Properties

Label 2541.2.a.bn.1.2
Level $2541$
Weight $2$
Character 2541.1
Self dual yes
Analytic conductor $20.290$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2541,2,Mod(1,2541)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2541, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2541.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2541.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: \(20.2899871536\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.35567\) of defining polynomial
Character \(\chi\) \(=\) 2541.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.162147 q^{2} -1.00000 q^{3} -1.97371 q^{4} -4.38705 q^{5} -0.162147 q^{6} -1.00000 q^{7} -0.644326 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.162147 q^{2} -1.00000 q^{3} -1.97371 q^{4} -4.38705 q^{5} -0.162147 q^{6} -1.00000 q^{7} -0.644326 q^{8} +1.00000 q^{9} -0.711349 q^{10} +1.97371 q^{12} +1.67571 q^{13} -0.162147 q^{14} +4.38705 q^{15} +3.84294 q^{16} +6.18663 q^{17} +0.162147 q^{18} -3.47528 q^{19} +8.65877 q^{20} +1.00000 q^{21} +0.568595 q^{23} +0.644326 q^{24} +14.2462 q^{25} +0.271711 q^{26} -1.00000 q^{27} +1.97371 q^{28} -8.86742 q^{29} +0.711349 q^{30} +4.33447 q^{31} +1.91177 q^{32} +1.00314 q^{34} +4.38705 q^{35} -1.97371 q^{36} +0.969445 q^{37} -0.563507 q^{38} -1.67571 q^{39} +2.82669 q^{40} +5.77725 q^{41} +0.162147 q^{42} -5.04388 q^{43} -4.38705 q^{45} +0.0921961 q^{46} +4.67256 q^{47} -3.84294 q^{48} +1.00000 q^{49} +2.30999 q^{50} -6.18663 q^{51} -3.30735 q^{52} -2.37882 q^{53} -0.162147 q^{54} +0.644326 q^{56} +3.47528 q^{57} -1.43783 q^{58} -2.84901 q^{59} -8.65877 q^{60} +9.29883 q^{61} +0.702822 q^{62} -1.00000 q^{63} -7.37589 q^{64} -7.35141 q^{65} -7.14275 q^{67} -12.2106 q^{68} -0.568595 q^{69} +0.711349 q^{70} -0.794487 q^{71} -0.644326 q^{72} +2.38705 q^{73} +0.157193 q^{74} -14.2462 q^{75} +6.85919 q^{76} -0.271711 q^{78} -0.670617 q^{79} -16.8592 q^{80} +1.00000 q^{81} +0.936765 q^{82} -9.28503 q^{83} -1.97371 q^{84} -27.1411 q^{85} -0.817850 q^{86} +8.86742 q^{87} -12.2106 q^{89} -0.711349 q^{90} -1.67571 q^{91} -1.12224 q^{92} -4.33447 q^{93} +0.757643 q^{94} +15.2462 q^{95} -1.91177 q^{96} +15.0596 q^{97} +0.162147 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{5} - q^{6} - 4 q^{7} - 9 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{5} - q^{6} - 4 q^{7} - 9 q^{8} + 4 q^{9} + 10 q^{10} - 3 q^{12} + 6 q^{13} - q^{14} + 4 q^{15} - 3 q^{16} + 8 q^{17} + q^{18} - 10 q^{19} + 4 q^{21} - 10 q^{23} + 9 q^{24} + 12 q^{25} - 20 q^{26} - 4 q^{27} - 3 q^{28} - 10 q^{30} - 18 q^{31} - 2 q^{32} + 18 q^{34} + 4 q^{35} + 3 q^{36} - 2 q^{37} - 8 q^{38} - 6 q^{39} + 6 q^{40} + 10 q^{41} + q^{42} - 4 q^{43} - 4 q^{45} + 11 q^{46} + 4 q^{47} + 3 q^{48} + 4 q^{49} - 9 q^{50} - 8 q^{51} + 20 q^{52} - q^{54} + 9 q^{56} + 10 q^{57} + 14 q^{58} - 16 q^{59} + 14 q^{61} - 4 q^{63} - 11 q^{64} - 28 q^{65} - 28 q^{67} - 16 q^{68} + 10 q^{69} - 10 q^{70} - 18 q^{71} - 9 q^{72} - 4 q^{73} - 41 q^{74} - 12 q^{75} - 4 q^{76} + 20 q^{78} - 20 q^{79} - 36 q^{80} + 4 q^{81} - 24 q^{82} + 6 q^{83} + 3 q^{84} - 20 q^{85} - 20 q^{86} - 16 q^{89} + 10 q^{90} - 6 q^{91} - 22 q^{92} + 18 q^{93} - 16 q^{94} + 16 q^{95} + 2 q^{96} + 32 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.162147 0.114655 0.0573277 0.998355i \(-0.481742\pi\)
0.0573277 + 0.998355i \(0.481742\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.97371 −0.986854
\(5\) −4.38705 −1.96195 −0.980975 0.194133i \(-0.937811\pi\)
−0.980975 + 0.194133i \(0.937811\pi\)
\(6\) −0.162147 −0.0661963
\(7\) −1.00000 −0.377964
\(8\) −0.644326 −0.227804
\(9\) 1.00000 0.333333
\(10\) −0.711349 −0.224948
\(11\) 0 0
\(12\) 1.97371 0.569761
\(13\) 1.67571 0.464757 0.232379 0.972625i \(-0.425349\pi\)
0.232379 + 0.972625i \(0.425349\pi\)
\(14\) −0.162147 −0.0433357
\(15\) 4.38705 1.13273
\(16\) 3.84294 0.960735
\(17\) 6.18663 1.50048 0.750239 0.661167i \(-0.229938\pi\)
0.750239 + 0.661167i \(0.229938\pi\)
\(18\) 0.162147 0.0382185
\(19\) −3.47528 −0.797284 −0.398642 0.917107i \(-0.630518\pi\)
−0.398642 + 0.917107i \(0.630518\pi\)
\(20\) 8.65877 1.93616
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 0.568595 0.118560 0.0592801 0.998241i \(-0.481119\pi\)
0.0592801 + 0.998241i \(0.481119\pi\)
\(24\) 0.644326 0.131522
\(25\) 14.2462 2.84925
\(26\) 0.271711 0.0532869
\(27\) −1.00000 −0.192450
\(28\) 1.97371 0.372996
\(29\) −8.86742 −1.64664 −0.823320 0.567578i \(-0.807880\pi\)
−0.823320 + 0.567578i \(0.807880\pi\)
\(30\) 0.711349 0.129874
\(31\) 4.33447 0.778494 0.389247 0.921133i \(-0.372735\pi\)
0.389247 + 0.921133i \(0.372735\pi\)
\(32\) 1.91177 0.337957
\(33\) 0 0
\(34\) 1.00314 0.172038
\(35\) 4.38705 0.741547
\(36\) −1.97371 −0.328951
\(37\) 0.969445 0.159376 0.0796879 0.996820i \(-0.474608\pi\)
0.0796879 + 0.996820i \(0.474608\pi\)
\(38\) −0.563507 −0.0914129
\(39\) −1.67571 −0.268328
\(40\) 2.82669 0.446939
\(41\) 5.77725 0.902255 0.451128 0.892459i \(-0.351022\pi\)
0.451128 + 0.892459i \(0.351022\pi\)
\(42\) 0.162147 0.0250199
\(43\) −5.04388 −0.769184 −0.384592 0.923087i \(-0.625658\pi\)
−0.384592 + 0.923087i \(0.625658\pi\)
\(44\) 0 0
\(45\) −4.38705 −0.653983
