Properties

Label 5070.2.a.bm.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 \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.19806 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.19806 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +0.554958 q^{11} -1.00000 q^{12} -1.19806 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.45473 q^{17} -1.00000 q^{18} -5.40581 q^{19} +1.00000 q^{20} -1.19806 q^{21} -0.554958 q^{22} +7.96077 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +1.19806 q^{28} -6.89977 q^{29} +1.00000 q^{30} +3.04892 q^{31} -1.00000 q^{32} -0.554958 q^{33} +6.45473 q^{34} +1.19806 q^{35} +1.00000 q^{36} -7.96615 q^{37} +5.40581 q^{38} -1.00000 q^{40} +3.08815 q^{41} +1.19806 q^{42} +1.15883 q^{43} +0.554958 q^{44} +1.00000 q^{45} -7.96077 q^{46} +12.2620 q^{47} -1.00000 q^{48} -5.56465 q^{49} -1.00000 q^{50} +6.45473 q^{51} +8.47219 q^{53} +1.00000 q^{54} +0.554958 q^{55} -1.19806 q^{56} +5.40581 q^{57} +6.89977 q^{58} +1.45712 q^{59} -1.00000 q^{60} +2.76271 q^{61} -3.04892 q^{62} +1.19806 q^{63} +1.00000 q^{64} +0.554958 q^{66} -3.34481 q^{67} -6.45473 q^{68} -7.96077 q^{69} -1.19806 q^{70} +2.35690 q^{71} -1.00000 q^{72} +13.3448 q^{73} +7.96615 q^{74} -1.00000 q^{75} -5.40581 q^{76} +0.664874 q^{77} -0.990311 q^{79} +1.00000 q^{80} +1.00000 q^{81} -3.08815 q^{82} +5.88471 q^{83} -1.19806 q^{84} -6.45473 q^{85} -1.15883 q^{86} +6.89977 q^{87} -0.554958 q^{88} +13.7845 q^{89} -1.00000 q^{90} +7.96077 q^{92} -3.04892 q^{93} -12.2620 q^{94} -5.40581 q^{95} +1.00000 q^{96} +7.07606 q^{97} +5.56465 q^{98} +0.554958 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} + 8q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} + 8q^{7} - 3q^{8} + 3q^{9} - 3q^{10} + 2q^{11} - 3q^{12} - 8q^{14} - 3q^{15} + 3q^{16} + 3q^{17} - 3q^{18} - 3q^{19} + 3q^{20} - 8q^{21} - 2q^{22} + 11q^{23} + 3q^{24} + 3q^{25} - 3q^{27} + 8q^{28} + 2q^{29} + 3q^{30} - 3q^{32} - 2q^{33} - 3q^{34} + 8q^{35} + 3q^{36} - 8q^{37} + 3q^{38} - 3q^{40} + 13q^{41} + 8q^{42} - 5q^{43} + 2q^{44} + 3q^{45} - 11q^{46} + 7q^{47} - 3q^{48} + 5q^{49} - 3q^{50} - 3q^{51} + 19q^{53} + 3q^{54} + 2q^{55} - 8q^{56} + 3q^{57} - 2q^{58} + 23q^{59} - 3q^{60} - 9q^{61} + 8q^{63} + 3q^{64} + 2q^{66} + 13q^{67} + 3q^{68} - 11q^{69} - 8q^{70} + 3q^{71} - 3q^{72} + 17q^{73} + 8q^{74} - 3q^{75} - 3q^{76} + 3q^{77} - 25q^{79} + 3q^{80} + 3q^{81} - 13q^{82} + 20q^{83} - 8q^{84} + 3q^{85} + 5q^{86} - 2q^{87} - 2q^{88} + 21q^{89} - 3q^{90} + 11q^{92} - 7q^{94} - 3q^{95} + 3q^{96} + 6q^{97} - 5q^{98} + 2q^{99} + O(q^{100}) \)

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
\(7\) 1.19806 0.452825 0.226412 0.974032i \(-0.427300\pi\)
0.226412 + 0.974032i \(0.427300\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0.554958 0.167326 0.0836631 0.996494i \(-0.473338\pi\)
0.0836631 + 0.996494i \(0.473338\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.19806 −0.320196
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −6.45473 −1.56550 −0.782751 0.622335i \(-0.786184\pi\)
−0.782751 + 0.622335i \(0.786184\pi\)
\(18\) −1.00000 −0.235702
\(19\) −5.40581 −1.24018 −0.620089 0.784531i \(-0.712904\pi\)
−0.620089 + 0.784531i \(0.712904\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.19806 −0.261439
\(22\) −0.554958 −0.118317
\(23\) 7.96077 1.65994 0.829968 0.557811i \(-0.188359\pi\)
0.829968 + 0.557811i \(0.188359\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.19806 0.226412
\(29\) −6.89977 −1.28126 −0.640628 0.767852i \(-0.721326\pi\)
−0.640628 + 0.767852i \(0.721326\pi\)
\(30\) 1.00000 0.182574
\(31\) 3.04892 0.547602 0.273801 0.961786i \(-0.411719\pi\)
0.273801 + 0.961786i \(0.411719\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.554958 −0.0966058
\(34\) 6.45473 1.10698
\(35\) 1.19806 0.202509
\(36\) 1.00000 0.166667
\(37\) −7.96615 −1.30963 −0.654813 0.755791i \(-0.727253\pi\)
−0.654813 + 0.755791i \(0.727253\pi\)
\(38\) 5.40581 0.876939
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 3.08815 0.482287 0.241144 0.970489i \(-0.422477\pi\)
0.241144 + 0.970489i \(0.422477\pi\)
\(42\) 1.19806 0.184865
\(43\) 1.15883 0.176720 0.0883602 0.996089i \(-0.471837\pi\)
0.0883602 + 0.996089i \(0.471837\pi\)
\(44\) 0.554958 0.0836631
\(45\) 1.00000 0.149071
\(46\) −7.96077 −1.17375
\(47\) 12.2620 1.78860 0.894302 0.447465i \(-0.147673\pi\)
0.894302 + 0.447465i \(0.147673\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.56465 −0.794950
\(50\) −1.00000 −0.141421
\(51\) 6.45473 0.903843
\(52\) 0 0
\(53\) 8.47219 1.16374 0.581872 0.813280i \(-0.302320\pi\)
0.581872 + 0.813280i \(0.302320\pi\)
\(54\) 1.00000 0.136083
\(55\) 0.554958 0.0748305
\(56\) −1.19806 −0.160098
\(57\) 5.40581 0.716017
\(58\) 6.89977 0.905985
\(59\) 1.45712 0.189701 0.0948507 0.995492i \(-0.469763\pi\)
0.0948507 + 0.995492i \(0.469763\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.76271 0.353729 0.176864 0.984235i \(-0.443405\pi\)
0.176864 + 0.984235i \(0.443405\pi\)
\(62\) −3.04892 −0.387213
\(63\) 1.19806 0.150942
