Properties

Label 405.2.a.h.1.2
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 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.73205 q^{2} +5.46410 q^{4} -1.00000 q^{5} -1.26795 q^{7} +9.46410 q^{8} +O(q^{10})\) \(q+2.73205 q^{2} +5.46410 q^{4} -1.00000 q^{5} -1.26795 q^{7} +9.46410 q^{8} -2.73205 q^{10} +2.26795 q^{11} -5.46410 q^{13} -3.46410 q^{14} +14.9282 q^{16} -0.732051 q^{17} -2.46410 q^{19} -5.46410 q^{20} +6.19615 q^{22} -3.46410 q^{23} +1.00000 q^{25} -14.9282 q^{26} -6.92820 q^{28} +7.19615 q^{29} -3.00000 q^{31} +21.8564 q^{32} -2.00000 q^{34} +1.26795 q^{35} +0.732051 q^{37} -6.73205 q^{38} -9.46410 q^{40} -3.19615 q^{41} -10.1962 q^{43} +12.3923 q^{44} -9.46410 q^{46} +5.26795 q^{47} -5.39230 q^{49} +2.73205 q^{50} -29.8564 q^{52} -3.26795 q^{53} -2.26795 q^{55} -12.0000 q^{56} +19.6603 q^{58} +11.7321 q^{59} +4.00000 q^{61} -8.19615 q^{62} +29.8564 q^{64} +5.46410 q^{65} -3.46410 q^{67} -4.00000 q^{68} +3.46410 q^{70} +0.267949 q^{71} +9.66025 q^{73} +2.00000 q^{74} -13.4641 q^{76} -2.87564 q^{77} +8.53590 q^{79} -14.9282 q^{80} -8.73205 q^{82} +8.19615 q^{83} +0.732051 q^{85} -27.8564 q^{86} +21.4641 q^{88} +5.19615 q^{89} +6.92820 q^{91} -18.9282 q^{92} +14.3923 q^{94} +2.46410 q^{95} +7.66025 q^{97} -14.7321 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.73205 1.93185 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(3\) 0 0
\(4\) 5.46410 2.73205
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.26795 −0.479240 −0.239620 0.970867i \(-0.577023\pi\)
−0.239620 + 0.970867i \(0.577023\pi\)
\(8\) 9.46410 3.34607
\(9\) 0 0
\(10\) −2.73205 −0.863950
\(11\) 2.26795 0.683812 0.341906 0.939734i \(-0.388927\pi\)
0.341906 + 0.939734i \(0.388927\pi\)
\(12\) 0 0
\(13\) −5.46410 −1.51547 −0.757735 0.652563i \(-0.773694\pi\)
−0.757735 + 0.652563i \(0.773694\pi\)
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) 14.9282 3.73205
\(17\) −0.732051 −0.177548 −0.0887742 0.996052i \(-0.528295\pi\)
−0.0887742 + 0.996052i \(0.528295\pi\)
\(18\) 0 0
\(19\) −2.46410 −0.565304 −0.282652 0.959223i \(-0.591214\pi\)
−0.282652 + 0.959223i \(0.591214\pi\)
\(20\) −5.46410 −1.22181
\(21\) 0 0
\(22\) 6.19615 1.32102
\(23\) −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
\(26\) −14.9282 −2.92766
\(27\) 0 0
\(28\) −6.92820 −1.30931
\(29\) 7.19615 1.33629 0.668146 0.744030i \(-0.267088\pi\)
0.668146 + 0.744030i \(0.267088\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 21.8564 3.86370
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 1.26795 0.214323
\(36\) 0 0
\(37\) 0.732051 0.120348 0.0601742 0.998188i \(-0.480834\pi\)
0.0601742 + 0.998188i \(0.480834\pi\)
\(38\) −6.73205 −1.09208
\(39\) 0 0
\(40\) −9.46410 −1.49641
\(41\) −3.19615 −0.499155 −0.249578 0.968355i \(-0.580292\pi\)
−0.249578 + 0.968355i \(0.580292\pi\)
\(42\) 0 0
\(43\) −10.1962 −1.55490 −0.777449 0.628946i \(-0.783487\pi\)
−0.777449 + 0.628946i \(0.783487\pi\)
\(44\) 12.3923 1.86821
\(45\) 0 0
\(46\) −9.46410 −1.39541
\(47\) 5.26795 0.768409 0.384205 0.923248i \(-0.374476\pi\)
0.384205 + 0.923248i \(0.374476\pi\)
\(48\) 0 0
\(49\) −5.39230 −0.770329
\(50\) 2.73205 0.386370
\(51\) 0 0
\(52\) −29.8564 −4.14034
\(53\) −3.26795 −0.448887 −0.224444 0.974487i \(-0.572056\pi\)
−0.224444 + 0.974487i \(0.572056\pi\)
\(54\) 0 0
\(55\) −2.26795 −0.305810
\(56\) −12.0000 −1.60357
\(57\) 0 0
\(58\) 19.6603 2.58152
\(59\) 11.7321 1.52738 0.763691 0.645581i \(-0.223385\pi\)
0.763691 + 0.645581i \(0.223385\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −8.19615 −1.04091
\(63\) 0 0
\(64\) 29.8564 3.73205
\(65\) 5.46410 0.677738
\(66\) 0 0
\(67\) −3.46410 −0.423207 −0.211604 0.977356i \(-0.567869\pi\)
−0.211604 + 0.977356i \(0.567869\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 3.46410 0.414039
\(71\) 0.267949 0.0317997 0.0158999 0.999874i \(-0.494939\pi\)
0.0158999 + 0.999874i \(0.494939\pi\)
\(72\) 0 0
\(73\) 9.66025 1.13065 0.565324 0.824869i \(-0.308751\pi\)
0.565324 + 0.824869i \(0.308751\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −13.4641 −1.54444
\(77\) −2.87564 −0.327710
\(78\) 0 0
\(79\) 8.53590 0.960364 0.480182 0.877169i \(-0.340571\pi\)
0.480182 + 0.877169i \(0.340571\pi\)
\(80\) −14.9282 −1.66902
\(81\) 0 0
\(82\) −8.73205 −0.964294
\(83\) 8.19615 0.899645 0.449822 0.893118i \(-0.351487\pi\)
0.449822 + 0.893118i \(0.351487\pi\)
\(84\) 0 0
\(85\) 0.732051 0.0794021
