Properties

Label 966.2.a.m.1.2
Level $966$
Weight $2$
Character 966.1
Self dual yes
Analytic conductor $7.714$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(1,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, 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("966.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.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: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 966.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.70156 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.70156 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.70156 q^{10} +1.00000 q^{12} -0.701562 q^{13} +1.00000 q^{14} +2.70156 q^{15} +1.00000 q^{16} -1.00000 q^{18} +7.40312 q^{19} +2.70156 q^{20} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{24} +2.29844 q^{25} +0.701562 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.70156 q^{29} -2.70156 q^{30} +6.00000 q^{31} -1.00000 q^{32} -2.70156 q^{35} +1.00000 q^{36} -2.70156 q^{37} -7.40312 q^{38} -0.701562 q^{39} -2.70156 q^{40} +1.29844 q^{41} +1.00000 q^{42} -0.701562 q^{43} +2.70156 q^{45} +1.00000 q^{46} -2.70156 q^{47} +1.00000 q^{48} +1.00000 q^{49} -2.29844 q^{50} -0.701562 q^{52} -3.40312 q^{53} -1.00000 q^{54} +1.00000 q^{56} +7.40312 q^{57} -6.70156 q^{58} -13.4031 q^{59} +2.70156 q^{60} +10.0000 q^{61} -6.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.89531 q^{65} +4.00000 q^{67} -1.00000 q^{69} +2.70156 q^{70} -5.40312 q^{71} -1.00000 q^{72} +7.40312 q^{73} +2.70156 q^{74} +2.29844 q^{75} +7.40312 q^{76} +0.701562 q^{78} +2.70156 q^{80} +1.00000 q^{81} -1.29844 q^{82} -15.4031 q^{83} -1.00000 q^{84} +0.701562 q^{86} +6.70156 q^{87} -2.59688 q^{89} -2.70156 q^{90} +0.701562 q^{91} -1.00000 q^{92} +6.00000 q^{93} +2.70156 q^{94} +20.0000 q^{95} -1.00000 q^{96} +18.1047 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} + q^{10} + 2 q^{12} + 5 q^{13} + 2 q^{14} - q^{15} + 2 q^{16} - 2 q^{18} + 2 q^{19} - q^{20} - 2 q^{21} - 2 q^{23} - 2 q^{24} + 11 q^{25} - 5 q^{26} + 2 q^{27} - 2 q^{28} + 7 q^{29} + q^{30} + 12 q^{31} - 2 q^{32} + q^{35} + 2 q^{36} + q^{37} - 2 q^{38} + 5 q^{39} + q^{40} + 9 q^{41} + 2 q^{42} + 5 q^{43} - q^{45} + 2 q^{46} + q^{47} + 2 q^{48} + 2 q^{49} - 11 q^{50} + 5 q^{52} + 6 q^{53} - 2 q^{54} + 2 q^{56} + 2 q^{57} - 7 q^{58} - 14 q^{59} - q^{60} + 20 q^{61} - 12 q^{62} - 2 q^{63} + 2 q^{64} - 23 q^{65} + 8 q^{67} - 2 q^{69} - q^{70} + 2 q^{71} - 2 q^{72} + 2 q^{73} - q^{74} + 11 q^{75} + 2 q^{76} - 5 q^{78} - q^{80} + 2 q^{81} - 9 q^{82} - 18 q^{83} - 2 q^{84} - 5 q^{86} + 7 q^{87} - 18 q^{89} + q^{90} - 5 q^{91} - 2 q^{92} + 12 q^{93} - q^{94} + 40 q^{95} - 2 q^{96} + 17 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.70156 1.20818 0.604088 0.796918i \(-0.293538\pi\)
0.604088 + 0.796918i \(0.293538\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.70156 −0.854309
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) −0.701562 −0.194578 −0.0972892 0.995256i \(-0.531017\pi\)
−0.0972892 + 0.995256i \(0.531017\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.70156 0.697540
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 7.40312 1.69839 0.849197 0.528077i \(-0.177087\pi\)
0.849197 + 0.528077i \(0.177087\pi\)
\(20\) 2.70156 0.604088
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 2.29844 0.459688
\(26\) 0.701562 0.137588
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 6.70156 1.24445 0.622224 0.782839i \(-0.286229\pi\)
0.622224 + 0.782839i \(0.286229\pi\)
\(30\) −2.70156 −0.493236
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −2.70156 −0.456647
\(36\) 1.00000 0.166667
\(37\) −2.70156 −0.444134 −0.222067 0.975031i \(-0.571280\pi\)
−0.222067 + 0.975031i \(0.571280\pi\)
\(38\) −7.40312 −1.20095
\(39\) −0.701562 −0.112340
\(40\) −2.70156 −0.427154
\(41\) 1.29844 0.202782 0.101391 0.994847i \(-0.467671\pi\)
0.101391 + 0.994847i \(0.467671\pi\)
\(42\) 1.00000 0.154303
\(43\) −0.701562 −0.106987 −0.0534936 0.998568i \(-0.517036\pi\)
−0.0534936 + 0.998568i \(0.517036\pi\)
\(44\) 0 0
\(45\) 2.70156 0.402725
\(46\) 1.00000 0.147442
\(47\) −2.70156 −0.394063 −0.197032 0.980397i \(-0.563130\pi\)
−0.197032 + 0.980397i \(0.563130\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −2.29844 −0.325048
\(51\) 0 0
\(52\) −0.701562 −0.0972892
\(53\) −3.40312 −0.467455 −0.233728 0.972302i \(-0.575092\pi\)
−0.233728 + 0.972302i \(0.575092\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 7.40312 0.980568
\(58\) −6.70156 −0.879958
\(59\) −13.4031 −1.74494 −0.872469 0.488669i \(-0.837482\pi\)
−0.872469 + 0.488669i \(0.837482\pi\)
\(60\) 2.70156 0.348770
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −6.00000 −0.762001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −1.89531 −0.235085
