Properties

Label 2205.4.a.e.1.1
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,4,Mod(1,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(130.099211563\)
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 \(\chi\) \(=\) 2205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{2} +1.00000 q^{4} -5.00000 q^{5} +21.0000 q^{8} +O(q^{10})\) \(q-3.00000 q^{2} +1.00000 q^{4} -5.00000 q^{5} +21.0000 q^{8} +15.0000 q^{10} +45.0000 q^{11} +59.0000 q^{13} -71.0000 q^{16} +54.0000 q^{17} -121.000 q^{19} -5.00000 q^{20} -135.000 q^{22} -69.0000 q^{23} +25.0000 q^{25} -177.000 q^{26} +162.000 q^{29} -88.0000 q^{31} +45.0000 q^{32} -162.000 q^{34} -259.000 q^{37} +363.000 q^{38} -105.000 q^{40} -195.000 q^{41} -286.000 q^{43} +45.0000 q^{44} +207.000 q^{46} -45.0000 q^{47} -75.0000 q^{50} +59.0000 q^{52} -597.000 q^{53} -225.000 q^{55} -486.000 q^{58} +360.000 q^{59} +392.000 q^{61} +264.000 q^{62} +433.000 q^{64} -295.000 q^{65} -280.000 q^{67} +54.0000 q^{68} -48.0000 q^{71} +668.000 q^{73} +777.000 q^{74} -121.000 q^{76} +782.000 q^{79} +355.000 q^{80} +585.000 q^{82} -768.000 q^{83} -270.000 q^{85} +858.000 q^{86} +945.000 q^{88} +1194.00 q^{89} -69.0000 q^{92} +135.000 q^{94} +605.000 q^{95} +902.000 q^{97} +O(q^{100})\)

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 \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) 0 0
\(4\) 1.00000 0.125000
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 21.0000 0.928078
\(9\) 0 0
\(10\) 15.0000 0.474342
\(11\) 45.0000 1.23346 0.616728 0.787177i \(-0.288458\pi\)
0.616728 + 0.787177i \(0.288458\pi\)
\(12\) 0 0
\(13\) 59.0000 1.25874 0.629371 0.777105i \(-0.283312\pi\)
0.629371 + 0.777105i \(0.283312\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 54.0000 0.770407 0.385204 0.922832i \(-0.374131\pi\)
0.385204 + 0.922832i \(0.374131\pi\)
\(18\) 0 0
\(19\) −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 \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −177.000 −1.33510
\(27\) 0 0
\(28\) 0 0
\(29\) 162.000 1.03733 0.518666 0.854977i \(-0.326429\pi\)
0.518666 + 0.854977i \(0.326429\pi\)
\(30\) 0 0
\(31\) −88.0000 −0.509847 −0.254924 0.966961i \(-0.582050\pi\)
−0.254924 + 0.966961i \(0.582050\pi\)
\(32\) 45.0000 0.248592
\(33\) 0 0
\(34\) −162.000 −0.817140
\(35\) 0 0
\(36\) 0 0
\(37\) −259.000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 363.000 1.54964
\(39\) 0 0
\(40\) −105.000 −0.415049
\(41\) −195.000 −0.742778 −0.371389 0.928477i \(-0.621118\pi\)
−0.371389 + 0.928477i \(0.621118\pi\)
\(42\) 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
\(45\) 0 0
\(46\) 207.000 0.663489
\(47\) −45.0000 −0.139658 −0.0698290 0.997559i \(-0.522245\pi\)
−0.0698290 + 0.997559i \(0.522245\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −75.0000 −0.212132
\(51\) 0 0
\(52\) 59.0000 0.157343
\(53\) −597.000 −1.54725 −0.773625 0.633644i \(-0.781559\pi\)
−0.773625 + 0.633644i \(0.781559\pi\)
\(54\) 0 0
\(55\) −225.000 −0.551618
\(56\) 0 0
\(57\) 0 0
\(58\) −486.000 −1.10026
\(59\) 360.000 0.794373 0.397187 0.917738i \(-0.369987\pi\)
0.397187 + 0.917738i \(0.369987\pi\)
\(60\) 0 0
\(61\) 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
\(66\) 0 0
\(67\) −280.000 −0.510559 −0.255279 0.966867i \(-0.582167\pi\)
−0.255279 + 0.966867i \(0.582167\pi\)
\(68\) 54.0000 0.0963009
\(69\) 0 0
\(70\) 0 0
\(71\) −48.0000 −0.0802331 −0.0401166 0.999195i \(-0.512773\pi\)
−0.0401166 + 0.999195i \(0.512773\pi\)
\(72\) 0 0
\(73\) 668.000 1.07101 0.535503 0.844533i \(-0.320122\pi\)
0.535503 + 0.844533i \(0.320122\pi\)
\(74\) 777.000 1.22060
\(75\) 0 0
\(76\) −121.000 −0.182627
\(77\) 0 0
\(78\) 0 0
\(79\) 782.000 1.11369 0.556847 0.830615i \(-0.312011\pi\)
0.556847 + 0.830615i \(0.312011\pi\)
\(80\) 355.000 0.496128
\(81\) 0 0
\(82\) 585.000 0.787835
\(83\) −768.000 −1.01565 −0.507825 0.861460i \(-0.669550\pi\)
−0.507825 + 0.861460i \(0.669550\pi\)
