Properties

Label 8281.2.a.be.1.2
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
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] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, 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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.688892 q^{2} +2.21432 q^{3} -1.52543 q^{4} +3.21432 q^{5} -1.52543 q^{6} +2.42864 q^{8} +1.90321 q^{9} -2.21432 q^{10} -2.68889 q^{11} -3.37778 q^{12} +7.11753 q^{15} +1.37778 q^{16} -3.59210 q^{17} -1.31111 q^{18} -8.54617 q^{19} -4.90321 q^{20} +1.85236 q^{22} -3.28100 q^{23} +5.37778 q^{24} +5.33185 q^{25} -2.42864 q^{27} +2.05086 q^{29} -4.90321 q^{30} +5.83654 q^{31} -5.80642 q^{32} -5.95407 q^{33} +2.47457 q^{34} -2.90321 q^{36} -3.93332 q^{37} +5.88739 q^{38} +7.80642 q^{40} +0.755569 q^{41} +8.80642 q^{43} +4.10171 q^{44} +6.11753 q^{45} +2.26025 q^{46} +1.88247 q^{47} +3.05086 q^{48} -3.67307 q^{50} -7.95407 q^{51} +2.52543 q^{53} +1.67307 q^{54} -8.64296 q^{55} -18.9240 q^{57} -1.41282 q^{58} +7.33185 q^{59} -10.8573 q^{60} -9.05086 q^{61} -4.02074 q^{62} +1.24443 q^{64} +4.10171 q^{66} -0.428639 q^{67} +5.47949 q^{68} -7.26517 q^{69} -8.98418 q^{71} +4.62222 q^{72} +5.79060 q^{73} +2.70964 q^{74} +11.8064 q^{75} +13.0366 q^{76} -4.47949 q^{79} +4.42864 q^{80} -11.0874 q^{81} -0.520505 q^{82} -10.8272 q^{83} -11.5462 q^{85} -6.06668 q^{86} +4.54125 q^{87} -6.53035 q^{88} +5.36196 q^{89} -4.21432 q^{90} +5.00492 q^{92} +12.9240 q^{93} -1.29682 q^{94} -27.4701 q^{95} -12.8573 q^{96} +9.62867 q^{97} -5.11753 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 6 q^{8} - q^{9} - 8 q^{11} - 10 q^{12} + 8 q^{15} + 4 q^{16} - 4 q^{17} - 4 q^{18} + q^{19} - 8 q^{20} + 12 q^{22} - 3 q^{23} + 16 q^{24} - 4 q^{25} + 6 q^{27}+ \cdots - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.688892 −0.487120 −0.243560 0.969886i \(-0.578315\pi\)
−0.243560 + 0.969886i \(0.578315\pi\)
\(3\) 2.21432 1.27844 0.639219 0.769025i \(-0.279258\pi\)
0.639219 + 0.769025i \(0.279258\pi\)
\(4\) −1.52543 −0.762714
\(5\) 3.21432 1.43749 0.718744 0.695275i \(-0.244717\pi\)
0.718744 + 0.695275i \(0.244717\pi\)
\(6\) −1.52543 −0.622753
\(7\) 0 0
\(8\) 2.42864 0.858654
\(9\) 1.90321 0.634404
\(10\) −2.21432 −0.700229
\(11\) −2.68889 −0.810731 −0.405366 0.914155i \(-0.632856\pi\)
−0.405366 + 0.914155i \(0.632856\pi\)
\(12\) −3.37778 −0.975082
\(13\) 0 0
\(14\) 0 0
\(15\) 7.11753 1.83774
\(16\) 1.37778 0.344446
\(17\) −3.59210 −0.871213 −0.435607 0.900137i \(-0.643466\pi\)
−0.435607 + 0.900137i \(0.643466\pi\)
\(18\) −1.31111 −0.309031
\(19\) −8.54617 −1.96063 −0.980313 0.197449i \(-0.936734\pi\)
−0.980313 + 0.197449i \(0.936734\pi\)
\(20\) −4.90321 −1.09639
\(21\) 0 0
\(22\) 1.85236 0.394924
\(23\) −3.28100 −0.684135 −0.342068 0.939675i \(-0.611127\pi\)
−0.342068 + 0.939675i \(0.611127\pi\)
\(24\) 5.37778 1.09774
\(25\) 5.33185 1.06637
\(26\) 0 0
\(27\) −2.42864 −0.467392
\(28\) 0 0
\(29\) 2.05086 0.380834 0.190417 0.981703i \(-0.439016\pi\)
0.190417 + 0.981703i \(0.439016\pi\)
\(30\) −4.90321 −0.895200
\(31\) 5.83654 1.04827 0.524136 0.851634i \(-0.324388\pi\)
0.524136 + 0.851634i \(0.324388\pi\)
\(32\) −5.80642 −1.02644
\(33\) −5.95407 −1.03647
\(34\) 2.47457 0.424386
\(35\) 0 0
\(36\) −2.90321 −0.483869
\(37\) −3.93332 −0.646634 −0.323317 0.946291i \(-0.604798\pi\)
−0.323317 + 0.946291i \(0.604798\pi\)
\(38\) 5.88739 0.955061
\(39\) 0 0
\(40\) 7.80642 1.23430
\(41\) 0.755569 0.118000 0.0590000 0.998258i \(-0.481209\pi\)
0.0590000 + 0.998258i \(0.481209\pi\)
\(42\) 0 0
\(43\) 8.80642 1.34297 0.671484 0.741019i \(-0.265657\pi\)
0.671484 + 0.741019i \(0.265657\pi\)
\(44\) 4.10171 0.618356
\(45\) 6.11753 0.911948
\(46\) 2.26025 0.333256
\(47\) 1.88247 0.274586 0.137293 0.990530i \(-0.456160\pi\)
0.137293 + 0.990530i \(0.456160\pi\)
\(48\) 3.05086 0.440353
\(49\) 0 0
\(50\) −3.67307 −0.519451
\(51\) −7.95407 −1.11379
\(52\) 0 0
\(53\) 2.52543 0.346894 0.173447 0.984843i \(-0.444509\pi\)
0.173447 + 0.984843i \(0.444509\pi\)
\(54\) 1.67307 0.227676
\(55\) −8.64296 −1.16542
\(56\) 0 0
\(57\) −18.9240 −2.50654
\(58\) −1.41282 −0.185512
\(59\) 7.33185 0.954526 0.477263 0.878761i \(-0.341629\pi\)
0.477263 + 0.878761i \(0.341629\pi\)
\(60\) −10.8573 −1.40167
\(61\) −9.05086 −1.15884 −0.579422 0.815028i \(-0.696722\pi\)
−0.579422 + 0.815028i \(0.696722\pi\)
\(62\) −4.02074 −0.510635
\(63\) 0 0
\(64\) 1.24443 0.155554
\(65\) 0 0
\(66\) 4.10171 0.504886
\(67\) −0.428639 −0.0523666 −0.0261833 0.999657i \(-0.508335\pi\)
−0.0261833 + 0.999657i \(0.508335\pi\)
\(68\) 5.47949 0.664486
\(69\) −7.26517 −0.874624
\(70\) 0 0
\(71\) −8.98418 −1.06623 −0.533113 0.846044i \(-0.678978\pi\)
−0.533113 + 0.846044i \(0.678978\pi\)
\(72\) 4.62222 0.544733
\(73\) 5.79060 0.677739 0.338869 0.940833i \(-0.389956\pi\)
0.338869 + 0.940833i \(0.389956\pi\)
