Properties

Label 5070.2.a.bz.1.4
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.131472.2
Defining polynomial: \(x^{4} - 2 x^{3} - 19 x^{2} + 20 x + 52\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-3.64466\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

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

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
\(7\) 3.64466 1.37755 0.688776 0.724974i \(-0.258148\pi\)
0.688776 + 0.724974i \(0.258148\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.66808 −0.502944 −0.251472 0.967865i \(-0.580915\pi\)
−0.251472 + 0.967865i \(0.580915\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −3.64466 −0.974076
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.31274 1.44824 0.724120 0.689674i \(-0.242246\pi\)
0.724120 + 0.689674i \(0.242246\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.64466 0.795330
\(22\) 1.66808 0.355635
\(23\) 1.24453 0.259503 0.129752 0.991547i \(-0.458582\pi\)
0.129752 + 0.991547i \(0.458582\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 3.64466 0.688776
\(29\) 10.0448 1.86527 0.932635 0.360821i \(-0.117504\pi\)
0.932635 + 0.360821i \(0.117504\pi\)
\(30\) −1.00000 −0.182574
\(31\) −4.21957 −0.757857 −0.378928 0.925426i \(-0.623707\pi\)
−0.378928 + 0.925426i \(0.623707\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.66808 −0.290375
\(34\) −4.00000 −0.685994
\(35\) 3.64466 0.616060
\(36\) 1.00000 0.166667
\(37\) 9.86423 1.62167 0.810835 0.585275i \(-0.199014\pi\)
0.810835 + 0.585275i \(0.199014\pi\)
\(38\) −6.31274 −1.02406
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −9.28932 −1.45075 −0.725374 0.688355i \(-0.758333\pi\)
−0.725374 + 0.688355i \(0.758333\pi\)
\(42\) −3.64466 −0.562383
\(43\) −7.57286 −1.15485 −0.577425 0.816443i \(-0.695943\pi\)
−0.577425 + 0.816443i \(0.695943\pi\)
\(44\) −1.66808 −0.251472
\(45\) 1.00000 0.149071
\(46\) −1.24453 −0.183496
\(47\) −6.82522 −0.995560 −0.497780 0.867303i \(-0.665851\pi\)
−0.497780 + 0.867303i \(0.665851\pi\)
\(48\) 1.00000 0.144338
\(49\) 6.28354 0.897649
\(50\) −1.00000 −0.141421
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) −0.848634 −0.116569 −0.0582844 0.998300i \(-0.518563\pi\)
−0.0582844 + 0.998300i \(0.518563\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.66808 −0.224923
\(56\) −3.64466 −0.487038
\(57\) 6.31274 0.836142
\(58\) −10.0448 −1.31895
\(59\) −6.10876 −0.795293 −0.397646 0.917539i \(-0.630173\pi\)
−0.397646 + 0.917539i \(0.630173\pi\)
\(60\) 1.00000 0.129099
\(61\) 7.46410 0.955680 0.477840 0.878447i \(-0.341420\pi\)
0.477840 + 0.878447i \(0.341420\pi\)
\(62\) 4.21957 0.535886
\(63\) 3.64466 0.459184
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.66808 0.205326
\(67\) −14.7534 −1.80242 −0.901209 0.433386i \(-0.857319\pi\)
−0.901209 + 0.433386i \(0.857319\pi\)
\(68\) 4.00000 0.485071
\(69\) 1.24453 0.149824
\(70\) −3.64466 −0.435620
\(71\) −3.51093 −0.416671 −0.208336 0.978057i \(-0.566805\pi\)
−0.208336 + 0.978057i \(0.566805\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.2175 1.42995 0.714976 0.699149i \(-0.246437\pi\)
0.714976 + 0.699149i \(0.246437\pi\)
\(74\) −9.86423 −1.14669
\(75\) 1.00000 0.115470
\(76\) 6.31274 0.724120
\(77\) −6.07957 −0.692831
\(78\) 0 0
\(79\) 9.93398 1.11766 0.558830 0.829282i \(-0.311250\pi\)
0.558830 + 0.829282i \(0.311250\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 9.28932 1.02583
\(83\) −7.95317 −0.872974 −0.436487 0.899711i \(-0.643777\pi\)
−0.436487 + 0.899711i \(0.643777\pi\)
\(84\) 3.64466 0.397665
\(85\) 4.00000 0.433861
\(86\) 7.57286 0.816603
\(87\) 10.0448 1.07691
\(88\) 1.66808 0.177817
\(89\) 5.95162 0.630870 0.315435 0.948947i \(-0.397850\pi\)
0.315435 + 0.948947i \(0.397850\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 1.24453 0.129752
\(93\) −4.21957 −0.437549
\(94\) 6.82522 0.703967
\(95\) 6.31274 0.647673
\(96\) −1.00000 −0.102062
\(97\) −2.75342 −0.279568 −0.139784 0.990182i \(-0.544641\pi\)
−0.139784 + 0.990182i \(0.544641\pi\)
\(98\) −6.28354 −0.634734
\(99\) −1.66808 −0.167648
\(100\) 1.00000 0.100000
\(101\) 5.33615 0.530967 0.265483 0.964115i \(-0.414468\pi\)
0.265483 + 0.964115i \(0.414468\pi\)
\(102\) −4.00000 −0.396059
\(103\) 7.51248 0.740227 0.370113 0.928987i \(-0.379319\pi\)
0.370113 + 0.928987i \(0.379319\pi\)
\(104\) 0 0
\(105\) 3.64466 0.355682
\(106\) 0.848634 0.0824266
\(107\) 16.9282 1.63651 0.818256 0.574855i \(-0.194941\pi\)
0.818256 + 0.574855i \(0.194941\pi\)
\(108\) 1.00000 0.0962250
\(109\) −0.663848 −0.0635851 −0.0317926 0.999494i \(-0.510122\pi\)
−0.0317926 + 0.999494i \(0.510122\pi\)
\(110\) 1.66808 0.159045
\(111\) 9.86423 0.936271
\(112\) 3.64466 0.344388
\(113\) −17.8700 −1.68107 −0.840534 0.541758i \(-0.817759\pi\)
−0.840534 + 0.541758i \(0.817759\pi\)
\(114\) −6.31274 −0.591242
\(115\) 1.24453 0.116053
