Properties

Label 4056.2.a.ba.1.3
Level $4056$
Weight $2$
Character 4056.1
Self dual yes
Analytic conductor $32.387$
Analytic rank $1$
Dimension $3$
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] = [4056,2,Mod(1,4056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4056.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.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: \(32.3873230598\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) 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 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 4056.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.24698 q^{5} -1.35690 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.24698 q^{5} -1.35690 q^{7} +1.00000 q^{9} -0.445042 q^{11} -2.24698 q^{15} -1.80194 q^{17} -2.69202 q^{19} +1.35690 q^{21} -2.24698 q^{23} +0.0489173 q^{25} -1.00000 q^{27} +4.65279 q^{29} +8.25667 q^{31} +0.445042 q^{33} -3.04892 q^{35} +1.41789 q^{37} -9.03684 q^{41} -9.09783 q^{43} +2.24698 q^{45} -7.58211 q^{47} -5.15883 q^{49} +1.80194 q^{51} +11.4330 q^{53} -1.00000 q^{55} +2.69202 q^{57} -6.93362 q^{59} -0.868313 q^{61} -1.35690 q^{63} -6.53319 q^{67} +2.24698 q^{69} -3.54288 q^{71} +3.35690 q^{73} -0.0489173 q^{75} +0.603875 q^{77} -12.6746 q^{79} +1.00000 q^{81} -3.19806 q^{83} -4.04892 q^{85} -4.65279 q^{87} -0.454731 q^{89} -8.25667 q^{93} -6.04892 q^{95} +15.4940 q^{97} -0.445042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 2 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 2 q^{5} + 3 q^{9} - q^{11} - 2 q^{15} - q^{17} - 3 q^{19} - 2 q^{23} - 9 q^{25} - 3 q^{27} - 4 q^{29} - 2 q^{31} + q^{33} + 10 q^{37} + q^{41} - 9 q^{43} + 2 q^{45} - 17 q^{47} - 7 q^{49} + q^{51} + 15 q^{53} - 3 q^{55} + 3 q^{57} - 14 q^{59} - 5 q^{61} - 23 q^{67} + 2 q^{69} + 8 q^{71} + 6 q^{73} + 9 q^{75} - 7 q^{77} - 17 q^{79} + 3 q^{81} - 14 q^{83} - 3 q^{85} + 4 q^{87} + 21 q^{89} + 2 q^{93} - 9 q^{95} + 37 q^{97} - q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.24698 1.00488 0.502440 0.864612i \(-0.332436\pi\)
0.502440 + 0.864612i \(0.332436\pi\)
\(6\) 0 0
\(7\) −1.35690 −0.512858 −0.256429 0.966563i \(-0.582546\pi\)
−0.256429 + 0.966563i \(0.582546\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.445042 −0.134185 −0.0670926 0.997747i \(-0.521372\pi\)
−0.0670926 + 0.997747i \(0.521372\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −2.24698 −0.580168
\(16\) 0 0
\(17\) −1.80194 −0.437034 −0.218517 0.975833i \(-0.570122\pi\)
−0.218517 + 0.975833i \(0.570122\pi\)
\(18\) 0 0
\(19\) −2.69202 −0.617592 −0.308796 0.951128i \(-0.599926\pi\)
−0.308796 + 0.951128i \(0.599926\pi\)
\(20\) 0 0
\(21\) 1.35690 0.296099
\(22\) 0 0
\(23\) −2.24698 −0.468528 −0.234264 0.972173i \(-0.575268\pi\)
−0.234264 + 0.972173i \(0.575268\pi\)
\(24\) 0 0
\(25\) 0.0489173 0.00978347
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.65279 0.864002 0.432001 0.901873i \(-0.357808\pi\)
0.432001 + 0.901873i \(0.357808\pi\)
\(30\) 0 0
\(31\) 8.25667 1.48294 0.741471 0.670985i \(-0.234129\pi\)
0.741471 + 0.670985i \(0.234129\pi\)
\(32\) 0 0
\(33\) 0.445042 0.0774718
\(34\) 0 0
\(35\) −3.04892 −0.515361
\(36\) 0 0
\(37\) 1.41789 0.233100 0.116550 0.993185i \(-0.462816\pi\)
0.116550 + 0.993185i \(0.462816\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.03684 −1.41132 −0.705658 0.708552i \(-0.749349\pi\)
−0.705658 + 0.708552i \(0.749349\pi\)
\(42\) 0 0
\(43\) −9.09783 −1.38741 −0.693703 0.720261i \(-0.744022\pi\)
−0.693703 + 0.720261i \(0.744022\pi\)
\(44\) 0 0
\(45\) 2.24698 0.334960
\(46\) 0 0
\(47\) −7.58211 −1.10596 −0.552982 0.833193i \(-0.686510\pi\)
−0.552982 + 0.833193i \(0.686510\pi\)
\(48\) 0 0
\(49\) −5.15883 −0.736976
\(50\) 0 0
\(51\) 1.80194 0.252322
\(52\) 0 0
\(53\) 11.4330 1.57044 0.785219 0.619218i \(-0.212550\pi\)
0.785219 + 0.619218i \(0.212550\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 2.69202 0.356567
\(58\) 0 0
\(59\) −6.93362 −0.902681 −0.451340 0.892352i \(-0.649054\pi\)
−0.451340 + 0.892352i \(0.649054\pi\)
\(60\) 0 0
\(61\) −0.868313 −0.111176 −0.0555881 0.998454i \(-0.517703\pi\)
−0.0555881 + 0.998454i \(0.517703\pi\)
\(62\) 0 0
\(63\) −1.35690 −0.170953
