Properties

Label 1849.2.a.o.1.11
Level $1849$
Weight $2$
Character 1849.1
Self dual yes
Analytic conductor $14.764$
Analytic rank $0$
Dimension $18$
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] = [1849,2,Mod(1,1849)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1849.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1849.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: \(14.7643393337\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} - 15 x^{16} + 106 x^{15} + 47 x^{14} - 897 x^{13} + 364 x^{12} + 3855 x^{11} - 3223 x^{10} - 8851 x^{9} + 9909 x^{8} + 10400 x^{7} - 14483 x^{6} - 5118 x^{5} + \cdots - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(1.04477\) of defining polynomial
Character \(\chi\) \(=\) 1849.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.04477 q^{2} -2.52412 q^{3} -0.908449 q^{4} +0.499809 q^{5} -2.63714 q^{6} -3.30668 q^{7} -3.03867 q^{8} +3.37119 q^{9} +O(q^{10})\) \(q+1.04477 q^{2} -2.52412 q^{3} -0.908449 q^{4} +0.499809 q^{5} -2.63714 q^{6} -3.30668 q^{7} -3.03867 q^{8} +3.37119 q^{9} +0.522187 q^{10} -3.72276 q^{11} +2.29304 q^{12} -4.68695 q^{13} -3.45473 q^{14} -1.26158 q^{15} -1.35782 q^{16} -4.59028 q^{17} +3.52213 q^{18} +6.05844 q^{19} -0.454051 q^{20} +8.34647 q^{21} -3.88944 q^{22} -0.388946 q^{23} +7.66997 q^{24} -4.75019 q^{25} -4.89680 q^{26} -0.936938 q^{27} +3.00395 q^{28} +3.53498 q^{29} -1.31806 q^{30} -0.870839 q^{31} +4.65872 q^{32} +9.39670 q^{33} -4.79580 q^{34} -1.65271 q^{35} -3.06256 q^{36} -3.93313 q^{37} +6.32970 q^{38} +11.8304 q^{39} -1.51875 q^{40} -10.7837 q^{41} +8.72017 q^{42} +3.38193 q^{44} +1.68495 q^{45} -0.406360 q^{46} +3.49088 q^{47} +3.42731 q^{48} +3.93414 q^{49} -4.96287 q^{50} +11.5864 q^{51} +4.25785 q^{52} +4.77361 q^{53} -0.978888 q^{54} -1.86067 q^{55} +10.0479 q^{56} -15.2923 q^{57} +3.69325 q^{58} +6.72259 q^{59} +1.14608 q^{60} -5.24696 q^{61} -0.909829 q^{62} -11.1475 q^{63} +7.58295 q^{64} -2.34258 q^{65} +9.81742 q^{66} +10.5533 q^{67} +4.17003 q^{68} +0.981747 q^{69} -1.72670 q^{70} -0.899542 q^{71} -10.2439 q^{72} +0.966562 q^{73} -4.10922 q^{74} +11.9901 q^{75} -5.50379 q^{76} +12.3100 q^{77} +12.3601 q^{78} -1.00197 q^{79} -0.678652 q^{80} -7.74863 q^{81} -11.2665 q^{82} -3.41592 q^{83} -7.58234 q^{84} -2.29426 q^{85} -8.92272 q^{87} +11.3122 q^{88} +1.84065 q^{89} +1.76039 q^{90} +15.4982 q^{91} +0.353338 q^{92} +2.19810 q^{93} +3.64718 q^{94} +3.02806 q^{95} -11.7592 q^{96} +0.960596 q^{97} +4.11028 q^{98} -12.5501 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9} - 7 q^{10} - 2 q^{11} + 16 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 9 q^{16} - 11 q^{17} + 25 q^{18} + 31 q^{19} + 25 q^{20} + 19 q^{22} - 11 q^{23} + 18 q^{24} + 9 q^{25} + 27 q^{26} + 23 q^{27} + 20 q^{28} + 37 q^{29} - 17 q^{30} - 12 q^{31} + 39 q^{32} + 38 q^{33} + 14 q^{34} + 16 q^{35} + 47 q^{36} - 19 q^{37} + 56 q^{38} + 46 q^{39} + 6 q^{40} - 7 q^{41} - q^{42} + 7 q^{44} + 23 q^{45} - 47 q^{46} - q^{47} + 15 q^{48} - 6 q^{49} - 3 q^{50} + 38 q^{51} + 15 q^{52} + 3 q^{53} - 67 q^{54} + 28 q^{55} - 81 q^{56} + 46 q^{57} + 34 q^{58} + 17 q^{59} - 83 q^{60} + 28 q^{61} + 33 q^{62} + 26 q^{63} + 10 q^{64} + 16 q^{65} - 72 q^{66} + 18 q^{67} + 53 q^{68} + 7 q^{69} - 34 q^{70} + 86 q^{71} - 2 q^{72} - 27 q^{73} - 79 q^{74} + 31 q^{75} + 59 q^{76} + 43 q^{77} + 91 q^{78} + 17 q^{79} + 8 q^{80} - 10 q^{81} + 13 q^{82} - 12 q^{83} - 32 q^{84} + 28 q^{85} - 43 q^{87} - 23 q^{88} + 51 q^{89} + 10 q^{90} - 20 q^{91} + 18 q^{92} - 30 q^{93} - 15 q^{94} - q^{95} - 20 q^{96} - 19 q^{97} - 5 q^{98} + 38 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.04477 0.738766 0.369383 0.929277i \(-0.379569\pi\)
0.369383 + 0.929277i \(0.379569\pi\)
\(3\) −2.52412 −1.45730 −0.728651 0.684885i \(-0.759853\pi\)
−0.728651 + 0.684885i \(0.759853\pi\)
\(4\) −0.908449 −0.454224
\(5\) 0.499809 0.223521 0.111761 0.993735i \(-0.464351\pi\)
0.111761 + 0.993735i \(0.464351\pi\)
\(6\) −2.63714 −1.07661
\(7\) −3.30668 −1.24981 −0.624904 0.780702i \(-0.714862\pi\)
−0.624904 + 0.780702i \(0.714862\pi\)
\(8\) −3.03867 −1.07433
\(9\) 3.37119 1.12373
\(10\) 0.522187 0.165130
\(11\) −3.72276 −1.12245 −0.561227 0.827662i \(-0.689670\pi\)
−0.561227 + 0.827662i \(0.689670\pi\)
\(12\) 2.29304 0.661942
\(13\) −4.68695 −1.29993 −0.649963 0.759966i \(-0.725216\pi\)
−0.649963 + 0.759966i \(0.725216\pi\)
\(14\) −3.45473 −0.923316
\(15\) −1.26158 −0.325738
\(16\) −1.35782 −0.339456
\(17\) −4.59028 −1.11331 −0.556653 0.830745i \(-0.687915\pi\)
−0.556653 + 0.830745i \(0.687915\pi\)
\(18\) 3.52213 0.830175
\(19\) 6.05844 1.38990 0.694951 0.719057i \(-0.255426\pi\)
0.694951 + 0.719057i \(0.255426\pi\)
\(20\) −0.454051 −0.101529
\(21\) 8.34647 1.82135
\(22\) −3.88944 −0.829231
\(23\) −0.388946 −0.0811008 −0.0405504 0.999177i \(-0.512911\pi\)
−0.0405504 + 0.999177i \(0.512911\pi\)
\(24\) 7.66997 1.56563
\(25\) −4.75019 −0.950038
\(26\) −4.89680 −0.960342
\(27\) −0.936938 −0.180314
\(28\) 3.00395 0.567693
\(29\) 3.53498 0.656429 0.328215 0.944603i \(-0.393553\pi\)
0.328215 + 0.944603i \(0.393553\pi\)
\(30\) −1.31806 −0.240644
\(31\) −0.870839 −0.156407 −0.0782037 0.996937i \(-0.524918\pi\)
−0.0782037 + 0.996937i \(0.524918\pi\)
\(32\) 4.65872 0.823553
\(33\) 9.39670 1.63575
\(34\) −4.79580 −0.822473
