Properties

Label 2475.4.a.k.1.1
Level $2475$
Weight $4$
Character 2475.1
Self dual yes
Analytic conductor $146.030$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(146.029727264\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.00000 q^{2} +17.0000 q^{4} -3.00000 q^{7} +45.0000 q^{8} +O(q^{10})\) \(q+5.00000 q^{2} +17.0000 q^{4} -3.00000 q^{7} +45.0000 q^{8} +11.0000 q^{11} -32.0000 q^{13} -15.0000 q^{14} +89.0000 q^{16} +33.0000 q^{17} +47.0000 q^{19} +55.0000 q^{22} +113.000 q^{23} -160.000 q^{26} -51.0000 q^{28} +54.0000 q^{29} +178.000 q^{31} +85.0000 q^{32} +165.000 q^{34} -19.0000 q^{37} +235.000 q^{38} -139.000 q^{41} +308.000 q^{43} +187.000 q^{44} +565.000 q^{46} +195.000 q^{47} -334.000 q^{49} -544.000 q^{52} +152.000 q^{53} -135.000 q^{56} +270.000 q^{58} +625.000 q^{59} +320.000 q^{61} +890.000 q^{62} -287.000 q^{64} -200.000 q^{67} +561.000 q^{68} +947.000 q^{71} +448.000 q^{73} -95.0000 q^{74} +799.000 q^{76} -33.0000 q^{77} -721.000 q^{79} -695.000 q^{82} +142.000 q^{83} +1540.00 q^{86} +495.000 q^{88} -404.000 q^{89} +96.0000 q^{91} +1921.00 q^{92} +975.000 q^{94} -79.0000 q^{97} -1670.00 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.00000 1.76777 0.883883 0.467707i \(-0.154920\pi\)
0.883883 + 0.467707i \(0.154920\pi\)
\(3\) 0 0
\(4\) 17.0000 2.12500
\(5\) 0 0
\(6\) 0 0
\(7\) −3.00000 −0.161985 −0.0809924 0.996715i \(-0.525809\pi\)
−0.0809924 + 0.996715i \(0.525809\pi\)
\(8\) 45.0000 1.98874
\(9\) 0 0
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) −32.0000 −0.682708 −0.341354 0.939935i \(-0.610885\pi\)
−0.341354 + 0.939935i \(0.610885\pi\)
\(14\) −15.0000 −0.286351
\(15\) 0 0
\(16\) 89.0000 1.39062
\(17\) 33.0000 0.470804 0.235402 0.971898i \(-0.424359\pi\)
0.235402 + 0.971898i \(0.424359\pi\)
\(18\) 0 0
\(19\) 47.0000 0.567502 0.283751 0.958898i \(-0.408421\pi\)
0.283751 + 0.958898i \(0.408421\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 55.0000 0.533002
\(23\) 113.000 1.02444 0.512220 0.858854i \(-0.328823\pi\)
0.512220 + 0.858854i \(0.328823\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −160.000 −1.20687
\(27\) 0 0
\(28\) −51.0000 −0.344218
\(29\) 54.0000 0.345778 0.172889 0.984941i \(-0.444690\pi\)
0.172889 + 0.984941i \(0.444690\pi\)
\(30\) 0 0
\(31\) 178.000 1.03128 0.515641 0.856805i \(-0.327554\pi\)
0.515641 + 0.856805i \(0.327554\pi\)
\(32\) 85.0000 0.469563
\(33\) 0 0
\(34\) 165.000 0.832273
\(35\) 0 0
\(36\) 0 0
\(37\) −19.0000 −0.0844211 −0.0422106 0.999109i \(-0.513440\pi\)
−0.0422106 + 0.999109i \(0.513440\pi\)
\(38\) 235.000 1.00321
\(39\) 0 0
\(40\) 0 0
\(41\) −139.000 −0.529467 −0.264734 0.964322i \(-0.585284\pi\)
−0.264734 + 0.964322i \(0.585284\pi\)
\(42\) 0 0
\(43\) 308.000 1.09232 0.546158 0.837682i \(-0.316090\pi\)
0.546158 + 0.837682i \(0.316090\pi\)
\(44\) 187.000 0.640712
\(45\) 0 0
\(46\) 565.000 1.81097
\(47\) 195.000 0.605185 0.302592 0.953120i \(-0.402148\pi\)
0.302592 + 0.953120i \(0.402148\pi\)
\(48\) 0 0
\(49\) −334.000 −0.973761
\(50\) 0 0
\(51\) 0 0
\(52\) −544.000 −1.45075
\(53\) 152.000 0.393940 0.196970 0.980410i \(-0.436890\pi\)
0.196970 + 0.980410i \(0.436890\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −135.000 −0.322145
\(57\) 0 0
\(58\) 270.000 0.611254
\(59\) 625.000 1.37912 0.689560 0.724229i \(-0.257804\pi\)
0.689560 + 0.724229i \(0.257804\pi\)
\(60\) 0 0
\(61\) 320.000 0.671669 0.335834 0.941921i \(-0.390982\pi\)
0.335834 + 0.941921i \(0.390982\pi\)
\(62\) 890.000 1.82307
\(63\) 0 0
\(64\) −287.000 −0.560547
\(65\) 0 0
\(66\) 0 0
\(67\) −200.000 −0.364685 −0.182342 0.983235i \(-0.558368\pi\)
−0.182342 + 0.983235i \(0.558368\pi\)
\(68\) 561.000 1.00046
\(69\) 0 0
\(70\) 0 0
\(71\) 947.000 1.58293 0.791466 0.611213i \(-0.209318\pi\)
0.791466 + 0.611213i \(0.209318\pi\)
\(72\) 0 0
\(73\) 448.000 0.718280 0.359140 0.933284i \(-0.383070\pi\)
0.359140 + 0.933284i \(0.383070\pi\)
\(74\) −95.0000 −0.149237
\(75\) 0 0
\(76\) 799.000 1.20594
\(77\) −33.0000 −0.0488402
\(78\) 0 0
\(79\) −721.000 −1.02682 −0.513410 0.858143i \(-0.671618\pi\)
−0.513410 + 0.858143i \(0.671618\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −695.000 −0.935975
\(83\) 142.000 0.187789 0.0938947 0.995582i \(-0.470068\pi\)
