Properties

Label 234.4.a.k.1.1
Level $234$
Weight $4$
Character 234.1
Self dual yes
Analytic conductor $13.806$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [234,4,Mod(1,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("234.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 234.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: \(13.8064469413\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +16.0000 q^{5} +28.0000 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +16.0000 q^{5} +28.0000 q^{7} +8.00000 q^{8} +32.0000 q^{10} -34.0000 q^{11} -13.0000 q^{13} +56.0000 q^{14} +16.0000 q^{16} -138.000 q^{17} +108.000 q^{19} +64.0000 q^{20} -68.0000 q^{22} +52.0000 q^{23} +131.000 q^{25} -26.0000 q^{26} +112.000 q^{28} +190.000 q^{29} -176.000 q^{31} +32.0000 q^{32} -276.000 q^{34} +448.000 q^{35} +342.000 q^{37} +216.000 q^{38} +128.000 q^{40} -240.000 q^{41} -140.000 q^{43} -136.000 q^{44} +104.000 q^{46} -454.000 q^{47} +441.000 q^{49} +262.000 q^{50} -52.0000 q^{52} -198.000 q^{53} -544.000 q^{55} +224.000 q^{56} +380.000 q^{58} +154.000 q^{59} +34.0000 q^{61} -352.000 q^{62} +64.0000 q^{64} -208.000 q^{65} -656.000 q^{67} -552.000 q^{68} +896.000 q^{70} -550.000 q^{71} +614.000 q^{73} +684.000 q^{74} +432.000 q^{76} -952.000 q^{77} +8.00000 q^{79} +256.000 q^{80} -480.000 q^{82} -762.000 q^{83} -2208.00 q^{85} -280.000 q^{86} -272.000 q^{88} +444.000 q^{89} -364.000 q^{91} +208.000 q^{92} -908.000 q^{94} +1728.00 q^{95} +1022.00 q^{97} +882.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 16.0000 1.43108 0.715542 0.698570i \(-0.246180\pi\)
0.715542 + 0.698570i \(0.246180\pi\)
\(6\) 0 0
\(7\) 28.0000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 32.0000 1.01193
\(11\) −34.0000 −0.931944 −0.465972 0.884799i \(-0.654295\pi\)
−0.465972 + 0.884799i \(0.654295\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 56.0000 1.06904
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −138.000 −1.96882 −0.984409 0.175893i \(-0.943719\pi\)
−0.984409 + 0.175893i \(0.943719\pi\)
\(18\) 0 0
\(19\) 108.000 1.30405 0.652024 0.758199i \(-0.273920\pi\)
0.652024 + 0.758199i \(0.273920\pi\)
\(20\) 64.0000 0.715542
\(21\) 0 0
\(22\) −68.0000 −0.658984
\(23\) 52.0000 0.471424 0.235712 0.971823i \(-0.424258\pi\)
0.235712 + 0.971823i \(0.424258\pi\)
\(24\) 0 0
\(25\) 131.000 1.04800
\(26\) −26.0000 −0.196116
\(27\) 0 0
\(28\) 112.000 0.755929
\(29\) 190.000 1.21662 0.608312 0.793698i \(-0.291847\pi\)
0.608312 + 0.793698i \(0.291847\pi\)
\(30\) 0 0
\(31\) −176.000 −1.01969 −0.509847 0.860265i \(-0.670298\pi\)
−0.509847 + 0.860265i \(0.670298\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) −276.000 −1.39216
\(35\) 448.000 2.16359
\(36\) 0 0
\(37\) 342.000 1.51958 0.759790 0.650169i \(-0.225302\pi\)
0.759790 + 0.650169i \(0.225302\pi\)
\(38\) 216.000 0.922101
\(39\) 0 0
\(40\) 128.000 0.505964
\(41\) −240.000 −0.914188 −0.457094 0.889418i \(-0.651110\pi\)
−0.457094 + 0.889418i \(0.651110\pi\)
\(42\) 0 0
\(43\) −140.000 −0.496507 −0.248253 0.968695i \(-0.579857\pi\)
−0.248253 + 0.968695i \(0.579857\pi\)
\(44\) −136.000 −0.465972
\(45\) 0 0
\(46\) 104.000 0.333347
\(47\) −454.000 −1.40899 −0.704497 0.709707i \(-0.748827\pi\)
−0.704497 + 0.709707i \(0.748827\pi\)
\(48\) 0 0
\(49\) 441.000 1.28571
\(50\) 262.000 0.741048
\(51\) 0 0
\(52\) −52.0000 −0.138675
\(53\) −198.000 −0.513158 −0.256579 0.966523i \(-0.582595\pi\)
−0.256579 + 0.966523i \(0.582595\pi\)
\(54\) 0 0
\(55\) −544.000 −1.33369
\(56\) 224.000 0.534522
\(57\) 0 0
\(58\) 380.000 0.860284
\(59\) 154.000 0.339815 0.169908 0.985460i \(-0.445653\pi\)
0.169908 + 0.985460i \(0.445653\pi\)
\(60\) 0 0
\(61\) 34.0000 0.0713648 0.0356824 0.999363i \(-0.488640\pi\)
0.0356824 + 0.999363i \(0.488640\pi\)
\(62\) −352.000 −0.721033
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −208.000 −0.396911
\(66\) 0 0
\(67\) −656.000 −1.19617 −0.598083 0.801434i \(-0.704071\pi\)
−0.598083 + 0.801434i \(0.704071\pi\)
\(68\) −552.000 −0.984409
\(69\) 0 0
\(70\) 896.000 1.52989
\(71\) −550.000 −0.919338 −0.459669 0.888090i \(-0.652032\pi\)
−0.459669 + 0.888090i \(0.652032\pi\)
\(72\) 0 0
\(73\) 614.000 0.984428 0.492214 0.870474i \(-0.336188\pi\)
0.492214 + 0.870474i \(0.336188\pi\)
\(74\) 684.000 1.07451
\(75\) 0 0
\(76\) 432.000 0.652024
\(77\) −952.000 −1.40897
\(78\) 0 0
\(79\) 8.00000 0.0113933 0.00569665 0.999984i \(-0.498187\pi\)
0.00569665 + 0.999984i \(0.498187\pi\)
