Properties

Label 6044.2.a.a.1.13
Level $6044$
Weight $2$
Character 6044.1
Self dual yes
Analytic conductor $48.262$
Analytic rank $1$
Dimension $63$
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] = [6044,2,Mod(1,6044)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6044, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6044.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6044 = 2^{2} \cdot 1511 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6044.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: \(48.2615829817\)
Analytic rank: \(1\)
Dimension: \(63\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 6044.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.33939 q^{3} +2.09632 q^{5} +1.25063 q^{7} +2.47276 q^{9} +O(q^{10})\) \(q-2.33939 q^{3} +2.09632 q^{5} +1.25063 q^{7} +2.47276 q^{9} +1.62252 q^{11} -2.29185 q^{13} -4.90411 q^{15} +2.81594 q^{17} -2.10341 q^{19} -2.92571 q^{21} -4.36031 q^{23} -0.605452 q^{25} +1.23341 q^{27} +1.61572 q^{29} +1.58233 q^{31} -3.79571 q^{33} +2.62171 q^{35} +2.61455 q^{37} +5.36154 q^{39} +1.86557 q^{41} -9.96037 q^{43} +5.18370 q^{45} -0.873354 q^{47} -5.43593 q^{49} -6.58759 q^{51} -4.32199 q^{53} +3.40132 q^{55} +4.92071 q^{57} -9.88548 q^{59} -6.72317 q^{61} +3.09251 q^{63} -4.80444 q^{65} -9.19005 q^{67} +10.2005 q^{69} -6.51180 q^{71} +10.2610 q^{73} +1.41639 q^{75} +2.02917 q^{77} +10.8872 q^{79} -10.3037 q^{81} -2.56323 q^{83} +5.90310 q^{85} -3.77980 q^{87} -8.69018 q^{89} -2.86625 q^{91} -3.70169 q^{93} -4.40942 q^{95} +13.9626 q^{97} +4.01211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.33939 −1.35065 −0.675325 0.737520i \(-0.735997\pi\)
−0.675325 + 0.737520i \(0.735997\pi\)
\(4\) 0 0
\(5\) 2.09632 0.937502 0.468751 0.883330i \(-0.344704\pi\)
0.468751 + 0.883330i \(0.344704\pi\)
\(6\) 0 0
\(7\) 1.25063 0.472693 0.236346 0.971669i \(-0.424050\pi\)
0.236346 + 0.971669i \(0.424050\pi\)
\(8\) 0 0
\(9\) 2.47276 0.824255
\(10\) 0 0
\(11\) 1.62252 0.489208 0.244604 0.969623i \(-0.421342\pi\)
0.244604 + 0.969623i \(0.421342\pi\)
\(12\) 0 0
\(13\) −2.29185 −0.635644 −0.317822 0.948150i \(-0.602952\pi\)
−0.317822 + 0.948150i \(0.602952\pi\)
\(14\) 0 0
\(15\) −4.90411 −1.26624
\(16\) 0 0
\(17\) 2.81594 0.682965 0.341483 0.939888i \(-0.389071\pi\)
0.341483 + 0.939888i \(0.389071\pi\)
\(18\) 0 0
\(19\) −2.10341 −0.482556 −0.241278 0.970456i \(-0.577567\pi\)
−0.241278 + 0.970456i \(0.577567\pi\)
\(20\) 0 0
\(21\) −2.92571 −0.638442
\(22\) 0 0
\(23\) −4.36031 −0.909187 −0.454593 0.890699i \(-0.650215\pi\)
−0.454593 + 0.890699i \(0.650215\pi\)
\(24\) 0 0
\(25\) −0.605452 −0.121090
\(26\) 0 0
\(27\) 1.23341 0.237370
\(28\) 0 0
\(29\) 1.61572 0.300031 0.150016 0.988684i \(-0.452068\pi\)
0.150016 + 0.988684i \(0.452068\pi\)
\(30\) 0 0
\(31\) 1.58233 0.284194 0.142097 0.989853i \(-0.454615\pi\)
0.142097 + 0.989853i \(0.454615\pi\)
\(32\) 0 0
\(33\) −3.79571 −0.660749
\(34\) 0 0
\(35\) 2.62171 0.443150
\(36\) 0 0
\(37\) 2.61455 0.429829 0.214914 0.976633i \(-0.431053\pi\)
0.214914 + 0.976633i \(0.431053\pi\)
\(38\) 0 0
\(39\) 5.36154 0.858533
\(40\) 0 0
\(41\) 1.86557 0.291352 0.145676 0.989332i \(-0.453464\pi\)
0.145676 + 0.989332i \(0.453464\pi\)
\(42\) 0 0
\(43\) −9.96037 −1.51894 −0.759471 0.650541i \(-0.774542\pi\)
−0.759471 + 0.650541i \(0.774542\pi\)
\(44\) 0 0
\(45\) 5.18370 0.772741
\(46\) 0 0
\(47\) −0.873354 −0.127392 −0.0636959 0.997969i \(-0.520289\pi\)
−0.0636959 + 0.997969i \(0.520289\pi\)
\(48\) 0 0
\(49\) −5.43593 −0.776562
\(50\) 0 0
\(51\) −6.58759 −0.922447
\(52\) 0 0
\(53\) −4.32199 −0.593671 −0.296836 0.954929i \(-0.595931\pi\)
−0.296836 + 0.954929i \(0.595931\pi\)
\(54\) 0 0
\(55\) 3.40132 0.458633
\(56\) 0 0
\(57\) 4.92071 0.651764
\(58\) 0 0
\(59\) −9.88548 −1.28698 −0.643490 0.765454i \(-0.722514\pi\)
−0.643490 + 0.765454i \(0.722514\pi\)
\(60\) 0 0
\(61\) −6.72317 −0.860814 −0.430407 0.902635i \(-0.641630\pi\)
−0.430407 + 0.902635i \(0.641630\pi\)
\(62\) 0 0
\(63\) 3.09251 0.389619
\(64\) 0 0
\(65\) −4.80444 −0.595918
\(66\) 0 0
\(67\) −9.19005 −1.12274 −0.561372 0.827564i \(-0.689726\pi\)
−0.561372 + 0.827564i \(0.689726\pi\)
\(68\) 0 0
