Properties

Label 5070.2.b.w.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.w.1351.3

$q$-expansion

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

Character values

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

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000i 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) 1.75302i 0.662579i 0.943529 + 0.331290i \(0.107484\pi\)
−0.943529 + 0.331290i \(0.892516\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.19806i 0.361229i 0.983554 + 0.180615i \(0.0578087\pi\)
−0.983554 + 0.180615i \(0.942191\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.75302 −0.468514
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 6.76271 1.64020 0.820099 0.572222i \(-0.193918\pi\)
0.820099 + 0.572222i \(0.193918\pi\)
\(18\) 1.00000i 0.235702i
\(19\) − 8.63102i − 1.98009i −0.140743 0.990046i \(-0.544949\pi\)
0.140743 0.990046i \(-0.455051\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) 1.75302i 0.382540i
\(22\) −1.19806 −0.255428
\(23\) 6.04892 1.26129 0.630643 0.776073i \(-0.282791\pi\)
0.630643 + 0.776073i \(0.282791\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 1.75302i − 0.331290i
\(29\) 4.07069 0.755908 0.377954 0.925824i \(-0.376628\pi\)
0.377954 + 0.925824i \(0.376628\pi\)
\(30\) −1.00000 −0.182574
\(31\) − 6.80194i − 1.22166i −0.791760 0.610832i \(-0.790835\pi\)
0.791760 0.610832i \(-0.209165\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 1.19806i 0.208556i
\(34\) 6.76271i 1.15980i
\(35\) −1.75302 −0.296315
\(36\) −1.00000 −0.166667
\(37\) − 4.34481i − 0.714283i −0.934050 0.357142i \(-0.883751\pi\)
0.934050 0.357142i \(-0.116249\pi\)
\(38\) 8.63102 1.40014
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) − 6.63102i − 1.03559i −0.855504 0.517796i \(-0.826753\pi\)
0.855504 0.517796i \(-0.173247\pi\)
\(42\) −1.75302 −0.270497
\(43\) −4.07606 −0.621594 −0.310797 0.950476i \(-0.600596\pi\)
−0.310797 + 0.950476i \(0.600596\pi\)
\(44\) − 1.19806i − 0.180615i
\(45\) 1.00000i 0.149071i
\(46\) 6.04892i 0.891864i
\(47\) − 11.1957i − 1.63306i −0.577306 0.816528i \(-0.695896\pi\)
0.577306 0.816528i \(-0.304104\pi\)
\(48\) 1.00000 0.144338
\(49\) 3.92692 0.560988
\(50\) − 1.00000i − 0.141421i
\(51\) 6.76271 0.946969
\(52\) 0 0
\(53\) −1.14914 −0.157847 −0.0789236 0.996881i \(-0.525148\pi\)
−0.0789236 + 0.996881i \(0.525148\pi\)
\(54\) 1.00000i 0.136083i
\(55\) −1.19806 −0.161547
\(56\) 1.75302 0.234257
\(57\) − 8.63102i − 1.14321i
\(58\) 4.07069i 0.534507i
\(59\) 13.7235i 1.78664i 0.449416 + 0.893322i \(0.351632\pi\)
−0.449416 + 0.893322i \(0.648368\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) −1.18598 −0.151849 −0.0759246 0.997114i \(-0.524191\pi\)
−0.0759246 + 0.997114i \(0.524191\pi\)
\(62\) 6.80194 0.863847
\(63\) 1.75302i 0.220860i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −1.19806 −0.147471
\(67\) − 14.1468i − 1.72830i −0.503234 0.864150i \(-0.667856\pi\)
0.503234 0.864150i \(-0.332144\pi\)
\(68\) −6.76271 −0.820099
\(69\) 6.04892 0.728204
\(70\) − 1.75302i − 0.209526i
\(71\) 2.32304i 0.275695i 0.990453 + 0.137847i \(0.0440183\pi\)
−0.990453 + 0.137847i \(0.955982\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 1.59850i 0.187090i 0.995615 + 0.0935451i \(0.0298199\pi\)
−0.995615 + 0.0935451i \(0.970180\pi\)
\(74\) 4.34481 0.505074
\(75\) −1.00000 −0.115470
\(76\) 8.63102i 0.990046i
\(77\) −2.10023 −0.239343
\(78\) 0 0
\(79\) 3.74094 0.420888 0.210444 0.977606i \(-0.432509\pi\)
0.210444 + 0.977606i \(0.432509\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 6.63102 0.732274
\(83\) − 8.40581i − 0.922658i −0.887229 0.461329i \(-0.847373\pi\)
0.887229 0.461329i \(-0.152627\pi\)
\(84\) − 1.75302i − 0.191270i
\(85\) 6.76271i 0.733519i
\(86\) − 4.07606i − 0.439533i
\(87\) 4.07069 0.436424
\(88\) 1.19806 0.127714
\(89\) 4.49934i 0.476929i 0.971151 + 0.238464i \(0.0766440\pi\)
−0.971151 + 0.238464i \(0.923356\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −6.04892 −0.630643
\(93\) − 6.80194i − 0.705328i
\(94\) 11.1957 1.15475
\(95\) 8.63102 0.885524
\(96\) 1.00000i 0.102062i
\(97\) 10.7952i 1.09609i 0.836449 + 0.548045i \(0.184628\pi\)
−0.836449 + 0.548045i \(0.815372\pi\)
\(98\) 3.92692i 0.396679i
\(99\) 1.19806i 0.120410i
\(100\) 1.00000 0.100000
\(101\) −16.7875 −1.67042 −0.835208 0.549935i \(-0.814653\pi\)
−0.835208 + 0.549935i \(0.814653\pi\)
\(102\) 6.76271i 0.669608i
\(103\) 4.16421 0.410312 0.205156 0.978729i \(-0.434230\pi\)
0.205156 + 0.978729i \(0.434230\pi\)
\(104\) 0 0
\(105\) −1.75302 −0.171077
\(106\) − 1.14914i − 0.111615i
\(107\) 12.9661 1.25348 0.626742 0.779226i \(-0.284388\pi\)
0.626742 + 0.779226i \(0.284388\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 19.4644i 1.86435i 0.362004 + 0.932177i \(0.382093\pi\)
−0.362004 + 0.932177i \(0.617907\pi\)
\(110\) − 1.19806i − 0.114231i
\(111\) − 4.34481i − 0.412392i
\(112\) 1.75302i 0.165645i
