Properties

Label 209.2.a.c.1.2
Level $209$
Weight $2$
Character 209.1
Self dual yes
Analytic conductor $1.669$
Analytic rank $0$
Dimension $5$
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] = [209,2,Mod(1,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("209.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.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: \(1.66887340224\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.71250\) of defining polynomial
Character \(\chi\) \(=\) 209.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.779856 q^{2} -2.98063 q^{3} -1.39182 q^{4} -3.49235 q^{5} +2.32446 q^{6} +1.06736 q^{7} +2.64513 q^{8} +5.88418 q^{9} +O(q^{10})\) \(q-0.779856 q^{2} -2.98063 q^{3} -1.39182 q^{4} -3.49235 q^{5} +2.32446 q^{6} +1.06736 q^{7} +2.64513 q^{8} +5.88418 q^{9} +2.72353 q^{10} +1.00000 q^{11} +4.14852 q^{12} -0.0563258 q^{13} -0.832387 q^{14} +10.4094 q^{15} +0.720827 q^{16} -4.53628 q^{17} -4.58881 q^{18} -1.00000 q^{19} +4.86074 q^{20} -3.18141 q^{21} -0.779856 q^{22} -1.07949 q^{23} -7.88418 q^{24} +7.19651 q^{25} +0.0439260 q^{26} -8.59667 q^{27} -1.48558 q^{28} +0.299905 q^{29} -8.11784 q^{30} +9.18548 q^{31} -5.85241 q^{32} -2.98063 q^{33} +3.53764 q^{34} -3.72760 q^{35} -8.18974 q^{36} +4.50448 q^{37} +0.779856 q^{38} +0.167887 q^{39} -9.23774 q^{40} +12.0009 q^{41} +2.48104 q^{42} +10.7260 q^{43} -1.39182 q^{44} -20.5496 q^{45} +0.841844 q^{46} +2.89630 q^{47} -2.14852 q^{48} -5.86074 q^{49} -5.61224 q^{50} +13.5210 q^{51} +0.0783957 q^{52} -12.3213 q^{53} +6.70416 q^{54} -3.49235 q^{55} +2.82331 q^{56} +2.98063 q^{57} -0.233882 q^{58} -1.14582 q^{59} -14.4881 q^{60} -8.09599 q^{61} -7.16335 q^{62} +6.28054 q^{63} +3.12238 q^{64} +0.196709 q^{65} +2.32446 q^{66} +11.2733 q^{67} +6.31370 q^{68} +3.21755 q^{69} +2.90699 q^{70} +13.4948 q^{71} +15.5644 q^{72} -11.1470 q^{73} -3.51284 q^{74} -21.4502 q^{75} +1.39182 q^{76} +1.06736 q^{77} -0.130927 q^{78} +11.4250 q^{79} -2.51738 q^{80} +7.97100 q^{81} -9.35899 q^{82} -13.9802 q^{83} +4.42797 q^{84} +15.8423 q^{85} -8.36475 q^{86} -0.893906 q^{87} +2.64513 q^{88} -0.183185 q^{89} +16.0257 q^{90} -0.0601200 q^{91} +1.50246 q^{92} -27.3785 q^{93} -2.25870 q^{94} +3.49235 q^{95} +17.4439 q^{96} +5.66263 q^{97} +4.57053 q^{98} +5.88418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{8} + 4 q^{9} + 12 q^{10} + 5 q^{11} + 6 q^{12} + 4 q^{13} - 14 q^{14} + 3 q^{15} + 8 q^{16} - 4 q^{17} - 20 q^{18} - 5 q^{19} - 8 q^{20} + 10 q^{21} + 2 q^{22} + 3 q^{23} - 14 q^{24} + 6 q^{25} - 6 q^{26} - 11 q^{27} - 10 q^{28} + 10 q^{29} + 6 q^{30} + 11 q^{31} + 14 q^{32} + q^{33} - 4 q^{34} - 8 q^{35} - 26 q^{36} + q^{37} - 2 q^{38} + 2 q^{39} - 16 q^{40} + 2 q^{41} - 16 q^{42} + 20 q^{43} + 6 q^{44} - 28 q^{45} - 4 q^{46} - 20 q^{47} + 4 q^{48} + 3 q^{49} - 32 q^{50} + 24 q^{51} + 6 q^{52} - 14 q^{53} + 16 q^{54} - 5 q^{55} - 38 q^{56} - q^{57} - 6 q^{58} + 3 q^{59} - 40 q^{60} - 10 q^{61} - 6 q^{62} + 24 q^{63} - 2 q^{66} + 9 q^{67} + 24 q^{68} - 5 q^{69} + 50 q^{70} + 23 q^{71} - 12 q^{72} + 8 q^{74} - 18 q^{75} - 6 q^{76} + 6 q^{77} - 22 q^{78} + 44 q^{79} - 18 q^{80} + q^{81} - 30 q^{82} - 14 q^{83} + 14 q^{84} - 12 q^{85} + 52 q^{86} + 28 q^{87} + 6 q^{88} - 27 q^{89} + 26 q^{90} + 24 q^{91} + 58 q^{92} - 27 q^{93} - 8 q^{94} + 5 q^{95} + 50 q^{96} + 15 q^{97} - 10 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.779856 −0.551441 −0.275721 0.961238i \(-0.588916\pi\)
−0.275721 + 0.961238i \(0.588916\pi\)
\(3\) −2.98063 −1.72087 −0.860435 0.509561i \(-0.829808\pi\)
−0.860435 + 0.509561i \(0.829808\pi\)
\(4\) −1.39182 −0.695912
\(5\) −3.49235 −1.56183 −0.780913 0.624639i \(-0.785246\pi\)
−0.780913 + 0.624639i \(0.785246\pi\)
\(6\) 2.32446 0.948959
\(7\) 1.06736 0.403424 0.201712 0.979445i \(-0.435349\pi\)
0.201712 + 0.979445i \(0.435349\pi\)
\(8\) 2.64513 0.935196
\(9\) 5.88418 1.96139
\(10\) 2.72353 0.861256
\(11\) 1.00000 0.301511
\(12\) 4.14852 1.19757
\(13\) −0.0563258 −0.0156220 −0.00781098 0.999969i \(-0.502486\pi\)
−0.00781098 + 0.999969i \(0.502486\pi\)
\(14\) −0.832387 −0.222465
\(15\) 10.4094 2.68770
\(16\) 0.720827 0.180207
\(17\) −4.53628 −1.10021 −0.550104 0.835096i \(-0.685412\pi\)
−0.550104 + 0.835096i \(0.685412\pi\)
\(18\) −4.58881 −1.08159
\(19\) −1.00000 −0.229416
\(20\) 4.86074 1.08689
\(21\) −3.18141 −0.694241
\(22\) −0.779856 −0.166266
\(23\) −1.07949 −0.225088 −0.112544 0.993647i \(-0.535900\pi\)
−0.112544 + 0.993647i \(0.535900\pi\)
\(24\) −7.88418 −1.60935
\(25\) 7.19651 1.43930
\(26\) 0.0439260 0.00861460
\(27\) −8.59667 −1.65443
\(28\) −1.48558 −0.280748
\(29\) 0.299905 0.0556909 0.0278455 0.999612i \(-0.491135\pi\)
0.0278455 + 0.999612i \(0.491135\pi\)
\(30\) −8.11784 −1.48211
\(31\) 9.18548 1.64976 0.824880 0.565307i \(-0.191242\pi\)
0.824880 + 0.565307i \(0.191242\pi\)
\(32\) −5.85241 −1.03457
\(33\) −2.98063 −0.518862
\(34\) 3.53764 0.606701
\(35\) −3.72760 −0.630079
