Properties

Label 5070.2.a.bp.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $1$
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: \(1\)
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 1.1
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.69202 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.69202 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.04892 q^{11} +1.00000 q^{12} +3.69202 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.85086 q^{17} -1.00000 q^{18} -0.911854 q^{19} +1.00000 q^{20} -3.69202 q^{21} -3.04892 q^{22} -0.356896 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -3.69202 q^{28} +10.5036 q^{29} -1.00000 q^{30} -2.06100 q^{31} -1.00000 q^{32} +3.04892 q^{33} +6.85086 q^{34} -3.69202 q^{35} +1.00000 q^{36} +0.899772 q^{37} +0.911854 q^{38} -1.00000 q^{40} -10.2959 q^{41} +3.69202 q^{42} +9.43296 q^{43} +3.04892 q^{44} +1.00000 q^{45} +0.356896 q^{46} -11.5918 q^{47} +1.00000 q^{48} +6.63102 q^{49} -1.00000 q^{50} -6.85086 q^{51} -3.40581 q^{53} -1.00000 q^{54} +3.04892 q^{55} +3.69202 q^{56} -0.911854 q^{57} -10.5036 q^{58} -8.54288 q^{59} +1.00000 q^{60} +1.55496 q^{61} +2.06100 q^{62} -3.69202 q^{63} +1.00000 q^{64} -3.04892 q^{66} -1.14914 q^{67} -6.85086 q^{68} -0.356896 q^{69} +3.69202 q^{70} +4.13706 q^{71} -1.00000 q^{72} -11.1685 q^{73} -0.899772 q^{74} +1.00000 q^{75} -0.911854 q^{76} -11.2567 q^{77} +9.62565 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.2959 q^{82} +9.86054 q^{83} -3.69202 q^{84} -6.85086 q^{85} -9.43296 q^{86} +10.5036 q^{87} -3.04892 q^{88} +6.41119 q^{89} -1.00000 q^{90} -0.356896 q^{92} -2.06100 q^{93} +11.5918 q^{94} -0.911854 q^{95} -1.00000 q^{96} -6.97823 q^{97} -6.63102 q^{98} +3.04892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} - 6q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} - 6q^{7} - 3q^{8} + 3q^{9} - 3q^{10} + 3q^{12} + 6q^{14} + 3q^{15} + 3q^{16} - 7q^{17} - 3q^{18} + q^{19} + 3q^{20} - 6q^{21} + 3q^{23} - 3q^{24} + 3q^{25} + 3q^{27} - 6q^{28} - 3q^{30} - 16q^{31} - 3q^{32} + 7q^{34} - 6q^{35} + 3q^{36} - 20q^{37} - q^{38} - 3q^{40} - 17q^{41} + 6q^{42} + 9q^{43} + 3q^{45} - 3q^{46} - 7q^{47} + 3q^{48} + 5q^{49} - 3q^{50} - 7q^{51} + 3q^{53} - 3q^{54} + 6q^{56} + q^{57} - 7q^{59} + 3q^{60} + 5q^{61} + 16q^{62} - 6q^{63} + 3q^{64} - 17q^{67} - 7q^{68} + 3q^{69} + 6q^{70} + 7q^{71} - 3q^{72} - 3q^{73} + 20q^{74} + 3q^{75} + q^{76} - 7q^{77} + 17q^{79} + 3q^{80} + 3q^{81} + 17q^{82} - 6q^{83} - 6q^{84} - 7q^{85} - 9q^{86} + 3q^{89} - 3q^{90} + 3q^{92} - 16q^{93} + 7q^{94} + q^{95} - 3q^{96} - 24q^{97} - 5q^{98} + O(q^{100}) \)

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
\(7\) −3.69202 −1.39545 −0.697726 0.716364i \(-0.745805\pi\)
−0.697726 + 0.716364i \(0.745805\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 3.04892 0.919283 0.459642 0.888105i \(-0.347978\pi\)
0.459642 + 0.888105i \(0.347978\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 3.69202 0.986734
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −6.85086 −1.66158 −0.830788 0.556589i \(-0.812110\pi\)
−0.830788 + 0.556589i \(0.812110\pi\)
\(18\) −1.00000 −0.235702
\(19\) −0.911854 −0.209194 −0.104597 0.994515i \(-0.533355\pi\)
−0.104597 + 0.994515i \(0.533355\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.69202 −0.805665
\(22\) −3.04892 −0.650031
\(23\) −0.356896 −0.0744179 −0.0372090 0.999308i \(-0.511847\pi\)
−0.0372090 + 0.999308i \(0.511847\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −3.69202 −0.697726
\(29\) 10.5036 1.95048 0.975239 0.221153i \(-0.0709819\pi\)
0.975239 + 0.221153i \(0.0709819\pi\)
\(30\) −1.00000 −0.182574
\(31\) −2.06100 −0.370166 −0.185083 0.982723i \(-0.559255\pi\)
−0.185083 + 0.982723i \(0.559255\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.04892 0.530748
\(34\) 6.85086 1.17491
\(35\) −3.69202 −0.624066
\(36\) 1.00000 0.166667
\(37\) 0.899772 0.147922 0.0739608 0.997261i \(-0.476436\pi\)
0.0739608 + 0.997261i \(0.476436\pi\)
\(38\) 0.911854 0.147922
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −10.2959 −1.60795 −0.803974 0.594664i \(-0.797285\pi\)
−0.803974 + 0.594664i \(0.797285\pi\)
\(42\) 3.69202 0.569691
\(43\) 9.43296 1.43851 0.719256 0.694745i \(-0.244483\pi\)
0.719256 + 0.694745i \(0.244483\pi\)
\(44\) 3.04892 0.459642
\(45\) 1.00000 0.149071
\(46\) 0.356896 0.0526214
\(47\) −11.5918 −1.69084 −0.845418 0.534105i \(-0.820649\pi\)
−0.845418 + 0.534105i \(0.820649\pi\)
\(48\) 1.00000 0.144338
\(49\) 6.63102 0.947289
\(50\) −1.00000 −0.141421
\(51\) −6.85086 −0.959312
\(52\) 0 0
\(53\) −3.40581 −0.467824 −0.233912 0.972258i \(-0.575153\pi\)
−0.233912 + 0.972258i \(0.575153\pi\)
\(54\) −1.00000 −0.136083
\(55\) 3.04892 0.411116
\(56\) 3.69202 0.493367
\(57\) −0.911854 −0.120778
\(58\) −10.5036 −1.37920
\(59\) −8.54288 −1.11219 −0.556094 0.831119i \(-0.687701\pi\)
−0.556094 + 0.831119i \(0.687701\pi\)
\(60\) 1.00000 0.129099
\(61\) 1.55496 0.199092 0.0995460 0.995033i \(-0.468261\pi\)
0.0995460 + 0.995033i \(0.468261\pi\)
\(62\) 2.06100 0.261747