\(80\) 256.000 0.357771
\(81\) 0 0
\(82\) −480.000 −0.646428
\(83\) −762.000 −1.00772 −0.503858 0.863787i \(-0.668086\pi\)
−0.503858 + 0.863787i \(0.668086\pi\)
\(84\) 0 0
\(85\) −2208.00 −2.81754
\(86\) −280.000 −0.351083
\(87\) 0 0
\(88\) −272.000 −0.329492
\(89\) 444.000 0.528808 0.264404 0.964412i \(-0.414825\pi\)
0.264404 + 0.964412i \(0.414825\pi\)
\(90\) 0 0
\(91\) −364.000 −0.419314
\(92\) 208.000 0.235712
\(93\) 0 0
\(94\) −908.000 −0.996309
\(95\) 1728.00 1.86620
\(96\) 0 0
\(97\) 1022.00 1.06978 0.534889 0.844923i \(-0.320354\pi\)
0.534889 + 0.844923i \(0.320354\pi\)
\(98\) 882.000 0.909137
\(99\) 0 0
\(100\) 524.000 0.524000
\(101\) 1190.00 1.17237 0.586185 0.810177i \(-0.300629\pi\)
0.586185 + 0.810177i \(0.300629\pi\)
\(102\) 0 0
\(103\) −224.000 −0.214285 −0.107143 0.994244i \(-0.534170\pi\)
−0.107143 + 0.994244i \(0.534170\pi\)
\(104\) −104.000 −0.0980581
\(105\) 0 0
\(106\) −396.000 −0.362858
\(107\) 640.000 0.578235 0.289117 0.957294i \(-0.406638\pi\)
0.289117 + 0.957294i \(0.406638\pi\)
\(108\) 0 0
\(109\) −1934.00 −1.69948 −0.849741 0.527200i \(-0.823242\pi\)
−0.849741 + 0.527200i \(0.823242\pi\)
\(110\) −1088.00 −0.943061
\(111\) 0 0
\(112\) 448.000 0.377964
\(113\) 418.000 0.347983 0.173992 0.984747i \(-0.444333\pi\)
0.173992 + 0.984747i \(0.444333\pi\)
\(114\) 0 0
\(115\) 832.000 0.674647
\(116\) 760.000 0.608312
\(117\) 0 0
\(118\) 308.000 0.240286
\(119\) −3864.00 −2.97657
\(120\) 0 0
\(121\) −175.000 −0.131480
\(122\) 68.0000 0.0504625
\(123\) 0 0
\(124\) −704.000 −0.509847
\(125\) 96.0000 0.0686920
\(126\) 0 0
\(127\) −1040.00 −0.726654 −0.363327 0.931662i \(-0.618359\pi\)
−0.363327 + 0.931662i \(0.618359\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) −416.000 −0.280659
\(131\) 568.000 0.378827 0.189414 0.981897i \(-0.439341\pi\)
0.189414 + 0.981897i \(0.439341\pi\)
\(132\) 0 0
\(133\) 3024.00 1.97153
\(134\) −1312.00 −0.845817
\(135\) 0 0
\(136\) −1104.00 −0.696082
\(137\) −528.000 −0.329271 −0.164635 0.986355i \(-0.552645\pi\)
−0.164635 + 0.986355i \(0.552645\pi\)
\(138\) 0 0
\(139\) −1556.00 −0.949483 −0.474742 0.880125i \(-0.657459\pi\)
−0.474742 + 0.880125i \(0.657459\pi\)
\(140\) 1792.00 1.08180
\(141\) 0 0
\(142\) −1100.00 −0.650070
\(143\) 442.000 0.258475
\(144\) 0 0
\(145\) 3040.00 1.74109
\(146\) 1228.00 0.696096
\(147\) 0 0
\(148\) 1368.00 0.759790
\(149\) −1524.00 −0.837926 −0.418963 0.908003i \(-0.637606\pi\)
−0.418963 + 0.908003i \(0.637606\pi\)
\(150\) 0 0
\(151\) −3024.00 −1.62973 −0.814866 0.579649i \(-0.803190\pi\)
−0.814866 + 0.579649i \(0.803190\pi\)
\(152\) 864.000 0.461050
\(153\) 0 0
\(154\) −1904.00 −0.996290
\(155\) −2816.00 −1.45927
\(156\) 0 0
\(157\) 2198.00 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 16.0000 0.00805628
\(159\) 0 0
\(160\) 512.000 0.252982
\(161\) 1456.00 0.712726
\(162\) 0 0
\(163\) −268.000 −0.128781 −0.0643907 0.997925i \(-0.520510\pi\)
−0.0643907 + 0.997925i \(0.520510\pi\)
\(164\) −960.000 −0.457094
\(165\) 0 0
\(166\) −1524.00 −0.712562
\(167\) −702.000 −0.325284 −0.162642 0.986685i \(-0.552002\pi\)
−0.162642 + 0.986685i \(0.552002\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) −4416.00 −1.99230
\(171\) 0 0
\(172\) −560.000 −0.248253
\(173\) −2066.00 −0.907948 −0.453974 0.891015i \(-0.649994\pi\)
−0.453974 + 0.891015i \(0.649994\pi\)
\(174\) 0 0
\(175\) 3668.00 1.58443
\(176\) −544.000 −0.232986
\(177\) 0 0
\(178\) 888.000 0.373924
\(179\) 276.000 0.115247 0.0576235 0.998338i \(-0.481648\pi\)
0.0576235 + 0.998338i \(0.481648\pi\)
\(180\) 0 0
\(181\) −3474.00 −1.42663 −0.713316 0.700843i \(-0.752808\pi\)
−0.713316 + 0.700843i \(0.752808\pi\)
\(182\) −728.000 −0.296500
\(183\) 0 0
\(184\) 416.000 0.166674
\(185\) 5472.00 2.17465
\(186\) 0 0
\(187\) 4692.00 1.83483
\(188\) −1816.00 −0.704497
\(189\) 0 0
\(190\) 3456.00 1.31960
\(191\) 3920.00 1.48503 0.742516 0.669828i \(-0.233632\pi\)
0.742516 + 0.669828i \(0.233632\pi\)
\(192\) 0 0
\(193\) 2186.00 0.815294 0.407647 0.913140i \(-0.366349\pi\)
0.407647 + 0.913140i \(0.366349\pi\)
\(194\) 2044.00 0.756447
\(195\) 0 0
\(196\) 1764.00 0.642857
\(197\) 1368.00 0.494751 0.247376 0.968920i \(-0.420432\pi\)
0.247376 + 0.968920i \(0.420432\pi\)
\(198\) 0 0
\(199\) −1072.00 −0.381870 −0.190935 0.981603i \(-0.561152\pi\)
−0.190935 + 0.981603i \(0.561152\pi\)
\(200\) 1048.00 0.370524
\(201\) 0 0
\(202\) 2380.00 0.828991
\(203\) 5320.00 1.83936
\(204\) 0 0
\(205\) −3840.00 −1.30828
\(206\) −448.000 −0.151523
