Properties

Label 8550.2.a.cu.1.3
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8550,2,Mod(1,8550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.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: \(68.2720937282\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1710)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.517638\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.03528 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.03528 q^{7} -1.00000 q^{8} +3.86370 q^{11} -1.03528 q^{13} -1.03528 q^{14} +1.00000 q^{16} +7.46410 q^{17} -1.00000 q^{19} -3.86370 q^{22} -1.46410 q^{23} +1.03528 q^{26} +1.03528 q^{28} -9.52056 q^{29} +2.92820 q^{31} -1.00000 q^{32} -7.46410 q^{34} +6.69213 q^{37} +1.00000 q^{38} -6.69213 q^{41} +1.79315 q^{43} +3.86370 q^{44} +1.46410 q^{46} +9.46410 q^{47} -5.92820 q^{49} -1.03528 q^{52} +6.00000 q^{53} -1.03528 q^{56} +9.52056 q^{58} +12.6264 q^{59} -2.00000 q^{61} -2.92820 q^{62} +1.00000 q^{64} -3.58630 q^{67} +7.46410 q^{68} +15.4548 q^{71} +13.3843 q^{73} -6.69213 q^{74} -1.00000 q^{76} +4.00000 q^{77} -4.00000 q^{79} +6.69213 q^{82} -4.39230 q^{83} -1.79315 q^{86} -3.86370 q^{88} -1.03528 q^{89} -1.07180 q^{91} -1.46410 q^{92} -9.46410 q^{94} -17.2480 q^{97} +5.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} + 16 q^{17} - 4 q^{19} + 8 q^{23} - 16 q^{31} - 4 q^{32} - 16 q^{34} + 4 q^{38} - 8 q^{46} + 24 q^{47} + 4 q^{49} + 24 q^{53} - 8 q^{61} + 16 q^{62} + 4 q^{64} + 16 q^{68} - 4 q^{76} + 16 q^{77} - 16 q^{79} + 24 q^{83} - 32 q^{91} + 8 q^{92} - 24 q^{94} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.03528 0.391298 0.195649 0.980674i \(-0.437319\pi\)
0.195649 + 0.980674i \(0.437319\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 3.86370 1.16495 0.582475 0.812848i \(-0.302084\pi\)
0.582475 + 0.812848i \(0.302084\pi\)
\(12\) 0 0
\(13\) −1.03528 −0.287134 −0.143567 0.989641i \(-0.545857\pi\)
−0.143567 + 0.989641i \(0.545857\pi\)
\(14\) −1.03528 −0.276689
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.46410 1.81031 0.905155 0.425081i \(-0.139754\pi\)
0.905155 + 0.425081i \(0.139754\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −3.86370 −0.823744
\(23\) −1.46410 −0.305286 −0.152643 0.988281i \(-0.548779\pi\)
−0.152643 + 0.988281i \(0.548779\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.03528 0.203034
\(27\) 0 0
\(28\) 1.03528 0.195649
\(29\) −9.52056 −1.76792 −0.883962 0.467560i \(-0.845133\pi\)
−0.883962 + 0.467560i \(0.845133\pi\)
\(30\) 0 0
\(31\) 2.92820 0.525921 0.262960 0.964807i \(-0.415301\pi\)
0.262960 + 0.964807i \(0.415301\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −7.46410 −1.28008
\(35\) 0 0
\(36\) 0 0
\(37\) 6.69213 1.10018 0.550090 0.835106i \(-0.314594\pi\)
0.550090 + 0.835106i \(0.314594\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −6.69213 −1.04514 −0.522568 0.852598i \(-0.675026\pi\)
−0.522568 + 0.852598i \(0.675026\pi\)
\(42\) 0 0
\(43\) 1.79315 0.273453 0.136726 0.990609i \(-0.456342\pi\)
0.136726 + 0.990609i \(0.456342\pi\)
\(44\) 3.86370 0.582475
\(45\) 0 0
\(46\) 1.46410 0.215870
\(47\) 9.46410 1.38048 0.690241 0.723580i \(-0.257505\pi\)
0.690241 + 0.723580i \(0.257505\pi\)
\(48\) 0 0
\(49\) −5.92820 −0.846886
\(50\) 0 0
\(51\) 0 0
\(52\) −1.03528 −0.143567
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.03528 −0.138345
\(57\) 0 0
\(58\) 9.52056 1.25011
\(59\) 12.6264 1.64382 0.821908 0.569621i \(-0.192910\pi\)
0.821908 + 0.569621i \(0.192910\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −2.92820 −0.371882
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −3.58630 −0.438137 −0.219068 0.975710i \(-0.570302\pi\)
−0.219068 + 0.975710i \(0.570302\pi\)
\(68\) 7.46410 0.905155
\(69\) 0 0
\(70\) 0 0
\(71\) 15.4548 1.83415 0.917074 0.398716i \(-0.130544\pi\)
0.917074 + 0.398716i \(0.130544\pi\)
\(72\) 0 0
\(73\) 13.3843 1.56651 0.783255 0.621701i \(-0.213558\pi\)
0.783255 + 0.621701i \(0.213558\pi\)
\(74\) −6.69213 −0.777944
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.69213 0.739022
\(83\) −4.39230 −0.482118 −0.241059 0.970510i \(-0.577495\pi\)
−0.241059 + 0.970510i \(0.577495\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.79315 −0.193360