\(35\) −1.65271 −0.279359
\(36\) −3.06256 −0.510426
\(37\) −3.93313 −0.646602 −0.323301 0.946296i \(-0.604793\pi\)
−0.323301 + 0.946296i \(0.604793\pi\)
\(38\) 6.32970 1.02681
\(39\) 11.8304 1.89439
\(40\) −1.51875 −0.240136
\(41\) −10.7837 −1.68412 −0.842062 0.539380i \(-0.818659\pi\)
−0.842062 + 0.539380i \(0.818659\pi\)
\(42\) 8.72017 1.34555
\(43\) 0 0
\(44\) 3.38193 0.509846
\(45\) 1.68495 0.251178
\(46\) −0.406360 −0.0599146
\(47\) 3.49088 0.509197 0.254599 0.967047i \(-0.418057\pi\)
0.254599 + 0.967047i \(0.418057\pi\)
\(48\) 3.42731 0.494690
\(49\) 3.93414 0.562020
\(50\) −4.96287 −0.701856
\(51\) 11.5864 1.62242
\(52\) 4.25785 0.590458
\(53\) 4.77361 0.655706 0.327853 0.944729i \(-0.393675\pi\)
0.327853 + 0.944729i \(0.393675\pi\)
\(54\) −0.978888 −0.133210
\(55\) −1.86067 −0.250892
\(56\) 10.0479 1.34271
\(57\) −15.2923 −2.02551
\(58\) 3.69325 0.484948
\(59\) 6.72259 0.875207 0.437603 0.899168i \(-0.355827\pi\)
0.437603 + 0.899168i \(0.355827\pi\)
\(60\) 1.14608 0.147958
\(61\) −5.24696 −0.671805 −0.335902 0.941897i \(-0.609041\pi\)
−0.335902 + 0.941897i \(0.609041\pi\)
\(62\) −0.909829 −0.115548
\(63\) −11.1475 −1.40445
\(64\) 7.58295 0.947869
\(65\) −2.34258 −0.290561
\(66\) 9.81742 1.20844
\(67\) 10.5533 1.28929 0.644645 0.764482i \(-0.277005\pi\)
0.644645 + 0.764482i \(0.277005\pi\)
\(68\) 4.17003 0.505691
\(69\) 0.981747 0.118188
\(70\) −1.72670 −0.206381
\(71\) −0.899542 −0.106756 −0.0533780 0.998574i \(-0.516999\pi\)
−0.0533780 + 0.998574i \(0.516999\pi\)
\(72\) −10.2439 −1.20726
\(73\) 0.966562 0.113128 0.0565638 0.998399i \(-0.481986\pi\)
0.0565638 + 0.998399i \(0.481986\pi\)
\(74\) −4.10922 −0.477688
\(75\) 11.9901 1.38449
\(76\) −5.50379 −0.631328
\(77\) 12.3100 1.40285
\(78\) 12.3601 1.39951
\(79\) −1.00197 −0.112730 −0.0563650 0.998410i \(-0.517951\pi\)
−0.0563650 + 0.998410i \(0.517951\pi\)
\(80\) −0.678652 −0.0758756
\(81\) −7.74863 −0.860959
\(82\) −11.2665 −1.24417
\(83\) −3.41592 −0.374946 −0.187473 0.982270i \(-0.560030\pi\)
−0.187473 + 0.982270i \(0.560030\pi\)
\(84\) −7.58234 −0.827301
\(85\) −2.29426 −0.248847
\(86\) 0 0
\(87\) −8.92272 −0.956616
\(88\) 11.3122 1.20589
\(89\) 1.84065 0.195108 0.0975542 0.995230i \(-0.468898\pi\)
0.0975542 + 0.995230i \(0.468898\pi\)
\(90\) 1.76039 0.185562
\(91\) 15.4982 1.62466
\(92\) 0.353338 0.0368380
\(93\) 2.19810 0.227933
\(94\) 3.64718 0.376178
\(95\) 3.02806 0.310673
\(96\) −11.7592 −1.20017
\(97\) 0.960596 0.0975337 0.0487669 0.998810i \(-0.484471\pi\)
0.0487669 + 0.998810i \(0.484471\pi\)
\(98\) 4.11028 0.415201
\(99\) −12.5501 −1.26134
\(100\) 4.31531 0.431531
\(101\) 16.4388 1.63572 0.817862 0.575414i \(-0.195159\pi\)
0.817862 + 0.575414i \(0.195159\pi\)
\(102\) 12.1052 1.19859
\(103\) −0.572128 −0.0563735 −0.0281867 0.999603i \(-0.508973\pi\)
−0.0281867 + 0.999603i \(0.508973\pi\)
\(104\) 14.2421 1.39655
\(105\) 4.17164 0.407110
\(106\) 4.98734 0.484413
\(107\) −16.1723 −1.56343 −0.781716 0.623634i \(-0.785656\pi\)
−0.781716 + 0.623634i \(0.785656\pi\)
\(108\) 0.851160 0.0819029
\(109\) −6.52710 −0.625182 −0.312591 0.949888i \(-0.601197\pi\)
−0.312591 + 0.949888i \(0.601197\pi\)
\(110\) −1.94397 −0.185351
\(111\) 9.92769 0.942295
\(112\) 4.48989 0.424255
\(113\) 1.70054 0.159974 0.0799868 0.996796i \(-0.474512\pi\)
0.0799868 + 0.996796i \(0.474512\pi\)
\(114\) −15.9769 −1.49638
\(115\) −0.194399 −0.0181278
\(116\) −3.21135 −0.298166
\(117\) −15.8006 −1.46077
\(118\) 7.02358 0.646573
\(119\) 15.1786 1.39142
\(120\) 3.83352 0.349951
\(121\) 2.85893 0.259902
\(122\) −5.48189 −0.496307
\(123\) 27.2193 2.45428
\(124\) 0.791113 0.0710440
\(125\) −4.87323 −0.435875
\(126\) −11.6466 −1.03756
\(127\) 15.6958 1.39277 0.696387 0.717666i \(-0.254790\pi\)
0.696387 + 0.717666i \(0.254790\pi\)
\(128\) −1.39498 −0.123300
\(129\) 0 0
\(130\) −2.44746 −0.214657
\(131\) 10.7262 0.937151 0.468576 0.883423i \(-0.344767\pi\)
0.468576 + 0.883423i \(0.344767\pi\)
\(132\) −8.53642 −0.743000
\(133\) −20.0333 −1.73711
\(134\) 11.0258 0.952484
\(135\) −0.468290 −0.0403040
\(136\) 13.9483 1.19606
\(137\) 8.08569 0.690807 0.345404 0.938454i \(-0.387742\pi\)
0.345404 + 0.938454i \(0.387742\pi\)
\(138\) 1.02570 0.0873137
\(139\) 12.8655 1.09124 0.545620 0.838032i \(-0.316294\pi\)
0.545620 + 0.838032i \(0.316294\pi\)
\(140\) 1.50140 0.126891
\(141\) −8.81141 −0.742054
\(142\) −0.939817 −0.0788677
\(143\) 17.4484 1.45911
\(144\) −4.57748 −0.381457
\(145\) 1.76681 0.146726
\(146\) 1.00984 0.0835748
\(147\) −9.93025 −0.819033
\(148\) 3.57304 0.293702
\(149\) 7.44008 0.609515 0.304757 0.952430i \(-0.401425\pi\)
0.304757 + 0.952430i \(0.401425\pi\)
\(150\) 12.5269 1.02282
\(151\) 4.77560 0.388633 0.194316 0.980939i \(-0.437751\pi\)
0.194316 + 0.980939i \(0.437751\pi\)
\(152\) −18.4096 −1.49322
\(153\) −15.4747 −1.25106
\(154\) 12.8611 1.03638
\(155\) −0.435253 −0.0349604
\(156\) −10.7473 −0.860476
\(157\) −13.4410 −1.07271 −0.536354 0.843993i \(-0.680199\pi\)
−0.536354 + 0.843993i \(0.680199\pi\)
\(158\) −1.04683 −0.0832811
\(159\) −12.0492 −0.955562
\(160\) 2.32847 0.184082
\(161\) 1.28612 0.101360
\(162\) −8.09557 −0.636048
\(163\) −18.7293 −1.46699 −0.733496 0.679694i \(-0.762113\pi\)
−0.733496 + 0.679694i \(0.762113\pi\)
\(164\) 9.79640 0.764970
\(165\) 4.69655 0.365626
\(166\) −3.56886 −0.276997
