Properties

Label 245.4.a.e.1.1
Level $245$
Weight $4$
Character 245.1
Self dual yes
Analytic conductor $14.455$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4554679514\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 245.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +5.00000 q^{5} -6.00000 q^{6} -21.0000 q^{8} -23.0000 q^{9} +O(q^{10})\) \(q+3.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +5.00000 q^{5} -6.00000 q^{6} -21.0000 q^{8} -23.0000 q^{9} +15.0000 q^{10} -45.0000 q^{11} -2.00000 q^{12} +59.0000 q^{13} -10.0000 q^{15} -71.0000 q^{16} -54.0000 q^{17} -69.0000 q^{18} -121.000 q^{19} +5.00000 q^{20} -135.000 q^{22} +69.0000 q^{23} +42.0000 q^{24} +25.0000 q^{25} +177.000 q^{26} +100.000 q^{27} -162.000 q^{29} -30.0000 q^{30} -88.0000 q^{31} -45.0000 q^{32} +90.0000 q^{33} -162.000 q^{34} -23.0000 q^{36} -259.000 q^{37} -363.000 q^{38} -118.000 q^{39} -105.000 q^{40} +195.000 q^{41} -286.000 q^{43} -45.0000 q^{44} -115.000 q^{45} +207.000 q^{46} +45.0000 q^{47} +142.000 q^{48} +75.0000 q^{50} +108.000 q^{51} +59.0000 q^{52} +597.000 q^{53} +300.000 q^{54} -225.000 q^{55} +242.000 q^{57} -486.000 q^{58} -360.000 q^{59} -10.0000 q^{60} +392.000 q^{61} -264.000 q^{62} +433.000 q^{64} +295.000 q^{65} +270.000 q^{66} -280.000 q^{67} -54.0000 q^{68} -138.000 q^{69} +48.0000 q^{71} +483.000 q^{72} +668.000 q^{73} -777.000 q^{74} -50.0000 q^{75} -121.000 q^{76} -354.000 q^{78} +782.000 q^{79} -355.000 q^{80} +421.000 q^{81} +585.000 q^{82} +768.000 q^{83} -270.000 q^{85} -858.000 q^{86} +324.000 q^{87} +945.000 q^{88} -1194.00 q^{89} -345.000 q^{90} +69.0000 q^{92} +176.000 q^{93} +135.000 q^{94} -605.000 q^{95} +90.0000 q^{96} +902.000 q^{97} +1035.00 q^{99} +O(q^{100})\)

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 \(0.322068\pi\)
\(3\) −2.00000 −0.384900 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) 1.00000 0.125000
\(5\) 5.00000 0.447214
\(6\) −6.00000 −0.408248
\(7\) 0 0
\(8\) −21.0000 −0.928078
\(9\) −23.0000 −0.851852
\(10\) 15.0000 0.474342
\(11\) −45.0000 −1.23346 −0.616728 0.787177i \(-0.711542\pi\)
−0.616728 + 0.787177i \(0.711542\pi\)
\(12\) −2.00000 −0.0481125
\(13\) 59.0000 1.25874 0.629371 0.777105i \(-0.283312\pi\)
0.629371 + 0.777105i \(0.283312\pi\)
\(14\) 0 0
\(15\) −10.0000 −0.172133
\(16\) −71.0000 −1.10938
\(17\) −54.0000 −0.770407 −0.385204 0.922832i \(-0.625869\pi\)
−0.385204 + 0.922832i \(0.625869\pi\)
\(18\) −69.0000 −0.903525
\(19\) −121.000 −1.46102 −0.730508 0.682904i \(-0.760717\pi\)
−0.730508 + 0.682904i \(0.760717\pi\)
\(20\) 5.00000 0.0559017
\(21\) 0 0
\(22\) −135.000 −1.30828
\(23\) 69.0000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 42.0000 0.357217
\(25\) 25.0000 0.200000
\(26\) 177.000 1.33510
\(27\) 100.000 0.712778
\(28\) 0 0
\(29\) −162.000 −1.03733 −0.518666 0.854977i \(-0.673571\pi\)
−0.518666 + 0.854977i \(0.673571\pi\)
\(30\) −30.0000 −0.182574
\(31\) −88.0000 −0.509847 −0.254924 0.966961i \(-0.582050\pi\)
−0.254924 + 0.966961i \(0.582050\pi\)
\(32\) −45.0000 −0.248592
\(33\) 90.0000 0.474757
\(34\) −162.000 −0.817140
\(35\) 0 0
\(36\) −23.0000 −0.106481
\(37\) −259.000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −363.000 −1.54964
\(39\) −118.000 −0.484490
\(40\) −105.000 −0.415049
\(41\) 195.000 0.742778 0.371389 0.928477i \(-0.378882\pi\)
0.371389 + 0.928477i \(0.378882\pi\)
\(42\) 0 0
\(43\) −286.000 −1.01429 −0.507146 0.861860i \(-0.669300\pi\)
−0.507146 + 0.861860i \(0.669300\pi\)
\(44\) −45.0000 −0.154182
\(45\) −115.000 −0.380960
\(46\) 207.000 0.663489
\(47\) 45.0000 0.139658 0.0698290 0.997559i \(-0.477755\pi\)
0.0698290 + 0.997559i \(0.477755\pi\)
\(48\) 142.000 0.426999
\(49\) 0 0
\(50\) 75.0000 0.212132
\(51\) 108.000 0.296530
\(52\) 59.0000 0.157343
\(53\) 597.000 1.54725 0.773625 0.633644i \(-0.218441\pi\)
0.773625 + 0.633644i \(0.218441\pi\)
\(54\) 300.000 0.756015
\(55\) −225.000 −0.551618
\(56\) 0 0
\(57\) 242.000 0.562345
\(58\) −486.000 −1.10026
\(59\) −360.000 −0.794373 −0.397187 0.917738i \(-0.630013\pi\)
−0.397187 + 0.917738i \(0.630013\pi\)
\(60\) −10.0000 −0.0215166
\(61\) 392.000 0.822794 0.411397 0.911456i \(-0.365041\pi\)
0.411397 + 0.911456i \(0.365041\pi\)
\(62\) −264.000 −0.540775
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) 295.000 0.562927
\(66\) 270.000 0.503556
\(67\) −280.000 −0.510559 −0.255279 0.966867i \(-0.582167\pi\)
−0.255279 + 0.966867i \(0.582167\pi\)
\(68\) −54.0000 −0.0963009
\(69\) −138.000 −0.240772
\(70\) 0 0
\(71\) 48.0000 0.0802331 0.0401166 0.999195i \(-0.487227\pi\)
0.0401166 + 0.999195i \(0.487227\pi\)
\(72\) 483.000 0.790585
\(73\) 668.000 1.07101 0.535503 0.844533i \(-0.320122\pi\)
0.535503 + 0.844533i \(0.320122\pi\)
\(74\) −777.000 −1.22060
\(75\) −50.0000 −0.0769800
\(76\) −121.000 −0.182627
\(77\) 0 0
\(78\) −354.000 −0.513880
\(79\) 782.000 1.11369 0.556847 0.830615i \(-0.312011\pi\)
0.556847 + 0.830615i \(0.312011\pi\)