\(46\) 0.0921961 0.0135936
\(47\) 4.67256 0.681563 0.340782 0.940143i \(-0.389308\pi\)
0.340782 + 0.940143i \(0.389308\pi\)
\(48\) −3.84294 −0.554681
\(49\) 1.00000 0.142857
\(50\) 2.30999 0.326682
\(51\) −6.18663 −0.866301
\(52\) −3.30735 −0.458647
\(53\) −2.37882 −0.326756 −0.163378 0.986564i \(-0.552239\pi\)
−0.163378 + 0.986564i \(0.552239\pi\)
\(54\) −0.162147 −0.0220654
\(55\) 0 0
\(56\) 0.644326 0.0861016
\(57\) 3.47528 0.460312
\(58\) −1.43783 −0.188796
\(59\) −2.84901 −0.370910 −0.185455 0.982653i \(-0.559376\pi\)
−0.185455 + 0.982653i \(0.559376\pi\)
\(60\) −8.65877 −1.11784
\(61\) 9.29883 1.19059 0.595296 0.803506i \(-0.297035\pi\)
0.595296 + 0.803506i \(0.297035\pi\)
\(62\) 0.702822 0.0892585
\(63\) −1.00000 −0.125988
\(64\) −7.37589 −0.921987
\(65\) −7.35141 −0.911830
\(66\) 0 0
\(67\) −7.14275 −0.872626 −0.436313 0.899795i \(-0.643716\pi\)
−0.436313 + 0.899795i \(0.643716\pi\)
\(68\) −12.2106 −1.48075
\(69\) −0.568595 −0.0684508
\(70\) 0.711349 0.0850224
\(71\) −0.794487 −0.0942882 −0.0471441 0.998888i \(-0.515012\pi\)
−0.0471441 + 0.998888i \(0.515012\pi\)
\(72\) −0.644326 −0.0759345
\(73\) 2.38705 0.279384 0.139692 0.990195i \(-0.455389\pi\)
0.139692 + 0.990195i \(0.455389\pi\)
\(74\) 0.157193 0.0182733
\(75\) −14.2462 −1.64501
\(76\) 6.85919 0.786803
\(77\) 0 0
\(78\) −0.271711 −0.0307652
\(79\) −0.670617 −0.0754504 −0.0377252 0.999288i \(-0.512011\pi\)
−0.0377252 + 0.999288i \(0.512011\pi\)
\(80\) −16.8592 −1.88491
\(81\) 1.00000 0.111111
\(82\) 0.936765 0.103448
\(83\) −9.28503 −1.01916 −0.509582 0.860422i \(-0.670200\pi\)
−0.509582 + 0.860422i \(0.670200\pi\)
\(84\) −1.97371 −0.215349
\(85\) −27.1411 −2.94386
\(86\) −0.817850 −0.0881911
\(87\) 8.86742 0.950688
\(88\) 0 0
\(89\) −12.2106 −1.29432 −0.647161 0.762354i \(-0.724044\pi\)
−0.647161 + 0.762354i \(0.724044\pi\)
\(90\) −0.711349 −0.0749827
\(91\) −1.67571 −0.175662
\(92\) −1.12224 −0.117002
\(93\) −4.33447 −0.449464
\(94\) 0.757643 0.0781449
\(95\) 15.2462 1.56423
\(96\) −1.91177 −0.195120
\(97\) 15.0596 1.52907 0.764536 0.644581i \(-0.222968\pi\)
0.764536 + 0.644581i \(0.222968\pi\)
\(98\) 0.162147 0.0163793
\(99\) 0 0
\(100\) −28.1179 −2.81179
\(101\) 5.76079 0.573220 0.286610 0.958047i \(-0.407472\pi\)
0.286610 + 0.958047i \(0.407472\pi\)
\(102\) −1.00314 −0.0993261
\(103\) 7.71449 0.760132 0.380066 0.924959i \(-0.375901\pi\)
0.380066 + 0.924959i \(0.375901\pi\)
\(104\) −1.07970 −0.105873
\(105\) −4.38705 −0.428133
\(106\) −0.385719 −0.0374644
\(107\) −16.4634 −1.59158 −0.795790 0.605573i \(-0.792944\pi\)
−0.795790 + 0.605573i \(0.792944\pi\)
\(108\) 1.97371 0.189920
\(109\) 9.74355 0.933263 0.466632 0.884452i \(-0.345467\pi\)
0.466632 + 0.884452i \(0.345467\pi\)
\(110\) 0 0
\(111\) −0.969445 −0.0920157
\(112\) −3.84294 −0.363124
\(113\) −2.85725 −0.268787 −0.134394 0.990928i \(-0.542909\pi\)
−0.134394 + 0.990928i \(0.542909\pi\)
\(114\) 0.563507 0.0527773
\(115\) −2.49446 −0.232609
\(116\) 17.5017 1.62499
\(117\) 1.67571 0.154919
\(118\) −0.461960 −0.0425268
\(119\) −6.18663 −0.567127
\(120\) −2.82669 −0.258040
\(121\) 0 0
\(122\) 1.50778 0.136508
\(123\) −5.77725 −0.520917
\(124\) −8.55498 −0.768260
\(125\) −40.5638 −3.62813
\(126\) −0.162147 −0.0144452
\(127\) −16.0514 −1.42433 −0.712165 0.702012i \(-0.752285\pi\)
−0.712165 + 0.702012i \(0.752285\pi\)
\(128\) −5.01953 −0.443668
\(129\) 5.04388 0.444089
\(130\) −1.19201 −0.104546
\(131\) 0.750136 0.0655397 0.0327699 0.999463i \(-0.489567\pi\)
0.0327699 + 0.999463i \(0.489567\pi\)
\(132\) 0 0
\(133\) 3.47528 0.301345
\(134\) −1.15818 −0.100051
\(135\) 4.38705 0.377578
\(136\) −3.98620 −0.341814
\(137\) −13.0082 −1.11137 −0.555684 0.831393i \(-0.687544\pi\)
−0.555684 + 0.831393i \(0.687544\pi\)
\(138\) −0.0921961 −0.00784826
\(139\) 3.90112 0.330889 0.165444 0.986219i \(-0.447094\pi\)
0.165444 + 0.986219i \(0.447094\pi\)
\(140\) −8.65877 −0.731799
\(141\) −4.67256 −0.393501
\(142\) −0.128824 −0.0108107
\(143\) 0 0
\(144\) 3.84294 0.320245
\(145\) 38.9019 3.23062
\(146\) 0.387054 0.0320328
\(147\) −1.00000 −0.0824786
\(148\) −1.91340 −0.157281
\(149\) 18.5434 1.51914 0.759568 0.650428i \(-0.225410\pi\)
0.759568 + 0.650428i \(0.225410\pi\)
\(150\) −2.30999 −0.188610
\(151\) 20.7979 1.69251 0.846255 0.532779i \(-0.178852\pi\)
0.846255 + 0.532779i \(0.178852\pi\)
\(152\) 2.23921 0.181624
\(153\) 6.18663 0.500159
\(154\) 0 0
\(155\) −19.0156 −1.52737
\(156\) 3.30735 0.264800
\(157\) 5.80975 0.463669 0.231834 0.972755i \(-0.425527\pi\)
0.231834 + 0.972755i \(0.425527\pi\)
\(158\) −0.108739 −0.00865079
\(159\) 2.37882 0.188653
\(160\) −8.38705 −0.663055
\(161\) −0.568595 −0.0448116
\(162\) 0.162147 0.0127395
\(163\) −8.90307 −0.697342 −0.348671 0.937245i \(-0.613367\pi\)
−0.348671 + 0.937245i \(0.613367\pi\)
\(164\) −11.4026 −0.890394
\(165\) 0 0
\(166\) −1.50554 −0.116853
\(167\) −3.82355 −0.295875 −0.147937 0.988997i \(-0.547263\pi\)
−0.147937 + 0.988997i \(0.547263\pi\)
\(168\) −0.644326 −0.0497108
\(169\) −10.1920 −0.784001
\(170\) −4.40085 −0.337530
\(171\) −3.47528 −0.265761
\(172\) 9.95514 0.759072
\(173\) 1.75061 0.133096 0.0665482 0.997783i \(-0.478801\pi\)
0.0665482 + 0.997783i \(0.478801\pi\)
