Properties

Label 5070.2.a.bq.1.2
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
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 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.643104 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.643104 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.40581 q^{11} +1.00000 q^{12} -0.643104 q^{14} +1.00000 q^{15} +1.00000 q^{16} +0.939001 q^{17} -1.00000 q^{18} +2.75302 q^{19} +1.00000 q^{20} +0.643104 q^{21} -4.40581 q^{22} -2.04892 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +0.643104 q^{28} -0.149145 q^{29} -1.00000 q^{30} +2.08815 q^{31} -1.00000 q^{32} +4.40581 q^{33} -0.939001 q^{34} +0.643104 q^{35} +1.00000 q^{36} +6.07069 q^{37} -2.75302 q^{38} -1.00000 q^{40} -5.74094 q^{41} -0.643104 q^{42} +4.69202 q^{43} +4.40581 q^{44} +1.00000 q^{45} +2.04892 q^{46} +7.00000 q^{47} +1.00000 q^{48} -6.58642 q^{49} -1.00000 q^{50} +0.939001 q^{51} +1.68664 q^{53} -1.00000 q^{54} +4.40581 q^{55} -0.643104 q^{56} +2.75302 q^{57} +0.149145 q^{58} +7.85623 q^{59} +1.00000 q^{60} -12.0761 q^{61} -2.08815 q^{62} +0.643104 q^{63} +1.00000 q^{64} -4.40581 q^{66} +1.45712 q^{67} +0.939001 q^{68} -2.04892 q^{69} -0.643104 q^{70} -7.03684 q^{71} -1.00000 q^{72} +14.8605 q^{73} -6.07069 q^{74} +1.00000 q^{75} +2.75302 q^{76} +2.83340 q^{77} +0.929312 q^{79} +1.00000 q^{80} +1.00000 q^{81} +5.74094 q^{82} +11.2959 q^{83} +0.643104 q^{84} +0.939001 q^{85} -4.69202 q^{86} -0.149145 q^{87} -4.40581 q^{88} -7.47650 q^{89} -1.00000 q^{90} -2.04892 q^{92} +2.08815 q^{93} -7.00000 q^{94} +2.75302 q^{95} -1.00000 q^{96} -5.05861 q^{97} +6.58642 q^{98} +4.40581 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 6q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 6q^{7} - 3q^{8} + 3q^{9} - 3q^{10} + 3q^{12} - 6q^{14} + 3q^{15} + 3q^{16} - 7q^{17} - 3q^{18} + 13q^{19} + 3q^{20} + 6q^{21} + 3q^{23} - 3q^{24} + 3q^{25} + 3q^{27} + 6q^{28} - 14q^{29} - 3q^{30} + 10q^{31} - 3q^{32} + 7q^{34} + 6q^{35} + 3q^{36} + 6q^{37} - 13q^{38} - 3q^{40} - 3q^{41} - 6q^{42} + 9q^{43} + 3q^{45} - 3q^{46} + 21q^{47} + 3q^{48} + 5q^{49} - 3q^{50} - 7q^{51} + 3q^{53} - 3q^{54} - 6q^{56} + 13q^{57} + 14q^{58} + 7q^{59} + 3q^{60} - 21q^{61} - 10q^{62} + 6q^{63} + 3q^{64} + 23q^{67} - 7q^{68} + 3q^{69} - 6q^{70} + 7q^{71} - 3q^{72} + 9q^{73} - 6q^{74} + 3q^{75} + 13q^{76} - 21q^{77} + 15q^{79} + 3q^{80} + 3q^{81} + 3q^{82} + 20q^{83} + 6q^{84} - 7q^{85} - 9q^{86} - 14q^{87} + 3q^{89} - 3q^{90} + 3q^{92} + 10q^{93} - 21q^{94} + 13q^{95} - 3q^{96} + 16q^{97} - 5q^{98} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 0.643104 0.243071 0.121535 0.992587i \(-0.461218\pi\)
0.121535 + 0.992587i \(0.461218\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 4.40581 1.32840 0.664201 0.747554i \(-0.268772\pi\)
0.664201 + 0.747554i \(0.268772\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −0.643104 −0.171877
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0.939001 0.227741 0.113871 0.993496i \(-0.463675\pi\)
0.113871 + 0.993496i \(0.463675\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.75302 0.631586 0.315793 0.948828i \(-0.397729\pi\)
0.315793 + 0.948828i \(0.397729\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.643104 0.140337
\(22\) −4.40581 −0.939323
\(23\) −2.04892 −0.427229 −0.213614 0.976918i \(-0.568524\pi\)
−0.213614 + 0.976918i \(0.568524\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0.643104 0.121535
\(29\) −0.149145 −0.0276955 −0.0138478 0.999904i \(-0.504408\pi\)
−0.0138478 + 0.999904i \(0.504408\pi\)
\(30\) −1.00000 −0.182574
\(31\) 2.08815 0.375042 0.187521 0.982261i \(-0.439955\pi\)
0.187521 + 0.982261i \(0.439955\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.40581 0.766954
\(34\) −0.939001 −0.161037
\(35\) 0.643104 0.108704
\(36\) 1.00000 0.166667
\(37\) 6.07069 0.998015 0.499007 0.866598i \(-0.333698\pi\)
0.499007 + 0.866598i \(0.333698\pi\)
\(38\) −2.75302 −0.446599
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −5.74094 −0.896584 −0.448292 0.893887i \(-0.647968\pi\)
−0.448292 + 0.893887i \(0.647968\pi\)
\(42\) −0.643104 −0.0992331
\(43\) 4.69202 0.715527 0.357763 0.933812i \(-0.383539\pi\)
0.357763 + 0.933812i \(0.383539\pi\)
\(44\) 4.40581 0.664201
\(45\) 1.00000 0.149071
\(46\) 2.04892 0.302096
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.58642 −0.940917
\(50\) −1.00000 −0.141421
\(51\) 0.939001 0.131486
\(52\) 0 0
\(53\) 1.68664 0.231678 0.115839 0.993268i \(-0.463044\pi\)
0.115839 + 0.993268i \(0.463044\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.40581 0.594080
\(56\) −0.643104 −0.0859384
\(57\) 2.75302 0.364646
\(58\) 0.149145 0.0195837
\(59\) 7.85623 1.02279 0.511397 0.859344i \(-0.329128\pi\)
0.511397 + 0.859344i \(0.329128\pi\)
\(60\) 1.00000 0.129099
\(61\) −12.0761 −1.54618 −0.773091 0.634295i \(-0.781290\pi\)
−0.773091 + 0.634295i \(0.781290\pi\)
\(62\) −2.08815 −0.265195
\(63\) 0.643104 0.0810235
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.40581 −0.542318
\(67\) 1.45712 0.178016 0.0890080 0.996031i \(-0.471630\pi\)
