Properties

Label 950.2.a.n.1.3
Level $950$
Weight $2$
Character 950.1
Self dual yes
Analytic conductor $7.586$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 950.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.76156 q^{3} +1.00000 q^{4} +2.76156 q^{6} -0.761557 q^{7} +1.00000 q^{8} +4.62620 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.76156 q^{3} +1.00000 q^{4} +2.76156 q^{6} -0.761557 q^{7} +1.00000 q^{8} +4.62620 q^{9} -0.864641 q^{11} +2.76156 q^{12} +5.62620 q^{13} -0.761557 q^{14} +1.00000 q^{16} -3.62620 q^{17} +4.62620 q^{18} +1.00000 q^{19} -2.10308 q^{21} -0.864641 q^{22} -8.01395 q^{23} +2.76156 q^{24} +5.62620 q^{26} +4.49084 q^{27} -0.761557 q^{28} -7.35548 q^{29} +8.11704 q^{31} +1.00000 q^{32} -2.38776 q^{33} -3.62620 q^{34} +4.62620 q^{36} +0.476886 q^{37} +1.00000 q^{38} +15.5371 q^{39} -2.65847 q^{41} -2.10308 q^{42} +6.86464 q^{43} -0.864641 q^{44} -8.01395 q^{46} +1.25240 q^{47} +2.76156 q^{48} -6.42003 q^{49} -10.0140 q^{51} +5.62620 q^{52} +2.37380 q^{53} +4.49084 q^{54} -0.761557 q^{56} +2.76156 q^{57} -7.35548 q^{58} +4.49084 q^{59} -10.8646 q^{61} +8.11704 q^{62} -3.52311 q^{63} +1.00000 q^{64} -2.38776 q^{66} +1.03228 q^{67} -3.62620 q^{68} -22.1310 q^{69} -10.1816 q^{71} +4.62620 q^{72} +16.4017 q^{73} +0.476886 q^{74} +1.00000 q^{76} +0.658473 q^{77} +15.5371 q^{78} -12.5693 q^{79} -1.47689 q^{81} -2.65847 q^{82} -0.270718 q^{83} -2.10308 q^{84} +6.86464 q^{86} -20.3126 q^{87} -0.864641 q^{88} +0.387755 q^{89} -4.28467 q^{91} -8.01395 q^{92} +22.4157 q^{93} +1.25240 q^{94} +2.76156 q^{96} -8.50479 q^{97} -6.42003 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + 4q^{7} + 3q^{8} + 5q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + 4q^{7} + 3q^{8} + 5q^{9} + 2q^{12} + 8q^{13} + 4q^{14} + 3q^{16} - 2q^{17} + 5q^{18} + 3q^{19} - 10q^{21} + 2q^{24} + 8q^{26} + 2q^{27} + 4q^{28} - 8q^{29} + 4q^{31} + 3q^{32} + 8q^{33} - 2q^{34} + 5q^{36} + 14q^{37} + 3q^{38} + 10q^{39} + 2q^{41} - 10q^{42} + 18q^{43} - 14q^{47} + 2q^{48} - 3q^{49} - 6q^{51} + 8q^{52} + 16q^{53} + 2q^{54} + 4q^{56} + 2q^{57} - 8q^{58} + 2q^{59} - 30q^{61} + 4q^{62} + 2q^{63} + 3q^{64} + 8q^{66} + 2q^{67} - 2q^{68} - 22q^{69} - 8q^{71} + 5q^{72} + 10q^{73} + 14q^{74} + 3q^{76} - 8q^{77} + 10q^{78} - 17q^{81} + 2q^{82} - 6q^{83} - 10q^{84} + 18q^{86} + 6q^{87} - 14q^{89} + 6q^{91} + 4q^{93} - 14q^{94} + 2q^{96} + 10q^{97} - 3q^{98} - 12q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.76156 1.59439 0.797193 0.603725i \(-0.206317\pi\)
0.797193 + 0.603725i \(0.206317\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.76156 1.12740
\(7\) −0.761557 −0.287842 −0.143921 0.989589i \(-0.545971\pi\)
−0.143921 + 0.989589i \(0.545971\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.62620 1.54207
\(10\) 0 0
\(11\) −0.864641 −0.260699 −0.130350 0.991468i \(-0.541610\pi\)
−0.130350 + 0.991468i \(0.541610\pi\)
\(12\) 2.76156 0.797193
\(13\) 5.62620 1.56043 0.780213 0.625514i \(-0.215111\pi\)
0.780213 + 0.625514i \(0.215111\pi\)
\(14\) −0.761557 −0.203535
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.62620 −0.879482 −0.439741 0.898125i \(-0.644930\pi\)
−0.439741 + 0.898125i \(0.644930\pi\)
\(18\) 4.62620 1.09041
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −2.10308 −0.458930
\(22\) −0.864641 −0.184342
\(23\) −8.01395 −1.67102 −0.835512 0.549472i \(-0.814829\pi\)
−0.835512 + 0.549472i \(0.814829\pi\)
\(24\) 2.76156 0.563700
\(25\) 0 0
\(26\) 5.62620 1.10339
\(27\) 4.49084 0.864262
\(28\) −0.761557 −0.143921
\(29\) −7.35548 −1.36588 −0.682939 0.730475i \(-0.739299\pi\)
−0.682939 + 0.730475i \(0.739299\pi\)
\(30\) 0 0
\(31\) 8.11704 1.45786 0.728931 0.684587i \(-0.240017\pi\)
0.728931 + 0.684587i \(0.240017\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.38776 −0.415655
\(34\) −3.62620 −0.621888
\(35\) 0 0
\(36\) 4.62620 0.771033
\(37\) 0.476886 0.0783995 0.0391998 0.999231i \(-0.487519\pi\)
0.0391998 + 0.999231i \(0.487519\pi\)
\(38\) 1.00000 0.162221
\(39\) 15.5371 2.48792
\(40\) 0 0
\(41\) −2.65847 −0.415184 −0.207592 0.978216i \(-0.566563\pi\)
−0.207592 + 0.978216i \(0.566563\pi\)
\(42\) −2.10308 −0.324513
\(43\) 6.86464 1.04685 0.523424 0.852072i \(-0.324654\pi\)
0.523424 + 0.852072i \(0.324654\pi\)
\(44\) −0.864641 −0.130350
\(45\) 0 0
\(46\) −8.01395 −1.18159
\(47\) 1.25240 0.182681 0.0913404 0.995820i \(-0.470885\pi\)
0.0913404 + 0.995820i \(0.470885\pi\)
\(48\) 2.76156 0.398596
\(49\) −6.42003 −0.917147
\(50\) 0 0
\(51\) −10.0140 −1.40223
\(52\) 5.62620 0.780213
\(53\) 2.37380 0.326067 0.163033 0.986621i \(-0.447872\pi\)
0.163033 + 0.986621i \(0.447872\pi\)
\(54\) 4.49084 0.611126
\(55\) 0 0
\(56\) −0.761557 −0.101767
\(57\) 2.76156 0.365777
\(58\) −7.35548 −0.965822
\(59\) 4.49084 0.584657 0.292329 0.956318i \(-0.405570\pi\)
0.292329 + 0.956318i \(0.405570\pi\)
\(60\) 0 0
