Properties

Label 722.2.a.n.1.4
Level $722$
Weight $2$
Character 722.1
Self dual yes
Analytic conductor $5.765$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \(x^{4} - 5 x^{2} + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.90211\) of defining polynomial
Character \(\chi\) \(=\) 722.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.79360 q^{3} +1.00000 q^{4} -2.34458 q^{5} +2.79360 q^{6} -1.28408 q^{7} +1.00000 q^{8} +4.80423 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.79360 q^{3} +1.00000 q^{4} -2.34458 q^{5} +2.79360 q^{6} -1.28408 q^{7} +1.00000 q^{8} +4.80423 q^{9} -2.34458 q^{10} +5.75621 q^{11} +2.79360 q^{12} +0.304282 q^{13} -1.28408 q^{14} -6.54982 q^{15} +1.00000 q^{16} +4.18619 q^{17} +4.80423 q^{18} -2.34458 q^{20} -3.58721 q^{21} +5.75621 q^{22} -6.47684 q^{23} +2.79360 q^{24} +0.497039 q^{25} +0.304282 q^{26} +5.04029 q^{27} -1.28408 q^{28} -3.12756 q^{29} -6.54982 q^{30} +6.44246 q^{31} +1.00000 q^{32} +16.0806 q^{33} +4.18619 q^{34} +3.01062 q^{35} +4.80423 q^{36} -3.97980 q^{37} +0.850045 q^{39} -2.34458 q^{40} -5.01719 q^{41} -3.58721 q^{42} -0.989378 q^{43} +5.75621 q^{44} -11.2639 q^{45} -6.47684 q^{46} -4.39445 q^{47} +2.79360 q^{48} -5.35114 q^{49} +0.497039 q^{50} +11.6946 q^{51} +0.304282 q^{52} +3.29064 q^{53} +5.04029 q^{54} -13.4959 q^{55} -1.28408 q^{56} -3.12756 q^{58} +3.31375 q^{59} -6.54982 q^{60} -10.9615 q^{61} +6.44246 q^{62} -6.16901 q^{63} +1.00000 q^{64} -0.713414 q^{65} +16.0806 q^{66} -4.38081 q^{67} +4.18619 q^{68} -18.0937 q^{69} +3.01062 q^{70} -4.41570 q^{71} +4.80423 q^{72} -2.26689 q^{73} -3.97980 q^{74} +1.38853 q^{75} -7.39144 q^{77} +0.850045 q^{78} +8.57659 q^{79} -2.34458 q^{80} -0.332090 q^{81} -5.01719 q^{82} -9.76464 q^{83} -3.58721 q^{84} -9.81485 q^{85} -0.989378 q^{86} -8.73716 q^{87} +5.75621 q^{88} -15.0765 q^{89} -11.2639 q^{90} -0.390723 q^{91} -6.47684 q^{92} +17.9977 q^{93} -4.39445 q^{94} +2.79360 q^{96} +17.3522 q^{97} -5.35114 q^{98} +27.6542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} + 4 q^{8} + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} + 4 q^{8} + 4 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} + 18 q^{13} - 2 q^{14} + 4 q^{15} + 4 q^{16} + 6 q^{17} + 4 q^{18} - 2 q^{20} + 4 q^{21} + 2 q^{22} - 10 q^{23} + 2 q^{24} + 6 q^{25} + 18 q^{26} - 4 q^{27} - 2 q^{28} - 2 q^{29} + 4 q^{30} + 26 q^{31} + 4 q^{32} + 16 q^{33} + 6 q^{34} + 6 q^{35} + 4 q^{36} + 4 q^{37} - 6 q^{39} - 2 q^{40} - 12 q^{41} + 4 q^{42} - 10 q^{43} + 2 q^{44} - 22 q^{45} - 10 q^{46} - 12 q^{47} + 2 q^{48} - 12 q^{49} + 6 q^{50} - 2 q^{51} + 18 q^{52} + 8 q^{53} - 4 q^{54} - 26 q^{55} - 2 q^{56} - 2 q^{58} - 8 q^{59} + 4 q^{60} + 26 q^{62} - 22 q^{63} + 4 q^{64} - 4 q^{65} + 16 q^{66} + 10 q^{67} + 6 q^{68} - 20 q^{69} + 6 q^{70} + 4 q^{72} - 14 q^{73} + 4 q^{74} + 8 q^{75} + 4 q^{77} - 6 q^{78} + 22 q^{79} - 2 q^{80} - 4 q^{81} - 12 q^{82} - 12 q^{83} + 4 q^{84} - 18 q^{85} - 10 q^{86} - 26 q^{87} + 2 q^{88} - 16 q^{89} - 22 q^{90} - 4 q^{91} - 10 q^{92} + 8 q^{93} - 12 q^{94} + 2 q^{96} + 28 q^{97} - 12 q^{98} + 22 q^{99} + O(q^{100}) \)

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
\(3\) 2.79360 1.61289 0.806444 0.591310i \(-0.201389\pi\)
0.806444 + 0.591310i \(0.201389\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.34458 −1.04853 −0.524263 0.851556i \(-0.675659\pi\)
−0.524263 + 0.851556i \(0.675659\pi\)
\(6\) 2.79360 1.14048
\(7\) −1.28408 −0.485336 −0.242668 0.970109i \(-0.578023\pi\)
−0.242668 + 0.970109i \(0.578023\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.80423 1.60141
\(10\) −2.34458 −0.741420
\(11\) 5.75621 1.73556 0.867782 0.496945i \(-0.165545\pi\)
0.867782 + 0.496945i \(0.165545\pi\)
\(12\) 2.79360 0.806444
\(13\) 0.304282 0.0843928 0.0421964 0.999109i \(-0.486564\pi\)
0.0421964 + 0.999109i \(0.486564\pi\)
\(14\) −1.28408 −0.343185
\(15\) −6.54982 −1.69116
\(16\) 1.00000 0.250000
\(17\) 4.18619 1.01530 0.507650 0.861563i \(-0.330514\pi\)
0.507650 + 0.861563i \(0.330514\pi\)
\(18\) 4.80423 1.13237
\(19\) 0 0
\(20\) −2.34458 −0.524263
\(21\) −3.58721 −0.782793
\(22\) 5.75621 1.22723
\(23\) −6.47684 −1.35051 −0.675257 0.737583i \(-0.735967\pi\)
−0.675257 + 0.737583i \(0.735967\pi\)
\(24\) 2.79360 0.570242
\(25\) 0.497039 0.0994078
\(26\) 0.304282 0.0596747
\(27\) 5.04029 0.970005
\(28\) −1.28408 −0.242668
\(29\) −3.12756 −0.580773 −0.290387 0.956909i \(-0.593784\pi\)
−0.290387 + 0.956909i \(0.593784\pi\)
\(30\) −6.54982 −1.19583
\(31\) 6.44246 1.15710 0.578550 0.815647i \(-0.303619\pi\)
0.578550 + 0.815647i \(0.303619\pi\)
\(32\) 1.00000 0.176777
\(33\) 16.0806 2.79927
\(34\) 4.18619 0.717926
\(35\) 3.01062 0.508888
\(36\) 4.80423 0.800704
\(37\) −3.97980 −0.654275 −0.327137 0.944977i \(-0.606084\pi\)
−0.327137 + 0.944977i \(0.606084\pi\)
\(38\) 0 0
\(39\) 0.850045 0.136116
\(40\) −2.34458 −0.370710
\(41\) −5.01719 −0.783553 −0.391776 0.920060i \(-0.628139\pi\)
−0.391776 + 0.920060i \(0.628139\pi\)
\(42\) −3.58721 −0.553518
\(43\) −0.989378 −0.150879 −0.0754394 0.997150i \(-0.524036\pi\)
−0.0754394 + 0.997150i \(0.524036\pi\)
\(44\) 5.75621 0.867782
\(45\) −11.2639 −1.67912
\(46\) −6.47684 −0.954957
\(47\) −4.39445 −0.640997 −0.320498 0.947249i \(-0.603850\pi\)
−0.320498 + 0.947249i \(0.603850\pi\)
\(48\) 2.79360 0.403222
