Properties

Label 5070.2.b.s.1351.4
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 1351.4
Root \(-1.24698i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.s.1351.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -3.04892i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -3.04892i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +6.24698i q^{11} +1.00000 q^{12} +3.04892 q^{14} -1.00000i q^{15} +1.00000 q^{16} -2.69202 q^{17} +1.00000i q^{18} -5.82908i q^{19} -1.00000i q^{20} +3.04892i q^{21} -6.24698 q^{22} +5.62565 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +3.04892i q^{28} -5.14675 q^{29} +1.00000 q^{30} -3.53319i q^{31} +1.00000i q^{32} -6.24698i q^{33} -2.69202i q^{34} +3.04892 q^{35} -1.00000 q^{36} +4.65279i q^{37} +5.82908 q^{38} +1.00000 q^{40} -3.77479i q^{41} -3.04892 q^{42} -2.85086 q^{43} -6.24698i q^{44} +1.00000i q^{45} +5.62565i q^{46} +3.61596i q^{47} -1.00000 q^{48} -2.29590 q^{49} -1.00000i q^{50} +2.69202 q^{51} +0.664874 q^{53} -1.00000i q^{54} -6.24698 q^{55} -3.04892 q^{56} +5.82908i q^{57} -5.14675i q^{58} +12.9487i q^{59} +1.00000i q^{60} -7.91723 q^{61} +3.53319 q^{62} -3.04892i q^{63} -1.00000 q^{64} +6.24698 q^{66} -0.198062i q^{67} +2.69202 q^{68} -5.62565 q^{69} +3.04892i q^{70} -0.374354i q^{71} -1.00000i q^{72} -14.6136i q^{73} -4.65279 q^{74} +1.00000 q^{75} +5.82908i q^{76} +19.0465 q^{77} -11.2567 q^{79} +1.00000i q^{80} +1.00000 q^{81} +3.77479 q^{82} +6.83877i q^{83} -3.04892i q^{84} -2.69202i q^{85} -2.85086i q^{86} +5.14675 q^{87} +6.24698 q^{88} -9.50365i q^{89} -1.00000 q^{90} -5.62565 q^{92} +3.53319i q^{93} -3.61596 q^{94} +5.82908 q^{95} -1.00000i q^{96} -0.335126i q^{97} -2.29590i q^{98} +6.24698i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} - 6q^{10} + 6q^{12} + 6q^{16} - 6q^{17} - 28q^{22} + 10q^{23} - 6q^{25} - 6q^{27} + 24q^{29} + 6q^{30} - 6q^{36} + 14q^{38} + 6q^{40} + 10q^{43} - 6q^{48} + 14q^{49} + 6q^{51} + 6q^{53} - 28q^{55} - 34q^{61} + 28q^{62} - 6q^{64} + 28q^{66} + 6q^{68} - 10q^{69} + 8q^{74} + 6q^{75} + 14q^{77} - 14q^{79} + 6q^{81} + 26q^{82} - 24q^{87} + 28q^{88} - 6q^{90} - 10q^{92} - 42q^{94} + 14q^{95} + O(q^{100}) \)

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
\(7\) − 3.04892i − 1.15238i −0.817315 0.576191i \(-0.804538\pi\)
0.817315 0.576191i \(-0.195462\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 6.24698i 1.88354i 0.336264 + 0.941768i \(0.390836\pi\)
−0.336264 + 0.941768i \(0.609164\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 3.04892 0.814857
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) −2.69202 −0.652911 −0.326456 0.945213i \(-0.605854\pi\)
−0.326456 + 0.945213i \(0.605854\pi\)
\(18\) 1.00000i 0.235702i
\(19\) − 5.82908i − 1.33728i −0.743585 0.668642i \(-0.766876\pi\)
0.743585 0.668642i \(-0.233124\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) 3.04892i 0.665328i
\(22\) −6.24698 −1.33186
\(23\) 5.62565 1.17303 0.586514 0.809939i \(-0.300500\pi\)
0.586514 + 0.809939i \(0.300500\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 3.04892i 0.576191i
\(29\) −5.14675 −0.955728 −0.477864 0.878434i \(-0.658589\pi\)
−0.477864 + 0.878434i \(0.658589\pi\)
\(30\) 1.00000 0.182574
\(31\) − 3.53319i − 0.634579i −0.948329 0.317290i \(-0.897227\pi\)
0.948329 0.317290i \(-0.102773\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 6.24698i − 1.08746i
\(34\) − 2.69202i − 0.461678i
\(35\) 3.04892 0.515361
\(36\) −1.00000 −0.166667
\(37\) 4.65279i 0.764914i 0.923973 + 0.382457i \(0.124922\pi\)
−0.923973 + 0.382457i \(0.875078\pi\)
\(38\) 5.82908 0.945602
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) − 3.77479i − 0.589523i −0.955571 0.294762i \(-0.904760\pi\)
0.955571 0.294762i \(-0.0952402\pi\)
\(42\) −3.04892 −0.470458
\(43\) −2.85086 −0.434751 −0.217376 0.976088i \(-0.569750\pi\)
−0.217376 + 0.976088i \(0.569750\pi\)
\(44\) − 6.24698i − 0.941768i
\(45\) 1.00000i 0.149071i
\(46\) 5.62565i 0.829456i
\(47\) 3.61596i 0.527442i 0.964599 + 0.263721i \(0.0849497\pi\)
−0.964599 + 0.263721i \(0.915050\pi\)
\(48\) −1.00000 −0.144338
\(49\) −2.29590 −0.327985
\(50\) − 1.00000i − 0.141421i
\(51\) 2.69202 0.376958
\(52\) 0 0
\(53\) 0.664874 0.0913275 0.0456638 0.998957i \(-0.485460\pi\)
0.0456638 + 0.998957i \(0.485460\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −6.24698 −0.842343
\(56\) −3.04892 −0.407429
\(57\) 5.82908i 0.772081i
\(58\) − 5.14675i − 0.675802i
\(59\) 12.9487i 1.68578i 0.538089 + 0.842888i \(0.319146\pi\)
−0.538089 + 0.842888i \(0.680854\pi\)
\(60\) 1.00000i 0.129099i
\(61\) −7.91723 −1.01370 −0.506849 0.862035i \(-0.669190\pi\)
−0.506849 + 0.862035i \(0.669190\pi\)
\(62\) 3.53319 0.448715
\(63\) − 3.04892i − 0.384127i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 6.24698 0.768950
\(67\) − 0.198062i − 0.0241972i −0.999927 0.0120986i \(-0.996149\pi\)
0.999927 0.0120986i \(-0.00385119\pi\)
\(68\) 2.69202 0.326456
\(69\) −5.62565 −0.677248
\(70\) 3.04892i 0.364415i
\(71\) − 0.374354i − 0.0444277i −0.999753 0.0222138i \(-0.992929\pi\)
0.999753 0.0222138i \(-0.00707147\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 14.6136i − 1.71039i −0.518308 0.855194i \(-0.673438\pi\)