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.554958 0.0683106
\(67\) −3.34481 −0.408634 −0.204317 0.978905i \(-0.565497\pi\)
−0.204317 + 0.978905i \(0.565497\pi\)
\(68\) −6.45473 −0.782751
\(69\) −7.96077 −0.958364
\(70\) −1.19806 −0.143196
\(71\) 2.35690 0.279712 0.139856 0.990172i \(-0.455336\pi\)
0.139856 + 0.990172i \(0.455336\pi\)
\(72\) −1.00000 −0.117851
\(73\) 13.3448 1.56189 0.780946 0.624598i \(-0.214737\pi\)
0.780946 + 0.624598i \(0.214737\pi\)
\(74\) 7.96615 0.926046
\(75\) −1.00000 −0.115470
\(76\) −5.40581 −0.620089
\(77\) 0.664874 0.0757695
\(78\) 0 0
\(79\) −0.990311 −0.111419 −0.0557094 0.998447i \(-0.517742\pi\)
−0.0557094 + 0.998447i \(0.517742\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −3.08815 −0.341029
\(83\) 5.88471 0.645931 0.322965 0.946411i \(-0.395320\pi\)
0.322965 + 0.946411i \(0.395320\pi\)
\(84\) −1.19806 −0.130719
\(85\) −6.45473 −0.700114
\(86\) −1.15883 −0.124960
\(87\) 6.89977 0.739733
\(88\) −0.554958 −0.0591587
\(89\) 13.7845 1.46115 0.730576 0.682831i \(-0.239252\pi\)
0.730576 + 0.682831i \(0.239252\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 7.96077 0.829968
\(93\) −3.04892 −0.316158
\(94\) −12.2620 −1.26473
\(95\) −5.40581 −0.554625
\(96\) 1.00000 0.102062
\(97\) 7.07606 0.718465 0.359233 0.933248i \(-0.383038\pi\)
0.359233 + 0.933248i \(0.383038\pi\)
\(98\) 5.56465 0.562114
\(99\) 0.554958 0.0557754
\(100\) 1.00000 0.100000
\(101\) 4.09783 0.407750 0.203875 0.978997i \(-0.434646\pi\)
0.203875 + 0.978997i \(0.434646\pi\)
\(102\) −6.45473 −0.639114
\(103\) −4.16421 −0.410312 −0.205156 0.978729i \(-0.565770\pi\)
−0.205156 + 0.978729i \(0.565770\pi\)
\(104\) 0 0
\(105\) −1.19806 −0.116919
\(106\) −8.47219 −0.822892
\(107\) −4.98254 −0.481680 −0.240840 0.970565i \(-0.577423\pi\)
−0.240840 + 0.970565i \(0.577423\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 1.31336 0.125797 0.0628983 0.998020i \(-0.479966\pi\)
0.0628983 + 0.998020i \(0.479966\pi\)
\(110\) −0.554958 −0.0529132
\(111\) 7.96615 0.756113
\(112\) 1.19806 0.113206
\(113\) 3.55496 0.334422 0.167211 0.985921i \(-0.446524\pi\)
0.167211 + 0.985921i \(0.446524\pi\)
\(114\) −5.40581 −0.506301
\(115\) 7.96077 0.742346
\(116\) −6.89977 −0.640628
\(117\) 0 0
\(118\) −1.45712 −0.134139
\(119\) −7.73317 −0.708898
\(120\) 1.00000 0.0912871
\(121\) −10.6920 −0.972002
\(122\) −2.76271 −0.250124
\(123\) −3.08815 −0.278449
\(124\) 3.04892 0.273801
\(125\) 1.00000 0.0894427
\(126\) −1.19806 −0.106732
\(127\) 21.8756 1.94115 0.970573 0.240806i \(-0.0774118\pi\)
0.970573 + 0.240806i \(0.0774118\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.15883 −0.102030
\(130\) 0 0
\(131\) −21.6528 −1.89181 −0.945907 0.324439i \(-0.894825\pi\)
−0.945907 + 0.324439i \(0.894825\pi\)
\(132\) −0.554958 −0.0483029
\(133\) −6.47650 −0.561584
\(134\) 3.34481 0.288948
\(135\) −1.00000 −0.0860663
\(136\) 6.45473 0.553489
\(137\) 7.08815 0.605581 0.302791 0.953057i \(-0.402082\pi\)
0.302791 + 0.953057i \(0.402082\pi\)
\(138\) 7.96077 0.677666
\(139\) 0.340502 0.0288810 0.0144405 0.999896i \(-0.495403\pi\)
0.0144405 + 0.999896i \(0.495403\pi\)
\(140\) 1.19806 0.101255
\(141\) −12.2620 −1.03265
\(142\) −2.35690 −0.197786
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.89977 −0.572995
\(146\) −13.3448 −1.10442
\(147\) 5.56465 0.458964
\(148\) −7.96615 −0.654813
\(149\) 7.40581 0.606708 0.303354 0.952878i \(-0.401894\pi\)
0.303354 + 0.952878i \(0.401894\pi\)
\(150\) 1.00000 0.0816497
\(151\) 10.4209 0.848039 0.424020 0.905653i \(-0.360619\pi\)
0.424020 + 0.905653i \(0.360619\pi\)
\(152\) 5.40581 0.438469
\(153\) −6.45473 −0.521834
\(154\) −0.664874 −0.0535771
\(155\) 3.04892 0.244895
\(156\) 0 0
\(157\) −19.0151 −1.51757 −0.758784 0.651343i \(-0.774206\pi\)
−0.758784 + 0.651343i \(0.774206\pi\)
\(158\) 0.990311 0.0787849
\(159\) −8.47219 −0.671888
\(160\) −1.00000 −0.0790569
\(161\) 9.53750 0.751660
\(162\) −1.00000 −0.0785674
\(163\) 5.65279 0.442761 0.221380 0.975188i \(-0.428944\pi\)
0.221380 + 0.975188i \(0.428944\pi\)
\(164\) 3.08815 0.241144
\(165\) −0.554958 −0.0432034
\(166\) −5.88471 −0.456742
\(167\) 2.15883 0.167056 0.0835278 0.996505i \(-0.473381\pi\)
0.0835278 + 0.996505i \(0.473381\pi\)
\(168\) 1.19806 0.0924325
\(169\) 0 0
\(170\) 6.45473 0.495055
\(171\) −5.40581 −0.413393
\(172\) 1.15883 0.0883602
\(173\) 21.6722 1.64770 0.823852 0.566805i \(-0.191821\pi\)
0.823852 + 0.566805i \(0.191821\pi\)
\(174\) −6.89977 −0.523070
\(175\) 1.19806 0.0905650
\(176\) 0.554958 0.0418315
\(177\) −1.45712 −0.109524
\(178\) −13.7845 −1.03319
\(179\) −0.956459 −0.0714891 −0.0357446 0.999361i \(-0.511380\pi\)
−0.0357446 + 0.999361i \(0.511380\pi\)
\(180\) 1.00000 0.0745356
\(181\) 0.405813 0.0301639 0.0150819 0.999886i \(-0.495199\pi\)
0.0150819 + 0.999886i \(0.495199\pi\)
\(182\) 0 0
\(183\) −2.76271 −0.204225
\(184\) −7.96077 −0.586876
\(185\) −7.96615 −0.585683
\(186\) 3.04892 0.223557
\(187\) −3.58211 −0.261949
\(188\) 12.2620 0.894302
\(189\) −1.19806 −0.0871462
\(190\) 5.40581 0.392179
\(191\) −19.0097 −1.37549 −0.687746 0.725951i \(-0.741400\pi\)
