Properties

Label 2023.4.a.b.1.1
Level $2023$
Weight $4$
Character 2023.1
Self dual yes
Analytic conductor $119.361$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 119)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +6.00000 q^{3} -7.00000 q^{4} +20.0000 q^{5} -6.00000 q^{6} +7.00000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +6.00000 q^{3} -7.00000 q^{4} +20.0000 q^{5} -6.00000 q^{6} +7.00000 q^{7} +15.0000 q^{8} +9.00000 q^{9} -20.0000 q^{10} -60.0000 q^{11} -42.0000 q^{12} -68.0000 q^{13} -7.00000 q^{14} +120.000 q^{15} +41.0000 q^{16} -9.00000 q^{18} -70.0000 q^{19} -140.000 q^{20} +42.0000 q^{21} +60.0000 q^{22} +176.000 q^{23} +90.0000 q^{24} +275.000 q^{25} +68.0000 q^{26} -108.000 q^{27} -49.0000 q^{28} +90.0000 q^{29} -120.000 q^{30} -196.000 q^{31} -161.000 q^{32} -360.000 q^{33} +140.000 q^{35} -63.0000 q^{36} -22.0000 q^{37} +70.0000 q^{38} -408.000 q^{39} +300.000 q^{40} +138.000 q^{41} -42.0000 q^{42} +328.000 q^{43} +420.000 q^{44} +180.000 q^{45} -176.000 q^{46} -12.0000 q^{47} +246.000 q^{48} +49.0000 q^{49} -275.000 q^{50} +476.000 q^{52} -234.000 q^{53} +108.000 q^{54} -1200.00 q^{55} +105.000 q^{56} -420.000 q^{57} -90.0000 q^{58} -54.0000 q^{59} -840.000 q^{60} -44.0000 q^{61} +196.000 q^{62} +63.0000 q^{63} -167.000 q^{64} -1360.00 q^{65} +360.000 q^{66} -596.000 q^{67} +1056.00 q^{69} -140.000 q^{70} -200.000 q^{71} +135.000 q^{72} -1122.00 q^{73} +22.0000 q^{74} +1650.00 q^{75} +490.000 q^{76} -420.000 q^{77} +408.000 q^{78} -480.000 q^{79} +820.000 q^{80} -891.000 q^{81} -138.000 q^{82} -838.000 q^{83} -294.000 q^{84} -328.000 q^{86} +540.000 q^{87} -900.000 q^{88} +778.000 q^{89} -180.000 q^{90} -476.000 q^{91} -1232.00 q^{92} -1176.00 q^{93} +12.0000 q^{94} -1400.00 q^{95} -966.000 q^{96} -1142.00 q^{97} -49.0000 q^{98} -540.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) 6.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −7.00000 −0.875000
\(5\) 20.0000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) −6.00000 −0.408248
\(7\) 7.00000 0.377964
\(8\) 15.0000 0.662913
\(9\) 9.00000 0.333333
\(10\) −20.0000 −0.632456
\(11\) −60.0000 −1.64461 −0.822304 0.569049i \(-0.807311\pi\)
−0.822304 + 0.569049i \(0.807311\pi\)
\(12\) −42.0000 −1.01036
\(13\) −68.0000 −1.45075 −0.725377 0.688352i \(-0.758335\pi\)
−0.725377 + 0.688352i \(0.758335\pi\)
\(14\) −7.00000 −0.133631
\(15\) 120.000 2.06559
\(16\) 41.0000 0.640625
\(17\) 0 0
\(18\) −9.00000 −0.117851
\(19\) −70.0000 −0.845216 −0.422608 0.906313i \(-0.638885\pi\)
−0.422608 + 0.906313i \(0.638885\pi\)
\(20\) −140.000 −1.56525
\(21\) 42.0000 0.436436
\(22\) 60.0000 0.581456
\(23\) 176.000 1.59559 0.797794 0.602930i \(-0.206000\pi\)
0.797794 + 0.602930i \(0.206000\pi\)
\(24\) 90.0000 0.765466
\(25\) 275.000 2.20000
\(26\) 68.0000 0.512919
\(27\) −108.000 −0.769800
\(28\) −49.0000 −0.330719
\(29\) 90.0000 0.576296 0.288148 0.957586i \(-0.406961\pi\)
0.288148 + 0.957586i \(0.406961\pi\)
\(30\) −120.000 −0.730297
\(31\) −196.000 −1.13557 −0.567785 0.823177i \(-0.692199\pi\)
−0.567785 + 0.823177i \(0.692199\pi\)
\(32\) −161.000 −0.889408
\(33\) −360.000 −1.89903
\(34\) 0 0
\(35\) 140.000 0.676123
\(36\) −63.0000 −0.291667
\(37\) −22.0000 −0.0977507 −0.0488754 0.998805i \(-0.515564\pi\)
−0.0488754 + 0.998805i \(0.515564\pi\)
\(38\) 70.0000 0.298829
\(39\) −408.000 −1.67519
\(40\) 300.000 1.18585
\(41\) 138.000 0.525658 0.262829 0.964842i \(-0.415344\pi\)
0.262829 + 0.964842i \(0.415344\pi\)
\(42\) −42.0000 −0.154303
\(43\) 328.000 1.16324 0.581622 0.813459i \(-0.302418\pi\)
0.581622 + 0.813459i \(0.302418\pi\)
\(44\) 420.000 1.43903
\(45\) 180.000 0.596285
\(46\) −176.000 −0.564126
\(47\) −12.0000 −0.0372421 −0.0186211 0.999827i \(-0.505928\pi\)
−0.0186211 + 0.999827i \(0.505928\pi\)
\(48\) 246.000 0.739730
\(49\) 49.0000 0.142857
\(50\) −275.000 −0.777817
\(51\) 0 0
\(52\) 476.000 1.26941
\(53\) −234.000 −0.606460 −0.303230 0.952917i \(-0.598065\pi\)
−0.303230 + 0.952917i \(0.598065\pi\)
\(54\) 108.000 0.272166
\(55\) −1200.00 −2.94196
\(56\) 105.000 0.250557
\(57\) −420.000 −0.975971
\(58\) −90.0000 −0.203751
\(59\) −54.0000 −0.119156 −0.0595780 0.998224i \(-0.518975\pi\)
−0.0595780 + 0.998224i \(0.518975\pi\)
\(60\) −840.000 −1.80739
\(61\) −44.0000 −0.0923545 −0.0461772 0.998933i \(-0.514704\pi\)
−0.0461772 + 0.998933i \(0.514704\pi\)
\(62\) 196.000 0.401484
\(63\) 63.0000 0.125988
\(64\) −167.000 −0.326172
\(65\) −1360.00 −2.59519
\(66\) 360.000 0.671408
\(67\) −596.000 −1.08676 −0.543381 0.839487i \(-0.682856\pi\)
−0.543381 + 0.839487i \(0.682856\pi\)
\(68\) 0 0
\(69\) 1056.00 1.84243
\(70\) −140.000 −0.239046
\(71\) −200.000 −0.334305 −0.167152 0.985931i \(-0.553457\pi\)
−0.167152 + 0.985931i \(0.553457\pi\)
\(72\) 135.000 0.220971
\(73\) −1122.00 −1.79891 −0.899453 0.437017i \(-0.856035\pi\)
−0.899453 + 0.437017i \(0.856035\pi\)
\(74\) 22.0000 0.0345601
\(75\) 1650.00 2.54034
\(76\) 490.000 0.739564
