Properties

Label 405.2.a.h.1.1
Level $405$
Weight $2$
Character 405.1
Self dual yes
Analytic conductor $3.234$
Analytic rank $0$
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: \(0\)
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 1.1
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 \(-0.583333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(3\) 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 \(-0.167864\pi\)
0.864139 + 0.503253i \(0.167864\pi\)
\(12\) 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 \(-0.392509\pi\)
0.331310 + 0.943522i \(0.392509\pi\)
\(18\) 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 \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 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 \(-0.595905\pi\)
−0.296755 + 0.954954i \(0.595905\pi\)
\(30\) 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 \(-0.310061\pi\)
0.561925 + 0.827188i \(0.310061\pi\)
\(42\) 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 \(-0.280237\pi\)
0.636850 + 0.770988i \(0.280237\pi\)
\(48\) 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 \(-0.652997\pi\)
−0.462359 + 0.886693i \(0.652997\pi\)
\(54\) 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 \(-0.319105\pi\)
0.538198 + 0.842819i \(0.319105\pi\)
\(60\) 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 \(-0.428919\pi\)
0.221456 + 0.975170i \(0.428919\pi\)
\(72\) 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 \(-0.538459\pi\)
−0.120530 + 0.992710i \(0.538459\pi\)
\(84\) 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 \(-0.588809\pi\)
−0.275396 + 0.961331i \(0.588809\pi\)
\(90\) 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 \(-0.457747\pi\)
0.132353 + 0.991203i \(0.457747\pi\)
\(102\) 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 \(-0.364621\pi\)
0.412598 + 0.910913i \(0.364621\pi\)
\(108\) 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 \(-0.143906\pi\)
0.899534 + 0.436851i \(0.143906\pi\)
\(114\) 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 \(-0.738448\pi\)
−0.680985 + 0.732297i \(0.738448\pi\)
\(132\) 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 \(-0.534550\pi\)
−0.108328 + 0.994115i \(0.534550\pi\)
\(138\) 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 \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(150\) 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 \(-0.260546\pi\)
0.683296 + 0.730142i \(0.260546\pi\)
\(168\) 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 \(-0.605191\pi\)
−0.324486 + 0.945890i \(0.605191\pi\)
\(174\) 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 \(-0.598196\pi\)
−0.303621 + 0.952793i \(0.598196\pi\)
\(180\) 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 \(-0.405037\pi\)
0.293929 + 0.955827i \(0.405037\pi\)
\(192\) 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 \(-0.335676\pi\)
0.493614 + 0.869681i \(0.335676\pi\)
\(198\) 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 \(-0.295527\pi\)
0.599095 + 0.800678i \(0.295527\pi\)
\(228\) 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 \(-0.129085\pi\)
0.918892 + 0.394509i \(0.129085\pi\)
\(234\) 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 \(-0.494483\pi\)
0.0173322 + 0.999850i \(0.494483\pi\)
\(240\) 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 \(-0.393633\pi\)
0.327978 + 0.944685i \(0.393633\pi\)
\(252\) 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 \(-0.605071\pi\)
−0.324127 + 0.946014i \(0.605071\pi\)
\(258\) 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 \(-0.728318\pi\)
−0.657339 + 0.753595i \(0.728318\pi\)
\(264\) 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 \(-0.605359\pi\)
−0.324984 + 0.945720i \(0.605359\pi\)
\(270\) 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 \(-0.652601\pi\)
−0.461255 + 0.887267i \(0.652601\pi\)
\(282\) 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 \(-0.817019\pi\)
−0.839272 + 0.543712i \(0.817019\pi\)
\(294\) 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 \(-0.688988\pi\)
−0.559451 + 0.828863i \(0.688988\pi\)
\(312\) 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 \(-0.555671\pi\)
−0.174005 + 0.984745i \(0.555671\pi\)
\(318\) 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 \(-0.221573\pi\)
0.767354 + 0.641223i \(0.221573\pi\)
\(348\) 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 \(-0.737611\pi\)
−0.679057 + 0.734085i \(0.737611\pi\)
\(354\) 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 \(-0.448330\pi\)
0.161616 + 0.986854i \(0.448330\pi\)
\(360\) 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 \(-0.520637\pi\)
−0.0647892 + 0.997899i \(0.520637\pi\)
\(384\) 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 \(-0.745146\pi\)
−0.696243 + 0.717807i \(0.745146\pi\)
\(390\) 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 \(-0.713853\pi\)
−0.622428 + 0.782677i \(0.713853\pi\)
\(402\) 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 \(-0.665974\pi\)
−0.498115 + 0.867111i \(0.665974\pi\)
\(420\) 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 \(-0.328182\pi\)
0.513950 + 0.857820i \(0.328182\pi\)
\(432\) 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 \(-0.497431\pi\)
0.00807091 + 0.999967i \(0.497431\pi\)
\(444\) 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 \(-0.561402\pi\)
−0.191706 + 0.981452i \(0.561402\pi\)
\(450\) 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 \(-0.168674\pi\)
0.862855 + 0.505452i \(0.168674\pi\)
\(462\) 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 \(-0.832498\pi\)
−0.864710 + 0.502272i \(0.832498\pi\)
