Properties

Label 1183.2.a.p.1.5
Level $1183$
Weight $2$
Character 1183.1
Self dual yes
Analytic conductor $9.446$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.7674048.1
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.10939\) of defining polynomial
Character \(\chi\) \(=\) 1183.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.10939 q^{2} +2.26165 q^{3} +2.44952 q^{4} +3.60178 q^{5} +4.77070 q^{6} -1.00000 q^{7} +0.948212 q^{8} +2.11505 q^{9} +O(q^{10})\) \(q+2.10939 q^{2} +2.26165 q^{3} +2.44952 q^{4} +3.60178 q^{5} +4.77070 q^{6} -1.00000 q^{7} +0.948212 q^{8} +2.11505 q^{9} +7.59755 q^{10} -0.886384 q^{11} +5.53995 q^{12} -2.10939 q^{14} +8.14596 q^{15} -2.89889 q^{16} -4.96016 q^{17} +4.46147 q^{18} -2.37878 q^{19} +8.82263 q^{20} -2.26165 q^{21} -1.86973 q^{22} -3.85851 q^{23} +2.14452 q^{24} +7.97282 q^{25} -2.00144 q^{27} -2.44952 q^{28} +1.28197 q^{29} +17.1830 q^{30} -8.46921 q^{31} -8.01131 q^{32} -2.00469 q^{33} -10.4629 q^{34} -3.60178 q^{35} +5.18087 q^{36} +9.63812 q^{37} -5.01776 q^{38} +3.41525 q^{40} +12.0841 q^{41} -4.77070 q^{42} -3.64250 q^{43} -2.17122 q^{44} +7.61796 q^{45} -8.13910 q^{46} -2.98229 q^{47} -6.55628 q^{48} +1.00000 q^{49} +16.8178 q^{50} -11.2181 q^{51} +4.92032 q^{53} -4.22181 q^{54} -3.19256 q^{55} -0.948212 q^{56} -5.37995 q^{57} +2.70418 q^{58} +7.32746 q^{59} +19.9537 q^{60} -1.53926 q^{61} -17.8648 q^{62} -2.11505 q^{63} -11.1012 q^{64} -4.22867 q^{66} +8.42649 q^{67} -12.1500 q^{68} -8.72660 q^{69} -7.59755 q^{70} +6.44888 q^{71} +2.00552 q^{72} +7.14859 q^{73} +20.3305 q^{74} +18.0317 q^{75} -5.82686 q^{76} +0.886384 q^{77} +0.757551 q^{79} -10.4412 q^{80} -10.8717 q^{81} +25.4901 q^{82} +4.76766 q^{83} -5.53995 q^{84} -17.8654 q^{85} -7.68344 q^{86} +2.89937 q^{87} -0.840480 q^{88} -3.61884 q^{89} +16.0692 q^{90} -9.45150 q^{92} -19.1544 q^{93} -6.29081 q^{94} -8.56783 q^{95} -18.1188 q^{96} +0.463300 q^{97} +2.10939 q^{98} -1.87475 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 6 q^{7} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 6 q^{7} + 12 q^{8} + 4 q^{9} + 12 q^{10} + 4 q^{11} + 2 q^{12} - 4 q^{14} + 20 q^{15} + 8 q^{16} - 4 q^{17} - 16 q^{18} + 2 q^{19} + 26 q^{20} - 6 q^{22} - 12 q^{23} + 2 q^{24} + 10 q^{25} + 6 q^{27} - 4 q^{28} - 8 q^{29} + 8 q^{30} - 14 q^{31} + 8 q^{32} + 16 q^{33} - 2 q^{34} - 6 q^{35} - 10 q^{36} + 12 q^{37} - 2 q^{38} + 46 q^{40} + 28 q^{41} - 4 q^{42} + 2 q^{43} - 20 q^{44} + 16 q^{45} + 20 q^{46} + 14 q^{47} + 2 q^{48} + 6 q^{49} + 32 q^{50} - 26 q^{51} - 22 q^{53} + 14 q^{54} + 6 q^{55} - 12 q^{56} + 4 q^{58} - 2 q^{59} - 14 q^{61} - 4 q^{62} - 4 q^{63} + 26 q^{64} - 26 q^{66} + 24 q^{67} + 8 q^{68} + 4 q^{69} - 12 q^{70} + 4 q^{71} + 8 q^{72} + 36 q^{73} - 6 q^{74} + 46 q^{75} - 26 q^{76} - 4 q^{77} - 28 q^{79} + 36 q^{80} - 2 q^{81} + 14 q^{82} + 26 q^{83} - 2 q^{84} - 20 q^{85} - 24 q^{86} + 2 q^{87} - 14 q^{88} + 42 q^{89} - 12 q^{90} + 12 q^{92} - 4 q^{94} - 22 q^{95} - 42 q^{96} + 24 q^{97} + 4 q^{98} + 16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.10939 1.49156 0.745781 0.666191i \(-0.232076\pi\)
0.745781 + 0.666191i \(0.232076\pi\)
\(3\) 2.26165 1.30576 0.652882 0.757460i \(-0.273560\pi\)
0.652882 + 0.757460i \(0.273560\pi\)
\(4\) 2.44952 1.22476
\(5\) 3.60178 1.61076 0.805382 0.592756i \(-0.201960\pi\)
0.805382 + 0.592756i \(0.201960\pi\)
\(6\) 4.77070 1.94763
\(7\) −1.00000 −0.377964
\(8\) 0.948212 0.335243
\(9\) 2.11505 0.705018
\(10\) 7.59755 2.40256
\(11\) −0.886384 −0.267255 −0.133627 0.991032i \(-0.542663\pi\)
−0.133627 + 0.991032i \(0.542663\pi\)
\(12\) 5.53995 1.59925
\(13\) 0 0
\(14\) −2.10939 −0.563758
\(15\) 8.14596 2.10328
\(16\) −2.89889 −0.724723
\(17\) −4.96016 −1.20302 −0.601508 0.798867i \(-0.705433\pi\)
−0.601508 + 0.798867i \(0.705433\pi\)
\(18\) 4.46147 1.05158
\(19\) −2.37878 −0.545729 −0.272864 0.962053i \(-0.587971\pi\)
−0.272864 + 0.962053i \(0.587971\pi\)
\(20\) 8.82263 1.97280
\(21\) −2.26165 −0.493532
\(22\) −1.86973 −0.398628
\(23\) −3.85851 −0.804555 −0.402278 0.915518i \(-0.631781\pi\)
−0.402278 + 0.915518i \(0.631781\pi\)
\(24\) 2.14452 0.437749
\(25\) 7.97282 1.59456
\(26\) 0 0
\(27\) −2.00144 −0.385177
\(28\) −2.44952 −0.462916
\(29\) 1.28197 0.238056 0.119028 0.992891i \(-0.462022\pi\)
0.119028 + 0.992891i \(0.462022\pi\)
\(30\) 17.1830 3.13717
\(31\) −8.46921 −1.52111 −0.760557 0.649271i \(-0.775074\pi\)
−0.760557 + 0.649271i \(0.775074\pi\)
\(32\) −8.01131 −1.41621
\(33\) −2.00469 −0.348972
\(34\) −10.4629 −1.79437
\(35\) −3.60178 −0.608812
\(36\) 5.18087 0.863478
\(37\) 9.63812 1.58450 0.792249 0.610198i \(-0.208910\pi\)
0.792249 + 0.610198i \(0.208910\pi\)
\(38\) −5.01776 −0.813988
\(39\) 0 0
\(40\) 3.41525 0.539998
\(41\) 12.0841 1.88722 0.943612 0.331053i \(-0.107404\pi\)
0.943612 + 0.331053i \(0.107404\pi\)
\(42\) −4.77070 −0.736134
\(43\) −3.64250 −0.555476 −0.277738 0.960657i \(-0.589585\pi\)
−0.277738 + 0.960657i \(0.589585\pi\)
\(44\) −2.17122 −0.327323
\(45\) 7.61796 1.13562
\(46\) −8.13910 −1.20004
\(47\) −2.98229 −0.435012 −0.217506 0.976059i \(-0.569792\pi\)
−0.217506 + 0.976059i \(0.569792\pi\)
\(48\) −6.55628 −0.946317
\(49\) 1.00000 0.142857
\(50\) 16.8178 2.37839
\(51\) −11.2181 −1.57085