−0.687746 + 0.725951i \(0.741400\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 0.599564 0.0431575 0.0215788 0.999767i \(-0.493131\pi\)
0.0215788 + 0.999767i \(0.493131\pi\)
\(194\) −7.07606 −0.508032
\(195\) 0 0
\(196\) −5.56465 −0.397475
\(197\) 3.78448 0.269633 0.134816 0.990871i \(-0.456956\pi\)
0.134816 + 0.990871i \(0.456956\pi\)
\(198\) −0.554958 −0.0394392
\(199\) −10.2349 −0.725533 −0.362766 0.931880i \(-0.618168\pi\)
−0.362766 + 0.931880i \(0.618168\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 3.34481 0.235925
\(202\) −4.09783 −0.288323
\(203\) −8.26636 −0.580185
\(204\) 6.45473 0.451922
\(205\) 3.08815 0.215685
\(206\) 4.16421 0.290134
\(207\) 7.96077 0.553312
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 1.19806 0.0826742
\(211\) −8.71379 −0.599882 −0.299941 0.953958i \(-0.596967\pi\)
−0.299941 + 0.953958i \(0.596967\pi\)
\(212\) 8.47219 0.581872
\(213\) −2.35690 −0.161492
\(214\) 4.98254 0.340600
\(215\) 1.15883 0.0790318
\(216\) 1.00000 0.0680414
\(217\) 3.65279 0.247968
\(218\) −1.31336 −0.0889516
\(219\) −13.3448 −0.901759
\(220\) 0.554958 0.0374153
\(221\) 0 0
\(222\) −7.96615 −0.534653
\(223\) −15.0694 −1.00912 −0.504559 0.863377i \(-0.668345\pi\)
−0.504559 + 0.863377i \(0.668345\pi\)
\(224\) −1.19806 −0.0800489
\(225\) 1.00000 0.0666667
\(226\) −3.55496 −0.236472
\(227\) −20.4577 −1.35783 −0.678913 0.734219i \(-0.737549\pi\)
−0.678913 + 0.734219i \(0.737549\pi\)
\(228\) 5.40581 0.358009
\(229\) 18.7657 1.24007 0.620036 0.784573i \(-0.287118\pi\)
0.620036 + 0.784573i \(0.287118\pi\)
\(230\) −7.96077 −0.524918
\(231\) −0.664874 −0.0437455
\(232\) 6.89977 0.452992
\(233\) 23.6353 1.54840 0.774201 0.632940i \(-0.218152\pi\)
0.774201 + 0.632940i \(0.218152\pi\)
\(234\) 0 0
\(235\) 12.2620 0.799888
\(236\) 1.45712 0.0948507
\(237\) 0.990311 0.0643276
\(238\) 7.73317 0.501267
\(239\) 28.7506 1.85972 0.929862 0.367909i \(-0.119926\pi\)
0.929862 + 0.367909i \(0.119926\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −3.89440 −0.250860 −0.125430 0.992102i \(-0.540031\pi\)
−0.125430 + 0.992102i \(0.540031\pi\)
\(242\) 10.6920 0.687309
\(243\) −1.00000 −0.0641500
\(244\) 2.76271 0.176864
\(245\) −5.56465 −0.355512
\(246\) 3.08815 0.196893
\(247\) 0 0
\(248\) −3.04892 −0.193606
\(249\) −5.88471 −0.372928
\(250\) −1.00000 −0.0632456
\(251\) −24.3327 −1.53587 −0.767934 0.640529i \(-0.778715\pi\)
−0.767934 + 0.640529i \(0.778715\pi\)
\(252\) 1.19806 0.0754708
\(253\) 4.41789 0.277751
\(254\) −21.8756 −1.37260
\(255\) 6.45473 0.404211
\(256\) 1.00000 0.0625000
\(257\) 30.4426 1.89896 0.949480 0.313827i \(-0.101611\pi\)
0.949480 + 0.313827i \(0.101611\pi\)
\(258\) 1.15883 0.0721458
\(259\) −9.54394 −0.593032
\(260\) 0 0
\(261\) −6.89977 −0.427085
\(262\) 21.6528 1.33771
\(263\) −9.33944 −0.575894 −0.287947 0.957646i \(-0.592973\pi\)
−0.287947 + 0.957646i \(0.592973\pi\)
\(264\) 0.554958 0.0341553
\(265\) 8.47219 0.520442
\(266\) 6.47650 0.397100
\(267\) −13.7845 −0.843596
\(268\) −3.34481 −0.204317
\(269\) 26.7821 1.63293 0.816466 0.577393i \(-0.195930\pi\)
0.816466 + 0.577393i \(0.195930\pi\)
\(270\) 1.00000 0.0608581
\(271\) −2.58748 −0.157178 −0.0785892 0.996907i \(-0.525042\pi\)
−0.0785892 + 0.996907i \(0.525042\pi\)
\(272\) −6.45473 −0.391376
\(273\) 0 0
\(274\) −7.08815 −0.428211
\(275\) 0.554958 0.0334652
\(276\) −7.96077 −0.479182
\(277\) 11.0532 0.664124 0.332062 0.943258i \(-0.392256\pi\)
0.332062 + 0.943258i \(0.392256\pi\)
\(278\) −0.340502 −0.0204220
\(279\) 3.04892 0.182534
\(280\) −1.19806 −0.0715979
\(281\) 10.5090 0.626916 0.313458 0.949602i \(-0.398513\pi\)
0.313458 + 0.949602i \(0.398513\pi\)
\(282\) 12.2620 0.730194
\(283\) −1.07069 −0.0636458 −0.0318229 0.999494i \(-0.510131\pi\)
−0.0318229 + 0.999494i \(0.510131\pi\)
\(284\) 2.35690 0.139856
\(285\) 5.40581 0.320213
\(286\) 0 0
\(287\) 3.69979 0.218392
\(288\) −1.00000 −0.0589256
\(289\) 24.6635 1.45080
\(290\) 6.89977 0.405169
\(291\) −7.07606 −0.414806
\(292\) 13.3448 0.780946
\(293\) 27.4142 1.60155 0.800777 0.598963i \(-0.204420\pi\)
0.800777 + 0.598963i \(0.204420\pi\)
\(294\) −5.56465 −0.324537
\(295\) 1.45712 0.0848370
\(296\) 7.96615 0.463023
\(297\) −0.554958 −0.0322019
\(298\) −7.40581 −0.429007
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 1.38835 0.0800234
\(302\) −10.4209 −0.599654
\(303\) −4.09783 −0.235414
\(304\) −5.40581 −0.310045
\(305\) 2.76271 0.158192
\(306\) 6.45473 0.368992
\(307\) 24.4969 1.39811 0.699057 0.715066i \(-0.253603\pi\)
0.699057 + 0.715066i \(0.253603\pi\)
\(308\) 0.664874 0.0378847
\(309\) 4.16421 0.236894
\(310\) −3.04892 −0.173167
\(311\) 16.4403 0.932241 0.466121 0.884721i \(-0.345651\pi\)
0.466121 + 0.884721i \(0.345651\pi\)
\(312\) 0 0
\(313\) −26.0320 −1.47142 −0.735709 0.677298i \(-0.763151\pi\)
−0.735709 + 0.677298i \(0.763151\pi\)
\(314\) 19.0151 1.07308
\(315\) 1.19806 0.0675032
\(316\) −0.990311 −0.0557094
\(317\) 19.3545 1.08706 0.543529 0.839391i \(-0.317088\pi\)
0.543529 + 0.839391i \(0.317088\pi\)
\(318\) 8.47219 0.475097
\(319\) −3.82908 −0.214388
\(320\) 1.00000 0.0559017
