.987932i \(0.549501\pi\)
\(20\) 0 0
\(21\) −2.96239 −0.646446
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 6.22425 1.24485
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.92478 −1.47159 −0.735797 0.677202i \(-0.763192\pi\)
−0.735797 + 0.677202i \(0.763192\pi\)
\(30\) 0 0
\(31\) 5.92478 1.06412 0.532061 0.846706i \(-0.321418\pi\)
0.532061 + 0.846706i \(0.321418\pi\)
\(32\) 0 0
\(33\) 1.61213 0.280635
\(34\) 0 0
\(35\) −9.92478 −1.67759
\(36\) 0 0
\(37\) −2.31265 −0.380197 −0.190099 0.981765i \(-0.560881\pi\)
−0.190099 + 0.981765i \(0.560881\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −1.22425 −0.191196 −0.0955982 0.995420i \(-0.530476\pi\)
−0.0955982 + 0.995420i \(0.530476\pi\)
\(42\) 0 0
\(43\) −4.57452 −0.697607 −0.348804 0.937196i \(-0.613412\pi\)
−0.348804 + 0.937196i \(0.613412\pi\)
\(44\) 0 0
\(45\) 3.35026 0.499428
\(46\) 0 0
\(47\) 1.92478 0.280758 0.140379 0.990098i \(-0.455168\pi\)
0.140379 + 0.990098i \(0.455168\pi\)
\(48\) 0 0
\(49\) 1.77575 0.253678
\(50\) 0 0
\(51\) 4.96239 0.694873
\(52\) 0 0
\(53\) −4.12601 −0.566751 −0.283375 0.959009i \(-0.591454\pi\)
−0.283375 + 0.959009i \(0.591454\pi\)
\(54\) 0 0
\(55\) 5.40105 0.728277
\(56\) 0 0
\(57\) −1.35026 −0.178847
\(58\) 0 0
\(59\) 2.70052 0.351578 0.175789 0.984428i \(-0.443752\pi\)
0.175789 + 0.984428i \(0.443752\pi\)
\(60\) 0 0
\(61\) 14.3127 1.83255 0.916274 0.400553i \(-0.131182\pi\)
0.916274 + 0.400553i \(0.131182\pi\)
\(62\) 0 0
\(63\) −2.96239 −0.373226
\(64\) 0 0
\(65\) 6.70052 0.831098
\(66\) 0 0
\(67\) 4.57452 0.558866 0.279433 0.960165i \(-0.409854\pi\)
0.279433 + 0.960165i \(0.409854\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −16.6253 −1.97306 −0.986530 0.163580i \(-0.947696\pi\)
−0.986530 + 0.163580i \(0.947696\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 6.22425 0.718715
\(76\) 0 0
\(77\) −4.77575 −0.544247
\(78\) 0 0
\(79\) −5.03761 −0.566776 −0.283388 0.959005i \(-0.591458\pi\)
−0.283388 + 0.959005i \(0.591458\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.0132 −1.64791 −0.823955 0.566655i \(-0.808237\pi\)
−0.823955 + 0.566655i \(0.808237\pi\)
\(84\) 0 0
\(85\) 16.6253 1.80327
\(86\) 0 0
\(87\) −7.92478 −0.849625
\(88\) 0 0
\(89\) −5.73813 −0.608241 −0.304121 0.952634i \(-0.598363\pi\)
−0.304121 + 0.952634i \(0.598363\pi\)
\(90\) 0 0
\(91\) −5.92478 −0.621085
\(92\) 0 0
\(93\) 5.92478 0.614371
\(94\) 0 0
\(95\) −4.52373 −0.464125
\(96\) 0 0
\(97\) 12.7005 1.28954 0.644771 0.764375i \(-0.276953\pi\)
0.644771 + 0.764375i \(0.276953\pi\)
\(98\) 0 0
\(99\) 1.61213 0.162025
\(100\) 0 0
\(101\) −15.4010 −1.53246 −0.766231 0.642566i \(-0.777870\pi\)
−0.766231 + 0.642566i \(0.777870\pi\)
\(102\) 0 0
\(103\) −13.6629 −1.34625 −0.673123 0.739530i \(-0.735048\pi\)
−0.673123 + 0.739530i \(0.735048\pi\)
\(104\) 0 0
\(105\) −9.92478 −0.968559
\(106\) 0 0
\(107\) 18.2374 1.76308 0.881539 0.472111i \(-0.156508\pi\)
0.881539 + 0.472111i \(0.156508\pi\)
\(108\) 0 0
\(109\) −19.4617 −1.86409 −0.932045 0.362341i \(-0.881977\pi\)
−0.932045 + 0.362341i \(0.881977\pi\)
\(110\) 0 0
\(111\) −2.31265 −0.219507
\(112\) 0 0
\(113\) 7.66291 0.720866 0.360433 0.932785i \(-0.382629\pi\)
0.360433 + 0.932785i \(0.382629\pi\)
\(114\) 0 0
\(115\) −3.35026 −0.312414
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −14.7005 −1.34759
\(120\) 0 0
\(121\) −8.40105 −0.763732
\(122\) 0 0
\(123\) −1.22425 −0.110387
\(124\) 0 0
\(125\) 4.10157 0.366856
\(126\) 0 0
\(127\) −15.4763 −1.37330 −0.686648 0.726990i \(-0.740919\pi\)
−0.686648 + 0.726990i \(0.740919\pi\)
\(128\) 0 0
\(129\) −4.57452 −0.402764
\(130\) 0 0
\(131\) 0.775746 0.0677773 0.0338886 0.999426i \(-0.489211\pi\)
0.0338886 + 0.999426i \(0.489211\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 3.35026 0.288345
\(136\) 0 0