0.0938947 + 0.995582i \(0.470068\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1540.00 1.93096
\(87\) 0 0
\(88\) 495.000 0.599627
\(89\) −404.000 −0.481168 −0.240584 0.970628i \(-0.577339\pi\)
−0.240584 + 0.970628i \(0.577339\pi\)
\(90\) 0 0
\(91\) 96.0000 0.110588
\(92\) 1921.00 2.17694
\(93\) 0 0
\(94\) 975.000 1.06983
\(95\) 0 0
\(96\) 0 0
\(97\) −79.0000 −0.0826931 −0.0413466 0.999145i \(-0.513165\pi\)
−0.0413466 + 0.999145i \(0.513165\pi\)
\(98\) −1670.00 −1.72138
\(99\) 0 0
\(100\) 0 0
\(101\) 545.000 0.536926 0.268463 0.963290i \(-0.413484\pi\)
0.268463 + 0.963290i \(0.413484\pi\)
\(102\) 0 0
\(103\) 1306.00 1.24936 0.624680 0.780881i \(-0.285230\pi\)
0.624680 + 0.780881i \(0.285230\pi\)
\(104\) −1440.00 −1.35773
\(105\) 0 0
\(106\) 760.000 0.696394
\(107\) 1938.00 1.75097 0.875484 0.483247i \(-0.160543\pi\)
0.875484 + 0.483247i \(0.160543\pi\)
\(108\) 0 0
\(109\) −576.000 −0.506154 −0.253077 0.967446i \(-0.581443\pi\)
−0.253077 + 0.967446i \(0.581443\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −267.000 −0.225260
\(113\) −1104.00 −0.919076 −0.459538 0.888158i \(-0.651985\pi\)
−0.459538 + 0.888158i \(0.651985\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 918.000 0.734777
\(117\) 0 0
\(118\) 3125.00 2.43796
\(119\) −99.0000 −0.0762632
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 1600.00 1.18735
\(123\) 0 0
\(124\) 3026.00 2.19147
\(125\) 0 0
\(126\) 0 0
\(127\) 1739.00 1.21505 0.607525 0.794301i \(-0.292163\pi\)
0.607525 + 0.794301i \(0.292163\pi\)
\(128\) −2115.00 −1.46048
\(129\) 0 0
\(130\) 0 0
\(131\) −1818.00 −1.21251 −0.606257 0.795269i \(-0.707330\pi\)
−0.606257 + 0.795269i \(0.707330\pi\)
\(132\) 0 0
\(133\) −141.000 −0.0919267
\(134\) −1000.00 −0.644678
\(135\) 0 0
\(136\) 1485.00 0.936307
\(137\) 870.000 0.542548 0.271274 0.962502i \(-0.412555\pi\)
0.271274 + 0.962502i \(0.412555\pi\)
\(138\) 0 0
\(139\) −636.000 −0.388092 −0.194046 0.980992i \(-0.562161\pi\)
−0.194046 + 0.980992i \(0.562161\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4735.00 2.79826
\(143\) −352.000 −0.205844
\(144\) 0 0
\(145\) 0 0
\(146\) 2240.00 1.26975
\(147\) 0 0
\(148\) −323.000 −0.179395
\(149\) 239.000 0.131407 0.0657035 0.997839i \(-0.479071\pi\)
0.0657035 + 0.997839i \(0.479071\pi\)
\(150\) 0 0
\(151\) 1208.00 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2115.00 1.12861
\(153\) 0 0
\(154\) −165.000 −0.0863382
\(155\) 0 0
\(156\) 0 0
\(157\) 1874.00 0.952621 0.476310 0.879277i \(-0.341974\pi\)
0.476310 + 0.879277i \(0.341974\pi\)
\(158\) −3605.00 −1.81518
\(159\) 0 0
\(160\) 0 0
\(161\) −339.000 −0.165944
\(162\) 0 0
\(163\) 1904.00 0.914925 0.457463 0.889229i \(-0.348758\pi\)
0.457463 + 0.889229i \(0.348758\pi\)
\(164\) −2363.00 −1.12512
\(165\) 0 0
\(166\) 710.000 0.331968
\(167\) −1180.00 −0.546773 −0.273387 0.961904i \(-0.588144\pi\)
−0.273387 + 0.961904i \(0.588144\pi\)
\(168\) 0 0
\(169\) −1173.00 −0.533910
\(170\) 0 0
\(171\) 0 0
\(172\) 5236.00 2.32117
\(173\) −3177.00 −1.39620 −0.698101 0.716000i \(-0.745971\pi\)
−0.698101 + 0.716000i \(0.745971\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 979.000 0.419289
\(177\) 0 0
\(178\) −2020.00 −0.850592
\(179\) −1787.00 −0.746182 −0.373091 0.927795i \(-0.621702\pi\)
−0.373091 + 0.927795i \(0.621702\pi\)
\(180\) 0 0
\(181\) −835.000 −0.342901 −0.171450 0.985193i \(-0.554845\pi\)
−0.171450 + 0.985193i \(0.554845\pi\)
\(182\) 480.000 0.195494
\(183\) 0 0
\(184\) 5085.00 2.03734
\(185\) 0 0
\(186\) 0 0
\(187\) 363.000 0.141953
\(188\) 3315.00 1.28602
\(189\) 0 0
\(190\) 0 0
\(191\) −3613.00 −1.36873 −0.684365 0.729139i \(-0.739921\pi\)
−0.684365 + 0.729139i \(0.739921\pi\)
\(192\) 0 0
\(193\) −4204.00 −1.56793 −0.783965 0.620805i \(-0.786806\pi\)
−0.783965 + 0.620805i \(0.786806\pi\)
\(194\) −395.000 −0.146182
\(195\) 0 0
\(196\) −5678.00 −2.06924
\(197\) −4517.00 −1.63362 −0.816809 0.576908i \(-0.804259\pi\)
−0.816809 + 0.576908i \(0.804259\pi\)
\(198\) 0 0
\(199\) 4164.00 1.48331 0.741654 0.670783i \(-0.234042\pi\)
0.741654 + 0.670783i \(0.234042\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2725.00 0.949160
\(203\) −162.000 −0.0560107
\(204\) 0 0
\(205\) 0 0
\(206\) 6530.00 2.20858
\(207\) 0 0
\(208\) −2848.00 −0.949391
\(209\) 517.000 0.171108
\(210\) 0 0