0.518308 0.855194i \(-0.326562\pi\)
\(74\) −4.65279 −0.540876
\(75\) 1.00000 0.115470
\(76\) 5.82908i 0.668642i
\(77\) 19.0465 2.17055
\(78\) 0 0
\(79\) −11.2567 −1.26647 −0.633237 0.773958i \(-0.718274\pi\)
−0.633237 + 0.773958i \(0.718274\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 3.77479 0.416856
\(83\) 6.83877i 0.750653i 0.926893 + 0.375326i \(0.122469\pi\)
−0.926893 + 0.375326i \(0.877531\pi\)
\(84\) − 3.04892i − 0.332664i
\(85\) − 2.69202i − 0.291991i
\(86\) − 2.85086i − 0.307416i
\(87\) 5.14675 0.551790
\(88\) 6.24698 0.665930
\(89\) − 9.50365i − 1.00738i −0.863883 0.503692i \(-0.831975\pi\)
0.863883 0.503692i \(-0.168025\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −5.62565 −0.586514
\(93\) 3.53319i 0.366375i
\(94\) −3.61596 −0.372957
\(95\) 5.82908 0.598051
\(96\) − 1.00000i − 0.102062i
\(97\) − 0.335126i − 0.0340268i −0.999855 0.0170134i \(-0.994584\pi\)
0.999855 0.0170134i \(-0.00541580\pi\)
\(98\) − 2.29590i − 0.231921i
\(99\) 6.24698i 0.627845i
\(100\) 1.00000 0.100000
\(101\) −0.274127 −0.0272766 −0.0136383 0.999907i \(-0.504341\pi\)
−0.0136383 + 0.999907i \(0.504341\pi\)
\(102\) 2.69202i 0.266550i
\(103\) 9.00000 0.886796 0.443398 0.896325i \(-0.353773\pi\)
0.443398 + 0.896325i \(0.353773\pi\)
\(104\) 0 0
\(105\) −3.04892 −0.297544
\(106\) 0.664874i 0.0645783i
\(107\) −10.8509 −1.04899 −0.524496 0.851413i \(-0.675746\pi\)
−0.524496 + 0.851413i \(0.675746\pi\)
\(108\) 1.00000 0.0962250
\(109\) − 1.02177i − 0.0978678i −0.998802 0.0489339i \(-0.984418\pi\)
0.998802 0.0489339i \(-0.0155824\pi\)
\(110\) − 6.24698i − 0.595626i
\(111\) − 4.65279i − 0.441624i
\(112\) − 3.04892i − 0.288096i
\(113\) 19.6383 1.84742 0.923709 0.383095i \(-0.125142\pi\)
0.923709 + 0.383095i \(0.125142\pi\)
\(114\) −5.82908 −0.545944
\(115\) 5.62565i 0.524594i
\(116\) 5.14675 0.477864
\(117\) 0 0
\(118\) −12.9487 −1.19202
\(119\) 8.20775i 0.752403i
\(120\) −1.00000 −0.0912871
\(121\) −28.0248 −2.54770
\(122\) − 7.91723i − 0.716792i
\(123\) 3.77479i 0.340361i
\(124\) 3.53319i 0.317290i
\(125\) − 1.00000i − 0.0894427i
\(126\) 3.04892 0.271619
\(127\) −18.5187 −1.64327 −0.821635 0.570014i \(-0.806938\pi\)
−0.821635 + 0.570014i \(0.806938\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 2.85086 0.251004
\(130\) 0 0
\(131\) −12.2349 −1.06897 −0.534484 0.845179i \(-0.679494\pi\)
−0.534484 + 0.845179i \(0.679494\pi\)
\(132\) 6.24698i 0.543730i
\(133\) −17.7724 −1.54106
\(134\) 0.198062 0.0171100
\(135\) − 1.00000i − 0.0860663i
\(136\) 2.69202i 0.230839i
\(137\) 19.3002i 1.64893i 0.565914 + 0.824464i \(0.308523\pi\)
−0.565914 + 0.824464i \(0.691477\pi\)
\(138\) − 5.62565i − 0.478887i
\(139\) 19.5133 1.65510 0.827550 0.561392i \(-0.189734\pi\)
0.827550 + 0.561392i \(0.189734\pi\)
\(140\) −3.04892 −0.257681
\(141\) − 3.61596i − 0.304519i
\(142\) 0.374354 0.0314151
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 5.14675i − 0.427414i
\(146\) 14.6136 1.20943
\(147\) 2.29590 0.189362
\(148\) − 4.65279i − 0.382457i
\(149\) − 18.0248i − 1.47665i −0.674448 0.738323i \(-0.735618\pi\)
0.674448 0.738323i \(-0.264382\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) − 5.30127i − 0.431412i −0.976458 0.215706i \(-0.930795\pi\)
0.976458 0.215706i \(-0.0692052\pi\)
\(152\) −5.82908 −0.472801
\(153\) −2.69202 −0.217637
\(154\) 19.0465i 1.53481i
\(155\) 3.53319 0.283792
\(156\) 0 0
\(157\) 7.41789 0.592012 0.296006 0.955186i \(-0.404345\pi\)
0.296006 + 0.955186i \(0.404345\pi\)
\(158\) − 11.2567i − 0.895532i
\(159\) −0.664874 −0.0527280
\(160\) −1.00000 −0.0790569
\(161\) − 17.1521i − 1.35178i
\(162\) 1.00000i 0.0785674i
\(163\) − 24.0151i − 1.88101i −0.339787 0.940503i \(-0.610355\pi\)
0.339787 0.940503i \(-0.389645\pi\)
\(164\) 3.77479i 0.294762i
\(165\) 6.24698 0.486327
\(166\) −6.83877 −0.530792
\(167\) − 11.9922i − 0.927987i −0.885839 0.463993i \(-0.846416\pi\)
0.885839 0.463993i \(-0.153584\pi\)
\(168\) 3.04892 0.235229
\(169\) 0 0
\(170\) 2.69202 0.206469
\(171\) − 5.82908i − 0.445761i
\(172\) 2.85086 0.217376
\(173\) −19.8823 −1.51162 −0.755812 0.654788i \(-0.772758\pi\)
−0.755812 + 0.654788i \(0.772758\pi\)
\(174\) 5.14675i 0.390174i
\(175\) 3.04892i 0.230476i
\(176\) 6.24698i 0.470884i
\(177\) − 12.9487i − 0.973283i
\(178\) 9.50365 0.712329
\(179\) −16.7453 −1.25160 −0.625799 0.779984i \(-0.715227\pi\)
−0.625799 + 0.779984i \(0.715227\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) 3.93900 0.292784 0.146392 0.989227i \(-0.453234\pi\)
0.146392 + 0.989227i \(0.453234\pi\)
\(182\) 0 0
\(183\) 7.91723 0.585259
\(184\) − 5.62565i − 0.414728i
\(185\) −4.65279 −0.342080
\(186\) −3.53319 −0.259066
\(187\) − 16.8170i − 1.22978i
\(188\) − 3.61596i − 0.263721i
\(189\) 3.04892i 0.221776i
\(190\) 5.82908i 0.422886i
\(191\) −27.1540 −1.96480 −0.982399 0.186795i \(-0.940190\pi\)
−0.982399 + 0.186795i \(0.940190\pi\)
\(192\) 1.00000 0.0721688
\(193\) − 15.3937i − 1.10807i −0.832495 0.554033i \(-0.813088\pi\)
0.832495 0.554033i \(-0.186912\pi\)
\(194\) 0.335126 0.0240606
\(195\) 0 0
\(196\) 2.29590 0.163993
\(197\) − 23.5036i − 1.67457i −0.546770 0.837283i \(-0.684143\pi\)
0.546770 0.837283i \(-0.315857\pi\)
\(198\) −6.24698 −0.443954
