Properties

Label 5225.2.a.z.1.4
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 19x^{14} + 144x^{12} - 552x^{10} + 1119x^{8} - 1146x^{6} + 524x^{4} - 83x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.78099\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.78099 q^{2} -0.213501 q^{3} +1.17191 q^{4} +0.380243 q^{6} -3.50934 q^{7} +1.47481 q^{8} -2.95442 q^{9} +O(q^{10})\) \(q-1.78099 q^{2} -0.213501 q^{3} +1.17191 q^{4} +0.380243 q^{6} -3.50934 q^{7} +1.47481 q^{8} -2.95442 q^{9} -1.00000 q^{11} -0.250205 q^{12} +1.11716 q^{13} +6.25008 q^{14} -4.97045 q^{16} +3.28198 q^{17} +5.26178 q^{18} +1.00000 q^{19} +0.749247 q^{21} +1.78099 q^{22} -0.303981 q^{23} -0.314874 q^{24} -1.98966 q^{26} +1.27127 q^{27} -4.11264 q^{28} -5.23841 q^{29} -0.126980 q^{31} +5.90267 q^{32} +0.213501 q^{33} -5.84516 q^{34} -3.46232 q^{36} +5.03915 q^{37} -1.78099 q^{38} -0.238516 q^{39} +0.293793 q^{41} -1.33440 q^{42} -0.180719 q^{43} -1.17191 q^{44} +0.541385 q^{46} +3.64879 q^{47} +1.06120 q^{48} +5.31545 q^{49} -0.700706 q^{51} +1.30922 q^{52} +1.17201 q^{53} -2.26412 q^{54} -5.17562 q^{56} -0.213501 q^{57} +9.32953 q^{58} +5.73363 q^{59} -3.61056 q^{61} +0.226150 q^{62} +10.3680 q^{63} -0.571685 q^{64} -0.380243 q^{66} +5.70818 q^{67} +3.84619 q^{68} +0.0649002 q^{69} -10.9423 q^{71} -4.35721 q^{72} +12.3731 q^{73} -8.97465 q^{74} +1.17191 q^{76} +3.50934 q^{77} +0.424794 q^{78} -2.32533 q^{79} +8.59183 q^{81} -0.523242 q^{82} +2.66052 q^{83} +0.878052 q^{84} +0.321858 q^{86} +1.11841 q^{87} -1.47481 q^{88} +0.0652748 q^{89} -3.92051 q^{91} -0.356239 q^{92} +0.0271104 q^{93} -6.49844 q^{94} -1.26023 q^{96} +17.5012 q^{97} -9.46675 q^{98} +2.95442 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{4} - 8 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{4} - 8 q^{6} + 8 q^{9} - 16 q^{11} - 4 q^{14} - 18 q^{16} + 16 q^{19} - 10 q^{21} - 10 q^{24} - 24 q^{26} - 2 q^{29} - 32 q^{31} + 16 q^{34} + 18 q^{36} - 40 q^{39} + 6 q^{41} - 6 q^{44} - 38 q^{49} - 16 q^{51} - 18 q^{54} + 12 q^{56} - 24 q^{59} - 42 q^{61} - 62 q^{64} + 8 q^{66} - 30 q^{69} - 46 q^{71} + 2 q^{74} + 6 q^{76} - 74 q^{79} - 56 q^{81} - 34 q^{84} + 8 q^{86} - 14 q^{89} - 24 q^{91} - 64 q^{94} + 54 q^{96} - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.78099 −1.25935 −0.629674 0.776860i \(-0.716811\pi\)
−0.629674 + 0.776860i \(0.716811\pi\)
\(3\) −0.213501 −0.123265 −0.0616325 0.998099i \(-0.519631\pi\)
−0.0616325 + 0.998099i \(0.519631\pi\)
\(4\) 1.17191 0.585956
\(5\) 0 0
\(6\) 0.380243 0.155233
\(7\) −3.50934 −1.32640 −0.663202 0.748440i \(-0.730803\pi\)
−0.663202 + 0.748440i \(0.730803\pi\)
\(8\) 1.47481 0.521425
\(9\) −2.95442 −0.984806
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −0.250205 −0.0722278
\(13\) 1.11716 0.309846 0.154923 0.987927i \(-0.450487\pi\)
0.154923 + 0.987927i \(0.450487\pi\)
\(14\) 6.25008 1.67040
\(15\) 0 0
\(16\) −4.97045 −1.24261
\(17\) 3.28198 0.795997 0.397999 0.917386i \(-0.369705\pi\)
0.397999 + 0.917386i \(0.369705\pi\)
\(18\) 5.26178 1.24021
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.749247 0.163499
\(22\) 1.78099 0.379708
\(23\) −0.303981 −0.0633844 −0.0316922 0.999498i \(-0.510090\pi\)
−0.0316922 + 0.999498i \(0.510090\pi\)
\(24\) −0.314874 −0.0642734
\(25\) 0 0
\(26\) −1.98966 −0.390203
\(27\) 1.27127 0.244657
\(28\) −4.11264 −0.777215
\(29\) −5.23841 −0.972748 −0.486374 0.873751i \(-0.661681\pi\)
−0.486374 + 0.873751i \(0.661681\pi\)
\(30\) 0 0
\(31\) −0.126980 −0.0228063 −0.0114031 0.999935i \(-0.503630\pi\)
−0.0114031 + 0.999935i \(0.503630\pi\)
\(32\) 5.90267 1.04345
\(33\) 0.213501 0.0371658
\(34\) −5.84516 −1.00244
\(35\) 0 0
\(36\) −3.46232 −0.577053
\(37\) 5.03915 0.828431 0.414215 0.910179i \(-0.364056\pi\)
0.414215 + 0.910179i \(0.364056\pi\)
\(38\) −1.78099 −0.288914
\(39\) −0.238516 −0.0381931
\(40\) 0 0
\(41\) 0.293793 0.0458828 0.0229414 0.999737i \(-0.492697\pi\)
0.0229414 + 0.999737i \(0.492697\pi\)
\(42\) −1.33440 −0.205902
\(43\) −0.180719 −0.0275594 −0.0137797 0.999905i \(-0.504386\pi\)
−0.0137797 + 0.999905i \(0.504386\pi\)
\(44\) −1.17191 −0.176672
\(45\) 0 0
\(46\) 0.541385 0.0798229
\(47\) 3.64879 0.532230 0.266115 0.963941i \(-0.414260\pi\)
0.266115 + 0.963941i \(0.414260\pi\)
\(48\) 1.06120 0.153170
\(49\) 5.31545 0.759350
\(50\) 0 0
\(51\) −0.700706 −0.0981185
\(52\) 1.30922 0.181556
\(53\) 1.17201 0.160988 0.0804939 0.996755i \(-0.474350\pi\)
0.0804939 + 0.996755i \(0.474350\pi\)
\(54\) −2.26412 −0.308108
\(55\) 0 0
\(56\) −5.17562 −0.691621
\(57\) −0.213501 −0.0282789
\(58\) 9.32953 1.22503
\(59\) 5.73363 0.746454 0.373227 0.927740i \(-0.378251\pi\)
0.373227 + 0.927740i \(0.378251\pi\)
\(60\) 0 0
\(61\) −3.61056 −0.462285 −0.231142 0.972920i \(-0.574246\pi\)
−0.231142 + 0.972920i \(0.574246\pi\)
\(62\) 0.226150 0.0287210
\(63\) 10.3680 1.30625
