Properties

Label 1350.4.a.w.1.1
Level $1350$
Weight $4$
Character 1350.1
Self dual yes
Analytic conductor $79.653$
Analytic rank $0$
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] = [1350,4,Mod(1,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, 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("1350.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.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: \(79.6525785077\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1350.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +13.0000 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +13.0000 q^{7} +8.00000 q^{8} +30.0000 q^{11} +61.0000 q^{13} +26.0000 q^{14} +16.0000 q^{16} +12.0000 q^{17} -49.0000 q^{19} +60.0000 q^{22} +18.0000 q^{23} +122.000 q^{26} +52.0000 q^{28} +186.000 q^{29} -160.000 q^{31} +32.0000 q^{32} +24.0000 q^{34} +91.0000 q^{37} -98.0000 q^{38} -378.000 q^{41} +268.000 q^{43} +120.000 q^{44} +36.0000 q^{46} +144.000 q^{47} -174.000 q^{49} +244.000 q^{52} +570.000 q^{53} +104.000 q^{56} +372.000 q^{58} -204.000 q^{59} -877.000 q^{61} -320.000 q^{62} +64.0000 q^{64} +187.000 q^{67} +48.0000 q^{68} +606.000 q^{71} -431.000 q^{73} +182.000 q^{74} -196.000 q^{76} +390.000 q^{77} +1151.00 q^{79} -756.000 q^{82} +102.000 q^{83} +536.000 q^{86} +240.000 q^{88} -984.000 q^{89} +793.000 q^{91} +72.0000 q^{92} +288.000 q^{94} +265.000 q^{97} -348.000 q^{98} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 13.0000 0.701934 0.350967 0.936388i \(-0.385853\pi\)
0.350967 + 0.936388i \(0.385853\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 30.0000 0.822304 0.411152 0.911567i \(-0.365127\pi\)
0.411152 + 0.911567i \(0.365127\pi\)
\(12\) 0 0
\(13\) 61.0000 1.30141 0.650706 0.759330i \(-0.274473\pi\)
0.650706 + 0.759330i \(0.274473\pi\)
\(14\) 26.0000 0.496342
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 12.0000 0.171202 0.0856008 0.996330i \(-0.472719\pi\)
0.0856008 + 0.996330i \(0.472719\pi\)
\(18\) 0 0
\(19\) −49.0000 −0.591651 −0.295826 0.955242i \(-0.595595\pi\)
−0.295826 + 0.955242i \(0.595595\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 60.0000 0.581456
\(23\) 18.0000 0.163185 0.0815926 0.996666i \(-0.473999\pi\)
0.0815926 + 0.996666i \(0.473999\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 122.000 0.920237
\(27\) 0 0
\(28\) 52.0000 0.350967
\(29\) 186.000 1.19101 0.595506 0.803351i \(-0.296952\pi\)
0.595506 + 0.803351i \(0.296952\pi\)
\(30\) 0 0
\(31\) −160.000 −0.926995 −0.463498 0.886098i \(-0.653406\pi\)
−0.463498 + 0.886098i \(0.653406\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) 24.0000 0.121058
\(35\) 0 0
\(36\) 0 0
\(37\) 91.0000 0.404333 0.202166 0.979351i \(-0.435202\pi\)
0.202166 + 0.979351i \(0.435202\pi\)
\(38\) −98.0000 −0.418361
\(39\) 0 0
\(40\) 0 0
\(41\) −378.000 −1.43985 −0.719923 0.694054i \(-0.755823\pi\)
−0.719923 + 0.694054i \(0.755823\pi\)
\(42\) 0 0
\(43\) 268.000 0.950456 0.475228 0.879863i \(-0.342366\pi\)
0.475228 + 0.879863i \(0.342366\pi\)
\(44\) 120.000 0.411152
\(45\) 0 0
\(46\) 36.0000 0.115389
\(47\) 144.000 0.446906 0.223453 0.974715i \(-0.428267\pi\)
0.223453 + 0.974715i \(0.428267\pi\)
\(48\) 0 0
\(49\) −174.000 −0.507289
\(50\) 0 0
\(51\) 0 0
\(52\) 244.000 0.650706
\(53\) 570.000 1.47727 0.738637 0.674103i \(-0.235470\pi\)
0.738637 + 0.674103i \(0.235470\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 104.000 0.248171
\(57\) 0 0
\(58\) 372.000 0.842172
\(59\) −204.000 −0.450145 −0.225072 0.974342i \(-0.572262\pi\)
−0.225072 + 0.974342i \(0.572262\pi\)
\(60\) 0 0
\(61\) −877.000 −1.84079 −0.920396 0.390987i \(-0.872134\pi\)
−0.920396 + 0.390987i \(0.872134\pi\)
\(62\) −320.000 −0.655485
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 187.000 0.340980 0.170490 0.985359i \(-0.445465\pi\)
0.170490 + 0.985359i \(0.445465\pi\)
\(68\) 48.0000 0.0856008
\(69\) 0 0
\(70\) 0 0
\(71\) 606.000 1.01294 0.506472 0.862257i \(-0.330950\pi\)
0.506472 + 0.862257i \(0.330950\pi\)
\(72\) 0 0
\(73\) −431.000 −0.691024 −0.345512 0.938414i \(-0.612295\pi\)
−0.345512 + 0.938414i \(0.612295\pi\)
\(74\) 182.000 0.285906
\(75\) 0 0
\(76\) −196.000 −0.295826
\(77\) 390.000 0.577203
\(78\) 0 0
\(79\) 1151.00 1.63921 0.819605 0.572929i \(-0.194193\pi\)
0.819605 + 0.572929i \(0.194193\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −756.000 −1.01812
