Properties

Label 1521.4.a.c.1.1
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{2} +1.00000 q^{4} +9.00000 q^{5} +2.00000 q^{7} +21.0000 q^{8} +O(q^{10})\) \(q-3.00000 q^{2} +1.00000 q^{4} +9.00000 q^{5} +2.00000 q^{7} +21.0000 q^{8} -27.0000 q^{10} -30.0000 q^{11} -6.00000 q^{14} -71.0000 q^{16} +111.000 q^{17} -46.0000 q^{19} +9.00000 q^{20} +90.0000 q^{22} +6.00000 q^{23} -44.0000 q^{25} +2.00000 q^{28} +105.000 q^{29} -100.000 q^{31} +45.0000 q^{32} -333.000 q^{34} +18.0000 q^{35} +17.0000 q^{37} +138.000 q^{38} +189.000 q^{40} +231.000 q^{41} -514.000 q^{43} -30.0000 q^{44} -18.0000 q^{46} +162.000 q^{47} -339.000 q^{49} +132.000 q^{50} -639.000 q^{53} -270.000 q^{55} +42.0000 q^{56} -315.000 q^{58} -600.000 q^{59} +233.000 q^{61} +300.000 q^{62} +433.000 q^{64} +926.000 q^{67} +111.000 q^{68} -54.0000 q^{70} +930.000 q^{71} -253.000 q^{73} -51.0000 q^{74} -46.0000 q^{76} -60.0000 q^{77} -1324.00 q^{79} -639.000 q^{80} -693.000 q^{82} -810.000 q^{83} +999.000 q^{85} +1542.00 q^{86} -630.000 q^{88} -498.000 q^{89} +6.00000 q^{92} -486.000 q^{94} -414.000 q^{95} +1358.00 q^{97} +1017.00 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −1.06066 −0.530330 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) 0 0
\(4\) 1.00000 0.125000
\(5\) 9.00000 0.804984 0.402492 0.915423i \(-0.368144\pi\)
0.402492 + 0.915423i \(0.368144\pi\)
\(6\) 0 0
\(7\) 2.00000 0.107990 0.0539949 0.998541i \(-0.482805\pi\)
0.0539949 + 0.998541i \(0.482805\pi\)
\(8\) 21.0000 0.928078
\(9\) 0 0
\(10\) −27.0000 −0.853815
\(11\) −30.0000 −0.822304 −0.411152 0.911567i \(-0.634873\pi\)
−0.411152 + 0.911567i \(0.634873\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −6.00000 −0.114541
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 111.000 1.58361 0.791807 0.610771i \(-0.209140\pi\)
0.791807 + 0.610771i \(0.209140\pi\)
\(18\) 0 0
\(19\) −46.0000 −0.555428 −0.277714 0.960664i \(-0.589577\pi\)
−0.277714 + 0.960664i \(0.589577\pi\)
\(20\) 9.00000 0.100623
\(21\) 0 0
\(22\) 90.0000 0.872185
\(23\) 6.00000 0.0543951 0.0271975 0.999630i \(-0.491342\pi\)
0.0271975 + 0.999630i \(0.491342\pi\)
\(24\) 0 0
\(25\) −44.0000 −0.352000
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 0.0134987
\(29\) 105.000 0.672345 0.336173 0.941800i \(-0.390867\pi\)
0.336173 + 0.941800i \(0.390867\pi\)
\(30\) 0 0
\(31\) −100.000 −0.579372 −0.289686 0.957122i \(-0.593551\pi\)
−0.289686 + 0.957122i \(0.593551\pi\)
\(32\) 45.0000 0.248592
\(33\) 0 0
\(34\) −333.000 −1.67968
\(35\) 18.0000 0.0869302
\(36\) 0 0
\(37\) 17.0000 0.0755347 0.0377673 0.999287i \(-0.487975\pi\)
0.0377673 + 0.999287i \(0.487975\pi\)
\(38\) 138.000 0.589120
\(39\) 0 0
\(40\) 189.000 0.747088
\(41\) 231.000 0.879906 0.439953 0.898021i \(-0.354995\pi\)
0.439953 + 0.898021i \(0.354995\pi\)
\(42\) 0 0
\(43\) −514.000 −1.82289 −0.911445 0.411422i \(-0.865032\pi\)
−0.911445 + 0.411422i \(0.865032\pi\)
\(44\) −30.0000 −0.102788
\(45\) 0 0
\(46\) −18.0000 −0.0576947
\(47\) 162.000 0.502769 0.251384 0.967887i \(-0.419114\pi\)
0.251384 + 0.967887i \(0.419114\pi\)
\(48\) 0 0
\(49\) −339.000 −0.988338
\(50\) 132.000 0.373352
\(51\) 0 0
\(52\) 0 0
\(53\) −639.000 −1.65610 −0.828051 0.560653i \(-0.810550\pi\)
−0.828051 + 0.560653i \(0.810550\pi\)
\(54\) 0 0
\(55\) −270.000 −0.661942
\(56\) 42.0000 0.100223
\(57\) 0 0
\(58\) −315.000 −0.713130
\(59\) −600.000 −1.32396 −0.661978 0.749524i \(-0.730283\pi\)
−0.661978 + 0.749524i \(0.730283\pi\)
\(60\) 0 0
\(61\) 233.000 0.489059 0.244529 0.969642i \(-0.421367\pi\)
0.244529 + 0.969642i \(0.421367\pi\)
\(62\) 300.000 0.614517
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) 0 0
\(66\) 0 0
\(67\) 926.000 1.68849 0.844246 0.535957i \(-0.180049\pi\)
0.844246 + 0.535957i \(0.180049\pi\)
\(68\) 111.000 0.197952
\(69\) 0 0
\(70\) −54.0000 −0.0922033
\(71\) 930.000 1.55452 0.777258 0.629182i \(-0.216610\pi\)
0.777258 + 0.629182i \(0.216610\pi\)
\(72\) 0 0
\(73\) −253.000 −0.405636 −0.202818 0.979216i \(-0.565010\pi\)
−0.202818 + 0.979216i \(0.565010\pi\)
\(74\) −51.0000 −0.0801166
\(75\) 0 0
\(76\) −46.0000 −0.0694284
\(77\) −60.0000 −0.0888004
\(78\) 0 0
\(79\) −1324.00 −1.88559 −0.942795 0.333373i \(-0.891813\pi\)
−0.942795 + 0.333373i \(0.891813\pi\)
\(80\) −639.000 −0.893030
\(81\) 0 0
\(82\) −693.000 −0.933281
\(83\) −810.000 −1.07119 −0.535597 0.844474i \(-0.679913\pi\)
−0.535597 + 0.844474i \(0.679913\pi\)
\(84\) 0 0