0.0890080 + 0.996031i \(0.471630\pi\)
\(68\) 0.939001 0.113871
\(69\) −2.04892 −0.246661
\(70\) −0.643104 −0.0768656
\(71\) −7.03684 −0.835119 −0.417559 0.908650i \(-0.637114\pi\)
−0.417559 + 0.908650i \(0.637114\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.8605 1.73930 0.869648 0.493673i \(-0.164346\pi\)
0.869648 + 0.493673i \(0.164346\pi\)
\(74\) −6.07069 −0.705703
\(75\) 1.00000 0.115470
\(76\) 2.75302 0.315793
\(77\) 2.83340 0.322896
\(78\) 0 0
\(79\) 0.929312 0.104556 0.0522779 0.998633i \(-0.483352\pi\)
0.0522779 + 0.998633i \(0.483352\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 5.74094 0.633981
\(83\) 11.2959 1.23989 0.619943 0.784647i \(-0.287156\pi\)
0.619943 + 0.784647i \(0.287156\pi\)
\(84\) 0.643104 0.0701684
\(85\) 0.939001 0.101849
\(86\) −4.69202 −0.505954
\(87\) −0.149145 −0.0159900
\(88\) −4.40581 −0.469661
\(89\) −7.47650 −0.792508 −0.396254 0.918141i \(-0.629690\pi\)
−0.396254 + 0.918141i \(0.629690\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −2.04892 −0.213614
\(93\) 2.08815 0.216531
\(94\) −7.00000 −0.721995
\(95\) 2.75302 0.282454
\(96\) −1.00000 −0.102062
\(97\) −5.05861 −0.513624 −0.256812 0.966461i \(-0.582672\pi\)
−0.256812 + 0.966461i \(0.582672\pi\)
\(98\) 6.58642 0.665329
\(99\) 4.40581 0.442801
\(100\) 1.00000 0.100000
\(101\) −6.81163 −0.677782 −0.338891 0.940826i \(-0.610052\pi\)
−0.338891 + 0.940826i \(0.610052\pi\)
\(102\) −0.939001 −0.0929750
\(103\) −12.0858 −1.19084 −0.595422 0.803413i \(-0.703015\pi\)
−0.595422 + 0.803413i \(0.703015\pi\)
\(104\) 0 0
\(105\) 0.643104 0.0627605
\(106\) −1.68664 −0.163821
\(107\) −14.6679 −1.41800 −0.708998 0.705211i \(-0.750852\pi\)
−0.708998 + 0.705211i \(0.750852\pi\)
\(108\) 1.00000 0.0962250
\(109\) 8.15883 0.781475 0.390737 0.920502i \(-0.372220\pi\)
0.390737 + 0.920502i \(0.372220\pi\)
\(110\) −4.40581 −0.420078
\(111\) 6.07069 0.576204
\(112\) 0.643104 0.0607676
\(113\) 16.8442 1.58456 0.792282 0.610155i \(-0.208893\pi\)
0.792282 + 0.610155i \(0.208893\pi\)
\(114\) −2.75302 −0.257844
\(115\) −2.04892 −0.191063
\(116\) −0.149145 −0.0138478
\(117\) 0 0
\(118\) −7.85623 −0.723225
\(119\) 0.603875 0.0553572
\(120\) −1.00000 −0.0912871
\(121\) 8.41119 0.764654
\(122\) 12.0761 1.09332
\(123\) −5.74094 −0.517643
\(124\) 2.08815 0.187521
\(125\) 1.00000 0.0894427
\(126\) −0.643104 −0.0572923
\(127\) 4.85086 0.430444 0.215222 0.976565i \(-0.430953\pi\)
0.215222 + 0.976565i \(0.430953\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.69202 0.413109
\(130\) 0 0
\(131\) 1.52781 0.133485 0.0667427 0.997770i \(-0.478739\pi\)
0.0667427 + 0.997770i \(0.478739\pi\)
\(132\) 4.40581 0.383477
\(133\) 1.77048 0.153520
\(134\) −1.45712 −0.125876
\(135\) 1.00000 0.0860663
\(136\) −0.939001 −0.0805187
\(137\) −6.41119 −0.547745 −0.273872 0.961766i \(-0.588305\pi\)
−0.273872 + 0.961766i \(0.588305\pi\)
\(138\) 2.04892 0.174415
\(139\) −8.78017 −0.744724 −0.372362 0.928088i \(-0.621452\pi\)
−0.372362 + 0.928088i \(0.621452\pi\)
\(140\) 0.643104 0.0543522
\(141\) 7.00000 0.589506
\(142\) 7.03684 0.590518
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −0.149145 −0.0123858
\(146\) −14.8605 −1.22987
\(147\) −6.58642 −0.543239
\(148\) 6.07069 0.499007
\(149\) −22.6504 −1.85559 −0.927797 0.373087i \(-0.878299\pi\)
−0.927797 + 0.373087i \(0.878299\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 10.2959 0.837868 0.418934 0.908017i \(-0.362404\pi\)
0.418934 + 0.908017i \(0.362404\pi\)
\(152\) −2.75302 −0.223299
\(153\) 0.939001 0.0759137
\(154\) −2.83340 −0.228322
\(155\) 2.08815 0.167724
\(156\) 0 0
\(157\) 0.994623 0.0793796 0.0396898 0.999212i \(-0.487363\pi\)
0.0396898 + 0.999212i \(0.487363\pi\)
\(158\) −0.929312 −0.0739321
\(159\) 1.68664 0.133760
\(160\) −1.00000 −0.0790569
\(161\) −1.31767 −0.103847
\(162\) −1.00000 −0.0785674
\(163\) 21.6015 1.69196 0.845979 0.533216i \(-0.179017\pi\)
0.845979 + 0.533216i \(0.179017\pi\)
\(164\) −5.74094 −0.448292
\(165\) 4.40581 0.342992
\(166\) −11.2959 −0.876732
\(167\) 7.96615 0.616439 0.308220 0.951315i \(-0.400267\pi\)
0.308220 + 0.951315i \(0.400267\pi\)
\(168\) −0.643104 −0.0496166
\(169\) 0 0
\(170\) −0.939001 −0.0720181
\(171\) 2.75302 0.210529
\(172\) 4.69202 0.357763
\(173\) 4.11960 0.313208 0.156604 0.987661i \(-0.449945\pi\)
0.156604 + 0.987661i \(0.449945\pi\)
\(174\) 0.149145 0.0113066
\(175\) 0.643104 0.0486141
\(176\) 4.40581 0.332101
\(177\) 7.85623 0.590511
\(178\) 7.47650 0.560387
\(179\) −8.78017 −0.656261 −0.328130 0.944632i \(-0.606418\pi\)
−0.328130 + 0.944632i \(0.606418\pi\)
\(180\) 1.00000 0.0745356
\(181\) 19.4741 1.44750 0.723750 0.690063i \(-0.242417\pi\)
0.723750 + 0.690063i \(0.242417\pi\)
\(182\) 0 0
\(183\) −12.0761 −0.892688
\(184\) 2.04892 0.151048
\(185\) 6.07069 0.446326
\(186\) −2.08815 −0.153110
\(187\) 4.13706 0.302532
\(188\) 7.00000 0.510527
\(189\) 0.643104 0.0467789
\(190\) −2.75302 −0.199725
\(191\) 0.753020 0.0544866 0.0272433 0.999629i \(-0.491327\pi\)
0.0272433 + 0.999629i \(0.491327\pi\)
\(192\) 1.00000 0.0721688