\(321\) 4.98254 0.278098
\(322\) −9.53750 −0.531504
\(323\) 34.8931 1.94150
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −5.65279 −0.313079
\(327\) −1.31336 −0.0726287
\(328\) −3.08815 −0.170514
\(329\) 14.6907 0.809924
\(330\) 0.554958 0.0305494
\(331\) −0.173899 −0.00955836 −0.00477918 0.999989i \(-0.501521\pi\)
−0.00477918 + 0.999989i \(0.501521\pi\)
\(332\) 5.88471 0.322965
\(333\) −7.96615 −0.436542
\(334\) −2.15883 −0.118126
\(335\) −3.34481 −0.182747
\(336\) −1.19806 −0.0653597
\(337\) −14.3793 −0.783288 −0.391644 0.920117i \(-0.628094\pi\)
−0.391644 + 0.920117i \(0.628094\pi\)
\(338\) 0 0
\(339\) −3.55496 −0.193079
\(340\) −6.45473 −0.350057
\(341\) 1.69202 0.0916281
\(342\) 5.40581 0.292313
\(343\) −15.0532 −0.812798
\(344\) −1.15883 −0.0624801
\(345\) −7.96077 −0.428594
\(346\) −21.6722 −1.16510
\(347\) −26.3400 −1.41401 −0.707003 0.707210i \(-0.749953\pi\)
−0.707003 + 0.707210i \(0.749953\pi\)
\(348\) 6.89977 0.369867
\(349\) −8.44371 −0.451982 −0.225991 0.974129i \(-0.572562\pi\)
−0.225991 + 0.974129i \(0.572562\pi\)
\(350\) −1.19806 −0.0640391
\(351\) 0 0
\(352\) −0.554958 −0.0295794
\(353\) −10.9420 −0.582383 −0.291192 0.956665i \(-0.594052\pi\)
−0.291192 + 0.956665i \(0.594052\pi\)
\(354\) 1.45712 0.0774452
\(355\) 2.35690 0.125091
\(356\) 13.7845 0.730576
\(357\) 7.73317 0.409283
\(358\) 0.956459 0.0505505
\(359\) 8.07606 0.426238 0.213119 0.977026i \(-0.431638\pi\)
0.213119 + 0.977026i \(0.431638\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 10.2228 0.538043
\(362\) −0.405813 −0.0213291
\(363\) 10.6920 0.561186
\(364\) 0 0
\(365\) 13.3448 0.698500
\(366\) 2.76271 0.144409
\(367\) 17.1575 0.895615 0.447807 0.894130i \(-0.352205\pi\)
0.447807 + 0.894130i \(0.352205\pi\)
\(368\) 7.96077 0.414984
\(369\) 3.08815 0.160762
\(370\) 7.96615 0.414140
\(371\) 10.1502 0.526973
\(372\) −3.04892 −0.158079
\(373\) 2.72455 0.141072 0.0705358 0.997509i \(-0.477529\pi\)
0.0705358 + 0.997509i \(0.477529\pi\)
\(374\) 3.58211 0.185226
\(375\) −1.00000 −0.0516398
\(376\) −12.2620 −0.632367
\(377\) 0 0
\(378\) 1.19806 0.0616217
\(379\) −26.8998 −1.38175 −0.690874 0.722975i \(-0.742774\pi\)
−0.690874 + 0.722975i \(0.742774\pi\)
\(380\) −5.40581 −0.277312
\(381\) −21.8756 −1.12072
\(382\) 19.0097 0.972620
\(383\) 29.9511 1.53043 0.765214 0.643776i \(-0.222633\pi\)
0.765214 + 0.643776i \(0.222633\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0.664874 0.0338851
\(386\) −0.599564 −0.0305170
\(387\) 1.15883 0.0589068
\(388\) 7.07606 0.359233
\(389\) 10.0140 0.507730 0.253865 0.967240i \(-0.418298\pi\)
0.253865 + 0.967240i \(0.418298\pi\)
\(390\) 0 0
\(391\) −51.3846 −2.59863
\(392\) 5.56465 0.281057
\(393\) 21.6528 1.09224
\(394\) −3.78448 −0.190659
\(395\) −0.990311 −0.0498280
\(396\) 0.554958 0.0278877
\(397\) 17.0489 0.855661 0.427830 0.903859i \(-0.359278\pi\)
0.427830 + 0.903859i \(0.359278\pi\)
\(398\) 10.2349 0.513029
\(399\) 6.47650 0.324231
\(400\) 1.00000 0.0500000
\(401\) 0.975246 0.0487015 0.0243507 0.999703i \(-0.492248\pi\)
0.0243507 + 0.999703i \(0.492248\pi\)
\(402\) −3.34481 −0.166824
\(403\) 0 0
\(404\) 4.09783 0.203875
\(405\) 1.00000 0.0496904
\(406\) 8.26636 0.410252
\(407\) −4.42088 −0.219135
\(408\) −6.45473 −0.319557
\(409\) −37.7439 −1.86632 −0.933158 0.359465i \(-0.882959\pi\)
−0.933158 + 0.359465i \(0.882959\pi\)
\(410\) −3.08815 −0.152513
\(411\) −7.08815 −0.349632
\(412\) −4.16421 −0.205156
\(413\) 1.74572 0.0859015
\(414\) −7.96077 −0.391251
\(415\) 5.88471 0.288869
\(416\) 0 0
\(417\) −0.340502 −0.0166745
\(418\) 3.00000 0.146735
\(419\) −21.5700 −1.05376 −0.526882 0.849938i \(-0.676639\pi\)
−0.526882 + 0.849938i \(0.676639\pi\)
\(420\) −1.19806 −0.0584595
\(421\) 33.2403 1.62003 0.810016 0.586408i \(-0.199458\pi\)
0.810016 + 0.586408i \(0.199458\pi\)
\(422\) 8.71379 0.424181
\(423\) 12.2620 0.596201
\(424\) −8.47219 −0.411446
\(425\) −6.45473 −0.313100
\(426\) 2.35690 0.114192
\(427\) 3.30990 0.160177
\(428\) −4.98254 −0.240840
\(429\) 0 0
\(430\) −1.15883 −0.0558839
\(431\) 20.9041 1.00691 0.503457 0.864020i \(-0.332061\pi\)
0.503457 + 0.864020i \(0.332061\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 36.5864 1.75823 0.879115 0.476609i \(-0.158134\pi\)
0.879115 + 0.476609i \(0.158134\pi\)
\(434\) −3.65279 −0.175340
\(435\) 6.89977 0.330819
\(436\) 1.31336 0.0628983
\(437\) −43.0344 −2.05862
\(438\) 13.3448 0.637640
\(439\) 27.5036 1.31268 0.656339 0.754466i \(-0.272104\pi\)
0.656339 + 0.754466i \(0.272104\pi\)
\(440\) −0.554958 −0.0264566
\(441\) −5.56465 −0.264983
\(442\) 0 0
\(443\) −4.34614 −0.206491 −0.103246 0.994656i \(-0.532923\pi\)
−0.103246 + 0.994656i \(0.532923\pi\)
\(444\) 7.96615 0.378057
\(445\) 13.7845 0.653447
\(446\) 15.0694 0.713555
\(447\) −7.40581 −0.350283
\(448\) 1.19806 0.0566031
\(449\) 0.423272 0.0199754 0.00998771 0.999950i \(-0.496821\pi\)
0.00998771 + 0.999950i \(0.496821\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 1.71379 0.0806993
\(452\) 3.55496 0.167211
\(453\) −10.4209 −0.489616
\(454\) 20.4577 0.960128
\(455\) 0 0
\(456\) −5.40581 −0.253150