\(36\) −8.18974 −1.36496
\(37\) 4.50448 0.740531 0.370266 0.928926i \(-0.379267\pi\)
0.370266 + 0.928926i \(0.379267\pi\)
\(38\) 0.779856 0.126509
\(39\) 0.167887 0.0268834
\(40\) −9.23774 −1.46061
\(41\) 12.0009 1.87423 0.937115 0.349020i \(-0.113486\pi\)
0.937115 + 0.349020i \(0.113486\pi\)
\(42\) 2.48104 0.382833
\(43\) 10.7260 1.63570 0.817851 0.575430i \(-0.195165\pi\)
0.817851 + 0.575430i \(0.195165\pi\)
\(44\) −1.39182 −0.209826
\(45\) −20.5496 −3.06335
\(46\) 0.841844 0.124123
\(47\) 2.89630 0.422469 0.211235 0.977435i \(-0.432252\pi\)
0.211235 + 0.977435i \(0.432252\pi\)
\(48\) −2.14852 −0.310112
\(49\) −5.86074 −0.837249
\(50\) −5.61224 −0.793691
\(51\) 13.5210 1.89332
\(52\) 0.0783957 0.0108715
\(53\) −12.3213 −1.69246 −0.846230 0.532818i \(-0.821133\pi\)
−0.846230 + 0.532818i \(0.821133\pi\)
\(54\) 6.70416 0.912321
\(55\) −3.49235 −0.470908
\(56\) 2.82331 0.377281
\(57\) 2.98063 0.394795
\(58\) −0.233882 −0.0307103
\(59\) −1.14582 −0.149173 −0.0745863 0.997215i \(-0.523764\pi\)
−0.0745863 + 0.997215i \(0.523764\pi\)
\(60\) −14.4881 −1.87040
\(61\) −8.09599 −1.03659 −0.518293 0.855203i \(-0.673432\pi\)
−0.518293 + 0.855203i \(0.673432\pi\)
\(62\) −7.16335 −0.909746
\(63\) 6.28054 0.791273
\(64\) 3.12238 0.390298
\(65\) 0.196709 0.0243988
\(66\) 2.32446 0.286122
\(67\) 11.2733 1.37726 0.688628 0.725115i \(-0.258213\pi\)
0.688628 + 0.725115i \(0.258213\pi\)
\(68\) 6.31370 0.765649
\(69\) 3.21755 0.387348
\(70\) 2.90699 0.347452
\(71\) 13.4948 1.60154 0.800771 0.598970i \(-0.204423\pi\)
0.800771 + 0.598970i \(0.204423\pi\)
\(72\) 15.5644 1.83429
\(73\) −11.1470 −1.30466 −0.652330 0.757935i \(-0.726208\pi\)
−0.652330 + 0.757935i \(0.726208\pi\)
\(74\) −3.51284 −0.408360
\(75\) −21.4502 −2.47685
\(76\) 1.39182 0.159653
\(77\) 1.06736 0.121637
\(78\) −0.130927 −0.0148246
\(79\) 11.4250 1.28541 0.642706 0.766113i \(-0.277812\pi\)
0.642706 + 0.766113i \(0.277812\pi\)
\(80\) −2.51738 −0.281452
\(81\) 7.97100 0.885666
\(82\) −9.35899 −1.03353
\(83\) −13.9802 −1.53452 −0.767261 0.641335i \(-0.778381\pi\)
−0.767261 + 0.641335i \(0.778381\pi\)
\(84\) 4.42797 0.483131
\(85\) 15.8423 1.71834
\(86\) −8.36475 −0.901994
\(87\) −0.893906 −0.0958368
\(88\) 2.64513 0.281972
\(89\) −0.183185 −0.0194176 −0.00970878 0.999953i \(-0.503090\pi\)
−0.00970878 + 0.999953i \(0.503090\pi\)
\(90\) 16.0257 1.68926
\(91\) −0.0601200 −0.00630228
\(92\) 1.50246 0.156642
\(93\) −27.3785 −2.83902
\(94\) −2.25870 −0.232967
\(95\) 3.49235 0.358308
\(96\) 17.4439 1.78036
\(97\) 5.66263 0.574953 0.287477 0.957788i \(-0.407184\pi\)
0.287477 + 0.957788i \(0.407184\pi\)
\(98\) 4.57053 0.461694
\(99\) 5.88418 0.591382
\(100\) −10.0163 −1.00163
\(101\) 8.00759 0.796785 0.398392 0.917215i \(-0.369568\pi\)
0.398392 + 0.917215i \(0.369568\pi\)
\(102\) −10.5444 −1.04405
\(103\) 6.18725 0.609648 0.304824 0.952409i \(-0.401402\pi\)
0.304824 + 0.952409i \(0.401402\pi\)
\(104\) −0.148989 −0.0146096
\(105\) 11.1106 1.08428
\(106\) 9.60883 0.933292
\(107\) −7.29027 −0.704777 −0.352388 0.935854i \(-0.614630\pi\)
−0.352388 + 0.935854i \(0.614630\pi\)
\(108\) 11.9651 1.15134
\(109\) 7.79895 0.747004 0.373502 0.927629i \(-0.378157\pi\)
0.373502 + 0.927629i \(0.378157\pi\)
\(110\) 2.72353 0.259678
\(111\) −13.4262 −1.27436
\(112\) 0.769382 0.0726998
\(113\) −0.430558 −0.0405035 −0.0202517 0.999795i \(-0.506447\pi\)
−0.0202517 + 0.999795i \(0.506447\pi\)
\(114\) −2.32446 −0.217706
\(115\) 3.76995 0.351549
\(116\) −0.417415 −0.0387560
\(117\) −0.331431 −0.0306408
\(118\) 0.893572 0.0822600
\(119\) −4.84184 −0.443851
\(120\) 27.5343 2.51353
\(121\) 1.00000 0.0909091
\(122\) 6.31370 0.571616
\(123\) −35.7704 −3.22531
\(124\) −12.7846 −1.14809
\(125\) −7.67100 −0.686115
\(126\) −4.89791 −0.436341
\(127\) −3.13888 −0.278531 −0.139265 0.990255i \(-0.544474\pi\)
−0.139265 + 0.990255i \(0.544474\pi\)
\(128\) 9.26981 0.819343
\(129\) −31.9703 −2.81483
\(130\) −0.153405 −0.0134545
\(131\) 12.4315 1.08615 0.543075 0.839684i \(-0.317260\pi\)
0.543075 + 0.839684i \(0.317260\pi\)
\(132\) 4.14852 0.361082
\(133\) −1.06736 −0.0925519
\(134\) −8.79157 −0.759476
\(135\) 30.0226 2.58393
\(136\) −11.9991 −1.02891
\(137\) 13.8301 1.18159 0.590794 0.806822i \(-0.298814\pi\)
0.590794 + 0.806822i \(0.298814\pi\)
\(138\) −2.50923 −0.213600
\(139\) 15.9968 1.35683 0.678417 0.734677i \(-0.262666\pi\)
0.678417 + 0.734677i \(0.262666\pi\)
\(140\) 5.18816 0.438480
\(141\) −8.63281 −0.727014
\(142\) −10.5240 −0.883157
\(143\) −0.0563258 −0.00471020
\(144\) 4.24147 0.353456
\(145\) −1.04737 −0.0869796
\(146\) 8.69307 0.719443
\(147\) 17.4687 1.44080
\(148\) −6.26944 −0.515345
\(149\) 11.3620 0.930812 0.465406 0.885097i \(-0.345908\pi\)
0.465406 + 0.885097i \(0.345908\pi\)
\(150\) 16.7280 1.36584
\(151\) 4.66341 0.379503 0.189751 0.981832i \(-0.439232\pi\)
0.189751 + 0.981832i \(0.439232\pi\)
\(152\) −2.64513 −0.214549
\(153\) −26.6923 −2.15794
\(154\) −0.832387 −0.0670757
\(155\) −32.0789 −2.57664
