Properties

Label 1386.2.a.r.1.3
Level $1386$
Weight $2$
Character 1386.1
Self dual yes
Analytic conductor $11.067$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1304.1
Defining polynomial: \(x^{3} - 11 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.22168\) of defining polynomial
Character \(\chi\) \(=\) 1386.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.80044 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.80044 q^{5} +1.00000 q^{7} +1.00000 q^{8} +3.80044 q^{10} +1.00000 q^{11} -6.44335 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.80044 q^{17} +6.64291 q^{19} +3.80044 q^{20} +1.00000 q^{22} +0.842472 q^{23} +9.44335 q^{25} -6.44335 q^{26} +1.00000 q^{28} -8.44335 q^{29} -9.40132 q^{31} +1.00000 q^{32} +3.80044 q^{34} +3.80044 q^{35} +9.60088 q^{37} +6.64291 q^{38} +3.80044 q^{40} -3.80044 q^{41} -8.04424 q^{43} +1.00000 q^{44} +0.842472 q^{46} +6.24380 q^{47} +1.00000 q^{49} +9.44335 q^{50} -6.44335 q^{52} -1.60088 q^{53} +3.80044 q^{55} +1.00000 q^{56} -8.44335 q^{58} -9.28583 q^{59} +8.75841 q^{61} -9.40132 q^{62} +1.00000 q^{64} -24.4876 q^{65} +1.15753 q^{67} +3.80044 q^{68} +3.80044 q^{70} -6.75841 q^{71} -7.80044 q^{73} +9.60088 q^{74} +6.64291 q^{76} +1.00000 q^{77} +2.84247 q^{79} +3.80044 q^{80} -3.80044 q^{82} -9.80044 q^{83} +14.4434 q^{85} -8.04424 q^{86} +1.00000 q^{88} -5.15753 q^{89} -6.44335 q^{91} +0.842472 q^{92} +6.24380 q^{94} +25.2460 q^{95} +18.8867 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} + 2q^{5} + 3q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} + 2q^{5} + 3q^{7} + 3q^{8} + 2q^{10} + 3q^{11} + 3q^{14} + 3q^{16} + 2q^{17} + 10q^{19} + 2q^{20} + 3q^{22} + 2q^{23} + 9q^{25} + 3q^{28} - 6q^{29} + 3q^{32} + 2q^{34} + 2q^{35} + 10q^{37} + 10q^{38} + 2q^{40} - 2q^{41} + 14q^{43} + 3q^{44} + 2q^{46} - 10q^{47} + 3q^{49} + 9q^{50} + 14q^{53} + 2q^{55} + 3q^{56} - 6q^{58} - 8q^{59} + 8q^{61} + 3q^{64} - 16q^{65} + 4q^{67} + 2q^{68} + 2q^{70} - 2q^{71} - 14q^{73} + 10q^{74} + 10q^{76} + 3q^{77} + 8q^{79} + 2q^{80} - 2q^{82} - 20q^{83} + 24q^{85} + 14q^{86} + 3q^{88} - 16q^{89} + 2q^{92} - 10q^{94} + 18q^{97} + 3q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.80044 1.69961 0.849805 0.527098i \(-0.176720\pi\)
0.849805 + 0.527098i \(0.176720\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.80044 1.20181
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −6.44335 −1.78707 −0.893533 0.448998i \(-0.851781\pi\)
−0.893533 + 0.448998i \(0.851781\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.80044 0.921742 0.460871 0.887467i \(-0.347537\pi\)
0.460871 + 0.887467i \(0.347537\pi\)
\(18\) 0 0
\(19\) 6.64291 1.52399 0.761994 0.647584i \(-0.224220\pi\)
0.761994 + 0.647584i \(0.224220\pi\)
\(20\) 3.80044 0.849805
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 0.842472 0.175668 0.0878338 0.996135i \(-0.472006\pi\)
0.0878338 + 0.996135i \(0.472006\pi\)
\(24\) 0 0
\(25\) 9.44335 1.88867
\(26\) −6.44335 −1.26365
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −8.44335 −1.56789 −0.783946 0.620829i \(-0.786796\pi\)
−0.783946 + 0.620829i \(0.786796\pi\)
\(30\) 0 0
\(31\) −9.40132 −1.68853 −0.844264 0.535928i \(-0.819962\pi\)
−0.844264 + 0.535928i \(0.819962\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.80044 0.651770
\(35\) 3.80044 0.642392
\(36\) 0 0
\(37\) 9.60088 1.57838 0.789188 0.614152i \(-0.210502\pi\)
0.789188 + 0.614152i \(0.210502\pi\)
\(38\) 6.64291 1.07762
\(39\) 0 0
\(40\) 3.80044 0.600903
\(41\) −3.80044 −0.593529 −0.296765 0.954951i \(-0.595908\pi\)
−0.296765 + 0.954951i \(0.595908\pi\)
\(42\) 0 0
\(43\) −8.04424 −1.22673 −0.613367 0.789798i \(-0.710185\pi\)
−0.613367 + 0.789798i \(0.710185\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 0.842472 0.124216
\(47\) 6.24380 0.910751 0.455376 0.890299i \(-0.349505\pi\)
0.455376 + 0.890299i \(0.349505\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 9.44335 1.33549
\(51\) 0 0
\(52\) −6.44335 −0.893533
\(53\) −1.60088 −0.219898 −0.109949 0.993937i \(-0.535069\pi\)
−0.109949 + 0.993937i \(0.535069\pi\)
\(54\) 0 0
\(55\) 3.80044 0.512451
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −8.44335 −1.10867
\(59\) −9.28583 −1.20891 −0.604456 0.796639i \(-0.706609\pi\)
−0.604456 + 0.796639i \(0.706609\pi\)
\(60\) 0 0
\(61\) 8.75841 1.12140 0.560700 0.828019i \(-0.310532\pi\)
0.560700 + 0.828019i \(0.310532\pi\)
\(62\) −9.40132 −1.19397
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −24.4876 −3.03731
\(66\) 0 0
