Properties

Label 450.4.a.c.1.1
Level $450$
Weight $4$
Character 450.1
Self dual yes
Analytic conductor $26.551$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 450.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -14.0000 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} -14.0000 q^{7} -8.00000 q^{8} -6.00000 q^{11} -68.0000 q^{13} +28.0000 q^{14} +16.0000 q^{16} +78.0000 q^{17} +44.0000 q^{19} +12.0000 q^{22} +120.000 q^{23} +136.000 q^{26} -56.0000 q^{28} -126.000 q^{29} -244.000 q^{31} -32.0000 q^{32} -156.000 q^{34} +304.000 q^{37} -88.0000 q^{38} +480.000 q^{41} -104.000 q^{43} -24.0000 q^{44} -240.000 q^{46} +600.000 q^{47} -147.000 q^{49} -272.000 q^{52} -258.000 q^{53} +112.000 q^{56} +252.000 q^{58} -534.000 q^{59} +362.000 q^{61} +488.000 q^{62} +64.0000 q^{64} +268.000 q^{67} +312.000 q^{68} +972.000 q^{71} -470.000 q^{73} -608.000 q^{74} +176.000 q^{76} +84.0000 q^{77} +1244.00 q^{79} -960.000 q^{82} +396.000 q^{83} +208.000 q^{86} +48.0000 q^{88} +972.000 q^{89} +952.000 q^{91} +480.000 q^{92} -1200.00 q^{94} +46.0000 q^{97} +294.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −14.0000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −6.00000 −0.164461 −0.0822304 0.996613i \(-0.526204\pi\)
−0.0822304 + 0.996613i \(0.526204\pi\)
\(12\) 0 0
\(13\) −68.0000 −1.45075 −0.725377 0.688352i \(-0.758335\pi\)
−0.725377 + 0.688352i \(0.758335\pi\)
\(14\) 28.0000 0.534522
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 78.0000 1.11281 0.556405 0.830911i \(-0.312180\pi\)
0.556405 + 0.830911i \(0.312180\pi\)
\(18\) 0 0
\(19\) 44.0000 0.531279 0.265639 0.964072i \(-0.414417\pi\)
0.265639 + 0.964072i \(0.414417\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 12.0000 0.116291
\(23\) 120.000 1.08790 0.543951 0.839117i \(-0.316928\pi\)
0.543951 + 0.839117i \(0.316928\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 136.000 1.02584
\(27\) 0 0
\(28\) −56.0000 −0.377964
\(29\) −126.000 −0.806814 −0.403407 0.915021i \(-0.632174\pi\)
−0.403407 + 0.915021i \(0.632174\pi\)
\(30\) 0 0
\(31\) −244.000 −1.41367 −0.706834 0.707380i \(-0.749877\pi\)
−0.706834 + 0.707380i \(0.749877\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −156.000 −0.786876
\(35\) 0 0
\(36\) 0 0
\(37\) 304.000 1.35074 0.675369 0.737480i \(-0.263984\pi\)
0.675369 + 0.737480i \(0.263984\pi\)
\(38\) −88.0000 −0.375671
\(39\) 0 0
\(40\) 0 0
\(41\) 480.000 1.82838 0.914188 0.405291i \(-0.132830\pi\)
0.914188 + 0.405291i \(0.132830\pi\)
\(42\) 0 0
\(43\) −104.000 −0.368834 −0.184417 0.982848i \(-0.559040\pi\)
−0.184417 + 0.982848i \(0.559040\pi\)
\(44\) −24.0000 −0.0822304
\(45\) 0 0
\(46\) −240.000 −0.769262
\(47\) 600.000 1.86211 0.931053 0.364884i \(-0.118891\pi\)
0.931053 + 0.364884i \(0.118891\pi\)
\(48\) 0 0
\(49\) −147.000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −272.000 −0.725377
\(53\) −258.000 −0.668661 −0.334330 0.942456i \(-0.608510\pi\)
−0.334330 + 0.942456i \(0.608510\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 112.000 0.267261
\(57\) 0 0
\(58\) 252.000 0.570504
\(59\) −534.000 −1.17832 −0.589160 0.808016i \(-0.700541\pi\)
−0.589160 + 0.808016i \(0.700541\pi\)
\(60\) 0 0
\(61\) 362.000 0.759825 0.379913 0.925022i \(-0.375954\pi\)
0.379913 + 0.925022i \(0.375954\pi\)
\(62\) 488.000 0.999614
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 268.000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 312.000 0.556405
\(69\) 0 0
\(70\) 0 0
\(71\) 972.000 1.62472 0.812360 0.583156i \(-0.198182\pi\)
0.812360 + 0.583156i \(0.198182\pi\)
\(72\) 0 0
\(73\) −470.000 −0.753553 −0.376776 0.926304i \(-0.622967\pi\)
−0.376776 + 0.926304i \(0.622967\pi\)
\(74\) −608.000 −0.955116
\(75\) 0 0
\(76\) 176.000 0.265639
\(77\) 84.0000 0.124321
\(78\) 0 0
\(79\) 1244.00 1.77166 0.885829 0.464012i \(-0.153591\pi\)
0.885829 + 0.464012i \(0.153591\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −960.000 −1.29286
\(83\) 396.000 0.523695 0.261847 0.965109i \(-0.415668\pi\)
0.261847 + 0.965109i \(0.415668\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 208.000 0.260805
\(87\) 0 0
\(88\) 48.0000 0.0581456
\(89\) 972.000 1.15766 0.578830 0.815448i \(-0.303509\pi\)
0.578830 + 0.815448i \(0.303509\pi\)
\(90\) 0 0
\(91\) 952.000 1.09667
\(92\) 480.000 0.543951