\(63\) −3.69202 −0.465151
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.04892 −0.375296
\(67\) −1.14914 −0.140390 −0.0701952 0.997533i \(-0.522362\pi\)
−0.0701952 + 0.997533i \(0.522362\pi\)
\(68\) −6.85086 −0.830788
\(69\) −0.356896 −0.0429652
\(70\) 3.69202 0.441281
\(71\) 4.13706 0.490979 0.245490 0.969399i \(-0.421051\pi\)
0.245490 + 0.969399i \(0.421051\pi\)
\(72\) −1.00000 −0.117851
\(73\) −11.1685 −1.30718 −0.653588 0.756850i \(-0.726737\pi\)
−0.653588 + 0.756850i \(0.726737\pi\)
\(74\) −0.899772 −0.104596
\(75\) 1.00000 0.115470
\(76\) −0.911854 −0.104597
\(77\) −11.2567 −1.28282
\(78\) 0 0
\(79\) 9.62565 1.08297 0.541485 0.840710i \(-0.317862\pi\)
0.541485 + 0.840710i \(0.317862\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.2959 1.13699
\(83\) 9.86054 1.08234 0.541168 0.840915i \(-0.317982\pi\)
0.541168 + 0.840915i \(0.317982\pi\)
\(84\) −3.69202 −0.402833
\(85\) −6.85086 −0.743080
\(86\) −9.43296 −1.01718
\(87\) 10.5036 1.12611
\(88\) −3.04892 −0.325016
\(89\) 6.41119 0.679585 0.339792 0.940500i \(-0.389643\pi\)
0.339792 + 0.940500i \(0.389643\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −0.356896 −0.0372090
\(93\) −2.06100 −0.213716
\(94\) 11.5918 1.19560
\(95\) −0.911854 −0.0935542
\(96\) −1.00000 −0.102062
\(97\) −6.97823 −0.708532 −0.354266 0.935145i \(-0.615269\pi\)
−0.354266 + 0.935145i \(0.615269\pi\)
\(98\) −6.63102 −0.669834
\(99\) 3.04892 0.306428
\(100\) 1.00000 0.100000
\(101\) −4.09783 −0.407750 −0.203875 0.978997i \(-0.565354\pi\)
−0.203875 + 0.978997i \(0.565354\pi\)
\(102\) 6.85086 0.678336
\(103\) −16.9215 −1.66733 −0.833665 0.552271i \(-0.813761\pi\)
−0.833665 + 0.552271i \(0.813761\pi\)
\(104\) 0 0
\(105\) −3.69202 −0.360304
\(106\) 3.40581 0.330802
\(107\) −12.7192 −1.22961 −0.614804 0.788680i \(-0.710765\pi\)
−0.614804 + 0.788680i \(0.710765\pi\)
\(108\) 1.00000 0.0962250
\(109\) −19.9922 −1.91491 −0.957454 0.288585i \(-0.906815\pi\)
−0.957454 + 0.288585i \(0.906815\pi\)
\(110\) −3.04892 −0.290703
\(111\) 0.899772 0.0854026
\(112\) −3.69202 −0.348863
\(113\) −4.49934 −0.423262 −0.211631 0.977350i \(-0.567877\pi\)
−0.211631 + 0.977350i \(0.567877\pi\)
\(114\) 0.911854 0.0854030
\(115\) −0.356896 −0.0332807
\(116\) 10.5036 0.975239
\(117\) 0 0
\(118\) 8.54288 0.786436
\(119\) 25.2935 2.31865
\(120\) −1.00000 −0.0912871
\(121\) −1.70410 −0.154918
\(122\) −1.55496 −0.140779
\(123\) −10.2959 −0.928350
\(124\) −2.06100 −0.185083
\(125\) 1.00000 0.0894427
\(126\) 3.69202 0.328911
\(127\) −7.08815 −0.628971 −0.314486 0.949262i \(-0.601832\pi\)
−0.314486 + 0.949262i \(0.601832\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 9.43296 0.830526
\(130\) 0 0
\(131\) 19.7506 1.72562 0.862810 0.505528i \(-0.168702\pi\)
0.862810 + 0.505528i \(0.168702\pi\)
\(132\) 3.04892 0.265374
\(133\) 3.36658 0.291920
\(134\) 1.14914 0.0992710
\(135\) 1.00000 0.0860663
\(136\) 6.85086 0.587456
\(137\) 7.94331 0.678643 0.339322 0.940670i \(-0.389803\pi\)
0.339322 + 0.940670i \(0.389803\pi\)
\(138\) 0.356896 0.0303810
\(139\) −1.71379 −0.145362 −0.0726810 0.997355i \(-0.523155\pi\)
−0.0726810 + 0.997355i \(0.523155\pi\)
\(140\) −3.69202 −0.312033
\(141\) −11.5918 −0.976205
\(142\) −4.13706 −0.347175
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 10.5036 0.872280
\(146\) 11.1685 0.924313
\(147\) 6.63102 0.546918
\(148\) 0.899772 0.0739608
\(149\) 6.91185 0.566241 0.283121 0.959084i \(-0.408630\pi\)
0.283121 + 0.959084i \(0.408630\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −5.43296 −0.442128 −0.221064 0.975259i \(-0.570953\pi\)
−0.221064 + 0.975259i \(0.570953\pi\)
\(152\) 0.911854 0.0739611
\(153\) −6.85086 −0.553859
\(154\) 11.2567 0.907088
\(155\) −2.06100 −0.165543
\(156\) 0 0
\(157\) −9.13706 −0.729217 −0.364609 0.931161i \(-0.618797\pi\)
−0.364609 + 0.931161i \(0.618797\pi\)
\(158\) −9.62565 −0.765775
\(159\) −3.40581 −0.270099
\(160\) −1.00000 −0.0790569
\(161\) 1.31767 0.103847
\(162\) −1.00000 −0.0785674
\(163\) −14.3773 −1.12612 −0.563060 0.826416i \(-0.690376\pi\)
−0.563060 + 0.826416i \(0.690376\pi\)
\(164\) −10.2959 −0.803974
\(165\) 3.04892 0.237358
\(166\) −9.86054 −0.765327
\(167\) 3.00538 0.232563 0.116282 0.993216i \(-0.462903\pi\)
0.116282 + 0.993216i \(0.462903\pi\)
\(168\) 3.69202 0.284846
\(169\) 0 0
\(170\) 6.85086 0.525437
\(171\) −0.911854 −0.0697312
\(172\) 9.43296 0.719256
\(173\) 6.89546 0.524252 0.262126 0.965034i \(-0.415576\pi\)
0.262126 + 0.965034i \(0.415576\pi\)
\(174\) −10.5036 −0.796279
\(175\) −3.69202 −0.279091
\(176\) 3.04892 0.229821
\(177\) −8.54288 −0.642122
\(178\) −6.41119 −0.480539
\(179\) −25.6461 −1.91688 −0.958439 0.285296i \(-0.907908\pi\)
−0.958439 + 0.285296i \(0.907908\pi\)
\(180\) 1.00000 0.0745356
\(181\) −25.8092 −1.91839 −0.959193 0.282753i \(-0.908752\pi\)
−0.959193 + 0.282753i \(0.908752\pi\)
\(182\) 0 0
\(183\) 1.55496 0.114946
\(184\) 0.356896 0.0263107
\(185\) 0.899772 0.0661526
\(186\) 2.06100 0.151120
\(187\) −20.8877 −1.52746
\(188\) −11.5918 −0.845418
\(189\) −3.69202 −0.268555
\(190\) 0.911854 0.0661528