\(116\) 10.0448 0.932635
\(117\) 0 0
\(118\) 6.10876 0.562357
\(119\) 14.5786 1.33642
\(120\) −1.00000 −0.0912871
\(121\) −8.21752 −0.747047
\(122\) −7.46410 −0.675768
\(123\) −9.28932 −0.837590
\(124\) −4.21957 −0.378928
\(125\) 1.00000 0.0894427
\(126\) −3.64466 −0.324692
\(127\) 14.4407 1.28140 0.640702 0.767790i \(-0.278643\pi\)
0.640702 + 0.767790i \(0.278643\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −7.57286 −0.666753
\(130\) 0 0
\(131\) −10.8892 −0.951393 −0.475697 0.879609i \(-0.657804\pi\)
−0.475697 + 0.879609i \(0.657804\pi\)
\(132\) −1.66808 −0.145187
\(133\) 23.0078 1.99503
\(134\) 14.7534 1.27450
\(135\) 1.00000 0.0860663
\(136\) −4.00000 −0.342997
\(137\) −7.03696 −0.601208 −0.300604 0.953749i \(-0.597188\pi\)
−0.300604 + 0.953749i \(0.597188\pi\)
\(138\) −1.24453 −0.105942
\(139\) 11.6447 0.987687 0.493844 0.869551i \(-0.335592\pi\)
0.493844 + 0.869551i \(0.335592\pi\)
\(140\) 3.64466 0.308030
\(141\) −6.82522 −0.574787
\(142\) 3.51093 0.294631
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 10.0448 0.834174
\(146\) −12.2175 −1.01113
\(147\) 6.28354 0.518258
\(148\) 9.86423 0.810835
\(149\) −0.772609 −0.0632946 −0.0316473 0.999499i \(-0.510075\pi\)
−0.0316473 + 0.999499i \(0.510075\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −9.77838 −0.795754 −0.397877 0.917439i \(-0.630253\pi\)
−0.397877 + 0.917439i \(0.630253\pi\)
\(152\) −6.31274 −0.512030
\(153\) 4.00000 0.323381
\(154\) 6.07957 0.489906
\(155\) −4.21957 −0.338924
\(156\) 0 0
\(157\) −12.0135 −0.958786 −0.479393 0.877600i \(-0.659143\pi\)
−0.479393 + 0.877600i \(0.659143\pi\)
\(158\) −9.93398 −0.790305
\(159\) −0.848634 −0.0673011
\(160\) −1.00000 −0.0790569
\(161\) 4.53590 0.357479
\(162\) −1.00000 −0.0785674
\(163\) 10.0916 0.790437 0.395218 0.918587i \(-0.370669\pi\)
0.395218 + 0.918587i \(0.370669\pi\)
\(164\) −9.28932 −0.725374
\(165\) −1.66808 −0.129860
\(166\) 7.95317 0.617285
\(167\) −7.89701 −0.611089 −0.305545 0.952178i \(-0.598839\pi\)
−0.305545 + 0.952178i \(0.598839\pi\)
\(168\) −3.64466 −0.281192
\(169\) 0 0
\(170\) −4.00000 −0.306786
\(171\) 6.31274 0.482747
\(172\) −7.57286 −0.577425
\(173\) −0.440685 −0.0335047 −0.0167523 0.999860i \(-0.505333\pi\)
−0.0167523 + 0.999860i \(0.505333\pi\)
\(174\) −10.0448 −0.761493
\(175\) 3.64466 0.275510
\(176\) −1.66808 −0.125736
\(177\) −6.10876 −0.459163
\(178\) −5.95162 −0.446093
\(179\) −19.6368 −1.46773 −0.733863 0.679297i \(-0.762285\pi\)
−0.733863 + 0.679297i \(0.762285\pi\)
\(180\) 1.00000 0.0745356
\(181\) −6.21752 −0.462145 −0.231072 0.972937i \(-0.574223\pi\)
−0.231072 + 0.972937i \(0.574223\pi\)
\(182\) 0 0
\(183\) 7.46410 0.551762
\(184\) −1.24453 −0.0917482
\(185\) 9.86423 0.725232
\(186\) 4.21957 0.309394
\(187\) −6.67230 −0.487927
\(188\) −6.82522 −0.497780
\(189\) 3.64466 0.265110
\(190\) −6.31274 −0.457974
\(191\) 15.6816 1.13468 0.567341 0.823483i \(-0.307972\pi\)
0.567341 + 0.823483i \(0.307972\pi\)
\(192\) 1.00000 0.0721688
\(193\) 8.12795 0.585063 0.292531 0.956256i \(-0.405502\pi\)
0.292531 + 0.956256i \(0.405502\pi\)
\(194\) 2.75342 0.197684
\(195\) 0 0
\(196\) 6.28354 0.448825
\(197\) −0.891239 −0.0634981 −0.0317491 0.999496i \(-0.510108\pi\)
−0.0317491 + 0.999496i \(0.510108\pi\)
\(198\) 1.66808 0.118545
\(199\) 0.361116 0.0255988 0.0127994 0.999918i \(-0.495926\pi\)
0.0127994 + 0.999918i \(0.495926\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −14.7534 −1.04063
\(202\) −5.33615 −0.375450
\(203\) 36.6098 2.56951
\(204\) 4.00000 0.280056
\(205\) −9.28932 −0.648794
\(206\) −7.51248 −0.523419
\(207\) 1.24453 0.0865010
\(208\) 0 0
\(209\) −10.5301 −0.728384
\(210\) −3.64466 −0.251505
\(211\) 2.22739 0.153340 0.0766700 0.997057i \(-0.475571\pi\)
0.0766700 + 0.997057i \(0.475571\pi\)
\(212\) −0.848634 −0.0582844
\(213\) −3.51093 −0.240565
\(214\) −16.9282 −1.15719
\(215\) −7.57286 −0.516465
\(216\) −1.00000 −0.0680414
\(217\) −15.3789 −1.04399
\(218\) 0.663848 0.0449615
\(219\) 12.2175 0.825584
\(220\) −1.66808 −0.112462
\(221\) 0 0
\(222\) −9.86423 −0.662044
\(223\) −6.08380 −0.407401 −0.203701 0.979033i \(-0.565297\pi\)
−0.203701 + 0.979033i \(0.565297\pi\)
\(224\) −3.64466 −0.243519
\(225\) 1.00000 0.0666667
\(226\) 17.8700 1.18869
\(227\) −15.3205 −1.01686 −0.508429 0.861104i \(-0.669774\pi\)
−0.508429 + 0.861104i \(0.669774\pi\)
\(228\) 6.31274 0.418071
\(229\) −22.2644 −1.47127 −0.735635 0.677378i \(-0.763116\pi\)
−0.735635 + 0.677378i \(0.763116\pi\)
\(230\) −1.24453 −0.0820621
\(231\) −6.07957 −0.400006
\(232\) −10.0448 −0.659473
\(233\) −10.8366 −0.709928 −0.354964 0.934880i \(-0.615507\pi\)
−0.354964 + 0.934880i \(0.615507\pi\)
\(234\) 0 0
\(235\) −6.82522 −0.445228
\(236\) −6.10876 −0.397646
\(237\) 9.93398 0.645281
\(238\) −14.5786 −0.944993
\(239\) −16.4975 −1.06714 −0.533568 0.845757i \(-0.679149\pi\)
