Properties

Label 5070.2.a.bw.1.1
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 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.04892 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.04892 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.35690 q^{11} +1.00000 q^{12} -1.04892 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.08815 q^{17} +1.00000 q^{18} +2.93900 q^{19} -1.00000 q^{20} -1.04892 q^{21} +1.35690 q^{22} -0.692021 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -1.04892 q^{28} -2.37867 q^{29} -1.00000 q^{30} +9.85086 q^{31} +1.00000 q^{32} +1.35690 q^{33} -1.08815 q^{34} +1.04892 q^{35} +1.00000 q^{36} +9.26875 q^{37} +2.93900 q^{38} -1.00000 q^{40} +2.84117 q^{41} -1.04892 q^{42} -4.45473 q^{43} +1.35690 q^{44} -1.00000 q^{45} -0.692021 q^{46} -3.31767 q^{47} +1.00000 q^{48} -5.89977 q^{49} +1.00000 q^{50} -1.08815 q^{51} +0.664874 q^{53} +1.00000 q^{54} -1.35690 q^{55} -1.04892 q^{56} +2.93900 q^{57} -2.37867 q^{58} -1.96077 q^{59} -1.00000 q^{60} +3.24698 q^{61} +9.85086 q^{62} -1.04892 q^{63} +1.00000 q^{64} +1.35690 q^{66} +6.91185 q^{67} -1.08815 q^{68} -0.692021 q^{69} +1.04892 q^{70} +2.29590 q^{71} +1.00000 q^{72} +3.36227 q^{73} +9.26875 q^{74} +1.00000 q^{75} +2.93900 q^{76} -1.42327 q^{77} +12.3230 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.84117 q^{82} +2.68664 q^{83} -1.04892 q^{84} +1.08815 q^{85} -4.45473 q^{86} -2.37867 q^{87} +1.35690 q^{88} +12.8877 q^{89} -1.00000 q^{90} -0.692021 q^{92} +9.85086 q^{93} -3.31767 q^{94} -2.93900 q^{95} +1.00000 q^{96} +1.56704 q^{97} -5.89977 q^{98} +1.35690 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} + 3q^{6} + 6q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} + 3q^{6} + 6q^{7} + 3q^{8} + 3q^{9} - 3q^{10} + 3q^{12} + 6q^{14} - 3q^{15} + 3q^{16} - 7q^{17} + 3q^{18} - q^{19} - 3q^{20} + 6q^{21} + 3q^{23} + 3q^{24} + 3q^{25} + 3q^{27} + 6q^{28} - 3q^{30} + 16q^{31} + 3q^{32} - 7q^{34} - 6q^{35} + 3q^{36} + 20q^{37} - q^{38} - 3q^{40} + 17q^{41} + 6q^{42} + 9q^{43} - 3q^{45} + 3q^{46} + 7q^{47} + 3q^{48} + 5q^{49} + 3q^{50} - 7q^{51} + 3q^{53} + 3q^{54} + 6q^{56} - q^{57} + 7q^{59} - 3q^{60} + 5q^{61} + 16q^{62} + 6q^{63} + 3q^{64} + 17q^{67} - 7q^{68} + 3q^{69} - 6q^{70} - 7q^{71} + 3q^{72} + 3q^{73} + 20q^{74} + 3q^{75} - q^{76} - 7q^{77} + 17q^{79} - 3q^{80} + 3q^{81} + 17q^{82} + 6q^{83} + 6q^{84} + 7q^{85} + 9q^{86} - 3q^{89} - 3q^{90} + 3q^{92} + 16q^{93} + 7q^{94} + q^{95} + 3q^{96} + 24q^{97} + 5q^{98} + O(q^{100}) \)

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
\(7\) −1.04892 −0.396453 −0.198227 0.980156i \(-0.563518\pi\)
−0.198227 + 0.980156i \(0.563518\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.35690 0.409119 0.204560 0.978854i \(-0.434424\pi\)
0.204560 + 0.978854i \(0.434424\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.04892 −0.280335
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −1.08815 −0.263914 −0.131957 0.991255i \(-0.542126\pi\)
−0.131957 + 0.991255i \(0.542126\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.93900 0.674253 0.337127 0.941459i \(-0.390545\pi\)
0.337127 + 0.941459i \(0.390545\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.04892 −0.228893
\(22\) 1.35690 0.289291
\(23\) −0.692021 −0.144296 −0.0721482 0.997394i \(-0.522985\pi\)
−0.0721482 + 0.997394i \(0.522985\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.04892 −0.198227
\(29\) −2.37867 −0.441707 −0.220854 0.975307i \(-0.570884\pi\)
−0.220854 + 0.975307i \(0.570884\pi\)
\(30\) −1.00000 −0.182574
\(31\) 9.85086 1.76927 0.884633 0.466288i \(-0.154409\pi\)
0.884633 + 0.466288i \(0.154409\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.35690 0.236205
\(34\) −1.08815 −0.186615
\(35\) 1.04892 0.177299
\(36\) 1.00000 0.166667
\(37\) 9.26875 1.52377 0.761887 0.647710i \(-0.224273\pi\)
0.761887 + 0.647710i \(0.224273\pi\)
\(38\) 2.93900 0.476769
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.84117 0.443716 0.221858 0.975079i \(-0.428788\pi\)
0.221858 + 0.975079i \(0.428788\pi\)
\(42\) −1.04892 −0.161851
\(43\) −4.45473 −0.679340 −0.339670 0.940545i \(-0.610315\pi\)
−0.339670 + 0.940545i \(0.610315\pi\)
\(44\) 1.35690 0.204560
\(45\) −1.00000 −0.149071
\(46\) −0.692021 −0.102033
\(47\) −3.31767 −0.483931 −0.241966 0.970285i \(-0.577792\pi\)
−0.241966 + 0.970285i \(0.577792\pi\)
\(48\) 1.00000 0.144338
\(49\) −5.89977 −0.842825
\(50\) 1.00000 0.141421
\(51\) −1.08815 −0.152371
\(52\) 0 0
\(53\) 0.664874 0.0913275 0.0456638 0.998957i \(-0.485460\pi\)
0.0456638 + 0.998957i \(0.485460\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.35690 −0.182964
\(56\) −1.04892 −0.140167
\(57\) 2.93900 0.389280
\(58\) −2.37867 −0.312334
\(59\) −1.96077 −0.255271 −0.127635 0.991821i \(-0.540739\pi\)
−0.127635 + 0.991821i \(0.540739\pi\)
\(60\) −1.00000 −0.129099
\(61\) 3.24698 0.415733 0.207867 0.978157i \(-0.433348\pi\)
0.207867 + 0.978157i \(0.433348\pi\)
\(62\) 9.85086 1.25106
\(63\) −1.04892 −0.132151
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.35690 0.167022