\(69\) 10.2005 1.22799
\(70\) 0 0
\(71\) −6.51180 −0.772809 −0.386404 0.922329i \(-0.626283\pi\)
−0.386404 + 0.922329i \(0.626283\pi\)
\(72\) 0 0
\(73\) 10.2610 1.20096 0.600481 0.799639i \(-0.294976\pi\)
0.600481 + 0.799639i \(0.294976\pi\)
\(74\) 0 0
\(75\) 1.41639 0.163551
\(76\) 0 0
\(77\) 2.02917 0.231245
\(78\) 0 0
\(79\) 10.8872 1.22491 0.612455 0.790506i \(-0.290182\pi\)
0.612455 + 0.790506i \(0.290182\pi\)
\(80\) 0 0
\(81\) −10.3037 −1.14486
\(82\) 0 0
\(83\) −2.56323 −0.281351 −0.140676 0.990056i \(-0.544928\pi\)
−0.140676 + 0.990056i \(0.544928\pi\)
\(84\) 0 0
\(85\) 5.90310 0.640281
\(86\) 0 0
\(87\) −3.77980 −0.405237
\(88\) 0 0
\(89\) −8.69018 −0.921158 −0.460579 0.887619i \(-0.652358\pi\)
−0.460579 + 0.887619i \(0.652358\pi\)
\(90\) 0 0
\(91\) −2.86625 −0.300465
\(92\) 0 0
\(93\) −3.70169 −0.383847
\(94\) 0 0
\(95\) −4.40942 −0.452397
\(96\) 0 0
\(97\) 13.9626 1.41769 0.708845 0.705365i \(-0.249217\pi\)
0.708845 + 0.705365i \(0.249217\pi\)
\(98\) 0 0
\(99\) 4.01211 0.403232
\(100\) 0 0
\(101\) 0.943829 0.0939145 0.0469572 0.998897i \(-0.485048\pi\)
0.0469572 + 0.998897i \(0.485048\pi\)
\(102\) 0 0
\(103\) −10.9923 −1.08310 −0.541550 0.840668i \(-0.682162\pi\)
−0.541550 + 0.840668i \(0.682162\pi\)
\(104\) 0 0
\(105\) −6.13322 −0.598541
\(106\) 0 0
\(107\) 1.67435 0.161866 0.0809328 0.996720i \(-0.474210\pi\)
0.0809328 + 0.996720i \(0.474210\pi\)
\(108\) 0 0
\(109\) 18.9052 1.81079 0.905393 0.424575i \(-0.139577\pi\)
0.905393 + 0.424575i \(0.139577\pi\)
\(110\) 0 0
\(111\) −6.11645 −0.580548
\(112\) 0 0
\(113\) −17.5153 −1.64770 −0.823850 0.566808i \(-0.808178\pi\)
−0.823850 + 0.566808i \(0.808178\pi\)
\(114\) 0 0
\(115\) −9.14059 −0.852364
\(116\) 0 0
\(117\) −5.66720 −0.523933
\(118\) 0 0
\(119\) 3.52169 0.322833
\(120\) 0 0
\(121\) −8.36743 −0.760676
\(122\) 0 0
\(123\) −4.36429 −0.393515
\(124\) 0 0
\(125\) −11.7508 −1.05102
\(126\) 0 0
\(127\) 14.8379 1.31665 0.658325 0.752734i \(-0.271265\pi\)
0.658325 + 0.752734i \(0.271265\pi\)
\(128\) 0 0
\(129\) 23.3012 2.05156
\(130\) 0 0
\(131\) 4.59201 0.401206 0.200603 0.979673i \(-0.435710\pi\)
0.200603 + 0.979673i \(0.435710\pi\)
\(132\) 0 0
\(133\) −2.63059 −0.228101
\(134\) 0 0
\(135\) 2.58562 0.222535
\(136\) 0 0
\(137\) 21.2387 1.81454 0.907271 0.420547i \(-0.138162\pi\)
0.907271 + 0.420547i \(0.138162\pi\)
\(138\) 0 0
\(139\) −10.6904 −0.906744 −0.453372 0.891321i \(-0.649779\pi\)
−0.453372 + 0.891321i \(0.649779\pi\)
\(140\) 0 0
\(141\) 2.04312 0.172062
\(142\) 0 0
\(143\) −3.71857 −0.310962
\(144\) 0 0
\(145\) 3.38706 0.281280
\(146\) 0 0
\(147\) 12.7168 1.04886
\(148\) 0 0
\(149\) 6.40860 0.525013 0.262507 0.964930i \(-0.415451\pi\)
0.262507 + 0.964930i \(0.415451\pi\)
\(150\) 0 0
\(151\) 4.70122 0.382579 0.191290 0.981534i \(-0.438733\pi\)
0.191290 + 0.981534i \(0.438733\pi\)
\(152\) 0 0
\(153\) 6.96315 0.562938
\(154\) 0 0
\(155\) 3.31706 0.266433
\(156\) 0 0
\(157\) 18.5259 1.47853 0.739263 0.673417i \(-0.235174\pi\)
0.739263 + 0.673417i \(0.235174\pi\)
\(158\) 0 0
\(159\) 10.1108 0.801842
\(160\) 0 0
\(161\) −5.45312 −0.429766
\(162\) 0 0
\(163\) 7.55440 0.591706 0.295853 0.955233i \(-0.404396\pi\)
0.295853 + 0.955233i \(0.404396\pi\)
\(164\) 0 0
\(165\) −7.95702 −0.619453
\(166\) 0 0
\(167\) −19.7889 −1.53131 −0.765656 0.643250i \(-0.777586\pi\)
−0.765656 + 0.643250i \(0.777586\pi\)
\(168\) 0 0
\(169\) −7.74743 −0.595956
\(170\) 0 0
\(171\) −5.20125 −0.397749
\(172\) 0 0
\(173\) 4.22823 0.321466 0.160733 0.986998i \(-0.448614\pi\)
0.160733 + 0.986998i \(0.448614\pi\)
\(174\) 0 0
\(175\) −0.757195 −0.0572386
\(176\) 0 0
\(177\) 23.1260 1.73826
\(178\) 0 0
\(179\) −15.6397 −1.16897 −0.584484 0.811405i \(-0.698703\pi\)
−0.584484 + 0.811405i \(0.698703\pi\)
\(180\) 0 0
\(181\) 2.49282 0.185290 0.0926449 0.995699i \(-0.470468\pi\)
0.0926449 + 0.995699i \(0.470468\pi\)
\(182\) 0 0
\(183\) 15.7282 1.16266
\(184\) 0 0
\(185\) 5.48092 0.402965
\(186\) 0 0
\(187\) 4.56891 0.334112
\(188\) 0 0
\(189\) 1.54254 0.112203
\(190\) 0 0
\(191\) −1.21349 −0.0878051 −0.0439025 0.999036i \(-0.513979\pi\)
−0.0439025 + 0.999036i \(0.513979\pi\)
\(192\) 0 0
\(193\) 12.4818 0.898458 0.449229 0.893417i \(-0.351699\pi\)