−0.533568 + 0.845757i \(0.679149\pi\)
\(240\) 1.00000 0.0645497
\(241\) −4.40435 −0.283709 −0.141855 0.989887i \(-0.545307\pi\)
−0.141855 + 0.989887i \(0.545307\pi\)
\(242\) 8.21752 0.528242
\(243\) 1.00000 0.0641500
\(244\) 7.46410 0.477840
\(245\) 6.28354 0.401441
\(246\) 9.28932 0.592265
\(247\) 0 0
\(248\) 4.21957 0.267943
\(249\) −7.95317 −0.504011
\(250\) −1.00000 −0.0632456
\(251\) 11.9453 0.753984 0.376992 0.926217i \(-0.376959\pi\)
0.376992 + 0.926217i \(0.376959\pi\)
\(252\) 3.64466 0.229592
\(253\) −2.07598 −0.130515
\(254\) −14.4407 −0.906089
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) 19.4621 1.21401 0.607005 0.794698i \(-0.292371\pi\)
0.607005 + 0.794698i \(0.292371\pi\)
\(258\) 7.57286 0.471466
\(259\) 35.9518 2.23393
\(260\) 0 0
\(261\) 10.0448 0.621757
\(262\) 10.8892 0.672737
\(263\) 2.03478 0.125470 0.0627350 0.998030i \(-0.480018\pi\)
0.0627350 + 0.998030i \(0.480018\pi\)
\(264\) 1.66808 0.102663
\(265\) −0.848634 −0.0521312
\(266\) −23.0078 −1.41070
\(267\) 5.95162 0.364233
\(268\) −14.7534 −0.901209
\(269\) 20.5287 1.25166 0.625829 0.779960i \(-0.284761\pi\)
0.625829 + 0.779960i \(0.284761\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 25.5401 1.55145 0.775725 0.631072i \(-0.217385\pi\)
0.775725 + 0.631072i \(0.217385\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 7.03696 0.425119
\(275\) −1.66808 −0.100589
\(276\) 1.24453 0.0749121
\(277\) 18.0604 1.08514 0.542572 0.840010i \(-0.317451\pi\)
0.542572 + 0.840010i \(0.317451\pi\)
\(278\) −11.6447 −0.698400
\(279\) −4.21957 −0.252619
\(280\) −3.64466 −0.217810
\(281\) 20.2175 1.20608 0.603038 0.797712i \(-0.293957\pi\)
0.603038 + 0.797712i \(0.293957\pi\)
\(282\) 6.82522 0.406436
\(283\) 8.69149 0.516656 0.258328 0.966057i \(-0.416829\pi\)
0.258328 + 0.966057i \(0.416829\pi\)
\(284\) −3.51093 −0.208336
\(285\) 6.31274 0.373934
\(286\) 0 0
\(287\) −33.8564 −1.99848
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) −10.0448 −0.589850
\(291\) −2.75342 −0.161408
\(292\) 12.2175 0.714976
\(293\) −7.54790 −0.440953 −0.220476 0.975392i \(-0.570761\pi\)
−0.220476 + 0.975392i \(0.570761\pi\)
\(294\) −6.28354 −0.366464
\(295\) −6.10876 −0.355666
\(296\) −9.86423 −0.573347
\(297\) −1.66808 −0.0967916
\(298\) 0.772609 0.0447560
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −27.6005 −1.59087
\(302\) 9.77838 0.562683
\(303\) 5.33615 0.306554
\(304\) 6.31274 0.362060
\(305\) 7.46410 0.427393
\(306\) −4.00000 −0.228665
\(307\) 26.0427 1.48634 0.743169 0.669104i \(-0.233322\pi\)
0.743169 + 0.669104i \(0.233322\pi\)
\(308\) −6.07957 −0.346416
\(309\) 7.51248 0.427370
\(310\) 4.21957 0.239655
\(311\) −25.3789 −1.43910 −0.719552 0.694438i \(-0.755653\pi\)
−0.719552 + 0.694438i \(0.755653\pi\)
\(312\) 0 0
\(313\) −31.4600 −1.77822 −0.889112 0.457689i \(-0.848677\pi\)
−0.889112 + 0.457689i \(0.848677\pi\)
\(314\) 12.0135 0.677964
\(315\) 3.64466 0.205353
\(316\) 9.93398 0.558830
\(317\) 24.7093 1.38781 0.693905 0.720066i \(-0.255889\pi\)
0.693905 + 0.720066i \(0.255889\pi\)
\(318\) 0.848634 0.0475890
\(319\) −16.7555 −0.938126
\(320\) 1.00000 0.0559017
\(321\) 16.9282 0.944840
\(322\) −4.53590 −0.252776
\(323\) 25.2509 1.40500
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −10.0916 −0.558923
\(327\) −0.663848 −0.0367109
\(328\) 9.28932 0.512917
\(329\) −24.8756 −1.37144
\(330\) 1.66808 0.0918246
\(331\) −5.60360 −0.308002 −0.154001 0.988071i \(-0.549216\pi\)
−0.154001 + 0.988071i \(0.549216\pi\)
\(332\) −7.95317 −0.436487
\(333\) 9.86423 0.540556
\(334\) 7.89701 0.432105
\(335\) −14.7534 −0.806065
\(336\) 3.64466 0.198832
\(337\) −21.7868 −1.18680 −0.593402 0.804906i \(-0.702216\pi\)
−0.593402 + 0.804906i \(0.702216\pi\)
\(338\) 0 0
\(339\) −17.8700 −0.970565
\(340\) 4.00000 0.216930
\(341\) 7.03856 0.381159
\(342\) −6.31274 −0.341354
\(343\) −2.61124 −0.140994
\(344\) 7.57286 0.408301
\(345\) 1.24453 0.0670034
\(346\) 0.440685 0.0236914
\(347\) 9.68162 0.519737 0.259868 0.965644i \(-0.416321\pi\)
0.259868 + 0.965644i \(0.416321\pi\)
\(348\) 10.0448 0.538457
\(349\) 19.3205 1.03420 0.517102 0.855924i \(-0.327011\pi\)
0.517102 + 0.855924i \(0.327011\pi\)
\(350\) −3.64466 −0.194815
\(351\) 0 0
\(352\) 1.66808 0.0889087
\(353\) 22.9750 1.22284 0.611419 0.791307i \(-0.290599\pi\)
0.611419 + 0.791307i \(0.290599\pi\)
\(354\) 6.10876 0.324677
\(355\) −3.51093 −0.186341
\(356\) 5.95162 0.315435
\(357\) 14.5786 0.771583
\(358\) 19.6368 1.03784
\(359\) −2.21752 −0.117036 −0.0585182 0.998286i \(-0.518638\pi\)
−0.0585182 + 0.998286i \(0.518638\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 20.8506 1.09740
\(362\) 6.21752 0.326786
\(363\) −8.21752 −0.431308
\(364\) 0 0
\(365\) 12.2175 0.639494
\(366\) −7.46410 −0.390155
\(367\) −6.08112 −0.317432 −0.158716 0.987324i \(-0.550735\pi\)
−0.158716 + 0.987324i \(0.550735\pi\)
