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Properties

Label 3330.2.a.u.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3330,2,Mod(1,3330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3330, 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("3330.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.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: \(26.5901838731\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3330.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.00000 q^{11} -7.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{19} +1.00000 q^{20} -3.00000 q^{22} +3.00000 q^{23} +1.00000 q^{25} -7.00000 q^{26} -1.00000 q^{28} -6.00000 q^{29} -10.0000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -1.00000 q^{35} +1.00000 q^{37} -1.00000 q^{38} +1.00000 q^{40} -4.00000 q^{43} -3.00000 q^{44} +3.00000 q^{46} +12.0000 q^{47} -6.00000 q^{49} +1.00000 q^{50} -7.00000 q^{52} -9.00000 q^{53} -3.00000 q^{55} -1.00000 q^{56} -6.00000 q^{58} -6.00000 q^{59} -10.0000 q^{61} -10.0000 q^{62} +1.00000 q^{64} -7.00000 q^{65} -10.0000 q^{67} +3.00000 q^{68} -1.00000 q^{70} +6.00000 q^{71} +11.0000 q^{73} +1.00000 q^{74} -1.00000 q^{76} +3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} -9.00000 q^{83} +3.00000 q^{85} -4.00000 q^{86} -3.00000 q^{88} +9.00000 q^{89} +7.00000 q^{91} +3.00000 q^{92} +12.0000 q^{94} -1.00000 q^{95} -16.0000 q^{97} -6.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −7.00000 −1.94145 −0.970725 − 0.240192i \(-0.922790\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 3.00000 0.625543 0.312772 − 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −7.00000 −1.37281
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −7.00000 −0.970725
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −10.0000 −1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −7.00000 −0.868243
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 − 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 0 0
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 9.00000 0.953998 0.476999 − 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 7.00000 0.733799
\(92\) 3.00000 0.312772
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −16.0000 −1.62455 −0.812277 − 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) −6.00000 −0.606092
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −7.00000 −0.686406
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) −3.00000 −0.290021 −0.145010 − 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 − 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) −3.00000 −0.286039
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −18.0000 −1.69330 −0.846649 − 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 − 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −7.00000 −0.613941
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) −10.0000 −0.863868
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 − 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 21.0000 1.75611
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 11.0000 0.910366
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 23.0000 1.87171 0.935857 − 0.352381i \(-0.114628\pi\)
0.935857 + 0.352381i \(0.114628\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 3.00000 0.241747
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 − 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) −1.00000 −0.0783260 −0.0391630 − 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) −3.00000 −0.232147 −0.116073 − 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 36.0000 2.76923
\(170\) 3.00000 0.230089
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 9.00000 0.684257 0.342129 − 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 9.00000 0.674579
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 7.00000 0.518875
\(183\) 0 0
\(184\) 3.00000 0.221163
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −9.00000 −0.658145
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −1.00000 −0.0725476
\(191\) −3.00000 −0.217072 −0.108536 − 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −3.00000 −0.213741 −0.106871 − 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −7.00000 −0.485363
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 − 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) −9.00000 −0.618123
\(213\) 0 0
\(214\) −3.00000 −0.205076
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) −7.00000 −0.474100
\(219\) 0 0
\(220\) −3.00000 −0.202260
\(221\) −21.0000 −1.41261
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) −24.0000 −1.59294 −0.796468 − 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 3.00000 0.197814