\(49\) −5.35114 −0.764449
\(50\) 0.497039 0.0702919
\(51\) 11.6946 1.63757
\(52\) 0.304282 0.0421964
\(53\) 3.29064 0.452005 0.226002 0.974127i \(-0.427434\pi\)
0.226002 + 0.974127i \(0.427434\pi\)
\(54\) 5.04029 0.685897
\(55\) −13.4959 −1.81978
\(56\) −1.28408 −0.171592
\(57\) 0 0
\(58\) −3.12756 −0.410669
\(59\) 3.31375 0.431414 0.215707 0.976458i \(-0.430794\pi\)
0.215707 + 0.976458i \(0.430794\pi\)
\(60\) −6.54982 −0.845578
\(61\) −10.9615 −1.40347 −0.701735 0.712438i \(-0.747591\pi\)
−0.701735 + 0.712438i \(0.747591\pi\)
\(62\) 6.44246 0.818194
\(63\) −6.16901 −0.777222
\(64\) 1.00000 0.125000
\(65\) −0.713414 −0.0884881
\(66\) 16.0806 1.97938
\(67\) −4.38081 −0.535202 −0.267601 0.963530i \(-0.586231\pi\)
−0.267601 + 0.963530i \(0.586231\pi\)
\(68\) 4.18619 0.507650
\(69\) −18.0937 −2.17823
\(70\) 3.01062 0.359838
\(71\) −4.41570 −0.524047 −0.262023 0.965062i \(-0.584390\pi\)
−0.262023 + 0.965062i \(0.584390\pi\)
\(72\) 4.80423 0.566183
\(73\) −2.26689 −0.265320 −0.132660 0.991162i \(-0.542352\pi\)
−0.132660 + 0.991162i \(0.542352\pi\)
\(74\) −3.97980 −0.462642
\(75\) 1.38853 0.160334
\(76\) 0 0
\(77\) −7.39144 −0.842332
\(78\) 0.850045 0.0962486
\(79\) 8.57659 0.964941 0.482471 0.875912i \(-0.339739\pi\)
0.482471 + 0.875912i \(0.339739\pi\)
\(80\) −2.34458 −0.262132
\(81\) −0.332090 −0.0368989
\(82\) −5.01719 −0.554056
\(83\) −9.76464 −1.07181 −0.535904 0.844279i \(-0.680029\pi\)
−0.535904 + 0.844279i \(0.680029\pi\)
\(84\) −3.58721 −0.391397
\(85\) −9.81485 −1.06457
\(86\) −0.989378 −0.106687
\(87\) −8.73716 −0.936722
\(88\) 5.75621 0.613615
\(89\) −15.0765 −1.59811 −0.799055 0.601259i \(-0.794666\pi\)
−0.799055 + 0.601259i \(0.794666\pi\)
\(90\) −11.2639 −1.18732
\(91\) −0.390723 −0.0409589
\(92\) −6.47684 −0.675257
\(93\) 17.9977 1.86627
\(94\) −4.39445 −0.453253
\(95\) 0 0
\(96\) 2.79360 0.285121
\(97\) 17.3522 1.76185 0.880924 0.473259i \(-0.156922\pi\)
0.880924 + 0.473259i \(0.156922\pi\)
\(98\) −5.35114 −0.540547
\(99\) 27.6542 2.77935
\(100\) 0.497039 0.0497039
\(101\) 3.67667 0.365842 0.182921 0.983128i \(-0.441445\pi\)
0.182921 + 0.983128i \(0.441445\pi\)
\(102\) 11.6946 1.15793
\(103\) 19.0135 1.87346 0.936729 0.350055i \(-0.113837\pi\)
0.936729 + 0.350055i \(0.113837\pi\)
\(104\) 0.304282 0.0298374
\(105\) 8.41049 0.820779
\(106\) 3.29064 0.319616
\(107\) 10.0373 0.970340 0.485170 0.874420i \(-0.338758\pi\)
0.485170 + 0.874420i \(0.338758\pi\)
\(108\) 5.04029 0.485003
\(109\) −8.84348 −0.847052 −0.423526 0.905884i \(-0.639208\pi\)
−0.423526 + 0.905884i \(0.639208\pi\)
\(110\) −13.4959 −1.28678
\(111\) −11.1180 −1.05527
\(112\) −1.28408 −0.121334
\(113\) −14.2502 −1.34055 −0.670275 0.742113i \(-0.733824\pi\)
−0.670275 + 0.742113i \(0.733824\pi\)
\(114\) 0 0
\(115\) 15.1854 1.41605
\(116\) −3.12756 −0.290387
\(117\) 1.46184 0.135147
\(118\) 3.31375 0.305056
\(119\) −5.37540 −0.492762
\(120\) −6.54982 −0.597914
\(121\) 22.1340 2.01218
\(122\) −10.9615 −0.992404
\(123\) −14.0160 −1.26378
\(124\) 6.44246 0.578550
\(125\) 10.5575 0.944295
\(126\) −6.16901 −0.549579
\(127\) 2.23015 0.197893 0.0989467 0.995093i \(-0.468453\pi\)
0.0989467 + 0.995093i \(0.468453\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.76393 −0.243351
\(130\) −0.713414 −0.0625705
\(131\) 1.52265 0.133035 0.0665175 0.997785i \(-0.478811\pi\)
0.0665175 + 0.997785i \(0.478811\pi\)
\(132\) 16.0806 1.39964
\(133\) 0 0
\(134\) −4.38081 −0.378445
\(135\) −11.8174 −1.01708
\(136\) 4.18619 0.358963
\(137\) 10.7118 0.915167 0.457583 0.889167i \(-0.348715\pi\)
0.457583 + 0.889167i \(0.348715\pi\)
\(138\) −18.0937 −1.54024
\(139\) −10.8143 −0.917260 −0.458630 0.888627i \(-0.651660\pi\)
−0.458630 + 0.888627i \(0.651660\pi\)
\(140\) 3.01062 0.254444
\(141\) −12.2764 −1.03386
\(142\) −4.41570 −0.370557
\(143\) 1.75152 0.146469
\(144\) 4.80423 0.400352
\(145\) 7.33280 0.608956
\(146\) −2.26689 −0.187609
\(147\) −14.9490 −1.23297
\(148\) −3.97980 −0.327137
\(149\) −9.33384 −0.764658 −0.382329 0.924026i \(-0.624878\pi\)
−0.382329 + 0.924026i \(0.624878\pi\)
\(150\) 1.38853 0.113373
\(151\) 19.8232 1.61319 0.806593 0.591107i \(-0.201309\pi\)
0.806593 + 0.591107i \(0.201309\pi\)
\(152\) 0 0
\(153\) 20.1114 1.62591
\(154\) −7.39144 −0.595619
\(155\) −15.1048 −1.21325
\(156\) 0.850045 0.0680581
\(157\) 13.8676 1.10676 0.553379 0.832930i \(-0.313338\pi\)
0.553379 + 0.832930i \(0.313338\pi\)
\(158\) 8.57659 0.682317
\(159\) 9.19276 0.729033
\(160\) −2.34458 −0.185355
\(161\) 8.31677 0.655453
\(162\) −0.332090 −0.0260915
\(163\) 14.2496 1.11611 0.558057 0.829803i \(-0.311547\pi\)
0.558057 + 0.829803i \(0.311547\pi\)
\(164\) −5.01719 −0.391776
\(165\) −37.7022 −2.93511
\(166\) −9.76464 −0.757883
\(167\) 11.5901 0.896870 0.448435 0.893815i \(-0.351982\pi\)
0.448435 + 0.893815i \(0.351982\pi\)
\(168\) −3.58721 −0.276759
\(169\) −12.9074 −0.992878
\(170\) −9.81485 −0.752764
\(171\) 0 0
\(172\) −0.989378 −0.0754394
\(173\) 3.98353 0.302862 0.151431 0.988468i \(-0.451612\pi\)
0.151431 + 0.988468i \(0.451612\pi\)
\(174\) −8.73716 −0.662363
\(175\) −0.638237 −0.0482462
\(176\) 5.75621 0.433891
\(177\) 9.25731 0.695822
\(178\) −15.0765 −1.13003
\(179\) −8.45089 −0.631649 −0.315825 0.948818i \(-0.602281\pi\)
−0.315825 + 0.948818i \(0.602281\pi\)
\(180\) −11.2639 −0.839560