\(167\) −13.8552 −1.07215 −0.536075 0.844170i \(-0.680094\pi\)
−0.536075 + 0.844170i \(0.680094\pi\)
\(168\) −25.3622 −1.95673
\(169\) 8.96750 0.689808
\(170\) −2.39698 −0.183840
\(171\) 20.4242 1.56188
\(172\) 0 0
\(173\) −13.8705 −1.05456 −0.527278 0.849693i \(-0.676787\pi\)
−0.527278 + 0.849693i \(0.676787\pi\)
\(174\) −9.32222 −0.706716
\(175\) 15.7074 1.18737
\(176\) 5.05485 0.381023
\(177\) −16.9686 −1.27544
\(178\) 1.92306 0.144140
\(179\) −6.18438 −0.462242 −0.231121 0.972925i \(-0.574239\pi\)
−0.231121 + 0.972925i \(0.574239\pi\)
\(180\) −1.53069 −0.114091
\(181\) 3.70276 0.275224 0.137612 0.990486i \(-0.456057\pi\)
0.137612 + 0.990486i \(0.456057\pi\)
\(182\) 16.1922 1.20024
\(183\) 13.2440 0.979023
\(184\) 1.18188 0.0871292
\(185\) −1.96581 −0.144529
\(186\) 2.29652 0.168389
\(187\) 17.0885 1.24963
\(188\) −3.17128 −0.231290
\(189\) 3.09815 0.225358
\(190\) 3.16364 0.229515
\(191\) −11.4154 −0.825991 −0.412995 0.910733i \(-0.635517\pi\)
−0.412995 + 0.910733i \(0.635517\pi\)
\(192\) −19.1403 −1.38133
\(193\) −8.44971 −0.608223 −0.304112 0.952636i \(-0.598360\pi\)
−0.304112 + 0.952636i \(0.598360\pi\)
\(194\) 1.00360 0.0720546
\(195\) 5.91295 0.423435
\(196\) −3.57396 −0.255283
\(197\) −21.6649 −1.54356 −0.771780 0.635890i \(-0.780633\pi\)
−0.771780 + 0.635890i \(0.780633\pi\)
\(198\) −13.1120 −0.931833
\(199\) 9.87849 0.700267 0.350134 0.936700i \(-0.386136\pi\)
0.350134 + 0.936700i \(0.386136\pi\)
\(200\) 14.4343 1.02066
\(201\) −26.6378 −1.87889
\(202\) 17.1748 1.20842
\(203\) −11.6890 −0.820410
\(204\) −10.5257 −0.736944
\(205\) −5.38977 −0.376438
\(206\) −0.597744 −0.0416468
\(207\) −1.31121 −0.0911356
\(208\) 6.36405 0.441267
\(209\) −22.5541 −1.56010
\(210\) 4.35841 0.300759
\(211\) 15.2999 1.05329 0.526644 0.850086i \(-0.323450\pi\)
0.526644 + 0.850086i \(0.323450\pi\)
\(212\) −4.33658 −0.297838
\(213\) 2.27055 0.155576
\(214\) −16.8964 −1.15501
\(215\) 0 0
\(216\) 2.84704 0.193717
\(217\) 2.87959 0.195479
\(218\) −6.81934 −0.461864
\(219\) −2.43972 −0.164861
\(220\) 1.69032 0.113961
\(221\) 21.5144 1.44722
\(222\) 10.3722 0.696135
\(223\) −11.7167 −0.784608 −0.392304 0.919836i \(-0.628322\pi\)
−0.392304 + 0.919836i \(0.628322\pi\)
\(224\) −15.4049 −1.02928
\(225\) −16.0138 −1.06759
\(226\) 1.77668 0.118183
\(227\) −9.03975 −0.599989 −0.299995 0.953941i \(-0.596985\pi\)
−0.299995 + 0.953941i \(0.596985\pi\)
\(228\) 13.8922 0.920035
\(229\) −15.5978 −1.03073 −0.515366 0.856970i \(-0.672344\pi\)
−0.515366 + 0.856970i \(0.672344\pi\)
\(230\) −0.203102 −0.0133922
\(231\) −31.0719 −2.04438
\(232\) −10.7416 −0.705223
\(233\) 4.69718 0.307722 0.153861 0.988092i \(-0.450829\pi\)
0.153861 + 0.988092i \(0.450829\pi\)
\(234\) −16.5081 −1.07917
\(235\) 1.74477 0.113816
\(236\) −6.10713 −0.397540
\(237\) 2.52908 0.164282
\(238\) 15.8582 1.02793
\(239\) −4.11467 −0.266156 −0.133078 0.991106i \(-0.542486\pi\)
−0.133078 + 0.991106i \(0.542486\pi\)
\(240\) 1.71300 0.110574
\(241\) −3.26436 −0.210276 −0.105138 0.994458i \(-0.533528\pi\)
−0.105138 + 0.994458i \(0.533528\pi\)
\(242\) 2.98693 0.192007
\(243\) 22.3693 1.43499
\(244\) 4.76660 0.305150
\(245\) 1.96632 0.125623
\(246\) 28.4380 1.81314
\(247\) −28.3956 −1.80677
\(248\) 2.64619 0.168033
\(249\) 8.62220 0.546410
\(250\) −5.09142 −0.322010
\(251\) 5.47251 0.345422 0.172711 0.984973i \(-0.444747\pi\)
0.172711 + 0.984973i \(0.444747\pi\)
\(252\) 10.1269 0.637935
\(253\) 1.44795 0.0910319
\(254\) 16.3985 1.02893
\(255\) 5.79099 0.362646
\(256\) −16.6233 −1.03896
\(257\) 8.41973 0.525208 0.262604 0.964904i \(-0.415419\pi\)
0.262604 + 0.964904i \(0.415419\pi\)
\(258\) 0 0
\(259\) 13.0056 0.808128
\(260\) 2.12811 0.131980
\(261\) 11.9171 0.737650
\(262\) 11.2064 0.692336
\(263\) −6.11997 −0.377374 −0.188687 0.982037i \(-0.560423\pi\)
−0.188687 + 0.982037i \(0.560423\pi\)
\(264\) −28.5535 −1.75734
\(265\) 2.38589 0.146564
\(266\) −20.9303 −1.28332
\(267\) −4.64602 −0.284332
\(268\) −9.58712 −0.585627
\(269\) 2.29453 0.139900 0.0699500 0.997550i \(-0.477716\pi\)
0.0699500 + 0.997550i \(0.477716\pi\)
\(270\) −0.489256 −0.0297752
\(271\) 27.3969 1.66425 0.832123 0.554591i \(-0.187125\pi\)
0.832123 + 0.554591i \(0.187125\pi\)
\(272\) 6.23279 0.377918
\(273\) −39.1195 −2.36762
\(274\) 8.44771 0.510345
\(275\) 17.6838 1.06637
\(276\) −0.891867 −0.0536841
\(277\) 26.5986 1.59815 0.799077 0.601229i \(-0.205322\pi\)
0.799077 + 0.601229i \(0.205322\pi\)
\(278\) 13.4416 0.806172
\(279\) −2.93577 −0.175760
\(280\) 5.02203 0.300124
\(281\) 16.6768 0.994852 0.497426 0.867506i \(-0.334279\pi\)
0.497426 + 0.867506i \(0.334279\pi\)
\(282\) −9.20592 −0.548205
\(283\) 17.1076 1.01694 0.508471 0.861079i \(-0.330211\pi\)
0.508471 + 0.861079i \(0.330211\pi\)
\(284\) 0.817188 0.0484912
\(285\) −7.64320 −0.452744
\(286\) 18.2296 1.07794
\(287\) 35.6581 2.10483
\(288\) 15.7055 0.925453
\(289\) 4.07065 0.239450
\(290\) 1.84592 0.108396
\(291\) −2.42466 −0.142136
\(292\) −0.878072 −0.0513853
\(293\) −29.9122 −1.74749 −0.873744 0.486387i \(-0.838315\pi\)
−0.873744 + 0.486387i \(0.838315\pi\)
\(294\) −10.3749 −0.605074
\(295\) 3.36001 0.195627
\(296\) 11.9515 0.694665
\(297\) 3.48799 0.202394
\(298\) 7.77319 0.450289
\(299\) 1.82297 0.105425
\(300\) −10.8924 −0.628871
\(301\) 0 0
\(302\) 4.98942 0.287109
\(303\) −41.4936 −2.38375
\(304\) −8.22630 −0.471810
