Properties

Label 5070.2.b.w.1351.1
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.1
Root \(-1.80194i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.w.1351.6

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) − 1.00000i − 0.408248i
\(7\) − 4.80194i − 1.81496i −0.420093 0.907481i \(-0.638003\pi\)
0.420093 0.907481i \(-0.361997\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) − 2.55496i − 0.770349i −0.922844 0.385174i \(-0.874141\pi\)
0.922844 0.385174i \(-0.125859\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −4.80194 −1.28337
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) 3.02715 0.734191 0.367096 0.930183i \(-0.380352\pi\)
0.367096 + 0.930183i \(0.380352\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) − 3.89977i − 0.894669i −0.894367 0.447335i \(-0.852373\pi\)
0.894367 0.447335i \(-0.147627\pi\)
\(20\) 1.00000i 0.223607i
\(21\) − 4.80194i − 1.04787i
\(22\) −2.55496 −0.544719
\(23\) 1.64310 0.342611 0.171305 0.985218i \(-0.445202\pi\)
0.171305 + 0.985218i \(0.445202\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 4.80194i 0.907481i
\(29\) 5.07606 0.942601 0.471301 0.881973i \(-0.343785\pi\)
0.471301 + 0.881973i \(0.343785\pi\)
\(30\) −1.00000 −0.182574
\(31\) 5.44504i 0.977958i 0.872295 + 0.488979i \(0.162631\pi\)
−0.872295 + 0.488979i \(0.837369\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 2.55496i − 0.444761i
\(34\) − 3.02715i − 0.519151i
\(35\) −4.80194 −0.811676
\(36\) −1.00000 −0.166667
\(37\) − 7.51573i − 1.23558i −0.786344 0.617789i \(-0.788029\pi\)
0.786344 0.617789i \(-0.211971\pi\)
\(38\) −3.89977 −0.632627
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) − 5.89977i − 0.921390i −0.887559 0.460695i \(-0.847600\pi\)
0.887559 0.460695i \(-0.152400\pi\)
\(42\) −4.80194 −0.740955
\(43\) 10.1468 1.54737 0.773683 0.633573i \(-0.218413\pi\)
0.773683 + 0.633573i \(0.218413\pi\)
\(44\) 2.55496i 0.385174i
\(45\) − 1.00000i − 0.149071i
\(46\) − 1.64310i − 0.242262i
\(47\) − 6.42758i − 0.937559i −0.883315 0.468780i \(-0.844694\pi\)
0.883315 0.468780i \(-0.155306\pi\)
\(48\) 1.00000 0.144338
\(49\) −16.0586 −2.29409
\(50\) 1.00000i 0.141421i
\(51\) 3.02715 0.423885
\(52\) 0 0
\(53\) −6.91185 −0.949416 −0.474708 0.880143i \(-0.657446\pi\)
−0.474708 + 0.880143i \(0.657446\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −2.55496 −0.344510
\(56\) 4.80194 0.641686
\(57\) − 3.89977i − 0.516537i
\(58\) − 5.07606i − 0.666520i
\(59\) − 7.60925i − 0.990640i −0.868711 0.495320i \(-0.835051\pi\)
0.868711 0.495320i \(-0.164949\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 9.65279 1.23591 0.617957 0.786212i \(-0.287961\pi\)
0.617957 + 0.786212i \(0.287961\pi\)
\(62\) 5.44504 0.691521
\(63\) − 4.80194i − 0.604987i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.55496 −0.314494
\(67\) 0.929312i 0.113534i 0.998387 + 0.0567668i \(0.0180791\pi\)
−0.998387 + 0.0567668i \(0.981921\pi\)
\(68\) −3.02715 −0.367096
\(69\) 1.64310 0.197806
\(70\) 4.80194i 0.573941i
\(71\) 14.9487i 1.77408i 0.461690 + 0.887042i \(0.347243\pi\)
−0.461690 + 0.887042i \(0.652757\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 14.1129i − 1.65179i −0.563824 0.825895i \(-0.690670\pi\)
0.563824 0.825895i \(-0.309330\pi\)
\(74\) −7.51573 −0.873686
\(75\) −1.00000 −0.115470
\(76\) 3.89977i 0.447335i
\(77\) −12.2687 −1.39815
\(78\) 0 0
\(79\) −5.40581 −0.608202 −0.304101 0.952640i \(-0.598356\pi\)
−0.304101 + 0.952640i \(0.598356\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) −5.89977 −0.651521
\(83\) 4.33513i 0.475842i 0.971285 + 0.237921i \(0.0764659\pi\)
−0.971285 + 0.237921i \(0.923534\pi\)
\(84\) 4.80194i 0.523934i
\(85\) − 3.02715i − 0.328340i
\(86\) − 10.1468i − 1.09415i
\(87\) 5.07606 0.544211
\(88\) 2.55496 0.272359
\(89\) 16.8267i 1.78363i 0.452404 + 0.891813i \(0.350566\pi\)
−0.452404 + 0.891813i \(0.649434\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −1.64310 −0.171305
\(93\) 5.44504i 0.564625i
\(94\) −6.42758 −0.662955
\(95\) −3.89977 −0.400108
\(96\) − 1.00000i − 0.102062i
\(97\) 17.9855i 1.82615i 0.407788 + 0.913077i \(0.366300\pi\)
−0.407788 + 0.913077i \(0.633700\pi\)
\(98\) 16.0586i 1.62216i
\(99\) − 2.55496i − 0.256783i
\(100\) 1.00000 0.100000
\(101\) 15.7453 1.56671 0.783355 0.621574i \(-0.213506\pi\)
0.783355 + 0.621574i \(0.213506\pi\)
\(102\) − 3.02715i − 0.299732i
\(103\) −12.0858 −1.19084 −0.595422 0.803413i \(-0.703015\pi\)
−0.595422 + 0.803413i \(0.703015\pi\)
\(104\) 0 0
\(105\) −4.80194 −0.468621
\(106\) 6.91185i 0.671339i
\(107\) −4.64071 −0.448634 −0.224317 0.974516i \(-0.572015\pi\)
−0.224317 + 0.974516i \(0.572015\pi\)
\(108\) −1.00000 −0.0962250
\(109\) − 4.20344i − 0.402616i −0.979528 0.201308i \(-0.935481\pi\)
0.979528 0.201308i \(-0.0645193\pi\)
\(110\) 2.55496i 0.243606i
\(111\) − 7.51573i − 0.713361i
\(112\) − 4.80194i − 0.453740i
\(113\) −0.170915 −0.0160783 −0.00803917 0.999968i \(-0.502559\pi\)
