Properties

Label 1755.2.a.q.1.3
Level $1755$
Weight $2$
Character 1755.1
Self dual yes
Analytic conductor $14.014$
Analytic rank $0$
Dimension $4$
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] = [1755,2,Mod(1,1755)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1755, 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("1755.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1755.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: \(14.0137455547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.12357.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 3x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.16027\) of defining polynomial
Character \(\chi\) \(=\) 1755.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.16027 q^{2} -0.653767 q^{4} +1.00000 q^{5} -1.26505 q^{7} -3.07909 q^{8} +O(q^{10})\) \(q+1.16027 q^{2} -0.653767 q^{4} +1.00000 q^{5} -1.26505 q^{7} -3.07909 q^{8} +1.16027 q^{10} +2.81404 q^{11} +1.00000 q^{13} -1.46781 q^{14} -2.26505 q^{16} -1.42533 q^{17} +6.05341 q^{19} -0.653767 q^{20} +3.26505 q^{22} +2.77156 q^{23} +1.00000 q^{25} +1.16027 q^{26} +0.827051 q^{28} -1.23937 q^{29} +0.653767 q^{31} +3.53011 q^{32} -1.65377 q^{34} -1.26505 q^{35} +4.39964 q^{37} +7.02360 q^{38} -3.07909 q^{40} +2.50651 q^{41} +6.27598 q^{43} -1.83973 q^{44} +3.21576 q^{46} +9.88012 q^{47} -5.39964 q^{49} +1.16027 q^{50} -0.653767 q^{52} +8.18387 q^{53} +2.81404 q^{55} +3.89522 q^{56} -1.43800 q^{58} -8.74379 q^{59} -0.957519 q^{61} +0.758548 q^{62} +8.62599 q^{64} +1.00000 q^{65} +10.7565 q^{67} +0.931832 q^{68} -1.46781 q^{70} +9.63901 q^{71} -5.09589 q^{73} +5.10478 q^{74} -3.95752 q^{76} -3.55991 q^{77} -10.8801 q^{79} -2.26505 q^{80} +2.90823 q^{82} +1.43119 q^{83} -1.42533 q^{85} +7.28185 q^{86} -8.66469 q^{88} +5.90581 q^{89} -1.26505 q^{91} -1.81195 q^{92} +11.4636 q^{94} +6.05341 q^{95} +11.8801 q^{97} -6.26505 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 3 q^{4} + 4 q^{5} + q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 3 q^{4} + 4 q^{5} + q^{7} + 3 q^{8} + q^{10} + 2 q^{11} + 4 q^{13} + 9 q^{14} - 3 q^{16} + 4 q^{17} - 4 q^{19} + 3 q^{20} + 7 q^{22} + 7 q^{23} + 4 q^{25} + q^{26} - 2 q^{28} + 14 q^{29} - 3 q^{31} + 2 q^{32} - q^{34} + q^{35} - 5 q^{37} + 14 q^{38} + 3 q^{40} + 12 q^{41} - 4 q^{43} - 11 q^{44} + 8 q^{46} + 11 q^{47} + q^{49} + q^{50} + 3 q^{52} + 15 q^{53} + 2 q^{55} + 18 q^{56} - 5 q^{58} + 9 q^{59} - 9 q^{61} - 5 q^{62} - 11 q^{64} + 4 q^{65} + 8 q^{67} - 4 q^{68} + 9 q^{70} - 3 q^{71} + 13 q^{73} + 18 q^{74} - 21 q^{76} + 12 q^{77} - 15 q^{79} - 3 q^{80} + 18 q^{82} + q^{83} + 4 q^{85} + 5 q^{86} - 6 q^{88} + 8 q^{89} + q^{91} + 29 q^{92} - 23 q^{94} - 4 q^{95} + 19 q^{97} - 19 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.16027 0.820437 0.410218 0.911987i \(-0.365452\pi\)
0.410218 + 0.911987i \(0.365452\pi\)
\(3\) 0 0
\(4\) −0.653767 −0.326884
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.26505 −0.478145 −0.239073 0.971002i \(-0.576843\pi\)
−0.239073 + 0.971002i \(0.576843\pi\)
\(8\) −3.07909 −1.08862
\(9\) 0 0
\(10\) 1.16027 0.366910
\(11\) 2.81404 0.848465 0.424232 0.905553i \(-0.360544\pi\)
0.424232 + 0.905553i \(0.360544\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −1.46781 −0.392288
\(15\) 0 0
\(16\) −2.26505 −0.566263
\(17\) −1.42533 −0.345692 −0.172846 0.984949i \(-0.555296\pi\)
−0.172846 + 0.984949i \(0.555296\pi\)
\(18\) 0 0
\(19\) 6.05341 1.38875 0.694373 0.719615i \(-0.255682\pi\)
0.694373 + 0.719615i \(0.255682\pi\)
\(20\) −0.653767 −0.146187
\(21\) 0 0
\(22\) 3.26505 0.696112
\(23\) 2.77156 0.577910 0.288955 0.957343i \(-0.406692\pi\)
0.288955 + 0.957343i \(0.406692\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.16027 0.227548
\(27\) 0 0
\(28\) 0.827051 0.156298
\(29\) −1.23937 −0.230145 −0.115072 0.993357i \(-0.536710\pi\)
−0.115072 + 0.993357i \(0.536710\pi\)
\(30\) 0 0
\(31\) 0.653767 0.117420 0.0587100 0.998275i \(-0.481301\pi\)
0.0587100 + 0.998275i \(0.481301\pi\)
\(32\) 3.53011 0.624041
\(33\) 0 0
\(34\) −1.65377 −0.283619
\(35\) −1.26505 −0.213833
\(36\) 0 0
\(37\) 4.39964 0.723296 0.361648 0.932315i \(-0.382214\pi\)
0.361648 + 0.932315i \(0.382214\pi\)
\(38\) 7.02360 1.13938
\(39\) 0 0
\(40\) −3.07909 −0.486847
\(41\) 2.50651 0.391450 0.195725 0.980659i \(-0.437294\pi\)
0.195725 + 0.980659i \(0.437294\pi\)
\(42\) 0 0
\(43\) 6.27598 0.957078 0.478539 0.878066i \(-0.341167\pi\)
0.478539 + 0.878066i \(0.341167\pi\)
\(44\) −1.83973 −0.277349
\(45\) 0 0
\(46\) 3.21576 0.474139
\(47\) 9.88012 1.44116 0.720582 0.693370i \(-0.243875\pi\)
0.720582 + 0.693370i \(0.243875\pi\)