\(84\) 0 0
\(85\) −270.000 −0.344537
\(86\) 858.000 1.07582
\(87\) 0 0
\(88\) 945.000 1.14474
\(89\) 1194.00 1.42206 0.711032 0.703159i \(-0.248228\pi\)
0.711032 + 0.703159i \(0.248228\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −69.0000 −0.0781929
\(93\) 0 0
\(94\) 135.000 0.148130
\(95\) 605.000 0.653386
\(96\) 0 0
\(97\) 902.000 0.944167 0.472084 0.881554i \(-0.343502\pi\)
0.472084 + 0.881554i \(0.343502\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 25.0000 0.0250000
\(101\) −684.000 −0.673867 −0.336933 0.941528i \(-0.609390\pi\)
−0.336933 + 0.941528i \(0.609390\pi\)
\(102\) 0 0
\(103\) −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 \(-0.392723\pi\)
0.330678 + 0.943744i \(0.392723\pi\)
\(108\) 0 0
\(109\) −1600.00 −1.40598 −0.702992 0.711198i \(-0.748153\pi\)
−0.702992 + 0.711198i \(0.748153\pi\)
\(110\) 675.000 0.585079
\(111\) 0 0
\(112\) 0 0
\(113\) 1392.00 1.15883 0.579417 0.815031i \(-0.303280\pi\)
0.579417 + 0.815031i \(0.303280\pi\)
\(114\) 0 0
\(115\) 345.000 0.279751
\(116\) 162.000 0.129667
\(117\) 0 0
\(118\) −1080.00 −0.842560
\(119\) 0 0
\(120\) 0 0
\(121\) 694.000 0.521412
\(122\) −1176.00 −0.872705
\(123\) 0 0
\(124\) −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
\(129\) 0 0
\(130\) 885.000 0.597074
\(131\) −2019.00 −1.34657 −0.673286 0.739382i \(-0.735118\pi\)
−0.673286 + 0.739382i \(0.735118\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 840.000 0.541529
\(135\) 0 0
\(136\) 1134.00 0.714998
\(137\) −60.0000 −0.0374171 −0.0187086 0.999825i \(-0.505955\pi\)
−0.0187086 + 0.999825i \(0.505955\pi\)
\(138\) 0 0
\(139\) −1708.00 −1.04224 −0.521118 0.853485i \(-0.674485\pi\)
−0.521118 + 0.853485i \(0.674485\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 144.000 0.0851001
\(143\) 2655.00 1.55260
\(144\) 0 0
\(145\) −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 \(-0.403496\pi\)
0.298552 + 0.954393i \(0.403496\pi\)
\(150\) 0 0
\(151\) −2866.00 −1.54458 −0.772291 0.635269i \(-0.780889\pi\)
−0.772291 + 0.635269i \(0.780889\pi\)
\(152\) −2541.00 −1.35594
\(153\) 0 0
\(154\) 0 0
\(155\) 440.000 0.228011
\(156\) 0 0
\(157\) −229.000 −0.116409 −0.0582044 0.998305i \(-0.518538\pi\)
−0.0582044 + 0.998305i \(0.518538\pi\)
\(158\) −2346.00 −1.18125
\(159\) 0 0
\(160\) −225.000 −0.111174
\(161\) 0 0
\(162\) 0 0
\(163\) −1228.00 −0.590088 −0.295044 0.955484i \(-0.595334\pi\)
−0.295044 + 0.955484i \(0.595334\pi\)
\(164\) −195.000 −0.0928472
\(165\) 0 0
\(166\) 2304.00 1.07726
\(167\) 1929.00 0.893835 0.446918 0.894575i \(-0.352522\pi\)
0.446918 + 0.894575i \(0.352522\pi\)
\(168\) 0 0
\(169\) 1284.00 0.584433
\(170\) 810.000 0.365436
\(171\) 0 0
\(172\) −286.000 −0.126787
\(173\) 699.000 0.307191 0.153595 0.988134i \(-0.450915\pi\)
0.153595 + 0.988134i \(0.450915\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3195.00 −1.36836
\(177\) 0 0
\(178\) −3582.00 −1.50833
\(179\) −3117.00 −1.30154 −0.650770 0.759275i \(-0.725554\pi\)
−0.650770 + 0.759275i \(0.725554\pi\)
\(180\) 0 0
\(181\) −1798.00 −0.738366 −0.369183 0.929357i \(-0.620362\pi\)
−0.369183 + 0.929357i \(0.620362\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1449.00 −0.580553
\(185\) 1295.00 0.514650
\(186\) 0 0
\(187\) 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 \(-0.350593\pi\)
0.452329 + 0.891851i \(0.350593\pi\)
\(192\) 0 0
\(193\) 272.000 0.101446 0.0507228 0.998713i \(-0.483848\pi\)
0.0507228 + 0.998713i \(0.483848\pi\)
\(194\) −2706.00 −1.00144
\(195\) 0 0
\(196\) 0 0
\(197\) 2109.00 0.762741 0.381371 0.924422i \(-0.375452\pi\)
0.381371 + 0.924422i \(0.375452\pi\)
\(198\) 0 0
\(199\) 1424.00 0.507260 0.253630 0.967301i \(-0.418375\pi\)
0.253630 + 0.967301i \(0.418375\pi\)
\(200\) 525.000 0.185616
\(201\) 0 0
\(202\) 2052.00 0.714744
\(203\) 0 0
\(204\) 0 0
\(205\) 975.000 0.332180
\(206\) 4548.00 1.53822
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) −2196.00 −0.701474
