Properties

Label 405.2.a.g.1.2
Level $405$
Weight $2$
Character 405.1
Self dual yes
Analytic conductor $3.234$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.23394128186\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} -4.73205 q^{7} -2.53590 q^{8} +O(q^{10})\) \(q+0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} -4.73205 q^{7} -2.53590 q^{8} +0.732051 q^{10} -5.73205 q^{11} +1.46410 q^{13} -3.46410 q^{14} +1.07180 q^{16} -2.73205 q^{17} +4.46410 q^{19} -1.46410 q^{20} -4.19615 q^{22} -3.46410 q^{23} +1.00000 q^{25} +1.07180 q^{26} +6.92820 q^{28} +3.19615 q^{29} -3.00000 q^{31} +5.85641 q^{32} -2.00000 q^{34} -4.73205 q^{35} -2.73205 q^{37} +3.26795 q^{38} -2.53590 q^{40} -7.19615 q^{41} +0.196152 q^{43} +8.39230 q^{44} -2.53590 q^{46} -8.73205 q^{47} +15.3923 q^{49} +0.732051 q^{50} -2.14359 q^{52} +6.73205 q^{53} -5.73205 q^{55} +12.0000 q^{56} +2.33975 q^{58} -8.26795 q^{59} +4.00000 q^{61} -2.19615 q^{62} +2.14359 q^{64} +1.46410 q^{65} +3.46410 q^{67} +4.00000 q^{68} -3.46410 q^{70} -3.73205 q^{71} -7.66025 q^{73} -2.00000 q^{74} -6.53590 q^{76} +27.1244 q^{77} +15.4641 q^{79} +1.07180 q^{80} -5.26795 q^{82} +2.19615 q^{83} -2.73205 q^{85} +0.143594 q^{86} +14.5359 q^{88} +5.19615 q^{89} -6.92820 q^{91} +5.07180 q^{92} -6.39230 q^{94} +4.46410 q^{95} -9.66025 q^{97} +11.2679 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} - 6 q^{7} - 12 q^{8} - 2 q^{10} - 8 q^{11} - 4 q^{13} + 16 q^{16} - 2 q^{17} + 2 q^{19} + 4 q^{20} + 2 q^{22} + 2 q^{25} + 16 q^{26} - 4 q^{29} - 6 q^{31} - 16 q^{32} - 4 q^{34} - 6 q^{35} - 2 q^{37} + 10 q^{38} - 12 q^{40} - 4 q^{41} - 10 q^{43} - 4 q^{44} - 12 q^{46} - 14 q^{47} + 10 q^{49} - 2 q^{50} - 32 q^{52} + 10 q^{53} - 8 q^{55} + 24 q^{56} + 22 q^{58} - 20 q^{59} + 8 q^{61} + 6 q^{62} + 32 q^{64} - 4 q^{65} + 8 q^{68} - 4 q^{71} + 2 q^{73} - 4 q^{74} - 20 q^{76} + 30 q^{77} + 24 q^{79} + 16 q^{80} - 14 q^{82} - 6 q^{83} - 2 q^{85} + 28 q^{86} + 36 q^{88} + 24 q^{92} + 8 q^{94} + 2 q^{95} - 2 q^{97} + 26 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.732051 0.517638 0.258819 0.965926i \(-0.416667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(3\) 0 0
\(4\) −1.46410 −0.732051
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.73205 −1.78855 −0.894274 0.447521i \(-0.852307\pi\)
−0.894274 + 0.447521i \(0.852307\pi\)
\(8\) −2.53590 −0.896575
\(9\) 0 0
\(10\) 0.732051 0.231495
\(11\) −5.73205 −1.72828 −0.864139 0.503253i \(-0.832136\pi\)
−0.864139 + 0.503253i \(0.832136\pi\)
\(12\) 0 0
\(13\) 1.46410 0.406069 0.203034 0.979172i \(-0.434920\pi\)
0.203034 + 0.979172i \(0.434920\pi\)
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) 1.07180 0.267949
\(17\) −2.73205 −0.662620 −0.331310 0.943522i \(-0.607491\pi\)
−0.331310 + 0.943522i \(0.607491\pi\)
\(18\) 0 0
\(19\) 4.46410 1.02414 0.512068 0.858945i \(-0.328880\pi\)
0.512068 + 0.858945i \(0.328880\pi\)
\(20\) −1.46410 −0.327383
\(21\) 0 0
\(22\) −4.19615 −0.894623
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.07180 0.210197
\(27\) 0 0
\(28\) 6.92820 1.30931
\(29\) 3.19615 0.593511 0.296755 0.954954i \(-0.404095\pi\)
0.296755 + 0.954954i \(0.404095\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 5.85641 1.03528
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −4.73205 −0.799863
\(36\) 0 0
\(37\) −2.73205 −0.449146 −0.224573 0.974457i \(-0.572099\pi\)
−0.224573 + 0.974457i \(0.572099\pi\)
\(38\) 3.26795 0.530131
\(39\) 0 0
\(40\) −2.53590 −0.400961
\(41\) −7.19615 −1.12385 −0.561925 0.827188i \(-0.689939\pi\)
−0.561925 + 0.827188i \(0.689939\pi\)
\(42\) 0 0
\(43\) 0.196152 0.0299130 0.0149565 0.999888i \(-0.495239\pi\)
0.0149565 + 0.999888i \(0.495239\pi\)
\(44\) 8.39230 1.26519
\(45\) 0 0
\(46\) −2.53590 −0.373898
\(47\) −8.73205 −1.27370 −0.636850 0.770988i \(-0.719763\pi\)
−0.636850 + 0.770988i \(0.719763\pi\)
\(48\) 0 0
\(49\) 15.3923 2.19890
\(50\) 0.732051 0.103528
\(51\) 0 0
\(52\) −2.14359 −0.297263
\(53\) 6.73205 0.924718 0.462359 0.886693i \(-0.347003\pi\)
0.462359 + 0.886693i \(0.347003\pi\)
\(54\) 0 0
\(55\) −5.73205 −0.772910
\(56\) 12.0000 1.60357
\(57\) 0 0
\(58\) 2.33975 0.307224
\(59\) −8.26795 −1.07640 −0.538198 0.842819i \(-0.680895\pi\)
−0.538198 + 0.842819i \(0.680895\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −2.19615 −0.278912
\(63\) 0 0
\(64\) 2.14359 0.267949
\(65\) 1.46410 0.181599
\(66\) 0 0
\(67\) 3.46410 0.423207 0.211604 0.977356i \(-0.432131\pi\)
0.211604 + 0.977356i \(0.432131\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) −3.46410 −0.414039