\(48\) 0 0
\(49\) −5.39964 −0.771377
\(50\) 1.16027 0.164087
\(51\) 0 0
\(52\) −0.653767 −0.0906612
\(53\) 8.18387 1.12414 0.562071 0.827089i \(-0.310005\pi\)
0.562071 + 0.827089i \(0.310005\pi\)
\(54\) 0 0
\(55\) 2.81404 0.379445
\(56\) 3.89522 0.520521
\(57\) 0 0
\(58\) −1.43800 −0.188819
\(59\) −8.74379 −1.13834 −0.569172 0.822218i \(-0.692736\pi\)
−0.569172 + 0.822218i \(0.692736\pi\)
\(60\) 0 0
\(61\) −0.957519 −0.122598 −0.0612989 0.998119i \(-0.519524\pi\)
−0.0612989 + 0.998119i \(0.519524\pi\)
\(62\) 0.758548 0.0963357
\(63\) 0 0
\(64\) 8.62599 1.07825
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 10.7565 1.31411 0.657056 0.753842i \(-0.271802\pi\)
0.657056 + 0.753842i \(0.271802\pi\)
\(68\) 0.931832 0.113001
\(69\) 0 0
\(70\) −1.46781 −0.175437
\(71\) 9.63901 1.14394 0.571970 0.820275i \(-0.306179\pi\)
0.571970 + 0.820275i \(0.306179\pi\)
\(72\) 0 0
\(73\) −5.09589 −0.596428 −0.298214 0.954499i \(-0.596391\pi\)
−0.298214 + 0.954499i \(0.596391\pi\)
\(74\) 5.10478 0.593419
\(75\) 0 0
\(76\) −3.95752 −0.453959
\(77\) −3.55991 −0.405690
\(78\) 0 0
\(79\) −10.8801 −1.22411 −0.612055 0.790815i \(-0.709657\pi\)
−0.612055 + 0.790815i \(0.709657\pi\)
\(80\) −2.26505 −0.253241
\(81\) 0 0
\(82\) 2.90823 0.321160
\(83\) 1.43119 0.157094 0.0785470 0.996910i \(-0.474972\pi\)
0.0785470 + 0.996910i \(0.474972\pi\)
\(84\) 0 0
\(85\) −1.42533 −0.154598
\(86\) 7.28185 0.785222
\(87\) 0 0
\(88\) −8.66469 −0.923659
\(89\) 5.90581 0.626015 0.313007 0.949751i \(-0.398664\pi\)
0.313007 + 0.949751i \(0.398664\pi\)
\(90\) 0 0
\(91\) −1.26505 −0.132614
\(92\) −1.81195 −0.188909
\(93\) 0 0
\(94\) 11.4636 1.18238
\(95\) 6.05341 0.621066
\(96\) 0 0
\(97\) 11.8801 1.20624 0.603122 0.797649i \(-0.293923\pi\)
0.603122 + 0.797649i \(0.293923\pi\)
\(98\) −6.26505 −0.632866
\(99\) 0 0
\(100\) −0.653767 −0.0653767
\(101\) 14.1712 1.41009 0.705043 0.709164i \(-0.250927\pi\)
0.705043 + 0.709164i \(0.250927\pi\)
\(102\) 0 0
\(103\) −10.4530 −1.02997 −0.514985 0.857199i \(-0.672202\pi\)
−0.514985 + 0.857199i \(0.672202\pi\)
\(104\) −3.07909 −0.301930
\(105\) 0 0
\(106\) 9.49553 0.922287
\(107\) −14.8229 −1.43298 −0.716491 0.697596i \(-0.754253\pi\)
−0.716491 + 0.697596i \(0.754253\pi\)
\(108\) 0 0
\(109\) −18.1561 −1.73904 −0.869520 0.493898i \(-0.835571\pi\)
−0.869520 + 0.493898i \(0.835571\pi\)
\(110\) 3.26505 0.311311
\(111\) 0 0
\(112\) 2.86541 0.270756
\(113\) 9.10890 0.856893 0.428447 0.903567i \(-0.359061\pi\)
0.428447 + 0.903567i \(0.359061\pi\)
\(114\) 0 0
\(115\) 2.77156 0.258449
\(116\) 0.810257 0.0752305
\(117\) 0 0
\(118\) −10.1452 −0.933939
\(119\) 1.80311 0.165291
\(120\) 0 0
\(121\) −3.08118 −0.280107
\(122\) −1.11098 −0.100584
\(123\) 0 0
\(124\) −0.427412 −0.0383827
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.66469 0.413925 0.206962 0.978349i \(-0.433642\pi\)
0.206962 + 0.978349i \(0.433642\pi\)
\(128\) 2.94829 0.260595
\(129\) 0 0
\(130\) 1.16027 0.101763
\(131\) 6.48082 0.566232 0.283116 0.959086i \(-0.408632\pi\)
0.283116 + 0.959086i \(0.408632\pi\)
\(132\) 0 0
\(133\) −7.65788 −0.664023
\(134\) 12.4804 1.07814
\(135\) 0 0
\(136\) 4.38871 0.376329
\(137\) 15.2500 1.30289 0.651446 0.758695i \(-0.274163\pi\)
0.651446 + 0.758695i \(0.274163\pi\)
\(138\) 0 0
\(139\) −19.9339 −1.69077 −0.845384 0.534159i \(-0.820628\pi\)
−0.845384 + 0.534159i \(0.820628\pi\)
\(140\) 0.827051 0.0698985
\(141\) 0 0
\(142\) 11.1839 0.938530
\(143\) 2.81404 0.235322
\(144\) 0 0
\(145\) −1.23937 −0.102924
\(146\) −5.91262 −0.489332
\(147\) 0 0
\(148\) −2.87634 −0.236434
\(149\) −4.90614 −0.401927 −0.200964 0.979599i \(-0.564407\pi\)
−0.200964 + 0.979599i \(0.564407\pi\)
\(150\) 0 0
\(151\) −15.9835 −1.30072 −0.650359 0.759627i \(-0.725381\pi\)
−0.650359 + 0.759627i \(0.725381\pi\)
\(152\) −18.6390 −1.51182
\(153\) 0 0
\(154\) −4.13047 −0.332843
\(155\) 0.653767 0.0525119
\(156\) 0 0
\(157\) −13.0260 −1.03959 −0.519793 0.854292i \(-0.673991\pi\)
−0.519793 + 0.854292i \(0.673991\pi\)
\(158\) −12.6239 −1.00430
\(159\) 0 0
\(160\) 3.53011 0.279079
\(161\) −3.50617 −0.276325
\(162\) 0 0
\(163\) 5.39586 0.422636 0.211318 0.977417i \(-0.432224\pi\)
0.211318 + 0.977417i \(0.432224\pi\)
\(164\) −1.63867 −0.127959
\(165\) 0 0
\(166\) 1.66058 0.128886
\(167\) −15.0732 −1.16640 −0.583201 0.812328i \(-0.698200\pi\)
−0.583201 + 0.812328i \(0.698200\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −1.65377 −0.126838
\(171\) 0 0
\(172\) −4.10303 −0.312853
\(173\) 8.74762 0.665069 0.332535 0.943091i \(-0.392096\pi\)
0.332535 + 0.943091i \(0.392096\pi\)
\(174\) 0 0
\(175\) −1.26505 −0.0956291
\(176\) −6.37395 −0.480455
