Properties

Label 99.2.d.a.98.1
Level $99$
Weight $2$
Character 99.98
Analytic conductor $0.791$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,2,Mod(98,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.98");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.d (of order \(2\), degree \(1\), 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: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 98.1
Root \(0.517638i\) of defining polynomial
Character \(\chi\) \(=\) 99.98
Dual form 99.2.d.a.98.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{2} +1.00000 q^{4} -1.41421i q^{5} -2.44949i q^{7} +1.73205 q^{8} +O(q^{10})\) \(q-1.73205 q^{2} +1.00000 q^{4} -1.41421i q^{5} -2.44949i q^{7} +1.73205 q^{8} +2.44949i q^{10} +(1.73205 - 2.82843i) q^{11} -4.89898i q^{13} +4.24264i q^{14} -5.00000 q^{16} +7.34847i q^{19} -1.41421i q^{20} +(-3.00000 + 4.89898i) q^{22} +2.82843i q^{23} +3.00000 q^{25} +8.48528i q^{26} -2.44949i q^{28} -6.92820 q^{29} -4.00000 q^{31} +5.19615 q^{32} -3.46410 q^{35} +8.00000 q^{37} -12.7279i q^{38} -2.44949i q^{40} +6.92820 q^{41} -2.44949i q^{43} +(1.73205 - 2.82843i) q^{44} -4.89898i q^{46} +2.82843i q^{47} +1.00000 q^{49} -5.19615 q^{50} -4.89898i q^{52} -9.89949i q^{53} +(-4.00000 - 2.44949i) q^{55} -4.24264i q^{56} +12.0000 q^{58} +11.3137i q^{59} +4.89898i q^{61} +6.92820 q^{62} +1.00000 q^{64} -6.92820 q^{65} -4.00000 q^{67} +6.00000 q^{70} +2.82843i q^{71} -13.8564 q^{74} +7.34847i q^{76} +(-6.92820 - 4.24264i) q^{77} +12.2474i q^{79} +7.07107i q^{80} -12.0000 q^{82} +13.8564 q^{83} +4.24264i q^{86} +(3.00000 - 4.89898i) q^{88} +7.07107i q^{89} -12.0000 q^{91} +2.82843i q^{92} -4.89898i q^{94} +10.3923 q^{95} -10.0000 q^{97} -1.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 20 q^{16} - 12 q^{22} + 12 q^{25} - 16 q^{31} + 32 q^{37} + 4 q^{49} - 16 q^{55} + 48 q^{58} + 4 q^{64} - 16 q^{67} + 24 q^{70} - 48 q^{82} + 12 q^{88} - 48 q^{91} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.41421i 0.632456i −0.948683 0.316228i \(-0.897584\pi\)
0.948683 0.316228i \(-0.102416\pi\)
\(6\) 0 0
\(7\) 2.44949i 0.925820i −0.886405 0.462910i \(-0.846805\pi\)
0.886405 0.462910i \(-0.153195\pi\)
\(8\) 1.73205 0.612372
\(9\) 0 0
\(10\) 2.44949i 0.774597i
\(11\) 1.73205 2.82843i 0.522233 0.852803i
\(12\) 0 0
\(13\) 4.89898i 1.35873i −0.733799 0.679366i \(-0.762255\pi\)
0.733799 0.679366i \(-0.237745\pi\)
\(14\) 4.24264i 1.13389i
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 7.34847i 1.68585i 0.538028 + 0.842927i \(0.319170\pi\)
−0.538028 + 0.842927i \(0.680830\pi\)
\(20\) 1.41421i 0.316228i
\(21\) 0 0
\(22\) −3.00000 + 4.89898i −0.639602 + 1.04447i
\(23\) 2.82843i 0.589768i 0.955533 + 0.294884i \(0.0952810\pi\)
−0.955533 + 0.294884i \(0.904719\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 8.48528i 1.66410i
\(27\) 0 0
\(28\) 2.44949i 0.462910i
\(29\) −6.92820 −1.28654 −0.643268 0.765641i \(-0.722422\pi\)
−0.643268 + 0.765641i \(0.722422\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 5.19615 0.918559
\(33\) 0 0
\(34\) 0 0
\(35\) −3.46410 −0.585540
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 12.7279i 2.06474i
\(39\) 0 0
\(40\) 2.44949i 0.387298i
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) 2.44949i 0.373544i −0.982403 0.186772i \(-0.940197\pi\)
0.982403 0.186772i \(-0.0598025\pi\)
\(44\) 1.73205 2.82843i 0.261116 0.426401i
\(45\) 0 0
\(46\) 4.89898i 0.722315i
\(47\) 2.82843i 0.412568i 0.978492 + 0.206284i \(0.0661372\pi\)
−0.978492 + 0.206284i \(0.933863\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −5.19615 −0.734847
\(51\) 0 0
\(52\) 4.89898i 0.679366i
\(53\) 9.89949i 1.35980i −0.733305 0.679900i \(-0.762023\pi\)
0.733305 0.679900i \(-0.237977\pi\)
\(54\) 0 0
\(55\) −4.00000 2.44949i −0.539360 0.330289i
\(56\) 4.24264i 0.566947i
\(57\) 0 0
\(58\) 12.0000 1.57568
\(59\) 11.3137i 1.47292i 0.676481 + 0.736460i \(0.263504\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) 4.89898i 0.627250i 0.949547 + 0.313625i \(0.101543\pi\)
−0.949547 + 0.313625i \(0.898457\pi\)
\(62\) 6.92820 0.879883
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.92820 −0.859338
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 6.00000 0.717137
\(71\) 2.82843i 0.335673i 0.985815 + 0.167836i \(0.0536780\pi\)
−0.985815 + 0.167836i \(0.946322\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −13.8564 −1.61077
\(75\) 0 0
\(76\) 7.34847i 0.842927i
\(77\) −6.92820 4.24264i −0.789542 0.483494i
\(78\) 0 0