\(113\) 0.515729 0.0485157 0.0242579 0.999706i \(-0.492278\pi\)
0.0242579 + 0.999706i \(0.492278\pi\)
\(114\) 8.63102 0.808369
\(115\) 6.04892i 0.564064i
\(116\) −4.07069 −0.377954
\(117\) 0 0
\(118\) −13.7235 −1.26335
\(119\) 11.8552i 1.08676i
\(120\) 1.00000 0.0912871
\(121\) 9.56465 0.869513
\(122\) − 1.18598i − 0.107374i
\(123\) − 6.63102i − 0.597899i
\(124\) 6.80194i 0.610832i
\(125\) − 1.00000i − 0.0894427i
\(126\) −1.75302 −0.156171
\(127\) −7.96077 −0.706404 −0.353202 0.935547i \(-0.614907\pi\)
−0.353202 + 0.935547i \(0.614907\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −4.07606 −0.358877
\(130\) 0 0
\(131\) −5.08815 −0.444553 −0.222277 0.974984i \(-0.571349\pi\)
−0.222277 + 0.974984i \(0.571349\pi\)
\(132\) − 1.19806i − 0.104278i
\(133\) 15.1304 1.31197
\(134\) 14.1468 1.22209
\(135\) 1.00000i 0.0860663i
\(136\) − 6.76271i − 0.579898i
\(137\) 0.674563i 0.0576318i 0.999585 + 0.0288159i \(0.00917366\pi\)
−0.999585 + 0.0288159i \(0.990826\pi\)
\(138\) 6.04892i 0.514918i
\(139\) 8.26205 0.700778 0.350389 0.936604i \(-0.386049\pi\)
0.350389 + 0.936604i \(0.386049\pi\)
\(140\) 1.75302 0.148157
\(141\) − 11.1957i − 0.942845i
\(142\) −2.32304 −0.194946
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.07069i 0.338052i
\(146\) −1.59850 −0.132293
\(147\) 3.92692 0.323887
\(148\) 4.34481i 0.357142i
\(149\) 8.19136i 0.671062i 0.942029 + 0.335531i \(0.108916\pi\)
−0.942029 + 0.335531i \(0.891084\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) − 5.72348i − 0.465770i −0.972504 0.232885i \(-0.925183\pi\)
0.972504 0.232885i \(-0.0748166\pi\)
\(152\) −8.63102 −0.700068
\(153\) 6.76271 0.546733
\(154\) − 2.10023i − 0.169241i
\(155\) 6.80194 0.546345
\(156\) 0 0
\(157\) 3.09246 0.246805 0.123403 0.992357i \(-0.460619\pi\)
0.123403 + 0.992357i \(0.460619\pi\)
\(158\) 3.74094i 0.297613i
\(159\) −1.14914 −0.0911331
\(160\) −1.00000 −0.0790569
\(161\) 10.6039i 0.835702i
\(162\) 1.00000i 0.0785674i
\(163\) 23.9541i 1.87623i 0.346328 + 0.938114i \(0.387429\pi\)
−0.346328 + 0.938114i \(0.612571\pi\)
\(164\) 6.63102i 0.517796i
\(165\) −1.19806 −0.0932690
\(166\) 8.40581 0.652418
\(167\) − 1.19806i − 0.0927088i −0.998925 0.0463544i \(-0.985240\pi\)
0.998925 0.0463544i \(-0.0147604\pi\)
\(168\) 1.75302 0.135248
\(169\) 0 0
\(170\) −6.76271 −0.518676
\(171\) − 8.63102i − 0.660031i
\(172\) 4.07606 0.310797
\(173\) 6.63773 0.504657 0.252329 0.967642i \(-0.418804\pi\)
0.252329 + 0.967642i \(0.418804\pi\)
\(174\) 4.07069i 0.308598i
\(175\) − 1.75302i − 0.132516i
\(176\) 1.19806i 0.0903073i
\(177\) 13.7235i 1.03152i
\(178\) −4.49934 −0.337239
\(179\) −19.6353 −1.46761 −0.733807 0.679358i \(-0.762258\pi\)
−0.733807 + 0.679358i \(0.762258\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) −5.25906 −0.390903 −0.195451 0.980713i \(-0.562617\pi\)
−0.195451 + 0.980713i \(0.562617\pi\)
\(182\) 0 0
\(183\) −1.18598 −0.0876702
\(184\) − 6.04892i − 0.445932i
\(185\) 4.34481 0.319437
\(186\) 6.80194 0.498742
\(187\) 8.10215i 0.592488i
\(188\) 11.1957i 0.816528i
\(189\) 1.75302i 0.127513i
\(190\) 8.63102i 0.626160i
\(191\) 4.45473 0.322333 0.161166 0.986927i \(-0.448474\pi\)
0.161166 + 0.986927i \(0.448474\pi\)
\(192\) −1.00000 −0.0721688
\(193\) − 0.818331i − 0.0589048i −0.999566 0.0294524i \(-0.990624\pi\)
0.999566 0.0294524i \(-0.00937634\pi\)
\(194\) −10.7952 −0.775053
\(195\) 0 0
\(196\) −3.92692 −0.280494
\(197\) − 12.7385i − 0.907584i −0.891108 0.453792i \(-0.850071\pi\)
0.891108 0.453792i \(-0.149929\pi\)
\(198\) −1.19806 −0.0851426
\(199\) 4.02475 0.285307 0.142654 0.989773i \(-0.454437\pi\)
0.142654 + 0.989773i \(0.454437\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) − 14.1468i − 0.997835i
\(202\) − 16.7875i − 1.18116i
\(203\) 7.13600i 0.500849i
\(204\) −6.76271 −0.473484
\(205\) 6.63102 0.463131
\(206\) 4.16421i 0.290134i
\(207\) 6.04892 0.420429
\(208\) 0 0
\(209\) 10.3405 0.715268
\(210\) − 1.75302i − 0.120970i
\(211\) 3.10992 0.214095 0.107048 0.994254i \(-0.465860\pi\)
0.107048 + 0.994254i \(0.465860\pi\)
\(212\) 1.14914 0.0789236
\(213\) 2.32304i 0.159172i
\(214\) 12.9661i 0.886348i
\(215\) − 4.07606i − 0.277985i
\(216\) − 1.00000i − 0.0680414i
\(217\) 11.9239 0.809449
\(218\) −19.4644 −1.31830
\(219\) 1.59850i 0.108017i
\(220\) 1.19806 0.0807733
\(221\) 0 0
\(222\) 4.34481 0.291605
\(223\) − 14.3787i − 0.962867i −0.876483 0.481433i \(-0.840116\pi\)
0.876483 0.481433i \(-0.159884\pi\)
\(224\) −1.75302 −0.117129
\(225\) −1.00000 −0.0666667
\(226\) 0.515729i 0.0343058i
\(227\) 5.97716i 0.396718i 0.980129 + 0.198359i \(0.0635612\pi\)
−0.980129 + 0.198359i \(0.936439\pi\)
\(228\) 8.63102i 0.571603i
\(229\) − 15.9172i − 1.05184i −0.850534 0.525920i \(-0.823721\pi\)
0.850534 0.525920i \(-0.176279\pi\)
\(230\) −6.04892 −0.398854
\(231\) −2.10023 −0.138185
\(232\) − 4.07069i − 0.267254i
\(233\) 6.92154 0.453445 0.226723 0.973959i \(-0.427199\pi\)
0.226723 + 0.973959i \(0.427199\pi\)
\(234\) 0 0
\(235\) 11.1957 0.730325
