Properties

Label 507.4.a.b.1.1
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9139683729\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} +7.00000 q^{5} -3.00000 q^{6} -10.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} +7.00000 q^{5} -3.00000 q^{6} -10.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} -7.00000 q^{10} -22.0000 q^{11} -21.0000 q^{12} +10.0000 q^{14} +21.0000 q^{15} +41.0000 q^{16} +37.0000 q^{17} -9.00000 q^{18} +30.0000 q^{19} -49.0000 q^{20} -30.0000 q^{21} +22.0000 q^{22} -162.000 q^{23} +45.0000 q^{24} -76.0000 q^{25} +27.0000 q^{27} +70.0000 q^{28} -113.000 q^{29} -21.0000 q^{30} +196.000 q^{31} -161.000 q^{32} -66.0000 q^{33} -37.0000 q^{34} -70.0000 q^{35} -63.0000 q^{36} +13.0000 q^{37} -30.0000 q^{38} +105.000 q^{40} +285.000 q^{41} +30.0000 q^{42} -246.000 q^{43} +154.000 q^{44} +63.0000 q^{45} +162.000 q^{46} -462.000 q^{47} +123.000 q^{48} -243.000 q^{49} +76.0000 q^{50} +111.000 q^{51} -537.000 q^{53} -27.0000 q^{54} -154.000 q^{55} -150.000 q^{56} +90.0000 q^{57} +113.000 q^{58} +576.000 q^{59} -147.000 q^{60} -635.000 q^{61} -196.000 q^{62} -90.0000 q^{63} -167.000 q^{64} +66.0000 q^{66} +202.000 q^{67} -259.000 q^{68} -486.000 q^{69} +70.0000 q^{70} -1086.00 q^{71} +135.000 q^{72} -805.000 q^{73} -13.0000 q^{74} -228.000 q^{75} -210.000 q^{76} +220.000 q^{77} +884.000 q^{79} +287.000 q^{80} +81.0000 q^{81} -285.000 q^{82} +518.000 q^{83} +210.000 q^{84} +259.000 q^{85} +246.000 q^{86} -339.000 q^{87} -330.000 q^{88} +194.000 q^{89} -63.0000 q^{90} +1134.00 q^{92} +588.000 q^{93} +462.000 q^{94} +210.000 q^{95} -483.000 q^{96} -1202.00 q^{97} +243.000 q^{98} -198.000 q^{99} +O(q^{100})\)

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 \(0.556567\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.00000 −0.875000
\(5\) 7.00000 0.626099 0.313050 0.949737i \(-0.398649\pi\)
0.313050 + 0.949737i \(0.398649\pi\)
\(6\) −3.00000 −0.204124
\(7\) −10.0000 −0.539949 −0.269975 0.962867i \(-0.587015\pi\)
−0.269975 + 0.962867i \(0.587015\pi\)
\(8\) 15.0000 0.662913
\(9\) 9.00000 0.333333
\(10\) −7.00000 −0.221359
\(11\) −22.0000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −21.0000 −0.505181
\(13\) 0 0
\(14\) 10.0000 0.190901
\(15\) 21.0000 0.361478
\(16\) 41.0000 0.640625
\(17\) 37.0000 0.527872 0.263936 0.964540i \(-0.414979\pi\)
0.263936 + 0.964540i \(0.414979\pi\)
\(18\) −9.00000 −0.117851
\(19\) 30.0000 0.362235 0.181118 0.983461i \(-0.442029\pi\)
0.181118 + 0.983461i \(0.442029\pi\)
\(20\) −49.0000 −0.547837
\(21\) −30.0000 −0.311740
\(22\) 22.0000 0.213201
\(23\) −162.000 −1.46867 −0.734333 0.678789i \(-0.762505\pi\)
−0.734333 + 0.678789i \(0.762505\pi\)
\(24\) 45.0000 0.382733
\(25\) −76.0000 −0.608000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 70.0000 0.472456
\(29\) −113.000 −0.723571 −0.361786 0.932261i \(-0.617833\pi\)
−0.361786 + 0.932261i \(0.617833\pi\)
\(30\) −21.0000 −0.127802
\(31\) 196.000 1.13557 0.567785 0.823177i \(-0.307801\pi\)
0.567785 + 0.823177i \(0.307801\pi\)
\(32\) −161.000 −0.889408
\(33\) −66.0000 −0.348155
\(34\) −37.0000 −0.186631
\(35\) −70.0000 −0.338062
\(36\) −63.0000 −0.291667
\(37\) 13.0000 0.0577618 0.0288809 0.999583i \(-0.490806\pi\)
0.0288809 + 0.999583i \(0.490806\pi\)
\(38\) −30.0000 −0.128070
\(39\) 0 0
\(40\) 105.000 0.415049
\(41\) 285.000 1.08560 0.542799 0.839863i \(-0.317365\pi\)
0.542799 + 0.839863i \(0.317365\pi\)
\(42\) 30.0000 0.110217
\(43\) −246.000 −0.872434 −0.436217 0.899842i \(-0.643682\pi\)
−0.436217 + 0.899842i \(0.643682\pi\)
\(44\) 154.000 0.527645
\(45\) 63.0000 0.208700
\(46\) 162.000 0.519252
\(47\) −462.000 −1.43382 −0.716911 0.697165i \(-0.754445\pi\)
−0.716911 + 0.697165i \(0.754445\pi\)
\(48\) 123.000 0.369865
\(49\) −243.000 −0.708455
\(50\) 76.0000 0.214960
\(51\) 111.000 0.304767
\(52\) 0 0
\(53\) −537.000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(54\) −27.0000 −0.0680414
\(55\) −154.000 −0.377552
\(56\) −150.000 −0.357939
\(57\) 90.0000 0.209137
\(58\) 113.000 0.255821
\(59\) 576.000 1.27100 0.635498 0.772102i \(-0.280795\pi\)
0.635498 + 0.772102i \(0.280795\pi\)
\(60\) −147.000 −0.316294
\(61\) −635.000 −1.33284 −0.666421 0.745575i \(-0.732175\pi\)
−0.666421 + 0.745575i \(0.732175\pi\)
\(62\) −196.000 −0.401484
\(63\) −90.0000 −0.179983
\(64\) −167.000 −0.326172
\(65\) 0 0
\(66\) 66.0000 0.123091
\(67\) 202.000 0.368332 0.184166 0.982895i \(-0.441042\pi\)
0.184166 + 0.982895i \(0.441042\pi\)
\(68\) −259.000 −0.461888
\(69\) −486.000 −0.847935
\(70\) 70.0000 0.119523
\(71\) −1086.00 −1.81527 −0.907637 0.419755i \(-0.862116\pi\)
−0.907637 + 0.419755i \(0.862116\pi\)
\(72\) 135.000 0.220971
\(73\) −805.000 −1.29066 −0.645330 0.763904i \(-0.723280\pi\)
−0.645330 + 0.763904i \(0.723280\pi\)
\(74\) −13.0000 −0.0204219
\(75\) −228.000 −0.351029
\(76\) −210.000 −0.316956
\(77\) 220.000 0.325602
\(78\) 0 0
\(79\) 884.000 1.25896 0.629480 0.777017i \(-0.283268\pi\)
0.629480 + 0.777017i \(0.283268\pi\)