\(215\) 1430.00 0.453606
\(216\) 0 0
\(217\) 0 0
\(218\) 4800.00 1.49127
\(219\) 0 0
\(220\) −225.000 −0.0689523
\(221\) 3186.00 0.969745
\(222\) 0 0
\(223\) −4960.00 −1.48944 −0.744722 0.667374i \(-0.767418\pi\)
−0.744722 + 0.667374i \(0.767418\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −4176.00 −1.22913
\(227\) 1500.00 0.438584 0.219292 0.975659i \(-0.429625\pi\)
0.219292 + 0.975659i \(0.429625\pi\)
\(228\) 0 0
\(229\) 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 \(-0.506176\pi\)
−0.0194006 + 0.999812i \(0.506176\pi\)
\(234\) 0 0
\(235\) 225.000 0.0624569
\(236\) 360.000 0.0992966
\(237\) 0 0
\(238\) 0 0
\(239\) 5502.00 1.48910 0.744550 0.667567i \(-0.232664\pi\)
0.744550 + 0.667567i \(0.232664\pi\)
\(240\) 0 0
\(241\) 3551.00 0.949129 0.474564 0.880221i \(-0.342606\pi\)
0.474564 + 0.880221i \(0.342606\pi\)
\(242\) −2082.00 −0.553041
\(243\) 0 0
\(244\) 392.000 0.102849
\(245\) 0 0
\(246\) 0 0
\(247\) −7139.00 −1.83904
\(248\) −1848.00 −0.473178
\(249\) 0 0
\(250\) 375.000 0.0948683
\(251\) −7065.00 −1.77665 −0.888324 0.459216i \(-0.848130\pi\)
−0.888324 + 0.459216i \(0.848130\pi\)
\(252\) 0 0
\(253\) −3105.00 −0.771580
\(254\) −2409.00 −0.595095
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) 4080.00 0.990286 0.495143 0.868812i \(-0.335116\pi\)
0.495143 + 0.868812i \(0.335116\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −295.000 −0.0703659
\(261\) 0 0
\(262\) 6057.00 1.42825
\(263\) 3288.00 0.770900 0.385450 0.922729i \(-0.374046\pi\)
0.385450 + 0.922729i \(0.374046\pi\)
\(264\) 0 0
\(265\) 2985.00 0.691951
\(266\) 0 0
\(267\) 0 0
\(268\) −280.000 −0.0638199
\(269\) 3264.00 0.739813 0.369906 0.929069i \(-0.379390\pi\)
0.369906 + 0.929069i \(0.379390\pi\)
\(270\) 0 0
\(271\) −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
\(276\) 0 0
\(277\) −4690.00 −1.01731 −0.508655 0.860971i \(-0.669857\pi\)
−0.508655 + 0.860971i \(0.669857\pi\)
\(278\) 5124.00 1.10546
\(279\) 0 0
\(280\) 0 0
\(281\) −7821.00 −1.66036 −0.830181 0.557494i \(-0.811763\pi\)
−0.830181 + 0.557494i \(0.811763\pi\)
\(282\) 0 0
\(283\) −658.000 −0.138212 −0.0691061 0.997609i \(-0.522015\pi\)
−0.0691061 + 0.997609i \(0.522015\pi\)
\(284\) −48.0000 −0.0100291
\(285\) 0 0
\(286\) −7965.00 −1.64678
\(287\) 0 0
\(288\) 0 0
\(289\) −1997.00 −0.406473
\(290\) 2430.00 0.492050
\(291\) 0 0
\(292\) 668.000 0.133876
\(293\) 5997.00 1.19573 0.597864 0.801597i \(-0.296016\pi\)
0.597864 + 0.801597i \(0.296016\pi\)
\(294\) 0 0
\(295\) −1800.00 −0.355254
\(296\) −5439.00 −1.06803
\(297\) 0 0
\(298\) −3258.00 −0.633325
\(299\) −4071.00 −0.787398
\(300\) 0 0
\(301\) 0 0
\(302\) 8598.00 1.63828
\(303\) 0 0
\(304\) 8591.00 1.62081
\(305\) −1960.00 −0.367965
\(306\) 0 0
\(307\) −6226.00 −1.15745 −0.578724 0.815523i \(-0.696449\pi\)
−0.578724 + 0.815523i \(0.696449\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1320.00 −0.241842
\(311\) −4680.00 −0.853307 −0.426653 0.904415i \(-0.640308\pi\)
−0.426653 + 0.904415i \(0.640308\pi\)
\(312\) 0 0
\(313\) 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 \(-0.776677\pi\)
−0.763817 + 0.645433i \(0.776677\pi\)
\(318\) 0 0
\(319\) 7290.00 1.27950
\(320\) −2165.00 −0.378210
\(321\) 0 0
\(322\) 0 0
\(323\) −6534.00 −1.12558
\(324\) 0 0
\(325\) 1475.00 0.251749
\(326\) 3684.00 0.625883
\(327\) 0 0
\(328\) −4095.00 −0.689355
\(329\) 0 0
\(330\) 0 0
\(331\) −1999.00 −0.331949 −0.165974 0.986130i \(-0.553077\pi\)
−0.165974 + 0.986130i \(0.553077\pi\)
\(332\) −768.000 −0.126956
\(333\) 0 0
\(334\) −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
\(339\) 0 0
\(340\) −270.000 −0.0430671
\(341\) −3960.00 −0.628874
\(342\) 0 0
\(343\) 0 0
\(344\) −6006.00 −0.941342
\(345\) 0 0
\(346\) −2097.00 −0.325825
\(347\) −4320.00 −0.668328 −0.334164 0.942515i \(-0.608454\pi\)
−0.334164 + 0.942515i \(0.608454\pi\)
\(348\) 0 0
\(349\) 7922.00 1.21506 0.607529 0.794298i \(-0.292161\pi\)