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.53319 −0.798156 −0.399078 0.916917i \(-0.630670\pi\)
−0.399078 + 0.916917i \(0.630670\pi\)
\(68\) 0 0
\(69\) 2.24698 0.270505
\(70\) 0 0
\(71\) −3.54288 −0.420462 −0.210231 0.977652i \(-0.567422\pi\)
−0.210231 + 0.977652i \(0.567422\pi\)
\(72\) 0 0
\(73\) 3.35690 0.392895 0.196447 0.980514i \(-0.437059\pi\)
0.196447 + 0.980514i \(0.437059\pi\)
\(74\) 0 0
\(75\) −0.0489173 −0.00564849
\(76\) 0 0
\(77\) 0.603875 0.0688180
\(78\) 0 0
\(79\) −12.6746 −1.42600 −0.713000 0.701164i \(-0.752664\pi\)
−0.713000 + 0.701164i \(0.752664\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.19806 −0.351033 −0.175516 0.984476i \(-0.556160\pi\)
−0.175516 + 0.984476i \(0.556160\pi\)
\(84\) 0 0
\(85\) −4.04892 −0.439167
\(86\) 0 0
\(87\) −4.65279 −0.498832
\(88\) 0 0
\(89\) −0.454731 −0.0482013 −0.0241007 0.999710i \(-0.507672\pi\)
−0.0241007 + 0.999710i \(0.507672\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.25667 −0.856177
\(94\) 0 0
\(95\) −6.04892 −0.620606
\(96\) 0 0
\(97\) 15.4940 1.57317 0.786587 0.617480i \(-0.211846\pi\)
0.786587 + 0.617480i \(0.211846\pi\)
\(98\) 0 0
\(99\) −0.445042 −0.0447284
\(100\) 0 0
\(101\) −9.77240 −0.972390 −0.486195 0.873850i \(-0.661616\pi\)
−0.486195 + 0.873850i \(0.661616\pi\)
\(102\) 0 0
\(103\) 15.1836 1.49608 0.748042 0.663652i \(-0.230994\pi\)
0.748042 + 0.663652i \(0.230994\pi\)
\(104\) 0 0
\(105\) 3.04892 0.297544
\(106\) 0 0
\(107\) 4.48858 0.433928 0.216964 0.976180i \(-0.430385\pi\)
0.216964 + 0.976180i \(0.430385\pi\)
\(108\) 0 0
\(109\) 17.1347 1.64120 0.820602 0.571500i \(-0.193638\pi\)
0.820602 + 0.571500i \(0.193638\pi\)
\(110\) 0 0
\(111\) −1.41789 −0.134581
\(112\) 0 0
\(113\) −20.0640 −1.88746 −0.943730 0.330716i \(-0.892710\pi\)
−0.943730 + 0.330716i \(0.892710\pi\)
\(114\) 0 0
\(115\) −5.04892 −0.470814
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.44504 0.224137
\(120\) 0 0
\(121\) −10.8019 −0.981994
\(122\) 0 0
\(123\) 9.03684 0.814824
\(124\) 0 0
\(125\) −11.1250 −0.995049
\(126\) 0 0
\(127\) 5.76809 0.511835 0.255917 0.966699i \(-0.417622\pi\)
0.255917 + 0.966699i \(0.417622\pi\)
\(128\) 0 0
\(129\) 9.09783 0.801020
\(130\) 0 0
\(131\) −1.07308 −0.0937555 −0.0468777 0.998901i \(-0.514927\pi\)
−0.0468777 + 0.998901i \(0.514927\pi\)
\(132\) 0 0
\(133\) 3.65279 0.316737
\(134\) 0 0
\(135\) −2.24698 −0.193389
\(136\) 0 0
\(137\) 19.0737 1.62957 0.814787 0.579761i \(-0.196854\pi\)
0.814787 + 0.579761i \(0.196854\pi\)
\(138\) 0 0
\(139\) −13.6093 −1.15432 −0.577161 0.816630i \(-0.695839\pi\)
−0.577161 + 0.816630i \(0.695839\pi\)
\(140\) 0 0
\(141\) 7.58211 0.638528
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 10.4547 0.868218
\(146\) 0 0
\(147\) 5.15883 0.425493
\(148\) 0 0
\(149\) 10.1914 0.834909 0.417454 0.908698i \(-0.362922\pi\)
0.417454 + 0.908698i \(0.362922\pi\)
\(150\) 0 0
\(151\) −11.5767 −0.942101 −0.471050 0.882106i \(-0.656125\pi\)
−0.471050 + 0.882106i \(0.656125\pi\)
\(152\) 0 0
\(153\) −1.80194 −0.145678
\(154\) 0 0
\(155\) 18.5526 1.49018
\(156\) 0 0
\(157\) −9.67025 −0.771770 −0.385885 0.922547i \(-0.626104\pi\)
−0.385885 + 0.922547i \(0.626104\pi\)
\(158\) 0 0
\(159\) −11.4330 −0.906693
\(160\) 0 0
\(161\) 3.04892 0.240288
\(162\) 0 0
\(163\) −12.4983 −0.978940 −0.489470 0.872020i \(-0.662810\pi\)
−0.489470 + 0.872020i \(0.662810\pi\)
\(164\) 0 0
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) 8.56033 0.662419 0.331209 0.943557i \(-0.392543\pi\)
0.331209 + 0.943557i \(0.392543\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −2.69202 −0.205864
\(172\) 0 0
\(173\) 4.70709 0.357873 0.178937 0.983861i \(-0.442734\pi\)
0.178937 + 0.983861i \(0.442734\pi\)
\(174\) 0 0
\(175\) −0.0663757 −0.00501753
\(176\) 0 0
\(177\) 6.93362 0.521163
\(178\) 0 0
\(179\) −16.7409 −1.25128 −0.625638 0.780113i \(-0.715161\pi\)
−0.625638 + 0.780113i \(0.715161\pi\)
\(180\) 0 0
\(181\) −19.0707 −1.41751 −0.708757 0.705453i \(-0.750744\pi\)
−0.708757 + 0.705453i \(0.750744\pi\)
\(182\) 0 0
\(183\) 0.868313 0.0641876
\(184\) 0 0
\(185\) 3.18598 0.234238
\(186\) 0 0
\(187\) 0.801938 0.0586435
\(188\) 0 0
\(189\) 1.35690 0.0986997
\(190\) 0 0
