Properties

Label 6047.2.a.b.1.16
Level $6047$
Weight $2$
Character 6047.1
Self dual yes
Analytic conductor $48.286$
Analytic rank $0$
Dimension $287$
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] = [6047,2,Mod(1,6047)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6047, 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("6047.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6047 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6047.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: \(48.2855381023\)
Analytic rank: \(0\)
Dimension: \(287\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 6047.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.58681 q^{2} +1.99218 q^{3} +4.69158 q^{4} +0.151026 q^{5} -5.15340 q^{6} -4.97386 q^{7} -6.96259 q^{8} +0.968794 q^{9} +O(q^{10})\) \(q-2.58681 q^{2} +1.99218 q^{3} +4.69158 q^{4} +0.151026 q^{5} -5.15340 q^{6} -4.97386 q^{7} -6.96259 q^{8} +0.968794 q^{9} -0.390675 q^{10} -0.968270 q^{11} +9.34648 q^{12} -6.17147 q^{13} +12.8664 q^{14} +0.300871 q^{15} +8.62773 q^{16} +3.00615 q^{17} -2.50608 q^{18} -5.05300 q^{19} +0.708549 q^{20} -9.90885 q^{21} +2.50473 q^{22} -6.46344 q^{23} -13.8708 q^{24} -4.97719 q^{25} +15.9644 q^{26} -4.04654 q^{27} -23.3353 q^{28} -5.49154 q^{29} -0.778296 q^{30} -7.05539 q^{31} -8.39310 q^{32} -1.92897 q^{33} -7.77633 q^{34} -0.751182 q^{35} +4.54517 q^{36} +8.19484 q^{37} +13.0711 q^{38} -12.2947 q^{39} -1.05153 q^{40} +0.0800108 q^{41} +25.6323 q^{42} -6.33730 q^{43} -4.54271 q^{44} +0.146313 q^{45} +16.7197 q^{46} +12.4672 q^{47} +17.1880 q^{48} +17.7393 q^{49} +12.8750 q^{50} +5.98880 q^{51} -28.9539 q^{52} -7.51380 q^{53} +10.4676 q^{54} -0.146234 q^{55} +34.6310 q^{56} -10.0665 q^{57} +14.2056 q^{58} -3.40256 q^{59} +1.41156 q^{60} +0.430487 q^{61} +18.2509 q^{62} -4.81865 q^{63} +4.45588 q^{64} -0.932052 q^{65} +4.98988 q^{66} +14.0613 q^{67} +14.1036 q^{68} -12.8764 q^{69} +1.94316 q^{70} +6.43164 q^{71} -6.74531 q^{72} -3.94156 q^{73} -21.1985 q^{74} -9.91548 q^{75} -23.7065 q^{76} +4.81604 q^{77} +31.8040 q^{78} -2.73184 q^{79} +1.30301 q^{80} -10.9678 q^{81} -0.206972 q^{82} +15.9217 q^{83} -46.4881 q^{84} +0.454006 q^{85} +16.3934 q^{86} -10.9402 q^{87} +6.74167 q^{88} -12.0433 q^{89} -0.378483 q^{90} +30.6961 q^{91} -30.3237 q^{92} -14.0556 q^{93} -32.2502 q^{94} -0.763134 q^{95} -16.7206 q^{96} -2.16350 q^{97} -45.8882 q^{98} -0.938054 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 287 q + 21 q^{2} + 29 q^{3} + 319 q^{4} + 19 q^{5} + 15 q^{6} + 52 q^{7} + 60 q^{8} + 352 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 287 q + 21 q^{2} + 29 q^{3} + 319 q^{4} + 19 q^{5} + 15 q^{6} + 52 q^{7} + 60 q^{8} + 352 q^{9} + 38 q^{10} + 32 q^{11} + 80 q^{12} + 86 q^{13} + 14 q^{14} + 41 q^{15} + 375 q^{16} + 59 q^{17} + 93 q^{18} + 39 q^{19} + 27 q^{20} + 51 q^{21} + 99 q^{22} + 68 q^{23} + 31 q^{24} + 492 q^{25} + 19 q^{26} + 107 q^{27} + 142 q^{28} + 39 q^{29} + 12 q^{30} + 104 q^{31} + 131 q^{32} + 139 q^{33} + 71 q^{34} - 5 q^{35} + 410 q^{36} + 298 q^{37} + 19 q^{38} + 37 q^{39} + 98 q^{40} + 90 q^{41} + 32 q^{42} + 105 q^{43} + 85 q^{44} + 73 q^{45} + 97 q^{46} + 66 q^{47} + 161 q^{48} + 473 q^{49} + 85 q^{50} + 34 q^{51} + 179 q^{52} + 95 q^{53} + 28 q^{54} + 62 q^{55} + 16 q^{56} + 247 q^{57} + 247 q^{58} + 32 q^{59} + 51 q^{60} + 106 q^{61} + 22 q^{62} + 104 q^{63} + 480 q^{64} + 150 q^{65} - 27 q^{66} + 232 q^{67} + 88 q^{68} + 57 q^{69} + 123 q^{70} + 46 q^{71} + 240 q^{72} + 372 q^{73} + 13 q^{74} + 81 q^{75} + 82 q^{76} + 65 q^{77} + 154 q^{78} + 143 q^{79} + 17 q^{80} + 519 q^{81} + 98 q^{82} + 49 q^{83} + 79 q^{84} + 236 q^{85} + 61 q^{86} + 31 q^{87} + 254 q^{88} + 114 q^{89} + 36 q^{90} + 96 q^{91} + 151 q^{92} + 189 q^{93} + 8 q^{94} + 30 q^{95} + 23 q^{96} + 503 q^{97} + 91 q^{98} + 130 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.58681 −1.82915 −0.914575 0.404417i \(-0.867474\pi\)
−0.914575 + 0.404417i \(0.867474\pi\)
\(3\) 1.99218 1.15019 0.575094 0.818088i \(-0.304966\pi\)
0.575094 + 0.818088i \(0.304966\pi\)
\(4\) 4.69158 2.34579
\(5\) 0.151026 0.0675408 0.0337704 0.999430i \(-0.489249\pi\)
0.0337704 + 0.999430i \(0.489249\pi\)
\(6\) −5.15340 −2.10386
\(7\) −4.97386 −1.87994 −0.939972 0.341252i \(-0.889149\pi\)
−0.939972 + 0.341252i \(0.889149\pi\)
\(8\) −6.96259 −2.46165
\(9\) 0.968794 0.322931
\(10\) −0.390675 −0.123542
\(11\) −0.968270 −0.291944 −0.145972 0.989289i \(-0.546631\pi\)
−0.145972 + 0.989289i \(0.546631\pi\)
\(12\) 9.34648 2.69810
\(13\) −6.17147 −1.71166 −0.855829 0.517258i \(-0.826953\pi\)
−0.855829 + 0.517258i \(0.826953\pi\)
\(14\) 12.8664 3.43870
\(15\) 0.300871 0.0776846
\(16\) 8.62773 2.15693
\(17\) 3.00615 0.729098 0.364549 0.931184i \(-0.381223\pi\)
0.364549 + 0.931184i \(0.381223\pi\)
\(18\) −2.50608 −0.590689
\(19\) −5.05300 −1.15924 −0.579619 0.814888i \(-0.696799\pi\)
−0.579619 + 0.814888i \(0.696799\pi\)
\(20\) 0.708549 0.158436
\(21\) −9.90885 −2.16229
\(22\) 2.50473 0.534010
\(23\) −6.46344 −1.34772 −0.673860 0.738859i \(-0.735365\pi\)
−0.673860 + 0.738859i \(0.735365\pi\)
\(24\) −13.8708 −2.83136
\(25\) −4.97719 −0.995438
\(26\) 15.9644 3.13088
\(27\) −4.04654 −0.778756
\(28\) −23.3353 −4.40995
\(29\) −5.49154 −1.01975 −0.509877 0.860248i \(-0.670309\pi\)
−0.509877 + 0.860248i \(0.670309\pi\)