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 2.70156 0.322898
\(71\) −5.40312 −0.641233 −0.320616 0.947209i \(-0.603890\pi\)
−0.320616 + 0.947209i \(0.603890\pi\)
\(72\) −1.00000 −0.117851
\(73\) 7.40312 0.866470 0.433235 0.901281i \(-0.357372\pi\)
0.433235 + 0.901281i \(0.357372\pi\)
\(74\) 2.70156 0.314050
\(75\) 2.29844 0.265401
\(76\) 7.40312 0.849197
\(77\) 0 0
\(78\) 0.701562 0.0794363
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.70156 0.302044
\(81\) 1.00000 0.111111
\(82\) −1.29844 −0.143388
\(83\) −15.4031 −1.69071 −0.845356 0.534203i \(-0.820612\pi\)
−0.845356 + 0.534203i \(0.820612\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 0.701562 0.0756514
\(87\) 6.70156 0.718483
\(88\) 0 0
\(89\) −2.59688 −0.275268 −0.137634 0.990483i \(-0.543950\pi\)
−0.137634 + 0.990483i \(0.543950\pi\)
\(90\) −2.70156 −0.284770
\(91\) 0.701562 0.0735437
\(92\) −1.00000 −0.104257
\(93\) 6.00000 0.622171
\(94\) 2.70156 0.278645
\(95\) 20.0000 2.05196
\(96\) −1.00000 −0.102062
\(97\) 18.1047 1.83825 0.919126 0.393963i \(-0.128896\pi\)
0.919126 + 0.393963i \(0.128896\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 2.29844 0.229844
\(101\) 17.4031 1.73168 0.865838 0.500325i \(-0.166786\pi\)
0.865838 + 0.500325i \(0.166786\pi\)
\(102\) 0 0
\(103\) 2.10469 0.207381 0.103690 0.994610i \(-0.466935\pi\)
0.103690 + 0.994610i \(0.466935\pi\)
\(104\) 0.701562 0.0687938
\(105\) −2.70156 −0.263645
\(106\) 3.40312 0.330541
\(107\) −10.8062 −1.04468 −0.522340 0.852737i \(-0.674941\pi\)
−0.522340 + 0.852737i \(0.674941\pi\)
\(108\) 1.00000 0.0962250
\(109\) 9.29844 0.890629 0.445314 0.895374i \(-0.353092\pi\)
0.445314 + 0.895374i \(0.353092\pi\)
\(110\) 0 0
\(111\) −2.70156 −0.256421
\(112\) −1.00000 −0.0944911
\(113\) −10.7016 −1.00672 −0.503359 0.864077i \(-0.667903\pi\)
−0.503359 + 0.864077i \(0.667903\pi\)
\(114\) −7.40312 −0.693366
\(115\) −2.70156 −0.251922
\(116\) 6.70156 0.622224
\(117\) −0.701562 −0.0648594
\(118\) 13.4031 1.23386
\(119\) 0 0
\(120\) −2.70156 −0.246618
\(121\) −11.0000 −1.00000
\(122\) −10.0000 −0.905357
\(123\) 1.29844 0.117076
\(124\) 6.00000 0.538816
\(125\) −7.29844 −0.652792
\(126\) 1.00000 0.0890871
\(127\) −2.10469 −0.186761 −0.0933804 0.995631i \(-0.529767\pi\)
−0.0933804 + 0.995631i \(0.529767\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.701562 −0.0617691
\(130\) 1.89531 0.166230
\(131\) 2.59688 0.226890 0.113445 0.993544i \(-0.463811\pi\)
0.113445 + 0.993544i \(0.463811\pi\)
\(132\) 0 0
\(133\) −7.40312 −0.641932
\(134\) −4.00000 −0.345547
\(135\) 2.70156 0.232513
\(136\) 0 0
\(137\) 20.1047 1.71766 0.858830 0.512261i \(-0.171192\pi\)
0.858830 + 0.512261i \(0.171192\pi\)
\(138\) 1.00000 0.0851257
\(139\) −1.89531 −0.160758 −0.0803792 0.996764i \(-0.525613\pi\)
−0.0803792 + 0.996764i \(0.525613\pi\)
\(140\) −2.70156 −0.228324
\(141\) −2.70156 −0.227513
\(142\) 5.40312 0.453420
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 18.1047 1.50351
\(146\) −7.40312 −0.612687
\(147\) 1.00000 0.0824786
\(148\) −2.70156 −0.222067
\(149\) −4.59688 −0.376591 −0.188295 0.982112i \(-0.560296\pi\)
−0.188295 + 0.982112i \(0.560296\pi\)
\(150\) −2.29844 −0.187667
\(151\) −10.1047 −0.822308 −0.411154 0.911566i \(-0.634874\pi\)
−0.411154 + 0.911566i \(0.634874\pi\)
\(152\) −7.40312 −0.600473
\(153\) 0 0
\(154\) 0 0
\(155\) 16.2094 1.30197
\(156\) −0.701562 −0.0561699
\(157\) −0.596876 −0.0476359 −0.0238179 0.999716i \(-0.507582\pi\)
−0.0238179 + 0.999716i \(0.507582\pi\)
\(158\) 0 0
\(159\) −3.40312 −0.269885
\(160\) −2.70156 −0.213577
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 1.29844 0.101391
\(165\) 0 0
\(166\) 15.4031 1.19551
\(167\) −8.59688 −0.665246 −0.332623 0.943060i \(-0.607934\pi\)
−0.332623 + 0.943060i \(0.607934\pi\)
\(168\) 1.00000 0.0771517
\(169\) −12.5078 −0.962139
\(170\) 0 0
\(171\) 7.40312 0.566131
\(172\) −0.701562 −0.0534936
\(173\) 9.40312 0.714906 0.357453 0.933931i \(-0.383645\pi\)
0.357453 + 0.933931i \(0.383645\pi\)
\(174\) −6.70156 −0.508044
\(175\) −2.29844 −0.173746
\(176\) 0 0
\(177\) −13.4031 −1.00744
\(178\) 2.59688 0.194644
\(179\) −0.701562 −0.0524372 −0.0262186 0.999656i \(-0.508347\pi\)
−0.0262186 + 0.999656i \(0.508347\pi\)
\(180\) 2.70156 0.201363
\(181\) −8.80625 −0.654563 −0.327282 0.944927i \(-0.606133\pi\)
−0.327282 + 0.944927i \(0.606133\pi\)
\(182\) −0.701562 −0.0520032
\(183\) 10.0000 0.739221
\(184\) 1.00000 0.0737210
\(185\) −7.29844 −0.536592
\(186\) −6.00000 −0.439941
\(187\) 0 0
\(188\) −2.70156 −0.197032
\(189\) −1.00000 −0.0727393
\(190\) −20.0000 −1.45095
\(191\) −2.80625 −0.203053 −0.101527 0.994833i \(-0.532373\pi\)
−0.101527 + 0.994833i \(0.532373\pi\)
\(192\) 1.00000 0.0721688
\(193\) 12.1047 0.871314 0.435657 0.900113i \(-0.356516\pi\)
0.435657 + 0.900113i \(0.356516\pi\)