\(193\) −7.84117 −0.564420 −0.282210 0.959353i \(-0.591067\pi\)
−0.282210 + 0.959353i \(0.591067\pi\)
\(194\) 5.05861 0.363187
\(195\) 0 0
\(196\) −6.58642 −0.470458
\(197\) 10.2010 0.726794 0.363397 0.931634i \(-0.381617\pi\)
0.363397 + 0.931634i \(0.381617\pi\)
\(198\) −4.40581 −0.313108
\(199\) 1.65279 0.117163 0.0585817 0.998283i \(-0.481342\pi\)
0.0585817 + 0.998283i \(0.481342\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 1.45712 0.102778
\(202\) 6.81163 0.479264
\(203\) −0.0959157 −0.00673196
\(204\) 0.939001 0.0657432
\(205\) −5.74094 −0.400965
\(206\) 12.0858 0.842054
\(207\) −2.04892 −0.142410
\(208\) 0 0
\(209\) 12.1293 0.839001
\(210\) −0.643104 −0.0443784
\(211\) 11.7453 0.808576 0.404288 0.914632i \(-0.367519\pi\)
0.404288 + 0.914632i \(0.367519\pi\)
\(212\) 1.68664 0.115839
\(213\) −7.03684 −0.482156
\(214\) 14.6679 1.00267
\(215\) 4.69202 0.319993
\(216\) −1.00000 −0.0680414
\(217\) 1.34290 0.0911617
\(218\) −8.15883 −0.552586
\(219\) 14.8605 1.00418
\(220\) 4.40581 0.297040
\(221\) 0 0
\(222\) −6.07069 −0.407438
\(223\) −9.73855 −0.652141 −0.326071 0.945345i \(-0.605725\pi\)
−0.326071 + 0.945345i \(0.605725\pi\)
\(224\) −0.643104 −0.0429692
\(225\) 1.00000 0.0666667
\(226\) −16.8442 −1.12046
\(227\) −25.2500 −1.67590 −0.837949 0.545748i \(-0.816246\pi\)
−0.837949 + 0.545748i \(0.816246\pi\)
\(228\) 2.75302 0.182323
\(229\) 23.7845 1.57172 0.785861 0.618403i \(-0.212220\pi\)
0.785861 + 0.618403i \(0.212220\pi\)
\(230\) 2.04892 0.135102
\(231\) 2.83340 0.186424
\(232\) 0.149145 0.00979184
\(233\) 0.603875 0.0395612 0.0197806 0.999804i \(-0.493703\pi\)
0.0197806 + 0.999804i \(0.493703\pi\)
\(234\) 0 0
\(235\) 7.00000 0.456630
\(236\) 7.85623 0.511397
\(237\) 0.929312 0.0603653
\(238\) −0.603875 −0.0391434
\(239\) −28.5579 −1.84726 −0.923630 0.383286i \(-0.874792\pi\)
−0.923630 + 0.383286i \(0.874792\pi\)
\(240\) 1.00000 0.0645497
\(241\) 6.31096 0.406525 0.203262 0.979124i \(-0.434846\pi\)
0.203262 + 0.979124i \(0.434846\pi\)
\(242\) −8.41119 −0.540692
\(243\) 1.00000 0.0641500
\(244\) −12.0761 −0.773091
\(245\) −6.58642 −0.420791
\(246\) 5.74094 0.366029
\(247\) 0 0
\(248\) −2.08815 −0.132597
\(249\) 11.2959 0.715848
\(250\) −1.00000 −0.0632456
\(251\) 24.5972 1.55256 0.776280 0.630388i \(-0.217104\pi\)
0.776280 + 0.630388i \(0.217104\pi\)
\(252\) 0.643104 0.0405118
\(253\) −9.02715 −0.567532
\(254\) −4.85086 −0.304370
\(255\) 0.939001 0.0588025
\(256\) 1.00000 0.0625000
\(257\) −26.9463 −1.68086 −0.840432 0.541917i \(-0.817699\pi\)
−0.840432 + 0.541917i \(0.817699\pi\)
\(258\) −4.69202 −0.292112
\(259\) 3.90408 0.242588
\(260\) 0 0
\(261\) −0.149145 −0.00923184
\(262\) −1.52781 −0.0943885
\(263\) −24.0465 −1.48277 −0.741386 0.671079i \(-0.765831\pi\)
−0.741386 + 0.671079i \(0.765831\pi\)
\(264\) −4.40581 −0.271159
\(265\) 1.68664 0.103610
\(266\) −1.77048 −0.108555
\(267\) −7.47650 −0.457554
\(268\) 1.45712 0.0890080
\(269\) −7.17523 −0.437481 −0.218741 0.975783i \(-0.570195\pi\)
−0.218741 + 0.975783i \(0.570195\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 10.9825 0.667142 0.333571 0.942725i \(-0.391746\pi\)
0.333571 + 0.942725i \(0.391746\pi\)
\(272\) 0.939001 0.0569353
\(273\) 0 0
\(274\) 6.41119 0.387314
\(275\) 4.40581 0.265681
\(276\) −2.04892 −0.123330
\(277\) 16.2107 0.974009 0.487004 0.873400i \(-0.338090\pi\)
0.487004 + 0.873400i \(0.338090\pi\)
\(278\) 8.78017 0.526599
\(279\) 2.08815 0.125014
\(280\) −0.643104 −0.0384328
\(281\) −12.9584 −0.773032 −0.386516 0.922283i \(-0.626322\pi\)
−0.386516 + 0.922283i \(0.626322\pi\)
\(282\) −7.00000 −0.416844
\(283\) 21.5415 1.28051 0.640256 0.768162i \(-0.278828\pi\)
0.640256 + 0.768162i \(0.278828\pi\)
\(284\) −7.03684 −0.417559
\(285\) 2.75302 0.163075
\(286\) 0 0
\(287\) −3.69202 −0.217933
\(288\) −1.00000 −0.0589256
\(289\) −16.1183 −0.948134
\(290\) 0.149145 0.00875809
\(291\) −5.05861 −0.296541
\(292\) 14.8605 0.869648
\(293\) 9.57971 0.559653 0.279826 0.960051i \(-0.409723\pi\)
0.279826 + 0.960051i \(0.409723\pi\)
\(294\) 6.58642 0.384128
\(295\) 7.85623 0.457408
\(296\) −6.07069 −0.352852
\(297\) 4.40581 0.255651
\(298\) 22.6504 1.31210
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 3.01746 0.173923
\(302\) −10.2959 −0.592462
\(303\) −6.81163 −0.391318
\(304\) 2.75302 0.157897
\(305\) −12.0761 −0.691473
\(306\) −0.939001 −0.0536791
\(307\) 19.0140 1.08519 0.542593 0.839996i \(-0.317443\pi\)
0.542593 + 0.839996i \(0.317443\pi\)
\(308\) 2.83340 0.161448
\(309\) −12.0858 −0.687534
\(310\) −2.08815 −0.118599
\(311\) −3.74764 −0.212509 −0.106255 0.994339i \(-0.533886\pi\)
−0.106255 + 0.994339i \(0.533886\pi\)
\(312\) 0 0
\(313\) 25.1196 1.41984 0.709922 0.704280i \(-0.248730\pi\)
0.709922 + 0.704280i \(0.248730\pi\)
\(314\) −0.994623 −0.0561298
\(315\) 0.643104 0.0362348
\(316\) 0.929312 0.0522779
\(317\) 12.6679 0.711498 0.355749 0.934582i \(-0.384226\pi\)
0.355749 + 0.934582i \(0.384226\pi\)
\(318\) −1.68664 −0.0945823
\(319\) −0.657105 −0.0367908
\(320\) 1.00000 0.0559017
\(321\) −14.6679 −0.818680
\(322\) 1.31767 0.0734307
