Properties

Label 5054.2.a.bg.1.2
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.19520000000.1
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 \) Copy content Toggle raw display
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 1.2
Root \(-0.761439\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.40760 q^{3} +1.00000 q^{4} -2.26420 q^{5} +1.40760 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.01865 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.40760 q^{3} +1.00000 q^{4} -2.26420 q^{5} +1.40760 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.01865 q^{9} +2.26420 q^{10} +2.53402 q^{11} -1.40760 q^{12} +1.31420 q^{13} -1.00000 q^{14} +3.18710 q^{15} +1.00000 q^{16} -4.50663 q^{17} +1.01865 q^{18} -2.26420 q^{20} -1.40760 q^{21} -2.53402 q^{22} -7.70050 q^{23} +1.40760 q^{24} +0.126599 q^{25} -1.31420 q^{26} +5.65667 q^{27} +1.00000 q^{28} +1.79740 q^{29} -3.18710 q^{30} +2.25306 q^{31} -1.00000 q^{32} -3.56689 q^{33} +4.50663 q^{34} -2.26420 q^{35} -1.01865 q^{36} +2.02693 q^{37} -1.84987 q^{39} +2.26420 q^{40} -8.23088 q^{41} +1.40760 q^{42} +1.11121 q^{43} +2.53402 q^{44} +2.30643 q^{45} +7.70050 q^{46} +5.77571 q^{47} -1.40760 q^{48} +1.00000 q^{49} -0.126599 q^{50} +6.34355 q^{51} +1.31420 q^{52} +12.0832 q^{53} -5.65667 q^{54} -5.73752 q^{55} -1.00000 q^{56} -1.79740 q^{58} -5.02694 q^{59} +3.18710 q^{60} -9.44679 q^{61} -2.25306 q^{62} -1.01865 q^{63} +1.00000 q^{64} -2.97560 q^{65} +3.56689 q^{66} +1.10809 q^{67} -4.50663 q^{68} +10.8393 q^{69} +2.26420 q^{70} +11.7694 q^{71} +1.01865 q^{72} -10.2135 q^{73} -2.02693 q^{74} -0.178202 q^{75} +2.53402 q^{77} +1.84987 q^{78} -2.75261 q^{79} -2.26420 q^{80} -4.90640 q^{81} +8.23088 q^{82} -11.7882 q^{83} -1.40760 q^{84} +10.2039 q^{85} -1.11121 q^{86} -2.53002 q^{87} -2.53402 q^{88} -13.4337 q^{89} -2.30643 q^{90} +1.31420 q^{91} -7.70050 q^{92} -3.17142 q^{93} -5.77571 q^{94} +1.40760 q^{96} +0.637330 q^{97} -1.00000 q^{98} -2.58128 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 2 q^{10} - 12 q^{11} + 4 q^{12} + 10 q^{13} - 8 q^{14} + 24 q^{15} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 20 q^{23} - 4 q^{24} + 8 q^{25} - 10 q^{26} + 22 q^{27} + 8 q^{28} + 8 q^{29} - 24 q^{30} + 18 q^{31} - 8 q^{32} - 16 q^{33} + 6 q^{34} - 2 q^{35} + 8 q^{36} + 36 q^{37} + 2 q^{40} + 4 q^{41} - 4 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} + 20 q^{46} - 10 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} + 12 q^{51} + 10 q^{52} + 32 q^{53} - 22 q^{54} - 22 q^{55} - 8 q^{56} - 8 q^{58} - 2 q^{59} + 24 q^{60} + 2 q^{61} - 18 q^{62} + 8 q^{63} + 8 q^{64} + 16 q^{66} + 44 q^{67} - 6 q^{68} + 2 q^{70} + 8 q^{71} - 8 q^{72} + 30 q^{73} - 36 q^{74} - 16 q^{75} - 12 q^{77} + 60 q^{79} - 2 q^{80} + 12 q^{81} - 4 q^{82} - 28 q^{83} + 4 q^{84} - 16 q^{85} + 16 q^{86} + 24 q^{87} + 12 q^{88} - 22 q^{89} - 8 q^{90} + 10 q^{91} - 20 q^{92} - 16 q^{93} + 10 q^{94} - 4 q^{96} - 6 q^{97} - 8 q^{98} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.40760 −0.812681 −0.406340 0.913722i \(-0.633195\pi\)
−0.406340 + 0.913722i \(0.633195\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.26420 −1.01258 −0.506290 0.862363i \(-0.668984\pi\)
−0.506290 + 0.862363i \(0.668984\pi\)
\(6\) 1.40760 0.574652
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −1.01865 −0.339550
\(10\) 2.26420 0.716003
\(11\) 2.53402 0.764035 0.382018 0.924155i \(-0.375229\pi\)
0.382018 + 0.924155i \(0.375229\pi\)
\(12\) −1.40760 −0.406340
\(13\) 1.31420 0.364492 0.182246 0.983253i \(-0.441663\pi\)
0.182246 + 0.983253i \(0.441663\pi\)
\(14\) −1.00000 −0.267261
\(15\) 3.18710 0.822905
\(16\) 1.00000 0.250000
\(17\) −4.50663 −1.09302 −0.546509 0.837453i \(-0.684044\pi\)
−0.546509 + 0.837453i \(0.684044\pi\)
\(18\) 1.01865 0.240098
\(19\) 0 0
\(20\) −2.26420 −0.506290
\(21\) −1.40760 −0.307164
\(22\) −2.53402 −0.540254
\(23\) −7.70050 −1.60566 −0.802832 0.596205i \(-0.796675\pi\)
−0.802832 + 0.596205i \(0.796675\pi\)
\(24\) 1.40760 0.287326
\(25\) 0.126599 0.0253199
\(26\) −1.31420 −0.257735
\(27\) 5.65667 1.08863
\(28\) 1.00000 0.188982
\(29\) 1.79740 0.333768 0.166884 0.985977i \(-0.446629\pi\)
0.166884 + 0.985977i \(0.446629\pi\)
\(30\) −3.18710 −0.581882
\(31\) 2.25306 0.404661 0.202331 0.979317i \(-0.435148\pi\)
0.202331 + 0.979317i \(0.435148\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.56689 −0.620917
\(34\) 4.50663 0.772881
\(35\) −2.26420 −0.382720
\(36\) −1.01865 −0.169775
\(37\) 2.02693 0.333225 0.166612 0.986022i \(-0.446717\pi\)
0.166612 + 0.986022i \(0.446717\pi\)
\(38\) 0 0
\(39\) −1.84987 −0.296216
\(40\) 2.26420 0.358001
\(41\) −8.23088 −1.28545 −0.642724 0.766098i \(-0.722196\pi\)
−0.642724 + 0.766098i \(0.722196\pi\)
\(42\) 1.40760 0.217198
\(43\) 1.11121 0.169459 0.0847293 0.996404i \(-0.472997\pi\)
0.0847293 + 0.996404i \(0.472997\pi\)
\(44\) 2.53402 0.382018
\(45\) 2.30643 0.343822
\(46\) 7.70050 1.13538
\(47\) 5.77571 0.842474 0.421237 0.906951i \(-0.361596\pi\)
0.421237 + 0.906951i \(0.361596\pi\)
\(48\) −1.40760 −0.203170
\(49\) 1.00000 0.142857
\(50\) −0.126599 −0.0179039
\(51\) 6.34355 0.888275
