Properties

Label 5070.2.a.bx.1.3
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.75302 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.75302 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.19806 q^{11} +1.00000 q^{12} -1.75302 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.76271 q^{17} +1.00000 q^{18} -8.63102 q^{19} +1.00000 q^{20} -1.75302 q^{21} -1.19806 q^{22} -6.04892 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -1.75302 q^{28} +4.07069 q^{29} +1.00000 q^{30} -6.80194 q^{31} +1.00000 q^{32} -1.19806 q^{33} -6.76271 q^{34} -1.75302 q^{35} +1.00000 q^{36} +4.34481 q^{37} -8.63102 q^{38} +1.00000 q^{40} -6.63102 q^{41} -1.75302 q^{42} +4.07606 q^{43} -1.19806 q^{44} +1.00000 q^{45} -6.04892 q^{46} +11.1957 q^{47} +1.00000 q^{48} -3.92692 q^{49} +1.00000 q^{50} -6.76271 q^{51} -1.14914 q^{53} +1.00000 q^{54} -1.19806 q^{55} -1.75302 q^{56} -8.63102 q^{57} +4.07069 q^{58} -13.7235 q^{59} +1.00000 q^{60} -1.18598 q^{61} -6.80194 q^{62} -1.75302 q^{63} +1.00000 q^{64} -1.19806 q^{66} -14.1468 q^{67} -6.76271 q^{68} -6.04892 q^{69} -1.75302 q^{70} +2.32304 q^{71} +1.00000 q^{72} -1.59850 q^{73} +4.34481 q^{74} +1.00000 q^{75} -8.63102 q^{76} +2.10023 q^{77} +3.74094 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.63102 q^{82} -8.40581 q^{83} -1.75302 q^{84} -6.76271 q^{85} +4.07606 q^{86} +4.07069 q^{87} -1.19806 q^{88} -4.49934 q^{89} +1.00000 q^{90} -6.04892 q^{92} -6.80194 q^{93} +11.1957 q^{94} -8.63102 q^{95} +1.00000 q^{96} +10.7952 q^{97} -3.92692 q^{98} -1.19806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} - 10q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} - 10q^{7} + 3q^{8} + 3q^{9} + 3q^{10} - 8q^{11} + 3q^{12} - 10q^{14} + 3q^{15} + 3q^{16} - 3q^{17} + 3q^{18} - 11q^{19} + 3q^{20} - 10q^{21} - 8q^{22} - 9q^{23} + 3q^{24} + 3q^{25} + 3q^{27} - 10q^{28} + 3q^{30} - 16q^{31} + 3q^{32} - 8q^{33} - 3q^{34} - 10q^{35} + 3q^{36} - 10q^{37} - 11q^{38} + 3q^{40} - 5q^{41} - 10q^{42} - 3q^{43} - 8q^{44} + 3q^{45} - 9q^{46} - 3q^{47} + 3q^{48} + 17q^{49} + 3q^{50} - 3q^{51} - 17q^{53} + 3q^{54} - 8q^{55} - 10q^{56} - 11q^{57} - 11q^{59} + 3q^{60} + 11q^{61} - 16q^{62} - 10q^{63} + 3q^{64} - 8q^{66} - 15q^{67} - 3q^{68} - 9q^{69} - 10q^{70} - 13q^{71} + 3q^{72} + q^{73} - 10q^{74} + 3q^{75} - 11q^{76} + 29q^{77} - 3q^{79} + 3q^{80} + 3q^{81} - 5q^{82} - 12q^{83} - 10q^{84} - 3q^{85} - 3q^{86} - 8q^{88} - q^{89} + 3q^{90} - 9q^{92} - 16q^{93} - 3q^{94} - 11q^{95} + 3q^{96} + 6q^{97} + 17q^{98} - 8q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −1.75302 −0.662579 −0.331290 0.943529i \(-0.607484\pi\)
−0.331290 + 0.943529i \(0.607484\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −1.19806 −0.361229 −0.180615 0.983554i \(-0.557809\pi\)
−0.180615 + 0.983554i \(0.557809\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.75302 −0.468514
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −6.76271 −1.64020 −0.820099 0.572222i \(-0.806082\pi\)
−0.820099 + 0.572222i \(0.806082\pi\)
\(18\) 1.00000 0.235702
\(19\) −8.63102 −1.98009 −0.990046 0.140743i \(-0.955051\pi\)
−0.990046 + 0.140743i \(0.955051\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.75302 −0.382540
\(22\) −1.19806 −0.255428
\(23\) −6.04892 −1.26129 −0.630643 0.776073i \(-0.717209\pi\)
−0.630643 + 0.776073i \(0.717209\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.75302 −0.331290
\(29\) 4.07069 0.755908 0.377954 0.925824i \(-0.376628\pi\)
0.377954 + 0.925824i \(0.376628\pi\)
\(30\) 1.00000 0.182574
\(31\) −6.80194 −1.22166 −0.610832 0.791760i \(-0.709165\pi\)
−0.610832 + 0.791760i \(0.709165\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.19806 −0.208556
\(34\) −6.76271 −1.15980
\(35\) −1.75302 −0.296315
\(36\) 1.00000 0.166667
\(37\) 4.34481 0.714283 0.357142 0.934050i \(-0.383751\pi\)
0.357142 + 0.934050i \(0.383751\pi\)
\(38\) −8.63102 −1.40014
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −6.63102 −1.03559 −0.517796 0.855504i \(-0.673247\pi\)
−0.517796 + 0.855504i \(0.673247\pi\)
\(42\) −1.75302 −0.270497
\(43\) 4.07606 0.621594 0.310797 0.950476i \(-0.399404\pi\)
0.310797 + 0.950476i \(0.399404\pi\)
\(44\) −1.19806 −0.180615
\(45\) 1.00000 0.149071
\(46\) −6.04892 −0.891864
\(47\) 11.1957 1.63306 0.816528 0.577306i \(-0.195896\pi\)
0.816528 + 0.577306i \(0.195896\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.92692 −0.560988
\(50\) 1.00000 0.141421
\(51\) −6.76271 −0.946969
\(52\) 0 0
\(53\) −1.14914 −0.157847 −0.0789236 0.996881i \(-0.525148\pi\)
−0.0789236 + 0.996881i \(0.525148\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.19806 −0.161547
\(56\) −1.75302 −0.234257
\(57\) −8.63102 −1.14321
\(58\) 4.07069 0.534507
\(59\) −13.7235 −1.78664 −0.893322 0.449416i \(-0.851632\pi\)
−0.893322 + 0.449416i \(0.851632\pi\)
\(60\) 1.00000 0.129099
\(61\) −1.18598 −0.151849 −0.0759246 0.997114i \(-0.524191\pi\)
−0.0759246 + 0.997114i \(0.524191\pi\)
\(62\) −6.80194 −0.863847
\(63\) −1.75302 −0.220860
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.19806 −0.147471
