Properties

Label 3525.2.a.bi.1.6
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
Dimension $13$
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] = [3525,2,Mod(1,3525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, 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("3525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.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: \(28.1472667125\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 3 x^{12} - 17 x^{11} + 51 x^{10} + 106 x^{9} - 316 x^{8} - 288 x^{7} + 852 x^{6} + 309 x^{5} - 923 x^{4} - 107 x^{3} + 293 x^{2} + 12 x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 705)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.105426\) of defining polynomial
Character \(\chi\) \(=\) 3525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.105426 q^{2} +1.00000 q^{3} -1.98889 q^{4} -0.105426 q^{6} -3.91364 q^{7} +0.420531 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.105426 q^{2} +1.00000 q^{3} -1.98889 q^{4} -0.105426 q^{6} -3.91364 q^{7} +0.420531 q^{8} +1.00000 q^{9} +6.01909 q^{11} -1.98889 q^{12} -5.26742 q^{13} +0.412598 q^{14} +3.93344 q^{16} -1.81336 q^{17} -0.105426 q^{18} +8.19937 q^{19} -3.91364 q^{21} -0.634566 q^{22} -4.74109 q^{23} +0.420531 q^{24} +0.555321 q^{26} +1.00000 q^{27} +7.78377 q^{28} -4.95306 q^{29} -5.06513 q^{31} -1.25575 q^{32} +6.01909 q^{33} +0.191174 q^{34} -1.98889 q^{36} +0.310810 q^{37} -0.864424 q^{38} -5.26742 q^{39} +2.61857 q^{41} +0.412598 q^{42} -4.72384 q^{43} -11.9713 q^{44} +0.499832 q^{46} +1.00000 q^{47} +3.93344 q^{48} +8.31654 q^{49} -1.81336 q^{51} +10.4763 q^{52} +3.05124 q^{53} -0.105426 q^{54} -1.64580 q^{56} +8.19937 q^{57} +0.522180 q^{58} -5.83634 q^{59} +10.0059 q^{61} +0.533994 q^{62} -3.91364 q^{63} -7.73448 q^{64} -0.634566 q^{66} +8.37059 q^{67} +3.60656 q^{68} -4.74109 q^{69} -9.96991 q^{71} +0.420531 q^{72} +3.65100 q^{73} -0.0327673 q^{74} -16.3076 q^{76} -23.5565 q^{77} +0.555321 q^{78} +9.00605 q^{79} +1.00000 q^{81} -0.276064 q^{82} +12.9270 q^{83} +7.78377 q^{84} +0.498013 q^{86} -4.95306 q^{87} +2.53121 q^{88} +8.60225 q^{89} +20.6148 q^{91} +9.42948 q^{92} -5.06513 q^{93} -0.105426 q^{94} -1.25575 q^{96} +8.42460 q^{97} -0.876777 q^{98} +6.01909 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 3 q^{2} + 13 q^{3} + 17 q^{4} + 3 q^{6} - 4 q^{7} + 15 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 3 q^{2} + 13 q^{3} + 17 q^{4} + 3 q^{6} - 4 q^{7} + 15 q^{8} + 13 q^{9} + 16 q^{11} + 17 q^{12} - 8 q^{13} - 4 q^{14} + 29 q^{16} + 12 q^{17} + 3 q^{18} + 28 q^{19} - 4 q^{21} + 6 q^{23} + 15 q^{24} + 4 q^{26} + 13 q^{27} - 20 q^{28} + 12 q^{29} + 26 q^{31} + 53 q^{32} + 16 q^{33} + 8 q^{34} + 17 q^{36} - 4 q^{37} + 2 q^{38} - 8 q^{39} + 24 q^{41} - 4 q^{42} - 6 q^{43} + 4 q^{44} + 16 q^{46} + 13 q^{47} + 29 q^{48} + 21 q^{49} + 12 q^{51} - 32 q^{52} + 6 q^{53} + 3 q^{54} + 28 q^{57} - 4 q^{58} + 34 q^{59} + 24 q^{61} + 30 q^{62} - 4 q^{63} + 13 q^{64} - 24 q^{67} + 44 q^{68} + 6 q^{69} + 20 q^{71} + 15 q^{72} - 6 q^{73} + 20 q^{74} + 66 q^{76} - 2 q^{77} + 4 q^{78} + 6 q^{79} + 13 q^{81} + 20 q^{82} + 14 q^{83} - 20 q^{84} + 48 q^{86} + 12 q^{87} - 22 q^{88} + 36 q^{89} + 4 q^{91} + 4 q^{92} + 26 q^{93} + 3 q^{94} + 53 q^{96} - 32 q^{97} - 39 q^{98} + 16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.105426 −0.0745472 −0.0372736 0.999305i \(-0.511867\pi\)
−0.0372736 + 0.999305i \(0.511867\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.98889 −0.994443
\(5\) 0 0
\(6\) −0.105426 −0.0430398
\(7\) −3.91364 −1.47922 −0.739608 0.673038i \(-0.764989\pi\)
−0.739608 + 0.673038i \(0.764989\pi\)
\(8\) 0.420531 0.148680
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.01909 1.81482 0.907412 0.420242i \(-0.138055\pi\)
0.907412 + 0.420242i \(0.138055\pi\)
\(12\) −1.98889 −0.574142
\(13\) −5.26742 −1.46092 −0.730460 0.682956i \(-0.760694\pi\)
−0.730460 + 0.682956i \(0.760694\pi\)
\(14\) 0.412598 0.110271
\(15\) 0 0
\(16\) 3.93344 0.983359
\(17\) −1.81336 −0.439803 −0.219902 0.975522i \(-0.570574\pi\)
−0.219902 + 0.975522i \(0.570574\pi\)
\(18\) −0.105426 −0.0248491
\(19\) 8.19937 1.88106 0.940532 0.339704i \(-0.110327\pi\)
0.940532 + 0.339704i \(0.110327\pi\)
\(20\) 0 0
\(21\) −3.91364 −0.854025
\(22\) −0.634566 −0.135290
\(23\) −4.74109 −0.988585 −0.494292 0.869296i \(-0.664573\pi\)
−0.494292 + 0.869296i \(0.664573\pi\)
\(24\) 0.420531 0.0858405
\(25\) 0 0
\(26\) 0.555321 0.108907
\(27\) 1.00000 0.192450
\(28\) 7.78377 1.47099
\(29\) −4.95306 −0.919760 −0.459880 0.887981i \(-0.652108\pi\)
−0.459880 + 0.887981i \(0.652108\pi\)
\(30\) 0 0
\(31\) −5.06513 −0.909723 −0.454862 0.890562i \(-0.650311\pi\)
−0.454862 + 0.890562i \(0.650311\pi\)
\(32\) −1.25575 −0.221987
\(33\) 6.01909 1.04779
\(34\) 0.191174 0.0327861
\(35\) 0 0
\(36\) −1.98889 −0.331481
\(37\) 0.310810 0.0510968 0.0255484 0.999674i \(-0.491867\pi\)
0.0255484 + 0.999674i \(0.491867\pi\)