\(86\) −27.8564 −3.00383
\(87\) 0 0
\(88\) 21.4641 2.28808
\(89\) 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
\(92\) −18.9282 −1.97340
\(93\) 0 0
\(94\) 14.3923 1.48445
\(95\) 2.46410 0.252811
\(96\) 0 0
\(97\) 7.66025 0.777781 0.388890 0.921284i \(-0.372858\pi\)
0.388890 + 0.921284i \(0.372858\pi\)
\(98\) −14.7321 −1.48816
\(99\) 0 0
\(100\) 5.46410 0.546410
\(101\) −14.6603 −1.45875 −0.729375 0.684114i \(-0.760189\pi\)
−0.729375 + 0.684114i \(0.760189\pi\)
\(102\) 0 0
\(103\) −7.46410 −0.735460 −0.367730 0.929933i \(-0.619865\pi\)
−0.367730 + 0.929933i \(0.619865\pi\)
\(104\) −51.7128 −5.07086
\(105\) 0 0
\(106\) −8.92820 −0.867184
\(107\) 15.4641 1.49497 0.747486 0.664278i \(-0.231261\pi\)
0.747486 + 0.664278i \(0.231261\pi\)
\(108\) 0 0
\(109\) −19.9282 −1.90878 −0.954388 0.298570i \(-0.903490\pi\)
−0.954388 + 0.298570i \(0.903490\pi\)
\(110\) −6.19615 −0.590780
\(111\) 0 0
\(112\) −18.9282 −1.78855
\(113\) −5.12436 −0.482059 −0.241029 0.970518i \(-0.577485\pi\)
−0.241029 + 0.970518i \(0.577485\pi\)
\(114\) 0 0
\(115\) 3.46410 0.323029
\(116\) 39.3205 3.65082
\(117\) 0 0
\(118\) 32.0526 2.95068
\(119\) 0.928203 0.0850883
\(120\) 0 0
\(121\) −5.85641 −0.532401
\(122\) 10.9282 0.989393
\(123\) 0 0
\(124\) −16.3923 −1.47207
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.5885 1.47199 0.735994 0.676988i \(-0.236715\pi\)
0.735994 + 0.676988i \(0.236715\pi\)
\(128\) 37.8564 3.34607
\(129\) 0 0
\(130\) 14.9282 1.30929
\(131\) 15.5885 1.36197 0.680985 0.732297i \(-0.261552\pi\)
0.680985 + 0.732297i \(0.261552\pi\)
\(132\) 0 0
\(133\) 3.12436 0.270916
\(134\) −9.46410 −0.817574
\(135\) 0 0
\(136\) −6.92820 −0.594089
\(137\) −9.46410 −0.808573 −0.404286 0.914632i \(-0.632480\pi\)
−0.404286 + 0.914632i \(0.632480\pi\)
\(138\) 0 0
\(139\) 21.3923 1.81447 0.907236 0.420622i \(-0.138188\pi\)
0.907236 + 0.420622i \(0.138188\pi\)
\(140\) 6.92820 0.585540
\(141\) 0 0
\(142\) 0.732051 0.0614323
\(143\) −12.3923 −1.03630
\(144\) 0 0
\(145\) −7.19615 −0.597608
\(146\) 26.3923 2.18424
\(147\) 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
\(151\) −15.3923 −1.25261 −0.626304 0.779579i \(-0.715433\pi\)
−0.626304 + 0.779579i \(0.715433\pi\)
\(152\) −23.3205 −1.89154
\(153\) 0 0
\(154\) −7.85641 −0.633087
\(155\) 3.00000 0.240966
\(156\) 0 0
\(157\) 5.12436 0.408968 0.204484 0.978870i \(-0.434448\pi\)
0.204484 + 0.978870i \(0.434448\pi\)
\(158\) 23.3205 1.85528
\(159\) 0 0
\(160\) −21.8564 −1.72790
\(161\) 4.39230 0.346162
\(162\) 0 0
\(163\) 9.26795 0.725922 0.362961 0.931804i \(-0.381766\pi\)
0.362961 + 0.931804i \(0.381766\pi\)
\(164\) −17.4641 −1.36372
\(165\) 0 0
\(166\) 22.3923 1.73798
\(167\) 0.339746 0.0262903 0.0131452 0.999914i \(-0.495816\pi\)
0.0131452 + 0.999914i \(0.495816\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −55.7128 −4.24806
\(173\) −15.4641 −1.17571 −0.587857 0.808965i \(-0.700028\pi\)
−0.587857 + 0.808965i \(0.700028\pi\)
\(174\) 0 0
\(175\) −1.26795 −0.0958479
\(176\) 33.8564 2.55202
\(177\) 0 0
\(178\) 14.1962 1.06405
\(179\) 16.1244 1.20519 0.602595 0.798047i \(-0.294133\pi\)
0.602595 + 0.798047i \(0.294133\pi\)
\(180\) 0 0
\(181\) 19.5359 1.45209 0.726046 0.687646i \(-0.241356\pi\)
0.726046 + 0.687646i \(0.241356\pi\)
\(182\) 18.9282 1.40305
\(183\) 0 0
\(184\) −32.7846 −2.41691
\(185\) −0.732051 −0.0538214
\(186\) 0 0
\(187\) −1.66025 −0.121410
\(188\) 28.7846 2.09933
\(189\) 0 0
\(190\) 6.73205 0.488394
\(191\) −16.1244 −1.16672 −0.583359 0.812215i \(-0.698262\pi\)
−0.583359 + 0.812215i \(0.698262\pi\)
\(192\) 0 0
\(193\) −8.73205 −0.628547 −0.314273 0.949333i \(-0.601761\pi\)
−0.314273 + 0.949333i \(0.601761\pi\)
\(194\) 20.9282 1.50256
\(195\) 0 0
\(196\) −29.4641 −2.10458
\(197\) −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 \(0.522583\pi\)
\(200\) 9.46410 0.669213
\(201\) 0 0
\(202\) −40.0526 −2.81809
\(203\) −9.12436 −0.640404
\(204\) 0 0
\(205\) 3.19615 0.223229
\(206\) −20.3923 −1.42080
\(207\) 0 0
\(208\) −81.5692 −5.65581
\(209\) −5.58846 −0.386562
\(210\) 0 0
\(211\) 18.8564 1.29813 0.649064 0.760734i \(-0.275161\pi\)
0.649064 + 0.760734i \(0.275161\pi\)
\(212\) −17.8564 −1.22638
\(213\) 0 0
\(214\) 42.2487 2.88806
\(215\) 10.1962 0.695372
\(216\) 0 0
\(217\) 3.80385 0.258222
\(218\) −54.4449 −3.68747
\(219\) 0 0
\(220\) −12.3923 −0.835489
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) −24.7846 −1.65970 −0.829850 0.557986i \(-0.811574\pi\)