\(211\) 4660.00 1.52042 0.760208 0.649680i \(-0.225097\pi\)
0.760208 + 0.649680i \(0.225097\pi\)
\(212\) 2584.00 0.837122
\(213\) 0 0
\(214\) 9690.00 3.09530
\(215\) 0 0
\(216\) 0 0
\(217\) −534.000 −0.167052
\(218\) −2880.00 −0.894762
\(219\) 0 0
\(220\) 0 0
\(221\) −1056.00 −0.321422
\(222\) 0 0
\(223\) −3560.00 −1.06904 −0.534518 0.845157i \(-0.679507\pi\)
−0.534518 + 0.845157i \(0.679507\pi\)
\(224\) −255.000 −0.0760621
\(225\) 0 0
\(226\) −5520.00 −1.62471
\(227\) −4678.00 −1.36780 −0.683898 0.729577i \(-0.739717\pi\)
−0.683898 + 0.729577i \(0.739717\pi\)
\(228\) 0 0
\(229\) −4447.00 −1.28326 −0.641629 0.767015i \(-0.721741\pi\)
−0.641629 + 0.767015i \(0.721741\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2430.00 0.687661
\(233\) 411.000 0.115560 0.0577801 0.998329i \(-0.481598\pi\)
0.0577801 + 0.998329i \(0.481598\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 10625.0 2.93063
\(237\) 0 0
\(238\) −495.000 −0.134815
\(239\) 6380.00 1.72673 0.863364 0.504582i \(-0.168353\pi\)
0.863364 + 0.504582i \(0.168353\pi\)
\(240\) 0 0
\(241\) 7282.00 1.94637 0.973184 0.230027i \(-0.0738813\pi\)
0.973184 + 0.230027i \(0.0738813\pi\)
\(242\) 605.000 0.160706
\(243\) 0 0
\(244\) 5440.00 1.42730
\(245\) 0 0
\(246\) 0 0
\(247\) −1504.00 −0.387438
\(248\) 8010.00 2.05095
\(249\) 0 0
\(250\) 0 0
\(251\) 4728.00 1.18896 0.594480 0.804111i \(-0.297358\pi\)
0.594480 + 0.804111i \(0.297358\pi\)
\(252\) 0 0
\(253\) 1243.00 0.308880
\(254\) 8695.00 2.14792
\(255\) 0 0
\(256\) −8279.00 −2.02124
\(257\) 5418.00 1.31504 0.657521 0.753437i \(-0.271605\pi\)
0.657521 + 0.753437i \(0.271605\pi\)
\(258\) 0 0
\(259\) 57.0000 0.0136749
\(260\) 0 0
\(261\) 0 0
\(262\) −9090.00 −2.14344
\(263\) −3354.00 −0.786375 −0.393187 0.919458i \(-0.628628\pi\)
−0.393187 + 0.919458i \(0.628628\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −705.000 −0.162505
\(267\) 0 0
\(268\) −3400.00 −0.774955
\(269\) −1062.00 −0.240711 −0.120356 0.992731i \(-0.538403\pi\)
−0.120356 + 0.992731i \(0.538403\pi\)
\(270\) 0 0
\(271\) −4821.00 −1.08065 −0.540323 0.841458i \(-0.681698\pi\)
−0.540323 + 0.841458i \(0.681698\pi\)
\(272\) 2937.00 0.654712
\(273\) 0 0
\(274\) 4350.00 0.959099
\(275\) 0 0
\(276\) 0 0
\(277\) −4.00000 −0.000867642 0 −0.000433821 1.00000i \(-0.500138\pi\)
−0.000433821 1.00000i \(0.500138\pi\)
\(278\) −3180.00 −0.686057
\(279\) 0 0
\(280\) 0 0
\(281\) −4647.00 −0.986537 −0.493268 0.869877i \(-0.664198\pi\)
−0.493268 + 0.869877i \(0.664198\pi\)
\(282\) 0 0
\(283\) 4283.00 0.899639 0.449820 0.893119i \(-0.351488\pi\)
0.449820 + 0.893119i \(0.351488\pi\)
\(284\) 16099.0 3.36373
\(285\) 0 0
\(286\) −1760.00 −0.363885
\(287\) 417.000 0.0857656
\(288\) 0 0
\(289\) −3824.00 −0.778343
\(290\) 0 0
\(291\) 0 0
\(292\) 7616.00 1.52634
\(293\) −6811.00 −1.35803 −0.679015 0.734124i \(-0.737593\pi\)
−0.679015 + 0.734124i \(0.737593\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −855.000 −0.167891
\(297\) 0 0
\(298\) 1195.00 0.232297
\(299\) −3616.00 −0.699394
\(300\) 0 0
\(301\) −924.000 −0.176938
\(302\) 6040.00 1.15087
\(303\) 0 0
\(304\) 4183.00 0.789183
\(305\) 0 0
\(306\) 0 0
\(307\) 460.000 0.0855166 0.0427583 0.999085i \(-0.486385\pi\)
0.0427583 + 0.999085i \(0.486385\pi\)
\(308\) −561.000 −0.103786
\(309\) 0 0
\(310\) 0 0
\(311\) −8328.00 −1.51845 −0.759224 0.650829i \(-0.774421\pi\)
−0.759224 + 0.650829i \(0.774421\pi\)
\(312\) 0 0
\(313\) 5929.00 1.07069 0.535346 0.844633i \(-0.320181\pi\)
0.535346 + 0.844633i \(0.320181\pi\)
\(314\) 9370.00 1.68401
\(315\) 0 0
\(316\) −12257.0 −2.18199
\(317\) 5040.00 0.892980 0.446490 0.894789i \(-0.352674\pi\)
0.446490 + 0.894789i \(0.352674\pi\)
\(318\) 0 0
\(319\) 594.000 0.104256
\(320\) 0 0
\(321\) 0 0
\(322\) −1695.00 −0.293350
\(323\) 1551.00 0.267183
\(324\) 0 0
\(325\) 0 0
\(326\) 9520.00 1.61737
\(327\) 0 0
\(328\) −6255.00 −1.05297
\(329\) −585.000 −0.0980307
\(330\) 0 0
\(331\) 10396.0 1.72633 0.863166 0.504920i \(-0.168478\pi\)
0.863166 + 0.504920i \(0.168478\pi\)
\(332\) 2414.00 0.399053
\(333\) 0 0
\(334\) −5900.00 −0.966568
\(335\) 0 0
\(336\) 0 0
\(337\) 7236.00 1.16964 0.584822 0.811162i \(-0.301164\pi\)
0.584822 + 0.811162i \(0.301164\pi\)
\(338\) −5865.00 −0.943828
\(339\) 0 0
\(340\) 0 0