\(80\) 287.000 0.401095
\(81\) 81.0000 0.111111
\(82\) −285.000 −0.383817
\(83\) 518.000 0.685035 0.342517 0.939511i \(-0.388720\pi\)
0.342517 + 0.939511i \(0.388720\pi\)
\(84\) 210.000 0.272772
\(85\) 259.000 0.330500
\(86\) 246.000 0.308452
\(87\) −339.000 −0.417754
\(88\) −330.000 −0.399751
\(89\) 194.000 0.231056 0.115528 0.993304i \(-0.463144\pi\)
0.115528 + 0.993304i \(0.463144\pi\)
\(90\) −63.0000 −0.0737865
\(91\) 0 0
\(92\) 1134.00 1.28508
\(93\) 588.000 0.655621
\(94\) 462.000 0.506933
\(95\) 210.000 0.226795
\(96\) −483.000 −0.513500
\(97\) −1202.00 −1.25819 −0.629096 0.777328i \(-0.716575\pi\)
−0.629096 + 0.777328i \(0.716575\pi\)
\(98\) 243.000 0.250477
\(99\) −198.000 −0.201008
\(100\) 532.000 0.532000
\(101\) −429.000 −0.422645 −0.211322 0.977416i \(-0.567777\pi\)
−0.211322 + 0.977416i \(0.567777\pi\)
\(102\) −111.000 −0.107751
\(103\) −1302.00 −1.24553 −0.622766 0.782408i \(-0.713991\pi\)
−0.622766 + 0.782408i \(0.713991\pi\)
\(104\) 0 0
\(105\) −210.000 −0.195180
\(106\) 537.000 0.492057
\(107\) −1338.00 −1.20887 −0.604436 0.796654i \(-0.706602\pi\)
−0.604436 + 0.796654i \(0.706602\pi\)
\(108\) −189.000 −0.168394
\(109\) −1034.00 −0.908617 −0.454308 0.890844i \(-0.650114\pi\)
−0.454308 + 0.890844i \(0.650114\pi\)
\(110\) 154.000 0.133485
\(111\) 39.0000 0.0333488
\(112\) −410.000 −0.345905
\(113\) 1077.00 0.896599 0.448299 0.893884i \(-0.352030\pi\)
0.448299 + 0.893884i \(0.352030\pi\)
\(114\) −90.0000 −0.0739410
\(115\) −1134.00 −0.919531
\(116\) 791.000 0.633125
\(117\) 0 0
\(118\) −576.000 −0.449365
\(119\) −370.000 −0.285024
\(120\) 315.000 0.239629
\(121\) −847.000 −0.636364
\(122\) 635.000 0.471231
\(123\) 855.000 0.626770
\(124\) −1372.00 −0.993623
\(125\) −1407.00 −1.00677
\(126\) 90.0000 0.0636336
\(127\) −988.000 −0.690321 −0.345161 0.938544i \(-0.612176\pi\)
−0.345161 + 0.938544i \(0.612176\pi\)
\(128\) 1455.00 1.00473
\(129\) −738.000 −0.503700
\(130\) 0 0
\(131\) 560.000 0.373492 0.186746 0.982408i \(-0.440206\pi\)
0.186746 + 0.982408i \(0.440206\pi\)
\(132\) 462.000 0.304636
\(133\) −300.000 −0.195589
\(134\) −202.000 −0.130225
\(135\) 189.000 0.120493
\(136\) 555.000 0.349933
\(137\) −519.000 −0.323658 −0.161829 0.986819i \(-0.551739\pi\)
−0.161829 + 0.986819i \(0.551739\pi\)
\(138\) 486.000 0.299790
\(139\) −348.000 −0.212352 −0.106176 0.994347i \(-0.533861\pi\)
−0.106176 + 0.994347i \(0.533861\pi\)
\(140\) 490.000 0.295804
\(141\) −1386.00 −0.827817
\(142\) 1086.00 0.641796
\(143\) 0 0
\(144\) 369.000 0.213542
\(145\) −791.000 −0.453027
\(146\) 805.000 0.456317
\(147\) −729.000 −0.409027
\(148\) −91.0000 −0.0505416
\(149\) −645.000 −0.354634 −0.177317 0.984154i \(-0.556742\pi\)
−0.177317 + 0.984154i \(0.556742\pi\)
\(150\) 228.000 0.124107
\(151\) 2914.00 1.57045 0.785225 0.619211i \(-0.212547\pi\)
0.785225 + 0.619211i \(0.212547\pi\)
\(152\) 450.000 0.240130
\(153\) 333.000 0.175957
\(154\) −220.000 −0.115118
\(155\) 1372.00 0.710979
\(156\) 0 0
\(157\) −2079.00 −1.05683 −0.528415 0.848986i \(-0.677213\pi\)
−0.528415 + 0.848986i \(0.677213\pi\)
\(158\) −884.000 −0.445109
\(159\) −1611.00 −0.803526
\(160\) −1127.00 −0.556857
\(161\) 1620.00 0.793006
\(162\) −81.0000 −0.0392837
\(163\) 1700.00 0.816897 0.408449 0.912781i \(-0.366070\pi\)
0.408449 + 0.912781i \(0.366070\pi\)
\(164\) −1995.00 −0.949898
\(165\) −462.000 −0.217980
\(166\) −518.000 −0.242196
\(167\) 3680.00 1.70519 0.852596 0.522571i \(-0.175027\pi\)
0.852596 + 0.522571i \(0.175027\pi\)
\(168\) −450.000 −0.206656
\(169\) 0 0
\(170\) −259.000 −0.116849
\(171\) 270.000 0.120745
\(172\) 1722.00 0.763379
\(173\) 4146.00 1.82205 0.911025 0.412352i \(-0.135293\pi\)
0.911025 + 0.412352i \(0.135293\pi\)
\(174\) 339.000 0.147698
\(175\) 760.000 0.328289
\(176\) −902.000 −0.386311
\(177\) 1728.00 0.733810
\(178\) −194.000 −0.0816905
\(179\) 3674.00 1.53412 0.767060 0.641575i \(-0.221719\pi\)
0.767060 + 0.641575i \(0.221719\pi\)
\(180\) −441.000 −0.182612
\(181\) −3283.00 −1.34820 −0.674098 0.738642i \(-0.735467\pi\)
−0.674098 + 0.738642i \(0.735467\pi\)
\(182\) 0 0
\(183\) −1905.00 −0.769517
\(184\) −2430.00 −0.973598
\(185\) 91.0000 0.0361646
\(186\) −588.000 −0.231797
\(187\) −814.000 −0.318319
\(188\) 3234.00 1.25459
\(189\) −270.000 −0.103913
\(190\) −210.000 −0.0801842
\(191\) −596.000 −0.225786 −0.112893 0.993607i \(-0.536012\pi\)
−0.112893 + 0.993607i \(0.536012\pi\)
\(192\) −501.000 −0.188315
\(193\) −393.000 −0.146574 −0.0732869 0.997311i \(-0.523349\pi\)
−0.0732869 + 0.997311i \(0.523349\pi\)
\(194\) 1202.00 0.444838
\(195\) 0 0
\(196\) 1701.00 0.619898
\(197\) −3522.00 −1.27377 −0.636884 0.770960i \(-0.719777\pi\)
−0.636884 + 0.770960i \(0.719777\pi\)
\(198\) 198.000 0.0710669
\(199\) 2018.00 0.718855 0.359428 0.933173i \(-0.382972\pi\)
0.359428 + 0.933173i \(0.382972\pi\)
\(200\) −1140.00 −0.403051
\(201\) 606.000 0.212656
\(202\) 429.000 0.149427
\(203\) 1130.00 0.390692
\(204\) −777.000 −0.266671
\(205\) 1995.00 0.679692
\(206\) 1302.00 0.440362
