Properties

Label 2151.2.a.k.1.15
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(1.61701\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.61701 q^{2} +0.614711 q^{4} +1.46591 q^{5} +3.83563 q^{7} -2.24002 q^{8} +O(q^{10})\) \(q+1.61701 q^{2} +0.614711 q^{4} +1.46591 q^{5} +3.83563 q^{7} -2.24002 q^{8} +2.37038 q^{10} +2.90374 q^{11} +2.33112 q^{13} +6.20223 q^{14} -4.85155 q^{16} +3.77139 q^{17} -1.21740 q^{19} +0.901109 q^{20} +4.69537 q^{22} -1.47665 q^{23} -2.85112 q^{25} +3.76944 q^{26} +2.35780 q^{28} +1.02391 q^{29} -1.72030 q^{31} -3.36495 q^{32} +6.09837 q^{34} +5.62267 q^{35} -1.08984 q^{37} -1.96855 q^{38} -3.28366 q^{40} -1.58037 q^{41} +0.433251 q^{43} +1.78496 q^{44} -2.38776 q^{46} +7.65919 q^{47} +7.71202 q^{49} -4.61028 q^{50} +1.43297 q^{52} +1.07471 q^{53} +4.25661 q^{55} -8.59188 q^{56} +1.65567 q^{58} -4.82708 q^{59} +10.7610 q^{61} -2.78174 q^{62} +4.26196 q^{64} +3.41720 q^{65} -6.11056 q^{67} +2.31832 q^{68} +9.09189 q^{70} +10.9308 q^{71} -12.1327 q^{73} -1.76228 q^{74} -0.748351 q^{76} +11.1377 q^{77} +14.7398 q^{79} -7.11192 q^{80} -2.55547 q^{82} -13.0905 q^{83} +5.52851 q^{85} +0.700570 q^{86} -6.50444 q^{88} -6.08868 q^{89} +8.94131 q^{91} -0.907716 q^{92} +12.3850 q^{94} -1.78460 q^{95} +7.99478 q^{97} +12.4704 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61701 1.14340 0.571698 0.820464i \(-0.306285\pi\)
0.571698 + 0.820464i \(0.306285\pi\)
\(3\) 0 0
\(4\) 0.614711 0.307356
\(5\) 1.46591 0.655573 0.327787 0.944752i \(-0.393697\pi\)
0.327787 + 0.944752i \(0.393697\pi\)
\(6\) 0 0
\(7\) 3.83563 1.44973 0.724865 0.688891i \(-0.241902\pi\)
0.724865 + 0.688891i \(0.241902\pi\)
\(8\) −2.24002 −0.791967
\(9\) 0 0
\(10\) 2.37038 0.749580
\(11\) 2.90374 0.875511 0.437755 0.899094i \(-0.355774\pi\)
0.437755 + 0.899094i \(0.355774\pi\)
\(12\) 0 0
\(13\) 2.33112 0.646537 0.323268 0.946307i \(-0.395218\pi\)
0.323268 + 0.946307i \(0.395218\pi\)
\(14\) 6.20223 1.65762
\(15\) 0 0
\(16\) −4.85155 −1.21289
\(17\) 3.77139 0.914697 0.457348 0.889288i \(-0.348799\pi\)
0.457348 + 0.889288i \(0.348799\pi\)
\(18\) 0 0
\(19\) −1.21740 −0.279291 −0.139646 0.990202i \(-0.544596\pi\)
−0.139646 + 0.990202i \(0.544596\pi\)
\(20\) 0.901109 0.201494
\(21\) 0 0
\(22\) 4.69537 1.00106
\(23\) −1.47665 −0.307904 −0.153952 0.988078i \(-0.549200\pi\)
−0.153952 + 0.988078i \(0.549200\pi\)
\(24\) 0 0
\(25\) −2.85112 −0.570224
\(26\) 3.76944 0.739248
\(27\) 0 0
\(28\) 2.35780 0.445583
\(29\) 1.02391 0.190136 0.0950678 0.995471i \(-0.469693\pi\)
0.0950678 + 0.995471i \(0.469693\pi\)
\(30\) 0 0
\(31\) −1.72030 −0.308975 −0.154487 0.987995i \(-0.549373\pi\)
−0.154487 + 0.987995i \(0.549373\pi\)
\(32\) −3.36495 −0.594845
\(33\) 0 0
\(34\) 6.09837 1.04586
\(35\) 5.62267 0.950404
\(36\) 0 0
\(37\) −1.08984 −0.179169 −0.0895845 0.995979i \(-0.528554\pi\)
−0.0895845 + 0.995979i \(0.528554\pi\)
\(38\) −1.96855 −0.319341
\(39\) 0 0
\(40\) −3.28366 −0.519192
\(41\) −1.58037 −0.246812 −0.123406 0.992356i \(-0.539382\pi\)
−0.123406 + 0.992356i \(0.539382\pi\)
\(42\) 0 0
\(43\) 0.433251 0.0660702 0.0330351 0.999454i \(-0.489483\pi\)
0.0330351 + 0.999454i \(0.489483\pi\)
\(44\) 1.78496 0.269093
\(45\) 0 0
\(46\) −2.38776 −0.352056
\(47\) 7.65919 1.11721 0.558604 0.829434i \(-0.311337\pi\)
0.558604 + 0.829434i \(0.311337\pi\)
\(48\) 0 0
\(49\) 7.71202 1.10172
\(50\) −4.61028 −0.651992
\(51\) 0 0
\(52\) 1.43297 0.198717
\(53\) 1.07471 0.147623 0.0738117 0.997272i \(-0.476484\pi\)
0.0738117 + 0.997272i \(0.476484\pi\)
\(54\) 0 0
\(55\) 4.25661 0.573961
\(56\) −8.59188 −1.14814
\(57\) 0 0
\(58\) 1.65567 0.217400
\(59\) −4.82708 −0.628432 −0.314216 0.949351i \(-0.601742\pi\)
−0.314216 + 0.949351i \(0.601742\pi\)
\(60\) 0 0
\(61\) 10.7610 1.37780 0.688902 0.724854i \(-0.258093\pi\)
0.688902 + 0.724854i \(0.258093\pi\)
\(62\) −2.78174 −0.353281
\(63\) 0 0
\(64\) 4.26196 0.532745
\(65\) 3.41720 0.423852
\(66\) 0 0
\(67\) −6.11056 −0.746523 −0.373262 0.927726i \(-0.621761\pi\)
−0.373262 + 0.927726i \(0.621761\pi\)
\(68\) 2.31832 0.281137
\(69\) 0 0
\(70\) 9.09189 1.08669
\(71\) 10.9308 1.29725 0.648626 0.761108i \(-0.275344\pi\)
0.648626 + 0.761108i \(0.275344\pi\)
\(72\) 0 0
\(73\) −12.1327 −1.42003 −0.710015 0.704187i \(-0.751312\pi\)
−0.710015 + 0.704187i \(0.751312\pi\)
\(74\) −1.76228 −0.204861
\(75\) 0 0
\(76\) −0.748351 −0.0858418
\(77\) 11.1377 1.26925
\(78\) 0 0
\(79\) 14.7398 1.65835 0.829177 0.558986i \(-0.188810\pi\)
