Properties

Label 675.4.b.l.649.6
Level $675$
Weight $4$
Character 675.649
Analytic conductor $39.826$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.8262892539\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.2033649216.1
Defining polynomial: \( x^{6} + 47x^{4} + 541x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.6
Root \(5.20067i\) of defining polynomial
Character \(\chi\) \(=\) 675.649
Dual form 675.4.b.l.649.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.20067i q^{2} -19.0470 q^{4} -24.4013i q^{7} -57.4517i q^{8} +O(q^{10})\) \(q+5.20067i q^{2} -19.0470 q^{4} -24.4013i q^{7} -57.4517i q^{8} +28.9839 q^{11} -65.3919i q^{13} +126.903 q^{14} +146.411 q^{16} +68.1718i q^{17} -104.424 q^{19} +150.736i q^{22} +154.807i q^{23} +340.082 q^{26} +464.772i q^{28} -205.658 q^{29} -18.2497 q^{31} +301.824i q^{32} -354.539 q^{34} +337.613i q^{37} -543.076i q^{38} +195.969 q^{41} +334.882i q^{43} -552.055 q^{44} -805.098 q^{46} -5.00398i q^{47} -252.425 q^{49} +1245.52i q^{52} +319.965i q^{53} -1401.90 q^{56} -1069.56i q^{58} +430.611 q^{59} +594.581 q^{61} -94.9106i q^{62} -398.396 q^{64} -195.876i q^{67} -1298.47i q^{68} -425.955 q^{71} +929.193i q^{73} -1755.82 q^{74} +1988.96 q^{76} -707.245i q^{77} -24.4296 q^{79} +1019.17i q^{82} +545.859i q^{83} -1741.61 q^{86} -1665.17i q^{88} -84.1332 q^{89} -1595.65 q^{91} -2948.60i q^{92} +26.0241 q^{94} -827.613i q^{97} -1312.78i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 46 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 46 q^{4} + 76 q^{11} + 216 q^{14} + 382 q^{16} - 374 q^{19} + 832 q^{26} - 320 q^{29} + 454 q^{31} - 34 q^{34} - 676 q^{41} - 3272 q^{44} - 2850 q^{46} + 394 q^{49} - 2508 q^{56} - 280 q^{59} + 1190 q^{61} + 1836 q^{64} - 1204 q^{71} - 5756 q^{74} + 1050 q^{76} - 1258 q^{79} - 7460 q^{86} - 4308 q^{89} - 880 q^{91} + 2216 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.20067i 1.83871i 0.393424 + 0.919357i \(0.371291\pi\)
−0.393424 + 0.919357i \(0.628709\pi\)
\(3\) 0 0
\(4\) −19.0470 −2.38087
\(5\) 0 0
\(6\) 0 0
\(7\) − 24.4013i − 1.31755i −0.752341 0.658774i \(-0.771075\pi\)
0.752341 0.658774i \(-0.228925\pi\)
\(8\) − 57.4517i − 2.53903i
\(9\) 0 0
\(10\) 0 0
\(11\) 28.9839 0.794451 0.397226 0.917721i \(-0.369973\pi\)
0.397226 + 0.917721i \(0.369973\pi\)
\(12\) 0 0
\(13\) − 65.3919i − 1.39511i −0.716530 0.697556i \(-0.754271\pi\)
0.716530 0.697556i \(-0.245729\pi\)
\(14\) 126.903 2.42260
\(15\) 0 0
\(16\) 146.411 2.28768
\(17\) 68.1718i 0.972593i 0.873794 + 0.486296i \(0.161653\pi\)
−0.873794 + 0.486296i \(0.838347\pi\)
\(18\) 0 0
\(19\) −104.424 −1.26087 −0.630435 0.776242i \(-0.717124\pi\)
−0.630435 + 0.776242i \(0.717124\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 150.736i 1.46077i
\(23\) 154.807i 1.40345i 0.712446 + 0.701727i \(0.247587\pi\)
−0.712446 + 0.701727i \(0.752413\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 340.082 2.56521
\(27\) 0 0
\(28\) 464.772i 3.13691i
\(29\) −205.658 −1.31689 −0.658443 0.752631i \(-0.728785\pi\)
−0.658443 + 0.752631i \(0.728785\pi\)
\(30\) 0 0
\(31\) −18.2497 −0.105734 −0.0528668 0.998602i \(-0.516836\pi\)
−0.0528668 + 0.998602i \(0.516836\pi\)
\(32\) 301.824i 1.66736i
\(33\) 0 0
\(34\) −354.539 −1.78832
\(35\) 0 0
\(36\) 0 0
\(37\) 337.613i 1.50009i 0.661387 + 0.750044i \(0.269968\pi\)
−0.661387 + 0.750044i \(0.730032\pi\)
\(38\) − 543.076i − 2.31838i
\(39\) 0 0
\(40\) 0 0
\(41\) 195.969 0.746469 0.373234 0.927737i \(-0.378249\pi\)
0.373234 + 0.927737i \(0.378249\pi\)
\(42\) 0 0
\(43\) 334.882i 1.18765i 0.804594 + 0.593826i \(0.202383\pi\)
−0.804594 + 0.593826i \(0.797617\pi\)
\(44\) −552.055 −1.89149
\(45\) 0 0
\(46\) −805.098 −2.58055
\(47\) − 5.00398i − 0.0155299i −0.999970 0.00776496i \(-0.997528\pi\)
0.999970 0.00776496i \(-0.00247169\pi\)
\(48\) 0 0
\(49\) −252.425 −0.735934
\(50\) 0 0
\(51\) 0 0
\(52\) 1245.52i 3.32158i
\(53\) 319.965i 0.829256i 0.909991 + 0.414628i \(0.136088\pi\)
−0.909991 + 0.414628i \(0.863912\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1401.90 −3.34529
\(57\) 0 0
\(58\) − 1069.56i − 2.42138i
\(59\) 430.611 0.950182 0.475091 0.879937i \(-0.342415\pi\)
0.475091 + 0.879937i \(0.342415\pi\)
\(60\) 0 0
\(61\) 594.581 1.24800 0.624002 0.781422i \(-0.285505\pi\)
0.624002 + 0.781422i \(0.285505\pi\)
\(62\) − 94.9106i − 0.194414i
\(63\) 0 0
\(64\) −398.396 −0.778118
\(65\) 0 0
\(66\) 0 0
\(67\) − 195.876i − 0.357166i −0.983925 0.178583i \(-0.942849\pi\)
0.983925 0.178583i \(-0.0571513\pi\)