\(74\) 2.70964 0.314989
\(75\) 11.8064 1.36329
\(76\) 13.0366 1.49540
\(77\) 0 0
\(78\) 0 0
\(79\) −4.47949 −0.503983 −0.251991 0.967730i \(-0.581085\pi\)
−0.251991 + 0.967730i \(0.581085\pi\)
\(80\) 4.42864 0.495137
\(81\) −11.0874 −1.23194
\(82\) −0.520505 −0.0574802
\(83\) −10.8272 −1.18844 −0.594218 0.804304i \(-0.702538\pi\)
−0.594218 + 0.804304i \(0.702538\pi\)
\(84\) 0 0
\(85\) −11.5462 −1.25236
\(86\) −6.06668 −0.654187
\(87\) 4.54125 0.486873
\(88\) −6.53035 −0.696138
\(89\) 5.36196 0.568367 0.284183 0.958770i \(-0.408278\pi\)
0.284183 + 0.958770i \(0.408278\pi\)
\(90\) −4.21432 −0.444228
\(91\) 0 0
\(92\) 5.00492 0.521799
\(93\) 12.9240 1.34015
\(94\) −1.29682 −0.133757
\(95\) −27.4701 −2.81838
\(96\) −12.8573 −1.31224
\(97\) 9.62867 0.977643 0.488822 0.872384i \(-0.337427\pi\)
0.488822 + 0.872384i \(0.337427\pi\)
\(98\) 0 0
\(99\) −5.11753 −0.514331
\(100\) −8.13335 −0.813335
\(101\) −13.6938 −1.36259 −0.681293 0.732011i \(-0.738582\pi\)
−0.681293 + 0.732011i \(0.738582\pi\)
\(102\) 5.47949 0.542551
\(103\) −12.2953 −1.21149 −0.605745 0.795659i \(-0.707125\pi\)
−0.605745 + 0.795659i \(0.707125\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1.73975 −0.168979
\(107\) −18.1891 −1.75841 −0.879205 0.476444i \(-0.841925\pi\)
−0.879205 + 0.476444i \(0.841925\pi\)
\(108\) 3.70471 0.356486
\(109\) −8.36196 −0.800931 −0.400465 0.916312i \(-0.631152\pi\)
−0.400465 + 0.916312i \(0.631152\pi\)
\(110\) 5.95407 0.567698
\(111\) −8.70964 −0.826682
\(112\) 0 0
\(113\) −8.46520 −0.796339 −0.398170 0.917312i \(-0.630354\pi\)
−0.398170 + 0.917312i \(0.630354\pi\)
\(114\) 13.0366 1.22099
\(115\) −10.5462 −0.983436
\(116\) −3.12843 −0.290468
\(117\) 0 0
\(118\) −5.05086 −0.464969
\(119\) 0 0
\(120\) 17.2859 1.57798
\(121\) −3.76986 −0.342714
\(122\) 6.23506 0.564496
\(123\) 1.67307 0.150856
\(124\) −8.90321 −0.799532
\(125\) 1.06668 0.0954065
\(126\) 0 0
\(127\) 4.08742 0.362700 0.181350 0.983419i \(-0.441953\pi\)
0.181350 + 0.983419i \(0.441953\pi\)
\(128\) 10.7556 0.950667
\(129\) 19.5002 1.71690
\(130\) 0 0
\(131\) −5.93978 −0.518961 −0.259480 0.965748i \(-0.583551\pi\)
−0.259480 + 0.965748i \(0.583551\pi\)
\(132\) 9.08250 0.790530
\(133\) 0 0
\(134\) 0.295286 0.0255089
\(135\) −7.80642 −0.671870
\(136\) −8.72393 −0.748070
\(137\) −16.3620 −1.39790 −0.698948 0.715172i \(-0.746348\pi\)
−0.698948 + 0.715172i \(0.746348\pi\)
\(138\) 5.00492 0.426047
\(139\) −3.03011 −0.257011 −0.128505 0.991709i \(-0.541018\pi\)
−0.128505 + 0.991709i \(0.541018\pi\)
\(140\) 0 0
\(141\) 4.16839 0.351041
\(142\) 6.18913 0.519380
\(143\) 0 0
\(144\) 2.62222 0.218518
\(145\) 6.59210 0.547444
\(146\) −3.98910 −0.330140
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) 14.5368 1.19090 0.595451 0.803392i \(-0.296974\pi\)
0.595451 + 0.803392i \(0.296974\pi\)
\(150\) −8.13335 −0.664085
\(151\) 19.9748 1.62553 0.812764 0.582594i \(-0.197962\pi\)
0.812764 + 0.582594i \(0.197962\pi\)
\(152\) −20.7556 −1.68350
\(153\) −6.83654 −0.552701
\(154\) 0 0
\(155\) 18.7605 1.50688
\(156\) 0 0
\(157\) −7.39853 −0.590467 −0.295233 0.955425i \(-0.595397\pi\)
−0.295233 + 0.955425i \(0.595397\pi\)
\(158\) 3.08589 0.245500
\(159\) 5.59210 0.443483
\(160\) −18.6637 −1.47550
\(161\) 0 0
\(162\) 7.63804 0.600101
\(163\) −2.32693 −0.182259 −0.0911296 0.995839i \(-0.529048\pi\)
−0.0911296 + 0.995839i \(0.529048\pi\)
\(164\) −1.15257 −0.0900002
\(165\) −19.1383 −1.48991
\(166\) 7.45875 0.578911
\(167\) −3.42219 −0.264817 −0.132408 0.991195i \(-0.542271\pi\)
−0.132408 + 0.991195i \(0.542271\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 7.95407 0.610049
\(171\) −16.2652 −1.24383
\(172\) −13.4336 −1.02430
\(173\) −8.27454 −0.629102 −0.314551 0.949241i \(-0.601854\pi\)
−0.314551 + 0.949241i \(0.601854\pi\)
\(174\) −3.12843 −0.237166
\(175\) 0 0
\(176\) −3.70471 −0.279253
\(177\) 16.2351 1.22030
\(178\) −3.69381 −0.276863
\(179\) −16.2257 −1.21277 −0.606383 0.795173i \(-0.707380\pi\)
−0.606383 + 0.795173i \(0.707380\pi\)
\(180\) −9.33185 −0.695555
\(181\) 9.20495 0.684199 0.342099 0.939664i \(-0.388862\pi\)
0.342099 + 0.939664i \(0.388862\pi\)
\(182\) 0 0
\(183\) −20.0415 −1.48151
\(184\) −7.96836 −0.587435
\(185\) −12.6430 −0.929529
\(186\) −8.90321 −0.652815
\(187\) 9.65878 0.706320
\(188\) −2.87157 −0.209431
\(189\) 0 0
\(190\) 18.9240 1.37289
\(191\) −6.66815 −0.482490 −0.241245 0.970464i \(-0.577556\pi\)
−0.241245 + 0.970464i \(0.577556\pi\)
\(192\) 2.75557 0.198866
\(193\) −20.6035 −1.48307 −0.741535 0.670914i \(-0.765902\pi\)
−0.741535 + 0.670914i \(0.765902\pi\)
\(194\) −6.63311 −0.476230
\(195\) 0 0
\(196\) 0 0
\(197\) 8.36842 0.596225 0.298112 0.954531i \(-0.403643\pi\)
0.298112 + 0.954531i \(0.403643\pi\)
\(198\) 3.52543 0.250541
\(199\) 0.601472 0.0426372 0.0213186 0.999773i \(-0.493214\pi\)
0.0213186 + 0.999773i \(0.493214\pi\)
\(200\) 12.9491 0.915643
