Properties

Label 109.2.a.a.1.1
Level $109$
Weight $2$
Character 109.1
Self dual yes
Analytic conductor $0.870$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [109,2,Mod(1,109)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 109.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: \(0.870369382032\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 109.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} -3.00000 q^{9} +3.00000 q^{10} +1.00000 q^{11} +2.00000 q^{14} -1.00000 q^{16} -8.00000 q^{17} -3.00000 q^{18} -5.00000 q^{19} -3.00000 q^{20} +1.00000 q^{22} +7.00000 q^{23} +4.00000 q^{25} -2.00000 q^{28} -5.00000 q^{29} +6.00000 q^{31} +5.00000 q^{32} -8.00000 q^{34} +6.00000 q^{35} +3.00000 q^{36} +2.00000 q^{37} -5.00000 q^{38} -9.00000 q^{40} +2.00000 q^{41} -4.00000 q^{43} -1.00000 q^{44} -9.00000 q^{45} +7.00000 q^{46} +9.00000 q^{47} -3.00000 q^{49} +4.00000 q^{50} +12.0000 q^{53} +3.00000 q^{55} -6.00000 q^{56} -5.00000 q^{58} +12.0000 q^{59} -5.00000 q^{61} +6.00000 q^{62} -6.00000 q^{63} +7.00000 q^{64} -12.0000 q^{67} +8.00000 q^{68} +6.00000 q^{70} -6.00000 q^{71} +9.00000 q^{72} -5.00000 q^{73} +2.00000 q^{74} +5.00000 q^{76} +2.00000 q^{77} +8.00000 q^{79} -3.00000 q^{80} +9.00000 q^{81} +2.00000 q^{82} -2.00000 q^{83} -24.0000 q^{85} -4.00000 q^{86} -3.00000 q^{88} +1.00000 q^{89} -9.00000 q^{90} -7.00000 q^{92} +9.00000 q^{94} -15.0000 q^{95} +1.00000 q^{97} -3.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1.00000 −0.500000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −3.00000 −1.06066
\(9\) −3.00000 −1.00000
\(10\) 3.00000 0.948683
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −8.00000 −1.94029 −0.970143 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) −3.00000 −0.707107
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 7.00000 1.45960 0.729800 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) −8.00000 −1.37199
\(35\) 6.00000 1.01419
\(36\) 3.00000 0.500000
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −5.00000 −0.811107
\(39\) 0 0
\(40\) −9.00000 −1.42302
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) −9.00000 −1.34164
\(46\) 7.00000 1.03209
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) −5.00000 −0.656532
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 6.00000 0.762001
\(63\) −6.00000 −0.755929
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 8.00000 0.970143
\(69\) 0 0
\(70\) 6.00000 0.717137
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 9.00000 1.06066
\(73\) −5.00000 −0.585206 −0.292603 0.956234i \(-0.594521\pi\)
−0.292603 + 0.956234i \(0.594521\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −3.00000 −0.335410
\(81\) 9.00000 1.00000
\(82\) 2.00000 0.220863
\(83\) −2.00000 −0.219529 −0.109764 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) −24.0000 −2.60317
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 1.00000 0.106000 0.0529999 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) −9.00000 −0.948683
\(91\) 0 0
\(92\) −7.00000 −0.729800
\(93\) 0 0
\(94\) 9.00000 0.928279
\(95\) −15.0000 −1.53897
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) −3.00000 −0.303046
\(99\) −3.00000 −0.301511
\(100\) −4.00000 −0.400000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −11.0000 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) 1.00000 0.0957826
\(110\) 3.00000 0.286039