0.449229 + 0.893417i \(0.351699\pi\)
\(194\) 0 0
\(195\) 11.2395 0.804876
\(196\) 0 0
\(197\) 6.03381 0.429891 0.214945 0.976626i \(-0.431043\pi\)
0.214945 + 0.976626i \(0.431043\pi\)
\(198\) 0 0
\(199\) −15.7229 −1.11457 −0.557283 0.830323i \(-0.688156\pi\)
−0.557283 + 0.830323i \(0.688156\pi\)
\(200\) 0 0
\(201\) 21.4991 1.51643
\(202\) 0 0
\(203\) 2.02066 0.141823
\(204\) 0 0
\(205\) 3.91082 0.273143
\(206\) 0 0
\(207\) −10.7820 −0.749402
\(208\) 0 0
\(209\) −3.41283 −0.236070
\(210\) 0 0
\(211\) −10.7874 −0.742634 −0.371317 0.928506i \(-0.621094\pi\)
−0.371317 + 0.928506i \(0.621094\pi\)
\(212\) 0 0
\(213\) 15.2337 1.04379
\(214\) 0 0
\(215\) −20.8801 −1.42401
\(216\) 0 0
\(217\) 1.97890 0.134337
\(218\) 0 0
\(219\) −24.0046 −1.62208
\(220\) 0 0
\(221\) −6.45370 −0.434123
\(222\) 0 0
\(223\) −20.5370 −1.37526 −0.687629 0.726062i \(-0.741349\pi\)
−0.687629 + 0.726062i \(0.741349\pi\)
\(224\) 0 0
\(225\) −1.49714 −0.0998094
\(226\) 0 0
\(227\) 15.5179 1.02996 0.514981 0.857202i \(-0.327799\pi\)
0.514981 + 0.857202i \(0.327799\pi\)
\(228\) 0 0
\(229\) −23.7606 −1.57014 −0.785072 0.619404i \(-0.787374\pi\)
−0.785072 + 0.619404i \(0.787374\pi\)
\(230\) 0 0
\(231\) −4.74702 −0.312331
\(232\) 0 0
\(233\) 4.15292 0.272067 0.136033 0.990704i \(-0.456565\pi\)
0.136033 + 0.990704i \(0.456565\pi\)
\(234\) 0 0
\(235\) −1.83083 −0.119430
\(236\) 0 0
\(237\) −25.4695 −1.65442
\(238\) 0 0
\(239\) −14.2605 −0.922436 −0.461218 0.887287i \(-0.652587\pi\)
−0.461218 + 0.887287i \(0.652587\pi\)
\(240\) 0 0
\(241\) −20.3800 −1.31279 −0.656397 0.754416i \(-0.727920\pi\)
−0.656397 + 0.754416i \(0.727920\pi\)
\(242\) 0 0
\(243\) 20.4043 1.30893
\(244\) 0 0
\(245\) −11.3954 −0.728028
\(246\) 0 0
\(247\) 4.82070 0.306734
\(248\) 0 0
\(249\) 5.99641 0.380007
\(250\) 0 0
\(251\) −24.9361 −1.57395 −0.786975 0.616985i \(-0.788354\pi\)
−0.786975 + 0.616985i \(0.788354\pi\)
\(252\) 0 0
\(253\) −7.07468 −0.444781
\(254\) 0 0
\(255\) −13.8097 −0.864796
\(256\) 0 0
\(257\) 20.7037 1.29146 0.645731 0.763565i \(-0.276553\pi\)
0.645731 + 0.763565i \(0.276553\pi\)
\(258\) 0 0
\(259\) 3.26982 0.203177
\(260\) 0 0
\(261\) 3.99529 0.247302
\(262\) 0 0
\(263\) 17.5536 1.08240 0.541200 0.840894i \(-0.317970\pi\)
0.541200 + 0.840894i \(0.317970\pi\)
\(264\) 0 0
\(265\) −9.06027 −0.556568
\(266\) 0 0
\(267\) 20.3298 1.24416
\(268\) 0 0
\(269\) −13.5851 −0.828299 −0.414149 0.910209i \(-0.635921\pi\)
−0.414149 + 0.910209i \(0.635921\pi\)
\(270\) 0 0
\(271\) 28.2551 1.71638 0.858189 0.513334i \(-0.171590\pi\)
0.858189 + 0.513334i \(0.171590\pi\)
\(272\) 0 0
\(273\) 6.70529 0.405822
\(274\) 0 0
\(275\) −0.982357 −0.0592384
\(276\) 0 0
\(277\) 0.648466 0.0389625 0.0194813 0.999810i \(-0.493799\pi\)
0.0194813 + 0.999810i \(0.493799\pi\)
\(278\) 0 0
\(279\) 3.91272 0.234249
\(280\) 0 0
\(281\) 24.4489 1.45850 0.729250 0.684248i \(-0.239869\pi\)
0.729250 + 0.684248i \(0.239869\pi\)
\(282\) 0 0
\(283\) −5.47153 −0.325249 −0.162624 0.986688i \(-0.551996\pi\)
−0.162624 + 0.986688i \(0.551996\pi\)
\(284\) 0 0
\(285\) 10.3154 0.611030
\(286\) 0 0
\(287\) 2.33313 0.137720
\(288\) 0 0
\(289\) −9.07049 −0.533558
\(290\) 0 0
\(291\) −32.6641 −1.91480
\(292\) 0 0
\(293\) −26.1161 −1.52572 −0.762860 0.646564i \(-0.776205\pi\)
−0.762860 + 0.646564i \(0.776205\pi\)
\(294\) 0 0
\(295\) −20.7231 −1.20655
\(296\) 0 0
\(297\) 2.00123 0.116123
\(298\) 0 0
\(299\) 9.99316 0.577919
\(300\) 0 0
\(301\) −12.4567 −0.717993
\(302\) 0 0
\(303\) −2.20799 −0.126846
\(304\) 0 0
\(305\) −14.0939 −0.807015
\(306\) 0 0
\(307\) −32.5909 −1.86006 −0.930031 0.367481i \(-0.880220\pi\)
−0.930031 + 0.367481i \(0.880220\pi\)
\(308\) 0 0
\(309\) 25.7152 1.46289
\(310\) 0 0
\(311\) −23.0139 −1.30500 −0.652501 0.757788i \(-0.726280\pi\)
−0.652501 + 0.757788i \(0.726280\pi\)
\(312\) 0 0
\(313\) −26.5159 −1.49877 −0.749383 0.662137i \(-0.769650\pi\)
−0.749383 + 0.662137i \(0.769650\pi\)
\(314\) 0 0
\(315\) 6.48288 0.365269
\(316\) 0 0
\(317\) −24.7282 −1.38888 −0.694438 0.719553i \(-0.744347\pi\)
−0.694438 + 0.719553i \(0.744347\pi\)
\(318\) 0 0
\(319\) 2.62153 0.146778
\(320\) 0 0
\(321\) −3.91697 −0.218624