\(30\) −0.778296 −0.142097
\(31\) −7.05539 −1.26719 −0.633593 0.773667i \(-0.718421\pi\)
−0.633593 + 0.773667i \(0.718421\pi\)
\(32\) −8.39310 −1.48370
\(33\) −1.92897 −0.335791
\(34\) −7.77633 −1.33363
\(35\) −0.751182 −0.126973
\(36\) 4.54517 0.757528
\(37\) 8.19484 1.34722 0.673612 0.739085i \(-0.264742\pi\)
0.673612 + 0.739085i \(0.264742\pi\)
\(38\) 13.0711 2.12042
\(39\) −12.2947 −1.96873
\(40\) −1.05153 −0.166262
\(41\) 0.0800108 0.0124956 0.00624779 0.999980i \(-0.498011\pi\)
0.00624779 + 0.999980i \(0.498011\pi\)
\(42\) 25.6323 3.95515
\(43\) −6.33730 −0.966429 −0.483214 0.875502i \(-0.660531\pi\)
−0.483214 + 0.875502i \(0.660531\pi\)
\(44\) −4.54271 −0.684840
\(45\) 0.146313 0.0218110
\(46\) 16.7197 2.46518
\(47\) 12.4672 1.81852 0.909261 0.416226i \(-0.136647\pi\)
0.909261 + 0.416226i \(0.136647\pi\)
\(48\) 17.1880 2.48088
\(49\) 17.7393 2.53419
\(50\) 12.8750 1.82081
\(51\) 5.98880 0.838600
\(52\) −28.9539 −4.01519
\(53\) −7.51380 −1.03210 −0.516050 0.856559i \(-0.672598\pi\)
−0.516050 + 0.856559i \(0.672598\pi\)
\(54\) 10.4676 1.42446
\(55\) −0.146234 −0.0197182
\(56\) 34.6310 4.62776
\(57\) −10.0665 −1.33334
\(58\) 14.2056 1.86528
\(59\) −3.40256 −0.442975 −0.221488 0.975163i \(-0.571091\pi\)
−0.221488 + 0.975163i \(0.571091\pi\)
\(60\) 1.41156 0.182232
\(61\) 0.430487 0.0551182 0.0275591 0.999620i \(-0.491227\pi\)
0.0275591 + 0.999620i \(0.491227\pi\)
\(62\) 18.2509 2.31787
\(63\) −4.81865 −0.607092
\(64\) 4.45588 0.556985
\(65\) −0.932052 −0.115607
\(66\) 4.98988 0.614211
\(67\) 14.0613 1.71786 0.858931 0.512091i \(-0.171129\pi\)
0.858931 + 0.512091i \(0.171129\pi\)
\(68\) 14.1036 1.71031
\(69\) −12.8764 −1.55013
\(70\) 1.94316 0.232252
\(71\) 6.43164 0.763296 0.381648 0.924308i \(-0.375357\pi\)
0.381648 + 0.924308i \(0.375357\pi\)
\(72\) −6.74531 −0.794942
\(73\) −3.94156 −0.461325 −0.230662 0.973034i \(-0.574089\pi\)
−0.230662 + 0.973034i \(0.574089\pi\)
\(74\) −21.1985 −2.46427
\(75\) −9.91548 −1.14494
\(76\) −23.7065 −2.71933
\(77\) 4.81604 0.548839
\(78\) 31.8040 3.60110
\(79\) −2.73184 −0.307356 −0.153678 0.988121i \(-0.549112\pi\)
−0.153678 + 0.988121i \(0.549112\pi\)
\(80\) 1.30301 0.145681
\(81\) −10.9678 −1.21865
\(82\) −0.206972 −0.0228563
\(83\) 15.9217 1.74764 0.873819 0.486251i \(-0.161636\pi\)
0.873819 + 0.486251i \(0.161636\pi\)
\(84\) −46.4881 −5.07227
\(85\) 0.454006 0.0492439
\(86\) 16.3934 1.76774
\(87\) −10.9402 −1.17291
\(88\) 6.74167 0.718664
\(89\) −12.0433 −1.27659 −0.638293 0.769794i \(-0.720359\pi\)
−0.638293 + 0.769794i \(0.720359\pi\)
\(90\) −0.378483 −0.0398956
\(91\) 30.6961 3.21782
\(92\) −30.3237 −3.16147
\(93\) −14.0556 −1.45750
\(94\) −32.2502 −3.32635
\(95\) −0.763134 −0.0782959
\(96\) −16.7206 −1.70654
\(97\) −2.16350 −0.219671 −0.109835 0.993950i \(-0.535032\pi\)
−0.109835 + 0.993950i \(0.535032\pi\)
\(98\) −45.8882 −4.63541
\(99\) −0.938054 −0.0942779
\(100\) −23.3509 −2.33509
\(101\) 0.313832 0.0312275 0.0156137 0.999878i \(-0.495030\pi\)
0.0156137 + 0.999878i \(0.495030\pi\)
\(102\) −15.4919 −1.53392
\(103\) 7.82414 0.770936 0.385468 0.922721i \(-0.374040\pi\)
0.385468 + 0.922721i \(0.374040\pi\)
\(104\) 42.9694 4.21350
\(105\) −1.49649 −0.146043
\(106\) 19.4367 1.88786
\(107\) −1.77036 −0.171147 −0.0855736 0.996332i \(-0.527272\pi\)
−0.0855736 + 0.996332i \(0.527272\pi\)
\(108\) −18.9846 −1.82680
\(109\) 14.8736 1.42463 0.712314 0.701861i \(-0.247647\pi\)
0.712314 + 0.701861i \(0.247647\pi\)
\(110\) 0.378279 0.0360675
\(111\) 16.3256 1.54956
\(112\) −42.9131 −4.05491
\(113\) −0.180437 −0.0169741 −0.00848705 0.999964i \(-0.502702\pi\)
−0.00848705 + 0.999964i \(0.502702\pi\)
\(114\) 26.0401 2.43888
\(115\) −0.976146 −0.0910261
\(116\) −25.7640 −2.39212
\(117\) −5.97888 −0.552748
\(118\) 8.80176 0.810268
\(119\) −14.9522 −1.37066
\(120\) −2.09484 −0.191232
\(121\) −10.0625 −0.914768
\(122\) −1.11359 −0.100819
\(123\) 0.159396 0.0143723
\(124\) −33.1009 −2.97255
\(125\) −1.50681 −0.134773
\(126\) 12.4649 1.11046
\(127\) −8.25832 −0.732807 −0.366404 0.930456i \(-0.619411\pi\)
−0.366404 + 0.930456i \(0.619411\pi\)
\(128\) 5.25970 0.464896
\(129\) −12.6251 −1.11157
\(130\) 2.41104 0.211462
\(131\) 4.51551 0.394522 0.197261 0.980351i \(-0.436795\pi\)
0.197261 + 0.980351i \(0.436795\pi\)
\(132\) −9.04991 −0.787694
\(133\) 25.1329 2.17930
\(134\) −36.3739 −3.14223
\(135\) −0.611131 −0.0525978
\(136\) −20.9306 −1.79478
\(137\) 18.7466 1.60163 0.800814 0.598913i \(-0.204400\pi\)
0.800814 + 0.598913i \(0.204400\pi\)
\(138\) 33.3087 2.83542
\(139\) 12.0992 1.02624 0.513121 0.858316i \(-0.328489\pi\)
0.513121 + 0.858316i \(0.328489\pi\)
\(140\) −3.52423 −0.297851
\(141\) 24.8369 2.09164
\(142\) −16.6374 −1.39618
\(143\) 5.97565 0.499709
\(144\) 8.35849 0.696541
\(145\) −0.829364 −0.0688749
\(146\) 10.1961 0.843832
\(147\) 35.3400 2.91479
\(148\) 38.4467 3.16030
\(149\) −18.6573 −1.52846 −0.764232 0.644941i \(-0.776882\pi\)
−0.764232 + 0.644941i \(0.776882\pi\)
\(150\) 25.6494 2.09427
\(151\) −18.9369 −1.54107 −0.770534 0.637399i \(-0.780010\pi\)
−0.770534 + 0.637399i \(0.780010\pi\)
\(152\) 35.1820 2.85363
\(153\) 2.91234 0.235449
\(154\) −12.4582 −1.00391
\(155\) −1.06555 −0.0855867
\(156\) −57.6815 −4.61822
\(157\) 23.4623 1.87250 0.936248 0.351341i \(-0.114274\pi\)
0.936248 + 0.351341i \(0.114274\pi\)