\(236\) − 13.7235i − 0.893322i
\(237\) 3.74094 0.243000
\(238\) −11.8552 −0.768456
\(239\) − 30.1075i − 1.94749i −0.227633 0.973747i \(-0.573099\pi\)
0.227633 0.973747i \(-0.426901\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 12.0562i 0.776609i 0.921531 + 0.388304i \(0.126939\pi\)
−0.921531 + 0.388304i \(0.873061\pi\)
\(242\) 9.56465i 0.614839i
\(243\) 1.00000 0.0641500
\(244\) 1.18598 0.0759246
\(245\) 3.92692i 0.250882i
\(246\) 6.63102 0.422779
\(247\) 0 0
\(248\) −6.80194 −0.431923
\(249\) − 8.40581i − 0.532697i
\(250\) 1.00000 0.0632456
\(251\) 14.8170 0.935241 0.467620 0.883929i \(-0.345111\pi\)
0.467620 + 0.883929i \(0.345111\pi\)
\(252\) − 1.75302i − 0.110430i
\(253\) 7.24698i 0.455614i
\(254\) − 7.96077i − 0.499503i
\(255\) 6.76271i 0.423497i
\(256\) 1.00000 0.0625000
\(257\) 10.5743 0.659609 0.329804 0.944049i \(-0.393017\pi\)
0.329804 + 0.944049i \(0.393017\pi\)
\(258\) − 4.07606i − 0.253765i
\(259\) 7.61655 0.473269
\(260\) 0 0
\(261\) 4.07069 0.251969
\(262\) − 5.08815i − 0.314347i
\(263\) 11.9922 0.739473 0.369736 0.929137i \(-0.379448\pi\)
0.369736 + 0.929137i \(0.379448\pi\)
\(264\) 1.19806 0.0737356
\(265\) − 1.14914i − 0.0705914i
\(266\) 15.1304i 0.927702i
\(267\) 4.49934i 0.275355i
\(268\) 14.1468i 0.864150i
\(269\) −25.4359 −1.55086 −0.775428 0.631435i \(-0.782466\pi\)
−0.775428 + 0.631435i \(0.782466\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 20.3666i 1.23718i 0.785713 + 0.618591i \(0.212296\pi\)
−0.785713 + 0.618591i \(0.787704\pi\)
\(272\) 6.76271 0.410049
\(273\) 0 0
\(274\) −0.674563 −0.0407518
\(275\) − 1.19806i − 0.0722459i
\(276\) −6.04892 −0.364102
\(277\) −20.1323 −1.20963 −0.604816 0.796366i \(-0.706753\pi\)
−0.604816 + 0.796366i \(0.706753\pi\)
\(278\) 8.26205i 0.495525i
\(279\) − 6.80194i − 0.407221i
\(280\) 1.75302i 0.104763i
\(281\) − 29.7265i − 1.77333i −0.462410 0.886666i \(-0.653015\pi\)
0.462410 0.886666i \(-0.346985\pi\)
\(282\) 11.1957 0.666692
\(283\) −8.84117 −0.525553 −0.262776 0.964857i \(-0.584638\pi\)
−0.262776 + 0.964857i \(0.584638\pi\)
\(284\) − 2.32304i − 0.137847i
\(285\) 8.63102 0.511258
\(286\) 0 0
\(287\) 11.6243 0.686162
\(288\) 1.00000i 0.0589256i
\(289\) 28.7342 1.69025
\(290\) −4.07069 −0.239039
\(291\) 10.7952i 0.632828i
\(292\) − 1.59850i − 0.0935451i
\(293\) 20.5918i 1.20299i 0.798878 + 0.601493i \(0.205427\pi\)
−0.798878 + 0.601493i \(0.794573\pi\)
\(294\) 3.92692i 0.229023i
\(295\) −13.7235 −0.799012
\(296\) −4.34481 −0.252537
\(297\) 1.19806i 0.0695186i
\(298\) −8.19136 −0.474513
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) − 7.14542i − 0.411855i
\(302\) 5.72348 0.329349
\(303\) −16.7875 −0.964415
\(304\) − 8.63102i − 0.495023i
\(305\) − 1.18598i − 0.0679091i
\(306\) 6.76271i 0.386598i
\(307\) 28.7265i 1.63951i 0.572717 + 0.819753i \(0.305889\pi\)
−0.572717 + 0.819753i \(0.694111\pi\)
\(308\) 2.10023 0.119672
\(309\) 4.16421 0.236894
\(310\) 6.80194i 0.386324i
\(311\) 18.6329 1.05658 0.528289 0.849065i \(-0.322834\pi\)
0.528289 + 0.849065i \(0.322834\pi\)
\(312\) 0 0
\(313\) −29.3696 −1.66007 −0.830033 0.557714i \(-0.811679\pi\)
−0.830033 + 0.557714i \(0.811679\pi\)
\(314\) 3.09246i 0.174517i
\(315\) −1.75302 −0.0987715
\(316\) −3.74094 −0.210444
\(317\) − 20.8092i − 1.16876i −0.811479 0.584381i \(-0.801337\pi\)
0.811479 0.584381i \(-0.198663\pi\)
\(318\) − 1.14914i − 0.0644408i
\(319\) 4.87694i 0.273056i
\(320\) − 1.00000i − 0.0559017i
\(321\) 12.9661 0.723700
\(322\) −10.6039 −0.590931
\(323\) − 58.3691i − 3.24774i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −23.9541 −1.32669
\(327\) 19.4644i 1.07638i
\(328\) −6.63102 −0.366137
\(329\) 19.6262 1.08203
\(330\) − 1.19806i − 0.0659512i
\(331\) 11.7603i 0.646405i 0.946330 + 0.323203i \(0.104760\pi\)
−0.946330 + 0.323203i \(0.895240\pi\)
\(332\) 8.40581i 0.461329i
\(333\) − 4.34481i − 0.238094i
\(334\) 1.19806 0.0655551
\(335\) 14.1468 0.772920
\(336\) 1.75302i 0.0956351i
\(337\) 26.8732 1.46388 0.731939 0.681371i \(-0.238616\pi\)
0.731939 + 0.681371i \(0.238616\pi\)
\(338\) 0 0
\(339\) 0.515729 0.0280106
\(340\) − 6.76271i − 0.366759i
\(341\) 8.14914 0.441301
\(342\) 8.63102 0.466712
\(343\) 19.1551i 1.03428i
\(344\) 4.07606i 0.219767i
\(345\) 6.04892i 0.325663i
\(346\) 6.63773i 0.356846i
\(347\) 29.5080 1.58407 0.792035 0.610476i \(-0.209022\pi\)
0.792035 + 0.610476i \(0.209022\pi\)
\(348\) −4.07069 −0.218212
\(349\) 17.0519i 0.912767i 0.889783 + 0.456384i \(0.150856\pi\)
−0.889783 + 0.456384i \(0.849144\pi\)
\(350\) 1.75302 0.0937029
\(351\) 0 0
\(352\) −1.19806 −0.0638569
\(353\) 21.8629i 1.16365i 0.813315 + 0.581823i \(0.197660\pi\)
−0.813315 + 0.581823i \(0.802340\pi\)
\(354\) −13.7235 −0.729395
\(355\) −2.32304 −0.123294
\(356\) − 4.49934i − 0.238464i
\(357\) 11.8552i 0.627442i
\(358\) − 19.6353i − 1.03776i
\(359\) 14.3026i 0.754862i 0.926038 + 0.377431i \(0.123193\pi\)
−0.926038 + 0.377431i \(0.876807\pi\)
\(360\) 1.00000 0.0527046
\(361\) −55.4946 −2.92077