0.829177 + 0.558986i \(0.188810\pi\)
\(80\) −7.11192 −0.795137
\(81\) 0 0
\(82\) −2.55547 −0.282204
\(83\) −13.0905 −1.43686 −0.718432 0.695597i \(-0.755140\pi\)
−0.718432 + 0.695597i \(0.755140\pi\)
\(84\) 0 0
\(85\) 5.52851 0.599651
\(86\) 0.700570 0.0755444
\(87\) 0 0
\(88\) −6.50444 −0.693376
\(89\) −6.08868 −0.645398 −0.322699 0.946502i \(-0.604590\pi\)
−0.322699 + 0.946502i \(0.604590\pi\)
\(90\) 0 0
\(91\) 8.94131 0.937304
\(92\) −0.907716 −0.0946359
\(93\) 0 0
\(94\) 12.3850 1.27741
\(95\) −1.78460 −0.183096
\(96\) 0 0
\(97\) 7.99478 0.811747 0.405873 0.913929i \(-0.366967\pi\)
0.405873 + 0.913929i \(0.366967\pi\)
\(98\) 12.4704 1.25970
\(99\) 0 0
\(100\) −1.75261 −0.175261
\(101\) 12.4894 1.24274 0.621371 0.783516i \(-0.286576\pi\)
0.621371 + 0.783516i \(0.286576\pi\)
\(102\) 0 0
\(103\) −6.88824 −0.678719 −0.339359 0.940657i \(-0.610210\pi\)
−0.339359 + 0.940657i \(0.610210\pi\)
\(104\) −5.22176 −0.512036
\(105\) 0 0
\(106\) 1.73782 0.168792
\(107\) 16.7073 1.61515 0.807577 0.589763i \(-0.200779\pi\)
0.807577 + 0.589763i \(0.200779\pi\)
\(108\) 0 0
\(109\) −15.9013 −1.52307 −0.761534 0.648125i \(-0.775554\pi\)
−0.761534 + 0.648125i \(0.775554\pi\)
\(110\) 6.88297 0.656265
\(111\) 0 0
\(112\) −18.6087 −1.75836
\(113\) 9.87403 0.928870 0.464435 0.885607i \(-0.346257\pi\)
0.464435 + 0.885607i \(0.346257\pi\)
\(114\) 0 0
\(115\) −2.16464 −0.201853
\(116\) 0.629410 0.0584393
\(117\) 0 0
\(118\) −7.80542 −0.718548
\(119\) 14.4656 1.32606
\(120\) 0 0
\(121\) −2.56829 −0.233481
\(122\) 17.4006 1.57538
\(123\) 0 0
\(124\) −1.05749 −0.0949652
\(125\) −11.5090 −1.02940
\(126\) 0 0
\(127\) −10.3172 −0.915505 −0.457753 0.889080i \(-0.651345\pi\)
−0.457753 + 0.889080i \(0.651345\pi\)
\(128\) 13.6215 1.20398
\(129\) 0 0
\(130\) 5.52564 0.484631
\(131\) −9.86768 −0.862143 −0.431072 0.902318i \(-0.641864\pi\)
−0.431072 + 0.902318i \(0.641864\pi\)
\(132\) 0 0
\(133\) −4.66950 −0.404897
\(134\) −9.88081 −0.853572
\(135\) 0 0
\(136\) −8.44800 −0.724410
\(137\) 3.24609 0.277332 0.138666 0.990339i \(-0.455718\pi\)
0.138666 + 0.990339i \(0.455718\pi\)
\(138\) 0 0
\(139\) −14.2318 −1.20713 −0.603564 0.797315i \(-0.706253\pi\)
−0.603564 + 0.797315i \(0.706253\pi\)
\(140\) 3.45632 0.292112
\(141\) 0 0
\(142\) 17.6752 1.48327
\(143\) 6.76897 0.566050
\(144\) 0 0
\(145\) 1.50096 0.124648
\(146\) −19.6187 −1.62366
\(147\) 0 0
\(148\) −0.669938 −0.0550686
\(149\) 3.90332 0.319772 0.159886 0.987135i \(-0.448887\pi\)
0.159886 + 0.987135i \(0.448887\pi\)
\(150\) 0 0
\(151\) −0.699764 −0.0569460 −0.0284730 0.999595i \(-0.509064\pi\)
−0.0284730 + 0.999595i \(0.509064\pi\)
\(152\) 2.72701 0.221190
\(153\) 0 0
\(154\) 18.0097 1.45126
\(155\) −2.52180 −0.202556
\(156\) 0 0
\(157\) 1.91363 0.152725 0.0763623 0.997080i \(-0.475669\pi\)
0.0763623 + 0.997080i \(0.475669\pi\)
\(158\) 23.8343 1.89616
\(159\) 0 0
\(160\) −4.93270 −0.389964
\(161\) −5.66389 −0.446377
\(162\) 0 0
\(163\) −17.1905 −1.34647 −0.673233 0.739430i \(-0.735095\pi\)
−0.673233 + 0.739430i \(0.735095\pi\)
\(164\) −0.971471 −0.0758592
\(165\) 0 0
\(166\) −21.1673 −1.64290
\(167\) 14.8924 1.15241 0.576205 0.817305i \(-0.304533\pi\)
0.576205 + 0.817305i \(0.304533\pi\)
\(168\) 0 0
\(169\) −7.56588 −0.581990
\(170\) 8.93963 0.685639
\(171\) 0 0
\(172\) 0.266324 0.0203070
\(173\) −11.7169 −0.890823 −0.445411 0.895326i \(-0.646943\pi\)
−0.445411 + 0.895326i \(0.646943\pi\)
\(174\) 0 0
\(175\) −10.9358 −0.826671
\(176\) −14.0876 −1.06190
\(177\) 0 0
\(178\) −9.84543 −0.737946
\(179\) −3.36761 −0.251707 −0.125854 0.992049i \(-0.540167\pi\)
−0.125854 + 0.992049i \(0.540167\pi\)
\(180\) 0 0
\(181\) −6.66617 −0.495493 −0.247746 0.968825i \(-0.579690\pi\)
−0.247746 + 0.968825i \(0.579690\pi\)
\(182\) 14.4582 1.07171
\(183\) 0 0
\(184\) 3.30774 0.243850
\(185\) −1.59761 −0.117458
\(186\) 0 0
\(187\) 10.9511 0.800827
\(188\) 4.70819 0.343380
\(189\) 0 0
\(190\) −2.88571 −0.209351
\(191\) −15.7911 −1.14260 −0.571300 0.820741i \(-0.693561\pi\)
−0.571300 + 0.820741i \(0.693561\pi\)
\(192\) 0 0
\(193\) 8.45917 0.608904 0.304452 0.952528i \(-0.401527\pi\)
0.304452 + 0.952528i \(0.401527\pi\)
\(194\) 12.9276 0.928149
\(195\) 0 0
\(196\) 4.74066 0.338619
\(197\) −4.68346 −0.333683 −0.166842 0.985984i \(-0.553357\pi\)
−0.166842 + 0.985984i \(0.553357\pi\)
\(198\) 0 0
\(199\) −25.8363 −1.83149 −0.915743 0.401764i \(-0.868397\pi\)
−0.915743 + 0.401764i \(0.868397\pi\)