\(93\) 0 0
\(94\) −1200.00 −1.31671
\(95\) 0 0
\(96\) 0 0
\(97\) 46.0000 0.0481504 0.0240752 0.999710i \(-0.492336\pi\)
0.0240752 + 0.999710i \(0.492336\pi\)
\(98\) 294.000 0.303046
\(99\) 0 0
\(100\) 0 0
\(101\) 1506.00 1.48369 0.741845 0.670572i \(-0.233951\pi\)
0.741845 + 0.670572i \(0.233951\pi\)
\(102\) 0 0
\(103\) 1474.00 1.41007 0.705037 0.709171i \(-0.250931\pi\)
0.705037 + 0.709171i \(0.250931\pi\)
\(104\) 544.000 0.512919
\(105\) 0 0
\(106\) 516.000 0.472815
\(107\) 924.000 0.834827 0.417413 0.908717i \(-0.362937\pi\)
0.417413 + 0.908717i \(0.362937\pi\)
\(108\) 0 0
\(109\) 698.000 0.613360 0.306680 0.951813i \(-0.400782\pi\)
0.306680 + 0.951813i \(0.400782\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −224.000 −0.188982
\(113\) −222.000 −0.184814 −0.0924071 0.995721i \(-0.529456\pi\)
−0.0924071 + 0.995721i \(0.529456\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −504.000 −0.403407
\(117\) 0 0
\(118\) 1068.00 0.833198
\(119\) −1092.00 −0.841206
\(120\) 0 0
\(121\) −1295.00 −0.972953
\(122\) −724.000 −0.537278
\(123\) 0 0
\(124\) −976.000 −0.706834
\(125\) 0 0
\(126\) 0 0
\(127\) 1906.00 1.33173 0.665867 0.746071i \(-0.268062\pi\)
0.665867 + 0.746071i \(0.268062\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −2874.00 −1.91681 −0.958407 0.285406i \(-0.907872\pi\)
−0.958407 + 0.285406i \(0.907872\pi\)
\(132\) 0 0
\(133\) −616.000 −0.401609
\(134\) −536.000 −0.345547
\(135\) 0 0
\(136\) −624.000 −0.393438
\(137\) −798.000 −0.497648 −0.248824 0.968549i \(-0.580044\pi\)
−0.248824 + 0.968549i \(0.580044\pi\)
\(138\) 0 0
\(139\) −700.000 −0.427146 −0.213573 0.976927i \(-0.568510\pi\)
−0.213573 + 0.976927i \(0.568510\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1944.00 −1.14885
\(143\) 408.000 0.238592
\(144\) 0 0
\(145\) 0 0
\(146\) 940.000 0.532842
\(147\) 0 0
\(148\) 1216.00 0.675369
\(149\) −114.000 −0.0626795 −0.0313397 0.999509i \(-0.509977\pi\)
−0.0313397 + 0.999509i \(0.509977\pi\)
\(150\) 0 0
\(151\) 1064.00 0.573424 0.286712 0.958017i \(-0.407438\pi\)
0.286712 + 0.958017i \(0.407438\pi\)
\(152\) −352.000 −0.187835
\(153\) 0 0
\(154\) −168.000 −0.0879080
\(155\) 0 0
\(156\) 0 0
\(157\) 1948.00 0.990238 0.495119 0.868825i \(-0.335125\pi\)
0.495119 + 0.868825i \(0.335125\pi\)
\(158\) −2488.00 −1.25275
\(159\) 0 0
\(160\) 0 0
\(161\) −1680.00 −0.822376
\(162\) 0 0
\(163\) −2060.00 −0.989887 −0.494944 0.868925i \(-0.664811\pi\)
−0.494944 + 0.868925i \(0.664811\pi\)
\(164\) 1920.00 0.914188
\(165\) 0 0
\(166\) −792.000 −0.370308
\(167\) −1248.00 −0.578282 −0.289141 0.957286i \(-0.593370\pi\)
−0.289141 + 0.957286i \(0.593370\pi\)
\(168\) 0 0
\(169\) 2427.00 1.10469
\(170\) 0 0
\(171\) 0 0
\(172\) −416.000 −0.184417
\(173\) −1146.00 −0.503634 −0.251817 0.967775i \(-0.581028\pi\)
−0.251817 + 0.967775i \(0.581028\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −96.0000 −0.0411152
\(177\) 0 0
\(178\) −1944.00 −0.818590
\(179\) 1146.00 0.478525 0.239263 0.970955i \(-0.423094\pi\)
0.239263 + 0.970955i \(0.423094\pi\)
\(180\) 0 0
\(181\) −118.000 −0.0484579 −0.0242289 0.999706i \(-0.507713\pi\)
−0.0242289 + 0.999706i \(0.507713\pi\)
\(182\) −1904.00 −0.775461
\(183\) 0 0
\(184\) −960.000 −0.384631
\(185\) 0 0
\(186\) 0 0
\(187\) −468.000 −0.183014
\(188\) 2400.00 0.931053
\(189\) 0 0
\(190\) 0 0
\(191\) −1692.00 −0.640989 −0.320494 0.947250i \(-0.603849\pi\)
−0.320494 + 0.947250i \(0.603849\pi\)
\(192\) 0 0
\(193\) −3350.00 −1.24942 −0.624711 0.780856i \(-0.714783\pi\)
−0.624711 + 0.780856i \(0.714783\pi\)
\(194\) −92.0000 −0.0340475
\(195\) 0 0
\(196\) −588.000 −0.214286
\(197\) −3606.00 −1.30415 −0.652073 0.758156i \(-0.726101\pi\)
−0.652073 + 0.758156i \(0.726101\pi\)
\(198\) 0 0
\(199\) 2696.00 0.960374 0.480187 0.877166i \(-0.340569\pi\)
0.480187 + 0.877166i \(0.340569\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −3012.00 −1.04913
\(203\) 1764.00 0.609894
\(204\) 0 0
\(205\) 0 0
\(206\) −2948.00 −0.997072
\(207\) 0 0
\(208\) −1088.00 −0.362689
\(209\) −264.000 −0.0873745
\(210\) 0 0
\(211\) −4.00000 −0.00130508 −0.000652539 1.00000i \(-0.500208\pi\)
−0.000652539 1.00000i \(0.500208\pi\)
\(212\) −1032.00 −0.334330
\(213\) 0 0
\(214\) −1848.00 −0.590312
\(215\) 0 0
\(216\) 0 0
\(217\) 3416.00 1.06863
\(218\) −1396.00 −0.433711
\(219\) 0 0