\(85\) 999.000 1.27479
\(86\) 1542.00 1.93347
\(87\) 0 0
\(88\) −630.000 −0.763162
\(89\) −498.000 −0.593122 −0.296561 0.955014i \(-0.595840\pi\)
−0.296561 + 0.955014i \(0.595840\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.00000 0.00679938
\(93\) 0 0
\(94\) −486.000 −0.533267
\(95\) −414.000 −0.447111
\(96\) 0 0
\(97\) 1358.00 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 1017.00 1.04829
\(99\) 0 0
\(100\) −44.0000 −0.0440000
\(101\) 357.000 0.351711 0.175856 0.984416i \(-0.443731\pi\)
0.175856 + 0.984416i \(0.443731\pi\)
\(102\) 0 0
\(103\) 1118.00 1.06951 0.534756 0.845006i \(-0.320403\pi\)
0.534756 + 0.845006i \(0.320403\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1917.00 1.75656
\(107\) −714.000 −0.645093 −0.322547 0.946554i \(-0.604539\pi\)
−0.322547 + 0.946554i \(0.604539\pi\)
\(108\) 0 0
\(109\) 2006.00 1.76275 0.881376 0.472416i \(-0.156618\pi\)
0.881376 + 0.472416i \(0.156618\pi\)
\(110\) 810.000 0.702095
\(111\) 0 0
\(112\) −142.000 −0.119801
\(113\) 1119.00 0.931563 0.465782 0.884900i \(-0.345773\pi\)
0.465782 + 0.884900i \(0.345773\pi\)
\(114\) 0 0
\(115\) 54.0000 0.0437872
\(116\) 105.000 0.0840431
\(117\) 0 0
\(118\) 1800.00 1.40427
\(119\) 222.000 0.171014
\(120\) 0 0
\(121\) −431.000 −0.323817
\(122\) −699.000 −0.518725
\(123\) 0 0
\(124\) −100.000 −0.0724215
\(125\) −1521.00 −1.08834
\(126\) 0 0
\(127\) −604.000 −0.422018 −0.211009 0.977484i \(-0.567675\pi\)
−0.211009 + 0.977484i \(0.567675\pi\)
\(128\) −1659.00 −1.14560
\(129\) 0 0
\(130\) 0 0
\(131\) 1584.00 1.05645 0.528224 0.849105i \(-0.322858\pi\)
0.528224 + 0.849105i \(0.322858\pi\)
\(132\) 0 0
\(133\) −92.0000 −0.0599805
\(134\) −2778.00 −1.79092
\(135\) 0 0
\(136\) 2331.00 1.46972
\(137\) −717.000 −0.447135 −0.223567 0.974688i \(-0.571770\pi\)
−0.223567 + 0.974688i \(0.571770\pi\)
\(138\) 0 0
\(139\) −820.000 −0.500370 −0.250185 0.968198i \(-0.580492\pi\)
−0.250185 + 0.968198i \(0.580492\pi\)
\(140\) 18.0000 0.0108663
\(141\) 0 0
\(142\) −2790.00 −1.64881
\(143\) 0 0
\(144\) 0 0
\(145\) 945.000 0.541227
\(146\) 759.000 0.430242
\(147\) 0 0
\(148\) 17.0000 0.00944183
\(149\) 1749.00 0.961635 0.480818 0.876821i \(-0.340340\pi\)
0.480818 + 0.876821i \(0.340340\pi\)
\(150\) 0 0
\(151\) −370.000 −0.199405 −0.0997026 0.995017i \(-0.531789\pi\)
−0.0997026 + 0.995017i \(0.531789\pi\)
\(152\) −966.000 −0.515480
\(153\) 0 0
\(154\) 180.000 0.0941871
\(155\) −900.000 −0.466385
\(156\) 0 0
\(157\) −2611.00 −1.32726 −0.663632 0.748059i \(-0.730986\pi\)
−0.663632 + 0.748059i \(0.730986\pi\)
\(158\) 3972.00 1.99997
\(159\) 0 0
\(160\) 405.000 0.200113
\(161\) 12.0000 0.00587411
\(162\) 0 0
\(163\) −1636.00 −0.786144 −0.393072 0.919508i \(-0.628588\pi\)
−0.393072 + 0.919508i \(0.628588\pi\)
\(164\) 231.000 0.109988
\(165\) 0 0
\(166\) 2430.00 1.13617
\(167\) −264.000 −0.122329 −0.0611645 0.998128i \(-0.519481\pi\)
−0.0611645 + 0.998128i \(0.519481\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −2997.00 −1.35211
\(171\) 0 0
\(172\) −514.000 −0.227861
\(173\) −1410.00 −0.619655 −0.309827 0.950793i \(-0.600271\pi\)
−0.309827 + 0.950793i \(0.600271\pi\)
\(174\) 0 0
\(175\) −88.0000 −0.0380124
\(176\) 2130.00 0.912243
\(177\) 0 0
\(178\) 1494.00 0.629101
\(179\) 474.000 0.197924 0.0989621 0.995091i \(-0.468448\pi\)
0.0989621 + 0.995091i \(0.468448\pi\)
\(180\) 0 0
\(181\) 2249.00 0.923574 0.461787 0.886991i \(-0.347208\pi\)
0.461787 + 0.886991i \(0.347208\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 126.000 0.0504828
\(185\) 153.000 0.0608042
\(186\) 0 0
\(187\) −3330.00 −1.30221
\(188\) 162.000 0.0628461
\(189\) 0 0
\(190\) 1242.00 0.474232
\(191\) −3444.00 −1.30471 −0.652354 0.757915i \(-0.726218\pi\)
−0.652354 + 0.757915i \(0.726218\pi\)
\(192\) 0 0
\(193\) −4273.00 −1.59366 −0.796832 0.604201i \(-0.793493\pi\)
−0.796832 + 0.604201i \(0.793493\pi\)
\(194\) −4074.00 −1.50771
\(195\) 0 0
\(196\) −339.000 −0.123542
\(197\) 1986.00 0.718257 0.359129 0.933288i \(-0.383074\pi\)
0.359129 + 0.933288i \(0.383074\pi\)
\(198\) 0 0
\(199\) −2386.00 −0.849945 −0.424973 0.905206i \(-0.639716\pi\)
−0.424973 + 0.905206i \(0.639716\pi\)
\(200\) −924.000 −0.326683
\(201\) 0 0
\(202\) −1071.00 −0.373046
\(203\) 210.000 0.0726065
\(204\) 0 0
\(205\) 2079.00 0.708311
\(206\) −3354.00 −1.13439
\(207\) 0 0
\(208\) 0 0
\(209\) 1380.00 0.456730
\(210\) 0 0
\(211\) −1600.00 −0.522031 −0.261016 0.965335i \(-0.584057\pi\)