\(64\) −0.571685 −0.0714606
\(65\) 0 0
\(66\) −0.380243 −0.0468046
\(67\) 5.70818 0.697365 0.348682 0.937241i \(-0.386629\pi\)
0.348682 + 0.937241i \(0.386629\pi\)
\(68\) 3.84619 0.466419
\(69\) 0.0649002 0.00781307
\(70\) 0 0
\(71\) −10.9423 −1.29861 −0.649306 0.760527i \(-0.724941\pi\)
−0.649306 + 0.760527i \(0.724941\pi\)
\(72\) −4.35721 −0.513502
\(73\) 12.3731 1.44816 0.724080 0.689716i \(-0.242265\pi\)
0.724080 + 0.689716i \(0.242265\pi\)
\(74\) −8.97465 −1.04328
\(75\) 0 0
\(76\) 1.17191 0.134428
\(77\) 3.50934 0.399926
\(78\) 0.424794 0.0480984
\(79\) −2.32533 −0.261620 −0.130810 0.991407i \(-0.541758\pi\)
−0.130810 + 0.991407i \(0.541758\pi\)
\(80\) 0 0
\(81\) 8.59183 0.954648
\(82\) −0.523242 −0.0577824
\(83\) 2.66052 0.292031 0.146015 0.989282i \(-0.453355\pi\)
0.146015 + 0.989282i \(0.453355\pi\)
\(84\) 0.878052 0.0958034
\(85\) 0 0
\(86\) 0.321858 0.0347069
\(87\) 1.11841 0.119906
\(88\) −1.47481 −0.157216
\(89\) 0.0652748 0.00691912 0.00345956 0.999994i \(-0.498899\pi\)
0.00345956 + 0.999994i \(0.498899\pi\)
\(90\) 0 0
\(91\) −3.92051 −0.410981
\(92\) −0.356239 −0.0371405
\(93\) 0.0271104 0.00281121
\(94\) −6.49844 −0.670263
\(95\) 0 0
\(96\) −1.26023 −0.128621
\(97\) 17.5012 1.77698 0.888489 0.458899i \(-0.151756\pi\)
0.888489 + 0.458899i \(0.151756\pi\)
\(98\) −9.46675 −0.956286
\(99\) 2.95442 0.296930
\(100\) 0 0
\(101\) 7.81681 0.777802 0.388901 0.921280i \(-0.372855\pi\)
0.388901 + 0.921280i \(0.372855\pi\)
\(102\) 1.24795 0.123565
\(103\) −5.42056 −0.534104 −0.267052 0.963682i \(-0.586050\pi\)
−0.267052 + 0.963682i \(0.586050\pi\)
\(104\) 1.64761 0.161561
\(105\) 0 0
\(106\) −2.08733 −0.202740
\(107\) −9.22338 −0.891658 −0.445829 0.895118i \(-0.647091\pi\)
−0.445829 + 0.895118i \(0.647091\pi\)
\(108\) 1.48982 0.143358
\(109\) 3.85920 0.369644 0.184822 0.982772i \(-0.440829\pi\)
0.184822 + 0.982772i \(0.440829\pi\)
\(110\) 0 0
\(111\) −1.07586 −0.102116
\(112\) 17.4430 1.64821
\(113\) −0.909757 −0.0855827 −0.0427914 0.999084i \(-0.513625\pi\)
−0.0427914 + 0.999084i \(0.513625\pi\)
\(114\) 0.380243 0.0356130
\(115\) 0 0
\(116\) −6.13895 −0.569988
\(117\) −3.30057 −0.305138
\(118\) −10.2115 −0.940046
\(119\) −11.5176 −1.05581
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 6.43036 0.582177
\(123\) −0.0627252 −0.00565574
\(124\) −0.148809 −0.0133635
\(125\) 0 0
\(126\) −18.4654 −1.64502
\(127\) −1.00655 −0.0893165 −0.0446582 0.999002i \(-0.514220\pi\)
−0.0446582 + 0.999002i \(0.514220\pi\)
\(128\) −10.7872 −0.953461
\(129\) 0.0385837 0.00339711
\(130\) 0 0
\(131\) 8.34802 0.729370 0.364685 0.931131i \(-0.381177\pi\)
0.364685 + 0.931131i \(0.381177\pi\)
\(132\) 0.250205 0.0217775
\(133\) −3.50934 −0.304298
\(134\) −10.1662 −0.878224
\(135\) 0 0
\(136\) 4.84031 0.415053
\(137\) −14.8998 −1.27297 −0.636486 0.771288i \(-0.719613\pi\)
−0.636486 + 0.771288i \(0.719613\pi\)
\(138\) −0.115586 −0.00983937
\(139\) −3.03344 −0.257293 −0.128646 0.991691i \(-0.541063\pi\)
−0.128646 + 0.991691i \(0.541063\pi\)
\(140\) 0 0
\(141\) −0.779020 −0.0656053
\(142\) 19.4881 1.63540
\(143\) −1.11716 −0.0934220
\(144\) 14.6848 1.22373
\(145\) 0 0
\(146\) −22.0363 −1.82374
\(147\) −1.13485 −0.0936012
\(148\) 5.90544 0.485424
\(149\) −23.5095 −1.92597 −0.962986 0.269553i \(-0.913124\pi\)
−0.962986 + 0.269553i \(0.913124\pi\)
\(150\) 0 0
\(151\) −5.05226 −0.411147 −0.205574 0.978642i \(-0.565906\pi\)
−0.205574 + 0.978642i \(0.565906\pi\)
\(152\) 1.47481 0.119623
\(153\) −9.69634 −0.783902
\(154\) −6.25008 −0.503646
\(155\) 0 0
\(156\) −0.279520 −0.0223795
\(157\) 14.0292 1.11965 0.559827 0.828609i \(-0.310867\pi\)
0.559827 + 0.828609i \(0.310867\pi\)
\(158\) 4.14138 0.329470
\(159\) −0.250225 −0.0198441
\(160\) 0 0
\(161\) 1.06677 0.0840733
\(162\) −15.3019 −1.20223
\(163\) −8.22502 −0.644233 −0.322117 0.946700i \(-0.604394\pi\)
−0.322117 + 0.946700i \(0.604394\pi\)
\(164\) 0.344300 0.0268853
\(165\) 0 0
\(166\) −4.73836 −0.367768
\(167\) −4.03652 −0.312356 −0.156178 0.987729i \(-0.549917\pi\)
−0.156178 + 0.987729i \(0.549917\pi\)
\(168\) 1.10500 0.0852526
\(169\) −11.7519 −0.903996
\(170\) 0 0
\(171\) −2.95442 −0.225930
\(172\) −0.211787 −0.0161486
\(173\) 3.01611 0.229311 0.114655 0.993405i \(-0.463424\pi\)
0.114655 + 0.993405i \(0.463424\pi\)
\(174\) −1.99187 −0.151003
\(175\) 0 0
\(176\) 4.97045 0.374661
\(177\) −1.22414 −0.0920116
\(178\) −0.116254 −0.00871357
\(179\) 8.24312 0.616120 0.308060 0.951367i \(-0.400320\pi\)
0.308060 + 0.951367i \(0.400320\pi\)
\(180\) 0 0
\(181\) −18.1982 −1.35266 −0.676331 0.736598i \(-0.736431\pi\)
−0.676331 + 0.736598i \(0.736431\pi\)
\(182\) 6.98237 0.517568
\(183\) 0.770858 0.0569835
\(184\) −0.448315 −0.0330502
\(185\) 0 0
\(186\) −0.0482832 −0.00354029
\(187\) −3.28198 −0.240002
\(188\) 4.27606 0.311864
\(189\) −4.46133 −0.324514
\(190\) 0 0
