Properties

Label 2300.2.c.f.1749.1
Level $2300$
Weight $2$
Character 2300.1749
Analytic conductor $18.366$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2300,2,Mod(1749,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2300.1749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(18.3655924649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 92)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1749.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2300.1749
Dual form 2300.2.c.f.1749.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +2.00000i q^{7} +2.00000 q^{9} +1.00000i q^{13} -6.00000i q^{17} -2.00000 q^{19} +2.00000 q^{21} +1.00000i q^{23} -5.00000i q^{27} +3.00000 q^{29} +5.00000 q^{31} +8.00000i q^{37} +1.00000 q^{39} +3.00000 q^{41} -8.00000i q^{43} +9.00000i q^{47} +3.00000 q^{49} -6.00000 q^{51} -6.00000i q^{53} +2.00000i q^{57} +12.0000 q^{59} +14.0000 q^{61} +4.00000i q^{63} +8.00000i q^{67} +1.00000 q^{69} -15.0000 q^{71} +7.00000i q^{73} +10.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -3.00000i q^{87} -2.00000 q^{91} -5.00000i q^{93} -10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9} - 4 q^{19} + 4 q^{21} + 6 q^{29} + 10 q^{31} + 2 q^{39} + 6 q^{41} + 6 q^{49} - 12 q^{51} + 24 q^{59} + 28 q^{61} + 2 q^{69} - 30 q^{71} + 20 q^{79} + 2 q^{81} - 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.00000i 1.31278i 0.754420 + 0.656392i \(0.227918\pi\)
−0.754420 + 0.656392i \(0.772082\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 4.00000i 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 0 0
\(73\) 7.00000i 0.819288i 0.912245 + 0.409644i \(0.134347\pi\)
−0.912245 + 0.409644i \(0.865653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 3.00000i − 0.321634i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) − 5.00000i − 0.518476i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) − 2.00000i − 0.197066i −0.995134 0.0985329i \(-0.968585\pi\)
0.995134 0.0985329i \(-0.0314150\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) − 18.0000i − 1.69330i −0.532152 0.846649i \(-0.678617\pi\)
0.532152 0.846649i \(-0.321383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) − 3.00000i − 0.270501i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.0000i 0.976092i 0.872818 + 0.488046i \(0.162290\pi\)
−0.872818 + 0.488046i \(0.837710\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 21.0000 1.83478 0.917389 0.397991i \(-0.130293\pi\)
0.917389 + 0.397991i \(0.130293\pi\)
\(132\) 0 0
\(133\) − 4.00000i − 0.346844i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) −11.0000 −0.933008 −0.466504 0.884519i \(-0.654487\pi\)
−0.466504 + 0.884519i \(0.654487\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 3.00000i − 0.247436i
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −7.00000 −0.569652 −0.284826 0.958579i \(-0.591936\pi\)
−0.284826 + 0.958579i \(0.591936\pi\)
\(152\) 0 0
\(153\) − 12.0000i − 0.970143i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.00000i − 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 1.00000i 0.0783260i 0.999233 + 0.0391630i \(0.0124692\pi\)
−0.999233 + 0.0391630i \(0.987531\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) 0 0
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) − 14.0000i − 1.03491i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 10.0000 0.727393
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 7.00000i 0.503871i 0.967744 + 0.251936i \(0.0810671\pi\)
−0.967744 + 0.251936i \(0.918933\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 15.0000i − 1.06871i −0.845262 0.534353i \(-0.820555\pi\)
0.845262 0.534353i \(-0.179445\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.00000i 0.139010i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 15.0000i 1.02778i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 10.0000i 0.678844i
\(218\) 0 0
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.00000i 0.589610i 0.955557 + 0.294805i \(0.0952546\pi\)
−0.955557 + 0.294805i \(0.904745\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 10.0000i − 0.649570i
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) − 16.0000i − 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 2.00000i − 0.127257i
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.0000i 1.68421i 0.539311 + 0.842107i \(0.318685\pi\)
−0.539311 + 0.842107i \(0.681315\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.0000 0.914566 0.457283 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 2.00000i 0.121046i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 19.0000i − 1.14160i −0.821089 0.570800i \(-0.806633\pi\)
0.821089 0.570800i \(-0.193367\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000i 0.354169i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) − 12.0000i − 0.701047i −0.936554 0.350524i \(-0.886004\pi\)
0.936554 0.350524i \(-0.113996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.00000 −0.0578315