\(191\) −20.2543 −1.46555 −0.732774 0.680472i \(-0.761775\pi\)
−0.732774 + 0.680472i \(0.761775\pi\)
\(192\) 0 0
\(193\) −14.1836 −1.02096 −0.510478 0.859891i \(-0.670532\pi\)
−0.510478 + 0.859891i \(0.670532\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.3502 −0.808668 −0.404334 0.914611i \(-0.632497\pi\)
−0.404334 + 0.914611i \(0.632497\pi\)
\(198\) 0 0
\(199\) −20.2784 −1.43750 −0.718750 0.695268i \(-0.755286\pi\)
−0.718750 + 0.695268i \(0.755286\pi\)
\(200\) 0 0
\(201\) 6.53319 0.460816
\(202\) 0 0
\(203\) −6.31336 −0.443111
\(204\) 0 0
\(205\) −20.3056 −1.41820
\(206\) 0 0
\(207\) −2.24698 −0.156176
\(208\) 0 0
\(209\) 1.19806 0.0828717
\(210\) 0 0
\(211\) 2.47650 0.170489 0.0852447 0.996360i \(-0.472833\pi\)
0.0852447 + 0.996360i \(0.472833\pi\)
\(212\) 0 0
\(213\) 3.54288 0.242754
\(214\) 0 0
\(215\) −20.4426 −1.39418
\(216\) 0 0
\(217\) −11.2034 −0.760539
\(218\) 0 0
\(219\) −3.35690 −0.226838
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 11.2567 0.753802 0.376901 0.926253i \(-0.376990\pi\)
0.376901 + 0.926253i \(0.376990\pi\)
\(224\) 0 0
\(225\) 0.0489173 0.00326116
\(226\) 0 0
\(227\) 26.9409 1.78813 0.894066 0.447936i \(-0.147841\pi\)
0.894066 + 0.447936i \(0.147841\pi\)
\(228\) 0 0
\(229\) −9.36658 −0.618961 −0.309481 0.950906i \(-0.600155\pi\)
−0.309481 + 0.950906i \(0.600155\pi\)
\(230\) 0 0
\(231\) −0.603875 −0.0397321
\(232\) 0 0
\(233\) −7.00969 −0.459220 −0.229610 0.973283i \(-0.573745\pi\)
−0.229610 + 0.973283i \(0.573745\pi\)
\(234\) 0 0
\(235\) −17.0368 −1.11136
\(236\) 0 0
\(237\) 12.6746 0.823301
\(238\) 0 0
\(239\) −8.58211 −0.555130 −0.277565 0.960707i \(-0.589527\pi\)
−0.277565 + 0.960707i \(0.589527\pi\)
\(240\) 0 0
\(241\) 15.2034 0.979340 0.489670 0.871908i \(-0.337117\pi\)
0.489670 + 0.871908i \(0.337117\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −11.5918 −0.740573
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 3.19806 0.202669
\(250\) 0 0
\(251\) −2.86831 −0.181046 −0.0905232 0.995894i \(-0.528854\pi\)
−0.0905232 + 0.995894i \(0.528854\pi\)
\(252\) 0 0
\(253\) 1.00000 0.0628695
\(254\) 0 0
\(255\) 4.04892 0.253553
\(256\) 0 0
\(257\) −10.7724 −0.671964 −0.335982 0.941868i \(-0.609068\pi\)
−0.335982 + 0.941868i \(0.609068\pi\)
\(258\) 0 0
\(259\) −1.92394 −0.119548
\(260\) 0 0
\(261\) 4.65279 0.288001
\(262\) 0 0
\(263\) −3.87369 −0.238862 −0.119431 0.992843i \(-0.538107\pi\)
−0.119431 + 0.992843i \(0.538107\pi\)
\(264\) 0 0
\(265\) 25.6896 1.57810
\(266\) 0 0
\(267\) 0.454731 0.0278291
\(268\) 0 0
\(269\) 3.67994 0.224370 0.112185 0.993687i \(-0.464215\pi\)
0.112185 + 0.993687i \(0.464215\pi\)
\(270\) 0 0
\(271\) −6.30798 −0.383182 −0.191591 0.981475i \(-0.561365\pi\)
−0.191591 + 0.981475i \(0.561365\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.0217703 −0.00131280
\(276\) 0 0
\(277\) −2.43429 −0.146262 −0.0731311 0.997322i \(-0.523299\pi\)
−0.0731311 + 0.997322i \(0.523299\pi\)
\(278\) 0 0
\(279\) 8.25667 0.494314
\(280\) 0 0
\(281\) 8.98493 0.535996 0.267998 0.963419i \(-0.413638\pi\)
0.267998 + 0.963419i \(0.413638\pi\)
\(282\) 0 0
\(283\) 15.9734 0.949523 0.474761 0.880115i \(-0.342534\pi\)
0.474761 + 0.880115i \(0.342534\pi\)
\(284\) 0 0
\(285\) 6.04892 0.358307
\(286\) 0 0
\(287\) 12.2620 0.723806
\(288\) 0 0
\(289\) −13.7530 −0.809001
\(290\) 0 0
\(291\) −15.4940 −0.908272
\(292\) 0 0
\(293\) −5.03923 −0.294395 −0.147197 0.989107i \(-0.547025\pi\)
−0.147197 + 0.989107i \(0.547025\pi\)
\(294\) 0 0
\(295\) −15.5797 −0.907086
\(296\) 0 0
\(297\) 0.445042 0.0258239
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 12.3448 0.711543
\(302\) 0 0
\(303\) 9.77240 0.561410
\(304\) 0 0
\(305\) −1.95108 −0.111719
\(306\) 0 0
\(307\) −16.8726 −0.962972 −0.481486 0.876454i \(-0.659903\pi\)
−0.481486 + 0.876454i \(0.659903\pi\)
\(308\) 0 0
\(309\) −15.1836 −0.863764
\(310\) 0 0
\(311\) 15.3207 0.868754 0.434377 0.900731i \(-0.356968\pi\)
0.434377 + 0.900731i \(0.356968\pi\)
\(312\) 0 0
\(313\) −18.4045 −1.04028 −0.520142 0.854080i \(-0.674121\pi\)
−0.520142 + 0.854080i \(0.674121\pi\)
\(314\) 0 0
\(315\) −3.04892 −0.171787
\(316\) 0 0
\(317\) 24.3521 1.36775 0.683875 0.729599i \(-0.260293\pi\)
0.683875 + 0.729599i \(0.260293\pi\)
\(318\) 0 0