\(158\) 7.06674 0.562200
\(159\) −14.9689 −1.18711
\(160\) −1.26757 −0.100211
\(161\) 32.1483 2.53364
\(162\) 28.3716 2.22909
\(163\) −11.8742 −0.930060 −0.465030 0.885295i \(-0.653956\pi\)
−0.465030 + 0.885295i \(0.653956\pi\)
\(164\) 0.375376 0.0293120
\(165\) −0.291324 −0.0226796
\(166\) −41.1865 −3.19669
\(167\) 3.72636 0.288355 0.144177 0.989552i \(-0.453946\pi\)
0.144177 + 0.989552i \(0.453946\pi\)
\(168\) 68.9912 5.32279
\(169\) 25.0871 1.92978
\(170\) −1.17443 −0.0900744
\(171\) −4.89532 −0.374354
\(172\) −29.7319 −2.26704
\(173\) −2.59867 −0.197573 −0.0987867 0.995109i \(-0.531496\pi\)
−0.0987867 + 0.995109i \(0.531496\pi\)
\(174\) 28.3001 2.14542
\(175\) 24.7559 1.87137
\(176\) −8.35397 −0.629704
\(177\) −6.77852 −0.509505
\(178\) 31.1537 2.33507
\(179\) −9.23214 −0.690042 −0.345021 0.938595i \(-0.612128\pi\)
−0.345021 + 0.938595i \(0.612128\pi\)
\(180\) 0.686438 0.0511640
\(181\) 5.00108 0.371727 0.185864 0.982576i \(-0.440492\pi\)
0.185864 + 0.982576i \(0.440492\pi\)
\(182\) −79.4048 −5.88588
\(183\) 0.857609 0.0633963
\(184\) 45.0023 3.31761
\(185\) 1.23763 0.0909926
\(186\) 36.3592 2.66599
\(187\) −2.91076 −0.212856
\(188\) 58.4906 4.26587
\(189\) 20.1269 1.46402
\(190\) 1.97408 0.143215
\(191\) −19.0187 −1.37615 −0.688073 0.725642i \(-0.741543\pi\)
−0.688073 + 0.725642i \(0.741543\pi\)
\(192\) 8.87693 0.640637
\(193\) 23.6376 1.70147 0.850737 0.525592i \(-0.176156\pi\)
0.850737 + 0.525592i \(0.176156\pi\)
\(194\) 5.59657 0.401810
\(195\) −1.85682 −0.132969
\(196\) 83.2253 5.94467
\(197\) −5.72476 −0.407873 −0.203936 0.978984i \(-0.565374\pi\)
−0.203936 + 0.978984i \(0.565374\pi\)
\(198\) 2.42656 0.172448
\(199\) −4.99872 −0.354350 −0.177175 0.984179i \(-0.556696\pi\)
−0.177175 + 0.984179i \(0.556696\pi\)
\(200\) 34.6541 2.45042
\(201\) 28.0127 1.97586
\(202\) −0.811824 −0.0571198
\(203\) 27.3142 1.91708
\(204\) 28.0969 1.96718
\(205\) 0.0120837 0.000843961 0
\(206\) −20.2395 −1.41016
\(207\) −6.26174 −0.435221
\(208\) −53.2458 −3.69193
\(209\) 4.89267 0.338433
\(210\) 3.87114 0.267134
\(211\) −24.0350 −1.65463 −0.827317 0.561735i \(-0.810134\pi\)
−0.827317 + 0.561735i \(0.810134\pi\)
\(212\) −35.2515 −2.42109
\(213\) 12.8130 0.877933
\(214\) 4.57958 0.313054
\(215\) −0.957095 −0.0652733
\(216\) 28.1744 1.91702
\(217\) 35.0926 2.38224
\(218\) −38.4750 −2.60586
\(219\) −7.85231 −0.530610
\(220\) −0.686067 −0.0462546
\(221\) −18.5524 −1.24797
\(222\) −42.2313 −2.83438
\(223\) 8.04524 0.538749 0.269375 0.963035i \(-0.413183\pi\)
0.269375 + 0.963035i \(0.413183\pi\)
\(224\) 41.7461 2.78928
\(225\) −4.82187 −0.321458
\(226\) 0.466756 0.0310482
\(227\) −6.79371 −0.450915 −0.225457 0.974253i \(-0.572388\pi\)
−0.225457 + 0.974253i \(0.572388\pi\)
\(228\) −47.2278 −3.12774
\(229\) 8.33769 0.550970 0.275485 0.961305i \(-0.411162\pi\)
0.275485 + 0.961305i \(0.411162\pi\)
\(230\) 2.52510 0.166500
\(231\) 9.59444 0.631268
\(232\) 38.2353 2.51027
\(233\) −11.0817 −0.725988 −0.362994 0.931791i \(-0.618245\pi\)
−0.362994 + 0.931791i \(0.618245\pi\)
\(234\) 15.4662 1.01106
\(235\) 1.88286 0.122824
\(236\) −15.9634 −1.03913
\(237\) −5.44232 −0.353517
\(238\) 38.6784 2.50715
\(239\) 19.3530 1.25184 0.625919 0.779888i \(-0.284724\pi\)
0.625919 + 0.779888i \(0.284724\pi\)
\(240\) 2.59583 0.167560
\(241\) 0.868809 0.0559649 0.0279824 0.999608i \(-0.491092\pi\)
0.0279824 + 0.999608i \(0.491092\pi\)
\(242\) 26.0296 1.67325
\(243\) −9.71030 −0.622916
\(244\) 2.01966 0.129296
\(245\) 2.67909 0.171161
\(246\) −0.412327 −0.0262890
\(247\) 31.1845 1.98422
\(248\) 49.1238 3.11936
\(249\) 31.7190 2.01011
\(250\) 3.89784 0.246521
\(251\) 29.4712 1.86021 0.930103 0.367299i \(-0.119717\pi\)
0.930103 + 0.367299i \(0.119717\pi\)
\(252\) −22.6070 −1.42411
\(253\) 6.25836 0.393459
\(254\) 21.3627 1.34041
\(255\) 0.904463 0.0566397
\(256\) −22.5176 −1.40735
\(257\) −19.3230 −1.20534 −0.602668 0.797992i \(-0.705896\pi\)
−0.602668 + 0.797992i \(0.705896\pi\)
\(258\) 32.6586 2.03323
\(259\) −40.7600 −2.53271
\(260\) −4.37279 −0.271189
\(261\) −5.32017 −0.329310
\(262\) −11.6807 −0.721639
\(263\) −18.6227 −1.14833 −0.574163 0.818741i \(-0.694672\pi\)
−0.574163 + 0.818741i \(0.694672\pi\)
\(264\) 13.4306 0.826598
\(265\) −1.13478 −0.0697088
\(266\) −65.0141 −3.98627
\(267\) −23.9924 −1.46831
\(268\) 65.9697 4.02974
\(269\) 4.29321 0.261762 0.130881 0.991398i \(-0.458219\pi\)
0.130881 + 0.991398i \(0.458219\pi\)
\(270\) 1.58088 0.0962092
\(271\) −12.8878 −0.782875 −0.391438 0.920205i \(-0.628022\pi\)
−0.391438 + 0.920205i \(0.628022\pi\)
\(272\) 25.9362 1.57262
\(273\) 61.1522 3.70110
\(274\) −48.4938 −2.92962
\(275\) 4.81926 0.290613
\(276\) −60.4104 −3.63628
\(277\) −10.4687 −0.629004 −0.314502 0.949257i \(-0.601838\pi\)
−0.314502 + 0.949257i \(0.601838\pi\)
\(278\) −31.2984 −1.87715
\(279\) −6.83522 −0.409214
\(280\) 5.23017 0.312562
\(281\) −11.2590 −0.671656 −0.335828 0.941923i \(-0.609016\pi\)
−0.335828 + 0.941923i \(0.609016\pi\)
\(282\) −64.2482 −3.82593
\(283\) 12.8406 0.763293 0.381646 0.924308i \(-0.375357\pi\)
0.381646 + 0.924308i \(0.375357\pi\)
\(284\) 30.1745 1.79053
\(285\) −1.52030 −0.0900549
\(286\) −15.4579 −0.914043
\(287\) −0.397963 −0.0234910
\(288\) −8.13118 −0.479134
\(289\) −7.96307 −0.468416
\(290\) 2.14541 0.125983
\(291\) −4.31010 −0.252662
\(292\) −18.4921 −1.08217
