Properties

Label 9075.2.a.cw.1.3
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.792287\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.792287 q^{2} -1.00000 q^{3} -1.37228 q^{4} -0.792287 q^{6} +2.52434 q^{7} -2.67181 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.792287 q^{2} -1.00000 q^{3} -1.37228 q^{4} -0.792287 q^{6} +2.52434 q^{7} -2.67181 q^{8} +1.00000 q^{9} +1.37228 q^{12} +6.78073 q^{13} +2.00000 q^{14} +0.627719 q^{16} +6.63325 q^{17} +0.792287 q^{18} -7.72049 q^{19} -2.52434 q^{21} +8.00000 q^{23} +2.67181 q^{24} +5.37228 q^{26} -1.00000 q^{27} -3.46410 q^{28} -3.16915 q^{29} -3.37228 q^{31} +5.84096 q^{32} +5.25544 q^{34} -1.37228 q^{36} +5.00000 q^{37} -6.11684 q^{38} -6.78073 q^{39} +6.92820 q^{41} -2.00000 q^{42} +9.94987 q^{43} +6.33830 q^{46} +2.74456 q^{47} -0.627719 q^{48} -0.627719 q^{49} -6.63325 q^{51} -9.30506 q^{52} -12.7446 q^{53} -0.792287 q^{54} -6.74456 q^{56} +7.72049 q^{57} -2.51087 q^{58} +4.00000 q^{59} +2.67181 q^{61} -2.67181 q^{62} +2.52434 q^{63} +3.37228 q^{64} +6.11684 q^{67} -9.10268 q^{68} -8.00000 q^{69} -0.744563 q^{71} -2.67181 q^{72} +5.19615 q^{73} +3.96143 q^{74} +10.5947 q^{76} -5.37228 q^{78} -0.147477 q^{79} +1.00000 q^{81} +5.48913 q^{82} -1.87953 q^{83} +3.46410 q^{84} +7.88316 q^{86} +3.16915 q^{87} -17.4891 q^{89} +17.1168 q^{91} -10.9783 q^{92} +3.37228 q^{93} +2.17448 q^{94} -5.84096 q^{96} -3.62772 q^{97} -0.497333 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 6 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 6 q^{4} + 4 q^{9} - 6 q^{12} + 8 q^{14} + 14 q^{16} + 32 q^{23} + 10 q^{26} - 4 q^{27} - 2 q^{31} + 44 q^{34} + 6 q^{36} + 20 q^{37} + 10 q^{38} - 8 q^{42} - 12 q^{47} - 14 q^{48} - 14 q^{49} - 28 q^{53} - 4 q^{56} - 56 q^{58} + 16 q^{59} + 2 q^{64} - 10 q^{67} - 32 q^{69} + 20 q^{71} - 10 q^{78} + 4 q^{81} - 24 q^{82} + 66 q^{86} - 24 q^{89} + 34 q^{91} + 48 q^{92} + 2 q^{93} - 26 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.792287 0.560232 0.280116 0.959966i \(-0.409627\pi\)
0.280116 + 0.959966i \(0.409627\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.37228 −0.686141
\(5\) 0 0
\(6\) −0.792287 −0.323450
\(7\) 2.52434 0.954110 0.477055 0.878873i \(-0.341704\pi\)
0.477055 + 0.878873i \(0.341704\pi\)
\(8\) −2.67181 −0.944629
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 1.37228 0.396143
\(13\) 6.78073 1.88064 0.940318 0.340298i \(-0.110528\pi\)
0.940318 + 0.340298i \(0.110528\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 0.627719 0.156930
\(17\) 6.63325 1.60880 0.804400 0.594089i \(-0.202487\pi\)
0.804400 + 0.594089i \(0.202487\pi\)
\(18\) 0.792287 0.186744
\(19\) −7.72049 −1.77120 −0.885601 0.464447i \(-0.846253\pi\)
−0.885601 + 0.464447i \(0.846253\pi\)
\(20\) 0 0
\(21\) −2.52434 −0.550856
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 2.67181 0.545382
\(25\) 0 0
\(26\) 5.37228 1.05359
\(27\) −1.00000 −0.192450
\(28\) −3.46410 −0.654654
\(29\) −3.16915 −0.588496 −0.294248 0.955729i \(-0.595069\pi\)
−0.294248 + 0.955729i \(0.595069\pi\)
\(30\) 0 0
\(31\) −3.37228 −0.605680 −0.302840 0.953041i \(-0.597935\pi\)
−0.302840 + 0.953041i \(0.597935\pi\)
\(32\) 5.84096 1.03255
\(33\) 0 0
\(34\) 5.25544 0.901300
\(35\) 0 0
\(36\) −1.37228 −0.228714
\(37\) 5.00000 0.821995 0.410997 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) −6.11684 −0.992283
\(39\) −6.78073 −1.08579
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) −2.00000 −0.308607
\(43\) 9.94987 1.51734 0.758671 0.651474i \(-0.225849\pi\)
0.758671 + 0.651474i \(0.225849\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.33830 0.934531
\(47\) 2.74456 0.400336 0.200168 0.979762i \(-0.435851\pi\)
0.200168 + 0.979762i \(0.435851\pi\)
\(48\) −0.627719 −0.0906034
\(49\) −0.627719 −0.0896741
\(50\) 0 0
\(51\) −6.63325 −0.928841
\(52\) −9.30506 −1.29038
\(53\) −12.7446 −1.75060 −0.875300 0.483580i \(-0.839336\pi\)
−0.875300 + 0.483580i \(0.839336\pi\)
\(54\) −0.792287 −0.107817
\(55\) 0 0
\(56\) −6.74456 −0.901280
\(57\) 7.72049 1.02260
\(58\) −2.51087 −0.329694
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 2.67181 0.342091 0.171045 0.985263i \(-0.445286\pi\)
0.171045 + 0.985263i \(0.445286\pi\)
\(62\) −2.67181 −0.339321
\(63\) 2.52434 0.318037
\(64\) 3.37228 0.421535
\(65\) 0 0
\(66\) 0 0
\(67\) 6.11684 0.747291 0.373646 0.927571i \(-0.378108\pi\)
0.373646 + 0.927571i \(0.378108\pi\)
\(68\) −9.10268 −1.10386
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −0.744563 −0.0883633 −0.0441817 0.999024i \(-0.514068\pi\)
−0.0441817 + 0.999024i \(0.514068\pi\)