\(341\) 1958.00 0.310943
\(342\) 0 0
\(343\) 2031.00 0.319719
\(344\) 13860.0 2.17233
\(345\) 0 0
\(346\) −15885.0 −2.46816
\(347\) 1468.00 0.227108 0.113554 0.993532i \(-0.463777\pi\)
0.113554 + 0.993532i \(0.463777\pi\)
\(348\) 0 0
\(349\) 5690.00 0.872718 0.436359 0.899773i \(-0.356268\pi\)
0.436359 + 0.899773i \(0.356268\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 935.000 0.141579
\(353\) 5376.00 0.810582 0.405291 0.914188i \(-0.367170\pi\)
0.405291 + 0.914188i \(0.367170\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6868.00 −1.02248
\(357\) 0 0
\(358\) −8935.00 −1.31908
\(359\) −3734.00 −0.548950 −0.274475 0.961594i \(-0.588504\pi\)
−0.274475 + 0.961594i \(0.588504\pi\)
\(360\) 0 0
\(361\) −4650.00 −0.677941
\(362\) −4175.00 −0.606169
\(363\) 0 0
\(364\) 1632.00 0.235000
\(365\) 0 0
\(366\) 0 0
\(367\) 10274.0 1.46130 0.730652 0.682750i \(-0.239216\pi\)
0.730652 + 0.682750i \(0.239216\pi\)
\(368\) 10057.0 1.42461
\(369\) 0 0
\(370\) 0 0
\(371\) −456.000 −0.0638122
\(372\) 0 0
\(373\) −13662.0 −1.89649 −0.948246 0.317537i \(-0.897144\pi\)
−0.948246 + 0.317537i \(0.897144\pi\)
\(374\) 1815.00 0.250940
\(375\) 0 0
\(376\) 8775.00 1.20355
\(377\) −1728.00 −0.236065
\(378\) 0 0
\(379\) −7906.00 −1.07151 −0.535757 0.844372i \(-0.679974\pi\)
−0.535757 + 0.844372i \(0.679974\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −18065.0 −2.41960
\(383\) −3168.00 −0.422656 −0.211328 0.977415i \(-0.567779\pi\)
−0.211328 + 0.977415i \(0.567779\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −21020.0 −2.77174
\(387\) 0 0
\(388\) −1343.00 −0.175723
\(389\) −10770.0 −1.40375 −0.701877 0.712298i \(-0.747655\pi\)
−0.701877 + 0.712298i \(0.747655\pi\)
\(390\) 0 0
\(391\) 3729.00 0.482311
\(392\) −15030.0 −1.93656
\(393\) 0 0
\(394\) −22585.0 −2.88786
\(395\) 0 0
\(396\) 0 0
\(397\) −5670.00 −0.716799 −0.358399 0.933568i \(-0.616677\pi\)
−0.358399 + 0.933568i \(0.616677\pi\)
\(398\) 20820.0 2.62214
\(399\) 0 0
\(400\) 0 0
\(401\) −832.000 −0.103611 −0.0518056 0.998657i \(-0.516498\pi\)
−0.0518056 + 0.998657i \(0.516498\pi\)
\(402\) 0 0
\(403\) −5696.00 −0.704064
\(404\) 9265.00 1.14097
\(405\) 0 0
\(406\) −810.000 −0.0990139
\(407\) −209.000 −0.0254539
\(408\) 0 0
\(409\) −5712.00 −0.690563 −0.345281 0.938499i \(-0.612217\pi\)
−0.345281 + 0.938499i \(0.612217\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 22202.0 2.65489
\(413\) −1875.00 −0.223396
\(414\) 0 0
\(415\) 0 0
\(416\) −2720.00 −0.320574
\(417\) 0 0
\(418\) 2585.00 0.302480
\(419\) 4559.00 0.531555 0.265778 0.964034i \(-0.414371\pi\)
0.265778 + 0.964034i \(0.414371\pi\)
\(420\) 0 0
\(421\) 6855.00 0.793568 0.396784 0.917912i \(-0.370126\pi\)
0.396784 + 0.917912i \(0.370126\pi\)
\(422\) 23300.0 2.68774
\(423\) 0 0
\(424\) 6840.00 0.783443
\(425\) 0 0
\(426\) 0 0
\(427\) −960.000 −0.108800
\(428\) 32946.0 3.72081
\(429\) 0 0
\(430\) 0 0
\(431\) −10770.0 −1.20365 −0.601824 0.798628i \(-0.705559\pi\)
−0.601824 + 0.798628i \(0.705559\pi\)
\(432\) 0 0
\(433\) −8498.00 −0.943159 −0.471579 0.881824i \(-0.656316\pi\)
−0.471579 + 0.881824i \(0.656316\pi\)
\(434\) −2670.00 −0.295309
\(435\) 0 0
\(436\) −9792.00 −1.07558
\(437\) 5311.00 0.581372
\(438\) 0 0
\(439\) 9835.00 1.06925 0.534623 0.845091i \(-0.320454\pi\)
0.534623 + 0.845091i \(0.320454\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −5280.00 −0.568199
\(443\) −10745.0 −1.15239 −0.576197 0.817311i \(-0.695464\pi\)
−0.576197 + 0.817311i \(0.695464\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −17800.0 −1.88981
\(447\) 0 0
\(448\) 861.000 0.0908001
\(449\) −8356.00 −0.878272 −0.439136 0.898421i \(-0.644715\pi\)
−0.439136 + 0.898421i \(0.644715\pi\)
\(450\) 0 0
\(451\) −1529.00 −0.159640
\(452\) −18768.0 −1.95304
\(453\) 0 0
\(454\) −23390.0 −2.41795
\(455\) 0 0
\(456\) 0 0
\(457\) 7058.00 0.722449 0.361225 0.932479i \(-0.382359\pi\)
0.361225 + 0.932479i \(0.382359\pi\)
\(458\) −22235.0 −2.26850
\(459\) 0 0
\(460\) 0 0
\(461\) −646.000 −0.0652651 −0.0326326 0.999467i \(-0.510389\pi\)
−0.0326326 + 0.999467i \(0.510389\pi\)
\(462\) 0 0
\(463\) 8982.00 0.901574 0.450787 0.892631i \(-0.351143\pi\)
0.450787 + 0.892631i \(0.351143\pi\)
\(464\) 4806.00 0.480847
\(465\) 0 0
\(466\) 2055.00 0.204283