\(67\) 6.91185 0.844417 0.422209 0.906499i \(-0.361255\pi\)
0.422209 + 0.906499i \(0.361255\pi\)
\(68\) −1.08815 −0.131957
\(69\) −0.692021 −0.0833096
\(70\) 1.04892 0.125370
\(71\) 2.29590 0.272473 0.136236 0.990676i \(-0.456499\pi\)
0.136236 + 0.990676i \(0.456499\pi\)
\(72\) 1.00000 0.117851
\(73\) 3.36227 0.393524 0.196762 0.980451i \(-0.436957\pi\)
0.196762 + 0.980451i \(0.436957\pi\)
\(74\) 9.26875 1.07747
\(75\) 1.00000 0.115470
\(76\) 2.93900 0.337127
\(77\) −1.42327 −0.162197
\(78\) 0 0
\(79\) 12.3230 1.38645 0.693225 0.720721i \(-0.256189\pi\)
0.693225 + 0.720721i \(0.256189\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 2.84117 0.313754
\(83\) 2.68664 0.294898 0.147449 0.989070i \(-0.452894\pi\)
0.147449 + 0.989070i \(0.452894\pi\)
\(84\) −1.04892 −0.114446
\(85\) 1.08815 0.118026
\(86\) −4.45473 −0.480366
\(87\) −2.37867 −0.255020
\(88\) 1.35690 0.144646
\(89\) 12.8877 1.36609 0.683046 0.730375i \(-0.260655\pi\)
0.683046 + 0.730375i \(0.260655\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −0.692021 −0.0721482
\(93\) 9.85086 1.02149
\(94\) −3.31767 −0.342191
\(95\) −2.93900 −0.301535
\(96\) 1.00000 0.102062
\(97\) 1.56704 0.159109 0.0795544 0.996831i \(-0.474650\pi\)
0.0795544 + 0.996831i \(0.474650\pi\)
\(98\) −5.89977 −0.595967
\(99\) 1.35690 0.136373
\(100\) 1.00000 0.100000
\(101\) 4.71379 0.469040 0.234520 0.972111i \(-0.424648\pi\)
0.234520 + 0.972111i \(0.424648\pi\)
\(102\) −1.08815 −0.107743
\(103\) −12.1642 −1.19858 −0.599288 0.800534i \(-0.704549\pi\)
−0.599288 + 0.800534i \(0.704549\pi\)
\(104\) 0 0
\(105\) 1.04892 0.102364
\(106\) 0.664874 0.0645783
\(107\) 1.83877 0.177761 0.0888805 0.996042i \(-0.471671\pi\)
0.0888805 + 0.996042i \(0.471671\pi\)
\(108\) 1.00000 0.0962250
\(109\) 16.2403 1.55554 0.777768 0.628551i \(-0.216352\pi\)
0.777768 + 0.628551i \(0.216352\pi\)
\(110\) −1.35690 −0.129375
\(111\) 9.26875 0.879751
\(112\) −1.04892 −0.0991134
\(113\) 16.8267 1.58292 0.791461 0.611220i \(-0.209321\pi\)
0.791461 + 0.611220i \(0.209321\pi\)
\(114\) 2.93900 0.275263
\(115\) 0.692021 0.0645313
\(116\) −2.37867 −0.220854
\(117\) 0 0
\(118\) −1.96077 −0.180504
\(119\) 1.14138 0.104630
\(120\) −1.00000 −0.0912871
\(121\) −9.15883 −0.832621
\(122\) 3.24698 0.293968
\(123\) 2.84117 0.256179
\(124\) 9.85086 0.884633
\(125\) −1.00000 −0.0894427
\(126\) −1.04892 −0.0934450
\(127\) −5.06100 −0.449091 −0.224546 0.974464i \(-0.572090\pi\)
−0.224546 + 0.974464i \(0.572090\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.45473 −0.392217
\(130\) 0 0
\(131\) 3.81940 0.333702 0.166851 0.985982i \(-0.446640\pi\)
0.166851 + 0.985982i \(0.446640\pi\)
\(132\) 1.35690 0.118103
\(133\) −3.08277 −0.267310
\(134\) 6.91185 0.597093
\(135\) −1.00000 −0.0860663
\(136\) −1.08815 −0.0933077
\(137\) −8.59717 −0.734506 −0.367253 0.930121i \(-0.619702\pi\)
−0.367253 + 0.930121i \(0.619702\pi\)
\(138\) −0.692021 −0.0589088
\(139\) −2.38404 −0.202212 −0.101106 0.994876i \(-0.532238\pi\)
−0.101106 + 0.994876i \(0.532238\pi\)
\(140\) 1.04892 0.0886497
\(141\) −3.31767 −0.279398
\(142\) 2.29590 0.192667
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 2.37867 0.197537
\(146\) 3.36227 0.278264
\(147\) −5.89977 −0.486605
\(148\) 9.26875 0.761887
\(149\) −8.93900 −0.732312 −0.366156 0.930554i \(-0.619326\pi\)
−0.366156 + 0.930554i \(0.619326\pi\)
\(150\) 1.00000 0.0816497
\(151\) −8.45473 −0.688036 −0.344018 0.938963i \(-0.611788\pi\)
−0.344018 + 0.938963i \(0.611788\pi\)
\(152\) 2.93900 0.238384
\(153\) −1.08815 −0.0879714
\(154\) −1.42327 −0.114690
\(155\) −9.85086 −0.791240
\(156\) 0 0
\(157\) −2.70410 −0.215811 −0.107905 0.994161i \(-0.534414\pi\)
−0.107905 + 0.994161i \(0.534414\pi\)
\(158\) 12.3230 0.980369
\(159\) 0.664874 0.0527280
\(160\) −1.00000 −0.0790569
\(161\) 0.725873 0.0572068
\(162\) 1.00000 0.0785674
\(163\) −22.5284 −1.76456 −0.882280 0.470724i \(-0.843993\pi\)
−0.882280 + 0.470724i \(0.843993\pi\)
\(164\) 2.84117 0.221858
\(165\) −1.35690 −0.105634
\(166\) 2.68664 0.208524
\(167\) 12.2228 0.945830 0.472915 0.881108i \(-0.343202\pi\)
0.472915 + 0.881108i \(0.343202\pi\)
\(168\) −1.04892 −0.0809257
\(169\) 0 0
\(170\) 1.08815 0.0834570
\(171\) 2.93900 0.224751
\(172\) −4.45473 −0.339670
\(173\) −11.7168 −0.890810 −0.445405 0.895329i \(-0.646940\pi\)
−0.445405 + 0.895329i \(0.646940\pi\)
\(174\) −2.37867 −0.180326
\(175\) −1.04892 −0.0792907
\(176\) 1.35690 0.102280
\(177\) −1.96077 −0.147381
\(178\) 12.8877 0.965973
\(179\) 8.89738 0.665021 0.332511 0.943099i \(-0.392104\pi\)
0.332511 + 0.943099i \(0.392104\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 1.31229 0.0975418 0.0487709 0.998810i \(-0.484470\pi\)
0.0487709 + 0.998810i \(0.484470\pi\)
\(182\) 0 0
\(183\) 3.24698 0.240024
\(184\) −0.692021 −0.0510165
\(185\) −9.26875 −0.681452
\(186\) 9.85086 0.722300
\(187\) −1.47650 −0.107972
\(188\) −3.31767 −0.241966
\(189\) −1.04892 −0.0762975
\(190\) −2.93900 −0.213218
\(191\) 5.75063 0.416101 0.208050 0.978118i \(-0.433288\pi\)
