Properties

Label 273.2.a.d.1.1
Level $273$
Weight $2$
Character 273.1
Self dual yes
Analytic conductor $2.180$
Analytic rank $1$
Dimension $3$
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] = [273,2,Mod(1,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, 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("273.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.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: \(2.17991597518\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 273.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.81361 q^{2} -1.00000 q^{3} +5.91638 q^{4} -1.28917 q^{5} +2.81361 q^{6} -1.00000 q^{7} -11.0192 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.81361 q^{2} -1.00000 q^{3} +5.91638 q^{4} -1.28917 q^{5} +2.81361 q^{6} -1.00000 q^{7} -11.0192 q^{8} +1.00000 q^{9} +3.62721 q^{10} +4.20555 q^{11} -5.91638 q^{12} -1.00000 q^{13} +2.81361 q^{14} +1.28917 q^{15} +19.1708 q^{16} +1.62721 q^{17} -2.81361 q^{18} -6.33804 q^{19} -7.62721 q^{20} +1.00000 q^{21} -11.8328 q^{22} -2.71083 q^{23} +11.0192 q^{24} -3.33804 q^{25} +2.81361 q^{26} -1.00000 q^{27} -5.91638 q^{28} -4.33804 q^{29} -3.62721 q^{30} -7.49472 q^{31} -31.9008 q^{32} -4.20555 q^{33} -4.57834 q^{34} +1.28917 q^{35} +5.91638 q^{36} +3.42166 q^{37} +17.8328 q^{38} +1.00000 q^{39} +14.2056 q^{40} -7.62721 q^{41} -2.81361 q^{42} -4.91638 q^{43} +24.8816 q^{44} -1.28917 q^{45} +7.62721 q^{46} -1.08362 q^{47} -19.1708 q^{48} +1.00000 q^{49} +9.39194 q^{50} -1.62721 q^{51} -5.91638 q^{52} +1.75971 q^{53} +2.81361 q^{54} -5.42166 q^{55} +11.0192 q^{56} +6.33804 q^{57} +12.2056 q^{58} -4.57834 q^{59} +7.62721 q^{60} -5.25443 q^{61} +21.0872 q^{62} -1.00000 q^{63} +51.4147 q^{64} +1.28917 q^{65} +11.8328 q^{66} +8.67609 q^{67} +9.62721 q^{68} +2.71083 q^{69} -3.62721 q^{70} -6.78389 q^{71} -11.0192 q^{72} +4.07306 q^{73} -9.62721 q^{74} +3.33804 q^{75} -37.4983 q^{76} -4.20555 q^{77} -2.81361 q^{78} -8.91638 q^{79} -24.7144 q^{80} +1.00000 q^{81} +21.4600 q^{82} -4.33804 q^{83} +5.91638 q^{84} -2.09775 q^{85} +13.8328 q^{86} +4.33804 q^{87} -46.3416 q^{88} +4.54359 q^{89} +3.62721 q^{90} +1.00000 q^{91} -16.0383 q^{92} +7.49472 q^{93} +3.04888 q^{94} +8.17081 q^{95} +31.9008 q^{96} +11.3275 q^{97} -2.81361 q^{98} +4.20555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 3 q^{3} + 4 q^{4} - 3 q^{5} + 2 q^{6} - 3 q^{7} - 12 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 3 q^{3} + 4 q^{4} - 3 q^{5} + 2 q^{6} - 3 q^{7} - 12 q^{8} + 3 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} - 3 q^{13} + 2 q^{14} + 3 q^{15} + 18 q^{16} - 8 q^{17} - 2 q^{18} - 7 q^{19} - 10 q^{20} + 3 q^{21} - 8 q^{22} - 9 q^{23} + 12 q^{24} + 2 q^{25} + 2 q^{26} - 3 q^{27} - 4 q^{28} - q^{29} + 2 q^{30} - 7 q^{31} - 36 q^{32} + 2 q^{33} - 12 q^{34} + 3 q^{35} + 4 q^{36} + 12 q^{37} + 26 q^{38} + 3 q^{39} + 28 q^{40} - 10 q^{41} - 2 q^{42} - q^{43} + 36 q^{44} - 3 q^{45} + 10 q^{46} - 17 q^{47} - 18 q^{48} + 3 q^{49} + 20 q^{50} + 8 q^{51} - 4 q^{52} - 5 q^{53} + 2 q^{54} - 18 q^{55} + 12 q^{56} + 7 q^{57} + 22 q^{58} - 12 q^{59} + 10 q^{60} + 10 q^{61} + 10 q^{62} - 3 q^{63} + 58 q^{64} + 3 q^{65} + 8 q^{66} + 2 q^{67} + 16 q^{68} + 9 q^{69} + 2 q^{70} - 4 q^{71} - 12 q^{72} - 5 q^{73} - 16 q^{74} - 2 q^{75} - 30 q^{76} + 2 q^{77} - 2 q^{78} - 13 q^{79} - 8 q^{80} + 3 q^{81} + 24 q^{82} - q^{83} + 4 q^{84} + 16 q^{85} + 14 q^{86} + q^{87} - 60 q^{88} - 13 q^{89} - 2 q^{90} + 3 q^{91} - 6 q^{92} + 7 q^{93} - 2 q^{94} - 15 q^{95} + 36 q^{96} - 9 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.81361 −1.98952 −0.994760 0.102237i \(-0.967400\pi\)
−0.994760 + 0.102237i \(0.967400\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.91638 2.95819
\(5\) −1.28917 −0.576534 −0.288267 0.957550i \(-0.593079\pi\)
−0.288267 + 0.957550i \(0.593079\pi\)
\(6\) 2.81361 1.14865
\(7\) −1.00000 −0.377964
\(8\) −11.0192 −3.89586
\(9\) 1.00000 0.333333
\(10\) 3.62721 1.14703
\(11\) 4.20555 1.26802 0.634011 0.773324i \(-0.281408\pi\)
0.634011 + 0.773324i \(0.281408\pi\)
\(12\) −5.91638 −1.70791
\(13\) −1.00000 −0.277350
\(14\) 2.81361 0.751968
\(15\) 1.28917 0.332862
\(16\) 19.1708 4.79270
\(17\) 1.62721 0.394657 0.197329 0.980337i \(-0.436773\pi\)
0.197329 + 0.980337i \(0.436773\pi\)
\(18\) −2.81361 −0.663173
\(19\) −6.33804 −1.45405 −0.727024 0.686613i \(-0.759097\pi\)
−0.727024 + 0.686613i \(0.759097\pi\)
\(20\) −7.62721 −1.70550
\(21\) 1.00000 0.218218
\(22\) −11.8328 −2.52275
\(23\) −2.71083 −0.565247 −0.282624 0.959231i \(-0.591205\pi\)
−0.282624 + 0.959231i \(0.591205\pi\)
\(24\) 11.0192 2.24928
\(25\) −3.33804 −0.667609
\(26\) 2.81361 0.551794
\(27\) −1.00000 −0.192450
\(28\) −5.91638 −1.11809
\(29\) −4.33804 −0.805555 −0.402777 0.915298i \(-0.631955\pi\)
−0.402777 + 0.915298i \(0.631955\pi\)
\(30\) −3.62721 −0.662235
\(31\) −7.49472 −1.34609 −0.673046 0.739601i \(-0.735014\pi\)
−0.673046 + 0.739601i \(0.735014\pi\)
\(32\) −31.9008 −5.63932
\(33\) −4.20555 −0.732092
\(34\) −4.57834 −0.785178
\(35\) 1.28917 0.217909
\(36\) 5.91638 0.986064
\(37\) 3.42166 0.562518 0.281259 0.959632i \(-0.409248\pi\)
0.281259 + 0.959632i \(0.409248\pi\)
\(38\) 17.8328 2.89286
\(39\) 1.00000 0.160128
\(40\) 14.2056 2.24609
\(41\) −7.62721 −1.19117 −0.595585 0.803292i \(-0.703080\pi\)