\(79\) 12.2474i 1.37795i 0.724787 + 0.688973i \(0.241938\pi\)
−0.724787 + 0.688973i \(0.758062\pi\)
\(80\) 7.07107i 0.790569i
\(81\) 0 0
\(82\) −12.0000 −1.32518
\(83\) 13.8564 1.52094 0.760469 0.649374i \(-0.224969\pi\)
0.760469 + 0.649374i \(0.224969\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.24264i 0.457496i
\(87\) 0 0
\(88\) 3.00000 4.89898i 0.319801 0.522233i
\(89\) 7.07107i 0.749532i 0.927119 + 0.374766i \(0.122277\pi\)
−0.927119 + 0.374766i \(0.877723\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 2.82843i 0.294884i
\(93\) 0 0
\(94\) 4.89898i 0.505291i
\(95\) 10.3923 1.06623
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −1.73205 −0.174964
\(99\) 0 0
\(100\) 3.00000 0.300000
\(101\) 13.8564 1.37876 0.689382 0.724398i \(-0.257882\pi\)
0.689382 + 0.724398i \(0.257882\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 8.48528i 0.832050i
\(105\) 0 0
\(106\) 17.1464i 1.66541i
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 14.6969i 1.40771i −0.710343 0.703856i \(-0.751460\pi\)
0.710343 0.703856i \(-0.248540\pi\)
\(110\) 6.92820 + 4.24264i 0.660578 + 0.404520i
\(111\) 0 0
\(112\) 12.2474i 1.15728i
\(113\) 1.41421i 0.133038i −0.997785 0.0665190i \(-0.978811\pi\)
0.997785 0.0665190i \(-0.0211893\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) −6.92820 −0.643268
\(117\) 0 0
\(118\) 19.5959i 1.80395i
\(119\) 0 0
\(120\) 0 0
\(121\) −5.00000 9.79796i −0.454545 0.890724i
\(122\) 8.48528i 0.768221i
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) 7.34847i 0.652071i 0.945357 + 0.326036i \(0.105713\pi\)
−0.945357 + 0.326036i \(0.894287\pi\)
\(128\) −12.1244 −1.07165
\(129\) 0 0
\(130\) 12.0000 1.05247
\(131\) −3.46410 −0.302660 −0.151330 0.988483i \(-0.548356\pi\)
−0.151330 + 0.988483i \(0.548356\pi\)
\(132\) 0 0
\(133\) 18.0000 1.56080
\(134\) 6.92820 0.598506
\(135\) 0 0
\(136\) 0 0
\(137\) 15.5563i 1.32907i 0.747258 + 0.664534i \(0.231370\pi\)
−0.747258 + 0.664534i \(0.768630\pi\)
\(138\) 0 0
\(139\) 2.44949i 0.207763i 0.994590 + 0.103882i \(0.0331263\pi\)
−0.994590 + 0.103882i \(0.966874\pi\)
\(140\) −3.46410 −0.292770
\(141\) 0 0
\(142\) 4.89898i 0.411113i
\(143\) −13.8564 8.48528i −1.15873 0.709575i
\(144\) 0 0
\(145\) 9.79796i 0.813676i
\(146\) 0 0
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −13.8564 −1.13516 −0.567581 0.823318i \(-0.692120\pi\)
−0.567581 + 0.823318i \(0.692120\pi\)
\(150\) 0 0
\(151\) 2.44949i 0.199337i −0.995021 0.0996683i \(-0.968222\pi\)
0.995021 0.0996683i \(-0.0317782\pi\)
\(152\) 12.7279i 1.03237i
\(153\) 0 0
\(154\) 12.0000 + 7.34847i 0.966988 + 0.592157i
\(155\) 5.65685i 0.454369i
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 21.2132i 1.68763i
\(159\) 0 0
\(160\) 7.34847i 0.580948i
\(161\) 6.92820 0.546019
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 6.92820 0.541002
\(165\) 0 0
\(166\) −24.0000 −1.86276
\(167\) −3.46410 −0.268060 −0.134030 0.990977i \(-0.542792\pi\)
−0.134030 + 0.990977i \(0.542792\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 0 0
\(172\) 2.44949i 0.186772i
\(173\) −6.92820 −0.526742 −0.263371 0.964695i \(-0.584834\pi\)
−0.263371 + 0.964695i \(0.584834\pi\)
\(174\) 0 0
\(175\) 7.34847i 0.555492i
\(176\) −8.66025 + 14.1421i −0.652791 + 1.06600i
\(177\) 0 0
\(178\) 12.2474i 0.917985i
\(179\) 22.6274i 1.69125i −0.533775 0.845626i \(-0.679227\pi\)
0.533775 0.845626i \(-0.320773\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 20.7846 1.54066
\(183\) 0 0
\(184\) 4.89898i 0.361158i
\(185\) 11.3137i 0.831800i
\(186\) 0 0
\(187\) 0 0
\(188\) 2.82843i 0.206284i
\(189\) 0 0
\(190\) −18.0000 −1.30586
\(191\) 2.82843i 0.204658i 0.994751 + 0.102329i \(0.0326294\pi\)
−0.994751 + 0.102329i \(0.967371\pi\)
\(192\) 0 0
\(193\) 9.79796i 0.705273i 0.935760 + 0.352636i \(0.114715\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) 17.3205 1.24354
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 5.19615 0.367423
\(201\) 0 0
\(202\) −24.0000 −1.68863
\(203\) 16.9706i 1.19110i
\(204\) 0 0
\(205\) 9.79796i 0.684319i
\(206\) 6.92820 0.482711
\(207\) 0 0
\(208\) 24.4949i 1.69842i
\(209\) 20.7846 + 12.7279i 1.43770 + 0.880409i
\(210\) 0 0
\(211\) 12.2474i 0.843149i −0.906794 0.421575i \(-0.861478\pi\)
0.906794 0.421575i \(-0.138522\pi\)
\(212\) 9.89949i 0.679900i
\(213\) 0 0
\(214\) 0 0
\(215\) −3.46410 −0.236250
\(216\) 0 0
\(217\) 9.79796i 0.665129i
\(218\) 25.4558i 1.72409i
\(219\) 0 0
\(220\) −4.00000 2.44949i −0.269680 0.165145i
\(221\) 0 0
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 12.7279i 0.850420i
