Properties

Label 2023.4.a.c.1.1
Level $2023$
Weight $4$
Character 2023.1
Self dual yes
Analytic conductor $119.361$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2023 = 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2023.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(119.360863942\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2023.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +6.00000 q^{5} -3.00000 q^{6} -7.00000 q^{7} -21.0000 q^{8} -26.0000 q^{9} +O(q^{10})\) \(q+3.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +6.00000 q^{5} -3.00000 q^{6} -7.00000 q^{7} -21.0000 q^{8} -26.0000 q^{9} +18.0000 q^{10} +30.0000 q^{11} -1.00000 q^{12} -22.0000 q^{13} -21.0000 q^{14} -6.00000 q^{15} -71.0000 q^{16} -78.0000 q^{18} +83.0000 q^{19} +6.00000 q^{20} +7.00000 q^{21} +90.0000 q^{22} +48.0000 q^{23} +21.0000 q^{24} -89.0000 q^{25} -66.0000 q^{26} +53.0000 q^{27} -7.00000 q^{28} -15.0000 q^{29} -18.0000 q^{30} +7.00000 q^{31} -45.0000 q^{32} -30.0000 q^{33} -42.0000 q^{35} -26.0000 q^{36} -50.0000 q^{37} +249.000 q^{38} +22.0000 q^{39} -126.000 q^{40} -360.000 q^{41} +21.0000 q^{42} +68.0000 q^{43} +30.0000 q^{44} -156.000 q^{45} +144.000 q^{46} -27.0000 q^{47} +71.0000 q^{48} +49.0000 q^{49} -267.000 q^{50} -22.0000 q^{52} +213.000 q^{53} +159.000 q^{54} +180.000 q^{55} +147.000 q^{56} -83.0000 q^{57} -45.0000 q^{58} +189.000 q^{59} -6.00000 q^{60} -314.000 q^{61} +21.0000 q^{62} +182.000 q^{63} +433.000 q^{64} -132.000 q^{65} -90.0000 q^{66} +314.000 q^{67} -48.0000 q^{69} -126.000 q^{70} +804.000 q^{71} +546.000 q^{72} +448.000 q^{73} -150.000 q^{74} +89.0000 q^{75} +83.0000 q^{76} -210.000 q^{77} +66.0000 q^{78} +1060.00 q^{79} -426.000 q^{80} +649.000 q^{81} -1080.00 q^{82} +873.000 q^{83} +7.00000 q^{84} +204.000 q^{86} +15.0000 q^{87} -630.000 q^{88} +270.000 q^{89} -468.000 q^{90} +154.000 q^{91} +48.0000 q^{92} -7.00000 q^{93} -81.0000 q^{94} +498.000 q^{95} +45.0000 q^{96} -1130.00 q^{97} +147.000 q^{98} -780.000 q^{99} +O(q^{100})\)

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 \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) −1.00000 −0.192450 −0.0962250 0.995360i \(-0.530677\pi\)
−0.0962250 + 0.995360i \(0.530677\pi\)
\(4\) 1.00000 0.125000
\(5\) 6.00000 0.536656 0.268328 0.963328i \(-0.413529\pi\)
0.268328 + 0.963328i \(0.413529\pi\)
\(6\) −3.00000 −0.204124
\(7\) −7.00000 −0.377964
\(8\) −21.0000 −0.928078
\(9\) −26.0000 −0.962963
\(10\) 18.0000 0.569210
\(11\) 30.0000 0.822304 0.411152 0.911567i \(-0.365127\pi\)
0.411152 + 0.911567i \(0.365127\pi\)
\(12\) −1.00000 −0.0240563
\(13\) −22.0000 −0.469362 −0.234681 0.972072i \(-0.575405\pi\)
−0.234681 + 0.972072i \(0.575405\pi\)
\(14\) −21.0000 −0.400892
\(15\) −6.00000 −0.103280
\(16\) −71.0000 −1.10938
\(17\) 0 0
\(18\) −78.0000 −1.02138
\(19\) 83.0000 1.00218 0.501092 0.865394i \(-0.332932\pi\)
0.501092 + 0.865394i \(0.332932\pi\)
\(20\) 6.00000 0.0670820
\(21\) 7.00000 0.0727393
\(22\) 90.0000 0.872185
\(23\) 48.0000 0.435161 0.217580 0.976042i \(-0.430184\pi\)
0.217580 + 0.976042i \(0.430184\pi\)
\(24\) 21.0000 0.178609
\(25\) −89.0000 −0.712000
\(26\) −66.0000 −0.497833
\(27\) 53.0000 0.377772
\(28\) −7.00000 −0.0472456
\(29\) −15.0000 −0.0960493 −0.0480247 0.998846i \(-0.515293\pi\)
−0.0480247 + 0.998846i \(0.515293\pi\)
\(30\) −18.0000 −0.109545
\(31\) 7.00000 0.0405560 0.0202780 0.999794i \(-0.493545\pi\)
0.0202780 + 0.999794i \(0.493545\pi\)
\(32\) −45.0000 −0.248592
\(33\) −30.0000 −0.158252
\(34\) 0 0
\(35\) −42.0000 −0.202837
\(36\) −26.0000 −0.120370
\(37\) −50.0000 −0.222161 −0.111080 0.993811i \(-0.535431\pi\)
−0.111080 + 0.993811i \(0.535431\pi\)
\(38\) 249.000 1.06298
\(39\) 22.0000 0.0903287
\(40\) −126.000 −0.498059
\(41\) −360.000 −1.37128 −0.685641 0.727940i \(-0.740478\pi\)
−0.685641 + 0.727940i \(0.740478\pi\)
\(42\) 21.0000 0.0771517
\(43\) 68.0000 0.241161 0.120580 0.992704i \(-0.461524\pi\)
0.120580 + 0.992704i \(0.461524\pi\)
\(44\) 30.0000 0.102788
\(45\) −156.000 −0.516780
\(46\) 144.000 0.461557
\(47\) −27.0000 −0.0837948 −0.0418974 0.999122i \(-0.513340\pi\)
−0.0418974 + 0.999122i \(0.513340\pi\)
\(48\) 71.0000 0.213499
\(49\) 49.0000 0.142857
\(50\) −267.000 −0.755190
\(51\) 0 0
\(52\) −22.0000 −0.0586702
\(53\) 213.000 0.552034 0.276017 0.961153i \(-0.410985\pi\)
0.276017 + 0.961153i \(0.410985\pi\)
\(54\) 159.000 0.400688
\(55\) 180.000 0.441294
\(56\) 147.000 0.350780
\(57\) −83.0000 −0.192871
\(58\) −45.0000 −0.101876
\(59\) 189.000 0.417046 0.208523 0.978017i \(-0.433134\pi\)
0.208523 + 0.978017i \(0.433134\pi\)
\(60\) −6.00000 −0.0129099
\(61\) −314.000 −0.659075 −0.329538 0.944142i \(-0.606893\pi\)
−0.329538 + 0.944142i \(0.606893\pi\)
\(62\) 21.0000 0.0430162
\(63\) 182.000 0.363966
\(64\) 433.000 0.845703
\(65\) −132.000 −0.251886
\(66\) −90.0000 −0.167852
\(67\) 314.000 0.572555 0.286278 0.958147i \(-0.407582\pi\)
0.286278 + 0.958147i \(0.407582\pi\)
\(68\) 0 0
\(69\) −48.0000 −0.0837467
\(70\) −126.000 −0.215141