\(80\) −355.000 −0.496128
\(81\) 421.000 0.577503
\(82\) 585.000 0.787835
\(83\) 768.000 1.01565 0.507825 0.861460i \(-0.330450\pi\)
0.507825 + 0.861460i \(0.330450\pi\)
\(84\) 0 0
\(85\) −270.000 −0.344537
\(86\) −858.000 −1.07582
\(87\) 324.000 0.399269
\(88\) 945.000 1.14474
\(89\) −1194.00 −1.42206 −0.711032 0.703159i \(-0.751772\pi\)
−0.711032 + 0.703159i \(0.751772\pi\)
\(90\) −345.000 −0.404069
\(91\) 0 0
\(92\) 69.0000 0.0781929
\(93\) 176.000 0.196240
\(94\) 135.000 0.148130
\(95\) −605.000 −0.653386
\(96\) 90.0000 0.0956832
\(97\) 902.000 0.944167 0.472084 0.881554i \(-0.343502\pi\)
0.472084 + 0.881554i \(0.343502\pi\)
\(98\) 0 0
\(99\) 1035.00 1.05072
\(100\) 25.0000 0.0250000
\(101\) 684.000 0.673867 0.336933 0.941528i \(-0.390610\pi\)
0.336933 + 0.941528i \(0.390610\pi\)
\(102\) 324.000 0.314517
\(103\) −1516.00 −1.45025 −0.725126 0.688616i \(-0.758218\pi\)
−0.725126 + 0.688616i \(0.758218\pi\)
\(104\) −1239.00 −1.16821
\(105\) 0 0
\(106\) 1791.00 1.64111
\(107\) −732.000 −0.661356 −0.330678 0.943744i \(-0.607277\pi\)
−0.330678 + 0.943744i \(0.607277\pi\)
\(108\) 100.000 0.0890973
\(109\) −1600.00 −1.40598 −0.702992 0.711198i \(-0.748153\pi\)
−0.702992 + 0.711198i \(0.748153\pi\)
\(110\) −675.000 −0.585079
\(111\) 518.000 0.442940
\(112\) 0 0
\(113\) −1392.00 −1.15883 −0.579417 0.815031i \(-0.696720\pi\)
−0.579417 + 0.815031i \(0.696720\pi\)
\(114\) 726.000 0.596457
\(115\) 345.000 0.279751
\(116\) −162.000 −0.129667
\(117\) −1357.00 −1.07226
\(118\) −1080.00 −0.842560
\(119\) 0 0
\(120\) 210.000 0.159752
\(121\) 694.000 0.521412
\(122\) 1176.00 0.872705
\(123\) −390.000 −0.285895
\(124\) −88.0000 −0.0637309
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 803.000 0.561061 0.280530 0.959845i \(-0.409490\pi\)
0.280530 + 0.959845i \(0.409490\pi\)
\(128\) 1659.00 1.14560
\(129\) 572.000 0.390401
\(130\) 885.000 0.597074
\(131\) 2019.00 1.34657 0.673286 0.739382i \(-0.264882\pi\)
0.673286 + 0.739382i \(0.264882\pi\)
\(132\) 90.0000 0.0593447
\(133\) 0 0
\(134\) −840.000 −0.541529
\(135\) 500.000 0.318764
\(136\) 1134.00 0.714998
\(137\) 60.0000 0.0374171 0.0187086 0.999825i \(-0.494045\pi\)
0.0187086 + 0.999825i \(0.494045\pi\)
\(138\) −414.000 −0.255377
\(139\) −1708.00 −1.04224 −0.521118 0.853485i \(-0.674485\pi\)
−0.521118 + 0.853485i \(0.674485\pi\)
\(140\) 0 0
\(141\) −90.0000 −0.0537544
\(142\) 144.000 0.0851001
\(143\) −2655.00 −1.55260
\(144\) 1633.00 0.945023
\(145\) −810.000 −0.463909
\(146\) 2004.00 1.13597
\(147\) 0 0
\(148\) −259.000 −0.143849
\(149\) −1086.00 −0.597105 −0.298552 0.954393i \(-0.596504\pi\)
−0.298552 + 0.954393i \(0.596504\pi\)
\(150\) −150.000 −0.0816497
\(151\) −2866.00 −1.54458 −0.772291 0.635269i \(-0.780889\pi\)
−0.772291 + 0.635269i \(0.780889\pi\)
\(152\) 2541.00 1.35594
\(153\) 1242.00 0.656273
\(154\) 0 0
\(155\) −440.000 −0.228011
\(156\) −118.000 −0.0605613
\(157\) −229.000 −0.116409 −0.0582044 0.998305i \(-0.518538\pi\)
−0.0582044 + 0.998305i \(0.518538\pi\)
\(158\) 2346.00 1.18125
\(159\) −1194.00 −0.595537
\(160\) −225.000 −0.111174
\(161\) 0 0
\(162\) 1263.00 0.612535
\(163\) −1228.00 −0.590088 −0.295044 0.955484i \(-0.595334\pi\)
−0.295044 + 0.955484i \(0.595334\pi\)
\(164\) 195.000 0.0928472
\(165\) 450.000 0.212318
\(166\) 2304.00 1.07726
\(167\) −1929.00 −0.893835 −0.446918 0.894575i \(-0.647478\pi\)
−0.446918 + 0.894575i \(0.647478\pi\)
\(168\) 0 0
\(169\) 1284.00 0.584433
\(170\) −810.000 −0.365436
\(171\) 2783.00 1.24457
\(172\) −286.000 −0.126787
\(173\) −699.000 −0.307191 −0.153595 0.988134i \(-0.549085\pi\)
−0.153595 + 0.988134i \(0.549085\pi\)
\(174\) 972.000 0.423489
\(175\) 0 0
\(176\) 3195.00 1.36836
\(177\) 720.000 0.305754
\(178\) −3582.00 −1.50833
\(179\) 3117.00 1.30154 0.650770 0.759275i \(-0.274446\pi\)
0.650770 + 0.759275i \(0.274446\pi\)
\(180\) −115.000 −0.0476200
\(181\) −1798.00 −0.738366 −0.369183 0.929357i \(-0.620362\pi\)
−0.369183 + 0.929357i \(0.620362\pi\)
\(182\) 0 0
\(183\) −784.000 −0.316694
\(184\) −1449.00 −0.580553
\(185\) −1295.00 −0.514650
\(186\) 528.000 0.208144
\(187\) 2430.00 0.950263
\(188\) 45.0000 0.0174572
\(189\) 0 0
\(190\) −1815.00 −0.693021
\(191\) −2388.00 −0.904658 −0.452329 0.891851i \(-0.649407\pi\)
−0.452329 + 0.891851i \(0.649407\pi\)
\(192\) −866.000 −0.325511
\(193\) 272.000 0.101446 0.0507228 0.998713i \(-0.483848\pi\)
0.0507228 + 0.998713i \(0.483848\pi\)
\(194\) 2706.00 1.00144
\(195\) −590.000 −0.216671
\(196\) 0 0
\(197\) −2109.00 −0.762741 −0.381371 0.924422i \(-0.624548\pi\)
−0.381371 + 0.924422i \(0.624548\pi\)
\(198\) 3105.00 1.11446
\(199\) 1424.00 0.507260 0.253630 0.967301i \(-0.418375\pi\)
0.253630 + 0.967301i \(0.418375\pi\)
\(200\) −525.000 −0.185616
\(201\) 560.000 0.196514
\(202\) 2052.00 0.714744
\(203\) 0 0
\(204\) 108.000 0.0370662
\(205\) 975.000 0.332180
\(206\) −4548.00 −1.53822
\(207\) −1587.00 −0.532870
\(208\) −4189.00 −1.39642