\(52\) 1.31420 0.182246
\(53\) 12.0832 1.65975 0.829876 0.557948i \(-0.188411\pi\)
0.829876 + 0.557948i \(0.188411\pi\)
\(54\) −5.65667 −0.769775
\(55\) −5.73752 −0.773647
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −1.79740 −0.236010
\(59\) −5.02694 −0.654452 −0.327226 0.944946i \(-0.606114\pi\)
−0.327226 + 0.944946i \(0.606114\pi\)
\(60\) 3.18710 0.411452
\(61\) −9.44679 −1.20954 −0.604769 0.796401i \(-0.706734\pi\)
−0.604769 + 0.796401i \(0.706734\pi\)
\(62\) −2.25306 −0.286139
\(63\) −1.01865 −0.128338
\(64\) 1.00000 0.125000
\(65\) −2.97560 −0.369078
\(66\) 3.56689 0.439054
\(67\) 1.10809 0.135375 0.0676876 0.997707i \(-0.478438\pi\)
0.0676876 + 0.997707i \(0.478438\pi\)
\(68\) −4.50663 −0.546509
\(69\) 10.8393 1.30489
\(70\) 2.26420 0.270624
\(71\) 11.7694 1.39678 0.698388 0.715720i \(-0.253901\pi\)
0.698388 + 0.715720i \(0.253901\pi\)
\(72\) 1.01865 0.120049
\(73\) −10.2135 −1.19540 −0.597699 0.801720i \(-0.703918\pi\)
−0.597699 + 0.801720i \(0.703918\pi\)
\(74\) −2.02693 −0.235625
\(75\) −0.178202 −0.0205770
\(76\) 0 0
\(77\) 2.53402 0.288778
\(78\) 1.84987 0.209456
\(79\) −2.75261 −0.309693 −0.154847 0.987939i \(-0.549488\pi\)
−0.154847 + 0.987939i \(0.549488\pi\)
\(80\) −2.26420 −0.253145
\(81\) −4.90640 −0.545156
\(82\) 8.23088 0.908949
\(83\) −11.7882 −1.29392 −0.646962 0.762522i \(-0.723961\pi\)
−0.646962 + 0.762522i \(0.723961\pi\)
\(84\) −1.40760 −0.153582
\(85\) 10.2039 1.10677
\(86\) −1.11121 −0.119825
\(87\) −2.53002 −0.271247
\(88\) −2.53402 −0.270127
\(89\) −13.4337 −1.42397 −0.711986 0.702194i \(-0.752204\pi\)
−0.711986 + 0.702194i \(0.752204\pi\)
\(90\) −2.30643 −0.243119
\(91\) 1.31420 0.137765
\(92\) −7.70050 −0.802832
\(93\) −3.17142 −0.328861
\(94\) −5.77571 −0.595719
\(95\) 0 0
\(96\) 1.40760 0.143663
\(97\) 0.637330 0.0647111 0.0323555 0.999476i \(-0.489699\pi\)
0.0323555 + 0.999476i \(0.489699\pi\)
\(98\) −1.00000 −0.101015
\(99\) −2.58128 −0.259428
\(100\) 0.126599 0.0126599
\(101\) −3.11677 −0.310130 −0.155065 0.987904i \(-0.549559\pi\)
−0.155065 + 0.987904i \(0.549559\pi\)
\(102\) −6.34355 −0.628105
\(103\) 1.21127 0.119350 0.0596750 0.998218i \(-0.480994\pi\)
0.0596750 + 0.998218i \(0.480994\pi\)
\(104\) −1.31420 −0.128867
\(105\) 3.18710 0.311029
\(106\) −12.0832 −1.17362
\(107\) −19.9061 −1.92439 −0.962195 0.272361i \(-0.912196\pi\)
−0.962195 + 0.272361i \(0.912196\pi\)
\(108\) 5.65667 0.544313
\(109\) 5.76507 0.552193 0.276097 0.961130i \(-0.410959\pi\)
0.276097 + 0.961130i \(0.410959\pi\)
\(110\) 5.73752 0.547051
\(111\) −2.85311 −0.270805
\(112\) 1.00000 0.0944911
\(113\) −10.7913 −1.01516 −0.507580 0.861604i \(-0.669460\pi\)
−0.507580 + 0.861604i \(0.669460\pi\)
\(114\) 0 0
\(115\) 17.4355 1.62587
\(116\) 1.79740 0.166884
\(117\) −1.33871 −0.123763
\(118\) 5.02694 0.462767
\(119\) −4.50663 −0.413122
\(120\) −3.18710 −0.290941
\(121\) −4.57875 −0.416250
\(122\) 9.44679 0.855272
\(123\) 11.5858 1.04466
\(124\) 2.25306 0.202331
\(125\) 11.0344 0.986942
\(126\) 1.01865 0.0907486
\(127\) 11.4309 1.01433 0.507165 0.861849i \(-0.330693\pi\)
0.507165 + 0.861849i \(0.330693\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.56415 −0.137716
\(130\) 2.97560 0.260978
\(131\) 19.7348 1.72424 0.862121 0.506703i \(-0.169136\pi\)
0.862121 + 0.506703i \(0.169136\pi\)
\(132\) −3.56689 −0.310458
\(133\) 0 0
\(134\) −1.10809 −0.0957248
\(135\) −12.8078 −1.10232
\(136\) 4.50663 0.386440
\(137\) −17.4210 −1.48838 −0.744189 0.667969i \(-0.767164\pi\)
−0.744189 + 0.667969i \(0.767164\pi\)
\(138\) −10.8393 −0.922698
\(139\) 6.38450 0.541526 0.270763 0.962646i \(-0.412724\pi\)
0.270763 + 0.962646i \(0.412724\pi\)
\(140\) −2.26420 −0.191360
\(141\) −8.12991 −0.684662
\(142\) −11.7694 −0.987669
\(143\) 3.33020 0.278485
\(144\) −1.01865 −0.0848875
\(145\) −4.06967 −0.337967
\(146\) 10.2135 0.845275
\(147\) −1.40760 −0.116097
\(148\) 2.02693 0.166612
\(149\) −18.3360 −1.50215 −0.751073 0.660219i \(-0.770463\pi\)
−0.751073 + 0.660219i \(0.770463\pi\)
\(150\) 0.178202 0.0145501
\(151\) 17.2670 1.40517 0.702586 0.711599i \(-0.252028\pi\)
0.702586 + 0.711599i \(0.252028\pi\)
\(152\) 0 0
\(153\) 4.59068 0.371135
\(154\) −2.53402 −0.204197
\(155\) −5.10138 −0.409752
\(156\) −1.84987 −0.148108
\(157\) 7.87418 0.628428 0.314214 0.949352i \(-0.398259\pi\)
0.314214 + 0.949352i \(0.398259\pi\)
\(158\) 2.75261 0.218986
\(159\) −17.0083 −1.34885
\(160\) 2.26420 0.179001
\(161\) −7.70050 −0.606884
\(162\) 4.90640 0.385483
\(163\) 0.922928 0.0722893 0.0361447 0.999347i \(-0.488492\pi\)
0.0361447 + 0.999347i \(0.488492\pi\)
\(164\) −8.23088 −0.642724
\(165\) 8.07616 0.628728
\(166\) 11.7882 0.914943
\(167\) −11.4713 −0.887678 −0.443839 0.896107i \(-0.646384\pi\)
−0.443839 + 0.896107i \(0.646384\pi\)
\(168\) 1.40760 0.108599
\(169\) −11.2729 −0.867145
\(170\) −10.2039 −0.782604
\(171\) 0 0
\(172\) 1.11121 0.0847293
\(173\) 1.01941 0.0775042 0.0387521 0.999249i \(-0.487662\pi\)
0.0387521 + 0.999249i \(0.487662\pi\)
\(174\) 2.53002 0.191801
\(175\) 0.126599 0.00957002
\(176\) 2.53402 0.191009
\(177\) 7.07594 0.531860
\(178\) 13.4337 1.00690
\(179\) 6.16884 0.461080 0.230540 0.973063i \(-0.425951\pi\)