\(38\) −0.864424 −0.140228
\(39\) −5.26742 −0.843462
\(40\) 0 0
\(41\) 2.61857 0.408952 0.204476 0.978872i \(-0.434451\pi\)
0.204476 + 0.978872i \(0.434451\pi\)
\(42\) 0.412598 0.0636652
\(43\) −4.72384 −0.720378 −0.360189 0.932879i \(-0.617288\pi\)
−0.360189 + 0.932879i \(0.617288\pi\)
\(44\) −11.9713 −1.80474
\(45\) 0 0
\(46\) 0.499832 0.0736962
\(47\) 1.00000 0.145865
\(48\) 3.93344 0.567743
\(49\) 8.31654 1.18808
\(50\) 0 0
\(51\) −1.81336 −0.253921
\(52\) 10.4763 1.45280
\(53\) 3.05124 0.419120 0.209560 0.977796i \(-0.432797\pi\)
0.209560 + 0.977796i \(0.432797\pi\)
\(54\) −0.105426 −0.0143466
\(55\) 0 0
\(56\) −1.64580 −0.219930
\(57\) 8.19937 1.08603
\(58\) 0.522180 0.0685655
\(59\) −5.83634 −0.759826 −0.379913 0.925022i \(-0.624046\pi\)
−0.379913 + 0.925022i \(0.624046\pi\)
\(60\) 0 0
\(61\) 10.0059 1.28113 0.640564 0.767905i \(-0.278701\pi\)
0.640564 + 0.767905i \(0.278701\pi\)
\(62\) 0.533994 0.0678173
\(63\) −3.91364 −0.493072
\(64\) −7.73448 −0.966811
\(65\) 0 0
\(66\) −0.634566 −0.0781097
\(67\) 8.37059 1.02263 0.511315 0.859393i \(-0.329159\pi\)
0.511315 + 0.859393i \(0.329159\pi\)
\(68\) 3.60656 0.437359
\(69\) −4.74109 −0.570760
\(70\) 0 0
\(71\) −9.96991 −1.18321 −0.591606 0.806228i \(-0.701506\pi\)
−0.591606 + 0.806228i \(0.701506\pi\)
\(72\) 0.420531 0.0495600
\(73\) 3.65100 0.427317 0.213659 0.976908i \(-0.431462\pi\)
0.213659 + 0.976908i \(0.431462\pi\)
\(74\) −0.0327673 −0.00380912
\(75\) 0 0
\(76\) −16.3076 −1.87061
\(77\) −23.5565 −2.68451
\(78\) 0.555321 0.0628777
\(79\) 9.00605 1.01326 0.506630 0.862163i \(-0.330891\pi\)
0.506630 + 0.862163i \(0.330891\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −0.276064 −0.0304862
\(83\) 12.9270 1.41892 0.709462 0.704744i \(-0.248938\pi\)
0.709462 + 0.704744i \(0.248938\pi\)
\(84\) 7.78377 0.849279
\(85\) 0 0
\(86\) 0.498013 0.0537022
\(87\) −4.95306 −0.531024
\(88\) 2.53121 0.269828
\(89\) 8.60225 0.911836 0.455918 0.890022i \(-0.349311\pi\)
0.455918 + 0.890022i \(0.349311\pi\)
\(90\) 0 0
\(91\) 20.6148 2.16101
\(92\) 9.42948 0.983091
\(93\) −5.06513 −0.525229
\(94\) −0.105426 −0.0108738
\(95\) 0 0
\(96\) −1.25575 −0.128164
\(97\) 8.42460 0.855388 0.427694 0.903924i \(-0.359326\pi\)
0.427694 + 0.903924i \(0.359326\pi\)
\(98\) −0.876777 −0.0885679
\(99\) 6.01909 0.604941
\(100\) 0 0
\(101\) 17.0893 1.70045 0.850225 0.526420i \(-0.176466\pi\)
0.850225 + 0.526420i \(0.176466\pi\)
\(102\) 0.191174 0.0189291
\(103\) 11.9079 1.17332 0.586662 0.809832i \(-0.300442\pi\)
0.586662 + 0.809832i \(0.300442\pi\)
\(104\) −2.21511 −0.217210
\(105\) 0 0
\(106\) −0.321679 −0.0312443
\(107\) 12.2483 1.18409 0.592045 0.805905i \(-0.298321\pi\)
0.592045 + 0.805905i \(0.298321\pi\)
\(108\) −1.98889 −0.191381
\(109\) 8.14773 0.780411 0.390205 0.920728i \(-0.372404\pi\)
0.390205 + 0.920728i \(0.372404\pi\)
\(110\) 0 0
\(111\) 0.310810 0.0295008
\(112\) −15.3940 −1.45460
\(113\) −5.65228 −0.531722 −0.265861 0.964011i \(-0.585656\pi\)
−0.265861 + 0.964011i \(0.585656\pi\)
\(114\) −0.864424 −0.0809607
\(115\) 0 0
\(116\) 9.85107 0.914649
\(117\) −5.26742 −0.486973
\(118\) 0.615299 0.0566429
\(119\) 7.09681 0.650564
\(120\) 0 0
\(121\) 25.2294 2.29359
\(122\) −1.05488 −0.0955045
\(123\) 2.61857 0.236108
\(124\) 10.0740 0.904668
\(125\) 0 0
\(126\) 0.412598 0.0367571
\(127\) −0.531624 −0.0471740 −0.0235870 0.999722i \(-0.507509\pi\)
−0.0235870 + 0.999722i \(0.507509\pi\)
\(128\) 3.32691 0.294060
\(129\) −4.72384 −0.415911
\(130\) 0 0
\(131\) −5.63792 −0.492587 −0.246294 0.969195i \(-0.579213\pi\)
−0.246294 + 0.969195i \(0.579213\pi\)
\(132\) −11.9713 −1.04197
\(133\) −32.0894 −2.78250
\(134\) −0.882475 −0.0762342
\(135\) 0 0
\(136\) −0.762572 −0.0653900
\(137\) 8.43334 0.720509 0.360254 0.932854i \(-0.382690\pi\)
0.360254 + 0.932854i \(0.382690\pi\)
\(138\) 0.499832 0.0425485
\(139\) −8.77831 −0.744566 −0.372283 0.928119i \(-0.621425\pi\)
−0.372283 + 0.928119i \(0.621425\pi\)
\(140\) 0 0
\(141\) 1.00000 0.0842152
\(142\) 1.05108 0.0882051
\(143\) −31.7051 −2.65131
\(144\) 3.93344 0.327786
\(145\) 0 0
\(146\) −0.384909 −0.0318553
\(147\) 8.31654 0.685937
\(148\) −0.618165 −0.0508128
\(149\) −5.39214 −0.441741 −0.220871 0.975303i \(-0.570890\pi\)
−0.220871 + 0.975303i \(0.570890\pi\)
\(150\) 0 0
\(151\) −2.79825 −0.227718 −0.113859 0.993497i \(-0.536321\pi\)
−0.113859 + 0.993497i \(0.536321\pi\)
\(152\) 3.44809 0.279677
\(153\) −1.81336 −0.146601
\(154\) 2.48346 0.200123
\(155\) 0 0
\(156\) 10.4763 0.838775
\(157\) 24.1331 1.92603 0.963017 0.269442i \(-0.0868392\pi\)
0.963017 + 0.269442i \(0.0868392\pi\)
\(158\) −0.949469 −0.0755357
\(159\) 3.05124 0.241979
\(160\) 0 0
\(161\) 18.5549 1.46233
\(162\) −0.105426 −0.00828302
\(163\) 17.8940 1.40157 0.700783 0.713374i \(-0.252834\pi\)
0.700783 + 0.713374i \(0.252834\pi\)
\(164\) −5.20803 −0.406679
\(165\) 0 0
\(166\) −1.36284 −0.105777
\(167\) −0.300971 −0.0232899 −0.0116449 0.999932i \(-0.503707\pi\)
−0.0116449 + 0.999932i \(0.503707\pi\)