\(68\) − 1298.47i − 2.31562i
\(69\) 0 0
\(70\) 0 0
\(71\) −425.955 −0.711994 −0.355997 0.934487i \(-0.615859\pi\)
−0.355997 + 0.934487i \(0.615859\pi\)
\(72\) 0 0
\(73\) 929.193i 1.48978i 0.667188 + 0.744889i \(0.267498\pi\)
−0.667188 + 0.744889i \(0.732502\pi\)
\(74\) −1755.82 −2.75824
\(75\) 0 0
\(76\) 1988.96 3.00197
\(77\) − 707.245i − 1.04673i
\(78\) 0 0
\(79\) −24.4296 −0.0347917 −0.0173959 0.999849i \(-0.505538\pi\)
−0.0173959 + 0.999849i \(0.505538\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1019.17i 1.37254i
\(83\) 545.859i 0.721877i 0.932590 + 0.360938i \(0.117544\pi\)
−0.932590 + 0.360938i \(0.882456\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1741.61 −2.18375
\(87\) 0 0
\(88\) − 1665.17i − 2.01714i
\(89\) −84.1332 −0.100203 −0.0501017 0.998744i \(-0.515955\pi\)
−0.0501017 + 0.998744i \(0.515955\pi\)
\(90\) 0 0
\(91\) −1595.65 −1.83813
\(92\) − 2948.60i − 3.34144i
\(93\) 0 0
\(94\) 26.0241 0.0285551
\(95\) 0 0
\(96\) 0 0
\(97\) − 827.613i − 0.866303i −0.901321 0.433152i \(-0.857401\pi\)
0.901321 0.433152i \(-0.142599\pi\)
\(98\) − 1312.78i − 1.35317i
\(99\) 0 0
\(100\) 0 0
\(101\) −823.576 −0.811375 −0.405688 0.914012i \(-0.632968\pi\)
−0.405688 + 0.914012i \(0.632968\pi\)
\(102\) 0 0
\(103\) 1171.19i 1.12040i 0.828359 + 0.560198i \(0.189275\pi\)
−0.828359 + 0.560198i \(0.810725\pi\)
\(104\) −3756.87 −3.54223
\(105\) 0 0
\(106\) −1664.03 −1.52477
\(107\) 1023.21i 0.924460i 0.886760 + 0.462230i \(0.152951\pi\)
−0.886760 + 0.462230i \(0.847049\pi\)
\(108\) 0 0
\(109\) 403.647 0.354700 0.177350 0.984148i \(-0.443247\pi\)
0.177350 + 0.984148i \(0.443247\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 3572.63i − 3.01413i
\(113\) − 1082.20i − 0.900931i −0.892794 0.450465i \(-0.851258\pi\)
0.892794 0.450465i \(-0.148742\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3917.16 3.13534
\(117\) 0 0
\(118\) 2239.46i 1.74711i
\(119\) 1663.48 1.28144
\(120\) 0 0
\(121\) −490.935 −0.368847
\(122\) 3092.22i 2.29473i
\(123\) 0 0
\(124\) 347.601 0.251738
\(125\) 0 0
\(126\) 0 0
\(127\) 774.132i 0.540890i 0.962735 + 0.270445i \(0.0871709\pi\)
−0.962735 + 0.270445i \(0.912829\pi\)
\(128\) 342.664i 0.236621i
\(129\) 0 0
\(130\) 0 0
\(131\) 1214.04 0.809702 0.404851 0.914383i \(-0.367323\pi\)
0.404851 + 0.914383i \(0.367323\pi\)
\(132\) 0 0
\(133\) 2548.09i 1.66126i
\(134\) 1018.69 0.656726
\(135\) 0 0
\(136\) 3916.58 2.46944
\(137\) 2300.15i 1.43441i 0.696860 + 0.717207i \(0.254580\pi\)
−0.696860 + 0.717207i \(0.745420\pi\)
\(138\) 0 0
\(139\) 1355.93 0.827396 0.413698 0.910414i \(-0.364237\pi\)
0.413698 + 0.910414i \(0.364237\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 2215.25i − 1.30915i
\(143\) − 1895.31i − 1.10835i
\(144\) 0 0
\(145\) 0 0
\(146\) −4832.43 −2.73928
\(147\) 0 0
\(148\) − 6430.51i − 3.57152i
\(149\) 259.845 0.142868 0.0714340 0.997445i \(-0.477242\pi\)
0.0714340 + 0.997445i \(0.477242\pi\)
\(150\) 0 0
\(151\) −508.304 −0.273941 −0.136971 0.990575i \(-0.543737\pi\)
−0.136971 + 0.990575i \(0.543737\pi\)
\(152\) 5999.34i 3.20139i
\(153\) 0 0
\(154\) 3678.15 1.92463
\(155\) 0 0
\(156\) 0 0
\(157\) 23.3052i 0.0118468i 0.999982 + 0.00592342i \(0.00188549\pi\)
−0.999982 + 0.00592342i \(0.998115\pi\)
\(158\) − 127.050i − 0.0639721i
\(159\) 0 0
\(160\) 0 0
\(161\) 3777.49 1.84912
\(162\) 0 0
\(163\) − 4032.10i − 1.93754i −0.247964 0.968769i \(-0.579762\pi\)
0.247964 0.968769i \(-0.420238\pi\)
\(164\) −3732.62 −1.77725
\(165\) 0 0
\(166\) −2838.83 −1.32733
\(167\) − 671.911i − 0.311341i −0.987809 0.155671i \(-0.950246\pi\)
0.987809 0.155671i \(-0.0497539\pi\)
\(168\) 0 0
\(169\) −2079.10 −0.946337
\(170\) 0 0
\(171\) 0 0
\(172\) − 6378.49i − 2.82765i
\(173\) − 1633.53i − 0.717889i −0.933359 0.358944i \(-0.883137\pi\)
0.933359 0.358944i \(-0.116863\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4243.57 1.81745
\(177\) 0 0
\(178\) − 437.549i − 0.184245i
\(179\) 341.260 0.142497 0.0712485 0.997459i \(-0.477302\pi\)
0.0712485 + 0.997459i \(0.477302\pi\)
\(180\) 0 0
\(181\) −1695.92 −0.696447 −0.348223 0.937412i \(-0.613215\pi\)
−0.348223 + 0.937412i \(0.613215\pi\)
\(182\) − 8298.45i − 3.37979i
\(183\) 0 0
\(184\) 8893.90 3.56341
\(185\) 0 0
\(186\) 0 0
\(187\) 1975.88i 0.772678i
\(188\) 95.3107i 0.0369747i
\(189\) 0 0
\(190\) 0 0
\(191\) 726.451 0.275205 0.137603 0.990488i \(-0.456060\pi\)
0.137603 + 0.990488i \(0.456060\pi\)
\(192\) 0 0
\(193\) 4247.26i 1.58406i 0.610479 + 0.792032i \(0.290977\pi\)
−0.610479 + 0.792032i \(0.709023\pi\)