\(293\) −5.88607 −0.343868 −0.171934 0.985108i \(-0.555002\pi\)
−0.171934 + 0.985108i \(0.555002\pi\)
\(294\) −91.4177 −5.33159
\(295\) −0.513874 −0.0299189
\(296\) −57.0573 −3.31639
\(297\) 3.91814 0.227353
\(298\) 48.2628 2.79579
\(299\) 39.8889 2.30684
\(300\) −46.5192 −2.68579
\(301\) 31.5208 1.81683
\(302\) 48.9863 2.81884
\(303\) 0.625212 0.0359175
\(304\) −43.5959 −2.50040
\(305\) 0.0650146 0.00372273
\(306\) −7.53366 −0.430671
\(307\) −8.74852 −0.499305 −0.249652 0.968336i \(-0.580316\pi\)
−0.249652 + 0.968336i \(0.580316\pi\)
\(308\) 22.5948 1.28746
\(309\) 15.5871 0.886720
\(310\) 2.75636 0.156551
\(311\) −19.0704 −1.08139 −0.540693 0.841220i \(-0.681838\pi\)
−0.540693 + 0.841220i \(0.681838\pi\)
\(312\) 85.6030 4.84631
\(313\) −6.17966 −0.349295 −0.174648 0.984631i \(-0.555879\pi\)
−0.174648 + 0.984631i \(0.555879\pi\)
\(314\) −60.6925 −3.42507
\(315\) −0.727740 −0.0410035
\(316\) −12.8166 −0.720991
\(317\) 5.33981 0.299913 0.149957 0.988693i \(-0.452087\pi\)
0.149957 + 0.988693i \(0.452087\pi\)
\(318\) 38.7216 2.17140
\(319\) 5.31729 0.297711
\(320\) 0.672953 0.0376192
\(321\) −3.52688 −0.196851
\(322\) −83.1614 −4.63440
\(323\) −15.1901 −0.845198
\(324\) −51.4564 −2.85869
\(325\) 30.7166 1.70385
\(326\) 30.7163 1.70122
\(327\) 29.6308 1.63859
\(328\) −0.557082 −0.0307597
\(329\) −62.0100 −3.41872
\(330\) 0.753600 0.0414843
\(331\) 18.2390 1.00250 0.501252 0.865301i \(-0.332873\pi\)
0.501252 + 0.865301i \(0.332873\pi\)
\(332\) 74.6981 4.09959
\(333\) 7.93911 0.435061
\(334\) −9.63938 −0.527444
\(335\) 2.12362 0.116026
\(336\) −85.4908 −4.66391
\(337\) 29.5450 1.60942 0.804710 0.593668i \(-0.202321\pi\)
0.804710 + 0.593668i \(0.202321\pi\)
\(338\) −64.8955 −3.52985
\(339\) −0.359464 −0.0195234
\(340\) 2.13000 0.115516
\(341\) 6.83152 0.369948
\(342\) 12.6632 0.684750
\(343\) −53.4159 −2.88419
\(344\) 44.1240 2.37901
\(345\) −1.94466 −0.104697
\(346\) 6.72226 0.361391
\(347\) −21.9194 −1.17669 −0.588347 0.808608i \(-0.700221\pi\)
−0.588347 + 0.808608i \(0.700221\pi\)
\(348\) −51.3265 −2.75139
\(349\) 5.27001 0.282097 0.141048 0.990003i \(-0.454953\pi\)
0.141048 + 0.990003i \(0.454953\pi\)
\(350\) −64.0387 −3.42301
\(351\) 24.9731 1.33296
\(352\) 8.12679 0.433159
\(353\) −33.0100 −1.75694 −0.878472 0.477795i \(-0.841436\pi\)
−0.878472 + 0.477795i \(0.841436\pi\)
\(354\) 17.5347 0.931960
\(355\) 0.971344 0.0515536
\(356\) −56.5020 −2.99460
\(357\) −29.7875 −1.57652
\(358\) 23.8818 1.26219
\(359\) −19.7262 −1.04111 −0.520554 0.853829i \(-0.674274\pi\)
−0.520554 + 0.853829i \(0.674274\pi\)
\(360\) −1.01872 −0.0536910
\(361\) 6.53282 0.343833
\(362\) −12.9368 −0.679945
\(363\) −20.0462 −1.05216
\(364\) 144.013 7.54833
\(365\) −0.595277 −0.0311582
\(366\) −2.21847 −0.115961
\(367\) −19.1526 −0.999756 −0.499878 0.866096i \(-0.666622\pi\)
−0.499878 + 0.866096i \(0.666622\pi\)
\(368\) −55.7648 −2.90694
\(369\) 0.0775139 0.00403521
\(370\) −3.20152 −0.166439
\(371\) 37.3726 1.94029
\(372\) −65.9431 −3.41899
\(373\) 15.8379 0.820054 0.410027 0.912073i \(-0.365519\pi\)
0.410027 + 0.912073i \(0.365519\pi\)
\(374\) 7.52959 0.389346
\(375\) −3.00185 −0.155015
\(376\) −86.8037 −4.47656
\(377\) 33.8909 1.74547
\(378\) −52.0645 −2.67791
\(379\) 9.25970 0.475639 0.237819 0.971309i \(-0.423567\pi\)
0.237819 + 0.971309i \(0.423567\pi\)
\(380\) −3.58030 −0.183665
\(381\) −16.4521 −0.842866
\(382\) 49.1977 2.51718
\(383\) −29.6271 −1.51387 −0.756936 0.653489i \(-0.773305\pi\)
−0.756936 + 0.653489i \(0.773305\pi\)
\(384\) 10.4783 0.534717
\(385\) 0.727347 0.0370690
\(386\) −61.1460 −3.11225
\(387\) −6.13953 −0.312090
\(388\) −10.1502 −0.515301
\(389\) −8.19494 −0.415500 −0.207750 0.978182i \(-0.566614\pi\)
−0.207750 + 0.978182i \(0.566614\pi\)
\(390\) 4.80323 0.243221
\(391\) −19.4301 −0.982621
\(392\) −123.512 −6.23828
\(393\) 8.99572 0.453774
\(394\) 14.8089 0.746060
\(395\) −0.412578 −0.0207591
\(396\) −4.40095 −0.221156
\(397\) 6.85428 0.344006 0.172003 0.985096i \(-0.444976\pi\)
0.172003 + 0.985096i \(0.444976\pi\)
\(398\) 12.9307 0.648159
\(399\) 50.0694 2.50661
\(400\) −42.9419 −2.14709
\(401\) 7.13203 0.356157 0.178078 0.984016i \(-0.443012\pi\)
0.178078 + 0.984016i \(0.443012\pi\)
\(402\) −72.4635 −3.61415
\(403\) 43.5422 2.16899
\(404\) 1.47237 0.0732531
\(405\) −1.65642 −0.0823084
\(406\) −70.6565 −3.50662
\(407\) −7.93482 −0.393314
\(408\) −41.6975 −2.06434
\(409\) 32.8102 1.62236 0.811179 0.584798i \(-0.198826\pi\)
0.811179 + 0.584798i \(0.198826\pi\)
\(410\) −0.0312582 −0.00154373
\(411\) 37.3466 1.84217
\(412\) 36.7075 1.80845
\(413\) 16.9239 0.832769
\(414\) 16.1979 0.796084
\(415\) 2.40459 0.118037
\(416\) 51.7978 2.53960
\(417\) 24.1039 1.18037
\(418\) −12.6564 −0.619045
\(419\) 17.6200 0.860795 0.430397 0.902640i \(-0.358373\pi\)
0.430397 + 0.902640i \(0.358373\pi\)
\(420\) −7.02090 −0.342585
\(421\) −37.8157 −1.84302 −0.921512 0.388350i \(-0.873045\pi\)
−0.921512 + 0.388350i \(0.873045\pi\)
\(422\) 62.1738 3.02657
\(423\) 12.0781 0.587258
\(424\) 52.3155 2.54066
\(425\) −14.9622 −0.725772
\(426\) −33.1448 −1.60587
\(427\) −2.14118 −0.103619
\(428\) −8.30578 −0.401475
\(429\) 11.9046 0.574759
\(430\) 2.47582 0.119395
\(431\) −17.0613 −0.821812 −0.410906 0.911678i \(-0.634788\pi\)
−0.410906 + 0.911678i \(0.634788\pi\)
\(432\) −34.9124 −1.67972