\(468\) 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 \(-0.412544\pi\)
0.271306 + 0.962493i \(0.412544\pi\)
\(480\) 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 \(-0.398633\pi\)
0.313100 + 0.949720i \(0.398633\pi\)
\(492\) 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 \(-0.447818\pi\)
0.163203 + 0.986593i \(0.447818\pi\)
\(504\) 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 \(-0.451956\pi\)
0.150361 + 0.988631i \(0.451956\pi\)
\(510\) 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 \(-0.640207\pi\)
−0.426369 + 0.904549i \(0.640207\pi\)
\(522\) 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 \(-0.482891\pi\)
0.0537247 + 0.998556i \(0.482891\pi\)
\(558\) 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 \(-0.385302\pi\)
0.352586 + 0.935779i \(0.385302\pi\)
\(564\) 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 \(-0.257695\pi\)
0.689808 + 0.723993i \(0.257695\pi\)
\(570\) 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 \(-0.422644\pi\)
0.240635 + 0.970616i \(0.422644\pi\)
\(588\) 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 \(-0.499062\pi\)
0.00294834 + 0.999996i \(0.499062\pi\)
\(594\) 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 \(-0.687512\pi\)
−0.555602 + 0.831448i \(0.687512\pi\)
\(600\) 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 \(-0.455464\pi\)
0.139459 + 0.990228i \(0.455464\pi\)
\(618\) 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 \(-0.559050\pi\)
−0.184449 + 0.982842i \(0.559050\pi\)
\(642\) 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 \(-0.572353\pi\)
−0.225350 + 0.974278i \(0.572353\pi\)
\(648\) 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 \(-0.388993\pi\)
0.341712 + 0.939805i \(0.388993\pi\)
\(654\) 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 \(-0.490922\pi\)
0.0285167 + 0.999593i \(0.490922\pi\)
\(660\) 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 \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 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 \(-0.377597\pi\)
0.375134 + 0.926971i \(0.377597\pi\)
\(684\) 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 \(-0.371480\pi\)
0.392875 + 0.919592i \(0.371480\pi\)
\(702\) 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 \(-0.550134\pi\)
−0.156849 + 0.987623i \(0.550134\pi\)
\(720\) 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 \(-0.618998\pi\)
−0.365195 + 0.930931i \(0.618998\pi\)
\(744\) 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 \(-0.639704\pi\)
−0.424938 + 0.905223i \(0.639704\pi\)
\(762\) 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 \(-0.508683\pi\)
−0.0272752 + 0.999628i \(0.508683\pi\)
\(774\) 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 \(-0.410850\pi\)
0.276426 + 0.961035i \(0.410850\pi\)
\(798\) 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 \(-0.794573\pi\)
−0.798878 + 0.601493i \(0.794573\pi\)
\(810\) 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 \(-0.271221\pi\)
0.658429 + 0.752643i \(0.271221\pi\)
\(822\) 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 \(-0.435315\pi\)
0.201820 + 0.979423i \(0.435315\pi\)
\(828\) 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 \(-0.694232\pi\)
−0.573029 + 0.819535i \(0.694232\pi\)
\(840\) 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 \(-0.774802\pi\)
−0.760002 + 0.649921i \(0.774802\pi\)
\(858\) 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 \(-0.371236\pi\)
0.393581 + 0.919290i \(0.371236\pi\)
\(864\) 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 \(-0.208716\pi\)
0.792620 + 0.609715i \(0.208716\pi\)
\(882\) 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 \(-0.692094\pi\)
−0.567511 + 0.823366i \(0.692094\pi\)
\(888\) 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 \(-0.627788\pi\)
−0.390760 + 0.920493i \(0.627788\pi\)
\(912\) 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 \(-0.240489\pi\)
0.727917 + 0.685666i \(0.240489\pi\)
\(930\) 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 \(-0.694363\pi\)
−0.573367 + 0.819298i \(0.694363\pi\)
\(942\) 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 \(-0.112944\pi\)
0.937707 + 0.347427i \(0.112944\pi\)
\(948\) 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 \(-0.700648\pi\)
−0.589431 + 0.807819i \(0.700648\pi\)
\(954\) 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 \(-0.590307\pi\)
−0.279916 + 0.960024i \(0.590307\pi\)
\(972\) 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 \(-0.527858\pi\)
−0.0874060 + 0.996173i \(0.527858\pi\)
\(978\) 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 \(-0.782182\pi\)
−0.774866 + 0.632126i \(0.782182\pi\)
\(984\) 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.h.1.1 yes 2
3.2 odd 2 405.2.a.g.1.2 2
4.3 odd 2 6480.2.a.bi.1.2 2
5.2 odd 4 2025.2.b.h.649.2 4
5.3 odd 4 2025.2.b.h.649.3 4
5.4 even 2 2025.2.a.g.1.2 2
9.2 odd 6 405.2.e.l.271.1 4
9.4 even 3 405.2.e.i.136.2 4
9.5 odd 6 405.2.e.l.136.1 4
9.7 even 3 405.2.e.i.271.2 4
12.11 even 2 6480.2.a.br.1.2 2
15.2 even 4 2025.2.b.g.649.3 4
15.8 even 4 2025.2.b.g.649.2 4
15.14 odd 2 2025.2.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.g.1.2 2 3.2 odd 2
405.2.a.h.1.1 yes 2 1.1 even 1 trivial
405.2.e.i.136.2 4 9.4 even 3
405.2.e.i.271.2 4 9.7 even 3
405.2.e.l.136.1 4 9.5 odd 6
405.2.e.l.271.1 4 9.2 odd 6
2025.2.a.g.1.2 2 5.4 even 2
2025.2.a.m.1.1 2 15.14 odd 2
2025.2.b.g.649.2 4 15.8 even 4
2025.2.b.g.649.3 4 15.2 even 4
2025.2.b.h.649.2 4 5.2 odd 4
2025.2.b.h.649.3 4 5.3 odd 4
6480.2.a.bi.1.2 2 4.3 odd 2
6480.2.a.br.1.2 2 12.11 even 2