\(191\) 22.9398 1.65986 0.829931 0.557866i \(-0.188380\pi\)
0.829931 + 0.557866i \(0.188380\pi\)
\(192\) 0.122055 0.00880858
\(193\) −4.53030 −0.326098 −0.163049 0.986618i \(-0.552133\pi\)
−0.163049 + 0.986618i \(0.552133\pi\)
\(194\) −31.1694 −2.23783
\(195\) 0 0
\(196\) 6.22924 0.444946
\(197\) 13.4791 0.960344 0.480172 0.877174i \(-0.340574\pi\)
0.480172 + 0.877174i \(0.340574\pi\)
\(198\) −5.26178 −0.373938
\(199\) −5.37014 −0.380679 −0.190340 0.981718i \(-0.560959\pi\)
−0.190340 + 0.981718i \(0.560959\pi\)
\(200\) 0 0
\(201\) −1.21870 −0.0859606
\(202\) −13.9216 −0.979523
\(203\) 18.3833 1.29026
\(204\) −0.821166 −0.0574931
\(205\) 0 0
\(206\) 9.65395 0.672623
\(207\) 0.898086 0.0624213
\(208\) −5.55281 −0.385018
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 21.6683 1.49171 0.745855 0.666108i \(-0.232041\pi\)
0.745855 + 0.666108i \(0.232041\pi\)
\(212\) 1.37349 0.0943318
\(213\) 2.33619 0.160073
\(214\) 16.4267 1.12291
\(215\) 0 0
\(216\) 1.87489 0.127570
\(217\) 0.445616 0.0302504
\(218\) −6.87318 −0.465510
\(219\) −2.64167 −0.178507
\(220\) 0 0
\(221\) 3.66651 0.246636
\(222\) 1.91610 0.128600
\(223\) −27.8124 −1.86246 −0.931228 0.364438i \(-0.881261\pi\)
−0.931228 + 0.364438i \(0.881261\pi\)
\(224\) −20.7145 −1.38404
\(225\) 0 0
\(226\) 1.62026 0.107778
\(227\) −7.61682 −0.505546 −0.252773 0.967526i \(-0.581343\pi\)
−0.252773 + 0.967526i \(0.581343\pi\)
\(228\) −0.250205 −0.0165702
\(229\) −16.4932 −1.08990 −0.544952 0.838467i \(-0.683452\pi\)
−0.544952 + 0.838467i \(0.683452\pi\)
\(230\) 0 0
\(231\) −0.749247 −0.0492969
\(232\) −7.72567 −0.507215
\(233\) 20.6984 1.35599 0.677997 0.735065i \(-0.262848\pi\)
0.677997 + 0.735065i \(0.262848\pi\)
\(234\) 5.87827 0.384275
\(235\) 0 0
\(236\) 6.71931 0.437390
\(237\) 0.496460 0.0322486
\(238\) 20.5126 1.32964
\(239\) 30.2075 1.95396 0.976981 0.213325i \(-0.0684295\pi\)
0.976981 + 0.213325i \(0.0684295\pi\)
\(240\) 0 0
\(241\) −13.5984 −0.875953 −0.437976 0.898986i \(-0.644305\pi\)
−0.437976 + 0.898986i \(0.644305\pi\)
\(242\) −1.78099 −0.114486
\(243\) −5.64819 −0.362332
\(244\) −4.23126 −0.270879
\(245\) 0 0
\(246\) 0.111713 0.00712254
\(247\) 1.11716 0.0710835
\(248\) −0.187272 −0.0118918
\(249\) −0.568025 −0.0359971
\(250\) 0 0
\(251\) 9.47667 0.598162 0.299081 0.954228i \(-0.403320\pi\)
0.299081 + 0.954228i \(0.403320\pi\)
\(252\) 12.1504 0.765406
\(253\) 0.303981 0.0191111
\(254\) 1.79264 0.112480
\(255\) 0 0
\(256\) 20.3552 1.27220
\(257\) −5.42346 −0.338306 −0.169153 0.985590i \(-0.554103\pi\)
−0.169153 + 0.985590i \(0.554103\pi\)
\(258\) −0.0687171 −0.00427814
\(259\) −17.6841 −1.09883
\(260\) 0 0
\(261\) 15.4764 0.957968
\(262\) −14.8677 −0.918531
\(263\) 10.1945 0.628618 0.314309 0.949321i \(-0.398227\pi\)
0.314309 + 0.949321i \(0.398227\pi\)
\(264\) 0.314874 0.0193792
\(265\) 0 0
\(266\) 6.25008 0.383217
\(267\) −0.0139362 −0.000852884 0
\(268\) 6.68948 0.408625
\(269\) −30.6046 −1.86600 −0.932999 0.359880i \(-0.882818\pi\)
−0.932999 + 0.359880i \(0.882818\pi\)
\(270\) 0 0
\(271\) −27.5296 −1.67230 −0.836152 0.548499i \(-0.815200\pi\)
−0.836152 + 0.548499i \(0.815200\pi\)
\(272\) −16.3129 −0.989115
\(273\) 0.837033 0.0506595
\(274\) 26.5363 1.60311
\(275\) 0 0
\(276\) 0.0760574 0.00457811
\(277\) 13.8247 0.830643 0.415321 0.909675i \(-0.363669\pi\)
0.415321 + 0.909675i \(0.363669\pi\)
\(278\) 5.40251 0.324021
\(279\) 0.375152 0.0224598
\(280\) 0 0
\(281\) 11.7359 0.700107 0.350054 0.936730i \(-0.386163\pi\)
0.350054 + 0.936730i \(0.386163\pi\)
\(282\) 1.38742 0.0826199
\(283\) −2.10305 −0.125013 −0.0625067 0.998045i \(-0.519909\pi\)
−0.0625067 + 0.998045i \(0.519909\pi\)
\(284\) −12.8234 −0.760930
\(285\) 0 0
\(286\) 1.98966 0.117651
\(287\) −1.03102 −0.0608592
\(288\) −17.4390 −1.02760
\(289\) −6.22861 −0.366389
\(290\) 0 0
\(291\) −3.73652 −0.219039
\(292\) 14.5002 0.848558
\(293\) −6.28988 −0.367458 −0.183729 0.982977i \(-0.558817\pi\)
−0.183729 + 0.982977i \(0.558817\pi\)
\(294\) 2.02116 0.117876
\(295\) 0 0
\(296\) 7.43180 0.431964
\(297\) −1.27127 −0.0737668
\(298\) 41.8701 2.42547
\(299\) −0.339596 −0.0196394
\(300\) 0 0
\(301\) 0.634204 0.0365549
\(302\) 8.99801 0.517777
\(303\) −1.66890 −0.0958757
\(304\) −4.97045 −0.285075
\(305\) 0 0
\(306\) 17.2690 0.987206
\(307\) −15.7997 −0.901738 −0.450869 0.892590i \(-0.648886\pi\)
−0.450869 + 0.892590i \(0.648886\pi\)
\(308\) 4.11264 0.234339
\(309\) 1.15730 0.0658363
\(310\) 0 0
\(311\) −7.80974 −0.442850 −0.221425 0.975177i \(-0.571071\pi\)
−0.221425 + 0.975177i \(0.571071\pi\)
\(312\) −0.351766 −0.0199148
\(313\) 12.0946 0.683629 0.341814 0.939768i \(-0.388959\pi\)
0.341814 + 0.939768i \(0.388959\pi\)
\(314\) −24.9859 −1.41003
\(315\) 0 0
\(316\) −2.72508 −0.153298
\(317\) −1.31596 −0.0739116 −0.0369558 0.999317i \(-0.511766\pi\)
−0.0369558 + 0.999317i \(0.511766\pi\)
\(318\) 0.445647 0.0249907