\(209\) 5445.00 1.80210
\(210\) 0 0
\(211\) −3625.00 −1.18273 −0.591363 0.806405i \(-0.701410\pi\)
−0.591363 + 0.806405i \(0.701410\pi\)
\(212\) 597.000 0.193406
\(213\) −96.0000 −0.0308817
\(214\) −2196.00 −0.701474
\(215\) −1430.00 −0.453606
\(216\) −2100.00 −0.661513
\(217\) 0 0
\(218\) −4800.00 −1.49127
\(219\) −1336.00 −0.412231
\(220\) −225.000 −0.0689523
\(221\) −3186.00 −0.969745
\(222\) 1554.00 0.469809
\(223\) −4960.00 −1.48944 −0.744722 0.667374i \(-0.767418\pi\)
−0.744722 + 0.667374i \(0.767418\pi\)
\(224\) 0 0
\(225\) −575.000 −0.170370
\(226\) −4176.00 −1.22913
\(227\) −1500.00 −0.438584 −0.219292 0.975659i \(-0.570375\pi\)
−0.219292 + 0.975659i \(0.570375\pi\)
\(228\) 242.000 0.0702932
\(229\) 6092.00 1.75795 0.878975 0.476867i \(-0.158228\pi\)
0.878975 + 0.476867i \(0.158228\pi\)
\(230\) 1035.00 0.296721
\(231\) 0 0
\(232\) 3402.00 0.962725
\(233\) 138.000 0.0388012 0.0194006 0.999812i \(-0.493824\pi\)
0.0194006 + 0.999812i \(0.493824\pi\)
\(234\) −4071.00 −1.13731
\(235\) 225.000 0.0624569
\(236\) −360.000 −0.0992966
\(237\) −1564.00 −0.428661
\(238\) 0 0
\(239\) −5502.00 −1.48910 −0.744550 0.667567i \(-0.767336\pi\)
−0.744550 + 0.667567i \(0.767336\pi\)
\(240\) 710.000 0.190960
\(241\) 3551.00 0.949129 0.474564 0.880221i \(-0.342606\pi\)
0.474564 + 0.880221i \(0.342606\pi\)
\(242\) 2082.00 0.553041
\(243\) −3542.00 −0.935059
\(244\) 392.000 0.102849
\(245\) 0 0
\(246\) −1170.00 −0.303238
\(247\) −7139.00 −1.83904
\(248\) 1848.00 0.473178
\(249\) −1536.00 −0.390924
\(250\) 375.000 0.0948683
\(251\) 7065.00 1.77665 0.888324 0.459216i \(-0.151870\pi\)
0.888324 + 0.459216i \(0.151870\pi\)
\(252\) 0 0
\(253\) −3105.00 −0.771580
\(254\) 2409.00 0.595095
\(255\) 540.000 0.132612
\(256\) 1513.00 0.369385
\(257\) −4080.00 −0.990286 −0.495143 0.868812i \(-0.664884\pi\)
−0.495143 + 0.868812i \(0.664884\pi\)
\(258\) 1716.00 0.414083
\(259\) 0 0
\(260\) 295.000 0.0703659
\(261\) 3726.00 0.883654
\(262\) 6057.00 1.42825
\(263\) −3288.00 −0.770900 −0.385450 0.922729i \(-0.625954\pi\)
−0.385450 + 0.922729i \(0.625954\pi\)
\(264\) −1890.00 −0.440612
\(265\) 2985.00 0.691951
\(266\) 0 0
\(267\) 2388.00 0.547353
\(268\) −280.000 −0.0638199
\(269\) −3264.00 −0.739813 −0.369906 0.929069i \(-0.620610\pi\)
−0.369906 + 0.929069i \(0.620610\pi\)
\(270\) 1500.00 0.338100
\(271\) −2752.00 −0.616871 −0.308436 0.951245i \(-0.599805\pi\)
−0.308436 + 0.951245i \(0.599805\pi\)
\(272\) 3834.00 0.854671
\(273\) 0 0
\(274\) 180.000 0.0396869
\(275\) −1125.00 −0.246691
\(276\) −138.000 −0.0300965
\(277\) −4690.00 −1.01731 −0.508655 0.860971i \(-0.669857\pi\)
−0.508655 + 0.860971i \(0.669857\pi\)
\(278\) −5124.00 −1.10546
\(279\) 2024.00 0.434314
\(280\) 0 0
\(281\) 7821.00 1.66036 0.830181 0.557494i \(-0.188237\pi\)
0.830181 + 0.557494i \(0.188237\pi\)
\(282\) −270.000 −0.0570151
\(283\) −658.000 −0.138212 −0.0691061 0.997609i \(-0.522015\pi\)
−0.0691061 + 0.997609i \(0.522015\pi\)
\(284\) 48.0000 0.0100291
\(285\) 1210.00 0.251488
\(286\) −7965.00 −1.64678
\(287\) 0 0
\(288\) 1035.00 0.211764
\(289\) −1997.00 −0.406473
\(290\) −2430.00 −0.492050
\(291\) −1804.00 −0.363410
\(292\) 668.000 0.133876
\(293\) −5997.00 −1.19573 −0.597864 0.801597i \(-0.703984\pi\)
−0.597864 + 0.801597i \(0.703984\pi\)
\(294\) 0 0
\(295\) −1800.00 −0.355254
\(296\) 5439.00 1.06803
\(297\) −4500.00 −0.879180
\(298\) −3258.00 −0.633325
\(299\) 4071.00 0.787398
\(300\) −50.0000 −0.00962250
\(301\) 0 0
\(302\) −8598.00 −1.63828
\(303\) −1368.00 −0.259371
\(304\) 8591.00 1.62081
\(305\) 1960.00 0.367965
\(306\) 3726.00 0.696082
\(307\) −6226.00 −1.15745 −0.578724 0.815523i \(-0.696449\pi\)
−0.578724 + 0.815523i \(0.696449\pi\)
\(308\) 0 0
\(309\) 3032.00 0.558202
\(310\) −1320.00 −0.241842
\(311\) 4680.00 0.853307 0.426653 0.904415i \(-0.359692\pi\)
0.426653 + 0.904415i \(0.359692\pi\)
\(312\) 2478.00 0.449645
\(313\) 1028.00 0.185642 0.0928211 0.995683i \(-0.470412\pi\)
0.0928211 + 0.995683i \(0.470412\pi\)
\(314\) −687.000 −0.123470
\(315\) 0 0
\(316\) 782.000 0.139212
\(317\) 8622.00 1.52763 0.763817 0.645433i \(-0.223323\pi\)
0.763817 + 0.645433i \(0.223323\pi\)
\(318\) −3582.00 −0.631662
\(319\) 7290.00 1.27950
\(320\) 2165.00 0.378210
\(321\) 1464.00 0.254556
\(322\) 0 0
\(323\) 6534.00 1.12558
\(324\) 421.000 0.0721879
\(325\) 1475.00 0.251749
\(326\) −3684.00 −0.625883
\(327\) 3200.00 0.541163
\(328\) −4095.00 −0.689355
\(329\) 0 0
\(330\) 1350.00 0.225197
\(331\) −1999.00 −0.331949 −0.165974 0.986130i \(-0.553077\pi\)
−0.165974 + 0.986130i \(0.553077\pi\)
\(332\) 768.000 0.126956
\(333\) 5957.00 0.980305
\(334\) −5787.00 −0.948056
\(335\) −1400.00 −0.228329
\(336\) 0 0
\(337\) 5114.00 0.826639 0.413319 0.910586i \(-0.364369\pi\)
0.413319 + 0.910586i \(0.364369\pi\)
\(338\) 3852.00 0.619885
\(339\) 2784.00 0.446036
\(340\) −270.000 −0.0430671
\(341\) 3960.00 0.628874
\(342\) 8349.00 1.32006
\(343\) 0 0
\(344\) 6006.00 0.941342