0.230540 + 0.973063i \(0.425951\pi\)
\(180\) 2.30643 0.171911
\(181\) 17.9933 1.33743 0.668716 0.743518i \(-0.266844\pi\)
0.668716 + 0.743518i \(0.266844\pi\)
\(182\) −1.31420 −0.0974147
\(183\) 13.2973 0.982968
\(184\) 7.70050 0.567688
\(185\) −4.58936 −0.337417
\(186\) 3.17142 0.232540
\(187\) −11.4199 −0.835105
\(188\) 5.77571 0.421237
\(189\) 5.65667 0.411462
\(190\) 0 0
\(191\) 3.40620 0.246464 0.123232 0.992378i \(-0.460674\pi\)
0.123232 + 0.992378i \(0.460674\pi\)
\(192\) −1.40760 −0.101585
\(193\) 3.90426 0.281035 0.140517 0.990078i \(-0.455123\pi\)
0.140517 + 0.990078i \(0.455123\pi\)
\(194\) −0.637330 −0.0457577
\(195\) 4.18847 0.299943
\(196\) 1.00000 0.0714286
\(197\) −2.65223 −0.188963 −0.0944817 0.995527i \(-0.530119\pi\)
−0.0944817 + 0.995527i \(0.530119\pi\)
\(198\) 2.58128 0.183443
\(199\) 12.4354 0.881520 0.440760 0.897625i \(-0.354709\pi\)
0.440760 + 0.897625i \(0.354709\pi\)
\(200\) −0.126599 −0.00895193
\(201\) −1.55976 −0.110017
\(202\) 3.11677 0.219295
\(203\) 1.79740 0.126153
\(204\) 6.34355 0.444138
\(205\) 18.6364 1.30162
\(206\) −1.21127 −0.0843932
\(207\) 7.84411 0.545204
\(208\) 1.31420 0.0911231
\(209\) 0 0
\(210\) −3.18710 −0.219931
\(211\) 14.5888 1.00433 0.502166 0.864771i \(-0.332537\pi\)
0.502166 + 0.864771i \(0.332537\pi\)
\(212\) 12.0832 0.829876
\(213\) −16.5667 −1.13513
\(214\) 19.9061 1.36075
\(215\) −2.51601 −0.171591
\(216\) −5.65667 −0.384888
\(217\) 2.25306 0.152948
\(218\) −5.76507 −0.390460
\(219\) 14.3766 0.971477
\(220\) −5.73752 −0.386824
\(221\) −5.92260 −0.398397
\(222\) 2.85311 0.191488
\(223\) −16.8285 −1.12692 −0.563459 0.826144i \(-0.690530\pi\)
−0.563459 + 0.826144i \(0.690530\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −0.128961 −0.00859737
\(226\) 10.7913 0.717827
\(227\) −4.45322 −0.295570 −0.147785 0.989019i \(-0.547214\pi\)
−0.147785 + 0.989019i \(0.547214\pi\)
\(228\) 0 0
\(229\) 20.1895 1.33416 0.667081 0.744985i \(-0.267544\pi\)
0.667081 + 0.744985i \(0.267544\pi\)
\(230\) −17.4355 −1.14966
\(231\) −3.56689 −0.234684
\(232\) −1.79740 −0.118005
\(233\) 20.3141 1.33082 0.665410 0.746478i \(-0.268257\pi\)
0.665410 + 0.746478i \(0.268257\pi\)
\(234\) 1.33871 0.0875139
\(235\) −13.0774 −0.853073
\(236\) −5.02694 −0.327226
\(237\) 3.87459 0.251682
\(238\) 4.50663 0.292122
\(239\) 25.8369 1.67125 0.835625 0.549300i \(-0.185105\pi\)
0.835625 + 0.549300i \(0.185105\pi\)
\(240\) 3.18710 0.205726
\(241\) 16.3177 1.05112 0.525558 0.850757i \(-0.323856\pi\)
0.525558 + 0.850757i \(0.323856\pi\)
\(242\) 4.57875 0.294333
\(243\) −10.0637 −0.645589
\(244\) −9.44679 −0.604769
\(245\) −2.26420 −0.144654
\(246\) −11.5858 −0.738685
\(247\) 0 0
\(248\) −2.25306 −0.143069
\(249\) 16.5931 1.05155
\(250\) −11.0344 −0.697874
\(251\) −13.4191 −0.847005 −0.423502 0.905895i \(-0.639199\pi\)
−0.423502 + 0.905895i \(0.639199\pi\)
\(252\) −1.01865 −0.0641689
\(253\) −19.5132 −1.22678
\(254\) −11.4309 −0.717240
\(255\) −14.3631 −0.899450
\(256\) 1.00000 0.0625000
\(257\) 6.56928 0.409780 0.204890 0.978785i \(-0.434316\pi\)
0.204890 + 0.978785i \(0.434316\pi\)
\(258\) 1.56415 0.0973797
\(259\) 2.02693 0.125947
\(260\) −2.97560 −0.184539
\(261\) −1.83092 −0.113331
\(262\) −19.7348 −1.21922
\(263\) 28.6089 1.76410 0.882051 0.471154i \(-0.156163\pi\)
0.882051 + 0.471154i \(0.156163\pi\)
\(264\) 3.56689 0.219527
\(265\) −27.3587 −1.68063
\(266\) 0 0
\(267\) 18.9094 1.15723
\(268\) 1.10809 0.0676876
\(269\) 19.7174 1.20219 0.601095 0.799178i \(-0.294731\pi\)
0.601095 + 0.799178i \(0.294731\pi\)
\(270\) 12.8078 0.779460
\(271\) 6.93759 0.421429 0.210714 0.977548i \(-0.432421\pi\)
0.210714 + 0.977548i \(0.432421\pi\)
\(272\) −4.50663 −0.273255
\(273\) −1.84987 −0.111959
\(274\) 17.4210 1.05244
\(275\) 0.320805 0.0193453
\(276\) 10.8393 0.652446
\(277\) −22.1848 −1.33296 −0.666478 0.745525i \(-0.732199\pi\)
−0.666478 + 0.745525i \(0.732199\pi\)
\(278\) −6.38450 −0.382917
\(279\) −2.29508 −0.137403
\(280\) 2.26420 0.135312
\(281\) 32.8994 1.96262 0.981308 0.192443i \(-0.0616410\pi\)
0.981308 + 0.192443i \(0.0616410\pi\)
\(282\) 8.12991 0.484129
\(283\) −2.48145 −0.147507 −0.0737533 0.997277i \(-0.523498\pi\)
−0.0737533 + 0.997277i \(0.523498\pi\)
\(284\) 11.7694 0.698388
\(285\) 0 0
\(286\) −3.33020 −0.196919
\(287\) −8.23088 −0.485854
\(288\) 1.01865 0.0600245
\(289\) 3.30973 0.194690
\(290\) 4.06967 0.238979
\(291\) −0.897109 −0.0525895
\(292\) −10.2135 −0.597699
\(293\) 0.143516 0.00838429 0.00419214 0.999991i \(-0.498666\pi\)
0.00419214 + 0.999991i \(0.498666\pi\)
\(294\) 1.40760 0.0820931
\(295\) 11.3820 0.662685
\(296\) −2.02693 −0.117813
\(297\) 14.3341 0.831749
\(298\) 18.3360 1.06218
\(299\) −10.1200 −0.585252
\(300\) −0.178202 −0.0102885
\(301\) 1.11121 0.0640493
\(302\) −17.2670 −0.993607
\(303\) 4.38717 0.252036
\(304\) 0 0
\(305\) 21.3894 1.22475
\(306\) −4.59068 −0.262432
\(307\) 23.3704 1.33382 0.666910 0.745138i \(-0.267616\pi\)
0.666910 + 0.745138i \(0.267616\pi\)
\(308\) 2.53402 0.144389
\(309\) −1.70499 −0.0969934
\(310\) 5.10138 0.289739
\(311\) −16.8078 −0.953086 −0.476543 0.879151i \(-0.658110\pi\)
−0.476543 + 0.879151i \(0.658110\pi\)