\(199\) 20.8442 1.47760 0.738801 0.673923i \(-0.235392\pi\)
0.738801 + 0.673923i \(0.235392\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 0.198062i 0.0139702i
\(202\) − 0.274127i − 0.0192875i
\(203\) 15.6920i 1.10136i
\(204\) −2.69202 −0.188479
\(205\) 3.77479 0.263643
\(206\) 9.00000i 0.627060i
\(207\) 5.62565 0.391009
\(208\) 0 0
\(209\) 36.4142 2.51882
\(210\) − 3.04892i − 0.210395i
\(211\) −19.0073 −1.30852 −0.654258 0.756271i \(-0.727019\pi\)
−0.654258 + 0.756271i \(0.727019\pi\)
\(212\) −0.664874 −0.0456638
\(213\) 0.374354i 0.0256503i
\(214\) − 10.8509i − 0.741749i
\(215\) − 2.85086i − 0.194427i
\(216\) 1.00000i 0.0680414i
\(217\) −10.7724 −0.731278
\(218\) 1.02177 0.0692030
\(219\) 14.6136i 0.987493i
\(220\) 6.24698 0.421171
\(221\) 0 0
\(222\) 4.65279 0.312275
\(223\) − 20.9433i − 1.40247i −0.712931 0.701234i \(-0.752633\pi\)
0.712931 0.701234i \(-0.247367\pi\)
\(224\) 3.04892 0.203714
\(225\) −1.00000 −0.0666667
\(226\) 19.6383i 1.30632i
\(227\) − 19.0978i − 1.26757i −0.773510 0.633784i \(-0.781501\pi\)
0.773510 0.633784i \(-0.218499\pi\)
\(228\) − 5.82908i − 0.386041i
\(229\) 9.33513i 0.616882i 0.951243 + 0.308441i \(0.0998073\pi\)
−0.951243 + 0.308441i \(0.900193\pi\)
\(230\) −5.62565 −0.370944
\(231\) −19.0465 −1.25317
\(232\) 5.14675i 0.337901i
\(233\) 14.3599 0.940747 0.470374 0.882467i \(-0.344119\pi\)
0.470374 + 0.882467i \(0.344119\pi\)
\(234\) 0 0
\(235\) −3.61596 −0.235879
\(236\) − 12.9487i − 0.842888i
\(237\) 11.2567 0.731199
\(238\) −8.20775 −0.532029
\(239\) − 27.7482i − 1.79488i −0.441132 0.897442i \(-0.645423\pi\)
0.441132 0.897442i \(-0.354577\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) − 14.5579i − 0.937759i −0.883262 0.468880i \(-0.844658\pi\)
0.883262 0.468880i \(-0.155342\pi\)
\(242\) − 28.0248i − 1.80150i
\(243\) −1.00000 −0.0641500
\(244\) 7.91723 0.506849
\(245\) − 2.29590i − 0.146679i
\(246\) −3.77479 −0.240672
\(247\) 0 0
\(248\) −3.53319 −0.224358
\(249\) − 6.83877i − 0.433390i
\(250\) 1.00000 0.0632456
\(251\) −27.8756 −1.75949 −0.879746 0.475443i \(-0.842288\pi\)
−0.879746 + 0.475443i \(0.842288\pi\)
\(252\) 3.04892i 0.192064i
\(253\) 35.1433i 2.20944i
\(254\) − 18.5187i − 1.16197i
\(255\) 2.69202i 0.168581i
\(256\) 1.00000 0.0625000
\(257\) 5.64848 0.352343 0.176171 0.984360i \(-0.443629\pi\)
0.176171 + 0.984360i \(0.443629\pi\)
\(258\) 2.85086i 0.177486i
\(259\) 14.1860 0.881474
\(260\) 0 0
\(261\) −5.14675 −0.318576
\(262\) − 12.2349i − 0.755875i
\(263\) 6.06100 0.373737 0.186869 0.982385i \(-0.440166\pi\)
0.186869 + 0.982385i \(0.440166\pi\)
\(264\) −6.24698 −0.384475
\(265\) 0.664874i 0.0408429i
\(266\) − 17.7724i − 1.08970i
\(267\) 9.50365i 0.581614i
\(268\) 0.198062i 0.0120986i
\(269\) 8.19136 0.499436 0.249718 0.968319i \(-0.419662\pi\)
0.249718 + 0.968319i \(0.419662\pi\)
\(270\) 1.00000 0.0608581
\(271\) − 12.8006i − 0.777582i −0.921326 0.388791i \(-0.872893\pi\)
0.921326 0.388791i \(-0.127107\pi\)
\(272\) −2.69202 −0.163228
\(273\) 0 0
\(274\) −19.3002 −1.16597
\(275\) − 6.24698i − 0.376707i
\(276\) 5.62565 0.338624
\(277\) 7.88338 0.473666 0.236833 0.971550i \(-0.423891\pi\)
0.236833 + 0.971550i \(0.423891\pi\)
\(278\) 19.5133i 1.17033i
\(279\) − 3.53319i − 0.211526i
\(280\) − 3.04892i − 0.182208i
\(281\) − 7.16182i − 0.427238i −0.976917 0.213619i \(-0.931475\pi\)
0.976917 0.213619i \(-0.0685251\pi\)
\(282\) 3.61596 0.215327
\(283\) −11.7342 −0.697528 −0.348764 0.937211i \(-0.613398\pi\)
−0.348764 + 0.937211i \(0.613398\pi\)
\(284\) 0.374354i 0.0222138i
\(285\) −5.82908 −0.345285
\(286\) 0 0
\(287\) −11.5090 −0.679356
\(288\) 1.00000i 0.0589256i
\(289\) −9.75302 −0.573707
\(290\) 5.14675 0.302228
\(291\) 0.335126i 0.0196454i
\(292\) 14.6136i 0.855194i
\(293\) 26.4456i 1.54497i 0.635033 + 0.772485i \(0.280987\pi\)
−0.635033 + 0.772485i \(0.719013\pi\)
\(294\) 2.29590i 0.133899i
\(295\) −12.9487 −0.753902
\(296\) 4.65279 0.270438
\(297\) − 6.24698i − 0.362487i
\(298\) 18.0248 1.04415
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 8.69202i 0.501000i
\(302\) 5.30127 0.305054
\(303\) 0.274127 0.0157482
\(304\) − 5.82908i − 0.334321i
\(305\) − 7.91723i − 0.453339i
\(306\) − 2.69202i − 0.153893i
\(307\) 10.8562i 0.619598i 0.950802 + 0.309799i \(0.100262\pi\)
−0.950802 + 0.309799i \(0.899738\pi\)
\(308\) −19.0465 −1.08528
\(309\) −9.00000 −0.511992
\(310\) 3.53319i 0.200672i
\(311\) −11.5453 −0.654672 −0.327336 0.944908i \(-0.606151\pi\)
−0.327336 + 0.944908i \(0.606151\pi\)
\(312\) 0 0
\(313\) 18.4862 1.04490 0.522451 0.852670i \(-0.325018\pi\)
0.522451 + 0.852670i \(0.325018\pi\)
\(314\) 7.41789i 0.418616i
\(315\) 3.04892 0.171787
\(316\) 11.2567 0.633237
\(317\) − 3.05131i − 0.171379i −0.996322 0.0856893i \(-0.972691\pi\)
0.996322 0.0856893i \(-0.0273093\pi\)
\(318\) − 0.664874i − 0.0372843i
\(319\) − 32.1517i − 1.80015i
\(320\) − 1.00000i − 0.0559017i
\(321\) 10.8509 0.605636
\(322\) 17.1521 0.955851
\(323\) 15.6920i 0.873127i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 24.0151 1.33007
\(327\) 1.02177i 0.0565040i
\(328\) −3.77479 −0.208428
\(329\) 11.0248 0.607814
\(330\) 6.24698i 0.343885i
\(331\) − 12.1575i − 0.668237i −0.942531 0.334118i \(-0.891561\pi\)