\(207\) −1458.00 −0.489556
\(208\) 0 0
\(209\) −660.000 −0.218436
\(210\) 210.000 0.0690066
\(211\) 160.000 0.0522031 0.0261016 0.999659i \(-0.491691\pi\)
0.0261016 + 0.999659i \(0.491691\pi\)
\(212\) 3759.00 1.21778
\(213\) −3258.00 −1.04805
\(214\) 1338.00 0.427401
\(215\) −1722.00 −0.546230
\(216\) 405.000 0.127578
\(217\) −1960.00 −0.613150
\(218\) 1034.00 0.321245
\(219\) −2415.00 −0.745162
\(220\) 1078.00 0.330358
\(221\) 0 0
\(222\) −39.0000 −0.0117906
\(223\) 4072.00 1.22279 0.611393 0.791327i \(-0.290609\pi\)
0.611393 + 0.791327i \(0.290609\pi\)
\(224\) 1610.00 0.480235
\(225\) −684.000 −0.202667
\(226\) −1077.00 −0.316995
\(227\) −5794.00 −1.69410 −0.847051 0.531511i \(-0.821624\pi\)
−0.847051 + 0.531511i \(0.821624\pi\)
\(228\) −630.000 −0.182995
\(229\) 6482.00 1.87049 0.935246 0.353999i \(-0.115178\pi\)
0.935246 + 0.353999i \(0.115178\pi\)
\(230\) 1134.00 0.325103
\(231\) 660.000 0.187986
\(232\) −1695.00 −0.479665
\(233\) 6890.00 1.93725 0.968624 0.248530i \(-0.0799474\pi\)
0.968624 + 0.248530i \(0.0799474\pi\)
\(234\) 0 0
\(235\) −3234.00 −0.897714
\(236\) −4032.00 −1.11212
\(237\) 2652.00 0.726860
\(238\) 370.000 0.100771
\(239\) 2466.00 0.667415 0.333708 0.942677i \(-0.391700\pi\)
0.333708 + 0.942677i \(0.391700\pi\)
\(240\) 861.000 0.231572
\(241\) −3617.00 −0.966770 −0.483385 0.875408i \(-0.660593\pi\)
−0.483385 + 0.875408i \(0.660593\pi\)
\(242\) 847.000 0.224989
\(243\) 243.000 0.0641500
\(244\) 4445.00 1.16624
\(245\) −1701.00 −0.443563
\(246\) −855.000 −0.221597
\(247\) 0 0
\(248\) 2940.00 0.752783
\(249\) 1554.00 0.395505
\(250\) 1407.00 0.355946
\(251\) 4860.00 1.22215 0.611077 0.791571i \(-0.290737\pi\)
0.611077 + 0.791571i \(0.290737\pi\)
\(252\) 630.000 0.157485
\(253\) 3564.00 0.885639
\(254\) 988.000 0.244065
\(255\) 777.000 0.190814
\(256\) −119.000 −0.0290527
\(257\) 565.000 0.137135 0.0685676 0.997646i \(-0.478157\pi\)
0.0685676 + 0.997646i \(0.478157\pi\)
\(258\) 738.000 0.178085
\(259\) −130.000 −0.0311884
\(260\) 0 0
\(261\) −1017.00 −0.241190
\(262\) −560.000 −0.132049
\(263\) −498.000 −0.116760 −0.0583802 0.998294i \(-0.518594\pi\)
−0.0583802 + 0.998294i \(0.518594\pi\)
\(264\) −990.000 −0.230797
\(265\) −3759.00 −0.871372
\(266\) 300.000 0.0691511
\(267\) 582.000 0.133400
\(268\) −1414.00 −0.322290
\(269\) 5546.00 1.25705 0.628523 0.777791i \(-0.283660\pi\)
0.628523 + 0.777791i \(0.283660\pi\)
\(270\) −189.000 −0.0426006
\(271\) −2256.00 −0.505691 −0.252845 0.967507i \(-0.581366\pi\)
−0.252845 + 0.967507i \(0.581366\pi\)
\(272\) 1517.00 0.338168
\(273\) 0 0
\(274\) 519.000 0.114430
\(275\) 1672.00 0.366638
\(276\) 3402.00 0.741943
\(277\) 2309.00 0.500846 0.250423 0.968137i \(-0.419430\pi\)
0.250423 + 0.968137i \(0.419430\pi\)
\(278\) 348.000 0.0750779
\(279\) 1764.00 0.378523
\(280\) −1050.00 −0.224105
\(281\) 5833.00 1.23832 0.619159 0.785265i \(-0.287473\pi\)
0.619159 + 0.785265i \(0.287473\pi\)
\(282\) 1386.00 0.292678
\(283\) 1650.00 0.346581 0.173290 0.984871i \(-0.444560\pi\)
0.173290 + 0.984871i \(0.444560\pi\)
\(284\) 7602.00 1.58837
\(285\) 630.000 0.130940
\(286\) 0 0
\(287\) −2850.00 −0.586168
\(288\) −1449.00 −0.296469
\(289\) −3544.00 −0.721352
\(290\) 791.000 0.160169
\(291\) −3606.00 −0.726417
\(292\) 5635.00 1.12933
\(293\) 2991.00 0.596369 0.298184 0.954508i \(-0.403619\pi\)
0.298184 + 0.954508i \(0.403619\pi\)
\(294\) 729.000 0.144613
\(295\) 4032.00 0.795770
\(296\) 195.000 0.0382910
\(297\) −594.000 −0.116052
\(298\) 645.000 0.125382
\(299\) 0 0
\(300\) 1596.00 0.307150
\(301\) 2460.00 0.471070
\(302\) −2914.00 −0.555238
\(303\) −1287.00 −0.244014
\(304\) 1230.00 0.232057
\(305\) −4445.00 −0.834492
\(306\) −333.000 −0.0622103
\(307\) −2422.00 −0.450263 −0.225132 0.974328i \(-0.572281\pi\)
−0.225132 + 0.974328i \(0.572281\pi\)
\(308\) −1540.00 −0.284901
\(309\) −3906.00 −0.719109
\(310\) −1372.00 −0.251369
\(311\) −3402.00 −0.620288 −0.310144 0.950690i \(-0.600377\pi\)
−0.310144 + 0.950690i \(0.600377\pi\)
\(312\) 0 0
\(313\) 2310.00 0.417153 0.208577 0.978006i \(-0.433117\pi\)
0.208577 + 0.978006i \(0.433117\pi\)
\(314\) 2079.00 0.373646
\(315\) −630.000 −0.112687
\(316\) −6188.00 −1.10159
\(317\) −257.000 −0.0455349 −0.0227674 0.999741i \(-0.507248\pi\)
−0.0227674 + 0.999741i \(0.507248\pi\)
\(318\) 1611.00 0.284089
\(319\) 2486.00 0.436330
\(320\) −1169.00 −0.204216
\(321\) −4014.00 −0.697943
\(322\) −1620.00 −0.280370
\(323\) 1110.00 0.191214
\(324\) −567.000 −0.0972222
\(325\) 0 0
\(326\) −1700.00 −0.288817
\(327\) −3102.00 −0.524590
\(328\) 4275.00 0.719657
\(329\) 4620.00 0.774191
\(330\) 462.000 0.0770675
\(331\) 1028.00 0.170707 0.0853535 0.996351i \(-0.472798\pi\)
0.0853535 + 0.996351i \(0.472798\pi\)
\(332\) −3626.00 −0.599405
\(333\) 117.000 0.0192539
\(334\) −3680.00 −0.602876
\(335\) 1414.00 0.230612
\(336\) −1230.00 −0.199708
\(337\) 2487.00 0.402005 0.201002 0.979591i \(-0.435580\pi\)
0.201002 + 0.979591i \(0.435580\pi\)
\(338\) 0 0
\(339\) 3231.00 0.517651
\(340\) −1813.00 −0.289187