\(467\) −13476.0 −1.33532 −0.667661 0.744466i \(-0.732704\pi\)
−0.667661 + 0.744466i \(0.732704\pi\)
\(468\) 0 0
\(469\) 600.000 0.0590734
\(470\) 0 0
\(471\) 0 0
\(472\) 28125.0 2.74271
\(473\) 3388.00 0.329345
\(474\) 0 0
\(475\) 0 0
\(476\) −1683.00 −0.162059
\(477\) 0 0
\(478\) 31900.0 3.05245
\(479\) −12996.0 −1.23967 −0.619835 0.784732i \(-0.712801\pi\)
−0.619835 + 0.784732i \(0.712801\pi\)
\(480\) 0 0
\(481\) 608.000 0.0576350
\(482\) 36410.0 3.44073
\(483\) 0 0
\(484\) 2057.00 0.193182
\(485\) 0 0
\(486\) 0 0
\(487\) 6026.00 0.560707 0.280353 0.959897i \(-0.409548\pi\)
0.280353 + 0.959897i \(0.409548\pi\)
\(488\) 14400.0 1.33577
\(489\) 0 0
\(490\) 0 0
\(491\) −11698.0 −1.07520 −0.537600 0.843200i \(-0.680669\pi\)
−0.537600 + 0.843200i \(0.680669\pi\)
\(492\) 0 0
\(493\) 1782.00 0.162794
\(494\) −7520.00 −0.684900
\(495\) 0 0
\(496\) 15842.0 1.43413
\(497\) −2841.00 −0.256411
\(498\) 0 0
\(499\) −17052.0 −1.52976 −0.764882 0.644170i \(-0.777203\pi\)
−0.764882 + 0.644170i \(0.777203\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 23640.0 2.10180
\(503\) −932.000 −0.0826160 −0.0413080 0.999146i \(-0.513152\pi\)
−0.0413080 + 0.999146i \(0.513152\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6215.00 0.546029
\(507\) 0 0
\(508\) 29563.0 2.58198
\(509\) −4384.00 −0.381763 −0.190882 0.981613i \(-0.561135\pi\)
−0.190882 + 0.981613i \(0.561135\pi\)
\(510\) 0 0
\(511\) −1344.00 −0.116350
\(512\) −24475.0 −2.11260
\(513\) 0 0
\(514\) 27090.0 2.32469
\(515\) 0 0
\(516\) 0 0
\(517\) 2145.00 0.182470
\(518\) 285.000 0.0241741
\(519\) 0 0
\(520\) 0 0
\(521\) 2322.00 0.195257 0.0976283 0.995223i \(-0.468874\pi\)
0.0976283 + 0.995223i \(0.468874\pi\)
\(522\) 0 0
\(523\) 9749.00 0.815094 0.407547 0.913184i \(-0.366384\pi\)
0.407547 + 0.913184i \(0.366384\pi\)
\(524\) −30906.0 −2.57659
\(525\) 0 0
\(526\) −16770.0 −1.39013
\(527\) 5874.00 0.485532
\(528\) 0 0
\(529\) 602.000 0.0494781
\(530\) 0 0
\(531\) 0 0
\(532\) −2397.00 −0.195344
\(533\) 4448.00 0.361471
\(534\) 0 0
\(535\) 0 0
\(536\) −9000.00 −0.725263
\(537\) 0 0
\(538\) −5310.00 −0.425521
\(539\) −3674.00 −0.293600
\(540\) 0 0
\(541\) 4208.00 0.334410 0.167205 0.985922i \(-0.446526\pi\)
0.167205 + 0.985922i \(0.446526\pi\)
\(542\) −24105.0 −1.91033
\(543\) 0 0
\(544\) 2805.00 0.221072
\(545\) 0 0
\(546\) 0 0
\(547\) −10179.0 −0.795654 −0.397827 0.917461i \(-0.630236\pi\)
−0.397827 + 0.917461i \(0.630236\pi\)
\(548\) 14790.0 1.15292
\(549\) 0 0
\(550\) 0 0
\(551\) 2538.00 0.196229
\(552\) 0 0
\(553\) 2163.00 0.166329
\(554\) −20.0000 −0.00153379
\(555\) 0 0
\(556\) −10812.0 −0.824696
\(557\) −2314.00 −0.176028 −0.0880138 0.996119i \(-0.528052\pi\)
−0.0880138 + 0.996119i \(0.528052\pi\)
\(558\) 0 0
\(559\) −9856.00 −0.745732
\(560\) 0 0
\(561\) 0 0
\(562\) −23235.0 −1.74397
\(563\) 24330.0 1.82129 0.910646 0.413188i \(-0.135585\pi\)
0.910646 + 0.413188i \(0.135585\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 21415.0 1.59035
\(567\) 0 0
\(568\) 42615.0 3.14804
\(569\) −3445.00 −0.253817 −0.126909 0.991914i \(-0.540505\pi\)
−0.126909 + 0.991914i \(0.540505\pi\)
\(570\) 0 0
\(571\) −13056.0 −0.956877 −0.478438 0.878121i \(-0.658797\pi\)
−0.478438 + 0.878121i \(0.658797\pi\)
\(572\) −5984.00 −0.437419
\(573\) 0 0
\(574\) 2085.00 0.151614
\(575\) 0 0
\(576\) 0 0
\(577\) 17347.0 1.25159 0.625793 0.779989i \(-0.284775\pi\)
0.625793 + 0.779989i \(0.284775\pi\)
\(578\) −19120.0 −1.37593
\(579\) 0 0
\(580\) 0 0
\(581\) −426.000 −0.0304190
\(582\) 0 0
\(583\) 1672.00 0.118777
\(584\) 20160.0 1.42847
\(585\) 0 0
\(586\) −34055.0 −2.40068
\(587\) 8379.00 0.589162 0.294581 0.955626i \(-0.404820\pi\)
0.294581 + 0.955626i \(0.404820\pi\)
\(588\) 0 0
\(589\) 8366.00 0.585255
\(590\) 0 0
\(591\) 0 0
\(592\) −1691.00 −0.117398
\(593\) 1958.00 0.135591 0.0677955 0.997699i \(-0.478403\pi\)
0.0677955 + 0.997699i \(0.478403\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4063.00 0.279240
\(597\) 0 0
\(598\) −18080.0 −1.23636
\(599\) −23583.0 −1.60864 −0.804320 0.594196i \(-0.797470\pi\)
−0.804320 + 0.594196i \(0.797470\pi\)
\(600\) 0 0
\(601\) −15328.0 −1.04034 −0.520168 0.854064i \(-0.674131\pi\)
−0.520168 + 0.854064i \(0.674131\pi\)
\(602\) −4620.00 −0.312786
\(603\) 0 0
\(604\) 20536.0 1.38344