−0.261016 + 0.965335i \(0.584057\pi\)
\(212\) −639.000 −0.207013
\(213\) 0 0
\(214\) 2142.00 0.684225
\(215\) −4626.00 −1.46740
\(216\) 0 0
\(217\) −200.000 −0.0625663
\(218\) −6018.00 −1.86968
\(219\) 0 0
\(220\) −270.000 −0.0827427
\(221\) 0 0
\(222\) 0 0
\(223\) −3832.00 −1.15072 −0.575358 0.817902i \(-0.695137\pi\)
−0.575358 + 0.817902i \(0.695137\pi\)
\(224\) 90.0000 0.0268454
\(225\) 0 0
\(226\) −3357.00 −0.988072
\(227\) 1398.00 0.408760 0.204380 0.978892i \(-0.434482\pi\)
0.204380 + 0.978892i \(0.434482\pi\)
\(228\) 0 0
\(229\) 4466.00 1.28874 0.644370 0.764714i \(-0.277120\pi\)
0.644370 + 0.764714i \(0.277120\pi\)
\(230\) −162.000 −0.0464433
\(231\) 0 0
\(232\) 2205.00 0.623989
\(233\) 1638.00 0.460553 0.230277 0.973125i \(-0.426037\pi\)
0.230277 + 0.973125i \(0.426037\pi\)
\(234\) 0 0
\(235\) 1458.00 0.404721
\(236\) −600.000 −0.165494
\(237\) 0 0
\(238\) −666.000 −0.181388
\(239\) 594.000 0.160764 0.0803821 0.996764i \(-0.474386\pi\)
0.0803821 + 0.996764i \(0.474386\pi\)
\(240\) 0 0
\(241\) 2303.00 0.615557 0.307779 0.951458i \(-0.400414\pi\)
0.307779 + 0.951458i \(0.400414\pi\)
\(242\) 1293.00 0.343459
\(243\) 0 0
\(244\) 233.000 0.0611324
\(245\) −3051.00 −0.795597
\(246\) 0 0
\(247\) 0 0
\(248\) −2100.00 −0.537702
\(249\) 0 0
\(250\) 4563.00 1.15436
\(251\) −6324.00 −1.59031 −0.795154 0.606407i \(-0.792610\pi\)
−0.795154 + 0.606407i \(0.792610\pi\)
\(252\) 0 0
\(253\) −180.000 −0.0447293
\(254\) 1812.00 0.447618
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) −7833.00 −1.90120 −0.950601 0.310414i \(-0.899532\pi\)
−0.950601 + 0.310414i \(0.899532\pi\)
\(258\) 0 0
\(259\) 34.0000 0.00815698
\(260\) 0 0
\(261\) 0 0
\(262\) −4752.00 −1.12053
\(263\) 3030.00 0.710410 0.355205 0.934788i \(-0.384411\pi\)
0.355205 + 0.934788i \(0.384411\pi\)
\(264\) 0 0
\(265\) −5751.00 −1.33314
\(266\) 276.000 0.0636190
\(267\) 0 0
\(268\) 926.000 0.211061
\(269\) 534.000 0.121036 0.0605178 0.998167i \(-0.480725\pi\)
0.0605178 + 0.998167i \(0.480725\pi\)
\(270\) 0 0
\(271\) −3688.00 −0.826679 −0.413340 0.910577i \(-0.635638\pi\)
−0.413340 + 0.910577i \(0.635638\pi\)
\(272\) −7881.00 −1.75682
\(273\) 0 0
\(274\) 2151.00 0.474258
\(275\) 1320.00 0.289451
\(276\) 0 0
\(277\) 1865.00 0.404538 0.202269 0.979330i \(-0.435168\pi\)
0.202269 + 0.979330i \(0.435168\pi\)
\(278\) 2460.00 0.530723
\(279\) 0 0
\(280\) 378.000 0.0806779
\(281\) −2997.00 −0.636249 −0.318125 0.948049i \(-0.603053\pi\)
−0.318125 + 0.948049i \(0.603053\pi\)
\(282\) 0 0
\(283\) −4114.00 −0.864141 −0.432071 0.901840i \(-0.642217\pi\)
−0.432071 + 0.901840i \(0.642217\pi\)
\(284\) 930.000 0.194315
\(285\) 0 0
\(286\) 0 0
\(287\) 462.000 0.0950209
\(288\) 0 0
\(289\) 7408.00 1.50784
\(290\) −2835.00 −0.574058
\(291\) 0 0
\(292\) −253.000 −0.0507045
\(293\) 4665.00 0.930144 0.465072 0.885273i \(-0.346028\pi\)
0.465072 + 0.885273i \(0.346028\pi\)
\(294\) 0 0
\(295\) −5400.00 −1.06576
\(296\) 357.000 0.0701020
\(297\) 0 0
\(298\) −5247.00 −1.01997
\(299\) 0 0
\(300\) 0 0
\(301\) −1028.00 −0.196854
\(302\) 1110.00 0.211501
\(303\) 0 0
\(304\) 3266.00 0.616177
\(305\) 2097.00 0.393685
\(306\) 0 0
\(307\) 1502.00 0.279230 0.139615 0.990206i \(-0.455413\pi\)
0.139615 + 0.990206i \(0.455413\pi\)
\(308\) −60.0000 −0.0111001
\(309\) 0 0
\(310\) 2700.00 0.494676
\(311\) −2106.00 −0.383988 −0.191994 0.981396i \(-0.561495\pi\)
−0.191994 + 0.981396i \(0.561495\pi\)
\(312\) 0 0
\(313\) −3898.00 −0.703923 −0.351962 0.936014i \(-0.614485\pi\)
−0.351962 + 0.936014i \(0.614485\pi\)
\(314\) 7833.00 1.40778
\(315\) 0 0
\(316\) −1324.00 −0.235699
\(317\) −9351.00 −1.65680 −0.828398 0.560140i \(-0.810747\pi\)
−0.828398 + 0.560140i \(0.810747\pi\)
\(318\) 0 0
\(319\) −3150.00 −0.552872
\(320\) 3897.00 0.680778
\(321\) 0 0
\(322\) −36.0000 −0.00623044
\(323\) −5106.00 −0.879583
\(324\) 0 0
\(325\) 0 0
\(326\) 4908.00 0.833831
\(327\) 0 0
\(328\) 4851.00 0.816621
\(329\) 324.000 0.0542939
\(330\) 0 0
\(331\) −9172.00 −1.52308 −0.761539 0.648119i \(-0.775556\pi\)
−0.761539 + 0.648119i \(0.775556\pi\)
\(332\) −810.000 −0.133899
\(333\) 0 0
\(334\) 792.000 0.129749
\(335\) 8334.00 1.35921
\(336\) 0 0
\(337\) −11089.0 −1.79245 −0.896226 0.443598i \(-0.853702\pi\)
−0.896226 + 0.443598i \(0.853702\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 999.000 0.159348
\(341\) 3000.00 0.476420
\(342\) 0 0
\(343\) −1364.00 −0.214720
\(344\) −10794.0 −1.69178