\(174\) 1.43783 0.109001
\(175\) −14.2462 −1.07691
\(176\) 0 0
\(177\) 2.84901 0.214145
\(178\) −1.97991 −0.148401
\(179\) −12.8541 −0.960761 −0.480380 0.877060i \(-0.659501\pi\)
−0.480380 + 0.877060i \(0.659501\pi\)
\(180\) 8.65877 0.645386
\(181\) −22.0831 −1.64142 −0.820712 0.571342i \(-0.806423\pi\)
−0.820712 + 0.571342i \(0.806423\pi\)
\(182\) −0.271711 −0.0201406
\(183\) −9.29883 −0.687389
\(184\) −0.366361 −0.0270085
\(185\) −4.25301 −0.312687
\(186\) −0.702822 −0.0515334
\(187\) 0 0
\(188\) −9.22227 −0.672603
\(189\) 1.00000 0.0727393
\(190\) 2.47214 0.179348
\(191\) 14.7160 1.06481 0.532405 0.846490i \(-0.321288\pi\)
0.532405 + 0.846490i \(0.321288\pi\)
\(192\) 7.37589 0.532309
\(193\) 8.51287 0.612770 0.306385 0.951908i \(-0.400881\pi\)
0.306385 + 0.951908i \(0.400881\pi\)
\(194\) 2.44187 0.175316
\(195\) 7.35141 0.526445
\(196\) −1.97371 −0.140979
\(197\) −4.72273 −0.336480 −0.168240 0.985746i \(-0.553808\pi\)
−0.168240 + 0.985746i \(0.553808\pi\)
\(198\) 0 0
\(199\) 16.7620 1.18822 0.594112 0.804382i \(-0.297504\pi\)
0.594112 + 0.804382i \(0.297504\pi\)
\(200\) −9.17922 −0.649069
\(201\) 7.14275 0.503811
\(202\) 0.934096 0.0657227
\(203\) 8.86742 0.622371
\(204\) 12.2106 0.854913
\(205\) −25.3451 −1.77018
\(206\) 1.25088 0.0871532
\(207\) 0.568595 0.0395201
\(208\) 6.43964 0.446509
\(209\) 0 0
\(210\) −0.711349 −0.0490877
\(211\) −10.1454 −0.698440 −0.349220 0.937041i \(-0.613553\pi\)
−0.349220 + 0.937041i \(0.613553\pi\)
\(212\) 4.69510 0.322461
\(213\) 0.794487 0.0544373
\(214\) −2.66950 −0.182483
\(215\) 22.1278 1.50910
\(216\) 0.644326 0.0438408
\(217\) −4.33447 −0.294243
\(218\) 1.57989 0.107004
\(219\) −2.38705 −0.161302
\(220\) 0 0
\(221\) 10.3670 0.697358
\(222\) −0.157193 −0.0105501
\(223\) −6.11172 −0.409271 −0.204636 0.978838i \(-0.565601\pi\)
−0.204636 + 0.978838i \(0.565601\pi\)
\(224\) −1.91177 −0.127736
\(225\) 14.2462 0.949750
\(226\) −0.463295 −0.0308179
\(227\) 14.2090 0.943081 0.471541 0.881844i \(-0.343698\pi\)
0.471541 + 0.881844i \(0.343698\pi\)
\(228\) −6.85919 −0.454261
\(229\) −2.72781 −0.180259 −0.0901295 0.995930i \(-0.528728\pi\)
−0.0901295 + 0.995930i \(0.528728\pi\)
\(230\) −0.404469 −0.0266699
\(231\) 0 0
\(232\) 5.71351 0.375110
\(233\) −17.7897 −1.16544 −0.582720 0.812673i \(-0.698012\pi\)
−0.582720 + 0.812673i \(0.698012\pi\)
\(234\) 0.271711 0.0177623
\(235\) −20.4988 −1.33719
\(236\) 5.62312 0.366034
\(237\) 0.670617 0.0435613
\(238\) −1.00314 −0.0650242
\(239\) 12.7058 0.821869 0.410934 0.911665i \(-0.365202\pi\)
0.410934 + 0.911665i \(0.365202\pi\)
\(240\) 16.8592 1.08826
\(241\) −18.6604 −1.20202 −0.601012 0.799240i \(-0.705235\pi\)
−0.601012 + 0.799240i \(0.705235\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −18.3532 −1.17494
\(245\) −4.38705 −0.280279
\(246\) −0.936765 −0.0597260
\(247\) −5.82355 −0.370543
\(248\) −2.79281 −0.177344
\(249\) 9.28503 0.588415
\(250\) −6.57730 −0.415985
\(251\) 18.8493 1.18976 0.594878 0.803816i \(-0.297200\pi\)
0.594878 + 0.803816i \(0.297200\pi\)
\(252\) 1.97371 0.124332
\(253\) 0 0
\(254\) −2.60269 −0.163307
\(255\) 27.1411 1.69964
\(256\) 13.9379 0.871118
\(257\) −10.3345 −0.644647 −0.322323 0.946630i \(-0.604464\pi\)
−0.322323 + 0.946630i \(0.604464\pi\)
\(258\) 0.817850 0.0509171
\(259\) −0.969445 −0.0602384
\(260\) 14.5095 0.899844
\(261\) −8.86742 −0.548880
\(262\) 0.121632 0.00751448
\(263\) −4.14979 −0.255887 −0.127943 0.991781i \(-0.540838\pi\)
−0.127943 + 0.991781i \(0.540838\pi\)
\(264\) 0 0
\(265\) 10.4360 0.641079
\(266\) 0.563507 0.0345508
\(267\) 12.2106 0.747277
\(268\) 14.0977 0.861155
\(269\) −5.06228 −0.308653 −0.154326 0.988020i \(-0.549321\pi\)
−0.154326 + 0.988020i \(0.549321\pi\)
\(270\) 0.711349 0.0432913
\(271\) −26.3160 −1.59859 −0.799293 0.600942i \(-0.794792\pi\)
−0.799293 + 0.600942i \(0.794792\pi\)
\(272\) 23.7749 1.44156
\(273\) 1.67571 0.101418
\(274\) −2.10925 −0.127424
\(275\) 0 0
\(276\) 1.12224 0.0675510
\(277\) 25.8187 1.55130 0.775648 0.631165i \(-0.217423\pi\)
0.775648 + 0.631165i \(0.217423\pi\)
\(278\) 0.632556 0.0379382
\(279\) 4.33447 0.259498
\(280\) −2.82669 −0.168927
\(281\) −5.43844 −0.324430 −0.162215 0.986755i \(-0.551864\pi\)
−0.162215 + 0.986755i \(0.551864\pi\)
\(282\) −0.757643 −0.0451170
\(283\) 25.3839 1.50892 0.754458 0.656348i \(-0.227900\pi\)
0.754458 + 0.656348i \(0.227900\pi\)
\(284\) 1.56809 0.0930487
\(285\) −15.2462 −0.903110
\(286\) 0 0
\(287\) −5.77725 −0.341020
\(288\) 1.91177 0.112652
\(289\) 21.2744 1.25143
\(290\) 6.30783 0.370408
\(291\) −15.0596 −0.882810
\(292\) −4.71135 −0.275711
\(293\) 6.75402 0.394574 0.197287 0.980346i \(-0.436787\pi\)
0.197287 + 0.980346i \(0.436787\pi\)
\(294\) −0.162147 −0.00945662
\(295\) 12.4988 0.727707
\(296\) −0.624638 −0.0363064
\(297\) 0 0
\(298\) 3.00676 0.174177
\(299\) 0.952798 0.0551017
\(300\) 28.1179 1.62339
\(301\) 5.04388 0.290724
\(302\) 3.37232 0.194055
\(303\) −5.76079 −0.330949
\(304\) −13.3553 −0.765979
\(305\) −40.7945 −2.33588
\(306\) 1.00314 0.0573460
\(307\) −17.6396 −1.00674 −0.503372 0.864070i \(-0.667908\pi\)
−0.503372 + 0.864070i \(0.667908\pi\)
\(308\) 0 0
\(309\) −7.71449 −0.438862