\(207\) 0 0
\(208\) −208.000 −0.0693375
\(209\) −3672.00 −1.21530
\(210\) 0 0
\(211\) 5444.00 1.77621 0.888105 0.459640i \(-0.152022\pi\)
0.888105 + 0.459640i \(0.152022\pi\)
\(212\) −792.000 −0.256579
\(213\) 0 0
\(214\) 1280.00 0.408874
\(215\) −2240.00 −0.710543
\(216\) 0 0
\(217\) −4928.00 −1.54163
\(218\) −3868.00 −1.20172
\(219\) 0 0
\(220\) −2176.00 −0.666845
\(221\) 1794.00 0.546052
\(222\) 0 0
\(223\) 96.0000 0.0288280 0.0144140 0.999896i \(-0.495412\pi\)
0.0144140 + 0.999896i \(0.495412\pi\)
\(224\) 896.000 0.267261
\(225\) 0 0
\(226\) 836.000 0.246061
\(227\) −198.000 −0.0578930 −0.0289465 0.999581i \(-0.509215\pi\)
−0.0289465 + 0.999581i \(0.509215\pi\)
\(228\) 0 0
\(229\) 5922.00 1.70889 0.854447 0.519538i \(-0.173896\pi\)
0.854447 + 0.519538i \(0.173896\pi\)
\(230\) 1664.00 0.477047
\(231\) 0 0
\(232\) 1520.00 0.430142
\(233\) 5114.00 1.43789 0.718947 0.695065i \(-0.244624\pi\)
0.718947 + 0.695065i \(0.244624\pi\)
\(234\) 0 0
\(235\) −7264.00 −2.01639
\(236\) 616.000 0.169908
\(237\) 0 0
\(238\) −7728.00 −2.10476
\(239\) 5226.00 1.41440 0.707200 0.707013i \(-0.249958\pi\)
0.707200 + 0.707013i \(0.249958\pi\)
\(240\) 0 0
\(241\) −762.000 −0.203671 −0.101836 0.994801i \(-0.532472\pi\)
−0.101836 + 0.994801i \(0.532472\pi\)
\(242\) −350.000 −0.0929705
\(243\) 0 0
\(244\) 136.000 0.0356824
\(245\) 7056.00 1.83996
\(246\) 0 0
\(247\) −1404.00 −0.361678
\(248\) −1408.00 −0.360516
\(249\) 0 0
\(250\) 192.000 0.0485726
\(251\) −3240.00 −0.814769 −0.407384 0.913257i \(-0.633559\pi\)
−0.407384 + 0.913257i \(0.633559\pi\)
\(252\) 0 0
\(253\) −1768.00 −0.439341
\(254\) −2080.00 −0.513822
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 1386.00 0.336406 0.168203 0.985752i \(-0.446204\pi\)
0.168203 + 0.985752i \(0.446204\pi\)
\(258\) 0 0
\(259\) 9576.00 2.29739
\(260\) −832.000 −0.198456
\(261\) 0 0
\(262\) 1136.00 0.267871
\(263\) −3300.00 −0.773714 −0.386857 0.922140i \(-0.626439\pi\)
−0.386857 + 0.922140i \(0.626439\pi\)
\(264\) 0 0
\(265\) −3168.00 −0.734372
\(266\) 6048.00 1.39409
\(267\) 0 0
\(268\) −2624.00 −0.598083
\(269\) −4290.00 −0.972364 −0.486182 0.873858i \(-0.661611\pi\)
−0.486182 + 0.873858i \(0.661611\pi\)
\(270\) 0 0
\(271\) 2452.00 0.549625 0.274813 0.961498i \(-0.411384\pi\)
0.274813 + 0.961498i \(0.411384\pi\)
\(272\) −2208.00 −0.492205
\(273\) 0 0
\(274\) −1056.00 −0.232830
\(275\) −4454.00 −0.976677
\(276\) 0 0
\(277\) −42.0000 −0.00911024 −0.00455512 0.999990i \(-0.501450\pi\)
−0.00455512 + 0.999990i \(0.501450\pi\)
\(278\) −3112.00 −0.671386
\(279\) 0 0
\(280\) 3584.00 0.764946
\(281\) 2288.00 0.485732 0.242866 0.970060i \(-0.421912\pi\)
0.242866 + 0.970060i \(0.421912\pi\)
\(282\) 0 0
\(283\) 1156.00 0.242816 0.121408 0.992603i \(-0.461259\pi\)
0.121408 + 0.992603i \(0.461259\pi\)
\(284\) −2200.00 −0.459669
\(285\) 0 0
\(286\) 884.000 0.182769
\(287\) −6720.00 −1.38212
\(288\) 0 0
\(289\) 14131.0 2.87625
\(290\) 6080.00 1.23114
\(291\) 0 0
\(292\) 2456.00 0.492214
\(293\) 8684.00 1.73148 0.865742 0.500491i \(-0.166847\pi\)
0.865742 + 0.500491i \(0.166847\pi\)
\(294\) 0 0
\(295\) 2464.00 0.486304
\(296\) 2736.00 0.537253
\(297\) 0 0
\(298\) −3048.00 −0.592503
\(299\) −676.000 −0.130749
\(300\) 0 0
\(301\) −3920.00 −0.750648
\(302\) −6048.00 −1.15240
\(303\) 0 0
\(304\) 1728.00 0.326012
\(305\) 544.000 0.102129
\(306\) 0 0
\(307\) −7552.00 −1.40396 −0.701979 0.712197i \(-0.747700\pi\)
−0.701979 + 0.712197i \(0.747700\pi\)
\(308\) −3808.00 −0.704484
\(309\) 0 0
\(310\) −5632.00 −1.03186
\(311\) −2652.00 −0.483541 −0.241770 0.970334i \(-0.577728\pi\)
−0.241770 + 0.970334i \(0.577728\pi\)
\(312\) 0 0
\(313\) −4426.00 −0.799273 −0.399636 0.916674i \(-0.630864\pi\)
−0.399636 + 0.916674i \(0.630864\pi\)
\(314\) 4396.00 0.790066
\(315\) 0 0
\(316\) 32.0000 0.00569665
\(317\) 4944.00 0.875971 0.437985 0.898982i \(-0.355692\pi\)
0.437985 + 0.898982i \(0.355692\pi\)
\(318\) 0 0
\(319\) −6460.00 −1.13383
\(320\) 1024.00 0.178885
\(321\) 0 0
\(322\) 2912.00 0.503973
\(323\) −14904.0 −2.56743
\(324\) 0 0
\(325\) −1703.00 −0.290663
\(326\) −536.000 −0.0910623
\(327\) 0 0
\(328\) −1920.00 −0.323214
\(329\) −12712.0 −2.13020
\(330\) 0 0
\(331\) −6088.00 −1.01096 −0.505478 0.862839i \(-0.668684\pi\)
−0.505478 + 0.862839i \(0.668684\pi\)
\(332\) −3048.00 −0.503858
\(333\) 0 0
\(334\) −1404.00 −0.230010
\(335\) −10496.0 −1.71181
\(336\) 0 0
\(337\) 6638.00 1.07298 0.536491 0.843906i \(-0.319750\pi\)
0.536491 + 0.843906i \(0.319750\pi\)