\(156\) −0.233669 −0.0187085
\(157\) −9.05206 −0.722433 −0.361217 0.932482i \(-0.617639\pi\)
−0.361217 + 0.932482i \(0.617639\pi\)
\(158\) −8.90984 −0.708829
\(159\) 36.7253 2.91250
\(160\) 20.4387 1.61582
\(161\) −1.15220 −0.0908062
\(162\) −6.21623 −0.488393
\(163\) 2.36761 0.185446 0.0927229 0.995692i \(-0.470443\pi\)
0.0927229 + 0.995692i \(0.470443\pi\)
\(164\) −16.7032 −1.30430
\(165\) 10.4094 0.810372
\(166\) 10.9025 0.846199
\(167\) −9.27361 −0.717613 −0.358807 0.933412i \(-0.616816\pi\)
−0.358807 + 0.933412i \(0.616816\pi\)
\(168\) −8.41526 −0.649251
\(169\) −12.9968 −0.999756
\(170\) −12.3547 −0.947561
\(171\) −5.88418 −0.449974
\(172\) −14.9287 −1.13831
\(173\) 10.7172 0.814812 0.407406 0.913247i \(-0.366433\pi\)
0.407406 + 0.913247i \(0.366433\pi\)
\(174\) 0.697118 0.0528484
\(175\) 7.68128 0.580650
\(176\) 0.720827 0.0543344
\(177\) 3.41526 0.256707
\(178\) 0.142858 0.0107076
\(179\) −22.9070 −1.71215 −0.856073 0.516854i \(-0.827103\pi\)
−0.856073 + 0.516854i \(0.827103\pi\)
\(180\) 28.6015 2.13183
\(181\) 5.90522 0.438931 0.219466 0.975620i \(-0.429569\pi\)
0.219466 + 0.975620i \(0.429569\pi\)
\(182\) 0.0468849 0.00347534
\(183\) 24.1312 1.78383
\(184\) −2.85539 −0.210502
\(185\) −15.7312 −1.15658
\(186\) 21.3513 1.56555
\(187\) −4.53628 −0.331725
\(188\) −4.03114 −0.294001
\(189\) −9.17575 −0.667438
\(190\) −2.72353 −0.197586
\(191\) 6.44628 0.466437 0.233218 0.972424i \(-0.425074\pi\)
0.233218 + 0.972424i \(0.425074\pi\)
\(192\) −9.30668 −0.671652
\(193\) 1.43606 0.103370 0.0516849 0.998663i \(-0.483541\pi\)
0.0516849 + 0.998663i \(0.483541\pi\)
\(194\) −4.41604 −0.317053
\(195\) −0.586319 −0.0419872
\(196\) 8.15713 0.582652
\(197\) −4.75399 −0.338708 −0.169354 0.985555i \(-0.554168\pi\)
−0.169354 + 0.985555i \(0.554168\pi\)
\(198\) −4.58881 −0.326112
\(199\) −2.36002 −0.167298 −0.0836489 0.996495i \(-0.526657\pi\)
−0.0836489 + 0.996495i \(0.526657\pi\)
\(200\) 19.0357 1.34603
\(201\) −33.6017 −2.37008
\(202\) −6.24476 −0.439380
\(203\) 0.320107 0.0224671
\(204\) −18.8188 −1.31758
\(205\) −41.9115 −2.92722
\(206\) −4.82517 −0.336185
\(207\) −6.35189 −0.441487
\(208\) −0.0406011 −0.00281518
\(209\) −1.00000 −0.0691714
\(210\) −8.66467 −0.597919
\(211\) −24.5133 −1.68756 −0.843781 0.536687i \(-0.819676\pi\)
−0.843781 + 0.536687i \(0.819676\pi\)
\(212\) 17.1491 1.17780
\(213\) −40.2232 −2.75605
\(214\) 5.68536 0.388643
\(215\) −37.4590 −2.55468
\(216\) −22.7393 −1.54722
\(217\) 9.80422 0.665554
\(218\) −6.08205 −0.411929
\(219\) 33.2252 2.24515
\(220\) 4.86074 0.327711
\(221\) 0.255509 0.0171874
\(222\) 10.4705 0.702734
\(223\) 8.47427 0.567479 0.283740 0.958901i \(-0.408425\pi\)
0.283740 + 0.958901i \(0.408425\pi\)
\(224\) −6.24663 −0.417371
\(225\) 42.3456 2.82304
\(226\) 0.335773 0.0223353
\(227\) 13.7491 0.912558 0.456279 0.889837i \(-0.349182\pi\)
0.456279 + 0.889837i \(0.349182\pi\)
\(228\) −4.14852 −0.274742
\(229\) −1.99640 −0.131926 −0.0659629 0.997822i \(-0.521012\pi\)
−0.0659629 + 0.997822i \(0.521012\pi\)
\(230\) −2.94001 −0.193859
\(231\) −3.18141 −0.209321
\(232\) 0.793288 0.0520819
\(233\) 11.3481 0.743437 0.371718 0.928346i \(-0.378769\pi\)
0.371718 + 0.928346i \(0.378769\pi\)
\(234\) 0.258468 0.0168966
\(235\) −10.1149 −0.659823
\(236\) 1.59478 0.103811
\(237\) −34.0537 −2.21203
\(238\) 3.77594 0.244758
\(239\) 5.21045 0.337036 0.168518 0.985699i \(-0.446102\pi\)
0.168518 + 0.985699i \(0.446102\pi\)
\(240\) 7.50339 0.484341
\(241\) 6.60827 0.425676 0.212838 0.977087i \(-0.431729\pi\)
0.212838 + 0.977087i \(0.431729\pi\)
\(242\) −0.779856 −0.0501310
\(243\) 2.03139 0.130314
\(244\) 11.2682 0.721373
\(245\) 20.4678 1.30764
\(246\) 27.8957 1.77857
\(247\) 0.0563258 0.00358393
\(248\) 24.2968 1.54285
\(249\) 41.6697 2.64071
\(250\) 5.98227 0.378352
\(251\) 24.0024 1.51502 0.757510 0.652823i \(-0.226415\pi\)
0.757510 + 0.652823i \(0.226415\pi\)
\(252\) −8.74141 −0.550657
\(253\) −1.07949 −0.0678667
\(254\) 2.44788 0.153593
\(255\) −47.2200 −2.95703
\(256\) −13.4739 −0.842117
\(257\) −5.68903 −0.354872 −0.177436 0.984132i \(-0.556780\pi\)
−0.177436 + 0.984132i \(0.556780\pi\)
\(258\) 24.9322 1.55221
\(259\) 4.80790 0.298748
\(260\) −0.273785 −0.0169794
\(261\) 1.76469 0.109232
\(262\) −9.69481 −0.598948
\(263\) 13.9857 0.862395 0.431197 0.902258i \(-0.358091\pi\)
0.431197 + 0.902258i \(0.358091\pi\)
\(264\) −7.88418 −0.485237
\(265\) 43.0303 2.64333
\(266\) 0.832387 0.0510369
\(267\) 0.546007 0.0334151
\(268\) −15.6905 −0.958450
\(269\) 15.4020 0.939075 0.469538 0.882912i \(-0.344421\pi\)
0.469538 + 0.882912i \(0.344421\pi\)
\(270\) −23.4133 −1.42489
\(271\) −25.3911 −1.54240 −0.771199 0.636595i \(-0.780342\pi\)
−0.771199 + 0.636595i \(0.780342\pi\)
\(272\) −3.26987 −0.198265
\(273\) 0.179196 0.0108454
\(274\) −10.7855 −0.651577
\(275\) 7.19651 0.433966
\(276\) −4.47827 −0.269560
\(277\) 19.9798 1.20047 0.600235 0.799824i \(-0.295074\pi\)
0.600235 + 0.799824i \(0.295074\pi\)