\(368\) 1.24453 0.0648758
\(369\) −9.28932 −0.483583
\(370\) −9.86423 −0.512817
\(371\) −3.09298 −0.160580
\(372\) −4.21957 −0.218774
\(373\) 15.6781 0.811780 0.405890 0.913922i \(-0.366962\pi\)
0.405890 + 0.913922i \(0.366962\pi\)
\(374\) 6.67230 0.345017
\(375\) 1.00000 0.0516398
\(376\) 6.82522 0.351984
\(377\) 0 0
\(378\) −3.64466 −0.187461
\(379\) −30.7166 −1.57781 −0.788903 0.614518i \(-0.789350\pi\)
−0.788903 + 0.614518i \(0.789350\pi\)
\(380\) 6.31274 0.323837
\(381\) 14.4407 0.739819
\(382\) −15.6816 −0.802342
\(383\) 20.0619 1.02512 0.512558 0.858652i \(-0.328698\pi\)
0.512558 + 0.858652i \(0.328698\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −6.07957 −0.309844
\(386\) −8.12795 −0.413702
\(387\) −7.57286 −0.384950
\(388\) −2.75342 −0.139784
\(389\) −27.0314 −1.37055 −0.685273 0.728287i \(-0.740317\pi\)
−0.685273 + 0.728287i \(0.740317\pi\)
\(390\) 0 0
\(391\) 4.97813 0.251755
\(392\) −6.28354 −0.317367
\(393\) −10.8892 −0.549287
\(394\) 0.891239 0.0449000
\(395\) 9.93398 0.499833
\(396\) −1.66808 −0.0838240
\(397\) 3.73628 0.187518 0.0937592 0.995595i \(-0.470112\pi\)
0.0937592 + 0.995595i \(0.470112\pi\)
\(398\) −0.361116 −0.0181011
\(399\) 23.0078 1.15183
\(400\) 1.00000 0.0500000
\(401\) 28.0911 1.40280 0.701402 0.712766i \(-0.252558\pi\)
0.701402 + 0.712766i \(0.252558\pi\)
\(402\) 14.7534 0.735834
\(403\) 0 0
\(404\) 5.33615 0.265483
\(405\) 1.00000 0.0496904
\(406\) −36.6098 −1.81692
\(407\) −16.4543 −0.815609
\(408\) −4.00000 −0.198030
\(409\) 27.3804 1.35388 0.676938 0.736040i \(-0.263307\pi\)
0.676938 + 0.736040i \(0.263307\pi\)
\(410\) 9.28932 0.458767
\(411\) −7.03696 −0.347108
\(412\) 7.51248 0.370113
\(413\) −22.2644 −1.09556
\(414\) −1.24453 −0.0611654
\(415\) −7.95317 −0.390406
\(416\) 0 0
\(417\) 11.6447 0.570241
\(418\) 10.5301 0.515045
\(419\) −13.1614 −0.642975 −0.321487 0.946914i \(-0.604183\pi\)
−0.321487 + 0.946914i \(0.604183\pi\)
\(420\) 3.64466 0.177841
\(421\) 1.29341 0.0630370 0.0315185 0.999503i \(-0.489966\pi\)
0.0315185 + 0.999503i \(0.489966\pi\)
\(422\) −2.22739 −0.108428
\(423\) −6.82522 −0.331853
\(424\) 0.848634 0.0412133
\(425\) 4.00000 0.194029
\(426\) 3.51093 0.170105
\(427\) 27.2041 1.31650
\(428\) 16.9282 0.818256
\(429\) 0 0
\(430\) 7.57286 0.365196
\(431\) −12.1279 −0.584183 −0.292091 0.956390i \(-0.594351\pi\)
−0.292091 + 0.956390i \(0.594351\pi\)
\(432\) 1.00000 0.0481125
\(433\) −2.07802 −0.0998633 −0.0499317 0.998753i \(-0.515900\pi\)
−0.0499317 + 0.998753i \(0.515900\pi\)
\(434\) 15.3789 0.738210
\(435\) 10.0448 0.481611
\(436\) −0.663848 −0.0317926
\(437\) 7.85641 0.375823
\(438\) −12.2175 −0.583776
\(439\) −2.39640 −0.114374 −0.0571869 0.998363i \(-0.518213\pi\)
−0.0571869 + 0.998363i \(0.518213\pi\)
\(440\) 1.66808 0.0795224
\(441\) 6.28354 0.299216
\(442\) 0 0
\(443\) 21.9959 1.04506 0.522529 0.852622i \(-0.324989\pi\)
0.522529 + 0.852622i \(0.324989\pi\)
\(444\) 9.86423 0.468136
\(445\) 5.95162 0.282134
\(446\) 6.08380 0.288076
\(447\) −0.772609 −0.0365432
\(448\) 3.64466 0.172194
\(449\) −29.2253 −1.37923 −0.689613 0.724178i \(-0.742220\pi\)
−0.689613 + 0.724178i \(0.742220\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 15.4953 0.729645
\(452\) −17.8700 −0.840534
\(453\) −9.77838 −0.459429
\(454\) 15.3205 0.719027
\(455\) 0 0
\(456\) −6.31274 −0.295621
\(457\) 39.6432 1.85443 0.927216 0.374526i \(-0.122195\pi\)
0.927216 + 0.374526i \(0.122195\pi\)
\(458\) 22.2644 1.04034
\(459\) 4.00000 0.186704
\(460\) 1.24453 0.0580266
\(461\) 16.6758 0.776672 0.388336 0.921518i \(-0.373050\pi\)
0.388336 + 0.921518i \(0.373050\pi\)
\(462\) 6.07957 0.282847
\(463\) 32.2175 1.49728 0.748638 0.662979i \(-0.230708\pi\)
0.748638 + 0.662979i \(0.230708\pi\)
\(464\) 10.0448 0.466318
\(465\) −4.21957 −0.195678
\(466\) 10.8366 0.501995
\(467\) 6.88137 0.318432 0.159216 0.987244i \(-0.449103\pi\)
0.159216 + 0.987244i \(0.449103\pi\)
\(468\) 0 0
\(469\) −53.7712 −2.48292
\(470\) 6.82522 0.314824
\(471\) −12.0135 −0.553555
\(472\) 6.10876 0.281179
\(473\) 12.6321 0.580825
\(474\) −9.93398 −0.456283
\(475\) 6.31274 0.289648
\(476\) 14.5786 0.668211
\(477\) −0.848634 −0.0388563
\(478\) 16.4975 0.754579
\(479\) −18.9709 −0.866805 −0.433402 0.901201i \(-0.642687\pi\)
−0.433402 + 0.901201i \(0.642687\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 4.40435 0.200613
\(483\) 4.53590 0.206391
\(484\) −8.21752 −0.373524
\(485\) −2.75342 −0.125026
\(486\) −1.00000 −0.0453609
\(487\) 1.91620 0.0868314 0.0434157 0.999057i \(-0.486176\pi\)
0.0434157 + 0.999057i \(0.486176\pi\)
\(488\) −7.46410 −0.337884
\(489\) 10.0916 0.456359
\(490\) −6.28354 −0.283862
\(491\) −33.6375 −1.51804 −0.759019 0.651069i \(-0.774321\pi\)
−0.759019 + 0.651069i \(0.774321\pi\)
\(492\) −9.28932 −0.418795
\(493\) 40.1791 1.80958
\(494\) 0 0
\(495\) −1.66808 −0.0749744
\(496\) −4.21957 −0.189464