\(345\) −690.000 −0.107676
\(346\) −2097.00 −0.325825
\(347\) 4320.00 0.668328 0.334164 0.942515i \(-0.391546\pi\)
0.334164 + 0.942515i \(0.391546\pi\)
\(348\) 324.000 0.0499087
\(349\) 7922.00 1.21506 0.607529 0.794298i \(-0.292161\pi\)
0.607529 + 0.794298i \(0.292161\pi\)
\(350\) 0 0
\(351\) 5900.00 0.897204
\(352\) 2025.00 0.306627
\(353\) 828.000 0.124844 0.0624221 0.998050i \(-0.480118\pi\)
0.0624221 + 0.998050i \(0.480118\pi\)
\(354\) 2160.00 0.324301
\(355\) 240.000 0.0358813
\(356\) −1194.00 −0.177758
\(357\) 0 0
\(358\) 9351.00 1.38049
\(359\) −1350.00 −0.198469 −0.0992344 0.995064i \(-0.531639\pi\)
−0.0992344 + 0.995064i \(0.531639\pi\)
\(360\) 2415.00 0.353560
\(361\) 7782.00 1.13457
\(362\) −5394.00 −0.783156
\(363\) −1388.00 −0.200692
\(364\) 0 0
\(365\) 3340.00 0.478969
\(366\) −2352.00 −0.335904
\(367\) 2801.00 0.398395 0.199198 0.979959i \(-0.436166\pi\)
0.199198 + 0.979959i \(0.436166\pi\)
\(368\) −4899.00 −0.693962
\(369\) −4485.00 −0.632737
\(370\) −3885.00 −0.545869
\(371\) 0 0
\(372\) 176.000 0.0245300
\(373\) 6602.00 0.916457 0.458229 0.888834i \(-0.348484\pi\)
0.458229 + 0.888834i \(0.348484\pi\)
\(374\) 7290.00 1.00791
\(375\) −250.000 −0.0344265
\(376\) −945.000 −0.129613
\(377\) −9558.00 −1.30573
\(378\) 0 0
\(379\) −8305.00 −1.12559 −0.562796 0.826596i \(-0.690274\pi\)
−0.562796 + 0.826596i \(0.690274\pi\)
\(380\) −605.000 −0.0816733
\(381\) −1606.00 −0.215952
\(382\) −7164.00 −0.959534
\(383\) 945.000 0.126076 0.0630382 0.998011i \(-0.479921\pi\)
0.0630382 + 0.998011i \(0.479921\pi\)
\(384\) −3318.00 −0.440940
\(385\) 0 0
\(386\) 816.000 0.107599
\(387\) 6578.00 0.864027
\(388\) 902.000 0.118021
\(389\) 12036.0 1.56876 0.784382 0.620278i \(-0.212980\pi\)
0.784382 + 0.620278i \(0.212980\pi\)
\(390\) −1770.00 −0.229814
\(391\) −3726.00 −0.481923
\(392\) 0 0
\(393\) −4038.00 −0.518296
\(394\) −6327.00 −0.809009
\(395\) 3910.00 0.498059
\(396\) 1035.00 0.131340
\(397\) −2698.00 −0.341080 −0.170540 0.985351i \(-0.554551\pi\)
−0.170540 + 0.985351i \(0.554551\pi\)
\(398\) 4272.00 0.538030
\(399\) 0 0
\(400\) −1775.00 −0.221875
\(401\) 7053.00 0.878329 0.439165 0.898407i \(-0.355274\pi\)
0.439165 + 0.898407i \(0.355274\pi\)
\(402\) 1680.00 0.208435
\(403\) −5192.00 −0.641767
\(404\) 684.000 0.0842333
\(405\) 2105.00 0.258267
\(406\) 0 0
\(407\) 11655.0 1.41945
\(408\) −2268.00 −0.275203
\(409\) −10870.0 −1.31415 −0.657074 0.753826i \(-0.728206\pi\)
−0.657074 + 0.753826i \(0.728206\pi\)
\(410\) 2925.00 0.352330
\(411\) −120.000 −0.0144019
\(412\) −1516.00 −0.181281
\(413\) 0 0
\(414\) −4761.00 −0.565194
\(415\) 3840.00 0.454212
\(416\) −2655.00 −0.312914
\(417\) 3416.00 0.401156
\(418\) 16335.0 1.91141
\(419\) −9729.00 −1.13435 −0.567175 0.823597i \(-0.691964\pi\)
−0.567175 + 0.823597i \(0.691964\pi\)
\(420\) 0 0
\(421\) −12550.0 −1.45285 −0.726425 0.687246i \(-0.758819\pi\)
−0.726425 + 0.687246i \(0.758819\pi\)
\(422\) −10875.0 −1.25447
\(423\) −1035.00 −0.118968
\(424\) −12537.0 −1.43597
\(425\) −1350.00 −0.154081
\(426\) −288.000 −0.0327550
\(427\) 0 0
\(428\) −732.000 −0.0826695
\(429\) 5310.00 0.597597
\(430\) −4290.00 −0.481121
\(431\) 2988.00 0.333937 0.166969 0.985962i \(-0.446602\pi\)
0.166969 + 0.985962i \(0.446602\pi\)
\(432\) −7100.00 −0.790738
\(433\) 16616.0 1.84414 0.922072 0.387019i \(-0.126495\pi\)
0.922072 + 0.387019i \(0.126495\pi\)
\(434\) 0 0
\(435\) 1620.00 0.178559
\(436\) −1600.00 −0.175748
\(437\) −8349.00 −0.913929
\(438\) −4008.00 −0.437237
\(439\) 7346.00 0.798646 0.399323 0.916810i \(-0.369245\pi\)
0.399323 + 0.916810i \(0.369245\pi\)
\(440\) 4725.00 0.511944
\(441\) 0 0
\(442\) −9558.00 −1.02857
\(443\) 12.0000 0.00128699 0.000643496 1.00000i \(-0.499795\pi\)
0.000643496 1.00000i \(0.499795\pi\)
\(444\) 518.000 0.0553675
\(445\) −5970.00 −0.635967
\(446\) −14880.0 −1.57979
\(447\) 2172.00 0.229826
\(448\) 0 0
\(449\) 9669.00 1.01628 0.508138 0.861275i \(-0.330334\pi\)
0.508138 + 0.861275i \(0.330334\pi\)
\(450\) −1725.00 −0.180705
\(451\) −8775.00 −0.916183
\(452\) −1392.00 −0.144854
\(453\) 5732.00 0.594510
\(454\) −4500.00 −0.465188
\(455\) 0 0
\(456\) −5082.00 −0.521900
\(457\) −9634.00 −0.986126 −0.493063 0.869994i \(-0.664123\pi\)
−0.493063 + 0.869994i \(0.664123\pi\)
\(458\) 18276.0 1.86459
\(459\) −5400.00 −0.549129
\(460\) 345.000 0.0349689
\(461\) −342.000 −0.0345521 −0.0172761 0.999851i \(-0.505499\pi\)
−0.0172761 + 0.999851i \(0.505499\pi\)
\(462\) 0 0
\(463\) 2411.00 0.242006 0.121003 0.992652i \(-0.461389\pi\)
0.121003 + 0.992652i \(0.461389\pi\)
\(464\) 11502.0 1.15079
\(465\) 880.000 0.0877613
\(466\) 414.000 0.0411549
\(467\) −1206.00 −0.119501 −0.0597506 0.998213i \(-0.519031\pi\)
−0.0597506 + 0.998213i \(0.519031\pi\)
\(468\) −1357.00 −0.134033
\(469\) 0 0
\(470\) 675.000 0.0662456
\(471\) 458.000 0.0448058
\(472\) 7560.00 0.737240
\(473\) 12870.0 1.25109
\(474\) −4692.00 −0.454664
\(475\) −3025.00 −0.292203
