Properties

Label 5070.2.b.ba.1351.8
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.17284886784.1
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 30 x^{5} + 185 x^{4} + 36 x^{3} + 8 x^{2} + 208 x + 2704\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.8
Root \(1.33404 - 1.33404i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.ba.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +3.64466i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +3.64466i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.66808i q^{11} -1.00000 q^{12} -3.64466 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000i q^{18} -6.31274i q^{19} +1.00000i q^{20} +3.64466i q^{21} +1.66808 q^{22} -1.24453 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -3.64466i q^{28} +10.0448 q^{29} +1.00000 q^{30} +4.21957i q^{31} +1.00000i q^{32} -1.66808i q^{33} -4.00000i q^{34} +3.64466 q^{35} -1.00000 q^{36} +9.86423i q^{37} +6.31274 q^{38} -1.00000 q^{40} +9.28932i q^{41} -3.64466 q^{42} +7.57286 q^{43} +1.66808i q^{44} -1.00000i q^{45} -1.24453i q^{46} -6.82522i q^{47} +1.00000 q^{48} -6.28354 q^{49} -1.00000i q^{50} -4.00000 q^{51} -0.848634 q^{53} +1.00000i q^{54} -1.66808 q^{55} +3.64466 q^{56} -6.31274i q^{57} +10.0448i q^{58} -6.10876i q^{59} +1.00000i q^{60} +7.46410 q^{61} -4.21957 q^{62} +3.64466i q^{63} -1.00000 q^{64} +1.66808 q^{66} +14.7534i q^{67} +4.00000 q^{68} -1.24453 q^{69} +3.64466i q^{70} +3.51093i q^{71} -1.00000i q^{72} +12.2175i q^{73} -9.86423 q^{74} -1.00000 q^{75} +6.31274i q^{76} +6.07957 q^{77} +9.93398 q^{79} -1.00000i q^{80} +1.00000 q^{81} -9.28932 q^{82} +7.95317i q^{83} -3.64466i q^{84} +4.00000i q^{85} +7.57286i q^{86} +10.0448 q^{87} -1.66808 q^{88} +5.95162i q^{89} +1.00000 q^{90} +1.24453 q^{92} +4.21957i q^{93} +6.82522 q^{94} -6.31274 q^{95} +1.00000i q^{96} +2.75342i q^{97} -6.28354i q^{98} -1.66808i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} - 8q^{4} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{3} - 8q^{4} + 8q^{9} + 8q^{10} - 8q^{12} + 4q^{14} + 8q^{16} - 32q^{17} - 4q^{22} - 8q^{23} - 8q^{25} + 8q^{27} + 16q^{29} + 8q^{30} - 4q^{35} - 8q^{36} - 8q^{40} + 4q^{42} - 28q^{43} + 8q^{48} - 28q^{49} - 32q^{51} + 16q^{53} + 4q^{55} - 4q^{56} + 32q^{61} - 8q^{62} - 8q^{64} - 4q^{66} + 32q^{68} - 8q^{69} - 20q^{74} - 8q^{75} + 16q^{77} - 20q^{79} + 8q^{81} - 8q^{82} + 16q^{87} + 4q^{88} + 8q^{90} + 8q^{92} + 16q^{94} + O(q^{100}) \)

Character values

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

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) 3.64466i 1.37755i 0.724974 + 0.688776i \(0.241852\pi\)
−0.724974 + 0.688776i \(0.758148\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) − 1.66808i − 0.502944i −0.967865 0.251472i \(-0.919085\pi\)
0.967865 0.251472i \(-0.0809146\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −3.64466 −0.974076
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 1.00000i 0.235702i
\(19\) − 6.31274i − 1.44824i −0.689674 0.724120i \(-0.742246\pi\)
0.689674 0.724120i \(-0.257754\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 3.64466i 0.795330i
\(22\) 1.66808 0.355635
\(23\) −1.24453 −0.259503 −0.129752 0.991547i \(-0.541418\pi\)
−0.129752 + 0.991547i \(0.541418\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 3.64466i − 0.688776i
\(29\) 10.0448 1.86527 0.932635 0.360821i \(-0.117504\pi\)
0.932635 + 0.360821i \(0.117504\pi\)
\(30\) 1.00000 0.182574
\(31\) 4.21957i 0.757857i 0.925426 + 0.378928i \(0.123707\pi\)
−0.925426 + 0.378928i \(0.876293\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 1.66808i − 0.290375i
\(34\) − 4.00000i − 0.685994i
\(35\) 3.64466 0.616060
\(36\) −1.00000 −0.166667
\(37\) 9.86423i 1.62167i 0.585275 + 0.810835i \(0.300986\pi\)
−0.585275 + 0.810835i \(0.699014\pi\)
\(38\) 6.31274 1.02406
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 9.28932i 1.45075i 0.688355 + 0.725374i \(0.258333\pi\)
−0.688355 + 0.725374i \(0.741667\pi\)
\(42\) −3.64466 −0.562383
\(43\) 7.57286 1.15485 0.577425 0.816443i \(-0.304057\pi\)
0.577425 + 0.816443i \(0.304057\pi\)
\(44\) 1.66808i 0.251472i
\(45\) − 1.00000i − 0.149071i
\(46\) − 1.24453i − 0.183496i
\(47\) − 6.82522i − 0.995560i −0.867303 0.497780i \(-0.834149\pi\)
0.867303 0.497780i \(-0.165851\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.28354 −0.897649
\(50\) − 1.00000i − 0.141421i
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −0.848634 −0.116569 −0.0582844 0.998300i \(-0.518563\pi\)
−0.0582844 + 0.998300i \(0.518563\pi\)
\(54\) 1.00000i 0.136083i
\(55\) −1.66808 −0.224923
\(56\) 3.64466 0.487038
\(57\) − 6.31274i − 0.836142i
\(58\) 10.0448i 1.31895i
\(59\) − 6.10876i − 0.795293i −0.917539 0.397646i \(-0.869827\pi\)
0.917539 0.397646i \(-0.130173\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 7.46410 0.955680 0.477840 0.878447i \(-0.341420\pi\)
0.477840 + 0.878447i \(0.341420\pi\)
\(62\) −4.21957 −0.535886
\(63\) 3.64466i 0.459184i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.66808 0.205326
\(67\) 14.7534i 1.80242i 0.433386 + 0.901209i \(0.357319\pi\)
−0.433386 + 0.901209i \(0.642681\pi\)
\(68\) 4.00000 0.485071
\(69\) −1.24453 −0.149824
\(70\) 3.64466i 0.435620i
\(71\) 3.51093i 0.416671i 0.978057 + 0.208336i \(0.0668046\pi\)