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 21.0000 1.95826
\(116\) 5.00000 0.464238
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) −16.0000 −1.46672
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −5.00000 −0.452679
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) −3.00000 −0.268328
\(126\) −6.00000 −0.534522
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −10.0000 −0.867110
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 24.0000 2.05798
\(137\) 1.00000 0.0854358 0.0427179 0.999087i \(-0.486398\pi\)
0.0427179 + 0.999087i \(0.486398\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 3.00000 0.250000
\(145\) −15.0000 −1.24568
\(146\) −5.00000 −0.413803
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 1.00000 0.0813788 0.0406894 0.999172i \(-0.487045\pi\)
0.0406894 + 0.999172i \(0.487045\pi\)
\(152\) 15.0000 1.21666
\(153\) 24.0000 1.94029
\(154\) 2.00000 0.161165
\(155\) 18.0000 1.44579
\(156\) 0 0
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 15.0000 1.18585
\(161\) 14.0000 1.10335
\(162\) 9.00000 0.707107
\(163\) −13.0000 −1.01824 −0.509119 0.860696i \(-0.670029\pi\)
−0.509119 + 0.860696i \(0.670029\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −24.0000 −1.84072
\(171\) 15.0000 1.14708
\(172\) 4.00000 0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 8.00000 0.604743
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 1.00000 0.0749532
\(179\) 11.0000 0.822179 0.411089 0.911595i \(-0.365148\pi\)
0.411089 + 0.911595i \(0.365148\pi\)
\(180\) 9.00000 0.670820
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −21.0000 −1.54814
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) −9.00000 −0.656392
\(189\) 0 0
\(190\) −15.0000 −1.08821
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −13.0000 −0.935760 −0.467880 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) 1.00000 0.0717958
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −17.0000 −1.21120 −0.605600 0.795769i \(-0.707067\pi\)
−0.605600 + 0.795769i \(0.707067\pi\)
\(198\) −3.00000 −0.213201
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −12.0000 −0.848528
\(201\) 0 0
\(202\) 12.0000 0.844317
\(203\) −10.0000 −0.701862
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) −11.0000 −0.766406
\(207\) −21.0000 −1.45960
\(208\) 0 0
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) 18.0000 1.23917 0.619586 0.784929i \(-0.287301\pi\)
0.619586 + 0.784929i \(0.287301\pi\)
\(212\) −12.0000 −0.824163
\(213\) 0 0
\(214\) 3.00000 0.205076
\(215\) −12.0000 −0.818393
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 1.00000 0.0677285
\(219\) 0 0
\(220\) −3.00000 −0.202260
\(221\) 0 0
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 10.0000 0.668153
\(225\) −12.0000 −0.800000
\(226\) −18.0000 −1.19734
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 21.0000 1.38470
\(231\) 0 0
\(232\) 15.0000 0.984798
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 0 0
\(235\) 27.0000 1.76129
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) −16.0000 −1.03713
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) 5.00000 0.320092
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) 0 0
\(248\) −18.0000 −1.14300
\(249\) 0 0
\(250\) −3.00000 −0.189737
\(251\) 23.0000 1.45175 0.725874 0.687828i \(-0.241436\pi\)
0.725874 + 0.687828i \(0.241436\pi\)
\(252\) 6.00000 0.377964