\(225\) 0 0
\(226\) 2.44949i 0.162938i
\(227\) −27.7128 −1.83936 −0.919682 0.392664i \(-0.871554\pi\)
−0.919682 + 0.392664i \(0.871554\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −6.92820 −0.456832
\(231\) 0 0
\(232\) −12.0000 −0.787839
\(233\) −20.7846 −1.36165 −0.680823 0.732448i \(-0.738378\pi\)
−0.680823 + 0.732448i \(0.738378\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 11.3137i 0.736460i
\(237\) 0 0
\(238\) 0 0
\(239\) 17.3205 1.12037 0.560185 0.828367i \(-0.310730\pi\)
0.560185 + 0.828367i \(0.310730\pi\)
\(240\) 0 0
\(241\) 19.5959i 1.26228i 0.775667 + 0.631142i \(0.217413\pi\)
−0.775667 + 0.631142i \(0.782587\pi\)
\(242\) 8.66025 + 16.9706i 0.556702 + 1.09091i
\(243\) 0 0
\(244\) 4.89898i 0.313625i
\(245\) 1.41421i 0.0903508i
\(246\) 0 0
\(247\) 36.0000 2.29063
\(248\) −6.92820 −0.439941
\(249\) 0 0
\(250\) 19.5959i 1.23935i
\(251\) 5.65685i 0.357057i −0.983935 0.178529i \(-0.942866\pi\)
0.983935 0.178529i \(-0.0571337\pi\)
\(252\) 0 0
\(253\) 8.00000 + 4.89898i 0.502956 + 0.307996i
\(254\) 12.7279i 0.798621i
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 9.89949i 0.617514i −0.951141 0.308757i \(-0.900087\pi\)
0.951141 0.308757i \(-0.0999129\pi\)
\(258\) 0 0
\(259\) 19.5959i 1.21763i
\(260\) −6.92820 −0.429669
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) 3.46410 0.213606 0.106803 0.994280i \(-0.465939\pi\)
0.106803 + 0.994280i \(0.465939\pi\)
\(264\) 0 0
\(265\) −14.0000 −0.860013
\(266\) −31.1769 −1.91158
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 15.5563i 0.948487i 0.880394 + 0.474244i \(0.157278\pi\)
−0.880394 + 0.474244i \(0.842722\pi\)
\(270\) 0 0
\(271\) 7.34847i 0.446388i −0.974774 0.223194i \(-0.928352\pi\)
0.974774 0.223194i \(-0.0716483\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 26.9444i 1.62777i
\(275\) 5.19615 8.48528i 0.313340 0.511682i
\(276\) 0 0
\(277\) 24.4949i 1.47176i −0.677114 0.735878i \(-0.736770\pi\)
0.677114 0.735878i \(-0.263230\pi\)
\(278\) 4.24264i 0.254457i
\(279\) 0 0
\(280\) −6.00000 −0.358569
\(281\) 13.8564 0.826604 0.413302 0.910594i \(-0.364375\pi\)
0.413302 + 0.910594i \(0.364375\pi\)
\(282\) 0 0
\(283\) 2.44949i 0.145607i 0.997346 + 0.0728035i \(0.0231946\pi\)
−0.997346 + 0.0728035i \(0.976805\pi\)
\(284\) 2.82843i 0.167836i
\(285\) 0 0
\(286\) 24.0000 + 14.6969i 1.41915 + 0.869048i
\(287\) 16.9706i 1.00174i
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 16.9706i 0.996546i
\(291\) 0 0
\(292\) 0 0
\(293\) 6.92820 0.404750 0.202375 0.979308i \(-0.435134\pi\)
0.202375 + 0.979308i \(0.435134\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 13.8564 0.805387
\(297\) 0 0
\(298\) 24.0000 1.39028
\(299\) 13.8564 0.801337
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 4.24264i 0.244137i
\(303\) 0 0
\(304\) 36.7423i 2.10732i
\(305\) 6.92820 0.396708
\(306\) 0 0
\(307\) 7.34847i 0.419399i 0.977766 + 0.209700i \(0.0672486\pi\)
−0.977766 + 0.209700i \(0.932751\pi\)
\(308\) −6.92820 4.24264i −0.394771 0.241747i
\(309\) 0 0
\(310\) 9.79796i 0.556487i
\(311\) 14.1421i 0.801927i −0.916094 0.400963i \(-0.868675\pi\)
0.916094 0.400963i \(-0.131325\pi\)
\(312\) 0 0
\(313\) −16.0000 −0.904373 −0.452187 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) −24.2487 −1.36843
\(315\) 0 0
\(316\) 12.2474i 0.688973i
\(317\) 18.3848i 1.03259i −0.856410 0.516296i \(-0.827310\pi\)
0.856410 0.516296i \(-0.172690\pi\)
\(318\) 0 0
\(319\) −12.0000 + 19.5959i −0.671871 + 1.09716i
\(320\) 1.41421i 0.0790569i
\(321\) 0 0
\(322\) −12.0000 −0.668734
\(323\) 0 0
\(324\) 0 0
\(325\) 14.6969i 0.815239i
\(326\) −13.8564 −0.767435
\(327\) 0 0
\(328\) 12.0000 0.662589
\(329\) 6.92820 0.381964
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 13.8564 0.760469
\(333\) 0 0
\(334\) 6.00000 0.328305
\(335\) 5.65685i 0.309067i
\(336\) 0 0
\(337\) 9.79796i 0.533729i 0.963734 + 0.266864i \(0.0859876\pi\)
−0.963734 + 0.266864i \(0.914012\pi\)
\(338\) 19.0526 1.03632
\(339\) 0 0
\(340\) 0 0
\(341\) −6.92820 + 11.3137i −0.375183 + 0.612672i
\(342\) 0 0
\(343\) 19.5959i 1.05808i
\(344\) 4.24264i 0.228748i
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) −13.8564 −0.743851 −0.371925 0.928263i \(-0.621302\pi\)
−0.371925 + 0.928263i \(0.621302\pi\)
\(348\) 0 0
\(349\) 4.89898i 0.262236i 0.991367 + 0.131118i \(0.0418567\pi\)
−0.991367 + 0.131118i \(0.958143\pi\)
\(350\) 12.7279i 0.680336i
\(351\) 0 0
\(352\) 9.00000 14.6969i 0.479702 0.783349i