\(319\) 5.23841 0.293295
\(320\) 0 0
\(321\) 1.96920 0.109910
\(322\) −1.89990 −0.105878
\(323\) 3.28198 0.182614
\(324\) 10.0689 0.559382
\(325\) 0 0
\(326\) 14.6486 0.811314
\(327\) −0.823943 −0.0455642
\(328\) 0.433290 0.0239244
\(329\) −12.8048 −0.705953
\(330\) 0 0
\(331\) −4.16094 −0.228706 −0.114353 0.993440i \(-0.536480\pi\)
−0.114353 + 0.993440i \(0.536480\pi\)
\(332\) 3.11790 0.171117
\(333\) −14.8877 −0.815843
\(334\) 7.18899 0.393364
\(335\) 0 0
\(336\) −3.72409 −0.203166
\(337\) 28.5927 1.55754 0.778771 0.627308i \(-0.215843\pi\)
0.778771 + 0.627308i \(0.215843\pi\)
\(338\) 20.9301 1.13844
\(339\) 0.194234 0.0105493
\(340\) 0 0
\(341\) 0.126980 0.00687635
\(342\) 5.26178 0.284524
\(343\) 5.91165 0.319199
\(344\) −0.266527 −0.0143702
\(345\) 0 0
\(346\) −5.37165 −0.288782
\(347\) −9.07959 −0.487418 −0.243709 0.969848i \(-0.578364\pi\)
−0.243709 + 0.969848i \(0.578364\pi\)
\(348\) 1.31067 0.0702595
\(349\) 18.6493 0.998273 0.499136 0.866523i \(-0.333651\pi\)
0.499136 + 0.866523i \(0.333651\pi\)
\(350\) 0 0
\(351\) 1.42022 0.0758059
\(352\) −5.90267 −0.314613
\(353\) −29.3091 −1.55997 −0.779984 0.625800i \(-0.784773\pi\)
−0.779984 + 0.625800i \(0.784773\pi\)
\(354\) 2.18017 0.115875
\(355\) 0 0
\(356\) 0.0764964 0.00405430
\(357\) 2.45902 0.130145
\(358\) −14.6809 −0.775909
\(359\) −26.4877 −1.39797 −0.698984 0.715137i \(-0.746364\pi\)
−0.698984 + 0.715137i \(0.746364\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 32.4107 1.70347
\(363\) −0.213501 −0.0112059
\(364\) −4.59449 −0.240817
\(365\) 0 0
\(366\) −1.37289 −0.0717620
\(367\) −27.3556 −1.42795 −0.713975 0.700171i \(-0.753107\pi\)
−0.713975 + 0.700171i \(0.753107\pi\)
\(368\) 1.51092 0.0787621
\(369\) −0.867988 −0.0451856
\(370\) 0 0
\(371\) −4.11297 −0.213535
\(372\) 0.0317710 0.00164725
\(373\) −37.5358 −1.94353 −0.971766 0.235947i \(-0.924181\pi\)
−0.971766 + 0.235947i \(0.924181\pi\)
\(374\) 5.84516 0.302246
\(375\) 0 0
\(376\) 5.38128 0.277518
\(377\) −5.85216 −0.301402
\(378\) 7.94557 0.408676
\(379\) −31.1607 −1.60062 −0.800308 0.599589i \(-0.795331\pi\)
−0.800308 + 0.599589i \(0.795331\pi\)
\(380\) 0 0
\(381\) 0.214899 0.0110096
\(382\) −40.8554 −2.09034
\(383\) −22.2016 −1.13445 −0.567225 0.823563i \(-0.691983\pi\)
−0.567225 + 0.823563i \(0.691983\pi\)
\(384\) 2.30307 0.117528
\(385\) 0 0
\(386\) 8.06841 0.410671
\(387\) 0.533919 0.0271406
\(388\) 20.5099 1.04123
\(389\) 29.7883 1.51033 0.755164 0.655535i \(-0.227557\pi\)
0.755164 + 0.655535i \(0.227557\pi\)
\(390\) 0 0
\(391\) −0.997659 −0.0504538
\(392\) 7.83929 0.395944
\(393\) −1.78231 −0.0899058
\(394\) −24.0060 −1.20941
\(395\) 0 0
\(396\) 3.46232 0.173988
\(397\) 31.8515 1.59858 0.799291 0.600944i \(-0.205209\pi\)
0.799291 + 0.600944i \(0.205209\pi\)
\(398\) 9.56415 0.479408
\(399\) 0.749247 0.0375093
\(400\) 0 0
\(401\) 26.7235 1.33451 0.667255 0.744830i \(-0.267469\pi\)
0.667255 + 0.744830i \(0.267469\pi\)
\(402\) 2.17049 0.108254
\(403\) −0.141858 −0.00706643
\(404\) 9.16062 0.455758
\(405\) 0 0
\(406\) −32.7405 −1.62488
\(407\) −5.03915 −0.249781
\(408\) −1.03341 −0.0511614
\(409\) 27.4840 1.35900 0.679498 0.733677i \(-0.262198\pi\)
0.679498 + 0.733677i \(0.262198\pi\)
\(410\) 0 0
\(411\) 3.18111 0.156913
\(412\) −6.35243 −0.312962
\(413\) −20.1212 −0.990101
\(414\) −1.59948 −0.0786101
\(415\) 0 0
\(416\) 6.59426 0.323310
\(417\) 0.647643 0.0317152
\(418\) 1.78099 0.0871109
\(419\) −18.4002 −0.898909 −0.449455 0.893303i \(-0.648382\pi\)
−0.449455 + 0.893303i \(0.648382\pi\)
\(420\) 0 0
\(421\) −36.7525 −1.79121 −0.895603 0.444855i \(-0.853255\pi\)
−0.895603 + 0.444855i \(0.853255\pi\)
\(422\) −38.5910 −1.87858
\(423\) −10.7800 −0.524143
\(424\) 1.72849 0.0839430
\(425\) 0 0
\(426\) −4.16073 −0.201588
\(427\) 12.6707 0.613177
\(428\) −10.8090 −0.522473
\(429\) 0.238516 0.0115157
\(430\) 0 0
\(431\) −5.14958 −0.248046 −0.124023 0.992279i \(-0.539580\pi\)
−0.124023 + 0.992279i \(0.539580\pi\)
\(432\) −6.31880 −0.304014
\(433\) 23.7703 1.14233 0.571164 0.820836i \(-0.306492\pi\)
0.571164 + 0.820836i \(0.306492\pi\)
\(434\) −0.793635 −0.0380957
\(435\) 0 0
\(436\) 4.52264 0.216595
\(437\) −0.303981 −0.0145414
\(438\) 4.70477 0.224803
\(439\) 0.861025 0.0410945 0.0205472 0.999789i \(-0.493459\pi\)
0.0205472 + 0.999789i \(0.493459\pi\)
\(440\) 0 0
\(441\) −15.7041 −0.747812
\(442\) −6.53001 −0.310601
\(443\) −25.1374 −1.19432 −0.597158 0.802124i \(-0.703703\pi\)
−0.597158 + 0.802124i \(0.703703\pi\)
\(444\) −1.26082 −0.0598358
\(445\) 0 0
\(446\) 49.5335 2.34548
\(447\) 5.01930 0.237405
\(448\) 2.00623 0.0947856
\(449\) −7.27114 −0.343146 −0.171573 0.985171i \(-0.554885\pi\)
−0.171573 + 0.985171i \(0.554885\pi\)
\(450\) 0 0
\(451\) −0.293793 −0.0138342
\(452\) −1.06616 −0.0501477
\(453\) 1.07866 0.0506800
\(454\) 13.5654 0.636658
\(455\) 0 0
\(456\) −0.314874 −0.0147453