\(319\) −2.07069 −0.115936
\(320\) 0 0
\(321\) −4.48858 −0.250528
\(322\) 0 0
\(323\) 4.85086 0.269909
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −17.1347 −0.947549
\(328\) 0 0
\(329\) 10.2881 0.567203
\(330\) 0 0
\(331\) −2.74333 −0.150787 −0.0753936 0.997154i \(-0.524021\pi\)
−0.0753936 + 0.997154i \(0.524021\pi\)
\(332\) 0 0
\(333\) 1.41789 0.0777002
\(334\) 0 0
\(335\) −14.6799 −0.802051
\(336\) 0 0
\(337\) −26.5743 −1.44760 −0.723798 0.690012i \(-0.757605\pi\)
−0.723798 + 0.690012i \(0.757605\pi\)
\(338\) 0 0
\(339\) 20.0640 1.08973
\(340\) 0 0
\(341\) −3.67456 −0.198989
\(342\) 0 0
\(343\) 16.4983 0.890823
\(344\) 0 0
\(345\) 5.04892 0.271825
\(346\) 0 0
\(347\) 17.3937 0.933744 0.466872 0.884325i \(-0.345381\pi\)
0.466872 + 0.884325i \(0.345381\pi\)
\(348\) 0 0
\(349\) −9.75600 −0.522227 −0.261113 0.965308i \(-0.584090\pi\)
−0.261113 + 0.965308i \(0.584090\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.7832 −0.680378 −0.340189 0.940357i \(-0.610491\pi\)
−0.340189 + 0.940357i \(0.610491\pi\)
\(354\) 0 0
\(355\) −7.96077 −0.422514
\(356\) 0 0
\(357\) −2.44504 −0.129405
\(358\) 0 0
\(359\) 14.2784 0.753587 0.376794 0.926297i \(-0.377027\pi\)
0.376794 + 0.926297i \(0.377027\pi\)
\(360\) 0 0
\(361\) −11.7530 −0.618580
\(362\) 0 0
\(363\) 10.8019 0.566955
\(364\) 0 0
\(365\) 7.54288 0.394812
\(366\) 0 0
\(367\) −15.1491 −0.790779 −0.395389 0.918514i \(-0.629390\pi\)
−0.395389 + 0.918514i \(0.629390\pi\)
\(368\) 0 0
\(369\) −9.03684 −0.470439
\(370\) 0 0
\(371\) −15.5133 −0.805412
\(372\) 0 0
\(373\) 18.7385 0.970245 0.485123 0.874446i \(-0.338775\pi\)
0.485123 + 0.874446i \(0.338775\pi\)
\(374\) 0 0
\(375\) 11.1250 0.574492
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −27.3937 −1.40712 −0.703561 0.710635i \(-0.748408\pi\)
−0.703561 + 0.710635i \(0.748408\pi\)
\(380\) 0 0
\(381\) −5.76809 −0.295508
\(382\) 0 0
\(383\) 17.3448 0.886279 0.443139 0.896453i \(-0.353865\pi\)
0.443139 + 0.896453i \(0.353865\pi\)
\(384\) 0 0
\(385\) 1.35690 0.0691538
\(386\) 0 0
\(387\) −9.09783 −0.462469
\(388\) 0 0
\(389\) −24.2620 −1.23013 −0.615067 0.788475i \(-0.710871\pi\)
−0.615067 + 0.788475i \(0.710871\pi\)
\(390\) 0 0
\(391\) 4.04892 0.204763
\(392\) 0 0
\(393\) 1.07308 0.0541298
\(394\) 0 0
\(395\) −28.4795 −1.43296
\(396\) 0 0
\(397\) −37.7536 −1.89480 −0.947400 0.320053i \(-0.896299\pi\)
−0.947400 + 0.320053i \(0.896299\pi\)
\(398\) 0 0
\(399\) −3.65279 −0.182868
\(400\) 0 0
\(401\) −9.01400 −0.450138 −0.225069 0.974343i \(-0.572261\pi\)
−0.225069 + 0.974343i \(0.572261\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2.24698 0.111653
\(406\) 0 0
\(407\) −0.631023 −0.0312786
\(408\) 0 0
\(409\) −15.0344 −0.743405 −0.371703 0.928352i \(-0.621226\pi\)
−0.371703 + 0.928352i \(0.621226\pi\)
\(410\) 0 0
\(411\) −19.0737 −0.940835
\(412\) 0 0
\(413\) 9.40821 0.462948
\(414\) 0 0
\(415\) −7.18598 −0.352746
\(416\) 0 0
\(417\) 13.6093 0.666448
\(418\) 0 0
\(419\) −23.8418 −1.16475 −0.582373 0.812922i \(-0.697876\pi\)
−0.582373 + 0.812922i \(0.697876\pi\)
\(420\) 0 0
\(421\) 26.4045 1.28688 0.643438 0.765498i \(-0.277507\pi\)
0.643438 + 0.765498i \(0.277507\pi\)
\(422\) 0 0
\(423\) −7.58211 −0.368655
\(424\) 0 0
\(425\) −0.0881460 −0.00427571
\(426\) 0 0
\(427\) 1.17821 0.0570176
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.47757 0.263845 0.131923 0.991260i \(-0.457885\pi\)
0.131923 + 0.991260i \(0.457885\pi\)
\(432\) 0 0
\(433\) −18.9071 −0.908616 −0.454308 0.890845i \(-0.650113\pi\)
−0.454308 + 0.890845i \(0.650113\pi\)
\(434\) 0 0
\(435\) −10.4547 −0.501266
\(436\) 0 0
\(437\) 6.04892 0.289359
\(438\) 0 0
\(439\) 28.6480 1.36729 0.683647 0.729812i \(-0.260393\pi\)
0.683647 + 0.729812i \(0.260393\pi\)
\(440\) 0 0
\(441\) −5.15883 −0.245659
\(442\) 0 0
\(443\) 14.3502 0.681798 0.340899 0.940100i \(-0.389269\pi\)
0.340899 + 0.940100i \(0.389269\pi\)
\(444\) 0 0
\(445\) −1.02177 −0.0484366
\(446\) 0 0
\(447\) −10.1914 −0.482035
\(448\) 0 0
\(449\) 33.7187 1.59128 0.795642 0.605767i \(-0.207134\pi\)
0.795642 + 0.605767i \(0.207134\pi\)
\(450\) 0 0
\(451\) 4.02177 0.189378
\(452\) 0 0
\(453\) 11.5767 0.543922
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 41.1758 1.92612 0.963062 0.269281i \(-0.0867859\pi\)