\(305\) −2.62248 −0.150163
\(306\) −16.1676 −0.924238
\(307\) −13.4430 −0.767230 −0.383615 0.923493i \(-0.625321\pi\)
−0.383615 + 0.923493i \(0.625321\pi\)
\(308\) −11.1830 −0.637209
\(309\) 1.44412 0.0821532
\(310\) −0.454741 −0.0258275
\(311\) 15.1900 0.861347 0.430673 0.902508i \(-0.358276\pi\)
0.430673 + 0.902508i \(0.358276\pi\)
\(312\) −35.9488 −2.03520
\(313\) −11.3920 −0.643913 −0.321956 0.946754i \(-0.604340\pi\)
−0.321956 + 0.946754i \(0.604340\pi\)
\(314\) −14.0428 −0.792480
\(315\) −5.57160 −0.313924
\(316\) 0.910235 0.0512047
\(317\) −22.8190 −1.28164 −0.640822 0.767689i \(-0.721406\pi\)
−0.640822 + 0.767689i \(0.721406\pi\)
\(318\) −12.5887 −0.705937
\(319\) −13.1599 −0.736811
\(320\) 3.79003 0.211869
\(321\) 40.8208 2.27840
\(322\) 1.34370 0.0748817
\(323\) −27.8099 −1.54739
\(324\) 7.03924 0.391069
\(325\) 22.2639 1.23498
\(326\) −19.5679 −1.08376
\(327\) 16.4752 0.911080
\(328\) 32.7680 1.80931
\(329\) −11.5432 −0.636399
\(330\) 4.90683 0.270112
\(331\) −15.5508 −0.854747 −0.427373 0.904075i \(-0.640561\pi\)
−0.427373 + 0.904075i \(0.640561\pi\)
\(332\) 3.10319 0.170310
\(333\) −13.2593 −0.726607
\(334\) −14.4756 −0.792068
\(335\) 5.27463 0.288184
\(336\) −11.3330 −0.618267
\(337\) 14.9040 0.811871 0.405935 0.913902i \(-0.366946\pi\)
0.405935 + 0.913902i \(0.366946\pi\)
\(338\) 9.36901 0.509607
\(339\) −4.29238 −0.233130
\(340\) 2.08422 0.113033
\(341\) 3.24192 0.175560
\(342\) 21.3386 1.15386
\(343\) 10.1378 0.547391
\(344\) 0 0
\(345\) 0.490686 0.0264176
\(346\) −14.4915 −0.779071
\(347\) −27.1661 −1.45835 −0.729177 0.684325i \(-0.760097\pi\)
−0.729177 + 0.684325i \(0.760097\pi\)
\(348\) 8.10583 0.434518
\(349\) −28.3556 −1.51784 −0.758920 0.651184i \(-0.774273\pi\)
−0.758920 + 0.651184i \(0.774273\pi\)
\(350\) 16.4106 0.877185
\(351\) 4.39138 0.234395
\(352\) −17.3433 −0.924401
\(353\) −19.7233 −1.04977 −0.524883 0.851175i \(-0.675891\pi\)
−0.524883 + 0.851175i \(0.675891\pi\)
\(354\) −17.7284 −0.942253
\(355\) −0.449599 −0.0238622
\(356\) −1.67214 −0.0886230
\(357\) −38.3126 −2.02772
\(358\) −6.46127 −0.341489
\(359\) −28.1781 −1.48718 −0.743592 0.668633i \(-0.766880\pi\)
−0.743592 + 0.668633i \(0.766880\pi\)
\(360\) −5.12001 −0.269848
\(361\) 17.7047 0.931829
\(362\) 3.86854 0.203326
\(363\) −7.21628 −0.378756
\(364\) −14.0794 −0.737959
\(365\) 0.483096 0.0252864
\(366\) 13.8370 0.723269
\(367\) −29.5394 −1.54194 −0.770972 0.636869i \(-0.780229\pi\)
−0.770972 + 0.636869i \(0.780229\pi\)
\(368\) 0.528120 0.0275302
\(369\) −36.3538 −1.89250
\(370\) −2.05383 −0.106773
\(371\) −15.7848 −0.819506
\(372\) −1.99687 −0.103533
\(373\) 32.1477 1.66454 0.832272 0.554367i \(-0.187040\pi\)
0.832272 + 0.554367i \(0.187040\pi\)
\(374\) 17.8536 0.923188
\(375\) 12.3006 0.635202
\(376\) −10.6076 −0.547047
\(377\) −16.5683 −0.853309
\(378\) 3.23687 0.166487
\(379\) −18.0279 −0.926033 −0.463017 0.886350i \(-0.653233\pi\)
−0.463017 + 0.886350i \(0.653233\pi\)
\(380\) −2.75084 −0.141115
\(381\) −39.6180 −2.02969
\(382\) −11.9265 −0.610214
\(383\) 22.9648 1.17345 0.586724 0.809787i \(-0.300417\pi\)
0.586724 + 0.809787i \(0.300417\pi\)
\(384\) 3.52109 0.179685
\(385\) 6.15263 0.313567
\(386\) −8.82803 −0.449335
\(387\) 0 0
\(388\) −0.872652 −0.0443022
\(389\) 22.8174 1.15689 0.578445 0.815721i \(-0.303660\pi\)
0.578445 + 0.815721i \(0.303660\pi\)
\(390\) 6.17770 0.312820
\(391\) 1.78537 0.0902901
\(392\) −11.9545 −0.603796
\(393\) −27.0742 −1.36571
\(394\) −22.6349 −1.14033
\(395\) −0.500791 −0.0251975
\(396\) 11.4012 0.572930
\(397\) −8.63189 −0.433222 −0.216611 0.976258i \(-0.569500\pi\)
−0.216611 + 0.976258i \(0.569500\pi\)
\(398\) 10.3208 0.517334
\(399\) 50.5666 2.53150
\(400\) 6.44992 0.322496
\(401\) 33.8253 1.68916 0.844578 0.535433i \(-0.179852\pi\)
0.844578 + 0.535433i \(0.179852\pi\)
\(402\) −27.8305 −1.38806
\(403\) 4.08158 0.203318
\(404\) −14.9338 −0.742986
\(405\) −3.87283 −0.192443
\(406\) −12.2124 −0.606091
\(407\) 14.6421 0.725781
\(408\) −35.2073 −1.74302
\(409\) 25.6569 1.26865 0.634326 0.773065i \(-0.281278\pi\)
0.634326 + 0.773065i \(0.281278\pi\)
\(410\) −5.63108 −0.278099
\(411\) −20.4093 −1.00672
\(412\) 0.519749 0.0256062
\(413\) −22.2295 −1.09384
\(414\) −1.36992 −0.0673279
\(415\) −1.70731 −0.0838083
\(416\) −21.8352 −1.07056
\(417\) −32.4742 −1.59027
\(418\) −23.5639 −1.15255
\(419\) 24.8323 1.21314 0.606570 0.795030i \(-0.292545\pi\)
0.606570 + 0.795030i \(0.292545\pi\)
\(420\) −3.78972 −0.184919
\(421\) −19.2945 −0.940356 −0.470178 0.882571i \(-0.655810\pi\)
−0.470178 + 0.882571i \(0.655810\pi\)
\(422\) 15.9849 0.778134
\(423\) 11.7684 0.572201
\(424\) −14.5054 −0.704446
\(425\) 21.8047 1.05768
\(426\) 2.37221 0.114934
\(427\) 17.3500 0.839627
\(428\) 14.6917 0.710149
\(429\) −44.0418 −2.12636
\(430\) 0 0
\(431\) 32.4273 1.56197 0.780983 0.624552i \(-0.214718\pi\)
0.780983 + 0.624552i \(0.214718\pi\)
\(432\) 1.27220 0.0612085
\(433\) 17.1636 0.824828 0.412414 0.910997i \(-0.364686\pi\)
0.412414 + 0.910997i \(0.364686\pi\)
\(434\) 3.00852 0.144413
\(435\) −4.45965 −0.213824
\(436\) 5.92953 0.283973
\(437\) −2.35641 −0.112722
\(438\) −2.54896 −0.121794
\(439\) 14.4303 0.688720 0.344360 0.938838i \(-0.388096\pi\)
0.344360 + 0.938838i \(0.388096\pi\)
\(440\) 5.65395 0.269542
\(441\) 13.2627 0.631559
\(442\) 22.4777 1.06915
