Properties

Label 722.2.a.m.1.1
Level $722$
Weight $2$
Character 722.1
Self dual yes
Analytic conductor $5.765$
Analytic rank $1$
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: \(1\)
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 1.1
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 \(-0.798611\pi\)
−0.806444 + 0.591310i \(0.798611\pi\)
\(4\) 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 \(-0.513436\pi\)
−0.0421964 + 0.999109i \(0.513436\pi\)
\(14\) 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 \(-0.406216\pi\)
0.290387 + 0.956909i \(0.406216\pi\)
\(30\) −6.54982 −1.19583
\(31\) −6.44246 −1.15710 −0.578550 0.815647i \(-0.696381\pi\)
−0.578550 + 0.815647i \(0.696381\pi\)
\(32\) −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 \(-0.393916\pi\)
0.327137 + 0.944977i \(0.393916\pi\)
\(38\) 0 0
\(39\) 0.850045 0.136116
\(40\) 2.34458 0.370710
\(41\) 5.01719 0.783553 0.391776 0.920060i \(-0.371861\pi\)
0.391776 + 0.920060i \(0.371861\pi\)
\(42\) −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 \(-0.572566\pi\)
−0.226002 + 0.974127i \(0.572566\pi\)
\(54\) 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 \(-0.569206\pi\)
−0.215707 + 0.976458i \(0.569206\pi\)
\(60\) 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 \(-0.413769\pi\)
0.267601 + 0.963530i \(0.413769\pi\)
\(68\) 4.18619 0.507650
\(69\) 18.0937 2.17823
\(70\) −3.01062 −0.359838
\(71\) 4.41570 0.524047 0.262023 0.965062i \(-0.415610\pi\)
0.262023 + 0.965062i \(0.415610\pi\)
\(72\) −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 \(-0.660261\pi\)
−0.482471 + 0.875912i \(0.660261\pi\)
\(80\) −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 \(-0.205334\pi\)
0.799055 + 0.601259i \(0.205334\pi\)
\(90\) 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 \(-0.843078\pi\)
−0.880924 + 0.473259i \(0.843078\pi\)
\(98\) 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 \(-0.886163\pi\)
−0.936729 + 0.350055i \(0.886163\pi\)
\(104\) 0.304282 0.0298374
\(105\) −8.41049 −0.820779
\(106\) 3.29064 0.319616
\(107\) −10.0373 −0.970340 −0.485170 0.874420i \(-0.661242\pi\)
−0.485170 + 0.874420i \(0.661242\pi\)
\(108\) −5.04029 −0.485003
\(109\) 8.84348 0.847052 0.423526 0.905884i \(-0.360792\pi\)
0.423526 + 0.905884i \(0.360792\pi\)
\(110\) 13.4959 1.28678
\(111\) −11.1180 −1.05527
\(112\) −1.28408 −0.121334
\(113\) 14.2502 1.34055 0.670275 0.742113i \(-0.266176\pi\)
0.670275 + 0.742113i \(0.266176\pi\)
\(114\) 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 \(-0.531547\pi\)
−0.0989467 + 0.995093i \(0.531547\pi\)
\(128\) −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 \(-0.798691\pi\)
−0.806593 + 0.591107i \(0.798691\pi\)
\(152\) 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 \(-0.648018\pi\)
−0.448435 + 0.893815i \(0.648018\pi\)
\(168\) −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 \(-0.548388\pi\)
−0.151431 + 0.988468i \(0.548388\pi\)
\(174\) 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 \(-0.397719\pi\)
0.315825 + 0.948818i \(0.397719\pi\)
\(180\) −11.2639 −0.839560
\(181\) −8.48526 −0.630705 −0.315352 0.948975i \(-0.602123\pi\)
−0.315352 + 0.948975i \(0.602123\pi\)
\(182\) −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 \(-0.586146\pi\)
−0.267344 + 0.963601i \(0.586146\pi\)
\(194\) 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 \(-0.340474\pi\)
0.480447 + 0.877024i \(0.340474\pi\)
\(212\) −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 \(-0.596586\pi\)
−0.298798 + 0.954317i \(0.596586\pi\)
\(224\) 1.28408 0.0857961
\(225\) 2.38789 0.159193
\(226\) −14.2502 −0.947912
\(227\) 0.659796 0.0437922 0.0218961 0.999760i \(-0.493030\pi\)
0.0218961 + 0.999760i \(0.493030\pi\)
\(228\) 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 \(-0.847632\pi\)
−0.887604 + 0.460607i \(0.847632\pi\)
\(242\) −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 \(-0.645296\pi\)
−0.440773 + 0.897619i \(0.645296\pi\)
\(258\) −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 \(-0.424869\pi\)