\(71\) −3.73205 −0.442913 −0.221456 0.975170i \(-0.571081\pi\)
−0.221456 + 0.975170i \(0.571081\pi\)
\(72\) 0 0
\(73\) −7.66025 −0.896565 −0.448282 0.893892i \(-0.647964\pi\)
−0.448282 + 0.893892i \(0.647964\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −6.53590 −0.749719
\(77\) 27.1244 3.09111
\(78\) 0 0
\(79\) 15.4641 1.73985 0.869924 0.493186i \(-0.164168\pi\)
0.869924 + 0.493186i \(0.164168\pi\)
\(80\) 1.07180 0.119831
\(81\) 0 0
\(82\) −5.26795 −0.581748
\(83\) 2.19615 0.241059 0.120530 0.992710i \(-0.461541\pi\)
0.120530 + 0.992710i \(0.461541\pi\)
\(84\) 0 0
\(85\) −2.73205 −0.296333
\(86\) 0.143594 0.0154841
\(87\) 0 0
\(88\) 14.5359 1.54953
\(89\) 5.19615 0.550791 0.275396 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) −6.92820 −0.726273
\(92\) 5.07180 0.528771
\(93\) 0 0
\(94\) −6.39230 −0.659316
\(95\) 4.46410 0.458007
\(96\) 0 0
\(97\) −9.66025 −0.980850 −0.490425 0.871483i \(-0.663158\pi\)
−0.490425 + 0.871483i \(0.663158\pi\)
\(98\) 11.2679 1.13823
\(99\) 0 0
\(100\) −1.46410 −0.146410
\(101\) −2.66025 −0.264705 −0.132353 0.991203i \(-0.542253\pi\)
−0.132353 + 0.991203i \(0.542253\pi\)
\(102\) 0 0
\(103\) −0.535898 −0.0528036 −0.0264018 0.999651i \(-0.508405\pi\)
−0.0264018 + 0.999651i \(0.508405\pi\)
\(104\) −3.71281 −0.364071
\(105\) 0 0
\(106\) 4.92820 0.478669
\(107\) −8.53590 −0.825196 −0.412598 0.910913i \(-0.635379\pi\)
−0.412598 + 0.910913i \(0.635379\pi\)
\(108\) 0 0
\(109\) −6.07180 −0.581573 −0.290786 0.956788i \(-0.593917\pi\)
−0.290786 + 0.956788i \(0.593917\pi\)
\(110\) −4.19615 −0.400087
\(111\) 0 0
\(112\) −5.07180 −0.479240
\(113\) −19.1244 −1.79907 −0.899534 0.436851i \(-0.856094\pi\)
−0.899534 + 0.436851i \(0.856094\pi\)
\(114\) 0 0
\(115\) −3.46410 −0.323029
\(116\) −4.67949 −0.434480
\(117\) 0 0
\(118\) −6.05256 −0.557183
\(119\) 12.9282 1.18513
\(120\) 0 0
\(121\) 21.8564 1.98695
\(122\) 2.92820 0.265107
\(123\) 0 0
\(124\) 4.39230 0.394441
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −14.5885 −1.29452 −0.647258 0.762271i \(-0.724084\pi\)
−0.647258 + 0.762271i \(0.724084\pi\)
\(128\) −10.1436 −0.896575
\(129\) 0 0
\(130\) 1.07180 0.0940028
\(131\) 15.5885 1.36197 0.680985 0.732297i \(-0.261552\pi\)
0.680985 + 0.732297i \(0.261552\pi\)
\(132\) 0 0
\(133\) −21.1244 −1.83171
\(134\) 2.53590 0.219068
\(135\) 0 0
\(136\) 6.92820 0.594089
\(137\) 2.53590 0.216656 0.108328 0.994115i \(-0.465450\pi\)
0.108328 + 0.994115i \(0.465450\pi\)
\(138\) 0 0
\(139\) 0.607695 0.0515440 0.0257720 0.999668i \(-0.491796\pi\)
0.0257720 + 0.999668i \(0.491796\pi\)
\(140\) 6.92820 0.585540
\(141\) 0 0
\(142\) −2.73205 −0.229269
\(143\) −8.39230 −0.701800
\(144\) 0 0
\(145\) 3.19615 0.265426
\(146\) −5.60770 −0.464096
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) −8.00000 −0.655386 −0.327693 0.944784i \(-0.606271\pi\)
−0.327693 + 0.944784i \(0.606271\pi\)
\(150\) 0 0
\(151\) 5.39230 0.438820 0.219410 0.975633i \(-0.429587\pi\)
0.219410 + 0.975633i \(0.429587\pi\)
\(152\) −11.3205 −0.918214
\(153\) 0 0
\(154\) 19.8564 1.60007
\(155\) −3.00000 −0.240966
\(156\) 0 0
\(157\) −19.1244 −1.52629 −0.763145 0.646227i \(-0.776346\pi\)
−0.763145 + 0.646227i \(0.776346\pi\)
\(158\) 11.3205 0.900611
\(159\) 0 0
\(160\) 5.85641 0.462990
\(161\) 16.3923 1.29189
\(162\) 0 0
\(163\) 12.7321 0.997251 0.498626 0.866817i \(-0.333838\pi\)
0.498626 + 0.866817i \(0.333838\pi\)
\(164\) 10.5359 0.822715
\(165\) 0 0
\(166\) 1.60770 0.124781
\(167\) −17.6603 −1.36659 −0.683296 0.730142i \(-0.739454\pi\)
−0.683296 + 0.730142i \(0.739454\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −0.287187 −0.0218978
\(173\) 8.53590 0.648972 0.324486 0.945890i \(-0.394809\pi\)
0.324486 + 0.945890i \(0.394809\pi\)
\(174\) 0 0
\(175\) −4.73205 −0.357709
\(176\) −6.14359 −0.463091
\(177\) 0 0
\(178\) 3.80385 0.285110
\(179\) 8.12436 0.607243 0.303621 0.952793i \(-0.401804\pi\)
0.303621 + 0.952793i \(0.401804\pi\)
\(180\) 0 0
\(181\) 26.4641 1.96706 0.983531 0.180742i \(-0.0578498\pi\)
0.983531 + 0.180742i \(0.0578498\pi\)
\(182\) −5.07180 −0.375947
\(183\) 0 0
\(184\) 8.78461 0.647610
\(185\) −2.73205 −0.200864
\(186\) 0 0
\(187\) 15.6603 1.14519
\(188\) 12.7846 0.932413
\(189\) 0 0
\(190\) 3.26795 0.237082
\(191\) −8.12436 −0.587858 −0.293929 0.955827i \(-0.594963\pi\)
−0.293929 + 0.955827i \(0.594963\pi\)
\(192\) 0 0
\(193\) −5.26795 −0.379195 −0.189598 0.981862i \(-0.560718\pi\)
−0.189598 + 0.981862i \(0.560718\pi\)
\(194\) −7.07180 −0.507725
\(195\) 0 0
\(196\) −22.5359 −1.60971
\(197\) −13.8564 −0.987228 −0.493614 0.869681i \(-0.664324\pi\)
