Properties

Label 503.2.a.e.1.1
Level $503$
Weight $2$
Character 503.1
Self dual yes
Analytic conductor $4.016$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 503.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.01647522167\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 9 x^{8} + 14 x^{7} + 27 x^{6} - 27 x^{5} - 34 x^{4} + 14 x^{3} + 17 x^{2} + x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.858231\) of defining polynomial
Character \(\chi\) \(=\) 503.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.58686 q^{2} -1.85823 q^{3} +4.69185 q^{4} +1.44291 q^{5} +4.80698 q^{6} -1.96509 q^{7} -6.96343 q^{8} +0.453023 q^{9} +O(q^{10})\) \(q-2.58686 q^{2} -1.85823 q^{3} +4.69185 q^{4} +1.44291 q^{5} +4.80698 q^{6} -1.96509 q^{7} -6.96343 q^{8} +0.453023 q^{9} -3.73261 q^{10} +2.85614 q^{11} -8.71853 q^{12} -3.84427 q^{13} +5.08341 q^{14} -2.68126 q^{15} +8.62972 q^{16} +1.30329 q^{17} -1.17191 q^{18} +3.53196 q^{19} +6.76991 q^{20} +3.65159 q^{21} -7.38845 q^{22} +4.20650 q^{23} +12.9397 q^{24} -2.91801 q^{25} +9.94458 q^{26} +4.73287 q^{27} -9.21989 q^{28} -1.10402 q^{29} +6.93604 q^{30} -3.53201 q^{31} -8.39703 q^{32} -5.30738 q^{33} -3.37144 q^{34} -2.83545 q^{35} +2.12551 q^{36} -6.61903 q^{37} -9.13668 q^{38} +7.14353 q^{39} -10.0476 q^{40} +0.671095 q^{41} -9.44615 q^{42} -4.32289 q^{43} +13.4006 q^{44} +0.653672 q^{45} -10.8816 q^{46} -0.378524 q^{47} -16.0360 q^{48} -3.13843 q^{49} +7.54849 q^{50} -2.42182 q^{51} -18.0367 q^{52} -7.13700 q^{53} -12.2433 q^{54} +4.12116 q^{55} +13.6838 q^{56} -6.56319 q^{57} +2.85594 q^{58} -13.8806 q^{59} -12.5801 q^{60} -6.51374 q^{61} +9.13682 q^{62} -0.890231 q^{63} +4.46249 q^{64} -5.54693 q^{65} +13.7294 q^{66} -7.55189 q^{67} +6.11485 q^{68} -7.81664 q^{69} +7.33490 q^{70} +0.0744967 q^{71} -3.15459 q^{72} +3.29052 q^{73} +17.1225 q^{74} +5.42234 q^{75} +16.5714 q^{76} -5.61258 q^{77} -18.4793 q^{78} +8.57578 q^{79} +12.4519 q^{80} -10.1538 q^{81} -1.73603 q^{82} +14.8165 q^{83} +17.1327 q^{84} +1.88053 q^{85} +11.1827 q^{86} +2.05152 q^{87} -19.8886 q^{88} -17.5415 q^{89} -1.69096 q^{90} +7.55432 q^{91} +19.7362 q^{92} +6.56329 q^{93} +0.979188 q^{94} +5.09630 q^{95} +15.6036 q^{96} -8.01702 q^{97} +8.11867 q^{98} +1.29390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} + O(q^{10}) \) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} - 4q^{10} - 3q^{11} - 7q^{12} - 18q^{13} + q^{14} - 2q^{15} - 4q^{16} - 11q^{17} - q^{18} - 3q^{20} + q^{21} - 18q^{22} - 2q^{23} + 10q^{24} - 27q^{25} + 11q^{26} - 2q^{27} - 22q^{28} - 9q^{29} + 12q^{30} - 22q^{31} - 10q^{32} - 10q^{33} - 10q^{34} - 6q^{35} + 2q^{36} - 35q^{37} + 2q^{38} + 8q^{39} - 19q^{40} - 4q^{41} + 4q^{42} - 20q^{43} + 9q^{44} + 2q^{45} - q^{46} + 7q^{47} - 27q^{49} + 16q^{50} + 9q^{51} - 7q^{52} - 24q^{53} + 17q^{54} - 11q^{55} + 12q^{56} - 23q^{57} + 2q^{58} + 17q^{59} - 4q^{61} + 8q^{62} + 10q^{63} + 3q^{64} - 16q^{65} + 46q^{66} - 6q^{67} + 28q^{68} - 2q^{69} + 26q^{70} - q^{71} - q^{72} - 31q^{73} + 11q^{74} + 30q^{75} + 20q^{76} + 3q^{77} + 11q^{78} - 10q^{79} + 24q^{80} - 6q^{81} - 9q^{82} + 22q^{83} + 22q^{84} - 6q^{85} + 38q^{86} + 25q^{87} - 3q^{88} + q^{89} + 2q^{90} + 10q^{91} + 27q^{92} - 6q^{93} + 33q^{94} + 39q^{95} + 46q^{96} - 57q^{97} + 40q^{98} + 35q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.58686 −1.82919 −0.914593 0.404375i \(-0.867489\pi\)
−0.914593 + 0.404375i \(0.867489\pi\)
\(3\) −1.85823 −1.07285 −0.536425 0.843948i \(-0.680226\pi\)
−0.536425 + 0.843948i \(0.680226\pi\)
\(4\) 4.69185 2.34592
\(5\) 1.44291 0.645289 0.322644 0.946520i \(-0.395428\pi\)
0.322644 + 0.946520i \(0.395428\pi\)
\(6\) 4.80698 1.96244
\(7\) −1.96509 −0.742734 −0.371367 0.928486i \(-0.621111\pi\)
−0.371367 + 0.928486i \(0.621111\pi\)
\(8\) −6.96343 −2.46194
\(9\) 0.453023 0.151008
\(10\) −3.73261 −1.18035
\(11\) 2.85614 0.861160 0.430580 0.902552i \(-0.358309\pi\)
0.430580 + 0.902552i \(0.358309\pi\)
\(12\) −8.71853 −2.51682
\(13\) −3.84427 −1.06621 −0.533104 0.846050i \(-0.678974\pi\)
−0.533104 + 0.846050i \(0.678974\pi\)
\(14\) 5.08341 1.35860
\(15\) −2.68126 −0.692298
\(16\) 8.62972 2.15743
\(17\) 1.30329 0.316095 0.158047 0.987432i \(-0.449480\pi\)
0.158047 + 0.987432i \(0.449480\pi\)
\(18\) −1.17191 −0.276221
\(19\) 3.53196 0.810287 0.405143 0.914253i \(-0.367222\pi\)
0.405143 + 0.914253i \(0.367222\pi\)
\(20\) 6.76991 1.51380
\(21\) 3.65159 0.796842
\(22\) −7.38845 −1.57522
\(23\) 4.20650 0.877115 0.438558 0.898703i \(-0.355490\pi\)
0.438558 + 0.898703i \(0.355490\pi\)
\(24\) 12.9397 2.64130
\(25\) −2.91801 −0.583602
\(26\) 9.94458 1.95029
\(27\) 4.73287 0.910842
\(28\) −9.21989 −1.74240
\(29\) −1.10402 −0.205011 −0.102506 0.994732i \(-0.532686\pi\)
−0.102506 + 0.994732i \(0.532686\pi\)
\(30\) 6.93604 1.26634
\(31\) −3.53201 −0.634368 −0.317184 0.948364i \(-0.602737\pi\)
−0.317184 + 0.948364i \(0.602737\pi\)
\(32\) −8.39703 −1.48440
\(33\) −5.30738 −0.923896
\(34\) −3.37144 −0.578196
\(35\) −2.83545 −0.479278
\(36\) 2.12551 0.354252
\(37\) −6.61903 −1.08816 −0.544081 0.839033i \(-0.683121\pi\)
−0.544081 + 0.839033i \(0.683121\pi\)
\(38\) −9.13668 −1.48217
\(39\) 7.14353 1.14388
\(40\) −10.0476 −1.58866
\(41\) 0.671095 0.104807 0.0524037 0.998626i \(-0.483312\pi\)