\(345\) 0 0
\(346\) 4230.00 0.657243
\(347\) −9762.00 −1.51024 −0.755118 0.655589i \(-0.772420\pi\)
−0.755118 + 0.655589i \(0.772420\pi\)
\(348\) 0 0
\(349\) −8290.00 −1.27150 −0.635750 0.771895i \(-0.719309\pi\)
−0.635750 + 0.771895i \(0.719309\pi\)
\(350\) 264.000 0.0403183
\(351\) 0 0
\(352\) −1350.00 −0.204418
\(353\) −12405.0 −1.87040 −0.935200 0.354119i \(-0.884781\pi\)
−0.935200 + 0.354119i \(0.884781\pi\)
\(354\) 0 0
\(355\) 8370.00 1.25136
\(356\) −498.000 −0.0741403
\(357\) 0 0
\(358\) −1422.00 −0.209930
\(359\) 1098.00 0.161421 0.0807106 0.996738i \(-0.474281\pi\)
0.0807106 + 0.996738i \(0.474281\pi\)
\(360\) 0 0
\(361\) −4743.00 −0.691500
\(362\) −6747.00 −0.979598
\(363\) 0 0
\(364\) 0 0
\(365\) −2277.00 −0.326530
\(366\) 0 0
\(367\) −5734.00 −0.815565 −0.407783 0.913079i \(-0.633698\pi\)
−0.407783 + 0.913079i \(0.633698\pi\)
\(368\) −426.000 −0.0603445
\(369\) 0 0
\(370\) −459.000 −0.0644926
\(371\) −1278.00 −0.178842
\(372\) 0 0
\(373\) −8971.00 −1.24531 −0.622655 0.782496i \(-0.713946\pi\)
−0.622655 + 0.782496i \(0.713946\pi\)
\(374\) 9990.00 1.38120
\(375\) 0 0
\(376\) 3402.00 0.466608
\(377\) 0 0
\(378\) 0 0
\(379\) 7244.00 0.981792 0.490896 0.871218i \(-0.336669\pi\)
0.490896 + 0.871218i \(0.336669\pi\)
\(380\) −414.000 −0.0558888
\(381\) 0 0
\(382\) 10332.0 1.38385
\(383\) 6312.00 0.842110 0.421055 0.907035i \(-0.361660\pi\)
0.421055 + 0.907035i \(0.361660\pi\)
\(384\) 0 0
\(385\) −540.000 −0.0714830
\(386\) 12819.0 1.69034
\(387\) 0 0
\(388\) 1358.00 0.177686
\(389\) −3627.00 −0.472741 −0.236370 0.971663i \(-0.575958\pi\)
−0.236370 + 0.971663i \(0.575958\pi\)
\(390\) 0 0
\(391\) 666.000 0.0861408
\(392\) −7119.00 −0.917255
\(393\) 0 0
\(394\) −5958.00 −0.761827
\(395\) −11916.0 −1.51787
\(396\) 0 0
\(397\) −3898.00 −0.492783 −0.246392 0.969170i \(-0.579245\pi\)
−0.246392 + 0.969170i \(0.579245\pi\)
\(398\) 7158.00 0.901503
\(399\) 0 0
\(400\) 3124.00 0.390500
\(401\) 5703.00 0.710210 0.355105 0.934826i \(-0.384445\pi\)
0.355105 + 0.934826i \(0.384445\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 357.000 0.0439639
\(405\) 0 0
\(406\) −630.000 −0.0770108
\(407\) −510.000 −0.0621124
\(408\) 0 0
\(409\) 6311.00 0.762980 0.381490 0.924373i \(-0.375411\pi\)
0.381490 + 0.924373i \(0.375411\pi\)
\(410\) −6237.00 −0.751277
\(411\) 0 0
\(412\) 1118.00 0.133689
\(413\) −1200.00 −0.142974
\(414\) 0 0
\(415\) −7290.00 −0.862294
\(416\) 0 0
\(417\) 0 0
\(418\) −4140.00 −0.484435
\(419\) 2328.00 0.271433 0.135716 0.990748i \(-0.456666\pi\)
0.135716 + 0.990748i \(0.456666\pi\)
\(420\) 0 0
\(421\) 2045.00 0.236739 0.118370 0.992970i \(-0.462233\pi\)
0.118370 + 0.992970i \(0.462233\pi\)
\(422\) 4800.00 0.553697
\(423\) 0 0
\(424\) −13419.0 −1.53699
\(425\) −4884.00 −0.557432
\(426\) 0 0
\(427\) 466.000 0.0528134
\(428\) −714.000 −0.0806367
\(429\) 0 0
\(430\) 13878.0 1.55641
\(431\) −5034.00 −0.562597 −0.281298 0.959620i \(-0.590765\pi\)
−0.281298 + 0.959620i \(0.590765\pi\)
\(432\) 0 0
\(433\) 4283.00 0.475353 0.237676 0.971344i \(-0.423614\pi\)
0.237676 + 0.971344i \(0.423614\pi\)
\(434\) 600.000 0.0663616
\(435\) 0 0
\(436\) 2006.00 0.220344
\(437\) −276.000 −0.0302125
\(438\) 0 0
\(439\) −1306.00 −0.141986 −0.0709931 0.997477i \(-0.522617\pi\)
−0.0709931 + 0.997477i \(0.522617\pi\)
\(440\) −5670.00 −0.614333
\(441\) 0 0
\(442\) 0 0
\(443\) 5796.00 0.621617 0.310808 0.950473i \(-0.399400\pi\)
0.310808 + 0.950473i \(0.399400\pi\)
\(444\) 0 0
\(445\) −4482.00 −0.477454
\(446\) 11496.0 1.22052
\(447\) 0 0
\(448\) 866.000 0.0913274
\(449\) −2706.00 −0.284419 −0.142209 0.989837i \(-0.545421\pi\)
−0.142209 + 0.989837i \(0.545421\pi\)
\(450\) 0 0
\(451\) −6930.00 −0.723550
\(452\) 1119.00 0.116445
\(453\) 0 0
\(454\) −4194.00 −0.433555
\(455\) 0 0
\(456\) 0 0
\(457\) −829.000 −0.0848555 −0.0424278 0.999100i \(-0.513509\pi\)
−0.0424278 + 0.999100i \(0.513509\pi\)
\(458\) −13398.0 −1.36692
\(459\) 0 0
\(460\) 54.0000 0.00547340
\(461\) 5493.00 0.554956 0.277478 0.960732i \(-0.410502\pi\)
0.277478 + 0.960732i \(0.410502\pi\)
\(462\) 0 0
\(463\) −15346.0 −1.54037 −0.770183 0.637823i \(-0.779835\pi\)
−0.770183 + 0.637823i \(0.779835\pi\)
\(464\) −7455.00 −0.745883
\(465\) 0 0
\(466\) −4914.00 −0.488491
\(467\) 9594.00 0.950658 0.475329 0.879808i \(-0.342329\pi\)
0.475329 + 0.879808i \(0.342329\pi\)
\(468\) 0 0
\(469\) 1852.00 0.182340
\(470\) −4374.00 −0.429271
\(471\) 0 0
\(472\) −12600.0 −1.22873