\(200\) 6.38657 0.451599
\(201\) 0 0
\(202\) 20.1955 1.42095
\(203\) 3.92734 0.275645
\(204\) 0 0
\(205\) −2.31667 −0.161804
\(206\) −11.1383 −0.776044
\(207\) 0 0
\(208\) −11.3096 −0.784177
\(209\) −3.53502 −0.244523
\(210\) 0 0
\(211\) 4.07485 0.280524 0.140262 0.990114i \(-0.455205\pi\)
0.140262 + 0.990114i \(0.455205\pi\)
\(212\) 0.660639 0.0453729
\(213\) 0 0
\(214\) 27.0158 1.84676
\(215\) 0.635106 0.0433138
\(216\) 0 0
\(217\) −6.59843 −0.447930
\(218\) −25.7125 −1.74147
\(219\) 0 0
\(220\) 2.61659 0.176410
\(221\) 8.79157 0.591385
\(222\) 0 0
\(223\) −2.44657 −0.163835 −0.0819173 0.996639i \(-0.526104\pi\)
−0.0819173 + 0.996639i \(0.526104\pi\)
\(224\) −12.9067 −0.862364
\(225\) 0 0
\(226\) 15.9664 1.06207
\(227\) 2.74547 0.182223 0.0911115 0.995841i \(-0.470958\pi\)
0.0911115 + 0.995841i \(0.470958\pi\)
\(228\) 0 0
\(229\) 13.7145 0.906277 0.453139 0.891440i \(-0.350304\pi\)
0.453139 + 0.891440i \(0.350304\pi\)
\(230\) −3.50023 −0.230798
\(231\) 0 0
\(232\) −2.29358 −0.150581
\(233\) −5.36762 −0.351645 −0.175822 0.984422i \(-0.556258\pi\)
−0.175822 + 0.984422i \(0.556258\pi\)
\(234\) 0 0
\(235\) 11.2277 0.732412
\(236\) −2.96726 −0.193152
\(237\) 0 0
\(238\) 23.3910 1.51622
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) −12.9180 −0.832120 −0.416060 0.909337i \(-0.636589\pi\)
−0.416060 + 0.909337i \(0.636589\pi\)
\(242\) −4.15295 −0.266962
\(243\) 0 0
\(244\) 6.61491 0.423476
\(245\) 11.3051 0.722256
\(246\) 0 0
\(247\) −2.83791 −0.180572
\(248\) 3.85351 0.244698
\(249\) 0 0
\(250\) −18.6101 −1.17701
\(251\) 18.9396 1.19546 0.597729 0.801698i \(-0.296070\pi\)
0.597729 + 0.801698i \(0.296070\pi\)
\(252\) 0 0
\(253\) −4.28782 −0.269573
\(254\) −16.6830 −1.04679
\(255\) 0 0
\(256\) 13.5022 0.843886
\(257\) −15.3234 −0.955847 −0.477923 0.878402i \(-0.658610\pi\)
−0.477923 + 0.878402i \(0.658610\pi\)
\(258\) 0 0
\(259\) −4.18023 −0.259747
\(260\) 2.10059 0.130273
\(261\) 0 0
\(262\) −15.9561 −0.985771
\(263\) 27.1435 1.67374 0.836870 0.547402i \(-0.184383\pi\)
0.836870 + 0.547402i \(0.184383\pi\)
\(264\) 0 0
\(265\) 1.57543 0.0967779
\(266\) −7.55062 −0.462958
\(267\) 0 0
\(268\) −3.75623 −0.229448
\(269\) 12.7939 0.780058 0.390029 0.920802i \(-0.372465\pi\)
0.390029 + 0.920802i \(0.372465\pi\)
\(270\) 0 0
\(271\) −28.4970 −1.73107 −0.865533 0.500851i \(-0.833020\pi\)
−0.865533 + 0.500851i \(0.833020\pi\)
\(272\) −18.2971 −1.10943
\(273\) 0 0
\(274\) 5.24895 0.317101
\(275\) −8.27891 −0.499237
\(276\) 0 0
\(277\) −2.41529 −0.145120 −0.0725602 0.997364i \(-0.523117\pi\)
−0.0725602 + 0.997364i \(0.523117\pi\)
\(278\) −23.0130 −1.38023
\(279\) 0 0
\(280\) −12.5949 −0.752689
\(281\) −8.27813 −0.493832 −0.246916 0.969037i \(-0.579417\pi\)
−0.246916 + 0.969037i \(0.579417\pi\)
\(282\) 0 0
\(283\) −0.142683 −0.00848163 −0.00424082 0.999991i \(-0.501350\pi\)
−0.00424082 + 0.999991i \(0.501350\pi\)
\(284\) 6.71930 0.398717
\(285\) 0 0
\(286\) 10.9455 0.647219
\(287\) −6.06171 −0.357811
\(288\) 0 0
\(289\) −2.77660 −0.163329
\(290\) 2.42706 0.142522
\(291\) 0 0
\(292\) −7.45813 −0.436454
\(293\) 12.9861 0.758657 0.379328 0.925262i \(-0.376155\pi\)
0.379328 + 0.925262i \(0.376155\pi\)
\(294\) 0 0
\(295\) −7.07605 −0.411983
\(296\) 2.44127 0.141896
\(297\) 0 0
\(298\) 6.31169 0.365627
\(299\) −3.44226 −0.199071
\(300\) 0 0
\(301\) 1.66179 0.0957839
\(302\) −1.13152 −0.0651119
\(303\) 0 0
\(304\) 5.90629 0.338749
\(305\) 15.7746 0.903252
\(306\) 0 0
\(307\) 7.69912 0.439412 0.219706 0.975566i \(-0.429490\pi\)
0.219706 + 0.975566i \(0.429490\pi\)
\(308\) 6.84644 0.390112
\(309\) 0 0
\(310\) −4.07776 −0.231601
\(311\) −0.0557085 −0.00315894 −0.00157947 0.999999i \(-0.500503\pi\)
−0.00157947 + 0.999999i \(0.500503\pi\)
\(312\) 0 0
\(313\) −2.73407 −0.154538 −0.0772692 0.997010i \(-0.524620\pi\)
−0.0772692 + 0.997010i \(0.524620\pi\)
\(314\) 3.09436 0.174625
\(315\) 0 0
\(316\) 9.06070 0.509704
\(317\) 5.74313 0.322566 0.161283 0.986908i \(-0.448437\pi\)
0.161283 + 0.986908i \(0.448437\pi\)
\(318\) 0 0
\(319\) 2.97317 0.166466
\(320\) 6.24763 0.349253
\(321\) 0 0
\(322\) −9.15855 −0.510386
\(323\) −4.59130 −0.255467
\(324\) 0 0
\(325\) −6.64630 −0.368671
\(326\) −27.7972 −1.53954
\(327\) 0 0
\(328\) 3.54006 0.195467
\(329\) 29.3778 1.61965
\(330\) 0 0
\(331\) −32.3302 −1.77703 −0.888514 0.458849i \(-0.848262\pi\)
−0.888514 + 0.458849i \(0.848262\pi\)
\(332\) −8.04685 −0.441628
\(333\) 0 0
\(334\) 24.0811 1.31766