−0.978057 + 0.208336i \(0.933195\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 12.2175i 1.42995i 0.699149 + 0.714976i \(0.253563\pi\)
−0.699149 + 0.714976i \(0.746437\pi\)
\(74\) −9.86423 −1.14669
\(75\) −1.00000 −0.115470
\(76\) 6.31274i 0.724120i
\(77\) 6.07957 0.692831
\(78\) 0 0
\(79\) 9.93398 1.11766 0.558830 0.829282i \(-0.311250\pi\)
0.558830 + 0.829282i \(0.311250\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) −9.28932 −1.02583
\(83\) 7.95317i 0.872974i 0.899711 + 0.436487i \(0.143777\pi\)
−0.899711 + 0.436487i \(0.856223\pi\)
\(84\) − 3.64466i − 0.397665i
\(85\) 4.00000i 0.433861i
\(86\) 7.57286i 0.816603i
\(87\) 10.0448 1.07691
\(88\) −1.66808 −0.177817
\(89\) 5.95162i 0.630870i 0.948947 + 0.315435i \(0.102150\pi\)
−0.948947 + 0.315435i \(0.897850\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 1.24453 0.129752
\(93\) 4.21957i 0.437549i
\(94\) 6.82522 0.703967
\(95\) −6.31274 −0.647673
\(96\) 1.00000i 0.102062i
\(97\) 2.75342i 0.279568i 0.990182 + 0.139784i \(0.0446407\pi\)
−0.990182 + 0.139784i \(0.955359\pi\)
\(98\) − 6.28354i − 0.634734i
\(99\) − 1.66808i − 0.167648i
\(100\) 1.00000 0.100000
\(101\) −5.33615 −0.530967 −0.265483 0.964115i \(-0.585532\pi\)
−0.265483 + 0.964115i \(0.585532\pi\)
\(102\) − 4.00000i − 0.396059i
\(103\) −7.51248 −0.740227 −0.370113 0.928987i \(-0.620681\pi\)
−0.370113 + 0.928987i \(0.620681\pi\)
\(104\) 0 0
\(105\) 3.64466 0.355682
\(106\) − 0.848634i − 0.0824266i
\(107\) 16.9282 1.63651 0.818256 0.574855i \(-0.194941\pi\)
0.818256 + 0.574855i \(0.194941\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 0.663848i 0.0635851i 0.999494 + 0.0317926i \(0.0101216\pi\)
−0.999494 + 0.0317926i \(0.989878\pi\)
\(110\) − 1.66808i − 0.159045i
\(111\) 9.86423i 0.936271i
\(112\) 3.64466i 0.344388i
\(113\) −17.8700 −1.68107 −0.840534 0.541758i \(-0.817759\pi\)
−0.840534 + 0.541758i \(0.817759\pi\)
\(114\) 6.31274 0.591242
\(115\) 1.24453i 0.116053i
\(116\) −10.0448 −0.932635
\(117\) 0 0
\(118\) 6.10876 0.562357
\(119\) − 14.5786i − 1.33642i
\(120\) −1.00000 −0.0912871
\(121\) 8.21752 0.747047
\(122\) 7.46410i 0.675768i
\(123\) 9.28932i 0.837590i
\(124\) − 4.21957i − 0.378928i
\(125\) 1.00000i 0.0894427i
\(126\) −3.64466 −0.324692
\(127\) −14.4407 −1.28140 −0.640702 0.767790i \(-0.721357\pi\)
−0.640702 + 0.767790i \(0.721357\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 7.57286 0.666753
\(130\) 0 0
\(131\) −10.8892 −0.951393 −0.475697 0.879609i \(-0.657804\pi\)
−0.475697 + 0.879609i \(0.657804\pi\)
\(132\) 1.66808i 0.145187i
\(133\) 23.0078 1.99503
\(134\) −14.7534 −1.27450
\(135\) − 1.00000i − 0.0860663i
\(136\) 4.00000i 0.342997i
\(137\) − 7.03696i − 0.601208i −0.953749 0.300604i \(-0.902812\pi\)
0.953749 0.300604i \(-0.0971883\pi\)
\(138\) − 1.24453i − 0.105942i
\(139\) 11.6447 0.987687 0.493844 0.869551i \(-0.335592\pi\)
0.493844 + 0.869551i \(0.335592\pi\)
\(140\) −3.64466 −0.308030
\(141\) − 6.82522i − 0.574787i
\(142\) −3.51093 −0.294631
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 10.0448i − 0.834174i
\(146\) −12.2175 −1.01113
\(147\) −6.28354 −0.518258
\(148\) − 9.86423i − 0.810835i
\(149\) 0.772609i 0.0632946i 0.999499 + 0.0316473i \(0.0100753\pi\)
−0.999499 + 0.0316473i \(0.989925\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) − 9.77838i − 0.795754i −0.917439 0.397877i \(-0.869747\pi\)
0.917439 0.397877i \(-0.130253\pi\)
\(152\) −6.31274 −0.512030
\(153\) −4.00000 −0.323381
\(154\) 6.07957i 0.489906i
\(155\) 4.21957 0.338924
\(156\) 0 0
\(157\) −12.0135 −0.958786 −0.479393 0.877600i \(-0.659143\pi\)
−0.479393 + 0.877600i \(0.659143\pi\)
\(158\) 9.93398i 0.790305i
\(159\) −0.848634 −0.0673011
\(160\) 1.00000 0.0790569
\(161\) − 4.53590i − 0.357479i
\(162\) 1.00000i 0.0785674i
\(163\) 10.0916i 0.790437i 0.918587 + 0.395218i \(0.129331\pi\)
−0.918587 + 0.395218i \(0.870669\pi\)
\(164\) − 9.28932i − 0.725374i
\(165\) −1.66808 −0.129860
\(166\) −7.95317 −0.617285
\(167\) − 7.89701i − 0.611089i −0.952178 0.305545i \(-0.901161\pi\)
0.952178 0.305545i \(-0.0988385\pi\)
\(168\) 3.64466 0.281192
\(169\) 0 0
\(170\) −4.00000 −0.306786
\(171\) − 6.31274i − 0.482747i
\(172\) −7.57286 −0.577425
\(173\) 0.440685 0.0335047 0.0167523 0.999860i \(-0.494667\pi\)
0.0167523 + 0.999860i \(0.494667\pi\)
\(174\) 10.0448i 0.761493i
\(175\) − 3.64466i − 0.275510i
\(176\) − 1.66808i − 0.125736i
\(177\) − 6.10876i − 0.459163i
\(178\) −5.95162 −0.446093
\(179\) 19.6368 1.46773 0.733863 0.679297i \(-0.237715\pi\)
0.733863 + 0.679297i \(0.237715\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 6.21752 0.462145 0.231072 0.972937i \(-0.425777\pi\)
0.231072 + 0.972937i \(0.425777\pi\)
\(182\) 0 0
\(183\) 7.46410 0.551762
\(184\) 1.24453i 0.0917482i
\(185\) 9.86423 0.725232
\(186\) −4.21957 −0.309394
\(187\) 6.67230i 0.487927i
\(188\) 6.82522i 0.497780i
\(189\) 3.64466i 0.265110i
\(190\) − 6.31274i − 0.457974i
\(191\) 15.6816 1.13468 0.567341 0.823483i \(-0.307972\pi\)
0.567341 + 0.823483i \(0.307972\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 8.12795i 0.585063i 0.956256 + 0.292531i \(0.0944976\pi\)
−0.956256 + 0.292531i \(0.905502\pi\)
\(194\) −2.75342 −0.197684
\(195\) 0 0
\(196\) 6.28354 0.448825
\(197\) 0.891239i 0.0634981i 0.999496 + 0.0317491i \(0.0101077\pi\)
−0.999496 + 0.0317491i \(0.989892\pi\)