−0.00803917 + 0.999968i \(0.502559\pi\)
\(114\) −3.89977 −0.365247
\(115\) − 1.64310i − 0.153220i
\(116\) −5.07606 −0.471301
\(117\) 0 0
\(118\) −7.60925 −0.700488
\(119\) − 14.5362i − 1.33253i
\(120\) 1.00000 0.0912871
\(121\) 4.47219 0.406563
\(122\) − 9.65279i − 0.873923i
\(123\) − 5.89977i − 0.531965i
\(124\) − 5.44504i − 0.488979i
\(125\) 1.00000i 0.0894427i
\(126\) −4.80194 −0.427791
\(127\) −5.58211 −0.495332 −0.247666 0.968846i \(-0.579664\pi\)
−0.247666 + 0.968846i \(0.579664\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 10.1468 0.893372
\(130\) 0 0
\(131\) −3.06100 −0.267441 −0.133720 0.991019i \(-0.542692\pi\)
−0.133720 + 0.991019i \(0.542692\pi\)
\(132\) 2.55496i 0.222381i
\(133\) −18.7265 −1.62379
\(134\) 0.929312 0.0802804
\(135\) − 1.00000i − 0.0860663i
\(136\) 3.02715i 0.259576i
\(137\) 1.03385i 0.0883279i 0.999024 + 0.0441640i \(0.0140624\pi\)
−0.999024 + 0.0441640i \(0.985938\pi\)
\(138\) − 1.64310i − 0.139870i
\(139\) −16.7995 −1.42492 −0.712459 0.701713i \(-0.752419\pi\)
−0.712459 + 0.701713i \(0.752419\pi\)
\(140\) 4.80194 0.405838
\(141\) − 6.42758i − 0.541300i
\(142\) 14.9487 1.25447
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 5.07606i − 0.421544i
\(146\) −14.1129 −1.16799
\(147\) −16.0586 −1.32449
\(148\) 7.51573i 0.617789i
\(149\) 17.8756i 1.46443i 0.681075 + 0.732213i \(0.261513\pi\)
−0.681075 + 0.732213i \(0.738487\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) − 0.390748i − 0.0317986i −0.999874 0.0158993i \(-0.994939\pi\)
0.999874 0.0158993i \(-0.00506112\pi\)
\(152\) 3.89977 0.316313
\(153\) 3.02715 0.244730
\(154\) 12.2687i 0.988644i
\(155\) 5.44504 0.437356
\(156\) 0 0
\(157\) 9.50902 0.758903 0.379451 0.925212i \(-0.376113\pi\)
0.379451 + 0.925212i \(0.376113\pi\)
\(158\) 5.40581i 0.430063i
\(159\) −6.91185 −0.548146
\(160\) −1.00000 −0.0790569
\(161\) − 7.89008i − 0.621826i
\(162\) − 1.00000i − 0.0785674i
\(163\) 5.84846i 0.458087i 0.973416 + 0.229043i \(0.0735598\pi\)
−0.973416 + 0.229043i \(0.926440\pi\)
\(164\) 5.89977i 0.460695i
\(165\) −2.55496 −0.198903
\(166\) 4.33513 0.336471
\(167\) 2.55496i 0.197709i 0.995102 + 0.0988543i \(0.0315178\pi\)
−0.995102 + 0.0988543i \(0.968482\pi\)
\(168\) 4.80194 0.370478
\(169\) 0 0
\(170\) −3.02715 −0.232172
\(171\) − 3.89977i − 0.298223i
\(172\) −10.1468 −0.773683
\(173\) 21.5308 1.63696 0.818478 0.574538i \(-0.194818\pi\)
0.818478 + 0.574538i \(0.194818\pi\)
\(174\) − 5.07606i − 0.384815i
\(175\) 4.80194i 0.362992i
\(176\) − 2.55496i − 0.192587i
\(177\) − 7.60925i − 0.571946i
\(178\) 16.8267 1.26121
\(179\) −15.5483 −1.16213 −0.581066 0.813857i \(-0.697364\pi\)
−0.581066 + 0.813857i \(0.697364\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) −14.4058 −1.07078 −0.535388 0.844606i \(-0.679835\pi\)
−0.535388 + 0.844606i \(0.679835\pi\)
\(182\) 0 0
\(183\) 9.65279 0.713555
\(184\) 1.64310i 0.121131i
\(185\) −7.51573 −0.552567
\(186\) 5.44504 0.399250
\(187\) − 7.73423i − 0.565583i
\(188\) 6.42758i 0.468780i
\(189\) − 4.80194i − 0.349290i
\(190\) 3.89977i 0.282919i
\(191\) −4.02177 −0.291005 −0.145503 0.989358i \(-0.546480\pi\)
−0.145503 + 0.989358i \(0.546480\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 20.1008i 1.44689i 0.690383 + 0.723444i \(0.257442\pi\)
−0.690383 + 0.723444i \(0.742558\pi\)
\(194\) 17.9855 1.29129
\(195\) 0 0
\(196\) 16.0586 1.14704
\(197\) − 15.3884i − 1.09637i −0.836356 0.548187i \(-0.815318\pi\)
0.836356 0.548187i \(-0.184682\pi\)
\(198\) −2.55496 −0.181573
\(199\) −24.7724 −1.75607 −0.878034 0.478598i \(-0.841145\pi\)
−0.878034 + 0.478598i \(0.841145\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) 0.929312i 0.0655486i
\(202\) − 15.7453i − 1.10783i
\(203\) − 24.3749i − 1.71079i
\(204\) −3.02715 −0.211943
\(205\) −5.89977 −0.412058
\(206\) 12.0858i 0.842054i
\(207\) 1.64310 0.114204
\(208\) 0 0
\(209\) −9.96376 −0.689207
\(210\) 4.80194i 0.331365i
\(211\) 6.49396 0.447063 0.223531 0.974697i \(-0.428242\pi\)
0.223531 + 0.974697i \(0.428242\pi\)
\(212\) 6.91185 0.474708
\(213\) 14.9487i 1.02427i
\(214\) 4.64071i 0.317232i
\(215\) − 10.1468i − 0.692003i
\(216\) 1.00000i 0.0680414i
\(217\) 26.1468 1.77496
\(218\) −4.20344 −0.284693
\(219\) − 14.1129i − 0.953661i
\(220\) 2.55496 0.172255
\(221\) 0 0
\(222\) −7.51573 −0.504423
\(223\) 20.1250i 1.34767i 0.738883 + 0.673834i \(0.235354\pi\)
−0.738883 + 0.673834i \(0.764646\pi\)
\(224\) −4.80194 −0.320843
\(225\) −1.00000 −0.0666667
\(226\) 0.170915i 0.0113691i
\(227\) − 24.2379i − 1.60872i −0.594139 0.804362i \(-0.702507\pi\)
0.594139 0.804362i \(-0.297493\pi\)
\(228\) 3.89977i 0.258269i
\(229\) 2.71618i 0.179491i 0.995965 + 0.0897453i \(0.0286053\pi\)
−0.995965 + 0.0897453i \(0.971395\pi\)
\(230\) −1.64310 −0.108343
\(231\) −12.2687 −0.807224
\(232\) 5.07606i 0.333260i
\(233\) 2.16421 0.141782 0.0708911 0.997484i \(-0.477416\pi\)
0.0708911 + 0.997484i \(0.477416\pi\)
\(234\) 0 0
\(235\) −6.42758 −0.419289