\(52\) 0 0
\(53\) 4.92032 0.675858 0.337929 0.941172i \(-0.390274\pi\)
0.337929 + 0.941172i \(0.390274\pi\)
\(54\) −4.22181 −0.574516
\(55\) −3.19256 −0.430485
\(56\) −0.948212 −0.126710
\(57\) −5.37995 −0.712592
\(58\) 2.70418 0.355076
\(59\) 7.32746 0.953954 0.476977 0.878916i \(-0.341732\pi\)
0.476977 + 0.878916i \(0.341732\pi\)
\(60\) 19.9537 2.57601
\(61\) −1.53926 −0.197082 −0.0985412 0.995133i \(-0.531418\pi\)
−0.0985412 + 0.995133i \(0.531418\pi\)
\(62\) −17.8648 −2.26884
\(63\) −2.11505 −0.266472
\(64\) −11.1012 −1.38765
\(65\) 0 0
\(66\) −4.22867 −0.520513
\(67\) 8.42649 1.02946 0.514730 0.857352i \(-0.327892\pi\)
0.514730 + 0.857352i \(0.327892\pi\)
\(68\) −12.1500 −1.47341
\(69\) −8.72660 −1.05056
\(70\) −7.59755 −0.908081
\(71\) 6.44888 0.765342 0.382671 0.923885i \(-0.375004\pi\)
0.382671 + 0.923885i \(0.375004\pi\)
\(72\) 2.00552 0.236353
\(73\) 7.14859 0.836679 0.418340 0.908291i \(-0.362612\pi\)
0.418340 + 0.908291i \(0.362612\pi\)
\(74\) 20.3305 2.36338
\(75\) 18.0317 2.08212
\(76\) −5.82686 −0.668386
\(77\) 0.886384 0.101013
\(78\) 0 0
\(79\) 0.757551 0.0852311 0.0426156 0.999092i \(-0.486431\pi\)
0.0426156 + 0.999092i \(0.486431\pi\)
\(80\) −10.4412 −1.16736
\(81\) −10.8717 −1.20797
\(82\) 25.4901 2.81491
\(83\) 4.76766 0.523319 0.261659 0.965160i \(-0.415730\pi\)
0.261659 + 0.965160i \(0.415730\pi\)
\(84\) −5.53995 −0.604458
\(85\) −17.8654 −1.93778
\(86\) −7.68344 −0.828527
\(87\) 2.89937 0.310845
\(88\) −0.840480 −0.0895955
\(89\) −3.61884 −0.383596 −0.191798 0.981434i \(-0.561432\pi\)
−0.191798 + 0.981434i \(0.561432\pi\)
\(90\) 16.0692 1.69385
\(91\) 0 0
\(92\) −9.45150 −0.985387
\(93\) −19.1544 −1.98622
\(94\) −6.29081 −0.648848
\(95\) −8.56783 −0.879040
\(96\) −18.1188 −1.84924
\(97\) 0.463300 0.0470409 0.0235205 0.999723i \(-0.492513\pi\)
0.0235205 + 0.999723i \(0.492513\pi\)
\(98\) 2.10939 0.213080
\(99\) −1.87475 −0.188419
\(100\) 19.5296 1.95296
\(101\) 5.82303 0.579413 0.289707 0.957115i \(-0.406442\pi\)
0.289707 + 0.957115i \(0.406442\pi\)
\(102\) −23.6634 −2.34303
\(103\) 8.23888 0.811801 0.405901 0.913917i \(-0.366958\pi\)
0.405901 + 0.913917i \(0.366958\pi\)
\(104\) 0 0
\(105\) −8.14596 −0.794964
\(106\) 10.3789 1.00809
\(107\) −3.83260 −0.370511 −0.185256 0.982690i \(-0.559311\pi\)
−0.185256 + 0.982690i \(0.559311\pi\)
\(108\) −4.90256 −0.471749
\(109\) 10.4180 0.997867 0.498934 0.866640i \(-0.333725\pi\)
0.498934 + 0.866640i \(0.333725\pi\)
\(110\) −6.73435 −0.642095
\(111\) 21.7980 2.06898
\(112\) 2.89889 0.273920
\(113\) −4.91011 −0.461904 −0.230952 0.972965i \(-0.574184\pi\)
−0.230952 + 0.972965i \(0.574184\pi\)
\(114\) −11.3484 −1.06288
\(115\) −13.8975 −1.29595
\(116\) 3.14022 0.291562
\(117\) 0 0
\(118\) 15.4565 1.42288
\(119\) 4.96016 0.454697
\(120\) 7.72409 0.705110
\(121\) −10.2143 −0.928575
\(122\) −3.24690 −0.293961
\(123\) 27.3301 2.46427
\(124\) −20.7455 −1.86300
\(125\) 10.7074 0.957702
\(126\) −4.46147 −0.397459
\(127\) −12.3102 −1.09235 −0.546175 0.837671i \(-0.683917\pi\)
−0.546175 + 0.837671i \(0.683917\pi\)
\(128\) −7.39409 −0.653551
\(129\) −8.23805 −0.725320
\(130\) 0 0
\(131\) −8.20265 −0.716669 −0.358335 0.933593i \(-0.616655\pi\)
−0.358335 + 0.933593i \(0.616655\pi\)
\(132\) −4.91053 −0.427406
\(133\) 2.37878 0.206266
\(134\) 17.7747 1.53550
\(135\) −7.20874 −0.620429
\(136\) −4.70328 −0.403303
\(137\) 7.45555 0.636971 0.318485 0.947928i \(-0.396826\pi\)
0.318485 + 0.947928i \(0.396826\pi\)
\(138\) −18.4078 −1.56697
\(139\) 16.6806 1.41483 0.707413 0.706800i \(-0.249862\pi\)
0.707413 + 0.706800i \(0.249862\pi\)
\(140\) −8.82263 −0.745648
\(141\) −6.74490 −0.568023
\(142\) 13.6032 1.14156
\(143\) 0 0
\(144\) −6.13131 −0.510943
\(145\) 4.61738 0.383453
\(146\) 15.0792 1.24796
\(147\) 2.26165 0.186538
\(148\) 23.6088 1.94063
\(149\) 2.52163 0.206580 0.103290 0.994651i \(-0.467063\pi\)
0.103290 + 0.994651i \(0.467063\pi\)
\(150\) 38.0359 3.10562
\(151\) 15.8972 1.29370 0.646849 0.762618i \(-0.276086\pi\)
0.646849 + 0.762618i \(0.276086\pi\)
\(152\) −2.25558 −0.182952
\(153\) −10.4910 −0.848148
\(154\) 1.86973 0.150667
\(155\) −30.5042 −2.45016
\(156\) 0 0
\(157\) 12.9831 1.03616 0.518082 0.855331i \(-0.326646\pi\)
0.518082 + 0.855331i \(0.326646\pi\)
\(158\) 1.59797 0.127128
\(159\) 11.1280 0.882511
\(160\) −28.8550 −2.28119
\(161\) 3.85851 0.304093
\(162\) −22.9327 −1.80176
\(163\) −2.31948 −0.181676 −0.0908378 0.995866i \(-0.528954\pi\)
−0.0908378 + 0.995866i \(0.528954\pi\)
\(164\) 29.6003 2.31140
\(165\) −7.22045 −0.562111
\(166\) 10.0569 0.780563
\(167\) 13.7918 1.06724 0.533622 0.845723i \(-0.320831\pi\)
0.533622 + 0.845723i \(0.320831\pi\)
\(168\) −2.14452 −0.165453
\(169\) 0 0
\(170\) −37.6851 −2.89031
\(171\) −5.03124 −0.384748
\(172\) −8.92237 −0.680324
\(173\) −3.68432 −0.280113 −0.140057 0.990143i \(-0.544728\pi\)
−0.140057 + 0.990143i \(0.544728\pi\)
\(174\) 6.11590 0.463645
\(175\) −7.97282 −0.602688
\(176\) 2.56953 0.193686
\(177\) 16.5721 1.24564
\(178\) −7.63353 −0.572157
\(179\) −5.89277 −0.440446 −0.220223 0.975450i \(-0.570678\pi\)
−0.220223 + 0.975450i \(0.570678\pi\)
\(180\) 18.6603 1.39086
\(181\) 2.11543 0.157239 0.0786193 0.996905i \(-0.474949\pi\)
0.0786193 + 0.996905i \(0.474949\pi\)