\(443\) 19.0065 0.903024 0.451512 0.892265i \(-0.350885\pi\)
0.451512 + 0.892265i \(0.350885\pi\)
\(444\) −9.01880 −0.428013
\(445\) 0.919972 0.0436109
\(446\) −12.2413 −0.579642
\(447\) −18.7797 −0.888248
\(448\) −25.0744 −1.18465
\(449\) −5.33116 −0.251593 −0.125797 0.992056i \(-0.540149\pi\)
−0.125797 + 0.992056i \(0.540149\pi\)
\(450\) −16.7308 −0.788698
\(451\) 40.1449 1.89035
\(452\) −1.54486 −0.0726639
\(453\) −12.0542 −0.566355
\(454\) −9.44449 −0.443252
\(455\) 7.74616 0.363146
\(456\) 46.4681 2.17607
\(457\) −7.95238 −0.371997 −0.185998 0.982550i \(-0.559552\pi\)
−0.185998 + 0.982550i \(0.559552\pi\)
\(458\) −16.2962 −0.761470
\(459\) 4.30080 0.200744
\(460\) 0.176601 0.00823407
\(461\) 42.5990 1.98403 0.992016 0.126115i \(-0.0402510\pi\)
0.992016 + 0.126115i \(0.0402510\pi\)
\(462\) −32.4631 −1.51032
\(463\) 5.69293 0.264573 0.132286 0.991212i \(-0.457768\pi\)
0.132286 + 0.991212i \(0.457768\pi\)
\(464\) −4.79988 −0.222829
\(465\) 1.09863 0.0509478
\(466\) 4.90749 0.227335
\(467\) 7.38749 0.341852 0.170926 0.985284i \(-0.445324\pi\)
0.170926 + 0.985284i \(0.445324\pi\)
\(468\) 14.3541 0.663516
\(469\) −34.8964 −1.61136
\(470\) 1.82289 0.0840837
\(471\) 33.9267 1.56326
\(472\) −20.4277 −0.940263
\(473\) 0 0
\(474\) 2.64232 0.121366
\(475\) −28.7788 −1.32046
\(476\) −13.7890 −0.632016
\(477\) 16.0928 0.736837
\(478\) −4.29889 −0.196627
\(479\) 20.7081 0.946177 0.473089 0.881015i \(-0.343139\pi\)
0.473089 + 0.881015i \(0.343139\pi\)
\(480\) −5.87734 −0.268263
\(481\) 18.4344 0.840535
\(482\) −3.41051 −0.155345
\(483\) −3.24633 −0.147713
\(484\) −2.59719 −0.118054
\(485\) 0.480114 0.0218009
\(486\) 23.3709 1.06012
\(487\) 35.6007 1.61322 0.806610 0.591084i \(-0.201300\pi\)
0.806610 + 0.591084i \(0.201300\pi\)
\(488\) 15.9438 0.721741
\(489\) 47.2750 2.13785
\(490\) 2.05436 0.0928063
\(491\) −0.665932 −0.0300531 −0.0150265 0.999887i \(-0.504783\pi\)
−0.0150265 + 0.999887i \(0.504783\pi\)
\(492\) −24.7273 −1.11479
\(493\) −16.2265 −0.730806
\(494\) −29.6670 −1.33478
\(495\) −6.27267 −0.281935
\(496\) 1.18245 0.0530934
\(497\) 2.97450 0.133424
\(498\) 9.00824 0.403669
\(499\) 7.19573 0.322125 0.161063 0.986944i \(-0.448508\pi\)
0.161063 + 0.986944i \(0.448508\pi\)
\(500\) 4.42708 0.197985
\(501\) 34.9723 1.56245
\(502\) 5.71753 0.255186
\(503\) 41.5468 1.85248 0.926240 0.376935i \(-0.123022\pi\)
0.926240 + 0.376935i \(0.123022\pi\)
\(504\) 33.8735 1.50884
\(505\) 8.21627 0.365619
\(506\) 1.51278 0.0672513
\(507\) −22.6351 −1.00526
\(508\) −14.2588 −0.632632
\(509\) −30.7587 −1.36335 −0.681677 0.731653i \(-0.738749\pi\)
−0.681677 + 0.731653i \(0.738749\pi\)
\(510\) 6.05028 0.267911
\(511\) −3.19611 −0.141388
\(512\) −14.5777 −0.644248
\(513\) −5.67639 −0.250619
\(514\) 8.79671 0.388006
\(515\) −0.285955 −0.0126007
\(516\) 0 0
\(517\) −12.9957 −0.571550
\(518\) 13.5879 0.597018
\(519\) 35.0109 1.53681
\(520\) 7.11832 0.312159
\(521\) 15.3022 0.670402 0.335201 0.942147i \(-0.391196\pi\)
0.335201 + 0.942147i \(0.391196\pi\)
\(522\) 12.4507 0.544951
\(523\) −1.34406 −0.0587716 −0.0293858 0.999568i \(-0.509355\pi\)
−0.0293858 + 0.999568i \(0.509355\pi\)
\(524\) −9.74419 −0.425677
\(525\) −39.6473 −1.73035
\(526\) −6.39399 −0.278791
\(527\) 3.99739 0.174129
\(528\) −12.7591 −0.555266
\(529\) −22.8487 −0.993423
\(530\) 2.49272 0.108277
\(531\) 22.6632 0.983497
\(532\) 18.1993 0.789038
\(533\) 50.5425 2.18924
\(534\) −4.85404 −0.210055
\(535\) −8.08304 −0.349460
\(536\) −32.0680 −1.38512
\(537\) 15.6101 0.673627
\(538\) 2.39727 0.103353
\(539\) −14.6458 −0.630841
\(540\) 0.425417 0.0183070
\(541\) −0.198489 −0.00853373 −0.00426686 0.999991i \(-0.501358\pi\)
−0.00426686 + 0.999991i \(0.501358\pi\)
\(542\) 28.6236 1.22949
\(543\) −9.34621 −0.401084
\(544\) −21.3848 −0.916867
\(545\) −3.26230 −0.139742
\(546\) −40.8710 −1.74912
\(547\) 7.93941 0.339465 0.169732 0.985490i \(-0.445710\pi\)
0.169732 + 0.985490i \(0.445710\pi\)
\(548\) −7.34544 −0.313782
\(549\) −17.6885 −0.754928
\(550\) 18.4756 0.787801
\(551\) 21.4165 0.912372
\(552\) −2.98321 −0.126974
\(553\) 3.31318 0.140891
\(554\) 27.7895 1.18066
\(555\) 4.96194 0.210623
\(556\) −11.6877 −0.495668
\(557\) −17.5130 −0.742048 −0.371024 0.928623i \(-0.620993\pi\)
−0.371024 + 0.928623i \(0.620993\pi\)
\(558\) −3.06721 −0.129845
\(559\) 0 0
\(560\) 2.24408 0.0948299
\(561\) −43.1334 −1.82110
\(562\) 17.4234 0.734963
\(563\) 24.8082 1.04554 0.522771 0.852473i \(-0.324898\pi\)
0.522771 + 0.852473i \(0.324898\pi\)
\(564\) 8.00471 0.337059
\(565\) 0.849946 0.0357575
\(566\) 17.8736 0.751283
\(567\) 25.6223 1.07603
\(568\) 2.73341 0.114691
\(569\) −28.8478 −1.20936 −0.604681 0.796468i \(-0.706699\pi\)
−0.604681 + 0.796468i \(0.706699\pi\)
\(570\) −7.98541 −0.334472
\(571\) −32.6060 −1.36452 −0.682259 0.731111i \(-0.739002\pi\)
−0.682259 + 0.731111i \(0.739002\pi\)
\(572\) −15.8510 −0.662762
\(573\) 28.8139 1.20372
\(574\) 37.2547 1.55498
\(575\) 1.84757 0.0770489
\(576\) 25.5636 1.06515
\(577\) 33.6229 1.39974 0.699870 0.714270i \(-0.253241\pi\)
0.699870 + 0.714270i \(0.253241\pi\)
\(578\) 4.25291 0.176898
\(579\) 21.3281 0.886366
\(580\) −1.60506 −0.0666465
\(581\) 11.2954 0.468610
\(582\) −2.53322 −0.105005
\(583\) −17.7710 −0.735999
\(584\) −2.93706 −0.121537
\(585\) −7.89728 −0.326513
\(586\) −31.2514 −1.29098