\(72\) −2.67181 −0.314876
\(73\) 5.19615 0.608164 0.304082 0.952646i \(-0.401650\pi\)
0.304082 + 0.952646i \(0.401650\pi\)
\(74\) 3.96143 0.460507
\(75\) 0 0
\(76\) 10.5947 1.21529
\(77\) 0 0
\(78\) −5.37228 −0.608291
\(79\) −0.147477 −0.0165924 −0.00829622 0.999966i \(-0.502641\pi\)
−0.00829622 + 0.999966i \(0.502641\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 5.48913 0.606172
\(83\) −1.87953 −0.206305 −0.103152 0.994666i \(-0.532893\pi\)
−0.103152 + 0.994666i \(0.532893\pi\)
\(84\) 3.46410 0.377964
\(85\) 0 0
\(86\) 7.88316 0.850063
\(87\) 3.16915 0.339768
\(88\) 0 0
\(89\) −17.4891 −1.85384 −0.926922 0.375255i \(-0.877555\pi\)
−0.926922 + 0.375255i \(0.877555\pi\)
\(90\) 0 0
\(91\) 17.1168 1.79433
\(92\) −10.9783 −1.14456
\(93\) 3.37228 0.349689
\(94\) 2.17448 0.224281
\(95\) 0 0
\(96\) −5.84096 −0.596141
\(97\) −3.62772 −0.368339 −0.184170 0.982894i \(-0.558960\pi\)
−0.184170 + 0.982894i \(0.558960\pi\)
\(98\) −0.497333 −0.0502383
\(99\) 0 0
\(100\) 0 0
\(101\) 13.2665 1.32007 0.660033 0.751237i \(-0.270542\pi\)
0.660033 + 0.751237i \(0.270542\pi\)
\(102\) −5.25544 −0.520366
\(103\) −7.74456 −0.763094 −0.381547 0.924349i \(-0.624609\pi\)
−0.381547 + 0.924349i \(0.624609\pi\)
\(104\) −18.1168 −1.77650
\(105\) 0 0
\(106\) −10.0974 −0.980741
\(107\) −11.6819 −1.12933 −0.564667 0.825319i \(-0.690995\pi\)
−0.564667 + 0.825319i \(0.690995\pi\)
\(108\) 1.37228 0.132048
\(109\) −5.98844 −0.573588 −0.286794 0.957992i \(-0.592590\pi\)
−0.286794 + 0.957992i \(0.592590\pi\)
\(110\) 0 0
\(111\) −5.00000 −0.474579
\(112\) 1.58457 0.149728
\(113\) −18.2337 −1.71528 −0.857641 0.514250i \(-0.828070\pi\)
−0.857641 + 0.514250i \(0.828070\pi\)
\(114\) 6.11684 0.572895
\(115\) 0 0
\(116\) 4.34896 0.403791
\(117\) 6.78073 0.626878
\(118\) 3.16915 0.291744
\(119\) 16.7446 1.53497
\(120\) 0 0
\(121\) 0 0
\(122\) 2.11684 0.191650
\(123\) −6.92820 −0.624695
\(124\) 4.62772 0.415581
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 3.61158 0.320476 0.160238 0.987078i \(-0.448774\pi\)
0.160238 + 0.987078i \(0.448774\pi\)
\(128\) −9.01011 −0.796389
\(129\) −9.94987 −0.876038
\(130\) 0 0
\(131\) 1.28962 0.112675 0.0563373 0.998412i \(-0.482058\pi\)
0.0563373 + 0.998412i \(0.482058\pi\)
\(132\) 0 0
\(133\) −19.4891 −1.68992
\(134\) 4.84630 0.418656
\(135\) 0 0
\(136\) −17.7228 −1.51972
\(137\) 10.7446 0.917970 0.458985 0.888444i \(-0.348213\pi\)
0.458985 + 0.888444i \(0.348213\pi\)
\(138\) −6.33830 −0.539552
\(139\) −10.3923 −0.881464 −0.440732 0.897639i \(-0.645281\pi\)
−0.440732 + 0.897639i \(0.645281\pi\)
\(140\) 0 0
\(141\) −2.74456 −0.231134
\(142\) −0.589907 −0.0495039
\(143\) 0 0
\(144\) 0.627719 0.0523099
\(145\) 0 0
\(146\) 4.11684 0.340712
\(147\) 0.627719 0.0517734
\(148\) −6.86141 −0.564004
\(149\) 18.6101 1.52460 0.762301 0.647223i \(-0.224070\pi\)
0.762301 + 0.647223i \(0.224070\pi\)
\(150\) 0 0
\(151\) −3.90653 −0.317909 −0.158955 0.987286i \(-0.550812\pi\)
−0.158955 + 0.987286i \(0.550812\pi\)
\(152\) 20.6277 1.67313
\(153\) 6.63325 0.536266
\(154\) 0 0
\(155\) 0 0
\(156\) 9.30506 0.745001
\(157\) 2.62772 0.209715 0.104857 0.994487i \(-0.466561\pi\)
0.104857 + 0.994487i \(0.466561\pi\)
\(158\) −0.116844 −0.00929561
\(159\) 12.7446 1.01071
\(160\) 0 0
\(161\) 20.1947 1.59157
\(162\) 0.792287 0.0622479
\(163\) 8.62772 0.675775 0.337888 0.941186i \(-0.390288\pi\)
0.337888 + 0.941186i \(0.390288\pi\)
\(164\) −9.50744 −0.742407
\(165\) 0 0
\(166\) −1.48913 −0.115579
\(167\) 1.58457 0.122618 0.0613090 0.998119i \(-0.480472\pi\)
0.0613090 + 0.998119i \(0.480472\pi\)
\(168\) 6.74456 0.520354
\(169\) 32.9783 2.53679
\(170\) 0 0
\(171\) −7.72049 −0.590401
\(172\) −13.6540 −1.04111
\(173\) 6.92820 0.526742 0.263371 0.964695i \(-0.415166\pi\)
0.263371 + 0.964695i \(0.415166\pi\)
\(174\) 2.51087 0.190349
\(175\) 0 0
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) −13.8564 −1.03858
\(179\) −3.48913 −0.260789 −0.130395 0.991462i \(-0.541624\pi\)
−0.130395 + 0.991462i \(0.541624\pi\)
\(180\) 0 0
\(181\) −14.8614 −1.10464 −0.552320 0.833632i \(-0.686257\pi\)
−0.552320 + 0.833632i \(0.686257\pi\)
\(182\) 13.5615 1.00524
\(183\) −2.67181 −0.197506
\(184\) −21.3745 −1.57575
\(185\) 0 0
\(186\) 2.67181 0.195907
\(187\) 0 0
\(188\) −3.76631 −0.274687
\(189\) −2.52434 −0.183619
\(190\) 0 0
\(191\) 5.48913 0.397179 0.198590 0.980083i \(-0.436364\pi\)
0.198590 + 0.980083i \(0.436364\pi\)
\(192\) −3.37228 −0.243373
\(193\) −2.08191 −0.149859 −0.0749295 0.997189i \(-0.523873\pi\)
−0.0749295 + 0.997189i \(0.523873\pi\)
\(194\) −2.87419 −0.206355
\(195\) 0 0
\(196\) 0.861407 0.0615290