\(457\) −18.6396 −0.871926 −0.435963 0.899965i \(-0.643592\pi\)
−0.435963 + 0.899965i \(0.643592\pi\)
\(458\) −18.7657 −0.876863
\(459\) 6.45473 0.301281
\(460\) 7.96077 0.371173
\(461\) 8.22223 0.382947 0.191474 0.981498i \(-0.438673\pi\)
0.191474 + 0.981498i \(0.438673\pi\)
\(462\) 0.664874 0.0309328
\(463\) 34.1118 1.58531 0.792656 0.609670i \(-0.208698\pi\)
0.792656 + 0.609670i \(0.208698\pi\)
\(464\) −6.89977 −0.320314
\(465\) −3.04892 −0.141390
\(466\) −23.6353 −1.09489
\(467\) 13.9245 0.644350 0.322175 0.946680i \(-0.395586\pi\)
0.322175 + 0.946680i \(0.395586\pi\)
\(468\) 0 0
\(469\) −4.00730 −0.185040
\(470\) −12.2620 −0.565606
\(471\) 19.0151 0.876168
\(472\) −1.45712 −0.0670695
\(473\) 0.643104 0.0295700
\(474\) −0.990311 −0.0454865
\(475\) −5.40581 −0.248036
\(476\) −7.73317 −0.354449
\(477\) 8.47219 0.387915
\(478\) −28.7506 −1.31502
\(479\) 1.79225 0.0818899 0.0409450 0.999161i \(-0.486963\pi\)
0.0409450 + 0.999161i \(0.486963\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 3.89440 0.177385
\(483\) −9.53750 −0.433971
\(484\) −10.6920 −0.486001
\(485\) 7.07606 0.321308
\(486\) 1.00000 0.0453609
\(487\) −0.126310 −0.00572364 −0.00286182 0.999996i \(-0.500911\pi\)
−0.00286182 + 0.999996i \(0.500911\pi\)
\(488\) −2.76271 −0.125062
\(489\) −5.65279 −0.255628
\(490\) 5.56465 0.251385
\(491\) −17.6823 −0.797993 −0.398996 0.916953i \(-0.630641\pi\)
−0.398996 + 0.916953i \(0.630641\pi\)
\(492\) −3.08815 −0.139224
\(493\) 44.5362 2.00581
\(494\) 0 0
\(495\) 0.554958 0.0249435
\(496\) 3.04892 0.136900
\(497\) 2.82371 0.126661
\(498\) 5.88471 0.263700
\(499\) −21.1535 −0.946959 −0.473479 0.880805i \(-0.657002\pi\)
−0.473479 + 0.880805i \(0.657002\pi\)
\(500\) 1.00000 0.0447214
\(501\) −2.15883 −0.0964496
\(502\) 24.3327 1.08602
\(503\) −29.7187 −1.32509 −0.662546 0.749022i \(-0.730524\pi\)
−0.662546 + 0.749022i \(0.730524\pi\)
\(504\) −1.19806 −0.0533659
\(505\) 4.09783 0.182351
\(506\) −4.41789 −0.196399
\(507\) 0 0
\(508\) 21.8756 0.970573
\(509\) 0.786872 0.0348775 0.0174387 0.999848i \(-0.494449\pi\)
0.0174387 + 0.999848i \(0.494449\pi\)
\(510\) −6.45473 −0.285820
\(511\) 15.9879 0.707264
\(512\) −1.00000 −0.0441942
\(513\) 5.40581 0.238672
\(514\) −30.4426 −1.34277
\(515\) −4.16421 −0.183497
\(516\) −1.15883 −0.0510148
\(517\) 6.80492 0.299280
\(518\) 9.54394 0.419337
\(519\) −21.6722 −0.951303
\(520\) 0 0
\(521\) 24.1782 1.05927 0.529633 0.848227i \(-0.322330\pi\)
0.529633 + 0.848227i \(0.322330\pi\)
\(522\) 6.89977 0.301995
\(523\) 23.0750 1.00900 0.504500 0.863412i \(-0.331677\pi\)
0.504500 + 0.863412i \(0.331677\pi\)
\(524\) −21.6528 −0.945907
\(525\) −1.19806 −0.0522877
\(526\) 9.33944 0.407219
\(527\) −19.6799 −0.857272
\(528\) −0.554958 −0.0241515
\(529\) 40.3739 1.75539
\(530\) −8.47219 −0.368008
\(531\) 1.45712 0.0632338
\(532\) −6.47650 −0.280792
\(533\) 0 0
\(534\) 13.7845 0.596513
\(535\) −4.98254 −0.215414
\(536\) 3.34481 0.144474
\(537\) 0.956459 0.0412743
\(538\) −26.7821 −1.15466
\(539\) −3.08815 −0.133016
\(540\) −1.00000 −0.0430331
\(541\) −24.6122 −1.05816 −0.529081 0.848571i \(-0.677463\pi\)
−0.529081 + 0.848571i \(0.677463\pi\)
\(542\) 2.58748 0.111142
\(543\) −0.405813 −0.0174151
\(544\) 6.45473 0.276744
\(545\) 1.31336 0.0562580
\(546\) 0 0
\(547\) −0.492894 −0.0210746 −0.0105373 0.999944i \(-0.503354\pi\)
−0.0105373 + 0.999944i \(0.503354\pi\)
\(548\) 7.08815 0.302791
\(549\) 2.76271 0.117910
\(550\) −0.554958 −0.0236635
\(551\) 37.2989 1.58899
\(552\) 7.96077 0.338833
\(553\) −1.18645 −0.0504532
\(554\) −11.0532 −0.469607
\(555\) 7.96615 0.338144
\(556\) 0.340502 0.0144405
\(557\) −12.6028 −0.533998 −0.266999 0.963697i \(-0.586032\pi\)
−0.266999 + 0.963697i \(0.586032\pi\)
\(558\) −3.04892 −0.129071
\(559\) 0 0
\(560\) 1.19806 0.0506274
\(561\) 3.58211 0.151237
\(562\) −10.5090 −0.443296
\(563\) 24.5743 1.03568 0.517842 0.855476i \(-0.326735\pi\)
0.517842 + 0.855476i \(0.326735\pi\)
\(564\) −12.2620 −0.516325
\(565\) 3.55496 0.149558
\(566\) 1.07069 0.0450044
\(567\) 1.19806 0.0503139
\(568\) −2.35690 −0.0988932
\(569\) 18.9414 0.794065 0.397032 0.917805i \(-0.370040\pi\)
0.397032 + 0.917805i \(0.370040\pi\)
\(570\) −5.40581 −0.226425
\(571\) −17.2978 −0.723891 −0.361946 0.932199i \(-0.617887\pi\)
−0.361946 + 0.932199i \(0.617887\pi\)
\(572\) 0 0
\(573\) 19.0097 0.794141
\(574\) −3.69979 −0.154426
\(575\) 7.96077 0.331987
\(576\) 1.00000 0.0416667
\(577\) −42.5628 −1.77191 −0.885957 0.463767i \(-0.846497\pi\)
−0.885957 + 0.463767i \(0.846497\pi\)
\(578\) −24.6635 −1.02587
\(579\) −0.599564 −0.0249170
\(580\) −6.89977 −0.286497
\(581\) 7.05025 0.292493
\(582\) 7.07606 0.293312
\(583\) 4.70171 0.194725
\(584\) −13.3448 −0.552212
\(585\) 0 0
\(586\) −27.4142 −1.13247
\(587\) 43.2737 1.78609 0.893047 0.449963i \(-0.148563\pi\)
0.893047 + 0.449963i \(0.148563\pi\)
\(588\) 5.56465 0.229482
\(589\) −16.4819 −0.679124
\(590\) −1.45712 −0.0599888
\(591\) −3.78448 −0.155673
\(592\) −7.96615 −0.327407
\(593\) 19.8672 0.815850 0.407925 0.913015i \(-0.366252\pi\)
0.407925 + 0.913015i \(0.366252\pi\)