\(338\) 338.000 0.0543928
\(339\) 0 0
\(340\) −8832.00 −1.40877
\(341\) 5984.00 0.950298
\(342\) 0 0
\(343\) 2744.00 0.431959
\(344\) −1120.00 −0.175542
\(345\) 0 0
\(346\) −4132.00 −0.642016
\(347\) 2292.00 0.354585 0.177293 0.984158i \(-0.443266\pi\)
0.177293 + 0.984158i \(0.443266\pi\)
\(348\) 0 0
\(349\) −9866.00 −1.51322 −0.756612 0.653865i \(-0.773147\pi\)
−0.756612 + 0.653865i \(0.773147\pi\)
\(350\) 7336.00 1.12036
\(351\) 0 0
\(352\) −1088.00 −0.164746
\(353\) −2368.00 −0.357042 −0.178521 0.983936i \(-0.557131\pi\)
−0.178521 + 0.983936i \(0.557131\pi\)
\(354\) 0 0
\(355\) −8800.00 −1.31565
\(356\) 1776.00 0.264404
\(357\) 0 0
\(358\) 552.000 0.0814919
\(359\) −5070.00 −0.745360 −0.372680 0.927960i \(-0.621561\pi\)
−0.372680 + 0.927960i \(0.621561\pi\)
\(360\) 0 0
\(361\) 4805.00 0.700539
\(362\) −6948.00 −1.00878
\(363\) 0 0
\(364\) −1456.00 −0.209657
\(365\) 9824.00 1.40880
\(366\) 0 0
\(367\) −8584.00 −1.22093 −0.610465 0.792043i \(-0.709017\pi\)
−0.610465 + 0.792043i \(0.709017\pi\)
\(368\) 832.000 0.117856
\(369\) 0 0
\(370\) 10944.0 1.53771
\(371\) −5544.00 −0.775822
\(372\) 0 0
\(373\) 4994.00 0.693243 0.346621 0.938005i \(-0.387329\pi\)
0.346621 + 0.938005i \(0.387329\pi\)
\(374\) 9384.00 1.29742
\(375\) 0 0
\(376\) −3632.00 −0.498155
\(377\) −2470.00 −0.337431
\(378\) 0 0
\(379\) 1300.00 0.176191 0.0880957 0.996112i \(-0.471922\pi\)
0.0880957 + 0.996112i \(0.471922\pi\)
\(380\) 6912.00 0.933100
\(381\) 0 0
\(382\) 7840.00 1.05008
\(383\) 4590.00 0.612371 0.306185 0.951972i \(-0.400947\pi\)
0.306185 + 0.951972i \(0.400947\pi\)
\(384\) 0 0
\(385\) −15232.0 −2.01635
\(386\) 4372.00 0.576500
\(387\) 0 0
\(388\) 4088.00 0.534889
\(389\) −3510.00 −0.457491 −0.228746 0.973486i \(-0.573462\pi\)
−0.228746 + 0.973486i \(0.573462\pi\)
\(390\) 0 0
\(391\) −7176.00 −0.928148
\(392\) 3528.00 0.454569
\(393\) 0 0
\(394\) 2736.00 0.349842
\(395\) 128.000 0.0163048
\(396\) 0 0
\(397\) 6230.00 0.787594 0.393797 0.919197i \(-0.371161\pi\)
0.393797 + 0.919197i \(0.371161\pi\)
\(398\) −2144.00 −0.270023
\(399\) 0 0
\(400\) 2096.00 0.262000
\(401\) 7500.00 0.933995 0.466998 0.884259i \(-0.345336\pi\)
0.466998 + 0.884259i \(0.345336\pi\)
\(402\) 0 0
\(403\) 2288.00 0.282812
\(404\) 4760.00 0.586185
\(405\) 0 0
\(406\) 10640.0 1.30063
\(407\) −11628.0 −1.41616
\(408\) 0 0
\(409\) 8254.00 0.997883 0.498941 0.866636i \(-0.333722\pi\)
0.498941 + 0.866636i \(0.333722\pi\)
\(410\) −7680.00 −0.925093
\(411\) 0 0
\(412\) −896.000 −0.107143
\(413\) 4312.00 0.513752
\(414\) 0 0
\(415\) −12192.0 −1.44212
\(416\) −416.000 −0.0490290
\(417\) 0 0
\(418\) −7344.00 −0.859346
\(419\) 14808.0 1.72653 0.863267 0.504747i \(-0.168414\pi\)
0.863267 + 0.504747i \(0.168414\pi\)
\(420\) 0 0
\(421\) 10354.0 1.19863 0.599315 0.800513i \(-0.295440\pi\)
0.599315 + 0.800513i \(0.295440\pi\)
\(422\) 10888.0 1.25597
\(423\) 0 0
\(424\) −1584.00 −0.181429
\(425\) −18078.0 −2.06332
\(426\) 0 0
\(427\) 952.000 0.107893
\(428\) 2560.00 0.289117
\(429\) 0 0
\(430\) −4480.00 −0.502430
\(431\) 15486.0 1.73071 0.865353 0.501163i \(-0.167094\pi\)
0.865353 + 0.501163i \(0.167094\pi\)
\(432\) 0 0
\(433\) −2018.00 −0.223970 −0.111985 0.993710i \(-0.535721\pi\)
−0.111985 + 0.993710i \(0.535721\pi\)
\(434\) −9856.00 −1.09010
\(435\) 0 0
\(436\) −7736.00 −0.849741
\(437\) 5616.00 0.614759
\(438\) 0 0
\(439\) 8792.00 0.955853 0.477926 0.878400i \(-0.341389\pi\)
0.477926 + 0.878400i \(0.341389\pi\)
\(440\) −4352.00 −0.471531
\(441\) 0 0
\(442\) 3588.00 0.386117
\(443\) 2760.00 0.296008 0.148004 0.988987i \(-0.452715\pi\)
0.148004 + 0.988987i \(0.452715\pi\)
\(444\) 0 0
\(445\) 7104.00 0.756768
\(446\) 192.000 0.0203844
\(447\) 0 0
\(448\) 1792.00 0.188982
\(449\) −9532.00 −1.00188 −0.500939 0.865483i \(-0.667012\pi\)
−0.500939 + 0.865483i \(0.667012\pi\)
\(450\) 0 0
\(451\) 8160.00 0.851972
\(452\) 1672.00 0.173992
\(453\) 0 0
\(454\) −396.000 −0.0409366
\(455\) −5824.00 −0.600073
\(456\) 0 0
\(457\) 12862.0 1.31654 0.658270 0.752782i \(-0.271288\pi\)
0.658270 + 0.752782i \(0.271288\pi\)
\(458\) 11844.0 1.20837
\(459\) 0 0
\(460\) 3328.00 0.337323
\(461\) 6744.00 0.681344 0.340672 0.940182i \(-0.389346\pi\)
0.340672 + 0.940182i \(0.389346\pi\)
\(462\) 0 0
\(463\) −9572.00 −0.960796 −0.480398 0.877051i \(-0.659508\pi\)
−0.480398 + 0.877051i \(0.659508\pi\)
\(464\) 3040.00 0.304156
\(465\) 0 0
\(466\) 10228.0 1.01674
\(467\) −9104.00 −0.902105 −0.451052 0.892498i \(-0.648951\pi\)