\(253\) 7.00000 0.440086
\(254\) −7.00000 −0.439219
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 15.0000 0.928477
\(262\) 0 0
\(263\) −26.0000 −1.60323 −0.801614 0.597841i \(-0.796025\pi\)
−0.801614 + 0.597841i \(0.796025\pi\)
\(264\) 0 0
\(265\) 36.0000 2.21146
\(266\) −10.0000 −0.613139
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 0 0
\(271\) −25.0000 −1.51864 −0.759321 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) 8.00000 0.485071
\(273\) 0 0
\(274\) 1.00000 0.0604122
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 16.0000 0.959616
\(279\) −18.0000 −1.07763
\(280\) −18.0000 −1.07571
\(281\) 27.0000 1.61068 0.805342 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) 0 0
\(283\) −23.0000 −1.36721 −0.683604 0.729853i \(-0.739588\pi\)
−0.683604 + 0.729853i \(0.739588\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) −15.0000 −0.883883
\(289\) 47.0000 2.76471
\(290\) −15.0000 −0.880830
\(291\) 0 0
\(292\) 5.00000 0.292603
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 36.0000 2.09600
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 1.00000 0.0575435
\(303\) 0 0
\(304\) 5.00000 0.286770
\(305\) −15.0000 −0.858898
\(306\) 24.0000 1.37199
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) 18.0000 1.02233
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) 28.0000 1.58265 0.791327 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) 17.0000 0.959366
\(315\) −18.0000 −1.01419
\(316\) −8.00000 −0.450035
\(317\) −24.0000 −1.34797 −0.673987 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) −5.00000 −0.279946
\(320\) 21.0000 1.17394
\(321\) 0 0
\(322\) 14.0000 0.780189
\(323\) 40.0000 2.22566
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) −13.0000 −0.720003
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 18.0000 0.992372
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 2.00000 0.109764
\(333\) −6.00000 −0.328798
\(334\) 8.00000 0.437741
\(335\) −36.0000 −1.96689
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) −13.0000 −0.707107
\(339\) 0 0
\(340\) 24.0000 1.30158
\(341\) 6.00000 0.324918
\(342\) 15.0000 0.811107
\(343\) −20.0000 −1.07990
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) 7.00000 0.374701 0.187351 0.982293i \(-0.440010\pi\)
0.187351 + 0.982293i \(0.440010\pi\)
\(350\) 8.00000 0.427618
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) 19.0000 1.01127 0.505634 0.862748i \(-0.331259\pi\)
0.505634 + 0.862748i \(0.331259\pi\)
\(354\) 0 0
\(355\) −18.0000 −0.955341
\(356\) −1.00000 −0.0529999
\(357\) 0 0
\(358\) 11.0000 0.581368
\(359\) −11.0000 −0.580558 −0.290279 0.956942i \(-0.593748\pi\)
−0.290279 + 0.956942i \(0.593748\pi\)
\(360\) 27.0000 1.42302
\(361\) 6.00000 0.315789
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 0 0
\(365\) −15.0000 −0.785136
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −7.00000 −0.364900
\(369\) −6.00000 −0.312348
\(370\) 6.00000 0.311925
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −15.0000 −0.776671 −0.388335 0.921518i \(-0.626950\pi\)
−0.388335 + 0.921518i \(0.626950\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −27.0000 −1.39242
\(377\) 0 0
\(378\) 0 0
\(379\) −31.0000 −1.59236 −0.796182 0.605058i \(-0.793150\pi\)
−0.796182 + 0.605058i \(0.793150\pi\)
\(380\) 15.0000 0.769484
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) −13.0000 −0.664269 −0.332134 0.943232i \(-0.607769\pi\)
