Properties

Label 4730.2.a.be.1.10
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(2.59957\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.59957 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.59957 q^{6} +1.31461 q^{7} +1.00000 q^{8} +3.75779 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.59957 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.59957 q^{6} +1.31461 q^{7} +1.00000 q^{8} +3.75779 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.59957 q^{12} -5.57356 q^{13} +1.31461 q^{14} +2.59957 q^{15} +1.00000 q^{16} +6.71697 q^{17} +3.75779 q^{18} -4.17687 q^{19} +1.00000 q^{20} +3.41742 q^{21} -1.00000 q^{22} +7.57356 q^{23} +2.59957 q^{24} +1.00000 q^{25} -5.57356 q^{26} +1.96993 q^{27} +1.31461 q^{28} +3.72343 q^{29} +2.59957 q^{30} +5.06624 q^{31} +1.00000 q^{32} -2.59957 q^{33} +6.71697 q^{34} +1.31461 q^{35} +3.75779 q^{36} -4.21701 q^{37} -4.17687 q^{38} -14.4889 q^{39} +1.00000 q^{40} +8.18305 q^{41} +3.41742 q^{42} +1.00000 q^{43} -1.00000 q^{44} +3.75779 q^{45} +7.57356 q^{46} +9.44879 q^{47} +2.59957 q^{48} -5.27180 q^{49} +1.00000 q^{50} +17.4613 q^{51} -5.57356 q^{52} -7.39275 q^{53} +1.96993 q^{54} -1.00000 q^{55} +1.31461 q^{56} -10.8581 q^{57} +3.72343 q^{58} -9.36203 q^{59} +2.59957 q^{60} -0.668250 q^{61} +5.06624 q^{62} +4.94002 q^{63} +1.00000 q^{64} -5.57356 q^{65} -2.59957 q^{66} -2.23127 q^{67} +6.71697 q^{68} +19.6880 q^{69} +1.31461 q^{70} -12.3895 q^{71} +3.75779 q^{72} +5.06235 q^{73} -4.21701 q^{74} +2.59957 q^{75} -4.17687 q^{76} -1.31461 q^{77} -14.4889 q^{78} +13.7458 q^{79} +1.00000 q^{80} -6.15238 q^{81} +8.18305 q^{82} +17.1925 q^{83} +3.41742 q^{84} +6.71697 q^{85} +1.00000 q^{86} +9.67935 q^{87} -1.00000 q^{88} +13.5657 q^{89} +3.75779 q^{90} -7.32705 q^{91} +7.57356 q^{92} +13.1701 q^{93} +9.44879 q^{94} -4.17687 q^{95} +2.59957 q^{96} -6.57793 q^{97} -5.27180 q^{98} -3.75779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 3q^{3} + 12q^{4} + 12q^{5} + 3q^{6} + 8q^{7} + 12q^{8} + 25q^{9} + O(q^{10}) \) \( 12q + 12q^{2} + 3q^{3} + 12q^{4} + 12q^{5} + 3q^{6} + 8q^{7} + 12q^{8} + 25q^{9} + 12q^{10} - 12q^{11} + 3q^{12} + 16q^{13} + 8q^{14} + 3q^{15} + 12q^{16} + 18q^{17} + 25q^{18} - 4q^{19} + 12q^{20} + 4q^{21} - 12q^{22} + 8q^{23} + 3q^{24} + 12q^{25} + 16q^{26} + 6q^{27} + 8q^{28} + 20q^{29} + 3q^{30} + 5q^{31} + 12q^{32} - 3q^{33} + 18q^{34} + 8q^{35} + 25q^{36} + 19q^{37} - 4q^{38} + 6q^{39} + 12q^{40} + 16q^{41} + 4q^{42} + 12q^{43} - 12q^{44} + 25q^{45} + 8q^{46} - q^{47} + 3q^{48} + 52q^{49} + 12q^{50} + q^{51} + 16q^{52} + 11q^{53} + 6q^{54} - 12q^{55} + 8q^{56} + 9q^{57} + 20q^{58} - 11q^{59} + 3q^{60} + 18q^{61} + 5q^{62} + 15q^{63} + 12q^{64} + 16q^{65} - 3q^{66} - 10q^{67} + 18q^{68} + 8q^{70} - 2q^{71} + 25q^{72} + 29q^{73} + 19q^{74} + 3q^{75} - 4q^{76} - 8q^{77} + 6q^{78} + 2q^{79} + 12q^{80} - 8q^{81} + 16q^{82} + 26q^{83} + 4q^{84} + 18q^{85} + 12q^{86} - 4q^{87} - 12q^{88} + 41q^{89} + 25q^{90} - 4q^{91} + 8q^{92} + 5q^{93} - q^{94} - 4q^{95} + 3q^{96} - 7q^{97} + 52q^{98} - 25q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.59957 1.50087 0.750433 0.660947i \(-0.229845\pi\)
0.750433 + 0.660947i \(0.229845\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.59957 1.06127
\(7\) 1.31461 0.496875 0.248438 0.968648i \(-0.420083\pi\)
0.248438 + 0.968648i \(0.420083\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.75779 1.25260
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) 2.59957 0.750433
\(13\) −5.57356 −1.54583 −0.772914 0.634511i \(-0.781202\pi\)
−0.772914 + 0.634511i \(0.781202\pi\)
\(14\) 1.31461 0.351344
\(15\) 2.59957 0.671207
\(16\) 1.00000 0.250000
\(17\) 6.71697 1.62910 0.814552 0.580090i \(-0.196983\pi\)
0.814552 + 0.580090i \(0.196983\pi\)
\(18\) 3.75779 0.885720
\(19\) −4.17687 −0.958239 −0.479120 0.877750i \(-0.659044\pi\)
−0.479120 + 0.877750i \(0.659044\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.41742 0.745743
\(22\) −1.00000 −0.213201
\(23\) 7.57356 1.57920 0.789599 0.613624i \(-0.210289\pi\)
0.789599 + 0.613624i \(0.210289\pi\)
\(24\) 2.59957 0.530636
\(25\) 1.00000 0.200000
\(26\) −5.57356 −1.09307
\(27\) 1.96993 0.379114
\(28\) 1.31461 0.248438
\(29\) 3.72343 0.691424 0.345712 0.938341i \(-0.387637\pi\)
0.345712 + 0.938341i \(0.387637\pi\)
\(30\) 2.59957 0.474615
\(31\) 5.06624 0.909923 0.454962 0.890511i \(-0.349653\pi\)
0.454962 + 0.890511i \(0.349653\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.59957 −0.452528
\(34\) 6.71697 1.15195
\(35\) 1.31461 0.222209
\(36\) 3.75779 0.626298
\(37\) −4.21701 −0.693273 −0.346636 0.938000i \(-0.612676\pi\)
−0.346636 + 0.938000i \(0.612676\pi\)
\(38\) −4.17687 −0.677578
\(39\) −14.4889 −2.32008
\(40\) 1.00000 0.158114
\(41\) 8.18305 1.27798 0.638989 0.769216i \(-0.279353\pi\)
0.638989 + 0.769216i \(0.279353\pi\)
\(42\) 3.41742 0.527320
\(43\) 1.00000 0.152499
\(44\) −1.00000 −0.150756
\(45\) 3.75779 0.560178
\(46\) 7.57356 1.11666
\(47\) 9.44879 1.37825 0.689124 0.724644i \(-0.257996\pi\)
0.689124 + 0.724644i \(0.257996\pi\)
\(48\) 2.59957 0.375216
\(49\) −5.27180 −0.753115
\(50\) 1.00000 0.141421
\(51\) 17.4613 2.44507
\(52\) −5.57356 −0.772914