\(594\) 0.554958 0.0227702
\(595\) −7.73317 −0.317029
\(596\) 7.40581 0.303354
\(597\) 10.2349 0.418886
\(598\) 0 0
\(599\) 22.3448 0.912984 0.456492 0.889727i \(-0.349106\pi\)
0.456492 + 0.889727i \(0.349106\pi\)
\(600\) 1.00000 0.0408248
\(601\) 38.3696 1.56513 0.782564 0.622571i \(-0.213912\pi\)
0.782564 + 0.622571i \(0.213912\pi\)
\(602\) −1.38835 −0.0565851
\(603\) −3.34481 −0.136211
\(604\) 10.4209 0.424020
\(605\) −10.6920 −0.434692
\(606\) 4.09783 0.166463
\(607\) 34.9778 1.41970 0.709852 0.704351i \(-0.248762\pi\)
0.709852 + 0.704351i \(0.248762\pi\)
\(608\) 5.40581 0.219235
\(609\) 8.26636 0.334970
\(610\) −2.76271 −0.111859
\(611\) 0 0
\(612\) −6.45473 −0.260917
\(613\) 28.1648 1.13757 0.568783 0.822488i \(-0.307414\pi\)
0.568783 + 0.822488i \(0.307414\pi\)
\(614\) −24.4969 −0.988616
\(615\) −3.08815 −0.124526
\(616\) −0.664874 −0.0267886
\(617\) −24.0640 −0.968779 −0.484390 0.874852i \(-0.660958\pi\)
−0.484390 + 0.874852i \(0.660958\pi\)
\(618\) −4.16421 −0.167509
\(619\) −14.7651 −0.593460 −0.296730 0.954961i \(-0.595896\pi\)
−0.296730 + 0.954961i \(0.595896\pi\)
\(620\) 3.04892 0.122447
\(621\) −7.96077 −0.319455
\(622\) −16.4403 −0.659194
\(623\) 16.5147 0.661646
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 26.0320 1.04045
\(627\) 3.00000 0.119808
\(628\) −19.0151 −0.758784
\(629\) 51.4193 2.05022
\(630\) −1.19806 −0.0477319
\(631\) −22.9788 −0.914772 −0.457386 0.889268i \(-0.651214\pi\)
−0.457386 + 0.889268i \(0.651214\pi\)
\(632\) 0.990311 0.0393925
\(633\) 8.71379 0.346342
\(634\) −19.3545 −0.768666
\(635\) 21.8756 0.868107
\(636\) −8.47219 −0.335944
\(637\) 0 0
\(638\) 3.82908 0.151595
\(639\) 2.35690 0.0932374
\(640\) −1.00000 −0.0395285
\(641\) −5.34913 −0.211278 −0.105639 0.994405i \(-0.533689\pi\)
−0.105639 + 0.994405i \(0.533689\pi\)
\(642\) −4.98254 −0.196645
\(643\) 39.2984 1.54978 0.774889 0.632097i \(-0.217806\pi\)
0.774889 + 0.632097i \(0.217806\pi\)
\(644\) 9.53750 0.375830
\(645\) −1.15883 −0.0456290
\(646\) −34.8931 −1.37285
\(647\) 0.328684 0.0129219 0.00646095 0.999979i \(-0.497943\pi\)
0.00646095 + 0.999979i \(0.497943\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0.808643 0.0317420
\(650\) 0 0
\(651\) −3.65279 −0.143164
\(652\) 5.65279 0.221380
\(653\) 41.6426 1.62960 0.814801 0.579741i \(-0.196846\pi\)
0.814801 + 0.579741i \(0.196846\pi\)
\(654\) 1.31336 0.0513563
\(655\) −21.6528 −0.846045
\(656\) 3.08815 0.120572
\(657\) 13.3448 0.520631
\(658\) −14.6907 −0.572703
\(659\) −4.24698 −0.165439 −0.0827194 0.996573i \(-0.526361\pi\)
−0.0827194 + 0.996573i \(0.526361\pi\)
\(660\) −0.554958 −0.0216017
\(661\) 29.7071 1.15547 0.577736 0.816224i \(-0.303936\pi\)
0.577736 + 0.816224i \(0.303936\pi\)
\(662\) 0.173899 0.00675878
\(663\) 0 0
\(664\) −5.88471 −0.228371
\(665\) −6.47650 −0.251148
\(666\) 7.96615 0.308682
\(667\) −54.9275 −2.12680
\(668\) 2.15883 0.0835278
\(669\) 15.0694 0.582615
\(670\) 3.34481 0.129221
\(671\) 1.53319 0.0591881
\(672\) 1.19806 0.0462163
\(673\) −21.7004 −0.836488 −0.418244 0.908335i \(-0.637354\pi\)
−0.418244 + 0.908335i \(0.637354\pi\)
\(674\) 14.3793 0.553868
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −12.5168 −0.481059 −0.240530 0.970642i \(-0.577321\pi\)
−0.240530 + 0.970642i \(0.577321\pi\)
\(678\) 3.55496 0.136527
\(679\) 8.47757 0.325339
\(680\) 6.45473 0.247528
\(681\) 20.4577 0.783941
\(682\) −1.69202 −0.0647909
\(683\) −3.82908 −0.146516 −0.0732579 0.997313i \(-0.523340\pi\)
−0.0732579 + 0.997313i \(0.523340\pi\)
\(684\) −5.40581 −0.206696
\(685\) 7.08815 0.270824
\(686\) 15.0532 0.574735
\(687\) −18.7657 −0.715956
\(688\) 1.15883 0.0441801
\(689\) 0 0
\(690\) 7.96077 0.303061
\(691\) 27.4239 1.04325 0.521626 0.853174i \(-0.325325\pi\)
0.521626 + 0.853174i \(0.325325\pi\)
\(692\) 21.6722 0.823852
\(693\) 0.664874 0.0252565
\(694\) 26.3400 0.999854
\(695\) 0.340502 0.0129160
\(696\) −6.89977 −0.261535
\(697\) −19.9332 −0.755022
\(698\) 8.44371 0.319599
\(699\) −23.6353 −0.893970
\(700\) 1.19806 0.0452825
\(701\) 31.5803 1.19277 0.596386 0.802698i \(-0.296603\pi\)
0.596386 + 0.802698i \(0.296603\pi\)
\(702\) 0 0
\(703\) 43.0635 1.62417
\(704\) 0.554958 0.0209158
\(705\) −12.2620 −0.461815
\(706\) 10.9420 0.411807
\(707\) 4.90946 0.184639
\(708\) −1.45712 −0.0547621
\(709\) 28.9788 1.08832 0.544161 0.838981i \(-0.316848\pi\)
0.544161 + 0.838981i \(0.316848\pi\)
\(710\) −2.35690 −0.0884527
\(711\) −0.990311 −0.0371396
\(712\) −13.7845 −0.516595
\(713\) 24.2717 0.908984
\(714\) −7.73317 −0.289407
\(715\) 0 0
\(716\) −0.956459 −0.0357446
\(717\) −28.7506 −1.07371
\(718\) −8.07606 −0.301396
\(719\) 27.0670 1.00943 0.504714 0.863287i \(-0.331598\pi\)
0.504714 + 0.863287i \(0.331598\pi\)
\(720\) 1.00000 0.0372678
\(721\) −4.98898 −0.185799
\(722\) −10.2228 −0.380454
\(723\) 3.89440 0.144834
\(724\) 0.405813 0.0150819
\(725\) −6.89977 −0.256251
\(726\) −10.6920 −0.396818
\(727\) −26.6612 −0.988807 −0.494404 0.869232i \(-0.664614\pi\)
−0.494404 + 0.869232i \(0.664614\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −13.3448 −0.493914