\(497\) −12.7962 −0.573986
\(498\) 7.95317 0.356390
\(499\) 1.82522 0.0817080 0.0408540 0.999165i \(-0.486992\pi\)
0.0408540 + 0.999165i \(0.486992\pi\)
\(500\) 1.00000 0.0447214
\(501\) −7.89701 −0.352813
\(502\) −11.9453 −0.533147
\(503\) −19.9985 −0.891687 −0.445843 0.895111i \(-0.647096\pi\)
−0.445843 + 0.895111i \(0.647096\pi\)
\(504\) −3.64466 −0.162346
\(505\) 5.33615 0.237456
\(506\) 2.07598 0.0922884
\(507\) 0 0
\(508\) 14.4407 0.640702
\(509\) −5.89124 −0.261125 −0.130562 0.991440i \(-0.541678\pi\)
−0.130562 + 0.991440i \(0.541678\pi\)
\(510\) −4.00000 −0.177123
\(511\) 44.5287 1.96983
\(512\) −1.00000 −0.0441942
\(513\) 6.31274 0.278714
\(514\) −19.4621 −0.858434
\(515\) 7.51248 0.331040
\(516\) −7.57286 −0.333377
\(517\) 11.3850 0.500711
\(518\) −35.9518 −1.57963
\(519\) −0.440685 −0.0193439
\(520\) 0 0
\(521\) 32.0370 1.40356 0.701782 0.712391i \(-0.252388\pi\)
0.701782 + 0.712391i \(0.252388\pi\)
\(522\) −10.0448 −0.439648
\(523\) 38.7186 1.69305 0.846523 0.532352i \(-0.178692\pi\)
0.846523 + 0.532352i \(0.178692\pi\)
\(524\) −10.8892 −0.475697
\(525\) 3.64466 0.159066
\(526\) −2.03478 −0.0887207
\(527\) −16.8783 −0.735229
\(528\) −1.66808 −0.0725937
\(529\) −21.4511 −0.932658
\(530\) 0.848634 0.0368623
\(531\) −6.10876 −0.265098
\(532\) 23.0078 0.997513
\(533\) 0 0
\(534\) −5.95162 −0.257552
\(535\) 16.9282 0.731870
\(536\) 14.7534 0.637251
\(537\) −19.6368 −0.847392
\(538\) −20.5287 −0.885056
\(539\) −10.4814 −0.451467
\(540\) 1.00000 0.0430331
\(541\) −25.9616 −1.11618 −0.558089 0.829781i \(-0.688465\pi\)
−0.558089 + 0.829781i \(0.688465\pi\)
\(542\) −25.5401 −1.09704
\(543\) −6.21752 −0.266819
\(544\) −4.00000 −0.171499
\(545\) −0.663848 −0.0284361
\(546\) 0 0
\(547\) −17.7596 −0.759348 −0.379674 0.925120i \(-0.623964\pi\)
−0.379674 + 0.925120i \(0.623964\pi\)
\(548\) −7.03696 −0.300604
\(549\) 7.46410 0.318560
\(550\) 1.66808 0.0711270
\(551\) 63.4101 2.70136
\(552\) −1.24453 −0.0529708
\(553\) 36.2060 1.53963
\(554\) −18.0604 −0.767312
\(555\) 9.86423 0.418713
\(556\) 11.6447 0.493844
\(557\) −26.2202 −1.11099 −0.555493 0.831521i \(-0.687470\pi\)
−0.555493 + 0.831521i \(0.687470\pi\)
\(558\) 4.21957 0.178629
\(559\) 0 0
\(560\) 3.64466 0.154015
\(561\) −6.67230 −0.281705
\(562\) −20.2175 −0.852825
\(563\) −25.9928 −1.09547 −0.547733 0.836653i \(-0.684509\pi\)
−0.547733 + 0.836653i \(0.684509\pi\)
\(564\) −6.82522 −0.287394
\(565\) −17.8700 −0.751797
\(566\) −8.69149 −0.365331
\(567\) 3.64466 0.153061
\(568\) 3.51093 0.147316
\(569\) −25.4699 −1.06775 −0.533876 0.845563i \(-0.679265\pi\)
−0.533876 + 0.845563i \(0.679265\pi\)
\(570\) −6.31274 −0.264411
\(571\) 8.44491 0.353409 0.176704 0.984264i \(-0.443456\pi\)
0.176704 + 0.984264i \(0.443456\pi\)
\(572\) 0 0
\(573\) 15.6816 0.655109
\(574\) 33.8564 1.41314
\(575\) 1.24453 0.0519006
\(576\) 1.00000 0.0416667
\(577\) −35.4216 −1.47462 −0.737311 0.675554i \(-0.763905\pi\)
−0.737311 + 0.675554i \(0.763905\pi\)
\(578\) 1.00000 0.0415945
\(579\) 8.12795 0.337786
\(580\) 10.0448 0.417087
\(581\) −28.9866 −1.20257
\(582\) 2.75342 0.114133
\(583\) 1.41559 0.0586276
\(584\) −12.2175 −0.505565
\(585\) 0 0
\(586\) 7.54790 0.311801
\(587\) −27.3789 −1.13005 −0.565024 0.825075i \(-0.691133\pi\)
−0.565024 + 0.825075i \(0.691133\pi\)
\(588\) 6.28354 0.259129
\(589\) −26.6370 −1.09756
\(590\) 6.10876 0.251494
\(591\) −0.891239 −0.0366607
\(592\) 9.86423 0.405417
\(593\) −12.0619 −0.495324 −0.247662 0.968846i \(-0.579662\pi\)
−0.247662 + 0.968846i \(0.579662\pi\)
\(594\) 1.66808 0.0684420
\(595\) 14.5786 0.597666
\(596\) −0.772609 −0.0316473
\(597\) 0.361116 0.0147795
\(598\) 0 0
\(599\) −28.6129 −1.16909 −0.584546 0.811360i \(-0.698727\pi\)
−0.584546 + 0.811360i \(0.698727\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −19.1676 −0.781862 −0.390931 0.920420i \(-0.627847\pi\)
−0.390931 + 0.920420i \(0.627847\pi\)
\(602\) 27.6005 1.12491
\(603\) −14.7534 −0.600806
\(604\) −9.77838 −0.397877
\(605\) −8.21752 −0.334090
\(606\) −5.33615 −0.216766
\(607\) 18.9382 0.768678 0.384339 0.923192i \(-0.374429\pi\)
0.384339 + 0.923192i \(0.374429\pi\)
\(608\) −6.31274 −0.256015
\(609\) 36.6098 1.48351
\(610\) −7.46410 −0.302213
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) 17.9694 0.725779 0.362890 0.931832i \(-0.381790\pi\)
0.362890 + 0.931832i \(0.381790\pi\)
\(614\) −26.0427 −1.05100
\(615\) −9.28932 −0.374582
\(616\) 6.07957 0.244953
\(617\) −18.0151 −0.725260 −0.362630 0.931933i \(-0.618121\pi\)
−0.362630 + 0.931933i \(0.618121\pi\)
\(618\) −7.51248 −0.302196
\(619\) −25.0505 −1.00687 −0.503433 0.864035i \(-0.667930\pi\)
−0.503433 + 0.864035i \(0.667930\pi\)
\(620\) −4.21957 −0.169462
\(621\) 1.24453 0.0499414
\(622\) 25.3789 1.01760
\(623\) 21.6916 0.869057
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 31.4600 1.25739
\(627\) −10.5301 −0.420533