\(71\) 804.000 1.34390 0.671952 0.740594i \(-0.265456\pi\)
0.671952 + 0.740594i \(0.265456\pi\)
\(72\) 546.000 0.893704
\(73\) 448.000 0.718280 0.359140 0.933284i \(-0.383070\pi\)
0.359140 + 0.933284i \(0.383070\pi\)
\(74\) −150.000 −0.235637
\(75\) 89.0000 0.137024
\(76\) 83.0000 0.125273
\(77\) −210.000 −0.310802
\(78\) 66.0000 0.0958081
\(79\) 1060.00 1.50961 0.754806 0.655948i \(-0.227731\pi\)
0.754806 + 0.655948i \(0.227731\pi\)
\(80\) −426.000 −0.595353
\(81\) 649.000 0.890261
\(82\) −1080.00 −1.45446
\(83\) 873.000 1.15451 0.577254 0.816564i \(-0.304124\pi\)
0.577254 + 0.816564i \(0.304124\pi\)
\(84\) 7.00000 0.00909241
\(85\) 0 0
\(86\) 204.000 0.255789
\(87\) 15.0000 0.0184847
\(88\) −630.000 −0.763162
\(89\) 270.000 0.321572 0.160786 0.986989i \(-0.448597\pi\)
0.160786 + 0.986989i \(0.448597\pi\)
\(90\) −468.000 −0.548128
\(91\) 154.000 0.177402
\(92\) 48.0000 0.0543951
\(93\) −7.00000 −0.00780501
\(94\) −81.0000 −0.0888778
\(95\) 498.000 0.537829
\(96\) 45.0000 0.0478416
\(97\) −1130.00 −1.18283 −0.591413 0.806369i \(-0.701430\pi\)
−0.591413 + 0.806369i \(0.701430\pi\)
\(98\) 147.000 0.151523
\(99\) −780.000 −0.791848
\(100\) −89.0000 −0.0890000
\(101\) 1668.00 1.64329 0.821645 0.570000i \(-0.193057\pi\)
0.821645 + 0.570000i \(0.193057\pi\)
\(102\) 0 0
\(103\) 575.000 0.550062 0.275031 0.961435i \(-0.411312\pi\)
0.275031 + 0.961435i \(0.411312\pi\)
\(104\) 462.000 0.435604
\(105\) 42.0000 0.0390360
\(106\) 639.000 0.585520
\(107\) 972.000 0.878194 0.439097 0.898440i \(-0.355298\pi\)
0.439097 + 0.898440i \(0.355298\pi\)
\(108\) 53.0000 0.0472215
\(109\) −407.000 −0.357647 −0.178824 0.983881i \(-0.557229\pi\)
−0.178824 + 0.983881i \(0.557229\pi\)
\(110\) 540.000 0.468063
\(111\) 50.0000 0.0427549
\(112\) 497.000 0.419304
\(113\) 1002.00 0.834161 0.417081 0.908869i \(-0.363053\pi\)
0.417081 + 0.908869i \(0.363053\pi\)
\(114\) −249.000 −0.204570
\(115\) 288.000 0.233532
\(116\) −15.0000 −0.0120062
\(117\) 572.000 0.451978
\(118\) 567.000 0.442344
\(119\) 0 0
\(120\) 126.000 0.0958514
\(121\) −431.000 −0.323817
\(122\) −942.000 −0.699055
\(123\) 360.000 0.263903
\(124\) 7.00000 0.00506950
\(125\) −1284.00 −0.918756
\(126\) 546.000 0.386044
\(127\) 1022.00 0.714077 0.357039 0.934090i \(-0.383786\pi\)
0.357039 + 0.934090i \(0.383786\pi\)
\(128\) 1659.00 1.14560
\(129\) −68.0000 −0.0464114
\(130\) −396.000 −0.267165
\(131\) −2685.00 −1.79076 −0.895380 0.445303i \(-0.853096\pi\)
−0.895380 + 0.445303i \(0.853096\pi\)
\(132\) −30.0000 −0.0197816
\(133\) −581.000 −0.378790
\(134\) 942.000 0.607287
\(135\) 318.000 0.202734
\(136\) 0 0
\(137\) −2547.00 −1.58836 −0.794178 0.607685i \(-0.792098\pi\)
−0.794178 + 0.607685i \(0.792098\pi\)
\(138\) −144.000 −0.0888268
\(139\) 628.000 0.383211 0.191605 0.981472i \(-0.438631\pi\)
0.191605 + 0.981472i \(0.438631\pi\)
\(140\) −42.0000 −0.0253546
\(141\) 27.0000 0.0161263
\(142\) 2412.00 1.42543
\(143\) −660.000 −0.385958
\(144\) 1846.00 1.06829
\(145\) −90.0000 −0.0515455
\(146\) 1344.00 0.761851
\(147\) −49.0000 −0.0274929
\(148\) −50.0000 −0.0277701
\(149\) −435.000 −0.239172 −0.119586 0.992824i \(-0.538157\pi\)
−0.119586 + 0.992824i \(0.538157\pi\)
\(150\) 267.000 0.145336
\(151\) 1826.00 0.984091 0.492046 0.870569i \(-0.336249\pi\)
0.492046 + 0.870569i \(0.336249\pi\)
\(152\) −1743.00 −0.930105
\(153\) 0 0
\(154\) −630.000 −0.329655
\(155\) 42.0000 0.0217647
\(156\) 22.0000 0.0112911
\(157\) 3332.00 1.69377 0.846887 0.531773i \(-0.178474\pi\)
0.846887 + 0.531773i \(0.178474\pi\)
\(158\) 3180.00 1.60118
\(159\) −213.000 −0.106239
\(160\) −270.000 −0.133409
\(161\) −336.000 −0.164475
\(162\) 1947.00 0.944264
\(163\) 988.000 0.474762 0.237381 0.971417i \(-0.423711\pi\)
0.237381 + 0.971417i \(0.423711\pi\)
\(164\) −360.000 −0.171410
\(165\) −180.000 −0.0849272
\(166\) 2619.00 1.22454
\(167\) 2211.00 1.02450 0.512252 0.858835i \(-0.328811\pi\)
0.512252 + 0.858835i \(0.328811\pi\)
\(168\) −147.000 −0.0675077
\(169\) −1713.00 −0.779700
\(170\) 0 0
\(171\) −2158.00 −0.965067
\(172\) 68.0000 0.0301451
\(173\) 2388.00 1.04946 0.524729 0.851269i \(-0.324167\pi\)
0.524729 + 0.851269i \(0.324167\pi\)
\(174\) 45.0000 0.0196060
\(175\) 623.000 0.269111
\(176\) −2130.00 −0.912243
\(177\) −189.000 −0.0802605
\(178\) 810.000 0.341079
\(179\) 1968.00 0.821761 0.410881 0.911689i \(-0.365221\pi\)
0.410881 + 0.911689i \(0.365221\pi\)
\(180\) −156.000 −0.0645975
\(181\) 652.000 0.267750 0.133875 0.990998i \(-0.457258\pi\)
0.133875 + 0.990998i \(0.457258\pi\)
\(182\) 462.000 0.188163
\(183\) 314.000 0.126839
\(184\) −1008.00 −0.403863
\(185\) −300.000 −0.119224
\(186\) −21.0000 −0.00827847
\(187\) 0 0
\(188\) −27.0000 −0.0104743
\(189\) −371.000 −0.142785
\(190\) 1494.00 0.570453
\(191\) 4134.00 1.56610 0.783052 0.621957i \(-0.213662\pi\)
0.783052 + 0.621957i \(0.213662\pi\)
\(192\) −433.000 −0.162756
\(193\) −1259.00 −0.469559 −0.234779 0.972049i \(-0.575437\pi\)
−0.234779 + 0.972049i \(0.575437\pi\)
\(194\) −3390.00 −1.25458
\(195\) 132.000 0.0484755
\(196\) 49.0000 0.0178571