\(278\) −12.4752 −0.748214
\(279\) 54.0490 3.23583
\(280\) −9.86000 −0.589248
\(281\) 6.18130 0.368745 0.184373 0.982856i \(-0.440975\pi\)
0.184373 + 0.982856i \(0.440975\pi\)
\(282\) 6.73235 0.400906
\(283\) −7.12127 −0.423316 −0.211658 0.977344i \(-0.567886\pi\)
−0.211658 + 0.977344i \(0.567886\pi\)
\(284\) −18.7825 −1.11453
\(285\) −10.4094 −0.616601
\(286\) 0.0439260 0.00259740
\(287\) 12.8093 0.756110
\(288\) −34.4366 −2.02920
\(289\) 3.57781 0.210459
\(290\) 0.816800 0.0479641
\(291\) −16.8782 −0.989420
\(292\) 15.5147 0.907929
\(293\) 19.1342 1.11783 0.558916 0.829224i \(-0.311217\pi\)
0.558916 + 0.829224i \(0.311217\pi\)
\(294\) −13.6231 −0.794514
\(295\) 4.00159 0.232982
\(296\) 11.9149 0.692542
\(297\) −8.59667 −0.498829
\(298\) −8.86073 −0.513288
\(299\) 0.0608030 0.00351633
\(300\) 29.8549 1.72367
\(301\) 11.4485 0.659882
\(302\) −3.63679 −0.209274
\(303\) −23.8677 −1.37116
\(304\) −0.720827 −0.0413422
\(305\) 28.2740 1.61897
\(306\) 20.8161 1.18998
\(307\) 6.82573 0.389565 0.194782 0.980846i \(-0.437600\pi\)
0.194782 + 0.980846i \(0.437600\pi\)
\(308\) −1.48558 −0.0846487
\(309\) −18.4419 −1.04912
\(310\) 25.0169 1.42087
\(311\) −12.8609 −0.729276 −0.364638 0.931149i \(-0.618807\pi\)
−0.364638 + 0.931149i \(0.618807\pi\)
\(312\) 0.444083 0.0251412
\(313\) −3.01707 −0.170535 −0.0852673 0.996358i \(-0.527174\pi\)
−0.0852673 + 0.996358i \(0.527174\pi\)
\(314\) 7.05930 0.398380
\(315\) −21.9338 −1.23583
\(316\) −15.9016 −0.894534
\(317\) 17.7857 0.998943 0.499471 0.866330i \(-0.333528\pi\)
0.499471 + 0.866330i \(0.333528\pi\)
\(318\) −28.6404 −1.60607
\(319\) 0.299905 0.0167914
\(320\) −10.9045 −0.609577
\(321\) 21.7296 1.21283
\(322\) 0.898551 0.0500743
\(323\) 4.53628 0.252405
\(324\) −11.0942 −0.616346
\(325\) −0.405349 −0.0224847
\(326\) −1.84640 −0.102262
\(327\) −23.2458 −1.28550
\(328\) 31.7441 1.75277
\(329\) 3.09140 0.170434
\(330\) −8.11784 −0.446873
\(331\) −26.3860 −1.45030 −0.725152 0.688589i \(-0.758230\pi\)
−0.725152 + 0.688589i \(0.758230\pi\)
\(332\) 19.4579 1.06789
\(333\) 26.5051 1.45247
\(334\) 7.23207 0.395722
\(335\) −39.3704 −2.15104
\(336\) −2.29325 −0.125107
\(337\) −13.2024 −0.719181 −0.359590 0.933110i \(-0.617084\pi\)
−0.359590 + 0.933110i \(0.617084\pi\)
\(338\) 10.1357 0.551307
\(339\) 1.28334 0.0697012
\(340\) −22.0497 −1.19581
\(341\) 9.18548 0.497422
\(342\) 4.58881 0.248134
\(343\) −13.7271 −0.741191
\(344\) 28.3718 1.52970
\(345\) −11.2368 −0.604970
\(346\) −8.35786 −0.449321
\(347\) 16.4809 0.884740 0.442370 0.896833i \(-0.354138\pi\)
0.442370 + 0.896833i \(0.354138\pi\)
\(348\) 1.24416 0.0666940
\(349\) −19.0165 −1.01793 −0.508966 0.860787i \(-0.669972\pi\)
−0.508966 + 0.860787i \(0.669972\pi\)
\(350\) −5.99029 −0.320194
\(351\) 0.484214 0.0258455
\(352\) −5.85241 −0.311934
\(353\) 13.1955 0.702325 0.351162 0.936315i \(-0.385787\pi\)
0.351162 + 0.936315i \(0.385787\pi\)
\(354\) −2.66341 −0.141559
\(355\) −47.1287 −2.50133
\(356\) 0.254961 0.0135129
\(357\) 14.4318 0.763810
\(358\) 17.8641 0.944148
\(359\) −2.88756 −0.152400 −0.0761998 0.997093i \(-0.524279\pi\)
−0.0761998 + 0.997093i \(0.524279\pi\)
\(360\) −54.3565 −2.86484
\(361\) 1.00000 0.0526316
\(362\) −4.60522 −0.242045
\(363\) −2.98063 −0.156443
\(364\) 0.0836765 0.00438584
\(365\) 38.9293 2.03765
\(366\) −18.8188 −0.983676
\(367\) −2.10833 −0.110054 −0.0550269 0.998485i \(-0.517524\pi\)
−0.0550269 + 0.998485i \(0.517524\pi\)
\(368\) −0.778123 −0.0405624
\(369\) 70.6156 3.67610
\(370\) 12.2681 0.637787
\(371\) −13.1513 −0.682780
\(372\) 38.1061 1.97571
\(373\) 8.07305 0.418007 0.209003 0.977915i \(-0.432978\pi\)
0.209003 + 0.977915i \(0.432978\pi\)
\(374\) 3.53764 0.182927
\(375\) 22.8644 1.18071
\(376\) 7.66111 0.395091
\(377\) −0.0168924 −0.000870002 0
\(378\) 7.15576 0.368053
\(379\) 14.3355 0.736365 0.368183 0.929754i \(-0.379980\pi\)
0.368183 + 0.929754i \(0.379980\pi\)
\(380\) −4.86074 −0.249351
\(381\) 9.35586 0.479315
\(382\) −5.02717 −0.257212
\(383\) −9.79867 −0.500689 −0.250344 0.968157i \(-0.580544\pi\)
−0.250344 + 0.968157i \(0.580544\pi\)
\(384\) −27.6299 −1.40998
\(385\) −3.72760 −0.189976
\(386\) −1.11992 −0.0570024
\(387\) 63.1138 3.20825
\(388\) −7.88139 −0.400117
\(389\) −19.9236 −1.01016 −0.505082 0.863071i \(-0.668538\pi\)
−0.505082 + 0.863071i \(0.668538\pi\)
\(390\) 0.457244 0.0231535
\(391\) 4.89685 0.247644
\(392\) −15.5024 −0.782992
\(393\) −37.0539 −1.86912
\(394\) 3.70743 0.186778
\(395\) −39.9001 −2.00759
\(396\) −8.18974 −0.411550
\(397\) 13.9941 0.702342 0.351171 0.936311i \(-0.385784\pi\)
0.351171 + 0.936311i \(0.385784\pi\)
\(398\) 1.84048 0.0922549
\(399\) 3.18141 0.159270
\(400\) 5.18744 0.259372
\(401\) −18.6293 −0.930301 −0.465151 0.885232i \(-0.654000\pi\)
−0.465151 + 0.885232i \(0.654000\pi\)
\(402\) 26.2045 1.30696
\(403\) −0.517380 −0.0257725
\(404\) −11.1452 −0.554492
\(405\) −27.8375 −1.38326
\(406\) −0.249637 −0.0123893
\(407\) 4.50448 0.223279