\(341\) −4312.00 −0.684774
\(342\) −270.000 −0.0426898
\(343\) 5860.00 0.922479
\(344\) −3690.00 −0.578347
\(345\) −3402.00 −0.530891
\(346\) −4146.00 −0.644192
\(347\) −2850.00 −0.440911 −0.220455 0.975397i \(-0.570754\pi\)
−0.220455 + 0.975397i \(0.570754\pi\)
\(348\) 2373.00 0.365535
\(349\) −2018.00 −0.309516 −0.154758 0.987952i \(-0.549460\pi\)
−0.154758 + 0.987952i \(0.549460\pi\)
\(350\) −760.000 −0.116068
\(351\) 0 0
\(352\) 3542.00 0.536333
\(353\) −5287.00 −0.797163 −0.398582 0.917133i \(-0.630497\pi\)
−0.398582 + 0.917133i \(0.630497\pi\)
\(354\) −1728.00 −0.259441
\(355\) −7602.00 −1.13654
\(356\) −1358.00 −0.202174
\(357\) −1110.00 −0.164559
\(358\) −3674.00 −0.542394
\(359\) −7278.00 −1.06997 −0.534983 0.844863i \(-0.679682\pi\)
−0.534983 + 0.844863i \(0.679682\pi\)
\(360\) 945.000 0.138350
\(361\) −5959.00 −0.868786
\(362\) 3283.00 0.476659
\(363\) −2541.00 −0.367405
\(364\) 0 0
\(365\) −5635.00 −0.808080
\(366\) 1905.00 0.272065
\(367\) −4202.00 −0.597664 −0.298832 0.954306i \(-0.596597\pi\)
−0.298832 + 0.954306i \(0.596597\pi\)
\(368\) −6642.00 −0.940865
\(369\) 2565.00 0.361866
\(370\) −91.0000 −0.0127861
\(371\) 5370.00 0.751473
\(372\) −4116.00 −0.573668
\(373\) −1583.00 −0.219744 −0.109872 0.993946i \(-0.535044\pi\)
−0.109872 + 0.993946i \(0.535044\pi\)
\(374\) 814.000 0.112543
\(375\) −4221.00 −0.581257
\(376\) −6930.00 −0.950499
\(377\) 0 0
\(378\) 270.000 0.0367389
\(379\) −2052.00 −0.278111 −0.139056 0.990285i \(-0.544407\pi\)
−0.139056 + 0.990285i \(0.544407\pi\)
\(380\) −1470.00 −0.198446
\(381\) −2964.00 −0.398557
\(382\) 596.000 0.0798273
\(383\) −6872.00 −0.916822 −0.458411 0.888740i \(-0.651581\pi\)
−0.458411 + 0.888740i \(0.651581\pi\)
\(384\) 4365.00 0.580079
\(385\) 1540.00 0.203859
\(386\) 393.000 0.0518217
\(387\) −2214.00 −0.290811
\(388\) 8414.00 1.10092
\(389\) −11653.0 −1.51884 −0.759422 0.650598i \(-0.774518\pi\)
−0.759422 + 0.650598i \(0.774518\pi\)
\(390\) 0 0
\(391\) −5994.00 −0.775268
\(392\) −3645.00 −0.469644
\(393\) 1680.00 0.215636
\(394\) 3522.00 0.450345
\(395\) 6188.00 0.788233
\(396\) 1386.00 0.175882
\(397\) 6134.00 0.775458 0.387729 0.921774i \(-0.373260\pi\)
0.387729 + 0.921774i \(0.373260\pi\)
\(398\) −2018.00 −0.254154
\(399\) −900.000 −0.112923
\(400\) −3116.00 −0.389500
\(401\) −10795.0 −1.34433 −0.672165 0.740401i \(-0.734636\pi\)
−0.672165 + 0.740401i \(0.734636\pi\)
\(402\) −606.000 −0.0751854
\(403\) 0 0
\(404\) 3003.00 0.369814
\(405\) 567.000 0.0695666
\(406\) −1130.00 −0.138130
\(407\) −286.000 −0.0348317
\(408\) 1665.00 0.202034
\(409\) −8489.00 −1.02629 −0.513147 0.858301i \(-0.671520\pi\)
−0.513147 + 0.858301i \(0.671520\pi\)
\(410\) −1995.00 −0.240307
\(411\) −1557.00 −0.186864
\(412\) 9114.00 1.08984
\(413\) −5760.00 −0.686274
\(414\) 1458.00 0.173084
\(415\) 3626.00 0.428900
\(416\) 0 0
\(417\) −1044.00 −0.122602
\(418\) 660.000 0.0772288
\(419\) 1496.00 0.174426 0.0872129 0.996190i \(-0.472204\pi\)
0.0872129 + 0.996190i \(0.472204\pi\)
\(420\) 1470.00 0.170783
\(421\) −11695.0 −1.35387 −0.676935 0.736043i \(-0.736692\pi\)
−0.676935 + 0.736043i \(0.736692\pi\)
\(422\) −160.000 −0.0184566
\(423\) −4158.00 −0.477941
\(424\) −8055.00 −0.922607
\(425\) −2812.00 −0.320946
\(426\) 3258.00 0.370541
\(427\) 6350.00 0.719668
\(428\) 9366.00 1.05776
\(429\) 0 0
\(430\) 1722.00 0.193121
\(431\) 10590.0 1.18353 0.591766 0.806110i \(-0.298431\pi\)
0.591766 + 0.806110i \(0.298431\pi\)
\(432\) 1107.00 0.123288
\(433\) −13949.0 −1.54814 −0.774072 0.633098i \(-0.781783\pi\)
−0.774072 + 0.633098i \(0.781783\pi\)
\(434\) 1960.00 0.216781
\(435\) −2373.00 −0.261555
\(436\) 7238.00 0.795040
\(437\) −4860.00 −0.532003
\(438\) 2415.00 0.263455
\(439\) −10726.0 −1.16611 −0.583057 0.812431i \(-0.698144\pi\)
−0.583057 + 0.812431i \(0.698144\pi\)
\(440\) −2310.00 −0.250284
\(441\) −2187.00 −0.236152
\(442\) 0 0
\(443\) 16228.0 1.74044 0.870221 0.492662i \(-0.163976\pi\)
0.870221 + 0.492662i \(0.163976\pi\)
\(444\) −273.000 −0.0291802
\(445\) 1358.00 0.144664
\(446\) −4072.00 −0.432320
\(447\) −1935.00 −0.204748
\(448\) 1670.00 0.176116
\(449\) 7538.00 0.792294 0.396147 0.918187i \(-0.370347\pi\)
0.396147 + 0.918187i \(0.370347\pi\)
\(450\) 684.000 0.0716535
\(451\) −6270.00 −0.654640
\(452\) −7539.00 −0.784524
\(453\) 8742.00 0.906700
\(454\) 5794.00 0.598956
\(455\) 0 0
\(456\) 1350.00 0.138639
\(457\) 15539.0 1.59056 0.795278 0.606245i \(-0.207325\pi\)
0.795278 + 0.606245i \(0.207325\pi\)
\(458\) −6482.00 −0.661319
\(459\) 999.000 0.101589
\(460\) 7938.00 0.804589
\(461\) 4811.00 0.486053 0.243027 0.970020i \(-0.421860\pi\)
0.243027 + 0.970020i \(0.421860\pi\)
\(462\) −660.000 −0.0664632
\(463\) 562.000 0.0564111 0.0282056 0.999602i \(-0.491021\pi\)
0.0282056 + 0.999602i \(0.491021\pi\)
\(464\) −4633.00 −0.463538
\(465\) 4116.00 0.410484
\(466\) −6890.00 −0.684921
\(467\) 4914.00 0.486922 0.243461 0.969911i \(-0.421717\pi\)
0.243461 + 0.969911i \(0.421717\pi\)
\(468\) 0 0
\(469\) −2020.00 −0.198880