\(83\) 102.000 0.134891 0.0674455 0.997723i \(-0.478515\pi\)
0.0674455 + 0.997723i \(0.478515\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 536.000 0.672074
\(87\) 0 0
\(88\) 240.000 0.290728
\(89\) −984.000 −1.17195 −0.585976 0.810328i \(-0.699289\pi\)
−0.585976 + 0.810328i \(0.699289\pi\)
\(90\) 0 0
\(91\) 793.000 0.913505
\(92\) 72.0000 0.0815926
\(93\) 0 0
\(94\) 288.000 0.316010
\(95\) 0 0
\(96\) 0 0
\(97\) 265.000 0.277388 0.138694 0.990335i \(-0.455709\pi\)
0.138694 + 0.990335i \(0.455709\pi\)
\(98\) −348.000 −0.358707
\(99\) 0 0
\(100\) 0 0
\(101\) 1248.00 1.22951 0.614756 0.788718i \(-0.289255\pi\)
0.614756 + 0.788718i \(0.289255\pi\)
\(102\) 0 0
\(103\) 1225.00 1.17187 0.585936 0.810357i \(-0.300727\pi\)
0.585936 + 0.810357i \(0.300727\pi\)
\(104\) 488.000 0.460119
\(105\) 0 0
\(106\) 1140.00 1.04459
\(107\) −78.0000 −0.0704724 −0.0352362 0.999379i \(-0.511218\pi\)
−0.0352362 + 0.999379i \(0.511218\pi\)
\(108\) 0 0
\(109\) 2198.00 1.93147 0.965735 0.259530i \(-0.0835678\pi\)
0.965735 + 0.259530i \(0.0835678\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 208.000 0.175484
\(113\) 1986.00 1.65334 0.826669 0.562689i \(-0.190233\pi\)
0.826669 + 0.562689i \(0.190233\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 744.000 0.595506
\(117\) 0 0
\(118\) −408.000 −0.318300
\(119\) 156.000 0.120172
\(120\) 0 0
\(121\) −431.000 −0.323817
\(122\) −1754.00 −1.30164
\(123\) 0 0
\(124\) −640.000 −0.463498
\(125\) 0 0
\(126\) 0 0
\(127\) −2792.00 −1.95079 −0.975393 0.220471i \(-0.929240\pi\)
−0.975393 + 0.220471i \(0.929240\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 708.000 0.472200 0.236100 0.971729i \(-0.424131\pi\)
0.236100 + 0.971729i \(0.424131\pi\)
\(132\) 0 0
\(133\) −637.000 −0.415300
\(134\) 374.000 0.241110
\(135\) 0 0
\(136\) 96.0000 0.0605289
\(137\) −1686.00 −1.05142 −0.525711 0.850663i \(-0.676201\pi\)
−0.525711 + 0.850663i \(0.676201\pi\)
\(138\) 0 0
\(139\) −307.000 −0.187334 −0.0936669 0.995604i \(-0.529859\pi\)
−0.0936669 + 0.995604i \(0.529859\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1212.00 0.716259
\(143\) 1830.00 1.07016
\(144\) 0 0
\(145\) 0 0
\(146\) −862.000 −0.488628
\(147\) 0 0
\(148\) 364.000 0.202166
\(149\) 1812.00 0.996274 0.498137 0.867098i \(-0.334018\pi\)
0.498137 + 0.867098i \(0.334018\pi\)
\(150\) 0 0
\(151\) 203.000 0.109403 0.0547017 0.998503i \(-0.482579\pi\)
0.0547017 + 0.998503i \(0.482579\pi\)
\(152\) −392.000 −0.209180
\(153\) 0 0
\(154\) 780.000 0.408144
\(155\) 0 0
\(156\) 0 0
\(157\) 214.000 0.108784 0.0543919 0.998520i \(-0.482678\pi\)
0.0543919 + 0.998520i \(0.482678\pi\)
\(158\) 2302.00 1.15910
\(159\) 0 0
\(160\) 0 0
\(161\) 234.000 0.114545
\(162\) 0 0
\(163\) 673.000 0.323395 0.161698 0.986840i \(-0.448303\pi\)
0.161698 + 0.986840i \(0.448303\pi\)
\(164\) −1512.00 −0.719923
\(165\) 0 0
\(166\) 204.000 0.0953824
\(167\) −3696.00 −1.71261 −0.856303 0.516474i \(-0.827244\pi\)
−0.856303 + 0.516474i \(0.827244\pi\)
\(168\) 0 0
\(169\) 1524.00 0.693673
\(170\) 0 0
\(171\) 0 0
\(172\) 1072.00 0.475228
\(173\) 3132.00 1.37643 0.688213 0.725509i \(-0.258396\pi\)
0.688213 + 0.725509i \(0.258396\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 480.000 0.205576
\(177\) 0 0
\(178\) −1968.00 −0.828696
\(179\) −510.000 −0.212956 −0.106478 0.994315i \(-0.533957\pi\)
−0.106478 + 0.994315i \(0.533957\pi\)
\(180\) 0 0
\(181\) −1087.00 −0.446387 −0.223194 0.974774i \(-0.571648\pi\)
−0.223194 + 0.974774i \(0.571648\pi\)
\(182\) 1586.00 0.645946
\(183\) 0 0
\(184\) 144.000 0.0576947
\(185\) 0 0
\(186\) 0 0
\(187\) 360.000 0.140780
\(188\) 576.000 0.223453
\(189\) 0 0
\(190\) 0 0
\(191\) 4056.00 1.53655 0.768277 0.640117i \(-0.221114\pi\)
0.768277 + 0.640117i \(0.221114\pi\)
\(192\) 0 0
\(193\) −473.000 −0.176411 −0.0882054 0.996102i \(-0.528113\pi\)
−0.0882054 + 0.996102i \(0.528113\pi\)
\(194\) 530.000 0.196143
\(195\) 0 0
\(196\) −696.000 −0.253644
\(197\) 2556.00 0.924403 0.462202 0.886775i \(-0.347060\pi\)
0.462202 + 0.886775i \(0.347060\pi\)
\(198\) 0 0
\(199\) −2923.00 −1.04124 −0.520618 0.853790i \(-0.674298\pi\)
−0.520618 + 0.853790i \(0.674298\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2496.00 0.869396
\(203\) 2418.00 0.836011
\(204\) 0 0
\(205\) 0 0
\(206\) 2450.00 0.828639
\(207\) 0 0
\(208\) 976.000 0.325353