\(476\) 0 0
\(477\) −13731.0 −1.31803
\(478\) −16506.0 −1.57943
\(479\) −432.000 −0.0412079 −0.0206039 0.999788i \(-0.506559\pi\)
−0.0206039 + 0.999788i \(0.506559\pi\)
\(480\) 450.000 0.0427908
\(481\) −15281.0 −1.44855
\(482\) 10653.0 1.00670
\(483\) 0 0
\(484\) 694.000 0.0651766
\(485\) 4510.00 0.422244
\(486\) −10626.0 −0.991780
\(487\) −11896.0 −1.10690 −0.553449 0.832883i \(-0.686689\pi\)
−0.553449 + 0.832883i \(0.686689\pi\)
\(488\) −8232.00 −0.763617
\(489\) 2456.00 0.227125
\(490\) 0 0
\(491\) −12276.0 −1.12833 −0.564163 0.825663i \(-0.690801\pi\)
−0.564163 + 0.825663i \(0.690801\pi\)
\(492\) −390.000 −0.0357369
\(493\) 8748.00 0.799169
\(494\) −21417.0 −1.95060
\(495\) 5175.00 0.469897
\(496\) 6248.00 0.565612
\(497\) 0 0
\(498\) −4608.00 −0.414637
\(499\) −10876.0 −0.975705 −0.487852 0.872926i \(-0.662220\pi\)
−0.487852 + 0.872926i \(0.662220\pi\)
\(500\) 125.000 0.0111803
\(501\) 3858.00 0.344037
\(502\) 21195.0 1.88442
\(503\) 12000.0 1.06372 0.531862 0.846831i \(-0.321492\pi\)
0.531862 + 0.846831i \(0.321492\pi\)
\(504\) 0 0
\(505\) 3420.00 0.301362
\(506\) −9315.00 −0.818384
\(507\) −2568.00 −0.224948
\(508\) 803.000 0.0701326
\(509\) −11682.0 −1.01728 −0.508640 0.860979i \(-0.669852\pi\)
−0.508640 + 0.860979i \(0.669852\pi\)
\(510\) 1620.00 0.140656
\(511\) 0 0
\(512\) −8733.00 −0.753804
\(513\) −12100.0 −1.04138
\(514\) −12240.0 −1.05036
\(515\) −7580.00 −0.648572
\(516\) 572.000 0.0488002
\(517\) −2025.00 −0.172262
\(518\) 0 0
\(519\) 1398.00 0.118238
\(520\) −6195.00 −0.522440
\(521\) 9609.00 0.808019 0.404010 0.914755i \(-0.367616\pi\)
0.404010 + 0.914755i \(0.367616\pi\)
\(522\) 11178.0 0.937256
\(523\) 21188.0 1.77148 0.885742 0.464177i \(-0.153650\pi\)
0.885742 + 0.464177i \(0.153650\pi\)
\(524\) 2019.00 0.168321
\(525\) 0 0
\(526\) −9864.00 −0.817663
\(527\) 4752.00 0.392790
\(528\) −6390.00 −0.526684
\(529\) −7406.00 −0.608696
\(530\) 8955.00 0.733925
\(531\) 8280.00 0.676688
\(532\) 0 0
\(533\) 11505.0 0.934966
\(534\) 7164.00 0.580555
\(535\) −3660.00 −0.295767
\(536\) 5880.00 0.473838
\(537\) −6234.00 −0.500963
\(538\) −9792.00 −0.784690
\(539\) 0 0
\(540\) 500.000 0.0398455
\(541\) 8072.00 0.641483 0.320742 0.947167i \(-0.396068\pi\)
0.320742 + 0.947167i \(0.396068\pi\)
\(542\) −8256.00 −0.654291
\(543\) 3596.00 0.284197
\(544\) 2430.00 0.191517
\(545\) −8000.00 −0.628775
\(546\) 0 0
\(547\) 344.000 0.0268892 0.0134446 0.999910i \(-0.495720\pi\)
0.0134446 + 0.999910i \(0.495720\pi\)
\(548\) 60.0000 0.00467714
\(549\) −9016.00 −0.700899
\(550\) −3375.00 −0.261655
\(551\) 19602.0 1.51556
\(552\) 2898.00 0.223455
\(553\) 0 0
\(554\) −14070.0 −1.07902
\(555\) 2590.00 0.198089
\(556\) −1708.00 −0.130279
\(557\) 18363.0 1.39689 0.698443 0.715666i \(-0.253877\pi\)
0.698443 + 0.715666i \(0.253877\pi\)
\(558\) 6072.00 0.460660
\(559\) −16874.0 −1.27673
\(560\) 0 0
\(561\) −4860.00 −0.365756
\(562\) 23463.0 1.76108
\(563\) −6294.00 −0.471155 −0.235578 0.971856i \(-0.575698\pi\)
−0.235578 + 0.971856i \(0.575698\pi\)
\(564\) −90.0000 −0.00671930
\(565\) −6960.00 −0.518247
\(566\) −1974.00 −0.146596
\(567\) 0 0
\(568\) −1008.00 −0.0744626
\(569\) 11733.0 0.864452 0.432226 0.901765i \(-0.357728\pi\)
0.432226 + 0.901765i \(0.357728\pi\)
\(570\) 3630.00 0.266744
\(571\) 1052.00 0.0771013 0.0385506 0.999257i \(-0.487726\pi\)
0.0385506 + 0.999257i \(0.487726\pi\)
\(572\) −2655.00 −0.194075
\(573\) 4776.00 0.348203
\(574\) 0 0
\(575\) 1725.00 0.125109
\(576\) −9959.00 −0.720414
\(577\) −13156.0 −0.949205 −0.474603 0.880200i \(-0.657408\pi\)
−0.474603 + 0.880200i \(0.657408\pi\)
\(578\) −5991.00 −0.431129
\(579\) −544.000 −0.0390464
\(580\) −810.000 −0.0579887
\(581\) 0 0
\(582\) −5412.00 −0.385455
\(583\) −26865.0 −1.90846
\(584\) −14028.0 −0.993977
\(585\) −6785.00 −0.479530
\(586\) −17991.0 −1.26826
\(587\) −13368.0 −0.939960 −0.469980 0.882677i \(-0.655739\pi\)
−0.469980 + 0.882677i \(0.655739\pi\)
\(588\) 0 0
\(589\) 10648.0 0.744895
\(590\) −5400.00 −0.376804
\(591\) 4218.00 0.293579
\(592\) 18389.0 1.27666
\(593\) 26664.0 1.84647 0.923237 0.384231i \(-0.125533\pi\)
0.923237 + 0.384231i \(0.125533\pi\)
\(594\) −13500.0 −0.932511
\(595\) 0 0
\(596\) −1086.00 −0.0746381
\(597\) −2848.00 −0.195244
\(598\) 12213.0 0.835162
\(599\) 7614.00 0.519365 0.259682 0.965694i \(-0.416382\pi\)
0.259682 + 0.965694i \(0.416382\pi\)
\(600\) 1050.00 0.0714435
\(601\) 6410.00 0.435057 0.217529 0.976054i \(-0.430200\pi\)
0.217529 + 0.976054i \(0.430200\pi\)
\(602\) 0 0
\(603\) 6440.00 0.434921
\(604\) −2866.00 −0.193073
\(605\) 3470.00 0.233183
\(606\) −4104.00 −0.275105
\(607\) −21469.0 −1.43558 −0.717792 0.696257i \(-0.754847\pi\)
−0.717792 + 0.696257i \(0.754847\pi\)
\(608\) 5445.00 0.363197
\(609\) 0 0
\(610\) 5880.00 0.390286
\(611\) 2655.00 0.175793
\(612\) 1242.00 0.0820341
\(613\) 3737.00 0.246225 0.123113 0.992393i \(-0.460712\pi\)
0.123113 + 0.992393i \(0.460712\pi\)
\(614\) −18678.0 −1.22766