\(194\) −18.1047 −1.29984
\(195\) −1.89531 −0.135726
\(196\) 1.00000 0.0714286
\(197\) −25.5078 −1.81736 −0.908678 0.417497i \(-0.862907\pi\)
−0.908678 + 0.417497i \(0.862907\pi\)
\(198\) 0 0
\(199\) −18.1047 −1.28341 −0.641704 0.766953i \(-0.721772\pi\)
−0.641704 + 0.766953i \(0.721772\pi\)
\(200\) −2.29844 −0.162524
\(201\) 4.00000 0.282138
\(202\) −17.4031 −1.22448
\(203\) −6.70156 −0.470357
\(204\) 0 0
\(205\) 3.50781 0.244996
\(206\) −2.10469 −0.146640
\(207\) −1.00000 −0.0695048
\(208\) −0.701562 −0.0486446
\(209\) 0 0
\(210\) 2.70156 0.186425
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −3.40312 −0.233728
\(213\) −5.40312 −0.370216
\(214\) 10.8062 0.738700
\(215\) −1.89531 −0.129259
\(216\) −1.00000 −0.0680414
\(217\) −6.00000 −0.407307
\(218\) −9.29844 −0.629770
\(219\) 7.40312 0.500257
\(220\) 0 0
\(221\) 0 0
\(222\) 2.70156 0.181317
\(223\) 4.80625 0.321850 0.160925 0.986967i \(-0.448552\pi\)
0.160925 + 0.986967i \(0.448552\pi\)
\(224\) 1.00000 0.0668153
\(225\) 2.29844 0.153229
\(226\) 10.7016 0.711857
\(227\) 5.50781 0.365566 0.182783 0.983153i \(-0.441489\pi\)
0.182783 + 0.983153i \(0.441489\pi\)
\(228\) 7.40312 0.490284
\(229\) −10.2094 −0.674654 −0.337327 0.941387i \(-0.609523\pi\)
−0.337327 + 0.941387i \(0.609523\pi\)
\(230\) 2.70156 0.178136
\(231\) 0 0
\(232\) −6.70156 −0.439979
\(233\) 3.19375 0.209230 0.104615 0.994513i \(-0.466639\pi\)
0.104615 + 0.994513i \(0.466639\pi\)
\(234\) 0.701562 0.0458626
\(235\) −7.29844 −0.476098
\(236\) −13.4031 −0.872469
\(237\) 0 0
\(238\) 0 0
\(239\) 6.80625 0.440260 0.220130 0.975471i \(-0.429352\pi\)
0.220130 + 0.975471i \(0.429352\pi\)
\(240\) 2.70156 0.174385
\(241\) −14.1047 −0.908563 −0.454281 0.890858i \(-0.650104\pi\)
−0.454281 + 0.890858i \(0.650104\pi\)
\(242\) 11.0000 0.707107
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) 2.70156 0.172596
\(246\) −1.29844 −0.0827854
\(247\) −5.19375 −0.330470
\(248\) −6.00000 −0.381000
\(249\) −15.4031 −0.976133
\(250\) 7.29844 0.461594
\(251\) −30.9109 −1.95108 −0.975540 0.219820i \(-0.929453\pi\)
−0.975540 + 0.219820i \(0.929453\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 2.10469 0.132060
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −27.6125 −1.72242 −0.861210 0.508249i \(-0.830293\pi\)
−0.861210 + 0.508249i \(0.830293\pi\)
\(258\) 0.701562 0.0436773
\(259\) 2.70156 0.167867
\(260\) −1.89531 −0.117542
\(261\) 6.70156 0.414816
\(262\) −2.59688 −0.160436
\(263\) 15.5078 0.956253 0.478126 0.878291i \(-0.341316\pi\)
0.478126 + 0.878291i \(0.341316\pi\)
\(264\) 0 0
\(265\) −9.19375 −0.564768
\(266\) 7.40312 0.453915
\(267\) −2.59688 −0.158926
\(268\) 4.00000 0.244339
\(269\) 9.40312 0.573319 0.286659 0.958033i \(-0.407455\pi\)
0.286659 + 0.958033i \(0.407455\pi\)
\(270\) −2.70156 −0.164412
\(271\) 4.59688 0.279240 0.139620 0.990205i \(-0.455412\pi\)
0.139620 + 0.990205i \(0.455412\pi\)
\(272\) 0 0
\(273\) 0.701562 0.0424605
\(274\) −20.1047 −1.21457
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −0.596876 −0.0358628 −0.0179314 0.999839i \(-0.505708\pi\)
−0.0179314 + 0.999839i \(0.505708\pi\)
\(278\) 1.89531 0.113673
\(279\) 6.00000 0.359211
\(280\) 2.70156 0.161449
\(281\) 10.7016 0.638402 0.319201 0.947687i \(-0.396586\pi\)
0.319201 + 0.947687i \(0.396586\pi\)
\(282\) 2.70156 0.160876
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) −5.40312 −0.320616
\(285\) 20.0000 1.18470
\(286\) 0 0
\(287\) −1.29844 −0.0766444
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) −18.1047 −1.06314
\(291\) 18.1047 1.06132
\(292\) 7.40312 0.433235
\(293\) −28.8062 −1.68288 −0.841440 0.540351i \(-0.818291\pi\)
−0.841440 + 0.540351i \(0.818291\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −36.2094 −2.10819
\(296\) 2.70156 0.157025
\(297\) 0 0
\(298\) 4.59688 0.266290
\(299\) 0.701562 0.0405724
\(300\) 2.29844 0.132700
\(301\) 0.701562 0.0404374
\(302\) 10.1047 0.581459
\(303\) 17.4031 0.999783
\(304\) 7.40312 0.424598
\(305\) 27.0156 1.54691
\(306\) 0 0
\(307\) −14.1047 −0.804997 −0.402498 0.915421i \(-0.631858\pi\)
−0.402498 + 0.915421i \(0.631858\pi\)
\(308\) 0 0
\(309\) 2.10469 0.119731
\(310\) −16.2094 −0.920631
\(311\) 22.2094 1.25938 0.629689 0.776847i \(-0.283182\pi\)
0.629689 + 0.776847i \(0.283182\pi\)
\(312\) 0.701562 0.0397181
\(313\) −17.4031 −0.983683 −0.491841 0.870685i \(-0.663676\pi\)
−0.491841 + 0.870685i \(0.663676\pi\)
\(314\) 0.596876 0.0336836
\(315\) −2.70156 −0.152216
\(316\) 0 0
\(317\) 1.29844 0.0729275 0.0364638 0.999335i \(-0.488391\pi\)
0.0364638 + 0.999335i \(0.488391\pi\)
\(318\) 3.40312 0.190838
\(319\) 0 0
\(320\) 2.70156 0.151022
\(321\) −10.8062 −0.603146
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −1.61250 −0.0894452
\(326\) 12.0000 0.664619
\(327\) 9.29844 0.514205
\(328\) −1.29844 −0.0716942