−0.451052 + 0.892498i \(0.648951\pi\)
\(468\) 0 0
\(469\) −18368.0 −1.80843
\(470\) −14528.0 −1.42580
\(471\) 0 0
\(472\) 1232.00 0.120143
\(473\) 4760.00 0.462717
\(474\) 0 0
\(475\) 14148.0 1.36664
\(476\) −15456.0 −1.48829
\(477\) 0 0
\(478\) 10452.0 1.00013
\(479\) 18870.0 1.79998 0.899992 0.435906i \(-0.143572\pi\)
0.899992 + 0.435906i \(0.143572\pi\)
\(480\) 0 0
\(481\) −4446.00 −0.421456
\(482\) −1524.00 −0.144017
\(483\) 0 0
\(484\) −700.000 −0.0657400
\(485\) 16352.0 1.53094
\(486\) 0 0
\(487\) −1744.00 −0.162276 −0.0811378 0.996703i \(-0.525855\pi\)
−0.0811378 + 0.996703i \(0.525855\pi\)
\(488\) 272.000 0.0252313
\(489\) 0 0
\(490\) 14112.0 1.30105
\(491\) −13360.0 −1.22796 −0.613980 0.789322i \(-0.710432\pi\)
−0.613980 + 0.789322i \(0.710432\pi\)
\(492\) 0 0
\(493\) −26220.0 −2.39531
\(494\) −2808.00 −0.255745
\(495\) 0 0
\(496\) −2816.00 −0.254924
\(497\) −15400.0 −1.38991
\(498\) 0 0
\(499\) 17368.0 1.55811 0.779057 0.626954i \(-0.215698\pi\)
0.779057 + 0.626954i \(0.215698\pi\)
\(500\) 384.000 0.0343460
\(501\) 0 0
\(502\) −6480.00 −0.576129
\(503\) 5828.00 0.516616 0.258308 0.966063i \(-0.416835\pi\)
0.258308 + 0.966063i \(0.416835\pi\)
\(504\) 0 0
\(505\) 19040.0 1.67776
\(506\) −3536.00 −0.310661
\(507\) 0 0
\(508\) −4160.00 −0.363327
\(509\) −10744.0 −0.935598 −0.467799 0.883835i \(-0.654953\pi\)
−0.467799 + 0.883835i \(0.654953\pi\)
\(510\) 0 0
\(511\) 17192.0 1.48832
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 2772.00 0.237875
\(515\) −3584.00 −0.306660
\(516\) 0 0
\(517\) 15436.0 1.31310
\(518\) 19152.0 1.62450
\(519\) 0 0
\(520\) −1664.00 −0.140329
\(521\) 12234.0 1.02875 0.514377 0.857564i \(-0.328023\pi\)
0.514377 + 0.857564i \(0.328023\pi\)
\(522\) 0 0
\(523\) 1812.00 0.151498 0.0757488 0.997127i \(-0.475865\pi\)
0.0757488 + 0.997127i \(0.475865\pi\)
\(524\) 2272.00 0.189414
\(525\) 0 0
\(526\) −6600.00 −0.547098
\(527\) 24288.0 2.00759
\(528\) 0 0
\(529\) −9463.00 −0.777760
\(530\) −6336.00 −0.519280
\(531\) 0 0
\(532\) 12096.0 0.985767
\(533\) 3120.00 0.253550
\(534\) 0 0
\(535\) 10240.0 0.827502
\(536\) −5248.00 −0.422909
\(537\) 0 0
\(538\) −8580.00 −0.687565
\(539\) −14994.0 −1.19821
\(540\) 0 0
\(541\) 6098.00 0.484609 0.242305 0.970200i \(-0.422097\pi\)
0.242305 + 0.970200i \(0.422097\pi\)
\(542\) 4904.00 0.388644
\(543\) 0 0
\(544\) −4416.00 −0.348041
\(545\) −30944.0 −2.43210
\(546\) 0 0
\(547\) −18332.0 −1.43294 −0.716471 0.697616i \(-0.754244\pi\)
−0.716471 + 0.697616i \(0.754244\pi\)
\(548\) −2112.00 −0.164635
\(549\) 0 0
\(550\) −8908.00 −0.690615
\(551\) 20520.0 1.58654
\(552\) 0 0
\(553\) 224.000 0.0172250
\(554\) −84.0000 −0.00644191
\(555\) 0 0
\(556\) −6224.00 −0.474742
\(557\) 20004.0 1.52172 0.760859 0.648917i \(-0.224778\pi\)
0.760859 + 0.648917i \(0.224778\pi\)
\(558\) 0 0
\(559\) 1820.00 0.137706
\(560\) 7168.00 0.540899
\(561\) 0 0
\(562\) 4576.00 0.343464
\(563\) −10988.0 −0.822538 −0.411269 0.911514i \(-0.634914\pi\)
−0.411269 + 0.911514i \(0.634914\pi\)
\(564\) 0 0
\(565\) 6688.00 0.497993
\(566\) 2312.00 0.171697
\(567\) 0 0
\(568\) −4400.00 −0.325035
\(569\) −11062.0 −0.815014 −0.407507 0.913202i \(-0.633602\pi\)
−0.407507 + 0.913202i \(0.633602\pi\)
\(570\) 0 0
\(571\) −708.000 −0.0518895 −0.0259447 0.999663i \(-0.508259\pi\)
−0.0259447 + 0.999663i \(0.508259\pi\)
\(572\) 1768.00 0.129237
\(573\) 0 0
\(574\) −13440.0 −0.977308
\(575\) 6812.00 0.494052
\(576\) 0 0
\(577\) −2094.00 −0.151082 −0.0755410 0.997143i \(-0.524068\pi\)
−0.0755410 + 0.997143i \(0.524068\pi\)
\(578\) 28262.0 2.03381
\(579\) 0 0
\(580\) 12160.0 0.870546
\(581\) −21336.0 −1.52352
\(582\) 0 0
\(583\) 6732.00 0.478235
\(584\) 4912.00 0.348048
\(585\) 0 0
\(586\) 17368.0 1.22434
\(587\) 17854.0 1.25539 0.627695 0.778460i \(-0.283999\pi\)
0.627695 + 0.778460i \(0.283999\pi\)
\(588\) 0 0
\(589\) −19008.0 −1.32973
\(590\) 4928.00 0.343869
\(591\) 0 0
\(592\) 5472.00 0.379895
\(593\) −23948.0 −1.65839 −0.829196 0.558958i \(-0.811201\pi\)
−0.829196 + 0.558958i \(0.811201\pi\)
\(594\) 0 0
\(595\) −61824.0 −4.25973
\(596\) −6096.00 −0.418963
\(597\) 0 0
\(598\) −1352.00 −0.0924538
\(599\) 18068.0 1.23245 0.616226 0.787570i \(-0.288661\pi\)
0.616226 + 0.787570i \(0.288661\pi\)
\(600\) 0 0
\(601\) 19942.0 1.35350 0.676748 0.736215i \(-0.263389\pi\)
0.676748 + 0.736215i \(0.263389\pi\)
\(602\) −7840.00 −0.530788
\(603\) 0 0
\(604\) −12096.0 −0.814866
\(605\) −2800.00 −0.188159
\(606\) 0 0
\(607\) 26376.0 1.76370 0.881852 0.471526i \(-0.156296\pi\)
