Properties

Label 5070.2.a.bo.1.3
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 1.3
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +4.80194 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +4.80194 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.55496 q^{11} +1.00000 q^{12} -4.80194 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.02715 q^{17} -1.00000 q^{18} -3.89977 q^{19} -1.00000 q^{20} +4.80194 q^{21} -2.55496 q^{22} -1.64310 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +4.80194 q^{28} +5.07606 q^{29} +1.00000 q^{30} +5.44504 q^{31} -1.00000 q^{32} +2.55496 q^{33} +3.02715 q^{34} -4.80194 q^{35} +1.00000 q^{36} +7.51573 q^{37} +3.89977 q^{38} +1.00000 q^{40} -5.89977 q^{41} -4.80194 q^{42} -10.1468 q^{43} +2.55496 q^{44} -1.00000 q^{45} +1.64310 q^{46} +6.42758 q^{47} +1.00000 q^{48} +16.0586 q^{49} -1.00000 q^{50} -3.02715 q^{51} -6.91185 q^{53} -1.00000 q^{54} -2.55496 q^{55} -4.80194 q^{56} -3.89977 q^{57} -5.07606 q^{58} +7.60925 q^{59} -1.00000 q^{60} +9.65279 q^{61} -5.44504 q^{62} +4.80194 q^{63} +1.00000 q^{64} -2.55496 q^{66} +0.929312 q^{67} -3.02715 q^{68} -1.64310 q^{69} +4.80194 q^{70} +14.9487 q^{71} -1.00000 q^{72} +14.1129 q^{73} -7.51573 q^{74} +1.00000 q^{75} -3.89977 q^{76} +12.2687 q^{77} -5.40581 q^{79} -1.00000 q^{80} +1.00000 q^{81} +5.89977 q^{82} +4.33513 q^{83} +4.80194 q^{84} +3.02715 q^{85} +10.1468 q^{86} +5.07606 q^{87} -2.55496 q^{88} -16.8267 q^{89} +1.00000 q^{90} -1.64310 q^{92} +5.44504 q^{93} -6.42758 q^{94} +3.89977 q^{95} -1.00000 q^{96} +17.9855 q^{97} -16.0586 q^{98} +2.55496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} + 10q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} + 10q^{7} - 3q^{8} + 3q^{9} + 3q^{10} + 8q^{11} + 3q^{12} - 10q^{14} - 3q^{15} + 3q^{16} - 3q^{17} - 3q^{18} + 11q^{19} - 3q^{20} + 10q^{21} - 8q^{22} - 9q^{23} - 3q^{24} + 3q^{25} + 3q^{27} + 10q^{28} + 3q^{30} + 16q^{31} - 3q^{32} + 8q^{33} + 3q^{34} - 10q^{35} + 3q^{36} + 10q^{37} - 11q^{38} + 3q^{40} + 5q^{41} - 10q^{42} - 3q^{43} + 8q^{44} - 3q^{45} + 9q^{46} + 3q^{47} + 3q^{48} + 17q^{49} - 3q^{50} - 3q^{51} - 17q^{53} - 3q^{54} - 8q^{55} - 10q^{56} + 11q^{57} + 11q^{59} - 3q^{60} + 11q^{61} - 16q^{62} + 10q^{63} + 3q^{64} - 8q^{66} + 15q^{67} - 3q^{68} - 9q^{69} + 10q^{70} + 13q^{71} - 3q^{72} - q^{73} - 10q^{74} + 3q^{75} + 11q^{76} + 29q^{77} - 3q^{79} - 3q^{80} + 3q^{81} - 5q^{82} + 12q^{83} + 10q^{84} + 3q^{85} + 3q^{86} - 8q^{88} + q^{89} + 3q^{90} - 9q^{92} + 16q^{93} - 3q^{94} - 11q^{95} - 3q^{96} - 6q^{97} - 17q^{98} + 8q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 4.80194 1.81496 0.907481 0.420093i \(-0.138003\pi\)
0.907481 + 0.420093i \(0.138003\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 2.55496 0.770349 0.385174 0.922844i \(-0.374141\pi\)
0.385174 + 0.922844i \(0.374141\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −4.80194 −1.28337
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −3.02715 −0.734191 −0.367096 0.930183i \(-0.619648\pi\)
−0.367096 + 0.930183i \(0.619648\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.89977 −0.894669 −0.447335 0.894367i \(-0.647627\pi\)
−0.447335 + 0.894367i \(0.647627\pi\)
\(20\) −1.00000 −0.223607
\(21\) 4.80194 1.04787
\(22\) −2.55496 −0.544719
\(23\) −1.64310 −0.342611 −0.171305 0.985218i \(-0.554798\pi\)
−0.171305 + 0.985218i \(0.554798\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 4.80194 0.907481
\(29\) 5.07606 0.942601 0.471301 0.881973i \(-0.343785\pi\)
0.471301 + 0.881973i \(0.343785\pi\)
\(30\) 1.00000 0.182574
\(31\) 5.44504 0.977958 0.488979 0.872295i \(-0.337369\pi\)
0.488979 + 0.872295i \(0.337369\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.55496 0.444761
\(34\) 3.02715 0.519151
\(35\) −4.80194 −0.811676
\(36\) 1.00000 0.166667
\(37\) 7.51573 1.23558 0.617789 0.786344i \(-0.288029\pi\)
0.617789 + 0.786344i \(0.288029\pi\)
\(38\) 3.89977 0.632627
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −5.89977 −0.921390 −0.460695 0.887559i \(-0.652400\pi\)
−0.460695 + 0.887559i \(0.652400\pi\)
\(42\) −4.80194 −0.740955
\(43\) −10.1468 −1.54737 −0.773683 0.633573i \(-0.781587\pi\)
−0.773683 + 0.633573i \(0.781587\pi\)
\(44\) 2.55496 0.385174
\(45\) −1.00000 −0.149071
\(46\) 1.64310 0.242262
\(47\) 6.42758 0.937559 0.468780 0.883315i \(-0.344694\pi\)
0.468780 + 0.883315i \(0.344694\pi\)
\(48\) 1.00000 0.144338
\(49\) 16.0586 2.29409
\(50\) −1.00000 −0.141421
\(51\) −3.02715 −0.423885
\(52\) 0 0
\(53\) −6.91185 −0.949416 −0.474708 0.880143i \(-0.657446\pi\)
−0.474708 + 0.880143i \(0.657446\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.55496 −0.344510
\(56\) −4.80194 −0.641686
\(57\) −3.89977 −0.516537
\(58\) −5.07606 −0.666520
\(59\) 7.60925 0.990640 0.495320 0.868711i \(-0.335051\pi\)
0.495320 + 0.868711i \(0.335051\pi\)
\(60\) −1.00000 −0.129099
\(61\) 9.65279 1.23591 0.617957 0.786212i \(-0.287961\pi\)
0.617957 + 0.786212i \(0.287961\pi\)
\(62\) −5.44504 −0.691521
\(63\) 4.80194 0.604987