\(181\) 8.48526 0.630705 0.315352 0.948975i \(-0.397877\pi\)
0.315352 + 0.948975i \(0.397877\pi\)
\(182\) −0.390723 −0.0289623
\(183\) −30.6220 −2.26364
\(184\) −6.47684 −0.477479
\(185\) 9.33094 0.686024
\(186\) 17.9977 1.31966
\(187\) 24.0966 1.76212
\(188\) −4.39445 −0.320498
\(189\) −6.47214 −0.470779
\(190\) 0 0
\(191\) −22.3790 −1.61929 −0.809644 0.586921i \(-0.800340\pi\)
−0.809644 + 0.586921i \(0.800340\pi\)
\(192\) 2.79360 0.201611
\(193\) 7.42811 0.534687 0.267344 0.963601i \(-0.413854\pi\)
0.267344 + 0.963601i \(0.413854\pi\)
\(194\) 17.3522 1.24581
\(195\) −1.99300 −0.142721
\(196\) −5.35114 −0.382224
\(197\) 8.53554 0.608132 0.304066 0.952651i \(-0.401656\pi\)
0.304066 + 0.952651i \(0.401656\pi\)
\(198\) 27.6542 1.96530
\(199\) −6.35926 −0.450796 −0.225398 0.974267i \(-0.572368\pi\)
−0.225398 + 0.974267i \(0.572368\pi\)
\(200\) 0.497039 0.0351460
\(201\) −12.2383 −0.863220
\(202\) 3.67667 0.258689
\(203\) 4.01603 0.281870
\(204\) 11.6946 0.818783
\(205\) 11.7632 0.821576
\(206\) 19.0135 1.32474
\(207\) −31.1162 −2.16272
\(208\) 0.304282 0.0210982
\(209\) 0 0
\(210\) 8.41049 0.580379
\(211\) −13.9578 −0.960894 −0.480447 0.877024i \(-0.659526\pi\)
−0.480447 + 0.877024i \(0.659526\pi\)
\(212\) 3.29064 0.226002
\(213\) −12.3357 −0.845229
\(214\) 10.0373 0.686134
\(215\) 2.31967 0.158200
\(216\) 5.04029 0.342949
\(217\) −8.27263 −0.561583
\(218\) −8.84348 −0.598956
\(219\) −6.33280 −0.427931
\(220\) −13.4959 −0.909892
\(221\) 1.27378 0.0856840
\(222\) −11.1180 −0.746190
\(223\) 8.92400 0.597595 0.298798 0.954317i \(-0.403414\pi\)
0.298798 + 0.954317i \(0.403414\pi\)
\(224\) −1.28408 −0.0857961
\(225\) 2.38789 0.159193
\(226\) −14.2502 −0.947912
\(227\) −0.659796 −0.0437922 −0.0218961 0.999760i \(-0.506970\pi\)
−0.0218961 + 0.999760i \(0.506970\pi\)
\(228\) 0 0
\(229\) −6.97208 −0.460728 −0.230364 0.973105i \(-0.573992\pi\)
−0.230364 + 0.973105i \(0.573992\pi\)
\(230\) 15.1854 1.00130
\(231\) −20.6487 −1.35859
\(232\) −3.12756 −0.205334
\(233\) −29.6097 −1.93980 −0.969898 0.243512i \(-0.921700\pi\)
−0.969898 + 0.243512i \(0.921700\pi\)
\(234\) 1.46184 0.0955636
\(235\) 10.3031 0.672102
\(236\) 3.31375 0.215707
\(237\) 23.9596 1.55634
\(238\) −5.37540 −0.348436
\(239\) −24.7034 −1.59793 −0.798965 0.601378i \(-0.794619\pi\)
−0.798965 + 0.601378i \(0.794619\pi\)
\(240\) −6.54982 −0.422789
\(241\) 27.5586 1.77521 0.887604 0.460607i \(-0.152368\pi\)
0.887604 + 0.460607i \(0.152368\pi\)
\(242\) 22.1340 1.42283
\(243\) −16.0486 −1.02952
\(244\) −10.9615 −0.701735
\(245\) 12.5462 0.801545
\(246\) −14.0160 −0.893630
\(247\) 0 0
\(248\) 6.44246 0.409097
\(249\) −27.2786 −1.72871
\(250\) 10.5575 0.667717
\(251\) −2.98855 −0.188636 −0.0943179 0.995542i \(-0.530067\pi\)
−0.0943179 + 0.995542i \(0.530067\pi\)
\(252\) −6.16901 −0.388611
\(253\) −37.2821 −2.34390
\(254\) 2.23015 0.139932
\(255\) −27.4188 −1.71703
\(256\) 1.00000 0.0625000
\(257\) 14.1323 0.881546 0.440773 0.897619i \(-0.354704\pi\)
0.440773 + 0.897619i \(0.354704\pi\)
\(258\) −2.76393 −0.172075
\(259\) 5.11037 0.317543
\(260\) −0.713414 −0.0442440
\(261\) −15.0255 −0.930055
\(262\) 1.52265 0.0940699
\(263\) 7.56887 0.466717 0.233358 0.972391i \(-0.425029\pi\)
0.233358 + 0.972391i \(0.425029\pi\)
\(264\) 16.0806 0.989692
\(265\) −7.71517 −0.473939
\(266\) 0 0
\(267\) −42.1179 −2.57757
\(268\) −4.38081 −0.267601
\(269\) −7.67074 −0.467694 −0.233847 0.972273i \(-0.575131\pi\)
−0.233847 + 0.972273i \(0.575131\pi\)
\(270\) −11.8174 −0.719181
\(271\) 11.6649 0.708592 0.354296 0.935133i \(-0.384721\pi\)
0.354296 + 0.935133i \(0.384721\pi\)
\(272\) 4.18619 0.253825
\(273\) −1.09152 −0.0660621
\(274\) 10.7118 0.647121
\(275\) 2.86106 0.172529
\(276\) −18.0937 −1.08911
\(277\) 27.4405 1.64874 0.824370 0.566052i \(-0.191530\pi\)
0.824370 + 0.566052i \(0.191530\pi\)
\(278\) −10.8143 −0.648601
\(279\) 30.9511 1.85299
\(280\) 3.01062 0.179919
\(281\) 16.2715 0.970675 0.485338 0.874327i \(-0.338697\pi\)
0.485338 + 0.874327i \(0.338697\pi\)
\(282\) −12.2764 −0.731047
\(283\) 3.39374 0.201737 0.100868 0.994900i \(-0.467838\pi\)
0.100868 + 0.994900i \(0.467838\pi\)
\(284\) −4.41570 −0.262023
\(285\) 0 0
\(286\) 1.75152 0.103569
\(287\) 6.44246 0.380287
\(288\) 4.80423 0.283092
\(289\) 0.524204 0.0308355
\(290\) 7.33280 0.430597
\(291\) 48.4751 2.84166
\(292\) −2.26689 −0.132660
\(293\) 23.3783 1.36578 0.682888 0.730523i \(-0.260724\pi\)
0.682888 + 0.730523i \(0.260724\pi\)
\(294\) −14.9490 −0.871842
\(295\) −7.76934 −0.452349
\(296\) −3.97980 −0.231321
\(297\) 29.0130 1.68351
\(298\) −9.33384 −0.540695
\(299\) −1.97079 −0.113974
\(300\) 1.38853 0.0801668
\(301\) 1.27044 0.0732270
\(302\) 19.8232 1.14069
\(303\) 10.2712 0.590062
\(304\) 0 0
\(305\) 25.7000 1.47158
\(306\) 20.1114 1.14969
\(307\) −1.38802 −0.0792185 −0.0396093 0.999215i \(-0.512611\pi\)
−0.0396093 + 0.999215i \(0.512611\pi\)
\(308\) −7.39144 −0.421166
\(309\) 53.1163 3.02168
\(310\) −15.1048 −0.857898
\(311\) −12.1902 −0.691246 −0.345623 0.938374i \(-0.612332\pi\)
−0.345623 + 0.938374i \(0.612332\pi\)
\(312\) 0.850045 0.0481243
\(313\) 16.5314 0.934411 0.467205 0.884149i \(-0.345261\pi\)
0.467205 + 0.884149i \(0.345261\pi\)
\(314\) 13.8676 0.782596
\(315\) 14.4637 0.814938
\(316\) 8.57659 0.482471
\(317\) 2.68811 0.150979 0.0754897 0.997147i \(-0.475948\pi\)