\(323\) 2.58509 0.143838
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −21.6015 −1.19640
\(327\) 8.15883 0.451185
\(328\) 5.74094 0.316990
\(329\) 4.50173 0.248188
\(330\) −4.40581 −0.242532
\(331\) −6.43057 −0.353456 −0.176728 0.984260i \(-0.556551\pi\)
−0.176728 + 0.984260i \(0.556551\pi\)
\(332\) 11.2959 0.619943
\(333\) 6.07069 0.332672
\(334\) −7.96615 −0.435888
\(335\) 1.45712 0.0796112
\(336\) 0.643104 0.0350842
\(337\) 10.5496 0.574672 0.287336 0.957830i \(-0.407230\pi\)
0.287336 + 0.957830i \(0.407230\pi\)
\(338\) 0 0
\(339\) 16.8442 0.914849
\(340\) 0.939001 0.0509245
\(341\) 9.19998 0.498207
\(342\) −2.75302 −0.148866
\(343\) −8.73748 −0.471780
\(344\) −4.69202 −0.252977
\(345\) −2.04892 −0.110310
\(346\) −4.11960 −0.221471
\(347\) 9.87130 0.529919 0.264960 0.964260i \(-0.414641\pi\)
0.264960 + 0.964260i \(0.414641\pi\)
\(348\) −0.149145 −0.00799501
\(349\) −30.5338 −1.63444 −0.817218 0.576329i \(-0.804485\pi\)
−0.817218 + 0.576329i \(0.804485\pi\)
\(350\) −0.643104 −0.0343754
\(351\) 0 0
\(352\) −4.40581 −0.234831
\(353\) −24.7047 −1.31490 −0.657449 0.753499i \(-0.728365\pi\)
−0.657449 + 0.753499i \(0.728365\pi\)
\(354\) −7.85623 −0.417554
\(355\) −7.03684 −0.373476
\(356\) −7.47650 −0.396254
\(357\) 0.603875 0.0319605
\(358\) 8.78017 0.464046
\(359\) −11.5888 −0.611634 −0.305817 0.952090i \(-0.598930\pi\)
−0.305817 + 0.952090i \(0.598930\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −11.4209 −0.601099
\(362\) −19.4741 −1.02354
\(363\) 8.41119 0.441473
\(364\) 0 0
\(365\) 14.8605 0.777836
\(366\) 12.0761 0.631226
\(367\) −20.5579 −1.07312 −0.536558 0.843863i \(-0.680276\pi\)
−0.536558 + 0.843863i \(0.680276\pi\)
\(368\) −2.04892 −0.106807
\(369\) −5.74094 −0.298861
\(370\) −6.07069 −0.315600
\(371\) 1.08469 0.0563142
\(372\) 2.08815 0.108265
\(373\) 1.31037 0.0678485 0.0339242 0.999424i \(-0.489200\pi\)
0.0339242 + 0.999424i \(0.489200\pi\)
\(374\) −4.13706 −0.213922
\(375\) 1.00000 0.0516398
\(376\) −7.00000 −0.360997
\(377\) 0 0
\(378\) −0.643104 −0.0330777
\(379\) 28.1551 1.44623 0.723115 0.690727i \(-0.242709\pi\)
0.723115 + 0.690727i \(0.242709\pi\)
\(380\) 2.75302 0.141227
\(381\) 4.85086 0.248517
\(382\) −0.753020 −0.0385279
\(383\) −18.6679 −0.953883 −0.476941 0.878935i \(-0.658255\pi\)
−0.476941 + 0.878935i \(0.658255\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 2.83340 0.144403
\(386\) 7.84117 0.399105
\(387\) 4.69202 0.238509
\(388\) −5.05861 −0.256812
\(389\) −29.6752 −1.50459 −0.752295 0.658826i \(-0.771053\pi\)
−0.752295 + 0.658826i \(0.771053\pi\)
\(390\) 0 0
\(391\) −1.92394 −0.0972976
\(392\) 6.58642 0.332664
\(393\) 1.52781 0.0770679
\(394\) −10.2010 −0.513921
\(395\) 0.929312 0.0467588
\(396\) 4.40581 0.221400
\(397\) −28.7536 −1.44310 −0.721551 0.692361i \(-0.756571\pi\)
−0.721551 + 0.692361i \(0.756571\pi\)
\(398\) −1.65279 −0.0828470
\(399\) 1.77048 0.0886348
\(400\) 1.00000 0.0500000
\(401\) 18.6993 0.933799 0.466900 0.884310i \(-0.345371\pi\)
0.466900 + 0.884310i \(0.345371\pi\)
\(402\) −1.45712 −0.0726747
\(403\) 0 0
\(404\) −6.81163 −0.338891
\(405\) 1.00000 0.0496904
\(406\) 0.0959157 0.00476022
\(407\) 26.7463 1.32577
\(408\) −0.939001 −0.0464875
\(409\) 10.6233 0.525286 0.262643 0.964893i \(-0.415406\pi\)
0.262643 + 0.964893i \(0.415406\pi\)
\(410\) 5.74094 0.283525
\(411\) −6.41119 −0.316241
\(412\) −12.0858 −0.595422
\(413\) 5.05238 0.248611
\(414\) 2.04892 0.100699
\(415\) 11.2959 0.554494
\(416\) 0 0
\(417\) −8.78017 −0.429967
\(418\) −12.1293 −0.593263
\(419\) 19.4198 0.948720 0.474360 0.880331i \(-0.342680\pi\)
0.474360 + 0.880331i \(0.342680\pi\)
\(420\) 0.643104 0.0313803
\(421\) 20.5719 1.00262 0.501308 0.865269i \(-0.332853\pi\)
0.501308 + 0.865269i \(0.332853\pi\)
\(422\) −11.7453 −0.571750
\(423\) 7.00000 0.340352
\(424\) −1.68664 −0.0819107
\(425\) 0.939001 0.0455482
\(426\) 7.03684 0.340936
\(427\) −7.76617 −0.375831
\(428\) −14.6679 −0.708998
\(429\) 0 0
\(430\) −4.69202 −0.226269
\(431\) 33.8079 1.62847 0.814235 0.580536i \(-0.197157\pi\)
0.814235 + 0.580536i \(0.197157\pi\)
\(432\) 1.00000 0.0481125
\(433\) 27.2838 1.31118 0.655588 0.755119i \(-0.272421\pi\)
0.655588 + 0.755119i \(0.272421\pi\)
\(434\) −1.34290 −0.0644610
\(435\) −0.149145 −0.00715095
\(436\) 8.15883 0.390737
\(437\) −5.64071 −0.269832
\(438\) −14.8605 −0.710064
\(439\) −1.56943 −0.0749049 −0.0374525 0.999298i \(-0.511924\pi\)
−0.0374525 + 0.999298i \(0.511924\pi\)
\(440\) −4.40581 −0.210039
\(441\) −6.58642 −0.313639
\(442\) 0 0
\(443\) 38.0489 1.80776 0.903879 0.427788i \(-0.140707\pi\)
0.903879 + 0.427788i \(0.140707\pi\)
\(444\) 6.07069 0.288102
\(445\) −7.47650 −0.354420
\(446\) 9.73855 0.461134
\(447\) −22.6504 −1.07133
\(448\) 0.643104 0.0303838
\(449\) −24.2373 −1.14383 −0.571914 0.820313i \(-0.693799\pi\)
−0.571914 + 0.820313i \(0.693799\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −25.2935 −1.19102
\(452\) 16.8442 0.792282
\(453\) 10.2959 0.483743
\(454\) 25.2500 1.18504
\(455\) 0 0
\(456\) −2.75302 −0.128922
\(457\) 9.75063 0.456115 0.228058 0.973648i \(-0.426763\pi\)