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.55496 −0.314494
\(67\) 0.929312 0.113534 0.0567668 0.998387i \(-0.481921\pi\)
0.0567668 + 0.998387i \(0.481921\pi\)
\(68\) −3.02715 −0.367096
\(69\) −1.64310 −0.197806
\(70\) 4.80194 0.573941
\(71\) 14.9487 1.77408 0.887042 0.461690i \(-0.152757\pi\)
0.887042 + 0.461690i \(0.152757\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.1129 1.65179 0.825895 0.563824i \(-0.190670\pi\)
0.825895 + 0.563824i \(0.190670\pi\)
\(74\) −7.51573 −0.873686
\(75\) 1.00000 0.115470
\(76\) −3.89977 −0.447335
\(77\) 12.2687 1.39815
\(78\) 0 0
\(79\) −5.40581 −0.608202 −0.304101 0.952640i \(-0.598356\pi\)
−0.304101 + 0.952640i \(0.598356\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 5.89977 0.651521
\(83\) 4.33513 0.475842 0.237921 0.971285i \(-0.423534\pi\)
0.237921 + 0.971285i \(0.423534\pi\)
\(84\) 4.80194 0.523934
\(85\) 3.02715 0.328340
\(86\) 10.1468 1.09415
\(87\) 5.07606 0.544211
\(88\) −2.55496 −0.272359
\(89\) −16.8267 −1.78363 −0.891813 0.452404i \(-0.850566\pi\)
−0.891813 + 0.452404i \(0.850566\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −1.64310 −0.171305
\(93\) 5.44504 0.564625
\(94\) −6.42758 −0.662955
\(95\) 3.89977 0.400108
\(96\) −1.00000 −0.102062
\(97\) 17.9855 1.82615 0.913077 0.407788i \(-0.133700\pi\)
0.913077 + 0.407788i \(0.133700\pi\)
\(98\) −16.0586 −1.62216
\(99\) 2.55496 0.256783
\(100\) 1.00000 0.100000
\(101\) −15.7453 −1.56671 −0.783355 0.621574i \(-0.786494\pi\)
−0.783355 + 0.621574i \(0.786494\pi\)
\(102\) 3.02715 0.299732
\(103\) 12.0858 1.19084 0.595422 0.803413i \(-0.296985\pi\)
0.595422 + 0.803413i \(0.296985\pi\)
\(104\) 0 0
\(105\) −4.80194 −0.468621
\(106\) 6.91185 0.671339
\(107\) −4.64071 −0.448634 −0.224317 0.974516i \(-0.572015\pi\)
−0.224317 + 0.974516i \(0.572015\pi\)
\(108\) 1.00000 0.0962250
\(109\) −4.20344 −0.402616 −0.201308 0.979528i \(-0.564519\pi\)
−0.201308 + 0.979528i \(0.564519\pi\)
\(110\) 2.55496 0.243606
\(111\) 7.51573 0.713361
\(112\) 4.80194 0.453740
\(113\) −0.170915 −0.0160783 −0.00803917 0.999968i \(-0.502559\pi\)
−0.00803917 + 0.999968i \(0.502559\pi\)
\(114\) 3.89977 0.365247
\(115\) 1.64310 0.153220
\(116\) 5.07606 0.471301
\(117\) 0 0
\(118\) −7.60925 −0.700488
\(119\) −14.5362 −1.33253
\(120\) 1.00000 0.0912871
\(121\) −4.47219 −0.406563
\(122\) −9.65279 −0.873923
\(123\) −5.89977 −0.531965
\(124\) 5.44504 0.488979
\(125\) −1.00000 −0.0894427
\(126\) −4.80194 −0.427791
\(127\) 5.58211 0.495332 0.247666 0.968846i \(-0.420336\pi\)
0.247666 + 0.968846i \(0.420336\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.1468 −0.893372
\(130\) 0 0
\(131\) −3.06100 −0.267441 −0.133720 0.991019i \(-0.542692\pi\)
−0.133720 + 0.991019i \(0.542692\pi\)
\(132\) 2.55496 0.222381
\(133\) −18.7265 −1.62379
\(134\) −0.929312 −0.0802804
\(135\) −1.00000 −0.0860663
\(136\) 3.02715 0.259576
\(137\) −1.03385 −0.0883279 −0.0441640 0.999024i \(-0.514062\pi\)
−0.0441640 + 0.999024i \(0.514062\pi\)
\(138\) 1.64310 0.139870
\(139\) −16.7995 −1.42492 −0.712459 0.701713i \(-0.752419\pi\)
−0.712459 + 0.701713i \(0.752419\pi\)
\(140\) −4.80194 −0.405838
\(141\) 6.42758 0.541300
\(142\) −14.9487 −1.25447
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −5.07606 −0.421544
\(146\) −14.1129 −1.16799
\(147\) 16.0586 1.32449
\(148\) 7.51573 0.617789
\(149\) 17.8756 1.46443 0.732213 0.681075i \(-0.238487\pi\)
0.732213 + 0.681075i \(0.238487\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0.390748 0.0317986 0.0158993 0.999874i \(-0.494939\pi\)
0.0158993 + 0.999874i \(0.494939\pi\)
\(152\) 3.89977 0.316313
\(153\) −3.02715 −0.244730
\(154\) −12.2687 −0.988644
\(155\) −5.44504 −0.437356
\(156\) 0 0
\(157\) 9.50902 0.758903 0.379451 0.925212i \(-0.376113\pi\)
0.379451 + 0.925212i \(0.376113\pi\)
\(158\) 5.40581 0.430063
\(159\) −6.91185 −0.548146
\(160\) 1.00000 0.0790569
\(161\) −7.89008 −0.621826
\(162\) −1.00000 −0.0785674
\(163\) −5.84846 −0.458087 −0.229043 0.973416i \(-0.573560\pi\)
−0.229043 + 0.973416i \(0.573560\pi\)
\(164\) −5.89977 −0.460695
\(165\) −2.55496 −0.198903
\(166\) −4.33513 −0.336471
\(167\) −2.55496 −0.197709 −0.0988543 0.995102i \(-0.531518\pi\)
−0.0988543 + 0.995102i \(0.531518\pi\)
\(168\) −4.80194 −0.370478
\(169\) 0 0
\(170\) −3.02715 −0.232172
\(171\) −3.89977 −0.298223
\(172\) −10.1468 −0.773683
\(173\) −21.5308 −1.63696 −0.818478 0.574538i \(-0.805182\pi\)
−0.818478 + 0.574538i \(0.805182\pi\)
\(174\) −5.07606 −0.384815
\(175\) 4.80194 0.362992
\(176\) 2.55496 0.192587
\(177\) 7.60925 0.571946
\(178\) 16.8267 1.26121
\(179\) 15.5483 1.16213 0.581066 0.813857i \(-0.302636\pi\)
0.581066 + 0.813857i \(0.302636\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 14.4058 1.07078 0.535388 0.844606i \(-0.320165\pi\)
0.535388 + 0.844606i \(0.320165\pi\)
\(182\) 0 0
\(183\) 9.65279 0.713555
\(184\) 1.64310 0.121131
\(185\) −7.51573 −0.552567
\(186\) −5.44504 −0.399250
\(187\) −7.73423 −0.565583
\(188\) 6.42758 0.468780
\(189\) 4.80194 0.349290
\(190\) −3.89977 −0.282919
\(191\) −4.02177 −0.291005 −0.145503 0.989358i \(-0.546480\pi\)