\(312\) 1.84987 0.104728
\(313\) 14.1649 0.800650 0.400325 0.916373i \(-0.368897\pi\)
0.400325 + 0.916373i \(0.368897\pi\)
\(314\) −7.87418 −0.444366
\(315\) 2.30643 0.129952
\(316\) −2.75261 −0.154847
\(317\) −15.7154 −0.882662 −0.441331 0.897344i \(-0.645494\pi\)
−0.441331 + 0.897344i \(0.645494\pi\)
\(318\) 17.0083 0.953780
\(319\) 4.55464 0.255011
\(320\) −2.26420 −0.126573
\(321\) 28.0198 1.56392
\(322\) 7.70050 0.429132
\(323\) 0 0
\(324\) −4.90640 −0.272578
\(325\) 0.166376 0.00922891
\(326\) −0.922928 −0.0511163
\(327\) −8.11493 −0.448757
\(328\) 8.23088 0.454474
\(329\) 5.77571 0.318425
\(330\) −8.07616 −0.444578
\(331\) −7.84789 −0.431359 −0.215680 0.976464i \(-0.569197\pi\)
−0.215680 + 0.976464i \(0.569197\pi\)
\(332\) −11.7882 −0.646962
\(333\) −2.06473 −0.113146
\(334\) 11.4713 0.627683
\(335\) −2.50895 −0.137078
\(336\) −1.40760 −0.0767911
\(337\) 32.7210 1.78243 0.891213 0.453584i \(-0.149855\pi\)
0.891213 + 0.453584i \(0.149855\pi\)
\(338\) 11.2729 0.613164
\(339\) 15.1899 0.825002
\(340\) 10.2039 0.553385
\(341\) 5.70929 0.309176
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −1.11121 −0.0599127
\(345\) −24.5422 −1.32131
\(346\) −1.01941 −0.0548038
\(347\) 33.6602 1.80697 0.903486 0.428617i \(-0.140999\pi\)
0.903486 + 0.428617i \(0.140999\pi\)
\(348\) −2.53002 −0.135624
\(349\) −2.56580 −0.137344 −0.0686721 0.997639i \(-0.521876\pi\)
−0.0686721 + 0.997639i \(0.521876\pi\)
\(350\) −0.126599 −0.00676703
\(351\) 7.43397 0.396796
\(352\) −2.53402 −0.135064
\(353\) −22.9673 −1.22243 −0.611213 0.791466i \(-0.709318\pi\)
−0.611213 + 0.791466i \(0.709318\pi\)
\(354\) −7.07594 −0.376082
\(355\) −26.6484 −1.41435
\(356\) −13.4337 −0.711986
\(357\) 6.34355 0.335736
\(358\) −6.16884 −0.326033
\(359\) −5.36264 −0.283029 −0.141515 0.989936i \(-0.545197\pi\)
−0.141515 + 0.989936i \(0.545197\pi\)
\(360\) −2.30643 −0.121559
\(361\) 0 0
\(362\) −17.9933 −0.945708
\(363\) 6.44507 0.338279
\(364\) 1.31420 0.0688826
\(365\) 23.1254 1.21044
\(366\) −13.2973 −0.695063
\(367\) −22.4786 −1.17337 −0.586687 0.809814i \(-0.699568\pi\)
−0.586687 + 0.809814i \(0.699568\pi\)
\(368\) −7.70050 −0.401416
\(369\) 8.38439 0.436474
\(370\) 4.58936 0.238590
\(371\) 12.0832 0.627327
\(372\) −3.17142 −0.164430
\(373\) −12.3028 −0.637015 −0.318507 0.947920i \(-0.603182\pi\)
−0.318507 + 0.947920i \(0.603182\pi\)
\(374\) 11.4199 0.590508
\(375\) −15.5320 −0.802069
\(376\) −5.77571 −0.297859
\(377\) 2.36213 0.121656
\(378\) −5.65667 −0.290948
\(379\) −9.72581 −0.499581 −0.249791 0.968300i \(-0.580362\pi\)
−0.249791 + 0.968300i \(0.580362\pi\)
\(380\) 0 0
\(381\) −16.0902 −0.824327
\(382\) −3.40620 −0.174277
\(383\) −15.4098 −0.787402 −0.393701 0.919239i \(-0.628806\pi\)
−0.393701 + 0.919239i \(0.628806\pi\)
\(384\) 1.40760 0.0718315
\(385\) −5.73752 −0.292411
\(386\) −3.90426 −0.198722
\(387\) −1.13194 −0.0575397
\(388\) 0.637330 0.0323555
\(389\) 2.58927 0.131281 0.0656405 0.997843i \(-0.479091\pi\)
0.0656405 + 0.997843i \(0.479091\pi\)
\(390\) −4.18847 −0.212091
\(391\) 34.7033 1.75502
\(392\) −1.00000 −0.0505076
\(393\) −27.7789 −1.40126
\(394\) 2.65223 0.133617
\(395\) 6.23246 0.313589
\(396\) −2.58128 −0.129714
\(397\) 11.2929 0.566772 0.283386 0.959006i \(-0.408542\pi\)
0.283386 + 0.959006i \(0.408542\pi\)
\(398\) −12.4354 −0.623329
\(399\) 0 0
\(400\) 0.126599 0.00632997
\(401\) 19.0880 0.953210 0.476605 0.879118i \(-0.341867\pi\)
0.476605 + 0.879118i \(0.341867\pi\)
\(402\) 1.55976 0.0777937
\(403\) 2.96096 0.147496
\(404\) −3.11677 −0.155065
\(405\) 11.1091 0.552014
\(406\) −1.79740 −0.0892033
\(407\) 5.13627 0.254595
\(408\) −6.34355 −0.314053
\(409\) 30.7422 1.52010 0.760052 0.649862i \(-0.225173\pi\)
0.760052 + 0.649862i \(0.225173\pi\)
\(410\) −18.6364 −0.920384
\(411\) 24.5219 1.20958
\(412\) 1.21127 0.0596750
\(413\) −5.02694 −0.247359
\(414\) −7.84411 −0.385517
\(415\) 26.6909 1.31020
\(416\) −1.31420 −0.0644337
\(417\) −8.98685 −0.440088
\(418\) 0 0
\(419\) −24.3506 −1.18961 −0.594803 0.803872i \(-0.702770\pi\)
−0.594803 + 0.803872i \(0.702770\pi\)
\(420\) 3.18710 0.155514
\(421\) 5.28504 0.257577 0.128789 0.991672i \(-0.458891\pi\)
0.128789 + 0.991672i \(0.458891\pi\)
\(422\) −14.5888 −0.710170
\(423\) −5.88343 −0.286062
\(424\) −12.0832 −0.586811
\(425\) −0.570537 −0.0276751
\(426\) 16.5667 0.802660
\(427\) −9.44679 −0.457162
\(428\) −19.9061 −0.962195
\(429\) −4.68760 −0.226319
\(430\) 2.51601 0.121333
\(431\) −0.618814 −0.0298072 −0.0149036 0.999889i \(-0.504744\pi\)
−0.0149036 + 0.999889i \(0.504744\pi\)
\(432\) 5.65667 0.272157
\(433\) −36.8806 −1.77237 −0.886185 0.463331i \(-0.846654\pi\)
−0.886185 + 0.463331i \(0.846654\pi\)
\(434\) −2.25306 −0.108150
\(435\) 5.72848 0.274660
\(436\) 5.76507 0.276097
\(437\) 0 0
\(438\) −14.3766 −0.686938
\(439\) −39.8288 −1.90092 −0.950462 0.310840i \(-0.899390\pi\)
−0.950462 + 0.310840i \(0.899390\pi\)
\(440\) 5.73752 0.273526
\(441\) −1.01865 −0.0485072
\(442\) 5.92260 0.281709
\(443\) 37.2011 1.76748 0.883739 0.467980i \(-0.155018\pi\)
0.883739 + 0.467980i \(0.155018\pi\)
\(444\) −2.85311 −0.135403
\(445\) 30.4166 1.44189
\(446\) 16.8285 0.796852