\(470\) 3234.00 0.317390
\(471\) −6237.00 −0.610161
\(472\) 8640.00 0.842560
\(473\) 5412.00 0.526097
\(474\) −2652.00 −0.256984
\(475\) −2280.00 −0.220239
\(476\) 2590.00 0.249396
\(477\) −4833.00 −0.463916
\(478\) −2466.00 −0.235967
\(479\) −3600.00 −0.343399 −0.171700 0.985149i \(-0.554926\pi\)
−0.171700 + 0.985149i \(0.554926\pi\)
\(480\) −3381.00 −0.321502
\(481\) 0 0
\(482\) 3617.00 0.341805
\(483\) 4860.00 0.457842
\(484\) 5929.00 0.556818
\(485\) −8414.00 −0.787753
\(486\) −243.000 −0.0226805
\(487\) −17130.0 −1.59391 −0.796955 0.604038i \(-0.793557\pi\)
−0.796955 + 0.604038i \(0.793557\pi\)
\(488\) −9525.00 −0.883558
\(489\) 5100.00 0.471636
\(490\) 1701.00 0.156823
\(491\) −11838.0 −1.08807 −0.544034 0.839063i \(-0.683104\pi\)
−0.544034 + 0.839063i \(0.683104\pi\)
\(492\) −5985.00 −0.548424
\(493\) −4181.00 −0.381953
\(494\) 0 0
\(495\) −1386.00 −0.125851
\(496\) 8036.00 0.727474
\(497\) 10860.0 0.980156
\(498\) −1554.00 −0.139832
\(499\) 8976.00 0.805252 0.402626 0.915364i \(-0.368097\pi\)
0.402626 + 0.915364i \(0.368097\pi\)
\(500\) 9849.00 0.880921
\(501\) 11040.0 0.984493
\(502\) −4860.00 −0.432096
\(503\) 1682.00 0.149099 0.0745494 0.997217i \(-0.476248\pi\)
0.0745494 + 0.997217i \(0.476248\pi\)
\(504\) −1350.00 −0.119313
\(505\) −3003.00 −0.264617
\(506\) −3564.00 −0.313121
\(507\) 0 0
\(508\) 6916.00 0.604031
\(509\) 15167.0 1.32076 0.660379 0.750933i \(-0.270396\pi\)
0.660379 + 0.750933i \(0.270396\pi\)
\(510\) −777.000 −0.0674630
\(511\) 8050.00 0.696890
\(512\) −11521.0 −0.994455
\(513\) 810.000 0.0697122
\(514\) −565.000 −0.0484846
\(515\) −9114.00 −0.779827
\(516\) 5166.00 0.440737
\(517\) 10164.0 0.864627
\(518\) 130.000 0.0110268
\(519\) 12438.0 1.05196
\(520\) 0 0
\(521\) −6783.00 −0.570381 −0.285191 0.958471i \(-0.592057\pi\)
−0.285191 + 0.958471i \(0.592057\pi\)
\(522\) 1017.00 0.0852737
\(523\) −13918.0 −1.16366 −0.581828 0.813312i \(-0.697662\pi\)
−0.581828 + 0.813312i \(0.697662\pi\)
\(524\) −3920.00 −0.326805
\(525\) 2280.00 0.189538
\(526\) 498.000 0.0412810
\(527\) 7252.00 0.599435
\(528\) −2706.00 −0.223037
\(529\) 14077.0 1.15698
\(530\) 3759.00 0.308076
\(531\) 5184.00 0.423666
\(532\) 2100.00 0.171140
\(533\) 0 0
\(534\) −582.000 −0.0471641
\(535\) −9366.00 −0.756874
\(536\) 3030.00 0.244172
\(537\) 11022.0 0.885725
\(538\) −5546.00 −0.444433
\(539\) 5346.00 0.427214
\(540\) −1323.00 −0.105431
\(541\) −1335.00 −0.106093 −0.0530463 0.998592i \(-0.516893\pi\)
−0.0530463 + 0.998592i \(0.516893\pi\)
\(542\) 2256.00 0.178789
\(543\) −9849.00 −0.778381
\(544\) −5957.00 −0.469493
\(545\) −7238.00 −0.568884
\(546\) 0 0
\(547\) −3806.00 −0.297501 −0.148750 0.988875i \(-0.547525\pi\)
−0.148750 + 0.988875i \(0.547525\pi\)
\(548\) 3633.00 0.283201
\(549\) −5715.00 −0.444281
\(550\) −1672.00 −0.129626
\(551\) −3390.00 −0.262103
\(552\) −7290.00 −0.562107
\(553\) −8840.00 −0.679774
\(554\) −2309.00 −0.177076
\(555\) 273.000 0.0208796
\(556\) 2436.00 0.185808
\(557\) −1905.00 −0.144915 −0.0724573 0.997372i \(-0.523084\pi\)
−0.0724573 + 0.997372i \(0.523084\pi\)
\(558\) −1764.00 −0.133828
\(559\) 0 0
\(560\) −2870.00 −0.216571
\(561\) −2442.00 −0.183781
\(562\) −5833.00 −0.437812
\(563\) −4800.00 −0.359318 −0.179659 0.983729i \(-0.557499\pi\)
−0.179659 + 0.983729i \(0.557499\pi\)
\(564\) 9702.00 0.724340
\(565\) 7539.00 0.561359
\(566\) −1650.00 −0.122535
\(567\) −810.000 −0.0599944
\(568\) −16290.0 −1.20337
\(569\) 14678.0 1.08143 0.540715 0.841206i \(-0.318154\pi\)
0.540715 + 0.841206i \(0.318154\pi\)
\(570\) −630.000 −0.0462944
\(571\) −586.000 −0.0429481 −0.0214740 0.999769i \(-0.506836\pi\)
−0.0214740 + 0.999769i \(0.506836\pi\)
\(572\) 0 0
\(573\) −1788.00 −0.130357
\(574\) 2850.00 0.207242
\(575\) 12312.0 0.892949
\(576\) −1503.00 −0.108724
\(577\) 8939.00 0.644949 0.322474 0.946578i \(-0.395485\pi\)
0.322474 + 0.946578i \(0.395485\pi\)
\(578\) 3544.00 0.255036
\(579\) −1179.00 −0.0846245
\(580\) 5537.00 0.396399
\(581\) −5180.00 −0.369884
\(582\) 3606.00 0.256827
\(583\) 11814.0 0.839255
\(584\) −12075.0 −0.855594
\(585\) 0 0
\(586\) −2991.00 −0.210848
\(587\) −13792.0 −0.969773 −0.484887 0.874577i \(-0.661139\pi\)
−0.484887 + 0.874577i \(0.661139\pi\)
\(588\) 5103.00 0.357898
\(589\) 5880.00 0.411343
\(590\) −4032.00 −0.281347
\(591\) −10566.0 −0.735410
\(592\) 533.000 0.0370037
\(593\) 9569.00 0.662650 0.331325 0.943517i \(-0.392504\pi\)
0.331325 + 0.943517i \(0.392504\pi\)
\(594\) 594.000 0.0410305
\(595\) −2590.00 −0.178453
\(596\) 4515.00 0.310305
\(597\) 6054.00 0.415031
\(598\) 0 0
\(599\) −5192.00 −0.354156 −0.177078 0.984197i \(-0.556664\pi\)
−0.177078 + 0.984197i \(0.556664\pi\)
\(600\) −3420.00 −0.232702
\(601\) −3677.00 −0.249564 −0.124782 0.992184i \(-0.539823\pi\)
−0.124782 + 0.992184i \(0.539823\pi\)
\(602\) −2460.00 −0.166548
\(603\) 1818.00 0.122777
\(604\) −20398.0 −1.37414
\(605\) −5929.00 −0.398427
\(606\) 1287.00 0.0862719
\(607\) −10960.0 −0.732871 −0.366435 0.930443i \(-0.619422\pi\)