\(67\) −14.1468 −1.72830 −0.864150 0.503234i \(-0.832144\pi\)
−0.864150 + 0.503234i \(0.832144\pi\)
\(68\) −6.76271 −0.820099
\(69\) −6.04892 −0.728204
\(70\) −1.75302 −0.209526
\(71\) 2.32304 0.275695 0.137847 0.990453i \(-0.455982\pi\)
0.137847 + 0.990453i \(0.455982\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.59850 −0.187090 −0.0935451 0.995615i \(-0.529820\pi\)
−0.0935451 + 0.995615i \(0.529820\pi\)
\(74\) 4.34481 0.505074
\(75\) 1.00000 0.115470
\(76\) −8.63102 −0.990046
\(77\) 2.10023 0.239343
\(78\) 0 0
\(79\) 3.74094 0.420888 0.210444 0.977606i \(-0.432509\pi\)
0.210444 + 0.977606i \(0.432509\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −6.63102 −0.732274
\(83\) −8.40581 −0.922658 −0.461329 0.887229i \(-0.652627\pi\)
−0.461329 + 0.887229i \(0.652627\pi\)
\(84\) −1.75302 −0.191270
\(85\) −6.76271 −0.733519
\(86\) 4.07606 0.439533
\(87\) 4.07069 0.436424
\(88\) −1.19806 −0.127714
\(89\) −4.49934 −0.476929 −0.238464 0.971151i \(-0.576644\pi\)
−0.238464 + 0.971151i \(0.576644\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −6.04892 −0.630643
\(93\) −6.80194 −0.705328
\(94\) 11.1957 1.15475
\(95\) −8.63102 −0.885524
\(96\) 1.00000 0.102062
\(97\) 10.7952 1.09609 0.548045 0.836449i \(-0.315372\pi\)
0.548045 + 0.836449i \(0.315372\pi\)
\(98\) −3.92692 −0.396679
\(99\) −1.19806 −0.120410
\(100\) 1.00000 0.100000
\(101\) 16.7875 1.67042 0.835208 0.549935i \(-0.185347\pi\)
0.835208 + 0.549935i \(0.185347\pi\)
\(102\) −6.76271 −0.669608
\(103\) −4.16421 −0.410312 −0.205156 0.978729i \(-0.565770\pi\)
−0.205156 + 0.978729i \(0.565770\pi\)
\(104\) 0 0
\(105\) −1.75302 −0.171077
\(106\) −1.14914 −0.111615
\(107\) 12.9661 1.25348 0.626742 0.779226i \(-0.284388\pi\)
0.626742 + 0.779226i \(0.284388\pi\)
\(108\) 1.00000 0.0962250
\(109\) 19.4644 1.86435 0.932177 0.362004i \(-0.117907\pi\)
0.932177 + 0.362004i \(0.117907\pi\)
\(110\) −1.19806 −0.114231
\(111\) 4.34481 0.412392
\(112\) −1.75302 −0.165645
\(113\) 0.515729 0.0485157 0.0242579 0.999706i \(-0.492278\pi\)
0.0242579 + 0.999706i \(0.492278\pi\)
\(114\) −8.63102 −0.808369
\(115\) −6.04892 −0.564064
\(116\) 4.07069 0.377954
\(117\) 0 0
\(118\) −13.7235 −1.26335
\(119\) 11.8552 1.08676
\(120\) 1.00000 0.0912871
\(121\) −9.56465 −0.869513
\(122\) −1.18598 −0.107374
\(123\) −6.63102 −0.597899
\(124\) −6.80194 −0.610832
\(125\) 1.00000 0.0894427
\(126\) −1.75302 −0.156171
\(127\) 7.96077 0.706404 0.353202 0.935547i \(-0.385093\pi\)
0.353202 + 0.935547i \(0.385093\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.07606 0.358877
\(130\) 0 0
\(131\) −5.08815 −0.444553 −0.222277 0.974984i \(-0.571349\pi\)
−0.222277 + 0.974984i \(0.571349\pi\)
\(132\) −1.19806 −0.104278
\(133\) 15.1304 1.31197
\(134\) −14.1468 −1.22209
\(135\) 1.00000 0.0860663
\(136\) −6.76271 −0.579898
\(137\) −0.674563 −0.0576318 −0.0288159 0.999585i \(-0.509174\pi\)
−0.0288159 + 0.999585i \(0.509174\pi\)
\(138\) −6.04892 −0.514918
\(139\) 8.26205 0.700778 0.350389 0.936604i \(-0.386049\pi\)
0.350389 + 0.936604i \(0.386049\pi\)
\(140\) −1.75302 −0.148157
\(141\) 11.1957 0.942845
\(142\) 2.32304 0.194946
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.07069 0.338052
\(146\) −1.59850 −0.132293
\(147\) −3.92692 −0.323887
\(148\) 4.34481 0.357142
\(149\) 8.19136 0.671062 0.335531 0.942029i \(-0.391084\pi\)
0.335531 + 0.942029i \(0.391084\pi\)
\(150\) 1.00000 0.0816497
\(151\) 5.72348 0.465770 0.232885 0.972504i \(-0.425183\pi\)
0.232885 + 0.972504i \(0.425183\pi\)
\(152\) −8.63102 −0.700068
\(153\) −6.76271 −0.546733
\(154\) 2.10023 0.169241
\(155\) −6.80194 −0.546345
\(156\) 0 0
\(157\) 3.09246 0.246805 0.123403 0.992357i \(-0.460619\pi\)
0.123403 + 0.992357i \(0.460619\pi\)
\(158\) 3.74094 0.297613
\(159\) −1.14914 −0.0911331
\(160\) 1.00000 0.0790569
\(161\) 10.6039 0.835702
\(162\) 1.00000 0.0785674
\(163\) −23.9541 −1.87623 −0.938114 0.346328i \(-0.887429\pi\)
−0.938114 + 0.346328i \(0.887429\pi\)
\(164\) −6.63102 −0.517796
\(165\) −1.19806 −0.0932690
\(166\) −8.40581 −0.652418
\(167\) 1.19806 0.0927088 0.0463544 0.998925i \(-0.485240\pi\)
0.0463544 + 0.998925i \(0.485240\pi\)
\(168\) −1.75302 −0.135248
\(169\) 0 0
\(170\) −6.76271 −0.518676
\(171\) −8.63102 −0.660031
\(172\) 4.07606 0.310797
\(173\) −6.63773 −0.504657 −0.252329 0.967642i \(-0.581196\pi\)
−0.252329 + 0.967642i \(0.581196\pi\)
\(174\) 4.07069 0.308598
\(175\) −1.75302 −0.132516
\(176\) −1.19806 −0.0903073
\(177\) −13.7235 −1.03152
\(178\) −4.49934 −0.337239
\(179\) 19.6353 1.46761 0.733807 0.679358i \(-0.237742\pi\)
0.733807 + 0.679358i \(0.237742\pi\)
\(180\) 1.00000 0.0745356
\(181\) 5.25906 0.390903 0.195451 0.980713i \(-0.437383\pi\)
0.195451 + 0.980713i \(0.437383\pi\)
\(182\) 0 0
\(183\) −1.18598 −0.0876702
\(184\) −6.04892 −0.445932
\(185\) 4.34481 0.319437
\(186\) −6.80194 −0.498742
\(187\) 8.10215 0.592488
\(188\) 11.1957 0.816528
\(189\) −1.75302 −0.127513
\(190\) −8.63102 −0.626160
\(191\) 4.45473 0.322333 0.161166 0.986927i \(-0.448474\pi\)