−0.493614 + 0.869681i \(0.664324\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) −2.53590 −0.179315
\(201\) 0 0
\(202\) −1.94744 −0.137021
\(203\) −15.1244 −1.06152
\(204\) 0 0
\(205\) −7.19615 −0.502601
\(206\) −0.392305 −0.0273332
\(207\) 0 0
\(208\) 1.56922 0.108806
\(209\) −25.5885 −1.76999
\(210\) 0 0
\(211\) −8.85641 −0.609700 −0.304850 0.952400i \(-0.598606\pi\)
−0.304850 + 0.952400i \(0.598606\pi\)
\(212\) −9.85641 −0.676941
\(213\) 0 0
\(214\) −6.24871 −0.427153
\(215\) 0.196152 0.0133775
\(216\) 0 0
\(217\) 14.1962 0.963698
\(218\) −4.44486 −0.301044
\(219\) 0 0
\(220\) 8.39230 0.565809
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 16.7846 1.12398 0.561990 0.827144i \(-0.310036\pi\)
0.561990 + 0.827144i \(0.310036\pi\)
\(224\) −27.7128 −1.85164
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) −18.0526 −1.19819 −0.599095 0.800678i \(-0.704473\pi\)
−0.599095 + 0.800678i \(0.704473\pi\)
\(228\) 0 0
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) −2.53590 −0.167212
\(231\) 0 0
\(232\) −8.10512 −0.532127
\(233\) −28.0526 −1.83778 −0.918892 0.394509i \(-0.870915\pi\)
−0.918892 + 0.394509i \(0.870915\pi\)
\(234\) 0 0
\(235\) −8.73205 −0.569616
\(236\) 12.1051 0.787976
\(237\) 0 0
\(238\) 9.46410 0.613467
\(239\) −0.535898 −0.0346644 −0.0173322 0.999850i \(-0.505517\pi\)
−0.0173322 + 0.999850i \(0.505517\pi\)
\(240\) 0 0
\(241\) −16.3205 −1.05130 −0.525648 0.850702i \(-0.676177\pi\)
−0.525648 + 0.850702i \(0.676177\pi\)
\(242\) 16.0000 1.02852
\(243\) 0 0
\(244\) −5.85641 −0.374918
\(245\) 15.3923 0.983378
\(246\) 0 0
\(247\) 6.53590 0.415869
\(248\) 7.60770 0.483089
\(249\) 0 0
\(250\) 0.732051 0.0462990
\(251\) −10.3923 −0.655956 −0.327978 0.944685i \(-0.606367\pi\)
−0.327978 + 0.944685i \(0.606367\pi\)
\(252\) 0 0
\(253\) 19.8564 1.24836
\(254\) −10.6795 −0.670091
\(255\) 0 0
\(256\) −11.7128 −0.732051
\(257\) 10.3923 0.648254 0.324127 0.946014i \(-0.394929\pi\)
0.324127 + 0.946014i \(0.394929\pi\)
\(258\) 0 0
\(259\) 12.9282 0.803319
\(260\) −2.14359 −0.132940
\(261\) 0 0
\(262\) 11.4115 0.705007
\(263\) 21.3205 1.31468 0.657339 0.753595i \(-0.271682\pi\)
0.657339 + 0.753595i \(0.271682\pi\)
\(264\) 0 0
\(265\) 6.73205 0.413547
\(266\) −15.4641 −0.948165
\(267\) 0 0
\(268\) −5.07180 −0.309809
\(269\) 10.6603 0.649967 0.324984 0.945720i \(-0.394641\pi\)
0.324984 + 0.945720i \(0.394641\pi\)
\(270\) 0 0
\(271\) −2.92820 −0.177876 −0.0889378 0.996037i \(-0.528347\pi\)
−0.0889378 + 0.996037i \(0.528347\pi\)
\(272\) −2.92820 −0.177548
\(273\) 0 0
\(274\) 1.85641 0.112150
\(275\) −5.73205 −0.345656
\(276\) 0 0
\(277\) 3.80385 0.228551 0.114276 0.993449i \(-0.463545\pi\)
0.114276 + 0.993449i \(0.463545\pi\)
\(278\) 0.444864 0.0266812
\(279\) 0 0
\(280\) 12.0000 0.717137
\(281\) 15.4641 0.922511 0.461255 0.887267i \(-0.347399\pi\)
0.461255 + 0.887267i \(0.347399\pi\)
\(282\) 0 0
\(283\) −29.3205 −1.74292 −0.871462 0.490464i \(-0.836827\pi\)
−0.871462 + 0.490464i \(0.836827\pi\)
\(284\) 5.46410 0.324235
\(285\) 0 0
\(286\) −6.14359 −0.363278
\(287\) 34.0526 2.01006
\(288\) 0 0
\(289\) −9.53590 −0.560935
\(290\) 2.33975 0.137395
\(291\) 0 0
\(292\) 11.2154 0.656331
\(293\) 28.7321 1.67854 0.839272 0.543712i \(-0.182981\pi\)
0.839272 + 0.543712i \(0.182981\pi\)
\(294\) 0 0
\(295\) −8.26795 −0.481379
\(296\) 6.92820 0.402694
\(297\) 0 0
\(298\) −5.85641 −0.339253
\(299\) −5.07180 −0.293310
\(300\) 0 0
\(301\) −0.928203 −0.0535007
\(302\) 3.94744 0.227150
\(303\) 0 0
\(304\) 4.78461 0.274416
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) 14.0526 0.802022 0.401011 0.916073i \(-0.368659\pi\)
0.401011 + 0.916073i \(0.368659\pi\)
\(308\) −39.7128 −2.26285
\(309\) 0 0
\(310\) −2.19615 −0.124733
\(311\) 19.7321 1.11890 0.559451 0.828863i \(-0.311012\pi\)
0.559451 + 0.828863i \(0.311012\pi\)
\(312\) 0 0
\(313\) −9.07180 −0.512768 −0.256384 0.966575i \(-0.582531\pi\)
−0.256384 + 0.966575i \(0.582531\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −22.6410 −1.27366
\(317\) 6.19615 0.348011 0.174005 0.984745i \(-0.444329\pi\)
0.174005 + 0.984745i \(0.444329\pi\)
\(318\) 0 0
\(319\) −18.3205 −1.02575
\(320\) 2.14359 0.119831
\(321\) 0 0
\(322\) 12.0000 0.668734
\(323\) −12.1962 −0.678612
\(324\) 0 0
\(325\) 1.46410 0.0812137
\(326\) 9.32051 0.516215
\(327\) 0 0
\(328\) 18.2487 1.00762
\(329\) 41.3205 2.27807
\(330\) 0 0
\(331\) −0.464102 −0.0255093 −0.0127547 0.999919i \(-0.504060\pi\)
−0.0127547 + 0.999919i \(0.504060\pi\)
\(332\) −3.21539 −0.176467
\(333\) 0 0
\(334\) −12.9282 −0.707400