\(197\) −1293.00 −0.467627 −0.233813 0.972282i \(-0.575120\pi\)
−0.233813 + 0.972282i \(0.575120\pi\)
\(198\) −2340.00 −0.839882
\(199\) 340.000 0.121115 0.0605577 0.998165i \(-0.480712\pi\)
0.0605577 + 0.998165i \(0.480712\pi\)
\(200\) 1869.00 0.660791
\(201\) −314.000 −0.110188
\(202\) 5004.00 1.74297
\(203\) 105.000 0.0363032
\(204\) 0 0
\(205\) −2160.00 −0.735907
\(206\) 1725.00 0.583429
\(207\) −1248.00 −0.419043
\(208\) 1562.00 0.520698
\(209\) 2490.00 0.824100
\(210\) 126.000 0.0414039
\(211\) 790.000 0.257753 0.128876 0.991661i \(-0.458863\pi\)
0.128876 + 0.991661i \(0.458863\pi\)
\(212\) 213.000 0.0690042
\(213\) −804.000 −0.258635
\(214\) 2916.00 0.931466
\(215\) 408.000 0.129420
\(216\) −1113.00 −0.350602
\(217\) −49.0000 −0.0153287
\(218\) −1221.00 −0.379342
\(219\) −448.000 −0.138233
\(220\) 180.000 0.0551618
\(221\) 0 0
\(222\) 150.000 0.0453484
\(223\) −97.0000 −0.0291283 −0.0145641 0.999894i \(-0.504636\pi\)
−0.0145641 + 0.999894i \(0.504636\pi\)
\(224\) 315.000 0.0939590
\(225\) 2314.00 0.685630
\(226\) 3006.00 0.884762
\(227\) −369.000 −0.107892 −0.0539458 0.998544i \(-0.517180\pi\)
−0.0539458 + 0.998544i \(0.517180\pi\)
\(228\) −83.0000 −0.0241088
\(229\) −1888.00 −0.544815 −0.272407 0.962182i \(-0.587820\pi\)
−0.272407 + 0.962182i \(0.587820\pi\)
\(230\) 864.000 0.247698
\(231\) 210.000 0.0598138
\(232\) 315.000 0.0891412
\(233\) −3633.00 −1.02148 −0.510742 0.859734i \(-0.670629\pi\)
−0.510742 + 0.859734i \(0.670629\pi\)
\(234\) 1716.00 0.479395
\(235\) −162.000 −0.0449690
\(236\) 189.000 0.0521307
\(237\) −1060.00 −0.290525
\(238\) 0 0
\(239\) −1728.00 −0.467678 −0.233839 0.972275i \(-0.575129\pi\)
−0.233839 + 0.972275i \(0.575129\pi\)
\(240\) 426.000 0.114576
\(241\) 5704.00 1.52459 0.762297 0.647228i \(-0.224072\pi\)
0.762297 + 0.647228i \(0.224072\pi\)
\(242\) −1293.00 −0.343459
\(243\) −2080.00 −0.549103
\(244\) −314.000 −0.0823844
\(245\) 294.000 0.0766652
\(246\) 1080.00 0.279912
\(247\) −1826.00 −0.470387
\(248\) −147.000 −0.0376392
\(249\) −873.000 −0.222185
\(250\) −3852.00 −0.974487
\(251\) 2928.00 0.736310 0.368155 0.929765i \(-0.379990\pi\)
0.368155 + 0.929765i \(0.379990\pi\)
\(252\) 182.000 0.0454957
\(253\) 1440.00 0.357834
\(254\) 3066.00 0.757394
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) −5586.00 −1.35582 −0.677909 0.735146i \(-0.737114\pi\)
−0.677909 + 0.735146i \(0.737114\pi\)
\(258\) −204.000 −0.0492267
\(259\) 350.000 0.0839689
\(260\) −132.000 −0.0314857
\(261\) 390.000 0.0924919
\(262\) −8055.00 −1.89939
\(263\) −432.000 −0.101286 −0.0506431 0.998717i \(-0.516127\pi\)
−0.0506431 + 0.998717i \(0.516127\pi\)
\(264\) 630.000 0.146871
\(265\) 1278.00 0.296253
\(266\) −1743.00 −0.401768
\(267\) −270.000 −0.0618866
\(268\) 314.000 0.0715694
\(269\) 228.000 0.0516781 0.0258390 0.999666i \(-0.491774\pi\)
0.0258390 + 0.999666i \(0.491774\pi\)
\(270\) 954.000 0.215032
\(271\) −3457.00 −0.774900 −0.387450 0.921891i \(-0.626644\pi\)
−0.387450 + 0.921891i \(0.626644\pi\)
\(272\) 0 0
\(273\) −154.000 −0.0341410
\(274\) −7641.00 −1.68471
\(275\) −2670.00 −0.585480
\(276\) −48.0000 −0.0104683
\(277\) 2947.00 0.639235 0.319617 0.947547i \(-0.396446\pi\)
0.319617 + 0.947547i \(0.396446\pi\)
\(278\) 1884.00 0.406456
\(279\) −182.000 −0.0390540
\(280\) 882.000 0.188249
\(281\) 6210.00 1.31835 0.659177 0.751988i \(-0.270905\pi\)
0.659177 + 0.751988i \(0.270905\pi\)
\(282\) 81.0000 0.0171045
\(283\) −6935.00 −1.45669 −0.728344 0.685211i \(-0.759710\pi\)
−0.728344 + 0.685211i \(0.759710\pi\)
\(284\) 804.000 0.167988
\(285\) −498.000 −0.103505
\(286\) −1980.00 −0.409370
\(287\) 2520.00 0.518296
\(288\) 1170.00 0.239385
\(289\) 0 0
\(290\) −270.000 −0.0546722
\(291\) 1130.00 0.227635
\(292\) 448.000 0.0897850
\(293\) 1614.00 0.321812 0.160906 0.986970i \(-0.448558\pi\)
0.160906 + 0.986970i \(0.448558\pi\)
\(294\) −147.000 −0.0291606
\(295\) 1134.00 0.223810
\(296\) 1050.00 0.206182
\(297\) 1590.00 0.310644
\(298\) −1305.00 −0.253680
\(299\) −1056.00 −0.204248
\(300\) 89.0000 0.0171281
\(301\) −476.000 −0.0911501
\(302\) 5478.00 1.04379
\(303\) −1668.00 −0.316251
\(304\) −5893.00 −1.11180
\(305\) −1884.00 −0.353697
\(306\) 0 0
\(307\) 10001.0 1.85924 0.929621 0.368517i \(-0.120134\pi\)
0.929621 + 0.368517i \(0.120134\pi\)
\(308\) −210.000 −0.0388502
\(309\) −575.000 −0.105860
\(310\) 126.000 0.0230849
\(311\) 5703.00 1.03983 0.519916 0.854218i \(-0.325963\pi\)
0.519916 + 0.854218i \(0.325963\pi\)
\(312\) −462.000 −0.0838320
\(313\) −10904.0 −1.96911 −0.984554 0.175084i \(-0.943980\pi\)
−0.984554 + 0.175084i \(0.943980\pi\)
\(314\) 9996.00 1.79652
\(315\) 1092.00 0.195325
\(316\) 1060.00 0.188701
\(317\) −9981.00 −1.76842 −0.884209 0.467091i \(-0.845302\pi\)
−0.884209 + 0.467091i \(0.845302\pi\)
\(318\) −639.000 −0.112683
\(319\) −450.000 −0.0789817
\(320\) 2598.00 0.453852
\(321\) −972.000 −0.169009
\(322\) −1008.00 −0.174452
\(323\) 0 0
\(324\) 649.000 0.111283
\(325\) 1958.00 0.334186
\(326\) 2964.00 0.503561
\(327\) 407.000 0.0688292
\(328\) 7560.00 1.27266