\(197\) −1.87953 −0.133911 −0.0669554 0.997756i \(-0.521329\pi\)
−0.0669554 + 0.997756i \(0.521329\pi\)
\(198\) 0 0
\(199\) −11.8614 −0.840833 −0.420416 0.907331i \(-0.638116\pi\)
−0.420416 + 0.907331i \(0.638116\pi\)
\(200\) 0 0
\(201\) −6.11684 −0.431449
\(202\) 10.5109 0.739543
\(203\) −8.00000 −0.561490
\(204\) 9.10268 0.637315
\(205\) 0 0
\(206\) −6.13592 −0.427510
\(207\) 8.00000 0.556038
\(208\) 4.25639 0.295127
\(209\) 0 0
\(210\) 0 0
\(211\) 14.5012 0.998305 0.499152 0.866514i \(-0.333645\pi\)
0.499152 + 0.866514i \(0.333645\pi\)
\(212\) 17.4891 1.20116
\(213\) 0.744563 0.0510166
\(214\) −9.25544 −0.632689
\(215\) 0 0
\(216\) 2.67181 0.181794
\(217\) −8.51278 −0.577885
\(218\) −4.74456 −0.321342
\(219\) −5.19615 −0.351123
\(220\) 0 0
\(221\) 44.9783 3.02556
\(222\) −3.96143 −0.265874
\(223\) 4.86141 0.325544 0.162772 0.986664i \(-0.447957\pi\)
0.162772 + 0.986664i \(0.447957\pi\)
\(224\) 14.7446 0.985163
\(225\) 0 0
\(226\) −14.4463 −0.960954
\(227\) 17.3205 1.14960 0.574801 0.818293i \(-0.305079\pi\)
0.574801 + 0.818293i \(0.305079\pi\)
\(228\) −10.5947 −0.701650
\(229\) 12.4891 0.825305 0.412652 0.910889i \(-0.364602\pi\)
0.412652 + 0.910889i \(0.364602\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.46738 0.555910
\(233\) 4.75372 0.311427 0.155713 0.987802i \(-0.450232\pi\)
0.155713 + 0.987802i \(0.450232\pi\)
\(234\) 5.37228 0.351197
\(235\) 0 0
\(236\) −5.48913 −0.357312
\(237\) 0.147477 0.00957965
\(238\) 13.2665 0.859939
\(239\) −13.2665 −0.858138 −0.429069 0.903272i \(-0.641158\pi\)
−0.429069 + 0.903272i \(0.641158\pi\)
\(240\) 0 0
\(241\) −20.6371 −1.32935 −0.664677 0.747131i \(-0.731431\pi\)
−0.664677 + 0.747131i \(0.731431\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −3.66648 −0.234722
\(245\) 0 0
\(246\) −5.48913 −0.349974
\(247\) −52.3505 −3.33098
\(248\) 9.01011 0.572143
\(249\) 1.87953 0.119110
\(250\) 0 0
\(251\) −23.4891 −1.48262 −0.741310 0.671163i \(-0.765795\pi\)
−0.741310 + 0.671163i \(0.765795\pi\)
\(252\) −3.46410 −0.218218
\(253\) 0 0
\(254\) 2.86141 0.179541
\(255\) 0 0
\(256\) −13.8832 −0.867697
\(257\) −5.48913 −0.342402 −0.171201 0.985236i \(-0.554765\pi\)
−0.171201 + 0.985236i \(0.554765\pi\)
\(258\) −7.88316 −0.490784
\(259\) 12.6217 0.784274
\(260\) 0 0
\(261\) −3.16915 −0.196165
\(262\) 1.02175 0.0631239
\(263\) −1.58457 −0.0977090 −0.0488545 0.998806i \(-0.515557\pi\)
−0.0488545 + 0.998806i \(0.515557\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −15.4410 −0.946747
\(267\) 17.4891 1.07032
\(268\) −8.39403 −0.512747
\(269\) 8.74456 0.533165 0.266583 0.963812i \(-0.414105\pi\)
0.266583 + 0.963812i \(0.414105\pi\)
\(270\) 0 0
\(271\) −10.3923 −0.631288 −0.315644 0.948878i \(-0.602220\pi\)
−0.315644 + 0.948878i \(0.602220\pi\)
\(272\) 4.16381 0.252468
\(273\) −17.1168 −1.03596
\(274\) 8.51278 0.514276
\(275\) 0 0
\(276\) 10.9783 0.660813
\(277\) 29.2048 1.75475 0.877374 0.479808i \(-0.159294\pi\)
0.877374 + 0.479808i \(0.159294\pi\)
\(278\) −8.23369 −0.493824
\(279\) −3.37228 −0.201893
\(280\) 0 0
\(281\) −1.87953 −0.112123 −0.0560616 0.998427i \(-0.517854\pi\)
−0.0560616 + 0.998427i \(0.517854\pi\)
\(282\) −2.17448 −0.129488
\(283\) 18.4077 1.09423 0.547114 0.837058i \(-0.315727\pi\)
0.547114 + 0.837058i \(0.315727\pi\)
\(284\) 1.02175 0.0606297
\(285\) 0 0
\(286\) 0 0
\(287\) 17.4891 1.03235
\(288\) 5.84096 0.344182
\(289\) 27.0000 1.58824
\(290\) 0 0
\(291\) 3.62772 0.212661
\(292\) −7.13058 −0.417286
\(293\) 26.1282 1.52643 0.763214 0.646146i \(-0.223620\pi\)
0.763214 + 0.646146i \(0.223620\pi\)
\(294\) 0.497333 0.0290051
\(295\) 0 0
\(296\) −13.3591 −0.776480
\(297\) 0 0
\(298\) 14.7446 0.854130
\(299\) 54.2458 3.13712
\(300\) 0 0
\(301\) 25.1168 1.44771
\(302\) −3.09509 −0.178103
\(303\) −13.2665 −0.762140
\(304\) −4.84630 −0.277954
\(305\) 0 0
\(306\) 5.25544 0.300433
\(307\) 9.89497 0.564736 0.282368 0.959306i \(-0.408880\pi\)
0.282368 + 0.959306i \(0.408880\pi\)
\(308\) 0 0
\(309\) 7.74456 0.440573
\(310\) 0 0
\(311\) 24.2337 1.37417 0.687083 0.726579i \(-0.258891\pi\)
0.687083 + 0.726579i \(0.258891\pi\)
\(312\) 18.1168 1.02566
\(313\) 19.9783 1.12924 0.564619 0.825352i \(-0.309023\pi\)
0.564619 + 0.825352i \(0.309023\pi\)
\(314\) 2.08191 0.117489
\(315\) 0 0
\(316\) 0.202380 0.0113847
\(317\) 26.9783 1.51525 0.757625 0.652690i \(-0.226360\pi\)
0.757625 + 0.652690i \(0.226360\pi\)
\(318\) 10.0974 0.566231
\(319\) 0 0
\(320\) 0 0
\(321\) 11.6819 0.652021
\(322\) 16.0000 0.891645
\(323\) −51.2119 −2.84951
\(324\) −1.37228 −0.0762379
\(325\) 0 0
\(326\) 6.83563 0.378590
\(327\) 5.98844 0.331161