0.942531 0.334118i \(-0.108439\pi\)
\(332\) − 6.83877i − 0.375326i
\(333\) 4.65279i 0.254971i
\(334\) 11.9922 0.656186
\(335\) 0.198062 0.0108213
\(336\) 3.04892i 0.166332i
\(337\) −6.48188 −0.353090 −0.176545 0.984293i \(-0.556492\pi\)
−0.176545 + 0.984293i \(0.556492\pi\)
\(338\) 0 0
\(339\) −19.6383 −1.06661
\(340\) 2.69202i 0.145995i
\(341\) 22.0718 1.19525
\(342\) 5.82908 0.315201
\(343\) − 14.3424i − 0.774418i
\(344\) 2.85086i 0.153708i
\(345\) − 5.62565i − 0.302875i
\(346\) − 19.8823i − 1.06888i
\(347\) 19.1933 1.03035 0.515175 0.857085i \(-0.327727\pi\)
0.515175 + 0.857085i \(0.327727\pi\)
\(348\) −5.14675 −0.275895
\(349\) 11.5767i 0.619688i 0.950787 + 0.309844i \(0.100277\pi\)
−0.950787 + 0.309844i \(0.899723\pi\)
\(350\) −3.04892 −0.162971
\(351\) 0 0
\(352\) −6.24698 −0.332965
\(353\) − 21.6963i − 1.15478i −0.816469 0.577390i \(-0.804071\pi\)
0.816469 0.577390i \(-0.195929\pi\)
\(354\) 12.9487 0.688215
\(355\) 0.374354 0.0198687
\(356\) 9.50365i 0.503692i
\(357\) − 8.20775i − 0.434400i
\(358\) − 16.7453i − 0.885014i
\(359\) − 13.0476i − 0.688625i −0.938855 0.344313i \(-0.888112\pi\)
0.938855 0.344313i \(-0.111888\pi\)
\(360\) 1.00000 0.0527046
\(361\) −14.9782 −0.788328
\(362\) 3.93900i 0.207029i
\(363\) 28.0248 1.47092
\(364\) 0 0
\(365\) 14.6136 0.764909
\(366\) 7.91723i 0.413840i
\(367\) 1.76377 0.0920683 0.0460341 0.998940i \(-0.485342\pi\)
0.0460341 + 0.998940i \(0.485342\pi\)
\(368\) 5.62565 0.293257
\(369\) − 3.77479i − 0.196508i
\(370\) − 4.65279i − 0.241887i
\(371\) − 2.02715i − 0.105244i
\(372\) − 3.53319i − 0.183187i
\(373\) 15.5013 0.802625 0.401312 0.915941i \(-0.368554\pi\)
0.401312 + 0.915941i \(0.368554\pi\)
\(374\) 16.8170 0.869587
\(375\) 1.00000i 0.0516398i
\(376\) 3.61596 0.186479
\(377\) 0 0
\(378\) −3.04892 −0.156819
\(379\) 8.62863i 0.443223i 0.975135 + 0.221611i \(0.0711317\pi\)
−0.975135 + 0.221611i \(0.928868\pi\)
\(380\) −5.82908 −0.299026
\(381\) 18.5187 0.948742
\(382\) − 27.1540i − 1.38932i
\(383\) 31.1008i 1.58918i 0.607148 + 0.794589i \(0.292314\pi\)
−0.607148 + 0.794589i \(0.707686\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 19.0465i 0.970701i
\(386\) 15.3937 0.783520
\(387\) −2.85086 −0.144917
\(388\) 0.335126i 0.0170134i
\(389\) −7.07069 −0.358498 −0.179249 0.983804i \(-0.557367\pi\)
−0.179249 + 0.983804i \(0.557367\pi\)
\(390\) 0 0
\(391\) −15.1444 −0.765883
\(392\) 2.29590i 0.115960i
\(393\) 12.2349 0.617169
\(394\) 23.5036 1.18410
\(395\) − 11.2567i − 0.566384i
\(396\) − 6.24698i − 0.313923i
\(397\) 28.5948i 1.43513i 0.696491 + 0.717565i \(0.254744\pi\)
−0.696491 + 0.717565i \(0.745256\pi\)
\(398\) 20.8442i 1.04482i
\(399\) 17.7724 0.889733
\(400\) −1.00000 −0.0500000
\(401\) 11.8267i 0.590597i 0.955405 + 0.295298i \(0.0954191\pi\)
−0.955405 + 0.295298i \(0.904581\pi\)
\(402\) −0.198062 −0.00987845
\(403\) 0 0
\(404\) 0.274127 0.0136383
\(405\) 1.00000i 0.0496904i
\(406\) −15.6920 −0.778782
\(407\) −29.0659 −1.44074
\(408\) − 2.69202i − 0.133275i
\(409\) − 9.00000i − 0.445021i −0.974930 0.222511i \(-0.928575\pi\)
0.974930 0.222511i \(-0.0714252\pi\)
\(410\) 3.77479i 0.186424i
\(411\) − 19.3002i − 0.952009i
\(412\) −9.00000 −0.443398
\(413\) 39.4795 1.94266
\(414\) 5.62565i 0.276485i
\(415\) −6.83877 −0.335702
\(416\) 0 0
\(417\) −19.5133 −0.955572
\(418\) 36.4142i 1.78108i
\(419\) 19.7735 0.965997 0.482998 0.875621i \(-0.339548\pi\)
0.482998 + 0.875621i \(0.339548\pi\)
\(420\) 3.04892 0.148772
\(421\) − 21.0127i − 1.02409i −0.858957 0.512047i \(-0.828887\pi\)
0.858957 0.512047i \(-0.171113\pi\)
\(422\) − 19.0073i − 0.925261i
\(423\) 3.61596i 0.175814i
\(424\) − 0.664874i − 0.0322892i
\(425\) 2.69202 0.130582
\(426\) −0.374354 −0.0181375
\(427\) 24.1390i 1.16817i
\(428\) 10.8509 0.524496
\(429\) 0 0
\(430\) 2.85086 0.137480
\(431\) − 26.5633i − 1.27951i −0.768579 0.639755i \(-0.779036\pi\)
0.768579 0.639755i \(-0.220964\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 14.7047 0.706662 0.353331 0.935498i \(-0.385049\pi\)
0.353331 + 0.935498i \(0.385049\pi\)
\(434\) − 10.7724i − 0.517092i
\(435\) 5.14675i 0.246768i
\(436\) 1.02177i 0.0489339i
\(437\) − 32.7924i − 1.56867i
\(438\) −14.6136 −0.698263
\(439\) 28.2640 1.34897 0.674483 0.738291i \(-0.264367\pi\)
0.674483 + 0.738291i \(0.264367\pi\)
\(440\) 6.24698i 0.297813i
\(441\) −2.29590 −0.109328
\(442\) 0 0
\(443\) 20.5109 0.974504 0.487252 0.873261i \(-0.337999\pi\)
0.487252 + 0.873261i \(0.337999\pi\)
\(444\) 4.65279i 0.220812i
\(445\) 9.50365 0.450516
\(446\) 20.9433 0.991695
\(447\) 18.0248i 0.852542i
\(448\) 3.04892i 0.144048i
\(449\) 3.19806i 0.150926i 0.997149 + 0.0754629i \(0.0240435\pi\)
−0.997149 + 0.0754629i \(0.975957\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) 23.5810 1.11039
\(452\) −19.6383 −0.923709
\(453\) 5.30127i 0.249076i
\(454\) 19.0978 0.896306
\(455\) 0 0
\(456\) 5.82908 0.272972
\(457\) 8.65710i 0.404962i 0.979286 + 0.202481i \(0.0649005\pi\)
−0.979286 + 0.202481i \(0.935100\pi\)
\(458\) −9.33513 −0.436202
\(459\) 2.69202 0.125653
\(460\) − 5.62565i − 0.262297i
\(461\) − 11.1239i − 0.518092i −0.965865 0.259046i \(-0.916592\pi\)
0.965865 0.259046i \(-0.0834082\pi\)