0.161166 + 0.986927i \(0.448474\pi\)
\(192\) 1.00000 0.0721688
\(193\) 0.818331 0.0589048 0.0294524 0.999566i \(-0.490624\pi\)
0.0294524 + 0.999566i \(0.490624\pi\)
\(194\) 10.7952 0.775053
\(195\) 0 0
\(196\) −3.92692 −0.280494
\(197\) −12.7385 −0.907584 −0.453792 0.891108i \(-0.649929\pi\)
−0.453792 + 0.891108i \(0.649929\pi\)
\(198\) −1.19806 −0.0851426
\(199\) −4.02475 −0.285307 −0.142654 0.989773i \(-0.545563\pi\)
−0.142654 + 0.989773i \(0.545563\pi\)
\(200\) 1.00000 0.0707107
\(201\) −14.1468 −0.997835
\(202\) 16.7875 1.18116
\(203\) −7.13600 −0.500849
\(204\) −6.76271 −0.473484
\(205\) −6.63102 −0.463131
\(206\) −4.16421 −0.290134
\(207\) −6.04892 −0.420429
\(208\) 0 0
\(209\) 10.3405 0.715268
\(210\) −1.75302 −0.120970
\(211\) 3.10992 0.214095 0.107048 0.994254i \(-0.465860\pi\)
0.107048 + 0.994254i \(0.465860\pi\)
\(212\) −1.14914 −0.0789236
\(213\) 2.32304 0.159172
\(214\) 12.9661 0.886348
\(215\) 4.07606 0.277985
\(216\) 1.00000 0.0680414
\(217\) 11.9239 0.809449
\(218\) 19.4644 1.31830
\(219\) −1.59850 −0.108017
\(220\) −1.19806 −0.0807733
\(221\) 0 0
\(222\) 4.34481 0.291605
\(223\) −14.3787 −0.962867 −0.481433 0.876483i \(-0.659884\pi\)
−0.481433 + 0.876483i \(0.659884\pi\)
\(224\) −1.75302 −0.117129
\(225\) 1.00000 0.0666667
\(226\) 0.515729 0.0343058
\(227\) 5.97716 0.396718 0.198359 0.980129i \(-0.436439\pi\)
0.198359 + 0.980129i \(0.436439\pi\)
\(228\) −8.63102 −0.571603
\(229\) 15.9172 1.05184 0.525920 0.850534i \(-0.323721\pi\)
0.525920 + 0.850534i \(0.323721\pi\)
\(230\) −6.04892 −0.398854
\(231\) 2.10023 0.138185
\(232\) 4.07069 0.267254
\(233\) −6.92154 −0.453445 −0.226723 0.973959i \(-0.572801\pi\)
−0.226723 + 0.973959i \(0.572801\pi\)
\(234\) 0 0
\(235\) 11.1957 0.730325
\(236\) −13.7235 −0.893322
\(237\) 3.74094 0.243000
\(238\) 11.8552 0.768456
\(239\) −30.1075 −1.94749 −0.973747 0.227633i \(-0.926901\pi\)
−0.973747 + 0.227633i \(0.926901\pi\)
\(240\) 1.00000 0.0645497
\(241\) −12.0562 −0.776609 −0.388304 0.921531i \(-0.626939\pi\)
−0.388304 + 0.921531i \(0.626939\pi\)
\(242\) −9.56465 −0.614839
\(243\) 1.00000 0.0641500
\(244\) −1.18598 −0.0759246
\(245\) −3.92692 −0.250882
\(246\) −6.63102 −0.422779
\(247\) 0 0
\(248\) −6.80194 −0.431923
\(249\) −8.40581 −0.532697
\(250\) 1.00000 0.0632456
\(251\) −14.8170 −0.935241 −0.467620 0.883929i \(-0.654889\pi\)
−0.467620 + 0.883929i \(0.654889\pi\)
\(252\) −1.75302 −0.110430
\(253\) 7.24698 0.455614
\(254\) 7.96077 0.499503
\(255\) −6.76271 −0.423497
\(256\) 1.00000 0.0625000
\(257\) −10.5743 −0.659609 −0.329804 0.944049i \(-0.606983\pi\)
−0.329804 + 0.944049i \(0.606983\pi\)
\(258\) 4.07606 0.253765
\(259\) −7.61655 −0.473269
\(260\) 0 0
\(261\) 4.07069 0.251969
\(262\) −5.08815 −0.314347
\(263\) 11.9922 0.739473 0.369736 0.929137i \(-0.379448\pi\)
0.369736 + 0.929137i \(0.379448\pi\)
\(264\) −1.19806 −0.0737356
\(265\) −1.14914 −0.0705914
\(266\) 15.1304 0.927702
\(267\) −4.49934 −0.275355
\(268\) −14.1468 −0.864150
\(269\) −25.4359 −1.55086 −0.775428 0.631435i \(-0.782466\pi\)
−0.775428 + 0.631435i \(0.782466\pi\)
\(270\) 1.00000 0.0608581
\(271\) −20.3666 −1.23718 −0.618591 0.785713i \(-0.712296\pi\)
−0.618591 + 0.785713i \(0.712296\pi\)
\(272\) −6.76271 −0.410049
\(273\) 0 0
\(274\) −0.674563 −0.0407518
\(275\) −1.19806 −0.0722459
\(276\) −6.04892 −0.364102
\(277\) 20.1323 1.20963 0.604816 0.796366i \(-0.293247\pi\)
0.604816 + 0.796366i \(0.293247\pi\)
\(278\) 8.26205 0.495525
\(279\) −6.80194 −0.407221
\(280\) −1.75302 −0.104763
\(281\) 29.7265 1.77333 0.886666 0.462410i \(-0.153015\pi\)
0.886666 + 0.462410i \(0.153015\pi\)
\(282\) 11.1957 0.666692
\(283\) 8.84117 0.525553 0.262776 0.964857i \(-0.415362\pi\)
0.262776 + 0.964857i \(0.415362\pi\)
\(284\) 2.32304 0.137847
\(285\) −8.63102 −0.511258
\(286\) 0 0
\(287\) 11.6243 0.686162
\(288\) 1.00000 0.0589256
\(289\) 28.7342 1.69025
\(290\) 4.07069 0.239039
\(291\) 10.7952 0.632828
\(292\) −1.59850 −0.0935451
\(293\) −20.5918 −1.20299 −0.601493 0.798878i \(-0.705427\pi\)
−0.601493 + 0.798878i \(0.705427\pi\)
\(294\) −3.92692 −0.229023
\(295\) −13.7235 −0.799012
\(296\) 4.34481 0.252537
\(297\) −1.19806 −0.0695186
\(298\) 8.19136 0.474513
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −7.14542 −0.411855
\(302\) 5.72348 0.329349
\(303\) 16.7875 0.964415
\(304\) −8.63102 −0.495023
\(305\) −1.18598 −0.0679091
\(306\) −6.76271 −0.386598
\(307\) −28.7265 −1.63951 −0.819753 0.572717i \(-0.805889\pi\)
−0.819753 + 0.572717i \(0.805889\pi\)
\(308\) 2.10023 0.119672
\(309\) −4.16421 −0.236894
\(310\) −6.80194 −0.386324
\(311\) −18.6329 −1.05658 −0.528289 0.849065i \(-0.677166\pi\)
−0.528289 + 0.849065i \(0.677166\pi\)
\(312\) 0 0
\(313\) −29.3696 −1.66007 −0.830033 0.557714i \(-0.811679\pi\)
−0.830033 + 0.557714i \(0.811679\pi\)
\(314\) 3.09246 0.174517
\(315\) −1.75302 −0.0987715
\(316\) 3.74094 0.210444
\(317\) −20.8092 −1.16876 −0.584381 0.811479i \(-0.698663\pi\)
−0.584381 + 0.811479i \(0.698663\pi\)
\(318\) −1.14914 −0.0644408
\(319\) −4.87694 −0.273056
\(320\) 1.00000 0.0559017