−0.829850 + 0.557986i \(0.811574\pi\)
\(224\) −27.7128 −1.85164
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) −20.0526 −1.33094 −0.665468 0.746427i \(-0.731768\pi\)
−0.665468 + 0.746427i \(0.731768\pi\)
\(228\) 0 0
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) 9.46410 0.624044
\(231\) 0 0
\(232\) 68.1051 4.47132
\(233\) −10.0526 −0.658565 −0.329283 0.944231i \(-0.606807\pi\)
−0.329283 + 0.944231i \(0.606807\pi\)
\(234\) 0 0
\(235\) −5.26795 −0.343643
\(236\) 64.1051 4.17289
\(237\) 0 0
\(238\) 2.53590 0.164378
\(239\) 7.46410 0.482813 0.241406 0.970424i \(-0.422391\pi\)
0.241406 + 0.970424i \(0.422391\pi\)
\(240\) 0 0
\(241\) 18.3205 1.18013 0.590064 0.807357i \(-0.299103\pi\)
0.590064 + 0.807357i \(0.299103\pi\)
\(242\) −16.0000 −1.02852
\(243\) 0 0
\(244\) 21.8564 1.39921
\(245\) 5.39230 0.344502
\(246\) 0 0
\(247\) 13.4641 0.856700
\(248\) −28.3923 −1.80291
\(249\) 0 0
\(250\) −2.73205 −0.172790
\(251\) −10.3923 −0.655956 −0.327978 0.944685i \(-0.606367\pi\)
−0.327978 + 0.944685i \(0.606367\pi\)
\(252\) 0 0
\(253\) −7.85641 −0.493928
\(254\) 45.3205 2.84366
\(255\) 0 0
\(256\) 43.7128 2.73205
\(257\) 10.3923 0.648254 0.324127 0.946014i \(-0.394929\pi\)
0.324127 + 0.946014i \(0.394929\pi\)
\(258\) 0 0
\(259\) −0.928203 −0.0576757
\(260\) 29.8564 1.85162
\(261\) 0 0
\(262\) 42.5885 2.63112
\(263\) 13.3205 0.821378 0.410689 0.911776i \(-0.365288\pi\)
0.410689 + 0.911776i \(0.365288\pi\)
\(264\) 0 0
\(265\) 3.26795 0.200749
\(266\) 8.53590 0.523370
\(267\) 0 0
\(268\) −18.9282 −1.15622
\(269\) 6.66025 0.406083 0.203041 0.979170i \(-0.434917\pi\)
0.203041 + 0.979170i \(0.434917\pi\)
\(270\) 0 0
\(271\) 10.9282 0.663841 0.331921 0.943307i \(-0.392303\pi\)
0.331921 + 0.943307i \(0.392303\pi\)
\(272\) −10.9282 −0.662620
\(273\) 0 0
\(274\) −25.8564 −1.56204
\(275\) 2.26795 0.136762
\(276\) 0 0
\(277\) 14.1962 0.852964 0.426482 0.904496i \(-0.359753\pi\)
0.426482 + 0.904496i \(0.359753\pi\)
\(278\) 58.4449 3.50529
\(279\) 0 0
\(280\) 12.0000 0.717137
\(281\) −8.53590 −0.509209 −0.254605 0.967045i \(-0.581945\pi\)
−0.254605 + 0.967045i \(0.581945\pi\)
\(282\) 0 0
\(283\) 5.32051 0.316271 0.158136 0.987417i \(-0.449452\pi\)
0.158136 + 0.987417i \(0.449452\pi\)
\(284\) 1.46410 0.0868784
\(285\) 0 0
\(286\) −33.8564 −2.00197
\(287\) 4.05256 0.239215
\(288\) 0 0
\(289\) −16.4641 −0.968477
\(290\) −19.6603 −1.15449
\(291\) 0 0
\(292\) 52.7846 3.08899
\(293\) −25.2679 −1.47617 −0.738085 0.674708i \(-0.764270\pi\)
−0.738085 + 0.674708i \(0.764270\pi\)
\(294\) 0 0
\(295\) −11.7321 −0.683066
\(296\) 6.92820 0.402694
\(297\) 0 0
\(298\) 21.8564 1.26611
\(299\) 18.9282 1.09465
\(300\) 0 0
\(301\) 12.9282 0.745169
\(302\) −42.0526 −2.41985
\(303\) 0 0
\(304\) −36.7846 −2.10974
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) −24.0526 −1.37275 −0.686376 0.727247i \(-0.740800\pi\)
−0.686376 + 0.727247i \(0.740800\pi\)
\(308\) −15.7128 −0.895321
\(309\) 0 0
\(310\) 8.19615 0.465510
\(311\) −16.2679 −0.922471 −0.461235 0.887278i \(-0.652594\pi\)
−0.461235 + 0.887278i \(0.652594\pi\)
\(312\) 0 0
\(313\) −22.9282 −1.29598 −0.647989 0.761649i \(-0.724390\pi\)
−0.647989 + 0.761649i \(0.724390\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 46.6410 2.62376
\(317\) 4.19615 0.235679 0.117840 0.993033i \(-0.462403\pi\)
0.117840 + 0.993033i \(0.462403\pi\)
\(318\) 0 0
\(319\) 16.3205 0.913773
\(320\) −29.8564 −1.66902
\(321\) 0 0
\(322\) 12.0000 0.668734
\(323\) 1.80385 0.100369
\(324\) 0 0
\(325\) −5.46410 −0.303094
\(326\) 25.3205 1.40237
\(327\) 0 0
\(328\) −30.2487 −1.67021
\(329\) −6.67949 −0.368252
\(330\) 0 0
\(331\) 6.46410 0.355299 0.177650 0.984094i \(-0.443151\pi\)
0.177650 + 0.984094i \(0.443151\pi\)
\(332\) 44.7846 2.45787
\(333\) 0 0
\(334\) 0.928203 0.0507890
\(335\) 3.46410 0.189264
\(336\) 0 0
\(337\) −7.32051 −0.398773 −0.199387 0.979921i \(-0.563895\pi\)
−0.199387 + 0.979921i \(0.563895\pi\)
\(338\) 46.0526 2.50493
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) −6.80385 −0.368449
\(342\) 0 0
\(343\) 15.7128 0.848412
\(344\) −96.4974 −5.20279
\(345\) 0 0
\(346\) −42.2487 −2.27130
\(347\) −2.58846 −0.138956 −0.0694778 0.997583i \(-0.522133\pi\)
−0.0694778 + 0.997583i \(0.522133\pi\)
\(348\) 0 0
\(349\) 8.85641 0.474073 0.237036 0.971501i \(-0.423824\pi\)
0.237036 + 0.971501i \(0.423824\pi\)
\(350\) −3.46410 −0.185164
\(351\) 0 0
\(352\) 49.5692 2.64205