\(209\) −1470.00 −0.486517
\(210\) 0 0
\(211\) −3175.00 −1.03591 −0.517953 0.855409i \(-0.673306\pi\)
−0.517953 + 0.855409i \(0.673306\pi\)
\(212\) 2280.00 0.738637
\(213\) 0 0
\(214\) −156.000 −0.0498315
\(215\) 0 0
\(216\) 0 0
\(217\) −2080.00 −0.650689
\(218\) 4396.00 1.36576
\(219\) 0 0
\(220\) 0 0
\(221\) 732.000 0.222804
\(222\) 0 0
\(223\) 2176.00 0.653434 0.326717 0.945122i \(-0.394058\pi\)
0.326717 + 0.945122i \(0.394058\pi\)
\(224\) 416.000 0.124086
\(225\) 0 0
\(226\) 3972.00 1.16909
\(227\) −3834.00 −1.12102 −0.560510 0.828148i \(-0.689395\pi\)
−0.560510 + 0.828148i \(0.689395\pi\)
\(228\) 0 0
\(229\) −3202.00 −0.923992 −0.461996 0.886882i \(-0.652867\pi\)
−0.461996 + 0.886882i \(0.652867\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1488.00 0.421086
\(233\) 4152.00 1.16741 0.583705 0.811966i \(-0.301602\pi\)
0.583705 + 0.811966i \(0.301602\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −816.000 −0.225072
\(237\) 0 0
\(238\) 312.000 0.0849746
\(239\) 5466.00 1.47936 0.739678 0.672961i \(-0.234978\pi\)
0.739678 + 0.672961i \(0.234978\pi\)
\(240\) 0 0
\(241\) −943.000 −0.252050 −0.126025 0.992027i \(-0.540222\pi\)
−0.126025 + 0.992027i \(0.540222\pi\)
\(242\) −862.000 −0.228973
\(243\) 0 0
\(244\) −3508.00 −0.920396
\(245\) 0 0
\(246\) 0 0
\(247\) −2989.00 −0.769982
\(248\) −1280.00 −0.327742
\(249\) 0 0
\(250\) 0 0
\(251\) 7290.00 1.83323 0.916615 0.399771i \(-0.130910\pi\)
0.916615 + 0.399771i \(0.130910\pi\)
\(252\) 0 0
\(253\) 540.000 0.134188
\(254\) −5584.00 −1.37941
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 312.000 0.0757277 0.0378639 0.999283i \(-0.487945\pi\)
0.0378639 + 0.999283i \(0.487945\pi\)
\(258\) 0 0
\(259\) 1183.00 0.283815
\(260\) 0 0
\(261\) 0 0
\(262\) 1416.00 0.333896
\(263\) −8004.00 −1.87661 −0.938304 0.345812i \(-0.887603\pi\)
−0.938304 + 0.345812i \(0.887603\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1274.00 −0.293661
\(267\) 0 0
\(268\) 748.000 0.170490
\(269\) 324.000 0.0734373 0.0367186 0.999326i \(-0.488309\pi\)
0.0367186 + 0.999326i \(0.488309\pi\)
\(270\) 0 0
\(271\) −7849.00 −1.75938 −0.879692 0.475545i \(-0.842251\pi\)
−0.879692 + 0.475545i \(0.842251\pi\)
\(272\) 192.000 0.0428004
\(273\) 0 0
\(274\) −3372.00 −0.743467
\(275\) 0 0
\(276\) 0 0
\(277\) 5758.00 1.24897 0.624485 0.781037i \(-0.285309\pi\)
0.624485 + 0.781037i \(0.285309\pi\)
\(278\) −614.000 −0.132465
\(279\) 0 0
\(280\) 0 0
\(281\) 2688.00 0.570650 0.285325 0.958431i \(-0.407898\pi\)
0.285325 + 0.958431i \(0.407898\pi\)
\(282\) 0 0
\(283\) −3260.00 −0.684759 −0.342380 0.939562i \(-0.611233\pi\)
−0.342380 + 0.939562i \(0.611233\pi\)
\(284\) 2424.00 0.506472
\(285\) 0 0
\(286\) 3660.00 0.756714
\(287\) −4914.00 −1.01068
\(288\) 0 0
\(289\) −4769.00 −0.970690
\(290\) 0 0
\(291\) 0 0
\(292\) −1724.00 −0.345512
\(293\) 5922.00 1.18077 0.590387 0.807120i \(-0.298975\pi\)
0.590387 + 0.807120i \(0.298975\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 728.000 0.142953
\(297\) 0 0
\(298\) 3624.00 0.704472
\(299\) 1098.00 0.212371
\(300\) 0 0
\(301\) 3484.00 0.667158
\(302\) 406.000 0.0773599
\(303\) 0 0
\(304\) −784.000 −0.147913
\(305\) 0 0
\(306\) 0 0
\(307\) −3728.00 −0.693056 −0.346528 0.938040i \(-0.612639\pi\)
−0.346528 + 0.938040i \(0.612639\pi\)
\(308\) 1560.00 0.288601
\(309\) 0 0
\(310\) 0 0
\(311\) 732.000 0.133466 0.0667330 0.997771i \(-0.478742\pi\)
0.0667330 + 0.997771i \(0.478742\pi\)
\(312\) 0 0
\(313\) −5357.00 −0.967398 −0.483699 0.875234i \(-0.660707\pi\)
−0.483699 + 0.875234i \(0.660707\pi\)
\(314\) 428.000 0.0769218
\(315\) 0 0
\(316\) 4604.00 0.819605
\(317\) −4572.00 −0.810060 −0.405030 0.914303i \(-0.632739\pi\)
−0.405030 + 0.914303i \(0.632739\pi\)
\(318\) 0 0
\(319\) 5580.00 0.979373
\(320\) 0 0
\(321\) 0 0
\(322\) 468.000 0.0809957
\(323\) −588.000 −0.101292
\(324\) 0 0
\(325\) 0 0
\(326\) 1346.00 0.228675
\(327\) 0 0
\(328\) −3024.00 −0.509062
\(329\) 1872.00 0.313698
\(330\) 0 0
\(331\) 845.000 0.140318 0.0701592 0.997536i \(-0.477649\pi\)
0.0701592 + 0.997536i \(0.477649\pi\)
\(332\) 408.000 0.0674455
\(333\) 0 0
\(334\) −7392.00 −1.21099
\(335\) 0 0
\(336\) 0 0
\(337\) −8723.00 −1.41001 −0.705003 0.709204i \(-0.749054\pi\)
−0.705003 + 0.709204i \(0.749054\pi\)
\(338\) 3048.00 0.490501
\(339\) 0 0
\(340\) 0 0
\(341\) −4800.00 −0.762271
\(342\) 0 0
