Properties

Label 675.4.b.k.649.1
Level $675$
Weight $4$
Character 675.649
Analytic conductor $39.826$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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

$q$-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 \(-0.628709\pi\)
0.393424 0.919357i \(-0.371291\pi\)
\(3\) 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 \(-0.630027\pi\)
−0.397226 + 0.917721i \(0.630027\pi\)
\(12\) 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 \(-0.838347\pi\)
0.873794 0.486296i \(-0.161653\pi\)
\(18\) 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 \(-0.752413\pi\)
0.712446 0.701727i \(-0.247587\pi\)
\(24\) 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 \(-0.271215\pi\)
0.658443 + 0.752631i \(0.271215\pi\)
\(30\) 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 \(-0.621751\pi\)
−0.373234 + 0.927737i \(0.621751\pi\)
\(42\) 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 \(0.00247169\pi\)
−0.999970 + 0.00776496i \(0.997528\pi\)
\(48\) 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 \(-0.863912\pi\)
0.909991 0.414628i \(-0.136088\pi\)
\(54\) 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 \(-0.657585\pi\)
−0.475091 + 0.879937i \(0.657585\pi\)
\(60\) 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 \(-0.384141\pi\)
0.355997 + 0.934487i \(0.384141\pi\)
\(72\) 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 \(-0.882456\pi\)
0.932590 0.360938i \(-0.117544\pi\)
\(84\) 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 \(-0.484045\pi\)
0.0501017 + 0.998744i \(0.484045\pi\)
\(90\) 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 \(-0.367032\pi\)
0.405688 + 0.914012i \(0.367032\pi\)
\(102\) 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 \(-0.847049\pi\)
0.886760 0.462230i \(-0.152951\pi\)
\(108\) 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 \(0.148742\pi\)
−0.892794 + 0.450465i \(0.851258\pi\)
\(114\) 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 \(-0.632677\pi\)
−0.404851 + 0.914383i \(0.632677\pi\)
\(132\) 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 \(-0.745420\pi\)
0.696860 0.717207i \(-0.254580\pi\)
\(138\) 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 \(-0.522758\pi\)
−0.0714340 + 0.997445i \(0.522758\pi\)
\(150\) 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 \(0.0497539\pi\)
−0.987809 + 0.155671i \(0.950246\pi\)
\(168\) 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 \(0.116863\pi\)
−0.933359 + 0.358944i \(0.883137\pi\)
\(174\) 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 \(-0.522698\pi\)
−0.0712485 + 0.997459i \(0.522698\pi\)
\(180\) 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 \(-0.543940\pi\)
−0.137603 + 0.990488i \(0.543940\pi\)
\(192\) 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 \(-0.839054\pi\)
0.874871 0.484357i \(-0.160946\pi\)
\(198\) 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 \(0.267927\pi\)
−0.666184 + 0.745788i \(0.732073\pi\)
\(228\) 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 \(0.0160097\pi\)
−0.998735 + 0.0502747i \(0.983990\pi\)
\(234\) 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 \(-0.484846\pi\)
0.0475909 + 0.998867i \(0.484846\pi\)
\(240\) 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 \(-0.366198\pi\)
0.408080 + 0.912946i \(0.366198\pi\)
\(252\) 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 \(-0.858129\pi\)
0.902309 0.431089i \(-0.141871\pi\)
\(258\) 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 \(0.173232\pi\)
−0.855530 + 0.517754i \(0.826768\pi\)
\(264\) 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 \(-0.373312\pi\)
0.387578 + 0.921837i \(0.373312\pi\)
\(270\) 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 \(-0.448625\pi\)
0.160698 + 0.987004i \(0.448625\pi\)
\(282\) 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 \(-0.703549\pi\)
0.596770 0.802413i \(-0.296451\pi\)
\(294\) 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 \(-0.747070\pi\)
−0.700567 + 0.713586i \(0.747070\pi\)
\(312\) 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 \(-0.764033\pi\)
0.737583 0.675257i \(-0.235967\pi\)
\(318\) 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 \(-0.771828\pi\)
0.753896 0.656994i \(-0.228172\pi\)
\(348\) 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 \(-0.970636\pi\)
0.995748 0.0921202i \(-0.0293644\pi\)
\(354\) 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 \(-0.401290\pi\)
0.305160 + 0.952301i \(0.401290\pi\)
\(360\) 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 \(0.0850646\pi\)
−0.964504 + 0.264069i \(0.914935\pi\)
\(384\) 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 \(-0.716596\pi\)
−0.629148 + 0.777285i \(0.716596\pi\)
\(390\) 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 \(-0.418124\pi\)
0.254393 + 0.967101i \(0.418124\pi\)
\(402\) 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 \(-0.876517\pi\)
−0.925693 + 0.378275i \(0.876517\pi\)
\(420\) 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 \(-0.534707\pi\)
−0.108819 + 0.994062i \(0.534707\pi\)
\(432\) 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 \(-0.614455\pi\)
0.351872 0.936048i \(-0.385545\pi\)
\(444\) 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 \(-0.179936\pi\)
0.844435 + 0.535658i \(0.179936\pi\)
\(450\) 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 \(-0.545210\pi\)
−0.141554 + 0.989931i \(0.545210\pi\)
\(462\) 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 \(-0.908077\pi\)
0.958591 0.284787i \(-0.0919228\pi\)