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 0 0
\(303\) − 6.00000i − 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) −2.00000 −0.113776
\(310\) 0 0
\(311\) −15.0000 −0.850572 −0.425286 0.905059i \(-0.639826\pi\)
−0.425286 + 0.905059i \(0.639826\pi\)
\(312\) 0 0
\(313\) 4.00000i 0.226093i 0.993590 + 0.113047i \(0.0360610\pi\)
−0.993590 + 0.113047i \(0.963939\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 16.0000i − 0.884802i
\(328\) 0 0
\(329\) −18.0000 −0.992372
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) 0 0
\(333\) 16.0000i 0.876795i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) −29.0000 −1.55233 −0.776167 0.630527i \(-0.782839\pi\)
−0.776167 + 0.630527i \(0.782839\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) 27.0000i 1.43706i 0.695493 + 0.718532i \(0.255186\pi\)
−0.695493 + 0.718532i \(0.744814\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 12.0000i − 0.635107i
\(358\) 0 0
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 11.0000i 0.577350i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 26.0000i 1.35719i 0.734513 + 0.678594i \(0.237411\pi\)
−0.734513 + 0.678594i \(0.762589\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) − 20.0000i − 1.03556i −0.855514 0.517780i \(-0.826758\pi\)
0.855514 0.517780i \(-0.173242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000i 0.154508i
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 11.0000 0.563547
\(382\) 0 0
\(383\) 36.0000i 1.83951i 0.392488 + 0.919757i \(0.371614\pi\)
−0.392488 + 0.919757i \(0.628386\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 16.0000i − 0.813326i
\(388\) 0 0
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) − 21.0000i − 1.05931i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.0000i 0.552074i 0.961147 + 0.276037i \(0.0890213\pi\)
−0.961147 + 0.276037i \(0.910979\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 5.00000i 0.249068i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −17.0000 −0.840596 −0.420298 0.907386i \(-0.638074\pi\)
−0.420298 + 0.907386i \(0.638074\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 0 0
\(413\) 24.0000i 1.18096i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 11.0000i 0.538672i
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) 18.0000i 0.875190i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 28.0000i 1.35501i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 16.0000i 0.768911i 0.923144 + 0.384455i \(0.125611\pi\)
−0.923144 + 0.384455i \(0.874389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 2.00000i − 0.0956730i
\(438\) 0 0
\(439\) −35.0000 −1.67046 −0.835229 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) − 21.0000i − 0.997740i −0.866677 0.498870i \(-0.833748\pi\)
0.866677 0.498870i \(-0.166252\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 7.00000i 0.328889i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 0 0
\(459\) −30.0000 −1.40028
\(460\) 0 0
\(461\) −27.0000 −1.25752 −0.628758 0.777601i \(-0.716436\pi\)
−0.628758 + 0.777601i \(0.716436\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 30.0000i − 1.38823i −0.719862 0.694117i \(-0.755795\pi\)
0.719862 0.694117i \(-0.244205\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 12.0000i − 0.549442i
\(478\) 0 0
\(479\) −42.0000 −1.91903 −0.959514 0.281659i \(-0.909115\pi\)
−0.959514 + 0.281659i \(0.909115\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) 2.00000i 0.0910032i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 23.0000i 1.04223i 0.853487 + 0.521115i \(0.174484\pi\)
−0.853487 + 0.521115i \(0.825516\pi\)
\(488\) 0 0
\(489\) 1.00000 0.0452216
\(490\) 0 0
\(491\) 3.00000 0.135388 0.0676941 0.997706i \(-0.478436\pi\)
0.0676941 + 0.997706i \(0.478436\pi\)
\(492\) 0 0
\(493\) − 18.0000i − 0.810679i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 30.0000i − 1.34568i
\(498\) 0 0
\(499\) −5.00000 −0.223831 −0.111915 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) − 6.00000i − 0.267527i −0.991013 0.133763i \(-0.957294\pi\)
0.991013 0.133763i \(-0.0427062\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 12.0000i − 0.532939i
\(508\) 0 0
\(509\) 3.00000 0.132973 0.0664863 0.997787i \(-0.478821\pi\)
0.0664863 + 0.997787i \(0.478821\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 0 0
\(513\) 10.0000i 0.441511i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −24.0000 −1.05146 −0.525730 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) 0 0
\(523\) − 38.0000i − 1.66162i −0.556553 0.830812i \(-0.687876\pi\)
0.556553 0.830812i \(-0.312124\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 30.0000i − 1.30682i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) 3.00000i 0.129944i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 9.00000i − 0.388379i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.00000 −0.300954 −0.150477 0.988614i \(-0.548081\pi\)
−0.150477 + 0.988614i \(0.548081\pi\)