\(343\) −6721.00 −1.05802
\(344\) 2144.00 0.336037
\(345\) 0 0
\(346\) 6264.00 0.973280
\(347\) 9018.00 1.39513 0.697567 0.716519i \(-0.254266\pi\)
0.697567 + 0.716519i \(0.254266\pi\)
\(348\) 0 0
\(349\) 5759.00 0.883301 0.441651 0.897187i \(-0.354393\pi\)
0.441651 + 0.897187i \(0.354393\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 960.000 0.145364
\(353\) −5772.00 −0.870291 −0.435145 0.900360i \(-0.643303\pi\)
−0.435145 + 0.900360i \(0.643303\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3936.00 −0.585976
\(357\) 0 0
\(358\) −1020.00 −0.150583
\(359\) 2046.00 0.300790 0.150395 0.988626i \(-0.451945\pi\)
0.150395 + 0.988626i \(0.451945\pi\)
\(360\) 0 0
\(361\) −4458.00 −0.649949
\(362\) −2174.00 −0.315643
\(363\) 0 0
\(364\) 3172.00 0.456753
\(365\) 0 0
\(366\) 0 0
\(367\) 1069.00 0.152047 0.0760236 0.997106i \(-0.475778\pi\)
0.0760236 + 0.997106i \(0.475778\pi\)
\(368\) 288.000 0.0407963
\(369\) 0 0
\(370\) 0 0
\(371\) 7410.00 1.03695
\(372\) 0 0
\(373\) −7133.00 −0.990168 −0.495084 0.868845i \(-0.664863\pi\)
−0.495084 + 0.868845i \(0.664863\pi\)
\(374\) 720.000 0.0995463
\(375\) 0 0
\(376\) 1152.00 0.158005
\(377\) 11346.0 1.55000
\(378\) 0 0
\(379\) −8557.00 −1.15975 −0.579873 0.814707i \(-0.696898\pi\)
−0.579873 + 0.814707i \(0.696898\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8112.00 1.08651
\(383\) 14328.0 1.91156 0.955779 0.294086i \(-0.0950153\pi\)
0.955779 + 0.294086i \(0.0950153\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −946.000 −0.124741
\(387\) 0 0
\(388\) 1060.00 0.138694
\(389\) −13500.0 −1.75958 −0.879791 0.475361i \(-0.842317\pi\)
−0.879791 + 0.475361i \(0.842317\pi\)
\(390\) 0 0
\(391\) 216.000 0.0279376
\(392\) −1392.00 −0.179354
\(393\) 0 0
\(394\) 5112.00 0.653652
\(395\) 0 0
\(396\) 0 0
\(397\) −1334.00 −0.168644 −0.0843218 0.996439i \(-0.526872\pi\)
−0.0843218 + 0.996439i \(0.526872\pi\)
\(398\) −5846.00 −0.736265
\(399\) 0 0
\(400\) 0 0
\(401\) −3474.00 −0.432627 −0.216313 0.976324i \(-0.569403\pi\)
−0.216313 + 0.976324i \(0.569403\pi\)
\(402\) 0 0
\(403\) −9760.00 −1.20640
\(404\) 4992.00 0.614756
\(405\) 0 0
\(406\) 4836.00 0.591149
\(407\) 2730.00 0.332484
\(408\) 0 0
\(409\) 569.000 0.0687903 0.0343952 0.999408i \(-0.489050\pi\)
0.0343952 + 0.999408i \(0.489050\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4900.00 0.585936
\(413\) −2652.00 −0.315972
\(414\) 0 0
\(415\) 0 0
\(416\) 1952.00 0.230059
\(417\) 0 0
\(418\) −2940.00 −0.344019
\(419\) 9132.00 1.06474 0.532372 0.846511i \(-0.321301\pi\)
0.532372 + 0.846511i \(0.321301\pi\)
\(420\) 0 0
\(421\) −2971.00 −0.343937 −0.171969 0.985102i \(-0.555013\pi\)
−0.171969 + 0.985102i \(0.555013\pi\)
\(422\) −6350.00 −0.732496
\(423\) 0 0
\(424\) 4560.00 0.522295
\(425\) 0 0
\(426\) 0 0
\(427\) −11401.0 −1.29211
\(428\) −312.000 −0.0352362
\(429\) 0 0
\(430\) 0 0
\(431\) −12042.0 −1.34581 −0.672903 0.739730i \(-0.734953\pi\)
−0.672903 + 0.739730i \(0.734953\pi\)
\(432\) 0 0
\(433\) 8566.00 0.950706 0.475353 0.879795i \(-0.342320\pi\)
0.475353 + 0.879795i \(0.342320\pi\)
\(434\) −4160.00 −0.460107
\(435\) 0 0
\(436\) 8792.00 0.965735
\(437\) −882.000 −0.0965487
\(438\) 0 0
\(439\) 7400.00 0.804516 0.402258 0.915526i \(-0.368225\pi\)
0.402258 + 0.915526i \(0.368225\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1464.00 0.157546
\(443\) −2580.00 −0.276703 −0.138352 0.990383i \(-0.544180\pi\)
−0.138352 + 0.990383i \(0.544180\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 4352.00 0.462047
\(447\) 0 0
\(448\) 832.000 0.0877418
\(449\) −13200.0 −1.38741 −0.693704 0.720260i \(-0.744023\pi\)
−0.693704 + 0.720260i \(0.744023\pi\)
\(450\) 0 0
\(451\) −11340.0 −1.18399
\(452\) 7944.00 0.826669
\(453\) 0 0
\(454\) −7668.00 −0.792681
\(455\) 0 0
\(456\) 0 0
\(457\) −18038.0 −1.84635 −0.923175 0.384380i \(-0.874415\pi\)
−0.923175 + 0.384380i \(0.874415\pi\)
\(458\) −6404.00 −0.653361
\(459\) 0 0
\(460\) 0 0
\(461\) −5544.00 −0.560108 −0.280054 0.959984i \(-0.590352\pi\)
−0.280054 + 0.959984i \(0.590352\pi\)
\(462\) 0 0
\(463\) 17137.0 1.72014 0.860069 0.510178i \(-0.170420\pi\)
0.860069 + 0.510178i \(0.170420\pi\)
\(464\) 2976.00 0.297753
\(465\) 0 0
\(466\) 8304.00 0.825484
\(467\) −15888.0 −1.57432 −0.787162 0.616747i \(-0.788450\pi\)
−0.787162 + 0.616747i \(0.788450\pi\)
\(468\) 0 0
\(469\) 2431.00 0.239346