\(67\) 1.15753 0.141415 0.0707073 0.997497i \(-0.477474\pi\)
0.0707073 + 0.997497i \(0.477474\pi\)
\(68\) 3.80044 0.460871
\(69\) 0 0
\(70\) 3.80044 0.454240
\(71\) −6.75841 −0.802076 −0.401038 0.916061i \(-0.631350\pi\)
−0.401038 + 0.916061i \(0.631350\pi\)
\(72\) 0 0
\(73\) −7.80044 −0.912973 −0.456486 0.889730i \(-0.650892\pi\)
−0.456486 + 0.889730i \(0.650892\pi\)
\(74\) 9.60088 1.11608
\(75\) 0 0
\(76\) 6.64291 0.761994
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 2.84247 0.319803 0.159902 0.987133i \(-0.448882\pi\)
0.159902 + 0.987133i \(0.448882\pi\)
\(80\) 3.80044 0.424902
\(81\) 0 0
\(82\) −3.80044 −0.419689
\(83\) −9.80044 −1.07574 −0.537869 0.843028i \(-0.680771\pi\)
−0.537869 + 0.843028i \(0.680771\pi\)
\(84\) 0 0
\(85\) 14.4434 1.56660
\(86\) −8.04424 −0.867432
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −5.15753 −0.546697 −0.273348 0.961915i \(-0.588131\pi\)
−0.273348 + 0.961915i \(0.588131\pi\)
\(90\) 0 0
\(91\) −6.44335 −0.675447
\(92\) 0.842472 0.0878338
\(93\) 0 0
\(94\) 6.24380 0.643998
\(95\) 25.2460 2.59019
\(96\) 0 0
\(97\) 18.8867 1.91765 0.958827 0.283989i \(-0.0916581\pi\)
0.958827 + 0.283989i \(0.0916581\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 9.44335 0.944335
\(101\) 10.8867 1.08327 0.541634 0.840614i \(-0.317806\pi\)
0.541634 + 0.840614i \(0.317806\pi\)
\(102\) 0 0
\(103\) 5.80044 0.571534 0.285767 0.958299i \(-0.407752\pi\)
0.285767 + 0.958299i \(0.407752\pi\)
\(104\) −6.44335 −0.631823
\(105\) 0 0
\(106\) −1.60088 −0.155491
\(107\) −7.60088 −0.734805 −0.367403 0.930062i \(-0.619753\pi\)
−0.367403 + 0.930062i \(0.619753\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 3.80044 0.362358
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 0.758411 0.0713453 0.0356726 0.999364i \(-0.488643\pi\)
0.0356726 + 0.999364i \(0.488643\pi\)
\(114\) 0 0
\(115\) 3.20177 0.298566
\(116\) −8.44335 −0.783946
\(117\) 0 0
\(118\) −9.28583 −0.854830
\(119\) 3.80044 0.348386
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 8.75841 0.792949
\(123\) 0 0
\(124\) −9.40132 −0.844264
\(125\) 16.8867 1.51039
\(126\) 0 0
\(127\) 2.84247 0.252229 0.126114 0.992016i \(-0.459749\pi\)
0.126114 + 0.992016i \(0.459749\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −24.4876 −2.14770
\(131\) −14.1996 −1.24062 −0.620311 0.784356i \(-0.712993\pi\)
−0.620311 + 0.784356i \(0.712993\pi\)
\(132\) 0 0
\(133\) 6.64291 0.576014
\(134\) 1.15753 0.0999952
\(135\) 0 0
\(136\) 3.80044 0.325885
\(137\) 12.7584 1.09002 0.545012 0.838428i \(-0.316525\pi\)
0.545012 + 0.838428i \(0.316525\pi\)
\(138\) 0 0
\(139\) −8.55885 −0.725952 −0.362976 0.931798i \(-0.618239\pi\)
−0.362976 + 0.931798i \(0.618239\pi\)
\(140\) 3.80044 0.321196
\(141\) 0 0
\(142\) −6.75841 −0.567153
\(143\) −6.44335 −0.538820
\(144\) 0 0
\(145\) −32.0885 −2.66480
\(146\) −7.80044 −0.645569
\(147\) 0 0
\(148\) 9.60088 0.789188
\(149\) 10.8867 0.891874 0.445937 0.895064i \(-0.352871\pi\)
0.445937 + 0.895064i \(0.352871\pi\)
\(150\) 0 0
\(151\) 6.31506 0.513912 0.256956 0.966423i \(-0.417280\pi\)
0.256956 + 0.966423i \(0.417280\pi\)
\(152\) 6.64291 0.538811
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −35.7292 −2.86984
\(156\) 0 0
\(157\) −12.1996 −0.973631 −0.486815 0.873505i \(-0.661842\pi\)
−0.486815 + 0.873505i \(0.661842\pi\)
\(158\) 2.84247 0.226135
\(159\) 0 0
\(160\) 3.80044 0.300451
\(161\) 0.842472 0.0663961
\(162\) 0 0
\(163\) −5.68494 −0.445279 −0.222640 0.974901i \(-0.571467\pi\)
−0.222640 + 0.974901i \(0.571467\pi\)
\(164\) −3.80044 −0.296765
\(165\) 0 0
\(166\) −9.80044 −0.760662
\(167\) −24.4876 −1.89491 −0.947453 0.319894i \(-0.896353\pi\)
−0.947453 + 0.319894i \(0.896353\pi\)
\(168\) 0 0
\(169\) 28.5168 2.19360
\(170\) 14.4434 1.10775
\(171\) 0 0
\(172\) −8.04424 −0.613367
\(173\) −15.2858 −1.16216 −0.581080 0.813846i \(-0.697370\pi\)
−0.581080 + 0.813846i \(0.697370\pi\)
\(174\) 0 0
\(175\) 9.44335 0.713851
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −5.15753 −0.386573
\(179\) −25.2460 −1.88697 −0.943487 0.331408i \(-0.892476\pi\)
−0.943487 + 0.331408i \(0.892476\pi\)
\(180\) 0 0
\(181\) −7.80044 −0.579802 −0.289901 0.957057i \(-0.593622\pi\)
−0.289901 + 0.957057i \(0.593622\pi\)
\(182\) −6.44335 −0.477613
\(183\) 0 0
\(184\) 0.842472 0.0621079
\(185\) 36.4876 2.68262
\(186\) 0 0
\(187\) 3.80044 0.277916
\(188\) 6.24380 0.455376
\(189\) 0 0
\(190\) 25.2460 1.83154
\(191\) 20.9309 1.51451 0.757255 0.653119i \(-0.226540\pi\)
0.757255 + 0.653119i \(0.226540\pi\)
\(192\) 0 0