\(408\) 35.7648 1.77062
\(409\) −30.8428 −1.52508 −0.762540 0.646941i \(-0.776048\pi\)
−0.762540 + 0.646941i \(0.776048\pi\)
\(410\) 32.6849 1.61419
\(411\) −41.2226 −2.03336
\(412\) −8.61157 −0.424262
\(413\) −1.22300 −0.0601799
\(414\) 4.95356 0.243454
\(415\) 48.8236 2.39666
\(416\) 0.329642 0.0161620
\(417\) −47.6807 −2.33493
\(418\) 0.779856 0.0381440
\(419\) 37.9347 1.85323 0.926616 0.376009i \(-0.122704\pi\)
0.926616 + 0.376009i \(0.122704\pi\)
\(420\) −15.4640 −0.754567
\(421\) −18.4642 −0.899891 −0.449945 0.893056i \(-0.648557\pi\)
−0.449945 + 0.893056i \(0.648557\pi\)
\(422\) 19.1168 0.930592
\(423\) 17.0423 0.828627
\(424\) −32.5915 −1.58278
\(425\) −32.6454 −1.58353
\(426\) 31.3683 1.51980
\(427\) −8.64134 −0.418184
\(428\) 10.1468 0.490463
\(429\) 0.167887 0.00810564
\(430\) 29.2126 1.40876
\(431\) 33.6742 1.62203 0.811013 0.585027i \(-0.198916\pi\)
0.811013 + 0.585027i \(0.198916\pi\)
\(432\) −6.19671 −0.298139
\(433\) 11.4041 0.548047 0.274024 0.961723i \(-0.411645\pi\)
0.274024 + 0.961723i \(0.411645\pi\)
\(434\) −7.64588 −0.367014
\(435\) 3.12183 0.149680
\(436\) −10.8548 −0.519849
\(437\) 1.07949 0.0516388
\(438\) −25.9108 −1.23807
\(439\) −3.76890 −0.179880 −0.0899399 0.995947i \(-0.528667\pi\)
−0.0899399 + 0.995947i \(0.528667\pi\)
\(440\) −9.23774 −0.440392
\(441\) −34.4856 −1.64217
\(442\) −0.199261 −0.00947786
\(443\) 4.51841 0.214676 0.107338 0.994223i \(-0.465767\pi\)
0.107338 + 0.994223i \(0.465767\pi\)
\(444\) 18.6869 0.886842
\(445\) 0.639746 0.0303269
\(446\) −6.60871 −0.312931
\(447\) −33.8660 −1.60181
\(448\) 3.33271 0.157456
\(449\) −37.3611 −1.76318 −0.881590 0.472016i \(-0.843526\pi\)
−0.881590 + 0.472016i \(0.843526\pi\)
\(450\) −33.0234 −1.55674
\(451\) 12.0009 0.565102
\(452\) 0.599261 0.0281869
\(453\) −13.8999 −0.653075
\(454\) −10.7223 −0.503222
\(455\) 0.209960 0.00984308
\(456\) 7.88418 0.369210
\(457\) 9.37773 0.438672 0.219336 0.975649i \(-0.429611\pi\)
0.219336 + 0.975649i \(0.429611\pi\)
\(458\) 1.55690 0.0727494
\(459\) 38.9969 1.82022
\(460\) −5.24710 −0.244648
\(461\) 31.6785 1.47542 0.737708 0.675120i \(-0.235908\pi\)
0.737708 + 0.675120i \(0.235908\pi\)
\(462\) 2.48104 0.115429
\(463\) −37.1284 −1.72550 −0.862752 0.505628i \(-0.831261\pi\)
−0.862752 + 0.505628i \(0.831261\pi\)
\(464\) 0.216179 0.0100359
\(465\) 95.6155 4.43406
\(466\) −8.84986 −0.409962
\(467\) −15.9678 −0.738902 −0.369451 0.929250i \(-0.620454\pi\)
−0.369451 + 0.929250i \(0.620454\pi\)
\(468\) 0.461294 0.0213233
\(469\) 12.0327 0.555619
\(470\) 7.88816 0.363854
\(471\) 26.9809 1.24321
\(472\) −3.03084 −0.139506
\(473\) 10.7260 0.493183
\(474\) 26.5570 1.21980
\(475\) −7.19651 −0.330199
\(476\) 6.73900 0.308882
\(477\) −72.5006 −3.31958
\(478\) −4.06340 −0.185855
\(479\) 28.1729 1.28725 0.643627 0.765340i \(-0.277429\pi\)
0.643627 + 0.765340i \(0.277429\pi\)
\(480\) −60.9202 −2.78061
\(481\) −0.253718 −0.0115686
\(482\) −5.15350 −0.234735
\(483\) 3.43429 0.156266
\(484\) −1.39182 −0.0632648
\(485\) −19.7759 −0.897977
\(486\) −1.58419 −0.0718604
\(487\) 24.0613 1.09032 0.545161 0.838331i \(-0.316468\pi\)
0.545161 + 0.838331i \(0.316468\pi\)
\(488\) −21.4150 −0.969410
\(489\) −7.05699 −0.319128
\(490\) −15.9619 −0.721085
\(491\) −29.0197 −1.30964 −0.654821 0.755784i \(-0.727256\pi\)
−0.654821 + 0.755784i \(0.727256\pi\)
\(492\) 49.7861 2.24453
\(493\) −1.36045 −0.0612716
\(494\) −0.0439260 −0.00197632
\(495\) −20.5496 −0.923636
\(496\) 6.62114 0.297298
\(497\) 14.4039 0.646102
\(498\) −32.4964 −1.45620
\(499\) −15.0257 −0.672642 −0.336321 0.941747i \(-0.609183\pi\)
−0.336321 + 0.941747i \(0.609183\pi\)
\(500\) 10.6767 0.477476
\(501\) 27.6412 1.23492
\(502\) −18.7184 −0.835445
\(503\) −28.7785 −1.28317 −0.641585 0.767052i \(-0.721723\pi\)
−0.641585 + 0.767052i \(0.721723\pi\)
\(504\) 16.6129 0.739996
\(505\) −27.9653 −1.24444
\(506\) 0.841844 0.0374245
\(507\) 38.7388 1.72045
\(508\) 4.36878 0.193833
\(509\) −10.4772 −0.464395 −0.232197 0.972669i \(-0.574592\pi\)
−0.232197 + 0.972669i \(0.574592\pi\)
\(510\) 36.8248 1.63063
\(511\) −11.8979 −0.526332
\(512\) −8.03194 −0.354965
\(513\) 8.59667 0.379552
\(514\) 4.43662 0.195691
\(515\) −21.6081 −0.952165
\(516\) 44.4971 1.95888
\(517\) 2.89630 0.127379
\(518\) −3.74947 −0.164742
\(519\) −31.9440 −1.40219
\(520\) 0.520323 0.0228177
\(521\) −24.7590 −1.08471 −0.542357 0.840148i \(-0.682468\pi\)
−0.542357 + 0.840148i \(0.682468\pi\)
\(522\) −1.37621 −0.0602349
\(523\) 14.8566 0.649635 0.324818 0.945777i \(-0.394697\pi\)
0.324818 + 0.945777i \(0.394697\pi\)
\(524\) −17.3025 −0.755865
\(525\) −22.8951 −0.999223
\(526\) −10.9068 −0.475560
\(527\) −41.6679 −1.81508
\(528\) −2.14852 −0.0935023
\(529\) −21.8347 −0.949335
\(530\) −33.5574 −1.45764
\(531\) −6.74219 −0.292586
\(532\) 1.48558 0.0644080
\(533\) −0.675962 −0.0292792
\(534\) −0.425807 −0.0184265
\(535\) 25.4602 1.10074
\(536\) 29.8195 1.28801
\(537\) 68.2773 2.94638
\(538\) −12.0113 −0.517845