\(310\) −3.08332 −0.175121
\(311\) −12.2414 −0.694149 −0.347074 0.937838i \(-0.612825\pi\)
−0.347074 + 0.937838i \(0.612825\pi\)
\(312\) 1.07970 0.0611260
\(313\) 3.15951 0.178586 0.0892931 0.996005i \(-0.471539\pi\)
0.0892931 + 0.996005i \(0.471539\pi\)
\(314\) 0.942035 0.0531621
\(315\) 4.38705 0.247182
\(316\) 1.32360 0.0744585
\(317\) 8.43364 0.473681 0.236840 0.971549i \(-0.423888\pi\)
0.236840 + 0.971549i \(0.423888\pi\)
\(318\) 0.385719 0.0216301
\(319\) 0 0
\(320\) 32.3584 1.80889
\(321\) 16.4634 0.918899
\(322\) −0.0921961 −0.00513789
\(323\) −21.5003 −1.19631
\(324\) −1.97371 −0.109650
\(325\) 23.8725 1.32421
\(326\) −1.44361 −0.0799540
\(327\) −9.74355 −0.538820
\(328\) −3.72243 −0.205537
\(329\) −4.67256 −0.257607
\(330\) 0 0
\(331\) 6.13284 0.337092 0.168546 0.985694i \(-0.446093\pi\)
0.168546 + 0.985694i \(0.446093\pi\)
\(332\) 18.3259 1.00577
\(333\) 0.969445 0.0531253
\(334\) −0.619977 −0.0339237
\(335\) 31.3356 1.71205
\(336\) 3.84294 0.209650
\(337\) −29.8792 −1.62763 −0.813813 0.581127i \(-0.802612\pi\)
−0.813813 + 0.581127i \(0.802612\pi\)
\(338\) −1.65261 −0.0898899
\(339\) 2.85725 0.155184
\(340\) 53.5686 2.90516
\(341\) 0 0
\(342\) −0.563507 −0.0304710
\(343\) −1.00000 −0.0539949
\(344\) 3.24990 0.175223
\(345\) 2.49446 0.134297
\(346\) 0.283857 0.0152602
\(347\) −29.1716 −1.56602 −0.783008 0.622012i \(-0.786315\pi\)
−0.783008 + 0.622012i \(0.786315\pi\)
\(348\) −17.5017 −0.938190
\(349\) 13.3924 0.716881 0.358440 0.933553i \(-0.383309\pi\)
0.358440 + 0.933553i \(0.383309\pi\)
\(350\) −2.30999 −0.123474
\(351\) −1.67571 −0.0894425
\(352\) 0 0
\(353\) −21.3175 −1.13462 −0.567309 0.823505i \(-0.692015\pi\)
−0.567309 + 0.823505i \(0.692015\pi\)
\(354\) 0.461960 0.0245529
\(355\) 3.48546 0.184989
\(356\) 24.1002 1.27731
\(357\) 6.18663 0.327431
\(358\) −2.08426 −0.110156
\(359\) −20.6234 −1.08846 −0.544231 0.838935i \(-0.683179\pi\)
−0.544231 + 0.838935i \(0.683179\pi\)
\(360\) 2.82669 0.148980
\(361\) −6.92242 −0.364338
\(362\) −3.58072 −0.188198
\(363\) 0 0
\(364\) 3.30735 0.173352
\(365\) −10.4721 −0.548137
\(366\) −1.50778 −0.0788129
\(367\) 32.8546 1.71499 0.857497 0.514489i \(-0.172018\pi\)
0.857497 + 0.514489i \(0.172018\pi\)
\(368\) 2.18508 0.113905
\(369\) 5.77725 0.300752
\(370\) −0.689613 −0.0358513
\(371\) 2.37882 0.123502
\(372\) 8.55498 0.443555
\(373\) 26.0979 1.35130 0.675650 0.737223i \(-0.263863\pi\)
0.675650 + 0.737223i \(0.263863\pi\)
\(374\) 0 0
\(375\) 40.5638 2.09470
\(376\) −3.01065 −0.155262
\(377\) −14.8592 −0.765287
\(378\) 0.162147 0.00833995
\(379\) −11.6733 −0.599616 −0.299808 0.953999i \(-0.596923\pi\)
−0.299808 + 0.953999i \(0.596923\pi\)
\(380\) −30.0916 −1.54367
\(381\) 16.0514 0.822337
\(382\) 2.38615 0.122086
\(383\) −15.3701 −0.785376 −0.392688 0.919672i \(-0.628455\pi\)
−0.392688 + 0.919672i \(0.628455\pi\)
\(384\) 5.01953 0.256152
\(385\) 0 0
\(386\) 1.38034 0.0702573
\(387\) −5.04388 −0.256395
\(388\) −29.7233 −1.50897
\(389\) −9.64424 −0.488983 −0.244491 0.969651i \(-0.578621\pi\)
−0.244491 + 0.969651i \(0.578621\pi\)
\(390\) 1.19201 0.0603598
\(391\) 3.51769 0.177897
\(392\) −0.644326 −0.0325434
\(393\) −0.750136 −0.0378394
\(394\) −0.765777 −0.0385793
\(395\) 2.94203 0.148030
\(396\) 0 0
\(397\) 20.2855 1.01810 0.509050 0.860737i \(-0.329997\pi\)
0.509050 + 0.860737i \(0.329997\pi\)
\(398\) 2.71791 0.136236
\(399\) −3.47528 −0.173982
\(400\) 54.7475 2.73737
\(401\) −8.15655 −0.407319 −0.203659 0.979042i \(-0.565283\pi\)
−0.203659 + 0.979042i \(0.565283\pi\)
\(402\) 1.15818 0.0577646
\(403\) 7.26330 0.361811
\(404\) −11.3701 −0.565684
\(405\) −4.38705 −0.217994
\(406\) 1.43783 0.0713582
\(407\) 0 0
\(408\) 3.98620 0.197347
\(409\) −9.67209 −0.478254 −0.239127 0.970988i \(-0.576861\pi\)
−0.239127 + 0.970988i \(0.576861\pi\)
\(410\) −4.10964 −0.202961
\(411\) 13.0082 0.641649
\(412\) −15.2262 −0.750139
\(413\) 2.84901 0.140191
\(414\) 0.0921961 0.00453119
\(415\) 40.7339 1.99955
\(416\) 3.20357 0.157068
\(417\) −3.90112 −0.191039
\(418\) 0 0
\(419\) −18.9357 −0.925072 −0.462536 0.886601i \(-0.653060\pi\)
−0.462536 + 0.886601i \(0.653060\pi\)
\(420\) 8.65877 0.422504
\(421\) −13.0950 −0.638210 −0.319105 0.947719i \(-0.603382\pi\)
−0.319105 + 0.947719i \(0.603382\pi\)
\(422\) −1.64505 −0.0800799
\(423\) 4.67256 0.227188
\(424\) 1.53274 0.0744362
\(425\) 88.1362 4.27524
\(426\) 0.128824 0.00624153
\(427\) −9.29883 −0.450002
\(428\) 32.4940 1.57066
\(429\) 0 0
\(430\) 3.58795 0.173027
\(431\) −16.3034 −0.785309 −0.392655 0.919686i \(-0.628443\pi\)
−0.392655 + 0.919686i \(0.628443\pi\)
\(432\) −3.84294 −0.184894
\(433\) 11.6842 0.561508 0.280754 0.959780i \(-0.409415\pi\)
0.280754 + 0.959780i \(0.409415\pi\)
\(434\) −0.702822 −0.0337366
\(435\) −38.9019 −1.86520
\(436\) −19.2309 −0.920995
\(437\) −1.97603 −0.0945262
\(438\) −0.387054 −0.0184942
\(439\) 11.7172 0.559230 0.279615 0.960112i \(-0.409793\pi\)
0.279615 + 0.960112i \(0.409793\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 1.68098 0.0799558
\(443\) −12.5671 −0.597081 −0.298540 0.954397i \(-0.596500\pi\)
−0.298540 + 0.954397i \(0.596500\pi\)
\(444\) 1.91340 0.0908060
\(445\) 53.5686 2.53939
\(446\) −0.990999 −0.0469252