\(335\) 3.46410 0.189264
\(336\) 0 0
\(337\) 27.3205 1.48824 0.744121 0.668044i \(-0.232868\pi\)
0.744121 + 0.668044i \(0.232868\pi\)
\(338\) −7.94744 −0.432284
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) 17.1962 0.931224
\(342\) 0 0
\(343\) −39.7128 −2.14429
\(344\) −0.497423 −0.0268192
\(345\) 0 0
\(346\) 6.24871 0.335933
\(347\) −28.5885 −1.53471 −0.767354 0.641223i \(-0.778427\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(348\) 0 0
\(349\) −18.8564 −1.00936 −0.504680 0.863306i \(-0.668390\pi\)
−0.504680 + 0.863306i \(0.668390\pi\)
\(350\) −3.46410 −0.185164
\(351\) 0 0
\(352\) −33.5692 −1.78925
\(353\) 25.5167 1.35811 0.679057 0.734085i \(-0.262389\pi\)
0.679057 + 0.734085i \(0.262389\pi\)
\(354\) 0 0
\(355\) −3.73205 −0.198077
\(356\) −7.60770 −0.403207
\(357\) 0 0
\(358\) 5.94744 0.314332
\(359\) −6.12436 −0.323231 −0.161616 0.986854i \(-0.551670\pi\)
−0.161616 + 0.986854i \(0.551670\pi\)
\(360\) 0 0
\(361\) 0.928203 0.0488528
\(362\) 19.3731 1.01823
\(363\) 0 0
\(364\) 10.1436 0.531669
\(365\) −7.66025 −0.400956
\(366\) 0 0
\(367\) 31.1769 1.62742 0.813711 0.581270i \(-0.197444\pi\)
0.813711 + 0.581270i \(0.197444\pi\)
\(368\) −3.71281 −0.193544
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) −31.8564 −1.65390
\(372\) 0 0
\(373\) 20.0526 1.03828 0.519141 0.854689i \(-0.326252\pi\)
0.519141 + 0.854689i \(0.326252\pi\)
\(374\) 11.4641 0.592795
\(375\) 0 0
\(376\) 22.1436 1.14197
\(377\) 4.67949 0.241006
\(378\) 0 0
\(379\) 2.39230 0.122884 0.0614422 0.998111i \(-0.480430\pi\)
0.0614422 + 0.998111i \(0.480430\pi\)
\(380\) −6.53590 −0.335285
\(381\) 0 0
\(382\) −5.94744 −0.304298
\(383\) 2.53590 0.129578 0.0647892 0.997899i \(-0.479363\pi\)
0.0647892 + 0.997899i \(0.479363\pi\)
\(384\) 0 0
\(385\) 27.1244 1.38239
\(386\) −3.85641 −0.196286
\(387\) 0 0
\(388\) 14.1436 0.718032
\(389\) 27.4641 1.39249 0.696243 0.717807i \(-0.254854\pi\)
0.696243 + 0.717807i \(0.254854\pi\)
\(390\) 0 0
\(391\) 9.46410 0.478620
\(392\) −39.0333 −1.97148
\(393\) 0 0
\(394\) −10.1436 −0.511027
\(395\) 15.4641 0.778083
\(396\) 0 0
\(397\) 14.3923 0.722329 0.361165 0.932502i \(-0.382379\pi\)
0.361165 + 0.932502i \(0.382379\pi\)
\(398\) −1.46410 −0.0733888
\(399\) 0 0
\(400\) 1.07180 0.0535898
\(401\) 24.9282 1.24486 0.622428 0.782677i \(-0.286147\pi\)
0.622428 + 0.782677i \(0.286147\pi\)
\(402\) 0 0
\(403\) −4.39230 −0.218796
\(404\) 3.89488 0.193778
\(405\) 0 0
\(406\) −11.0718 −0.549484
\(407\) 15.6603 0.776250
\(408\) 0 0
\(409\) −17.8564 −0.882942 −0.441471 0.897275i \(-0.645543\pi\)
−0.441471 + 0.897275i \(0.645543\pi\)
\(410\) −5.26795 −0.260165
\(411\) 0 0
\(412\) 0.784610 0.0386549
\(413\) 39.1244 1.92518
\(414\) 0 0
\(415\) 2.19615 0.107805
\(416\) 8.57437 0.420393
\(417\) 0 0
\(418\) −18.7321 −0.916215
\(419\) 20.3923 0.996229 0.498115 0.867111i \(-0.334026\pi\)
0.498115 + 0.867111i \(0.334026\pi\)
\(420\) 0 0
\(421\) 33.7846 1.64656 0.823281 0.567635i \(-0.192141\pi\)
0.823281 + 0.567635i \(0.192141\pi\)
\(422\) −6.48334 −0.315604
\(423\) 0 0
\(424\) −17.0718 −0.829080
\(425\) −2.73205 −0.132524
\(426\) 0 0
\(427\) −18.9282 −0.916000
\(428\) 12.4974 0.604086
\(429\) 0 0
\(430\) 0.143594 0.00692470
\(431\) −21.3397 −1.02790 −0.513950 0.857820i \(-0.671818\pi\)
−0.513950 + 0.857820i \(0.671818\pi\)
\(432\) 0 0
\(433\) −35.4641 −1.70430 −0.852148 0.523301i \(-0.824700\pi\)
−0.852148 + 0.523301i \(0.824700\pi\)
\(434\) 10.3923 0.498847
\(435\) 0 0
\(436\) 8.88973 0.425741
\(437\) −15.4641 −0.739748
\(438\) 0 0
\(439\) 5.39230 0.257361 0.128680 0.991686i \(-0.458926\pi\)
0.128680 + 0.991686i \(0.458926\pi\)
\(440\) 14.5359 0.692972
\(441\) 0 0
\(442\) −2.92820 −0.139280
\(443\) −0.339746 −0.0161418 −0.00807091 0.999967i \(-0.502569\pi\)
−0.00807091 + 0.999967i \(0.502569\pi\)
\(444\) 0 0
\(445\) 5.19615 0.246321
\(446\) 12.2872 0.581815
\(447\) 0 0
\(448\) −10.1436 −0.479240
\(449\) 8.12436 0.383412 0.191706 0.981452i \(-0.438598\pi\)
0.191706 + 0.981452i \(0.438598\pi\)
\(450\) 0 0
\(451\) 41.2487 1.94233
\(452\) 28.0000 1.31701
\(453\) 0 0
\(454\) −13.2154 −0.620229
\(455\) −6.92820 −0.324799
\(456\) 0 0
\(457\) 2.73205 0.127800 0.0639000 0.997956i \(-0.479646\pi\)
0.0639000 + 0.997956i \(0.479646\pi\)
\(458\) −8.78461 −0.410478
\(459\) 0 0
\(460\) 5.07180 0.236474
\(461\) −37.0526 −1.72571 −0.862855 0.505452i \(-0.831326\pi\)
−0.862855 + 0.505452i \(0.831326\pi\)
\(462\) 0 0
\(463\) −10.3923 −0.482971 −0.241486 0.970404i \(-0.577635\pi\)
−0.241486 + 0.970404i \(0.577635\pi\)
\(464\) 3.42563 0.159031