\(329\) 2.70156 0.148942
\(330\) 0 0
\(331\) 33.6125 1.84751 0.923755 0.382984i \(-0.125104\pi\)
0.923755 + 0.382984i \(0.125104\pi\)
\(332\) −15.4031 −0.845356
\(333\) −2.70156 −0.148045
\(334\) 8.59688 0.470400
\(335\) 10.8062 0.590408
\(336\) −1.00000 −0.0545545
\(337\) 4.59688 0.250408 0.125204 0.992131i \(-0.460041\pi\)
0.125204 + 0.992131i \(0.460041\pi\)
\(338\) 12.5078 0.680335
\(339\) −10.7016 −0.581229
\(340\) 0 0
\(341\) 0 0
\(342\) −7.40312 −0.400315
\(343\) −1.00000 −0.0539949
\(344\) 0.701562 0.0378257
\(345\) −2.70156 −0.145447
\(346\) −9.40312 −0.505515
\(347\) −11.5078 −0.617772 −0.308886 0.951099i \(-0.599956\pi\)
−0.308886 + 0.951099i \(0.599956\pi\)
\(348\) 6.70156 0.359241
\(349\) −4.20937 −0.225323 −0.112661 0.993633i \(-0.535937\pi\)
−0.112661 + 0.993633i \(0.535937\pi\)
\(350\) 2.29844 0.122857
\(351\) −0.701562 −0.0374466
\(352\) 0 0
\(353\) 24.3141 1.29411 0.647053 0.762445i \(-0.276001\pi\)
0.647053 + 0.762445i \(0.276001\pi\)
\(354\) 13.4031 0.712368
\(355\) −14.5969 −0.774722
\(356\) −2.59688 −0.137634
\(357\) 0 0
\(358\) 0.701562 0.0370787
\(359\) 3.29844 0.174085 0.0870424 0.996205i \(-0.472258\pi\)
0.0870424 + 0.996205i \(0.472258\pi\)
\(360\) −2.70156 −0.142385
\(361\) 35.8062 1.88454
\(362\) 8.80625 0.462846
\(363\) −11.0000 −0.577350
\(364\) 0.701562 0.0367718
\(365\) 20.0000 1.04685
\(366\) −10.0000 −0.522708
\(367\) −28.9109 −1.50914 −0.754569 0.656220i \(-0.772154\pi\)
−0.754569 + 0.656220i \(0.772154\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 1.29844 0.0675940
\(370\) 7.29844 0.379428
\(371\) 3.40312 0.176681
\(372\) 6.00000 0.311086
\(373\) 20.8062 1.07731 0.538653 0.842527i \(-0.318933\pi\)
0.538653 + 0.842527i \(0.318933\pi\)
\(374\) 0 0
\(375\) −7.29844 −0.376890
\(376\) 2.70156 0.139322
\(377\) −4.70156 −0.242143
\(378\) 1.00000 0.0514344
\(379\) −11.2984 −0.580362 −0.290181 0.956972i \(-0.593715\pi\)
−0.290181 + 0.956972i \(0.593715\pi\)
\(380\) 20.0000 1.02598
\(381\) −2.10469 −0.107826
\(382\) 2.80625 0.143580
\(383\) −18.8062 −0.960954 −0.480477 0.877007i \(-0.659537\pi\)
−0.480477 + 0.877007i \(0.659537\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −12.1047 −0.616112
\(387\) −0.701562 −0.0356624
\(388\) 18.1047 0.919126
\(389\) 11.1938 0.567546 0.283773 0.958892i \(-0.408414\pi\)
0.283773 + 0.958892i \(0.408414\pi\)
\(390\) 1.89531 0.0959729
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 2.59688 0.130995
\(394\) 25.5078 1.28506
\(395\) 0 0
\(396\) 0 0
\(397\) −32.2094 −1.61654 −0.808271 0.588811i \(-0.799596\pi\)
−0.808271 + 0.588811i \(0.799596\pi\)
\(398\) 18.1047 0.907506
\(399\) −7.40312 −0.370620
\(400\) 2.29844 0.114922
\(401\) 3.19375 0.159488 0.0797442 0.996815i \(-0.474590\pi\)
0.0797442 + 0.996815i \(0.474590\pi\)
\(402\) −4.00000 −0.199502
\(403\) −4.20937 −0.209684
\(404\) 17.4031 0.865838
\(405\) 2.70156 0.134242
\(406\) 6.70156 0.332593
\(407\) 0 0
\(408\) 0 0
\(409\) 6.20937 0.307034 0.153517 0.988146i \(-0.450940\pi\)
0.153517 + 0.988146i \(0.450940\pi\)
\(410\) −3.50781 −0.173238
\(411\) 20.1047 0.991691
\(412\) 2.10469 0.103690
\(413\) 13.4031 0.659525
\(414\) 1.00000 0.0491473
\(415\) −41.6125 −2.04268
\(416\) 0.701562 0.0343969
\(417\) −1.89531 −0.0928139
\(418\) 0 0
\(419\) −15.4031 −0.752492 −0.376246 0.926520i \(-0.622785\pi\)
−0.376246 + 0.926520i \(0.622785\pi\)
\(420\) −2.70156 −0.131823
\(421\) 9.29844 0.453178 0.226589 0.973990i \(-0.427243\pi\)
0.226589 + 0.973990i \(0.427243\pi\)
\(422\) 12.0000 0.584151
\(423\) −2.70156 −0.131354
\(424\) 3.40312 0.165270
\(425\) 0 0
\(426\) 5.40312 0.261782
\(427\) −10.0000 −0.483934
\(428\) −10.8062 −0.522340
\(429\) 0 0
\(430\) 1.89531 0.0914001
\(431\) −14.1047 −0.679399 −0.339699 0.940534i \(-0.610325\pi\)
−0.339699 + 0.940534i \(0.610325\pi\)
\(432\) 1.00000 0.0481125
\(433\) −2.10469 −0.101145 −0.0505724 0.998720i \(-0.516105\pi\)
−0.0505724 + 0.998720i \(0.516105\pi\)
\(434\) 6.00000 0.288009
\(435\) 18.1047 0.868053
\(436\) 9.29844 0.445314
\(437\) −7.40312 −0.354139
\(438\) −7.40312 −0.353735
\(439\) 28.8062 1.37485 0.687424 0.726257i \(-0.258742\pi\)
0.687424 + 0.726257i \(0.258742\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −2.10469 −0.0999967 −0.0499983 0.998749i \(-0.515922\pi\)
−0.0499983 + 0.998749i \(0.515922\pi\)
\(444\) −2.70156 −0.128210
\(445\) −7.01562 −0.332572
\(446\) −4.80625 −0.227582
\(447\) −4.59688 −0.217425
\(448\) −1.00000 −0.0472456
\(449\) −23.6125 −1.11434 −0.557171 0.830398i \(-0.688113\pi\)
−0.557171 + 0.830398i \(0.688113\pi\)
\(450\) −2.29844 −0.108349
\(451\) 0 0
\(452\) −10.7016 −0.503359
\(453\) −10.1047 −0.474760
\(454\) −5.50781 −0.258494
\(455\) 1.89531 0.0888537
\(456\) −7.40312 −0.346683
\(457\) 22.2094 1.03891 0.519455 0.854498i \(-0.326135\pi\)
0.519455 + 0.854498i \(0.326135\pi\)