\(353\) 19.5167 1.03877 0.519384 0.854541i \(-0.326162\pi\)
0.519384 + 0.854541i \(0.326162\pi\)
\(354\) 0 0
\(355\) −0.267949 −0.0142213
\(356\) 28.3923 1.50479
\(357\) 0 0
\(358\) 44.0526 2.32825
\(359\) −18.1244 −0.956567 −0.478283 0.878206i \(-0.658741\pi\)
−0.478283 + 0.878206i \(0.658741\pi\)
\(360\) 0 0
\(361\) −12.9282 −0.680432
\(362\) 53.3731 2.80523
\(363\) 0 0
\(364\) 37.8564 1.98421
\(365\) −9.66025 −0.505641
\(366\) 0 0
\(367\) −31.1769 −1.62742 −0.813711 0.581270i \(-0.802556\pi\)
−0.813711 + 0.581270i \(0.802556\pi\)
\(368\) −51.7128 −2.69572
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) 4.14359 0.215125
\(372\) 0 0
\(373\) −18.0526 −0.934726 −0.467363 0.884065i \(-0.654796\pi\)
−0.467363 + 0.884065i \(0.654796\pi\)
\(374\) −4.53590 −0.234546
\(375\) 0 0
\(376\) 49.8564 2.57115
\(377\) −39.3205 −2.02511
\(378\) 0 0
\(379\) −18.3923 −0.944749 −0.472375 0.881398i \(-0.656603\pi\)
−0.472375 + 0.881398i \(0.656603\pi\)
\(380\) 13.4641 0.690694
\(381\) 0 0
\(382\) −44.0526 −2.25392
\(383\) −9.46410 −0.483593 −0.241797 0.970327i \(-0.577737\pi\)
−0.241797 + 0.970327i \(0.577737\pi\)
\(384\) 0 0
\(385\) 2.87564 0.146556
\(386\) −23.8564 −1.21426
\(387\) 0 0
\(388\) 41.8564 2.12494
\(389\) −20.5359 −1.04121 −0.520606 0.853797i \(-0.674294\pi\)
−0.520606 + 0.853797i \(0.674294\pi\)
\(390\) 0 0
\(391\) 2.53590 0.128246
\(392\) −51.0333 −2.57757
\(393\) 0 0
\(394\) −37.8564 −1.90718
\(395\) −8.53590 −0.429488
\(396\) 0 0
\(397\) −6.39230 −0.320821 −0.160410 0.987050i \(-0.551282\pi\)
−0.160410 + 0.987050i \(0.551282\pi\)
\(398\) −5.46410 −0.273891
\(399\) 0 0
\(400\) 14.9282 0.746410
\(401\) −11.0718 −0.552899 −0.276450 0.961028i \(-0.589158\pi\)
−0.276450 + 0.961028i \(0.589158\pi\)
\(402\) 0 0
\(403\) 16.3923 0.816559
\(404\) −80.1051 −3.98538
\(405\) 0 0
\(406\) −24.9282 −1.23717
\(407\) 1.66025 0.0822957
\(408\) 0 0
\(409\) 9.85641 0.487368 0.243684 0.969855i \(-0.421644\pi\)
0.243684 + 0.969855i \(0.421644\pi\)
\(410\) 8.73205 0.431245
\(411\) 0 0
\(412\) −40.7846 −2.00931
\(413\) −14.8756 −0.731983
\(414\) 0 0
\(415\) −8.19615 −0.402333
\(416\) −119.426 −5.85532
\(417\) 0 0
\(418\) −15.2679 −0.746780
\(419\) 0.392305 0.0191653 0.00958267 0.999954i \(-0.496950\pi\)
0.00958267 + 0.999954i \(0.496950\pi\)
\(420\) 0 0
\(421\) −7.78461 −0.379399 −0.189699 0.981842i \(-0.560751\pi\)
−0.189699 + 0.981842i \(0.560751\pi\)
\(422\) 51.5167 2.50779
\(423\) 0 0
\(424\) −30.9282 −1.50201
\(425\) −0.732051 −0.0355097
\(426\) 0 0
\(427\) −5.07180 −0.245441
\(428\) 84.4974 4.08434
\(429\) 0 0
\(430\) 27.8564 1.34336
\(431\) 38.6603 1.86220 0.931099 0.364765i \(-0.118851\pi\)
0.931099 + 0.364765i \(0.118851\pi\)
\(432\) 0 0
\(433\) −28.5359 −1.37135 −0.685674 0.727909i \(-0.740492\pi\)
−0.685674 + 0.727909i \(0.740492\pi\)
\(434\) 10.3923 0.498847
\(435\) 0 0
\(436\) −108.890 −5.21487
\(437\) 8.53590 0.408327
\(438\) 0 0
\(439\) −15.3923 −0.734635 −0.367317 0.930096i \(-0.619724\pi\)
−0.367317 + 0.930096i \(0.619724\pi\)
\(440\) −21.4641 −1.02326
\(441\) 0 0
\(442\) 10.9282 0.519802
\(443\) 17.6603 0.839064 0.419532 0.907741i \(-0.362194\pi\)
0.419532 + 0.907741i \(0.362194\pi\)
\(444\) 0 0
\(445\) −5.19615 −0.246321
\(446\) −67.7128 −3.20629
\(447\) 0 0
\(448\) −37.8564 −1.78855
\(449\) 16.1244 0.760955 0.380478 0.924790i \(-0.375760\pi\)
0.380478 + 0.924790i \(0.375760\pi\)
\(450\) 0 0
\(451\) −7.24871 −0.341328
\(452\) −28.0000 −1.31701
\(453\) 0 0
\(454\) −54.7846 −2.57117
\(455\) −6.92820 −0.324799
\(456\) 0 0
\(457\) −0.732051 −0.0342439 −0.0171219 0.999853i \(-0.505450\pi\)
−0.0171219 + 0.999853i \(0.505450\pi\)
\(458\) −32.7846 −1.53192
\(459\) 0 0
\(460\) 18.9282 0.882532
\(461\) −1.05256 −0.0490226 −0.0245113 0.999700i \(-0.507803\pi\)
−0.0245113 + 0.999700i \(0.507803\pi\)
\(462\) 0 0
\(463\) 10.3923 0.482971 0.241486 0.970404i \(-0.422365\pi\)
0.241486 + 0.970404i \(0.422365\pi\)
\(464\) 107.426 4.98711
\(465\) 0 0
\(466\) −27.4641 −1.27225
\(467\) 35.3731 1.63687 0.818435 0.574599i \(-0.194842\pi\)
0.818435 + 0.574599i \(0.194842\pi\)
\(468\) 0 0
\(469\) 4.39230 0.202818
\(470\) −14.3923 −0.663868
\(471\) 0 0
\(472\) 111.033 5.11072
\(473\) −23.1244 −1.06326
\(474\) 0 0
\(475\) −2.46410 −0.113061
\(476\) 5.07180 0.232465
\(477\) 0 0
\(478\) 20.3923 0.932722
\(479\) 36.1244 1.65056 0.825282 0.564721i \(-0.191016\pi\)