0.607529 + 0.794298i \(0.292161\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2025.00 0.306627
\(353\) −828.000 −0.124844 −0.0624221 0.998050i \(-0.519882\pi\)
−0.0624221 + 0.998050i \(0.519882\pi\)
\(354\) 0 0
\(355\) 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 \(-0.468361\pi\)
0.0992344 + 0.995064i \(0.468361\pi\)
\(360\) 0 0
\(361\) 7782.00 1.13457
\(362\) 5394.00 0.783156
\(363\) 0 0
\(364\) 0 0
\(365\) −3340.00 −0.478969
\(366\) 0 0
\(367\) 2801.00 0.398395 0.199198 0.979959i \(-0.436166\pi\)
0.199198 + 0.979959i \(0.436166\pi\)
\(368\) 4899.00 0.693962
\(369\) 0 0
\(370\) −3885.00 −0.545869
\(371\) 0 0
\(372\) 0 0
\(373\) 6602.00 0.916457 0.458229 0.888834i \(-0.348484\pi\)
0.458229 + 0.888834i \(0.348484\pi\)
\(374\) −7290.00 −1.00791
\(375\) 0 0
\(376\) −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
\(381\) 0 0
\(382\) −7164.00 −0.959534
\(383\) −945.000 −0.126076 −0.0630382 0.998011i \(-0.520079\pi\)
−0.0630382 + 0.998011i \(0.520079\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −816.000 −0.107599
\(387\) 0 0
\(388\) 902.000 0.118021
\(389\) −12036.0 −1.56876 −0.784382 0.620278i \(-0.787020\pi\)
−0.784382 + 0.620278i \(0.787020\pi\)
\(390\) 0 0
\(391\) −3726.00 −0.481923
\(392\) 0 0
\(393\) 0 0
\(394\) −6327.00 −0.809009
\(395\) −3910.00 −0.498059
\(396\) 0 0
\(397\) −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 \(-0.644726\pi\)
−0.439165 + 0.898407i \(0.644726\pi\)
\(402\) 0 0
\(403\) −5192.00 −0.641767
\(404\) −684.000 −0.0842333
\(405\) 0 0
\(406\) 0 0
\(407\) −11655.0 −1.41945
\(408\) 0 0
\(409\) −10870.0 −1.31415 −0.657074 0.753826i \(-0.728206\pi\)
−0.657074 + 0.753826i \(0.728206\pi\)
\(410\) −2925.00 −0.352330
\(411\) 0 0
\(412\) −1516.00 −0.181281
\(413\) 0 0
\(414\) 0 0
\(415\) 3840.00 0.454212
\(416\) 2655.00 0.312914
\(417\) 0 0
\(418\) 16335.0 1.91141
\(419\) 9729.00 1.13435 0.567175 0.823597i \(-0.308036\pi\)
0.567175 + 0.823597i \(0.308036\pi\)
\(420\) 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
\(423\) 0 0
\(424\) −12537.0 −1.43597
\(425\) 1350.00 0.154081
\(426\) 0 0
\(427\) 0 0
\(428\) 732.000 0.0826695
\(429\) 0 0
\(430\) −4290.00 −0.481121
\(431\) −2988.00 −0.333937 −0.166969 0.985962i \(-0.553398\pi\)
−0.166969 + 0.985962i \(0.553398\pi\)
\(432\) 0 0
\(433\) 16616.0 1.84414 0.922072 0.387019i \(-0.126495\pi\)
0.922072 + 0.387019i \(0.126495\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1600.00 −0.175748
\(437\) 8349.00 0.913929
\(438\) 0 0
\(439\) 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 \(-0.500205\pi\)
−0.000643496 1.00000i \(0.500205\pi\)
\(444\) 0 0
\(445\) −5970.00 −0.635967
\(446\) 14880.0 1.57979
\(447\) 0 0
\(448\) 0 0
\(449\) −9669.00 −1.01628 −0.508138 0.861275i \(-0.669666\pi\)
−0.508138 + 0.861275i \(0.669666\pi\)
\(450\) 0 0
\(451\) −8775.00 −0.916183
\(452\) 1392.00 0.144854
\(453\) 0 0
\(454\) −4500.00 −0.465188
\(455\) 0 0
\(456\) 0 0
\(457\) −9634.00 −0.986126 −0.493063 0.869994i \(-0.664123\pi\)
−0.493063 + 0.869994i \(0.664123\pi\)
\(458\) −18276.0 −1.86459
\(459\) 0 0
\(460\) 345.000 0.0349689
\(461\) 342.000 0.0345521 0.0172761 0.999851i \(-0.494501\pi\)
0.0172761 + 0.999851i \(0.494501\pi\)
\(462\) 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
\(465\) 0 0
\(466\) 414.000 0.0411549
\(467\) 1206.00 0.119501 0.0597506 0.998213i \(-0.480969\pi\)
0.0597506 + 0.998213i \(0.480969\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −675.000 −0.0662456
\(471\) 0 0
\(472\) 7560.00 0.737240
\(473\) −12870.0 −1.25109
\(474\) 0 0
\(475\) −3025.00 −0.292203
\(476\) 0 0
\(477\) 0 0
\(478\) −16506.0 −1.57943
\(479\) 432.000 0.0412079 0.0206039 0.999788i \(-0.493441\pi\)
0.0206039 + 0.999788i \(0.493441\pi\)
\(480\) 0 0
\(481\) −15281.0 −1.44855
\(482\) −10653.0 −1.00670
\(483\) 0 0
\(484\) 694.000 0.0651766
\(485\) −4510.00 −0.422244
\(486\) 0 0
\(487\) −11896.0 −1.10690 −0.553449 0.832883i \(-0.686689\pi\)