\(137\) −8.96239 −0.765709 −0.382854 0.923809i \(-0.625059\pi\)
−0.382854 + 0.923809i \(0.625059\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 1.92478 0.162095
\(142\) 0 0
\(143\) 3.22425 0.269626
\(144\) 0 0
\(145\) −26.5501 −2.20486
\(146\) 0 0
\(147\) 1.77575 0.146461
\(148\) 0 0
\(149\) −10.0508 −0.823392 −0.411696 0.911321i \(-0.635064\pi\)
−0.411696 + 0.911321i \(0.635064\pi\)
\(150\) 0 0
\(151\) 11.8496 0.964303 0.482152 0.876088i \(-0.339855\pi\)
0.482152 + 0.876088i \(0.339855\pi\)
\(152\) 0 0
\(153\) 4.96239 0.401185
\(154\) 0 0
\(155\) 19.8496 1.59435
\(156\) 0 0
\(157\) 22.9380 1.83065 0.915324 0.402718i \(-0.131935\pi\)
0.915324 + 0.402718i \(0.131935\pi\)
\(158\) 0 0
\(159\) −4.12601 −0.327214
\(160\) 0 0
\(161\) 2.96239 0.233469
\(162\) 0 0
\(163\) 6.70052 0.524826 0.262413 0.964956i \(-0.415482\pi\)
0.262413 + 0.964956i \(0.415482\pi\)
\(164\) 0 0
\(165\) 5.40105 0.420471
\(166\) 0 0
\(167\) 24.6253 1.90556 0.952781 0.303657i \(-0.0982076\pi\)
0.952781 + 0.303657i \(0.0982076\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −1.35026 −0.103257
\(172\) 0 0
\(173\) −23.9248 −1.81897 −0.909484 0.415740i \(-0.863523\pi\)
−0.909484 + 0.415740i \(0.863523\pi\)
\(174\) 0 0
\(175\) −18.4387 −1.39383
\(176\) 0 0
\(177\) 2.70052 0.202984
\(178\) 0 0
\(179\) 19.3258 1.44448 0.722240 0.691643i \(-0.243113\pi\)
0.722240 + 0.691643i \(0.243113\pi\)
\(180\) 0 0
\(181\) 20.2374 1.50424 0.752118 0.659028i \(-0.229032\pi\)
0.752118 + 0.659028i \(0.229032\pi\)
\(182\) 0 0
\(183\) 14.3127 1.05802
\(184\) 0 0
\(185\) −7.74798 −0.569643
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) −2.96239 −0.215482
\(190\) 0 0
\(191\) −9.14903 −0.662001 −0.331000 0.943631i \(-0.607386\pi\)
−0.331000 + 0.943631i \(0.607386\pi\)
\(192\) 0 0
\(193\) −6.77575 −0.487729 −0.243864 0.969809i \(-0.578415\pi\)
−0.243864 + 0.969809i \(0.578415\pi\)
\(194\) 0 0
\(195\) 6.70052 0.479834
\(196\) 0 0
\(197\) −24.5501 −1.74912 −0.874560 0.484917i \(-0.838850\pi\)
−0.874560 + 0.484917i \(0.838850\pi\)
\(198\) 0 0
\(199\) −15.3357 −1.08712 −0.543559 0.839371i \(-0.682923\pi\)
−0.543559 + 0.839371i \(0.682923\pi\)
\(200\) 0 0
\(201\) 4.57452 0.322661
\(202\) 0 0
\(203\) 23.4763 1.64771
\(204\) 0 0
\(205\) −4.10157 −0.286466
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −2.17679 −0.150572
\(210\) 0 0
\(211\) −5.14903 −0.354474 −0.177237 0.984168i \(-0.556716\pi\)
−0.177237 + 0.984168i \(0.556716\pi\)
\(212\) 0 0
\(213\) −16.6253 −1.13915
\(214\) 0 0
\(215\) −15.3258 −1.04521
\(216\) 0 0
\(217\) −17.5515 −1.19147
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 9.92478 0.667613
\(222\) 0 0
\(223\) 9.55149 0.639615 0.319808 0.947482i \(-0.396382\pi\)
0.319808 + 0.947482i \(0.396382\pi\)
\(224\) 0 0
\(225\) 6.22425 0.414950
\(226\) 0 0
\(227\) 3.68735 0.244738 0.122369 0.992485i \(-0.460951\pi\)
0.122369 + 0.992485i \(0.460951\pi\)
\(228\) 0 0
\(229\) 12.2374 0.808672 0.404336 0.914611i \(-0.367503\pi\)
0.404336 + 0.914611i \(0.367503\pi\)
\(230\) 0 0
\(231\) −4.77575 −0.314221
\(232\) 0 0
\(233\) −3.40105 −0.222810 −0.111405 0.993775i \(-0.535535\pi\)
−0.111405 + 0.993775i \(0.535535\pi\)
\(234\) 0 0
\(235\) 6.44851 0.420654
\(236\) 0 0
\(237\) −5.03761 −0.327228
\(238\) 0 0
\(239\) −1.55149 −0.100358 −0.0501789 0.998740i \(-0.515979\pi\)
−0.0501789 + 0.998740i \(0.515979\pi\)
\(240\) 0 0
\(241\) 20.7005 1.33344 0.666719 0.745309i \(-0.267698\pi\)
0.666719 + 0.745309i \(0.267698\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 5.94921 0.380081
\(246\) 0 0
\(247\) −2.70052 −0.171830
\(248\) 0 0
\(249\) −15.0132 −0.951421
\(250\) 0 0
\(251\) 8.46310 0.534186 0.267093 0.963671i \(-0.413937\pi\)
0.267093 + 0.963671i \(0.413937\pi\)
\(252\) 0 0
\(253\) −1.61213 −0.101354
\(254\) 0 0