0.825282 + 0.564721i \(0.191016\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 50.0526 2.27983
\(483\) 0 0
\(484\) −32.0000 −1.45455
\(485\) −7.66025 −0.347834
\(486\) 0 0
\(487\) 3.60770 0.163480 0.0817401 0.996654i \(-0.473952\pi\)
0.0817401 + 0.996654i \(0.473952\pi\)
\(488\) 37.8564 1.71368
\(489\) 0 0
\(490\) 14.7321 0.665526
\(491\) 38.1244 1.72053 0.860264 0.509849i \(-0.170299\pi\)
0.860264 + 0.509849i \(0.170299\pi\)
\(492\) 0 0
\(493\) −5.26795 −0.237256
\(494\) 36.7846 1.65502
\(495\) 0 0
\(496\) −44.7846 −2.01089
\(497\) −0.339746 −0.0152397
\(498\) 0 0
\(499\) −10.3205 −0.462009 −0.231005 0.972953i \(-0.574201\pi\)
−0.231005 + 0.972953i \(0.574201\pi\)
\(500\) −5.46410 −0.244362
\(501\) 0 0
\(502\) −28.3923 −1.26721
\(503\) −27.3205 −1.21816 −0.609081 0.793108i \(-0.708461\pi\)
−0.609081 + 0.793108i \(0.708461\pi\)
\(504\) 0 0
\(505\) 14.6603 0.652373
\(506\) −21.4641 −0.954196
\(507\) 0 0
\(508\) 90.6410 4.02154
\(509\) −34.7846 −1.54180 −0.770900 0.636956i \(-0.780193\pi\)
−0.770900 + 0.636956i \(0.780193\pi\)
\(510\) 0 0
\(511\) −12.2487 −0.541851
\(512\) 43.7128 1.93185
\(513\) 0 0
\(514\) 28.3923 1.25233
\(515\) 7.46410 0.328908
\(516\) 0 0
\(517\) 11.9474 0.525448
\(518\) −2.53590 −0.111421
\(519\) 0 0
\(520\) 51.7128 2.26776
\(521\) −12.5359 −0.549208 −0.274604 0.961557i \(-0.588547\pi\)
−0.274604 + 0.961557i \(0.588547\pi\)
\(522\) 0 0
\(523\) −26.2487 −1.14778 −0.573888 0.818934i \(-0.694566\pi\)
−0.573888 + 0.818934i \(0.694566\pi\)
\(524\) 85.1769 3.72097
\(525\) 0 0
\(526\) 36.3923 1.58678
\(527\) 2.19615 0.0956659
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 8.92820 0.387816
\(531\) 0 0
\(532\) 17.0718 0.740156
\(533\) 17.4641 0.756454
\(534\) 0 0
\(535\) −15.4641 −0.668571
\(536\) −32.7846 −1.41608
\(537\) 0 0
\(538\) 18.1962 0.784492
\(539\) −12.2295 −0.526761
\(540\) 0 0
\(541\) −17.5359 −0.753927 −0.376964 0.926228i \(-0.623032\pi\)
−0.376964 + 0.926228i \(0.623032\pi\)
\(542\) 29.8564 1.28244
\(543\) 0 0
\(544\) −16.0000 −0.685994
\(545\) 19.9282 0.853630
\(546\) 0 0
\(547\) −6.14359 −0.262681 −0.131341 0.991337i \(-0.541928\pi\)
−0.131341 + 0.991337i \(0.541928\pi\)
\(548\) −51.7128 −2.20906
\(549\) 0 0
\(550\) 6.19615 0.264205
\(551\) −17.7321 −0.755411
\(552\) 0 0
\(553\) −10.8231 −0.460244
\(554\) 38.7846 1.64780
\(555\) 0 0
\(556\) 116.890 4.95723
\(557\) 9.46410 0.401007 0.200503 0.979693i \(-0.435742\pi\)
0.200503 + 0.979693i \(0.435742\pi\)
\(558\) 0 0
\(559\) 55.7128 2.35640
\(560\) 18.9282 0.799863
\(561\) 0 0
\(562\) −23.3205 −0.983716
\(563\) 13.2679 0.559177 0.279589 0.960120i \(-0.409802\pi\)
0.279589 + 0.960120i \(0.409802\pi\)
\(564\) 0 0
\(565\) 5.12436 0.215583
\(566\) 14.5359 0.610989
\(567\) 0 0
\(568\) 2.53590 0.106404
\(569\) −32.9090 −1.37962 −0.689808 0.723993i \(-0.742305\pi\)
−0.689808 + 0.723993i \(0.742305\pi\)
\(570\) 0 0
\(571\) 1.78461 0.0746836 0.0373418 0.999303i \(-0.488111\pi\)
0.0373418 + 0.999303i \(0.488111\pi\)
\(572\) −67.7128 −2.83121
\(573\) 0 0
\(574\) 11.0718 0.462128
\(575\) −3.46410 −0.144463
\(576\) 0 0
\(577\) 18.7321 0.779825 0.389913 0.920852i \(-0.372505\pi\)
0.389913 + 0.920852i \(0.372505\pi\)
\(578\) −44.9808 −1.87095
\(579\) 0 0
\(580\) −39.3205 −1.63270
\(581\) −10.3923 −0.431145
\(582\) 0 0
\(583\) −7.41154 −0.306955
\(584\) 91.4256 3.78322
\(585\) 0 0
\(586\) −69.0333 −2.85174
\(587\) −5.66025 −0.233624 −0.116812 0.993154i \(-0.537267\pi\)
−0.116812 + 0.993154i \(0.537267\pi\)
\(588\) 0 0
\(589\) 7.39230 0.304595
\(590\) −32.0526 −1.31958
\(591\) 0 0
\(592\) 10.9282 0.449146
\(593\) 27.8564 1.14393 0.571963 0.820280i \(-0.306182\pi\)
0.571963 + 0.820280i \(0.306182\pi\)
\(594\) 0 0
\(595\) −0.928203 −0.0380526
\(596\) 43.7128 1.79055
\(597\) 0 0
\(598\) 51.7128 2.11469
\(599\) −16.8038 −0.686587 −0.343293 0.939228i \(-0.611542\pi\)
−0.343293 + 0.939228i \(0.611542\pi\)
\(600\) 0 0
\(601\) 17.2487 0.703590 0.351795 0.936077i \(-0.385571\pi\)
0.351795 + 0.936077i \(0.385571\pi\)
\(602\) 35.3205 1.43956
\(603\) 0 0
\(604\) −84.1051 −3.42219
\(605\) 5.85641 0.238097
\(606\) 0 0
\(607\) −5.80385 −0.235571 −0.117785 0.993039i \(-0.537580\pi\)
−0.117785 + 0.993039i \(0.537580\pi\)
\(608\) −53.8564 −2.18417
\(609\) 0 0
\(610\) −10.9282 −0.442470
\(611\) −28.7846 −1.16450
\(612\) 0 0
\(613\) −5.46410 −0.220693 −0.110346 0.993893i \(-0.535196\pi\)
−0.110346 + 0.993893i \(0.535196\pi\)
\(614\) −65.7128 −2.65195
\(615\) 0 0
