Properties

Label 7942.2.a.bc.1.3
Level $7942$
Weight $2$
Character 7942.1
Self dual yes
Analytic conductor $63.417$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.4171892853\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.16425\) of defining polynomial
Character \(\chi\) \(=\) 7942.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.16425 q^{3} +1.00000 q^{4} +4.16425 q^{5} -2.16425 q^{6} -0.683969 q^{7} -1.00000 q^{8} +1.68397 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.16425 q^{3} +1.00000 q^{4} +4.16425 q^{5} -2.16425 q^{6} -0.683969 q^{7} -1.00000 q^{8} +1.68397 q^{9} -4.16425 q^{10} -1.00000 q^{11} +2.16425 q^{12} -2.68397 q^{13} +0.683969 q^{14} +9.01247 q^{15} +1.00000 q^{16} +3.36794 q^{17} -1.68397 q^{18} +4.16425 q^{20} -1.48028 q^{21} +1.00000 q^{22} -1.36794 q^{23} -2.16425 q^{24} +12.3410 q^{25} +2.68397 q^{26} -2.84822 q^{27} -0.683969 q^{28} -2.79631 q^{29} -9.01247 q^{30} -5.01247 q^{31} -1.00000 q^{32} -2.16425 q^{33} -3.36794 q^{34} -2.84822 q^{35} +1.68397 q^{36} +11.6964 q^{37} -5.80877 q^{39} -4.16425 q^{40} +7.01247 q^{41} +1.48028 q^{42} +7.86068 q^{43} -1.00000 q^{44} +7.01247 q^{45} +1.36794 q^{46} +2.96056 q^{47} +2.16425 q^{48} -6.53219 q^{49} -12.3410 q^{50} +7.28905 q^{51} -2.68397 q^{52} +10.3285 q^{53} +2.84822 q^{54} -4.16425 q^{55} +0.683969 q^{56} +2.79631 q^{58} +9.92112 q^{59} +9.01247 q^{60} +2.00000 q^{61} +5.01247 q^{62} -1.15178 q^{63} +1.00000 q^{64} -11.1767 q^{65} +2.16425 q^{66} +0.683969 q^{67} +3.36794 q^{68} -2.96056 q^{69} +2.84822 q^{70} -3.53219 q^{71} -1.68397 q^{72} +0.407381 q^{73} -11.6964 q^{74} +26.7089 q^{75} +0.683969 q^{77} +5.80877 q^{78} +7.28905 q^{79} +4.16425 q^{80} -11.2162 q^{81} -7.01247 q^{82} -12.1892 q^{83} -1.48028 q^{84} +14.0249 q^{85} -7.86068 q^{86} -6.05191 q^{87} +1.00000 q^{88} +16.6819 q^{89} -7.01247 q^{90} +1.83575 q^{91} -1.36794 q^{92} -10.8482 q^{93} -2.96056 q^{94} -2.16425 q^{96} +11.6964 q^{97} +6.53219 q^{98} -1.68397 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} - 3 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} - 3 q^{8} + 2 q^{9} - 5 q^{10} - 3 q^{11} - q^{12} - 5 q^{13} - q^{14} + 9 q^{15} + 3 q^{16} + 4 q^{17} - 2 q^{18} + 5 q^{20} + 3 q^{22} + 2 q^{23} + q^{24} + 4 q^{25} + 5 q^{26} + 2 q^{27} + q^{28} - 7 q^{29} - 9 q^{30} + 3 q^{31} - 3 q^{32} + q^{33} - 4 q^{34} + 2 q^{35} + 2 q^{36} + 14 q^{37} + 2 q^{39} - 5 q^{40} + 3 q^{41} - 5 q^{43} - 3 q^{44} + 3 q^{45} - 2 q^{46} - q^{48} - 6 q^{49} - 4 q^{50} - 2 q^{51} - 5 q^{52} + 16 q^{53} - 2 q^{54} - 5 q^{55} - q^{56} + 7 q^{58} + 12 q^{59} + 9 q^{60} + 6 q^{61} - 3 q^{62} - 14 q^{63} + 3 q^{64} - 8 q^{65} - q^{66} - q^{67} + 4 q^{68} - 2 q^{70} + 3 q^{71} - 2 q^{72} + 4 q^{73} - 14 q^{74} + 41 q^{75} - q^{77} - 2 q^{78} - 2 q^{79} + 5 q^{80} - 17 q^{81} - 3 q^{82} + 7 q^{83} + 6 q^{85} + 5 q^{86} - 9 q^{87} + 3 q^{88} - 16 q^{89} - 3 q^{90} + 13 q^{91} + 2 q^{92} - 22 q^{93} + q^{96} + 14 q^{97} + 6 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2.16425 1.24953 0.624765 0.780813i \(-0.285195\pi\)
0.624765 + 0.780813i \(0.285195\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.16425 1.86231 0.931154 0.364626i \(-0.118803\pi\)
0.931154 + 0.364626i \(0.118803\pi\)
\(6\) −2.16425 −0.883551
\(7\) −0.683969 −0.258516 −0.129258 0.991611i \(-0.541260\pi\)
−0.129258 + 0.991611i \(0.541260\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.68397 0.561323
\(10\) −4.16425 −1.31685
\(11\) −1.00000 −0.301511
\(12\) 2.16425 0.624765
\(13\) −2.68397 −0.744399 −0.372200 0.928153i \(-0.621396\pi\)
−0.372200 + 0.928153i \(0.621396\pi\)
\(14\) 0.683969 0.182798
\(15\) 9.01247 2.32701
\(16\) 1.00000 0.250000
\(17\) 3.36794 0.816845 0.408423 0.912793i \(-0.366079\pi\)
0.408423 + 0.912793i \(0.366079\pi\)
\(18\) −1.68397 −0.396915
\(19\) 0 0
\(20\) 4.16425 0.931154
\(21\) −1.48028 −0.323023
\(22\) 1.00000 0.213201
\(23\) −1.36794 −0.285235 −0.142617 0.989778i \(-0.545552\pi\)
−0.142617 + 0.989778i \(0.545552\pi\)
\(24\) −2.16425 −0.441775
\(25\) 12.3410 2.46819
\(26\) 2.68397 0.526370
\(27\) −2.84822 −0.548140
\(28\) −0.683969 −0.129258
\(29\) −2.79631 −0.519262 −0.259631 0.965708i \(-0.583601\pi\)
−0.259631 + 0.965708i \(0.583601\pi\)
\(30\) −9.01247 −1.64544
\(31\) −5.01247 −0.900265 −0.450133 0.892962i \(-0.648623\pi\)
−0.450133 + 0.892962i \(0.648623\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.16425 −0.376747
\(34\) −3.36794 −0.577597
\(35\) −2.84822 −0.481437
\(36\) 1.68397 0.280662
\(37\) 11.6964 1.92288 0.961441 0.275011i \(-0.0886816\pi\)
0.961441 + 0.275011i \(0.0886816\pi\)
\(38\) 0 0
\(39\) −5.80877 −0.930148
\(40\) −4.16425 −0.658425
\(41\) 7.01247 1.09516 0.547582 0.836752i \(-0.315549\pi\)
0.547582 + 0.836752i \(0.315549\pi\)
\(42\) 1.48028 0.228412
\(43\) 7.86068 1.19874 0.599371 0.800471i \(-0.295417\pi\)
0.599371 + 0.800471i \(0.295417\pi\)
\(44\) −1.00000 −0.150756
\(45\) 7.01247 1.04536
\(46\) 1.36794 0.201691
\(47\) 2.96056 0.431842 0.215921 0.976411i \(-0.430725\pi\)
0.215921 + 0.976411i \(0.430725\pi\)
\(48\) 2.16425 0.312382
\(49\) −6.53219 −0.933169
\(50\) −12.3410 −1.74528
