Properties

Label 2023.4.a.d.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 $1$

Related objects

Downloads

Learn more

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 \(-0.469323\pi\)
0.0962250 + 0.995360i \(0.469323\pi\)
\(4\) 1.00000 0.125000
\(5\) −6.00000 −0.536656 −0.268328 0.963328i \(-0.586471\pi\)
−0.268328 + 0.963328i \(0.586471\pi\)
\(6\) 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 \(-0.634873\pi\)
−0.411152 + 0.911567i \(0.634873\pi\)
\(12\) 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 \(-0.569816\pi\)
−0.217580 + 0.976042i \(0.569816\pi\)
\(24\) −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 \(-0.484707\pi\)
0.0480247 + 0.998846i \(0.484707\pi\)
\(30\) −18.0000 −0.109545
\(31\) −7.00000 −0.0405560 −0.0202780 0.999794i \(-0.506455\pi\)
−0.0202780 + 0.999794i \(0.506455\pi\)
\(32\) −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 \(-0.464569\pi\)
0.111080 + 0.993811i \(0.464569\pi\)
\(38\) 249.000 1.06298
\(39\) −22.0000 −0.0903287
\(40\) 126.000 0.498059
\(41\) 360.000 1.37128 0.685641 0.727940i \(-0.259522\pi\)
0.685641 + 0.727940i \(0.259522\pi\)
\(42\) 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 \(-0.393107\pi\)
0.329538 + 0.944142i \(0.393107\pi\)
\(62\) −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 \(-0.734544\pi\)
−0.671952 + 0.740594i \(0.734544\pi\)
\(72\) 546.000 0.893704
\(73\) −448.000 −0.718280 −0.359140 0.933284i \(-0.616930\pi\)
−0.359140 + 0.933284i \(0.616930\pi\)
\(74\) 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 \(-0.772269\pi\)
−0.754806 + 0.655948i \(0.772269\pi\)
\(80\) 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 \(-0.298570\pi\)
0.591413 + 0.806369i \(0.298570\pi\)
\(98\) 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 \(-0.644702\pi\)
−0.439097 + 0.898440i \(0.644702\pi\)
\(108\) −53.0000 −0.0472215
\(109\) 407.000 0.357647 0.178824 0.983881i \(-0.442771\pi\)
0.178824 + 0.983881i \(0.442771\pi\)
\(110\) 540.000 0.468063
\(111\) 50.0000 0.0427549
\(112\) −497.000 −0.419304
\(113\) −1002.00 −0.834161 −0.417081 0.908869i \(-0.636947\pi\)
−0.417081 + 0.908869i \(0.636947\pi\)
\(114\) 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 \(-0.146904\pi\)
0.895380 + 0.445303i \(0.146904\pi\)
\(132\) −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 \(-0.561369\pi\)
−0.191605 + 0.981472i \(0.561369\pi\)
\(140\) −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 \(-0.576289\pi\)
−0.237381 + 0.971417i \(0.576289\pi\)
\(164\) 360.000 0.171410
\(165\) 180.000 0.0849272
\(166\) 2619.00 1.22454
\(167\) −2211.00 −1.02450 −0.512252 0.858835i \(-0.671189\pi\)
−0.512252 + 0.858835i \(0.671189\pi\)
\(168\) −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 \(-0.675833\pi\)
−0.524729 + 0.851269i \(0.675833\pi\)
\(174\) 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 \(-0.542742\pi\)
−0.133875 + 0.990998i \(0.542742\pi\)
\(182\) −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 \(-0.424563\pi\)
0.234779 + 0.972049i \(0.424563\pi\)
\(194\) 3390.00 1.25458
\(195\) 132.000 0.0484755
\(196\) 49.0000 0.0178571
\(197\) 1293.00 0.467627 0.233813 0.972282i \(-0.424880\pi\)
0.233813 + 0.972282i \(0.424880\pi\)
\(198\) 2340.00 0.839882
\(199\) −340.000 −0.121115 −0.0605577 0.998165i \(-0.519288\pi\)
−0.0605577 + 0.998165i \(0.519288\pi\)
\(200\) 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 \(-0.541137\pi\)
−0.128876 + 0.991661i \(0.541137\pi\)
\(212\) 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 \(-0.482820\pi\)
0.0539458 + 0.998544i \(0.482820\pi\)
\(228\) 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 \(-0.329371\pi\)
0.510742 + 0.859734i \(0.329371\pi\)
\(234\) 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 \(-0.775928\pi\)
−0.762297 + 0.647228i \(0.775928\pi\)