\(322\) 0 0
\(323\) −5.92308 −0.329569
\(324\) 0 0
\(325\) 1.38760 0.0769704
\(326\) 0 0
\(327\) −44.2266 −2.44574
\(328\) 0 0
\(329\) −1.09224 −0.0602172
\(330\) 0 0
\(331\) 31.5628 1.73485 0.867423 0.497571i \(-0.165775\pi\)
0.867423 + 0.497571i \(0.165775\pi\)
\(332\) 0 0
\(333\) 6.46516 0.354288
\(334\) 0 0
\(335\) −19.2653 −1.05257
\(336\) 0 0
\(337\) −11.1570 −0.607760 −0.303880 0.952710i \(-0.598282\pi\)
−0.303880 + 0.952710i \(0.598282\pi\)
\(338\) 0 0
\(339\) 40.9752 2.22547
\(340\) 0 0
\(341\) 2.56736 0.139030
\(342\) 0 0
\(343\) −15.5527 −0.839768
\(344\) 0 0
\(345\) 21.3834 1.15125
\(346\) 0 0
\(347\) −32.8636 −1.76421 −0.882106 0.471051i \(-0.843875\pi\)
−0.882106 + 0.471051i \(0.843875\pi\)
\(348\) 0 0
\(349\) 23.9749 1.28335 0.641673 0.766978i \(-0.278241\pi\)
0.641673 + 0.766978i \(0.278241\pi\)
\(350\) 0 0
\(351\) −2.82679 −0.150883
\(352\) 0 0
\(353\) 22.3419 1.18914 0.594571 0.804043i \(-0.297322\pi\)
0.594571 + 0.804043i \(0.297322\pi\)
\(354\) 0 0
\(355\) −13.6508 −0.724509
\(356\) 0 0
\(357\) −8.23862 −0.436034
\(358\) 0 0
\(359\) 20.7579 1.09556 0.547781 0.836622i \(-0.315473\pi\)
0.547781 + 0.836622i \(0.315473\pi\)
\(360\) 0 0
\(361\) −14.5757 −0.767140
\(362\) 0 0
\(363\) 19.5747 1.02741
\(364\) 0 0
\(365\) 21.5104 1.12590
\(366\) 0 0
\(367\) −13.1741 −0.687680 −0.343840 0.939028i \(-0.611728\pi\)
−0.343840 + 0.939028i \(0.611728\pi\)
\(368\) 0 0
\(369\) 4.61311 0.240149
\(370\) 0 0
\(371\) −5.40520 −0.280624
\(372\) 0 0
\(373\) 14.6258 0.757297 0.378648 0.925541i \(-0.376389\pi\)
0.378648 + 0.925541i \(0.376389\pi\)
\(374\) 0 0
\(375\) 27.4898 1.41957
\(376\) 0 0
\(377\) −3.70298 −0.190713
\(378\) 0 0
\(379\) −27.5163 −1.41342 −0.706710 0.707504i \(-0.749821\pi\)
−0.706710 + 0.707504i \(0.749821\pi\)
\(380\) 0 0
\(381\) −34.7117 −1.77833
\(382\) 0 0
\(383\) −21.3191 −1.08936 −0.544678 0.838645i \(-0.683348\pi\)
−0.544678 + 0.838645i \(0.683348\pi\)
\(384\) 0 0
\(385\) 4.25378 0.216793
\(386\) 0 0
\(387\) −24.6297 −1.25200
\(388\) 0 0
\(389\) 7.39267 0.374823 0.187412 0.982281i \(-0.439990\pi\)
0.187412 + 0.982281i \(0.439990\pi\)
\(390\) 0 0
\(391\) −12.2784 −0.620943
\(392\) 0 0
\(393\) −10.7425 −0.541888
\(394\) 0 0
\(395\) 22.8231 1.14835
\(396\) 0 0
\(397\) −17.6897 −0.887819 −0.443910 0.896072i \(-0.646409\pi\)
−0.443910 + 0.896072i \(0.646409\pi\)
\(398\) 0 0
\(399\) 6.15398 0.308084
\(400\) 0 0
\(401\) −7.31176 −0.365132 −0.182566 0.983194i \(-0.558440\pi\)
−0.182566 + 0.983194i \(0.558440\pi\)
\(402\) 0 0
\(403\) −3.62645 −0.180646
\(404\) 0 0
\(405\) −21.5999 −1.07331
\(406\) 0 0
\(407\) 4.24215 0.210276
\(408\) 0 0
\(409\) 6.69476 0.331034 0.165517 0.986207i \(-0.447071\pi\)
0.165517 + 0.986207i \(0.447071\pi\)
\(410\) 0 0
\(411\) −49.6856 −2.45081
\(412\) 0 0
\(413\) −12.3631 −0.608346
\(414\) 0 0
\(415\) −5.37335 −0.263768
\(416\) 0 0
\(417\) 25.0090 1.22469
\(418\) 0 0
\(419\) 10.8104 0.528122 0.264061 0.964506i \(-0.414938\pi\)
0.264061 + 0.964506i \(0.414938\pi\)
\(420\) 0 0
\(421\) −30.8037 −1.50128 −0.750641 0.660710i \(-0.770255\pi\)
−0.750641 + 0.660710i \(0.770255\pi\)
\(422\) 0 0
\(423\) −2.15960 −0.105003
\(424\) 0 0
\(425\) −1.70492 −0.0827006
\(426\) 0 0
\(427\) −8.40819 −0.406901
\(428\) 0 0
\(429\) 8.69920 0.420001
\(430\) 0 0
\(431\) −31.7273 −1.52825 −0.764124 0.645069i \(-0.776828\pi\)
−0.764124 + 0.645069i \(0.776828\pi\)
\(432\) 0 0
\(433\) 26.3277 1.26523 0.632614 0.774467i \(-0.281982\pi\)
0.632614 + 0.774467i \(0.281982\pi\)
\(434\) 0 0
\(435\) −7.92367 −0.379911
\(436\) 0 0
\(437\) 9.17152 0.438734
\(438\) 0 0
\(439\) 30.6301 1.46189 0.730947 0.682434i \(-0.239078\pi\)
0.730947 + 0.682434i \(0.239078\pi\)
\(440\) 0 0
\(441\) −13.4418 −0.640085
\(442\) 0 0
\(443\) −9.55685 −0.454060 −0.227030 0.973888i \(-0.572902\pi\)
−0.227030 + 0.973888i \(0.572902\pi\)
\(444\) 0 0
\(445\) −18.2174 −0.863587
\(446\) 0 0
\(447\) −14.9922 −0.709109
\(448\) 0 0
\(449\) −20.4566 −0.965407 −0.482703 0.875784i \(-0.660345\pi\)
−0.482703 + 0.875784i \(0.660345\pi\)
\(450\) 0 0
\(451\) 3.02692 0.142532
\(452\) 0 0
\(453\) −10.9980 −0.516731
\(454\) 0 0
\(455\) −6.00857 −0.281686
\(456\) 0 0