−0.553449 + 0.832883i \(0.686689\pi\)
\(488\) 8232.00 0.763617
\(489\) 0 0
\(490\) 0 0
\(491\) 12276.0 1.12833 0.564163 0.825663i \(-0.309199\pi\)
0.564163 + 0.825663i \(0.309199\pi\)
\(492\) 0 0
\(493\) 8748.00 0.799169
\(494\) 21417.0 1.95060
\(495\) 0 0
\(496\) 6248.00 0.565612
\(497\) 0 0
\(498\) 0 0
\(499\) −10876.0 −0.975705 −0.487852 0.872926i \(-0.662220\pi\)
−0.487852 + 0.872926i \(0.662220\pi\)
\(500\) −125.000 −0.0111803
\(501\) 0 0
\(502\) 21195.0 1.88442
\(503\) −12000.0 −1.06372 −0.531862 0.846831i \(-0.678508\pi\)
−0.531862 + 0.846831i \(0.678508\pi\)
\(504\) 0 0
\(505\) 3420.00 0.301362
\(506\) 9315.00 0.818384
\(507\) 0 0
\(508\) 803.000 0.0701326
\(509\) 11682.0 1.01728 0.508640 0.860979i \(-0.330148\pi\)
0.508640 + 0.860979i \(0.330148\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 8733.00 0.753804
\(513\) 0 0
\(514\) −12240.0 −1.05036
\(515\) 7580.00 0.648572
\(516\) 0 0
\(517\) −2025.00 −0.172262
\(518\) 0 0
\(519\) 0 0
\(520\) −6195.00 −0.522440
\(521\) −9609.00 −0.808019 −0.404010 0.914755i \(-0.632384\pi\)
−0.404010 + 0.914755i \(0.632384\pi\)
\(522\) 0 0
\(523\) 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
\(528\) 0 0
\(529\) −7406.00 −0.608696
\(530\) −8955.00 −0.733925
\(531\) 0 0
\(532\) 0 0
\(533\) −11505.0 −0.934966
\(534\) 0 0
\(535\) −3660.00 −0.295767
\(536\) −5880.00 −0.473838
\(537\) 0 0
\(538\) −9792.00 −0.784690
\(539\) 0 0
\(540\) 0 0
\(541\) 8072.00 0.641483 0.320742 0.947167i \(-0.396068\pi\)
0.320742 + 0.947167i \(0.396068\pi\)
\(542\) 8256.00 0.654291
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) −3375.00 −0.261655
\(551\) −19602.0 −1.51556
\(552\) 0 0
\(553\) 0 0
\(554\) 14070.0 1.07902
\(555\) 0 0
\(556\) −1708.00 −0.130279
\(557\) −18363.0 −1.39689 −0.698443 0.715666i \(-0.746123\pi\)
−0.698443 + 0.715666i \(0.746123\pi\)
\(558\) 0 0
\(559\) −16874.0 −1.27673
\(560\) 0 0
\(561\) 0 0
\(562\) 23463.0 1.76108
\(563\) 6294.00 0.471155 0.235578 0.971856i \(-0.424302\pi\)
0.235578 + 0.971856i \(0.424302\pi\)
\(564\) 0 0
\(565\) −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 \(-0.642272\pi\)
−0.432226 + 0.901765i \(0.642272\pi\)
\(570\) 0 0
\(571\) 1052.00 0.0771013 0.0385506 0.999257i \(-0.487726\pi\)
0.0385506 + 0.999257i \(0.487726\pi\)
\(572\) 2655.00 0.194075
\(573\) 0 0
\(574\) 0 0
\(575\) −1725.00 −0.125109
\(576\) 0 0
\(577\) −13156.0 −0.949205 −0.474603 0.880200i \(-0.657408\pi\)
−0.474603 + 0.880200i \(0.657408\pi\)
\(578\) 5991.00 0.431129
\(579\) 0 0
\(580\) −810.000 −0.0579887
\(581\) 0 0
\(582\) 0 0
\(583\) −26865.0 −1.90846
\(584\) 14028.0 0.993977
\(585\) 0 0
\(586\) −17991.0 −1.26826
\(587\) 13368.0 0.939960 0.469980 0.882677i \(-0.344261\pi\)
0.469980 + 0.882677i \(0.344261\pi\)
\(588\) 0 0
\(589\) 10648.0 0.744895
\(590\) 5400.00 0.376804
\(591\) 0 0
\(592\) 18389.0 1.27666
\(593\) −26664.0 −1.84647 −0.923237 0.384231i \(-0.874467\pi\)
−0.923237 + 0.384231i \(0.874467\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1086.00 0.0746381
\(597\) 0 0
\(598\) 12213.0 0.835162
\(599\) −7614.00 −0.519365 −0.259682 0.965694i \(-0.583618\pi\)
−0.259682 + 0.965694i \(0.583618\pi\)
\(600\) 0 0
\(601\) 6410.00 0.435057 0.217529 0.976054i \(-0.430200\pi\)
0.217529 + 0.976054i \(0.430200\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2866.00 −0.193073
\(605\) −3470.00 −0.233183
\(606\) 0 0
\(607\) −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
\(612\) 0 0
\(613\) 3737.00 0.246225 0.123113 0.992393i \(-0.460712\pi\)
0.123113 + 0.992393i \(0.460712\pi\)
\(614\) 18678.0 1.22766
\(615\) 0 0
\(616\) 0 0
\(617\) −18078.0 −1.17957 −0.589784 0.807561i \(-0.700787\pi\)
−0.589784 + 0.807561i \(0.700787\pi\)
\(618\) 0 0
\(619\) 12287.0 0.797829 0.398915 0.916988i \(-0.369387\pi\)
0.398915 + 0.916988i \(0.369387\pi\)
\(620\) 440.000 0.0285013
\(621\) 0 0
\(622\) 14040.0 0.905069