−0.145503 + 0.989358i \(0.546480\pi\)
\(192\) 1.00000 0.0721688
\(193\) −20.1008 −1.44689 −0.723444 0.690383i \(-0.757442\pi\)
−0.723444 + 0.690383i \(0.757442\pi\)
\(194\) −17.9855 −1.29129
\(195\) 0 0
\(196\) 16.0586 1.14704
\(197\) −15.3884 −1.09637 −0.548187 0.836356i \(-0.684682\pi\)
−0.548187 + 0.836356i \(0.684682\pi\)
\(198\) −2.55496 −0.181573
\(199\) 24.7724 1.75607 0.878034 0.478598i \(-0.158855\pi\)
0.878034 + 0.478598i \(0.158855\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0.929312 0.0655486
\(202\) 15.7453 1.10783
\(203\) 24.3749 1.71079
\(204\) −3.02715 −0.211943
\(205\) 5.89977 0.412058
\(206\) −12.0858 −0.842054
\(207\) −1.64310 −0.114204
\(208\) 0 0
\(209\) −9.96376 −0.689207
\(210\) 4.80194 0.331365
\(211\) 6.49396 0.447063 0.223531 0.974697i \(-0.428242\pi\)
0.223531 + 0.974697i \(0.428242\pi\)
\(212\) −6.91185 −0.474708
\(213\) 14.9487 1.02427
\(214\) 4.64071 0.317232
\(215\) 10.1468 0.692003
\(216\) −1.00000 −0.0680414
\(217\) 26.1468 1.77496
\(218\) 4.20344 0.284693
\(219\) 14.1129 0.953661
\(220\) −2.55496 −0.172255
\(221\) 0 0
\(222\) −7.51573 −0.504423
\(223\) 20.1250 1.34767 0.673834 0.738883i \(-0.264646\pi\)
0.673834 + 0.738883i \(0.264646\pi\)
\(224\) −4.80194 −0.320843
\(225\) 1.00000 0.0666667
\(226\) 0.170915 0.0113691
\(227\) −24.2379 −1.60872 −0.804362 0.594139i \(-0.797493\pi\)
−0.804362 + 0.594139i \(0.797493\pi\)
\(228\) −3.89977 −0.258269
\(229\) −2.71618 −0.179491 −0.0897453 0.995965i \(-0.528605\pi\)
−0.0897453 + 0.995965i \(0.528605\pi\)
\(230\) −1.64310 −0.108343
\(231\) 12.2687 0.807224
\(232\) −5.07606 −0.333260
\(233\) −2.16421 −0.141782 −0.0708911 0.997484i \(-0.522584\pi\)
−0.0708911 + 0.997484i \(0.522584\pi\)
\(234\) 0 0
\(235\) −6.42758 −0.419289
\(236\) 7.60925 0.495320
\(237\) −5.40581 −0.351145
\(238\) 14.5362 0.942240
\(239\) 14.5114 0.938666 0.469333 0.883021i \(-0.344494\pi\)
0.469333 + 0.883021i \(0.344494\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −18.1142 −1.16684 −0.583420 0.812171i \(-0.698286\pi\)
−0.583420 + 0.812171i \(0.698286\pi\)
\(242\) 4.47219 0.287483
\(243\) 1.00000 0.0641500
\(244\) 9.65279 0.617957
\(245\) −16.0586 −1.02595
\(246\) 5.89977 0.376156
\(247\) 0 0
\(248\) −5.44504 −0.345761
\(249\) 4.33513 0.274727
\(250\) 1.00000 0.0632456
\(251\) 8.55257 0.539833 0.269917 0.962884i \(-0.413004\pi\)
0.269917 + 0.962884i \(0.413004\pi\)
\(252\) 4.80194 0.302494
\(253\) −4.19806 −0.263930
\(254\) −5.58211 −0.350252
\(255\) 3.02715 0.189567
\(256\) 1.00000 0.0625000
\(257\) 1.30260 0.0812541 0.0406270 0.999174i \(-0.487064\pi\)
0.0406270 + 0.999174i \(0.487064\pi\)
\(258\) 10.1468 0.631709
\(259\) 36.0901 2.24253
\(260\) 0 0
\(261\) 5.07606 0.314200
\(262\) 3.06100 0.189109
\(263\) 8.24027 0.508117 0.254059 0.967189i \(-0.418234\pi\)
0.254059 + 0.967189i \(0.418234\pi\)
\(264\) −2.55496 −0.157247
\(265\) 6.91185 0.424592
\(266\) 18.7265 1.14819
\(267\) −16.8267 −1.02978
\(268\) 0.929312 0.0567668
\(269\) 22.6601 1.38161 0.690805 0.723041i \(-0.257256\pi\)
0.690805 + 0.723041i \(0.257256\pi\)
\(270\) 1.00000 0.0608581
\(271\) 13.9172 0.845412 0.422706 0.906267i \(-0.361080\pi\)
0.422706 + 0.906267i \(0.361080\pi\)
\(272\) −3.02715 −0.183548
\(273\) 0 0
\(274\) 1.03385 0.0624573
\(275\) 2.55496 0.154070
\(276\) −1.64310 −0.0989032
\(277\) −24.2610 −1.45770 −0.728851 0.684673i \(-0.759945\pi\)
−0.728851 + 0.684673i \(0.759945\pi\)
\(278\) 16.7995 1.00757
\(279\) 5.44504 0.325986
\(280\) 4.80194 0.286971
\(281\) 10.5961 0.632111 0.316055 0.948741i \(-0.397641\pi\)
0.316055 + 0.948741i \(0.397641\pi\)
\(282\) −6.42758 −0.382757
\(283\) 9.86294 0.586291 0.293145 0.956068i \(-0.405298\pi\)
0.293145 + 0.956068i \(0.405298\pi\)
\(284\) 14.9487 0.887042
\(285\) 3.89977 0.231003
\(286\) 0 0
\(287\) −28.3303 −1.67229
\(288\) −1.00000 −0.0589256
\(289\) −7.83638 −0.460964
\(290\) 5.07606 0.298077
\(291\) 17.9855 1.05433
\(292\) 14.1129 0.825895
\(293\) 5.68233 0.331965 0.165983 0.986129i \(-0.446920\pi\)
0.165983 + 0.986129i \(0.446920\pi\)
\(294\) −16.0586 −0.936557
\(295\) −7.60925 −0.443028
\(296\) −7.51573 −0.436843
\(297\) 2.55496 0.148254
\(298\) −17.8756 −1.03551
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −48.7241 −2.80841
\(302\) −0.390748 −0.0224850
\(303\) −15.7453 −0.904541
\(304\) −3.89977 −0.223667
\(305\) −9.65279 −0.552717
\(306\) 3.02715 0.173050
\(307\) −11.5961 −0.661825 −0.330912 0.943661i \(-0.607356\pi\)
−0.330912 + 0.943661i \(0.607356\pi\)
\(308\) 12.2687 0.699077
\(309\) 12.0858 0.687534
\(310\) 5.44504 0.309258
\(311\) 4.43429 0.251445 0.125723 0.992065i \(-0.459875\pi\)
0.125723 + 0.992065i \(0.459875\pi\)
\(312\) 0 0
\(313\) 11.2881 0.638043 0.319021 0.947748i \(-0.396646\pi\)
0.319021 + 0.947748i \(0.396646\pi\)
\(314\) −9.50902 −0.536625
\(315\) −4.80194 −0.270559
\(316\) −5.40581 −0.304101
\(317\) −6.31229 −0.354534 −0.177267 0.984163i \(-0.556726\pi\)
−0.177267 + 0.984163i \(0.556726\pi\)
\(318\) 6.91185 0.387598
\(319\) 12.9691 0.726132
\(320\) −1.00000 −0.0559017