0.0754897 + 0.997147i \(0.475948\pi\)
\(318\) 9.19276 0.515504
\(319\) −18.0029 −1.00797
\(320\) −2.34458 −0.131066
\(321\) 28.0402 1.56505
\(322\) 8.31677 0.463475
\(323\) 0 0
\(324\) −0.332090 −0.0184495
\(325\) 0.151240 0.00838930
\(326\) 14.2496 0.789212
\(327\) −24.7052 −1.36620
\(328\) −5.01719 −0.277028
\(329\) 5.64282 0.311099
\(330\) −37.7022 −2.07544
\(331\) 5.21303 0.286534 0.143267 0.989684i \(-0.454239\pi\)
0.143267 + 0.989684i \(0.454239\pi\)
\(332\) −9.76464 −0.535904
\(333\) −19.1198 −1.04776
\(334\) 11.5901 0.634183
\(335\) 10.2712 0.561173
\(336\) −3.58721 −0.195698
\(337\) −26.7275 −1.45594 −0.727969 0.685610i \(-0.759536\pi\)
−0.727969 + 0.685610i \(0.759536\pi\)
\(338\) −12.9074 −0.702071
\(339\) −39.8095 −2.16216
\(340\) −9.81485 −0.532285
\(341\) 37.0842 2.00822
\(342\) 0 0
\(343\) 15.8598 0.856351
\(344\) −0.989378 −0.0533437
\(345\) 42.4221 2.28393
\(346\) 3.98353 0.214156
\(347\) −32.6682 −1.75372 −0.876860 0.480745i \(-0.840366\pi\)
−0.876860 + 0.480745i \(0.840366\pi\)
\(348\) −8.73716 −0.468361
\(349\) −29.1058 −1.55800 −0.778998 0.627026i \(-0.784272\pi\)
−0.778998 + 0.627026i \(0.784272\pi\)
\(350\) −0.638237 −0.0341152
\(351\) 1.53367 0.0818614
\(352\) 5.75621 0.306807
\(353\) −14.3049 −0.761372 −0.380686 0.924704i \(-0.624312\pi\)
−0.380686 + 0.924704i \(0.624312\pi\)
\(354\) 9.25731 0.492021
\(355\) 10.3529 0.549477
\(356\) −15.0765 −0.799055
\(357\) −15.0167 −0.794770
\(358\) −8.45089 −0.446644
\(359\) 19.6707 1.03818 0.519090 0.854720i \(-0.326271\pi\)
0.519090 + 0.854720i \(0.326271\pi\)
\(360\) −11.2639 −0.593658
\(361\) 0 0
\(362\) 8.48526 0.445976
\(363\) 61.8337 3.24543
\(364\) −0.390723 −0.0204794
\(365\) 5.31490 0.278195
\(366\) −30.6220 −1.60064
\(367\) 9.55053 0.498534 0.249267 0.968435i \(-0.419810\pi\)
0.249267 + 0.968435i \(0.419810\pi\)
\(368\) −6.47684 −0.337628
\(369\) −24.1037 −1.25479
\(370\) 9.33094 0.485092
\(371\) −4.22545 −0.219374
\(372\) 17.9977 0.933137
\(373\) 3.64010 0.188477 0.0942387 0.995550i \(-0.469958\pi\)
0.0942387 + 0.995550i \(0.469958\pi\)
\(374\) 24.0966 1.24601
\(375\) 29.4936 1.52304
\(376\) −4.39445 −0.226627
\(377\) −0.951662 −0.0490131
\(378\) −6.47214 −0.332891
\(379\) 8.34741 0.428778 0.214389 0.976748i \(-0.431224\pi\)
0.214389 + 0.976748i \(0.431224\pi\)
\(380\) 0 0
\(381\) 6.23015 0.319180
\(382\) −22.3790 −1.14501
\(383\) −9.48566 −0.484695 −0.242347 0.970190i \(-0.577917\pi\)
−0.242347 + 0.970190i \(0.577917\pi\)
\(384\) 2.79360 0.142561
\(385\) 17.3298 0.883208
\(386\) 7.42811 0.378081
\(387\) −4.75320 −0.241619
\(388\) 17.3522 0.880924
\(389\) 36.0480 1.82770 0.913852 0.406047i \(-0.133093\pi\)
0.913852 + 0.406047i \(0.133093\pi\)
\(390\) −1.99300 −0.100919
\(391\) −27.1133 −1.37118
\(392\) −5.35114 −0.270273
\(393\) 4.25369 0.214570
\(394\) 8.53554 0.430014
\(395\) −20.1085 −1.01177
\(396\) 27.6542 1.38967
\(397\) −6.40906 −0.321662 −0.160831 0.986982i \(-0.551417\pi\)
−0.160831 + 0.986982i \(0.551417\pi\)
\(398\) −6.35926 −0.318761
\(399\) 0 0
\(400\) 0.497039 0.0248520
\(401\) 0.187660 0.00937128 0.00468564 0.999989i \(-0.498509\pi\)
0.00468564 + 0.999989i \(0.498509\pi\)
\(402\) −12.2383 −0.610389
\(403\) 1.96033 0.0976509
\(404\) 3.67667 0.182921
\(405\) 0.778611 0.0386895
\(406\) 4.01603 0.199312
\(407\) −22.9086 −1.13554
\(408\) 11.6946 0.578967
\(409\) 17.2663 0.853762 0.426881 0.904308i \(-0.359612\pi\)
0.426881 + 0.904308i \(0.359612\pi\)
\(410\) 11.7632 0.580942
\(411\) 29.9244 1.47606
\(412\) 19.0135 0.936729
\(413\) −4.25512 −0.209381
\(414\) −31.1162 −1.52928
\(415\) 22.8940 1.12382
\(416\) 0.304282 0.0149187
\(417\) −30.2110 −1.47944
\(418\) 0 0
\(419\) −1.72572 −0.0843068 −0.0421534 0.999111i \(-0.513422\pi\)
−0.0421534 + 0.999111i \(0.513422\pi\)
\(420\) 8.41049 0.410390
\(421\) 32.7546 1.59636 0.798180 0.602420i \(-0.205797\pi\)
0.798180 + 0.602420i \(0.205797\pi\)
\(422\) −13.9578 −0.679455
\(423\) −21.1119 −1.02650
\(424\) 3.29064 0.159808
\(425\) 2.08070 0.100929
\(426\) −12.3357 −0.597667
\(427\) 14.0754 0.681155
\(428\) 10.0373 0.485170
\(429\) 4.89304 0.236238
\(430\) 2.31967 0.111865
\(431\) −25.5119 −1.22887 −0.614433 0.788969i \(-0.710615\pi\)
−0.614433 + 0.788969i \(0.710615\pi\)
\(432\) 5.04029 0.242501
\(433\) 24.4834 1.17660 0.588299 0.808644i \(-0.299798\pi\)
0.588299 + 0.808644i \(0.299798\pi\)
\(434\) −8.27263 −0.397099
\(435\) 20.4849 0.982178
\(436\) −8.84348 −0.423526
\(437\) 0 0
\(438\) −6.33280 −0.302593
\(439\) −30.8388 −1.47186 −0.735928 0.677060i \(-0.763254\pi\)
−0.735928 + 0.677060i \(0.763254\pi\)
\(440\) −13.4959 −0.643391
\(441\) −25.7081 −1.22419
\(442\) 1.27378 0.0605878
\(443\) 10.7047 0.508595 0.254297 0.967126i \(-0.418156\pi\)
0.254297 + 0.967126i \(0.418156\pi\)
\(444\) −11.1180 −0.527636
\(445\) 35.3481 1.67566
\(446\) 8.92400 0.422564
\(447\) −26.0751 −1.23331
\(448\) −1.28408 −0.0606670
\(449\) 2.31562 0.109281 0.0546403 0.998506i \(-0.482599\pi\)
0.0546403 + 0.998506i \(0.482599\pi\)
\(450\) 2.38789 0.112566
\(451\) −28.8800 −1.35991
\(452\) −14.2502 −0.670275
\(453\) 55.3781 2.60189
\(454\) −0.659796 −0.0309657
\(455\) 0.916079 0.0429465
\(456\) 0 0
\(457\) −0.0521810 −0.00244092 −0.00122046 0.999999i \(-0.500388\pi\)
−0.00122046 + 0.999999i \(0.500388\pi\)
\(458\) −6.97208 −0.325784
\(459\) 21.0996 0.984847
\(460\) 15.1854 0.708025