−0.366435 + 0.930443i \(0.619422\pi\)
\(608\) −4830.00 −0.322175
\(609\) 3390.00 0.225566
\(610\) 4445.00 0.295037
\(611\) 0 0
\(612\) −2331.00 −0.153963
\(613\) −26027.0 −1.71488 −0.857439 0.514585i \(-0.827946\pi\)
−0.857439 + 0.514585i \(0.827946\pi\)
\(614\) 2422.00 0.159192
\(615\) 5985.00 0.392420
\(616\) 3300.00 0.215845
\(617\) 17681.0 1.15366 0.576832 0.816863i \(-0.304289\pi\)
0.576832 + 0.816863i \(0.304289\pi\)
\(618\) 3906.00 0.254243
\(619\) 3192.00 0.207265 0.103633 0.994616i \(-0.466953\pi\)
0.103633 + 0.994616i \(0.466953\pi\)
\(620\) −9604.00 −0.622106
\(621\) −4374.00 −0.282645
\(622\) 3402.00 0.219305
\(623\) −1940.00 −0.124758
\(624\) 0 0
\(625\) −349.000 −0.0223360
\(626\) −2310.00 −0.147486
\(627\) −1980.00 −0.126114
\(628\) 14553.0 0.924726
\(629\) 481.000 0.0304908
\(630\) 630.000 0.0398410
\(631\) 7580.00 0.478217 0.239109 0.970993i \(-0.423145\pi\)
0.239109 + 0.970993i \(0.423145\pi\)
\(632\) 13260.0 0.834580
\(633\) 480.000 0.0301395
\(634\) 257.000 0.0160990
\(635\) −6916.00 −0.432210
\(636\) 11277.0 0.703085
\(637\) 0 0
\(638\) −2486.00 −0.154266
\(639\) −9774.00 −0.605091
\(640\) 10185.0 0.629059
\(641\) −27707.0 −1.70727 −0.853635 0.520871i \(-0.825607\pi\)
−0.853635 + 0.520871i \(0.825607\pi\)
\(642\) 4014.00 0.246760
\(643\) 11216.0 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(644\) −11340.0 −0.693880
\(645\) −5166.00 −0.315366
\(646\) −1110.00 −0.0676043
\(647\) −2536.00 −0.154097 −0.0770483 0.997027i \(-0.524550\pi\)
−0.0770483 + 0.997027i \(0.524550\pi\)
\(648\) 1215.00 0.0736570
\(649\) −12672.0 −0.766440
\(650\) 0 0
\(651\) −5880.00 −0.354002
\(652\) −11900.0 −0.714785
\(653\) 17730.0 1.06252 0.531262 0.847207i \(-0.321718\pi\)
0.531262 + 0.847207i \(0.321718\pi\)
\(654\) 3102.00 0.185471
\(655\) 3920.00 0.233843
\(656\) 11685.0 0.695461
\(657\) −7245.00 −0.430220
\(658\) −4620.00 −0.273718
\(659\) 18920.0 1.11839 0.559195 0.829036i \(-0.311110\pi\)
0.559195 + 0.829036i \(0.311110\pi\)
\(660\) 3234.00 0.190732
\(661\) 5241.00 0.308398 0.154199 0.988040i \(-0.450720\pi\)
0.154199 + 0.988040i \(0.450720\pi\)
\(662\) −1028.00 −0.0603540
\(663\) 0 0
\(664\) 7770.00 0.454118
\(665\) −2100.00 −0.122458
\(666\) −117.000 −0.00680729
\(667\) 18306.0 1.06269
\(668\) −25760.0 −1.49204
\(669\) 12216.0 0.705976
\(670\) −1414.00 −0.0815337
\(671\) 13970.0 0.803735
\(672\) 4830.00 0.277264
\(673\) 20467.0 1.17228 0.586140 0.810210i \(-0.300647\pi\)
0.586140 + 0.810210i \(0.300647\pi\)
\(674\) −2487.00 −0.142130
\(675\) −2052.00 −0.117010
\(676\) 0 0
\(677\) −70.0000 −0.00397388 −0.00198694 0.999998i \(-0.500632\pi\)
−0.00198694 + 0.999998i \(0.500632\pi\)
\(678\) −3231.00 −0.183017
\(679\) 12020.0 0.679360
\(680\) 3885.00 0.219093
\(681\) −17382.0 −0.978091
\(682\) 4312.00 0.242104
\(683\) 6432.00 0.360342 0.180171 0.983635i \(-0.442335\pi\)
0.180171 + 0.983635i \(0.442335\pi\)
\(684\) −1890.00 −0.105652
\(685\) −3633.00 −0.202642
\(686\) −5860.00 −0.326146
\(687\) 19446.0 1.07993
\(688\) −10086.0 −0.558903
\(689\) 0 0
\(690\) 3402.00 0.187698
\(691\) −6666.00 −0.366985 −0.183492 0.983021i \(-0.558740\pi\)
−0.183492 + 0.983021i \(0.558740\pi\)
\(692\) −29022.0 −1.59429
\(693\) 1980.00 0.108534
\(694\) 2850.00 0.155885
\(695\) −2436.00 −0.132954
\(696\) −5085.00 −0.276935
\(697\) 10545.0 0.573056
\(698\) 2018.00 0.109430
\(699\) 20670.0 1.11847
\(700\) −5320.00 −0.287253
\(701\) −14054.0 −0.757221 −0.378611 0.925556i \(-0.623598\pi\)
−0.378611 + 0.925556i \(0.623598\pi\)
\(702\) 0 0
\(703\) 390.000 0.0209234
\(704\) 3674.00 0.196689
\(705\) −9702.00 −0.518296
\(706\) 5287.00 0.281840
\(707\) 4290.00 0.228207
\(708\) −12096.0 −0.642084
\(709\) −71.0000 −0.00376088 −0.00188044 0.999998i \(-0.500599\pi\)
−0.00188044 + 0.999998i \(0.500599\pi\)
\(710\) 7602.00 0.401828
\(711\) 7956.00 0.419653
\(712\) 2910.00 0.153170
\(713\) −31752.0 −1.66777
\(714\) 1110.00 0.0581803
\(715\) 0 0
\(716\) −25718.0 −1.34236
\(717\) 7398.00 0.385332
\(718\) 7278.00 0.378290
\(719\) 3936.00 0.204156 0.102078 0.994776i \(-0.467451\pi\)
0.102078 + 0.994776i \(0.467451\pi\)
\(720\) 2583.00 0.133698
\(721\) 13020.0 0.672524
\(722\) 5959.00 0.307162
\(723\) −10851.0 −0.558165
\(724\) 22981.0 1.17967
\(725\) 8588.00 0.439931
\(726\) 2541.00 0.129897
\(727\) 34202.0 1.74482 0.872409 0.488777i \(-0.162557\pi\)
0.872409 + 0.488777i \(0.162557\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 5635.00 0.285700
\(731\) −9102.00 −0.460533
\(732\) 13335.0 0.673328
\(733\) −27363.0 −1.37882 −0.689410 0.724371i \(-0.742130\pi\)
−0.689410 + 0.724371i \(0.742130\pi\)
\(734\) 4202.00 0.211306
\(735\) −5103.00 −0.256091
\(736\) 26082.0 1.30624
\(737\) −4444.00 −0.222112
\(738\) −2565.00 −0.127939
\(739\) 21776.0 1.08396 0.541978 0.840393i \(-0.317676\pi\)
0.541978 + 0.840393i \(0.317676\pi\)
\(740\) −637.000 −0.0316440
\(741\) 0 0
\(742\) −5370.00 −0.265686
\(743\) 2484.00 0.122650 0.0613251 0.998118i \(-0.480467\pi\)
0.0613251 + 0.998118i \(0.480467\pi\)