\(87\) 0 0
\(88\) −3.86370 −0.411872
\(89\) −1.03528 −0.109739 −0.0548695 0.998494i \(-0.517474\pi\)
−0.0548695 + 0.998494i \(0.517474\pi\)
\(90\) 0 0
\(91\) −1.07180 −0.112355
\(92\) −1.46410 −0.152643
\(93\) 0 0
\(94\) −9.46410 −0.976148
\(95\) 0 0
\(96\) 0 0
\(97\) −17.2480 −1.75127 −0.875633 0.482978i \(-0.839555\pi\)
−0.875633 + 0.482978i \(0.839555\pi\)
\(98\) 5.92820 0.598839
\(99\) 0 0
\(100\) 0 0
\(101\) 0.757875 0.0754114 0.0377057 0.999289i \(-0.487995\pi\)
0.0377057 + 0.999289i \(0.487995\pi\)
\(102\) 0 0
\(103\) 4.89898 0.482711 0.241355 0.970437i \(-0.422408\pi\)
0.241355 + 0.970437i \(0.422408\pi\)
\(104\) 1.03528 0.101517
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 10.9282 1.05647 0.528235 0.849098i \(-0.322854\pi\)
0.528235 + 0.849098i \(0.322854\pi\)
\(108\) 0 0
\(109\) −4.92820 −0.472036 −0.236018 0.971749i \(-0.575842\pi\)
−0.236018 + 0.971749i \(0.575842\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.03528 0.0978244
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −9.52056 −0.883962
\(117\) 0 0
\(118\) −12.6264 −1.16235
\(119\) 7.72741 0.708370
\(120\) 0 0
\(121\) 3.92820 0.357109
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 2.92820 0.262960
\(125\) 0 0
\(126\) 0 0
\(127\) −8.48528 −0.752947 −0.376473 0.926427i \(-0.622863\pi\)
−0.376473 + 0.926427i \(0.622863\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 11.5911 1.01272 0.506360 0.862322i \(-0.330991\pi\)
0.506360 + 0.862322i \(0.330991\pi\)
\(132\) 0 0
\(133\) −1.03528 −0.0897698
\(134\) 3.58630 0.309809
\(135\) 0 0
\(136\) −7.46410 −0.640041
\(137\) 7.46410 0.637701 0.318851 0.947805i \(-0.396703\pi\)
0.318851 + 0.947805i \(0.396703\pi\)
\(138\) 0 0
\(139\) −6.92820 −0.587643 −0.293821 0.955860i \(-0.594927\pi\)
−0.293821 + 0.955860i \(0.594927\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −15.4548 −1.29694
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) −13.3843 −1.10769
\(147\) 0 0
\(148\) 6.69213 0.550090
\(149\) 9.04008 0.740593 0.370296 0.928914i \(-0.379256\pi\)
0.370296 + 0.928914i \(0.379256\pi\)
\(150\) 0 0
\(151\) −21.8564 −1.77865 −0.889325 0.457277i \(-0.848825\pi\)
−0.889325 + 0.457277i \(0.848825\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) −14.1421 −1.12867 −0.564333 0.825547i \(-0.690866\pi\)
−0.564333 + 0.825547i \(0.690866\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) 0 0
\(161\) −1.51575 −0.119458
\(162\) 0 0
\(163\) −18.7637 −1.46969 −0.734844 0.678236i \(-0.762745\pi\)
−0.734844 + 0.678236i \(0.762745\pi\)
\(164\) −6.69213 −0.522568
\(165\) 0 0
\(166\) 4.39230 0.340909
\(167\) −2.92820 −0.226591 −0.113296 0.993561i \(-0.536141\pi\)
−0.113296 + 0.993561i \(0.536141\pi\)
\(168\) 0 0
\(169\) −11.9282 −0.917554
\(170\) 0 0
\(171\) 0 0
\(172\) 1.79315 0.136726
\(173\) 18.7846 1.42817 0.714084 0.700060i \(-0.246844\pi\)
0.714084 + 0.700060i \(0.246844\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.86370 0.291238
\(177\) 0 0
\(178\) 1.03528 0.0775972
\(179\) −22.4243 −1.67607 −0.838037 0.545613i \(-0.816297\pi\)
−0.838037 + 0.545613i \(0.816297\pi\)
\(180\) 0 0
\(181\) −7.85641 −0.583962 −0.291981 0.956424i \(-0.594314\pi\)
−0.291981 + 0.956424i \(0.594314\pi\)
\(182\) 1.07180 0.0794469
\(183\) 0 0
\(184\) 1.46410 0.107935
\(185\) 0 0
\(186\) 0 0
\(187\) 28.8391 2.10892
\(188\) 9.46410 0.690241
\(189\) 0 0
\(190\) 0 0
\(191\) 2.55103 0.184586 0.0922929 0.995732i \(-0.470580\pi\)
0.0922929 + 0.995732i \(0.470580\pi\)
\(192\) 0 0
\(193\) 5.93426 0.427157 0.213579 0.976926i \(-0.431488\pi\)
0.213579 + 0.976926i \(0.431488\pi\)
\(194\) 17.2480 1.23833
\(195\) 0 0
\(196\) −5.92820 −0.423443
\(197\) 16.5359 1.17813 0.589067 0.808084i \(-0.299495\pi\)
0.589067 + 0.808084i \(0.299495\pi\)
\(198\) 0 0
\(199\) 26.9282 1.90889 0.954445 0.298387i \(-0.0964487\pi\)
0.954445 + 0.298387i \(0.0964487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −0.757875 −0.0533239
\(203\) −9.85641 −0.691784
\(204\) 0 0
\(205\) 0 0
\(206\) −4.89898 −0.341328
\(207\) 0 0
\(208\) −1.03528 −0.0717835
\(209\) −3.86370 −0.267258
\(210\) 0 0
\(211\) 9.85641 0.678543 0.339272 0.940688i \(-0.389819\pi\)