0.228058 + 0.973648i \(0.426763\pi\)
\(458\) −23.7845 −1.11138
\(459\) 0.939001 0.0438288
\(460\) −2.04892 −0.0955313
\(461\) 31.0358 1.44548 0.722740 0.691120i \(-0.242882\pi\)
0.722740 + 0.691120i \(0.242882\pi\)
\(462\) −2.83340 −0.131822
\(463\) 33.1618 1.54116 0.770580 0.637343i \(-0.219967\pi\)
0.770580 + 0.637343i \(0.219967\pi\)
\(464\) −0.149145 −0.00692388
\(465\) 2.08815 0.0968355
\(466\) −0.603875 −0.0279740
\(467\) −16.1153 −0.745727 −0.372863 0.927886i \(-0.621624\pi\)
−0.372863 + 0.927886i \(0.621624\pi\)
\(468\) 0 0
\(469\) 0.937082 0.0432704
\(470\) −7.00000 −0.322886
\(471\) 0.994623 0.0458298
\(472\) −7.85623 −0.361612
\(473\) 20.6722 0.950507
\(474\) −0.929312 −0.0426847
\(475\) 2.75302 0.126317
\(476\) 0.603875 0.0276786
\(477\) 1.68664 0.0772262
\(478\) 28.5579 1.30621
\(479\) −30.6474 −1.40032 −0.700158 0.713988i \(-0.746887\pi\)
−0.700158 + 0.713988i \(0.746887\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −6.31096 −0.287456
\(483\) −1.31767 −0.0599559
\(484\) 8.41119 0.382327
\(485\) −5.05861 −0.229699
\(486\) −1.00000 −0.0453609
\(487\) 17.6829 0.801290 0.400645 0.916233i \(-0.368786\pi\)
0.400645 + 0.916233i \(0.368786\pi\)
\(488\) 12.0761 0.546658
\(489\) 21.6015 0.976853
\(490\) 6.58642 0.297544
\(491\) −28.5435 −1.28815 −0.644074 0.764963i \(-0.722757\pi\)
−0.644074 + 0.764963i \(0.722757\pi\)
\(492\) −5.74094 −0.258822
\(493\) −0.140047 −0.00630741
\(494\) 0 0
\(495\) 4.40581 0.198027
\(496\) 2.08815 0.0937605
\(497\) −4.52542 −0.202993
\(498\) −11.2959 −0.506181
\(499\) −27.9409 −1.25081 −0.625404 0.780301i \(-0.715066\pi\)
−0.625404 + 0.780301i \(0.715066\pi\)
\(500\) 1.00000 0.0447214
\(501\) 7.96615 0.355901
\(502\) −24.5972 −1.09783
\(503\) 31.0901 1.38624 0.693119 0.720823i \(-0.256236\pi\)
0.693119 + 0.720823i \(0.256236\pi\)
\(504\) −0.643104 −0.0286461
\(505\) −6.81163 −0.303113
\(506\) 9.02715 0.401306
\(507\) 0 0
\(508\) 4.85086 0.215222
\(509\) 0.0217703 0.000964950 0 0.000482475 1.00000i \(-0.499846\pi\)
0.000482475 1.00000i \(0.499846\pi\)
\(510\) −0.939001 −0.0415797
\(511\) 9.55688 0.422771
\(512\) −1.00000 −0.0441942
\(513\) 2.75302 0.121549
\(514\) 26.9463 1.18855
\(515\) −12.0858 −0.532562
\(516\) 4.69202 0.206555
\(517\) 30.8407 1.35637
\(518\) −3.90408 −0.171536
\(519\) 4.11960 0.180831
\(520\) 0 0
\(521\) 18.0747 0.791869 0.395934 0.918279i \(-0.370421\pi\)
0.395934 + 0.918279i \(0.370421\pi\)
\(522\) 0.149145 0.00652790
\(523\) −15.3793 −0.672488 −0.336244 0.941775i \(-0.609157\pi\)
−0.336244 + 0.941775i \(0.609157\pi\)
\(524\) 1.52781 0.0667427
\(525\) 0.643104 0.0280674
\(526\) 24.0465 1.04848
\(527\) 1.96077 0.0854125
\(528\) 4.40581 0.191738
\(529\) −18.8019 −0.817476
\(530\) −1.68664 −0.0732632
\(531\) 7.85623 0.340931
\(532\) 1.77048 0.0767600
\(533\) 0 0
\(534\) 7.47650 0.323540
\(535\) −14.6679 −0.634147
\(536\) −1.45712 −0.0629381
\(537\) −8.78017 −0.378892
\(538\) 7.17523 0.309346
\(539\) −29.0185 −1.24992
\(540\) 1.00000 0.0430331
\(541\) 5.06339 0.217692 0.108846 0.994059i \(-0.465284\pi\)
0.108846 + 0.994059i \(0.465284\pi\)
\(542\) −10.9825 −0.471741
\(543\) 19.4741 0.835714
\(544\) −0.939001 −0.0402593
\(545\) 8.15883 0.349486
\(546\) 0 0
\(547\) 8.77107 0.375024 0.187512 0.982262i \(-0.439958\pi\)
0.187512 + 0.982262i \(0.439958\pi\)
\(548\) −6.41119 −0.273872
\(549\) −12.0761 −0.515394
\(550\) −4.40581 −0.187865
\(551\) −0.410599 −0.0174921
\(552\) 2.04892 0.0872077
\(553\) 0.597645 0.0254144
\(554\) −16.2107 −0.688728
\(555\) 6.07069 0.257686
\(556\) −8.78017 −0.372362
\(557\) −8.10215 −0.343299 −0.171649 0.985158i \(-0.554910\pi\)
−0.171649 + 0.985158i \(0.554910\pi\)
\(558\) −2.08815 −0.0883983
\(559\) 0 0
\(560\) 0.643104 0.0271761
\(561\) 4.13706 0.174667
\(562\) 12.9584 0.546616
\(563\) −6.86699 −0.289409 −0.144704 0.989475i \(-0.546223\pi\)
−0.144704 + 0.989475i \(0.546223\pi\)
\(564\) 7.00000 0.294753
\(565\) 16.8442 0.708639
\(566\) −21.5415 −0.905459
\(567\) 0.643104 0.0270078
\(568\) 7.03684 0.295259
\(569\) 6.93171 0.290592 0.145296 0.989388i \(-0.453587\pi\)
0.145296 + 0.989388i \(0.453587\pi\)
\(570\) −2.75302 −0.115311
\(571\) −24.2760 −1.01592 −0.507960 0.861380i \(-0.669600\pi\)
−0.507960 + 0.861380i \(0.669600\pi\)
\(572\) 0 0
\(573\) 0.753020 0.0314579
\(574\) 3.69202 0.154102
\(575\) −2.04892 −0.0854458
\(576\) 1.00000 0.0416667
\(577\) 13.8189 0.575289 0.287645 0.957737i \(-0.407128\pi\)
0.287645 + 0.957737i \(0.407128\pi\)
\(578\) 16.1183 0.670432
\(579\) −7.84117 −0.325868
\(580\) −0.149145 −0.00619291
\(581\) 7.26444 0.301380
\(582\) 5.05861 0.209686
\(583\) 7.43104 0.307762
\(584\) −14.8605 −0.614934
\(585\) 0 0
\(586\) −9.57971 −0.395734
\(587\) 27.6165 1.13986 0.569928 0.821694i \(-0.306971\pi\)
0.569928 + 0.821694i \(0.306971\pi\)
\(588\) −6.58642 −0.271619
\(589\) 5.74871 0.236871
\(590\) −7.85623 −0.323436
\(591\) 10.2010 0.419615
\(592\) 6.07069 0.249504
\(593\) −30.3913 −1.24802 −0.624011 0.781415i \(-0.714498\pi\)
−0.624011 + 0.781415i \(0.714498\pi\)
\(594\) −4.40581 −0.180773
\(595\) 0.603875 0.0247565