\(191\) −14.5700 −1.05425 −0.527125 0.849788i \(-0.676730\pi\)
−0.527125 + 0.849788i \(0.676730\pi\)
\(192\) 1.00000 0.0721688
\(193\) −8.32544 −0.599278 −0.299639 0.954053i \(-0.596866\pi\)
−0.299639 + 0.954053i \(0.596866\pi\)
\(194\) 6.97823 0.501008
\(195\) 0 0
\(196\) 6.63102 0.473644
\(197\) 26.8810 1.91519 0.957595 0.288116i \(-0.0930289\pi\)
0.957595 + 0.288116i \(0.0930289\pi\)
\(198\) −3.04892 −0.216677
\(199\) −18.7047 −1.32594 −0.662970 0.748646i \(-0.730704\pi\)
−0.662970 + 0.748646i \(0.730704\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −1.14914 −0.0810544
\(202\) 4.09783 0.288323
\(203\) −38.7797 −2.72180
\(204\) −6.85086 −0.479656
\(205\) −10.2959 −0.719097
\(206\) 16.9215 1.17898
\(207\) −0.356896 −0.0248060
\(208\) 0 0
\(209\) −2.78017 −0.192308
\(210\) 3.69202 0.254774
\(211\) 2.93362 0.201959 0.100980 0.994889i \(-0.467802\pi\)
0.100980 + 0.994889i \(0.467802\pi\)
\(212\) −3.40581 −0.233912
\(213\) 4.13706 0.283467
\(214\) 12.7192 0.869464
\(215\) 9.43296 0.643323
\(216\) −1.00000 −0.0680414
\(217\) 7.60925 0.516550
\(218\) 19.9922 1.35404
\(219\) −11.1685 −0.754699
\(220\) 3.04892 0.205558
\(221\) 0 0
\(222\) −0.899772 −0.0603888
\(223\) 18.1444 1.21504 0.607518 0.794306i \(-0.292165\pi\)
0.607518 + 0.794306i \(0.292165\pi\)
\(224\) 3.69202 0.246684
\(225\) 1.00000 0.0666667
\(226\) 4.49934 0.299291
\(227\) −12.3056 −0.816750 −0.408375 0.912814i \(-0.633904\pi\)
−0.408375 + 0.912814i \(0.633904\pi\)
\(228\) −0.911854 −0.0603890
\(229\) −4.64848 −0.307180 −0.153590 0.988135i \(-0.549084\pi\)
−0.153590 + 0.988135i \(0.549084\pi\)
\(230\) 0.356896 0.0235330
\(231\) −11.2567 −0.740634
\(232\) −10.5036 −0.689598
\(233\) 0.658170 0.0431181 0.0215591 0.999768i \(-0.493137\pi\)
0.0215591 + 0.999768i \(0.493137\pi\)
\(234\) 0 0
\(235\) −11.5918 −0.756165
\(236\) −8.54288 −0.556094
\(237\) 9.62565 0.625253
\(238\) −25.2935 −1.63953
\(239\) 18.7748 1.21444 0.607220 0.794534i \(-0.292285\pi\)
0.607220 + 0.794534i \(0.292285\pi\)
\(240\) 1.00000 0.0645497
\(241\) −18.0271 −1.16123 −0.580615 0.814178i \(-0.697188\pi\)
−0.580615 + 0.814178i \(0.697188\pi\)
\(242\) 1.70410 0.109544
\(243\) 1.00000 0.0641500
\(244\) 1.55496 0.0995460
\(245\) 6.63102 0.423640
\(246\) 10.2959 0.656442
\(247\) 0 0
\(248\) 2.06100 0.130874
\(249\) 9.86054 0.624887
\(250\) −1.00000 −0.0632456
\(251\) −2.18060 −0.137638 −0.0688192 0.997629i \(-0.521923\pi\)
−0.0688192 + 0.997629i \(0.521923\pi\)
\(252\) −3.69202 −0.232575
\(253\) −1.08815 −0.0684112
\(254\) 7.08815 0.444750
\(255\) −6.85086 −0.429017
\(256\) 1.00000 0.0625000
\(257\) −9.58748 −0.598051 −0.299025 0.954245i \(-0.596662\pi\)
−0.299025 + 0.954245i \(0.596662\pi\)
\(258\) −9.43296 −0.587270
\(259\) −3.32198 −0.206418
\(260\) 0 0
\(261\) 10.5036 0.650159
\(262\) −19.7506 −1.22020
\(263\) −1.41789 −0.0874311 −0.0437156 0.999044i \(-0.513920\pi\)
−0.0437156 + 0.999044i \(0.513920\pi\)
\(264\) −3.04892 −0.187648
\(265\) −3.40581 −0.209217
\(266\) −3.36658 −0.206419
\(267\) 6.41119 0.392358
\(268\) −1.14914 −0.0701952
\(269\) −4.14675 −0.252832 −0.126416 0.991977i \(-0.540347\pi\)
−0.126416 + 0.991977i \(0.540347\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −6.40044 −0.388799 −0.194399 0.980922i \(-0.562276\pi\)
−0.194399 + 0.980922i \(0.562276\pi\)
\(272\) −6.85086 −0.415394
\(273\) 0 0
\(274\) −7.94331 −0.479873
\(275\) 3.04892 0.183857
\(276\) −0.356896 −0.0214826
\(277\) 10.3394 0.621237 0.310618 0.950535i \(-0.399464\pi\)
0.310618 + 0.950535i \(0.399464\pi\)
\(278\) 1.71379 0.102786
\(279\) −2.06100 −0.123389
\(280\) 3.69202 0.220640
\(281\) −5.08144 −0.303133 −0.151567 0.988447i \(-0.548432\pi\)
−0.151567 + 0.988447i \(0.548432\pi\)
\(282\) 11.5918 0.690281
\(283\) 24.5284 1.45806 0.729031 0.684481i \(-0.239971\pi\)
0.729031 + 0.684481i \(0.239971\pi\)
\(284\) 4.13706 0.245490
\(285\) −0.911854 −0.0540136
\(286\) 0 0
\(287\) 38.0127 2.24382
\(288\) −1.00000 −0.0589256
\(289\) 29.9342 1.76084
\(290\) −10.5036 −0.616795
\(291\) −6.97823 −0.409071
\(292\) −11.1685 −0.653588
\(293\) −15.8538 −0.926191 −0.463096 0.886308i \(-0.653261\pi\)
−0.463096 + 0.886308i \(0.653261\pi\)
\(294\) −6.63102 −0.386729
\(295\) −8.54288 −0.497386
\(296\) −0.899772 −0.0522982
\(297\) 3.04892 0.176916
\(298\) −6.91185 −0.400393
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −34.8267 −2.00738
\(302\) 5.43296 0.312632
\(303\) −4.09783 −0.235414
\(304\) −0.911854 −0.0522984
\(305\) 1.55496 0.0890366
\(306\) 6.85086 0.391637
\(307\) 14.2121 0.811125 0.405563 0.914067i \(-0.367076\pi\)
0.405563 + 0.914067i \(0.367076\pi\)
\(308\) −11.2567 −0.641408
\(309\) −16.9215 −0.962633
\(310\) 2.06100 0.117057
\(311\) 15.0127 0.851291 0.425645 0.904890i \(-0.360047\pi\)
0.425645 + 0.904890i \(0.360047\pi\)
\(312\) 0 0
\(313\) −13.4993 −0.763028 −0.381514 0.924363i \(-0.624597\pi\)
−0.381514 + 0.924363i \(0.624597\pi\)
\(314\) 9.13706 0.515634
\(315\) −3.69202 −0.208022
\(316\) 9.62565 0.541485
\(317\) −33.9463 −1.90661 −0.953307 0.302003i \(-0.902345\pi\)
−0.953307 + 0.302003i \(0.902345\pi\)
\(318\) 3.40581 0.190989
