Properties

Label 4014.2.a.u.1.2
Level $4014$
Weight $2$
Character 4014.1
Self dual yes
Analytic conductor $32.052$
Analytic rank $0$
Dimension $6$
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] = [4014,2,Mod(1,4014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4014, 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("4014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4014.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: \(32.0519513713\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.103354048.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 13x^{2} - 16x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.605879\) of defining polynomial
Character \(\chi\) \(=\) 4014.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.63291 q^{5} -2.24656 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.63291 q^{5} -2.24656 q^{7} +1.00000 q^{8} -2.63291 q^{10} -4.09429 q^{11} +0.00776648 q^{13} -2.24656 q^{14} +1.00000 q^{16} +0.673658 q^{17} -3.57290 q^{19} -2.63291 q^{20} -4.09429 q^{22} +3.39412 q^{23} +1.93222 q^{25} +0.00776648 q^{26} -2.24656 q^{28} +8.00809 q^{29} -6.93425 q^{31} +1.00000 q^{32} +0.673658 q^{34} +5.91498 q^{35} +0.651332 q^{37} -3.57290 q^{38} -2.63291 q^{40} +4.43181 q^{41} +9.59164 q^{43} -4.09429 q^{44} +3.39412 q^{46} +10.6264 q^{47} -1.95299 q^{49} +1.93222 q^{50} +0.00776648 q^{52} +4.64357 q^{53} +10.7799 q^{55} -2.24656 q^{56} +8.00809 q^{58} +7.95351 q^{59} -11.0517 q^{61} -6.93425 q^{62} +1.00000 q^{64} -0.0204485 q^{65} +15.2010 q^{67} +0.673658 q^{68} +5.91498 q^{70} -9.64824 q^{71} +0.995761 q^{73} +0.651332 q^{74} -3.57290 q^{76} +9.19804 q^{77} -1.27648 q^{79} -2.63291 q^{80} +4.43181 q^{82} +5.75391 q^{83} -1.77368 q^{85} +9.59164 q^{86} -4.09429 q^{88} -2.07485 q^{89} -0.0174478 q^{91} +3.39412 q^{92} +10.6264 q^{94} +9.40712 q^{95} -17.5825 q^{97} -1.95299 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8} + 2 q^{10} + 2 q^{11} - 2 q^{13} + 8 q^{14} + 6 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{20} + 2 q^{22} + 22 q^{23} + 12 q^{25} - 2 q^{26} + 8 q^{28} + 8 q^{29} - 12 q^{31} + 6 q^{32} + 2 q^{34} + 20 q^{35} - 2 q^{38} + 2 q^{40} + 28 q^{41} + 14 q^{43} + 2 q^{44} + 22 q^{46} + 2 q^{47} + 12 q^{50} - 2 q^{52} + 26 q^{53} + 6 q^{55} + 8 q^{56} + 8 q^{58} + 4 q^{59} - 6 q^{61} - 12 q^{62} + 6 q^{64} + 16 q^{65} + 18 q^{67} + 2 q^{68} + 20 q^{70} + 12 q^{71} - 20 q^{73} - 2 q^{76} + 20 q^{77} + 12 q^{79} + 2 q^{80} + 28 q^{82} + 12 q^{83} - 10 q^{85} + 14 q^{86} + 2 q^{88} - 2 q^{89} - 22 q^{91} + 22 q^{92} + 2 q^{94} + 6 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.63291 −1.17747 −0.588737 0.808325i \(-0.700375\pi\)
−0.588737 + 0.808325i \(0.700375\pi\)
\(6\) 0 0
\(7\) −2.24656 −0.849118 −0.424559 0.905400i \(-0.639571\pi\)
−0.424559 + 0.905400i \(0.639571\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.63291 −0.832600
\(11\) −4.09429 −1.23447 −0.617237 0.786777i \(-0.711748\pi\)
−0.617237 + 0.786777i \(0.711748\pi\)
\(12\) 0 0
\(13\) 0.00776648 0.00215403 0.00107702 0.999999i \(-0.499657\pi\)
0.00107702 + 0.999999i \(0.499657\pi\)
\(14\) −2.24656 −0.600417
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.673658 0.163386 0.0816931 0.996658i \(-0.473967\pi\)
0.0816931 + 0.996658i \(0.473967\pi\)
\(18\) 0 0
\(19\) −3.57290 −0.819679 −0.409839 0.912158i \(-0.634415\pi\)
−0.409839 + 0.912158i \(0.634415\pi\)
\(20\) −2.63291 −0.588737
\(21\) 0 0
\(22\) −4.09429 −0.872905
\(23\) 3.39412 0.707723 0.353862 0.935298i \(-0.384868\pi\)
0.353862 + 0.935298i \(0.384868\pi\)
\(24\) 0 0
\(25\) 1.93222 0.386444
\(26\) 0.00776648 0.00152313
\(27\) 0 0
\(28\) −2.24656 −0.424559
\(29\) 8.00809 1.48706 0.743532 0.668700i \(-0.233149\pi\)
0.743532 + 0.668700i \(0.233149\pi\)
\(30\) 0 0
\(31\) −6.93425 −1.24543 −0.622714 0.782450i \(-0.713970\pi\)
−0.622714 + 0.782450i \(0.713970\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.673658 0.115531
\(35\) 5.91498 0.999815
\(36\) 0 0
\(37\) 0.651332 0.107078 0.0535391 0.998566i \(-0.482950\pi\)
0.0535391 + 0.998566i \(0.482950\pi\)
\(38\) −3.57290 −0.579601
\(39\) 0 0
\(40\) −2.63291 −0.416300
\(41\) 4.43181 0.692132 0.346066 0.938210i \(-0.387517\pi\)
0.346066 + 0.938210i \(0.387517\pi\)
\(42\) 0 0
\(43\) 9.59164 1.46271 0.731355 0.681996i \(-0.238888\pi\)
0.731355 + 0.681996i \(0.238888\pi\)
\(44\) −4.09429 −0.617237
\(45\) 0 0
\(46\) 3.39412 0.500436
\(47\) 10.6264 1.55003 0.775013 0.631946i \(-0.217744\pi\)
0.775013 + 0.631946i \(0.217744\pi\)
\(48\) 0 0
\(49\) −1.95299 −0.278998
\(50\) 1.93222 0.273257
\(51\) 0 0