0.0524037 + 0.998626i \(0.483312\pi\)
\(42\) −9.44615 −1.45757
\(43\) −4.32289 −0.659235 −0.329617 0.944115i \(-0.606920\pi\)
−0.329617 + 0.944115i \(0.606920\pi\)
\(44\) 13.4006 2.02021
\(45\) 0.653672 0.0974436
\(46\) −10.8816 −1.60441
\(47\) −0.378524 −0.0552134 −0.0276067 0.999619i \(-0.508789\pi\)
−0.0276067 + 0.999619i \(0.508789\pi\)
\(48\) −16.0360 −2.31460
\(49\) −3.13843 −0.448347
\(50\) 7.54849 1.06752
\(51\) −2.42182 −0.339122
\(52\) −18.0367 −2.50124
\(53\) −7.13700 −0.980342 −0.490171 0.871626i \(-0.663066\pi\)
−0.490171 + 0.871626i \(0.663066\pi\)
\(54\) −12.2433 −1.66610
\(55\) 4.12116 0.555697
\(56\) 13.6838 1.82857
\(57\) −6.56319 −0.869316
\(58\) 2.85594 0.375004
\(59\) −13.8806 −1.80709 −0.903547 0.428489i \(-0.859046\pi\)
−0.903547 + 0.428489i \(0.859046\pi\)
\(60\) −12.5801 −1.62408
\(61\) −6.51374 −0.833999 −0.417000 0.908907i \(-0.636918\pi\)
−0.417000 + 0.908907i \(0.636918\pi\)
\(62\) 9.13682 1.16038
\(63\) −0.890231 −0.112159
\(64\) 4.46249 0.557812
\(65\) −5.54693 −0.688012
\(66\) 13.7294 1.68998
\(67\) −7.55189 −0.922610 −0.461305 0.887242i \(-0.652619\pi\)
−0.461305 + 0.887242i \(0.652619\pi\)
\(68\) 6.11485 0.741534
\(69\) −7.81664 −0.941013
\(70\) 7.33490 0.876688
\(71\) 0.0744967 0.00884113 0.00442057 0.999990i \(-0.498593\pi\)
0.00442057 + 0.999990i \(0.498593\pi\)
\(72\) −3.15459 −0.371772
\(73\) 3.29052 0.385126 0.192563 0.981285i \(-0.438320\pi\)
0.192563 + 0.981285i \(0.438320\pi\)
\(74\) 17.1225 1.99045
\(75\) 5.42234 0.626118
\(76\) 16.5714 1.90087
\(77\) −5.61258 −0.639612
\(78\) −18.4793 −2.09237
\(79\) 8.57578 0.964850 0.482425 0.875937i \(-0.339756\pi\)
0.482425 + 0.875937i \(0.339756\pi\)
\(80\) 12.4519 1.39217
\(81\) −10.1538 −1.12820
\(82\) −1.73603 −0.191712
\(83\) 14.8165 1.62632 0.813158 0.582042i \(-0.197746\pi\)
0.813158 + 0.582042i \(0.197746\pi\)
\(84\) 17.1327 1.86933
\(85\) 1.88053 0.203973
\(86\) 11.1827 1.20586
\(87\) 2.05152 0.219946
\(88\) −19.8886 −2.12013
\(89\) −17.5415 −1.85940 −0.929700 0.368318i \(-0.879934\pi\)
−0.929700 + 0.368318i \(0.879934\pi\)
\(90\) −1.69096 −0.178243
\(91\) 7.55432 0.791908
\(92\) 19.7362 2.05764
\(93\) 6.56329 0.680582
\(94\) 0.979188 0.100996
\(95\) 5.09630 0.522869
\(96\) 15.6036 1.59254
\(97\) −8.01702 −0.814005 −0.407003 0.913427i \(-0.633426\pi\)
−0.407003 + 0.913427i \(0.633426\pi\)
\(98\) 8.11867 0.820110
\(99\) 1.29390 0.130042
\(100\) −13.6909 −1.36909
\(101\) 9.11148 0.906626 0.453313 0.891351i \(-0.350242\pi\)
0.453313 + 0.891351i \(0.350242\pi\)
\(102\) 6.26491 0.620318
\(103\) −15.2943 −1.50699 −0.753494 0.657455i \(-0.771633\pi\)
−0.753494 + 0.657455i \(0.771633\pi\)
\(104\) 26.7693 2.62494
\(105\) 5.26891 0.514193
\(106\) 18.4624 1.79323
\(107\) 3.50420 0.338764 0.169382 0.985550i \(-0.445823\pi\)
0.169382 + 0.985550i \(0.445823\pi\)
\(108\) 22.2059 2.13676
\(109\) 2.49301 0.238787 0.119393 0.992847i \(-0.461905\pi\)
0.119393 + 0.992847i \(0.461905\pi\)
\(110\) −10.6609 −1.01647
\(111\) 12.2997 1.16743
\(112\) −16.9582 −1.60240
\(113\) −14.5343 −1.36727 −0.683634 0.729825i \(-0.739601\pi\)
−0.683634 + 0.729825i \(0.739601\pi\)
\(114\) 16.9781 1.59014
\(115\) 6.06959 0.565993
\(116\) −5.17989 −0.480941
\(117\) −1.74154 −0.161006
\(118\) 35.9070 3.30551
\(119\) −2.56109 −0.234774
\(120\) 18.6708 1.70440
\(121\) −2.84244 −0.258404
\(122\) 16.8501 1.52554
\(123\) −1.24705 −0.112443
\(124\) −16.5716 −1.48818
\(125\) −11.4250 −1.02188
\(126\) 2.30290 0.205159
\(127\) 5.52702 0.490443 0.245222 0.969467i \(-0.421139\pi\)
0.245222 + 0.969467i \(0.421139\pi\)
\(128\) 5.25021 0.464057
\(129\) 8.03293 0.707260
\(130\) 14.3491 1.25850
\(131\) −21.2174 −1.85377 −0.926885 0.375346i \(-0.877524\pi\)
−0.926885 + 0.375346i \(0.877524\pi\)
\(132\) −24.9014 −2.16739
\(133\) −6.94061 −0.601827
\(134\) 19.5357 1.68763
\(135\) 6.82911 0.587756
\(136\) −9.07538 −0.778208
\(137\) 11.6794 0.997835 0.498917 0.866650i \(-0.333731\pi\)
0.498917 + 0.866650i \(0.333731\pi\)
\(138\) 20.2206 1.72129
\(139\) −10.8676 −0.921778 −0.460889 0.887458i \(-0.652469\pi\)
−0.460889 + 0.887458i \(0.652469\pi\)
\(140\) −13.3035 −1.12435
\(141\) 0.703385 0.0592357
\(142\) −0.192713 −0.0161721
\(143\) −10.9798 −0.918175
\(144\) 3.90946 0.325789
\(145\) −1.59300 −0.132292
\(146\) −8.51212 −0.704468
\(147\) 5.83192 0.481009
\(148\) −31.0555 −2.55274
\(149\) −7.40209 −0.606403 −0.303202 0.952926i \(-0.598056\pi\)
−0.303202 + 0.952926i \(0.598056\pi\)
\(150\) −14.0268 −1.14529
\(151\) 6.09188 0.495750 0.247875 0.968792i \(-0.420268\pi\)
0.247875 + 0.968792i \(0.420268\pi\)
\(152\) −24.5945 −1.99488
\(153\) 0.590422 0.0477328
\(154\) 14.5189 1.16997
\(155\) −5.09637 −0.409350
\(156\) 33.5164 2.68346
\(157\) 2.22790 0.177806 0.0889028 0.996040i \(-0.471664\pi\)
0.0889028 + 0.996040i \(0.471664\pi\)
\(158\) −22.1843 −1.76489
\(159\) 13.2622 1.05176
\(160\) −12.1162 −0.957866
\(161\) −8.26614 −0.651463
\(162\) 26.2666 2.06370
\(163\) −18.7370 −1.46759 −0.733797 0.679368i \(-0.762254\pi\)
−0.733797 + 0.679368i \(0.762254\pi\)
\(164\) 3.14867 0.245870
\(165\) −7.65807 −0.596180
\(166\) −38.3281 −2.97484
\(167\) 22.7237 1.75841 0.879205 0.476443i \(-0.158074\pi\)
0.879205 + 0.476443i \(0.158074\pi\)
\(168\) −25.4276 −1.96178