\(319\) 32.0248 1.79304
\(320\) 1.00000 0.0559017
\(321\) −12.7192 −0.709915
\(322\) −1.31767 −0.0734307
\(323\) 6.24698 0.347591
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 14.3773 0.796287
\(327\) −19.9922 −1.10557
\(328\) 10.2959 0.568496
\(329\) 42.7972 2.35948
\(330\) −3.04892 −0.167837
\(331\) 6.84415 0.376189 0.188094 0.982151i \(-0.439769\pi\)
0.188094 + 0.982151i \(0.439769\pi\)
\(332\) 9.86054 0.541168
\(333\) 0.899772 0.0493072
\(334\) −3.00538 −0.164447
\(335\) −1.14914 −0.0627845
\(336\) −3.69202 −0.201416
\(337\) 27.7332 1.51072 0.755361 0.655309i \(-0.227462\pi\)
0.755361 + 0.655309i \(0.227462\pi\)
\(338\) 0 0
\(339\) −4.49934 −0.244370
\(340\) −6.85086 −0.371540
\(341\) −6.28382 −0.340288
\(342\) 0.911854 0.0493074
\(343\) 1.36227 0.0735558
\(344\) −9.43296 −0.508591
\(345\) −0.356896 −0.0192146
\(346\) −6.89546 −0.370702
\(347\) −10.4862 −0.562928 −0.281464 0.959572i \(-0.590820\pi\)
−0.281464 + 0.959572i \(0.590820\pi\)
\(348\) 10.5036 0.563055
\(349\) 8.13467 0.435439 0.217719 0.976011i \(-0.430138\pi\)
0.217719 + 0.976011i \(0.430138\pi\)
\(350\) 3.69202 0.197347
\(351\) 0 0
\(352\) −3.04892 −0.162508
\(353\) 25.8649 1.37665 0.688324 0.725404i \(-0.258347\pi\)
0.688324 + 0.725404i \(0.258347\pi\)
\(354\) 8.54288 0.454049
\(355\) 4.13706 0.219573
\(356\) 6.41119 0.339792
\(357\) 25.2935 1.33867
\(358\) 25.6461 1.35544
\(359\) 30.2959 1.59896 0.799478 0.600695i \(-0.205109\pi\)
0.799478 + 0.600695i \(0.205109\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −18.1685 −0.956238
\(362\) 25.8092 1.35650
\(363\) −1.70410 −0.0894422
\(364\) 0 0
\(365\) −11.1685 −0.584587
\(366\) −1.55496 −0.0812790
\(367\) −3.41013 −0.178007 −0.0890035 0.996031i \(-0.528368\pi\)
−0.0890035 + 0.996031i \(0.528368\pi\)
\(368\) −0.356896 −0.0186045
\(369\) −10.2959 −0.535983
\(370\) −0.899772 −0.0467769
\(371\) 12.5743 0.652827
\(372\) −2.06100 −0.106858
\(373\) 3.35988 0.173968 0.0869840 0.996210i \(-0.472277\pi\)
0.0869840 + 0.996210i \(0.472277\pi\)
\(374\) 20.8877 1.08008
\(375\) 1.00000 0.0516398
\(376\) 11.5918 0.597801
\(377\) 0 0
\(378\) 3.69202 0.189897
\(379\) 31.4916 1.61761 0.808807 0.588075i \(-0.200114\pi\)
0.808807 + 0.588075i \(0.200114\pi\)
\(380\) −0.911854 −0.0467771
\(381\) −7.08815 −0.363137
\(382\) 14.5700 0.745467
\(383\) −21.8485 −1.11640 −0.558202 0.829705i \(-0.688509\pi\)
−0.558202 + 0.829705i \(0.688509\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −11.2567 −0.573693
\(386\) 8.32544 0.423754
\(387\) 9.43296 0.479504
\(388\) −6.97823 −0.354266
\(389\) −11.3787 −0.576921 −0.288461 0.957492i \(-0.593143\pi\)
−0.288461 + 0.957492i \(0.593143\pi\)
\(390\) 0 0
\(391\) 2.44504 0.123651
\(392\) −6.63102 −0.334917
\(393\) 19.7506 0.996287
\(394\) −26.8810 −1.35424
\(395\) 9.62565 0.484319
\(396\) 3.04892 0.153214
\(397\) −22.0476 −1.10654 −0.553268 0.833003i \(-0.686620\pi\)
−0.553268 + 0.833003i \(0.686620\pi\)
\(398\) 18.7047 0.937582
\(399\) 3.36658 0.168540
\(400\) 1.00000 0.0500000
\(401\) −10.2459 −0.511657 −0.255828 0.966722i \(-0.582348\pi\)
−0.255828 + 0.966722i \(0.582348\pi\)
\(402\) 1.14914 0.0573141
\(403\) 0 0
\(404\) −4.09783 −0.203875
\(405\) 1.00000 0.0496904
\(406\) 38.7797 1.92460
\(407\) 2.74333 0.135982
\(408\) 6.85086 0.339168
\(409\) −22.8189 −1.12832 −0.564162 0.825664i \(-0.690800\pi\)
−0.564162 + 0.825664i \(0.690800\pi\)
\(410\) 10.2959 0.508478
\(411\) 7.94331 0.391815
\(412\) −16.9215 −0.833665
\(413\) 31.5405 1.55201
\(414\) 0.356896 0.0175405
\(415\) 9.86054 0.484035
\(416\) 0 0
\(417\) −1.71379 −0.0839247
\(418\) 2.78017 0.135982
\(419\) −28.5773 −1.39609 −0.698047 0.716052i \(-0.745947\pi\)
−0.698047 + 0.716052i \(0.745947\pi\)
\(420\) −3.69202 −0.180152
\(421\) −32.3937 −1.57877 −0.789387 0.613896i \(-0.789601\pi\)
−0.789387 + 0.613896i \(0.789601\pi\)
\(422\) −2.93362 −0.142807
\(423\) −11.5918 −0.563612
\(424\) 3.40581 0.165401
\(425\) −6.85086 −0.332315
\(426\) −4.13706 −0.200441
\(427\) −5.74094 −0.277824
\(428\) −12.7192 −0.614804
\(429\) 0 0
\(430\) −9.43296 −0.454898
\(431\) 14.6455 0.705449 0.352724 0.935727i \(-0.385255\pi\)
0.352724 + 0.935727i \(0.385255\pi\)
\(432\) 1.00000 0.0481125
\(433\) −35.8485 −1.72277 −0.861384 0.507955i \(-0.830402\pi\)
−0.861384 + 0.507955i \(0.830402\pi\)
\(434\) −7.60925 −0.365256
\(435\) 10.5036 0.503611
\(436\) −19.9922 −0.957454
\(437\) 0.325437 0.0155678
\(438\) 11.1685 0.533653
\(439\) 2.55927 0.122147 0.0610736 0.998133i \(-0.480548\pi\)
0.0610736 + 0.998133i \(0.480548\pi\)
\(440\) −3.04892 −0.145351
\(441\) 6.63102 0.315763
\(442\) 0 0
\(443\) 23.2857 1.10634 0.553169 0.833069i \(-0.313418\pi\)
0.553169 + 0.833069i \(0.313418\pi\)
\(444\) 0.899772 0.0427013
\(445\) 6.41119 0.303920
\(446\) −18.1444 −0.859160
\(447\) 6.91185 0.326919
\(448\) −3.69202 −0.174432
\(449\) 27.3327 1.28991 0.644956 0.764220i \(-0.276876\pi\)
0.644956 + 0.764220i \(0.276876\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −31.3913 −1.47816
\(452\) −4.49934 −0.211631
\(453\) −5.43296 −0.255263
\(454\) 12.3056 0.577530