\(457\) −19.7002 −0.921536 −0.460768 0.887521i \(-0.652426\pi\)
−0.460768 + 0.887521i \(0.652426\pi\)
\(458\) 29.3742 1.37257
\(459\) 4.17230 0.194746
\(460\) 0 0
\(461\) −11.4754 −0.534461 −0.267231 0.963633i \(-0.586109\pi\)
−0.267231 + 0.963633i \(0.586109\pi\)
\(462\) 1.33440 0.0620819
\(463\) −17.6025 −0.818058 −0.409029 0.912521i \(-0.634132\pi\)
−0.409029 + 0.912521i \(0.634132\pi\)
\(464\) 26.0372 1.20875
\(465\) 0 0
\(466\) −36.8635 −1.70767
\(467\) −5.49601 −0.254325 −0.127163 0.991882i \(-0.540587\pi\)
−0.127163 + 0.991882i \(0.540587\pi\)
\(468\) −3.86798 −0.178797
\(469\) −20.0319 −0.924988
\(470\) 0 0
\(471\) −2.99526 −0.138014
\(472\) 8.45602 0.389220
\(473\) 0.180719 0.00830947
\(474\) −0.884189 −0.0406121
\(475\) 0 0
\(476\) −13.4976 −0.618661
\(477\) −3.46260 −0.158542
\(478\) −53.7992 −2.46072
\(479\) −11.8626 −0.542018 −0.271009 0.962577i \(-0.587357\pi\)
−0.271009 + 0.962577i \(0.587357\pi\)
\(480\) 0 0
\(481\) 5.62956 0.256686
\(482\) 24.2186 1.10313
\(483\) −0.227757 −0.0103633
\(484\) 1.17191 0.0532687
\(485\) 0 0
\(486\) 10.0593 0.456301
\(487\) −33.1690 −1.50303 −0.751516 0.659714i \(-0.770677\pi\)
−0.751516 + 0.659714i \(0.770677\pi\)
\(488\) −5.32490 −0.241047
\(489\) 1.75605 0.0794114
\(490\) 0 0
\(491\) 8.69377 0.392344 0.196172 0.980569i \(-0.437149\pi\)
0.196172 + 0.980569i \(0.437149\pi\)
\(492\) −0.0735084 −0.00331401
\(493\) −17.1923 −0.774304
\(494\) −1.98966 −0.0895188
\(495\) 0 0
\(496\) 0.631147 0.0283393
\(497\) 38.4002 1.72249
\(498\) 1.01164 0.0453329
\(499\) 2.21585 0.0991951 0.0495975 0.998769i \(-0.484206\pi\)
0.0495975 + 0.998769i \(0.484206\pi\)
\(500\) 0 0
\(501\) 0.861802 0.0385025
\(502\) −16.8778 −0.753294
\(503\) −25.4694 −1.13562 −0.567812 0.823159i \(-0.692210\pi\)
−0.567812 + 0.823159i \(0.692210\pi\)
\(504\) 15.2909 0.681112
\(505\) 0 0
\(506\) −0.541385 −0.0240675
\(507\) 2.50905 0.111431
\(508\) −1.17958 −0.0523355
\(509\) 31.0920 1.37813 0.689063 0.724701i \(-0.258022\pi\)
0.689063 + 0.724701i \(0.258022\pi\)
\(510\) 0 0
\(511\) −43.4213 −1.92085
\(512\) −14.6780 −0.648680
\(513\) 1.27127 0.0561281
\(514\) 9.65910 0.426045
\(515\) 0 0
\(516\) 0.0452167 0.00199056
\(517\) −3.64879 −0.160473
\(518\) 31.4951 1.38381
\(519\) −0.643943 −0.0282660
\(520\) 0 0
\(521\) −34.0870 −1.49338 −0.746688 0.665174i \(-0.768357\pi\)
−0.746688 + 0.665174i \(0.768357\pi\)
\(522\) −27.5633 −1.20641
\(523\) −26.0002 −1.13691 −0.568456 0.822714i \(-0.692459\pi\)
−0.568456 + 0.822714i \(0.692459\pi\)
\(524\) 9.78315 0.427379
\(525\) 0 0
\(526\) −18.1562 −0.791649
\(527\) −0.416746 −0.0181537
\(528\) −1.06120 −0.0461826
\(529\) −22.9076 −0.995982
\(530\) 0 0
\(531\) −16.9395 −0.735113
\(532\) −4.11264 −0.178305
\(533\) 0.328215 0.0142166
\(534\) 0.0248203 0.00107408
\(535\) 0 0
\(536\) 8.41849 0.363623
\(537\) −1.75992 −0.0759460
\(538\) 54.5064 2.34994
\(539\) −5.31545 −0.228953
\(540\) 0 0
\(541\) −5.06892 −0.217930 −0.108965 0.994046i \(-0.534754\pi\)
−0.108965 + 0.994046i \(0.534754\pi\)
\(542\) 49.0298 2.10601
\(543\) 3.88534 0.166736
\(544\) 19.3724 0.830587
\(545\) 0 0
\(546\) −1.49074 −0.0637980
\(547\) 10.9056 0.466289 0.233145 0.972442i \(-0.425098\pi\)
0.233145 + 0.972442i \(0.425098\pi\)
\(548\) −17.4612 −0.745906
\(549\) 10.6671 0.455261
\(550\) 0 0
\(551\) −5.23841 −0.223164
\(552\) 0.0957157 0.00407393
\(553\) 8.16036 0.347014
\(554\) −24.6215 −1.04607
\(555\) 0 0
\(556\) −3.55493 −0.150762
\(557\) 16.6435 0.705209 0.352605 0.935772i \(-0.385296\pi\)
0.352605 + 0.935772i \(0.385296\pi\)
\(558\) −0.668140 −0.0282846
\(559\) −0.201893 −0.00853916
\(560\) 0 0
\(561\) 0.700706 0.0295838
\(562\) −20.9015 −0.881678
\(563\) 32.9233 1.38755 0.693776 0.720190i \(-0.255946\pi\)
0.693776 + 0.720190i \(0.255946\pi\)
\(564\) −0.912943 −0.0384418
\(565\) 0 0
\(566\) 3.74550 0.157435
\(567\) −30.1516 −1.26625
\(568\) −16.1378 −0.677129
\(569\) 5.35168 0.224354 0.112177 0.993688i \(-0.464218\pi\)
0.112177 + 0.993688i \(0.464218\pi\)
\(570\) 0 0
\(571\) −32.1025 −1.34345 −0.671724 0.740802i \(-0.734446\pi\)
−0.671724 + 0.740802i \(0.734446\pi\)
\(572\) −1.30922 −0.0547412
\(573\) −4.89766 −0.204603
\(574\) 1.83623 0.0766428
\(575\) 0 0
\(576\) 1.68899 0.0703748
\(577\) −26.5229 −1.10416 −0.552082 0.833790i \(-0.686166\pi\)
−0.552082 + 0.833790i \(0.686166\pi\)
\(578\) 11.0931 0.461411
\(579\) 0.967224 0.0401965
\(580\) 0 0
\(581\) −9.33668 −0.387351
\(582\) 6.65470 0.275846
\(583\) −1.17201 −0.0485396
\(584\) 18.2480 0.755107
\(585\) 0 0
\(586\) 11.2022 0.462758
\(587\) −38.3527 −1.58298 −0.791492 0.611180i \(-0.790695\pi\)
−0.791492 + 0.611180i \(0.790695\pi\)
\(588\) −1.32995 −0.0548462
\(589\) −0.126980 −0.00523212
\(590\) 0 0
\(591\) −2.87780 −0.118377
\(592\) −25.0468 −1.02942
\(593\) 10.8794 0.446763 0.223381 0.974731i \(-0.428291\pi\)