\(461\) 39.4720 1.83840 0.919198 0.393796i \(-0.128838\pi\)
0.919198 + 0.393796i \(0.128838\pi\)
\(462\) −20.6487 −0.960667
\(463\) −6.19056 −0.287700 −0.143850 0.989600i \(-0.545948\pi\)
−0.143850 + 0.989600i \(0.545948\pi\)
\(464\) −3.12756 −0.145193
\(465\) −42.1970 −1.95684
\(466\) −29.6097 −1.37164
\(467\) −20.9962 −0.971586 −0.485793 0.874074i \(-0.661469\pi\)
−0.485793 + 0.874074i \(0.661469\pi\)
\(468\) 1.46184 0.0675737
\(469\) 5.62531 0.259753
\(470\) 10.3031 0.475248
\(471\) 38.7407 1.78508
\(472\) 3.31375 0.152528
\(473\) −5.69507 −0.261860
\(474\) 23.9596 1.10050
\(475\) 0 0
\(476\) −5.37540 −0.246381
\(477\) 15.8090 0.723844
\(478\) −24.7034 −1.12991
\(479\) 15.5229 0.709261 0.354631 0.935006i \(-0.384607\pi\)
0.354631 + 0.935006i \(0.384607\pi\)
\(480\) −6.54982 −0.298957
\(481\) −1.21098 −0.0552160
\(482\) 27.5586 1.25526
\(483\) 23.2338 1.05717
\(484\) 22.1340 1.00609
\(485\) −40.6835 −1.84734
\(486\) −16.0486 −0.727980
\(487\) −28.6065 −1.29629 −0.648143 0.761519i \(-0.724454\pi\)
−0.648143 + 0.761519i \(0.724454\pi\)
\(488\) −10.9615 −0.496202
\(489\) 39.8077 1.80017
\(490\) 12.5462 0.566778
\(491\) −30.3488 −1.36962 −0.684812 0.728720i \(-0.740116\pi\)
−0.684812 + 0.728720i \(0.740116\pi\)
\(492\) −14.0160 −0.631892
\(493\) −13.0926 −0.589659
\(494\) 0 0
\(495\) −64.8373 −2.91422
\(496\) 6.44246 0.289275
\(497\) 5.67010 0.254339
\(498\) −27.2786 −1.22238
\(499\) 8.77624 0.392878 0.196439 0.980516i \(-0.437062\pi\)
0.196439 + 0.980516i \(0.437062\pi\)
\(500\) 10.5575 0.472147
\(501\) 32.3782 1.44655
\(502\) −2.98855 −0.133386
\(503\) −21.1043 −0.940996 −0.470498 0.882401i \(-0.655926\pi\)
−0.470498 + 0.882401i \(0.655926\pi\)
\(504\) −6.16901 −0.274789
\(505\) −8.62023 −0.383595
\(506\) −37.2821 −1.65739
\(507\) −36.0582 −1.60140
\(508\) 2.23015 0.0989467
\(509\) 22.2212 0.984936 0.492468 0.870331i \(-0.336095\pi\)
0.492468 + 0.870331i \(0.336095\pi\)
\(510\) −27.4188 −1.21412
\(511\) 2.91087 0.128769
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 14.1323 0.623347
\(515\) −44.5787 −1.96437
\(516\) −2.76393 −0.121675
\(517\) −25.2954 −1.11249
\(518\) 5.11037 0.224537
\(519\) 11.1284 0.488482
\(520\) −0.713414 −0.0312853
\(521\) 10.0332 0.439563 0.219782 0.975549i \(-0.429466\pi\)
0.219782 + 0.975549i \(0.429466\pi\)
\(522\) −15.0255 −0.657648
\(523\) 29.2823 1.28042 0.640212 0.768198i \(-0.278846\pi\)
0.640212 + 0.768198i \(0.278846\pi\)
\(524\) 1.52265 0.0665175
\(525\) −1.78298 −0.0778158
\(526\) 7.56887 0.330018
\(527\) 26.9694 1.17481
\(528\) 16.0806 0.699818
\(529\) 18.9494 0.823887
\(530\) −7.71517 −0.335125
\(531\) 15.9200 0.690870
\(532\) 0 0
\(533\) −1.52664 −0.0661262
\(534\) −42.1179 −1.82262
\(535\) −23.5332 −1.01743
\(536\) −4.38081 −0.189222
\(537\) −23.6085 −1.01878
\(538\) −7.67074 −0.330709
\(539\) −30.8023 −1.32675
\(540\) −11.8174 −0.508538
\(541\) −37.8404 −1.62688 −0.813442 0.581646i \(-0.802409\pi\)
−0.813442 + 0.581646i \(0.802409\pi\)
\(542\) 11.6649 0.501050
\(543\) 23.7045 1.01726
\(544\) 4.18619 0.179482
\(545\) 20.7342 0.888156
\(546\) −1.09152 −0.0467130
\(547\) −14.4976 −0.619871 −0.309936 0.950758i \(-0.600308\pi\)
−0.309936 + 0.950758i \(0.600308\pi\)
\(548\) 10.7118 0.457583
\(549\) −52.6613 −2.24753
\(550\) 2.86106 0.121996
\(551\) 0 0
\(552\) −18.0937 −0.770120
\(553\) −11.0130 −0.468321
\(554\) 27.4405 1.16583
\(555\) 26.0669 1.10648
\(556\) −10.8143 −0.458630
\(557\) −4.33801 −0.183807 −0.0919037 0.995768i \(-0.529295\pi\)
−0.0919037 + 0.995768i \(0.529295\pi\)
\(558\) 30.9511 1.31026
\(559\) −0.301051 −0.0127331
\(560\) 3.01062 0.127222
\(561\) 67.3164 2.84210
\(562\) 16.2715 0.686371
\(563\) −27.2936 −1.15029 −0.575144 0.818052i \(-0.695054\pi\)
−0.575144 + 0.818052i \(0.695054\pi\)
\(564\) −12.2764 −0.516928
\(565\) 33.4108 1.40560
\(566\) 3.39374 0.142650
\(567\) 0.426430 0.0179084
\(568\) −4.41570 −0.185278
\(569\) −7.19043 −0.301439 −0.150719 0.988577i \(-0.548159\pi\)
−0.150719 + 0.988577i \(0.548159\pi\)
\(570\) 0 0
\(571\) 42.0134 1.75821 0.879103 0.476631i \(-0.158142\pi\)
0.879103 + 0.476631i \(0.158142\pi\)
\(572\) 1.75152 0.0732345
\(573\) −62.5181 −2.61173
\(574\) 6.44246 0.268903
\(575\) −3.21924 −0.134252
\(576\) 4.80423 0.200176
\(577\) 21.6713 0.902190 0.451095 0.892476i \(-0.351034\pi\)
0.451095 + 0.892476i \(0.351034\pi\)
\(578\) 0.524204 0.0218040
\(579\) 20.7512 0.862391
\(580\) 7.33280 0.304478
\(581\) 12.5386 0.520188
\(582\) 48.4751 2.00936
\(583\) 18.9417 0.784483
\(584\) −2.26689 −0.0938047
\(585\) −3.42740 −0.141706
\(586\) 23.3783 0.965749
\(587\) −1.26864 −0.0523626 −0.0261813 0.999657i \(-0.508335\pi\)
−0.0261813 + 0.999657i \(0.508335\pi\)
\(588\) −14.9490 −0.616485
\(589\) 0 0
\(590\) −7.76934 −0.319859
\(591\) 23.8449 0.980849
\(592\) −3.97980 −0.163569
\(593\) 21.5600 0.885363 0.442682 0.896679i \(-0.354027\pi\)
0.442682 + 0.896679i \(0.354027\pi\)
\(594\) 29.0130 1.19042
\(595\) 12.6030 0.516674
\(596\) −9.33384 −0.382329
\(597\) −17.7652 −0.727083
\(598\) −1.97079 −0.0805915
\(599\) −28.6417 −1.17027 −0.585135 0.810936i \(-0.698958\pi\)
−0.585135 + 0.810936i \(0.698958\pi\)
\(600\) 1.38853 0.0566865
\(601\) 7.55865 0.308324 0.154162 0.988046i \(-0.450732\pi\)
0.154162 + 0.988046i \(0.450732\pi\)
\(602\) 1.27044 0.0517793
\(603\) −21.0464 −0.857076