\(329\) 189.000 0.0316715
\(330\) −540.000 −0.0900789
\(331\) 10118.0 1.68017 0.840084 0.542456i \(-0.182506\pi\)
0.840084 + 0.542456i \(0.182506\pi\)
\(332\) 873.000 0.144314
\(333\) 1300.00 0.213933
\(334\) 6633.00 1.08665
\(335\) 1884.00 0.307265
\(336\) −497.000 −0.0806952
\(337\) 7855.00 1.26970 0.634850 0.772635i \(-0.281062\pi\)
0.634850 + 0.772635i \(0.281062\pi\)
\(338\) −5139.00 −0.826996
\(339\) −1002.00 −0.160534
\(340\) 0 0
\(341\) 210.000 0.0333494
\(342\) −6474.00 −1.02361
\(343\) −343.000 −0.0539949
\(344\) −1428.00 −0.223816
\(345\) −288.000 −0.0449432
\(346\) 7164.00 1.11312
\(347\) 6714.00 1.03869 0.519346 0.854564i \(-0.326175\pi\)
0.519346 + 0.854564i \(0.326175\pi\)
\(348\) 15.0000 0.00231059
\(349\) 2306.00 0.353689 0.176844 0.984239i \(-0.443411\pi\)
0.176844 + 0.984239i \(0.443411\pi\)
\(350\) 1869.00 0.285435
\(351\) −1166.00 −0.177312
\(352\) −1350.00 −0.204418
\(353\) 2490.00 0.375437 0.187719 0.982223i \(-0.439891\pi\)
0.187719 + 0.982223i \(0.439891\pi\)
\(354\) −567.000 −0.0851291
\(355\) 4824.00 0.721215
\(356\) 270.000 0.0401965
\(357\) 0 0
\(358\) 5904.00 0.871609
\(359\) −4194.00 −0.616576 −0.308288 0.951293i \(-0.599756\pi\)
−0.308288 + 0.951293i \(0.599756\pi\)
\(360\) 3276.00 0.479612
\(361\) 30.0000 0.00437382
\(362\) 1956.00 0.283992
\(363\) 431.000 0.0623185
\(364\) 154.000 0.0221753
\(365\) 2688.00 0.385469
\(366\) 942.000 0.134533
\(367\) −1223.00 −0.173951 −0.0869756 0.996210i \(-0.527720\pi\)
−0.0869756 + 0.996210i \(0.527720\pi\)
\(368\) −3408.00 −0.482756
\(369\) 9360.00 1.32049
\(370\) −900.000 −0.126456
\(371\) −1491.00 −0.208649
\(372\) −7.00000 −0.000975627 0
\(373\) 1577.00 0.218911 0.109456 0.993992i \(-0.465089\pi\)
0.109456 + 0.993992i \(0.465089\pi\)
\(374\) 0 0
\(375\) 1284.00 0.176815
\(376\) 567.000 0.0777681
\(377\) 330.000 0.0450819
\(378\) −1113.00 −0.151446
\(379\) 5620.00 0.761689 0.380844 0.924639i \(-0.375633\pi\)
0.380844 + 0.924639i \(0.375633\pi\)
\(380\) 498.000 0.0672286
\(381\) −1022.00 −0.137424
\(382\) 12402.0 1.66110
\(383\) 5967.00 0.796082 0.398041 0.917368i \(-0.369690\pi\)
0.398041 + 0.917368i \(0.369690\pi\)
\(384\) −1659.00 −0.220470
\(385\) −1260.00 −0.166794
\(386\) −3777.00 −0.498042
\(387\) −1768.00 −0.232229
\(388\) −1130.00 −0.147853
\(389\) −5181.00 −0.675288 −0.337644 0.941274i \(-0.609630\pi\)
−0.337644 + 0.941274i \(0.609630\pi\)
\(390\) 396.000 0.0514160
\(391\) 0 0
\(392\) −1029.00 −0.132583
\(393\) 2685.00 0.344632
\(394\) −3879.00 −0.495993
\(395\) 6360.00 0.810143
\(396\) −780.000 −0.0989810
\(397\) 11050.0 1.39694 0.698468 0.715641i \(-0.253865\pi\)
0.698468 + 0.715641i \(0.253865\pi\)
\(398\) 1020.00 0.128462
\(399\) 581.000 0.0728982
\(400\) 6319.00 0.789875
\(401\) −8265.00 −1.02926 −0.514631 0.857412i \(-0.672071\pi\)
−0.514631 + 0.857412i \(0.672071\pi\)
\(402\) −942.000 −0.116872
\(403\) −154.000 −0.0190355
\(404\) 1668.00 0.205411
\(405\) 3894.00 0.477764
\(406\) 315.000 0.0385054
\(407\) −1500.00 −0.182684
\(408\) 0 0
\(409\) 3038.00 0.367285 0.183642 0.982993i \(-0.441211\pi\)
0.183642 + 0.982993i \(0.441211\pi\)
\(410\) −6480.00 −0.780547
\(411\) 2547.00 0.305679
\(412\) 575.000 0.0687578
\(413\) −1323.00 −0.157629
\(414\) −3744.00 −0.444463
\(415\) 5238.00 0.619574
\(416\) 990.000 0.116680
\(417\) −628.000 −0.0737489
\(418\) 7470.00 0.874090
\(419\) −5628.00 −0.656195 −0.328098 0.944644i \(-0.606407\pi\)
−0.328098 + 0.944644i \(0.606407\pi\)
\(420\) 42.0000 0.00487950
\(421\) −13855.0 −1.60392 −0.801961 0.597376i \(-0.796210\pi\)
−0.801961 + 0.597376i \(0.796210\pi\)
\(422\) 2370.00 0.273388
\(423\) 702.000 0.0806913
\(424\) −4473.00 −0.512330
\(425\) 0 0
\(426\) −2412.00 −0.274323
\(427\) 2198.00 0.249107
\(428\) 972.000 0.109774
\(429\) 660.000 0.0742776
\(430\) 1224.00 0.137271
\(431\) −3714.00 −0.415074 −0.207537 0.978227i \(-0.566545\pi\)
−0.207537 + 0.978227i \(0.566545\pi\)
\(432\) −3763.00 −0.419091
\(433\) −13246.0 −1.47012 −0.735060 0.678002i \(-0.762846\pi\)
−0.735060 + 0.678002i \(0.762846\pi\)
\(434\) −147.000 −0.0162586
\(435\) 90.0000 0.00991993
\(436\) −407.000 −0.0447059
\(437\) 3984.00 0.436111
\(438\) −1344.00 −0.146618
\(439\) 16453.0 1.78874 0.894372 0.447323i \(-0.147623\pi\)
0.894372 + 0.447323i \(0.147623\pi\)
\(440\) −3780.00 −0.409556
\(441\) −1274.00 −0.137566
\(442\) 0 0
\(443\) 8088.00 0.867432 0.433716 0.901050i \(-0.357202\pi\)
0.433716 + 0.901050i \(0.357202\pi\)
\(444\) 50.0000 0.00534436
\(445\) 1620.00 0.172574
\(446\) −291.000 −0.0308952
\(447\) 435.000 0.0460286
\(448\) −3031.00 −0.319646
\(449\) −5199.00 −0.546450 −0.273225 0.961950i \(-0.588090\pi\)
−0.273225 + 0.961950i \(0.588090\pi\)
\(450\) 6942.00 0.727220
\(451\) −10800.0 −1.12761
\(452\) 1002.00 0.104270
\(453\) −1826.00 −0.189388
\(454\) −1107.00 −0.114436
\(455\) 924.000 0.0952039
\(456\) 1743.00 0.178999
\(457\) 11459.0 1.17293 0.586465 0.809974i \(-0.300519\pi\)
0.586465 + 0.809974i \(0.300519\pi\)
\(458\) −5664.00 −0.577863
\(459\) 0 0
\(460\) 288.000 0.0291915