\(458\) 10.2094 0.477053
\(459\) 0 0
\(460\) −2.70156 −0.125961
\(461\) 14.5969 0.679844 0.339922 0.940454i \(-0.389599\pi\)
0.339922 + 0.940454i \(0.389599\pi\)
\(462\) 0 0
\(463\) 6.10469 0.283709 0.141854 0.989888i \(-0.454694\pi\)
0.141854 + 0.989888i \(0.454694\pi\)
\(464\) 6.70156 0.311112
\(465\) 16.2094 0.751692
\(466\) −3.19375 −0.147948
\(467\) −14.7016 −0.680307 −0.340154 0.940370i \(-0.610479\pi\)
−0.340154 + 0.940370i \(0.610479\pi\)
\(468\) −0.701562 −0.0324297
\(469\) −4.00000 −0.184703
\(470\) 7.29844 0.336652
\(471\) −0.596876 −0.0275026
\(472\) 13.4031 0.616929
\(473\) 0 0
\(474\) 0 0
\(475\) 17.0156 0.780730
\(476\) 0 0
\(477\) −3.40312 −0.155818
\(478\) −6.80625 −0.311311
\(479\) −5.40312 −0.246875 −0.123438 0.992352i \(-0.539392\pi\)
−0.123438 + 0.992352i \(0.539392\pi\)
\(480\) −2.70156 −0.123309
\(481\) 1.89531 0.0864189
\(482\) 14.1047 0.642451
\(483\) 1.00000 0.0455016
\(484\) −11.0000 −0.500000
\(485\) 48.9109 2.22093
\(486\) −1.00000 −0.0453609
\(487\) 6.10469 0.276630 0.138315 0.990388i \(-0.455831\pi\)
0.138315 + 0.990388i \(0.455831\pi\)
\(488\) −10.0000 −0.452679
\(489\) −12.0000 −0.542659
\(490\) −2.70156 −0.122044
\(491\) 33.6125 1.51691 0.758455 0.651725i \(-0.225954\pi\)
0.758455 + 0.651725i \(0.225954\pi\)
\(492\) 1.29844 0.0585381
\(493\) 0 0
\(494\) 5.19375 0.233678
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 5.40312 0.242363
\(498\) 15.4031 0.690231
\(499\) −27.0156 −1.20939 −0.604693 0.796459i \(-0.706704\pi\)
−0.604693 + 0.796459i \(0.706704\pi\)
\(500\) −7.29844 −0.326396
\(501\) −8.59688 −0.384080
\(502\) 30.9109 1.37962
\(503\) 1.40312 0.0625622 0.0312811 0.999511i \(-0.490041\pi\)
0.0312811 + 0.999511i \(0.490041\pi\)
\(504\) 1.00000 0.0445435
\(505\) 47.0156 2.09217
\(506\) 0 0
\(507\) −12.5078 −0.555491
\(508\) −2.10469 −0.0933804
\(509\) 10.8062 0.478979 0.239489 0.970899i \(-0.423020\pi\)
0.239489 + 0.970899i \(0.423020\pi\)
\(510\) 0 0
\(511\) −7.40312 −0.327495
\(512\) −1.00000 −0.0441942
\(513\) 7.40312 0.326856
\(514\) 27.6125 1.21794
\(515\) 5.68594 0.250552
\(516\) −0.701562 −0.0308846
\(517\) 0 0
\(518\) −2.70156 −0.118700
\(519\) 9.40312 0.412751
\(520\) 1.89531 0.0831150
\(521\) 2.59688 0.113771 0.0568856 0.998381i \(-0.481883\pi\)
0.0568856 + 0.998381i \(0.481883\pi\)
\(522\) −6.70156 −0.293319
\(523\) 38.4187 1.67993 0.839967 0.542637i \(-0.182574\pi\)
0.839967 + 0.542637i \(0.182574\pi\)
\(524\) 2.59688 0.113445
\(525\) −2.29844 −0.100312
\(526\) −15.5078 −0.676173
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 9.19375 0.399351
\(531\) −13.4031 −0.581646
\(532\) −7.40312 −0.320966
\(533\) −0.910935 −0.0394570
\(534\) 2.59688 0.112378
\(535\) −29.1938 −1.26216
\(536\) −4.00000 −0.172774
\(537\) −0.701562 −0.0302746
\(538\) −9.40312 −0.405397
\(539\) 0 0
\(540\) 2.70156 0.116257
\(541\) 18.2094 0.782882 0.391441 0.920203i \(-0.371977\pi\)
0.391441 + 0.920203i \(0.371977\pi\)
\(542\) −4.59688 −0.197453
\(543\) −8.80625 −0.377912
\(544\) 0 0
\(545\) 25.1203 1.07604
\(546\) −0.701562 −0.0300241
\(547\) 24.2094 1.03512 0.517559 0.855648i \(-0.326841\pi\)
0.517559 + 0.855648i \(0.326841\pi\)
\(548\) 20.1047 0.858830
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 49.6125 2.11356
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 0.596876 0.0253588
\(555\) −7.29844 −0.309801
\(556\) −1.89531 −0.0803792
\(557\) 8.59688 0.364261 0.182131 0.983274i \(-0.441701\pi\)
0.182131 + 0.983274i \(0.441701\pi\)
\(558\) −6.00000 −0.254000
\(559\) 0.492189 0.0208174
\(560\) −2.70156 −0.114162
\(561\) 0 0
\(562\) −10.7016 −0.451418
\(563\) −25.7172 −1.08385 −0.541925 0.840427i \(-0.682304\pi\)
−0.541925 + 0.840427i \(0.682304\pi\)
\(564\) −2.70156 −0.113756
\(565\) −28.9109 −1.21629
\(566\) 10.0000 0.420331
\(567\) −1.00000 −0.0419961
\(568\) 5.40312 0.226710
\(569\) −8.10469 −0.339766 −0.169883 0.985464i \(-0.554339\pi\)
−0.169883 + 0.985464i \(0.554339\pi\)
\(570\) −20.0000 −0.837708
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) −2.80625 −0.117233
\(574\) 1.29844 0.0541958
\(575\) −2.29844 −0.0958515
\(576\) 1.00000 0.0416667
\(577\) 27.6125 1.14952 0.574762 0.818321i \(-0.305095\pi\)
0.574762 + 0.818321i \(0.305095\pi\)
\(578\) 17.0000 0.707107
\(579\) 12.1047 0.503054
\(580\) 18.1047 0.751756
\(581\) 15.4031 0.639029
\(582\) −18.1047 −0.750463
\(583\) 0 0
\(584\) −7.40312 −0.306343
\(585\) −1.89531 −0.0783616
\(586\) 28.8062 1.18998
\(587\) −26.8062 −1.10641 −0.553206 0.833044i \(-0.686596\pi\)
−0.553206 + 0.833044i \(0.686596\pi\)
\(588\) 1.00000 0.0412393
\(589\) 44.4187 1.83024
\(590\) 36.2094 1.49072
\(591\) −25.5078 −1.04925
\(592\) −2.70156 −0.111034
\(593\) 2.91093 0.119538 0.0597689 0.998212i \(-0.480964\pi\)
0.0597689 + 0.998212i \(0.480964\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.59688 −0.188295