\(193\) 5.20177 0.374431 0.187216 0.982319i \(-0.440054\pi\)
0.187216 + 0.982319i \(0.440054\pi\)
\(194\) 18.8867 1.35599
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 13.3301 0.949728 0.474864 0.880059i \(-0.342497\pi\)
0.474864 + 0.880059i \(0.342497\pi\)
\(198\) 0 0
\(199\) −13.8004 −0.978287 −0.489144 0.872203i \(-0.662691\pi\)
−0.489144 + 0.872203i \(0.662691\pi\)
\(200\) 9.44335 0.667746
\(201\) 0 0
\(202\) 10.8867 0.765986
\(203\) −8.44335 −0.592607
\(204\) 0 0
\(205\) −14.4434 −1.00877
\(206\) 5.80044 0.404136
\(207\) 0 0
\(208\) −6.44335 −0.446766
\(209\) 6.64291 0.459500
\(210\) 0 0
\(211\) 13.7292 0.945156 0.472578 0.881289i \(-0.343324\pi\)
0.472578 + 0.881289i \(0.343324\pi\)
\(212\) −1.60088 −0.109949
\(213\) 0 0
\(214\) −7.60088 −0.519586
\(215\) −30.5717 −2.08497
\(216\) 0 0
\(217\) −9.40132 −0.638203
\(218\) 8.00000 0.541828
\(219\) 0 0
\(220\) 3.80044 0.256226
\(221\) −24.4876 −1.64721
\(222\) 0 0
\(223\) −24.6031 −1.64754 −0.823772 0.566921i \(-0.808135\pi\)
−0.823772 + 0.566921i \(0.808135\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 0.758411 0.0504487
\(227\) −25.0022 −1.65945 −0.829727 0.558169i \(-0.811504\pi\)
−0.829727 + 0.558169i \(0.811504\pi\)
\(228\) 0 0
\(229\) 16.6872 1.10272 0.551359 0.834268i \(-0.314109\pi\)
0.551359 + 0.834268i \(0.314109\pi\)
\(230\) 3.20177 0.211118
\(231\) 0 0
\(232\) −8.44335 −0.554333
\(233\) −27.7734 −1.81950 −0.909749 0.415160i \(-0.863726\pi\)
−0.909749 + 0.415160i \(0.863726\pi\)
\(234\) 0 0
\(235\) 23.7292 1.54792
\(236\) −9.28583 −0.604456
\(237\) 0 0
\(238\) 3.80044 0.246346
\(239\) 1.68494 0.108990 0.0544950 0.998514i \(-0.482645\pi\)
0.0544950 + 0.998514i \(0.482645\pi\)
\(240\) 0 0
\(241\) −15.4013 −0.992087 −0.496043 0.868298i \(-0.665214\pi\)
−0.496043 + 0.868298i \(0.665214\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 8.75841 0.560700
\(245\) 3.80044 0.242801
\(246\) 0 0
\(247\) −42.8026 −2.72347
\(248\) −9.40132 −0.596985
\(249\) 0 0
\(250\) 16.8867 1.06801
\(251\) 17.9159 1.13084 0.565422 0.824802i \(-0.308713\pi\)
0.565422 + 0.824802i \(0.308713\pi\)
\(252\) 0 0
\(253\) 0.842472 0.0529658
\(254\) 2.84247 0.178353
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −10.0442 −0.626542 −0.313271 0.949664i \(-0.601425\pi\)
−0.313271 + 0.949664i \(0.601425\pi\)
\(258\) 0 0
\(259\) 9.60088 0.596570
\(260\) −24.4876 −1.51866
\(261\) 0 0
\(262\) −14.1996 −0.877252
\(263\) 11.7292 0.723252 0.361626 0.932323i \(-0.382222\pi\)
0.361626 + 0.932323i \(0.382222\pi\)
\(264\) 0 0
\(265\) −6.08406 −0.373741
\(266\) 6.64291 0.407303
\(267\) 0 0
\(268\) 1.15753 0.0707073
\(269\) 20.1996 1.23159 0.615794 0.787907i \(-0.288835\pi\)
0.615794 + 0.787907i \(0.288835\pi\)
\(270\) 0 0
\(271\) −8.39912 −0.510210 −0.255105 0.966913i \(-0.582110\pi\)
−0.255105 + 0.966913i \(0.582110\pi\)
\(272\) 3.80044 0.230436
\(273\) 0 0
\(274\) 12.7584 0.770764
\(275\) 9.44335 0.569456
\(276\) 0 0
\(277\) −8.88671 −0.533951 −0.266975 0.963703i \(-0.586024\pi\)
−0.266975 + 0.963703i \(0.586024\pi\)
\(278\) −8.55885 −0.513326
\(279\) 0 0
\(280\) 3.80044 0.227120
\(281\) −4.31506 −0.257415 −0.128707 0.991683i \(-0.541083\pi\)
−0.128707 + 0.991683i \(0.541083\pi\)
\(282\) 0 0
\(283\) −2.15532 −0.128121 −0.0640603 0.997946i \(-0.520405\pi\)
−0.0640603 + 0.997946i \(0.520405\pi\)
\(284\) −6.75841 −0.401038
\(285\) 0 0
\(286\) −6.44335 −0.381004
\(287\) −3.80044 −0.224333
\(288\) 0 0
\(289\) −2.55665 −0.150391
\(290\) −32.0885 −1.88430
\(291\) 0 0
\(292\) −7.80044 −0.456486
\(293\) −1.60088 −0.0935246 −0.0467623 0.998906i \(-0.514890\pi\)
−0.0467623 + 0.998906i \(0.514890\pi\)
\(294\) 0 0
\(295\) −35.2902 −2.05468
\(296\) 9.60088 0.558040
\(297\) 0 0
\(298\) 10.8867 0.630650
\(299\) −5.42835 −0.313929
\(300\) 0 0
\(301\) −8.04424 −0.463662
\(302\) 6.31506 0.363391
\(303\) 0 0
\(304\) 6.64291 0.380997
\(305\) 33.2858 1.90594
\(306\) 0 0
\(307\) −7.52962 −0.429738 −0.214869 0.976643i \(-0.568933\pi\)
−0.214869 + 0.976643i \(0.568933\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −35.7292 −2.02928
\(311\) 6.24380 0.354053 0.177027 0.984206i \(-0.443352\pi\)
0.177027 + 0.984206i \(0.443352\pi\)
\(312\) 0 0
\(313\) −0.714173 −0.0403675 −0.0201837 0.999796i \(-0.506425\pi\)
−0.0201837 + 0.999796i \(0.506425\pi\)
\(314\) −12.1996 −0.688461
\(315\) 0 0
\(316\) 2.84247 0.159902
\(317\) 8.71417 0.489437 0.244718 0.969594i \(-0.421304\pi\)
0.244718 + 0.969594i \(0.421304\pi\)
\(318\) 0 0
\(319\) −8.44335 −0.472737
\(320\) 3.80044 0.212451