\(53\) −7.39275 −1.01547 −0.507736 0.861513i \(-0.669517\pi\)
−0.507736 + 0.861513i \(0.669517\pi\)
\(54\) 1.96993 0.268074
\(55\) −1.00000 −0.134840
\(56\) 1.31461 0.175672
\(57\) −10.8581 −1.43819
\(58\) 3.72343 0.488911
\(59\) −9.36203 −1.21883 −0.609416 0.792850i \(-0.708596\pi\)
−0.609416 + 0.792850i \(0.708596\pi\)
\(60\) 2.59957 0.335604
\(61\) −0.668250 −0.0855607 −0.0427803 0.999085i \(-0.513622\pi\)
−0.0427803 + 0.999085i \(0.513622\pi\)
\(62\) 5.06624 0.643413
\(63\) 4.94002 0.622384
\(64\) 1.00000 0.125000
\(65\) −5.57356 −0.691315
\(66\) −2.59957 −0.319986
\(67\) −2.23127 −0.272593 −0.136296 0.990668i \(-0.543520\pi\)
−0.136296 + 0.990668i \(0.543520\pi\)
\(68\) 6.71697 0.814552
\(69\) 19.6880 2.37016
\(70\) 1.31461 0.157126
\(71\) −12.3895 −1.47036 −0.735179 0.677873i \(-0.762902\pi\)
−0.735179 + 0.677873i \(0.762902\pi\)
\(72\) 3.75779 0.442860
\(73\) 5.06235 0.592503 0.296251 0.955110i \(-0.404263\pi\)
0.296251 + 0.955110i \(0.404263\pi\)
\(74\) −4.21701 −0.490218
\(75\) 2.59957 0.300173
\(76\) −4.17687 −0.479120
\(77\) −1.31461 −0.149814
\(78\) −14.4889 −1.64054
\(79\) 13.7458 1.54652 0.773259 0.634090i \(-0.218625\pi\)
0.773259 + 0.634090i \(0.218625\pi\)
\(80\) 1.00000 0.111803
\(81\) −6.15238 −0.683598
\(82\) 8.18305 0.903667
\(83\) 17.1925 1.88712 0.943560 0.331203i \(-0.107454\pi\)
0.943560 + 0.331203i \(0.107454\pi\)
\(84\) 3.41742 0.372872
\(85\) 6.71697 0.728558
\(86\) 1.00000 0.107833
\(87\) 9.67935 1.03773
\(88\) −1.00000 −0.106600
\(89\) 13.5657 1.43796 0.718982 0.695029i \(-0.244608\pi\)
0.718982 + 0.695029i \(0.244608\pi\)
\(90\) 3.75779 0.396106
\(91\) −7.32705 −0.768084
\(92\) 7.57356 0.789599
\(93\) 13.1701 1.36567
\(94\) 9.44879 0.974568
\(95\) −4.17687 −0.428538
\(96\) 2.59957 0.265318
\(97\) −6.57793 −0.667887 −0.333944 0.942593i \(-0.608380\pi\)
−0.333944 + 0.942593i \(0.608380\pi\)
\(98\) −5.27180 −0.532533
\(99\) −3.75779 −0.377672
\(100\) 1.00000 0.100000
\(101\) 17.3922 1.73059 0.865293 0.501266i \(-0.167132\pi\)
0.865293 + 0.501266i \(0.167132\pi\)
\(102\) 17.4613 1.72892
\(103\) −11.8833 −1.17090 −0.585450 0.810708i \(-0.699082\pi\)
−0.585450 + 0.810708i \(0.699082\pi\)
\(104\) −5.57356 −0.546533
\(105\) 3.41742 0.333506
\(106\) −7.39275 −0.718047
\(107\) −6.38220 −0.616991 −0.308495 0.951226i \(-0.599825\pi\)
−0.308495 + 0.951226i \(0.599825\pi\)
\(108\) 1.96993 0.189557
\(109\) −13.9123 −1.33256 −0.666279 0.745703i \(-0.732114\pi\)
−0.666279 + 0.745703i \(0.732114\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −10.9624 −1.04051
\(112\) 1.31461 0.124219
\(113\) −11.8699 −1.11662 −0.558312 0.829631i \(-0.688551\pi\)
−0.558312 + 0.829631i \(0.688551\pi\)
\(114\) −10.8581 −1.01695
\(115\) 7.57356 0.706238
\(116\) 3.72343 0.345712
\(117\) −20.9443 −1.93630
\(118\) −9.36203 −0.861845
\(119\) 8.83019 0.809462
\(120\) 2.59957 0.237308
\(121\) 1.00000 0.0909091
\(122\) −0.668250 −0.0605005
\(123\) 21.2725 1.91807
\(124\) 5.06624 0.454962
\(125\) 1.00000 0.0894427
\(126\) 4.94002 0.440092
\(127\) −14.7236 −1.30651 −0.653254 0.757139i \(-0.726597\pi\)
−0.653254 + 0.757139i \(0.726597\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.59957 0.228880
\(130\) −5.57356 −0.488834
\(131\) 4.49008 0.392300 0.196150 0.980574i \(-0.437156\pi\)
0.196150 + 0.980574i \(0.437156\pi\)
\(132\) −2.59957 −0.226264
\(133\) −5.49095 −0.476126
\(134\) −2.23127 −0.192752
\(135\) 1.96993 0.169545
\(136\) 6.71697 0.575976
\(137\) −9.15835 −0.782450 −0.391225 0.920295i \(-0.627949\pi\)
−0.391225 + 0.920295i \(0.627949\pi\)
\(138\) 19.6880 1.67596
\(139\) −1.70694 −0.144781 −0.0723905 0.997376i \(-0.523063\pi\)
−0.0723905 + 0.997376i \(0.523063\pi\)
\(140\) 1.31461 0.111105
\(141\) 24.5628 2.06856
\(142\) −12.3895 −1.03970
\(143\) 5.57356 0.466085
\(144\) 3.75779 0.313149
\(145\) 3.72343 0.309214
\(146\) 5.06235 0.418963
\(147\) −13.7044 −1.13032
\(148\) −4.21701 −0.346636
\(149\) 0.849654 0.0696064 0.0348032 0.999394i \(-0.488920\pi\)
0.0348032 + 0.999394i \(0.488920\pi\)
\(150\) 2.59957 0.212254
\(151\) −7.85240 −0.639019 −0.319510 0.947583i \(-0.603518\pi\)
−0.319510 + 0.947583i \(0.603518\pi\)
\(152\) −4.17687 −0.338789
\(153\) 25.2410 2.04061
\(154\) −1.31461 −0.105934
\(155\) 5.06624 0.406930
\(156\) −14.4889 −1.16004
\(157\) 2.62743 0.209692 0.104846 0.994488i \(-0.466565\pi\)
0.104846 + 0.994488i \(0.466565\pi\)
\(158\) 13.7458 1.09355
\(159\) −19.2180 −1.52409
\(160\) 1.00000 0.0790569
\(161\) 9.95627 0.784664
\(162\) −6.15238 −0.483377
\(163\) −17.3419 −1.35832 −0.679159 0.733991i \(-0.737655\pi\)
−0.679159 + 0.733991i \(0.737655\pi\)
\(164\) 8.18305 0.638989
\(165\) −2.59957 −0.202377
\(166\) 17.1925 1.33439
\(167\) 15.9304 1.23273 0.616367 0.787459i \(-0.288604\pi\)
0.616367 + 0.787459i \(0.288604\pi\)
\(168\) 3.41742 0.263660
\(169\) 18.0646 1.38958
\(170\) 6.71697 0.515168
\(171\) −15.6958 −1.20029
\(172\) 1.00000 0.0762493
\(173\) 6.45313 0.490622 0.245311 0.969444i \(-0.421110\pi\)
0.245311 + 0.969444i \(0.421110\pi\)
\(174\) 9.67935 0.733789
\(175\) 1.31461 0.0993751
\(176\) −1.00000 −0.0753778
\(177\) −24.3373 −1.82930
\(178\) 13.5657 1.01679
\(179\) −17.1890 −1.28477 −0.642384 0.766383i \(-0.722055\pi\)
−0.642384 + 0.766383i \(0.722055\pi\)