0.339272 + 0.940688i \(0.389819\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −10.9282 −0.747037
\(215\) 0 0
\(216\) 0 0
\(217\) 3.03150 0.205792
\(218\) 4.92820 0.333780
\(219\) 0 0
\(220\) 0 0
\(221\) −7.72741 −0.519802
\(222\) 0 0
\(223\) 0.757875 0.0507510 0.0253755 0.999678i \(-0.491922\pi\)
0.0253755 + 0.999678i \(0.491922\pi\)
\(224\) −1.03528 −0.0691723
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 25.8564 1.71615 0.858075 0.513524i \(-0.171660\pi\)
0.858075 + 0.513524i \(0.171660\pi\)
\(228\) 0 0
\(229\) 23.8564 1.57648 0.788238 0.615371i \(-0.210994\pi\)
0.788238 + 0.615371i \(0.210994\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 9.52056 0.625055
\(233\) 19.4641 1.27514 0.637568 0.770394i \(-0.279941\pi\)
0.637568 + 0.770394i \(0.279941\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.6264 0.821908
\(237\) 0 0
\(238\) −7.72741 −0.500893
\(239\) −18.0058 −1.16470 −0.582350 0.812938i \(-0.697867\pi\)
−0.582350 + 0.812938i \(0.697867\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −3.92820 −0.252514
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 1.03528 0.0658730
\(248\) −2.92820 −0.185941
\(249\) 0 0
\(250\) 0 0
\(251\) −25.5302 −1.61145 −0.805725 0.592290i \(-0.798224\pi\)
−0.805725 + 0.592290i \(0.798224\pi\)
\(252\) 0 0
\(253\) −5.65685 −0.355643
\(254\) 8.48528 0.532414
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.7846 −1.17175 −0.585876 0.810401i \(-0.699249\pi\)
−0.585876 + 0.810401i \(0.699249\pi\)
\(258\) 0 0
\(259\) 6.92820 0.430498
\(260\) 0 0
\(261\) 0 0
\(262\) −11.5911 −0.716101
\(263\) −15.3205 −0.944703 −0.472351 0.881410i \(-0.656595\pi\)
−0.472351 + 0.881410i \(0.656595\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.03528 0.0634769
\(267\) 0 0
\(268\) −3.58630 −0.219068
\(269\) −12.1459 −0.740549 −0.370275 0.928922i \(-0.620736\pi\)
−0.370275 + 0.928922i \(0.620736\pi\)
\(270\) 0 0
\(271\) −2.92820 −0.177876 −0.0889378 0.996037i \(-0.528347\pi\)
−0.0889378 + 0.996037i \(0.528347\pi\)
\(272\) 7.46410 0.452578
\(273\) 0 0
\(274\) −7.46410 −0.450923
\(275\) 0 0
\(276\) 0 0
\(277\) −1.31268 −0.0788712 −0.0394356 0.999222i \(-0.512556\pi\)
−0.0394356 + 0.999222i \(0.512556\pi\)
\(278\) 6.92820 0.415526
\(279\) 0 0
\(280\) 0 0
\(281\) −22.7017 −1.35427 −0.677136 0.735858i \(-0.736779\pi\)
−0.677136 + 0.735858i \(0.736779\pi\)
\(282\) 0 0
\(283\) 19.3185 1.14837 0.574183 0.818727i \(-0.305320\pi\)
0.574183 + 0.818727i \(0.305320\pi\)
\(284\) 15.4548 0.917074
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) −6.92820 −0.408959
\(288\) 0 0
\(289\) 38.7128 2.27722
\(290\) 0 0
\(291\) 0 0
\(292\) 13.3843 0.783255
\(293\) 7.85641 0.458976 0.229488 0.973311i \(-0.426295\pi\)
0.229488 + 0.973311i \(0.426295\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.69213 −0.388972
\(297\) 0 0
\(298\) −9.04008 −0.523678
\(299\) 1.51575 0.0876581
\(300\) 0 0
\(301\) 1.85641 0.107001
\(302\) 21.8564 1.25769
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) 11.8685 0.677372 0.338686 0.940900i \(-0.390018\pi\)
0.338686 + 0.940900i \(0.390018\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) 0 0
\(311\) 6.69213 0.379476 0.189738 0.981835i \(-0.439236\pi\)
0.189738 + 0.981835i \(0.439236\pi\)
\(312\) 0 0
\(313\) −19.5959 −1.10763 −0.553813 0.832641i \(-0.686828\pi\)
−0.553813 + 0.832641i \(0.686828\pi\)
\(314\) 14.1421 0.798087
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 0 0
\(319\) −36.7846 −2.05954
\(320\) 0 0
\(321\) 0 0
\(322\) 1.51575 0.0844694
\(323\) −7.46410 −0.415314
\(324\) 0 0
\(325\) 0 0
\(326\) 18.7637 1.03923
\(327\) 0 0
\(328\) 6.69213 0.369511
\(329\) 9.79796 0.540179
\(330\) 0 0
\(331\) −13.8564 −0.761617 −0.380808 0.924654i \(-0.624354\pi\)
−0.380808 + 0.924654i \(0.624354\pi\)
\(332\) −4.39230 −0.241059
\(333\) 0 0
\(334\) 2.92820 0.160224
\(335\) 0 0
\(336\) 0 0
\(337\) 3.86370 0.210469 0.105235 0.994447i \(-0.466441\pi\)
0.105235 + 0.994447i \(0.466441\pi\)
\(338\) 11.9282 0.648809
\(339\) 0 0
\(340\) 0 0
\(341\) 11.3137 0.612672
\(342\) 0 0
\(343\) −13.3843 −0.722682
\(344\) −1.79315 −0.0966802
\(345\) 0 0
\(346\) −18.7846 −1.00987
\(347\) 25.4641 1.36698 0.683492 0.729958i \(-0.260460\pi\)
0.683492 + 0.729958i \(0.260460\pi\)