\(473\) 15420.0 1.49897
\(474\) 0 0
\(475\) 2024.00 0.195511
\(476\) 222.000 0.0213768
\(477\) 0 0
\(478\) −1782.00 −0.170516
\(479\) 12840.0 1.22479 0.612395 0.790552i \(-0.290206\pi\)
0.612395 + 0.790552i \(0.290206\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −6909.00 −0.652897
\(483\) 0 0
\(484\) −431.000 −0.0404771
\(485\) 12222.0 1.14427
\(486\) 0 0
\(487\) −14086.0 −1.31067 −0.655336 0.755337i \(-0.727473\pi\)
−0.655336 + 0.755337i \(0.727473\pi\)
\(488\) 4893.00 0.453885
\(489\) 0 0
\(490\) 9153.00 0.843858
\(491\) −11694.0 −1.07483 −0.537416 0.843317i \(-0.680600\pi\)
−0.537416 + 0.843317i \(0.680600\pi\)
\(492\) 0 0
\(493\) 11655.0 1.06474
\(494\) 0 0
\(495\) 0 0
\(496\) 7100.00 0.642741
\(497\) 1860.00 0.167872
\(498\) 0 0
\(499\) −3688.00 −0.330857 −0.165428 0.986222i \(-0.552901\pi\)
−0.165428 + 0.986222i \(0.552901\pi\)
\(500\) −1521.00 −0.136042
\(501\) 0 0
\(502\) 18972.0 1.68678
\(503\) 4746.00 0.420703 0.210352 0.977626i \(-0.432539\pi\)
0.210352 + 0.977626i \(0.432539\pi\)
\(504\) 0 0
\(505\) 3213.00 0.283122
\(506\) 540.000 0.0474425
\(507\) 0 0
\(508\) −604.000 −0.0527523
\(509\) 14505.0 1.26311 0.631555 0.775331i \(-0.282417\pi\)
0.631555 + 0.775331i \(0.282417\pi\)
\(510\) 0 0
\(511\) −506.000 −0.0438045
\(512\) 8733.00 0.753804
\(513\) 0 0
\(514\) 23499.0 2.01653
\(515\) 10062.0 0.860941
\(516\) 0 0
\(517\) −4860.00 −0.413429
\(518\) −102.000 −0.00865178
\(519\) 0 0
\(520\) 0 0
\(521\) −5085.00 −0.427597 −0.213798 0.976878i \(-0.568584\pi\)
−0.213798 + 0.976878i \(0.568584\pi\)
\(522\) 0 0
\(523\) −10882.0 −0.909821 −0.454911 0.890537i \(-0.650329\pi\)
−0.454911 + 0.890537i \(0.650329\pi\)
\(524\) 1584.00 0.132056
\(525\) 0 0
\(526\) −9090.00 −0.753503
\(527\) −11100.0 −0.917502
\(528\) 0 0
\(529\) −12131.0 −0.997041
\(530\) 17253.0 1.41400
\(531\) 0 0
\(532\) −92.0000 −0.00749757
\(533\) 0 0
\(534\) 0 0
\(535\) −6426.00 −0.519290
\(536\) 19446.0 1.56705
\(537\) 0 0
\(538\) −1602.00 −0.128378
\(539\) 10170.0 0.812714
\(540\) 0 0
\(541\) −4699.00 −0.373430 −0.186715 0.982414i \(-0.559784\pi\)
−0.186715 + 0.982414i \(0.559784\pi\)
\(542\) 11064.0 0.876826
\(543\) 0 0
\(544\) 4995.00 0.393674
\(545\) 18054.0 1.41899
\(546\) 0 0
\(547\) 8270.00 0.646434 0.323217 0.946325i \(-0.395236\pi\)
0.323217 + 0.946325i \(0.395236\pi\)
\(548\) −717.000 −0.0558918
\(549\) 0 0
\(550\) −3960.00 −0.307009
\(551\) −4830.00 −0.373439
\(552\) 0 0
\(553\) −2648.00 −0.203625
\(554\) −5595.00 −0.429077
\(555\) 0 0
\(556\) −820.000 −0.0625463
\(557\) 22785.0 1.73327 0.866635 0.498943i \(-0.166278\pi\)
0.866635 + 0.498943i \(0.166278\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1278.00 −0.0964381
\(561\) 0 0
\(562\) 8991.00 0.674844
\(563\) 11928.0 0.892905 0.446452 0.894807i \(-0.352687\pi\)
0.446452 + 0.894807i \(0.352687\pi\)
\(564\) 0 0
\(565\) 10071.0 0.749894
\(566\) 12342.0 0.916560
\(567\) 0 0
\(568\) 19530.0 1.44271
\(569\) 7962.00 0.586616 0.293308 0.956018i \(-0.405244\pi\)
0.293308 + 0.956018i \(0.405244\pi\)
\(570\) 0 0
\(571\) 20618.0 1.51110 0.755549 0.655093i \(-0.227370\pi\)
0.755549 + 0.655093i \(0.227370\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1386.00 −0.100785
\(575\) −264.000 −0.0191471
\(576\) 0 0
\(577\) −3493.00 −0.252020 −0.126010 0.992029i \(-0.540217\pi\)
−0.126010 + 0.992029i \(0.540217\pi\)
\(578\) −22224.0 −1.59930
\(579\) 0 0
\(580\) 945.000 0.0676534
\(581\) −1620.00 −0.115678
\(582\) 0 0
\(583\) 19170.0 1.36182
\(584\) −5313.00 −0.376461
\(585\) 0 0
\(586\) −13995.0 −0.986567
\(587\) −10416.0 −0.732392 −0.366196 0.930538i \(-0.619340\pi\)
−0.366196 + 0.930538i \(0.619340\pi\)
\(588\) 0 0
\(589\) 4600.00 0.321799
\(590\) 16200.0 1.13041
\(591\) 0 0
\(592\) −1207.00 −0.0837963
\(593\) −2061.00 −0.142724 −0.0713618 0.997450i \(-0.522734\pi\)
−0.0713618 + 0.997450i \(0.522734\pi\)
\(594\) 0 0
\(595\) 1998.00 0.137664
\(596\) 1749.00 0.120204
\(597\) 0 0
\(598\) 0 0
\(599\) −12456.0 −0.849647 −0.424823 0.905276i \(-0.639664\pi\)
−0.424823 + 0.905276i \(0.639664\pi\)
\(600\) 0 0
\(601\) −781.000 −0.0530077 −0.0265039 0.999649i \(-0.508437\pi\)
−0.0265039 + 0.999649i \(0.508437\pi\)
\(602\) 3084.00 0.208795
\(603\) 0 0
\(604\) −370.000 −0.0249256
\(605\) −3879.00 −0.260667
\(606\) 0 0
\(607\) 19304.0 1.29082 0.645408 0.763838i \(-0.276687\pi\)
0.645408 + 0.763838i \(0.276687\pi\)
\(608\) −2070.00 −0.138075
\(609\) 0 0
\(610\) −6291.00 −0.417566