\(52\) 0.00776648 0.00107702
\(53\) 4.64357 0.637843 0.318921 0.947781i \(-0.396679\pi\)
0.318921 + 0.947781i \(0.396679\pi\)
\(54\) 0 0
\(55\) 10.7799 1.45356
\(56\) −2.24656 −0.300209
\(57\) 0 0
\(58\) 8.00809 1.05151
\(59\) 7.95351 1.03546 0.517730 0.855544i \(-0.326777\pi\)
0.517730 + 0.855544i \(0.326777\pi\)
\(60\) 0 0
\(61\) −11.0517 −1.41502 −0.707510 0.706703i \(-0.750182\pi\)
−0.707510 + 0.706703i \(0.750182\pi\)
\(62\) −6.93425 −0.880650
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.0204485 −0.00253632
\(66\) 0 0
\(67\) 15.2010 1.85710 0.928551 0.371204i \(-0.121055\pi\)
0.928551 + 0.371204i \(0.121055\pi\)
\(68\) 0.673658 0.0816931
\(69\) 0 0
\(70\) 5.91498 0.706976
\(71\) −9.64824 −1.14504 −0.572518 0.819892i \(-0.694033\pi\)
−0.572518 + 0.819892i \(0.694033\pi\)
\(72\) 0 0
\(73\) 0.995761 0.116545 0.0582725 0.998301i \(-0.481441\pi\)
0.0582725 + 0.998301i \(0.481441\pi\)
\(74\) 0.651332 0.0757157
\(75\) 0 0
\(76\) −3.57290 −0.409839
\(77\) 9.19804 1.04821
\(78\) 0 0
\(79\) −1.27648 −0.143615 −0.0718074 0.997419i \(-0.522877\pi\)
−0.0718074 + 0.997419i \(0.522877\pi\)
\(80\) −2.63291 −0.294368
\(81\) 0 0
\(82\) 4.43181 0.489411
\(83\) 5.75391 0.631574 0.315787 0.948830i \(-0.397732\pi\)
0.315787 + 0.948830i \(0.397732\pi\)
\(84\) 0 0
\(85\) −1.77368 −0.192383
\(86\) 9.59164 1.03429
\(87\) 0 0
\(88\) −4.09429 −0.436452
\(89\) −2.07485 −0.219933 −0.109967 0.993935i \(-0.535074\pi\)
−0.109967 + 0.993935i \(0.535074\pi\)
\(90\) 0 0
\(91\) −0.0174478 −0.00182903
\(92\) 3.39412 0.353862
\(93\) 0 0
\(94\) 10.6264 1.09603
\(95\) 9.40712 0.965150
\(96\) 0 0
\(97\) −17.5825 −1.78523 −0.892615 0.450819i \(-0.851132\pi\)
−0.892615 + 0.450819i \(0.851132\pi\)
\(98\) −1.95299 −0.197281
\(99\) 0 0
\(100\) 1.93222 0.193222
\(101\) −8.56611 −0.852359 −0.426180 0.904639i \(-0.640141\pi\)
−0.426180 + 0.904639i \(0.640141\pi\)
\(102\) 0 0
\(103\) 19.9320 1.96396 0.981978 0.188996i \(-0.0605234\pi\)
0.981978 + 0.188996i \(0.0605234\pi\)
\(104\) 0.00776648 0.000761566 0
\(105\) 0 0
\(106\) 4.64357 0.451023
\(107\) 13.1728 1.27347 0.636733 0.771085i \(-0.280285\pi\)
0.636733 + 0.771085i \(0.280285\pi\)
\(108\) 0 0
\(109\) −5.10676 −0.489139 −0.244569 0.969632i \(-0.578647\pi\)
−0.244569 + 0.969632i \(0.578647\pi\)
\(110\) 10.7799 1.02782
\(111\) 0 0
\(112\) −2.24656 −0.212280
\(113\) 15.5553 1.46332 0.731659 0.681671i \(-0.238746\pi\)
0.731659 + 0.681671i \(0.238746\pi\)
\(114\) 0 0
\(115\) −8.93642 −0.833325
\(116\) 8.00809 0.743532
\(117\) 0 0
\(118\) 7.95351 0.732180
\(119\) −1.51341 −0.138734
\(120\) 0 0
\(121\) 5.76317 0.523925
\(122\) −11.0517 −1.00057
\(123\) 0 0
\(124\) −6.93425 −0.622714
\(125\) 8.07719 0.722446
\(126\) 0 0
\(127\) 15.4106 1.36747 0.683737 0.729729i \(-0.260354\pi\)
0.683737 + 0.729729i \(0.260354\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −0.0204485 −0.00179345
\(131\) −21.0190 −1.83644 −0.918219 0.396073i \(-0.870373\pi\)
−0.918219 + 0.396073i \(0.870373\pi\)
\(132\) 0 0
\(133\) 8.02672 0.696005
\(134\) 15.2010 1.31317
\(135\) 0 0
\(136\) 0.673658 0.0577657
\(137\) 12.5823 1.07498 0.537490 0.843270i \(-0.319373\pi\)
0.537490 + 0.843270i \(0.319373\pi\)
\(138\) 0 0
\(139\) −9.51306 −0.806887 −0.403444 0.915004i \(-0.632187\pi\)
−0.403444 + 0.915004i \(0.632187\pi\)
\(140\) 5.91498 0.499907
\(141\) 0 0
\(142\) −9.64824 −0.809662
\(143\) −0.0317982 −0.00265910
\(144\) 0 0
\(145\) −21.0846 −1.75098
\(146\) 0.995761 0.0824097
\(147\) 0 0
\(148\) 0.651332 0.0535391
\(149\) −2.73070 −0.223708 −0.111854 0.993725i \(-0.535679\pi\)
−0.111854 + 0.993725i \(0.535679\pi\)
\(150\) 0 0
\(151\) 7.96037 0.647806 0.323903 0.946090i \(-0.395005\pi\)
0.323903 + 0.946090i \(0.395005\pi\)
\(152\) −3.57290 −0.289800
\(153\) 0 0
\(154\) 9.19804 0.741199
\(155\) 18.2573 1.46646
\(156\) 0 0
\(157\) 12.4903 0.996831 0.498416 0.866938i \(-0.333915\pi\)
0.498416 + 0.866938i \(0.333915\pi\)
\(158\) −1.27648 −0.101551
\(159\) 0 0
\(160\) −2.63291 −0.208150
\(161\) −7.62508 −0.600941
\(162\) 0 0
\(163\) −6.27007 −0.491109 −0.245555 0.969383i \(-0.578970\pi\)
−0.245555 + 0.969383i \(0.578970\pi\)
\(164\) 4.43181 0.346066
\(165\) 0 0
\(166\) 5.75391 0.446590
\(167\) 2.90654 0.224915 0.112458 0.993657i \(-0.464128\pi\)
0.112458 + 0.993657i \(0.464128\pi\)
\(168\) 0 0
\(169\) −12.9999 −0.999995
\(170\) −1.77368 −0.136035
\(171\) 0 0
\(172\) 9.59164 0.731355
\(173\) −9.28984 −0.706293 −0.353147 0.935568i \(-0.614888\pi\)
−0.353147 + 0.935568i \(0.614888\pi\)
\(174\) 0 0
\(175\) −4.34084 −0.328137
\(176\) −4.09429 −0.308618
\(177\) 0 0
\(178\) −2.07485 −0.155516