\(194\) 4304.14 1.59288
\(195\) 0 0
\(196\) 4807.94 1.75216
\(197\) 2678.52i 0.968713i 0.874871 + 0.484357i \(0.160946\pi\)
−0.874871 + 0.484357i \(0.839054\pi\)
\(198\) 0 0
\(199\) −1486.48 −0.529517 −0.264759 0.964315i \(-0.585292\pi\)
−0.264759 + 0.964315i \(0.585292\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 4283.15i − 1.49189i
\(203\) 5018.32i 1.73506i
\(204\) 0 0
\(205\) 0 0
\(206\) −6090.97 −2.06009
\(207\) 0 0
\(208\) − 9574.12i − 3.19157i
\(209\) −3026.62 −1.00170
\(210\) 0 0
\(211\) −4827.41 −1.57504 −0.787519 0.616291i \(-0.788635\pi\)
−0.787519 + 0.616291i \(0.788635\pi\)
\(212\) − 6094.37i − 1.97435i
\(213\) 0 0
\(214\) −5321.37 −1.69982
\(215\) 0 0
\(216\) 0 0
\(217\) 445.317i 0.139309i
\(218\) 2099.23i 0.652193i
\(219\) 0 0
\(220\) 0 0
\(221\) 4457.88 1.35688
\(222\) 0 0
\(223\) 2774.48i 0.833153i 0.909101 + 0.416576i \(0.136770\pi\)
−0.909101 + 0.416576i \(0.863230\pi\)
\(224\) 7364.91 2.19683
\(225\) 0 0
\(226\) 5628.19 1.65655
\(227\) − 5101.34i − 1.49158i −0.666184 0.745788i \(-0.732073\pi\)
0.666184 0.745788i \(-0.267927\pi\)
\(228\) 0 0
\(229\) 4097.83 1.18250 0.591249 0.806489i \(-0.298635\pi\)
0.591249 + 0.806489i \(0.298635\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 11815.4i 3.34361i
\(233\) − 357.613i − 0.100549i −0.998735 0.0502747i \(-0.983990\pi\)
0.998735 0.0502747i \(-0.0160097\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −8201.83 −2.26226
\(237\) 0 0
\(238\) 8651.22i 2.35620i
\(239\) −351.682 −0.0951818 −0.0475909 0.998867i \(-0.515154\pi\)
−0.0475909 + 0.998867i \(0.515154\pi\)
\(240\) 0 0
\(241\) −6165.53 −1.64795 −0.823976 0.566624i \(-0.808249\pi\)
−0.823976 + 0.566624i \(0.808249\pi\)
\(242\) − 2553.19i − 0.678204i
\(243\) 0 0
\(244\) −11325.0 −2.97134
\(245\) 0 0
\(246\) 0 0
\(247\) 6828.50i 1.75906i
\(248\) 1048.48i 0.268461i
\(249\) 0 0
\(250\) 0 0
\(251\) −3245.53 −0.816160 −0.408080 0.912946i \(-0.633802\pi\)
−0.408080 + 0.912946i \(0.633802\pi\)
\(252\) 0 0
\(253\) 4486.90i 1.11498i
\(254\) −4026.00 −0.994543
\(255\) 0 0
\(256\) −4969.25 −1.21320
\(257\) 3552.19i 0.862178i 0.902309 + 0.431089i \(0.141871\pi\)
−0.902309 + 0.431089i \(0.858129\pi\)
\(258\) 0 0
\(259\) 8238.22 1.97644
\(260\) 0 0
\(261\) 0 0
\(262\) 6313.81i 1.48881i
\(263\) − 4416.59i − 1.03551i −0.855530 0.517754i \(-0.826768\pi\)
0.855530 0.517754i \(-0.173232\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −13251.8 −3.05458
\(267\) 0 0
\(268\) 3730.85i 0.850366i
\(269\) −3419.93 −0.775155 −0.387578 0.921837i \(-0.626688\pi\)
−0.387578 + 0.921837i \(0.626688\pi\)
\(270\) 0 0
\(271\) 716.407 0.160585 0.0802927 0.996771i \(-0.474415\pi\)
0.0802927 + 0.996771i \(0.474415\pi\)
\(272\) 9981.12i 2.22498i
\(273\) 0 0
\(274\) −11962.3 −2.63748
\(275\) 0 0
\(276\) 0 0
\(277\) − 657.529i − 0.142625i −0.997454 0.0713124i \(-0.977281\pi\)
0.997454 0.0713124i \(-0.0227187\pi\)
\(278\) 7051.72i 1.52135i
\(279\) 0 0
\(280\) 0 0
\(281\) −1513.91 −0.321397 −0.160698 0.987004i \(-0.551375\pi\)
−0.160698 + 0.987004i \(0.551375\pi\)
\(282\) 0 0
\(283\) 3906.38i 0.820532i 0.911966 + 0.410266i \(0.134564\pi\)
−0.911966 + 0.410266i \(0.865436\pi\)
\(284\) 8113.16 1.69517
\(285\) 0 0
\(286\) 9856.89 2.03794
\(287\) − 4781.91i − 0.983509i
\(288\) 0 0
\(289\) 265.611 0.0540629
\(290\) 0 0
\(291\) 0 0
\(292\) − 17698.3i − 3.54697i
\(293\) 8048.76i 1.60483i 0.596770 + 0.802413i \(0.296451\pi\)
−0.596770 + 0.802413i \(0.703549\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 19396.4 3.80877
\(297\) 0 0
\(298\) 1351.37i 0.262693i
\(299\) 10123.1 1.95797
\(300\) 0 0
\(301\) 8171.57 1.56479
\(302\) − 2643.52i − 0.503700i
\(303\) 0 0
\(304\) −15288.9 −2.88447
\(305\) 0 0
\(306\) 0 0
\(307\) − 101.564i − 0.0188814i −0.999955 0.00944068i \(-0.996995\pi\)
0.999955 0.00944068i \(-0.00300511\pi\)
\(308\) 13470.9i 2.49213i
\(309\) 0 0
\(310\) 0 0
\(311\) 7684.59 1.40113 0.700567 0.713586i \(-0.252930\pi\)
0.700567 + 0.713586i \(0.252930\pi\)
\(312\) 0 0
\(313\) − 1345.15i − 0.242915i −0.992597 0.121457i \(-0.961243\pi\)
0.992597 0.121457i \(-0.0387568\pi\)
\(314\) −121.202 −0.0217830
\(315\) 0 0
\(316\) 465.310 0.0828346
\(317\) 7622.33i 1.35051i 0.737583 + 0.675257i \(0.235967\pi\)
−0.737583 + 0.675257i \(0.764033\pi\)
\(318\) 0 0
\(319\) −5960.76 −1.04620
\(320\) 0 0
\(321\) 0 0
\(322\) 19645.5i 3.40000i
\(323\) − 7118.78i − 1.22631i
\(324\) 0 0
\(325\) 0 0
\(326\) 20969.6 3.56258
\(327\) 0 0
\(328\) − 11258.7i − 1.89531i
\(329\) −122.104 −0.0204614
\(330\) 0 0