\(455\) 0 0
\(456\) 0.911854 0.0427015
\(457\) 7.74094 0.362106 0.181053 0.983473i \(-0.442049\pi\)
0.181053 + 0.983473i \(0.442049\pi\)
\(458\) 4.64848 0.217209
\(459\) −6.85086 −0.319771
\(460\) −0.356896 −0.0166404
\(461\) −17.1511 −0.798805 −0.399402 0.916776i \(-0.630782\pi\)
−0.399402 + 0.916776i \(0.630782\pi\)
\(462\) 11.2567 0.523708
\(463\) 34.2747 1.59288 0.796441 0.604717i \(-0.206714\pi\)
0.796441 + 0.604717i \(0.206714\pi\)
\(464\) 10.5036 0.487620
\(465\) −2.06100 −0.0955765
\(466\) −0.658170 −0.0304891
\(467\) 20.9474 0.969328 0.484664 0.874700i \(-0.338942\pi\)
0.484664 + 0.874700i \(0.338942\pi\)
\(468\) 0 0
\(469\) 4.24267 0.195908
\(470\) 11.5918 0.534690
\(471\) −9.13706 −0.421014
\(472\) 8.54288 0.393218
\(473\) 28.7603 1.32240
\(474\) −9.62565 −0.442121
\(475\) −0.911854 −0.0418387
\(476\) 25.2935 1.15933
\(477\) −3.40581 −0.155941
\(478\) −18.7748 −0.858739
\(479\) −15.8310 −0.723337 −0.361669 0.932307i \(-0.617793\pi\)
−0.361669 + 0.932307i \(0.617793\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 18.0271 0.821114
\(483\) 1.31767 0.0599559
\(484\) −1.70410 −0.0774592
\(485\) −6.97823 −0.316865
\(486\) −1.00000 −0.0453609
\(487\) −33.7060 −1.52737 −0.763683 0.645592i \(-0.776611\pi\)
−0.763683 + 0.645592i \(0.776611\pi\)
\(488\) −1.55496 −0.0703896
\(489\) −14.3773 −0.650166
\(490\) −6.63102 −0.299559
\(491\) 3.98925 0.180032 0.0900161 0.995940i \(-0.471308\pi\)
0.0900161 + 0.995940i \(0.471308\pi\)
\(492\) −10.2959 −0.464175
\(493\) −71.9590 −3.24087
\(494\) 0 0
\(495\) 3.04892 0.137039
\(496\) −2.06100 −0.0925416
\(497\) −15.2741 −0.685138
\(498\) −9.86054 −0.441862
\(499\) −20.4698 −0.916354 −0.458177 0.888861i \(-0.651497\pi\)
−0.458177 + 0.888861i \(0.651497\pi\)
\(500\) 1.00000 0.0447214
\(501\) 3.00538 0.134270
\(502\) 2.18060 0.0973251
\(503\) 22.5351 1.00479 0.502395 0.864638i \(-0.332452\pi\)
0.502395 + 0.864638i \(0.332452\pi\)
\(504\) 3.69202 0.164456
\(505\) −4.09783 −0.182351
\(506\) 1.08815 0.0483740
\(507\) 0 0
\(508\) −7.08815 −0.314486
\(509\) 10.9718 0.486316 0.243158 0.969987i \(-0.421817\pi\)
0.243158 + 0.969987i \(0.421817\pi\)
\(510\) 6.85086 0.303361
\(511\) 41.2344 1.82410
\(512\) −1.00000 −0.0441942
\(513\) −0.911854 −0.0402593
\(514\) 9.58748 0.422886
\(515\) −16.9215 −0.745652
\(516\) 9.43296 0.415263
\(517\) −35.3424 −1.55436
\(518\) 3.32198 0.145959
\(519\) 6.89546 0.302677
\(520\) 0 0
\(521\) 15.9691 0.699620 0.349810 0.936821i \(-0.386246\pi\)
0.349810 + 0.936821i \(0.386246\pi\)
\(522\) −10.5036 −0.459732
\(523\) −24.7681 −1.08303 −0.541516 0.840690i \(-0.682150\pi\)
−0.541516 + 0.840690i \(0.682150\pi\)
\(524\) 19.7506 0.862810
\(525\) −3.69202 −0.161133
\(526\) 1.41789 0.0618232
\(527\) 14.1196 0.615060
\(528\) 3.04892 0.132687
\(529\) −22.8726 −0.994462
\(530\) 3.40581 0.147939
\(531\) −8.54288 −0.370729
\(532\) 3.36658 0.145960
\(533\) 0 0
\(534\) −6.41119 −0.277439
\(535\) −12.7192 −0.549898
\(536\) 1.14914 0.0496355
\(537\) −25.6461 −1.10671
\(538\) 4.14675 0.178779
\(539\) 20.2174 0.870827
\(540\) 1.00000 0.0430331
\(541\) −0.460107 −0.0197816 −0.00989078 0.999951i \(-0.503148\pi\)
−0.00989078 + 0.999951i \(0.503148\pi\)
\(542\) 6.40044 0.274922
\(543\) −25.8092 −1.10758
\(544\) 6.85086 0.293728
\(545\) −19.9922 −0.856373
\(546\) 0 0
\(547\) −8.70410 −0.372161 −0.186080 0.982535i \(-0.559578\pi\)
−0.186080 + 0.982535i \(0.559578\pi\)
\(548\) 7.94331 0.339322
\(549\) 1.55496 0.0663640
\(550\) −3.04892 −0.130006
\(551\) −9.57779 −0.408028
\(552\) 0.356896 0.0151905
\(553\) −35.5381 −1.51123
\(554\) −10.3394 −0.439281
\(555\) 0.899772 0.0381932
\(556\) −1.71379 −0.0726810
\(557\) 36.7415 1.55679 0.778394 0.627776i \(-0.216034\pi\)
0.778394 + 0.627776i \(0.216034\pi\)
\(558\) 2.06100 0.0872490
\(559\) 0 0
\(560\) −3.69202 −0.156016
\(561\) −20.8877 −0.881879
\(562\) 5.08144 0.214348
\(563\) −27.7713 −1.17042 −0.585211 0.810881i \(-0.698988\pi\)
−0.585211 + 0.810881i \(0.698988\pi\)
\(564\) −11.5918 −0.488103
\(565\) −4.49934 −0.189288
\(566\) −24.5284 −1.03101
\(567\) −3.69202 −0.155050
\(568\) −4.13706 −0.173587
\(569\) 38.4379 1.61140 0.805700 0.592324i \(-0.201790\pi\)
0.805700 + 0.592324i \(0.201790\pi\)
\(570\) 0.911854 0.0381934
\(571\) 10.8086 0.452328 0.226164 0.974089i \(-0.427382\pi\)
0.226164 + 0.974089i \(0.427382\pi\)
\(572\) 0 0
\(573\) −14.5700 −0.608671
\(574\) −38.0127 −1.58662
\(575\) −0.356896 −0.0148836
\(576\) 1.00000 0.0416667
\(577\) −35.5991 −1.48201 −0.741005 0.671500i \(-0.765650\pi\)
−0.741005 + 0.671500i \(0.765650\pi\)
\(578\) −29.9342 −1.24510
\(579\) −8.32544 −0.345993
\(580\) 10.5036 0.436140
\(581\) −36.4053 −1.51035
\(582\) 6.97823 0.289257
\(583\) −10.3840 −0.430063
\(584\) 11.1685 0.462157
\(585\) 0 0
\(586\) 15.8538 0.654916
\(587\) −11.4741 −0.473587 −0.236794 0.971560i \(-0.576097\pi\)
−0.236794 + 0.971560i \(0.576097\pi\)
\(588\) 6.63102 0.273459
\(589\) 1.87933 0.0774364
\(590\) 8.54288 0.351705
\(591\) 26.8810 1.10574
\(592\) 0.899772 0.0369804
\(593\) −15.5147 −0.637111 −0.318555 0.947904i \(-0.603198\pi\)