\(77\) −420.000 −0.621603
\(78\) 408.000 0.592268
\(79\) −480.000 −0.683598 −0.341799 0.939773i \(-0.611036\pi\)
−0.341799 + 0.939773i \(0.611036\pi\)
\(80\) 820.000 1.14598
\(81\) −891.000 −1.22222
\(82\) −138.000 −0.185848
\(83\) −838.000 −1.10822 −0.554111 0.832443i \(-0.686942\pi\)
−0.554111 + 0.832443i \(0.686942\pi\)
\(84\) −294.000 −0.381881
\(85\) 0 0
\(86\) −328.000 −0.411269
\(87\) 540.000 0.665449
\(88\) −900.000 −1.09023
\(89\) 778.000 0.926605 0.463302 0.886200i \(-0.346664\pi\)
0.463302 + 0.886200i \(0.346664\pi\)
\(90\) −180.000 −0.210819
\(91\) −476.000 −0.548334
\(92\) −1232.00 −1.39614
\(93\) −1176.00 −1.31124
\(94\) 12.0000 0.0131671
\(95\) −1400.00 −1.51197
\(96\) −966.000 −1.02700
\(97\) −1142.00 −1.19539 −0.597693 0.801725i \(-0.703916\pi\)
−0.597693 + 0.801725i \(0.703916\pi\)
\(98\) −49.0000 −0.0505076
\(99\) −540.000 −0.548202
\(100\) −1925.00 −1.92500
\(101\) 32.0000 0.0315259 0.0157630 0.999876i \(-0.494982\pi\)
0.0157630 + 0.999876i \(0.494982\pi\)
\(102\) 0 0
\(103\) −1300.00 −1.24362 −0.621810 0.783168i \(-0.713602\pi\)
−0.621810 + 0.783168i \(0.713602\pi\)
\(104\) −1020.00 −0.961723
\(105\) 840.000 0.780720
\(106\) 234.000 0.214416
\(107\) 256.000 0.231294 0.115647 0.993290i \(-0.463106\pi\)
0.115647 + 0.993290i \(0.463106\pi\)
\(108\) 756.000 0.673575
\(109\) −318.000 −0.279439 −0.139720 0.990191i \(-0.544620\pi\)
−0.139720 + 0.990191i \(0.544620\pi\)
\(110\) 1200.00 1.04014
\(111\) −132.000 −0.112873
\(112\) 287.000 0.242133
\(113\) −1918.00 −1.59673 −0.798364 0.602175i \(-0.794301\pi\)
−0.798364 + 0.602175i \(0.794301\pi\)
\(114\) 420.000 0.345058
\(115\) 3520.00 2.85428
\(116\) −630.000 −0.504259
\(117\) −612.000 −0.483585
\(118\) 54.0000 0.0421280
\(119\) 0 0
\(120\) 1800.00 1.36931
\(121\) 2269.00 1.70473
\(122\) 44.0000 0.0326522
\(123\) 828.000 0.606978
\(124\) 1372.00 0.993623
\(125\) 3000.00 2.14663
\(126\) −63.0000 −0.0445435
\(127\) 1832.00 1.28003 0.640015 0.768363i \(-0.278928\pi\)
0.640015 + 0.768363i \(0.278928\pi\)
\(128\) 1455.00 1.00473
\(129\) 1968.00 1.34320
\(130\) 1360.00 0.917538
\(131\) −262.000 −0.174741 −0.0873704 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) 2520.00 1.66165
\(133\) −490.000 −0.319462
\(134\) 596.000 0.384228
\(135\) −2160.00 −1.37706
\(136\) 0 0
\(137\) −1902.00 −1.18612 −0.593061 0.805157i \(-0.702081\pi\)
−0.593061 + 0.805157i \(0.702081\pi\)
\(138\) −1056.00 −0.651396
\(139\) −82.0000 −0.0500370 −0.0250185 0.999687i \(-0.507964\pi\)
−0.0250185 + 0.999687i \(0.507964\pi\)
\(140\) −980.000 −0.591608
\(141\) −72.0000 −0.0430035
\(142\) 200.000 0.118195
\(143\) 4080.00 2.38592
\(144\) 369.000 0.213542
\(145\) 1800.00 1.03091
\(146\) 1122.00 0.636009
\(147\) 294.000 0.164957
\(148\) 154.000 0.0855319
\(149\) −1258.00 −0.691674 −0.345837 0.938295i \(-0.612405\pi\)
−0.345837 + 0.938295i \(0.612405\pi\)
\(150\) −1650.00 −0.898146
\(151\) −1840.00 −0.991636 −0.495818 0.868426i \(-0.665132\pi\)
−0.495818 + 0.868426i \(0.665132\pi\)
\(152\) −1050.00 −0.560304
\(153\) 0 0
\(154\) 420.000 0.219770
\(155\) −3920.00 −2.03137
\(156\) 2856.00 1.46579
\(157\) −2156.00 −1.09597 −0.547986 0.836488i \(-0.684605\pi\)
−0.547986 + 0.836488i \(0.684605\pi\)
\(158\) 480.000 0.241688
\(159\) −1404.00 −0.700280
\(160\) −3220.00 −1.59102
\(161\) 1232.00 0.603076
\(162\) 891.000 0.432121
\(163\) 1692.00 0.813053 0.406527 0.913639i \(-0.366740\pi\)
0.406527 + 0.913639i \(0.366740\pi\)
\(164\) −966.000 −0.459951
\(165\) −7200.00 −3.39709
\(166\) 838.000 0.391816
\(167\) 2644.00 1.22514 0.612571 0.790415i \(-0.290135\pi\)
0.612571 + 0.790415i \(0.290135\pi\)
\(168\) 630.000 0.289319
\(169\) 2427.00 1.10469
\(170\) 0 0
\(171\) −630.000 −0.281739
\(172\) −2296.00 −1.01784
\(173\) 1808.00 0.794565 0.397282 0.917696i \(-0.369953\pi\)
0.397282 + 0.917696i \(0.369953\pi\)
\(174\) −540.000 −0.235272
\(175\) 1925.00 0.831522
\(176\) −2460.00 −1.05358
\(177\) −324.000 −0.137589
\(178\) −778.000 −0.327604
\(179\) −1716.00 −0.716536 −0.358268 0.933619i \(-0.616633\pi\)
−0.358268 + 0.933619i \(0.616633\pi\)
\(180\) −1260.00 −0.521749
\(181\) −3304.00 −1.35682 −0.678410 0.734684i \(-0.737331\pi\)
−0.678410 + 0.734684i \(0.737331\pi\)
\(182\) 476.000 0.193865
\(183\) −264.000 −0.106642
\(184\) 2640.00 1.05774
\(185\) −440.000 −0.174862
\(186\) 1176.00 0.463594
\(187\) 0 0
\(188\) 84.0000 0.0325869
\(189\) −756.000 −0.290957
\(190\) 1400.00 0.534561
\(191\) 4088.00 1.54868 0.774338 0.632772i \(-0.218083\pi\)
0.774338 + 0.632772i \(0.218083\pi\)
\(192\) −1002.00 −0.376631
\(193\) −3038.00 −1.13306 −0.566529 0.824042i \(-0.691714\pi\)
−0.566529 + 0.824042i \(0.691714\pi\)
\(194\) 1142.00 0.422633
\(195\) −8160.00 −2.99667
\(196\) −343.000 −0.125000
\(197\) 1230.00 0.444842 0.222421 0.974951i \(-0.428604\pi\)
0.222421 + 0.974951i \(0.428604\pi\)
\(198\) 540.000 0.193819
\(199\) −1276.00 −0.454539 −0.227269 0.973832i \(-0.572980\pi\)
−0.227269 + 0.973832i \(0.572980\pi\)