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −3084.00 −0.196903
\(627\) 0 0
\(628\) −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
\(633\) 0 0
\(634\) 25866.0 1.62030
\(635\) −4015.00 −0.250914
\(636\) 0 0
\(637\) 0 0
\(638\) −21870.0 −1.35712
\(639\) 0 0
\(640\) 8295.00 0.512326
\(641\) −10779.0 −0.664189 −0.332094 0.943246i \(-0.607755\pi\)
−0.332094 + 0.943246i \(0.607755\pi\)
\(642\) 0 0
\(643\) 8882.00 0.544746 0.272373 0.962192i \(-0.412191\pi\)
0.272373 + 0.962192i \(0.412191\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 19602.0 1.19386
\(647\) 11019.0 0.669554 0.334777 0.942297i \(-0.391339\pi\)
0.334777 + 0.942297i \(0.391339\pi\)
\(648\) 0 0
\(649\) 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 \(-0.733229\pi\)
−0.668887 + 0.743364i \(0.733229\pi\)
\(654\) 0 0
\(655\) 10095.0 0.602205
\(656\) 13845.0 0.824019
\(657\) 0 0
\(658\) 0 0
\(659\) 11856.0 0.700826 0.350413 0.936595i \(-0.386041\pi\)
0.350413 + 0.936595i \(0.386041\pi\)
\(660\) 0 0
\(661\) −33244.0 −1.95619 −0.978095 0.208158i \(-0.933253\pi\)
−0.978095 + 0.208158i \(0.933253\pi\)
\(662\) 5997.00 0.352085
\(663\) 0 0
\(664\) −16128.0 −0.942602
\(665\) 0 0
\(666\) 0 0
\(667\) −11178.0 −0.648896
\(668\) 1929.00 0.111729
\(669\) 0 0
\(670\) −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
\(675\) 0 0
\(676\) 1284.00 0.0730542
\(677\) 12597.0 0.715129 0.357564 0.933889i \(-0.383607\pi\)
0.357564 + 0.933889i \(0.383607\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −5670.00 −0.319757
\(681\) 0 0
\(682\) 11880.0 0.667022
\(683\) 8340.00 0.467235 0.233617 0.972329i \(-0.424944\pi\)
0.233617 + 0.972329i \(0.424944\pi\)
\(684\) 0 0
\(685\) 300.000 0.0167334
\(686\) 0 0
\(687\) 0 0
\(688\) 20306.0 1.12523
\(689\) −35223.0 −1.94759
\(690\) 0 0
\(691\) −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
\(696\) 0 0
\(697\) −10530.0 −0.572241
\(698\) −23766.0 −1.28876
\(699\) 0 0
\(700\) 0 0
\(701\) −474.000 −0.0255388 −0.0127694 0.999918i \(-0.504065\pi\)
−0.0127694 + 0.999918i \(0.504065\pi\)
\(702\) 0 0
\(703\) 31339.0 1.68133
\(704\) 19485.0 1.04314
\(705\) 0 0
\(706\) 2484.00 0.132417
\(707\) 0 0
\(708\) 0 0
\(709\) −25126.0 −1.33093 −0.665463 0.746431i \(-0.731766\pi\)
−0.665463 + 0.746431i \(0.731766\pi\)
\(710\) −720.000 −0.0380579
\(711\) 0 0
\(712\) 25074.0 1.31979
\(713\) 6072.00 0.318932
\(714\) 0 0
\(715\) −13275.0 −0.694345
\(716\) −3117.00 −0.162692
\(717\) 0 0
\(718\) −4050.00 −0.210508
\(719\) 7296.00 0.378435 0.189218 0.981935i \(-0.439405\pi\)
0.189218 + 0.981935i \(0.439405\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −23346.0 −1.20339
\(723\) 0 0
\(724\) −1798.00 −0.0922958
\(725\) 4050.00 0.207467
\(726\) 0 0
\(727\) −15421.0 −0.786703 −0.393352 0.919388i \(-0.628684\pi\)
−0.393352 + 0.919388i \(0.628684\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 10020.0 0.508023
\(731\) −15444.0 −0.781419
\(732\) 0 0
\(733\) −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
\(738\) 0 0
\(739\) −13381.0 −0.666073 −0.333037 0.942914i \(-0.608073\pi\)
−0.333037 + 0.942914i \(0.608073\pi\)
\(740\) 1295.00 0.0643313
\(741\) 0 0
\(742\) 0 0
\(743\) −5487.00 −0.270927 −0.135463 0.990782i \(-0.543252\pi\)
−0.135463 + 0.990782i \(0.543252\pi\)
\(744\) 0 0
\(745\) −5430.00 −0.267033
\(746\) −19806.0 −0.972050
\(747\) 0 0
\(748\) 2430.00 0.118783
\(749\) 0 0
\(750\) 0 0
\(751\) 6638.00 0.322535 0.161268 0.986911i \(-0.448442\pi\)
0.161268 + 0.986911i \(0.448442\pi\)
\(752\) 3195.00 0.154933
\(753\) 0 0
\(754\) −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
\(759\) 0 0
\(760\) 12705.0 0.606393
\(761\) 3651.00 0.173914 0.0869571 0.996212i \(-0.472286\pi\)
0.0869571 + 0.996212i \(0.472286\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 2388.00 0.113082
\(765\) 0 0
\(766\) 2835.00 0.133724
\(767\) 21240.0 0.999911
\(768\) 0 0
\(769\) 29855.0 1.40000 0.699999 0.714144i \(-0.253184\pi\)