\(362\) − 5.25906i − 0.276410i
\(363\) 9.56465 0.502014
\(364\) 0 0
\(365\) −1.59850 −0.0836692
\(366\) − 1.18598i − 0.0619922i
\(367\) −11.2416 −0.586807 −0.293403 0.955989i \(-0.594788\pi\)
−0.293403 + 0.955989i \(0.594788\pi\)
\(368\) 6.04892 0.315322
\(369\) − 6.63102i − 0.345197i
\(370\) 4.34481i 0.225876i
\(371\) − 2.01447i − 0.104586i
\(372\) 6.80194i 0.352664i
\(373\) −9.70171 −0.502336 −0.251168 0.967944i \(-0.580815\pi\)
−0.251168 + 0.967944i \(0.580815\pi\)
\(374\) −8.10215 −0.418952
\(375\) − 1.00000i − 0.0516398i
\(376\) −11.1957 −0.577373
\(377\) 0 0
\(378\) −1.75302 −0.0901656
\(379\) 2.82802i 0.145266i 0.997359 + 0.0726328i \(0.0231401\pi\)
−0.997359 + 0.0726328i \(0.976860\pi\)
\(380\) −8.63102 −0.442762
\(381\) −7.96077 −0.407843
\(382\) 4.45473i 0.227924i
\(383\) − 11.6558i − 0.595582i −0.954631 0.297791i \(-0.903750\pi\)
0.954631 0.297791i \(-0.0962499\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) − 2.10023i − 0.107038i
\(386\) 0.818331 0.0416520
\(387\) −4.07606 −0.207198
\(388\) − 10.7952i − 0.548045i
\(389\) −12.5157 −0.634573 −0.317286 0.948330i \(-0.602772\pi\)
−0.317286 + 0.948330i \(0.602772\pi\)
\(390\) 0 0
\(391\) 40.9071 2.06876
\(392\) − 3.92692i − 0.198339i
\(393\) −5.08815 −0.256663
\(394\) 12.7385 0.641759
\(395\) 3.74094i 0.188227i
\(396\) − 1.19806i − 0.0602049i
\(397\) 21.6136i 1.08475i 0.840135 + 0.542377i \(0.182475\pi\)
−0.840135 + 0.542377i \(0.817525\pi\)
\(398\) 4.02475i 0.201743i
\(399\) 15.1304 0.757465
\(400\) −1.00000 −0.0500000
\(401\) − 11.5157i − 0.575068i −0.957770 0.287534i \(-0.907165\pi\)
0.957770 0.287534i \(-0.0928354\pi\)
\(402\) 14.1468 0.705576
\(403\) 0 0
\(404\) 16.7875 0.835208
\(405\) 1.00000i 0.0496904i
\(406\) −7.13600 −0.354154
\(407\) 5.20536 0.258020
\(408\) − 6.76271i − 0.334804i
\(409\) 0.0991626i 0.00490328i 0.999997 + 0.00245164i \(0.000780382\pi\)
−0.999997 + 0.00245164i \(0.999220\pi\)
\(410\) 6.63102i 0.327483i
\(411\) 0.674563i 0.0332737i
\(412\) −4.16421 −0.205156
\(413\) −24.0575 −1.18379
\(414\) 6.04892i 0.297288i
\(415\) 8.40581 0.412625
\(416\) 0 0
\(417\) 8.26205 0.404594
\(418\) 10.3405i 0.505771i
\(419\) 31.2784 1.52805 0.764026 0.645186i \(-0.223220\pi\)
0.764026 + 0.645186i \(0.223220\pi\)
\(420\) 1.75302 0.0855386
\(421\) 38.0452i 1.85421i 0.374802 + 0.927105i \(0.377711\pi\)
−0.374802 + 0.927105i \(0.622289\pi\)
\(422\) 3.10992i 0.151388i
\(423\) − 11.1957i − 0.544352i
\(424\) 1.14914i 0.0558074i
\(425\) −6.76271 −0.328040
\(426\) −2.32304 −0.112552
\(427\) − 2.07905i − 0.100612i
\(428\) −12.9661 −0.626742
\(429\) 0 0
\(430\) 4.07606 0.196565
\(431\) 14.4614i 0.696583i 0.937386 + 0.348291i \(0.113238\pi\)
−0.937386 + 0.348291i \(0.886762\pi\)
\(432\) 1.00000 0.0481125
\(433\) −21.6209 −1.03903 −0.519516 0.854461i \(-0.673888\pi\)
−0.519516 + 0.854461i \(0.673888\pi\)
\(434\) 11.9239i 0.572367i
\(435\) 4.07069i 0.195175i
\(436\) − 19.4644i − 0.932177i
\(437\) − 52.2083i − 2.49746i
\(438\) −1.59850 −0.0763792
\(439\) 3.23623 0.154457 0.0772283 0.997013i \(-0.475393\pi\)
0.0772283 + 0.997013i \(0.475393\pi\)
\(440\) 1.19806i 0.0571154i
\(441\) 3.92692 0.186996
\(442\) 0 0
\(443\) −14.6649 −0.696749 −0.348375 0.937355i \(-0.613266\pi\)
−0.348375 + 0.937355i \(0.613266\pi\)
\(444\) 4.34481i 0.206196i
\(445\) −4.49934 −0.213289
\(446\) 14.3787 0.680850
\(447\) 8.19136i 0.387438i
\(448\) − 1.75302i − 0.0828224i
\(449\) 4.08516i 0.192791i 0.995343 + 0.0963954i \(0.0307313\pi\)
−0.995343 + 0.0963954i \(0.969269\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) 7.94438 0.374086
\(452\) −0.515729 −0.0242579
\(453\) − 5.72348i − 0.268913i
\(454\) −5.97716 −0.280522
\(455\) 0 0
\(456\) −8.63102 −0.404185
\(457\) − 41.8678i − 1.95849i −0.202668 0.979247i \(-0.564961\pi\)
0.202668 0.979247i \(-0.435039\pi\)
\(458\) 15.9172 0.743763
\(459\) 6.76271 0.315656
\(460\) − 6.04892i − 0.282032i
\(461\) 0.215521i 0.0100378i 0.999987 + 0.00501890i \(0.00159757\pi\)
−0.999987 + 0.00501890i \(0.998402\pi\)
\(462\) − 2.10023i − 0.0977114i
\(463\) 23.4795i 1.09118i 0.838051 + 0.545592i \(0.183695\pi\)
−0.838051 + 0.545592i \(0.816305\pi\)
\(464\) 4.07069 0.188977
\(465\) 6.80194 0.315432
\(466\) 6.92154i 0.320634i
\(467\) −10.1153 −0.468080 −0.234040 0.972227i \(-0.575195\pi\)
−0.234040 + 0.972227i \(0.575195\pi\)
\(468\) 0 0
\(469\) 24.7995 1.14514
\(470\) 11.1957i 0.516418i
\(471\) 3.09246 0.142493
\(472\) 13.7235 0.631674
\(473\) − 4.88338i − 0.224538i
\(474\) 3.74094i 0.171827i
\(475\) 8.63102i 0.396018i
\(476\) − 11.8552i − 0.543381i
\(477\) −1.14914 −0.0526157
\(478\) 30.1075 1.37709
\(479\) 0.980623i 0.0448058i 0.999749 + 0.0224029i \(0.00713166\pi\)
−0.999749 + 0.0224029i \(0.992868\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −12.0562 −0.549145
\(483\) 10.6039i 0.482493i
\(484\) −9.56465 −0.434757
\(485\) −10.7952 −0.490186
\(486\) 1.00000i 0.0453609i
\(487\) − 7.18300i − 0.325493i −0.986668 0.162746i \(-0.947965\pi\)