\(731\) −7.47996 −0.276656
\(732\) −2.76271 −0.102113
\(733\) −51.2984 −1.89475 −0.947375 0.320126i \(-0.896275\pi\)
−0.947375 + 0.320126i \(0.896275\pi\)
\(734\) −17.1575 −0.633295
\(735\) 5.56465 0.205255
\(736\) −7.96077 −0.293438
\(737\) −1.85623 −0.0683752
\(738\) −3.08815 −0.113676
\(739\) 28.6456 1.05375 0.526873 0.849944i \(-0.323364\pi\)
0.526873 + 0.849944i \(0.323364\pi\)
\(740\) −7.96615 −0.292841
\(741\) 0 0
\(742\) −10.1502 −0.372626
\(743\) −43.6674 −1.60200 −0.801000 0.598664i \(-0.795699\pi\)
−0.801000 + 0.598664i \(0.795699\pi\)
\(744\) 3.04892 0.111779
\(745\) 7.40581 0.271328
\(746\) −2.72455 −0.0997527
\(747\) 5.88471 0.215310
\(748\) −3.58211 −0.130975
\(749\) −5.96940 −0.218117
\(750\) 1.00000 0.0365148
\(751\) −20.1535 −0.735410 −0.367705 0.929942i \(-0.619856\pi\)
−0.367705 + 0.929942i \(0.619856\pi\)
\(752\) 12.2620 0.447151
\(753\) 24.3327 0.886734
\(754\) 0 0
\(755\) 10.4209 0.379255
\(756\) −1.19806 −0.0435731
\(757\) −41.8562 −1.52129 −0.760645 0.649168i \(-0.775117\pi\)
−0.760645 + 0.649168i \(0.775117\pi\)
\(758\) 26.8998 0.977044
\(759\) −4.41789 −0.160359
\(760\) 5.40581 0.196089
\(761\) −32.8665 −1.19141 −0.595705 0.803203i \(-0.703127\pi\)
−0.595705 + 0.803203i \(0.703127\pi\)
\(762\) 21.8756 0.792470
\(763\) 1.57348 0.0569639
\(764\) −19.0097 −0.687746
\(765\) −6.45473 −0.233371
\(766\) −29.9511 −1.08218
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −37.8877 −1.36627 −0.683133 0.730294i \(-0.739383\pi\)
−0.683133 + 0.730294i \(0.739383\pi\)
\(770\) −0.664874 −0.0239604
\(771\) −30.4426 −1.09637
\(772\) 0.599564 0.0215788
\(773\) −12.4523 −0.447879 −0.223940 0.974603i \(-0.571892\pi\)
−0.223940 + 0.974603i \(0.571892\pi\)
\(774\) −1.15883 −0.0416534
\(775\) 3.04892 0.109520
\(776\) −7.07606 −0.254016
\(777\) 9.54394 0.342387
\(778\) −10.0140 −0.359019
\(779\) −16.6939 −0.598122
\(780\) 0 0
\(781\) 1.30798 0.0468032
\(782\) 51.3846 1.83751
\(783\) 6.89977 0.246578
\(784\) −5.56465 −0.198737
\(785\) −19.0151 −0.678677
\(786\) −21.6528 −0.772330
\(787\) −6.20536 −0.221197 −0.110599 0.993865i \(-0.535277\pi\)
−0.110599 + 0.993865i \(0.535277\pi\)
\(788\) 3.78448 0.134816
\(789\) 9.33944 0.332493
\(790\) 0.990311 0.0352337
\(791\) 4.25906 0.151435
\(792\) −0.554958 −0.0197196
\(793\) 0 0
\(794\) −17.0489 −0.605043
\(795\) −8.47219 −0.300478
\(796\) −10.2349 −0.362766
\(797\) 33.0627 1.17114 0.585570 0.810622i \(-0.300871\pi\)
0.585570 + 0.810622i \(0.300871\pi\)
\(798\) −6.47650 −0.229266
\(799\) −79.1482 −2.80006
\(800\) −1.00000 −0.0353553
\(801\) 13.7845 0.487051
\(802\) −0.975246 −0.0344371
\(803\) 7.40581 0.261345
\(804\) 3.34481 0.117963
\(805\) 9.53750 0.336153
\(806\) 0 0
\(807\) −26.7821 −0.942774
\(808\) −4.09783 −0.144161
\(809\) −52.5454 −1.84740 −0.923699 0.383120i \(-0.874850\pi\)
−0.923699 + 0.383120i \(0.874850\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −9.96508 −0.349921 −0.174961 0.984575i \(-0.555980\pi\)
−0.174961 + 0.984575i \(0.555980\pi\)
\(812\) −8.26636 −0.290092
\(813\) 2.58748 0.0907470
\(814\) 4.42088 0.154952
\(815\) 5.65279 0.198009
\(816\) 6.45473 0.225961
\(817\) −6.26444 −0.219165
\(818\) 37.7439 1.31969
\(819\) 0 0
\(820\) 3.08815 0.107843
\(821\) 28.6950 1.00146 0.500731 0.865603i \(-0.333064\pi\)
0.500731 + 0.865603i \(0.333064\pi\)
\(822\) 7.08815 0.247227
\(823\) 37.4053 1.30387 0.651934 0.758276i \(-0.273958\pi\)
0.651934 + 0.758276i \(0.273958\pi\)
\(824\) 4.16421 0.145067
\(825\) −0.554958 −0.0193212
\(826\) −1.74572 −0.0607415
\(827\) −36.3666 −1.26459 −0.632295 0.774728i \(-0.717887\pi\)
−0.632295 + 0.774728i \(0.717887\pi\)
\(828\) 7.96077 0.276656
\(829\) −32.6370 −1.13353 −0.566765 0.823880i \(-0.691805\pi\)
−0.566765 + 0.823880i \(0.691805\pi\)
\(830\) −5.88471 −0.204261
\(831\) −11.0532 −0.383432
\(832\) 0 0
\(833\) 35.9183 1.24450
\(834\) 0.340502 0.0117906
\(835\) 2.15883 0.0747095
\(836\) −3.00000 −0.103757
\(837\) −3.04892 −0.105386
\(838\) 21.5700 0.745124
\(839\) −11.6810 −0.403273 −0.201637 0.979460i \(-0.564626\pi\)
−0.201637 + 0.979460i \(0.564626\pi\)
\(840\) 1.19806 0.0413371
\(841\) 18.6069 0.641616
\(842\) −33.2403 −1.14554
\(843\) −10.5090 −0.361950
\(844\) −8.71379 −0.299941
\(845\) 0 0
\(846\) −12.2620 −0.421578
\(847\) −12.8097 −0.440147
\(848\) 8.47219 0.290936
\(849\) 1.07069 0.0367459
\(850\) 6.45473 0.221395
\(851\) −63.4167 −2.17390
\(852\) −2.35690 −0.0807459
\(853\) −45.1756 −1.54678 −0.773391 0.633930i \(-0.781441\pi\)
−0.773391 + 0.633930i \(0.781441\pi\)
\(854\) −3.30990 −0.113262
\(855\) −5.40581 −0.184875
\(856\) 4.98254 0.170300
\(857\) −16.1602 −0.552021 −0.276010 0.961155i \(-0.589012\pi\)
−0.276010 + 0.961155i \(0.589012\pi\)
\(858\) 0 0
\(859\) 24.9071 0.849818 0.424909 0.905236i \(-0.360306\pi\)
0.424909 + 0.905236i \(0.360306\pi\)
\(860\) 1.15883 0.0395159
\(861\) −3.69979 −0.126089
\(862\) −20.9041 −0.711996
\(863\) −25.9030 −0.881749 −0.440875 0.897569i \(-0.645332\pi\)
−0.440875 + 0.897569i \(0.645332\pi\)
\(864\) 1.00000 0.0340207
\(865\) 21.6722 0.736876
\(866\) −36.5864 −1.24326