\(321\) −4.64071 −0.259019
\(322\) 7.89008 0.439697
\(323\) 11.8052 0.656858
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 5.84846 0.323916
\(327\) −4.20344 −0.232451
\(328\) 5.89977 0.325760
\(329\) 30.8649 1.70163
\(330\) 2.55496 0.140646
\(331\) 10.9554 0.602163 0.301081 0.953598i \(-0.402652\pi\)
0.301081 + 0.953598i \(0.402652\pi\)
\(332\) 4.33513 0.237921
\(333\) 7.51573 0.411859
\(334\) 2.55496 0.139801
\(335\) −0.929312 −0.0507738
\(336\) 4.80194 0.261967
\(337\) 26.6668 1.45263 0.726316 0.687361i \(-0.241231\pi\)
0.726316 + 0.687361i \(0.241231\pi\)
\(338\) 0 0
\(339\) −0.170915 −0.00928284
\(340\) 3.02715 0.164170
\(341\) 13.9119 0.753369
\(342\) 3.89977 0.210876
\(343\) 43.4989 2.34872
\(344\) 10.1468 0.547076
\(345\) 1.64310 0.0884618
\(346\) 21.5308 1.15750
\(347\) 25.0694 1.34579 0.672897 0.739736i \(-0.265050\pi\)
0.672897 + 0.739736i \(0.265050\pi\)
\(348\) 5.07606 0.272106
\(349\) −21.5623 −1.15420 −0.577100 0.816673i \(-0.695816\pi\)
−0.577100 + 0.816673i \(0.695816\pi\)
\(350\) −4.80194 −0.256674
\(351\) 0 0
\(352\) −2.55496 −0.136180
\(353\) −28.2959 −1.50604 −0.753019 0.657998i \(-0.771403\pi\)
−0.753019 + 0.657998i \(0.771403\pi\)
\(354\) −7.60925 −0.404427
\(355\) −14.9487 −0.793394
\(356\) −16.8267 −0.891813
\(357\) −14.5362 −0.769336
\(358\) −15.5483 −0.821751
\(359\) 34.2717 1.80879 0.904396 0.426693i \(-0.140322\pi\)
0.904396 + 0.426693i \(0.140322\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.79178 −0.199567
\(362\) −14.4058 −0.757153
\(363\) −4.47219 −0.234729
\(364\) 0 0
\(365\) −14.1129 −0.738703
\(366\) −9.65279 −0.504560
\(367\) −23.4209 −1.22256 −0.611280 0.791414i \(-0.709345\pi\)
−0.611280 + 0.791414i \(0.709345\pi\)
\(368\) −1.64310 −0.0856527
\(369\) −5.89977 −0.307130
\(370\) 7.51573 0.390724
\(371\) −33.1903 −1.72315
\(372\) 5.44504 0.282312
\(373\) 1.82371 0.0944280 0.0472140 0.998885i \(-0.484966\pi\)
0.0472140 + 0.998885i \(0.484966\pi\)
\(374\) 7.73423 0.399928
\(375\) −1.00000 −0.0516398
\(376\) −6.42758 −0.331477
\(377\) 0 0
\(378\) −4.80194 −0.246985
\(379\) −15.3260 −0.787245 −0.393623 0.919272i \(-0.628778\pi\)
−0.393623 + 0.919272i \(0.628778\pi\)
\(380\) 3.89977 0.200054
\(381\) 5.58211 0.285980
\(382\) 4.02177 0.205772
\(383\) −29.6722 −1.51618 −0.758089 0.652152i \(-0.773867\pi\)
−0.758089 + 0.652152i \(0.773867\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −12.2687 −0.625273
\(386\) 20.1008 1.02310
\(387\) −10.1468 −0.515788
\(388\) 17.9855 0.913077
\(389\) 11.8291 0.599758 0.299879 0.953977i \(-0.403054\pi\)
0.299879 + 0.953977i \(0.403054\pi\)
\(390\) 0 0
\(391\) 4.97392 0.251542
\(392\) −16.0586 −0.811082
\(393\) −3.06100 −0.154407
\(394\) 15.3884 0.775254
\(395\) 5.40581 0.271996
\(396\) 2.55496 0.128391
\(397\) 12.1153 0.608049 0.304025 0.952664i \(-0.401670\pi\)
0.304025 + 0.952664i \(0.401670\pi\)
\(398\) −24.7724 −1.24173
\(399\) −18.7265 −0.937496
\(400\) 1.00000 0.0500000
\(401\) −10.8291 −0.540779 −0.270389 0.962751i \(-0.587152\pi\)
−0.270389 + 0.962751i \(0.587152\pi\)
\(402\) −0.929312 −0.0463499
\(403\) 0 0
\(404\) −15.7453 −0.783355
\(405\) −1.00000 −0.0496904
\(406\) −24.3749 −1.20971
\(407\) 19.2024 0.951826
\(408\) 3.02715 0.149866
\(409\) −33.9396 −1.67820 −0.839102 0.543973i \(-0.816919\pi\)
−0.839102 + 0.543973i \(0.816919\pi\)
\(410\) −5.89977 −0.291369
\(411\) −1.03385 −0.0509962
\(412\) 12.0858 0.595422
\(413\) 36.5392 1.79797
\(414\) 1.64310 0.0807542
\(415\) −4.33513 −0.212803
\(416\) 0 0
\(417\) −16.7995 −0.822677
\(418\) 9.96376 0.487343
\(419\) 26.8562 1.31201 0.656006 0.754755i \(-0.272244\pi\)
0.656006 + 0.754755i \(0.272244\pi\)
\(420\) −4.80194 −0.234311
\(421\) 27.9928 1.36429 0.682143 0.731219i \(-0.261048\pi\)
0.682143 + 0.731219i \(0.261048\pi\)
\(422\) −6.49396 −0.316121
\(423\) 6.42758 0.312520
\(424\) 6.91185 0.335669
\(425\) −3.02715 −0.146838
\(426\) −14.9487 −0.724266
\(427\) 46.3521 2.24314
\(428\) −4.64071 −0.224317
\(429\) 0 0
\(430\) −10.1468 −0.489320
\(431\) −33.4088 −1.60925 −0.804623 0.593787i \(-0.797632\pi\)
−0.804623 + 0.593787i \(0.797632\pi\)
\(432\) 1.00000 0.0481125
\(433\) −13.6420 −0.655595 −0.327797 0.944748i \(-0.606306\pi\)
−0.327797 + 0.944748i \(0.606306\pi\)
\(434\) −26.1468 −1.25508
\(435\) −5.07606 −0.243379
\(436\) −4.20344 −0.201308
\(437\) 6.40773 0.306523
\(438\) −14.1129 −0.674340
\(439\) −30.6437 −1.46254 −0.731272 0.682086i \(-0.761073\pi\)
−0.731272 + 0.682086i \(0.761073\pi\)
\(440\) 2.55496 0.121803
\(441\) 16.0586 0.764696
\(442\) 0 0
\(443\) −19.7409 −0.937920 −0.468960 0.883219i \(-0.655371\pi\)
−0.468960 + 0.883219i \(0.655371\pi\)
\(444\) 7.51573 0.356681
\(445\) 16.8267 0.797662
\(446\) −20.1250 −0.952946
\(447\) 17.8756 0.845487
\(448\) 4.80194 0.226870
\(449\) 36.2664 1.71152 0.855758 0.517377i \(-0.173091\pi\)
0.855758 + 0.517377i \(0.173091\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −15.0737 −0.709791
\(452\) −0.170915 −0.00803917
\(453\) 0.390748 0.0183589
\(454\) 24.2379 1.13754
\(455\) 0 0
\(456\) 3.89977 0.182624