\(447\) −18.5434 −0.877074
\(448\) 7.37589 0.348478
\(449\) −15.9360 −0.752068 −0.376034 0.926606i \(-0.622712\pi\)
−0.376034 + 0.926606i \(0.622712\pi\)
\(450\) 2.30999 0.108894
\(451\) 0 0
\(452\) 5.63937 0.265254
\(453\) −20.7979 −0.977171
\(454\) 2.30394 0.108129
\(455\) 7.35141 0.344639
\(456\) −2.23921 −0.104861
\(457\) −23.7945 −1.11306 −0.556530 0.830828i \(-0.687867\pi\)
−0.556530 + 0.830828i \(0.687867\pi\)
\(458\) −0.442307 −0.0206677
\(459\) −6.18663 −0.288767
\(460\) 4.92333 0.229552
\(461\) −22.6035 −1.05275 −0.526375 0.850252i \(-0.676449\pi\)
−0.526375 + 0.850252i \(0.676449\pi\)
\(462\) 0 0
\(463\) 3.03759 0.141169 0.0705843 0.997506i \(-0.477514\pi\)
0.0705843 + 0.997506i \(0.477514\pi\)
\(464\) −34.0770 −1.58198
\(465\) 19.0156 0.881825
\(466\) −2.88454 −0.133624
\(467\) −23.0051 −1.06455 −0.532274 0.846572i \(-0.678662\pi\)
−0.532274 + 0.846572i \(0.678662\pi\)
\(468\) −3.30735 −0.152882
\(469\) 7.14275 0.329822
\(470\) −3.32382 −0.153316
\(471\) −5.80975 −0.267699
\(472\) 1.83569 0.0844946
\(473\) 0 0
\(474\) 0.108739 0.00499454
\(475\) −49.5097 −2.27166
\(476\) 12.2106 0.559672
\(477\) −2.37882 −0.108919
\(478\) 2.06021 0.0942317
\(479\) 8.17283 0.373426 0.186713 0.982414i \(-0.440217\pi\)
0.186713 + 0.982414i \(0.440217\pi\)
\(480\) 8.38705 0.382815
\(481\) 1.62450 0.0740710
\(482\) −3.02573 −0.137818
\(483\) 0.568595 0.0258720
\(484\) 0 0
\(485\) −66.0673 −2.99996
\(486\) −0.162147 −0.00735515
\(487\) −0.999232 −0.0452795 −0.0226398 0.999744i \(-0.507207\pi\)
−0.0226398 + 0.999744i \(0.507207\pi\)
\(488\) −5.99147 −0.271221
\(489\) 8.90307 0.402611
\(490\) −0.711349 −0.0321355
\(491\) 24.2957 1.09645 0.548224 0.836331i \(-0.315304\pi\)
0.548224 + 0.836331i \(0.315304\pi\)
\(492\) 11.4026 0.514069
\(493\) −54.8595 −2.47075
\(494\) −0.944272 −0.0424848
\(495\) 0 0
\(496\) 16.6571 0.747927
\(497\) 0.794487 0.0356376
\(498\) 1.50554 0.0674650
\(499\) −41.0807 −1.83902 −0.919512 0.393061i \(-0.871416\pi\)
−0.919512 + 0.393061i \(0.871416\pi\)
\(500\) 80.0611 3.58044
\(501\) 3.82355 0.170823
\(502\) 3.05636 0.136412
\(503\) −10.3270 −0.460457 −0.230228 0.973137i \(-0.573947\pi\)
−0.230228 + 0.973137i \(0.573947\pi\)
\(504\) 0.644326 0.0287005
\(505\) −25.2729 −1.12463
\(506\) 0 0
\(507\) 10.1920 0.452643
\(508\) 31.6807 1.40561
\(509\) −40.6105 −1.80003 −0.900014 0.435860i \(-0.856444\pi\)
−0.900014 + 0.435860i \(0.856444\pi\)
\(510\) 4.40085 0.194873
\(511\) −2.38705 −0.105597
\(512\) 12.2990 0.543546
\(513\) 3.47528 0.153437
\(514\) −1.67571 −0.0739122
\(515\) −33.8439 −1.49134
\(516\) −9.95514 −0.438251
\(517\) 0 0
\(518\) −0.157193 −0.00690666
\(519\) −1.75061 −0.0768433
\(520\) 4.73670 0.207718
\(521\) −22.2145 −0.973234 −0.486617 0.873615i \(-0.661769\pi\)
−0.486617 + 0.873615i \(0.661769\pi\)
\(522\) −1.43783 −0.0629320
\(523\) −22.6653 −0.991085 −0.495543 0.868584i \(-0.665031\pi\)
−0.495543 + 0.868584i \(0.665031\pi\)
\(524\) −1.48055 −0.0646781
\(525\) 14.2462 0.621757
\(526\) −0.672876 −0.0293388
\(527\) 26.8158 1.16811
\(528\) 0 0
\(529\) −22.6767 −0.985943
\(530\) 1.69217 0.0735032
\(531\) −2.84901 −0.123637
\(532\) −6.85919 −0.297384
\(533\) 9.68098 0.419330
\(534\) 1.97991 0.0856793
\(535\) 72.2260 3.12260
\(536\) 4.60226 0.198787
\(537\) 12.8541 0.554695
\(538\) −0.820835 −0.0353887
\(539\) 0 0
\(540\) −8.65877 −0.372614
\(541\) −6.26662 −0.269423 −0.134712 0.990885i \(-0.543011\pi\)
−0.134712 + 0.990885i \(0.543011\pi\)
\(542\) −4.26707 −0.183286
\(543\) 22.0831 0.947677
\(544\) 11.8274 0.507097
\(545\) −42.7455 −1.83102
\(546\) 0.271711 0.0116282
\(547\) 34.0916 1.45765 0.728827 0.684698i \(-0.240066\pi\)
0.728827 + 0.684698i \(0.240066\pi\)
\(548\) 25.6745 1.09676
\(549\) 9.29883 0.396864
\(550\) 0 0
\(551\) 30.8168 1.31284
\(552\) 0.366361 0.0155933
\(553\) 0.670617 0.0285176
\(554\) 4.18643 0.177865
\(555\) 4.25301 0.180530
\(556\) −7.69968 −0.326539
\(557\) −26.3992 −1.11857 −0.559284 0.828976i \(-0.688924\pi\)
−0.559284 + 0.828976i \(0.688924\pi\)
\(558\) 0.702822 0.0297528
\(559\) −8.45205 −0.357484
\(560\) 16.8592 0.712431
\(561\) 0 0
\(562\) −0.881827 −0.0371976
\(563\) −6.85125 −0.288746 −0.144373 0.989523i \(-0.546116\pi\)
−0.144373 + 0.989523i \(0.546116\pi\)
\(564\) 9.22227 0.388328
\(565\) 12.5349 0.527347
\(566\) 4.11593 0.173005
\(567\) −1.00000 −0.0419961
\(568\) 0.511908 0.0214792
\(569\) 18.4008 0.771404 0.385702 0.922623i \(-0.373959\pi\)
0.385702 + 0.922623i \(0.373959\pi\)
\(570\) −2.47214 −0.103546
\(571\) 0.336889 0.0140984 0.00704918 0.999975i \(-0.497756\pi\)
0.00704918 + 0.999975i \(0.497756\pi\)
\(572\) 0 0
\(573\) −14.7160 −0.614768
\(574\) −0.936765 −0.0390998
\(575\) 8.10035 0.337808
\(576\) −7.37589 −0.307329
\(577\) −25.8347 −1.07551 −0.537757 0.843100i \(-0.680728\pi\)
−0.537757 + 0.843100i \(0.680728\pi\)
\(578\) 3.44958 0.143484
\(579\) −8.51287 −0.353783
\(580\) −76.7809 −3.18815
\(581\) 9.28503 0.385208
\(582\) −2.44187 −0.101219
\(583\) 0 0
\(584\) −1.53804 −0.0636446
\(585\) −7.35141 −0.303943
\(586\) 1.09515 0.0452401
\(587\) −3.73415 −0.154125 −0.0770623 0.997026i \(-0.524554\pi\)
−0.0770623 + 0.997026i \(0.524554\pi\)
\(588\) 1.97371 0.0813944