\(168\) −1.64580 −0.126977
\(169\) 14.7457 1.13429
\(170\) 0 0
\(171\) 8.19937 0.627022
\(172\) 9.39517 0.716375
\(173\) 11.3416 0.862285 0.431143 0.902284i \(-0.358111\pi\)
0.431143 + 0.902284i \(0.358111\pi\)
\(174\) 0.522180 0.0395863
\(175\) 0 0
\(176\) 23.6757 1.78462
\(177\) −5.83634 −0.438686
\(178\) −0.906898 −0.0679748
\(179\) −2.95113 −0.220577 −0.110289 0.993900i \(-0.535178\pi\)
−0.110289 + 0.993900i \(0.535178\pi\)
\(180\) 0 0
\(181\) −12.8185 −0.952791 −0.476396 0.879231i \(-0.658057\pi\)
−0.476396 + 0.879231i \(0.658057\pi\)
\(182\) −2.17332 −0.161098
\(183\) 10.0059 0.739659
\(184\) −1.99377 −0.146983
\(185\) 0 0
\(186\) 0.533994 0.0391544
\(187\) −10.9148 −0.798166
\(188\) −1.98889 −0.145054
\(189\) −3.91364 −0.284675
\(190\) 0 0
\(191\) 15.8899 1.14975 0.574876 0.818241i \(-0.305050\pi\)
0.574876 + 0.818241i \(0.305050\pi\)
\(192\) −7.73448 −0.558188
\(193\) −15.7194 −1.13151 −0.565753 0.824575i \(-0.691415\pi\)
−0.565753 + 0.824575i \(0.691415\pi\)
\(194\) −0.888169 −0.0637668
\(195\) 0 0
\(196\) −16.5407 −1.18148
\(197\) 18.7134 1.33327 0.666636 0.745383i \(-0.267733\pi\)
0.666636 + 0.745383i \(0.267733\pi\)
\(198\) −0.634566 −0.0450967
\(199\) −10.8238 −0.767276 −0.383638 0.923484i \(-0.625329\pi\)
−0.383638 + 0.923484i \(0.625329\pi\)
\(200\) 0 0
\(201\) 8.37059 0.590416
\(202\) −1.80165 −0.126764
\(203\) 19.3845 1.36052
\(204\) 3.60656 0.252510
\(205\) 0 0
\(206\) −1.25540 −0.0874680
\(207\) −4.74109 −0.329528
\(208\) −20.7191 −1.43661
\(209\) 49.3528 3.41380
\(210\) 0 0
\(211\) 8.37231 0.576373 0.288187 0.957574i \(-0.406948\pi\)
0.288187 + 0.957574i \(0.406948\pi\)
\(212\) −6.06857 −0.416791
\(213\) −9.96991 −0.683127
\(214\) −1.29129 −0.0882706
\(215\) 0 0
\(216\) 0.420531 0.0286135
\(217\) 19.8231 1.34568
\(218\) −0.858979 −0.0581774
\(219\) 3.65100 0.246712
\(220\) 0 0
\(221\) 9.55171 0.642517
\(222\) −0.0327673 −0.00219920
\(223\) −1.53200 −0.102590 −0.0512950 0.998684i \(-0.516335\pi\)
−0.0512950 + 0.998684i \(0.516335\pi\)
\(224\) 4.91454 0.328366
\(225\) 0 0
\(226\) 0.595895 0.0396384
\(227\) −18.1538 −1.20491 −0.602454 0.798154i \(-0.705810\pi\)
−0.602454 + 0.798154i \(0.705810\pi\)
\(228\) −16.3076 −1.08000
\(229\) 15.1187 0.999071 0.499536 0.866293i \(-0.333504\pi\)
0.499536 + 0.866293i \(0.333504\pi\)
\(230\) 0 0
\(231\) −23.5565 −1.54991
\(232\) −2.08291 −0.136750
\(233\) 19.5817 1.28284 0.641419 0.767191i \(-0.278346\pi\)
0.641419 + 0.767191i \(0.278346\pi\)
\(234\) 0.555321 0.0363025
\(235\) 0 0
\(236\) 11.6078 0.755604
\(237\) 9.00605 0.585006
\(238\) −0.748186 −0.0484977
\(239\) −9.11127 −0.589359 −0.294680 0.955596i \(-0.595213\pi\)
−0.294680 + 0.955596i \(0.595213\pi\)
\(240\) 0 0
\(241\) −26.0653 −1.67902 −0.839508 0.543348i \(-0.817156\pi\)
−0.839508 + 0.543348i \(0.817156\pi\)
\(242\) −2.65983 −0.170980
\(243\) 1.00000 0.0641500
\(244\) −19.9006 −1.27401
\(245\) 0 0
\(246\) −0.276064 −0.0176012
\(247\) −43.1895 −2.74808
\(248\) −2.13004 −0.135258
\(249\) 12.9270 0.819216
\(250\) 0 0
\(251\) −23.4279 −1.47876 −0.739378 0.673290i \(-0.764880\pi\)
−0.739378 + 0.673290i \(0.764880\pi\)
\(252\) 7.78377 0.490332
\(253\) −28.5370 −1.79411
\(254\) 0.0560468 0.00351669
\(255\) 0 0
\(256\) 15.1182 0.944889
\(257\) −1.10336 −0.0688257 −0.0344128 0.999408i \(-0.510956\pi\)
−0.0344128 + 0.999408i \(0.510956\pi\)
\(258\) 0.498013 0.0310050
\(259\) −1.21640 −0.0755832
\(260\) 0 0
\(261\) −4.95306 −0.306587
\(262\) 0.594381 0.0367210
\(263\) 9.70681 0.598548 0.299274 0.954167i \(-0.403256\pi\)
0.299274 + 0.954167i \(0.403256\pi\)
\(264\) 2.53121 0.155785
\(265\) 0 0
\(266\) 3.38304 0.207428
\(267\) 8.60225 0.526449
\(268\) −16.6481 −1.01695
\(269\) −19.1818 −1.16954 −0.584768 0.811200i \(-0.698815\pi\)
−0.584768 + 0.811200i \(0.698815\pi\)
\(270\) 0 0
\(271\) 7.39437 0.449176 0.224588 0.974454i \(-0.427896\pi\)
0.224588 + 0.974454i \(0.427896\pi\)
\(272\) −7.13272 −0.432485
\(273\) 20.6148 1.24766
\(274\) −0.889090 −0.0537119
\(275\) 0 0
\(276\) 9.42948 0.567588
\(277\) −4.95059 −0.297452 −0.148726 0.988878i \(-0.547517\pi\)
−0.148726 + 0.988878i \(0.547517\pi\)
\(278\) 0.925459 0.0555053
\(279\) −5.06513 −0.303241
\(280\) 0 0
\(281\) −2.11672 −0.126273 −0.0631365 0.998005i \(-0.520110\pi\)
−0.0631365 + 0.998005i \(0.520110\pi\)
\(282\) −0.105426 −0.00627801
\(283\) −8.64610 −0.513957 −0.256979 0.966417i \(-0.582727\pi\)
−0.256979 + 0.966417i \(0.582727\pi\)
\(284\) 19.8290 1.17664
\(285\) 0 0
\(286\) 3.34253 0.197648
\(287\) −10.2481 −0.604928
\(288\) −1.25575 −0.0739956
\(289\) −13.7117 −0.806573
\(290\) 0 0
\(291\) 8.42460 0.493859
\(292\) −7.26142 −0.424942
\(293\) 6.47739 0.378413 0.189207 0.981937i \(-0.439408\pi\)
0.189207 + 0.981937i \(0.439408\pi\)
\(294\) −0.876777 −0.0511347
\(295\) 0 0
\(296\) 0.130705 0.00759708
\(297\) 6.01909 0.349263
\(298\) 0.568470 0.0329306
\(299\) 24.9733 1.44424
\(300\) 0 0
\(301\) 18.4874 1.06559
\(302\) 0.295007 0.0169757
\(303\) 17.0893 0.981755