\(61\) −10.8646 −1.39107 −0.695537 0.718490i \(-0.744834\pi\)
−0.695537 + 0.718490i \(0.744834\pi\)
\(62\) 8.11704 1.03086
\(63\) −3.52311 −0.443871
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.38776 −0.293912
\(67\) 1.03228 0.126113 0.0630563 0.998010i \(-0.479915\pi\)
0.0630563 + 0.998010i \(0.479915\pi\)
\(68\) −3.62620 −0.439741
\(69\) −22.1310 −2.66426
\(70\) 0 0
\(71\) −10.1816 −1.20833 −0.604166 0.796858i \(-0.706494\pi\)
−0.604166 + 0.796858i \(0.706494\pi\)
\(72\) 4.62620 0.545203
\(73\) 16.4017 1.91967 0.959837 0.280557i \(-0.0905191\pi\)
0.959837 + 0.280557i \(0.0905191\pi\)
\(74\) 0.476886 0.0554368
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0.658473 0.0750400
\(78\) 15.5371 1.75923
\(79\) −12.5693 −1.41416 −0.707081 0.707133i \(-0.749988\pi\)
−0.707081 + 0.707133i \(0.749988\pi\)
\(80\) 0 0
\(81\) −1.47689 −0.164098
\(82\) −2.65847 −0.293579
\(83\) −0.270718 −0.0297152 −0.0148576 0.999890i \(-0.504729\pi\)
−0.0148576 + 0.999890i \(0.504729\pi\)
\(84\) −2.10308 −0.229465
\(85\) 0 0
\(86\) 6.86464 0.740233
\(87\) −20.3126 −2.17774
\(88\) −0.864641 −0.0921710
\(89\) 0.387755 0.0411020 0.0205510 0.999789i \(-0.493458\pi\)
0.0205510 + 0.999789i \(0.493458\pi\)
\(90\) 0 0
\(91\) −4.28467 −0.449156
\(92\) −8.01395 −0.835512
\(93\) 22.4157 2.32440
\(94\) 1.25240 0.129175
\(95\) 0 0
\(96\) 2.76156 0.281850
\(97\) −8.50479 −0.863531 −0.431765 0.901986i \(-0.642109\pi\)
−0.431765 + 0.901986i \(0.642109\pi\)
\(98\) −6.42003 −0.648521
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 16.4157 1.63342 0.816710 0.577049i \(-0.195796\pi\)
0.816710 + 0.577049i \(0.195796\pi\)
\(102\) −10.0140 −0.991529
\(103\) −9.64015 −0.949872 −0.474936 0.880020i \(-0.657529\pi\)
−0.474936 + 0.880020i \(0.657529\pi\)
\(104\) 5.62620 0.551694
\(105\) 0 0
\(106\) 2.37380 0.230564
\(107\) 4.28467 0.414215 0.207107 0.978318i \(-0.433595\pi\)
0.207107 + 0.978318i \(0.433595\pi\)
\(108\) 4.49084 0.432131
\(109\) −13.4200 −1.28541 −0.642703 0.766116i \(-0.722187\pi\)
−0.642703 + 0.766116i \(0.722187\pi\)
\(110\) 0 0
\(111\) 1.31695 0.124999
\(112\) −0.761557 −0.0719604
\(113\) −10.3232 −0.971125 −0.485563 0.874202i \(-0.661385\pi\)
−0.485563 + 0.874202i \(0.661385\pi\)
\(114\) 2.76156 0.258644
\(115\) 0 0
\(116\) −7.35548 −0.682939
\(117\) 26.0279 2.40628
\(118\) 4.49084 0.413415
\(119\) 2.76156 0.253152
\(120\) 0 0
\(121\) −10.2524 −0.932036
\(122\) −10.8646 −0.983638
\(123\) −7.34153 −0.661963
\(124\) 8.11704 0.728931
\(125\) 0 0
\(126\) −3.52311 −0.313864
\(127\) 16.9817 1.50688 0.753440 0.657517i \(-0.228393\pi\)
0.753440 + 0.657517i \(0.228393\pi\)
\(128\) 1.00000 0.0883883
\(129\) 18.9571 1.66908
\(130\) 0 0
\(131\) 0.541436 0.0473055 0.0236528 0.999720i \(-0.492470\pi\)
0.0236528 + 0.999720i \(0.492470\pi\)
\(132\) −2.38776 −0.207827
\(133\) −0.761557 −0.0660354
\(134\) 1.03228 0.0891750
\(135\) 0 0
\(136\) −3.62620 −0.310944
\(137\) 2.87859 0.245935 0.122967 0.992411i \(-0.460759\pi\)
0.122967 + 0.992411i \(0.460759\pi\)
\(138\) −22.1310 −1.88392
\(139\) 3.58767 0.304302 0.152151 0.988357i \(-0.451380\pi\)
0.152151 + 0.988357i \(0.451380\pi\)
\(140\) 0 0
\(141\) 3.45856 0.291264
\(142\) −10.1816 −0.854420
\(143\) −4.86464 −0.406802
\(144\) 4.62620 0.385517
\(145\) 0 0
\(146\) 16.4017 1.35742
\(147\) −17.7293 −1.46229
\(148\) 0.476886 0.0391998
\(149\) −16.8401 −1.37959 −0.689796 0.724004i \(-0.742300\pi\)
−0.689796 + 0.724004i \(0.742300\pi\)
\(150\) 0 0
\(151\) −16.9817 −1.38195 −0.690975 0.722879i \(-0.742818\pi\)
−0.690975 + 0.722879i \(0.742818\pi\)
\(152\) 1.00000 0.0811107
\(153\) −16.7755 −1.35622
\(154\) 0.658473 0.0530613
\(155\) 0 0
\(156\) 15.5371 1.24396
\(157\) 14.8401 1.18437 0.592183 0.805804i \(-0.298266\pi\)
0.592183 + 0.805804i \(0.298266\pi\)
\(158\) −12.5693 −0.999963
\(159\) 6.55539 0.519876
\(160\) 0 0
\(161\) 6.10308 0.480990
\(162\) −1.47689 −0.116035
\(163\) 13.3694 1.04717 0.523587 0.851972i \(-0.324593\pi\)
0.523587 + 0.851972i \(0.324593\pi\)
\(164\) −2.65847 −0.207592
\(165\) 0 0
\(166\) −0.270718 −0.0210118
\(167\) 9.84632 0.761931 0.380966 0.924589i \(-0.375592\pi\)
0.380966 + 0.924589i \(0.375592\pi\)
\(168\) −2.10308 −0.162256
\(169\) 18.6541 1.43493
\(170\) 0 0
\(171\) 4.62620 0.353774
\(172\) 6.86464 0.523424
\(173\) 2.98168 0.226693 0.113346 0.993556i \(-0.463843\pi\)
0.113346 + 0.993556i \(0.463843\pi\)
\(174\) −20.3126 −1.53989
\(175\) 0 0
\(176\) −0.864641 −0.0651748
\(177\) 12.4017 0.932169
\(178\) 0.387755 0.0290635
\(179\) 11.7938 0.881512 0.440756 0.897627i \(-0.354710\pi\)
0.440756 + 0.897627i \(0.354710\pi\)
\(180\) 0 0
\(181\) 14.5693 1.08293 0.541465 0.840723i \(-0.317870\pi\)
0.541465 + 0.840723i \(0.317870\pi\)
\(182\) −4.28467 −0.317601
\(183\) −30.0033 −2.21791
\(184\) −8.01395 −0.590796
\(185\) 0 0
\(186\) 22.4157 1.64360
\(187\) 3.13536 0.229280
\(188\) 1.25240 0.0913404
\(189\) −3.42003 −0.248771
\(190\) 0 0