\(328\) −18.5109 −1.02209
\(329\) 6.92820 0.381964
\(330\) 0 0
\(331\) −33.2337 −1.82669 −0.913344 0.407188i \(-0.866509\pi\)
−0.913344 + 0.407188i \(0.866509\pi\)
\(332\) 2.57924 0.141554
\(333\) 5.00000 0.273998
\(334\) 1.25544 0.0686945
\(335\) 0 0
\(336\) −1.58457 −0.0864456
\(337\) 19.2549 1.04888 0.524442 0.851446i \(-0.324274\pi\)
0.524442 + 0.851446i \(0.324274\pi\)
\(338\) 26.1282 1.42119
\(339\) 18.2337 0.990318
\(340\) 0 0
\(341\) 0 0
\(342\) −6.11684 −0.330761
\(343\) −19.2549 −1.03967
\(344\) −26.5842 −1.43333
\(345\) 0 0
\(346\) 5.48913 0.295097
\(347\) 22.0742 1.18501 0.592503 0.805568i \(-0.298140\pi\)
0.592503 + 0.805568i \(0.298140\pi\)
\(348\) −4.34896 −0.233129
\(349\) 8.66025 0.463573 0.231786 0.972767i \(-0.425543\pi\)
0.231786 + 0.972767i \(0.425543\pi\)
\(350\) 0 0
\(351\) −6.78073 −0.361928
\(352\) 0 0
\(353\) 18.2337 0.970481 0.485241 0.874381i \(-0.338732\pi\)
0.485241 + 0.874381i \(0.338732\pi\)
\(354\) −3.16915 −0.168438
\(355\) 0 0
\(356\) 24.0000 1.27200
\(357\) −16.7446 −0.886216
\(358\) −2.76439 −0.146102
\(359\) −6.63325 −0.350090 −0.175045 0.984560i \(-0.556007\pi\)
−0.175045 + 0.984560i \(0.556007\pi\)
\(360\) 0 0
\(361\) 40.6060 2.13716
\(362\) −11.7745 −0.618854
\(363\) 0 0
\(364\) −23.4891 −1.23116
\(365\) 0 0
\(366\) −2.11684 −0.110649
\(367\) −15.0000 −0.782994 −0.391497 0.920179i \(-0.628043\pi\)
−0.391497 + 0.920179i \(0.628043\pi\)
\(368\) 5.02175 0.261777
\(369\) 6.92820 0.360668
\(370\) 0 0
\(371\) −32.1716 −1.67027
\(372\) −4.62772 −0.239936
\(373\) 10.4472 0.540936 0.270468 0.962729i \(-0.412822\pi\)
0.270468 + 0.962729i \(0.412822\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −7.33296 −0.378169
\(377\) −21.4891 −1.10675
\(378\) −2.00000 −0.102869
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) −3.61158 −0.185027
\(382\) 4.34896 0.222512
\(383\) 24.2337 1.23828 0.619142 0.785279i \(-0.287481\pi\)
0.619142 + 0.785279i \(0.287481\pi\)
\(384\) 9.01011 0.459795
\(385\) 0 0
\(386\) −1.64947 −0.0839557
\(387\) 9.94987 0.505781
\(388\) 4.97825 0.252732
\(389\) 21.2554 1.07769 0.538847 0.842404i \(-0.318860\pi\)
0.538847 + 0.842404i \(0.318860\pi\)
\(390\) 0 0
\(391\) 53.0660 2.68366
\(392\) 1.67715 0.0847088
\(393\) −1.28962 −0.0650527
\(394\) −1.48913 −0.0750210
\(395\) 0 0
\(396\) 0 0
\(397\) −8.37228 −0.420193 −0.210096 0.977681i \(-0.567378\pi\)
−0.210096 + 0.977681i \(0.567378\pi\)
\(398\) −9.39764 −0.471061
\(399\) 19.4891 0.975677
\(400\) 0 0
\(401\) 14.9783 0.747978 0.373989 0.927433i \(-0.377990\pi\)
0.373989 + 0.927433i \(0.377990\pi\)
\(402\) −4.84630 −0.241711
\(403\) −22.8665 −1.13906
\(404\) −18.2054 −0.905751
\(405\) 0 0
\(406\) −6.33830 −0.314564
\(407\) 0 0
\(408\) 17.7228 0.877410
\(409\) 8.45787 0.418215 0.209107 0.977893i \(-0.432944\pi\)
0.209107 + 0.977893i \(0.432944\pi\)
\(410\) 0 0
\(411\) −10.7446 −0.529990
\(412\) 10.6277 0.523590
\(413\) 10.0974 0.496858
\(414\) 6.33830 0.311510
\(415\) 0 0
\(416\) 39.6060 1.94184
\(417\) 10.3923 0.508913
\(418\) 0 0
\(419\) 32.2337 1.57472 0.787359 0.616494i \(-0.211448\pi\)
0.787359 + 0.616494i \(0.211448\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 11.4891 0.559282
\(423\) 2.74456 0.133445
\(424\) 34.0511 1.65367
\(425\) 0 0
\(426\) 0.589907 0.0285811
\(427\) 6.74456 0.326392
\(428\) 16.0309 0.774882
\(429\) 0 0
\(430\) 0 0
\(431\) −2.87419 −0.138445 −0.0692225 0.997601i \(-0.522052\pi\)
−0.0692225 + 0.997601i \(0.522052\pi\)
\(432\) −0.627719 −0.0302011
\(433\) −23.0951 −1.10988 −0.554940 0.831891i \(-0.687259\pi\)
−0.554940 + 0.831891i \(0.687259\pi\)
\(434\) −6.74456 −0.323749
\(435\) 0 0
\(436\) 8.21782 0.393562
\(437\) −61.7639 −2.95457
\(438\) −4.11684 −0.196710
\(439\) −22.7190 −1.08432 −0.542160 0.840275i \(-0.682393\pi\)
−0.542160 + 0.840275i \(0.682393\pi\)
\(440\) 0 0
\(441\) −0.627719 −0.0298914
\(442\) 35.6357 1.69502
\(443\) 20.7446 0.985604 0.492802 0.870142i \(-0.335973\pi\)
0.492802 + 0.870142i \(0.335973\pi\)
\(444\) 6.86141 0.325628
\(445\) 0 0
\(446\) 3.85163 0.182380
\(447\) −18.6101 −0.880229
\(448\) 8.51278 0.402191
\(449\) 1.25544 0.0592478 0.0296239 0.999561i \(-0.490569\pi\)
0.0296239 + 0.999561i \(0.490569\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 25.0217 1.17692
\(453\) 3.90653 0.183545
\(454\) 13.7228 0.644044
\(455\) 0 0
\(456\) −20.6277 −0.965981
\(457\) −33.4612 −1.56525 −0.782624 0.622494i \(-0.786119\pi\)
−0.782624 + 0.622494i \(0.786119\pi\)
\(458\) 9.89497 0.462362
\(459\) −6.63325 −0.309614
\(460\) 0 0
\(461\) 8.51278 0.396480 0.198240 0.980154i \(-0.436478\pi\)