0.986668 0.162746i \(-0.0520352\pi\)
\(488\) 1.18598i 0.0536868i
\(489\) 23.9541i 1.08324i
\(490\) −3.92692 −0.177400
\(491\) −24.5918 −1.10981 −0.554906 0.831913i \(-0.687246\pi\)
−0.554906 + 0.831913i \(0.687246\pi\)
\(492\) 6.63102i 0.298950i
\(493\) 27.5289 1.23984
\(494\) 0 0
\(495\) −1.19806 −0.0538489
\(496\) − 6.80194i − 0.305416i
\(497\) −4.07234 −0.182670
\(498\) 8.40581 0.376673
\(499\) − 30.1608i − 1.35018i −0.737735 0.675090i \(-0.764105\pi\)
0.737735 0.675090i \(-0.235895\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) − 1.19806i − 0.0535255i
\(502\) 14.8170i 0.661315i
\(503\) −33.8883 −1.51100 −0.755502 0.655146i \(-0.772607\pi\)
−0.755502 + 0.655146i \(0.772607\pi\)
\(504\) 1.75302 0.0780857
\(505\) − 16.7875i − 0.747032i
\(506\) −7.24698 −0.322168
\(507\) 0 0
\(508\) 7.96077 0.353202
\(509\) − 14.1089i − 0.625364i −0.949858 0.312682i \(-0.898773\pi\)
0.949858 0.312682i \(-0.101227\pi\)
\(510\) −6.76271 −0.299458
\(511\) −2.80220 −0.123962
\(512\) 1.00000i 0.0441942i
\(513\) − 8.63102i − 0.381069i
\(514\) 10.5743i 0.466414i
\(515\) 4.16421i 0.183497i
\(516\) 4.07606 0.179439
\(517\) 13.4131 0.589908
\(518\) 7.61655i 0.334652i
\(519\) 6.63773 0.291364
\(520\) 0 0
\(521\) 17.3937 0.762033 0.381017 0.924568i \(-0.375574\pi\)
0.381017 + 0.924568i \(0.375574\pi\)
\(522\) 4.07069i 0.178169i
\(523\) −34.9879 −1.52991 −0.764957 0.644081i \(-0.777240\pi\)
−0.764957 + 0.644081i \(0.777240\pi\)
\(524\) 5.08815 0.222277
\(525\) − 1.75302i − 0.0765081i
\(526\) 11.9922i 0.522886i
\(527\) − 45.9995i − 2.00377i
\(528\) 1.19806i 0.0521390i
\(529\) 13.5894 0.590844
\(530\) 1.14914 0.0499157
\(531\) 13.7235i 0.595548i
\(532\) −15.1304 −0.655984
\(533\) 0 0
\(534\) −4.49934 −0.194705
\(535\) 12.9661i 0.560575i
\(536\) −14.1468 −0.611047
\(537\) −19.6353 −0.847327
\(538\) − 25.4359i − 1.09662i
\(539\) 4.70469i 0.202646i
\(540\) − 1.00000i − 0.0430331i
\(541\) 20.8689i 0.897224i 0.893727 + 0.448612i \(0.148082\pi\)
−0.893727 + 0.448612i \(0.851918\pi\)
\(542\) −20.3666 −0.874820
\(543\) −5.25906 −0.225688
\(544\) 6.76271i 0.289949i
\(545\) −19.4644 −0.833764
\(546\) 0 0
\(547\) 5.67456 0.242627 0.121313 0.992614i \(-0.461289\pi\)
0.121313 + 0.992614i \(0.461289\pi\)
\(548\) − 0.674563i − 0.0288159i
\(549\) −1.18598 −0.0506164
\(550\) 1.19806 0.0510855
\(551\) − 35.1342i − 1.49677i
\(552\) − 6.04892i − 0.257459i
\(553\) 6.55794i 0.278872i
\(554\) − 20.1323i − 0.855339i
\(555\) 4.34481 0.184427
\(556\) −8.26205 −0.350389
\(557\) − 14.7724i − 0.625927i −0.949765 0.312963i \(-0.898678\pi\)
0.949765 0.312963i \(-0.101322\pi\)
\(558\) 6.80194 0.287949
\(559\) 0 0
\(560\) −1.75302 −0.0740786
\(561\) 8.10215i 0.342073i
\(562\) 29.7265 1.25394
\(563\) −18.0398 −0.760288 −0.380144 0.924927i \(-0.624126\pi\)
−0.380144 + 0.924927i \(0.624126\pi\)
\(564\) 11.1957i 0.471423i
\(565\) 0.515729i 0.0216969i
\(566\) − 8.84117i − 0.371622i
\(567\) 1.75302i 0.0736199i
\(568\) 2.32304 0.0974728
\(569\) 11.4886 0.481626 0.240813 0.970571i \(-0.422586\pi\)
0.240813 + 0.970571i \(0.422586\pi\)
\(570\) 8.63102i 0.361514i
\(571\) 44.3062 1.85416 0.927078 0.374869i \(-0.122312\pi\)
0.927078 + 0.374869i \(0.122312\pi\)
\(572\) 0 0
\(573\) 4.45473 0.186099
\(574\) 11.6243i 0.485190i
\(575\) −6.04892 −0.252257
\(576\) −1.00000 −0.0416667
\(577\) − 28.0000i − 1.16566i −0.812596 0.582828i \(-0.801946\pi\)
0.812596 0.582828i \(-0.198054\pi\)
\(578\) 28.7342i 1.19519i
\(579\) − 0.818331i − 0.0340087i
\(580\) − 4.07069i − 0.169026i
\(581\) 14.7356 0.611334
\(582\) −10.7952 −0.447477
\(583\) − 1.37675i − 0.0570190i
\(584\) 1.59850 0.0661463
\(585\) 0 0
\(586\) −20.5918 −0.850639
\(587\) 35.4316i 1.46242i 0.682152 + 0.731210i \(0.261044\pi\)
−0.682152 + 0.731210i \(0.738956\pi\)
\(588\) −3.92692 −0.161943
\(589\) −58.7077 −2.41901
\(590\) − 13.7235i − 0.564987i
\(591\) − 12.7385i − 0.523994i
\(592\) − 4.34481i − 0.178571i
\(593\) 2.19567i 0.0901653i 0.998983 + 0.0450827i \(0.0143551\pi\)
−0.998983 + 0.0450827i \(0.985645\pi\)
\(594\) −1.19806 −0.0491571
\(595\) −11.8552 −0.486014
\(596\) − 8.19136i − 0.335531i
\(597\) 4.02475 0.164722
\(598\) 0 0
\(599\) −31.1855 −1.27421 −0.637103 0.770779i \(-0.719867\pi\)
−0.637103 + 0.770779i \(0.719867\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −24.2435 −0.988914 −0.494457 0.869202i \(-0.664633\pi\)
−0.494457 + 0.869202i \(0.664633\pi\)
\(602\) 7.14542 0.291226
\(603\) − 14.1468i − 0.576100i
\(604\) 5.72348i 0.232885i
\(605\) 9.56465i 0.388858i
\(606\) − 16.7875i − 0.681944i
\(607\) −17.7235 −0.719374 −0.359687 0.933073i \(-0.617117\pi\)
−0.359687 + 0.933073i \(0.617117\pi\)
\(608\) 8.63102 0.350034
\(609\) 7.13600i 0.289165i
\(610\) 1.18598 0.0480190
\(611\) 0 0
\(612\) −6.76271 −0.273366
\(613\) 1.71140i 0.0691227i 0.999403 + 0.0345614i \(0.0110034\pi\)
−0.999403 + 0.0345614i \(0.988997\pi\)
\(614\) −28.7265 −1.15931
\(615\) 6.63102 0.267389
\(616\) 2.10023i 0.0846206i