\(304\) 32.2517 1.84976
\(305\) 0 0
\(306\) 0.191174 0.0109287
\(307\) −28.5347 −1.62856 −0.814279 0.580473i \(-0.802868\pi\)
−0.814279 + 0.580473i \(0.802868\pi\)
\(308\) 46.8512 2.66960
\(309\) 11.9079 0.677419
\(310\) 0 0
\(311\) 5.78357 0.327956 0.163978 0.986464i \(-0.447567\pi\)
0.163978 + 0.986464i \(0.447567\pi\)
\(312\) −2.21511 −0.125406
\(313\) −7.35209 −0.415565 −0.207782 0.978175i \(-0.566625\pi\)
−0.207782 + 0.978175i \(0.566625\pi\)
\(314\) −2.54425 −0.143580
\(315\) 0 0
\(316\) −17.9120 −1.00763
\(317\) 34.6205 1.94448 0.972242 0.233979i \(-0.0751747\pi\)
0.972242 + 0.233979i \(0.0751747\pi\)
\(318\) −0.321679 −0.0180389
\(319\) −29.8129 −1.66920
\(320\) 0 0
\(321\) 12.2483 0.683635
\(322\) −1.95616 −0.109013
\(323\) −14.8684 −0.827299
\(324\) −1.98889 −0.110494
\(325\) 0 0
\(326\) −1.88649 −0.104483
\(327\) 8.14773 0.450570
\(328\) 1.10119 0.0608030
\(329\) −3.91364 −0.215766
\(330\) 0 0
\(331\) −16.8888 −0.928294 −0.464147 0.885758i \(-0.653639\pi\)
−0.464147 + 0.885758i \(0.653639\pi\)
\(332\) −25.7103 −1.41104
\(333\) 0.310810 0.0170323
\(334\) 0.0317301 0.00173619
\(335\) 0 0
\(336\) −15.3940 −0.839814
\(337\) −21.8081 −1.18797 −0.593983 0.804478i \(-0.702445\pi\)
−0.593983 + 0.804478i \(0.702445\pi\)
\(338\) −1.55458 −0.0845578
\(339\) −5.65228 −0.306990
\(340\) 0 0
\(341\) −30.4874 −1.65099
\(342\) −0.864424 −0.0467427
\(343\) −5.15248 −0.278208
\(344\) −1.98652 −0.107106
\(345\) 0 0
\(346\) −1.19569 −0.0642809
\(347\) 8.30860 0.446029 0.223015 0.974815i \(-0.428410\pi\)
0.223015 + 0.974815i \(0.428410\pi\)
\(348\) 9.85107 0.528073
\(349\) −31.0289 −1.66094 −0.830469 0.557065i \(-0.811927\pi\)
−0.830469 + 0.557065i \(0.811927\pi\)
\(350\) 0 0
\(351\) −5.26742 −0.281154
\(352\) −7.55845 −0.402867
\(353\) 5.20699 0.277140 0.138570 0.990353i \(-0.455749\pi\)
0.138570 + 0.990353i \(0.455749\pi\)
\(354\) 0.615299 0.0327028
\(355\) 0 0
\(356\) −17.1089 −0.906769
\(357\) 7.09681 0.375603
\(358\) 0.311124 0.0164434
\(359\) 10.4951 0.553908 0.276954 0.960883i \(-0.410675\pi\)
0.276954 + 0.960883i \(0.410675\pi\)
\(360\) 0 0
\(361\) 48.2297 2.53841
\(362\) 1.35140 0.0710279
\(363\) 25.2294 1.32420
\(364\) −41.0004 −2.14900
\(365\) 0 0
\(366\) −1.05488 −0.0551395
\(367\) 9.10793 0.475430 0.237715 0.971335i \(-0.423602\pi\)
0.237715 + 0.971335i \(0.423602\pi\)
\(368\) −18.6488 −0.972134
\(369\) 2.61857 0.136317
\(370\) 0 0
\(371\) −11.9415 −0.619969
\(372\) 10.0740 0.522310
\(373\) −5.56531 −0.288161 −0.144080 0.989566i \(-0.546022\pi\)
−0.144080 + 0.989566i \(0.546022\pi\)
\(374\) 1.15069 0.0595010
\(375\) 0 0
\(376\) 0.420531 0.0216872
\(377\) 26.0898 1.34370
\(378\) 0.412598 0.0212217
\(379\) 29.4344 1.51194 0.755971 0.654605i \(-0.227165\pi\)
0.755971 + 0.654605i \(0.227165\pi\)
\(380\) 0 0
\(381\) −0.531624 −0.0272359
\(382\) −1.67520 −0.0857108
\(383\) −20.1438 −1.02930 −0.514649 0.857401i \(-0.672078\pi\)
−0.514649 + 0.857401i \(0.672078\pi\)
\(384\) 3.32691 0.169776
\(385\) 0 0
\(386\) 1.65723 0.0843506
\(387\) −4.72384 −0.240126
\(388\) −16.7556 −0.850635
\(389\) −7.57089 −0.383860 −0.191930 0.981409i \(-0.561475\pi\)
−0.191930 + 0.981409i \(0.561475\pi\)
\(390\) 0 0
\(391\) 8.59728 0.434783
\(392\) 3.49736 0.176644
\(393\) −5.63792 −0.284395
\(394\) −1.97287 −0.0993918
\(395\) 0 0
\(396\) −11.9713 −0.601579
\(397\) 9.90230 0.496982 0.248491 0.968634i \(-0.420065\pi\)
0.248491 + 0.968634i \(0.420065\pi\)
\(398\) 1.14110 0.0571983
\(399\) −32.0894 −1.60648
\(400\) 0 0
\(401\) −4.11343 −0.205415 −0.102707 0.994712i \(-0.532751\pi\)
−0.102707 + 0.994712i \(0.532751\pi\)
\(402\) −0.882475 −0.0440138
\(403\) 26.6801 1.32903
\(404\) −33.9887 −1.69100
\(405\) 0 0
\(406\) −2.04362 −0.101423
\(407\) 1.87079 0.0927317
\(408\) −0.762572 −0.0377529
\(409\) 9.70852 0.480056 0.240028 0.970766i \(-0.422843\pi\)
0.240028 + 0.970766i \(0.422843\pi\)
\(410\) 0 0
\(411\) 8.43334 0.415986
\(412\) −23.6835 −1.16680
\(413\) 22.8413 1.12395
\(414\) 0.499832 0.0245654
\(415\) 0 0
\(416\) 6.61455 0.324305
\(417\) −8.77831 −0.429876
\(418\) −5.20305 −0.254489
\(419\) 11.7035 0.571753 0.285877 0.958266i \(-0.407715\pi\)
0.285877 + 0.958266i \(0.407715\pi\)
\(420\) 0 0
\(421\) 4.14482 0.202006 0.101003 0.994886i \(-0.467795\pi\)
0.101003 + 0.994886i \(0.467795\pi\)
\(422\) −0.882656 −0.0429670
\(423\) 1.00000 0.0486217
\(424\) 1.28314 0.0623149
\(425\) 0 0
\(426\) 1.05108 0.0509252
\(427\) −39.1596 −1.89506
\(428\) −24.3605 −1.17751
\(429\) −31.7051 −1.53074
\(430\) 0 0
\(431\) 6.28815 0.302890 0.151445 0.988466i \(-0.451607\pi\)
0.151445 + 0.988466i \(0.451607\pi\)
\(432\) 3.93344 0.189248
\(433\) 9.10564 0.437589 0.218795 0.975771i \(-0.429788\pi\)
0.218795 + 0.975771i \(0.429788\pi\)
\(434\) −2.08986 −0.100316
\(435\) 0 0
\(436\) −16.2049 −0.776074
\(437\) −38.8739 −1.85959
\(438\) −0.384909 −0.0183917
\(439\) 26.1154 1.24642 0.623210 0.782054i \(-0.285828\pi\)
0.623210 + 0.782054i \(0.285828\pi\)
\(440\) 0 0