\(589\) −15.0635 −0.620681
\(590\) 2.02664 0.0834355
\(591\) 4.72273 0.194267
\(592\) 3.72552 0.153118
\(593\) 17.5647 0.721295 0.360648 0.932702i \(-0.382556\pi\)
0.360648 + 0.932702i \(0.382556\pi\)
\(594\) 0 0
\(595\) 27.1411 1.11268
\(596\) −36.5993 −1.49917
\(597\) −16.7620 −0.686021
\(598\) 0.154494 0.00631771
\(599\) 11.3088 0.462066 0.231033 0.972946i \(-0.425789\pi\)
0.231033 + 0.972946i \(0.425789\pi\)
\(600\) 9.17922 0.374740
\(601\) −28.9821 −1.18220 −0.591102 0.806596i \(-0.701307\pi\)
−0.591102 + 0.806596i \(0.701307\pi\)
\(602\) 0.817850 0.0333331
\(603\) −7.14275 −0.290875
\(604\) −41.0490 −1.67026
\(605\) 0 0
\(606\) −0.934096 −0.0379450
\(607\) 7.34347 0.298062 0.149031 0.988832i \(-0.452385\pi\)
0.149031 + 0.988832i \(0.452385\pi\)
\(608\) −6.64395 −0.269448
\(609\) −8.86742 −0.359326
\(610\) −6.61471 −0.267822
\(611\) 7.82984 0.316761
\(612\) −12.2106 −0.493584
\(613\) 18.9535 0.765526 0.382763 0.923847i \(-0.374973\pi\)
0.382763 + 0.923847i \(0.374973\pi\)
\(614\) −2.86021 −0.115429
\(615\) 25.3451 1.02201
\(616\) 0 0
\(617\) 25.2571 1.01681 0.508407 0.861117i \(-0.330234\pi\)
0.508407 + 0.861117i \(0.330234\pi\)
\(618\) −1.25088 −0.0503179
\(619\) −15.5725 −0.625912 −0.312956 0.949768i \(-0.601319\pi\)
−0.312956 + 0.949768i \(0.601319\pi\)
\(620\) 37.5312 1.50729
\(621\) −0.568595 −0.0228169
\(622\) −1.98492 −0.0795879
\(623\) 12.2106 0.489207
\(624\) −6.43964 −0.257792
\(625\) 106.724 4.26897
\(626\) 0.512306 0.0204759
\(627\) 0 0
\(628\) −11.4668 −0.457573
\(629\) 5.99760 0.239140
\(630\) 0.711349 0.0283408
\(631\) −36.0169 −1.43381 −0.716905 0.697170i \(-0.754442\pi\)
−0.716905 + 0.697170i \(0.754442\pi\)
\(632\) 0.432096 0.0171879
\(633\) 10.1454 0.403244
\(634\) 1.36749 0.0543100
\(635\) 70.4183 2.79446
\(636\) −4.69510 −0.186173
\(637\) 1.67571 0.0663939
\(638\) 0 0
\(639\) −0.794487 −0.0314294
\(640\) 22.0209 0.870454
\(641\) −26.7410 −1.05620 −0.528102 0.849181i \(-0.677096\pi\)
−0.528102 + 0.849181i \(0.677096\pi\)
\(642\) 2.66950 0.105357
\(643\) 7.31364 0.288422 0.144211 0.989547i \(-0.453936\pi\)
0.144211 + 0.989547i \(0.453936\pi\)
\(644\) 1.12224 0.0442225
\(645\) −22.1278 −0.871280
\(646\) −3.48621 −0.137163
\(647\) −35.0221 −1.37686 −0.688431 0.725302i \(-0.741700\pi\)
−0.688431 + 0.725302i \(0.741700\pi\)
\(648\) −0.644326 −0.0253115
\(649\) 0 0
\(650\) 3.87086 0.151828
\(651\) 4.33447 0.169881
\(652\) 17.5721 0.688175
\(653\) −4.05138 −0.158543 −0.0792714 0.996853i \(-0.525259\pi\)
−0.0792714 + 0.996853i \(0.525259\pi\)
\(654\) −1.57989 −0.0617786
\(655\) −3.29089 −0.128586
\(656\) 22.2016 0.866828
\(657\) 2.38705 0.0931279
\(658\) −0.757643 −0.0295360
\(659\) 28.6360 1.11550 0.557750 0.830009i \(-0.311665\pi\)
0.557750 + 0.830009i \(0.311665\pi\)
\(660\) 0 0
\(661\) 45.2741 1.76096 0.880479 0.474085i \(-0.157221\pi\)
0.880479 + 0.474085i \(0.157221\pi\)
\(662\) 0.994424 0.0386494
\(663\) −10.3670 −0.402620
\(664\) 5.98258 0.232169
\(665\) −15.2462 −0.591224
\(666\) 0.157193 0.00609110
\(667\) −5.04197 −0.195226
\(668\) 7.54657 0.291985
\(669\) 6.11172 0.236293
\(670\) 5.08099 0.196296
\(671\) 0 0
\(672\) 1.91177 0.0737483
\(673\) 0.0509075 0.00196234 0.000981170 1.00000i \(-0.499688\pi\)
0.000981170 1.00000i \(0.499688\pi\)
\(674\) −4.84484 −0.186616
\(675\) −14.2462 −0.548338
\(676\) 20.1161 0.773694
\(677\) 39.1437 1.50442 0.752208 0.658925i \(-0.228989\pi\)
0.752208 + 0.658925i \(0.228989\pi\)
\(678\) 0.463295 0.0177927
\(679\) −15.0596 −0.577935
\(680\) 17.4877 0.670622
\(681\) −14.2090 −0.544488
\(682\) 0 0
\(683\) 15.4140 0.589800 0.294900 0.955528i \(-0.404714\pi\)
0.294900 + 0.955528i \(0.404714\pi\)
\(684\) 6.85919 0.262268
\(685\) 57.0678 2.18045
\(686\) −0.162147 −0.00619081
\(687\) 2.72781 0.104073
\(688\) −19.3833 −0.738982
\(689\) −3.98620 −0.151862
\(690\) 0.404469 0.0153979
\(691\) −38.2746 −1.45603 −0.728017 0.685559i \(-0.759558\pi\)
−0.728017 + 0.685559i \(0.759558\pi\)
\(692\) −3.45520 −0.131347
\(693\) 0 0
\(694\) −4.73010 −0.179552
\(695\) −17.1144 −0.649188
\(696\) −5.71351 −0.216570
\(697\) 35.7417 1.35381
\(698\) 2.17155 0.0821942
\(699\) 17.7897 0.672867
\(700\) 28.1179 1.06276
\(701\) −1.38873 −0.0524516 −0.0262258 0.999656i \(-0.508349\pi\)
−0.0262258 + 0.999656i \(0.508349\pi\)
\(702\) −0.271711 −0.0102551
\(703\) −3.36909 −0.127068
\(704\) 0 0
\(705\) 20.4988 0.772029
\(706\) −3.45658 −0.130090
\(707\) −5.76079 −0.216657
\(708\) −5.62312 −0.211330
\(709\) −12.8361 −0.482071 −0.241036 0.970516i \(-0.577487\pi\)
−0.241036 + 0.970516i \(0.577487\pi\)
\(710\) 0.565157 0.0212100
\(711\) −0.670617 −0.0251501
\(712\) 7.86760 0.294851
\(713\) 2.46456 0.0922985
\(714\) 1.00314 0.0375417
\(715\) 0 0
\(716\) 25.3702 0.948131
\(717\) −12.7058 −0.474506
\(718\) −3.34403 −0.124798
\(719\) −21.5055 −0.802021 −0.401011 0.916073i \(-0.631341\pi\)
−0.401011 + 0.916073i \(0.631341\pi\)
\(720\) −16.8592 −0.628305
\(721\) −7.71449 −0.287303
\(722\) −1.12245 −0.0417733
\(723\) 18.6604 0.693988
\(724\) 43.5856 1.61985
\(725\) −126.327 −4.69168
\(726\) 0 0
\(727\) −5.35770 −0.198706 −0.0993531 0.995052i \(-0.531677\pi\)
−0.0993531 + 0.995052i \(0.531677\pi\)