\(177\) 0 0
\(178\) 6.85235 0.513605
\(179\) −11.3777 −0.850412 −0.425206 0.905097i \(-0.639798\pi\)
−0.425206 + 0.905097i \(0.639798\pi\)
\(180\) 0 0
\(181\) 13.1136 0.974728 0.487364 0.873199i \(-0.337959\pi\)
0.487364 + 0.873199i \(0.337959\pi\)
\(182\) −1.46781 −0.108801
\(183\) 0 0
\(184\) −8.53389 −0.629127
\(185\) 4.39964 0.323468
\(186\) 0 0
\(187\) −4.01093 −0.293308
\(188\) −6.45930 −0.471093
\(189\) 0 0
\(190\) 7.02360 0.509546
\(191\) 18.8191 1.36170 0.680851 0.732422i \(-0.261610\pi\)
0.680851 + 0.732422i \(0.261610\pi\)
\(192\) 0 0
\(193\) −6.08530 −0.438029 −0.219015 0.975722i \(-0.570284\pi\)
−0.219015 + 0.975722i \(0.570284\pi\)
\(194\) 13.7842 0.989647
\(195\) 0 0
\(196\) 3.53011 0.252151
\(197\) −2.71304 −0.193296 −0.0966481 0.995319i \(-0.530812\pi\)
−0.0966481 + 0.995319i \(0.530812\pi\)
\(198\) 0 0
\(199\) 3.66848 0.260051 0.130026 0.991511i \(-0.458494\pi\)
0.130026 + 0.991511i \(0.458494\pi\)
\(200\) −3.07909 −0.217725
\(201\) 0 0
\(202\) 16.4425 1.15689
\(203\) 1.56787 0.110043
\(204\) 0 0
\(205\) 2.50651 0.175062
\(206\) −12.1284 −0.845024
\(207\) 0 0
\(208\) −2.26505 −0.157053
\(209\) 17.0345 1.17830
\(210\) 0 0
\(211\) −14.2199 −0.978937 −0.489469 0.872021i \(-0.662809\pi\)
−0.489469 + 0.872021i \(0.662809\pi\)
\(212\) −5.35035 −0.367464
\(213\) 0 0
\(214\) −17.1986 −1.17567
\(215\) 6.27598 0.428018
\(216\) 0 0
\(217\) −0.827051 −0.0561439
\(218\) −21.0660 −1.42677
\(219\) 0 0
\(220\) −1.83973 −0.124034
\(221\) −1.42533 −0.0958778
\(222\) 0 0
\(223\) 21.3756 1.43142 0.715710 0.698398i \(-0.246103\pi\)
0.715710 + 0.698398i \(0.246103\pi\)
\(224\) −4.46578 −0.298382
\(225\) 0 0
\(226\) 10.5688 0.703026
\(227\) −4.24523 −0.281766 −0.140883 0.990026i \(-0.544994\pi\)
−0.140883 + 0.990026i \(0.544994\pi\)
\(228\) 0 0
\(229\) −7.91916 −0.523313 −0.261656 0.965161i \(-0.584269\pi\)
−0.261656 + 0.965161i \(0.584269\pi\)
\(230\) 3.21576 0.212041
\(231\) 0 0
\(232\) 3.81613 0.250541
\(233\) −0.657550 −0.0430775 −0.0215388 0.999768i \(-0.506857\pi\)
−0.0215388 + 0.999768i \(0.506857\pi\)
\(234\) 0 0
\(235\) 9.88012 0.644508
\(236\) 5.71640 0.372106
\(237\) 0 0
\(238\) 2.09210 0.135611
\(239\) −1.98227 −0.128222 −0.0641111 0.997943i \(-0.520421\pi\)
−0.0641111 + 0.997943i \(0.520421\pi\)
\(240\) 0 0
\(241\) 1.35716 0.0874222 0.0437111 0.999044i \(-0.486082\pi\)
0.0437111 + 0.999044i \(0.486082\pi\)
\(242\) −3.57501 −0.229810
\(243\) 0 0
\(244\) 0.625995 0.0400752
\(245\) −5.39964 −0.344970
\(246\) 0 0
\(247\) 6.05341 0.385169
\(248\) −2.01301 −0.127826
\(249\) 0 0
\(250\) 1.16027 0.0733821
\(251\) 10.3668 0.654347 0.327174 0.944964i \(-0.393904\pi\)
0.327174 + 0.944964i \(0.393904\pi\)
\(252\) 0 0
\(253\) 7.79928 0.490336
\(254\) 5.41232 0.339599
\(255\) 0 0
\(256\) −13.8312 −0.864448
\(257\) 9.28070 0.578914 0.289457 0.957191i \(-0.406525\pi\)
0.289457 + 0.957191i \(0.406525\pi\)
\(258\) 0 0
\(259\) −5.56578 −0.345841
\(260\) −0.653767 −0.0405449
\(261\) 0 0
\(262\) 7.51952 0.464557
\(263\) −9.62094 −0.593252 −0.296626 0.954994i \(-0.595862\pi\)
−0.296626 + 0.954994i \(0.595862\pi\)
\(264\) 0 0
\(265\) 8.18387 0.502731
\(266\) −8.88523 −0.544789
\(267\) 0 0
\(268\) −7.03222 −0.429561
\(269\) 4.77628 0.291215 0.145608 0.989342i \(-0.453486\pi\)
0.145608 + 0.989342i \(0.453486\pi\)
\(270\) 0 0
\(271\) −15.3787 −0.934188 −0.467094 0.884208i \(-0.654699\pi\)
−0.467094 + 0.884208i \(0.654699\pi\)
\(272\) 3.22844 0.195753
\(273\) 0 0
\(274\) 17.6941 1.06894
\(275\) 2.81404 0.169693
\(276\) 0 0
\(277\) 6.33262 0.380490 0.190245 0.981737i \(-0.439072\pi\)
0.190245 + 0.981737i \(0.439072\pi\)
\(278\) −23.1287 −1.38717
\(279\) 0 0
\(280\) 3.89522 0.232784
\(281\) 17.3928 1.03757 0.518785 0.854905i \(-0.326385\pi\)
0.518785 + 0.854905i \(0.326385\pi\)
\(282\) 0 0
\(283\) −24.6012 −1.46239 −0.731196 0.682167i \(-0.761038\pi\)
−0.731196 + 0.682167i \(0.761038\pi\)
\(284\) −6.30167 −0.373935
\(285\) 0 0
\(286\) 3.26505 0.193067
\(287\) −3.17086 −0.187170
\(288\) 0 0
\(289\) −14.9684 −0.880497
\(290\) −1.43800 −0.0844424
\(291\) 0 0
\(292\) 3.33152 0.194963
\(293\) 19.7462 1.15358 0.576791 0.816892i \(-0.304305\pi\)
0.576791 + 0.816892i \(0.304305\pi\)
\(294\) 0 0
\(295\) −8.74379 −0.509083
\(296\) −13.5469 −0.787398
\(297\) 0 0
\(298\) −5.69247 −0.329756
\(299\) 2.77156 0.160283
\(300\) 0 0
\(301\) −7.93945 −0.457622
\(302\) −18.5452 −1.06716
\(303\) 0 0
\(304\) −13.7113 −0.786396
\(305\) −0.957519 −0.0548274
\(306\) 0 0
\(307\) −24.1249 −1.37688 −0.688440 0.725293i \(-0.741704\pi\)
−0.688440 + 0.725293i \(0.741704\pi\)
\(308\) 2.32735 0.132613
\(309\) 0 0
\(310\) 0.758548 0.0430827