\(321\) 12.9661 0.723700
\(322\) 10.6039 0.590931
\(323\) 58.3691 3.24774
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −23.9541 −1.32669
\(327\) 19.4644 1.07638
\(328\) −6.63102 −0.366137
\(329\) −19.6262 −1.08203
\(330\) −1.19806 −0.0659512
\(331\) 11.7603 0.646405 0.323203 0.946330i \(-0.395240\pi\)
0.323203 + 0.946330i \(0.395240\pi\)
\(332\) −8.40581 −0.461329
\(333\) 4.34481 0.238094
\(334\) 1.19806 0.0655551
\(335\) −14.1468 −0.772920
\(336\) −1.75302 −0.0956351
\(337\) −26.8732 −1.46388 −0.731939 0.681371i \(-0.761384\pi\)
−0.731939 + 0.681371i \(0.761384\pi\)
\(338\) 0 0
\(339\) 0.515729 0.0280106
\(340\) −6.76271 −0.366759
\(341\) 8.14914 0.441301
\(342\) −8.63102 −0.466712
\(343\) 19.1551 1.03428
\(344\) 4.07606 0.219767
\(345\) −6.04892 −0.325663
\(346\) −6.63773 −0.356846
\(347\) 29.5080 1.58407 0.792035 0.610476i \(-0.209022\pi\)
0.792035 + 0.610476i \(0.209022\pi\)
\(348\) 4.07069 0.218212
\(349\) −17.0519 −0.912767 −0.456384 0.889783i \(-0.650856\pi\)
−0.456384 + 0.889783i \(0.650856\pi\)
\(350\) −1.75302 −0.0937029
\(351\) 0 0
\(352\) −1.19806 −0.0638569
\(353\) 21.8629 1.16365 0.581823 0.813315i \(-0.302340\pi\)
0.581823 + 0.813315i \(0.302340\pi\)
\(354\) −13.7235 −0.729395
\(355\) 2.32304 0.123294
\(356\) −4.49934 −0.238464
\(357\) 11.8552 0.627442
\(358\) 19.6353 1.03776
\(359\) −14.3026 −0.754862 −0.377431 0.926038i \(-0.623193\pi\)
−0.377431 + 0.926038i \(0.623193\pi\)
\(360\) 1.00000 0.0527046
\(361\) 55.4946 2.92077
\(362\) 5.25906 0.276410
\(363\) −9.56465 −0.502014
\(364\) 0 0
\(365\) −1.59850 −0.0836692
\(366\) −1.18598 −0.0619922
\(367\) −11.2416 −0.586807 −0.293403 0.955989i \(-0.594788\pi\)
−0.293403 + 0.955989i \(0.594788\pi\)
\(368\) −6.04892 −0.315322
\(369\) −6.63102 −0.345197
\(370\) 4.34481 0.225876
\(371\) 2.01447 0.104586
\(372\) −6.80194 −0.352664
\(373\) −9.70171 −0.502336 −0.251168 0.967944i \(-0.580815\pi\)
−0.251168 + 0.967944i \(0.580815\pi\)
\(374\) 8.10215 0.418952
\(375\) 1.00000 0.0516398
\(376\) 11.1957 0.577373
\(377\) 0 0
\(378\) −1.75302 −0.0901656
\(379\) 2.82802 0.145266 0.0726328 0.997359i \(-0.476860\pi\)
0.0726328 + 0.997359i \(0.476860\pi\)
\(380\) −8.63102 −0.442762
\(381\) 7.96077 0.407843
\(382\) 4.45473 0.227924
\(383\) −11.6558 −0.595582 −0.297791 0.954631i \(-0.596250\pi\)
−0.297791 + 0.954631i \(0.596250\pi\)
\(384\) 1.00000 0.0510310
\(385\) 2.10023 0.107038
\(386\) 0.818331 0.0416520
\(387\) 4.07606 0.207198
\(388\) 10.7952 0.548045
\(389\) 12.5157 0.634573 0.317286 0.948330i \(-0.397228\pi\)
0.317286 + 0.948330i \(0.397228\pi\)
\(390\) 0 0
\(391\) 40.9071 2.06876
\(392\) −3.92692 −0.198339
\(393\) −5.08815 −0.256663
\(394\) −12.7385 −0.641759
\(395\) 3.74094 0.188227
\(396\) −1.19806 −0.0602049
\(397\) −21.6136 −1.08475 −0.542377 0.840135i \(-0.682475\pi\)
−0.542377 + 0.840135i \(0.682475\pi\)
\(398\) −4.02475 −0.201743
\(399\) 15.1304 0.757465
\(400\) 1.00000 0.0500000
\(401\) 11.5157 0.575068 0.287534 0.957770i \(-0.407165\pi\)
0.287534 + 0.957770i \(0.407165\pi\)
\(402\) −14.1468 −0.705576
\(403\) 0 0
\(404\) 16.7875 0.835208
\(405\) 1.00000 0.0496904
\(406\) −7.13600 −0.354154
\(407\) −5.20536 −0.258020
\(408\) −6.76271 −0.334804
\(409\) 0.0991626 0.00490328 0.00245164 0.999997i \(-0.499220\pi\)
0.00245164 + 0.999997i \(0.499220\pi\)
\(410\) −6.63102 −0.327483
\(411\) −0.674563 −0.0332737
\(412\) −4.16421 −0.205156
\(413\) 24.0575 1.18379
\(414\) −6.04892 −0.297288
\(415\) −8.40581 −0.412625
\(416\) 0 0
\(417\) 8.26205 0.404594
\(418\) 10.3405 0.505771
\(419\) 31.2784 1.52805 0.764026 0.645186i \(-0.223220\pi\)
0.764026 + 0.645186i \(0.223220\pi\)
\(420\) −1.75302 −0.0855386
\(421\) 38.0452 1.85421 0.927105 0.374802i \(-0.122289\pi\)
0.927105 + 0.374802i \(0.122289\pi\)
\(422\) 3.10992 0.151388
\(423\) 11.1957 0.544352
\(424\) −1.14914 −0.0558074
\(425\) −6.76271 −0.328040
\(426\) 2.32304 0.112552
\(427\) 2.07905 0.100612
\(428\) 12.9661 0.626742
\(429\) 0 0
\(430\) 4.07606 0.196565
\(431\) 14.4614 0.696583 0.348291 0.937386i \(-0.386762\pi\)
0.348291 + 0.937386i \(0.386762\pi\)
\(432\) 1.00000 0.0481125
\(433\) 21.6209 1.03903 0.519516 0.854461i \(-0.326112\pi\)
0.519516 + 0.854461i \(0.326112\pi\)
\(434\) 11.9239 0.572367
\(435\) 4.07069 0.195175
\(436\) 19.4644 0.932177
\(437\) 52.2083 2.49746
\(438\) −1.59850 −0.0763792
\(439\) −3.23623 −0.154457 −0.0772283 0.997013i \(-0.524607\pi\)
−0.0772283 + 0.997013i \(0.524607\pi\)
\(440\) −1.19806 −0.0571154
\(441\) −3.92692 −0.186996
\(442\) 0 0
\(443\) −14.6649 −0.696749 −0.348375 0.937355i \(-0.613266\pi\)
−0.348375 + 0.937355i \(0.613266\pi\)
\(444\) 4.34481 0.206196
\(445\) −4.49934 −0.213289
\(446\) −14.3787 −0.680850
\(447\) 8.19136 0.387438
\(448\) −1.75302 −0.0828224
\(449\) −4.08516 −0.192791 −0.0963954 0.995343i \(-0.530731\pi\)
−0.0963954 + 0.995343i \(0.530731\pi\)
\(450\) 1.00000 0.0471405
\(451\) 7.94438 0.374086
\(452\) 0.515729 0.0242579
\(453\) 5.72348 0.268913
\(454\) 5.97716 0.280522
\(455\) 0 0
\(456\) −8.63102 −0.404185
\(457\) −41.8678 −1.95849 −0.979247 0.202668i \(-0.935039\pi\)