\(51\) 7.28905 1.02067
\(52\) −2.68397 −0.372200
\(53\) 10.3285 1.41873 0.709364 0.704842i \(-0.248982\pi\)
0.709364 + 0.704842i \(0.248982\pi\)
\(54\) 2.84822 0.387593
\(55\) −4.16425 −0.561507
\(56\) 0.683969 0.0913992
\(57\) 0 0
\(58\) 2.79631 0.367173
\(59\) 9.92112 1.29162 0.645810 0.763499i \(-0.276520\pi\)
0.645810 + 0.763499i \(0.276520\pi\)
\(60\) 9.01247 1.16350
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 5.01247 0.636584
\(63\) −1.15178 −0.145111
\(64\) 1.00000 0.125000
\(65\) −11.1767 −1.38630
\(66\) 2.16425 0.266401
\(67\) 0.683969 0.0835601 0.0417801 0.999127i \(-0.486697\pi\)
0.0417801 + 0.999127i \(0.486697\pi\)
\(68\) 3.36794 0.408423
\(69\) −2.96056 −0.356409
\(70\) 2.84822 0.340427
\(71\) −3.53219 −0.419193 −0.209597 0.977788i \(-0.567215\pi\)
−0.209597 + 0.977788i \(0.567215\pi\)
\(72\) −1.68397 −0.198458
\(73\) 0.407381 0.0476803 0.0238402 0.999716i \(-0.492411\pi\)
0.0238402 + 0.999716i \(0.492411\pi\)
\(74\) −11.6964 −1.35968
\(75\) 26.7089 3.08408
\(76\) 0 0
\(77\) 0.683969 0.0779455
\(78\) 5.80877 0.657714
\(79\) 7.28905 0.820083 0.410041 0.912067i \(-0.365514\pi\)
0.410041 + 0.912067i \(0.365514\pi\)
\(80\) 4.16425 0.465577
\(81\) −11.2162 −1.24624
\(82\) −7.01247 −0.774397
\(83\) −12.1892 −1.33794 −0.668968 0.743291i \(-0.733264\pi\)
−0.668968 + 0.743291i \(0.733264\pi\)
\(84\) −1.48028 −0.161512
\(85\) 14.0249 1.52122
\(86\) −7.86068 −0.847639
\(87\) −6.05191 −0.648833
\(88\) 1.00000 0.106600
\(89\) 16.6819 1.76828 0.884140 0.467222i \(-0.154745\pi\)
0.884140 + 0.467222i \(0.154745\pi\)
\(90\) −7.01247 −0.739179
\(91\) 1.83575 0.192439
\(92\) −1.36794 −0.142617
\(93\) −10.8482 −1.12491
\(94\) −2.96056 −0.305358
\(95\) 0 0
\(96\) −2.16425 −0.220888
\(97\) 11.6964 1.18759 0.593796 0.804615i \(-0.297628\pi\)
0.593796 + 0.804615i \(0.297628\pi\)
\(98\) 6.53219 0.659850
\(99\) −1.68397 −0.169245
\(100\) 12.3410 1.23410
\(101\) 3.59262 0.357479 0.178739 0.983896i \(-0.442798\pi\)
0.178739 + 0.983896i \(0.442798\pi\)
\(102\) −7.28905 −0.721724
\(103\) 4.46781 0.440227 0.220113 0.975474i \(-0.429357\pi\)
0.220113 + 0.975474i \(0.429357\pi\)
\(104\) 2.68397 0.263185
\(105\) −6.16425 −0.601569
\(106\) −10.3285 −1.00319
\(107\) −9.69643 −0.937390 −0.468695 0.883360i \(-0.655276\pi\)
−0.468695 + 0.883360i \(0.655276\pi\)
\(108\) −2.84822 −0.274070
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 4.16425 0.397045
\(111\) 25.3140 2.40270
\(112\) −0.683969 −0.0646290
\(113\) −10.4323 −0.981389 −0.490695 0.871332i \(-0.663257\pi\)
−0.490695 + 0.871332i \(0.663257\pi\)
\(114\) 0 0
\(115\) −5.69643 −0.531195
\(116\) −2.79631 −0.259631
\(117\) −4.51972 −0.417848
\(118\) −9.92112 −0.913313
\(119\) −2.30357 −0.211168
\(120\) −9.01247 −0.822722
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) 15.1767 1.36844
\(124\) −5.01247 −0.450133
\(125\) 30.5696 2.73423
\(126\) 1.15178 0.102609
\(127\) −15.2891 −1.35668 −0.678342 0.734746i \(-0.737301\pi\)
−0.678342 + 0.734746i \(0.737301\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 17.0125 1.49786
\(130\) 11.1767 0.980263
\(131\) 3.64453 0.318424 0.159212 0.987244i \(-0.449105\pi\)
0.159212 + 0.987244i \(0.449105\pi\)
\(132\) −2.16425 −0.188374
\(133\) 0 0
\(134\) −0.683969 −0.0590859
\(135\) −11.8607 −1.02080
\(136\) −3.36794 −0.288798
\(137\) 14.6840 1.25454 0.627268 0.778803i \(-0.284173\pi\)
0.627268 + 0.778803i \(0.284173\pi\)
\(138\) 2.96056 0.252019
\(139\) −18.5966 −1.57734 −0.788670 0.614817i \(-0.789230\pi\)
−0.788670 + 0.614817i \(0.789230\pi\)
\(140\) −2.84822 −0.240718
\(141\) 6.40738 0.539599
\(142\) 3.53219 0.296414
\(143\) 2.68397 0.224445
\(144\) 1.68397 0.140331
\(145\) −11.6445 −0.967025
\(146\) −0.407381 −0.0337151
\(147\) −14.1373 −1.16602
\(148\) 11.6964 0.961441
\(149\) 4.30357 0.352562 0.176281 0.984340i \(-0.443593\pi\)
0.176281 + 0.984340i \(0.443593\pi\)
\(150\) −26.7089 −2.18077
\(151\) 21.6964 1.76563 0.882815 0.469720i \(-0.155645\pi\)
0.882815 + 0.469720i \(0.155645\pi\)
\(152\) 0 0
\(153\) 5.67150 0.458514
\(154\) −0.683969 −0.0551158
\(155\) −20.8731 −1.67657
\(156\) −5.80877 −0.465074
\(157\) −10.9001 −0.869925 −0.434962 0.900449i \(-0.643238\pi\)
−0.434962 + 0.900449i \(0.643238\pi\)
\(158\) −7.28905 −0.579886
\(159\) 22.3534 1.77274
\(160\) −4.16425 −0.329213
\(161\) 0.935628 0.0737378
\(162\) 11.2162 0.881224
\(163\) 19.7214 1.54470 0.772348 0.635199i \(-0.219082\pi\)
0.772348 + 0.635199i \(0.219082\pi\)
\(164\) 7.01247 0.547582
\(165\) −9.01247 −0.701619
\(166\) 12.1892 0.946064
\(167\) −7.77532 −0.601672 −0.300836 0.953676i \(-0.597266\pi\)
−0.300836 + 0.953676i \(0.597266\pi\)
\(168\) 1.48028 0.114206
\(169\) −5.79631 −0.445870
\(170\) −14.0249 −1.07566
\(171\) 0 0
\(172\) 7.86068 0.599371
\(173\) −15.7818 −1.19987 −0.599934 0.800050i \(-0.704806\pi\)
−0.599934 + 0.800050i \(0.704806\pi\)
\(174\) 6.05191 0.458794
\(175\) −8.44084 −0.638067
\(176\) −1.00000 −0.0753778
\(177\) 21.4718 1.61392
\(178\) −16.6819 −1.25036
\(179\) 7.97302 0.595932 0.297966 0.954577i \(-0.403692\pi\)
0.297966 + 0.954577i \(0.403692\pi\)
\(180\) 7.01247 0.522678