\(441\) 8.31654 0.396026
\(442\) −1.00700 −0.0478979
\(443\) 22.6260 1.07499 0.537497 0.843266i \(-0.319370\pi\)
0.537497 + 0.843266i \(0.319370\pi\)
\(444\) −0.618165 −0.0293368
\(445\) 0 0
\(446\) 0.161512 0.00764780
\(447\) −5.39214 −0.255039
\(448\) 30.2700 1.43012
\(449\) 3.25384 0.153558 0.0767790 0.997048i \(-0.475536\pi\)
0.0767790 + 0.997048i \(0.475536\pi\)
\(450\) 0 0
\(451\) 15.7614 0.742175
\(452\) 11.2417 0.528767
\(453\) −2.79825 −0.131473
\(454\) 1.91387 0.0898225
\(455\) 0 0
\(456\) 3.44809 0.161472
\(457\) −7.68301 −0.359396 −0.179698 0.983722i \(-0.557512\pi\)
−0.179698 + 0.983722i \(0.557512\pi\)
\(458\) −1.59390 −0.0744780
\(459\) −1.81336 −0.0846402
\(460\) 0 0
\(461\) −34.6542 −1.61401 −0.807004 0.590546i \(-0.798913\pi\)
−0.807004 + 0.590546i \(0.798913\pi\)
\(462\) 2.48346 0.115541
\(463\) 3.09033 0.143620 0.0718098 0.997418i \(-0.477123\pi\)
0.0718098 + 0.997418i \(0.477123\pi\)
\(464\) −19.4825 −0.904454
\(465\) 0 0
\(466\) −2.06441 −0.0956319
\(467\) −37.3216 −1.72704 −0.863520 0.504315i \(-0.831745\pi\)
−0.863520 + 0.504315i \(0.831745\pi\)
\(468\) 10.4763 0.484267
\(469\) −32.7594 −1.51269
\(470\) 0 0
\(471\) 24.1331 1.11200
\(472\) −2.45436 −0.112971
\(473\) −28.4332 −1.30736
\(474\) −0.949469 −0.0436106
\(475\) 0 0
\(476\) −14.1148 −0.646949
\(477\) 3.05124 0.139707
\(478\) 0.960562 0.0439351
\(479\) 23.6474 1.08048 0.540239 0.841512i \(-0.318334\pi\)
0.540239 + 0.841512i \(0.318334\pi\)
\(480\) 0 0
\(481\) −1.63717 −0.0746483
\(482\) 2.74795 0.125166
\(483\) 18.5549 0.844276
\(484\) −50.1785 −2.28084
\(485\) 0 0
\(486\) −0.105426 −0.00478220
\(487\) 20.9636 0.949953 0.474977 0.879998i \(-0.342457\pi\)
0.474977 + 0.879998i \(0.342457\pi\)
\(488\) 4.20780 0.190478
\(489\) 17.8940 0.809195
\(490\) 0 0
\(491\) 11.3991 0.514432 0.257216 0.966354i \(-0.417195\pi\)
0.257216 + 0.966354i \(0.417195\pi\)
\(492\) −5.20803 −0.234796
\(493\) 8.98166 0.404514
\(494\) 4.55329 0.204862
\(495\) 0 0
\(496\) −19.9233 −0.894585
\(497\) 39.0186 1.75022
\(498\) −1.36284 −0.0610702
\(499\) 5.18330 0.232036 0.116018 0.993247i \(-0.462987\pi\)
0.116018 + 0.993247i \(0.462987\pi\)
\(500\) 0 0
\(501\) −0.300971 −0.0134464
\(502\) 2.46990 0.110237
\(503\) 31.1393 1.38843 0.694217 0.719766i \(-0.255751\pi\)
0.694217 + 0.719766i \(0.255751\pi\)
\(504\) −1.64580 −0.0733100
\(505\) 0 0
\(506\) 3.00853 0.133746
\(507\) 14.7457 0.654880
\(508\) 1.05734 0.0469119
\(509\) 23.8471 1.05700 0.528502 0.848932i \(-0.322754\pi\)
0.528502 + 0.848932i \(0.322754\pi\)
\(510\) 0 0
\(511\) −14.2887 −0.632094
\(512\) −8.24766 −0.364499
\(513\) 8.19937 0.362011
\(514\) 0.116322 0.00513076
\(515\) 0 0
\(516\) 9.39517 0.413599
\(517\) 6.01909 0.264719
\(518\) 0.128239 0.00563451
\(519\) 11.3416 0.497841
\(520\) 0 0
\(521\) −35.8990 −1.57276 −0.786382 0.617741i \(-0.788048\pi\)
−0.786382 + 0.617741i \(0.788048\pi\)
\(522\) 0.522180 0.0228552
\(523\) −1.71087 −0.0748113 −0.0374057 0.999300i \(-0.511909\pi\)
−0.0374057 + 0.999300i \(0.511909\pi\)
\(524\) 11.2132 0.489850
\(525\) 0 0
\(526\) −1.02335 −0.0446200
\(527\) 9.18487 0.400099
\(528\) 23.6757 1.03035
\(529\) −0.522109 −0.0227004
\(530\) 0 0
\(531\) −5.83634 −0.253275
\(532\) 63.8221 2.76704
\(533\) −13.7931 −0.597446
\(534\) −0.906898 −0.0392453
\(535\) 0 0
\(536\) 3.52009 0.152045
\(537\) −2.95113 −0.127350
\(538\) 2.02226 0.0871857
\(539\) 50.0580 2.15615
\(540\) 0 0
\(541\) 11.3408 0.487577 0.243788 0.969828i \(-0.421610\pi\)
0.243788 + 0.969828i \(0.421610\pi\)
\(542\) −0.779556 −0.0334848
\(543\) −12.8185 −0.550094
\(544\) 2.27712 0.0976305
\(545\) 0 0
\(546\) −2.17332 −0.0930097
\(547\) −0.484399 −0.0207114 −0.0103557 0.999946i \(-0.503296\pi\)
−0.0103557 + 0.999946i \(0.503296\pi\)
\(548\) −16.7729 −0.716505
\(549\) 10.0059 0.427043
\(550\) 0 0
\(551\) −40.6120 −1.73013
\(552\) −1.99377 −0.0848606
\(553\) −35.2464 −1.49883
\(554\) 0.521920 0.0221742
\(555\) 0 0
\(556\) 17.4590 0.740428
\(557\) −21.3581 −0.904970 −0.452485 0.891772i \(-0.649462\pi\)
−0.452485 + 0.891772i \(0.649462\pi\)
\(558\) 0.533994 0.0226058
\(559\) 24.8824 1.05241
\(560\) 0 0
\(561\) −10.9148 −0.460821
\(562\) 0.223157 0.00941330
\(563\) 11.0818 0.467044 0.233522 0.972352i \(-0.424975\pi\)
0.233522 + 0.972352i \(0.424975\pi\)
\(564\) −1.98889 −0.0837472
\(565\) 0 0
\(566\) 0.911521 0.0383141
\(567\) −3.91364 −0.164357
\(568\) −4.19266 −0.175920
\(569\) −25.7144 −1.07800 −0.539001 0.842305i \(-0.681198\pi\)
−0.539001 + 0.842305i \(0.681198\pi\)
\(570\) 0 0
\(571\) 4.26461 0.178468 0.0892342 0.996011i \(-0.471558\pi\)
0.0892342 + 0.996011i \(0.471558\pi\)
\(572\) 63.0578 2.63658
\(573\) 15.8899 0.663810
\(574\) 1.08042 0.0450957
\(575\) 0 0
\(576\) −7.73448 −0.322270
\(577\) 16.2722 0.677422 0.338711 0.940890i \(-0.390009\pi\)
0.338711 + 0.940890i \(0.390009\pi\)
\(578\) 1.44557 0.0601278
\(579\) −15.7194 −0.653276
\(580\) 0 0
\(581\) −50.5916 −2.09889
\(582\) −0.888169 −0.0368158
\(583\) 18.3657 0.760630
