Properties

Label 5070.2.a.bu.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.15883 q^{11} -1.00000 q^{12} -1.69202 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.35690 q^{17} +1.00000 q^{18} +0.198062 q^{19} +1.00000 q^{20} +1.69202 q^{21} +2.15883 q^{22} -3.74094 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.69202 q^{28} +1.29590 q^{29} -1.00000 q^{30} +1.44504 q^{31} +1.00000 q^{32} -2.15883 q^{33} +2.35690 q^{34} -1.69202 q^{35} +1.00000 q^{36} +0.801938 q^{37} +0.198062 q^{38} +1.00000 q^{40} -1.89977 q^{41} +1.69202 q^{42} +12.5429 q^{43} +2.15883 q^{44} +1.00000 q^{45} -3.74094 q^{46} +8.87800 q^{47} -1.00000 q^{48} -4.13706 q^{49} +1.00000 q^{50} -2.35690 q^{51} -1.00969 q^{53} -1.00000 q^{54} +2.15883 q^{55} -1.69202 q^{56} -0.198062 q^{57} +1.29590 q^{58} +3.73125 q^{59} -1.00000 q^{60} -6.32304 q^{61} +1.44504 q^{62} -1.69202 q^{63} +1.00000 q^{64} -2.15883 q^{66} -7.56465 q^{67} +2.35690 q^{68} +3.74094 q^{69} -1.69202 q^{70} -4.18060 q^{71} +1.00000 q^{72} +11.9366 q^{73} +0.801938 q^{74} -1.00000 q^{75} +0.198062 q^{76} -3.65279 q^{77} +9.40581 q^{79} +1.00000 q^{80} +1.00000 q^{81} -1.89977 q^{82} +8.43296 q^{83} +1.69202 q^{84} +2.35690 q^{85} +12.5429 q^{86} -1.29590 q^{87} +2.15883 q^{88} -2.41119 q^{89} +1.00000 q^{90} -3.74094 q^{92} -1.44504 q^{93} +8.87800 q^{94} +0.198062 q^{95} -1.00000 q^{96} -0.0881460 q^{97} -4.13706 q^{98} +2.15883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 3q^{8} + 3q^{9} + 3q^{10} - 2q^{11} - 3q^{12} - 3q^{15} + 3q^{16} + 3q^{17} + 3q^{18} + 5q^{19} + 3q^{20} - 2q^{22} + 3q^{23} - 3q^{24} + 3q^{25} - 3q^{27} - 10q^{29} - 3q^{30} + 4q^{31} + 3q^{32} + 2q^{33} + 3q^{34} + 3q^{36} - 2q^{37} + 5q^{38} + 3q^{40} + 17q^{41} + 19q^{43} - 2q^{44} + 3q^{45} + 3q^{46} + 7q^{47} - 3q^{48} - 7q^{49} + 3q^{50} - 3q^{51} + 19q^{53} - 3q^{54} - 2q^{55} - 5q^{57} - 10q^{58} + 19q^{59} - 3q^{60} + q^{61} + 4q^{62} + 3q^{64} + 2q^{66} - q^{67} + 3q^{68} - 3q^{69} - q^{71} + 3q^{72} - 15q^{73} - 2q^{74} - 3q^{75} + 5q^{76} + 7q^{77} + 15q^{79} + 3q^{80} + 3q^{81} + 17q^{82} + 6q^{83} + 3q^{85} + 19q^{86} + 10q^{87} - 2q^{88} + 9q^{89} + 3q^{90} + 3q^{92} - 4q^{93} + 7q^{94} + 5q^{95} - 3q^{96} - 4q^{97} - 7q^{98} - 2q^{99} + O(q^{100}) \)

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
\(7\) −1.69202 −0.639524 −0.319762 0.947498i \(-0.603603\pi\)
−0.319762 + 0.947498i \(0.603603\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 2.15883 0.650913 0.325456 0.945557i \(-0.394482\pi\)
0.325456 + 0.945557i \(0.394482\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.69202 −0.452212
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 2.35690 0.571631 0.285816 0.958285i \(-0.407736\pi\)
0.285816 + 0.958285i \(0.407736\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.198062 0.0454386 0.0227193 0.999742i \(-0.492768\pi\)
0.0227193 + 0.999742i \(0.492768\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.69202 0.369229
\(22\) 2.15883 0.460265
\(23\) −3.74094 −0.780040 −0.390020 0.920806i \(-0.627532\pi\)
−0.390020 + 0.920806i \(0.627532\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.69202 −0.319762
\(29\) 1.29590 0.240642 0.120321 0.992735i \(-0.461608\pi\)
0.120321 + 0.992735i \(0.461608\pi\)
\(30\) −1.00000 −0.182574
\(31\) 1.44504 0.259537 0.129769 0.991544i \(-0.458577\pi\)
0.129769 + 0.991544i \(0.458577\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.15883 −0.375805
\(34\) 2.35690 0.404204
\(35\) −1.69202 −0.286004
\(36\) 1.00000 0.166667
\(37\) 0.801938 0.131838 0.0659189 0.997825i \(-0.479002\pi\)
0.0659189 + 0.997825i \(0.479002\pi\)
\(38\) 0.198062 0.0321299
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −1.89977 −0.296695 −0.148347 0.988935i \(-0.547395\pi\)
−0.148347 + 0.988935i \(0.547395\pi\)
\(42\) 1.69202 0.261085
\(43\) 12.5429 1.91277 0.956385 0.292108i \(-0.0943566\pi\)
0.956385 + 0.292108i \(0.0943566\pi\)
\(44\) 2.15883 0.325456
\(45\) 1.00000 0.149071
\(46\) −3.74094 −0.551571
\(47\) 8.87800 1.29499 0.647495 0.762070i \(-0.275817\pi\)
0.647495 + 0.762070i \(0.275817\pi\)
\(48\) −1.00000 −0.144338
\(49\) −4.13706 −0.591009
\(50\) 1.00000 0.141421
\(51\) −2.35690 −0.330031
\(52\) 0 0
\(53\) −1.00969 −0.138691 −0.0693457 0.997593i \(-0.522091\pi\)
−0.0693457 + 0.997593i \(0.522091\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.15883 0.291097
\(56\) −1.69202 −0.226106
\(57\) −0.198062 −0.0262340
\(58\) 1.29590 0.170160
\(59\) 3.73125 0.485767 0.242884 0.970055i \(-0.421907\pi\)
0.242884 + 0.970055i \(0.421907\pi\)
\(60\) −1.00000 −0.129099
\(61\) −6.32304 −0.809583 −0.404791 0.914409i \(-0.632656\pi\)
−0.404791 + 0.914409i \(0.632656\pi\)
\(62\) 1.44504 0.183521
\(63\) −1.69202 −0.213175
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.15883 −0.265734
\(67\) −7.56465 −0.924169 −0.462084 0.886836i \(-0.652898\pi\)
−0.462084 + 0.886836i \(0.652898\pi\)
\(68\) 2.35690 0.285816
\(69\) 3.74094 0.450356
\(70\) −1.69202 −0.202235
\(71\) −4.18060 −0.496146 −0.248073 0.968741i \(-0.579797\pi\)