0.233847 + 0.972273i \(0.424869\pi\)
\(270\) −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 \(-0.661303\pi\)
−0.485338 + 0.874327i \(0.661303\pi\)
\(282\) −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 \(-0.739276\pi\)
−0.682888 + 0.730523i \(0.739276\pi\)
\(294\) −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 \(-0.487389\pi\)
0.0396093 + 0.999215i \(0.487389\pi\)
\(308\) −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 \(-0.524052\pi\)
−0.0754897 + 0.997147i \(0.524052\pi\)
\(318\) −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 \(-0.545761\pi\)
−0.143267 + 0.989684i \(0.545761\pi\)
\(332\) −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 \(-0.240464\pi\)
0.727969 + 0.685610i \(0.240464\pi\)
\(338\) 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 \(-0.530042\pi\)
−0.0942387 + 0.995550i \(0.530042\pi\)
\(374\) −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 \(-0.568776\pi\)
−0.214389 + 0.976748i \(0.568776\pi\)
\(380\) 0 0
\(381\) 6.23015 0.319180
\(382\) 22.3790 1.14501
\(383\) 9.48566 0.484695 0.242347 0.970190i \(-0.422083\pi\)
0.242347 + 0.970190i \(0.422083\pi\)
\(384\) 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 \(-0.501491\pi\)
−0.00468564 + 0.999989i \(0.501491\pi\)
\(402\) 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 \(-0.640388\pi\)
−0.426881 + 0.904308i \(0.640388\pi\)
\(410\) 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 \(-0.794203\pi\)
−0.798180 + 0.602420i \(0.794203\pi\)
\(422\) −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 \(-0.289385\pi\)
0.614433 + 0.788969i \(0.289385\pi\)
\(432\) −5.04029 −0.242501
\(433\) −24.4834 −1.17660 −0.588299 0.808644i \(-0.700202\pi\)
−0.588299 + 0.808644i \(0.700202\pi\)
\(434\) −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 \(-0.236746\pi\)
0.735928 + 0.677060i \(0.236746\pi\)
\(440\) 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 \(-0.517401\pi\)
−0.0546403 + 0.998506i \(0.517401\pi\)
\(450\) −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 \(-0.275546\pi\)
0.648143 + 0.761519i \(0.275546\pi\)
\(488\) 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 \(-0.663905\pi\)
−0.492468 + 0.870331i \(0.663905\pi\)
\(510\) −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 \(-0.570534\pi\)
−0.219782 + 0.975549i \(0.570534\pi\)
\(522\) −15.0255 −0.657648
\(523\) −29.2823 −1.28042 −0.640212 0.768198i \(-0.721154\pi\)
−0.640212 + 0.768198i \(0.721154\pi\)
\(524\) 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 \(-0.399692\pi\)
0.309936 + 0.950758i \(0.399692\pi\)
\(548\) 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 \(-0.304946\pi\)
0.575144 + 0.818052i \(0.304946\pi\)
\(564\) 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 \(-0.451841\pi\)
0.150719 + 0.988577i \(0.451841\pi\)
\(570\) 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 \(-0.301042\pi\)
0.585135 + 0.810936i \(0.301042\pi\)
\(600\) 1.38853 0.0566865
\(601\) −7.55865 −0.308324 −0.154162 0.988046i \(-0.549268\pi\)
−0.154162 + 0.988046i \(0.549268\pi\)
\(602\) −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 \(-0.511161\pi\)
−0.0350558 + 0.999385i \(0.511161\pi\)
\(608\) 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 \(-0.456028\pi\)
0.137703 + 0.990474i \(0.456028\pi\)
\(642\) −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 \(-0.328266\pi\)
0.513724 + 0.857956i \(0.328266\pi\)
\(660\) 37.7022 1.46755
\(661\) 24.6939 0.960483 0.480242 0.877136i \(-0.340549\pi\)
0.480242 + 0.877136i \(0.340549\pi\)
\(662\) 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 \(-0.439678\pi\)
0.188376 + 0.982097i \(0.439678\pi\)
\(674\) −26.7275 −1.02950
\(675\) −2.50522 −0.0964261
\(676\) −12.9074 −0.496439
\(677\) 21.5747 0.829184 0.414592 0.910007i \(-0.363924\pi\)
0.414592 + 0.910007i \(0.363924\pi\)
\(678\) 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 \(-0.719476\pi\)
−0.636155 + 0.771561i \(0.719476\pi\)