0.198240 + 0.980154i \(0.436478\pi\)
\(462\) 0 0
\(463\) −18.9783 −0.881994 −0.440997 0.897509i \(-0.645375\pi\)
−0.440997 + 0.897509i \(0.645375\pi\)
\(464\) −1.98933 −0.0923525
\(465\) 0 0
\(466\) 3.76631 0.174471
\(467\) 21.4891 0.994398 0.497199 0.867636i \(-0.334362\pi\)
0.497199 + 0.867636i \(0.334362\pi\)
\(468\) −9.30506 −0.430127
\(469\) 15.4410 0.712998
\(470\) 0 0
\(471\) −2.62772 −0.121079
\(472\) −10.6873 −0.491921
\(473\) 0 0
\(474\) 0.116844 0.00536682
\(475\) 0 0
\(476\) −22.9783 −1.05321
\(477\) −12.7446 −0.583533
\(478\) −10.5109 −0.480756
\(479\) −23.6588 −1.08100 −0.540499 0.841345i \(-0.681765\pi\)
−0.540499 + 0.841345i \(0.681765\pi\)
\(480\) 0 0
\(481\) 33.9036 1.54587
\(482\) −16.3505 −0.744746
\(483\) −20.1947 −0.918891
\(484\) 0 0
\(485\) 0 0
\(486\) −0.792287 −0.0359389
\(487\) 9.97825 0.452158 0.226079 0.974109i \(-0.427409\pi\)
0.226079 + 0.974109i \(0.427409\pi\)
\(488\) −7.13859 −0.323149
\(489\) −8.62772 −0.390159
\(490\) 0 0
\(491\) 6.63325 0.299354 0.149677 0.988735i \(-0.452177\pi\)
0.149677 + 0.988735i \(0.452177\pi\)
\(492\) 9.50744 0.428629
\(493\) −21.0217 −0.946772
\(494\) −41.4766 −1.86612
\(495\) 0 0
\(496\) −2.11684 −0.0950491
\(497\) −1.87953 −0.0843083
\(498\) 1.48913 0.0667293
\(499\) 3.11684 0.139529 0.0697645 0.997563i \(-0.477775\pi\)
0.0697645 + 0.997563i \(0.477775\pi\)
\(500\) 0 0
\(501\) −1.58457 −0.0707935
\(502\) −18.6101 −0.830611
\(503\) −10.0974 −0.450219 −0.225109 0.974334i \(-0.572274\pi\)
−0.225109 + 0.974334i \(0.572274\pi\)
\(504\) −6.74456 −0.300427
\(505\) 0 0
\(506\) 0 0
\(507\) −32.9783 −1.46462
\(508\) −4.95610 −0.219891
\(509\) 14.9783 0.663899 0.331950 0.943297i \(-0.392294\pi\)
0.331950 + 0.943297i \(0.392294\pi\)
\(510\) 0 0
\(511\) 13.1168 0.580255
\(512\) 7.02078 0.310277
\(513\) 7.72049 0.340868
\(514\) −4.34896 −0.191825
\(515\) 0 0
\(516\) 13.6540 0.601085
\(517\) 0 0
\(518\) 10.0000 0.439375
\(519\) −6.92820 −0.304114
\(520\) 0 0
\(521\) −25.2554 −1.10646 −0.553230 0.833028i \(-0.686605\pi\)
−0.553230 + 0.833028i \(0.686605\pi\)
\(522\) −2.51087 −0.109898
\(523\) −37.0179 −1.61868 −0.809339 0.587341i \(-0.800175\pi\)
−0.809339 + 0.587341i \(0.800175\pi\)
\(524\) −1.76972 −0.0773107
\(525\) 0 0
\(526\) −1.25544 −0.0547397
\(527\) −22.3692 −0.974417
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 26.7446 1.15952
\(533\) 46.9783 2.03485
\(534\) 13.8564 0.599625
\(535\) 0 0
\(536\) −16.3431 −0.705913
\(537\) 3.48913 0.150567
\(538\) 6.92820 0.298696
\(539\) 0 0
\(540\) 0 0
\(541\) −9.65492 −0.415097 −0.207549 0.978225i \(-0.566549\pi\)
−0.207549 + 0.978225i \(0.566549\pi\)
\(542\) −8.23369 −0.353667
\(543\) 14.8614 0.637764
\(544\) 38.7446 1.66116
\(545\) 0 0
\(546\) −13.5615 −0.580377
\(547\) −40.6844 −1.73954 −0.869769 0.493459i \(-0.835732\pi\)
−0.869769 + 0.493459i \(0.835732\pi\)
\(548\) −14.7446 −0.629857
\(549\) 2.67181 0.114030
\(550\) 0 0
\(551\) 24.4674 1.04235
\(552\) 21.3745 0.909760
\(553\) −0.372281 −0.0158310
\(554\) 23.1386 0.983065
\(555\) 0 0
\(556\) 14.2612 0.604808
\(557\) −20.1947 −0.855677 −0.427839 0.903855i \(-0.640725\pi\)
−0.427839 + 0.903855i \(0.640725\pi\)
\(558\) −2.67181 −0.113107
\(559\) 67.4674 2.85357
\(560\) 0 0
\(561\) 0 0
\(562\) −1.48913 −0.0628150
\(563\) −9.21249 −0.388260 −0.194130 0.980976i \(-0.562188\pi\)
−0.194130 + 0.980976i \(0.562188\pi\)
\(564\) 3.76631 0.158590
\(565\) 0 0
\(566\) 14.5842 0.613020
\(567\) 2.52434 0.106012
\(568\) 1.98933 0.0834706
\(569\) −29.9971 −1.25754 −0.628772 0.777590i \(-0.716442\pi\)
−0.628772 + 0.777590i \(0.716442\pi\)
\(570\) 0 0
\(571\) 39.7446 1.66326 0.831630 0.555330i \(-0.187408\pi\)
0.831630 + 0.555330i \(0.187408\pi\)
\(572\) 0 0
\(573\) −5.48913 −0.229311
\(574\) 13.8564 0.578355
\(575\) 0 0
\(576\) 3.37228 0.140512
\(577\) 39.0951 1.62755 0.813775 0.581180i \(-0.197409\pi\)
0.813775 + 0.581180i \(0.197409\pi\)
\(578\) 21.3917 0.889779
\(579\) 2.08191 0.0865211
\(580\) 0 0
\(581\) −4.74456 −0.196838
\(582\) 2.87419 0.119139
\(583\) 0 0
\(584\) −13.8832 −0.574489
\(585\) 0 0
\(586\) 20.7011 0.855153
\(587\) 23.2554 0.959855 0.479927 0.877308i \(-0.340663\pi\)
0.479927 + 0.877308i \(0.340663\pi\)
\(588\) −0.861407 −0.0355238
\(589\) 26.0357 1.07278
\(590\) 0 0
\(591\) 1.87953 0.0773134
\(592\) 3.13859 0.128995
\(593\) −37.8102 −1.55268 −0.776339 0.630316i \(-0.782925\pi\)
−0.776339 + 0.630316i \(0.782925\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −25.5383 −1.04609
\(597\) 11.8614 0.485455
\(598\) 42.9783 1.75751
\(599\) −44.7446 −1.82821 −0.914107 0.405474i \(-0.867107\pi\)
−0.914107 + 0.405474i \(0.867107\pi\)