−0.248073 + 0.968741i \(0.579797\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.9366 1.39707 0.698537 0.715574i \(-0.253835\pi\)
0.698537 + 0.715574i \(0.253835\pi\)
\(74\) 0.801938 0.0932234
\(75\) −1.00000 −0.115470
\(76\) 0.198062 0.0227193
\(77\) −3.65279 −0.416274
\(78\) 0 0
\(79\) 9.40581 1.05824 0.529118 0.848548i \(-0.322523\pi\)
0.529118 + 0.848548i \(0.322523\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −1.89977 −0.209795
\(83\) 8.43296 0.925638 0.462819 0.886453i \(-0.346838\pi\)
0.462819 + 0.886453i \(0.346838\pi\)
\(84\) 1.69202 0.184615
\(85\) 2.35690 0.255641
\(86\) 12.5429 1.35253
\(87\) −1.29590 −0.138935
\(88\) 2.15883 0.230132
\(89\) −2.41119 −0.255586 −0.127793 0.991801i \(-0.540789\pi\)
−0.127793 + 0.991801i \(0.540789\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −3.74094 −0.390020
\(93\) −1.44504 −0.149844
\(94\) 8.87800 0.915696
\(95\) 0.198062 0.0203208
\(96\) −1.00000 −0.102062
\(97\) −0.0881460 −0.00894987 −0.00447494 0.999990i \(-0.501424\pi\)
−0.00447494 + 0.999990i \(0.501424\pi\)
\(98\) −4.13706 −0.417907
\(99\) 2.15883 0.216971
\(100\) 1.00000 0.100000
\(101\) −11.1099 −1.10548 −0.552739 0.833354i \(-0.686417\pi\)
−0.552739 + 0.833354i \(0.686417\pi\)
\(102\) −2.35690 −0.233367
\(103\) 9.54825 0.940817 0.470409 0.882449i \(-0.344107\pi\)
0.470409 + 0.882449i \(0.344107\pi\)
\(104\) 0 0
\(105\) 1.69202 0.165124
\(106\) −1.00969 −0.0980696
\(107\) 18.1468 1.75431 0.877156 0.480205i \(-0.159438\pi\)
0.877156 + 0.480205i \(0.159438\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.24698 −0.215222 −0.107611 0.994193i \(-0.534320\pi\)
−0.107611 + 0.994193i \(0.534320\pi\)
\(110\) 2.15883 0.205837
\(111\) −0.801938 −0.0761166
\(112\) −1.69202 −0.159881
\(113\) −11.8726 −1.11688 −0.558441 0.829544i \(-0.688600\pi\)
−0.558441 + 0.829544i \(0.688600\pi\)
\(114\) −0.198062 −0.0185502
\(115\) −3.74094 −0.348844
\(116\) 1.29590 0.120321
\(117\) 0 0
\(118\) 3.73125 0.343489
\(119\) −3.98792 −0.365572
\(120\) −1.00000 −0.0912871
\(121\) −6.33944 −0.576312
\(122\) −6.32304 −0.572462
\(123\) 1.89977 0.171297
\(124\) 1.44504 0.129769
\(125\) 1.00000 0.0894427
\(126\) −1.69202 −0.150737
\(127\) 13.0097 1.15442 0.577212 0.816595i \(-0.304141\pi\)
0.577212 + 0.816595i \(0.304141\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.5429 −1.10434
\(130\) 0 0
\(131\) −7.82908 −0.684030 −0.342015 0.939694i \(-0.611109\pi\)
−0.342015 + 0.939694i \(0.611109\pi\)
\(132\) −2.15883 −0.187902
\(133\) −0.335126 −0.0290591
\(134\) −7.56465 −0.653486
\(135\) −1.00000 −0.0860663
\(136\) 2.35690 0.202102
\(137\) 5.36227 0.458130 0.229065 0.973411i \(-0.426433\pi\)
0.229065 + 0.973411i \(0.426433\pi\)
\(138\) 3.74094 0.318450
\(139\) 3.12200 0.264804 0.132402 0.991196i \(-0.457731\pi\)
0.132402 + 0.991196i \(0.457731\pi\)
\(140\) −1.69202 −0.143002
\(141\) −8.87800 −0.747663
\(142\) −4.18060 −0.350828
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 1.29590 0.107618
\(146\) 11.9366 0.987881
\(147\) 4.13706 0.341219
\(148\) 0.801938 0.0659189
\(149\) 4.24160 0.347486 0.173743 0.984791i \(-0.444414\pi\)
0.173743 + 0.984791i \(0.444414\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 22.9148 1.86478 0.932392 0.361450i \(-0.117718\pi\)
0.932392 + 0.361450i \(0.117718\pi\)
\(152\) 0.198062 0.0160650
\(153\) 2.35690 0.190544
\(154\) −3.65279 −0.294350
\(155\) 1.44504 0.116069
\(156\) 0 0
\(157\) 16.0707 1.28258 0.641290 0.767298i \(-0.278399\pi\)
0.641290 + 0.767298i \(0.278399\pi\)
\(158\) 9.40581 0.748286
\(159\) 1.00969 0.0800735
\(160\) 1.00000 0.0790569
\(161\) 6.32975 0.498854
\(162\) 1.00000 0.0785674
\(163\) −0.488582 −0.0382687 −0.0191344 0.999817i \(-0.506091\pi\)
−0.0191344 + 0.999817i \(0.506091\pi\)
\(164\) −1.89977 −0.148347
\(165\) −2.15883 −0.168065
\(166\) 8.43296 0.654525
\(167\) 14.0610 1.08807 0.544036 0.839062i \(-0.316895\pi\)
0.544036 + 0.839062i \(0.316895\pi\)
\(168\) 1.69202 0.130542
\(169\) 0 0
\(170\) 2.35690 0.180766
\(171\) 0.198062 0.0151462
\(172\) 12.5429 0.956385
\(173\) 21.7942 1.65698 0.828490 0.560004i \(-0.189200\pi\)
0.828490 + 0.560004i \(0.189200\pi\)
\(174\) −1.29590 −0.0982417
\(175\) −1.69202 −0.127905
\(176\) 2.15883 0.162728
\(177\) −3.73125 −0.280458
\(178\) −2.41119 −0.180726
\(179\) −14.1293 −1.05607 −0.528036 0.849222i \(-0.677072\pi\)
−0.528036 + 0.849222i \(0.677072\pi\)
\(180\) 1.00000 0.0745356
\(181\) −5.63773 −0.419049 −0.209524 0.977803i \(-0.567192\pi\)
−0.209524 + 0.977803i \(0.567192\pi\)
\(182\) 0 0
\(183\) 6.32304 0.467413
\(184\) −3.74094 −0.275786
\(185\) 0.801938 0.0589596
\(186\) −1.44504 −0.105956
\(187\) 5.08815 0.372082
\(188\) 8.87800 0.647495
\(189\) 1.69202 0.123076
\(190\) 0.198062 0.0143689
\(191\) −2.59419 −0.187709 −0.0938544 0.995586i \(-0.529919\pi\)
−0.0938544 + 0.995586i \(0.529919\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −3.98361 −0.286746 −0.143373 0.989669i \(-0.545795\pi\)
−0.143373 + 0.989669i \(0.545795\pi\)
\(194\) −0.0881460 −0.00632851
\(195\) 0 0
\(196\) −4.13706 −0.295505