\(180\) 3.75779 0.280089
\(181\) −26.1524 −1.94389 −0.971947 0.235199i \(-0.924426\pi\)
−0.971947 + 0.235199i \(0.924426\pi\)
\(182\) −7.32705 −0.543117
\(183\) −1.73717 −0.128415
\(184\) 7.57356 0.558330
\(185\) −4.21701 −0.310041
\(186\) 13.1701 0.965676
\(187\) −6.71697 −0.491194
\(188\) 9.44879 0.689124
\(189\) 2.58969 0.188372
\(190\) −4.17687 −0.303022
\(191\) 1.97353 0.142800 0.0713999 0.997448i \(-0.477253\pi\)
0.0713999 + 0.997448i \(0.477253\pi\)
\(192\) 2.59957 0.187608
\(193\) −18.8219 −1.35483 −0.677414 0.735602i \(-0.736899\pi\)
−0.677414 + 0.735602i \(0.736899\pi\)
\(194\) −6.57793 −0.472268
\(195\) −14.4889 −1.03757
\(196\) −5.27180 −0.376557
\(197\) −2.64046 −0.188125 −0.0940625 0.995566i \(-0.529985\pi\)
−0.0940625 + 0.995566i \(0.529985\pi\)
\(198\) −3.75779 −0.267055
\(199\) −27.2645 −1.93273 −0.966365 0.257173i \(-0.917209\pi\)
−0.966365 + 0.257173i \(0.917209\pi\)
\(200\) 1.00000 0.0707107
\(201\) −5.80035 −0.409125
\(202\) 17.3922 1.22371
\(203\) 4.89486 0.343552
\(204\) 17.4613 1.22253
\(205\) 8.18305 0.571529
\(206\) −11.8833 −0.827952
\(207\) 28.4599 1.97810
\(208\) −5.57356 −0.386457
\(209\) 4.17687 0.288920
\(210\) 3.41742 0.235825
\(211\) −3.07497 −0.211690 −0.105845 0.994383i \(-0.533755\pi\)
−0.105845 + 0.994383i \(0.533755\pi\)
\(212\) −7.39275 −0.507736
\(213\) −32.2073 −2.20681
\(214\) −6.38220 −0.436278
\(215\) 1.00000 0.0681994
\(216\) 1.96993 0.134037
\(217\) 6.66012 0.452119
\(218\) −13.9123 −0.942261
\(219\) 13.1600 0.889267
\(220\) −1.00000 −0.0674200
\(221\) −37.4375 −2.51832
\(222\) −10.9624 −0.735751
\(223\) −17.0264 −1.14017 −0.570086 0.821585i \(-0.693090\pi\)
−0.570086 + 0.821585i \(0.693090\pi\)
\(224\) 1.31461 0.0878360
\(225\) 3.75779 0.250519
\(226\) −11.8699 −0.789572
\(227\) 5.54356 0.367939 0.183969 0.982932i \(-0.441105\pi\)
0.183969 + 0.982932i \(0.441105\pi\)
\(228\) −10.8581 −0.719094
\(229\) −7.85340 −0.518967 −0.259484 0.965747i \(-0.583552\pi\)
−0.259484 + 0.965747i \(0.583552\pi\)
\(230\) 7.57356 0.499386
\(231\) −3.41742 −0.224850
\(232\) 3.72343 0.244455
\(233\) −19.4216 −1.27235 −0.636174 0.771546i \(-0.719484\pi\)
−0.636174 + 0.771546i \(0.719484\pi\)
\(234\) −20.9443 −1.36917
\(235\) 9.44879 0.616371
\(236\) −9.36203 −0.609416
\(237\) 35.7331 2.32112
\(238\) 8.83019 0.572376
\(239\) −3.24394 −0.209833 −0.104917 0.994481i \(-0.533458\pi\)
−0.104917 + 0.994481i \(0.533458\pi\)
\(240\) 2.59957 0.167802
\(241\) −5.23092 −0.336953 −0.168477 0.985706i \(-0.553885\pi\)
−0.168477 + 0.985706i \(0.553885\pi\)
\(242\) 1.00000 0.0642824
\(243\) −21.9034 −1.40510
\(244\) −0.668250 −0.0427803
\(245\) −5.27180 −0.336803
\(246\) 21.2725 1.35628
\(247\) 23.2800 1.48127
\(248\) 5.06624 0.321706
\(249\) 44.6931 2.83231
\(250\) 1.00000 0.0632456
\(251\) 10.8220 0.683080 0.341540 0.939867i \(-0.389052\pi\)
0.341540 + 0.939867i \(0.389052\pi\)
\(252\) 4.94002 0.311192
\(253\) −7.57356 −0.476146
\(254\) −14.7236 −0.923841
\(255\) 17.4613 1.09347
\(256\) 1.00000 0.0625000
\(257\) 17.4637 1.08935 0.544677 0.838646i \(-0.316652\pi\)
0.544677 + 0.838646i \(0.316652\pi\)
\(258\) 2.59957 0.161842
\(259\) −5.54372 −0.344470
\(260\) −5.57356 −0.345658
\(261\) 13.9919 0.866076
\(262\) 4.49008 0.277398
\(263\) 5.85841 0.361245 0.180622 0.983552i \(-0.442189\pi\)
0.180622 + 0.983552i \(0.442189\pi\)
\(264\) −2.59957 −0.159993
\(265\) −7.39275 −0.454133
\(266\) −5.49095 −0.336672
\(267\) 35.2651 2.15819
\(268\) −2.23127 −0.136296
\(269\) −22.5619 −1.37563 −0.687813 0.725888i \(-0.741429\pi\)
−0.687813 + 0.725888i \(0.741429\pi\)
\(270\) 1.96993 0.119886
\(271\) 0.0467252 0.00283835 0.00141918 0.999999i \(-0.499548\pi\)
0.00141918 + 0.999999i \(0.499548\pi\)
\(272\) 6.71697 0.407276
\(273\) −19.0472 −1.15279
\(274\) −9.15835 −0.553276
\(275\) −1.00000 −0.0603023
\(276\) 19.6880 1.18508
\(277\) −7.87535 −0.473184 −0.236592 0.971609i \(-0.576030\pi\)
−0.236592 + 0.971609i \(0.576030\pi\)
\(278\) −1.70694 −0.102376
\(279\) 19.0379 1.13977
\(280\) 1.31461 0.0785629
\(281\) −18.3908 −1.09710 −0.548551 0.836117i \(-0.684820\pi\)
−0.548551 + 0.836117i \(0.684820\pi\)
\(282\) 24.5628 1.46270
\(283\) −26.2707 −1.56163 −0.780815 0.624762i \(-0.785196\pi\)
−0.780815 + 0.624762i \(0.785196\pi\)
\(284\) −12.3895 −0.735179
\(285\) −10.8581 −0.643177
\(286\) 5.57356 0.329572
\(287\) 10.7575 0.634996
\(288\) 3.75779 0.221430
\(289\) 28.1177 1.65398
\(290\) 3.72343 0.218648
\(291\) −17.0998 −1.00241
\(292\) 5.06235 0.296251
\(293\) 24.2493 1.41666 0.708330 0.705881i \(-0.249449\pi\)
0.708330 + 0.705881i \(0.249449\pi\)
\(294\) −13.7044 −0.799260
\(295\) −9.36203 −0.545079
\(296\) −4.21701 −0.245109
\(297\) −1.96993 −0.114307
\(298\) 0.849654 0.0492191
\(299\) −42.2117 −2.44117
\(300\) 2.59957 0.150087
\(301\) 1.31461 0.0757728
\(302\) −7.85240 −0.451855
\(303\) 45.2123 2.59738
\(304\) −4.17687 −0.239560
\(305\) −0.668250 −0.0382639
\(306\) 25.2410 1.44293
\(307\) 3.00747 0.171646 0.0858228 0.996310i \(-0.472648\pi\)
0.0858228 + 0.996310i \(0.472648\pi\)
\(308\) −1.31461 −0.0749068
\(309\) −30.8916 −1.75736
\(310\) 5.06624 0.287743
\(311\) 15.9087 0.902098 0.451049 0.892499i \(-0.351050\pi\)
0.451049 + 0.892499i \(0.351050\pi\)