\(242\) −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 \(-0.508226\pi\)
−0.0258390 + 0.999666i \(0.508226\pi\)
\(270\) 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 \(-0.603554\pi\)
−0.319617 + 0.947547i \(0.603554\pi\)
\(278\) −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 \(-0.240290\pi\)
0.728344 + 0.685211i \(0.240290\pi\)
\(284\) −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 \(-0.674037\pi\)
−0.519916 + 0.854218i \(0.674037\pi\)
\(312\) 462.000 0.0838320
\(313\) 10904.0 1.96911 0.984554 0.175084i \(-0.0560196\pi\)
0.984554 + 0.175084i \(0.0560196\pi\)
\(314\) 9996.00 1.79652
\(315\) 1092.00 0.195325
\(316\) −1060.00 −0.188701
\(317\) 9981.00 1.76842 0.884209 0.467091i \(-0.154698\pi\)
0.884209 + 0.467091i \(0.154698\pi\)
\(318\) 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 \(-0.718938\pi\)
−0.634850 + 0.772635i \(0.718938\pi\)
\(338\) −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 \(-0.673825\pi\)
−0.519346 + 0.854564i \(0.673825\pi\)
\(348\) 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 \(-0.472280\pi\)
0.0869756 + 0.996210i \(0.472280\pi\)
\(368\) 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 \(-0.624367\pi\)
−0.380844 + 0.924639i \(0.624367\pi\)
\(380\) −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 \(-0.746135\pi\)
−0.698468 + 0.715641i \(0.746135\pi\)
\(398\) −1020.00 −0.128462
\(399\) 581.000 0.0728982
\(400\) 6319.00 0.789875
\(401\) 8265.00 1.02926 0.514631 0.857412i \(-0.327929\pi\)
0.514631 + 0.857412i \(0.327929\pi\)
\(402\) 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 \(-0.393593\pi\)
0.328098 + 0.944644i \(0.393593\pi\)
\(420\) −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 \(-0.433455\pi\)
0.207537 + 0.978227i \(0.433455\pi\)
\(432\) 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 \(-0.852377\pi\)
−0.894372 + 0.447323i \(0.852377\pi\)
\(440\) −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 \(-0.411910\pi\)
0.273225 + 0.961950i \(0.411910\pi\)
\(450\) 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 \(-0.474650\pi\)
0.0795541 + 0.996831i \(0.474650\pi\)
\(480\) 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 \(-0.179730\pi\)
0.844782 + 0.535111i \(0.179730\pi\)
\(488\) −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 \(-0.359403\pi\)
0.427476 + 0.904027i \(0.359403\pi\)
\(500\) 1284.00 0.114844
\(501\) −2211.00 −0.197166
\(502\) 8784.00 0.780974
\(503\) 10875.0 0.964001 0.482000 0.876171i \(-0.339910\pi\)
0.482000 + 0.876171i \(0.339910\pi\)
\(504\) 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 \(-0.418574\pi\)
0.253026 + 0.967459i \(0.418574\pi\)
\(522\) −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 \(-0.527442\pi\)
−0.0861059 + 0.996286i \(0.527442\pi\)
\(542\) −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 \(-0.421862\pi\)
0.243019 + 0.970022i \(0.421862\pi\)
\(548\) −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 \(-0.424597\pi\)
0.234675 + 0.972074i \(0.424597\pi\)
\(572\) 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 \(-0.412013\pi\)
0.272912 + 0.962039i \(0.412013\pi\)
\(602\) 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 \(-0.617058\pi\)
−0.359515 + 0.933139i \(0.617058\pi\)
\(608\) −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 \(-0.446822\pi\)
0.166286 + 0.986077i \(0.446822\pi\)
\(618\) 1725.00 0.112281
\(619\) −11095.0 −0.720429 −0.360215 0.932869i \(-0.617297\pi\)
−0.360215 + 0.932869i \(0.617297\pi\)
\(620\) 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 \(-0.398648\pi\)
0.313054 + 0.949735i \(0.398648\pi\)
\(642\) −2916.00 −0.179261
\(643\) −17524.0 −1.07477 −0.537387 0.843336i \(-0.680588\pi\)
−0.537387 + 0.843336i \(0.680588\pi\)
\(644\) −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 \(-0.606198\pi\)
−0.327477 + 0.944859i \(0.606198\pi\)