\(461\) 1242.00 0.125479 0.0627394 0.998030i \(-0.480016\pi\)
0.0627394 + 0.998030i \(0.480016\pi\)
\(462\) 630.000 0.0634421
\(463\) −11596.0 −1.16396 −0.581978 0.813204i \(-0.697721\pi\)
−0.581978 + 0.813204i \(0.697721\pi\)
\(464\) 1065.00 0.106555
\(465\) −42.0000 −0.00418861
\(466\) −10899.0 −1.08345
\(467\) −2652.00 −0.262784 −0.131392 0.991331i \(-0.541945\pi\)
−0.131392 + 0.991331i \(0.541945\pi\)
\(468\) 572.000 0.0564972
\(469\) −2198.00 −0.216406
\(470\) −486.000 −0.0476968
\(471\) −3332.00 −0.325967
\(472\) −3969.00 −0.387051
\(473\) 2040.00 0.198307
\(474\) −3180.00 −0.308148
\(475\) −7387.00 −0.713555
\(476\) 0 0
\(477\) −5538.00 −0.531588
\(478\) −5184.00 −0.496047
\(479\) −1668.00 −0.159108 −0.0795541 0.996831i \(-0.525350\pi\)
−0.0795541 + 0.996831i \(0.525350\pi\)
\(480\) 270.000 0.0256745
\(481\) 1100.00 0.104274
\(482\) 17112.0 1.61708
\(483\) 336.000 0.0316533
\(484\) −431.000 −0.0404771
\(485\) −6780.00 −0.634771
\(486\) −6240.00 −0.582412
\(487\) −18158.0 −1.68956 −0.844782 0.535111i \(-0.820270\pi\)
−0.844782 + 0.535111i \(0.820270\pi\)
\(488\) 6594.00 0.611673
\(489\) −988.000 −0.0913679
\(490\) 882.000 0.0813157
\(491\) 17022.0 1.56455 0.782273 0.622936i \(-0.214060\pi\)
0.782273 + 0.622936i \(0.214060\pi\)
\(492\) 360.000 0.0329879
\(493\) 0 0
\(494\) −5478.00 −0.498921
\(495\) −4680.00 −0.424950
\(496\) −497.000 −0.0449919
\(497\) −5628.00 −0.507948
\(498\) −2619.00 −0.235663
\(499\) −9530.00 −0.854953 −0.427476 0.904027i \(-0.640597\pi\)
−0.427476 + 0.904027i \(0.640597\pi\)
\(500\) −1284.00 −0.114844
\(501\) −2211.00 −0.197166
\(502\) 8784.00 0.780974
\(503\) −10875.0 −0.964001 −0.482000 0.876171i \(-0.660090\pi\)
−0.482000 + 0.876171i \(0.660090\pi\)
\(504\) −3822.00 −0.337789
\(505\) 10008.0 0.881881
\(506\) 4320.00 0.379540
\(507\) 1713.00 0.150053
\(508\) 1022.00 0.0892597
\(509\) −15798.0 −1.37571 −0.687853 0.725850i \(-0.741447\pi\)
−0.687853 + 0.725850i \(0.741447\pi\)
\(510\) 0 0
\(511\) −3136.00 −0.271484
\(512\) −8733.00 −0.753804
\(513\) 4399.00 0.378598
\(514\) −16758.0 −1.43806
\(515\) 3450.00 0.295194
\(516\) −68.0000 −0.00580142
\(517\) −810.000 −0.0689048
\(518\) 1050.00 0.0890625
\(519\) −2388.00 −0.201968
\(520\) 2772.00 0.233770
\(521\) −6018.00 −0.506053 −0.253026 0.967459i \(-0.581426\pi\)
−0.253026 + 0.967459i \(0.581426\pi\)
\(522\) 1170.00 0.0981025
\(523\) −11896.0 −0.994600 −0.497300 0.867579i \(-0.665675\pi\)
−0.497300 + 0.867579i \(0.665675\pi\)
\(524\) −2685.00 −0.223845
\(525\) −623.000 −0.0517904
\(526\) −1296.00 −0.107430
\(527\) 0 0
\(528\) 2130.00 0.175561
\(529\) −9863.00 −0.810635
\(530\) 3834.00 0.314223
\(531\) −4914.00 −0.401600
\(532\) −581.000 −0.0473488
\(533\) 7920.00 0.643627
\(534\) −810.000 −0.0656407
\(535\) 5832.00 0.471288
\(536\) −6594.00 −0.531376
\(537\) −1968.00 −0.158148
\(538\) 684.000 0.0548129
\(539\) 1470.00 0.117472
\(540\) 318.000 0.0253417
\(541\) 2167.00 0.172212 0.0861059 0.996286i \(-0.472558\pi\)
0.0861059 + 0.996286i \(0.472558\pi\)
\(542\) −10371.0 −0.821905
\(543\) −652.000 −0.0515285
\(544\) 0 0
\(545\) −2442.00 −0.191934
\(546\) −462.000 −0.0362120
\(547\) −6218.00 −0.486037 −0.243019 0.970022i \(-0.578138\pi\)
−0.243019 + 0.970022i \(0.578138\pi\)
\(548\) −2547.00 −0.198545
\(549\) 8164.00 0.634665
\(550\) −8010.00 −0.620996
\(551\) −1245.00 −0.0962591
\(552\) 1008.00 0.0777234
\(553\) −7420.00 −0.570580
\(554\) 8841.00 0.678011
\(555\) 300.000 0.0229447
\(556\) 628.000 0.0479013
\(557\) 17130.0 1.30309 0.651545 0.758610i \(-0.274121\pi\)
0.651545 + 0.758610i \(0.274121\pi\)
\(558\) −546.000 −0.0414230
\(559\) −1496.00 −0.113192
\(560\) 2982.00 0.225022
\(561\) 0 0
\(562\) 18630.0 1.39833
\(563\) −14859.0 −1.11231 −0.556156 0.831078i \(-0.687724\pi\)
−0.556156 + 0.831078i \(0.687724\pi\)
\(564\) 27.0000 0.00201579
\(565\) 6012.00 0.447658
\(566\) −20805.0 −1.54505
\(567\) −4543.00 −0.336487
\(568\) −16884.0 −1.24725
\(569\) 2601.00 0.191634 0.0958169 0.995399i \(-0.469454\pi\)
0.0958169 + 0.995399i \(0.469454\pi\)
\(570\) −1494.00 −0.109784
\(571\) −6404.00 −0.469350 −0.234675 0.972074i \(-0.575403\pi\)
−0.234675 + 0.972074i \(0.575403\pi\)
\(572\) −660.000 −0.0482447
\(573\) −4134.00 −0.301397
\(574\) 7560.00 0.549736
\(575\) −4272.00 −0.309834
\(576\) −11258.0 −0.814381
\(577\) 4166.00 0.300577 0.150288 0.988642i \(-0.451980\pi\)
0.150288 + 0.988642i \(0.451980\pi\)
\(578\) 0 0
\(579\) 1259.00 0.0903666
\(580\) −90.0000 −0.00644318
\(581\) −6111.00 −0.436363
\(582\) 3390.00 0.241443
\(583\) 6390.00 0.453940
\(584\) −9408.00 −0.666620
\(585\) 3432.00 0.242557
\(586\) 4842.00 0.341333
\(587\) 25653.0 1.80377 0.901885 0.431977i \(-0.142184\pi\)
0.901885 + 0.431977i \(0.142184\pi\)
\(588\) −49.0000 −0.00343661
\(589\) 581.000 0.0406446
\(590\) 3402.00 0.237387
\(591\) 1293.00 0.0899948
\(592\) 3550.00 0.246460
\(593\) 2268.00 0.157058 0.0785292 0.996912i \(-0.474978\pi\)
0.0785292 + 0.996912i \(0.474978\pi\)
\(594\) 4770.00 0.329487
\(595\) 0 0
\(596\) −435.000 −0.0298965