\(335\) −8.95750 −0.489401
\(336\) 0 0
\(337\) −4.91558 −0.267769 −0.133884 0.990997i \(-0.542745\pi\)
−0.133884 + 0.990997i \(0.542745\pi\)
\(338\) −12.2341 −0.665446
\(339\) 0 0
\(340\) 3.39843 0.184306
\(341\) −4.99530 −0.270511
\(342\) 0 0
\(343\) 2.73104 0.147462
\(344\) −0.970492 −0.0523254
\(345\) 0 0
\(346\) −18.9464 −1.01856
\(347\) −28.7741 −1.54467 −0.772337 0.635213i \(-0.780912\pi\)
−0.772337 + 0.635213i \(0.780912\pi\)
\(348\) 0 0
\(349\) −12.6879 −0.679168 −0.339584 0.940576i \(-0.610286\pi\)
−0.339584 + 0.940576i \(0.610286\pi\)
\(350\) −17.6833 −0.945212
\(351\) 0 0
\(352\) −9.77094 −0.520793
\(353\) −6.88724 −0.366571 −0.183285 0.983060i \(-0.558673\pi\)
−0.183285 + 0.983060i \(0.558673\pi\)
\(354\) 0 0
\(355\) 16.0236 0.850443
\(356\) −3.74278 −0.198367
\(357\) 0 0
\(358\) −5.44545 −0.287801
\(359\) −4.46094 −0.235439 −0.117720 0.993047i \(-0.537558\pi\)
−0.117720 + 0.993047i \(0.537558\pi\)
\(360\) 0 0
\(361\) −17.5179 −0.921996
\(362\) −10.7792 −0.566545
\(363\) 0 0
\(364\) 5.49632 0.288085
\(365\) −17.7855 −0.930934
\(366\) 0 0
\(367\) 4.13856 0.216031 0.108016 0.994149i \(-0.465550\pi\)
0.108016 + 0.994149i \(0.465550\pi\)
\(368\) 7.16406 0.373453
\(369\) 0 0
\(370\) −2.58334 −0.134302
\(371\) 4.12220 0.214014
\(372\) 0 0
\(373\) −15.9172 −0.824161 −0.412080 0.911148i \(-0.635198\pi\)
−0.412080 + 0.911148i \(0.635198\pi\)
\(374\) 17.7081 0.915663
\(375\) 0 0
\(376\) −17.1568 −0.884792
\(377\) 2.38686 0.122930
\(378\) 0 0
\(379\) 35.7486 1.83628 0.918142 0.396252i \(-0.129689\pi\)
0.918142 + 0.396252i \(0.129689\pi\)
\(380\) −1.09701 −0.0562756
\(381\) 0 0
\(382\) −25.5342 −1.30645
\(383\) −22.7410 −1.16201 −0.581005 0.813900i \(-0.697340\pi\)
−0.581005 + 0.813900i \(0.697340\pi\)
\(384\) 0 0
\(385\) 16.3268 0.832089
\(386\) 13.6785 0.696219
\(387\) 0 0
\(388\) 4.91448 0.249495
\(389\) −2.53628 −0.128594 −0.0642972 0.997931i \(-0.520481\pi\)
−0.0642972 + 0.997931i \(0.520481\pi\)
\(390\) 0 0
\(391\) −5.56904 −0.281639
\(392\) −17.2751 −0.872524
\(393\) 0 0
\(394\) −7.57319 −0.381532
\(395\) 21.6071 1.08717
\(396\) 0 0
\(397\) −13.3410 −0.669564 −0.334782 0.942296i \(-0.608663\pi\)
−0.334782 + 0.942296i \(0.608663\pi\)
\(398\) −41.7775 −2.09412
\(399\) 0 0
\(400\) 13.8324 0.691618
\(401\) −0.833758 −0.0416359 −0.0208179 0.999783i \(-0.506627\pi\)
−0.0208179 + 0.999783i \(0.506627\pi\)
\(402\) 0 0
\(403\) −4.01023 −0.199764
\(404\) 7.67738 0.381964
\(405\) 0 0
\(406\) 6.35054 0.315172
\(407\) −3.16462 −0.156864
\(408\) 0 0
\(409\) 12.4602 0.616118 0.308059 0.951367i \(-0.400321\pi\)
0.308059 + 0.951367i \(0.400321\pi\)
\(410\) −3.74608 −0.185006
\(411\) 0 0
\(412\) −4.23428 −0.208608
\(413\) −18.5149 −0.911057
\(414\) 0 0
\(415\) −19.1894 −0.941969
\(416\) −7.84411 −0.384589
\(417\) 0 0
\(418\) −5.71615 −0.279586
\(419\) −9.75869 −0.476743 −0.238372 0.971174i \(-0.576614\pi\)
−0.238372 + 0.971174i \(0.576614\pi\)
\(420\) 0 0
\(421\) 16.1165 0.785470 0.392735 0.919652i \(-0.371529\pi\)
0.392735 + 0.919652i \(0.371529\pi\)
\(422\) 6.58906 0.320750
\(423\) 0 0
\(424\) −2.40738 −0.116913
\(425\) −10.7527 −0.521582
\(426\) 0 0
\(427\) 41.2752 1.99744
\(428\) 10.2701 0.496426
\(429\) 0 0
\(430\) 1.02697 0.0495249
\(431\) −23.3638 −1.12539 −0.562697 0.826663i \(-0.690236\pi\)
−0.562697 + 0.826663i \(0.690236\pi\)
\(432\) 0 0
\(433\) −16.8084 −0.807758 −0.403879 0.914812i \(-0.632338\pi\)
−0.403879 + 0.914812i \(0.632338\pi\)
\(434\) −10.6697 −0.512162
\(435\) 0 0
\(436\) −9.77471 −0.468124
\(437\) 1.79768 0.0859948
\(438\) 0 0
\(439\) 4.57370 0.218291 0.109145 0.994026i \(-0.465189\pi\)
0.109145 + 0.994026i \(0.465189\pi\)
\(440\) −9.53490 −0.454558
\(441\) 0 0
\(442\) 14.2160 0.676188
\(443\) −8.43805 −0.400904 −0.200452 0.979704i \(-0.564241\pi\)
−0.200452 + 0.979704i \(0.564241\pi\)
\(444\) 0 0
\(445\) −8.92543 −0.423106
\(446\) −3.95612 −0.187328
\(447\) 0 0
\(448\) 16.3473 0.772336
\(449\) 11.3585 0.536043 0.268021 0.963413i \(-0.413630\pi\)
0.268021 + 0.963413i \(0.413630\pi\)
\(450\) 0 0
\(451\) −4.58898 −0.216087
\(452\) 6.06967 0.285493
\(453\) 0 0
\(454\) 4.43944 0.208353
\(455\) 13.1071 0.614471
\(456\) 0 0
\(457\) 13.8039 0.645720 0.322860 0.946447i \(-0.395356\pi\)
0.322860 + 0.946447i \(0.395356\pi\)
\(458\) 22.1764 1.03623
\(459\) 0 0
\(460\) −1.33063 −0.0620408
\(461\) 12.8220 0.597179 0.298589 0.954382i \(-0.403484\pi\)
0.298589 + 0.954382i \(0.403484\pi\)
\(462\) 0 0