−0.979247 + 0.202668i \(0.935039\pi\)
\(458\) 15.9172 0.743763
\(459\) −6.76271 −0.315656
\(460\) −6.04892 −0.282032
\(461\) 0.215521 0.0100378 0.00501890 0.999987i \(-0.498402\pi\)
0.00501890 + 0.999987i \(0.498402\pi\)
\(462\) 2.10023 0.0977114
\(463\) −23.4795 −1.09118 −0.545592 0.838051i \(-0.683695\pi\)
−0.545592 + 0.838051i \(0.683695\pi\)
\(464\) 4.07069 0.188977
\(465\) −6.80194 −0.315432
\(466\) −6.92154 −0.320634
\(467\) 10.1153 0.468080 0.234040 0.972227i \(-0.424805\pi\)
0.234040 + 0.972227i \(0.424805\pi\)
\(468\) 0 0
\(469\) 24.7995 1.14514
\(470\) 11.1957 0.516418
\(471\) 3.09246 0.142493
\(472\) −13.7235 −0.631674
\(473\) −4.88338 −0.224538
\(474\) 3.74094 0.171827
\(475\) −8.63102 −0.396018
\(476\) 11.8552 0.543381
\(477\) −1.14914 −0.0526157
\(478\) −30.1075 −1.37709
\(479\) −0.980623 −0.0448058 −0.0224029 0.999749i \(-0.507132\pi\)
−0.0224029 + 0.999749i \(0.507132\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −12.0562 −0.549145
\(483\) 10.6039 0.482493
\(484\) −9.56465 −0.434757
\(485\) 10.7952 0.490186
\(486\) 1.00000 0.0453609
\(487\) −7.18300 −0.325493 −0.162746 0.986668i \(-0.552035\pi\)
−0.162746 + 0.986668i \(0.552035\pi\)
\(488\) −1.18598 −0.0536868
\(489\) −23.9541 −1.08324
\(490\) −3.92692 −0.177400
\(491\) 24.5918 1.10981 0.554906 0.831913i \(-0.312754\pi\)
0.554906 + 0.831913i \(0.312754\pi\)
\(492\) −6.63102 −0.298950
\(493\) −27.5289 −1.23984
\(494\) 0 0
\(495\) −1.19806 −0.0538489
\(496\) −6.80194 −0.305416
\(497\) −4.07234 −0.182670
\(498\) −8.40581 −0.376673
\(499\) −30.1608 −1.35018 −0.675090 0.737735i \(-0.735895\pi\)
−0.675090 + 0.737735i \(0.735895\pi\)
\(500\) 1.00000 0.0447214
\(501\) 1.19806 0.0535255
\(502\) −14.8170 −0.661315
\(503\) −33.8883 −1.51100 −0.755502 0.655146i \(-0.772607\pi\)
−0.755502 + 0.655146i \(0.772607\pi\)
\(504\) −1.75302 −0.0780857
\(505\) 16.7875 0.747032
\(506\) 7.24698 0.322168
\(507\) 0 0
\(508\) 7.96077 0.353202
\(509\) −14.1089 −0.625364 −0.312682 0.949858i \(-0.601227\pi\)
−0.312682 + 0.949858i \(0.601227\pi\)
\(510\) −6.76271 −0.299458
\(511\) 2.80220 0.123962
\(512\) 1.00000 0.0441942
\(513\) −8.63102 −0.381069
\(514\) −10.5743 −0.466414
\(515\) −4.16421 −0.183497
\(516\) 4.07606 0.179439
\(517\) −13.4131 −0.589908
\(518\) −7.61655 −0.334652
\(519\) −6.63773 −0.291364
\(520\) 0 0
\(521\) 17.3937 0.762033 0.381017 0.924568i \(-0.375574\pi\)
0.381017 + 0.924568i \(0.375574\pi\)
\(522\) 4.07069 0.178169
\(523\) −34.9879 −1.52991 −0.764957 0.644081i \(-0.777240\pi\)
−0.764957 + 0.644081i \(0.777240\pi\)
\(524\) −5.08815 −0.222277
\(525\) −1.75302 −0.0765081
\(526\) 11.9922 0.522886
\(527\) 45.9995 2.00377
\(528\) −1.19806 −0.0521390
\(529\) 13.5894 0.590844
\(530\) −1.14914 −0.0499157
\(531\) −13.7235 −0.595548
\(532\) 15.1304 0.655984
\(533\) 0 0
\(534\) −4.49934 −0.194705
\(535\) 12.9661 0.560575
\(536\) −14.1468 −0.611047
\(537\) 19.6353 0.847327
\(538\) −25.4359 −1.09662
\(539\) 4.70469 0.202646
\(540\) 1.00000 0.0430331
\(541\) −20.8689 −0.897224 −0.448612 0.893727i \(-0.648082\pi\)
−0.448612 + 0.893727i \(0.648082\pi\)
\(542\) −20.3666 −0.874820
\(543\) 5.25906 0.225688
\(544\) −6.76271 −0.289949
\(545\) 19.4644 0.833764
\(546\) 0 0
\(547\) 5.67456 0.242627 0.121313 0.992614i \(-0.461289\pi\)
0.121313 + 0.992614i \(0.461289\pi\)
\(548\) −0.674563 −0.0288159
\(549\) −1.18598 −0.0506164
\(550\) −1.19806 −0.0510855
\(551\) −35.1342 −1.49677
\(552\) −6.04892 −0.257459
\(553\) −6.55794 −0.278872
\(554\) 20.1323 0.855339
\(555\) 4.34481 0.184427
\(556\) 8.26205 0.350389
\(557\) 14.7724 0.625927 0.312963 0.949765i \(-0.398678\pi\)
0.312963 + 0.949765i \(0.398678\pi\)
\(558\) −6.80194 −0.287949
\(559\) 0 0
\(560\) −1.75302 −0.0740786
\(561\) 8.10215 0.342073
\(562\) 29.7265 1.25394
\(563\) 18.0398 0.760288 0.380144 0.924927i \(-0.375874\pi\)
0.380144 + 0.924927i \(0.375874\pi\)
\(564\) 11.1957 0.471423
\(565\) 0.515729 0.0216969
\(566\) 8.84117 0.371622
\(567\) −1.75302 −0.0736199
\(568\) 2.32304 0.0974728
\(569\) −11.4886 −0.481626 −0.240813 0.970571i \(-0.577414\pi\)
−0.240813 + 0.970571i \(0.577414\pi\)
\(570\) −8.63102 −0.361514
\(571\) −44.3062 −1.85416 −0.927078 0.374869i \(-0.877688\pi\)
−0.927078 + 0.374869i \(0.877688\pi\)
\(572\) 0 0
\(573\) 4.45473 0.186099
\(574\) 11.6243 0.485190
\(575\) −6.04892 −0.252257
\(576\) 1.00000 0.0416667
\(577\) −28.0000 −1.16566 −0.582828 0.812596i \(-0.698054\pi\)
−0.582828 + 0.812596i \(0.698054\pi\)
\(578\) 28.7342 1.19519
\(579\) 0.818331 0.0340087
\(580\) 4.07069 0.169026
\(581\) 14.7356 0.611334
\(582\) 10.7952 0.447477
\(583\) 1.37675 0.0570190
\(584\) −1.59850 −0.0661463
\(585\) 0 0
\(586\) −20.5918 −0.850639
\(587\) 35.4316 1.46242 0.731210 0.682152i \(-0.238956\pi\)
0.731210 + 0.682152i \(0.238956\pi\)
\(588\) −3.92692 −0.161943
\(589\) 58.7077 2.41901
\(590\) −13.7235 −0.564987
\(591\) −12.7385 −0.523994
\(592\) 4.34481 0.178571
\(593\) −2.19567 −0.0901653 −0.0450827 0.998983i \(-0.514355\pi\)
−0.0450827 + 0.998983i \(0.514355\pi\)