\(198\) 1.66808 0.118545
\(199\) −0.361116 −0.0255988 −0.0127994 0.999918i \(-0.504074\pi\)
−0.0127994 + 0.999918i \(0.504074\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 14.7534i 1.04063i
\(202\) − 5.33615i − 0.375450i
\(203\) 36.6098i 2.56951i
\(204\) 4.00000 0.280056
\(205\) 9.28932 0.648794
\(206\) − 7.51248i − 0.523419i
\(207\) −1.24453 −0.0865010
\(208\) 0 0
\(209\) −10.5301 −0.728384
\(210\) 3.64466i 0.251505i
\(211\) 2.22739 0.153340 0.0766700 0.997057i \(-0.475571\pi\)
0.0766700 + 0.997057i \(0.475571\pi\)
\(212\) 0.848634 0.0582844
\(213\) 3.51093i 0.240565i
\(214\) 16.9282i 1.15719i
\(215\) − 7.57286i − 0.516465i
\(216\) − 1.00000i − 0.0680414i
\(217\) −15.3789 −1.04399
\(218\) −0.663848 −0.0449615
\(219\) 12.2175i 0.825584i
\(220\) 1.66808 0.112462
\(221\) 0 0
\(222\) −9.86423 −0.662044
\(223\) 6.08380i 0.407401i 0.979033 + 0.203701i \(0.0652969\pi\)
−0.979033 + 0.203701i \(0.934703\pi\)
\(224\) −3.64466 −0.243519
\(225\) −1.00000 −0.0666667
\(226\) − 17.8700i − 1.18869i
\(227\) 15.3205i 1.01686i 0.861104 + 0.508429i \(0.169774\pi\)
−0.861104 + 0.508429i \(0.830226\pi\)
\(228\) 6.31274i 0.418071i
\(229\) − 22.2644i − 1.47127i −0.677378 0.735635i \(-0.736884\pi\)
0.677378 0.735635i \(-0.263116\pi\)
\(230\) −1.24453 −0.0820621
\(231\) 6.07957 0.400006
\(232\) − 10.0448i − 0.659473i
\(233\) 10.8366 0.709928 0.354964 0.934880i \(-0.384493\pi\)
0.354964 + 0.934880i \(0.384493\pi\)
\(234\) 0 0
\(235\) −6.82522 −0.445228
\(236\) 6.10876i 0.397646i
\(237\) 9.93398 0.645281
\(238\) 14.5786 0.944993
\(239\) 16.4975i 1.06714i 0.845757 + 0.533568i \(0.179149\pi\)
−0.845757 + 0.533568i \(0.820851\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) − 4.40435i − 0.283709i −0.989887 0.141855i \(-0.954693\pi\)
0.989887 0.141855i \(-0.0453066\pi\)
\(242\) 8.21752i 0.528242i
\(243\) 1.00000 0.0641500
\(244\) −7.46410 −0.477840
\(245\) 6.28354i 0.401441i
\(246\) −9.28932 −0.592265
\(247\) 0 0
\(248\) 4.21957 0.267943
\(249\) 7.95317i 0.504011i
\(250\) −1.00000 −0.0632456
\(251\) −11.9453 −0.753984 −0.376992 0.926217i \(-0.623041\pi\)
−0.376992 + 0.926217i \(0.623041\pi\)
\(252\) − 3.64466i − 0.229592i
\(253\) 2.07598i 0.130515i
\(254\) − 14.4407i − 0.906089i
\(255\) 4.00000i 0.250490i
\(256\) 1.00000 0.0625000
\(257\) −19.4621 −1.21401 −0.607005 0.794698i \(-0.707629\pi\)
−0.607005 + 0.794698i \(0.707629\pi\)
\(258\) 7.57286i 0.471466i
\(259\) −35.9518 −2.23393
\(260\) 0 0
\(261\) 10.0448 0.621757
\(262\) − 10.8892i − 0.672737i
\(263\) 2.03478 0.125470 0.0627350 0.998030i \(-0.480018\pi\)
0.0627350 + 0.998030i \(0.480018\pi\)
\(264\) −1.66808 −0.102663
\(265\) 0.848634i 0.0521312i
\(266\) 23.0078i 1.41070i
\(267\) 5.95162i 0.364233i
\(268\) − 14.7534i − 0.901209i
\(269\) 20.5287 1.25166 0.625829 0.779960i \(-0.284761\pi\)
0.625829 + 0.779960i \(0.284761\pi\)
\(270\) 1.00000 0.0608581
\(271\) 25.5401i 1.55145i 0.631072 + 0.775725i \(0.282615\pi\)
−0.631072 + 0.775725i \(0.717385\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 7.03696 0.425119
\(275\) 1.66808i 0.100589i
\(276\) 1.24453 0.0749121
\(277\) −18.0604 −1.08514 −0.542572 0.840010i \(-0.682549\pi\)
−0.542572 + 0.840010i \(0.682549\pi\)
\(278\) 11.6447i 0.698400i
\(279\) 4.21957i 0.252619i
\(280\) − 3.64466i − 0.217810i
\(281\) 20.2175i 1.20608i 0.797712 + 0.603038i \(0.206043\pi\)
−0.797712 + 0.603038i \(0.793957\pi\)
\(282\) 6.82522 0.406436
\(283\) −8.69149 −0.516656 −0.258328 0.966057i \(-0.583171\pi\)
−0.258328 + 0.966057i \(0.583171\pi\)
\(284\) − 3.51093i − 0.208336i
\(285\) −6.31274 −0.373934
\(286\) 0 0
\(287\) −33.8564 −1.99848
\(288\) 1.00000i 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) 10.0448 0.589850
\(291\) 2.75342i 0.161408i
\(292\) − 12.2175i − 0.714976i
\(293\) − 7.54790i − 0.440953i −0.975392 0.220476i \(-0.929239\pi\)
0.975392 0.220476i \(-0.0707612\pi\)
\(294\) − 6.28354i − 0.366464i
\(295\) −6.10876 −0.355666
\(296\) 9.86423 0.573347
\(297\) − 1.66808i − 0.0967916i
\(298\) −0.772609 −0.0447560
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 27.6005i 1.59087i
\(302\) 9.77838 0.562683
\(303\) −5.33615 −0.306554
\(304\) − 6.31274i − 0.362060i
\(305\) − 7.46410i − 0.427393i
\(306\) − 4.00000i − 0.228665i
\(307\) 26.0427i 1.48634i 0.669104 + 0.743169i \(0.266678\pi\)
−0.669104 + 0.743169i \(0.733322\pi\)
\(308\) −6.07957 −0.346416
\(309\) −7.51248 −0.427370
\(310\) 4.21957i 0.239655i
\(311\) 25.3789 1.43910 0.719552 0.694438i \(-0.244347\pi\)
0.719552 + 0.694438i \(0.244347\pi\)
\(312\) 0 0
\(313\) −31.4600 −1.77822 −0.889112 0.457689i \(-0.848677\pi\)
−0.889112 + 0.457689i \(0.848677\pi\)
\(314\) − 12.0135i − 0.677964i
\(315\) 3.64466 0.205353
\(316\) −9.93398 −0.558830
\(317\) − 24.7093i − 1.38781i −0.720066 0.693905i \(-0.755889\pi\)
0.720066 0.693905i \(-0.244111\pi\)
\(318\) − 0.848634i − 0.0475890i
\(319\) − 16.7555i − 0.938126i
\(320\) 1.00000i 0.0559017i
\(321\) 16.9282 0.944840
\(322\) 4.53590 0.252776
\(323\) 25.2509i 1.40500i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −10.0916 −0.558923
\(327\) 0.663848i 0.0367109i
\(328\) 9.28932 0.512917
\(329\) 24.8756 1.37144
\(330\) − 1.66808i − 0.0918246i
\(331\) 5.60360i 0.308002i 0.988071 + 0.154001i \(0.0492159\pi\)
−0.988071 + 0.154001i \(0.950784\pi\)
\(332\) − 7.95317i − 0.436487i