\(654\) 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 \(-0.509254\pi\)
−0.0290679 + 0.999577i \(0.509254\pi\)
\(674\) −23565.0 −1.34672
\(675\) 4717.00 0.268974
\(676\) −1713.00 −0.0974624
\(677\) 264.000 0.0149872 0.00749361 0.999972i \(-0.497615\pi\)
0.00749361 + 0.999972i \(0.497615\pi\)
\(678\) −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 \(-0.407991\pi\)
0.285047 + 0.958514i \(0.407991\pi\)
\(684\) −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 \(-0.775008\pi\)
−0.760423 + 0.649428i \(0.775008\pi\)
\(692\) −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 \(-0.715461\pi\)
−0.626371 + 0.779525i \(0.715461\pi\)
\(710\) 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 \(-0.464661\pi\)
0.110792 + 0.993844i \(0.464661\pi\)
\(720\) −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 \(-0.151869\pi\)
0.888325 + 0.459215i \(0.151869\pi\)
\(744\) 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 \(-0.564076\pi\)
−0.199945 + 0.979807i \(0.564076\pi\)
\(752\) 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 \(-0.754982\pi\)
−0.718086 + 0.695954i \(0.754982\pi\)
\(788\) 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 \(-0.201706\pi\)
0.805856 + 0.592112i \(0.201706\pi\)
\(810\) −11682.0 −0.506745
\(811\) 20063.0 0.868690 0.434345 0.900747i \(-0.356980\pi\)
0.434345 + 0.900747i \(0.356980\pi\)
\(812\) 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 \(-0.629682\pi\)
−0.396230 + 0.918151i \(0.629682\pi\)
\(822\) −7641.00 −0.324222
\(823\) −10606.0 −0.449213 −0.224606 0.974450i \(-0.572110\pi\)
−0.224606 + 0.974450i \(0.572110\pi\)
\(824\) −12075.0 −0.510501
\(825\) 2670.00 0.112676
\(826\) 3969.00 0.167190
\(827\) −32154.0 −1.35200 −0.676000 0.736902i \(-0.736288\pi\)
−0.676000 + 0.736902i \(0.736288\pi\)
\(828\) 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 \(-0.374808\pi\)
0.383239 + 0.923649i \(0.374808\pi\)
\(840\) 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 \(-0.457274\pi\)
0.133826 + 0.991005i \(0.457274\pi\)
\(854\) 6594.00 0.264218
\(855\) 12948.0 0.517909
\(856\) 20412.0 0.815032
\(857\) −2328.00 −0.0927923 −0.0463961 0.998923i \(-0.514774\pi\)
−0.0463961 + 0.998923i \(0.514774\pi\)
\(858\) 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 \(-0.311852\pi\)
0.557262 + 0.830337i \(0.311852\pi\)
\(878\) −49359.0 −1.89725
\(879\) 1614.00 0.0619327
\(880\) −12780.0 −0.489561
\(881\) 4806.00 0.183789 0.0918946 0.995769i \(-0.470708\pi\)
0.0918946 + 0.995769i \(0.470708\pi\)
\(882\) −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 \(-0.774227\pi\)
−0.758826 + 0.651293i \(0.774227\pi\)
\(888\) −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 \(-0.244062\pi\)
0.720174 + 0.693793i \(0.244062\pi\)
\(908\) 369.000 0.0134864
\(909\) −43368.0 −1.58243
\(910\) 2772.00 0.100979
\(911\) 3342.00 0.121543 0.0607714 0.998152i \(-0.480644\pi\)
0.0607714 + 0.998152i \(0.480644\pi\)
\(912\) −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 \(-0.811542\pi\)
−0.829794 + 0.558070i \(0.811542\pi\)
\(930\) 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 \(-0.204919\pi\)
0.799838 + 0.600216i \(0.204919\pi\)
\(942\) 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 \(-0.504948\pi\)
−0.0155444 + 0.999879i \(0.504948\pi\)
\(948\) −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 \(-0.135871\pi\)
0.910274 + 0.414006i \(0.135871\pi\)
\(984\) −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 \(-0.470562\pi\)
0.0923491 + 0.995727i \(0.470562\pi\)
\(992\) 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 \(-0.644465\pi\)
−0.438429 + 0.898766i \(0.644465\pi\)
\(998\) 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.d.1.1 yes 1
17.16 even 2 2023.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2023.4.a.c.1.1 1 17.16 even 2
2023.4.a.d.1.1 yes 1 1.1 even 1 trivial