\(433\) 35.9074 1.72560 0.862801 0.505544i \(-0.168708\pi\)
0.862801 + 0.505544i \(0.168708\pi\)
\(434\) −90.7777 −4.35747
\(435\) −1.65225 −0.0792191
\(436\) 69.7804 3.34187
\(437\) 32.6598 1.56233
\(438\) 20.3124 0.970565
\(439\) −17.3935 −0.830147 −0.415074 0.909788i \(-0.636244\pi\)
−0.415074 + 0.909788i \(0.636244\pi\)
\(440\) 1.01817 0.0485391
\(441\) 17.1857 0.818368
\(442\) 47.9914 2.28272
\(443\) 16.5288 0.785307 0.392654 0.919686i \(-0.371557\pi\)
0.392654 + 0.919686i \(0.371557\pi\)
\(444\) 76.5929 3.63494
\(445\) −1.81885 −0.0862216
\(446\) −20.8115 −0.985453
\(447\) −37.1687 −1.75802
\(448\) −22.1629 −1.04710
\(449\) −7.12487 −0.336243 −0.168122 0.985766i \(-0.553770\pi\)
−0.168122 + 0.985766i \(0.553770\pi\)
\(450\) 12.4733 0.587995
\(451\) −0.0774720 −0.00364801
\(452\) −0.846534 −0.0398176
\(453\) −37.7259 −1.77252
\(454\) 17.5740 0.824790
\(455\) 4.63590 0.217334
\(456\) 70.0889 3.28221
\(457\) −1.36874 −0.0640268 −0.0320134 0.999487i \(-0.510192\pi\)
−0.0320134 + 0.999487i \(0.510192\pi\)
\(458\) −21.5680 −1.00781
\(459\) −12.1645 −0.567790
\(460\) −4.57966 −0.213528
\(461\) 17.6511 0.822094 0.411047 0.911614i \(-0.365163\pi\)
0.411047 + 0.911614i \(0.365163\pi\)
\(462\) −24.8190 −1.15468
\(463\) 13.7068 0.637009 0.318505 0.947921i \(-0.396819\pi\)
0.318505 + 0.947921i \(0.396819\pi\)
\(464\) −47.3795 −2.19954
\(465\) −2.12276 −0.0984408
\(466\) 28.6663 1.32794
\(467\) 10.3100 0.477088 0.238544 0.971132i \(-0.423330\pi\)
0.238544 + 0.971132i \(0.423330\pi\)
\(468\) −28.0504 −1.29663
\(469\) −69.9391 −3.22949
\(470\) −4.87061 −0.224664
\(471\) 46.7412 2.15372
\(472\) 23.6906 1.09045
\(473\) 6.13621 0.282143
\(474\) 14.0782 0.646635
\(475\) 25.1498 1.15395
\(476\) −70.1492 −3.21529
\(477\) −7.27932 −0.333297
\(478\) −50.0624 −2.28980
\(479\) −37.8772 −1.73066 −0.865328 0.501206i \(-0.832890\pi\)
−0.865328 + 0.501206i \(0.832890\pi\)
\(480\) −2.52524 −0.115261
\(481\) −50.5743 −2.30599
\(482\) −2.24744 −0.102368
\(483\) 64.0452 2.91416
\(484\) −47.2088 −2.14585
\(485\) −0.326745 −0.0148367
\(486\) 25.1187 1.13941
\(487\) −12.1461 −0.550391 −0.275196 0.961388i \(-0.588743\pi\)
−0.275196 + 0.961388i \(0.588743\pi\)
\(488\) −2.99730 −0.135682
\(489\) −23.6556 −1.06974
\(490\) −6.93030 −0.313079
\(491\) −26.3989 −1.19136 −0.595682 0.803221i \(-0.703118\pi\)
−0.595682 + 0.803221i \(0.703118\pi\)
\(492\) 0.747819 0.0337143
\(493\) −16.5084 −0.743500
\(494\) −80.6682 −3.62943
\(495\) −0.141670 −0.00636761
\(496\) −60.8720 −2.73323
\(497\) −31.9901 −1.43495
\(498\) −82.0511 −3.67680
\(499\) −16.6196 −0.743997 −0.371999 0.928233i \(-0.621327\pi\)
−0.371999 + 0.928233i \(0.621327\pi\)
\(500\) −7.06933 −0.316150
\(501\) 7.42360 0.331662
\(502\) −76.2364 −3.40260
\(503\) −3.29795 −0.147048 −0.0735241 0.997293i \(-0.523425\pi\)
−0.0735241 + 0.997293i \(0.523425\pi\)
\(504\) 33.5503 1.49445
\(505\) 0.0473968 0.00210913
\(506\) −16.1892 −0.719696
\(507\) 49.9781 2.21960
\(508\) −38.7445 −1.71901
\(509\) −38.2183 −1.69400 −0.846999 0.531595i \(-0.821593\pi\)
−0.846999 + 0.531595i \(0.821593\pi\)
\(510\) −2.33967 −0.103602
\(511\) 19.6048 0.867264
\(512\) 47.7293 2.10936
\(513\) 20.4471 0.902764
\(514\) 49.9849 2.20474
\(515\) 1.18165 0.0520696
\(516\) −59.2314 −2.60752
\(517\) −12.0716 −0.530907
\(518\) 105.438 4.63270
\(519\) −5.17703 −0.227246
\(520\) 6.48949 0.284583
\(521\) 22.0306 0.965180 0.482590 0.875846i \(-0.339696\pi\)
0.482590 + 0.875846i \(0.339696\pi\)
\(522\) 13.7623 0.602357
\(523\) 38.2309 1.67172 0.835861 0.548942i \(-0.184969\pi\)
0.835861 + 0.548942i \(0.184969\pi\)
\(524\) 21.1848 0.925464
\(525\) 49.3182 2.15242
\(526\) 48.1734 2.10046
\(527\) −21.2096 −0.923903
\(528\) −16.6426 −0.724278
\(529\) 18.7761 0.816351
\(530\) 2.93545 0.127508
\(531\) −3.29638 −0.143051
\(532\) 117.913 5.11218
\(533\) −0.493784 −0.0213882
\(534\) 62.0638 2.68576
\(535\) −0.267370 −0.0115594
\(536\) −97.9031 −4.22877
\(537\) −18.3921 −0.793678
\(538\) −11.1057 −0.478801
\(539\) −17.1764 −0.739842
\(540\) −2.86717 −0.123383
\(541\) −35.2959 −1.51749 −0.758744 0.651389i \(-0.774186\pi\)
−0.758744 + 0.651389i \(0.774186\pi\)
\(542\) 33.3382 1.43200
\(543\) 9.96306 0.427556
\(544\) −25.2309 −1.08177
\(545\) 2.24629 0.0962205
\(546\) −158.189 −6.76986
\(547\) −32.7415 −1.39992 −0.699962 0.714180i \(-0.746800\pi\)
−0.699962 + 0.714180i \(0.746800\pi\)
\(548\) 87.9510 3.75708
\(549\) 0.417053 0.0177994
\(550\) −12.4665 −0.531574
\(551\) 27.7488 1.18214
\(552\) 89.6528 3.81588
\(553\) 13.5878 0.577812
\(554\) 27.0806 1.15054
\(555\) 2.46559 0.104659
\(556\) 56.7644 2.40735
\(557\) 27.7016 1.17375 0.586876 0.809677i \(-0.300357\pi\)
0.586876 + 0.809677i \(0.300357\pi\)
\(558\) 17.6814 0.748513
\(559\) 39.1105 1.65420
\(560\) −6.48099 −0.273872
\(561\) −5.79877 −0.244824
\(562\) 29.1249 1.22856
\(563\) 13.9198 0.586649 0.293325 0.956013i \(-0.405238\pi\)
0.293325 + 0.956013i \(0.405238\pi\)
\(564\) 116.524 4.90655
\(565\) −0.0272507 −0.00114644
\(566\) −33.2161 −1.39618
\(567\) 54.5524 2.29099
\(568\) −44.7809 −1.87896
\(569\) −36.3945 −1.52574 −0.762869 0.646553i \(-0.776210\pi\)
−0.762869 + 0.646553i \(0.776210\pi\)
\(570\) 3.93273 0.164724
\(571\) −45.4291 −1.90115 −0.950574 0.310498i \(-0.899504\pi\)
−0.950574 + 0.310498i \(0.899504\pi\)
\(572\) 28.0352 1.17221
\(573\) −37.8887 −1.58282