\(465\) 0 0
\(466\) −20.5359 −0.951307
\(467\) 37.3731 1.72942 0.864710 0.502272i \(-0.167502\pi\)
0.864710 + 0.502272i \(0.167502\pi\)
\(468\) 0 0
\(469\) −16.3923 −0.756926
\(470\) −6.39230 −0.294855
\(471\) 0 0
\(472\) 20.9667 0.965070
\(473\) −1.12436 −0.0516979
\(474\) 0 0
\(475\) 4.46410 0.204827
\(476\) −18.9282 −0.867573
\(477\) 0 0
\(478\) −0.392305 −0.0179436
\(479\) −11.8756 −0.542612 −0.271306 0.962493i \(-0.587456\pi\)
−0.271306 + 0.962493i \(0.587456\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −11.9474 −0.544191
\(483\) 0 0
\(484\) −32.0000 −1.45455
\(485\) −9.66025 −0.438650
\(486\) 0 0
\(487\) 24.3923 1.10532 0.552660 0.833407i \(-0.313613\pi\)
0.552660 + 0.833407i \(0.313613\pi\)
\(488\) −10.1436 −0.459179
\(489\) 0 0
\(490\) 11.2679 0.509034
\(491\) −13.8756 −0.626199 −0.313100 0.949720i \(-0.601367\pi\)
−0.313100 + 0.949720i \(0.601367\pi\)
\(492\) 0 0
\(493\) −8.73205 −0.393272
\(494\) 4.78461 0.215270
\(495\) 0 0
\(496\) −3.21539 −0.144375
\(497\) 17.6603 0.792171
\(498\) 0 0
\(499\) 24.3205 1.08874 0.544368 0.838847i \(-0.316770\pi\)
0.544368 + 0.838847i \(0.316770\pi\)
\(500\) −1.46410 −0.0654766
\(501\) 0 0
\(502\) −7.60770 −0.339548
\(503\) −7.32051 −0.326405 −0.163203 0.986593i \(-0.552182\pi\)
−0.163203 + 0.986593i \(0.552182\pi\)
\(504\) 0 0
\(505\) −2.66025 −0.118380
\(506\) 14.5359 0.646200
\(507\) 0 0
\(508\) 21.3590 0.947652
\(509\) −6.78461 −0.300723 −0.150361 0.988631i \(-0.548044\pi\)
−0.150361 + 0.988631i \(0.548044\pi\)
\(510\) 0 0
\(511\) 36.2487 1.60355
\(512\) 11.7128 0.517638
\(513\) 0 0
\(514\) 7.60770 0.335561
\(515\) −0.535898 −0.0236145
\(516\) 0 0
\(517\) 50.0526 2.20131
\(518\) 9.46410 0.415829
\(519\) 0 0
\(520\) −3.71281 −0.162818
\(521\) 19.4641 0.852738 0.426369 0.904549i \(-0.359793\pi\)
0.426369 + 0.904549i \(0.359793\pi\)
\(522\) 0 0
\(523\) 22.2487 0.972868 0.486434 0.873717i \(-0.338297\pi\)
0.486434 + 0.873717i \(0.338297\pi\)
\(524\) −22.8231 −0.997031
\(525\) 0 0
\(526\) 15.6077 0.680528
\(527\) 8.19615 0.357030
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 4.92820 0.214067
\(531\) 0 0
\(532\) 30.9282 1.34091
\(533\) −10.5359 −0.456360
\(534\) 0 0
\(535\) −8.53590 −0.369039
\(536\) −8.78461 −0.379437
\(537\) 0 0
\(538\) 7.80385 0.336448
\(539\) −88.2295 −3.80031
\(540\) 0 0
\(541\) −24.4641 −1.05179 −0.525897 0.850548i \(-0.676270\pi\)
−0.525897 + 0.850548i \(0.676270\pi\)
\(542\) −2.14359 −0.0920752
\(543\) 0 0
\(544\) −16.0000 −0.685994
\(545\) −6.07180 −0.260087
\(546\) 0 0
\(547\) −33.8564 −1.44760 −0.723798 0.690012i \(-0.757605\pi\)
−0.723798 + 0.690012i \(0.757605\pi\)
\(548\) −3.71281 −0.158604
\(549\) 0 0
\(550\) −4.19615 −0.178925
\(551\) 14.2679 0.607835
\(552\) 0 0
\(553\) −73.1769 −3.11180
\(554\) 2.78461 0.118307
\(555\) 0 0
\(556\) −0.889727 −0.0377328
\(557\) −2.53590 −0.107449 −0.0537247 0.998556i \(-0.517109\pi\)
−0.0537247 + 0.998556i \(0.517109\pi\)
\(558\) 0 0
\(559\) 0.287187 0.0121467
\(560\) −5.07180 −0.214323
\(561\) 0 0
\(562\) 11.3205 0.477527
\(563\) −16.7321 −0.705172 −0.352586 0.935779i \(-0.614698\pi\)
−0.352586 + 0.935779i \(0.614698\pi\)
\(564\) 0 0
\(565\) −19.1244 −0.804568
\(566\) −21.4641 −0.902203
\(567\) 0 0
\(568\) 9.46410 0.397105
\(569\) −32.9090 −1.37962 −0.689808 0.723993i \(-0.742305\pi\)
−0.689808 + 0.723993i \(0.742305\pi\)
\(570\) 0 0
\(571\) −39.7846 −1.66493 −0.832467 0.554075i \(-0.813072\pi\)
−0.832467 + 0.554075i \(0.813072\pi\)
\(572\) 12.2872 0.513753
\(573\) 0 0
\(574\) 24.9282 1.04048
\(575\) −3.46410 −0.144463
\(576\) 0 0
\(577\) 15.2679 0.635613 0.317807 0.948156i \(-0.397054\pi\)
0.317807 + 0.948156i \(0.397054\pi\)
\(578\) −6.98076 −0.290361
\(579\) 0 0
\(580\) −4.67949 −0.194305
\(581\) −10.3923 −0.431145
\(582\) 0 0
\(583\) −38.5885 −1.59817
\(584\) 19.4256 0.803838
\(585\) 0 0
\(586\) 21.0333 0.868878
\(587\) −11.6603 −0.481270 −0.240635 0.970616i \(-0.577356\pi\)
−0.240635 + 0.970616i \(0.577356\pi\)
\(588\) 0 0
\(589\) −13.3923 −0.551820
\(590\) −6.05256 −0.249180
\(591\) 0 0
\(592\) −2.92820 −0.120348
\(593\) −0.143594 −0.00589668 −0.00294834 0.999996i \(-0.500938\pi\)
−0.00294834 + 0.999996i \(0.500938\pi\)
\(594\) 0 0
\(595\) 12.9282 0.530005
\(596\) 11.7128 0.479776
\(597\) 0 0
\(598\) −3.71281 −0.151828
\(599\) 27.1962 1.11120 0.555602 0.831448i \(-0.312488\pi\)
0.555602 + 0.831448i \(0.312488\pi\)
\(600\) 0 0
\(601\) −31.2487 −1.27466 −0.637331 0.770590i \(-0.719961\pi\)
−0.637331 + 0.770590i \(0.719961\pi\)
\(602\) −0.679492 −0.0276940
\(603\) 0 0
\(604\) −7.89488 −0.321238