\(600\) 0 0
\(601\) −1.08724 −0.0443495 −0.0221747 0.999754i \(-0.507059\pi\)
−0.0221747 + 0.999754i \(0.507059\pi\)
\(602\) 19.8997 0.811053
\(603\) 6.11684 0.249097
\(604\) 5.36086 0.218130
\(605\) 0 0
\(606\) −10.5109 −0.426975
\(607\) −6.04334 −0.245292 −0.122646 0.992451i \(-0.539138\pi\)
−0.122646 + 0.992451i \(0.539138\pi\)
\(608\) −45.0951 −1.82885
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 18.6101 0.752885
\(612\) −9.10268 −0.367954
\(613\) 11.5344 0.465872 0.232936 0.972492i \(-0.425167\pi\)
0.232936 + 0.972492i \(0.425167\pi\)
\(614\) 7.83966 0.316383
\(615\) 0 0
\(616\) 0 0
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 6.13592 0.246823
\(619\) −25.5109 −1.02537 −0.512684 0.858577i \(-0.671349\pi\)
−0.512684 + 0.858577i \(0.671349\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 19.2000 0.769851
\(623\) −44.1485 −1.76877
\(624\) −4.25639 −0.170392
\(625\) 0 0
\(626\) 15.8285 0.632634
\(627\) 0 0
\(628\) −3.60597 −0.143894
\(629\) 33.1662 1.32242
\(630\) 0 0
\(631\) −13.5109 −0.537859 −0.268930 0.963160i \(-0.586670\pi\)
−0.268930 + 0.963160i \(0.586670\pi\)
\(632\) 0.394031 0.0156737
\(633\) −14.5012 −0.576372
\(634\) 21.3745 0.848891
\(635\) 0 0
\(636\) −17.4891 −0.693489
\(637\) −4.25639 −0.168644
\(638\) 0 0
\(639\) −0.744563 −0.0294544
\(640\) 0 0
\(641\) −16.9783 −0.670601 −0.335300 0.942111i \(-0.608838\pi\)
−0.335300 + 0.942111i \(0.608838\pi\)
\(642\) 9.25544 0.365283
\(643\) 25.2337 0.995120 0.497560 0.867430i \(-0.334230\pi\)
0.497560 + 0.867430i \(0.334230\pi\)
\(644\) −27.7128 −1.09204
\(645\) 0 0
\(646\) −40.5746 −1.59638
\(647\) 2.00000 0.0786281 0.0393141 0.999227i \(-0.487483\pi\)
0.0393141 + 0.999227i \(0.487483\pi\)
\(648\) −2.67181 −0.104959
\(649\) 0 0
\(650\) 0 0
\(651\) 8.51278 0.333642
\(652\) −11.8397 −0.463677
\(653\) 22.5109 0.880919 0.440459 0.897773i \(-0.354816\pi\)
0.440459 + 0.897773i \(0.354816\pi\)
\(654\) 4.74456 0.185527
\(655\) 0 0
\(656\) 4.34896 0.169798
\(657\) 5.19615 0.202721
\(658\) 5.48913 0.213988
\(659\) −18.9051 −0.736437 −0.368219 0.929739i \(-0.620032\pi\)
−0.368219 + 0.929739i \(0.620032\pi\)
\(660\) 0 0
\(661\) −3.00000 −0.116686 −0.0583432 0.998297i \(-0.518582\pi\)
−0.0583432 + 0.998297i \(0.518582\pi\)
\(662\) −26.3306 −1.02337
\(663\) −44.9783 −1.74681
\(664\) 5.02175 0.194882
\(665\) 0 0
\(666\) 3.96143 0.153502
\(667\) −25.3532 −0.981679
\(668\) −2.17448 −0.0841332
\(669\) −4.86141 −0.187953
\(670\) 0 0
\(671\) 0 0
\(672\) −14.7446 −0.568784
\(673\) −2.32196 −0.0895049 −0.0447525 0.998998i \(-0.514250\pi\)
−0.0447525 + 0.998998i \(0.514250\pi\)
\(674\) 15.2554 0.587617
\(675\) 0 0
\(676\) −45.2554 −1.74059
\(677\) 37.5152 1.44183 0.720913 0.693025i \(-0.243723\pi\)
0.720913 + 0.693025i \(0.243723\pi\)
\(678\) 14.4463 0.554807
\(679\) −9.15759 −0.351436
\(680\) 0 0
\(681\) −17.3205 −0.663723
\(682\) 0 0
\(683\) 18.0000 0.688751 0.344375 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(684\) 10.5947 0.405098
\(685\) 0 0
\(686\) −15.2554 −0.582455
\(687\) −12.4891 −0.476490
\(688\) 6.24572 0.238116
\(689\) −86.4174 −3.29224
\(690\) 0 0
\(691\) −13.7446 −0.522868 −0.261434 0.965221i \(-0.584195\pi\)
−0.261434 + 0.965221i \(0.584195\pi\)
\(692\) −9.50744 −0.361419
\(693\) 0 0
\(694\) 17.4891 0.663878
\(695\) 0 0
\(696\) −8.46738 −0.320955
\(697\) 45.9565 1.74073
\(698\) 6.86141 0.259708
\(699\) −4.75372 −0.179802
\(700\) 0 0
\(701\) −30.5870 −1.15526 −0.577628 0.816300i \(-0.696021\pi\)
−0.577628 + 0.816300i \(0.696021\pi\)
\(702\) −5.37228 −0.202764
\(703\) −38.6025 −1.45592
\(704\) 0 0
\(705\) 0 0
\(706\) 14.4463 0.543694
\(707\) 33.4891 1.25949
\(708\) 5.48913 0.206294
\(709\) 31.0000 1.16423 0.582115 0.813107i \(-0.302225\pi\)
0.582115 + 0.813107i \(0.302225\pi\)
\(710\) 0 0
\(711\) −0.147477 −0.00553081
\(712\) 46.7277 1.75119
\(713\) −26.9783 −1.01034
\(714\) −13.2665 −0.496486
\(715\) 0 0
\(716\) 4.78806 0.178938
\(717\) 13.2665 0.495446
\(718\) −5.25544 −0.196131
\(719\) 14.0000 0.522112 0.261056 0.965324i \(-0.415929\pi\)
0.261056 + 0.965324i \(0.415929\pi\)
\(720\) 0 0
\(721\) −19.5499 −0.728076
\(722\) 32.1716 1.19730
\(723\) 20.6371 0.767503
\(724\) 20.3940 0.757938
\(725\) 0 0
\(726\) 0 0
\(727\) −47.4674 −1.76047 −0.880234 0.474540i \(-0.842614\pi\)
−0.880234 + 0.474540i \(0.842614\pi\)
\(728\) −45.7330 −1.69498
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 66.0000 2.44110
\(732\) 3.66648 0.135517
\(733\) 13.8564 0.511798 0.255899 0.966704i \(-0.417629\pi\)
0.255899 + 0.966704i \(0.417629\pi\)
\(734\) −11.8843 −0.438658
\(735\) 0 0