\(628\) −12.0135 −0.479393
\(629\) 39.4569 1.57325
\(630\) −3.64466 −0.145207
\(631\) 1.41727 0.0564206 0.0282103 0.999602i \(-0.491019\pi\)
0.0282103 + 0.999602i \(0.491019\pi\)
\(632\) −9.93398 −0.395152
\(633\) 2.22739 0.0885308
\(634\) −24.7093 −0.981330
\(635\) 14.4407 0.573061
\(636\) −0.848634 −0.0336505
\(637\) 0 0
\(638\) 16.7555 0.663355
\(639\) −3.51093 −0.138890
\(640\) −1.00000 −0.0395285
\(641\) 4.61970 0.182467 0.0912335 0.995830i \(-0.470919\pi\)
0.0912335 + 0.995830i \(0.470919\pi\)
\(642\) −16.9282 −0.668103
\(643\) 48.0896 1.89647 0.948234 0.317573i \(-0.102868\pi\)
0.948234 + 0.317573i \(0.102868\pi\)
\(644\) 4.53590 0.178739
\(645\) −7.57286 −0.298181
\(646\) −25.2509 −0.993485
\(647\) 3.74565 0.147257 0.0736283 0.997286i \(-0.476542\pi\)
0.0736283 + 0.997286i \(0.476542\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 10.1899 0.399988
\(650\) 0 0
\(651\) −15.3789 −0.602746
\(652\) 10.0916 0.395218
\(653\) 42.2899 1.65493 0.827466 0.561516i \(-0.189782\pi\)
0.827466 + 0.561516i \(0.189782\pi\)
\(654\) 0.663848 0.0259585
\(655\) −10.8892 −0.425476
\(656\) −9.28932 −0.362687
\(657\) 12.2175 0.476651
\(658\) 24.8756 0.969752
\(659\) −29.1750 −1.13650 −0.568248 0.822858i \(-0.692378\pi\)
−0.568248 + 0.822858i \(0.692378\pi\)
\(660\) −1.66808 −0.0649298
\(661\) −44.4662 −1.72954 −0.864768 0.502172i \(-0.832535\pi\)
−0.864768 + 0.502172i \(0.832535\pi\)
\(662\) 5.60360 0.217790
\(663\) 0 0
\(664\) 7.95317 0.308643
\(665\) 23.0078 0.892203
\(666\) −9.86423 −0.382231
\(667\) 12.5011 0.484043
\(668\) −7.89701 −0.305545
\(669\) −6.08380 −0.235213
\(670\) 14.7534 0.569974
\(671\) −12.4507 −0.480654
\(672\) −3.64466 −0.140596
\(673\) 7.90633 0.304767 0.152383 0.988321i \(-0.451305\pi\)
0.152383 + 0.988321i \(0.451305\pi\)
\(674\) 21.7868 0.839198
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −7.05615 −0.271190 −0.135595 0.990764i \(-0.543295\pi\)
−0.135595 + 0.990764i \(0.543295\pi\)
\(678\) 17.8700 0.686293
\(679\) −10.0353 −0.385119
\(680\) −4.00000 −0.153393
\(681\) −15.3205 −0.587083
\(682\) −7.03856 −0.269520
\(683\) 5.95317 0.227792 0.113896 0.993493i \(-0.463667\pi\)
0.113896 + 0.993493i \(0.463667\pi\)
\(684\) 6.31274 0.241373
\(685\) −7.03696 −0.268869
\(686\) 2.61124 0.0996976
\(687\) −22.2644 −0.849438
\(688\) −7.57286 −0.288713
\(689\) 0 0
\(690\) −1.24453 −0.0473786
\(691\) −18.7591 −0.713628 −0.356814 0.934175i \(-0.616137\pi\)
−0.356814 + 0.934175i \(0.616137\pi\)
\(692\) −0.440685 −0.0167523
\(693\) −6.07957 −0.230944
\(694\) −9.68162 −0.367509
\(695\) 11.6447 0.441707
\(696\) −10.0448 −0.380747
\(697\) −37.1573 −1.40743
\(698\) −19.3205 −0.731292
\(699\) −10.8366 −0.409877
\(700\) 3.64466 0.137755
\(701\) 28.5298 1.07755 0.538777 0.842448i \(-0.318887\pi\)
0.538777 + 0.842448i \(0.318887\pi\)
\(702\) 0 0
\(703\) 62.2703 2.34857
\(704\) −1.66808 −0.0628680
\(705\) −6.82522 −0.257053
\(706\) −22.9750 −0.864677
\(707\) 19.4485 0.731435
\(708\) −6.10876 −0.229581
\(709\) 23.1926 0.871015 0.435507 0.900185i \(-0.356569\pi\)
0.435507 + 0.900185i \(0.356569\pi\)
\(710\) 3.51093 0.131763
\(711\) 9.93398 0.372553
\(712\) −5.95162 −0.223046
\(713\) −5.25139 −0.196666
\(714\) −14.5786 −0.545592
\(715\) 0 0
\(716\) −19.6368 −0.733863
\(717\) −16.4975 −0.616111
\(718\) 2.21752 0.0827572
\(719\) 11.7128 0.436814 0.218407 0.975858i \(-0.429914\pi\)
0.218407 + 0.975858i \(0.429914\pi\)
\(720\) 1.00000 0.0372678
\(721\) 27.3804 1.01970
\(722\) −20.8506 −0.775980
\(723\) −4.40435 −0.163800
\(724\) −6.21752 −0.231072
\(725\) 10.0448 0.373054
\(726\) 8.21752 0.304981
\(727\) −3.82677 −0.141927 −0.0709634 0.997479i \(-0.522607\pi\)
−0.0709634 + 0.997479i \(0.522607\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −12.2175 −0.452191
\(731\) −30.2915 −1.12037
\(732\) 7.46410 0.275881
\(733\) 12.9340 0.477727 0.238864 0.971053i \(-0.423225\pi\)
0.238864 + 0.971053i \(0.423225\pi\)
\(734\) 6.08112 0.224458
\(735\) 6.28354 0.231772
\(736\) −1.24453 −0.0458741
\(737\) 24.6098 0.906515
\(738\) 9.28932 0.341945
\(739\) −35.3462 −1.30023 −0.650115 0.759836i \(-0.725279\pi\)
−0.650115 + 0.759836i \(0.725279\pi\)
\(740\) 9.86423 0.362616
\(741\) 0 0
\(742\) 3.09298 0.113547
\(743\) 37.0953 1.36090 0.680448 0.732796i \(-0.261785\pi\)
0.680448 + 0.732796i \(0.261785\pi\)
\(744\) 4.21957 0.154697
\(745\) −0.772609 −0.0283062
\(746\) −15.6781 −0.574015
\(747\) −7.95317 −0.290991
\(748\) −6.67230 −0.243964
\(749\) 61.6975 2.25438
\(750\) −1.00000 −0.0365148
\(751\) −9.65807 −0.352428 −0.176214 0.984352i \(-0.556385\pi\)
−0.176214 + 0.984352i \(0.556385\pi\)
\(752\) −6.82522 −0.248890
\(753\) 11.9453 0.435313
\(754\) 0 0
\(755\) −9.77838 −0.355872
\(756\) 3.64466 0.132555
\(757\) −43.6885 −1.58789 −0.793943 0.607992i \(-0.791975\pi\)
−0.793943 + 0.607992i \(0.791975\pi\)
\(758\) 30.7166 1.11568