\(594\) −1.19806 −0.0491571
\(595\) 11.8552 0.486014
\(596\) 8.19136 0.335531
\(597\) −4.02475 −0.164722
\(598\) 0 0
\(599\) −31.1855 −1.27421 −0.637103 0.770779i \(-0.719867\pi\)
−0.637103 + 0.770779i \(0.719867\pi\)
\(600\) 1.00000 0.0408248
\(601\) −24.2435 −0.988914 −0.494457 0.869202i \(-0.664633\pi\)
−0.494457 + 0.869202i \(0.664633\pi\)
\(602\) −7.14542 −0.291226
\(603\) −14.1468 −0.576100
\(604\) 5.72348 0.232885
\(605\) −9.56465 −0.388858
\(606\) 16.7875 0.681944
\(607\) −17.7235 −0.719374 −0.359687 0.933073i \(-0.617117\pi\)
−0.359687 + 0.933073i \(0.617117\pi\)
\(608\) −8.63102 −0.350034
\(609\) −7.13600 −0.289165
\(610\) −1.18598 −0.0480190
\(611\) 0 0
\(612\) −6.76271 −0.273366
\(613\) 1.71140 0.0691227 0.0345614 0.999403i \(-0.488997\pi\)
0.0345614 + 0.999403i \(0.488997\pi\)
\(614\) −28.7265 −1.15931
\(615\) −6.63102 −0.267389
\(616\) 2.10023 0.0846206
\(617\) 0.569433 0.0229245 0.0114622 0.999934i \(-0.496351\pi\)
0.0114622 + 0.999934i \(0.496351\pi\)
\(618\) −4.16421 −0.167509
\(619\) 26.4282 1.06224 0.531119 0.847297i \(-0.321772\pi\)
0.531119 + 0.847297i \(0.321772\pi\)
\(620\) −6.80194 −0.273172
\(621\) −6.04892 −0.242735
\(622\) −18.6329 −0.747113
\(623\) 7.88743 0.316003
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −29.3696 −1.17384
\(627\) 10.3405 0.412960
\(628\) 3.09246 0.123403
\(629\) −29.3827 −1.17157
\(630\) −1.75302 −0.0698420
\(631\) 38.0006 1.51278 0.756390 0.654121i \(-0.226961\pi\)
0.756390 + 0.654121i \(0.226961\pi\)
\(632\) 3.74094 0.148807
\(633\) 3.10992 0.123608
\(634\) −20.8092 −0.826440
\(635\) 7.96077 0.315914
\(636\) −1.14914 −0.0455666
\(637\) 0 0
\(638\) −4.87694 −0.193080
\(639\) 2.32304 0.0918982
\(640\) 1.00000 0.0395285
\(641\) −25.9168 −1.02365 −0.511825 0.859090i \(-0.671030\pi\)
−0.511825 + 0.859090i \(0.671030\pi\)
\(642\) 12.9661 0.511733
\(643\) 8.52243 0.336092 0.168046 0.985779i \(-0.446254\pi\)
0.168046 + 0.985779i \(0.446254\pi\)
\(644\) 10.6039 0.417851
\(645\) 4.07606 0.160495
\(646\) 58.3691 2.29650
\(647\) 37.9318 1.49125 0.745627 0.666364i \(-0.232150\pi\)
0.745627 + 0.666364i \(0.232150\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.4416 0.645389
\(650\) 0 0
\(651\) 11.9239 0.467336
\(652\) −23.9541 −0.938114
\(653\) −6.07846 −0.237868 −0.118934 0.992902i \(-0.537948\pi\)
−0.118934 + 0.992902i \(0.537948\pi\)
\(654\) 19.4644 0.761119
\(655\) −5.08815 −0.198810
\(656\) −6.63102 −0.258898
\(657\) −1.59850 −0.0623634
\(658\) −19.6262 −0.765110
\(659\) 38.9162 1.51596 0.757979 0.652279i \(-0.226187\pi\)
0.757979 + 0.652279i \(0.226187\pi\)
\(660\) −1.19806 −0.0466345
\(661\) −25.9191 −1.00814 −0.504069 0.863663i \(-0.668164\pi\)
−0.504069 + 0.863663i \(0.668164\pi\)
\(662\) 11.7603 0.457078
\(663\) 0 0
\(664\) −8.40581 −0.326209
\(665\) 15.1304 0.586730
\(666\) 4.34481 0.168358
\(667\) −24.6233 −0.953416
\(668\) 1.19806 0.0463544
\(669\) −14.3787 −0.555911
\(670\) −14.1468 −0.546537
\(671\) 1.42088 0.0548524
\(672\) −1.75302 −0.0676242
\(673\) −25.5724 −0.985744 −0.492872 0.870102i \(-0.664053\pi\)
−0.492872 + 0.870102i \(0.664053\pi\)
\(674\) −26.8732 −1.03512
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 24.7453 0.951037 0.475519 0.879706i \(-0.342260\pi\)
0.475519 + 0.879706i \(0.342260\pi\)
\(678\) 0.515729 0.0198065
\(679\) −18.9243 −0.726247
\(680\) −6.76271 −0.259338
\(681\) 5.97716 0.229045
\(682\) 8.14914 0.312047
\(683\) −22.4045 −0.857284 −0.428642 0.903474i \(-0.641008\pi\)
−0.428642 + 0.903474i \(0.641008\pi\)
\(684\) −8.63102 −0.330015
\(685\) −0.674563 −0.0257737
\(686\) 19.1551 0.731346
\(687\) 15.9172 0.607280
\(688\) 4.07606 0.155398
\(689\) 0 0
\(690\) −6.04892 −0.230278
\(691\) 16.0030 0.608782 0.304391 0.952547i \(-0.401547\pi\)
0.304391 + 0.952547i \(0.401547\pi\)
\(692\) −6.63773 −0.252329
\(693\) 2.10023 0.0797810
\(694\) 29.5080 1.12011
\(695\) 8.26205 0.313397
\(696\) 4.07069 0.154299
\(697\) 44.8437 1.69858
\(698\) −17.0519 −0.645424
\(699\) −6.92154 −0.261797
\(700\) −1.75302 −0.0662579
\(701\) 32.7222 1.23590 0.617949 0.786218i \(-0.287964\pi\)
0.617949 + 0.786218i \(0.287964\pi\)
\(702\) 0 0
\(703\) −37.5002 −1.41435
\(704\) −1.19806 −0.0451537
\(705\) 11.1957 0.421653
\(706\) 21.8629 0.822822
\(707\) −29.4288 −1.10678
\(708\) −13.7235 −0.515760
\(709\) −14.0597 −0.528022 −0.264011 0.964520i \(-0.585046\pi\)
−0.264011 + 0.964520i \(0.585046\pi\)
\(710\) 2.32304 0.0871823
\(711\) 3.74094 0.140296
\(712\) −4.49934 −0.168620
\(713\) 41.1444 1.54087
\(714\) 11.8552 0.443668
\(715\) 0 0
\(716\) 19.6353 0.733807
\(717\) −30.1075 −1.12439
\(718\) −14.3026 −0.533768
\(719\) −7.15585 −0.266868 −0.133434 0.991058i \(-0.542600\pi\)
−0.133434 + 0.991058i \(0.542600\pi\)
\(720\) 1.00000 0.0372678
\(721\) 7.29995 0.271864
\(722\) 55.4946 2.06529
\(723\) −12.0562 −0.448375
\(724\) 5.25906 0.195451
\(725\) 4.07069 0.151182
\(726\) −9.56465 −0.354977
\(727\) 36.4704 1.35261 0.676306 0.736621i \(-0.263580\pi\)
0.676306 + 0.736621i \(0.263580\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −1.59850 −0.0591631