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −18.0000 −1.17922 −0.589610 − 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) −3.00000 −0.194461
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 − 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 7.00000 0.445399
\(248\) −10.0000 −0.635001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −6.00000 −0.378717 −0.189358 − 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) −7.00000 −0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.00000 −0.561405 −0.280702 − 0.959795i \(-0.590567\pi\)
−0.280702 + 0.959795i \(0.590567\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) −7.00000 −0.434122
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 1.00000 0.0613139
\(267\) 0 0
\(268\) −10.0000 −0.610847
\(269\) −9.00000 −0.548740 −0.274370 − 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) 23.0000 1.38194 0.690968 − 0.722885i \(-0.257185\pi\)
0.690968 + 0.722885i \(0.257185\pi\)
\(278\) 14.0000 0.839664
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −27.0000 −1.61068 −0.805342 − 0.592810i \(-0.798019\pi\)
−0.805342 + 0.592810i \(0.798019\pi\)
\(282\) 0 0
\(283\) −13.0000 −0.772770 −0.386385 − 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 21.0000 1.24176
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) 11.0000 0.643726
\(293\) 15.0000 0.876309 0.438155 − 0.898900i \(-0.355632\pi\)
0.438155 + 0.898900i \(0.355632\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −21.0000 −1.21446
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 23.0000 1.32350
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −22.0000 −1.25561 −0.627803 − 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) −10.0000 −0.567962
\(311\) 24.0000 1.36092 0.680458 − 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 − 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −3.00000 −0.167183
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) −7.00000 −0.388290
\(326\) −1.00000 −0.0553849
\(327\) 0 0
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 − 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −9.00000 −0.493939
\(333\) 0 0
\(334\) −3.00000 −0.164153
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 − 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) 36.0000 1.95814
\(339\) 0 0
\(340\) 3.00000 0.162698
\(341\) 30.0000 1.62459
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 9.00000 0.483843
\(347\) −36.0000 −1.93258 −0.966291 − 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 0 0
\(349\) 8.00000 0.428230 0.214115 − 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 9.00000 0.476999
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 − 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −22.0000 −1.15629
\(363\) 0 0
\(364\) 7.00000 0.366900
\(365\) 11.0000 0.575766
\(366\) 0 0
\(367\) 23.0000 1.20059 0.600295 − 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) 3.00000 0.156386
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) 9.00000 0.467257
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −9.00000 −0.465379
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 42.0000 2.16311
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 − 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 0 0
\(382\) −3.00000 −0.153493
\(383\) 9.00000 0.459879 0.229939 − 0.973205i \(-0.426147\pi\)
0.229939 + 0.973205i \(0.426147\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) −16.0000 −0.812277
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) −6.00000 −0.303046
\(393\) 0 0
\(394\) −3.00000 −0.151138
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −4.00000 −0.200754 −0.100377 − 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −15.0000 −0.749064 −0.374532 − 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) 70.0000 3.48695
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) −3.00000 −0.148704
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 − 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) −9.00000 −0.441793
\(416\) −7.00000 −0.343203
\(417\) 0 0
\(418\) 3.00000 0.146735
\(419\) 27.0000 1.31904 0.659518 − 0.751689i \(-0.270760\pi\)
0.659518 + 0.751689i \(0.270760\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 2.00000 0.0973585
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) −3.00000 −0.145010
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 3.00000 0.144505 0.0722525 − 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) 0 0
\(433\) −31.0000 −1.48976 −0.744882 − 0.667196i \(-0.767494\pi\)
−0.744882 + 0.667196i \(0.767494\pi\)
\(434\) 10.0000 0.480015
\(435\) 0 0
\(436\) −7.00000 −0.335239
\(437\) −3.00000 −0.143509
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) −21.0000 −0.998868