\(353\) 32.5269i 1.73123i 0.500708 + 0.865616i \(0.333073\pi\)
−0.500708 + 0.865616i \(0.666927\pi\)
\(354\) 0 0
\(355\) 4.00000 0.212298
\(356\) 7.07107i 0.374766i
\(357\) 0 0
\(358\) 39.1918i 2.07135i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −35.0000 −1.84211
\(362\) −13.8564 −0.728277
\(363\) 0 0
\(364\) −12.0000 −0.628971
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 14.1421i 0.737210i
\(369\) 0 0
\(370\) 19.5959i 1.01874i
\(371\) −24.2487 −1.25893
\(372\) 0 0
\(373\) 24.4949i 1.26830i 0.773211 + 0.634149i \(0.218649\pi\)
−0.773211 + 0.634149i \(0.781351\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 4.89898i 0.252646i
\(377\) 33.9411i 1.74806i
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 10.3923 0.533114
\(381\) 0 0
\(382\) 4.89898i 0.250654i
\(383\) 36.7696i 1.87884i 0.342773 + 0.939418i \(0.388634\pi\)
−0.342773 + 0.939418i \(0.611366\pi\)
\(384\) 0 0
\(385\) −6.00000 + 9.79796i −0.305788 + 0.499350i
\(386\) 16.9706i 0.863779i
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) 9.89949i 0.501924i −0.967997 0.250962i \(-0.919253\pi\)
0.967997 0.250962i \(-0.0807470\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.73205 0.0874818
\(393\) 0 0
\(394\) 0 0
\(395\) 17.3205 0.871489
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) −13.8564 −0.694559
\(399\) 0 0
\(400\) −15.0000 −0.750000
\(401\) 1.41421i 0.0706225i −0.999376 0.0353112i \(-0.988758\pi\)
0.999376 0.0353112i \(-0.0112422\pi\)
\(402\) 0 0
\(403\) 19.5959i 0.976142i
\(404\) 13.8564 0.689382
\(405\) 0 0
\(406\) 29.3939i 1.45879i
\(407\) 13.8564 22.6274i 0.686837 1.12160i
\(408\) 0 0
\(409\) 39.1918i 1.93791i 0.247234 + 0.968956i \(0.420478\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 16.9706i 0.838116i
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 27.7128 1.36366
\(414\) 0 0
\(415\) 19.5959i 0.961926i
\(416\) 25.4558i 1.24808i
\(417\) 0 0
\(418\) −36.0000 22.0454i −1.76082 1.07828i
\(419\) 11.3137i 0.552711i 0.961056 + 0.276355i \(0.0891267\pi\)
−0.961056 + 0.276355i \(0.910873\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 21.2132i 1.03264i
\(423\) 0 0
\(424\) 17.1464i 0.832704i
\(425\) 0 0
\(426\) 0 0
\(427\) 12.0000 0.580721
\(428\) 0 0
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) −10.3923 −0.500580 −0.250290 0.968171i \(-0.580526\pi\)
−0.250290 + 0.968171i \(0.580526\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 16.9706i 0.814613i
\(435\) 0 0
\(436\) 14.6969i 0.703856i
\(437\) −20.7846 −0.994263
\(438\) 0 0
\(439\) 2.44949i 0.116908i −0.998290 0.0584539i \(-0.981383\pi\)
0.998290 0.0584539i \(-0.0186171\pi\)
\(440\) −6.92820 4.24264i −0.330289 0.202260i
\(441\) 0 0
\(442\) 0 0
\(443\) 22.6274i 1.07506i −0.843244 0.537531i \(-0.819357\pi\)
0.843244 0.537531i \(-0.180643\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 27.7128 1.31224
\(447\) 0 0
\(448\) 2.44949i 0.115728i
\(449\) 9.89949i 0.467186i −0.972334 0.233593i \(-0.924952\pi\)
0.972334 0.233593i \(-0.0750483\pi\)
\(450\) 0 0
\(451\) 12.0000 19.5959i 0.565058 0.922736i
\(452\) 1.41421i 0.0665190i
\(453\) 0 0
\(454\) 48.0000 2.25275
\(455\) 16.9706i 0.795592i
\(456\) 0 0
\(457\) 9.79796i 0.458329i −0.973388 0.229165i \(-0.926401\pi\)
0.973388 0.229165i \(-0.0735994\pi\)
\(458\) 17.3205 0.809334
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) −6.92820 −0.322679 −0.161339 0.986899i \(-0.551581\pi\)
−0.161339 + 0.986899i \(0.551581\pi\)
\(462\) 0 0
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) 34.6410 1.60817
\(465\) 0 0
\(466\) 36.0000 1.66767
\(467\) 22.6274i 1.04707i −0.852004 0.523536i \(-0.824613\pi\)
0.852004 0.523536i \(-0.175387\pi\)
\(468\) 0 0
\(469\) 9.79796i 0.452428i
\(470\) −6.92820 −0.319574
\(471\) 0 0
\(472\) 19.5959i 0.901975i
\(473\) −6.92820 4.24264i −0.318559 0.195077i
\(474\) 0 0
\(475\) 22.0454i 1.01151i
\(476\) 0 0
\(477\) 0 0
\(478\) −30.0000 −1.37217
\(479\) −38.1051 −1.74107 −0.870534 0.492109i \(-0.836226\pi\)
−0.870534 + 0.492109i \(0.836226\pi\)
\(480\) 0 0
\(481\) 39.1918i 1.78699i
\(482\) 33.9411i 1.54598i
\(483\) 0 0
\(484\) −5.00000 9.79796i −0.227273 0.445362i
\(485\) 14.1421i 0.642161i
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 8.48528i 0.384111i
\(489\) 0 0
\(490\) 2.44949i 0.110657i
\(491\) 27.7128 1.25066 0.625331 0.780360i \(-0.284964\pi\)
0.625331 + 0.780360i \(0.284964\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −62.3538 −2.80543
\(495\) 0 0
\(496\) 20.0000 0.898027
\(497\) 6.92820 0.310772