\(759\) −2.07598 −0.0753531
\(760\) −6.31274 −0.228987
\(761\) −39.9007 −1.44640 −0.723200 0.690639i \(-0.757329\pi\)
−0.723200 + 0.690639i \(0.757329\pi\)
\(762\) −14.4407 −0.523131
\(763\) −2.41950 −0.0875918
\(764\) 15.6816 0.567341
\(765\) 4.00000 0.144620
\(766\) −20.0619 −0.724867
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 17.5588 0.633187 0.316594 0.948561i \(-0.397461\pi\)
0.316594 + 0.948561i \(0.397461\pi\)
\(770\) 6.07957 0.219092
\(771\) 19.4621 0.700909
\(772\) 8.12795 0.292531
\(773\) 12.8372 0.461723 0.230861 0.972987i \(-0.425846\pi\)
0.230861 + 0.972987i \(0.425846\pi\)
\(774\) 7.57286 0.272201
\(775\) −4.21957 −0.151571
\(776\) 2.75342 0.0988420
\(777\) 35.9518 1.28976
\(778\) 27.0314 0.969122
\(779\) −58.6410 −2.10103
\(780\) 0 0
\(781\) 5.85651 0.209562
\(782\) −4.97813 −0.178018
\(783\) 10.0448 0.358971
\(784\) 6.28354 0.224412
\(785\) −12.0135 −0.428782
\(786\) 10.8892 0.388405
\(787\) 6.95735 0.248003 0.124001 0.992282i \(-0.460427\pi\)
0.124001 + 0.992282i \(0.460427\pi\)
\(788\) −0.891239 −0.0317491
\(789\) 2.03478 0.0724402
\(790\) −9.93398 −0.353435
\(791\) −65.1301 −2.31576
\(792\) 1.66808 0.0592725
\(793\) 0 0
\(794\) −3.73628 −0.132596
\(795\) −0.848634 −0.0300979
\(796\) 0.361116 0.0127994
\(797\) 34.1354 1.20914 0.604569 0.796553i \(-0.293345\pi\)
0.604569 + 0.796553i \(0.293345\pi\)
\(798\) −23.0078 −0.814466
\(799\) −27.3009 −0.965835
\(800\) −1.00000 −0.0353553
\(801\) 5.95162 0.210290
\(802\) −28.0911 −0.991932
\(803\) −20.3798 −0.719186
\(804\) −14.7534 −0.520313
\(805\) 4.53590 0.159869
\(806\) 0 0
\(807\) 20.5287 0.722645
\(808\) −5.33615 −0.187725
\(809\) −3.32770 −0.116996 −0.0584978 0.998288i \(-0.518631\pi\)
−0.0584978 + 0.998288i \(0.518631\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −20.6083 −0.723655 −0.361827 0.932245i \(-0.617847\pi\)
−0.361827 + 0.932245i \(0.617847\pi\)
\(812\) 36.6098 1.28475
\(813\) 25.5401 0.895730
\(814\) 16.4543 0.576722
\(815\) 10.0916 0.353494
\(816\) 4.00000 0.140028
\(817\) −47.8055 −1.67250
\(818\) −27.3804 −0.957335
\(819\) 0 0
\(820\) −9.28932 −0.324397
\(821\) −1.58005 −0.0551442 −0.0275721 0.999620i \(-0.508778\pi\)
−0.0275721 + 0.999620i \(0.508778\pi\)
\(822\) 7.03696 0.245442
\(823\) 37.5168 1.30776 0.653878 0.756600i \(-0.273141\pi\)
0.653878 + 0.756600i \(0.273141\pi\)
\(824\) −7.51248 −0.261710
\(825\) −1.66808 −0.0580750
\(826\) 22.2644 0.774676
\(827\) 49.4912 1.72098 0.860489 0.509469i \(-0.170158\pi\)
0.860489 + 0.509469i \(0.170158\pi\)
\(828\) 1.24453 0.0432505
\(829\) 49.1385 1.70665 0.853326 0.521378i \(-0.174582\pi\)
0.853326 + 0.521378i \(0.174582\pi\)
\(830\) 7.95317 0.276058
\(831\) 18.0604 0.626508
\(832\) 0 0
\(833\) 25.1342 0.870848
\(834\) −11.6447 −0.403222
\(835\) −7.89701 −0.273287
\(836\) −10.5301 −0.364192
\(837\) −4.21957 −0.145850
\(838\) 13.1614 0.454652
\(839\) 21.5421 0.743717 0.371858 0.928289i \(-0.378721\pi\)
0.371858 + 0.928289i \(0.378721\pi\)
\(840\) −3.64466 −0.125753
\(841\) 71.8977 2.47923
\(842\) −1.29341 −0.0445739
\(843\) 20.2175 0.696328
\(844\) 2.22739 0.0766700
\(845\) 0 0
\(846\) 6.82522 0.234656
\(847\) −29.9501 −1.02910
\(848\) −0.848634 −0.0291422
\(849\) 8.69149 0.298291
\(850\) −4.00000 −0.137199
\(851\) 12.2764 0.420828
\(852\) −3.51093 −0.120283
\(853\) −2.79821 −0.0958088 −0.0479044 0.998852i \(-0.515254\pi\)
−0.0479044 + 0.998852i \(0.515254\pi\)
\(854\) −27.2041 −0.930905
\(855\) 6.31274 0.215891
\(856\) −16.9282 −0.578594
\(857\) 48.9658 1.67264 0.836320 0.548242i \(-0.184703\pi\)
0.836320 + 0.548242i \(0.184703\pi\)
\(858\) 0 0
\(859\) 4.22739 0.144237 0.0721184 0.997396i \(-0.477024\pi\)
0.0721184 + 0.997396i \(0.477024\pi\)
\(860\) −7.57286 −0.258232
\(861\) −33.8564 −1.15382
\(862\) 12.1279 0.413080
\(863\) −19.0285 −0.647738 −0.323869 0.946102i \(-0.604984\pi\)
−0.323869 + 0.946102i \(0.604984\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −0.440685 −0.0149837
\(866\) 2.07802 0.0706141
\(867\) −1.00000 −0.0339618
\(868\) −15.3789 −0.521994
\(869\) −16.5706 −0.562120
\(870\) −10.0448 −0.340550
\(871\) 0 0
\(872\) 0.663848 0.0224807
\(873\) −2.75342 −0.0931892
\(874\) −7.85641 −0.265747
\(875\) 3.64466 0.123212
\(876\) 12.2175 0.412792
\(877\) 13.5473 0.457459 0.228729 0.973490i \(-0.426543\pi\)
0.228729 + 0.973490i \(0.426543\pi\)
\(878\) 2.39640 0.0808745
\(879\) −7.54790 −0.254584
\(880\) −1.66808 −0.0562308
\(881\) 4.98081 0.167808 0.0839039 0.996474i \(-0.473261\pi\)
0.0839039 + 0.996474i \(0.473261\pi\)
\(882\) −6.28354 −0.211578
\(883\) −30.9829 −1.04266 −0.521330 0.853355i \(-0.674564\pi\)
−0.521330 + 0.853355i \(0.674564\pi\)
\(884\) 0 0
\(885\) −6.10876 −0.205344
\(886\) −21.9959 −0.738967
\(887\) 17.1765 0.576731 0.288365 0.957520i \(-0.406888\pi\)
0.288365 + 0.957520i \(0.406888\pi\)
\(888\) −9.86423 −0.331022
\(889\) 52.6314 1.76520