\(615\) −1950.00 −0.127856
\(616\) 0 0
\(617\) 18078.0 1.17957 0.589784 0.807561i \(-0.299213\pi\)
0.589784 + 0.807561i \(0.299213\pi\)
\(618\) 9096.00 0.592063
\(619\) 12287.0 0.797829 0.398915 0.916988i \(-0.369387\pi\)
0.398915 + 0.916988i \(0.369387\pi\)
\(620\) −440.000 −0.0285013
\(621\) 6900.00 0.445874
\(622\) 14040.0 0.905069
\(623\) 0 0
\(624\) 8378.00 0.537481
\(625\) 625.000 0.0400000
\(626\) 3084.00 0.196903
\(627\) −10890.0 −0.693628
\(628\) −229.000 −0.0145511
\(629\) 13986.0 0.886579
\(630\) 0 0
\(631\) −9580.00 −0.604396 −0.302198 0.953245i \(-0.597720\pi\)
−0.302198 + 0.953245i \(0.597720\pi\)
\(632\) −16422.0 −1.03360
\(633\) 7250.00 0.455232
\(634\) 25866.0 1.62030
\(635\) 4015.00 0.250914
\(636\) −1194.00 −0.0744421
\(637\) 0 0
\(638\) 21870.0 1.35712
\(639\) −1104.00 −0.0683467
\(640\) 8295.00 0.512326
\(641\) 10779.0 0.664189 0.332094 0.943246i \(-0.392245\pi\)
0.332094 + 0.943246i \(0.392245\pi\)
\(642\) 4392.00 0.269998
\(643\) 8882.00 0.544746 0.272373 0.962192i \(-0.412191\pi\)
0.272373 + 0.962192i \(0.412191\pi\)
\(644\) 0 0
\(645\) 2860.00 0.174593
\(646\) 19602.0 1.19386
\(647\) −11019.0 −0.669554 −0.334777 0.942297i \(-0.608661\pi\)
−0.334777 + 0.942297i \(0.608661\pi\)
\(648\) −8841.00 −0.535968
\(649\) 16200.0 0.979824
\(650\) 4425.00 0.267020
\(651\) 0 0
\(652\) −1228.00 −0.0737610
\(653\) 22323.0 1.33777 0.668887 0.743364i \(-0.266771\pi\)
0.668887 + 0.743364i \(0.266771\pi\)
\(654\) 9600.00 0.573990
\(655\) 10095.0 0.602205
\(656\) −13845.0 −0.824019
\(657\) −15364.0 −0.912339
\(658\) 0 0
\(659\) −11856.0 −0.700826 −0.350413 0.936595i \(-0.613959\pi\)
−0.350413 + 0.936595i \(0.613959\pi\)
\(660\) 450.000 0.0265397
\(661\) −33244.0 −1.95619 −0.978095 0.208158i \(-0.933253\pi\)
−0.978095 + 0.208158i \(0.933253\pi\)
\(662\) −5997.00 −0.352085
\(663\) 6372.00 0.373255
\(664\) −16128.0 −0.942602
\(665\) 0 0
\(666\) 17871.0 1.03977
\(667\) −11178.0 −0.648896
\(668\) −1929.00 −0.111729
\(669\) 9920.00 0.573288
\(670\) −4200.00 −0.242179
\(671\) −17640.0 −1.01488
\(672\) 0 0
\(673\) −12322.0 −0.705763 −0.352881 0.935668i \(-0.614798\pi\)
−0.352881 + 0.935668i \(0.614798\pi\)
\(674\) 15342.0 0.876783
\(675\) 2500.00 0.142556
\(676\) 1284.00 0.0730542
\(677\) −12597.0 −0.715129 −0.357564 0.933889i \(-0.616393\pi\)
−0.357564 + 0.933889i \(0.616393\pi\)
\(678\) 8352.00 0.473092
\(679\) 0 0
\(680\) 5670.00 0.319757
\(681\) 3000.00 0.168811
\(682\) 11880.0 0.667022
\(683\) −8340.00 −0.467235 −0.233617 0.972329i \(-0.575056\pi\)
−0.233617 + 0.972329i \(0.575056\pi\)
\(684\) 2783.00 0.155571
\(685\) 300.000 0.0167334
\(686\) 0 0
\(687\) −12184.0 −0.676636
\(688\) 20306.0 1.12523
\(689\) 35223.0 1.94759
\(690\) −2070.00 −0.114208
\(691\) −20200.0 −1.11208 −0.556038 0.831157i \(-0.687679\pi\)
−0.556038 + 0.831157i \(0.687679\pi\)
\(692\) −699.000 −0.0383988
\(693\) 0 0
\(694\) 12960.0 0.708869
\(695\) −8540.00 −0.466102
\(696\) −6804.00 −0.370553
\(697\) −10530.0 −0.572241
\(698\) 23766.0 1.28876
\(699\) −276.000 −0.0149346
\(700\) 0 0
\(701\) 474.000 0.0255388 0.0127694 0.999918i \(-0.495935\pi\)
0.0127694 + 0.999918i \(0.495935\pi\)
\(702\) 17700.0 0.951629
\(703\) 31339.0 1.68133
\(704\) −19485.0 −1.04314
\(705\) −450.000 −0.0240397
\(706\) 2484.00 0.132417
\(707\) 0 0
\(708\) 720.000 0.0382193
\(709\) −25126.0 −1.33093 −0.665463 0.746431i \(-0.731766\pi\)
−0.665463 + 0.746431i \(0.731766\pi\)
\(710\) 720.000 0.0380579
\(711\) −17986.0 −0.948703
\(712\) 25074.0 1.31979
\(713\) −6072.00 −0.318932
\(714\) 0 0
\(715\) −13275.0 −0.694345
\(716\) 3117.00 0.162692
\(717\) 11004.0 0.573155
\(718\) −4050.00 −0.210508
\(719\) −7296.00 −0.378435 −0.189218 0.981935i \(-0.560595\pi\)
−0.189218 + 0.981935i \(0.560595\pi\)
\(720\) 8165.00 0.422627
\(721\) 0 0
\(722\) 23346.0 1.20339
\(723\) −7102.00 −0.365320
\(724\) −1798.00 −0.0922958
\(725\) −4050.00 −0.207467
\(726\) −4164.00 −0.212866
\(727\) −15421.0 −0.786703 −0.393352 0.919388i \(-0.628684\pi\)
−0.393352 + 0.919388i \(0.628684\pi\)
\(728\) 0 0
\(729\) −4283.00 −0.217599
\(730\) 10020.0 0.508023
\(731\) 15444.0 0.781419
\(732\) −784.000 −0.0395867
\(733\) −29167.0 −1.46972 −0.734862 0.678217i \(-0.762753\pi\)
−0.734862 + 0.678217i \(0.762753\pi\)
\(734\) 8403.00 0.422562
\(735\) 0 0
\(736\) −3105.00 −0.155505
\(737\) 12600.0 0.629752
\(738\) −13455.0 −0.671118
\(739\) −13381.0 −0.666073 −0.333037 0.942914i \(-0.608073\pi\)
−0.333037 + 0.942914i \(0.608073\pi\)
\(740\) −1295.00 −0.0643313
\(741\) 14278.0 0.707848
\(742\) 0 0
\(743\) 5487.00 0.270927 0.135463 0.990782i \(-0.456748\pi\)
0.135463 + 0.990782i \(0.456748\pi\)
\(744\) −3696.00 −0.182126
\(745\) −5430.00 −0.267033
\(746\) 19806.0 0.972050
\(747\) −17664.0 −0.865183
\(748\) 2430.00 0.118783
\(749\) 0 0
\(750\) −750.000 −0.0365148
\(751\) 6638.00 0.322535 0.161268 0.986911i \(-0.448442\pi\)
0.161268 + 0.986911i \(0.448442\pi\)
\(752\) −3195.00 −0.154933