\(574\) 1.02945 0.0429685
\(575\) 32.1698 1.34157
\(576\) 4.31683 0.179868
\(577\) −31.5444 −1.31321 −0.656605 0.754235i \(-0.728008\pi\)
−0.656605 + 0.754235i \(0.728008\pi\)
\(578\) 20.5989 0.856803
\(579\) 47.0905 1.95701
\(580\) −3.89102 −0.161566
\(581\) −79.1926 −3.28546
\(582\) 11.1494 0.462157
\(583\) 7.27538 0.301316
\(584\) 27.4435 1.13562
\(585\) −0.902966 −0.0373330
\(586\) 15.2261 0.628986
\(587\) −5.50999 −0.227421 −0.113711 0.993514i \(-0.536274\pi\)
−0.113711 + 0.993514i \(0.536274\pi\)
\(588\) 165.800 6.83748
\(589\) 35.6509 1.46897
\(590\) 1.32929 0.0547261
\(591\) −11.4048 −0.469130
\(592\) 70.7029 2.90587
\(593\) 14.6019 0.599628 0.299814 0.953998i \(-0.403075\pi\)
0.299814 + 0.953998i \(0.403075\pi\)
\(594\) −10.1355 −0.415863
\(595\) −2.25816 −0.0925757
\(596\) −87.5321 −3.58545
\(597\) −9.95836 −0.407569
\(598\) −103.185 −4.21955
\(599\) 20.5528 0.839765 0.419882 0.907579i \(-0.362071\pi\)
0.419882 + 0.907579i \(0.362071\pi\)
\(600\) 69.0374 2.81844
\(601\) −3.79199 −0.154678 −0.0773392 0.997005i \(-0.524642\pi\)
−0.0773392 + 0.997005i \(0.524642\pi\)
\(602\) −81.5384 −3.32326
\(603\) 13.6225 0.554751
\(604\) −88.8441 −3.61502
\(605\) −1.51969 −0.0617842
\(606\) −1.61730 −0.0656984
\(607\) −33.8412 −1.37357 −0.686786 0.726860i \(-0.740979\pi\)
−0.686786 + 0.726860i \(0.740979\pi\)
\(608\) 42.4103 1.71997
\(609\) 54.4148 2.20500
\(610\) −0.168180 −0.00680943
\(611\) −76.9408 −3.11269
\(612\) 13.6635 0.552312
\(613\) 7.81334 0.315578 0.157789 0.987473i \(-0.449563\pi\)
0.157789 + 0.987473i \(0.449563\pi\)
\(614\) 22.6308 0.913303
\(615\) 0.0240729 0.000970714 0
\(616\) −33.5321 −1.35105
\(617\) 10.3077 0.414973 0.207487 0.978238i \(-0.433472\pi\)
0.207487 + 0.978238i \(0.433472\pi\)
\(618\) −40.3209 −1.62194
\(619\) 25.6000 1.02895 0.514475 0.857505i \(-0.327987\pi\)
0.514475 + 0.857505i \(0.327987\pi\)
\(620\) −4.99909 −0.200768
\(621\) 26.1545 1.04955
\(622\) 49.3316 1.97802
\(623\) 59.9016 2.39991
\(624\) −106.075 −4.24641
\(625\) 24.6584 0.986336
\(626\) 15.9856 0.638913
\(627\) 9.74709 0.389261
\(628\) 110.075 4.39248
\(629\) 24.6349 0.982259
\(630\) 1.88252 0.0750015
\(631\) 15.6003 0.621038 0.310519 0.950567i \(-0.399497\pi\)
0.310519 + 0.950567i \(0.399497\pi\)
\(632\) 19.0207 0.756601
\(633\) −47.8820 −1.90314
\(634\) −13.8131 −0.548587
\(635\) −1.24722 −0.0494944
\(636\) −70.2275 −2.78470
\(637\) −109.478 −4.33767
\(638\) −13.7548 −0.544558
\(639\) 6.23094 0.246492
\(640\) 0.794350 0.0313994
\(641\) 39.1275 1.54544 0.772722 0.634745i \(-0.218895\pi\)
0.772722 + 0.634745i \(0.218895\pi\)
\(642\) 9.12337 0.360071
\(643\) 43.3090 1.70794 0.853971 0.520321i \(-0.174188\pi\)
0.853971 + 0.520321i \(0.174188\pi\)
\(644\) 150.826 5.94338
\(645\) −1.90671 −0.0750766
\(646\) 39.2938 1.54599
\(647\) −34.3027 −1.34858 −0.674289 0.738468i \(-0.735550\pi\)
−0.674289 + 0.738468i \(0.735550\pi\)
\(648\) 76.3644 2.99988
\(649\) 3.29459 0.129324
\(650\) −79.4580 −3.11660
\(651\) 69.9108 2.74002
\(652\) −55.7087 −2.18172
\(653\) −14.7413 −0.576871 −0.288436 0.957499i \(-0.593135\pi\)
−0.288436 + 0.957499i \(0.593135\pi\)
\(654\) −76.6493 −2.99722
\(655\) 0.681958 0.0266463
\(656\) 0.690311 0.0269521
\(657\) −3.81856 −0.148976
\(658\) 160.408 6.25335
\(659\) −8.39053 −0.326849 −0.163424 0.986556i \(-0.552254\pi\)
−0.163424 + 0.986556i \(0.552254\pi\)
\(660\) −1.36677 −0.0532015
\(661\) 12.7159 0.494591 0.247296 0.968940i \(-0.420458\pi\)
0.247296 + 0.968940i \(0.420458\pi\)
\(662\) −47.1807 −1.83373
\(663\) −36.9597 −1.43540
\(664\) −110.857 −4.30207
\(665\) 3.79572 0.147192
\(666\) −20.5370 −0.795791
\(667\) 35.4942 1.37434
\(668\) 17.4825 0.676419
\(669\) 16.0276 0.619663
\(670\) −5.49340 −0.212229
\(671\) −0.416828 −0.0160915
\(672\) 83.1659 3.20820
\(673\) 12.7901 0.493020 0.246510 0.969140i \(-0.420716\pi\)
0.246510 + 0.969140i \(0.420716\pi\)
\(674\) −76.4273 −2.94387
\(675\) 20.1404 0.775204
\(676\) 117.698 4.52684
\(677\) 20.0158 0.769270 0.384635 0.923069i \(-0.374327\pi\)
0.384635 + 0.923069i \(0.374327\pi\)
\(678\) 0.929864 0.0357112
\(679\) 10.7610 0.412968
\(680\) −3.16106 −0.121221
\(681\) −13.5343 −0.518636
\(682\) −17.6718 −0.676690
\(683\) −32.0046 −1.22462 −0.612312 0.790616i \(-0.709760\pi\)
−0.612312 + 0.790616i \(0.709760\pi\)
\(684\) −22.9667 −0.878155
\(685\) 2.83122 0.108175
\(686\) 138.177 5.27561
\(687\) 16.6102 0.633719
\(688\) −54.6765 −2.08452
\(689\) 46.3712 1.76660
\(690\) 5.03047 0.191507
\(691\) −50.7021 −1.92880 −0.964399 0.264453i \(-0.914809\pi\)
−0.964399 + 0.264453i \(0.914809\pi\)
\(692\) −12.1919 −0.463465
\(693\) 4.66575 0.177237
\(694\) 56.7013 2.15235
\(695\) 1.82729 0.0693132
\(696\) 76.1718 2.88728
\(697\) 0.240524 0.00911050
\(698\) −13.6325 −0.515998
\(699\) −22.0768 −0.835022
\(700\) 116.144 4.38983
\(701\) −25.6171 −0.967543 −0.483771 0.875194i \(-0.660733\pi\)
−0.483771 + 0.875194i \(0.660733\pi\)
\(702\) −64.6006 −2.43819
\(703\) −41.4086 −1.56175
\(704\) −4.31449 −0.162609
\(705\) 3.75101 0.141271
\(706\) 85.3904 3.21371
\(707\) −1.56096 −0.0587059
\(708\) −31.8019 −1.19519
\(709\) 23.4172 0.879450 0.439725 0.898132i \(-0.355076\pi\)
0.439725 + 0.898132i \(0.355076\pi\)
\(710\) −2.51268 −0.0942992
\(711\) −2.64659 −0.0992548
\(712\) 83.8524 3.14250
\(713\) 45.6021 1.70781
\(714\) 77.0545 2.88369