\(200\) 4125.00 1.45841
\(201\) −3576.00 −1.25488
\(202\) −32.0000 −0.0111461
\(203\) 630.000 0.217819
\(204\) 0 0
\(205\) 2760.00 0.940326
\(206\) 1300.00 0.439686
\(207\) 1584.00 0.531863
\(208\) −2788.00 −0.929390
\(209\) 4200.00 1.39005
\(210\) −840.000 −0.276026
\(211\) −632.000 −0.206202 −0.103101 0.994671i \(-0.532877\pi\)
−0.103101 + 0.994671i \(0.532877\pi\)
\(212\) 1638.00 0.530652
\(213\) −1200.00 −0.386022
\(214\) −256.000 −0.0817748
\(215\) 6560.00 2.08088
\(216\) −1620.00 −0.510310
\(217\) −1372.00 −0.429205
\(218\) 318.000 0.0987967
\(219\) −6732.00 −2.07720
\(220\) 8400.00 2.57422
\(221\) 0 0
\(222\) 132.000 0.0399066
\(223\) −840.000 −0.252245 −0.126122 0.992015i \(-0.540253\pi\)
−0.126122 + 0.992015i \(0.540253\pi\)
\(224\) −1127.00 −0.336165
\(225\) 2475.00 0.733333
\(226\) 1918.00 0.564529
\(227\) −4566.00 −1.33505 −0.667524 0.744588i \(-0.732646\pi\)
−0.667524 + 0.744588i \(0.732646\pi\)
\(228\) 2940.00 0.853975
\(229\) 1236.00 0.356669 0.178334 0.983970i \(-0.442929\pi\)
0.178334 + 0.983970i \(0.442929\pi\)
\(230\) −3520.00 −1.00914
\(231\) −2520.00 −0.717765
\(232\) 1350.00 0.382034
\(233\) −1386.00 −0.389699 −0.194850 0.980833i \(-0.562422\pi\)
−0.194850 + 0.980833i \(0.562422\pi\)
\(234\) 612.000 0.170973
\(235\) −240.000 −0.0666207
\(236\) 378.000 0.104261
\(237\) −2880.00 −0.789351
\(238\) 0 0
\(239\) 6392.00 1.72998 0.864988 0.501793i \(-0.167326\pi\)
0.864988 + 0.501793i \(0.167326\pi\)
\(240\) 4920.00 1.32327
\(241\) −990.000 −0.264612 −0.132306 0.991209i \(-0.542238\pi\)
−0.132306 + 0.991209i \(0.542238\pi\)
\(242\) −2269.00 −0.602714
\(243\) −2430.00 −0.641500
\(244\) 308.000 0.0808102
\(245\) 980.000 0.255551
\(246\) −828.000 −0.214599
\(247\) 4760.00 1.22620
\(248\) −2940.00 −0.752783
\(249\) −5028.00 −1.27966
\(250\) −3000.00 −0.758947
\(251\) 4398.00 1.10597 0.552987 0.833190i \(-0.313488\pi\)
0.552987 + 0.833190i \(0.313488\pi\)
\(252\) −441.000 −0.110240
\(253\) −10560.0 −2.62412
\(254\) −1832.00 −0.452559
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) 3018.00 0.732520 0.366260 0.930513i \(-0.380638\pi\)
0.366260 + 0.930513i \(0.380638\pi\)
\(258\) −1968.00 −0.474893
\(259\) −154.000 −0.0369463
\(260\) 9520.00 2.27079
\(261\) 810.000 0.192099
\(262\) 262.000 0.0617802
\(263\) −7384.00 −1.73124 −0.865622 0.500699i \(-0.833076\pi\)
−0.865622 + 0.500699i \(0.833076\pi\)
\(264\) −5400.00 −1.25889
\(265\) −4680.00 −1.08487
\(266\) 490.000 0.112947
\(267\) 4668.00 1.06995
\(268\) 4172.00 0.950916
\(269\) −1408.00 −0.319135 −0.159567 0.987187i \(-0.551010\pi\)
−0.159567 + 0.987187i \(0.551010\pi\)
\(270\) 2160.00 0.486864
\(271\) 1952.00 0.437548 0.218774 0.975776i \(-0.429794\pi\)
0.218774 + 0.975776i \(0.429794\pi\)
\(272\) 0 0
\(273\) −2856.00 −0.633161
\(274\) 1902.00 0.419358
\(275\) −16500.0 −3.61814
\(276\) −7392.00 −1.61212
\(277\) −4274.00 −0.927075 −0.463537 0.886077i \(-0.653420\pi\)
−0.463537 + 0.886077i \(0.653420\pi\)
\(278\) 82.0000 0.0176908
\(279\) −1764.00 −0.378523
\(280\) 2100.00 0.448211
\(281\) −150.000 −0.0318443 −0.0159222 0.999873i \(-0.505068\pi\)
−0.0159222 + 0.999873i \(0.505068\pi\)
\(282\) 72.0000 0.0152040
\(283\) 586.000 0.123089 0.0615443 0.998104i \(-0.480397\pi\)
0.0615443 + 0.998104i \(0.480397\pi\)
\(284\) 1400.00 0.292517
\(285\) −8400.00 −1.74587
\(286\) −4080.00 −0.843551
\(287\) 966.000 0.198680
\(288\) −1449.00 −0.296469
\(289\) 0 0
\(290\) −1800.00 −0.364482
\(291\) −6852.00 −1.38031
\(292\) 7854.00 1.57404
\(293\) 1248.00 0.248836 0.124418 0.992230i \(-0.460294\pi\)
0.124418 + 0.992230i \(0.460294\pi\)
\(294\) −294.000 −0.0583212
\(295\) −1080.00 −0.213153
\(296\) −330.000 −0.0648002
\(297\) 6480.00 1.26602
\(298\) 1258.00 0.244544
\(299\) −11968.0 −2.31481
\(300\) −11550.0 −2.22280
\(301\) 2296.00 0.439665
\(302\) 1840.00 0.350596
\(303\) 192.000 0.0364030
\(304\) −2870.00 −0.541466
\(305\) −880.000 −0.165209
\(306\) 0 0
\(307\) −6390.00 −1.18794 −0.593968 0.804488i \(-0.702440\pi\)
−0.593968 + 0.804488i \(0.702440\pi\)
\(308\) 2940.00 0.543903
\(309\) −7800.00 −1.43601
\(310\) 3920.00 0.718197
\(311\) 3800.00 0.692856 0.346428 0.938077i \(-0.387394\pi\)
0.346428 + 0.938077i \(0.387394\pi\)
\(312\) −6120.00 −1.11050
\(313\) 3274.00 0.591238 0.295619 0.955306i \(-0.404474\pi\)
0.295619 + 0.955306i \(0.404474\pi\)
\(314\) 2156.00 0.387484
\(315\) 1260.00 0.225374
\(316\) 3360.00 0.598148
\(317\) 6838.00 1.21155 0.605773 0.795637i \(-0.292864\pi\)
0.605773 + 0.795637i \(0.292864\pi\)
\(318\) 1404.00 0.247586
\(319\) −5400.00 −0.947780
\(320\) −3340.00 −0.583474
\(321\) 1536.00 0.267075
\(322\) −1232.00 −0.213219
\(323\) 0 0
\(324\) 6237.00 1.06944
\(325\) −18700.0 −3.19166
\(326\) −1692.00 −0.287458
\(327\) −1908.00 −0.322669
\(328\) 2070.00 0.348465
\(329\) −84.0000 −0.0140762
\(330\) 7200.00 1.20105
\(331\) −6104.00 −1.01361 −0.506807 0.862060i \(-0.669174\pi\)
−0.506807 + 0.862060i \(0.669174\pi\)
\(332\) 5866.00 0.969695