\(331\) −6585.09 −1.09350 −0.546751 0.837295i \(-0.684136\pi\)
−0.546751 + 0.837295i \(0.684136\pi\)
\(332\) − 10397.0i − 1.71870i
\(333\) 0 0
\(334\) 3494.39 0.572468
\(335\) 0 0
\(336\) 0 0
\(337\) 2946.94i 0.476351i 0.971222 + 0.238175i \(0.0765493\pi\)
−0.971222 + 0.238175i \(0.923451\pi\)
\(338\) − 10812.7i − 1.74004i
\(339\) 0 0
\(340\) 0 0
\(341\) −528.947 −0.0840002
\(342\) 0 0
\(343\) − 2210.14i − 0.347919i
\(344\) 19239.5 3.01548
\(345\) 0 0
\(346\) 8495.44 1.31999
\(347\) 8493.48i 1.31399i 0.753896 + 0.656994i \(0.228172\pi\)
−0.753896 + 0.656994i \(0.771828\pi\)
\(348\) 0 0
\(349\) −5646.54 −0.866053 −0.433027 0.901381i \(-0.642554\pi\)
−0.433027 + 0.901381i \(0.642554\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 8748.03i 1.32464i
\(353\) 1221.93i 0.184240i 0.995748 + 0.0921202i \(0.0293644\pi\)
−0.995748 + 0.0921202i \(0.970636\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1602.48 0.238571
\(357\) 0 0
\(358\) 1774.78i 0.262011i
\(359\) −4151.44 −0.610319 −0.305160 0.952301i \(-0.598710\pi\)
−0.305160 + 0.952301i \(0.598710\pi\)
\(360\) 0 0
\(361\) 4045.41 0.589796
\(362\) − 8819.93i − 1.28057i
\(363\) 0 0
\(364\) 30392.3 4.37635
\(365\) 0 0
\(366\) 0 0
\(367\) 7038.71i 1.00114i 0.865696 + 0.500569i \(0.166876\pi\)
−0.865696 + 0.500569i \(0.833124\pi\)
\(368\) 22665.5i 3.21065i
\(369\) 0 0
\(370\) 0 0
\(371\) 7807.58 1.09259
\(372\) 0 0
\(373\) − 7119.57i − 0.988303i −0.869376 0.494152i \(-0.835479\pi\)
0.869376 0.494152i \(-0.164521\pi\)
\(374\) −10275.9 −1.42073
\(375\) 0 0
\(376\) −287.487 −0.0394309
\(377\) 13448.4i 1.83720i
\(378\) 0 0
\(379\) −3372.29 −0.457053 −0.228526 0.973538i \(-0.573391\pi\)
−0.228526 + 0.973538i \(0.573391\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 3778.03i 0.506024i
\(383\) − 3958.63i − 0.528138i −0.964504 0.264069i \(-0.914935\pi\)
0.964504 0.264069i \(-0.0850646\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −22088.6 −2.91264
\(387\) 0 0
\(388\) 15763.5i 2.06256i
\(389\) 9654.01 1.25830 0.629148 0.777285i \(-0.283404\pi\)
0.629148 + 0.777285i \(0.283404\pi\)
\(390\) 0 0
\(391\) −10553.4 −1.36499
\(392\) 14502.3i 1.86856i
\(393\) 0 0
\(394\) −13930.1 −1.78119
\(395\) 0 0
\(396\) 0 0
\(397\) − 10928.3i − 1.38155i −0.723068 0.690776i \(-0.757269\pi\)
0.723068 0.690776i \(-0.242731\pi\)
\(398\) − 7730.70i − 0.973631i
\(399\) 0 0
\(400\) 0 0
\(401\) −4085.57 −0.508787 −0.254393 0.967101i \(-0.581876\pi\)
−0.254393 + 0.967101i \(0.581876\pi\)
\(402\) 0 0
\(403\) 1193.38i 0.147510i
\(404\) 15686.6 1.93178
\(405\) 0 0
\(406\) −26098.7 −3.19028
\(407\) 9785.34i 1.19175i
\(408\) 0 0
\(409\) −10156.3 −1.22786 −0.613930 0.789361i \(-0.710412\pi\)
−0.613930 + 0.789361i \(0.710412\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 22307.6i − 2.66752i
\(413\) − 10507.5i − 1.25191i
\(414\) 0 0
\(415\) 0 0
\(416\) 19736.9 2.32615
\(417\) 0 0
\(418\) − 15740.4i − 1.84184i
\(419\) 15878.8 1.85139 0.925693 0.378275i \(-0.123483\pi\)
0.925693 + 0.378275i \(0.123483\pi\)
\(420\) 0 0
\(421\) −2279.85 −0.263926 −0.131963 0.991255i \(-0.542128\pi\)
−0.131963 + 0.991255i \(0.542128\pi\)
\(422\) − 25105.8i − 2.89604i
\(423\) 0 0
\(424\) 18382.5 2.10551
\(425\) 0 0
\(426\) 0 0
\(427\) − 14508.6i − 1.64431i
\(428\) − 19489.0i − 2.20102i
\(429\) 0 0
\(430\) 0 0
\(431\) 1947.38 0.217638 0.108819 0.994062i \(-0.465293\pi\)
0.108819 + 0.994062i \(0.465293\pi\)
\(432\) 0 0
\(433\) 12636.2i 1.40244i 0.712946 + 0.701219i \(0.247361\pi\)
−0.712946 + 0.701219i \(0.752639\pi\)
\(434\) −2315.95 −0.256150
\(435\) 0 0
\(436\) −7688.25 −0.844496
\(437\) − 16165.6i − 1.76957i
\(438\) 0 0
\(439\) 15849.8 1.72317 0.861585 0.507614i \(-0.169472\pi\)
0.861585 + 0.507614i \(0.169472\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 23184.0i 2.49491i
\(443\) 17455.6i 1.87210i 0.351872 + 0.936048i \(0.385545\pi\)
−0.351872 + 0.936048i \(0.614455\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −14429.2 −1.53193
\(447\) 0 0
\(448\) 9721.41i 1.02521i
\(449\) −16068.1 −1.68887 −0.844435 0.535658i \(-0.820064\pi\)
−0.844435 + 0.535658i \(0.820064\pi\)
\(450\) 0 0
\(451\) 5679.94 0.593033
\(452\) 20612.7i 2.14500i
\(453\) 0 0
\(454\) 26530.4 2.74258
\(455\) 0 0
\(456\) 0 0
\(457\) − 11891.7i − 1.21722i −0.793468 0.608612i \(-0.791727\pi\)
0.793468 0.608612i \(-0.208273\pi\)
\(458\) 21311.4i 2.17428i
\(459\) 0 0
\(460\) 0 0
\(461\) 2802.23 0.283108 0.141554 0.989931i \(-0.454790\pi\)
0.141554 + 0.989931i \(0.454790\pi\)
\(462\) 0 0