\(220\) 0 0
\(221\) −5304.00 −1.61441
\(222\) 0 0
\(223\) 1162.00 0.348938 0.174469 0.984663i \(-0.444179\pi\)
0.174469 + 0.984663i \(0.444179\pi\)
\(224\) 448.000 0.133631
\(225\) 0 0
\(226\) 444.000 0.130683
\(227\) −2400.00 −0.701734 −0.350867 0.936425i \(-0.614113\pi\)
−0.350867 + 0.936425i \(0.614113\pi\)
\(228\) 0 0
\(229\) −2314.00 −0.667744 −0.333872 0.942618i \(-0.608355\pi\)
−0.333872 + 0.942618i \(0.608355\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1008.00 0.285252
\(233\) −18.0000 −0.00506103 −0.00253051 0.999997i \(-0.500805\pi\)
−0.00253051 + 0.999997i \(0.500805\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2136.00 −0.589160
\(237\) 0 0
\(238\) 2184.00 0.594822
\(239\) 5868.00 1.58816 0.794078 0.607816i \(-0.207954\pi\)
0.794078 + 0.607816i \(0.207954\pi\)
\(240\) 0 0
\(241\) −4330.00 −1.15734 −0.578672 0.815560i \(-0.696429\pi\)
−0.578672 + 0.815560i \(0.696429\pi\)
\(242\) 2590.00 0.687981
\(243\) 0 0
\(244\) 1448.00 0.379913
\(245\) 0 0
\(246\) 0 0
\(247\) −2992.00 −0.770755
\(248\) 1952.00 0.499807
\(249\) 0 0
\(250\) 0 0
\(251\) −498.000 −0.125233 −0.0626165 0.998038i \(-0.519944\pi\)
−0.0626165 + 0.998038i \(0.519944\pi\)
\(252\) 0 0
\(253\) −720.000 −0.178917
\(254\) −3812.00 −0.941678
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 642.000 0.155824 0.0779122 0.996960i \(-0.475175\pi\)
0.0779122 + 0.996960i \(0.475175\pi\)
\(258\) 0 0
\(259\) −4256.00 −1.02106
\(260\) 0 0
\(261\) 0 0
\(262\) 5748.00 1.35539
\(263\) −7968.00 −1.86817 −0.934084 0.357055i \(-0.883781\pi\)
−0.934084 + 0.357055i \(0.883781\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1232.00 0.283980
\(267\) 0 0
\(268\) 1072.00 0.244339
\(269\) 4218.00 0.956045 0.478022 0.878348i \(-0.341354\pi\)
0.478022 + 0.878348i \(0.341354\pi\)
\(270\) 0 0
\(271\) 848.000 0.190082 0.0950412 0.995473i \(-0.469702\pi\)
0.0950412 + 0.995473i \(0.469702\pi\)
\(272\) 1248.00 0.278203
\(273\) 0 0
\(274\) 1596.00 0.351890
\(275\) 0 0
\(276\) 0 0
\(277\) 1504.00 0.326233 0.163117 0.986607i \(-0.447845\pi\)
0.163117 + 0.986607i \(0.447845\pi\)
\(278\) 1400.00 0.302037
\(279\) 0 0
\(280\) 0 0
\(281\) 1308.00 0.277682 0.138841 0.990315i \(-0.455662\pi\)
0.138841 + 0.990315i \(0.455662\pi\)
\(282\) 0 0
\(283\) 5932.00 1.24601 0.623005 0.782218i \(-0.285912\pi\)
0.623005 + 0.782218i \(0.285912\pi\)
\(284\) 3888.00 0.812360
\(285\) 0 0
\(286\) −816.000 −0.168710
\(287\) −6720.00 −1.38212
\(288\) 0 0
\(289\) 1171.00 0.238347
\(290\) 0 0
\(291\) 0 0
\(292\) −1880.00 −0.376776
\(293\) 5226.00 1.04200 0.521000 0.853556i \(-0.325559\pi\)
0.521000 + 0.853556i \(0.325559\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2432.00 −0.477558
\(297\) 0 0
\(298\) 228.000 0.0443211
\(299\) −8160.00 −1.57828
\(300\) 0 0
\(301\) 1456.00 0.278812
\(302\) −2128.00 −0.405472
\(303\) 0 0
\(304\) 704.000 0.132820
\(305\) 0 0
\(306\) 0 0
\(307\) −4448.00 −0.826908 −0.413454 0.910525i \(-0.635678\pi\)
−0.413454 + 0.910525i \(0.635678\pi\)
\(308\) 336.000 0.0621603
\(309\) 0 0
\(310\) 0 0
\(311\) 9132.00 1.66504 0.832521 0.553993i \(-0.186897\pi\)
0.832521 + 0.553993i \(0.186897\pi\)
\(312\) 0 0
\(313\) 2170.00 0.391871 0.195936 0.980617i \(-0.437226\pi\)
0.195936 + 0.980617i \(0.437226\pi\)
\(314\) −3896.00 −0.700204
\(315\) 0 0
\(316\) 4976.00 0.885829
\(317\) 7674.00 1.35967 0.679834 0.733366i \(-0.262052\pi\)
0.679834 + 0.733366i \(0.262052\pi\)
\(318\) 0 0
\(319\) 756.000 0.132689
\(320\) 0 0
\(321\) 0 0
\(322\) 3360.00 0.581508
\(323\) 3432.00 0.591212
\(324\) 0 0
\(325\) 0 0
\(326\) 4120.00 0.699956
\(327\) 0 0
\(328\) −3840.00 −0.646428
\(329\) −8400.00 −1.40762
\(330\) 0 0
\(331\) 9596.00 1.59349 0.796743 0.604318i \(-0.206554\pi\)
0.796743 + 0.604318i \(0.206554\pi\)
\(332\) 1584.00 0.261847
\(333\) 0 0
\(334\) 2496.00 0.408907
\(335\) 0 0
\(336\) 0 0
\(337\) −12158.0 −1.96525 −0.982624 0.185608i \(-0.940574\pi\)
−0.982624 + 0.185608i \(0.940574\pi\)
\(338\) −4854.00 −0.781133
\(339\) 0 0
\(340\) 0 0
\(341\) 1464.00 0.232493
\(342\) 0 0
\(343\) 6860.00 1.07990
\(344\) 832.000 0.130402
\(345\) 0 0
\(346\) 2292.00 0.356123
\(347\) 10320.0 1.59656 0.798280 0.602286i \(-0.205743\pi\)
0.798280 + 0.602286i \(0.205743\pi\)
\(348\) 0 0
\(349\) −2158.00 −0.330989 −0.165494 0.986211i \(-0.552922\pi\)