\(197\) −6.65040 −0.473821 −0.236911 0.971531i \(-0.576135\pi\)
−0.236911 + 0.971531i \(0.576135\pi\)
\(198\) 2.15883 0.153422
\(199\) −1.01639 −0.0720502 −0.0360251 0.999351i \(-0.511470\pi\)
−0.0360251 + 0.999351i \(0.511470\pi\)
\(200\) 1.00000 0.0707107
\(201\) 7.56465 0.533569
\(202\) −11.1099 −0.781691
\(203\) −2.19269 −0.153896
\(204\) −2.35690 −0.165016
\(205\) −1.89977 −0.132686
\(206\) 9.54825 0.665258
\(207\) −3.74094 −0.260013
\(208\) 0 0
\(209\) 0.427583 0.0295766
\(210\) 1.69202 0.116761
\(211\) 10.1414 0.698161 0.349081 0.937093i \(-0.386494\pi\)
0.349081 + 0.937093i \(0.386494\pi\)
\(212\) −1.00969 −0.0693457
\(213\) 4.18060 0.286450
\(214\) 18.1468 1.24049
\(215\) 12.5429 0.855417
\(216\) −1.00000 −0.0680414
\(217\) −2.44504 −0.165980
\(218\) −2.24698 −0.152185
\(219\) −11.9366 −0.806601
\(220\) 2.15883 0.145549
\(221\) 0 0
\(222\) −0.801938 −0.0538225
\(223\) −16.8388 −1.12761 −0.563804 0.825909i \(-0.690663\pi\)
−0.563804 + 0.825909i \(0.690663\pi\)
\(224\) −1.69202 −0.113053
\(225\) 1.00000 0.0666667
\(226\) −11.8726 −0.789755
\(227\) 18.3056 1.21498 0.607492 0.794326i \(-0.292176\pi\)
0.607492 + 0.794326i \(0.292176\pi\)
\(228\) −0.198062 −0.0131170
\(229\) 5.14244 0.339822 0.169911 0.985459i \(-0.445652\pi\)
0.169911 + 0.985459i \(0.445652\pi\)
\(230\) −3.74094 −0.246670
\(231\) 3.65279 0.240336
\(232\) 1.29590 0.0850798
\(233\) −10.9215 −0.715494 −0.357747 0.933819i \(-0.616455\pi\)
−0.357747 + 0.933819i \(0.616455\pi\)
\(234\) 0 0
\(235\) 8.87800 0.579137
\(236\) 3.73125 0.242884
\(237\) −9.40581 −0.610973
\(238\) −3.98792 −0.258498
\(239\) 19.9041 1.28749 0.643744 0.765241i \(-0.277380\pi\)
0.643744 + 0.765241i \(0.277380\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 18.2228 1.17383 0.586917 0.809647i \(-0.300341\pi\)
0.586917 + 0.809647i \(0.300341\pi\)
\(242\) −6.33944 −0.407514
\(243\) −1.00000 −0.0641500
\(244\) −6.32304 −0.404791
\(245\) −4.13706 −0.264307
\(246\) 1.89977 0.121125
\(247\) 0 0
\(248\) 1.44504 0.0917603
\(249\) −8.43296 −0.534417
\(250\) 1.00000 0.0632456
\(251\) −17.0030 −1.07322 −0.536609 0.843831i \(-0.680295\pi\)
−0.536609 + 0.843831i \(0.680295\pi\)
\(252\) −1.69202 −0.106587
\(253\) −8.07606 −0.507738
\(254\) 13.0097 0.816300
\(255\) −2.35690 −0.147595
\(256\) 1.00000 0.0625000
\(257\) −7.65519 −0.477517 −0.238759 0.971079i \(-0.576740\pi\)
−0.238759 + 0.971079i \(0.576740\pi\)
\(258\) −12.5429 −0.780885
\(259\) −1.35690 −0.0843134
\(260\) 0 0
\(261\) 1.29590 0.0802140
\(262\) −7.82908 −0.483682
\(263\) 21.0411 1.29745 0.648726 0.761022i \(-0.275302\pi\)
0.648726 + 0.761022i \(0.275302\pi\)
\(264\) −2.15883 −0.132867
\(265\) −1.00969 −0.0620247
\(266\) −0.335126 −0.0205479
\(267\) 2.41119 0.147562
\(268\) −7.56465 −0.462084
\(269\) 24.3183 1.48271 0.741355 0.671113i \(-0.234183\pi\)
0.741355 + 0.671113i \(0.234183\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 10.9922 0.667730 0.333865 0.942621i \(-0.391647\pi\)
0.333865 + 0.942621i \(0.391647\pi\)
\(272\) 2.35690 0.142908
\(273\) 0 0
\(274\) 5.36227 0.323947
\(275\) 2.15883 0.130183
\(276\) 3.74094 0.225178
\(277\) −11.4601 −0.688571 −0.344286 0.938865i \(-0.611879\pi\)
−0.344286 + 0.938865i \(0.611879\pi\)
\(278\) 3.12200 0.187245
\(279\) 1.44504 0.0865124
\(280\) −1.69202 −0.101118
\(281\) −9.88231 −0.589529 −0.294765 0.955570i \(-0.595241\pi\)
−0.294765 + 0.955570i \(0.595241\pi\)
\(282\) −8.87800 −0.528677
\(283\) −21.3207 −1.26738 −0.633691 0.773587i \(-0.718461\pi\)
−0.633691 + 0.773587i \(0.718461\pi\)
\(284\) −4.18060 −0.248073
\(285\) −0.198062 −0.0117322
\(286\) 0 0
\(287\) 3.21446 0.189743
\(288\) 1.00000 0.0589256
\(289\) −11.4450 −0.673238
\(290\) 1.29590 0.0760977
\(291\) 0.0881460 0.00516721
\(292\) 11.9366 0.698537
\(293\) −18.7090 −1.09299 −0.546496 0.837462i \(-0.684039\pi\)
−0.546496 + 0.837462i \(0.684039\pi\)
\(294\) 4.13706 0.241278
\(295\) 3.73125 0.217242
\(296\) 0.801938 0.0466117
\(297\) −2.15883 −0.125268
\(298\) 4.24160 0.245709
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −21.2228 −1.22326
\(302\) 22.9148 1.31860
\(303\) 11.1099 0.638248
\(304\) 0.198062 0.0113596
\(305\) −6.32304 −0.362056
\(306\) 2.35690 0.134735
\(307\) 17.2241 0.983034 0.491517 0.870868i \(-0.336443\pi\)
0.491517 + 0.870868i \(0.336443\pi\)
\(308\) −3.65279 −0.208137
\(309\) −9.54825 −0.543181
\(310\) 1.44504 0.0820729
\(311\) −13.5200 −0.766651 −0.383326 0.923613i \(-0.625221\pi\)
−0.383326 + 0.923613i \(0.625221\pi\)
\(312\) 0 0
\(313\) 24.3303 1.37523 0.687616 0.726074i \(-0.258657\pi\)
0.687616 + 0.726074i \(0.258657\pi\)
\(314\) 16.0707 0.906921
\(315\) −1.69202 −0.0953346
\(316\) 9.40581 0.529118
\(317\) 34.7821 1.95356 0.976778 0.214253i \(-0.0687316\pi\)
0.976778 + 0.214253i \(0.0687316\pi\)
\(318\) 1.00969 0.0566205
\(319\) 2.79763 0.156637
\(320\) 1.00000 0.0559017
\(321\) −18.1468 −1.01285
\(322\) 6.32975 0.352743
\(323\) 0.466812 0.0259741
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −0.488582 −0.0270601
\(327\) 2.24698 0.124258
\(328\) −1.89977 −0.104897