\(597\) −340.000 −0.0233087
\(598\) −3168.00 −0.216637
\(599\) −4938.00 −0.336830 −0.168415 0.985716i \(-0.553865\pi\)
−0.168415 + 0.985716i \(0.553865\pi\)
\(600\) −1869.00 −0.127169
\(601\) −8042.00 −0.545824 −0.272912 0.962039i \(-0.587987\pi\)
−0.272912 + 0.962039i \(0.587987\pi\)
\(602\) −1428.00 −0.0966793
\(603\) −8164.00 −0.551350
\(604\) 1826.00 0.123011
\(605\) −2586.00 −0.173778
\(606\) −5004.00 −0.335435
\(607\) 10753.0 0.719029 0.359515 0.933139i \(-0.382942\pi\)
0.359515 + 0.933139i \(0.382942\pi\)
\(608\) −3735.00 −0.249135
\(609\) −105.000 −0.00698656
\(610\) −5652.00 −0.375152
\(611\) 594.000 0.0393301
\(612\) 0 0
\(613\) −17218.0 −1.13447 −0.567234 0.823557i \(-0.691986\pi\)
−0.567234 + 0.823557i \(0.691986\pi\)
\(614\) 30003.0 1.97202
\(615\) 2160.00 0.141625
\(616\) 4410.00 0.288448
\(617\) −5097.00 −0.332573 −0.166286 0.986077i \(-0.553178\pi\)
−0.166286 + 0.986077i \(0.553178\pi\)
\(618\) −1725.00 −0.112281
\(619\) 11095.0 0.720429 0.360215 0.932869i \(-0.382703\pi\)
0.360215 + 0.932869i \(0.382703\pi\)
\(620\) 42.0000 0.00272058
\(621\) 2544.00 0.164392
\(622\) 17109.0 1.10291
\(623\) −1890.00 −0.121543
\(624\) −1562.00 −0.100208
\(625\) 3421.00 0.218944
\(626\) −32712.0 −2.08855
\(627\) −2490.00 −0.158598
\(628\) 3332.00 0.211722
\(629\) 0 0
\(630\) 3276.00 0.207173
\(631\) 21458.0 1.35377 0.676885 0.736088i \(-0.263329\pi\)
0.676885 + 0.736088i \(0.263329\pi\)
\(632\) −22260.0 −1.40104
\(633\) −790.000 −0.0496046
\(634\) −29943.0 −1.87569
\(635\) 6132.00 0.383214
\(636\) −213.000 −0.0132799
\(637\) −1078.00 −0.0670517
\(638\) −1350.00 −0.0837727
\(639\) −20904.0 −1.29413
\(640\) 9954.00 0.614791
\(641\) −10161.0 −0.626108 −0.313054 0.949735i \(-0.601352\pi\)
−0.313054 + 0.949735i \(0.601352\pi\)
\(642\) −2916.00 −0.179261
\(643\) 17524.0 1.07477 0.537387 0.843336i \(-0.319412\pi\)
0.537387 + 0.843336i \(0.319412\pi\)
\(644\) −336.000 −0.0205594
\(645\) −408.000 −0.0249070
\(646\) 0 0
\(647\) 4896.00 0.297499 0.148749 0.988875i \(-0.452475\pi\)
0.148749 + 0.988875i \(0.452475\pi\)
\(648\) −13629.0 −0.826231
\(649\) 5670.00 0.342938
\(650\) 5874.00 0.354457
\(651\) 49.0000 0.00295002
\(652\) 988.000 0.0593452
\(653\) 10929.0 0.654954 0.327477 0.944859i \(-0.393802\pi\)
0.327477 + 0.944859i \(0.393802\pi\)
\(654\) 1221.00 0.0730044
\(655\) −16110.0 −0.961023
\(656\) 25560.0 1.52127
\(657\) −11648.0 −0.691677
\(658\) 567.000 0.0335926
\(659\) −28632.0 −1.69248 −0.846240 0.532802i \(-0.821139\pi\)
−0.846240 + 0.532802i \(0.821139\pi\)
\(660\) −180.000 −0.0106159
\(661\) 7982.00 0.469688 0.234844 0.972033i \(-0.424542\pi\)
0.234844 + 0.972033i \(0.424542\pi\)
\(662\) 30354.0 1.78209
\(663\) 0 0
\(664\) −18333.0 −1.07147
\(665\) −3486.00 −0.203280
\(666\) 3900.00 0.226910
\(667\) −720.000 −0.0417969
\(668\) 2211.00 0.128063
\(669\) 97.0000 0.00560573
\(670\) 5652.00 0.325904
\(671\) −9420.00 −0.541960
\(672\) −315.000 −0.0180824
\(673\) 1015.00 0.0581358 0.0290679 0.999577i \(-0.490746\pi\)
0.0290679 + 0.999577i \(0.490746\pi\)
\(674\) 23565.0 1.34672
\(675\) −4717.00 −0.268974
\(676\) −1713.00 −0.0974624
\(677\) −264.000 −0.0149872 −0.00749361 0.999972i \(-0.502385\pi\)
−0.00749361 + 0.999972i \(0.502385\pi\)
\(678\) −3006.00 −0.170272
\(679\) 7910.00 0.447066
\(680\) 0 0
\(681\) 369.000 0.0207637
\(682\) 630.000 0.0353724
\(683\) −10176.0 −0.570093 −0.285047 0.958514i \(-0.592009\pi\)
−0.285047 + 0.958514i \(0.592009\pi\)
\(684\) −2158.00 −0.120633
\(685\) −15282.0 −0.852402
\(686\) −1029.00 −0.0572703
\(687\) 1888.00 0.104850
\(688\) −4828.00 −0.267537
\(689\) −4686.00 −0.259104
\(690\) −864.000 −0.0476694
\(691\) 27625.0 1.52085 0.760423 0.649428i \(-0.224992\pi\)
0.760423 + 0.649428i \(0.224992\pi\)
\(692\) 2388.00 0.131182
\(693\) 5460.00 0.299290
\(694\) 20142.0 1.10170
\(695\) 3768.00 0.205652
\(696\) −315.000 −0.0171552
\(697\) 0 0
\(698\) 6918.00 0.375143
\(699\) 3633.00 0.196585
\(700\) 623.000 0.0336388
\(701\) 5187.00 0.279473 0.139736 0.990189i \(-0.455375\pi\)
0.139736 + 0.990189i \(0.455375\pi\)
\(702\) −3498.00 −0.188068
\(703\) −4150.00 −0.222646
\(704\) 12990.0 0.695425
\(705\) 162.000 0.00865429
\(706\) 7470.00 0.398211
\(707\) −11676.0 −0.621105
\(708\) −189.000 −0.0100326
\(709\) 23650.0 1.25274 0.626371 0.779525i \(-0.284539\pi\)
0.626371 + 0.779525i \(0.284539\pi\)
\(710\) 14472.0 0.764964
\(711\) −27560.0 −1.45370
\(712\) −5670.00 −0.298444
\(713\) 336.000 0.0176484
\(714\) 0 0
\(715\) −3960.00 −0.207127
\(716\) 1968.00 0.102720
\(717\) 1728.00 0.0900047
\(718\) −12582.0 −0.653978
\(719\) −4272.00 −0.221584 −0.110792 0.993844i \(-0.535339\pi\)
−0.110792 + 0.993844i \(0.535339\pi\)
\(720\) 11076.0 0.573303
\(721\) −4025.00 −0.207904
\(722\) 90.0000 0.00463913
\(723\) −5704.00 −0.293408
\(724\) 652.000 0.0334688
\(725\) 1335.00 0.0683871
\(726\) 1293.00 0.0660988
\(727\) −10504.0 −0.535862 −0.267931 0.963438i \(-0.586340\pi\)
−0.267931 + 0.963438i \(0.586340\pi\)
\(728\) −3234.00 −0.164643
\(729\) −15443.0 −0.784586
\(730\) 8064.00 0.408852
\(731\) 0 0