\(169\) 1.77838 0.136798
\(170\) −4.86468 −0.373104
\(171\) 1.60006 0.122360
\(172\) −20.2823 −1.54651
\(173\) −2.25141 −0.171172 −0.0855859 0.996331i \(-0.527276\pi\)
−0.0855859 + 0.996331i \(0.527276\pi\)
\(174\) −5.30701 −0.402323
\(175\) 5.73415 0.433461
\(176\) 24.6477 1.85789
\(177\) 25.7933 1.93874
\(178\) 45.3775 3.40119
\(179\) 15.8986 1.18832 0.594159 0.804348i \(-0.297485\pi\)
0.594159 + 0.804348i \(0.297485\pi\)
\(180\) 3.06693 0.228595
\(181\) 16.5942 1.23344 0.616718 0.787184i \(-0.288462\pi\)
0.616718 + 0.787184i \(0.288462\pi\)
\(182\) −19.5420 −1.44855
\(183\) 12.1040 0.894756
\(184\) −29.2916 −2.15941
\(185\) −9.55066 −0.702179
\(186\) −16.9783 −1.24491
\(187\) 3.72239 0.272208
\(188\) −1.77598 −0.129526
\(189\) −9.30051 −0.676513
\(190\) −13.1834 −0.956425
\(191\) −2.03640 −0.147348 −0.0736742 0.997282i \(-0.523473\pi\)
−0.0736742 + 0.997282i \(0.523473\pi\)
\(192\) −8.29235 −0.598448
\(193\) −15.1384 −1.08968 −0.544841 0.838539i \(-0.683410\pi\)
−0.544841 + 0.838539i \(0.683410\pi\)
\(194\) 20.7389 1.48897
\(195\) 10.3075 0.738134
\(196\) −14.7250 −1.05179
\(197\) −6.01285 −0.428398 −0.214199 0.976790i \(-0.568714\pi\)
−0.214199 + 0.976790i \(0.568714\pi\)
\(198\) −3.34714 −0.237871
\(199\) 23.5513 1.66951 0.834753 0.550624i \(-0.185610\pi\)
0.834753 + 0.550624i \(0.185610\pi\)
\(200\) 20.3194 1.43680
\(201\) 14.0332 0.989823
\(202\) −23.5701 −1.65839
\(203\) 2.16950 0.152269
\(204\) −11.3628 −0.795555
\(205\) 0.968329 0.0676311
\(206\) 39.5641 2.75656
\(207\) 1.90564 0.132451
\(208\) −33.1749 −2.30027
\(209\) 10.0878 0.697786
\(210\) −13.6299 −0.940555
\(211\) 2.35555 0.162163 0.0810814 0.996707i \(-0.474163\pi\)
0.0810814 + 0.996707i \(0.474163\pi\)
\(212\) −33.4857 −2.29981
\(213\) −0.138432 −0.00948521
\(214\) −9.06488 −0.619662
\(215\) −6.23754 −0.425397
\(216\) −32.9570 −2.24244
\(217\) 6.94071 0.471166
\(218\) −6.44906 −0.436785
\(219\) −6.11455 −0.413183
\(220\) 19.3358 1.30362
\(221\) −5.01020 −0.337023
\(222\) −31.8176 −2.13546
\(223\) 13.6268 0.912519 0.456260 0.889847i \(-0.349189\pi\)
0.456260 + 0.889847i \(0.349189\pi\)
\(224\) 16.5009 1.10251
\(225\) −1.32193 −0.0881285
\(226\) 37.5981 2.50099
\(227\) 11.9812 0.795218 0.397609 0.917555i \(-0.369840\pi\)
0.397609 + 0.917555i \(0.369840\pi\)
\(228\) −30.7935 −2.03935
\(229\) 27.2066 1.79786 0.898932 0.438087i \(-0.144344\pi\)
0.898932 + 0.438087i \(0.144344\pi\)
\(230\) −15.7012 −1.03531
\(231\) 10.4295 0.686208
\(232\) 7.68776 0.504726
\(233\) 14.4504 0.946679 0.473339 0.880880i \(-0.343048\pi\)
0.473339 + 0.880880i \(0.343048\pi\)
\(234\) 4.50512 0.294509
\(235\) −0.546176 −0.0356286
\(236\) −65.1254 −4.23930
\(237\) −15.9358 −1.03514
\(238\) 6.62517 0.429446
\(239\) −23.9361 −1.54830 −0.774148 0.633005i \(-0.781821\pi\)
−0.774148 + 0.633005i \(0.781821\pi\)
\(240\) −23.1385 −1.49359
\(241\) −13.7734 −0.887224 −0.443612 0.896219i \(-0.646303\pi\)
−0.443612 + 0.896219i \(0.646303\pi\)
\(242\) 7.35300 0.472668
\(243\) 4.66957 0.299553
\(244\) −30.5615 −1.95650
\(245\) −4.52847 −0.289313
\(246\) 3.22594 0.205679
\(247\) −13.5778 −0.863934
\(248\) 24.5949 1.56178
\(249\) −27.5324 −1.74479
\(250\) 29.5548 1.86921
\(251\) −8.26975 −0.521982 −0.260991 0.965341i \(-0.584049\pi\)
−0.260991 + 0.965341i \(0.584049\pi\)
\(252\) −4.17682 −0.263115
\(253\) 12.0144 0.755336
\(254\) −14.2976 −0.897112
\(255\) −3.49447 −0.218832
\(256\) −22.5065 −1.40666
\(257\) −20.1486 −1.25684 −0.628418 0.777876i \(-0.716297\pi\)
−0.628418 + 0.777876i \(0.716297\pi\)
\(258\) −20.7801 −1.29371
\(259\) 13.0070 0.808214
\(260\) −26.0253 −1.61402
\(261\) −0.500147 −0.0309583
\(262\) 54.8863 3.39089
\(263\) −27.9229 −1.72180 −0.860900 0.508774i \(-0.830099\pi\)
−0.860900 + 0.508774i \(0.830099\pi\)
\(264\) 36.9575 2.27458
\(265\) −10.2980 −0.632604
\(266\) 17.9544 1.10085
\(267\) 32.5962 1.99486
\(268\) −35.4323 −2.16437
\(269\) 11.6165 0.708269 0.354135 0.935194i \(-0.384775\pi\)
0.354135 + 0.935194i \(0.384775\pi\)
\(270\) −17.6659 −1.07512
\(271\) −3.27787 −0.199117 −0.0995583 0.995032i \(-0.531743\pi\)
−0.0995583 + 0.995032i \(0.531743\pi\)
\(272\) 11.2471 0.681953
\(273\) −14.0377 −0.849599
\(274\) −30.2129 −1.82523
\(275\) −8.33426 −0.502575
\(276\) −36.6745 −2.20754
\(277\) 10.1157 0.607795 0.303897 0.952705i \(-0.401712\pi\)
0.303897 + 0.952705i \(0.401712\pi\)
\(278\) 28.1130 1.68610
\(279\) −1.60008 −0.0957944
\(280\) 19.7444 1.17995
\(281\) 16.7656 1.00015 0.500077 0.865981i \(-0.333305\pi\)
0.500077 + 0.865981i \(0.333305\pi\)
\(282\) −1.81956 −0.108353
\(283\) 15.4382 0.917705 0.458852 0.888513i \(-0.348261\pi\)
0.458852 + 0.888513i \(0.348261\pi\)
\(284\) 0.349527 0.0207406
\(285\) −9.47010 −0.560960
\(286\) 28.4031 1.67951
\(287\) −1.31876 −0.0778440
\(288\) −3.80405 −0.224156
\(289\) −15.3014 −0.900084
\(290\) 4.12087 0.241986
\(291\) 14.8975 0.873306
\(292\) 15.4386 0.903477
\(293\) −7.52887 −0.439841 −0.219921 0.975518i \(-0.570580\pi\)
−0.219921 + 0.975518i \(0.570580\pi\)
\(294\) −15.0864 −0.879855
\(295\) −20.0284 −1.16610
\(296\) 46.0911 2.67899
\(297\) 13.5178 0.784380
\(298\) 19.1482 1.10922
\(299\) −16.1709 −0.935187
\(300\) 25.4408 1.46882
\(301\) 8.49487 0.489636
\(302\) −15.7588 −0.906819
\(303\) −16.9312 −0.972674