\(715\) 0.902478 0.0337508
\(716\) −43.3133 −1.61869
\(717\) 38.5546 1.43985
\(718\) 51.0278 1.90434
\(719\) −0.169207 −0.00631036 −0.00315518 0.999995i \(-0.501004\pi\)
−0.00315518 + 0.999995i \(0.501004\pi\)
\(720\) 1.26235 0.0470449
\(721\) −38.9162 −1.44932
\(722\) −16.8992 −0.628922
\(723\) 1.73083 0.0643701
\(724\) 23.4629 0.871993
\(725\) 27.3324 1.01510
\(726\) 51.8558 1.92455
\(727\) −5.76002 −0.213627 −0.106814 0.994279i \(-0.534065\pi\)
−0.106814 + 0.994279i \(0.534065\pi\)
\(728\) −213.724 −7.92114
\(729\) 13.5588 0.502176
\(730\) 1.53987 0.0569931
\(731\) −19.0509 −0.704621
\(732\) 4.02354 0.148714
\(733\) −11.7182 −0.432820 −0.216410 0.976303i \(-0.569435\pi\)
−0.216410 + 0.976303i \(0.569435\pi\)
\(734\) 49.5440 1.82870
\(735\) 5.33725 0.196867
\(736\) 54.2483 1.99962
\(737\) −13.6151 −0.501520
\(738\) −0.200514 −0.00738101
\(739\) 24.9178 0.916617 0.458309 0.888793i \(-0.348455\pi\)
0.458309 + 0.888793i \(0.348455\pi\)
\(740\) 5.80645 0.213449
\(741\) 62.1252 2.28222
\(742\) −96.6757 −3.54908
\(743\) 37.3941 1.37186 0.685928 0.727669i \(-0.259396\pi\)
0.685928 + 0.727669i \(0.259396\pi\)
\(744\) 97.8636 3.58785
\(745\) −2.81773 −0.103234
\(746\) −40.9695 −1.50000
\(747\) 15.4249 0.564367
\(748\) −13.6561 −0.499315
\(749\) 8.80553 0.321747
\(750\) 7.76520 0.283545
\(751\) 34.6167 1.26318 0.631590 0.775303i \(-0.282403\pi\)
0.631590 + 0.775303i \(0.282403\pi\)
\(752\) 107.563 3.92243
\(753\) 58.7120 2.13959
\(754\) −87.6692 −3.19272
\(755\) −2.85997 −0.104085
\(756\) 94.4269 3.43427
\(757\) 7.41207 0.269396 0.134698 0.990887i \(-0.456994\pi\)
0.134698 + 0.990887i \(0.456994\pi\)
\(758\) −23.9531 −0.870014
\(759\) 12.4678 0.452552
\(760\) 5.31339 0.192737
\(761\) 9.61285 0.348465 0.174233 0.984705i \(-0.444256\pi\)
0.174233 + 0.984705i \(0.444256\pi\)
\(762\) 42.5584 1.54173
\(763\) −73.9790 −2.67822
\(764\) −89.2277 −3.22814
\(765\) 0.439838 0.0159024
\(766\) 76.6395 2.76910
\(767\) 20.9988 0.758222
\(768\) −44.8591 −1.61871
\(769\) 38.0448 1.37193 0.685965 0.727634i \(-0.259380\pi\)
0.685965 + 0.727634i \(0.259380\pi\)
\(770\) −1.88151 −0.0678048
\(771\) −38.4950 −1.38636
\(772\) 110.898 3.99130
\(773\) −24.3451 −0.875634 −0.437817 0.899064i \(-0.644248\pi\)
−0.437817 + 0.899064i \(0.644248\pi\)
\(774\) 15.8818 0.570859
\(775\) 35.1160 1.26141
\(776\) 15.0636 0.540751
\(777\) −81.2015 −2.91309
\(778\) 21.1987 0.760011
\(779\) −0.404294 −0.0144854
\(780\) −8.71140 −0.311918
\(781\) −6.22757 −0.222840
\(782\) 50.2618 1.79736
\(783\) 22.2217 0.794139
\(784\) 153.050 5.46607
\(785\) 3.54341 0.126470
\(786\) −23.2702 −0.830020
\(787\) 34.9320 1.24519 0.622596 0.782543i \(-0.286078\pi\)
0.622596 + 0.782543i \(0.286078\pi\)
\(788\) −26.8582 −0.956782
\(789\) −37.0998 −1.32079
\(790\) 1.06726 0.0379714
\(791\) 0.897470 0.0319103
\(792\) 6.53128 0.232079
\(793\) −2.65674 −0.0943436
\(794\) −17.7307 −0.629239
\(795\) −2.26068 −0.0801782
\(796\) −23.4519 −0.831229
\(797\) −4.40735 −0.156116 −0.0780581 0.996949i \(-0.524872\pi\)
−0.0780581 + 0.996949i \(0.524872\pi\)
\(798\) −129.520 −4.58496
\(799\) 37.4781 1.32588
\(800\) 41.7741 1.47694
\(801\) −11.6675 −0.412249
\(802\) −18.4492 −0.651464
\(803\) 3.81649 0.134681
\(804\) 131.424 4.63496
\(805\) 4.85522 0.171124
\(806\) −112.635 −3.96741
\(807\) 8.55286 0.301075
\(808\) −2.18509 −0.0768711
\(809\) 14.2117 0.499658 0.249829 0.968290i \(-0.419626\pi\)
0.249829 + 0.968290i \(0.419626\pi\)
\(810\) 4.28485 0.150554
\(811\) 15.4198 0.541463 0.270731 0.962655i \(-0.412734\pi\)
0.270731 + 0.962655i \(0.412734\pi\)
\(812\) 128.146 4.49706
\(813\) −25.6748 −0.900454
\(814\) 20.5259 0.719431
\(815\) −1.79331 −0.0628170
\(816\) 51.6697 1.80880
\(817\) 32.0224 1.12032
\(818\) −84.8736 −2.96754
\(819\) 29.7381 1.03914
\(820\) 0.0566915 0.00197975
\(821\) −7.63569 −0.266487 −0.133244 0.991083i \(-0.542539\pi\)
−0.133244 + 0.991083i \(0.542539\pi\)
\(822\) −96.6086 −3.36961
\(823\) 7.03484 0.245219 0.122610 0.992455i \(-0.460874\pi\)
0.122610 + 0.992455i \(0.460874\pi\)
\(824\) −54.4763 −1.89777
\(825\) 9.60086 0.334259
\(826\) −43.7788 −1.52326
\(827\) 11.6860 0.406361 0.203181 0.979141i \(-0.434872\pi\)
0.203181 + 0.979141i \(0.434872\pi\)
\(828\) −29.3774 −1.02094
\(829\) 10.9805 0.381368 0.190684 0.981651i \(-0.438929\pi\)
0.190684 + 0.981651i \(0.438929\pi\)
\(830\) −6.22023 −0.215907
\(831\) −20.8556 −0.723473
\(832\) −27.4993 −0.953368
\(833\) 53.3270 1.84767
\(834\) −62.3521 −2.15908
\(835\) 0.562777 0.0194757
\(836\) 22.9543 0.793892
\(837\) 28.5499 0.986829
\(838\) −45.5796 −1.57452
\(839\) 12.1386 0.419073 0.209536 0.977801i \(-0.432805\pi\)
0.209536 + 0.977801i \(0.432805\pi\)
\(840\) 10.4195 0.359505
\(841\) 1.15700 0.0398965
\(842\) 97.8219 3.37117
\(843\) −22.4300 −0.772531
\(844\) −112.762 −3.88142
\(845\) 3.78880 0.130339
\(846\) −31.2437 −1.07418
\(847\) 50.0493 1.71971
\(848\) −64.8270 −2.22617
\(849\) 25.5808 0.877930
\(850\) 38.7043 1.32755
\(851\) −52.9669 −1.81568
\(852\) 60.1132 2.05944
\(853\) 21.1229 0.723235 0.361617 0.932327i \(-0.382225\pi\)
0.361617 + 0.932327i \(0.382225\pi\)
\(854\) 5.53883 0.189535
\(855\) −0.739319 −0.0252842
\(856\) 12.3263 0.421304
\(857\) 3.55034 0.121277 0.0606387 0.998160i \(-0.480686\pi\)
0.0606387 + 0.998160i \(0.480686\pi\)
\(858\) −30.7949 −1.05132