\(605\) 0 0
\(606\) 0 0
\(607\) −160.000 −0.0106988 −0.00534942 0.999986i \(-0.501703\pi\)
−0.00534942 + 0.999986i \(0.501703\pi\)
\(608\) 3995.00 0.266478
\(609\) 0 0
\(610\) 0 0
\(611\) −6240.00 −0.413164
\(612\) 0 0
\(613\) 5948.00 0.391904 0.195952 0.980613i \(-0.437220\pi\)
0.195952 + 0.980613i \(0.437220\pi\)
\(614\) 2300.00 0.151173
\(615\) 0 0
\(616\) −1485.00 −0.0971304
\(617\) 334.000 0.0217931 0.0108965 0.999941i \(-0.496531\pi\)
0.0108965 + 0.999941i \(0.496531\pi\)
\(618\) 0 0
\(619\) −7202.00 −0.467646 −0.233823 0.972279i \(-0.575124\pi\)
−0.233823 + 0.972279i \(0.575124\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −41640.0 −2.68426
\(623\) 1212.00 0.0779418
\(624\) 0 0
\(625\) 0 0
\(626\) 29645.0 1.89274
\(627\) 0 0
\(628\) 31858.0 2.02432
\(629\) −627.000 −0.0397458
\(630\) 0 0
\(631\) 10306.0 0.650199 0.325099 0.945680i \(-0.394602\pi\)
0.325099 + 0.945680i \(0.394602\pi\)
\(632\) −32445.0 −2.04208
\(633\) 0 0
\(634\) 25200.0 1.57858
\(635\) 0 0
\(636\) 0 0
\(637\) 10688.0 0.664794
\(638\) 2970.00 0.184300
\(639\) 0 0
\(640\) 0 0
\(641\) 1228.00 0.0756678 0.0378339 0.999284i \(-0.487954\pi\)
0.0378339 + 0.999284i \(0.487954\pi\)
\(642\) 0 0
\(643\) 18454.0 1.13181 0.565906 0.824470i \(-0.308527\pi\)
0.565906 + 0.824470i \(0.308527\pi\)
\(644\) −5763.00 −0.352630
\(645\) 0 0
\(646\) 7755.00 0.472316
\(647\) 17647.0 1.07230 0.536148 0.844124i \(-0.319879\pi\)
0.536148 + 0.844124i \(0.319879\pi\)
\(648\) 0 0
\(649\) 6875.00 0.415820
\(650\) 0 0
\(651\) 0 0
\(652\) 32368.0 1.94422
\(653\) 25918.0 1.55322 0.776608 0.629984i \(-0.216939\pi\)
0.776608 + 0.629984i \(0.216939\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −12371.0 −0.736290
\(657\) 0 0
\(658\) −2925.00 −0.173295
\(659\) −12864.0 −0.760410 −0.380205 0.924902i \(-0.624147\pi\)
−0.380205 + 0.924902i \(0.624147\pi\)
\(660\) 0 0
\(661\) −11419.0 −0.671933 −0.335966 0.941874i \(-0.609063\pi\)
−0.335966 + 0.941874i \(0.609063\pi\)
\(662\) 51980.0 3.05175
\(663\) 0 0
\(664\) 6390.00 0.373464
\(665\) 0 0
\(666\) 0 0
\(667\) 6102.00 0.354228
\(668\) −20060.0 −1.16189
\(669\) 0 0
\(670\) 0 0
\(671\) 3520.00 0.202516
\(672\) 0 0
\(673\) −15784.0 −0.904054 −0.452027 0.892004i \(-0.649299\pi\)
−0.452027 + 0.892004i \(0.649299\pi\)
\(674\) 36180.0 2.06766
\(675\) 0 0
\(676\) −19941.0 −1.13456
\(677\) −26050.0 −1.47885 −0.739426 0.673238i \(-0.764903\pi\)
−0.739426 + 0.673238i \(0.764903\pi\)
\(678\) 0 0
\(679\) 237.000 0.0133950
\(680\) 0 0
\(681\) 0 0
\(682\) 9790.00 0.549675
\(683\) −15095.0 −0.845672 −0.422836 0.906206i \(-0.638965\pi\)
−0.422836 + 0.906206i \(0.638965\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 10155.0 0.565189
\(687\) 0 0
\(688\) 27412.0 1.51900
\(689\) −4864.00 −0.268946
\(690\) 0 0
\(691\) 15896.0 0.875126 0.437563 0.899188i \(-0.355842\pi\)
0.437563 + 0.899188i \(0.355842\pi\)
\(692\) −54009.0 −2.96693
\(693\) 0 0
\(694\) 7340.00 0.401473
\(695\) 0 0
\(696\) 0 0
\(697\) −4587.00 −0.249275
\(698\) 28450.0 1.54276
\(699\) 0 0
\(700\) 0 0
\(701\) −10529.0 −0.567296 −0.283648 0.958928i \(-0.591545\pi\)
−0.283648 + 0.958928i \(0.591545\pi\)
\(702\) 0 0
\(703\) −893.000 −0.0479092
\(704\) −3157.00 −0.169011
\(705\) 0 0
\(706\) 26880.0 1.43292
\(707\) −1635.00 −0.0869738
\(708\) 0 0
\(709\) −16087.0 −0.852130 −0.426065 0.904693i \(-0.640100\pi\)
−0.426065 + 0.904693i \(0.640100\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −18180.0 −0.956916
\(713\) 20114.0 1.05649
\(714\) 0 0
\(715\) 0 0
\(716\) −30379.0 −1.58564
\(717\) 0 0
\(718\) −18670.0 −0.970415
\(719\) −24336.0 −1.26228 −0.631140 0.775669i \(-0.717413\pi\)
−0.631140 + 0.775669i \(0.717413\pi\)
\(720\) 0 0
\(721\) −3918.00 −0.202377
\(722\) −23250.0 −1.19844
\(723\) 0 0
\(724\) −14195.0 −0.728664
\(725\) 0 0
\(726\) 0 0
\(727\) −13960.0 −0.712170 −0.356085 0.934454i \(-0.615889\pi\)
−0.356085 + 0.934454i \(0.615889\pi\)
\(728\) 4320.00 0.219931
\(729\) 0 0
\(730\) 0 0
\(731\) 10164.0 0.514267
\(732\) 0 0
\(733\) −9252.00 −0.466208 −0.233104 0.972452i \(-0.574888\pi\)
−0.233104 + 0.972452i \(0.574888\pi\)
\(734\) 51370.0 2.58324
\(735\) 0 0
\(736\) 9605.00 0.481039
\(737\) −2200.00 −0.109957
\(738\) 0 0
\(739\) 28453.0 1.41632 0.708160 0.706052i \(-0.249526\pi\)