\(596\) −22.6504 −0.927797
\(597\) 1.65279 0.0676443
\(598\) 0 0
\(599\) 10.6480 0.435066 0.217533 0.976053i \(-0.430199\pi\)
0.217533 + 0.976053i \(0.430199\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 42.2495 1.72339 0.861696 0.507424i \(-0.169402\pi\)
0.861696 + 0.507424i \(0.169402\pi\)
\(602\) −3.01746 −0.122982
\(603\) 1.45712 0.0593387
\(604\) 10.2959 0.418934
\(605\) 8.41119 0.341964
\(606\) 6.81163 0.276703
\(607\) −17.6209 −0.715209 −0.357604 0.933873i \(-0.616406\pi\)
−0.357604 + 0.933873i \(0.616406\pi\)
\(608\) −2.75302 −0.111650
\(609\) −0.0959157 −0.00388670
\(610\) 12.0761 0.488946
\(611\) 0 0
\(612\) 0.939001 0.0379569
\(613\) −27.9527 −1.12900 −0.564500 0.825433i \(-0.690931\pi\)
−0.564500 + 0.825433i \(0.690931\pi\)
\(614\) −19.0140 −0.767343
\(615\) −5.74094 −0.231497
\(616\) −2.83340 −0.114161
\(617\) 14.9065 0.600112 0.300056 0.953922i \(-0.402995\pi\)
0.300056 + 0.953922i \(0.402995\pi\)
\(618\) 12.0858 0.486160
\(619\) −33.7493 −1.35650 −0.678249 0.734832i \(-0.737261\pi\)
−0.678249 + 0.734832i \(0.737261\pi\)
\(620\) 2.08815 0.0838620
\(621\) −2.04892 −0.0822202
\(622\) 3.74764 0.150267
\(623\) −4.80817 −0.192635
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −25.1196 −1.00398
\(627\) 12.1293 0.484397
\(628\) 0.994623 0.0396898
\(629\) 5.70038 0.227289
\(630\) −0.643104 −0.0256219
\(631\) −37.2519 −1.48297 −0.741487 0.670967i \(-0.765879\pi\)
−0.741487 + 0.670967i \(0.765879\pi\)
\(632\) −0.929312 −0.0369661
\(633\) 11.7453 0.466832
\(634\) −12.6679 −0.503105
\(635\) 4.85086 0.192500
\(636\) 1.68664 0.0668798
\(637\) 0 0
\(638\) 0.657105 0.0260150
\(639\) −7.03684 −0.278373
\(640\) −1.00000 −0.0395285
\(641\) 24.8203 0.980341 0.490170 0.871627i \(-0.336935\pi\)
0.490170 + 0.871627i \(0.336935\pi\)
\(642\) 14.6679 0.578894
\(643\) −5.66248 −0.223306 −0.111653 0.993747i \(-0.535615\pi\)
−0.111653 + 0.993747i \(0.535615\pi\)
\(644\) −1.31767 −0.0519234
\(645\) 4.69202 0.184748
\(646\) −2.58509 −0.101709
\(647\) −27.0465 −1.06331 −0.531654 0.846961i \(-0.678429\pi\)
−0.531654 + 0.846961i \(0.678429\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 34.6131 1.35868
\(650\) 0 0
\(651\) 1.34290 0.0526322
\(652\) 21.6015 0.845979
\(653\) 42.7741 1.67388 0.836939 0.547296i \(-0.184343\pi\)
0.836939 + 0.547296i \(0.184343\pi\)
\(654\) −8.15883 −0.319036
\(655\) 1.52781 0.0596965
\(656\) −5.74094 −0.224146
\(657\) 14.8605 0.579765
\(658\) −4.50173 −0.175496
\(659\) 21.5623 0.839946 0.419973 0.907537i \(-0.362040\pi\)
0.419973 + 0.907537i \(0.362040\pi\)
\(660\) 4.40581 0.171496
\(661\) 48.5002 1.88644 0.943219 0.332170i \(-0.107781\pi\)
0.943219 + 0.332170i \(0.107781\pi\)
\(662\) 6.43057 0.249931
\(663\) 0 0
\(664\) −11.2959 −0.438366
\(665\) 1.77048 0.0686562
\(666\) −6.07069 −0.235234
\(667\) 0.305586 0.0118323
\(668\) 7.96615 0.308220
\(669\) −9.73855 −0.376514
\(670\) −1.45712 −0.0562936
\(671\) −53.2049 −2.05395
\(672\) −0.643104 −0.0248083
\(673\) 30.4711 1.17458 0.587288 0.809378i \(-0.300196\pi\)
0.587288 + 0.809378i \(0.300196\pi\)
\(674\) −10.5496 −0.406355
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −8.57971 −0.329745 −0.164873 0.986315i \(-0.552721\pi\)
−0.164873 + 0.986315i \(0.552721\pi\)
\(678\) −16.8442 −0.646896
\(679\) −3.25321 −0.124847
\(680\) −0.939001 −0.0360090
\(681\) −25.2500 −0.967581
\(682\) −9.19998 −0.352285
\(683\) 27.8890 1.06714 0.533572 0.845755i \(-0.320849\pi\)
0.533572 + 0.845755i \(0.320849\pi\)
\(684\) 2.75302 0.105264
\(685\) −6.41119 −0.244959
\(686\) 8.73748 0.333599
\(687\) 23.7845 0.907434
\(688\) 4.69202 0.178882
\(689\) 0 0
\(690\) 2.04892 0.0780010
\(691\) 39.4198 1.49960 0.749800 0.661664i \(-0.230150\pi\)
0.749800 + 0.661664i \(0.230150\pi\)
\(692\) 4.11960 0.156604
\(693\) 2.83340 0.107632
\(694\) −9.87130 −0.374709
\(695\) −8.78017 −0.333051
\(696\) 0.149145 0.00565332
\(697\) −5.39075 −0.204189
\(698\) 30.5338 1.15572
\(699\) 0.603875 0.0228407
\(700\) 0.643104 0.0243071
\(701\) −51.7101 −1.95306 −0.976531 0.215376i \(-0.930902\pi\)
−0.976531 + 0.215376i \(0.930902\pi\)
\(702\) 0 0
\(703\) 16.7127 0.630332
\(704\) 4.40581 0.166050
\(705\) 7.00000 0.263635
\(706\) 24.7047 0.929773
\(707\) −4.38059 −0.164749
\(708\) 7.85623 0.295255
\(709\) −12.7278 −0.478002 −0.239001 0.971019i \(-0.576820\pi\)
−0.239001 + 0.971019i \(0.576820\pi\)
\(710\) 7.03684 0.264088
\(711\) 0.929312 0.0348519
\(712\) 7.47650 0.280194
\(713\) −4.27844 −0.160229
\(714\) −0.603875 −0.0225995
\(715\) 0 0
\(716\) −8.78017 −0.328130
\(717\) −28.5579 −1.06652
\(718\) 11.5888 0.432491
\(719\) −11.7342 −0.437613 −0.218807 0.975768i \(-0.570216\pi\)
−0.218807 + 0.975768i \(0.570216\pi\)
\(720\) 1.00000 0.0372678
\(721\) −7.77240 −0.289459
\(722\) 11.4209 0.425041
\(723\) 6.31096 0.234707
\(724\) 19.4741 0.723750
\(725\) −0.149145 −0.00553910
\(726\) −8.41119 −0.312169
\(727\) −6.84979 −0.254045 −0.127022 0.991900i \(-0.540542\pi\)
−0.127022 + 0.991900i \(0.540542\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −14.8605 −0.550013
\(731\) 4.40581 0.162955