\(312\) −14.4889 −0.820272
\(313\) 11.4679 0.648204 0.324102 0.946022i \(-0.394938\pi\)
0.324102 + 0.946022i \(0.394938\pi\)
\(314\) 2.62743 0.148275
\(315\) 4.94002 0.278339
\(316\) 13.7458 0.773259
\(317\) 19.5543 1.09828 0.549141 0.835730i \(-0.314955\pi\)
0.549141 + 0.835730i \(0.314955\pi\)
\(318\) −19.2180 −1.07769
\(319\) −3.72343 −0.208472
\(320\) 1.00000 0.0559017
\(321\) −16.5910 −0.926020
\(322\) 9.95627 0.554841
\(323\) −28.0559 −1.56107
\(324\) −6.15238 −0.341799
\(325\) −5.57356 −0.309166
\(326\) −17.3419 −0.960476
\(327\) −36.1661 −1.99999
\(328\) 8.18305 0.451834
\(329\) 12.4215 0.684817
\(330\) −2.59957 −0.143102
\(331\) 28.3899 1.56045 0.780224 0.625500i \(-0.215105\pi\)
0.780224 + 0.625500i \(0.215105\pi\)
\(332\) 17.1925 0.943560
\(333\) −15.8466 −0.868391
\(334\) 15.9304 0.871674
\(335\) −2.23127 −0.121907
\(336\) 3.41742 0.186436
\(337\) −11.4319 −0.622733 −0.311366 0.950290i \(-0.600787\pi\)
−0.311366 + 0.950290i \(0.600787\pi\)
\(338\) 18.0646 0.982585
\(339\) −30.8566 −1.67590
\(340\) 6.71697 0.364279
\(341\) −5.06624 −0.274352
\(342\) −15.6958 −0.848732
\(343\) −16.1326 −0.871080
\(344\) 1.00000 0.0539164
\(345\) 19.6880 1.05997
\(346\) 6.45313 0.346922
\(347\) −0.216577 −0.0116265 −0.00581323 0.999983i \(-0.501850\pi\)
−0.00581323 + 0.999983i \(0.501850\pi\)
\(348\) 9.67935 0.518867
\(349\) 28.8333 1.54341 0.771706 0.635979i \(-0.219404\pi\)
0.771706 + 0.635979i \(0.219404\pi\)
\(350\) 1.31461 0.0702688
\(351\) −10.9795 −0.586045
\(352\) −1.00000 −0.0533002
\(353\) 8.43021 0.448695 0.224347 0.974509i \(-0.427975\pi\)
0.224347 + 0.974509i \(0.427975\pi\)
\(354\) −24.3373 −1.29351
\(355\) −12.3895 −0.657564
\(356\) 13.5657 0.718982
\(357\) 22.9547 1.21489
\(358\) −17.1890 −0.908468
\(359\) 13.8343 0.730149 0.365074 0.930978i \(-0.381044\pi\)
0.365074 + 0.930978i \(0.381044\pi\)
\(360\) 3.75779 0.198053
\(361\) −1.55376 −0.0817771
\(362\) −26.1524 −1.37454
\(363\) 2.59957 0.136442
\(364\) −7.32705 −0.384042
\(365\) 5.06235 0.264975
\(366\) −1.73717 −0.0908031
\(367\) 30.3158 1.58247 0.791237 0.611510i \(-0.209438\pi\)
0.791237 + 0.611510i \(0.209438\pi\)
\(368\) 7.57356 0.394799
\(369\) 30.7502 1.60079
\(370\) −4.21701 −0.219232
\(371\) −9.71857 −0.504563
\(372\) 13.1701 0.682836
\(373\) −14.0392 −0.726923 −0.363462 0.931609i \(-0.618405\pi\)
−0.363462 + 0.931609i \(0.618405\pi\)
\(374\) −6.71697 −0.347326
\(375\) 2.59957 0.134241
\(376\) 9.44879 0.487284
\(377\) −20.7528 −1.06882
\(378\) 2.58969 0.133199
\(379\) 16.3978 0.842300 0.421150 0.906991i \(-0.361627\pi\)
0.421150 + 0.906991i \(0.361627\pi\)
\(380\) −4.17687 −0.214269
\(381\) −38.2751 −1.96089
\(382\) 1.97353 0.100975
\(383\) −10.9455 −0.559287 −0.279643 0.960104i \(-0.590216\pi\)
−0.279643 + 0.960104i \(0.590216\pi\)
\(384\) 2.59957 0.132659
\(385\) −1.31461 −0.0669987
\(386\) −18.8219 −0.958008
\(387\) 3.75779 0.191019
\(388\) −6.57793 −0.333944
\(389\) 5.20236 0.263770 0.131885 0.991265i \(-0.457897\pi\)
0.131885 + 0.991265i \(0.457897\pi\)
\(390\) −14.4889 −0.733674
\(391\) 50.8714 2.57268
\(392\) −5.27180 −0.266266
\(393\) 11.6723 0.588790
\(394\) −2.64046 −0.133024
\(395\) 13.7458 0.691624
\(396\) −3.75779 −0.188836
\(397\) −6.79903 −0.341234 −0.170617 0.985337i \(-0.554576\pi\)
−0.170617 + 0.985337i \(0.554576\pi\)
\(398\) −27.2645 −1.36665
\(399\) −14.2741 −0.714600
\(400\) 1.00000 0.0500000
\(401\) 5.06500 0.252934 0.126467 0.991971i \(-0.459636\pi\)
0.126467 + 0.991971i \(0.459636\pi\)
\(402\) −5.80035 −0.289295
\(403\) −28.2370 −1.40659
\(404\) 17.3922 0.865293
\(405\) −6.15238 −0.305714
\(406\) 4.89486 0.242928
\(407\) 4.21701 0.209030
\(408\) 17.4613 0.864462
\(409\) 20.5308 1.01518 0.507592 0.861598i \(-0.330536\pi\)
0.507592 + 0.861598i \(0.330536\pi\)
\(410\) 8.18305 0.404132
\(411\) −23.8078 −1.17435
\(412\) −11.8833 −0.585450
\(413\) −12.3074 −0.605608
\(414\) 28.4599 1.39873
\(415\) 17.1925 0.843945
\(416\) −5.57356 −0.273266
\(417\) −4.43733 −0.217297
\(418\) 4.17687 0.204297
\(419\) 8.48337 0.414440 0.207220 0.978294i \(-0.433558\pi\)
0.207220 + 0.978294i \(0.433558\pi\)
\(420\) 3.41742 0.166753
\(421\) 0.573257 0.0279388 0.0139694 0.999902i \(-0.495553\pi\)
0.0139694 + 0.999902i \(0.495553\pi\)
\(422\) −3.07497 −0.149687
\(423\) 35.5066 1.72639
\(424\) −7.39275 −0.359023
\(425\) 6.71697 0.325821
\(426\) −32.2073 −1.56045
\(427\) −0.878487 −0.0425130
\(428\) −6.38220 −0.308495
\(429\) 14.4889 0.699530
\(430\) 1.00000 0.0482243
\(431\) 36.2286 1.74507 0.872536 0.488550i \(-0.162474\pi\)
0.872536 + 0.488550i \(0.162474\pi\)
\(432\) 1.96993 0.0947784
\(433\) −18.0485 −0.867356 −0.433678 0.901068i \(-0.642784\pi\)
−0.433678 + 0.901068i \(0.642784\pi\)
\(434\) 6.66012 0.319696
\(435\) 9.67935 0.464089
\(436\) −13.9123 −0.666279
\(437\) −31.6338 −1.51325
\(438\) 13.1600 0.628807
\(439\) 15.7171 0.750137 0.375069 0.926997i \(-0.377619\pi\)
0.375069 + 0.926997i \(0.377619\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −19.8103 −0.943349
\(442\) −37.4375 −1.78072
\(443\) 30.9404 1.47002 0.735012 0.678054i \(-0.237176\pi\)
0.735012 + 0.678054i \(0.237176\pi\)
\(444\) −10.9624 −0.520254
\(445\) 13.5657 0.643077
\(446\) −17.0264 −0.806223