\(744\) 8820.00 0.434619
\(745\) −4515.00 −0.222036
\(746\) 1583.00 0.0776914
\(747\) 4662.00 0.228345
\(748\) 5698.00 0.278529
\(749\) 13380.0 0.652730
\(750\) 4221.00 0.205506
\(751\) 32906.0 1.59888 0.799439 0.600748i \(-0.205130\pi\)
0.799439 + 0.600748i \(0.205130\pi\)
\(752\) −18942.0 −0.918542
\(753\) 14580.0 0.705611
\(754\) 0 0
\(755\) 20398.0 0.983257
\(756\) 1890.00 0.0909241
\(757\) −3914.00 −0.187922 −0.0939609 0.995576i \(-0.529953\pi\)
−0.0939609 + 0.995576i \(0.529953\pi\)
\(758\) 2052.00 0.0983272
\(759\) 10692.0 0.511324
\(760\) 3150.00 0.150345
\(761\) −33038.0 −1.57375 −0.786877 0.617110i \(-0.788303\pi\)
−0.786877 + 0.617110i \(0.788303\pi\)
\(762\) 2964.00 0.140911
\(763\) 10340.0 0.490607
\(764\) 4172.00 0.197562
\(765\) 2331.00 0.110167
\(766\) 6872.00 0.324145
\(767\) 0 0
\(768\) −357.000 −0.0167736
\(769\) −17586.0 −0.824665 −0.412332 0.911033i \(-0.635286\pi\)
−0.412332 + 0.911033i \(0.635286\pi\)
\(770\) −1540.00 −0.0720750
\(771\) 1695.00 0.0791750
\(772\) 2751.00 0.128252
\(773\) 18314.0 0.852146 0.426073 0.904689i \(-0.359897\pi\)
0.426073 + 0.904689i \(0.359897\pi\)
\(774\) 2214.00 0.102817
\(775\) −14896.0 −0.690426
\(776\) −18030.0 −0.834071
\(777\) −390.000 −0.0180067
\(778\) 11653.0 0.536993
\(779\) 8550.00 0.393242
\(780\) 0 0
\(781\) 23892.0 1.09465
\(782\) 5994.00 0.274098
\(783\) −3051.00 −0.139251
\(784\) −9963.00 −0.453854
\(785\) −14553.0 −0.661680
\(786\) −1680.00 −0.0762387
\(787\) −42068.0 −1.90542 −0.952708 0.303888i \(-0.901715\pi\)
−0.952708 + 0.303888i \(0.901715\pi\)
\(788\) 24654.0 1.11455
\(789\) −1494.00 −0.0674117
\(790\) −6188.00 −0.278682
\(791\) −10770.0 −0.484118
\(792\) −2970.00 −0.133250
\(793\) 0 0
\(794\) −6134.00 −0.274166
\(795\) −11277.0 −0.503087
\(796\) −14126.0 −0.628998
\(797\) −4282.00 −0.190309 −0.0951545 0.995463i \(-0.530335\pi\)
−0.0951545 + 0.995463i \(0.530335\pi\)
\(798\) 900.000 0.0399244
\(799\) −17094.0 −0.756874
\(800\) 12236.0 0.540760
\(801\) 1746.00 0.0770186
\(802\) 10795.0 0.475293
\(803\) 17710.0 0.778297
\(804\) −4242.00 −0.186074
\(805\) 11340.0 0.496500
\(806\) 0 0
\(807\) 16638.0 0.725756
\(808\) −6435.00 −0.280176
\(809\) 40221.0 1.74795 0.873977 0.485967i \(-0.161532\pi\)
0.873977 + 0.485967i \(0.161532\pi\)
\(810\) −567.000 −0.0245955
\(811\) −7084.00 −0.306724 −0.153362 0.988170i \(-0.549010\pi\)
−0.153362 + 0.988170i \(0.549010\pi\)
\(812\) −7910.00 −0.341855
\(813\) −6768.00 −0.291961
\(814\) 286.000 0.0123149
\(815\) 11900.0 0.511459
\(816\) 4551.00 0.195241
\(817\) −7380.00 −0.316026
\(818\) 8489.00 0.362850
\(819\) 0 0
\(820\) −13965.0 −0.594730
\(821\) 17338.0 0.737028 0.368514 0.929622i \(-0.379867\pi\)
0.368514 + 0.929622i \(0.379867\pi\)
\(822\) 1557.00 0.0660664
\(823\) 35496.0 1.50342 0.751709 0.659495i \(-0.229230\pi\)
0.751709 + 0.659495i \(0.229230\pi\)
\(824\) −19530.0 −0.825679
\(825\) 5016.00 0.211678
\(826\) 5760.00 0.242634
\(827\) −14992.0 −0.630378 −0.315189 0.949029i \(-0.602068\pi\)
−0.315189 + 0.949029i \(0.602068\pi\)
\(828\) 10206.0 0.428361
\(829\) −20659.0 −0.865521 −0.432760 0.901509i \(-0.642460\pi\)
−0.432760 + 0.901509i \(0.642460\pi\)
\(830\) −3626.00 −0.151639
\(831\) 6927.00 0.289164
\(832\) 0 0
\(833\) −8991.00 −0.373973
\(834\) 1044.00 0.0433462
\(835\) 25760.0 1.06762
\(836\) 4620.00 0.191132
\(837\) 5292.00 0.218540
\(838\) −1496.00 −0.0616688
\(839\) 28716.0 1.18163 0.590814 0.806808i \(-0.298807\pi\)
0.590814 + 0.806808i \(0.298807\pi\)
\(840\) −3150.00 −0.129387
\(841\) −11620.0 −0.476444
\(842\) 11695.0 0.478665
\(843\) 17499.0 0.714944
\(844\) −1120.00 −0.0456777
\(845\) 0 0
\(846\) 4158.00 0.168978
\(847\) 8470.00 0.343604
\(848\) −22017.0 −0.891588
\(849\) 4950.00 0.200098
\(850\) 2812.00 0.113472
\(851\) −2106.00 −0.0848328
\(852\) 22806.0 0.917043
\(853\) 13377.0 0.536952 0.268476 0.963286i \(-0.413480\pi\)
0.268476 + 0.963286i \(0.413480\pi\)
\(854\) −6350.00 −0.254441
\(855\) 1890.00 0.0755984
\(856\) −20070.0 −0.801377
\(857\) −27419.0 −1.09290 −0.546450 0.837492i \(-0.684021\pi\)
−0.546450 + 0.837492i \(0.684021\pi\)
\(858\) 0 0
\(859\) 2422.00 0.0962021 0.0481010 0.998842i \(-0.484683\pi\)
0.0481010 + 0.998842i \(0.484683\pi\)
\(860\) 12054.0 0.477951
\(861\) −8550.00 −0.338424
\(862\) −10590.0 −0.418442
\(863\) −34522.0 −1.36169 −0.680847 0.732425i \(-0.738388\pi\)
−0.680847 + 0.732425i \(0.738388\pi\)
\(864\) −4347.00 −0.171167
\(865\) 29022.0 1.14078
\(866\) 13949.0 0.547351
\(867\) −10632.0 −0.416472
\(868\) 13720.0 0.536506
\(869\) −19448.0 −0.759181
\(870\) 2373.00 0.0924738
\(871\) 0 0
\(872\) −15510.0 −0.602334
\(873\) −10818.0 −0.419397
\(874\) 4860.00 0.188091
\(875\) 14070.0 0.543603
\(876\) 16905.0 0.652017
\(877\) 13733.0 0.528769 0.264385 0.964417i \(-0.414831\pi\)
0.264385 + 0.964417i \(0.414831\pi\)
\(878\) 10726.0 0.412284
\(879\) 8973.00 0.344314
\(880\) −6314.00 −0.241869
\(881\) −22759.0 −0.870341 −0.435170 0.900348i \(-0.643312\pi\)
−0.435170 + 0.900348i \(0.643312\pi\)