\(304\) 30.4798 1.74814
\(305\) −9.39874 −0.538170
\(306\) −1.52734 −0.0873121
\(307\) 17.1932 0.981265 0.490632 0.871367i \(-0.336766\pi\)
0.490632 + 0.871367i \(0.336766\pi\)
\(308\) −26.3333 −1.50048
\(309\) 28.4203 1.61677
\(310\) 13.1836 0.748778
\(311\) 10.1595 0.576094 0.288047 0.957616i \(-0.406994\pi\)
0.288047 + 0.957616i \(0.406994\pi\)
\(312\) −49.7435 −2.81617
\(313\) 26.6909 1.50866 0.754329 0.656496i \(-0.227962\pi\)
0.754329 + 0.656496i \(0.227962\pi\)
\(314\) −5.76326 −0.325240
\(315\) −1.28452 −0.0723746
\(316\) 40.2362 2.26346
\(317\) 19.3855 1.08880 0.544398 0.838827i \(-0.316758\pi\)
0.544398 + 0.838827i \(0.316758\pi\)
\(318\) −34.3074 −1.92386
\(319\) −3.15324 −0.176548
\(320\) 6.43898 0.359950
\(321\) −6.51161 −0.363443
\(322\) 21.3833 1.19165
\(323\) 4.60317 0.256127
\(324\) −47.6402 −2.64668
\(325\) 11.2176 0.622241
\(326\) 48.4700 2.68450
\(327\) −4.63258 −0.256182
\(328\) −4.67312 −0.258030
\(329\) 0.743833 0.0410088
\(330\) 19.8103 1.09052
\(331\) −26.0579 −1.43227 −0.716135 0.697962i \(-0.754090\pi\)
−0.716135 + 0.697962i \(0.754090\pi\)
\(332\) 69.5165 3.81521
\(333\) −2.99857 −0.164321
\(334\) −58.7830 −3.21646
\(335\) −10.8967 −0.595350
\(336\) 31.5122 1.71913
\(337\) −11.6240 −0.633199 −0.316600 0.948559i \(-0.602541\pi\)
−0.316600 + 0.948559i \(0.602541\pi\)
\(338\) −4.60042 −0.250230
\(339\) 27.0080 1.46687
\(340\) 8.82317 0.478504
\(341\) −10.0879 −0.546292
\(342\) −4.13913 −0.223818
\(343\) 19.9229 1.07574
\(344\) 30.1021 1.62300
\(345\) −11.2787 −0.607225
\(346\) 5.82409 0.313105
\(347\) 24.5102 1.31578 0.657888 0.753115i \(-0.271450\pi\)
0.657888 + 0.753115i \(0.271450\pi\)
\(348\) 9.62543 0.515977
\(349\) 10.6510 0.570133 0.285066 0.958508i \(-0.407984\pi\)
0.285066 + 0.958508i \(0.407984\pi\)
\(350\) −14.8334 −0.792881
\(351\) −18.1944 −0.971146
\(352\) −23.9831 −1.27830
\(353\) 1.04651 0.0557003 0.0278501 0.999612i \(-0.491134\pi\)
0.0278501 + 0.999612i \(0.491134\pi\)
\(354\) −66.7236 −3.54632
\(355\) 0.107492 0.00570508
\(356\) −82.3022 −4.36201
\(357\) 4.75909 0.251878
\(358\) −41.1275 −2.17365
\(359\) −24.0194 −1.26769 −0.633847 0.773459i \(-0.718525\pi\)
−0.633847 + 0.773459i \(0.718525\pi\)
\(360\) −4.55179 −0.239901
\(361\) −6.52528 −0.343436
\(362\) −42.9268 −2.25618
\(363\) 5.28191 0.277228
\(364\) 35.4437 1.85776
\(365\) 4.74793 0.248518
\(366\) −31.3115 −1.63668
\(367\) −18.6963 −0.975937 −0.487969 0.872861i \(-0.662262\pi\)
−0.487969 + 0.872861i \(0.662262\pi\)
\(368\) 36.3009 1.89231
\(369\) 0.304022 0.0158267
\(370\) 24.7062 1.28442
\(371\) 14.0248 0.728133
\(372\) 30.7939 1.59659
\(373\) −23.3008 −1.20647 −0.603235 0.797564i \(-0.706122\pi\)
−0.603235 + 0.797564i \(0.706122\pi\)
\(374\) −9.62931 −0.497920
\(375\) 21.2302 1.09633
\(376\) 2.63582 0.135932
\(377\) 4.24415 0.218585
\(378\) 24.0591 1.23747
\(379\) −9.79951 −0.503367 −0.251683 0.967810i \(-0.580984\pi\)
−0.251683 + 0.967810i \(0.580984\pi\)
\(380\) 23.9110 1.22661
\(381\) −10.2705 −0.526172
\(382\) 5.26787 0.269528
\(383\) −1.97295 −0.100813 −0.0504066 0.998729i \(-0.516052\pi\)
−0.0504066 + 0.998729i \(0.516052\pi\)
\(384\) −9.75610 −0.497864
\(385\) −8.09844 −0.412735
\(386\) 39.1608 1.99323
\(387\) −1.95837 −0.0995496
\(388\) −37.6146 −1.90959
\(389\) −6.61202 −0.335243 −0.167621 0.985851i \(-0.553609\pi\)
−0.167621 + 0.985851i \(0.553609\pi\)
\(390\) −26.6640 −1.35018
\(391\) 5.48230 0.277252
\(392\) 21.8542 1.10380
\(393\) 39.4268 1.98882
\(394\) 15.5544 0.783619
\(395\) 12.3741 0.622607
\(396\) 6.07078 0.305068
\(397\) −7.61750 −0.382311 −0.191156 0.981560i \(-0.561224\pi\)
−0.191156 + 0.981560i \(0.561224\pi\)
\(398\) −60.9239 −3.05384
\(399\) 12.8973 0.645670
\(400\) −25.1816 −1.25908
\(401\) 25.8162 1.28920 0.644601 0.764519i \(-0.277024\pi\)
0.644601 + 0.764519i \(0.277024\pi\)
\(402\) −36.3018 −1.81057
\(403\) 13.5780 0.676368
\(404\) 42.7497 2.12687
\(405\) −14.6511 −0.728018
\(406\) −5.61218 −0.278528
\(407\) −18.9049 −0.937081
\(408\) 16.8642 0.834900
\(409\) −13.6924 −0.677044 −0.338522 0.940958i \(-0.609927\pi\)
−0.338522 + 0.940958i \(0.609927\pi\)
\(410\) −2.50493 −0.123710
\(411\) −21.7029 −1.07053
\(412\) −71.7583 −3.53528
\(413\) 27.2765 1.34219
\(414\) −4.92962 −0.242278
\(415\) 21.3788 1.04944
\(416\) 32.2804 1.58268
\(417\) 20.1945 0.988930
\(418\) −26.0957 −1.27638
\(419\) 24.0027 1.17261 0.586305 0.810091i \(-0.300582\pi\)
0.586305 + 0.810091i \(0.300582\pi\)
\(420\) 24.7209 1.20626
\(421\) 31.1293 1.51715 0.758573 0.651588i \(-0.225897\pi\)
0.758573 + 0.651588i \(0.225897\pi\)
\(422\) −6.09348 −0.296626
\(423\) −0.171480 −0.00833765
\(424\) 49.6979 2.41355
\(425\) −3.80302 −0.184474
\(426\) 0.358104 0.0173502
\(427\) 12.8001 0.619439
\(428\) 16.4412 0.794714
\(429\) 20.4030 0.985064
\(430\) 16.1357 0.778130
\(431\) 28.0585 1.35153 0.675766 0.737116i \(-0.263813\pi\)
0.675766 + 0.737116i \(0.263813\pi\)
\(432\) 40.8434 1.96508
\(433\) −20.6011 −0.990026 −0.495013 0.868886i \(-0.664837\pi\)
−0.495013 + 0.868886i \(0.664837\pi\)
\(434\) −17.9547 −0.861851
\(435\) 2.96016 0.141929
\(436\) 11.6968 0.560175
\(437\) 14.8572 0.710715
\(438\) 15.8175 0.755789
\(439\) −36.4050 −1.73752 −0.868758 0.495238i \(-0.835081\pi\)
−0.868758 + 0.495238i \(0.835081\pi\)