\(597\) −18.1047 −0.740975
\(598\) −0.701562 −0.0286890
\(599\) 21.6125 0.883063 0.441531 0.897246i \(-0.354435\pi\)
0.441531 + 0.897246i \(0.354435\pi\)
\(600\) −2.29844 −0.0938333
\(601\) 12.5969 0.513837 0.256919 0.966433i \(-0.417293\pi\)
0.256919 + 0.966433i \(0.417293\pi\)
\(602\) −0.701562 −0.0285935
\(603\) 4.00000 0.162893
\(604\) −10.1047 −0.411154
\(605\) −29.7172 −1.20818
\(606\) −17.4031 −0.706954
\(607\) −2.00000 −0.0811775 −0.0405887 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) −7.40312 −0.300236
\(609\) −6.70156 −0.271561
\(610\) −27.0156 −1.09383
\(611\) 1.89531 0.0766762
\(612\) 0 0
\(613\) 32.3141 1.30515 0.652576 0.757723i \(-0.273688\pi\)
0.652576 + 0.757723i \(0.273688\pi\)
\(614\) 14.1047 0.569219
\(615\) 3.50781 0.141449
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) −2.10469 −0.0846629
\(619\) −30.0000 −1.20580 −0.602901 0.797816i \(-0.705989\pi\)
−0.602901 + 0.797816i \(0.705989\pi\)
\(620\) 16.2094 0.650984
\(621\) −1.00000 −0.0401286
\(622\) −22.2094 −0.890515
\(623\) 2.59688 0.104042
\(624\) −0.701562 −0.0280850
\(625\) −31.2094 −1.24837
\(626\) 17.4031 0.695569
\(627\) 0 0
\(628\) −0.596876 −0.0238179
\(629\) 0 0
\(630\) 2.70156 0.107633
\(631\) 34.8062 1.38561 0.692807 0.721123i \(-0.256374\pi\)
0.692807 + 0.721123i \(0.256374\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) −1.29844 −0.0515676
\(635\) −5.68594 −0.225640
\(636\) −3.40312 −0.134943
\(637\) −0.701562 −0.0277969
\(638\) 0 0
\(639\) −5.40312 −0.213744
\(640\) −2.70156 −0.106789
\(641\) −5.29844 −0.209276 −0.104638 0.994510i \(-0.533368\pi\)
−0.104638 + 0.994510i \(0.533368\pi\)
\(642\) 10.8062 0.426489
\(643\) 28.8062 1.13601 0.568004 0.823026i \(-0.307716\pi\)
0.568004 + 0.823026i \(0.307716\pi\)
\(644\) 1.00000 0.0394055
\(645\) −1.89531 −0.0746279
\(646\) 0 0
\(647\) 34.2094 1.34491 0.672455 0.740138i \(-0.265240\pi\)
0.672455 + 0.740138i \(0.265240\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 1.61250 0.0632473
\(651\) −6.00000 −0.235159
\(652\) −12.0000 −0.469956
\(653\) 27.8953 1.09163 0.545814 0.837906i \(-0.316221\pi\)
0.545814 + 0.837906i \(0.316221\pi\)
\(654\) −9.29844 −0.363598
\(655\) 7.01562 0.274123
\(656\) 1.29844 0.0506955
\(657\) 7.40312 0.288823
\(658\) −2.70156 −0.105318
\(659\) −25.6125 −0.997721 −0.498861 0.866682i \(-0.666248\pi\)
−0.498861 + 0.866682i \(0.666248\pi\)
\(660\) 0 0
\(661\) 4.80625 0.186941 0.0934707 0.995622i \(-0.470204\pi\)
0.0934707 + 0.995622i \(0.470204\pi\)
\(662\) −33.6125 −1.30639
\(663\) 0 0
\(664\) 15.4031 0.597757
\(665\) −20.0000 −0.775567
\(666\) 2.70156 0.104683
\(667\) −6.70156 −0.259486
\(668\) −8.59688 −0.332623
\(669\) 4.80625 0.185820
\(670\) −10.8062 −0.417482
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) −14.7016 −0.566704 −0.283352 0.959016i \(-0.591446\pi\)
−0.283352 + 0.959016i \(0.591446\pi\)
\(674\) −4.59688 −0.177065
\(675\) 2.29844 0.0884669
\(676\) −12.5078 −0.481070
\(677\) −28.8062 −1.10711 −0.553557 0.832811i \(-0.686730\pi\)
−0.553557 + 0.832811i \(0.686730\pi\)
\(678\) 10.7016 0.410991
\(679\) −18.1047 −0.694794
\(680\) 0 0
\(681\) 5.50781 0.211060
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 7.40312 0.283066
\(685\) 54.3141 2.07523
\(686\) 1.00000 0.0381802
\(687\) −10.2094 −0.389512
\(688\) −0.701562 −0.0267468
\(689\) 2.38750 0.0909566
\(690\) 2.70156 0.102847
\(691\) −35.2984 −1.34282 −0.671408 0.741088i \(-0.734310\pi\)
−0.671408 + 0.741088i \(0.734310\pi\)
\(692\) 9.40312 0.357453
\(693\) 0 0
\(694\) 11.5078 0.436831
\(695\) −5.12031 −0.194224
\(696\) −6.70156 −0.254022
\(697\) 0 0
\(698\) 4.20937 0.159327
\(699\) 3.19375 0.120799
\(700\) −2.29844 −0.0868728
\(701\) 27.4031 1.03500 0.517501 0.855683i \(-0.326862\pi\)
0.517501 + 0.855683i \(0.326862\pi\)
\(702\) 0.701562 0.0264788
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) −7.29844 −0.274875
\(706\) −24.3141 −0.915072
\(707\) −17.4031 −0.654512
\(708\) −13.4031 −0.503720
\(709\) −40.8062 −1.53251 −0.766255 0.642536i \(-0.777882\pi\)
−0.766255 + 0.642536i \(0.777882\pi\)
\(710\) 14.5969 0.547811
\(711\) 0 0
\(712\) 2.59688 0.0973220
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) −0.701562 −0.0262186
\(717\) 6.80625 0.254184
\(718\) −3.29844 −0.123097
\(719\) −17.5078 −0.652931 −0.326466 0.945209i \(-0.605858\pi\)
−0.326466 + 0.945209i \(0.605858\pi\)
\(720\) 2.70156 0.100681
\(721\) −2.10469 −0.0783826
\(722\) −35.8062 −1.33257
\(723\) −14.1047 −0.524559
\(724\) −8.80625 −0.327282
\(725\) 15.4031 0.572058
\(726\) 11.0000 0.408248
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) −0.701562 −0.0260016
\(729\) 1.00000 0.0370370
\(730\) −20.0000 −0.740233
\(731\) 0 0
\(732\) 10.0000 0.369611
\(733\) −33.0156 −1.21946 −0.609730 0.792609i \(-0.708722\pi\)