0.208050 + 0.978118i \(0.433288\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.0339 0.722252 0.361126 0.932517i \(-0.382392\pi\)
0.361126 + 0.932517i \(0.382392\pi\)
\(194\) 1.56704 0.112507
\(195\) 0 0
\(196\) −5.89977 −0.421412
\(197\) 22.9071 1.63206 0.816031 0.578009i \(-0.196170\pi\)
0.816031 + 0.578009i \(0.196170\pi\)
\(198\) 1.35690 0.0964304
\(199\) 27.0291 1.91604 0.958020 0.286702i \(-0.0925589\pi\)
0.958020 + 0.286702i \(0.0925589\pi\)
\(200\) 1.00000 0.0707107
\(201\) 6.91185 0.487525
\(202\) 4.71379 0.331661
\(203\) 2.49502 0.175116
\(204\) −1.08815 −0.0761855
\(205\) −2.84117 −0.198436
\(206\) −12.1642 −0.847521
\(207\) −0.692021 −0.0480988
\(208\) 0 0
\(209\) 3.98792 0.275850
\(210\) 1.04892 0.0723822
\(211\) 10.3720 0.714035 0.357018 0.934098i \(-0.383794\pi\)
0.357018 + 0.934098i \(0.383794\pi\)
\(212\) 0.664874 0.0456638
\(213\) 2.29590 0.157312
\(214\) 1.83877 0.125696
\(215\) 4.45473 0.303810
\(216\) 1.00000 0.0680414
\(217\) −10.3327 −0.701432
\(218\) 16.2403 1.09993
\(219\) 3.36227 0.227201
\(220\) −1.35690 −0.0914819
\(221\) 0 0
\(222\) 9.26875 0.622078
\(223\) 14.0532 0.941074 0.470537 0.882380i \(-0.344060\pi\)
0.470537 + 0.882380i \(0.344060\pi\)
\(224\) −1.04892 −0.0700837
\(225\) 1.00000 0.0666667
\(226\) 16.8267 1.11929
\(227\) −1.93362 −0.128339 −0.0641696 0.997939i \(-0.520440\pi\)
−0.0641696 + 0.997939i \(0.520440\pi\)
\(228\) 2.93900 0.194640
\(229\) −10.9148 −0.721273 −0.360636 0.932706i \(-0.617440\pi\)
−0.360636 + 0.932706i \(0.617440\pi\)
\(230\) 0.692021 0.0456305
\(231\) −1.42327 −0.0936444
\(232\) −2.37867 −0.156167
\(233\) −21.6896 −1.42093 −0.710467 0.703730i \(-0.751516\pi\)
−0.710467 + 0.703730i \(0.751516\pi\)
\(234\) 0 0
\(235\) 3.31767 0.216421
\(236\) −1.96077 −0.127635
\(237\) 12.3230 0.800468
\(238\) 1.14138 0.0739844
\(239\) −27.2349 −1.76168 −0.880840 0.473415i \(-0.843021\pi\)
−0.880840 + 0.473415i \(0.843021\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 8.21014 0.528862 0.264431 0.964405i \(-0.414816\pi\)
0.264431 + 0.964405i \(0.414816\pi\)
\(242\) −9.15883 −0.588752
\(243\) 1.00000 0.0641500
\(244\) 3.24698 0.207867
\(245\) 5.89977 0.376923
\(246\) 2.84117 0.181146
\(247\) 0 0
\(248\) 9.85086 0.625530
\(249\) 2.68664 0.170259
\(250\) −1.00000 −0.0632456
\(251\) −6.57002 −0.414696 −0.207348 0.978267i \(-0.566483\pi\)
−0.207348 + 0.978267i \(0.566483\pi\)
\(252\) −1.04892 −0.0660756
\(253\) −0.939001 −0.0590345
\(254\) −5.06100 −0.317555
\(255\) 1.08815 0.0681423
\(256\) 1.00000 0.0625000
\(257\) 13.7657 0.858680 0.429340 0.903143i \(-0.358746\pi\)
0.429340 + 0.903143i \(0.358746\pi\)
\(258\) −4.45473 −0.277339
\(259\) −9.72215 −0.604105
\(260\) 0 0
\(261\) −2.37867 −0.147236
\(262\) 3.81940 0.235963
\(263\) −9.54288 −0.588439 −0.294219 0.955738i \(-0.595060\pi\)
−0.294219 + 0.955738i \(0.595060\pi\)
\(264\) 1.35690 0.0835112
\(265\) −0.664874 −0.0408429
\(266\) −3.08277 −0.189017
\(267\) 12.8877 0.788714
\(268\) 6.91185 0.422209
\(269\) 9.07069 0.553050 0.276525 0.961007i \(-0.410817\pi\)
0.276525 + 0.961007i \(0.410817\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 17.5579 1.06657 0.533285 0.845936i \(-0.320958\pi\)
0.533285 + 0.845936i \(0.320958\pi\)
\(272\) −1.08815 −0.0659785
\(273\) 0 0
\(274\) −8.59717 −0.519374
\(275\) 1.35690 0.0818239
\(276\) −0.692021 −0.0416548
\(277\) 13.7071 0.823579 0.411790 0.911279i \(-0.364904\pi\)
0.411790 + 0.911279i \(0.364904\pi\)
\(278\) −2.38404 −0.142985
\(279\) 9.85086 0.589755
\(280\) 1.04892 0.0626848
\(281\) −24.3696 −1.45377 −0.726883 0.686761i \(-0.759032\pi\)
−0.726883 + 0.686761i \(0.759032\pi\)
\(282\) −3.31767 −0.197564
\(283\) −17.1511 −1.01952 −0.509762 0.860315i \(-0.670267\pi\)
−0.509762 + 0.860315i \(0.670267\pi\)
\(284\) 2.29590 0.136236
\(285\) −2.93900 −0.174091
\(286\) 0 0
\(287\) −2.98015 −0.175913
\(288\) 1.00000 0.0589256
\(289\) −15.8159 −0.930349
\(290\) 2.37867 0.139680
\(291\) 1.56704 0.0918615
\(292\) 3.36227 0.196762
\(293\) −24.1172 −1.40894 −0.704471 0.709732i \(-0.748816\pi\)
−0.704471 + 0.709732i \(0.748816\pi\)
\(294\) −5.89977 −0.344082
\(295\) 1.96077 0.114161
\(296\) 9.26875 0.538735
\(297\) 1.35690 0.0787351
\(298\) −8.93900 −0.517822
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 4.67264 0.269327
\(302\) −8.45473 −0.486515
\(303\) 4.71379 0.270800
\(304\) 2.93900 0.168563
\(305\) −3.24698 −0.185922
\(306\) −1.08815 −0.0622052
\(307\) −17.2282 −0.983265 −0.491632 0.870803i \(-0.663600\pi\)
−0.491632 + 0.870803i \(0.663600\pi\)
\(308\) −1.42327 −0.0810984
\(309\) −12.1642 −0.691998
\(310\) −9.85086 −0.559491
\(311\) −25.9801 −1.47320 −0.736600 0.676329i \(-0.763570\pi\)
−0.736600 + 0.676329i \(0.763570\pi\)
\(312\) 0 0
\(313\) 7.82669 0.442391 0.221195 0.975230i \(-0.429004\pi\)
0.221195 + 0.975230i \(0.429004\pi\)
\(314\) −2.70410 −0.152601
\(315\) 1.04892 0.0590998
\(316\) 12.3230 0.693225
\(317\) 0.391813 0.0220064 0.0110032 0.999939i \(-0.496498\pi\)
0.0110032 + 0.999939i \(0.496498\pi\)
\(318\) 0.664874 0.0372843
\(319\) −3.22760 −0.180711
\(320\) −1.00000 −0.0559017
\(321\) 1.83877 0.102630
\(322\) 0.725873 0.0404513