\(463\) − 12933.3i − 1.29819i −0.760707 0.649096i \(-0.775147\pi\)
0.760707 0.649096i \(-0.224853\pi\)
\(464\) −30110.6 −3.01261
\(465\) 0 0
\(466\) 1859.83 0.184882
\(467\) 5748.11i 0.569573i 0.958591 + 0.284787i \(0.0919228\pi\)
−0.958591 + 0.284787i \(0.908077\pi\)
\(468\) 0 0
\(469\) −4779.65 −0.470583
\(470\) 0 0
\(471\) 0 0
\(472\) − 24739.3i − 2.41254i
\(473\) 9706.17i 0.943531i
\(474\) 0 0
\(475\) 0 0
\(476\) −31684.3 −3.05094
\(477\) 0 0
\(478\) − 1828.98i − 0.175012i
\(479\) −11217.3 −1.07000 −0.535002 0.844851i \(-0.679689\pi\)
−0.535002 + 0.844851i \(0.679689\pi\)
\(480\) 0 0
\(481\) 22077.2 2.09279
\(482\) − 32064.9i − 3.03012i
\(483\) 0 0
\(484\) 9350.83 0.878177
\(485\) 0 0
\(486\) 0 0
\(487\) 8905.12i 0.828603i 0.910140 + 0.414301i \(0.135974\pi\)
−0.910140 + 0.414301i \(0.864026\pi\)
\(488\) − 34159.7i − 3.16872i
\(489\) 0 0
\(490\) 0 0
\(491\) 6553.10 0.602316 0.301158 0.953574i \(-0.402627\pi\)
0.301158 + 0.953574i \(0.402627\pi\)
\(492\) 0 0
\(493\) − 14020.1i − 1.28079i
\(494\) −35512.8 −3.23440
\(495\) 0 0
\(496\) −2671.96 −0.241884
\(497\) 10393.9i 0.938087i
\(498\) 0 0
\(499\) 4610.09 0.413579 0.206789 0.978385i \(-0.433698\pi\)
0.206789 + 0.978385i \(0.433698\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 16878.9i − 1.50069i
\(503\) − 13069.1i − 1.15850i −0.815151 0.579249i \(-0.803346\pi\)
0.815151 0.579249i \(-0.196654\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −23334.9 −2.05012
\(507\) 0 0
\(508\) − 14744.9i − 1.28779i
\(509\) 15930.8 1.38727 0.693635 0.720327i \(-0.256008\pi\)
0.693635 + 0.720327i \(0.256008\pi\)
\(510\) 0 0
\(511\) 22673.6 1.96286
\(512\) − 23102.1i − 1.99410i
\(513\) 0 0
\(514\) −18473.8 −1.58530
\(515\) 0 0
\(516\) 0 0
\(517\) − 145.035i − 0.0123378i
\(518\) 42844.3i 3.63411i
\(519\) 0 0
\(520\) 0 0
\(521\) −3654.38 −0.307296 −0.153648 0.988126i \(-0.549102\pi\)
−0.153648 + 0.988126i \(0.549102\pi\)
\(522\) 0 0
\(523\) − 5138.66i − 0.429633i −0.976654 0.214816i \(-0.931085\pi\)
0.976654 0.214816i \(-0.0689153\pi\)
\(524\) −23123.7 −1.92780
\(525\) 0 0
\(526\) 22969.2 1.90400
\(527\) − 1244.11i − 0.102836i
\(528\) 0 0
\(529\) −11798.1 −0.969681
\(530\) 0 0
\(531\) 0 0
\(532\) − 48533.4i − 3.95524i
\(533\) − 12814.8i − 1.04141i
\(534\) 0 0
\(535\) 0 0
\(536\) −11253.4 −0.906854
\(537\) 0 0
\(538\) − 17785.9i − 1.42529i
\(539\) −7316.27 −0.584664
\(540\) 0 0
\(541\) 6932.06 0.550892 0.275446 0.961317i \(-0.411175\pi\)
0.275446 + 0.961317i \(0.411175\pi\)
\(542\) 3725.80i 0.295271i
\(543\) 0 0
\(544\) −20575.9 −1.62166
\(545\) 0 0
\(546\) 0 0
\(547\) 3423.11i 0.267572i 0.991010 + 0.133786i \(0.0427135\pi\)
−0.991010 + 0.133786i \(0.957287\pi\)
\(548\) − 43810.8i − 3.41516i
\(549\) 0 0
\(550\) 0 0
\(551\) 21475.6 1.66042
\(552\) 0 0
\(553\) 596.115i 0.0458398i
\(554\) 3419.59 0.262246
\(555\) 0 0
\(556\) −25826.3 −1.96992
\(557\) − 24489.2i − 1.86291i −0.363856 0.931455i \(-0.618540\pi\)
0.363856 0.931455i \(-0.381460\pi\)
\(558\) 0 0
\(559\) 21898.6 1.65691
\(560\) 0 0
\(561\) 0 0
\(562\) − 7873.37i − 0.590957i
\(563\) 10053.1i 0.752552i 0.926508 + 0.376276i \(0.122796\pi\)
−0.926508 + 0.376276i \(0.877204\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −20315.8 −1.50872
\(567\) 0 0
\(568\) 24471.8i 1.80777i
\(569\) −6670.45 −0.491459 −0.245729 0.969338i \(-0.579027\pi\)
−0.245729 + 0.969338i \(0.579027\pi\)
\(570\) 0 0
\(571\) 4633.55 0.339594 0.169797 0.985479i \(-0.445689\pi\)
0.169797 + 0.985479i \(0.445689\pi\)
\(572\) 36099.9i 2.63884i
\(573\) 0 0
\(574\) 24869.1 1.80839
\(575\) 0 0
\(576\) 0 0
\(577\) − 7045.15i − 0.508307i −0.967164 0.254154i \(-0.918203\pi\)
0.967164 0.254154i \(-0.0817969\pi\)
\(578\) 1381.35i 0.0994062i
\(579\) 0 0
\(580\) 0 0
\(581\) 13319.7 0.951108
\(582\) 0 0
\(583\) 9273.82i 0.658804i
\(584\) 53383.7 3.78259
\(585\) 0 0
\(586\) −41859.0 −2.95082
\(587\) − 8001.06i − 0.562588i −0.959622 0.281294i \(-0.909236\pi\)
0.959622 0.281294i \(-0.0907636\pi\)
\(588\) 0 0
\(589\) 1905.71 0.133316
\(590\) 0 0
\(591\) 0 0
\(592\) 49430.4i 3.43172i
\(593\) − 6747.53i − 0.467265i −0.972325 0.233632i \(-0.924939\pi\)
0.972325 0.233632i \(-0.0750612\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4949.26 −0.340150
\(597\) 0 0
\(598\) 52646.9i 3.60016i
\(599\) 21547.3 1.46978 0.734890 0.678186i \(-0.237234\pi\)
0.734890 + 0.678186i \(0.237234\pi\)
\(600\) 0 0
\(601\) −12155.1 −0.824983 −0.412492 0.910961i \(-0.635341\pi\)
−0.412492 + 0.910961i \(0.635341\pi\)
\(602\) 42497.6i 2.87720i
\(603\) 0 0
\(604\) 9681.64 0.652219