0.699999 + 0.714144i \(0.253184\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 272.000 0.0126807
\(773\) 6519.00 0.303327 0.151664 0.988432i \(-0.451537\pi\)
0.151664 + 0.988432i \(0.451537\pi\)
\(774\) 0 0
\(775\) −2200.00 −0.101969
\(776\) 18942.0 0.876261
\(777\) 0 0
\(778\) 36108.0 1.66393
\(779\) 23595.0 1.08521
\(780\) 0 0
\(781\) −2160.00 −0.0989640
\(782\) 11178.0 0.511157
\(783\) 0 0
\(784\) 0 0
\(785\) 1145.00 0.0520596
\(786\) 0 0
\(787\) 35114.0 1.59044 0.795222 0.606319i \(-0.207354\pi\)
0.795222 + 0.606319i \(0.207354\pi\)
\(788\) 2109.00 0.0953427
\(789\) 0 0
\(790\) 11730.0 0.528272
\(791\) 0 0
\(792\) 0 0
\(793\) 23128.0 1.03569
\(794\) 8094.00 0.361770
\(795\) 0 0
\(796\) 1424.00 0.0634075
\(797\) −20910.0 −0.929323 −0.464661 0.885488i \(-0.653824\pi\)
−0.464661 + 0.885488i \(0.653824\pi\)
\(798\) 0 0
\(799\) −2430.00 −0.107594
\(800\) 1125.00 0.0497184
\(801\) 0 0
\(802\) 21159.0 0.931609
\(803\) 30060.0 1.32104
\(804\) 0 0
\(805\) 0 0
\(806\) 15576.0 0.680696
\(807\) 0 0
\(808\) −14364.0 −0.625401
\(809\) −4431.00 −0.192566 −0.0962829 0.995354i \(-0.530695\pi\)
−0.0962829 + 0.995354i \(0.530695\pi\)
\(810\) 0 0
\(811\) −9577.00 −0.414666 −0.207333 0.978270i \(-0.566478\pi\)
−0.207333 + 0.978270i \(0.566478\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 34965.0 1.50556
\(815\) 6140.00 0.263895
\(816\) 0 0
\(817\) 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 \(-0.425315\pi\)
0.232484 + 0.972600i \(0.425315\pi\)
\(822\) 0 0
\(823\) 11540.0 0.488772 0.244386 0.969678i \(-0.421414\pi\)
0.244386 + 0.969678i \(0.421414\pi\)
\(824\) −31836.0 −1.34595
\(825\) 0 0
\(826\) 0 0
\(827\) 18762.0 0.788898 0.394449 0.918918i \(-0.370935\pi\)
0.394449 + 0.918918i \(0.370935\pi\)
\(828\) 0 0
\(829\) −39610.0 −1.65948 −0.829742 0.558147i \(-0.811512\pi\)
−0.829742 + 0.558147i \(0.811512\pi\)
\(830\) −11520.0 −0.481765
\(831\) 0 0
\(832\) 25547.0 1.06452
\(833\) 0 0
\(834\) 0 0
\(835\) −9645.00 −0.399735
\(836\) −5445.00 −0.225262
\(837\) 0 0
\(838\) −29187.0 −1.20316
\(839\) −39162.0 −1.61147 −0.805734 0.592277i \(-0.798229\pi\)
−0.805734 + 0.592277i \(0.798229\pi\)
\(840\) 0 0
\(841\) 1855.00 0.0760589
\(842\) 37650.0 1.54098
\(843\) 0 0
\(844\) −3625.00 −0.147841
\(845\) −6420.00 −0.261367
\(846\) 0 0
\(847\) 0 0
\(848\) 42387.0 1.71648
\(849\) 0 0
\(850\) −4050.00 −0.163428
\(851\) 17871.0 0.719871
\(852\) 0 0
\(853\) −11527.0 −0.462693 −0.231346 0.972871i \(-0.574313\pi\)
−0.231346 + 0.972871i \(0.574313\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 15372.0 0.613790
\(857\) 41826.0 1.66715 0.833576 0.552405i \(-0.186290\pi\)
0.833576 + 0.552405i \(0.186290\pi\)
\(858\) 0 0
\(859\) 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 \(-0.442797\pi\)
0.178742 + 0.983896i \(0.442797\pi\)
\(864\) 0 0
\(865\) −3495.00 −0.137380
\(866\) −49848.0 −1.95601
\(867\) 0 0
\(868\) 0 0
\(869\) 35190.0 1.37369
\(870\) 0 0
\(871\) −16520.0 −0.642662
\(872\) −33600.0 −1.30486
\(873\) 0 0
\(874\) −25047.0 −0.969368
\(875\) 0 0
\(876\) 0 0
\(877\) 28439.0 1.09500 0.547501 0.836805i \(-0.315579\pi\)
0.547501 + 0.836805i \(0.315579\pi\)
\(878\) −22038.0 −0.847092
\(879\) 0 0
\(880\) 15975.0 0.611951
\(881\) 9303.00 0.355762 0.177881 0.984052i \(-0.443076\pi\)
0.177881 + 0.984052i \(0.443076\pi\)
\(882\) 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
\(885\) 0 0
\(886\) 36.0000 0.00136506
\(887\) 17016.0 0.644128 0.322064 0.946718i \(-0.395623\pi\)
0.322064 + 0.946718i \(0.395623\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 17910.0 0.674544
\(891\) 0 0
\(892\) −4960.00 −0.186181
\(893\) 5445.00 0.204043
\(894\) 0 0
\(895\) 15585.0 0.582066
\(896\) 0 0
\(897\) 0 0
\(898\) 29007.0 1.07792
\(899\) −14256.0 −0.528881
\(900\) 0 0
\(901\) −32238.0 −1.19201
\(902\) 26325.0 0.971759
\(903\) 0 0
\(904\) 29232.0 1.07549
\(905\) 8990.00 0.330207
\(906\) 0 0
\(907\) −24922.0 −0.912372 −0.456186 0.889884i \(-0.650785\pi\)