\(539\) −5.86074 −0.252440
\(540\) −41.7862 −1.79819
\(541\) 3.88960 0.167227 0.0836134 0.996498i \(-0.473354\pi\)
0.0836134 + 0.996498i \(0.473354\pi\)
\(542\) 19.8014 0.850541
\(543\) −17.6013 −0.755343
\(544\) 26.5481 1.13824
\(545\) −27.2367 −1.16669
\(546\) −0.139747 −0.00598061
\(547\) 17.5180 0.749015 0.374507 0.927224i \(-0.377812\pi\)
0.374507 + 0.927224i \(0.377812\pi\)
\(548\) −19.2491 −0.822283
\(549\) −47.6382 −2.03315
\(550\) −5.61224 −0.239307
\(551\) −0.299905 −0.0127764
\(552\) 8.51086 0.362246
\(553\) 12.1946 0.518567
\(554\) −15.5814 −0.661988
\(555\) 46.8890 1.99033
\(556\) −22.2648 −0.944237
\(557\) 28.9860 1.22818 0.614088 0.789237i \(-0.289524\pi\)
0.614088 + 0.789237i \(0.289524\pi\)
\(558\) −42.1504 −1.78437
\(559\) −0.604152 −0.0255529
\(560\) −2.68695 −0.113544
\(561\) 13.5210 0.570856
\(562\) −4.82052 −0.203341
\(563\) 29.6112 1.24796 0.623982 0.781438i \(-0.285514\pi\)
0.623982 + 0.781438i \(0.285514\pi\)
\(564\) 12.0154 0.505938
\(565\) 1.50366 0.0632594
\(566\) 5.55356 0.233434
\(567\) 8.50793 0.357299
\(568\) 35.6957 1.49776
\(569\) −4.05818 −0.170128 −0.0850640 0.996375i \(-0.527109\pi\)
−0.0850640 + 0.996375i \(0.527109\pi\)
\(570\) 8.11784 0.340019
\(571\) 20.3380 0.851117 0.425559 0.904931i \(-0.360078\pi\)
0.425559 + 0.904931i \(0.360078\pi\)
\(572\) 0.0783957 0.00327789
\(573\) −19.2140 −0.802677
\(574\) −9.98942 −0.416950
\(575\) −7.76854 −0.323971
\(576\) 18.3726 0.765527
\(577\) 24.4768 1.01898 0.509492 0.860476i \(-0.329833\pi\)
0.509492 + 0.860476i \(0.329833\pi\)
\(578\) −2.79017 −0.116056
\(579\) −4.28037 −0.177886
\(580\) 1.45776 0.0605302
\(581\) −14.9219 −0.619064
\(582\) 13.1626 0.545607
\(583\) −12.3213 −0.510296
\(584\) −29.4854 −1.22011
\(585\) 1.15747 0.0478556
\(586\) −14.9219 −0.616419
\(587\) −3.34628 −0.138116 −0.0690579 0.997613i \(-0.521999\pi\)
−0.0690579 + 0.997613i \(0.521999\pi\)
\(588\) −24.3134 −1.00267
\(589\) −9.18548 −0.378481
\(590\) −3.12067 −0.128476
\(591\) 14.1699 0.582872
\(592\) 3.24695 0.133449
\(593\) −39.2063 −1.61001 −0.805006 0.593267i \(-0.797838\pi\)
−0.805006 + 0.593267i \(0.797838\pi\)
\(594\) 6.70416 0.275075
\(595\) 16.9094 0.693219
\(596\) −15.8139 −0.647764
\(597\) 7.03437 0.287898
\(598\) −0.0474175 −0.00193905
\(599\) 5.72987 0.234116 0.117058 0.993125i \(-0.462654\pi\)
0.117058 + 0.993125i \(0.462654\pi\)
\(600\) −56.7386 −2.31634
\(601\) 13.9163 0.567656 0.283828 0.958875i \(-0.408395\pi\)
0.283828 + 0.958875i \(0.408395\pi\)
\(602\) −8.92820 −0.363886
\(603\) 66.3343 2.70134
\(604\) −6.49065 −0.264101
\(605\) −3.49235 −0.141984
\(606\) 18.6134 0.756116
\(607\) 0.156175 0.00633897 0.00316948 0.999995i \(-0.498991\pi\)
0.00316948 + 0.999995i \(0.498991\pi\)
\(608\) 5.85241 0.237347
\(609\) −0.954120 −0.0386629
\(610\) −22.0497 −0.892765
\(611\) −0.163137 −0.00659980
\(612\) 37.1509 1.50174
\(613\) 40.1902 1.62327 0.811634 0.584166i \(-0.198578\pi\)
0.811634 + 0.584166i \(0.198578\pi\)
\(614\) −5.32308 −0.214822
\(615\) 124.923 5.03737
\(616\) 2.82331 0.113755
\(617\) 12.8606 0.517747 0.258874 0.965911i \(-0.416649\pi\)
0.258874 + 0.965911i \(0.416649\pi\)
\(618\) 14.3820 0.578531
\(619\) −2.03398 −0.0817526 −0.0408763 0.999164i \(-0.513015\pi\)
−0.0408763 + 0.999164i \(0.513015\pi\)
\(620\) 44.6482 1.79312
\(621\) 9.27999 0.372393
\(622\) 10.0297 0.402153
\(623\) −0.195524 −0.00783352
\(624\) 0.121017 0.00484456
\(625\) −9.19275 −0.367710
\(626\) 2.35288 0.0940398
\(627\) 2.98063 0.119035
\(628\) 12.5989 0.502750
\(629\) −20.4336 −0.814739
\(630\) 17.1052 0.681489
\(631\) 37.6984 1.50075 0.750375 0.661012i \(-0.229873\pi\)
0.750375 + 0.661012i \(0.229873\pi\)
\(632\) 30.2206 1.20211
\(633\) 73.0651 2.90408
\(634\) −13.8703 −0.550858
\(635\) 10.9621 0.435017
\(636\) −51.1151 −2.02685
\(637\) 0.330111 0.0130795
\(638\) −0.233882 −0.00925950
\(639\) 79.4060 3.14125
\(640\) −32.3734 −1.27967
\(641\) 2.84360 0.112315 0.0561576 0.998422i \(-0.482115\pi\)
0.0561576 + 0.998422i \(0.482115\pi\)
\(642\) −16.9460 −0.668804
\(643\) 30.6389 1.20828 0.604140 0.796878i \(-0.293517\pi\)
0.604140 + 0.796878i \(0.293517\pi\)
\(644\) 1.60366 0.0631932
\(645\) 111.652 4.39628
\(646\) −3.53764 −0.139187
\(647\) −6.76617 −0.266006 −0.133003 0.991116i \(-0.542462\pi\)
−0.133003 + 0.991116i \(0.542462\pi\)
\(648\) 21.0844 0.828272
\(649\) −1.14582 −0.0449772
\(650\) 0.316114 0.0123990
\(651\) −29.2228 −1.14533
\(652\) −3.29530 −0.129054
\(653\) 11.9015 0.465743 0.232872 0.972508i \(-0.425188\pi\)
0.232872 + 0.972508i \(0.425188\pi\)
\(654\) 18.1284 0.708876
\(655\) −43.4153 −1.69638
\(656\) 8.65059 0.337749
\(657\) −65.5910 −2.55895
\(658\) −2.41085 −0.0939845
\(659\) −11.2353 −0.437664 −0.218832 0.975763i \(-0.570225\pi\)
−0.218832 + 0.975763i \(0.570225\pi\)
\(660\) −14.4881 −0.563948
\(661\) −22.9273 −0.891768 −0.445884 0.895091i \(-0.647111\pi\)
−0.445884 + 0.895091i \(0.647111\pi\)
\(662\) 20.5772 0.799757
\(663\) −0.761580 −0.0295773