−0.595585 + 0.803292i \(0.703080\pi\)
\(42\) −2.81361 −0.434149
\(43\) −4.91638 −0.749741 −0.374871 0.927077i \(-0.622313\pi\)
−0.374871 + 0.927077i \(0.622313\pi\)
\(44\) 24.8816 3.75105
\(45\) −1.28917 −0.192178
\(46\) 7.62721 1.12457
\(47\) −1.08362 −0.158062 −0.0790310 0.996872i \(-0.525183\pi\)
−0.0790310 + 0.996872i \(0.525183\pi\)
\(48\) −19.1708 −2.76707
\(49\) 1.00000 0.142857
\(50\) 9.39194 1.32822
\(51\) −1.62721 −0.227855
\(52\) −5.91638 −0.820455
\(53\) 1.75971 0.241714 0.120857 0.992670i \(-0.461436\pi\)
0.120857 + 0.992670i \(0.461436\pi\)
\(54\) 2.81361 0.382883
\(55\) −5.42166 −0.731057
\(56\) 11.0192 1.47250
\(57\) 6.33804 0.839494
\(58\) 12.2056 1.60267
\(59\) −4.57834 −0.596049 −0.298024 0.954558i \(-0.596328\pi\)
−0.298024 + 0.954558i \(0.596328\pi\)
\(60\) 7.62721 0.984669
\(61\) −5.25443 −0.672760 −0.336380 0.941726i \(-0.609203\pi\)
−0.336380 + 0.941726i \(0.609203\pi\)
\(62\) 21.0872 2.67808
\(63\) −1.00000 −0.125988
\(64\) 51.4147 6.42683
\(65\) 1.28917 0.159902
\(66\) 11.8328 1.45651
\(67\) 8.67609 1.05995 0.529976 0.848012i \(-0.322201\pi\)
0.529976 + 0.848012i \(0.322201\pi\)
\(68\) 9.62721 1.16747
\(69\) 2.71083 0.326346
\(70\) −3.62721 −0.433535
\(71\) −6.78389 −0.805099 −0.402550 0.915398i \(-0.631876\pi\)
−0.402550 + 0.915398i \(0.631876\pi\)
\(72\) −11.0192 −1.29862
\(73\) 4.07306 0.476715 0.238358 0.971177i \(-0.423391\pi\)
0.238358 + 0.971177i \(0.423391\pi\)
\(74\) −9.62721 −1.11914
\(75\) 3.33804 0.385444
\(76\) −37.4983 −4.30135
\(77\) −4.20555 −0.479267
\(78\) −2.81361 −0.318578
\(79\) −8.91638 −1.00317 −0.501586 0.865108i \(-0.667250\pi\)
−0.501586 + 0.865108i \(0.667250\pi\)
\(80\) −24.7144 −2.76315
\(81\) 1.00000 0.111111
\(82\) 21.4600 2.36986
\(83\) −4.33804 −0.476162 −0.238081 0.971245i \(-0.576518\pi\)
−0.238081 + 0.971245i \(0.576518\pi\)
\(84\) 5.91638 0.645530
\(85\) −2.09775 −0.227533
\(86\) 13.8328 1.49163
\(87\) 4.33804 0.465087
\(88\) −46.3416 −4.94003
\(89\) 4.54359 0.481620 0.240810 0.970572i \(-0.422587\pi\)
0.240810 + 0.970572i \(0.422587\pi\)
\(90\) 3.62721 0.382342
\(91\) 1.00000 0.104828
\(92\) −16.0383 −1.67211
\(93\) 7.49472 0.777166
\(94\) 3.04888 0.314468
\(95\) 8.17081 0.838307
\(96\) 31.9008 3.25586
\(97\) 11.3275 1.15013 0.575066 0.818107i \(-0.304976\pi\)
0.575066 + 0.818107i \(0.304976\pi\)
\(98\) −2.81361 −0.284217
\(99\) 4.20555 0.422674
\(100\) −19.7491 −1.97491
\(101\) 14.3033 1.42323 0.711616 0.702569i \(-0.247964\pi\)
0.711616 + 0.702569i \(0.247964\pi\)
\(102\) 4.57834 0.453323
\(103\) −19.6655 −1.93770 −0.968851 0.247645i \(-0.920343\pi\)
−0.968851 + 0.247645i \(0.920343\pi\)
\(104\) 11.0192 1.08052
\(105\) −1.28917 −0.125810
\(106\) −4.95112 −0.480896
\(107\) 2.20555 0.213219 0.106609 0.994301i \(-0.466001\pi\)
0.106609 + 0.994301i \(0.466001\pi\)
\(108\) −5.91638 −0.569304
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 15.2544 1.45445
\(111\) −3.42166 −0.324770
\(112\) −19.1708 −1.81147
\(113\) −2.17081 −0.204212 −0.102106 0.994774i \(-0.532558\pi\)
−0.102106 + 0.994774i \(0.532558\pi\)
\(114\) −17.8328 −1.67019
\(115\) 3.49472 0.325884
\(116\) −25.6655 −2.38298
\(117\) −1.00000 −0.0924500
\(118\) 12.8816 1.18585
\(119\) −1.62721 −0.149166
\(120\) −14.2056 −1.29678
\(121\) 6.68665 0.607877
\(122\) 14.7839 1.33847
\(123\) 7.62721 0.687723
\(124\) −44.3416 −3.98199
\(125\) 10.7491 0.961433
\(126\) 2.81361 0.250656
\(127\) −0.745574 −0.0661590 −0.0330795 0.999453i \(-0.510531\pi\)
−0.0330795 + 0.999453i \(0.510531\pi\)
\(128\) −80.8591 −7.14700
\(129\) 4.91638 0.432863
\(130\) −3.62721 −0.318128
\(131\) −14.5089 −1.26764 −0.633822 0.773479i \(-0.718515\pi\)
−0.633822 + 0.773479i \(0.718515\pi\)
\(132\) −24.8816 −2.16567
\(133\) 6.33804 0.549578
\(134\) −24.4111 −2.10880
\(135\) 1.28917 0.110954
\(136\) −17.9305 −1.53753
\(137\) 4.41110 0.376866 0.188433 0.982086i \(-0.439659\pi\)
0.188433 + 0.982086i \(0.439659\pi\)
\(138\) −7.62721 −0.649271
\(139\) 18.9894 1.61066 0.805332 0.592825i \(-0.201987\pi\)
0.805332 + 0.592825i \(0.201987\pi\)
\(140\) 7.62721 0.644617
\(141\) 1.08362 0.0912571
\(142\) 19.0872 1.60176
\(143\) −4.20555 −0.351686
\(144\) 19.1708 1.59757
\(145\) 5.59247 0.464429
\(146\) −11.4600 −0.948434
\(147\) −1.00000 −0.0824786
\(148\) 20.2439 1.66404
\(149\) −9.42166 −0.771853 −0.385926 0.922530i \(-0.626118\pi\)
−0.385926 + 0.922530i \(0.626118\pi\)
\(150\) −9.39194 −0.766849
\(151\) −16.4111 −1.33552 −0.667758 0.744378i \(-0.732746\pi\)
−0.667758 + 0.744378i \(0.732746\pi\)
\(152\) 69.8399 5.66476
\(153\) 1.62721 0.131552
\(154\) 11.8328 0.953511
\(155\) 9.66196 0.776067
\(156\) 5.91638 0.473690
\(157\) −9.51941 −0.759732 −0.379866 0.925042i \(-0.624030\pi\)
−0.379866 + 0.925042i \(0.624030\pi\)
\(158\) 25.0872 1.99583
\(159\) −1.75971 −0.139554
\(160\) 41.1255 3.25126
\(161\) 2.71083 0.213643
\(162\) −2.81361 −0.221058
\(163\) −0.745574 −0.0583979 −0.0291989 0.999574i \(-0.509296\pi\)
−0.0291989 + 0.999574i \(0.509296\pi\)
\(164\) −45.1255 −3.52371
\(165\) 5.42166 0.422076
\(166\) 12.2056 0.947334
\(167\) 3.59247 0.277994 0.138997 0.990293i \(-0.455612\pi\)
0.138997 + 0.990293i \(0.455612\pi\)
\(168\) −11.0192 −0.850146
\(169\) 1.00000 0.0769231
\(170\) 5.90225 0.452682
\(171\) −6.33804 −0.484682
\(172\) −29.0872 −2.21788