\(191\) 13.2384 0.957900 0.478950 0.877842i \(-0.341017\pi\)
0.478950 + 0.877842i \(0.341017\pi\)
\(192\) 2.76156 0.199298
\(193\) 2.54144 0.182937 0.0914683 0.995808i \(-0.470844\pi\)
0.0914683 + 0.995808i \(0.470844\pi\)
\(194\) −8.50479 −0.610609
\(195\) 0 0
\(196\) −6.42003 −0.458574
\(197\) −19.9109 −1.41859 −0.709295 0.704911i \(-0.750987\pi\)
−0.709295 + 0.704911i \(0.750987\pi\)
\(198\) −4.00000 −0.284268
\(199\) −20.3126 −1.43992 −0.719960 0.694015i \(-0.755840\pi\)
−0.719960 + 0.694015i \(0.755840\pi\)
\(200\) 0 0
\(201\) 2.85069 0.201072
\(202\) 16.4157 1.15500
\(203\) 5.60162 0.393157
\(204\) −10.0140 −0.701117
\(205\) 0 0
\(206\) −9.64015 −0.671661
\(207\) −37.0741 −2.57683
\(208\) 5.62620 0.390107
\(209\) −0.864641 −0.0598085
\(210\) 0 0
\(211\) 18.0419 1.24205 0.621026 0.783790i \(-0.286716\pi\)
0.621026 + 0.783790i \(0.286716\pi\)
\(212\) 2.37380 0.163033
\(213\) −28.1170 −1.92655
\(214\) 4.28467 0.292894
\(215\) 0 0
\(216\) 4.49084 0.305563
\(217\) −6.18159 −0.419634
\(218\) −13.4200 −0.908919
\(219\) 45.2943 3.06070
\(220\) 0 0
\(221\) −20.4017 −1.37237
\(222\) 1.31695 0.0883877
\(223\) −13.5231 −0.905575 −0.452787 0.891619i \(-0.649570\pi\)
−0.452787 + 0.891619i \(0.649570\pi\)
\(224\) −0.761557 −0.0508837
\(225\) 0 0
\(226\) −10.3232 −0.686689
\(227\) 13.6016 0.902771 0.451386 0.892329i \(-0.350930\pi\)
0.451386 + 0.892329i \(0.350930\pi\)
\(228\) 2.76156 0.182889
\(229\) 13.5877 0.897898 0.448949 0.893557i \(-0.351798\pi\)
0.448949 + 0.893557i \(0.351798\pi\)
\(230\) 0 0
\(231\) 1.81841 0.119643
\(232\) −7.35548 −0.482911
\(233\) 25.5510 1.67390 0.836952 0.547277i \(-0.184336\pi\)
0.836952 + 0.547277i \(0.184336\pi\)
\(234\) 26.0279 1.70150
\(235\) 0 0
\(236\) 4.49084 0.292329
\(237\) −34.7110 −2.25472
\(238\) 2.76156 0.179005
\(239\) −11.3309 −0.732935 −0.366468 0.930431i \(-0.619433\pi\)
−0.366468 + 0.930431i \(0.619433\pi\)
\(240\) 0 0
\(241\) −1.25240 −0.0806739 −0.0403370 0.999186i \(-0.512843\pi\)
−0.0403370 + 0.999186i \(0.512843\pi\)
\(242\) −10.2524 −0.659049
\(243\) −17.5510 −1.12590
\(244\) −10.8646 −0.695537
\(245\) 0 0
\(246\) −7.34153 −0.468079
\(247\) 5.62620 0.357986
\(248\) 8.11704 0.515432
\(249\) −0.747604 −0.0473775
\(250\) 0 0
\(251\) 10.5939 0.668682 0.334341 0.942452i \(-0.391486\pi\)
0.334341 + 0.942452i \(0.391486\pi\)
\(252\) −3.52311 −0.221935
\(253\) 6.92919 0.435635
\(254\) 16.9817 1.06553
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −0.153681 −0.00958637 −0.00479319 0.999989i \(-0.501526\pi\)
−0.00479319 + 0.999989i \(0.501526\pi\)
\(258\) 18.9571 1.18022
\(259\) −0.363176 −0.0225666
\(260\) 0 0
\(261\) −34.0279 −2.10627
\(262\) 0.541436 0.0334501
\(263\) 0.504792 0.0311268 0.0155634 0.999879i \(-0.495046\pi\)
0.0155634 + 0.999879i \(0.495046\pi\)
\(264\) −2.38776 −0.146956
\(265\) 0 0
\(266\) −0.761557 −0.0466941
\(267\) 1.07081 0.0655324
\(268\) 1.03228 0.0630563
\(269\) −3.49521 −0.213107 −0.106553 0.994307i \(-0.533981\pi\)
−0.106553 + 0.994307i \(0.533981\pi\)
\(270\) 0 0
\(271\) 5.47252 0.332432 0.166216 0.986089i \(-0.446845\pi\)
0.166216 + 0.986089i \(0.446845\pi\)
\(272\) −3.62620 −0.219871
\(273\) −11.8324 −0.716127
\(274\) 2.87859 0.173902
\(275\) 0 0
\(276\) −22.1310 −1.33213
\(277\) 12.9538 0.778317 0.389158 0.921171i \(-0.372766\pi\)
0.389158 + 0.921171i \(0.372766\pi\)
\(278\) 3.58767 0.215174
\(279\) 37.5510 2.24812
\(280\) 0 0
\(281\) 0.153681 0.00916785 0.00458393 0.999989i \(-0.498541\pi\)
0.00458393 + 0.999989i \(0.498541\pi\)
\(282\) 3.45856 0.205954
\(283\) −18.2341 −1.08390 −0.541952 0.840410i \(-0.682314\pi\)
−0.541952 + 0.840410i \(0.682314\pi\)
\(284\) −10.1816 −0.604166
\(285\) 0 0
\(286\) −4.86464 −0.287652
\(287\) 2.02458 0.119507
\(288\) 4.62620 0.272601
\(289\) −3.85069 −0.226511
\(290\) 0 0
\(291\) −23.4865 −1.37680
\(292\) 16.4017 0.959837
\(293\) 2.03853 0.119092 0.0595462 0.998226i \(-0.481035\pi\)
0.0595462 + 0.998226i \(0.481035\pi\)
\(294\) −17.7293 −1.03399
\(295\) 0 0
\(296\) 0.476886 0.0277184
\(297\) −3.88296 −0.225312
\(298\) −16.8401 −0.975519
\(299\) −45.0881 −2.60751
\(300\) 0 0
\(301\) −5.22782 −0.301326
\(302\) −16.9817 −0.977186
\(303\) 45.3328 2.60430
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −16.7755 −0.958992
\(307\) 16.5414 0.944070 0.472035 0.881580i \(-0.343520\pi\)
0.472035 + 0.881580i \(0.343520\pi\)
\(308\) 0.658473 0.0375200
\(309\) −26.6218 −1.51446
\(310\) 0 0
\(311\) −21.4725 −1.21759 −0.608797 0.793326i \(-0.708348\pi\)
−0.608797 + 0.793326i \(0.708348\pi\)
\(312\) 15.5371 0.879613
\(313\) −1.12141 −0.0633856 −0.0316928 0.999498i \(-0.510090\pi\)
−0.0316928 + 0.999498i \(0.510090\pi\)
\(314\) 14.8401 0.837473
\(315\) 0 0
\(316\) −12.5693 −0.707081
\(317\) −29.8882 −1.67869 −0.839344 0.543601i \(-0.817060\pi\)
−0.839344 + 0.543601i \(0.817060\pi\)
\(318\) 6.55539 0.367608
\(319\) 6.35985 0.356083
\(320\) 0 0