−0.318555 + 0.947904i \(0.603198\pi\)
\(594\) −3.04892 −0.125099
\(595\) 25.2935 1.03693
\(596\) 6.91185 0.283121
\(597\) −18.7047 −0.765532
\(598\) 0 0
\(599\) −14.2034 −0.580337 −0.290168 0.956976i \(-0.593711\pi\)
−0.290168 + 0.956976i \(0.593711\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −39.5579 −1.61360 −0.806801 0.590823i \(-0.798803\pi\)
−0.806801 + 0.590823i \(0.798803\pi\)
\(602\) 34.8267 1.41943
\(603\) −1.14914 −0.0467968
\(604\) −5.43296 −0.221064
\(605\) −1.70410 −0.0692816
\(606\) 4.09783 0.166463
\(607\) 4.77346 0.193749 0.0968744 0.995297i \(-0.469115\pi\)
0.0968744 + 0.995297i \(0.469115\pi\)
\(608\) 0.911854 0.0369806
\(609\) −38.7797 −1.57143
\(610\) −1.55496 −0.0629584
\(611\) 0 0
\(612\) −6.85086 −0.276929
\(613\) −5.43429 −0.219489 −0.109744 0.993960i \(-0.535003\pi\)
−0.109744 + 0.993960i \(0.535003\pi\)
\(614\) −14.2121 −0.573552
\(615\) −10.2959 −0.415171
\(616\) 11.2567 0.453544
\(617\) 20.6692 0.832110 0.416055 0.909339i \(-0.363412\pi\)
0.416055 + 0.909339i \(0.363412\pi\)
\(618\) 16.9215 0.680684
\(619\) −24.4077 −0.981030 −0.490515 0.871433i \(-0.663191\pi\)
−0.490515 + 0.871433i \(0.663191\pi\)
\(620\) −2.06100 −0.0827717
\(621\) −0.356896 −0.0143217
\(622\) −15.0127 −0.601953
\(623\) −23.6703 −0.948329
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 13.4993 0.539542
\(627\) −2.78017 −0.111029
\(628\) −9.13706 −0.364609
\(629\) −6.16421 −0.245783
\(630\) 3.69202 0.147094
\(631\) 32.0538 1.27604 0.638021 0.770019i \(-0.279753\pi\)
0.638021 + 0.770019i \(0.279753\pi\)
\(632\) −9.62565 −0.382888
\(633\) 2.93362 0.116601
\(634\) 33.9463 1.34818
\(635\) −7.08815 −0.281284
\(636\) −3.40581 −0.135049
\(637\) 0 0
\(638\) −32.0248 −1.26787
\(639\) 4.13706 0.163660
\(640\) −1.00000 −0.0395285
\(641\) 18.5327 0.731998 0.365999 0.930615i \(-0.380727\pi\)
0.365999 + 0.930615i \(0.380727\pi\)
\(642\) 12.7192 0.501986
\(643\) 22.3454 0.881217 0.440608 0.897699i \(-0.354763\pi\)
0.440608 + 0.897699i \(0.354763\pi\)
\(644\) 1.31767 0.0519234
\(645\) 9.43296 0.371422
\(646\) −6.24698 −0.245784
\(647\) 7.86964 0.309388 0.154694 0.987962i \(-0.450561\pi\)
0.154694 + 0.987962i \(0.450561\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −26.0465 −1.02242
\(650\) 0 0
\(651\) 7.60925 0.298230
\(652\) −14.3773 −0.563060
\(653\) −47.2271 −1.84814 −0.924070 0.382223i \(-0.875159\pi\)
−0.924070 + 0.382223i \(0.875159\pi\)
\(654\) 19.9922 0.781758
\(655\) 19.7506 0.771721
\(656\) −10.2959 −0.401987
\(657\) −11.1685 −0.435726
\(658\) −42.7972 −1.66841
\(659\) −46.6276 −1.81635 −0.908176 0.418588i \(-0.862525\pi\)
−0.908176 + 0.418588i \(0.862525\pi\)
\(660\) 3.04892 0.118679
\(661\) 39.9415 1.55354 0.776772 0.629781i \(-0.216856\pi\)
0.776772 + 0.629781i \(0.216856\pi\)
\(662\) −6.84415 −0.266005
\(663\) 0 0
\(664\) −9.86054 −0.382663
\(665\) 3.36658 0.130551
\(666\) −0.899772 −0.0348655
\(667\) −3.74871 −0.145151
\(668\) 3.00538 0.116282
\(669\) 18.1444 0.701501
\(670\) 1.14914 0.0443953
\(671\) 4.74094 0.183022
\(672\) 3.69202 0.142423
\(673\) −35.3612 −1.36307 −0.681537 0.731783i \(-0.738688\pi\)
−0.681537 + 0.731783i \(0.738688\pi\)
\(674\) −27.7332 −1.06824
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 19.1715 0.736821 0.368410 0.929663i \(-0.379902\pi\)
0.368410 + 0.929663i \(0.379902\pi\)
\(678\) 4.49934 0.172796
\(679\) 25.7638 0.988723
\(680\) 6.85086 0.262718
\(681\) −12.3056 −0.471551
\(682\) 6.28382 0.240620
\(683\) 1.13946 0.0436001 0.0218000 0.999762i \(-0.493060\pi\)
0.0218000 + 0.999762i \(0.493060\pi\)
\(684\) −0.911854 −0.0348656
\(685\) 7.94331 0.303498
\(686\) −1.36227 −0.0520118
\(687\) −4.64848 −0.177351
\(688\) 9.43296 0.359628
\(689\) 0 0
\(690\) 0.356896 0.0135868
\(691\) −11.9071 −0.452966 −0.226483 0.974015i \(-0.572723\pi\)
−0.226483 + 0.974015i \(0.572723\pi\)
\(692\) 6.89546 0.262126
\(693\) −11.2567 −0.427605
\(694\) 10.4862 0.398050
\(695\) −1.71379 −0.0650078
\(696\) −10.5036 −0.398140
\(697\) 70.5357 2.67173
\(698\) −8.13467 −0.307902
\(699\) 0.658170 0.0248943
\(700\) −3.69202 −0.139545
\(701\) 38.0575 1.43741 0.718707 0.695313i \(-0.244734\pi\)
0.718707 + 0.695313i \(0.244734\pi\)
\(702\) 0 0
\(703\) −0.820461 −0.0309443
\(704\) 3.04892 0.114910
\(705\) −11.5918 −0.436572
\(706\) −25.8649 −0.973437
\(707\) 15.1293 0.568996
\(708\) −8.54288 −0.321061
\(709\) 26.1215 0.981014 0.490507 0.871437i \(-0.336812\pi\)
0.490507 + 0.871437i \(0.336812\pi\)
\(710\) −4.13706 −0.155261
\(711\) 9.62565 0.360990
\(712\) −6.41119 −0.240270
\(713\) 0.735562 0.0275470
\(714\) −25.2935 −0.946586
\(715\) 0 0
\(716\) −25.6461 −0.958439
\(717\) 18.7748 0.701157
\(718\) −30.2959 −1.13063
\(719\) 21.3739 0.797111 0.398556 0.917144i \(-0.369512\pi\)
0.398556 + 0.917144i \(0.369512\pi\)
\(720\) 1.00000 0.0372678
\(721\) 62.4747 2.32668
\(722\) 18.1685 0.676162
\(723\) −18.0271 −0.670437
\(724\) −25.8092 −0.959193
\(725\) 10.5036 0.390096
\(726\) 1.70410 0.0632452
\(727\) −17.9124 −0.664336 −0.332168 0.943220i \(-0.607780\pi\)
−0.332168 + 0.943220i \(0.607780\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 11.1685 0.413366