\(333\) 9.86423i 0.540556i
\(334\) 7.89701 0.432105
\(335\) 14.7534 0.806065
\(336\) 3.64466i 0.198832i
\(337\) 21.7868 1.18680 0.593402 0.804906i \(-0.297784\pi\)
0.593402 + 0.804906i \(0.297784\pi\)
\(338\) 0 0
\(339\) −17.8700 −0.970565
\(340\) − 4.00000i − 0.216930i
\(341\) 7.03856 0.381159
\(342\) 6.31274 0.341354
\(343\) 2.61124i 0.140994i
\(344\) − 7.57286i − 0.408301i
\(345\) 1.24453i 0.0670034i
\(346\) 0.440685i 0.0236914i
\(347\) 9.68162 0.519737 0.259868 0.965644i \(-0.416321\pi\)
0.259868 + 0.965644i \(0.416321\pi\)
\(348\) −10.0448 −0.538457
\(349\) 19.3205i 1.03420i 0.855924 + 0.517102i \(0.172989\pi\)
−0.855924 + 0.517102i \(0.827011\pi\)
\(350\) 3.64466 0.194815
\(351\) 0 0
\(352\) 1.66808 0.0889087
\(353\) − 22.9750i − 1.22284i −0.791307 0.611419i \(-0.790599\pi\)
0.791307 0.611419i \(-0.209401\pi\)
\(354\) 6.10876 0.324677
\(355\) 3.51093 0.186341
\(356\) − 5.95162i − 0.315435i
\(357\) − 14.5786i − 0.771583i
\(358\) 19.6368i 1.03784i
\(359\) − 2.21752i − 0.117036i −0.998286 0.0585182i \(-0.981362\pi\)
0.998286 0.0585182i \(-0.0186376\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −20.8506 −1.09740
\(362\) 6.21752i 0.326786i
\(363\) 8.21752 0.431308
\(364\) 0 0
\(365\) 12.2175 0.639494
\(366\) 7.46410i 0.390155i
\(367\) −6.08112 −0.317432 −0.158716 0.987324i \(-0.550735\pi\)
−0.158716 + 0.987324i \(0.550735\pi\)
\(368\) −1.24453 −0.0648758
\(369\) 9.28932i 0.483583i
\(370\) 9.86423i 0.512817i
\(371\) − 3.09298i − 0.160580i
\(372\) − 4.21957i − 0.218774i
\(373\) 15.6781 0.811780 0.405890 0.913922i \(-0.366962\pi\)
0.405890 + 0.913922i \(0.366962\pi\)
\(374\) −6.67230 −0.345017
\(375\) 1.00000i 0.0516398i
\(376\) −6.82522 −0.351984
\(377\) 0 0
\(378\) −3.64466 −0.187461
\(379\) 30.7166i 1.57781i 0.614518 + 0.788903i \(0.289350\pi\)
−0.614518 + 0.788903i \(0.710650\pi\)
\(380\) 6.31274 0.323837
\(381\) −14.4407 −0.739819
\(382\) 15.6816i 0.802342i
\(383\) − 20.0619i − 1.02512i −0.858652 0.512558i \(-0.828698\pi\)
0.858652 0.512558i \(-0.171302\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) − 6.07957i − 0.309844i
\(386\) −8.12795 −0.413702
\(387\) 7.57286 0.384950
\(388\) − 2.75342i − 0.139784i
\(389\) 27.0314 1.37055 0.685273 0.728287i \(-0.259683\pi\)
0.685273 + 0.728287i \(0.259683\pi\)
\(390\) 0 0
\(391\) 4.97813 0.251755
\(392\) 6.28354i 0.317367i
\(393\) −10.8892 −0.549287
\(394\) −0.891239 −0.0449000
\(395\) − 9.93398i − 0.499833i
\(396\) 1.66808i 0.0838240i
\(397\) 3.73628i 0.187518i 0.995595 + 0.0937592i \(0.0298884\pi\)
−0.995595 + 0.0937592i \(0.970112\pi\)
\(398\) − 0.361116i − 0.0181011i
\(399\) 23.0078 1.15183
\(400\) −1.00000 −0.0500000
\(401\) 28.0911i 1.40280i 0.712766 + 0.701402i \(0.247442\pi\)
−0.712766 + 0.701402i \(0.752558\pi\)
\(402\) −14.7534 −0.735834
\(403\) 0 0
\(404\) 5.33615 0.265483
\(405\) − 1.00000i − 0.0496904i
\(406\) −36.6098 −1.81692
\(407\) 16.4543 0.815609
\(408\) 4.00000i 0.198030i
\(409\) − 27.3804i − 1.35388i −0.736040 0.676938i \(-0.763307\pi\)
0.736040 0.676938i \(-0.236693\pi\)
\(410\) 9.28932i 0.458767i
\(411\) − 7.03696i − 0.347108i
\(412\) 7.51248 0.370113
\(413\) 22.2644 1.09556
\(414\) − 1.24453i − 0.0611654i
\(415\) 7.95317 0.390406
\(416\) 0 0
\(417\) 11.6447 0.570241
\(418\) − 10.5301i − 0.515045i
\(419\) −13.1614 −0.642975 −0.321487 0.946914i \(-0.604183\pi\)
−0.321487 + 0.946914i \(0.604183\pi\)
\(420\) −3.64466 −0.177841
\(421\) − 1.29341i − 0.0630370i −0.999503 0.0315185i \(-0.989966\pi\)
0.999503 0.0315185i \(-0.0100343\pi\)
\(422\) 2.22739i 0.108428i
\(423\) − 6.82522i − 0.331853i
\(424\) 0.848634i 0.0412133i
\(425\) 4.00000 0.194029
\(426\) −3.51093 −0.170105
\(427\) 27.2041i 1.31650i
\(428\) −16.9282 −0.818256
\(429\) 0 0
\(430\) 7.57286 0.365196
\(431\) 12.1279i 0.584183i 0.956390 + 0.292091i \(0.0943511\pi\)
−0.956390 + 0.292091i \(0.905649\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.07802 0.0998633 0.0499317 0.998753i \(-0.484100\pi\)
0.0499317 + 0.998753i \(0.484100\pi\)
\(434\) − 15.3789i − 0.738210i
\(435\) − 10.0448i − 0.481611i
\(436\) − 0.663848i − 0.0317926i
\(437\) 7.85641i 0.375823i
\(438\) −12.2175 −0.583776
\(439\) 2.39640 0.114374 0.0571869 0.998363i \(-0.481787\pi\)
0.0571869 + 0.998363i \(0.481787\pi\)
\(440\) 1.66808i 0.0795224i
\(441\) −6.28354 −0.299216
\(442\) 0 0
\(443\) 21.9959 1.04506 0.522529 0.852622i \(-0.324989\pi\)
0.522529 + 0.852622i \(0.324989\pi\)
\(444\) − 9.86423i − 0.468136i
\(445\) 5.95162 0.282134
\(446\) −6.08380 −0.288076
\(447\) 0.772609i 0.0365432i
\(448\) − 3.64466i − 0.172194i
\(449\) − 29.2253i − 1.37923i −0.724178 0.689613i \(-0.757780\pi\)
0.724178 0.689613i \(-0.242220\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) 15.4953 0.729645
\(452\) 17.8700 0.840534
\(453\) − 9.77838i − 0.459429i
\(454\) −15.3205 −0.719027
\(455\) 0 0
\(456\) −6.31274 −0.295621
\(457\) − 39.6432i − 1.85443i −0.374526 0.927216i \(-0.622195\pi\)
0.374526 0.927216i \(-0.377805\pi\)
\(458\) 22.2644 1.04034
\(459\) −4.00000 −0.186704
\(460\) − 1.24453i − 0.0580266i
\(461\) − 16.6758i − 0.776672i −0.921518 0.388336i \(-0.873050\pi\)
0.921518 0.388336i \(-0.126950\pi\)
\(462\) 6.07957i 0.282847i
\(463\) 32.2175i 1.49728i 0.662979 + 0.748638i \(0.269292\pi\)
−0.662979 + 0.748638i \(0.730708\pi\)
\(464\) 10.0448 0.466318
\(465\) 4.21957 0.195678