\(173\) −15.7250 −1.19555 −0.597773 0.801665i \(-0.703948\pi\)
−0.597773 + 0.801665i \(0.703948\pi\)
\(174\) −12.2056 −0.925300
\(175\) 3.33804 0.252332
\(176\) 80.6238 6.07725
\(177\) 4.57834 0.344129
\(178\) −12.7839 −0.958193
\(179\) 24.9547 1.86520 0.932601 0.360910i \(-0.117534\pi\)
0.932601 + 0.360910i \(0.117534\pi\)
\(180\) −7.62721 −0.568499
\(181\) 10.9411 0.813244 0.406622 0.913597i \(-0.366707\pi\)
0.406622 + 0.913597i \(0.366707\pi\)
\(182\) −2.81361 −0.208558
\(183\) 5.25443 0.388418
\(184\) 29.8711 2.20212
\(185\) −4.41110 −0.324311
\(186\) −21.0872 −1.54619
\(187\) 6.84333 0.500434
\(188\) −6.41110 −0.467578
\(189\) 1.00000 0.0727393
\(190\) −22.9894 −1.66783
\(191\) 5.04888 0.365324 0.182662 0.983176i \(-0.441529\pi\)
0.182662 + 0.983176i \(0.441529\pi\)
\(192\) −51.4147 −3.71053
\(193\) 2.74557 0.197631 0.0988154 0.995106i \(-0.468495\pi\)
0.0988154 + 0.995106i \(0.468495\pi\)
\(194\) −31.8711 −2.28821
\(195\) −1.28917 −0.0923193
\(196\) 5.91638 0.422599
\(197\) 5.42166 0.386277 0.193139 0.981171i \(-0.438133\pi\)
0.193139 + 0.981171i \(0.438133\pi\)
\(198\) −11.8328 −0.840918
\(199\) −20.4111 −1.44690 −0.723452 0.690374i \(-0.757446\pi\)
−0.723452 + 0.690374i \(0.757446\pi\)
\(200\) 36.7824 2.60091
\(201\) −8.67609 −0.611964
\(202\) −40.2439 −2.83155
\(203\) 4.33804 0.304471
\(204\) −9.62721 −0.674040
\(205\) 9.83276 0.686750
\(206\) 55.3311 3.85510
\(207\) −2.71083 −0.188416
\(208\) −19.1708 −1.32926
\(209\) −26.6550 −1.84376
\(210\) 3.62721 0.250301
\(211\) −14.4842 −0.997130 −0.498565 0.866852i \(-0.666140\pi\)
−0.498565 + 0.866852i \(0.666140\pi\)
\(212\) 10.4111 0.715037
\(213\) 6.78389 0.464824
\(214\) −6.20555 −0.424203
\(215\) 6.33804 0.432251
\(216\) 11.0192 0.749759
\(217\) 7.49472 0.508775
\(218\) −28.1361 −1.90561
\(219\) −4.07306 −0.275232
\(220\) −32.0766 −2.16261
\(221\) −1.62721 −0.109458
\(222\) 9.62721 0.646136
\(223\) 16.8469 1.12815 0.564076 0.825723i \(-0.309233\pi\)
0.564076 + 0.825723i \(0.309233\pi\)
\(224\) 31.9008 2.13146
\(225\) −3.33804 −0.222536
\(226\) 6.10780 0.406285
\(227\) −21.9305 −1.45558 −0.727790 0.685800i \(-0.759452\pi\)
−0.727790 + 0.685800i \(0.759452\pi\)
\(228\) 37.4983 2.48338
\(229\) −5.25443 −0.347222 −0.173611 0.984814i \(-0.555544\pi\)
−0.173611 + 0.984814i \(0.555544\pi\)
\(230\) −9.83276 −0.648353
\(231\) 4.20555 0.276705
\(232\) 47.8016 3.13833
\(233\) 4.07306 0.266835 0.133417 0.991060i \(-0.457405\pi\)
0.133417 + 0.991060i \(0.457405\pi\)
\(234\) 2.81361 0.183931
\(235\) 1.39697 0.0911281
\(236\) −27.0872 −1.76323
\(237\) 8.91638 0.579181
\(238\) 4.57834 0.296770
\(239\) −2.37279 −0.153483 −0.0767414 0.997051i \(-0.524452\pi\)
−0.0767414 + 0.997051i \(0.524452\pi\)
\(240\) 24.7144 1.59531
\(241\) 14.9164 0.960849 0.480424 0.877036i \(-0.340483\pi\)
0.480424 + 0.877036i \(0.340483\pi\)
\(242\) −18.8136 −1.20938
\(243\) −1.00000 −0.0641500
\(244\) −31.0872 −1.99015
\(245\) −1.28917 −0.0823620
\(246\) −21.4600 −1.36824
\(247\) 6.33804 0.403280
\(248\) 82.5855 5.24418
\(249\) 4.33804 0.274912
\(250\) −30.2439 −1.91279
\(251\) −1.08719 −0.0686228 −0.0343114 0.999411i \(-0.510924\pi\)
−0.0343114 + 0.999411i \(0.510924\pi\)
\(252\) −5.91638 −0.372697
\(253\) −11.4005 −0.716746
\(254\) 2.09775 0.131625
\(255\) 2.09775 0.131366
\(256\) 124.676 7.79226
\(257\) −16.6167 −1.03652 −0.518259 0.855224i \(-0.673420\pi\)
−0.518259 + 0.855224i \(0.673420\pi\)
\(258\) −13.8328 −0.861190
\(259\) −3.42166 −0.212612
\(260\) 7.62721 0.473020
\(261\) −4.33804 −0.268518
\(262\) 40.8222 2.52200
\(263\) −27.0524 −1.66813 −0.834063 0.551670i \(-0.813991\pi\)
−0.834063 + 0.551670i \(0.813991\pi\)
\(264\) 46.3416 2.85213
\(265\) −2.26856 −0.139356
\(266\) −17.8328 −1.09340
\(267\) −4.54359 −0.278063
\(268\) 51.3311 3.13554
\(269\) 7.52946 0.459079 0.229540 0.973299i \(-0.426278\pi\)
0.229540 + 0.973299i \(0.426278\pi\)
\(270\) −3.62721 −0.220745
\(271\) 16.8222 1.02188 0.510938 0.859618i \(-0.329298\pi\)
0.510938 + 0.859618i \(0.329298\pi\)
\(272\) 31.1950 1.89147
\(273\) −1.00000 −0.0605228
\(274\) −12.4111 −0.749782
\(275\) −14.0383 −0.846542
\(276\) 16.0383 0.965393
\(277\) 17.4947 1.05116 0.525578 0.850745i \(-0.323849\pi\)
0.525578 + 0.850745i \(0.323849\pi\)
\(278\) −53.4288 −3.20445
\(279\) −7.49472 −0.448697
\(280\) −14.2056 −0.848944
\(281\) 27.4005 1.63458 0.817290 0.576227i \(-0.195476\pi\)
0.817290 + 0.576227i \(0.195476\pi\)
\(282\) −3.04888 −0.181558
\(283\) 20.0766 1.19343 0.596716 0.802453i \(-0.296472\pi\)
0.596716 + 0.802453i \(0.296472\pi\)
\(284\) −40.1361 −2.38164
\(285\) −8.17081 −0.483997
\(286\) 11.8328 0.699686
\(287\) 7.62721 0.450220
\(288\) −31.9008 −1.87977
\(289\) −14.3522 −0.844246
\(290\) −15.7350 −0.923992
\(291\) −11.3275 −0.664029
\(292\) 24.0978 1.41021
\(293\) −27.0524 −1.58042 −0.790210 0.612836i \(-0.790029\pi\)
−0.790210 + 0.612836i \(0.790029\pi\)
\(294\) 2.81361 0.164093
\(295\) 5.90225 0.343642
\(296\) −37.7038 −2.19149
\(297\) −4.20555 −0.244031
\(298\) 26.5089 1.53562
\(299\) 2.71083 0.156771
\(300\) 19.7491 1.14022
\(301\) 4.91638 0.283376
\(302\) 46.1744 2.65704
\(303\) −14.3033 −0.821703
\(304\) −121.505 −6.96881
\(305\) 6.77384 0.387869
\(306\) −4.57834 −0.261726
\(307\) −20.1708 −1.15121 −0.575604 0.817728i \(-0.695233\pi\)
−0.575604 + 0.817728i \(0.695233\pi\)