\(731\) −27.5652 −1.01954
\(732\) −1.18598 −0.0438351
\(733\) 38.0646 1.40595 0.702974 0.711216i \(-0.251855\pi\)
0.702974 + 0.711216i \(0.251855\pi\)
\(734\) −11.2416 −0.414935
\(735\) −3.92692 −0.144847
\(736\) −6.04892 −0.222966
\(737\) 16.9487 0.624313
\(738\) −6.63102 −0.244091
\(739\) −35.5114 −1.30631 −0.653154 0.757225i \(-0.726555\pi\)
−0.653154 + 0.757225i \(0.726555\pi\)
\(740\) 4.34481 0.159719
\(741\) 0 0
\(742\) 2.01447 0.0739537
\(743\) −20.3696 −0.747287 −0.373643 0.927572i \(-0.621892\pi\)
−0.373643 + 0.927572i \(0.621892\pi\)
\(744\) −6.80194 −0.249371
\(745\) 8.19136 0.300108
\(746\) −9.70171 −0.355205
\(747\) −8.40581 −0.307553
\(748\) 8.10215 0.296244
\(749\) −22.7299 −0.830533
\(750\) 1.00000 0.0365148
\(751\) −5.58104 −0.203655 −0.101828 0.994802i \(-0.532469\pi\)
−0.101828 + 0.994802i \(0.532469\pi\)
\(752\) 11.1957 0.408264
\(753\) −14.8170 −0.539962
\(754\) 0 0
\(755\) 5.72348 0.208299
\(756\) −1.75302 −0.0637567
\(757\) 12.8267 0.466194 0.233097 0.972453i \(-0.425114\pi\)
0.233097 + 0.972453i \(0.425114\pi\)
\(758\) 2.82802 0.102718
\(759\) 7.24698 0.263049
\(760\) −8.63102 −0.313080
\(761\) 14.0084 0.507803 0.253901 0.967230i \(-0.418286\pi\)
0.253901 + 0.967230i \(0.418286\pi\)
\(762\) 7.96077 0.288388
\(763\) −34.1215 −1.23528
\(764\) 4.45473 0.161166
\(765\) −6.76271 −0.244506
\(766\) −11.6558 −0.421140
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 50.4051 1.81765 0.908827 0.417174i \(-0.136979\pi\)
0.908827 + 0.417174i \(0.136979\pi\)
\(770\) 2.10023 0.0756869
\(771\) −10.5743 −0.380825
\(772\) 0.818331 0.0294524
\(773\) 42.5532 1.53053 0.765265 0.643715i \(-0.222608\pi\)
0.765265 + 0.643715i \(0.222608\pi\)
\(774\) 4.07606 0.146511
\(775\) −6.80194 −0.244333
\(776\) 10.7952 0.387526
\(777\) −7.61655 −0.273242
\(778\) 12.5157 0.448711
\(779\) 57.2325 2.05057
\(780\) 0 0
\(781\) −2.78315 −0.0995890
\(782\) 40.9071 1.46283
\(783\) 4.07069 0.145475
\(784\) −3.92692 −0.140247
\(785\) 3.09246 0.110375
\(786\) −5.08815 −0.181488
\(787\) −12.1927 −0.434622 −0.217311 0.976102i \(-0.569729\pi\)
−0.217311 + 0.976102i \(0.569729\pi\)
\(788\) −12.7385 −0.453792
\(789\) 11.9922 0.426935
\(790\) 3.74094 0.133097
\(791\) −0.904084 −0.0321455
\(792\) −1.19806 −0.0425713
\(793\) 0 0
\(794\) −21.6136 −0.767037
\(795\) −1.14914 −0.0407560
\(796\) −4.02475 −0.142654
\(797\) 36.4534 1.29125 0.645623 0.763656i \(-0.276598\pi\)
0.645623 + 0.763656i \(0.276598\pi\)
\(798\) 15.1304 0.535609
\(799\) −75.7131 −2.67854
\(800\) 1.00000 0.0353553
\(801\) −4.49934 −0.158976
\(802\) 11.5157 0.406635
\(803\) 1.91510 0.0675824
\(804\) −14.1468 −0.498917
\(805\) 10.6039 0.373738
\(806\) 0 0
\(807\) −25.4359 −0.895388
\(808\) 16.7875 0.590581
\(809\) 9.70278 0.341131 0.170566 0.985346i \(-0.445440\pi\)
0.170566 + 0.985346i \(0.445440\pi\)
\(810\) 1.00000 0.0351364
\(811\) 26.6547 0.935974 0.467987 0.883735i \(-0.344979\pi\)
0.467987 + 0.883735i \(0.344979\pi\)
\(812\) −7.13600 −0.250424
\(813\) −20.3666 −0.714287
\(814\) −5.20536 −0.182448
\(815\) −23.9541 −0.839074
\(816\) −6.76271 −0.236742
\(817\) −35.1806 −1.23081
\(818\) 0.0991626 0.00346714
\(819\) 0 0
\(820\) −6.63102 −0.231565
\(821\) 2.58343 0.0901624 0.0450812 0.998983i \(-0.485645\pi\)
0.0450812 + 0.998983i \(0.485645\pi\)
\(822\) −0.674563 −0.0235281
\(823\) 19.9855 0.696652 0.348326 0.937374i \(-0.386750\pi\)
0.348326 + 0.937374i \(0.386750\pi\)
\(824\) −4.16421 −0.145067
\(825\) −1.19806 −0.0417112
\(826\) 24.0575 0.837069
\(827\) 1.39506 0.0485110 0.0242555 0.999706i \(-0.492278\pi\)
0.0242555 + 0.999706i \(0.492278\pi\)
\(828\) −6.04892 −0.210214
\(829\) −27.6179 −0.959208 −0.479604 0.877485i \(-0.659220\pi\)
−0.479604 + 0.877485i \(0.659220\pi\)
\(830\) −8.40581 −0.291770
\(831\) 20.1323 0.698381
\(832\) 0 0
\(833\) 26.5566 0.920132
\(834\) 8.26205 0.286091
\(835\) 1.19806 0.0414607
\(836\) 10.3405 0.357634
\(837\) −6.80194 −0.235109
\(838\) 31.2784 1.08050
\(839\) −54.3357 −1.87588 −0.937939 0.346801i \(-0.887268\pi\)
−0.937939 + 0.346801i \(0.887268\pi\)
\(840\) −1.75302 −0.0604850
\(841\) −12.4295 −0.428604
\(842\) 38.0452 1.31112
\(843\) 29.7265 1.02383
\(844\) 3.10992 0.107048
\(845\) 0 0
\(846\) 11.1957 0.384915
\(847\) 16.7670 0.576122
\(848\) −1.14914 −0.0394618
\(849\) 8.84117 0.303428
\(850\) −6.76271 −0.231959
\(851\) −26.2814 −0.900916
\(852\) 2.32304 0.0795862
\(853\) −13.4553 −0.460701 −0.230351 0.973108i \(-0.573987\pi\)
−0.230351 + 0.973108i \(0.573987\pi\)
\(854\) 2.07905 0.0711436
\(855\) −8.63102 −0.295175
\(856\) 12.9661 0.443174
\(857\) 42.7200 1.45929 0.729644 0.683827i \(-0.239686\pi\)
0.729644 + 0.683827i \(0.239686\pi\)
\(858\) 0 0
\(859\) 44.3038 1.51163 0.755813 0.654788i \(-0.227242\pi\)
0.755813 + 0.654788i \(0.227242\pi\)
\(860\) 4.07606 0.138993
\(861\) 11.6243 0.396156
\(862\) 14.4614 0.492558
\(863\) −5.47783 −0.186467 −0.0932337 0.995644i \(-0.529720\pi\)
−0.0932337 + 0.995644i \(0.529720\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.63773 −0.225689
\(866\) 21.6209 0.734707