\(311\) −20.6794 −1.17262 −0.586311 0.810086i \(-0.699420\pi\)
−0.586311 + 0.810086i \(0.699420\pi\)
\(312\) 0 0
\(313\) −9.27565 −0.524290 −0.262145 0.965028i \(-0.584430\pi\)
−0.262145 + 0.965028i \(0.584430\pi\)
\(314\) −15.1137 −0.852914
\(315\) 0 0
\(316\) 7.11307 0.400141
\(317\) −25.8276 −1.45062 −0.725311 0.688421i \(-0.758304\pi\)
−0.725311 + 0.688421i \(0.758304\pi\)
\(318\) 0 0
\(319\) −3.48763 −0.195270
\(320\) 8.62599 0.482208
\(321\) 0 0
\(322\) −4.06811 −0.226707
\(323\) −8.62808 −0.480079
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 6.26066 0.346746
\(327\) 0 0
\(328\) −7.71776 −0.426142
\(329\) −12.4989 −0.689086
\(330\) 0 0
\(331\) −2.57225 −0.141384 −0.0706919 0.997498i \(-0.522521\pi\)
−0.0706919 + 0.997498i \(0.522521\pi\)
\(332\) −0.935668 −0.0513515
\(333\) 0 0
\(334\) −17.4891 −0.956958
\(335\) 10.7565 0.587688
\(336\) 0 0
\(337\) 15.0812 0.821524 0.410762 0.911743i \(-0.365263\pi\)
0.410762 + 0.911743i \(0.365263\pi\)
\(338\) 1.16027 0.0631105
\(339\) 0 0
\(340\) 0.931832 0.0505357
\(341\) 1.83973 0.0996268
\(342\) 0 0
\(343\) 15.6862 0.846976
\(344\) −19.3243 −1.04190
\(345\) 0 0
\(346\) 10.1496 0.545647
\(347\) −10.1438 −0.544546 −0.272273 0.962220i \(-0.587775\pi\)
−0.272273 + 0.962220i \(0.587775\pi\)
\(348\) 0 0
\(349\) 9.88760 0.529271 0.264636 0.964348i \(-0.414748\pi\)
0.264636 + 0.964348i \(0.414748\pi\)
\(350\) −1.46781 −0.0784576
\(351\) 0 0
\(352\) 9.93386 0.529477
\(353\) −24.0256 −1.27875 −0.639376 0.768894i \(-0.720807\pi\)
−0.639376 + 0.768894i \(0.720807\pi\)
\(354\) 0 0
\(355\) 9.63901 0.511585
\(356\) −3.86103 −0.204634
\(357\) 0 0
\(358\) −13.2013 −0.697709
\(359\) −10.5948 −0.559171 −0.279585 0.960121i \(-0.590197\pi\)
−0.279585 + 0.960121i \(0.590197\pi\)
\(360\) 0 0
\(361\) 17.6437 0.928617
\(362\) 15.2154 0.799702
\(363\) 0 0
\(364\) 0.827051 0.0433492
\(365\) −5.09589 −0.266731
\(366\) 0 0
\(367\) 3.44481 0.179818 0.0899088 0.995950i \(-0.471342\pi\)
0.0899088 + 0.995950i \(0.471342\pi\)
\(368\) −6.27773 −0.327249
\(369\) 0 0
\(370\) 5.10478 0.265385
\(371\) −10.3530 −0.537503
\(372\) 0 0
\(373\) −11.9726 −0.619916 −0.309958 0.950750i \(-0.600315\pi\)
−0.309958 + 0.950750i \(0.600315\pi\)
\(374\) −4.65377 −0.240641
\(375\) 0 0
\(376\) −30.4218 −1.56889
\(377\) −1.23937 −0.0638306
\(378\) 0 0
\(379\) −14.1877 −0.728771 −0.364386 0.931248i \(-0.618721\pi\)
−0.364386 + 0.931248i \(0.618721\pi\)
\(380\) −3.95752 −0.203016
\(381\) 0 0
\(382\) 21.8353 1.11719
\(383\) −19.0968 −0.975802 −0.487901 0.872899i \(-0.662237\pi\)
−0.487901 + 0.872899i \(0.662237\pi\)
\(384\) 0 0
\(385\) −3.55991 −0.181430
\(386\) −7.06060 −0.359375
\(387\) 0 0
\(388\) −7.76684 −0.394301
\(389\) 27.3794 1.38819 0.694096 0.719882i \(-0.255804\pi\)
0.694096 + 0.719882i \(0.255804\pi\)
\(390\) 0 0
\(391\) −3.95038 −0.199779
\(392\) 16.6260 0.839740
\(393\) 0 0
\(394\) −3.14787 −0.158587
\(395\) −10.8801 −0.547438
\(396\) 0 0
\(397\) 11.0181 0.552981 0.276490 0.961017i \(-0.410829\pi\)
0.276490 + 0.961017i \(0.410829\pi\)
\(398\) 4.25643 0.213356
\(399\) 0 0
\(400\) −2.26505 −0.113253
\(401\) −1.33058 −0.0664462 −0.0332231 0.999448i \(-0.510577\pi\)
−0.0332231 + 0.999448i \(0.510577\pi\)
\(402\) 0 0
\(403\) 0.653767 0.0325665
\(404\) −9.26467 −0.460934
\(405\) 0 0
\(406\) 1.81915 0.0902829
\(407\) 12.3808 0.613692
\(408\) 0 0
\(409\) 16.6691 0.824236 0.412118 0.911130i \(-0.364789\pi\)
0.412118 + 0.911130i \(0.364789\pi\)
\(410\) 2.90823 0.143627
\(411\) 0 0
\(412\) 6.83386 0.336680
\(413\) 11.0614 0.544294
\(414\) 0 0
\(415\) 1.43119 0.0702546
\(416\) 3.53011 0.173078
\(417\) 0 0
\(418\) 19.7647 0.966723
\(419\) −22.0593 −1.07767 −0.538833 0.842413i \(-0.681135\pi\)
−0.538833 + 0.842413i \(0.681135\pi\)
\(420\) 0 0
\(421\) 18.2907 0.891434 0.445717 0.895174i \(-0.352949\pi\)
0.445717 + 0.895174i \(0.352949\pi\)
\(422\) −16.4989 −0.803156
\(423\) 0 0
\(424\) −25.1989 −1.22377
\(425\) −1.42533 −0.0691385
\(426\) 0 0
\(427\) 1.21131 0.0586196
\(428\) 9.69071 0.468418
\(429\) 0 0
\(430\) 7.28185 0.351162
\(431\) −5.83939 −0.281274 −0.140637 0.990061i \(-0.544915\pi\)
−0.140637 + 0.990061i \(0.544915\pi\)
\(432\) 0 0
\(433\) 9.49174 0.456144 0.228072 0.973644i \(-0.426758\pi\)
0.228072 + 0.973644i \(0.426758\pi\)
\(434\) −0.959604 −0.0460625
\(435\) 0 0
\(436\) 11.8699 0.568463
\(437\) 16.7774 0.802571
\(438\) 0 0
\(439\) 17.1568 0.818848 0.409424 0.912344i \(-0.365730\pi\)
0.409424 + 0.912344i \(0.365730\pi\)
\(440\) −8.66469 −0.413073
\(441\) 0 0
\(442\) −1.65377 −0.0786617
\(443\) −2.21610 −0.105290 −0.0526450 0.998613i \(-0.516765\pi\)
−0.0526450 + 0.998613i \(0.516765\pi\)