\(542\) 0 0
\(543\) 16.0000i 0.686626i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 1.00000i − 0.0427569i −0.999771 0.0213785i \(-0.993195\pi\)
0.999771 0.0213785i \(-0.00680549\pi\)
\(548\) 0 0
\(549\) 28.0000 1.19501
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 20.0000i 0.850487i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.0000i 1.01691i 0.861088 + 0.508456i \(0.169784\pi\)
−0.861088 + 0.508456i \(0.830216\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) −34.0000 −1.42286 −0.711428 0.702759i \(-0.751951\pi\)
−0.711428 + 0.702759i \(0.751951\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 7.00000i − 0.291414i −0.989328 0.145707i \(-0.953454\pi\)
0.989328 0.145707i \(-0.0465456\pi\)
\(578\) 0 0
\(579\) 7.00000 0.290910
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.0000i 0.866763i 0.901211 + 0.433381i \(0.142680\pi\)
−0.901211 + 0.433381i \(0.857320\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) −15.0000 −0.617018
\(592\) 0 0
\(593\) 42.0000i 1.72473i 0.506284 + 0.862367i \(0.331019\pi\)
−0.506284 + 0.862367i \(0.668981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 4.00000i − 0.163709i
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 0 0
\(603\) 16.0000i 0.651570i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 8.00000i 0.324710i 0.986732 + 0.162355i \(0.0519090\pi\)
−0.986732 + 0.162355i \(0.948091\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) −9.00000 −0.364101
\(612\) 0 0
\(613\) − 26.0000i − 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 48.0000 1.91389
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 16.0000i 0.635943i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.00000i 0.118864i
\(638\) 0 0
\(639\) −30.0000 −1.18678
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 33.0000i 1.29736i 0.761060 + 0.648682i \(0.224679\pi\)
−0.761060 + 0.648682i \(0.775321\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 0 0
\(653\) − 27.0000i − 1.05659i −0.849060 0.528296i \(-0.822831\pi\)
0.849060 0.528296i \(-0.177169\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14.0000i 0.546192i
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) − 6.00000i − 0.233021i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.00000i 0.116160i
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 13.0000i 0.501113i 0.968102 + 0.250557i \(0.0806136\pi\)
−0.968102 + 0.250557i \(0.919386\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) − 39.0000i − 1.49229i −0.665782 0.746147i \(-0.731902\pi\)
0.665782 0.746147i \(-0.268098\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 4.00000i − 0.152610i
\(688\) 0 0
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 18.0000i − 0.681799i
\(698\) 0 0
\(699\) 9.00000 0.340411
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) − 16.0000i − 0.603451i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.0000i 0.451306i
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) 5.00000i 0.187251i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 21.0000i 0.784259i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) 4.00000i 0.148762i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 28.0000i − 1.03846i −0.854634 0.519231i \(-0.826218\pi\)
0.854634 0.519231i \(-0.173782\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 25.0000 0.919640 0.459820 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) − 6.00000i − 0.220119i −0.993925 0.110059i \(-0.964896\pi\)
0.993925 0.110059i \(-0.0351041\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 18.0000i 0.655956i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 20.0000i 0.726912i 0.931611 + 0.363456i \(0.118403\pi\)
−0.931611 + 0.363456i \(0.881597\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) 0 0
\(763\) 32.0000i 1.15848i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000i 0.433295i
\(768\) 0 0
\(769\) 28.0000 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(770\) 0 0
\(771\) 27.0000 0.972381
\(772\) 0 0
\(773\) − 24.0000i − 0.863220i −0.902060 0.431610i \(-0.857946\pi\)
0.902060 0.431610i \(-0.142054\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 16.0000i 0.573997i
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 15.0000i − 0.536056i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 40.0000i − 1.42585i −0.701242 0.712923i \(-0.747371\pi\)
0.701242 0.712923i \(-0.252629\pi\)
\(788\) 0 0
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) 36.0000 1.28001
\(792\) 0 0
\(793\) 14.0000i 0.497155i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 0 0
\(799\) 54.0000 1.91038
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 15.0000i − 0.528025i
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −7.00000 −0.245803 −0.122902 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(812\) 0 0
\(813\) − 8.00000i − 0.280572i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000i 0.559769i