\(457\) 3.55602 0.166344 0.0831719 0.996535i \(-0.473495\pi\)
0.0831719 + 0.996535i \(0.473495\pi\)
\(458\) 2.71618 0.126919
\(459\) −3.02715 −0.141295
\(460\) 1.64310 0.0766101
\(461\) 1.46011 0.0680040 0.0340020 0.999422i \(-0.489175\pi\)
0.0340020 + 0.999422i \(0.489175\pi\)
\(462\) −12.2687 −0.570794
\(463\) −13.7942 −0.641069 −0.320535 0.947237i \(-0.603863\pi\)
−0.320535 + 0.947237i \(0.603863\pi\)
\(464\) 5.07606 0.235650
\(465\) −5.44504 −0.252508
\(466\) 2.16421 0.100255
\(467\) −1.72886 −0.0800020 −0.0400010 0.999200i \(-0.512736\pi\)
−0.0400010 + 0.999200i \(0.512736\pi\)
\(468\) 0 0
\(469\) 4.46250 0.206059
\(470\) 6.42758 0.296482
\(471\) 9.50902 0.438153
\(472\) −7.60925 −0.350244
\(473\) −25.9245 −1.19201
\(474\) 5.40581 0.248297
\(475\) −3.89977 −0.178934
\(476\) −14.5362 −0.666264
\(477\) −6.91185 −0.316472
\(478\) −14.5114 −0.663737
\(479\) 14.5496 0.664787 0.332394 0.943141i \(-0.392144\pi\)
0.332394 + 0.943141i \(0.392144\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 18.1142 0.825080
\(483\) −7.89008 −0.359011
\(484\) −4.47219 −0.203281
\(485\) −17.9855 −0.816681
\(486\) −1.00000 −0.0453609
\(487\) 30.5526 1.38447 0.692234 0.721673i \(-0.256626\pi\)
0.692234 + 0.721673i \(0.256626\pi\)
\(488\) −9.65279 −0.436961
\(489\) −5.84846 −0.264477
\(490\) 16.0586 0.725454
\(491\) 9.68233 0.436958 0.218479 0.975842i \(-0.429891\pi\)
0.218479 + 0.975842i \(0.429891\pi\)
\(492\) −5.89977 −0.265982
\(493\) −15.3660 −0.692050
\(494\) 0 0
\(495\) −2.55496 −0.114837
\(496\) 5.44504 0.244490
\(497\) 71.7827 3.21989
\(498\) −4.33513 −0.194262
\(499\) 18.6025 0.832764 0.416382 0.909190i \(-0.363298\pi\)
0.416382 + 0.909190i \(0.363298\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −2.55496 −0.114147
\(502\) −8.55257 −0.381720
\(503\) 41.6644 1.85772 0.928862 0.370426i \(-0.120788\pi\)
0.928862 + 0.370426i \(0.120788\pi\)
\(504\) −4.80194 −0.213895
\(505\) 15.7453 0.700654
\(506\) 4.19806 0.186627
\(507\) 0 0
\(508\) 5.58211 0.247666
\(509\) 41.1648 1.82460 0.912299 0.409525i \(-0.134306\pi\)
0.912299 + 0.409525i \(0.134306\pi\)
\(510\) −3.02715 −0.134044
\(511\) 67.7693 2.99794
\(512\) −1.00000 −0.0441942
\(513\) −3.89977 −0.172179
\(514\) −1.30260 −0.0574553
\(515\) −12.0858 −0.532562
\(516\) −10.1468 −0.446686
\(517\) 16.4222 0.722248
\(518\) −36.0901 −1.58571
\(519\) −21.5308 −0.945097
\(520\) 0 0
\(521\) 1.12737 0.0493912 0.0246956 0.999695i \(-0.492138\pi\)
0.0246956 + 0.999695i \(0.492138\pi\)
\(522\) −5.07606 −0.222173
\(523\) −22.7922 −0.996635 −0.498318 0.866994i \(-0.666049\pi\)
−0.498318 + 0.866994i \(0.666049\pi\)
\(524\) −3.06100 −0.133720
\(525\) 4.80194 0.209574
\(526\) −8.24027 −0.359293
\(527\) −16.4829 −0.718008
\(528\) 2.55496 0.111190
\(529\) −20.3002 −0.882618
\(530\) −6.91185 −0.300232
\(531\) 7.60925 0.330213
\(532\) −18.7265 −0.811895
\(533\) 0 0
\(534\) 16.8267 0.728162
\(535\) 4.64071 0.200635
\(536\) −0.929312 −0.0401402
\(537\) 15.5483 0.670957
\(538\) −22.6601 −0.976946
\(539\) 41.0291 1.76725
\(540\) −1.00000 −0.0430331
\(541\) −41.1148 −1.76766 −0.883832 0.467804i \(-0.845045\pi\)
−0.883832 + 0.467804i \(0.845045\pi\)
\(542\) −13.9172 −0.597796
\(543\) 14.4058 0.618213
\(544\) 3.02715 0.129788
\(545\) 4.20344 0.180056
\(546\) 0 0
\(547\) 3.96615 0.169580 0.0847901 0.996399i \(-0.472978\pi\)
0.0847901 + 0.996399i \(0.472978\pi\)
\(548\) −1.03385 −0.0441640
\(549\) 9.65279 0.411971
\(550\) −2.55496 −0.108944
\(551\) −19.7955 −0.843316
\(552\) 1.64310 0.0699352
\(553\) −25.9584 −1.10386
\(554\) 24.2610 1.03075
\(555\) −7.51573 −0.319025
\(556\) −16.7995 −0.712459
\(557\) −4.25236 −0.180178 −0.0900891 0.995934i \(-0.528715\pi\)
−0.0900891 + 0.995934i \(0.528715\pi\)
\(558\) −5.44504 −0.230507
\(559\) 0 0
\(560\) −4.80194 −0.202919
\(561\) −7.73423 −0.326540
\(562\) −10.5961 −0.446970
\(563\) −32.7700 −1.38109 −0.690546 0.723289i \(-0.742629\pi\)
−0.690546 + 0.723289i \(0.742629\pi\)
\(564\) 6.42758 0.270650
\(565\) 0.170915 0.00719046
\(566\) −9.86294 −0.414570
\(567\) 4.80194 0.201662
\(568\) −14.9487 −0.627233
\(569\) −20.6189 −0.864391 −0.432195 0.901780i \(-0.642261\pi\)
−0.432195 + 0.901780i \(0.642261\pi\)
\(570\) −3.89977 −0.163343
\(571\) 23.1215 0.967606 0.483803 0.875177i \(-0.339255\pi\)
0.483803 + 0.875177i \(0.339255\pi\)
\(572\) 0 0
\(573\) −4.02177 −0.168012
\(574\) 28.3303 1.18249
\(575\) −1.64310 −0.0685222
\(576\) 1.00000 0.0416667
\(577\) 28.0000 1.16566 0.582828 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) 7.83638 0.325950
\(579\) −20.1008 −0.835362
\(580\) −5.07606 −0.210772
\(581\) 20.8170 0.863635
\(582\) −17.9855 −0.745524
\(583\) −17.6595 −0.731382
\(584\) −14.1129 −0.583996
\(585\) 0 0
\(586\) −5.68233 −0.234735
\(587\) 21.1081 0.871225 0.435613 0.900134i \(-0.356532\pi\)
0.435613 + 0.900134i \(0.356532\pi\)
\(588\) 16.0586 0.662246
\(589\) −21.2344 −0.874949
\(590\) 7.60925 0.313268
\(591\) −15.3884 −0.632992
\(592\) 7.51573 0.308895
\(593\) −15.4276 −0.633535 −0.316767 0.948503i \(-0.602597\pi\)
−0.316767 + 0.948503i \(0.602597\pi\)