\(329\) −15.0218 −0.828177
\(330\) −2.15883 −0.118840
\(331\) −9.66919 −0.531467 −0.265733 0.964047i \(-0.585614\pi\)
−0.265733 + 0.964047i \(0.585614\pi\)
\(332\) 8.43296 0.462819
\(333\) 0.801938 0.0439459
\(334\) 14.0610 0.769384
\(335\) −7.56465 −0.413301
\(336\) 1.69202 0.0923073
\(337\) −17.3612 −0.945725 −0.472863 0.881136i \(-0.656779\pi\)
−0.472863 + 0.881136i \(0.656779\pi\)
\(338\) 0 0
\(339\) 11.8726 0.644832
\(340\) 2.35690 0.127821
\(341\) 3.11960 0.168936
\(342\) 0.198062 0.0107100
\(343\) 18.8442 1.01749
\(344\) 12.5429 0.676267
\(345\) 3.74094 0.201405
\(346\) 21.7942 1.17166
\(347\) 8.86294 0.475787 0.237894 0.971291i \(-0.423543\pi\)
0.237894 + 0.971291i \(0.423543\pi\)
\(348\) −1.29590 −0.0694674
\(349\) 2.78554 0.149107 0.0745534 0.997217i \(-0.476247\pi\)
0.0745534 + 0.997217i \(0.476247\pi\)
\(350\) −1.69202 −0.0904424
\(351\) 0 0
\(352\) 2.15883 0.115066
\(353\) 21.0640 1.12112 0.560561 0.828113i \(-0.310585\pi\)
0.560561 + 0.828113i \(0.310585\pi\)
\(354\) −3.73125 −0.198314
\(355\) −4.18060 −0.221883
\(356\) −2.41119 −0.127793
\(357\) 3.98792 0.211063
\(358\) −14.1293 −0.746756
\(359\) 17.3274 0.914503 0.457251 0.889337i \(-0.348834\pi\)
0.457251 + 0.889337i \(0.348834\pi\)
\(360\) 1.00000 0.0527046
\(361\) −18.9608 −0.997935
\(362\) −5.63773 −0.296312
\(363\) 6.33944 0.332734
\(364\) 0 0
\(365\) 11.9366 0.624791
\(366\) 6.32304 0.330511
\(367\) −30.8049 −1.60800 −0.804002 0.594627i \(-0.797300\pi\)
−0.804002 + 0.594627i \(0.797300\pi\)
\(368\) −3.74094 −0.195010
\(369\) −1.89977 −0.0988982
\(370\) 0.801938 0.0416908
\(371\) 1.70841 0.0886965
\(372\) −1.44504 −0.0749219
\(373\) −25.6668 −1.32898 −0.664488 0.747299i \(-0.731350\pi\)
−0.664488 + 0.747299i \(0.731350\pi\)
\(374\) 5.08815 0.263102
\(375\) −1.00000 −0.0516398
\(376\) 8.87800 0.457848
\(377\) 0 0
\(378\) 1.69202 0.0870282
\(379\) 5.46144 0.280535 0.140268 0.990114i \(-0.455204\pi\)
0.140268 + 0.990114i \(0.455204\pi\)
\(380\) 0.198062 0.0101604
\(381\) −13.0097 −0.666507
\(382\) −2.59419 −0.132730
\(383\) −25.9221 −1.32456 −0.662280 0.749257i \(-0.730411\pi\)
−0.662280 + 0.749257i \(0.730411\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −3.65279 −0.186164
\(386\) −3.98361 −0.202760
\(387\) 12.5429 0.637590
\(388\) −0.0881460 −0.00447494
\(389\) −32.0006 −1.62249 −0.811247 0.584703i \(-0.801211\pi\)
−0.811247 + 0.584703i \(0.801211\pi\)
\(390\) 0 0
\(391\) −8.81700 −0.445895
\(392\) −4.13706 −0.208953
\(393\) 7.82908 0.394925
\(394\) −6.65040 −0.335042
\(395\) 9.40581 0.473258
\(396\) 2.15883 0.108485
\(397\) 0.457123 0.0229424 0.0114712 0.999934i \(-0.496349\pi\)
0.0114712 + 0.999934i \(0.496349\pi\)
\(398\) −1.01639 −0.0509472
\(399\) 0.335126 0.0167773
\(400\) 1.00000 0.0500000
\(401\) 23.2553 1.16132 0.580658 0.814147i \(-0.302795\pi\)
0.580658 + 0.814147i \(0.302795\pi\)
\(402\) 7.56465 0.377290
\(403\) 0 0
\(404\) −11.1099 −0.552739
\(405\) 1.00000 0.0496904
\(406\) −2.19269 −0.108821
\(407\) 1.73125 0.0858149
\(408\) −2.35690 −0.116684
\(409\) 39.0629 1.93154 0.965768 0.259406i \(-0.0835266\pi\)
0.965768 + 0.259406i \(0.0835266\pi\)
\(410\) −1.89977 −0.0938231
\(411\) −5.36227 −0.264501
\(412\) 9.54825 0.470409
\(413\) −6.31336 −0.310660
\(414\) −3.74094 −0.183857
\(415\) 8.43296 0.413958
\(416\) 0 0
\(417\) −3.12200 −0.152885
\(418\) 0.427583 0.0209138
\(419\) −18.1269 −0.885557 −0.442779 0.896631i \(-0.646007\pi\)
−0.442779 + 0.896631i \(0.646007\pi\)
\(420\) 1.69202 0.0825622
\(421\) −20.4916 −0.998698 −0.499349 0.866401i \(-0.666427\pi\)
−0.499349 + 0.866401i \(0.666427\pi\)
\(422\) 10.1414 0.493674
\(423\) 8.87800 0.431663
\(424\) −1.00969 −0.0490348
\(425\) 2.35690 0.114326
\(426\) 4.18060 0.202551
\(427\) 10.6987 0.517748
\(428\) 18.1468 0.877156
\(429\) 0 0
\(430\) 12.5429 0.604871
\(431\) −22.2989 −1.07410 −0.537050 0.843551i \(-0.680461\pi\)
−0.537050 + 0.843551i \(0.680461\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 7.47650 0.359298 0.179649 0.983731i \(-0.442504\pi\)
0.179649 + 0.983731i \(0.442504\pi\)
\(434\) −2.44504 −0.117366
\(435\) −1.29590 −0.0621335
\(436\) −2.24698 −0.107611
\(437\) −0.740939 −0.0354439
\(438\) −11.9366 −0.570353
\(439\) −19.3924 −0.925549 −0.462774 0.886476i \(-0.653146\pi\)
−0.462774 + 0.886476i \(0.653146\pi\)
\(440\) 2.15883 0.102918
\(441\) −4.13706 −0.197003
\(442\) 0 0
\(443\) 10.0935 0.479558 0.239779 0.970828i \(-0.422925\pi\)
0.239779 + 0.970828i \(0.422925\pi\)
\(444\) −0.801938 −0.0380583
\(445\) −2.41119 −0.114301
\(446\) −16.8388 −0.797339
\(447\) −4.24160 −0.200621
\(448\) −1.69202 −0.0799405
\(449\) −2.78448 −0.131408 −0.0657039 0.997839i \(-0.520929\pi\)
−0.0657039 + 0.997839i \(0.520929\pi\)
\(450\) 1.00000 0.0471405
\(451\) −4.10129 −0.193122
\(452\) −11.8726 −0.558441
\(453\) −22.9148 −1.07663
\(454\) 18.3056 0.859124
\(455\) 0 0
\(456\) −0.198062 −0.00927512
\(457\) −13.1056 −0.613054 −0.306527 0.951862i \(-0.599167\pi\)
−0.306527 + 0.951862i \(0.599167\pi\)
\(458\) 5.14244 0.240290
\(459\) −2.35690 −0.110010