\(470\) 0 0
\(471\) 0 0
\(472\) −1632.00 −0.159150
\(473\) 8040.00 0.781564
\(474\) 0 0
\(475\) 0 0
\(476\) 624.000 0.0600861
\(477\) 0 0
\(478\) 10932.0 1.04606
\(479\) −3942.00 −0.376022 −0.188011 0.982167i \(-0.560204\pi\)
−0.188011 + 0.982167i \(0.560204\pi\)
\(480\) 0 0
\(481\) 5551.00 0.526203
\(482\) −1886.00 −0.178226
\(483\) 0 0
\(484\) −1724.00 −0.161908
\(485\) 0 0
\(486\) 0 0
\(487\) −1379.00 −0.128313 −0.0641565 0.997940i \(-0.520436\pi\)
−0.0641565 + 0.997940i \(0.520436\pi\)
\(488\) −7016.00 −0.650818
\(489\) 0 0
\(490\) 0 0
\(491\) −14214.0 −1.30645 −0.653227 0.757162i \(-0.726585\pi\)
−0.653227 + 0.757162i \(0.726585\pi\)
\(492\) 0 0
\(493\) 2232.00 0.203903
\(494\) −5978.00 −0.544459
\(495\) 0 0
\(496\) −2560.00 −0.231749
\(497\) 7878.00 0.711019
\(498\) 0 0
\(499\) 9992.00 0.896400 0.448200 0.893933i \(-0.352065\pi\)
0.448200 + 0.893933i \(0.352065\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 14580.0 1.29629
\(503\) −21258.0 −1.88439 −0.942194 0.335067i \(-0.891241\pi\)
−0.942194 + 0.335067i \(0.891241\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1080.00 0.0948851
\(507\) 0 0
\(508\) −11168.0 −0.975393
\(509\) 16614.0 1.44676 0.723382 0.690448i \(-0.242587\pi\)
0.723382 + 0.690448i \(0.242587\pi\)
\(510\) 0 0
\(511\) −5603.00 −0.485053
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 624.000 0.0535476
\(515\) 0 0
\(516\) 0 0
\(517\) 4320.00 0.367492
\(518\) 2366.00 0.200687
\(519\) 0 0
\(520\) 0 0
\(521\) −11838.0 −0.995455 −0.497728 0.867333i \(-0.665832\pi\)
−0.497728 + 0.867333i \(0.665832\pi\)
\(522\) 0 0
\(523\) −2201.00 −0.184021 −0.0920105 0.995758i \(-0.529329\pi\)
−0.0920105 + 0.995758i \(0.529329\pi\)
\(524\) 2832.00 0.236100
\(525\) 0 0
\(526\) −16008.0 −1.32696
\(527\) −1920.00 −0.158703
\(528\) 0 0
\(529\) −11843.0 −0.973371
\(530\) 0 0
\(531\) 0 0
\(532\) −2548.00 −0.207650
\(533\) −23058.0 −1.87383
\(534\) 0 0
\(535\) 0 0
\(536\) 1496.00 0.120555
\(537\) 0 0
\(538\) 648.000 0.0519280
\(539\) −5220.00 −0.417145
\(540\) 0 0
\(541\) −10795.0 −0.857880 −0.428940 0.903333i \(-0.641113\pi\)
−0.428940 + 0.903333i \(0.641113\pi\)
\(542\) −15698.0 −1.24407
\(543\) 0 0
\(544\) 384.000 0.0302645
\(545\) 0 0
\(546\) 0 0
\(547\) 14185.0 1.10879 0.554394 0.832254i \(-0.312950\pi\)
0.554394 + 0.832254i \(0.312950\pi\)
\(548\) −6744.00 −0.525711
\(549\) 0 0
\(550\) 0 0
\(551\) −9114.00 −0.704663
\(552\) 0 0
\(553\) 14963.0 1.15062
\(554\) 11516.0 0.883155
\(555\) 0 0
\(556\) −1228.00 −0.0936669
\(557\) −3576.00 −0.272029 −0.136014 0.990707i \(-0.543429\pi\)
−0.136014 + 0.990707i \(0.543429\pi\)
\(558\) 0 0
\(559\) 16348.0 1.23694
\(560\) 0 0
\(561\) 0 0
\(562\) 5376.00 0.403510
\(563\) −9132.00 −0.683602 −0.341801 0.939772i \(-0.611037\pi\)
−0.341801 + 0.939772i \(0.611037\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −6520.00 −0.484198
\(567\) 0 0
\(568\) 4848.00 0.358130
\(569\) −25020.0 −1.84340 −0.921699 0.387907i \(-0.873198\pi\)
−0.921699 + 0.387907i \(0.873198\pi\)
\(570\) 0 0
\(571\) −15997.0 −1.17242 −0.586212 0.810158i \(-0.699381\pi\)
−0.586212 + 0.810158i \(0.699381\pi\)
\(572\) 7320.00 0.535078
\(573\) 0 0
\(574\) −9828.00 −0.714656
\(575\) 0 0
\(576\) 0 0
\(577\) 5971.00 0.430808 0.215404 0.976525i \(-0.430893\pi\)
0.215404 + 0.976525i \(0.430893\pi\)
\(578\) −9538.00 −0.686381
\(579\) 0 0
\(580\) 0 0
\(581\) 1326.00 0.0946846
\(582\) 0 0
\(583\) 17100.0 1.21477
\(584\) −3448.00 −0.244314
\(585\) 0 0
\(586\) 11844.0 0.834934
\(587\) 19242.0 1.35299 0.676493 0.736449i \(-0.263499\pi\)
0.676493 + 0.736449i \(0.263499\pi\)
\(588\) 0 0
\(589\) 7840.00 0.548458
\(590\) 0 0
\(591\) 0 0
\(592\) 1456.00 0.101083
\(593\) 8118.00 0.562169 0.281085 0.959683i \(-0.409306\pi\)
0.281085 + 0.959683i \(0.409306\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7248.00 0.498137
\(597\) 0 0
\(598\) 2196.00 0.150169
\(599\) −1902.00 −0.129739 −0.0648695 0.997894i \(-0.520663\pi\)
−0.0648695 + 0.997894i \(0.520663\pi\)
\(600\) 0 0
\(601\) −14074.0 −0.955225 −0.477613 0.878571i \(-0.658498\pi\)
−0.477613 + 0.878571i \(0.658498\pi\)
\(602\) 6968.00 0.471752
\(603\) 0 0
\(604\) 812.000 0.0547017
\(605\) 0 0
\(606\) 0 0
\(607\) 13825.0 0.924447 0.462224 0.886763i \(-0.347052\pi\)
0.462224 + 0.886763i \(0.347052\pi\)
\(608\) −1568.00 −0.104590
\(609\) 0 0
\(610\) 0 0
\(611\) 8784.00 0.581608