\(616\) −27.2154 −1.09654
\(617\) −6.92820 −0.278919 −0.139459 0.990228i \(-0.544536\pi\)
−0.139459 + 0.990228i \(0.544536\pi\)
\(618\) 0 0
\(619\) 15.8564 0.637323 0.318661 0.947869i \(-0.396767\pi\)
0.318661 + 0.947869i \(0.396767\pi\)
\(620\) 16.3923 0.658331
\(621\) 0 0
\(622\) −44.4449 −1.78208
\(623\) −6.58846 −0.263961
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −62.6410 −2.50364
\(627\) 0 0
\(628\) 28.0000 1.11732
\(629\) −0.535898 −0.0213677
\(630\) 0 0
\(631\) 22.7128 0.904183 0.452091 0.891972i \(-0.350678\pi\)
0.452091 + 0.891972i \(0.350678\pi\)
\(632\) 80.7846 3.21344
\(633\) 0 0
\(634\) 11.4641 0.455298
\(635\) −16.5885 −0.658293
\(636\) 0 0
\(637\) 29.4641 1.16741
\(638\) 44.5885 1.76527
\(639\) 0 0
\(640\) −37.8564 −1.49641
\(641\) −26.6603 −1.05302 −0.526508 0.850170i \(-0.676499\pi\)
−0.526508 + 0.850170i \(0.676499\pi\)
\(642\) 0 0
\(643\) 24.3397 0.959866 0.479933 0.877305i \(-0.340661\pi\)
0.479933 + 0.877305i \(0.340661\pi\)
\(644\) 24.0000 0.945732
\(645\) 0 0
\(646\) 4.92820 0.193898
\(647\) −4.53590 −0.178325 −0.0891623 0.996017i \(-0.528419\pi\)
−0.0891623 + 0.996017i \(0.528419\pi\)
\(648\) 0 0
\(649\) 26.6077 1.04444
\(650\) −14.9282 −0.585532
\(651\) 0 0
\(652\) 50.6410 1.98326
\(653\) 10.5359 0.412302 0.206151 0.978520i \(-0.433906\pi\)
0.206151 + 0.978520i \(0.433906\pi\)
\(654\) 0 0
\(655\) −15.5885 −0.609091
\(656\) −47.7128 −1.86287
\(657\) 0 0
\(658\) −18.2487 −0.711409
\(659\) −5.46410 −0.212851 −0.106426 0.994321i \(-0.533941\pi\)
−0.106426 + 0.994321i \(0.533941\pi\)
\(660\) 0 0
\(661\) −16.3205 −0.634794 −0.317397 0.948293i \(-0.602809\pi\)
−0.317397 + 0.948293i \(0.602809\pi\)
\(662\) 17.6603 0.686385
\(663\) 0 0
\(664\) 77.5692 3.01027
\(665\) −3.12436 −0.121157
\(666\) 0 0
\(667\) −24.9282 −0.965224
\(668\) 1.85641 0.0718265
\(669\) 0 0
\(670\) 9.46410 0.365630
\(671\) 9.07180 0.350213
\(672\) 0 0
\(673\) 10.3923 0.400594 0.200297 0.979735i \(-0.435809\pi\)
0.200297 + 0.979735i \(0.435809\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) 92.1051 3.54250
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) −9.71281 −0.372744
\(680\) 6.92820 0.265684
\(681\) 0 0
\(682\) −18.5885 −0.711789
\(683\) 40.3923 1.54557 0.772784 0.634669i \(-0.218863\pi\)
0.772784 + 0.634669i \(0.218863\pi\)
\(684\) 0 0
\(685\) 9.46410 0.361605
\(686\) 42.9282 1.63901
\(687\) 0 0
\(688\) −152.210 −5.80296
\(689\) 17.8564 0.680275
\(690\) 0 0
\(691\) 37.7128 1.43466 0.717332 0.696732i \(-0.245363\pi\)
0.717332 + 0.696732i \(0.245363\pi\)
\(692\) −84.4974 −3.21211
\(693\) 0 0
\(694\) −7.07180 −0.268442
\(695\) −21.3923 −0.811456
\(696\) 0 0
\(697\) 2.33975 0.0886242
\(698\) 24.1962 0.915838
\(699\) 0 0
\(700\) −6.92820 −0.261861
\(701\) 31.1962 1.17826 0.589131 0.808037i \(-0.299470\pi\)
0.589131 + 0.808037i \(0.299470\pi\)
\(702\) 0 0
\(703\) −1.80385 −0.0680334
\(704\) 67.7128 2.55202
\(705\) 0 0
\(706\) 53.3205 2.00674
\(707\) 18.5885 0.699091
\(708\) 0 0
\(709\) −29.4641 −1.10655 −0.553274 0.832999i \(-0.686622\pi\)
−0.553274 + 0.832999i \(0.686622\pi\)
\(710\) −0.732051 −0.0274734
\(711\) 0 0
\(712\) 49.1769 1.84298
\(713\) 10.3923 0.389195
\(714\) 0 0
\(715\) 12.3923 0.463446
\(716\) 88.1051 3.29264
\(717\) 0 0
\(718\) −49.5167 −1.84795
\(719\) −39.5885 −1.47640 −0.738200 0.674582i \(-0.764324\pi\)
−0.738200 + 0.674582i \(0.764324\pi\)
\(720\) 0 0
\(721\) 9.46410 0.352462
\(722\) −35.3205 −1.31449
\(723\) 0 0
\(724\) 106.746 3.96719
\(725\) 7.19615 0.267258
\(726\) 0 0
\(727\) 12.3923 0.459605 0.229803 0.973237i \(-0.426192\pi\)
0.229803 + 0.973237i \(0.426192\pi\)
\(728\) 65.5692 2.43016
\(729\) 0 0
\(730\) −26.3923 −0.976823
\(731\) 7.46410 0.276070
\(732\) 0 0
\(733\) −6.78461 −0.250595 −0.125298 0.992119i \(-0.539989\pi\)
−0.125298 + 0.992119i \(0.539989\pi\)
\(734\) −85.1769 −3.14394
\(735\) 0 0
\(736\) −75.7128 −2.79081
\(737\) −7.85641 −0.289394
\(738\) 0 0
\(739\) 15.5359 0.571497 0.285749 0.958305i \(-0.407758\pi\)
0.285749 + 0.958305i \(0.407758\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 11.3205 0.415589
\(743\) 45.9090 1.68424 0.842118 0.539293i \(-0.181308\pi\)
0.842118 + 0.539293i \(0.181308\pi\)
\(744\) 0 0
\(745\) −8.00000 −0.293097
\(746\) −49.3205 −1.80575
\(747\) 0 0
\(748\) −9.07180 −0.331698
\(749\) −19.6077 −0.716450
\(750\) 0 0
\(751\) 7.21539 0.263293 0.131647 0.991297i \(-0.457974\pi\)
0.131647 + 0.991297i \(0.457974\pi\)