\(867\) −24.6635 −0.837618
\(868\) 3.65279 0.123984
\(869\) −0.549581 −0.0186433
\(870\) −6.89977 −0.233924
\(871\) 0 0
\(872\) −1.31336 −0.0444758
\(873\) 7.07606 0.239488
\(874\) 43.0344 1.45566
\(875\) 1.19806 0.0405019
\(876\) −13.3448 −0.450879
\(877\) −21.5875 −0.728957 −0.364479 0.931212i \(-0.618753\pi\)
−0.364479 + 0.931212i \(0.618753\pi\)
\(878\) −27.5036 −0.928203
\(879\) −27.4142 −0.924657
\(880\) 0.554958 0.0187076
\(881\) 0.468140 0.0157720 0.00788602 0.999969i \(-0.497490\pi\)
0.00788602 + 0.999969i \(0.497490\pi\)
\(882\) 5.56465 0.187371
\(883\) −43.6034 −1.46737 −0.733686 0.679489i \(-0.762202\pi\)
−0.733686 + 0.679489i \(0.762202\pi\)
\(884\) 0 0
\(885\) −1.45712 −0.0489807
\(886\) 4.34614 0.146012
\(887\) 24.1038 0.809326 0.404663 0.914466i \(-0.367389\pi\)
0.404663 + 0.914466i \(0.367389\pi\)
\(888\) −7.96615 −0.267326
\(889\) 26.2083 0.879000
\(890\) −13.7845 −0.462057
\(891\) 0.554958 0.0185918
\(892\) −15.0694 −0.504559
\(893\) −66.2863 −2.21819
\(894\) 7.40581 0.247687
\(895\) −0.956459 −0.0319709
\(896\) −1.19806 −0.0400245
\(897\) 0 0
\(898\) −0.423272 −0.0141248
\(899\) −21.0368 −0.701618
\(900\) 1.00000 0.0333333
\(901\) −54.6857 −1.82184
\(902\) −1.71379 −0.0570630
\(903\) −1.38835 −0.0462016
\(904\) −3.55496 −0.118236
\(905\) 0.405813 0.0134897
\(906\) 10.4209 0.346211
\(907\) 55.0635 1.82835 0.914177 0.405315i \(-0.132838\pi\)
0.914177 + 0.405315i \(0.132838\pi\)
\(908\) −20.4577 −0.678913
\(909\) 4.09783 0.135917
\(910\) 0 0
\(911\) 7.17151 0.237603 0.118801 0.992918i \(-0.462095\pi\)
0.118801 + 0.992918i \(0.462095\pi\)
\(912\) 5.40581 0.179004
\(913\) 3.26577 0.108081
\(914\) 18.6396 0.616545
\(915\) −2.76271 −0.0913323
\(916\) 18.7657 0.620036
\(917\) −25.9414 −0.856660
\(918\) −6.45473 −0.213038
\(919\) −32.1081 −1.05915 −0.529574 0.848263i \(-0.677648\pi\)
−0.529574 + 0.848263i \(0.677648\pi\)
\(920\) −7.96077 −0.262459
\(921\) −24.4969 −0.807202
\(922\) −8.22223 −0.270785
\(923\) 0 0
\(924\) −0.664874 −0.0218728
\(925\) −7.96615 −0.261925
\(926\) −34.1118 −1.12098
\(927\) −4.16421 −0.136771
\(928\) 6.89977 0.226496
\(929\) 3.46011 0.113522 0.0567612 0.998388i \(-0.481923\pi\)
0.0567612 + 0.998388i \(0.481923\pi\)
\(930\) 3.04892 0.0999779
\(931\) 30.0814 0.985879
\(932\) 23.6353 0.774201
\(933\) −16.4403 −0.538230
\(934\) −13.9245 −0.455624
\(935\) −3.58211 −0.117147
\(936\) 0 0
\(937\) 13.3918 0.437491 0.218746 0.975782i \(-0.429803\pi\)
0.218746 + 0.975782i \(0.429803\pi\)
\(938\) 4.00730 0.130843
\(939\) 26.0320 0.849524
\(940\) 12.2620 0.399944
\(941\) 47.0060 1.53235 0.766175 0.642632i \(-0.222157\pi\)
0.766175 + 0.642632i \(0.222157\pi\)
\(942\) −19.0151 −0.619544
\(943\) 24.5840 0.800566
\(944\) 1.45712 0.0474253
\(945\) −1.19806 −0.0389730
\(946\) −0.643104 −0.0209091
\(947\) 44.9687 1.46129 0.730643 0.682760i \(-0.239221\pi\)
0.730643 + 0.682760i \(0.239221\pi\)
\(948\) 0.990311 0.0321638
\(949\) 0 0
\(950\) 5.40581 0.175388
\(951\) −19.3545 −0.627613
\(952\) 7.73317 0.250633
\(953\) −12.1691 −0.394196 −0.197098 0.980384i \(-0.563152\pi\)
−0.197098 + 0.980384i \(0.563152\pi\)
\(954\) −8.47219 −0.274297
\(955\) −19.0097 −0.615139
\(956\) 28.7506 0.929862
\(957\) 3.82908 0.123777
\(958\) −1.79225 −0.0579049
\(959\) 8.49204 0.274222
\(960\) −1.00000 −0.0322749
\(961\) −21.7041 −0.700132
\(962\) 0 0
\(963\) −4.98254 −0.160560
\(964\) −3.89440 −0.125430
\(965\) 0.599564 0.0193006
\(966\) 9.53750 0.306864
\(967\) 21.8799 0.703611 0.351805 0.936073i \(-0.385568\pi\)
0.351805 + 0.936073i \(0.385568\pi\)
\(968\) 10.6920 0.343655
\(969\) −34.8931 −1.12093
\(970\) −7.07606 −0.227199
\(971\) −36.6450 −1.17599 −0.587997 0.808863i \(-0.700083\pi\)
−0.587997 + 0.808863i \(0.700083\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0.407943 0.0130781
\(974\) 0.126310 0.00404722
\(975\) 0 0
\(976\) 2.76271 0.0884322
\(977\) 7.36957 0.235773 0.117887 0.993027i \(-0.462388\pi\)
0.117887 + 0.993027i \(0.462388\pi\)
\(978\) 5.65279 0.180756
\(979\) 7.64981 0.244489
\(980\) −5.56465 −0.177756
\(981\) 1.31336 0.0419322
\(982\) 17.6823 0.564266
\(983\) 45.5612 1.45318 0.726588 0.687073i \(-0.241105\pi\)
0.726588 + 0.687073i \(0.241105\pi\)
\(984\) 3.08815 0.0984465
\(985\) 3.78448 0.120584
\(986\) −44.5362 −1.41832
\(987\) −14.6907 −0.467610
\(988\) 0 0
\(989\) 9.22521 0.293345
\(990\) −0.554958 −0.0176377
\(991\) 35.4693 1.12672 0.563360 0.826211i \(-0.309508\pi\)
0.563360 + 0.826211i \(0.309508\pi\)
\(992\) −3.04892 −0.0968032
\(993\) 0.173899 0.00551852
\(994\) −2.82371 −0.0895626
\(995\) −10.2349 −0.324468
\(996\) −5.88471 −0.186464
\(997\) −3.74632 −0.118647 −0.0593235 0.998239i \(-0.518894\pi\)
−0.0593235 + 0.998239i \(0.518894\pi\)
\(998\) 21.1535 0.669601
\(999\) 7.96615 0.252038
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.bm.1.1 3
13.5 odd 4 5070.2.b.u.1351.4 6
13.8 odd 4 5070.2.b.u.1351.3 6
13.12 even 2 5070.2.a.br.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bm.1.1 3 1.1 even 1 trivial
5070.2.a.br.1.3 yes 3 13.12 even 2
5070.2.b.u.1351.3 6 13.8 odd 4
5070.2.b.u.1351.4 6 13.5 odd 4