\(882\) 2187.00 0.0834922
\(883\) −2168.00 −0.0826263 −0.0413131 0.999146i \(-0.513154\pi\)
−0.0413131 + 0.999146i \(0.513154\pi\)
\(884\) 0 0
\(885\) 12096.0 0.459438
\(886\) −16228.0 −0.615339
\(887\) 15888.0 0.601428 0.300714 0.953714i \(-0.402775\pi\)
0.300714 + 0.953714i \(0.402775\pi\)
\(888\) 585.000 0.0221073
\(889\) 9880.00 0.372739
\(890\) −1358.00 −0.0511464
\(891\) −1782.00 −0.0670025
\(892\) −28504.0 −1.06994
\(893\) −13860.0 −0.519381
\(894\) 1935.00 0.0723894
\(895\) 25718.0 0.960512
\(896\) −14550.0 −0.542502
\(897\) 0 0
\(898\) −7538.00 −0.280118
\(899\) −22148.0 −0.821665
\(900\) 4788.00 0.177333
\(901\) −19869.0 −0.734664
\(902\) 6270.00 0.231450
\(903\) 7380.00 0.271972
\(904\) 16155.0 0.594366
\(905\) −22981.0 −0.844104
\(906\) −8742.00 −0.320567
\(907\) −11628.0 −0.425691 −0.212845 0.977086i \(-0.568273\pi\)
−0.212845 + 0.977086i \(0.568273\pi\)
\(908\) 40558.0 1.48234
\(909\) −3861.00 −0.140882
\(910\) 0 0
\(911\) −12584.0 −0.457658 −0.228829 0.973467i \(-0.573490\pi\)
−0.228829 + 0.973467i \(0.573490\pi\)
\(912\) 3690.00 0.133978
\(913\) −11396.0 −0.413092
\(914\) −15539.0 −0.562346
\(915\) −13335.0 −0.481794
\(916\) −45374.0 −1.63668
\(917\) −5600.00 −0.201667
\(918\) −999.000 −0.0359171
\(919\) 17184.0 0.616809 0.308405 0.951255i \(-0.400205\pi\)
0.308405 + 0.951255i \(0.400205\pi\)
\(920\) −17010.0 −0.609569
\(921\) −7266.00 −0.259960
\(922\) −4811.00 −0.171846
\(923\) 0 0
\(924\) −4620.00 −0.164488
\(925\) −988.000 −0.0351192
\(926\) −562.000 −0.0199443
\(927\) −11718.0 −0.415178
\(928\) 18193.0 0.643550
\(929\) 12777.0 0.451238 0.225619 0.974216i \(-0.427560\pi\)
0.225619 + 0.974216i \(0.427560\pi\)
\(930\) −4116.00 −0.145128
\(931\) −7290.00 −0.256627
\(932\) −48230.0 −1.69509
\(933\) −10206.0 −0.358124
\(934\) −4914.00 −0.172153
\(935\) −5698.00 −0.199299
\(936\) 0 0
\(937\) 9191.00 0.320445 0.160222 0.987081i \(-0.448779\pi\)
0.160222 + 0.987081i \(0.448779\pi\)
\(938\) 2020.00 0.0703149
\(939\) 6930.00 0.240843
\(940\) 22638.0 0.785500
\(941\) 50498.0 1.74940 0.874701 0.484662i \(-0.161058\pi\)
0.874701 + 0.484662i \(0.161058\pi\)
\(942\) 6237.00 0.215724
\(943\) −46170.0 −1.59438
\(944\) 23616.0 0.814232
\(945\) −1890.00 −0.0650600
\(946\) −5412.00 −0.186003
\(947\) 1560.00 0.0535303 0.0267651 0.999642i \(-0.491479\pi\)
0.0267651 + 0.999642i \(0.491479\pi\)
\(948\) −18564.0 −0.636003
\(949\) 0 0
\(950\) 2280.00 0.0778663
\(951\) −771.000 −0.0262896
\(952\) −5550.00 −0.188946
\(953\) −21498.0 −0.730733 −0.365366 0.930864i \(-0.619056\pi\)
−0.365366 + 0.930864i \(0.619056\pi\)
\(954\) 4833.00 0.164019
\(955\) −4172.00 −0.141364
\(956\) −17262.0 −0.583988
\(957\) 7458.00 0.251915
\(958\) 3600.00 0.121410
\(959\) 5190.00 0.174759
\(960\) −3507.00 −0.117904
\(961\) 8625.00 0.289517
\(962\) 0 0
\(963\) −12042.0 −0.402957
\(964\) 25319.0 0.845923
\(965\) −2751.00 −0.0917698
\(966\) −4860.00 −0.161872
\(967\) −418.000 −0.0139007 −0.00695035 0.999976i \(-0.502212\pi\)
−0.00695035 + 0.999976i \(0.502212\pi\)
\(968\) −12705.0 −0.421853
\(969\) 3330.00 0.110397
\(970\) 8414.00 0.278513
\(971\) 18132.0 0.599262 0.299631 0.954055i \(-0.403136\pi\)
0.299631 + 0.954055i \(0.403136\pi\)
\(972\) −1701.00 −0.0561313
\(973\) 3480.00 0.114659
\(974\) 17130.0 0.563532
\(975\) 0 0
\(976\) −26035.0 −0.853853
\(977\) 12501.0 0.409358 0.204679 0.978829i \(-0.434385\pi\)
0.204679 + 0.978829i \(0.434385\pi\)
\(978\) −5100.00 −0.166748
\(979\) −4268.00 −0.139332
\(980\) 11907.0 0.388118
\(981\) −9306.00 −0.302872
\(982\) 11838.0 0.384690
\(983\) −43708.0 −1.41818 −0.709089 0.705119i \(-0.750894\pi\)
−0.709089 + 0.705119i \(0.750894\pi\)
\(984\) 12825.0 0.415494
\(985\) −24654.0 −0.797504
\(986\) 4181.00 0.135041
\(987\) 13860.0 0.446979
\(988\) 0 0
\(989\) 39852.0 1.28131
\(990\) 1386.00 0.0444949
\(991\) −39614.0 −1.26981 −0.634904 0.772591i \(-0.718960\pi\)
−0.634904 + 0.772591i \(0.718960\pi\)
\(992\) −31556.0 −1.00998
\(993\) 3084.00 0.0985577
\(994\) −10860.0 −0.346538
\(995\) 14126.0 0.450075
\(996\) −10878.0 −0.346067
\(997\) −36503.0 −1.15954 −0.579770 0.814780i \(-0.696858\pi\)
−0.579770 + 0.814780i \(0.696858\pi\)
\(998\) −8976.00 −0.284700
\(999\) 351.000 0.0111163
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 507.4.a.b.1.1 1
3.2 odd 2 1521.4.a.h.1.1 1
13.3 even 3 39.4.e.b.22.1 yes 2
13.5 odd 4 507.4.b.d.337.2 2
13.8 odd 4 507.4.b.d.337.1 2
13.9 even 3 39.4.e.b.16.1 2
13.12 even 2 507.4.a.d.1.1 1
39.29 odd 6 117.4.g.a.100.1 2
39.35 odd 6 117.4.g.a.55.1 2
39.38 odd 2 1521.4.a.e.1.1 1
52.3 odd 6 624.4.q.c.529.1 2
52.35 odd 6 624.4.q.c.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.b.16.1 2 13.9 even 3
39.4.e.b.22.1 yes 2 13.3 even 3
117.4.g.a.55.1 2 39.35 odd 6
117.4.g.a.100.1 2 39.29 odd 6
507.4.a.b.1.1 1 1.1 even 1 trivial
507.4.a.d.1.1 1 13.12 even 2
507.4.b.d.337.1 2 13.8 odd 4
507.4.b.d.337.2 2 13.5 odd 4
624.4.q.c.289.1 2 52.35 odd 6
624.4.q.c.529.1 2 52.3 odd 6
1521.4.a.e.1.1 1 39.38 odd 2
1521.4.a.h.1.1 1 3.2 odd 2