\(752\) 78.6410 2.86774
\(753\) 0 0
\(754\) −107.426 −3.91221
\(755\) 15.3923 0.560183
\(756\) 0 0
\(757\) 53.1769 1.93275 0.966374 0.257141i \(-0.0827805\pi\)
0.966374 + 0.257141i \(0.0827805\pi\)
\(758\) −50.2487 −1.82512
\(759\) 0 0
\(760\) 23.3205 0.845924
\(761\) 35.4449 1.28488 0.642438 0.766338i \(-0.277923\pi\)
0.642438 + 0.766338i \(0.277923\pi\)
\(762\) 0 0
\(763\) 25.2679 0.914761
\(764\) −88.1051 −3.18753
\(765\) 0 0
\(766\) −25.8564 −0.934230
\(767\) −64.1051 −2.31470
\(768\) 0 0
\(769\) −16.4641 −0.593711 −0.296855 0.954922i \(-0.595938\pi\)
−0.296855 + 0.954922i \(0.595938\pi\)
\(770\) 7.85641 0.283125
\(771\) 0 0
\(772\) −47.7128 −1.71722
\(773\) 43.5167 1.56519 0.782593 0.622534i \(-0.213897\pi\)
0.782593 + 0.622534i \(0.213897\pi\)
\(774\) 0 0
\(775\) −3.00000 −0.107763
\(776\) 72.4974 2.60251
\(777\) 0 0
\(778\) −56.1051 −2.01147
\(779\) 7.87564 0.282174
\(780\) 0 0
\(781\) 0.607695 0.0217450
\(782\) 6.92820 0.247752
\(783\) 0 0
\(784\) −80.4974 −2.87491
\(785\) −5.12436 −0.182896
\(786\) 0 0
\(787\) −9.94744 −0.354588 −0.177294 0.984158i \(-0.556734\pi\)
−0.177294 + 0.984158i \(0.556734\pi\)
\(788\) −75.7128 −2.69716
\(789\) 0 0
\(790\) −23.3205 −0.829706
\(791\) 6.49742 0.231022
\(792\) 0 0
\(793\) −21.8564 −0.776144
\(794\) −17.4641 −0.619778
\(795\) 0 0
\(796\) −10.9282 −0.387340
\(797\) 36.3923 1.28908 0.644541 0.764570i \(-0.277049\pi\)
0.644541 + 0.764570i \(0.277049\pi\)
\(798\) 0 0
\(799\) −3.85641 −0.136430
\(800\) 21.8564 0.772741
\(801\) 0 0
\(802\) −30.2487 −1.06812
\(803\) 21.9090 0.773151
\(804\) 0 0
\(805\) −4.39230 −0.154808
\(806\) 44.7846 1.57747
\(807\) 0 0
\(808\) −138.746 −4.88107
\(809\) 13.4449 0.472696 0.236348 0.971668i \(-0.424049\pi\)
0.236348 + 0.971668i \(0.424049\pi\)
\(810\) 0 0
\(811\) −11.5359 −0.405080 −0.202540 0.979274i \(-0.564920\pi\)
−0.202540 + 0.979274i \(0.564920\pi\)
\(812\) −49.8564 −1.74962
\(813\) 0 0
\(814\) 4.53590 0.158983
\(815\) −9.26795 −0.324642
\(816\) 0 0
\(817\) 25.1244 0.878990
\(818\) 26.9282 0.941523
\(819\) 0 0
\(820\) 17.4641 0.609873
\(821\) 34.2679 1.19596 0.597980 0.801511i \(-0.295970\pi\)
0.597980 + 0.801511i \(0.295970\pi\)
\(822\) 0 0
\(823\) −23.8564 −0.831582 −0.415791 0.909460i \(-0.636495\pi\)
−0.415791 + 0.909460i \(0.636495\pi\)
\(824\) −70.6410 −2.46090
\(825\) 0 0
\(826\) −40.6410 −1.41408
\(827\) 32.3923 1.12639 0.563195 0.826324i \(-0.309572\pi\)
0.563195 + 0.826324i \(0.309572\pi\)
\(828\) 0 0
\(829\) 17.7846 0.617685 0.308843 0.951113i \(-0.400058\pi\)
0.308843 + 0.951113i \(0.400058\pi\)
\(830\) −22.3923 −0.777248
\(831\) 0 0
\(832\) −163.138 −5.65581
\(833\) 3.94744 0.136771
\(834\) 0 0
\(835\) −0.339746 −0.0117574
\(836\) −30.5359 −1.05611
\(837\) 0 0
\(838\) 1.07180 0.0370246
\(839\) −22.8038 −0.787276 −0.393638 0.919265i \(-0.628784\pi\)
−0.393638 + 0.919265i \(0.628784\pi\)
\(840\) 0 0
\(841\) 22.7846 0.785676
\(842\) −21.2679 −0.732942
\(843\) 0 0
\(844\) 103.033 3.54655
\(845\) −16.8564 −0.579878
\(846\) 0 0
\(847\) 7.42563 0.255148
\(848\) −48.7846 −1.67527
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) −2.53590 −0.0869295
\(852\) 0 0
\(853\) 55.5167 1.90085 0.950427 0.310947i \(-0.100646\pi\)
0.950427 + 0.310947i \(0.100646\pi\)
\(854\) −13.8564 −0.474156
\(855\) 0 0
\(856\) 146.354 5.00227
\(857\) 52.4974 1.79328 0.896639 0.442763i \(-0.146002\pi\)
0.896639 + 0.442763i \(0.146002\pi\)
\(858\) 0 0
\(859\) −11.0000 −0.375315 −0.187658 0.982235i \(-0.560090\pi\)
−0.187658 + 0.982235i \(0.560090\pi\)
\(860\) 55.7128 1.89979
\(861\) 0 0
\(862\) 105.622 3.59749
\(863\) −1.12436 −0.0382735 −0.0191368 0.999817i \(-0.506092\pi\)
−0.0191368 + 0.999817i \(0.506092\pi\)
\(864\) 0 0
\(865\) 15.4641 0.525795
\(866\) −77.9615 −2.64924
\(867\) 0 0
\(868\) 20.7846 0.705476
\(869\) 19.3590 0.656709
\(870\) 0 0
\(871\) 18.9282 0.641358
\(872\) −188.603 −6.38689
\(873\) 0 0
\(874\) 23.3205 0.788828
\(875\) 1.26795 0.0428645
\(876\) 0 0
\(877\) 26.5359 0.896054 0.448027 0.894020i \(-0.352127\pi\)
0.448027 + 0.894020i \(0.352127\pi\)
\(878\) −42.0526 −1.41921
\(879\) 0 0
\(880\) −33.8564 −1.14130
\(881\) 8.94744 0.301447 0.150723 0.988576i \(-0.451840\pi\)
0.150723 + 0.988576i \(0.451840\pi\)
\(882\) 0 0
\(883\) −19.8038 −0.666453 −0.333226 0.942847i \(-0.608137\pi\)
−0.333226 + 0.942847i \(0.608137\pi\)
\(884\) 21.8564 0.735111
\(885\) 0 0
\(886\) 48.2487 1.62095