\(728\) 1.07970 0.0400164
\(729\) 1.00000 0.0370370
\(730\) −1.69803 −0.0628468
\(731\) −31.2046 −1.15414
\(732\) 18.3532 0.678353
\(733\) 41.4506 1.53101 0.765506 0.643429i \(-0.222489\pi\)
0.765506 + 0.643429i \(0.222489\pi\)
\(734\) 5.32727 0.196633
\(735\) 4.38705 0.161819
\(736\) 1.08703 0.0400683
\(737\) 0 0
\(738\) 0.936765 0.0344828
\(739\) 42.6086 1.56738 0.783691 0.621151i \(-0.213335\pi\)
0.783691 + 0.621151i \(0.213335\pi\)
\(740\) 8.39420 0.308577
\(741\) 5.82355 0.213933
\(742\) 0.385719 0.0141602
\(743\) −30.4479 −1.11702 −0.558512 0.829496i \(-0.688628\pi\)
−0.558512 + 0.829496i \(0.688628\pi\)
\(744\) 2.79281 0.102389
\(745\) −81.3510 −2.98047
\(746\) 4.23171 0.154934
\(747\) −9.28503 −0.339722
\(748\) 0 0
\(749\) 16.4634 0.601561
\(750\) 6.57730 0.240169
\(751\) 9.06635 0.330836 0.165418 0.986224i \(-0.447103\pi\)
0.165418 + 0.986224i \(0.447103\pi\)
\(752\) 17.9564 0.654802
\(753\) −18.8493 −0.686906
\(754\) −2.40938 −0.0877443
\(755\) −91.2415 −3.32062
\(756\) −1.97371 −0.0717831
\(757\) 7.83076 0.284614 0.142307 0.989823i \(-0.454548\pi\)
0.142307 + 0.989823i \(0.454548\pi\)
\(758\) −1.89279 −0.0687493
\(759\) 0 0
\(760\) −9.82355 −0.356338
\(761\) 26.4983 0.960564 0.480282 0.877114i \(-0.340534\pi\)
0.480282 + 0.877114i \(0.340534\pi\)
\(762\) 2.60269 0.0942854
\(763\) −9.74355 −0.352740
\(764\) −29.0450 −1.05081
\(765\) −27.1411 −0.981288
\(766\) −2.49222 −0.0900476
\(767\) −4.77411 −0.172383
\(768\) −13.9379 −0.502940
\(769\) −26.3995 −0.951989 −0.475995 0.879448i \(-0.657912\pi\)
−0.475995 + 0.879448i \(0.657912\pi\)
\(770\) 0 0
\(771\) 10.3345 0.372187
\(772\) −16.8019 −0.604714
\(773\) 34.1566 1.22853 0.614264 0.789100i \(-0.289453\pi\)
0.614264 + 0.789100i \(0.289453\pi\)
\(774\) −0.817850 −0.0293970
\(775\) 61.7499 2.21812
\(776\) −9.70330 −0.348328
\(777\) 0.969445 0.0347787
\(778\) −1.56379 −0.0560645
\(779\) −20.0776 −0.719354
\(780\) −14.5095 −0.519525
\(781\) 0 0
\(782\) 0.570383 0.0203969
\(783\) 8.86742 0.316896
\(784\) 3.84294 0.137248
\(785\) −25.4877 −0.909695
\(786\) −0.121632 −0.00433849
\(787\) 11.7503 0.418853 0.209426 0.977824i \(-0.432840\pi\)
0.209426 + 0.977824i \(0.432840\pi\)
\(788\) 9.32128 0.332057
\(789\) 4.14979 0.147736
\(790\) 0.477043 0.0169724
\(791\) 2.85725 0.101592
\(792\) 0 0
\(793\) 15.5821 0.553337
\(794\) 3.28924 0.116731
\(795\) −10.4360 −0.370127
\(796\) −33.0832 −1.17260
\(797\) 15.3027 0.542050 0.271025 0.962572i \(-0.412637\pi\)
0.271025 + 0.962572i \(0.412637\pi\)
\(798\) −0.563507 −0.0199479
\(799\) 28.9074 1.02267
\(800\) 27.2356 0.962924
\(801\) −12.2106 −0.431440
\(802\) −1.32256 −0.0467013
\(803\) 0 0
\(804\) −14.0977 −0.497188
\(805\) 2.49446 0.0879181
\(806\) 1.17772 0.0414835
\(807\) 5.06228 0.178201
\(808\) −3.71182 −0.130581
\(809\) 18.2311 0.640973 0.320486 0.947253i \(-0.396154\pi\)
0.320486 + 0.947253i \(0.396154\pi\)
\(810\) −0.711349 −0.0249942
\(811\) 6.23292 0.218868 0.109434 0.993994i \(-0.465096\pi\)
0.109434 + 0.993994i \(0.465096\pi\)
\(812\) −17.5017 −0.614189
\(813\) 26.3160 0.922944
\(814\) 0 0
\(815\) 39.0582 1.36815
\(816\) −23.7749 −0.832286
\(817\) 17.5289 0.613258
\(818\) −1.56830 −0.0548344
\(819\) −1.67571 −0.0585539
\(820\) 50.0239 1.74691
\(821\) −29.6399 −1.03444 −0.517219 0.855853i \(-0.673033\pi\)
−0.517219 + 0.855853i \(0.673033\pi\)
\(822\) 2.10925 0.0735685
\(823\) −19.7302 −0.687753 −0.343876 0.939015i \(-0.611740\pi\)
−0.343876 + 0.939015i \(0.611740\pi\)
\(824\) −4.97065 −0.173161
\(825\) 0 0
\(826\) 0.461960 0.0160736
\(827\) 41.1710 1.43166 0.715828 0.698276i \(-0.246049\pi\)
0.715828 + 0.698276i \(0.246049\pi\)
\(828\) −1.12224 −0.0390006
\(829\) 22.7961 0.791742 0.395871 0.918306i \(-0.370443\pi\)
0.395871 + 0.918306i \(0.370443\pi\)
\(830\) 6.60489 0.229259
\(831\) −25.8187 −0.895642
\(832\) −12.3598 −0.428500
\(833\) 6.18663 0.214354
\(834\) −0.632556 −0.0219036
\(835\) 16.7741 0.580492
\(836\) 0 0
\(837\) −4.33447 −0.149821
\(838\) −3.07038 −0.106064
\(839\) −47.2725 −1.63203 −0.816013 0.578033i \(-0.803820\pi\)
−0.816013 + 0.578033i \(0.803820\pi\)
\(840\) 2.82669 0.0975301
\(841\) 49.6312 1.71142
\(842\) −2.12331 −0.0731742
\(843\) 5.43844 0.187310
\(844\) 20.0241 0.689258
\(845\) 44.7129 1.53817
\(846\) 0.757643 0.0260483
\(847\) 0 0
\(848\) −9.14167 −0.313926
\(849\) −25.3839 −0.871174
\(850\) 14.2910 0.490179
\(851\) 0.551222 0.0188956
\(852\) −1.56809 −0.0537217
\(853\) 10.8549 0.371664 0.185832 0.982582i \(-0.440502\pi\)
0.185832 + 0.982582i \(0.440502\pi\)
\(854\) −1.50778 −0.0515951
\(855\) 15.2462 0.521411
\(856\) 10.6078 0.362567
\(857\) 28.5431 0.975014 0.487507 0.873119i \(-0.337906\pi\)
0.487507 + 0.873119i \(0.337906\pi\)
\(858\) 0 0
\(859\) −36.8034 −1.25572 −0.627858 0.778328i \(-0.716068\pi\)
−0.627858 + 0.778328i \(0.716068\pi\)
\(860\) −43.6737 −1.48926
\(861\) 5.77725 0.196888
\(862\) −2.64356 −0.0900399
\(863\) −45.1551 −1.53710 −0.768549 0.639790i \(-0.779021\pi\)
−0.768549 + 0.639790i \(0.779021\pi\)
\(864\) −1.91177 −0.0650399
\(865\) −7.68003 −0.261129
\(866\) 1.89457 0.0643800
\(867\) −21.2744 −0.722516
\(868\) 8.55498 0.290375
\(869\) 0 0
\(870\) −6.30783 −0.213855
\(871\) −11.9692 −0.405559