\(181\) 2.55318 0.189776 0.0948881 0.995488i \(-0.469751\pi\)
0.0948881 + 0.995488i \(0.469751\pi\)
\(182\) −1.83575 −0.136075
\(183\) 4.32850 0.319972
\(184\) 1.36794 0.100846
\(185\) 48.7069 3.58100
\(186\) 10.8482 0.795430
\(187\) −3.36794 −0.246288
\(188\) 2.96056 0.215921
\(189\) 1.94809 0.141703
\(190\) 0 0
\(191\) 8.22468 0.595117 0.297559 0.954704i \(-0.403828\pi\)
0.297559 + 0.954704i \(0.403828\pi\)
\(192\) 2.16425 0.156191
\(193\) 24.2141 1.74297 0.871485 0.490422i \(-0.163158\pi\)
0.871485 + 0.490422i \(0.163158\pi\)
\(194\) −11.6964 −0.839755
\(195\) −24.1892 −1.73222
\(196\) −6.53219 −0.466585
\(197\) 26.6570 1.89923 0.949616 0.313416i \(-0.101473\pi\)
0.949616 + 0.313416i \(0.101473\pi\)
\(198\) 1.68397 0.119674
\(199\) −7.06437 −0.500780 −0.250390 0.968145i \(-0.580559\pi\)
−0.250390 + 0.968145i \(0.580559\pi\)
\(200\) −12.3410 −0.872638
\(201\) 1.48028 0.104411
\(202\) −3.59262 −0.252776
\(203\) 1.91259 0.134237
\(204\) 7.28905 0.510336
\(205\) 29.2016 2.03953
\(206\) −4.46781 −0.311287
\(207\) −2.30357 −0.160109
\(208\) −2.68397 −0.186100
\(209\) 0 0
\(210\) 6.16425 0.425374
\(211\) 11.0644 0.761703 0.380851 0.924636i \(-0.375631\pi\)
0.380851 + 0.924636i \(0.375631\pi\)
\(212\) 10.3285 0.709364
\(213\) −7.64453 −0.523794
\(214\) 9.69643 0.662835
\(215\) 32.7338 2.23243
\(216\) 2.84822 0.193797
\(217\) 3.42837 0.232733
\(218\) 6.00000 0.406371
\(219\) 0.881673 0.0595779
\(220\) −4.16425 −0.280754
\(221\) −9.03944 −0.608059
\(222\) −25.3140 −1.69896
\(223\) 14.7359 0.986787 0.493394 0.869806i \(-0.335756\pi\)
0.493394 + 0.869806i \(0.335756\pi\)
\(224\) 0.683969 0.0456996
\(225\) 20.7818 1.38545
\(226\) 10.4323 0.693947
\(227\) −21.8002 −1.44693 −0.723467 0.690359i \(-0.757452\pi\)
−0.723467 + 0.690359i \(0.757452\pi\)
\(228\) 0 0
\(229\) 17.7483 1.17284 0.586422 0.810006i \(-0.300536\pi\)
0.586422 + 0.810006i \(0.300536\pi\)
\(230\) 5.69643 0.375612
\(231\) 1.48028 0.0973952
\(232\) 2.79631 0.183587
\(233\) −20.3534 −1.33340 −0.666699 0.745327i \(-0.732293\pi\)
−0.666699 + 0.745327i \(0.732293\pi\)
\(234\) 4.51972 0.295463
\(235\) 12.3285 0.804222
\(236\) 9.92112 0.645810
\(237\) 15.7753 1.02472
\(238\) 2.30357 0.149318
\(239\) 3.75687 0.243012 0.121506 0.992591i \(-0.461228\pi\)
0.121506 + 0.992591i \(0.461228\pi\)
\(240\) 9.01247 0.581752
\(241\) 9.14974 0.589386 0.294693 0.955592i \(-0.404783\pi\)
0.294693 + 0.955592i \(0.404783\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −15.7299 −1.00907
\(244\) 2.00000 0.128037
\(245\) −27.2016 −1.73785
\(246\) −15.1767 −0.967632
\(247\) 0 0
\(248\) 5.01247 0.318292
\(249\) −26.3804 −1.67179
\(250\) −30.5696 −1.93339
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −1.15178 −0.0725555
\(253\) 1.36794 0.0860015
\(254\) 15.2891 0.959321
\(255\) 30.3534 1.90081
\(256\) 1.00000 0.0625000
\(257\) −23.2102 −1.44781 −0.723905 0.689899i \(-0.757655\pi\)
−0.723905 + 0.689899i \(0.757655\pi\)
\(258\) −17.0125 −1.05915
\(259\) −8.00000 −0.497096
\(260\) −11.1767 −0.693150
\(261\) −4.70890 −0.291474
\(262\) −3.64453 −0.225160
\(263\) −22.2141 −1.36978 −0.684890 0.728646i \(-0.740150\pi\)
−0.684890 + 0.728646i \(0.740150\pi\)
\(264\) 2.16425 0.133200
\(265\) 43.0104 2.64211
\(266\) 0 0
\(267\) 36.1038 2.20952
\(268\) 0.683969 0.0417801
\(269\) 18.9855 1.15757 0.578783 0.815482i \(-0.303528\pi\)
0.578783 + 0.815482i \(0.303528\pi\)
\(270\) 11.8607 0.721818
\(271\) 20.7338 1.25949 0.629745 0.776802i \(-0.283159\pi\)
0.629745 + 0.776802i \(0.283159\pi\)
\(272\) 3.36794 0.204211
\(273\) 3.97302 0.240458
\(274\) −14.6840 −0.887091
\(275\) −12.3410 −0.744188
\(276\) −2.96056 −0.178205
\(277\) 21.8422 1.31237 0.656186 0.754599i \(-0.272169\pi\)
0.656186 + 0.754599i \(0.272169\pi\)
\(278\) 18.5966 1.11535
\(279\) −8.44084 −0.505340
\(280\) 2.84822 0.170214
\(281\) −0.164248 −0.00979821 −0.00489911 0.999988i \(-0.501559\pi\)
−0.00489911 + 0.999988i \(0.501559\pi\)
\(282\) −6.40738 −0.381554
\(283\) 4.30152 0.255699 0.127849 0.991794i \(-0.459193\pi\)
0.127849 + 0.991794i \(0.459193\pi\)
\(284\) −3.53219 −0.209597
\(285\) 0 0
\(286\) −2.68397 −0.158706
\(287\) −4.79631 −0.283117
\(288\) −1.68397 −0.0992288
\(289\) −5.65699 −0.332764
\(290\) 11.6445 0.683790
\(291\) 25.3140 1.48393
\(292\) 0.407381 0.0238402
\(293\) −32.5511 −1.90166 −0.950829 0.309717i \(-0.899766\pi\)
−0.950829 + 0.309717i \(0.899766\pi\)
\(294\) 14.1373 0.824502
\(295\) 41.3140 2.40539
\(296\) −11.6964 −0.679841
\(297\) 2.84822 0.165270
\(298\) −4.30357 −0.249299
\(299\) 3.67150 0.212329
\(300\) 26.7089 1.54204
\(301\) −5.37646 −0.309894
\(302\) −21.6964 −1.24849
\(303\) 7.77532 0.446680
\(304\) 0 0
\(305\) 8.32850 0.476888
\(306\) −5.67150 −0.324218
\(307\) −22.2496 −1.26985 −0.634926 0.772573i \(-0.718970\pi\)
−0.634926 + 0.772573i \(0.718970\pi\)
\(308\) 0.683969 0.0389728
\(309\) 9.66946 0.550076
\(310\) 20.8731 1.18552
\(311\) −20.9855 −1.18998 −0.594989 0.803734i \(-0.702844\pi\)
−0.594989 + 0.803734i \(0.702844\pi\)
\(312\) 5.80877 0.328857
\(313\) −11.4533 −0.647379 −0.323689 0.946163i \(-0.604923\pi\)
−0.323689 + 0.946163i \(0.604923\pi\)
\(314\) 10.9001 0.615130