\(612\) 0 0
\(613\) −15569.0 −1.02582 −0.512909 0.858443i \(-0.671432\pi\)
−0.512909 + 0.858443i \(0.671432\pi\)
\(614\) −7456.00 −0.490065
\(615\) 0 0
\(616\) 3120.00 0.204072
\(617\) 11922.0 0.777896 0.388948 0.921260i \(-0.372839\pi\)
0.388948 + 0.921260i \(0.372839\pi\)
\(618\) 0 0
\(619\) 6899.00 0.447971 0.223986 0.974592i \(-0.428093\pi\)
0.223986 + 0.974592i \(0.428093\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 1464.00 0.0943747
\(623\) −12792.0 −0.822633
\(624\) 0 0
\(625\) 0 0
\(626\) −10714.0 −0.684054
\(627\) 0 0
\(628\) 856.000 0.0543919
\(629\) 1092.00 0.0692224
\(630\) 0 0
\(631\) 11711.0 0.738839 0.369420 0.929263i \(-0.379557\pi\)
0.369420 + 0.929263i \(0.379557\pi\)
\(632\) 9208.00 0.579548
\(633\) 0 0
\(634\) −9144.00 −0.572799
\(635\) 0 0
\(636\) 0 0
\(637\) −10614.0 −0.660192
\(638\) 11160.0 0.692521
\(639\) 0 0
\(640\) 0 0
\(641\) −9240.00 −0.569357 −0.284679 0.958623i \(-0.591887\pi\)
−0.284679 + 0.958623i \(0.591887\pi\)
\(642\) 0 0
\(643\) 17908.0 1.09832 0.549162 0.835716i \(-0.314947\pi\)
0.549162 + 0.835716i \(0.314947\pi\)
\(644\) 936.000 0.0572726
\(645\) 0 0
\(646\) −1176.00 −0.0716240
\(647\) 7530.00 0.457550 0.228775 0.973479i \(-0.426528\pi\)
0.228775 + 0.973479i \(0.426528\pi\)
\(648\) 0 0
\(649\) −6120.00 −0.370156
\(650\) 0 0
\(651\) 0 0
\(652\) 2692.00 0.161698
\(653\) −22788.0 −1.36564 −0.682820 0.730586i \(-0.739247\pi\)
−0.682820 + 0.730586i \(0.739247\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −6048.00 −0.359961
\(657\) 0 0
\(658\) 3744.00 0.221818
\(659\) −16440.0 −0.971793 −0.485896 0.874016i \(-0.661507\pi\)
−0.485896 + 0.874016i \(0.661507\pi\)
\(660\) 0 0
\(661\) −12421.0 −0.730894 −0.365447 0.930832i \(-0.619084\pi\)
−0.365447 + 0.930832i \(0.619084\pi\)
\(662\) 1690.00 0.0992201
\(663\) 0 0
\(664\) 816.000 0.0476912
\(665\) 0 0
\(666\) 0 0
\(667\) 3348.00 0.194355
\(668\) −14784.0 −0.856303
\(669\) 0 0
\(670\) 0 0
\(671\) −26310.0 −1.51369
\(672\) 0 0
\(673\) −6461.00 −0.370064 −0.185032 0.982732i \(-0.559239\pi\)
−0.185032 + 0.982732i \(0.559239\pi\)
\(674\) −17446.0 −0.997025
\(675\) 0 0
\(676\) 6096.00 0.346837
\(677\) −912.000 −0.0517740 −0.0258870 0.999665i \(-0.508241\pi\)
−0.0258870 + 0.999665i \(0.508241\pi\)
\(678\) 0 0
\(679\) 3445.00 0.194708
\(680\) 0 0
\(681\) 0 0
\(682\) −9600.00 −0.539007
\(683\) −14442.0 −0.809089 −0.404544 0.914518i \(-0.632570\pi\)
−0.404544 + 0.914518i \(0.632570\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13442.0 −0.748131
\(687\) 0 0
\(688\) 4288.00 0.237614
\(689\) 34770.0 1.92254
\(690\) 0 0
\(691\) 1892.00 0.104161 0.0520804 0.998643i \(-0.483415\pi\)
0.0520804 + 0.998643i \(0.483415\pi\)
\(692\) 12528.0 0.688213
\(693\) 0 0
\(694\) 18036.0 0.986509
\(695\) 0 0
\(696\) 0 0
\(697\) −4536.00 −0.246504
\(698\) 11518.0 0.624588
\(699\) 0 0
\(700\) 0 0
\(701\) 7914.00 0.426402 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(702\) 0 0
\(703\) −4459.00 −0.239224
\(704\) 1920.00 0.102788
\(705\) 0 0
\(706\) −11544.0 −0.615388
\(707\) 16224.0 0.863036
\(708\) 0 0
\(709\) −1291.00 −0.0683844 −0.0341922 0.999415i \(-0.510886\pi\)
−0.0341922 + 0.999415i \(0.510886\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −7872.00 −0.414348
\(713\) −2880.00 −0.151272
\(714\) 0 0
\(715\) 0 0
\(716\) −2040.00 −0.106478
\(717\) 0 0
\(718\) 4092.00 0.212691
\(719\) 30210.0 1.56696 0.783479 0.621418i \(-0.213443\pi\)
0.783479 + 0.621418i \(0.213443\pi\)
\(720\) 0 0
\(721\) 15925.0 0.822577
\(722\) −8916.00 −0.459583
\(723\) 0 0
\(724\) −4348.00 −0.223194
\(725\) 0 0
\(726\) 0 0
\(727\) −15680.0 −0.799916 −0.399958 0.916533i \(-0.630975\pi\)
−0.399958 + 0.916533i \(0.630975\pi\)
\(728\) 6344.00 0.322973
\(729\) 0 0
\(730\) 0 0
\(731\) 3216.00 0.162720
\(732\) 0 0
\(733\) −13898.0 −0.700320 −0.350160 0.936690i \(-0.613873\pi\)
−0.350160 + 0.936690i \(0.613873\pi\)
\(734\) 2138.00 0.107514
\(735\) 0 0
\(736\) 576.000 0.0288473
\(737\) 5610.00 0.280389
\(738\) 0 0
\(739\) 35300.0 1.75715 0.878573 0.477607i \(-0.158496\pi\)
0.878573 + 0.477607i \(0.158496\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 14820.0 0.733234
\(743\) 28188.0 1.39181 0.695907 0.718132i \(-0.255003\pi\)
0.695907 + 0.718132i \(0.255003\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −14266.0 −0.700155
\(747\) 0 0
\(748\) 1440.00 0.0703899