\(255\) 16.6253 1.04112
\(256\) 0 0
\(257\) 18.6253 1.16181 0.580907 0.813970i \(-0.302698\pi\)
0.580907 + 0.813970i \(0.302698\pi\)
\(258\) 0 0
\(259\) 6.85097 0.425699
\(260\) 0 0
\(261\) −7.92478 −0.490531
\(262\) 0 0
\(263\) −0.523730 −0.0322946 −0.0161473 0.999870i \(-0.505140\pi\)
−0.0161473 + 0.999870i \(0.505140\pi\)
\(264\) 0 0
\(265\) −13.8232 −0.849153
\(266\) 0 0
\(267\) −5.73813 −0.351168
\(268\) 0 0
\(269\) 0.700523 0.0427117 0.0213558 0.999772i \(-0.493202\pi\)
0.0213558 + 0.999772i \(0.493202\pi\)
\(270\) 0 0
\(271\) −10.0752 −0.612026 −0.306013 0.952027i \(-0.598995\pi\)
−0.306013 + 0.952027i \(0.598995\pi\)
\(272\) 0 0
\(273\) −5.92478 −0.358584
\(274\) 0 0
\(275\) 10.0343 0.605090
\(276\) 0 0
\(277\) −22.4749 −1.35038 −0.675192 0.737642i \(-0.735939\pi\)
−0.675192 + 0.737642i \(0.735939\pi\)
\(278\) 0 0
\(279\) 5.92478 0.354707
\(280\) 0 0
\(281\) 11.8134 0.704726 0.352363 0.935863i \(-0.385378\pi\)
0.352363 + 0.935863i \(0.385378\pi\)
\(282\) 0 0
\(283\) 29.1998 1.73575 0.867874 0.496784i \(-0.165486\pi\)
0.867874 + 0.496784i \(0.165486\pi\)
\(284\) 0 0
\(285\) −4.52373 −0.267963
\(286\) 0 0
\(287\) 3.62672 0.214078
\(288\) 0 0
\(289\) 7.62530 0.448547
\(290\) 0 0
\(291\) 12.7005 0.744518
\(292\) 0 0
\(293\) 9.27504 0.541854 0.270927 0.962600i \(-0.412670\pi\)
0.270927 + 0.962600i \(0.412670\pi\)
\(294\) 0 0
\(295\) 9.04746 0.526764
\(296\) 0 0
\(297\) 1.61213 0.0935451
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 13.5515 0.781095
\(302\) 0 0
\(303\) −15.4010 −0.884767
\(304\) 0 0
\(305\) 47.9511 2.74567
\(306\) 0 0
\(307\) 31.9511 1.82355 0.911774 0.410693i \(-0.134713\pi\)
0.911774 + 0.410693i \(0.134713\pi\)
\(308\) 0 0
\(309\) −13.6629 −0.777256
\(310\) 0 0
\(311\) 13.1490 0.745613 0.372807 0.927909i \(-0.378395\pi\)
0.372807 + 0.927909i \(0.378395\pi\)
\(312\) 0 0
\(313\) 24.0263 1.35805 0.679025 0.734115i \(-0.262403\pi\)
0.679025 + 0.734115i \(0.262403\pi\)
\(314\) 0 0
\(315\) −9.92478 −0.559198
\(316\) 0 0
\(317\) 27.9248 1.56841 0.784206 0.620501i \(-0.213071\pi\)
0.784206 + 0.620501i \(0.213071\pi\)
\(318\) 0 0
\(319\) −12.7757 −0.715304
\(320\) 0 0
\(321\) 18.2374 1.01791
\(322\) 0 0
\(323\) −6.70052 −0.372827
\(324\) 0 0
\(325\) 12.4485 0.690519
\(326\) 0 0
\(327\) −19.4617 −1.07623
\(328\) 0 0
\(329\) −5.70194 −0.314358
\(330\) 0 0
\(331\) 6.70052 0.368294 0.184147 0.982899i \(-0.441048\pi\)
0.184147 + 0.982899i \(0.441048\pi\)
\(332\) 0 0
\(333\) −2.31265 −0.126732
\(334\) 0 0
\(335\) 15.3258 0.837339
\(336\) 0 0
\(337\) −10.3733 −0.565069 −0.282534 0.959257i \(-0.591175\pi\)
−0.282534 + 0.959257i \(0.591175\pi\)
\(338\) 0 0
\(339\) 7.66291 0.416192
\(340\) 0 0
\(341\) 9.55149 0.517242
\(342\) 0 0
\(343\) 15.4763 0.835640
\(344\) 0 0
\(345\) −3.35026 −0.180372
\(346\) 0 0
\(347\) −17.7743 −0.954176 −0.477088 0.878855i \(-0.658308\pi\)
−0.477088 + 0.878855i \(0.658308\pi\)
\(348\) 0 0
\(349\) 18.6253 0.996989 0.498495 0.866893i \(-0.333886\pi\)
0.498495 + 0.866893i \(0.333886\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −15.7743 −0.839583 −0.419791 0.907621i \(-0.637897\pi\)
−0.419791 + 0.907621i \(0.637897\pi\)
\(354\) 0 0
\(355\) −55.6991 −2.95620
\(356\) 0 0
\(357\) −14.7005 −0.778034
\(358\) 0 0
\(359\) −5.29948 −0.279696 −0.139848 0.990173i \(-0.544661\pi\)
−0.139848 + 0.990173i \(0.544661\pi\)
\(360\) 0 0
\(361\) −17.1768 −0.904042
\(362\) 0 0
\(363\) −8.40105 −0.440941
\(364\) 0 0
\(365\) 6.70052 0.350721
\(366\) 0 0
\(367\) 4.51388 0.235623 0.117811 0.993036i \(-0.462412\pi\)
0.117811 + 0.993036i \(0.462412\pi\)
\(368\) 0 0
\(369\) −1.22425 −0.0637321
\(370\) 0 0
\(371\) 12.2228 0.634578
\(372\) 0 0
\(373\) 13.6873 0.708704 0.354352 0.935112i \(-0.384701\pi\)
0.354352 + 0.935112i \(0.384701\pi\)
\(374\) 0 0
\(375\) 4.10157 0.211804