\(315\) −4.79631 −0.270241
\(316\) 7.28905 0.410041
\(317\) −10.7109 −0.601587 −0.300793 0.953689i \(-0.597251\pi\)
−0.300793 + 0.953689i \(0.597251\pi\)
\(318\) −22.3534 −1.25352
\(319\) 2.79631 0.156563
\(320\) 4.16425 0.232789
\(321\) −20.9855 −1.17130
\(322\) −0.935628 −0.0521405
\(323\) 0 0
\(324\) −11.2162 −0.623120
\(325\) −33.1228 −1.83732
\(326\) −19.7214 −1.09227
\(327\) −12.9855 −0.718099
\(328\) −7.01247 −0.387199
\(329\) −2.02493 −0.111638
\(330\) 9.01247 0.496120
\(331\) 6.77138 0.372189 0.186094 0.982532i \(-0.440417\pi\)
0.186094 + 0.982532i \(0.440417\pi\)
\(332\) −12.1892 −0.668968
\(333\) 19.6964 1.07936
\(334\) 7.77532 0.425447
\(335\) 2.84822 0.155615
\(336\) −1.48028 −0.0807558
\(337\) −21.0374 −1.14598 −0.572990 0.819562i \(-0.694217\pi\)
−0.572990 + 0.819562i \(0.694217\pi\)
\(338\) 5.79631 0.315278
\(339\) −22.5781 −1.22627
\(340\) 14.0249 0.760609
\(341\) 5.01247 0.271440
\(342\) 0 0
\(343\) 9.25560 0.499755
\(344\) −7.86068 −0.423820
\(345\) −12.3285 −0.663744
\(346\) 15.7818 0.848435
\(347\) −29.9710 −1.60893 −0.804463 0.594003i \(-0.797547\pi\)
−0.804463 + 0.594003i \(0.797547\pi\)
\(348\) −6.05191 −0.324416
\(349\) −22.6570 −1.21280 −0.606400 0.795159i \(-0.707387\pi\)
−0.606400 + 0.795159i \(0.707387\pi\)
\(350\) 8.44084 0.451182
\(351\) 7.64453 0.408035
\(352\) 1.00000 0.0533002
\(353\) −1.18524 −0.0630839 −0.0315419 0.999502i \(-0.510042\pi\)
−0.0315419 + 0.999502i \(0.510042\pi\)
\(354\) −21.4718 −1.14121
\(355\) −14.7089 −0.780667
\(356\) 16.6819 0.884140
\(357\) −4.98549 −0.263860
\(358\) −7.97302 −0.421387
\(359\) 19.3659 1.02209 0.511046 0.859553i \(-0.329258\pi\)
0.511046 + 0.859553i \(0.329258\pi\)
\(360\) −7.01247 −0.369589
\(361\) 0 0
\(362\) −2.55318 −0.134192
\(363\) 2.16425 0.113594
\(364\) 1.83575 0.0962196
\(365\) 1.69643 0.0887954
\(366\) −4.32850 −0.226254
\(367\) −18.3036 −0.955438 −0.477719 0.878513i \(-0.658536\pi\)
−0.477719 + 0.878513i \(0.658536\pi\)
\(368\) −1.36794 −0.0713087
\(369\) 11.8088 0.614740
\(370\) −48.7069 −2.53215
\(371\) −7.06437 −0.366764
\(372\) −10.8482 −0.562454
\(373\) −6.46781 −0.334891 −0.167445 0.985881i \(-0.553552\pi\)
−0.167445 + 0.985881i \(0.553552\pi\)
\(374\) 3.36794 0.174152
\(375\) 66.1602 3.41650
\(376\) −2.96056 −0.152679
\(377\) 7.50521 0.386538
\(378\) −1.94809 −0.100199
\(379\) −31.9191 −1.63957 −0.819786 0.572670i \(-0.805908\pi\)
−0.819786 + 0.572670i \(0.805908\pi\)
\(380\) 0 0
\(381\) −33.0893 −1.69522
\(382\) −8.22468 −0.420811
\(383\) −5.66946 −0.289696 −0.144848 0.989454i \(-0.546269\pi\)
−0.144848 + 0.989454i \(0.546269\pi\)
\(384\) −2.16425 −0.110444
\(385\) 2.84822 0.145159
\(386\) −24.2141 −1.23247
\(387\) 13.2371 0.672882
\(388\) 11.6964 0.593796
\(389\) 20.1642 1.02237 0.511184 0.859471i \(-0.329207\pi\)
0.511184 + 0.859471i \(0.329207\pi\)
\(390\) 24.1892 1.22487
\(391\) −4.60713 −0.232993
\(392\) 6.53219 0.329925
\(393\) 7.88766 0.397880
\(394\) −26.6570 −1.34296
\(395\) 30.3534 1.52725
\(396\) −1.68397 −0.0846226
\(397\) −15.8272 −0.794346 −0.397173 0.917744i \(-0.630009\pi\)
−0.397173 + 0.917744i \(0.630009\pi\)
\(398\) 7.06437 0.354105
\(399\) 0 0
\(400\) 12.3410 0.617048
\(401\) −9.06437 −0.452653 −0.226327 0.974051i \(-0.572672\pi\)
−0.226327 + 0.974051i \(0.572672\pi\)
\(402\) −1.48028 −0.0738296
\(403\) 13.4533 0.670157
\(404\) 3.59262 0.178739
\(405\) −46.7069 −2.32088
\(406\) −1.91259 −0.0949202
\(407\) −11.6964 −0.579771
\(408\) −7.28905 −0.360862
\(409\) −2.90012 −0.143402 −0.0717010 0.997426i \(-0.522843\pi\)
−0.0717010 + 0.997426i \(0.522843\pi\)
\(410\) −29.2016 −1.44217
\(411\) 31.7797 1.56758
\(412\) 4.46781 0.220113
\(413\) −6.78574 −0.333904
\(414\) 2.30357 0.113214
\(415\) −50.7588 −2.49165
\(416\) 2.68397 0.131592
\(417\) −40.2476 −1.97093
\(418\) 0 0
\(419\) 19.7753 0.966088 0.483044 0.875596i \(-0.339531\pi\)
0.483044 + 0.875596i \(0.339531\pi\)
\(420\) −6.16425 −0.300785
\(421\) −14.9855 −0.730348 −0.365174 0.930939i \(-0.618991\pi\)
−0.365174 + 0.930939i \(0.618991\pi\)
\(422\) −11.0644 −0.538605
\(423\) 4.98549 0.242403
\(424\) −10.3285 −0.501596
\(425\) 41.5636 2.01613
\(426\) 7.64453 0.370379
\(427\) −1.36794 −0.0661992
\(428\) −9.69643 −0.468695
\(429\) 5.80877 0.280450
\(430\) −32.7338 −1.57857
\(431\) −39.4427 −1.89989 −0.949945 0.312418i \(-0.898861\pi\)
−0.949945 + 0.312418i \(0.898861\pi\)
\(432\) −2.84822 −0.137035
\(433\) 1.23919 0.0595518 0.0297759 0.999557i \(-0.490521\pi\)
0.0297759 + 0.999557i \(0.490521\pi\)
\(434\) −3.42837 −0.164567
\(435\) −25.2016 −1.20833
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) −0.881673 −0.0421280
\(439\) 11.1852 0.533842 0.266921 0.963718i \(-0.413994\pi\)
0.266921 + 0.963718i \(0.413994\pi\)
\(440\) 4.16425 0.198523
\(441\) −11.0000 −0.523810
\(442\) 9.03944 0.429962
\(443\) 23.8422 1.13278 0.566389 0.824138i \(-0.308340\pi\)
0.566389 + 0.824138i \(0.308340\pi\)
\(444\) 25.3140 1.20135
\(445\) 69.4677 3.29308
\(446\) −14.7359 −0.697764
\(447\) 9.31398 0.440536
\(448\) −0.683969 −0.0323145
\(449\) −20.9066 −0.986644 −0.493322 0.869847i \(-0.664217\pi\)
−0.493322 + 0.869847i \(0.664217\pi\)