\(584\) 1.53536 0.0635336
\(585\) 0 0
\(586\) −0.682883 −0.0282097
\(587\) 16.0920 0.664189 0.332094 0.943246i \(-0.392245\pi\)
0.332094 + 0.943246i \(0.392245\pi\)
\(588\) −16.5407 −0.682125
\(589\) −41.5308 −1.71125
\(590\) 0 0
\(591\) 18.7134 0.769765
\(592\) 1.22255 0.0502465
\(593\) 34.4845 1.41611 0.708055 0.706157i \(-0.249573\pi\)
0.708055 + 0.706157i \(0.249573\pi\)
\(594\) −0.634566 −0.0260366
\(595\) 0 0
\(596\) 10.7243 0.439286
\(597\) −10.8238 −0.442987
\(598\) −2.63283 −0.107664
\(599\) −25.1565 −1.02787 −0.513934 0.857830i \(-0.671812\pi\)
−0.513934 + 0.857830i \(0.671812\pi\)
\(600\) 0 0
\(601\) 20.3707 0.830937 0.415469 0.909608i \(-0.363618\pi\)
0.415469 + 0.909608i \(0.363618\pi\)
\(602\) −1.94904 −0.0794371
\(603\) 8.37059 0.340877
\(604\) 5.56539 0.226453
\(605\) 0 0
\(606\) −1.80165 −0.0731871
\(607\) 35.8381 1.45462 0.727312 0.686307i \(-0.240769\pi\)
0.727312 + 0.686307i \(0.240769\pi\)
\(608\) −10.2963 −0.417572
\(609\) 19.3845 0.785499
\(610\) 0 0
\(611\) −5.26742 −0.213097
\(612\) 3.60656 0.145786
\(613\) −15.4848 −0.625426 −0.312713 0.949848i \(-0.601238\pi\)
−0.312713 + 0.949848i \(0.601238\pi\)
\(614\) 3.00828 0.121404
\(615\) 0 0
\(616\) −9.90625 −0.399134
\(617\) −2.17201 −0.0874418 −0.0437209 0.999044i \(-0.513921\pi\)
−0.0437209 + 0.999044i \(0.513921\pi\)
\(618\) −1.25540 −0.0504997
\(619\) 48.9908 1.96911 0.984553 0.175088i \(-0.0560210\pi\)
0.984553 + 0.175088i \(0.0560210\pi\)
\(620\) 0 0
\(621\) −4.74109 −0.190253
\(622\) −0.609736 −0.0244482
\(623\) −33.6661 −1.34880
\(624\) −20.7191 −0.829426
\(625\) 0 0
\(626\) 0.775099 0.0309792
\(627\) 49.3528 1.97096
\(628\) −47.9980 −1.91533
\(629\) −0.563609 −0.0224725
\(630\) 0 0
\(631\) −39.3727 −1.56740 −0.783701 0.621138i \(-0.786670\pi\)
−0.783701 + 0.621138i \(0.786670\pi\)
\(632\) 3.78732 0.150652
\(633\) 8.37231 0.332769
\(634\) −3.64989 −0.144956
\(635\) 0 0
\(636\) −6.06857 −0.240635
\(637\) −43.8067 −1.73569
\(638\) 3.14305 0.124434
\(639\) −9.96991 −0.394404
\(640\) 0 0
\(641\) 6.26333 0.247387 0.123693 0.992320i \(-0.460526\pi\)
0.123693 + 0.992320i \(0.460526\pi\)
\(642\) −1.29129 −0.0509631
\(643\) −9.33050 −0.367959 −0.183980 0.982930i \(-0.558898\pi\)
−0.183980 + 0.982930i \(0.558898\pi\)
\(644\) −36.9035 −1.45420
\(645\) 0 0
\(646\) 1.56751 0.0616728
\(647\) 39.0295 1.53441 0.767204 0.641404i \(-0.221648\pi\)
0.767204 + 0.641404i \(0.221648\pi\)
\(648\) 0.420531 0.0165200
\(649\) −35.1294 −1.37895
\(650\) 0 0
\(651\) 19.8231 0.776927
\(652\) −35.5891 −1.39378
\(653\) 0.459927 0.0179983 0.00899916 0.999960i \(-0.497135\pi\)
0.00899916 + 0.999960i \(0.497135\pi\)
\(654\) −0.858979 −0.0335888
\(655\) 0 0
\(656\) 10.3000 0.402146
\(657\) 3.65100 0.142439
\(658\) 0.412598 0.0160847
\(659\) 24.7401 0.963738 0.481869 0.876243i \(-0.339958\pi\)
0.481869 + 0.876243i \(0.339958\pi\)
\(660\) 0 0
\(661\) −10.4421 −0.406149 −0.203075 0.979163i \(-0.565093\pi\)
−0.203075 + 0.979163i \(0.565093\pi\)
\(662\) 1.78052 0.0692017
\(663\) 9.55171 0.370958
\(664\) 5.43621 0.210966
\(665\) 0 0
\(666\) −0.0327673 −0.00126971
\(667\) 23.4829 0.909261
\(668\) 0.598598 0.0231604
\(669\) −1.53200 −0.0592304
\(670\) 0 0
\(671\) 60.2266 2.32502
\(672\) 4.91454 0.189582
\(673\) 0.153505 0.00591720 0.00295860 0.999996i \(-0.499058\pi\)
0.00295860 + 0.999996i \(0.499058\pi\)
\(674\) 2.29914 0.0885595
\(675\) 0 0
\(676\) −29.3275 −1.12798
\(677\) 35.9544 1.38184 0.690920 0.722931i \(-0.257206\pi\)
0.690920 + 0.722931i \(0.257206\pi\)
\(678\) 0.595895 0.0228852
\(679\) −32.9708 −1.26530
\(680\) 0 0
\(681\) −18.1538 −0.695654
\(682\) 3.21416 0.123076
\(683\) 2.63818 0.100947 0.0504736 0.998725i \(-0.483927\pi\)
0.0504736 + 0.998725i \(0.483927\pi\)
\(684\) −16.3076 −0.623537
\(685\) 0 0
\(686\) 0.543203 0.0207396
\(687\) 15.1187 0.576814
\(688\) −18.5809 −0.708390
\(689\) −16.0722 −0.612301
\(690\) 0 0
\(691\) 5.92106 0.225248 0.112624 0.993638i \(-0.464074\pi\)
0.112624 + 0.993638i \(0.464074\pi\)
\(692\) −22.5571 −0.857493
\(693\) −23.5565 −0.894838
\(694\) −0.875940 −0.0332502
\(695\) 0 0
\(696\) −2.08291 −0.0789527
\(697\) −4.74840 −0.179858
\(698\) 3.27124 0.123818
\(699\) 19.5817 0.740646
\(700\) 0 0
\(701\) −30.9641 −1.16950 −0.584749 0.811214i \(-0.698807\pi\)
−0.584749 + 0.811214i \(0.698807\pi\)
\(702\) 0.555321 0.0209592
\(703\) 2.54844 0.0961164
\(704\) −46.5546 −1.75459
\(705\) 0 0
\(706\) −0.548950 −0.0206600
\(707\) −66.8813 −2.51533
\(708\) 11.6078 0.436248
\(709\) 15.6593 0.588096 0.294048 0.955791i \(-0.404998\pi\)
0.294048 + 0.955791i \(0.404998\pi\)
\(710\) 0 0
\(711\) 9.00605 0.337753
\(712\) 3.61751 0.135572
\(713\) 24.0142 0.899339
\(714\) −0.748186 −0.0280002
\(715\) 0 0
\(716\) 5.86945 0.219352
\(717\) −9.11127 −0.340267
\(718\) −1.10645 −0.0412923
\(719\) 31.2286 1.16463 0.582315 0.812963i \(-0.302147\pi\)
0.582315 + 0.812963i \(0.302147\pi\)
\(720\) 0 0
\(721\) −46.6033 −1.73560
\(722\) −5.08465 −0.189231
\(723\) −26.0653 −0.969380