\(179\) 21.1054 1.57749 0.788745 0.614721i \(-0.210731\pi\)
0.788745 + 0.614721i \(0.210731\pi\)
\(180\) 0 0
\(181\) 0.543334 0.0403857 0.0201928 0.999796i \(-0.493572\pi\)
0.0201928 + 0.999796i \(0.493572\pi\)
\(182\) −0.0174478 −0.00129332
\(183\) 0 0
\(184\) 3.39412 0.250218
\(185\) −1.71490 −0.126082
\(186\) 0 0
\(187\) −2.75815 −0.201696
\(188\) 10.6264 0.775013
\(189\) 0 0
\(190\) 9.40712 0.682464
\(191\) 14.0400 1.01590 0.507950 0.861387i \(-0.330404\pi\)
0.507950 + 0.861387i \(0.330404\pi\)
\(192\) 0 0
\(193\) −9.77786 −0.703826 −0.351913 0.936033i \(-0.614469\pi\)
−0.351913 + 0.936033i \(0.614469\pi\)
\(194\) −17.5825 −1.26235
\(195\) 0 0
\(196\) −1.95299 −0.139499
\(197\) −14.5046 −1.03341 −0.516705 0.856163i \(-0.672842\pi\)
−0.516705 + 0.856163i \(0.672842\pi\)
\(198\) 0 0
\(199\) 22.6203 1.60351 0.801755 0.597653i \(-0.203900\pi\)
0.801755 + 0.597653i \(0.203900\pi\)
\(200\) 1.93222 0.136629
\(201\) 0 0
\(202\) −8.56611 −0.602709
\(203\) −17.9906 −1.26269
\(204\) 0 0
\(205\) −11.6686 −0.814967
\(206\) 19.9320 1.38873
\(207\) 0 0
\(208\) 0.00776648 0.000538508 0
\(209\) 14.6285 1.01187
\(210\) 0 0
\(211\) −10.0332 −0.690715 −0.345358 0.938471i \(-0.612242\pi\)
−0.345358 + 0.938471i \(0.612242\pi\)
\(212\) 4.64357 0.318921
\(213\) 0 0
\(214\) 13.1728 0.900476
\(215\) −25.2539 −1.72230
\(216\) 0 0
\(217\) 15.5782 1.05752
\(218\) −5.10676 −0.345873
\(219\) 0 0
\(220\) 10.7799 0.726780
\(221\) 0.00523195 0.000351939 0
\(222\) 0 0
\(223\) −1.00000 −0.0669650
\(224\) −2.24656 −0.150104
\(225\) 0 0
\(226\) 15.5553 1.03472
\(227\) −20.8914 −1.38661 −0.693305 0.720645i \(-0.743846\pi\)
−0.693305 + 0.720645i \(0.743846\pi\)
\(228\) 0 0
\(229\) 20.5829 1.36015 0.680077 0.733141i \(-0.261946\pi\)
0.680077 + 0.733141i \(0.261946\pi\)
\(230\) −8.93642 −0.589250
\(231\) 0 0
\(232\) 8.00809 0.525757
\(233\) 9.41577 0.616848 0.308424 0.951249i \(-0.400198\pi\)
0.308424 + 0.951249i \(0.400198\pi\)
\(234\) 0 0
\(235\) −27.9785 −1.82511
\(236\) 7.95351 0.517730
\(237\) 0 0
\(238\) −1.51341 −0.0980999
\(239\) −14.6538 −0.947877 −0.473939 0.880558i \(-0.657168\pi\)
−0.473939 + 0.880558i \(0.657168\pi\)
\(240\) 0 0
\(241\) 17.1216 1.10290 0.551448 0.834209i \(-0.314075\pi\)
0.551448 + 0.834209i \(0.314075\pi\)
\(242\) 5.76317 0.370471
\(243\) 0 0
\(244\) −11.0517 −0.707510
\(245\) 5.14204 0.328513
\(246\) 0 0
\(247\) −0.0277488 −0.00176562
\(248\) −6.93425 −0.440325
\(249\) 0 0
\(250\) 8.07719 0.510846
\(251\) 22.7295 1.43467 0.717337 0.696727i \(-0.245361\pi\)
0.717337 + 0.696727i \(0.245361\pi\)
\(252\) 0 0
\(253\) −13.8965 −0.873666
\(254\) 15.4106 0.966950
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −7.48452 −0.466872 −0.233436 0.972372i \(-0.574997\pi\)
−0.233436 + 0.972372i \(0.574997\pi\)
\(258\) 0 0
\(259\) −1.46325 −0.0909221
\(260\) −0.0204485 −0.00126816
\(261\) 0 0
\(262\) −21.0190 −1.29856
\(263\) −7.82448 −0.482478 −0.241239 0.970466i \(-0.577554\pi\)
−0.241239 + 0.970466i \(0.577554\pi\)
\(264\) 0 0
\(265\) −12.2261 −0.751043
\(266\) 8.02672 0.492150
\(267\) 0 0
\(268\) 15.2010 0.928551
\(269\) 31.3794 1.91324 0.956619 0.291342i \(-0.0941016\pi\)
0.956619 + 0.291342i \(0.0941016\pi\)
\(270\) 0 0
\(271\) −1.63798 −0.0995000 −0.0497500 0.998762i \(-0.515842\pi\)
−0.0497500 + 0.998762i \(0.515842\pi\)
\(272\) 0.673658 0.0408465
\(273\) 0 0
\(274\) 12.5823 0.760125
\(275\) −7.91106 −0.477055
\(276\) 0 0
\(277\) 8.78445 0.527806 0.263903 0.964549i \(-0.414990\pi\)
0.263903 + 0.964549i \(0.414990\pi\)
\(278\) −9.51306 −0.570555
\(279\) 0 0
\(280\) 5.91498 0.353488
\(281\) −13.1236 −0.782887 −0.391444 0.920202i \(-0.628024\pi\)
−0.391444 + 0.920202i \(0.628024\pi\)
\(282\) 0 0
\(283\) 21.8961 1.30159 0.650795 0.759253i \(-0.274436\pi\)
0.650795 + 0.759253i \(0.274436\pi\)
\(284\) −9.64824 −0.572518
\(285\) 0 0
\(286\) −0.0317982 −0.00188027
\(287\) −9.95630 −0.587702
\(288\) 0 0
\(289\) −16.5462 −0.973305
\(290\) −21.0846 −1.23813
\(291\) 0 0
\(292\) 0.995761 0.0582725
\(293\) −15.4295 −0.901402 −0.450701 0.892675i \(-0.648826\pi\)
−0.450701 + 0.892675i \(0.648826\pi\)
\(294\) 0 0
\(295\) −20.9409 −1.21923
\(296\) 0.651332 0.0378579
\(297\) 0 0
\(298\) −2.73070 −0.158185
\(299\) 0.0263604 0.00152446
\(300\) 0 0
\(301\) −21.5482 −1.24201
\(302\) 7.96037 0.458068
\(303\) 0 0
\(304\) −3.57290 −0.204920
\(305\) 29.0980 1.66615
\(306\) 0 0
\(307\) −18.6854 −1.06643 −0.533216 0.845979i \(-0.679017\pi\)
−0.533216 + 0.845979i \(0.679017\pi\)
\(308\) 9.19804 0.524107
\(309\) 0 0
\(310\) 18.2573 1.03694
\(311\) 21.9347 1.24380 0.621902 0.783095i \(-0.286360\pi\)
0.621902 + 0.783095i \(0.286360\pi\)