\(323\) −3.19806 −0.177945
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −22.5284 −1.24773
\(327\) 16.2403 0.898089
\(328\) 2.84117 0.156877
\(329\) 3.47996 0.191856
\(330\) −1.35690 −0.0746947
\(331\) 26.3424 1.44791 0.723955 0.689847i \(-0.242322\pi\)
0.723955 + 0.689847i \(0.242322\pi\)
\(332\) 2.68664 0.147449
\(333\) 9.26875 0.507924
\(334\) 12.2228 0.668803
\(335\) −6.91185 −0.377635
\(336\) −1.04892 −0.0572231
\(337\) 14.8345 0.808085 0.404042 0.914740i \(-0.367605\pi\)
0.404042 + 0.914740i \(0.367605\pi\)
\(338\) 0 0
\(339\) 16.8267 0.913900
\(340\) 1.08815 0.0590130
\(341\) 13.3666 0.723841
\(342\) 2.93900 0.158923
\(343\) 13.5308 0.730594
\(344\) −4.45473 −0.240183
\(345\) 0.692021 0.0372572
\(346\) −11.7168 −0.629898
\(347\) −0.636399 −0.0341637 −0.0170819 0.999854i \(-0.505438\pi\)
−0.0170819 + 0.999854i \(0.505438\pi\)
\(348\) −2.37867 −0.127510
\(349\) 17.2784 0.924894 0.462447 0.886647i \(-0.346972\pi\)
0.462447 + 0.886647i \(0.346972\pi\)
\(350\) −1.04892 −0.0560670
\(351\) 0 0
\(352\) 1.35690 0.0723228
\(353\) −21.7614 −1.15824 −0.579121 0.815242i \(-0.696604\pi\)
−0.579121 + 0.815242i \(0.696604\pi\)
\(354\) −1.96077 −0.104214
\(355\) −2.29590 −0.121854
\(356\) 12.8877 0.683046
\(357\) 1.14138 0.0604080
\(358\) 8.89738 0.470241
\(359\) −22.8412 −1.20551 −0.602755 0.797926i \(-0.705930\pi\)
−0.602755 + 0.797926i \(0.705930\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −10.3623 −0.545383
\(362\) 1.31229 0.0689725
\(363\) −9.15883 −0.480714
\(364\) 0 0
\(365\) −3.36227 −0.175989
\(366\) 3.24698 0.169722
\(367\) −7.78315 −0.406277 −0.203139 0.979150i \(-0.565114\pi\)
−0.203139 + 0.979150i \(0.565114\pi\)
\(368\) −0.692021 −0.0360741
\(369\) 2.84117 0.147905
\(370\) −9.26875 −0.481859
\(371\) −0.697398 −0.0362071
\(372\) 9.85086 0.510743
\(373\) −30.5133 −1.57992 −0.789960 0.613158i \(-0.789899\pi\)
−0.789960 + 0.613158i \(0.789899\pi\)
\(374\) −1.47650 −0.0763480
\(375\) −1.00000 −0.0516398
\(376\) −3.31767 −0.171096
\(377\) 0 0
\(378\) −1.04892 −0.0539505
\(379\) −6.41358 −0.329444 −0.164722 0.986340i \(-0.552673\pi\)
−0.164722 + 0.986340i \(0.552673\pi\)
\(380\) −2.93900 −0.150768
\(381\) −5.06100 −0.259283
\(382\) 5.75063 0.294228
\(383\) −2.89440 −0.147897 −0.0739484 0.997262i \(-0.523560\pi\)
−0.0739484 + 0.997262i \(0.523560\pi\)
\(384\) 1.00000 0.0510310
\(385\) 1.42327 0.0725366
\(386\) 10.0339 0.510710
\(387\) −4.45473 −0.226447
\(388\) 1.56704 0.0795544
\(389\) −17.1250 −0.868271 −0.434136 0.900848i \(-0.642946\pi\)
−0.434136 + 0.900848i \(0.642946\pi\)
\(390\) 0 0
\(391\) 0.753020 0.0380819
\(392\) −5.89977 −0.297984
\(393\) 3.81940 0.192663
\(394\) 22.9071 1.15404
\(395\) −12.3230 −0.620040
\(396\) 1.35690 0.0681866
\(397\) −25.0103 −1.25523 −0.627615 0.778524i \(-0.715969\pi\)
−0.627615 + 0.778524i \(0.715969\pi\)
\(398\) 27.0291 1.35484
\(399\) −3.08277 −0.154331
\(400\) 1.00000 0.0500000
\(401\) 30.8689 1.54152 0.770760 0.637126i \(-0.219877\pi\)
0.770760 + 0.637126i \(0.219877\pi\)
\(402\) 6.91185 0.344732
\(403\) 0 0
\(404\) 4.71379 0.234520
\(405\) −1.00000 −0.0496904
\(406\) 2.49502 0.123826
\(407\) 12.5767 0.623405
\(408\) −1.08815 −0.0538713
\(409\) −11.0871 −0.548221 −0.274110 0.961698i \(-0.588383\pi\)
−0.274110 + 0.961698i \(0.588383\pi\)
\(410\) −2.84117 −0.140315
\(411\) −8.59717 −0.424067
\(412\) −12.1642 −0.599288
\(413\) 2.05669 0.101203
\(414\) −0.692021 −0.0340110
\(415\) −2.68664 −0.131882
\(416\) 0 0
\(417\) −2.38404 −0.116747
\(418\) 3.98792 0.195055
\(419\) 17.5080 0.855320 0.427660 0.903940i \(-0.359338\pi\)
0.427660 + 0.903940i \(0.359338\pi\)
\(420\) 1.04892 0.0511819
\(421\) 16.1274 0.786000 0.393000 0.919538i \(-0.371437\pi\)
0.393000 + 0.919538i \(0.371437\pi\)
\(422\) 10.3720 0.504899
\(423\) −3.31767 −0.161310
\(424\) 0.664874 0.0322892
\(425\) −1.08815 −0.0527828
\(426\) 2.29590 0.111237
\(427\) −3.40581 −0.164819
\(428\) 1.83877 0.0888805
\(429\) 0 0
\(430\) 4.45473 0.214826
\(431\) −33.2905 −1.60355 −0.801774 0.597627i \(-0.796110\pi\)
−0.801774 + 0.597627i \(0.796110\pi\)
\(432\) 1.00000 0.0481125
\(433\) −11.1056 −0.533701 −0.266851 0.963738i \(-0.585983\pi\)
−0.266851 + 0.963738i \(0.585983\pi\)
\(434\) −10.3327 −0.495987
\(435\) 2.37867 0.114048
\(436\) 16.2403 0.777768
\(437\) −2.03385 −0.0972923
\(438\) 3.36227 0.160656
\(439\) 12.6950 0.605900 0.302950 0.953007i \(-0.402029\pi\)
0.302950 + 0.953007i \(0.402029\pi\)
\(440\) −1.35690 −0.0646875
\(441\) −5.89977 −0.280942
\(442\) 0 0
\(443\) −6.90110 −0.327881 −0.163941 0.986470i \(-0.552421\pi\)
−0.163941 + 0.986470i \(0.552421\pi\)
\(444\) 9.26875 0.439875
\(445\) −12.8877 −0.610935
\(446\) 14.0532 0.665440
\(447\) −8.93900 −0.422800
\(448\) −1.04892 −0.0495567
\(449\) −3.27652 −0.154629 −0.0773143 0.997007i \(-0.524634\pi\)
−0.0773143 + 0.997007i \(0.524634\pi\)
\(450\) 1.00000 0.0471405
\(451\) 3.85517 0.181533
\(452\) 16.8267 0.791461
\(453\) −8.45473 −0.397238
\(454\) −1.93362 −0.0907495
\(455\) 0 0
\(456\) 2.93900 0.137631
\(457\) 1.40581 0.0657612 0.0328806 0.999459i \(-0.489532\pi\)