\(466\) 10.8366i 0.501995i
\(467\) −6.88137 −0.318432 −0.159216 0.987244i \(-0.550897\pi\)
−0.159216 + 0.987244i \(0.550897\pi\)
\(468\) 0 0
\(469\) −53.7712 −2.48292
\(470\) − 6.82522i − 0.314824i
\(471\) −12.0135 −0.553555
\(472\) −6.10876 −0.281179
\(473\) − 12.6321i − 0.580825i
\(474\) 9.93398i 0.456283i
\(475\) 6.31274i 0.289648i
\(476\) 14.5786i 0.668211i
\(477\) −0.848634 −0.0388563
\(478\) −16.4975 −0.754579
\(479\) − 18.9709i − 0.866805i −0.901201 0.433402i \(-0.857313\pi\)
0.901201 0.433402i \(-0.142687\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 4.40435 0.200613
\(483\) − 4.53590i − 0.206391i
\(484\) −8.21752 −0.373524
\(485\) 2.75342 0.125026
\(486\) 1.00000i 0.0453609i
\(487\) − 1.91620i − 0.0868314i −0.999057 0.0434157i \(-0.986176\pi\)
0.999057 0.0434157i \(-0.0138240\pi\)
\(488\) − 7.46410i − 0.337884i
\(489\) 10.0916i 0.456359i
\(490\) −6.28354 −0.283862
\(491\) 33.6375 1.51804 0.759019 0.651069i \(-0.225679\pi\)
0.759019 + 0.651069i \(0.225679\pi\)
\(492\) − 9.28932i − 0.418795i
\(493\) −40.1791 −1.80958
\(494\) 0 0
\(495\) −1.66808 −0.0749744
\(496\) 4.21957i 0.189464i
\(497\) −12.7962 −0.573986
\(498\) −7.95317 −0.356390
\(499\) − 1.82522i − 0.0817080i −0.999165 0.0408540i \(-0.986992\pi\)
0.999165 0.0408540i \(-0.0130078\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 7.89701i − 0.352813i
\(502\) − 11.9453i − 0.533147i
\(503\) −19.9985 −0.891687 −0.445843 0.895111i \(-0.647096\pi\)
−0.445843 + 0.895111i \(0.647096\pi\)
\(504\) 3.64466 0.162346
\(505\) 5.33615i 0.237456i
\(506\) −2.07598 −0.0922884
\(507\) 0 0
\(508\) 14.4407 0.640702
\(509\) 5.89124i 0.261125i 0.991440 + 0.130562i \(0.0416783\pi\)
−0.991440 + 0.130562i \(0.958322\pi\)
\(510\) −4.00000 −0.177123
\(511\) −44.5287 −1.96983
\(512\) 1.00000i 0.0441942i
\(513\) − 6.31274i − 0.278714i
\(514\) − 19.4621i − 0.858434i
\(515\) 7.51248i 0.331040i
\(516\) −7.57286 −0.333377
\(517\) −11.3850 −0.500711
\(518\) − 35.9518i − 1.57963i
\(519\) 0.440685 0.0193439
\(520\) 0 0
\(521\) 32.0370 1.40356 0.701782 0.712391i \(-0.252388\pi\)
0.701782 + 0.712391i \(0.252388\pi\)
\(522\) 10.0448i 0.439648i
\(523\) 38.7186 1.69305 0.846523 0.532352i \(-0.178692\pi\)
0.846523 + 0.532352i \(0.178692\pi\)
\(524\) 10.8892 0.475697
\(525\) − 3.64466i − 0.159066i
\(526\) 2.03478i 0.0887207i
\(527\) − 16.8783i − 0.735229i
\(528\) − 1.66808i − 0.0725937i
\(529\) −21.4511 −0.932658
\(530\) −0.848634 −0.0368623
\(531\) − 6.10876i − 0.265098i
\(532\) −23.0078 −0.997513
\(533\) 0 0
\(534\) −5.95162 −0.257552
\(535\) − 16.9282i − 0.731870i
\(536\) 14.7534 0.637251
\(537\) 19.6368 0.847392
\(538\) 20.5287i 0.885056i
\(539\) 10.4814i 0.451467i
\(540\) 1.00000i 0.0430331i
\(541\) − 25.9616i − 1.11618i −0.829781 0.558089i \(-0.811535\pi\)
0.829781 0.558089i \(-0.188465\pi\)
\(542\) −25.5401 −1.09704
\(543\) 6.21752 0.266819
\(544\) − 4.00000i − 0.171499i
\(545\) 0.663848 0.0284361
\(546\) 0 0
\(547\) −17.7596 −0.759348 −0.379674 0.925120i \(-0.623964\pi\)
−0.379674 + 0.925120i \(0.623964\pi\)
\(548\) 7.03696i 0.300604i
\(549\) 7.46410 0.318560
\(550\) −1.66808 −0.0711270
\(551\) − 63.4101i − 2.70136i
\(552\) 1.24453i 0.0529708i
\(553\) 36.2060i 1.53963i
\(554\) − 18.0604i − 0.767312i
\(555\) 9.86423 0.418713
\(556\) −11.6447 −0.493844
\(557\) − 26.2202i − 1.11099i −0.831521 0.555493i \(-0.812530\pi\)
0.831521 0.555493i \(-0.187470\pi\)
\(558\) −4.21957 −0.178629
\(559\) 0 0
\(560\) 3.64466 0.154015
\(561\) 6.67230i 0.281705i
\(562\) −20.2175 −0.852825
\(563\) 25.9928 1.09547 0.547733 0.836653i \(-0.315491\pi\)
0.547733 + 0.836653i \(0.315491\pi\)
\(564\) 6.82522i 0.287394i
\(565\) 17.8700i 0.751797i
\(566\) − 8.69149i − 0.365331i
\(567\) 3.64466i 0.153061i
\(568\) 3.51093 0.147316
\(569\) 25.4699 1.06775 0.533876 0.845563i \(-0.320735\pi\)
0.533876 + 0.845563i \(0.320735\pi\)
\(570\) − 6.31274i − 0.264411i
\(571\) −8.44491 −0.353409 −0.176704 0.984264i \(-0.556544\pi\)
−0.176704 + 0.984264i \(0.556544\pi\)
\(572\) 0 0
\(573\) 15.6816 0.655109
\(574\) − 33.8564i − 1.41314i
\(575\) 1.24453 0.0519006
\(576\) −1.00000 −0.0416667
\(577\) 35.4216i 1.47462i 0.675554 + 0.737311i \(0.263905\pi\)
−0.675554 + 0.737311i \(0.736095\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) 8.12795i 0.337786i
\(580\) 10.0448i 0.417087i
\(581\) −28.9866 −1.20257
\(582\) −2.75342 −0.114133
\(583\) 1.41559i 0.0586276i
\(584\) 12.2175 0.505565
\(585\) 0 0
\(586\) 7.54790 0.311801
\(587\) 27.3789i 1.13005i 0.825075 + 0.565024i \(0.191133\pi\)
−0.825075 + 0.565024i \(0.808867\pi\)
\(588\) 6.28354 0.259129
\(589\) 26.6370 1.09756
\(590\) − 6.10876i − 0.251494i
\(591\) 0.891239i 0.0366607i
\(592\) 9.86423i 0.405417i
\(593\) − 12.0619i − 0.495324i −0.968846 0.247662i \(-0.920338\pi\)
0.968846 0.247662i \(-0.0796623\pi\)
\(594\) 1.66808 0.0684420
\(595\) −14.5786 −0.597666
\(596\) − 0.772609i − 0.0316473i
\(597\) −0.361116 −0.0147795
\(598\) 0 0
\(599\) −28.6129 −1.16909 −0.584546 0.811360i \(-0.698727\pi\)
−0.584546 + 0.811360i \(0.698727\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −19.1676 −0.781862 −0.390931 0.920420i \(-0.627847\pi\)
−0.390931 + 0.920420i \(0.627847\pi\)
\(602\) −27.6005 −1.12491
\(603\) 14.7534i 0.600806i
\(604\) 9.77838i 0.397877i
\(605\) − 8.21752i − 0.334090i
\(606\) − 5.33615i − 0.216766i