\(308\) −24.8816 −1.41776
\(309\) 19.6655 1.11873
\(310\) −27.1849 −1.54400
\(311\) −6.57834 −0.373023 −0.186512 0.982453i \(-0.559718\pi\)
−0.186512 + 0.982453i \(0.559718\pi\)
\(312\) −11.0192 −0.623837
\(313\) −10.7456 −0.607376 −0.303688 0.952772i \(-0.598218\pi\)
−0.303688 + 0.952772i \(0.598218\pi\)
\(314\) 26.7839 1.51150
\(315\) 1.28917 0.0726364
\(316\) −52.7527 −2.96757
\(317\) 24.6066 1.38204 0.691022 0.722833i \(-0.257161\pi\)
0.691022 + 0.722833i \(0.257161\pi\)
\(318\) 4.95112 0.277645
\(319\) −18.2439 −1.02146
\(320\) −66.2822 −3.70529
\(321\) −2.20555 −0.123102
\(322\) −7.62721 −0.425048
\(323\) −10.3133 −0.573850
\(324\) 5.91638 0.328688
\(325\) 3.33804 0.185161
\(326\) 2.09775 0.116184
\(327\) −10.0000 −0.553001
\(328\) 84.0455 4.64063
\(329\) 1.08362 0.0597418
\(330\) −15.2544 −0.839729
\(331\) 30.6550 1.68495 0.842475 0.538736i \(-0.181098\pi\)
0.842475 + 0.538736i \(0.181098\pi\)
\(332\) −25.6655 −1.40858
\(333\) 3.42166 0.187506
\(334\) −10.1078 −0.553074
\(335\) −11.1849 −0.611099
\(336\) 19.1708 1.04585
\(337\) −6.50528 −0.354365 −0.177183 0.984178i \(-0.556698\pi\)
−0.177183 + 0.984178i \(0.556698\pi\)
\(338\) −2.81361 −0.153040
\(339\) 2.17081 0.117902
\(340\) −12.4111 −0.673086
\(341\) −31.5194 −1.70687
\(342\) 17.8328 0.964285
\(343\) −1.00000 −0.0539949
\(344\) 54.1744 2.92089
\(345\) −3.49472 −0.188149
\(346\) 44.2439 2.37856
\(347\) 24.3033 1.30467 0.652335 0.757931i \(-0.273790\pi\)
0.652335 + 0.757931i \(0.273790\pi\)
\(348\) 25.6655 1.37582
\(349\) 17.7597 0.950655 0.475328 0.879809i \(-0.342330\pi\)
0.475328 + 0.879809i \(0.342330\pi\)
\(350\) −9.39194 −0.502021
\(351\) 1.00000 0.0533761
\(352\) −134.160 −7.15077
\(353\) 1.94056 0.103286 0.0516428 0.998666i \(-0.483554\pi\)
0.0516428 + 0.998666i \(0.483554\pi\)
\(354\) −12.8816 −0.684651
\(355\) 8.74557 0.464167
\(356\) 26.8816 1.42472
\(357\) 1.62721 0.0861212
\(358\) −70.2127 −3.71086
\(359\) −33.6272 −1.77478 −0.887388 0.461023i \(-0.847483\pi\)
−0.887388 + 0.461023i \(0.847483\pi\)
\(360\) 14.2056 0.748698
\(361\) 21.1708 1.11425
\(362\) −30.7839 −1.61797
\(363\) −6.68665 −0.350958
\(364\) 5.91638 0.310103
\(365\) −5.25086 −0.274842
\(366\) −14.7839 −0.772766
\(367\) −10.5783 −0.552185 −0.276092 0.961131i \(-0.589040\pi\)
−0.276092 + 0.961131i \(0.589040\pi\)
\(368\) −51.9688 −2.70906
\(369\) −7.62721 −0.397057
\(370\) 12.4111 0.645222
\(371\) −1.75971 −0.0913595
\(372\) 44.3416 2.29901
\(373\) −5.47002 −0.283227 −0.141614 0.989922i \(-0.545229\pi\)
−0.141614 + 0.989922i \(0.545229\pi\)
\(374\) −19.2544 −0.995623
\(375\) −10.7491 −0.555083
\(376\) 11.9406 0.615787
\(377\) 4.33804 0.223421
\(378\) −2.81361 −0.144716
\(379\) −5.42166 −0.278492 −0.139246 0.990258i \(-0.544468\pi\)
−0.139246 + 0.990258i \(0.544468\pi\)
\(380\) 48.3416 2.47987
\(381\) 0.745574 0.0381969
\(382\) −14.2056 −0.726819
\(383\) 14.1461 0.722833 0.361416 0.932405i \(-0.382293\pi\)
0.361416 + 0.932405i \(0.382293\pi\)
\(384\) 80.8591 4.12632
\(385\) 5.42166 0.276314
\(386\) −7.72496 −0.393190
\(387\) −4.91638 −0.249914
\(388\) 67.0177 3.40231
\(389\) 0.313348 0.0158874 0.00794370 0.999968i \(-0.497471\pi\)
0.00794370 + 0.999968i \(0.497471\pi\)
\(390\) 3.62721 0.183671
\(391\) −4.41110 −0.223079
\(392\) −11.0192 −0.556551
\(393\) 14.5089 0.731875
\(394\) −15.2544 −0.768507
\(395\) 11.4947 0.578362
\(396\) 24.8816 1.25035
\(397\) 20.6797 1.03788 0.518941 0.854810i \(-0.326326\pi\)
0.518941 + 0.854810i \(0.326326\pi\)
\(398\) 57.4288 2.87865
\(399\) −6.33804 −0.317299
\(400\) −63.9930 −3.19965
\(401\) −12.7456 −0.636484 −0.318242 0.948010i \(-0.603092\pi\)
−0.318242 + 0.948010i \(0.603092\pi\)
\(402\) 24.4111 1.21752
\(403\) 7.49472 0.373339
\(404\) 84.6238 4.21019
\(405\) −1.28917 −0.0640593
\(406\) −12.2056 −0.605751
\(407\) 14.3900 0.713285
\(408\) 17.9305 0.887693
\(409\) −18.6514 −0.922252 −0.461126 0.887335i \(-0.652554\pi\)
−0.461126 + 0.887335i \(0.652554\pi\)
\(410\) −27.6655 −1.36630
\(411\) −4.41110 −0.217584
\(412\) −116.349 −5.73209
\(413\) 4.57834 0.225285
\(414\) 7.62721 0.374857
\(415\) 5.59247 0.274524
\(416\) 31.9008 1.56407
\(417\) −18.9894 −0.929917
\(418\) 74.9966 3.66820
\(419\) 3.10831 0.151851 0.0759256 0.997113i \(-0.475809\pi\)
0.0759256 + 0.997113i \(0.475809\pi\)
\(420\) −7.62721 −0.372170
\(421\) 13.4005 0.653102 0.326551 0.945180i \(-0.394113\pi\)
0.326551 + 0.945180i \(0.394113\pi\)
\(422\) 40.7527 1.98381
\(423\) −1.08362 −0.0526873
\(424\) −19.3905 −0.941685
\(425\) −5.43171 −0.263477
\(426\) −19.0872 −0.924777
\(427\) 5.25443 0.254279
\(428\) 13.0489 0.630741
\(429\) 4.20555 0.203046
\(430\) −17.8328 −0.859972
\(431\) 9.89220 0.476491 0.238245 0.971205i \(-0.423428\pi\)
0.238245 + 0.971205i \(0.423428\pi\)
\(432\) −19.1708 −0.922356
\(433\) 0.578337 0.0277931 0.0138966 0.999903i \(-0.495576\pi\)
0.0138966 + 0.999903i \(0.495576\pi\)
\(434\) −21.0872 −1.01222
\(435\) −5.59247 −0.268138
\(436\) 59.1638 2.83343
\(437\) 17.1814 0.821896
\(438\) 11.4600 0.547579
\(439\) 23.7350 1.13281 0.566405 0.824127i \(-0.308334\pi\)
0.566405 + 0.824127i \(0.308334\pi\)
\(440\) 59.7422 2.84810
\(441\) 1.00000 0.0476190
\(442\) 4.57834 0.217769
\(443\) −3.86751 −0.183751 −0.0918754 0.995771i \(-0.529286\pi\)
−0.0918754 + 0.995771i \(0.529286\pi\)