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 9.00000 0.426641
\(446\) 8.00000 0.378811
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 30.0000 1.41579 0.707894 − 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) −24.0000 −1.12638
\(455\) 7.00000 0.328165
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 3.00000 0.139876
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 − 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 10.0000 0.461757
\(470\) 12.0000 0.553519
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) −3.00000 −0.137505
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) 39.0000 1.78196 0.890978 − 0.454047i \(-0.150020\pi\)
0.890978 + 0.454047i \(0.150020\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 26.0000 1.18427
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −16.0000 −0.726523
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 − 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) −6.00000 −0.271052
\(491\) 9.00000 0.406164 0.203082 − 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) 7.00000 0.314945
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 41.0000 1.83541 0.917706 − 0.397260i \(-0.130039\pi\)
0.917706 + 0.397260i \(0.130039\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −6.00000 −0.267793
\(503\) −36.0000 −1.60516 −0.802580 − 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) −9.00000 −0.400099
\(507\) 0 0
\(508\) −7.00000 −0.310575
\(509\) 33.0000 1.46270 0.731350 − 0.682003i \(-0.238891\pi\)
0.731350 + 0.682003i \(0.238891\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −9.00000 −0.396973
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) −36.0000 −1.58328
\(518\) −1.00000 −0.0439375
\(519\) 0 0
\(520\) −7.00000 −0.306970
\(521\) 12.0000 0.525730 0.262865 − 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) 44.0000 1.92399 0.961993 − 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) −30.0000 −1.30682
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −9.00000 −0.390935
\(531\) 0 0
\(532\) 1.00000 0.0433555
\(533\) 0 0
\(534\) 0 0
\(535\) −3.00000 −0.129701
\(536\) −10.0000 −0.431934
\(537\) 0 0
\(538\) −9.00000 −0.388018
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 − 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 20.0000 0.859074
\(543\) 0 0
\(544\) 3.00000 0.128624
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) −37.0000 −1.58201 −0.791003 − 0.611812i \(-0.790441\pi\)
−0.791003 + 0.611812i \(0.790441\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) −3.00000 −0.127920
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 23.0000 0.977176
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 24.0000 1.01691 0.508456 − 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 28.0000 1.18427
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −27.0000 −1.13893
\(563\) 18.0000 0.758610 0.379305 − 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) −13.0000 −0.546431
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 39.0000 1.63497 0.817483 − 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) 0 0
\(571\) 14.0000 0.585882 0.292941 − 0.956131i \(-0.405366\pi\)
0.292941 + 0.956131i \(0.405366\pi\)
\(572\) 21.0000 0.878054
\(573\) 0 0
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 − 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 9.00000 0.373383
\(582\) 0 0
\(583\) 27.0000 1.11823
\(584\) 11.0000 0.455183
\(585\) 0 0
\(586\) 15.0000 0.619644
\(587\) 36.0000 1.48588 0.742940 − 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) 10.0000 0.412043
\(590\) −6.00000 −0.247016
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) 36.0000 1.47834 0.739171 − 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −21.0000 −0.858754
\(599\) 12.0000 0.490307 0.245153 − 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 − 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) 23.0000 0.935857
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 − 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) −84.0000 −3.39828
\(612\) 0 0
\(613\) −34.0000 −1.37325 −0.686624 − 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) −22.0000 −0.887848
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −42.0000 −1.69086 −0.845428 − 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 − 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) −10.0000 −0.401610
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) −9.00000 −0.360577
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 − 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) −7.00000 −0.277787
\(636\) 0 0
\(637\) 42.0000 1.66410
\(638\) 18.0000 0.712627