\(312\) 0 0
\(313\) −5.51529 −0.311743 −0.155871 0.987777i \(-0.549819\pi\)
−0.155871 + 0.987777i \(0.549819\pi\)
\(314\) 12.4903 0.704866
\(315\) 0 0
\(316\) −1.27648 −0.0718074
\(317\) 10.2843 0.577622 0.288811 0.957386i \(-0.406740\pi\)
0.288811 + 0.957386i \(0.406740\pi\)
\(318\) 0 0
\(319\) −32.7874 −1.83574
\(320\) −2.63291 −0.147184
\(321\) 0 0
\(322\) −7.62508 −0.424929
\(323\) −2.40691 −0.133924
\(324\) 0 0
\(325\) 0.0150066 0.000832414 0
\(326\) −6.27007 −0.347267
\(327\) 0 0
\(328\) 4.43181 0.244706
\(329\) −23.8729 −1.31615
\(330\) 0 0
\(331\) −23.9394 −1.31583 −0.657913 0.753094i \(-0.728561\pi\)
−0.657913 + 0.753094i \(0.728561\pi\)
\(332\) 5.75391 0.315787
\(333\) 0 0
\(334\) 2.90654 0.159039
\(335\) −40.0230 −2.18669
\(336\) 0 0
\(337\) 4.45595 0.242731 0.121366 0.992608i \(-0.461273\pi\)
0.121366 + 0.992608i \(0.461273\pi\)
\(338\) −12.9999 −0.707104
\(339\) 0 0
\(340\) −1.77368 −0.0961914
\(341\) 28.3908 1.53745
\(342\) 0 0
\(343\) 20.1134 1.08602
\(344\) 9.59164 0.517146
\(345\) 0 0
\(346\) −9.28984 −0.499425
\(347\) 30.7768 1.65219 0.826093 0.563534i \(-0.190558\pi\)
0.826093 + 0.563534i \(0.190558\pi\)
\(348\) 0 0
\(349\) 10.6268 0.568837 0.284419 0.958700i \(-0.408199\pi\)
0.284419 + 0.958700i \(0.408199\pi\)
\(350\) −4.34084 −0.232028
\(351\) 0 0
\(352\) −4.09429 −0.218226
\(353\) −23.9565 −1.27508 −0.637538 0.770419i \(-0.720047\pi\)
−0.637538 + 0.770419i \(0.720047\pi\)
\(354\) 0 0
\(355\) 25.4029 1.34825
\(356\) −2.07485 −0.109967
\(357\) 0 0
\(358\) 21.1054 1.11545
\(359\) −17.1370 −0.904457 −0.452229 0.891902i \(-0.649371\pi\)
−0.452229 + 0.891902i \(0.649371\pi\)
\(360\) 0 0
\(361\) −6.23440 −0.328126
\(362\) 0.543334 0.0285570
\(363\) 0 0
\(364\) −0.0174478 −0.000914515 0
\(365\) −2.62175 −0.137229
\(366\) 0 0
\(367\) −11.9256 −0.622511 −0.311255 0.950326i \(-0.600749\pi\)
−0.311255 + 0.950326i \(0.600749\pi\)
\(368\) 3.39412 0.176931
\(369\) 0 0
\(370\) −1.71490 −0.0891533
\(371\) −10.4320 −0.541604
\(372\) 0 0
\(373\) −3.91021 −0.202463 −0.101232 0.994863i \(-0.532278\pi\)
−0.101232 + 0.994863i \(0.532278\pi\)
\(374\) −2.75815 −0.142621
\(375\) 0 0
\(376\) 10.6264 0.548017
\(377\) 0.0621946 0.00320319
\(378\) 0 0
\(379\) 10.4984 0.539268 0.269634 0.962963i \(-0.413097\pi\)
0.269634 + 0.962963i \(0.413097\pi\)
\(380\) 9.40712 0.482575
\(381\) 0 0
\(382\) 14.0400 0.718350
\(383\) 8.52206 0.435457 0.217729 0.976009i \(-0.430135\pi\)
0.217729 + 0.976009i \(0.430135\pi\)
\(384\) 0 0
\(385\) −24.2176 −1.23424
\(386\) −9.77786 −0.497680
\(387\) 0 0
\(388\) −17.5825 −0.892615
\(389\) 31.7085 1.60768 0.803842 0.594843i \(-0.202786\pi\)
0.803842 + 0.594843i \(0.202786\pi\)
\(390\) 0 0
\(391\) 2.28648 0.115632
\(392\) −1.95299 −0.0986407
\(393\) 0 0
\(394\) −14.5046 −0.730732
\(395\) 3.36085 0.169103
\(396\) 0 0
\(397\) −5.92081 −0.297157 −0.148578 0.988901i \(-0.547470\pi\)
−0.148578 + 0.988901i \(0.547470\pi\)
\(398\) 22.6203 1.13385
\(399\) 0 0
\(400\) 1.93222 0.0966110
\(401\) 8.12974 0.405980 0.202990 0.979181i \(-0.434934\pi\)
0.202990 + 0.979181i \(0.434934\pi\)
\(402\) 0 0
\(403\) −0.0538547 −0.00268269
\(404\) −8.56611 −0.426180
\(405\) 0 0
\(406\) −17.9906 −0.892859
\(407\) −2.66674 −0.132185
\(408\) 0 0
\(409\) 12.9592 0.640791 0.320395 0.947284i \(-0.396184\pi\)
0.320395 + 0.947284i \(0.396184\pi\)
\(410\) −11.6686 −0.576269
\(411\) 0 0
\(412\) 19.9320 0.981978
\(413\) −17.8680 −0.879228
\(414\) 0 0
\(415\) −15.1495 −0.743661
\(416\) 0.00776648 0.000380783 0
\(417\) 0 0
\(418\) 14.6285 0.715502
\(419\) 30.4806 1.48907 0.744537 0.667581i \(-0.232670\pi\)
0.744537 + 0.667581i \(0.232670\pi\)
\(420\) 0 0
\(421\) −38.0654 −1.85520 −0.927598 0.373581i \(-0.878130\pi\)
−0.927598 + 0.373581i \(0.878130\pi\)
\(422\) −10.0332 −0.488409
\(423\) 0 0
\(424\) 4.64357 0.225511
\(425\) 1.30166 0.0631396
\(426\) 0 0
\(427\) 24.8282 1.20152
\(428\) 13.1728 0.636733
\(429\) 0 0
\(430\) −25.2539 −1.21785
\(431\) 21.5664 1.03881 0.519407 0.854527i \(-0.326153\pi\)
0.519407 + 0.854527i \(0.326153\pi\)
\(432\) 0 0
\(433\) −8.92010 −0.428673 −0.214336 0.976760i \(-0.568759\pi\)
−0.214336 + 0.976760i \(0.568759\pi\)
\(434\) 15.5782 0.747776
\(435\) 0 0
\(436\) −5.10676 −0.244569
\(437\) −12.1268 −0.580106
\(438\) 0 0
\(439\) −26.0102 −1.24140 −0.620700 0.784048i \(-0.713152\pi\)
−0.620700 + 0.784048i \(0.713152\pi\)
\(440\) 10.7799 0.513911
\(441\) 0 0
\(442\) 0.00523195 0.000248859 0
\(443\) −10.7806 −0.512200 −0.256100 0.966650i \(-0.582438\pi\)
−0.256100 + 0.966650i \(0.582438\pi\)
\(444\) 0 0
\(445\) 5.46289 0.258966
\(446\) −1.00000 −0.0473514