\(890\) −5.95162 −0.199499
\(891\) −1.66808 −0.0558826
\(892\) −6.08380 −0.203701
\(893\) −43.0858 −1.44181
\(894\) 0.772609 0.0258399
\(895\) −19.6368 −0.656387
\(896\) −3.64466 −0.121760
\(897\) 0 0
\(898\) 29.2253 0.975261
\(899\) −42.3847 −1.41361
\(900\) 1.00000 0.0333333
\(901\) −3.39454 −0.113088
\(902\) −15.4953 −0.515937
\(903\) −27.6005 −0.918487
\(904\) 17.8700 0.594347
\(905\) −6.21752 −0.206677
\(906\) 9.77838 0.324865
\(907\) −25.0797 −0.832758 −0.416379 0.909191i \(-0.636701\pi\)
−0.416379 + 0.909191i \(0.636701\pi\)
\(908\) −15.3205 −0.508429
\(909\) 5.33615 0.176989
\(910\) 0 0
\(911\) 18.2332 0.604092 0.302046 0.953293i \(-0.402330\pi\)
0.302046 + 0.953293i \(0.402330\pi\)
\(912\) 6.31274 0.209036
\(913\) 13.2665 0.439057
\(914\) −39.6432 −1.31128
\(915\) 7.46410 0.246756
\(916\) −22.2644 −0.735635
\(917\) −39.6874 −1.31059
\(918\) −4.00000 −0.132020
\(919\) −34.3882 −1.13436 −0.567181 0.823593i \(-0.691966\pi\)
−0.567181 + 0.823593i \(0.691966\pi\)
\(920\) −1.24453 −0.0410310
\(921\) 26.0427 0.858137
\(922\) −16.6758 −0.549190
\(923\) 0 0
\(924\) −6.07957 −0.200003
\(925\) 9.86423 0.324334
\(926\) −32.2175 −1.05873
\(927\) 7.51248 0.246742
\(928\) −10.0448 −0.329736
\(929\) 31.0904 1.02004 0.510022 0.860161i \(-0.329637\pi\)
0.510022 + 0.860161i \(0.329637\pi\)
\(930\) 4.21957 0.138365
\(931\) 39.6663 1.30001
\(932\) −10.8366 −0.354964
\(933\) −25.3789 −0.830868
\(934\) −6.88137 −0.225165
\(935\) −6.67230 −0.218208
\(936\) 0 0
\(937\) −18.8783 −0.616726 −0.308363 0.951269i \(-0.599781\pi\)
−0.308363 + 0.951269i \(0.599781\pi\)
\(938\) 53.7712 1.75569
\(939\) −31.4600 −1.02666
\(940\) −6.82522 −0.222614
\(941\) −28.9398 −0.943409 −0.471705 0.881757i \(-0.656361\pi\)
−0.471705 + 0.881757i \(0.656361\pi\)
\(942\) 12.0135 0.391423
\(943\) −11.5609 −0.376473
\(944\) −6.10876 −0.198823
\(945\) 3.64466 0.118561
\(946\) −12.6321 −0.410705
\(947\) 7.86896 0.255707 0.127853 0.991793i \(-0.459191\pi\)
0.127853 + 0.991793i \(0.459191\pi\)
\(948\) 9.93398 0.322641
\(949\) 0 0
\(950\) −6.31274 −0.204812
\(951\) 24.7093 0.801253
\(952\) −14.5786 −0.472496
\(953\) −55.0968 −1.78476 −0.892381 0.451283i \(-0.850967\pi\)
−0.892381 + 0.451283i \(0.850967\pi\)
\(954\) 0.848634 0.0274755
\(955\) 15.6816 0.507445
\(956\) −16.4975 −0.533568
\(957\) −16.7555 −0.541627
\(958\) 18.9709 0.612923
\(959\) −25.6473 −0.828196
\(960\) 1.00000 0.0322749
\(961\) −13.1952 −0.425653
\(962\) 0 0
\(963\) 16.9282 0.545504
\(964\) −4.40435 −0.141855
\(965\) 8.12795 0.261648
\(966\) −4.53590 −0.145940
\(967\) 31.5523 1.01465 0.507326 0.861754i \(-0.330634\pi\)
0.507326 + 0.861754i \(0.330634\pi\)
\(968\) 8.21752 0.264121
\(969\) 25.2509 0.811177
\(970\) 2.75342 0.0884070
\(971\) 3.67585 0.117964 0.0589818 0.998259i \(-0.481215\pi\)
0.0589818 + 0.998259i \(0.481215\pi\)
\(972\) 1.00000 0.0320750
\(973\) 42.4408 1.36059
\(974\) −1.91620 −0.0613991
\(975\) 0 0
\(976\) 7.46410 0.238920
\(977\) −42.0205 −1.34435 −0.672177 0.740391i \(-0.734641\pi\)
−0.672177 + 0.740391i \(0.734641\pi\)
\(978\) −10.0916 −0.322694
\(979\) −9.92775 −0.317292
\(980\) 6.28354 0.200720
\(981\) −0.663848 −0.0211950
\(982\) 33.6375 1.07341
\(983\) 13.4307 0.428372 0.214186 0.976793i \(-0.431290\pi\)
0.214186 + 0.976793i \(0.431290\pi\)
\(984\) 9.28932 0.296133
\(985\) −0.891239 −0.0283972
\(986\) −40.1791 −1.27956
\(987\) −24.8756 −0.791799
\(988\) 0 0
\(989\) −9.42468 −0.299687
\(990\) 1.66808 0.0530149
\(991\) 43.7988 1.39132 0.695658 0.718373i \(-0.255113\pi\)
0.695658 + 0.718373i \(0.255113\pi\)
\(992\) 4.21957 0.133971
\(993\) −5.60360 −0.177825
\(994\) 12.7962 0.405870
\(995\) 0.361116 0.0114481
\(996\) −7.95317 −0.252006
\(997\) −23.5125 −0.744648 −0.372324 0.928103i \(-0.621439\pi\)
−0.372324 + 0.928103i \(0.621439\pi\)
\(998\) −1.82522 −0.0577763
\(999\) 9.86423 0.312090
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.bz.1.4 4
13.5 odd 4 5070.2.b.ba.1351.8 8
13.6 odd 12 390.2.bb.c.361.3 yes 8
13.8 odd 4 5070.2.b.ba.1351.1 8
13.11 odd 12 390.2.bb.c.121.3 8
13.12 even 2 5070.2.a.ca.1.1 4
39.11 even 12 1170.2.bs.f.901.1 8
39.32 even 12 1170.2.bs.f.361.1 8
65.19 odd 12 1950.2.bc.g.751.2 8
65.24 odd 12 1950.2.bc.g.901.2 8
65.32 even 12 1950.2.y.k.49.2 8
65.37 even 12 1950.2.y.j.199.3 8
65.58 even 12 1950.2.y.j.49.3 8
65.63 even 12 1950.2.y.k.199.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.c.121.3 8 13.11 odd 12
390.2.bb.c.361.3 yes 8 13.6 odd 12
1170.2.bs.f.361.1 8 39.32 even 12
1170.2.bs.f.901.1 8 39.11 even 12
1950.2.y.j.49.3 8 65.58 even 12
1950.2.y.j.199.3 8 65.37 even 12
1950.2.y.k.49.2 8 65.32 even 12
1950.2.y.k.199.2 8 65.63 even 12
1950.2.bc.g.751.2 8 65.19 odd 12
1950.2.bc.g.901.2 8 65.24 odd 12
5070.2.a.bz.1.4 4 1.1 even 1 trivial
5070.2.a.ca.1.1 4 13.12 even 2
5070.2.b.ba.1351.1 8 13.8 odd 4
5070.2.b.ba.1351.8 8 13.5 odd 4