\(664\) −36.9794 −1.43508
\(665\) 3.72760 0.144550
\(666\) −20.6702 −0.800953
\(667\) −0.323743 −0.0125354
\(668\) 12.9072 0.499396
\(669\) −25.2587 −0.976558
\(670\) 30.7033 1.18617
\(671\) −8.09599 −0.312542
\(672\) 18.6189 0.718240
\(673\) −3.09828 −0.119430 −0.0597150 0.998215i \(-0.519019\pi\)
−0.0597150 + 0.998215i \(0.519019\pi\)
\(674\) 10.2960 0.396586
\(675\) −61.8661 −2.38123
\(676\) 18.0893 0.695743
\(677\) 33.3985 1.28361 0.641804 0.766868i \(-0.278186\pi\)
0.641804 + 0.766868i \(0.278186\pi\)
\(678\) −1.00082 −0.0384361
\(679\) 6.04407 0.231950
\(680\) 41.9049 1.60698
\(681\) −40.9810 −1.57039
\(682\) −7.16335 −0.274299
\(683\) 6.01634 0.230209 0.115104 0.993353i \(-0.463280\pi\)
0.115104 + 0.993353i \(0.463280\pi\)
\(684\) 8.18974 0.313143
\(685\) −48.2997 −1.84544
\(686\) 10.7051 0.408723
\(687\) 5.95054 0.227027
\(688\) 7.73160 0.294765
\(689\) 0.694007 0.0264396
\(690\) 8.76310 0.333606
\(691\) −15.6730 −0.596227 −0.298114 0.954530i \(-0.596357\pi\)
−0.298114 + 0.954530i \(0.596357\pi\)
\(692\) −14.9164 −0.567038
\(693\) 6.28054 0.238578
\(694\) −12.8527 −0.487882
\(695\) −55.8665 −2.11914
\(696\) −2.36450 −0.0896262
\(697\) −54.4395 −2.06204
\(698\) 14.8302 0.561330
\(699\) −33.8244 −1.27936
\(700\) −10.6910 −0.404082
\(701\) 18.0567 0.681992 0.340996 0.940065i \(-0.389236\pi\)
0.340996 + 0.940065i \(0.389236\pi\)
\(702\) −0.377617 −0.0142523
\(703\) −4.50448 −0.169890
\(704\) 3.12238 0.117679
\(705\) 30.1488 1.13547
\(706\) −10.2906 −0.387291
\(707\) 8.54699 0.321442
\(708\) −4.75344 −0.178645
\(709\) 16.2048 0.608586 0.304293 0.952579i \(-0.401580\pi\)
0.304293 + 0.952579i \(0.401580\pi\)
\(710\) 36.7536 1.37934
\(711\) 67.2266 2.52120
\(712\) −0.484549 −0.0181592
\(713\) −9.91560 −0.371342
\(714\) −11.2547 −0.421196
\(715\) 0.196709 0.00735652
\(716\) 31.8825 1.19150
\(717\) −15.5304 −0.579995
\(718\) 2.25188 0.0840395
\(719\) −27.9403 −1.04200 −0.520998 0.853558i \(-0.674440\pi\)
−0.520998 + 0.853558i \(0.674440\pi\)
\(720\) −14.8127 −0.552037
\(721\) 6.60403 0.245947
\(722\) −0.779856 −0.0290232
\(723\) −19.6968 −0.732533
\(724\) −8.21903 −0.305458
\(725\) 2.15827 0.0801561
\(726\) 2.32446 0.0862690
\(727\) −30.7020 −1.13867 −0.569337 0.822104i \(-0.692800\pi\)
−0.569337 + 0.822104i \(0.692800\pi\)
\(728\) −0.159025 −0.00589387
\(729\) −29.9678 −1.10992
\(730\) −30.3592 −1.12365
\(731\) −48.6562 −1.79961
\(732\) −33.5864 −1.24139
\(733\) 44.9333 1.65965 0.829826 0.558023i \(-0.188440\pi\)
0.829826 + 0.558023i \(0.188440\pi\)
\(734\) 1.64419 0.0606882
\(735\) −61.0069 −2.25027
\(736\) 6.31760 0.232870
\(737\) 11.2733 0.415258
\(738\) −55.0700 −2.02715
\(739\) −3.19172 −0.117409 −0.0587047 0.998275i \(-0.518697\pi\)
−0.0587047 + 0.998275i \(0.518697\pi\)
\(740\) 21.8951 0.804880
\(741\) −0.167887 −0.00616747
\(742\) 10.2561 0.376513
\(743\) 17.4348 0.639619 0.319810 0.947482i \(-0.396381\pi\)
0.319810 + 0.947482i \(0.396381\pi\)
\(744\) −72.4199 −2.65504
\(745\) −39.6801 −1.45377
\(746\) −6.29581 −0.230506
\(747\) −82.2617 −3.00980
\(748\) 6.31370 0.230852
\(749\) −7.78135 −0.284324
\(750\) −17.8310 −0.651095
\(751\) 9.55633 0.348715 0.174358 0.984682i \(-0.444215\pi\)
0.174358 + 0.984682i \(0.444215\pi\)
\(752\) 2.08773 0.0761317
\(753\) −71.5425 −2.60715
\(754\) 0.0131736 0.000479755 0
\(755\) −16.2863 −0.592718
\(756\) 12.7710 0.464478
\(757\) −48.5259 −1.76370 −0.881852 0.471527i \(-0.843703\pi\)
−0.881852 + 0.471527i \(0.843703\pi\)
\(758\) −11.1796 −0.406062
\(759\) 3.21755 0.116790
\(760\) 9.23774 0.335088
\(761\) 38.6035 1.39937 0.699687 0.714449i \(-0.253323\pi\)
0.699687 + 0.714449i \(0.253323\pi\)
\(762\) −7.29622 −0.264314
\(763\) 8.32429 0.301360
\(764\) −8.97210 −0.324599
\(765\) 93.2187 3.37033
\(766\) 7.64155 0.276101
\(767\) 0.0645391 0.00233037
\(768\) 40.1607 1.44917
\(769\) 53.2658 1.92081 0.960406 0.278603i \(-0.0898712\pi\)
0.960406 + 0.278603i \(0.0898712\pi\)
\(770\) 2.90699 0.104761
\(771\) 16.9569 0.610688
\(772\) −1.99874 −0.0719363
\(773\) 9.31748 0.335126 0.167563 0.985861i \(-0.446410\pi\)
0.167563 + 0.985861i \(0.446410\pi\)
\(774\) −49.2196 −1.76916
\(775\) 66.1034 2.37451
\(776\) 14.9784 0.537694
\(777\) −14.3306 −0.514107
\(778\) 15.5375 0.557047
\(779\) −12.0009 −0.429978
\(780\) 0.816053 0.0292194
\(781\) 13.4948 0.482883
\(782\) −3.81884 −0.136561
\(783\) −2.57818 −0.0921367
\(784\) −4.22458 −0.150878
\(785\) 31.6130 1.12832
\(786\) 28.8967 1.03071
\(787\) −15.0616 −0.536889 −0.268444 0.963295i \(-0.586510\pi\)
−0.268444 + 0.963295i \(0.586510\pi\)
\(788\) 6.61672 0.235711
\(789\) −41.6862 −1.48407
\(790\) 31.1163 1.10707
\(791\) −0.459561 −0.0163401
\(792\) 15.5644 0.553058
\(793\) 0.456013 0.0161935
\(794\) −10.9133 −0.387300
\(795\) −128.257 −4.54882
\(796\) 3.28474 0.116425
\(797\) −19.0593 −0.675114 −0.337557 0.941305i \(-0.609601\pi\)
−0.337557 + 0.941305i \(0.609601\pi\)
\(798\) −2.48104 −0.0878279
\(799\) −13.1384 −0.464804