\(447\) 25.8099 1.22077
\(448\) 1.00000 0.0472456
\(449\) 27.2063 1.28395 0.641973 0.766728i \(-0.278116\pi\)
0.641973 + 0.766728i \(0.278116\pi\)
\(450\) 0.128961 0.00607926
\(451\) −20.8572 −0.982127
\(452\) −10.7913 −0.507580
\(453\) −24.3052 −1.14196
\(454\) 4.45322 0.209000
\(455\) −2.97560 −0.139498
\(456\) 0 0
\(457\) 5.70914 0.267062 0.133531 0.991045i \(-0.457368\pi\)
0.133531 + 0.991045i \(0.457368\pi\)
\(458\) −20.1895 −0.943395
\(459\) −25.4925 −1.18989
\(460\) 17.4355 0.812933
\(461\) 18.4972 0.861499 0.430749 0.902472i \(-0.358249\pi\)
0.430749 + 0.902472i \(0.358249\pi\)
\(462\) 3.56689 0.165947
\(463\) 29.8307 1.38635 0.693175 0.720769i \(-0.256211\pi\)
0.693175 + 0.720769i \(0.256211\pi\)
\(464\) 1.79740 0.0834421
\(465\) 7.18072 0.332998
\(466\) −20.3141 −0.941032
\(467\) 34.9102 1.61545 0.807726 0.589559i \(-0.200698\pi\)
0.807726 + 0.589559i \(0.200698\pi\)
\(468\) −1.33871 −0.0618817
\(469\) 1.10809 0.0511670
\(470\) 13.0774 0.603213
\(471\) −11.0837 −0.510711
\(472\) 5.02694 0.231384
\(473\) 2.81584 0.129472
\(474\) −3.87459 −0.177966
\(475\) 0 0
\(476\) −4.50663 −0.206561
\(477\) −12.3085 −0.563569
\(478\) −25.8369 −1.18175
\(479\) 1.48027 0.0676354 0.0338177 0.999428i \(-0.489233\pi\)
0.0338177 + 0.999428i \(0.489233\pi\)
\(480\) −3.18710 −0.145470
\(481\) 2.66378 0.121458
\(482\) −16.3177 −0.743252
\(483\) 10.8393 0.493203
\(484\) −4.57875 −0.208125
\(485\) −1.44304 −0.0655252
\(486\) 10.0637 0.456500
\(487\) 29.1423 1.32057 0.660283 0.751017i \(-0.270436\pi\)
0.660283 + 0.751017i \(0.270436\pi\)
\(488\) 9.44679 0.427636
\(489\) −1.29912 −0.0587481
\(490\) 2.26420 0.102286
\(491\) −7.07800 −0.319426 −0.159713 0.987164i \(-0.551057\pi\)
−0.159713 + 0.987164i \(0.551057\pi\)
\(492\) 11.5858 0.522329
\(493\) −8.10021 −0.364815
\(494\) 0 0
\(495\) 5.84453 0.262692
\(496\) 2.25306 0.101165
\(497\) 11.7694 0.527931
\(498\) −16.5931 −0.743557
\(499\) 11.0798 0.496000 0.248000 0.968760i \(-0.420227\pi\)
0.248000 + 0.968760i \(0.420227\pi\)
\(500\) 11.0344 0.493471
\(501\) 16.1471 0.721399
\(502\) 13.4191 0.598923
\(503\) 23.3118 1.03942 0.519711 0.854342i \(-0.326040\pi\)
0.519711 + 0.854342i \(0.326040\pi\)
\(504\) 1.01865 0.0453743
\(505\) 7.05698 0.314031
\(506\) 19.5132 0.867467
\(507\) 15.8678 0.704712
\(508\) 11.4309 0.507165
\(509\) 25.2949 1.12118 0.560588 0.828095i \(-0.310575\pi\)
0.560588 + 0.828095i \(0.310575\pi\)
\(510\) 14.3631 0.636007
\(511\) −10.2135 −0.451818
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.56928 −0.289758
\(515\) −2.74256 −0.120852
\(516\) −1.56415 −0.0688579
\(517\) 14.6357 0.643679
\(518\) −2.02693 −0.0890580
\(519\) −1.43492 −0.0629862
\(520\) 2.97560 0.130489
\(521\) −5.59439 −0.245095 −0.122547 0.992463i \(-0.539106\pi\)
−0.122547 + 0.992463i \(0.539106\pi\)
\(522\) 1.83092 0.0801372
\(523\) −38.7757 −1.69554 −0.847771 0.530362i \(-0.822056\pi\)
−0.847771 + 0.530362i \(0.822056\pi\)
\(524\) 19.7348 0.862121
\(525\) −0.178202 −0.00777737
\(526\) −28.6089 −1.24741
\(527\) −10.1537 −0.442302
\(528\) −3.56689 −0.155229
\(529\) 36.2977 1.57816
\(530\) 27.3587 1.18839
\(531\) 5.12069 0.222219
\(532\) 0 0
\(533\) −10.8170 −0.468536
\(534\) −18.9094 −0.818288
\(535\) 45.0713 1.94860
\(536\) −1.10809 −0.0478624
\(537\) −8.68328 −0.374711
\(538\) −19.7174 −0.850077
\(539\) 2.53402 0.109148
\(540\) −12.8078 −0.551161
\(541\) 18.8155 0.808943 0.404472 0.914551i \(-0.367455\pi\)
0.404472 + 0.914551i \(0.367455\pi\)
\(542\) −6.93759 −0.297995
\(543\) −25.3275 −1.08691
\(544\) 4.50663 0.193220
\(545\) −13.0533 −0.559140
\(546\) 1.84987 0.0791670
\(547\) 5.85379 0.250290 0.125145 0.992138i \(-0.460060\pi\)
0.125145 + 0.992138i \(0.460060\pi\)
\(548\) −17.4210 −0.744189
\(549\) 9.62297 0.410699
\(550\) −0.320805 −0.0136792
\(551\) 0 0
\(552\) −10.8393 −0.461349
\(553\) −2.75261 −0.117053
\(554\) 22.1848 0.942542
\(555\) 6.46001 0.274212
\(556\) 6.38450 0.270763
\(557\) 43.6742 1.85053 0.925267 0.379316i \(-0.123841\pi\)
0.925267 + 0.379316i \(0.123841\pi\)
\(558\) 2.29508 0.0971585
\(559\) 1.46035 0.0617664
\(560\) −2.26420 −0.0956799
\(561\) 16.0747 0.678673
\(562\) −32.8994 −1.38778
\(563\) −3.05365 −0.128696 −0.0643479 0.997928i \(-0.520497\pi\)
−0.0643479 + 0.997928i \(0.520497\pi\)
\(564\) −8.12991 −0.342331
\(565\) 24.4337 1.02793
\(566\) 2.48145 0.104303
\(567\) −4.90640 −0.206049
\(568\) −11.7694 −0.493835
\(569\) 33.6088 1.40895 0.704477 0.709726i \(-0.251181\pi\)
0.704477 + 0.709726i \(0.251181\pi\)
\(570\) 0 0
\(571\) 21.8909 0.916106 0.458053 0.888925i \(-0.348547\pi\)
0.458053 + 0.888925i \(0.348547\pi\)
\(572\) 3.33020 0.139242
\(573\) −4.79459 −0.200297
\(574\) 8.23088 0.343550
\(575\) −0.974879 −0.0406553
\(576\) −1.01865 −0.0424438
\(577\) 41.7821 1.73941 0.869705 0.493571i \(-0.164309\pi\)
0.869705 + 0.493571i \(0.164309\pi\)
\(578\) −3.30973 −0.137666
\(579\) −5.49565 −0.228392
\(580\) −4.06967 −0.168984
\(581\) −11.7882 −0.489058
\(582\) 0.897109 0.0371864
\(583\) 30.6190 1.26811
\(584\) 10.2135 0.422637
\(585\) 3.03110 0.125320
\(586\) −0.143516 −0.00592859
\(587\) −32.6303 −1.34680 −0.673398 0.739280i \(-0.735166\pi\)