\(732\) 314.000 0.0158549
\(733\) 36992.0 1.86403 0.932013 0.362425i \(-0.118051\pi\)
0.932013 + 0.362425i \(0.118051\pi\)
\(734\) −3669.00 −0.184503
\(735\) −294.000 −0.0147542
\(736\) −2160.00 −0.108178
\(737\) 9420.00 0.470814
\(738\) 28080.0 1.40059
\(739\) −35566.0 −1.77039 −0.885194 0.465222i \(-0.845974\pi\)
−0.885194 + 0.465222i \(0.845974\pi\)
\(740\) −300.000 −0.0149030
\(741\) 1826.00 0.0905260
\(742\) −4473.00 −0.221306
\(743\) −35982.0 −1.77665 −0.888325 0.459215i \(-0.848131\pi\)
−0.888325 + 0.459215i \(0.848131\pi\)
\(744\) 147.000 0.00724366
\(745\) −2610.00 −0.128353
\(746\) 4731.00 0.232191
\(747\) −22698.0 −1.11175
\(748\) 0 0
\(749\) −6804.00 −0.331926
\(750\) 3852.00 0.187540
\(751\) 8230.00 0.399889 0.199945 0.979807i \(-0.435924\pi\)
0.199945 + 0.979807i \(0.435924\pi\)
\(752\) 1917.00 0.0929598
\(753\) −2928.00 −0.141703
\(754\) 990.000 0.0478165
\(755\) 10956.0 0.528119
\(756\) −371.000 −0.0178481
\(757\) 12887.0 0.618740 0.309370 0.950942i \(-0.399882\pi\)
0.309370 + 0.950942i \(0.399882\pi\)
\(758\) 16860.0 0.807893
\(759\) −1440.00 −0.0688652
\(760\) −10458.0 −0.499147
\(761\) −36042.0 −1.71685 −0.858424 0.512941i \(-0.828556\pi\)
−0.858424 + 0.512941i \(0.828556\pi\)
\(762\) −3066.00 −0.145760
\(763\) 2849.00 0.135178
\(764\) 4134.00 0.195763
\(765\) 0 0
\(766\) 17901.0 0.844373
\(767\) −4158.00 −0.195745
\(768\) −1513.00 −0.0710881
\(769\) 3170.00 0.148652 0.0743258 0.997234i \(-0.476320\pi\)
0.0743258 + 0.997234i \(0.476320\pi\)
\(770\) −3780.00 −0.176911
\(771\) 5586.00 0.260927
\(772\) −1259.00 −0.0586948
\(773\) −21390.0 −0.995271 −0.497636 0.867386i \(-0.665798\pi\)
−0.497636 + 0.867386i \(0.665798\pi\)
\(774\) −5304.00 −0.246316
\(775\) −623.000 −0.0288759
\(776\) 23730.0 1.09775
\(777\) −350.000 −0.0161598
\(778\) −15543.0 −0.716251
\(779\) −29880.0 −1.37428
\(780\) 132.000 0.00605943
\(781\) 24120.0 1.10510
\(782\) 0 0
\(783\) −795.000 −0.0362848
\(784\) −3479.00 −0.158482
\(785\) 19992.0 0.908975
\(786\) 8055.00 0.365537
\(787\) 31708.0 1.43617 0.718086 0.695954i \(-0.245018\pi\)
0.718086 + 0.695954i \(0.245018\pi\)
\(788\) −1293.00 −0.0584533
\(789\) 432.000 0.0194925
\(790\) 19080.0 0.859286
\(791\) −7014.00 −0.315283
\(792\) 16380.0 0.734896
\(793\) 6908.00 0.309345
\(794\) 33150.0 1.48167
\(795\) −1278.00 −0.0570138
\(796\) 340.000 0.0151394
\(797\) 12114.0 0.538394 0.269197 0.963085i \(-0.413242\pi\)
0.269197 + 0.963085i \(0.413242\pi\)
\(798\) 1743.00 0.0773202
\(799\) 0 0
\(800\) 4005.00 0.176998
\(801\) −7020.00 −0.309662
\(802\) −24795.0 −1.09170
\(803\) 13440.0 0.590644
\(804\) −314.000 −0.0137735
\(805\) −2016.00 −0.0882667
\(806\) −462.000 −0.0201901
\(807\) −228.000 −0.00994545
\(808\) −35028.0 −1.52510
\(809\) −37086.0 −1.61171 −0.805856 0.592112i \(-0.798294\pi\)
−0.805856 + 0.592112i \(0.798294\pi\)
\(810\) 11682.0 0.506745
\(811\) −20063.0 −0.868690 −0.434345 0.900747i \(-0.643020\pi\)
−0.434345 + 0.900747i \(0.643020\pi\)
\(812\) 105.000 0.00453790
\(813\) 3457.00 0.149129
\(814\) −4500.00 −0.193765
\(815\) 5928.00 0.254784
\(816\) 0 0
\(817\) 5644.00 0.241687
\(818\) 9114.00 0.389564
\(819\) −4004.00 −0.170832
\(820\) −2160.00 −0.0919884
\(821\) 18642.0 0.792461 0.396230 0.918151i \(-0.370318\pi\)
0.396230 + 0.918151i \(0.370318\pi\)
\(822\) 7641.00 0.324222
\(823\) 10606.0 0.449213 0.224606 0.974450i \(-0.427890\pi\)
0.224606 + 0.974450i \(0.427890\pi\)
\(824\) −12075.0 −0.510501
\(825\) 2670.00 0.112676
\(826\) −3969.00 −0.167190
\(827\) 32154.0 1.35200 0.676000 0.736902i \(-0.263712\pi\)
0.676000 + 0.736902i \(0.263712\pi\)
\(828\) −1248.00 −0.0523804
\(829\) 32198.0 1.34895 0.674477 0.738296i \(-0.264369\pi\)
0.674477 + 0.738296i \(0.264369\pi\)
\(830\) 15714.0 0.657158
\(831\) −2947.00 −0.123021
\(832\) −9526.00 −0.396941
\(833\) 0 0
\(834\) −1884.00 −0.0782225
\(835\) 13266.0 0.549807
\(836\) 2490.00 0.103013
\(837\) 371.000 0.0153210
\(838\) −16884.0 −0.696000
\(839\) −18627.0 −0.766478 −0.383239 0.923649i \(-0.625192\pi\)
−0.383239 + 0.923649i \(0.625192\pi\)
\(840\) −882.000 −0.0362284
\(841\) −24164.0 −0.990775
\(842\) −41565.0 −1.70122
\(843\) −6210.00 −0.253717
\(844\) 790.000 0.0322191
\(845\) −10278.0 −0.418431
\(846\) 2106.00 0.0855860
\(847\) 3017.00 0.122391
\(848\) −15123.0 −0.612413
\(849\) 6935.00 0.280340
\(850\) 0 0
\(851\) −2400.00 −0.0966756
\(852\) −804.000 −0.0323293
\(853\) −6668.00 −0.267653 −0.133826 0.991005i \(-0.542726\pi\)
−0.133826 + 0.991005i \(0.542726\pi\)
\(854\) 6594.00 0.264218
\(855\) −12948.0 −0.517909
\(856\) −20412.0 −0.815032
\(857\) 2328.00 0.0927923 0.0463961 0.998923i \(-0.485226\pi\)
0.0463961 + 0.998923i \(0.485226\pi\)
\(858\) 1980.00 0.0787833
\(859\) 33761.0 1.34099 0.670495 0.741914i \(-0.266082\pi\)
0.670495 + 0.741914i \(0.266082\pi\)
\(860\) 408.000 0.0161775
\(861\) −2520.00 −0.0997461
\(862\) −11142.0 −0.440253
\(863\) 19740.0 0.778630 0.389315 0.921105i \(-0.372712\pi\)
0.389315 + 0.921105i \(0.372712\pi\)
\(864\) −2385.00 −0.0939113
\(865\) 14328.0 0.563198