\(447\) 2.20874 0.104470
\(448\) 1.31461 0.0621094
\(449\) −10.3368 −0.487823 −0.243911 0.969798i \(-0.578431\pi\)
−0.243911 + 0.969798i \(0.578431\pi\)
\(450\) 3.75779 0.177144
\(451\) −8.18305 −0.385325
\(452\) −11.8699 −0.558312
\(453\) −20.4129 −0.959082
\(454\) 5.54356 0.260172
\(455\) −7.32705 −0.343498
\(456\) −10.8581 −0.508476
\(457\) 11.5843 0.541889 0.270945 0.962595i \(-0.412664\pi\)
0.270945 + 0.962595i \(0.412664\pi\)
\(458\) −7.85340 −0.366965
\(459\) 13.2320 0.617616
\(460\) 7.57356 0.353119
\(461\) −17.0675 −0.794915 −0.397457 0.917621i \(-0.630107\pi\)
−0.397457 + 0.917621i \(0.630107\pi\)
\(462\) −3.41742 −0.158993
\(463\) −36.7991 −1.71020 −0.855099 0.518464i \(-0.826504\pi\)
−0.855099 + 0.518464i \(0.826504\pi\)
\(464\) 3.72343 0.172856
\(465\) 13.1701 0.610747
\(466\) −19.4216 −0.899686
\(467\) −32.8303 −1.51920 −0.759602 0.650388i \(-0.774606\pi\)
−0.759602 + 0.650388i \(0.774606\pi\)
\(468\) −20.9443 −0.968150
\(469\) −2.93325 −0.135445
\(470\) 9.44879 0.435840
\(471\) 6.83021 0.314719
\(472\) −9.36203 −0.430922
\(473\) −1.00000 −0.0459800
\(474\) 35.7331 1.64128
\(475\) −4.17687 −0.191648
\(476\) 8.83019 0.404731
\(477\) −27.7804 −1.27198
\(478\) −3.24394 −0.148374
\(479\) −17.6284 −0.805464 −0.402732 0.915318i \(-0.631939\pi\)
−0.402732 + 0.915318i \(0.631939\pi\)
\(480\) 2.59957 0.118654
\(481\) 23.5038 1.07168
\(482\) −5.23092 −0.238262
\(483\) 25.8821 1.17768
\(484\) 1.00000 0.0454545
\(485\) −6.57793 −0.298688
\(486\) −21.9034 −0.993557
\(487\) 14.3735 0.651327 0.325663 0.945486i \(-0.394412\pi\)
0.325663 + 0.945486i \(0.394412\pi\)
\(488\) −0.668250 −0.0302503
\(489\) −45.0814 −2.03865
\(490\) −5.27180 −0.238156
\(491\) 0.294792 0.0133038 0.00665188 0.999978i \(-0.497883\pi\)
0.00665188 + 0.999978i \(0.497883\pi\)
\(492\) 21.2725 0.959037
\(493\) 25.0102 1.12640
\(494\) 23.2800 1.04742
\(495\) −3.75779 −0.168900
\(496\) 5.06624 0.227481
\(497\) −16.2873 −0.730584
\(498\) 44.6931 2.00275
\(499\) −21.3048 −0.953736 −0.476868 0.878975i \(-0.658228\pi\)
−0.476868 + 0.878975i \(0.658228\pi\)
\(500\) 1.00000 0.0447214
\(501\) 41.4123 1.85017
\(502\) 10.8220 0.483010
\(503\) 32.0649 1.42970 0.714852 0.699276i \(-0.246494\pi\)
0.714852 + 0.699276i \(0.246494\pi\)
\(504\) 4.94002 0.220046
\(505\) 17.3922 0.773942
\(506\) −7.57356 −0.336686
\(507\) 46.9603 2.08558
\(508\) −14.7236 −0.653254
\(509\) 25.5094 1.13068 0.565342 0.824857i \(-0.308744\pi\)
0.565342 + 0.824857i \(0.308744\pi\)
\(510\) 17.4613 0.773198
\(511\) 6.65501 0.294400
\(512\) 1.00000 0.0441942
\(513\) −8.22815 −0.363282
\(514\) 17.4637 0.770289
\(515\) −11.8833 −0.523643
\(516\) 2.59957 0.114440
\(517\) −9.44879 −0.415557
\(518\) −5.54372 −0.243577
\(519\) 16.7754 0.736358
\(520\) −5.57356 −0.244417
\(521\) −22.9894 −1.00718 −0.503592 0.863942i \(-0.667988\pi\)
−0.503592 + 0.863942i \(0.667988\pi\)
\(522\) 13.9919 0.612408
\(523\) 0.126856 0.00554702 0.00277351 0.999996i \(-0.499117\pi\)
0.00277351 + 0.999996i \(0.499117\pi\)
\(524\) 4.49008 0.196150
\(525\) 3.41742 0.149149
\(526\) 5.85841 0.255439
\(527\) 34.0298 1.48236
\(528\) −2.59957 −0.113132
\(529\) 34.3589 1.49386
\(530\) −7.39275 −0.321120
\(531\) −35.1806 −1.52671
\(532\) −5.49095 −0.238063
\(533\) −45.6088 −1.97554
\(534\) 35.2651 1.52607
\(535\) −6.38220 −0.275927
\(536\) −2.23127 −0.0963762
\(537\) −44.6842 −1.92826
\(538\) −22.5619 −0.972714
\(539\) 5.27180 0.227073
\(540\) 1.96993 0.0847724
\(541\) 17.5008 0.752418 0.376209 0.926535i \(-0.377228\pi\)
0.376209 + 0.926535i \(0.377228\pi\)
\(542\) 0.0467252 0.00200702
\(543\) −67.9852 −2.91752
\(544\) 6.71697 0.287988
\(545\) −13.9123 −0.595938
\(546\) −19.0472 −0.815146
\(547\) 22.3258 0.954584 0.477292 0.878745i \(-0.341618\pi\)
0.477292 + 0.878745i \(0.341618\pi\)
\(548\) −9.15835 −0.391225
\(549\) −2.51114 −0.107173
\(550\) −1.00000 −0.0426401
\(551\) −15.5523 −0.662550
\(552\) 19.6880 0.837979
\(553\) 18.0703 0.768427
\(554\) −7.87535 −0.334591
\(555\) −10.9624 −0.465330
\(556\) −1.70694 −0.0723905
\(557\) 29.4251 1.24678 0.623391 0.781910i \(-0.285754\pi\)
0.623391 + 0.781910i \(0.285754\pi\)
\(558\) 19.0379 0.805937
\(559\) −5.57356 −0.235737
\(560\) 1.31461 0.0555524
\(561\) −17.4613 −0.737215
\(562\) −18.3908 −0.775769
\(563\) −28.9048 −1.21819 −0.609096 0.793097i \(-0.708468\pi\)
−0.609096 + 0.793097i \(0.708468\pi\)
\(564\) 24.5628 1.03428
\(565\) −11.8699 −0.499369
\(566\) −26.2707 −1.10424
\(567\) −8.08798 −0.339663
\(568\) −12.3895 −0.519850
\(569\) −21.5069 −0.901614 −0.450807 0.892621i \(-0.648864\pi\)
−0.450807 + 0.892621i \(0.648864\pi\)
\(570\) −10.8581 −0.454795
\(571\) −35.2843 −1.47660 −0.738301 0.674471i \(-0.764372\pi\)
−0.738301 + 0.674471i \(0.764372\pi\)
\(572\) 5.57356 0.233042
\(573\) 5.13035 0.214323
\(574\) 10.7575 0.449010
\(575\) 7.57356 0.315839
\(576\) 3.75779 0.156575
\(577\) 15.7311 0.654893 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(578\) 28.1177 1.16954
\(579\) −48.9289 −2.03341
\(580\) 3.72343 0.154607
\(581\) 22.6014 0.937663
\(582\) −17.0998 −0.708810
\(583\) 7.39275 0.306176
\(584\) 5.06235 0.209481
\(585\) −20.9443 −0.865939
\(586\) 24.2493 1.00173
\(587\) −18.1412 −0.748770 −0.374385 0.927273i \(-0.622146\pi\)