−0.673398 + 0.739280i \(0.735166\pi\)
\(588\) −1.40760 −0.0580486
\(589\) 0 0
\(590\) −11.3820 −0.468589
\(591\) 3.73329 0.153567
\(592\) 2.02693 0.0833061
\(593\) 0.644712 0.0264752 0.0132376 0.999912i \(-0.495786\pi\)
0.0132376 + 0.999912i \(0.495786\pi\)
\(594\) −14.3341 −0.588135
\(595\) 10.2039 0.418320
\(596\) −18.3360 −0.751073
\(597\) −17.5041 −0.716395
\(598\) 10.1200 0.413836
\(599\) 4.92246 0.201126 0.100563 0.994931i \(-0.467936\pi\)
0.100563 + 0.994931i \(0.467936\pi\)
\(600\) 0.178202 0.00727506
\(601\) 13.5325 0.552004 0.276002 0.961157i \(-0.410990\pi\)
0.276002 + 0.961157i \(0.410990\pi\)
\(602\) −1.11121 −0.0452897
\(603\) −1.12876 −0.0459667
\(604\) 17.2670 0.702586
\(605\) 10.3672 0.421487
\(606\) −4.38717 −0.178217
\(607\) 7.47174 0.303269 0.151634 0.988437i \(-0.451546\pi\)
0.151634 + 0.988437i \(0.451546\pi\)
\(608\) 0 0
\(609\) −2.53002 −0.102522
\(610\) −21.3894 −0.866032
\(611\) 7.59041 0.307075
\(612\) 4.59068 0.185567
\(613\) 8.98220 0.362788 0.181394 0.983411i \(-0.441939\pi\)
0.181394 + 0.983411i \(0.441939\pi\)
\(614\) −23.3704 −0.943153
\(615\) −26.2326 −1.05780
\(616\) −2.53402 −0.102098
\(617\) 4.62851 0.186337 0.0931683 0.995650i \(-0.470301\pi\)
0.0931683 + 0.995650i \(0.470301\pi\)
\(618\) 1.70499 0.0685847
\(619\) −0.578478 −0.0232510 −0.0116255 0.999932i \(-0.503701\pi\)
−0.0116255 + 0.999932i \(0.503701\pi\)
\(620\) −5.10138 −0.204876
\(621\) −43.5592 −1.74797
\(622\) 16.8078 0.673933
\(623\) −13.4337 −0.538210
\(624\) −1.84987 −0.0740540
\(625\) −25.6170 −1.02468
\(626\) −14.1649 −0.566145
\(627\) 0 0
\(628\) 7.87418 0.314214
\(629\) −9.13461 −0.364221
\(630\) −2.30643 −0.0918903
\(631\) −44.3940 −1.76730 −0.883648 0.468152i \(-0.844920\pi\)
−0.883648 + 0.468152i \(0.844920\pi\)
\(632\) 2.75261 0.109493
\(633\) −20.5352 −0.816201
\(634\) 15.7154 0.624136
\(635\) −25.8819 −1.02709
\(636\) −17.0083 −0.674424
\(637\) 1.31420 0.0520703
\(638\) −4.55464 −0.180320
\(639\) −11.9889 −0.474275
\(640\) 2.26420 0.0895003
\(641\) −21.8734 −0.863948 −0.431974 0.901886i \(-0.642183\pi\)
−0.431974 + 0.901886i \(0.642183\pi\)
\(642\) −28.0198 −1.10585
\(643\) 27.8528 1.09841 0.549204 0.835689i \(-0.314931\pi\)
0.549204 + 0.835689i \(0.314931\pi\)
\(644\) −7.70050 −0.303442
\(645\) 3.54155 0.139448
\(646\) 0 0
\(647\) −32.8926 −1.29314 −0.646570 0.762855i \(-0.723797\pi\)
−0.646570 + 0.762855i \(0.723797\pi\)
\(648\) 4.90640 0.192742
\(649\) −12.7383 −0.500024
\(650\) −0.166376 −0.00652582
\(651\) −3.17142 −0.124298
\(652\) 0.922928 0.0361447
\(653\) 28.3623 1.10990 0.554951 0.831883i \(-0.312737\pi\)
0.554951 + 0.831883i \(0.312737\pi\)
\(654\) 8.11493 0.317319
\(655\) −44.6836 −1.74593
\(656\) −8.23088 −0.321362
\(657\) 10.4040 0.405898
\(658\) −5.77571 −0.225161
\(659\) −4.04510 −0.157575 −0.0787874 0.996891i \(-0.525105\pi\)
−0.0787874 + 0.996891i \(0.525105\pi\)
\(660\) 8.07616 0.314364
\(661\) 45.1752 1.75711 0.878556 0.477639i \(-0.158507\pi\)
0.878556 + 0.477639i \(0.158507\pi\)
\(662\) 7.84789 0.305017
\(663\) 8.33667 0.323769
\(664\) 11.7882 0.457472
\(665\) 0 0
\(666\) 2.06473 0.0800066
\(667\) −13.8409 −0.535920
\(668\) −11.4713 −0.443839
\(669\) 23.6878 0.915825
\(670\) 2.50895 0.0969291
\(671\) −23.9383 −0.924129
\(672\) 1.40760 0.0542995
\(673\) −24.4991 −0.944370 −0.472185 0.881500i \(-0.656534\pi\)
−0.472185 + 0.881500i \(0.656534\pi\)
\(674\) −32.7210 −1.26037
\(675\) 0.716131 0.0275639
\(676\) −11.2729 −0.433573
\(677\) −51.1922 −1.96748 −0.983738 0.179610i \(-0.942516\pi\)
−0.983738 + 0.179610i \(0.942516\pi\)
\(678\) −15.1899 −0.583364
\(679\) 0.637330 0.0244585
\(680\) −10.2039 −0.391302
\(681\) 6.26837 0.240204
\(682\) −5.70929 −0.218620
\(683\) 24.6139 0.941825 0.470912 0.882180i \(-0.343925\pi\)
0.470912 + 0.882180i \(0.343925\pi\)
\(684\) 0 0
\(685\) 39.4447 1.50710
\(686\) −1.00000 −0.0381802
\(687\) −28.4189 −1.08425
\(688\) 1.11121 0.0423646
\(689\) 15.8797 0.604967
\(690\) 24.5422 0.934307
\(691\) 27.6269 1.05098 0.525488 0.850801i \(-0.323883\pi\)
0.525488 + 0.850801i \(0.323883\pi\)
\(692\) 1.01941 0.0387521
\(693\) −2.58128 −0.0980546
\(694\) −33.6602 −1.27772
\(695\) −14.4558 −0.548339
\(696\) 2.53002 0.0959003
\(697\) 37.0935 1.40502
\(698\) 2.56580 0.0971170
\(699\) −28.5942 −1.08153
\(700\) 0.126599 0.00478501
\(701\) −26.1374 −0.987198 −0.493599 0.869690i \(-0.664319\pi\)
−0.493599 + 0.869690i \(0.664319\pi\)
\(702\) −7.43397 −0.280577
\(703\) 0 0
\(704\) 2.53402 0.0955044
\(705\) 18.4077 0.693276
\(706\) 22.9673 0.864386
\(707\) −3.11677 −0.117218
\(708\) 7.07594 0.265930
\(709\) 0.802841 0.0301513 0.0150757 0.999886i \(-0.495201\pi\)
0.0150757 + 0.999886i \(0.495201\pi\)
\(710\) 26.6484 1.00009
\(711\) 2.80395 0.105156
\(712\) 13.4337 0.503450
\(713\) −17.3497 −0.649750
\(714\) −6.34355 −0.237402
\(715\) −7.54023 −0.281988
\(716\) 6.16884 0.230540
\(717\) −36.3681 −1.35819
\(718\) 5.36264 0.200132
\(719\) −33.3899 −1.24523 −0.622616 0.782527i \(-0.713930\pi\)
−0.622616 + 0.782527i \(0.713930\pi\)
\(720\) 2.30643 0.0859555
\(721\) 1.21127 0.0451101
\(722\) 0 0
\(723\) −22.9689 −0.854223
\(724\) 17.9933 0.668716
\(725\) 0.227550 0.00845098