\(587\) −16.2192 −0.669439 −0.334720 0.942318i \(-0.608642\pi\)
−0.334720 + 0.942318i \(0.608642\pi\)
\(588\) 9.02112 0.372025
\(589\) −5.27593 −0.217391
\(590\) 3.51045 0.144523
\(591\) 54.6848 2.24943
\(592\) 5.34049 0.219493
\(593\) 44.2180 1.81582 0.907909 0.419168i \(-0.137678\pi\)
0.907909 + 0.419168i \(0.137678\pi\)
\(594\) 3.64416 0.149522
\(595\) 7.58639 0.311012
\(596\) −6.75893 −0.276857
\(597\) −24.9345 −1.02050
\(598\) 1.90459 0.0778845
\(599\) 4.14316 0.169285 0.0846425 0.996411i \(-0.473025\pi\)
0.0846425 + 0.996411i \(0.473025\pi\)
\(600\) −36.4338 −1.48741
\(601\) −19.9921 −0.815496 −0.407748 0.913094i \(-0.633686\pi\)
−0.407748 + 0.913094i \(0.633686\pi\)
\(602\) 0 0
\(603\) 35.5772 1.44881
\(604\) −4.33839 −0.176526
\(605\) 1.42892 0.0580937
\(606\) −43.3514 −1.76103
\(607\) 18.0883 0.734183 0.367091 0.930185i \(-0.380354\pi\)
0.367091 + 0.930185i \(0.380354\pi\)
\(608\) 28.2246 1.14466
\(609\) 29.5046 1.19559
\(610\) −2.73989 −0.110935
\(611\) −16.3616 −0.661919
\(612\) 14.0580 0.568260
\(613\) 37.5137 1.51516 0.757582 0.652740i \(-0.226380\pi\)
0.757582 + 0.652740i \(0.226380\pi\)
\(614\) −14.0448 −0.566804
\(615\) 13.6044 0.548583
\(616\) −37.4059 −1.50713
\(617\) −9.05821 −0.364670 −0.182335 0.983236i \(-0.558366\pi\)
−0.182335 + 0.983236i \(0.558366\pi\)
\(618\) 1.50878 0.0606920
\(619\) 4.96217 0.199446 0.0997232 0.995015i \(-0.468204\pi\)
0.0997232 + 0.995015i \(0.468204\pi\)
\(620\) 0.395405 0.0158798
\(621\) 0.364418 0.0146236
\(622\) 15.8701 0.636334
\(623\) −6.08644 −0.243848
\(624\) −16.0636 −0.643060
\(625\) 21.3153 0.852611
\(626\) −11.9020 −0.475701
\(627\) 56.9294 2.27354
\(628\) 12.2104 0.487250
\(629\) 18.0541 0.719866
\(630\) −5.82106 −0.231916
\(631\) −34.8666 −1.38802 −0.694009 0.719966i \(-0.744157\pi\)
−0.694009 + 0.719966i \(0.744157\pi\)
\(632\) 3.04464 0.121109
\(633\) −38.6188 −1.53496
\(634\) −23.8407 −0.946836
\(635\) 7.84488 0.311315
\(636\) 10.9461 0.434039
\(637\) −18.4391 −0.730584
\(638\) −13.7491 −0.544331
\(639\) −3.03253 −0.119965
\(640\) −0.697221 −0.0275601
\(641\) 43.9244 1.73491 0.867454 0.497518i \(-0.165755\pi\)
0.867454 + 0.497518i \(0.165755\pi\)
\(642\) 42.6485 1.68320
\(643\) 23.0813 0.910236 0.455118 0.890431i \(-0.349597\pi\)
0.455118 + 0.890431i \(0.349597\pi\)
\(644\) −1.16837 −0.0460404
\(645\) 0 0
\(646\) −29.0551 −1.14316
\(647\) 44.1509 1.73575 0.867876 0.496780i \(-0.165485\pi\)
0.867876 + 0.496780i \(0.165485\pi\)
\(648\) 23.5455 0.924956
\(649\) −25.0266 −0.982379
\(650\) 23.2607 0.912361
\(651\) −7.26843 −0.284872
\(652\) 17.0146 0.666344
\(653\) −6.65908 −0.260590 −0.130295 0.991475i \(-0.541592\pi\)
−0.130295 + 0.991475i \(0.541592\pi\)
\(654\) 17.2128 0.673075
\(655\) 5.36104 0.209473
\(656\) 14.6423 0.571686
\(657\) 3.25847 0.127125
\(658\) −12.0601 −0.470150
\(659\) −9.47092 −0.368935 −0.184467 0.982839i \(-0.559056\pi\)
−0.184467 + 0.982839i \(0.559056\pi\)
\(660\) −4.26658 −0.166076
\(661\) 16.8401 0.655005 0.327503 0.944850i \(-0.393793\pi\)
0.327503 + 0.944850i \(0.393793\pi\)
\(662\) −16.2470 −0.631458
\(663\) −54.3050 −2.10903
\(664\) 10.3798 0.402816
\(665\) −10.0128 −0.388281
\(666\) −13.8530 −0.536792
\(667\) −1.37492 −0.0532370
\(668\) 12.5868 0.486997
\(669\) 29.5744 1.14341
\(670\) 5.51079 0.212900
\(671\) 19.5332 0.754070
\(672\) 38.8839 1.49998
\(673\) −18.0156 −0.694449 −0.347225 0.937782i \(-0.612876\pi\)
−0.347225 + 0.937782i \(0.612876\pi\)
\(674\) 15.5713 0.599783
\(675\) 4.45063 0.171305
\(676\) −8.14652 −0.313328
\(677\) −5.40686 −0.207802 −0.103901 0.994588i \(-0.533133\pi\)
−0.103901 + 0.994588i \(0.533133\pi\)
\(678\) −4.48456 −0.172229
\(679\) −3.17638 −0.121898
\(680\) 6.97150 0.267345
\(681\) 22.8174 0.874366
\(682\) 3.38707 0.129698
\(683\) −36.8545 −1.41020 −0.705099 0.709109i \(-0.749098\pi\)
−0.705099 + 0.709109i \(0.749098\pi\)
\(684\) −18.5543 −0.709443
\(685\) 4.04130 0.154410
\(686\) 10.5917 0.404394
\(687\) 39.3708 1.50209
\(688\) 0 0
\(689\) −22.3737 −0.852369
\(690\) 0.512655 0.0195165
\(691\) −9.97585 −0.379499 −0.189750 0.981833i \(-0.560768\pi\)
−0.189750 + 0.981833i \(0.560768\pi\)
\(692\) 12.6007 0.479005
\(693\) 41.4993 1.57643
\(694\) −28.3824 −1.07738
\(695\) 6.43031 0.243915
\(696\) 27.1132 1.02772
\(697\) 49.5000 1.87495
\(698\) −29.6252 −1.12133
\(699\) −11.8562 −0.448445
\(700\) −14.2693 −0.539330
\(701\) 11.6450 0.439827 0.219913 0.975519i \(-0.429422\pi\)
0.219913 + 0.975519i \(0.429422\pi\)
\(702\) 4.58800 0.173163
\(703\) −23.8286 −0.898713
\(704\) −28.2295 −1.06394
\(705\) −4.40402 −0.165865
\(706\) −20.6064 −0.775531
\(707\) −54.3580 −2.04434
\(708\) 15.4151 0.579337
\(709\) 16.6947 0.626981 0.313491 0.949591i \(-0.398502\pi\)
0.313491 + 0.949591i \(0.398502\pi\)
\(710\) −0.469729 −0.0176286
\(711\) −3.37782 −0.126678
\(712\) −5.59312 −0.209611
\(713\) 0.338709 0.0126848
\(714\) −40.0280 −1.49801
\(715\) 8.72085 0.326141
\(716\) 5.61819 0.209962
\(717\) 10.3859 0.387869
\(718\) −29.4398 −1.09868
\(719\) −32.2362 −1.20221 −0.601104 0.799170i \(-0.705272\pi\)
−0.601104 + 0.799170i \(0.705272\pi\)
\(720\) −2.28787 −0.0852637
\(721\) 1.89184 0.0704560
\(722\) 18.4974 0.688404
\(723\) 8.23964 0.306435
\(724\) −3.36376 −0.125013
\(725\) −16.7918 −0.623633
\(726\) −7.53937 −0.279812
\(727\) −21.0335 −0.780088 −0.390044 0.920796i \(-0.627540\pi\)