\(321\) 11.8324 0.660418
\(322\) 6.10308 0.340112
\(323\) −3.62620 −0.201767
\(324\) −1.47689 −0.0820492
\(325\) 0 0
\(326\) 13.3694 0.740464
\(327\) −37.0602 −2.04943
\(328\) −2.65847 −0.146790
\(329\) −0.953771 −0.0525831
\(330\) 0 0
\(331\) 32.3126 1.77606 0.888030 0.459786i \(-0.152074\pi\)
0.888030 + 0.459786i \(0.152074\pi\)
\(332\) −0.270718 −0.0148576
\(333\) 2.20617 0.120897
\(334\) 9.84632 0.538767
\(335\) 0 0
\(336\) −2.10308 −0.114733
\(337\) 26.3511 1.43544 0.717718 0.696334i \(-0.245187\pi\)
0.717718 + 0.696334i \(0.245187\pi\)
\(338\) 18.6541 1.01465
\(339\) −28.5081 −1.54835
\(340\) 0 0
\(341\) −7.01832 −0.380063
\(342\) 4.62620 0.250156
\(343\) 10.2201 0.551835
\(344\) 6.86464 0.370117
\(345\) 0 0
\(346\) 2.98168 0.160296
\(347\) −2.77551 −0.148997 −0.0744986 0.997221i \(-0.523736\pi\)
−0.0744986 + 0.997221i \(0.523736\pi\)
\(348\) −20.3126 −1.08887
\(349\) 11.5510 0.618312 0.309156 0.951011i \(-0.399953\pi\)
0.309156 + 0.951011i \(0.399953\pi\)
\(350\) 0 0
\(351\) 25.2663 1.34862
\(352\) −0.864641 −0.0460855
\(353\) −8.40171 −0.447178 −0.223589 0.974684i \(-0.571777\pi\)
−0.223589 + 0.974684i \(0.571777\pi\)
\(354\) 12.4017 0.659143
\(355\) 0 0
\(356\) 0.387755 0.0205510
\(357\) 7.62620 0.403621
\(358\) 11.7938 0.623323
\(359\) 22.7895 1.20278 0.601391 0.798955i \(-0.294613\pi\)
0.601391 + 0.798955i \(0.294613\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 14.5693 0.765748
\(363\) −28.3126 −1.48602
\(364\) −4.28467 −0.224578
\(365\) 0 0
\(366\) −30.0033 −1.56830
\(367\) 4.06455 0.212168 0.106084 0.994357i \(-0.466169\pi\)
0.106084 + 0.994357i \(0.466169\pi\)
\(368\) −8.01395 −0.417756
\(369\) −12.2986 −0.640241
\(370\) 0 0
\(371\) −1.80779 −0.0938556
\(372\) 22.4157 1.16220
\(373\) −18.4017 −0.952804 −0.476402 0.879227i \(-0.658059\pi\)
−0.476402 + 0.879227i \(0.658059\pi\)
\(374\) 3.13536 0.162126
\(375\) 0 0
\(376\) 1.25240 0.0645874
\(377\) −41.3834 −2.13135
\(378\) −3.42003 −0.175907
\(379\) 1.23844 0.0636145 0.0318073 0.999494i \(-0.489874\pi\)
0.0318073 + 0.999494i \(0.489874\pi\)
\(380\) 0 0
\(381\) 46.8959 2.40255
\(382\) 13.2384 0.677338
\(383\) 16.8646 0.861743 0.430871 0.902413i \(-0.358206\pi\)
0.430871 + 0.902413i \(0.358206\pi\)
\(384\) 2.76156 0.140925
\(385\) 0 0
\(386\) 2.54144 0.129356
\(387\) 31.7572 1.61431
\(388\) −8.50479 −0.431765
\(389\) 8.59392 0.435729 0.217865 0.975979i \(-0.430091\pi\)
0.217865 + 0.975979i \(0.430091\pi\)
\(390\) 0 0
\(391\) 29.0602 1.46964
\(392\) −6.42003 −0.324261
\(393\) 1.49521 0.0754233
\(394\) −19.9109 −1.00310
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 16.0558 0.805818 0.402909 0.915240i \(-0.367999\pi\)
0.402909 + 0.915240i \(0.367999\pi\)
\(398\) −20.3126 −1.01818
\(399\) −2.10308 −0.105286
\(400\) 0 0
\(401\) −14.8925 −0.743698 −0.371849 0.928293i \(-0.621276\pi\)
−0.371849 + 0.928293i \(0.621276\pi\)
\(402\) 2.85069 0.142179
\(403\) 45.6681 2.27489
\(404\) 16.4157 0.816710
\(405\) 0 0
\(406\) 5.60162 0.278004
\(407\) −0.412335 −0.0204387
\(408\) −10.0140 −0.495765
\(409\) −18.3511 −0.907404 −0.453702 0.891153i \(-0.649897\pi\)
−0.453702 + 0.891153i \(0.649897\pi\)
\(410\) 0 0
\(411\) 7.94940 0.392115
\(412\) −9.64015 −0.474936
\(413\) −3.42003 −0.168289
\(414\) −37.0741 −1.82209
\(415\) 0 0
\(416\) 5.62620 0.275847
\(417\) 9.90754 0.485174
\(418\) −0.864641 −0.0422910
\(419\) 34.7509 1.69769 0.848847 0.528639i \(-0.177297\pi\)
0.848847 + 0.528639i \(0.177297\pi\)
\(420\) 0 0
\(421\) −40.1589 −1.95722 −0.978612 0.205713i \(-0.934049\pi\)
−0.978612 + 0.205713i \(0.934049\pi\)
\(422\) 18.0419 0.878264
\(423\) 5.79383 0.281706
\(424\) 2.37380 0.115282
\(425\) 0 0
\(426\) −28.1170 −1.36227
\(427\) 8.27405 0.400409
\(428\) 4.28467 0.207107
\(429\) −13.4340 −0.648599
\(430\) 0 0
\(431\) 34.9571 1.68382 0.841912 0.539615i \(-0.181430\pi\)
0.841912 + 0.539615i \(0.181430\pi\)
\(432\) 4.49084 0.216066
\(433\) 1.13536 0.0545619 0.0272809 0.999628i \(-0.491315\pi\)
0.0272809 + 0.999628i \(0.491315\pi\)
\(434\) −6.18159 −0.296726
\(435\) 0 0
\(436\) −13.4200 −0.642703
\(437\) −8.01395 −0.383359
\(438\) 45.2943 2.16424
\(439\) −6.80009 −0.324551 −0.162275 0.986746i \(-0.551883\pi\)
−0.162275 + 0.986746i \(0.551883\pi\)
\(440\) 0 0
\(441\) −29.7003 −1.41430
\(442\) −20.4017 −0.970410
\(443\) 38.0679 1.80866 0.904330 0.426835i \(-0.140371\pi\)
0.904330 + 0.426835i \(0.140371\pi\)
\(444\) 1.31695 0.0624995
\(445\) 0 0
\(446\) −13.5231 −0.640338
\(447\) −46.5048 −2.19960
\(448\) −0.761557 −0.0359802
\(449\) −18.5414 −0.875024 −0.437512 0.899212i \(-0.644140\pi\)
−0.437512 + 0.899212i \(0.644140\pi\)
\(450\) 0 0
\(451\) 2.29862 0.108238
\(452\) −10.3232 −0.485563
\(453\) −46.8959 −2.20336
\(454\) 13.6016 0.638356
\(455\) 0 0
\(456\) 2.76156 0.129322
\(457\) 16.3738 0.765934 0.382967 0.923762i \(-0.374902\pi\)
0.382967 + 0.923762i \(0.374902\pi\)
\(458\) 13.5877 0.634910