−0.374385 + 0.927273i \(0.622146\pi\)
\(588\) −13.7044 −0.565162
\(589\) −21.1610 −0.871925
\(590\) −9.36203 −0.385429
\(591\) −6.86407 −0.282350
\(592\) −4.21701 −0.173318
\(593\) −35.2994 −1.44957 −0.724785 0.688975i \(-0.758061\pi\)
−0.724785 + 0.688975i \(0.758061\pi\)
\(594\) −1.96993 −0.0808273
\(595\) 8.83019 0.362002
\(596\) 0.849654 0.0348032
\(597\) −70.8762 −2.90077
\(598\) −42.2117 −1.72617
\(599\) 29.7558 1.21579 0.607894 0.794018i \(-0.292014\pi\)
0.607894 + 0.794018i \(0.292014\pi\)
\(600\) 2.59957 0.106127
\(601\) −25.6905 −1.04794 −0.523968 0.851738i \(-0.675549\pi\)
−0.523968 + 0.851738i \(0.675549\pi\)
\(602\) 1.31461 0.0535795
\(603\) −8.38464 −0.341449
\(604\) −7.85240 −0.319510
\(605\) 1.00000 0.0406558
\(606\) 45.2123 1.83662
\(607\) −24.5178 −0.995146 −0.497573 0.867422i \(-0.665775\pi\)
−0.497573 + 0.867422i \(0.665775\pi\)
\(608\) −4.17687 −0.169394
\(609\) 12.7246 0.515625
\(610\) −0.668250 −0.0270567
\(611\) −52.6634 −2.13053
\(612\) 25.2410 1.02031
\(613\) 21.9544 0.886732 0.443366 0.896341i \(-0.353784\pi\)
0.443366 + 0.896341i \(0.353784\pi\)
\(614\) 3.00747 0.121372
\(615\) 21.2725 0.857789
\(616\) −1.31461 −0.0529671
\(617\) −2.80351 −0.112865 −0.0564325 0.998406i \(-0.517973\pi\)
−0.0564325 + 0.998406i \(0.517973\pi\)
\(618\) −30.8916 −1.24264
\(619\) −42.9736 −1.72725 −0.863627 0.504132i \(-0.831812\pi\)
−0.863627 + 0.504132i \(0.831812\pi\)
\(620\) 5.06624 0.203465
\(621\) 14.9194 0.598695
\(622\) 15.9087 0.637879
\(623\) 17.8336 0.714489
\(624\) −14.4889 −0.580020
\(625\) 1.00000 0.0400000
\(626\) 11.4679 0.458349
\(627\) 10.8581 0.433630
\(628\) 2.62743 0.104846
\(629\) −28.3256 −1.12941
\(630\) 4.94002 0.196815
\(631\) 47.3795 1.88615 0.943074 0.332584i \(-0.107921\pi\)
0.943074 + 0.332584i \(0.107921\pi\)
\(632\) 13.7458 0.546777
\(633\) −7.99362 −0.317718
\(634\) 19.5543 0.776602
\(635\) −14.7236 −0.584288
\(636\) −19.2180 −0.762043
\(637\) 29.3827 1.16419
\(638\) −3.72343 −0.147412
\(639\) −46.5570 −1.84177
\(640\) 1.00000 0.0395285
\(641\) 32.3462 1.27760 0.638799 0.769374i \(-0.279432\pi\)
0.638799 + 0.769374i \(0.279432\pi\)
\(642\) −16.5910 −0.654795
\(643\) −16.7661 −0.661191 −0.330595 0.943773i \(-0.607250\pi\)
−0.330595 + 0.943773i \(0.607250\pi\)
\(644\) 9.95627 0.392332
\(645\) 2.59957 0.102358
\(646\) −28.0559 −1.10385
\(647\) −4.18826 −0.164657 −0.0823287 0.996605i \(-0.526236\pi\)
−0.0823287 + 0.996605i \(0.526236\pi\)
\(648\) −6.15238 −0.241688
\(649\) 9.36203 0.367492
\(650\) −5.57356 −0.218613
\(651\) 17.3135 0.678569
\(652\) −17.3419 −0.679159
\(653\) −33.8154 −1.32330 −0.661650 0.749813i \(-0.730143\pi\)
−0.661650 + 0.749813i \(0.730143\pi\)
\(654\) −36.1661 −1.41421
\(655\) 4.49008 0.175442
\(656\) 8.18305 0.319495
\(657\) 19.0232 0.742167
\(658\) 12.4215 0.484239
\(659\) 13.8008 0.537603 0.268801 0.963196i \(-0.413373\pi\)
0.268801 + 0.963196i \(0.413373\pi\)
\(660\) −2.59957 −0.101188
\(661\) 37.5014 1.45864 0.729318 0.684175i \(-0.239837\pi\)
0.729318 + 0.684175i \(0.239837\pi\)
\(662\) 28.3899 1.10340
\(663\) −97.3215 −3.77965
\(664\) 17.1925 0.667197
\(665\) −5.49095 −0.212930
\(666\) −15.8466 −0.614045
\(667\) 28.1997 1.09190
\(668\) 15.9304 0.616367
\(669\) −44.2614 −1.71124
\(670\) −2.23127 −0.0862015
\(671\) 0.668250 0.0257975
\(672\) 3.41742 0.131830
\(673\) 5.07386 0.195583 0.0977914 0.995207i \(-0.468822\pi\)
0.0977914 + 0.995207i \(0.468822\pi\)
\(674\) −11.4319 −0.440339
\(675\) 1.96993 0.0758227
\(676\) 18.0646 0.694792
\(677\) −38.5192 −1.48041 −0.740207 0.672379i \(-0.765272\pi\)
−0.740207 + 0.672379i \(0.765272\pi\)
\(678\) −30.8566 −1.18504
\(679\) −8.64740 −0.331857
\(680\) 6.71697 0.257584
\(681\) 14.4109 0.552227
\(682\) −5.06624 −0.193996
\(683\) −28.0937 −1.07498 −0.537488 0.843271i \(-0.680627\pi\)
−0.537488 + 0.843271i \(0.680627\pi\)
\(684\) −15.6958 −0.600144
\(685\) −9.15835 −0.349922
\(686\) −16.1326 −0.615946
\(687\) −20.4155 −0.778900
\(688\) 1.00000 0.0381246
\(689\) 41.2039 1.56974
\(690\) 19.6880 0.749511
\(691\) 11.9943 0.456284 0.228142 0.973628i \(-0.426735\pi\)
0.228142 + 0.973628i \(0.426735\pi\)
\(692\) 6.45313 0.245311
\(693\) −4.94002 −0.187656
\(694\) −0.216577 −0.00822115
\(695\) −1.70694 −0.0647480
\(696\) 9.67935 0.366895
\(697\) 54.9653 2.08196
\(698\) 28.8333 1.09136
\(699\) −50.4878 −1.90962
\(700\) 1.31461 0.0496875
\(701\) 3.21670 0.121493 0.0607465 0.998153i \(-0.480652\pi\)
0.0607465 + 0.998153i \(0.480652\pi\)
\(702\) −10.9795 −0.414396
\(703\) 17.6139 0.664321
\(704\) −1.00000 −0.0376889
\(705\) 24.5628 0.925090
\(706\) 8.43021 0.317275
\(707\) 22.8639 0.859886
\(708\) −24.3373 −0.914652
\(709\) 32.6871 1.22759 0.613794 0.789466i \(-0.289642\pi\)
0.613794 + 0.789466i \(0.289642\pi\)
\(710\) −12.3895 −0.464968
\(711\) 51.6536 1.93716
\(712\) 13.5657 0.508397
\(713\) 38.3695 1.43695
\(714\) 22.9547 0.859060
\(715\) 5.57356 0.208439
\(716\) −17.1890 −0.642384
\(717\) −8.43287 −0.314931
\(718\) 13.8343 0.516293
\(719\) 49.1563 1.83322 0.916611 0.399781i \(-0.130914\pi\)
0.916611 + 0.399781i \(0.130914\pi\)
\(720\) 3.75779 0.140045
\(721\) −15.6219 −0.581792
\(722\) −1.55376 −0.0578251
\(723\) −13.5982 −0.505721