0.881852 + 0.471526i \(0.156296\pi\)
\(608\) 3456.00 0.230525
\(609\) 0 0
\(610\) 1088.00 0.0722161
\(611\) 5902.00 0.390785
\(612\) 0 0
\(613\) −19426.0 −1.27995 −0.639975 0.768396i \(-0.721055\pi\)
−0.639975 + 0.768396i \(0.721055\pi\)
\(614\) −15104.0 −0.992749
\(615\) 0 0
\(616\) −7616.00 −0.498145
\(617\) 8024.00 0.523556 0.261778 0.965128i \(-0.415691\pi\)
0.261778 + 0.965128i \(0.415691\pi\)
\(618\) 0 0
\(619\) −20648.0 −1.34073 −0.670366 0.742031i \(-0.733863\pi\)
−0.670366 + 0.742031i \(0.733863\pi\)
\(620\) −11264.0 −0.729634
\(621\) 0 0
\(622\) −5304.00 −0.341915
\(623\) 12432.0 0.799482
\(624\) 0 0
\(625\) −14839.0 −0.949696
\(626\) −8852.00 −0.565171
\(627\) 0 0
\(628\) 8792.00 0.558661
\(629\) −47196.0 −2.99178
\(630\) 0 0
\(631\) 12280.0 0.774737 0.387369 0.921925i \(-0.373384\pi\)
0.387369 + 0.921925i \(0.373384\pi\)
\(632\) 64.0000 0.00402814
\(633\) 0 0
\(634\) 9888.00 0.619405
\(635\) −16640.0 −1.03990
\(636\) 0 0
\(637\) −5733.00 −0.356593
\(638\) −12920.0 −0.801736
\(639\) 0 0
\(640\) 2048.00 0.126491
\(641\) 15878.0 0.978383 0.489191 0.872176i \(-0.337292\pi\)
0.489191 + 0.872176i \(0.337292\pi\)
\(642\) 0 0
\(643\) −21520.0 −1.31985 −0.659927 0.751330i \(-0.729413\pi\)
−0.659927 + 0.751330i \(0.729413\pi\)
\(644\) 5824.00 0.356363
\(645\) 0 0
\(646\) −29808.0 −1.81545
\(647\) −7312.00 −0.444304 −0.222152 0.975012i \(-0.571308\pi\)
−0.222152 + 0.975012i \(0.571308\pi\)
\(648\) 0 0
\(649\) −5236.00 −0.316689
\(650\) −3406.00 −0.205530
\(651\) 0 0
\(652\) −1072.00 −0.0643907
\(653\) −3090.00 −0.185178 −0.0925889 0.995704i \(-0.529514\pi\)
−0.0925889 + 0.995704i \(0.529514\pi\)
\(654\) 0 0
\(655\) 9088.00 0.542134
\(656\) −3840.00 −0.228547
\(657\) 0 0
\(658\) −25424.0 −1.50628
\(659\) 13428.0 0.793749 0.396875 0.917873i \(-0.370095\pi\)
0.396875 + 0.917873i \(0.370095\pi\)
\(660\) 0 0
\(661\) 22598.0 1.32974 0.664872 0.746958i \(-0.268486\pi\)
0.664872 + 0.746958i \(0.268486\pi\)
\(662\) −12176.0 −0.714854
\(663\) 0 0
\(664\) −6096.00 −0.356281
\(665\) 48384.0 2.82143
\(666\) 0 0
\(667\) 9880.00 0.573546
\(668\) −2808.00 −0.162642
\(669\) 0 0
\(670\) −20992.0 −1.21044
\(671\) −1156.00 −0.0665080
\(672\) 0 0
\(673\) 6178.00 0.353855 0.176927 0.984224i \(-0.443384\pi\)
0.176927 + 0.984224i \(0.443384\pi\)
\(674\) 13276.0 0.758713
\(675\) 0 0
\(676\) 676.000 0.0384615
\(677\) −22398.0 −1.27153 −0.635764 0.771883i \(-0.719315\pi\)
−0.635764 + 0.771883i \(0.719315\pi\)
\(678\) 0 0
\(679\) 28616.0 1.61735
\(680\) −17664.0 −0.996152
\(681\) 0 0
\(682\) 11968.0 0.671962
\(683\) −11410.0 −0.639226 −0.319613 0.947548i \(-0.603553\pi\)
−0.319613 + 0.947548i \(0.603553\pi\)
\(684\) 0 0
\(685\) −8448.00 −0.471214
\(686\) 5488.00 0.305441
\(687\) 0 0
\(688\) −2240.00 −0.124127
\(689\) 2574.00 0.142325
\(690\) 0 0
\(691\) 32488.0 1.78857 0.894285 0.447498i \(-0.147685\pi\)
0.894285 + 0.447498i \(0.147685\pi\)
\(692\) −8264.00 −0.453974
\(693\) 0 0
\(694\) 4584.00 0.250729
\(695\) −24896.0 −1.35879
\(696\) 0 0
\(697\) 33120.0 1.79987
\(698\) −19732.0 −1.07001
\(699\) 0 0
\(700\) 14672.0 0.792214
\(701\) −5094.00 −0.274462 −0.137231 0.990539i \(-0.543820\pi\)
−0.137231 + 0.990539i \(0.543820\pi\)
\(702\) 0 0
\(703\) 36936.0 1.98160
\(704\) −2176.00 −0.116493
\(705\) 0 0
\(706\) −4736.00 −0.252467
\(707\) 33320.0 1.77246
\(708\) 0 0
\(709\) 25418.0 1.34639 0.673197 0.739463i \(-0.264921\pi\)
0.673197 + 0.739463i \(0.264921\pi\)
\(710\) −17600.0 −0.930305
\(711\) 0 0
\(712\) 3552.00 0.186962
\(713\) −9152.00 −0.480708
\(714\) 0 0
\(715\) 7072.00 0.369899
\(716\) 1104.00 0.0576235
\(717\) 0 0
\(718\) −10140.0 −0.527049
\(719\) 20428.0 1.05958 0.529788 0.848130i \(-0.322271\pi\)
0.529788 + 0.848130i \(0.322271\pi\)
\(720\) 0 0
\(721\) −6272.00 −0.323969
\(722\) 9610.00 0.495356
\(723\) 0 0
\(724\) −13896.0 −0.713316
\(725\) 24890.0 1.27502
\(726\) 0 0
\(727\) −38336.0 −1.95571 −0.977857 0.209276i \(-0.932889\pi\)
−0.977857 + 0.209276i \(0.932889\pi\)
\(728\) −2912.00 −0.148250
\(729\) 0 0
\(730\) 19648.0 0.996171
\(731\) 19320.0 0.977532
\(732\) 0 0
\(733\) −166.000 −0.00836473 −0.00418237 0.999991i \(-0.501331\pi\)
−0.00418237 + 0.999991i \(0.501331\pi\)
\(734\) −17168.0 −0.863328
\(735\) 0 0
\(736\) 1664.00 0.0833368
\(737\) 22304.0 1.11476
\(738\) 0 0
\(739\) −25248.0 −1.25678 −0.628392 0.777897i \(-0.716286\pi\)
−0.628392 + 0.777897i \(0.716286\pi\)
\(740\) 21888.0 1.08732
\(741\) 0 0
\(742\) −11088.0 −0.548589