\(182\) 0 0
\(183\) −3.48127 −0.257343
\(184\) −3.65869 −0.269722
\(185\) 34.7144 2.55225
\(186\) −40.4040 −2.96257
\(187\) 4.39661 0.321512
\(188\) −7.30518 −0.532785
\(189\) 2.00144 0.145583
\(190\) −18.0729 −1.31114
\(191\) −11.3667 −0.822462 −0.411231 0.911531i \(-0.634901\pi\)
−0.411231 + 0.911531i \(0.634901\pi\)
\(192\) −25.1070 −1.81194
\(193\) −14.0894 −1.01417 −0.507087 0.861895i \(-0.669278\pi\)
−0.507087 + 0.861895i \(0.669278\pi\)
\(194\) 0.977279 0.0701645
\(195\) 0 0
\(196\) 2.44952 0.174966
\(197\) −22.9571 −1.63563 −0.817814 0.575482i \(-0.804814\pi\)
−0.817814 + 0.575482i \(0.804814\pi\)
\(198\) −3.95458 −0.281040
\(199\) −3.14985 −0.223287 −0.111643 0.993748i \(-0.535611\pi\)
−0.111643 + 0.993748i \(0.535611\pi\)
\(200\) 7.55992 0.534567
\(201\) 19.0578 1.34423
\(202\) 12.2830 0.864231
\(203\) −1.28197 −0.0899768
\(204\) −27.4791 −1.92392
\(205\) 43.5244 3.03987
\(206\) 17.3790 1.21085
\(207\) −8.16096 −0.567226
\(208\) 0 0
\(209\) 2.10851 0.145849
\(210\) −17.1830 −1.18574
\(211\) −14.8638 −1.02327 −0.511634 0.859203i \(-0.670960\pi\)
−0.511634 + 0.859203i \(0.670960\pi\)
\(212\) 12.0524 0.827764
\(213\) 14.5851 0.999355
\(214\) −8.08444 −0.552641
\(215\) −13.1195 −0.894741
\(216\) −1.89779 −0.129128
\(217\) 8.46921 0.574927
\(218\) 21.9757 1.48838
\(219\) 16.1676 1.09251
\(220\) −7.82024 −0.527241
\(221\) 0 0
\(222\) 45.9805 3.08601
\(223\) 4.38089 0.293366 0.146683 0.989184i \(-0.453140\pi\)
0.146683 + 0.989184i \(0.453140\pi\)
\(224\) 8.01131 0.535278
\(225\) 16.8629 1.12420
\(226\) −10.3573 −0.688959
\(227\) −13.5663 −0.900428 −0.450214 0.892921i \(-0.648652\pi\)
−0.450214 + 0.892921i \(0.648652\pi\)
\(228\) −13.1783 −0.872754
\(229\) −16.5180 −1.09154 −0.545770 0.837935i \(-0.683763\pi\)
−0.545770 + 0.837935i \(0.683763\pi\)
\(230\) −29.3152 −1.93299
\(231\) 2.00469 0.131899
\(232\) 1.21558 0.0798068
\(233\) −16.5026 −1.08112 −0.540561 0.841305i \(-0.681788\pi\)
−0.540561 + 0.841305i \(0.681788\pi\)
\(234\) 0 0
\(235\) −10.7416 −0.700702
\(236\) 17.9488 1.16836
\(237\) 1.71331 0.111292
\(238\) 10.4629 0.678210
\(239\) −30.4210 −1.96777 −0.983886 0.178796i \(-0.942780\pi\)
−0.983886 + 0.178796i \(0.942780\pi\)
\(240\) −23.6143 −1.52429
\(241\) −29.5143 −1.90119 −0.950593 0.310440i \(-0.899524\pi\)
−0.950593 + 0.310440i \(0.899524\pi\)
\(242\) −21.5460 −1.38503
\(243\) −18.5837 −1.19214
\(244\) −3.77046 −0.241379
\(245\) 3.60178 0.230109
\(246\) 57.6497 3.67561
\(247\) 0 0
\(248\) −8.03060 −0.509944
\(249\) 10.7828 0.683331
\(250\) 22.5861 1.42847
\(251\) −12.9827 −0.819459 −0.409730 0.912207i \(-0.634377\pi\)
−0.409730 + 0.912207i \(0.634377\pi\)
\(252\) −5.18087 −0.326364
\(253\) 3.42012 0.215021
\(254\) −25.9669 −1.62931
\(255\) −40.4053 −2.53028
\(256\) 6.60537 0.412836
\(257\) −4.58521 −0.286018 −0.143009 0.989721i \(-0.545678\pi\)
−0.143009 + 0.989721i \(0.545678\pi\)
\(258\) −17.3772 −1.08186
\(259\) −9.63812 −0.598884
\(260\) 0 0
\(261\) 2.71144 0.167834
\(262\) −17.3026 −1.06896
\(263\) −2.66499 −0.164330 −0.0821652 0.996619i \(-0.526184\pi\)
−0.0821652 + 0.996619i \(0.526184\pi\)
\(264\) −1.90087 −0.116990
\(265\) 17.7219 1.08865
\(266\) 5.01776 0.307659
\(267\) −8.18453 −0.500885
\(268\) 20.6409 1.26084
\(269\) 11.9256 0.727119 0.363559 0.931571i \(-0.381561\pi\)
0.363559 + 0.931571i \(0.381561\pi\)
\(270\) −15.2060 −0.925409
\(271\) 13.0283 0.791414 0.395707 0.918377i \(-0.370500\pi\)
0.395707 + 0.918377i \(0.370500\pi\)
\(272\) 14.3790 0.871853
\(273\) 0 0
\(274\) 15.7267 0.950082
\(275\) −7.06698 −0.426155
\(276\) −21.3760 −1.28668
\(277\) 21.3649 1.28369 0.641846 0.766833i \(-0.278169\pi\)
0.641846 + 0.766833i \(0.278169\pi\)
\(278\) 35.1858 2.11030
\(279\) −17.9128 −1.07241
\(280\) −3.41525 −0.204100
\(281\) −17.2678 −1.03011 −0.515054 0.857158i \(-0.672228\pi\)
−0.515054 + 0.857158i \(0.672228\pi\)
\(282\) −14.2276 −0.847242
\(283\) −21.2402 −1.26260 −0.631299 0.775539i \(-0.717478\pi\)
−0.631299 + 0.775539i \(0.717478\pi\)
\(284\) 15.7967 0.937360
\(285\) −19.3774 −1.14782
\(286\) 0 0
\(287\) −12.0841 −0.713304
\(288\) −16.9444 −0.998456
\(289\) 7.60320 0.447247
\(290\) 9.73985 0.571944
\(291\) 1.04782 0.0614243
\(292\) 17.5106 1.02473
\(293\) −0.420060 −0.0245402 −0.0122701 0.999925i \(-0.503906\pi\)
−0.0122701 + 0.999925i \(0.503906\pi\)
\(294\) 4.77070 0.278233
\(295\) 26.3919 1.53660
\(296\) 9.13898 0.531192
\(297\) 1.77404 0.102940
\(298\) 5.31910 0.308127
\(299\) 0 0
\(300\) 44.1690 2.55010
\(301\) 3.64250 0.209950
\(302\) 33.5335 1.92963
\(303\) 13.1696 0.756577
\(304\) 6.89581 0.395502
\(305\) −5.54409 −0.317453
\(306\) −22.1296 −1.26507
\(307\) 14.0807 0.803628 0.401814 0.915721i \(-0.368380\pi\)
0.401814 + 0.915721i \(0.368380\pi\)
\(308\) 2.17122 0.123717
\(309\) 18.6335 1.06002
\(310\) −64.3453 −3.65456
\(311\) 10.3848 0.588867 0.294434 0.955672i \(-0.404869\pi\)
0.294434 + 0.955672i \(0.404869\pi\)
\(312\) 0 0
\(313\) 6.84759 0.387048 0.193524 0.981096i \(-0.438008\pi\)
0.193524 + 0.981096i \(0.438008\pi\)
\(314\) 27.3864 1.54550
\(315\) −7.61796 −0.429223
\(316\) 1.85564 0.104388
\(317\) 0.701249 0.0393861 0.0196930 0.999806i \(-0.493731\pi\)
0.0196930 + 0.999806i \(0.493731\pi\)
\(318\) 23.4734 1.31632
\(319\) −1.13632 −0.0636217