\(604\) 19.8232 0.806593
\(605\) −51.8949 −2.10983
\(606\) 10.2712 0.417237
\(607\) 1.72737 0.0701117 0.0350558 0.999385i \(-0.488839\pi\)
0.0350558 + 0.999385i \(0.488839\pi\)
\(608\) 0 0
\(609\) 11.2192 0.454625
\(610\) 25.7000 1.04056
\(611\) −1.33715 −0.0540955
\(612\) 20.1114 0.812956
\(613\) −27.4169 −1.10736 −0.553680 0.832730i \(-0.686777\pi\)
−0.553680 + 0.832730i \(0.686777\pi\)
\(614\) −1.38802 −0.0560160
\(615\) 32.8617 1.32511
\(616\) −7.39144 −0.297809
\(617\) 10.6275 0.427847 0.213924 0.976850i \(-0.431376\pi\)
0.213924 + 0.976850i \(0.431376\pi\)
\(618\) 53.1163 2.13665
\(619\) 8.36008 0.336020 0.168010 0.985785i \(-0.446266\pi\)
0.168010 + 0.985785i \(0.446266\pi\)
\(620\) −15.1048 −0.606625
\(621\) −32.6452 −1.31000
\(622\) −12.1902 −0.488784
\(623\) 19.3595 0.775620
\(624\) 0.850045 0.0340290
\(625\) −27.2381 −1.08953
\(626\) 16.5314 0.660728
\(627\) 0 0
\(628\) 13.8676 0.553379
\(629\) −16.6602 −0.664285
\(630\) 14.4637 0.576248
\(631\) 24.5301 0.976530 0.488265 0.872696i \(-0.337630\pi\)
0.488265 + 0.872696i \(0.337630\pi\)
\(632\) 8.57659 0.341158
\(633\) −38.9926 −1.54982
\(634\) 2.68811 0.106759
\(635\) −5.22875 −0.207497
\(636\) 9.19276 0.364517
\(637\) −1.62826 −0.0645139
\(638\) −18.0029 −0.712742
\(639\) −21.2140 −0.839213
\(640\) −2.34458 −0.0926775
\(641\) −6.97270 −0.275405 −0.137703 0.990474i \(-0.543972\pi\)
−0.137703 + 0.990474i \(0.543972\pi\)
\(642\) 28.0402 1.10666
\(643\) −9.09024 −0.358484 −0.179242 0.983805i \(-0.557365\pi\)
−0.179242 + 0.983805i \(0.557365\pi\)
\(644\) 8.31677 0.327727
\(645\) 6.48025 0.255160
\(646\) 0 0
\(647\) 30.0074 1.17971 0.589856 0.807509i \(-0.299185\pi\)
0.589856 + 0.807509i \(0.299185\pi\)
\(648\) −0.332090 −0.0130457
\(649\) 19.0747 0.748746
\(650\) 0.151240 0.00593213
\(651\) −23.1105 −0.905770
\(652\) 14.2496 0.558057
\(653\) 10.4191 0.407732 0.203866 0.978999i \(-0.434649\pi\)
0.203866 + 0.978999i \(0.434649\pi\)
\(654\) −24.7052 −0.966049
\(655\) −3.56998 −0.139491
\(656\) −5.01719 −0.195888
\(657\) −10.8907 −0.424885
\(658\) 5.64282 0.219980
\(659\) −26.3756 −1.02745 −0.513724 0.857956i \(-0.671734\pi\)
−0.513724 + 0.857956i \(0.671734\pi\)
\(660\) −37.7022 −1.46755
\(661\) −24.6939 −0.960483 −0.480242 0.877136i \(-0.659451\pi\)
−0.480242 + 0.877136i \(0.659451\pi\)
\(662\) 5.21303 0.202610
\(663\) 3.55845 0.138199
\(664\) −9.76464 −0.378942
\(665\) 0 0
\(666\) −19.1198 −0.740879
\(667\) 20.2567 0.784342
\(668\) 11.5901 0.448435
\(669\) 24.9301 0.963854
\(670\) 10.2712 0.396809
\(671\) −63.0965 −2.43581
\(672\) −3.58721 −0.138380
\(673\) −9.77380 −0.376752 −0.188376 0.982097i \(-0.560322\pi\)
−0.188376 + 0.982097i \(0.560322\pi\)
\(674\) −26.7275 −1.02950
\(675\) 2.50522 0.0964261
\(676\) −12.9074 −0.496439
\(677\) −21.5747 −0.829184 −0.414592 0.910007i \(-0.636076\pi\)
−0.414592 + 0.910007i \(0.636076\pi\)
\(678\) −39.8095 −1.52888
\(679\) −22.2816 −0.855088
\(680\) −9.81485 −0.376382
\(681\) −1.84321 −0.0706319
\(682\) 37.0842 1.42003
\(683\) 33.2509 1.27231 0.636155 0.771561i \(-0.280524\pi\)
0.636155 + 0.771561i \(0.280524\pi\)
\(684\) 0 0
\(685\) −25.1145 −0.959576
\(686\) 15.8598 0.605532
\(687\) −19.4772 −0.743103
\(688\) −0.989378 −0.0377197
\(689\) 1.00129 0.0381459
\(690\) 42.4221 1.61498
\(691\) 4.80932 0.182955 0.0914776 0.995807i \(-0.470841\pi\)
0.0914776 + 0.995807i \(0.470841\pi\)
\(692\) 3.98353 0.151431
\(693\) −35.5101 −1.34892
\(694\) −32.6682 −1.24007
\(695\) 25.3550 0.961772
\(696\) −8.73716 −0.331181
\(697\) −21.0029 −0.795542
\(698\) −29.1058 −1.10167
\(699\) −82.7178 −3.12867
\(700\) −0.638237 −0.0241231
\(701\) 3.46466 0.130859 0.0654293 0.997857i \(-0.479158\pi\)
0.0654293 + 0.997857i \(0.479158\pi\)
\(702\) 1.53367 0.0578848
\(703\) 0 0
\(704\) 5.75621 0.216946
\(705\) 28.7829 1.08403
\(706\) −14.3049 −0.538371
\(707\) −4.72113 −0.177556
\(708\) 9.25731 0.347911
\(709\) 20.5743 0.772684 0.386342 0.922356i \(-0.373738\pi\)
0.386342 + 0.922356i \(0.373738\pi\)
\(710\) 10.3529 0.388539
\(711\) 41.2039 1.54527
\(712\) −15.0765 −0.565017
\(713\) −41.7268 −1.56268
\(714\) −15.0167 −0.561988
\(715\) −4.10656 −0.153577
\(716\) −8.45089 −0.315825
\(717\) −69.0115 −2.57728
\(718\) 19.6707 0.734104
\(719\) 24.1885 0.902080 0.451040 0.892504i \(-0.351053\pi\)
0.451040 + 0.892504i \(0.351053\pi\)
\(720\) −11.2639 −0.419780
\(721\) −24.4149 −0.909257
\(722\) 0 0
\(723\) 76.9880 2.86321
\(724\) 8.48526 0.315352
\(725\) −1.55452 −0.0577334
\(726\) 61.8337 2.29486
\(727\) 18.1276 0.672317 0.336158 0.941806i \(-0.390872\pi\)
0.336158 + 0.941806i \(0.390872\pi\)
\(728\) −0.390723 −0.0144811
\(729\) −43.8372 −1.62360
\(730\) 5.31490 0.196713
\(731\) −4.14173 −0.153187
\(732\) −30.6220 −1.13182
\(733\) 22.7363 0.839785 0.419893 0.907574i \(-0.362068\pi\)
0.419893 + 0.907574i \(0.362068\pi\)
\(734\) 9.55053 0.352517
\(735\) 35.0490 1.29280
\(736\) −6.47684 −0.238739
\(737\) −25.2169 −0.928877
\(738\) −24.1037 −0.887269
\(739\) 15.1787 0.558356 0.279178 0.960239i \(-0.409938\pi\)
0.279178 + 0.960239i \(0.409938\pi\)
\(740\) 9.33094 0.343012
\(741\) 0 0
\(742\) −4.22545 −0.155121
\(743\) −3.27625 −0.120194 −0.0600970 0.998193i \(-0.519141\pi\)
−0.0600970 + 0.998193i \(0.519141\pi\)
\(744\) 17.9977 0.659828
\(745\) 21.8839 0.801764
\(746\) 3.64010 0.133274
\(747\) −46.9116 −1.71640