\(236\) 7.60925i 0.495320i
\(237\) −5.40581 −0.351145
\(238\) −14.5362 −0.942240
\(239\) 14.5114i 0.938666i 0.883021 + 0.469333i \(0.155506\pi\)
−0.883021 + 0.469333i \(0.844494\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) 18.1142i 1.16684i 0.812171 + 0.583420i \(0.198286\pi\)
−0.812171 + 0.583420i \(0.801714\pi\)
\(242\) − 4.47219i − 0.287483i
\(243\) 1.00000 0.0641500
\(244\) −9.65279 −0.617957
\(245\) 16.0586i 1.02595i
\(246\) −5.89977 −0.376156
\(247\) 0 0
\(248\) −5.44504 −0.345761
\(249\) 4.33513i 0.274727i
\(250\) 1.00000 0.0632456
\(251\) −8.55257 −0.539833 −0.269917 0.962884i \(-0.586996\pi\)
−0.269917 + 0.962884i \(0.586996\pi\)
\(252\) 4.80194i 0.302494i
\(253\) − 4.19806i − 0.263930i
\(254\) 5.58211i 0.350252i
\(255\) − 3.02715i − 0.189567i
\(256\) 1.00000 0.0625000
\(257\) −1.30260 −0.0812541 −0.0406270 0.999174i \(-0.512936\pi\)
−0.0406270 + 0.999174i \(0.512936\pi\)
\(258\) − 10.1468i − 0.631709i
\(259\) −36.0901 −2.24253
\(260\) 0 0
\(261\) 5.07606 0.314200
\(262\) 3.06100i 0.189109i
\(263\) 8.24027 0.508117 0.254059 0.967189i \(-0.418234\pi\)
0.254059 + 0.967189i \(0.418234\pi\)
\(264\) 2.55496 0.157247
\(265\) 6.91185i 0.424592i
\(266\) 18.7265i 1.14819i
\(267\) 16.8267i 1.02978i
\(268\) − 0.929312i − 0.0567668i
\(269\) 22.6601 1.38161 0.690805 0.723041i \(-0.257256\pi\)
0.690805 + 0.723041i \(0.257256\pi\)
\(270\) −1.00000 −0.0608581
\(271\) − 13.9172i − 0.845412i −0.906267 0.422706i \(-0.861080\pi\)
0.906267 0.422706i \(-0.138920\pi\)
\(272\) 3.02715 0.183548
\(273\) 0 0
\(274\) 1.03385 0.0624573
\(275\) 2.55496i 0.154070i
\(276\) −1.64310 −0.0989032
\(277\) 24.2610 1.45770 0.728851 0.684673i \(-0.240055\pi\)
0.728851 + 0.684673i \(0.240055\pi\)
\(278\) 16.7995i 1.00757i
\(279\) 5.44504i 0.325986i
\(280\) − 4.80194i − 0.286971i
\(281\) − 10.5961i − 0.632111i −0.948741 0.316055i \(-0.897641\pi\)
0.948741 0.316055i \(-0.102359\pi\)
\(282\) −6.42758 −0.382757
\(283\) −9.86294 −0.586291 −0.293145 0.956068i \(-0.594702\pi\)
−0.293145 + 0.956068i \(0.594702\pi\)
\(284\) − 14.9487i − 0.887042i
\(285\) −3.89977 −0.231003
\(286\) 0 0
\(287\) −28.3303 −1.67229
\(288\) − 1.00000i − 0.0589256i
\(289\) −7.83638 −0.460964
\(290\) −5.07606 −0.298077
\(291\) 17.9855i 1.05433i
\(292\) 14.1129i 0.825895i
\(293\) − 5.68233i − 0.331965i −0.986129 0.165983i \(-0.946920\pi\)
0.986129 0.165983i \(-0.0530796\pi\)
\(294\) 16.0586i 0.936557i
\(295\) −7.60925 −0.443028
\(296\) 7.51573 0.436843
\(297\) − 2.55496i − 0.148254i
\(298\) 17.8756 1.03551
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) − 48.7241i − 2.80841i
\(302\) −0.390748 −0.0224850
\(303\) 15.7453 0.904541
\(304\) − 3.89977i − 0.223667i
\(305\) − 9.65279i − 0.552717i
\(306\) − 3.02715i − 0.173050i
\(307\) 11.5961i 0.661825i 0.943661 + 0.330912i \(0.107356\pi\)
−0.943661 + 0.330912i \(0.892644\pi\)
\(308\) 12.2687 0.699077
\(309\) −12.0858 −0.687534
\(310\) − 5.44504i − 0.309258i
\(311\) −4.43429 −0.251445 −0.125723 0.992065i \(-0.540125\pi\)
−0.125723 + 0.992065i \(0.540125\pi\)
\(312\) 0 0
\(313\) 11.2881 0.638043 0.319021 0.947748i \(-0.396646\pi\)
0.319021 + 0.947748i \(0.396646\pi\)
\(314\) − 9.50902i − 0.536625i
\(315\) −4.80194 −0.270559
\(316\) 5.40581 0.304101
\(317\) − 6.31229i − 0.354534i −0.984163 0.177267i \(-0.943274\pi\)
0.984163 0.177267i \(-0.0567255\pi\)
\(318\) 6.91185i 0.387598i
\(319\) − 12.9691i − 0.726132i
\(320\) 1.00000i 0.0559017i
\(321\) −4.64071 −0.259019
\(322\) −7.89008 −0.439697
\(323\) − 11.8052i − 0.656858i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 5.84846 0.323916
\(327\) − 4.20344i − 0.232451i
\(328\) 5.89977 0.325760
\(329\) −30.8649 −1.70163
\(330\) 2.55496i 0.140646i
\(331\) 10.9554i 0.602163i 0.953598 + 0.301081i \(0.0973476\pi\)
−0.953598 + 0.301081i \(0.902652\pi\)
\(332\) − 4.33513i − 0.237921i
\(333\) − 7.51573i − 0.411859i
\(334\) 2.55496 0.139801
\(335\) 0.929312 0.0507738
\(336\) − 4.80194i − 0.261967i
\(337\) −26.6668 −1.45263 −0.726316 0.687361i \(-0.758769\pi\)
−0.726316 + 0.687361i \(0.758769\pi\)
\(338\) 0 0
\(339\) −0.170915 −0.00928284
\(340\) 3.02715i 0.164170i
\(341\) 13.9119 0.753369
\(342\) −3.89977 −0.210876
\(343\) 43.4989i 2.34872i
\(344\) 10.1468i 0.547076i
\(345\) − 1.64310i − 0.0884618i
\(346\) − 21.5308i − 1.15750i
\(347\) 25.0694 1.34579 0.672897 0.739736i \(-0.265050\pi\)
0.672897 + 0.739736i \(0.265050\pi\)
\(348\) −5.07606 −0.272106
\(349\) 21.5623i 1.15420i 0.816673 + 0.577100i \(0.195816\pi\)
−0.816673 + 0.577100i \(0.804184\pi\)
\(350\) 4.80194 0.256674
\(351\) 0 0
\(352\) −2.55496 −0.136180
\(353\) − 28.2959i − 1.50604i −0.657998 0.753019i \(-0.728597\pi\)
0.657998 0.753019i \(-0.271403\pi\)
\(354\) −7.60925 −0.404427
\(355\) 14.9487 0.793394
\(356\) − 16.8267i − 0.891813i
\(357\) − 14.5362i − 0.769336i
\(358\) 15.5483i 0.821751i
\(359\) − 34.2717i − 1.80879i −0.426693 0.904396i \(-0.640322\pi\)
0.426693 0.904396i \(-0.359678\pi\)