−0.165494 + 0.986211i \(0.552922\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 192.000 0.0290728
\(353\) −330.000 −0.0497567 −0.0248784 0.999690i \(-0.507920\pi\)
−0.0248784 + 0.999690i \(0.507920\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3888.00 0.578830
\(357\) 0 0
\(358\) −2292.00 −0.338369
\(359\) 8664.00 1.27373 0.636864 0.770976i \(-0.280231\pi\)
0.636864 + 0.770976i \(0.280231\pi\)
\(360\) 0 0
\(361\) −4923.00 −0.717743
\(362\) 236.000 0.0342649
\(363\) 0 0
\(364\) 3808.00 0.548334
\(365\) 0 0
\(366\) 0 0
\(367\) −3782.00 −0.537926 −0.268963 0.963151i \(-0.586681\pi\)
−0.268963 + 0.963151i \(0.586681\pi\)
\(368\) 1920.00 0.271975
\(369\) 0 0
\(370\) 0 0
\(371\) 3612.00 0.505460
\(372\) 0 0
\(373\) −11276.0 −1.56528 −0.782640 0.622475i \(-0.786127\pi\)
−0.782640 + 0.622475i \(0.786127\pi\)
\(374\) 936.000 0.129410
\(375\) 0 0
\(376\) −4800.00 −0.658354
\(377\) 8568.00 1.17049
\(378\) 0 0
\(379\) 980.000 0.132821 0.0664106 0.997792i \(-0.478845\pi\)
0.0664106 + 0.997792i \(0.478845\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 3384.00 0.453247
\(383\) −4200.00 −0.560339 −0.280170 0.959950i \(-0.590391\pi\)
−0.280170 + 0.959950i \(0.590391\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 6700.00 0.883474
\(387\) 0 0
\(388\) 184.000 0.0240752
\(389\) −13338.0 −1.73847 −0.869233 0.494402i \(-0.835387\pi\)
−0.869233 + 0.494402i \(0.835387\pi\)
\(390\) 0 0
\(391\) 9360.00 1.21063
\(392\) 1176.00 0.151523
\(393\) 0 0
\(394\) 7212.00 0.922171
\(395\) 0 0
\(396\) 0 0
\(397\) 7192.00 0.909209 0.454605 0.890693i \(-0.349781\pi\)
0.454605 + 0.890693i \(0.349781\pi\)
\(398\) −5392.00 −0.679087
\(399\) 0 0
\(400\) 0 0
\(401\) 2316.00 0.288418 0.144209 0.989547i \(-0.453936\pi\)
0.144209 + 0.989547i \(0.453936\pi\)
\(402\) 0 0
\(403\) 16592.0 2.05088
\(404\) 6024.00 0.741845
\(405\) 0 0
\(406\) −3528.00 −0.431260
\(407\) −1824.00 −0.222143
\(408\) 0 0
\(409\) −12358.0 −1.49404 −0.747022 0.664800i \(-0.768517\pi\)
−0.747022 + 0.664800i \(0.768517\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 5896.00 0.705037
\(413\) 7476.00 0.890726
\(414\) 0 0
\(415\) 0 0
\(416\) 2176.00 0.256460
\(417\) 0 0
\(418\) 528.000 0.0617831
\(419\) −3306.00 −0.385462 −0.192731 0.981252i \(-0.561735\pi\)
−0.192731 + 0.981252i \(0.561735\pi\)
\(420\) 0 0
\(421\) −14506.0 −1.67929 −0.839643 0.543139i \(-0.817236\pi\)
−0.839643 + 0.543139i \(0.817236\pi\)
\(422\) 8.00000 0.000922829 0
\(423\) 0 0
\(424\) 2064.00 0.236407
\(425\) 0 0
\(426\) 0 0
\(427\) −5068.00 −0.574374
\(428\) 3696.00 0.417413
\(429\) 0 0
\(430\) 0 0
\(431\) 6480.00 0.724201 0.362100 0.932139i \(-0.382060\pi\)
0.362100 + 0.932139i \(0.382060\pi\)
\(432\) 0 0
\(433\) −11894.0 −1.32007 −0.660034 0.751236i \(-0.729458\pi\)
−0.660034 + 0.751236i \(0.729458\pi\)
\(434\) −6832.00 −0.755637
\(435\) 0 0
\(436\) 2792.00 0.306680
\(437\) 5280.00 0.577979
\(438\) 0 0
\(439\) −12688.0 −1.37942 −0.689710 0.724086i \(-0.742262\pi\)
−0.689710 + 0.724086i \(0.742262\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 10608.0 1.14156
\(443\) 4968.00 0.532814 0.266407 0.963861i \(-0.414163\pi\)
0.266407 + 0.963861i \(0.414163\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2324.00 −0.246737
\(447\) 0 0
\(448\) −896.000 −0.0944911
\(449\) 11508.0 1.20957 0.604784 0.796389i \(-0.293259\pi\)
0.604784 + 0.796389i \(0.293259\pi\)
\(450\) 0 0
\(451\) −2880.00 −0.300696
\(452\) −888.000 −0.0924071
\(453\) 0 0
\(454\) 4800.00 0.496201
\(455\) 0 0
\(456\) 0 0
\(457\) −1082.00 −0.110752 −0.0553762 0.998466i \(-0.517636\pi\)
−0.0553762 + 0.998466i \(0.517636\pi\)
\(458\) 4628.00 0.472166
\(459\) 0 0
\(460\) 0 0
\(461\) 11238.0 1.13537 0.567685 0.823246i \(-0.307839\pi\)
0.567685 + 0.823246i \(0.307839\pi\)
\(462\) 0 0
\(463\) 2302.00 0.231065 0.115532 0.993304i \(-0.463143\pi\)
0.115532 + 0.993304i \(0.463143\pi\)
\(464\) −2016.00 −0.201704
\(465\) 0 0
\(466\) 36.0000 0.00357869
\(467\) 15876.0 1.57313 0.786567 0.617505i \(-0.211856\pi\)
0.786567 + 0.617505i \(0.211856\pi\)
\(468\) 0 0
\(469\) −3752.00 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 4272.00 0.416599
\(473\) 624.000 0.0606587
\(474\) 0 0
\(475\) 0 0
\(476\) −4368.00 −0.420603
\(477\) 0 0
\(478\) −11736.0 −1.12300
\(479\) −4644.00 −0.442985 −0.221492 0.975162i \(-0.571093\pi\)