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 11.3137i 0.505964i
\(501\) 0 0
\(502\) 9.79796i 0.437304i
\(503\) −31.1769 −1.39011 −0.695055 0.718957i \(-0.744620\pi\)
−0.695055 + 0.718957i \(0.744620\pi\)
\(504\) 0 0
\(505\) 19.5959i 0.872007i
\(506\) −13.8564 8.48528i −0.615992 0.377217i
\(507\) 0 0
\(508\) 7.34847i 0.326036i
\(509\) 15.5563i 0.689523i 0.938690 + 0.344762i \(0.112040\pi\)
−0.938690 + 0.344762i \(0.887960\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −8.66025 −0.382733
\(513\) 0 0
\(514\) 17.1464i 0.756297i
\(515\) 5.65685i 0.249271i
\(516\) 0 0
\(517\) 8.00000 + 4.89898i 0.351840 + 0.215457i
\(518\) 33.9411i 1.49129i
\(519\) 0 0
\(520\) −12.0000 −0.526235
\(521\) 15.5563i 0.681536i 0.940147 + 0.340768i \(0.110687\pi\)
−0.940147 + 0.340768i \(0.889313\pi\)
\(522\) 0 0
\(523\) 22.0454i 0.963978i 0.876177 + 0.481989i \(0.160086\pi\)
−0.876177 + 0.481989i \(0.839914\pi\)
\(524\) −3.46410 −0.151330
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) 0 0
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 24.2487 1.05330
\(531\) 0 0
\(532\) 18.0000 0.780399
\(533\) 33.9411i 1.47015i
\(534\) 0 0
\(535\) 0 0
\(536\) −6.92820 −0.299253
\(537\) 0 0
\(538\) 26.9444i 1.16166i
\(539\) 1.73205 2.82843i 0.0746047 0.121829i
\(540\) 0 0
\(541\) 14.6969i 0.631871i 0.948781 + 0.315935i \(0.102318\pi\)
−0.948781 + 0.315935i \(0.897682\pi\)
\(542\) 12.7279i 0.546711i
\(543\) 0 0
\(544\) 0 0
\(545\) −20.7846 −0.890315
\(546\) 0 0
\(547\) 31.8434i 1.36152i −0.732505 0.680762i \(-0.761649\pi\)
0.732505 0.680762i \(-0.238351\pi\)
\(548\) 15.5563i 0.664534i
\(549\) 0 0
\(550\) −9.00000 + 14.6969i −0.383761 + 0.626680i
\(551\) 50.9117i 2.16891i
\(552\) 0 0
\(553\) 30.0000 1.27573
\(554\) 42.4264i 1.80253i
\(555\) 0 0
\(556\) 2.44949i 0.103882i
\(557\) 20.7846 0.880672 0.440336 0.897833i \(-0.354859\pi\)
0.440336 + 0.897833i \(0.354859\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 17.3205 0.731925
\(561\) 0 0
\(562\) −24.0000 −1.01238
\(563\) 17.3205 0.729972 0.364986 0.931013i \(-0.381074\pi\)
0.364986 + 0.931013i \(0.381074\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 4.24264i 0.178331i
\(567\) 0 0
\(568\) 4.89898i 0.205557i
\(569\) 13.8564 0.580891 0.290445 0.956892i \(-0.406197\pi\)
0.290445 + 0.956892i \(0.406197\pi\)
\(570\) 0 0
\(571\) 2.44949i 0.102508i 0.998686 + 0.0512540i \(0.0163218\pi\)
−0.998686 + 0.0512540i \(0.983678\pi\)
\(572\) −13.8564 8.48528i −0.579365 0.354787i
\(573\) 0 0
\(574\) 29.3939i 1.22688i
\(575\) 8.48528i 0.353861i
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 29.4449 1.22474
\(579\) 0 0
\(580\) 9.79796i 0.406838i
\(581\) 33.9411i 1.40812i
\(582\) 0 0
\(583\) −28.0000 17.1464i −1.15964 0.710132i
\(584\) 0 0
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) 28.2843i 1.16742i 0.811963 + 0.583708i \(0.198399\pi\)
−0.811963 + 0.583708i \(0.801601\pi\)
\(588\) 0 0
\(589\) 29.3939i 1.21115i
\(590\) −27.7128 −1.14092
\(591\) 0 0
\(592\) −40.0000 −1.64399
\(593\) 41.5692 1.70704 0.853522 0.521057i \(-0.174462\pi\)
0.853522 + 0.521057i \(0.174462\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −13.8564 −0.567581
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) 2.82843i 0.115566i 0.998329 + 0.0577832i \(0.0184032\pi\)
−0.998329 + 0.0577832i \(0.981597\pi\)
\(600\) 0 0
\(601\) 19.5959i 0.799334i 0.916660 + 0.399667i \(0.130874\pi\)
−0.916660 + 0.399667i \(0.869126\pi\)
\(602\) 10.3923 0.423559
\(603\) 0 0
\(604\) 2.44949i 0.0996683i
\(605\) −13.8564 + 7.07107i −0.563343 + 0.287480i
\(606\) 0 0
\(607\) 12.2474i 0.497109i −0.968618 0.248554i \(-0.920045\pi\)
0.968618 0.248554i \(-0.0799554\pi\)
\(608\) 38.1838i 1.54856i
\(609\) 0 0
\(610\) −12.0000 −0.485866
\(611\) 13.8564 0.560570
\(612\) 0 0
\(613\) 14.6969i 0.593604i −0.954939 0.296802i \(-0.904080\pi\)
0.954939 0.296802i \(-0.0959201\pi\)
\(614\) 12.7279i 0.513657i
\(615\) 0 0
\(616\) −12.0000 7.34847i −0.483494 0.296078i
\(617\) 26.8701i 1.08175i −0.841104 0.540874i \(-0.818094\pi\)
0.841104 0.540874i \(-0.181906\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 5.65685i 0.227185i
\(621\) 0 0
\(622\) 24.4949i 0.982156i
\(623\) 17.3205 0.693932
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 27.7128 1.10763
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 0 0
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 21.2132i 0.843816i
\(633\) 0 0
\(634\) 31.8434i 1.26466i
\(635\) 10.3923 0.412406
\(636\) 0 0
\(637\) 4.89898i 0.194105i
\(638\) 20.7846 33.9411i 0.822871 1.34374i