\(450\) −20.7818 −0.979663
\(451\) −7.01247 −0.330204
\(452\) −10.4323 −0.490695
\(453\) 46.9565 2.20621
\(454\) 21.8002 1.02314
\(455\) 7.64453 0.358381
\(456\) 0 0
\(457\) −4.18270 −0.195658 −0.0978292 0.995203i \(-0.531190\pi\)
−0.0978292 + 0.995203i \(0.531190\pi\)
\(458\) −17.7483 −0.829326
\(459\) −9.59262 −0.447745
\(460\) −5.69643 −0.265598
\(461\) −31.5387 −1.46890 −0.734451 0.678662i \(-0.762560\pi\)
−0.734451 + 0.678662i \(0.762560\pi\)
\(462\) −1.48028 −0.0688688
\(463\) −4.60713 −0.214112 −0.107056 0.994253i \(-0.534142\pi\)
−0.107056 + 0.994253i \(0.534142\pi\)
\(464\) −2.79631 −0.129815
\(465\) −45.1747 −2.09492
\(466\) 20.3534 0.942854
\(467\) −3.94605 −0.182601 −0.0913006 0.995823i \(-0.529102\pi\)
−0.0913006 + 0.995823i \(0.529102\pi\)
\(468\) −4.51972 −0.208924
\(469\) −0.467814 −0.0216016
\(470\) −12.3285 −0.568671
\(471\) −23.5906 −1.08700
\(472\) −9.92112 −0.456656
\(473\) −7.86068 −0.361435
\(474\) −15.7753 −0.724584
\(475\) 0 0
\(476\) −2.30357 −0.105584
\(477\) 17.3929 0.796365
\(478\) −3.75687 −0.171835
\(479\) −16.6840 −0.762310 −0.381155 0.924511i \(-0.624474\pi\)
−0.381155 + 0.924511i \(0.624474\pi\)
\(480\) −9.01247 −0.411361
\(481\) −31.3929 −1.43139
\(482\) −9.14974 −0.416759
\(483\) 2.02493 0.0921375
\(484\) 1.00000 0.0454545
\(485\) 48.7069 2.21166
\(486\) 15.7299 0.713522
\(487\) 14.4388 0.654284 0.327142 0.944975i \(-0.393914\pi\)
0.327142 + 0.944975i \(0.393914\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 42.6819 1.93014
\(490\) 27.2016 1.22884
\(491\) 6.38040 0.287944 0.143972 0.989582i \(-0.454013\pi\)
0.143972 + 0.989582i \(0.454013\pi\)
\(492\) 15.1767 0.684219
\(493\) −9.41780 −0.424156
\(494\) 0 0
\(495\) −7.01247 −0.315187
\(496\) −5.01247 −0.225066
\(497\) 2.41591 0.108368
\(498\) 26.3804 1.18213
\(499\) −31.8252 −1.42469 −0.712345 0.701829i \(-0.752367\pi\)
−0.712345 + 0.701829i \(0.752367\pi\)
\(500\) 30.5696 1.36711
\(501\) −16.8277 −0.751807
\(502\) 12.0000 0.535586
\(503\) 24.5716 1.09559 0.547797 0.836611i \(-0.315466\pi\)
0.547797 + 0.836611i \(0.315466\pi\)
\(504\) 1.15178 0.0513045
\(505\) 14.9606 0.665736
\(506\) −1.36794 −0.0608123
\(507\) −12.5447 −0.557128
\(508\) −15.2891 −0.678342
\(509\) 5.56769 0.246783 0.123392 0.992358i \(-0.460623\pi\)
0.123392 + 0.992358i \(0.460623\pi\)
\(510\) −30.3534 −1.34407
\(511\) −0.278636 −0.0123261
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 23.2102 1.02376
\(515\) 18.6051 0.819838
\(516\) 17.0125 0.748932
\(517\) −2.96056 −0.130205
\(518\) 8.00000 0.351500
\(519\) −34.1557 −1.49927
\(520\) 11.1767 0.490131
\(521\) 10.8148 0.473803 0.236902 0.971534i \(-0.423868\pi\)
0.236902 + 0.971534i \(0.423868\pi\)
\(522\) 4.70890 0.206103
\(523\) 18.4074 0.804899 0.402449 0.915442i \(-0.368159\pi\)
0.402449 + 0.915442i \(0.368159\pi\)
\(524\) 3.64453 0.159212
\(525\) −18.2681 −0.797284
\(526\) 22.2141 0.968581
\(527\) −16.8817 −0.735377
\(528\) −2.16425 −0.0941868
\(529\) −21.1287 −0.918641
\(530\) −43.0104 −1.86825
\(531\) 16.7069 0.725016
\(532\) 0 0
\(533\) −18.8212 −0.815238
\(534\) −36.1038 −1.56236
\(535\) −40.3784 −1.74571
\(536\) −0.683969 −0.0295430
\(537\) 17.2556 0.744634
\(538\) −18.9855 −0.818522
\(539\) 6.53219 0.281361
\(540\) −11.8607 −0.510402
\(541\) 33.0644 1.42155 0.710774 0.703420i \(-0.248345\pi\)
0.710774 + 0.703420i \(0.248345\pi\)
\(542\) −20.7338 −0.890594
\(543\) 5.52571 0.237131
\(544\) −3.36794 −0.144399
\(545\) −24.9855 −1.07026
\(546\) −3.97302 −0.170030
\(547\) −34.1287 −1.45924 −0.729620 0.683853i \(-0.760303\pi\)
−0.729620 + 0.683853i \(0.760303\pi\)
\(548\) 14.6840 0.627268
\(549\) 3.36794 0.143740
\(550\) 12.3410 0.526220
\(551\) 0 0
\(552\) 2.96056 0.126010
\(553\) −4.98549 −0.212004
\(554\) −21.8422 −0.927987
\(555\) 105.414 4.47456
\(556\) −18.5966 −0.788670
\(557\) 20.5112 0.869087 0.434544 0.900651i \(-0.356910\pi\)
0.434544 + 0.900651i \(0.356910\pi\)
\(558\) 8.44084 0.357329
\(559\) −21.0978 −0.892343
\(560\) −2.84822 −0.120359
\(561\) −7.28905 −0.307744
\(562\) 0.164248 0.00692838
\(563\) 27.0104 1.13835 0.569177 0.822215i \(-0.307262\pi\)
0.569177 + 0.822215i \(0.307262\pi\)
\(564\) 6.40738 0.269799
\(565\) −43.4427 −1.82765
\(566\) −4.30152 −0.180806
\(567\) 7.67150 0.322173
\(568\) 3.53219 0.148207
\(569\) −7.56564 −0.317168 −0.158584 0.987345i \(-0.550693\pi\)
−0.158584 + 0.987345i \(0.550693\pi\)
\(570\) 0 0
\(571\) 42.2226 1.76696 0.883481 0.468467i \(-0.155193\pi\)
0.883481 + 0.468467i \(0.155193\pi\)
\(572\) 2.68397 0.112222
\(573\) 17.8002 0.743616
\(574\) 4.79631 0.200194
\(575\) −16.8817 −0.704014
\(576\) 1.68397 0.0701654
\(577\) −25.9191 −1.07902 −0.539512 0.841978i \(-0.681391\pi\)
−0.539512 + 0.841978i \(0.681391\pi\)
\(578\) 5.65699 0.235300
\(579\) 52.4053 2.17789
\(580\) −11.6445 −0.483513
\(581\) 8.33702 0.345878
\(582\) −25.3140 −1.04930
\(583\) −10.3285 −0.427763
\(584\) −0.407381 −0.0168575
\(585\) −18.8212 −0.778162
\(586\) 32.5511 1.34467
\(587\) 20.6570 0.852605 0.426303 0.904581i \(-0.359816\pi\)
0.426303 + 0.904581i \(0.359816\pi\)
\(588\) −14.1373 −0.583011
\(589\) 0 0
\(590\) −41.3140 −1.70087