\(444\) 0 0
\(445\) 5.90581 0.279962
\(446\) 24.8016 1.17439
\(447\) 0 0
\(448\) −10.9123 −0.515560
\(449\) −31.1619 −1.47062 −0.735311 0.677730i \(-0.762964\pi\)
−0.735311 + 0.677730i \(0.762964\pi\)
\(450\) 0 0
\(451\) 7.05341 0.332132
\(452\) −5.95510 −0.280104
\(453\) 0 0
\(454\) −4.92563 −0.231171
\(455\) −1.26505 −0.0593066
\(456\) 0 0
\(457\) 29.1812 1.36504 0.682519 0.730868i \(-0.260884\pi\)
0.682519 + 0.730868i \(0.260884\pi\)
\(458\) −9.18838 −0.429345
\(459\) 0 0
\(460\) −1.81195 −0.0844828
\(461\) 24.6927 1.15006 0.575028 0.818134i \(-0.304991\pi\)
0.575028 + 0.818134i \(0.304991\pi\)
\(462\) 0 0
\(463\) 36.2844 1.68628 0.843140 0.537694i \(-0.180704\pi\)
0.843140 + 0.537694i \(0.180704\pi\)
\(464\) 2.80723 0.130322
\(465\) 0 0
\(466\) −0.762937 −0.0353424
\(467\) 8.19803 0.379360 0.189680 0.981846i \(-0.439255\pi\)
0.189680 + 0.981846i \(0.439255\pi\)
\(468\) 0 0
\(469\) −13.6075 −0.628336
\(470\) 11.4636 0.528778
\(471\) 0 0
\(472\) 26.9229 1.23923
\(473\) 17.6609 0.812047
\(474\) 0 0
\(475\) 6.05341 0.277749
\(476\) −1.17882 −0.0540310
\(477\) 0 0
\(478\) −2.29997 −0.105198
\(479\) −12.6946 −0.580029 −0.290014 0.957022i \(-0.593660\pi\)
−0.290014 + 0.957022i \(0.593660\pi\)
\(480\) 0 0
\(481\) 4.39964 0.200606
\(482\) 1.57467 0.0717244
\(483\) 0 0
\(484\) 2.01437 0.0915625
\(485\) 11.8801 0.539449
\(486\) 0 0
\(487\) 12.1140 0.548936 0.274468 0.961596i \(-0.411498\pi\)
0.274468 + 0.961596i \(0.411498\pi\)
\(488\) 2.94829 0.133463
\(489\) 0 0
\(490\) −6.26505 −0.283026
\(491\) −37.7496 −1.70362 −0.851808 0.523854i \(-0.824494\pi\)
−0.851808 + 0.523854i \(0.824494\pi\)
\(492\) 0 0
\(493\) 1.76650 0.0795592
\(494\) 7.02360 0.316007
\(495\) 0 0
\(496\) −1.48082 −0.0664907
\(497\) −12.1939 −0.546969
\(498\) 0 0
\(499\) 10.9688 0.491030 0.245515 0.969393i \(-0.421043\pi\)
0.245515 + 0.969393i \(0.421043\pi\)
\(500\) −0.653767 −0.0292374
\(501\) 0 0
\(502\) 12.0283 0.536850
\(503\) −4.15582 −0.185299 −0.0926495 0.995699i \(-0.529534\pi\)
−0.0926495 + 0.995699i \(0.529534\pi\)
\(504\) 0 0
\(505\) 14.1712 0.630610
\(506\) 9.04929 0.402290
\(507\) 0 0
\(508\) −3.04962 −0.135305
\(509\) 0.895219 0.0396799 0.0198399 0.999803i \(-0.493684\pi\)
0.0198399 + 0.999803i \(0.493684\pi\)
\(510\) 0 0
\(511\) 6.44657 0.285180
\(512\) −21.9445 −0.969819
\(513\) 0 0
\(514\) 10.7681 0.474963
\(515\) −10.4530 −0.460616
\(516\) 0 0
\(517\) 27.8031 1.22278
\(518\) −6.45782 −0.283740
\(519\) 0 0
\(520\) −3.07909 −0.135027
\(521\) 41.6448 1.82449 0.912246 0.409642i \(-0.134346\pi\)
0.912246 + 0.409642i \(0.134346\pi\)
\(522\) 0 0
\(523\) −35.0731 −1.53364 −0.766820 0.641862i \(-0.778162\pi\)
−0.766820 + 0.641862i \(0.778162\pi\)
\(524\) −4.23695 −0.185092
\(525\) 0 0
\(526\) −11.1629 −0.486726
\(527\) −0.931832 −0.0405912
\(528\) 0 0
\(529\) −15.3185 −0.666020
\(530\) 9.49553 0.412459
\(531\) 0 0
\(532\) 5.00647 0.217058
\(533\) 2.50651 0.108569
\(534\) 0 0
\(535\) −14.8229 −0.640849
\(536\) −33.1202 −1.43057
\(537\) 0 0
\(538\) 5.54179 0.238924
\(539\) −15.1948 −0.654486
\(540\) 0 0
\(541\) −11.3444 −0.487735 −0.243868 0.969809i \(-0.578416\pi\)
−0.243868 + 0.969809i \(0.578416\pi\)
\(542\) −17.8435 −0.766442
\(543\) 0 0
\(544\) −5.03156 −0.215726
\(545\) −18.1561 −0.777722
\(546\) 0 0
\(547\) 32.3170 1.38177 0.690887 0.722962i \(-0.257220\pi\)
0.690887 + 0.722962i \(0.257220\pi\)
\(548\) −9.96992 −0.425894
\(549\) 0 0
\(550\) 3.26505 0.139222
\(551\) −7.50239 −0.319612
\(552\) 0 0
\(553\) 13.7639 0.585302
\(554\) 7.34756 0.312168
\(555\) 0 0
\(556\) 13.0321 0.552684
\(557\) 41.1683 1.74436 0.872178 0.489188i \(-0.162707\pi\)
0.872178 + 0.489188i \(0.162707\pi\)
\(558\) 0 0
\(559\) 6.27598 0.265446
\(560\) 2.86541 0.121086
\(561\) 0 0
\(562\) 20.1804 0.851260
\(563\) 36.2940 1.52961 0.764805 0.644262i \(-0.222835\pi\)
0.764805 + 0.644262i \(0.222835\pi\)
\(564\) 0 0
\(565\) 9.10890 0.383214
\(566\) −28.5442 −1.19980
\(567\) 0 0
\(568\) −29.6794 −1.24532
\(569\) −16.7700 −0.703036 −0.351518 0.936181i \(-0.614334\pi\)
−0.351518 + 0.936181i \(0.614334\pi\)
\(570\) 0 0
\(571\) −21.2366 −0.888725 −0.444362 0.895847i \(-0.646570\pi\)
−0.444362 + 0.895847i \(0.646570\pi\)
\(572\) −1.83973 −0.0769229
\(573\) 0 0
\(574\) −3.67907 −0.153561
\(575\) 2.77156 0.115582
\(576\) 0 0
\(577\) −32.4640 −1.35149 −0.675746 0.737134i \(-0.736179\pi\)
−0.675746 + 0.737134i \(0.736179\pi\)
\(578\) −17.3675 −0.722392
\(579\) 0 0
\(580\) 0.810257 0.0336441
\(581\) −1.81054 −0.0751138
\(582\) 0 0
\(583\) 23.0298 0.953795
\(584\) 15.6907 0.649286
\(585\) 0 0
\(586\) 22.9109 0.946441
\(587\) 12.3701 0.510569 0.255285 0.966866i \(-0.417831\pi\)