\(724\) 25.4945 0.947496
\(725\) 0 0
\(726\) −2.65983 −0.0987155
\(727\) −0.383501 −0.0142233 −0.00711163 0.999975i \(-0.502264\pi\)
−0.00711163 + 0.999975i \(0.502264\pi\)
\(728\) 8.66914 0.321300
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.56600 0.316825
\(732\) −19.9006 −0.735549
\(733\) −41.5815 −1.53585 −0.767924 0.640541i \(-0.778710\pi\)
−0.767924 + 0.640541i \(0.778710\pi\)
\(734\) −0.960210 −0.0354420
\(735\) 0 0
\(736\) 5.95360 0.219453
\(737\) 50.3833 1.85589
\(738\) −0.276064 −0.0101621
\(739\) 16.4098 0.603645 0.301822 0.953364i \(-0.402405\pi\)
0.301822 + 0.953364i \(0.402405\pi\)
\(740\) 0 0
\(741\) −43.1895 −1.58661
\(742\) 1.25894 0.0462170
\(743\) −5.59200 −0.205151 −0.102575 0.994725i \(-0.532708\pi\)
−0.102575 + 0.994725i \(0.532708\pi\)
\(744\) −2.13004 −0.0780911
\(745\) 0 0
\(746\) 0.586727 0.0214816
\(747\) 12.9270 0.472975
\(748\) 21.7082 0.793730
\(749\) −47.9355 −1.75152
\(750\) 0 0
\(751\) 33.6322 1.22725 0.613627 0.789596i \(-0.289710\pi\)
0.613627 + 0.789596i \(0.289710\pi\)
\(752\) 3.93344 0.143438
\(753\) −23.4279 −0.853761
\(754\) −2.75054 −0.100169
\(755\) 0 0
\(756\) 7.78377 0.283093
\(757\) −19.9225 −0.724095 −0.362048 0.932160i \(-0.617922\pi\)
−0.362048 + 0.932160i \(0.617922\pi\)
\(758\) −3.10314 −0.112711
\(759\) −28.5370 −1.03583
\(760\) 0 0
\(761\) 5.15807 0.186980 0.0934899 0.995620i \(-0.470198\pi\)
0.0934899 + 0.995620i \(0.470198\pi\)
\(762\) 0.0560468 0.00203036
\(763\) −31.8872 −1.15440
\(764\) −31.6032 −1.14336
\(765\) 0 0
\(766\) 2.12367 0.0767313
\(767\) 30.7424 1.11004
\(768\) 15.1182 0.545532
\(769\) 32.8776 1.18560 0.592799 0.805351i \(-0.298023\pi\)
0.592799 + 0.805351i \(0.298023\pi\)
\(770\) 0 0
\(771\) −1.10336 −0.0397365
\(772\) 31.2641 1.12522
\(773\) −15.8051 −0.568471 −0.284236 0.958754i \(-0.591740\pi\)
−0.284236 + 0.958754i \(0.591740\pi\)
\(774\) 0.498013 0.0179007
\(775\) 0 0
\(776\) 3.54280 0.127179
\(777\) −1.21640 −0.0436380
\(778\) 0.798166 0.0286157
\(779\) 21.4706 0.769265
\(780\) 0 0
\(781\) −60.0098 −2.14732
\(782\) −0.906373 −0.0324118
\(783\) −4.95306 −0.177008
\(784\) 32.7126 1.16831
\(785\) 0 0
\(786\) 0.594381 0.0212009
\(787\) −41.2581 −1.47069 −0.735346 0.677692i \(-0.762980\pi\)
−0.735346 + 0.677692i \(0.762980\pi\)
\(788\) −37.2188 −1.32586
\(789\) 9.70681 0.345572
\(790\) 0 0
\(791\) 22.1210 0.786531
\(792\) 2.53121 0.0899427
\(793\) −52.7054 −1.87162
\(794\) −1.04396 −0.0370486
\(795\) 0 0
\(796\) 21.5272 0.763012
\(797\) −12.4799 −0.442060 −0.221030 0.975267i \(-0.570942\pi\)
−0.221030 + 0.975267i \(0.570942\pi\)
\(798\) 3.38304 0.119758
\(799\) −1.81336 −0.0641519
\(800\) 0 0
\(801\) 8.60225 0.303945
\(802\) 0.433661 0.0153131
\(803\) 21.9757 0.775505
\(804\) −16.6481 −0.587135
\(805\) 0 0
\(806\) −2.81277 −0.0990756
\(807\) −19.1818 −0.675232
\(808\) 7.18658 0.252823
\(809\) −24.9414 −0.876893 −0.438446 0.898757i \(-0.644471\pi\)
−0.438446 + 0.898757i \(0.644471\pi\)
\(810\) 0 0
\(811\) −18.1982 −0.639026 −0.319513 0.947582i \(-0.603519\pi\)
−0.319513 + 0.947582i \(0.603519\pi\)
\(812\) −38.5535 −1.35296
\(813\) 7.39437 0.259332
\(814\) −0.197229 −0.00691289
\(815\) 0 0
\(816\) −7.13272 −0.249695
\(817\) −38.7325 −1.35508
\(818\) −1.02353 −0.0357868
\(819\) 20.6148 0.720338
\(820\) 0 0
\(821\) 0.753591 0.0263005 0.0131502 0.999914i \(-0.495814\pi\)
0.0131502 + 0.999914i \(0.495814\pi\)
\(822\) −0.889090 −0.0310106
\(823\) −15.9465 −0.555860 −0.277930 0.960601i \(-0.589648\pi\)
−0.277930 + 0.960601i \(0.589648\pi\)
\(824\) 5.00765 0.174450
\(825\) 0 0
\(826\) −2.40806 −0.0837871
\(827\) −33.2977 −1.15788 −0.578938 0.815372i \(-0.696533\pi\)
−0.578938 + 0.815372i \(0.696533\pi\)
\(828\) 9.42948 0.327697
\(829\) 22.2704 0.773484 0.386742 0.922188i \(-0.373600\pi\)
0.386742 + 0.922188i \(0.373600\pi\)
\(830\) 0 0
\(831\) −4.95059 −0.171734
\(832\) 40.7408 1.41243
\(833\) −15.0809 −0.522521
\(834\) 0.925459 0.0320460
\(835\) 0 0
\(836\) −98.1570 −3.39483
\(837\) −5.06513 −0.175076
\(838\) −1.23385 −0.0426226
\(839\) −33.0485 −1.14096 −0.570480 0.821311i \(-0.693243\pi\)
−0.570480 + 0.821311i \(0.693243\pi\)
\(840\) 0 0
\(841\) −4.46719 −0.154041
\(842\) −0.436970 −0.0150590
\(843\) −2.11672 −0.0729038
\(844\) −16.6516 −0.573170
\(845\) 0 0
\(846\) −0.105426 −0.00362461
\(847\) −98.7388 −3.39271
\(848\) 12.0019 0.412146
\(849\) −8.64610 −0.296733
\(850\) 0 0
\(851\) −1.47358 −0.0505135
\(852\) 19.8290 0.679331
\(853\) −54.0166 −1.84949 −0.924746 0.380584i \(-0.875723\pi\)
−0.924746 + 0.380584i \(0.875723\pi\)
\(854\) 4.12842 0.141272
\(855\) 0 0
\(856\) 5.15080 0.176051
\(857\) −5.71466 −0.195209 −0.0976045 0.995225i \(-0.531118\pi\)
−0.0976045 + 0.995225i \(0.531118\pi\)
\(858\) 3.34253 0.114112
\(859\) −8.70298 −0.296942 −0.148471 0.988917i \(-0.547435\pi\)
−0.148471 + 0.988917i \(0.547435\pi\)
\(860\) 0 0
\(861\) −10.2481 −0.349255
\(862\) −0.662933 −0.0225796
\(863\) 31.9407 1.08727 0.543637 0.839321i \(-0.317047\pi\)