\(743\) 4442.00 0.219329 0.109664 0.993969i \(-0.465022\pi\)
0.109664 + 0.993969i \(0.465022\pi\)
\(744\) 0 0
\(745\) −24384.0 −1.19914
\(746\) 9988.00 0.490197
\(747\) 0 0
\(748\) 18768.0 0.917414
\(749\) 17920.0 0.874209
\(750\) 0 0
\(751\) −19848.0 −0.964399 −0.482200 0.876061i \(-0.660162\pi\)
−0.482200 + 0.876061i \(0.660162\pi\)
\(752\) −7264.00 −0.352248
\(753\) 0 0
\(754\) −4940.00 −0.238600
\(755\) −48384.0 −2.33228
\(756\) 0 0
\(757\) −29166.0 −1.40034 −0.700169 0.713977i \(-0.746892\pi\)
−0.700169 + 0.713977i \(0.746892\pi\)
\(758\) 2600.00 0.124586
\(759\) 0 0
\(760\) 13824.0 0.659802
\(761\) 6240.00 0.297240 0.148620 0.988894i \(-0.452517\pi\)
0.148620 + 0.988894i \(0.452517\pi\)
\(762\) 0 0
\(763\) −54152.0 −2.56938
\(764\) 15680.0 0.742516
\(765\) 0 0
\(766\) 9180.00 0.433012
\(767\) −2002.00 −0.0942478
\(768\) 0 0
\(769\) −39750.0 −1.86401 −0.932004 0.362449i \(-0.881941\pi\)
−0.932004 + 0.362449i \(0.881941\pi\)
\(770\) −30464.0 −1.42577
\(771\) 0 0
\(772\) 8744.00 0.407647
\(773\) −9764.00 −0.454317 −0.227158 0.973858i \(-0.572943\pi\)
−0.227158 + 0.973858i \(0.572943\pi\)
\(774\) 0 0
\(775\) −23056.0 −1.06864
\(776\) 8176.00 0.378223
\(777\) 0 0
\(778\) −7020.00 −0.323495
\(779\) −25920.0 −1.19214
\(780\) 0 0
\(781\) 18700.0 0.856772
\(782\) −14352.0 −0.656300
\(783\) 0 0
\(784\) 7056.00 0.321429
\(785\) 35168.0 1.59898
\(786\) 0 0
\(787\) −36016.0 −1.63130 −0.815649 0.578547i \(-0.803620\pi\)
−0.815649 + 0.578547i \(0.803620\pi\)
\(788\) 5472.00 0.247376
\(789\) 0 0
\(790\) 256.000 0.0115292
\(791\) 11704.0 0.526102
\(792\) 0 0
\(793\) −442.000 −0.0197930
\(794\) 12460.0 0.556913
\(795\) 0 0
\(796\) −4288.00 −0.190935
\(797\) 22290.0 0.990655 0.495328 0.868706i \(-0.335048\pi\)
0.495328 + 0.868706i \(0.335048\pi\)
\(798\) 0 0
\(799\) 62652.0 2.77405
\(800\) 4192.00 0.185262
\(801\) 0 0
\(802\) 15000.0 0.660434
\(803\) −20876.0 −0.917432
\(804\) 0 0
\(805\) 23296.0 1.01997
\(806\) 4576.00 0.199979
\(807\) 0 0
\(808\) 9520.00 0.414496
\(809\) 25578.0 1.11159 0.555794 0.831320i \(-0.312414\pi\)
0.555794 + 0.831320i \(0.312414\pi\)
\(810\) 0 0
\(811\) 29900.0 1.29461 0.647306 0.762230i \(-0.275895\pi\)
0.647306 + 0.762230i \(0.275895\pi\)
\(812\) 21280.0 0.919682
\(813\) 0 0
\(814\) −23256.0 −1.00138
\(815\) −4288.00 −0.184297
\(816\) 0 0
\(817\) −15120.0 −0.647469
\(818\) 16508.0 0.705610
\(819\) 0 0
\(820\) −15360.0 −0.654140
\(821\) −16412.0 −0.697665 −0.348832 0.937185i \(-0.613422\pi\)
−0.348832 + 0.937185i \(0.613422\pi\)
\(822\) 0 0
\(823\) 18552.0 0.785762 0.392881 0.919589i \(-0.371478\pi\)
0.392881 + 0.919589i \(0.371478\pi\)
\(824\) −1792.00 −0.0757613
\(825\) 0 0
\(826\) 8624.00 0.363278
\(827\) 28662.0 1.20517 0.602585 0.798055i \(-0.294137\pi\)
0.602585 + 0.798055i \(0.294137\pi\)
\(828\) 0 0
\(829\) −3686.00 −0.154427 −0.0772136 0.997015i \(-0.524602\pi\)
−0.0772136 + 0.997015i \(0.524602\pi\)
\(830\) −24384.0 −1.01974
\(831\) 0 0
\(832\) −832.000 −0.0346688
\(833\) −60858.0 −2.53134
\(834\) 0 0
\(835\) −11232.0 −0.465508
\(836\) −14688.0 −0.607650
\(837\) 0 0
\(838\) 29616.0 1.22084
\(839\) −13370.0 −0.550159 −0.275080 0.961421i \(-0.588704\pi\)
−0.275080 + 0.961421i \(0.588704\pi\)
\(840\) 0 0
\(841\) 11711.0 0.480175
\(842\) 20708.0 0.847559
\(843\) 0 0
\(844\) 21776.0 0.888105
\(845\) 2704.00 0.110083
\(846\) 0 0
\(847\) −4900.00 −0.198779
\(848\) −3168.00 −0.128290
\(849\) 0 0
\(850\) −36156.0 −1.45899
\(851\) 17784.0 0.716366
\(852\) 0 0
\(853\) 11398.0 0.457515 0.228757 0.973483i \(-0.426534\pi\)
0.228757 + 0.973483i \(0.426534\pi\)
\(854\) 1904.00 0.0762922
\(855\) 0 0
\(856\) 5120.00 0.204437
\(857\) −7990.00 −0.318475 −0.159238 0.987240i \(-0.550904\pi\)
−0.159238 + 0.987240i \(0.550904\pi\)
\(858\) 0 0
\(859\) 7652.00 0.303938 0.151969 0.988385i \(-0.451439\pi\)
0.151969 + 0.988385i \(0.451439\pi\)
\(860\) −8960.00 −0.355271
\(861\) 0 0
\(862\) 30972.0 1.22379
\(863\) 1022.00 0.0403120 0.0201560 0.999797i \(-0.493584\pi\)
0.0201560 + 0.999797i \(0.493584\pi\)
\(864\) 0 0
\(865\) −33056.0 −1.29935
\(866\) −4036.00 −0.158371
\(867\) 0 0
\(868\) −19712.0 −0.770817
\(869\) −272.000 −0.0106179
\(870\) 0 0
\(871\) 8528.00 0.331757
\(872\) −15472.0 −0.600858
\(873\) 0 0
\(874\) 11232.0 0.434700
\(875\) 2688.00 0.103853
\(876\) 0 0
\(877\) −15546.0 −0.598576 −0.299288 0.954163i \(-0.596749\pi\)
−0.299288 + 0.954163i \(0.596749\pi\)
\(878\) 17584.0 0.675890
\(879\) 0 0
\(880\) −8704.00 −0.333422