0.0328806 + 0.999459i \(0.489532\pi\)
\(458\) −10.9148 −0.510017
\(459\) −1.08815 −0.0507903
\(460\) 0.692021 0.0322657
\(461\) 12.3773 0.576470 0.288235 0.957560i \(-0.406932\pi\)
0.288235 + 0.957560i \(0.406932\pi\)
\(462\) −1.42327 −0.0662166
\(463\) 31.7797 1.47693 0.738464 0.674293i \(-0.235552\pi\)
0.738464 + 0.674293i \(0.235552\pi\)
\(464\) −2.37867 −0.110427
\(465\) −9.85086 −0.456822
\(466\) −21.6896 −1.00475
\(467\) −36.2790 −1.67879 −0.839397 0.543519i \(-0.817091\pi\)
−0.839397 + 0.543519i \(0.817091\pi\)
\(468\) 0 0
\(469\) −7.24996 −0.334772
\(470\) 3.31767 0.153033
\(471\) −2.70410 −0.124598
\(472\) −1.96077 −0.0902518
\(473\) −6.04461 −0.277931
\(474\) 12.3230 0.566016
\(475\) 2.93900 0.134851
\(476\) 1.14138 0.0523148
\(477\) 0.664874 0.0304425
\(478\) −27.2349 −1.24570
\(479\) −5.87933 −0.268633 −0.134317 0.990938i \(-0.542884\pi\)
−0.134317 + 0.990938i \(0.542884\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 8.21014 0.373962
\(483\) 0.725873 0.0330284
\(484\) −9.15883 −0.416311
\(485\) −1.56704 −0.0711556
\(486\) 1.00000 0.0453609
\(487\) 30.6243 1.38772 0.693860 0.720110i \(-0.255909\pi\)
0.693860 + 0.720110i \(0.255909\pi\)
\(488\) 3.24698 0.146984
\(489\) −22.5284 −1.01877
\(490\) 5.89977 0.266525
\(491\) 34.4456 1.55451 0.777255 0.629186i \(-0.216612\pi\)
0.777255 + 0.629186i \(0.216612\pi\)
\(492\) 2.84117 0.128090
\(493\) 2.58834 0.116573
\(494\) 0 0
\(495\) −1.35690 −0.0609879
\(496\) 9.85086 0.442316
\(497\) −2.40821 −0.108023
\(498\) 2.68664 0.120391
\(499\) −10.0194 −0.448529 −0.224264 0.974528i \(-0.571998\pi\)
−0.224264 + 0.974528i \(0.571998\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 12.2228 0.546075
\(502\) −6.57002 −0.293235
\(503\) 8.27950 0.369165 0.184582 0.982817i \(-0.440907\pi\)
0.184582 + 0.982817i \(0.440907\pi\)
\(504\) −1.04892 −0.0467225
\(505\) −4.71379 −0.209761
\(506\) −0.939001 −0.0417437
\(507\) 0 0
\(508\) −5.06100 −0.224546
\(509\) −44.4607 −1.97069 −0.985343 0.170585i \(-0.945434\pi\)
−0.985343 + 0.170585i \(0.945434\pi\)
\(510\) 1.08815 0.0481839
\(511\) −3.52675 −0.156014
\(512\) 1.00000 0.0441942
\(513\) 2.93900 0.129760
\(514\) 13.7657 0.607179
\(515\) 12.1642 0.536019
\(516\) −4.45473 −0.196109
\(517\) −4.50173 −0.197986
\(518\) −9.72215 −0.427167
\(519\) −11.7168 −0.514309
\(520\) 0 0
\(521\) −35.8461 −1.57044 −0.785222 0.619214i \(-0.787451\pi\)
−0.785222 + 0.619214i \(0.787451\pi\)
\(522\) −2.37867 −0.104111
\(523\) −5.80433 −0.253806 −0.126903 0.991915i \(-0.540504\pi\)
−0.126903 + 0.991915i \(0.540504\pi\)
\(524\) 3.81940 0.166851
\(525\) −1.04892 −0.0457785
\(526\) −9.54288 −0.416089
\(527\) −10.7192 −0.466934
\(528\) 1.35690 0.0590513
\(529\) −22.5211 −0.979179
\(530\) −0.664874 −0.0288803
\(531\) −1.96077 −0.0850902
\(532\) −3.08277 −0.133655
\(533\) 0 0
\(534\) 12.8877 0.557705
\(535\) −1.83877 −0.0794971
\(536\) 6.91185 0.298547
\(537\) 8.89738 0.383950
\(538\) 9.07069 0.391065
\(539\) −8.00538 −0.344816
\(540\) −1.00000 −0.0430331
\(541\) −23.2446 −0.999363 −0.499681 0.866209i \(-0.666550\pi\)
−0.499681 + 0.866209i \(0.666550\pi\)
\(542\) 17.5579 0.754178
\(543\) 1.31229 0.0563158
\(544\) −1.08815 −0.0466539
\(545\) −16.2403 −0.695657
\(546\) 0 0
\(547\) −16.1588 −0.690902 −0.345451 0.938437i \(-0.612274\pi\)
−0.345451 + 0.938437i \(0.612274\pi\)
\(548\) −8.59717 −0.367253
\(549\) 3.24698 0.138578
\(550\) 1.35690 0.0578582
\(551\) −6.99090 −0.297822
\(552\) −0.692021 −0.0294544
\(553\) −12.9259 −0.549663
\(554\) 13.7071 0.582358
\(555\) −9.26875 −0.393437
\(556\) −2.38404 −0.101106
\(557\) 25.5937 1.08444 0.542220 0.840236i \(-0.317584\pi\)
0.542220 + 0.840236i \(0.317584\pi\)
\(558\) 9.85086 0.417020
\(559\) 0 0
\(560\) 1.04892 0.0443248
\(561\) −1.47650 −0.0623379
\(562\) −24.3696 −1.02797
\(563\) −40.9232 −1.72471 −0.862354 0.506307i \(-0.831010\pi\)
−0.862354 + 0.506307i \(0.831010\pi\)
\(564\) −3.31767 −0.139699
\(565\) −16.8267 −0.707904
\(566\) −17.1511 −0.720913
\(567\) −1.04892 −0.0440504
\(568\) 2.29590 0.0963337
\(569\) −20.1946 −0.846602 −0.423301 0.905989i \(-0.639129\pi\)
−0.423301 + 0.905989i \(0.639129\pi\)
\(570\) −2.93900 −0.123101
\(571\) 36.8756 1.54320 0.771598 0.636110i \(-0.219458\pi\)
0.771598 + 0.636110i \(0.219458\pi\)
\(572\) 0 0
\(573\) 5.75063 0.240236
\(574\) −2.98015 −0.124389
\(575\) −0.692021 −0.0288593
\(576\) 1.00000 0.0416667
\(577\) −5.07500 −0.211275 −0.105637 0.994405i \(-0.533688\pi\)
−0.105637 + 0.994405i \(0.533688\pi\)
\(578\) −15.8159 −0.657856
\(579\) 10.0339 0.416993
\(580\) 2.37867 0.0987687
\(581\) −2.81807 −0.116913
\(582\) 1.56704 0.0649559
\(583\) 0.902165 0.0373639
\(584\) 3.36227 0.139132
\(585\) 0 0
\(586\) −24.1172 −0.996273
\(587\) −10.5714 −0.436326 −0.218163 0.975912i \(-0.570007\pi\)
−0.218163 + 0.975912i \(0.570007\pi\)
\(588\) −5.89977 −0.243303
\(589\) 28.9517 1.19293
\(590\) 1.96077 0.0807237
\(591\) 22.9071 0.942271
\(592\) 9.26875 0.380943
\(593\) 38.4999 1.58100 0.790501 0.612460i \(-0.209820\pi\)
0.790501 + 0.612460i \(0.209820\pi\)
\(594\) 1.35690 0.0556741