\(360\) 1.00000 0.0527046
\(361\) 3.79178 0.199567
\(362\) 14.4058i 0.757153i
\(363\) 4.47219 0.234729
\(364\) 0 0
\(365\) −14.1129 −0.738703
\(366\) − 9.65279i − 0.504560i
\(367\) −23.4209 −1.22256 −0.611280 0.791414i \(-0.709345\pi\)
−0.611280 + 0.791414i \(0.709345\pi\)
\(368\) 1.64310 0.0856527
\(369\) − 5.89977i − 0.307130i
\(370\) 7.51573i 0.390724i
\(371\) 33.1903i 1.72315i
\(372\) − 5.44504i − 0.282312i
\(373\) 1.82371 0.0944280 0.0472140 0.998885i \(-0.484966\pi\)
0.0472140 + 0.998885i \(0.484966\pi\)
\(374\) −7.73423 −0.399928
\(375\) 1.00000i 0.0516398i
\(376\) 6.42758 0.331477
\(377\) 0 0
\(378\) −4.80194 −0.246985
\(379\) − 15.3260i − 0.787245i −0.919272 0.393623i \(-0.871222\pi\)
0.919272 0.393623i \(-0.128778\pi\)
\(380\) 3.89977 0.200054
\(381\) −5.58211 −0.285980
\(382\) 4.02177i 0.205772i
\(383\) − 29.6722i − 1.51618i −0.652152 0.758089i \(-0.726133\pi\)
0.652152 0.758089i \(-0.273867\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 12.2687i 0.625273i
\(386\) 20.1008 1.02310
\(387\) 10.1468 0.515788
\(388\) − 17.9855i − 0.913077i
\(389\) −11.8291 −0.599758 −0.299879 0.953977i \(-0.596946\pi\)
−0.299879 + 0.953977i \(0.596946\pi\)
\(390\) 0 0
\(391\) 4.97392 0.251542
\(392\) − 16.0586i − 0.811082i
\(393\) −3.06100 −0.154407
\(394\) −15.3884 −0.775254
\(395\) 5.40581i 0.271996i
\(396\) 2.55496i 0.128391i
\(397\) − 12.1153i − 0.608049i −0.952664 0.304025i \(-0.901670\pi\)
0.952664 0.304025i \(-0.0983305\pi\)
\(398\) 24.7724i 1.24173i
\(399\) −18.7265 −0.937496
\(400\) −1.00000 −0.0500000
\(401\) 10.8291i 0.540779i 0.962751 + 0.270389i \(0.0871524\pi\)
−0.962751 + 0.270389i \(0.912848\pi\)
\(402\) 0.929312 0.0463499
\(403\) 0 0
\(404\) −15.7453 −0.783355
\(405\) − 1.00000i − 0.0496904i
\(406\) −24.3749 −1.20971
\(407\) −19.2024 −0.951826
\(408\) 3.02715i 0.149866i
\(409\) − 33.9396i − 1.67820i −0.543973 0.839102i \(-0.683081\pi\)
0.543973 0.839102i \(-0.316919\pi\)
\(410\) 5.89977i 0.291369i
\(411\) 1.03385i 0.0509962i
\(412\) 12.0858 0.595422
\(413\) −36.5392 −1.79797
\(414\) − 1.64310i − 0.0807542i
\(415\) 4.33513 0.212803
\(416\) 0 0
\(417\) −16.7995 −0.822677
\(418\) 9.96376i 0.487343i
\(419\) 26.8562 1.31201 0.656006 0.754755i \(-0.272244\pi\)
0.656006 + 0.754755i \(0.272244\pi\)
\(420\) 4.80194 0.234311
\(421\) 27.9928i 1.36429i 0.731219 + 0.682143i \(0.238952\pi\)
−0.731219 + 0.682143i \(0.761048\pi\)
\(422\) − 6.49396i − 0.316121i
\(423\) − 6.42758i − 0.312520i
\(424\) − 6.91185i − 0.335669i
\(425\) −3.02715 −0.146838
\(426\) 14.9487 0.724266
\(427\) − 46.3521i − 2.24314i
\(428\) 4.64071 0.224317
\(429\) 0 0
\(430\) −10.1468 −0.489320
\(431\) − 33.4088i − 1.60925i −0.593787 0.804623i \(-0.702368\pi\)
0.593787 0.804623i \(-0.297632\pi\)
\(432\) 1.00000 0.0481125
\(433\) 13.6420 0.655595 0.327797 0.944748i \(-0.393694\pi\)
0.327797 + 0.944748i \(0.393694\pi\)
\(434\) − 26.1468i − 1.25508i
\(435\) − 5.07606i − 0.243379i
\(436\) 4.20344i 0.201308i
\(437\) − 6.40773i − 0.306523i
\(438\) −14.1129 −0.674340
\(439\) 30.6437 1.46254 0.731272 0.682086i \(-0.238927\pi\)
0.731272 + 0.682086i \(0.238927\pi\)
\(440\) − 2.55496i − 0.121803i
\(441\) −16.0586 −0.764696
\(442\) 0 0
\(443\) −19.7409 −0.937920 −0.468960 0.883219i \(-0.655371\pi\)
−0.468960 + 0.883219i \(0.655371\pi\)
\(444\) 7.51573i 0.356681i
\(445\) 16.8267 0.797662
\(446\) 20.1250 0.952946
\(447\) 17.8756i 0.845487i
\(448\) 4.80194i 0.226870i
\(449\) − 36.2664i − 1.71152i −0.517377 0.855758i \(-0.673091\pi\)
0.517377 0.855758i \(-0.326909\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) −15.0737 −0.709791
\(452\) 0.170915 0.00803917
\(453\) − 0.390748i − 0.0183589i
\(454\) −24.2379 −1.13754
\(455\) 0 0
\(456\) 3.89977 0.182624
\(457\) 3.55602i 0.166344i 0.996535 + 0.0831719i \(0.0265050\pi\)
−0.996535 + 0.0831719i \(0.973495\pi\)
\(458\) 2.71618 0.126919
\(459\) 3.02715 0.141295
\(460\) 1.64310i 0.0766101i
\(461\) 1.46011i 0.0680040i 0.999422 + 0.0340020i \(0.0108253\pi\)
−0.999422 + 0.0340020i \(0.989175\pi\)
\(462\) 12.2687i 0.570794i
\(463\) 13.7942i 0.641069i 0.947237 + 0.320535i \(0.103863\pi\)
−0.947237 + 0.320535i \(0.896137\pi\)
\(464\) 5.07606 0.235650
\(465\) 5.44504 0.252508
\(466\) − 2.16421i − 0.100255i
\(467\) 1.72886 0.0800020 0.0400010 0.999200i \(-0.487264\pi\)
0.0400010 + 0.999200i \(0.487264\pi\)
\(468\) 0 0
\(469\) 4.46250 0.206059
\(470\) 6.42758i 0.296482i
\(471\) 9.50902 0.438153
\(472\) 7.60925 0.350244
\(473\) − 25.9245i − 1.19201i
\(474\) 5.40581i 0.248297i
\(475\) 3.89977i 0.178934i
\(476\) 14.5362i 0.666264i
\(477\) −6.91185 −0.316472
\(478\) 14.5114 0.663737
\(479\) − 14.5496i − 0.664787i −0.943141 0.332394i \(-0.892144\pi\)
0.943141 0.332394i \(-0.107856\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 18.1142 0.825080
\(483\) − 7.89008i − 0.359011i
\(484\) −4.47219 −0.203281
\(485\) 17.9855 0.816681
\(486\) − 1.00000i − 0.0453609i
\(487\) 30.5526i 1.38447i 0.721673 + 0.692234i \(0.243374\pi\)