0.255285 + 0.966866i \(0.417831\pi\)
\(588\) 0 0
\(589\) 3.95752 0.163067
\(590\) −10.1452 −0.417670
\(591\) 0 0
\(592\) −9.96542 −0.409576
\(593\) 36.0294 1.47955 0.739775 0.672854i \(-0.234932\pi\)
0.739775 + 0.672854i \(0.234932\pi\)
\(594\) 0 0
\(595\) 1.80311 0.0739205
\(596\) 3.20748 0.131383
\(597\) 0 0
\(598\) 3.21576 0.131502
\(599\) 30.2656 1.23662 0.618310 0.785935i \(-0.287818\pi\)
0.618310 + 0.785935i \(0.287818\pi\)
\(600\) 0 0
\(601\) −26.7110 −1.08956 −0.544781 0.838578i \(-0.683387\pi\)
−0.544781 + 0.838578i \(0.683387\pi\)
\(602\) −9.21193 −0.375450
\(603\) 0 0
\(604\) 10.4495 0.425183
\(605\) −3.08118 −0.125268
\(606\) 0 0
\(607\) 1.11543 0.0452737 0.0226369 0.999744i \(-0.492794\pi\)
0.0226369 + 0.999744i \(0.492794\pi\)
\(608\) 21.3692 0.866634
\(609\) 0 0
\(610\) −1.11098 −0.0449824
\(611\) 9.88012 0.399707
\(612\) 0 0
\(613\) −38.6446 −1.56084 −0.780421 0.625255i \(-0.784995\pi\)
−0.780421 + 0.625255i \(0.784995\pi\)
\(614\) −27.9914 −1.12964
\(615\) 0 0
\(616\) 10.9613 0.441643
\(617\) −34.0379 −1.37032 −0.685158 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(618\) 0 0
\(619\) −30.2665 −1.21651 −0.608256 0.793741i \(-0.708131\pi\)
−0.608256 + 0.793741i \(0.708131\pi\)
\(620\) −0.427412 −0.0171653
\(621\) 0 0
\(622\) −23.9937 −0.962061
\(623\) −7.47117 −0.299326
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −10.7623 −0.430147
\(627\) 0 0
\(628\) 8.51595 0.339823
\(629\) −6.27092 −0.250038
\(630\) 0 0
\(631\) 3.44426 0.137114 0.0685569 0.997647i \(-0.478161\pi\)
0.0685569 + 0.997647i \(0.478161\pi\)
\(632\) 33.5009 1.33259
\(633\) 0 0
\(634\) −29.9671 −1.19014
\(635\) 4.66469 0.185113
\(636\) 0 0
\(637\) −5.39964 −0.213941
\(638\) −4.04660 −0.160206
\(639\) 0 0
\(640\) 2.94829 0.116541
\(641\) −25.0135 −0.987974 −0.493987 0.869469i \(-0.664461\pi\)
−0.493987 + 0.869469i \(0.664461\pi\)
\(642\) 0 0
\(643\) 17.7996 0.701948 0.350974 0.936385i \(-0.385851\pi\)
0.350974 + 0.936385i \(0.385851\pi\)
\(644\) 2.29222 0.0903261
\(645\) 0 0
\(646\) −10.0109 −0.393875
\(647\) 36.6601 1.44126 0.720629 0.693321i \(-0.243853\pi\)
0.720629 + 0.693321i \(0.243853\pi\)
\(648\) 0 0
\(649\) −24.6054 −0.965845
\(650\) 1.16027 0.0455096
\(651\) 0 0
\(652\) −3.52763 −0.138153
\(653\) −18.1192 −0.709057 −0.354529 0.935045i \(-0.615359\pi\)
−0.354529 + 0.935045i \(0.615359\pi\)
\(654\) 0 0
\(655\) 6.48082 0.253227
\(656\) −5.67737 −0.221664
\(657\) 0 0
\(658\) −14.5021 −0.565351
\(659\) −8.26297 −0.321880 −0.160940 0.986964i \(-0.551452\pi\)
−0.160940 + 0.986964i \(0.551452\pi\)
\(660\) 0 0
\(661\) −30.0331 −1.16815 −0.584076 0.811699i \(-0.698543\pi\)
−0.584076 + 0.811699i \(0.698543\pi\)
\(662\) −2.98452 −0.115997
\(663\) 0 0
\(664\) −4.40678 −0.171016
\(665\) −7.65788 −0.296960
\(666\) 0 0
\(667\) −3.43498 −0.133003
\(668\) 9.85438 0.381277
\(669\) 0 0
\(670\) 12.4804 0.482161
\(671\) −2.69450 −0.104020
\(672\) 0 0
\(673\) 36.1463 1.39334 0.696668 0.717394i \(-0.254665\pi\)
0.696668 + 0.717394i \(0.254665\pi\)
\(674\) 17.4983 0.674009
\(675\) 0 0
\(676\) −0.653767 −0.0251449
\(677\) 4.27098 0.164147 0.0820735 0.996626i \(-0.473846\pi\)
0.0820735 + 0.996626i \(0.473846\pi\)
\(678\) 0 0
\(679\) −15.0290 −0.576760
\(680\) 4.38871 0.168299
\(681\) 0 0
\(682\) 2.13459 0.0817375
\(683\) 7.91443 0.302837 0.151419 0.988470i \(-0.451616\pi\)
0.151419 + 0.988470i \(0.451616\pi\)
\(684\) 0 0
\(685\) 15.2500 0.582671
\(686\) 18.2003 0.694890
\(687\) 0 0
\(688\) −14.2154 −0.541958
\(689\) 8.18387 0.311781
\(690\) 0 0
\(691\) −32.5914 −1.23984 −0.619918 0.784667i \(-0.712834\pi\)
−0.619918 + 0.784667i \(0.712834\pi\)
\(692\) −5.71891 −0.217400
\(693\) 0 0
\(694\) −11.7695 −0.446765
\(695\) −19.9339 −0.756135
\(696\) 0 0
\(697\) −3.57259 −0.135321
\(698\) 11.4723 0.434233
\(699\) 0 0
\(700\) 0.827051 0.0312596
\(701\) 29.5797 1.11721 0.558606 0.829433i \(-0.311337\pi\)
0.558606 + 0.829433i \(0.311337\pi\)
\(702\) 0 0
\(703\) 26.6328 1.00448
\(704\) 24.2739 0.914857
\(705\) 0 0
\(706\) −27.8762 −1.04914
\(707\) −17.9273 −0.674227
\(708\) 0 0
\(709\) 18.5817 0.697851 0.348925 0.937151i \(-0.386547\pi\)
0.348925 + 0.937151i \(0.386547\pi\)
\(710\) 11.1839 0.419723
\(711\) 0 0
\(712\) −18.1845 −0.681495
\(713\) 1.81195 0.0678582
\(714\) 0 0
\(715\) 2.81404 0.105239
\(716\) 7.43839 0.277986
\(717\) 0 0
\(718\) −12.2928 −0.458764
\(719\) 9.75410 0.363766 0.181883 0.983320i \(-0.441781\pi\)
0.181883 + 0.983320i \(0.441781\pi\)
\(720\) 0 0
\(721\) 13.2237 0.492475
\(722\) 20.4715 0.761872
\(723\) 0 0
\(724\) −8.57326 −0.318623
\(725\) −1.23937 −0.0460289
\(726\) 0 0