\(736\) 46.7277 1.72241
\(737\) 0 0
\(738\) 5.48913 0.202057
\(739\) −12.8241 −0.471741 −0.235870 0.971784i \(-0.575794\pi\)
−0.235870 + 0.971784i \(0.575794\pi\)
\(740\) 0 0
\(741\) 52.3505 1.92314
\(742\) −25.4891 −0.935735
\(743\) −17.7253 −0.650277 −0.325138 0.945666i \(-0.605411\pi\)
−0.325138 + 0.945666i \(0.605411\pi\)
\(744\) −9.01011 −0.330327
\(745\) 0 0
\(746\) 8.27719 0.303049
\(747\) −1.87953 −0.0687683
\(748\) 0 0
\(749\) −29.4891 −1.07751
\(750\) 0 0
\(751\) −5.23369 −0.190980 −0.0954900 0.995430i \(-0.530442\pi\)
−0.0954900 + 0.995430i \(0.530442\pi\)
\(752\) 1.72281 0.0628245
\(753\) 23.4891 0.855991
\(754\) −17.0256 −0.620034
\(755\) 0 0
\(756\) 3.46410 0.125988
\(757\) −19.3505 −0.703307 −0.351654 0.936130i \(-0.614380\pi\)
−0.351654 + 0.936130i \(0.614380\pi\)
\(758\) −3.96143 −0.143886
\(759\) 0 0
\(760\) 0 0
\(761\) −1.28962 −0.0467487 −0.0233744 0.999727i \(-0.507441\pi\)
−0.0233744 + 0.999727i \(0.507441\pi\)
\(762\) −2.86141 −0.103658
\(763\) −15.1168 −0.547266
\(764\) −7.53262 −0.272521
\(765\) 0 0
\(766\) 19.2000 0.693725
\(767\) 27.1229 0.979351
\(768\) 13.8832 0.500965
\(769\) −13.7638 −0.496336 −0.248168 0.968717i \(-0.579829\pi\)
−0.248168 + 0.968717i \(0.579829\pi\)
\(770\) 0 0
\(771\) 5.48913 0.197686
\(772\) 2.85696 0.102824
\(773\) 23.4891 0.844845 0.422423 0.906399i \(-0.361180\pi\)
0.422423 + 0.906399i \(0.361180\pi\)
\(774\) 7.88316 0.283354
\(775\) 0 0
\(776\) 9.69259 0.347944
\(777\) −12.6217 −0.452801
\(778\) 16.8404 0.603758
\(779\) −53.4891 −1.91645
\(780\) 0 0
\(781\) 0 0
\(782\) 42.0435 1.50347
\(783\) 3.16915 0.113256
\(784\) −0.394031 −0.0140725
\(785\) 0 0
\(786\) −1.02175 −0.0364446
\(787\) −6.48577 −0.231193 −0.115596 0.993296i \(-0.536878\pi\)
−0.115596 + 0.993296i \(0.536878\pi\)
\(788\) 2.57924 0.0918816
\(789\) 1.58457 0.0564123
\(790\) 0 0
\(791\) −46.0280 −1.63657
\(792\) 0 0
\(793\) 18.1168 0.643348
\(794\) −6.63325 −0.235405
\(795\) 0 0
\(796\) 16.2772 0.576930
\(797\) 27.7228 0.981992 0.490996 0.871162i \(-0.336633\pi\)
0.490996 + 0.871162i \(0.336633\pi\)
\(798\) 15.4410 0.546605
\(799\) 18.2054 0.644060
\(800\) 0 0
\(801\) −17.4891 −0.617948
\(802\) 11.8671 0.419041
\(803\) 0 0
\(804\) 8.39403 0.296035
\(805\) 0 0
\(806\) −18.1168 −0.638139
\(807\) −8.74456 −0.307823
\(808\) −35.4456 −1.24697
\(809\) −7.22316 −0.253953 −0.126976 0.991906i \(-0.540527\pi\)
−0.126976 + 0.991906i \(0.540527\pi\)
\(810\) 0 0
\(811\) 44.7933 1.57290 0.786452 0.617651i \(-0.211916\pi\)
0.786452 + 0.617651i \(0.211916\pi\)
\(812\) 10.9783 0.385261
\(813\) 10.3923 0.364474
\(814\) 0 0
\(815\) 0 0
\(816\) −4.16381 −0.145763
\(817\) −76.8179 −2.68752
\(818\) 6.70106 0.234297
\(819\) 17.1168 0.598111
\(820\) 0 0
\(821\) −12.8617 −0.448878 −0.224439 0.974488i \(-0.572055\pi\)
−0.224439 + 0.974488i \(0.572055\pi\)
\(822\) −8.51278 −0.296917
\(823\) −49.3505 −1.72025 −0.860126 0.510082i \(-0.829615\pi\)
−0.860126 + 0.510082i \(0.829615\pi\)
\(824\) 20.6920 0.720841
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) −36.9253 −1.28402 −0.642009 0.766697i \(-0.721899\pi\)
−0.642009 + 0.766697i \(0.721899\pi\)
\(828\) −10.9783 −0.381521
\(829\) −27.4674 −0.953981 −0.476991 0.878908i \(-0.658272\pi\)
−0.476991 + 0.878908i \(0.658272\pi\)
\(830\) 0 0
\(831\) −29.2048 −1.01310
\(832\) 22.8665 0.792754
\(833\) −4.16381 −0.144268
\(834\) 8.23369 0.285109
\(835\) 0 0
\(836\) 0 0
\(837\) 3.37228 0.116563
\(838\) 25.5383 0.882207
\(839\) 34.7446 1.19952 0.599758 0.800182i \(-0.295264\pi\)
0.599758 + 0.800182i \(0.295264\pi\)
\(840\) 0 0
\(841\) −18.9565 −0.653672
\(842\) 14.2612 0.491472
\(843\) 1.87953 0.0647344
\(844\) −19.8997 −0.684978
\(845\) 0 0
\(846\) 2.17448 0.0747602
\(847\) 0 0
\(848\) −8.00000 −0.274721
\(849\) −18.4077 −0.631752
\(850\) 0 0
\(851\) 40.0000 1.37118
\(852\) −1.02175 −0.0350046
\(853\) 26.0357 0.891444 0.445722 0.895171i \(-0.352947\pi\)
0.445722 + 0.895171i \(0.352947\pi\)
\(854\) 5.34363 0.182855
\(855\) 0 0
\(856\) 31.2119 1.06680
\(857\) 9.39764 0.321017 0.160509 0.987034i \(-0.448687\pi\)
0.160509 + 0.987034i \(0.448687\pi\)
\(858\) 0 0
\(859\) −21.7446 −0.741915 −0.370957 0.928650i \(-0.620970\pi\)
−0.370957 + 0.928650i \(0.620970\pi\)
\(860\) 0 0
\(861\) −17.4891 −0.596028
\(862\) −2.27719 −0.0775613
\(863\) 14.2337 0.484520 0.242260 0.970211i \(-0.422111\pi\)
0.242260 + 0.970211i \(0.422111\pi\)
\(864\) −5.84096 −0.198714
\(865\) 0 0
\(866\) −18.2979 −0.621789
\(867\) −27.0000 −0.916968
\(868\) 11.6819 0.396510
\(869\) 0 0
\(870\) 0 0
\(871\) 41.4766 1.40538
\(872\) 16.0000 0.541828
\(873\) −3.62772 −0.122780