−0.221492 + 0.975162i \(0.571093\pi\)
\(480\) 0 0
\(481\) −20672.0 −1.95959
\(482\) 8660.00 0.818366
\(483\) 0 0
\(484\) −5180.00 −0.486476
\(485\) 0 0
\(486\) 0 0
\(487\) −2426.00 −0.225734 −0.112867 0.993610i \(-0.536003\pi\)
−0.112867 + 0.993610i \(0.536003\pi\)
\(488\) −2896.00 −0.268639
\(489\) 0 0
\(490\) 0 0
\(491\) −234.000 −0.0215077 −0.0107538 0.999942i \(-0.503423\pi\)
−0.0107538 + 0.999942i \(0.503423\pi\)
\(492\) 0 0
\(493\) −9828.00 −0.897831
\(494\) 5984.00 0.545006
\(495\) 0 0
\(496\) −3904.00 −0.353417
\(497\) −13608.0 −1.22817
\(498\) 0 0
\(499\) 14204.0 1.27427 0.637133 0.770754i \(-0.280120\pi\)
0.637133 + 0.770754i \(0.280120\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 996.000 0.0885531
\(503\) 4920.00 0.436127 0.218064 0.975935i \(-0.430026\pi\)
0.218064 + 0.975935i \(0.430026\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1440.00 0.126513
\(507\) 0 0
\(508\) 7624.00 0.665867
\(509\) 4458.00 0.388207 0.194104 0.980981i \(-0.437820\pi\)
0.194104 + 0.980981i \(0.437820\pi\)
\(510\) 0 0
\(511\) 6580.00 0.569632
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −1284.00 −0.110184
\(515\) 0 0
\(516\) 0 0
\(517\) −3600.00 −0.306243
\(518\) 8512.00 0.722000
\(519\) 0 0
\(520\) 0 0
\(521\) −4212.00 −0.354186 −0.177093 0.984194i \(-0.556669\pi\)
−0.177093 + 0.984194i \(0.556669\pi\)
\(522\) 0 0
\(523\) 11212.0 0.937412 0.468706 0.883354i \(-0.344720\pi\)
0.468706 + 0.883354i \(0.344720\pi\)
\(524\) −11496.0 −0.958407
\(525\) 0 0
\(526\) 15936.0 1.32099
\(527\) −19032.0 −1.57314
\(528\) 0 0
\(529\) 2233.00 0.183529
\(530\) 0 0
\(531\) 0 0
\(532\) −2464.00 −0.200804
\(533\) −32640.0 −2.65252
\(534\) 0 0
\(535\) 0 0
\(536\) −2144.00 −0.172774
\(537\) 0 0
\(538\) −8436.00 −0.676026
\(539\) 882.000 0.0704832
\(540\) 0 0
\(541\) 14018.0 1.11401 0.557006 0.830508i \(-0.311950\pi\)
0.557006 + 0.830508i \(0.311950\pi\)
\(542\) −1696.00 −0.134409
\(543\) 0 0
\(544\) −2496.00 −0.196719
\(545\) 0 0
\(546\) 0 0
\(547\) −18200.0 −1.42262 −0.711312 0.702876i \(-0.751899\pi\)
−0.711312 + 0.702876i \(0.751899\pi\)
\(548\) −3192.00 −0.248824
\(549\) 0 0
\(550\) 0 0
\(551\) −5544.00 −0.428643
\(552\) 0 0
\(553\) −17416.0 −1.33925
\(554\) −3008.00 −0.230682
\(555\) 0 0
\(556\) −2800.00 −0.213573
\(557\) −11826.0 −0.899612 −0.449806 0.893126i \(-0.648507\pi\)
−0.449806 + 0.893126i \(0.648507\pi\)
\(558\) 0 0
\(559\) 7072.00 0.535087
\(560\) 0 0
\(561\) 0 0
\(562\) −2616.00 −0.196351
\(563\) −2952.00 −0.220980 −0.110490 0.993877i \(-0.535242\pi\)
−0.110490 + 0.993877i \(0.535242\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −11864.0 −0.881062
\(567\) 0 0
\(568\) −7776.00 −0.574426
\(569\) −3084.00 −0.227220 −0.113610 0.993525i \(-0.536241\pi\)
−0.113610 + 0.993525i \(0.536241\pi\)
\(570\) 0 0
\(571\) −4756.00 −0.348568 −0.174284 0.984695i \(-0.555761\pi\)
−0.174284 + 0.984695i \(0.555761\pi\)
\(572\) 1632.00 0.119296
\(573\) 0 0
\(574\) 13440.0 0.977308
\(575\) 0 0
\(576\) 0 0
\(577\) 11014.0 0.794660 0.397330 0.917676i \(-0.369937\pi\)
0.397330 + 0.917676i \(0.369937\pi\)
\(578\) −2342.00 −0.168537
\(579\) 0 0
\(580\) 0 0
\(581\) −5544.00 −0.395876
\(582\) 0 0
\(583\) 1548.00 0.109968
\(584\) 3760.00 0.266421
\(585\) 0 0
\(586\) −10452.0 −0.736806
\(587\) −852.000 −0.0599077 −0.0299538 0.999551i \(-0.509536\pi\)
−0.0299538 + 0.999551i \(0.509536\pi\)
\(588\) 0 0
\(589\) −10736.0 −0.751051
\(590\) 0 0
\(591\) 0 0
\(592\) 4864.00 0.337684
\(593\) 15546.0 1.07656 0.538278 0.842767i \(-0.319075\pi\)
0.538278 + 0.842767i \(0.319075\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −456.000 −0.0313397
\(597\) 0 0
\(598\) 16320.0 1.11601
\(599\) 8616.00 0.587713 0.293857 0.955850i \(-0.405061\pi\)
0.293857 + 0.955850i \(0.405061\pi\)
\(600\) 0 0
\(601\) 17510.0 1.18843 0.594216 0.804305i \(-0.297462\pi\)
0.594216 + 0.804305i \(0.297462\pi\)
\(602\) −2912.00 −0.197150
\(603\) 0 0
\(604\) 4256.00 0.286712
\(605\) 0 0
\(606\) 0 0
\(607\) 13894.0 0.929061 0.464531 0.885557i \(-0.346223\pi\)
0.464531 + 0.885557i \(0.346223\pi\)
\(608\) −1408.00 −0.0939177
\(609\) 0 0
\(610\) 0 0
\(611\) −40800.0 −2.70146
\(612\) 0 0
\(613\) 6496.00 0.428011 0.214006 0.976832i \(-0.431349\pi\)
0.214006 + 0.976832i \(0.431349\pi\)
\(614\) 8896.00 0.584712
\(615\) 0 0