−0.332134 + 0.943232i \(0.607769\pi\)
\(384\) 0 0
\(385\) 6.00000 0.305788
\(386\) −13.0000 −0.661683
\(387\) 12.0000 0.609994
\(388\) −1.00000 −0.0507673
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 0 0
\(391\) −56.0000 −2.83204
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) −17.0000 −0.856448
\(395\) 24.0000 1.20757
\(396\) 3.00000 0.150756
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −19.0000 −0.948815 −0.474407 0.880305i \(-0.657338\pi\)
−0.474407 + 0.880305i \(0.657338\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) 27.0000 1.34164
\(406\) −10.0000 −0.496292
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) 17.0000 0.840596 0.420298 0.907386i \(-0.361926\pi\)
0.420298 + 0.907386i \(0.361926\pi\)
\(410\) 6.00000 0.296319
\(411\) 0 0
\(412\) 11.0000 0.541931
\(413\) 24.0000 1.18096
\(414\) −21.0000 −1.03209
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 0 0
\(418\) −5.00000 −0.244558
\(419\) 40.0000 1.95413 0.977064 0.212946i \(-0.0683059\pi\)
0.977064 + 0.212946i \(0.0683059\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) 18.0000 0.876226
\(423\) −27.0000 −1.31278
\(424\) −36.0000 −1.74831
\(425\) −32.0000 −1.55223
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) −3.00000 −0.145010
\(429\) 0 0
\(430\) −12.0000 −0.578691
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) −1.00000 −0.0478913
\(437\) −35.0000 −1.67428
\(438\) 0 0
\(439\) −6.00000 −0.286364 −0.143182 0.989696i \(-0.545733\pi\)
−0.143182 + 0.989696i \(0.545733\pi\)
\(440\) −9.00000 −0.429058
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −10.0000 −0.475114 −0.237557 0.971374i \(-0.576347\pi\)
−0.237557 + 0.971374i \(0.576347\pi\)
\(444\) 0 0
\(445\) 3.00000 0.142214
\(446\) −2.00000 −0.0947027
\(447\) 0 0
\(448\) 14.0000 0.661438
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) −12.0000 −0.565685
\(451\) 2.00000 0.0941763
\(452\) 18.0000 0.846649
\(453\) 0 0
\(454\) −14.0000 −0.657053
\(455\) 0 0
\(456\) 0 0
\(457\) −27.0000 −1.26301 −0.631503 0.775373i \(-0.717562\pi\)
−0.631503 + 0.775373i \(0.717562\pi\)
\(458\) −4.00000 −0.186908
\(459\) 0 0
\(460\) −21.0000 −0.979130
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) 9.00000 0.416917
\(467\) 18.0000 0.832941 0.416470 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 0 0
\(469\) −24.0000 −1.10822
\(470\) 27.0000 1.24542
\(471\) 0 0
\(472\) −36.0000 −1.65703
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) −20.0000 −0.917663
\(476\) 16.0000 0.733359
\(477\) −36.0000 −1.64833
\(478\) −6.00000 −0.274434
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 10.0000 0.454545
\(485\) 3.00000 0.136223
\(486\) 0 0
\(487\) 19.0000 0.860972 0.430486 0.902597i \(-0.358342\pi\)
0.430486 + 0.902597i \(0.358342\pi\)
\(488\) 15.0000 0.679018
\(489\) 0 0
\(490\) −9.00000 −0.406579
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 40.0000 1.80151
\(494\) 0 0
\(495\) −9.00000 −0.404520
\(496\) −6.00000 −0.269408
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 3.00000 0.134164
\(501\) 0 0
\(502\) 23.0000 1.02654
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 18.0000 0.801784
\(505\) 36.0000 1.60198
\(506\) 7.00000 0.311188
\(507\) 0 0
\(508\) 7.00000 0.310575
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) −24.0000 −1.05859
\(515\) −33.0000 −1.45415