\(320\) −39.9840 −2.23518
\(321\) −8.66799 −0.483800
\(322\) 8.13910 0.453574
\(323\) 11.7991 0.656520
\(324\) −26.6305 −1.47947
\(325\) 0 0
\(326\) −4.89268 −0.270980
\(327\) 23.5619 1.30298
\(328\) 11.4583 0.632680
\(329\) 2.98229 0.164419
\(330\) −15.2307 −0.838424
\(331\) −4.19865 −0.230778 −0.115389 0.993320i \(-0.536812\pi\)
−0.115389 + 0.993320i \(0.536812\pi\)
\(332\) 11.6785 0.640940
\(333\) 20.3851 1.11710
\(334\) 29.0923 1.59186
\(335\) 30.3504 1.65822
\(336\) 6.55628 0.357674
\(337\) 20.4278 1.11278 0.556388 0.830923i \(-0.312187\pi\)
0.556388 + 0.830923i \(0.312187\pi\)
\(338\) 0 0
\(339\) −11.1049 −0.603138
\(340\) −43.7617 −2.37331
\(341\) 7.50697 0.406525
\(342\) −10.6128 −0.573876
\(343\) −1.00000 −0.0539949
\(344\) −3.45386 −0.186220
\(345\) −31.4313 −1.69220
\(346\) −7.77165 −0.417807
\(347\) 7.97000 0.427852 0.213926 0.976850i \(-0.431375\pi\)
0.213926 + 0.976850i \(0.431375\pi\)
\(348\) 7.10207 0.380711
\(349\) −21.5972 −1.15607 −0.578037 0.816011i \(-0.696181\pi\)
−0.578037 + 0.816011i \(0.696181\pi\)
\(350\) −16.8178 −0.898947
\(351\) 0 0
\(352\) 7.10110 0.378490
\(353\) 21.6176 1.15059 0.575295 0.817946i \(-0.304887\pi\)
0.575295 + 0.817946i \(0.304887\pi\)
\(354\) 34.9571 1.85795
\(355\) 23.2275 1.23279
\(356\) −8.86441 −0.469813
\(357\) 11.2181 0.593727
\(358\) −12.4301 −0.656953
\(359\) −13.6834 −0.722180 −0.361090 0.932531i \(-0.617595\pi\)
−0.361090 + 0.932531i \(0.617595\pi\)
\(360\) 7.22344 0.380708
\(361\) −13.3414 −0.702180
\(362\) 4.46226 0.234531
\(363\) −23.1012 −1.21250
\(364\) 0 0
\(365\) 25.7477 1.34769
\(366\) −7.34335 −0.383843
\(367\) −11.4128 −0.595741 −0.297871 0.954606i \(-0.596276\pi\)
−0.297871 + 0.954606i \(0.596276\pi\)
\(368\) 11.1854 0.583080
\(369\) 25.5586 1.33053
\(370\) 73.2261 3.80685
\(371\) −4.92032 −0.255450
\(372\) −46.9190 −2.43264
\(373\) −31.2808 −1.61966 −0.809830 0.586664i \(-0.800441\pi\)
−0.809830 + 0.586664i \(0.800441\pi\)
\(374\) 9.27416 0.479555
\(375\) 24.2164 1.25053
\(376\) −2.82784 −0.145835
\(377\) 0 0
\(378\) 4.22181 0.217146
\(379\) 27.4151 1.40822 0.704108 0.710093i \(-0.251347\pi\)
0.704108 + 0.710093i \(0.251347\pi\)
\(380\) −20.9871 −1.07661
\(381\) −27.8412 −1.42635
\(382\) −23.9767 −1.22675
\(383\) −16.1006 −0.822705 −0.411352 0.911476i \(-0.634943\pi\)
−0.411352 + 0.911476i \(0.634943\pi\)
\(384\) −16.7228 −0.853383
\(385\) 3.19256 0.162708
\(386\) −29.7199 −1.51271
\(387\) −7.70408 −0.391620
\(388\) 1.13486 0.0576139
\(389\) 21.1380 1.07174 0.535870 0.844301i \(-0.319984\pi\)
0.535870 + 0.844301i \(0.319984\pi\)
\(390\) 0 0
\(391\) 19.1388 0.967893
\(392\) 0.948212 0.0478919
\(393\) −18.5515 −0.935800
\(394\) −48.4255 −2.43964
\(395\) 2.72853 0.137287
\(396\) −4.59224 −0.230769
\(397\) 13.0984 0.657390 0.328695 0.944436i \(-0.393391\pi\)
0.328695 + 0.944436i \(0.393391\pi\)
\(398\) −6.64426 −0.333046
\(399\) 5.37995 0.269335
\(400\) −23.1123 −1.15562
\(401\) −19.4447 −0.971022 −0.485511 0.874230i \(-0.661367\pi\)
−0.485511 + 0.874230i \(0.661367\pi\)
\(402\) 40.2002 2.00501
\(403\) 0 0
\(404\) 14.2636 0.709642
\(405\) −39.1575 −1.94575
\(406\) −2.70418 −0.134206
\(407\) −8.54308 −0.423465
\(408\) −10.6372 −0.526618
\(409\) −24.0559 −1.18949 −0.594743 0.803916i \(-0.702746\pi\)
−0.594743 + 0.803916i \(0.702746\pi\)
\(410\) 91.8098 4.53416
\(411\) 16.8618 0.831733
\(412\) 20.1813 0.994261
\(413\) −7.32746 −0.360561
\(414\) −17.2146 −0.846053
\(415\) 17.1721 0.842944
\(416\) 0 0
\(417\) 37.7256 1.84743
\(418\) 4.44767 0.217542
\(419\) 39.0238 1.90644 0.953218 0.302283i \(-0.0977487\pi\)
0.953218 + 0.302283i \(0.0977487\pi\)
\(420\) −19.9537 −0.973640
\(421\) −22.0284 −1.07360 −0.536799 0.843710i \(-0.680367\pi\)
−0.536799 + 0.843710i \(0.680367\pi\)
\(422\) −31.3536 −1.52627
\(423\) −6.30771 −0.306691
\(424\) 4.66551 0.226577
\(425\) −39.5465 −1.91828
\(426\) 30.7657 1.49060
\(427\) 1.53926 0.0744902
\(428\) −9.38803 −0.453787
\(429\) 0 0
\(430\) −27.6741 −1.33456
\(431\) −35.8797 −1.72826 −0.864131 0.503267i \(-0.832131\pi\)
−0.864131 + 0.503267i \(0.832131\pi\)
\(432\) 5.80195 0.279147
\(433\) 12.2136 0.586946 0.293473 0.955967i \(-0.405189\pi\)
0.293473 + 0.955967i \(0.405189\pi\)
\(434\) 17.8648 0.857540
\(435\) 10.4429 0.500699
\(436\) 25.5192 1.22215
\(437\) 9.17853 0.439069
\(438\) 34.1037 1.62954
\(439\) −15.7553 −0.751960 −0.375980 0.926628i \(-0.622694\pi\)
−0.375980 + 0.926628i \(0.622694\pi\)
\(440\) −3.02722 −0.144317
\(441\) 2.11505 0.100717
\(442\) 0 0
\(443\) −15.0706 −0.716028 −0.358014 0.933716i \(-0.616546\pi\)
−0.358014 + 0.933716i \(0.616546\pi\)
\(444\) 53.3947 2.53400
\(445\) −13.0342 −0.617883
\(446\) 9.24099 0.437574
\(447\) 5.70305 0.269745
\(448\) 11.1012 0.524482
\(449\) 30.7826 1.45272 0.726360 0.687315i \(-0.241211\pi\)
0.726360 + 0.687315i \(0.241211\pi\)
\(450\) 35.5705 1.67681
\(451\) −10.7112 −0.504370
\(452\) −12.0274 −0.565722
\(453\) 35.9540 1.68926
\(454\) −28.6166 −1.34304
\(455\) 0 0
\(456\) −5.10133 −0.238892
\(457\) −7.75597 −0.362809 −0.181405 0.983409i \(-0.558064\pi\)
−0.181405 + 0.983409i \(0.558064\pi\)
\(458\) −34.8429 −1.62810
\(459\) 9.92745 0.463374
\(460\) −34.0422 −1.58723
\(461\) −1.47222 −0.0685681 −0.0342840 0.999412i \(-0.510915\pi\)