0.708160 + 0.706052i \(0.249526\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2280.00 −0.112805
\(743\) 512.000 0.0252806 0.0126403 0.999920i \(-0.495976\pi\)
0.0126403 + 0.999920i \(0.495976\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −68310.0 −3.35256
\(747\) 0 0
\(748\) 6171.00 0.301650
\(749\) −5814.00 −0.283630
\(750\) 0 0
\(751\) 772.000 0.0375109 0.0187554 0.999824i \(-0.494030\pi\)
0.0187554 + 0.999824i \(0.494030\pi\)
\(752\) 17355.0 0.841585
\(753\) 0 0
\(754\) −8640.00 −0.417308
\(755\) 0 0
\(756\) 0 0
\(757\) −8058.00 −0.386886 −0.193443 0.981111i \(-0.561966\pi\)
−0.193443 + 0.981111i \(0.561966\pi\)
\(758\) −39530.0 −1.89419
\(759\) 0 0
\(760\) 0 0
\(761\) −18650.0 −0.888386 −0.444193 0.895931i \(-0.646510\pi\)
−0.444193 + 0.895931i \(0.646510\pi\)
\(762\) 0 0
\(763\) 1728.00 0.0819893
\(764\) −61421.0 −2.90855
\(765\) 0 0
\(766\) −15840.0 −0.747157
\(767\) −20000.0 −0.941536
\(768\) 0 0
\(769\) −7144.00 −0.335005 −0.167503 0.985872i \(-0.553570\pi\)
−0.167503 + 0.985872i \(0.553570\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −71468.0 −3.33185
\(773\) −1904.00 −0.0885927 −0.0442963 0.999018i \(-0.514105\pi\)
−0.0442963 + 0.999018i \(0.514105\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3555.00 −0.164455
\(777\) 0 0
\(778\) −53850.0 −2.48151
\(779\) −6533.00 −0.300474
\(780\) 0 0
\(781\) 10417.0 0.477272
\(782\) 18645.0 0.852614
\(783\) 0 0
\(784\) −29726.0 −1.35414
\(785\) 0 0
\(786\) 0 0
\(787\) −7555.00 −0.342194 −0.171097 0.985254i \(-0.554731\pi\)
−0.171097 + 0.985254i \(0.554731\pi\)
\(788\) −76789.0 −3.47144
\(789\) 0 0
\(790\) 0 0
\(791\) 3312.00 0.148876
\(792\) 0 0
\(793\) −10240.0 −0.458554
\(794\) −28350.0 −1.26713
\(795\) 0 0
\(796\) 70788.0 3.15203
\(797\) 24950.0 1.10888 0.554438 0.832225i \(-0.312933\pi\)
0.554438 + 0.832225i \(0.312933\pi\)
\(798\) 0 0
\(799\) 6435.00 0.284924
\(800\) 0 0
\(801\) 0 0
\(802\) −4160.00 −0.183160
\(803\) 4928.00 0.216570
\(804\) 0 0
\(805\) 0 0
\(806\) −28480.0 −1.24462
\(807\) 0 0
\(808\) 24525.0 1.06781
\(809\) −19893.0 −0.864525 −0.432262 0.901748i \(-0.642285\pi\)
−0.432262 + 0.901748i \(0.642285\pi\)
\(810\) 0 0
\(811\) 34503.0 1.49391 0.746957 0.664872i \(-0.231514\pi\)
0.746957 + 0.664872i \(0.231514\pi\)
\(812\) −2754.00 −0.119023
\(813\) 0 0
\(814\) −1045.00 −0.0449966
\(815\) 0 0
\(816\) 0 0
\(817\) 14476.0 0.619891
\(818\) −28560.0 −1.22075
\(819\) 0 0
\(820\) 0 0
\(821\) −16890.0 −0.717984 −0.358992 0.933341i \(-0.616880\pi\)
−0.358992 + 0.933341i \(0.616880\pi\)
\(822\) 0 0
\(823\) −34692.0 −1.46936 −0.734682 0.678411i \(-0.762669\pi\)
−0.734682 + 0.678411i \(0.762669\pi\)
\(824\) 58770.0 2.48465
\(825\) 0 0
\(826\) −9375.00 −0.394913
\(827\) 41424.0 1.74178 0.870891 0.491476i \(-0.163543\pi\)
0.870891 + 0.491476i \(0.163543\pi\)
\(828\) 0 0
\(829\) −18494.0 −0.774817 −0.387408 0.921908i \(-0.626630\pi\)
−0.387408 + 0.921908i \(0.626630\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 9184.00 0.382690
\(833\) −11022.0 −0.458451
\(834\) 0 0
\(835\) 0 0
\(836\) 8789.00 0.363605
\(837\) 0 0
\(838\) 22795.0 0.939666
\(839\) −6680.00 −0.274874 −0.137437 0.990511i \(-0.543886\pi\)
−0.137437 + 0.990511i \(0.543886\pi\)
\(840\) 0 0
\(841\) −21473.0 −0.880438
\(842\) 34275.0 1.40284
\(843\) 0 0
\(844\) 79220.0 3.23088
\(845\) 0 0
\(846\) 0 0
\(847\) −363.000 −0.0147259
\(848\) 13528.0 0.547822
\(849\) 0 0
\(850\) 0 0
\(851\) −2147.00 −0.0864844
\(852\) 0 0
\(853\) −43358.0 −1.74039 −0.870193 0.492711i \(-0.836006\pi\)
−0.870193 + 0.492711i \(0.836006\pi\)
\(854\) −4800.00 −0.192333
\(855\) 0 0
\(856\) 87210.0 3.48222
\(857\) 15585.0 0.621206 0.310603 0.950540i \(-0.399469\pi\)
0.310603 + 0.950540i \(0.399469\pi\)
\(858\) 0 0
\(859\) −17036.0 −0.676672 −0.338336 0.941025i \(-0.609864\pi\)
−0.338336 + 0.941025i \(0.609864\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −53850.0 −2.12777
\(863\) 28064.0 1.10696 0.553482 0.832861i \(-0.313299\pi\)
0.553482 + 0.832861i \(0.313299\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −42490.0 −1.66729
\(867\) 0 0
\(868\) −9078.00 −0.354985
\(869\) −7931.00 −0.309598
\(870\) 0 0
\(871\) 6400.00 0.248973
\(872\) −25920.0 −1.00661
\(873\) 0 0
\(874\) 26555.0 1.02773
\(875\) 0 0