\(731\) −64.6238 −2.39020
\(732\) 1.55496 0.0574729
\(733\) 27.5386 1.01716 0.508580 0.861015i \(-0.330171\pi\)
0.508580 + 0.861015i \(0.330171\pi\)
\(734\) 3.41013 0.125870
\(735\) 6.63102 0.244589
\(736\) 0.356896 0.0131554
\(737\) −3.50365 −0.129059
\(738\) 10.2959 0.378997
\(739\) −7.99090 −0.293950 −0.146975 0.989140i \(-0.546954\pi\)
−0.146975 + 0.989140i \(0.546954\pi\)
\(740\) 0.899772 0.0330763
\(741\) 0 0
\(742\) −12.5743 −0.461618
\(743\) 19.8398 0.727853 0.363927 0.931428i \(-0.381436\pi\)
0.363927 + 0.931428i \(0.381436\pi\)
\(744\) 2.06100 0.0755599
\(745\) 6.91185 0.253231
\(746\) −3.35988 −0.123014
\(747\) 9.86054 0.360778
\(748\) −20.8877 −0.763730
\(749\) 46.9594 1.71586
\(750\) −1.00000 −0.0365148
\(751\) −25.7525 −0.939724 −0.469862 0.882740i \(-0.655696\pi\)
−0.469862 + 0.882740i \(0.655696\pi\)
\(752\) −11.5918 −0.422709
\(753\) −2.18060 −0.0794656
\(754\) 0 0
\(755\) −5.43296 −0.197726
\(756\) −3.69202 −0.134278
\(757\) −51.5096 −1.87215 −0.936074 0.351802i \(-0.885569\pi\)
−0.936074 + 0.351802i \(0.885569\pi\)
\(758\) −31.4916 −1.14383
\(759\) −1.08815 −0.0394972
\(760\) 0.911854 0.0330764
\(761\) −31.4765 −1.14102 −0.570511 0.821290i \(-0.693255\pi\)
−0.570511 + 0.821290i \(0.693255\pi\)
\(762\) 7.08815 0.256776
\(763\) 73.8117 2.67216
\(764\) −14.5700 −0.527125
\(765\) −6.85086 −0.247693
\(766\) 21.8485 0.789417
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −47.1584 −1.70057 −0.850287 0.526319i \(-0.823572\pi\)
−0.850287 + 0.526319i \(0.823572\pi\)
\(770\) 11.2567 0.405662
\(771\) −9.58748 −0.345285
\(772\) −8.32544 −0.299639
\(773\) −3.71917 −0.133769 −0.0668846 0.997761i \(-0.521306\pi\)
−0.0668846 + 0.997761i \(0.521306\pi\)
\(774\) −9.43296 −0.339061
\(775\) −2.06100 −0.0740333
\(776\) 6.97823 0.250504
\(777\) −3.32198 −0.119175
\(778\) 11.3787 0.407945
\(779\) 9.38835 0.336373
\(780\) 0 0
\(781\) 12.6136 0.451349
\(782\) −2.44504 −0.0874345
\(783\) 10.5036 0.375370
\(784\) 6.63102 0.236822
\(785\) −9.13706 −0.326116
\(786\) −19.7506 −0.704482
\(787\) 47.2282 1.68350 0.841752 0.539865i \(-0.181525\pi\)
0.841752 + 0.539865i \(0.181525\pi\)
\(788\) 26.8810 0.957595
\(789\) −1.41789 −0.0504784
\(790\) −9.62565 −0.342465
\(791\) 16.6116 0.590642
\(792\) −3.04892 −0.108339
\(793\) 0 0
\(794\) 22.0476 0.782440
\(795\) −3.40581 −0.120792
\(796\) −18.7047 −0.662970
\(797\) 34.4359 1.21978 0.609892 0.792485i \(-0.291213\pi\)
0.609892 + 0.792485i \(0.291213\pi\)
\(798\) −3.36658 −0.119176
\(799\) 79.4137 2.80945
\(800\) −1.00000 −0.0353553
\(801\) 6.41119 0.226528
\(802\) 10.2459 0.361796
\(803\) −34.0519 −1.20167
\(804\) −1.14914 −0.0405272
\(805\) 1.31767 0.0464417
\(806\) 0 0
\(807\) −4.14675 −0.145973
\(808\) 4.09783 0.144161
\(809\) −32.3846 −1.13858 −0.569292 0.822136i \(-0.692782\pi\)
−0.569292 + 0.822136i \(0.692782\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −37.8297 −1.32838 −0.664190 0.747564i \(-0.731223\pi\)
−0.664190 + 0.747564i \(0.731223\pi\)
\(812\) −38.7797 −1.36090
\(813\) −6.40044 −0.224473
\(814\) −2.74333 −0.0961537
\(815\) −14.3773 −0.503616
\(816\) −6.85086 −0.239828
\(817\) −8.60148 −0.300928
\(818\) 22.8189 0.797845
\(819\) 0 0
\(820\) −10.2959 −0.359548
\(821\) −43.2820 −1.51055 −0.755276 0.655406i \(-0.772497\pi\)
−0.755276 + 0.655406i \(0.772497\pi\)
\(822\) −7.94331 −0.277055
\(823\) −26.7399 −0.932093 −0.466047 0.884760i \(-0.654322\pi\)
−0.466047 + 0.884760i \(0.654322\pi\)
\(824\) 16.9215 0.589490
\(825\) 3.04892 0.106150
\(826\) −31.5405 −1.09743
\(827\) −21.4437 −0.745671 −0.372835 0.927897i \(-0.621614\pi\)
−0.372835 + 0.927897i \(0.621614\pi\)
\(828\) −0.356896 −0.0124030
\(829\) 22.4547 0.779885 0.389943 0.920839i \(-0.372495\pi\)
0.389943 + 0.920839i \(0.372495\pi\)
\(830\) −9.86054 −0.342264
\(831\) 10.3394 0.358671
\(832\) 0 0
\(833\) −45.4282 −1.57399
\(834\) 1.71379 0.0593438
\(835\) 3.00538 0.104005
\(836\) −2.78017 −0.0961541
\(837\) −2.06100 −0.0712385
\(838\) 28.5773 0.987187
\(839\) −24.4034 −0.842500 −0.421250 0.906945i \(-0.638408\pi\)
−0.421250 + 0.906945i \(0.638408\pi\)
\(840\) 3.69202 0.127387
\(841\) 81.3266 2.80437
\(842\) 32.3937 1.11636
\(843\) −5.08144 −0.175014
\(844\) 2.93362 0.100980
\(845\) 0 0
\(846\) 11.5918 0.398534
\(847\) 6.29159 0.216181
\(848\) −3.40581 −0.116956
\(849\) 24.5284 0.841813
\(850\) 6.85086 0.234982
\(851\) −0.321125 −0.0110080
\(852\) 4.13706 0.141733
\(853\) 7.96184 0.272608 0.136304 0.990667i \(-0.456478\pi\)
0.136304 + 0.990667i \(0.456478\pi\)
\(854\) 5.74094 0.196451
\(855\) −0.911854 −0.0311847
\(856\) 12.7192 0.434732
\(857\) −8.12737 −0.277626 −0.138813 0.990319i \(-0.544329\pi\)
−0.138813 + 0.990319i \(0.544329\pi\)
\(858\) 0 0
\(859\) 17.1666 0.585717 0.292858 0.956156i \(-0.405394\pi\)
0.292858 + 0.956156i \(0.405394\pi\)
\(860\) 9.43296 0.321661
\(861\) 38.0127 1.29547
\(862\) −14.6455 −0.498828
\(863\) 46.2857 1.57558 0.787792 0.615941i \(-0.211224\pi\)
0.787792 + 0.615941i \(0.211224\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.89546 0.234453