\(321\) 0 0
\(322\) 0.842472 0.0469491
\(323\) 25.2460 1.40473
\(324\) 0 0
\(325\) −60.8469 −3.37518
\(326\) −5.68494 −0.314860
\(327\) 0 0
\(328\) −3.80044 −0.209844
\(329\) 6.24380 0.344232
\(330\) 0 0
\(331\) 6.04424 0.332221 0.166111 0.986107i \(-0.446879\pi\)
0.166111 + 0.986107i \(0.446879\pi\)
\(332\) −9.80044 −0.537869
\(333\) 0 0
\(334\) −24.4876 −1.33990
\(335\) 4.39912 0.240349
\(336\) 0 0
\(337\) −34.4876 −1.87866 −0.939329 0.343016i \(-0.888551\pi\)
−0.939329 + 0.343016i \(0.888551\pi\)
\(338\) 28.5168 1.55111
\(339\) 0 0
\(340\) 14.4434 0.783301
\(341\) −9.40132 −0.509110
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −8.04424 −0.433716
\(345\) 0 0
\(346\) −15.2858 −0.821771
\(347\) −7.60088 −0.408037 −0.204018 0.978967i \(-0.565400\pi\)
−0.204018 + 0.978967i \(0.565400\pi\)
\(348\) 0 0
\(349\) 14.8425 0.794499 0.397250 0.917711i \(-0.369965\pi\)
0.397250 + 0.917711i \(0.369965\pi\)
\(350\) 9.44335 0.504769
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −10.0442 −0.534601 −0.267300 0.963613i \(-0.586132\pi\)
−0.267300 + 0.963613i \(0.586132\pi\)
\(354\) 0 0
\(355\) −25.6849 −1.36322
\(356\) −5.15753 −0.273348
\(357\) 0 0
\(358\) −25.2460 −1.33429
\(359\) 3.47258 0.183276 0.0916380 0.995792i \(-0.470790\pi\)
0.0916380 + 0.995792i \(0.470790\pi\)
\(360\) 0 0
\(361\) 25.1283 1.32254
\(362\) −7.80044 −0.409982
\(363\) 0 0
\(364\) −6.44335 −0.337724
\(365\) −29.6451 −1.55170
\(366\) 0 0
\(367\) 1.40132 0.0731485 0.0365743 0.999331i \(-0.488355\pi\)
0.0365743 + 0.999331i \(0.488355\pi\)
\(368\) 0.842472 0.0439169
\(369\) 0 0
\(370\) 36.4876 1.89690
\(371\) −1.60088 −0.0831137
\(372\) 0 0
\(373\) 13.6451 0.706518 0.353259 0.935526i \(-0.385074\pi\)
0.353259 + 0.935526i \(0.385074\pi\)
\(374\) 3.80044 0.196516
\(375\) 0 0
\(376\) 6.24380 0.321999
\(377\) 54.4035 2.80192
\(378\) 0 0
\(379\) 9.51682 0.488846 0.244423 0.969669i \(-0.421401\pi\)
0.244423 + 0.969669i \(0.421401\pi\)
\(380\) 25.2460 1.29509
\(381\) 0 0
\(382\) 20.9309 1.07092
\(383\) 15.0420 0.768612 0.384306 0.923206i \(-0.374441\pi\)
0.384306 + 0.923206i \(0.374441\pi\)
\(384\) 0 0
\(385\) 3.80044 0.193688
\(386\) 5.20177 0.264763
\(387\) 0 0
\(388\) 18.8867 0.958827
\(389\) 2.63011 0.133352 0.0666760 0.997775i \(-0.478761\pi\)
0.0666760 + 0.997775i \(0.478761\pi\)
\(390\) 0 0
\(391\) 3.20177 0.161920
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 13.3301 0.671559
\(395\) 10.8026 0.543540
\(396\) 0 0
\(397\) −2.91373 −0.146236 −0.0731180 0.997323i \(-0.523295\pi\)
−0.0731180 + 0.997323i \(0.523295\pi\)
\(398\) −13.8004 −0.691754
\(399\) 0 0
\(400\) 9.44335 0.472168
\(401\) 12.7584 0.637125 0.318562 0.947902i \(-0.396800\pi\)
0.318562 + 0.947902i \(0.396800\pi\)
\(402\) 0 0
\(403\) 60.5761 3.01751
\(404\) 10.8867 0.541634
\(405\) 0 0
\(406\) −8.44335 −0.419037
\(407\) 9.60088 0.475898
\(408\) 0 0
\(409\) −4.59868 −0.227390 −0.113695 0.993516i \(-0.536269\pi\)
−0.113695 + 0.993516i \(0.536269\pi\)
\(410\) −14.4434 −0.713306
\(411\) 0 0
\(412\) 5.80044 0.285767
\(413\) −9.28583 −0.456926
\(414\) 0 0
\(415\) −37.2460 −1.82833
\(416\) −6.44335 −0.315911
\(417\) 0 0
\(418\) 6.64291 0.324916
\(419\) 21.7734 1.06370 0.531851 0.846838i \(-0.321497\pi\)
0.531851 + 0.846838i \(0.321497\pi\)
\(420\) 0 0
\(421\) 29.2018 1.42321 0.711603 0.702581i \(-0.247969\pi\)
0.711603 + 0.702581i \(0.247969\pi\)
\(422\) 13.7292 0.668326
\(423\) 0 0
\(424\) −1.60088 −0.0777457
\(425\) 35.8889 1.74087
\(426\) 0 0
\(427\) 8.75841 0.423849
\(428\) −7.60088 −0.367403
\(429\) 0 0
\(430\) −30.5717 −1.47430
\(431\) 22.0442 1.06183 0.530917 0.847424i \(-0.321848\pi\)
0.530917 + 0.847424i \(0.321848\pi\)
\(432\) 0 0
\(433\) −16.4035 −0.788303 −0.394152 0.919045i \(-0.628962\pi\)
−0.394152 + 0.919045i \(0.628962\pi\)
\(434\) −9.40132 −0.451278
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) 5.59647 0.267715
\(438\) 0 0
\(439\) 17.2858 0.825008 0.412504 0.910956i \(-0.364654\pi\)
0.412504 + 0.910956i \(0.364654\pi\)
\(440\) 3.80044 0.181179
\(441\) 0 0
\(442\) −24.4876 −1.16476
\(443\) −1.95576 −0.0929211 −0.0464605 0.998920i \(-0.514794\pi\)
−0.0464605 + 0.998920i \(0.514794\pi\)
\(444\) 0 0
\(445\) −19.6009 −0.929171
\(446\) −24.6031 −1.16499
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 24.7584 1.16842 0.584211 0.811602i \(-0.301404\pi\)
0.584211 + 0.811602i \(0.301404\pi\)
\(450\) 0 0
\(451\) −3.80044 −0.178956
\(452\) 0.758411 0.0356726
\(453\) 0 0
\(454\) −25.0022 −1.17341
\(455\) −24.4876 −1.14800
\(456\) 0 0