\(516\) 0 0
\(517\) 9.00000 0.395820
\(518\) 4.00000 0.175750
\(519\) 0 0
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 15.0000 0.656532
\(523\) 42.0000 1.83653 0.918266 0.395964i \(-0.129590\pi\)
0.918266 + 0.395964i \(0.129590\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −26.0000 −1.13365
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 36.0000 1.56374
\(531\) −36.0000 −1.56227
\(532\) 10.0000 0.433555
\(533\) 0 0
\(534\) 0 0
\(535\) 9.00000 0.389104
\(536\) 36.0000 1.55496
\(537\) 0 0
\(538\) 4.00000 0.172452
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 19.0000 0.816874 0.408437 0.912787i \(-0.366074\pi\)
0.408437 + 0.912787i \(0.366074\pi\)
\(542\) −25.0000 −1.07384
\(543\) 0 0
\(544\) −40.0000 −1.71499
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) −23.0000 −0.983409 −0.491704 0.870762i \(-0.663626\pi\)
−0.491704 + 0.870762i \(0.663626\pi\)
\(548\) −1.00000 −0.0427179
\(549\) 15.0000 0.640184
\(550\) 4.00000 0.170561
\(551\) 25.0000 1.06504
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 12.0000 0.509831
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) −18.0000 −0.762001
\(559\) 0 0
\(560\) −6.00000 −0.253546
\(561\) 0 0
\(562\) 27.0000 1.13893
\(563\) 25.0000 1.05362 0.526812 0.849982i \(-0.323387\pi\)
0.526812 + 0.849982i \(0.323387\pi\)
\(564\) 0 0
\(565\) −54.0000 −2.27180
\(566\) −23.0000 −0.966762
\(567\) 18.0000 0.755929
\(568\) 18.0000 0.755263
\(569\) 2.00000 0.0838444 0.0419222 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 0 0
\(571\) 38.0000 1.59025 0.795125 0.606445i \(-0.207405\pi\)
0.795125 + 0.606445i \(0.207405\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 4.00000 0.166957
\(575\) 28.0000 1.16768
\(576\) −21.0000 −0.875000
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 47.0000 1.95494
\(579\) 0 0
\(580\) 15.0000 0.622841
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 12.0000 0.496989
\(584\) 15.0000 0.620704
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) −3.00000 −0.123823 −0.0619116 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 0 0
\(589\) −30.0000 −1.23613
\(590\) 36.0000 1.48210
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) −48.0000 −1.96781
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) −8.00000 −0.326056
\(603\) 36.0000 1.46603
\(604\) −1.00000 −0.0406894
\(605\) −30.0000 −1.21967
\(606\) 0 0
\(607\) 15.0000 0.608831 0.304416 0.952539i \(-0.401539\pi\)
0.304416 + 0.952539i \(0.401539\pi\)
\(608\) −25.0000 −1.01388
\(609\) 0 0
\(610\) −15.0000 −0.607332
\(611\) 0 0
\(612\) −24.0000 −0.970143
\(613\) 46.0000 1.85792 0.928961 0.370177i \(-0.120703\pi\)
0.928961 + 0.370177i \(0.120703\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) −18.0000 −0.722897
\(621\) 0 0
\(622\) −30.0000 −1.20289
\(623\) 2.00000 0.0801283
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 28.0000 1.11911
\(627\) 0 0
\(628\) −17.0000 −0.678374
\(629\) −16.0000 −0.637962
\(630\) −18.0000 −0.717137
\(631\) 1.00000 0.0398094 0.0199047 0.999802i \(-0.493664\pi\)
0.0199047 + 0.999802i \(0.493664\pi\)
\(632\) −24.0000 −0.954669
\(633\) 0 0
\(634\) −24.0000 −0.953162
\(635\) −21.0000 −0.833360
\(636\) 0 0
\(637\) 0 0
\(638\) −5.00000 −0.197952
\(639\) 18.0000 0.712069
\(640\) −9.00000 −0.355756
\(641\) −28.0000 −1.10593 −0.552967 0.833203i \(-0.686504\pi\)
−0.552967 + 0.833203i \(0.686504\pi\)
\(642\) 0 0