\(463\) −5.22053 −0.242618 −0.121309 0.992615i \(-0.538709\pi\)
−0.121309 + 0.992615i \(0.538709\pi\)
\(464\) −4.96756 −0.230613
\(465\) 0 0
\(466\) −8.67948 −0.402069
\(467\) −8.47835 −0.392331 −0.196166 0.980571i \(-0.562849\pi\)
−0.196166 + 0.980571i \(0.562849\pi\)
\(468\) 0 0
\(469\) −23.4378 −1.08226
\(470\) 18.1552 0.837437
\(471\) 0 0
\(472\) 10.8128 0.497698
\(473\) 1.25805 0.0578451
\(474\) 0 0
\(475\) 3.47096 0.159259
\(476\) 8.89219 0.407573
\(477\) 0 0
\(478\) 1.61701 0.0739602
\(479\) 37.0880 1.69459 0.847297 0.531119i \(-0.178228\pi\)
0.847297 + 0.531119i \(0.178228\pi\)
\(480\) 0 0
\(481\) −2.54056 −0.115839
\(482\) −20.8885 −0.951443
\(483\) 0 0
\(484\) −1.57876 −0.0717618
\(485\) 11.7196 0.532159
\(486\) 0 0
\(487\) 34.2956 1.55408 0.777041 0.629450i \(-0.216720\pi\)
0.777041 + 0.629450i \(0.216720\pi\)
\(488\) −24.1049 −1.09118
\(489\) 0 0
\(490\) 18.2804 0.825825
\(491\) −1.07986 −0.0487335 −0.0243667 0.999703i \(-0.507757\pi\)
−0.0243667 + 0.999703i \(0.507757\pi\)
\(492\) 0 0
\(493\) 3.86157 0.173916
\(494\) −4.58893 −0.206466
\(495\) 0 0
\(496\) 8.34613 0.374752
\(497\) 41.9266 1.88066
\(498\) 0 0
\(499\) 0.972126 0.0435183 0.0217592 0.999763i \(-0.493073\pi\)
0.0217592 + 0.999763i \(0.493073\pi\)
\(500\) −7.07471 −0.316391
\(501\) 0 0
\(502\) 30.6255 1.36688
\(503\) 30.5203 1.36083 0.680417 0.732825i \(-0.261799\pi\)
0.680417 + 0.732825i \(0.261799\pi\)
\(504\) 0 0
\(505\) 18.3083 0.814708
\(506\) −6.93343 −0.308229
\(507\) 0 0
\(508\) −6.34211 −0.281386
\(509\) −20.2337 −0.896845 −0.448422 0.893822i \(-0.648014\pi\)
−0.448422 + 0.893822i \(0.648014\pi\)
\(510\) 0 0
\(511\) −46.5367 −2.05866
\(512\) −5.40993 −0.239087
\(513\) 0 0
\(514\) −24.7780 −1.09291
\(515\) −10.0975 −0.444950
\(516\) 0 0
\(517\) 22.2403 0.978127
\(518\) −6.75946 −0.296993
\(519\) 0 0
\(520\) −7.65461 −0.335677
\(521\) −17.2171 −0.754296 −0.377148 0.926153i \(-0.623095\pi\)
−0.377148 + 0.926153i \(0.623095\pi\)
\(522\) 0 0
\(523\) −2.23753 −0.0978405 −0.0489202 0.998803i \(-0.515578\pi\)
−0.0489202 + 0.998803i \(0.515578\pi\)
\(524\) −6.06577 −0.264985
\(525\) 0 0
\(526\) 43.8912 1.91375
\(527\) −6.48793 −0.282618
\(528\) 0 0
\(529\) −20.8195 −0.905195
\(530\) 2.54748 0.110656
\(531\) 0 0
\(532\) −2.87039 −0.124447
\(533\) −3.68403 −0.159573
\(534\) 0 0
\(535\) 24.4913 1.05885
\(536\) 13.6878 0.591222
\(537\) 0 0
\(538\) 20.6878 0.891916
\(539\) 22.3937 0.964565
\(540\) 0 0
\(541\) −4.02257 −0.172944 −0.0864719 0.996254i \(-0.527559\pi\)
−0.0864719 + 0.996254i \(0.527559\pi\)
\(542\) −46.0798 −1.97930
\(543\) 0 0
\(544\) −12.6905 −0.544103
\(545\) −23.3098 −0.998483
\(546\) 0 0
\(547\) −16.4728 −0.704327 −0.352163 0.935939i \(-0.614554\pi\)
−0.352163 + 0.935939i \(0.614554\pi\)
\(548\) 1.99541 0.0852397
\(549\) 0 0
\(550\) −13.3871 −0.570826
\(551\) −1.24651 −0.0531033
\(552\) 0 0
\(553\) 56.5362 2.40417
\(554\) −3.90553 −0.165930
\(555\) 0 0
\(556\) −8.74846 −0.371018
\(557\) 28.4924 1.20726 0.603631 0.797264i \(-0.293720\pi\)
0.603631 + 0.797264i \(0.293720\pi\)
\(558\) 0 0
\(559\) 1.00996 0.0427168
\(560\) −27.2787 −1.15273
\(561\) 0 0
\(562\) −13.3858 −0.564646
\(563\) −19.7584 −0.832716 −0.416358 0.909201i \(-0.636694\pi\)
−0.416358 + 0.909201i \(0.636694\pi\)
\(564\) 0 0
\(565\) 14.4744 0.608942
\(566\) −0.230720 −0.00969787
\(567\) 0 0
\(568\) −24.4853 −1.02738
\(569\) −9.62616 −0.403550 −0.201775 0.979432i \(-0.564671\pi\)
−0.201775 + 0.979432i \(0.564671\pi\)
\(570\) 0 0
\(571\) 17.4099 0.728583 0.364292 0.931285i \(-0.381311\pi\)
0.364292 + 0.931285i \(0.381311\pi\)
\(572\) 4.16096 0.173979
\(573\) 0 0
\(574\) −9.80182 −0.409120
\(575\) 4.21012 0.175574
\(576\) 0 0
\(577\) 39.3271 1.63721 0.818605 0.574357i \(-0.194748\pi\)
0.818605 + 0.574357i \(0.194748\pi\)
\(578\) −4.48978 −0.186750
\(579\) 0 0
\(580\) 0.922656 0.0383112
\(581\) −50.2101 −2.08306
\(582\) 0 0
\(583\) 3.12069 0.129246
\(584\) 27.1776 1.12462
\(585\) 0 0
\(586\) 20.9986 0.867445
\(587\) −26.7724 −1.10502 −0.552508 0.833507i \(-0.686329\pi\)
−0.552508 + 0.833507i \(0.686329\pi\)
\(588\) 0 0
\(589\) 2.09430 0.0862941
\(590\) −11.4420 −0.471060
\(591\) 0 0
\(592\) 5.28743 0.217312
\(593\) 44.4733 1.82630 0.913150 0.407623i \(-0.133642\pi\)
0.913150 + 0.407623i \(0.133642\pi\)
\(594\) 0 0
\(595\) 21.2053 0.869332
\(596\) 2.39941 0.0982838
\(597\) 0 0
\(598\) −5.56616 −0.227617
\(599\) −19.6240 −0.801815 −0.400908 0.916118i \(-0.631305\pi\)