\(376\) 0 0
\(377\) −15.8496 −0.816294
\(378\) 0 0
\(379\) 30.3488 1.55892 0.779458 0.626455i \(-0.215495\pi\)
0.779458 + 0.626455i \(0.215495\pi\)
\(380\) 0 0
\(381\) −15.4763 −0.792873
\(382\) 0 0
\(383\) 28.3733 1.44981 0.724904 0.688850i \(-0.241884\pi\)
0.724904 + 0.688850i \(0.241884\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) 0 0
\(387\) −4.57452 −0.232536
\(388\) 0 0
\(389\) 19.4518 0.986247 0.493124 0.869959i \(-0.335855\pi\)
0.493124 + 0.869959i \(0.335855\pi\)
\(390\) 0 0
\(391\) −4.96239 −0.250959
\(392\) 0 0
\(393\) 0.775746 0.0391312
\(394\) 0 0
\(395\) −16.8773 −0.849190
\(396\) 0 0
\(397\) −28.8021 −1.44554 −0.722768 0.691091i \(-0.757130\pi\)
−0.722768 + 0.691091i \(0.757130\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −24.1378 −1.20538 −0.602691 0.797974i \(-0.705905\pi\)
−0.602691 + 0.797974i \(0.705905\pi\)
\(402\) 0 0
\(403\) 11.8496 0.590268
\(404\) 0 0
\(405\) 3.35026 0.166476
\(406\) 0 0
\(407\) −3.72829 −0.184804
\(408\) 0 0
\(409\) −28.1768 −1.39325 −0.696626 0.717434i \(-0.745316\pi\)
−0.696626 + 0.717434i \(0.745316\pi\)
\(410\) 0 0
\(411\) −8.96239 −0.442082
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) −50.2981 −2.46903
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −21.4617 −1.04847 −0.524236 0.851573i \(-0.675649\pi\)
−0.524236 + 0.851573i \(0.675649\pi\)
\(420\) 0 0
\(421\) 13.1636 0.641556 0.320778 0.947154i \(-0.396056\pi\)
0.320778 + 0.947154i \(0.396056\pi\)
\(422\) 0 0
\(423\) 1.92478 0.0935859
\(424\) 0 0
\(425\) 30.8872 1.49825
\(426\) 0 0
\(427\) −42.3996 −2.05186
\(428\) 0 0
\(429\) 3.22425 0.155668
\(430\) 0 0
\(431\) −21.5223 −1.03669 −0.518347 0.855171i \(-0.673452\pi\)
−0.518347 + 0.855171i \(0.673452\pi\)
\(432\) 0 0
\(433\) −2.77575 −0.133394 −0.0666969 0.997773i \(-0.521246\pi\)
−0.0666969 + 0.997773i \(0.521246\pi\)
\(434\) 0 0
\(435\) −26.5501 −1.27298
\(436\) 0 0
\(437\) 1.35026 0.0645918
\(438\) 0 0
\(439\) 3.84955 0.183729 0.0918646 0.995772i \(-0.470717\pi\)
0.0918646 + 0.995772i \(0.470717\pi\)
\(440\) 0 0
\(441\) 1.77575 0.0845593
\(442\) 0 0
\(443\) 26.9525 1.28055 0.640277 0.768144i \(-0.278820\pi\)
0.640277 + 0.768144i \(0.278820\pi\)
\(444\) 0 0
\(445\) −19.2243 −0.911317
\(446\) 0 0
\(447\) −10.0508 −0.475386
\(448\) 0 0
\(449\) 1.07381 0.0506761 0.0253381 0.999679i \(-0.491934\pi\)
0.0253381 + 0.999679i \(0.491934\pi\)
\(450\) 0 0
\(451\) −1.97365 −0.0929357
\(452\) 0 0
\(453\) 11.8496 0.556741
\(454\) 0 0
\(455\) −19.8496 −0.930561
\(456\) 0 0
\(457\) 7.29948 0.341455 0.170728 0.985318i \(-0.445388\pi\)
0.170728 + 0.985318i \(0.445388\pi\)
\(458\) 0 0
\(459\) 4.96239 0.231624
\(460\) 0 0
\(461\) 5.59754 0.260703 0.130352 0.991468i \(-0.458389\pi\)
0.130352 + 0.991468i \(0.458389\pi\)
\(462\) 0 0
\(463\) −3.84955 −0.178904 −0.0894520 0.995991i \(-0.528512\pi\)
−0.0894520 + 0.995991i \(0.528512\pi\)
\(464\) 0 0
\(465\) 19.8496 0.920501
\(466\) 0 0
\(467\) −3.68735 −0.170630 −0.0853151 0.996354i \(-0.527190\pi\)
−0.0853151 + 0.996354i \(0.527190\pi\)
\(468\) 0 0
\(469\) −13.5515 −0.625750
\(470\) 0 0
\(471\) 22.9380 1.05692
\(472\) 0 0
\(473\) −7.37470 −0.339089
\(474\) 0 0
\(475\) −8.40437 −0.385619
\(476\) 0 0
\(477\) −4.12601 −0.188917
\(478\) 0 0
\(479\) −2.59895 −0.118749 −0.0593746 0.998236i \(-0.518911\pi\)
−0.0593746 + 0.998236i \(0.518911\pi\)
\(480\) 0 0
\(481\) −4.62530 −0.210896
\(482\) 0 0
\(483\) 2.96239 0.134793
\(484\) 0 0
\(485\) 42.5501 1.93210
\(486\) 0 0
\(487\) 33.2506 1.50673 0.753364 0.657603i \(-0.228430\pi\)
0.753364 + 0.657603i \(0.228430\pi\)
\(488\) 0 0
\(489\) 6.70052 0.303008
\(490\) 0 0
\(491\) −13.9248 −0.628416 −0.314208 0.949354i \(-0.601739\pi\)
−0.314208 + 0.949354i \(0.601739\pi\)
\(492\) 0 0
\(493\) −39.3258 −1.77115
\(494\) 0 0
\(495\) 5.40105 0.242759