\(595\) −1.14138 −0.0467918
\(596\) −8.93900 −0.366156
\(597\) 27.0291 1.10623
\(598\) 0 0
\(599\) −0.332142 −0.0135709 −0.00678547 0.999977i \(-0.502160\pi\)
−0.00678547 + 0.999977i \(0.502160\pi\)
\(600\) 1.00000 0.0408248
\(601\) −7.04162 −0.287234 −0.143617 0.989633i \(-0.545873\pi\)
−0.143617 + 0.989633i \(0.545873\pi\)
\(602\) 4.67264 0.190443
\(603\) 6.91185 0.281472
\(604\) −8.45473 −0.344018
\(605\) 9.15883 0.372360
\(606\) 4.71379 0.191485
\(607\) −29.4185 −1.19406 −0.597030 0.802219i \(-0.703653\pi\)
−0.597030 + 0.802219i \(0.703653\pi\)
\(608\) 2.93900 0.119192
\(609\) 2.49502 0.101103
\(610\) −3.24698 −0.131466
\(611\) 0 0
\(612\) −1.08815 −0.0439857
\(613\) 34.1987 1.38127 0.690635 0.723203i \(-0.257331\pi\)
0.690635 + 0.723203i \(0.257331\pi\)
\(614\) −17.2282 −0.695273
\(615\) −2.84117 −0.114567
\(616\) −1.42327 −0.0573452
\(617\) −34.1890 −1.37640 −0.688198 0.725523i \(-0.741598\pi\)
−0.688198 + 0.725523i \(0.741598\pi\)
\(618\) −12.1642 −0.489316
\(619\) 9.80061 0.393920 0.196960 0.980412i \(-0.436893\pi\)
0.196960 + 0.980412i \(0.436893\pi\)
\(620\) −9.85086 −0.395620
\(621\) −0.692021 −0.0277699
\(622\) −25.9801 −1.04171
\(623\) −13.5181 −0.541592
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 7.82669 0.312818
\(627\) 3.98792 0.159262
\(628\) −2.70410 −0.107905
\(629\) −10.0858 −0.402145
\(630\) 1.04892 0.0417899
\(631\) 17.0968 0.680612 0.340306 0.940315i \(-0.389469\pi\)
0.340306 + 0.940315i \(0.389469\pi\)
\(632\) 12.3230 0.490184
\(633\) 10.3720 0.412248
\(634\) 0.391813 0.0155609
\(635\) 5.06100 0.200840
\(636\) 0.664874 0.0263640
\(637\) 0 0
\(638\) −3.22760 −0.127782
\(639\) 2.29590 0.0908243
\(640\) −1.00000 −0.0395285
\(641\) −14.7030 −0.580735 −0.290368 0.956915i \(-0.593778\pi\)
−0.290368 + 0.956915i \(0.593778\pi\)
\(642\) 1.83877 0.0725706
\(643\) 42.7036 1.68407 0.842033 0.539425i \(-0.181359\pi\)
0.842033 + 0.539425i \(0.181359\pi\)
\(644\) 0.725873 0.0286034
\(645\) 4.45473 0.175405
\(646\) −3.19806 −0.125826
\(647\) 41.7265 1.64044 0.820218 0.572051i \(-0.193852\pi\)
0.820218 + 0.572051i \(0.193852\pi\)
\(648\) 1.00000 0.0392837
\(649\) −2.66056 −0.104436
\(650\) 0 0
\(651\) −10.3327 −0.404972
\(652\) −22.5284 −0.882280
\(653\) −28.2306 −1.10475 −0.552374 0.833596i \(-0.686278\pi\)
−0.552374 + 0.833596i \(0.686278\pi\)
\(654\) 16.2403 0.635045
\(655\) −3.81940 −0.149236
\(656\) 2.84117 0.110929
\(657\) 3.36227 0.131175
\(658\) 3.47996 0.135663
\(659\) −38.7885 −1.51099 −0.755493 0.655156i \(-0.772603\pi\)
−0.755493 + 0.655156i \(0.772603\pi\)
\(660\) −1.35690 −0.0528171
\(661\) 31.5733 1.22806 0.614029 0.789284i \(-0.289548\pi\)
0.614029 + 0.789284i \(0.289548\pi\)
\(662\) 26.3424 1.02383
\(663\) 0 0
\(664\) 2.68664 0.104262
\(665\) 3.08277 0.119545
\(666\) 9.26875 0.359157
\(667\) 1.64609 0.0637368
\(668\) 12.2228 0.472915
\(669\) 14.0532 0.543329
\(670\) −6.91185 −0.267028
\(671\) 4.40581 0.170085
\(672\) −1.04892 −0.0404629
\(673\) −44.1400 −1.70147 −0.850737 0.525592i \(-0.823844\pi\)
−0.850737 + 0.525592i \(0.823844\pi\)
\(674\) 14.8345 0.571402
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −22.8431 −0.877931 −0.438966 0.898504i \(-0.644655\pi\)
−0.438966 + 0.898504i \(0.644655\pi\)
\(678\) 16.8267 0.646225
\(679\) −1.64370 −0.0630792
\(680\) 1.08815 0.0417285
\(681\) −1.93362 −0.0740966
\(682\) 13.3666 0.511833
\(683\) −13.6866 −0.523705 −0.261852 0.965108i \(-0.584333\pi\)
−0.261852 + 0.965108i \(0.584333\pi\)
\(684\) 2.93900 0.112376
\(685\) 8.59717 0.328481
\(686\) 13.5308 0.516608
\(687\) −10.9148 −0.416427
\(688\) −4.45473 −0.169835
\(689\) 0 0
\(690\) 0.692021 0.0263448
\(691\) −24.0261 −0.913995 −0.456998 0.889468i \(-0.651075\pi\)
−0.456998 + 0.889468i \(0.651075\pi\)
\(692\) −11.7168 −0.445405
\(693\) −1.42327 −0.0540656
\(694\) −0.636399 −0.0241574
\(695\) 2.38404 0.0904319
\(696\) −2.37867 −0.0901631
\(697\) −3.09160 −0.117103
\(698\) 17.2784 0.653999
\(699\) −21.6896 −0.820377
\(700\) −1.04892 −0.0396453
\(701\) 50.5392 1.90884 0.954419 0.298471i \(-0.0964765\pi\)
0.954419 + 0.298471i \(0.0964765\pi\)
\(702\) 0 0
\(703\) 27.2409 1.02741
\(704\) 1.35690 0.0511399
\(705\) 3.31767 0.124951
\(706\) −21.7614 −0.819000
\(707\) −4.94438 −0.185952
\(708\) −1.96077 −0.0736903
\(709\) −12.1847 −0.457604 −0.228802 0.973473i \(-0.573481\pi\)
−0.228802 + 0.973473i \(0.573481\pi\)
\(710\) −2.29590 −0.0861635
\(711\) 12.3230 0.462150
\(712\) 12.8877 0.482987
\(713\) −6.81700 −0.255299
\(714\) 1.14138 0.0427149
\(715\) 0 0
\(716\) 8.89738 0.332511
\(717\) −27.2349 −1.01711
\(718\) −22.8412 −0.852425
\(719\) −10.8401 −0.404268 −0.202134 0.979358i \(-0.564788\pi\)
−0.202134 + 0.979358i \(0.564788\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 12.7593 0.475179
\(722\) −10.3623 −0.385644
\(723\) 8.21014 0.305339
\(724\) 1.31229 0.0487709
\(725\) −2.37867 −0.0883414
\(726\) −9.15883 −0.339916
\(727\) 33.2489 1.23313 0.616567 0.787303i \(-0.288523\pi\)
0.616567 + 0.787303i \(0.288523\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −3.36227 −0.124443