\(457\) 22.0885 1.03326 0.516628 0.856210i \(-0.327187\pi\)
0.516628 + 0.856210i \(0.327187\pi\)
\(458\) 16.6872 0.779739
\(459\) 0 0
\(460\) 3.20177 0.149283
\(461\) −39.7734 −1.85243 −0.926216 0.376992i \(-0.876958\pi\)
−0.926216 + 0.376992i \(0.876958\pi\)
\(462\) 0 0
\(463\) −7.20177 −0.334694 −0.167347 0.985898i \(-0.553520\pi\)
−0.167347 + 0.985898i \(0.553520\pi\)
\(464\) −8.44335 −0.391973
\(465\) 0 0
\(466\) −27.7734 −1.28658
\(467\) 1.68494 0.0779699 0.0389850 0.999240i \(-0.487588\pi\)
0.0389850 + 0.999240i \(0.487588\pi\)
\(468\) 0 0
\(469\) 1.15753 0.0534497
\(470\) 23.7292 1.09455
\(471\) 0 0
\(472\) −9.28583 −0.427415
\(473\) −8.04424 −0.369874
\(474\) 0 0
\(475\) 62.7314 2.87831
\(476\) 3.80044 0.174193
\(477\) 0 0
\(478\) 1.68494 0.0770675
\(479\) −27.2018 −1.24288 −0.621440 0.783462i \(-0.713452\pi\)
−0.621440 + 0.783462i \(0.713452\pi\)
\(480\) 0 0
\(481\) −61.8619 −2.82066
\(482\) −15.4013 −0.701511
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 71.7778 3.25926
\(486\) 0 0
\(487\) 13.9159 0.630591 0.315296 0.948993i \(-0.397896\pi\)
0.315296 + 0.948993i \(0.397896\pi\)
\(488\) 8.75841 0.396475
\(489\) 0 0
\(490\) 3.80044 0.171686
\(491\) 3.20177 0.144494 0.0722468 0.997387i \(-0.476983\pi\)
0.0722468 + 0.997387i \(0.476983\pi\)
\(492\) 0 0
\(493\) −32.0885 −1.44519
\(494\) −42.8026 −1.92578
\(495\) 0 0
\(496\) −9.40132 −0.422132
\(497\) −6.75841 −0.303156
\(498\) 0 0
\(499\) −17.2460 −0.772037 −0.386019 0.922491i \(-0.626150\pi\)
−0.386019 + 0.922491i \(0.626150\pi\)
\(500\) 16.8867 0.755197
\(501\) 0 0
\(502\) 17.9159 0.799627
\(503\) −0.487592 −0.0217407 −0.0108703 0.999941i \(-0.503460\pi\)
−0.0108703 + 0.999941i \(0.503460\pi\)
\(504\) 0 0
\(505\) 41.3743 1.84113
\(506\) 0.842472 0.0374525
\(507\) 0 0
\(508\) 2.84247 0.126114
\(509\) −28.7756 −1.27546 −0.637729 0.770261i \(-0.720126\pi\)
−0.637729 + 0.770261i \(0.720126\pi\)
\(510\) 0 0
\(511\) −7.80044 −0.345071
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −10.0442 −0.443032
\(515\) 22.0442 0.971385
\(516\) 0 0
\(517\) 6.24380 0.274602
\(518\) 9.60088 0.421839
\(519\) 0 0
\(520\) −24.4876 −1.07385
\(521\) −25.2460 −1.10605 −0.553024 0.833165i \(-0.686526\pi\)
−0.553024 + 0.833165i \(0.686526\pi\)
\(522\) 0 0
\(523\) 37.0464 1.61993 0.809964 0.586480i \(-0.199487\pi\)
0.809964 + 0.586480i \(0.199487\pi\)
\(524\) −14.1996 −0.620311
\(525\) 0 0
\(526\) 11.7292 0.511417
\(527\) −35.7292 −1.55639
\(528\) 0 0
\(529\) −22.2902 −0.969141
\(530\) −6.08406 −0.264275
\(531\) 0 0
\(532\) 6.64291 0.288007
\(533\) 24.4876 1.06068
\(534\) 0 0
\(535\) −28.8867 −1.24888
\(536\) 1.15753 0.0499976
\(537\) 0 0
\(538\) 20.1996 0.870865
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 30.5318 1.31267 0.656333 0.754471i \(-0.272107\pi\)
0.656333 + 0.754471i \(0.272107\pi\)
\(542\) −8.39912 −0.360773
\(543\) 0 0
\(544\) 3.80044 0.162943
\(545\) 30.4035 1.30234
\(546\) 0 0
\(547\) 45.1619 1.93099 0.965493 0.260430i \(-0.0838645\pi\)
0.965493 + 0.260430i \(0.0838645\pi\)
\(548\) 12.7584 0.545012
\(549\) 0 0
\(550\) 9.44335 0.402666
\(551\) −56.0885 −2.38945
\(552\) 0 0
\(553\) 2.84247 0.120874
\(554\) −8.88671 −0.377560
\(555\) 0 0
\(556\) −8.55885 −0.362976
\(557\) −9.96018 −0.422026 −0.211013 0.977483i \(-0.567676\pi\)
−0.211013 + 0.977483i \(0.567676\pi\)
\(558\) 0 0
\(559\) 51.8319 2.19225
\(560\) 3.80044 0.160598
\(561\) 0 0
\(562\) −4.31506 −0.182020
\(563\) −33.8004 −1.42452 −0.712259 0.701916i \(-0.752328\pi\)
−0.712259 + 0.701916i \(0.752328\pi\)
\(564\) 0 0
\(565\) 2.88230 0.121259
\(566\) −2.15532 −0.0905949
\(567\) 0 0
\(568\) −6.75841 −0.283577
\(569\) 31.6849 1.32830 0.664151 0.747598i \(-0.268793\pi\)
0.664151 + 0.747598i \(0.268793\pi\)
\(570\) 0 0
\(571\) −0.443355 −0.0185538 −0.00927691 0.999957i \(-0.502953\pi\)
−0.00927691 + 0.999957i \(0.502953\pi\)
\(572\) −6.44335 −0.269410
\(573\) 0 0
\(574\) −3.80044 −0.158627
\(575\) 7.95576 0.331778
\(576\) 0 0
\(577\) −0.714173 −0.0297314 −0.0148657 0.999889i \(-0.504732\pi\)
−0.0148657 + 0.999889i \(0.504732\pi\)
\(578\) −2.55665 −0.106342
\(579\) 0 0
\(580\) −32.0885 −1.33240
\(581\) −9.80044 −0.406591
\(582\) 0 0
\(583\) −1.60088 −0.0663018
\(584\) −7.80044 −0.322785
\(585\) 0 0
\(586\) −1.60088 −0.0661319
\(587\) 4.39912 0.181571 0.0907855 0.995870i \(-0.471062\pi\)
0.0907855 + 0.995870i \(0.471062\pi\)
\(588\) 0 0
\(589\) −62.4522 −2.57330
\(590\) −35.2902 −1.45288
\(591\) 0 0
\(592\) 9.60088 0.394594
\(593\) −34.2040 −1.40459 −0.702294 0.711887i \(-0.747841\pi\)