\(591\) 57.6923 2.37315
\(592\) 11.6964 0.480720
\(593\) −28.1287 −1.15511 −0.577555 0.816352i \(-0.695993\pi\)
−0.577555 + 0.816352i \(0.695993\pi\)
\(594\) −2.84822 −0.116864
\(595\) −9.59262 −0.393259
\(596\) 4.30357 0.176281
\(597\) −15.2891 −0.625739
\(598\) −3.67150 −0.150139
\(599\) 26.3179 1.07532 0.537661 0.843161i \(-0.319308\pi\)
0.537661 + 0.843161i \(0.319308\pi\)
\(600\) −26.7089 −1.09039
\(601\) −38.4053 −1.56659 −0.783293 0.621653i \(-0.786462\pi\)
−0.783293 + 0.621653i \(0.786462\pi\)
\(602\) 5.37646 0.219128
\(603\) 1.15178 0.0469042
\(604\) 21.6964 0.882815
\(605\) 4.16425 0.169301
\(606\) −7.77532 −0.315851
\(607\) 33.2600 1.34998 0.674991 0.737826i \(-0.264147\pi\)
0.674991 + 0.737826i \(0.264147\pi\)
\(608\) 0 0
\(609\) 4.13932 0.167734
\(610\) −8.32850 −0.337211
\(611\) −7.94605 −0.321463
\(612\) 5.67150 0.229257
\(613\) 17.5506 0.708864 0.354432 0.935082i \(-0.384674\pi\)
0.354432 + 0.935082i \(0.384674\pi\)
\(614\) 22.2496 0.897921
\(615\) 63.1996 2.54845
\(616\) −0.683969 −0.0275579
\(617\) 14.6840 0.591154 0.295577 0.955319i \(-0.404488\pi\)
0.295577 + 0.955319i \(0.404488\pi\)
\(618\) −9.66946 −0.388963
\(619\) 11.1183 0.446883 0.223442 0.974717i \(-0.428271\pi\)
0.223442 + 0.974717i \(0.428271\pi\)
\(620\) −20.8731 −0.838286
\(621\) 3.89619 0.156349
\(622\) 20.9855 0.841441
\(623\) −11.4099 −0.457129
\(624\) −5.80877 −0.232537
\(625\) 65.5945 2.62378
\(626\) 11.4533 0.457766
\(627\) 0 0
\(628\) −10.9001 −0.434962
\(629\) 39.3929 1.57070
\(630\) 4.79631 0.191090
\(631\) −46.0748 −1.83421 −0.917104 0.398648i \(-0.869480\pi\)
−0.917104 + 0.398648i \(0.869480\pi\)
\(632\) −7.28905 −0.289943
\(633\) 23.9460 0.951770
\(634\) 10.7109 0.425386
\(635\) −63.6674 −2.52656
\(636\) 22.3534 0.886371
\(637\) 17.5322 0.694651
\(638\) −2.79631 −0.110707
\(639\) −5.94809 −0.235303
\(640\) −4.16425 −0.164606
\(641\) 11.3181 0.447037 0.223519 0.974700i \(-0.428246\pi\)
0.223519 + 0.974700i \(0.428246\pi\)
\(642\) 20.9855 0.828231
\(643\) −30.6280 −1.20785 −0.603925 0.797041i \(-0.706397\pi\)
−0.603925 + 0.797041i \(0.706397\pi\)
\(644\) 0.935628 0.0368689
\(645\) 70.8441 2.78948
\(646\) 0 0
\(647\) −31.2720 −1.22943 −0.614715 0.788750i \(-0.710729\pi\)
−0.614715 + 0.788750i \(0.710729\pi\)
\(648\) 11.2162 0.440612
\(649\) −9.92112 −0.389438
\(650\) 33.1228 1.29918
\(651\) 7.41985 0.290807
\(652\) 19.7214 0.772348
\(653\) −2.09389 −0.0819402 −0.0409701 0.999160i \(-0.513045\pi\)
−0.0409701 + 0.999160i \(0.513045\pi\)
\(654\) 12.9855 0.507773
\(655\) 15.1767 0.593003
\(656\) 7.01247 0.273791
\(657\) 0.686016 0.0267641
\(658\) 2.02493 0.0789400
\(659\) −24.7608 −0.964544 −0.482272 0.876022i \(-0.660188\pi\)
−0.482272 + 0.876022i \(0.660188\pi\)
\(660\) −9.01247 −0.350810
\(661\) −36.4033 −1.41592 −0.707962 0.706251i \(-0.750385\pi\)
−0.707962 + 0.706251i \(0.750385\pi\)
\(662\) −6.77138 −0.263177
\(663\) −19.5636 −0.759787
\(664\) 12.1892 0.473032
\(665\) 0 0
\(666\) −19.6964 −0.763221
\(667\) 3.82518 0.148112
\(668\) −7.77532 −0.300836
\(669\) 31.8921 1.23302
\(670\) −2.84822 −0.110036
\(671\) −2.00000 −0.0772091
\(672\) 1.48028 0.0571030
\(673\) 16.1728 0.623415 0.311707 0.950178i \(-0.399099\pi\)
0.311707 + 0.950178i \(0.399099\pi\)
\(674\) 21.0374 0.810330
\(675\) −35.1497 −1.35291
\(676\) −5.79631 −0.222935
\(677\) −5.32456 −0.204639 −0.102320 0.994752i \(-0.532626\pi\)
−0.102320 + 0.994752i \(0.532626\pi\)
\(678\) 22.5781 0.867107
\(679\) −8.00000 −0.307012
\(680\) −14.0249 −0.537832
\(681\) −47.1811 −1.80799
\(682\) −5.01247 −0.191937
\(683\) −49.1562 −1.88091 −0.940455 0.339918i \(-0.889601\pi\)
−0.940455 + 0.339918i \(0.889601\pi\)
\(684\) 0 0
\(685\) 61.1477 2.33633
\(686\) −9.25560 −0.353380
\(687\) 38.4118 1.46550
\(688\) 7.86068 0.299686
\(689\) −27.7214 −1.05610
\(690\) 12.3285 0.469338
\(691\) −6.51120 −0.247698 −0.123849 0.992301i \(-0.539524\pi\)
−0.123849 + 0.992301i \(0.539524\pi\)
\(692\) −15.7818 −0.599934
\(693\) 1.15178 0.0437526
\(694\) 29.9710 1.13768
\(695\) −77.4407 −2.93749
\(696\) 6.05191 0.229397
\(697\) 23.6175 0.894578
\(698\) 22.6570 0.857580
\(699\) −44.0499 −1.66612
\(700\) −8.44084 −0.319034
\(701\) −5.77532 −0.218131 −0.109065 0.994035i \(-0.534786\pi\)
−0.109065 + 0.994035i \(0.534786\pi\)
\(702\) −7.64453 −0.288524
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 26.6819 1.00490
\(706\) 1.18524 0.0446070
\(707\) −2.45724 −0.0924140
\(708\) 21.4718 0.806958
\(709\) −35.6155 −1.33757 −0.668784 0.743457i \(-0.733185\pi\)
−0.668784 + 0.743457i \(0.733185\pi\)
\(710\) 14.7089 0.552015
\(711\) 12.2745 0.460331
\(712\) −16.6819 −0.625181
\(713\) 6.85674 0.256787
\(714\) 4.98549 0.186577
\(715\) 11.1767 0.417985
\(716\) 7.97302 0.297966
\(717\) 8.13079 0.303650
\(718\) −19.3659 −0.722729
\(719\) −8.93563 −0.333243 −0.166621 0.986021i \(-0.553286\pi\)
−0.166621 + 0.986021i \(0.553286\pi\)
\(720\) 7.01247 0.261339
\(721\) −3.05585 −0.113806
\(722\) 0 0
\(723\) 19.8023 0.736455
\(724\) 2.55318 0.0948881
\(725\) −34.5091 −1.28164
\(726\) −2.16425 −0.0803228
\(727\) −1.42189 −0.0527351 −0.0263675 0.999652i \(-0.508394\pi\)