\(440\) −28.6974 −1.36809
\(441\) −1.42178 −0.0677038
\(442\) 12.9607 0.616477
\(443\) −28.1192 −1.33598 −0.667992 0.744169i \(-0.732846\pi\)
−0.667992 + 0.744169i \(0.732846\pi\)
\(444\) 57.7082 2.73871
\(445\) −25.3109 −1.19985
\(446\) −35.2507 −1.66917
\(447\) 13.7548 0.650580
\(448\) −8.76920 −0.414306
\(449\) −12.9493 −0.611116 −0.305558 0.952173i \(-0.598843\pi\)
−0.305558 + 0.952173i \(0.598843\pi\)
\(450\) 3.41964 0.161203
\(451\) 1.91674 0.0902559
\(452\) −68.1925 −3.20750
\(453\) −11.3201 −0.531866
\(454\) −30.9936 −1.45460
\(455\) 10.9002 0.511010
\(456\) 45.7023 2.14021
\(457\) 0.149626 0.00699919 0.00349960 0.999994i \(-0.498886\pi\)
0.00349960 + 0.999994i \(0.498886\pi\)
\(458\) −70.3797 −3.28863
\(459\) 6.16832 0.287912
\(460\) 28.4776 1.32777
\(461\) 38.9917 1.81603 0.908013 0.418942i \(-0.137599\pi\)
0.908013 + 0.418942i \(0.137599\pi\)
\(462\) −26.9796 −1.25520
\(463\) −20.1386 −0.935918 −0.467959 0.883750i \(-0.655011\pi\)
−0.467959 + 0.883750i \(0.655011\pi\)
\(464\) −9.52738 −0.442298
\(465\) 9.47024 0.439172
\(466\) −37.3812 −1.73165
\(467\) 14.8664 0.687936 0.343968 0.938981i \(-0.388229\pi\)
0.343968 + 0.938981i \(0.388229\pi\)
\(468\) −8.17104 −0.377707
\(469\) 14.8401 0.685254
\(470\) 1.41288 0.0651713
\(471\) −4.13995 −0.190759
\(472\) 96.6562 4.44896
\(473\) −12.3468 −0.567707
\(474\) 41.2236 1.89346
\(475\) −10.3063 −0.472885
\(476\) −12.0162 −0.550762
\(477\) −3.23322 −0.148039
\(478\) 61.9193 2.83212
\(479\) −35.2515 −1.61068 −0.805342 0.592810i \(-0.798018\pi\)
−0.805342 + 0.592810i \(0.798018\pi\)
\(480\) 22.5146 1.02765
\(481\) 25.4453 1.16021
\(482\) 35.6299 1.62290
\(483\) 15.3604 0.698922
\(484\) −13.3363 −0.606195
\(485\) −11.5678 −0.525268
\(486\) −12.0795 −0.547938
\(487\) 27.4362 1.24325 0.621626 0.783314i \(-0.286472\pi\)
0.621626 + 0.783314i \(0.286472\pi\)
\(488\) 45.3580 2.05326
\(489\) 34.8177 1.57451
\(490\) 11.7145 0.529208
\(491\) −4.83937 −0.218398 −0.109199 0.994020i \(-0.534829\pi\)
−0.109199 + 0.994020i \(0.534829\pi\)
\(492\) −5.85096 −0.263782
\(493\) −1.43886 −0.0648030
\(494\) 35.1238 1.58030
\(495\) 1.86698 0.0839145
\(496\) −30.4803 −1.36860
\(497\) −0.146393 −0.00656661
\(498\) 71.2225 3.19155
\(499\) 16.7739 0.750904 0.375452 0.926842i \(-0.377488\pi\)
0.375452 + 0.926842i \(0.377488\pi\)
\(500\) −53.6042 −2.39725
\(501\) −42.2258 −1.88651
\(502\) 21.3927 0.954803
\(503\) −1.00000 −0.0445878
\(504\) 6.19906 0.276128
\(505\) 13.1470 0.585036
\(506\) −31.0795 −1.38165
\(507\) −3.30464 −0.146764
\(508\) 25.9319 1.15054
\(509\) −27.5401 −1.22069 −0.610347 0.792134i \(-0.708970\pi\)
−0.610347 + 0.792134i \(0.708970\pi\)
\(510\) 9.03970 0.400284
\(511\) −6.46617 −0.286046
\(512\) 47.7209 2.10898
\(513\) 16.7163 0.738043
\(514\) 52.1216 2.29899
\(515\) −22.0682 −0.972443
\(516\) 37.6893 1.65918
\(517\) −1.08112 −0.0475475
\(518\) −33.6472 −1.47837
\(519\) 4.18365 0.183642
\(520\) 38.6256 1.69385
\(521\) 25.7250 1.12703 0.563515 0.826106i \(-0.309449\pi\)
0.563515 + 0.826106i \(0.309449\pi\)
\(522\) 1.29381 0.0566285
\(523\) −29.5746 −1.29321 −0.646605 0.762825i \(-0.723812\pi\)
−0.646605 + 0.762825i \(0.723812\pi\)
\(524\) −99.5486 −4.34880
\(525\) −10.6554 −0.465039
\(526\) 72.2327 3.14949
\(527\) −4.60324 −0.200520
\(528\) −45.8012 −1.99324
\(529\) −5.30539 −0.230669
\(530\) 26.6396 1.15715
\(531\) −6.28821 −0.272885
\(532\) −32.5643 −1.41184
\(533\) −2.57987 −0.111746
\(534\) −84.3219 −3.64897
\(535\) 5.05624 0.218601
\(536\) 52.5870 2.27141
\(537\) −29.5433 −1.27489
\(538\) −30.0502 −1.29556
\(539\) −8.96380 −0.386098
\(540\) 32.0411 1.37883
\(541\) −13.1889 −0.567035 −0.283517 0.958967i \(-0.591501\pi\)
−0.283517 + 0.958967i \(0.591501\pi\)
\(542\) 8.47940 0.364222
\(543\) −30.8358 −1.32329
\(544\) −10.9438 −0.469211
\(545\) 3.59718 0.154086
\(546\) 36.3135 1.55407
\(547\) −17.9543 −0.767673 −0.383836 0.923401i \(-0.625397\pi\)
−0.383836 + 0.923401i \(0.625397\pi\)
\(548\) 54.7977 2.34084
\(549\) −2.95088 −0.125940
\(550\) 21.5596 0.919303
\(551\) −3.89935 −0.166118
\(552\) 54.4306 2.31672
\(553\) −16.8522 −0.716627
\(554\) −26.1680 −1.11177
\(555\) 17.7473 0.753332
\(556\) −50.9891 −2.16242
\(557\) 9.47412 0.401431 0.200716 0.979650i \(-0.435673\pi\)
0.200716 + 0.979650i \(0.435673\pi\)
\(558\) 4.13919 0.175226
\(559\) 16.6183 0.702881
\(560\) −24.4691 −1.03401
\(561\) −6.91706 −0.292039
\(562\) −43.3703 −1.82947
\(563\) −33.2995 −1.40341 −0.701704 0.712469i \(-0.747577\pi\)
−0.701704 + 0.712469i \(0.747577\pi\)
\(564\) 3.30017 0.138962
\(565\) −20.9716 −0.882283
\(566\) −39.9364 −1.67865
\(567\) 19.9532 0.837955
\(568\) −0.518752 −0.0217664
\(569\) −28.8525 −1.20956 −0.604780 0.796393i \(-0.706739\pi\)
−0.604780 + 0.796393i \(0.706739\pi\)
\(570\) 24.4978 1.02610
\(571\) 22.7379 0.951550 0.475775 0.879567i \(-0.342168\pi\)
0.475775 + 0.879567i \(0.342168\pi\)
\(572\) −51.5154 −2.15397
\(573\) 3.78409 0.158083
\(574\) 3.41145 0.142391
\(575\) −12.2746 −0.511886
\(576\) 2.02161 0.0842339
\(577\) −4.60031 −0.191513 −0.0957567 0.995405i \(-0.530527\pi\)
−0.0957567 + 0.995405i \(0.530527\pi\)
\(578\) 39.5827 1.64642
\(579\) 28.1306 1.16907
\(580\) −7.47411 −0.310346
\(581\) −29.1156 −1.20792
\(582\) −38.5377 −1.59744