\(727\) −34.8851 −1.29382 −0.646909 0.762567i \(-0.723939\pi\)
−0.646909 + 0.762567i \(0.723939\pi\)
\(728\) 3.89522 0.144366
\(729\) 0 0
\(730\) −5.91262 −0.218836
\(731\) −8.94532 −0.330855
\(732\) 0 0
\(733\) 13.0790 0.483085 0.241543 0.970390i \(-0.422347\pi\)
0.241543 + 0.970390i \(0.422347\pi\)
\(734\) 3.99692 0.147529
\(735\) 0 0
\(736\) 9.78390 0.360639
\(737\) 30.2691 1.11498
\(738\) 0 0
\(739\) −16.5102 −0.607339 −0.303669 0.952777i \(-0.598212\pi\)
−0.303669 + 0.952777i \(0.598212\pi\)
\(740\) −2.87634 −0.105736
\(741\) 0 0
\(742\) −12.0124 −0.440987
\(743\) 49.4943 1.81577 0.907885 0.419218i \(-0.137696\pi\)
0.907885 + 0.419218i \(0.137696\pi\)
\(744\) 0 0
\(745\) −4.90614 −0.179747
\(746\) −13.8914 −0.508602
\(747\) 0 0
\(748\) 2.62221 0.0958776
\(749\) 18.7517 0.685174
\(750\) 0 0
\(751\) −31.6331 −1.15431 −0.577155 0.816635i \(-0.695837\pi\)
−0.577155 + 0.816635i \(0.695837\pi\)
\(752\) −22.3790 −0.816078
\(753\) 0 0
\(754\) −1.43800 −0.0523690
\(755\) −15.9835 −0.581699
\(756\) 0 0
\(757\) 1.80575 0.0656312 0.0328156 0.999461i \(-0.489553\pi\)
0.0328156 + 0.999461i \(0.489553\pi\)
\(758\) −16.4616 −0.597910
\(759\) 0 0
\(760\) −18.6390 −0.676108
\(761\) −36.2291 −1.31330 −0.656651 0.754194i \(-0.728028\pi\)
−0.656651 + 0.754194i \(0.728028\pi\)
\(762\) 0 0
\(763\) 22.9684 0.831514
\(764\) −12.3033 −0.445118
\(765\) 0 0
\(766\) −22.1575 −0.800584
\(767\) −8.74379 −0.315720
\(768\) 0 0
\(769\) 51.1502 1.84452 0.922261 0.386567i \(-0.126339\pi\)
0.922261 + 0.386567i \(0.126339\pi\)
\(770\) −4.13047 −0.148852
\(771\) 0 0
\(772\) 3.97837 0.143185
\(773\) −32.5307 −1.17005 −0.585023 0.811016i \(-0.698915\pi\)
−0.585023 + 0.811016i \(0.698915\pi\)
\(774\) 0 0
\(775\) 0.653767 0.0234840
\(776\) −36.5800 −1.31315
\(777\) 0 0
\(778\) 31.7676 1.13892
\(779\) 15.1729 0.543625
\(780\) 0 0
\(781\) 27.1245 0.970593
\(782\) −4.58351 −0.163906
\(783\) 0 0
\(784\) 12.2305 0.436803
\(785\) −13.0260 −0.464917
\(786\) 0 0
\(787\) 44.4269 1.58365 0.791824 0.610749i \(-0.209132\pi\)
0.791824 + 0.610749i \(0.209132\pi\)
\(788\) 1.77370 0.0631854
\(789\) 0 0
\(790\) −12.6239 −0.449139
\(791\) −11.5232 −0.409719
\(792\) 0 0
\(793\) −0.957519 −0.0340025
\(794\) 12.7840 0.453686
\(795\) 0 0
\(796\) −2.39833 −0.0850065
\(797\) −11.7800 −0.417269 −0.208635 0.977994i \(-0.566902\pi\)
−0.208635 + 0.977994i \(0.566902\pi\)
\(798\) 0 0
\(799\) −14.0824 −0.498200
\(800\) 3.53011 0.124808
\(801\) 0 0
\(802\) −1.54384 −0.0545149
\(803\) −14.3400 −0.506049
\(804\) 0 0
\(805\) −3.50617 −0.123576
\(806\) 0.758548 0.0267187
\(807\) 0 0
\(808\) −43.6344 −1.53505
\(809\) 27.9603 0.983032 0.491516 0.870869i \(-0.336443\pi\)
0.491516 + 0.870869i \(0.336443\pi\)
\(810\) 0 0
\(811\) 25.6760 0.901608 0.450804 0.892623i \(-0.351137\pi\)
0.450804 + 0.892623i \(0.351137\pi\)
\(812\) −1.02502 −0.0359711
\(813\) 0 0
\(814\) 14.3651 0.503495
\(815\) 5.39586 0.189009
\(816\) 0 0
\(817\) 37.9911 1.32914
\(818\) 19.3408 0.676234
\(819\) 0 0
\(820\) −1.63867 −0.0572249
\(821\) 44.8582 1.56556 0.782780 0.622298i \(-0.213801\pi\)
0.782780 + 0.622298i \(0.213801\pi\)
\(822\) 0 0
\(823\) −13.9798 −0.487305 −0.243652 0.969863i \(-0.578346\pi\)
−0.243652 + 0.969863i \(0.578346\pi\)
\(824\) 32.1859 1.12125
\(825\) 0 0
\(826\) 12.8342 0.446559
\(827\) 38.0478 1.32305 0.661526 0.749922i \(-0.269909\pi\)
0.661526 + 0.749922i \(0.269909\pi\)
\(828\) 0 0
\(829\) −31.8983 −1.10787 −0.553937 0.832559i \(-0.686875\pi\)
−0.553937 + 0.832559i \(0.686875\pi\)
\(830\) 1.66058 0.0576394
\(831\) 0 0
\(832\) 8.62599 0.299053
\(833\) 7.69625 0.266659
\(834\) 0 0
\(835\) −15.0732 −0.521630
\(836\) −11.1366 −0.385168
\(837\) 0 0
\(838\) −25.5948 −0.884157
\(839\) 4.74170 0.163702 0.0818509 0.996645i \(-0.473917\pi\)
0.0818509 + 0.996645i \(0.473917\pi\)
\(840\) 0 0
\(841\) −27.4640 −0.947033
\(842\) 21.2222 0.731365
\(843\) 0 0
\(844\) 9.29649 0.319998
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 3.89786 0.133932
\(848\) −18.5369 −0.636560
\(849\) 0 0
\(850\) −1.65377 −0.0567238
\(851\) 12.1939 0.418000
\(852\) 0 0
\(853\) −47.4163 −1.62350 −0.811752 0.584002i \(-0.801486\pi\)
−0.811752 + 0.584002i \(0.801486\pi\)
\(854\) 1.40545 0.0480936
\(855\) 0 0
\(856\) 45.6410 1.55998
\(857\) −6.78726 −0.231848 −0.115924 0.993258i \(-0.536983\pi\)
−0.115924 + 0.993258i \(0.536983\pi\)
\(858\) 0 0
\(859\) −2.73973 −0.0934784 −0.0467392 0.998907i \(-0.514883\pi\)
−0.0467392 + 0.998907i \(0.514883\pi\)
\(860\) −4.10303 −0.139912
\(861\) 0 0
\(862\) −6.77529 −0.230767
\(863\) −16.4176 −0.558861 −0.279430 0.960166i \(-0.590146\pi\)
−0.279430 + 0.960166i \(0.590146\pi\)