\(605\) 21.8564 0.888589
\(606\) 0 0
\(607\) −16.1962 −0.657382 −0.328691 0.944438i \(-0.606607\pi\)
−0.328691 + 0.944438i \(0.606607\pi\)
\(608\) 26.1436 1.06026
\(609\) 0 0
\(610\) 2.92820 0.118559
\(611\) −12.7846 −0.517210
\(612\) 0 0
\(613\) 1.46410 0.0591345 0.0295673 0.999563i \(-0.490587\pi\)
0.0295673 + 0.999563i \(0.490587\pi\)
\(614\) 10.2872 0.415157
\(615\) 0 0
\(616\) −68.7846 −2.77141
\(617\) −6.92820 −0.278919 −0.139459 0.990228i \(-0.544536\pi\)
−0.139459 + 0.990228i \(0.544536\pi\)
\(618\) 0 0
\(619\) −11.8564 −0.476549 −0.238275 0.971198i \(-0.576582\pi\)
−0.238275 + 0.971198i \(0.576582\pi\)
\(620\) 4.39230 0.176399
\(621\) 0 0
\(622\) 14.4449 0.579186
\(623\) −24.5885 −0.985116
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −6.64102 −0.265428
\(627\) 0 0
\(628\) 28.0000 1.11732
\(629\) 7.46410 0.297613
\(630\) 0 0
\(631\) −32.7128 −1.30228 −0.651138 0.758959i \(-0.725708\pi\)
−0.651138 + 0.758959i \(0.725708\pi\)
\(632\) −39.2154 −1.55990
\(633\) 0 0
\(634\) 4.53590 0.180144
\(635\) −14.5885 −0.578925
\(636\) 0 0
\(637\) 22.5359 0.892905
\(638\) −13.4115 −0.530968
\(639\) 0 0
\(640\) −10.1436 −0.400961
\(641\) 9.33975 0.368898 0.184449 0.982842i \(-0.440950\pi\)
0.184449 + 0.982842i \(0.440950\pi\)
\(642\) 0 0
\(643\) 41.6603 1.64292 0.821460 0.570266i \(-0.193160\pi\)
0.821460 + 0.570266i \(0.193160\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) −8.92820 −0.351275
\(647\) 11.4641 0.450700 0.225350 0.974278i \(-0.427647\pi\)
0.225350 + 0.974278i \(0.427647\pi\)
\(648\) 0 0
\(649\) 47.3923 1.86031
\(650\) 1.07180 0.0420393
\(651\) 0 0
\(652\) −18.6410 −0.730039
\(653\) −17.4641 −0.683423 −0.341712 0.939805i \(-0.611007\pi\)
−0.341712 + 0.939805i \(0.611007\pi\)
\(654\) 0 0
\(655\) 15.5885 0.609091
\(656\) −7.71281 −0.301135
\(657\) 0 0
\(658\) 30.2487 1.17922
\(659\) −1.46410 −0.0570333 −0.0285167 0.999593i \(-0.509078\pi\)
−0.0285167 + 0.999593i \(0.509078\pi\)
\(660\) 0 0
\(661\) 18.3205 0.712585 0.356293 0.934374i \(-0.384041\pi\)
0.356293 + 0.934374i \(0.384041\pi\)
\(662\) −0.339746 −0.0132046
\(663\) 0 0
\(664\) −5.56922 −0.216128
\(665\) −21.1244 −0.819167
\(666\) 0 0
\(667\) −11.0718 −0.428702
\(668\) 25.8564 1.00041
\(669\) 0 0
\(670\) 2.53590 0.0979703
\(671\) −22.9282 −0.885133
\(672\) 0 0
\(673\) −10.3923 −0.400594 −0.200297 0.979735i \(-0.564191\pi\)
−0.200297 + 0.979735i \(0.564191\pi\)
\(674\) 20.0000 0.770371
\(675\) 0 0
\(676\) 15.8949 0.611342
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 45.7128 1.75430
\(680\) 6.92820 0.265684
\(681\) 0 0
\(682\) 12.5885 0.482037
\(683\) −19.6077 −0.750268 −0.375134 0.926971i \(-0.622403\pi\)
−0.375134 + 0.926971i \(0.622403\pi\)
\(684\) 0 0
\(685\) 2.53590 0.0968917
\(686\) −29.0718 −1.10997
\(687\) 0 0
\(688\) 0.210236 0.00801515
\(689\) 9.85641 0.375499
\(690\) 0 0
\(691\) −17.7128 −0.673827 −0.336914 0.941536i \(-0.609383\pi\)
−0.336914 + 0.941536i \(0.609383\pi\)
\(692\) −12.4974 −0.475081
\(693\) 0 0
\(694\) −20.9282 −0.794424
\(695\) 0.607695 0.0230512
\(696\) 0 0
\(697\) 19.6603 0.744685
\(698\) −13.8038 −0.522483
\(699\) 0 0
\(700\) 6.92820 0.261861
\(701\) −20.8038 −0.785750 −0.392875 0.919592i \(-0.628520\pi\)
−0.392875 + 0.919592i \(0.628520\pi\)
\(702\) 0 0
\(703\) −12.1962 −0.459987
\(704\) −12.2872 −0.463091
\(705\) 0 0
\(706\) 18.6795 0.703012
\(707\) 12.5885 0.473438
\(708\) 0 0
\(709\) −22.5359 −0.846353 −0.423177 0.906047i \(-0.639085\pi\)
−0.423177 + 0.906047i \(0.639085\pi\)
\(710\) −2.73205 −0.102532
\(711\) 0 0
\(712\) −13.1769 −0.493826
\(713\) 10.3923 0.389195
\(714\) 0 0
\(715\) −8.39230 −0.313854
\(716\) −11.8949 −0.444533
\(717\) 0 0
\(718\) −4.48334 −0.167317
\(719\) 8.41154 0.313698 0.156849 0.987623i \(-0.449866\pi\)
0.156849 + 0.987623i \(0.449866\pi\)
\(720\) 0 0
\(721\) 2.53590 0.0944418
\(722\) 0.679492 0.0252881
\(723\) 0 0
\(724\) −38.7461 −1.43999
\(725\) 3.19615 0.118702
\(726\) 0 0
\(727\) −8.39230 −0.311253 −0.155627 0.987816i \(-0.549740\pi\)
−0.155627 + 0.987816i \(0.549740\pi\)
\(728\) 17.5692 0.651159
\(729\) 0 0
\(730\) −5.60770 −0.207550
\(731\) −0.535898 −0.0198209
\(732\) 0 0
\(733\) 34.7846 1.28480 0.642399 0.766370i \(-0.277939\pi\)
0.642399 + 0.766370i \(0.277939\pi\)
\(734\) 22.8231 0.842415
\(735\) 0 0
\(736\) −20.2872 −0.747796
\(737\) −19.8564 −0.731420
\(738\) 0 0
\(739\) 22.4641 0.826355 0.413178 0.910650i \(-0.364419\pi\)
0.413178 + 0.910650i \(0.364419\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) −23.3205 −0.856123
\(743\) 19.9090 0.730389 0.365195 0.930931i \(-0.381002\pi\)