\(731\) 4.84740 0.179287
\(732\) 3.24698 0.120012
\(733\) −8.59120 −0.317323 −0.158662 0.987333i \(-0.550718\pi\)
−0.158662 + 0.987333i \(0.550718\pi\)
\(734\) −7.78315 −0.287281
\(735\) 5.89977 0.217616
\(736\) −0.692021 −0.0255082
\(737\) 9.37867 0.345468
\(738\) 2.84117 0.104585
\(739\) −38.4131 −1.41305 −0.706525 0.707689i \(-0.749738\pi\)
−0.706525 + 0.707689i \(0.749738\pi\)
\(740\) −9.26875 −0.340726
\(741\) 0 0
\(742\) −0.697398 −0.0256023
\(743\) 21.7904 0.799414 0.399707 0.916643i \(-0.369112\pi\)
0.399707 + 0.916643i \(0.369112\pi\)
\(744\) 9.85086 0.361150
\(745\) 8.93900 0.327500
\(746\) −30.5133 −1.11717
\(747\) 2.68664 0.0982992
\(748\) −1.47650 −0.0539862
\(749\) −1.92872 −0.0704739
\(750\) −1.00000 −0.0365148
\(751\) 0.715120 0.0260951 0.0130475 0.999915i \(-0.495847\pi\)
0.0130475 + 0.999915i \(0.495847\pi\)
\(752\) −3.31767 −0.120983
\(753\) −6.57002 −0.239425
\(754\) 0 0
\(755\) 8.45473 0.307699
\(756\) −1.04892 −0.0381488
\(757\) 29.7894 1.08271 0.541357 0.840793i \(-0.317911\pi\)
0.541357 + 0.840793i \(0.317911\pi\)
\(758\) −6.41358 −0.232952
\(759\) −0.939001 −0.0340836
\(760\) −2.93900 −0.106609
\(761\) 28.4112 1.02990 0.514952 0.857219i \(-0.327810\pi\)
0.514952 + 0.857219i \(0.327810\pi\)
\(762\) −5.06100 −0.183341
\(763\) −17.0347 −0.616698
\(764\) 5.75063 0.208050
\(765\) 1.08815 0.0393420
\(766\) −2.89440 −0.104579
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 16.6200 0.599333 0.299666 0.954044i \(-0.403125\pi\)
0.299666 + 0.954044i \(0.403125\pi\)
\(770\) 1.42327 0.0512911
\(771\) 13.7657 0.495759
\(772\) 10.0339 0.361126
\(773\) −10.8388 −0.389844 −0.194922 0.980819i \(-0.562445\pi\)
−0.194922 + 0.980819i \(0.562445\pi\)
\(774\) −4.45473 −0.160122
\(775\) 9.85086 0.353853
\(776\) 1.56704 0.0562534
\(777\) −9.72215 −0.348780
\(778\) −17.1250 −0.613960
\(779\) 8.35019 0.299177
\(780\) 0 0
\(781\) 3.11529 0.111474
\(782\) 0.753020 0.0269280
\(783\) −2.37867 −0.0850066
\(784\) −5.89977 −0.210706
\(785\) 2.70410 0.0965136
\(786\) 3.81940 0.136233
\(787\) −4.55974 −0.162537 −0.0812687 0.996692i \(-0.525897\pi\)
−0.0812687 + 0.996692i \(0.525897\pi\)
\(788\) 22.9071 0.816031
\(789\) −9.54288 −0.339735
\(790\) −12.3230 −0.438434
\(791\) −17.6498 −0.627555
\(792\) 1.35690 0.0482152
\(793\) 0 0
\(794\) −25.0103 −0.887582
\(795\) −0.664874 −0.0235807
\(796\) 27.0291 0.958020
\(797\) −13.6601 −0.483865 −0.241933 0.970293i \(-0.577781\pi\)
−0.241933 + 0.970293i \(0.577781\pi\)
\(798\) −3.08277 −0.109129
\(799\) 3.61011 0.127716
\(800\) 1.00000 0.0353553
\(801\) 12.8877 0.455364
\(802\) 30.8689 1.09002
\(803\) 4.56225 0.160998
\(804\) 6.91185 0.243762
\(805\) −0.725873 −0.0255837
\(806\) 0 0
\(807\) 9.07069 0.319303
\(808\) 4.71379 0.165831
\(809\) 30.2857 1.06479 0.532395 0.846496i \(-0.321292\pi\)
0.532395 + 0.846496i \(0.321292\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −26.5327 −0.931690 −0.465845 0.884866i \(-0.654249\pi\)
−0.465845 + 0.884866i \(0.654249\pi\)
\(812\) 2.49502 0.0875582
\(813\) 17.5579 0.615784
\(814\) 12.5767 0.440814
\(815\) 22.5284 0.789136
\(816\) −1.08815 −0.0380927
\(817\) −13.0925 −0.458047
\(818\) −11.0871 −0.387651
\(819\) 0 0
\(820\) −2.84117 −0.0992178
\(821\) −48.5370 −1.69395 −0.846977 0.531630i \(-0.821580\pi\)
−0.846977 + 0.531630i \(0.821580\pi\)
\(822\) −8.59717 −0.299861
\(823\) −41.2650 −1.43841 −0.719204 0.694799i \(-0.755493\pi\)
−0.719204 + 0.694799i \(0.755493\pi\)
\(824\) −12.1642 −0.423760
\(825\) 1.35690 0.0472411
\(826\) 2.05669 0.0715613
\(827\) −22.9004 −0.796324 −0.398162 0.917315i \(-0.630352\pi\)
−0.398162 + 0.917315i \(0.630352\pi\)
\(828\) −0.692021 −0.0240494
\(829\) 13.9782 0.485484 0.242742 0.970091i \(-0.421953\pi\)
0.242742 + 0.970091i \(0.421953\pi\)
\(830\) −2.68664 −0.0932548
\(831\) 13.7071 0.475494
\(832\) 0 0
\(833\) 6.41981 0.222433
\(834\) −2.38404 −0.0825527
\(835\) −12.2228 −0.422988
\(836\) 3.98792 0.137925
\(837\) 9.85086 0.340495
\(838\) 17.5080 0.604802
\(839\) 1.35258 0.0466964 0.0233482 0.999727i \(-0.492567\pi\)
0.0233482 + 0.999727i \(0.492567\pi\)
\(840\) 1.04892 0.0361911
\(841\) −23.3419 −0.804895
\(842\) 16.1274 0.555786
\(843\) −24.3696 −0.839333
\(844\) 10.3720 0.357018
\(845\) 0 0
\(846\) −3.31767 −0.114064
\(847\) 9.60686 0.330096
\(848\) 0.664874 0.0228319
\(849\) −17.1511 −0.588623
\(850\) −1.08815 −0.0373231
\(851\) −6.41417 −0.219875
\(852\) 2.29590 0.0786561
\(853\) 18.0887 0.619347 0.309673 0.950843i \(-0.399780\pi\)
0.309673 + 0.950843i \(0.399780\pi\)
\(854\) −3.40581 −0.116545
\(855\) −2.93900 −0.100512
\(856\) 1.83877 0.0628480
\(857\) −8.47889 −0.289633 −0.144817 0.989458i \(-0.546259\pi\)
−0.144817 + 0.989458i \(0.546259\pi\)
\(858\) 0 0
\(859\) 19.8968 0.678870 0.339435 0.940630i \(-0.389764\pi\)
0.339435 + 0.940630i \(0.389764\pi\)
\(860\) 4.45473 0.151905
\(861\) −2.98015 −0.101563
\(862\) −33.2905 −1.13388
\(863\) −16.0989 −0.548013 −0.274006 0.961728i \(-0.588349\pi\)
−0.274006 + 0.961728i \(0.588349\pi\)
\(864\) 1.00000 0.0340207
\(865\) 11.7168 0.398382
\(866\) −11.1056 −0.377384