\(459\) −16.2847 −0.760103
\(460\) 0 0
\(461\) 1.70470 0.0793959 0.0396979 0.999212i \(-0.487360\pi\)
0.0396979 + 0.999212i \(0.487360\pi\)
\(462\) 1.81841 0.0846002
\(463\) −10.0279 −0.466036 −0.233018 0.972472i \(-0.574860\pi\)
−0.233018 + 0.972472i \(0.574860\pi\)
\(464\) −7.35548 −0.341470
\(465\) 0 0
\(466\) 25.5510 1.18363
\(467\) 32.7509 1.51553 0.757766 0.652526i \(-0.226291\pi\)
0.757766 + 0.652526i \(0.226291\pi\)
\(468\) 26.0279 1.20314
\(469\) −0.786137 −0.0363004
\(470\) 0 0
\(471\) 40.9817 1.88834
\(472\) 4.49084 0.206708
\(473\) −5.93545 −0.272912
\(474\) −34.7110 −1.59433
\(475\) 0 0
\(476\) 2.76156 0.126576
\(477\) 10.9817 0.502816
\(478\) −11.3309 −0.518263
\(479\) 27.2803 1.24647 0.623234 0.782035i \(-0.285818\pi\)
0.623234 + 0.782035i \(0.285818\pi\)
\(480\) 0 0
\(481\) 2.68305 0.122337
\(482\) −1.25240 −0.0570451
\(483\) 16.8540 0.766884
\(484\) −10.2524 −0.466018
\(485\) 0 0
\(486\) −17.5510 −0.796130
\(487\) −11.0741 −0.501817 −0.250908 0.968011i \(-0.580729\pi\)
−0.250908 + 0.968011i \(0.580729\pi\)
\(488\) −10.8646 −0.491819
\(489\) 36.9205 1.66960
\(490\) 0 0
\(491\) 8.11704 0.366317 0.183158 0.983083i \(-0.441368\pi\)
0.183158 + 0.983083i \(0.441368\pi\)
\(492\) −7.34153 −0.330982
\(493\) 26.6724 1.20127
\(494\) 5.62620 0.253135
\(495\) 0 0
\(496\) 8.11704 0.364466
\(497\) 7.75386 0.347808
\(498\) −0.747604 −0.0335009
\(499\) 0.295298 0.0132193 0.00660967 0.999978i \(-0.497896\pi\)
0.00660967 + 0.999978i \(0.497896\pi\)
\(500\) 0 0
\(501\) 27.1912 1.21481
\(502\) 10.5939 0.472830
\(503\) 19.6016 0.873993 0.436996 0.899463i \(-0.356042\pi\)
0.436996 + 0.899463i \(0.356042\pi\)
\(504\) −3.52311 −0.156932
\(505\) 0 0
\(506\) 6.92919 0.308040
\(507\) 51.5144 2.28783
\(508\) 16.9817 0.753440
\(509\) −1.79383 −0.0795102 −0.0397551 0.999209i \(-0.512658\pi\)
−0.0397551 + 0.999209i \(0.512658\pi\)
\(510\) 0 0
\(511\) −12.4908 −0.552562
\(512\) 1.00000 0.0441942
\(513\) 4.49084 0.198275
\(514\) −0.153681 −0.00677859
\(515\) 0 0
\(516\) 18.9571 0.834540
\(517\) −1.08287 −0.0476247
\(518\) −0.363176 −0.0159570
\(519\) 8.23407 0.361436
\(520\) 0 0
\(521\) −2.61850 −0.114719 −0.0573593 0.998354i \(-0.518268\pi\)
−0.0573593 + 0.998354i \(0.518268\pi\)
\(522\) −34.0279 −1.48936
\(523\) 14.9956 0.655713 0.327857 0.944728i \(-0.393674\pi\)
0.327857 + 0.944728i \(0.393674\pi\)
\(524\) 0.541436 0.0236528
\(525\) 0 0
\(526\) 0.504792 0.0220100
\(527\) −29.4340 −1.28216
\(528\) −2.38776 −0.103914
\(529\) 41.2234 1.79232
\(530\) 0 0
\(531\) 20.7755 0.901580
\(532\) −0.761557 −0.0330177
\(533\) −14.9571 −0.647864
\(534\) 1.07081 0.0463384
\(535\) 0 0
\(536\) 1.03228 0.0445875
\(537\) 32.5693 1.40547
\(538\) −3.49521 −0.150689
\(539\) 5.55102 0.239099
\(540\) 0 0
\(541\) 3.40608 0.146439 0.0732194 0.997316i \(-0.476673\pi\)
0.0732194 + 0.997316i \(0.476673\pi\)
\(542\) 5.47252 0.235065
\(543\) 40.2341 1.72661
\(544\) −3.62620 −0.155472
\(545\) 0 0
\(546\) −11.8324 −0.506378
\(547\) 4.74760 0.202993 0.101496 0.994836i \(-0.467637\pi\)
0.101496 + 0.994836i \(0.467637\pi\)
\(548\) 2.87859 0.122967
\(549\) −50.2620 −2.14513
\(550\) 0 0
\(551\) −7.35548 −0.313354
\(552\) −22.1310 −0.941958
\(553\) 9.57227 0.407054
\(554\) 12.9538 0.550353
\(555\) 0 0
\(556\) 3.58767 0.152151
\(557\) 43.0462 1.82393 0.911964 0.410271i \(-0.134566\pi\)
0.911964 + 0.410271i \(0.134566\pi\)
\(558\) 37.5510 1.58966
\(559\) 38.6218 1.63353
\(560\) 0 0
\(561\) 8.65847 0.365561
\(562\) 0.153681 0.00648265
\(563\) 17.0096 0.716869 0.358434 0.933555i \(-0.383311\pi\)
0.358434 + 0.933555i \(0.383311\pi\)
\(564\) 3.45856 0.145632
\(565\) 0 0
\(566\) −18.2341 −0.766435
\(567\) 1.12473 0.0472343
\(568\) −10.1816 −0.427210
\(569\) −19.7572 −0.828264 −0.414132 0.910217i \(-0.635915\pi\)
−0.414132 + 0.910217i \(0.635915\pi\)
\(570\) 0 0
\(571\) −11.3973 −0.476964 −0.238482 0.971147i \(-0.576650\pi\)
−0.238482 + 0.971147i \(0.576650\pi\)
\(572\) −4.86464 −0.203401
\(573\) 36.5587 1.52726
\(574\) 2.02458 0.0845043
\(575\) 0 0
\(576\) 4.62620 0.192758
\(577\) −18.3372 −0.763386 −0.381693 0.924289i \(-0.624659\pi\)
−0.381693 + 0.924289i \(0.624659\pi\)
\(578\) −3.85069 −0.160167
\(579\) 7.01832 0.291672
\(580\) 0 0
\(581\) 0.206167 0.00855327
\(582\) −23.4865 −0.973546
\(583\) −2.05249 −0.0850053
\(584\) 16.4017 0.678708
\(585\) 0 0
\(586\) 2.03853 0.0842110
\(587\) −11.9475 −0.493127 −0.246563 0.969127i \(-0.579301\pi\)
−0.246563 + 0.969127i \(0.579301\pi\)
\(588\) −17.7293 −0.731143
\(589\) 8.11704 0.334457
\(590\) 0 0
\(591\) −54.9850 −2.26178
\(592\) 0.476886 0.0195999
\(593\) −24.3911 −1.00162 −0.500811 0.865557i \(-0.666965\pi\)
−0.500811 + 0.865557i \(0.666965\pi\)
\(594\) −3.88296 −0.159320
\(595\) 0 0
\(596\) −16.8401 −0.689796
\(597\) −56.0943 −2.29579
\(598\) −45.0881 −1.84379
\(599\) −21.0708 −0.860930 −0.430465 0.902607i \(-0.641650\pi\)
−0.430465 + 0.902607i \(0.641650\pi\)