\(616\) −672.000 −0.0439540
\(617\) −570.000 −0.0371918 −0.0185959 0.999827i \(-0.505920\pi\)
−0.0185959 + 0.999827i \(0.505920\pi\)
\(618\) 0 0
\(619\) −2140.00 −0.138956 −0.0694781 0.997583i \(-0.522133\pi\)
−0.0694781 + 0.997583i \(0.522133\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −18264.0 −1.17736
\(623\) −13608.0 −0.875109
\(624\) 0 0
\(625\) 0 0
\(626\) −4340.00 −0.277095
\(627\) 0 0
\(628\) 7792.00 0.495119
\(629\) 23712.0 1.50312
\(630\) 0 0
\(631\) 14660.0 0.924890 0.462445 0.886648i \(-0.346972\pi\)
0.462445 + 0.886648i \(0.346972\pi\)
\(632\) −9952.00 −0.626375
\(633\) 0 0
\(634\) −15348.0 −0.961431
\(635\) 0 0
\(636\) 0 0
\(637\) 9996.00 0.621752
\(638\) −1512.00 −0.0938255
\(639\) 0 0
\(640\) 0 0
\(641\) 456.000 0.0280982 0.0140491 0.999901i \(-0.495528\pi\)
0.0140491 + 0.999901i \(0.495528\pi\)
\(642\) 0 0
\(643\) 23452.0 1.43835 0.719173 0.694831i \(-0.244521\pi\)
0.719173 + 0.694831i \(0.244521\pi\)
\(644\) −6720.00 −0.411188
\(645\) 0 0
\(646\) −6864.00 −0.418050
\(647\) 7224.00 0.438956 0.219478 0.975617i \(-0.429565\pi\)
0.219478 + 0.975617i \(0.429565\pi\)
\(648\) 0 0
\(649\) 3204.00 0.193787
\(650\) 0 0
\(651\) 0 0
\(652\) −8240.00 −0.494944
\(653\) 19146.0 1.14738 0.573691 0.819072i \(-0.305511\pi\)
0.573691 + 0.819072i \(0.305511\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 7680.00 0.457094
\(657\) 0 0
\(658\) 16800.0 0.995338
\(659\) −27810.0 −1.64389 −0.821945 0.569567i \(-0.807111\pi\)
−0.821945 + 0.569567i \(0.807111\pi\)
\(660\) 0 0
\(661\) −30598.0 −1.80049 −0.900245 0.435383i \(-0.856613\pi\)
−0.900245 + 0.435383i \(0.856613\pi\)
\(662\) −19192.0 −1.12676
\(663\) 0 0
\(664\) −3168.00 −0.185154
\(665\) 0 0
\(666\) 0 0
\(667\) −15120.0 −0.877734
\(668\) −4992.00 −0.289141
\(669\) 0 0
\(670\) 0 0
\(671\) −2172.00 −0.124961
\(672\) 0 0
\(673\) 3778.00 0.216391 0.108196 0.994130i \(-0.465493\pi\)
0.108196 + 0.994130i \(0.465493\pi\)
\(674\) 24316.0 1.38964
\(675\) 0 0
\(676\) 9708.00 0.552344
\(677\) −27198.0 −1.54402 −0.772012 0.635608i \(-0.780749\pi\)
−0.772012 + 0.635608i \(0.780749\pi\)
\(678\) 0 0
\(679\) −644.000 −0.0363983
\(680\) 0 0
\(681\) 0 0
\(682\) −2928.00 −0.164397
\(683\) 32316.0 1.81045 0.905225 0.424933i \(-0.139702\pi\)
0.905225 + 0.424933i \(0.139702\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13720.0 −0.763604
\(687\) 0 0
\(688\) −1664.00 −0.0922084
\(689\) 17544.0 0.970063
\(690\) 0 0
\(691\) 29324.0 1.61438 0.807191 0.590291i \(-0.200987\pi\)
0.807191 + 0.590291i \(0.200987\pi\)
\(692\) −4584.00 −0.251817
\(693\) 0 0
\(694\) −20640.0 −1.12894
\(695\) 0 0
\(696\) 0 0
\(697\) 37440.0 2.03464
\(698\) 4316.00 0.234044
\(699\) 0 0
\(700\) 0 0
\(701\) −22782.0 −1.22748 −0.613741 0.789508i \(-0.710336\pi\)
−0.613741 + 0.789508i \(0.710336\pi\)
\(702\) 0 0
\(703\) 13376.0 0.717618
\(704\) −384.000 −0.0205576
\(705\) 0 0
\(706\) 660.000 0.0351833
\(707\) −21084.0 −1.12156
\(708\) 0 0
\(709\) 26054.0 1.38008 0.690041 0.723770i \(-0.257592\pi\)
0.690041 + 0.723770i \(0.257592\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −7776.00 −0.409295
\(713\) −29280.0 −1.53793
\(714\) 0 0
\(715\) 0 0
\(716\) 4584.00 0.239263
\(717\) 0 0
\(718\) −17328.0 −0.900662
\(719\) 5976.00 0.309968 0.154984 0.987917i \(-0.450467\pi\)
0.154984 + 0.987917i \(0.450467\pi\)
\(720\) 0 0
\(721\) −20636.0 −1.06592
\(722\) 9846.00 0.507521
\(723\) 0 0
\(724\) −472.000 −0.0242289
\(725\) 0 0
\(726\) 0 0
\(727\) 5110.00 0.260687 0.130343 0.991469i \(-0.458392\pi\)
0.130343 + 0.991469i \(0.458392\pi\)
\(728\) −7616.00 −0.387730
\(729\) 0 0
\(730\) 0 0
\(731\) −8112.00 −0.410442
\(732\) 0 0
\(733\) −17336.0 −0.873560 −0.436780 0.899568i \(-0.643881\pi\)
−0.436780 + 0.899568i \(0.643881\pi\)
\(734\) 7564.00 0.380371
\(735\) 0 0
\(736\) −3840.00 −0.192316
\(737\) −1608.00 −0.0803683
\(738\) 0 0
\(739\) −13660.0 −0.679961 −0.339981 0.940432i \(-0.610420\pi\)
−0.339981 + 0.940432i \(0.610420\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −7224.00 −0.357414
\(743\) 1320.00 0.0651765 0.0325882 0.999469i \(-0.489625\pi\)
0.0325882 + 0.999469i \(0.489625\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22552.0 1.10682
\(747\) 0 0
\(748\) −1872.00 −0.0915068
\(749\) −12936.0 −0.631070
\(750\) 0 0
\(751\) 15860.0 0.770625 0.385313 0.922786i \(-0.374094\pi\)