\(753\) −14130.0 −0.683832
\(754\) −28674.0 −1.38494
\(755\) −14330.0 −0.690758
\(756\) 0 0
\(757\) 14846.0 0.712797 0.356398 0.934334i \(-0.384005\pi\)
0.356398 + 0.934334i \(0.384005\pi\)
\(758\) −24915.0 −1.19387
\(759\) 6210.00 0.296981
\(760\) 12705.0 0.606393
\(761\) −3651.00 −0.173914 −0.0869571 0.996212i \(-0.527714\pi\)
−0.0869571 + 0.996212i \(0.527714\pi\)
\(762\) −4818.00 −0.229052
\(763\) 0 0
\(764\) −2388.00 −0.113082
\(765\) 6210.00 0.293494
\(766\) 2835.00 0.133724
\(767\) −21240.0 −0.999911
\(768\) −3026.00 −0.142176
\(769\) 29855.0 1.40000 0.699999 0.714144i \(-0.253184\pi\)
0.699999 + 0.714144i \(0.253184\pi\)
\(770\) 0 0
\(771\) 8160.00 0.381161
\(772\) 272.000 0.0126807
\(773\) −6519.00 −0.303327 −0.151664 0.988432i \(-0.548463\pi\)
−0.151664 + 0.988432i \(0.548463\pi\)
\(774\) 19734.0 0.916439
\(775\) −2200.00 −0.101969
\(776\) −18942.0 −0.876261
\(777\) 0 0
\(778\) 36108.0 1.66393
\(779\) −23595.0 −1.08521
\(780\) −590.000 −0.0270838
\(781\) −2160.00 −0.0989640
\(782\) −11178.0 −0.511157
\(783\) −16200.0 −0.739388
\(784\) 0 0
\(785\) −1145.00 −0.0520596
\(786\) −12114.0 −0.549735
\(787\) 35114.0 1.59044 0.795222 0.606319i \(-0.207354\pi\)
0.795222 + 0.606319i \(0.207354\pi\)
\(788\) −2109.00 −0.0953427
\(789\) 6576.00 0.296720
\(790\) 11730.0 0.528272
\(791\) 0 0
\(792\) −21735.0 −0.975151
\(793\) 23128.0 1.03569
\(794\) −8094.00 −0.361770
\(795\) −5970.00 −0.266332
\(796\) 1424.00 0.0634075
\(797\) 20910.0 0.929323 0.464661 0.885488i \(-0.346176\pi\)
0.464661 + 0.885488i \(0.346176\pi\)
\(798\) 0 0
\(799\) −2430.00 −0.107594
\(800\) −1125.00 −0.0497184
\(801\) 27462.0 1.21139
\(802\) 21159.0 0.931609
\(803\) −30060.0 −1.32104
\(804\) 560.000 0.0245643
\(805\) 0 0
\(806\) −15576.0 −0.680696
\(807\) 6528.00 0.284754
\(808\) −14364.0 −0.625401
\(809\) 4431.00 0.192566 0.0962829 0.995354i \(-0.469305\pi\)
0.0962829 + 0.995354i \(0.469305\pi\)
\(810\) 6315.00 0.273934
\(811\) −9577.00 −0.414666 −0.207333 0.978270i \(-0.566478\pi\)
−0.207333 + 0.978270i \(0.566478\pi\)
\(812\) 0 0
\(813\) 5504.00 0.237434
\(814\) 34965.0 1.50556
\(815\) −6140.00 −0.263895
\(816\) −7668.00 −0.328963
\(817\) 34606.0 1.48190
\(818\) −32610.0 −1.39387
\(819\) 0 0
\(820\) 975.000 0.0415225
\(821\) −10938.0 −0.464968 −0.232484 0.972600i \(-0.574685\pi\)
−0.232484 + 0.972600i \(0.574685\pi\)
\(822\) −360.000 −0.0152755
\(823\) 11540.0 0.488772 0.244386 0.969678i \(-0.421414\pi\)
0.244386 + 0.969678i \(0.421414\pi\)
\(824\) 31836.0 1.34595
\(825\) 2250.00 0.0949514
\(826\) 0 0
\(827\) −18762.0 −0.788898 −0.394449 0.918918i \(-0.629065\pi\)
−0.394449 + 0.918918i \(0.629065\pi\)
\(828\) −1587.00 −0.0666088
\(829\) −39610.0 −1.65948 −0.829742 0.558147i \(-0.811512\pi\)
−0.829742 + 0.558147i \(0.811512\pi\)
\(830\) 11520.0 0.481765
\(831\) 9380.00 0.391563
\(832\) 25547.0 1.06452
\(833\) 0 0
\(834\) 10248.0 0.425491
\(835\) −9645.00 −0.399735
\(836\) 5445.00 0.225262
\(837\) −8800.00 −0.363408
\(838\) −29187.0 −1.20316
\(839\) 39162.0 1.61147 0.805734 0.592277i \(-0.201771\pi\)
0.805734 + 0.592277i \(0.201771\pi\)
\(840\) 0 0
\(841\) 1855.00 0.0760589
\(842\) −37650.0 −1.54098
\(843\) −15642.0 −0.639074
\(844\) −3625.00 −0.147841
\(845\) 6420.00 0.261367
\(846\) −3105.00 −0.126185
\(847\) 0 0
\(848\) −42387.0 −1.71648
\(849\) 1316.00 0.0531979
\(850\) −4050.00 −0.163428
\(851\) −17871.0 −0.719871
\(852\) −96.0000 −0.00386022
\(853\) −11527.0 −0.462693 −0.231346 0.972871i \(-0.574313\pi\)
−0.231346 + 0.972871i \(0.574313\pi\)
\(854\) 0 0
\(855\) 13915.0 0.556588
\(856\) 15372.0 0.613790
\(857\) −41826.0 −1.66715 −0.833576 0.552405i \(-0.813710\pi\)
−0.833576 + 0.552405i \(0.813710\pi\)
\(858\) 15930.0 0.633848
\(859\) 35192.0 1.39783 0.698915 0.715205i \(-0.253667\pi\)
0.698915 + 0.715205i \(0.253667\pi\)
\(860\) −1430.00 −0.0567007
\(861\) 0 0
\(862\) 8964.00 0.354194
\(863\) −9063.00 −0.357483 −0.178742 0.983896i \(-0.557203\pi\)
−0.178742 + 0.983896i \(0.557203\pi\)
\(864\) −4500.00 −0.177191
\(865\) −3495.00 −0.137380
\(866\) 49848.0 1.95601
\(867\) 3994.00 0.156451
\(868\) 0 0
\(869\) −35190.0 −1.37369
\(870\) 4860.00 0.189390
\(871\) −16520.0 −0.642662
\(872\) 33600.0 1.30486
\(873\) −20746.0 −0.804291
\(874\) −25047.0 −0.969368
\(875\) 0 0
\(876\) −1336.00 −0.0515288
\(877\) 28439.0 1.09500 0.547501 0.836805i \(-0.315579\pi\)
0.547501 + 0.836805i \(0.315579\pi\)
\(878\) 22038.0 0.847092
\(879\) 11994.0 0.460236
\(880\) 15975.0 0.611951
\(881\) −9303.00 −0.355762 −0.177881 0.984052i \(-0.556924\pi\)
−0.177881 + 0.984052i \(0.556924\pi\)
\(882\) 0 0
\(883\) −14728.0 −0.561310 −0.280655 0.959809i \(-0.590552\pi\)
−0.280655 + 0.959809i \(0.590552\pi\)
\(884\) −3186.00 −0.121218
\(885\) 3600.00 0.136737
\(886\) 36.0000 0.00136506
\(887\) −17016.0 −0.644128 −0.322064 0.946718i \(-0.604377\pi\)
−0.322064 + 0.946718i \(0.604377\pi\)
\(888\) −10878.0 −0.411083
\(889\) 0 0
\(890\) −17910.0 −0.674544