0.963062 + 0.269281i \(0.0867859\pi\)
\(458\) 0 0
\(459\) 1.80194 0.0841073
\(460\) 0 0
\(461\) −2.94571 −0.137195 −0.0685976 0.997644i \(-0.521852\pi\)
−0.0685976 + 0.997644i \(0.521852\pi\)
\(462\) 0 0
\(463\) −28.2828 −1.31441 −0.657205 0.753711i \(-0.728262\pi\)
−0.657205 + 0.753711i \(0.728262\pi\)
\(464\) 0 0
\(465\) −18.5526 −0.860355
\(466\) 0 0
\(467\) 22.5060 1.04146 0.520728 0.853723i \(-0.325661\pi\)
0.520728 + 0.853723i \(0.325661\pi\)
\(468\) 0 0
\(469\) 8.86486 0.409341
\(470\) 0 0
\(471\) 9.67025 0.445582
\(472\) 0 0
\(473\) 4.04892 0.186169
\(474\) 0 0
\(475\) −0.131687 −0.00604219
\(476\) 0 0
\(477\) 11.4330 0.523479
\(478\) 0 0
\(479\) 35.1933 1.60802 0.804011 0.594615i \(-0.202695\pi\)
0.804011 + 0.594615i \(0.202695\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −3.04892 −0.138731
\(484\) 0 0
\(485\) 34.8146 1.58085
\(486\) 0 0
\(487\) −22.9517 −1.04004 −0.520020 0.854154i \(-0.674075\pi\)
−0.520020 + 0.854154i \(0.674075\pi\)
\(488\) 0 0
\(489\) 12.4983 0.565191
\(490\) 0 0
\(491\) 13.5657 0.612212 0.306106 0.951997i \(-0.400974\pi\)
0.306106 + 0.951997i \(0.400974\pi\)
\(492\) 0 0
\(493\) −8.38404 −0.377598
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) 4.80731 0.215638
\(498\) 0 0
\(499\) −20.5676 −0.920734 −0.460367 0.887729i \(-0.652282\pi\)
−0.460367 + 0.887729i \(0.652282\pi\)
\(500\) 0 0
\(501\) −8.56033 −0.382448
\(502\) 0 0
\(503\) 17.0935 0.762163 0.381081 0.924542i \(-0.375552\pi\)
0.381081 + 0.924542i \(0.375552\pi\)
\(504\) 0 0
\(505\) −21.9584 −0.977135
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −36.9288 −1.63684 −0.818421 0.574619i \(-0.805150\pi\)
−0.818421 + 0.574619i \(0.805150\pi\)
\(510\) 0 0
\(511\) −4.55496 −0.201499
\(512\) 0 0
\(513\) 2.69202 0.118856
\(514\) 0 0
\(515\) 34.1172 1.50338
\(516\) 0 0
\(517\) 3.37435 0.148404
\(518\) 0 0
\(519\) −4.70709 −0.206618
\(520\) 0 0
\(521\) −8.63401 −0.378263 −0.189131 0.981952i \(-0.560567\pi\)
−0.189131 + 0.981952i \(0.560567\pi\)
\(522\) 0 0
\(523\) 13.7832 0.602695 0.301348 0.953514i \(-0.402564\pi\)
0.301348 + 0.953514i \(0.402564\pi\)
\(524\) 0 0
\(525\) 0.0663757 0.00289687
\(526\) 0 0
\(527\) −14.8780 −0.648096
\(528\) 0 0
\(529\) −17.9511 −0.780482
\(530\) 0 0
\(531\) −6.93362 −0.300894
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 10.0858 0.436045
\(536\) 0 0
\(537\) 16.7409 0.722425
\(538\) 0 0
\(539\) 2.29590 0.0988913
\(540\) 0 0
\(541\) 0.377338 0.0162230 0.00811152 0.999967i \(-0.497418\pi\)
0.00811152 + 0.999967i \(0.497418\pi\)
\(542\) 0 0
\(543\) 19.0707 0.818402
\(544\) 0 0
\(545\) 38.5013 1.64921
\(546\) 0 0
\(547\) −39.8635 −1.70444 −0.852221 0.523182i \(-0.824745\pi\)
−0.852221 + 0.523182i \(0.824745\pi\)
\(548\) 0 0
\(549\) −0.868313 −0.0370587
\(550\) 0 0
\(551\) −12.5254 −0.533601
\(552\) 0 0
\(553\) 17.1981 0.731336
\(554\) 0 0
\(555\) −3.18598 −0.135237
\(556\) 0 0
\(557\) 1.34242 0.0568802 0.0284401 0.999595i \(-0.490946\pi\)
0.0284401 + 0.999595i \(0.490946\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −0.801938 −0.0338578
\(562\) 0 0
\(563\) −34.9493 −1.47294 −0.736468 0.676472i \(-0.763508\pi\)
−0.736468 + 0.676472i \(0.763508\pi\)
\(564\) 0 0
\(565\) −45.0834 −1.89667
\(566\) 0 0
\(567\) −1.35690 −0.0569843
\(568\) 0 0
\(569\) 16.0575 0.673167 0.336584 0.941654i \(-0.390729\pi\)
0.336584 + 0.941654i \(0.390729\pi\)
\(570\) 0 0
\(571\) −34.1444 −1.42890 −0.714448 0.699688i \(-0.753322\pi\)
−0.714448 + 0.699688i \(0.753322\pi\)
\(572\) 0 0
\(573\) 20.2543 0.846134
\(574\) 0 0
\(575\) −0.109916 −0.00458383
\(576\) 0 0
\(577\) −11.2368 −0.467795 −0.233897 0.972261i \(-0.575148\pi\)
−0.233897 + 0.972261i \(0.575148\pi\)
\(578\) 0 0
\(579\) 14.1836 0.589450
\(580\) 0 0
\(581\) 4.33944 0.180030
\(582\) 0 0
\(583\) −5.08815 −0.210729
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.53750 −0.352380 −0.176190 0.984356i \(-0.556377\pi\)
−0.176190 + 0.984356i \(0.556377\pi\)
\(588\) 0 0
\(589\) −22.2271 −0.915853
\(590\) 0 0
\(591\) 11.3502 0.466884
\(592\) 0 0
\(593\) 15.4383 0.633977 0.316988 0.948429i \(-0.397328\pi\)
0.316988 + 0.948429i \(0.397328\pi\)
\(594\) 0 0
\(595\) 5.49396 0.225230