−0.609730 + 0.792609i \(0.708722\pi\)
\(734\) 28.9109 1.06712
\(735\) 2.70156 0.0996486
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) −1.29844 −0.0477962
\(739\) −0.209373 −0.00770190 −0.00385095 0.999993i \(-0.501226\pi\)
−0.00385095 + 0.999993i \(0.501226\pi\)
\(740\) −7.29844 −0.268296
\(741\) −5.19375 −0.190797
\(742\) −3.40312 −0.124933
\(743\) 29.6125 1.08638 0.543189 0.839611i \(-0.317217\pi\)
0.543189 + 0.839611i \(0.317217\pi\)
\(744\) −6.00000 −0.219971
\(745\) −12.4187 −0.454988
\(746\) −20.8062 −0.761771
\(747\) −15.4031 −0.563571
\(748\) 0 0
\(749\) 10.8062 0.394852
\(750\) 7.29844 0.266501
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −2.70156 −0.0985158
\(753\) −30.9109 −1.12646
\(754\) 4.70156 0.171221
\(755\) −27.2984 −0.993492
\(756\) −1.00000 −0.0363696
\(757\) 32.8062 1.19236 0.596182 0.802850i \(-0.296684\pi\)
0.596182 + 0.802850i \(0.296684\pi\)
\(758\) 11.2984 0.410378
\(759\) 0 0
\(760\) −20.0000 −0.725476
\(761\) −35.6125 −1.29095 −0.645476 0.763781i \(-0.723341\pi\)
−0.645476 + 0.763781i \(0.723341\pi\)
\(762\) 2.10469 0.0762447
\(763\) −9.29844 −0.336626
\(764\) −2.80625 −0.101527
\(765\) 0 0
\(766\) 18.8062 0.679497
\(767\) 9.40312 0.339527
\(768\) 1.00000 0.0360844
\(769\) −9.89531 −0.356834 −0.178417 0.983955i \(-0.557098\pi\)
−0.178417 + 0.983955i \(0.557098\pi\)
\(770\) 0 0
\(771\) −27.6125 −0.994440
\(772\) 12.1047 0.435657
\(773\) −25.7172 −0.924983 −0.462491 0.886624i \(-0.653044\pi\)
−0.462491 + 0.886624i \(0.653044\pi\)
\(774\) 0.701562 0.0252171
\(775\) 13.7906 0.495374
\(776\) −18.1047 −0.649920
\(777\) 2.70156 0.0969180
\(778\) −11.1938 −0.401315
\(779\) 9.61250 0.344403
\(780\) −1.89531 −0.0678631
\(781\) 0 0
\(782\) 0 0
\(783\) 6.70156 0.239494
\(784\) 1.00000 0.0357143
\(785\) −1.61250 −0.0575525
\(786\) −2.59688 −0.0926275
\(787\) −39.8219 −1.41950 −0.709748 0.704455i \(-0.751191\pi\)
−0.709748 + 0.704455i \(0.751191\pi\)
\(788\) −25.5078 −0.908678
\(789\) 15.5078 0.552093
\(790\) 0 0
\(791\) 10.7016 0.380504
\(792\) 0 0
\(793\) −7.01562 −0.249132
\(794\) 32.2094 1.14307
\(795\) −9.19375 −0.326069
\(796\) −18.1047 −0.641704
\(797\) −25.2984 −0.896117 −0.448058 0.894004i \(-0.647884\pi\)
−0.448058 + 0.894004i \(0.647884\pi\)
\(798\) 7.40312 0.262068
\(799\) 0 0
\(800\) −2.29844 −0.0812621
\(801\) −2.59688 −0.0917561
\(802\) −3.19375 −0.112775
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 2.70156 0.0952176
\(806\) 4.20937 0.148269
\(807\) 9.40312 0.331006
\(808\) −17.4031 −0.612240
\(809\) −8.59688 −0.302250 −0.151125 0.988515i \(-0.548290\pi\)
−0.151125 + 0.988515i \(0.548290\pi\)
\(810\) −2.70156 −0.0949232
\(811\) −46.3141 −1.62631 −0.813153 0.582050i \(-0.802251\pi\)
−0.813153 + 0.582050i \(0.802251\pi\)
\(812\) −6.70156 −0.235179
\(813\) 4.59688 0.161219
\(814\) 0 0
\(815\) −32.4187 −1.13558
\(816\) 0 0
\(817\) −5.19375 −0.181706
\(818\) −6.20937 −0.217106
\(819\) 0.701562 0.0245146
\(820\) 3.50781 0.122498
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −20.1047 −0.701231
\(823\) −23.7172 −0.826729 −0.413365 0.910566i \(-0.635646\pi\)
−0.413365 + 0.910566i \(0.635646\pi\)
\(824\) −2.10469 −0.0733202
\(825\) 0 0
\(826\) −13.4031 −0.466354
\(827\) 31.0156 1.07852 0.539259 0.842140i \(-0.318704\pi\)
0.539259 + 0.842140i \(0.318704\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 8.20937 0.285123 0.142562 0.989786i \(-0.454466\pi\)
0.142562 + 0.989786i \(0.454466\pi\)
\(830\) 41.6125 1.44439
\(831\) −0.596876 −0.0207054
\(832\) −0.701562 −0.0243223
\(833\) 0 0
\(834\) 1.89531 0.0656293
\(835\) −23.2250 −0.803734
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 15.4031 0.532092
\(839\) −6.59688 −0.227749 −0.113875 0.993495i \(-0.536326\pi\)
−0.113875 + 0.993495i \(0.536326\pi\)
\(840\) 2.70156 0.0932127
\(841\) 15.9109 0.548653
\(842\) −9.29844 −0.320445
\(843\) 10.7016 0.368581
\(844\) −12.0000 −0.413057
\(845\) −33.7906 −1.16243
\(846\) 2.70156 0.0928816
\(847\) 11.0000 0.377964
\(848\) −3.40312 −0.116864
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) 2.70156 0.0926084
\(852\) −5.40312 −0.185108
\(853\) 21.8953 0.749681 0.374841 0.927089i \(-0.377697\pi\)
0.374841 + 0.927089i \(0.377697\pi\)
\(854\) 10.0000 0.342193
\(855\) 20.0000 0.683986
\(856\) 10.8062 0.369350
\(857\) 46.9109 1.60245 0.801224 0.598365i \(-0.204183\pi\)
0.801224 + 0.598365i \(0.204183\pi\)
\(858\) 0 0
\(859\) 40.9109 1.39586 0.697932 0.716164i \(-0.254104\pi\)
0.697932 + 0.716164i \(0.254104\pi\)
\(860\) −1.89531 −0.0646297
\(861\) −1.29844 −0.0442506
\(862\) 14.1047 0.480408
\(863\) 14.8062 0.504011 0.252005 0.967726i \(-0.418910\pi\)
0.252005 + 0.967726i \(0.418910\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 25.4031 0.863732
\(866\) 2.10469 0.0715202
\(867\) −17.0000 −0.577350
\(868\) −6.00000 −0.203653
\(869\) 0 0