\(444\) −20.2439 −0.960731
\(445\) −5.85746 −0.277670
\(446\) −47.4005 −2.24448
\(447\) 9.42166 0.445629
\(448\) −51.4147 −2.42911
\(449\) 1.68665 0.0795980 0.0397990 0.999208i \(-0.487328\pi\)
0.0397990 + 0.999208i \(0.487328\pi\)
\(450\) 9.39194 0.442740
\(451\) −32.0766 −1.51043
\(452\) −12.8433 −0.604099
\(453\) 16.4111 0.771061
\(454\) 61.7038 2.89590
\(455\) −1.28917 −0.0604372
\(456\) −69.8399 −3.27055
\(457\) −3.83276 −0.179289 −0.0896445 0.995974i \(-0.528573\pi\)
−0.0896445 + 0.995974i \(0.528573\pi\)
\(458\) 14.7839 0.690806
\(459\) −1.62721 −0.0759518
\(460\) 20.6761 0.964028
\(461\) −11.2161 −0.522386 −0.261193 0.965287i \(-0.584116\pi\)
−0.261193 + 0.965287i \(0.584116\pi\)
\(462\) −11.8328 −0.550510
\(463\) −25.8328 −1.20055 −0.600275 0.799794i \(-0.704942\pi\)
−0.600275 + 0.799794i \(0.704942\pi\)
\(464\) −83.1638 −3.86078
\(465\) −9.66196 −0.448062
\(466\) −11.4600 −0.530873
\(467\) 41.0872 1.90129 0.950644 0.310283i \(-0.100424\pi\)
0.950644 + 0.310283i \(0.100424\pi\)
\(468\) −5.91638 −0.273485
\(469\) −8.67609 −0.400625
\(470\) −3.93051 −0.181301
\(471\) 9.51941 0.438631
\(472\) 50.4494 2.32212
\(473\) −20.6761 −0.950688
\(474\) −25.0872 −1.15229
\(475\) 21.1567 0.970735
\(476\) −9.62721 −0.441263
\(477\) 1.75971 0.0805715
\(478\) 6.67609 0.305357
\(479\) 2.43580 0.111294 0.0556472 0.998450i \(-0.482278\pi\)
0.0556472 + 0.998450i \(0.482278\pi\)
\(480\) −41.1255 −1.87711
\(481\) −3.42166 −0.156014
\(482\) −41.9688 −1.91163
\(483\) −2.71083 −0.123347
\(484\) 39.5608 1.79822
\(485\) −14.6030 −0.663090
\(486\) 2.81361 0.127628
\(487\) −6.57834 −0.298093 −0.149046 0.988830i \(-0.547620\pi\)
−0.149046 + 0.988830i \(0.547620\pi\)
\(488\) 57.8993 2.62098
\(489\) 0.745574 0.0337160
\(490\) 3.62721 0.163861
\(491\) 28.3033 1.27731 0.638655 0.769493i \(-0.279491\pi\)
0.638655 + 0.769493i \(0.279491\pi\)
\(492\) 45.1255 2.03441
\(493\) −7.05892 −0.317918
\(494\) −17.8328 −0.802334
\(495\) −5.42166 −0.243686
\(496\) −143.680 −6.45141
\(497\) 6.78389 0.304299
\(498\) −12.2056 −0.546944
\(499\) 28.8222 1.29026 0.645129 0.764073i \(-0.276803\pi\)
0.645129 + 0.764073i \(0.276803\pi\)
\(500\) 63.5960 2.84410
\(501\) −3.59247 −0.160500
\(502\) 3.05892 0.136526
\(503\) 8.67609 0.386848 0.193424 0.981115i \(-0.438041\pi\)
0.193424 + 0.981115i \(0.438041\pi\)
\(504\) 11.0192 0.490832
\(505\) −18.4394 −0.820541
\(506\) 32.0766 1.42598
\(507\) −1.00000 −0.0444116
\(508\) −4.41110 −0.195711
\(509\) −9.28917 −0.411735 −0.205868 0.978580i \(-0.566002\pi\)
−0.205868 + 0.978580i \(0.566002\pi\)
\(510\) −5.90225 −0.261356
\(511\) −4.07306 −0.180181
\(512\) −189.072 −8.35587
\(513\) 6.33804 0.279831
\(514\) 46.7527 2.06217
\(515\) 25.3522 1.11715
\(516\) 29.0872 1.28049
\(517\) −4.55721 −0.200426
\(518\) 9.62721 0.422995
\(519\) 15.7250 0.690249
\(520\) −14.2056 −0.622955
\(521\) −11.7250 −0.513680 −0.256840 0.966454i \(-0.582681\pi\)
−0.256840 + 0.966454i \(0.582681\pi\)
\(522\) 12.2056 0.534222
\(523\) 0.676089 0.0295633 0.0147817 0.999891i \(-0.495295\pi\)
0.0147817 + 0.999891i \(0.495295\pi\)
\(524\) −85.8399 −3.74993
\(525\) −3.33804 −0.145684
\(526\) 76.1149 3.31877
\(527\) −12.1955 −0.531244
\(528\) −80.6238 −3.50870
\(529\) −15.6514 −0.680495
\(530\) 6.38283 0.277253
\(531\) −4.57834 −0.198683
\(532\) 37.4983 1.62576
\(533\) 7.62721 0.330371
\(534\) 12.7839 0.553213
\(535\) −2.84333 −0.122928
\(536\) −95.6032 −4.12943
\(537\) −24.9547 −1.07687
\(538\) −21.1849 −0.913348
\(539\) 4.20555 0.181146
\(540\) 7.62721 0.328223
\(541\) −0.578337 −0.0248647 −0.0124323 0.999923i \(-0.503957\pi\)
−0.0124323 + 0.999923i \(0.503957\pi\)
\(542\) −47.3311 −2.03304
\(543\) −10.9411 −0.469527
\(544\) −51.9094 −2.22560
\(545\) −12.8917 −0.552219
\(546\) 2.81361 0.120411
\(547\) −0.985867 −0.0421526 −0.0210763 0.999778i \(-0.506709\pi\)
−0.0210763 + 0.999778i \(0.506709\pi\)
\(548\) 26.0978 1.11484
\(549\) −5.25443 −0.224253
\(550\) 39.4983 1.68421
\(551\) 27.4947 1.17131
\(552\) −29.8711 −1.27140
\(553\) 8.91638 0.379163
\(554\) −49.2233 −2.09130
\(555\) 4.41110 0.187241
\(556\) 112.349 4.76465
\(557\) 41.2333 1.74711 0.873556 0.486725i \(-0.161808\pi\)
0.873556 + 0.486725i \(0.161808\pi\)
\(558\) 21.0872 0.892692
\(559\) 4.91638 0.207941
\(560\) 24.7144 1.04437
\(561\) −6.84333 −0.288925
\(562\) −77.0943 −3.25203
\(563\) 6.91995 0.291641 0.145821 0.989311i \(-0.453418\pi\)
0.145821 + 0.989311i \(0.453418\pi\)
\(564\) 6.41110 0.269956
\(565\) 2.79854 0.117735
\(566\) −56.4877 −2.37436
\(567\) −1.00000 −0.0419961
\(568\) 74.7527 3.13655
\(569\) −29.5713 −1.23970 −0.619848 0.784722i \(-0.712806\pi\)
−0.619848 + 0.784722i \(0.712806\pi\)
\(570\) 22.9894 0.962922
\(571\) −19.4252 −0.812921 −0.406460 0.913668i \(-0.633237\pi\)
−0.406460 + 0.913668i \(0.633237\pi\)
\(572\) −24.8816 −1.04035
\(573\) −5.04888 −0.210920
\(574\) −21.4600 −0.895722
\(575\) 9.04888 0.377364
\(576\) 51.4147 2.14228
\(577\) −38.8222 −1.61619 −0.808095 0.589053i \(-0.799501\pi\)
−0.808095 + 0.589053i \(0.799501\pi\)
\(578\) 40.3814 1.67964
\(579\) −2.74557 −0.114102
\(580\) 33.0872 1.37387
\(581\) 4.33804 0.179972
\(582\) 31.8711 1.32110
\(583\) 7.40054 0.306499
\(584\) −44.8816 −1.85722
\(585\) 1.28917 0.0533006
\(586\) 76.1149 3.14428
\(587\) 24.0731 0.993601 0.496801 0.867865i \(-0.334508\pi\)