0.543637 + 0.839321i \(0.317047\pi\)
\(864\) −1.25575 −0.0427214
\(865\) 0 0
\(866\) −0.959968 −0.0326210
\(867\) −13.7117 −0.465675
\(868\) −39.4258 −1.33820
\(869\) 54.2082 1.83889
\(870\) 0 0
\(871\) −44.0914 −1.49398
\(872\) 3.42637 0.116032
\(873\) 8.42460 0.285129
\(874\) 4.09831 0.138627
\(875\) 0 0
\(876\) −7.26142 −0.245341
\(877\) −3.18040 −0.107395 −0.0536973 0.998557i \(-0.517101\pi\)
−0.0536973 + 0.998557i \(0.517101\pi\)
\(878\) −2.75323 −0.0929172
\(879\) 6.47739 0.218477
\(880\) 0 0
\(881\) 29.8243 1.00481 0.502403 0.864634i \(-0.332449\pi\)
0.502403 + 0.864634i \(0.332449\pi\)
\(882\) −0.876777 −0.0295226
\(883\) 16.0274 0.539365 0.269682 0.962949i \(-0.413081\pi\)
0.269682 + 0.962949i \(0.413081\pi\)
\(884\) −18.9973 −0.638947
\(885\) 0 0
\(886\) −2.38536 −0.0801378
\(887\) 43.7250 1.46814 0.734071 0.679073i \(-0.237618\pi\)
0.734071 + 0.679073i \(0.237618\pi\)
\(888\) 0.130705 0.00438618
\(889\) 2.08058 0.0697805
\(890\) 0 0
\(891\) 6.01909 0.201647
\(892\) 3.04697 0.102020
\(893\) 8.19937 0.274382
\(894\) 0.568470 0.0190125
\(895\) 0 0
\(896\) −13.0203 −0.434978
\(897\) 24.9733 0.833834
\(898\) −0.343038 −0.0114473
\(899\) 25.0879 0.836727
\(900\) 0 0
\(901\) −5.53299 −0.184331
\(902\) −1.66166 −0.0553271
\(903\) 18.4874 0.615221
\(904\) −2.37696 −0.0790564
\(905\) 0 0
\(906\) 0.295007 0.00980095
\(907\) −32.2224 −1.06993 −0.534963 0.844875i \(-0.679675\pi\)
−0.534963 + 0.844875i \(0.679675\pi\)
\(908\) 36.1058 1.19821
\(909\) 17.0893 0.566816
\(910\) 0 0
\(911\) −51.9641 −1.72165 −0.860823 0.508904i \(-0.830051\pi\)
−0.860823 + 0.508904i \(0.830051\pi\)
\(912\) 32.2517 1.06796
\(913\) 77.8088 2.57510
\(914\) 0.809987 0.0267920
\(915\) 0 0
\(916\) −30.0693 −0.993519
\(917\) 22.0647 0.728642
\(918\) 0.191174 0.00630969
\(919\) 56.4042 1.86060 0.930301 0.366796i \(-0.119545\pi\)
0.930301 + 0.366796i \(0.119545\pi\)
\(920\) 0 0
\(921\) −28.5347 −0.940249
\(922\) 3.65345 0.120320
\(923\) 52.5157 1.72858
\(924\) 46.8512 1.54129
\(925\) 0 0
\(926\) −0.325800 −0.0107064
\(927\) 11.9079 0.391108
\(928\) 6.21979 0.204175
\(929\) 10.3110 0.338292 0.169146 0.985591i \(-0.445899\pi\)
0.169146 + 0.985591i \(0.445899\pi\)
\(930\) 0 0
\(931\) 68.1904 2.23485
\(932\) −38.9457 −1.27571
\(933\) 5.78357 0.189345
\(934\) 3.93466 0.128746
\(935\) 0 0
\(936\) −2.21511 −0.0724032
\(937\) −2.05321 −0.0670756 −0.0335378 0.999437i \(-0.510677\pi\)
−0.0335378 + 0.999437i \(0.510677\pi\)
\(938\) 3.45368 0.112767
\(939\) −7.35209 −0.239926
\(940\) 0 0
\(941\) 37.5244 1.22326 0.611631 0.791143i \(-0.290514\pi\)
0.611631 + 0.791143i \(0.290514\pi\)
\(942\) −2.54425 −0.0828962
\(943\) −12.4149 −0.404283
\(944\) −22.9569 −0.747182
\(945\) 0 0
\(946\) 2.99759 0.0974600
\(947\) −15.9626 −0.518714 −0.259357 0.965782i \(-0.583511\pi\)
−0.259357 + 0.965782i \(0.583511\pi\)
\(948\) −17.9120 −0.581755
\(949\) −19.2313 −0.624276
\(950\) 0 0
\(951\) 34.6205 1.12265
\(952\) 2.98443 0.0967259
\(953\) 12.3043 0.398575 0.199287 0.979941i \(-0.436137\pi\)
0.199287 + 0.979941i \(0.436137\pi\)
\(954\) −0.321679 −0.0104148
\(955\) 0 0
\(956\) 18.1213 0.586084
\(957\) −29.8129 −0.963715
\(958\) −2.49304 −0.0805465
\(959\) −33.0050 −1.06579
\(960\) 0 0
\(961\) −5.34451 −0.172403
\(962\) 0.172599 0.00556482
\(963\) 12.2483 0.394697
\(964\) 51.8410 1.66968
\(965\) 0 0
\(966\) −1.95616 −0.0629384
\(967\) −24.7346 −0.795411 −0.397705 0.917513i \(-0.630193\pi\)
−0.397705 + 0.917513i \(0.630193\pi\)
\(968\) 10.6098 0.341010
\(969\) −14.8684 −0.477641
\(970\) 0 0
\(971\) −15.5319 −0.498442 −0.249221 0.968447i \(-0.580175\pi\)
−0.249221 + 0.968447i \(0.580175\pi\)
\(972\) −1.98889 −0.0637935
\(973\) 34.3551 1.10137
\(974\) −2.21011 −0.0708163
\(975\) 0 0
\(976\) 39.3577 1.25981
\(977\) 60.0176 1.92013 0.960066 0.279773i \(-0.0902593\pi\)
0.960066 + 0.279773i \(0.0902593\pi\)
\(978\) −1.88649 −0.0603232
\(979\) 51.7777 1.65482
\(980\) 0 0
\(981\) 8.14773 0.260137
\(982\) −1.20175 −0.0383495
\(983\) −59.1158 −1.88550 −0.942751 0.333498i \(-0.891771\pi\)
−0.942751 + 0.333498i \(0.891771\pi\)
\(984\) 1.10119 0.0351046
\(985\) 0 0
\(986\) −0.946898 −0.0301554
\(987\) −3.91364 −0.124572
\(988\) 85.8990 2.73281
\(989\) 22.3961 0.712155
\(990\) 0 0
\(991\) 9.32558 0.296237 0.148118 0.988970i \(-0.452678\pi\)
0.148118 + 0.988970i \(0.452678\pi\)
\(992\) 6.36051 0.201947
\(993\) −16.8888 −0.535951
\(994\) −4.11356 −0.130474
\(995\) 0 0
\(996\) −25.7103 −0.814663
\(997\) −25.8977 −0.820188 −0.410094 0.912043i \(-0.634504\pi\)
−0.410094 + 0.912043i \(0.634504\pi\)
\(998\) −0.546453 −0.0172977
\(999\) 0.310810 0.00983358
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 3525.2.a.bi.1.6 13
5.2 odd 4 705.2.c.c.424.12 26
5.3 odd 4 705.2.c.c.424.15 yes 26
5.4 even 2 3525.2.a.bh.1.8 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
705.2.c.c.424.12 26 5.2 odd 4
705.2.c.c.424.15 yes 26 5.3 odd 4
3525.2.a.bh.1.8 13 5.4 even 2
3525.2.a.bi.1.6 13 1.1 even 1 trivial