\(870\) −18.1047 −0.613806
\(871\) −2.80625 −0.0950861
\(872\) −9.29844 −0.314885
\(873\) 18.1047 0.612751
\(874\) 7.40312 0.250414
\(875\) 7.29844 0.246732
\(876\) 7.40312 0.250128
\(877\) −9.79063 −0.330606 −0.165303 0.986243i \(-0.552860\pi\)
−0.165303 + 0.986243i \(0.552860\pi\)
\(878\) −28.8062 −0.972164
\(879\) −28.8062 −0.971611
\(880\) 0 0
\(881\) 28.2094 0.950398 0.475199 0.879878i \(-0.342376\pi\)
0.475199 + 0.879878i \(0.342376\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 29.4031 0.989494 0.494747 0.869037i \(-0.335261\pi\)
0.494747 + 0.869037i \(0.335261\pi\)
\(884\) 0 0
\(885\) −36.2094 −1.21716
\(886\) 2.10469 0.0707083
\(887\) −13.7906 −0.463044 −0.231522 0.972830i \(-0.574371\pi\)
−0.231522 + 0.972830i \(0.574371\pi\)
\(888\) 2.70156 0.0906585
\(889\) 2.10469 0.0705889
\(890\) 7.01562 0.235164
\(891\) 0 0
\(892\) 4.80625 0.160925
\(893\) −20.0000 −0.669274
\(894\) 4.59688 0.153743
\(895\) −1.89531 −0.0633533
\(896\) 1.00000 0.0334077
\(897\) 0.701562 0.0234245
\(898\) 23.6125 0.787959
\(899\) 40.2094 1.34106
\(900\) 2.29844 0.0766146
\(901\) 0 0
\(902\) 0 0
\(903\) 0.701562 0.0233465
\(904\) 10.7016 0.355929
\(905\) −23.7906 −0.790827
\(906\) 10.1047 0.335706
\(907\) −22.1047 −0.733974 −0.366987 0.930226i \(-0.619611\pi\)
−0.366987 + 0.930226i \(0.619611\pi\)
\(908\) 5.50781 0.182783
\(909\) 17.4031 0.577225
\(910\) −1.89531 −0.0628290
\(911\) −4.70156 −0.155770 −0.0778849 0.996962i \(-0.524817\pi\)
−0.0778849 + 0.996962i \(0.524817\pi\)
\(912\) 7.40312 0.245142
\(913\) 0 0
\(914\) −22.2094 −0.734621
\(915\) 27.0156 0.893109
\(916\) −10.2094 −0.337327
\(917\) −2.59688 −0.0857564
\(918\) 0 0
\(919\) 26.8062 0.884257 0.442128 0.896952i \(-0.354224\pi\)
0.442128 + 0.896952i \(0.354224\pi\)
\(920\) 2.70156 0.0890679
\(921\) −14.1047 −0.464765
\(922\) −14.5969 −0.480723
\(923\) 3.79063 0.124770
\(924\) 0 0
\(925\) −6.20937 −0.204163
\(926\) −6.10469 −0.200612
\(927\) 2.10469 0.0691270
\(928\) −6.70156 −0.219990
\(929\) 17.2984 0.567543 0.283772 0.958892i \(-0.408414\pi\)
0.283772 + 0.958892i \(0.408414\pi\)
\(930\) −16.2094 −0.531526
\(931\) 7.40312 0.242628
\(932\) 3.19375 0.104615
\(933\) 22.2094 0.727102
\(934\) 14.7016 0.481050
\(935\) 0 0
\(936\) 0.701562 0.0229313
\(937\) −10.3141 −0.336946 −0.168473 0.985706i \(-0.553884\pi\)
−0.168473 + 0.985706i \(0.553884\pi\)
\(938\) 4.00000 0.130605
\(939\) −17.4031 −0.567929
\(940\) −7.29844 −0.238049
\(941\) 48.3141 1.57499 0.787497 0.616319i \(-0.211377\pi\)
0.787497 + 0.616319i \(0.211377\pi\)
\(942\) 0.596876 0.0194473
\(943\) −1.29844 −0.0422830
\(944\) −13.4031 −0.436235
\(945\) −2.70156 −0.0878818
\(946\) 0 0
\(947\) 35.5078 1.15385 0.576924 0.816798i \(-0.304253\pi\)
0.576924 + 0.816798i \(0.304253\pi\)
\(948\) 0 0
\(949\) −5.19375 −0.168596
\(950\) −17.0156 −0.552060
\(951\) 1.29844 0.0421047
\(952\) 0 0
\(953\) 4.80625 0.155690 0.0778448 0.996965i \(-0.475196\pi\)
0.0778448 + 0.996965i \(0.475196\pi\)
\(954\) 3.40312 0.110180
\(955\) −7.58125 −0.245324
\(956\) 6.80625 0.220130
\(957\) 0 0
\(958\) 5.40312 0.174567
\(959\) −20.1047 −0.649214
\(960\) 2.70156 0.0871925
\(961\) 5.00000 0.161290
\(962\) −1.89531 −0.0611074
\(963\) −10.8062 −0.348226
\(964\) −14.1047 −0.454281
\(965\) 32.7016 1.05270
\(966\) −1.00000 −0.0321745
\(967\) −56.4187 −1.81430 −0.907152 0.420803i \(-0.861749\pi\)
−0.907152 + 0.420803i \(0.861749\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) −48.9109 −1.57044
\(971\) −20.5969 −0.660985 −0.330493 0.943809i \(-0.607215\pi\)
−0.330493 + 0.943809i \(0.607215\pi\)
\(972\) 1.00000 0.0320750
\(973\) 1.89531 0.0607610
\(974\) −6.10469 −0.195607
\(975\) −1.61250 −0.0516412
\(976\) 10.0000 0.320092
\(977\) −46.7016 −1.49412 −0.747058 0.664759i \(-0.768534\pi\)
−0.747058 + 0.664759i \(0.768534\pi\)
\(978\) 12.0000 0.383718
\(979\) 0 0
\(980\) 2.70156 0.0862982
\(981\) 9.29844 0.296876
\(982\) −33.6125 −1.07262
\(983\) 33.8219 1.07875 0.539375 0.842066i \(-0.318661\pi\)
0.539375 + 0.842066i \(0.318661\pi\)
\(984\) −1.29844 −0.0413927
\(985\) −68.9109 −2.19568
\(986\) 0 0
\(987\) 2.70156 0.0859917
\(988\) −5.19375 −0.165235
\(989\) 0.701562 0.0223084
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −6.00000 −0.190500
\(993\) 33.6125 1.06666
\(994\) −5.40312 −0.171377
\(995\) −48.9109 −1.55058
\(996\) −15.4031 −0.488067
\(997\) 61.4031 1.94466 0.972328 0.233619i \(-0.0750568\pi\)
0.972328 + 0.233619i \(0.0750568\pi\)
\(998\) 27.0156 0.855165
\(999\) −2.70156 −0.0854736
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 966.2.a.m.1.2 2
3.2 odd 2 2898.2.a.bc.1.1 2
4.3 odd 2 7728.2.a.z.1.2 2
7.6 odd 2 6762.2.a.bq.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.m.1.2 2 1.1 even 1 trivial
2898.2.a.bc.1.1 2 3.2 odd 2
6762.2.a.bq.1.1 2 7.6 odd 2
7728.2.a.z.1.2 2 4.3 odd 2