\(460\) −3.74094 −0.174422
\(461\) 16.2174 0.755321 0.377661 0.925944i \(-0.376729\pi\)
0.377661 + 0.925944i \(0.376729\pi\)
\(462\) 3.65279 0.169943
\(463\) 15.5415 0.722277 0.361139 0.932512i \(-0.382388\pi\)
0.361139 + 0.932512i \(0.382388\pi\)
\(464\) 1.29590 0.0601605
\(465\) −1.44504 −0.0670122
\(466\) −10.9215 −0.505931
\(467\) −10.1578 −0.470045 −0.235023 0.971990i \(-0.575516\pi\)
−0.235023 + 0.971990i \(0.575516\pi\)
\(468\) 0 0
\(469\) 12.7995 0.591028
\(470\) 8.87800 0.409512
\(471\) −16.0707 −0.740498
\(472\) 3.73125 0.171745
\(473\) 27.0780 1.24505
\(474\) −9.40581 −0.432023
\(475\) 0.198062 0.00908772
\(476\) −3.98792 −0.182786
\(477\) −1.00969 −0.0462305
\(478\) 19.9041 0.910392
\(479\) 31.4263 1.43590 0.717951 0.696094i \(-0.245080\pi\)
0.717951 + 0.696094i \(0.245080\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 18.2228 0.830027
\(483\) −6.32975 −0.288014
\(484\) −6.33944 −0.288156
\(485\) −0.0881460 −0.00400250
\(486\) −1.00000 −0.0453609
\(487\) −24.1323 −1.09354 −0.546769 0.837284i \(-0.684142\pi\)
−0.546769 + 0.837284i \(0.684142\pi\)
\(488\) −6.32304 −0.286231
\(489\) 0.488582 0.0220945
\(490\) −4.13706 −0.186893
\(491\) −19.1099 −0.862418 −0.431209 0.902252i \(-0.641913\pi\)
−0.431209 + 0.902252i \(0.641913\pi\)
\(492\) 1.89977 0.0856484
\(493\) 3.05429 0.137558
\(494\) 0 0
\(495\) 2.15883 0.0970324
\(496\) 1.44504 0.0648843
\(497\) 7.07367 0.317298
\(498\) −8.43296 −0.377890
\(499\) −34.9724 −1.56558 −0.782789 0.622287i \(-0.786204\pi\)
−0.782789 + 0.622287i \(0.786204\pi\)
\(500\) 1.00000 0.0447214
\(501\) −14.0610 −0.628199
\(502\) −17.0030 −0.758880
\(503\) 29.5579 1.31792 0.658962 0.752176i \(-0.270996\pi\)
0.658962 + 0.752176i \(0.270996\pi\)
\(504\) −1.69202 −0.0753686
\(505\) −11.1099 −0.494385
\(506\) −8.07606 −0.359025
\(507\) 0 0
\(508\) 13.0097 0.577212
\(509\) 24.4838 1.08523 0.542613 0.839983i \(-0.317435\pi\)
0.542613 + 0.839983i \(0.317435\pi\)
\(510\) −2.35690 −0.104365
\(511\) −20.1970 −0.893463
\(512\) 1.00000 0.0441942
\(513\) −0.198062 −0.00874466
\(514\) −7.65519 −0.337656
\(515\) 9.54825 0.420746
\(516\) −12.5429 −0.552169
\(517\) 19.1661 0.842925
\(518\) −1.35690 −0.0596186
\(519\) −21.7942 −0.956658
\(520\) 0 0
\(521\) −33.8713 −1.48393 −0.741964 0.670439i \(-0.766106\pi\)
−0.741964 + 0.670439i \(0.766106\pi\)
\(522\) 1.29590 0.0567199
\(523\) −42.5870 −1.86220 −0.931100 0.364764i \(-0.881150\pi\)
−0.931100 + 0.364764i \(0.881150\pi\)
\(524\) −7.82908 −0.342015
\(525\) 1.69202 0.0738459
\(526\) 21.0411 0.917438
\(527\) 3.40581 0.148360
\(528\) −2.15883 −0.0939512
\(529\) −9.00538 −0.391538
\(530\) −1.00969 −0.0438581
\(531\) 3.73125 0.161922
\(532\) −0.335126 −0.0145295
\(533\) 0 0
\(534\) 2.41119 0.104342
\(535\) 18.1468 0.784553
\(536\) −7.56465 −0.326743
\(537\) 14.1293 0.609724
\(538\) 24.3183 1.04843
\(539\) −8.93123 −0.384695
\(540\) −1.00000 −0.0430331
\(541\) −3.23298 −0.138997 −0.0694983 0.997582i \(-0.522140\pi\)
−0.0694983 + 0.997582i \(0.522140\pi\)
\(542\) 10.9922 0.472157
\(543\) 5.63773 0.241938
\(544\) 2.35690 0.101051
\(545\) −2.24698 −0.0962500
\(546\) 0 0
\(547\) −5.51573 −0.235836 −0.117918 0.993023i \(-0.537622\pi\)
−0.117918 + 0.993023i \(0.537622\pi\)
\(548\) 5.36227 0.229065
\(549\) −6.32304 −0.269861
\(550\) 2.15883 0.0920530
\(551\) 0.256668 0.0109344
\(552\) 3.74094 0.159225
\(553\) −15.9148 −0.676768
\(554\) −11.4601 −0.486893
\(555\) −0.801938 −0.0340404
\(556\) 3.12200 0.132402
\(557\) 30.3502 1.28598 0.642989 0.765875i \(-0.277694\pi\)
0.642989 + 0.765875i \(0.277694\pi\)
\(558\) 1.44504 0.0611735
\(559\) 0 0
\(560\) −1.69202 −0.0715010
\(561\) −5.08815 −0.214822
\(562\) −9.88231 −0.416860
\(563\) 6.34375 0.267357 0.133679 0.991025i \(-0.457321\pi\)
0.133679 + 0.991025i \(0.457321\pi\)
\(564\) −8.87800 −0.373831
\(565\) −11.8726 −0.499485
\(566\) −21.3207 −0.896174
\(567\) −1.69202 −0.0710582
\(568\) −4.18060 −0.175414
\(569\) 24.7275 1.03663 0.518316 0.855189i \(-0.326559\pi\)
0.518316 + 0.855189i \(0.326559\pi\)
\(570\) −0.198062 −0.00829592
\(571\) 13.6262 0.570240 0.285120 0.958492i \(-0.407967\pi\)
0.285120 + 0.958492i \(0.407967\pi\)
\(572\) 0 0
\(573\) 2.59419 0.108374
\(574\) 3.21446 0.134169
\(575\) −3.74094 −0.156008
\(576\) 1.00000 0.0416667
\(577\) −16.2828 −0.677860 −0.338930 0.940812i \(-0.610065\pi\)
−0.338930 + 0.940812i \(0.610065\pi\)
\(578\) −11.4450 −0.476051
\(579\) 3.98361 0.165553
\(580\) 1.29590 0.0538092
\(581\) −14.2687 −0.591967
\(582\) 0.0881460 0.00365377
\(583\) −2.17975 −0.0902760
\(584\) 11.9366 0.493940
\(585\) 0 0
\(586\) −18.7090 −0.772862
\(587\) −11.0344 −0.455440 −0.227720 0.973727i \(-0.573127\pi\)
−0.227720 + 0.973727i \(0.573127\pi\)
\(588\) 4.13706 0.170610
\(589\) 0.286208 0.0117930
\(590\) 3.73125 0.153613
\(591\) 6.65040 0.273561
\(592\) 0.801938 0.0329594
\(593\) −25.4905 −1.04677 −0.523385 0.852096i \(-0.675331\pi\)
−0.523385 + 0.852096i \(0.675331\pi\)
\(594\) −2.15883 −0.0885780
\(595\) −3.98792 −0.163489
\(596\) 4.24160 0.173743
\(597\) 1.01639 0.0415982
\(598\) 0 0