\(462\) − 19.0465i − 0.886125i
\(463\) 33.0659i 1.53670i 0.640028 + 0.768351i \(0.278923\pi\)
−0.640028 + 0.768351i \(0.721077\pi\)
\(464\) −5.14675 −0.238932
\(465\) −3.53319 −0.163848
\(466\) 14.3599i 0.665209i
\(467\) −15.3937 −0.712337 −0.356168 0.934422i \(-0.615917\pi\)
−0.356168 + 0.934422i \(0.615917\pi\)
\(468\) 0 0
\(469\) −0.603875 −0.0278844
\(470\) − 3.61596i − 0.166792i
\(471\) −7.41789 −0.341799
\(472\) 12.9487 0.596012
\(473\) − 17.8092i − 0.818869i
\(474\) 11.2567i 0.517036i
\(475\) 5.82908i 0.267457i
\(476\) − 8.20775i − 0.376202i
\(477\) 0.664874 0.0304425
\(478\) 27.7482 1.26917
\(479\) − 42.2137i − 1.92879i −0.264459 0.964397i \(-0.585193\pi\)
0.264459 0.964397i \(-0.414807\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 14.5579 0.663096
\(483\) 17.1521i 0.780449i
\(484\) 28.0248 1.27385
\(485\) 0.335126 0.0152173
\(486\) − 1.00000i − 0.0453609i
\(487\) 14.9903i 0.679276i 0.940556 + 0.339638i \(0.110305\pi\)
−0.940556 + 0.339638i \(0.889695\pi\)
\(488\) 7.91723i 0.358396i
\(489\) 24.0151i 1.08600i
\(490\) 2.29590 0.103718
\(491\) 21.3706 0.964443 0.482222 0.876049i \(-0.339830\pi\)
0.482222 + 0.876049i \(0.339830\pi\)
\(492\) − 3.77479i − 0.170181i
\(493\) 13.8552 0.624005
\(494\) 0 0
\(495\) −6.24698 −0.280781
\(496\) − 3.53319i − 0.158645i
\(497\) −1.14138 −0.0511977
\(498\) 6.83877 0.306453
\(499\) − 26.6160i − 1.19149i −0.803172 0.595747i \(-0.796856\pi\)
0.803172 0.595747i \(-0.203144\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 11.9922i 0.535773i
\(502\) − 27.8756i − 1.24415i
\(503\) −25.2218 −1.12458 −0.562291 0.826939i \(-0.690080\pi\)
−0.562291 + 0.826939i \(0.690080\pi\)
\(504\) −3.04892 −0.135810
\(505\) − 0.274127i − 0.0121985i
\(506\) −35.1433 −1.56231
\(507\) 0 0
\(508\) 18.5187 0.821635
\(509\) − 17.3797i − 0.770343i −0.922845 0.385172i \(-0.874142\pi\)
0.922845 0.385172i \(-0.125858\pi\)
\(510\) −2.69202 −0.119205
\(511\) −44.5555 −1.97102
\(512\) 1.00000i 0.0441942i
\(513\) 5.82908i 0.257360i
\(514\) 5.64848i 0.249144i
\(515\) 9.00000i 0.396587i
\(516\) −2.85086 −0.125502
\(517\) −22.5888 −0.993455
\(518\) 14.1860i 0.623296i
\(519\) 19.8823 0.872737
\(520\) 0 0
\(521\) 8.27114 0.362365 0.181183 0.983449i \(-0.442007\pi\)
0.181183 + 0.983449i \(0.442007\pi\)
\(522\) − 5.14675i − 0.225267i
\(523\) −40.9154 −1.78911 −0.894553 0.446961i \(-0.852506\pi\)
−0.894553 + 0.446961i \(0.852506\pi\)
\(524\) 12.2349 0.534484
\(525\) − 3.04892i − 0.133066i
\(526\) 6.06100i 0.264272i
\(527\) 9.51142i 0.414324i
\(528\) − 6.24698i − 0.271865i
\(529\) 8.64789 0.375995
\(530\) −0.664874 −0.0288803
\(531\) 12.9487i 0.561925i
\(532\) 17.7724 0.770531
\(533\) 0 0
\(534\) −9.50365 −0.411263
\(535\) − 10.8509i − 0.469123i
\(536\) −0.198062 −0.00855499
\(537\) 16.7453 0.722611
\(538\) 8.19136i 0.353154i
\(539\) − 14.3424i − 0.617772i
\(540\) 1.00000i 0.0430331i
\(541\) − 27.2252i − 1.17050i −0.810852 0.585252i \(-0.800996\pi\)
0.810852 0.585252i \(-0.199004\pi\)
\(542\) 12.8006 0.549833
\(543\) −3.93900 −0.169039
\(544\) − 2.69202i − 0.115419i
\(545\) 1.02177 0.0437678
\(546\) 0 0
\(547\) −32.7023 −1.39825 −0.699125 0.715000i \(-0.746427\pi\)
−0.699125 + 0.715000i \(0.746427\pi\)
\(548\) − 19.3002i − 0.824464i
\(549\) −7.91723 −0.337899
\(550\) 6.24698 0.266372
\(551\) 30.0009i 1.27808i
\(552\) 5.62565i 0.239443i
\(553\) 34.3207i 1.45946i
\(554\) 7.88338i 0.334933i
\(555\) 4.65279 0.197500
\(556\) −19.5133 −0.827550
\(557\) − 0.225209i − 0.00954243i −0.999989 0.00477121i \(-0.998481\pi\)
0.999989 0.00477121i \(-0.00151873\pi\)
\(558\) 3.53319 0.149572
\(559\) 0 0
\(560\) 3.04892 0.128840
\(561\) 16.8170i 0.710014i
\(562\) 7.16182 0.302103
\(563\) 9.55735 0.402794 0.201397 0.979510i \(-0.435452\pi\)
0.201397 + 0.979510i \(0.435452\pi\)
\(564\) 3.61596i 0.152259i
\(565\) 19.6383i 0.826190i
\(566\) − 11.7342i − 0.493227i
\(567\) − 3.04892i − 0.128042i
\(568\) −0.374354 −0.0157076
\(569\) −34.8455 −1.46080 −0.730399 0.683020i \(-0.760666\pi\)
−0.730399 + 0.683020i \(0.760666\pi\)
\(570\) − 5.82908i − 0.244153i
\(571\) −36.5472 −1.52945 −0.764726 0.644355i \(-0.777126\pi\)
−0.764726 + 0.644355i \(0.777126\pi\)
\(572\) 0 0
\(573\) 27.1540 1.13438
\(574\) − 11.5090i − 0.480377i
\(575\) −5.62565 −0.234606
\(576\) −1.00000 −0.0416667
\(577\) 18.3284i 0.763022i 0.924364 + 0.381511i \(0.124596\pi\)
−0.924364 + 0.381511i \(0.875404\pi\)
\(578\) − 9.75302i − 0.405672i
\(579\) 15.3937i 0.639742i
\(580\) 5.14675i 0.213707i
\(581\) 20.8509 0.865039
\(582\) −0.335126 −0.0138914
\(583\) 4.15346i 0.172019i
\(584\) −14.6136 −0.604714
\(585\) 0 0
\(586\) −26.4456 −1.09246
\(587\) − 17.3937i − 0.717916i −0.933354 0.358958i \(-0.883132\pi\)
0.933354 0.358958i \(-0.116868\pi\)
\(588\) −2.29590 −0.0946812
\(589\) −20.5953 −0.848613
\(590\) − 12.9487i − 0.533089i
\(591\) 23.5036i 0.966811i
\(592\) 4.65279i 0.191229i
\(593\) 19.5362i 0.802254i 0.916022 + 0.401127i \(0.131381\pi\)
−0.916022 + 0.401127i \(0.868619\pi\)
\(594\) 6.24698 0.256317
\(595\) −8.20775 −0.336485
\(596\) 18.0248i 0.738323i
\(597\) −20.8442 −0.853094
\(598\) 0 0
\(599\) −14.6364 −0.598027 −0.299014 0.954249i \(-0.596658\pi\)