\(639\) 0 0
\(640\) 17.1464i 0.677772i
\(641\) 32.5269i 1.28474i 0.766396 + 0.642368i \(0.222048\pi\)
−0.766396 + 0.642368i \(0.777952\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 6.92820 0.273009
\(645\) 0 0
\(646\) 0 0
\(647\) 19.7990i 0.778379i 0.921158 + 0.389189i \(0.127245\pi\)
−0.921158 + 0.389189i \(0.872755\pi\)
\(648\) 0 0
\(649\) 32.0000 + 19.5959i 1.25611 + 0.769207i
\(650\) 25.4558i 0.998460i
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 26.8701i 1.05151i −0.850637 0.525753i \(-0.823784\pi\)
0.850637 0.525753i \(-0.176216\pi\)
\(654\) 0 0
\(655\) 4.89898i 0.191419i
\(656\) −34.6410 −1.35250
\(657\) 0 0
\(658\) −12.0000 −0.467809
\(659\) −38.1051 −1.48436 −0.742182 0.670198i \(-0.766209\pi\)
−0.742182 + 0.670198i \(0.766209\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) −13.8564 −0.538545
\(663\) 0 0
\(664\) 24.0000 0.931381
\(665\) 25.4558i 0.987135i
\(666\) 0 0
\(667\) 19.5959i 0.758757i
\(668\) −3.46410 −0.134030
\(669\) 0 0
\(670\) 9.79796i 0.378528i
\(671\) 13.8564 + 8.48528i 0.534921 + 0.327571i
\(672\) 0 0
\(673\) 39.1918i 1.51073i −0.655302 0.755367i \(-0.727459\pi\)
0.655302 0.755367i \(-0.272541\pi\)
\(674\) 16.9706i 0.653682i
\(675\) 0 0
\(676\) −11.0000 −0.423077
\(677\) −27.7128 −1.06509 −0.532545 0.846402i \(-0.678764\pi\)
−0.532545 + 0.846402i \(0.678764\pi\)
\(678\) 0 0
\(679\) 24.4949i 0.940028i
\(680\) 0 0
\(681\) 0 0
\(682\) 12.0000 19.5959i 0.459504 0.750366i
\(683\) 28.2843i 1.08227i 0.840937 + 0.541134i \(0.182005\pi\)
−0.840937 + 0.541134i \(0.817995\pi\)
\(684\) 0 0
\(685\) 22.0000 0.840577
\(686\) 33.9411i 1.29588i
\(687\) 0 0
\(688\) 12.2474i 0.466930i
\(689\) −48.4974 −1.84760
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) −6.92820 −0.263371
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) 3.46410 0.131401
\(696\) 0 0
\(697\) 0 0
\(698\) 8.48528i 0.321173i
\(699\) 0 0
\(700\) 7.34847i 0.277746i
\(701\) −20.7846 −0.785024 −0.392512 0.919747i \(-0.628394\pi\)
−0.392512 + 0.919747i \(0.628394\pi\)
\(702\) 0 0
\(703\) 58.7878i 2.21722i
\(704\) 1.73205 2.82843i 0.0652791 0.106600i
\(705\) 0 0
\(706\) 56.3383i 2.12032i
\(707\) 33.9411i 1.27649i
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −6.92820 −0.260011
\(711\) 0 0
\(712\) 12.2474i 0.458993i
\(713\) 11.3137i 0.423702i
\(714\) 0 0
\(715\) −12.0000 + 19.5959i −0.448775 + 0.732846i
\(716\) 22.6274i 0.845626i
\(717\) 0 0
\(718\) 0 0
\(719\) 2.82843i 0.105483i 0.998608 + 0.0527413i \(0.0167959\pi\)
−0.998608 + 0.0527413i \(0.983204\pi\)
\(720\) 0 0
\(721\) 9.79796i 0.364895i
\(722\) 60.6218 2.25611
\(723\) 0 0
\(724\) 8.00000 0.297318
\(725\) −20.7846 −0.771921
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −20.7846 −0.770329
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 34.2929i 1.26664i −0.773892 0.633318i \(-0.781693\pi\)
0.773892 0.633318i \(-0.218307\pi\)
\(734\) −13.8564 −0.511449
\(735\) 0 0
\(736\) 14.6969i 0.541736i
\(737\) −6.92820 + 11.3137i −0.255204 + 0.416746i
\(738\) 0 0
\(739\) 7.34847i 0.270318i −0.990824 0.135159i \(-0.956846\pi\)
0.990824 0.135159i \(-0.0431545\pi\)
\(740\) 11.3137i 0.415900i
\(741\) 0 0
\(742\) 42.0000 1.54187
\(743\) 27.7128 1.01668 0.508342 0.861155i \(-0.330258\pi\)
0.508342 + 0.861155i \(0.330258\pi\)
\(744\) 0 0
\(745\) 19.5959i 0.717939i
\(746\) 42.4264i 1.55334i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 14.1421i 0.515711i
\(753\) 0 0
\(754\) 58.7878i 2.14092i
\(755\) −3.46410 −0.126072
\(756\) 0 0
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 27.7128 1.00657
\(759\) 0 0
\(760\) 18.0000 0.652929
\(761\) −13.8564 −0.502294 −0.251147 0.967949i \(-0.580808\pi\)
−0.251147 + 0.967949i \(0.580808\pi\)
\(762\) 0 0
\(763\) −36.0000 −1.30329
\(764\) 2.82843i 0.102329i
\(765\) 0 0
\(766\) 63.6867i 2.30110i
\(767\) 55.4256 2.00130
\(768\) 0 0
\(769\) 19.5959i 0.706647i −0.935501 0.353323i \(-0.885052\pi\)
0.935501 0.353323i \(-0.114948\pi\)
\(770\) 10.3923 16.9706i 0.374513 0.611577i
\(771\) 0 0
\(772\) 9.79796i 0.352636i
\(773\) 32.5269i 1.16991i 0.811065 + 0.584956i \(0.198888\pi\)
−0.811065 + 0.584956i \(0.801112\pi\)
\(774\) 0 0
\(775\) −12.0000 −0.431053
\(776\) −17.3205 −0.621770
\(777\) 0 0
\(778\) 17.1464i 0.614729i
\(779\) 50.9117i 1.82410i
\(780\) 0 0
\(781\) 8.00000 + 4.89898i 0.286263 + 0.175299i
\(782\) 0 0
\(783\) 0 0
\(784\) −5.00000 −0.178571
\(785\) 19.7990i 0.706656i
\(786\) 0 0
\(787\) 12.2474i 0.436574i −0.975885 0.218287i \(-0.929953\pi\)