\(726\) −6.44507 −0.239199
\(727\) −0.512039 −0.0189905 −0.00949524 0.999955i \(-0.503022\pi\)
−0.00949524 + 0.999955i \(0.503022\pi\)
\(728\) −1.31420 −0.0487073
\(729\) 28.8850 1.06981
\(730\) −23.1254 −0.855909
\(731\) −5.00783 −0.185221
\(732\) 13.2973 0.491484
\(733\) 0.898494 0.0331866 0.0165933 0.999862i \(-0.494718\pi\)
0.0165933 + 0.999862i \(0.494718\pi\)
\(734\) 22.4786 0.829700
\(735\) 3.18710 0.117558
\(736\) 7.70050 0.283844
\(737\) 2.80793 0.103431
\(738\) −8.38439 −0.308634
\(739\) 24.0173 0.883490 0.441745 0.897141i \(-0.354360\pi\)
0.441745 + 0.897141i \(0.354360\pi\)
\(740\) −4.58936 −0.168708
\(741\) 0 0
\(742\) −12.0832 −0.443587
\(743\) −5.73799 −0.210507 −0.105253 0.994445i \(-0.533565\pi\)
−0.105253 + 0.994445i \(0.533565\pi\)
\(744\) 3.17142 0.116270
\(745\) 41.5164 1.52104
\(746\) 12.3028 0.450437
\(747\) 12.0081 0.439352
\(748\) −11.4199 −0.417552
\(749\) −19.9061 −0.727351
\(750\) 15.5320 0.567148
\(751\) 37.6807 1.37499 0.687495 0.726189i \(-0.258710\pi\)
0.687495 + 0.726189i \(0.258710\pi\)
\(752\) 5.77571 0.210618
\(753\) 18.8888 0.688344
\(754\) −2.36213 −0.0860238
\(755\) −39.0960 −1.42285
\(756\) 5.65667 0.205731
\(757\) −39.0772 −1.42028 −0.710142 0.704059i \(-0.751369\pi\)
−0.710142 + 0.704059i \(0.751369\pi\)
\(758\) 9.72581 0.353257
\(759\) 27.4669 0.996984
\(760\) 0 0
\(761\) 13.4457 0.487407 0.243703 0.969850i \(-0.421638\pi\)
0.243703 + 0.969850i \(0.421638\pi\)
\(762\) 16.0902 0.582887
\(763\) 5.76507 0.208709
\(764\) 3.40620 0.123232
\(765\) −10.3942 −0.375804
\(766\) 15.4098 0.556777
\(767\) −6.60638 −0.238543
\(768\) −1.40760 −0.0507925
\(769\) −1.43300 −0.0516752 −0.0258376 0.999666i \(-0.508225\pi\)
−0.0258376 + 0.999666i \(0.508225\pi\)
\(770\) 5.73752 0.206766
\(771\) −9.24694 −0.333020
\(772\) 3.90426 0.140517
\(773\) −33.2573 −1.19618 −0.598090 0.801429i \(-0.704074\pi\)
−0.598090 + 0.801429i \(0.704074\pi\)
\(774\) 1.13194 0.0406867
\(775\) 0.285236 0.0102460
\(776\) −0.637330 −0.0228788
\(777\) −2.85311 −0.102355
\(778\) −2.58927 −0.0928297
\(779\) 0 0
\(780\) 4.18847 0.149971
\(781\) 29.8240 1.06719
\(782\) −34.7033 −1.24099
\(783\) 10.1673 0.363349
\(784\) 1.00000 0.0357143
\(785\) −17.8287 −0.636334
\(786\) 27.7789 0.990839
\(787\) −12.5378 −0.446925 −0.223463 0.974713i \(-0.571736\pi\)
−0.223463 + 0.974713i \(0.571736\pi\)
\(788\) −2.65223 −0.0944817
\(789\) −40.2700 −1.43365
\(790\) −6.23246 −0.221741
\(791\) −10.7913 −0.383695
\(792\) 2.58128 0.0917217
\(793\) −12.4149 −0.440867
\(794\) −11.2929 −0.400769
\(795\) 38.5102 1.36582
\(796\) 12.4354 0.440760
\(797\) 2.86425 0.101457 0.0507285 0.998712i \(-0.483846\pi\)
0.0507285 + 0.998712i \(0.483846\pi\)
\(798\) 0 0
\(799\) −26.0290 −0.920839
\(800\) −0.126599 −0.00447597
\(801\) 13.6843 0.483509
\(802\) −19.0880 −0.674021
\(803\) −25.8812 −0.913327
\(804\) −1.55976 −0.0550084
\(805\) 17.4355 0.614519
\(806\) −2.96096 −0.104295
\(807\) −27.7543 −0.976996
\(808\) 3.11677 0.109647
\(809\) 12.8448 0.451599 0.225799 0.974174i \(-0.427501\pi\)
0.225799 + 0.974174i \(0.427501\pi\)
\(810\) −11.1091 −0.390333
\(811\) −7.85076 −0.275678 −0.137839 0.990455i \(-0.544016\pi\)
−0.137839 + 0.990455i \(0.544016\pi\)
\(812\) 1.79740 0.0630763
\(813\) −9.76539 −0.342487
\(814\) −5.13627 −0.180026
\(815\) −2.08969 −0.0731988
\(816\) 6.34355 0.222069
\(817\) 0 0
\(818\) −30.7422 −1.07488
\(819\) −1.33871 −0.0467782
\(820\) 18.6364 0.650810
\(821\) −16.9053 −0.590000 −0.295000 0.955497i \(-0.595320\pi\)
−0.295000 + 0.955497i \(0.595320\pi\)
\(822\) −24.5219 −0.855300
\(823\) 0.523333 0.0182422 0.00912112 0.999958i \(-0.497097\pi\)
0.00912112 + 0.999958i \(0.497097\pi\)
\(824\) −1.21127 −0.0421966
\(825\) −0.451567 −0.0157215
\(826\) 5.02694 0.174910
\(827\) −31.6821 −1.10170 −0.550848 0.834606i \(-0.685695\pi\)
−0.550848 + 0.834606i \(0.685695\pi\)
\(828\) 7.84411 0.272602
\(829\) 7.83907 0.272262 0.136131 0.990691i \(-0.456533\pi\)
0.136131 + 0.990691i \(0.456533\pi\)
\(830\) −26.6909 −0.926454
\(831\) 31.2274 1.08327
\(832\) 1.31420 0.0455615
\(833\) −4.50663 −0.156146
\(834\) 8.98685 0.311189
\(835\) 25.9734 0.898846
\(836\) 0 0
\(837\) 12.7448 0.440525
\(838\) 24.3506 0.841178
\(839\) 13.7773 0.475646 0.237823 0.971308i \(-0.423566\pi\)
0.237823 + 0.971308i \(0.423566\pi\)
\(840\) −3.18710 −0.109965
\(841\) −25.7694 −0.888599
\(842\) −5.28504 −0.182134
\(843\) −46.3094 −1.59498
\(844\) 14.5888 0.502166
\(845\) 25.5241 0.878055
\(846\) 5.88343 0.202276
\(847\) −4.57875 −0.157328
\(848\) 12.0832 0.414938
\(849\) 3.49289 0.119876
\(850\) 0.570537 0.0195693
\(851\) −15.6083 −0.535047
\(852\) −16.5667 −0.567566
\(853\) −46.1988 −1.58182 −0.790909 0.611934i \(-0.790392\pi\)
−0.790909 + 0.611934i \(0.790392\pi\)
\(854\) 9.44679 0.323262
\(855\) 0 0
\(856\) 19.9061 0.680375
\(857\) 24.2885 0.829681 0.414840 0.909894i \(-0.363837\pi\)
0.414840 + 0.909894i \(0.363837\pi\)
\(858\) 4.68760 0.160032
\(859\) −52.9802 −1.80766 −0.903830 0.427892i \(-0.859257\pi\)
−0.903830 + 0.427892i \(0.859257\pi\)
\(860\) −2.51601 −0.0857953
\(861\) 11.5858 0.394844
\(862\) 0.618814 0.0210769