−0.299014 + 0.954249i \(0.596658\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) 33.3980 1.36233 0.681167 0.732128i \(-0.261473\pi\)
0.681167 + 0.732128i \(0.261473\pi\)
\(602\) −8.69202 −0.354260
\(603\) − 0.198062i − 0.00806572i
\(604\) 5.30127i 0.215706i
\(605\) − 28.0248i − 1.13937i
\(606\) 0.274127i 0.0111356i
\(607\) 29.9694 1.21642 0.608210 0.793776i \(-0.291888\pi\)
0.608210 + 0.793776i \(0.291888\pi\)
\(608\) 5.82908 0.236401
\(609\) − 15.6920i − 0.635873i
\(610\) 7.91723 0.320559
\(611\) 0 0
\(612\) 2.69202 0.108819
\(613\) 1.58748i 0.0641178i 0.999486 + 0.0320589i \(0.0102064\pi\)
−0.999486 + 0.0320589i \(0.989794\pi\)
\(614\) −10.8562 −0.438122
\(615\) −3.77479 −0.152214
\(616\) − 19.0465i − 0.767406i
\(617\) − 12.9511i − 0.521391i −0.965421 0.260695i \(-0.916048\pi\)
0.965421 0.260695i \(-0.0839519\pi\)
\(618\) − 9.00000i − 0.362033i
\(619\) − 23.9554i − 0.962849i −0.876488 0.481424i \(-0.840120\pi\)
0.876488 0.481424i \(-0.159880\pi\)
\(620\) −3.53319 −0.141896
\(621\) −5.62565 −0.225749
\(622\) − 11.5453i − 0.462923i
\(623\) −28.9758 −1.16089
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 18.4862i 0.738857i
\(627\) −36.4142 −1.45424
\(628\) −7.41789 −0.296006
\(629\) − 12.5254i − 0.499421i
\(630\) 3.04892i 0.121472i
\(631\) − 35.4601i − 1.41164i −0.708389 0.705822i \(-0.750578\pi\)
0.708389 0.705822i \(-0.249422\pi\)
\(632\) 11.2567i 0.447766i
\(633\) 19.0073 0.755472
\(634\) 3.05131 0.121183
\(635\) − 18.5187i − 0.734893i
\(636\) 0.664874 0.0263640
\(637\) 0 0
\(638\) 32.1517 1.27290
\(639\) − 0.374354i − 0.0148092i
\(640\) 1.00000 0.0395285
\(641\) −16.8358 −0.664974 −0.332487 0.943108i \(-0.607888\pi\)
−0.332487 + 0.943108i \(0.607888\pi\)
\(642\) 10.8509i 0.428249i
\(643\) 18.4838i 0.728930i 0.931217 + 0.364465i \(0.118748\pi\)
−0.931217 + 0.364465i \(0.881252\pi\)
\(644\) 17.1521i 0.675889i
\(645\) 2.85086i 0.112252i
\(646\) −15.6920 −0.617394
\(647\) 1.20237 0.0472702 0.0236351 0.999721i \(-0.492476\pi\)
0.0236351 + 0.999721i \(0.492476\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −80.8902 −3.17522
\(650\) 0 0
\(651\) 10.7724 0.422204
\(652\) 24.0151i 0.940503i
\(653\) −36.9530 −1.44608 −0.723041 0.690805i \(-0.757256\pi\)
−0.723041 + 0.690805i \(0.757256\pi\)
\(654\) −1.02177 −0.0399544
\(655\) − 12.2349i − 0.478057i
\(656\) − 3.77479i − 0.147381i
\(657\) − 14.6136i − 0.570129i
\(658\) 11.0248i 0.429790i
\(659\) −26.4494 −1.03032 −0.515160 0.857094i \(-0.672268\pi\)
−0.515160 + 0.857094i \(0.672268\pi\)
\(660\) −6.24698 −0.243163
\(661\) 2.21073i 0.0859876i 0.999075 + 0.0429938i \(0.0136896\pi\)
−0.999075 + 0.0429938i \(0.986310\pi\)
\(662\) 12.1575 0.472515
\(663\) 0 0
\(664\) 6.83877 0.265396
\(665\) − 17.7724i − 0.689184i
\(666\) −4.65279 −0.180292
\(667\) −28.9538 −1.12110
\(668\) 11.9922i 0.463993i
\(669\) 20.9433i 0.809715i
\(670\) 0.198062i 0.00765181i
\(671\) − 49.4588i − 1.90934i
\(672\) −3.04892 −0.117615
\(673\) −16.5555 −0.638170 −0.319085 0.947726i \(-0.603375\pi\)
−0.319085 + 0.947726i \(0.603375\pi\)
\(674\) − 6.48188i − 0.249673i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −8.53617 −0.328072 −0.164036 0.986454i \(-0.552451\pi\)
−0.164036 + 0.986454i \(0.552451\pi\)
\(678\) − 19.6383i − 0.754205i
\(679\) −1.02177 −0.0392119
\(680\) −2.69202 −0.103234
\(681\) 19.0978i 0.731831i
\(682\) 22.0718i 0.845171i
\(683\) − 0.204767i − 0.00783519i −0.999992 0.00391760i \(-0.998753\pi\)
0.999992 0.00391760i \(-0.00124701\pi\)
\(684\) 5.82908i 0.222881i
\(685\) −19.3002 −0.737423
\(686\) 14.3424 0.547596
\(687\) − 9.33513i − 0.356157i
\(688\) −2.85086 −0.108688
\(689\) 0 0
\(690\) 5.62565 0.214165
\(691\) 33.5187i 1.27511i 0.770404 + 0.637556i \(0.220055\pi\)
−0.770404 + 0.637556i \(0.779945\pi\)
\(692\) 19.8823 0.755812
\(693\) 19.0465 0.723518
\(694\) 19.1933i 0.728567i
\(695\) 19.5133i 0.740183i
\(696\) − 5.14675i − 0.195087i
\(697\) 10.1618i 0.384906i
\(698\) −11.5767 −0.438186
\(699\) −14.3599 −0.543141
\(700\) − 3.04892i − 0.115238i
\(701\) −19.0271 −0.718645 −0.359323 0.933213i \(-0.616992\pi\)
−0.359323 + 0.933213i \(0.616992\pi\)
\(702\) 0 0
\(703\) 27.1215 1.02291
\(704\) − 6.24698i − 0.235442i
\(705\) 3.61596 0.136185
\(706\) 21.6963 0.816552
\(707\) 0.835790i 0.0314331i
\(708\) 12.9487i 0.486642i
\(709\) − 42.5760i − 1.59897i −0.600683 0.799487i \(-0.705105\pi\)
0.600683 0.799487i \(-0.294895\pi\)
\(710\) 0.374354i 0.0140493i
\(711\) −11.2567 −0.422158
\(712\) −9.50365 −0.356164
\(713\) − 19.8765i − 0.744379i
\(714\) 8.20775 0.307167
\(715\) 0 0
\(716\) 16.7453 0.625799
\(717\) 27.7482i 1.03628i
\(718\) 13.0476 0.486932
\(719\) −7.51632 −0.280311 −0.140156 0.990129i \(-0.544760\pi\)
−0.140156 + 0.990129i \(0.544760\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) − 27.4403i − 1.02193i
\(722\) − 14.9782i − 0.557432i
\(723\) 14.5579i 0.541416i
\(724\) −3.93900 −0.146392
\(725\) 5.14675 0.191146
\(726\) 28.0248i 1.04010i
\(727\) −39.6558 −1.47075 −0.735376 0.677660i \(-0.762994\pi\)
−0.735376 + 0.677660i \(0.762994\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 14.6136i 0.540872i
\(731\) 7.67456 0.283854
\(732\) −7.91723 −0.292629
\(733\) 24.4886i 0.904506i 0.891890 + 0.452253i \(0.149380\pi\)