\(457\) −14.5119 −0.678837 −0.339418 0.940636i \(-0.610230\pi\)
−0.339418 + 0.940636i \(0.610230\pi\)
\(458\) 0 0
\(459\) 3.47321 0.162115
\(460\) 0 0
\(461\) 12.5993 0.586808 0.293404 0.955989i \(-0.405212\pi\)
0.293404 + 0.955989i \(0.405212\pi\)
\(462\) 0 0
\(463\) 25.8428 1.20102 0.600508 0.799619i \(-0.294965\pi\)
0.600508 + 0.799619i \(0.294965\pi\)
\(464\) 0 0
\(465\) −7.75991 −0.359857
\(466\) 0 0
\(467\) 11.9066 0.550970 0.275485 0.961305i \(-0.411162\pi\)
0.275485 + 0.961305i \(0.411162\pi\)
\(468\) 0 0
\(469\) −11.4933 −0.530712
\(470\) 0 0
\(471\) −43.3393 −1.99697
\(472\) 0 0
\(473\) −16.1609 −0.743078
\(474\) 0 0
\(475\) 1.27352 0.0584329
\(476\) 0 0
\(477\) −10.6873 −0.489336
\(478\) 0 0
\(479\) −7.56415 −0.345615 −0.172807 0.984956i \(-0.555284\pi\)
−0.172807 + 0.984956i \(0.555284\pi\)
\(480\) 0 0
\(481\) −5.99214 −0.273218
\(482\) 0 0
\(483\) 12.7570 0.580463
\(484\) 0 0
\(485\) 29.2701 1.32909
\(486\) 0 0
\(487\) −32.7371 −1.48346 −0.741729 0.670700i \(-0.765994\pi\)
−0.741729 + 0.670700i \(0.765994\pi\)
\(488\) 0 0
\(489\) −17.6727 −0.799188
\(490\) 0 0
\(491\) 20.8368 0.940349 0.470175 0.882573i \(-0.344191\pi\)
0.470175 + 0.882573i \(0.344191\pi\)
\(492\) 0 0
\(493\) 4.54976 0.204911
\(494\) 0 0
\(495\) 8.41065 0.378031
\(496\) 0 0
\(497\) −8.14384 −0.365301
\(498\) 0 0
\(499\) −0.136028 −0.00608946 −0.00304473 0.999995i \(-0.500969\pi\)
−0.00304473 + 0.999995i \(0.500969\pi\)
\(500\) 0 0
\(501\) 46.2941 2.06827
\(502\) 0 0
\(503\) −5.33516 −0.237883 −0.118942 0.992901i \(-0.537950\pi\)
−0.118942 + 0.992901i \(0.537950\pi\)
\(504\) 0 0
\(505\) 1.97857 0.0880450
\(506\) 0 0
\(507\) 18.1243 0.804928
\(508\) 0 0
\(509\) 19.5868 0.868169 0.434085 0.900872i \(-0.357072\pi\)
0.434085 + 0.900872i \(0.357072\pi\)
\(510\) 0 0
\(511\) 12.8327 0.567686
\(512\) 0 0
\(513\) −2.59437 −0.114544
\(514\) 0 0
\(515\) −23.0433 −1.01541
\(516\) 0 0
\(517\) −1.41703 −0.0623211
\(518\) 0 0
\(519\) −9.89149 −0.434188
\(520\) 0 0
\(521\) −4.41975 −0.193633 −0.0968164 0.995302i \(-0.530866\pi\)
−0.0968164 + 0.995302i \(0.530866\pi\)
\(522\) 0 0
\(523\) −36.5492 −1.59818 −0.799091 0.601209i \(-0.794686\pi\)
−0.799091 + 0.601209i \(0.794686\pi\)
\(524\) 0 0
\(525\) 1.77138 0.0773092
\(526\) 0 0
\(527\) 4.45573 0.194095
\(528\) 0 0
\(529\) −3.98774 −0.173380
\(530\) 0 0
\(531\) −24.4445 −1.06080
\(532\) 0 0
\(533\) −4.27559 −0.185197
\(534\) 0 0
\(535\) 3.50997 0.151749
\(536\) 0 0
\(537\) 36.5875 1.57887
\(538\) 0 0
\(539\) −8.81990 −0.379900
\(540\) 0 0
\(541\) −34.1501 −1.46823 −0.734114 0.679026i \(-0.762402\pi\)
−0.734114 + 0.679026i \(0.762402\pi\)
\(542\) 0 0
\(543\) −5.83169 −0.250262
\(544\) 0 0
\(545\) 39.6312 1.69761
\(546\) 0 0
\(547\) 38.5083 1.64650 0.823248 0.567681i \(-0.192159\pi\)
0.823248 + 0.567681i \(0.192159\pi\)
\(548\) 0 0
\(549\) −16.6248 −0.709530
\(550\) 0 0
\(551\) −3.39852 −0.144782
\(552\) 0 0
\(553\) 13.6159 0.579006
\(554\) 0 0
\(555\) −12.8220 −0.544265
\(556\) 0 0
\(557\) 29.6617 1.25681 0.628403 0.777888i \(-0.283709\pi\)
0.628403 + 0.777888i \(0.283709\pi\)
\(558\) 0 0
\(559\) 22.8277 0.965507
\(560\) 0 0
\(561\) −10.6885 −0.451268
\(562\) 0 0
\(563\) −40.7633 −1.71797 −0.858984 0.512002i \(-0.828904\pi\)
−0.858984 + 0.512002i \(0.828904\pi\)
\(564\) 0 0
\(565\) −36.7176 −1.54472
\(566\) 0 0
\(567\) −12.8861 −0.541166
\(568\) 0 0
\(569\) −30.6807 −1.28620 −0.643100 0.765782i \(-0.722352\pi\)
−0.643100 + 0.765782i \(0.722352\pi\)
\(570\) 0 0
\(571\) 15.8969 0.665263 0.332631 0.943057i \(-0.392064\pi\)
0.332631 + 0.943057i \(0.392064\pi\)
\(572\) 0 0
\(573\) 2.83883 0.118594
\(574\) 0 0
\(575\) 2.63996 0.110094
\(576\) 0 0
\(577\) 29.3106 1.22022 0.610109 0.792317i \(-0.291126\pi\)
0.610109 + 0.792317i \(0.291126\pi\)
\(578\) 0 0
\(579\) −29.1998 −1.21350
\(580\) 0 0
\(581\) −3.20565 −0.132993
\(582\) 0 0
\(583\) −7.01251 −0.290429
\(584\) 0 0
\(585\) −11.8803 −0.491188
\(586\) 0 0
\(587\) −12.2565 −0.505880 −0.252940 0.967482i \(-0.581398\pi\)
−0.252940 + 0.967482i \(0.581398\pi\)
\(588\) 0 0
\(589\) −3.32829 −0.137140
\(590\) 0 0
\(591\) −14.1154 −0.580632
\(592\) 0 0
\(593\) 20.7824 0.853429 0.426715 0.904386i \(-0.359671\pi\)