\(599\) −0.501729 −0.0205001 −0.0102500 0.999947i \(-0.503263\pi\)
−0.0102500 + 0.999947i \(0.503263\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −18.3744 −0.749506 −0.374753 0.927125i \(-0.622272\pi\)
−0.374753 + 0.927125i \(0.622272\pi\)
\(602\) −21.2228 −0.864977
\(603\) −7.56465 −0.308056
\(604\) 22.9148 0.932392
\(605\) −6.33944 −0.257735
\(606\) 11.1099 0.451309
\(607\) −17.1395 −0.695669 −0.347835 0.937556i \(-0.613083\pi\)
−0.347835 + 0.937556i \(0.613083\pi\)
\(608\) 0.198062 0.00803249
\(609\) 2.19269 0.0888521
\(610\) −6.32304 −0.256013
\(611\) 0 0
\(612\) 2.35690 0.0952719
\(613\) 1.15751 0.0467512 0.0233756 0.999727i \(-0.492559\pi\)
0.0233756 + 0.999727i \(0.492559\pi\)
\(614\) 17.2241 0.695110
\(615\) 1.89977 0.0766062
\(616\) −3.65279 −0.147175
\(617\) 7.71618 0.310642 0.155321 0.987864i \(-0.450359\pi\)
0.155321 + 0.987864i \(0.450359\pi\)
\(618\) −9.54825 −0.384087
\(619\) 16.1142 0.647686 0.323843 0.946111i \(-0.395025\pi\)
0.323843 + 0.946111i \(0.395025\pi\)
\(620\) 1.44504 0.0580343
\(621\) 3.74094 0.150119
\(622\) −13.5200 −0.542104
\(623\) 4.07979 0.163453
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 24.3303 0.972436
\(627\) −0.427583 −0.0170760
\(628\) 16.0707 0.641290
\(629\) 1.89008 0.0753626
\(630\) −1.69202 −0.0674117
\(631\) −7.77586 −0.309552 −0.154776 0.987950i \(-0.549466\pi\)
−0.154776 + 0.987950i \(0.549466\pi\)
\(632\) 9.40581 0.374143
\(633\) −10.1414 −0.403083
\(634\) 34.7821 1.38137
\(635\) 13.0097 0.516274
\(636\) 1.00969 0.0400368
\(637\) 0 0
\(638\) 2.79763 0.110759
\(639\) −4.18060 −0.165382
\(640\) 1.00000 0.0395285
\(641\) −35.1293 −1.38752 −0.693762 0.720204i \(-0.744048\pi\)
−0.693762 + 0.720204i \(0.744048\pi\)
\(642\) −18.1468 −0.716195
\(643\) −14.5459 −0.573633 −0.286816 0.957986i \(-0.592597\pi\)
−0.286816 + 0.957986i \(0.592597\pi\)
\(644\) 6.32975 0.249427
\(645\) −12.5429 −0.493875
\(646\) 0.466812 0.0183665
\(647\) 36.2234 1.42409 0.712045 0.702134i \(-0.247769\pi\)
0.712045 + 0.702134i \(0.247769\pi\)
\(648\) 1.00000 0.0392837
\(649\) 8.05515 0.316192
\(650\) 0 0
\(651\) 2.44504 0.0958287
\(652\) −0.488582 −0.0191344
\(653\) 19.4926 0.762806 0.381403 0.924409i \(-0.375441\pi\)
0.381403 + 0.924409i \(0.375441\pi\)
\(654\) 2.24698 0.0878639
\(655\) −7.82908 −0.305908
\(656\) −1.89977 −0.0741737
\(657\) 11.9366 0.465691
\(658\) −15.0218 −0.585610
\(659\) 38.3400 1.49352 0.746758 0.665096i \(-0.231609\pi\)
0.746758 + 0.665096i \(0.231609\pi\)
\(660\) −2.15883 −0.0840325
\(661\) 13.5985 0.528920 0.264460 0.964397i \(-0.414806\pi\)
0.264460 + 0.964397i \(0.414806\pi\)
\(662\) −9.66919 −0.375804
\(663\) 0 0
\(664\) 8.43296 0.327262
\(665\) −0.335126 −0.0129956
\(666\) 0.801938 0.0310745
\(667\) −4.84787 −0.187710
\(668\) 14.0610 0.544036
\(669\) 16.8388 0.651025
\(670\) −7.56465 −0.292248
\(671\) −13.6504 −0.526968
\(672\) 1.69202 0.0652711
\(673\) −49.7090 −1.91614 −0.958071 0.286532i \(-0.907498\pi\)
−0.958071 + 0.286532i \(0.907498\pi\)
\(674\) −17.3612 −0.668729
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −50.2573 −1.93154 −0.965772 0.259394i \(-0.916477\pi\)
−0.965772 + 0.259394i \(0.916477\pi\)
\(678\) 11.8726 0.455965
\(679\) 0.149145 0.00572366
\(680\) 2.35690 0.0903828
\(681\) −18.3056 −0.701472
\(682\) 3.11960 0.119456
\(683\) −21.5415 −0.824264 −0.412132 0.911124i \(-0.635216\pi\)
−0.412132 + 0.911124i \(0.635216\pi\)
\(684\) 0.198062 0.00757310
\(685\) 5.36227 0.204882
\(686\) 18.8442 0.719473
\(687\) −5.14244 −0.196196
\(688\) 12.5429 0.478193
\(689\) 0 0
\(690\) 3.74094 0.142415
\(691\) −50.8165 −1.93315 −0.966576 0.256380i \(-0.917470\pi\)
−0.966576 + 0.256380i \(0.917470\pi\)
\(692\) 21.7942 0.828490
\(693\) −3.65279 −0.138758
\(694\) 8.86294 0.336432
\(695\) 3.12200 0.118424
\(696\) −1.29590 −0.0491208
\(697\) −4.47757 −0.169600
\(698\) 2.78554 0.105434
\(699\) 10.9215 0.413091
\(700\) −1.69202 −0.0639524
\(701\) −13.2024 −0.498647 −0.249323 0.968420i \(-0.580208\pi\)
−0.249323 + 0.968420i \(0.580208\pi\)
\(702\) 0 0
\(703\) 0.158834 0.00599052
\(704\) 2.15883 0.0813641
\(705\) −8.87800 −0.334365
\(706\) 21.0640 0.792753
\(707\) 18.7982 0.706980
\(708\) −3.73125 −0.140229
\(709\) 0.430567 0.0161703 0.00808515 0.999967i \(-0.497426\pi\)
0.00808515 + 0.999967i \(0.497426\pi\)
\(710\) −4.18060 −0.156895
\(711\) 9.40581 0.352746
\(712\) −2.41119 −0.0903632
\(713\) −5.40581 −0.202449
\(714\) 3.98792 0.149244
\(715\) 0 0
\(716\) −14.1293 −0.528036
\(717\) −19.9041 −0.743332
\(718\) 17.3274 0.646651
\(719\) 8.24459 0.307471 0.153736 0.988112i \(-0.450870\pi\)
0.153736 + 0.988112i \(0.450870\pi\)
\(720\) 1.00000 0.0372678
\(721\) −16.1558 −0.601675
\(722\) −18.9608 −0.705647
\(723\) −18.2228 −0.677714
\(724\) −5.63773 −0.209524
\(725\) 1.29590 0.0481284
\(726\) 6.33944 0.235279
\(727\) 21.2664 0.788726 0.394363 0.918955i \(-0.370965\pi\)
0.394363 + 0.918955i \(0.370965\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 11.9366 0.441794
\(731\) 29.5623 1.09340
\(732\) 6.32304 0.233706
\(733\) 38.9178 1.43746 0.718731 0.695288i \(-0.244723\pi\)