\(594\) −2.55496 −0.104831
\(595\) 14.5362 0.595925
\(596\) 17.8756 0.732213
\(597\) 24.7724 1.01387
\(598\) 0 0
\(599\) 9.16985 0.374670 0.187335 0.982296i \(-0.440015\pi\)
0.187335 + 0.982296i \(0.440015\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −25.8864 −1.05593 −0.527963 0.849267i \(-0.677044\pi\)
−0.527963 + 0.849267i \(0.677044\pi\)
\(602\) 48.7241 1.98584
\(603\) 0.929312 0.0378445
\(604\) 0.390748 0.0158993
\(605\) 4.47219 0.181820
\(606\) 15.7453 0.639607
\(607\) −11.6093 −0.471205 −0.235603 0.971850i \(-0.575706\pi\)
−0.235603 + 0.971850i \(0.575706\pi\)
\(608\) 3.89977 0.158157
\(609\) 24.3749 0.987723
\(610\) 9.65279 0.390830
\(611\) 0 0
\(612\) −3.02715 −0.122365
\(613\) 16.5985 0.670407 0.335204 0.942146i \(-0.391195\pi\)
0.335204 + 0.942146i \(0.391195\pi\)
\(614\) 11.5961 0.467981
\(615\) 5.89977 0.237902
\(616\) −12.2687 −0.494322
\(617\) −33.4373 −1.34613 −0.673067 0.739582i \(-0.735023\pi\)
−0.673067 + 0.739582i \(0.735023\pi\)
\(618\) −12.0858 −0.486160
\(619\) 25.4198 1.02171 0.510854 0.859667i \(-0.329329\pi\)
0.510854 + 0.859667i \(0.329329\pi\)
\(620\) −5.44504 −0.218678
\(621\) −1.64310 −0.0659355
\(622\) −4.43429 −0.177799
\(623\) −80.8007 −3.23721
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −11.2881 −0.451164
\(627\) −9.96376 −0.397914
\(628\) 9.50902 0.379451
\(629\) −22.7512 −0.907150
\(630\) 4.80194 0.191314
\(631\) 15.1879 0.604621 0.302310 0.953210i \(-0.402242\pi\)
0.302310 + 0.953210i \(0.402242\pi\)
\(632\) 5.40581 0.215032
\(633\) 6.49396 0.258112
\(634\) 6.31229 0.250693
\(635\) −5.58211 −0.221519
\(636\) −6.91185 −0.274073
\(637\) 0 0
\(638\) −12.9691 −0.513453
\(639\) 14.9487 0.591361
\(640\) 1.00000 0.0395285
\(641\) 16.8009 0.663595 0.331797 0.943351i \(-0.392345\pi\)
0.331797 + 0.943351i \(0.392345\pi\)
\(642\) 4.64071 0.183154
\(643\) −35.2597 −1.39050 −0.695252 0.718766i \(-0.744707\pi\)
−0.695252 + 0.718766i \(0.744707\pi\)
\(644\) −7.89008 −0.310913
\(645\) 10.1468 0.399528
\(646\) −11.8052 −0.464469
\(647\) −26.7985 −1.05356 −0.526778 0.850003i \(-0.676600\pi\)
−0.526778 + 0.850003i \(0.676600\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 19.4413 0.763139
\(650\) 0 0
\(651\) 26.1468 1.02477
\(652\) −5.84846 −0.229043
\(653\) −10.8358 −0.424037 −0.212019 0.977266i \(-0.568004\pi\)
−0.212019 + 0.977266i \(0.568004\pi\)
\(654\) 4.20344 0.164367
\(655\) 3.06100 0.119603
\(656\) −5.89977 −0.230347
\(657\) 14.1129 0.550597
\(658\) −30.8649 −1.20324
\(659\) 49.3870 1.92385 0.961923 0.273322i \(-0.0881223\pi\)
0.961923 + 0.273322i \(0.0881223\pi\)
\(660\) −2.55496 −0.0994516
\(661\) 2.18167 0.0848571 0.0424285 0.999100i \(-0.486491\pi\)
0.0424285 + 0.999100i \(0.486491\pi\)
\(662\) −10.9554 −0.425793
\(663\) 0 0
\(664\) −4.33513 −0.168236
\(665\) 18.7265 0.726181
\(666\) −7.51573 −0.291229
\(667\) −8.34050 −0.322946
\(668\) −2.55496 −0.0988543
\(669\) 20.1250 0.778077
\(670\) 0.929312 0.0359025
\(671\) 24.6625 0.952085
\(672\) −4.80194 −0.185239
\(673\) −24.2319 −0.934072 −0.467036 0.884238i \(-0.654678\pi\)
−0.467036 + 0.884238i \(0.654678\pi\)
\(674\) −26.6668 −1.02717
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 24.0422 0.924017 0.462009 0.886875i \(-0.347129\pi\)
0.462009 + 0.886875i \(0.347129\pi\)
\(678\) 0.170915 0.00656396
\(679\) 86.3654 3.31440
\(680\) −3.02715 −0.116086
\(681\) −24.2379 −0.928798
\(682\) −13.9119 −0.532712
\(683\) −24.3183 −0.930512 −0.465256 0.885176i \(-0.654038\pi\)
−0.465256 + 0.885176i \(0.654038\pi\)
\(684\) −3.89977 −0.149112
\(685\) 1.03385 0.0395014
\(686\) −43.4989 −1.66079
\(687\) −2.71618 −0.103629
\(688\) −10.1468 −0.386841
\(689\) 0 0
\(690\) −1.64310 −0.0625519
\(691\) 18.2054 0.692564 0.346282 0.938130i \(-0.387444\pi\)
0.346282 + 0.938130i \(0.387444\pi\)
\(692\) −21.5308 −0.818478
\(693\) 12.2687 0.466051
\(694\) −25.0694 −0.951620
\(695\) 16.7995 0.637243
\(696\) −5.07606 −0.192408
\(697\) 17.8595 0.676476
\(698\) 21.5623 0.816143
\(699\) −2.16421 −0.0818580
\(700\) 4.80194 0.181496
\(701\) −16.0441 −0.605978 −0.302989 0.952994i \(-0.597985\pi\)
−0.302989 + 0.952994i \(0.597985\pi\)
\(702\) 0 0
\(703\) −29.3096 −1.10543
\(704\) 2.55496 0.0962936
\(705\) −6.42758 −0.242077
\(706\) 28.2959 1.06493
\(707\) −75.6077 −2.84352
\(708\) 7.60925 0.285973
\(709\) −20.8025 −0.781255 −0.390628 0.920549i \(-0.627742\pi\)
−0.390628 + 0.920549i \(0.627742\pi\)
\(710\) 14.9487 0.561014
\(711\) −5.40581 −0.202734
\(712\) 16.8267 0.630607
\(713\) −8.94677 −0.335059
\(714\) 14.5362 0.544003
\(715\) 0 0
\(716\) 15.5483 0.581066
\(717\) 14.5114 0.541939
\(718\) −34.2717 −1.27901
\(719\) −40.3424 −1.50452 −0.752259 0.658867i \(-0.771036\pi\)
−0.752259 + 0.658867i \(0.771036\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 58.0350 2.16134
\(722\) 3.79178 0.141115
\(723\) −18.1142 −0.673675
\(724\) 14.4058 0.535388
\(725\) 5.07606 0.188520
\(726\) 4.47219 0.165978
\(727\) −47.2073 −1.75082 −0.875410 0.483380i \(-0.839409\pi\)
−0.875410 + 0.483380i \(0.839409\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 14.1129 0.522342