\(866\) −39738.0 −1.55930
\(867\) 0 0
\(868\) −49.0000 −0.00191609
\(869\) 31800.0 1.24136
\(870\) 270.000 0.0105217
\(871\) −6908.00 −0.268736
\(872\) 8547.00 0.331924
\(873\) 29380.0 1.13902
\(874\) 11952.0 0.462566
\(875\) 8988.00 0.347257
\(876\) −448.000 −0.0172791
\(877\) −28946.0 −1.11452 −0.557262 0.830337i \(-0.688148\pi\)
−0.557262 + 0.830337i \(0.688148\pi\)
\(878\) 49359.0 1.89725
\(879\) −1614.00 −0.0619327
\(880\) −12780.0 −0.489561
\(881\) −4806.00 −0.183789 −0.0918946 0.995769i \(-0.529292\pi\)
−0.0918946 + 0.995769i \(0.529292\pi\)
\(882\) −3822.00 −0.145911
\(883\) 12638.0 0.481656 0.240828 0.970568i \(-0.422581\pi\)
0.240828 + 0.970568i \(0.422581\pi\)
\(884\) 0 0
\(885\) −1134.00 −0.0430723
\(886\) 24264.0 0.920051
\(887\) 40092.0 1.51765 0.758826 0.651293i \(-0.225773\pi\)
0.758826 + 0.651293i \(0.225773\pi\)
\(888\) −1050.00 −0.0396798
\(889\) −7154.00 −0.269896
\(890\) 4860.00 0.183042
\(891\) 19470.0 0.732065
\(892\) −97.0000 −0.00364103
\(893\) −2241.00 −0.0839778
\(894\) 1305.00 0.0488207
\(895\) 11808.0 0.441003
\(896\) −11613.0 −0.432995
\(897\) 1056.00 0.0393075
\(898\) −15597.0 −0.579598
\(899\) −105.000 −0.00389538
\(900\) 2314.00 0.0857037
\(901\) 0 0
\(902\) −32400.0 −1.19601
\(903\) 476.000 0.0175418
\(904\) −21042.0 −0.774166
\(905\) 3912.00 0.143690
\(906\) −5478.00 −0.200877
\(907\) −39344.0 −1.44035 −0.720174 0.693793i \(-0.755938\pi\)
−0.720174 + 0.693793i \(0.755938\pi\)
\(908\) −369.000 −0.0134864
\(909\) −43368.0 −1.58243
\(910\) 2772.00 0.100979
\(911\) −3342.00 −0.121543 −0.0607714 0.998152i \(-0.519356\pi\)
−0.0607714 + 0.998152i \(0.519356\pi\)
\(912\) 5893.00 0.213966
\(913\) 26190.0 0.949357
\(914\) 34377.0 1.24408
\(915\) 1884.00 0.0680690
\(916\) −1888.00 −0.0681018
\(917\) 18795.0 0.676844
\(918\) 0 0
\(919\) −21454.0 −0.770079 −0.385039 0.922900i \(-0.625812\pi\)
−0.385039 + 0.922900i \(0.625812\pi\)
\(920\) −6048.00 −0.216735
\(921\) −10001.0 −0.357811
\(922\) 3726.00 0.133090
\(923\) −17688.0 −0.630777
\(924\) 210.000 0.00747672
\(925\) 4450.00 0.158178
\(926\) −34788.0 −1.23456
\(927\) −14950.0 −0.529690
\(928\) 675.000 0.0238771
\(929\) 46992.0 1.65959 0.829794 0.558070i \(-0.188458\pi\)
0.829794 + 0.558070i \(0.188458\pi\)
\(930\) −126.000 −0.00444269
\(931\) 4067.00 0.143169
\(932\) −3633.00 −0.127685
\(933\) −5703.00 −0.200116
\(934\) −7956.00 −0.278724
\(935\) 0 0
\(936\) −12012.0 −0.419471
\(937\) 33584.0 1.17091 0.585454 0.810705i \(-0.300916\pi\)
0.585454 + 0.810705i \(0.300916\pi\)
\(938\) −6594.00 −0.229533
\(939\) 10904.0 0.378955
\(940\) −162.000 −0.00562112
\(941\) −46176.0 −1.59968 −0.799838 0.600216i \(-0.795081\pi\)
−0.799838 + 0.600216i \(0.795081\pi\)
\(942\) −9996.00 −0.345740
\(943\) −17280.0 −0.596728
\(944\) −13419.0 −0.462660
\(945\) −2226.00 −0.0766262
\(946\) 6120.00 0.210337
\(947\) 906.000 0.0310887 0.0155444 0.999879i \(-0.495052\pi\)
0.0155444 + 0.999879i \(0.495052\pi\)
\(948\) −1060.00 −0.0363156
\(949\) −9856.00 −0.337133
\(950\) −22161.0 −0.756840
\(951\) 9981.00 0.340332
\(952\) 0 0
\(953\) −35937.0 −1.22153 −0.610763 0.791814i \(-0.709137\pi\)
−0.610763 + 0.791814i \(0.709137\pi\)
\(954\) −16614.0 −0.563834
\(955\) 24804.0 0.840459
\(956\) −1728.00 −0.0584597
\(957\) 450.000 0.0152000
\(958\) −5004.00 −0.168760
\(959\) 17829.0 0.600342
\(960\) −2598.00 −0.0873438
\(961\) −29742.0 −0.998355
\(962\) 3300.00 0.110599
\(963\) −25272.0 −0.845669
\(964\) 5704.00 0.190574
\(965\) −7554.00 −0.251992
\(966\) 1008.00 0.0335734
\(967\) 29990.0 0.997325 0.498663 0.866796i \(-0.333825\pi\)
0.498663 + 0.866796i \(0.333825\pi\)
\(968\) 9051.00 0.300527
\(969\) 0 0
\(970\) −20340.0 −0.673276
\(971\) 34896.0 1.15331 0.576656 0.816987i \(-0.304357\pi\)
0.576656 + 0.816987i \(0.304357\pi\)
\(972\) −2080.00 −0.0686379
\(973\) −4396.00 −0.144840
\(974\) −54474.0 −1.79205
\(975\) −1958.00 −0.0643140
\(976\) 22294.0 0.731161
\(977\) −14931.0 −0.488930 −0.244465 0.969658i \(-0.578612\pi\)
−0.244465 + 0.969658i \(0.578612\pi\)
\(978\) −2964.00 −0.0969103
\(979\) 8100.00 0.264430
\(980\) 294.000 0.00958315
\(981\) 10582.0 0.344401
\(982\) 51066.0 1.65945
\(983\) −56109.0 −1.82055 −0.910274 0.414006i \(-0.864129\pi\)
−0.910274 + 0.414006i \(0.864129\pi\)
\(984\) −7560.00 −0.244923
\(985\) −7758.00 −0.250955
\(986\) 0 0
\(987\) −189.000 −0.00609517
\(988\) −1826.00 −0.0587984
\(989\) 3264.00 0.104944
\(990\) −14040.0 −0.450728
\(991\) −5762.00 −0.184698 −0.0923491 0.995727i \(-0.529438\pi\)
−0.0923491 + 0.995727i \(0.529438\pi\)
\(992\) −315.000 −0.0100819
\(993\) −10118.0 −0.323348
\(994\) −16884.0 −0.538761
\(995\) 2040.00 0.0649973
\(996\) −873.000 −0.0277732
\(997\) 27604.0 0.876858 0.438429 0.898766i \(-0.355535\pi\)
0.438429 + 0.898766i \(0.355535\pi\)
\(998\) −28590.0 −0.906814
\(999\) −2650.00 −0.0839262
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 2023.4.a.c.1.1 1
17.16 even 2 2023.4.a.d.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2023.4.a.c.1.1 1 1.1 even 1 trivial
2023.4.a.d.1.1 yes 1 17.16 even 2