\(864\) 0 0
\(865\) 8.74762 0.297428
\(866\) 11.0130 0.374237
\(867\) 0 0
\(868\) 0.540699 0.0183525
\(869\) −30.6171 −1.03861
\(870\) 0 0
\(871\) 10.7565 0.364469
\(872\) 55.9043 1.89316
\(873\) 0 0
\(874\) 19.4663 0.658458
\(875\) −1.26505 −0.0427666
\(876\) 0 0
\(877\) 8.85625 0.299054 0.149527 0.988758i \(-0.452225\pi\)
0.149527 + 0.988758i \(0.452225\pi\)
\(878\) 19.9065 0.671813
\(879\) 0 0
\(880\) −6.37395 −0.214866
\(881\) −4.18037 −0.140840 −0.0704202 0.997517i \(-0.522434\pi\)
−0.0704202 + 0.997517i \(0.522434\pi\)
\(882\) 0 0
\(883\) 6.21879 0.209279 0.104639 0.994510i \(-0.466631\pi\)
0.104639 + 0.994510i \(0.466631\pi\)
\(884\) 0.931832 0.0313409
\(885\) 0 0
\(886\) −2.57128 −0.0863838
\(887\) −16.2949 −0.547130 −0.273565 0.961854i \(-0.588203\pi\)
−0.273565 + 0.961854i \(0.588203\pi\)
\(888\) 0 0
\(889\) −5.90109 −0.197916
\(890\) 6.85235 0.229691
\(891\) 0 0
\(892\) −13.9747 −0.467908
\(893\) 59.8084 2.00141
\(894\) 0 0
\(895\) −11.3777 −0.380316
\(896\) −3.72975 −0.124602
\(897\) 0 0
\(898\) −36.1563 −1.20655
\(899\) −0.810257 −0.0270236
\(900\) 0 0
\(901\) −11.6647 −0.388607
\(902\) 8.18387 0.272493
\(903\) 0 0
\(904\) −28.0472 −0.932834
\(905\) 13.1136 0.435912
\(906\) 0 0
\(907\) −0.0236556 −0.000785472 0 −0.000392736 1.00000i \(-0.500125\pi\)
−0.000392736 1.00000i \(0.500125\pi\)
\(908\) 2.77540 0.0921047
\(909\) 0 0
\(910\) −1.46781 −0.0486573
\(911\) 33.7724 1.11893 0.559465 0.828854i \(-0.311007\pi\)
0.559465 + 0.828854i \(0.311007\pi\)
\(912\) 0 0
\(913\) 4.02744 0.133289
\(914\) 33.8581 1.11993
\(915\) 0 0
\(916\) 5.17728 0.171062
\(917\) −8.19858 −0.270741
\(918\) 0 0
\(919\) 30.0482 0.991197 0.495599 0.868552i \(-0.334949\pi\)
0.495599 + 0.868552i \(0.334949\pi\)
\(920\) −8.53389 −0.281354
\(921\) 0 0
\(922\) 28.6503 0.943548
\(923\) 9.63901 0.317272
\(924\) 0 0
\(925\) 4.39964 0.144659
\(926\) 42.0998 1.38349
\(927\) 0 0
\(928\) −4.37510 −0.143620
\(929\) 14.4372 0.473669 0.236834 0.971550i \(-0.423890\pi\)
0.236834 + 0.971550i \(0.423890\pi\)
\(930\) 0 0
\(931\) −32.6862 −1.07125
\(932\) 0.429885 0.0140813
\(933\) 0 0
\(934\) 9.51195 0.311241
\(935\) −4.01093 −0.131171
\(936\) 0 0
\(937\) −24.0678 −0.786260 −0.393130 0.919483i \(-0.628608\pi\)
−0.393130 + 0.919483i \(0.628608\pi\)
\(938\) −15.7884 −0.515510
\(939\) 0 0
\(940\) −6.45930 −0.210679
\(941\) −31.1287 −1.01477 −0.507383 0.861721i \(-0.669387\pi\)
−0.507383 + 0.861721i \(0.669387\pi\)
\(942\) 0 0
\(943\) 6.94693 0.226223
\(944\) 19.8051 0.644603
\(945\) 0 0
\(946\) 20.4914 0.666233
\(947\) −10.2949 −0.334540 −0.167270 0.985911i \(-0.553495\pi\)
−0.167270 + 0.985911i \(0.553495\pi\)
\(948\) 0 0
\(949\) −5.09589 −0.165419
\(950\) 7.02360 0.227876
\(951\) 0 0
\(952\) −5.55196 −0.179940
\(953\) 40.4483 1.31025 0.655125 0.755521i \(-0.272616\pi\)
0.655125 + 0.755521i \(0.272616\pi\)
\(954\) 0 0
\(955\) 18.8191 0.608972
\(956\) 1.29594 0.0419137
\(957\) 0 0
\(958\) −14.7291 −0.475877
\(959\) −19.2920 −0.622972
\(960\) 0 0
\(961\) −30.5726 −0.986213
\(962\) 5.10478 0.164585
\(963\) 0 0
\(964\) −0.887266 −0.0285769
\(965\) −6.08530 −0.195893
\(966\) 0 0
\(967\) −18.9049 −0.607940 −0.303970 0.952682i \(-0.598312\pi\)
−0.303970 + 0.952682i \(0.598312\pi\)
\(968\) 9.48724 0.304931
\(969\) 0 0
\(970\) 13.7842 0.442583
\(971\) −22.3318 −0.716660 −0.358330 0.933595i \(-0.616654\pi\)
−0.358330 + 0.933595i \(0.616654\pi\)
\(972\) 0 0
\(973\) 25.2174 0.808433
\(974\) 14.0555 0.450367
\(975\) 0 0
\(976\) 2.16883 0.0694226
\(977\) 51.0252 1.63244 0.816220 0.577741i \(-0.196066\pi\)
0.816220 + 0.577741i \(0.196066\pi\)
\(978\) 0 0
\(979\) 16.6192 0.531151
\(980\) 3.53011 0.112765
\(981\) 0 0
\(982\) −43.7998 −1.39771
\(983\) −60.9224 −1.94312 −0.971562 0.236785i \(-0.923906\pi\)
−0.971562 + 0.236785i \(0.923906\pi\)
\(984\) 0 0
\(985\) −2.71304 −0.0864447
\(986\) 2.04962 0.0652733
\(987\) 0 0
\(988\) −3.95752 −0.125905
\(989\) 17.3942 0.553105
\(990\) 0 0
\(991\) 10.8267 0.343922 0.171961 0.985104i \(-0.444990\pi\)
0.171961 + 0.985104i \(0.444990\pi\)
\(992\) 2.30787 0.0732749
\(993\) 0 0
\(994\) −14.1482 −0.448754
\(995\) 3.66848 0.116298
\(996\) 0 0
\(997\) −21.8396 −0.691668 −0.345834 0.938296i \(-0.612404\pi\)
−0.345834 + 0.938296i \(0.612404\pi\)
\(998\) 12.7268 0.402859
\(999\) 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 1755.2.a.q.1.3 yes 4
3.2 odd 2 1755.2.a.o.1.2 4
5.4 even 2 8775.2.a.bj.1.2 4
15.14 odd 2 8775.2.a.br.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1755.2.a.o.1.2 4 3.2 odd 2
1755.2.a.q.1.3 yes 4 1.1 even 1 trivial
8775.2.a.bj.1.2 4 5.4 even 2
8775.2.a.br.1.3 4 15.14 odd 2