\(348\) 0 0
\(349\) 31.8564 1.70523 0.852617 0.522536i \(-0.175014\pi\)
0.852617 + 0.522536i \(0.175014\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.86370 −0.205936
\(353\) −13.3205 −0.708979 −0.354490 0.935060i \(-0.615345\pi\)
−0.354490 + 0.935060i \(0.615345\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.03528 −0.0548695
\(357\) 0 0
\(358\) 22.4243 1.18516
\(359\) 9.31749 0.491758 0.245879 0.969301i \(-0.420923\pi\)
0.245879 + 0.969301i \(0.420923\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 7.85641 0.412924
\(363\) 0 0
\(364\) −1.07180 −0.0561774
\(365\) 0 0
\(366\) 0 0
\(367\) 20.0764 1.04798 0.523990 0.851725i \(-0.324443\pi\)
0.523990 + 0.851725i \(0.324443\pi\)
\(368\) −1.46410 −0.0763216
\(369\) 0 0
\(370\) 0 0
\(371\) 6.21166 0.322493
\(372\) 0 0
\(373\) 0.480473 0.0248780 0.0124390 0.999923i \(-0.496040\pi\)
0.0124390 + 0.999923i \(0.496040\pi\)
\(374\) −28.8391 −1.49123
\(375\) 0 0
\(376\) −9.46410 −0.488074
\(377\) 9.85641 0.507631
\(378\) 0 0
\(379\) 19.7128 1.01258 0.506290 0.862364i \(-0.331017\pi\)
0.506290 + 0.862364i \(0.331017\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2.55103 −0.130522
\(383\) −9.07180 −0.463547 −0.231774 0.972770i \(-0.574453\pi\)
−0.231774 + 0.972770i \(0.574453\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.93426 −0.302046
\(387\) 0 0
\(388\) −17.2480 −0.875633
\(389\) 19.7990 1.00385 0.501924 0.864912i \(-0.332626\pi\)
0.501924 + 0.864912i \(0.332626\pi\)
\(390\) 0 0
\(391\) −10.9282 −0.552663
\(392\) 5.92820 0.299419
\(393\) 0 0
\(394\) −16.5359 −0.833067
\(395\) 0 0
\(396\) 0 0
\(397\) 1.31268 0.0658814 0.0329407 0.999457i \(-0.489513\pi\)
0.0329407 + 0.999457i \(0.489513\pi\)
\(398\) −26.9282 −1.34979
\(399\) 0 0
\(400\) 0 0
\(401\) −4.62158 −0.230791 −0.115395 0.993320i \(-0.536813\pi\)
−0.115395 + 0.993320i \(0.536813\pi\)
\(402\) 0 0
\(403\) −3.03150 −0.151010
\(404\) 0.757875 0.0377057
\(405\) 0 0
\(406\) 9.85641 0.489165
\(407\) 25.8564 1.28165
\(408\) 0 0
\(409\) −7.07180 −0.349678 −0.174839 0.984597i \(-0.555940\pi\)
−0.174839 + 0.984597i \(0.555940\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.89898 0.241355
\(413\) 13.0718 0.643221
\(414\) 0 0
\(415\) 0 0
\(416\) 1.03528 0.0507586
\(417\) 0 0
\(418\) 3.86370 0.188980
\(419\) −24.4206 −1.19302 −0.596511 0.802605i \(-0.703447\pi\)
−0.596511 + 0.802605i \(0.703447\pi\)
\(420\) 0 0
\(421\) −25.7128 −1.25317 −0.626583 0.779355i \(-0.715547\pi\)
−0.626583 + 0.779355i \(0.715547\pi\)
\(422\) −9.85641 −0.479802
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −2.07055 −0.100201
\(428\) 10.9282 0.528235
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0117 1.73462 0.867311 0.497767i \(-0.165847\pi\)
0.867311 + 0.497767i \(0.165847\pi\)
\(432\) 0 0
\(433\) −13.6617 −0.656538 −0.328269 0.944584i \(-0.606465\pi\)
−0.328269 + 0.944584i \(0.606465\pi\)
\(434\) −3.03150 −0.145517
\(435\) 0 0
\(436\) −4.92820 −0.236018
\(437\) 1.46410 0.0700375
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 7.72741 0.367555
\(443\) −4.39230 −0.208685 −0.104342 0.994541i \(-0.533274\pi\)
−0.104342 + 0.994541i \(0.533274\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −0.757875 −0.0358864
\(447\) 0 0
\(448\) 1.03528 0.0489122
\(449\) −3.66063 −0.172756 −0.0863779 0.996262i \(-0.527529\pi\)
−0.0863779 + 0.996262i \(0.527529\pi\)
\(450\) 0 0
\(451\) −25.8564 −1.21753
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −25.8564 −1.21350
\(455\) 0 0
\(456\) 0 0
\(457\) 33.9411 1.58770 0.793849 0.608114i \(-0.208074\pi\)
0.793849 + 0.608114i \(0.208074\pi\)
\(458\) −23.8564 −1.11474
\(459\) 0 0
\(460\) 0 0
\(461\) −17.7284 −0.825696 −0.412848 0.910800i \(-0.635466\pi\)
−0.412848 + 0.910800i \(0.635466\pi\)
\(462\) 0 0
\(463\) 34.0155 1.58083 0.790416 0.612570i \(-0.209864\pi\)
0.790416 + 0.612570i \(0.209864\pi\)
\(464\) −9.52056 −0.441981
\(465\) 0 0
\(466\) −19.4641 −0.901657
\(467\) 9.46410 0.437946 0.218973 0.975731i \(-0.429729\pi\)
0.218973 + 0.975731i \(0.429729\pi\)
\(468\) 0 0
\(469\) −3.71281 −0.171442
\(470\) 0 0
\(471\) 0 0
\(472\) −12.6264 −0.581177
\(473\) 6.92820 0.318559
\(474\) 0 0
\(475\) 0 0
\(476\) 7.72741 0.354185
\(477\) 0 0