\(596\) 0 0
\(597\) 20.2784 0.829941
\(598\) 0 0
\(599\) −6.82072 −0.278687 −0.139344 0.990244i \(-0.544499\pi\)
−0.139344 + 0.990244i \(0.544499\pi\)
\(600\) 0 0
\(601\) 44.4142 1.81169 0.905846 0.423607i \(-0.139236\pi\)
0.905846 + 0.423607i \(0.139236\pi\)
\(602\) 0 0
\(603\) −6.53319 −0.266052
\(604\) 0 0
\(605\) −24.2717 −0.986786
\(606\) 0 0
\(607\) 26.2218 1.06431 0.532154 0.846648i \(-0.321383\pi\)
0.532154 + 0.846648i \(0.321383\pi\)
\(608\) 0 0
\(609\) 6.31336 0.255830
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.02177 −0.0412689 −0.0206345 0.999787i \(-0.506569\pi\)
−0.0206345 + 0.999787i \(0.506569\pi\)
\(614\) 0 0
\(615\) 20.3056 0.818800
\(616\) 0 0
\(617\) 24.5724 0.989248 0.494624 0.869107i \(-0.335306\pi\)
0.494624 + 0.869107i \(0.335306\pi\)
\(618\) 0 0
\(619\) −16.4614 −0.661641 −0.330820 0.943694i \(-0.607325\pi\)
−0.330820 + 0.943694i \(0.607325\pi\)
\(620\) 0 0
\(621\) 2.24698 0.0901682
\(622\) 0 0
\(623\) 0.617022 0.0247205
\(624\) 0 0
\(625\) −25.2422 −1.00969
\(626\) 0 0
\(627\) −1.19806 −0.0478460
\(628\) 0 0
\(629\) −2.55496 −0.101873
\(630\) 0 0
\(631\) 3.39852 0.135293 0.0676464 0.997709i \(-0.478451\pi\)
0.0676464 + 0.997709i \(0.478451\pi\)
\(632\) 0 0
\(633\) −2.47650 −0.0984321
\(634\) 0 0
\(635\) 12.9608 0.514333
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −3.54288 −0.140154
\(640\) 0 0
\(641\) 29.4634 1.16373 0.581866 0.813284i \(-0.302323\pi\)
0.581866 + 0.813284i \(0.302323\pi\)
\(642\) 0 0
\(643\) −13.7235 −0.541201 −0.270601 0.962692i \(-0.587222\pi\)
−0.270601 + 0.962692i \(0.587222\pi\)
\(644\) 0 0
\(645\) 20.4426 0.804929
\(646\) 0 0
\(647\) 21.5200 0.846040 0.423020 0.906120i \(-0.360970\pi\)
0.423020 + 0.906120i \(0.360970\pi\)
\(648\) 0 0
\(649\) 3.08575 0.121126
\(650\) 0 0
\(651\) 11.2034 0.439097
\(652\) 0 0
\(653\) −26.4935 −1.03677 −0.518385 0.855147i \(-0.673467\pi\)
−0.518385 + 0.855147i \(0.673467\pi\)
\(654\) 0 0
\(655\) −2.41119 −0.0942130
\(656\) 0 0
\(657\) 3.35690 0.130965
\(658\) 0 0
\(659\) 37.9135 1.47690 0.738450 0.674308i \(-0.235558\pi\)
0.738450 + 0.674308i \(0.235558\pi\)
\(660\) 0 0
\(661\) −0.940066 −0.0365643 −0.0182822 0.999833i \(-0.505820\pi\)
−0.0182822 + 0.999833i \(0.505820\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.20775 0.318283
\(666\) 0 0
\(667\) −10.4547 −0.404809
\(668\) 0 0
\(669\) −11.2567 −0.435208
\(670\) 0 0
\(671\) 0.386436 0.0149182
\(672\) 0 0
\(673\) 23.1540 0.892523 0.446261 0.894903i \(-0.352755\pi\)
0.446261 + 0.894903i \(0.352755\pi\)
\(674\) 0 0
\(675\) −0.0489173 −0.00188283
\(676\) 0 0
\(677\) 2.41060 0.0926468 0.0463234 0.998926i \(-0.485250\pi\)
0.0463234 + 0.998926i \(0.485250\pi\)
\(678\) 0 0
\(679\) −21.0237 −0.806815
\(680\) 0 0
\(681\) −26.9409 −1.03238
\(682\) 0 0
\(683\) 42.8823 1.64085 0.820423 0.571757i \(-0.193738\pi\)
0.820423 + 0.571757i \(0.193738\pi\)
\(684\) 0 0
\(685\) 42.8582 1.63753
\(686\) 0 0
\(687\) 9.36658 0.357357
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 49.2820 1.87478 0.937388 0.348287i \(-0.113236\pi\)
0.937388 + 0.348287i \(0.113236\pi\)
\(692\) 0 0
\(693\) 0.603875 0.0229393
\(694\) 0 0
\(695\) −30.5797 −1.15995
\(696\) 0 0
\(697\) 16.2838 0.616793
\(698\) 0 0
\(699\) 7.00969 0.265131
\(700\) 0 0
\(701\) −31.1213 −1.17543 −0.587717 0.809067i \(-0.699973\pi\)
−0.587717 + 0.809067i \(0.699973\pi\)
\(702\) 0 0
\(703\) −3.81700 −0.143961
\(704\) 0 0
\(705\) 17.0368 0.641644
\(706\) 0 0
\(707\) 13.2601 0.498698
\(708\) 0 0
\(709\) 40.4416 1.51882 0.759408 0.650615i \(-0.225489\pi\)
0.759408 + 0.650615i \(0.225489\pi\)
\(710\) 0 0
\(711\) −12.6746 −0.475333
\(712\) 0 0
\(713\) −18.5526 −0.694799
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 8.58211 0.320505
\(718\) 0 0
\(719\) 8.59658 0.320598 0.160299 0.987068i \(-0.448754\pi\)
0.160299 + 0.987068i \(0.448754\pi\)
\(720\) 0 0
\(721\) −20.6025 −0.767279
\(722\) 0 0
\(723\) −15.2034 −0.565422
\(724\) 0 0
\(725\) 0.227602 0.00845294
\(726\) 0 0
\(727\) 14.4300 0.535178 0.267589 0.963533i \(-0.413773\pi\)
0.267589 + 0.963533i \(0.413773\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.3937 0.606344
\(732\) 0 0
\(733\) 7.26636 0.268389 0.134195 0.990955i \(-0.457155\pi\)