\(643\) 19.0000 0.749287 0.374643 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) −14.0000 −0.551677
\(645\) 0 0
\(646\) 40.0000 1.57378
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −27.0000 −1.06066
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 13.0000 0.509119
\(653\) 21.0000 0.821794 0.410897 0.911682i \(-0.365216\pi\)
0.410897 + 0.911682i \(0.365216\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 15.0000 0.585206
\(658\) 18.0000 0.701713
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) −30.0000 −1.16335
\(666\) −6.00000 −0.232495
\(667\) −35.0000 −1.35521
\(668\) −8.00000 −0.309529
\(669\) 0 0
\(670\) −36.0000 −1.39080
\(671\) −5.00000 −0.193023
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −32.0000 −1.23259
\(675\) 0 0
\(676\) 13.0000 0.500000
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 72.0000 2.76107
\(681\) 0 0
\(682\) 6.00000 0.229752
\(683\) 2.00000 0.0765279 0.0382639 0.999268i \(-0.487817\pi\)
0.0382639 + 0.999268i \(0.487817\pi\)
\(684\) −15.0000 −0.573539
\(685\) 3.00000 0.114624
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) −17.0000 −0.646710 −0.323355 0.946278i \(-0.604811\pi\)
−0.323355 + 0.946278i \(0.604811\pi\)
\(692\) −6.00000 −0.228086
\(693\) −6.00000 −0.227921
\(694\) 6.00000 0.227757
\(695\) 48.0000 1.82074
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) 7.00000 0.264954
\(699\) 0 0
\(700\) −8.00000 −0.302372
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 0 0
\(703\) −10.0000 −0.377157
\(704\) 7.00000 0.263822
\(705\) 0 0
\(706\) 19.0000 0.715074
\(707\) 24.0000 0.902613
\(708\) 0 0
\(709\) 28.0000 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) −18.0000 −0.675528
\(711\) −24.0000 −0.900070
\(712\) −3.00000 −0.112430
\(713\) 42.0000 1.57291
\(714\) 0 0
\(715\) 0 0
\(716\) −11.0000 −0.411089
\(717\) 0 0
\(718\) −11.0000 −0.410516
\(719\) 37.0000 1.37987 0.689934 0.723873i \(-0.257640\pi\)
0.689934 + 0.723873i \(0.257640\pi\)
\(720\) 9.00000 0.335410
\(721\) −22.0000 −0.819323
\(722\) 6.00000 0.223297
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) −15.0000 −0.555175
\(731\) 32.0000 1.18356
\(732\) 0 0
\(733\) −12.0000 −0.443230 −0.221615 0.975134i \(-0.571133\pi\)
−0.221615 + 0.975134i \(0.571133\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 35.0000 1.29012
\(737\) −12.0000 −0.442026
\(738\) −6.00000 −0.220863
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) −6.00000 −0.220564
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) −38.0000 −1.39408 −0.697042 0.717030i \(-0.745501\pi\)
−0.697042 + 0.717030i \(0.745501\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) −15.0000 −0.549189
\(747\) 6.00000 0.219529
\(748\) 8.00000 0.292509
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −9.00000 −0.328196
\(753\) 0 0
\(754\) 0 0
\(755\) 3.00000 0.109181
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) −31.0000 −1.12597
\(759\) 0 0
\(760\) 45.0000 1.63232
\(761\) 8.00000 0.290000 0.145000 0.989432i \(-0.453682\pi\)
0.145000 + 0.989432i \(0.453682\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) −12.0000 −0.434145
\(765\) 72.0000 2.60317
\(766\) −13.0000 −0.469709
\(767\) 0 0
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 6.00000 0.216225
\(771\) 0 0
\(772\) 13.0000 0.467880
\(773\) −4.00000 −0.143870 −0.0719350 0.997409i \(-0.522917\pi\)