−0.721673 + 0.692234i \(0.756626\pi\)
\(488\) 9.65279i 0.436961i
\(489\) 5.84846i 0.264477i
\(490\) 16.0586 0.725454
\(491\) −9.68233 −0.436958 −0.218479 0.975842i \(-0.570109\pi\)
−0.218479 + 0.975842i \(0.570109\pi\)
\(492\) 5.89977i 0.265982i
\(493\) 15.3660 0.692050
\(494\) 0 0
\(495\) −2.55496 −0.114837
\(496\) 5.44504i 0.244490i
\(497\) 71.7827 3.21989
\(498\) 4.33513 0.194262
\(499\) 18.6025i 0.832764i 0.909190 + 0.416382i \(0.136702\pi\)
−0.909190 + 0.416382i \(0.863298\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) 2.55496i 0.114147i
\(502\) 8.55257i 0.381720i
\(503\) 41.6644 1.85772 0.928862 0.370426i \(-0.120788\pi\)
0.928862 + 0.370426i \(0.120788\pi\)
\(504\) 4.80194 0.213895
\(505\) − 15.7453i − 0.700654i
\(506\) −4.19806 −0.186627
\(507\) 0 0
\(508\) 5.58211 0.247666
\(509\) 41.1648i 1.82460i 0.409525 + 0.912299i \(0.365694\pi\)
−0.409525 + 0.912299i \(0.634306\pi\)
\(510\) −3.02715 −0.134044
\(511\) −67.7693 −2.99794
\(512\) − 1.00000i − 0.0441942i
\(513\) − 3.89977i − 0.172179i
\(514\) 1.30260i 0.0574553i
\(515\) 12.0858i 0.532562i
\(516\) −10.1468 −0.446686
\(517\) −16.4222 −0.722248
\(518\) 36.0901i 1.58571i
\(519\) 21.5308 0.945097
\(520\) 0 0
\(521\) 1.12737 0.0493912 0.0246956 0.999695i \(-0.492138\pi\)
0.0246956 + 0.999695i \(0.492138\pi\)
\(522\) − 5.07606i − 0.222173i
\(523\) −22.7922 −0.996635 −0.498318 0.866994i \(-0.666049\pi\)
−0.498318 + 0.866994i \(0.666049\pi\)
\(524\) 3.06100 0.133720
\(525\) 4.80194i 0.209574i
\(526\) − 8.24027i − 0.359293i
\(527\) 16.4829i 0.718008i
\(528\) − 2.55496i − 0.111190i
\(529\) −20.3002 −0.882618
\(530\) 6.91185 0.300232
\(531\) − 7.60925i − 0.330213i
\(532\) 18.7265 0.811895
\(533\) 0 0
\(534\) 16.8267 0.728162
\(535\) 4.64071i 0.200635i
\(536\) −0.929312 −0.0401402
\(537\) −15.5483 −0.670957
\(538\) − 22.6601i − 0.976946i
\(539\) 41.0291i 1.76725i
\(540\) 1.00000i 0.0430331i
\(541\) 41.1148i 1.76766i 0.467804 + 0.883832i \(0.345045\pi\)
−0.467804 + 0.883832i \(0.654955\pi\)
\(542\) −13.9172 −0.597796
\(543\) −14.4058 −0.618213
\(544\) − 3.02715i − 0.129788i
\(545\) −4.20344 −0.180056
\(546\) 0 0
\(547\) 3.96615 0.169580 0.0847901 0.996399i \(-0.472978\pi\)
0.0847901 + 0.996399i \(0.472978\pi\)
\(548\) − 1.03385i − 0.0441640i
\(549\) 9.65279 0.411971
\(550\) 2.55496 0.108944
\(551\) − 19.7955i − 0.843316i
\(552\) 1.64310i 0.0699352i
\(553\) 25.9584i 1.10386i
\(554\) − 24.2610i − 1.03075i
\(555\) −7.51573 −0.319025
\(556\) 16.7995 0.712459
\(557\) 4.25236i 0.180178i 0.995934 + 0.0900891i \(0.0287152\pi\)
−0.995934 + 0.0900891i \(0.971285\pi\)
\(558\) 5.44504 0.230507
\(559\) 0 0
\(560\) −4.80194 −0.202919
\(561\) − 7.73423i − 0.326540i
\(562\) −10.5961 −0.446970
\(563\) 32.7700 1.38109 0.690546 0.723289i \(-0.257371\pi\)
0.690546 + 0.723289i \(0.257371\pi\)
\(564\) 6.42758i 0.270650i
\(565\) 0.170915i 0.00719046i
\(566\) 9.86294i 0.414570i
\(567\) − 4.80194i − 0.201662i
\(568\) −14.9487 −0.627233
\(569\) 20.6189 0.864391 0.432195 0.901780i \(-0.357739\pi\)
0.432195 + 0.901780i \(0.357739\pi\)
\(570\) 3.89977i 0.163343i
\(571\) −23.1215 −0.967606 −0.483803 0.875177i \(-0.660745\pi\)
−0.483803 + 0.875177i \(0.660745\pi\)
\(572\) 0 0
\(573\) −4.02177 −0.168012
\(574\) 28.3303i 1.18249i
\(575\) −1.64310 −0.0685222
\(576\) −1.00000 −0.0416667
\(577\) 28.0000i 1.16566i 0.812596 + 0.582828i \(0.198054\pi\)
−0.812596 + 0.582828i \(0.801946\pi\)
\(578\) 7.83638i 0.325950i
\(579\) 20.1008i 0.835362i
\(580\) 5.07606i 0.210772i
\(581\) 20.8170 0.863635
\(582\) 17.9855 0.745524
\(583\) 17.6595i 0.731382i
\(584\) 14.1129 0.583996
\(585\) 0 0
\(586\) −5.68233 −0.234735
\(587\) 21.1081i 0.871225i 0.900134 + 0.435613i \(0.143468\pi\)
−0.900134 + 0.435613i \(0.856532\pi\)
\(588\) 16.0586 0.662246
\(589\) 21.2344 0.874949
\(590\) 7.60925i 0.313268i
\(591\) − 15.3884i − 0.632992i
\(592\) − 7.51573i − 0.308895i
\(593\) 15.4276i 0.633535i 0.948503 + 0.316767i \(0.102597\pi\)
−0.948503 + 0.316767i \(0.897403\pi\)
\(594\) −2.55496 −0.104831
\(595\) −14.5362 −0.595925
\(596\) − 17.8756i − 0.732213i
\(597\) −24.7724 −1.01387
\(598\) 0 0
\(599\) 9.16985 0.374670 0.187335 0.982296i \(-0.440015\pi\)
0.187335 + 0.982296i \(0.440015\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −25.8864 −1.05593 −0.527963 0.849267i \(-0.677044\pi\)
−0.527963 + 0.849267i \(0.677044\pi\)
\(602\) −48.7241 −1.98584
\(603\) 0.929312i 0.0378445i
\(604\) 0.390748i 0.0158993i
\(605\) − 4.47219i − 0.181820i
\(606\) − 15.7453i − 0.639607i
\(607\) −11.6093 −0.471205 −0.235603 0.971850i \(-0.575706\pi\)
−0.235603 + 0.971850i \(0.575706\pi\)
\(608\) −3.89977 −0.158157
\(609\) − 24.3749i − 0.987723i
\(610\) −9.65279 −0.390830
\(611\) 0 0
\(612\) −3.02715 −0.122365
\(613\) 16.5985i 0.670407i 0.942146 + 0.335204i \(0.108805\pi\)
−0.942146 + 0.335204i \(0.891195\pi\)
\(614\) 11.5961 0.467981
\(615\) −5.89977 −0.237902
\(616\) − 12.2687i − 0.494322i