\(607\) 18.9382 0.768678 0.384339 0.923192i \(-0.374429\pi\)
0.384339 + 0.923192i \(0.374429\pi\)
\(608\) 6.31274 0.256015
\(609\) 36.6098i 1.48351i
\(610\) 7.46410 0.302213
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) − 17.9694i − 0.725779i −0.931832 0.362890i \(-0.881790\pi\)
0.931832 0.362890i \(-0.118210\pi\)
\(614\) −26.0427 −1.05100
\(615\) 9.28932 0.374582
\(616\) − 6.07957i − 0.244953i
\(617\) 18.0151i 0.725260i 0.931933 + 0.362630i \(0.118121\pi\)
−0.931933 + 0.362630i \(0.881879\pi\)
\(618\) − 7.51248i − 0.302196i
\(619\) − 25.0505i − 1.00687i −0.864035 0.503433i \(-0.832070\pi\)
0.864035 0.503433i \(-0.167930\pi\)
\(620\) −4.21957 −0.169462
\(621\) −1.24453 −0.0499414
\(622\) 25.3789i 1.01760i
\(623\) −21.6916 −0.869057
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 31.4600i − 1.25739i
\(627\) −10.5301 −0.420533
\(628\) 12.0135 0.479393
\(629\) − 39.4569i − 1.57325i
\(630\) 3.64466i 0.145207i
\(631\) 1.41727i 0.0564206i 0.999602 + 0.0282103i \(0.00898081\pi\)
−0.999602 + 0.0282103i \(0.991019\pi\)
\(632\) − 9.93398i − 0.395152i
\(633\) 2.22739 0.0885308
\(634\) 24.7093 0.981330
\(635\) 14.4407i 0.573061i
\(636\) 0.848634 0.0336505
\(637\) 0 0
\(638\) 16.7555 0.663355
\(639\) 3.51093i 0.138890i
\(640\) −1.00000 −0.0395285
\(641\) −4.61970 −0.182467 −0.0912335 0.995830i \(-0.529081\pi\)
−0.0912335 + 0.995830i \(0.529081\pi\)
\(642\) 16.9282i 0.668103i
\(643\) − 48.0896i − 1.89647i −0.317573 0.948234i \(-0.602868\pi\)
0.317573 0.948234i \(-0.397132\pi\)
\(644\) 4.53590i 0.178739i
\(645\) − 7.57286i − 0.298181i
\(646\) −25.2509 −0.993485
\(647\) −3.74565 −0.147257 −0.0736283 0.997286i \(-0.523458\pi\)
−0.0736283 + 0.997286i \(0.523458\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −10.1899 −0.399988
\(650\) 0 0
\(651\) −15.3789 −0.602746
\(652\) − 10.0916i − 0.395218i
\(653\) 42.2899 1.65493 0.827466 0.561516i \(-0.189782\pi\)
0.827466 + 0.561516i \(0.189782\pi\)
\(654\) −0.663848 −0.0259585
\(655\) 10.8892i 0.425476i
\(656\) 9.28932i 0.362687i
\(657\) 12.2175i 0.476651i
\(658\) 24.8756i 0.969752i
\(659\) −29.1750 −1.13650 −0.568248 0.822858i \(-0.692378\pi\)
−0.568248 + 0.822858i \(0.692378\pi\)
\(660\) 1.66808 0.0649298
\(661\) − 44.4662i − 1.72954i −0.502172 0.864768i \(-0.667465\pi\)
0.502172 0.864768i \(-0.332535\pi\)
\(662\) −5.60360 −0.217790
\(663\) 0 0
\(664\) 7.95317 0.308643
\(665\) − 23.0078i − 0.892203i
\(666\) −9.86423 −0.382231
\(667\) −12.5011 −0.484043
\(668\) 7.89701i 0.305545i
\(669\) 6.08380i 0.235213i
\(670\) 14.7534i 0.569974i
\(671\) − 12.4507i − 0.480654i
\(672\) −3.64466 −0.140596
\(673\) −7.90633 −0.304767 −0.152383 0.988321i \(-0.548695\pi\)
−0.152383 + 0.988321i \(0.548695\pi\)
\(674\) 21.7868i 0.839198i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −7.05615 −0.271190 −0.135595 0.990764i \(-0.543295\pi\)
−0.135595 + 0.990764i \(0.543295\pi\)
\(678\) − 17.8700i − 0.686293i
\(679\) −10.0353 −0.385119
\(680\) 4.00000 0.153393
\(681\) 15.3205i 0.587083i
\(682\) 7.03856i 0.269520i
\(683\) 5.95317i 0.227792i 0.993493 + 0.113896i \(0.0363330\pi\)
−0.993493 + 0.113896i \(0.963667\pi\)
\(684\) 6.31274i 0.241373i
\(685\) −7.03696 −0.268869
\(686\) −2.61124 −0.0996976
\(687\) − 22.2644i − 0.849438i
\(688\) 7.57286 0.288713
\(689\) 0 0
\(690\) −1.24453 −0.0473786
\(691\) 18.7591i 0.713628i 0.934175 + 0.356814i \(0.116137\pi\)
−0.934175 + 0.356814i \(0.883863\pi\)
\(692\) −0.440685 −0.0167523
\(693\) 6.07957 0.230944
\(694\) 9.68162i 0.367509i
\(695\) − 11.6447i − 0.441707i
\(696\) − 10.0448i − 0.380747i
\(697\) − 37.1573i − 1.40743i
\(698\) −19.3205 −0.731292
\(699\) 10.8366 0.409877
\(700\) 3.64466i 0.137755i
\(701\) −28.5298 −1.07755 −0.538777 0.842448i \(-0.681113\pi\)
−0.538777 + 0.842448i \(0.681113\pi\)
\(702\) 0 0
\(703\) 62.2703 2.34857
\(704\) 1.66808i 0.0628680i
\(705\) −6.82522 −0.257053
\(706\) 22.9750 0.864677
\(707\) − 19.4485i − 0.731435i
\(708\) 6.10876i 0.229581i
\(709\) 23.1926i 0.871015i 0.900185 + 0.435507i \(0.143431\pi\)
−0.900185 + 0.435507i \(0.856569\pi\)
\(710\) 3.51093i 0.131763i
\(711\) 9.93398 0.372553
\(712\) 5.95162 0.223046
\(713\) − 5.25139i − 0.196666i
\(714\) 14.5786 0.545592
\(715\) 0 0
\(716\) −19.6368 −0.733863
\(717\) 16.4975i 0.616111i
\(718\) 2.21752 0.0827572
\(719\) −11.7128 −0.436814 −0.218407 0.975858i \(-0.570086\pi\)
−0.218407 + 0.975858i \(0.570086\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) − 27.3804i − 1.01970i
\(722\) − 20.8506i − 0.775980i
\(723\) − 4.40435i − 0.163800i
\(724\) −6.21752 −0.231072
\(725\) −10.0448 −0.373054
\(726\) 8.21752i 0.304981i
\(727\) 3.82677 0.141927 0.0709634 0.997479i \(-0.477393\pi\)
0.0709634 + 0.997479i \(0.477393\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 12.2175i 0.452191i
\(731\) −30.2915 −1.12037
\(732\) −7.46410 −0.275881
\(733\) − 12.9340i − 0.477727i −0.971053 0.238864i \(-0.923225\pi\)
0.971053 0.238864i \(-0.0767749\pi\)
\(734\) − 6.08112i − 0.224458i
\(735\) 6.28354i 0.231772i
\(736\) − 1.24453i − 0.0458741i
\(737\) 24.6098 0.906515
\(738\) −9.28932 −0.341945
\(739\) − 35.3462i − 1.30023i −0.759836 0.650115i \(-0.774721\pi\)
0.759836 0.650115i \(-0.225279\pi\)
\(740\) −9.86423 −0.362616
\(741\) 0 0
\(742\) 3.09298 0.113547
\(743\) − 37.0953i − 1.36090i −0.732796 0.680448i \(-0.761785\pi\)