0.718731 + 0.695288i \(0.244723\pi\)
\(734\) −30.8049 −1.13703
\(735\) 4.13706 0.152598
\(736\) −3.74094 −0.137893
\(737\) −16.3308 −0.601553
\(738\) −1.89977 −0.0699316
\(739\) 4.48486 0.164978 0.0824891 0.996592i \(-0.473713\pi\)
0.0824891 + 0.996592i \(0.473713\pi\)
\(740\) 0.801938 0.0294798
\(741\) 0 0
\(742\) 1.70841 0.0627179
\(743\) 26.8558 0.985242 0.492621 0.870244i \(-0.336039\pi\)
0.492621 + 0.870244i \(0.336039\pi\)
\(744\) −1.44504 −0.0529778
\(745\) 4.24160 0.155400
\(746\) −25.6668 −0.939728
\(747\) 8.43296 0.308546
\(748\) 5.08815 0.186041
\(749\) −30.7047 −1.12193
\(750\) −1.00000 −0.0365148
\(751\) −5.64012 −0.205811 −0.102905 0.994691i \(-0.532814\pi\)
−0.102905 + 0.994691i \(0.532814\pi\)
\(752\) 8.87800 0.323747
\(753\) 17.0030 0.619623
\(754\) 0 0
\(755\) 22.9148 0.833956
\(756\) 1.69202 0.0615382
\(757\) −35.6644 −1.29624 −0.648122 0.761536i \(-0.724445\pi\)
−0.648122 + 0.761536i \(0.724445\pi\)
\(758\) 5.46144 0.198368
\(759\) 8.07606 0.293143
\(760\) 0.198062 0.00718447
\(761\) 2.68425 0.0973041 0.0486520 0.998816i \(-0.484507\pi\)
0.0486520 + 0.998816i \(0.484507\pi\)
\(762\) −13.0097 −0.471291
\(763\) 3.80194 0.137639
\(764\) −2.59419 −0.0938544
\(765\) 2.35690 0.0852137
\(766\) −25.9221 −0.936605
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 48.7644 1.75849 0.879244 0.476372i \(-0.158048\pi\)
0.879244 + 0.476372i \(0.158048\pi\)
\(770\) −3.65279 −0.131638
\(771\) 7.65519 0.275695
\(772\) −3.98361 −0.143373
\(773\) −46.6152 −1.67663 −0.838316 0.545184i \(-0.816460\pi\)
−0.838316 + 0.545184i \(0.816460\pi\)
\(774\) 12.5429 0.450844
\(775\) 1.44504 0.0519074
\(776\) −0.0881460 −0.00316426
\(777\) 1.35690 0.0486784
\(778\) −32.0006 −1.14728
\(779\) −0.376273 −0.0134814
\(780\) 0 0
\(781\) −9.02523 −0.322948
\(782\) −8.81700 −0.315295
\(783\) −1.29590 −0.0463116
\(784\) −4.13706 −0.147752
\(785\) 16.0707 0.573587
\(786\) 7.82908 0.279254
\(787\) −27.4045 −0.976864 −0.488432 0.872602i \(-0.662431\pi\)
−0.488432 + 0.872602i \(0.662431\pi\)
\(788\) −6.65040 −0.236911
\(789\) −21.0411 −0.749085
\(790\) 9.40581 0.334644
\(791\) 20.0887 0.714273
\(792\) 2.15883 0.0767108
\(793\) 0 0
\(794\) 0.457123 0.0162227
\(795\) 1.00969 0.0358100
\(796\) −1.01639 −0.0360251
\(797\) −5.70278 −0.202003 −0.101001 0.994886i \(-0.532205\pi\)
−0.101001 + 0.994886i \(0.532205\pi\)
\(798\) 0.335126 0.0118633
\(799\) 20.9245 0.740257
\(800\) 1.00000 0.0353553
\(801\) −2.41119 −0.0851952
\(802\) 23.2553 0.821175
\(803\) 25.7692 0.909374
\(804\) 7.56465 0.266785
\(805\) 6.32975 0.223094
\(806\) 0 0
\(807\) −24.3183 −0.856043
\(808\) −11.1099 −0.390845
\(809\) −29.6165 −1.04126 −0.520631 0.853782i \(-0.674303\pi\)
−0.520631 + 0.853782i \(0.674303\pi\)
\(810\) 1.00000 0.0351364
\(811\) −48.8901 −1.71676 −0.858382 0.513012i \(-0.828530\pi\)
−0.858382 + 0.513012i \(0.828530\pi\)
\(812\) −2.19269 −0.0769482
\(813\) −10.9922 −0.385514
\(814\) 1.73125 0.0606803
\(815\) −0.488582 −0.0171143
\(816\) −2.35690 −0.0825079
\(817\) 2.48427 0.0869136
\(818\) 39.0629 1.36580
\(819\) 0 0
\(820\) −1.89977 −0.0663429
\(821\) −19.7942 −0.690821 −0.345411 0.938452i \(-0.612260\pi\)
−0.345411 + 0.938452i \(0.612260\pi\)
\(822\) −5.36227 −0.187031
\(823\) 49.6118 1.72936 0.864679 0.502325i \(-0.167522\pi\)
0.864679 + 0.502325i \(0.167522\pi\)
\(824\) 9.54825 0.332629
\(825\) −2.15883 −0.0751609
\(826\) −6.31336 −0.219670
\(827\) 26.0683 0.906483 0.453242 0.891388i \(-0.350267\pi\)
0.453242 + 0.891388i \(0.350267\pi\)
\(828\) −3.74094 −0.130007
\(829\) 46.2215 1.60534 0.802669 0.596424i \(-0.203412\pi\)
0.802669 + 0.596424i \(0.203412\pi\)
\(830\) 8.43296 0.292712
\(831\) 11.4601 0.397547
\(832\) 0 0
\(833\) −9.75063 −0.337839
\(834\) −3.12200 −0.108106
\(835\) 14.0610 0.486601
\(836\) 0.427583 0.0147883
\(837\) −1.44504 −0.0499480
\(838\) −18.1269 −0.626183
\(839\) 33.2693 1.14859 0.574293 0.818650i \(-0.305277\pi\)
0.574293 + 0.818650i \(0.305277\pi\)
\(840\) 1.69202 0.0583803
\(841\) −27.3207 −0.942091
\(842\) −20.4916 −0.706186
\(843\) 9.88231 0.340365
\(844\) 10.1414 0.349081
\(845\) 0 0
\(846\) 8.87800 0.305232
\(847\) 10.7265 0.368566
\(848\) −1.00969 −0.0346729
\(849\) 21.3207 0.731723
\(850\) 2.35690 0.0808409
\(851\) −3.00000 −0.102839
\(852\) 4.18060 0.143225
\(853\) −39.3002 −1.34561 −0.672807 0.739818i \(-0.734911\pi\)
−0.672807 + 0.739818i \(0.734911\pi\)
\(854\) 10.6987 0.366103
\(855\) 0.198062 0.00677359
\(856\) 18.1468 0.620243
\(857\) −39.7566 −1.35806 −0.679030 0.734111i \(-0.737599\pi\)
−0.679030 + 0.734111i \(0.737599\pi\)
\(858\) 0 0
\(859\) −3.43860 −0.117324 −0.0586618 0.998278i \(-0.518683\pi\)
−0.0586618 + 0.998278i \(0.518683\pi\)
\(860\) 12.5429 0.427709
\(861\) −3.21446 −0.109548
\(862\) −22.2989 −0.759503
\(863\) 21.2301 0.722681 0.361341 0.932434i \(-0.382319\pi\)
0.361341 + 0.932434i \(0.382319\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 21.7942 0.741024
\(866\) 7.47650 0.254062
\(867\) 11.4450 0.388694
\(868\) −2.44504 −0.0829901
\(869\) 20.3056 0.688820