\(496\) 0 0
\(497\) 49.2506 2.20919
\(498\) 0 0
\(499\) 19.5975 0.877306 0.438653 0.898656i \(-0.355456\pi\)
0.438653 + 0.898656i \(0.355456\pi\)
\(500\) 0 0
\(501\) 24.6253 1.10018
\(502\) 0 0
\(503\) 28.4749 1.26963 0.634816 0.772664i \(-0.281076\pi\)
0.634816 + 0.772664i \(0.281076\pi\)
\(504\) 0 0
\(505\) −51.5975 −2.29606
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −39.6239 −1.75630 −0.878149 0.478387i \(-0.841222\pi\)
−0.878149 + 0.478387i \(0.841222\pi\)
\(510\) 0 0
\(511\) −5.92478 −0.262097
\(512\) 0 0
\(513\) −1.35026 −0.0596155
\(514\) 0 0
\(515\) −45.7743 −2.01706
\(516\) 0 0
\(517\) 3.10299 0.136469
\(518\) 0 0
\(519\) −23.9248 −1.05018
\(520\) 0 0
\(521\) −18.2130 −0.797926 −0.398963 0.916967i \(-0.630630\pi\)
−0.398963 + 0.916967i \(0.630630\pi\)
\(522\) 0 0
\(523\) −24.8265 −1.08559 −0.542794 0.839866i \(-0.682634\pi\)
−0.542794 + 0.839866i \(0.682634\pi\)
\(524\) 0 0
\(525\) −18.4387 −0.804729
\(526\) 0 0
\(527\) 29.4010 1.28073
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 2.70052 0.117193
\(532\) 0 0
\(533\) −2.44851 −0.106057
\(534\) 0 0
\(535\) 61.1002 2.64159
\(536\) 0 0
\(537\) 19.3258 0.833971
\(538\) 0 0
\(539\) 2.86273 0.123306
\(540\) 0 0
\(541\) 6.47486 0.278376 0.139188 0.990266i \(-0.455551\pi\)
0.139188 + 0.990266i \(0.455551\pi\)
\(542\) 0 0
\(543\) 20.2374 0.868471
\(544\) 0 0
\(545\) −65.2017 −2.79294
\(546\) 0 0
\(547\) 2.55008 0.109033 0.0545167 0.998513i \(-0.482638\pi\)
0.0545167 + 0.998513i \(0.482638\pi\)
\(548\) 0 0
\(549\) 14.3127 0.610849
\(550\) 0 0
\(551\) 10.7005 0.455858
\(552\) 0 0
\(553\) 14.9234 0.634606
\(554\) 0 0
\(555\) −7.74798 −0.328884
\(556\) 0 0
\(557\) −4.75131 −0.201319 −0.100660 0.994921i \(-0.532095\pi\)
−0.100660 + 0.994921i \(0.532095\pi\)
\(558\) 0 0
\(559\) −9.14903 −0.386963
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 19.6873 0.829723 0.414861 0.909885i \(-0.363830\pi\)
0.414861 + 0.909885i \(0.363830\pi\)
\(564\) 0 0
\(565\) 25.6728 1.08006
\(566\) 0 0
\(567\) −2.96239 −0.124409
\(568\) 0 0
\(569\) −1.48612 −0.0623013 −0.0311507 0.999515i \(-0.509917\pi\)
−0.0311507 + 0.999515i \(0.509917\pi\)
\(570\) 0 0
\(571\) −11.1246 −0.465550 −0.232775 0.972531i \(-0.574781\pi\)
−0.232775 + 0.972531i \(0.574781\pi\)
\(572\) 0 0
\(573\) −9.14903 −0.382206
\(574\) 0 0
\(575\) −6.22425 −0.259569
\(576\) 0 0
\(577\) 0.176793 0.00736000 0.00368000 0.999993i \(-0.498829\pi\)
0.00368000 + 0.999993i \(0.498829\pi\)
\(578\) 0 0
\(579\) −6.77575 −0.281590
\(580\) 0 0
\(581\) 44.4749 1.84513
\(582\) 0 0
\(583\) −6.65165 −0.275483
\(584\) 0 0
\(585\) 6.70052 0.277033
\(586\) 0 0
\(587\) −13.6728 −0.564335 −0.282168 0.959365i \(-0.591053\pi\)
−0.282168 + 0.959365i \(0.591053\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −24.5501 −1.00986
\(592\) 0 0
\(593\) 21.2243 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(594\) 0 0
\(595\) −49.2506 −2.01908
\(596\) 0 0
\(597\) −15.3357 −0.627647
\(598\) 0 0
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) 40.1768 1.63885 0.819423 0.573190i \(-0.194294\pi\)
0.819423 + 0.573190i \(0.194294\pi\)
\(602\) 0 0
\(603\) 4.57452 0.186289
\(604\) 0 0
\(605\) −28.1457 −1.14429
\(606\) 0 0
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 0 0
\(609\) 23.4763 0.951306
\(610\) 0 0
\(611\) 3.84955 0.155736
\(612\) 0 0
\(613\) 14.3127 0.578083 0.289041 0.957317i \(-0.406664\pi\)
0.289041 + 0.957317i \(0.406664\pi\)
\(614\) 0 0
\(615\) −4.10157 −0.165391
\(616\) 0 0
\(617\) −20.6907 −0.832975 −0.416488 0.909141i \(-0.636739\pi\)
−0.416488 + 0.909141i \(0.636739\pi\)
\(618\) 0 0
\(619\) 32.3028 1.29836 0.649180 0.760635i \(-0.275112\pi\)
0.649180 + 0.760635i \(0.275112\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 16.9986 0.681034
\(624\) 0 0
\(625\) −17.3799 −0.695197