\(748\) 24.0966 0.881060
\(749\) −12.8887 −0.470941
\(750\) 29.4936 1.07695
\(751\) 52.2323 1.90598 0.952992 0.302995i \(-0.0979866\pi\)
0.952992 + 0.302995i \(0.0979866\pi\)
\(752\) −4.39445 −0.160249
\(753\) −8.34884 −0.304248
\(754\) −0.951662 −0.0346575
\(755\) −46.4769 −1.69147
\(756\) −6.47214 −0.235389
\(757\) −24.2342 −0.880807 −0.440404 0.897800i \(-0.645165\pi\)
−0.440404 + 0.897800i \(0.645165\pi\)
\(758\) 8.34741 0.303192
\(759\) −104.151 −3.78045
\(760\) 0 0
\(761\) −24.8054 −0.899195 −0.449597 0.893231i \(-0.648433\pi\)
−0.449597 + 0.893231i \(0.648433\pi\)
\(762\) 6.23015 0.225694
\(763\) 11.3557 0.411105
\(764\) −22.3790 −0.809644
\(765\) −47.1527 −1.70481
\(766\) −9.48566 −0.342731
\(767\) 1.00832 0.0364082
\(768\) 2.79360 0.100806
\(769\) 9.64289 0.347732 0.173866 0.984769i \(-0.444374\pi\)
0.173866 + 0.984769i \(0.444374\pi\)
\(770\) 17.3298 0.624522
\(771\) 39.4799 1.42184
\(772\) 7.42811 0.267344
\(773\) 1.28731 0.0463014 0.0231507 0.999732i \(-0.492630\pi\)
0.0231507 + 0.999732i \(0.492630\pi\)
\(774\) −4.75320 −0.170850
\(775\) 3.20216 0.115025
\(776\) 17.3522 0.622907
\(777\) 14.2764 0.512162
\(778\) 36.0480 1.29238
\(779\) 0 0
\(780\) −1.99300 −0.0713607
\(781\) −25.4177 −0.909517
\(782\) −27.1133 −0.969569
\(783\) −15.7638 −0.563353
\(784\) −5.35114 −0.191112
\(785\) −32.5137 −1.16046
\(786\) 4.25369 0.151724
\(787\) −16.3446 −0.582624 −0.291312 0.956628i \(-0.594092\pi\)
−0.291312 + 0.956628i \(0.594092\pi\)
\(788\) 8.53554 0.304066
\(789\) 21.1444 0.752762
\(790\) −20.1085 −0.715427
\(791\) 18.2984 0.650617
\(792\) 27.6542 0.982648
\(793\) −3.33538 −0.118443
\(794\) −6.40906 −0.227449
\(795\) −21.5531 −0.764410
\(796\) −6.35926 −0.225398
\(797\) −15.6884 −0.555712 −0.277856 0.960623i \(-0.589624\pi\)
−0.277856 + 0.960623i \(0.589624\pi\)
\(798\) 0 0
\(799\) −18.3960 −0.650804
\(800\) 0.497039 0.0175730
\(801\) −72.4311 −2.55923
\(802\) 0.187660 0.00662650
\(803\) −13.0487 −0.460479
\(804\) −12.2383 −0.431610
\(805\) −19.4993 −0.687260
\(806\) 1.96033 0.0690496
\(807\) −21.4290 −0.754337
\(808\) 3.67667 0.129345
\(809\) −11.5248 −0.405192 −0.202596 0.979262i \(-0.564938\pi\)
−0.202596 + 0.979262i \(0.564938\pi\)
\(810\) 0.778611 0.0273576
\(811\) −8.30102 −0.291488 −0.145744 0.989322i \(-0.546558\pi\)
−0.145744 + 0.989322i \(0.546558\pi\)
\(812\) 4.01603 0.140935
\(813\) 32.5871 1.14288
\(814\) −22.9086 −0.802945
\(815\) −33.4093 −1.17028
\(816\) 11.6946 0.409392
\(817\) 0 0
\(818\) 17.2663 0.603701
\(819\) −1.87712 −0.0655919
\(820\) 11.7632 0.410788
\(821\) −33.7112 −1.17653 −0.588265 0.808668i \(-0.700189\pi\)
−0.588265 + 0.808668i \(0.700189\pi\)
\(822\) 29.9244 1.04373
\(823\) 21.8678 0.762262 0.381131 0.924521i \(-0.375535\pi\)
0.381131 + 0.924521i \(0.375535\pi\)
\(824\) 19.0135 0.662368
\(825\) 7.99268 0.278269
\(826\) −4.25512 −0.148055
\(827\) 39.1695 1.36206 0.681028 0.732257i \(-0.261533\pi\)
0.681028 + 0.732257i \(0.261533\pi\)
\(828\) −31.1162 −1.08136
\(829\) −14.4176 −0.500743 −0.250371 0.968150i \(-0.580553\pi\)
−0.250371 + 0.968150i \(0.580553\pi\)
\(830\) 22.8940 0.794661
\(831\) 76.6579 2.65923
\(832\) 0.304282 0.0105491
\(833\) −22.4009 −0.776145
\(834\) −30.2110 −1.04612
\(835\) −27.1739 −0.940392
\(836\) 0 0
\(837\) 32.4719 1.12239
\(838\) −1.72572 −0.0596139
\(839\) 20.9532 0.723385 0.361693 0.932297i \(-0.382199\pi\)
0.361693 + 0.932297i \(0.382199\pi\)
\(840\) 8.41049 0.290189
\(841\) −19.2184 −0.662702
\(842\) 32.7546 1.12880
\(843\) 45.4561 1.56559
\(844\) −13.9578 −0.480447
\(845\) 30.2624 1.04106
\(846\) −21.1119 −0.725844
\(847\) −28.4218 −0.976585
\(848\) 3.29064 0.113001
\(849\) 9.48077 0.325379
\(850\) 2.08070 0.0713675
\(851\) 25.7765 0.883607
\(852\) −12.3357 −0.422614
\(853\) 51.9553 1.77892 0.889458 0.457017i \(-0.151082\pi\)
0.889458 + 0.457017i \(0.151082\pi\)
\(854\) 14.0754 0.481650
\(855\) 0 0
\(856\) 10.0373 0.343067
\(857\) 29.1835 0.996888 0.498444 0.866922i \(-0.333905\pi\)
0.498444 + 0.866922i \(0.333905\pi\)
\(858\) 4.89304 0.167046
\(859\) 21.9913 0.750335 0.375168 0.926957i \(-0.377585\pi\)
0.375168 + 0.926957i \(0.377585\pi\)
\(860\) 2.31967 0.0791002
\(861\) 17.9977 0.613360
\(862\) −25.5119 −0.868939
\(863\) −35.8157 −1.21918 −0.609590 0.792717i \(-0.708666\pi\)
−0.609590 + 0.792717i \(0.708666\pi\)
\(864\) 5.04029 0.171474
\(865\) −9.33968 −0.317559
\(866\) 24.4834 0.831980
\(867\) 1.46442 0.0497343
\(868\) −8.27263 −0.280791
\(869\) 49.3687 1.67472
\(870\) 20.4849 0.694505
\(871\) −1.33300 −0.0451671
\(872\) −8.84348 −0.299478
\(873\) 83.3638 2.82144
\(874\) 0 0
\(875\) −13.5567 −0.458300
\(876\) −6.33280 −0.213966
\(877\) 7.82483 0.264226 0.132113 0.991235i \(-0.457824\pi\)
0.132113 + 0.991235i \(0.457824\pi\)
\(878\) −30.8388 −1.04076
\(879\) 65.3097 2.20284
\(880\) −13.4959 −0.454946
\(881\) −14.4829 −0.487941 −0.243970 0.969783i \(-0.578450\pi\)
−0.243970 + 0.969783i \(0.578450\pi\)
\(882\) −25.7081 −0.865636
\(883\) −23.5529 −0.792619 −0.396310 0.918117i \(-0.629709\pi\)
−0.396310 + 0.918117i \(0.629709\pi\)
\(884\) 1.27378 0.0428420
\(885\) −21.7045 −0.729588
\(886\) 10.7047 0.359631
\(887\) −15.8736 −0.532982 −0.266491 0.963837i \(-0.585864\pi\)
−0.266491 + 0.963837i \(0.585864\pi\)
\(888\) −11.1180 −0.373095
\(889\) −2.86368 −0.0960449
\(890\) 35.3481 1.18487