\(866\) 35.8485 1.21818
\(867\) 29.9342 1.01662
\(868\) 7.60925 0.258275
\(869\) 29.3478 0.995556
\(870\) −10.5036 −0.356107
\(871\) 0 0
\(872\) 19.9922 0.677022
\(873\) −6.97823 −0.236177
\(874\) −0.325437 −0.0110081
\(875\) −3.69202 −0.124813
\(876\) −11.1685 −0.377349
\(877\) −2.72156 −0.0919006 −0.0459503 0.998944i \(-0.514632\pi\)
−0.0459503 + 0.998944i \(0.514632\pi\)
\(878\) −2.55927 −0.0863712
\(879\) −15.8538 −0.534737
\(880\) 3.04892 0.102779
\(881\) −16.7241 −0.563448 −0.281724 0.959495i \(-0.590906\pi\)
−0.281724 + 0.959495i \(0.590906\pi\)
\(882\) −6.63102 −0.223278
\(883\) −2.88364 −0.0970423 −0.0485211 0.998822i \(-0.515451\pi\)
−0.0485211 + 0.998822i \(0.515451\pi\)
\(884\) 0 0
\(885\) −8.54288 −0.287166
\(886\) −23.2857 −0.782300
\(887\) 55.8866 1.87649 0.938245 0.345973i \(-0.112451\pi\)
0.938245 + 0.345973i \(0.112451\pi\)
\(888\) −0.899772 −0.0301944
\(889\) 26.1696 0.877700
\(890\) −6.41119 −0.214904
\(891\) 3.04892 0.102143
\(892\) 18.1444 0.607518
\(893\) 10.5700 0.353712
\(894\) −6.91185 −0.231167
\(895\) −25.6461 −0.857254
\(896\) 3.69202 0.123342
\(897\) 0 0
\(898\) −27.3327 −0.912105
\(899\) −21.6480 −0.722001
\(900\) 1.00000 0.0333333
\(901\) 23.3327 0.777326
\(902\) 31.3913 1.04522
\(903\) −34.8267 −1.15896
\(904\) 4.49934 0.149646
\(905\) −25.8092 −0.857928
\(906\) 5.43296 0.180498
\(907\) 49.5381 1.64489 0.822443 0.568848i \(-0.192611\pi\)
0.822443 + 0.568848i \(0.192611\pi\)
\(908\) −12.3056 −0.408375
\(909\) −4.09783 −0.135917
\(910\) 0 0
\(911\) 37.0603 1.22786 0.613931 0.789360i \(-0.289587\pi\)
0.613931 + 0.789360i \(0.289587\pi\)
\(912\) −0.911854 −0.0301945
\(913\) 30.0640 0.994973
\(914\) −7.74094 −0.256047
\(915\) 1.55496 0.0514053
\(916\) −4.64848 −0.153590
\(917\) −72.9197 −2.40802
\(918\) 6.85086 0.226112
\(919\) 53.6228 1.76885 0.884426 0.466680i \(-0.154550\pi\)
0.884426 + 0.466680i \(0.154550\pi\)
\(920\) 0.356896 0.0117665
\(921\) 14.2121 0.468303
\(922\) 17.1511 0.564840
\(923\) 0 0
\(924\) −11.2567 −0.370317
\(925\) 0.899772 0.0295843
\(926\) −34.2747 −1.12634
\(927\) −16.9215 −0.555776
\(928\) −10.5036 −0.344799
\(929\) 10.6726 0.350158 0.175079 0.984554i \(-0.443982\pi\)
0.175079 + 0.984554i \(0.443982\pi\)
\(930\) 2.06100 0.0675828
\(931\) −6.04652 −0.198167
\(932\) 0.658170 0.0215591
\(933\) 15.0127 0.491493
\(934\) −20.9474 −0.685419
\(935\) −20.8877 −0.683101
\(936\) 0 0
\(937\) 17.0538 0.557124 0.278562 0.960418i \(-0.410142\pi\)
0.278562 + 0.960418i \(0.410142\pi\)
\(938\) −4.24267 −0.138528
\(939\) −13.4993 −0.440534
\(940\) −11.5918 −0.378083
\(941\) 46.7706 1.52468 0.762339 0.647178i \(-0.224051\pi\)
0.762339 + 0.647178i \(0.224051\pi\)
\(942\) 9.13706 0.297702
\(943\) 3.67456 0.119660
\(944\) −8.54288 −0.278047
\(945\) −3.69202 −0.120101
\(946\) −28.7603 −0.935079
\(947\) 30.0398 0.976163 0.488081 0.872798i \(-0.337697\pi\)
0.488081 + 0.872798i \(0.337697\pi\)
\(948\) 9.62565 0.312626
\(949\) 0 0
\(950\) 0.911854 0.0295845
\(951\) −33.9463 −1.10078
\(952\) −25.2935 −0.819767
\(953\) −29.4252 −0.953175 −0.476588 0.879127i \(-0.658127\pi\)
−0.476588 + 0.879127i \(0.658127\pi\)
\(954\) 3.40581 0.110267
\(955\) −14.5700 −0.471475
\(956\) 18.7748 0.607220
\(957\) 32.0248 1.03521
\(958\) 15.8310 0.511477
\(959\) −29.3269 −0.947014
\(960\) 1.00000 0.0322749
\(961\) −26.7523 −0.862977
\(962\) 0 0
\(963\) −12.7192 −0.409869
\(964\) −18.0271 −0.580615
\(965\) −8.32544 −0.268005
\(966\) −1.31767 −0.0423952
\(967\) −12.7023 −0.408478 −0.204239 0.978921i \(-0.565472\pi\)
−0.204239 + 0.978921i \(0.565472\pi\)
\(968\) 1.70410 0.0547719
\(969\) 6.24698 0.200682
\(970\) 6.97823 0.224057
\(971\) 22.9105 0.735234 0.367617 0.929977i \(-0.380174\pi\)
0.367617 + 0.929977i \(0.380174\pi\)
\(972\) 1.00000 0.0320750
\(973\) 6.32736 0.202846
\(974\) 33.7060 1.08001
\(975\) 0 0
\(976\) 1.55496 0.0497730
\(977\) 25.1390 0.804267 0.402134 0.915581i \(-0.368269\pi\)
0.402134 + 0.915581i \(0.368269\pi\)
\(978\) 14.3773 0.459737
\(979\) 19.5472 0.624731
\(980\) 6.63102 0.211820
\(981\) −19.9922 −0.638303
\(982\) −3.98925 −0.127302
\(983\) 50.3895 1.60718 0.803588 0.595186i \(-0.202921\pi\)
0.803588 + 0.595186i \(0.202921\pi\)
\(984\) 10.2959 0.328221
\(985\) 26.8810 0.856499
\(986\) 71.9590 2.29164
\(987\) 42.7972 1.36225
\(988\) 0 0
\(989\) −3.36658 −0.107051
\(990\) −3.04892 −0.0969010
\(991\) 23.3375 0.741341 0.370670 0.928764i \(-0.379128\pi\)
0.370670 + 0.928764i \(0.379128\pi\)
\(992\) 2.06100 0.0654368
\(993\) 6.84415 0.217193
\(994\) 15.2741 0.484466
\(995\) −18.7047 −0.592979
\(996\) 9.86054 0.312443
\(997\) 22.8127 0.722485 0.361243 0.932472i \(-0.382353\pi\)
0.361243 + 0.932472i \(0.382353\pi\)
\(998\) 20.4698 0.647960
\(999\) 0.899772 0.0284675
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.bp.1.1 3
13.5 odd 4 5070.2.b.z.1351.4 6
13.8 odd 4 5070.2.b.z.1351.3 6
13.12 even 2 5070.2.a.bw.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bp.1.1 3 1.1 even 1 trivial
5070.2.a.bw.1.3 yes 3 13.12 even 2
5070.2.b.z.1351.3 6 13.8 odd 4
5070.2.b.z.1351.4 6 13.5 odd 4