\(881\) −11310.0 −0.432513 −0.216256 0.976337i \(-0.569385\pi\)
−0.216256 + 0.976337i \(0.569385\pi\)
\(882\) 0 0
\(883\) 17260.0 0.657809 0.328904 0.944363i \(-0.393321\pi\)
0.328904 + 0.944363i \(0.393321\pi\)
\(884\) 7176.00 0.273026
\(885\) 0 0
\(886\) 5520.00 0.209309
\(887\) −832.000 −0.0314947 −0.0157474 0.999876i \(-0.505013\pi\)
−0.0157474 + 0.999876i \(0.505013\pi\)
\(888\) 0 0
\(889\) −29120.0 −1.09860
\(890\) 14208.0 0.535116
\(891\) 0 0
\(892\) 384.000 0.0144140
\(893\) −49032.0 −1.83739
\(894\) 0 0
\(895\) 4416.00 0.164928
\(896\) 3584.00 0.133631
\(897\) 0 0
\(898\) −19064.0 −0.708434
\(899\) −33440.0 −1.24059
\(900\) 0 0
\(901\) 27324.0 1.01032
\(902\) 16320.0 0.602435
\(903\) 0 0
\(904\) 3344.00 0.123031
\(905\) −55584.0 −2.04163
\(906\) 0 0
\(907\) 31740.0 1.16197 0.580986 0.813913i \(-0.302667\pi\)
0.580986 + 0.813913i \(0.302667\pi\)
\(908\) −792.000 −0.0289465
\(909\) 0 0
\(910\) −11648.0 −0.424316
\(911\) 23568.0 0.857127 0.428563 0.903512i \(-0.359020\pi\)
0.428563 + 0.903512i \(0.359020\pi\)
\(912\) 0 0
\(913\) 25908.0 0.939134
\(914\) 25724.0 0.930935
\(915\) 0 0
\(916\) 23688.0 0.854447
\(917\) 15904.0 0.572733
\(918\) 0 0
\(919\) −18864.0 −0.677112 −0.338556 0.940946i \(-0.609938\pi\)
−0.338556 + 0.940946i \(0.609938\pi\)
\(920\) 6656.00 0.238524
\(921\) 0 0
\(922\) 13488.0 0.481783
\(923\) 7150.00 0.254978
\(924\) 0 0
\(925\) 44802.0 1.59252
\(926\) −19144.0 −0.679385
\(927\) 0 0
\(928\) 6080.00 0.215071
\(929\) 19536.0 0.689941 0.344971 0.938613i \(-0.387889\pi\)
0.344971 + 0.938613i \(0.387889\pi\)
\(930\) 0 0
\(931\) 47628.0 1.67663
\(932\) 20456.0 0.718947
\(933\) 0 0
\(934\) −18208.0 −0.637884
\(935\) 75072.0 2.62579
\(936\) 0 0
\(937\) 18174.0 0.633638 0.316819 0.948486i \(-0.397385\pi\)
0.316819 + 0.948486i \(0.397385\pi\)
\(938\) −36736.0 −1.27876
\(939\) 0 0
\(940\) −29056.0 −1.00819
\(941\) −51172.0 −1.77275 −0.886376 0.462966i \(-0.846785\pi\)
−0.886376 + 0.462966i \(0.846785\pi\)
\(942\) 0 0
\(943\) −12480.0 −0.430970
\(944\) 2464.00 0.0849538
\(945\) 0 0
\(946\) 9520.00 0.327190
\(947\) −3726.00 −0.127855 −0.0639275 0.997955i \(-0.520363\pi\)
−0.0639275 + 0.997955i \(0.520363\pi\)
\(948\) 0 0
\(949\) −7982.00 −0.273031
\(950\) 28296.0 0.966362
\(951\) 0 0
\(952\) −30912.0 −1.05238
\(953\) 40498.0 1.37656 0.688279 0.725447i \(-0.258367\pi\)
0.688279 + 0.725447i \(0.258367\pi\)
\(954\) 0 0
\(955\) 62720.0 2.12521
\(956\) 20904.0 0.707200
\(957\) 0 0
\(958\) 37740.0 1.27278
\(959\) −14784.0 −0.497810
\(960\) 0 0
\(961\) 1185.00 0.0397771
\(962\) −8892.00 −0.298014
\(963\) 0 0
\(964\) −3048.00 −0.101836
\(965\) 34976.0 1.16675
\(966\) 0 0
\(967\) 28568.0 0.950036 0.475018 0.879976i \(-0.342442\pi\)
0.475018 + 0.879976i \(0.342442\pi\)
\(968\) −1400.00 −0.0464852
\(969\) 0 0
\(970\) 32704.0 1.08254
\(971\) 8676.00 0.286742 0.143371 0.989669i \(-0.454206\pi\)
0.143371 + 0.989669i \(0.454206\pi\)
\(972\) 0 0
\(973\) −43568.0 −1.43548
\(974\) −3488.00 −0.114746
\(975\) 0 0
\(976\) 544.000 0.0178412
\(977\) 2796.00 0.0915578 0.0457789 0.998952i \(-0.485423\pi\)
0.0457789 + 0.998952i \(0.485423\pi\)
\(978\) 0 0
\(979\) −15096.0 −0.492819
\(980\) 28224.0 0.919982
\(981\) 0 0
\(982\) −26720.0 −0.868299
\(983\) 406.000 0.0131733 0.00658667 0.999978i \(-0.497903\pi\)
0.00658667 + 0.999978i \(0.497903\pi\)
\(984\) 0 0
\(985\) 21888.0 0.708030
\(986\) −52440.0 −1.69374
\(987\) 0 0
\(988\) −5616.00 −0.180839
\(989\) −7280.00 −0.234065
\(990\) 0 0
\(991\) −23232.0 −0.744691 −0.372346 0.928094i \(-0.621446\pi\)
−0.372346 + 0.928094i \(0.621446\pi\)
\(992\) −5632.00 −0.180258
\(993\) 0 0
\(994\) −30800.0 −0.982814
\(995\) −17152.0 −0.546487
\(996\) 0 0
\(997\) 6110.00 0.194088 0.0970440 0.995280i \(-0.469061\pi\)
0.0970440 + 0.995280i \(0.469061\pi\)
\(998\) 34736.0 1.10175
\(999\) 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 234.4.a.k.1.1 1
3.2 odd 2 78.4.a.a.1.1 1
4.3 odd 2 1872.4.a.o.1.1 1
12.11 even 2 624.4.a.f.1.1 1
15.14 odd 2 1950.4.a.o.1.1 1
24.5 odd 2 2496.4.a.q.1.1 1
24.11 even 2 2496.4.a.g.1.1 1
39.5 even 4 1014.4.b.a.337.2 2
39.8 even 4 1014.4.b.a.337.1 2
39.38 odd 2 1014.4.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.a.1.1 1 3.2 odd 2
234.4.a.k.1.1 1 1.1 even 1 trivial
624.4.a.f.1.1 1 12.11 even 2
1014.4.a.i.1.1 1 39.38 odd 2
1014.4.b.a.337.1 2 39.8 even 4
1014.4.b.a.337.2 2 39.5 even 4
1872.4.a.o.1.1 1 4.3 odd 2
1950.4.a.o.1.1 1 15.14 odd 2
2496.4.a.g.1.1 1 24.11 even 2
2496.4.a.q.1.1 1 24.5 odd 2