−0.891890 + 0.452253i \(0.850620\pi\)
\(734\) 1.76377i 0.0651021i
\(735\) 2.29590i 0.0846854i
\(736\) 5.62565i 0.207364i
\(737\) 1.23729 0.0455762
\(738\) 3.77479 0.138952
\(739\) 41.9842i 1.54441i 0.635371 + 0.772207i \(0.280847\pi\)
−0.635371 + 0.772207i \(0.719153\pi\)
\(740\) 4.65279 0.171040
\(741\) 0 0
\(742\) 2.02715 0.0744189
\(743\) 29.7711i 1.09219i 0.837722 + 0.546097i \(0.183887\pi\)
−0.837722 + 0.546097i \(0.816113\pi\)
\(744\) 3.53319 0.129533
\(745\) 18.0248 0.660376
\(746\) 15.5013i 0.567541i
\(747\) 6.83877i 0.250218i
\(748\) 16.8170i 0.614891i
\(749\) 33.0834i 1.20884i
\(750\) −1.00000 −0.0365148
\(751\) −35.4999 −1.29541 −0.647705 0.761891i \(-0.724271\pi\)
−0.647705 + 0.761891i \(0.724271\pi\)
\(752\) 3.61596i 0.131860i
\(753\) 27.8756 1.01584
\(754\) 0 0
\(755\) 5.30127 0.192933
\(756\) − 3.04892i − 0.110888i
\(757\) 2.73258 0.0993172 0.0496586 0.998766i \(-0.484187\pi\)
0.0496586 + 0.998766i \(0.484187\pi\)
\(758\) −8.62863 −0.313406
\(759\) − 35.1433i − 1.27562i
\(760\) − 5.82908i − 0.211443i
\(761\) 30.2349i 1.09601i 0.836474 + 0.548007i \(0.184613\pi\)
−0.836474 + 0.548007i \(0.815387\pi\)
\(762\) 18.5187i 0.670862i
\(763\) −3.11529 −0.112781
\(764\) 27.1540 0.982399
\(765\) − 2.69202i − 0.0973302i
\(766\) −31.1008 −1.12372
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) − 35.3183i − 1.27361i −0.771025 0.636804i \(-0.780256\pi\)
0.771025 0.636804i \(-0.219744\pi\)
\(770\) −19.0465 −0.686389
\(771\) −5.64848 −0.203425
\(772\) 15.3937i 0.554033i
\(773\) − 32.0551i − 1.15294i −0.817117 0.576472i \(-0.804429\pi\)
0.817117 0.576472i \(-0.195571\pi\)
\(774\) − 2.85086i − 0.102472i
\(775\) 3.53319i 0.126916i
\(776\) −0.335126 −0.0120303
\(777\) −14.1860 −0.508919
\(778\) − 7.07069i − 0.253496i
\(779\) −22.0036 −0.788360
\(780\) 0 0
\(781\) 2.33858 0.0836811
\(782\) − 15.1444i − 0.541561i
\(783\) 5.14675 0.183930
\(784\) −2.29590 −0.0819963
\(785\) 7.41789i 0.264756i
\(786\) 12.2349i 0.436404i
\(787\) 43.9235i 1.56570i 0.622209 + 0.782851i \(0.286235\pi\)
−0.622209 + 0.782851i \(0.713765\pi\)
\(788\) 23.5036i 0.837283i
\(789\) −6.06100 −0.215777
\(790\) 11.2567 0.400494
\(791\) − 59.8756i − 2.12893i
\(792\) 6.24698 0.221977
\(793\) 0 0
\(794\) −28.5948 −1.01479
\(795\) − 0.664874i − 0.0235807i
\(796\) −20.8442 −0.738801
\(797\) 26.4523 0.936990 0.468495 0.883466i \(-0.344796\pi\)
0.468495 + 0.883466i \(0.344796\pi\)
\(798\) 17.7724i 0.629136i
\(799\) − 9.73423i − 0.344372i
\(800\) − 1.00000i − 0.0353553i
\(801\) − 9.50365i − 0.335795i
\(802\) −11.8267 −0.417615
\(803\) 91.2906 3.22158
\(804\) − 0.198062i − 0.00698512i
\(805\) 17.1521 0.604533
\(806\) 0 0
\(807\) −8.19136 −0.288349
\(808\) 0.274127i 0.00964374i
\(809\) 41.4634 1.45777 0.728887 0.684634i \(-0.240038\pi\)
0.728887 + 0.684634i \(0.240038\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 24.9095i 0.874689i 0.899294 + 0.437345i \(0.144081\pi\)
−0.899294 + 0.437345i \(0.855919\pi\)
\(812\) − 15.6920i − 0.550682i
\(813\) 12.8006i 0.448937i
\(814\) − 29.0659i − 1.01876i
\(815\) 24.0151 0.841211
\(816\) 2.69202 0.0942396
\(817\) 16.6179i 0.581386i
\(818\) 9.00000 0.314678
\(819\) 0 0
\(820\) −3.77479 −0.131821
\(821\) 3.11901i 0.108854i 0.998518 + 0.0544272i \(0.0173333\pi\)
−0.998518 + 0.0544272i \(0.982667\pi\)
\(822\) 19.3002 0.673172
\(823\) −36.4077 −1.26909 −0.634547 0.772884i \(-0.718813\pi\)
−0.634547 + 0.772884i \(0.718813\pi\)
\(824\) − 9.00000i − 0.313530i
\(825\) 6.24698i 0.217492i
\(826\) 39.4795i 1.37367i
\(827\) 8.04785i 0.279851i 0.990162 + 0.139926i \(0.0446863\pi\)
−0.990162 + 0.139926i \(0.955314\pi\)
\(828\) −5.62565 −0.195505
\(829\) −28.7006 −0.996815 −0.498407 0.866943i \(-0.666082\pi\)
−0.498407 + 0.866943i \(0.666082\pi\)
\(830\) − 6.83877i − 0.237377i
\(831\) −7.88338 −0.273471
\(832\) 0 0
\(833\) 6.18060 0.214145
\(834\) − 19.5133i − 0.675692i
\(835\) 11.9922 0.415008
\(836\) −36.4142 −1.25941
\(837\) 3.53319i 0.122125i
\(838\) 19.7735i 0.683063i
\(839\) 34.8491i 1.20312i 0.798827 + 0.601561i \(0.205455\pi\)
−0.798827 + 0.601561i \(0.794545\pi\)
\(840\) 3.04892i 0.105198i
\(841\) −2.51094 −0.0865843
\(842\) 21.0127 0.724145
\(843\) 7.16182i 0.246666i
\(844\) 19.0073 0.654258
\(845\) 0 0
\(846\) −3.61596 −0.124319
\(847\) 85.4452i 2.93593i
\(848\) 0.664874 0.0228319
\(849\) 11.7342 0.402718
\(850\) 2.69202i 0.0923356i
\(851\) 26.1750i 0.897266i
\(852\) − 0.374354i − 0.0128252i
\(853\) − 0.0779834i − 0.00267010i −0.999999 0.00133505i \(-0.999575\pi\)
0.999999 0.00133505i \(-0.000424960\pi\)
\(854\) −24.1390 −0.826019
\(855\) 5.82908 0.199350
\(856\) 10.8509i 0.370875i
\(857\) −18.3811 −0.627885 −0.313943 0.949442i \(-0.601650\pi\)
−0.313943 + 0.949442i \(0.601650\pi\)
\(858\) 0 0
\(859\) 0.871297 0.0297283 0.0148641 0.999890i \(-0.495268\pi\)
0.0148641 + 0.999890i \(0.495268\pi\)
\(860\) 2.85086i 0.0972134i
\(861\) 11.5090 0.392227
\(862\) 26.5633 0.904750
\(863\) 1.88172i 0.0640546i 0.999487 + 0.0320273i \(0.0101963\pi\)
−0.999487 + 0.0320273i \(0.989804\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) − 19.8823i − 0.676019i
\(866\) 14.7047i 0.499686i
\(867\) 9.75302 0.331230
\(868\) 10.7724 0.365639