0.134195 + 0.990955i \(0.457155\pi\)
\(734\) 0 0
\(735\) 11.5918 0.427570
\(736\) 0 0
\(737\) 2.90754 0.107101
\(738\) 0 0
\(739\) 3.60281 0.132532 0.0662658 0.997802i \(-0.478891\pi\)
0.0662658 + 0.997802i \(0.478891\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 33.1812 1.21730 0.608650 0.793439i \(-0.291711\pi\)
0.608650 + 0.793439i \(0.291711\pi\)
\(744\) 0 0
\(745\) 22.8998 0.838983
\(746\) 0 0
\(747\) −3.19806 −0.117011
\(748\) 0 0
\(749\) −6.09054 −0.222543
\(750\) 0 0
\(751\) 32.4077 1.18257 0.591287 0.806461i \(-0.298620\pi\)
0.591287 + 0.806461i \(0.298620\pi\)
\(752\) 0 0
\(753\) 2.86831 0.104527
\(754\) 0 0
\(755\) −26.0127 −0.946698
\(756\) 0 0
\(757\) 13.8726 0.504209 0.252105 0.967700i \(-0.418877\pi\)
0.252105 + 0.967700i \(0.418877\pi\)
\(758\) 0 0
\(759\) −1.00000 −0.0362977
\(760\) 0 0
\(761\) 35.3943 1.28304 0.641522 0.767105i \(-0.278303\pi\)
0.641522 + 0.767105i \(0.278303\pi\)
\(762\) 0 0
\(763\) −23.2500 −0.841705
\(764\) 0 0
\(765\) −4.04892 −0.146389
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 32.1909 1.16083 0.580416 0.814320i \(-0.302890\pi\)
0.580416 + 0.814320i \(0.302890\pi\)
\(770\) 0 0
\(771\) 10.7724 0.387958
\(772\) 0 0
\(773\) −15.9186 −0.572551 −0.286275 0.958147i \(-0.592417\pi\)
−0.286275 + 0.958147i \(0.592417\pi\)
\(774\) 0 0
\(775\) 0.403894 0.0145083
\(776\) 0 0
\(777\) 1.92394 0.0690208
\(778\) 0 0
\(779\) 24.3274 0.871618
\(780\) 0 0
\(781\) 1.57673 0.0564198
\(782\) 0 0
\(783\) −4.65279 −0.166277
\(784\) 0 0
\(785\) −21.7289 −0.775536
\(786\) 0 0
\(787\) −21.0476 −0.750266 −0.375133 0.926971i \(-0.622403\pi\)
−0.375133 + 0.926971i \(0.622403\pi\)
\(788\) 0 0
\(789\) 3.87369 0.137907
\(790\) 0 0
\(791\) 27.2247 0.968000
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −25.6896 −0.911117
\(796\) 0 0
\(797\) 49.0917 1.73892 0.869459 0.494005i \(-0.164468\pi\)
0.869459 + 0.494005i \(0.164468\pi\)
\(798\) 0 0
\(799\) 13.6625 0.483344
\(800\) 0 0
\(801\) −0.454731 −0.0160671
\(802\) 0 0
\(803\) −1.49396 −0.0527207
\(804\) 0 0
\(805\) 6.85086 0.241461
\(806\) 0 0
\(807\) −3.67994 −0.129540
\(808\) 0 0
\(809\) 17.4370 0.613053 0.306526 0.951862i \(-0.400833\pi\)
0.306526 + 0.951862i \(0.400833\pi\)
\(810\) 0 0
\(811\) 12.0411 0.422822 0.211411 0.977397i \(-0.432194\pi\)
0.211411 + 0.977397i \(0.432194\pi\)
\(812\) 0 0
\(813\) 6.30798 0.221230
\(814\) 0 0
\(815\) −28.0834 −0.983717
\(816\) 0 0
\(817\) 24.4916 0.856851
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −30.4403 −1.06237 −0.531186 0.847255i \(-0.678254\pi\)
−0.531186 + 0.847255i \(0.678254\pi\)
\(822\) 0 0
\(823\) −2.22282 −0.0774825 −0.0387413 0.999249i \(-0.512335\pi\)
−0.0387413 + 0.999249i \(0.512335\pi\)
\(824\) 0 0
\(825\) 0.0217703 0.000757943 0
\(826\) 0 0
\(827\) 36.9028 1.28323 0.641617 0.767025i \(-0.278264\pi\)
0.641617 + 0.767025i \(0.278264\pi\)
\(828\) 0 0
\(829\) 21.2978 0.739704 0.369852 0.929091i \(-0.379408\pi\)
0.369852 + 0.929091i \(0.379408\pi\)
\(830\) 0 0
\(831\) 2.43429 0.0844445
\(832\) 0 0
\(833\) 9.29590 0.322084
\(834\) 0 0
\(835\) 19.2349 0.665651
\(836\) 0 0
\(837\) −8.25667 −0.285392
\(838\) 0 0
\(839\) 39.6353 1.36836 0.684182 0.729311i \(-0.260159\pi\)
0.684182 + 0.729311i \(0.260159\pi\)
\(840\) 0 0
\(841\) −7.35152 −0.253501
\(842\) 0 0
\(843\) −8.98493 −0.309458
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 14.6571 0.503624
\(848\) 0 0
\(849\) −15.9734 −0.548207
\(850\) 0 0
\(851\) −3.18598 −0.109214
\(852\) 0 0
\(853\) 49.8133 1.70557 0.852787 0.522259i \(-0.174910\pi\)
0.852787 + 0.522259i \(0.174910\pi\)
\(854\) 0 0
\(855\) −6.04892 −0.206869
\(856\) 0 0
\(857\) −15.6969 −0.536197 −0.268098 0.963392i \(-0.586395\pi\)
−0.268098 + 0.963392i \(0.586395\pi\)
\(858\) 0 0
\(859\) −25.5478 −0.871679 −0.435839 0.900024i \(-0.643548\pi\)
−0.435839 + 0.900024i \(0.643548\pi\)
\(860\) 0 0
\(861\) −12.2620 −0.417889
\(862\) 0 0
\(863\) −42.6088 −1.45042 −0.725210 0.688528i \(-0.758257\pi\)
−0.725210 + 0.688528i \(0.758257\pi\)
\(864\) 0 0
\(865\) 10.5767 0.359620
\(866\) 0 0
\(867\) 13.7530 0.467077
\(868\) 0 0
\(869\) 5.64071 0.191348
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 15.4940 0.524391