\(872\) −6.27802 −0.212601
\(873\) 15.0596 0.509691
\(874\) −0.320407 −0.0108379
\(875\) 40.5638 1.37131
\(876\) 4.71135 0.159182
\(877\) 34.8517 1.17686 0.588428 0.808549i \(-0.299747\pi\)
0.588428 + 0.808549i \(0.299747\pi\)
\(878\) 1.89991 0.0641187
\(879\) −6.75402 −0.227808
\(880\) 0 0
\(881\) −35.3563 −1.19118 −0.595592 0.803287i \(-0.703083\pi\)
−0.595592 + 0.803287i \(0.703083\pi\)
\(882\) 0.162147 0.00545978
\(883\) −24.8870 −0.837516 −0.418758 0.908098i \(-0.637534\pi\)
−0.418758 + 0.908098i \(0.637534\pi\)
\(884\) −20.4614 −0.688190
\(885\) −12.4988 −0.420142
\(886\) −2.03772 −0.0684586
\(887\) −49.5708 −1.66442 −0.832212 0.554457i \(-0.812926\pi\)
−0.832212 + 0.554457i \(0.812926\pi\)
\(888\) 0.624638 0.0209615
\(889\) 16.0514 0.538346
\(890\) 8.68599 0.291155
\(891\) 0 0
\(892\) 12.0628 0.403891
\(893\) −16.2385 −0.543399
\(894\) −3.00676 −0.100561
\(895\) 56.3916 1.88496
\(896\) 5.01953 0.167691
\(897\) −0.952798 −0.0318130
\(898\) −2.58398 −0.0862287
\(899\) −38.4356 −1.28190
\(900\) −28.1179 −0.937264
\(901\) −14.7169 −0.490291
\(902\) 0 0
\(903\) −5.04388 −0.167850
\(904\) 1.84100 0.0612307
\(905\) 96.8798 3.22039
\(906\) −3.37232 −0.112038
\(907\) −1.42826 −0.0474246 −0.0237123 0.999719i \(-0.507549\pi\)
−0.0237123 + 0.999719i \(0.507549\pi\)
\(908\) −28.0443 −0.930684
\(909\) 5.76079 0.191073
\(910\) 1.19201 0.0395148
\(911\) 54.1725 1.79482 0.897408 0.441202i \(-0.145448\pi\)
0.897408 + 0.441202i \(0.145448\pi\)
\(912\) 13.3553 0.442238
\(913\) 0 0
\(914\) −3.85821 −0.127618
\(915\) 40.7945 1.34862
\(916\) 5.38391 0.177889
\(917\) −0.750136 −0.0247717
\(918\) −1.00314 −0.0331087
\(919\) −22.9009 −0.755432 −0.377716 0.925921i \(-0.623290\pi\)
−0.377716 + 0.925921i \(0.623290\pi\)
\(920\) 1.60724 0.0529892
\(921\) 17.6396 0.581244
\(922\) −3.66510 −0.120704
\(923\) −1.33133 −0.0438211
\(924\) 0 0
\(925\) 13.8110 0.454101
\(926\) 0.492536 0.0161857
\(927\) 7.71449 0.253377
\(928\) −16.9525 −0.556493
\(929\) −14.5306 −0.476733 −0.238366 0.971175i \(-0.576612\pi\)
−0.238366 + 0.971175i \(0.576612\pi\)
\(930\) 3.08332 0.101106
\(931\) −3.47528 −0.113898
\(932\) 35.1116 1.15012
\(933\) 12.2414 0.400767
\(934\) −3.73021 −0.122056
\(935\) 0 0
\(936\) −1.07970 −0.0352911
\(937\) 30.6351 1.00080 0.500402 0.865793i \(-0.333186\pi\)
0.500402 + 0.865793i \(0.333186\pi\)
\(938\) 1.15818 0.0378158
\(939\) −3.15951 −0.103107
\(940\) 40.4586 1.31961
\(941\) 21.0533 0.686319 0.343159 0.939277i \(-0.388503\pi\)
0.343159 + 0.939277i \(0.388503\pi\)
\(942\) −0.942035 −0.0306932
\(943\) 3.28492 0.106972
\(944\) −10.9486 −0.356346
\(945\) −4.38705 −0.142711
\(946\) 0 0
\(947\) 0.729464 0.0237044 0.0118522 0.999930i \(-0.496227\pi\)
0.0118522 + 0.999930i \(0.496227\pi\)
\(948\) −1.32360 −0.0429886
\(949\) 4.00000 0.129845
\(950\) −8.02786 −0.260458
\(951\) −8.43364 −0.273480
\(952\) 3.98620 0.129194
\(953\) 39.0151 1.26382 0.631911 0.775041i \(-0.282271\pi\)
0.631911 + 0.775041i \(0.282271\pi\)
\(954\) −0.385719 −0.0124881
\(955\) −64.5597 −2.08910
\(956\) −25.0775 −0.811065
\(957\) 0 0
\(958\) 1.32520 0.0428153
\(959\) 13.0082 0.420058
\(960\) −32.3584 −1.04436
\(961\) −12.2124 −0.393947
\(962\) 0.263409 0.00849264
\(963\) −16.4634 −0.530527
\(964\) 36.8302 1.18622
\(965\) −37.3464 −1.20222
\(966\) 0.0921961 0.00296636
\(967\) 2.46386 0.0792324 0.0396162 0.999215i \(-0.487386\pi\)
0.0396162 + 0.999215i \(0.487386\pi\)
\(968\) 0 0
\(969\) 21.5003 0.690688
\(970\) −10.7126 −0.343962
\(971\) 34.6794 1.11291 0.556457 0.830876i \(-0.312160\pi\)
0.556457 + 0.830876i \(0.312160\pi\)
\(972\) 1.97371 0.0633067
\(973\) −3.90112 −0.125064
\(974\) −0.162023 −0.00519154
\(975\) −23.8725 −0.764532
\(976\) 35.7348 1.14384
\(977\) −25.1338 −0.804100 −0.402050 0.915618i \(-0.631702\pi\)
−0.402050 + 0.915618i \(0.631702\pi\)
\(978\) 1.44361 0.0461615
\(979\) 0 0
\(980\) 8.65877 0.276594
\(981\) 9.74355 0.311088
\(982\) 3.93948 0.125714
\(983\) 30.3900 0.969290 0.484645 0.874711i \(-0.338949\pi\)
0.484645 + 0.874711i \(0.338949\pi\)
\(984\) 3.72243 0.118667
\(985\) 20.7189 0.660158
\(986\) −8.89531 −0.283284
\(987\) 4.67256 0.148729
\(988\) 11.4940 0.365672
\(989\) −2.86792 −0.0911947
\(990\) 0 0
\(991\) −8.09793 −0.257239 −0.128620 0.991694i \(-0.541055\pi\)
−0.128620 + 0.991694i \(0.541055\pi\)
\(992\) 8.28653 0.263097
\(993\) −6.13284 −0.194620
\(994\) 0.128824 0.00408604
\(995\) −73.5356 −2.33124
\(996\) −18.3259 −0.580680
\(997\) −45.8866 −1.45324 −0.726621 0.687038i \(-0.758910\pi\)
−0.726621 + 0.687038i \(0.758910\pi\)
\(998\) −6.66112 −0.210854
\(999\) −0.969445 −0.0306719
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 2541.2.a.bn.1.2 4
3.2 odd 2 7623.2.a.ci.1.3 4
11.3 even 5 231.2.j.f.64.1 8
11.4 even 5 231.2.j.f.148.1 yes 8
11.10 odd 2 2541.2.a.bm.1.3 4
33.14 odd 10 693.2.m.f.64.2 8
33.26 odd 10 693.2.m.f.379.2 8
33.32 even 2 7623.2.a.cl.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.f.64.1 8 11.3 even 5
231.2.j.f.148.1 yes 8 11.4 even 5
693.2.m.f.64.2 8 33.14 odd 10
693.2.m.f.379.2 8 33.26 odd 10
2541.2.a.bm.1.3 4 11.10 odd 2
2541.2.a.bn.1.2 4 1.1 even 1 trivial
7623.2.a.ci.1.3 4 3.2 odd 2
7623.2.a.cl.1.2 4 33.32 even 2