−0.0263675 + 0.999652i \(0.508394\pi\)
\(728\) −1.83575 −0.0680375
\(729\) −0.394916 −0.0146265
\(730\) −1.69643 −0.0627878
\(731\) 26.4743 0.979187
\(732\) 4.32850 0.159986
\(733\) −46.7567 −1.72700 −0.863499 0.504350i \(-0.831732\pi\)
−0.863499 + 0.504350i \(0.831732\pi\)
\(734\) 18.3036 0.675597
\(735\) −58.8711 −2.17149
\(736\) 1.36794 0.0504229
\(737\) −0.683969 −0.0251943
\(738\) −11.8088 −0.434687
\(739\) −34.4886 −1.26869 −0.634343 0.773052i \(-0.718729\pi\)
−0.634343 + 0.773052i \(0.718729\pi\)
\(740\) 48.7069 1.79050
\(741\) 0 0
\(742\) 7.06437 0.259341
\(743\) 10.9606 0.402104 0.201052 0.979581i \(-0.435564\pi\)
0.201052 + 0.979581i \(0.435564\pi\)
\(744\) 10.8482 0.397715
\(745\) 17.9211 0.656579
\(746\) 6.46781 0.236803
\(747\) −20.5262 −0.751014
\(748\) −3.36794 −0.123144
\(749\) 6.63206 0.242330
\(750\) −66.1602 −2.41583
\(751\) −13.3140 −0.485834 −0.242917 0.970047i \(-0.578104\pi\)
−0.242917 + 0.970047i \(0.578104\pi\)
\(752\) 2.96056 0.107960
\(753\) −25.9710 −0.946435
\(754\) −7.50521 −0.273324
\(755\) 90.3493 3.28815
\(756\) 1.94809 0.0708514
\(757\) −8.54260 −0.310486 −0.155243 0.987876i \(-0.549616\pi\)
−0.155243 + 0.987876i \(0.549616\pi\)
\(758\) 31.9191 1.15935
\(759\) 2.96056 0.107461
\(760\) 0 0
\(761\) 32.7318 1.18653 0.593263 0.805009i \(-0.297839\pi\)
0.593263 + 0.805009i \(0.297839\pi\)
\(762\) 33.0893 1.19870
\(763\) 4.10381 0.148568
\(764\) 8.22468 0.297559
\(765\) 23.6175 0.853894
\(766\) 5.66946 0.204846
\(767\) −26.6280 −0.961480
\(768\) 2.16425 0.0780956
\(769\) 50.5491 1.82285 0.911423 0.411470i \(-0.134985\pi\)
0.911423 + 0.411470i \(0.134985\pi\)
\(770\) −2.84822 −0.102643
\(771\) −50.2326 −1.80908
\(772\) 24.2141 0.871485
\(773\) 40.7359 1.46517 0.732584 0.680677i \(-0.238314\pi\)
0.732584 + 0.680677i \(0.238314\pi\)
\(774\) −13.2371 −0.475799
\(775\) −61.8586 −2.22203
\(776\) −11.6964 −0.419878
\(777\) −17.3140 −0.621136
\(778\) −20.1642 −0.722923
\(779\) 0 0
\(780\) −24.1892 −0.866112
\(781\) 3.53219 0.126392
\(782\) 4.60713 0.164751
\(783\) 7.96450 0.284628
\(784\) −6.53219 −0.233292
\(785\) −45.3908 −1.62007
\(786\) −7.88766 −0.281343
\(787\) −8.53613 −0.304280 −0.152140 0.988359i \(-0.548616\pi\)
−0.152140 + 0.988359i \(0.548616\pi\)
\(788\) 26.6570 0.949616
\(789\) −48.0768 −1.71158
\(790\) −30.3534 −1.07993
\(791\) 7.13538 0.253705
\(792\) 1.68397 0.0598372
\(793\) −5.36794 −0.190621
\(794\) 15.8272 0.561687
\(795\) 93.0852 3.30139
\(796\) −7.06437 −0.250390
\(797\) 16.0249 0.567632 0.283816 0.958879i \(-0.408399\pi\)
0.283816 + 0.958879i \(0.408399\pi\)
\(798\) 0 0
\(799\) 9.97098 0.352748
\(800\) −12.3410 −0.436319
\(801\) 28.0918 0.992576
\(802\) 9.06437 0.320074
\(803\) −0.407381 −0.0143762
\(804\) 1.48028 0.0522054
\(805\) 3.89619 0.137322
\(806\) −13.4533 −0.473872
\(807\) 41.0893 1.44641
\(808\) −3.59262 −0.126388
\(809\) −20.1458 −0.708288 −0.354144 0.935191i \(-0.615228\pi\)
−0.354144 + 0.935191i \(0.615228\pi\)
\(810\) 46.7069 1.64111
\(811\) −8.27454 −0.290558 −0.145279 0.989391i \(-0.546408\pi\)
−0.145279 + 0.989391i \(0.546408\pi\)
\(812\) 1.91259 0.0671187
\(813\) 44.8731 1.57377
\(814\) 11.6964 0.409960
\(815\) 82.1247 2.87670
\(816\) 7.28905 0.255168
\(817\) 0 0
\(818\) 2.90012 0.101400
\(819\) 3.09135 0.108021
\(820\) 29.2016 1.01977
\(821\) −21.5137 −0.750835 −0.375417 0.926856i \(-0.622501\pi\)
−0.375417 + 0.926856i \(0.622501\pi\)
\(822\) −31.7797 −1.10845
\(823\) −5.26412 −0.183496 −0.0917479 0.995782i \(-0.529245\pi\)
−0.0917479 + 0.995782i \(0.529245\pi\)
\(824\) −4.46781 −0.155644
\(825\) −26.7089 −0.929885
\(826\) 6.78574 0.236106
\(827\) −16.7069 −0.580954 −0.290477 0.956882i \(-0.593814\pi\)
−0.290477 + 0.956882i \(0.593814\pi\)
\(828\) −2.30357 −0.0800544
\(829\) 5.55064 0.192782 0.0963908 0.995344i \(-0.469270\pi\)
0.0963908 + 0.995344i \(0.469270\pi\)
\(830\) 50.7588 1.76186
\(831\) 47.2720 1.63985
\(832\) −2.68397 −0.0930499
\(833\) −22.0000 −0.762255
\(834\) 40.2476 1.39366
\(835\) −32.3784 −1.12050
\(836\) 0 0
\(837\) 14.2766 0.493471
\(838\) −19.7753 −0.683127
\(839\) −15.3744 −0.530784 −0.265392 0.964141i \(-0.585501\pi\)
−0.265392 + 0.964141i \(0.585501\pi\)
\(840\) 6.16425 0.212687
\(841\) −21.1807 −0.730367
\(842\) 14.9855 0.516434
\(843\) −0.355473 −0.0122431
\(844\) 11.0644 0.380851
\(845\) −24.1373 −0.830347
\(846\) −4.98549 −0.171405
\(847\) −0.683969 −0.0235015
\(848\) 10.3285 0.354682
\(849\) 9.30955 0.319503
\(850\) −41.5636 −1.42562
\(851\) −16.0000 −0.548473
\(852\) −7.64453 −0.261897
\(853\) 24.8028 0.849231 0.424616 0.905374i \(-0.360409\pi\)
0.424616 + 0.905374i \(0.360409\pi\)
\(854\) 1.36794 0.0468099
\(855\) 0 0
\(856\) 9.69643 0.331417
\(857\) 45.6445 1.55919 0.779594 0.626286i \(-0.215426\pi\)
0.779594 + 0.626286i \(0.215426\pi\)
\(858\) −5.80877 −0.198308
\(859\) 54.8356 1.87097 0.935483 0.353371i \(-0.114965\pi\)
0.935483 + 0.353371i \(0.114965\pi\)
\(860\) 32.7338 1.11621
\(861\) −10.3804 −0.353763
\(862\) 39.4427 1.34342
\(863\) −45.8771 −1.56167 −0.780837 0.624735i \(-0.785207\pi\)
−0.780837 + 0.624735i \(0.785207\pi\)
\(864\) 2.84822 0.0968983
\(865\) −65.7193 −2.23452