\(611\) 0 0
\(612\) 0 0
\(613\) 12041.0 0.793363 0.396681 0.917956i \(-0.370162\pi\)
0.396681 + 0.917956i \(0.370162\pi\)
\(614\) −4506.00 −0.296168
\(615\) 0 0
\(616\) −1260.00 −0.0824137
\(617\) −9717.00 −0.634022 −0.317011 0.948422i \(-0.602679\pi\)
−0.317011 + 0.948422i \(0.602679\pi\)
\(618\) 0 0
\(619\) −21040.0 −1.36619 −0.683093 0.730332i \(-0.739366\pi\)
−0.683093 + 0.730332i \(0.739366\pi\)
\(620\) −900.000 −0.0582982
\(621\) 0 0
\(622\) 6318.00 0.407281
\(623\) −996.000 −0.0640512
\(624\) 0 0
\(625\) −8189.00 −0.524096
\(626\) 11694.0 0.746623
\(627\) 0 0
\(628\) −2611.00 −0.165908
\(629\) 1887.00 0.119618
\(630\) 0 0
\(631\) −5068.00 −0.319737 −0.159868 0.987138i \(-0.551107\pi\)
−0.159868 + 0.987138i \(0.551107\pi\)
\(632\) −27804.0 −1.74997
\(633\) 0 0
\(634\) 28053.0 1.75730
\(635\) −5436.00 −0.339718
\(636\) 0 0
\(637\) 0 0
\(638\) 9450.00 0.586409
\(639\) 0 0
\(640\) −14931.0 −0.922187
\(641\) −10185.0 −0.627587 −0.313794 0.949491i \(-0.601600\pi\)
−0.313794 + 0.949491i \(0.601600\pi\)
\(642\) 0 0
\(643\) 25928.0 1.59020 0.795101 0.606476i \(-0.207418\pi\)
0.795101 + 0.606476i \(0.207418\pi\)
\(644\) 12.0000 0.000734264 0
\(645\) 0 0
\(646\) 15318.0 0.932939
\(647\) −23160.0 −1.40729 −0.703643 0.710554i \(-0.748444\pi\)
−0.703643 + 0.710554i \(0.748444\pi\)
\(648\) 0 0
\(649\) 18000.0 1.08869
\(650\) 0 0
\(651\) 0 0
\(652\) −1636.00 −0.0982680
\(653\) −16626.0 −0.996364 −0.498182 0.867073i \(-0.665999\pi\)
−0.498182 + 0.867073i \(0.665999\pi\)
\(654\) 0 0
\(655\) 14256.0 0.850424
\(656\) −16401.0 −0.976146
\(657\) 0 0
\(658\) −972.000 −0.0575874
\(659\) 14808.0 0.875323 0.437661 0.899140i \(-0.355807\pi\)
0.437661 + 0.899140i \(0.355807\pi\)
\(660\) 0 0
\(661\) 4853.00 0.285567 0.142784 0.989754i \(-0.454395\pi\)
0.142784 + 0.989754i \(0.454395\pi\)
\(662\) 27516.0 1.61547
\(663\) 0 0
\(664\) −17010.0 −0.994151
\(665\) −828.000 −0.0482834
\(666\) 0 0
\(667\) 630.000 0.0365723
\(668\) −264.000 −0.0152911
\(669\) 0 0
\(670\) −25002.0 −1.44166
\(671\) −6990.00 −0.402155
\(672\) 0 0
\(673\) −16165.0 −0.925877 −0.462938 0.886391i \(-0.653205\pi\)
−0.462938 + 0.886391i \(0.653205\pi\)
\(674\) 33267.0 1.90118
\(675\) 0 0
\(676\) 0 0
\(677\) 25686.0 1.45819 0.729094 0.684414i \(-0.239942\pi\)
0.729094 + 0.684414i \(0.239942\pi\)
\(678\) 0 0
\(679\) 2716.00 0.153506
\(680\) 20979.0 1.18310
\(681\) 0 0
\(682\) −9000.00 −0.505319
\(683\) 19056.0 1.06758 0.533790 0.845617i \(-0.320767\pi\)
0.533790 + 0.845617i \(0.320767\pi\)
\(684\) 0 0
\(685\) −6453.00 −0.359936
\(686\) 4092.00 0.227745
\(687\) 0 0
\(688\) 36494.0 2.02227
\(689\) 0 0
\(690\) 0 0
\(691\) −16390.0 −0.902323 −0.451161 0.892442i \(-0.648990\pi\)
−0.451161 + 0.892442i \(0.648990\pi\)
\(692\) −1410.00 −0.0774569
\(693\) 0 0
\(694\) 29286.0 1.60185
\(695\) −7380.00 −0.402790
\(696\) 0 0
\(697\) 25641.0 1.39343
\(698\) 24870.0 1.34863
\(699\) 0 0
\(700\) −88.0000 −0.00475155
\(701\) 27846.0 1.50033 0.750163 0.661253i \(-0.229975\pi\)
0.750163 + 0.661253i \(0.229975\pi\)
\(702\) 0 0
\(703\) −782.000 −0.0419540
\(704\) −12990.0 −0.695425
\(705\) 0 0
\(706\) 37215.0 1.98386
\(707\) 714.000 0.0379812
\(708\) 0 0
\(709\) −12283.0 −0.650632 −0.325316 0.945605i \(-0.605471\pi\)
−0.325316 + 0.945605i \(0.605471\pi\)
\(710\) −25110.0 −1.32727
\(711\) 0 0
\(712\) −10458.0 −0.550464
\(713\) −600.000 −0.0315150
\(714\) 0 0
\(715\) 0 0
\(716\) 474.000 0.0247405
\(717\) 0 0
\(718\) −3294.00 −0.171213
\(719\) −25512.0 −1.32328 −0.661639 0.749822i \(-0.730139\pi\)
−0.661639 + 0.749822i \(0.730139\pi\)
\(720\) 0 0
\(721\) 2236.00 0.115497
\(722\) 14229.0 0.733447
\(723\) 0 0
\(724\) 2249.00 0.115447
\(725\) −4620.00 −0.236666
\(726\) 0 0
\(727\) 6110.00 0.311702 0.155851 0.987781i \(-0.450188\pi\)
0.155851 + 0.987781i \(0.450188\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 6831.00 0.346338
\(731\) −57054.0 −2.88676
\(732\) 0 0
\(733\) −27127.0 −1.36693 −0.683464 0.729984i \(-0.739527\pi\)
−0.683464 + 0.729984i \(0.739527\pi\)
\(734\) 17202.0 0.865037
\(735\) 0 0
\(736\) 270.000 0.0135222
\(737\) −27780.0 −1.38845
\(738\) 0 0
\(739\) −880.000 −0.0438042 −0.0219021 0.999760i \(-0.506972\pi\)
−0.0219021 + 0.999760i \(0.506972\pi\)
\(740\) 153.000 0.00760053
\(741\) 0 0
\(742\) 3834.00 0.189691
\(743\) 21876.0 1.08015 0.540076 0.841616i \(-0.318396\pi\)
0.540076 + 0.841616i \(0.318396\pi\)
\(744\) 0 0
\(745\) 15741.0 0.774102