\(333\) −198.000 −0.0325836
\(334\) −2644.00 −0.433153
\(335\) −11920.0 −1.94406
\(336\) 1722.00 0.279592
\(337\) −4678.00 −0.756163 −0.378081 0.925772i \(-0.623416\pi\)
−0.378081 + 0.925772i \(0.623416\pi\)
\(338\) −2427.00 −0.390566
\(339\) −11508.0 −1.84374
\(340\) 0 0
\(341\) 11760.0 1.86757
\(342\) 630.000 0.0996096
\(343\) 343.000 0.0539949
\(344\) 4920.00 0.771130
\(345\) 21120.0 3.29583
\(346\) −1808.00 −0.280921
\(347\) 7396.00 1.14420 0.572101 0.820183i \(-0.306129\pi\)
0.572101 + 0.820183i \(0.306129\pi\)
\(348\) −3780.00 −0.582268
\(349\) 4020.00 0.616578 0.308289 0.951293i \(-0.400244\pi\)
0.308289 + 0.951293i \(0.400244\pi\)
\(350\) −1925.00 −0.293987
\(351\) 7344.00 1.11679
\(352\) 9660.00 1.46273
\(353\) −5622.00 −0.847674 −0.423837 0.905739i \(-0.639317\pi\)
−0.423837 + 0.905739i \(0.639317\pi\)
\(354\) 324.000 0.0486452
\(355\) −4000.00 −0.598022
\(356\) −5446.00 −0.810779
\(357\) 0 0
\(358\) 1716.00 0.253334
\(359\) 7368.00 1.08320 0.541599 0.840637i \(-0.317819\pi\)
0.541599 + 0.840637i \(0.317819\pi\)
\(360\) 2700.00 0.395285
\(361\) −1959.00 −0.285610
\(362\) 3304.00 0.479708
\(363\) 13614.0 1.96846
\(364\) 3332.00 0.479792
\(365\) −22440.0 −3.21798
\(366\) 264.000 0.0377036
\(367\) −11080.0 −1.57594 −0.787972 0.615711i \(-0.788869\pi\)
−0.787972 + 0.615711i \(0.788869\pi\)
\(368\) 7216.00 1.02217
\(369\) 1242.00 0.175219
\(370\) 440.000 0.0618230
\(371\) −1638.00 −0.229220
\(372\) 8232.00 1.14734
\(373\) 2142.00 0.297342 0.148671 0.988887i \(-0.452500\pi\)
0.148671 + 0.988887i \(0.452500\pi\)
\(374\) 0 0
\(375\) 18000.0 2.47871
\(376\) −180.000 −0.0246883
\(377\) −6120.00 −0.836064
\(378\) 756.000 0.102869
\(379\) −7612.00 −1.03167 −0.515834 0.856689i \(-0.672518\pi\)
−0.515834 + 0.856689i \(0.672518\pi\)
\(380\) 9800.00 1.32297
\(381\) 10992.0 1.47805
\(382\) −4088.00 −0.547540
\(383\) 7572.00 1.01021 0.505106 0.863057i \(-0.331453\pi\)
0.505106 + 0.863057i \(0.331453\pi\)
\(384\) 8730.00 1.16016
\(385\) −8400.00 −1.11196
\(386\) 3038.00 0.400596
\(387\) 2952.00 0.387748
\(388\) 7994.00 1.04596
\(389\) 6858.00 0.893867 0.446934 0.894567i \(-0.352516\pi\)
0.446934 + 0.894567i \(0.352516\pi\)
\(390\) 8160.00 1.05948
\(391\) 0 0
\(392\) 735.000 0.0947018
\(393\) −1572.00 −0.201773
\(394\) −1230.00 −0.157275
\(395\) −9600.00 −1.22286
\(396\) 3780.00 0.479677
\(397\) −10008.0 −1.26521 −0.632603 0.774476i \(-0.718014\pi\)
−0.632603 + 0.774476i \(0.718014\pi\)
\(398\) 1276.00 0.160704
\(399\) −2940.00 −0.368882
\(400\) 11275.0 1.40938
\(401\) 8402.00 1.04632 0.523162 0.852233i \(-0.324752\pi\)
0.523162 + 0.852233i \(0.324752\pi\)
\(402\) 3576.00 0.443668
\(403\) 13328.0 1.64743
\(404\) −224.000 −0.0275852
\(405\) −17820.0 −2.18638
\(406\) −630.000 −0.0770108
\(407\) 1320.00 0.160762
\(408\) 0 0
\(409\) 13382.0 1.61784 0.808921 0.587917i \(-0.200052\pi\)
0.808921 + 0.587917i \(0.200052\pi\)
\(410\) −2760.00 −0.332455
\(411\) −11412.0 −1.36962
\(412\) 9100.00 1.08817
\(413\) −378.000 −0.0450367
\(414\) −1584.00 −0.188042
\(415\) −16760.0 −1.98245
\(416\) 10948.0 1.29031
\(417\) −492.000 −0.0577778
\(418\) −4200.00 −0.491456
\(419\) 14662.0 1.70951 0.854756 0.519030i \(-0.173707\pi\)
0.854756 + 0.519030i \(0.173707\pi\)
\(420\) −5880.00 −0.683130
\(421\) −7370.00 −0.853187 −0.426594 0.904443i \(-0.640287\pi\)
−0.426594 + 0.904443i \(0.640287\pi\)
\(422\) 632.000 0.0729035
\(423\) −108.000 −0.0124140
\(424\) −3510.00 −0.402030
\(425\) 0 0
\(426\) 1200.00 0.136479
\(427\) −308.000 −0.0349067
\(428\) −1792.00 −0.202382
\(429\) 24480.0 2.75502
\(430\) −6560.00 −0.735701
\(431\) −3480.00 −0.388923 −0.194461 0.980910i \(-0.562296\pi\)
−0.194461 + 0.980910i \(0.562296\pi\)
\(432\) −4428.00 −0.493153
\(433\) 13070.0 1.45059 0.725293 0.688440i \(-0.241704\pi\)
0.725293 + 0.688440i \(0.241704\pi\)
\(434\) 1372.00 0.151747
\(435\) 10800.0 1.19039
\(436\) 2226.00 0.244509
\(437\) −12320.0 −1.34862
\(438\) 6732.00 0.734400
\(439\) −9352.00 −1.01673 −0.508367 0.861140i \(-0.669751\pi\)
−0.508367 + 0.861140i \(0.669751\pi\)
\(440\) −18000.0 −1.95026
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) 1828.00 0.196052 0.0980258 0.995184i \(-0.468747\pi\)
0.0980258 + 0.995184i \(0.468747\pi\)
\(444\) 924.000 0.0987637
\(445\) 15560.0 1.65756
\(446\) 840.000 0.0891820
\(447\) −7548.00 −0.798676
\(448\) −1169.00 −0.123281
\(449\) 7334.00 0.770853 0.385426 0.922739i \(-0.374054\pi\)
0.385426 + 0.922739i \(0.374054\pi\)
\(450\) −2475.00 −0.259272
\(451\) −8280.00 −0.864501
\(452\) 13426.0 1.39714
\(453\) −11040.0 −1.14504
\(454\) 4566.00 0.472011
\(455\) −9520.00 −0.980889
\(456\) −6300.00 −0.646984
\(457\) −5370.00 −0.549667 −0.274834 0.961492i \(-0.588623\pi\)
−0.274834 + 0.961492i \(0.588623\pi\)
\(458\) −1236.00 −0.126102
\(459\) 0 0
\(460\) −24640.0 −2.49749
\(461\) 14284.0 1.44311 0.721553 0.692359i \(-0.243428\pi\)
0.721553 + 0.692359i \(0.243428\pi\)
\(462\) 2520.00 0.253768
\(463\) −11272.0 −1.13143 −0.565717 0.824599i \(-0.691401\pi\)