0.496801 + 0.867865i \(0.334508\pi\)
\(588\) −5.91638 −0.243987
\(589\) 47.5019 1.95728
\(590\) −16.6066 −0.683683
\(591\) −5.42166 −0.223017
\(592\) 65.5960 2.69598
\(593\) 40.4741 1.66207 0.831036 0.556218i \(-0.187748\pi\)
0.831036 + 0.556218i \(0.187748\pi\)
\(594\) 11.8328 0.485504
\(595\) 2.09775 0.0859994
\(596\) −55.7422 −2.28329
\(597\) 20.4111 0.835371
\(598\) −7.62721 −0.311900
\(599\) −2.49523 −0.101953 −0.0509763 0.998700i \(-0.516233\pi\)
−0.0509763 + 0.998700i \(0.516233\pi\)
\(600\) −36.7824 −1.50164
\(601\) 35.2333 1.43720 0.718598 0.695426i \(-0.244784\pi\)
0.718598 + 0.695426i \(0.244784\pi\)
\(602\) −13.8328 −0.563781
\(603\) 8.67609 0.353318
\(604\) −97.0943 −3.95071
\(605\) −8.62022 −0.350462
\(606\) 40.2439 1.63480
\(607\) 4.89169 0.198547 0.0992737 0.995060i \(-0.468348\pi\)
0.0992737 + 0.995060i \(0.468348\pi\)
\(608\) 202.189 8.19983
\(609\) −4.33804 −0.175786
\(610\) −19.0589 −0.771673
\(611\) 1.08362 0.0438385
\(612\) 9.62721 0.389157
\(613\) 7.83276 0.316362 0.158181 0.987410i \(-0.449437\pi\)
0.158181 + 0.987410i \(0.449437\pi\)
\(614\) 56.7527 2.29035
\(615\) −9.83276 −0.396495
\(616\) 46.3416 1.86716
\(617\) 15.7350 0.633468 0.316734 0.948514i \(-0.397414\pi\)
0.316734 + 0.948514i \(0.397414\pi\)
\(618\) −55.3311 −2.22574
\(619\) −41.0177 −1.64864 −0.824320 0.566124i \(-0.808442\pi\)
−0.824320 + 0.566124i \(0.808442\pi\)
\(620\) 57.1638 2.29575
\(621\) 2.71083 0.108782
\(622\) 18.5089 0.742137
\(623\) −4.54359 −0.182035
\(624\) 19.1708 0.767447
\(625\) 2.83276 0.113311
\(626\) 30.2338 1.20839
\(627\) 26.6550 1.06450
\(628\) −56.3205 −2.24743
\(629\) 5.56777 0.222002
\(630\) −3.62721 −0.144512
\(631\) −19.5960 −0.780106 −0.390053 0.920792i \(-0.627543\pi\)
−0.390053 + 0.920792i \(0.627543\pi\)
\(632\) 98.2510 3.90822
\(633\) 14.4842 0.575694
\(634\) −69.2333 −2.74961
\(635\) 0.961171 0.0381429
\(636\) −10.4111 −0.412827
\(637\) −1.00000 −0.0396214
\(638\) 51.3311 2.03222
\(639\) −6.78389 −0.268366
\(640\) 104.241 4.12049
\(641\) −20.8953 −0.825313 −0.412656 0.910887i \(-0.635399\pi\)
−0.412656 + 0.910887i \(0.635399\pi\)
\(642\) 6.20555 0.244914
\(643\) −9.15667 −0.361104 −0.180552 0.983565i \(-0.557788\pi\)
−0.180552 + 0.983565i \(0.557788\pi\)
\(644\) 16.0383 0.631998
\(645\) −6.33804 −0.249560
\(646\) 29.0177 1.14169
\(647\) −6.24386 −0.245472 −0.122736 0.992439i \(-0.539167\pi\)
−0.122736 + 0.992439i \(0.539167\pi\)
\(648\) −11.0192 −0.432873
\(649\) −19.2544 −0.755802
\(650\) −9.39194 −0.368382
\(651\) −7.49472 −0.293741
\(652\) −4.41110 −0.172752
\(653\) −38.8222 −1.51923 −0.759615 0.650373i \(-0.774613\pi\)
−0.759615 + 0.650373i \(0.774613\pi\)
\(654\) 28.1361 1.10021
\(655\) 18.7044 0.730840
\(656\) −146.220 −5.70893
\(657\) 4.07306 0.158905
\(658\) −3.04888 −0.118858
\(659\) −0.954695 −0.0371896 −0.0185948 0.999827i \(-0.505919\pi\)
−0.0185948 + 0.999827i \(0.505919\pi\)
\(660\) 32.0766 1.24858
\(661\) 2.28968 0.0890584 0.0445292 0.999008i \(-0.485821\pi\)
0.0445292 + 0.999008i \(0.485821\pi\)
\(662\) −86.2510 −3.35224
\(663\) 1.62721 0.0631957
\(664\) 47.8016 1.85506
\(665\) −8.17081 −0.316850
\(666\) −9.62721 −0.373047
\(667\) 11.7597 0.455338
\(668\) 21.2544 0.822358
\(669\) −16.8469 −0.651339
\(670\) 31.4700 1.21579
\(671\) −22.0978 −0.853074
\(672\) −31.9008 −1.23060
\(673\) −44.6797 −1.72227 −0.861137 0.508373i \(-0.830247\pi\)
−0.861137 + 0.508373i \(0.830247\pi\)
\(674\) 18.3033 0.705017
\(675\) 3.33804 0.128481
\(676\) 5.91638 0.227553
\(677\) −37.0278 −1.42309 −0.711546 0.702639i \(-0.752005\pi\)
−0.711546 + 0.702639i \(0.752005\pi\)
\(678\) −6.10780 −0.234569
\(679\) −11.3275 −0.434709
\(680\) 23.1155 0.886437
\(681\) 21.9305 0.840379
\(682\) 88.6832 3.39586
\(683\) −26.8605 −1.02779 −0.513894 0.857853i \(-0.671798\pi\)
−0.513894 + 0.857853i \(0.671798\pi\)
\(684\) −37.4983 −1.43378
\(685\) −5.68665 −0.217276
\(686\) 2.81361 0.107424
\(687\) 5.25443 0.200469
\(688\) −94.2510 −3.59329
\(689\) −1.75971 −0.0670395
\(690\) 9.83276 0.374327
\(691\) 2.86802 0.109105 0.0545523 0.998511i \(-0.482627\pi\)
0.0545523 + 0.998511i \(0.482627\pi\)
\(692\) −93.0349 −3.53666
\(693\) −4.20555 −0.159756
\(694\) −68.3799 −2.59567
\(695\) −24.4806 −0.928602
\(696\) −47.8016 −1.81191
\(697\) −12.4111 −0.470104
\(698\) −49.9688 −1.89135
\(699\) −4.07306 −0.154057
\(700\) 19.7491 0.746448
\(701\) −42.9930 −1.62382 −0.811912 0.583780i \(-0.801573\pi\)
−0.811912 + 0.583780i \(0.801573\pi\)
\(702\) −2.81361 −0.106193
\(703\) −21.6867 −0.817928
\(704\) 216.227 8.14936
\(705\) −1.39697 −0.0526128
\(706\) −5.45998 −0.205489
\(707\) −14.3033 −0.537931
\(708\) 27.0872 1.01800
\(709\) 3.49115 0.131113 0.0655564 0.997849i \(-0.479118\pi\)
0.0655564 + 0.997849i \(0.479118\pi\)
\(710\) −24.6066 −0.923469
\(711\) −8.91638 −0.334390
\(712\) −50.0666 −1.87632
\(713\) 20.3169 0.760875
\(714\) −4.57834 −0.171340
\(715\) 5.42166 0.202759
\(716\) 147.641 5.51762
\(717\) 2.37279 0.0886134
\(718\) 94.6137 3.53095
\(719\) 27.5194 1.02630 0.513150 0.858299i \(-0.328478\pi\)
0.513150 + 0.858299i \(0.328478\pi\)
\(720\) −24.7144 −0.921051
\(721\) 19.6655 0.732382
\(722\) −59.5663 −2.21683
\(723\) −14.9164 −0.554746
\(724\) 64.7316 2.40573
\(725\) 14.4806 0.537795
\(726\) 18.8136 0.698238
\(727\) 13.4983 0.500624 0.250312 0.968165i \(-0.419467\pi\)