\(201\) −0.949145 −0.0669475
\(202\) 9.43356 0.663743
\(203\) 0 0
\(204\) 12.1334 0.849505
\(205\) 2.42864 0.169624
\(206\) 8.47013 0.590142
\(207\) −6.24443 −0.434018
\(208\) 0 0
\(209\) 22.9797 1.58954
\(210\) 0 0
\(211\) −1.90321 −0.131023 −0.0655113 0.997852i \(-0.520868\pi\)
−0.0655113 + 0.997852i \(0.520868\pi\)
\(212\) −3.85236 −0.264581
\(213\) −19.8938 −1.36310
\(214\) 12.5303 0.856557
\(215\) 28.3067 1.93050
\(216\) −5.89829 −0.401328
\(217\) 0 0
\(218\) 5.76049 0.390150
\(219\) 12.8222 0.866447
\(220\) 13.1842 0.888879
\(221\) 0 0
\(222\) 6.00000 0.402694
\(223\) 10.6336 0.712078 0.356039 0.934471i \(-0.384127\pi\)
0.356039 + 0.934471i \(0.384127\pi\)
\(224\) 0 0
\(225\) 10.1476 0.676510
\(226\) 5.83161 0.387913
\(227\) 15.0509 0.998960 0.499480 0.866325i \(-0.333524\pi\)
0.499480 + 0.866325i \(0.333524\pi\)
\(228\) 28.8671 1.91177
\(229\) −6.53480 −0.431831 −0.215916 0.976412i \(-0.569274\pi\)
−0.215916 + 0.976412i \(0.569274\pi\)
\(230\) 7.26517 0.479051
\(231\) 0 0
\(232\) 4.98079 0.327005
\(233\) −27.9590 −1.83165 −0.915827 0.401573i \(-0.868464\pi\)
−0.915827 + 0.401573i \(0.868464\pi\)
\(234\) 0 0
\(235\) 6.05086 0.394714
\(236\) −11.1842 −0.728030
\(237\) −9.91903 −0.644310
\(238\) 0 0
\(239\) 19.5812 1.26660 0.633301 0.773905i \(-0.281699\pi\)
0.633301 + 0.773905i \(0.281699\pi\)
\(240\) 9.80642 0.633002
\(241\) −13.3575 −0.860433 −0.430217 0.902726i \(-0.641563\pi\)
−0.430217 + 0.902726i \(0.641563\pi\)
\(242\) 2.59703 0.166943
\(243\) −17.2652 −1.10756
\(244\) 13.8064 0.883866
\(245\) 0 0
\(246\) −1.15257 −0.0734849
\(247\) 0 0
\(248\) 14.1748 0.900103
\(249\) −23.9748 −1.51934
\(250\) −0.734825 −0.0464744
\(251\) −3.29682 −0.208093 −0.104047 0.994572i \(-0.533179\pi\)
−0.104047 + 0.994572i \(0.533179\pi\)
\(252\) 0 0
\(253\) 8.82225 0.554650
\(254\) −2.81579 −0.176678
\(255\) −25.5669 −1.60106
\(256\) −9.89829 −0.618643
\(257\) 23.6938 1.47798 0.738990 0.673717i \(-0.235303\pi\)
0.738990 + 0.673717i \(0.235303\pi\)
\(258\) −13.4336 −0.836337
\(259\) 0 0
\(260\) 0 0
\(261\) 3.90321 0.241603
\(262\) 4.09187 0.252796
\(263\) 9.99063 0.616049 0.308024 0.951378i \(-0.400332\pi\)
0.308024 + 0.951378i \(0.400332\pi\)
\(264\) −14.4603 −0.889969
\(265\) 8.11753 0.498656
\(266\) 0 0
\(267\) 11.8731 0.726622
\(268\) 0.653858 0.0399408
\(269\) 18.2034 1.10988 0.554941 0.831890i \(-0.312741\pi\)
0.554941 + 0.831890i \(0.312741\pi\)
\(270\) 5.37778 0.327282
\(271\) −24.1748 −1.46852 −0.734258 0.678870i \(-0.762470\pi\)
−0.734258 + 0.678870i \(0.762470\pi\)
\(272\) −4.94914 −0.300086
\(273\) 0 0
\(274\) 11.2716 0.680944
\(275\) −14.3368 −0.864540
\(276\) 11.0825 0.667088
\(277\) 1.69535 0.101863 0.0509317 0.998702i \(-0.483781\pi\)
0.0509317 + 0.998702i \(0.483781\pi\)
\(278\) 2.08742 0.125195
\(279\) 11.1082 0.665028
\(280\) 0 0
\(281\) −11.6479 −0.694854 −0.347427 0.937707i \(-0.612945\pi\)
−0.347427 + 0.937707i \(0.612945\pi\)
\(282\) −2.87157 −0.170999
\(283\) −12.1334 −0.721253 −0.360626 0.932710i \(-0.617437\pi\)
−0.360626 + 0.932710i \(0.617437\pi\)
\(284\) 13.7047 0.813225
\(285\) −60.8276 −3.60312
\(286\) 0 0
\(287\) 0 0
\(288\) −11.0509 −0.651178
\(289\) −4.09679 −0.240988
\(290\) −4.54125 −0.266671
\(291\) 21.3210 1.24986
\(292\) −8.83314 −0.516921
\(293\) 11.4538 0.669140 0.334570 0.942371i \(-0.391409\pi\)
0.334570 + 0.942371i \(0.391409\pi\)
\(294\) 0 0
\(295\) 23.5669 1.37212
\(296\) −9.55262 −0.555235
\(297\) 6.53035 0.378929
\(298\) −10.0143 −0.580112
\(299\) 0 0
\(300\) −18.0098 −1.03980
\(301\) 0 0
\(302\) −13.7605 −0.791827
\(303\) −30.3225 −1.74198
\(304\) −11.7748 −0.675330
\(305\) −29.0923 −1.66582
\(306\) 4.70964 0.269232
\(307\) 3.96989 0.226574 0.113287 0.993562i \(-0.463862\pi\)
0.113287 + 0.993562i \(0.463862\pi\)
\(308\) 0 0
\(309\) −27.2257 −1.54882
\(310\) −12.9240 −0.734031
\(311\) −27.3481 −1.55077 −0.775386 0.631488i \(-0.782444\pi\)
−0.775386 + 0.631488i \(0.782444\pi\)
\(312\) 0 0
\(313\) 19.1032 1.07978 0.539890 0.841736i \(-0.318466\pi\)
0.539890 + 0.841736i \(0.318466\pi\)
\(314\) 5.09679 0.287628
\(315\) 0 0
\(316\) 6.83314 0.384394
\(317\) −9.90813 −0.556496 −0.278248 0.960509i \(-0.589754\pi\)
−0.278248 + 0.960509i \(0.589754\pi\)
\(318\) −3.85236 −0.216029
\(319\) −5.51453 −0.308754
\(320\) 4.00000 0.223607
\(321\) −40.2766 −2.24802
\(322\) 0 0
\(323\) 30.6987 1.70812
\(324\) 16.9131 0.939614
\(325\) 0 0
\(326\) 1.60300 0.0887821
\(327\) −18.5161 −1.02394
\(328\) 1.83500 0.101321
\(329\) 0 0
\(330\) 13.1842 0.725767
\(331\) −23.4193 −1.28724 −0.643620 0.765345i \(-0.722568\pi\)
−0.643620 + 0.765345i \(0.722568\pi\)
\(332\) 16.5161 0.906437
\(333\) −7.48595 −0.410227
\(334\) 2.35752 0.128998
\(335\) −1.37778 −0.0752764
\(336\) 0 0
\(337\) −7.51606 −0.409426 −0.204713 0.978822i \(-0.565626\pi\)
−0.204713 + 0.978822i \(0.565626\pi\)
\(338\) 0 0