−0.565717 + 0.824599i \(0.691401\pi\)
\(464\) 3690.00 0.369190
\(465\) −23520.0 −2.34562
\(466\) 1386.00 0.137779
\(467\) 16722.0 1.65696 0.828482 0.560016i \(-0.189205\pi\)
0.828482 + 0.560016i \(0.189205\pi\)
\(468\) 4284.00 0.423137
\(469\) −4172.00 −0.410757
\(470\) 240.000 0.0235540
\(471\) −12936.0 −1.26552
\(472\) −810.000 −0.0789900
\(473\) −19680.0 −1.91308
\(474\) 2880.00 0.279078
\(475\) −19250.0 −1.85947
\(476\) 0 0
\(477\) −2106.00 −0.202153
\(478\) −6392.00 −0.611639
\(479\) 6972.00 0.665050 0.332525 0.943094i \(-0.392099\pi\)
0.332525 + 0.943094i \(0.392099\pi\)
\(480\) −19320.0 −1.83715
\(481\) 1496.00 0.141812
\(482\) 990.000 0.0935545
\(483\) 7392.00 0.696372
\(484\) −15883.0 −1.49164
\(485\) −22840.0 −2.13837
\(486\) 2430.00 0.226805
\(487\) −17936.0 −1.66891 −0.834454 0.551078i \(-0.814217\pi\)
−0.834454 + 0.551078i \(0.814217\pi\)
\(488\) −660.000 −0.0612229
\(489\) 10152.0 0.938833
\(490\) −980.000 −0.0903508
\(491\) 8780.00 0.806998 0.403499 0.914980i \(-0.367794\pi\)
0.403499 + 0.914980i \(0.367794\pi\)
\(492\) −5796.00 −0.531105
\(493\) 0 0
\(494\) −4760.00 −0.433527
\(495\) −10800.0 −0.980654
\(496\) −8036.00 −0.727474
\(497\) −1400.00 −0.126355
\(498\) 5028.00 0.452430
\(499\) 5080.00 0.455736 0.227868 0.973692i \(-0.426825\pi\)
0.227868 + 0.973692i \(0.426825\pi\)
\(500\) −21000.0 −1.87830
\(501\) 15864.0 1.41467
\(502\) −4398.00 −0.391021
\(503\) 1392.00 0.123392 0.0616960 0.998095i \(-0.480349\pi\)
0.0616960 + 0.998095i \(0.480349\pi\)
\(504\) 945.000 0.0835191
\(505\) 640.000 0.0563953
\(506\) 10560.0 0.927765
\(507\) 14562.0 1.27558
\(508\) −12824.0 −1.12003
\(509\) −11204.0 −0.975655 −0.487828 0.872940i \(-0.662211\pi\)
−0.487828 + 0.872940i \(0.662211\pi\)
\(510\) 0 0
\(511\) −7854.00 −0.679923
\(512\) −11521.0 −0.994455
\(513\) 7560.00 0.650647
\(514\) −3018.00 −0.258985
\(515\) −26000.0 −2.22465
\(516\) −13776.0 −1.17530
\(517\) 720.000 0.0612487
\(518\) 154.000 0.0130625
\(519\) 10848.0 0.917484
\(520\) −20400.0 −1.72038
\(521\) 2962.00 0.249074 0.124537 0.992215i \(-0.460255\pi\)
0.124537 + 0.992215i \(0.460255\pi\)
\(522\) −810.000 −0.0679171
\(523\) −4014.00 −0.335602 −0.167801 0.985821i \(-0.553667\pi\)
−0.167801 + 0.985821i \(0.553667\pi\)
\(524\) 1834.00 0.152898
\(525\) 11550.0 0.960159
\(526\) 7384.00 0.612087
\(527\) 0 0
\(528\) −14760.0 −1.21657
\(529\) 18809.0 1.54590
\(530\) 4680.00 0.383559
\(531\) −486.000 −0.0397187
\(532\) 3430.00 0.279529
\(533\) −9384.00 −0.762601
\(534\) −4668.00 −0.378285
\(535\) 5120.00 0.413751
\(536\) −8940.00 −0.720428
\(537\) −10296.0 −0.827384
\(538\) 1408.00 0.112831
\(539\) −2940.00 −0.234944
\(540\) 15120.0 1.20493
\(541\) −19866.0 −1.57875 −0.789377 0.613909i \(-0.789596\pi\)
−0.789377 + 0.613909i \(0.789596\pi\)
\(542\) −1952.00 −0.154697
\(543\) −19824.0 −1.56672
\(544\) 0 0
\(545\) −6360.00 −0.499876
\(546\) 2856.00 0.223856
\(547\) 14084.0 1.10089 0.550446 0.834871i \(-0.314458\pi\)
0.550446 + 0.834871i \(0.314458\pi\)
\(548\) 13314.0 1.03786
\(549\) −396.000 −0.0307848
\(550\) 16500.0 1.27920
\(551\) −6300.00 −0.487094
\(552\) 15840.0 1.22137
\(553\) −3360.00 −0.258376
\(554\) 4274.00 0.327771
\(555\) −2640.00 −0.201913
\(556\) 574.000 0.0437824
\(557\) 5382.00 0.409412 0.204706 0.978823i \(-0.434376\pi\)
0.204706 + 0.978823i \(0.434376\pi\)
\(558\) 1764.00 0.133828
\(559\) −22304.0 −1.68758
\(560\) 5740.00 0.433142
\(561\) 0 0
\(562\) 150.000 0.0112587
\(563\) 3662.00 0.274129 0.137065 0.990562i \(-0.456233\pi\)
0.137065 + 0.990562i \(0.456233\pi\)
\(564\) 504.000 0.0376281
\(565\) −38360.0 −2.85631
\(566\) −586.000 −0.0435184
\(567\) −6237.00 −0.461957
\(568\) −3000.00 −0.221615
\(569\) −22154.0 −1.63224 −0.816120 0.577883i \(-0.803879\pi\)
−0.816120 + 0.577883i \(0.803879\pi\)
\(570\) 8400.00 0.617258
\(571\) 6764.00 0.495735 0.247867 0.968794i \(-0.420270\pi\)
0.247867 + 0.968794i \(0.420270\pi\)
\(572\) −28560.0 −2.08768
\(573\) 24528.0 1.78826
\(574\) −966.000 −0.0702440
\(575\) 48400.0 3.51029
\(576\) −1503.00 −0.108724
\(577\) 7738.00 0.558297 0.279148 0.960248i \(-0.409948\pi\)
0.279148 + 0.960248i \(0.409948\pi\)
\(578\) 0 0
\(579\) −18228.0 −1.30834
\(580\) −12600.0 −0.902046
\(581\) −5866.00 −0.418869
\(582\) 6852.00 0.488015
\(583\) 14040.0 0.997388
\(584\) −16830.0 −1.19252
\(585\) −12240.0 −0.865063
\(586\) −1248.00 −0.0879768
\(587\) 18334.0 1.28914 0.644570 0.764545i \(-0.277036\pi\)
0.644570 + 0.764545i \(0.277036\pi\)
\(588\) −2058.00 −0.144338
\(589\) 13720.0 0.959801
\(590\) 1080.00 0.0753608
\(591\) 7380.00 0.513659
\(592\) −902.000 −0.0626216
\(593\) −7206.00 −0.499013 −0.249507 0.968373i \(-0.580268\pi\)
−0.249507 + 0.968373i \(0.580268\pi\)
\(594\) −6480.00 −0.447605
\(595\) 0 0
\(596\) 8806.00 0.605214
\(597\) −7656.00 −0.524856
\(598\) 11968.0 0.818408
\(599\) 4176.00 0.284853 0.142426 0.989805i \(-0.454510\pi\)
0.142426 + 0.989805i \(0.454510\pi\)
\(600\) 24750.0 1.68402