\(874\) −48.9348 −1.65524
\(875\) 0 0
\(876\) 7.13058 0.240920
\(877\) −10.0048 −0.337837 −0.168919 0.985630i \(-0.554028\pi\)
−0.168919 + 0.985630i \(0.554028\pi\)
\(878\) −18.0000 −0.607471
\(879\) −26.1282 −0.881284
\(880\) 0 0
\(881\) −22.4674 −0.756945 −0.378473 0.925613i \(-0.623551\pi\)
−0.378473 + 0.925613i \(0.623551\pi\)
\(882\) −0.497333 −0.0167461
\(883\) 39.8614 1.34144 0.670722 0.741709i \(-0.265985\pi\)
0.670722 + 0.741709i \(0.265985\pi\)
\(884\) −61.7228 −2.07596
\(885\) 0 0
\(886\) 16.4356 0.552166
\(887\) 24.9484 0.837686 0.418843 0.908059i \(-0.362436\pi\)
0.418843 + 0.908059i \(0.362436\pi\)
\(888\) 13.3591 0.448301
\(889\) 9.11684 0.305769
\(890\) 0 0
\(891\) 0 0
\(892\) −6.67122 −0.223369
\(893\) −21.1894 −0.709075
\(894\) −14.7446 −0.493132
\(895\) 0 0
\(896\) −22.7446 −0.759843
\(897\) −54.2458 −1.81121
\(898\) 0.994667 0.0331925
\(899\) 10.6873 0.356440
\(900\) 0 0
\(901\) −84.5379 −2.81636
\(902\) 0 0
\(903\) −25.1168 −0.835836
\(904\) 48.7170 1.62030
\(905\) 0 0
\(906\) 3.09509 0.102828
\(907\) 15.2337 0.505826 0.252913 0.967489i \(-0.418611\pi\)
0.252913 + 0.967489i \(0.418611\pi\)
\(908\) −23.7686 −0.788789
\(909\) 13.2665 0.440022
\(910\) 0 0
\(911\) 27.4891 0.910755 0.455378 0.890298i \(-0.349504\pi\)
0.455378 + 0.890298i \(0.349504\pi\)
\(912\) 4.84630 0.160477
\(913\) 0 0
\(914\) −26.5109 −0.876902
\(915\) 0 0
\(916\) −17.1386 −0.566275
\(917\) 3.25544 0.107504
\(918\) −5.25544 −0.173455
\(919\) −5.69349 −0.187811 −0.0939054 0.995581i \(-0.529935\pi\)
−0.0939054 + 0.995581i \(0.529935\pi\)
\(920\) 0 0
\(921\) −9.89497 −0.326050
\(922\) 6.74456 0.222120
\(923\) −5.04868 −0.166179
\(924\) 0 0
\(925\) 0 0
\(926\) −15.0362 −0.494121
\(927\) −7.74456 −0.254365
\(928\) −18.5109 −0.607649
\(929\) 24.7446 0.811843 0.405921 0.913908i \(-0.366951\pi\)
0.405921 + 0.913908i \(0.366951\pi\)
\(930\) 0 0
\(931\) 4.84630 0.158831
\(932\) −6.52344 −0.213683
\(933\) −24.2337 −0.793375
\(934\) 17.0256 0.557093
\(935\) 0 0
\(936\) −18.1168 −0.592168
\(937\) 24.8935 0.813236 0.406618 0.913598i \(-0.366708\pi\)
0.406618 + 0.913598i \(0.366708\pi\)
\(938\) 12.2337 0.399444
\(939\) −19.9783 −0.651966
\(940\) 0 0
\(941\) 17.0256 0.555017 0.277509 0.960723i \(-0.410491\pi\)
0.277509 + 0.960723i \(0.410491\pi\)
\(942\) −2.08191 −0.0678322
\(943\) 55.4256 1.80491
\(944\) 2.51087 0.0817220
\(945\) 0 0
\(946\) 0 0
\(947\) −1.48913 −0.0483901 −0.0241950 0.999707i \(-0.507702\pi\)
−0.0241950 + 0.999707i \(0.507702\pi\)
\(948\) −0.202380 −0.00657299
\(949\) 35.2337 1.14373
\(950\) 0 0
\(951\) −26.9783 −0.874830
\(952\) −44.7384 −1.44998
\(953\) −7.92287 −0.256647 −0.128323 0.991732i \(-0.540960\pi\)
−0.128323 + 0.991732i \(0.540960\pi\)
\(954\) −10.0974 −0.326914
\(955\) 0 0
\(956\) 18.2054 0.588804
\(957\) 0 0
\(958\) −18.7446 −0.605609
\(959\) 27.1229 0.875844
\(960\) 0 0
\(961\) −19.6277 −0.633152
\(962\) 26.8614 0.866047
\(963\) −11.6819 −0.376445
\(964\) 28.3200 0.912124
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) −54.3933 −1.74917 −0.874585 0.484872i \(-0.838866\pi\)
−0.874585 + 0.484872i \(0.838866\pi\)
\(968\) 0 0
\(969\) 51.2119 1.64516
\(970\) 0 0
\(971\) 26.5109 0.850774 0.425387 0.905011i \(-0.360138\pi\)
0.425387 + 0.905011i \(0.360138\pi\)
\(972\) 1.37228 0.0440159
\(973\) −26.2337 −0.841013
\(974\) 7.90564 0.253313
\(975\) 0 0
\(976\) 1.67715 0.0536842
\(977\) 27.2554 0.871979 0.435989 0.899952i \(-0.356399\pi\)
0.435989 + 0.899952i \(0.356399\pi\)
\(978\) −6.83563 −0.218579
\(979\) 0 0
\(980\) 0 0
\(981\) −5.98844 −0.191196
\(982\) 5.25544 0.167708
\(983\) −14.5109 −0.462825 −0.231413 0.972856i \(-0.574335\pi\)
−0.231413 + 0.972856i \(0.574335\pi\)
\(984\) 18.5109 0.590105
\(985\) 0 0
\(986\) −16.6553 −0.530411
\(987\) −6.92820 −0.220527
\(988\) 71.8397 2.28552
\(989\) 79.5990 2.53110
\(990\) 0 0
\(991\) −44.9565 −1.42809 −0.714045 0.700100i \(-0.753139\pi\)
−0.714045 + 0.700100i \(0.753139\pi\)
\(992\) −19.6974 −0.625392
\(993\) 33.2337 1.05464
\(994\) −1.48913 −0.0472322
\(995\) 0 0
\(996\) −2.57924 −0.0817264
\(997\) −52.5138 −1.66313 −0.831564 0.555429i \(-0.812554\pi\)
−0.831564 + 0.555429i \(0.812554\pi\)
\(998\) 2.46943 0.0781686
\(999\) −5.00000 −0.158193
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 9075.2.a.cw.1.3 yes 4
5.4 even 2 9075.2.a.db.1.2 yes 4
11.10 odd 2 inner 9075.2.a.cw.1.2 4
55.54 odd 2 9075.2.a.db.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9075.2.a.cw.1.2 4 11.10 odd 2 inner
9075.2.a.cw.1.3 yes 4 1.1 even 1 trivial
9075.2.a.db.1.2 yes 4 5.4 even 2
9075.2.a.db.1.3 yes 4 55.54 odd 2