\(583\) −20.3843 −0.844231
\(584\) −22.9133 −0.948159
\(585\) −2.51289 −0.103895
\(586\) 19.4761 0.804552
\(587\) 32.6537 1.34776 0.673881 0.738839i \(-0.264626\pi\)
0.673881 + 0.738839i \(0.264626\pi\)
\(588\) 27.3625 1.12841
\(589\) −12.4749 −0.514020
\(590\) 51.8106 2.13301
\(591\) 11.1733 0.459607
\(592\) −57.1204 −2.34763
\(593\) −36.4077 −1.49509 −0.747543 0.664214i \(-0.768767\pi\)
−0.747543 + 0.664214i \(0.768767\pi\)
\(594\) −34.9686 −1.43478
\(595\) −3.69542 −0.151497
\(596\) −34.7295 −1.42257
\(597\) −43.7637 −1.79113
\(598\) 41.8318 1.71063
\(599\) −40.2375 −1.64406 −0.822031 0.569443i \(-0.807159\pi\)
−0.822031 + 0.569443i \(0.807159\pi\)
\(600\) −37.7581 −1.54147
\(601\) 8.80257 0.359065 0.179532 0.983752i \(-0.442542\pi\)
0.179532 + 0.983752i \(0.442542\pi\)
\(602\) −21.9750 −0.895635
\(603\) −3.42118 −0.139321
\(604\) 28.5822 1.16299
\(605\) −4.10139 −0.166745
\(606\) 43.7987 1.77920
\(607\) 13.3824 0.543176 0.271588 0.962414i \(-0.412451\pi\)
0.271588 + 0.962414i \(0.412451\pi\)
\(608\) −29.6579 −1.20279
\(609\) −4.03143 −0.163362
\(610\) 24.3132 0.984414
\(611\) 1.45515 0.0588689
\(612\) 2.77017 0.111977
\(613\) −15.6160 −0.630725 −0.315363 0.948971i \(-0.602126\pi\)
−0.315363 + 0.948971i \(0.602126\pi\)
\(614\) −44.4763 −1.79492
\(615\) −1.79938 −0.0725580
\(616\) 39.0828 1.57469
\(617\) 4.29634 0.172964 0.0864820 0.996253i \(-0.472437\pi\)
0.0864820 + 0.996253i \(0.472437\pi\)
\(618\) −73.5193 −2.95738
\(619\) −10.8380 −0.435614 −0.217807 0.975992i \(-0.569890\pi\)
−0.217807 + 0.975992i \(0.569890\pi\)
\(620\) −23.9114 −0.960305
\(621\) 19.9088 0.798913
\(622\) −26.2813 −1.05378
\(623\) 34.4707 1.38104
\(624\) 61.6467 2.46784
\(625\) −1.89515 −0.0758062
\(626\) −69.0456 −2.75962
\(627\) −18.7454 −0.748620
\(628\) 10.4529 0.417118
\(629\) −8.62653 −0.343962
\(630\) 3.32288 0.132387
\(631\) 29.5073 1.17467 0.587334 0.809345i \(-0.300178\pi\)
0.587334 + 0.809345i \(0.300178\pi\)
\(632\) −59.7168 −2.37541
\(633\) −4.37716 −0.173976
\(634\) −50.1475 −1.99161
\(635\) 7.97499 0.316478
\(636\) 62.2241 2.46735
\(637\) 12.0649 0.478031
\(638\) 8.15699 0.322938
\(639\) 0.0337487 0.00133508
\(640\) 7.57558 0.299451
\(641\) 7.72732 0.305211 0.152605 0.988287i \(-0.451234\pi\)
0.152605 + 0.988287i \(0.451234\pi\)
\(642\) 16.8446 0.664805
\(643\) 10.0167 0.395021 0.197511 0.980301i \(-0.436714\pi\)
0.197511 + 0.980301i \(0.436714\pi\)
\(644\) −38.7834 −1.52828
\(645\) 11.5908 0.456387
\(646\) −11.9078 −0.468505
\(647\) −13.6884 −0.538147 −0.269074 0.963120i \(-0.586718\pi\)
−0.269074 + 0.963120i \(0.586718\pi\)
\(648\) 70.7055 2.77758
\(649\) −39.6449 −1.55620
\(650\) −29.0184 −1.13819
\(651\) −12.8974 −0.505491
\(652\) −87.9111 −3.44286
\(653\) 38.1399 1.49253 0.746264 0.665650i \(-0.231845\pi\)
0.746264 + 0.665650i \(0.231845\pi\)
\(654\) 11.9838 0.468605
\(655\) −30.6147 −1.19622
\(656\) 5.79136 0.226115
\(657\) 1.49068 0.0581571
\(658\) −1.92419 −0.0750128
\(659\) 31.5253 1.22805 0.614026 0.789286i \(-0.289549\pi\)
0.614026 + 0.789286i \(0.289549\pi\)
\(660\) −35.9305 −1.39859
\(661\) −30.1122 −1.17123 −0.585614 0.810590i \(-0.699146\pi\)
−0.585614 + 0.810590i \(0.699146\pi\)
\(662\) 67.4080 2.61989
\(663\) 9.31012 0.361575
\(664\) −103.173 −4.00390
\(665\) −10.0147 −0.388352
\(666\) 7.75689 0.300573
\(667\) −4.64405 −0.179818
\(668\) 106.616 4.12510
\(669\) −25.3218 −0.978996
\(670\) 28.1882 1.08901
\(671\) −18.6042 −0.718207
\(672\) −30.6625 −1.18283
\(673\) −40.7982 −1.57265 −0.786327 0.617810i \(-0.788020\pi\)
−0.786327 + 0.617810i \(0.788020\pi\)
\(674\) 30.0697 1.15824
\(675\) −13.8106 −0.531569
\(676\) 8.34388 0.320918
\(677\) 3.18994 0.122599 0.0612997 0.998119i \(-0.480475\pi\)
0.0612997 + 0.998119i \(0.480475\pi\)
\(678\) −69.8659 −2.68319
\(679\) 15.7542 0.604589
\(680\) −13.0950 −0.502169
\(681\) −22.2638 −0.853150
\(682\) 26.0961 0.999270
\(683\) 25.0239 0.957512 0.478756 0.877948i \(-0.341088\pi\)
0.478756 + 0.877948i \(0.341088\pi\)
\(684\) 7.50723 0.287046
\(685\) 16.8523 0.643892
\(686\) −51.5378 −1.96772
\(687\) −50.5562 −1.92884
\(688\) −37.3054 −1.42225
\(689\) 27.4365 1.04525
\(690\) 29.1764 1.11073
\(691\) 36.5349 1.38985 0.694927 0.719081i \(-0.255437\pi\)
0.694927 + 0.719081i \(0.255437\pi\)
\(692\) −10.5633 −0.401556
\(693\) −2.54263 −0.0965864
\(694\) −63.4045 −2.40680
\(695\) −15.6810 −0.594813
\(696\) −14.2856 −0.541496
\(697\) 0.874633 0.0331291
\(698\) −27.5525 −1.04288
\(699\) −26.8522 −1.01564
\(700\) 26.9037 1.01687
\(701\) 40.6252 1.53439 0.767196 0.641413i \(-0.221652\pi\)
0.767196 + 0.641413i \(0.221652\pi\)
\(702\) 47.0664 1.77641
\(703\) −23.3781 −0.881723
\(704\) 12.7455 0.480365
\(705\) 1.01492 0.0382241
\(706\) −2.70718 −0.101886
\(707\) −17.9049 −0.673382
\(708\) 121.018 4.54814
\(709\) −29.3401 −1.10189 −0.550945 0.834542i \(-0.685733\pi\)
−0.550945 + 0.834542i \(0.685733\pi\)
\(710\) −0.278067 −0.0104357
\(711\) 3.88503 0.145700
\(712\) 122.149 4.57774
\(713\) −14.8574 −0.556414
\(714\) −12.3111 −0.460731
\(715\) −15.8428 −0.592488
\(716\) 74.5938 2.78770
\(717\) 44.4788 1.66109
\(718\) 62.1347 2.31885
\(719\) −5.69273 −0.212303 −0.106152 0.994350i \(-0.533853\pi\)
−0.106152 + 0.994350i \(0.533853\pi\)
\(720\) 5.64100 0.210228