−0.400908 + 0.916118i \(0.631305\pi\)
\(600\) 0 0
\(601\) 15.4787 0.631389 0.315695 0.948861i \(-0.397763\pi\)
0.315695 + 0.948861i \(0.397763\pi\)
\(602\) 2.68712 0.109519
\(603\) 0 0
\(604\) −0.430153 −0.0175027
\(605\) −3.76488 −0.153064
\(606\) 0 0
\(607\) 20.5122 0.832566 0.416283 0.909235i \(-0.363333\pi\)
0.416283 + 0.909235i \(0.363333\pi\)
\(608\) 4.09650 0.166135
\(609\) 0 0
\(610\) 25.5077 1.03277
\(611\) 17.8545 0.722316
\(612\) 0 0
\(613\) 32.1077 1.29682 0.648409 0.761292i \(-0.275435\pi\)
0.648409 + 0.761292i \(0.275435\pi\)
\(614\) 12.4495 0.502422
\(615\) 0 0
\(616\) −24.9486 −1.00521
\(617\) −16.7973 −0.676233 −0.338117 0.941104i \(-0.609790\pi\)
−0.338117 + 0.941104i \(0.609790\pi\)
\(618\) 0 0
\(619\) 21.4017 0.860206 0.430103 0.902780i \(-0.358477\pi\)
0.430103 + 0.902780i \(0.358477\pi\)
\(620\) −1.55018 −0.0622566
\(621\) 0 0
\(622\) −0.0900810 −0.00361192
\(623\) −23.3539 −0.935653
\(624\) 0 0
\(625\) −2.61552 −0.104621
\(626\) −4.42100 −0.176699
\(627\) 0 0
\(628\) 1.17633 0.0469407
\(629\) −4.11022 −0.163885
\(630\) 0 0
\(631\) −16.0747 −0.639924 −0.319962 0.947430i \(-0.603670\pi\)
−0.319962 + 0.947430i \(0.603670\pi\)
\(632\) −33.0174 −1.31336
\(633\) 0 0
\(634\) 9.28668 0.368821
\(635\) −15.1241 −0.600181
\(636\) 0 0
\(637\) 17.9777 0.712300
\(638\) 4.80764 0.190336
\(639\) 0 0
\(640\) 19.9679 0.789299
\(641\) 35.1303 1.38756 0.693782 0.720185i \(-0.255943\pi\)
0.693782 + 0.720185i \(0.255943\pi\)
\(642\) 0 0
\(643\) −1.14336 −0.0450898 −0.0225449 0.999746i \(-0.507177\pi\)
−0.0225449 + 0.999746i \(0.507177\pi\)
\(644\) −3.48166 −0.137197
\(645\) 0 0
\(646\) −7.42417 −0.292100
\(647\) 11.6354 0.457433 0.228716 0.973493i \(-0.426547\pi\)
0.228716 + 0.973493i \(0.426547\pi\)
\(648\) 0 0
\(649\) −14.0166 −0.550199
\(650\) −10.7471 −0.421537
\(651\) 0 0
\(652\) −10.5672 −0.413844
\(653\) −2.96261 −0.115936 −0.0579680 0.998318i \(-0.518462\pi\)
−0.0579680 + 0.998318i \(0.518462\pi\)
\(654\) 0 0
\(655\) −14.4651 −0.565198
\(656\) 7.66725 0.299356
\(657\) 0 0
\(658\) 47.5041 1.85190
\(659\) −35.0365 −1.36483 −0.682415 0.730965i \(-0.739070\pi\)
−0.682415 + 0.730965i \(0.739070\pi\)
\(660\) 0 0
\(661\) −30.3727 −1.18136 −0.590681 0.806905i \(-0.701141\pi\)
−0.590681 + 0.806905i \(0.701141\pi\)
\(662\) −52.2782 −2.03185
\(663\) 0 0
\(664\) 29.3229 1.13795
\(665\) −6.84505 −0.265440
\(666\) 0 0
\(667\) −1.51196 −0.0585435
\(668\) 9.15453 0.354199
\(669\) 0 0
\(670\) −14.4843 −0.559579
\(671\) 31.2471 1.20628
\(672\) 0 0
\(673\) 0.848634 0.0327124 0.0163562 0.999866i \(-0.494793\pi\)
0.0163562 + 0.999866i \(0.494793\pi\)
\(674\) −7.94852 −0.306166
\(675\) 0 0
\(676\) −4.65083 −0.178878
\(677\) −1.77815 −0.0683400 −0.0341700 0.999416i \(-0.510879\pi\)
−0.0341700 + 0.999416i \(0.510879\pi\)
\(678\) 0 0
\(679\) 30.6650 1.17681
\(680\) −12.3840 −0.474904
\(681\) 0 0
\(682\) −8.07744 −0.309301
\(683\) −34.3363 −1.31384 −0.656922 0.753959i \(-0.728142\pi\)
−0.656922 + 0.753959i \(0.728142\pi\)
\(684\) 0 0
\(685\) 4.75847 0.181812
\(686\) 4.41611 0.168608
\(687\) 0 0
\(688\) −2.10194 −0.0801357
\(689\) 2.50529 0.0954439
\(690\) 0 0
\(691\) 30.6872 1.16740 0.583698 0.811971i \(-0.301605\pi\)
0.583698 + 0.811971i \(0.301605\pi\)
\(692\) −7.20254 −0.273799
\(693\) 0 0
\(694\) −46.5279 −1.76618
\(695\) −20.8625 −0.791361
\(696\) 0 0
\(697\) −5.96020 −0.225759
\(698\) −20.5164 −0.776558
\(699\) 0 0
\(700\) −6.72237 −0.254082
\(701\) 1.99537 0.0753641 0.0376820 0.999290i \(-0.488003\pi\)
0.0376820 + 0.999290i \(0.488003\pi\)
\(702\) 0 0
\(703\) 1.32678 0.0500404
\(704\) 12.3756 0.466424
\(705\) 0 0
\(706\) −11.1367 −0.419136
\(707\) 47.9047 1.80164
\(708\) 0 0
\(709\) 33.9090 1.27348 0.636739 0.771080i \(-0.280283\pi\)
0.636739 + 0.771080i \(0.280283\pi\)
\(710\) 25.9102 0.972394
\(711\) 0 0
\(712\) 13.6388 0.511134
\(713\) 2.54029 0.0951345
\(714\) 0 0
\(715\) 9.92267 0.371087
\(716\) −2.07011 −0.0773636
\(717\) 0 0
\(718\) −7.21337 −0.269201
\(719\) 24.1685 0.901331 0.450666 0.892693i \(-0.351187\pi\)
0.450666 + 0.892693i \(0.351187\pi\)
\(720\) 0 0
\(721\) −26.4207 −0.983959
\(722\) −28.3266 −1.05421
\(723\) 0 0
\(724\) −4.09777 −0.152292
\(725\) −2.91929 −0.108420
\(726\) 0 0
\(727\) −46.2043 −1.71362 −0.856811 0.515631i \(-0.827558\pi\)
−0.856811 + 0.515631i \(0.827558\pi\)
\(728\) −20.0287 −0.742314
\(729\) 0 0
\(730\) −28.7592 −1.06443
\(731\) 1.63396 0.0604342
\(732\) 0 0