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) − 41.0000i − 1.42917i −0.699549 0.714585i \(-0.746616\pi\)
0.699549 0.714585i \(-0.253384\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) −19.0000 −0.659103
\(832\) 0 0
\(833\) − 18.0000i − 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 25.0000i − 0.864126i
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) − 30.0000i − 1.03325i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 22.0000i − 0.755929i
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) − 2.00000i − 0.0684787i −0.999414 0.0342393i \(-0.989099\pi\)
0.999414 0.0342393i \(-0.0109009\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 33.0000i − 1.12726i −0.826028 0.563629i \(-0.809405\pi\)
0.826028 0.563629i \(-0.190595\pi\)
\(858\) 0 0
\(859\) 19.0000 0.648272 0.324136 0.946011i \(-0.394927\pi\)
0.324136 + 0.946011i \(0.394927\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 0 0
\(863\) 9.00000i 0.306364i 0.988198 + 0.153182i \(0.0489520\pi\)
−0.988198 + 0.153182i \(0.951048\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 19.0000i 0.645274i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) − 20.0000i − 0.676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 50.0000i 1.68838i 0.536044 + 0.844190i \(0.319918\pi\)
−0.536044 + 0.844190i \(0.680082\pi\)
\(878\) 0 0
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 52.0000i 1.74994i 0.484178 + 0.874970i \(0.339119\pi\)
−0.484178 + 0.874970i \(0.660881\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 51.0000i − 1.71241i −0.516634 0.856206i \(-0.672815\pi\)
0.516634 0.856206i \(-0.327185\pi\)
\(888\) 0 0
\(889\) −22.0000 −0.737856
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 18.0000i − 0.602347i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1.00000i 0.0333890i
\(898\) 0 0
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) − 16.0000i − 0.532447i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 50.0000i 1.66022i 0.557598 + 0.830111i \(0.311723\pi\)
−0.557598 + 0.830111i \(0.688277\pi\)
\(908\) 0 0
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 42.0000i 1.38696i
\(918\) 0 0
\(919\) −38.0000 −1.25350 −0.626752 0.779219i \(-0.715616\pi\)
−0.626752 + 0.779219i \(0.715616\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) − 15.0000i − 0.493731i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4.00000i − 0.131377i
\(928\) 0 0
\(929\) −15.0000 −0.492134 −0.246067 0.969253i \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) 15.0000i 0.491078i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 28.0000i − 0.914720i −0.889282 0.457360i \(-0.848795\pi\)
0.889282 0.457360i \(-0.151205\pi\)
\(938\) 0 0
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) 3.00000i 0.0976934i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.00000i − 0.0974869i −0.998811 0.0487435i \(-0.984478\pi\)
0.998811 0.0487435i \(-0.0155217\pi\)
\(948\) 0 0
\(949\) −7.00000 −0.227230
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 6.00000i 0.194359i 0.995267 + 0.0971795i \(0.0309821\pi\)
−0.995267 + 0.0971795i \(0.969018\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) − 12.0000i − 0.386695i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 47.0000i 1.51142i 0.654907 + 0.755709i \(0.272708\pi\)
−0.654907 + 0.755709i \(0.727292\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 0 0
\(973\) − 22.0000i − 0.705288i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 48.0000i 1.53566i 0.640656 + 0.767828i \(0.278662\pi\)
−0.640656 + 0.767828i \(0.721338\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 32.0000 1.02168
\(982\) 0 0
\(983\) 18.0000i 0.574111i 0.957914 + 0.287055i \(0.0926764\pi\)
−0.957914 + 0.287055i \(0.907324\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 18.0000i 0.572946i
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 25.0000i 0.793351i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 62.0000i 1.96356i 0.190022 + 0.981780i \(0.439144\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) 0 0
\(999\) 40.0000 1.26554
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 2300.2.c.f.1749.1 2
5.2 odd 4 2300.2.a.c.1.1 1
5.3 odd 4 92.2.a.b.1.1 1
5.4 even 2 inner 2300.2.c.f.1749.2 2
15.8 even 4 828.2.a.b.1.1 1
20.3 even 4 368.2.a.b.1.1 1
20.7 even 4 9200.2.a.ba.1.1 1
35.13 even 4 4508.2.a.a.1.1 1
40.3 even 4 1472.2.a.j.1.1 1
40.13 odd 4 1472.2.a.c.1.1 1
60.23 odd 4 3312.2.a.g.1.1 1
115.68 even 4 2116.2.a.d.1.1 1
460.183 odd 4 8464.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
92.2.a.b.1.1 1 5.3 odd 4
368.2.a.b.1.1 1 20.3 even 4
828.2.a.b.1.1 1 15.8 even 4
1472.2.a.c.1.1 1 40.13 odd 4
1472.2.a.j.1.1 1 40.3 even 4
2116.2.a.d.1.1 1 115.68 even 4
2300.2.a.c.1.1 1 5.2 odd 4
2300.2.c.f.1749.1 2 1.1 even 1 trivial
2300.2.c.f.1749.2 2 5.4 even 2 inner
3312.2.a.g.1.1 1 60.23 odd 4
4508.2.a.a.1.1 1 35.13 even 4
8464.2.a.f.1.1 1 460.183 odd 4
9200.2.a.ba.1.1 1 20.7 even 4