\(749\) −1014.00 −0.0494670
\(750\) 0 0
\(751\) 25163.0 1.22265 0.611326 0.791379i \(-0.290637\pi\)
0.611326 + 0.791379i \(0.290637\pi\)
\(752\) 2304.00 0.111726
\(753\) 0 0
\(754\) 22692.0 1.09601
\(755\) 0 0
\(756\) 0 0
\(757\) −7979.00 −0.383093 −0.191547 0.981484i \(-0.561350\pi\)
−0.191547 + 0.981484i \(0.561350\pi\)
\(758\) −17114.0 −0.820064
\(759\) 0 0
\(760\) 0 0
\(761\) −26622.0 −1.26813 −0.634065 0.773280i \(-0.718615\pi\)
−0.634065 + 0.773280i \(0.718615\pi\)
\(762\) 0 0
\(763\) 28574.0 1.35576
\(764\) 16224.0 0.768277
\(765\) 0 0
\(766\) 28656.0 1.35168
\(767\) −12444.0 −0.585824
\(768\) 0 0
\(769\) −35413.0 −1.66063 −0.830316 0.557293i \(-0.811840\pi\)
−0.830316 + 0.557293i \(0.811840\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1892.00 −0.0882054
\(773\) −33834.0 −1.57429 −0.787144 0.616769i \(-0.788441\pi\)
−0.787144 + 0.616769i \(0.788441\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2120.00 0.0980716
\(777\) 0 0
\(778\) −27000.0 −1.24421
\(779\) 18522.0 0.851886
\(780\) 0 0
\(781\) 18180.0 0.832947
\(782\) 432.000 0.0197548
\(783\) 0 0
\(784\) −2784.00 −0.126822
\(785\) 0 0
\(786\) 0 0
\(787\) −13469.0 −0.610061 −0.305030 0.952343i \(-0.598667\pi\)
−0.305030 + 0.952343i \(0.598667\pi\)
\(788\) 10224.0 0.462202
\(789\) 0 0
\(790\) 0 0
\(791\) 25818.0 1.16053
\(792\) 0 0
\(793\) −53497.0 −2.39563
\(794\) −2668.00 −0.119249
\(795\) 0 0
\(796\) −11692.0 −0.520618
\(797\) −21168.0 −0.940789 −0.470395 0.882456i \(-0.655888\pi\)
−0.470395 + 0.882456i \(0.655888\pi\)
\(798\) 0 0
\(799\) 1728.00 0.0765109
\(800\) 0 0
\(801\) 0 0
\(802\) −6948.00 −0.305913
\(803\) −12930.0 −0.568231
\(804\) 0 0
\(805\) 0 0
\(806\) −19520.0 −0.853055
\(807\) 0 0
\(808\) 9984.00 0.434698
\(809\) −408.000 −0.0177312 −0.00886558 0.999961i \(-0.502822\pi\)
−0.00886558 + 0.999961i \(0.502822\pi\)
\(810\) 0 0
\(811\) −36916.0 −1.59839 −0.799196 0.601070i \(-0.794741\pi\)
−0.799196 + 0.601070i \(0.794741\pi\)
\(812\) 9672.00 0.418006
\(813\) 0 0
\(814\) 5460.00 0.235102
\(815\) 0 0
\(816\) 0 0
\(817\) −13132.0 −0.562338
\(818\) 1138.00 0.0486421
\(819\) 0 0
\(820\) 0 0
\(821\) 24636.0 1.04726 0.523631 0.851945i \(-0.324577\pi\)
0.523631 + 0.851945i \(0.324577\pi\)
\(822\) 0 0
\(823\) 18187.0 0.770303 0.385151 0.922853i \(-0.374149\pi\)
0.385151 + 0.922853i \(0.374149\pi\)
\(824\) 9800.00 0.414319
\(825\) 0 0
\(826\) −5304.00 −0.223426
\(827\) 1464.00 0.0615578 0.0307789 0.999526i \(-0.490201\pi\)
0.0307789 + 0.999526i \(0.490201\pi\)
\(828\) 0 0
\(829\) −12295.0 −0.515106 −0.257553 0.966264i \(-0.582916\pi\)
−0.257553 + 0.966264i \(0.582916\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3904.00 0.162676
\(833\) −2088.00 −0.0868486
\(834\) 0 0
\(835\) 0 0
\(836\) −5880.00 −0.243258
\(837\) 0 0
\(838\) 18264.0 0.752887
\(839\) −46884.0 −1.92922 −0.964610 0.263681i \(-0.915063\pi\)
−0.964610 + 0.263681i \(0.915063\pi\)
\(840\) 0 0
\(841\) 10207.0 0.418508
\(842\) −5942.00 −0.243201
\(843\) 0 0
\(844\) −12700.0 −0.517953
\(845\) 0 0
\(846\) 0 0
\(847\) −5603.00 −0.227298
\(848\) 9120.00 0.369318
\(849\) 0 0
\(850\) 0 0
\(851\) 1638.00 0.0659811
\(852\) 0 0
\(853\) −12197.0 −0.489587 −0.244793 0.969575i \(-0.578720\pi\)
−0.244793 + 0.969575i \(0.578720\pi\)
\(854\) −22802.0 −0.913663
\(855\) 0 0
\(856\) −624.000 −0.0249157
\(857\) −28182.0 −1.12331 −0.561656 0.827371i \(-0.689836\pi\)
−0.561656 + 0.827371i \(0.689836\pi\)
\(858\) 0 0
\(859\) 25433.0 1.01020 0.505101 0.863060i \(-0.331455\pi\)
0.505101 + 0.863060i \(0.331455\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −24084.0 −0.951629
\(863\) −16968.0 −0.669290 −0.334645 0.942344i \(-0.608616\pi\)
−0.334645 + 0.942344i \(0.608616\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 17132.0 0.672251
\(867\) 0 0
\(868\) −8320.00 −0.325345
\(869\) 34530.0 1.34793
\(870\) 0 0
\(871\) 11407.0 0.443756
\(872\) 17584.0 0.682878
\(873\) 0 0
\(874\) −1764.00 −0.0682702
\(875\) 0 0
\(876\) 0 0
\(877\) 22423.0 0.863365 0.431682 0.902026i \(-0.357920\pi\)
0.431682 + 0.902026i \(0.357920\pi\)
\(878\) 14800.0 0.568879
\(879\) 0 0
\(880\) 0 0
\(881\) −8442.00 −0.322836 −0.161418 0.986886i \(-0.551607\pi\)
−0.161418 + 0.986886i \(0.551607\pi\)
\(882\) 0 0
\(883\) −41207.0 −1.57047 −0.785236 0.619197i \(-0.787458\pi\)
−0.785236 + 0.619197i \(0.787458\pi\)
\(884\) 2928.00 0.111402