\(447\) 0 0
\(448\) −2.24656 −0.106140
\(449\) 20.0819 0.947722 0.473861 0.880600i \(-0.342860\pi\)
0.473861 + 0.880600i \(0.342860\pi\)
\(450\) 0 0
\(451\) −18.1451 −0.854419
\(452\) 15.5553 0.731659
\(453\) 0 0
\(454\) −20.8914 −0.980481
\(455\) 0.0459386 0.00215363
\(456\) 0 0
\(457\) −35.5020 −1.66071 −0.830357 0.557232i \(-0.811863\pi\)
−0.830357 + 0.557232i \(0.811863\pi\)
\(458\) 20.5829 0.961774
\(459\) 0 0
\(460\) −8.93642 −0.416663
\(461\) 3.26003 0.151835 0.0759173 0.997114i \(-0.475811\pi\)
0.0759173 + 0.997114i \(0.475811\pi\)
\(462\) 0 0
\(463\) 5.67209 0.263604 0.131802 0.991276i \(-0.457924\pi\)
0.131802 + 0.991276i \(0.457924\pi\)
\(464\) 8.00809 0.371766
\(465\) 0 0
\(466\) 9.41577 0.436177
\(467\) 10.7885 0.499231 0.249616 0.968345i \(-0.419696\pi\)
0.249616 + 0.968345i \(0.419696\pi\)
\(468\) 0 0
\(469\) −34.1500 −1.57690
\(470\) −27.9785 −1.29055
\(471\) 0 0
\(472\) 7.95351 0.366090
\(473\) −39.2709 −1.80568
\(474\) 0 0
\(475\) −6.90363 −0.316760
\(476\) −1.51341 −0.0693671
\(477\) 0 0
\(478\) −14.6538 −0.670250
\(479\) −17.5354 −0.801211 −0.400605 0.916251i \(-0.631200\pi\)
−0.400605 + 0.916251i \(0.631200\pi\)
\(480\) 0 0
\(481\) 0.00505855 0.000230650 0
\(482\) 17.1216 0.779866
\(483\) 0 0
\(484\) 5.76317 0.261962
\(485\) 46.2931 2.10206
\(486\) 0 0
\(487\) 21.6167 0.979547 0.489774 0.871850i \(-0.337079\pi\)
0.489774 + 0.871850i \(0.337079\pi\)
\(488\) −11.0517 −0.500285
\(489\) 0 0
\(490\) 5.14204 0.232294
\(491\) 11.0429 0.498359 0.249180 0.968457i \(-0.419839\pi\)
0.249180 + 0.968457i \(0.419839\pi\)
\(492\) 0 0
\(493\) 5.39471 0.242966
\(494\) −0.0277488 −0.00124848
\(495\) 0 0
\(496\) −6.93425 −0.311357
\(497\) 21.6753 0.972270
\(498\) 0 0
\(499\) −9.24468 −0.413849 −0.206924 0.978357i \(-0.566345\pi\)
−0.206924 + 0.978357i \(0.566345\pi\)
\(500\) 8.07719 0.361223
\(501\) 0 0
\(502\) 22.7295 1.01447
\(503\) −42.2295 −1.88292 −0.941461 0.337123i \(-0.890546\pi\)
−0.941461 + 0.337123i \(0.890546\pi\)
\(504\) 0 0
\(505\) 22.5538 1.00363
\(506\) −13.8965 −0.617775
\(507\) 0 0
\(508\) 15.4106 0.683737
\(509\) 38.2695 1.69627 0.848134 0.529782i \(-0.177726\pi\)
0.848134 + 0.529782i \(0.177726\pi\)
\(510\) 0 0
\(511\) −2.23703 −0.0989605
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −7.48452 −0.330128
\(515\) −52.4791 −2.31251
\(516\) 0 0
\(517\) −43.5077 −1.91346
\(518\) −1.46325 −0.0642916
\(519\) 0 0
\(520\) −0.0204485 −0.000896724 0
\(521\) −20.1266 −0.881760 −0.440880 0.897566i \(-0.645334\pi\)
−0.440880 + 0.897566i \(0.645334\pi\)
\(522\) 0 0
\(523\) −3.72087 −0.162702 −0.0813512 0.996686i \(-0.525924\pi\)
−0.0813512 + 0.996686i \(0.525924\pi\)
\(524\) −21.0190 −0.918219
\(525\) 0 0
\(526\) −7.82448 −0.341163
\(527\) −4.67131 −0.203486
\(528\) 0 0
\(529\) −11.4799 −0.499128
\(530\) −12.2261 −0.531068
\(531\) 0 0
\(532\) 8.02672 0.348002
\(533\) 0.0344195 0.00149088
\(534\) 0 0
\(535\) −34.6829 −1.49947
\(536\) 15.2010 0.656585
\(537\) 0 0
\(538\) 31.3794 1.35286
\(539\) 7.99608 0.344416
\(540\) 0 0
\(541\) −22.9511 −0.986745 −0.493373 0.869818i \(-0.664236\pi\)
−0.493373 + 0.869818i \(0.664236\pi\)
\(542\) −1.63798 −0.0703571
\(543\) 0 0
\(544\) 0.673658 0.0288829
\(545\) 13.4456 0.575948
\(546\) 0 0
\(547\) −2.69925 −0.115412 −0.0577058 0.998334i \(-0.518379\pi\)
−0.0577058 + 0.998334i \(0.518379\pi\)
\(548\) 12.5823 0.537490
\(549\) 0 0
\(550\) −7.91106 −0.337329
\(551\) −28.6121 −1.21892
\(552\) 0 0
\(553\) 2.86768 0.121946
\(554\) 8.78445 0.373215
\(555\) 0 0
\(556\) −9.51306 −0.403444
\(557\) 19.9531 0.845440 0.422720 0.906260i \(-0.361075\pi\)
0.422720 + 0.906260i \(0.361075\pi\)
\(558\) 0 0
\(559\) 0.0744933 0.00315073
\(560\) 5.91498 0.249954
\(561\) 0 0
\(562\) −13.1236 −0.553585
\(563\) 1.38362 0.0583125 0.0291563 0.999575i \(-0.490718\pi\)
0.0291563 + 0.999575i \(0.490718\pi\)
\(564\) 0 0
\(565\) −40.9557 −1.72302
\(566\) 21.8961 0.920364
\(567\) 0 0
\(568\) −9.64824 −0.404831
\(569\) 12.6605 0.530757 0.265379 0.964144i \(-0.414503\pi\)
0.265379 + 0.964144i \(0.414503\pi\)
\(570\) 0 0
\(571\) 46.7306 1.95562 0.977808 0.209503i \(-0.0671846\pi\)
0.977808 + 0.209503i \(0.0671846\pi\)
\(572\) −0.0317982 −0.00132955
\(573\) 0 0
\(574\) −9.95630 −0.415568
\(575\) 6.55819 0.273495
\(576\) 0 0
\(577\) −18.8790 −0.785944 −0.392972 0.919550i \(-0.628553\pi\)
−0.392972 + 0.919550i \(0.628553\pi\)
\(578\) −16.5462 −0.688231
\(579\) 0 0
\(580\) −21.0846 −0.875489
\(581\) −12.9265 −0.536281
\(582\) 0 0
\(583\) −19.0121 −0.787400
\(584\) 0.995761 0.0412049
\(585\) 0 0
\(586\) −15.4295 −0.637387