\(863\) −32.1746 −1.09524 −0.547618 0.836728i \(-0.684465\pi\)
−0.547618 + 0.836728i \(0.684465\pi\)
\(864\) −5.65667 −0.192444
\(865\) −2.30815 −0.0784793
\(866\) 36.8806 1.25326
\(867\) −4.65879 −0.158221
\(868\) 2.25306 0.0764738
\(869\) −6.97517 −0.236616
\(870\) −5.72848 −0.194214
\(871\) 1.45625 0.0493432
\(872\) −5.76507 −0.195230
\(873\) −0.649217 −0.0219727
\(874\) 0 0
\(875\) 11.0344 0.373029
\(876\) 14.3766 0.485739
\(877\) 45.6282 1.54075 0.770377 0.637588i \(-0.220068\pi\)
0.770377 + 0.637588i \(0.220068\pi\)
\(878\) 39.8288 1.34416
\(879\) −0.202014 −0.00681375
\(880\) −5.73752 −0.193412
\(881\) 29.3106 0.987500 0.493750 0.869604i \(-0.335626\pi\)
0.493750 + 0.869604i \(0.335626\pi\)
\(882\) 1.01865 0.0342997
\(883\) −39.9499 −1.34442 −0.672211 0.740360i \(-0.734655\pi\)
−0.672211 + 0.740360i \(0.734655\pi\)
\(884\) −5.92260 −0.199198
\(885\) −16.0213 −0.538551
\(886\) −37.2011 −1.24980
\(887\) −36.1050 −1.21229 −0.606144 0.795355i \(-0.707284\pi\)
−0.606144 + 0.795355i \(0.707284\pi\)
\(888\) 2.85311 0.0957441
\(889\) 11.4309 0.383381
\(890\) −30.4166 −1.01957
\(891\) −12.4329 −0.416518
\(892\) −16.8285 −0.563459
\(893\) 0 0
\(894\) −25.8099 −0.863211
\(895\) −13.9675 −0.466881
\(896\) −1.00000 −0.0334077
\(897\) 14.2449 0.475623
\(898\) −27.2063 −0.907886
\(899\) 4.04964 0.135063
\(900\) −0.128961 −0.00429869
\(901\) −54.4544 −1.81414
\(902\) 20.8572 0.694469
\(903\) −1.56415 −0.0520517
\(904\) 10.7913 0.358914
\(905\) −40.7405 −1.35426
\(906\) 24.3052 0.807485
\(907\) 10.7645 0.357428 0.178714 0.983901i \(-0.442806\pi\)
0.178714 + 0.983901i \(0.442806\pi\)
\(908\) −4.45322 −0.147785
\(909\) 3.17489 0.105305
\(910\) 2.97560 0.0986402
\(911\) 52.6695 1.74502 0.872509 0.488599i \(-0.162492\pi\)
0.872509 + 0.488599i \(0.162492\pi\)
\(912\) 0 0
\(913\) −29.8715 −0.988604
\(914\) −5.70914 −0.188842
\(915\) −30.1078 −0.995334
\(916\) 20.1895 0.667081
\(917\) 19.7348 0.651702
\(918\) 25.4925 0.841379
\(919\) 47.3855 1.56310 0.781552 0.623840i \(-0.214428\pi\)
0.781552 + 0.623840i \(0.214428\pi\)
\(920\) −17.4355 −0.574830
\(921\) −32.8963 −1.08397
\(922\) −18.4972 −0.609172
\(923\) 15.4673 0.509114
\(924\) −3.56689 −0.117342
\(925\) 0.256608 0.00843721
\(926\) −29.8307 −0.980298
\(927\) −1.23386 −0.0405253
\(928\) −1.79740 −0.0590025
\(929\) 46.9608 1.54073 0.770366 0.637602i \(-0.220073\pi\)
0.770366 + 0.637602i \(0.220073\pi\)
\(930\) −7.18072 −0.235465
\(931\) 0 0
\(932\) 20.3141 0.665410
\(933\) 23.6588 0.774554
\(934\) −34.9102 −1.14230
\(935\) 25.8569 0.845611
\(936\) 1.33871 0.0437570
\(937\) 4.08922 0.133589 0.0667945 0.997767i \(-0.478723\pi\)
0.0667945 + 0.997767i \(0.478723\pi\)
\(938\) −1.10809 −0.0361806
\(939\) −19.9386 −0.650673
\(940\) −13.0774 −0.426536
\(941\) 5.65921 0.184485 0.0922425 0.995737i \(-0.470597\pi\)
0.0922425 + 0.995737i \(0.470597\pi\)
\(942\) 11.0837 0.361127
\(943\) 63.3819 2.06400
\(944\) −5.02694 −0.163613
\(945\) −12.8078 −0.416639
\(946\) −2.81584 −0.0915507
\(947\) −20.0115 −0.650286 −0.325143 0.945665i \(-0.605413\pi\)
−0.325143 + 0.945665i \(0.605413\pi\)
\(948\) 3.87459 0.125841
\(949\) −13.4225 −0.435714
\(950\) 0 0
\(951\) 22.1210 0.717323
\(952\) 4.50663 0.146061
\(953\) −24.3912 −0.790107 −0.395054 0.918658i \(-0.629274\pi\)
−0.395054 + 0.918658i \(0.629274\pi\)
\(954\) 12.3085 0.398503
\(955\) −7.71233 −0.249565
\(956\) 25.8369 0.835625
\(957\) −6.41113 −0.207242
\(958\) −1.48027 −0.0478255
\(959\) −17.4210 −0.562554
\(960\) 3.18710 0.102863
\(961\) −25.9237 −0.836249
\(962\) −2.66378 −0.0858836
\(963\) 20.2773 0.653427
\(964\) 16.3177 0.525558
\(965\) −8.84002 −0.284570
\(966\) −10.8393 −0.348747
\(967\) 34.3208 1.10368 0.551842 0.833949i \(-0.313925\pi\)
0.551842 + 0.833949i \(0.313925\pi\)
\(968\) 4.57875 0.147167
\(969\) 0 0
\(970\) 1.44304 0.0463333
\(971\) 8.68641 0.278760 0.139380 0.990239i \(-0.455489\pi\)
0.139380 + 0.990239i \(0.455489\pi\)
\(972\) −10.0637 −0.322794
\(973\) 6.38450 0.204678
\(974\) −29.1423 −0.933781
\(975\) −0.234192 −0.00750016
\(976\) −9.44679 −0.302384
\(977\) 33.2315 1.06317 0.531584 0.847005i \(-0.321597\pi\)
0.531584 + 0.847005i \(0.321597\pi\)
\(978\) 1.29912 0.0415412
\(979\) −34.0413 −1.08796
\(980\) −2.26420 −0.0723272
\(981\) −5.87259 −0.187497
\(982\) 7.07800 0.225868
\(983\) −13.4827 −0.430032 −0.215016 0.976611i \(-0.568980\pi\)
−0.215016 + 0.976611i \(0.568980\pi\)
\(984\) −11.5858 −0.369343
\(985\) 6.00518 0.191341
\(986\) 8.10021 0.257963
\(987\) −8.12991 −0.258778
\(988\) 0 0
\(989\) −8.55690 −0.272094
\(990\) −5.84453 −0.185751
\(991\) 1.59631 0.0507084 0.0253542 0.999679i \(-0.491929\pi\)
0.0253542 + 0.999679i \(0.491929\pi\)
\(992\) −2.25306 −0.0715347
\(993\) 11.0467 0.350557
\(994\) −11.7694 −0.373304
\(995\) −28.1562 −0.892610
\(996\) 16.5931 0.525774
\(997\) −49.0430 −1.55321 −0.776603 0.629990i \(-0.783059\pi\)
−0.776603 + 0.629990i \(0.783059\pi\)
\(998\) −11.0798 −0.350725
\(999\) 11.4656 0.362757
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 5054.2.a.bg.1.2 8
19.18 odd 2 5054.2.a.bh.1.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.bg.1.2 8 1.1 even 1 trivial
5054.2.a.bh.1.7 yes 8 19.18 odd 2