\(874\) 0 0
\(875\) 15.0954 0.510319
\(876\) 0 0
\(877\) −15.6461 −0.528331 −0.264165 0.964477i \(-0.585096\pi\)
−0.264165 + 0.964477i \(0.585096\pi\)
\(878\) 0 0
\(879\) 5.03923 0.169969
\(880\) 0 0
\(881\) 49.0484 1.65248 0.826242 0.563315i \(-0.190474\pi\)
0.826242 + 0.563315i \(0.190474\pi\)
\(882\) 0 0
\(883\) −57.8165 −1.94568 −0.972841 0.231476i \(-0.925644\pi\)
−0.972841 + 0.231476i \(0.925644\pi\)
\(884\) 0 0
\(885\) 15.5797 0.523706
\(886\) 0 0
\(887\) 54.2847 1.82270 0.911350 0.411631i \(-0.135041\pi\)
0.911350 + 0.411631i \(0.135041\pi\)
\(888\) 0 0
\(889\) −7.82669 −0.262499
\(890\) 0 0
\(891\) −0.445042 −0.0149095
\(892\) 0 0
\(893\) 20.4112 0.683034
\(894\) 0 0
\(895\) −37.6165 −1.25738
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 38.4166 1.28126
\(900\) 0 0
\(901\) −20.6015 −0.686335
\(902\) 0 0
\(903\) −12.3448 −0.410810
\(904\) 0 0
\(905\) −42.8514 −1.42443
\(906\) 0 0
\(907\) 27.7700 0.922088 0.461044 0.887377i \(-0.347475\pi\)
0.461044 + 0.887377i \(0.347475\pi\)
\(908\) 0 0
\(909\) −9.77240 −0.324130
\(910\) 0 0
\(911\) 43.4403 1.43924 0.719620 0.694368i \(-0.244316\pi\)
0.719620 + 0.694368i \(0.244316\pi\)
\(912\) 0 0
\(913\) 1.42327 0.0471034
\(914\) 0 0
\(915\) 1.95108 0.0645008
\(916\) 0 0
\(917\) 1.45606 0.0480833
\(918\) 0 0
\(919\) 46.6661 1.53937 0.769686 0.638423i \(-0.220413\pi\)
0.769686 + 0.638423i \(0.220413\pi\)
\(920\) 0 0
\(921\) 16.8726 0.555972
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0.0693596 0.00228053
\(926\) 0 0
\(927\) 15.1836 0.498694
\(928\) 0 0
\(929\) −6.36467 −0.208818 −0.104409 0.994534i \(-0.533295\pi\)
−0.104409 + 0.994534i \(0.533295\pi\)
\(930\) 0 0
\(931\) 13.8877 0.455151
\(932\) 0 0
\(933\) −15.3207 −0.501576
\(934\) 0 0
\(935\) 1.80194 0.0589297
\(936\) 0 0
\(937\) −9.95599 −0.325248 −0.162624 0.986688i \(-0.551996\pi\)
−0.162624 + 0.986688i \(0.551996\pi\)
\(938\) 0 0
\(939\) 18.4045 0.600608
\(940\) 0 0
\(941\) −52.6950 −1.71781 −0.858904 0.512137i \(-0.828854\pi\)
−0.858904 + 0.512137i \(0.828854\pi\)
\(942\) 0 0
\(943\) 20.3056 0.661241
\(944\) 0 0
\(945\) 3.04892 0.0991813
\(946\) 0 0
\(947\) −7.11051 −0.231060 −0.115530 0.993304i \(-0.536857\pi\)
−0.115530 + 0.993304i \(0.536857\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −24.3521 −0.789671
\(952\) 0 0
\(953\) 60.7569 1.96811 0.984054 0.177871i \(-0.0569209\pi\)
0.984054 + 0.177871i \(0.0569209\pi\)
\(954\) 0 0
\(955\) −45.5109 −1.47270
\(956\) 0 0
\(957\) 2.07069 0.0669358
\(958\) 0 0
\(959\) −25.8810 −0.835741
\(960\) 0 0
\(961\) 37.1726 1.19912
\(962\) 0 0
\(963\) 4.48858 0.144643
\(964\) 0 0
\(965\) −31.8702 −1.02594
\(966\) 0 0
\(967\) 29.6002 0.951877 0.475938 0.879479i \(-0.342109\pi\)
0.475938 + 0.879479i \(0.342109\pi\)
\(968\) 0 0
\(969\) −4.85086 −0.155832
\(970\) 0 0
\(971\) 53.8525 1.72821 0.864105 0.503312i \(-0.167885\pi\)
0.864105 + 0.503312i \(0.167885\pi\)
\(972\) 0 0
\(973\) 18.4663 0.592004
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −7.55065 −0.241567 −0.120783 0.992679i \(-0.538541\pi\)
−0.120783 + 0.992679i \(0.538541\pi\)
\(978\) 0 0
\(979\) 0.202374 0.00646791
\(980\) 0 0
\(981\) 17.1347 0.547068
\(982\) 0 0
\(983\) 28.7375 0.916583 0.458292 0.888802i \(-0.348462\pi\)
0.458292 + 0.888802i \(0.348462\pi\)
\(984\) 0 0
\(985\) −25.5036 −0.812614
\(986\) 0 0
\(987\) −10.2881 −0.327475
\(988\) 0 0
\(989\) 20.4426 0.650038
\(990\) 0 0
\(991\) −57.4368 −1.82454 −0.912270 0.409589i \(-0.865672\pi\)
−0.912270 + 0.409589i \(0.865672\pi\)
\(992\) 0 0
\(993\) 2.74333 0.0870570
\(994\) 0 0
\(995\) −45.5652 −1.44452
\(996\) 0 0
\(997\) 24.5633 0.777928 0.388964 0.921253i \(-0.372833\pi\)
0.388964 + 0.921253i \(0.372833\pi\)
\(998\) 0 0
\(999\) −1.41789 −0.0448602
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 4056.2.a.ba.1.3 yes 3
4.3 odd 2 8112.2.a.cn.1.3 3
13.5 odd 4 4056.2.c.n.337.1 6
13.8 odd 4 4056.2.c.n.337.6 6
13.12 even 2 4056.2.a.x.1.1 3
52.51 odd 2 8112.2.a.ci.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4056.2.a.x.1.1 3 13.12 even 2
4056.2.a.ba.1.3 yes 3 1.1 even 1 trivial
4056.2.c.n.337.1 6 13.5 odd 4
4056.2.c.n.337.6 6 13.8 odd 4
8112.2.a.ci.1.1 3 52.51 odd 2
8112.2.a.cn.1.3 3 4.3 odd 2