\(721\) 30.0546 1.11929
\(722\) 16.8800 0.628208
\(723\) 25.5942 0.951858
\(724\) 77.8574 2.89355
\(725\) 3.22154 0.119645
\(726\) −13.6636 −0.507102
\(727\) 31.4072 1.16483 0.582414 0.812892i \(-0.302108\pi\)
0.582414 + 0.812892i \(0.302108\pi\)
\(728\) −52.6040 −1.94963
\(729\) 21.7844 0.806829
\(730\) −12.2822 −0.454585
\(731\) −5.63399 −0.208381
\(732\) 56.7903 2.09903
\(733\) 47.6365 1.75949 0.879747 0.475441i \(-0.157712\pi\)
0.879747 + 0.475441i \(0.157712\pi\)
\(734\) 48.3646 1.78517
\(735\) 8.41494 0.310390
\(736\) −35.3221 −1.30199
\(737\) −21.5693 −0.794515
\(738\) −0.786461 −0.0289500
\(739\) 9.88794 0.363734 0.181867 0.983323i \(-0.441786\pi\)
0.181867 + 0.983323i \(0.441786\pi\)
\(740\) −44.8102 −1.64726
\(741\) 25.2307 0.926871
\(742\) −36.2803 −1.33189
\(743\) 17.7935 0.652782 0.326391 0.945235i \(-0.394167\pi\)
0.326391 + 0.945235i \(0.394167\pi\)
\(744\) −45.7030 −1.67555
\(745\) −10.6806 −0.391305
\(746\) 60.2759 2.20686
\(747\) 6.71220 0.245586
\(748\) 17.4649 0.638579
\(749\) −6.88606 −0.251611
\(750\) −54.9197 −2.00538
\(751\) −43.7934 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(752\) −3.26656 −0.119119
\(753\) 15.3671 0.560009
\(754\) −10.9790 −0.399832
\(755\) 8.79003 0.319902
\(756\) −43.6366 −1.58705
\(757\) 0.950190 0.0345353 0.0172676 0.999851i \(-0.494503\pi\)
0.0172676 + 0.999851i \(0.494503\pi\)
\(758\) 25.3499 0.920752
\(759\) −22.3255 −0.810363
\(760\) −35.4877 −1.28727
\(761\) −5.00434 −0.181407 −0.0907035 0.995878i \(-0.528912\pi\)
−0.0907035 + 0.995878i \(0.528912\pi\)
\(762\) 26.5683 0.962467
\(763\) −4.89898 −0.177355
\(764\) −9.55445 −0.345668
\(765\) 0.851925 0.0308014
\(766\) 5.10375 0.184406
\(767\) 53.3605 1.92674
\(768\) 41.8224 1.50913
\(769\) 16.5794 0.597870 0.298935 0.954274i \(-0.403369\pi\)
0.298935 + 0.954274i \(0.403369\pi\)
\(770\) 20.9495 0.754969
\(771\) 37.4408 1.34840
\(772\) −71.0268 −2.55631
\(773\) −30.4696 −1.09592 −0.547958 0.836506i \(-0.684595\pi\)
−0.547958 + 0.836506i \(0.684595\pi\)
\(774\) 5.06603 0.182095
\(775\) 10.3064 0.370218
\(776\) 55.8259 2.00403
\(777\) −24.1700 −0.867093
\(778\) 17.1044 0.613221
\(779\) 2.37028 0.0849240
\(780\) 48.3611 1.73160
\(781\) 0.212773 0.00761363
\(782\) −14.1819 −0.507145
\(783\) −5.22518 −0.186733
\(784\) −27.0837 −0.967277
\(785\) 3.21465 0.114736
\(786\) −101.992 −3.63792
\(787\) 29.7903 1.06191 0.530954 0.847400i \(-0.321834\pi\)
0.530954 + 0.847400i \(0.321834\pi\)
\(788\) −28.2114 −1.00499
\(789\) 51.8872 1.84723
\(790\) −32.0100 −1.13886
\(791\) 28.5611 1.01552
\(792\) −9.00998 −0.320156
\(793\) 25.0406 0.889216
\(794\) 19.7054 0.699318
\(795\) 19.1361 0.678689
\(796\) 110.499 3.91653
\(797\) 0.503125 0.0178216 0.00891081 0.999960i \(-0.497164\pi\)
0.00891081 + 0.999960i \(0.497164\pi\)
\(798\) −33.3634 −1.18105
\(799\) −0.493327 −0.0174527
\(800\) 24.5026 0.866299
\(801\) −7.94673 −0.280784
\(802\) −66.7830 −2.35819
\(803\) 9.39820 0.331655
\(804\) 65.8414 2.32205
\(805\) −11.9273 −0.420382
\(806\) −35.1244 −1.23720
\(807\) −21.5861 −0.759867
\(808\) −63.4471 −2.23206
\(809\) −21.1371 −0.743142 −0.371571 0.928405i \(-0.621181\pi\)
−0.371571 + 0.928405i \(0.621181\pi\)
\(810\) 37.9003 1.33168
\(811\) 15.7238 0.552139 0.276069 0.961138i \(-0.410968\pi\)
0.276069 + 0.961138i \(0.410968\pi\)
\(812\) 10.1789 0.357211
\(813\) 6.09105 0.213622
\(814\) 48.9043 1.71410
\(815\) −27.0358 −0.947023
\(816\) −20.8996 −0.731633
\(817\) −15.2683 −0.534169
\(818\) 35.4202 1.23844
\(819\) 3.42228 0.119584
\(820\) 4.54325 0.158657
\(821\) −42.7588 −1.49229 −0.746147 0.665781i \(-0.768098\pi\)
−0.746147 + 0.665781i \(0.768098\pi\)
\(822\) 56.1425 1.95819
\(823\) −33.2920 −1.16049 −0.580243 0.814444i \(-0.697042\pi\)
−0.580243 + 0.814444i \(0.697042\pi\)
\(824\) 106.500 3.71012
\(825\) 15.4870 0.539188
\(826\) −70.5605 −2.45511
\(827\) 14.2343 0.494975 0.247487 0.968891i \(-0.420395\pi\)
0.247487 + 0.968891i \(0.420395\pi\)
\(828\) 8.94097 0.310720
\(829\) −12.3766 −0.429855 −0.214928 0.976630i \(-0.568952\pi\)
−0.214928 + 0.976630i \(0.568952\pi\)
\(830\) −55.3040 −1.91963
\(831\) −18.7973 −0.652073
\(832\) −17.1550 −0.594743
\(833\) −4.09029 −0.141720
\(834\) −52.2404 −1.80894
\(835\) 32.7882 1.13468
\(836\) 47.3303 1.63695
\(837\) −16.7166 −0.577809
\(838\) −62.0917 −2.14492
\(839\) 31.6864 1.09394 0.546969 0.837153i \(-0.315782\pi\)
0.546969 + 0.837153i \(0.315782\pi\)
\(840\) −36.6897 −1.26591
\(841\) −27.7811 −0.957970
\(842\) −80.5270 −2.77514
\(843\) −31.1544 −1.07301
\(844\) 11.0519 0.380421
\(845\) 2.56604 0.0882745
\(846\) 0.443595 0.0152511
\(847\) 5.58565 0.191925
\(848\) −61.5903 −2.11502
\(849\) −28.6877 −0.984560
\(850\) 9.83789 0.337437
\(851\) −27.8429 −0.954443
\(852\) −0.649502 −0.0222516
\(853\) 30.2234 1.03483 0.517416 0.855734i \(-0.326894\pi\)
0.517416 + 0.855734i \(0.326894\pi\)
\(854\) −33.1120 −1.13307
\(855\) 2.30874 0.0789573
\(856\) −24.4012 −0.834017
\(857\) −31.4206 −1.07331 −0.536653 0.843803i \(-0.680312\pi\)
−0.536653 + 0.843803i \(0.680312\pi\)
\(858\) −52.7796 −1.80187
\(859\) 27.5866 0.941244 0.470622 0.882335i \(-0.344030\pi\)
0.470622 + 0.882335i \(0.344030\pi\)
\(860\) −29.2656 −0.997948
\(861\) 2.45056 0.0835149
\(862\) −72.5835 −2.47220