\(617\) 0.569433i 0.0229245i 0.999934 + 0.0114622i \(0.00364863\pi\)
−0.999934 + 0.0114622i \(0.996351\pi\)
\(618\) 4.16421i 0.167509i
\(619\) − 26.4282i − 1.06224i −0.847297 0.531119i \(-0.821772\pi\)
0.847297 0.531119i \(-0.178228\pi\)
\(620\) −6.80194 −0.273172
\(621\) 6.04892 0.242735
\(622\) 18.6329i 0.747113i
\(623\) −7.88743 −0.316003
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 29.3696i − 1.17384i
\(627\) 10.3405 0.412960
\(628\) −3.09246 −0.123403
\(629\) − 29.3827i − 1.17157i
\(630\) − 1.75302i − 0.0698420i
\(631\) − 38.0006i − 1.51278i −0.654121 0.756390i \(-0.726961\pi\)
0.654121 0.756390i \(-0.273039\pi\)
\(632\) − 3.74094i − 0.148807i
\(633\) 3.10992 0.123608
\(634\) 20.8092 0.826440
\(635\) − 7.96077i − 0.315914i
\(636\) 1.14914 0.0455666
\(637\) 0 0
\(638\) −4.87694 −0.193080
\(639\) 2.32304i 0.0918982i
\(640\) 1.00000 0.0395285
\(641\) 25.9168 1.02365 0.511825 0.859090i \(-0.328970\pi\)
0.511825 + 0.859090i \(0.328970\pi\)
\(642\) 12.9661i 0.511733i
\(643\) 8.52243i 0.336092i 0.985779 + 0.168046i \(0.0537457\pi\)
−0.985779 + 0.168046i \(0.946254\pi\)
\(644\) − 10.6039i − 0.417851i
\(645\) − 4.07606i − 0.160495i
\(646\) 58.3691 2.29650
\(647\) −37.9318 −1.49125 −0.745627 0.666364i \(-0.767850\pi\)
−0.745627 + 0.666364i \(0.767850\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −16.4416 −0.645389
\(650\) 0 0
\(651\) 11.9239 0.467336
\(652\) − 23.9541i − 0.938114i
\(653\) −6.07846 −0.237868 −0.118934 0.992902i \(-0.537948\pi\)
−0.118934 + 0.992902i \(0.537948\pi\)
\(654\) −19.4644 −0.761119
\(655\) − 5.08815i − 0.198810i
\(656\) − 6.63102i − 0.258898i
\(657\) 1.59850i 0.0623634i
\(658\) 19.6262i 0.765110i
\(659\) 38.9162 1.51596 0.757979 0.652279i \(-0.226187\pi\)
0.757979 + 0.652279i \(0.226187\pi\)
\(660\) 1.19806 0.0466345
\(661\) 25.9191i 1.00814i 0.863663 + 0.504069i \(0.168164\pi\)
−0.863663 + 0.504069i \(0.831836\pi\)
\(662\) −11.7603 −0.457078
\(663\) 0 0
\(664\) −8.40581 −0.326209
\(665\) 15.1304i 0.586730i
\(666\) 4.34481 0.168358
\(667\) 24.6233 0.953416
\(668\) 1.19806i 0.0463544i
\(669\) − 14.3787i − 0.555911i
\(670\) 14.1468i 0.546537i
\(671\) − 1.42088i − 0.0548524i
\(672\) −1.75302 −0.0676242
\(673\) 25.5724 0.985744 0.492872 0.870102i \(-0.335947\pi\)
0.492872 + 0.870102i \(0.335947\pi\)
\(674\) 26.8732i 1.03512i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 24.7453 0.951037 0.475519 0.879706i \(-0.342260\pi\)
0.475519 + 0.879706i \(0.342260\pi\)
\(678\) 0.515729i 0.0198065i
\(679\) −18.9243 −0.726247
\(680\) 6.76271 0.259338
\(681\) 5.97716i 0.229045i
\(682\) 8.14914i 0.312047i
\(683\) 22.4045i 0.857284i 0.903474 + 0.428642i \(0.141008\pi\)
−0.903474 + 0.428642i \(0.858992\pi\)
\(684\) 8.63102i 0.330015i
\(685\) −0.674563 −0.0257737
\(686\) −19.1551 −0.731346
\(687\) − 15.9172i − 0.607280i
\(688\) −4.07606 −0.155398
\(689\) 0 0
\(690\) −6.04892 −0.230278
\(691\) 16.0030i 0.608782i 0.952547 + 0.304391i \(0.0984530\pi\)
−0.952547 + 0.304391i \(0.901547\pi\)
\(692\) −6.63773 −0.252329
\(693\) −2.10023 −0.0797810
\(694\) 29.5080i 1.12011i
\(695\) 8.26205i 0.313397i
\(696\) − 4.07069i − 0.154299i
\(697\) − 44.8437i − 1.69858i
\(698\) −17.0519 −0.645424
\(699\) 6.92154 0.261797
\(700\) 1.75302i 0.0662579i
\(701\) −32.7222 −1.23590 −0.617949 0.786218i \(-0.712036\pi\)
−0.617949 + 0.786218i \(0.712036\pi\)
\(702\) 0 0
\(703\) −37.5002 −1.41435
\(704\) − 1.19806i − 0.0451537i
\(705\) 11.1957 0.421653
\(706\) −21.8629 −0.822822
\(707\) − 29.4288i − 1.10678i
\(708\) − 13.7235i − 0.515760i
\(709\) 14.0597i 0.528022i 0.964520 + 0.264011i \(0.0850455\pi\)
−0.964520 + 0.264011i \(0.914954\pi\)
\(710\) − 2.32304i − 0.0871823i
\(711\) 3.74094 0.140296
\(712\) 4.49934 0.168620
\(713\) − 41.1444i − 1.54087i
\(714\) −11.8552 −0.443668
\(715\) 0 0
\(716\) 19.6353 0.733807
\(717\) − 30.1075i − 1.12439i
\(718\) −14.3026 −0.533768
\(719\) 7.15585 0.266868 0.133434 0.991058i \(-0.457400\pi\)
0.133434 + 0.991058i \(0.457400\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 7.29995i 0.271864i
\(722\) − 55.4946i − 2.06529i
\(723\) 12.0562i 0.448375i
\(724\) 5.25906 0.195451
\(725\) −4.07069 −0.151182
\(726\) 9.56465i 0.354977i
\(727\) −36.4704 −1.35261 −0.676306 0.736621i \(-0.736420\pi\)
−0.676306 + 0.736621i \(0.736420\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 1.59850i − 0.0591631i
\(731\) −27.5652 −1.01954
\(732\) 1.18598 0.0438351
\(733\) 38.0646i 1.40595i 0.711216 + 0.702974i \(0.248145\pi\)
−0.711216 + 0.702974i \(0.751855\pi\)
\(734\) − 11.2416i − 0.414935i
\(735\) 3.92692i 0.144847i
\(736\) 6.04892i 0.222966i
\(737\) 16.9487 0.624313
\(738\) 6.63102 0.244091
\(739\) 35.5114i 1.30631i 0.757225 + 0.653154i \(0.226555\pi\)
−0.757225 + 0.653154i \(0.773445\pi\)
\(740\) −4.34481 −0.159719
\(741\) 0 0
\(742\) 2.01447 0.0739537
\(743\) − 20.3696i − 0.747287i −0.927572 0.373643i \(-0.878108\pi\)
0.927572 0.373643i \(-0.121892\pi\)
\(744\) −6.80194 −0.249371
\(745\) −8.19136 −0.300108