−0.390044 + 0.920796i \(0.627540\pi\)
\(728\) −47.0941 −1.74542
\(729\) −33.2170 −1.23026
\(730\) 0.504726 0.0186807
\(731\) 0 0
\(732\) −12.0315 −0.444696
\(733\) −4.52071 −0.166976 −0.0834881 0.996509i \(-0.526606\pi\)
−0.0834881 + 0.996509i \(0.526606\pi\)
\(734\) −30.8620 −1.13914
\(735\) −4.96322 −0.183071
\(736\) −1.81199 −0.0667909
\(737\) −39.2873 −1.44717
\(738\) −37.9815 −1.39812
\(739\) 6.99980 0.257492 0.128746 0.991678i \(-0.458905\pi\)
0.128746 + 0.991678i \(0.458905\pi\)
\(740\) 1.78584 0.0656487
\(741\) 71.6740 2.63301
\(742\) −16.4915 −0.605424
\(743\) 23.0203 0.844532 0.422266 0.906472i \(-0.361235\pi\)
0.422266 + 0.906472i \(0.361235\pi\)
\(744\) −6.67931 −0.244876
\(745\) 3.71861 0.136240
\(746\) 33.5870 1.22971
\(747\) −11.5157 −0.421338
\(748\) −15.5240 −0.567614
\(749\) 53.4766 1.95399
\(750\) 12.8514 0.469266
\(751\) −14.3313 −0.522957 −0.261478 0.965209i \(-0.584210\pi\)
−0.261478 + 0.965209i \(0.584210\pi\)
\(752\) −4.74000 −0.172850
\(753\) −13.8133 −0.503384
\(754\) −17.3101 −0.630396
\(755\) 2.38689 0.0868676
\(756\) −2.81451 −0.102363
\(757\) −36.4497 −1.32479 −0.662393 0.749156i \(-0.730459\pi\)
−0.662393 + 0.749156i \(0.730459\pi\)
\(758\) −18.8351 −0.684122
\(759\) −3.65481 −0.132661
\(760\) −9.20128 −0.333766
\(761\) 44.4716 1.61209 0.806047 0.591852i \(-0.201603\pi\)
0.806047 + 0.591852i \(0.201603\pi\)
\(762\) −41.3919 −1.49947
\(763\) 21.5830 0.781358
\(764\) 10.3703 0.375185
\(765\) −7.73440 −0.279638
\(766\) 23.9931 0.866904
\(767\) −31.5084 −1.13770
\(768\) 41.9593 1.51408
\(769\) 2.51601 0.0907298 0.0453649 0.998970i \(-0.485555\pi\)
0.0453649 + 0.998970i \(0.485555\pi\)
\(770\) 6.42810 0.231653
\(771\) −21.2524 −0.765387
\(772\) 7.67613 0.276270
\(773\) 44.8085 1.61165 0.805824 0.592155i \(-0.201723\pi\)
0.805824 + 0.592155i \(0.201723\pi\)
\(774\) 0 0
\(775\) 4.13665 0.148593
\(776\) −2.91893 −0.104784
\(777\) −32.8277 −1.17769
\(778\) 23.8391 0.854672
\(779\) −65.3322 −2.34077
\(780\) −5.37162 −0.192335
\(781\) 3.34878 0.119829
\(782\) 1.86531 0.0667032
\(783\) −3.31206 −0.118363
\(784\) −5.34187 −0.190781
\(785\) −6.71792 −0.239773
\(786\) −28.2864 −1.00894
\(787\) 2.59659 0.0925583 0.0462792 0.998929i \(-0.485264\pi\)
0.0462792 + 0.998929i \(0.485264\pi\)
\(788\) 19.6814 0.701122
\(789\) 15.4476 0.549948
\(790\) −0.523213 −0.0186151
\(791\) −5.62315 −0.199936
\(792\) 38.1357 1.35509
\(793\) 24.5922 0.873296
\(794\) −9.01836 −0.320050
\(795\) −6.02228 −0.213588
\(796\) −8.97410 −0.318078
\(797\) 23.6408 0.837400 0.418700 0.908124i \(-0.362486\pi\)
0.418700 + 0.908124i \(0.362486\pi\)
\(798\) 52.8306 1.87018
\(799\) −16.0241 −0.566892
\(800\) −22.1298 −0.782407
\(801\) 6.20518 0.219249
\(802\) 35.3398 1.24789
\(803\) −3.59828 −0.126980
\(804\) 24.1991 0.853435
\(805\) 0.642814 0.0226562
\(806\) 4.26433 0.150204
\(807\) −5.79168 −0.203877
\(808\) −49.9522 −1.75731
\(809\) 40.0359 1.40759 0.703793 0.710405i \(-0.251488\pi\)
0.703793 + 0.710405i \(0.251488\pi\)
\(810\) −4.04623 −0.142170
\(811\) 9.32389 0.327406 0.163703 0.986510i \(-0.447656\pi\)
0.163703 + 0.986510i \(0.447656\pi\)
\(812\) 10.6189 0.372650
\(813\) −69.1533 −2.42531
\(814\) 15.2976 0.536182
\(815\) −9.36107 −0.327904
\(816\) −15.7323 −0.550741
\(817\) 0 0
\(818\) 26.8057 0.937238
\(819\) 52.2476 1.82568
\(820\) 4.89633 0.170987
\(821\) −1.87089 −0.0652944 −0.0326472 0.999467i \(-0.510394\pi\)
−0.0326472 + 0.999467i \(0.510394\pi\)
\(822\) −21.3231 −0.743727
\(823\) 7.72253 0.269191 0.134595 0.990901i \(-0.457027\pi\)
0.134595 + 0.990901i \(0.457027\pi\)
\(824\) 1.73851 0.0605638
\(825\) −44.6361 −1.55403
\(826\) −23.2247 −0.808092
\(827\) 10.8269 0.376490 0.188245 0.982122i \(-0.439720\pi\)
0.188245 + 0.982122i \(0.439720\pi\)
\(828\) 1.19117 0.0413960
\(829\) 16.7060 0.580223 0.290111 0.956993i \(-0.406308\pi\)
0.290111 + 0.956993i \(0.406308\pi\)
\(830\) −1.78375 −0.0619148
\(831\) −67.1380 −2.32899
\(832\) −35.5409 −1.23216
\(833\) −18.0588 −0.625700
\(834\) −33.9282 −1.17484
\(835\) −6.92497 −0.239648
\(836\) 20.4893 0.708636
\(837\) 0.815922 0.0282024
\(838\) 25.9442 0.896226
\(839\) −2.28441 −0.0788666 −0.0394333 0.999222i \(-0.512555\pi\)
−0.0394333 + 0.999222i \(0.512555\pi\)
\(840\) −12.6762 −0.437371
\(841\) −16.5039 −0.569101
\(842\) −20.1584 −0.694704
\(843\) −42.0942 −1.44980
\(844\) −13.8992 −0.478429
\(845\) 4.48203 0.154187
\(846\) 12.2953 0.422723
\(847\) −9.45356 −0.324828
\(848\) −6.48172 −0.222583
\(849\) −43.1817 −1.48199
\(850\) 22.7810 0.781381
\(851\) 1.52977 0.0524400
\(852\) −2.06268 −0.0706663
\(853\) 14.0427 0.480811 0.240406 0.970673i \(-0.422720\pi\)
0.240406 + 0.970673i \(0.422720\pi\)
\(854\) 18.1268 0.620288
\(855\) 10.2082 0.349113
\(856\) 49.1422 1.67965
\(857\) −40.2306 −1.37425 −0.687126 0.726538i \(-0.741128\pi\)
−0.687126 + 0.726538i \(0.741128\pi\)
\(858\) −46.0137 −1.57088
\(859\) 34.4229 1.17449 0.587247 0.809407i \(-0.300212\pi\)
0.587247 + 0.809407i \(0.300212\pi\)
\(860\) 0 0
\(861\) −90.0055 −3.06738
\(862\) 33.8791 1.15393
\(863\) −20.2575 −0.689575 −0.344787 0.938681i \(-0.612049\pi\)
−0.344787 + 0.938681i \(0.612049\pi\)
\(864\) −4.36493 −0.148498
\(865\) −6.93261 −0.235716
\(866\) 17.9320 0.609355
\(867\) −10.2748 −0.348951
\(868\) −2.61596 −0.0887914
\(869\) 3.73008 0.126534
\(870\) −4.65933 −0.157966