−0.0342840 + 0.999412i \(0.510915\pi\)
\(462\) 4.22867 0.196735
\(463\) 14.0366 0.652335 0.326168 0.945312i \(-0.394243\pi\)
0.326168 + 0.945312i \(0.394243\pi\)
\(464\) −3.71630 −0.172525
\(465\) −68.9898 −3.19933
\(466\) −34.8104 −1.61256
\(467\) 31.3806 1.45212 0.726060 0.687631i \(-0.241349\pi\)
0.726060 + 0.687631i \(0.241349\pi\)
\(468\) 0 0
\(469\) −8.42649 −0.389099
\(470\) −22.6581 −1.04514
\(471\) 29.3632 1.35298
\(472\) 6.94798 0.319807
\(473\) 3.22865 0.148454
\(474\) 3.61404 0.165999
\(475\) −18.9655 −0.870199
\(476\) 12.1500 0.556895
\(477\) 10.4067 0.476492
\(478\) −64.1698 −2.93506
\(479\) −41.1951 −1.88225 −0.941125 0.338059i \(-0.890230\pi\)
−0.941125 + 0.338059i \(0.890230\pi\)
\(480\) −65.2598 −2.97869
\(481\) 0 0
\(482\) −62.2572 −2.83574
\(483\) 8.72660 0.397074
\(484\) −25.0202 −1.13728
\(485\) 1.66870 0.0757719
\(486\) −39.2002 −1.77816
\(487\) −28.3265 −1.28360 −0.641798 0.766874i \(-0.721811\pi\)
−0.641798 + 0.766874i \(0.721811\pi\)
\(488\) −1.45955 −0.0660706
\(489\) −5.24584 −0.237225
\(490\) 7.59755 0.343222
\(491\) 34.7863 1.56988 0.784941 0.619571i \(-0.212693\pi\)
0.784941 + 0.619571i \(0.212693\pi\)
\(492\) 66.9455 3.01814
\(493\) −6.35879 −0.286386
\(494\) 0 0
\(495\) −6.75244 −0.303499
\(496\) 24.5513 1.10239
\(497\) −6.44888 −0.289272
\(498\) 22.7451 1.01923
\(499\) −0.0694885 −0.00311073 −0.00155537 0.999999i \(-0.500495\pi\)
−0.00155537 + 0.999999i \(0.500495\pi\)
\(500\) 26.2281 1.17295
\(501\) 31.1923 1.39357
\(502\) −27.3855 −1.22228
\(503\) 25.7372 1.14756 0.573782 0.819008i \(-0.305476\pi\)
0.573782 + 0.819008i \(0.305476\pi\)
\(504\) −2.00552 −0.0893329
\(505\) 20.9733 0.933299
\(506\) 7.21437 0.320718
\(507\) 0 0
\(508\) −30.1540 −1.33787
\(509\) −7.04000 −0.312042 −0.156021 0.987754i \(-0.549867\pi\)
−0.156021 + 0.987754i \(0.549867\pi\)
\(510\) −85.2304 −3.77407
\(511\) −7.14859 −0.316235
\(512\) 28.7215 1.26932
\(513\) 4.76097 0.210202
\(514\) −9.67199 −0.426613
\(515\) 29.6746 1.30762
\(516\) −20.1793 −0.888342
\(517\) 2.64346 0.116259
\(518\) −20.3305 −0.893273
\(519\) −8.33263 −0.365762
\(520\) 0 0
\(521\) 16.3253 0.715225 0.357613 0.933870i \(-0.383591\pi\)
0.357613 + 0.933870i \(0.383591\pi\)
\(522\) 5.71948 0.250335
\(523\) −7.08946 −0.310000 −0.155000 0.987914i \(-0.549538\pi\)
−0.155000 + 0.987914i \(0.549538\pi\)
\(524\) −20.0926 −0.877748
\(525\) −18.0317 −0.786968
\(526\) −5.62150 −0.245109
\(527\) 42.0086 1.82993
\(528\) 5.81138 0.252908
\(529\) −8.11189 −0.352691
\(530\) 37.3824 1.62379
\(531\) 15.4980 0.672555
\(532\) 5.82686 0.252626
\(533\) 0 0
\(534\) −17.2644 −0.747102
\(535\) −13.8042 −0.596807
\(536\) 7.99010 0.345120
\(537\) −13.3274 −0.575118
\(538\) 25.1558 1.08454
\(539\) −0.886384 −0.0381793
\(540\) −17.6579 −0.759877
\(541\) 25.5162 1.09703 0.548515 0.836141i \(-0.315194\pi\)
0.548515 + 0.836141i \(0.315194\pi\)
\(542\) 27.4818 1.18044
\(543\) 4.78436 0.205316
\(544\) 39.7374 1.70373
\(545\) 37.5235 1.60733
\(546\) 0 0
\(547\) −13.3073 −0.568978 −0.284489 0.958679i \(-0.591824\pi\)
−0.284489 + 0.958679i \(0.591824\pi\)
\(548\) 18.2625 0.780136
\(549\) −3.25562 −0.138947
\(550\) −14.9070 −0.635637
\(551\) −3.04952 −0.129914
\(552\) −8.27466 −0.352193
\(553\) −0.757551 −0.0322143
\(554\) 45.0669 1.91471
\(555\) 78.5118 3.33264
\(556\) 40.8594 1.73282
\(557\) 17.0071 0.720612 0.360306 0.932834i \(-0.382672\pi\)
0.360306 + 0.932834i \(0.382672\pi\)
\(558\) −37.7851 −1.59957
\(559\) 0 0
\(560\) 10.4412 0.441220
\(561\) 9.94358 0.419818
\(562\) −36.4244 −1.53647
\(563\) −24.9193 −1.05022 −0.525111 0.851034i \(-0.675976\pi\)
−0.525111 + 0.851034i \(0.675976\pi\)
\(564\) −16.5218 −0.695691
\(565\) −17.6851 −0.744019
\(566\) −44.8038 −1.88325
\(567\) 10.8717 0.456569
\(568\) 6.11491 0.256576
\(569\) 5.88129 0.246557 0.123278 0.992372i \(-0.460659\pi\)
0.123278 + 0.992372i \(0.460659\pi\)
\(570\) −40.8745 −1.71204
\(571\) 8.92622 0.373551 0.186775 0.982403i \(-0.440196\pi\)
0.186775 + 0.982403i \(0.440196\pi\)
\(572\) 0 0
\(573\) −25.7074 −1.07394
\(574\) −25.4901 −1.06394
\(575\) −30.7632 −1.28291
\(576\) −23.4796 −0.978317
\(577\) 36.1933 1.50675 0.753374 0.657592i \(-0.228425\pi\)
0.753374 + 0.657592i \(0.228425\pi\)
\(578\) 16.0381 0.667097
\(579\) −31.8652 −1.32427
\(580\) 11.3104 0.469638
\(581\) −4.76766 −0.197796
\(582\) 2.21026 0.0916183
\(583\) −4.36130 −0.180626
\(584\) 6.77838 0.280491
\(585\) 0 0
\(586\) −0.886069 −0.0366032
\(587\) 36.4895 1.50608 0.753041 0.657974i \(-0.228586\pi\)
0.753041 + 0.657974i \(0.228586\pi\)
\(588\) 5.53995 0.228464
\(589\) 20.1463 0.830116
\(590\) 55.6708 2.29193
\(591\) −51.9210 −2.13574
\(592\) −27.9399 −1.14832
\(593\) 34.9930 1.43699 0.718495 0.695533i \(-0.244832\pi\)
0.718495 + 0.695533i \(0.244832\pi\)
\(594\) 3.74215 0.153542
\(595\) 17.8654 0.732410
\(596\) 6.17679 0.253011
\(597\) −7.12385 −0.291560
\(598\) 0 0
\(599\) −32.5052 −1.32812 −0.664062 0.747677i \(-0.731169\pi\)
−0.664062 + 0.747677i \(0.731169\pi\)
\(600\) 17.0979 0.698018
\(601\) 20.0780 0.819000 0.409500 0.912310i \(-0.365703\pi\)
0.409500 + 0.912310i \(0.365703\pi\)
\(602\) 7.68344 0.313154
\(603\) 17.8225 0.725788
\(604\) 38.9406 1.58447
\(605\) −36.7897 −1.49572
\(606\) 27.7799 1.12848