\(339\) −18.7447 −1.01807
\(340\) 17.6128 0.955191
\(341\) −15.6938 −0.849868
\(342\) 11.2050 0.605894
\(343\) 0 0
\(344\) 21.3876 1.15314
\(345\) −23.3526 −1.25726
\(346\) 5.70027 0.306448
\(347\) 5.64449 0.303012 0.151506 0.988456i \(-0.451588\pi\)
0.151506 + 0.988456i \(0.451588\pi\)
\(348\) −6.92735 −0.371345
\(349\) 24.1590 1.29320 0.646601 0.762828i \(-0.276190\pi\)
0.646601 + 0.762828i \(0.276190\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 15.6128 0.832168
\(353\) 36.1289 1.92295 0.961474 0.274896i \(-0.0886436\pi\)
0.961474 + 0.274896i \(0.0886436\pi\)
\(354\) −11.1842 −0.594434
\(355\) −28.8780 −1.53269
\(356\) −8.17929 −0.433501
\(357\) 0 0
\(358\) 11.1778 0.590763
\(359\) −17.4128 −0.919013 −0.459507 0.888174i \(-0.651974\pi\)
−0.459507 + 0.888174i \(0.651974\pi\)
\(360\) 14.8573 0.783047
\(361\) 54.0370 2.84405
\(362\) −6.34122 −0.333287
\(363\) −8.34767 −0.438139
\(364\) 0 0
\(365\) 18.6128 0.974241
\(366\) 13.8064 0.721673
\(367\) −2.93825 −0.153375 −0.0766876 0.997055i \(-0.524434\pi\)
−0.0766876 + 0.997055i \(0.524434\pi\)
\(368\) −4.52051 −0.235648
\(369\) 1.43801 0.0748597
\(370\) 8.70964 0.452792
\(371\) 0 0
\(372\) −19.7146 −1.02215
\(373\) −18.7699 −0.971866 −0.485933 0.873996i \(-0.661520\pi\)
−0.485933 + 0.873996i \(0.661520\pi\)
\(374\) −6.65386 −0.344063
\(375\) 2.36196 0.121971
\(376\) 4.57184 0.235774
\(377\) 0 0
\(378\) 0 0
\(379\) −23.6894 −1.21684 −0.608421 0.793615i \(-0.708197\pi\)
−0.608421 + 0.793615i \(0.708197\pi\)
\(380\) 41.9037 2.14961
\(381\) 9.05086 0.463689
\(382\) 4.59364 0.235031
\(383\) 31.6128 1.61534 0.807671 0.589634i \(-0.200728\pi\)
0.807671 + 0.589634i \(0.200728\pi\)
\(384\) 23.8163 1.21537
\(385\) 0 0
\(386\) 14.1936 0.722434
\(387\) 16.7605 0.851984
\(388\) −14.6878 −0.745662
\(389\) −20.0558 −1.01687 −0.508434 0.861101i \(-0.669775\pi\)
−0.508434 + 0.861101i \(0.669775\pi\)
\(390\) 0 0
\(391\) 11.7857 0.596027
\(392\) 0 0
\(393\) −13.1526 −0.663459
\(394\) −5.76494 −0.290433
\(395\) −14.3985 −0.724469
\(396\) 7.80642 0.392288
\(397\) 22.8731 1.14797 0.573984 0.818867i \(-0.305397\pi\)
0.573984 + 0.818867i \(0.305397\pi\)
\(398\) −0.414349 −0.0207695
\(399\) 0 0
\(400\) 7.34614 0.367307
\(401\) 5.61285 0.280292 0.140146 0.990131i \(-0.455243\pi\)
0.140146 + 0.990131i \(0.455243\pi\)
\(402\) 0.653858 0.0326115
\(403\) 0 0
\(404\) 20.8889 1.03926
\(405\) −35.6385 −1.77089
\(406\) 0 0
\(407\) 10.5763 0.524247
\(408\) −19.3176 −0.956362
\(409\) −26.1175 −1.29143 −0.645714 0.763579i \(-0.723440\pi\)
−0.645714 + 0.763579i \(0.723440\pi\)
\(410\) −1.67307 −0.0826271
\(411\) −36.2306 −1.78712
\(412\) 18.7556 0.924021
\(413\) 0 0
\(414\) 4.30174 0.211419
\(415\) −34.8020 −1.70836
\(416\) 0 0
\(417\) −6.70964 −0.328572
\(418\) −15.8306 −0.774298
\(419\) 25.2464 1.23337 0.616685 0.787210i \(-0.288475\pi\)
0.616685 + 0.787210i \(0.288475\pi\)
\(420\) 0 0
\(421\) 8.22861 0.401038 0.200519 0.979690i \(-0.435737\pi\)
0.200519 + 0.979690i \(0.435737\pi\)
\(422\) 1.31111 0.0638237
\(423\) 3.58274 0.174199
\(424\) 6.13335 0.297862
\(425\) −19.1526 −0.929036
\(426\) 13.7047 0.663996
\(427\) 0 0
\(428\) 27.7462 1.34116
\(429\) 0 0
\(430\) −19.5002 −0.940385
\(431\) 1.43801 0.0692664 0.0346332 0.999400i \(-0.488974\pi\)
0.0346332 + 0.999400i \(0.488974\pi\)
\(432\) −3.34614 −0.160991
\(433\) 16.5018 0.793024 0.396512 0.918029i \(-0.370220\pi\)
0.396512 + 0.918029i \(0.370220\pi\)
\(434\) 0 0
\(435\) 14.5970 0.699874
\(436\) 12.7556 0.610881
\(437\) 28.0400 1.34133
\(438\) −8.83314 −0.422064
\(439\) 10.1619 0.485003 0.242501 0.970151i \(-0.422032\pi\)
0.242501 + 0.970151i \(0.422032\pi\)
\(440\) −20.9906 −1.00069
\(441\) 0 0
\(442\) 0 0
\(443\) 3.20787 0.152410 0.0762052 0.997092i \(-0.475720\pi\)
0.0762052 + 0.997092i \(0.475720\pi\)
\(444\) 13.2859 0.630522
\(445\) 17.2351 0.817020
\(446\) −7.32540 −0.346868
\(447\) 32.1891 1.52249
\(448\) 0 0
\(449\) 18.4099 0.868817 0.434409 0.900716i \(-0.356957\pi\)
0.434409 + 0.900716i \(0.356957\pi\)
\(450\) −6.99063 −0.329542
\(451\) −2.03164 −0.0956663
\(452\) 12.9131 0.607379
\(453\) 44.2306 2.07814
\(454\) −10.3684 −0.486614
\(455\) 0 0
\(456\) −45.9595 −2.15225
\(457\) −3.40297 −0.159184 −0.0795922 0.996828i \(-0.525362\pi\)
−0.0795922 + 0.996828i \(0.525362\pi\)
\(458\) 4.50177 0.210354
\(459\) 8.72393 0.407198
\(460\) 16.0874 0.750080
\(461\) 17.5714 0.818380 0.409190 0.912449i \(-0.365811\pi\)
0.409190 + 0.912449i \(0.365811\pi\)
\(462\) 0 0
\(463\) −15.7714 −0.732959 −0.366479 0.930426i \(-0.619437\pi\)
−0.366479 + 0.930426i \(0.619437\pi\)
\(464\) 2.82564 0.131177
\(465\) 41.5417 1.92645
\(466\) 19.2607 0.892236
\(467\) 3.76694 0.174313 0.0871567 0.996195i \(-0.472222\pi\)
0.0871567 + 0.996195i \(0.472222\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4.16839 −0.192273
\(471\) −16.3827 −0.754875
\(472\) 17.8064 0.819607