\(731\) 30.7157 1.13606
\(732\) 9.65279 0.356777
\(733\) 41.5424 1.53440 0.767202 0.641406i \(-0.221648\pi\)
0.767202 + 0.641406i \(0.221648\pi\)
\(734\) 23.4209 0.864480
\(735\) −16.0586 −0.592331
\(736\) 1.64310 0.0605656
\(737\) 2.37435 0.0874605
\(738\) 5.89977 0.217174
\(739\) 26.3811 0.970443 0.485221 0.874391i \(-0.338739\pi\)
0.485221 + 0.874391i \(0.338739\pi\)
\(740\) −7.51573 −0.276284
\(741\) 0 0
\(742\) 33.1903 1.21845
\(743\) −20.2881 −0.744299 −0.372150 0.928173i \(-0.621379\pi\)
−0.372150 + 0.928173i \(0.621379\pi\)
\(744\) −5.44504 −0.199625
\(745\) −17.8756 −0.654912
\(746\) −1.82371 −0.0667707
\(747\) 4.33513 0.158614
\(748\) −7.73423 −0.282792
\(749\) −22.2844 −0.814254
\(750\) 1.00000 0.0365148
\(751\) −21.1280 −0.770970 −0.385485 0.922714i \(-0.625966\pi\)
−0.385485 + 0.922714i \(0.625966\pi\)
\(752\) 6.42758 0.234390
\(753\) 8.55257 0.311673
\(754\) 0 0
\(755\) −0.390748 −0.0142208
\(756\) 4.80194 0.174645
\(757\) −17.3274 −0.629773 −0.314887 0.949129i \(-0.601967\pi\)
−0.314887 + 0.949129i \(0.601967\pi\)
\(758\) 15.3260 0.556666
\(759\) −4.19806 −0.152380
\(760\) −3.89977 −0.141460
\(761\) 35.4282 1.28427 0.642135 0.766591i \(-0.278049\pi\)
0.642135 + 0.766591i \(0.278049\pi\)
\(762\) −5.58211 −0.202218
\(763\) −20.1847 −0.730733
\(764\) −4.02177 −0.145503
\(765\) 3.02715 0.109447
\(766\) 29.6722 1.07210
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 49.5062 1.78524 0.892619 0.450812i \(-0.148866\pi\)
0.892619 + 0.450812i \(0.148866\pi\)
\(770\) 12.2687 0.442135
\(771\) 1.30260 0.0469121
\(772\) −20.1008 −0.723444
\(773\) 27.9235 1.00434 0.502169 0.864770i \(-0.332536\pi\)
0.502169 + 0.864770i \(0.332536\pi\)
\(774\) 10.1468 0.364717
\(775\) 5.44504 0.195592
\(776\) −17.9855 −0.645643
\(777\) 36.0901 1.29472
\(778\) −11.8291 −0.424093
\(779\) 23.0078 0.824339
\(780\) 0 0
\(781\) 38.1933 1.36666
\(782\) −4.97392 −0.177867
\(783\) 5.07606 0.181404
\(784\) 16.0586 0.573522
\(785\) −9.50902 −0.339392
\(786\) 3.06100 0.109182
\(787\) 28.7778 1.02582 0.512908 0.858443i \(-0.328568\pi\)
0.512908 + 0.858443i \(0.328568\pi\)
\(788\) −15.3884 −0.548187
\(789\) 8.24027 0.293362
\(790\) −5.40581 −0.192330
\(791\) −0.820724 −0.0291816
\(792\) −2.55496 −0.0907865
\(793\) 0 0
\(794\) −12.1153 −0.429956
\(795\) 6.91185 0.245138
\(796\) 24.7724 0.878034
\(797\) −14.6752 −0.519821 −0.259910 0.965633i \(-0.583693\pi\)
−0.259910 + 0.965633i \(0.583693\pi\)
\(798\) 18.7265 0.662910
\(799\) −19.4572 −0.688348
\(800\) −1.00000 −0.0353553
\(801\) −16.8267 −0.594542
\(802\) 10.8291 0.382388
\(803\) 36.0579 1.27245
\(804\) 0.929312 0.0327743
\(805\) 7.89008 0.278089
\(806\) 0 0
\(807\) 22.6601 0.797673
\(808\) 15.7453 0.553916
\(809\) −25.4946 −0.896341 −0.448170 0.893948i \(-0.647924\pi\)
−0.448170 + 0.893948i \(0.647924\pi\)
\(810\) 1.00000 0.0351364
\(811\) −8.99867 −0.315986 −0.157993 0.987440i \(-0.550502\pi\)
−0.157993 + 0.987440i \(0.550502\pi\)
\(812\) 24.3749 0.855393
\(813\) 13.9172 0.488099
\(814\) −19.2024 −0.673043
\(815\) 5.84846 0.204863
\(816\) −3.02715 −0.105971
\(817\) 39.5700 1.38438
\(818\) 33.9396 1.18667
\(819\) 0 0
\(820\) 5.89977 0.206029
\(821\) −37.1105 −1.29517 −0.647583 0.761995i \(-0.724220\pi\)
−0.647583 + 0.761995i \(0.724220\pi\)
\(822\) 1.03385 0.0360597
\(823\) −11.1903 −0.390069 −0.195035 0.980796i \(-0.562482\pi\)
−0.195035 + 0.980796i \(0.562482\pi\)
\(824\) −12.0858 −0.421027
\(825\) 2.55496 0.0889522
\(826\) −36.5392 −1.27136
\(827\) −27.7808 −0.966032 −0.483016 0.875612i \(-0.660459\pi\)
−0.483016 + 0.875612i \(0.660459\pi\)
\(828\) −1.64310 −0.0571018
\(829\) −26.5633 −0.922582 −0.461291 0.887249i \(-0.652614\pi\)
−0.461291 + 0.887249i \(0.652614\pi\)
\(830\) 4.33513 0.150474
\(831\) −24.2610 −0.841604
\(832\) 0 0
\(833\) −48.6118 −1.68430
\(834\) 16.7995 0.581721
\(835\) 2.55496 0.0884180
\(836\) −9.96376 −0.344604
\(837\) 5.44504 0.188208
\(838\) −26.8562 −0.927733
\(839\) −3.92884 −0.135639 −0.0678193 0.997698i \(-0.521604\pi\)
−0.0678193 + 0.997698i \(0.521604\pi\)
\(840\) 4.80194 0.165683
\(841\) −3.23357 −0.111502
\(842\) −27.9928 −0.964696
\(843\) 10.5961 0.364949
\(844\) 6.49396 0.223531
\(845\) 0 0
\(846\) −6.42758 −0.220985
\(847\) −21.4752 −0.737896
\(848\) −6.91185 −0.237354
\(849\) 9.86294 0.338495
\(850\) 3.02715 0.103830
\(851\) −12.3491 −0.423323
\(852\) 14.9487 0.512134
\(853\) −48.2097 −1.65067 −0.825334 0.564645i \(-0.809013\pi\)
−0.825334 + 0.564645i \(0.809013\pi\)
\(854\) −46.3521 −1.58614
\(855\) 3.89977 0.133369
\(856\) 4.64071 0.158616
\(857\) 41.2976 1.41070 0.705349 0.708860i \(-0.250790\pi\)
0.705349 + 0.708860i \(0.250790\pi\)
\(858\) 0 0
\(859\) −42.1041 −1.43657 −0.718286 0.695748i \(-0.755073\pi\)
−0.718286 + 0.695748i \(0.755073\pi\)
\(860\) 10.1468 0.346001
\(861\) −28.3303 −0.965495
\(862\) 33.4088 1.13791
\(863\) 45.0646 1.53402 0.767008 0.641638i \(-0.221745\pi\)
0.767008 + 0.641638i \(0.221745\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 21.5308 0.732069
\(866\) 13.6420 0.463575