\(607\) 9.71601 0.394361 0.197180 0.980367i \(-0.436822\pi\)
0.197180 + 0.980367i \(0.436822\pi\)
\(608\) 19.0571 0.772868
\(609\) −2.89937 −0.117488
\(610\) −11.6946 −0.473502
\(611\) 0 0
\(612\) −25.6979 −1.03878
\(613\) −11.8816 −0.479893 −0.239947 0.970786i \(-0.577130\pi\)
−0.239947 + 0.970786i \(0.577130\pi\)
\(614\) 29.7017 1.19866
\(615\) 98.4368 3.96936
\(616\) 0.840480 0.0338639
\(617\) 19.9884 0.804705 0.402352 0.915485i \(-0.368193\pi\)
0.402352 + 0.915485i \(0.368193\pi\)
\(618\) 39.3052 1.58109
\(619\) 41.7176 1.67677 0.838386 0.545078i \(-0.183500\pi\)
0.838386 + 0.545078i \(0.183500\pi\)
\(620\) −74.7207 −3.00086
\(621\) 7.72257 0.309896
\(622\) 21.9056 0.878333
\(623\) 3.61884 0.144986
\(624\) 0 0
\(625\) −1.29828 −0.0519312
\(626\) 14.4442 0.577307
\(627\) 4.76871 0.190444
\(628\) 31.8023 1.26905
\(629\) −47.8066 −1.90618
\(630\) −16.0692 −0.640213
\(631\) 17.6415 0.702296 0.351148 0.936320i \(-0.385791\pi\)
0.351148 + 0.936320i \(0.385791\pi\)
\(632\) 0.718319 0.0285732
\(633\) −33.6168 −1.33615
\(634\) 1.47921 0.0587468
\(635\) −44.3385 −1.75952
\(636\) 27.2584 1.08086
\(637\) 0 0
\(638\) −2.39694 −0.0948958
\(639\) 13.6397 0.539580
\(640\) −26.6319 −1.05272
\(641\) −10.9202 −0.431324 −0.215662 0.976468i \(-0.569191\pi\)
−0.215662 + 0.976468i \(0.569191\pi\)
\(642\) −18.2842 −0.721618
\(643\) −17.6351 −0.695462 −0.347731 0.937594i \(-0.613048\pi\)
−0.347731 + 0.937594i \(0.613048\pi\)
\(644\) 9.45150 0.372441
\(645\) −29.6716 −1.16832
\(646\) 24.8889 0.979241
\(647\) 16.6726 0.655469 0.327735 0.944770i \(-0.393715\pi\)
0.327735 + 0.944770i \(0.393715\pi\)
\(648\) −10.3087 −0.404963
\(649\) −6.49495 −0.254949
\(650\) 0 0
\(651\) 19.1544 0.750719
\(652\) −5.68161 −0.222509
\(653\) −6.77329 −0.265059 −0.132530 0.991179i \(-0.542310\pi\)
−0.132530 + 0.991179i \(0.542310\pi\)
\(654\) 49.7013 1.94347
\(655\) −29.5441 −1.15439
\(656\) −35.0306 −1.36772
\(657\) 15.1197 0.589874
\(658\) 6.29081 0.245241
\(659\) 33.5361 1.30638 0.653190 0.757194i \(-0.273430\pi\)
0.653190 + 0.757194i \(0.273430\pi\)
\(660\) −17.6866 −0.688451
\(661\) 25.1661 0.978848 0.489424 0.872046i \(-0.337207\pi\)
0.489424 + 0.872046i \(0.337207\pi\)
\(662\) −8.85657 −0.344221
\(663\) 0 0
\(664\) 4.52075 0.175439
\(665\) 8.56783 0.332246
\(666\) 43.0002 1.66622
\(667\) −4.94651 −0.191529
\(668\) 33.7833 1.30712
\(669\) 9.90802 0.383066
\(670\) 64.0207 2.47334
\(671\) 1.36438 0.0526713
\(672\) 18.1188 0.698947
\(673\) −1.85468 −0.0714927 −0.0357464 0.999361i \(-0.511381\pi\)
−0.0357464 + 0.999361i \(0.511381\pi\)
\(674\) 43.0902 1.65977
\(675\) −15.9571 −0.614189
\(676\) 0 0
\(677\) −14.7209 −0.565770 −0.282885 0.959154i \(-0.591291\pi\)
−0.282885 + 0.959154i \(0.591291\pi\)
\(678\) −23.4246 −0.899618
\(679\) −0.463300 −0.0177798
\(680\) −16.9402 −0.649627
\(681\) −30.6822 −1.17575
\(682\) 15.8351 0.606358
\(683\) 7.94353 0.303951 0.151975 0.988384i \(-0.451437\pi\)
0.151975 + 0.988384i \(0.451437\pi\)
\(684\) −12.3241 −0.471224
\(685\) 26.8533 1.02601
\(686\) −2.10939 −0.0805368
\(687\) −37.3579 −1.42529
\(688\) 10.5592 0.402566
\(689\) 0 0
\(690\) −66.3008 −2.52403
\(691\) 10.2307 0.389193 0.194597 0.980883i \(-0.437660\pi\)
0.194597 + 0.980883i \(0.437660\pi\)
\(692\) −9.02481 −0.343072
\(693\) 1.87475 0.0712159
\(694\) 16.8118 0.638168
\(695\) 60.0797 2.27895
\(696\) 2.74922 0.104209
\(697\) −59.9392 −2.27036
\(698\) −45.5570 −1.72436
\(699\) −37.3231 −1.41169
\(700\) −19.5296 −0.738148
\(701\) 16.5978 0.626891 0.313445 0.949606i \(-0.398517\pi\)
0.313445 + 0.949606i \(0.398517\pi\)
\(702\) 0 0
\(703\) −22.9269 −0.864706
\(704\) 9.83992 0.370856
\(705\) −24.2936 −0.914951
\(706\) 45.5999 1.71618
\(707\) −5.82303 −0.218998
\(708\) 40.5938 1.52561
\(709\) 47.8659 1.79764 0.898820 0.438318i \(-0.144426\pi\)
0.898820 + 0.438318i \(0.144426\pi\)
\(710\) 48.9957 1.83878
\(711\) 1.60226 0.0600895
\(712\) −3.43142 −0.128598
\(713\) 32.6785 1.22382
\(714\) 23.6634 0.885581
\(715\) 0 0
\(716\) −14.4344 −0.539441
\(717\) −68.8017 −2.56944
\(718\) −28.8635 −1.07718
\(719\) −38.0922 −1.42060 −0.710300 0.703899i \(-0.751441\pi\)
−0.710300 + 0.703899i \(0.751441\pi\)
\(720\) −22.0836 −0.823009
\(721\) −8.23888 −0.306832
\(722\) −28.1423 −1.04735
\(723\) −66.7511 −2.48250
\(724\) 5.18179 0.192580
\(725\) 10.2209 0.379596
\(726\) −48.7294 −1.80852
\(727\) −15.4059 −0.571374 −0.285687 0.958323i \(-0.592222\pi\)
−0.285687 + 0.958323i \(0.592222\pi\)
\(728\) 0 0
\(729\) −9.41460 −0.348689
\(730\) 54.3118 2.01017
\(731\) 18.0674 0.668246
\(732\) −8.52744 −0.315183
\(733\) −11.6298 −0.429557 −0.214778 0.976663i \(-0.568903\pi\)
−0.214778 + 0.976663i \(0.568903\pi\)
\(734\) −24.0740 −0.888586
\(735\) 8.14596 0.300468
\(736\) 30.9117 1.13942
\(737\) −7.46911 −0.275128
\(738\) 53.9130 1.98456
\(739\) −2.68901 −0.0989168 −0.0494584 0.998776i \(-0.515750\pi\)
−0.0494584 + 0.998776i \(0.515750\pi\)
\(740\) 85.0336 3.12590
\(741\) 0 0
\(742\) −10.3789 −0.381020
\(743\) −2.46720 −0.0905126 −0.0452563 0.998975i \(-0.514410\pi\)
−0.0452563 + 0.998975i \(0.514410\pi\)
\(744\) −18.1624 −0.665866
\(745\) 9.08236 0.332752
\(746\) −65.9835 −2.41583
\(747\) 10.0839 0.368949
\(748\) 10.7696 0.393775
\(749\) 3.83260 0.140040