\(859\) −20.8298 −0.710703 −0.355352 0.934733i \(-0.615639\pi\)
−0.355352 + 0.934733i \(0.615639\pi\)
\(860\) −4.49028 −0.153117
\(861\) −0.792814 −0.0270190
\(862\) 44.1342 1.50322
\(863\) −1.77956 −0.0605769 −0.0302885 0.999541i \(-0.509643\pi\)
−0.0302885 + 0.999541i \(0.509643\pi\)
\(864\) 33.9630 1.15544
\(865\) −0.392466 −0.0133443
\(866\) −92.8857 −3.15638
\(867\) −15.8639 −0.538766
\(868\) 164.639 5.58823
\(869\) 2.64516 0.0897308
\(870\) 4.27404 0.144904
\(871\) −86.7790 −2.94039
\(872\) −103.558 −3.50693
\(873\) −2.09599 −0.0709385
\(874\) −84.4846 −2.85773
\(875\) 7.49468 0.253367
\(876\) −36.8397 −1.24470
\(877\) 25.2856 0.853836 0.426918 0.904290i \(-0.359599\pi\)
0.426918 + 0.904290i \(0.359599\pi\)
\(878\) 44.9937 1.51846
\(879\) −11.7261 −0.395513
\(880\) −1.26167 −0.0425307
\(881\) 53.0433 1.78707 0.893537 0.448989i \(-0.148216\pi\)
0.893537 + 0.448989i \(0.148216\pi\)
\(882\) −44.4562 −1.49692
\(883\) 20.4028 0.686609 0.343305 0.939224i \(-0.388454\pi\)
0.343305 + 0.939224i \(0.388454\pi\)
\(884\) −87.0398 −2.92747
\(885\) −1.02373 −0.0344123
\(886\) −42.7569 −1.43644
\(887\) 27.4801 0.922690 0.461345 0.887221i \(-0.347367\pi\)
0.461345 + 0.887221i \(0.347367\pi\)
\(888\) −113.669 −3.81447
\(889\) 41.0757 1.37764
\(890\) 4.70501 0.157712
\(891\) 10.6198 0.355777
\(892\) 37.7449 1.26379
\(893\) −62.9966 −2.10810
\(894\) 96.1484 3.21568
\(895\) −1.39429 −0.0466060
\(896\) −26.1610 −0.873978
\(897\) 79.4661 2.65330
\(898\) 18.4307 0.615040
\(899\) 38.7450 1.29222
\(900\) −22.6222 −0.754072
\(901\) −22.5876 −0.752502
\(902\) 0.200405 0.00667276
\(903\) 62.7953 2.08970
\(904\) 1.25631 0.0417842
\(905\) 0.755292 0.0251067
\(906\) 97.5896 3.24220
\(907\) −46.8641 −1.55610 −0.778048 0.628204i \(-0.783790\pi\)
−0.778048 + 0.628204i \(0.783790\pi\)
\(908\) −31.8732 −1.05775
\(909\) 0.304039 0.0100843
\(910\) −11.9922 −0.397537
\(911\) −5.05152 −0.167364 −0.0836822 0.996492i \(-0.526668\pi\)
−0.0836822 + 0.996492i \(0.526668\pi\)
\(912\) −86.8511 −2.87593
\(913\) −15.4166 −0.510213
\(914\) 3.54066 0.117115
\(915\) 0.129521 0.00428183
\(916\) 39.1169 1.29246
\(917\) −22.4595 −0.741678
\(918\) 31.4672 1.03857
\(919\) −49.9846 −1.64884 −0.824419 0.565979i \(-0.808498\pi\)
−0.824419 + 0.565979i \(0.808498\pi\)
\(920\) 6.79650 0.224074
\(921\) −17.4287 −0.574294
\(922\) −45.6600 −1.50373
\(923\) −39.6927 −1.30650
\(924\) 45.0130 1.48082
\(925\) −40.7873 −1.34108
\(926\) −35.4569 −1.16518
\(927\) 7.57998 0.248959
\(928\) 46.0910 1.51301
\(929\) 23.3799 0.767071 0.383536 0.923526i \(-0.374706\pi\)
0.383536 + 0.923526i \(0.374706\pi\)
\(930\) 5.49118 0.180063
\(931\) −89.6368 −2.93773
\(932\) −51.9907 −1.70301
\(933\) −37.9918 −1.24380
\(934\) −26.6699 −0.872665
\(935\) −0.439600 −0.0143765
\(936\) 41.6285 1.36067
\(937\) −38.6593 −1.26294 −0.631472 0.775399i \(-0.717549\pi\)
−0.631472 + 0.775399i \(0.717549\pi\)
\(938\) 180.919 5.90721
\(939\) −12.3110 −0.401755
\(940\) 8.83359 0.288120
\(941\) −38.6825 −1.26101 −0.630506 0.776184i \(-0.717153\pi\)
−0.630506 + 0.776184i \(0.717153\pi\)
\(942\) −120.911 −3.93948
\(943\) −0.517145 −0.0168405
\(944\) −29.3563 −0.955468
\(945\) 3.03968 0.0988809
\(946\) −15.8732 −0.516082
\(947\) 1.52713 0.0496250 0.0248125 0.999692i \(-0.492101\pi\)
0.0248125 + 0.999692i \(0.492101\pi\)
\(948\) −25.5331 −0.829275
\(949\) 24.3252 0.789630
\(950\) −65.0576 −2.11075
\(951\) 10.6379 0.344957
\(952\) 104.106 3.37409
\(953\) −6.44281 −0.208703 −0.104352 0.994540i \(-0.533277\pi\)
−0.104352 + 0.994540i \(0.533277\pi\)
\(954\) 18.8302 0.609650
\(955\) −2.87232 −0.0929459
\(956\) 90.7959 2.93655
\(957\) 10.5930 0.342424
\(958\) 97.9812 3.16563
\(959\) −93.2430 −3.01097
\(960\) 1.34064 0.0432691
\(961\) 18.7786 0.605760
\(962\) 130.826 4.21800
\(963\) −1.71511 −0.0552688
\(964\) 4.07608 0.131282
\(965\) 3.56989 0.114919
\(966\) −165.673 −5.33043
\(967\) −21.2147 −0.682217 −0.341109 0.940024i \(-0.610802\pi\)
−0.341109 + 0.940024i \(0.610802\pi\)
\(968\) 70.0607 2.25184
\(969\) −30.2614 −0.972137
\(970\) 0.845227 0.0271386
\(971\) 32.7107 1.04974 0.524868 0.851184i \(-0.324115\pi\)
0.524868 + 0.851184i \(0.324115\pi\)
\(972\) −45.5566 −1.46123
\(973\) −60.1799 −1.92928
\(974\) 31.4196 1.00675
\(975\) 61.1931 1.95975
\(976\) 3.71412 0.118886
\(977\) 33.8823 1.08399 0.541995 0.840382i \(-0.317669\pi\)
0.541995 + 0.840382i \(0.317669\pi\)
\(978\) 61.1925 1.95672
\(979\) 11.6611 0.372692
\(980\) 12.5692 0.401508
\(981\) 14.4094 0.460057
\(982\) 68.2888 2.17918
\(983\) −33.8288 −1.07897 −0.539485 0.841995i \(-0.681381\pi\)
−0.539485 + 0.841995i \(0.681381\pi\)
\(984\) −1.10981 −0.0353794
\(985\) −0.864587 −0.0275480
\(986\) 42.7040 1.35997
\(987\) −123.535 −3.93217
\(988\) 146.304 4.65456
\(989\) 40.9607 1.30248
\(990\) 0.366474 0.0116473
\(991\) 13.6248 0.432806 0.216403 0.976304i \(-0.430568\pi\)
0.216403 + 0.976304i \(0.430568\pi\)
\(992\) 59.2166 1.88013
\(993\) 36.3353 1.15307
\(994\) 82.7523 2.62474
\(995\) −0.754935 −0.0239331
\(996\) 148.812 4.71530
\(997\) 45.8380 1.45170 0.725852 0.687851i \(-0.241446\pi\)
0.725852 + 0.687851i \(0.241446\pi\)
\(998\) 42.9918 1.36088
\(999\) −33.1607 −1.04916
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 6047.2.a.b.1.16 287
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6047.2.a.b.1.16 287 1.1 even 1 trivial