\(887\) −44.1962 −1.48396 −0.741981 0.670421i \(-0.766113\pi\)
−0.741981 + 0.670421i \(0.766113\pi\)
\(888\) 0 0
\(889\) −21.0333 −0.705435
\(890\) −14.1962 −0.475856
\(891\) 0 0
\(892\) −135.426 −4.53439
\(893\) −12.9808 −0.434385
\(894\) 0 0
\(895\) −16.1244 −0.538978
\(896\) −48.0000 −1.60357
\(897\) 0 0
\(898\) 44.0526 1.47005
\(899\) −21.5885 −0.720015
\(900\) 0 0
\(901\) 2.39230 0.0796992
\(902\) −19.8038 −0.659396
\(903\) 0 0
\(904\) −48.4974 −1.61300
\(905\) −19.5359 −0.649395
\(906\) 0 0
\(907\) −30.0000 −0.996134 −0.498067 0.867139i \(-0.665957\pi\)
−0.498067 + 0.867139i \(0.665957\pi\)
\(908\) −109.569 −3.63618
\(909\) 0 0
\(910\) −18.9282 −0.627464
\(911\) 7.58846 0.251417 0.125708 0.992067i \(-0.459880\pi\)
0.125708 + 0.992067i \(0.459880\pi\)
\(912\) 0 0
\(913\) 18.5885 0.615188
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −65.5692 −2.16647
\(917\) −19.7654 −0.652710
\(918\) 0 0
\(919\) 46.9615 1.54912 0.774559 0.632502i \(-0.217972\pi\)
0.774559 + 0.632502i \(0.217972\pi\)
\(920\) 32.7846 1.08088
\(921\) 0 0
\(922\) −2.87564 −0.0947043
\(923\) −1.46410 −0.0481915
\(924\) 0 0
\(925\) 0.732051 0.0240697
\(926\) 28.3923 0.933029
\(927\) 0 0
\(928\) 157.282 5.16304
\(929\) −28.3731 −0.930890 −0.465445 0.885077i \(-0.654106\pi\)
−0.465445 + 0.885077i \(0.654106\pi\)
\(930\) 0 0
\(931\) 13.2872 0.435470
\(932\) −54.9282 −1.79923
\(933\) 0 0
\(934\) 96.6410 3.16219
\(935\) 1.66025 0.0542961
\(936\) 0 0
\(937\) −31.8564 −1.04070 −0.520352 0.853952i \(-0.674199\pi\)
−0.520352 + 0.853952i \(0.674199\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) −28.7846 −0.938850
\(941\) 27.1769 0.885942 0.442971 0.896536i \(-0.353924\pi\)
0.442971 + 0.896536i \(0.353924\pi\)
\(942\) 0 0
\(943\) 11.0718 0.360547
\(944\) 175.138 5.70027
\(945\) 0 0
\(946\) −63.1769 −2.05406
\(947\) 2.28719 0.0743236 0.0371618 0.999309i \(-0.488168\pi\)
0.0371618 + 0.999309i \(0.488168\pi\)
\(948\) 0 0
\(949\) −52.7846 −1.71346
\(950\) −6.73205 −0.218417
\(951\) 0 0
\(952\) 8.78461 0.284711
\(953\) −15.6077 −0.505583 −0.252791 0.967521i \(-0.581349\pi\)
−0.252791 + 0.967521i \(0.581349\pi\)
\(954\) 0 0
\(955\) 16.1244 0.521772
\(956\) 40.7846 1.31907
\(957\) 0 0
\(958\) 98.6936 3.18864
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −10.9282 −0.352339
\(963\) 0 0
\(964\) 100.105 3.22417
\(965\) 8.73205 0.281095
\(966\) 0 0
\(967\) −36.1962 −1.16399 −0.581995 0.813192i \(-0.697728\pi\)
−0.581995 + 0.813192i \(0.697728\pi\)
\(968\) −55.4256 −1.78145
\(969\) 0 0
\(970\) −20.9282 −0.671964
\(971\) 41.4449 1.33003 0.665014 0.746830i \(-0.268425\pi\)
0.665014 + 0.746830i \(0.268425\pi\)
\(972\) 0 0
\(973\) −27.1244 −0.869567
\(974\) 9.85641 0.315820
\(975\) 0 0
\(976\) 59.7128 1.91136
\(977\) 1.46410 0.0468408 0.0234204 0.999726i \(-0.492544\pi\)
0.0234204 + 0.999726i \(0.492544\pi\)
\(978\) 0 0
\(979\) 11.7846 0.376638
\(980\) 29.4641 0.941196
\(981\) 0 0
\(982\) 104.158 3.32381
\(983\) −17.4115 −0.555342 −0.277671 0.960676i \(-0.589563\pi\)
−0.277671 + 0.960676i \(0.589563\pi\)
\(984\) 0 0
\(985\) 13.8564 0.441502
\(986\) −14.3923 −0.458344
\(987\) 0 0
\(988\) 73.5692 2.34055
\(989\) 35.3205 1.12313
\(990\) 0 0
\(991\) 3.14359 0.0998595 0.0499298 0.998753i \(-0.484100\pi\)
0.0499298 + 0.998753i \(0.484100\pi\)
\(992\) −65.5692 −2.08182
\(993\) 0 0
\(994\) −0.928203 −0.0294408
\(995\) 2.00000 0.0634043
\(996\) 0 0
\(997\) −19.5167 −0.618099 −0.309049 0.951046i \(-0.600011\pi\)
−0.309049 + 0.951046i \(0.600011\pi\)
\(998\) −28.1962 −0.892534
\(999\) 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.1.2 yes 2
3.2 odd 2 405.2.a.g.1.1 2
4.3 odd 2 6480.2.a.bi.1.1 2
5.2 odd 4 2025.2.b.h.649.4 4
5.3 odd 4 2025.2.b.h.649.1 4
5.4 even 2 2025.2.a.g.1.1 2
9.2 odd 6 405.2.e.l.271.2 4
9.4 even 3 405.2.e.i.136.1 4
9.5 odd 6 405.2.e.l.136.2 4
9.7 even 3 405.2.e.i.271.1 4
12.11 even 2 6480.2.a.br.1.1 2
15.2 even 4 2025.2.b.g.649.1 4
15.8 even 4 2025.2.b.g.649.4 4
15.14 odd 2 2025.2.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.g.1.1 2 3.2 odd 2
405.2.a.h.1.2 yes 2 1.1 even 1 trivial
405.2.e.i.136.1 4 9.4 even 3
405.2.e.i.271.1 4 9.7 even 3
405.2.e.l.136.2 4 9.5 odd 6
405.2.e.l.271.2 4 9.2 odd 6
2025.2.a.g.1.1 2 5.4 even 2
2025.2.a.m.1.2 2 15.14 odd 2
2025.2.b.g.649.1 4 15.2 even 4
2025.2.b.g.649.4 4 15.8 even 4
2025.2.b.h.649.1 4 5.3 odd 4
2025.2.b.h.649.4 4 5.2 odd 4
6480.2.a.bi.1.1 2 4.3 odd 2
6480.2.a.br.1.1 2 12.11 even 2