\(626\) 0 0
\(627\) −2.17679 −0.0869327
\(628\) 0 0
\(629\) −11.4763 −0.457589
\(630\) 0 0
\(631\) 3.58769 0.142824 0.0714118 0.997447i \(-0.477250\pi\)
0.0714118 + 0.997447i \(0.477250\pi\)
\(632\) 0 0
\(633\) −5.14903 −0.204656
\(634\) 0 0
\(635\) −51.8496 −2.05759
\(636\) 0 0
\(637\) 3.55149 0.140715
\(638\) 0 0
\(639\) −16.6253 −0.657687
\(640\) 0 0
\(641\) 3.71179 0.146607 0.0733034 0.997310i \(-0.476646\pi\)
0.0733034 + 0.997310i \(0.476646\pi\)
\(642\) 0 0
\(643\) 43.1246 1.70067 0.850334 0.526243i \(-0.176400\pi\)
0.850334 + 0.526243i \(0.176400\pi\)
\(644\) 0 0
\(645\) −15.3258 −0.603454
\(646\) 0 0
\(647\) −28.1016 −1.10479 −0.552393 0.833584i \(-0.686285\pi\)
−0.552393 + 0.833584i \(0.686285\pi\)
\(648\) 0 0
\(649\) 4.35359 0.170893
\(650\) 0 0
\(651\) −17.5515 −0.687897
\(652\) 0 0
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 0 0
\(655\) 2.59895 0.101549
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −33.8350 −1.31802 −0.659012 0.752133i \(-0.729025\pi\)
−0.659012 + 0.752133i \(0.729025\pi\)
\(660\) 0 0
\(661\) 16.6107 0.646082 0.323041 0.946385i \(-0.395295\pi\)
0.323041 + 0.946385i \(0.395295\pi\)
\(662\) 0 0
\(663\) 9.92478 0.385446
\(664\) 0 0
\(665\) 13.4010 0.519670
\(666\) 0 0
\(667\) 7.92478 0.306849
\(668\) 0 0
\(669\) 9.55149 0.369282
\(670\) 0 0
\(671\) 23.0738 0.890754
\(672\) 0 0
\(673\) 8.17679 0.315192 0.157596 0.987504i \(-0.449626\pi\)
0.157596 + 0.987504i \(0.449626\pi\)
\(674\) 0 0
\(675\) 6.22425 0.239572
\(676\) 0 0
\(677\) 46.8989 1.80247 0.901236 0.433328i \(-0.142661\pi\)
0.901236 + 0.433328i \(0.142661\pi\)
\(678\) 0 0
\(679\) −37.6239 −1.44387
\(680\) 0 0
\(681\) 3.68735 0.141300
\(682\) 0 0
\(683\) −4.92619 −0.188495 −0.0942477 0.995549i \(-0.530045\pi\)
−0.0942477 + 0.995549i \(0.530045\pi\)
\(684\) 0 0
\(685\) −30.0263 −1.14725
\(686\) 0 0
\(687\) 12.2374 0.466887
\(688\) 0 0
\(689\) −8.25202 −0.314377
\(690\) 0 0
\(691\) −36.1016 −1.37337 −0.686684 0.726956i \(-0.740934\pi\)
−0.686684 + 0.726956i \(0.740934\pi\)
\(692\) 0 0
\(693\) −4.77575 −0.181416
\(694\) 0 0
\(695\) −13.4010 −0.508331
\(696\) 0 0
\(697\) −6.07522 −0.230115
\(698\) 0 0
\(699\) −3.40105 −0.128639
\(700\) 0 0
\(701\) 46.2736 1.74773 0.873865 0.486168i \(-0.161606\pi\)
0.873865 + 0.486168i \(0.161606\pi\)
\(702\) 0 0
\(703\) 3.12268 0.117774
\(704\) 0 0
\(705\) 6.44851 0.242865
\(706\) 0 0
\(707\) 45.6239 1.71586
\(708\) 0 0
\(709\) 26.5647 0.997657 0.498828 0.866701i \(-0.333764\pi\)
0.498828 + 0.866701i \(0.333764\pi\)
\(710\) 0 0
\(711\) −5.03761 −0.188925
\(712\) 0 0
\(713\) −5.92478 −0.221885
\(714\) 0 0
\(715\) 10.8021 0.403975
\(716\) 0 0
\(717\) −1.55149 −0.0579416
\(718\) 0 0
\(719\) −26.5501 −0.990151 −0.495075 0.868850i \(-0.664860\pi\)
−0.495075 + 0.868850i \(0.664860\pi\)
\(720\) 0 0
\(721\) 40.4749 1.50736
\(722\) 0 0
\(723\) 20.7005 0.769861
\(724\) 0 0
\(725\) −49.3258 −1.83192
\(726\) 0 0
\(727\) 18.4387 0.683852 0.341926 0.939727i \(-0.388921\pi\)
0.341926 + 0.939727i \(0.388921\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −22.7005 −0.839609
\(732\) 0 0
\(733\) 20.1162 0.743007 0.371504 0.928431i \(-0.378842\pi\)
0.371504 + 0.928431i \(0.378842\pi\)
\(734\) 0 0
\(735\) 5.94921 0.219440
\(736\) 0 0
\(737\) 7.37470 0.271651
\(738\) 0 0
\(739\) −34.6516 −1.27468 −0.637341 0.770582i \(-0.719966\pi\)
−0.637341 + 0.770582i \(0.719966\pi\)
\(740\) 0 0
\(741\) −2.70052 −0.0992062
\(742\) 0 0
\(743\) −3.32582 −0.122013 −0.0610063 0.998137i \(-0.519431\pi\)
−0.0610063 + 0.998137i \(0.519431\pi\)
\(744\) 0 0
\(745\) −33.6728 −1.23367
\(746\) 0 0
\(747\) −15.0132 −0.549303
\(748\) 0 0
\(749\) −54.0263 −1.97408
\(750\) 0 0
\(751\) 28.1114 1.02580 0.512900 0.858448i \(-0.328571\pi\)
0.512900 + 0.858448i \(0.328571\pi\)
\(752\) 0 0