\(601\) 18158.0 1.23241 0.616207 0.787585i \(-0.288669\pi\)
0.616207 + 0.787585i \(0.288669\pi\)
\(602\) −2296.00 −0.155445
\(603\) −5364.00 −0.362254
\(604\) 12880.0 0.867682
\(605\) 45380.0 3.04952
\(606\) −192.000 −0.0128704
\(607\) 2896.00 0.193649 0.0968246 0.995301i \(-0.469131\pi\)
0.0968246 + 0.995301i \(0.469131\pi\)
\(608\) 11270.0 0.751742
\(609\) 3780.00 0.251516
\(610\) 880.000 0.0584101
\(611\) 816.000 0.0540292
\(612\) 0 0
\(613\) 4762.00 0.313761 0.156880 0.987618i \(-0.449856\pi\)
0.156880 + 0.987618i \(0.449856\pi\)
\(614\) 6390.00 0.419999
\(615\) 16560.0 1.08579
\(616\) −6300.00 −0.412069
\(617\) −10158.0 −0.662797 −0.331398 0.943491i \(-0.607520\pi\)
−0.331398 + 0.943491i \(0.607520\pi\)
\(618\) 7800.00 0.507706
\(619\) 10050.0 0.652574 0.326287 0.945271i \(-0.394202\pi\)
0.326287 + 0.945271i \(0.394202\pi\)
\(620\) 27440.0 1.77745
\(621\) −19008.0 −1.22828
\(622\) −3800.00 −0.244962
\(623\) 5446.00 0.350224
\(624\) −16728.0 −1.07317
\(625\) 25625.0 1.64000
\(626\) −3274.00 −0.209034
\(627\) 25200.0 1.60509
\(628\) 15092.0 0.958975
\(629\) 0 0
\(630\) −1260.00 −0.0796819
\(631\) 6352.00 0.400743 0.200372 0.979720i \(-0.435785\pi\)
0.200372 + 0.979720i \(0.435785\pi\)
\(632\) −7200.00 −0.453166
\(633\) −3792.00 −0.238102
\(634\) −6838.00 −0.428346
\(635\) 36640.0 2.28979
\(636\) 9828.00 0.612745
\(637\) −3332.00 −0.207251
\(638\) 5400.00 0.335091
\(639\) −1800.00 −0.111435
\(640\) 29100.0 1.79731
\(641\) −8334.00 −0.513531 −0.256765 0.966474i \(-0.582657\pi\)
−0.256765 + 0.966474i \(0.582657\pi\)
\(642\) −1536.00 −0.0944254
\(643\) −4634.00 −0.284210 −0.142105 0.989852i \(-0.545387\pi\)
−0.142105 + 0.989852i \(0.545387\pi\)
\(644\) −8624.00 −0.527691
\(645\) 39360.0 2.40279
\(646\) 0 0
\(647\) 6164.00 0.374547 0.187273 0.982308i \(-0.440035\pi\)
0.187273 + 0.982308i \(0.440035\pi\)
\(648\) −13365.0 −0.810227
\(649\) 3240.00 0.195965
\(650\) 18700.0 1.12842
\(651\) −8232.00 −0.495603
\(652\) −11844.0 −0.711422
\(653\) −31566.0 −1.89169 −0.945845 0.324620i \(-0.894764\pi\)
−0.945845 + 0.324620i \(0.894764\pi\)
\(654\) 1908.00 0.114081
\(655\) −5240.00 −0.312586
\(656\) 5658.00 0.336750
\(657\) −10098.0 −0.599635
\(658\) 84.0000 0.00497669
\(659\) 18352.0 1.08481 0.542407 0.840116i \(-0.317513\pi\)
0.542407 + 0.840116i \(0.317513\pi\)
\(660\) 50400.0 2.97245
\(661\) 296.000 0.0174176 0.00870882 0.999962i \(-0.497228\pi\)
0.00870882 + 0.999962i \(0.497228\pi\)
\(662\) 6104.00 0.358367
\(663\) 0 0
\(664\) −12570.0 −0.734655
\(665\) −9800.00 −0.571470
\(666\) 198.000 0.0115200
\(667\) 15840.0 0.919531
\(668\) −18508.0 −1.07200
\(669\) −5040.00 −0.291267
\(670\) 11920.0 0.687328
\(671\) 2640.00 0.151887
\(672\) −6762.00 −0.388169
\(673\) −15794.0 −0.904627 −0.452313 0.891859i \(-0.649401\pi\)
−0.452313 + 0.891859i \(0.649401\pi\)
\(674\) 4678.00 0.267344
\(675\) −29700.0 −1.69356
\(676\) −16989.0 −0.966602
\(677\) 16392.0 0.930570 0.465285 0.885161i \(-0.345952\pi\)
0.465285 + 0.885161i \(0.345952\pi\)
\(678\) 11508.0 0.651861
\(679\) −7994.00 −0.451814
\(680\) 0 0
\(681\) −27396.0 −1.54158
\(682\) −11760.0 −0.660284
\(683\) −10880.0 −0.609534 −0.304767 0.952427i \(-0.598579\pi\)
−0.304767 + 0.952427i \(0.598579\pi\)
\(684\) 4410.00 0.246521
\(685\) −38040.0 −2.12180
\(686\) −343.000 −0.0190901
\(687\) 7416.00 0.411846
\(688\) 13448.0 0.745204
\(689\) 15912.0 0.879824
\(690\) −21120.0 −1.16525
\(691\) 10130.0 0.557689 0.278845 0.960336i \(-0.410049\pi\)
0.278845 + 0.960336i \(0.410049\pi\)
\(692\) −12656.0 −0.695244
\(693\) −3780.00 −0.207201
\(694\) −7396.00 −0.404536
\(695\) −1640.00 −0.0895090
\(696\) 8100.00 0.441135
\(697\) 0 0
\(698\) −4020.00 −0.217993
\(699\) −8316.00 −0.449986
\(700\) −13475.0 −0.727582
\(701\) −12558.0 −0.676618 −0.338309 0.941035i \(-0.609855\pi\)
−0.338309 + 0.941035i \(0.609855\pi\)
\(702\) −7344.00 −0.394845
\(703\) 1540.00 0.0826205
\(704\) 10020.0 0.536425
\(705\) −1440.00 −0.0769270
\(706\) 5622.00 0.299698
\(707\) 224.000 0.0119157
\(708\) 2268.00 0.120391
\(709\) 19282.0 1.02137 0.510685 0.859768i \(-0.329392\pi\)
0.510685 + 0.859768i \(0.329392\pi\)
\(710\) 4000.00 0.211433
\(711\) −4320.00 −0.227866
\(712\) 11670.0 0.614258
\(713\) −34496.0 −1.81190
\(714\) 0 0
\(715\) 81600.0 4.26807
\(716\) 12012.0 0.626969
\(717\) 38352.0 1.99760
\(718\) −7368.00 −0.382968
\(719\) −19012.0 −0.986131 −0.493065 0.869992i \(-0.664124\pi\)
−0.493065 + 0.869992i \(0.664124\pi\)
\(720\) 7380.00 0.381995
\(721\) −9100.00 −0.470044
\(722\) 1959.00 0.100978
\(723\) −5940.00 −0.305548
\(724\) 23128.0 1.18722
\(725\) 24750.0 1.26785
\(726\) −13614.0 −0.695954
\(727\) −32228.0 −1.64411 −0.822057 0.569406i \(-0.807173\pi\)
−0.822057 + 0.569406i \(0.807173\pi\)
\(728\) −7140.00 −0.363497
\(729\) 9477.00 0.481481
\(730\) 22440.0 1.13773
\(731\) 0 0
\(732\) 1848.00 0.0933115
\(733\) 31096.0 1.56693 0.783463 0.621438i \(-0.213451\pi\)
0.783463 + 0.621438i \(0.213451\pi\)