\(478\) 18.0058 0.823568
\(479\) −31.3901 −1.43425 −0.717125 0.696944i \(-0.754542\pi\)
−0.717125 + 0.696944i \(0.754542\pi\)
\(480\) 0 0
\(481\) −6.92820 −0.315899
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 3.92820 0.178555
\(485\) 0 0
\(486\) 0 0
\(487\) −26.0106 −1.17865 −0.589327 0.807894i \(-0.700607\pi\)
−0.589327 + 0.807894i \(0.700607\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 0 0
\(491\) 15.7322 0.709985 0.354992 0.934869i \(-0.384483\pi\)
0.354992 + 0.934869i \(0.384483\pi\)
\(492\) 0 0
\(493\) −71.0624 −3.20049
\(494\) −1.03528 −0.0465793
\(495\) 0 0
\(496\) 2.92820 0.131480
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) −1.85641 −0.0831042 −0.0415521 0.999136i \(-0.513230\pi\)
−0.0415521 + 0.999136i \(0.513230\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 25.5302 1.13947
\(503\) 23.3205 1.03981 0.519905 0.854224i \(-0.325967\pi\)
0.519905 + 0.854224i \(0.325967\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 5.65685 0.251478
\(507\) 0 0
\(508\) −8.48528 −0.376473
\(509\) 22.3500 0.990647 0.495324 0.868709i \(-0.335050\pi\)
0.495324 + 0.868709i \(0.335050\pi\)
\(510\) 0 0
\(511\) 13.8564 0.612971
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 18.7846 0.828554
\(515\) 0 0
\(516\) 0 0
\(517\) 36.5665 1.60819
\(518\) −6.92820 −0.304408
\(519\) 0 0
\(520\) 0 0
\(521\) −4.62158 −0.202475 −0.101238 0.994862i \(-0.532280\pi\)
−0.101238 + 0.994862i \(0.532280\pi\)
\(522\) 0 0
\(523\) −7.17260 −0.313636 −0.156818 0.987628i \(-0.550124\pi\)
−0.156818 + 0.987628i \(0.550124\pi\)
\(524\) 11.5911 0.506360
\(525\) 0 0
\(526\) 15.3205 0.668006
\(527\) 21.8564 0.952080
\(528\) 0 0
\(529\) −20.8564 −0.906800
\(530\) 0 0
\(531\) 0 0
\(532\) −1.03528 −0.0448849
\(533\) 6.92820 0.300094
\(534\) 0 0
\(535\) 0 0
\(536\) 3.58630 0.154905
\(537\) 0 0
\(538\) 12.1459 0.523647
\(539\) −22.9048 −0.986580
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 2.92820 0.125777
\(543\) 0 0
\(544\) −7.46410 −0.320021
\(545\) 0 0
\(546\) 0 0
\(547\) −38.6370 −1.65200 −0.826000 0.563670i \(-0.809389\pi\)
−0.826000 + 0.563670i \(0.809389\pi\)
\(548\) 7.46410 0.318851
\(549\) 0 0
\(550\) 0 0
\(551\) 9.52056 0.405589
\(552\) 0 0
\(553\) −4.14110 −0.176098
\(554\) 1.31268 0.0557703
\(555\) 0 0
\(556\) −6.92820 −0.293821
\(557\) 24.5359 1.03962 0.519810 0.854282i \(-0.326003\pi\)
0.519810 + 0.854282i \(0.326003\pi\)
\(558\) 0 0
\(559\) −1.85641 −0.0785176
\(560\) 0 0
\(561\) 0 0
\(562\) 22.7017 0.957615
\(563\) −1.85641 −0.0782382 −0.0391191 0.999235i \(-0.512455\pi\)
−0.0391191 + 0.999235i \(0.512455\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −19.3185 −0.812018
\(567\) 0 0
\(568\) −15.4548 −0.648470
\(569\) −9.72363 −0.407636 −0.203818 0.979009i \(-0.565335\pi\)
−0.203818 + 0.979009i \(0.565335\pi\)
\(570\) 0 0
\(571\) 1.07180 0.0448533 0.0224266 0.999748i \(-0.492861\pi\)
0.0224266 + 0.999748i \(0.492861\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) 6.92820 0.289178
\(575\) 0 0
\(576\) 0 0
\(577\) −5.65685 −0.235498 −0.117749 0.993043i \(-0.537568\pi\)
−0.117749 + 0.993043i \(0.537568\pi\)
\(578\) −38.7128 −1.61024
\(579\) 0 0
\(580\) 0 0
\(581\) −4.54725 −0.188652
\(582\) 0 0
\(583\) 23.1822 0.960109
\(584\) −13.3843 −0.553845
\(585\) 0 0
\(586\) −7.85641 −0.324545
\(587\) 3.60770 0.148906 0.0744528 0.997225i \(-0.476279\pi\)
0.0744528 + 0.997225i \(0.476279\pi\)
\(588\) 0 0
\(589\) −2.92820 −0.120655
\(590\) 0 0
\(591\) 0 0
\(592\) 6.69213 0.275045
\(593\) −11.4641 −0.470774 −0.235387 0.971902i \(-0.575636\pi\)
−0.235387 + 0.971902i \(0.575636\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9.04008 0.370296
\(597\) 0 0
\(598\) −1.51575 −0.0619836
\(599\) 30.3548 1.24026 0.620132 0.784497i \(-0.287079\pi\)
0.620132 + 0.784497i \(0.287079\pi\)
\(600\) 0 0
\(601\) 22.7846 0.929404 0.464702 0.885467i \(-0.346162\pi\)
0.464702 + 0.885467i \(0.346162\pi\)
\(602\) −1.85641 −0.0756615
\(603\) 0 0
\(604\) −21.8564 −0.889325
\(605\) 0 0
\(606\) 0 0
\(607\) −22.9791 −0.932695 −0.466347 0.884602i \(-0.654430\pi\)
−0.466347 + 0.884602i \(0.654430\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −9.79796 −0.396383
\(612\) 0 0