0.365195 + 0.930931i \(0.381002\pi\)
\(744\) 0 0
\(745\) −8.00000 −0.293097
\(746\) 14.6795 0.537454
\(747\) 0 0
\(748\) −22.9282 −0.838338
\(749\) 40.3923 1.47590
\(750\) 0 0
\(751\) 48.7846 1.78018 0.890088 0.455789i \(-0.150643\pi\)
0.890088 + 0.455789i \(0.150643\pi\)
\(752\) −9.35898 −0.341287
\(753\) 0 0
\(754\) 3.42563 0.124754
\(755\) 5.39230 0.196246
\(756\) 0 0
\(757\) −9.17691 −0.333541 −0.166770 0.985996i \(-0.553334\pi\)
−0.166770 + 0.985996i \(0.553334\pi\)
\(758\) 1.75129 0.0636097
\(759\) 0 0
\(760\) −11.3205 −0.410638
\(761\) 23.4449 0.849876 0.424938 0.905223i \(-0.360296\pi\)
0.424938 + 0.905223i \(0.360296\pi\)
\(762\) 0 0
\(763\) 28.7321 1.04017
\(764\) 11.8949 0.430342
\(765\) 0 0
\(766\) 1.85641 0.0670747
\(767\) −12.1051 −0.437090
\(768\) 0 0
\(769\) −9.53590 −0.343873 −0.171937 0.985108i \(-0.555002\pi\)
−0.171937 + 0.985108i \(0.555002\pi\)
\(770\) 19.8564 0.715575
\(771\) 0 0
\(772\) 7.71281 0.277590
\(773\) 1.51666 0.0545505 0.0272752 0.999628i \(-0.491317\pi\)
0.0272752 + 0.999628i \(0.491317\pi\)
\(774\) 0 0
\(775\) −3.00000 −0.107763
\(776\) 24.4974 0.879406
\(777\) 0 0
\(778\) 20.1051 0.720803
\(779\) −32.1244 −1.15097
\(780\) 0 0
\(781\) 21.3923 0.765477
\(782\) 6.92820 0.247752
\(783\) 0 0
\(784\) 16.4974 0.589194
\(785\) −19.1244 −0.682578
\(786\) 0 0
\(787\) −48.0526 −1.71289 −0.856444 0.516239i \(-0.827332\pi\)
−0.856444 + 0.516239i \(0.827332\pi\)
\(788\) 20.2872 0.722701
\(789\) 0 0
\(790\) 11.3205 0.402766
\(791\) 90.4974 3.21772
\(792\) 0 0
\(793\) 5.85641 0.207967
\(794\) 10.5359 0.373905
\(795\) 0 0
\(796\) 2.92820 0.103787
\(797\) −15.6077 −0.552853 −0.276426 0.961035i \(-0.589150\pi\)
−0.276426 + 0.961035i \(0.589150\pi\)
\(798\) 0 0
\(799\) 23.8564 0.843979
\(800\) 5.85641 0.207055
\(801\) 0 0
\(802\) 18.2487 0.644384
\(803\) 43.9090 1.54951
\(804\) 0 0
\(805\) 16.3923 0.577753
\(806\) −3.21539 −0.113257
\(807\) 0 0
\(808\) 6.74613 0.237328
\(809\) 45.4449 1.59776 0.798878 0.601493i \(-0.205427\pi\)
0.798878 + 0.601493i \(0.205427\pi\)
\(810\) 0 0
\(811\) −18.4641 −0.648362 −0.324181 0.945995i \(-0.605089\pi\)
−0.324181 + 0.945995i \(0.605089\pi\)
\(812\) 22.1436 0.777088
\(813\) 0 0
\(814\) 11.4641 0.401817
\(815\) 12.7321 0.445984
\(816\) 0 0
\(817\) 0.875644 0.0306349
\(818\) −13.0718 −0.457045
\(819\) 0 0
\(820\) 10.5359 0.367930
\(821\) −37.7321 −1.31686 −0.658429 0.752643i \(-0.728779\pi\)
−0.658429 + 0.752643i \(0.728779\pi\)
\(822\) 0 0
\(823\) 3.85641 0.134426 0.0672129 0.997739i \(-0.478589\pi\)
0.0672129 + 0.997739i \(0.478589\pi\)
\(824\) 1.35898 0.0473424
\(825\) 0 0
\(826\) 28.6410 0.996548
\(827\) −11.6077 −0.403639 −0.201820 0.979423i \(-0.564685\pi\)
−0.201820 + 0.979423i \(0.564685\pi\)
\(828\) 0 0
\(829\) −23.7846 −0.826074 −0.413037 0.910714i \(-0.635532\pi\)
−0.413037 + 0.910714i \(0.635532\pi\)
\(830\) 1.60770 0.0558039
\(831\) 0 0
\(832\) 3.13844 0.108806
\(833\) −42.0526 −1.45703
\(834\) 0 0
\(835\) −17.6603 −0.611158
\(836\) 37.4641 1.29572
\(837\) 0 0
\(838\) 14.9282 0.515686
\(839\) 33.1962 1.14606 0.573029 0.819535i \(-0.305768\pi\)
0.573029 + 0.819535i \(0.305768\pi\)
\(840\) 0 0
\(841\) −18.7846 −0.647745
\(842\) 24.7321 0.852323
\(843\) 0 0
\(844\) 12.9667 0.446331
\(845\) −10.8564 −0.373472
\(846\) 0 0
\(847\) −103.426 −3.55375
\(848\) 7.21539 0.247778
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 9.46410 0.324425
\(852\) 0 0
\(853\) 10.4833 0.358943 0.179471 0.983763i \(-0.442561\pi\)
0.179471 + 0.983763i \(0.442561\pi\)
\(854\) −13.8564 −0.474156
\(855\) 0 0
\(856\) 21.6462 0.739851
\(857\) 44.4974 1.52000 0.760002 0.649921i \(-0.225198\pi\)
0.760002 + 0.649921i \(0.225198\pi\)
\(858\) 0 0
\(859\) −11.0000 −0.375315 −0.187658 0.982235i \(-0.560090\pi\)
−0.187658 + 0.982235i \(0.560090\pi\)
\(860\) −0.287187 −0.00979300
\(861\) 0 0
\(862\) −15.6218 −0.532080
\(863\) −23.1244 −0.787162 −0.393581 0.919290i \(-0.628764\pi\)
−0.393581 + 0.919290i \(0.628764\pi\)
\(864\) 0 0
\(865\) 8.53590 0.290229
\(866\) −25.9615 −0.882209
\(867\) 0 0
\(868\) −20.7846 −0.705476
\(869\) −88.6410 −3.00694
\(870\) 0 0
\(871\) 5.07180 0.171851
\(872\) 15.3975 0.521424
\(873\) 0 0
\(874\) −11.3205 −0.382922
\(875\) −4.73205 −0.159973
\(876\) 0 0
\(877\) 33.4641 1.13000 0.565001 0.825090i \(-0.308876\pi\)
0.565001 + 0.825090i \(0.308876\pi\)
\(878\) 3.94744 0.133220
\(879\) 0 0
\(880\) −6.14359 −0.207100
\(881\) −47.0526 −1.58524 −0.792620 0.609715i \(-0.791284\pi\)
−0.792620 + 0.609715i \(0.791284\pi\)
\(882\) 0 0
\(883\) −30.1962 −1.01618 −0.508091 0.861304i \(-0.669649\pi\)