\(867\) −7.83638 −0.266137
\(868\) 26.1468 0.887479
\(869\) −13.8116 −0.468527
\(870\) 5.07606 0.172095
\(871\) 0 0
\(872\) 4.20344 0.142346
\(873\) 17.9855 0.608718
\(874\) −6.40773 −0.216745
\(875\) −4.80194 −0.162335
\(876\) 14.1129 0.476831
\(877\) −33.4935 −1.13099 −0.565497 0.824750i \(-0.691316\pi\)
−0.565497 + 0.824750i \(0.691316\pi\)
\(878\) 30.6437 1.03417
\(879\) 5.68233 0.191660
\(880\) −2.55496 −0.0861276
\(881\) 46.0431 1.55123 0.775615 0.631206i \(-0.217440\pi\)
0.775615 + 0.631206i \(0.217440\pi\)
\(882\) −16.0586 −0.540721
\(883\) −41.8629 −1.40880 −0.704400 0.709803i \(-0.748784\pi\)
−0.704400 + 0.709803i \(0.748784\pi\)
\(884\) 0 0
\(885\) −7.60925 −0.255782
\(886\) 19.7409 0.663210
\(887\) 23.8866 0.802034 0.401017 0.916071i \(-0.368657\pi\)
0.401017 + 0.916071i \(0.368657\pi\)
\(888\) −7.51573 −0.252211
\(889\) 26.8049 0.899008
\(890\) −16.8267 −0.564032
\(891\) 2.55496 0.0855943
\(892\) 20.1250 0.673834
\(893\) −25.0661 −0.838805
\(894\) −17.8756 −0.597850
\(895\) −15.5483 −0.519721
\(896\) −4.80194 −0.160421
\(897\) 0 0
\(898\) −36.2664 −1.21022
\(899\) 27.6394 0.921825
\(900\) 1.00000 0.0333333
\(901\) 20.9232 0.697053
\(902\) 15.0737 0.501898
\(903\) −48.7241 −1.62144
\(904\) 0.170915 0.00568455
\(905\) −14.4058 −0.478865
\(906\) −0.390748 −0.0129817
\(907\) −20.4136 −0.677822 −0.338911 0.940818i \(-0.610059\pi\)
−0.338911 + 0.940818i \(0.610059\pi\)
\(908\) −24.2379 −0.804362
\(909\) −15.7453 −0.522237
\(910\) 0 0
\(911\) 23.0108 0.762380 0.381190 0.924497i \(-0.375514\pi\)
0.381190 + 0.924497i \(0.375514\pi\)
\(912\) −3.89977 −0.129134
\(913\) 11.0761 0.366564
\(914\) −3.55602 −0.117623
\(915\) −9.65279 −0.319111
\(916\) −2.71618 −0.0897453
\(917\) −14.6987 −0.485395
\(918\) 3.02715 0.0999107
\(919\) −38.3327 −1.26448 −0.632240 0.774773i \(-0.717864\pi\)
−0.632240 + 0.774773i \(0.717864\pi\)
\(920\) −1.64310 −0.0541715
\(921\) −11.5961 −0.382105
\(922\) −1.46011 −0.0480861
\(923\) 0 0
\(924\) 12.2687 0.403612
\(925\) 7.51573 0.247116
\(926\) 13.7942 0.453304
\(927\) 12.0858 0.396948
\(928\) −5.07606 −0.166630
\(929\) 15.3515 0.503667 0.251834 0.967771i \(-0.418966\pi\)
0.251834 + 0.967771i \(0.418966\pi\)
\(930\) 5.44504 0.178550
\(931\) −62.6249 −2.05245
\(932\) −2.16421 −0.0708911
\(933\) 4.43429 0.145172
\(934\) 1.72886 0.0565699
\(935\) 7.73423 0.252936
\(936\) 0 0
\(937\) 51.3629 1.67795 0.838976 0.544169i \(-0.183155\pi\)
0.838976 + 0.544169i \(0.183155\pi\)
\(938\) −4.46250 −0.145706
\(939\) 11.2881 0.368374
\(940\) −6.42758 −0.209645
\(941\) −17.2413 −0.562052 −0.281026 0.959700i \(-0.590675\pi\)
−0.281026 + 0.959700i \(0.590675\pi\)
\(942\) −9.50902 −0.309821
\(943\) 9.69394 0.315678
\(944\) 7.60925 0.247660
\(945\) −4.80194 −0.156207
\(946\) 25.9245 0.842879
\(947\) 5.54586 0.180216 0.0901081 0.995932i \(-0.471279\pi\)
0.0901081 + 0.995932i \(0.471279\pi\)
\(948\) −5.40581 −0.175573
\(949\) 0 0
\(950\) 3.89977 0.126525
\(951\) −6.31229 −0.204690
\(952\) 14.5362 0.471120
\(953\) −50.1584 −1.62479 −0.812394 0.583108i \(-0.801836\pi\)
−0.812394 + 0.583108i \(0.801836\pi\)
\(954\) 6.91185 0.223780
\(955\) 4.02177 0.130141
\(956\) 14.5114 0.469333
\(957\) 12.9691 0.419233
\(958\) −14.5496 −0.470076
\(959\) −4.96449 −0.160312
\(960\) −1.00000 −0.0322749
\(961\) −1.35152 −0.0435974
\(962\) 0 0
\(963\) −4.64071 −0.149545
\(964\) −18.1142 −0.583420
\(965\) 20.1008 0.647068
\(966\) 7.89008 0.253859
\(967\) −13.6890 −0.440210 −0.220105 0.975476i \(-0.570640\pi\)
−0.220105 + 0.975476i \(0.570640\pi\)
\(968\) 4.47219 0.143742
\(969\) 11.8052 0.379237
\(970\) 17.9855 0.577480
\(971\) −50.6075 −1.62407 −0.812035 0.583608i \(-0.801640\pi\)
−0.812035 + 0.583608i \(0.801640\pi\)
\(972\) 1.00000 0.0320750
\(973\) −80.6704 −2.58617
\(974\) −30.5526 −0.978967
\(975\) 0 0
\(976\) 9.65279 0.308978
\(977\) −11.2959 −0.361388 −0.180694 0.983539i \(-0.557834\pi\)
−0.180694 + 0.983539i \(0.557834\pi\)
\(978\) 5.84846 0.187013
\(979\) −42.9915 −1.37401
\(980\) −16.0586 −0.512973
\(981\) −4.20344 −0.134205
\(982\) −9.68233 −0.308976
\(983\) −36.8194 −1.17436 −0.587178 0.809458i \(-0.699761\pi\)
−0.587178 + 0.809458i \(0.699761\pi\)
\(984\) 5.89977 0.188078
\(985\) 15.3884 0.490314
\(986\) 15.3660 0.489353
\(987\) 30.8649 0.982439
\(988\) 0 0
\(989\) 16.6722 0.530144
\(990\) 2.55496 0.0812019
\(991\) 10.8254 0.343879 0.171940 0.985108i \(-0.444997\pi\)
0.171940 + 0.985108i \(0.444997\pi\)
\(992\) −5.44504 −0.172880
\(993\) 10.9554 0.347659
\(994\) −71.7827 −2.27681
\(995\) −24.7724 −0.785338
\(996\) 4.33513 0.137364
\(997\) −4.30319 −0.136283 −0.0681417 0.997676i \(-0.521707\pi\)
−0.0681417 + 0.997676i \(0.521707\pi\)
\(998\) −18.6025 −0.588853
\(999\) 7.51573 0.237787
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.bo.1.3 3
13.5 odd 4 5070.2.b.w.1351.6 6
13.8 odd 4 5070.2.b.w.1351.1 6
13.12 even 2 5070.2.a.bx.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bo.1.3 3 1.1 even 1 trivial
5070.2.a.bx.1.1 yes 3 13.12 even 2
5070.2.b.w.1351.1 6 13.8 odd 4
5070.2.b.w.1351.6 6 13.5 odd 4