\(724\) −26.1524 −0.971947
\(725\) 3.72343 0.138285
\(726\) 2.59957 0.0964793
\(727\) 13.6910 0.507770 0.253885 0.967234i \(-0.418291\pi\)
0.253885 + 0.967234i \(0.418291\pi\)
\(728\) −7.32705 −0.271559
\(729\) −38.4823 −1.42527
\(730\) 5.06235 0.187366
\(731\) 6.71697 0.248436
\(732\) −1.73717 −0.0642075
\(733\) −19.3546 −0.714877 −0.357438 0.933937i \(-0.616350\pi\)
−0.357438 + 0.933937i \(0.616350\pi\)
\(734\) 30.3158 1.11898
\(735\) −13.7044 −0.505496
\(736\) 7.57356 0.279165
\(737\) 2.23127 0.0821899
\(738\) 30.7502 1.13193
\(739\) −11.7315 −0.431550 −0.215775 0.976443i \(-0.569228\pi\)
−0.215775 + 0.976443i \(0.569228\pi\)
\(740\) −4.21701 −0.155020
\(741\) 60.5182 2.22319
\(742\) −9.71857 −0.356780
\(743\) −5.47258 −0.200770 −0.100385 0.994949i \(-0.532007\pi\)
−0.100385 + 0.994949i \(0.532007\pi\)
\(744\) 13.1701 0.482838
\(745\) 0.849654 0.0311289
\(746\) −14.0392 −0.514012
\(747\) 64.6057 2.36380
\(748\) −6.71697 −0.245597
\(749\) −8.39010 −0.306567
\(750\) 2.59957 0.0949231
\(751\) 53.8834 1.96623 0.983117 0.182980i \(-0.0585743\pi\)
0.983117 + 0.182980i \(0.0585743\pi\)
\(752\) 9.44879 0.344562
\(753\) 28.1327 1.02521
\(754\) −20.7528 −0.755772
\(755\) −7.85240 −0.285778
\(756\) 2.58969 0.0941861
\(757\) 46.0984 1.67547 0.837737 0.546074i \(-0.183878\pi\)
0.837737 + 0.546074i \(0.183878\pi\)
\(758\) 16.3978 0.595596
\(759\) −19.6880 −0.714631
\(760\) −4.17687 −0.151511
\(761\) 19.9270 0.722353 0.361177 0.932497i \(-0.382375\pi\)
0.361177 + 0.932497i \(0.382375\pi\)
\(762\) −38.2751 −1.38656
\(763\) −18.2892 −0.662115
\(764\) 1.97353 0.0713999
\(765\) 25.2410 0.912589
\(766\) −10.9455 −0.395475
\(767\) 52.1799 1.88411
\(768\) 2.59957 0.0938041
\(769\) −24.2110 −0.873070 −0.436535 0.899687i \(-0.643795\pi\)
−0.436535 + 0.899687i \(0.643795\pi\)
\(770\) −1.31461 −0.0473752
\(771\) 45.3981 1.63497
\(772\) −18.8219 −0.677414
\(773\) 19.7406 0.710019 0.355009 0.934863i \(-0.384478\pi\)
0.355009 + 0.934863i \(0.384478\pi\)
\(774\) 3.75779 0.135071
\(775\) 5.06624 0.181985
\(776\) −6.57793 −0.236134
\(777\) −14.4113 −0.517003
\(778\) 5.20236 0.186514
\(779\) −34.1795 −1.22461
\(780\) −14.4889 −0.518786
\(781\) 12.3895 0.443329
\(782\) 50.8714 1.81916
\(783\) 7.33491 0.262128
\(784\) −5.27180 −0.188279
\(785\) 2.62743 0.0937771
\(786\) 11.6723 0.416337
\(787\) −33.9623 −1.21063 −0.605313 0.795988i \(-0.706952\pi\)
−0.605313 + 0.795988i \(0.706952\pi\)
\(788\) −2.64046 −0.0940625
\(789\) 15.2294 0.542180
\(790\) 13.7458 0.489052
\(791\) −15.6042 −0.554823
\(792\) −3.75779 −0.133527
\(793\) 3.72453 0.132262
\(794\) −6.79903 −0.241289
\(795\) −19.2180 −0.681592
\(796\) −27.2645 −0.966365
\(797\) −18.5917 −0.658552 −0.329276 0.944234i \(-0.606805\pi\)
−0.329276 + 0.944234i \(0.606805\pi\)
\(798\) −14.2741 −0.505299
\(799\) 63.4673 2.24531
\(800\) 1.00000 0.0353553
\(801\) 50.9771 1.80119
\(802\) 5.06500 0.178851
\(803\) −5.06235 −0.178646
\(804\) −5.80035 −0.204563
\(805\) 9.95627 0.350912
\(806\) −28.2370 −0.994606
\(807\) −58.6515 −2.06463
\(808\) 17.3922 0.611855
\(809\) 17.5314 0.616372 0.308186 0.951326i \(-0.400278\pi\)
0.308186 + 0.951326i \(0.400278\pi\)
\(810\) −6.15238 −0.216173
\(811\) 28.8875 1.01438 0.507188 0.861836i \(-0.330685\pi\)
0.507188 + 0.861836i \(0.330685\pi\)
\(812\) 4.89486 0.171776
\(813\) 0.121466 0.00425999
\(814\) 4.21701 0.147806
\(815\) −17.3419 −0.607459
\(816\) 17.4613 0.611267
\(817\) −4.17687 −0.146130
\(818\) 20.5308 0.717843
\(819\) −27.5335 −0.962099
\(820\) 8.18305 0.285765
\(821\) −2.11062 −0.0736611 −0.0368305 0.999322i \(-0.511726\pi\)
−0.0368305 + 0.999322i \(0.511726\pi\)
\(822\) −23.8078 −0.830393
\(823\) −20.3993 −0.711074 −0.355537 0.934662i \(-0.615702\pi\)
−0.355537 + 0.934662i \(0.615702\pi\)
\(824\) −11.8833 −0.413976
\(825\) −2.59957 −0.0905056
\(826\) −12.3074 −0.428229
\(827\) 27.7088 0.963528 0.481764 0.876301i \(-0.339996\pi\)
0.481764 + 0.876301i \(0.339996\pi\)
\(828\) 28.4599 0.989048
\(829\) −43.7080 −1.51804 −0.759021 0.651066i \(-0.774322\pi\)
−0.759021 + 0.651066i \(0.774322\pi\)
\(830\) 17.1925 0.596760
\(831\) −20.4726 −0.710185
\(832\) −5.57356 −0.193229
\(833\) −35.4106 −1.22690
\(834\) −4.43733 −0.153652
\(835\) 15.9304 0.551295
\(836\) 4.17687 0.144460
\(837\) 9.98015 0.344964
\(838\) 8.48337 0.293053
\(839\) −50.1817 −1.73247 −0.866233 0.499640i \(-0.833466\pi\)
−0.866233 + 0.499640i \(0.833466\pi\)
\(840\) 3.41742 0.117912
\(841\) −15.1360 −0.521932
\(842\) 0.573257 0.0197557
\(843\) −47.8082 −1.64660
\(844\) −3.07497 −0.105845
\(845\) 18.0646 0.621441
\(846\) 35.5066 1.22074
\(847\) 1.31461 0.0451705
\(848\) −7.39275 −0.253868
\(849\) −68.2926 −2.34380
\(850\) 6.71697 0.230390
\(851\) −31.9378 −1.09481
\(852\) −32.2073 −1.10340
\(853\) 13.2315 0.453037 0.226518 0.974007i \(-0.427266\pi\)
0.226518 + 0.974007i \(0.427266\pi\)
\(854\) −0.878487 −0.0300612
\(855\) −15.6958 −0.536785
\(856\) −6.38220 −0.218139
\(857\) −8.15827 −0.278681 −0.139341 0.990245i \(-0.544498\pi\)
−0.139341 + 0.990245i \(0.544498\pi\)
\(858\) 14.4889 0.494643
\(859\) 41.8387 1.42752 0.713759 0.700391i \(-0.246991\pi\)
0.713759 + 0.700391i \(0.246991\pi\)
\(860\) 1.00000 0.0340997
\(861\) 27.9650 0.953043