\(600\) 0 0
\(601\) 32.3878 1.32112 0.660562 0.750771i \(-0.270318\pi\)
0.660562 + 0.750771i \(0.270318\pi\)
\(602\) −5.22782 −0.213070
\(603\) 4.77551 0.194474
\(604\) −16.9817 −0.690975
\(605\) 0 0
\(606\) 45.3328 1.84152
\(607\) 19.0183 0.771930 0.385965 0.922513i \(-0.373869\pi\)
0.385965 + 0.922513i \(0.373869\pi\)
\(608\) 1.00000 0.0405554
\(609\) 15.4692 0.626843
\(610\) 0 0
\(611\) 7.04623 0.285060
\(612\) −16.7755 −0.678110
\(613\) −23.5756 −0.952210 −0.476105 0.879389i \(-0.657952\pi\)
−0.476105 + 0.879389i \(0.657952\pi\)
\(614\) 16.5414 0.667558
\(615\) 0 0
\(616\) 0.658473 0.0265307
\(617\) −26.2707 −1.05762 −0.528810 0.848740i \(-0.677361\pi\)
−0.528810 + 0.848740i \(0.677361\pi\)
\(618\) −26.6218 −1.07089
\(619\) 11.4985 0.462165 0.231083 0.972934i \(-0.425773\pi\)
0.231083 + 0.972934i \(0.425773\pi\)
\(620\) 0 0
\(621\) −35.9894 −1.44420
\(622\) −21.4725 −0.860969
\(623\) −0.295298 −0.0118309
\(624\) 15.5371 0.621980
\(625\) 0 0
\(626\) −1.12141 −0.0448204
\(627\) −2.38776 −0.0953578
\(628\) 14.8401 0.592183
\(629\) −1.72928 −0.0689510
\(630\) 0 0
\(631\) −45.8130 −1.82379 −0.911893 0.410427i \(-0.865380\pi\)
−0.911893 + 0.410427i \(0.865380\pi\)
\(632\) −12.5693 −0.499982
\(633\) 49.8236 1.98031
\(634\) −29.8882 −1.18701
\(635\) 0 0
\(636\) 6.55539 0.259938
\(637\) −36.1204 −1.43114
\(638\) 6.35985 0.251789
\(639\) −47.1020 −1.86333
\(640\) 0 0
\(641\) 1.36943 0.0540894 0.0270447 0.999634i \(-0.491390\pi\)
0.0270447 + 0.999634i \(0.491390\pi\)
\(642\) 11.8324 0.466986
\(643\) −24.7389 −0.975606 −0.487803 0.872954i \(-0.662201\pi\)
−0.487803 + 0.872954i \(0.662201\pi\)
\(644\) 6.10308 0.240495
\(645\) 0 0
\(646\) −3.62620 −0.142671
\(647\) −6.82611 −0.268362 −0.134181 0.990957i \(-0.542840\pi\)
−0.134181 + 0.990957i \(0.542840\pi\)
\(648\) −1.47689 −0.0580175
\(649\) −3.88296 −0.152420
\(650\) 0 0
\(651\) −17.0708 −0.669058
\(652\) 13.3694 0.523587
\(653\) 6.91713 0.270688 0.135344 0.990799i \(-0.456786\pi\)
0.135344 + 0.990799i \(0.456786\pi\)
\(654\) −37.0602 −1.44917
\(655\) 0 0
\(656\) −2.65847 −0.103796
\(657\) 75.8776 2.96027
\(658\) −0.953771 −0.0371819
\(659\) 9.44461 0.367910 0.183955 0.982935i \(-0.441110\pi\)
0.183955 + 0.982935i \(0.441110\pi\)
\(660\) 0 0
\(661\) −22.1955 −0.863306 −0.431653 0.902040i \(-0.642070\pi\)
−0.431653 + 0.902040i \(0.642070\pi\)
\(662\) 32.3126 1.25586
\(663\) −56.3405 −2.18808
\(664\) −0.270718 −0.0105059
\(665\) 0 0
\(666\) 2.20617 0.0854873
\(667\) 58.9465 2.28242
\(668\) 9.84632 0.380966
\(669\) −37.3449 −1.44384
\(670\) 0 0
\(671\) 9.39401 0.362652
\(672\) −2.10308 −0.0811282
\(673\) 12.2986 0.474077 0.237039 0.971500i \(-0.423823\pi\)
0.237039 + 0.971500i \(0.423823\pi\)
\(674\) 26.3511 1.01501
\(675\) 0 0
\(676\) 18.6541 0.717466
\(677\) −35.9527 −1.38178 −0.690888 0.722962i \(-0.742780\pi\)
−0.690888 + 0.722962i \(0.742780\pi\)
\(678\) −28.5081 −1.09485
\(679\) 6.47689 0.248560
\(680\) 0 0
\(681\) 37.5616 1.43937
\(682\) −7.01832 −0.268745
\(683\) −9.00958 −0.344742 −0.172371 0.985032i \(-0.555143\pi\)
−0.172371 + 0.985032i \(0.555143\pi\)
\(684\) 4.62620 0.176887
\(685\) 0 0
\(686\) 10.2201 0.390206
\(687\) 37.5231 1.43160
\(688\) 6.86464 0.261712
\(689\) 13.3555 0.508803
\(690\) 0 0
\(691\) −9.11078 −0.346590 −0.173295 0.984870i \(-0.555441\pi\)
−0.173295 + 0.984870i \(0.555441\pi\)
\(692\) 2.98168 0.113346
\(693\) 3.04623 0.115717
\(694\) −2.77551 −0.105357
\(695\) 0 0
\(696\) −20.3126 −0.769946
\(697\) 9.64015 0.365147
\(698\) 11.5510 0.437213
\(699\) 70.5606 2.66885
\(700\) 0 0
\(701\) −14.7476 −0.557009 −0.278505 0.960435i \(-0.589839\pi\)
−0.278505 + 0.960435i \(0.589839\pi\)
\(702\) 25.2663 0.953617
\(703\) 0.476886 0.0179861
\(704\) −0.864641 −0.0325874
\(705\) 0 0
\(706\) −8.40171 −0.316202
\(707\) −12.5015 −0.470166
\(708\) 12.4017 0.466085
\(709\) −8.63389 −0.324253 −0.162126 0.986770i \(-0.551835\pi\)
−0.162126 + 0.986770i \(0.551835\pi\)
\(710\) 0 0
\(711\) −58.1483 −2.18073
\(712\) 0.387755 0.0145317
\(713\) −65.0496 −2.43613
\(714\) 7.62620 0.285403
\(715\) 0 0
\(716\) 11.7938 0.440756
\(717\) −31.2909 −1.16858
\(718\) 22.7895 0.850495
\(719\) −38.2759 −1.42745 −0.713726 0.700425i \(-0.752994\pi\)
−0.713726 + 0.700425i \(0.752994\pi\)
\(720\) 0 0
\(721\) 7.34153 0.273413
\(722\) 1.00000 0.0372161
\(723\) −3.45856 −0.128625
\(724\) 14.5693 0.541465
\(725\) 0 0
\(726\) −28.3126 −1.05078
\(727\) −31.1893 −1.15675 −0.578373 0.815772i \(-0.696312\pi\)
−0.578373 + 0.815772i \(0.696312\pi\)
\(728\) −4.28467 −0.158800
\(729\) −44.0375 −1.63102
\(730\) 0 0
\(731\) −24.8925 −0.920684
\(732\) −30.0033 −1.10895
\(733\) −13.9634 −0.515748 −0.257874 0.966179i \(-0.583022\pi\)
−0.257874 + 0.966179i \(0.583022\pi\)
\(734\) 4.06455 0.150025
\(735\) 0 0
\(736\) −8.01395 −0.295398
\(737\) −0.892548 −0.0328774
\(738\) −12.2986 −0.452719