\(891\) −18945.0 −0.712325
\(892\) −4960.00 −0.186181
\(893\) −5445.00 −0.204043
\(894\) 6516.00 0.243767
\(895\) 15585.0 0.582066
\(896\) 0 0
\(897\) −8142.00 −0.303070
\(898\) 29007.0 1.07792
\(899\) 14256.0 0.528881
\(900\) −575.000 −0.0212963
\(901\) −32238.0 −1.19201
\(902\) −26325.0 −0.971759
\(903\) 0 0
\(904\) 29232.0 1.07549
\(905\) −8990.00 −0.330207
\(906\) 17196.0 0.630573
\(907\) −24922.0 −0.912372 −0.456186 0.889884i \(-0.650785\pi\)
−0.456186 + 0.889884i \(0.650785\pi\)
\(908\) −1500.00 −0.0548230
\(909\) −15732.0 −0.574035
\(910\) 0 0
\(911\) 30714.0 1.11701 0.558507 0.829500i \(-0.311374\pi\)
0.558507 + 0.829500i \(0.311374\pi\)
\(912\) −17182.0 −0.623852
\(913\) −34560.0 −1.25276
\(914\) −28902.0 −1.04594
\(915\) −3920.00 −0.141630
\(916\) 6092.00 0.219744
\(917\) 0 0
\(918\) −16200.0 −0.582440
\(919\) 17426.0 0.625496 0.312748 0.949836i \(-0.398750\pi\)
0.312748 + 0.949836i \(0.398750\pi\)
\(920\) −7245.00 −0.259631
\(921\) 12452.0 0.445502
\(922\) −1026.00 −0.0366481
\(923\) 2832.00 0.100993
\(924\) 0 0
\(925\) −6475.00 −0.230159
\(926\) 7233.00 0.256686
\(927\) 34868.0 1.23540
\(928\) 7290.00 0.257873
\(929\) 26649.0 0.941147 0.470573 0.882361i \(-0.344047\pi\)
0.470573 + 0.882361i \(0.344047\pi\)
\(930\) 2640.00 0.0930850
\(931\) 0 0
\(932\) 138.000 0.00485015
\(933\) −9360.00 −0.328438
\(934\) −3618.00 −0.126750
\(935\) 12150.0 0.424971
\(936\) 28497.0 0.995143
\(937\) 27686.0 0.965274 0.482637 0.875820i \(-0.339679\pi\)
0.482637 + 0.875820i \(0.339679\pi\)
\(938\) 0 0
\(939\) −2056.00 −0.0714537
\(940\) 225.000 0.00780712
\(941\) −17808.0 −0.616923 −0.308461 0.951237i \(-0.599814\pi\)
−0.308461 + 0.951237i \(0.599814\pi\)
\(942\) 1374.00 0.0475237
\(943\) 13455.0 0.464640
\(944\) 25560.0 0.881258
\(945\) 0 0
\(946\) 38610.0 1.32698
\(947\) −6906.00 −0.236974 −0.118487 0.992956i \(-0.537804\pi\)
−0.118487 + 0.992956i \(0.537804\pi\)
\(948\) −1564.00 −0.0535827
\(949\) 39412.0 1.34812
\(950\) −9075.00 −0.309928
\(951\) −17244.0 −0.587986
\(952\) 0 0
\(953\) −20940.0 −0.711766 −0.355883 0.934530i \(-0.615820\pi\)
−0.355883 + 0.934530i \(0.615820\pi\)
\(954\) −41193.0 −1.39798
\(955\) −11940.0 −0.404575
\(956\) −5502.00 −0.186137
\(957\) −14580.0 −0.492481
\(958\) −1296.00 −0.0437076
\(959\) 0 0
\(960\) −4330.00 −0.145573
\(961\) −22047.0 −0.740056
\(962\) −45843.0 −1.53642
\(963\) 16836.0 0.563377
\(964\) 3551.00 0.118641
\(965\) 1360.00 0.0453678
\(966\) 0 0
\(967\) 9176.00 0.305150 0.152575 0.988292i \(-0.451243\pi\)
0.152575 + 0.988292i \(0.451243\pi\)
\(968\) −14574.0 −0.483911
\(969\) −13068.0 −0.433235
\(970\) 13530.0 0.447858
\(971\) 29763.0 0.983666 0.491833 0.870689i \(-0.336327\pi\)
0.491833 + 0.870689i \(0.336327\pi\)
\(972\) −3542.00 −0.116882
\(973\) 0 0
\(974\) −35688.0 −1.17404
\(975\) −2950.00 −0.0968981
\(976\) −27832.0 −0.912788
\(977\) −38490.0 −1.26039 −0.630197 0.776436i \(-0.717026\pi\)
−0.630197 + 0.776436i \(0.717026\pi\)
\(978\) 7368.00 0.240903
\(979\) 53730.0 1.75405
\(980\) 0 0
\(981\) 36800.0 1.19769
\(982\) −36828.0 −1.19677
\(983\) −12609.0 −0.409120 −0.204560 0.978854i \(-0.565576\pi\)
−0.204560 + 0.978854i \(0.565576\pi\)
\(984\) 8190.00 0.265333
\(985\) −10545.0 −0.341108
\(986\) 26244.0 0.847646
\(987\) 0 0
\(988\) −7139.00 −0.229880
\(989\) −19734.0 −0.634484
\(990\) 15525.0 0.498401
\(991\) 19820.0 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 3960.00 0.126744
\(993\) 3998.00 0.127767
\(994\) 0 0
\(995\) 7120.00 0.226853
\(996\) −1536.00 −0.0488655
\(997\) 46034.0 1.46230 0.731149 0.682218i \(-0.238984\pi\)
0.731149 + 0.682218i \(0.238984\pi\)
\(998\) −32628.0 −1.03489
\(999\) −25900.0 −0.820260
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 245.4.a.e.1.1 1
3.2 odd 2 2205.4.a.e.1.1 1
5.4 even 2 1225.4.a.b.1.1 1
7.2 even 3 35.4.e.a.11.1 2
7.3 odd 6 245.4.e.a.226.1 2
7.4 even 3 35.4.e.a.16.1 yes 2
7.5 odd 6 245.4.e.a.116.1 2
7.6 odd 2 245.4.a.f.1.1 1
21.2 odd 6 315.4.j.b.46.1 2
21.11 odd 6 315.4.j.b.226.1 2
21.20 even 2 2205.4.a.g.1.1 1
28.11 odd 6 560.4.q.b.401.1 2
28.23 odd 6 560.4.q.b.81.1 2
35.2 odd 12 175.4.k.b.74.2 4
35.4 even 6 175.4.e.b.51.1 2
35.9 even 6 175.4.e.b.151.1 2
35.18 odd 12 175.4.k.b.149.2 4
35.23 odd 12 175.4.k.b.74.1 4
35.32 odd 12 175.4.k.b.149.1 4
35.34 odd 2 1225.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.e.a.11.1 2 7.2 even 3
35.4.e.a.16.1 yes 2 7.4 even 3
175.4.e.b.51.1 2 35.4 even 6
175.4.e.b.151.1 2 35.9 even 6
175.4.k.b.74.1 4 35.23 odd 12
175.4.k.b.74.2 4 35.2 odd 12
175.4.k.b.149.1 4 35.32 odd 12
175.4.k.b.149.2 4 35.18 odd 12
245.4.a.e.1.1 1 1.1 even 1 trivial
245.4.a.f.1.1 1 7.6 odd 2
245.4.e.a.116.1 2 7.5 odd 6
245.4.e.a.226.1 2 7.3 odd 6
315.4.j.b.46.1 2 21.2 odd 6
315.4.j.b.226.1 2 21.11 odd 6
560.4.q.b.81.1 2 28.23 odd 6
560.4.q.b.401.1 2 28.11 odd 6
1225.4.a.a.1.1 1 35.34 odd 2
1225.4.a.b.1.1 1 5.4 even 2
2205.4.a.e.1.1 1 3.2 odd 2
2205.4.a.g.1.1 1 21.20 even 2