−0.508091 + 0.861304i \(0.669649\pi\)
\(884\) 5.85641 0.196972
\(885\) 0 0
\(886\) −0.248711 −0.00835562
\(887\) 33.8038 1.13502 0.567511 0.823366i \(-0.307906\pi\)
0.567511 + 0.823366i \(0.307906\pi\)
\(888\) 0 0
\(889\) 69.0333 2.31530
\(890\) 3.80385 0.127505
\(891\) 0 0
\(892\) −24.5744 −0.822811
\(893\) −38.9808 −1.30444
\(894\) 0 0
\(895\) 8.12436 0.271567
\(896\) 48.0000 1.60357
\(897\) 0 0
\(898\) 5.94744 0.198469
\(899\) −9.58846 −0.319793
\(900\) 0 0
\(901\) −18.3923 −0.612737
\(902\) 30.1962 1.00542
\(903\) 0 0
\(904\) 48.4974 1.61300
\(905\) 26.4641 0.879697
\(906\) 0 0
\(907\) −30.0000 −0.996134 −0.498067 0.867139i \(-0.665957\pi\)
−0.498067 + 0.867139i \(0.665957\pi\)
\(908\) 26.4308 0.877136
\(909\) 0 0
\(910\) −5.07180 −0.168128
\(911\) 23.5885 0.781520 0.390760 0.920493i \(-0.372212\pi\)
0.390760 + 0.920493i \(0.372212\pi\)
\(912\) 0 0
\(913\) −12.5885 −0.416617
\(914\) 2.00000 0.0661541
\(915\) 0 0
\(916\) 17.5692 0.580503
\(917\) −73.7654 −2.43595
\(918\) 0 0
\(919\) −56.9615 −1.87899 −0.939494 0.342566i \(-0.888704\pi\)
−0.939494 + 0.342566i \(0.888704\pi\)
\(920\) 8.78461 0.289620
\(921\) 0 0
\(922\) −27.1244 −0.893293
\(923\) −5.46410 −0.179853
\(924\) 0 0
\(925\) −2.73205 −0.0898293
\(926\) −7.60770 −0.250004
\(927\) 0 0
\(928\) 18.7180 0.614447
\(929\) −44.3731 −1.45583 −0.727917 0.685666i \(-0.759511\pi\)
−0.727917 + 0.685666i \(0.759511\pi\)
\(930\) 0 0
\(931\) 68.7128 2.25197
\(932\) 41.0718 1.34535
\(933\) 0 0
\(934\) 27.3590 0.895213
\(935\) 15.6603 0.512145
\(936\) 0 0
\(937\) −4.14359 −0.135365 −0.0676827 0.997707i \(-0.521561\pi\)
−0.0676827 + 0.997707i \(0.521561\pi\)
\(938\) −12.0000 −0.391814
\(939\) 0 0
\(940\) 12.7846 0.416988
\(941\) 35.1769 1.14673 0.573367 0.819298i \(-0.305637\pi\)
0.573367 + 0.819298i \(0.305637\pi\)
\(942\) 0 0
\(943\) 24.9282 0.811774
\(944\) −8.86156 −0.288419
\(945\) 0 0
\(946\) −0.823085 −0.0267608
\(947\) −57.7128 −1.87541 −0.937707 0.347427i \(-0.887056\pi\)
−0.937707 + 0.347427i \(0.887056\pi\)
\(948\) 0 0
\(949\) −11.2154 −0.364067
\(950\) 3.26795 0.106026
\(951\) 0 0
\(952\) −32.7846 −1.06256
\(953\) 36.3923 1.17886 0.589431 0.807819i \(-0.299352\pi\)
0.589431 + 0.807819i \(0.299352\pi\)
\(954\) 0 0
\(955\) −8.12436 −0.262898
\(956\) 0.784610 0.0253761
\(957\) 0 0
\(958\) −8.69358 −0.280877
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −2.92820 −0.0944091
\(963\) 0 0
\(964\) 23.8949 0.769602
\(965\) −5.26795 −0.169581
\(966\) 0 0
\(967\) −25.8038 −0.829796 −0.414898 0.909868i \(-0.636183\pi\)
−0.414898 + 0.909868i \(0.636183\pi\)
\(968\) −55.4256 −1.78145
\(969\) 0 0
\(970\) −7.07180 −0.227062
\(971\) 17.4449 0.559832 0.279916 0.960024i \(-0.409693\pi\)
0.279916 + 0.960024i \(0.409693\pi\)
\(972\) 0 0
\(973\) −2.87564 −0.0921889
\(974\) 17.8564 0.572156
\(975\) 0 0
\(976\) 4.28719 0.137230
\(977\) 5.46410 0.174812 0.0874060 0.996173i \(-0.472142\pi\)
0.0874060 + 0.996173i \(0.472142\pi\)
\(978\) 0 0
\(979\) −29.7846 −0.951920
\(980\) −22.5359 −0.719883
\(981\) 0 0
\(982\) −10.1577 −0.324144
\(983\) 48.5885 1.54973 0.774866 0.632126i \(-0.217818\pi\)
0.774866 + 0.632126i \(0.217818\pi\)
\(984\) 0 0
\(985\) −13.8564 −0.441502
\(986\) −6.39230 −0.203572
\(987\) 0 0
\(988\) −9.56922 −0.304437
\(989\) −0.679492 −0.0216066
\(990\) 0 0
\(991\) 30.8564 0.980186 0.490093 0.871670i \(-0.336963\pi\)
0.490093 + 0.871670i \(0.336963\pi\)
\(992\) −17.5692 −0.557823
\(993\) 0 0
\(994\) 12.9282 0.410058
\(995\) −2.00000 −0.0634043
\(996\) 0 0
\(997\) 25.5167 0.808121 0.404060 0.914732i \(-0.367599\pi\)
0.404060 + 0.914732i \(0.367599\pi\)
\(998\) 17.8038 0.563571
\(999\) 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 405.2.a.g.1.2 2
3.2 odd 2 405.2.a.h.1.1 yes 2
4.3 odd 2 6480.2.a.br.1.2 2
5.2 odd 4 2025.2.b.g.649.3 4
5.3 odd 4 2025.2.b.g.649.2 4
5.4 even 2 2025.2.a.m.1.1 2
9.2 odd 6 405.2.e.i.271.2 4
9.4 even 3 405.2.e.l.136.1 4
9.5 odd 6 405.2.e.i.136.2 4
9.7 even 3 405.2.e.l.271.1 4
12.11 even 2 6480.2.a.bi.1.2 2
15.2 even 4 2025.2.b.h.649.2 4
15.8 even 4 2025.2.b.h.649.3 4
15.14 odd 2 2025.2.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.g.1.2 2 1.1 even 1 trivial
405.2.a.h.1.1 yes 2 3.2 odd 2
405.2.e.i.136.2 4 9.5 odd 6
405.2.e.i.271.2 4 9.2 odd 6
405.2.e.l.136.1 4 9.4 even 3
405.2.e.l.271.1 4 9.7 even 3
2025.2.a.g.1.2 2 15.14 odd 2
2025.2.a.m.1.1 2 5.4 even 2
2025.2.b.g.649.2 4 5.3 odd 4
2025.2.b.g.649.3 4 5.2 odd 4
2025.2.b.h.649.2 4 15.2 even 4
2025.2.b.h.649.3 4 15.8 even 4
6480.2.a.bi.1.2 2 12.11 even 2
6480.2.a.br.1.2 2 4.3 odd 2