\(869\) − 70.3202i − 2.38545i
\(870\) −5.14675 −0.174491
\(871\) 0 0
\(872\) −1.02177 −0.0346015
\(873\) − 0.335126i − 0.0113423i
\(874\) 32.7924 1.10922
\(875\) −3.04892 −0.103072
\(876\) − 14.6136i − 0.493747i
\(877\) 46.6859i 1.57647i 0.615374 + 0.788236i \(0.289005\pi\)
−0.615374 + 0.788236i \(0.710995\pi\)
\(878\) 28.2640i 0.953863i
\(879\) − 26.4456i − 0.891989i
\(880\) −6.24698 −0.210586
\(881\) −51.9821 −1.75132 −0.875660 0.482928i \(-0.839573\pi\)
−0.875660 + 0.482928i \(0.839573\pi\)
\(882\) − 2.29590i − 0.0773069i
\(883\) −22.3889 −0.753448 −0.376724 0.926326i \(-0.622949\pi\)
−0.376724 + 0.926326i \(0.622949\pi\)
\(884\) 0 0
\(885\) 12.9487 0.435265
\(886\) 20.5109i 0.689079i
\(887\) −20.7332 −0.696152 −0.348076 0.937466i \(-0.613165\pi\)
−0.348076 + 0.937466i \(0.613165\pi\)
\(888\) −4.65279 −0.156138
\(889\) 56.4620i 1.89368i
\(890\) 9.50365i 0.318563i
\(891\) 6.24698i 0.209282i
\(892\) 20.9433i 0.701234i
\(893\) 21.0777 0.705339
\(894\) −18.0248 −0.602838
\(895\) − 16.7453i − 0.559732i
\(896\) −3.04892 −0.101857
\(897\) 0 0
\(898\) −3.19806 −0.106721
\(899\) 18.1844i 0.606485i
\(900\) 1.00000 0.0333333
\(901\) −1.78986 −0.0596288
\(902\) 23.5810i 0.785163i
\(903\) − 8.69202i − 0.289252i
\(904\) − 19.6383i − 0.653161i
\(905\) 3.93900i 0.130937i
\(906\) −5.30127 −0.176123
\(907\) 9.27365 0.307927 0.153963 0.988077i \(-0.450796\pi\)
0.153963 + 0.988077i \(0.450796\pi\)
\(908\) 19.0978i 0.633784i
\(909\) −0.274127 −0.00909221
\(910\) 0 0
\(911\) −3.87071 −0.128242 −0.0641211 0.997942i \(-0.520424\pi\)
−0.0641211 + 0.997942i \(0.520424\pi\)
\(912\) 5.82908i 0.193020i
\(913\) −42.7217 −1.41388
\(914\) −8.65710 −0.286352
\(915\) 7.91723i 0.261736i
\(916\) − 9.33513i − 0.308441i
\(917\) 37.3032i 1.23186i
\(918\) 2.69202i 0.0888499i
\(919\) −33.2644 −1.09729 −0.548646 0.836055i \(-0.684857\pi\)
−0.548646 + 0.836055i \(0.684857\pi\)
\(920\) 5.62565 0.185472
\(921\) − 10.8562i − 0.357725i
\(922\) 11.1239 0.366347
\(923\) 0 0
\(924\) 19.0465 0.626585
\(925\) − 4.65279i − 0.152983i
\(926\) −33.0659 −1.08661
\(927\) 9.00000 0.295599
\(928\) − 5.14675i − 0.168950i
\(929\) − 1.92095i − 0.0630244i −0.999503 0.0315122i \(-0.989968\pi\)
0.999503 0.0315122i \(-0.0100323\pi\)
\(930\) − 3.53319i − 0.115858i
\(931\) 13.3830i 0.438609i
\(932\) −14.3599 −0.470374
\(933\) 11.5453 0.377975
\(934\) − 15.3937i − 0.503698i
\(935\) 16.8170 0.549975
\(936\) 0 0
\(937\) 26.6886 0.871877 0.435939 0.899976i \(-0.356416\pi\)
0.435939 + 0.899976i \(0.356416\pi\)
\(938\) − 0.603875i − 0.0197172i
\(939\) −18.4862 −0.603274
\(940\) 3.61596 0.117940
\(941\) 2.69309i 0.0877921i 0.999036 + 0.0438961i \(0.0139770\pi\)
−0.999036 + 0.0438961i \(0.986023\pi\)
\(942\) − 7.41789i − 0.241688i
\(943\) − 21.2356i − 0.691527i
\(944\) 12.9487i 0.421444i
\(945\) −3.04892 −0.0991813
\(946\) 17.8092 0.579028
\(947\) 23.1226i 0.751383i 0.926745 + 0.375692i \(0.122595\pi\)
−0.926745 + 0.375692i \(0.877405\pi\)
\(948\) −11.2567 −0.365600
\(949\) 0 0
\(950\) −5.82908 −0.189120
\(951\) 3.05131i 0.0989455i
\(952\) 8.20775 0.266015
\(953\) 31.7313 1.02788 0.513938 0.857827i \(-0.328186\pi\)
0.513938 + 0.857827i \(0.328186\pi\)
\(954\) 0.664874i 0.0215261i
\(955\) − 27.1540i − 0.878684i
\(956\) 27.7482i 0.897442i
\(957\) 32.1517i 1.03932i
\(958\) 42.2137 1.36386
\(959\) 58.8447 1.90020
\(960\) 1.00000i 0.0322749i
\(961\) 18.5166 0.597309
\(962\) 0 0
\(963\) −10.8509 −0.349664
\(964\) 14.5579i 0.468880i
\(965\) 15.3937 0.495542
\(966\) −17.1521 −0.551861
\(967\) − 29.7095i − 0.955392i −0.878525 0.477696i \(-0.841472\pi\)
0.878525 0.477696i \(-0.158528\pi\)
\(968\) 28.0248i 0.900750i
\(969\) − 15.6920i − 0.504100i
\(970\) 0.335126i 0.0107602i
\(971\) −50.0428 −1.60595 −0.802975 0.596013i \(-0.796751\pi\)
−0.802975 + 0.596013i \(0.796751\pi\)
\(972\) 1.00000 0.0320750
\(973\) − 59.4946i − 1.90731i
\(974\) −14.9903 −0.480321
\(975\) 0 0
\(976\) −7.91723 −0.253424
\(977\) − 15.4800i − 0.495248i −0.968856 0.247624i \(-0.920350\pi\)
0.968856 0.247624i \(-0.0796497\pi\)
\(978\) −24.0151 −0.767917
\(979\) 59.3691 1.89744
\(980\) 2.29590i 0.0733397i
\(981\) − 1.02177i − 0.0326226i
\(982\) 21.3706i 0.681964i
\(983\) − 24.3080i − 0.775304i −0.921806 0.387652i \(-0.873286\pi\)
0.921806 0.387652i \(-0.126714\pi\)
\(984\) 3.77479 0.120336
\(985\) 23.5036 0.748888
\(986\) 13.8552i 0.441238i
\(987\) −11.0248 −0.350922
\(988\) 0 0
\(989\) −16.0379 −0.509976
\(990\) − 6.24698i − 0.198542i
\(991\) 41.1196 1.30621 0.653104 0.757269i \(-0.273467\pi\)
0.653104 + 0.757269i \(0.273467\pi\)
\(992\) 3.53319 0.112179
\(993\) 12.1575i 0.385807i
\(994\) − 1.14138i − 0.0362022i
\(995\) 20.8442i 0.660804i
\(996\) 6.83877i 0.216695i
\(997\) 8.75063 0.277135 0.138568 0.990353i \(-0.455750\pi\)
0.138568 + 0.990353i \(0.455750\pi\)
\(998\) 26.6160 0.842513
\(999\) − 4.65279i − 0.147208i
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.s.1351.4 6
13.5 odd 4 5070.2.a.bt.1.3 yes 3
13.8 odd 4 5070.2.a.bk.1.1 3
13.12 even 2 inner 5070.2.b.s.1351.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bk.1.1 3 13.8 odd 4
5070.2.a.bt.1.3 yes 3 13.5 odd 4
5070.2.b.s.1351.3 6 13.12 even 2 inner
5070.2.b.s.1351.4 6 1.1 even 1 trivial