0.426715 + 0.904386i \(0.359671\pi\)
\(594\) 0 0
\(595\) 7.38258 0.302656
\(596\) 0 0
\(597\) 36.7820 1.50539
\(598\) 0 0
\(599\) −30.3717 −1.24095 −0.620476 0.784225i \(-0.713061\pi\)
−0.620476 + 0.784225i \(0.713061\pi\)
\(600\) 0 0
\(601\) 33.4762 1.36552 0.682762 0.730641i \(-0.260779\pi\)
0.682762 + 0.730641i \(0.260779\pi\)
\(602\) 0 0
\(603\) −22.7248 −0.925427
\(604\) 0 0
\(605\) −17.5408 −0.713135
\(606\) 0 0
\(607\) 44.4310 1.80340 0.901699 0.432365i \(-0.142321\pi\)
0.901699 + 0.432365i \(0.142321\pi\)
\(608\) 0 0
\(609\) −4.72712 −0.191553
\(610\) 0 0
\(611\) 2.00160 0.0809759
\(612\) 0 0
\(613\) −22.0681 −0.891321 −0.445660 0.895202i \(-0.647031\pi\)
−0.445660 + 0.895202i \(0.647031\pi\)
\(614\) 0 0
\(615\) −9.14895 −0.368921
\(616\) 0 0
\(617\) −5.50264 −0.221528 −0.110764 0.993847i \(-0.535330\pi\)
−0.110764 + 0.993847i \(0.535330\pi\)
\(618\) 0 0
\(619\) −10.2191 −0.410742 −0.205371 0.978684i \(-0.565840\pi\)
−0.205371 + 0.978684i \(0.565840\pi\)
\(620\) 0 0
\(621\) −5.37805 −0.215814
\(622\) 0 0
\(623\) −10.8682 −0.435425
\(624\) 0 0
\(625\) −21.6062 −0.864247
\(626\) 0 0
\(627\) 7.98395 0.318848
\(628\) 0 0
\(629\) 7.36240 0.293558
\(630\) 0 0
\(631\) 1.94835 0.0775628 0.0387814 0.999248i \(-0.487652\pi\)
0.0387814 + 0.999248i \(0.487652\pi\)
\(632\) 0 0
\(633\) 25.2359 1.00304
\(634\) 0 0
\(635\) 31.1049 1.23436
\(636\) 0 0
\(637\) 12.4583 0.493617
\(638\) 0 0
\(639\) −16.1022 −0.636991
\(640\) 0 0
\(641\) 36.8027 1.45362 0.726810 0.686839i \(-0.241002\pi\)
0.726810 + 0.686839i \(0.241002\pi\)
\(642\) 0 0
\(643\) −35.6800 −1.40708 −0.703540 0.710656i \(-0.748398\pi\)
−0.703540 + 0.710656i \(0.748398\pi\)
\(644\) 0 0
\(645\) 48.8468 1.92334
\(646\) 0 0
\(647\) −33.3499 −1.31112 −0.655560 0.755143i \(-0.727567\pi\)
−0.655560 + 0.755143i \(0.727567\pi\)
\(648\) 0 0
\(649\) −16.0394 −0.629601
\(650\) 0 0
\(651\) −4.62943 −0.181442
\(652\) 0 0
\(653\) −31.1440 −1.21876 −0.609379 0.792879i \(-0.708581\pi\)
−0.609379 + 0.792879i \(0.708581\pi\)
\(654\) 0 0
\(655\) 9.62631 0.376131
\(656\) 0 0
\(657\) 25.3731 0.989899
\(658\) 0 0
\(659\) 13.4428 0.523659 0.261829 0.965114i \(-0.415674\pi\)
0.261829 + 0.965114i \(0.415674\pi\)
\(660\) 0 0
\(661\) −34.8663 −1.35614 −0.678071 0.734996i \(-0.737184\pi\)
−0.678071 + 0.734996i \(0.737184\pi\)
\(662\) 0 0
\(663\) 15.0978 0.586348
\(664\) 0 0
\(665\) −5.51455 −0.213845
\(666\) 0 0
\(667\) −7.04503 −0.272784
\(668\) 0 0
\(669\) 48.0441 1.85749
\(670\) 0 0
\(671\) −10.9085 −0.421117
\(672\) 0 0
\(673\) 6.97888 0.269016 0.134508 0.990913i \(-0.457055\pi\)
0.134508 + 0.990913i \(0.457055\pi\)
\(674\) 0 0
\(675\) −0.746771 −0.0287432
\(676\) 0 0
\(677\) 3.63867 0.139846 0.0699228 0.997552i \(-0.477725\pi\)
0.0699228 + 0.997552i \(0.477725\pi\)
\(678\) 0 0
\(679\) 17.4620 0.670132
\(680\) 0 0
\(681\) −36.3026 −1.39112
\(682\) 0 0
\(683\) 10.6467 0.407385 0.203693 0.979035i \(-0.434706\pi\)
0.203693 + 0.979035i \(0.434706\pi\)
\(684\) 0 0
\(685\) 44.5230 1.70114
\(686\) 0 0
\(687\) 55.5854 2.12072
\(688\) 0 0
\(689\) 9.90535 0.377364
\(690\) 0 0
\(691\) 1.93492 0.0736079 0.0368039 0.999323i \(-0.488282\pi\)
0.0368039 + 0.999323i \(0.488282\pi\)
\(692\) 0 0
\(693\) 5.01765 0.190605
\(694\) 0 0
\(695\) −22.4104 −0.850074
\(696\) 0 0
\(697\) 5.25332 0.198984
\(698\) 0 0
\(699\) −9.71531 −0.367467
\(700\) 0 0
\(701\) −14.1282 −0.533615 −0.266807 0.963750i \(-0.585969\pi\)
−0.266807 + 0.963750i \(0.585969\pi\)
\(702\) 0 0
\(703\) −5.49947 −0.207416
\(704\) 0 0
\(705\) 4.28303 0.161308
\(706\) 0 0
\(707\) 1.18038 0.0443927
\(708\) 0 0
\(709\) −25.0245 −0.939814 −0.469907 0.882716i \(-0.655713\pi\)
−0.469907 + 0.882716i \(0.655713\pi\)
\(710\) 0 0
\(711\) 26.9216 1.00964
\(712\) 0 0
\(713\) −6.89943 −0.258386
\(714\) 0 0
\(715\) −7.79530 −0.291528
\(716\) 0 0
\(717\) 33.3610 1.24589
\(718\) 0 0
\(719\) −6.05269 −0.225727 −0.112864 0.993610i \(-0.536002\pi\)
−0.112864 + 0.993610i \(0.536002\pi\)
\(720\) 0 0
\(721\) −13.7472 −0.511974
\(722\) 0 0
\(723\) 47.6769 1.77312
\(724\) 0 0
\(725\) −0.978240 −0.0363309
\(726\) 0 0
\(727\) 2.89602 0.107407 0.0537036 0.998557i \(-0.482897\pi\)