\(734\) 11080.0 0.557180
\(735\) 5880.00 0.295084
\(736\) −28336.0 −1.41913
\(737\) 35760.0 1.78730
\(738\) −1242.00 −0.0619494
\(739\) −7928.00 −0.394636 −0.197318 0.980340i \(-0.563223\pi\)
−0.197318 + 0.980340i \(0.563223\pi\)
\(740\) 3080.00 0.153004
\(741\) 28560.0 1.41589
\(742\) 1638.00 0.0810416
\(743\) 26640.0 1.31538 0.657690 0.753289i \(-0.271534\pi\)
0.657690 + 0.753289i \(0.271534\pi\)
\(744\) −17640.0 −0.869239
\(745\) −25160.0 −1.23730
\(746\) −2142.00 −0.105126
\(747\) −7542.00 −0.369407
\(748\) 0 0
\(749\) 1792.00 0.0874209
\(750\) −18000.0 −0.876356
\(751\) −10904.0 −0.529817 −0.264909 0.964274i \(-0.585342\pi\)
−0.264909 + 0.964274i \(0.585342\pi\)
\(752\) −492.000 −0.0238582
\(753\) 26388.0 1.27707
\(754\) 6120.00 0.295593
\(755\) −36800.0 −1.77389
\(756\) 5292.00 0.254588
\(757\) −28406.0 −1.36385 −0.681924 0.731423i \(-0.738857\pi\)
−0.681924 + 0.731423i \(0.738857\pi\)
\(758\) 7612.00 0.364750
\(759\) −63360.0 −3.03007
\(760\) −21000.0 −1.00230
\(761\) 9590.00 0.456816 0.228408 0.973565i \(-0.426648\pi\)
0.228408 + 0.973565i \(0.426648\pi\)
\(762\) −10992.0 −0.522570
\(763\) −2226.00 −0.105618
\(764\) −28616.0 −1.35509
\(765\) 0 0
\(766\) −7572.00 −0.357164
\(767\) 3672.00 0.172866
\(768\) −714.000 −0.0335472
\(769\) −33338.0 −1.56333 −0.781664 0.623700i \(-0.785629\pi\)
−0.781664 + 0.623700i \(0.785629\pi\)
\(770\) 8400.00 0.393136
\(771\) 18108.0 0.845841
\(772\) 21266.0 0.991425
\(773\) −34496.0 −1.60509 −0.802545 0.596591i \(-0.796521\pi\)
−0.802545 + 0.596591i \(0.796521\pi\)
\(774\) −2952.00 −0.137090
\(775\) −53900.0 −2.49825
\(776\) −17130.0 −0.792437
\(777\) −924.000 −0.0426619
\(778\) −6858.00 −0.316030
\(779\) −9660.00 −0.444295
\(780\) 57120.0 2.62208
\(781\) 12000.0 0.549800
\(782\) 0 0
\(783\) −9720.00 −0.443633
\(784\) 2009.00 0.0915179
\(785\) −43120.0 −1.96053
\(786\) 1572.00 0.0713376
\(787\) −13582.0 −0.615179 −0.307590 0.951519i \(-0.599522\pi\)
−0.307590 + 0.951519i \(0.599522\pi\)
\(788\) −8610.00 −0.389237
\(789\) −44304.0 −1.99907
\(790\) 9600.00 0.432345
\(791\) −13426.0 −0.603506
\(792\) −8100.00 −0.363410
\(793\) 2992.00 0.133984
\(794\) 10008.0 0.447318
\(795\) −28080.0 −1.25270
\(796\) 8932.00 0.397722
\(797\) 30244.0 1.34416 0.672081 0.740477i \(-0.265401\pi\)
0.672081 + 0.740477i \(0.265401\pi\)
\(798\) 2940.00 0.130420
\(799\) 0 0
\(800\) −44275.0 −1.95670
\(801\) 7002.00 0.308868
\(802\) −8402.00 −0.369931
\(803\) 67320.0 2.95849
\(804\) 25032.0 1.09802
\(805\) 24640.0 1.07881
\(806\) −13328.0 −0.582455
\(807\) −8448.00 −0.368505
\(808\) 480.000 0.0208989
\(809\) 28906.0 1.25622 0.628109 0.778125i \(-0.283829\pi\)
0.628109 + 0.778125i \(0.283829\pi\)
\(810\) 17820.0 0.773001
\(811\) 5906.00 0.255719 0.127859 0.991792i \(-0.459189\pi\)
0.127859 + 0.991792i \(0.459189\pi\)
\(812\) −4410.00 −0.190592
\(813\) 11712.0 0.505237
\(814\) −1320.00 −0.0568378
\(815\) 33840.0 1.45443
\(816\) 0 0
\(817\) −22960.0 −0.983193
\(818\) −13382.0 −0.571993
\(819\) −4284.00 −0.182778
\(820\) −19320.0 −0.822785
\(821\) 11430.0 0.485883 0.242941 0.970041i \(-0.421888\pi\)
0.242941 + 0.970041i \(0.421888\pi\)
\(822\) 11412.0 0.484233
\(823\) 13168.0 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −19500.0 −0.824411
\(825\) −99000.0 −4.17786
\(826\) 378.000 0.0159229
\(827\) 24600.0 1.03437 0.517186 0.855873i \(-0.326980\pi\)
0.517186 + 0.855873i \(0.326980\pi\)
\(828\) −11088.0 −0.465380
\(829\) 25328.0 1.06113 0.530566 0.847644i \(-0.321980\pi\)
0.530566 + 0.847644i \(0.321980\pi\)
\(830\) 16760.0 0.700901
\(831\) −25644.0 −1.07049
\(832\) 11356.0 0.473195
\(833\) 0 0
\(834\) 492.000 0.0204275
\(835\) 52880.0 2.19160
\(836\) −29400.0 −1.21629
\(837\) 21168.0 0.874161
\(838\) −14662.0 −0.604404
\(839\) 11596.0 0.477161 0.238581 0.971123i \(-0.423318\pi\)
0.238581 + 0.971123i \(0.423318\pi\)
\(840\) 12600.0 0.517549
\(841\) −16289.0 −0.667883
\(842\) 7370.00 0.301647
\(843\) −900.000 −0.0367706
\(844\) 4424.00 0.180427
\(845\) 48540.0 1.97613
\(846\) 108.000 0.00438903
\(847\) 15883.0 0.644329
\(848\) −9594.00 −0.388513
\(849\) 3516.00 0.142131
\(850\) 0 0
\(851\) −3872.00 −0.155970
\(852\) 8400.00 0.337769
\(853\) −39176.0 −1.57252 −0.786261 0.617895i \(-0.787986\pi\)
−0.786261 + 0.617895i \(0.787986\pi\)
\(854\) 308.000 0.0123414
\(855\) −12600.0 −0.503989
\(856\) 3840.00 0.153328
\(857\) 25106.0 1.00071 0.500353 0.865822i \(-0.333204\pi\)
0.500353 + 0.865822i \(0.333204\pi\)
\(858\) −24480.0 −0.974048
\(859\) 9406.00 0.373607 0.186804 0.982397i \(-0.440187\pi\)
0.186804 + 0.982397i \(0.440187\pi\)
\(860\) −45920.0 −1.82077
\(861\) 5796.00 0.229416
\(862\) 3480.00 0.137505
\(863\) −32240.0 −1.27168 −0.635841 0.771820i \(-0.719347\pi\)
−0.635841 + 0.771820i \(0.719347\pi\)
\(864\) 17388.0 0.684666
\(865\) 36160.0 1.42136
\(866\) −13070.0 −0.512860
\(867\) 0 0
\(868\) 9604.00 0.375554
\(869\) 28800.0 1.12425
\(870\) −10800.0 −0.420867
\(871\) 40528.0 1.57662
\(872\) −4770.00 −0.185244