0.223381 + 0.974731i \(0.428291\pi\)
\(594\) 2.26412 0.0928981
\(595\) 0 0
\(596\) −27.5510 −1.12853
\(597\) 1.14653 0.0469244
\(598\) 0.604817 0.0247328
\(599\) 25.8612 1.05666 0.528329 0.849040i \(-0.322819\pi\)
0.528329 + 0.849040i \(0.322819\pi\)
\(600\) 0 0
\(601\) −41.9127 −1.70965 −0.854827 0.518914i \(-0.826337\pi\)
−0.854827 + 0.518914i \(0.826337\pi\)
\(602\) −1.12951 −0.0460353
\(603\) −16.8643 −0.686769
\(604\) −5.92081 −0.240914
\(605\) 0 0
\(606\) 2.97228 0.120741
\(607\) 41.4401 1.68200 0.841002 0.541032i \(-0.181966\pi\)
0.841002 + 0.541032i \(0.181966\pi\)
\(608\) 5.90267 0.239385
\(609\) −3.92486 −0.159043
\(610\) 0 0
\(611\) 4.07629 0.164909
\(612\) −11.3633 −0.459333
\(613\) 5.66031 0.228618 0.114309 0.993445i \(-0.463535\pi\)
0.114309 + 0.993445i \(0.463535\pi\)
\(614\) 28.1391 1.13560
\(615\) 0 0
\(616\) 5.17562 0.208532
\(617\) −21.2337 −0.854837 −0.427419 0.904054i \(-0.640577\pi\)
−0.427419 + 0.904054i \(0.640577\pi\)
\(618\) −2.06113 −0.0829108
\(619\) 15.5489 0.624962 0.312481 0.949924i \(-0.398840\pi\)
0.312481 + 0.949924i \(0.398840\pi\)
\(620\) 0 0
\(621\) −0.386443 −0.0155074
\(622\) 13.9090 0.557701
\(623\) −0.229071 −0.00917755
\(624\) 1.18553 0.0474592
\(625\) 0 0
\(626\) −21.5404 −0.860926
\(627\) 0.213501 0.00852641
\(628\) 16.4410 0.656069
\(629\) 16.5384 0.659428
\(630\) 0 0
\(631\) −36.0897 −1.43671 −0.718354 0.695678i \(-0.755104\pi\)
−0.718354 + 0.695678i \(0.755104\pi\)
\(632\) −3.42942 −0.136415
\(633\) −4.62622 −0.183876
\(634\) 2.34370 0.0930804
\(635\) 0 0
\(636\) −0.293242 −0.0116278
\(637\) 5.93823 0.235281
\(638\) −9.32953 −0.369360
\(639\) 32.3281 1.27888
\(640\) 0 0
\(641\) −3.17128 −0.125258 −0.0626290 0.998037i \(-0.519948\pi\)
−0.0626290 + 0.998037i \(0.519948\pi\)
\(642\) −3.50712 −0.138415
\(643\) 17.2355 0.679702 0.339851 0.940479i \(-0.389623\pi\)
0.339851 + 0.940479i \(0.389623\pi\)
\(644\) 1.25016 0.0492633
\(645\) 0 0
\(646\) −5.84516 −0.229975
\(647\) −21.6081 −0.849504 −0.424752 0.905310i \(-0.639639\pi\)
−0.424752 + 0.905310i \(0.639639\pi\)
\(648\) 12.6713 0.497777
\(649\) −5.73363 −0.225064
\(650\) 0 0
\(651\) −0.0951394 −0.00372881
\(652\) −9.63900 −0.377492
\(653\) 1.82065 0.0712477 0.0356239 0.999365i \(-0.488658\pi\)
0.0356239 + 0.999365i \(0.488658\pi\)
\(654\) 1.46743 0.0573811
\(655\) 0 0
\(656\) −1.46028 −0.0570145
\(657\) −36.5553 −1.42616
\(658\) 22.8052 0.889040
\(659\) −7.52543 −0.293149 −0.146575 0.989200i \(-0.546825\pi\)
−0.146575 + 0.989200i \(0.546825\pi\)
\(660\) 0 0
\(661\) −4.05494 −0.157719 −0.0788595 0.996886i \(-0.525128\pi\)
−0.0788595 + 0.996886i \(0.525128\pi\)
\(662\) 7.41058 0.288020
\(663\) −0.782804 −0.0304016
\(664\) 3.92378 0.152272
\(665\) 0 0
\(666\) 26.5149 1.02743
\(667\) 1.59237 0.0616570
\(668\) −4.73045 −0.183027
\(669\) 5.93797 0.229575
\(670\) 0 0
\(671\) 3.61056 0.139384
\(672\) 4.42256 0.170604
\(673\) −5.99814 −0.231211 −0.115606 0.993295i \(-0.536881\pi\)
−0.115606 + 0.993295i \(0.536881\pi\)
\(674\) −50.9232 −1.96149
\(675\) 0 0
\(676\) −13.7722 −0.529702
\(677\) 4.64183 0.178400 0.0891999 0.996014i \(-0.471569\pi\)
0.0891999 + 0.996014i \(0.471569\pi\)
\(678\) −0.345928 −0.0132853
\(679\) −61.4176 −2.35699
\(680\) 0 0
\(681\) 1.62620 0.0623161
\(682\) −0.226150 −0.00865972
\(683\) −9.58079 −0.366599 −0.183299 0.983057i \(-0.558678\pi\)
−0.183299 + 0.983057i \(0.558678\pi\)
\(684\) −3.46232 −0.132385
\(685\) 0 0
\(686\) −10.5286 −0.401983
\(687\) 3.52132 0.134347
\(688\) 0.898254 0.0342456
\(689\) 1.30933 0.0498814
\(690\) 0 0
\(691\) −22.4231 −0.853014 −0.426507 0.904484i \(-0.640256\pi\)
−0.426507 + 0.904484i \(0.640256\pi\)
\(692\) 3.53462 0.134366
\(693\) −10.3680 −0.393850
\(694\) 16.1706 0.613828
\(695\) 0 0
\(696\) 1.64944 0.0625218
\(697\) 0.964223 0.0365226
\(698\) −33.2141 −1.25717
\(699\) −4.41912 −0.167147
\(700\) 0 0
\(701\) 5.29648 0.200045 0.100023 0.994985i \(-0.468108\pi\)
0.100023 + 0.994985i \(0.468108\pi\)
\(702\) −2.52940 −0.0954660
\(703\) 5.03915 0.190055
\(704\) 0.571685 0.0215462
\(705\) 0 0
\(706\) 52.1992 1.96454
\(707\) −27.4318 −1.03168
\(708\) −1.43458 −0.0539148
\(709\) −32.7803 −1.23109 −0.615545 0.788102i \(-0.711064\pi\)
−0.615545 + 0.788102i \(0.711064\pi\)
\(710\) 0 0
\(711\) 6.86999 0.257645
\(712\) 0.0962681 0.00360780
\(713\) 0.0385995 0.00144556
\(714\) −4.37947 −0.163898
\(715\) 0 0
\(716\) 9.66022 0.361019
\(717\) −6.44934 −0.240855
\(718\) 47.1743 1.76053
\(719\) −3.60564 −0.134468 −0.0672338 0.997737i \(-0.521417\pi\)
−0.0672338 + 0.997737i \(0.521417\pi\)
\(720\) 0 0
\(721\) 19.0226 0.708438
\(722\) −1.78099 −0.0662814
\(723\) 2.90328 0.107974
\(724\) −21.3267 −0.792600
\(725\) 0 0
\(726\) 0.380243 0.0141121
\(727\) −18.4897 −0.685744 −0.342872 0.939382i \(-0.611400\pi\)
−0.342872 + 0.939382i \(0.611400\pi\)
\(728\) −5.78202 −0.214296
\(729\) −24.5696 −0.909985