\(605\) 0 0
\(606\) 0 0
\(607\) − 16348.9i − 1.09322i −0.837388 0.546608i \(-0.815919\pi\)
0.837388 0.546608i \(-0.184081\pi\)
\(608\) − 31517.7i − 2.10232i
\(609\) 0 0
\(610\) 0 0
\(611\) −327.220 −0.0216660
\(612\) 0 0
\(613\) − 29955.5i − 1.97372i −0.161581 0.986859i \(-0.551659\pi\)
0.161581 0.986859i \(-0.448341\pi\)
\(614\) 528.202 0.0347174
\(615\) 0 0
\(616\) −40632.4 −2.65767
\(617\) 2159.74i 0.140921i 0.997515 + 0.0704603i \(0.0224468\pi\)
−0.997515 + 0.0704603i \(0.977553\pi\)
\(618\) 0 0
\(619\) −22100.8 −1.43507 −0.717535 0.696523i \(-0.754730\pi\)
−0.717535 + 0.696523i \(0.754730\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 39965.0i 2.57629i
\(623\) 2052.96i 0.132023i
\(624\) 0 0
\(625\) 0 0
\(626\) 6995.68 0.446651
\(627\) 0 0
\(628\) − 443.893i − 0.0282058i
\(629\) −23015.7 −1.45898
\(630\) 0 0
\(631\) 18360.1 1.15833 0.579164 0.815211i \(-0.303379\pi\)
0.579164 + 0.815211i \(0.303379\pi\)
\(632\) 1403.52i 0.0883372i
\(633\) 0 0
\(634\) −39641.2 −2.48321
\(635\) 0 0
\(636\) 0 0
\(637\) 16506.6i 1.02671i
\(638\) − 30999.9i − 1.92367i
\(639\) 0 0
\(640\) 0 0
\(641\) 21064.5 1.29797 0.648985 0.760802i \(-0.275194\pi\)
0.648985 + 0.760802i \(0.275194\pi\)
\(642\) 0 0
\(643\) − 10539.1i − 0.646381i −0.946334 0.323190i \(-0.895245\pi\)
0.946334 0.323190i \(-0.104755\pi\)
\(644\) −71949.8 −4.40251
\(645\) 0 0
\(646\) 37022.4 2.25484
\(647\) 22553.4i 1.37043i 0.728343 + 0.685213i \(0.240291\pi\)
−0.728343 + 0.685213i \(0.759709\pi\)
\(648\) 0 0
\(649\) 12480.8 0.754873
\(650\) 0 0
\(651\) 0 0
\(652\) 76799.4i 4.61303i
\(653\) − 22624.0i − 1.35582i −0.735147 0.677908i \(-0.762887\pi\)
0.735147 0.677908i \(-0.237113\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 28692.1 1.70768
\(657\) 0 0
\(658\) − 635.022i − 0.0376227i
\(659\) 6376.60 0.376930 0.188465 0.982080i \(-0.439649\pi\)
0.188465 + 0.982080i \(0.439649\pi\)
\(660\) 0 0
\(661\) −22097.5 −1.30029 −0.650146 0.759809i \(-0.725292\pi\)
−0.650146 + 0.759809i \(0.725292\pi\)
\(662\) − 34246.9i − 2.01064i
\(663\) 0 0
\(664\) 31360.5 1.83287
\(665\) 0 0
\(666\) 0 0
\(667\) − 31837.2i − 1.84819i
\(668\) 12797.9i 0.741264i
\(669\) 0 0
\(670\) 0 0
\(671\) 17233.3 0.991479
\(672\) 0 0
\(673\) 24033.0i 1.37653i 0.725461 + 0.688264i \(0.241627\pi\)
−0.725461 + 0.688264i \(0.758373\pi\)
\(674\) −15326.1 −0.875873
\(675\) 0 0
\(676\) 39600.6 2.25311
\(677\) 179.638i 0.0101980i 0.999987 + 0.00509901i \(0.00162307\pi\)
−0.999987 + 0.00509901i \(0.998377\pi\)
\(678\) 0 0
\(679\) −20194.9 −1.14140
\(680\) 0 0
\(681\) 0 0
\(682\) − 2750.88i − 0.154452i
\(683\) 30434.2i 1.70502i 0.522709 + 0.852511i \(0.324921\pi\)
−0.522709 + 0.852511i \(0.675079\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 11494.2 0.639725
\(687\) 0 0
\(688\) 49030.5i 2.71696i
\(689\) 20923.1 1.15691
\(690\) 0 0
\(691\) 9792.73 0.539122 0.269561 0.962983i \(-0.413121\pi\)
0.269561 + 0.962983i \(0.413121\pi\)
\(692\) 31113.7i 1.70920i
\(693\) 0 0
\(694\) −44171.8 −2.41605
\(695\) 0 0
\(696\) 0 0
\(697\) 13359.6i 0.726010i
\(698\) − 29365.8i − 1.59242i
\(699\) 0 0
\(700\) 0 0
\(701\) −8130.47 −0.438065 −0.219032 0.975718i \(-0.570290\pi\)
−0.219032 + 0.975718i \(0.570290\pi\)
\(702\) 0 0
\(703\) − 35255.0i − 1.89142i
\(704\) −11547.1 −0.618177
\(705\) 0 0
\(706\) −6354.86 −0.338765
\(707\) 20096.4i 1.06903i
\(708\) 0 0
\(709\) 4859.95 0.257432 0.128716 0.991681i \(-0.458914\pi\)
0.128716 + 0.991681i \(0.458914\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 4833.59i 0.254419i
\(713\) − 2825.17i − 0.148392i
\(714\) 0 0
\(715\) 0 0
\(716\) −6499.96 −0.339267
\(717\) 0 0
\(718\) − 21590.3i − 1.12220i
\(719\) 19463.0 1.00952 0.504762 0.863259i \(-0.331580\pi\)
0.504762 + 0.863259i \(0.331580\pi\)
\(720\) 0 0
\(721\) 28578.6 1.47618
\(722\) 21038.8i 1.08447i
\(723\) 0 0
\(724\) 32302.2 1.65815
\(725\) 0 0
\(726\) 0 0
\(727\) − 2432.66i − 0.124102i −0.998073 0.0620512i \(-0.980236\pi\)
0.998073 0.0620512i \(-0.0197642\pi\)
\(728\) 91672.8i 4.66706i
\(729\) 0 0
\(730\) 0 0
\(731\) −22829.5 −1.15510
\(732\) 0 0
\(733\) − 17967.6i − 0.905386i −0.891666 0.452693i \(-0.850463\pi\)
0.891666 0.452693i \(-0.149537\pi\)
\(734\) −36606.0 −1.84081
\(735\) 0 0
\(736\) −46724.4 −2.34006
\(737\) − 5677.26i − 0.283751i
\(738\) 0 0
\(739\) −23473.0 −1.16843 −0.584214 0.811599i \(-0.698597\pi\)
−0.584214 + 0.811599i \(0.698597\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 40604.6i 2.00895i
\(743\) 33559.2i 1.65702i 0.559971 + 0.828512i \(0.310812\pi\)