0.250312 + 0.968165i \(0.419467\pi\)
\(728\) −11.0192 −0.408397
\(729\) 1.00000 0.0370370
\(730\) 14.7738 0.546804
\(731\) −8.00000 −0.295891
\(732\) 31.0872 1.14902
\(733\) −5.08362 −0.187768 −0.0938839 0.995583i \(-0.529928\pi\)
−0.0938839 + 0.995583i \(0.529928\pi\)
\(734\) 29.7633 1.09858
\(735\) 1.28917 0.0475517
\(736\) 86.4777 3.18761
\(737\) 36.4877 1.34404
\(738\) 21.4600 0.789953
\(739\) −53.1638 −1.95566 −0.977831 0.209394i \(-0.932851\pi\)
−0.977831 + 0.209394i \(0.932851\pi\)
\(740\) −26.0978 −0.959372
\(741\) −6.33804 −0.232834
\(742\) 4.95112 0.181761
\(743\) −46.1149 −1.69179 −0.845897 0.533347i \(-0.820934\pi\)
−0.845897 + 0.533347i \(0.820934\pi\)
\(744\) −82.5855 −3.02773
\(745\) 12.1461 0.444999
\(746\) 15.3905 0.563486
\(747\) −4.33804 −0.158721
\(748\) 40.4877 1.48038
\(749\) −2.20555 −0.0805890
\(750\) 30.2439 1.10435
\(751\) −32.7774 −1.19606 −0.598032 0.801472i \(-0.704051\pi\)
−0.598032 + 0.801472i \(0.704051\pi\)
\(752\) −20.7738 −0.757544
\(753\) 1.08719 0.0396194
\(754\) −12.2056 −0.444500
\(755\) 21.1567 0.769970
\(756\) 5.91638 0.215177
\(757\) −30.1708 −1.09658 −0.548288 0.836289i \(-0.684720\pi\)
−0.548288 + 0.836289i \(0.684720\pi\)
\(758\) 15.2544 0.554066
\(759\) 11.4005 0.413813
\(760\) −90.0354 −3.26593
\(761\) −1.81915 −0.0659440 −0.0329720 0.999456i \(-0.510497\pi\)
−0.0329720 + 0.999456i \(0.510497\pi\)
\(762\) −2.09775 −0.0759935
\(763\) −10.0000 −0.362024
\(764\) 29.8711 1.08070
\(765\) −2.09775 −0.0758444
\(766\) −39.8016 −1.43809
\(767\) 4.57834 0.165314
\(768\) −124.676 −4.49887
\(769\) −0.937507 −0.0338074 −0.0169037 0.999857i \(-0.505381\pi\)
−0.0169037 + 0.999857i \(0.505381\pi\)
\(770\) −15.2544 −0.549731
\(771\) 16.6167 0.598434
\(772\) 16.2439 0.584629
\(773\) −8.03831 −0.289118 −0.144559 0.989496i \(-0.546176\pi\)
−0.144559 + 0.989496i \(0.546176\pi\)
\(774\) 13.8328 0.497208
\(775\) 25.0177 0.898662
\(776\) −124.819 −4.48075
\(777\) 3.42166 0.122751
\(778\) −0.881639 −0.0316083
\(779\) 48.3416 1.73202
\(780\) −7.62721 −0.273098
\(781\) −28.5300 −1.02088
\(782\) 12.4111 0.443820
\(783\) 4.33804 0.155029
\(784\) 19.1708 0.684672
\(785\) 12.2721 0.438011
\(786\) −40.8222 −1.45608
\(787\) 25.3275 0.902827 0.451414 0.892315i \(-0.350920\pi\)
0.451414 + 0.892315i \(0.350920\pi\)
\(788\) 32.0766 1.14268
\(789\) 27.0524 0.963093
\(790\) −32.3416 −1.15066
\(791\) 2.17081 0.0771850
\(792\) −46.3416 −1.64668
\(793\) 5.25443 0.186590
\(794\) −58.1844 −2.06489
\(795\) 2.26856 0.0804575
\(796\) −120.760 −4.28022
\(797\) 40.8122 1.44564 0.722820 0.691036i \(-0.242845\pi\)
0.722820 + 0.691036i \(0.242845\pi\)
\(798\) 17.8328 0.631273
\(799\) −1.76328 −0.0623803
\(800\) 106.486 3.76486
\(801\) 4.54359 0.160540
\(802\) 35.8610 1.26630
\(803\) 17.1294 0.604485
\(804\) −51.3311 −1.81031
\(805\) −3.49472 −0.123173
\(806\) −21.0872 −0.742765
\(807\) −7.52946 −0.265050
\(808\) −157.610 −5.54471
\(809\) 25.4947 0.896347 0.448173 0.893947i \(-0.352075\pi\)
0.448173 + 0.893947i \(0.352075\pi\)
\(810\) 3.62721 0.127447
\(811\) −13.1567 −0.461993 −0.230997 0.972955i \(-0.574199\pi\)
−0.230997 + 0.972955i \(0.574199\pi\)
\(812\) 25.6655 0.900683
\(813\) −16.8222 −0.589980
\(814\) −40.4877 −1.41909
\(815\) 0.961171 0.0336683
\(816\) −31.1950 −1.09204
\(817\) 31.1602 1.09016
\(818\) 52.4777 1.83484
\(819\) 1.00000 0.0349428
\(820\) 58.1744 2.03154
\(821\) −23.1083 −0.806486 −0.403243 0.915093i \(-0.632117\pi\)
−0.403243 + 0.915093i \(0.632117\pi\)
\(822\) 12.4111 0.432887
\(823\) −17.4911 −0.609703 −0.304852 0.952400i \(-0.598607\pi\)
−0.304852 + 0.952400i \(0.598607\pi\)
\(824\) 216.698 7.54902
\(825\) 14.0383 0.488751
\(826\) −12.8816 −0.448210
\(827\) 39.3905 1.36974 0.684871 0.728665i \(-0.259859\pi\)
0.684871 + 0.728665i \(0.259859\pi\)
\(828\) −16.0383 −0.557370
\(829\) 10.8222 0.375871 0.187935 0.982181i \(-0.439820\pi\)
0.187935 + 0.982181i \(0.439820\pi\)
\(830\) −15.7350 −0.546170
\(831\) −17.4947 −0.606885
\(832\) −51.4147 −1.78248
\(833\) 1.62721 0.0563796
\(834\) 53.4288 1.85009
\(835\) −4.63130 −0.160273
\(836\) −157.701 −5.45420
\(837\) 7.49472 0.259055
\(838\) −8.74557 −0.302111
\(839\) 4.57834 0.158062 0.0790309 0.996872i \(-0.474817\pi\)
0.0790309 + 0.996872i \(0.474817\pi\)
\(840\) 14.2056 0.490138
\(841\) −10.1814 −0.351082
\(842\) −37.7038 −1.29936
\(843\) −27.4005 −0.943725
\(844\) −85.6938 −2.94970
\(845\) −1.28917 −0.0443487
\(846\) 3.04888 0.104823
\(847\) −6.68665 −0.229756
\(848\) 33.7350 1.15847
\(849\) −20.0766 −0.689028
\(850\) 15.2827 0.524192
\(851\) −9.27555 −0.317962
\(852\) 40.1361 1.37504
\(853\) −54.6585 −1.87147 −0.935736 0.352700i \(-0.885263\pi\)
−0.935736 + 0.352700i \(0.885263\pi\)
\(854\) −14.7839 −0.505894
\(855\) 8.17081 0.279436
\(856\) −24.3033 −0.830670
\(857\) 8.20555 0.280296 0.140148 0.990131i \(-0.455242\pi\)
0.140148 + 0.990131i \(0.455242\pi\)
\(858\) −11.8328 −0.403964
\(859\) −3.20503 −0.109354 −0.0546772 0.998504i \(-0.517413\pi\)
−0.0546772 + 0.998504i \(0.517413\pi\)
\(860\) 37.4983 1.27868
\(861\) −7.62721 −0.259935
\(862\) −27.8328 −0.947988
\(863\) −21.7038 −0.738807 −0.369404 0.929269i \(-0.620438\pi\)
−0.369404 + 0.929269i \(0.620438\pi\)
\(864\) 31.9008 1.08529
\(865\) 20.2721 0.689273
\(866\) −1.62721 −0.0552949