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −24.0000 −0.947943 −0.473972 − 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) −31.0000 −1.22252 −0.611260 − 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) −3.00000 −0.118033
\(647\) −45.0000 −1.76913 −0.884566 − 0.466415i \(-0.845546\pi\)
−0.884566 + 0.466415i \(0.845546\pi\)
\(648\) 0 0
\(649\) 18.0000 0.706562
\(650\) −7.00000 −0.274563
\(651\) 0 0
\(652\) −1.00000 −0.0391630
\(653\) 36.0000 1.40879 0.704394 − 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 0 0
\(658\) −12.0000 −0.467809
\(659\) −12.0000 −0.467454 −0.233727 − 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 − 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) −9.00000 −0.349268
\(665\) 1.00000 0.0387783
\(666\) 0 0
\(667\) −18.0000 −0.696963
\(668\) −3.00000 −0.116073
\(669\) 0 0
\(670\) −10.0000 −0.386334
\(671\) 30.0000 1.15814
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 − 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −13.0000 −0.500741
\(675\) 0 0
\(676\) 36.0000 1.38462
\(677\) −3.00000 −0.115299 −0.0576497 − 0.998337i \(-0.518361\pi\)
−0.0576497 + 0.998337i \(0.518361\pi\)
\(678\) 0 0
\(679\) 16.0000 0.614024
\(680\) 3.00000 0.115045
\(681\) 0 0
\(682\) 30.0000 1.14876
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 63.0000 2.40011
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 − 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 9.00000 0.342129
\(693\) 0 0
\(694\) −36.0000 −1.36654
\(695\) 14.0000 0.531050
\(696\) 0 0
\(697\) 0 0
\(698\) 8.00000 0.302804
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) 48.0000 1.81293 0.906467 − 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) 0 0
\(703\) −1.00000 −0.0377157
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 − 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 6.00000 0.225176
\(711\) 0 0
\(712\) 9.00000 0.337289
\(713\) −30.0000 −1.12351
\(714\) 0 0
\(715\) 21.0000 0.785355
\(716\) 0 0
\(717\) 0 0
\(718\) 12.0000 0.447836
\(719\) −6.00000 −0.223762 −0.111881 − 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −18.0000 −0.669891
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 2.00000 0.0741759 0.0370879 − 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 7.00000 0.259437
\(729\) 0 0
\(730\) 11.0000 0.407128
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 − 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 23.0000 0.848945
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) 30.0000 1.10506
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 − 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 9.00000 0.330400
\(743\) −6.00000 −0.220119 −0.110059 − 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) −9.00000 −0.329073
\(749\) 3.00000 0.109618
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 − 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 42.0000 1.52955
\(755\) 23.0000 0.837056
\(756\) 0 0
\(757\) −13.0000 −0.472493 −0.236247 − 0.971693i \(-0.575917\pi\)
−0.236247 + 0.971693i \(0.575917\pi\)
\(758\) 26.0000 0.944363
\(759\) 0 0
\(760\) −1.00000 −0.0362738
\(761\) −6.00000 −0.217500 −0.108750 − 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 7.00000 0.253417
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) 9.00000 0.325183
\(767\) 42.0000 1.51653
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 − 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 3.00000 0.108112
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) −15.0000 −0.539513 −0.269756 − 0.962929i \(-0.586943\pi\)
−0.269756 + 0.962929i \(0.586943\pi\)
\(774\) 0 0
\(775\) −10.0000 −0.359211
\(776\) −16.0000 −0.574367
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) 0 0
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 9.00000 0.321839
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) 2.00000 0.0712923 0.0356462 − 0.999364i \(-0.488651\pi\)
0.0356462 + 0.999364i \(0.488651\pi\)
\(788\) −3.00000 −0.106871
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 70.0000 2.48577
\(794\) −4.00000 −0.141955
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −30.0000 −1.06265 −0.531327 − 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −15.0000 −0.529668
\(803\) −33.0000 −1.16454
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) 70.0000 2.46564
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 21.0000 0.738321 0.369160 − 0.929366i \(-0.379645\pi\)
0.369160 + 0.929366i \(0.379645\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 6.00000 0.210559
\(813\) 0 0
\(814\) −3.00000 −0.105150
\(815\) −1.00000 −0.0350285
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) −4.00000 −0.139857
\(819\) 0 0
\(820\) 0 0
\(821\) 9.00000 0.314102 0.157051 − 0.987590i \(-0.449801\pi\)
0.157051 + 0.987590i \(0.449801\pi\)
\(822\) 0 0