\(468\) 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 \(-0.320311\pi\)
0.535002 + 0.844851i \(0.320311\pi\)
\(480\) 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 \(-0.597373\pi\)
−0.301158 + 0.953574i \(0.597373\pi\)
\(492\) 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 \(0.196654\pi\)
−0.815151 + 0.579249i \(0.803346\pi\)
\(504\) 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 \(-0.743992\pi\)
−0.693635 + 0.720327i \(0.743992\pi\)
\(510\) 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 \(-0.450898\pi\)
0.153648 + 0.988126i \(0.450898\pi\)
\(522\) 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 \(0.381460\pi\)
−0.363856 + 0.931455i \(0.618540\pi\)
\(558\) 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 \(-0.877204\pi\)
0.926508 0.376276i \(-0.122796\pi\)
\(564\) 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 \(-0.420973\pi\)
0.245729 + 0.969338i \(0.420973\pi\)
\(570\) 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 \(0.0907636\pi\)
−0.959622 + 0.281294i \(0.909236\pi\)
\(588\) 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 \(0.0750612\pi\)
−0.972325 + 0.233632i \(0.924939\pi\)
\(594\) 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 \(-0.762766\pi\)
−0.734890 + 0.678186i \(0.762766\pi\)
\(600\) 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 \(-0.977553\pi\)
0.997515 0.0704603i \(-0.0224468\pi\)
\(618\) 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 \(-0.724806\pi\)
−0.648985 + 0.760802i \(0.724806\pi\)
\(642\) 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 \(-0.759709\pi\)
0.728343 0.685213i \(-0.240291\pi\)
\(648\) 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 \(0.237113\pi\)
−0.735147 + 0.677908i \(0.762887\pi\)
\(654\) 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 \(-0.560351\pi\)
−0.188465 + 0.982080i \(0.560351\pi\)
\(660\) 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 \(-0.998377\pi\)
0.999987 0.00509901i \(-0.00162307\pi\)
\(678\) 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 \(-0.675079\pi\)
0.522709 0.852511i \(-0.324921\pi\)
\(684\) 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 \(-0.429710\pi\)
0.219032 + 0.975718i \(0.429710\pi\)
\(702\) 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 \(-0.668420\pi\)
−0.504762 + 0.863259i \(0.668420\pi\)
\(720\) 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 \(-0.689188\pi\)
0.559971 0.828512i \(-0.310812\pi\)
\(744\) 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 \(-0.343105\pi\)
0.473182 + 0.880965i \(0.343105\pi\)
\(762\) 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 \(0.205036\pi\)
−0.799617 + 0.600510i \(0.794964\pi\)
\(774\) 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 \(-0.742370\pi\)
0.689955 0.723853i \(-0.257630\pi\)
\(798\) 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 \(-0.228697\pi\)
0.752812 + 0.658236i \(0.228697\pi\)
\(810\) 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 \(-0.0382259\pi\)
0.992798 + 0.119802i \(0.0382259\pi\)
\(822\) 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 \(-0.648904\pi\)
0.450920 0.892564i \(-0.351096\pi\)
\(828\) 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 \(-0.262794\pi\)
0.678123 + 0.734948i \(0.262794\pi\)
\(840\) 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 \(0.198530\pi\)
−0.811723 + 0.584043i \(0.801470\pi\)
\(858\) 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 \(0.395529\pi\)
−0.322346 + 0.946622i \(0.604471\pi\)
\(864\) 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 \(-0.720344\pi\)
−0.638256 + 0.769824i \(0.720344\pi\)
\(882\) 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 \(0.0375534\pi\)
−0.993049 + 0.117704i \(0.962447\pi\)
\(888\) 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 \(-0.684670\pi\)
−0.548157 + 0.836376i \(0.684670\pi\)
\(912\) 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 \(-0.500708\pi\)
−0.00222467 + 0.999998i \(0.500708\pi\)
\(930\) 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 \(-0.824689\pi\)
−0.852130 + 0.523330i \(0.824689\pi\)
\(942\) 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 \(0.0800181\pi\)
−0.968569 + 0.248745i \(0.919982\pi\)
\(948\) 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 \(0.0173296\pi\)
−0.998518 + 0.0544155i \(0.982670\pi\)
\(954\) 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 \(-0.486259\pi\)
0.0431566 + 0.999068i \(0.486259\pi\)
\(972\) 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 \(0.185412\pi\)
−0.835096 + 0.550104i \(0.814588\pi\)
\(978\) 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 \(-0.778696\pi\)
0.767895 0.640575i \(-0.221304\pi\)
\(984\) 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.k.649.1 6
3.2 odd 2 675.4.b.l.649.6 6
5.2 odd 4 135.4.a.g.1.3 yes 3
5.3 odd 4 675.4.a.q.1.1 3
5.4 even 2 inner 675.4.b.k.649.6 6
15.2 even 4 135.4.a.f.1.1 3
15.8 even 4 675.4.a.r.1.3 3
15.14 odd 2 675.4.b.l.649.1 6
20.7 even 4 2160.4.a.be.1.1 3
45.2 even 12 405.4.e.t.271.3 6
45.7 odd 12 405.4.e.r.271.1 6
45.22 odd 12 405.4.e.r.136.1 6
45.32 even 12 405.4.e.t.136.3 6
60.47 odd 4 2160.4.a.bm.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.1 3 15.2 even 4
135.4.a.g.1.3 yes 3 5.2 odd 4
405.4.e.r.136.1 6 45.22 odd 12
405.4.e.r.271.1 6 45.7 odd 12
405.4.e.t.136.3 6 45.32 even 12
405.4.e.t.271.3 6 45.2 even 12
675.4.a.q.1.1 3 5.3 odd 4
675.4.a.r.1.3 3 15.8 even 4
675.4.b.k.649.1 6 1.1 even 1 trivial
675.4.b.k.649.6 6 5.4 even 2 inner
675.4.b.l.649.1 6 15.14 odd 2
675.4.b.l.649.6 6 3.2 odd 2
2160.4.a.be.1.1 3 20.7 even 4
2160.4.a.bm.1.1 3 60.47 odd 4