\(862\) 36.2286 1.23395
\(863\) −56.7578 −1.93206 −0.966029 0.258432i \(-0.916794\pi\)
−0.966029 + 0.258432i \(0.916794\pi\)
\(864\) 1.96993 0.0670185
\(865\) 6.45313 0.219413
\(866\) −18.0485 −0.613313
\(867\) 73.0941 2.48241
\(868\) 6.66012 0.226059
\(869\) −13.7458 −0.466293
\(870\) 9.67935 0.328161
\(871\) 12.4361 0.421382
\(872\) −13.9123 −0.471130
\(873\) −24.7185 −0.836594
\(874\) −31.6338 −1.07003
\(875\) 1.31461 0.0444419
\(876\) 13.1600 0.444634
\(877\) 47.5158 1.60449 0.802247 0.596992i \(-0.203637\pi\)
0.802247 + 0.596992i \(0.203637\pi\)
\(878\) 15.7171 0.530427
\(879\) 63.0379 2.12622
\(880\) −1.00000 −0.0337100
\(881\) 45.6699 1.53866 0.769328 0.638854i \(-0.220591\pi\)
0.769328 + 0.638854i \(0.220591\pi\)
\(882\) −19.8103 −0.667049
\(883\) 56.9027 1.91493 0.957464 0.288553i \(-0.0931743\pi\)
0.957464 + 0.288553i \(0.0931743\pi\)
\(884\) −37.4375 −1.25916
\(885\) −24.3373 −0.818090
\(886\) 30.9404 1.03946
\(887\) 23.3035 0.782456 0.391228 0.920294i \(-0.372050\pi\)
0.391228 + 0.920294i \(0.372050\pi\)
\(888\) −10.9624 −0.367875
\(889\) −19.3558 −0.649172
\(890\) 13.5657 0.454724
\(891\) 6.15238 0.206113
\(892\) −17.0264 −0.570086
\(893\) −39.4664 −1.32069
\(894\) 2.20874 0.0738713
\(895\) −17.1890 −0.574566
\(896\) 1.31461 0.0439180
\(897\) −109.733 −3.66386
\(898\) −10.3368 −0.344943
\(899\) 18.8638 0.629143
\(900\) 3.75779 0.125260
\(901\) −49.6569 −1.65431
\(902\) −8.18305 −0.272466
\(903\) 3.41742 0.113725
\(904\) −11.8699 −0.394786
\(905\) −26.1524 −0.869336
\(906\) −20.4129 −0.678173
\(907\) 32.4461 1.07736 0.538678 0.842512i \(-0.318924\pi\)
0.538678 + 0.842512i \(0.318924\pi\)
\(908\) 5.54356 0.183969
\(909\) 65.3562 2.16773
\(910\) −7.32705 −0.242889
\(911\) −41.2146 −1.36550 −0.682750 0.730652i \(-0.739216\pi\)
−0.682750 + 0.730652i \(0.739216\pi\)
\(912\) −10.8581 −0.359547
\(913\) −17.1925 −0.568988
\(914\) 11.5843 0.383174
\(915\) −1.73717 −0.0574289
\(916\) −7.85340 −0.259484
\(917\) 5.90270 0.194924
\(918\) 13.2320 0.436720
\(919\) −36.9652 −1.21937 −0.609684 0.792644i \(-0.708704\pi\)
−0.609684 + 0.792644i \(0.708704\pi\)
\(920\) 7.57356 0.249693
\(921\) 7.81815 0.257617
\(922\) −17.0675 −0.562090
\(923\) 69.0534 2.27292
\(924\) −3.41742 −0.112425
\(925\) −4.21701 −0.138655
\(926\) −36.7991 −1.20929
\(927\) −44.6551 −1.46667
\(928\) 3.72343 0.122228
\(929\) −8.40193 −0.275659 −0.137829 0.990456i \(-0.544013\pi\)
−0.137829 + 0.990456i \(0.544013\pi\)
\(930\) 13.1701 0.431864
\(931\) 22.0196 0.721664
\(932\) −19.4216 −0.636174
\(933\) 41.3558 1.35393
\(934\) −32.8303 −1.07424
\(935\) −6.71697 −0.219668
\(936\) −20.9443 −0.684585
\(937\) 0.466767 0.0152486 0.00762431 0.999971i \(-0.497573\pi\)
0.00762431 + 0.999971i \(0.497573\pi\)
\(938\) −2.93325 −0.0957739
\(939\) 29.8117 0.972866
\(940\) 9.44879 0.308186
\(941\) −16.4977 −0.537809 −0.268905 0.963167i \(-0.586662\pi\)
−0.268905 + 0.963167i \(0.586662\pi\)
\(942\) 6.83021 0.222540
\(943\) 61.9749 2.01818
\(944\) −9.36203 −0.304708
\(945\) 2.58969 0.0842426
\(946\) −1.00000 −0.0325128
\(947\) −14.0154 −0.455440 −0.227720 0.973727i \(-0.573127\pi\)
−0.227720 + 0.973727i \(0.573127\pi\)
\(948\) 35.7331 1.16056
\(949\) −28.2153 −0.915908
\(950\) −4.17687 −0.135516
\(951\) 50.8330 1.64837
\(952\) 8.83019 0.286188
\(953\) 9.28262 0.300694 0.150347 0.988633i \(-0.451961\pi\)
0.150347 + 0.988633i \(0.451961\pi\)
\(954\) −27.7804 −0.899423
\(955\) 1.97353 0.0638620
\(956\) −3.24394 −0.104917
\(957\) −9.67935 −0.312889
\(958\) −17.6284 −0.569549
\(959\) −12.0396 −0.388780
\(960\) 2.59957 0.0839009
\(961\) −5.33322 −0.172039
\(962\) 23.5038 0.757792
\(963\) −23.9830 −0.772840
\(964\) −5.23092 −0.168477
\(965\) −18.8219 −0.605898
\(966\) 25.8821 0.832742
\(967\) 19.6189 0.630903 0.315451 0.948942i \(-0.397844\pi\)
0.315451 + 0.948942i \(0.397844\pi\)
\(968\) 1.00000 0.0321412
\(969\) −72.9334 −2.34296
\(970\) −6.57793 −0.211205
\(971\) −32.6615 −1.04816 −0.524079 0.851670i \(-0.675590\pi\)
−0.524079 + 0.851670i \(0.675590\pi\)
\(972\) −21.9034 −0.702551
\(973\) −2.24396 −0.0719381
\(974\) 14.3735 0.460558
\(975\) −14.4889 −0.464016
\(976\) −0.668250 −0.0213902
\(977\) 36.8075 1.17758 0.588788 0.808287i \(-0.299605\pi\)
0.588788 + 0.808287i \(0.299605\pi\)
\(978\) −45.0814 −1.44155
\(979\) −13.5657 −0.433562
\(980\) −5.27180 −0.168402
\(981\) −52.2795 −1.66916
\(982\) 0.294792 0.00940718
\(983\) 52.4941 1.67430 0.837151 0.546972i \(-0.184220\pi\)
0.837151 + 0.546972i \(0.184220\pi\)
\(984\) 21.2725 0.678141
\(985\) −2.64046 −0.0841320
\(986\) 25.0102 0.796487
\(987\) 32.2905 1.02782
\(988\) 23.2800 0.740637
\(989\) 7.57356 0.240825
\(990\) −3.75779 −0.119430
\(991\) −6.94321 −0.220558 −0.110279 0.993901i \(-0.535174\pi\)
−0.110279 + 0.993901i \(0.535174\pi\)
\(992\) 5.06624 0.160853
\(993\) 73.8016 2.34202
\(994\) −16.2873 −0.516601
\(995\) −27.2645 −0.864343
\(996\) 44.6931 1.41616
\(997\) −45.2883 −1.43429 −0.717147 0.696922i \(-0.754553\pi\)
−0.717147 + 0.696922i \(0.754553\pi\)
\(998\) −21.3048 −0.674393
\(999\) −8.30723 −0.262829
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 4730.2.a.be.1.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.be.1.10 12 1.1 even 1 trivial