\(587\) −7.31271 −0.301828 −0.150914 0.988547i \(-0.548222\pi\)
−0.150914 + 0.988547i \(0.548222\pi\)
\(588\) 0 0
\(589\) 24.7754 1.02085
\(590\) −20.9409 −0.862123
\(591\) 0 0
\(592\) 0.651332 0.0267696
\(593\) 0.674078 0.0276811 0.0138405 0.999904i \(-0.495594\pi\)
0.0138405 + 0.999904i \(0.495594\pi\)
\(594\) 0 0
\(595\) 3.98468 0.163356
\(596\) −2.73070 −0.111854
\(597\) 0 0
\(598\) 0.0263604 0.00107796
\(599\) −9.80274 −0.400529 −0.200265 0.979742i \(-0.564180\pi\)
−0.200265 + 0.979742i \(0.564180\pi\)
\(600\) 0 0
\(601\) 22.8526 0.932178 0.466089 0.884738i \(-0.345663\pi\)
0.466089 + 0.884738i \(0.345663\pi\)
\(602\) −21.5482 −0.878237
\(603\) 0 0
\(604\) 7.96037 0.323903
\(605\) −15.1739 −0.616908
\(606\) 0 0
\(607\) 44.0179 1.78663 0.893316 0.449428i \(-0.148372\pi\)
0.893316 + 0.449428i \(0.148372\pi\)
\(608\) −3.57290 −0.144900
\(609\) 0 0
\(610\) 29.0980 1.17815
\(611\) 0.0825300 0.00333881
\(612\) 0 0
\(613\) 23.4729 0.948063 0.474031 0.880508i \(-0.342798\pi\)
0.474031 + 0.880508i \(0.342798\pi\)
\(614\) −18.6854 −0.754082
\(615\) 0 0
\(616\) 9.19804 0.370600
\(617\) −24.9763 −1.00551 −0.502753 0.864430i \(-0.667679\pi\)
−0.502753 + 0.864430i \(0.667679\pi\)
\(618\) 0 0
\(619\) −9.73956 −0.391466 −0.195733 0.980657i \(-0.562709\pi\)
−0.195733 + 0.980657i \(0.562709\pi\)
\(620\) 18.2573 0.733229
\(621\) 0 0
\(622\) 21.9347 0.879502
\(623\) 4.66126 0.186749
\(624\) 0 0
\(625\) −30.9276 −1.23711
\(626\) −5.51529 −0.220435
\(627\) 0 0
\(628\) 12.4903 0.498416
\(629\) 0.438775 0.0174951
\(630\) 0 0
\(631\) −23.1790 −0.922740 −0.461370 0.887208i \(-0.652642\pi\)
−0.461370 + 0.887208i \(0.652642\pi\)
\(632\) −1.27648 −0.0507755
\(633\) 0 0
\(634\) 10.2843 0.408440
\(635\) −40.5749 −1.61016
\(636\) 0 0
\(637\) −0.0151678 −0.000600971 0
\(638\) −32.7874 −1.29807
\(639\) 0 0
\(640\) −2.63291 −0.104075
\(641\) −11.3880 −0.449799 −0.224900 0.974382i \(-0.572205\pi\)
−0.224900 + 0.974382i \(0.572205\pi\)
\(642\) 0 0
\(643\) 7.09359 0.279744 0.139872 0.990170i \(-0.455331\pi\)
0.139872 + 0.990170i \(0.455331\pi\)
\(644\) −7.62508 −0.300470
\(645\) 0 0
\(646\) −2.40691 −0.0946987
\(647\) −26.4990 −1.04178 −0.520892 0.853623i \(-0.674400\pi\)
−0.520892 + 0.853623i \(0.674400\pi\)
\(648\) 0 0
\(649\) −32.5640 −1.27825
\(650\) 0.0150066 0.000588605 0
\(651\) 0 0
\(652\) −6.27007 −0.245555
\(653\) −11.3962 −0.445966 −0.222983 0.974822i \(-0.571579\pi\)
−0.222983 + 0.974822i \(0.571579\pi\)
\(654\) 0 0
\(655\) 55.3412 2.16236
\(656\) 4.43181 0.173033
\(657\) 0 0
\(658\) −23.8729 −0.930662
\(659\) 41.0879 1.60056 0.800278 0.599629i \(-0.204685\pi\)
0.800278 + 0.599629i \(0.204685\pi\)
\(660\) 0 0
\(661\) 43.6409 1.69743 0.848716 0.528849i \(-0.177376\pi\)
0.848716 + 0.528849i \(0.177376\pi\)
\(662\) −23.9394 −0.930430
\(663\) 0 0
\(664\) 5.75391 0.223295
\(665\) −21.1336 −0.819527
\(666\) 0 0
\(667\) 27.1804 1.05243
\(668\) 2.90654 0.112458
\(669\) 0 0
\(670\) −40.0230 −1.54622
\(671\) 45.2487 1.74680
\(672\) 0 0
\(673\) 22.3619 0.861987 0.430994 0.902355i \(-0.358163\pi\)
0.430994 + 0.902355i \(0.358163\pi\)
\(674\) 4.45595 0.171637
\(675\) 0 0
\(676\) −12.9999 −0.499998
\(677\) −34.4169 −1.32275 −0.661374 0.750056i \(-0.730026\pi\)
−0.661374 + 0.750056i \(0.730026\pi\)
\(678\) 0 0
\(679\) 39.5000 1.51587
\(680\) −1.77368 −0.0680176
\(681\) 0 0
\(682\) 28.3908 1.08714
\(683\) 10.4429 0.399588 0.199794 0.979838i \(-0.435973\pi\)
0.199794 + 0.979838i \(0.435973\pi\)
\(684\) 0 0
\(685\) −33.1281 −1.26576
\(686\) 20.1134 0.767933
\(687\) 0 0
\(688\) 9.59164 0.365678
\(689\) 0.0360642 0.00137393
\(690\) 0 0
\(691\) 45.0269 1.71290 0.856452 0.516227i \(-0.172664\pi\)
0.856452 + 0.516227i \(0.172664\pi\)
\(692\) −9.28984 −0.353147
\(693\) 0 0
\(694\) 30.7768 1.16827
\(695\) 25.0470 0.950088
\(696\) 0 0
\(697\) 2.98552 0.113085
\(698\) 10.6268 0.402229
\(699\) 0 0
\(700\) −4.34084 −0.164068
\(701\) −19.7764 −0.746944 −0.373472 0.927641i \(-0.621833\pi\)
−0.373472 + 0.927641i \(0.621833\pi\)
\(702\) 0 0
\(703\) −2.32714 −0.0877698
\(704\) −4.09429 −0.154309
\(705\) 0 0
\(706\) −23.9565 −0.901615
\(707\) 19.2442 0.723754
\(708\) 0 0
\(709\) 11.3226 0.425230 0.212615 0.977136i \(-0.431802\pi\)
0.212615 + 0.977136i \(0.431802\pi\)
\(710\) 25.4029 0.953356
\(711\) 0 0
\(712\) −2.07485 −0.0777582
\(713\) −23.5357 −0.881418
\(714\) 0 0
\(715\) 0.0837218 0.00313102
\(716\) 21.1054 0.788745
\(717\) 0 0
\(718\) −17.1370 −0.639548
\(719\) −42.4365 −1.58261 −0.791307 0.611419i \(-0.790599\pi\)
−0.791307 + 0.611419i \(0.790599\pi\)
\(720\) 0 0
\(721\) −44.7783 −1.66763
\(722\) −6.23440 −0.232020
\(723\) 0 0