\(617\) − 33.4373i − 1.34613i −0.739582 0.673067i \(-0.764977\pi\)
0.739582 0.673067i \(-0.235023\pi\)
\(618\) 12.0858i 0.486160i
\(619\) − 25.4198i − 1.02171i −0.859667 0.510854i \(-0.829329\pi\)
0.859667 0.510854i \(-0.170671\pi\)
\(620\) −5.44504 −0.218678
\(621\) 1.64310 0.0659355
\(622\) 4.43429i 0.177799i
\(623\) 80.8007 3.23721
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 11.2881i − 0.451164i
\(627\) −9.96376 −0.397914
\(628\) −9.50902 −0.379451
\(629\) − 22.7512i − 0.907150i
\(630\) 4.80194i 0.191314i
\(631\) − 15.1879i − 0.604621i −0.953210 0.302310i \(-0.902242\pi\)
0.953210 0.302310i \(-0.0977579\pi\)
\(632\) − 5.40581i − 0.215032i
\(633\) 6.49396 0.258112
\(634\) −6.31229 −0.250693
\(635\) 5.58211i 0.221519i
\(636\) 6.91185 0.274073
\(637\) 0 0
\(638\) −12.9691 −0.513453
\(639\) 14.9487i 0.591361i
\(640\) 1.00000 0.0395285
\(641\) −16.8009 −0.663595 −0.331797 0.943351i \(-0.607655\pi\)
−0.331797 + 0.943351i \(0.607655\pi\)
\(642\) 4.64071i 0.183154i
\(643\) − 35.2597i − 1.39050i −0.718766 0.695252i \(-0.755293\pi\)
0.718766 0.695252i \(-0.244707\pi\)
\(644\) 7.89008i 0.310913i
\(645\) − 10.1468i − 0.399528i
\(646\) −11.8052 −0.464469
\(647\) 26.7985 1.05356 0.526778 0.850003i \(-0.323400\pi\)
0.526778 + 0.850003i \(0.323400\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −19.4413 −0.763139
\(650\) 0 0
\(651\) 26.1468 1.02477
\(652\) − 5.84846i − 0.229043i
\(653\) −10.8358 −0.424037 −0.212019 0.977266i \(-0.568004\pi\)
−0.212019 + 0.977266i \(0.568004\pi\)
\(654\) −4.20344 −0.164367
\(655\) 3.06100i 0.119603i
\(656\) − 5.89977i − 0.230347i
\(657\) − 14.1129i − 0.550597i
\(658\) 30.8649i 1.20324i
\(659\) 49.3870 1.92385 0.961923 0.273322i \(-0.0881223\pi\)
0.961923 + 0.273322i \(0.0881223\pi\)
\(660\) 2.55496 0.0994516
\(661\) − 2.18167i − 0.0848571i −0.999100 0.0424285i \(-0.986491\pi\)
0.999100 0.0424285i \(-0.0135095\pi\)
\(662\) 10.9554 0.425793
\(663\) 0 0
\(664\) −4.33513 −0.168236
\(665\) 18.7265i 0.726181i
\(666\) −7.51573 −0.291229
\(667\) 8.34050 0.322946
\(668\) − 2.55496i − 0.0988543i
\(669\) 20.1250i 0.778077i
\(670\) − 0.929312i − 0.0359025i
\(671\) − 24.6625i − 0.952085i
\(672\) −4.80194 −0.185239
\(673\) 24.2319 0.934072 0.467036 0.884238i \(-0.345322\pi\)
0.467036 + 0.884238i \(0.345322\pi\)
\(674\) 26.6668i 1.02717i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 24.0422 0.924017 0.462009 0.886875i \(-0.347129\pi\)
0.462009 + 0.886875i \(0.347129\pi\)
\(678\) 0.170915i 0.00656396i
\(679\) 86.3654 3.31440
\(680\) 3.02715 0.116086
\(681\) − 24.2379i − 0.928798i
\(682\) − 13.9119i − 0.532712i
\(683\) 24.3183i 0.930512i 0.885176 + 0.465256i \(0.154038\pi\)
−0.885176 + 0.465256i \(0.845962\pi\)
\(684\) 3.89977i 0.149112i
\(685\) 1.03385 0.0395014
\(686\) 43.4989 1.66079
\(687\) 2.71618i 0.103629i
\(688\) 10.1468 0.386841
\(689\) 0 0
\(690\) −1.64310 −0.0625519
\(691\) 18.2054i 0.692564i 0.938130 + 0.346282i \(0.112556\pi\)
−0.938130 + 0.346282i \(0.887444\pi\)
\(692\) −21.5308 −0.818478
\(693\) −12.2687 −0.466051
\(694\) − 25.0694i − 0.951620i
\(695\) 16.7995i 0.637243i
\(696\) 5.07606i 0.192408i
\(697\) − 17.8595i − 0.676476i
\(698\) 21.5623 0.816143
\(699\) 2.16421 0.0818580
\(700\) − 4.80194i − 0.181496i
\(701\) 16.0441 0.605978 0.302989 0.952994i \(-0.402015\pi\)
0.302989 + 0.952994i \(0.402015\pi\)
\(702\) 0 0
\(703\) −29.3096 −1.10543
\(704\) 2.55496i 0.0962936i
\(705\) −6.42758 −0.242077
\(706\) −28.2959 −1.06493
\(707\) − 75.6077i − 2.84352i
\(708\) 7.60925i 0.285973i
\(709\) 20.8025i 0.781255i 0.920549 + 0.390628i \(0.127742\pi\)
−0.920549 + 0.390628i \(0.872258\pi\)
\(710\) − 14.9487i − 0.561014i
\(711\) −5.40581 −0.202734
\(712\) −16.8267 −0.630607
\(713\) 8.94677i 0.335059i
\(714\) −14.5362 −0.544003
\(715\) 0 0
\(716\) 15.5483 0.581066
\(717\) 14.5114i 0.541939i
\(718\) −34.2717 −1.27901
\(719\) 40.3424 1.50452 0.752259 0.658867i \(-0.228964\pi\)
0.752259 + 0.658867i \(0.228964\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) 58.0350i 2.16134i
\(722\) − 3.79178i − 0.141115i
\(723\) 18.1142i 0.673675i
\(724\) 14.4058 0.535388
\(725\) −5.07606 −0.188520
\(726\) − 4.47219i − 0.165978i
\(727\) 47.2073 1.75082 0.875410 0.483380i \(-0.160591\pi\)
0.875410 + 0.483380i \(0.160591\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 14.1129i 0.522342i
\(731\) 30.7157 1.13606
\(732\) −9.65279 −0.356777
\(733\) 41.5424i 1.53440i 0.641406 + 0.767202i \(0.278352\pi\)
−0.641406 + 0.767202i \(0.721648\pi\)
\(734\) 23.4209i 0.864480i
\(735\) 16.0586i 0.592331i
\(736\) − 1.64310i − 0.0605656i
\(737\) 2.37435 0.0874605
\(738\) −5.89977 −0.217174
\(739\) − 26.3811i − 0.970443i −0.874391 0.485221i \(-0.838739\pi\)
0.874391 0.485221i \(-0.161261\pi\)
\(740\) 7.51573 0.276284
\(741\) 0 0
\(742\) 33.1903 1.21845
\(743\) − 20.2881i − 0.744299i −0.928173 0.372150i \(-0.878621\pi\)
0.928173 0.372150i \(-0.121379\pi\)
\(744\) −5.44504 −0.199625
\(745\) 17.8756 0.654912