−0.0719350 + 0.997409i \(0.522917\pi\)
\(774\) 12.0000 0.431331
\(775\) 24.0000 0.862105
\(776\) −3.00000 −0.107694
\(777\) 0 0
\(778\) 4.00000 0.143407
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) −56.0000 −2.00256
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 51.0000 1.82027
\(786\) 0 0
\(787\) −11.0000 −0.392108 −0.196054 0.980593i \(-0.562813\pi\)
−0.196054 + 0.980593i \(0.562813\pi\)
\(788\) 17.0000 0.605600
\(789\) 0 0
\(790\) 24.0000 0.853882
\(791\) −36.0000 −1.28001
\(792\) 9.00000 0.319801
\(793\) 0 0
\(794\) 28.0000 0.993683
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −38.0000 −1.34603 −0.673015 0.739629i \(-0.735001\pi\)
−0.673015 + 0.739629i \(0.735001\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 20.0000 0.707107
\(801\) −3.00000 −0.106000
\(802\) −19.0000 −0.670913
\(803\) −5.00000 −0.176446
\(804\) 0 0
\(805\) 42.0000 1.48031
\(806\) 0 0
\(807\) 0 0
\(808\) −36.0000 −1.26648
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 27.0000 0.948683
\(811\) 22.0000 0.772524 0.386262 0.922389i \(-0.373766\pi\)
0.386262 + 0.922389i \(0.373766\pi\)
\(812\) 10.0000 0.350931
\(813\) 0 0
\(814\) 2.00000 0.0701000
\(815\) −39.0000 −1.36611
\(816\) 0 0
\(817\) 20.0000 0.699711
\(818\) 17.0000 0.594391
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 10.0000 0.348578 0.174289 0.984695i \(-0.444237\pi\)
0.174289 + 0.984695i \(0.444237\pi\)
\(824\) 33.0000 1.14961
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 26.0000 0.904109 0.452054 0.891990i \(-0.350691\pi\)
0.452054 + 0.891990i \(0.350691\pi\)
\(828\) 21.0000 0.729800
\(829\) 11.0000 0.382046 0.191023 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(830\) −6.00000 −0.208263
\(831\) 0 0
\(832\) 0 0
\(833\) 24.0000 0.831551
\(834\) 0 0
\(835\) 24.0000 0.830554
\(836\) 5.00000 0.172929
\(837\) 0 0
\(838\) 40.0000 1.38178
\(839\) 7.00000 0.241667 0.120833 0.992673i \(-0.461443\pi\)
0.120833 + 0.992673i \(0.461443\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 17.0000 0.585859
\(843\) 0 0
\(844\) −18.0000 −0.619586
\(845\) −39.0000 −1.34164
\(846\) −27.0000 −0.928279
\(847\) −20.0000 −0.687208
\(848\) −12.0000 −0.412082
\(849\) 0 0
\(850\) −32.0000 −1.09759
\(851\) 14.0000 0.479914
\(852\) 0 0
\(853\) −40.0000 −1.36957 −0.684787 0.728743i \(-0.740105\pi\)
−0.684787 + 0.728743i \(0.740105\pi\)
\(854\) −10.0000 −0.342193
\(855\) 45.0000 1.53897
\(856\) −9.00000 −0.307614
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) 21.0000 0.716511 0.358255 0.933624i \(-0.383372\pi\)
0.358255 + 0.933624i \(0.383372\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) 32.0000 1.08992
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) −26.0000 −0.883516
\(867\) 0 0
\(868\) −12.0000 −0.407307
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 0 0
\(872\) −3.00000 −0.101593
\(873\) −3.00000 −0.101535
\(874\) −35.0000 −1.18389
\(875\) −6.00000 −0.202837
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −6.00000 −0.202490
\(879\) 0 0
\(880\) −3.00000 −0.101130
\(881\) −39.0000 −1.31394 −0.656972 0.753915i \(-0.728163\pi\)
−0.656972 + 0.753915i \(0.728163\pi\)
\(882\) 9.00000 0.303046
\(883\) 19.0000 0.639401 0.319700 0.947519i \(-0.396418\pi\)
0.319700 + 0.947519i \(0.396418\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −10.0000 −0.335957
\(887\) −2.00000 −0.0671534 −0.0335767 0.999436i \(-0.510690\pi\)