\(739\) 9.02165 0.331867 0.165933 0.986137i \(-0.446936\pi\)
0.165933 + 0.986137i \(0.446936\pi\)
\(740\) 0 0
\(741\) 15.5371 0.570768
\(742\) −1.80779 −0.0663659
\(743\) −15.0342 −0.551550 −0.275775 0.961222i \(-0.588934\pi\)
−0.275775 + 0.961222i \(0.588934\pi\)
\(744\) 22.4157 0.821798
\(745\) 0 0
\(746\) −18.4017 −0.673734
\(747\) −1.25240 −0.0458228
\(748\) 3.13536 0.114640
\(749\) −3.26302 −0.119228
\(750\) 0 0
\(751\) 29.6681 1.08260 0.541301 0.840829i \(-0.317932\pi\)
0.541301 + 0.840829i \(0.317932\pi\)
\(752\) 1.25240 0.0456702
\(753\) 29.2557 1.06614
\(754\) −41.3834 −1.50709
\(755\) 0 0
\(756\) −3.42003 −0.124385
\(757\) 10.5819 0.384604 0.192302 0.981336i \(-0.438405\pi\)
0.192302 + 0.981336i \(0.438405\pi\)
\(758\) 1.23844 0.0449823
\(759\) 19.1354 0.694570
\(760\) 0 0
\(761\) −0.979789 −0.0355173 −0.0177587 0.999842i \(-0.505653\pi\)
−0.0177587 + 0.999842i \(0.505653\pi\)
\(762\) 46.8959 1.69886
\(763\) 10.2201 0.369993
\(764\) 13.2384 0.478950
\(765\) 0 0
\(766\) 16.8646 0.609344
\(767\) 25.2663 0.912315
\(768\) 2.76156 0.0996491
\(769\) 43.1772 1.55701 0.778505 0.627638i \(-0.215978\pi\)
0.778505 + 0.627638i \(0.215978\pi\)
\(770\) 0 0
\(771\) −0.424399 −0.0152844
\(772\) 2.54144 0.0914683
\(773\) 37.5250 1.34968 0.674840 0.737964i \(-0.264213\pi\)
0.674840 + 0.737964i \(0.264213\pi\)
\(774\) 31.7572 1.14149
\(775\) 0 0
\(776\) −8.50479 −0.305304
\(777\) −1.00293 −0.0359799
\(778\) 8.59392 0.308107
\(779\) −2.65847 −0.0952497
\(780\) 0 0
\(781\) 8.80342 0.315011
\(782\) 29.0602 1.03919
\(783\) −33.0323 −1.18048
\(784\) −6.42003 −0.229287
\(785\) 0 0
\(786\) 1.49521 0.0533323
\(787\) 22.5833 0.805008 0.402504 0.915418i \(-0.368140\pi\)
0.402504 + 0.915418i \(0.368140\pi\)
\(788\) −19.9109 −0.709295
\(789\) 1.39401 0.0496282
\(790\) 0 0
\(791\) 7.86171 0.279530
\(792\) −4.00000 −0.142134
\(793\) −61.1266 −2.17067
\(794\) 16.0558 0.569799
\(795\) 0 0
\(796\) −20.3126 −0.719960
\(797\) 35.9806 1.27450 0.637250 0.770657i \(-0.280072\pi\)
0.637250 + 0.770657i \(0.280072\pi\)
\(798\) −2.10308 −0.0744484
\(799\) −4.54144 −0.160664
\(800\) 0 0
\(801\) 1.79383 0.0633820
\(802\) −14.8925 −0.525874
\(803\) −14.1816 −0.500457
\(804\) 2.85069 0.100536
\(805\) 0 0
\(806\) 45.6681 1.60859
\(807\) −9.65222 −0.339774
\(808\) 16.4157 0.577501
\(809\) −0.955660 −0.0335992 −0.0167996 0.999859i \(-0.505348\pi\)
−0.0167996 + 0.999859i \(0.505348\pi\)
\(810\) 0 0
\(811\) −7.53707 −0.264662 −0.132331 0.991206i \(-0.542246\pi\)
−0.132331 + 0.991206i \(0.542246\pi\)
\(812\) 5.60162 0.196578
\(813\) 15.1127 0.530024
\(814\) −0.412335 −0.0144523
\(815\) 0 0
\(816\) −10.0140 −0.350558
\(817\) 6.86464 0.240163
\(818\) −18.3511 −0.641632
\(819\) −19.8217 −0.692628
\(820\) 0 0
\(821\) 13.3082 0.464460 0.232230 0.972661i \(-0.425398\pi\)
0.232230 + 0.972661i \(0.425398\pi\)
\(822\) 7.94940 0.277267
\(823\) 24.4050 0.850706 0.425353 0.905028i \(-0.360150\pi\)
0.425353 + 0.905028i \(0.360150\pi\)
\(824\) −9.64015 −0.335831
\(825\) 0 0
\(826\) −3.42003 −0.118998
\(827\) −11.6874 −0.406411 −0.203206 0.979136i \(-0.565136\pi\)
−0.203206 + 0.979136i \(0.565136\pi\)
\(828\) −37.0741 −1.28842
\(829\) 25.6541 0.891004 0.445502 0.895281i \(-0.353025\pi\)
0.445502 + 0.895281i \(0.353025\pi\)
\(830\) 0 0
\(831\) 35.7726 1.24094
\(832\) 5.62620 0.195053
\(833\) 23.2803 0.806615
\(834\) 9.90754 0.343070
\(835\) 0 0
\(836\) −0.864641 −0.0299042
\(837\) 36.4523 1.25998
\(838\) 34.7509 1.20045
\(839\) −2.91713 −0.100710 −0.0503552 0.998731i \(-0.516035\pi\)
−0.0503552 + 0.998731i \(0.516035\pi\)
\(840\) 0 0
\(841\) 25.1031 0.865624
\(842\) −40.1589 −1.38397
\(843\) 0.424399 0.0146171
\(844\) 18.0419 0.621026
\(845\) 0 0
\(846\) 5.79383 0.199196
\(847\) 7.80779 0.268279
\(848\) 2.37380 0.0815167
\(849\) −50.3544 −1.72816
\(850\) 0 0
\(851\) −3.82174 −0.131008
\(852\) −28.1170 −0.963274
\(853\) −6.24281 −0.213750 −0.106875 0.994272i \(-0.534084\pi\)
−0.106875 + 0.994272i \(0.534084\pi\)
\(854\) 8.27405 0.283132
\(855\) 0 0
\(856\) 4.28467 0.146447
\(857\) −23.2158 −0.793035 −0.396517 0.918027i \(-0.629781\pi\)
−0.396517 + 0.918027i \(0.629781\pi\)
\(858\) −13.4340 −0.458629
\(859\) −7.13536 −0.243455 −0.121728 0.992564i \(-0.538843\pi\)
−0.121728 + 0.992564i \(0.538843\pi\)
\(860\) 0 0
\(861\) 5.59099 0.190541
\(862\) 34.9571 1.19064
\(863\) −7.31362 −0.248959 −0.124479 0.992222i \(-0.539726\pi\)
−0.124479 + 0.992222i \(0.539726\pi\)
\(864\) 4.49084 0.152781
\(865\) 0 0
\(866\) 1.13536 0.0385811
\(867\) −10.6339 −0.361146
\(868\) −6.18159 −0.209817
\(869\) 10.8680 0.368671
\(870\) 0 0
\(871\) 5.80779 0.196789
\(872\) −13.4200 −0.454460
\(873\) −39.3449 −1.33162
\(874\) −8.01395 −0.271076
\(875\) 0 0
\(876\) 45.2943 1.53035
\(877\) −22.0173 −0.743471 −0.371735 0.928339i \(-0.621237\pi\)
−0.371735 + 0.928339i \(0.621237\pi\)
\(878\) −6.80009 −0.229492
\(879\) 5.62953 0.189879
\(880\) 0 0