\(613\) 45.0518 1.81962 0.909812 0.415021i \(-0.136226\pi\)
0.909812 + 0.415021i \(0.136226\pi\)
\(614\) −11.8685 −0.478974
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) −20.2487 −0.815182 −0.407591 0.913165i \(-0.633631\pi\)
−0.407591 + 0.913165i \(0.633631\pi\)
\(618\) 0 0
\(619\) 34.6410 1.39234 0.696170 0.717877i \(-0.254886\pi\)
0.696170 + 0.717877i \(0.254886\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.69213 −0.268330
\(623\) −1.07180 −0.0429406
\(624\) 0 0
\(625\) 0 0
\(626\) 19.5959 0.783210
\(627\) 0 0
\(628\) −14.1421 −0.564333
\(629\) 49.9507 1.99167
\(630\) 0 0
\(631\) −5.85641 −0.233140 −0.116570 0.993182i \(-0.537190\pi\)
−0.116570 + 0.993182i \(0.537190\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) −14.0000 −0.556011
\(635\) 0 0
\(636\) 0 0
\(637\) 6.13733 0.243170
\(638\) 36.7846 1.45632
\(639\) 0 0
\(640\) 0 0
\(641\) −2.55103 −0.100759 −0.0503797 0.998730i \(-0.516043\pi\)
−0.0503797 + 0.998730i \(0.516043\pi\)
\(642\) 0 0
\(643\) 22.3500 0.881399 0.440699 0.897655i \(-0.354731\pi\)
0.440699 + 0.897655i \(0.354731\pi\)
\(644\) −1.51575 −0.0597289
\(645\) 0 0
\(646\) 7.46410 0.293671
\(647\) 6.53590 0.256953 0.128476 0.991713i \(-0.458991\pi\)
0.128476 + 0.991713i \(0.458991\pi\)
\(648\) 0 0
\(649\) 48.7846 1.91496
\(650\) 0 0
\(651\) 0 0
\(652\) −18.7637 −0.734844
\(653\) 24.2487 0.948925 0.474463 0.880276i \(-0.342642\pi\)
0.474463 + 0.880276i \(0.342642\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −6.69213 −0.261284
\(657\) 0 0
\(658\) −9.79796 −0.381964
\(659\) 9.04008 0.352152 0.176076 0.984377i \(-0.443660\pi\)
0.176076 + 0.984377i \(0.443660\pi\)
\(660\) 0 0
\(661\) 30.7846 1.19738 0.598691 0.800980i \(-0.295688\pi\)
0.598691 + 0.800980i \(0.295688\pi\)
\(662\) 13.8564 0.538545
\(663\) 0 0
\(664\) 4.39230 0.170454
\(665\) 0 0
\(666\) 0 0
\(667\) 13.9391 0.539723
\(668\) −2.92820 −0.113296
\(669\) 0 0
\(670\) 0 0
\(671\) −7.72741 −0.298313
\(672\) 0 0
\(673\) 32.1480 1.23921 0.619607 0.784912i \(-0.287292\pi\)
0.619607 + 0.784912i \(0.287292\pi\)
\(674\) −3.86370 −0.148824
\(675\) 0 0
\(676\) −11.9282 −0.458777
\(677\) −32.6410 −1.25450 −0.627248 0.778820i \(-0.715819\pi\)
−0.627248 + 0.778820i \(0.715819\pi\)
\(678\) 0 0
\(679\) −17.8564 −0.685266
\(680\) 0 0
\(681\) 0 0
\(682\) −11.3137 −0.433224
\(683\) 5.07180 0.194067 0.0970335 0.995281i \(-0.469065\pi\)
0.0970335 + 0.995281i \(0.469065\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13.3843 0.511013
\(687\) 0 0
\(688\) 1.79315 0.0683632
\(689\) −6.21166 −0.236645
\(690\) 0 0
\(691\) 25.0718 0.953776 0.476888 0.878964i \(-0.341765\pi\)
0.476888 + 0.878964i \(0.341765\pi\)
\(692\) 18.7846 0.714084
\(693\) 0 0
\(694\) −25.4641 −0.966604
\(695\) 0 0
\(696\) 0 0
\(697\) −49.9507 −1.89202
\(698\) −31.8564 −1.20578
\(699\) 0 0
\(700\) 0 0
\(701\) −41.4655 −1.56613 −0.783064 0.621941i \(-0.786345\pi\)
−0.783064 + 0.621941i \(0.786345\pi\)
\(702\) 0 0
\(703\) −6.69213 −0.252398
\(704\) 3.86370 0.145619
\(705\) 0 0
\(706\) 13.3205 0.501324
\(707\) 0.784610 0.0295083
\(708\) 0 0
\(709\) 15.8564 0.595500 0.297750 0.954644i \(-0.403764\pi\)
0.297750 + 0.954644i \(0.403764\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.03528 0.0387986
\(713\) −4.28719 −0.160556
\(714\) 0 0
\(715\) 0 0
\(716\) −22.4243 −0.838037
\(717\) 0 0
\(718\) −9.31749 −0.347725
\(719\) 26.8429 1.00107 0.500535 0.865716i \(-0.333137\pi\)
0.500535 + 0.865716i \(0.333137\pi\)
\(720\) 0 0
\(721\) 5.07180 0.188884
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) −7.85641 −0.291981
\(725\) 0 0
\(726\) 0 0
\(727\) 10.2784 0.381206 0.190603 0.981667i \(-0.438956\pi\)
0.190603 + 0.981667i \(0.438956\pi\)
\(728\) 1.07180 0.0397234
\(729\) 0 0
\(730\) 0 0
\(731\) 13.3843 0.495035
\(732\) 0 0
\(733\) 12.0716 0.445874 0.222937 0.974833i \(-0.428436\pi\)
0.222937 + 0.974833i \(0.428436\pi\)
\(734\) −20.0764 −0.741033
\(735\) 0 0
\(736\) 1.46410 0.0539675
\(737\) −13.8564 −0.510407
\(738\) 0 0
\(739\) −31.7128 −1.16657 −0.583287 0.812266i \(-0.698234\pi\)
−0.583287 + 0.812266i \(0.698234\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −6.21166 −0.228037
\(743\) 34.6410 1.27086 0.635428 0.772160i \(-0.280824\pi\)