\(870\) −1.29590 −0.0439350
\(871\) 0 0
\(872\) −2.24698 −0.0760923
\(873\) −0.0881460 −0.00298329
\(874\) −0.740939 −0.0250626
\(875\) −1.69202 −0.0572008
\(876\) −11.9366 −0.403301
\(877\) −5.74227 −0.193903 −0.0969513 0.995289i \(-0.530909\pi\)
−0.0969513 + 0.995289i \(0.530909\pi\)
\(878\) −19.3924 −0.654462
\(879\) 18.7090 0.631039
\(880\) 2.15883 0.0727743
\(881\) 12.6069 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(882\) −4.13706 −0.139302
\(883\) 25.8086 0.868530 0.434265 0.900785i \(-0.357008\pi\)
0.434265 + 0.900785i \(0.357008\pi\)
\(884\) 0 0
\(885\) −3.73125 −0.125425
\(886\) 10.0935 0.339099
\(887\) −20.2392 −0.679566 −0.339783 0.940504i \(-0.610354\pi\)
−0.339783 + 0.940504i \(0.610354\pi\)
\(888\) −0.801938 −0.0269113
\(889\) −22.0127 −0.738281
\(890\) −2.41119 −0.0808233
\(891\) 2.15883 0.0723236
\(892\) −16.8388 −0.563804
\(893\) 1.75840 0.0588425
\(894\) −4.24160 −0.141860
\(895\) −14.1293 −0.472290
\(896\) −1.69202 −0.0565265
\(897\) 0 0
\(898\) −2.78448 −0.0929193
\(899\) 1.87263 0.0624556
\(900\) 1.00000 0.0333333
\(901\) −2.37973 −0.0792803
\(902\) −4.10129 −0.136558
\(903\) 21.2228 0.706251
\(904\) −11.8726 −0.394878
\(905\) −5.63773 −0.187404
\(906\) −22.9148 −0.761294
\(907\) −3.52350 −0.116996 −0.0584979 0.998288i \(-0.518631\pi\)
−0.0584979 + 0.998288i \(0.518631\pi\)
\(908\) 18.3056 0.607492
\(909\) −11.1099 −0.368493
\(910\) 0 0
\(911\) −34.7426 −1.15107 −0.575537 0.817776i \(-0.695207\pi\)
−0.575537 + 0.817776i \(0.695207\pi\)
\(912\) −0.198062 −0.00655850
\(913\) 18.2054 0.602509
\(914\) −13.1056 −0.433495
\(915\) 6.32304 0.209033
\(916\) 5.14244 0.169911
\(917\) 13.2470 0.437454
\(918\) −2.35690 −0.0777892
\(919\) −24.1430 −0.796405 −0.398203 0.917298i \(-0.630366\pi\)
−0.398203 + 0.917298i \(0.630366\pi\)
\(920\) −3.74094 −0.123335
\(921\) −17.2241 −0.567555
\(922\) 16.2174 0.534093
\(923\) 0 0
\(924\) 3.65279 0.120168
\(925\) 0.801938 0.0263676
\(926\) 15.5415 0.510727
\(927\) 9.54825 0.313606
\(928\) 1.29590 0.0425399
\(929\) −56.9700 −1.86912 −0.934562 0.355800i \(-0.884209\pi\)
−0.934562 + 0.355800i \(0.884209\pi\)
\(930\) −1.44504 −0.0473848
\(931\) −0.819396 −0.0268546
\(932\) −10.9215 −0.357747
\(933\) 13.5200 0.442626
\(934\) −10.1578 −0.332372
\(935\) 5.08815 0.166400
\(936\) 0 0
\(937\) −45.7966 −1.49611 −0.748054 0.663638i \(-0.769012\pi\)
−0.748054 + 0.663638i \(0.769012\pi\)
\(938\) 12.7995 0.417920
\(939\) −24.3303 −0.793991
\(940\) 8.87800 0.289569
\(941\) −7.10262 −0.231539 −0.115769 0.993276i \(-0.536933\pi\)
−0.115769 + 0.993276i \(0.536933\pi\)
\(942\) −16.0707 −0.523611
\(943\) 7.10693 0.231434
\(944\) 3.73125 0.121442
\(945\) 1.69202 0.0550415
\(946\) 27.0780 0.880381
\(947\) −11.2524 −0.365652 −0.182826 0.983145i \(-0.558525\pi\)
−0.182826 + 0.983145i \(0.558525\pi\)
\(948\) −9.40581 −0.305487
\(949\) 0 0
\(950\) 0.198062 0.00642599
\(951\) −34.7821 −1.12789
\(952\) −3.98792 −0.129249
\(953\) −1.61702 −0.0523805 −0.0261902 0.999657i \(-0.508338\pi\)
−0.0261902 + 0.999657i \(0.508338\pi\)
\(954\) −1.00969 −0.0326899
\(955\) −2.59419 −0.0839459
\(956\) 19.9041 0.643744
\(957\) −2.79763 −0.0904344
\(958\) 31.4263 1.01534
\(959\) −9.07308 −0.292985
\(960\) −1.00000 −0.0322749
\(961\) −28.9119 −0.932640
\(962\) 0 0
\(963\) 18.1468 0.584771
\(964\) 18.2228 0.586917
\(965\) −3.98361 −0.128237
\(966\) −6.32975 −0.203656
\(967\) −24.1588 −0.776896 −0.388448 0.921471i \(-0.626989\pi\)
−0.388448 + 0.921471i \(0.626989\pi\)
\(968\) −6.33944 −0.203757
\(969\) −0.466812 −0.0149962
\(970\) −0.0881460 −0.00283020
\(971\) 43.2669 1.38850 0.694251 0.719733i \(-0.255736\pi\)
0.694251 + 0.719733i \(0.255736\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −5.28249 −0.169349
\(974\) −24.1323 −0.773248
\(975\) 0 0
\(976\) −6.32304 −0.202396
\(977\) 24.9782 0.799124 0.399562 0.916706i \(-0.369162\pi\)
0.399562 + 0.916706i \(0.369162\pi\)
\(978\) 0.488582 0.0156231
\(979\) −5.20536 −0.166364
\(980\) −4.13706 −0.132154
\(981\) −2.24698 −0.0717405
\(982\) −19.1099 −0.609822
\(983\) −19.3134 −0.616000 −0.308000 0.951386i \(-0.599660\pi\)
−0.308000 + 0.951386i \(0.599660\pi\)
\(984\) 1.89977 0.0605625
\(985\) −6.65040 −0.211899
\(986\) 3.05429 0.0972685
\(987\) 15.0218 0.478148
\(988\) 0 0
\(989\) −46.9221 −1.49204
\(990\) 2.15883 0.0686122
\(991\) −27.5090 −0.873853 −0.436926 0.899497i \(-0.643933\pi\)
−0.436926 + 0.899497i \(0.643933\pi\)
\(992\) 1.44504 0.0458801
\(993\) 9.66919 0.306842
\(994\) 7.07367 0.224363
\(995\) −1.01639 −0.0322218
\(996\) −8.43296 −0.267209
\(997\) −23.1933 −0.734538 −0.367269 0.930115i \(-0.619707\pi\)
−0.367269 + 0.930115i \(0.619707\pi\)
\(998\) −34.9724 −1.10703
\(999\) −0.801938 −0.0253722
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.bu.1.1 yes 3
13.5 odd 4 5070.2.b.t.1351.1 6
13.8 odd 4 5070.2.b.t.1351.6 6
13.12 even 2 5070.2.a.bj.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bj.1.3 3 13.12 even 2
5070.2.a.bu.1.1 yes 3 1.1 even 1 trivial
5070.2.b.t.1351.1 6 13.5 odd 4
5070.2.b.t.1351.6 6 13.8 odd 4