\(867\) 14.3522 0.487426
\(868\) 44.3416 1.50505
\(869\) −37.4983 −1.27204
\(870\) 15.7350 0.533467
\(871\) −8.67609 −0.293978
\(872\) −110.192 −3.73156
\(873\) 11.3275 0.383377
\(874\) −48.3416 −1.63518
\(875\) −10.7491 −0.363387
\(876\) −24.0978 −0.814188
\(877\) −38.5371 −1.30131 −0.650653 0.759375i \(-0.725505\pi\)
−0.650653 + 0.759375i \(0.725505\pi\)
\(878\) −66.7810 −2.25375
\(879\) 27.0524 0.912456
\(880\) −103.938 −3.50374
\(881\) −20.2056 −0.680742 −0.340371 0.940291i \(-0.610553\pi\)
−0.340371 + 0.940291i \(0.610553\pi\)
\(882\) −2.81361 −0.0947391
\(883\) 5.49115 0.184792 0.0923959 0.995722i \(-0.470547\pi\)
0.0923959 + 0.995722i \(0.470547\pi\)
\(884\) −9.62721 −0.323798
\(885\) −5.90225 −0.198402
\(886\) 10.8816 0.365576
\(887\) 45.7633 1.53658 0.768290 0.640102i \(-0.221108\pi\)
0.768290 + 0.640102i \(0.221108\pi\)
\(888\) 37.7038 1.26526
\(889\) 0.745574 0.0250057
\(890\) 16.4806 0.552430
\(891\) 4.20555 0.140891
\(892\) 99.6727 3.33729
\(893\) 6.86802 0.229830
\(894\) −26.5089 −0.886589
\(895\) −32.1708 −1.07535
\(896\) 80.8591 2.70131
\(897\) −2.71083 −0.0905120
\(898\) −4.74557 −0.158362
\(899\) 32.5124 1.08435
\(900\) −19.7491 −0.658305
\(901\) 2.86342 0.0953943
\(902\) 90.2510 3.00503
\(903\) −4.91638 −0.163607
\(904\) 23.9205 0.795583
\(905\) −14.1049 −0.468863
\(906\) −46.1744 −1.53404
\(907\) 6.67252 0.221557 0.110779 0.993845i \(-0.464666\pi\)
0.110779 + 0.993845i \(0.464666\pi\)
\(908\) −129.749 −4.30588
\(909\) 14.3033 0.474411
\(910\) 3.62721 0.120241
\(911\) 23.5330 0.779684 0.389842 0.920882i \(-0.372530\pi\)
0.389842 + 0.920882i \(0.372530\pi\)
\(912\) 121.505 4.02345
\(913\) −18.2439 −0.603784
\(914\) 10.7839 0.356699
\(915\) −6.77384 −0.223936
\(916\) −31.0872 −1.02715
\(917\) 14.5089 0.479125
\(918\) 4.57834 0.151108
\(919\) −28.1955 −0.930084 −0.465042 0.885289i \(-0.653961\pi\)
−0.465042 + 0.885289i \(0.653961\pi\)
\(920\) −38.5089 −1.26960
\(921\) 20.1708 0.664651
\(922\) 31.5577 1.03930
\(923\) 6.78389 0.223294
\(924\) 24.8816 0.818546
\(925\) −11.4217 −0.375542
\(926\) 72.6832 2.38852
\(927\) −19.6655 −0.645901
\(928\) 138.387 4.54278
\(929\) −6.97582 −0.228869 −0.114435 0.993431i \(-0.536506\pi\)
−0.114435 + 0.993431i \(0.536506\pi\)
\(930\) 27.1849 0.891429
\(931\) −6.33804 −0.207721
\(932\) 24.0978 0.789348
\(933\) 6.57834 0.215365
\(934\) −115.603 −3.78265
\(935\) −8.82220 −0.288517
\(936\) 11.0192 0.360172
\(937\) 45.5960 1.48956 0.744779 0.667311i \(-0.232555\pi\)
0.744779 + 0.667311i \(0.232555\pi\)
\(938\) 24.4111 0.797051
\(939\) 10.7456 0.350669
\(940\) 8.26499 0.269574
\(941\) −52.8086 −1.72151 −0.860755 0.509019i \(-0.830008\pi\)
−0.860755 + 0.509019i \(0.830008\pi\)
\(942\) −26.7839 −0.872666
\(943\) 20.6761 0.673306
\(944\) −87.7704 −2.85668
\(945\) −1.28917 −0.0419367
\(946\) 58.1744 1.89141
\(947\) 8.73553 0.283867 0.141933 0.989876i \(-0.454668\pi\)
0.141933 + 0.989876i \(0.454668\pi\)
\(948\) 52.7527 1.71333
\(949\) −4.07306 −0.132217
\(950\) −59.5266 −1.93130
\(951\) −24.6066 −0.797924
\(952\) 17.9305 0.581131
\(953\) 56.0036 1.81413 0.907067 0.420987i \(-0.138316\pi\)
0.907067 + 0.420987i \(0.138316\pi\)
\(954\) −4.95112 −0.160299
\(955\) −6.50885 −0.210622
\(956\) −14.0383 −0.454031
\(957\) 18.2439 0.589740
\(958\) −6.85337 −0.221422
\(959\) −4.41110 −0.142442
\(960\) 66.2822 2.13925
\(961\) 25.1708 0.811962
\(962\) 9.62721 0.310394
\(963\) 2.20555 0.0710729
\(964\) 88.2510 2.84237
\(965\) −3.53951 −0.113941
\(966\) 7.62721 0.245402
\(967\) 29.9094 0.961821 0.480911 0.876770i \(-0.340306\pi\)
0.480911 + 0.876770i \(0.340306\pi\)
\(968\) −73.6813 −2.36821
\(969\) 10.3133 0.331312
\(970\) 41.0872 1.31923
\(971\) 11.5194 0.369676 0.184838 0.982769i \(-0.440824\pi\)
0.184838 + 0.982769i \(0.440824\pi\)
\(972\) −5.91638 −0.189768
\(973\) −18.9894 −0.608773
\(974\) 18.5089 0.593062
\(975\) −3.33804 −0.106903
\(976\) −100.732 −3.22434
\(977\) −36.4182 −1.16512 −0.582561 0.812787i \(-0.697949\pi\)
−0.582561 + 0.812787i \(0.697949\pi\)
\(978\) −2.09775 −0.0670787
\(979\) 19.1083 0.610704
\(980\) −7.62721 −0.243642
\(981\) 10.0000 0.319275
\(982\) −79.6344 −2.54123
\(983\) 17.6902 0.564230 0.282115 0.959381i \(-0.408964\pi\)
0.282115 + 0.959381i \(0.408964\pi\)
\(984\) −84.0455 −2.67927
\(985\) −6.98944 −0.222702
\(986\) 19.8610 0.632504
\(987\) −1.08362 −0.0344920
\(988\) 37.4983 1.19298
\(989\) 13.3275 0.423789
\(990\) 15.2544 0.484817
\(991\) −11.0589 −0.351298 −0.175649 0.984453i \(-0.556202\pi\)
−0.175649 + 0.984453i \(0.556202\pi\)
\(992\) 239.087 7.59104
\(993\) −30.6550 −0.972806
\(994\) −19.0872 −0.605409
\(995\) 26.3133 0.834189
\(996\) 25.6655 0.813243
\(997\) −49.6727 −1.57315 −0.786575 0.617495i \(-0.788147\pi\)
−0.786575 + 0.617495i \(0.788147\pi\)
\(998\) −81.0943 −2.56700
\(999\) −3.42166 −0.108257
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 273.2.a.d.1.1 3
3.2 odd 2 819.2.a.j.1.3 3
4.3 odd 2 4368.2.a.bq.1.2 3
5.4 even 2 6825.2.a.bd.1.3 3
7.6 odd 2 1911.2.a.n.1.1 3
13.12 even 2 3549.2.a.t.1.3 3
21.20 even 2 5733.2.a.bc.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.d.1.1 3 1.1 even 1 trivial
819.2.a.j.1.3 3 3.2 odd 2
1911.2.a.n.1.1 3 7.6 odd 2
3549.2.a.t.1.3 3 13.12 even 2
4368.2.a.bq.1.2 3 4.3 odd 2
5733.2.a.bc.1.3 3 21.20 even 2
6825.2.a.bd.1.3 3 5.4 even 2