\(724\) 0.543334 0.0201928
\(725\) 15.4734 0.574667
\(726\) 0 0
\(727\) 16.3993 0.608218 0.304109 0.952637i \(-0.401641\pi\)
0.304109 + 0.952637i \(0.401641\pi\)
\(728\) −0.0174478 −0.000646660 0
\(729\) 0 0
\(730\) −2.62175 −0.0970353
\(731\) 6.46149 0.238987
\(732\) 0 0
\(733\) −21.4011 −0.790468 −0.395234 0.918580i \(-0.629337\pi\)
−0.395234 + 0.918580i \(0.629337\pi\)
\(734\) −11.9256 −0.440182
\(735\) 0 0
\(736\) 3.39412 0.125109
\(737\) −62.2374 −2.29254
\(738\) 0 0
\(739\) 5.23739 0.192661 0.0963303 0.995349i \(-0.469289\pi\)
0.0963303 + 0.995349i \(0.469289\pi\)
\(740\) −1.71490 −0.0630409
\(741\) 0 0
\(742\) −10.4320 −0.382972
\(743\) 37.6115 1.37983 0.689916 0.723889i \(-0.257647\pi\)
0.689916 + 0.723889i \(0.257647\pi\)
\(744\) 0 0
\(745\) 7.18969 0.263410
\(746\) −3.91021 −0.143163
\(747\) 0 0
\(748\) −2.75815 −0.100848
\(749\) −29.5935 −1.08132
\(750\) 0 0
\(751\) 15.3955 0.561790 0.280895 0.959738i \(-0.409369\pi\)
0.280895 + 0.959738i \(0.409369\pi\)
\(752\) 10.6264 0.387506
\(753\) 0 0
\(754\) 0.0621946 0.00226500
\(755\) −20.9589 −0.762774
\(756\) 0 0
\(757\) 45.9995 1.67188 0.835941 0.548819i \(-0.184923\pi\)
0.835941 + 0.548819i \(0.184923\pi\)
\(758\) 10.4984 0.381320
\(759\) 0 0
\(760\) 9.40712 0.341232
\(761\) 23.9738 0.869050 0.434525 0.900660i \(-0.356916\pi\)
0.434525 + 0.900660i \(0.356916\pi\)
\(762\) 0 0
\(763\) 11.4726 0.415337
\(764\) 14.0400 0.507950
\(765\) 0 0
\(766\) 8.52206 0.307915
\(767\) 0.0617708 0.00223041
\(768\) 0 0
\(769\) 31.3460 1.13037 0.565183 0.824966i \(-0.308806\pi\)
0.565183 + 0.824966i \(0.308806\pi\)
\(770\) −24.2176 −0.872743
\(771\) 0 0
\(772\) −9.77786 −0.351913
\(773\) 43.8102 1.57574 0.787871 0.615840i \(-0.211183\pi\)
0.787871 + 0.615840i \(0.211183\pi\)
\(774\) 0 0
\(775\) −13.3985 −0.481288
\(776\) −17.5825 −0.631174
\(777\) 0 0
\(778\) 31.7085 1.13680
\(779\) −15.8344 −0.567326
\(780\) 0 0
\(781\) 39.5026 1.41352
\(782\) 2.28648 0.0817643
\(783\) 0 0
\(784\) −1.95299 −0.0697495
\(785\) −32.8857 −1.17374
\(786\) 0 0
\(787\) 8.20309 0.292409 0.146204 0.989254i \(-0.453294\pi\)
0.146204 + 0.989254i \(0.453294\pi\)
\(788\) −14.5046 −0.516705
\(789\) 0 0
\(790\) 3.36085 0.119574
\(791\) −34.9458 −1.24253
\(792\) 0 0
\(793\) −0.0858325 −0.00304800
\(794\) −5.92081 −0.210122
\(795\) 0 0
\(796\) 22.6203 0.801755
\(797\) −22.7577 −0.806118 −0.403059 0.915174i \(-0.632053\pi\)
−0.403059 + 0.915174i \(0.632053\pi\)
\(798\) 0 0
\(799\) 7.15859 0.253253
\(800\) 1.93222 0.0683143
\(801\) 0 0
\(802\) 8.12974 0.287071
\(803\) −4.07693 −0.143872
\(804\) 0 0
\(805\) 20.0762 0.707592
\(806\) −0.0538547 −0.00189695
\(807\) 0 0
\(808\) −8.56611 −0.301355
\(809\) 27.2179 0.956930 0.478465 0.878107i \(-0.341193\pi\)
0.478465 + 0.878107i \(0.341193\pi\)
\(810\) 0 0
\(811\) −9.81625 −0.344695 −0.172348 0.985036i \(-0.555135\pi\)
−0.172348 + 0.985036i \(0.555135\pi\)
\(812\) −17.9906 −0.631347
\(813\) 0 0
\(814\) −2.66674 −0.0934691
\(815\) 16.5085 0.578268
\(816\) 0 0
\(817\) −34.2699 −1.19895
\(818\) 12.9592 0.453108
\(819\) 0 0
\(820\) −11.6686 −0.407484
\(821\) 17.8972 0.624616 0.312308 0.949981i \(-0.398898\pi\)
0.312308 + 0.949981i \(0.398898\pi\)
\(822\) 0 0
\(823\) 16.0259 0.558627 0.279313 0.960200i \(-0.409893\pi\)
0.279313 + 0.960200i \(0.409893\pi\)
\(824\) 19.9320 0.694363
\(825\) 0 0
\(826\) −17.8680 −0.621708
\(827\) 18.7380 0.651584 0.325792 0.945441i \(-0.394369\pi\)
0.325792 + 0.945441i \(0.394369\pi\)
\(828\) 0 0
\(829\) −0.931833 −0.0323639 −0.0161820 0.999869i \(-0.505151\pi\)
−0.0161820 + 0.999869i \(0.505151\pi\)
\(830\) −15.1495 −0.525848
\(831\) 0 0
\(832\) 0.00776648 0.000269254 0
\(833\) −1.31564 −0.0455844
\(834\) 0 0
\(835\) −7.65267 −0.264832
\(836\) 14.6285 0.505936
\(837\) 0 0
\(838\) 30.4806 1.05293
\(839\) −10.4652 −0.361300 −0.180650 0.983547i \(-0.557820\pi\)
−0.180650 + 0.983547i \(0.557820\pi\)
\(840\) 0 0
\(841\) 35.1294 1.21136
\(842\) −38.0654 −1.31182
\(843\) 0 0
\(844\) −10.0332 −0.345358
\(845\) 34.2277 1.17747
\(846\) 0 0
\(847\) −12.9473 −0.444874
\(848\) 4.64357 0.159461
\(849\) 0 0
\(850\) 1.30166 0.0446464
\(851\) 2.21070 0.0757818
\(852\) 0 0
\(853\) 28.7039 0.982801 0.491401 0.870934i \(-0.336485\pi\)
0.491401 + 0.870934i \(0.336485\pi\)
\(854\) 24.8282 0.849603
\(855\) 0 0
\(856\) 13.1728 0.450238
\(857\) 10.0329 0.342717 0.171359 0.985209i \(-0.445184\pi\)
0.171359 + 0.985209i \(0.445184\pi\)
\(858\) 0 0
\(859\) −32.0157 −1.09236 −0.546181 0.837667i \(-0.683919\pi\)
−0.546181 + 0.837667i \(0.683919\pi\)
\(860\) −25.2539 −0.861152
\(861\) 0 0
\(862\) 21.5664 0.734553