\(866\) −1.23919 −0.0421095
\(867\) −12.2431 −0.415799
\(868\) 3.42837 0.116367
\(869\) −7.28905 −0.247264
\(870\) 25.2016 0.854416
\(871\) −1.83575 −0.0622021
\(872\) 6.00000 0.203186
\(873\) 19.6964 0.666623
\(874\) 0 0
\(875\) −20.9087 −0.706841
\(876\) 0.881673 0.0297890
\(877\) 15.6609 0.528832 0.264416 0.964409i \(-0.414821\pi\)
0.264416 + 0.964409i \(0.414821\pi\)
\(878\) −11.1852 −0.377484
\(879\) −70.4487 −2.37618
\(880\) −4.16425 −0.140377
\(881\) −3.39492 −0.114378 −0.0571888 0.998363i \(-0.518214\pi\)
−0.0571888 + 0.998363i \(0.518214\pi\)
\(882\) 11.0000 0.370389
\(883\) −33.6425 −1.13216 −0.566080 0.824350i \(-0.691541\pi\)
−0.566080 + 0.824350i \(0.691541\pi\)
\(884\) −9.03944 −0.304029
\(885\) 89.4137 3.00561
\(886\) −23.8422 −0.800995
\(887\) 22.1287 0.743011 0.371505 0.928431i \(-0.378842\pi\)
0.371505 + 0.928431i \(0.378842\pi\)
\(888\) −25.3140 −0.849482
\(889\) 10.4572 0.350725
\(890\) −69.4677 −2.32856
\(891\) 11.2162 0.375755
\(892\) 14.7359 0.493394
\(893\) 0 0
\(894\) −9.31398 −0.311506
\(895\) 33.2016 1.10981
\(896\) 0.683969 0.0228498
\(897\) 7.94605 0.265311
\(898\) 20.9066 0.697662
\(899\) 14.0164 0.467473
\(900\) 20.7818 0.692727
\(901\) 34.7857 1.15888
\(902\) 7.01247 0.233490
\(903\) −11.6360 −0.387222
\(904\) 10.4323 0.346973
\(905\) 10.6321 0.353422
\(906\) −46.9565 −1.56002
\(907\) 21.7633 0.722640 0.361320 0.932442i \(-0.382326\pi\)
0.361320 + 0.932442i \(0.382326\pi\)
\(908\) −21.8002 −0.723467
\(909\) 6.04986 0.200661
\(910\) −7.64453 −0.253414
\(911\) −46.8356 −1.55173 −0.775866 0.630897i \(-0.782687\pi\)
−0.775866 + 0.630897i \(0.782687\pi\)
\(912\) 0 0
\(913\) 12.1892 0.403403
\(914\) 4.18270 0.138351
\(915\) 18.0249 0.595886
\(916\) 17.7483 0.586422
\(917\) −2.49274 −0.0823177
\(918\) 9.59262 0.316604
\(919\) −42.0563 −1.38731 −0.693655 0.720307i \(-0.744001\pi\)
−0.693655 + 0.720307i \(0.744001\pi\)
\(920\) 5.69643 0.187806
\(921\) −48.1537 −1.58672
\(922\) 31.5387 1.03867
\(923\) 9.48028 0.312047
\(924\) 1.48028 0.0486976
\(925\) 144.345 4.74604
\(926\) 4.60713 0.151400
\(927\) 7.52366 0.247109
\(928\) 2.79631 0.0917934
\(929\) 36.1018 1.18446 0.592230 0.805769i \(-0.298248\pi\)
0.592230 + 0.805769i \(0.298248\pi\)
\(930\) 45.1747 1.48134
\(931\) 0 0
\(932\) −20.3534 −0.666699
\(933\) −45.4178 −1.48691
\(934\) 3.94605 0.129119
\(935\) −14.0249 −0.458664
\(936\) 4.51972 0.147732
\(937\) 47.3638 1.54731 0.773655 0.633607i \(-0.218427\pi\)
0.773655 + 0.633607i \(0.218427\pi\)
\(938\) 0.467814 0.0152747
\(939\) −24.7878 −0.808919
\(940\) 12.3285 0.402111
\(941\) 9.34301 0.304573 0.152287 0.988336i \(-0.451336\pi\)
0.152287 + 0.988336i \(0.451336\pi\)
\(942\) 23.5906 0.768622
\(943\) −9.59262 −0.312379
\(944\) 9.92112 0.322905
\(945\) 8.11234 0.263894
\(946\) 7.86068 0.255573
\(947\) 22.2496 0.723015 0.361508 0.932369i \(-0.382262\pi\)
0.361508 + 0.932369i \(0.382262\pi\)
\(948\) 15.7753 0.512359
\(949\) −1.09340 −0.0354932
\(950\) 0 0
\(951\) −23.1811 −0.751700
\(952\) 2.30357 0.0746590
\(953\) 31.9710 1.03564 0.517821 0.855489i \(-0.326743\pi\)
0.517821 + 0.855489i \(0.326743\pi\)
\(954\) −17.3929 −0.563115
\(955\) 34.2496 1.10829
\(956\) 3.75687 0.121506
\(957\) 6.05191 0.195630
\(958\) 16.6840 0.539035
\(959\) −10.0434 −0.324318
\(960\) 9.01247 0.290876
\(961\) −5.87519 −0.189522
\(962\) 31.3929 1.01215
\(963\) −16.3285 −0.526178
\(964\) 9.14974 0.294693
\(965\) 100.834 3.24595
\(966\) −2.02493 −0.0651511
\(967\) 5.97098 0.192014 0.0960068 0.995381i \(-0.469393\pi\)
0.0960068 + 0.995381i \(0.469393\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −48.7069 −1.56388
\(971\) 19.0999 0.612944 0.306472 0.951880i \(-0.400851\pi\)
0.306472 + 0.951880i \(0.400851\pi\)
\(972\) −15.7299 −0.504536
\(973\) 12.7195 0.407768
\(974\) −14.4388 −0.462649
\(975\) −71.6859 −2.29578
\(976\) 2.00000 0.0640184
\(977\) 10.2745 0.328712 0.164356 0.986401i \(-0.447445\pi\)
0.164356 + 0.986401i \(0.447445\pi\)
\(978\) −42.6819 −1.36482
\(979\) −16.6819 −0.533157
\(980\) −27.2016 −0.868925
\(981\) −10.1038 −0.322590
\(982\) −6.38040 −0.203607
\(983\) −33.0084 −1.05280 −0.526402 0.850236i \(-0.676459\pi\)
−0.526402 + 0.850236i \(0.676459\pi\)
\(984\) −15.1767 −0.483816
\(985\) 111.006 3.53696
\(986\) 9.41780 0.299924
\(987\) −4.38245 −0.139495
\(988\) 0 0
\(989\) −10.7529 −0.341923
\(990\) 7.01247 0.222871
\(991\) −1.56564 −0.0497343 −0.0248671 0.999691i \(-0.507916\pi\)
−0.0248671 + 0.999691i \(0.507916\pi\)
\(992\) 5.01247 0.159146
\(993\) 14.6549 0.465061
\(994\) −2.41591 −0.0766279
\(995\) −29.4178 −0.932607
\(996\) −26.3804 −0.835895
\(997\) 46.5361 1.47381 0.736907 0.675994i \(-0.236286\pi\)
0.736907 + 0.675994i \(0.236286\pi\)
\(998\) 31.8252 1.00741
\(999\) −33.3140 −1.05401
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 7942.2.a.bc.1.3 3
19.18 odd 2 418.2.a.h.1.1 3
57.56 even 2 3762.2.a.bd.1.1 3
76.75 even 2 3344.2.a.p.1.3 3
209.208 even 2 4598.2.a.bm.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.h.1.1 3 19.18 odd 2
3344.2.a.p.1.3 3 76.75 even 2
3762.2.a.bd.1.1 3 57.56 even 2
4598.2.a.bm.1.1 3 209.208 even 2
7942.2.a.bc.1.3 3 1.1 even 1 trivial