\(473\) −23.6795 −1.08879
\(474\) 6.83314 0.313857
\(475\) −45.5669 −2.09075
\(476\) 0 0
\(477\) 4.80642 0.220071
\(478\) −13.4893 −0.616988
\(479\) −12.1032 −0.553011 −0.276506 0.961012i \(-0.589176\pi\)
−0.276506 + 0.961012i \(0.589176\pi\)
\(480\) −41.3274 −1.88633
\(481\) 0 0
\(482\) 9.20189 0.419135
\(483\) 0 0
\(484\) 5.75065 0.261393
\(485\) 30.9496 1.40535
\(486\) 11.8938 0.539516
\(487\) 17.3778 0.787463 0.393731 0.919226i \(-0.371184\pi\)
0.393731 + 0.919226i \(0.371184\pi\)
\(488\) −21.9813 −0.995045
\(489\) −5.15257 −0.233007
\(490\) 0 0
\(491\) −19.3921 −0.875152 −0.437576 0.899181i \(-0.644163\pi\)
−0.437576 + 0.899181i \(0.644163\pi\)
\(492\) −2.55215 −0.115060
\(493\) −7.36689 −0.331788
\(494\) 0 0
\(495\) −16.4494 −0.739345
\(496\) 8.04149 0.361073
\(497\) 0 0
\(498\) 16.5161 0.740102
\(499\) 10.6702 0.477662 0.238831 0.971061i \(-0.423236\pi\)
0.238831 + 0.971061i \(0.423236\pi\)
\(500\) −1.62714 −0.0727678
\(501\) −7.57781 −0.338552
\(502\) 2.27115 0.101366
\(503\) 4.27655 0.190682 0.0953410 0.995445i \(-0.469606\pi\)
0.0953410 + 0.995445i \(0.469606\pi\)
\(504\) 0 0
\(505\) −44.0163 −1.95870
\(506\) −6.07758 −0.270181
\(507\) 0 0
\(508\) −6.23506 −0.276636
\(509\) 33.0765 1.46609 0.733046 0.680180i \(-0.238098\pi\)
0.733046 + 0.680180i \(0.238098\pi\)
\(510\) 17.6128 0.779910
\(511\) 0 0
\(512\) −14.6923 −0.649313
\(513\) 20.7556 0.916381
\(514\) −16.3225 −0.719954
\(515\) −39.5210 −1.74150
\(516\) −29.7462 −1.30950
\(517\) −5.06175 −0.222616
\(518\) 0 0
\(519\) −18.3225 −0.804268
\(520\) 0 0
\(521\) −24.3783 −1.06803 −0.534015 0.845475i \(-0.679318\pi\)
−0.534015 + 0.845475i \(0.679318\pi\)
\(522\) −2.68889 −0.117690
\(523\) −34.9403 −1.52783 −0.763915 0.645317i \(-0.776725\pi\)
−0.763915 + 0.645317i \(0.776725\pi\)
\(524\) 9.06070 0.395818
\(525\) 0 0
\(526\) −6.88247 −0.300090
\(527\) −20.9654 −0.913269
\(528\) −8.20342 −0.357008
\(529\) −12.2351 −0.531959
\(530\) −5.59210 −0.242905
\(531\) 13.9541 0.605555
\(532\) 0 0
\(533\) 0 0
\(534\) −8.17929 −0.353952
\(535\) −58.4657 −2.52769
\(536\) −1.04101 −0.0449648
\(537\) −35.9289 −1.55045
\(538\) −12.5402 −0.540646
\(539\) 0 0
\(540\) 11.9081 0.512445
\(541\) 30.4953 1.31110 0.655548 0.755153i \(-0.272438\pi\)
0.655548 + 0.755153i \(0.272438\pi\)
\(542\) 16.6539 0.715344
\(543\) 20.3827 0.874706
\(544\) 20.8573 0.894248
\(545\) −26.8780 −1.15133
\(546\) 0 0
\(547\) −10.0049 −0.427780 −0.213890 0.976858i \(-0.568613\pi\)
−0.213890 + 0.976858i \(0.568613\pi\)
\(548\) 24.9590 1.06620
\(549\) −17.2257 −0.735175
\(550\) 9.87649 0.421135
\(551\) −17.5270 −0.746674
\(552\) −17.6445 −0.750999
\(553\) 0 0
\(554\) −1.16791 −0.0496198
\(555\) −27.9956 −1.18835
\(556\) 4.62222 0.196026
\(557\) −14.1936 −0.601401 −0.300701 0.953719i \(-0.597220\pi\)
−0.300701 + 0.953719i \(0.597220\pi\)
\(558\) −7.65233 −0.323949
\(559\) 0 0
\(560\) 0 0
\(561\) 21.3876 0.902986
\(562\) 8.02413 0.338478
\(563\) 21.3590 0.900177 0.450088 0.892984i \(-0.351393\pi\)
0.450088 + 0.892984i \(0.351393\pi\)
\(564\) −6.35857 −0.267744
\(565\) −27.2099 −1.14473
\(566\) 8.35857 0.351337
\(567\) 0 0
\(568\) −21.8193 −0.915519
\(569\) 34.2672 1.43656 0.718278 0.695757i \(-0.244931\pi\)
0.718278 + 0.695757i \(0.244931\pi\)
\(570\) 41.9037 1.75515
\(571\) −37.6494 −1.57558 −0.787789 0.615945i \(-0.788774\pi\)
−0.787789 + 0.615945i \(0.788774\pi\)
\(572\) 0 0
\(573\) −14.7654 −0.616834
\(574\) 0 0
\(575\) −17.4938 −0.729541
\(576\) 2.36842 0.0986840
\(577\) 28.3970 1.18218 0.591091 0.806605i \(-0.298697\pi\)
0.591091 + 0.806605i \(0.298697\pi\)
\(578\) 2.82225 0.117390
\(579\) −45.6227 −1.89601
\(580\) −10.0558 −0.417543
\(581\) 0 0
\(582\) −14.6878 −0.608830
\(583\) −6.79060 −0.281238
\(584\) 14.0633 0.581943
\(585\) 0 0
\(586\) −7.89045 −0.325952
\(587\) −6.23659 −0.257412 −0.128706 0.991683i \(-0.541082\pi\)
−0.128706 + 0.991683i \(0.541082\pi\)
\(588\) 0 0
\(589\) −49.8800 −2.05527
\(590\) −16.2351 −0.668387
\(591\) 18.5303 0.762237
\(592\) −5.41927 −0.222731
\(593\) 17.5698 0.721506 0.360753 0.932661i \(-0.382520\pi\)
0.360753 + 0.932661i \(0.382520\pi\)
\(594\) −4.49871 −0.184584
\(595\) 0 0
\(596\) −22.1748 −0.908317
\(597\) 1.33185 0.0545090
\(598\) 0 0
\(599\) 16.9813 0.693836 0.346918 0.937896i \(-0.387228\pi\)
0.346918 + 0.937896i \(0.387228\pi\)
\(600\) 28.6735 1.17059
\(601\) 18.7052 0.763001 0.381500 0.924369i \(-0.375408\pi\)
0.381500 + 0.924369i \(0.375408\pi\)
\(602\) 0 0
\(603\) −0.815792 −0.0332216
\(604\) −30.4701 −1.23981
\(605\) −12.1175 −0.492648
\(606\) 20.8889 0.848554
\(607\) 10.3575 0.420399 0.210199 0.977659i \(-0.432589\pi\)
0.210199 + 0.977659i \(0.432589\pi\)
\(608\) 49.6227 2.01247
\(609\) 0 0
\(610\) 20.0415 0.811456
\(611\) 0 0
\(612\) 10.4286 0.421553
\(613\) 7.02227 0.283627 0.141814 0.989893i \(-0.454707\pi\)
0.141814 + 0.989893i \(0.454707\pi\)