\(746\) − 9.70171i − 0.355205i
\(747\) − 8.40581i − 0.307553i
\(748\) − 8.10215i − 0.296244i
\(749\) 22.7299i 0.830533i
\(750\) 1.00000 0.0365148
\(751\) 5.58104 0.203655 0.101828 0.994802i \(-0.467531\pi\)
0.101828 + 0.994802i \(0.467531\pi\)
\(752\) − 11.1957i − 0.408264i
\(753\) 14.8170 0.539962
\(754\) 0 0
\(755\) 5.72348 0.208299
\(756\) − 1.75302i − 0.0637567i
\(757\) 12.8267 0.466194 0.233097 0.972453i \(-0.425114\pi\)
0.233097 + 0.972453i \(0.425114\pi\)
\(758\) −2.82802 −0.102718
\(759\) 7.24698i 0.263049i
\(760\) − 8.63102i − 0.313080i
\(761\) − 14.0084i − 0.507803i −0.967230 0.253901i \(-0.918286\pi\)
0.967230 0.253901i \(-0.0817139\pi\)
\(762\) − 7.96077i − 0.288388i
\(763\) −34.1215 −1.23528
\(764\) −4.45473 −0.161166
\(765\) 6.76271i 0.244506i
\(766\) 11.6558 0.421140
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 50.4051i 1.81765i 0.417174 + 0.908827i \(0.363021\pi\)
−0.417174 + 0.908827i \(0.636979\pi\)
\(770\) 2.10023 0.0756869
\(771\) 10.5743 0.380825
\(772\) 0.818331i 0.0294524i
\(773\) 42.5532i 1.53053i 0.643715 + 0.765265i \(0.277392\pi\)
−0.643715 + 0.765265i \(0.722608\pi\)
\(774\) − 4.07606i − 0.146511i
\(775\) 6.80194i 0.244333i
\(776\) 10.7952 0.387526
\(777\) 7.61655 0.273242
\(778\) − 12.5157i − 0.448711i
\(779\) −57.2325 −2.05057
\(780\) 0 0
\(781\) −2.78315 −0.0995890
\(782\) 40.9071i 1.46283i
\(783\) 4.07069 0.145475
\(784\) 3.92692 0.140247
\(785\) 3.09246i 0.110375i
\(786\) − 5.08815i − 0.181488i
\(787\) 12.1927i 0.434622i 0.976102 + 0.217311i \(0.0697286\pi\)
−0.976102 + 0.217311i \(0.930271\pi\)
\(788\) 12.7385i 0.453792i
\(789\) 11.9922 0.426935
\(790\) −3.74094 −0.133097
\(791\) 0.904084i 0.0321455i
\(792\) 1.19806 0.0425713
\(793\) 0 0
\(794\) −21.6136 −0.767037
\(795\) − 1.14914i − 0.0407560i
\(796\) −4.02475 −0.142654
\(797\) −36.4534 −1.29125 −0.645623 0.763656i \(-0.723402\pi\)
−0.645623 + 0.763656i \(0.723402\pi\)
\(798\) 15.1304i 0.535609i
\(799\) − 75.7131i − 2.67854i
\(800\) − 1.00000i − 0.0353553i
\(801\) 4.49934i 0.158976i
\(802\) 11.5157 0.406635
\(803\) −1.91510 −0.0675824
\(804\) 14.1468i 0.498917i
\(805\) −10.6039 −0.373738
\(806\) 0 0
\(807\) −25.4359 −0.895388
\(808\) 16.7875i 0.590581i
\(809\) 9.70278 0.341131 0.170566 0.985346i \(-0.445440\pi\)
0.170566 + 0.985346i \(0.445440\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 26.6547i 0.935974i 0.883735 + 0.467987i \(0.155021\pi\)
−0.883735 + 0.467987i \(0.844979\pi\)
\(812\) − 7.13600i − 0.250424i
\(813\) 20.3666i 0.714287i
\(814\) 5.20536i 0.182448i
\(815\) −23.9541 −0.839074
\(816\) 6.76271 0.236742
\(817\) 35.1806i 1.23081i
\(818\) −0.0991626 −0.00346714
\(819\) 0 0
\(820\) −6.63102 −0.231565
\(821\) 2.58343i 0.0901624i 0.998983 + 0.0450812i \(0.0143547\pi\)
−0.998983 + 0.0450812i \(0.985645\pi\)
\(822\) −0.674563 −0.0235281
\(823\) −19.9855 −0.696652 −0.348326 0.937374i \(-0.613250\pi\)
−0.348326 + 0.937374i \(0.613250\pi\)
\(824\) − 4.16421i − 0.145067i
\(825\) − 1.19806i − 0.0417112i
\(826\) − 24.0575i − 0.837069i
\(827\) − 1.39506i − 0.0485110i −0.999706 0.0242555i \(-0.992278\pi\)
0.999706 0.0242555i \(-0.00772152\pi\)
\(828\) −6.04892 −0.210214
\(829\) 27.6179 0.959208 0.479604 0.877485i \(-0.340780\pi\)
0.479604 + 0.877485i \(0.340780\pi\)
\(830\) 8.40581i 0.291770i
\(831\) −20.1323 −0.698381
\(832\) 0 0
\(833\) 26.5566 0.920132
\(834\) 8.26205i 0.286091i
\(835\) 1.19806 0.0414607
\(836\) −10.3405 −0.357634
\(837\) − 6.80194i − 0.235109i
\(838\) 31.2784i 1.08050i
\(839\) 54.3357i 1.87588i 0.346801 + 0.937939i \(0.387268\pi\)
−0.346801 + 0.937939i \(0.612732\pi\)
\(840\) 1.75302i 0.0604850i
\(841\) −12.4295 −0.428604
\(842\) −38.0452 −1.31112
\(843\) − 29.7265i − 1.02383i
\(844\) −3.10992 −0.107048
\(845\) 0 0
\(846\) 11.1957 0.384915
\(847\) 16.7670i 0.576122i
\(848\) −1.14914 −0.0394618
\(849\) −8.84117 −0.303428
\(850\) − 6.76271i − 0.231959i
\(851\) − 26.2814i − 0.900916i
\(852\) − 2.32304i − 0.0795862i
\(853\) 13.4553i 0.460701i 0.973108 + 0.230351i \(0.0739873\pi\)
−0.973108 + 0.230351i \(0.926013\pi\)
\(854\) 2.07905 0.0711436
\(855\) 8.63102 0.295175
\(856\) − 12.9661i − 0.443174i
\(857\) −42.7200 −1.45929 −0.729644 0.683827i \(-0.760314\pi\)
−0.729644 + 0.683827i \(0.760314\pi\)
\(858\) 0 0
\(859\) 44.3038 1.51163 0.755813 0.654788i \(-0.227242\pi\)
0.755813 + 0.654788i \(0.227242\pi\)
\(860\) 4.07606i 0.138993i
\(861\) 11.6243 0.396156
\(862\) −14.4614 −0.492558
\(863\) − 5.47783i − 0.186467i −0.995644 0.0932337i \(-0.970280\pi\)
0.995644 0.0932337i \(-0.0297204\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 6.63773i 0.225689i
\(866\) − 21.6209i − 0.734707i
\(867\) 28.7342 0.975866
\(868\) −11.9239 −0.404725
\(869\) 4.48188i 0.152037i
\(870\) −4.07069 −0.138009
\(871\) 0 0
\(872\) 19.4644 0.659148
\(873\) 10.7952i 0.365363i
\(874\) 52.2083 1.76597
\(875\) 1.75302 0.0592629
\(876\) − 1.59850i − 0.0540083i
\(877\) − 41.3726i − 1.39705i −0.715585 0.698526i \(-0.753840\pi\)