\(746\) − 1.82371i − 0.0667707i
\(747\) 4.33513i 0.158614i
\(748\) 7.73423i 0.282792i
\(749\) 22.2844i 0.814254i
\(750\) 1.00000 0.0365148
\(751\) 21.1280 0.770970 0.385485 0.922714i \(-0.374034\pi\)
0.385485 + 0.922714i \(0.374034\pi\)
\(752\) − 6.42758i − 0.234390i
\(753\) −8.55257 −0.311673
\(754\) 0 0
\(755\) −0.390748 −0.0142208
\(756\) 4.80194i 0.174645i
\(757\) −17.3274 −0.629773 −0.314887 0.949129i \(-0.601967\pi\)
−0.314887 + 0.949129i \(0.601967\pi\)
\(758\) −15.3260 −0.556666
\(759\) − 4.19806i − 0.152380i
\(760\) − 3.89977i − 0.141460i
\(761\) − 35.4282i − 1.28427i −0.766591 0.642135i \(-0.778049\pi\)
0.766591 0.642135i \(-0.221951\pi\)
\(762\) 5.58211i 0.202218i
\(763\) −20.1847 −0.730733
\(764\) 4.02177 0.145503
\(765\) − 3.02715i − 0.109447i
\(766\) −29.6722 −1.07210
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 49.5062i 1.78524i 0.450812 + 0.892619i \(0.351134\pi\)
−0.450812 + 0.892619i \(0.648866\pi\)
\(770\) 12.2687 0.442135
\(771\) −1.30260 −0.0469121
\(772\) − 20.1008i − 0.723444i
\(773\) 27.9235i 1.00434i 0.864770 + 0.502169i \(0.167464\pi\)
−0.864770 + 0.502169i \(0.832536\pi\)
\(774\) − 10.1468i − 0.364717i
\(775\) − 5.44504i − 0.195592i
\(776\) −17.9855 −0.645643
\(777\) −36.0901 −1.29472
\(778\) 11.8291i 0.424093i
\(779\) −23.0078 −0.824339
\(780\) 0 0
\(781\) 38.1933 1.36666
\(782\) − 4.97392i − 0.177867i
\(783\) 5.07606 0.181404
\(784\) −16.0586 −0.573522
\(785\) − 9.50902i − 0.339392i
\(786\) 3.06100i 0.109182i
\(787\) − 28.7778i − 1.02582i −0.858443 0.512908i \(-0.828568\pi\)
0.858443 0.512908i \(-0.171432\pi\)
\(788\) 15.3884i 0.548187i
\(789\) 8.24027 0.293362
\(790\) 5.40581 0.192330
\(791\) 0.820724i 0.0291816i
\(792\) 2.55496 0.0907865
\(793\) 0 0
\(794\) −12.1153 −0.429956
\(795\) 6.91185i 0.245138i
\(796\) 24.7724 0.878034
\(797\) 14.6752 0.519821 0.259910 0.965633i \(-0.416307\pi\)
0.259910 + 0.965633i \(0.416307\pi\)
\(798\) 18.7265i 0.662910i
\(799\) − 19.4572i − 0.688348i
\(800\) 1.00000i 0.0353553i
\(801\) 16.8267i 0.594542i
\(802\) 10.8291 0.382388
\(803\) −36.0579 −1.27245
\(804\) − 0.929312i − 0.0327743i
\(805\) −7.89008 −0.278089
\(806\) 0 0
\(807\) 22.6601 0.797673
\(808\) 15.7453i 0.553916i
\(809\) −25.4946 −0.896341 −0.448170 0.893948i \(-0.647924\pi\)
−0.448170 + 0.893948i \(0.647924\pi\)
\(810\) −1.00000 −0.0351364
\(811\) − 8.99867i − 0.315986i −0.987440 0.157993i \(-0.949498\pi\)
0.987440 0.157993i \(-0.0505024\pi\)
\(812\) 24.3749i 0.855393i
\(813\) − 13.9172i − 0.488099i
\(814\) 19.2024i 0.673043i
\(815\) 5.84846 0.204863
\(816\) 3.02715 0.105971
\(817\) − 39.5700i − 1.38438i
\(818\) −33.9396 −1.18667
\(819\) 0 0
\(820\) 5.89977 0.206029
\(821\) − 37.1105i − 1.29517i −0.761995 0.647583i \(-0.775780\pi\)
0.761995 0.647583i \(-0.224220\pi\)
\(822\) 1.03385 0.0360597
\(823\) 11.1903 0.390069 0.195035 0.980796i \(-0.437518\pi\)
0.195035 + 0.980796i \(0.437518\pi\)
\(824\) − 12.0858i − 0.421027i
\(825\) 2.55496i 0.0889522i
\(826\) 36.5392i 1.27136i
\(827\) 27.7808i 0.966032i 0.875612 + 0.483016i \(0.160459\pi\)
−0.875612 + 0.483016i \(0.839541\pi\)
\(828\) −1.64310 −0.0571018
\(829\) 26.5633 0.922582 0.461291 0.887249i \(-0.347386\pi\)
0.461291 + 0.887249i \(0.347386\pi\)
\(830\) − 4.33513i − 0.150474i
\(831\) 24.2610 0.841604
\(832\) 0 0
\(833\) −48.6118 −1.68430
\(834\) 16.7995i 0.581721i
\(835\) 2.55496 0.0884180
\(836\) 9.96376 0.344604
\(837\) 5.44504i 0.188208i
\(838\) − 26.8562i − 0.927733i
\(839\) 3.92884i 0.135639i 0.997698 + 0.0678193i \(0.0216041\pi\)
−0.997698 + 0.0678193i \(0.978396\pi\)
\(840\) − 4.80194i − 0.165683i
\(841\) −3.23357 −0.111502
\(842\) 27.9928 0.964696
\(843\) − 10.5961i − 0.364949i
\(844\) −6.49396 −0.223531
\(845\) 0 0
\(846\) −6.42758 −0.220985
\(847\) − 21.4752i − 0.737896i
\(848\) −6.91185 −0.237354
\(849\) −9.86294 −0.338495
\(850\) 3.02715i 0.103830i
\(851\) − 12.3491i − 0.423323i
\(852\) − 14.9487i − 0.512134i
\(853\) 48.2097i 1.65067i 0.564645 + 0.825334i \(0.309013\pi\)
−0.564645 + 0.825334i \(0.690987\pi\)
\(854\) −46.3521 −1.58614
\(855\) −3.89977 −0.133369
\(856\) − 4.64071i − 0.158616i
\(857\) −41.2976 −1.41070 −0.705349 0.708860i \(-0.749210\pi\)
−0.705349 + 0.708860i \(0.749210\pi\)
\(858\) 0 0
\(859\) −42.1041 −1.43657 −0.718286 0.695748i \(-0.755073\pi\)
−0.718286 + 0.695748i \(0.755073\pi\)
\(860\) 10.1468i 0.346001i
\(861\) −28.3303 −0.965495
\(862\) −33.4088 −1.13791
\(863\) 45.0646i 1.53402i 0.641638 + 0.767008i \(0.278255\pi\)
−0.641638 + 0.767008i \(0.721745\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) − 21.5308i − 0.732069i
\(866\) − 13.6420i − 0.463575i
\(867\) −7.83638 −0.266137
\(868\) −26.1468 −0.887479
\(869\) 13.8116i 0.468527i
\(870\) −5.07606 −0.172095
\(871\) 0 0
\(872\) 4.20344 0.142346
\(873\) 17.9855i 0.608718i
\(874\) −6.40773 −0.216745
\(875\) 4.80194 0.162335
\(876\) 14.1129i 0.476831i
\(877\) − 33.4935i − 1.13099i −0.824750 0.565497i \(-0.808684\pi\)