\(863\) −4.53904 −0.154511 −0.0772553 0.997011i \(-0.524616\pi\)
−0.0772553 + 0.997011i \(0.524616\pi\)
\(864\) 0 0
\(865\) 24.4593 0.831642
\(866\) −8.92010 −0.303117
\(867\) 0 0
\(868\) 15.5782 0.528758
\(869\) 5.22626 0.177289
\(870\) 0 0
\(871\) 0.118059 0.00400026
\(872\) −5.10676 −0.172937
\(873\) 0 0
\(874\) −12.1268 −0.410197
\(875\) −18.1459 −0.613442
\(876\) 0 0
\(877\) 40.9646 1.38328 0.691639 0.722243i \(-0.256889\pi\)
0.691639 + 0.722243i \(0.256889\pi\)
\(878\) −26.0102 −0.877802
\(879\) 0 0
\(880\) 10.7799 0.363390
\(881\) 0.505607 0.0170343 0.00851716 0.999964i \(-0.497289\pi\)
0.00851716 + 0.999964i \(0.497289\pi\)
\(882\) 0 0
\(883\) 27.7967 0.935434 0.467717 0.883878i \(-0.345077\pi\)
0.467717 + 0.883878i \(0.345077\pi\)
\(884\) 0.00523195 0.000175970 0
\(885\) 0 0
\(886\) −10.7806 −0.362180
\(887\) 6.73644 0.226188 0.113094 0.993584i \(-0.463924\pi\)
0.113094 + 0.993584i \(0.463924\pi\)
\(888\) 0 0
\(889\) −34.6209 −1.16115
\(890\) 5.46289 0.183116
\(891\) 0 0
\(892\) −1.00000 −0.0334825
\(893\) −37.9672 −1.27052
\(894\) 0 0
\(895\) −55.5686 −1.85745
\(896\) −2.24656 −0.0750522
\(897\) 0 0
\(898\) 20.0819 0.670140
\(899\) −55.5301 −1.85203
\(900\) 0 0
\(901\) 3.12818 0.104215
\(902\) −18.1451 −0.604165
\(903\) 0 0
\(904\) 15.5553 0.517361
\(905\) −1.43055 −0.0475531
\(906\) 0 0
\(907\) −21.5625 −0.715972 −0.357986 0.933727i \(-0.616536\pi\)
−0.357986 + 0.933727i \(0.616536\pi\)
\(908\) −20.8914 −0.693305
\(909\) 0 0
\(910\) 0.0459386 0.00152285
\(911\) −59.1815 −1.96077 −0.980386 0.197088i \(-0.936852\pi\)
−0.980386 + 0.197088i \(0.936852\pi\)
\(912\) 0 0
\(913\) −23.5581 −0.779661
\(914\) −35.5020 −1.17430
\(915\) 0 0
\(916\) 20.5829 0.680077
\(917\) 47.2204 1.55935
\(918\) 0 0
\(919\) −28.4787 −0.939426 −0.469713 0.882819i \(-0.655643\pi\)
−0.469713 + 0.882819i \(0.655643\pi\)
\(920\) −8.93642 −0.294625
\(921\) 0 0
\(922\) 3.26003 0.107363
\(923\) −0.0749328 −0.00246644
\(924\) 0 0
\(925\) 1.25852 0.0413798
\(926\) 5.67209 0.186397
\(927\) 0 0
\(928\) 8.00809 0.262878
\(929\) −32.6469 −1.07111 −0.535555 0.844500i \(-0.679898\pi\)
−0.535555 + 0.844500i \(0.679898\pi\)
\(930\) 0 0
\(931\) 6.97782 0.228689
\(932\) 9.41577 0.308424
\(933\) 0 0
\(934\) 10.7885 0.353010
\(935\) 7.26196 0.237492
\(936\) 0 0
\(937\) −26.4204 −0.863118 −0.431559 0.902085i \(-0.642036\pi\)
−0.431559 + 0.902085i \(0.642036\pi\)
\(938\) −34.1500 −1.11504
\(939\) 0 0
\(940\) −27.9785 −0.912557
\(941\) −27.0639 −0.882259 −0.441130 0.897443i \(-0.645422\pi\)
−0.441130 + 0.897443i \(0.645422\pi\)
\(942\) 0 0
\(943\) 15.0421 0.489838
\(944\) 7.95351 0.258865
\(945\) 0 0
\(946\) −39.2709 −1.27681
\(947\) −15.7697 −0.512447 −0.256223 0.966618i \(-0.582478\pi\)
−0.256223 + 0.966618i \(0.582478\pi\)
\(948\) 0 0
\(949\) 0.00773355 0.000251042 0
\(950\) −6.90363 −0.223983
\(951\) 0 0
\(952\) −1.51341 −0.0490499
\(953\) −25.0914 −0.812791 −0.406395 0.913697i \(-0.633214\pi\)
−0.406395 + 0.913697i \(0.633214\pi\)
\(954\) 0 0
\(955\) −36.9661 −1.19620
\(956\) −14.6538 −0.473939
\(957\) 0 0
\(958\) −17.5354 −0.566542
\(959\) −28.2669 −0.912785
\(960\) 0 0
\(961\) 17.0838 0.551090
\(962\) 0.00505855 0.000163094 0
\(963\) 0 0
\(964\) 17.1216 0.551448
\(965\) 25.7442 0.828737
\(966\) 0 0
\(967\) 40.5703 1.30465 0.652327 0.757938i \(-0.273793\pi\)
0.652327 + 0.757938i \(0.273793\pi\)
\(968\) 5.76317 0.185235
\(969\) 0 0
\(970\) 46.2931 1.48638
\(971\) −32.4433 −1.04116 −0.520578 0.853814i \(-0.674283\pi\)
−0.520578 + 0.853814i \(0.674283\pi\)
\(972\) 0 0
\(973\) 21.3716 0.685143
\(974\) 21.6167 0.692645
\(975\) 0 0
\(976\) −11.0517 −0.353755
\(977\) 32.0199 1.02441 0.512204 0.858864i \(-0.328829\pi\)
0.512204 + 0.858864i \(0.328829\pi\)
\(978\) 0 0
\(979\) 8.49501 0.271502
\(980\) 5.14204 0.164256
\(981\) 0 0
\(982\) 11.0429 0.352393
\(983\) 11.3623 0.362400 0.181200 0.983446i \(-0.442002\pi\)
0.181200 + 0.983446i \(0.442002\pi\)
\(984\) 0 0
\(985\) 38.1894 1.21681
\(986\) 5.39471 0.171803
\(987\) 0 0
\(988\) −0.0277488 −0.000882808 0
\(989\) 32.5552 1.03519
\(990\) 0 0
\(991\) 57.7380 1.83411 0.917054 0.398764i \(-0.130561\pi\)
0.917054 + 0.398764i \(0.130561\pi\)
\(992\) −6.93425 −0.220163
\(993\) 0 0
\(994\) 21.6753 0.687499
\(995\) −59.5572 −1.88809
\(996\) 0 0
\(997\) −48.5327 −1.53705 −0.768524 0.639821i \(-0.779008\pi\)
−0.768524 + 0.639821i \(0.779008\pi\)
\(998\) −9.24468 −0.292635
\(999\) 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 4014.2.a.u.1.2 yes 6
3.2 odd 2 4014.2.a.t.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4014.2.a.t.1.5 6 3.2 odd 2
4014.2.a.u.1.2 yes 6 1.1 even 1 trivial