\(614\) −2.73483 −0.110369
\(615\) 5.37778 0.216853
\(616\) 0 0
\(617\) −29.9813 −1.20700 −0.603500 0.797363i \(-0.706228\pi\)
−0.603500 + 0.797363i \(0.706228\pi\)
\(618\) 18.7556 0.754460
\(619\) 0.0285802 0.00114874 0.000574368 1.00000i \(-0.499817\pi\)
0.000574368 1.00000i \(0.499817\pi\)
\(620\) −28.6178 −1.14932
\(621\) 7.96836 0.319759
\(622\) 18.8399 0.755412
\(623\) 0 0
\(624\) 0 0
\(625\) −23.2306 −0.929225
\(626\) −13.1601 −0.525982
\(627\) 50.8845 2.03213
\(628\) 11.2859 0.450357
\(629\) 14.1289 0.563356
\(630\) 0 0
\(631\) 12.1936 0.485419 0.242709 0.970099i \(-0.421964\pi\)
0.242709 + 0.970099i \(0.421964\pi\)
\(632\) −10.8791 −0.432746
\(633\) −4.21432 −0.167504
\(634\) 6.82564 0.271081
\(635\) 13.1383 0.521377
\(636\) −8.53035 −0.338250
\(637\) 0 0
\(638\) 3.79892 0.150401
\(639\) −17.0988 −0.676418
\(640\) 34.5718 1.36657
\(641\) −2.82516 −0.111587 −0.0557935 0.998442i \(-0.517769\pi\)
−0.0557935 + 0.998442i \(0.517769\pi\)
\(642\) 27.7462 1.09506
\(643\) −37.7275 −1.48783 −0.743913 0.668276i \(-0.767032\pi\)
−0.743913 + 0.668276i \(0.767032\pi\)
\(644\) 0 0
\(645\) 62.6800 2.46802
\(646\) −21.1481 −0.832062
\(647\) −12.3664 −0.486174 −0.243087 0.970005i \(-0.578160\pi\)
−0.243087 + 0.970005i \(0.578160\pi\)
\(648\) −26.9273 −1.05781
\(649\) −19.7146 −0.773864
\(650\) 0 0
\(651\) 0 0
\(652\) 3.54956 0.139012
\(653\) −33.0005 −1.29141 −0.645704 0.763588i \(-0.723436\pi\)
−0.645704 + 0.763588i \(0.723436\pi\)
\(654\) 12.7556 0.498782
\(655\) −19.0923 −0.746000
\(656\) 1.04101 0.0406446
\(657\) 11.0207 0.429960
\(658\) 0 0
\(659\) 32.8118 1.27817 0.639084 0.769137i \(-0.279314\pi\)
0.639084 + 0.769137i \(0.279314\pi\)
\(660\) 29.1941 1.13638
\(661\) −14.7067 −0.572025 −0.286013 0.958226i \(-0.592330\pi\)
−0.286013 + 0.958226i \(0.592330\pi\)
\(662\) 16.1334 0.627041
\(663\) 0 0
\(664\) −26.2953 −1.02046
\(665\) 0 0
\(666\) 5.15701 0.199830
\(667\) −6.72885 −0.260542
\(668\) 5.22030 0.201979
\(669\) 23.5462 0.910348
\(670\) 0.949145 0.0366687
\(671\) 24.3368 0.939511
\(672\) 0 0
\(673\) 21.2908 0.820702 0.410351 0.911928i \(-0.365406\pi\)
0.410351 + 0.911928i \(0.365406\pi\)
\(674\) 5.17775 0.199440
\(675\) −12.9491 −0.498413
\(676\) 0 0
\(677\) 3.07160 0.118051 0.0590256 0.998256i \(-0.481201\pi\)
0.0590256 + 0.998256i \(0.481201\pi\)
\(678\) 12.9131 0.495923
\(679\) 0 0
\(680\) −28.0415 −1.07534
\(681\) 33.3274 1.27711
\(682\) 10.8113 0.413988
\(683\) 24.7971 0.948833 0.474416 0.880301i \(-0.342659\pi\)
0.474416 + 0.880301i \(0.342659\pi\)
\(684\) 24.8113 0.948686
\(685\) −52.5926 −2.00946
\(686\) 0 0
\(687\) −14.4701 −0.552070
\(688\) 12.1334 0.462580
\(689\) 0 0
\(690\) 16.0874 0.612438
\(691\) −27.0953 −1.03075 −0.515376 0.856964i \(-0.672348\pi\)
−0.515376 + 0.856964i \(0.672348\pi\)
\(692\) 12.6222 0.479825
\(693\) 0 0
\(694\) −3.88845 −0.147603
\(695\) −9.73975 −0.369450
\(696\) 11.0291 0.418055
\(697\) −2.71408 −0.102803
\(698\) −16.6430 −0.629945
\(699\) −61.9101 −2.34166
\(700\) 0 0
\(701\) 13.5205 0.510662 0.255331 0.966854i \(-0.417815\pi\)
0.255331 + 0.966854i \(0.417815\pi\)
\(702\) 0 0
\(703\) 33.6149 1.26781
\(704\) −3.34614 −0.126112
\(705\) 13.3985 0.504618
\(706\) −24.8889 −0.936707
\(707\) 0 0
\(708\) −24.7654 −0.930741
\(709\) 50.9753 1.91442 0.957209 0.289399i \(-0.0934554\pi\)
0.957209 + 0.289399i \(0.0934554\pi\)
\(710\) 19.8938 0.746603
\(711\) −8.52543 −0.319729
\(712\) 13.0223 0.488030
\(713\) −19.1497 −0.717160
\(714\) 0 0
\(715\) 0 0
\(716\) 24.7511 0.924993
\(717\) 43.3590 1.61927
\(718\) 11.9956 0.447670
\(719\) −41.5417 −1.54924 −0.774622 0.632424i \(-0.782060\pi\)
−0.774622 + 0.632424i \(0.782060\pi\)
\(720\) 8.42864 0.314117
\(721\) 0 0
\(722\) −37.2257 −1.38540
\(723\) −29.5778 −1.10001
\(724\) −14.0415 −0.521848
\(725\) 10.9349 0.406110
\(726\) 5.75065 0.213427
\(727\) 20.8988 0.775092 0.387546 0.921850i \(-0.373323\pi\)
0.387546 + 0.921850i \(0.373323\pi\)
\(728\) 0 0
\(729\) −4.96836 −0.184013
\(730\) −12.8222 −0.474573
\(731\) −31.6336 −1.17001
\(732\) 30.5718 1.12997
\(733\) 19.3575 0.714986 0.357493 0.933916i \(-0.383632\pi\)
0.357493 + 0.933916i \(0.383632\pi\)
\(734\) 2.02413 0.0747122
\(735\) 0 0
\(736\) 19.0509 0.702224
\(737\) 1.15257 0.0424553
\(738\) −0.990632 −0.0364657
\(739\) 1.31111 0.0482299 0.0241149 0.999709i \(-0.492323\pi\)
0.0241149 + 0.999709i \(0.492323\pi\)
\(740\) 19.2859 0.708964
\(741\) 0 0
\(742\) 0 0
\(743\) 21.5210 0.789528 0.394764 0.918783i \(-0.370826\pi\)
0.394764 + 0.918783i \(0.370826\pi\)
\(744\) 31.3876 1.15073
\(745\) 46.7259 1.71191
\(746\) 12.9304 0.473416
\(747\) −20.6064 −0.753949
\(748\) −14.7338 −0.538720
\(749\) 0 0
\(750\) −1.62714 −0.0594147
\(751\) −16.3176 −0.595436 −0.297718 0.954654i \(-0.596226\pi\)
−0.297718 + 0.954654i \(0.596226\pi\)
\(752\) 2.59364 0.0945802
\(753\) −7.30021 −0.266034
\(754\) 0 0