\(873\) −10278.0 −0.398462
\(874\) 12320.0 0.476808
\(875\) 21000.0 0.811348
\(876\) 47124.0 1.81755
\(877\) 14394.0 0.554220 0.277110 0.960838i \(-0.410623\pi\)
0.277110 + 0.960838i \(0.410623\pi\)
\(878\) 9352.00 0.359470
\(879\) 7488.00 0.287331
\(880\) −49200.0 −1.88470
\(881\) −10754.0 −0.411250 −0.205625 0.978631i \(-0.565923\pi\)
−0.205625 + 0.978631i \(0.565923\pi\)
\(882\) −441.000 −0.0168359
\(883\) 37284.0 1.42096 0.710479 0.703718i \(-0.248478\pi\)
0.710479 + 0.703718i \(0.248478\pi\)
\(884\) 0 0
\(885\) −6480.00 −0.246127
\(886\) −1828.00 −0.0693147
\(887\) 29388.0 1.11246 0.556230 0.831028i \(-0.312247\pi\)
0.556230 + 0.831028i \(0.312247\pi\)
\(888\) −1980.00 −0.0748248
\(889\) 12824.0 0.483806
\(890\) −15560.0 −0.586036
\(891\) 53460.0 2.01008
\(892\) 5880.00 0.220714
\(893\) 840.000 0.0314776
\(894\) 7548.00 0.282375
\(895\) −34320.0 −1.28178
\(896\) 10185.0 0.379751
\(897\) −71808.0 −2.67291
\(898\) −7334.00 −0.272538
\(899\) −17640.0 −0.654424
\(900\) −17325.0 −0.641667
\(901\) 0 0
\(902\) 8280.00 0.305647
\(903\) 13776.0 0.507682
\(904\) −28770.0 −1.05849
\(905\) −66080.0 −2.42715
\(906\) 11040.0 0.404834
\(907\) 27544.0 1.00836 0.504181 0.863598i \(-0.331795\pi\)
0.504181 + 0.863598i \(0.331795\pi\)
\(908\) 31962.0 1.16817
\(909\) 288.000 0.0105086
\(910\) 9520.00 0.346797
\(911\) −344.000 −0.0125107 −0.00625534 0.999980i \(-0.501991\pi\)
−0.00625534 + 0.999980i \(0.501991\pi\)
\(912\) −17220.0 −0.625232
\(913\) 50280.0 1.82259
\(914\) 5370.00 0.194337
\(915\) −5280.00 −0.190767
\(916\) −8652.00 −0.312085
\(917\) −1834.00 −0.0660458
\(918\) 0 0
\(919\) 16152.0 0.579766 0.289883 0.957062i \(-0.406384\pi\)
0.289883 + 0.957062i \(0.406384\pi\)
\(920\) 52800.0 1.89214
\(921\) −38340.0 −1.37171
\(922\) −14284.0 −0.510215
\(923\) 13600.0 0.484994
\(924\) 17640.0 0.628045
\(925\) −6050.00 −0.215052
\(926\) 11272.0 0.400023
\(927\) −11700.0 −0.414540
\(928\) −14490.0 −0.512562
\(929\) −20910.0 −0.738466 −0.369233 0.929337i \(-0.620380\pi\)
−0.369233 + 0.929337i \(0.620380\pi\)
\(930\) 23520.0 0.829302
\(931\) −3430.00 −0.120745
\(932\) 9702.00 0.340987
\(933\) 22800.0 0.800041
\(934\) −16722.0 −0.585825
\(935\) 0 0
\(936\) −9180.00 −0.320574
\(937\) 26074.0 0.909072 0.454536 0.890728i \(-0.349805\pi\)
0.454536 + 0.890728i \(0.349805\pi\)
\(938\) 4172.00 0.145225
\(939\) 19644.0 0.682703
\(940\) 1680.00 0.0582931
\(941\) −36204.0 −1.25422 −0.627108 0.778933i \(-0.715761\pi\)
−0.627108 + 0.778933i \(0.715761\pi\)
\(942\) 12936.0 0.447429
\(943\) 24288.0 0.838734
\(944\) −2214.00 −0.0763343
\(945\) −15120.0 −0.520480
\(946\) 19680.0 0.676376
\(947\) −12956.0 −0.444576 −0.222288 0.974981i \(-0.571353\pi\)
−0.222288 + 0.974981i \(0.571353\pi\)
\(948\) 20160.0 0.690682
\(949\) 76296.0 2.60977
\(950\) 19250.0 0.657424
\(951\) 41028.0 1.39897
\(952\) 0 0
\(953\) 51050.0 1.73523 0.867614 0.497239i \(-0.165653\pi\)
0.867614 + 0.497239i \(0.165653\pi\)
\(954\) 2106.00 0.0714720
\(955\) 81760.0 2.77036
\(956\) −44744.0 −1.51373
\(957\) −32400.0 −1.09440
\(958\) −6972.00 −0.235131
\(959\) −13314.0 −0.448312
\(960\) −20040.0 −0.673738
\(961\) 8625.00 0.289517
\(962\) −1496.00 −0.0501382
\(963\) 2304.00 0.0770980
\(964\) 6930.00 0.231536
\(965\) −60760.0 −2.02687
\(966\) −7392.00 −0.246205
\(967\) 28848.0 0.959348 0.479674 0.877447i \(-0.340755\pi\)
0.479674 + 0.877447i \(0.340755\pi\)
\(968\) 34035.0 1.13009
\(969\) 0 0
\(970\) 22840.0 0.756029
\(971\) −24294.0 −0.802916 −0.401458 0.915877i \(-0.631496\pi\)
−0.401458 + 0.915877i \(0.631496\pi\)
\(972\) 17010.0 0.561313
\(973\) −574.000 −0.0189122
\(974\) 17936.0 0.590048
\(975\) −112200. −3.68541
\(976\) −1804.00 −0.0591646
\(977\) 18090.0 0.592375 0.296187 0.955130i \(-0.404285\pi\)
0.296187 + 0.955130i \(0.404285\pi\)
\(978\) −10152.0 −0.331928
\(979\) −46680.0 −1.52390
\(980\) −6860.00 −0.223607
\(981\) −2862.00 −0.0931464
\(982\) −8780.00 −0.285317
\(983\) 44988.0 1.45971 0.729855 0.683602i \(-0.239588\pi\)
0.729855 + 0.683602i \(0.239588\pi\)
\(984\) 12420.0 0.402373
\(985\) 24600.0 0.795758
\(986\) 0 0
\(987\) −504.000 −0.0162538
\(988\) −33320.0 −1.07293
\(989\) 57728.0 1.85606
\(990\) 10800.0 0.346714
\(991\) −22632.0 −0.725458 −0.362729 0.931895i \(-0.618155\pi\)
−0.362729 + 0.931895i \(0.618155\pi\)
\(992\) 31556.0 1.00998
\(993\) −36624.0 −1.17042
\(994\) 1400.00 0.0446733
\(995\) −25520.0 −0.813104
\(996\) 35196.0 1.11971
\(997\) 30788.0 0.978000 0.489000 0.872284i \(-0.337362\pi\)
0.489000 + 0.872284i \(0.337362\pi\)
\(998\) −5080.00 −0.161127
\(999\) 2376.00 0.0752486
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.b.1.1 1
17.16 even 2 119.4.a.a.1.1 1
51.50 odd 2 1071.4.a.b.1.1 1
68.67 odd 2 1904.4.a.a.1.1 1
119.118 odd 2 833.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
119.4.a.a.1.1 1 17.16 even 2
833.4.a.b.1.1 1 119.118 odd 2
1071.4.a.b.1.1 1 51.50 odd 2
1904.4.a.a.1.1 1 68.67 odd 2
2023.4.a.b.1.1 1 1.1 even 1 trivial