\(753\) 8.46310 0.308412
\(754\) 0 0
\(755\) 39.6991 1.44480
\(756\) 0 0
\(757\) 42.6859 1.55145 0.775723 0.631073i \(-0.217385\pi\)
0.775723 + 0.631073i \(0.217385\pi\)
\(758\) 0 0
\(759\) −1.61213 −0.0585165
\(760\) 0 0
\(761\) −23.7743 −0.861819 −0.430909 0.902395i \(-0.641807\pi\)
−0.430909 + 0.902395i \(0.641807\pi\)
\(762\) 0 0
\(763\) 57.6531 2.08718
\(764\) 0 0
\(765\) 16.6253 0.601089
\(766\) 0 0
\(767\) 5.40105 0.195021
\(768\) 0 0
\(769\) −33.2243 −1.19810 −0.599049 0.800713i \(-0.704454\pi\)
−0.599049 + 0.800713i \(0.704454\pi\)
\(770\) 0 0
\(771\) 18.6253 0.670774
\(772\) 0 0
\(773\) 4.27645 0.153813 0.0769067 0.997038i \(-0.475496\pi\)
0.0769067 + 0.997038i \(0.475496\pi\)
\(774\) 0 0
\(775\) 36.8773 1.32467
\(776\) 0 0
\(777\) 6.85097 0.245777
\(778\) 0 0
\(779\) 1.65306 0.0592271
\(780\) 0 0
\(781\) −26.8021 −0.959054
\(782\) 0 0
\(783\) −7.92478 −0.283208
\(784\) 0 0
\(785\) 76.8481 2.74283
\(786\) 0 0
\(787\) −19.0230 −0.678098 −0.339049 0.940769i \(-0.610105\pi\)
−0.339049 + 0.940769i \(0.610105\pi\)
\(788\) 0 0
\(789\) −0.523730 −0.0186453
\(790\) 0 0
\(791\) −22.7005 −0.807138
\(792\) 0 0
\(793\) 28.6253 1.01651
\(794\) 0 0
\(795\) −13.8232 −0.490259
\(796\) 0 0
\(797\) −7.35026 −0.260360 −0.130180 0.991490i \(-0.541555\pi\)
−0.130180 + 0.991490i \(0.541555\pi\)
\(798\) 0 0
\(799\) 9.55149 0.337908
\(800\) 0 0
\(801\) −5.73813 −0.202747
\(802\) 0 0
\(803\) 3.22425 0.113781
\(804\) 0 0
\(805\) 9.92478 0.349802
\(806\) 0 0
\(807\) 0.700523 0.0246596
\(808\) 0 0
\(809\) −26.3733 −0.927235 −0.463618 0.886035i \(-0.653449\pi\)
−0.463618 + 0.886035i \(0.653449\pi\)
\(810\) 0 0
\(811\) −8.99859 −0.315983 −0.157992 0.987440i \(-0.550502\pi\)
−0.157992 + 0.987440i \(0.550502\pi\)
\(812\) 0 0
\(813\) −10.0752 −0.353353
\(814\) 0 0
\(815\) 22.4485 0.786337
\(816\) 0 0
\(817\) 6.17679 0.216099
\(818\) 0 0
\(819\) −5.92478 −0.207028
\(820\) 0 0
\(821\) 9.10299 0.317696 0.158848 0.987303i \(-0.449222\pi\)
0.158848 + 0.987303i \(0.449222\pi\)
\(822\) 0 0
\(823\) −33.7743 −1.17730 −0.588650 0.808388i \(-0.700340\pi\)
−0.588650 + 0.808388i \(0.700340\pi\)
\(824\) 0 0
\(825\) 10.0343 0.349349
\(826\) 0 0
\(827\) 23.4156 0.814241 0.407121 0.913374i \(-0.366533\pi\)
0.407121 + 0.913374i \(0.366533\pi\)
\(828\) 0 0
\(829\) −10.7757 −0.374257 −0.187129 0.982335i \(-0.559918\pi\)
−0.187129 + 0.982335i \(0.559918\pi\)
\(830\) 0 0
\(831\) −22.4749 −0.779644
\(832\) 0 0
\(833\) 8.81194 0.305316
\(834\) 0 0
\(835\) 82.5012 2.85507
\(836\) 0 0
\(837\) 5.92478 0.204790
\(838\) 0 0
\(839\) −5.92478 −0.204546 −0.102273 0.994756i \(-0.532612\pi\)
−0.102273 + 0.994756i \(0.532612\pi\)
\(840\) 0 0
\(841\) 33.8021 1.16559
\(842\) 0 0
\(843\) 11.8134 0.406874
\(844\) 0 0
\(845\) −30.1524 −1.03727
\(846\) 0 0
\(847\) 24.8872 0.855133
\(848\) 0 0
\(849\) 29.1998 1.00214
\(850\) 0 0
\(851\) 2.31265 0.0792766
\(852\) 0 0
\(853\) 38.0000 1.30110 0.650548 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) 0 0
\(855\) −4.52373 −0.154708
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 3.62672 0.123598
\(862\) 0 0
\(863\) 25.2995 0.861204 0.430602 0.902542i \(-0.358301\pi\)
0.430602 + 0.902542i \(0.358301\pi\)
\(864\) 0 0
\(865\) −80.1543 −2.72533
\(866\) 0 0
\(867\) 7.62530 0.258969
\(868\) 0 0
\(869\) −8.12127 −0.275495
\(870\) 0 0
\(871\) 9.14903 0.310003
\(872\) 0 0
\(873\) 12.7005 0.429848
\(874\) 0 0
\(875\) −12.1504 −0.410760
\(876\) 0 0
\(877\) 30.6253 1.03414 0.517071 0.855942i \(-0.327022\pi\)
0.517071 + 0.855942i \(0.327022\pi\)
\(878\) 0 0
\(879\) 9.27504 0.312839
\(880\) 0 0
\(881\) −44.1866 −1.48869 −0.744343 0.667798i \(-0.767237\pi\)
−0.744343 + 0.667798i \(0.767237\pi\)
\(882\) 0 0
\(883\) −6.70052 −0.225491 −0.112745 0.993624i \(-0.535964\pi\)
−0.112745 + 0.993624i \(0.535964\pi\)