\(755\) 64.2054 2.33667
\(756\) 0 0
\(757\) −18.7462 −0.681342 −0.340671 0.940183i \(-0.610654\pi\)
−0.340671 + 0.940183i \(0.610654\pi\)
\(758\) 16.3194 0.592748
\(759\) 19.5353 0.709085
\(760\) −66.7150 −2.42001
\(761\) −10.2968 −0.373259 −0.186630 0.982430i \(-0.559756\pi\)
−0.186630 + 0.982430i \(0.559756\pi\)
\(762\) −6.23506 −0.225873
\(763\) 0 0
\(764\) 10.1718 0.368002
\(765\) −21.9748 −0.794501
\(766\) −21.7778 −0.786865
\(767\) 0 0
\(768\) −21.9180 −0.790897
\(769\) 9.36641 0.337761 0.168881 0.985637i \(-0.445985\pi\)
0.168881 + 0.985637i \(0.445985\pi\)
\(770\) 0 0
\(771\) 52.4657 1.88951
\(772\) 31.4291 1.13116
\(773\) −3.71456 −0.133603 −0.0668017 0.997766i \(-0.521279\pi\)
−0.0668017 + 0.997766i \(0.521279\pi\)
\(774\) −11.5462 −0.415019
\(775\) 31.1195 1.11785
\(776\) 23.3846 0.839457
\(777\) 0 0
\(778\) 13.8163 0.495337
\(779\) −6.45722 −0.231354
\(780\) 0 0
\(781\) 24.1575 0.864423
\(782\) −8.11906 −0.290337
\(783\) −4.98079 −0.177999
\(784\) 0 0
\(785\) −23.7812 −0.848789
\(786\) 9.06070 0.323184
\(787\) 6.23659 0.222311 0.111155 0.993803i \(-0.464545\pi\)
0.111155 + 0.993803i \(0.464545\pi\)
\(788\) −12.7654 −0.454749
\(789\) 22.1225 0.787580
\(790\) 9.91903 0.352903
\(791\) 0 0
\(792\) −12.4286 −0.441632
\(793\) 0 0
\(794\) −15.7571 −0.559199
\(795\) 17.9748 0.637501
\(796\) −0.917502 −0.0325200
\(797\) 40.0701 1.41935 0.709677 0.704527i \(-0.248841\pi\)
0.709677 + 0.704527i \(0.248841\pi\)
\(798\) 0 0
\(799\) −6.76202 −0.239223
\(800\) −30.9590 −1.09457
\(801\) 10.2050 0.360574
\(802\) −3.86665 −0.136536
\(803\) −15.5703 −0.549464
\(804\) 1.44785 0.0510618
\(805\) 0 0
\(806\) 0 0
\(807\) 40.3082 1.41892
\(808\) −33.2573 −1.16999
\(809\) −2.57136 −0.0904042 −0.0452021 0.998978i \(-0.514393\pi\)
−0.0452021 + 0.998978i \(0.514393\pi\)
\(810\) 24.5511 0.862637
\(811\) −28.3654 −0.996042 −0.498021 0.867165i \(-0.665940\pi\)
−0.498021 + 0.867165i \(0.665940\pi\)
\(812\) 0 0
\(813\) −53.5308 −1.87741
\(814\) −7.28592 −0.255371
\(815\) −7.47949 −0.261995
\(816\) −10.9590 −0.383641
\(817\) −75.2612 −2.63306
\(818\) 17.9922 0.629081
\(819\) 0 0
\(820\) −3.70471 −0.129374
\(821\) 31.7846 1.10929 0.554646 0.832087i \(-0.312854\pi\)
0.554646 + 0.832087i \(0.312854\pi\)
\(822\) 24.9590 0.870545
\(823\) 30.9131 1.07756 0.538781 0.842446i \(-0.318885\pi\)
0.538781 + 0.842446i \(0.318885\pi\)
\(824\) −29.8608 −1.04025
\(825\) −31.7462 −1.10526
\(826\) 0 0
\(827\) −49.7560 −1.73019 −0.865094 0.501610i \(-0.832741\pi\)
−0.865094 + 0.501610i \(0.832741\pi\)
\(828\) 9.52543 0.331031
\(829\) 19.9190 0.691817 0.345908 0.938268i \(-0.387571\pi\)
0.345908 + 0.938268i \(0.387571\pi\)
\(830\) 23.9748 0.832178
\(831\) 3.75404 0.130226
\(832\) 0 0
\(833\) 0 0
\(834\) 4.62222 0.160054
\(835\) −11.0000 −0.380671
\(836\) −35.0539 −1.21237
\(837\) −14.1748 −0.489954
\(838\) −17.3921 −0.600799
\(839\) 11.5625 0.399181 0.199590 0.979879i \(-0.436039\pi\)
0.199590 + 0.979879i \(0.436039\pi\)
\(840\) 0 0
\(841\) −24.7940 −0.854965
\(842\) −5.66862 −0.195354
\(843\) −25.7921 −0.888328
\(844\) 2.90321 0.0999327
\(845\) 0 0
\(846\) −2.46812 −0.0848557
\(847\) 0 0
\(848\) 3.47949 0.119486
\(849\) −26.8671 −0.922077
\(850\) 13.1941 0.452552
\(851\) 12.9052 0.442385
\(852\) 30.3466 1.03966
\(853\) 1.74467 0.0597363 0.0298682 0.999554i \(-0.490491\pi\)
0.0298682 + 0.999554i \(0.490491\pi\)
\(854\) 0 0
\(855\) −52.2815 −1.78799
\(856\) −44.1748 −1.50986
\(857\) 42.3368 1.44620 0.723098 0.690745i \(-0.242717\pi\)
0.723098 + 0.690745i \(0.242717\pi\)
\(858\) 0 0
\(859\) 5.37778 0.183488 0.0917438 0.995783i \(-0.470756\pi\)
0.0917438 + 0.995783i \(0.470756\pi\)
\(860\) −43.1798 −1.47242
\(861\) 0 0
\(862\) −0.990632 −0.0337411
\(863\) −15.4291 −0.525213 −0.262607 0.964903i \(-0.584582\pi\)
−0.262607 + 0.964903i \(0.584582\pi\)
\(864\) 14.1017 0.479750
\(865\) −26.5970 −0.904326
\(866\) −11.3679 −0.386298
\(867\) −9.07160 −0.308088
\(868\) 0 0
\(869\) 12.0449 0.408595
\(870\) −10.0558 −0.340923
\(871\) 0 0
\(872\) −20.3082 −0.687722
\(873\) 18.3254 0.620221
\(874\) −19.3165 −0.653391
\(875\) 0 0
\(876\) −19.5594 −0.660851
\(877\) −38.5181 −1.30066 −0.650331 0.759651i \(-0.725370\pi\)
−0.650331 + 0.759651i \(0.725370\pi\)
\(878\) −7.00048 −0.236255
\(879\) 25.3624 0.855454
\(880\) −11.9081 −0.401423
\(881\) 41.7373 1.40617 0.703083 0.711108i \(-0.251806\pi\)
0.703083 + 0.711108i \(0.251806\pi\)
\(882\) 0 0
\(883\) 52.8439 1.77834 0.889170 0.457577i \(-0.151283\pi\)
0.889170 + 0.457577i \(0.151283\pi\)
\(884\) 0 0
\(885\) 52.1847 1.75417
\(886\) −2.20987 −0.0742422
\(887\) −8.99063 −0.301876 −0.150938 0.988543i \(-0.548229\pi\)
−0.150938 + 0.988543i \(0.548229\pi\)
\(888\) −21.1526 −0.709834
\(889\) 0 0
\(890\) −11.8731 −0.397987
\(891\) 29.8129 0.998769
\(892\) −16.2208 −0.543112
\(893\) −16.0879 −0.538361
\(894\) −22.1748 −0.741638