−0.702294 + 0.711887i \(0.747841\pi\)
\(594\) 0 0
\(595\) 14.4434 0.592120
\(596\) 10.8867 0.445937
\(597\) 0 0
\(598\) −5.42835 −0.221982
\(599\) 5.24159 0.214166 0.107083 0.994250i \(-0.465849\pi\)
0.107083 + 0.994250i \(0.465849\pi\)
\(600\) 0 0
\(601\) 16.0314 0.653936 0.326968 0.945035i \(-0.393973\pi\)
0.326968 + 0.945035i \(0.393973\pi\)
\(602\) −8.04424 −0.327859
\(603\) 0 0
\(604\) 6.31506 0.256956
\(605\) 3.80044 0.154510
\(606\) 0 0
\(607\) 31.2902 1.27003 0.635016 0.772499i \(-0.280994\pi\)
0.635016 + 0.772499i \(0.280994\pi\)
\(608\) 6.64291 0.269406
\(609\) 0 0
\(610\) 33.2858 1.34770
\(611\) −40.2310 −1.62757
\(612\) 0 0
\(613\) 15.8717 0.641052 0.320526 0.947240i \(-0.396140\pi\)
0.320526 + 0.947240i \(0.396140\pi\)
\(614\) −7.52962 −0.303871
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −22.3151 −0.898370 −0.449185 0.893439i \(-0.648286\pi\)
−0.449185 + 0.893439i \(0.648286\pi\)
\(618\) 0 0
\(619\) −5.68494 −0.228497 −0.114249 0.993452i \(-0.536446\pi\)
−0.114249 + 0.993452i \(0.536446\pi\)
\(620\) −35.7292 −1.43492
\(621\) 0 0
\(622\) 6.24380 0.250353
\(623\) −5.15753 −0.206632
\(624\) 0 0
\(625\) 16.9602 0.678407
\(626\) −0.714173 −0.0285441
\(627\) 0 0
\(628\) −12.1996 −0.486815
\(629\) 36.4876 1.45486
\(630\) 0 0
\(631\) 46.1725 1.83810 0.919050 0.394141i \(-0.128958\pi\)
0.919050 + 0.394141i \(0.128958\pi\)
\(632\) 2.84247 0.113067
\(633\) 0 0
\(634\) 8.71417 0.346084
\(635\) 10.8026 0.428690
\(636\) 0 0
\(637\) −6.44335 −0.255295
\(638\) −8.44335 −0.334276
\(639\) 0 0
\(640\) 3.80044 0.150226
\(641\) 35.4584 1.40052 0.700261 0.713887i \(-0.253067\pi\)
0.700261 + 0.713887i \(0.253067\pi\)
\(642\) 0 0
\(643\) −6.71417 −0.264781 −0.132391 0.991198i \(-0.542265\pi\)
−0.132391 + 0.991198i \(0.542265\pi\)
\(644\) 0.842472 0.0331980
\(645\) 0 0
\(646\) 25.2460 0.993291
\(647\) 30.2438 1.18901 0.594503 0.804093i \(-0.297349\pi\)
0.594503 + 0.804093i \(0.297349\pi\)
\(648\) 0 0
\(649\) −9.28583 −0.364501
\(650\) −60.8469 −2.38661
\(651\) 0 0
\(652\) −5.68494 −0.222640
\(653\) 15.7734 0.617262 0.308631 0.951182i \(-0.400129\pi\)
0.308631 + 0.951182i \(0.400129\pi\)
\(654\) 0 0
\(655\) −53.9646 −2.10857
\(656\) −3.80044 −0.148382
\(657\) 0 0
\(658\) 6.24380 0.243409
\(659\) −39.8575 −1.55263 −0.776314 0.630347i \(-0.782913\pi\)
−0.776314 + 0.630347i \(0.782913\pi\)
\(660\) 0 0
\(661\) 35.2588 1.37141 0.685704 0.727880i \(-0.259494\pi\)
0.685704 + 0.727880i \(0.259494\pi\)
\(662\) 6.04424 0.234916
\(663\) 0 0
\(664\) −9.80044 −0.380331
\(665\) 25.2460 0.978998
\(666\) 0 0
\(667\) −7.11329 −0.275428
\(668\) −24.4876 −0.947453
\(669\) 0 0
\(670\) 4.39912 0.169953
\(671\) 8.75841 0.338115
\(672\) 0 0
\(673\) 19.9159 0.767703 0.383852 0.923395i \(-0.374597\pi\)
0.383852 + 0.923395i \(0.374597\pi\)
\(674\) −34.4876 −1.32841
\(675\) 0 0
\(676\) 28.5168 1.09680
\(677\) 45.6894 1.75598 0.877992 0.478675i \(-0.158883\pi\)
0.877992 + 0.478675i \(0.158883\pi\)
\(678\) 0 0
\(679\) 18.8867 0.724805
\(680\) 14.4434 0.553877
\(681\) 0 0
\(682\) −9.40132 −0.359995
\(683\) −20.3593 −0.779027 −0.389513 0.921021i \(-0.627357\pi\)
−0.389513 + 0.921021i \(0.627357\pi\)
\(684\) 0 0
\(685\) 48.4876 1.85262
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −8.04424 −0.306684
\(689\) 10.3151 0.392972
\(690\) 0 0
\(691\) 24.2310 0.921790 0.460895 0.887455i \(-0.347528\pi\)
0.460895 + 0.887455i \(0.347528\pi\)
\(692\) −15.2858 −0.581080
\(693\) 0 0
\(694\) −7.60088 −0.288526
\(695\) −32.5274 −1.23384
\(696\) 0 0
\(697\) −14.4434 −0.547081
\(698\) 14.8425 0.561796
\(699\) 0 0
\(700\) 9.44335 0.356925
\(701\) −14.2566 −0.538464 −0.269232 0.963075i \(-0.586770\pi\)
−0.269232 + 0.963075i \(0.586770\pi\)
\(702\) 0 0
\(703\) 63.7778 2.40543
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −10.0442 −0.378020
\(707\) 10.8867 0.409437
\(708\) 0 0
\(709\) −10.6557 −0.400184 −0.200092 0.979777i \(-0.564124\pi\)
−0.200092 + 0.979777i \(0.564124\pi\)
\(710\) −25.6849 −0.963939
\(711\) 0 0
\(712\) −5.15753 −0.193287
\(713\) −7.92035 −0.296620
\(714\) 0 0
\(715\) −24.4876 −0.915784
\(716\) −25.2460 −0.943487
\(717\) 0 0
\(718\) 3.47258 0.129596
\(719\) 28.5589 1.06507 0.532533 0.846409i \(-0.321240\pi\)
0.532533 + 0.846409i \(0.321240\pi\)
\(720\) 0 0
\(721\) 5.80044 0.216020
\(722\) 25.1283 0.935178
\(723\) 0 0
\(724\) −7.80044 −0.289901
\(725\) −79.7336 −2.96123
\(726\) 0 0
\(727\) −6.03144 −0.223694 −0.111847 0.993725i \(-0.535677\pi\)
−0.111847 + 0.993725i \(0.535677\pi\)
\(728\) −6.44335 −0.238807