−0.456186 + 0.889884i \(0.650785\pi\)
\(908\) 1500.00 0.0548230
\(909\) 0 0
\(910\) 0 0
\(911\) −30714.0 −1.11701 −0.558507 0.829500i \(-0.688626\pi\)
−0.558507 + 0.829500i \(0.688626\pi\)
\(912\) 0 0
\(913\) −34560.0 −1.25276
\(914\) 28902.0 1.04594
\(915\) 0 0
\(916\) 6092.00 0.219744
\(917\) 0 0
\(918\) 0 0
\(919\) 17426.0 0.625496 0.312748 0.949836i \(-0.398750\pi\)
0.312748 + 0.949836i \(0.398750\pi\)
\(920\) 7245.00 0.259631
\(921\) 0 0
\(922\) −1026.00 −0.0366481
\(923\) −2832.00 −0.100993
\(924\) 0 0
\(925\) −6475.00 −0.230159
\(926\) −7233.00 −0.256686
\(927\) 0 0
\(928\) 7290.00 0.257873
\(929\) −26649.0 −0.941147 −0.470573 0.882361i \(-0.655953\pi\)
−0.470573 + 0.882361i \(0.655953\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −138.000 −0.00485015
\(933\) 0 0
\(934\) −3618.00 −0.126750
\(935\) −12150.0 −0.424971
\(936\) 0 0
\(937\) 27686.0 0.965274 0.482637 0.875820i \(-0.339679\pi\)
0.482637 + 0.875820i \(0.339679\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 225.000 0.00780712
\(941\) 17808.0 0.616923 0.308461 0.951237i \(-0.400186\pi\)
0.308461 + 0.951237i \(0.400186\pi\)
\(942\) 0 0
\(943\) 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 \(-0.462196\pi\)
0.118487 + 0.992956i \(0.462196\pi\)
\(948\) 0 0
\(949\) 39412.0 1.34812
\(950\) 9075.00 0.309928
\(951\) 0 0
\(952\) 0 0
\(953\) 20940.0 0.711766 0.355883 0.934530i \(-0.384180\pi\)
0.355883 + 0.934530i \(0.384180\pi\)
\(954\) 0 0
\(955\) −11940.0 −0.404575
\(956\) 5502.00 0.186137
\(957\) 0 0
\(958\) −1296.00 −0.0437076
\(959\) 0 0
\(960\) 0 0
\(961\) −22047.0 −0.740056
\(962\) 45843.0 1.53642
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) 13530.0 0.447858
\(971\) −29763.0 −0.983666 −0.491833 0.870689i \(-0.663673\pi\)
−0.491833 + 0.870689i \(0.663673\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 35688.0 1.17404
\(975\) 0 0
\(976\) −27832.0 −0.912788
\(977\) 38490.0 1.26039 0.630197 0.776436i \(-0.282974\pi\)
0.630197 + 0.776436i \(0.282974\pi\)
\(978\) 0 0
\(979\) 53730.0 1.75405
\(980\) 0 0
\(981\) 0 0
\(982\) −36828.0 −1.19677
\(983\) 12609.0 0.409120 0.204560 0.978854i \(-0.434424\pi\)
0.204560 + 0.978854i \(0.434424\pi\)
\(984\) 0 0
\(985\) −10545.0 −0.341108
\(986\) −26244.0 −0.847646
\(987\) 0 0
\(988\) −7139.00 −0.229880
\(989\) 19734.0 0.634484
\(990\) 0 0
\(991\) 19820.0 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −3960.00 −0.126744
\(993\) 0 0
\(994\) 0 0
\(995\) −7120.00 −0.226853
\(996\) 0 0
\(997\) 46034.0 1.46230 0.731149 0.682218i \(-0.238984\pi\)
0.731149 + 0.682218i \(0.238984\pi\)
\(998\) 32628.0 1.03489
\(999\) 0 0
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 2205.4.a.e.1.1 1
3.2 odd 2 245.4.a.e.1.1 1
7.2 even 3 315.4.j.b.46.1 2
7.4 even 3 315.4.j.b.226.1 2
7.6 odd 2 2205.4.a.g.1.1 1
15.14 odd 2 1225.4.a.b.1.1 1
21.2 odd 6 35.4.e.a.11.1 2
21.5 even 6 245.4.e.a.116.1 2
21.11 odd 6 35.4.e.a.16.1 yes 2
21.17 even 6 245.4.e.a.226.1 2
21.20 even 2 245.4.a.f.1.1 1
84.11 even 6 560.4.q.b.401.1 2
84.23 even 6 560.4.q.b.81.1 2
105.2 even 12 175.4.k.b.74.2 4
105.23 even 12 175.4.k.b.74.1 4
105.32 even 12 175.4.k.b.149.1 4
105.44 odd 6 175.4.e.b.151.1 2
105.53 even 12 175.4.k.b.149.2 4
105.74 odd 6 175.4.e.b.51.1 2
105.104 even 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 21.2 odd 6
35.4.e.a.16.1 yes 2 21.11 odd 6
175.4.e.b.51.1 2 105.74 odd 6
175.4.e.b.151.1 2 105.44 odd 6
175.4.k.b.74.1 4 105.23 even 12
175.4.k.b.74.2 4 105.2 even 12
175.4.k.b.149.1 4 105.32 even 12
175.4.k.b.149.2 4 105.53 even 12
245.4.a.e.1.1 1 3.2 odd 2
245.4.a.f.1.1 1 21.20 even 2
245.4.e.a.116.1 2 21.5 even 6
245.4.e.a.226.1 2 21.17 even 6
315.4.j.b.46.1 2 7.2 even 3
315.4.j.b.226.1 2 7.4 even 3
560.4.q.b.81.1 2 84.23 even 6
560.4.q.b.401.1 2 84.11 even 6
1225.4.a.a.1.1 1 105.104 even 2
1225.4.a.b.1.1 1 15.14 odd 2
2205.4.a.e.1.1 1 1.1 even 1 trivial
2205.4.a.g.1.1 1 7.6 odd 2