0.975885 0.218287i \(-0.0700470\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) −30.0000 −1.06735
\(791\) −3.46410 −0.123169
\(792\) 0 0
\(793\) 24.0000 0.852265
\(794\) −13.8564 −0.491745
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 26.8701i 0.951786i −0.879503 0.475893i \(-0.842125\pi\)
0.879503 0.475893i \(-0.157875\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 15.5885 0.551135
\(801\) 0 0
\(802\) 2.44949i 0.0864945i
\(803\) 0 0
\(804\) 0 0
\(805\) 9.79796i 0.345333i
\(806\) 33.9411i 1.19553i
\(807\) 0 0
\(808\) 24.0000 0.844317
\(809\) 20.7846 0.730748 0.365374 0.930861i \(-0.380941\pi\)
0.365374 + 0.930861i \(0.380941\pi\)
\(810\) 0 0
\(811\) 22.0454i 0.774119i 0.922055 + 0.387059i \(0.126509\pi\)
−0.922055 + 0.387059i \(0.873491\pi\)
\(812\) 16.9706i 0.595550i
\(813\) 0 0
\(814\) −24.0000 + 39.1918i −0.841200 + 1.37367i
\(815\) 11.3137i 0.396302i
\(816\) 0 0
\(817\) 18.0000 0.629740
\(818\) 67.8823i 2.37345i
\(819\) 0 0
\(820\) 9.79796i 0.342160i
\(821\) 13.8564 0.483592 0.241796 0.970327i \(-0.422264\pi\)
0.241796 + 0.970327i \(0.422264\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −6.92820 −0.241355
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) −41.5692 −1.44550 −0.722752 0.691108i \(-0.757123\pi\)
−0.722752 + 0.691108i \(0.757123\pi\)
\(828\) 0 0
\(829\) −46.0000 −1.59765 −0.798823 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 33.9411i 1.17811i
\(831\) 0 0
\(832\) 4.89898i 0.169842i
\(833\) 0 0
\(834\) 0 0
\(835\) 4.89898i 0.169536i
\(836\) 20.7846 + 12.7279i 0.718851 + 0.440204i
\(837\) 0 0
\(838\) 19.5959i 0.676930i
\(839\) 19.7990i 0.683537i 0.939784 + 0.341769i \(0.111026\pi\)
−0.939784 + 0.341769i \(0.888974\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) −13.8564 −0.477523
\(843\) 0 0
\(844\) 12.2474i 0.421575i
\(845\) 15.5563i 0.535155i
\(846\) 0 0
\(847\) −24.0000 + 12.2474i −0.824650 + 0.420827i
\(848\) 49.4975i 1.69975i
\(849\) 0 0
\(850\) 0 0
\(851\) 22.6274i 0.775658i
\(852\) 0 0
\(853\) 34.2929i 1.17417i 0.809527 + 0.587083i \(0.199724\pi\)
−0.809527 + 0.587083i \(0.800276\pi\)
\(854\) −20.7846 −0.711235
\(855\) 0 0
\(856\) 0 0
\(857\) 34.6410 1.18331 0.591657 0.806190i \(-0.298474\pi\)
0.591657 + 0.806190i \(0.298474\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) −3.46410 −0.118125
\(861\) 0 0
\(862\) 18.0000 0.613082
\(863\) 19.7990i 0.673965i 0.941511 + 0.336983i \(0.109406\pi\)
−0.941511 + 0.336983i \(0.890594\pi\)
\(864\) 0 0
\(865\) 9.79796i 0.333141i
\(866\) 27.7128 0.941720
\(867\) 0 0
\(868\) 9.79796i 0.332564i
\(869\) 34.6410 + 21.2132i 1.17512 + 0.719609i
\(870\) 0 0
\(871\) 19.5959i 0.663982i
\(872\) 25.4558i 0.862044i
\(873\) 0 0
\(874\) 36.0000 1.21772
\(875\) −27.7128 −0.936864
\(876\) 0 0
\(877\) 53.8888i 1.81969i 0.414943 + 0.909847i \(0.363801\pi\)
−0.414943 + 0.909847i \(0.636199\pi\)
\(878\) 4.24264i 0.143182i
\(879\) 0 0
\(880\) 20.0000 + 12.2474i 0.674200 + 0.412861i
\(881\) 9.89949i 0.333522i −0.985997 0.166761i \(-0.946669\pi\)
0.985997 0.166761i \(-0.0533309\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 39.1918i 1.31668i
\(887\) −55.4256 −1.86101 −0.930505 0.366279i \(-0.880632\pi\)
−0.930505 + 0.366279i \(0.880632\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) −17.3205 −0.580585
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) −20.7846 −0.695530
\(894\) 0 0
\(895\) −32.0000 −1.06964
\(896\) 29.6985i 0.992157i
\(897\) 0 0
\(898\) 17.1464i 0.572184i
\(899\) 27.7128 0.924274
\(900\) 0 0
\(901\) 0 0
\(902\) −20.7846 + 33.9411i −0.692052 + 1.13012i
\(903\) 0 0
\(904\) 2.44949i 0.0814688i
\(905\) 11.3137i 0.376080i
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) −27.7128 −0.919682
\(909\) 0 0
\(910\) 29.3939i 0.974398i
\(911\) 53.7401i 1.78049i 0.455483 + 0.890245i \(0.349467\pi\)
−0.455483 + 0.890245i \(0.650533\pi\)
\(912\) 0 0
\(913\) 24.0000 39.1918i 0.794284 1.29706i
\(914\) 16.9706i 0.561336i
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 8.48528i 0.280209i
\(918\) 0 0
\(919\) 36.7423i 1.21202i −0.795458 0.606009i \(-0.792770\pi\)
0.795458 0.606009i \(-0.207230\pi\)
\(920\) 6.92820 0.228416
\(921\) 0 0
\(922\) 12.0000 0.395199
\(923\) 13.8564 0.456089
\(924\) 0 0
\(925\) 24.0000 0.789115
\(926\) 48.4974 1.59372
\(927\) 0 0
\(928\) −36.0000 −1.18176
\(929\) 9.89949i 0.324792i −0.986726 0.162396i \(-0.948078\pi\)
0.986726 0.162396i \(-0.0519222\pi\)
\(930\) 0 0
\(931\) 7.34847i 0.240836i
\(932\) −20.7846 −0.680823
\(933\) 0 0
\(934\) 39.1918i 1.28240i