\(750\) 51.0819 1.86525
\(751\) −37.9185 −1.38366 −0.691832 0.722058i \(-0.743196\pi\)
−0.691832 + 0.722058i \(0.743196\pi\)
\(752\) 8.64534 0.315263
\(753\) −29.3623 −1.07002
\(754\) 0 0
\(755\) 57.2583 2.08384
\(756\) 4.90256 0.178304
\(757\) 34.6451 1.25920 0.629598 0.776921i \(-0.283219\pi\)
0.629598 + 0.776921i \(0.283219\pi\)
\(758\) 57.8290 2.10044
\(759\) 7.73512 0.280767
\(760\) −8.12411 −0.294692
\(761\) 22.8595 0.828655 0.414328 0.910128i \(-0.364017\pi\)
0.414328 + 0.910128i \(0.364017\pi\)
\(762\) −58.7280 −2.12749
\(763\) −10.4180 −0.377158
\(764\) −27.8428 −1.00732
\(765\) −37.7863 −1.36617
\(766\) −33.9625 −1.22712
\(767\) 0 0
\(768\) 14.9390 0.539065
\(769\) −51.8275 −1.86895 −0.934473 0.356034i \(-0.884129\pi\)
−0.934473 + 0.356034i \(0.884129\pi\)
\(770\) 6.73435 0.242689
\(771\) −10.3701 −0.373471
\(772\) −34.5122 −1.24212
\(773\) 5.69966 0.205003 0.102501 0.994733i \(-0.467315\pi\)
0.102501 + 0.994733i \(0.467315\pi\)
\(774\) −16.2509 −0.584126
\(775\) −67.5234 −2.42551
\(776\) 0.439306 0.0157702
\(777\) −21.7980 −0.782001
\(778\) 44.5883 1.59857
\(779\) −28.7454 −1.02991
\(780\) 0 0
\(781\) −5.71619 −0.204541
\(782\) 40.3712 1.44367
\(783\) −2.56579 −0.0916938
\(784\) −2.89889 −0.103532
\(785\) 46.7622 1.66902
\(786\) −39.1324 −1.39580
\(787\) −16.8141 −0.599358 −0.299679 0.954040i \(-0.596880\pi\)
−0.299679 + 0.954040i \(0.596880\pi\)
\(788\) −56.2340 −2.00325
\(789\) −6.02727 −0.214577
\(790\) 5.75553 0.204773
\(791\) 4.91011 0.174583
\(792\) −1.77766 −0.0631664
\(793\) 0 0
\(794\) 27.6296 0.980539
\(795\) 40.0807 1.42152
\(796\) −7.71562 −0.273473
\(797\) 42.1301 1.49233 0.746163 0.665764i \(-0.231894\pi\)
0.746163 + 0.665764i \(0.231894\pi\)
\(798\) 11.3484 0.401729
\(799\) 14.7926 0.523326
\(800\) −63.8727 −2.25824
\(801\) −7.65403 −0.270442
\(802\) −41.0164 −1.44834
\(803\) −6.33640 −0.223607
\(804\) 46.6824 1.64636
\(805\) 13.8975 0.489823
\(806\) 0 0
\(807\) 26.9716 0.949445
\(808\) 5.52147 0.194245
\(809\) −30.1686 −1.06067 −0.530336 0.847787i \(-0.677934\pi\)
−0.530336 + 0.847787i \(0.677934\pi\)
\(810\) −82.5984 −2.90221
\(811\) 23.7929 0.835480 0.417740 0.908567i \(-0.362822\pi\)
0.417740 + 0.908567i \(0.362822\pi\)
\(812\) −3.14022 −0.110200
\(813\) 29.4655 1.03340
\(814\) −18.0207 −0.631624
\(815\) −8.35425 −0.292637
\(816\) 32.5202 1.13843
\(817\) 8.66468 0.303139
\(818\) −50.7432 −1.77419
\(819\) 0 0
\(820\) 106.614 3.72312
\(821\) −35.8847 −1.25239 −0.626193 0.779668i \(-0.715388\pi\)
−0.626193 + 0.779668i \(0.715388\pi\)
\(822\) 35.5682 1.24058
\(823\) −12.2346 −0.426470 −0.213235 0.977001i \(-0.568400\pi\)
−0.213235 + 0.977001i \(0.568400\pi\)
\(824\) 7.81220 0.272151
\(825\) −15.9830 −0.556457
\(826\) −15.4565 −0.537799
\(827\) 27.3474 0.950962 0.475481 0.879726i \(-0.342274\pi\)
0.475481 + 0.879726i \(0.342274\pi\)
\(828\) −19.9904 −0.694715
\(829\) 23.5738 0.818751 0.409376 0.912366i \(-0.365747\pi\)
0.409376 + 0.912366i \(0.365747\pi\)
\(830\) 36.2226 1.25730
\(831\) 48.3199 1.67620
\(832\) 0 0
\(833\) −4.96016 −0.171859
\(834\) 79.5779 2.75556
\(835\) 49.6751 1.71908
\(836\) 5.16483 0.178630
\(837\) 16.9506 0.585898
\(838\) 82.3163 2.84357
\(839\) 10.5883 0.365549 0.182775 0.983155i \(-0.441492\pi\)
0.182775 + 0.983155i \(0.441492\pi\)
\(840\) −7.72409 −0.266507
\(841\) −27.3565 −0.943329
\(842\) −46.4664 −1.60134
\(843\) −39.0536 −1.34508
\(844\) −36.4092 −1.25326
\(845\) 0 0
\(846\) −13.3054 −0.457449
\(847\) 10.2143 0.350968
\(848\) −14.2635 −0.489810
\(849\) −48.0379 −1.64866
\(850\) −83.4188 −2.86124
\(851\) −37.1888 −1.27482
\(852\) 35.7265 1.22397
\(853\) 21.3925 0.732464 0.366232 0.930524i \(-0.380648\pi\)
0.366232 + 0.930524i \(0.380648\pi\)
\(854\) 3.24690 0.111107
\(855\) −18.1214 −0.619739
\(856\) −3.63412 −0.124212
\(857\) 7.22129 0.246675 0.123337 0.992365i \(-0.460640\pi\)
0.123337 + 0.992365i \(0.460640\pi\)
\(858\) 0 0
\(859\) 57.1073 1.94848 0.974238 0.225524i \(-0.0724095\pi\)
0.974238 + 0.225524i \(0.0724095\pi\)
\(860\) −32.1364 −1.09584
\(861\) −27.3301 −0.931406
\(862\) −75.6841 −2.57781
\(863\) 51.3361 1.74750 0.873751 0.486374i \(-0.161681\pi\)
0.873751 + 0.486374i \(0.161681\pi\)
\(864\) 16.0341 0.545493
\(865\) −13.2701 −0.451197
\(866\) 25.7631 0.875467
\(867\) 17.1958 0.583999
\(868\) 20.7455 0.704148
\(869\) −0.671481 −0.0227784
\(870\) 22.0281 0.746823
\(871\) 0 0
\(872\) 9.87851 0.334528
\(873\) 0.979903 0.0331647
\(874\) 19.3611 0.654899
\(875\) −10.7074 −0.361977
\(876\) 39.6029 1.33806
\(877\) 21.4277 0.723563 0.361781 0.932263i \(-0.382169\pi\)
0.361781 + 0.932263i \(0.382169\pi\)
\(878\) −33.2341 −1.12160
\(879\) −0.950027 −0.0320436
\(880\) 9.25489 0.311982
\(881\) 29.0619 0.979120 0.489560 0.871970i \(-0.337157\pi\)
0.489560 + 0.871970i \(0.337157\pi\)
\(882\) 4.46147 0.150225
\(883\) −4.83594 −0.162742 −0.0813711 0.996684i \(-0.525930\pi\)
−0.0813711 + 0.996684i \(0.525930\pi\)
\(884\) 0 0
\(885\) 59.6892 2.00643
\(886\) −31.7899 −1.06800
\(887\) −24.9898 −0.839075 −0.419538 0.907738i \(-0.637808\pi\)
−0.419538 + 0.907738i \(0.637808\pi\)
\(888\) 20.6692 0.693612
\(889\) 12.3102 0.412869
\(890\) −27.4943 −0.921611
\(891\) 9.63651 0.322835
\(892\) 10.7311 0.359303
\(893\) 7.09420 0.237398