0.385313 + 0.922786i \(0.374094\pi\)
\(752\) 9600.00 0.465527
\(753\) 0 0
\(754\) −17136.0 −0.827661
\(755\) 0 0
\(756\) 0 0
\(757\) −22160.0 −1.06396 −0.531981 0.846756i \(-0.678552\pi\)
−0.531981 + 0.846756i \(0.678552\pi\)
\(758\) −1960.00 −0.0939187
\(759\) 0 0
\(760\) 0 0
\(761\) −13116.0 −0.624776 −0.312388 0.949955i \(-0.601129\pi\)
−0.312388 + 0.949955i \(0.601129\pi\)
\(762\) 0 0
\(763\) −9772.00 −0.463657
\(764\) −6768.00 −0.320494
\(765\) 0 0
\(766\) 8400.00 0.396220
\(767\) 36312.0 1.70945
\(768\) 0 0
\(769\) 32846.0 1.54026 0.770128 0.637889i \(-0.220192\pi\)
0.770128 + 0.637889i \(0.220192\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −13400.0 −0.624711
\(773\) −11982.0 −0.557520 −0.278760 0.960361i \(-0.589923\pi\)
−0.278760 + 0.960361i \(0.589923\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −368.000 −0.0170238
\(777\) 0 0
\(778\) 26676.0 1.22928
\(779\) 21120.0 0.971377
\(780\) 0 0
\(781\) −5832.00 −0.267203
\(782\) −18720.0 −0.856043
\(783\) 0 0
\(784\) −2352.00 −0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) 21076.0 0.954610 0.477305 0.878738i \(-0.341614\pi\)
0.477305 + 0.878738i \(0.341614\pi\)
\(788\) −14424.0 −0.652073
\(789\) 0 0
\(790\) 0 0
\(791\) 3108.00 0.139706
\(792\) 0 0
\(793\) −24616.0 −1.10232
\(794\) −14384.0 −0.642908
\(795\) 0 0
\(796\) 10784.0 0.480187
\(797\) 22086.0 0.981589 0.490794 0.871275i \(-0.336707\pi\)
0.490794 + 0.871275i \(0.336707\pi\)
\(798\) 0 0
\(799\) 46800.0 2.07217
\(800\) 0 0
\(801\) 0 0
\(802\) −4632.00 −0.203942
\(803\) 2820.00 0.123930
\(804\) 0 0
\(805\) 0 0
\(806\) −33184.0 −1.45019
\(807\) 0 0
\(808\) −12048.0 −0.524563
\(809\) −21384.0 −0.929322 −0.464661 0.885489i \(-0.653824\pi\)
−0.464661 + 0.885489i \(0.653824\pi\)
\(810\) 0 0
\(811\) 5228.00 0.226362 0.113181 0.993574i \(-0.463896\pi\)
0.113181 + 0.993574i \(0.463896\pi\)
\(812\) 7056.00 0.304947
\(813\) 0 0
\(814\) 3648.00 0.157079
\(815\) 0 0
\(816\) 0 0
\(817\) −4576.00 −0.195953
\(818\) 24716.0 1.05645
\(819\) 0 0
\(820\) 0 0
\(821\) 38010.0 1.61578 0.807892 0.589331i \(-0.200609\pi\)
0.807892 + 0.589331i \(0.200609\pi\)
\(822\) 0 0
\(823\) −38642.0 −1.63667 −0.818333 0.574745i \(-0.805101\pi\)
−0.818333 + 0.574745i \(0.805101\pi\)
\(824\) −11792.0 −0.498536
\(825\) 0 0
\(826\) −14952.0 −0.629839
\(827\) −15432.0 −0.648879 −0.324440 0.945906i \(-0.605176\pi\)
−0.324440 + 0.945906i \(0.605176\pi\)
\(828\) 0 0
\(829\) −3886.00 −0.162806 −0.0814031 0.996681i \(-0.525940\pi\)
−0.0814031 + 0.996681i \(0.525940\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4352.00 −0.181344
\(833\) −11466.0 −0.476919
\(834\) 0 0
\(835\) 0 0
\(836\) −1056.00 −0.0436872
\(837\) 0 0
\(838\) 6612.00 0.272563
\(839\) −27552.0 −1.13373 −0.566866 0.823810i \(-0.691844\pi\)
−0.566866 + 0.823810i \(0.691844\pi\)
\(840\) 0 0
\(841\) −8513.00 −0.349051
\(842\) 29012.0 1.18743
\(843\) 0 0
\(844\) −16.0000 −0.000652539 0
\(845\) 0 0
\(846\) 0 0
\(847\) 18130.0 0.735483
\(848\) −4128.00 −0.167165
\(849\) 0 0
\(850\) 0 0
\(851\) 36480.0 1.46947
\(852\) 0 0
\(853\) −15104.0 −0.606273 −0.303137 0.952947i \(-0.598034\pi\)
−0.303137 + 0.952947i \(0.598034\pi\)
\(854\) 10136.0 0.406144
\(855\) 0 0
\(856\) −7392.00 −0.295156
\(857\) −12306.0 −0.490508 −0.245254 0.969459i \(-0.578871\pi\)
−0.245254 + 0.969459i \(0.578871\pi\)
\(858\) 0 0
\(859\) −47500.0 −1.88670 −0.943352 0.331793i \(-0.892346\pi\)
−0.943352 + 0.331793i \(0.892346\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −12960.0 −0.512087
\(863\) 4272.00 0.168506 0.0842529 0.996444i \(-0.473150\pi\)
0.0842529 + 0.996444i \(0.473150\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 23788.0 0.933429
\(867\) 0 0
\(868\) 13664.0 0.534316
\(869\) −7464.00 −0.291368
\(870\) 0 0
\(871\) −18224.0 −0.708951
\(872\) −5584.00 −0.216856
\(873\) 0 0
\(874\) −10560.0 −0.408693
\(875\) 0 0
\(876\) 0 0
\(877\) 27796.0 1.07024 0.535122 0.844775i \(-0.320266\pi\)
0.535122 + 0.844775i \(0.320266\pi\)
\(878\) 25376.0 0.975397
\(879\) 0 0
\(880\) 0 0
\(881\) 39996.0 1.52951 0.764756 0.644320i \(-0.222860\pi\)
0.764756 + 0.644320i \(0.222860\pi\)
\(882\) 0 0
\(883\) 3772.00 0.143758 0.0718788 0.997413i \(-0.477101\pi\)
0.0718788 + 0.997413i \(0.477101\pi\)
\(884\) −21216.0 −0.807207
\(885\) 0 0