\(935\) 0 0
\(936\) 0 0
\(937\) 29.3939i 0.960256i −0.877198 0.480128i \(-0.840590\pi\)
0.877198 0.480128i \(-0.159410\pi\)
\(938\) 16.9706i 0.554109i
\(939\) 0 0
\(940\) 4.00000 0.130466
\(941\) 27.7128 0.903412 0.451706 0.892167i \(-0.350816\pi\)
0.451706 + 0.892167i \(0.350816\pi\)
\(942\) 0 0
\(943\) 19.5959i 0.638131i
\(944\) 56.5685i 1.84115i
\(945\) 0 0
\(946\) 12.0000 + 7.34847i 0.390154 + 0.238919i
\(947\) 22.6274i 0.735292i −0.929966 0.367646i \(-0.880164\pi\)
0.929966 0.367646i \(-0.119836\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 38.1838i 1.23884i
\(951\) 0 0
\(952\) 0 0
\(953\) −20.7846 −0.673280 −0.336640 0.941634i \(-0.609290\pi\)
−0.336640 + 0.941634i \(0.609290\pi\)
\(954\) 0 0
\(955\) 4.00000 0.129437
\(956\) 17.3205 0.560185
\(957\) 0 0
\(958\) 66.0000 2.13236
\(959\) 38.1051 1.23048
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 67.8823i 2.18861i
\(963\) 0 0
\(964\) 19.5959i 0.631142i
\(965\) 13.8564 0.446054
\(966\) 0 0
\(967\) 41.6413i 1.33909i −0.742769 0.669547i \(-0.766488\pi\)
0.742769 0.669547i \(-0.233512\pi\)
\(968\) −8.66025 16.9706i −0.278351 0.545455i
\(969\) 0 0
\(970\) 24.4949i 0.786484i
\(971\) 56.5685i 1.81537i −0.419651 0.907685i \(-0.637848\pi\)
0.419651 0.907685i \(-0.362152\pi\)
\(972\) 0 0
\(973\) 6.00000 0.192351
\(974\) −13.8564 −0.443988
\(975\) 0 0
\(976\) 24.4949i 0.784063i
\(977\) 35.3553i 1.13112i −0.824708 0.565559i \(-0.808661\pi\)
0.824708 0.565559i \(-0.191339\pi\)
\(978\) 0 0
\(979\) 20.0000 + 12.2474i 0.639203 + 0.391430i
\(980\) 1.41421i 0.0451754i
\(981\) 0 0
\(982\) −48.0000 −1.53174
\(983\) 48.0833i 1.53362i −0.641875 0.766809i \(-0.721843\pi\)
0.641875 0.766809i \(-0.278157\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 36.0000 1.14531
\(989\) 6.92820 0.220304
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) −20.7846 −0.659912
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 11.3137i 0.358669i
\(996\) 0 0
\(997\) 24.4949i 0.775761i −0.921710 0.387881i \(-0.873207\pi\)
0.921710 0.387881i \(-0.126793\pi\)
\(998\) −55.4256 −1.75447
\(999\) 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 99.2.d.a.98.1 4
3.2 odd 2 inner 99.2.d.a.98.4 yes 4
4.3 odd 2 1584.2.b.e.593.2 4
5.2 odd 4 2475.2.d.a.2474.4 8
5.3 odd 4 2475.2.d.a.2474.6 8
5.4 even 2 2475.2.f.e.2276.4 4
8.3 odd 2 6336.2.b.t.2177.4 4
8.5 even 2 6336.2.b.s.2177.3 4
9.2 odd 6 891.2.g.c.296.1 8
9.4 even 3 891.2.g.c.593.3 8
9.5 odd 6 891.2.g.c.593.2 8
9.7 even 3 891.2.g.c.296.4 8
11.10 odd 2 inner 99.2.d.a.98.3 yes 4
12.11 even 2 1584.2.b.e.593.4 4
15.2 even 4 2475.2.d.a.2474.7 8
15.8 even 4 2475.2.d.a.2474.1 8
15.14 odd 2 2475.2.f.e.2276.2 4
24.5 odd 2 6336.2.b.s.2177.1 4
24.11 even 2 6336.2.b.t.2177.2 4
33.32 even 2 inner 99.2.d.a.98.2 yes 4
44.43 even 2 1584.2.b.e.593.1 4
55.32 even 4 2475.2.d.a.2474.5 8
55.43 even 4 2475.2.d.a.2474.3 8
55.54 odd 2 2475.2.f.e.2276.1 4
88.21 odd 2 6336.2.b.s.2177.4 4
88.43 even 2 6336.2.b.t.2177.3 4
99.32 even 6 891.2.g.c.593.4 8
99.43 odd 6 891.2.g.c.296.2 8
99.65 even 6 891.2.g.c.296.3 8
99.76 odd 6 891.2.g.c.593.1 8
132.131 odd 2 1584.2.b.e.593.3 4
165.32 odd 4 2475.2.d.a.2474.2 8
165.98 odd 4 2475.2.d.a.2474.8 8
165.164 even 2 2475.2.f.e.2276.3 4
264.131 odd 2 6336.2.b.t.2177.1 4
264.197 even 2 6336.2.b.s.2177.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.d.a.98.1 4 1.1 even 1 trivial
99.2.d.a.98.2 yes 4 33.32 even 2 inner
99.2.d.a.98.3 yes 4 11.10 odd 2 inner
99.2.d.a.98.4 yes 4 3.2 odd 2 inner
891.2.g.c.296.1 8 9.2 odd 6
891.2.g.c.296.2 8 99.43 odd 6
891.2.g.c.296.3 8 99.65 even 6
891.2.g.c.296.4 8 9.7 even 3
891.2.g.c.593.1 8 99.76 odd 6
891.2.g.c.593.2 8 9.5 odd 6
891.2.g.c.593.3 8 9.4 even 3
891.2.g.c.593.4 8 99.32 even 6
1584.2.b.e.593.1 4 44.43 even 2
1584.2.b.e.593.2 4 4.3 odd 2
1584.2.b.e.593.3 4 132.131 odd 2
1584.2.b.e.593.4 4 12.11 even 2
2475.2.d.a.2474.1 8 15.8 even 4
2475.2.d.a.2474.2 8 165.32 odd 4
2475.2.d.a.2474.3 8 55.43 even 4
2475.2.d.a.2474.4 8 5.2 odd 4
2475.2.d.a.2474.5 8 55.32 even 4
2475.2.d.a.2474.6 8 5.3 odd 4
2475.2.d.a.2474.7 8 15.2 even 4
2475.2.d.a.2474.8 8 165.98 odd 4
2475.2.f.e.2276.1 4 55.54 odd 2
2475.2.f.e.2276.2 4 15.14 odd 2
2475.2.f.e.2276.3 4 165.164 even 2
2475.2.f.e.2276.4 4 5.4 even 2
6336.2.b.s.2177.1 4 24.5 odd 2
6336.2.b.s.2177.2 4 264.197 even 2
6336.2.b.s.2177.3 4 8.5 even 2
6336.2.b.s.2177.4 4 88.21 odd 2
6336.2.b.t.2177.1 4 264.131 odd 2
6336.2.b.t.2177.2 4 24.11 even 2
6336.2.b.t.2177.3 4 88.43 even 2
6336.2.b.t.2177.4 4 8.3 odd 2