\(733\) −3.13031 −0.115621 −0.0578104 0.998328i \(-0.518412\pi\)
−0.0578104 + 0.998328i \(0.518412\pi\)
\(734\) 6.69208 0.247009
\(735\) 0 0
\(736\) 4.96887 0.183155
\(737\) −17.7435 −0.653589
\(738\) 0 0
\(739\) −46.7714 −1.72052 −0.860258 0.509859i \(-0.829697\pi\)
−0.860258 + 0.509859i \(0.829697\pi\)
\(740\) −0.982067 −0.0361015
\(741\) 0 0
\(742\) 6.66563 0.244703
\(743\) 41.7626 1.53212 0.766061 0.642768i \(-0.222214\pi\)
0.766061 + 0.642768i \(0.222214\pi\)
\(744\) 0 0
\(745\) 5.72190 0.209634
\(746\) −25.7382 −0.942342
\(747\) 0 0
\(748\) 6.73179 0.246139
\(749\) 64.0828 2.34154
\(750\) 0 0
\(751\) 4.01683 0.146576 0.0732881 0.997311i \(-0.476651\pi\)
0.0732881 + 0.997311i \(0.476651\pi\)
\(752\) −37.1590 −1.35505
\(753\) 0 0
\(754\) 3.85957 0.140557
\(755\) −1.02579 −0.0373323
\(756\) 0 0
\(757\) 34.0626 1.23803 0.619014 0.785380i \(-0.287532\pi\)
0.619014 + 0.785380i \(0.287532\pi\)
\(758\) 57.8058 2.09960
\(759\) 0 0
\(760\) 3.99754 0.145006
\(761\) 17.7142 0.642139 0.321070 0.947056i \(-0.395958\pi\)
0.321070 + 0.947056i \(0.395958\pi\)
\(762\) 0 0
\(763\) −60.9914 −2.20804
\(764\) −9.70694 −0.351185
\(765\) 0 0
\(766\) −36.7723 −1.32864
\(767\) −11.2525 −0.406305
\(768\) 0 0
\(769\) 19.4368 0.700907 0.350454 0.936580i \(-0.386027\pi\)
0.350454 + 0.936580i \(0.386027\pi\)
\(770\) 26.4005 0.951407
\(771\) 0 0
\(772\) 5.19994 0.187150
\(773\) 15.5259 0.558428 0.279214 0.960229i \(-0.409926\pi\)
0.279214 + 0.960229i \(0.409926\pi\)
\(774\) 0 0
\(775\) 4.90478 0.176185
\(776\) −17.9085 −0.642877
\(777\) 0 0
\(778\) −4.10118 −0.147034
\(779\) 1.92395 0.0689326
\(780\) 0 0
\(781\) 31.7403 1.13576
\(782\) −9.00518 −0.322024
\(783\) 0 0
\(784\) −37.4153 −1.33626
\(785\) 2.80521 0.100122
\(786\) 0 0
\(787\) −25.2433 −0.899828 −0.449914 0.893072i \(-0.648545\pi\)
−0.449914 + 0.893072i \(0.648545\pi\)
\(788\) −2.87898 −0.102559
\(789\) 0 0
\(790\) 34.9389 1.24307
\(791\) 37.8731 1.34661
\(792\) 0 0
\(793\) 25.0852 0.890801
\(794\) −21.5724 −0.765577
\(795\) 0 0
\(796\) −15.8819 −0.562918
\(797\) 39.2256 1.38944 0.694721 0.719279i \(-0.255528\pi\)
0.694721 + 0.719279i \(0.255528\pi\)
\(798\) 0 0
\(799\) 28.8858 1.02191
\(800\) 9.59388 0.339195
\(801\) 0 0
\(802\) −1.34819 −0.0476063
\(803\) −35.2303 −1.24325
\(804\) 0 0
\(805\) −8.30273 −0.292633
\(806\) −6.48456 −0.228409
\(807\) 0 0
\(808\) −27.9765 −0.984211
\(809\) 21.5713 0.758405 0.379203 0.925314i \(-0.376198\pi\)
0.379203 + 0.925314i \(0.376198\pi\)
\(810\) 0 0
\(811\) 2.52560 0.0886858 0.0443429 0.999016i \(-0.485881\pi\)
0.0443429 + 0.999016i \(0.485881\pi\)
\(812\) 2.41418 0.0847211
\(813\) 0 0
\(814\) −5.11721 −0.179358
\(815\) −25.1997 −0.882707
\(816\) 0 0
\(817\) −0.527441 −0.0184528
\(818\) 20.1482 0.704467
\(819\) 0 0
\(820\) −1.42409 −0.0497312
\(821\) 21.8608 0.762949 0.381474 0.924379i \(-0.375416\pi\)
0.381474 + 0.924379i \(0.375416\pi\)
\(822\) 0 0
\(823\) 15.6288 0.544785 0.272393 0.962186i \(-0.412185\pi\)
0.272393 + 0.962186i \(0.412185\pi\)
\(824\) 15.4298 0.537523
\(825\) 0 0
\(826\) −29.9387 −1.04170
\(827\) 1.15424 0.0401368 0.0200684 0.999799i \(-0.493612\pi\)
0.0200684 + 0.999799i \(0.493612\pi\)
\(828\) 0 0
\(829\) 22.5972 0.784832 0.392416 0.919788i \(-0.371639\pi\)
0.392416 + 0.919788i \(0.371639\pi\)
\(830\) −31.0293 −1.07704
\(831\) 0 0
\(832\) 9.93514 0.344439
\(833\) 29.0851 1.00774
\(834\) 0 0
\(835\) 21.8309 0.755489
\(836\) −2.17302 −0.0751554
\(837\) 0 0
\(838\) −15.7799 −0.545106
\(839\) −26.1778 −0.903757 −0.451878 0.892080i \(-0.649246\pi\)
−0.451878 + 0.892080i \(0.649246\pi\)
\(840\) 0 0
\(841\) −27.9516 −0.963848
\(842\) 26.0605 0.898103
\(843\) 0 0
\(844\) 2.50486 0.0862207
\(845\) −11.0909 −0.381537
\(846\) 0 0
\(847\) −9.85102 −0.338485
\(848\) −5.21403 −0.179051
\(849\) 0 0
\(850\) −17.3872 −0.596375
\(851\) 1.60932 0.0551668
\(852\) 0 0
\(853\) 20.2252 0.692497 0.346249 0.938143i \(-0.387455\pi\)
0.346249 + 0.938143i \(0.387455\pi\)
\(854\) 66.7422 2.28387
\(855\) 0 0
\(856\) −37.4247 −1.27915
\(857\) 24.4336 0.834636 0.417318 0.908761i \(-0.362970\pi\)
0.417318 + 0.908761i \(0.362970\pi\)
\(858\) 0 0
\(859\) 36.4278 1.24290 0.621451 0.783453i \(-0.286543\pi\)
0.621451 + 0.783453i \(0.286543\pi\)
\(860\) 0.390406 0.0133128
\(861\) 0 0
\(862\) −37.7794 −1.28677
\(863\) −18.5506 −0.631471 −0.315735 0.948847i \(-0.602251\pi\)
−0.315735 + 0.948847i \(0.602251\pi\)
\(864\) 0 0
\(865\) −17.1759 −0.583999
\(866\) −27.1792 −0.923588