\(895\) −52.1546 −1.74334
\(896\) 0 0
\(897\) 0 0
\(898\) −12.6824 −0.423218
\(899\) 11.9699 0.399218
\(900\) −15.4795 −0.515983
\(901\) −9.07160 −0.302219
\(902\) 1.39958 0.0466010
\(903\) 0 0
\(904\) −20.5589 −0.683780
\(905\) 29.5877 0.983527
\(906\) −30.4701 −1.01230
\(907\) −20.7003 −0.687341 −0.343671 0.939090i \(-0.611670\pi\)
−0.343671 + 0.939090i \(0.611670\pi\)
\(908\) −22.9590 −0.761921
\(909\) −26.0622 −0.864430
\(910\) 0 0
\(911\) 29.8988 0.990590 0.495295 0.868725i \(-0.335060\pi\)
0.495295 + 0.868725i \(0.335060\pi\)
\(912\) −26.0731 −0.863368
\(913\) 29.1131 0.963503
\(914\) 2.34428 0.0775420
\(915\) −64.4197 −2.12965
\(916\) 9.96836 0.329364
\(917\) 0 0
\(918\) −6.00984 −0.198354
\(919\) 38.5847 1.27279 0.636397 0.771362i \(-0.280424\pi\)
0.636397 + 0.771362i \(0.280424\pi\)
\(920\) −25.6128 −0.844431
\(921\) 8.79060 0.289660
\(922\) −12.1048 −0.398649
\(923\) 0 0
\(924\) 0 0
\(925\) −20.9719 −0.689552
\(926\) 10.8648 0.357039
\(927\) −23.4005 −0.768574
\(928\) −11.9081 −0.390904
\(929\) 3.25581 0.106820 0.0534098 0.998573i \(-0.482991\pi\)
0.0534098 + 0.998573i \(0.482991\pi\)
\(930\) −28.6178 −0.938414
\(931\) 0 0
\(932\) 42.6494 1.39703
\(933\) −60.5575 −1.98257
\(934\) −2.59502 −0.0849116
\(935\) 31.0464 1.01533
\(936\) 0 0
\(937\) 11.6840 0.381699 0.190849 0.981619i \(-0.438876\pi\)
0.190849 + 0.981619i \(0.438876\pi\)
\(938\) 0 0
\(939\) 42.3007 1.38043
\(940\) −9.23014 −0.301054
\(941\) −41.9699 −1.36818 −0.684090 0.729398i \(-0.739800\pi\)
−0.684090 + 0.729398i \(0.739800\pi\)
\(942\) 11.2859 0.367715
\(943\) −2.47902 −0.0807279
\(944\) 10.1017 0.328783
\(945\) 0 0
\(946\) 16.3126 0.530370
\(947\) 20.3555 0.661465 0.330733 0.943725i \(-0.392704\pi\)
0.330733 + 0.943725i \(0.392704\pi\)
\(948\) 15.1308 0.491424
\(949\) 0 0
\(950\) 31.3907 1.01845
\(951\) −21.9398 −0.711446
\(952\) 0 0
\(953\) −30.5496 −0.989597 −0.494799 0.869008i \(-0.664758\pi\)
−0.494799 + 0.869008i \(0.664758\pi\)
\(954\) −3.31111 −0.107201
\(955\) −21.4336 −0.693574
\(956\) −29.8697 −0.966055
\(957\) −12.2109 −0.394723
\(958\) 8.33783 0.269383
\(959\) 0 0
\(960\) 8.85728 0.285867
\(961\) 3.06515 0.0988757
\(962\) 0 0
\(963\) −34.6178 −1.11554
\(964\) 20.3759 0.656264
\(965\) −66.2262 −2.13190
\(966\) 0 0
\(967\) −54.4548 −1.75115 −0.875574 0.483084i \(-0.839516\pi\)
−0.875574 + 0.483084i \(0.839516\pi\)
\(968\) −9.15563 −0.294273
\(969\) 67.9768 2.18373
\(970\) −21.3210 −0.684575
\(971\) 54.6035 1.75231 0.876155 0.482030i \(-0.160101\pi\)
0.876155 + 0.482030i \(0.160101\pi\)
\(972\) 26.3368 0.844752
\(973\) 0 0
\(974\) −11.9714 −0.383589
\(975\) 0 0
\(976\) −12.4701 −0.399159
\(977\) −32.6474 −1.04448 −0.522242 0.852798i \(-0.674904\pi\)
−0.522242 + 0.852798i \(0.674904\pi\)
\(978\) 3.54956 0.113502
\(979\) −14.4177 −0.460793
\(980\) 0 0
\(981\) −15.9146 −0.508114
\(982\) 13.3590 0.426304
\(983\) −27.5052 −0.877278 −0.438639 0.898663i \(-0.644539\pi\)
−0.438639 + 0.898663i \(0.644539\pi\)
\(984\) 4.06329 0.129533
\(985\) 26.8988 0.857066
\(986\) 5.07499 0.161621
\(987\) 0 0
\(988\) 0 0
\(989\) −28.8938 −0.918771
\(990\) 11.3319 0.360150
\(991\) −6.29390 −0.199932 −0.0999662 0.994991i \(-0.531873\pi\)
−0.0999662 + 0.994991i \(0.531873\pi\)
\(992\) −33.8894 −1.07599
\(993\) −51.8578 −1.64566
\(994\) 0 0
\(995\) 1.93332 0.0612905
\(996\) 36.5718 1.15882
\(997\) 20.8702 0.660965 0.330483 0.943812i \(-0.392788\pi\)
0.330483 + 0.943812i \(0.392788\pi\)
\(998\) −7.35059 −0.232679
\(999\) 9.55262 0.302232
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 8281.2.a.be.1.2 3
7.6 odd 2 1183.2.a.h.1.2 3
13.5 odd 4 637.2.c.d.246.4 6
13.8 odd 4 637.2.c.d.246.3 6
13.12 even 2 8281.2.a.bi.1.2 3
91.5 even 12 637.2.r.e.116.4 12
91.18 odd 12 637.2.r.d.324.3 12
91.31 even 12 637.2.r.e.324.3 12
91.34 even 4 91.2.c.a.64.3 6
91.44 odd 12 637.2.r.d.116.4 12
91.47 even 12 637.2.r.e.116.3 12
91.60 odd 12 637.2.r.d.324.4 12
91.73 even 12 637.2.r.e.324.4 12
91.83 even 4 91.2.c.a.64.4 yes 6
91.86 odd 12 637.2.r.d.116.3 12
91.90 odd 2 1183.2.a.j.1.2 3
273.83 odd 4 819.2.c.b.64.3 6
273.125 odd 4 819.2.c.b.64.4 6
364.83 odd 4 1456.2.k.c.337.6 6
364.307 odd 4 1456.2.k.c.337.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.c.a.64.3 6 91.34 even 4
91.2.c.a.64.4 yes 6 91.83 even 4
637.2.c.d.246.3 6 13.8 odd 4
637.2.c.d.246.4 6 13.5 odd 4
637.2.r.d.116.3 12 91.86 odd 12
637.2.r.d.116.4 12 91.44 odd 12
637.2.r.d.324.3 12 91.18 odd 12
637.2.r.d.324.4 12 91.60 odd 12
637.2.r.e.116.3 12 91.47 even 12
637.2.r.e.116.4 12 91.5 even 12
637.2.r.e.324.3 12 91.31 even 12
637.2.r.e.324.4 12 91.73 even 12
819.2.c.b.64.3 6 273.83 odd 4
819.2.c.b.64.4 6 273.125 odd 4
1183.2.a.h.1.2 3 7.6 odd 2
1183.2.a.j.1.2 3 91.90 odd 2
1456.2.k.c.337.5 6 364.307 odd 4
1456.2.k.c.337.6 6 364.83 odd 4
8281.2.a.be.1.2 3 1.1 even 1 trivial
8281.2.a.bi.1.2 3 13.12 even 2