\(876\) 0 0
\(877\) 22654.0 0.872259 0.436130 0.899884i \(-0.356349\pi\)
0.436130 + 0.899884i \(0.356349\pi\)
\(878\) 49175.0 1.89018
\(879\) 0 0
\(880\) 0 0
\(881\) 22380.0 0.855847 0.427924 0.903815i \(-0.359245\pi\)
0.427924 + 0.903815i \(0.359245\pi\)
\(882\) 0 0
\(883\) −35174.0 −1.34054 −0.670271 0.742116i \(-0.733822\pi\)
−0.670271 + 0.742116i \(0.733822\pi\)
\(884\) −17952.0 −0.683022
\(885\) 0 0
\(886\) −53725.0 −2.03716
\(887\) 30868.0 1.16848 0.584242 0.811579i \(-0.301392\pi\)
0.584242 + 0.811579i \(0.301392\pi\)
\(888\) 0 0
\(889\) −5217.00 −0.196820
\(890\) 0 0
\(891\) 0 0
\(892\) −60520.0 −2.27170
\(893\) 9165.00 0.343443
\(894\) 0 0
\(895\) 0 0
\(896\) 6345.00 0.236575
\(897\) 0 0
\(898\) −41780.0 −1.55258
\(899\) 9612.00 0.356594
\(900\) 0 0
\(901\) 5016.00 0.185469
\(902\) −7645.00 −0.282207
\(903\) 0 0
\(904\) −49680.0 −1.82780
\(905\) 0 0
\(906\) 0 0
\(907\) −10070.0 −0.368654 −0.184327 0.982865i \(-0.559010\pi\)
−0.184327 + 0.982865i \(0.559010\pi\)
\(908\) −79526.0 −2.90657
\(909\) 0 0
\(910\) 0 0
\(911\) −1885.00 −0.0685542 −0.0342771 0.999412i \(-0.510913\pi\)
−0.0342771 + 0.999412i \(0.510913\pi\)
\(912\) 0 0
\(913\) 1562.00 0.0566207
\(914\) 35290.0 1.27712
\(915\) 0 0
\(916\) −75599.0 −2.72692
\(917\) 5454.00 0.196409
\(918\) 0 0
\(919\) 23703.0 0.850805 0.425403 0.905004i \(-0.360133\pi\)
0.425403 + 0.905004i \(0.360133\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −3230.00 −0.115374
\(923\) −30304.0 −1.08068
\(924\) 0 0
\(925\) 0 0
\(926\) 44910.0 1.59377
\(927\) 0 0
\(928\) 4590.00 0.162364
\(929\) −53804.0 −1.90016 −0.950082 0.312001i \(-0.899001\pi\)
−0.950082 + 0.312001i \(0.899001\pi\)
\(930\) 0 0
\(931\) −15698.0 −0.552611
\(932\) 6987.00 0.245565
\(933\) 0 0
\(934\) −67380.0 −2.36054
\(935\) 0 0
\(936\) 0 0
\(937\) −1326.00 −0.0462311 −0.0231155 0.999733i \(-0.507359\pi\)
−0.0231155 + 0.999733i \(0.507359\pi\)
\(938\) 3000.00 0.104428
\(939\) 0 0
\(940\) 0 0
\(941\) 27109.0 0.939137 0.469569 0.882896i \(-0.344409\pi\)
0.469569 + 0.882896i \(0.344409\pi\)
\(942\) 0 0
\(943\) −15707.0 −0.542408
\(944\) 55625.0 1.91784
\(945\) 0 0
\(946\) 16940.0 0.582206
\(947\) 31143.0 1.06865 0.534325 0.845279i \(-0.320566\pi\)
0.534325 + 0.845279i \(0.320566\pi\)
\(948\) 0 0
\(949\) −14336.0 −0.490375
\(950\) 0 0
\(951\) 0 0
\(952\) −4455.00 −0.151667
\(953\) −879.000 −0.0298779 −0.0149389 0.999888i \(-0.504755\pi\)
−0.0149389 + 0.999888i \(0.504755\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 108460. 3.66930
\(957\) 0 0
\(958\) −64980.0 −2.19145
\(959\) −2610.00 −0.0878846
\(960\) 0 0
\(961\) 1893.00 0.0635427
\(962\) 3040.00 0.101885
\(963\) 0 0
\(964\) 123794. 4.13603
\(965\) 0 0
\(966\) 0 0
\(967\) 14824.0 0.492976 0.246488 0.969146i \(-0.420723\pi\)
0.246488 + 0.969146i \(0.420723\pi\)
\(968\) 5445.00 0.180794
\(969\) 0 0
\(970\) 0 0
\(971\) −34089.0 −1.12664 −0.563320 0.826239i \(-0.690476\pi\)
−0.563320 + 0.826239i \(0.690476\pi\)
\(972\) 0 0
\(973\) 1908.00 0.0628650
\(974\) 30130.0 0.991199
\(975\) 0 0
\(976\) 28480.0 0.934040
\(977\) 33446.0 1.09522 0.547611 0.836733i \(-0.315537\pi\)
0.547611 + 0.836733i \(0.315537\pi\)
\(978\) 0 0
\(979\) −4444.00 −0.145077
\(980\) 0 0
\(981\) 0 0
\(982\) −58490.0 −1.90070
\(983\) −52025.0 −1.68804 −0.844018 0.536315i \(-0.819816\pi\)
−0.844018 + 0.536315i \(0.819816\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 8910.00 0.287781
\(987\) 0 0
\(988\) −25568.0 −0.823306
\(989\) 34804.0 1.11901
\(990\) 0 0
\(991\) −41260.0 −1.32257 −0.661285 0.750135i \(-0.729989\pi\)
−0.661285 + 0.750135i \(0.729989\pi\)
\(992\) 15130.0 0.484252
\(993\) 0 0
\(994\) −14205.0 −0.453275
\(995\) 0 0
\(996\) 0 0
\(997\) 190.000 0.00603547 0.00301773 0.999995i \(-0.499039\pi\)
0.00301773 + 0.999995i \(0.499039\pi\)
\(998\) −85260.0 −2.70427
\(999\) 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 2475.4.a.k.1.1 1
3.2 odd 2 825.4.a.a.1.1 1
5.4 even 2 2475.4.a.a.1.1 1
15.2 even 4 825.4.c.b.199.1 2
15.8 even 4 825.4.c.b.199.2 2
15.14 odd 2 825.4.a.j.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.a.1.1 1 3.2 odd 2
825.4.a.j.1.1 yes 1 15.14 odd 2
825.4.c.b.199.1 2 15.2 even 4
825.4.c.b.199.2 2 15.8 even 4
2475.4.a.a.1.1 1 5.4 even 2
2475.4.a.k.1.1 1 1.1 even 1 trivial