\(867\) 28.7342 0.975866
\(868\) 11.9239 0.404725
\(869\) −4.48188 −0.152037
\(870\) 4.07069 0.138009
\(871\) 0 0
\(872\) 19.4644 0.659148
\(873\) 10.7952 0.365363
\(874\) 52.2083 1.76597
\(875\) −1.75302 −0.0592629
\(876\) −1.59850 −0.0540083
\(877\) −41.3726 −1.39705 −0.698526 0.715585i \(-0.746160\pi\)
−0.698526 + 0.715585i \(0.746160\pi\)
\(878\) −3.23623 −0.109217
\(879\) −20.5918 −0.694544
\(880\) −1.19806 −0.0403867
\(881\) −42.3919 −1.42822 −0.714110 0.700033i \(-0.753168\pi\)
−0.714110 + 0.700033i \(0.753168\pi\)
\(882\) −3.92692 −0.132226
\(883\) −40.8412 −1.37441 −0.687207 0.726461i \(-0.741164\pi\)
−0.687207 + 0.726461i \(0.741164\pi\)
\(884\) 0 0
\(885\) −13.7235 −0.461310
\(886\) −14.6649 −0.492676
\(887\) −35.0810 −1.17790 −0.588952 0.808168i \(-0.700459\pi\)
−0.588952 + 0.808168i \(0.700459\pi\)
\(888\) 4.34481 0.145802
\(889\) −13.9554 −0.468049
\(890\) −4.49934 −0.150818
\(891\) −1.19806 −0.0401366
\(892\) −14.3787 −0.481433
\(893\) −96.6301 −3.23360
\(894\) 8.19136 0.273960
\(895\) 19.6353 0.656337
\(896\) −1.75302 −0.0585643
\(897\) 0 0
\(898\) −4.08516 −0.136324
\(899\) −27.6886 −0.923465
\(900\) 1.00000 0.0333333
\(901\) 7.77133 0.258901
\(902\) 7.94438 0.264519
\(903\) −7.14542 −0.237785
\(904\) 0.515729 0.0171529
\(905\) 5.25906 0.174817
\(906\) 5.72348 0.190150
\(907\) −45.4916 −1.51052 −0.755261 0.655424i \(-0.772490\pi\)
−0.755261 + 0.655424i \(0.772490\pi\)
\(908\) 5.97716 0.198359
\(909\) 16.7875 0.556805
\(910\) 0 0
\(911\) 47.4349 1.57159 0.785794 0.618489i \(-0.212255\pi\)
0.785794 + 0.618489i \(0.212255\pi\)
\(912\) −8.63102 −0.285802
\(913\) 10.0707 0.333291
\(914\) −41.8678 −1.38487
\(915\) −1.18598 −0.0392073
\(916\) 15.9172 0.525920
\(917\) 8.91962 0.294552
\(918\) −6.76271 −0.223203
\(919\) −20.3907 −0.672629 −0.336314 0.941750i \(-0.609180\pi\)
−0.336314 + 0.941750i \(0.609180\pi\)
\(920\) −6.04892 −0.199427
\(921\) −28.7265 −0.946569
\(922\) 0.215521 0.00709779
\(923\) 0 0
\(924\) 2.10023 0.0690924
\(925\) 4.34481 0.142857
\(926\) −23.4795 −0.771584
\(927\) −4.16421 −0.136771
\(928\) 4.07069 0.133627
\(929\) 1.26636 0.0415478 0.0207739 0.999784i \(-0.493387\pi\)
0.0207739 + 0.999784i \(0.493387\pi\)
\(930\) −6.80194 −0.223044
\(931\) 33.8933 1.11081
\(932\) −6.92154 −0.226723
\(933\) −18.6329 −0.610015
\(934\) 10.1153 0.330983
\(935\) 8.10215 0.264969
\(936\) 0 0
\(937\) 27.3558 0.893676 0.446838 0.894615i \(-0.352550\pi\)
0.446838 + 0.894615i \(0.352550\pi\)
\(938\) 24.7995 0.809734
\(939\) −29.3696 −0.958440
\(940\) 11.1957 0.365162
\(941\) 60.6620 1.97752 0.988762 0.149495i \(-0.0477648\pi\)
0.988762 + 0.149495i \(0.0477648\pi\)
\(942\) 3.09246 0.100758
\(943\) 40.1105 1.30618
\(944\) −13.7235 −0.446661
\(945\) −1.75302 −0.0570258
\(946\) −4.88338 −0.158772
\(947\) 12.3797 0.402287 0.201144 0.979562i \(-0.435534\pi\)
0.201144 + 0.979562i \(0.435534\pi\)
\(948\) 3.74094 0.121500
\(949\) 0 0
\(950\) −8.63102 −0.280027
\(951\) −20.8092 −0.674786
\(952\) 11.8552 0.384228
\(953\) 28.7784 0.932223 0.466111 0.884726i \(-0.345655\pi\)
0.466111 + 0.884726i \(0.345655\pi\)
\(954\) −1.14914 −0.0372049
\(955\) 4.45473 0.144152
\(956\) −30.1075 −0.973747
\(957\) −4.87694 −0.157649
\(958\) −0.980623 −0.0316825
\(959\) 1.18252 0.0381857
\(960\) 1.00000 0.0322749
\(961\) 15.2664 0.492463
\(962\) 0 0
\(963\) 12.9661 0.417828
\(964\) −12.0562 −0.388304
\(965\) 0.818331 0.0263430
\(966\) 10.6039 0.341174
\(967\) 23.1545 0.744599 0.372300 0.928113i \(-0.378569\pi\)
0.372300 + 0.928113i \(0.378569\pi\)
\(968\) −9.56465 −0.307419
\(969\) 58.3691 1.87509
\(970\) 10.7952 0.346614
\(971\) −2.89572 −0.0929282 −0.0464641 0.998920i \(-0.514795\pi\)
−0.0464641 + 0.998920i \(0.514795\pi\)
\(972\) 1.00000 0.0320750
\(973\) −14.4835 −0.464321
\(974\) −7.18300 −0.230158
\(975\) 0 0
\(976\) −1.18598 −0.0379623
\(977\) 4.86294 0.155579 0.0777896 0.996970i \(-0.475214\pi\)
0.0777896 + 0.996970i \(0.475214\pi\)
\(978\) −23.9541 −0.765967
\(979\) 5.39048 0.172281
\(980\) −3.92692 −0.125441
\(981\) 19.4644 0.621451
\(982\) 24.5918 0.784756
\(983\) 52.7506 1.68248 0.841242 0.540659i \(-0.181825\pi\)
0.841242 + 0.540659i \(0.181825\pi\)
\(984\) −6.63102 −0.211389
\(985\) −12.7385 −0.405884
\(986\) −27.5289 −0.876698
\(987\) −19.6262 −0.624710
\(988\) 0 0
\(989\) −24.6558 −0.784008
\(990\) −1.19806 −0.0380769
\(991\) 7.15538 0.227298 0.113649 0.993521i \(-0.463746\pi\)
0.113649 + 0.993521i \(0.463746\pi\)
\(992\) −6.80194 −0.215962
\(993\) 11.7603 0.373202
\(994\) −4.07234 −0.129167
\(995\) −4.02475 −0.127593
\(996\) −8.40581 −0.266348
\(997\) 39.3870 1.24740 0.623700 0.781664i \(-0.285629\pi\)
0.623700 + 0.781664i \(0.285629\pi\)
\(998\) −30.1608 −0.954722
\(999\) 4.34481 0.137464
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.bx.1.3 yes 3
13.5 odd 4 5070.2.b.w.1351.3 6
13.8 odd 4 5070.2.b.w.1351.4 6
13.12 even 2 5070.2.a.bo.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bo.1.1 3 13.12 even 2
5070.2.a.bx.1.3 yes 3 1.1 even 1 trivial
5070.2.b.w.1351.3 6 13.5 odd 4
5070.2.b.w.1351.4 6 13.8 odd 4