0.732796 0.680448i \(-0.238215\pi\)
\(744\) 4.21957 0.154697
\(745\) 0.772609 0.0283062
\(746\) 15.6781i 0.574015i
\(747\) 7.95317i 0.290991i
\(748\) − 6.67230i − 0.243964i
\(749\) 61.6975i 2.25438i
\(750\) −1.00000 −0.0365148
\(751\) 9.65807 0.352428 0.176214 0.984352i \(-0.443615\pi\)
0.176214 + 0.984352i \(0.443615\pi\)
\(752\) − 6.82522i − 0.248890i
\(753\) −11.9453 −0.435313
\(754\) 0 0
\(755\) −9.77838 −0.355872
\(756\) − 3.64466i − 0.132555i
\(757\) −43.6885 −1.58789 −0.793943 0.607992i \(-0.791975\pi\)
−0.793943 + 0.607992i \(0.791975\pi\)
\(758\) −30.7166 −1.11568
\(759\) 2.07598i 0.0753531i
\(760\) 6.31274i 0.228987i
\(761\) − 39.9007i − 1.44640i −0.690639 0.723200i \(-0.742671\pi\)
0.690639 0.723200i \(-0.257329\pi\)
\(762\) − 14.4407i − 0.523131i
\(763\) −2.41950 −0.0875918
\(764\) −15.6816 −0.567341
\(765\) 4.00000i 0.144620i
\(766\) 20.0619 0.724867
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) − 17.5588i − 0.633187i −0.948561 0.316594i \(-0.897461\pi\)
0.948561 0.316594i \(-0.102539\pi\)
\(770\) 6.07957 0.219092
\(771\) −19.4621 −0.700909
\(772\) − 8.12795i − 0.292531i
\(773\) − 12.8372i − 0.461723i −0.972987 0.230861i \(-0.925846\pi\)
0.972987 0.230861i \(-0.0741544\pi\)
\(774\) 7.57286i 0.272201i
\(775\) − 4.21957i − 0.151571i
\(776\) 2.75342 0.0988420
\(777\) −35.9518 −1.28976
\(778\) 27.0314i 0.969122i
\(779\) 58.6410 2.10103
\(780\) 0 0
\(781\) 5.85651 0.209562
\(782\) 4.97813i 0.178018i
\(783\) 10.0448 0.358971
\(784\) −6.28354 −0.224412
\(785\) 12.0135i 0.428782i
\(786\) − 10.8892i − 0.388405i
\(787\) 6.95735i 0.248003i 0.992282 + 0.124001i \(0.0395727\pi\)
−0.992282 + 0.124001i \(0.960427\pi\)
\(788\) − 0.891239i − 0.0317491i
\(789\) 2.03478 0.0724402
\(790\) 9.93398 0.353435
\(791\) − 65.1301i − 2.31576i
\(792\) −1.66808 −0.0592725
\(793\) 0 0
\(794\) −3.73628 −0.132596
\(795\) 0.848634i 0.0300979i
\(796\) 0.361116 0.0127994
\(797\) −34.1354 −1.20914 −0.604569 0.796553i \(-0.706655\pi\)
−0.604569 + 0.796553i \(0.706655\pi\)
\(798\) 23.0078i 0.814466i
\(799\) 27.3009i 0.965835i
\(800\) − 1.00000i − 0.0353553i
\(801\) 5.95162i 0.210290i
\(802\) −28.0911 −0.991932
\(803\) 20.3798 0.719186
\(804\) − 14.7534i − 0.520313i
\(805\) −4.53590 −0.159869
\(806\) 0 0
\(807\) 20.5287 0.722645
\(808\) 5.33615i 0.187725i
\(809\) −3.32770 −0.116996 −0.0584978 0.998288i \(-0.518631\pi\)
−0.0584978 + 0.998288i \(0.518631\pi\)
\(810\) 1.00000 0.0351364
\(811\) 20.6083i 0.723655i 0.932245 + 0.361827i \(0.117847\pi\)
−0.932245 + 0.361827i \(0.882153\pi\)
\(812\) − 36.6098i − 1.28475i
\(813\) 25.5401i 0.895730i
\(814\) 16.4543i 0.576722i
\(815\) 10.0916 0.353494
\(816\) −4.00000 −0.140028
\(817\) − 47.8055i − 1.67250i
\(818\) 27.3804 0.957335
\(819\) 0 0
\(820\) −9.28932 −0.324397
\(821\) 1.58005i 0.0551442i 0.999620 + 0.0275721i \(0.00877758\pi\)
−0.999620 + 0.0275721i \(0.991222\pi\)
\(822\) 7.03696 0.245442
\(823\) −37.5168 −1.30776 −0.653878 0.756600i \(-0.726859\pi\)
−0.653878 + 0.756600i \(0.726859\pi\)
\(824\) 7.51248i 0.261710i
\(825\) 1.66808i 0.0580750i
\(826\) 22.2644i 0.774676i
\(827\) 49.4912i 1.72098i 0.509469 + 0.860489i \(0.329842\pi\)
−0.509469 + 0.860489i \(0.670158\pi\)
\(828\) 1.24453 0.0432505
\(829\) −49.1385 −1.70665 −0.853326 0.521378i \(-0.825418\pi\)
−0.853326 + 0.521378i \(0.825418\pi\)
\(830\) 7.95317i 0.276058i
\(831\) −18.0604 −0.626508
\(832\) 0 0
\(833\) 25.1342 0.870848
\(834\) 11.6447i 0.403222i
\(835\) −7.89701 −0.273287
\(836\) 10.5301 0.364192
\(837\) 4.21957i 0.145850i
\(838\) − 13.1614i − 0.454652i
\(839\) 21.5421i 0.743717i 0.928289 + 0.371858i \(0.121279\pi\)
−0.928289 + 0.371858i \(0.878721\pi\)
\(840\) − 3.64466i − 0.125753i
\(841\) 71.8977 2.47923
\(842\) 1.29341 0.0445739
\(843\) 20.2175i 0.696328i
\(844\) −2.22739 −0.0766700
\(845\) 0 0
\(846\) 6.82522 0.234656
\(847\) 29.9501i 1.02910i
\(848\) −0.848634 −0.0291422
\(849\) −8.69149 −0.298291
\(850\) 4.00000i 0.137199i
\(851\) − 12.2764i − 0.420828i
\(852\) − 3.51093i − 0.120283i
\(853\) − 2.79821i − 0.0958088i −0.998852 0.0479044i \(-0.984746\pi\)
0.998852 0.0479044i \(-0.0152543\pi\)
\(854\) −27.2041 −0.930905
\(855\) −6.31274 −0.215891
\(856\) − 16.9282i − 0.578594i
\(857\) −48.9658 −1.67264 −0.836320 0.548242i \(-0.815297\pi\)
−0.836320 + 0.548242i \(0.815297\pi\)
\(858\) 0 0
\(859\) 4.22739 0.144237 0.0721184 0.997396i \(-0.477024\pi\)
0.0721184 + 0.997396i \(0.477024\pi\)
\(860\) 7.57286i 0.258232i
\(861\) −33.8564 −1.15382
\(862\) −12.1279 −0.413080
\(863\) 19.0285i 0.647738i 0.946102 + 0.323869i \(0.104984\pi\)
−0.946102 + 0.323869i \(0.895016\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) − 0.440685i − 0.0149837i
\(866\) 2.07802i 0.0706141i
\(867\) −1.00000 −0.0339618
\(868\) 15.3789 0.521994
\(869\) − 16.5706i − 0.562120i
\(870\) 10.0448 0.340550
\(871\) 0 0
\(872\) 0.663848 0.0224807
\(873\) 2.75342i 0.0931892i
\(874\) −7.85641 −0.265747
\(875\) −3.64466 −0.123212
\(876\) − 12.2175i − 0.412792i
\(877\) − 13.5473i − 0.457459i −0.973490 0.228729i \(-0.926543\pi\)
0.973490 0.228729i \(-0.0734571\pi\)
\(878\) 2.39640i 0.0808745i
\(879\) − 7.54790i − 0.254584i
\(880\) −1.66808 −0.0562308
\(881\) −4.98081 −0.167808 −0.0839039 0.996474i \(-0.526739\pi\)
−0.0839039 + 0.996474i \(0.526739\pi\)
\(882\) − 6.28354i − 0.211578i