0.635428 + 0.772160i \(0.280824\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −0.480473 −0.0175914
\(747\) 0 0
\(748\) 28.8391 1.05446
\(749\) 11.3137 0.413394
\(750\) 0 0
\(751\) −21.8564 −0.797552 −0.398776 0.917048i \(-0.630565\pi\)
−0.398776 + 0.917048i \(0.630565\pi\)
\(752\) 9.46410 0.345120
\(753\) 0 0
\(754\) −9.85641 −0.358949
\(755\) 0 0
\(756\) 0 0
\(757\) −50.1538 −1.82287 −0.911436 0.411443i \(-0.865025\pi\)
−0.911436 + 0.411443i \(0.865025\pi\)
\(758\) −19.7128 −0.716002
\(759\) 0 0
\(760\) 0 0
\(761\) 6.21166 0.225172 0.112586 0.993642i \(-0.464087\pi\)
0.112586 + 0.993642i \(0.464087\pi\)
\(762\) 0 0
\(763\) −5.10205 −0.184707
\(764\) 2.55103 0.0922929
\(765\) 0 0
\(766\) 9.07180 0.327777
\(767\) −13.0718 −0.471995
\(768\) 0 0
\(769\) 8.14359 0.293665 0.146833 0.989161i \(-0.453092\pi\)
0.146833 + 0.989161i \(0.453092\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.93426 0.213579
\(773\) 19.8564 0.714185 0.357093 0.934069i \(-0.383768\pi\)
0.357093 + 0.934069i \(0.383768\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 17.2480 0.619166
\(777\) 0 0
\(778\) −19.7990 −0.709828
\(779\) 6.69213 0.239770
\(780\) 0 0
\(781\) 59.7128 2.13669
\(782\) 10.9282 0.390792
\(783\) 0 0
\(784\) −5.92820 −0.211722
\(785\) 0 0
\(786\) 0 0
\(787\) −26.7685 −0.954195 −0.477097 0.878850i \(-0.658311\pi\)
−0.477097 + 0.878850i \(0.658311\pi\)
\(788\) 16.5359 0.589067
\(789\) 0 0
\(790\) 0 0
\(791\) 6.21166 0.220861
\(792\) 0 0
\(793\) 2.07055 0.0735275
\(794\) −1.31268 −0.0465852
\(795\) 0 0
\(796\) 26.9282 0.954445
\(797\) −28.6410 −1.01452 −0.507258 0.861794i \(-0.669341\pi\)
−0.507258 + 0.861794i \(0.669341\pi\)
\(798\) 0 0
\(799\) 70.6410 2.49910
\(800\) 0 0
\(801\) 0 0
\(802\) 4.62158 0.163194
\(803\) 51.7128 1.82491
\(804\) 0 0
\(805\) 0 0
\(806\) 3.03150 0.106780
\(807\) 0 0
\(808\) −0.757875 −0.0266619
\(809\) 33.5350 1.17903 0.589514 0.807758i \(-0.299319\pi\)
0.589514 + 0.807758i \(0.299319\pi\)
\(810\) 0 0
\(811\) 56.4974 1.98389 0.991946 0.126658i \(-0.0404251\pi\)
0.991946 + 0.126658i \(0.0404251\pi\)
\(812\) −9.85641 −0.345892
\(813\) 0 0
\(814\) −25.8564 −0.906267
\(815\) 0 0
\(816\) 0 0
\(817\) −1.79315 −0.0627344
\(818\) 7.07180 0.247260
\(819\) 0 0
\(820\) 0 0
\(821\) 37.3244 1.30263 0.651314 0.758808i \(-0.274218\pi\)
0.651314 + 0.758808i \(0.274218\pi\)
\(822\) 0 0
\(823\) −55.6819 −1.94095 −0.970475 0.241202i \(-0.922458\pi\)
−0.970475 + 0.241202i \(0.922458\pi\)
\(824\) −4.89898 −0.170664
\(825\) 0 0
\(826\) −13.0718 −0.454826
\(827\) −6.92820 −0.240917 −0.120459 0.992718i \(-0.538437\pi\)
−0.120459 + 0.992718i \(0.538437\pi\)
\(828\) 0 0
\(829\) −26.7846 −0.930268 −0.465134 0.885240i \(-0.653994\pi\)
−0.465134 + 0.885240i \(0.653994\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.03528 −0.0358917
\(833\) −44.2487 −1.53313
\(834\) 0 0
\(835\) 0 0
\(836\) −3.86370 −0.133629
\(837\) 0 0
\(838\) 24.4206 0.843595
\(839\) 28.2843 0.976481 0.488241 0.872709i \(-0.337639\pi\)
0.488241 + 0.872709i \(0.337639\pi\)
\(840\) 0 0
\(841\) 61.6410 2.12555
\(842\) 25.7128 0.886122
\(843\) 0 0
\(844\) 9.85641 0.339272
\(845\) 0 0
\(846\) 0 0
\(847\) 4.06678 0.139736
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 0 0
\(851\) −9.79796 −0.335870
\(852\) 0 0
\(853\) −14.6969 −0.503214 −0.251607 0.967830i \(-0.580959\pi\)
−0.251607 + 0.967830i \(0.580959\pi\)
\(854\) 2.07055 0.0708528
\(855\) 0 0
\(856\) −10.9282 −0.373518
\(857\) −52.6410 −1.79818 −0.899091 0.437761i \(-0.855772\pi\)
−0.899091 + 0.437761i \(0.855772\pi\)
\(858\) 0 0
\(859\) 36.7846 1.25507 0.627537 0.778586i \(-0.284063\pi\)
0.627537 + 0.778586i \(0.284063\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −36.0117 −1.22656
\(863\) −20.7846 −0.707516 −0.353758 0.935337i \(-0.615096\pi\)
−0.353758 + 0.935337i \(0.615096\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 13.6617 0.464242
\(867\) 0 0
\(868\) 3.03150 0.102896
\(869\) −15.4548 −0.524269
\(870\) 0 0
\(871\) 3.71281 0.125804
\(872\) 4.92820 0.166890
\(873\) 0 0
\(874\) −1.46410 −0.0495240
\(875\) 0 0
\(876\) 0 0
\(877\) 43.8134 1.47947 0.739737 0.672896i \(-0.234950\pi\)
0.739737 + 0.672896i \(0.234950\pi\)
\(878\) −20.0000 −0.674967
\(879\) 0 0