\(730\) 0 0
\(731\) −0.593116 −0.0219372
\(732\) 0.903378 0.0333898
\(733\) −29.3667 −1.08468 −0.542341 0.840158i \(-0.682462\pi\)
−0.542341 + 0.840158i \(0.682462\pi\)
\(734\) 48.7199 1.79829
\(735\) 0 0
\(736\) −1.79430 −0.0661387
\(737\) −5.70818 −0.210263
\(738\) 1.54587 0.0569044
\(739\) 22.8091 0.839048 0.419524 0.907744i \(-0.362197\pi\)
0.419524 + 0.907744i \(0.362197\pi\)
\(740\) 0 0
\(741\) −0.238516 −0.00876210
\(742\) 7.32515 0.268915
\(743\) 40.2104 1.47518 0.737589 0.675250i \(-0.235964\pi\)
0.737589 + 0.675250i \(0.235964\pi\)
\(744\) 0.0399827 0.00146584
\(745\) 0 0
\(746\) 66.8508 2.44758
\(747\) −7.86030 −0.287593
\(748\) −3.84619 −0.140631
\(749\) 32.3680 1.18270
\(750\) 0 0
\(751\) 45.2958 1.65287 0.826434 0.563034i \(-0.190366\pi\)
0.826434 + 0.563034i \(0.190366\pi\)
\(752\) −18.1361 −0.661355
\(753\) −2.02328 −0.0737324
\(754\) 10.4226 0.379570
\(755\) 0 0
\(756\) −5.22829 −0.190151
\(757\) −38.7580 −1.40868 −0.704341 0.709862i \(-0.748757\pi\)
−0.704341 + 0.709862i \(0.748757\pi\)
\(758\) 55.4967 2.01573
\(759\) −0.0649002 −0.00235573
\(760\) 0 0
\(761\) 29.0483 1.05300 0.526500 0.850175i \(-0.323504\pi\)
0.526500 + 0.850175i \(0.323504\pi\)
\(762\) −0.382731 −0.0138649
\(763\) −13.5432 −0.490298
\(764\) 26.8834 0.972607
\(765\) 0 0
\(766\) 39.5408 1.42867
\(767\) 6.40540 0.231286
\(768\) −4.34586 −0.156818
\(769\) −31.9246 −1.15123 −0.575616 0.817720i \(-0.695238\pi\)
−0.575616 + 0.817720i \(0.695238\pi\)
\(770\) 0 0
\(771\) 1.15791 0.0417013
\(772\) −5.30912 −0.191079
\(773\) 4.83464 0.173890 0.0869449 0.996213i \(-0.472290\pi\)
0.0869449 + 0.996213i \(0.472290\pi\)
\(774\) −0.950903 −0.0341795
\(775\) 0 0
\(776\) 25.8110 0.926560
\(777\) 3.77557 0.135448
\(778\) −53.0526 −1.90203
\(779\) 0.293793 0.0105262
\(780\) 0 0
\(781\) 10.9423 0.391546
\(782\) 1.77682 0.0635388
\(783\) −6.65945 −0.237989
\(784\) −26.4202 −0.943577
\(785\) 0 0
\(786\) 3.17427 0.113223
\(787\) −34.5482 −1.23151 −0.615755 0.787938i \(-0.711149\pi\)
−0.615755 + 0.787938i \(0.711149\pi\)
\(788\) 15.7963 0.562720
\(789\) −2.17653 −0.0774866
\(790\) 0 0
\(791\) 3.19264 0.113517
\(792\) 4.35721 0.154827
\(793\) −4.03359 −0.143237
\(794\) −56.7271 −2.01317
\(795\) 0 0
\(796\) −6.29334 −0.223061
\(797\) −49.3536 −1.74819 −0.874097 0.485751i \(-0.838546\pi\)
−0.874097 + 0.485751i \(0.838546\pi\)
\(798\) −1.33440 −0.0472372
\(799\) 11.9752 0.423654
\(800\) 0 0
\(801\) −0.192849 −0.00681399
\(802\) −47.5942 −1.68061
\(803\) −12.3731 −0.436637
\(804\) −1.42821 −0.0503691
\(805\) 0 0
\(806\) 0.252646 0.00889909
\(807\) 6.53412 0.230012
\(808\) 11.5283 0.405565
\(809\) 22.5147 0.791575 0.395787 0.918342i \(-0.370472\pi\)
0.395787 + 0.918342i \(0.370472\pi\)
\(810\) 0 0
\(811\) −29.8110 −1.04680 −0.523402 0.852086i \(-0.675337\pi\)
−0.523402 + 0.852086i \(0.675337\pi\)
\(812\) 21.5437 0.756034
\(813\) 5.87760 0.206136
\(814\) 8.97465 0.314561
\(815\) 0 0
\(816\) 3.48282 0.121923
\(817\) −0.180719 −0.00632256
\(818\) −48.9486 −1.71145
\(819\) 11.5828 0.404736
\(820\) 0 0
\(821\) 23.5479 0.821828 0.410914 0.911674i \(-0.365210\pi\)
0.410914 + 0.911674i \(0.365210\pi\)
\(822\) −5.66552 −0.197608
\(823\) −5.77193 −0.201197 −0.100598 0.994927i \(-0.532076\pi\)
−0.100598 + 0.994927i \(0.532076\pi\)
\(824\) −7.99432 −0.278495
\(825\) 0 0
\(826\) 35.8356 1.24688
\(827\) −34.9143 −1.21409 −0.607045 0.794668i \(-0.707645\pi\)
−0.607045 + 0.794668i \(0.707645\pi\)
\(828\) 1.05248 0.0365761
\(829\) 12.1015 0.420303 0.210152 0.977669i \(-0.432604\pi\)
0.210152 + 0.977669i \(0.432604\pi\)
\(830\) 0 0
\(831\) −2.95158 −0.102389
\(832\) −0.638666 −0.0221418
\(833\) 17.4452 0.604440
\(834\) −1.15344 −0.0399405
\(835\) 0 0
\(836\) −1.17191 −0.0405314
\(837\) −0.161426 −0.00557971
\(838\) 32.7705 1.13204
\(839\) 18.3333 0.632937 0.316469 0.948603i \(-0.397503\pi\)
0.316469 + 0.948603i \(0.397503\pi\)
\(840\) 0 0
\(841\) −1.55909 −0.0537616
\(842\) 65.4556 2.25575
\(843\) −2.50564 −0.0862987
\(844\) 25.3934 0.874077
\(845\) 0 0
\(846\) 19.1991 0.660079
\(847\) −3.50934 −0.120582
\(848\) −5.82540 −0.200045
\(849\) 0.449003 0.0154098
\(850\) 0 0
\(851\) −1.53180 −0.0525095
\(852\) 2.73781 0.0937959
\(853\) −26.7354 −0.915402 −0.457701 0.889106i \(-0.651327\pi\)
−0.457701 + 0.889106i \(0.651327\pi\)
\(854\) −22.5663 −0.772203
\(855\) 0 0
\(856\) −13.6028 −0.464933
\(857\) 16.8931 0.577057 0.288529 0.957471i \(-0.406834\pi\)
0.288529 + 0.957471i \(0.406834\pi\)
\(858\) −0.424794 −0.0145022
\(859\) 5.29172 0.180551 0.0902755 0.995917i \(-0.471225\pi\)
0.0902755 + 0.995917i \(0.471225\pi\)
\(860\) 0 0
\(861\) 0.220124 0.00750180
\(862\) 9.17133 0.312377
\(863\) 26.4397 0.900018 0.450009 0.893024i \(-0.351421\pi\)
0.450009 + 0.893024i \(0.351421\pi\)
\(864\) 7.50392 0.255288
\(865\) 0 0
\(866\) −42.3346 −1.43859
\(867\) 1.32981 0.0451629