\(732\) −12.0761 −0.446344
\(733\) −0.224144 −0.00827896 −0.00413948 0.999991i \(-0.501318\pi\)
−0.00413948 + 0.999991i \(0.501318\pi\)
\(734\) 20.5579 0.758807
\(735\) −6.58642 −0.242944
\(736\) 2.04892 0.0755241
\(737\) 6.41981 0.236477
\(738\) 5.74094 0.211327
\(739\) −2.39075 −0.0879451 −0.0439725 0.999033i \(-0.514001\pi\)
−0.0439725 + 0.999033i \(0.514001\pi\)
\(740\) 6.07069 0.223163
\(741\) 0 0
\(742\) −1.08469 −0.0398202
\(743\) −25.5144 −0.936033 −0.468016 0.883720i \(-0.655031\pi\)
−0.468016 + 0.883720i \(0.655031\pi\)
\(744\) −2.08815 −0.0765551
\(745\) −22.6504 −0.829846
\(746\) −1.31037 −0.0479761
\(747\) 11.2959 0.413295
\(748\) 4.13706 0.151266
\(749\) −9.43296 −0.344673
\(750\) −1.00000 −0.0365148
\(751\) 18.2755 0.666881 0.333440 0.942771i \(-0.391790\pi\)
0.333440 + 0.942771i \(0.391790\pi\)
\(752\) 7.00000 0.255264
\(753\) 24.5972 0.896371
\(754\) 0 0
\(755\) 10.2959 0.374706
\(756\) 0.643104 0.0233895
\(757\) −20.2301 −0.735276 −0.367638 0.929969i \(-0.619833\pi\)
−0.367638 + 0.929969i \(0.619833\pi\)
\(758\) −28.1551 −1.02264
\(759\) −9.02715 −0.327665
\(760\) −2.75302 −0.0998625
\(761\) −0.217440 −0.00788218 −0.00394109 0.999992i \(-0.501254\pi\)
−0.00394109 + 0.999992i \(0.501254\pi\)
\(762\) −4.85086 −0.175728
\(763\) 5.24698 0.189953
\(764\) 0.753020 0.0272433
\(765\) 0.939001 0.0339497
\(766\) 18.6679 0.674497
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 0.124982 0.00450696 0.00225348 0.999997i \(-0.499283\pi\)
0.00225348 + 0.999997i \(0.499283\pi\)
\(770\) −2.83340 −0.102109
\(771\) −26.9463 −0.970447
\(772\) −7.84117 −0.282210
\(773\) 11.7028 0.420920 0.210460 0.977603i \(-0.432504\pi\)
0.210460 + 0.977603i \(0.432504\pi\)
\(774\) −4.69202 −0.168651
\(775\) 2.08815 0.0750084
\(776\) 5.05861 0.181593
\(777\) 3.90408 0.140058
\(778\) 29.6752 1.06391
\(779\) −15.8049 −0.566270
\(780\) 0 0
\(781\) −31.0030 −1.10937
\(782\) 1.92394 0.0687998
\(783\) −0.149145 −0.00533000
\(784\) −6.58642 −0.235229
\(785\) 0.994623 0.0354996
\(786\) −1.52781 −0.0544952
\(787\) −50.6926 −1.80700 −0.903498 0.428592i \(-0.859010\pi\)
−0.903498 + 0.428592i \(0.859010\pi\)
\(788\) 10.2010 0.363397
\(789\) −24.0465 −0.856079
\(790\) −0.929312 −0.0330635
\(791\) 10.8325 0.385161
\(792\) −4.40581 −0.156554
\(793\) 0 0
\(794\) 28.7536 1.02043
\(795\) 1.68664 0.0598191
\(796\) 1.65279 0.0585817
\(797\) 45.7977 1.62224 0.811120 0.584880i \(-0.198859\pi\)
0.811120 + 0.584880i \(0.198859\pi\)
\(798\) −1.77048 −0.0626743
\(799\) 6.57301 0.232536
\(800\) −1.00000 −0.0353553
\(801\) −7.47650 −0.264169
\(802\) −18.6993 −0.660296
\(803\) 65.4728 2.31048
\(804\) 1.45712 0.0513888
\(805\) −1.31767 −0.0464417
\(806\) 0 0
\(807\) −7.17523 −0.252580
\(808\) 6.81163 0.239632
\(809\) 22.5579 0.793095 0.396548 0.918014i \(-0.370208\pi\)
0.396548 + 0.918014i \(0.370208\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 19.2728 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(812\) −0.0959157 −0.00336598
\(813\) 10.9825 0.385175
\(814\) −26.7463 −0.937458
\(815\) 21.6015 0.756667
\(816\) 0.939001 0.0328716
\(817\) 12.9172 0.451917
\(818\) −10.6233 −0.371433
\(819\) 0 0
\(820\) −5.74094 −0.200482
\(821\) 11.5036 0.401480 0.200740 0.979645i \(-0.435665\pi\)
0.200740 + 0.979645i \(0.435665\pi\)
\(822\) 6.41119 0.223616
\(823\) −25.2610 −0.880542 −0.440271 0.897865i \(-0.645118\pi\)
−0.440271 + 0.897865i \(0.645118\pi\)
\(824\) 12.0858 0.421027
\(825\) 4.40581 0.153391
\(826\) −5.05238 −0.175795
\(827\) −7.09677 −0.246779 −0.123389 0.992358i \(-0.539376\pi\)
−0.123389 + 0.992358i \(0.539376\pi\)
\(828\) −2.04892 −0.0712048
\(829\) 10.1226 0.351572 0.175786 0.984428i \(-0.443753\pi\)
0.175786 + 0.984428i \(0.443753\pi\)
\(830\) −11.2959 −0.392086
\(831\) 16.2107 0.562344
\(832\) 0 0
\(833\) −6.18465 −0.214286
\(834\) 8.78017 0.304032
\(835\) 7.96615 0.275680
\(836\) 12.1293 0.419500
\(837\) 2.08815 0.0721769
\(838\) −19.4198 −0.670846
\(839\) −43.0844 −1.48744 −0.743720 0.668492i \(-0.766940\pi\)
−0.743720 + 0.668492i \(0.766940\pi\)
\(840\) −0.643104 −0.0221892
\(841\) −28.9778 −0.999233
\(842\) −20.5719 −0.708956
\(843\) −12.9584 −0.446310
\(844\) 11.7453 0.404288
\(845\) 0 0
\(846\) −7.00000 −0.240665
\(847\) 5.40927 0.185865
\(848\) 1.68664 0.0579196
\(849\) 21.5415 0.739304
\(850\) −0.939001 −0.0322075
\(851\) −12.4383 −0.426381
\(852\) −7.03684 −0.241078
\(853\) 43.7837 1.49913 0.749563 0.661933i \(-0.230263\pi\)
0.749563 + 0.661933i \(0.230263\pi\)
\(854\) 7.76617 0.265753
\(855\) 2.75302 0.0941513
\(856\) 14.6679 0.501337
\(857\) −16.3067 −0.557025 −0.278512 0.960433i \(-0.589841\pi\)
−0.278512 + 0.960433i \(0.589841\pi\)
\(858\) 0 0
\(859\) −31.8629 −1.08715 −0.543575 0.839361i \(-0.682930\pi\)
−0.543575 + 0.839361i \(0.682930\pi\)
\(860\) 4.69202 0.159997
\(861\) −3.69202 −0.125824
\(862\) −33.8079 −1.15150
\(863\) −0.511418 −0.0174089 −0.00870443 0.999962i \(-0.502771\pi\)
−0.00870443 + 0.999962i \(0.502771\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 4.11960 0.140071
\(866\) −27.2838 −0.927142
\(867\) −16.1183 −0.547405