\(885\) 0 0
\(886\) −5160.00 −0.195659
\(887\) 8472.00 0.320701 0.160351 0.987060i \(-0.448738\pi\)
0.160351 + 0.987060i \(0.448738\pi\)
\(888\) 0 0
\(889\) −36296.0 −1.36932
\(890\) 0 0
\(891\) 0 0
\(892\) 8704.00 0.326717
\(893\) −7056.00 −0.264412
\(894\) 0 0
\(895\) 0 0
\(896\) 1664.00 0.0620428
\(897\) 0 0
\(898\) −26400.0 −0.981046
\(899\) −29760.0 −1.10406
\(900\) 0 0
\(901\) 6840.00 0.252912
\(902\) −22680.0 −0.837208
\(903\) 0 0
\(904\) 15888.0 0.584543
\(905\) 0 0
\(906\) 0 0
\(907\) 21799.0 0.798042 0.399021 0.916942i \(-0.369350\pi\)
0.399021 + 0.916942i \(0.369350\pi\)
\(908\) −15336.0 −0.560510
\(909\) 0 0
\(910\) 0 0
\(911\) −23544.0 −0.856254 −0.428127 0.903719i \(-0.640826\pi\)
−0.428127 + 0.903719i \(0.640826\pi\)
\(912\) 0 0
\(913\) 3060.00 0.110921
\(914\) −36076.0 −1.30557
\(915\) 0 0
\(916\) −12808.0 −0.461996
\(917\) 9204.00 0.331453
\(918\) 0 0
\(919\) 11072.0 0.397423 0.198711 0.980058i \(-0.436324\pi\)
0.198711 + 0.980058i \(0.436324\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −11088.0 −0.396056
\(923\) 36966.0 1.31826
\(924\) 0 0
\(925\) 0 0
\(926\) 34274.0 1.21632
\(927\) 0 0
\(928\) 5952.00 0.210543
\(929\) 21654.0 0.764741 0.382371 0.924009i \(-0.375108\pi\)
0.382371 + 0.924009i \(0.375108\pi\)
\(930\) 0 0
\(931\) 8526.00 0.300138
\(932\) 16608.0 0.583705
\(933\) 0 0
\(934\) −31776.0 −1.11321
\(935\) 0 0
\(936\) 0 0
\(937\) 42835.0 1.49345 0.746723 0.665135i \(-0.231626\pi\)
0.746723 + 0.665135i \(0.231626\pi\)
\(938\) 4862.00 0.169243
\(939\) 0 0
\(940\) 0 0
\(941\) 48534.0 1.68136 0.840682 0.541529i \(-0.182155\pi\)
0.840682 + 0.541529i \(0.182155\pi\)
\(942\) 0 0
\(943\) −6804.00 −0.234962
\(944\) −3264.00 −0.112536
\(945\) 0 0
\(946\) 16080.0 0.552649
\(947\) 14676.0 0.503597 0.251798 0.967780i \(-0.418978\pi\)
0.251798 + 0.967780i \(0.418978\pi\)
\(948\) 0 0
\(949\) −26291.0 −0.899307
\(950\) 0 0
\(951\) 0 0
\(952\) 1248.00 0.0424873
\(953\) 27372.0 0.930395 0.465197 0.885207i \(-0.345983\pi\)
0.465197 + 0.885207i \(0.345983\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 21864.0 0.739678
\(957\) 0 0
\(958\) −7884.00 −0.265888
\(959\) −21918.0 −0.738028
\(960\) 0 0
\(961\) −4191.00 −0.140680
\(962\) 11102.0 0.372082
\(963\) 0 0
\(964\) −3772.00 −0.126025
\(965\) 0 0
\(966\) 0 0
\(967\) −3581.00 −0.119087 −0.0595435 0.998226i \(-0.518965\pi\)
−0.0595435 + 0.998226i \(0.518965\pi\)
\(968\) −3448.00 −0.114486
\(969\) 0 0
\(970\) 0 0
\(971\) 1824.00 0.0602832 0.0301416 0.999546i \(-0.490404\pi\)
0.0301416 + 0.999546i \(0.490404\pi\)
\(972\) 0 0
\(973\) −3991.00 −0.131496
\(974\) −2758.00 −0.0907310
\(975\) 0 0
\(976\) −14032.0 −0.460198
\(977\) −29778.0 −0.975110 −0.487555 0.873092i \(-0.662111\pi\)
−0.487555 + 0.873092i \(0.662111\pi\)
\(978\) 0 0
\(979\) −29520.0 −0.963701
\(980\) 0 0
\(981\) 0 0
\(982\) −28428.0 −0.923802
\(983\) 9402.00 0.305063 0.152532 0.988299i \(-0.451257\pi\)
0.152532 + 0.988299i \(0.451257\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 4464.00 0.144181
\(987\) 0 0
\(988\) −11956.0 −0.384991
\(989\) 4824.00 0.155100
\(990\) 0 0
\(991\) −24907.0 −0.798382 −0.399191 0.916868i \(-0.630709\pi\)
−0.399191 + 0.916868i \(0.630709\pi\)
\(992\) −5120.00 −0.163871
\(993\) 0 0
\(994\) 15756.0 0.502767
\(995\) 0 0
\(996\) 0 0
\(997\) −27830.0 −0.884037 −0.442019 0.897006i \(-0.645737\pi\)
−0.442019 + 0.897006i \(0.645737\pi\)
\(998\) 19984.0 0.633850
\(999\) 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 1350.4.a.w.1.1 1
3.2 odd 2 1350.4.a.i.1.1 1
5.2 odd 4 1350.4.c.o.649.2 2
5.3 odd 4 1350.4.c.o.649.1 2
5.4 even 2 270.4.a.d.1.1 1
15.2 even 4 1350.4.c.f.649.1 2
15.8 even 4 1350.4.c.f.649.2 2
15.14 odd 2 270.4.a.h.1.1 yes 1
20.19 odd 2 2160.4.a.q.1.1 1
45.4 even 6 810.4.e.r.541.1 2
45.14 odd 6 810.4.e.j.541.1 2
45.29 odd 6 810.4.e.j.271.1 2
45.34 even 6 810.4.e.r.271.1 2
60.59 even 2 2160.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.4.a.d.1.1 1 5.4 even 2
270.4.a.h.1.1 yes 1 15.14 odd 2
810.4.e.j.271.1 2 45.29 odd 6
810.4.e.j.541.1 2 45.14 odd 6
810.4.e.r.271.1 2 45.34 even 6
810.4.e.r.541.1 2 45.4 even 6
1350.4.a.i.1.1 1 3.2 odd 2
1350.4.a.w.1.1 1 1.1 even 1 trivial
1350.4.c.f.649.1 2 15.2 even 4
1350.4.c.f.649.2 2 15.8 even 4
1350.4.c.o.649.1 2 5.3 odd 4
1350.4.c.o.649.2 2 5.2 odd 4
2160.4.a.g.1.1 1 60.59 even 2
2160.4.a.q.1.1 1 20.19 odd 2