\(894\) 12.0299 0.402341
\(895\) −21.2244 −0.709455
\(896\) 7.39409 0.247019
\(897\) 0 0
\(898\) 64.9324 2.16682
\(899\) −10.8573 −0.362111
\(900\) 41.3061 1.37687
\(901\) −24.4056 −0.813068
\(902\) −22.5940 −0.752300
\(903\) 8.23805 0.274145
\(904\) −4.65582 −0.154850
\(905\) 7.61931 0.253274
\(906\) 75.8409 2.51964
\(907\) 15.0412 0.499435 0.249717 0.968319i \(-0.419662\pi\)
0.249717 + 0.968319i \(0.419662\pi\)
\(908\) −33.2310 −1.10281
\(909\) 12.3160 0.408497
\(910\) 0 0
\(911\) −9.22150 −0.305522 −0.152761 0.988263i \(-0.548816\pi\)
−0.152761 + 0.988263i \(0.548816\pi\)
\(912\) 15.5959 0.516432
\(913\) −4.22598 −0.139860
\(914\) −16.3604 −0.541153
\(915\) −12.5388 −0.414519
\(916\) −40.4612 −1.33687
\(917\) 8.20265 0.270875
\(918\) 20.9409 0.691151
\(919\) −45.0803 −1.48706 −0.743531 0.668701i \(-0.766851\pi\)
−0.743531 + 0.668701i \(0.766851\pi\)
\(920\) −13.1778 −0.434459
\(921\) 31.8456 1.04935
\(922\) −3.10548 −0.102274
\(923\) 0 0
\(924\) 4.91053 0.161544
\(925\) 76.8430 2.52658
\(926\) 29.6086 0.972999
\(927\) 17.4257 0.572334
\(928\) −10.2703 −0.337139
\(929\) 46.9169 1.53929 0.769647 0.638469i \(-0.220432\pi\)
0.769647 + 0.638469i \(0.220432\pi\)
\(930\) −145.526 −4.77200
\(931\) −2.37878 −0.0779612
\(932\) −40.4235 −1.32412
\(933\) 23.4867 0.768921
\(934\) 66.1939 2.16593
\(935\) 15.8356 0.517880
\(936\) 0 0
\(937\) −28.3912 −0.927501 −0.463750 0.885966i \(-0.653497\pi\)
−0.463750 + 0.885966i \(0.653497\pi\)
\(938\) −17.7747 −0.580366
\(939\) 15.4868 0.505394
\(940\) −26.3117 −0.858192
\(941\) 17.8718 0.582603 0.291302 0.956631i \(-0.405912\pi\)
0.291302 + 0.956631i \(0.405912\pi\)
\(942\) 61.9384 2.01806
\(943\) −46.6268 −1.51838
\(944\) −21.2415 −0.691353
\(945\) 7.20874 0.234500
\(946\) 6.81048 0.221428
\(947\) −9.52234 −0.309435 −0.154717 0.987959i \(-0.549447\pi\)
−0.154717 + 0.987959i \(0.549447\pi\)
\(948\) 4.19680 0.136306
\(949\) 0 0
\(950\) −40.0057 −1.29796
\(951\) 1.58598 0.0514289
\(952\) 4.70328 0.152434
\(953\) −13.4180 −0.434652 −0.217326 0.976099i \(-0.569733\pi\)
−0.217326 + 0.976099i \(0.569733\pi\)
\(954\) 21.9519 0.710718
\(955\) −40.9402 −1.32479
\(956\) −74.5169 −2.41005
\(957\) −2.56996 −0.0830749
\(958\) −86.8964 −2.80749
\(959\) −7.45555 −0.240752
\(960\) −90.4298 −2.91861
\(961\) 40.7275 1.31379
\(962\) 0 0
\(963\) −8.10615 −0.261217
\(964\) −72.2960 −2.32850
\(965\) −50.7468 −1.63360
\(966\) 18.4078 0.592261
\(967\) −12.9316 −0.415851 −0.207926 0.978145i \(-0.566671\pi\)
−0.207926 + 0.978145i \(0.566671\pi\)
\(968\) −9.68534 −0.311299
\(969\) 26.6854 0.857260
\(970\) 3.51994 0.113019
\(971\) 47.5213 1.52503 0.762516 0.646969i \(-0.223964\pi\)
0.762516 + 0.646969i \(0.223964\pi\)
\(972\) −45.5211 −1.46009
\(973\) −16.6806 −0.534754
\(974\) −59.7515 −1.91456
\(975\) 0 0
\(976\) 4.46216 0.142830
\(977\) 36.4942 1.16755 0.583776 0.811915i \(-0.301575\pi\)
0.583776 + 0.811915i \(0.301575\pi\)
\(978\) −11.0655 −0.353836
\(979\) 3.20768 0.102518
\(980\) 8.82263 0.281829
\(981\) 22.0347 0.703514
\(982\) 73.3777 2.34158
\(983\) −44.1843 −1.40926 −0.704629 0.709576i \(-0.748887\pi\)
−0.704629 + 0.709576i \(0.748887\pi\)
\(984\) 25.9147 0.826130
\(985\) −82.6865 −2.63461
\(986\) −13.4132 −0.427162
\(987\) 6.74490 0.214692
\(988\) 0 0
\(989\) 14.0546 0.446911
\(990\) −14.2435 −0.452689
\(991\) −50.7097 −1.61085 −0.805424 0.592699i \(-0.798062\pi\)
−0.805424 + 0.592699i \(0.798062\pi\)
\(992\) 67.8495 2.15422
\(993\) −9.49586 −0.301342
\(994\) −13.6032 −0.431467
\(995\) −11.3451 −0.359663
\(996\) 26.4126 0.836916
\(997\) −50.2768 −1.59228 −0.796141 0.605111i \(-0.793129\pi\)
−0.796141 + 0.605111i \(0.793129\pi\)
\(998\) −0.146578 −0.00463985
\(999\) −19.2901 −0.610312
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 1183.2.a.p.1.5 6
7.6 odd 2 8281.2.a.ch.1.5 6
13.2 odd 12 91.2.q.a.43.6 yes 12
13.5 odd 4 1183.2.c.i.337.2 12
13.7 odd 12 91.2.q.a.36.6 12
13.8 odd 4 1183.2.c.i.337.11 12
13.12 even 2 1183.2.a.m.1.2 6
39.2 even 12 819.2.ct.a.316.1 12
39.20 even 12 819.2.ct.a.127.1 12
52.7 even 12 1456.2.cc.c.673.5 12
52.15 even 12 1456.2.cc.c.225.5 12
91.2 odd 12 637.2.k.h.459.1 12
91.20 even 12 637.2.q.h.491.6 12
91.33 even 12 637.2.u.i.361.1 12
91.41 even 12 637.2.q.h.589.6 12
91.46 odd 12 637.2.k.h.569.6 12
91.54 even 12 637.2.k.g.459.1 12
91.59 even 12 637.2.k.g.569.6 12
91.67 odd 12 637.2.u.h.30.1 12
91.72 odd 12 637.2.u.h.361.1 12
91.80 even 12 637.2.u.i.30.1 12
91.90 odd 2 8281.2.a.by.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.q.a.36.6 12 13.7 odd 12
91.2.q.a.43.6 yes 12 13.2 odd 12
637.2.k.g.459.1 12 91.54 even 12
637.2.k.g.569.6 12 91.59 even 12
637.2.k.h.459.1 12 91.2 odd 12
637.2.k.h.569.6 12 91.46 odd 12
637.2.q.h.491.6 12 91.20 even 12
637.2.q.h.589.6 12 91.41 even 12
637.2.u.h.30.1 12 91.67 odd 12
637.2.u.h.361.1 12 91.72 odd 12
637.2.u.i.30.1 12 91.80 even 12
637.2.u.i.361.1 12 91.33 even 12
819.2.ct.a.127.1 12 39.20 even 12
819.2.ct.a.316.1 12 39.2 even 12
1183.2.a.m.1.2 6 13.12 even 2
1183.2.a.p.1.5 6 1.1 even 1 trivial
1183.2.c.i.337.2 12 13.5 odd 4
1183.2.c.i.337.11 12 13.8 odd 4
1456.2.cc.c.225.5 12 52.15 even 12
1456.2.cc.c.673.5 12 52.7 even 12
8281.2.a.by.1.2 6 91.90 odd 2
8281.2.a.ch.1.5 6 7.6 odd 2