−0.0335767 + 0.999436i \(0.510690\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.469545
\(890\) 3.00000 0.100560
\(891\) 9.00000 0.301511
\(892\) 2.00000 0.0669650
\(893\) −45.0000 −1.50587
\(894\) 0 0
\(895\) 33.0000 1.10307
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) −24.0000 −0.800890
\(899\) −30.0000 −1.00056
\(900\) 12.0000 0.400000
\(901\) −96.0000 −3.19822
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) 54.0000 1.79601
\(905\) −30.0000 −0.997234
\(906\) 0 0
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 14.0000 0.464606
\(909\) −36.0000 −1.19404
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 0 0
\(913\) −2.00000 −0.0661903
\(914\) −27.0000 −0.893081
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) 0 0
\(918\) 0 0
\(919\) 35.0000 1.15454 0.577272 0.816552i \(-0.304117\pi\)
0.577272 + 0.816552i \(0.304117\pi\)
\(920\) −63.0000 −2.07705
\(921\) 0 0
\(922\) 18.0000 0.592798
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 4.00000 0.131448
\(927\) 33.0000 1.08386
\(928\) −25.0000 −0.820665
\(929\) 12.0000 0.393707 0.196854 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 0 0
\(931\) 15.0000 0.491605
\(932\) −9.00000 −0.294805
\(933\) 0 0
\(934\) 18.0000 0.588978
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) −24.0000 −0.783628
\(939\) 0 0
\(940\) −27.0000 −0.880643
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 0 0
\(943\) 14.0000 0.455903
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −40.0000 −1.29983 −0.649913 0.760009i \(-0.725195\pi\)
−0.649913 + 0.760009i \(0.725195\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −20.0000 −0.648886
\(951\) 0 0
\(952\) 48.0000 1.55569
\(953\) −3.00000 −0.0971795 −0.0485898 0.998819i \(-0.515473\pi\)
−0.0485898 + 0.998819i \(0.515473\pi\)
\(954\) −36.0000 −1.16554
\(955\) 36.0000 1.16493
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −9.00000 −0.290021
\(964\) 10.0000 0.322078
\(965\) −39.0000 −1.25545
\(966\) 0 0
\(967\) 27.0000 0.868261 0.434131 0.900850i \(-0.357056\pi\)
0.434131 + 0.900850i \(0.357056\pi\)
\(968\) 30.0000 0.964237
\(969\) 0 0
\(970\) 3.00000 0.0963242
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 32.0000 1.02587
\(974\) 19.0000 0.608799
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) −47.0000 −1.50366 −0.751832 0.659355i \(-0.770829\pi\)
−0.751832 + 0.659355i \(0.770829\pi\)
\(978\) 0 0
\(979\) 1.00000 0.0319601
\(980\) 9.00000 0.287494
\(981\) −3.00000 −0.0957826
\(982\) 0 0
\(983\) 45.0000 1.43528 0.717639 0.696416i \(-0.245223\pi\)
0.717639 + 0.696416i \(0.245223\pi\)
\(984\) 0 0
\(985\) −51.0000 −1.62500
\(986\) 40.0000 1.27386
\(987\) 0 0
\(988\) 0 0
\(989\) −28.0000 −0.890348
\(990\) −9.00000 −0.286039
\(991\) 49.0000 1.55654 0.778268 0.627932i \(-0.216098\pi\)
0.778268 + 0.627932i \(0.216098\pi\)
\(992\) 30.0000 0.952501
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 48.0000 1.52170
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 109.2.a.a.1.1 1
3.2 odd 2 981.2.a.a.1.1 1
4.3 odd 2 1744.2.a.a.1.1 1
5.4 even 2 2725.2.a.b.1.1 1
7.6 odd 2 5341.2.a.b.1.1 1
8.3 odd 2 6976.2.a.c.1.1 1
8.5 even 2 6976.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
109.2.a.a.1.1 1 1.1 even 1 trivial
981.2.a.a.1.1 1 3.2 odd 2
1744.2.a.a.1.1 1 4.3 odd 2
2725.2.a.b.1.1 1 5.4 even 2
5341.2.a.b.1.1 1 7.6 odd 2
6976.2.a.c.1.1 1 8.3 odd 2
6976.2.a.f.1.1 1 8.5 even 2