0.715585 0.698526i \(-0.246160\pi\)
\(878\) 3.23623i 0.109217i
\(879\) 20.5918i 0.694544i
\(880\) −1.19806 −0.0403867
\(881\) 42.3919 1.42822 0.714110 0.700033i \(-0.246832\pi\)
0.714110 + 0.700033i \(0.246832\pi\)
\(882\) 3.92692i 0.132226i
\(883\) 40.8412 1.37441 0.687207 0.726461i \(-0.258836\pi\)
0.687207 + 0.726461i \(0.258836\pi\)
\(884\) 0 0
\(885\) −13.7235 −0.461310
\(886\) − 14.6649i − 0.492676i
\(887\) −35.0810 −1.17790 −0.588952 0.808168i \(-0.700459\pi\)
−0.588952 + 0.808168i \(0.700459\pi\)
\(888\) −4.34481 −0.145802
\(889\) − 13.9554i − 0.468049i
\(890\) − 4.49934i − 0.150818i
\(891\) 1.19806i 0.0401366i
\(892\) 14.3787i 0.481433i
\(893\) −96.6301 −3.23360
\(894\) −8.19136 −0.273960
\(895\) − 19.6353i − 0.656337i
\(896\) 1.75302 0.0585643
\(897\) 0 0
\(898\) −4.08516 −0.136324
\(899\) − 27.6886i − 0.923465i
\(900\) 1.00000 0.0333333
\(901\) −7.77133 −0.258901
\(902\) 7.94438i 0.264519i
\(903\) − 7.14542i − 0.237785i
\(904\) − 0.515729i − 0.0171529i
\(905\) − 5.25906i − 0.174817i
\(906\) 5.72348 0.190150
\(907\) 45.4916 1.51052 0.755261 0.655424i \(-0.227510\pi\)
0.755261 + 0.655424i \(0.227510\pi\)
\(908\) − 5.97716i − 0.198359i
\(909\) −16.7875 −0.556805
\(910\) 0 0
\(911\) 47.4349 1.57159 0.785794 0.618489i \(-0.212255\pi\)
0.785794 + 0.618489i \(0.212255\pi\)
\(912\) − 8.63102i − 0.285802i
\(913\) 10.0707 0.333291
\(914\) 41.8678 1.38487
\(915\) − 1.18598i − 0.0392073i
\(916\) 15.9172i 0.525920i
\(917\) − 8.91962i − 0.294552i
\(918\) 6.76271i 0.223203i
\(919\) −20.3907 −0.672629 −0.336314 0.941750i \(-0.609180\pi\)
−0.336314 + 0.941750i \(0.609180\pi\)
\(920\) 6.04892 0.199427
\(921\) 28.7265i 0.946569i
\(922\) −0.215521 −0.00709779
\(923\) 0 0
\(924\) 2.10023 0.0690924
\(925\) 4.34481i 0.142857i
\(926\) −23.4795 −0.771584
\(927\) 4.16421 0.136771
\(928\) 4.07069i 0.133627i
\(929\) 1.26636i 0.0415478i 0.999784 + 0.0207739i \(0.00661302\pi\)
−0.999784 + 0.0207739i \(0.993387\pi\)
\(930\) 6.80194i 0.223044i
\(931\) − 33.8933i − 1.11081i
\(932\) −6.92154 −0.226723
\(933\) 18.6329 0.610015
\(934\) − 10.1153i − 0.330983i
\(935\) −8.10215 −0.264969
\(936\) 0 0
\(937\) 27.3558 0.893676 0.446838 0.894615i \(-0.352550\pi\)
0.446838 + 0.894615i \(0.352550\pi\)
\(938\) 24.7995i 0.809734i
\(939\) −29.3696 −0.958440
\(940\) −11.1957 −0.365162
\(941\) 60.6620i 1.97752i 0.149495 + 0.988762i \(0.452235\pi\)
−0.149495 + 0.988762i \(0.547765\pi\)
\(942\) 3.09246i 0.100758i
\(943\) − 40.1105i − 1.30618i
\(944\) 13.7235i 0.446661i
\(945\) −1.75302 −0.0570258
\(946\) 4.88338 0.158772
\(947\) − 12.3797i − 0.402287i −0.979562 0.201144i \(-0.935534\pi\)
0.979562 0.201144i \(-0.0644658\pi\)
\(948\) −3.74094 −0.121500
\(949\) 0 0
\(950\) −8.63102 −0.280027
\(951\) − 20.8092i − 0.674786i
\(952\) 11.8552 0.384228
\(953\) −28.7784 −0.932223 −0.466111 0.884726i \(-0.654345\pi\)
−0.466111 + 0.884726i \(0.654345\pi\)
\(954\) − 1.14914i − 0.0372049i
\(955\) 4.45473i 0.144152i
\(956\) 30.1075i 0.973747i
\(957\) 4.87694i 0.157649i
\(958\) −0.980623 −0.0316825
\(959\) −1.18252 −0.0381857
\(960\) − 1.00000i − 0.0322749i
\(961\) −15.2664 −0.492463
\(962\) 0 0
\(963\) 12.9661 0.417828
\(964\) − 12.0562i − 0.388304i
\(965\) 0.818331 0.0263430
\(966\) −10.6039 −0.341174
\(967\) 23.1545i 0.744599i 0.928113 + 0.372300i \(0.121431\pi\)
−0.928113 + 0.372300i \(0.878569\pi\)
\(968\) − 9.56465i − 0.307419i
\(969\) − 58.3691i − 1.87509i
\(970\) − 10.7952i − 0.346614i
\(971\) −2.89572 −0.0929282 −0.0464641 0.998920i \(-0.514795\pi\)
−0.0464641 + 0.998920i \(0.514795\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 14.4835i 0.464321i
\(974\) 7.18300 0.230158
\(975\) 0 0
\(976\) −1.18598 −0.0379623
\(977\) 4.86294i 0.155579i 0.996970 + 0.0777896i \(0.0247862\pi\)
−0.996970 + 0.0777896i \(0.975214\pi\)
\(978\) −23.9541 −0.765967
\(979\) −5.39048 −0.172281
\(980\) − 3.92692i − 0.125441i
\(981\) 19.4644i 0.621451i
\(982\) − 24.5918i − 0.784756i
\(983\) − 52.7506i − 1.68248i −0.540659 0.841242i \(-0.681825\pi\)
0.540659 0.841242i \(-0.318175\pi\)
\(984\) −6.63102 −0.211389
\(985\) 12.7385 0.405884
\(986\) 27.5289i 0.876698i
\(987\) 19.6262 0.624710
\(988\) 0 0
\(989\) −24.6558 −0.784008
\(990\) − 1.19806i − 0.0380769i
\(991\) 7.15538 0.227298 0.113649 0.993521i \(-0.463746\pi\)
0.113649 + 0.993521i \(0.463746\pi\)
\(992\) 6.80194 0.215962
\(993\) 11.7603i 0.373202i
\(994\) − 4.07234i − 0.129167i
\(995\) 4.02475i 0.127593i
\(996\) 8.40581i 0.266348i
\(997\) 39.3870 1.24740 0.623700 0.781664i \(-0.285629\pi\)
0.623700 + 0.781664i \(0.285629\pi\)
\(998\) 30.1608 0.954722
\(999\) − 4.34481i − 0.137464i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5070.2.b.w.1351.4 6
13.5 odd 4 5070.2.a.bx.1.3 yes 3
13.8 odd 4 5070.2.a.bo.1.1 3
13.12 even 2 inner 5070.2.b.w.1351.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bo.1.1 3 13.8 odd 4
5070.2.a.bx.1.3 yes 3 13.5 odd 4
5070.2.b.w.1351.3 6 13.12 even 2 inner
5070.2.b.w.1351.4 6 1.1 even 1 trivial