\(868\) 0.522222 0.0177254
\(869\) 2.32533 0.0788814
\(870\) 0 0
\(871\) 6.37697 0.216075
\(872\) 5.69159 0.192742
\(873\) −51.7058 −1.74998
\(874\) 0.541385 0.0183126
\(875\) 0 0
\(876\) −3.09580 −0.104597
\(877\) −5.61130 −0.189480 −0.0947401 0.995502i \(-0.530202\pi\)
−0.0947401 + 0.995502i \(0.530202\pi\)
\(878\) −1.53347 −0.0517522
\(879\) 1.34290 0.0452947
\(880\) 0 0
\(881\) −7.21640 −0.243127 −0.121563 0.992584i \(-0.538791\pi\)
−0.121563 + 0.992584i \(0.538791\pi\)
\(882\) 27.9687 0.941756
\(883\) −36.5678 −1.23061 −0.615303 0.788291i \(-0.710966\pi\)
−0.615303 + 0.788291i \(0.710966\pi\)
\(884\) 4.29683 0.144518
\(885\) 0 0
\(886\) 44.7694 1.50406
\(887\) 47.3649 1.59036 0.795179 0.606374i \(-0.207377\pi\)
0.795179 + 0.606374i \(0.207377\pi\)
\(888\) −1.58670 −0.0532461
\(889\) 3.53231 0.118470
\(890\) 0 0
\(891\) −8.59183 −0.287837
\(892\) −32.5937 −1.09132
\(893\) 3.64879 0.122102
\(894\) −8.93930 −0.298975
\(895\) 0 0
\(896\) 37.8559 1.26468
\(897\) 0.0725042 0.00242085
\(898\) 12.9498 0.432140
\(899\) 0.665173 0.0221848
\(900\) 0 0
\(901\) 3.84651 0.128146
\(902\) 0.523242 0.0174220
\(903\) −0.135403 −0.00450594
\(904\) −1.34172 −0.0446250
\(905\) 0 0
\(906\) −1.92109 −0.0638238
\(907\) 48.1508 1.59882 0.799411 0.600784i \(-0.205145\pi\)
0.799411 + 0.600784i \(0.205145\pi\)
\(908\) −8.92624 −0.296228
\(909\) −23.0941 −0.765984
\(910\) 0 0
\(911\) −20.7792 −0.688446 −0.344223 0.938888i \(-0.611858\pi\)
−0.344223 + 0.938888i \(0.611858\pi\)
\(912\) 1.06120 0.0351397
\(913\) −2.66052 −0.0880505
\(914\) 35.0858 1.16053
\(915\) 0 0
\(916\) −19.3286 −0.638636
\(917\) −29.2960 −0.967440
\(918\) −7.43081 −0.245253
\(919\) 9.76309 0.322055 0.161027 0.986950i \(-0.448519\pi\)
0.161027 + 0.986950i \(0.448519\pi\)
\(920\) 0 0
\(921\) 3.37326 0.111153
\(922\) 20.4375 0.673073
\(923\) −12.2244 −0.402369
\(924\) −0.878052 −0.0288858
\(925\) 0 0
\(926\) 31.3498 1.03022
\(927\) 16.0146 0.525989
\(928\) −30.9206 −1.01502
\(929\) 47.0624 1.54407 0.772034 0.635582i \(-0.219240\pi\)
0.772034 + 0.635582i \(0.219240\pi\)
\(930\) 0 0
\(931\) 5.31545 0.174207
\(932\) 24.2567 0.794553
\(933\) 1.66739 0.0545878
\(934\) 9.78832 0.320284
\(935\) 0 0
\(936\) −4.86772 −0.159107
\(937\) 37.3958 1.22167 0.610833 0.791759i \(-0.290835\pi\)
0.610833 + 0.791759i \(0.290835\pi\)
\(938\) 35.6766 1.16488
\(939\) −2.58222 −0.0842674
\(940\) 0 0
\(941\) 16.4787 0.537191 0.268595 0.963253i \(-0.413441\pi\)
0.268595 + 0.963253i \(0.413441\pi\)
\(942\) 5.33451 0.173808
\(943\) −0.0893075 −0.00290825
\(944\) −28.4987 −0.927553
\(945\) 0 0
\(946\) −0.321858 −0.0104645
\(947\) −15.8450 −0.514893 −0.257447 0.966292i \(-0.582881\pi\)
−0.257447 + 0.966292i \(0.582881\pi\)
\(948\) 0.581808 0.0188962
\(949\) 13.8228 0.448706
\(950\) 0 0
\(951\) 0.280959 0.00911071
\(952\) −16.9863 −0.550528
\(953\) 29.0991 0.942612 0.471306 0.881970i \(-0.343783\pi\)
0.471306 + 0.881970i \(0.343783\pi\)
\(954\) 6.16685 0.199659
\(955\) 0 0
\(956\) 35.4006 1.14494
\(957\) −1.11841 −0.0361529
\(958\) 21.1272 0.682589
\(959\) 52.2883 1.68848
\(960\) 0 0
\(961\) −30.9839 −0.999480
\(962\) −10.0262 −0.323256
\(963\) 27.2497 0.878110
\(964\) −15.9362 −0.513270
\(965\) 0 0
\(966\) 0.405632 0.0130510
\(967\) −27.6570 −0.889390 −0.444695 0.895682i \(-0.646688\pi\)
−0.444695 + 0.895682i \(0.646688\pi\)
\(968\) 1.47481 0.0474023
\(969\) −0.700706 −0.0225099
\(970\) 0 0
\(971\) 29.9538 0.961264 0.480632 0.876922i \(-0.340407\pi\)
0.480632 + 0.876922i \(0.340407\pi\)
\(972\) −6.61918 −0.212310
\(973\) 10.6454 0.341275
\(974\) 59.0736 1.89284
\(975\) 0 0
\(976\) 17.9461 0.574440
\(977\) 11.1701 0.357364 0.178682 0.983907i \(-0.442817\pi\)
0.178682 + 0.983907i \(0.442817\pi\)
\(978\) −3.12750 −0.100006
\(979\) −0.0652748 −0.00208619
\(980\) 0 0
\(981\) −11.4017 −0.364028
\(982\) −15.4835 −0.494098
\(983\) −22.5894 −0.720491 −0.360245 0.932858i \(-0.617307\pi\)
−0.360245 + 0.932858i \(0.617307\pi\)
\(984\) −0.0925079 −0.00294904
\(985\) 0 0
\(986\) 30.6193 0.975118
\(987\) 2.73384 0.0870192
\(988\) 1.30922 0.0416518
\(989\) 0.0549351 0.00174683
\(990\) 0 0
\(991\) −26.8461 −0.852796 −0.426398 0.904536i \(-0.640218\pi\)
−0.426398 + 0.904536i \(0.640218\pi\)
\(992\) −0.749521 −0.0237973
\(993\) 0.888366 0.0281914
\(994\) −68.3903 −2.16921
\(995\) 0 0
\(996\) −0.665675 −0.0210927
\(997\) 31.2387 0.989339 0.494669 0.869081i \(-0.335289\pi\)
0.494669 + 0.869081i \(0.335289\pi\)
\(998\) −3.94640 −0.124921
\(999\) 6.40614 0.202681
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 5225.2.a.z.1.4 16
5.2 odd 4 1045.2.b.b.419.4 16
5.3 odd 4 1045.2.b.b.419.13 yes 16
5.4 even 2 inner 5225.2.a.z.1.13 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.b.419.4 16 5.2 odd 4
1045.2.b.b.419.13 yes 16 5.3 odd 4
5225.2.a.z.1.4 16 1.1 even 1 trivial
5225.2.a.z.1.13 16 5.4 even 2 inner