\(729\) 0 0
\(730\) −29.6451 −1.09722
\(731\) −30.5717 −1.13073
\(732\) 0 0
\(733\) 3.33006 0.122999 0.0614994 0.998107i \(-0.480412\pi\)
0.0614994 + 0.998107i \(0.480412\pi\)
\(734\) 1.40132 0.0517238
\(735\) 0 0
\(736\) 0.842472 0.0310539
\(737\) 1.15753 0.0426381
\(738\) 0 0
\(739\) 6.66994 0.245358 0.122679 0.992446i \(-0.460852\pi\)
0.122679 + 0.992446i \(0.460852\pi\)
\(740\) 36.4876 1.34131
\(741\) 0 0
\(742\) −1.60088 −0.0587703
\(743\) 15.2018 0.557699 0.278849 0.960335i \(-0.410047\pi\)
0.278849 + 0.960335i \(0.410047\pi\)
\(744\) 0 0
\(745\) 41.3743 1.51584
\(746\) 13.6451 0.499583
\(747\) 0 0
\(748\) 3.80044 0.138958
\(749\) −7.60088 −0.277730
\(750\) 0 0
\(751\) −8.23099 −0.300353 −0.150177 0.988659i \(-0.547984\pi\)
−0.150177 + 0.988659i \(0.547984\pi\)
\(752\) 6.24380 0.227688
\(753\) 0 0
\(754\) 54.4035 1.98126
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) −49.6894 −1.80599 −0.902995 0.429651i \(-0.858637\pi\)
−0.902995 + 0.429651i \(0.858637\pi\)
\(758\) 9.51682 0.345667
\(759\) 0 0
\(760\) 25.2460 0.915769
\(761\) −0.111083 −0.00402677 −0.00201338 0.999998i \(-0.500641\pi\)
−0.00201338 + 0.999998i \(0.500641\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 20.9309 0.757255
\(765\) 0 0
\(766\) 15.0420 0.543491
\(767\) 59.8319 2.16040
\(768\) 0 0
\(769\) 45.5739 1.64344 0.821718 0.569895i \(-0.193016\pi\)
0.821718 + 0.569895i \(0.193016\pi\)
\(770\) 3.80044 0.136958
\(771\) 0 0
\(772\) 5.20177 0.187216
\(773\) 22.2040 0.798621 0.399311 0.916816i \(-0.369249\pi\)
0.399311 + 0.916816i \(0.369249\pi\)
\(774\) 0 0
\(775\) −88.7800 −3.18907
\(776\) 18.8867 0.677993
\(777\) 0 0
\(778\) 2.63011 0.0942941
\(779\) −25.2460 −0.904532
\(780\) 0 0
\(781\) −6.75841 −0.241835
\(782\) 3.20177 0.114495
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −46.3637 −1.65479
\(786\) 0 0
\(787\) −51.1305 −1.82261 −0.911303 0.411737i \(-0.864922\pi\)
−0.911303 + 0.411737i \(0.864922\pi\)
\(788\) 13.3301 0.474864
\(789\) 0 0
\(790\) 10.8026 0.384341
\(791\) 0.758411 0.0269660
\(792\) 0 0
\(793\) −56.4335 −2.00401
\(794\) −2.91373 −0.103404
\(795\) 0 0
\(796\) −13.8004 −0.489144
\(797\) 32.6872 1.15784 0.578919 0.815385i \(-0.303475\pi\)
0.578919 + 0.815385i \(0.303475\pi\)
\(798\) 0 0
\(799\) 23.7292 0.839478
\(800\) 9.44335 0.333873
\(801\) 0 0
\(802\) 12.7584 0.450515
\(803\) −7.80044 −0.275272
\(804\) 0 0
\(805\) 3.20177 0.112847
\(806\) 60.5761 2.13370
\(807\) 0 0
\(808\) 10.8867 0.382993
\(809\) −0.0840613 −0.00295544 −0.00147772 0.999999i \(-0.500470\pi\)
−0.00147772 + 0.999999i \(0.500470\pi\)
\(810\) 0 0
\(811\) 11.6977 0.410763 0.205382 0.978682i \(-0.434156\pi\)
0.205382 + 0.978682i \(0.434156\pi\)
\(812\) −8.44335 −0.296304
\(813\) 0 0
\(814\) 9.60088 0.336511
\(815\) −21.6053 −0.756801
\(816\) 0 0
\(817\) −53.4372 −1.86953
\(818\) −4.59868 −0.160789
\(819\) 0 0
\(820\) −14.4434 −0.504384
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 46.1725 1.60947 0.804737 0.593632i \(-0.202306\pi\)
0.804737 + 0.593632i \(0.202306\pi\)
\(824\) 5.80044 0.202068
\(825\) 0 0
\(826\) −9.28583 −0.323095
\(827\) −27.2018 −0.945898 −0.472949 0.881090i \(-0.656811\pi\)
−0.472949 + 0.881090i \(0.656811\pi\)
\(828\) 0 0
\(829\) −19.2588 −0.668886 −0.334443 0.942416i \(-0.608548\pi\)
−0.334443 + 0.942416i \(0.608548\pi\)
\(830\) −37.2460 −1.29283
\(831\) 0 0
\(832\) −6.44335 −0.223383
\(833\) 3.80044 0.131677
\(834\) 0 0
\(835\) −93.0637 −3.22060
\(836\) 6.64291 0.229750
\(837\) 0 0
\(838\) 21.7734 0.752150
\(839\) 28.5589 0.985961 0.492981 0.870040i \(-0.335907\pi\)
0.492981 + 0.870040i \(0.335907\pi\)
\(840\) 0 0
\(841\) 42.2902 1.45828
\(842\) 29.2018 1.00636
\(843\) 0 0
\(844\) 13.7292 0.472578
\(845\) 108.377 3.72827
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −1.60088 −0.0549745
\(849\) 0 0
\(850\) 35.8889 1.23098
\(851\) 8.08848 0.277269
\(852\) 0 0
\(853\) 10.2752 0.351817 0.175909 0.984406i \(-0.443714\pi\)
0.175909 + 0.984406i \(0.443714\pi\)
\(854\) 8.75841 0.299707
\(855\) 0 0
\(856\) −7.60088 −0.259793
\(857\) 57.1747 1.95305 0.976526 0.215399i \(-0.0691054\pi\)
0.976526 + 0.215399i \(0.0691054\pi\)
\(858\) 0 0
\(859\) 5.82746 0.198830 0.0994152 0.995046i \(-0.468303\pi\)
0.0994152 + 0.995046i \(0.468303\pi\)
\(860\) −30.5717 −1.04248
\(861\) 0 0
\(862\) 22.0442 0.750830
\(863\) −46.4478 −1.58110 −0.790550 0.612397i \(-0.790205\pi\)
−0.790550 + 0.612397i \(0.790205\pi\)
\(864\) 0 0
\(865\) −58.0929 −1.97522
\(866\) −16.4035 −0.557415
\(867\) 0 0
\(868\) −9.40132 −0.319102