\(881\) 11.7572 0.396110 0.198055 0.980191i \(-0.436538\pi\)
0.198055 + 0.980191i \(0.436538\pi\)
\(882\) −29.7003 −1.00006
\(883\) 55.6560 1.87297 0.936487 0.350703i \(-0.114057\pi\)
0.936487 + 0.350703i \(0.114057\pi\)
\(884\) −20.4017 −0.686184
\(885\) 0 0
\(886\) 38.0679 1.27892
\(887\) −6.41566 −0.215417 −0.107708 0.994183i \(-0.534351\pi\)
−0.107708 + 0.994183i \(0.534351\pi\)
\(888\) 1.31695 0.0441938
\(889\) −12.9325 −0.433743
\(890\) 0 0
\(891\) 1.27698 0.0427803
\(892\) −13.5231 −0.452787
\(893\) 1.25240 0.0419098
\(894\) −46.5048 −1.55535
\(895\) 0 0
\(896\) −0.761557 −0.0254418
\(897\) −124.513 −4.15738
\(898\) −18.5414 −0.618736
\(899\) −59.7047 −1.99126
\(900\) 0 0
\(901\) −8.60788 −0.286770
\(902\) 2.29862 0.0765358
\(903\) −14.4369 −0.480430
\(904\) −10.3232 −0.343345
\(905\) 0 0
\(906\) −46.8959 −1.55801
\(907\) −57.1160 −1.89651 −0.948253 0.317516i \(-0.897151\pi\)
−0.948253 + 0.317516i \(0.897151\pi\)
\(908\) 13.6016 0.451386
\(909\) 75.9421 2.51884
\(910\) 0 0
\(911\) −26.6339 −0.882420 −0.441210 0.897404i \(-0.645451\pi\)
−0.441210 + 0.897404i \(0.645451\pi\)
\(912\) 2.76156 0.0914443
\(913\) 0.234074 0.00774672
\(914\) 16.3738 0.541597
\(915\) 0 0
\(916\) 13.5877 0.448949
\(917\) −0.412335 −0.0136165
\(918\) −16.2847 −0.537474
\(919\) 6.63246 0.218785 0.109392 0.993999i \(-0.465110\pi\)
0.109392 + 0.993999i \(0.465110\pi\)
\(920\) 0 0
\(921\) 45.6801 1.50521
\(922\) 1.70470 0.0561414
\(923\) −57.2836 −1.88551
\(924\) 1.81841 0.0598214
\(925\) 0 0
\(926\) −10.0279 −0.329537
\(927\) −44.5972 −1.46477
\(928\) −7.35548 −0.241455
\(929\) −50.0173 −1.64101 −0.820507 0.571637i \(-0.806309\pi\)
−0.820507 + 0.571637i \(0.806309\pi\)
\(930\) 0 0
\(931\) −6.42003 −0.210408
\(932\) 25.5510 0.836952
\(933\) −59.2976 −1.94132
\(934\) 32.7509 1.07164
\(935\) 0 0
\(936\) 26.0279 0.850749
\(937\) −39.8882 −1.30309 −0.651545 0.758610i \(-0.725879\pi\)
−0.651545 + 0.758610i \(0.725879\pi\)
\(938\) −0.786137 −0.0256683
\(939\) −3.09683 −0.101061
\(940\) 0 0
\(941\) −5.59829 −0.182499 −0.0912495 0.995828i \(-0.529086\pi\)
−0.0912495 + 0.995828i \(0.529086\pi\)
\(942\) 40.9817 1.33526
\(943\) 21.3049 0.693782
\(944\) 4.49084 0.146164
\(945\) 0 0
\(946\) −5.93545 −0.192978
\(947\) 12.7110 0.413051 0.206525 0.978441i \(-0.433784\pi\)
0.206525 + 0.978441i \(0.433784\pi\)
\(948\) −34.7110 −1.12736
\(949\) 92.2793 2.99551
\(950\) 0 0
\(951\) −82.5379 −2.67648
\(952\) 2.76156 0.0895026
\(953\) −57.0129 −1.84683 −0.923415 0.383804i \(-0.874614\pi\)
−0.923415 + 0.383804i \(0.874614\pi\)
\(954\) 10.9817 0.355545
\(955\) 0 0
\(956\) −11.3309 −0.366468
\(957\) 17.5631 0.567734
\(958\) 27.2803 0.881387
\(959\) −2.19221 −0.0707903
\(960\) 0 0
\(961\) 34.8863 1.12536
\(962\) 2.68305 0.0865051
\(963\) 19.8217 0.638747
\(964\) −1.25240 −0.0403370
\(965\) 0 0
\(966\) 16.8540 0.542269
\(967\) −33.0183 −1.06180 −0.530899 0.847435i \(-0.678146\pi\)
−0.530899 + 0.847435i \(0.678146\pi\)
\(968\) −10.2524 −0.329524
\(969\) −10.0140 −0.321695
\(970\) 0 0
\(971\) −3.04623 −0.0977581 −0.0488791 0.998805i \(-0.515565\pi\)
−0.0488791 + 0.998805i \(0.515565\pi\)
\(972\) −17.5510 −0.562949
\(973\) −2.73221 −0.0875907
\(974\) −11.0741 −0.354838
\(975\) 0 0
\(976\) −10.8646 −0.347769
\(977\) 13.4465 0.430192 0.215096 0.976593i \(-0.430994\pi\)
0.215096 + 0.976593i \(0.430994\pi\)
\(978\) 36.9205 1.18059
\(979\) −0.335269 −0.0107152
\(980\) 0 0
\(981\) −62.0837 −1.98218
\(982\) 8.11704 0.259025
\(983\) −22.0646 −0.703750 −0.351875 0.936047i \(-0.614456\pi\)
−0.351875 + 0.936047i \(0.614456\pi\)
\(984\) −7.34153 −0.234039
\(985\) 0 0
\(986\) 26.6724 0.849423
\(987\) −2.63389 −0.0838378
\(988\) 5.62620 0.178993
\(989\) −55.0129 −1.74931
\(990\) 0 0
\(991\) 51.9946 1.65166 0.825831 0.563917i \(-0.190706\pi\)
0.825831 + 0.563917i \(0.190706\pi\)
\(992\) 8.11704 0.257716
\(993\) 89.2330 2.83172
\(994\) 7.75386 0.245938
\(995\) 0 0
\(996\) −0.747604 −0.0236887
\(997\) −37.7693 −1.19616 −0.598082 0.801435i \(-0.704070\pi\)
−0.598082 + 0.801435i \(0.704070\pi\)
\(998\) 0.295298 0.00934749
\(999\) 2.14162 0.0677578
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 950.2.a.n.1.3 3
3.2 odd 2 8550.2.a.ck.1.1 3
4.3 odd 2 7600.2.a.bi.1.1 3
5.2 odd 4 190.2.b.b.39.4 yes 6
5.3 odd 4 190.2.b.b.39.3 6
5.4 even 2 950.2.a.i.1.1 3
15.2 even 4 1710.2.d.d.1369.3 6
15.8 even 4 1710.2.d.d.1369.6 6
15.14 odd 2 8550.2.a.cl.1.3 3
20.3 even 4 1520.2.d.j.609.1 6
20.7 even 4 1520.2.d.j.609.6 6
20.19 odd 2 7600.2.a.cd.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.b.39.3 6 5.3 odd 4
190.2.b.b.39.4 yes 6 5.2 odd 4
950.2.a.i.1.1 3 5.4 even 2
950.2.a.n.1.3 3 1.1 even 1 trivial
1520.2.d.j.609.1 6 20.3 even 4
1520.2.d.j.609.6 6 20.7 even 4
1710.2.d.d.1369.3 6 15.2 even 4
1710.2.d.d.1369.6 6 15.8 even 4
7600.2.a.bi.1.1 3 4.3 odd 2
7600.2.a.cd.1.3 3 20.19 odd 2
8550.2.a.ck.1.1 3 3.2 odd 2
8550.2.a.cl.1.3 3 15.14 odd 2