\(880\) 0 0
\(881\) −10.3528 −0.348793 −0.174397 0.984675i \(-0.555798\pi\)
−0.174397 + 0.984675i \(0.555798\pi\)
\(882\) 0 0
\(883\) −24.0144 −0.808150 −0.404075 0.914726i \(-0.632406\pi\)
−0.404075 + 0.914726i \(0.632406\pi\)
\(884\) −7.72741 −0.259901
\(885\) 0 0
\(886\) 4.39230 0.147562
\(887\) 39.7128 1.33343 0.666713 0.745315i \(-0.267701\pi\)
0.666713 + 0.745315i \(0.267701\pi\)
\(888\) 0 0
\(889\) −8.78461 −0.294626
\(890\) 0 0
\(891\) 0 0
\(892\) 0.757875 0.0253755
\(893\) −9.46410 −0.316704
\(894\) 0 0
\(895\) 0 0
\(896\) −1.03528 −0.0345861
\(897\) 0 0
\(898\) 3.66063 0.122157
\(899\) −27.8781 −0.929788
\(900\) 0 0
\(901\) 44.7846 1.49199
\(902\) 25.8564 0.860924
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) 2.62536 0.0871735 0.0435867 0.999050i \(-0.486122\pi\)
0.0435867 + 0.999050i \(0.486122\pi\)
\(908\) 25.8564 0.858075
\(909\) 0 0
\(910\) 0 0
\(911\) −45.2548 −1.49936 −0.749680 0.661801i \(-0.769792\pi\)
−0.749680 + 0.661801i \(0.769792\pi\)
\(912\) 0 0
\(913\) −16.9706 −0.561644
\(914\) −33.9411 −1.12267
\(915\) 0 0
\(916\) 23.8564 0.788238
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −27.7128 −0.914161 −0.457081 0.889425i \(-0.651105\pi\)
−0.457081 + 0.889425i \(0.651105\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 17.7284 0.583855
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 0 0
\(926\) −34.0155 −1.11782
\(927\) 0 0
\(928\) 9.52056 0.312528
\(929\) −11.3137 −0.371191 −0.185595 0.982626i \(-0.559421\pi\)
−0.185595 + 0.982626i \(0.559421\pi\)
\(930\) 0 0
\(931\) 5.92820 0.194289
\(932\) 19.4641 0.637568
\(933\) 0 0
\(934\) −9.46410 −0.309675
\(935\) 0 0
\(936\) 0 0
\(937\) −30.3548 −0.991649 −0.495824 0.868423i \(-0.665134\pi\)
−0.495824 + 0.868423i \(0.665134\pi\)
\(938\) 3.71281 0.121228
\(939\) 0 0
\(940\) 0 0
\(941\) −20.8343 −0.679178 −0.339589 0.940574i \(-0.610288\pi\)
−0.339589 + 0.940574i \(0.610288\pi\)
\(942\) 0 0
\(943\) 9.79796 0.319065
\(944\) 12.6264 0.410954
\(945\) 0 0
\(946\) −6.92820 −0.225255
\(947\) 37.1769 1.20809 0.604044 0.796951i \(-0.293555\pi\)
0.604044 + 0.796951i \(0.293555\pi\)
\(948\) 0 0
\(949\) −13.8564 −0.449798
\(950\) 0 0
\(951\) 0 0
\(952\) −7.72741 −0.250447
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −18.0058 −0.582350
\(957\) 0 0
\(958\) 31.3901 1.01417
\(959\) 7.72741 0.249531
\(960\) 0 0
\(961\) −22.4256 −0.723407
\(962\) 6.92820 0.223374
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −2.55103 −0.0820355 −0.0410177 0.999158i \(-0.513060\pi\)
−0.0410177 + 0.999158i \(0.513060\pi\)
\(968\) −3.92820 −0.126257
\(969\) 0 0
\(970\) 0 0
\(971\) −24.9010 −0.799112 −0.399556 0.916709i \(-0.630836\pi\)
−0.399556 + 0.916709i \(0.630836\pi\)
\(972\) 0 0
\(973\) −7.17260 −0.229943
\(974\) 26.0106 0.833435
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −49.7128 −1.59045 −0.795227 0.606312i \(-0.792648\pi\)
−0.795227 + 0.606312i \(0.792648\pi\)
\(978\) 0 0
\(979\) −4.00000 −0.127841
\(980\) 0 0
\(981\) 0 0
\(982\) −15.7322 −0.502035
\(983\) 48.7846 1.55599 0.777994 0.628272i \(-0.216238\pi\)
0.777994 + 0.628272i \(0.216238\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 71.0624 2.26309
\(987\) 0 0
\(988\) 1.03528 0.0329365
\(989\) −2.62536 −0.0834814
\(990\) 0 0
\(991\) 20.7846 0.660245 0.330122 0.943938i \(-0.392910\pi\)
0.330122 + 0.943938i \(0.392910\pi\)
\(992\) −2.92820 −0.0929705
\(993\) 0 0
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) 1.31268 0.0415729 0.0207865 0.999784i \(-0.493383\pi\)
0.0207865 + 0.999784i \(0.493383\pi\)
\(998\) 1.85641 0.0587635
\(999\) 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 8550.2.a.cu.1.3 4
3.2 odd 2 8550.2.a.cv.1.3 4
5.2 odd 4 1710.2.d.g.1369.3 yes 8
5.3 odd 4 1710.2.d.g.1369.8 yes 8
5.4 even 2 8550.2.a.cv.1.2 4
15.2 even 4 1710.2.d.g.1369.6 yes 8
15.8 even 4 1710.2.d.g.1369.1 8
15.14 odd 2 inner 8550.2.a.cu.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1710.2.d.g.1369.1 8 15.8 even 4
1710.2.d.g.1369.3 yes 8 5.2 odd 4
1710.2.d.g.1369.6 yes 8 15.2 even 4
1710.2.d.g.1369.8 yes 8 5.3 odd 4
8550.2.a.cu.1.2 4 15.14 odd 2 inner
8550.2.a.cu.1.3 4 1.1 even 1 trivial
8550.2.a.cv.1.2 4 5.4 even 2
8550.2.a.cv.1.3 4 3.2 odd 2