\(883\) 30.9829 1.04266 0.521330 0.853355i \(-0.325436\pi\)
0.521330 + 0.853355i \(0.325436\pi\)
\(884\) 0 0
\(885\) −6.10876 −0.205344
\(886\) 21.9959i 0.738967i
\(887\) 17.1765 0.576731 0.288365 0.957520i \(-0.406888\pi\)
0.288365 + 0.957520i \(0.406888\pi\)
\(888\) 9.86423 0.331022
\(889\) − 52.6314i − 1.76520i
\(890\) 5.95162i 0.199499i
\(891\) − 1.66808i − 0.0558826i
\(892\) − 6.08380i − 0.203701i
\(893\) −43.0858 −1.44181
\(894\) −0.772609 −0.0258399
\(895\) − 19.6368i − 0.656387i
\(896\) 3.64466 0.121760
\(897\) 0 0
\(898\) 29.2253 0.975261
\(899\) 42.3847i 1.41361i
\(900\) 1.00000 0.0333333
\(901\) 3.39454 0.113088
\(902\) 15.4953i 0.515937i
\(903\) 27.6005i 0.918487i
\(904\) 17.8700i 0.594347i
\(905\) − 6.21752i − 0.206677i
\(906\) 9.77838 0.324865
\(907\) 25.0797 0.832758 0.416379 0.909191i \(-0.363299\pi\)
0.416379 + 0.909191i \(0.363299\pi\)
\(908\) − 15.3205i − 0.508429i
\(909\) −5.33615 −0.176989
\(910\) 0 0
\(911\) 18.2332 0.604092 0.302046 0.953293i \(-0.402330\pi\)
0.302046 + 0.953293i \(0.402330\pi\)
\(912\) − 6.31274i − 0.209036i
\(913\) 13.2665 0.439057
\(914\) 39.6432 1.31128
\(915\) − 7.46410i − 0.246756i
\(916\) 22.2644i 0.735635i
\(917\) − 39.6874i − 1.31059i
\(918\) − 4.00000i − 0.132020i
\(919\) −34.3882 −1.13436 −0.567181 0.823593i \(-0.691966\pi\)
−0.567181 + 0.823593i \(0.691966\pi\)
\(920\) 1.24453 0.0410310
\(921\) 26.0427i 0.858137i
\(922\) 16.6758 0.549190
\(923\) 0 0
\(924\) −6.07957 −0.200003
\(925\) − 9.86423i − 0.324334i
\(926\) −32.2175 −1.05873
\(927\) −7.51248 −0.246742
\(928\) 10.0448i 0.329736i
\(929\) − 31.0904i − 1.02004i −0.860161 0.510022i \(-0.829637\pi\)
0.860161 0.510022i \(-0.170363\pi\)
\(930\) 4.21957i 0.138365i
\(931\) 39.6663i 1.30001i
\(932\) −10.8366 −0.354964
\(933\) 25.3789 0.830868
\(934\) − 6.88137i − 0.225165i
\(935\) 6.67230 0.218208
\(936\) 0 0
\(937\) −18.8783 −0.616726 −0.308363 0.951269i \(-0.599781\pi\)
−0.308363 + 0.951269i \(0.599781\pi\)
\(938\) − 53.7712i − 1.75569i
\(939\) −31.4600 −1.02666
\(940\) 6.82522 0.222614
\(941\) 28.9398i 0.943409i 0.881757 + 0.471705i \(0.156361\pi\)
−0.881757 + 0.471705i \(0.843639\pi\)
\(942\) − 12.0135i − 0.391423i
\(943\) − 11.5609i − 0.376473i
\(944\) − 6.10876i − 0.198823i
\(945\) 3.64466 0.118561
\(946\) 12.6321 0.410705
\(947\) 7.86896i 0.255707i 0.991793 + 0.127853i \(0.0408087\pi\)
−0.991793 + 0.127853i \(0.959191\pi\)
\(948\) −9.93398 −0.322641
\(949\) 0 0
\(950\) −6.31274 −0.204812
\(951\) − 24.7093i − 0.801253i
\(952\) −14.5786 −0.472496
\(953\) 55.0968 1.78476 0.892381 0.451283i \(-0.149033\pi\)
0.892381 + 0.451283i \(0.149033\pi\)
\(954\) − 0.848634i − 0.0274755i
\(955\) − 15.6816i − 0.507445i
\(956\) − 16.4975i − 0.533568i
\(957\) − 16.7555i − 0.541627i
\(958\) 18.9709 0.612923
\(959\) 25.6473 0.828196
\(960\) 1.00000i 0.0322749i
\(961\) 13.1952 0.425653
\(962\) 0 0
\(963\) 16.9282 0.545504
\(964\) 4.40435i 0.141855i
\(965\) 8.12795 0.261648
\(966\) 4.53590 0.145940
\(967\) − 31.5523i − 1.01465i −0.861754 0.507326i \(-0.830634\pi\)
0.861754 0.507326i \(-0.169366\pi\)
\(968\) − 8.21752i − 0.264121i
\(969\) 25.2509i 0.811177i
\(970\) 2.75342i 0.0884070i
\(971\) 3.67585 0.117964 0.0589818 0.998259i \(-0.481215\pi\)
0.0589818 + 0.998259i \(0.481215\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 42.4408i 1.36059i
\(974\) 1.91620 0.0613991
\(975\) 0 0
\(976\) 7.46410 0.238920
\(977\) 42.0205i 1.34435i 0.740391 + 0.672177i \(0.234641\pi\)
−0.740391 + 0.672177i \(0.765359\pi\)
\(978\) −10.0916 −0.322694
\(979\) 9.92775 0.317292
\(980\) − 6.28354i − 0.200720i
\(981\) 0.663848i 0.0211950i
\(982\) 33.6375i 1.07341i
\(983\) 13.4307i 0.428372i 0.976793 + 0.214186i \(0.0687099\pi\)
−0.976793 + 0.214186i \(0.931290\pi\)
\(984\) 9.28932 0.296133
\(985\) 0.891239 0.0283972
\(986\) − 40.1791i − 1.27956i
\(987\) 24.8756 0.791799
\(988\) 0 0
\(989\) −9.42468 −0.299687
\(990\) − 1.66808i − 0.0530149i
\(991\) 43.7988 1.39132 0.695658 0.718373i \(-0.255113\pi\)
0.695658 + 0.718373i \(0.255113\pi\)
\(992\) −4.21957 −0.133971
\(993\) 5.60360i 0.177825i
\(994\) − 12.7962i − 0.405870i
\(995\) 0.361116i 0.0114481i
\(996\) − 7.95317i − 0.252006i
\(997\) −23.5125 −0.744648 −0.372324 0.928103i \(-0.621439\pi\)
−0.372324 + 0.928103i \(0.621439\pi\)
\(998\) 1.82522 0.0577763
\(999\) 9.86423i 0.312090i
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.ba.1351.8 8
13.5 odd 4 5070.2.a.ca.1.1 4
13.8 odd 4 5070.2.a.bz.1.4 4
13.9 even 3 390.2.bb.c.361.3 yes 8
13.10 even 6 390.2.bb.c.121.3 8
13.12 even 2 inner 5070.2.b.ba.1351.1 8
39.23 odd 6 1170.2.bs.f.901.1 8
39.35 odd 6 1170.2.bs.f.361.1 8
65.9 even 6 1950.2.bc.g.751.2 8
65.22 odd 12 1950.2.y.k.49.2 8
65.23 odd 12 1950.2.y.k.199.2 8
65.48 odd 12 1950.2.y.j.49.3 8
65.49 even 6 1950.2.bc.g.901.2 8
65.62 odd 12 1950.2.y.j.199.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.c.121.3 8 13.10 even 6
390.2.bb.c.361.3 yes 8 13.9 even 3
1170.2.bs.f.361.1 8 39.35 odd 6
1170.2.bs.f.901.1 8 39.23 odd 6
1950.2.y.j.49.3 8 65.48 odd 12
1950.2.y.j.199.3 8 65.62 odd 12
1950.2.y.k.49.2 8 65.22 odd 12
1950.2.y.k.199.2 8 65.23 odd 12
1950.2.bc.g.751.2 8 65.9 even 6
1950.2.bc.g.901.2 8 65.49 even 6
5070.2.a.bz.1.4 4 13.8 odd 4
5070.2.a.ca.1.1 4 13.5 odd 4
5070.2.b.ba.1351.1 8 13.12 even 2 inner
5070.2.b.ba.1351.8 8 1.1 even 1 trivial