\(867\) −15.8159 −0.537137
\(868\) −10.3327 −0.350716
\(869\) 16.7211 0.567224
\(870\) 2.37867 0.0806443
\(871\) 0 0
\(872\) 16.2403 0.549965
\(873\) 1.56704 0.0530363
\(874\) −2.03385 −0.0687961
\(875\) 1.04892 0.0354599
\(876\) 3.36227 0.113601
\(877\) 7.14377 0.241228 0.120614 0.992699i \(-0.461514\pi\)
0.120614 + 0.992699i \(0.461514\pi\)
\(878\) 12.6950 0.428436
\(879\) −24.1172 −0.813453
\(880\) −1.35690 −0.0457410
\(881\) 42.5786 1.43451 0.717256 0.696810i \(-0.245398\pi\)
0.717256 + 0.696810i \(0.245398\pi\)
\(882\) −5.89977 −0.198656
\(883\) −38.3997 −1.29225 −0.646126 0.763230i \(-0.723612\pi\)
−0.646126 + 0.763230i \(0.723612\pi\)
\(884\) 0 0
\(885\) 1.96077 0.0659106
\(886\) −6.90110 −0.231847
\(887\) 57.1943 1.92040 0.960199 0.279317i \(-0.0901079\pi\)
0.960199 + 0.279317i \(0.0901079\pi\)
\(888\) 9.26875 0.311039
\(889\) 5.30857 0.178044
\(890\) −12.8877 −0.431996
\(891\) 1.35690 0.0454577
\(892\) 14.0532 0.470537
\(893\) −9.75063 −0.326292
\(894\) −8.93900 −0.298965
\(895\) −8.89738 −0.297407
\(896\) −1.04892 −0.0350419
\(897\) 0 0
\(898\) −3.27652 −0.109339
\(899\) −23.4319 −0.781497
\(900\) 1.00000 0.0333333
\(901\) −0.723480 −0.0241026
\(902\) 3.85517 0.128363
\(903\) 4.67264 0.155496
\(904\) 16.8267 0.559647
\(905\) −1.31229 −0.0436220
\(906\) −8.45473 −0.280890
\(907\) 1.07415 0.0356664 0.0178332 0.999841i \(-0.494323\pi\)
0.0178332 + 0.999841i \(0.494323\pi\)
\(908\) −1.93362 −0.0641696
\(909\) 4.71379 0.156347
\(910\) 0 0
\(911\) −50.9904 −1.68939 −0.844694 0.535249i \(-0.820218\pi\)
−0.844694 + 0.535249i \(0.820218\pi\)
\(912\) 2.93900 0.0973201
\(913\) 3.64550 0.120648
\(914\) 1.40581 0.0465002
\(915\) −3.24698 −0.107342
\(916\) −10.9148 −0.360636
\(917\) −4.00623 −0.132297
\(918\) −1.08815 −0.0359142
\(919\) 7.82344 0.258072 0.129036 0.991640i \(-0.458812\pi\)
0.129036 + 0.991640i \(0.458812\pi\)
\(920\) 0.692021 0.0228153
\(921\) −17.2282 −0.567688
\(922\) 12.3773 0.407626
\(923\) 0 0
\(924\) −1.42327 −0.0468222
\(925\) 9.26875 0.304755
\(926\) 31.7797 1.04435
\(927\) −12.1642 −0.399525
\(928\) −2.37867 −0.0780835
\(929\) −19.5007 −0.639796 −0.319898 0.947452i \(-0.603649\pi\)
−0.319898 + 0.947452i \(0.603649\pi\)
\(930\) −9.85086 −0.323022
\(931\) −17.3394 −0.568277
\(932\) −21.6896 −0.710467
\(933\) −25.9801 −0.850552
\(934\) −36.2790 −1.18709
\(935\) 1.47650 0.0482867
\(936\) 0 0
\(937\) −32.0968 −1.04856 −0.524278 0.851547i \(-0.675665\pi\)
−0.524278 + 0.851547i \(0.675665\pi\)
\(938\) −7.24996 −0.236720
\(939\) 7.82669 0.255414
\(940\) 3.31767 0.108210
\(941\) 35.9181 1.17090 0.585448 0.810710i \(-0.300919\pi\)
0.585448 + 0.810710i \(0.300919\pi\)
\(942\) −2.70410 −0.0881044
\(943\) −1.96615 −0.0640266
\(944\) −1.96077 −0.0638177
\(945\) 1.04892 0.0341213
\(946\) −6.04461 −0.196527
\(947\) 20.7700 0.674934 0.337467 0.941337i \(-0.390430\pi\)
0.337467 + 0.941337i \(0.390430\pi\)
\(948\) 12.3230 0.400234
\(949\) 0 0
\(950\) 2.93900 0.0953538
\(951\) 0.391813 0.0127054
\(952\) 1.14138 0.0369922
\(953\) −11.7855 −0.381771 −0.190886 0.981612i \(-0.561136\pi\)
−0.190886 + 0.981612i \(0.561136\pi\)
\(954\) 0.664874 0.0215261
\(955\) −5.75063 −0.186086
\(956\) −27.2349 −0.880840
\(957\) −3.22760 −0.104334
\(958\) −5.87933 −0.189953
\(959\) 9.01772 0.291197
\(960\) −1.00000 −0.0322749
\(961\) 66.0393 2.13030
\(962\) 0 0
\(963\) 1.83877 0.0592536
\(964\) 8.21014 0.264431
\(965\) −10.0339 −0.323001
\(966\) 0.725873 0.0233546
\(967\) −52.0116 −1.67258 −0.836290 0.548287i \(-0.815280\pi\)
−0.836290 + 0.548287i \(0.815280\pi\)
\(968\) −9.15883 −0.294376
\(969\) −3.19806 −0.102737
\(970\) −1.56704 −0.0503146
\(971\) −17.7144 −0.568482 −0.284241 0.958753i \(-0.591742\pi\)
−0.284241 + 0.958753i \(0.591742\pi\)
\(972\) 1.00000 0.0320750
\(973\) 2.50066 0.0801676
\(974\) 30.6243 0.981266
\(975\) 0 0
\(976\) 3.24698 0.103933
\(977\) −8.16959 −0.261368 −0.130684 0.991424i \(-0.541717\pi\)
−0.130684 + 0.991424i \(0.541717\pi\)
\(978\) −22.5284 −0.720379
\(979\) 17.4873 0.558895
\(980\) 5.89977 0.188461
\(981\) 16.2403 0.518512
\(982\) 34.4456 1.09920
\(983\) 57.0256 1.81883 0.909417 0.415885i \(-0.136528\pi\)
0.909417 + 0.415885i \(0.136528\pi\)
\(984\) 2.84117 0.0905731
\(985\) −22.9071 −0.729880
\(986\) 2.58834 0.0824294
\(987\) 3.47996 0.110768
\(988\) 0 0
\(989\) 3.08277 0.0980264
\(990\) −1.35690 −0.0431250
\(991\) 37.2416 1.18302 0.591509 0.806298i \(-0.298532\pi\)
0.591509 + 0.806298i \(0.298532\pi\)
\(992\) 9.85086 0.312765
\(993\) 26.3424 0.835951
\(994\) −2.40821 −0.0763837
\(995\) −27.0291 −0.856879
\(996\) 2.68664 0.0851296
\(997\) −9.00059 −0.285052 −0.142526 0.989791i \(-0.545522\pi\)
−0.142526 + 0.989791i \(0.545522\pi\)
\(998\) −10.0194 −0.317158
\(999\) 9.26875 0.293250
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.bw.1.1 yes 3
13.5 odd 4 5070.2.b.z.1351.1 6
13.8 odd 4 5070.2.b.z.1351.6 6
13.12 even 2 5070.2.a.bp.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bp.1.3 3 13.12 even 2
5070.2.a.bw.1.1 yes 3 1.1 even 1 trivial
5070.2.b.z.1351.1 6 13.5 odd 4
5070.2.b.z.1351.6 6 13.8 odd 4