−0.559971 + 0.828512i \(0.689188\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 37026.5 1.81721
\(747\) 0 0
\(748\) − 37634.6i − 1.83965i
\(749\) 24967.6 1.21802
\(750\) 0 0
\(751\) −7782.75 −0.378158 −0.189079 0.981962i \(-0.560550\pi\)
−0.189079 + 0.981962i \(0.560550\pi\)
\(752\) − 732.640i − 0.0355274i
\(753\) 0 0
\(754\) −69940.5 −3.37809
\(755\) 0 0
\(756\) 0 0
\(757\) 38154.3i 1.83189i 0.401300 + 0.915946i \(0.368558\pi\)
−0.401300 + 0.915946i \(0.631442\pi\)
\(758\) − 17538.2i − 0.840390i
\(759\) 0 0
\(760\) 0 0
\(761\) −19867.1 −0.946363 −0.473182 0.880965i \(-0.656895\pi\)
−0.473182 + 0.880965i \(0.656895\pi\)
\(762\) 0 0
\(763\) − 9849.52i − 0.467335i
\(764\) −13836.7 −0.655228
\(765\) 0 0
\(766\) 20587.5 0.971094
\(767\) − 28158.5i − 1.32561i
\(768\) 0 0
\(769\) 15710.8 0.736730 0.368365 0.929681i \(-0.379918\pi\)
0.368365 + 0.929681i \(0.379918\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 80897.5i − 3.77146i
\(773\) − 25811.9i − 1.20102i −0.799617 0.600510i \(-0.794964\pi\)
0.799617 0.600510i \(-0.205036\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −47547.8 −2.19957
\(777\) 0 0
\(778\) 50207.3i 2.31365i
\(779\) −20463.9 −0.941201
\(780\) 0 0
\(781\) −12345.8 −0.565645
\(782\) − 54885.0i − 2.50982i
\(783\) 0 0
\(784\) −36958.0 −1.68358
\(785\) 0 0
\(786\) 0 0
\(787\) 29242.6i 1.32450i 0.749281 + 0.662252i \(0.230399\pi\)
−0.749281 + 0.662252i \(0.769601\pi\)
\(788\) − 51017.7i − 2.30638i
\(789\) 0 0
\(790\) 0 0
\(791\) −26407.2 −1.18702
\(792\) 0 0
\(793\) − 38880.8i − 1.74111i
\(794\) 56834.6 2.54028
\(795\) 0 0
\(796\) 28313.0 1.26071
\(797\) 32573.7i 1.44771i 0.689955 + 0.723853i \(0.257630\pi\)
−0.689955 + 0.723853i \(0.742370\pi\)
\(798\) 0 0
\(799\) 341.130 0.0151043
\(800\) 0 0
\(801\) 0 0
\(802\) − 21247.7i − 0.935514i
\(803\) 26931.6i 1.18356i
\(804\) 0 0
\(805\) 0 0
\(806\) −6206.39 −0.271229
\(807\) 0 0
\(808\) 47315.8i 2.06010i
\(809\) −34644.9 −1.50562 −0.752812 0.658236i \(-0.771303\pi\)
−0.752812 + 0.658236i \(0.771303\pi\)
\(810\) 0 0
\(811\) −29057.9 −1.25815 −0.629077 0.777343i \(-0.716567\pi\)
−0.629077 + 0.777343i \(0.716567\pi\)
\(812\) − 95583.9i − 4.13096i
\(813\) 0 0
\(814\) −50890.3 −2.19128
\(815\) 0 0
\(816\) 0 0
\(817\) − 34969.8i − 1.49748i
\(818\) − 52819.3i − 2.25768i
\(819\) 0 0
\(820\) 0 0
\(821\) −46709.5 −1.98560 −0.992798 0.119802i \(-0.961774\pi\)
−0.992798 + 0.119802i \(0.961774\pi\)
\(822\) 0 0
\(823\) − 3468.10i − 0.146890i −0.997299 0.0734450i \(-0.976601\pi\)
0.997299 0.0734450i \(-0.0233993\pi\)
\(824\) 67286.8 2.84472
\(825\) 0 0
\(826\) 54645.9 2.30191
\(827\) 42454.9i 1.78513i 0.450920 + 0.892564i \(0.351096\pi\)
−0.450920 + 0.892564i \(0.648904\pi\)
\(828\) 0 0
\(829\) 3933.47 0.164795 0.0823975 0.996600i \(-0.473742\pi\)
0.0823975 + 0.996600i \(0.473742\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 26051.9i 1.08556i
\(833\) − 17208.3i − 0.715764i
\(834\) 0 0
\(835\) 0 0
\(836\) 57647.9 2.38492
\(837\) 0 0
\(838\) 82580.5i 3.40417i
\(839\) −32959.6 −1.35625 −0.678123 0.734948i \(-0.737206\pi\)
−0.678123 + 0.734948i \(0.737206\pi\)
\(840\) 0 0
\(841\) 17906.1 0.734188
\(842\) − 11856.7i − 0.485285i
\(843\) 0 0
\(844\) 91947.6 3.74996
\(845\) 0 0
\(846\) 0 0
\(847\) 11979.5i 0.485974i
\(848\) 46846.5i 1.89707i
\(849\) 0 0
\(850\) 0 0
\(851\) −52264.8 −2.10530
\(852\) 0 0
\(853\) 38845.8i 1.55927i 0.626235 + 0.779634i \(0.284595\pi\)
−0.626235 + 0.779634i \(0.715405\pi\)
\(854\) 75454.3 3.02341
\(855\) 0 0
\(856\) 58785.0 2.34723
\(857\) − 29305.3i − 1.16809i −0.811723 0.584043i \(-0.801470\pi\)
0.811723 0.584043i \(-0.198530\pi\)
\(858\) 0 0
\(859\) 909.659 0.0361318 0.0180659 0.999837i \(-0.494249\pi\)
0.0180659 + 0.999837i \(0.494249\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 10127.7i 0.400173i
\(863\) − 47998.0i − 1.89324i −0.322346 0.946622i \(-0.604471\pi\)
0.322346 0.946622i \(-0.395529\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −65716.6 −2.57868
\(867\) 0 0
\(868\) − 8481.94i − 0.331677i
\(869\) −708.065 −0.0276403
\(870\) 0 0
\(871\) −12808.7 −0.498286
\(872\) − 23190.2i − 0.900594i
\(873\) 0 0
\(874\) 84071.7 3.25374
\(875\) 0 0
\(876\) 0 0
\(877\) − 3258.01i − 0.125445i −0.998031 0.0627225i \(-0.980022\pi\)
0.998031 0.0627225i \(-0.0199783\pi\)
\(878\) 82429.8i 3.16842i
\(879\) 0 0
\(880\) 0 0
\(881\) 33380.2 1.27651 0.638256 0.769824i \(-0.279656\pi\)
0.638256 + 0.769824i \(0.279656\pi\)
\(882\) 0 0
\(883\) 33714.6i 1.28492i 0.766319 + 0.642460i \(0.222086\pi\)