\(867\) 0 0
\(868\) −4.05613 −0.137674
\(869\) 42.8005 1.45191
\(870\) 0 0
\(871\) −14.2444 −0.482655
\(872\) 35.6193 1.20622
\(873\) 0 0
\(874\) 2.90687 0.0983262
\(875\) −44.1442 −1.49235
\(876\) 0 0
\(877\) 39.5945 1.33701 0.668505 0.743708i \(-0.266934\pi\)
0.668505 + 0.743708i \(0.266934\pi\)
\(878\) 7.39570 0.249593
\(879\) 0 0
\(880\) −20.6512 −0.696151
\(881\) 32.1516 1.08321 0.541607 0.840632i \(-0.317816\pi\)
0.541607 + 0.840632i \(0.317816\pi\)
\(882\) 0 0
\(883\) 20.8275 0.700900 0.350450 0.936581i \(-0.386029\pi\)
0.350450 + 0.936581i \(0.386029\pi\)
\(884\) 5.40428 0.181766
\(885\) 0 0
\(886\) −13.6444 −0.458392
\(887\) 34.9748 1.17434 0.587170 0.809464i \(-0.300242\pi\)
0.587170 + 0.809464i \(0.300242\pi\)
\(888\) 0 0
\(889\) −39.5730 −1.32724
\(890\) −14.4325 −0.483778
\(891\) 0 0
\(892\) −1.50394 −0.0503555
\(893\) −9.32432 −0.312027
\(894\) 0 0
\(895\) −4.93660 −0.165012
\(896\) 52.2470 1.74545
\(897\) 0 0
\(898\) 18.3668 0.612909
\(899\) −1.76144 −0.0587472
\(900\) 0 0
\(901\) 4.05317 0.135031
\(902\) −7.42042 −0.247073
\(903\) 0 0
\(904\) −22.1180 −0.735635
\(905\) −9.77198 −0.324832
\(906\) 0 0
\(907\) −28.3094 −0.939999 −0.470000 0.882667i \(-0.655746\pi\)
−0.470000 + 0.882667i \(0.655746\pi\)
\(908\) 1.68767 0.0560073
\(909\) 0 0
\(910\) 21.1943 0.702584
\(911\) −47.1094 −1.56081 −0.780403 0.625277i \(-0.784986\pi\)
−0.780403 + 0.625277i \(0.784986\pi\)
\(912\) 0 0
\(913\) −38.0113 −1.25799
\(914\) 22.3210 0.738314
\(915\) 0 0
\(916\) 8.43044 0.278549
\(917\) −37.8487 −1.24987
\(918\) 0 0
\(919\) −17.0107 −0.561131 −0.280565 0.959835i \(-0.590522\pi\)
−0.280565 + 0.959835i \(0.590522\pi\)
\(920\) 4.84883 0.159861
\(921\) 0 0
\(922\) 20.7332 0.682812
\(923\) 25.4811 0.838720
\(924\) 0 0
\(925\) 3.10727 0.102166
\(926\) −8.44163 −0.277409
\(927\) 0 0
\(928\) −3.44541 −0.113101
\(929\) −0.496932 −0.0163038 −0.00815190 0.999967i \(-0.502595\pi\)
−0.00815190 + 0.999967i \(0.502595\pi\)
\(930\) 0 0
\(931\) −9.38864 −0.307700
\(932\) −3.29954 −0.108080
\(933\) 0 0
\(934\) −13.7095 −0.448590
\(935\) 16.0533 0.525001
\(936\) 0 0
\(937\) 8.22697 0.268763 0.134382 0.990930i \(-0.457095\pi\)
0.134382 + 0.990930i \(0.457095\pi\)
\(938\) −37.8991 −1.23745
\(939\) 0 0
\(940\) 6.90177 0.225111
\(941\) 50.1600 1.63517 0.817584 0.575809i \(-0.195313\pi\)
0.817584 + 0.575809i \(0.195313\pi\)
\(942\) 0 0
\(943\) 2.33366 0.0759944
\(944\) 23.4188 0.762218
\(945\) 0 0
\(946\) 2.03427 0.0661399
\(947\) 11.6253 0.377773 0.188886 0.981999i \(-0.439512\pi\)
0.188886 + 0.981999i \(0.439512\pi\)
\(948\) 0 0
\(949\) −28.2829 −0.918101
\(950\) 5.61257 0.182096
\(951\) 0 0
\(952\) −32.4034 −1.05020
\(953\) 27.5219 0.891522 0.445761 0.895152i \(-0.352933\pi\)
0.445761 + 0.895152i \(0.352933\pi\)
\(954\) 0 0
\(955\) −23.1482 −0.749058
\(956\) 0.614711 0.0198812
\(957\) 0 0
\(958\) 59.9716 1.93759
\(959\) 12.4508 0.402057
\(960\) 0 0
\(961\) −28.0406 −0.904534
\(962\) −4.10809 −0.132450
\(963\) 0 0
\(964\) −7.94082 −0.255757
\(965\) 12.4003 0.399181
\(966\) 0 0
\(967\) −41.9315 −1.34843 −0.674213 0.738537i \(-0.735517\pi\)
−0.674213 + 0.738537i \(0.735517\pi\)
\(968\) 5.75304 0.184910
\(969\) 0 0
\(970\) 18.9507 0.608469
\(971\) −44.8246 −1.43849 −0.719244 0.694757i \(-0.755512\pi\)
−0.719244 + 0.694757i \(0.755512\pi\)
\(972\) 0 0
\(973\) −54.5880 −1.75001
\(974\) 55.4562 1.77693
\(975\) 0 0
\(976\) −52.2075 −1.67112
\(977\) −36.6672 −1.17309 −0.586544 0.809917i \(-0.699512\pi\)
−0.586544 + 0.809917i \(0.699512\pi\)
\(978\) 0 0
\(979\) −17.6799 −0.565053
\(980\) 6.94937 0.221989
\(981\) 0 0
\(982\) −1.74614 −0.0557217
\(983\) −43.4827 −1.38688 −0.693441 0.720513i \(-0.743906\pi\)
−0.693441 + 0.720513i \(0.743906\pi\)
\(984\) 0 0
\(985\) −6.86552 −0.218754
\(986\) 6.24419 0.198856
\(987\) 0 0
\(988\) −1.74450 −0.0554998
\(989\) −0.639762 −0.0203433
\(990\) 0 0
\(991\) −4.85590 −0.154253 −0.0771263 0.997021i \(-0.524574\pi\)
−0.0771263 + 0.997021i \(0.524574\pi\)
\(992\) 5.78872 0.183792
\(993\) 0 0
\(994\) 67.7956 2.15034
\(995\) −37.8736 −1.20067
\(996\) 0 0
\(997\) −10.2069 −0.323257 −0.161628 0.986852i \(-0.551675\pi\)
−0.161628 + 0.986852i \(0.551675\pi\)
\(998\) 1.57193 0.0497587
\(999\) 0 0
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 2151.2.a.k.1.15 yes 20
3.2 odd 2 2151.2.a.j.1.6 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.6 20 3.2 odd 2
2151.2.a.k.1.15 yes 20 1.1 even 1 trivial