\(884\) 0 0
\(885\) 9.04746 0.304127
\(886\) 0 0
\(887\) −32.5764 −1.09381 −0.546905 0.837195i \(-0.684194\pi\)
−0.546905 + 0.837195i \(0.684194\pi\)
\(888\) 0 0
\(889\) 45.8467 1.53765
\(890\) 0 0
\(891\) 1.61213 0.0540083
\(892\) 0 0
\(893\) −2.59895 −0.0869706
\(894\) 0 0
\(895\) 64.7466 2.16424
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) 0 0
\(899\) −46.9525 −1.56595
\(900\) 0 0
\(901\) −20.4749 −0.682116
\(902\) 0 0
\(903\) 13.5515 0.450965
\(904\) 0 0
\(905\) 67.8007 2.25377
\(906\) 0 0
\(907\) −1.75272 −0.0581982 −0.0290991 0.999577i \(-0.509264\pi\)
−0.0290991 + 0.999577i \(0.509264\pi\)
\(908\) 0 0
\(909\) −15.4010 −0.510820
\(910\) 0 0
\(911\) −9.67276 −0.320473 −0.160236 0.987079i \(-0.551226\pi\)
−0.160236 + 0.987079i \(0.551226\pi\)
\(912\) 0 0
\(913\) −24.2031 −0.801007
\(914\) 0 0
\(915\) 47.9511 1.58522
\(916\) 0 0
\(917\) −2.29806 −0.0758887
\(918\) 0 0
\(919\) 25.4109 0.838228 0.419114 0.907934i \(-0.362341\pi\)
0.419114 + 0.907934i \(0.362341\pi\)
\(920\) 0 0
\(921\) 31.9511 1.05283
\(922\) 0 0
\(923\) −33.2506 −1.09446
\(924\) 0 0
\(925\) −14.3945 −0.473289
\(926\) 0 0
\(927\) −13.6629 −0.448749
\(928\) 0 0
\(929\) 24.0263 0.788279 0.394139 0.919051i \(-0.371043\pi\)
0.394139 + 0.919051i \(0.371043\pi\)
\(930\) 0 0
\(931\) −2.39772 −0.0785822
\(932\) 0 0
\(933\) 13.1490 0.430480
\(934\) 0 0
\(935\) 26.8021 0.876522
\(936\) 0 0
\(937\) −49.8496 −1.62851 −0.814257 0.580505i \(-0.802855\pi\)
−0.814257 + 0.580505i \(0.802855\pi\)
\(938\) 0 0
\(939\) 24.0263 0.784070
\(940\) 0 0
\(941\) −30.2012 −0.984532 −0.492266 0.870445i \(-0.663831\pi\)
−0.492266 + 0.870445i \(0.663831\pi\)
\(942\) 0 0
\(943\) 1.22425 0.0398672
\(944\) 0 0
\(945\) −9.92478 −0.322853
\(946\) 0 0
\(947\) −10.8218 −0.351661 −0.175830 0.984420i \(-0.556261\pi\)
−0.175830 + 0.984420i \(0.556261\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 27.9248 0.905523
\(952\) 0 0
\(953\) −33.6893 −1.09130 −0.545651 0.838012i \(-0.683718\pi\)
−0.545651 + 0.838012i \(0.683718\pi\)
\(954\) 0 0
\(955\) −30.6516 −0.991864
\(956\) 0 0
\(957\) −12.7757 −0.412981
\(958\) 0 0
\(959\) 26.5501 0.857347
\(960\) 0 0
\(961\) 4.10299 0.132354
\(962\) 0 0
\(963\) 18.2374 0.587693
\(964\) 0 0
\(965\) −22.7005 −0.730756
\(966\) 0 0
\(967\) 43.0249 1.38359 0.691794 0.722095i \(-0.256820\pi\)
0.691794 + 0.722095i \(0.256820\pi\)
\(968\) 0 0
\(969\) −6.70052 −0.215252
\(970\) 0 0
\(971\) 42.0381 1.34907 0.674534 0.738244i \(-0.264345\pi\)
0.674534 + 0.738244i \(0.264345\pi\)
\(972\) 0 0
\(973\) 11.8496 0.379879
\(974\) 0 0
\(975\) 12.4485 0.398671
\(976\) 0 0
\(977\) 11.1881 0.357938 0.178969 0.983855i \(-0.442724\pi\)
0.178969 + 0.983855i \(0.442724\pi\)
\(978\) 0 0
\(979\) −9.25060 −0.295651
\(980\) 0 0
\(981\) −19.4617 −0.621364
\(982\) 0 0
\(983\) −18.9234 −0.603562 −0.301781 0.953377i \(-0.597581\pi\)
−0.301781 + 0.953377i \(0.597581\pi\)
\(984\) 0 0
\(985\) −82.2492 −2.62068
\(986\) 0 0
\(987\) −5.70194 −0.181495
\(988\) 0 0
\(989\) 4.57452 0.145461
\(990\) 0 0
\(991\) −62.1279 −1.97356 −0.986779 0.162071i \(-0.948183\pi\)
−0.986779 + 0.162071i \(0.948183\pi\)
\(992\) 0 0
\(993\) 6.70052 0.212635
\(994\) 0 0
\(995\) −51.3785 −1.62881
\(996\) 0 0
\(997\) −55.8759 −1.76961 −0.884804 0.465964i \(-0.845708\pi\)
−0.884804 + 0.465964i \(0.845708\pi\)
\(998\) 0 0
\(999\) −2.31265 −0.0731690
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 552.2.a.g.1.3 3
3.2 odd 2 1656.2.a.n.1.1 3
4.3 odd 2 1104.2.a.o.1.3 3
8.3 odd 2 4416.2.a.bs.1.1 3
8.5 even 2 4416.2.a.bp.1.1 3
12.11 even 2 3312.2.a.bf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
552.2.a.g.1.3 3 1.1 even 1 trivial
1104.2.a.o.1.3 3 4.3 odd 2
1656.2.a.n.1.1 3 3.2 odd 2
3312.2.a.bf.1.1 3 12.11 even 2
4416.2.a.bp.1.1 3 8.5 even 2
4416.2.a.bs.1.1 3 8.3 odd 2