\(891\) −1.91158 −0.0640404
\(892\) 8.92400 0.298798
\(893\) 0 0
\(894\) −26.0751 −0.872081
\(895\) 19.8138 0.662301
\(896\) −1.28408 −0.0428981
\(897\) −5.50560 −0.183827
\(898\) 2.31562 0.0772731
\(899\) −20.1492 −0.672013
\(900\) 2.38789 0.0795963
\(901\) 13.7753 0.458921
\(902\) −28.8800 −0.961599
\(903\) 3.54911 0.118107
\(904\) −14.2502 −0.473956
\(905\) −19.8944 −0.661311
\(906\) 55.3781 1.83981
\(907\) 45.0127 1.49462 0.747311 0.664474i \(-0.231344\pi\)
0.747311 + 0.664474i \(0.231344\pi\)
\(908\) −0.659796 −0.0218961
\(909\) 17.6635 0.585863
\(910\) 0.916079 0.0303677
\(911\) −2.75901 −0.0914100 −0.0457050 0.998955i \(-0.514553\pi\)
−0.0457050 + 0.998955i \(0.514553\pi\)
\(912\) 0 0
\(913\) −56.2074 −1.86019
\(914\) −0.0521810 −0.00172599
\(915\) 71.7956 2.37349
\(916\) −6.97208 −0.230364
\(917\) −1.95521 −0.0645667
\(918\) 21.0996 0.696392
\(919\) −39.3083 −1.29666 −0.648331 0.761359i \(-0.724533\pi\)
−0.648331 + 0.761359i \(0.724533\pi\)
\(920\) 15.1854 0.500649
\(921\) −3.87758 −0.127771
\(922\) 39.4720 1.29994
\(923\) −1.34362 −0.0442258
\(924\) −20.6487 −0.679294
\(925\) −1.97811 −0.0650400
\(926\) −6.19056 −0.203435
\(927\) 91.3453 3.00017
\(928\) −3.12756 −0.102667
\(929\) −29.7417 −0.975795 −0.487897 0.872901i \(-0.662236\pi\)
−0.487897 + 0.872901i \(0.662236\pi\)
\(930\) −42.1970 −1.38369
\(931\) 0 0
\(932\) −29.6097 −0.969898
\(933\) −34.0547 −1.11490
\(934\) −20.9962 −0.687015
\(935\) −56.4964 −1.84763
\(936\) 1.46184 0.0477818
\(937\) 37.7211 1.23229 0.616147 0.787631i \(-0.288693\pi\)
0.616147 + 0.787631i \(0.288693\pi\)
\(938\) 5.62531 0.183673
\(939\) 46.1822 1.50710
\(940\) 10.3031 0.336051
\(941\) −6.47652 −0.211129 −0.105564 0.994412i \(-0.533665\pi\)
−0.105564 + 0.994412i \(0.533665\pi\)
\(942\) 38.7407 1.26224
\(943\) 32.4955 1.05820
\(944\) 3.31375 0.107853
\(945\) 15.1744 0.493624
\(946\) −5.69507 −0.185163
\(947\) 16.2913 0.529396 0.264698 0.964331i \(-0.414728\pi\)
0.264698 + 0.964331i \(0.414728\pi\)
\(948\) 23.9596 0.778171
\(949\) −0.689776 −0.0223911
\(950\) 0 0
\(951\) 7.50953 0.243513
\(952\) −5.37540 −0.174218
\(953\) −34.6213 −1.12150 −0.560748 0.827987i \(-0.689486\pi\)
−0.560748 + 0.827987i \(0.689486\pi\)
\(954\) 15.8090 0.511835
\(955\) 52.4693 1.69787
\(956\) −24.7034 −0.798965
\(957\) −50.2930 −1.62574
\(958\) 15.5229 0.501523
\(959\) −13.7547 −0.444164
\(960\) −6.54982 −0.211395
\(961\) 10.5053 0.338882
\(962\) −1.21098 −0.0390436
\(963\) 48.2213 1.55391
\(964\) 27.5586 0.887604
\(965\) −17.4158 −0.560634
\(966\) 23.2338 0.747534
\(967\) −48.9105 −1.57286 −0.786428 0.617682i \(-0.788072\pi\)
−0.786428 + 0.617682i \(0.788072\pi\)
\(968\) 22.1340 0.711414
\(969\) 0 0
\(970\) −40.6835 −1.30627
\(971\) 6.07592 0.194986 0.0974928 0.995236i \(-0.468918\pi\)
0.0974928 + 0.995236i \(0.468918\pi\)
\(972\) −16.0486 −0.514759
\(973\) 13.8865 0.445180
\(974\) −28.6065 −0.916613
\(975\) 0.422505 0.0135310
\(976\) −10.9615 −0.350868
\(977\) 57.0477 1.82512 0.912559 0.408944i \(-0.134103\pi\)
0.912559 + 0.408944i \(0.134103\pi\)
\(978\) 39.8077 1.27291
\(979\) −86.7838 −2.77362
\(980\) 12.5462 0.400772
\(981\) −42.4861 −1.35648
\(982\) −30.3488 −0.968470
\(983\) 0.186519 0.00594905 0.00297452 0.999996i \(-0.499053\pi\)
0.00297452 + 0.999996i \(0.499053\pi\)
\(984\) −14.0160 −0.446815
\(985\) −20.0122 −0.637642
\(986\) −13.0926 −0.416952
\(987\) 15.7638 0.501768
\(988\) 0 0
\(989\) 6.40804 0.203764
\(990\) −64.8373 −2.06066
\(991\) −46.2398 −1.46886 −0.734428 0.678686i \(-0.762550\pi\)
−0.734428 + 0.678686i \(0.762550\pi\)
\(992\) 6.44246 0.204548
\(993\) 14.5631 0.462147
\(994\) 5.67010 0.179845
\(995\) 14.9098 0.472671
\(996\) −27.2786 −0.864354
\(997\) −20.8490 −0.660295 −0.330147 0.943929i \(-0.607098\pi\)
−0.330147 + 0.943929i \(0.607098\pi\)
\(998\) 8.77624 0.277807
\(999\) −20.0593 −0.634650
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 722.2.a.n.1.4 yes 4
3.2 odd 2 6498.2.a.bx.1.3 4
4.3 odd 2 5776.2.a.bt.1.1 4
19.2 odd 18 722.2.e.s.99.1 24
19.3 odd 18 722.2.e.s.389.1 24
19.4 even 9 722.2.e.r.415.1 24
19.5 even 9 722.2.e.r.595.1 24
19.6 even 9 722.2.e.r.245.4 24
19.7 even 3 722.2.c.m.429.1 8
19.8 odd 6 722.2.c.n.653.4 8
19.9 even 9 722.2.e.r.423.4 24
19.10 odd 18 722.2.e.s.423.1 24
19.11 even 3 722.2.c.m.653.1 8
19.12 odd 6 722.2.c.n.429.4 8
19.13 odd 18 722.2.e.s.245.1 24
19.14 odd 18 722.2.e.s.595.4 24
19.15 odd 18 722.2.e.s.415.4 24
19.16 even 9 722.2.e.r.389.4 24
19.17 even 9 722.2.e.r.99.4 24
19.18 odd 2 722.2.a.m.1.1 4
57.56 even 2 6498.2.a.ca.1.3 4
76.75 even 2 5776.2.a.bv.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
722.2.a.m.1.1 4 19.18 odd 2
722.2.a.n.1.4 yes 4 1.1 even 1 trivial
722.2.c.m.429.1 8 19.7 even 3
722.2.c.m.653.1 8 19.11 even 3
722.2.c.n.429.4 8 19.12 odd 6
722.2.c.n.653.4 8 19.8 odd 6
722.2.e.r.99.4 24 19.17 even 9
722.2.e.r.245.4 24 19.6 even 9
722.2.e.r.389.4 24 19.16 even 9
722.2.e.r.415.1 24 19.4 even 9
722.2.e.r.423.4 24 19.9 even 9
722.2.e.r.595.1 24 19.5 even 9
722.2.e.s.99.1 24 19.2 odd 18
722.2.e.s.245.1 24 19.13 odd 18
722.2.e.s.389.1 24 19.3 odd 18
722.2.e.s.415.4 24 19.15 odd 18
722.2.e.s.423.1 24 19.10 odd 18
722.2.e.s.595.4 24 19.14 odd 18
5776.2.a.bt.1.1 4 4.3 odd 2
5776.2.a.bv.1.4 4 76.75 even 2
6498.2.a.bx.1.3 4 3.2 odd 2
6498.2.a.ca.1.3 4 57.56 even 2