0.0537036 + 0.998557i \(0.482897\pi\)
\(728\) 0 0
\(729\) −16.8224 −0.623052
\(730\) 0 0
\(731\) −28.0478 −1.03738
\(732\) 0 0
\(733\) 33.5724 1.24003 0.620013 0.784591i \(-0.287127\pi\)
0.620013 + 0.784591i \(0.287127\pi\)
\(734\) 0 0
\(735\) 26.6584 0.983311
\(736\) 0 0
\(737\) −14.9110 −0.549255
\(738\) 0 0
\(739\) −45.9758 −1.69125 −0.845623 0.533780i \(-0.820771\pi\)
−0.845623 + 0.533780i \(0.820771\pi\)
\(740\) 0 0
\(741\) −11.2775 −0.414290
\(742\) 0 0
\(743\) 19.2913 0.707730 0.353865 0.935297i \(-0.384867\pi\)
0.353865 + 0.935297i \(0.384867\pi\)
\(744\) 0 0
\(745\) 13.4345 0.492201
\(746\) 0 0
\(747\) −6.33828 −0.231905
\(748\) 0 0
\(749\) 2.09399 0.0765127
\(750\) 0 0
\(751\) −10.4782 −0.382356 −0.191178 0.981555i \(-0.561231\pi\)
−0.191178 + 0.981555i \(0.561231\pi\)
\(752\) 0 0
\(753\) 58.3353 2.12585
\(754\) 0 0
\(755\) 9.85524 0.358669
\(756\) 0 0
\(757\) −7.99185 −0.290469 −0.145234 0.989397i \(-0.546394\pi\)
−0.145234 + 0.989397i \(0.546394\pi\)
\(758\) 0 0
\(759\) 16.5505 0.600744
\(760\) 0 0
\(761\) −32.2015 −1.16730 −0.583651 0.812005i \(-0.698376\pi\)
−0.583651 + 0.812005i \(0.698376\pi\)
\(762\) 0 0
\(763\) 23.6433 0.855945
\(764\) 0 0
\(765\) 14.5970 0.527755
\(766\) 0 0
\(767\) 22.6560 0.818062
\(768\) 0 0
\(769\) 53.4278 1.92665 0.963327 0.268330i \(-0.0864716\pi\)
0.963327 + 0.268330i \(0.0864716\pi\)
\(770\) 0 0
\(771\) −48.4341 −1.74431
\(772\) 0 0
\(773\) 23.3664 0.840431 0.420215 0.907424i \(-0.361955\pi\)
0.420215 + 0.907424i \(0.361955\pi\)
\(774\) 0 0
\(775\) −0.958023 −0.0344132
\(776\) 0 0
\(777\) −7.64940 −0.274421
\(778\) 0 0
\(779\) −3.92406 −0.140594
\(780\) 0 0
\(781\) −10.5655 −0.378064
\(782\) 0 0
\(783\) 1.99284 0.0712184
\(784\) 0 0
\(785\) 38.8361 1.38612
\(786\) 0 0
\(787\) −13.2603 −0.472680 −0.236340 0.971670i \(-0.575948\pi\)
−0.236340 + 0.971670i \(0.575948\pi\)
\(788\) 0 0
\(789\) −41.0648 −1.46194
\(790\) 0 0
\(791\) −21.9051 −0.778856
\(792\) 0 0
\(793\) 15.4085 0.547172
\(794\) 0 0
\(795\) 21.1955 0.751728
\(796\) 0 0
\(797\) 46.1666 1.63531 0.817653 0.575712i \(-0.195275\pi\)
0.817653 + 0.575712i \(0.195275\pi\)
\(798\) 0 0
\(799\) −2.45931 −0.0870042
\(800\) 0 0
\(801\) −21.4888 −0.759269
\(802\) 0 0
\(803\) 16.6487 0.587520
\(804\) 0 0
\(805\) −11.4315 −0.402906
\(806\) 0 0
\(807\) 31.7809 1.11874
\(808\) 0 0
\(809\) 8.37430 0.294425 0.147212 0.989105i \(-0.452970\pi\)
0.147212 + 0.989105i \(0.452970\pi\)
\(810\) 0 0
\(811\) −1.15993 −0.0407306 −0.0203653 0.999793i \(-0.506483\pi\)
−0.0203653 + 0.999793i \(0.506483\pi\)
\(812\) 0 0
\(813\) −66.0999 −2.31822
\(814\) 0 0
\(815\) 15.8364 0.554726
\(816\) 0 0
\(817\) 20.9508 0.732975
\(818\) 0 0
\(819\) −7.08756 −0.247659
\(820\) 0 0
\(821\) 17.9098 0.625055 0.312527 0.949909i \(-0.398824\pi\)
0.312527 + 0.949909i \(0.398824\pi\)
\(822\) 0 0
\(823\) −32.8323 −1.14446 −0.572232 0.820092i \(-0.693922\pi\)
−0.572232 + 0.820092i \(0.693922\pi\)
\(824\) 0 0
\(825\) 2.29812 0.0800103
\(826\) 0 0
\(827\) 30.3422 1.05510 0.527552 0.849523i \(-0.323110\pi\)
0.527552 + 0.849523i \(0.323110\pi\)
\(828\) 0 0
\(829\) 10.5415 0.366121 0.183061 0.983102i \(-0.441400\pi\)
0.183061 + 0.983102i \(0.441400\pi\)
\(830\) 0 0
\(831\) −1.51702 −0.0526247
\(832\) 0 0
\(833\) −15.3072 −0.530365
\(834\) 0 0
\(835\) −41.4838 −1.43561
\(836\) 0 0
\(837\) 1.95166 0.0674592
\(838\) 0 0
\(839\) 27.1469 0.937215 0.468608 0.883406i \(-0.344756\pi\)
0.468608 + 0.883406i \(0.344756\pi\)
\(840\) 0 0
\(841\) −26.3895 −0.909981
\(842\) 0 0
\(843\) −57.1956 −1.96992
\(844\) 0 0
\(845\) −16.2411 −0.558710
\(846\) 0 0
\(847\) −10.4645 −0.359566
\(848\) 0 0
\(849\) 12.8001 0.439297
\(850\) 0 0
\(851\) −11.4002 −0.390794
\(852\) 0 0
\(853\) 38.6156 1.32217 0.661086 0.750310i \(-0.270096\pi\)
0.661086 + 0.750310i \(0.270096\pi\)
\(854\) 0 0
\(855\) −10.9035 −0.372891
\(856\) 0 0
\(857\) −23.7952 −0.812830 −0.406415 0.913689i \(-0.633221\pi\)
−0.406415 + 0.913689i \(0.633221\pi\)
\(858\) 0 0
\(859\) −40.0722 −1.36724 −0.683622 0.729836i \(-0.739596\pi\)
−0.683622 + 0.729836i \(0.739596\pi\)
\(860\) 0 0
\(861\) −5.45811 −0.186012
\(862\) 0 0
\(863\) −54.1693 −1.84394 −0.921972 0.387257i \(-0.873423\pi\)