\(746\) 26913.0 1.32085
\(747\) 0 0
\(748\) −3330.00 −0.162777
\(749\) −1428.00 −0.0696635
\(750\) 0 0
\(751\) 11798.0 0.573256 0.286628 0.958042i \(-0.407466\pi\)
0.286628 + 0.958042i \(0.407466\pi\)
\(752\) −11502.0 −0.557759
\(753\) 0 0
\(754\) 0 0
\(755\) −3330.00 −0.160518
\(756\) 0 0
\(757\) −8074.00 −0.387655 −0.193827 0.981036i \(-0.562090\pi\)
−0.193827 + 0.981036i \(0.562090\pi\)
\(758\) −21732.0 −1.04135
\(759\) 0 0
\(760\) −8694.00 −0.414953
\(761\) −19554.0 −0.931448 −0.465724 0.884930i \(-0.654206\pi\)
−0.465724 + 0.884930i \(0.654206\pi\)
\(762\) 0 0
\(763\) 4012.00 0.190359
\(764\) −3444.00 −0.163088
\(765\) 0 0
\(766\) −18936.0 −0.893193
\(767\) 0 0
\(768\) 0 0
\(769\) 14030.0 0.657913 0.328956 0.944345i \(-0.393303\pi\)
0.328956 + 0.944345i \(0.393303\pi\)
\(770\) 1620.00 0.0758192
\(771\) 0 0
\(772\) −4273.00 −0.199208
\(773\) −36042.0 −1.67703 −0.838513 0.544882i \(-0.816574\pi\)
−0.838513 + 0.544882i \(0.816574\pi\)
\(774\) 0 0
\(775\) 4400.00 0.203939
\(776\) 28518.0 1.31925
\(777\) 0 0
\(778\) 10881.0 0.501417
\(779\) −10626.0 −0.488724
\(780\) 0 0
\(781\) −27900.0 −1.27828
\(782\) −1998.00 −0.0913662
\(783\) 0 0
\(784\) 24069.0 1.09644
\(785\) −23499.0 −1.06843
\(786\) 0 0
\(787\) 28628.0 1.29667 0.648334 0.761356i \(-0.275466\pi\)
0.648334 + 0.761356i \(0.275466\pi\)
\(788\) 1986.00 0.0897821
\(789\) 0 0
\(790\) 35748.0 1.60995
\(791\) 2238.00 0.100599
\(792\) 0 0
\(793\) 0 0
\(794\) 11694.0 0.522676
\(795\) 0 0
\(796\) −2386.00 −0.106243
\(797\) 37434.0 1.66371 0.831857 0.554990i \(-0.187278\pi\)
0.831857 + 0.554990i \(0.187278\pi\)
\(798\) 0 0
\(799\) 17982.0 0.796192
\(800\) −1980.00 −0.0875045
\(801\) 0 0
\(802\) −17109.0 −0.753292
\(803\) 7590.00 0.333556
\(804\) 0 0
\(805\) 108.000 0.00472857
\(806\) 0 0
\(807\) 0 0
\(808\) 7497.00 0.326415
\(809\) −37569.0 −1.63270 −0.816351 0.577556i \(-0.804006\pi\)
−0.816351 + 0.577556i \(0.804006\pi\)
\(810\) 0 0
\(811\) 5516.00 0.238832 0.119416 0.992844i \(-0.461898\pi\)
0.119416 + 0.992844i \(0.461898\pi\)
\(812\) 210.000 0.00907581
\(813\) 0 0
\(814\) 1530.00 0.0658802
\(815\) −14724.0 −0.632833
\(816\) 0 0
\(817\) 23644.0 1.01248
\(818\) −18933.0 −0.809263
\(819\) 0 0
\(820\) 2079.00 0.0885388
\(821\) −8778.00 −0.373148 −0.186574 0.982441i \(-0.559738\pi\)
−0.186574 + 0.982441i \(0.559738\pi\)
\(822\) 0 0
\(823\) −3088.00 −0.130791 −0.0653955 0.997859i \(-0.520831\pi\)
−0.0653955 + 0.997859i \(0.520831\pi\)
\(824\) 23478.0 0.992591
\(825\) 0 0
\(826\) 3600.00 0.151647
\(827\) −13176.0 −0.554020 −0.277010 0.960867i \(-0.589343\pi\)
−0.277010 + 0.960867i \(0.589343\pi\)
\(828\) 0 0
\(829\) −2359.00 −0.0988317 −0.0494158 0.998778i \(-0.515736\pi\)
−0.0494158 + 0.998778i \(0.515736\pi\)
\(830\) 21870.0 0.914601
\(831\) 0 0
\(832\) 0 0
\(833\) −37629.0 −1.56515
\(834\) 0 0
\(835\) −2376.00 −0.0984729
\(836\) 1380.00 0.0570913
\(837\) 0 0
\(838\) −6984.00 −0.287898
\(839\) 2676.00 0.110114 0.0550571 0.998483i \(-0.482466\pi\)
0.0550571 + 0.998483i \(0.482466\pi\)
\(840\) 0 0
\(841\) −13364.0 −0.547952
\(842\) −6135.00 −0.251100
\(843\) 0 0
\(844\) −1600.00 −0.0652539
\(845\) 0 0
\(846\) 0 0
\(847\) −862.000 −0.0349689
\(848\) 45369.0 1.83724
\(849\) 0 0
\(850\) 14652.0 0.591246
\(851\) 102.000 0.00410871
\(852\) 0 0
\(853\) 2477.00 0.0994266 0.0497133 0.998764i \(-0.484169\pi\)
0.0497133 + 0.998764i \(0.484169\pi\)
\(854\) −1398.00 −0.0560171
\(855\) 0 0
\(856\) −14994.0 −0.598697
\(857\) 17199.0 0.685539 0.342769 0.939420i \(-0.388635\pi\)
0.342769 + 0.939420i \(0.388635\pi\)
\(858\) 0 0
\(859\) 24338.0 0.966708 0.483354 0.875425i \(-0.339418\pi\)
0.483354 + 0.875425i \(0.339418\pi\)
\(860\) −4626.00 −0.183425
\(861\) 0 0
\(862\) 15102.0 0.596724
\(863\) −25146.0 −0.991865 −0.495933 0.868361i \(-0.665174\pi\)
−0.495933 + 0.868361i \(0.665174\pi\)
\(864\) 0 0
\(865\) −12690.0 −0.498813
\(866\) −12849.0 −0.504188
\(867\) 0 0
\(868\) −200.000 −0.00782079
\(869\) 39720.0 1.55053
\(870\) 0 0
\(871\) 0 0
\(872\) 42126.0 1.63597
\(873\) 0 0
\(874\) 828.000 0.0320452
\(875\) −3042.00 −0.117530
\(876\) 0 0
\(877\) 18089.0 0.696490 0.348245 0.937403i \(-0.386778\pi\)
0.348245 + 0.937403i \(0.386778\pi\)
\(878\) 3918.00 0.150599
\(879\) 0 0
\(880\) 19170.0 0.734342
\(881\) 15099.0 0.577410 0.288705 0.957418i \(-0.406775\pi\)
0.288705 + 0.957418i \(0.406775\pi\)
\(882\) 0 0
\(883\) 33488.0 1.27629 0.638143 0.769918i \(-0.279703\pi\)