\(863\) 40.3690 1.37418 0.687088 0.726574i \(-0.258889\pi\)
0.687088 + 0.726574i \(0.258889\pi\)
\(864\) −39.7421 −1.35205
\(865\) −3.24859 −0.110455
\(866\) 53.2922 1.81094
\(867\) 28.4336 0.965655
\(868\) 32.5648 1.10532
\(869\) 24.4937 0.830890
\(870\) −7.65753 −0.259615
\(871\) 29.0315 0.983694
\(872\) −17.3599 −0.587879
\(873\) −3.63190 −0.122921
\(874\) −38.4334 −1.30003
\(875\) 22.4511 0.758985
\(876\) −28.6885 −0.969295
\(877\) −0.330616 −0.0111641 −0.00558205 0.999984i \(-0.501777\pi\)
−0.00558205 + 0.999984i \(0.501777\pi\)
\(878\) 94.1746 3.17824
\(879\) 13.9904 0.471884
\(880\) 35.5644 1.19888
\(881\) −4.51734 −0.152193 −0.0760965 0.997100i \(-0.524246\pi\)
−0.0760965 + 0.997100i \(0.524246\pi\)
\(882\) 3.67795 0.123843
\(883\) −46.7445 −1.57308 −0.786539 0.617540i \(-0.788129\pi\)
−0.786539 + 0.617540i \(0.788129\pi\)
\(884\) −23.5071 −0.790629
\(885\) 37.2174 1.25105
\(886\) 72.7404 2.44376
\(887\) −9.29057 −0.311947 −0.155973 0.987761i \(-0.549851\pi\)
−0.155973 + 0.987761i \(0.549851\pi\)
\(888\) −85.6480 −2.87416
\(889\) −10.8611 −0.364269
\(890\) 65.4757 2.19475
\(891\) −29.0008 −0.971564
\(892\) 63.9349 2.14070
\(893\) −1.33693 −0.0447387
\(894\) −35.5817 −1.19003
\(895\) 22.9403 0.766808
\(896\) −10.3171 −0.344671
\(897\) 30.0492 1.00332
\(898\) 33.4981 1.11785
\(899\) 3.89941 0.130053
\(900\) −6.20228 −0.206743
\(901\) −9.30159 −0.309881
\(902\) −4.95835 −0.165095
\(903\) −15.7854 −0.525306
\(904\) 101.208 3.36614
\(905\) 23.9439 0.795923
\(906\) 29.2836 0.972881
\(907\) −24.0423 −0.798313 −0.399156 0.916883i \(-0.630697\pi\)
−0.399156 + 0.916883i \(0.630697\pi\)
\(908\) 56.2138 1.86552
\(909\) 4.12771 0.136908
\(910\) −28.1973 −0.934732
\(911\) 6.04735 0.200358 0.100179 0.994969i \(-0.468059\pi\)
0.100179 + 0.994969i \(0.468059\pi\)
\(912\) −56.6385 −1.87549
\(913\) 42.3179 1.40052
\(914\) −0.387061 −0.0128028
\(915\) 17.4650 0.577376
\(916\) 127.649 4.21765
\(917\) 41.6940 1.37686
\(918\) −15.9566 −0.526645
\(919\) −34.8990 −1.15121 −0.575606 0.817727i \(-0.695234\pi\)
−0.575606 + 0.817727i \(0.695234\pi\)
\(920\) −42.2652 −1.39344
\(921\) −31.9488 −1.05275
\(922\) −100.866 −3.32185
\(923\) −0.286385 −0.00942648
\(924\) 48.9334 1.60979
\(925\) 19.3144 0.635054
\(926\) 52.0956 1.71197
\(927\) −6.92866 −0.227567
\(928\) 9.27048 0.304319
\(929\) 39.8343 1.30692 0.653460 0.756961i \(-0.273317\pi\)
0.653460 + 0.756961i \(0.273317\pi\)
\(930\) −24.4982 −0.803327
\(931\) −11.0848 −0.363289
\(932\) 67.7992 2.22084
\(933\) −18.8788 −0.618063
\(934\) −38.4573 −1.25836
\(935\) 5.37108 0.175653
\(936\) 12.1271 0.396387
\(937\) 10.6822 0.348971 0.174485 0.984660i \(-0.444174\pi\)
0.174485 + 0.984660i \(0.444174\pi\)
\(938\) −38.3894 −1.25346
\(939\) −49.5978 −1.61856
\(940\) −2.56257 −0.0835819
\(941\) −1.40329 −0.0457459 −0.0228729 0.999738i \(-0.507281\pi\)
−0.0228729 + 0.999738i \(0.507281\pi\)
\(942\) 10.7095 0.348933
\(943\) 2.82296 0.0919281
\(944\) −119.785 −3.89868
\(945\) −13.4198 −0.436546
\(946\) 31.9395 1.03844
\(947\) 9.93625 0.322885 0.161442 0.986882i \(-0.448385\pi\)
0.161442 + 0.986882i \(0.448385\pi\)
\(948\) −74.7682 −2.42836
\(949\) −12.6496 −0.410625
\(950\) 26.6609 0.864995
\(951\) −36.0227 −1.16812
\(952\) 17.8339 0.578001
\(953\) 11.4560 0.371097 0.185548 0.982635i \(-0.440594\pi\)
0.185548 + 0.982635i \(0.440594\pi\)
\(954\) 8.36390 0.270791
\(955\) −2.93834 −0.0950823
\(956\) −112.304 −3.63218
\(957\) 5.85945 0.189409
\(958\) 91.1908 2.94624
\(959\) −22.9510 −0.741126
\(960\) −11.9651 −0.386172
\(961\) −18.5249 −0.597578
\(962\) −65.8234 −2.12223
\(963\) 1.58748 0.0511560
\(964\) −64.6227 −2.08136
\(965\) −21.8433 −0.703160
\(966\) −39.7352 −1.27846
\(967\) −32.0337 −1.03014 −0.515068 0.857150i \(-0.672233\pi\)
−0.515068 + 0.857150i \(0.672233\pi\)
\(968\) 19.7931 0.636175
\(969\) −8.55376 −0.274786
\(970\) 29.9244 0.960814
\(971\) −21.7771 −0.698859 −0.349429 0.936963i \(-0.613625\pi\)
−0.349429 + 0.936963i \(0.613625\pi\)
\(972\) 21.9089 0.702728
\(973\) 21.3558 0.684636
\(974\) −70.9736 −2.27414
\(975\) −20.8449 −0.667572
\(976\) −56.2118 −1.79930
\(977\) 48.8640 1.56330 0.781649 0.623719i \(-0.214379\pi\)
0.781649 + 0.623719i \(0.214379\pi\)
\(978\) −90.0684 −2.88007
\(979\) −50.1012 −1.60124
\(980\) −21.2469 −0.678706
\(981\) 1.12939 0.0360586
\(982\) 12.5188 0.399490
\(983\) 43.9288 1.40111 0.700555 0.713598i \(-0.252936\pi\)
0.700555 + 0.713598i \(0.252936\pi\)
\(984\) 8.68374 0.276827
\(985\) −8.67600 −0.276440
\(986\) 3.72213 0.118537
\(987\) −1.38221 −0.0439963
\(988\) −63.7049 −2.02672
\(989\) −18.1842 −0.578225
\(990\) −4.82962 −0.153495
\(991\) 18.0715 0.574060 0.287030 0.957922i \(-0.407332\pi\)
0.287030 + 0.957922i \(0.407332\pi\)
\(992\) 29.6584 0.941655
\(993\) 48.4215 1.53661
\(994\) 0.378697 0.0120115
\(995\) 33.9824 1.07731
\(996\) −129.178 −4.09315
\(997\) −36.5966 −1.15903 −0.579514 0.814963i \(-0.696757\pi\)
−0.579514 + 0.814963i \(0.696757\pi\)
\(998\) −43.3918 −1.37354
\(999\) −31.3270 −0.991143
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 503.2.a.e.1.1 10
3.2 odd 2 4527.2.a.k.1.10 10
4.3 odd 2 8048.2.a.p.1.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.1 10 1.1 even 1 trivial
4527.2.a.k.1.10 10 3.2 odd 2
8048.2.a.p.1.8 10 4.3 odd 2