−0.921972 + 0.387257i \(0.873423\pi\)
\(864\) 0 0
\(865\) 8.86371 0.301375
\(866\) 0 0
\(867\) 21.2195 0.720650
\(868\) 0 0
\(869\) 17.6647 0.599235
\(870\) 0 0
\(871\) 21.0622 0.713665
\(872\) 0 0
\(873\) 34.5263 1.16854
\(874\) 0 0
\(875\) −14.6959 −0.496812
\(876\) 0 0
\(877\) −28.8631 −0.974636 −0.487318 0.873224i \(-0.662025\pi\)
−0.487318 + 0.873224i \(0.662025\pi\)
\(878\) 0 0
\(879\) 61.0959 2.06071
\(880\) 0 0
\(881\) 17.0933 0.575887 0.287943 0.957647i \(-0.407028\pi\)
0.287943 + 0.957647i \(0.407028\pi\)
\(882\) 0 0
\(883\) −4.46810 −0.150363 −0.0751817 0.997170i \(-0.523954\pi\)
−0.0751817 + 0.997170i \(0.523954\pi\)
\(884\) 0 0
\(885\) 48.4795 1.62962
\(886\) 0 0
\(887\) 42.1368 1.41481 0.707407 0.706807i \(-0.249865\pi\)
0.707407 + 0.706807i \(0.249865\pi\)
\(888\) 0 0
\(889\) 18.5567 0.622371
\(890\) 0 0
\(891\) −16.7180 −0.560074
\(892\) 0 0
\(893\) 1.83703 0.0614737
\(894\) 0 0
\(895\) −32.7858 −1.09591
\(896\) 0 0
\(897\) −23.3779 −0.780567
\(898\) 0 0
\(899\) 2.55659 0.0852672
\(900\) 0 0
\(901\) −12.1705 −0.405457
\(902\) 0 0
\(903\) 29.1412 0.969757
\(904\) 0 0
\(905\) 5.22574 0.173710
\(906\) 0 0
\(907\) −48.0919 −1.59686 −0.798432 0.602084i \(-0.794337\pi\)
−0.798432 + 0.602084i \(0.794337\pi\)
\(908\) 0 0
\(909\) 2.33387 0.0774095
\(910\) 0 0
\(911\) 36.4297 1.20697 0.603484 0.797375i \(-0.293779\pi\)
0.603484 + 0.797375i \(0.293779\pi\)
\(912\) 0 0
\(913\) −4.15890 −0.137639
\(914\) 0 0
\(915\) 32.9712 1.08999
\(916\) 0 0
\(917\) 5.74289 0.189647
\(918\) 0 0
\(919\) 12.6771 0.418180 0.209090 0.977896i \(-0.432950\pi\)
0.209090 + 0.977896i \(0.432950\pi\)
\(920\) 0 0
\(921\) 76.2430 2.51229
\(922\) 0 0
\(923\) 14.9241 0.491231
\(924\) 0 0
\(925\) −1.58298 −0.0520481
\(926\) 0 0
\(927\) −27.1813 −0.892751
\(928\) 0 0
\(929\) −53.4959 −1.75514 −0.877572 0.479445i \(-0.840838\pi\)
−0.877572 + 0.479445i \(0.840838\pi\)
\(930\) 0 0
\(931\) 11.4340 0.374735
\(932\) 0 0
\(933\) 53.8387 1.76260
\(934\) 0 0
\(935\) 9.57789 0.313231
\(936\) 0 0
\(937\) −10.4650 −0.341875 −0.170938 0.985282i \(-0.554680\pi\)
−0.170938 + 0.985282i \(0.554680\pi\)
\(938\) 0 0
\(939\) 62.0311 2.02431
\(940\) 0 0
\(941\) −3.13648 −0.102246 −0.0511232 0.998692i \(-0.516280\pi\)
−0.0511232 + 0.998692i \(0.516280\pi\)
\(942\) 0 0
\(943\) −8.13444 −0.264894
\(944\) 0 0
\(945\) 3.23365 0.105191
\(946\) 0 0
\(947\) −4.84338 −0.157389 −0.0786943 0.996899i \(-0.525075\pi\)
−0.0786943 + 0.996899i \(0.525075\pi\)
\(948\) 0 0
\(949\) −23.5167 −0.763385
\(950\) 0 0
\(951\) 57.8491 1.87589
\(952\) 0 0
\(953\) 26.5101 0.858748 0.429374 0.903127i \(-0.358734\pi\)
0.429374 + 0.903127i \(0.358734\pi\)
\(954\) 0 0
\(955\) −2.54386 −0.0823174
\(956\) 0 0
\(957\) −6.13280 −0.198245
\(958\) 0 0
\(959\) 26.5617 0.857721
\(960\) 0 0
\(961\) −28.4962 −0.919234
\(962\) 0 0
\(963\) 4.14028 0.133419
\(964\) 0 0
\(965\) 26.1658 0.842306
\(966\) 0 0
\(967\) 11.0463 0.355225 0.177612 0.984101i \(-0.443163\pi\)
0.177612 + 0.984101i \(0.443163\pi\)
\(968\) 0 0
\(969\) 13.8564 0.445133
\(970\) 0 0
\(971\) 36.3228 1.16565 0.582827 0.812596i \(-0.301946\pi\)
0.582827 + 0.812596i \(0.301946\pi\)
\(972\) 0 0
\(973\) −13.3697 −0.428611
\(974\) 0 0
\(975\) −3.24615 −0.103960
\(976\) 0 0
\(977\) 12.0389 0.385157 0.192579 0.981282i \(-0.438315\pi\)
0.192579 + 0.981282i \(0.438315\pi\)
\(978\) 0 0
\(979\) −14.1000 −0.450638
\(980\) 0 0
\(981\) 46.7480 1.49255
\(982\) 0 0
\(983\) 46.2927 1.47651 0.738254 0.674523i \(-0.235651\pi\)
0.738254 + 0.674523i \(0.235651\pi\)
\(984\) 0 0
\(985\) 12.6488 0.403023
\(986\) 0 0
\(987\) 2.55518 0.0813323
\(988\) 0 0
\(989\) 43.4303 1.38100
\(990\) 0 0
\(991\) 28.4998 0.905325 0.452663 0.891682i \(-0.350474\pi\)
0.452663 + 0.891682i \(0.350474\pi\)
\(992\) 0 0
\(993\) −73.8378 −2.34317
\(994\) 0 0
\(995\) −32.9601 −1.04491
\(996\) 0 0
\(997\) −15.9609 −0.505488 −0.252744 0.967533i \(-0.581333\pi\)
−0.252744 + 0.967533i \(0.581333\pi\)
\(998\) 0 0
\(999\) 3.22481 0.102028
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 6044.2.a.a.1.13 63
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6044.2.a.a.1.13 63 1.1 even 1 trivial