\(869\) 2.84247 0.0964243
\(870\) 0 0
\(871\) −7.45836 −0.252717
\(872\) 8.00000 0.270914
\(873\) 0 0
\(874\) 5.59647 0.189303
\(875\) 16.8867 0.570875
\(876\) 0 0
\(877\) 11.2018 0.378257 0.189128 0.981952i \(-0.439434\pi\)
0.189128 + 0.981952i \(0.439434\pi\)
\(878\) 17.2858 0.583368
\(879\) 0 0
\(880\) 3.80044 0.128113
\(881\) −34.7000 −1.16907 −0.584536 0.811368i \(-0.698723\pi\)
−0.584536 + 0.811368i \(0.698723\pi\)
\(882\) 0 0
\(883\) 39.9204 1.34343 0.671713 0.740811i \(-0.265559\pi\)
0.671713 + 0.740811i \(0.265559\pi\)
\(884\) −24.4876 −0.823607
\(885\) 0 0
\(886\) −1.95576 −0.0657051
\(887\) −8.08848 −0.271584 −0.135792 0.990737i \(-0.543358\pi\)
−0.135792 + 0.990737i \(0.543358\pi\)
\(888\) 0 0
\(889\) 2.84247 0.0953335
\(890\) −19.6009 −0.657023
\(891\) 0 0
\(892\) −24.6031 −0.823772
\(893\) 41.4770 1.38797
\(894\) 0 0
\(895\) −95.9460 −3.20712
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 24.7584 0.826199
\(899\) 79.3787 2.64743
\(900\) 0 0
\(901\) −6.08406 −0.202689
\(902\) −3.80044 −0.126541
\(903\) 0 0
\(904\) 0.758411 0.0252244
\(905\) −29.6451 −0.985437
\(906\) 0 0
\(907\) −5.95576 −0.197758 −0.0988789 0.995099i \(-0.531526\pi\)
−0.0988789 + 0.995099i \(0.531526\pi\)
\(908\) −25.0022 −0.829727
\(909\) 0 0
\(910\) −24.4876 −0.811756
\(911\) −22.4478 −0.743728 −0.371864 0.928287i \(-0.621281\pi\)
−0.371864 + 0.928287i \(0.621281\pi\)
\(912\) 0 0
\(913\) −9.80044 −0.324347
\(914\) 22.0885 0.730622
\(915\) 0 0
\(916\) 16.6872 0.551359
\(917\) −14.1996 −0.468911
\(918\) 0 0
\(919\) 24.8867 0.820937 0.410468 0.911875i \(-0.365365\pi\)
0.410468 + 0.911875i \(0.365365\pi\)
\(920\) 3.20177 0.105559
\(921\) 0 0
\(922\) −39.7734 −1.30987
\(923\) 43.5468 1.43336
\(924\) 0 0
\(925\) 90.6645 2.98103
\(926\) −7.20177 −0.236665
\(927\) 0 0
\(928\) −8.44335 −0.277167
\(929\) 15.4726 0.507639 0.253820 0.967252i \(-0.418313\pi\)
0.253820 + 0.967252i \(0.418313\pi\)
\(930\) 0 0
\(931\) 6.64291 0.217713
\(932\) −27.7734 −0.909749
\(933\) 0 0
\(934\) 1.68494 0.0551331
\(935\) 14.4434 0.472348
\(936\) 0 0
\(937\) 17.8845 0.584261 0.292131 0.956378i \(-0.405636\pi\)
0.292131 + 0.956378i \(0.405636\pi\)
\(938\) 1.15753 0.0377946
\(939\) 0 0
\(940\) 23.7292 0.773961
\(941\) −6.65572 −0.216970 −0.108485 0.994098i \(-0.534600\pi\)
−0.108485 + 0.994098i \(0.534600\pi\)
\(942\) 0 0
\(943\) −3.20177 −0.104264
\(944\) −9.28583 −0.302228
\(945\) 0 0
\(946\) −8.04424 −0.261541
\(947\) −32.2566 −1.04820 −0.524099 0.851657i \(-0.675598\pi\)
−0.524099 + 0.851657i \(0.675598\pi\)
\(948\) 0 0
\(949\) 50.2610 1.63154
\(950\) 62.7314 2.03528
\(951\) 0 0
\(952\) 3.80044 0.123173
\(953\) 37.6009 1.21801 0.609006 0.793166i \(-0.291569\pi\)
0.609006 + 0.793166i \(0.291569\pi\)
\(954\) 0 0
\(955\) 79.5468 2.57408
\(956\) 1.68494 0.0544950
\(957\) 0 0
\(958\) −27.2018 −0.878849
\(959\) 12.7584 0.411991
\(960\) 0 0
\(961\) 57.3849 1.85113
\(962\) −61.8619 −1.99451
\(963\) 0 0
\(964\) −15.4013 −0.496043
\(965\) 19.7690 0.636387
\(966\) 0 0
\(967\) −2.48318 −0.0798536 −0.0399268 0.999203i \(-0.512712\pi\)
−0.0399268 + 0.999203i \(0.512712\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 71.7778 2.30465
\(971\) 13.6849 0.439171 0.219585 0.975593i \(-0.429530\pi\)
0.219585 + 0.975593i \(0.429530\pi\)
\(972\) 0 0
\(973\) −8.55885 −0.274384
\(974\) 13.9159 0.445895
\(975\) 0 0
\(976\) 8.75841 0.280350
\(977\) −13.6849 −0.437820 −0.218910 0.975745i \(-0.570250\pi\)
−0.218910 + 0.975745i \(0.570250\pi\)
\(978\) 0 0
\(979\) −5.15753 −0.164835
\(980\) 3.80044 0.121401
\(981\) 0 0
\(982\) 3.20177 0.102172
\(983\) 31.2190 0.995731 0.497865 0.867254i \(-0.334117\pi\)
0.497865 + 0.867254i \(0.334117\pi\)
\(984\) 0 0
\(985\) 50.6601 1.61417
\(986\) −32.0885 −1.02191
\(987\) 0 0
\(988\) −42.8026 −1.36173
\(989\) −6.77705 −0.215498
\(990\) 0 0
\(991\) 9.51682 0.302312 0.151156 0.988510i \(-0.451700\pi\)
0.151156 + 0.988510i \(0.451700\pi\)
\(992\) −9.40132 −0.298492
\(993\) 0 0
\(994\) −6.75841 −0.214364
\(995\) −52.4478 −1.66271
\(996\) 0 0
\(997\) −17.9558 −0.568665 −0.284332 0.958726i \(-0.591772\pi\)
−0.284332 + 0.958726i \(0.591772\pi\)
\(998\) −17.2460 −0.545913
\(999\) 0 0
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 1386.2.a.r.1.3 yes 3
3.2 odd 2 1386.2.a.q.1.1 3
7.6 odd 2 9702.2.a.dy.1.1 3
21.20 even 2 9702.2.a.dx.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1386.2.a.q.1.1 3 3.2 odd 2
1386.2.a.r.1.3 yes 3 1.1 even 1 trivial
9702.2.a.dx.1.3 3 21.20 even 2
9702.2.a.dy.1.1 3 7.6 odd 2