0.824750 0.565497i \(-0.191316\pi\)
\(878\) − 30.6437i − 1.03417i
\(879\) − 5.68233i − 0.191660i
\(880\) −2.55496 −0.0861276
\(881\) −46.0431 −1.55123 −0.775615 0.631206i \(-0.782560\pi\)
−0.775615 + 0.631206i \(0.782560\pi\)
\(882\) 16.0586i 0.540721i
\(883\) 41.8629 1.40880 0.704400 0.709803i \(-0.251216\pi\)
0.704400 + 0.709803i \(0.251216\pi\)
\(884\) 0 0
\(885\) −7.60925 −0.255782
\(886\) 19.7409i 0.663210i
\(887\) 23.8866 0.802034 0.401017 0.916071i \(-0.368657\pi\)
0.401017 + 0.916071i \(0.368657\pi\)
\(888\) 7.51573 0.252211
\(889\) 26.8049i 0.899008i
\(890\) − 16.8267i − 0.564032i
\(891\) − 2.55496i − 0.0855943i
\(892\) − 20.1250i − 0.673834i
\(893\) −25.0661 −0.838805
\(894\) 17.8756 0.597850
\(895\) 15.5483i 0.519721i
\(896\) 4.80194 0.160421
\(897\) 0 0
\(898\) −36.2664 −1.21022
\(899\) 27.6394i 0.921825i
\(900\) 1.00000 0.0333333
\(901\) −20.9232 −0.697053
\(902\) 15.0737i 0.501898i
\(903\) − 48.7241i − 1.62144i
\(904\) − 0.170915i − 0.00568455i
\(905\) 14.4058i 0.478865i
\(906\) −0.390748 −0.0129817
\(907\) 20.4136 0.677822 0.338911 0.940818i \(-0.389941\pi\)
0.338911 + 0.940818i \(0.389941\pi\)
\(908\) 24.2379i 0.804362i
\(909\) 15.7453 0.522237
\(910\) 0 0
\(911\) 23.0108 0.762380 0.381190 0.924497i \(-0.375514\pi\)
0.381190 + 0.924497i \(0.375514\pi\)
\(912\) − 3.89977i − 0.129134i
\(913\) 11.0761 0.366564
\(914\) 3.55602 0.117623
\(915\) − 9.65279i − 0.319111i
\(916\) − 2.71618i − 0.0897453i
\(917\) 14.6987i 0.485395i
\(918\) − 3.02715i − 0.0999107i
\(919\) −38.3327 −1.26448 −0.632240 0.774773i \(-0.717864\pi\)
−0.632240 + 0.774773i \(0.717864\pi\)
\(920\) 1.64310 0.0541715
\(921\) 11.5961i 0.382105i
\(922\) 1.46011 0.0480861
\(923\) 0 0
\(924\) 12.2687 0.403612
\(925\) 7.51573i 0.247116i
\(926\) 13.7942 0.453304
\(927\) −12.0858 −0.396948
\(928\) − 5.07606i − 0.166630i
\(929\) 15.3515i 0.503667i 0.967771 + 0.251834i \(0.0810335\pi\)
−0.967771 + 0.251834i \(0.918966\pi\)
\(930\) − 5.44504i − 0.178550i
\(931\) 62.6249i 2.05245i
\(932\) −2.16421 −0.0708911
\(933\) −4.43429 −0.145172
\(934\) − 1.72886i − 0.0565699i
\(935\) −7.73423 −0.252936
\(936\) 0 0
\(937\) 51.3629 1.67795 0.838976 0.544169i \(-0.183155\pi\)
0.838976 + 0.544169i \(0.183155\pi\)
\(938\) − 4.46250i − 0.145706i
\(939\) 11.2881 0.368374
\(940\) 6.42758 0.209645
\(941\) − 17.2413i − 0.562052i −0.959700 0.281026i \(-0.909325\pi\)
0.959700 0.281026i \(-0.0906747\pi\)
\(942\) − 9.50902i − 0.309821i
\(943\) − 9.69394i − 0.315678i
\(944\) − 7.60925i − 0.247660i
\(945\) −4.80194 −0.156207
\(946\) −25.9245 −0.842879
\(947\) − 5.54586i − 0.180216i −0.995932 0.0901081i \(-0.971279\pi\)
0.995932 0.0901081i \(-0.0287213\pi\)
\(948\) 5.40581 0.175573
\(949\) 0 0
\(950\) 3.89977 0.126525
\(951\) − 6.31229i − 0.204690i
\(952\) 14.5362 0.471120
\(953\) 50.1584 1.62479 0.812394 0.583108i \(-0.198164\pi\)
0.812394 + 0.583108i \(0.198164\pi\)
\(954\) 6.91185i 0.223780i
\(955\) 4.02177i 0.130141i
\(956\) − 14.5114i − 0.469333i
\(957\) − 12.9691i − 0.419233i
\(958\) −14.5496 −0.470076
\(959\) 4.96449 0.160312
\(960\) 1.00000i 0.0322749i
\(961\) 1.35152 0.0435974
\(962\) 0 0
\(963\) −4.64071 −0.149545
\(964\) − 18.1142i − 0.583420i
\(965\) 20.1008 0.647068
\(966\) −7.89008 −0.253859
\(967\) − 13.6890i − 0.440210i −0.975476 0.220105i \(-0.929360\pi\)
0.975476 0.220105i \(-0.0706400\pi\)
\(968\) 4.47219i 0.143742i
\(969\) − 11.8052i − 0.379237i
\(970\) − 17.9855i − 0.577480i
\(971\) −50.6075 −1.62407 −0.812035 0.583608i \(-0.801640\pi\)
−0.812035 + 0.583608i \(0.801640\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 80.6704i 2.58617i
\(974\) 30.5526 0.978967
\(975\) 0 0
\(976\) 9.65279 0.308978
\(977\) − 11.2959i − 0.361388i −0.983539 0.180694i \(-0.942166\pi\)
0.983539 0.180694i \(-0.0578343\pi\)
\(978\) 5.84846 0.187013
\(979\) 42.9915 1.37401
\(980\) − 16.0586i − 0.512973i
\(981\) − 4.20344i − 0.134205i
\(982\) 9.68233i 0.308976i
\(983\) 36.8194i 1.17436i 0.809458 + 0.587178i \(0.199761\pi\)
−0.809458 + 0.587178i \(0.800239\pi\)
\(984\) 5.89977 0.188078
\(985\) −15.3884 −0.490314
\(986\) − 15.3660i − 0.489353i
\(987\) −30.8649 −0.982439
\(988\) 0 0
\(989\) 16.6722 0.530144
\(990\) 2.55496i 0.0812019i
\(991\) 10.8254 0.343879 0.171940 0.985108i \(-0.444997\pi\)
0.171940 + 0.985108i \(0.444997\pi\)
\(992\) 5.44504 0.172880
\(993\) 10.9554i 0.347659i
\(994\) − 71.7827i − 2.27681i
\(995\) 24.7724i 0.785338i
\(996\) − 4.33513i − 0.137364i
\(997\) −4.30319 −0.136283 −0.0681417 0.997676i \(-0.521707\pi\)
−0.0681417 + 0.997676i \(0.521707\pi\)
\(998\) 18.6025 0.588853
\(999\) − 7.51573i − 0.237787i
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.b.w.1351.1 6
13.5 odd 4 5070.2.a.bo.1.3 3
13.8 odd 4 5070.2.a.bx.1.1 yes 3
13.12 even 2 inner 5070.2.b.w.1351.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bo.1.3 3 13.5 odd 4
5070.2.a.bx.1.1 yes 3 13.8 odd 4
5070.2.b.w.1351.1 6 1.1 even 1 trivial
5070.2.b.w.1351.6 6 13.12 even 2 inner