0.638143 + 0.769918i \(0.279703\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −17388.0 −0.659324
\(887\) 39768.0 1.50539 0.752694 0.658371i \(-0.228754\pi\)
0.752694 + 0.658371i \(0.228754\pi\)
\(888\) 0 0
\(889\) −1208.00 −0.0455737
\(890\) 13446.0 0.506417
\(891\) 0 0
\(892\) −3832.00 −0.143840
\(893\) −7452.00 −0.279252
\(894\) 0 0
\(895\) 4266.00 0.159326
\(896\) −3318.00 −0.123713
\(897\) 0 0
\(898\) 8118.00 0.301672
\(899\) −10500.0 −0.389538
\(900\) 0 0
\(901\) −70929.0 −2.62263
\(902\) 20790.0 0.767440
\(903\) 0 0
\(904\) 23499.0 0.864563
\(905\) 20241.0 0.743463
\(906\) 0 0
\(907\) 32156.0 1.17720 0.588601 0.808424i \(-0.299679\pi\)
0.588601 + 0.808424i \(0.299679\pi\)
\(908\) 1398.00 0.0510950
\(909\) 0 0
\(910\) 0 0
\(911\) −11520.0 −0.418962 −0.209481 0.977813i \(-0.567177\pi\)
−0.209481 + 0.977813i \(0.567177\pi\)
\(912\) 0 0
\(913\) 24300.0 0.880846
\(914\) 2487.00 0.0900029
\(915\) 0 0
\(916\) 4466.00 0.161093
\(917\) 3168.00 0.114086
\(918\) 0 0
\(919\) 4952.00 0.177749 0.0888745 0.996043i \(-0.471673\pi\)
0.0888745 + 0.996043i \(0.471673\pi\)
\(920\) 1134.00 0.0406379
\(921\) 0 0
\(922\) −16479.0 −0.588619
\(923\) 0 0
\(924\) 0 0
\(925\) −748.000 −0.0265882
\(926\) 46038.0 1.63380
\(927\) 0 0
\(928\) 4725.00 0.167140
\(929\) −8781.00 −0.310113 −0.155057 0.987906i \(-0.549556\pi\)
−0.155057 + 0.987906i \(0.549556\pi\)
\(930\) 0 0
\(931\) 15594.0 0.548950
\(932\) 1638.00 0.0575692
\(933\) 0 0
\(934\) −28782.0 −1.00833
\(935\) −29970.0 −1.04826
\(936\) 0 0
\(937\) 50039.0 1.74461 0.872307 0.488959i \(-0.162623\pi\)
0.872307 + 0.488959i \(0.162623\pi\)
\(938\) −5556.00 −0.193401
\(939\) 0 0
\(940\) 1458.00 0.0505901
\(941\) 50670.0 1.75536 0.877681 0.479246i \(-0.159090\pi\)
0.877681 + 0.479246i \(0.159090\pi\)
\(942\) 0 0
\(943\) 1386.00 0.0478625
\(944\) 42600.0 1.46876
\(945\) 0 0
\(946\) −46260.0 −1.58990
\(947\) −42384.0 −1.45438 −0.727188 0.686438i \(-0.759173\pi\)
−0.727188 + 0.686438i \(0.759173\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −6072.00 −0.207370
\(951\) 0 0
\(952\) 4662.00 0.158715
\(953\) 50538.0 1.71782 0.858912 0.512123i \(-0.171141\pi\)
0.858912 + 0.512123i \(0.171141\pi\)
\(954\) 0 0
\(955\) −30996.0 −1.05027
\(956\) 594.000 0.0200955
\(957\) 0 0
\(958\) −38520.0 −1.29909
\(959\) −1434.00 −0.0482860
\(960\) 0 0
\(961\) −19791.0 −0.664328
\(962\) 0 0
\(963\) 0 0
\(964\) 2303.00 0.0769446
\(965\) −38457.0 −1.28288
\(966\) 0 0
\(967\) −6886.00 −0.228996 −0.114498 0.993423i \(-0.536526\pi\)
−0.114498 + 0.993423i \(0.536526\pi\)
\(968\) −9051.00 −0.300527
\(969\) 0 0
\(970\) −36666.0 −1.21368
\(971\) −9060.00 −0.299433 −0.149716 0.988729i \(-0.547836\pi\)
−0.149716 + 0.988729i \(0.547836\pi\)
\(972\) 0 0
\(973\) −1640.00 −0.0540349
\(974\) 42258.0 1.39018
\(975\) 0 0
\(976\) −16543.0 −0.542550
\(977\) 28311.0 0.927072 0.463536 0.886078i \(-0.346581\pi\)
0.463536 + 0.886078i \(0.346581\pi\)
\(978\) 0 0
\(979\) 14940.0 0.487727
\(980\) −3051.00 −0.0994496
\(981\) 0 0
\(982\) 35082.0 1.14003
\(983\) −4284.00 −0.139001 −0.0695007 0.997582i \(-0.522141\pi\)
−0.0695007 + 0.997582i \(0.522141\pi\)
\(984\) 0 0
\(985\) 17874.0 0.578186
\(986\) −34965.0 −1.12932
\(987\) 0 0
\(988\) 0 0
\(989\) −3084.00 −0.0991562
\(990\) 0 0
\(991\) −2458.00 −0.0787901 −0.0393950 0.999224i \(-0.512543\pi\)
−0.0393950 + 0.999224i \(0.512543\pi\)
\(992\) −4500.00 −0.144027
\(993\) 0 0
\(994\) −5580.00 −0.178055
\(995\) −21474.0 −0.684193
\(996\) 0 0
\(997\) 24101.0 0.765583 0.382792 0.923835i \(-0.374963\pi\)
0.382792 + 0.923835i \(0.374963\pi\)
\(998\) 11064.0 0.350927
\(999\) 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 1521.4.a.c.1.1 1
3.2 odd 2 507.4.a.e.1.1 1
13.3 even 3 117.4.g.b.100.1 2
13.9 even 3 117.4.g.b.55.1 2
13.12 even 2 1521.4.a.j.1.1 1
39.5 even 4 507.4.b.c.337.1 2
39.8 even 4 507.4.b.c.337.2 2
39.29 odd 6 39.4.e.a.22.1 yes 2
39.35 odd 6 39.4.e.a.16.1 2
39.38 odd 2 507.4.a.a.1.1 1
156.35 even 6 624.4.q.b.289.1 2
156.107 even 6 624.4.q.b.529.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.a.16.1 2 39.35 odd 6
39.4.e.a.22.1 yes 2 39.29 odd 6
117.4.g.b.55.1 2 13.9 even 3
117.4.g.b.100.1 2 13.3 even 3
507.4.a.a.1.1 1 39.38 odd 2
507.4.a.e.1.1 1 3.2 odd 2
507.4.b.c.337.1 2 39.5 even 4
507.4.b.c.337.2 2 39.8 even 4
624.4.q.b.289.1 2 156.35 even 6
624.4.q.b.529.1 2 156.107 even 6
1521.4.a.c.1.1 1 1.1 even 1 trivial
1521.4.a.j.1.1 1 13.12 even 2