\(871\) −49.4627 −1.67598
\(872\) 19.8337 0.671653
\(873\) 3.23835 0.109602
\(874\) −2.46191 −0.0832754
\(875\) 16.1142 0.544760
\(876\) 2.21636 0.0748839
\(877\) −43.7768 −1.47824 −0.739119 0.673575i \(-0.764758\pi\)
−0.739119 + 0.673575i \(0.764758\pi\)
\(878\) 15.0764 0.508803
\(879\) 75.5020 2.54662
\(880\) 2.52646 0.0851668
\(881\) 3.24793 0.109426 0.0547128 0.998502i \(-0.482576\pi\)
0.0547128 + 0.998502i \(0.482576\pi\)
\(882\) 13.8566 0.466575
\(883\) −15.3073 −0.515131 −0.257565 0.966261i \(-0.582920\pi\)
−0.257565 + 0.966261i \(0.582920\pi\)
\(884\) −19.5447 −0.657361
\(885\) −8.48107 −0.285088
\(886\) 19.8574 0.667124
\(887\) 10.3041 0.345977 0.172988 0.984924i \(-0.444658\pi\)
0.172988 + 0.984924i \(0.444658\pi\)
\(888\) −30.1670 −1.01234
\(889\) −51.9009 −1.74070
\(890\) 0.961162 0.0322182
\(891\) 28.8463 0.966387
\(892\) 10.6440 0.356388
\(893\) 21.1493 0.707734
\(894\) −19.6205 −0.656207
\(895\) −3.09101 −0.103321
\(896\) 4.61274 0.154101
\(897\) −4.60140 −0.153636
\(898\) −5.56986 −0.185868
\(899\) −3.07840 −0.102670
\(900\) 14.5477 0.484924
\(901\) −21.9122 −0.730001
\(902\) 41.9424 1.39653
\(903\) 0 0
\(904\) −5.16739 −0.171865
\(905\) 1.85067 0.0615183
\(906\) −12.5939 −0.418404
\(907\) −44.5746 −1.48008 −0.740038 0.672565i \(-0.765192\pi\)
−0.740038 + 0.672565i \(0.765192\pi\)
\(908\) 8.21215 0.272530
\(909\) 55.4185 1.83811
\(910\) 8.09298 0.268280
\(911\) −9.12024 −0.302167 −0.151084 0.988521i \(-0.548276\pi\)
−0.151084 + 0.988521i \(0.548276\pi\)
\(912\) 20.7642 0.687571
\(913\) 12.7166 0.420859
\(914\) −8.30843 −0.274819
\(915\) 6.61945 0.218832
\(916\) 14.1698 0.468184
\(917\) −35.4681 −1.17126
\(918\) 4.49337 0.148303
\(919\) −27.4060 −0.904039 −0.452020 0.892008i \(-0.649296\pi\)
−0.452020 + 0.892008i \(0.649296\pi\)
\(920\) 0.590713 0.0194752
\(921\) 33.9317 1.11809
\(922\) 44.5062 1.46574
\(923\) 4.21611 0.138775
\(924\) 28.2272 0.928607
\(925\) 18.6831 0.614296
\(926\) 5.94782 0.195458
\(927\) −1.92875 −0.0633486
\(928\) 16.4685 0.540604
\(929\) −8.08472 −0.265251 −0.132625 0.991166i \(-0.542341\pi\)
−0.132625 + 0.991166i \(0.542341\pi\)
\(930\) 1.14782 0.0376385
\(931\) 23.8348 0.781153
\(932\) −4.26714 −0.139775
\(933\) −38.3415 −1.25524
\(934\) 7.71825 0.252549
\(935\) 8.54098 0.279320
\(936\) 48.0129 1.56935
\(937\) −37.2600 −1.21723 −0.608615 0.793465i \(-0.708275\pi\)
−0.608615 + 0.793465i \(0.708275\pi\)
\(938\) −36.4588 −1.19042
\(939\) 28.7548 0.938376
\(940\) −1.58504 −0.0516982
\(941\) 36.3058 1.18354 0.591768 0.806108i \(-0.298430\pi\)
0.591768 + 0.806108i \(0.298430\pi\)
\(942\) 35.4457 1.15488
\(943\) 4.19426 0.136584
\(944\) −9.12809 −0.297094
\(945\) 1.54848 0.0503722
\(946\) 0 0
\(947\) 46.5566 1.51288 0.756442 0.654060i \(-0.226936\pi\)
0.756442 + 0.654060i \(0.226936\pi\)
\(948\) −2.29754 −0.0746207
\(949\) −4.53023 −0.147057
\(950\) −30.0673 −0.975512
\(951\) 57.5980 1.86774
\(952\) −46.1227 −1.49485
\(953\) −31.4936 −1.02018 −0.510089 0.860122i \(-0.670388\pi\)
−0.510089 + 0.860122i \(0.670388\pi\)
\(954\) 16.8133 0.544350
\(955\) −5.70552 −0.184626
\(956\) 3.73796 0.120894
\(957\) 33.2171 1.07376
\(958\) 21.6353 0.699004
\(959\) −26.7368 −0.863376
\(960\) −9.56649 −0.308757
\(961\) −30.2416 −0.975537
\(962\) 19.2597 0.620959
\(963\) −54.5199 −1.75688
\(964\) 2.96550 0.0955124
\(965\) −4.22324 −0.135951
\(966\) −3.39167 −0.109125
\(967\) −1.18018 −0.0379519 −0.0189760 0.999820i \(-0.506041\pi\)
−0.0189760 + 0.999820i \(0.506041\pi\)
\(968\) −8.68733 −0.279221
\(969\) 70.1957 2.25501
\(970\) 0.501610 0.0161057
\(971\) −39.8983 −1.28040 −0.640199 0.768209i \(-0.721148\pi\)
−0.640199 + 0.768209i \(0.721148\pi\)
\(972\) −20.3214 −0.651809
\(973\) −42.5422 −1.36384
\(974\) 37.1946 1.19179
\(975\) −56.1968 −1.79974
\(976\) 7.12445 0.228048
\(977\) −31.6088 −1.01125 −0.505627 0.862752i \(-0.668739\pi\)
−0.505627 + 0.862752i \(0.668739\pi\)
\(978\) 49.3917 1.57937
\(979\) −6.85229 −0.219000
\(980\) −1.78630 −0.0570612
\(981\) −22.0041 −0.702537
\(982\) −0.695748 −0.0222022
\(983\) 24.3164 0.775573 0.387787 0.921749i \(-0.373240\pi\)
0.387787 + 0.921749i \(0.373240\pi\)
\(984\) −82.7104 −2.63671
\(985\) −10.8283 −0.345018
\(986\) −16.9531 −0.539895
\(987\) 29.1365 0.927425
\(988\) 25.7960 0.820679
\(989\) 0 0
\(990\) −6.55351 −0.208284
\(991\) 3.22998 0.102604 0.0513019 0.998683i \(-0.483663\pi\)
0.0513019 + 0.998683i \(0.483663\pi\)
\(992\) −4.05700 −0.128810
\(993\) 39.2520 1.24562
\(994\) 3.10768 0.0985695
\(995\) 4.93735 0.156525
\(996\) −7.83283 −0.248193
\(997\) 14.7265 0.466394 0.233197 0.972430i \(-0.425081\pi\)
0.233197 + 0.972430i \(0.425081\pi\)
\(998\) 7.51791 0.237975
\(999\) 3.68509 0.116591
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 1849.2.a.o.1.11 18
43.12 odd 42 43.2.g.a.15.2 36
43.18 odd 42 43.2.g.a.23.2 yes 36
43.42 odd 2 1849.2.a.n.1.8 18
129.98 even 42 387.2.y.c.316.2 36
129.104 even 42 387.2.y.c.109.2 36
172.55 even 42 688.2.bg.c.273.1 36
172.147 even 42 688.2.bg.c.625.1 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.2.g.a.15.2 36 43.12 odd 42
43.2.g.a.23.2 yes 36 43.18 odd 42
387.2.y.c.109.2 36 129.104 even 42
387.2.y.c.316.2 36 129.98 even 42
688.2.bg.c.273.1 36 172.55 even 42
688.2.bg.c.625.1 36 172.147 even 42
1849.2.a.n.1.8 18 43.42 odd 2
1849.2.a.o.1.11 18 1.1 even 1 trivial