\(823\) −7.00000 −0.244005 −0.122002 − 0.992530i \(-0.538932\pi\)
−0.122002 + 0.992530i \(0.538932\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 29.0000 1.00721 0.503606 − 0.863934i \(-0.332006\pi\)
0.503606 + 0.863934i \(0.332006\pi\)
\(830\) −9.00000 −0.312395
\(831\) 0 0
\(832\) −7.00000 −0.242681
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) −3.00000 −0.103819
\(836\) 3.00000 0.103757
\(837\) 0 0
\(838\) 27.0000 0.932700
\(839\) −30.0000 −1.03572 −0.517858 − 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) 0 0
\(844\) 2.00000 0.0688428
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) −9.00000 −0.309061
\(849\) 0 0
\(850\) 3.00000 0.102899
\(851\) 3.00000 0.102839
\(852\) 0 0
\(853\) −19.0000 −0.650548 −0.325274 − 0.945620i \(-0.605456\pi\)
−0.325274 + 0.945620i \(0.605456\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) −3.00000 −0.102538
\(857\) 9.00000 0.307434 0.153717 − 0.988115i \(-0.450876\pi\)
0.153717 + 0.988115i \(0.450876\pi\)
\(858\) 0 0
\(859\) 5.00000 0.170598 0.0852989 − 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 3.00000 0.102180
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 9.00000 0.306009
\(866\) −31.0000 −1.05342
\(867\) 0 0
\(868\) 10.0000 0.339422
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 70.0000 2.37186
\(872\) −7.00000 −0.237050
\(873\) 0 0
\(874\) −3.00000 −0.101477
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 32.0000 1.08056 0.540282 − 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) −3.00000 −0.101130
\(881\) 12.0000 0.404290 0.202145 − 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) 0 0
\(883\) 35.0000 1.17784 0.588922 − 0.808190i \(-0.299553\pi\)
0.588922 + 0.808190i \(0.299553\pi\)
\(884\) −21.0000 −0.706306
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 18.0000 0.604381 0.302190 − 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) 0 0
\(889\) 7.00000 0.234772
\(890\) 9.00000 0.301681
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) 60.0000 2.00111
\(900\) 0 0
\(901\) −27.0000 −0.899500
\(902\) 0 0
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) −37.0000 −1.22856 −0.614282 − 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) −24.0000 −0.796468
\(909\) 0 0
\(910\) 7.00000 0.232048
\(911\) −48.0000 −1.59031 −0.795155 − 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) 27.0000 0.893570
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −4.00000 −0.131948 −0.0659739 − 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 3.00000 0.0989071
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) −42.0000 −1.38245
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 0 0
\(935\) −9.00000 −0.294331
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 − 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 10.0000 0.326512
\(939\) 0 0
\(940\) 12.0000 0.391397
\(941\) −18.0000 −0.586783 −0.293392 − 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 24.0000 0.779895 0.389948 − 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 0 0
\(949\) −77.0000 −2.49953
\(950\) −1.00000 −0.0324443
\(951\) 0 0
\(952\) −3.00000 −0.0972306
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −3.00000 −0.0970777
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 39.0000 1.26003
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) −7.00000 −0.225689
\(963\) 0 0
\(964\) 26.0000 0.837404
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 − 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) −16.0000 −0.513729
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) −14.0000 −0.448819
\(974\) −22.0000 −0.704925
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −9.00000 −0.287936 −0.143968 − 0.989582i \(-0.545986\pi\)
−0.143968 + 0.989582i \(0.545986\pi\)
\(978\) 0 0
\(979\) −27.0000 −0.862924
\(980\) −6.00000 −0.191663
\(981\) 0 0
\(982\) 9.00000 0.287202
\(983\) −42.0000 −1.33959 −0.669796 − 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 0 0
\(985\) −3.00000 −0.0955879
\(986\) −18.0000 −0.573237
\(987\) 0 0
\(988\) 7.00000 0.222700
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 − 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) −10.0000 −0.317500
\(993\) 0 0
\(994\) −6.00000 −0.190308
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) −1.00000 −0.0316703 −0.0158352 − 0.999875i \(-0.505041\pi\)
−0.0158352 + 0.999875i \(0.505041\pi\)
\(998\) 41.0000 1.29783
\(999\) 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 3330.2.a.u.1.1 1
3.2 odd 2 1110.2.a.e.1.1 ✓ 1
12.11 even 2 8880.2.a.f.1.1 1
15.14 odd 2 5550.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.e.1.1 ✓ 1 3.2 odd 2
3330.2.a.u.1.1 1 1.1 even 1 trivial
5550.2.a.ba.1.1 1 15.14 odd 2
8880.2.a.f.1.1 1 12.11 even 2