\(868\) 1.34290 0.0455808
\(869\) 4.09438 0.138892
\(870\) 0.149145 0.00505649
\(871\) 0 0
\(872\) −8.15883 −0.276293
\(873\) −5.05861 −0.171208
\(874\) 5.64071 0.190800
\(875\) 0.643104 0.0217409
\(876\) 14.8605 0.502091
\(877\) 3.83638 0.129545 0.0647727 0.997900i \(-0.479368\pi\)
0.0647727 + 0.997900i \(0.479368\pi\)
\(878\) 1.56943 0.0529658
\(879\) 9.57971 0.323116
\(880\) 4.40581 0.148520
\(881\) −0.499336 −0.0168231 −0.00841153 0.999965i \(-0.502678\pi\)
−0.00841153 + 0.999965i \(0.502678\pi\)
\(882\) 6.58642 0.221776
\(883\) −38.8993 −1.30907 −0.654533 0.756034i \(-0.727135\pi\)
−0.654533 + 0.756034i \(0.727135\pi\)
\(884\) 0 0
\(885\) 7.85623 0.264084
\(886\) −38.0489 −1.27828
\(887\) −8.32245 −0.279441 −0.139720 0.990191i \(-0.544620\pi\)
−0.139720 + 0.990191i \(0.544620\pi\)
\(888\) −6.07069 −0.203719
\(889\) 3.11960 0.104628
\(890\) 7.47650 0.250613
\(891\) 4.40581 0.147600
\(892\) −9.73855 −0.326071
\(893\) 19.2711 0.644884
\(894\) 22.6504 0.757543
\(895\) −8.78017 −0.293489
\(896\) −0.643104 −0.0214846
\(897\) 0 0
\(898\) 24.2373 0.808809
\(899\) −0.311436 −0.0103870
\(900\) 1.00000 0.0333333
\(901\) 1.58376 0.0527627
\(902\) 25.2935 0.842182
\(903\) 3.01746 0.100415
\(904\) −16.8442 −0.560228
\(905\) 19.4741 0.647341
\(906\) −10.2959 −0.342058
\(907\) 27.4566 0.911683 0.455842 0.890061i \(-0.349338\pi\)
0.455842 + 0.890061i \(0.349338\pi\)
\(908\) −25.2500 −0.837949
\(909\) −6.81163 −0.225927
\(910\) 0 0
\(911\) −29.7224 −0.984748 −0.492374 0.870384i \(-0.663871\pi\)
−0.492374 + 0.870384i \(0.663871\pi\)
\(912\) 2.75302 0.0911616
\(913\) 49.7676 1.64707
\(914\) −9.75063 −0.322522
\(915\) −12.0761 −0.399222
\(916\) 23.7845 0.785861
\(917\) 0.982542 0.0324464
\(918\) −0.939001 −0.0309917
\(919\) −2.02954 −0.0669483 −0.0334742 0.999440i \(-0.510657\pi\)
−0.0334742 + 0.999440i \(0.510657\pi\)
\(920\) 2.04892 0.0675508
\(921\) 19.0140 0.626533
\(922\) −31.0358 −1.02211
\(923\) 0 0
\(924\) 2.83340 0.0932119
\(925\) 6.07069 0.199603
\(926\) −33.1618 −1.08976
\(927\) −12.0858 −0.396948
\(928\) 0.149145 0.00489592
\(929\) 0.815938 0.0267701 0.0133850 0.999910i \(-0.495739\pi\)
0.0133850 + 0.999910i \(0.495739\pi\)
\(930\) −2.08815 −0.0684730
\(931\) −18.1325 −0.594270
\(932\) 0.603875 0.0197806
\(933\) −3.74764 −0.122692
\(934\) 16.1153 0.527308
\(935\) 4.13706 0.135296
\(936\) 0 0
\(937\) −29.5864 −0.966546 −0.483273 0.875470i \(-0.660552\pi\)
−0.483273 + 0.875470i \(0.660552\pi\)
\(938\) −0.937082 −0.0305968
\(939\) 25.1196 0.819747
\(940\) 7.00000 0.228315
\(941\) 30.2573 0.986358 0.493179 0.869928i \(-0.335835\pi\)
0.493179 + 0.869928i \(0.335835\pi\)
\(942\) −0.994623 −0.0324066
\(943\) 11.7627 0.383047
\(944\) 7.85623 0.255699
\(945\) 0.643104 0.0209202
\(946\) −20.6722 −0.672110
\(947\) 7.12152 0.231418 0.115709 0.993283i \(-0.463086\pi\)
0.115709 + 0.993283i \(0.463086\pi\)
\(948\) 0.929312 0.0301827
\(949\) 0 0
\(950\) −2.75302 −0.0893198
\(951\) 12.6679 0.410783
\(952\) −0.603875 −0.0195717
\(953\) −38.3279 −1.24156 −0.620782 0.783983i \(-0.713185\pi\)
−0.620782 + 0.783983i \(0.713185\pi\)
\(954\) −1.68664 −0.0546071
\(955\) 0.753020 0.0243672
\(956\) −28.5579 −0.923630
\(957\) −0.657105 −0.0212412
\(958\) 30.6474 0.990173
\(959\) −4.12306 −0.133141
\(960\) 1.00000 0.0322749
\(961\) −26.6396 −0.859343
\(962\) 0 0
\(963\) −14.6679 −0.472665
\(964\) 6.31096 0.203262
\(965\) −7.84117 −0.252416
\(966\) 1.31767 0.0423952
\(967\) −50.6015 −1.62723 −0.813617 0.581401i \(-0.802505\pi\)
−0.813617 + 0.581401i \(0.802505\pi\)
\(968\) −8.41119 −0.270346
\(969\) 2.58509 0.0830450
\(970\) 5.05861 0.162422
\(971\) −27.3026 −0.876182 −0.438091 0.898931i \(-0.644345\pi\)
−0.438091 + 0.898931i \(0.644345\pi\)
\(972\) 1.00000 0.0320750
\(973\) −5.64656 −0.181020
\(974\) −17.6829 −0.566597
\(975\) 0 0
\(976\) −12.0761 −0.386545
\(977\) −46.5491 −1.48924 −0.744619 0.667490i \(-0.767369\pi\)
−0.744619 + 0.667490i \(0.767369\pi\)
\(978\) −21.6015 −0.690739
\(979\) −32.9401 −1.05277
\(980\) −6.58642 −0.210395
\(981\) 8.15883 0.260492
\(982\) 28.5435 0.910859
\(983\) 1.10321 0.0351870 0.0175935 0.999845i \(-0.494400\pi\)
0.0175935 + 0.999845i \(0.494400\pi\)
\(984\) 5.74094 0.183014
\(985\) 10.2010 0.325032
\(986\) 0.140047 0.00446001
\(987\) 4.50173 0.143292
\(988\) 0 0
\(989\) −9.61356 −0.305694
\(990\) −4.40581 −0.140026
\(991\) −51.4752 −1.63516 −0.817581 0.575813i \(-0.804686\pi\)
−0.817581 + 0.575813i \(0.804686\pi\)
\(992\) −2.08815 −0.0662987
\(993\) −6.43057 −0.204068
\(994\) 4.52542 0.143538
\(995\) 1.65279 0.0523971
\(996\) 11.2959 0.357924
\(997\) −28.9715 −0.917537 −0.458769 0.888556i \(-0.651709\pi\)
−0.458769 + 0.888556i \(0.651709\pi\)
\(998\) 27.9409 0.884454
\(999\) 6.07069 0.192068
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 5070.2.a.bq.1.2 3
13.5 odd 4 5070.2.b.y.1351.5 6
13.8 odd 4 5070.2.b.y.1351.2 6
13.12 even 2 5070.2.a.bv.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bq.1.2 3 1.1 even 1 trivial
5070.2.a.bv.1.2 yes 3 13.12 even 2
5070.2.b.y.1351.2 6 13.8 odd 4
5070.2.b.y.1351.5 6 13.5 odd 4