−0.766319 + 0.642460i \(0.777914\pi\)
\(884\) −84909.2 −3.23055
\(885\) 0 0
\(886\) −90780.6 −3.44225
\(887\) − 6218.80i − 0.235408i −0.993049 0.117704i \(-0.962447\pi\)
0.993049 0.117704i \(-0.0375534\pi\)
\(888\) 0 0
\(889\) 18889.8 0.712649
\(890\) 0 0
\(891\) 0 0
\(892\) − 52845.5i − 1.98363i
\(893\) 522.537i 0.0195812i
\(894\) 0 0
\(895\) 0 0
\(896\) 8361.46 0.311760
\(897\) 0 0
\(898\) − 83565.1i − 3.10535i
\(899\) 3753.19 0.139239
\(900\) 0 0
\(901\) −21812.6 −0.806529
\(902\) 29539.5i 1.09042i
\(903\) 0 0
\(904\) −62174.4 −2.28749
\(905\) 0 0
\(906\) 0 0
\(907\) 22878.6i 0.837566i 0.908086 + 0.418783i \(0.137543\pi\)
−0.908086 + 0.418783i \(0.862457\pi\)
\(908\) 97165.0i 3.55125i
\(909\) 0 0
\(910\) 0 0
\(911\) 30144.8 1.09631 0.548157 0.836376i \(-0.315330\pi\)
0.548157 + 0.836376i \(0.315330\pi\)
\(912\) 0 0
\(913\) 15821.1i 0.573496i
\(914\) 61844.9 2.23813
\(915\) 0 0
\(916\) −78051.2 −2.81538
\(917\) − 29624.1i − 1.06682i
\(918\) 0 0
\(919\) −3803.52 −0.136525 −0.0682625 0.997667i \(-0.521746\pi\)
−0.0682625 + 0.997667i \(0.521746\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 14573.5i 0.520555i
\(923\) 27854.0i 0.993312i
\(924\) 0 0
\(925\) 0 0
\(926\) 67262.0 2.38700
\(927\) 0 0
\(928\) − 62072.5i − 2.19572i
\(929\) 125.985 0.00444934 0.00222467 0.999998i \(-0.499292\pi\)
0.00222467 + 0.999998i \(0.499292\pi\)
\(930\) 0 0
\(931\) 26359.3 0.927918
\(932\) 6811.45i 0.239395i
\(933\) 0 0
\(934\) −29894.0 −1.04728
\(935\) 0 0
\(936\) 0 0
\(937\) − 28107.9i − 0.979984i −0.871727 0.489992i \(-0.837000\pi\)
0.871727 0.489992i \(-0.163000\pi\)
\(938\) − 24857.4i − 0.865268i
\(939\) 0 0
\(940\) 0 0
\(941\) 49194.9 1.70426 0.852130 0.523330i \(-0.175311\pi\)
0.852130 + 0.523330i \(0.175311\pi\)
\(942\) 0 0
\(943\) 30337.3i 1.04763i
\(944\) 63046.3 2.17371
\(945\) 0 0
\(946\) −50478.6 −1.73488
\(947\) − 14498.0i − 0.497490i −0.968569 0.248745i \(-0.919982\pi\)
0.968569 0.248745i \(-0.0800181\pi\)
\(948\) 0 0
\(949\) 60761.7 2.07841
\(950\) 0 0
\(951\) 0 0
\(952\) − 95569.8i − 3.25361i
\(953\) − 3201.79i − 0.108831i −0.998518 0.0544155i \(-0.982670\pi\)
0.998518 0.0544155i \(-0.0173296\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 6698.48 0.226616
\(957\) 0 0
\(958\) − 58337.6i − 1.96743i
\(959\) 56126.7 1.88991
\(960\) 0 0
\(961\) −29457.9 −0.988820
\(962\) 114816.i 3.84805i
\(963\) 0 0
\(964\) 117435. 3.92356
\(965\) 0 0
\(966\) 0 0
\(967\) − 57781.9i − 1.92155i −0.277326 0.960776i \(-0.589448\pi\)
0.277326 0.960776i \(-0.410552\pi\)
\(968\) 28205.1i 0.936513i
\(969\) 0 0
\(970\) 0 0
\(971\) −2611.60 −0.0863133 −0.0431566 0.999068i \(-0.513741\pi\)
−0.0431566 + 0.999068i \(0.513741\pi\)
\(972\) 0 0
\(973\) − 33086.4i − 1.09013i
\(974\) −46312.6 −1.52356
\(975\) 0 0
\(976\) 87053.4 2.85503
\(977\) − 33598.3i − 1.10021i −0.835096 0.550104i \(-0.814588\pi\)
0.835096 0.550104i \(-0.185412\pi\)
\(978\) 0 0
\(979\) −2438.51 −0.0796067
\(980\) 0 0
\(981\) 0 0
\(982\) 34080.5i 1.10749i
\(983\) 39484.8i 1.28115i 0.767895 + 0.640575i \(0.221304\pi\)
−0.767895 + 0.640575i \(0.778696\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 72913.7 2.35501
\(987\) 0 0
\(988\) − 130062.i − 4.18809i
\(989\) −51841.9 −1.66681
\(990\) 0 0
\(991\) 39918.6 1.27957 0.639786 0.768553i \(-0.279023\pi\)
0.639786 + 0.768553i \(0.279023\pi\)
\(992\) − 5508.20i − 0.176296i
\(993\) 0 0
\(994\) −54055.2 −1.72487
\(995\) 0 0
\(996\) 0 0
\(997\) 25670.3i 0.815432i 0.913109 + 0.407716i \(0.133675\pi\)
−0.913109 + 0.407716i \(0.866325\pi\)
\(998\) 23975.6i 0.760454i
\(999\) 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 675.4.b.l.649.6 6
3.2 odd 2 675.4.b.k.649.1 6
5.2 odd 4 135.4.a.f.1.1 3
5.3 odd 4 675.4.a.r.1.3 3
5.4 even 2 inner 675.4.b.l.649.1 6
15.2 even 4 135.4.a.g.1.3 yes 3
15.8 even 4 675.4.a.q.1.1 3
15.14 odd 2 675.4.b.k.649.6 6
20.7 even 4 2160.4.a.bm.1.1 3
45.2 even 12 405.4.e.r.271.1 6
45.7 odd 12 405.4.e.t.271.3 6
45.22 odd 12 405.4.e.t.136.3 6
45.32 even 12 405.4.e.r.136.1 6
60.47 odd 4 2160.4.a.be.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.1 3 5.2 odd 4
135.4.a.g.1.3 yes 3 15.2 even 4
405.4.e.r.136.1 6 45.32 even 12
405.4.e.r.271.1 6 45.2 even 12
405.4.e.t.136.3 6 45.22 odd 12
405.4.e.t.271.3 6 45.7 odd 12
675.4.a.q.1.1 3 15.8 even 4
675.4.a.r.1.3 3 5.3 odd 4
675.4.b.k.649.1 6 3.2 odd 2
675.4.b.k.649.6 6 15.14 odd 2
675.4.b.l.649.1 6 5.4 even 2 inner
675.4.b.l.649.6 6 1.1 even 1 trivial
2160.4.a.be.1.1 3 60.47 odd 4
2160.4.a.bm.1.1 3 20.7 even 4