\(684\) 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 \(-0.480859\pi\)
0.0600970 + 0.998193i \(0.480859\pi\)
\(744\) −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 \(-0.902013\pi\)
−0.952992 + 0.302995i \(0.902013\pi\)
\(752\) −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 \(-0.507370\pi\)
−0.0231507 + 0.999732i \(0.507370\pi\)
\(774\) 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 \(-0.405908\pi\)
0.291312 + 0.956628i \(0.405908\pi\)
\(788\) 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 \(-0.410376\pi\)
0.277856 + 0.960623i \(0.410376\pi\)
\(798\) 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 \(-0.453442\pi\)
0.145744 + 0.989322i \(0.453442\pi\)
\(812\) −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 \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(828\) −31.1162 −1.08136
\(829\) 14.4176 0.500743 0.250371 0.968150i \(-0.419447\pi\)
0.250371 + 0.968150i \(0.419447\pi\)
\(830\) −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 \(-0.617801\pi\)
−0.361693 + 0.932297i \(0.617801\pi\)
\(840\) 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 \(-0.666095\pi\)
−0.498444 + 0.866922i \(0.666095\pi\)
\(858\) −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 \(-0.291334\pi\)
0.609590 + 0.792717i \(0.291334\pi\)
\(864\) 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 \(-0.542176\pi\)
−0.132113 + 0.991235i \(0.542176\pi\)
\(878\) −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 \(-0.414136\pi\)
0.266491 + 0.963837i \(0.414136\pi\)
\(888\) 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 \(-0.768656\pi\)
−0.747311 + 0.664474i \(0.768656\pi\)
\(908\) 0.659796 0.0218961
\(909\) 17.6635 0.585863
\(910\) 0.916079 0.0303677
\(911\) 2.75901 0.0914100 0.0457050 0.998955i \(-0.485447\pi\)
0.0457050 + 0.998955i \(0.485447\pi\)
\(912\) 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 \(-0.466335\pi\)
0.105564 + 0.994412i \(0.466335\pi\)
\(942\) 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 \(-0.310514\pi\)
0.560748 + 0.827987i \(0.310514\pi\)
\(954\) 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 \(-0.531082\pi\)
−0.0974928 + 0.995236i \(0.531082\pi\)
\(972\) 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 \(-0.865897\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(978\) 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 \(-0.500947\pi\)
−0.00297452 + 0.999996i \(0.500947\pi\)
\(984\) 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 \(-0.237450\pi\)
0.734428 + 0.678686i \(0.237450\pi\)
\(992\) 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.m.1.1 4
3.2 odd 2 6498.2.a.ca.1.3 4
4.3 odd 2 5776.2.a.bv.1.4 4
19.2 odd 18 722.2.e.r.99.4 24
19.3 odd 18 722.2.e.r.389.4 24
19.4 even 9 722.2.e.s.415.4 24
19.5 even 9 722.2.e.s.595.4 24
19.6 even 9 722.2.e.s.245.1 24
19.7 even 3 722.2.c.n.429.4 8
19.8 odd 6 722.2.c.m.653.1 8
19.9 even 9 722.2.e.s.423.1 24
19.10 odd 18 722.2.e.r.423.4 24
19.11 even 3 722.2.c.n.653.4 8
19.12 odd 6 722.2.c.m.429.1 8
19.13 odd 18 722.2.e.r.245.4 24
19.14 odd 18 722.2.e.r.595.1 24
19.15 odd 18 722.2.e.r.415.1 24
19.16 even 9 722.2.e.s.389.1 24
19.17 even 9 722.2.e.s.99.1 24
19.18 odd 2 722.2.a.n.1.4 yes 4
57.56 even 2 6498.2.a.bx.1.3 4
76.75 even 2 5776.2.a.bt.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
722.2.a.m.1.1 4 1.1 even 1 trivial
722.2.a.n.1.4 yes 4 19.18 odd 2
722.2.c.m.429.1 8 19.12 odd 6
722.2.c.m.653.1 8 19.8 odd 6
722.2.c.n.429.4 8 19.7 even 3
722.2.c.n.653.4 8 19.11 even 3
722.2.e.r.99.4 24 19.2 odd 18
722.2.e.r.245.4 24 19.13 odd 18
722.2.e.r.389.4 24 19.3 odd 18
722.2.e.r.415.1 24 19.15 odd 18
722.2.e.r.423.4 24 19.10 odd 18
722.2.e.r.595.1 24 19.14 odd 18
722.2.e.s.99.1 24 19.17 even 9
722.2.e.s.245.1 24 19.6 even 9
722.2.e.s.389.1 24 19.16 even 9
722.2.e.s.415.4 24 19.4 even 9
722.2.e.s.423.1 24 19.9 even 9
722.2.e.s.595.4 24 19.5 even 9
5776.2.a.bt.1.1 4 76.75 even 2
5776.2.a.bv.1.4 4 4.3 odd 2
6498.2.a.bx.1.3 4 57.56 even 2
6498.2.a.ca.1.3 4 3.2 odd 2