\(886\) −9936.00 −0.376757
\(887\) −5784.00 −0.218949 −0.109474 0.993990i \(-0.534917\pi\)
−0.109474 + 0.993990i \(0.534917\pi\)
\(888\) 0 0
\(889\) −26684.0 −1.00670
\(890\) 0 0
\(891\) 0 0
\(892\) 4648.00 0.174469
\(893\) 26400.0 0.989297
\(894\) 0 0
\(895\) 0 0
\(896\) 1792.00 0.0668153
\(897\) 0 0
\(898\) −23016.0 −0.855294
\(899\) 30744.0 1.14057
\(900\) 0 0
\(901\) −20124.0 −0.744093
\(902\) 5760.00 0.212624
\(903\) 0 0
\(904\) 1776.00 0.0653417
\(905\) 0 0
\(906\) 0 0
\(907\) 8440.00 0.308981 0.154490 0.987994i \(-0.450626\pi\)
0.154490 + 0.987994i \(0.450626\pi\)
\(908\) −9600.00 −0.350867
\(909\) 0 0
\(910\) 0 0
\(911\) 31920.0 1.16087 0.580437 0.814305i \(-0.302882\pi\)
0.580437 + 0.814305i \(0.302882\pi\)
\(912\) 0 0
\(913\) −2376.00 −0.0861272
\(914\) 2164.00 0.0783137
\(915\) 0 0
\(916\) −9256.00 −0.333872
\(917\) 40236.0 1.44897
\(918\) 0 0
\(919\) 34652.0 1.24381 0.621906 0.783092i \(-0.286358\pi\)
0.621906 + 0.783092i \(0.286358\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −22476.0 −0.802828
\(923\) −66096.0 −2.35707
\(924\) 0 0
\(925\) 0 0
\(926\) −4604.00 −0.163388
\(927\) 0 0
\(928\) 4032.00 0.142626
\(929\) −1404.00 −0.0495842 −0.0247921 0.999693i \(-0.507892\pi\)
−0.0247921 + 0.999693i \(0.507892\pi\)
\(930\) 0 0
\(931\) −6468.00 −0.227691
\(932\) −72.0000 −0.00253051
\(933\) 0 0
\(934\) −31752.0 −1.11237
\(935\) 0 0
\(936\) 0 0
\(937\) 7654.00 0.266857 0.133429 0.991058i \(-0.457401\pi\)
0.133429 + 0.991058i \(0.457401\pi\)
\(938\) 7504.00 0.261209
\(939\) 0 0
\(940\) 0 0
\(941\) −11298.0 −0.391397 −0.195698 0.980664i \(-0.562697\pi\)
−0.195698 + 0.980664i \(0.562697\pi\)
\(942\) 0 0
\(943\) 57600.0 1.98909
\(944\) −8544.00 −0.294580
\(945\) 0 0
\(946\) −1248.00 −0.0428922
\(947\) 28968.0 0.994016 0.497008 0.867746i \(-0.334432\pi\)
0.497008 + 0.867746i \(0.334432\pi\)
\(948\) 0 0
\(949\) 31960.0 1.09322
\(950\) 0 0
\(951\) 0 0
\(952\) 8736.00 0.297411
\(953\) 46410.0 1.57751 0.788755 0.614707i \(-0.210726\pi\)
0.788755 + 0.614707i \(0.210726\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 23472.0 0.794078
\(957\) 0 0
\(958\) 9288.00 0.313238
\(959\) 11172.0 0.376186
\(960\) 0 0
\(961\) 29745.0 0.998456
\(962\) 41344.0 1.38564
\(963\) 0 0
\(964\) −17320.0 −0.578672
\(965\) 0 0
\(966\) 0 0
\(967\) 41506.0 1.38029 0.690146 0.723670i \(-0.257546\pi\)
0.690146 + 0.723670i \(0.257546\pi\)
\(968\) 10360.0 0.343991
\(969\) 0 0
\(970\) 0 0
\(971\) 18246.0 0.603030 0.301515 0.953461i \(-0.402508\pi\)
0.301515 + 0.953461i \(0.402508\pi\)
\(972\) 0 0
\(973\) 9800.00 0.322892
\(974\) 4852.00 0.159618
\(975\) 0 0
\(976\) 5792.00 0.189956
\(977\) 25998.0 0.851330 0.425665 0.904881i \(-0.360040\pi\)
0.425665 + 0.904881i \(0.360040\pi\)
\(978\) 0 0
\(979\) −5832.00 −0.190390
\(980\) 0 0
\(981\) 0 0
\(982\) 468.000 0.0152082
\(983\) 14616.0 0.474240 0.237120 0.971480i \(-0.423797\pi\)
0.237120 + 0.971480i \(0.423797\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 19656.0 0.634863
\(987\) 0 0
\(988\) −11968.0 −0.385377
\(989\) −12480.0 −0.401255
\(990\) 0 0
\(991\) −2968.00 −0.0951379 −0.0475689 0.998868i \(-0.515147\pi\)
−0.0475689 + 0.998868i \(0.515147\pi\)
\(992\) 7808.00 0.249903
\(993\) 0 0
\(994\) 27216.0 0.868450
\(995\) 0 0
\(996\) 0 0
\(997\) 9052.00 0.287542 0.143771 0.989611i \(-0.454077\pi\)
0.143771 + 0.989611i \(0.454077\pi\)
\(998\) −28408.0 −0.901042
\(999\) 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 450.4.a.c.1.1 1
3.2 odd 2 450.4.a.m.1.1 1
5.2 odd 4 450.4.c.f.199.1 2
5.3 odd 4 450.4.c.f.199.2 2
5.4 even 2 90.4.a.e.1.1 yes 1
15.2 even 4 450.4.c.g.199.2 2
15.8 even 4 450.4.c.g.199.1 2
15.14 odd 2 90.4.a.b.1.1 1
20.19 odd 2 720.4.a.t.1.1 1
45.4 even 6 810.4.e.a.541.1 2
45.14 odd 6 810.4.e.u.541.1 2
45.29 odd 6 810.4.e.u.271.1 2
45.34 even 6 810.4.e.a.271.1 2
60.59 even 2 720.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.4.a.b.1.1 1 15.14 odd 2
90.4.a.e.1.1 yes 1 5.4 even 2
450.4.a.c.1.1 1 1.1 even 1 trivial
450.4.a.m.1.1 1 3.2 odd 2
450.4.c.f.199.1 2 5.2 odd 4
450.4.c.f.199.2 2 5.3 odd 4
450.4.c.g.199.1 2 15.8 even 4
450.4.c.g.199.2 2 15.2 even 4
720.4.a.e.1.1 1 60.59 even 2
720.4.a.t.1.1 1 20.19 odd 2
810.4.e.a.271.1 2 45.34 even 6
810.4.e.a.541.1 2 45.4 even 6
810.4.e.u.271.1 2 45.29 odd 6
810.4.e.u.541.1 2 45.14 odd 6