\(800\) −42.1169 −1.48906
\(801\) −1.07789 −0.0380855
\(802\) 14.5281 0.513006
\(803\) −11.1470 −0.393370
\(804\) 46.7676 1.64937
\(805\) 4.02389 0.141824
\(806\) 0.403481 0.0142120
\(807\) −45.9077 −1.61603
\(808\) 21.1811 0.745150
\(809\) −15.2273 −0.535363 −0.267681 0.963507i \(-0.586257\pi\)
−0.267681 + 0.963507i \(0.586257\pi\)
\(810\) 21.7093 0.762785
\(811\) −43.3354 −1.52171 −0.760857 0.648920i \(-0.775221\pi\)
−0.760857 + 0.648920i \(0.775221\pi\)
\(812\) −0.445532 −0.0156351
\(813\) 75.6814 2.65426
\(814\) −3.51284 −0.123125
\(815\) −8.26854 −0.289634
\(816\) 9.74628 0.341188
\(817\) −10.7260 −0.375256
\(818\) 24.0530 0.840992
\(819\) −0.353756 −0.0123612
\(820\) 58.3334 2.03709
\(821\) 19.4279 0.678040 0.339020 0.940779i \(-0.389905\pi\)
0.339020 + 0.940779i \(0.389905\pi\)
\(822\) 32.1477 1.12128
\(823\) 2.35683 0.0821539 0.0410769 0.999156i \(-0.486921\pi\)
0.0410769 + 0.999156i \(0.486921\pi\)
\(824\) 16.3661 0.570141
\(825\) −21.4502 −0.746799
\(826\) 0.953763 0.0331857
\(827\) −29.8127 −1.03669 −0.518344 0.855172i \(-0.673451\pi\)
−0.518344 + 0.855172i \(0.673451\pi\)
\(828\) 8.84072 0.307236
\(829\) 52.6510 1.82864 0.914322 0.404988i \(-0.132724\pi\)
0.914322 + 0.404988i \(0.132724\pi\)
\(830\) −38.0754 −1.32162
\(831\) −59.5524 −2.06585
\(832\) −0.175871 −0.00609722
\(833\) 26.5859 0.921148
\(834\) 37.1841 1.28758
\(835\) 32.3867 1.12079
\(836\) 1.39182 0.0481373
\(837\) −78.9645 −2.72941
\(838\) −29.5836 −1.02195
\(839\) 29.7892 1.02844 0.514218 0.857660i \(-0.328082\pi\)
0.514218 + 0.857660i \(0.328082\pi\)
\(840\) 29.3890 1.01402
\(841\) −28.9101 −0.996899
\(842\) 14.3994 0.496237
\(843\) −18.4242 −0.634563
\(844\) 34.1182 1.17440
\(845\) 45.3895 1.56145
\(846\) −13.2906 −0.456939
\(847\) 1.06736 0.0366749
\(848\) −8.88152 −0.304992
\(849\) 21.2259 0.728471
\(850\) 25.4587 0.873226
\(851\) −4.86252 −0.166685
\(852\) 55.9836 1.91797
\(853\) −35.2393 −1.20657 −0.603285 0.797526i \(-0.706142\pi\)
−0.603285 + 0.797526i \(0.706142\pi\)
\(854\) 6.73900 0.230604
\(855\) 20.5496 0.702782
\(856\) −19.2837 −0.659105
\(857\) −21.5877 −0.737421 −0.368711 0.929544i \(-0.620201\pi\)
−0.368711 + 0.929544i \(0.620201\pi\)
\(858\) −0.130927 −0.00446979
\(859\) 27.7669 0.947396 0.473698 0.880687i \(-0.342919\pi\)
0.473698 + 0.880687i \(0.342919\pi\)
\(860\) 52.1364 1.77784
\(861\) −38.1799 −1.30117
\(862\) −26.2610 −0.894453
\(863\) 7.11774 0.242291 0.121145 0.992635i \(-0.461343\pi\)
0.121145 + 0.992635i \(0.461343\pi\)
\(864\) 50.3112 1.71162
\(865\) −37.4282 −1.27260
\(866\) −8.89357 −0.302216
\(867\) −10.6641 −0.362173
\(868\) −13.6458 −0.463167
\(869\) 11.4250 0.387566
\(870\) −2.43458 −0.0825400
\(871\) −0.634980 −0.0215155
\(872\) 20.6293 0.698595
\(873\) 33.3199 1.12771
\(874\) −0.841844 −0.0284758
\(875\) −8.18772 −0.276796
\(876\) −46.2436 −1.56243
\(877\) −44.1031 −1.48925 −0.744627 0.667481i \(-0.767373\pi\)
−0.744627 + 0.667481i \(0.767373\pi\)
\(878\) 2.93920 0.0991931
\(879\) −57.0321 −1.92364
\(880\) −2.51738 −0.0848608
\(881\) −37.9204 −1.27757 −0.638785 0.769385i \(-0.720563\pi\)
−0.638785 + 0.769385i \(0.720563\pi\)
\(882\) 26.8938 0.905562
\(883\) 4.85252 0.163300 0.0816502 0.996661i \(-0.473981\pi\)
0.0816502 + 0.996661i \(0.473981\pi\)
\(884\) −0.355624 −0.0119609
\(885\) −11.9273 −0.400931
\(886\) −3.52371 −0.118381
\(887\) 11.2754 0.378591 0.189295 0.981920i \(-0.439380\pi\)
0.189295 + 0.981920i \(0.439380\pi\)
\(888\) −35.5141 −1.19177
\(889\) −3.35032 −0.112366
\(890\) −0.498910 −0.0167235
\(891\) 7.97100 0.267038
\(892\) −11.7947 −0.394916
\(893\) −2.89630 −0.0969210
\(894\) 26.4106 0.883302
\(895\) 79.9991 2.67408
\(896\) 9.89423 0.330543
\(897\) −0.181231 −0.00605114
\(898\) 29.1363 0.972290
\(899\) 2.75477 0.0918767
\(900\) −58.9376 −1.96459
\(901\) 55.8928 1.86206
\(902\) −9.35899 −0.311620
\(903\) −34.1239 −1.13557
\(904\) −1.13888 −0.0378787
\(905\) −20.6231 −0.685534
\(906\) 10.8399 0.360133
\(907\) 24.0191 0.797540 0.398770 0.917051i \(-0.369437\pi\)
0.398770 + 0.917051i \(0.369437\pi\)
\(908\) −19.1363 −0.635061
\(909\) 47.1181 1.56281
\(910\) −0.163739 −0.00542788
\(911\) 7.48540 0.248002 0.124001 0.992282i \(-0.460427\pi\)
0.124001 + 0.992282i \(0.460427\pi\)
\(912\) 2.14852 0.0711446
\(913\) −13.9802 −0.462676
\(914\) −7.31328 −0.241902
\(915\) −84.2745 −2.78603
\(916\) 2.77864 0.0918089
\(917\) 13.2689 0.438179
\(918\) −30.4119 −1.00374
\(919\) 40.0187 1.32009 0.660047 0.751224i \(-0.270536\pi\)
0.660047 + 0.751224i \(0.270536\pi\)
\(920\) 9.97201 0.328768
\(921\) −20.3450 −0.670390
\(922\) −24.7047 −0.813605
\(923\) −0.760108 −0.0250193
\(924\) 4.42797 0.145669
\(925\) 32.4165 1.06585
\(926\) 28.9548 0.951514
\(927\) 36.4069 1.19576
\(928\) −1.75517 −0.0576161
\(929\) −32.3099 −1.06005 −0.530027 0.847981i \(-0.677818\pi\)
−0.530027 + 0.847981i \(0.677818\pi\)
\(930\) −74.5663 −2.44513
\(931\) 5.86074 0.192078
\(932\) −15.7945 −0.517367
\(933\) 38.3337 1.25499
\(934\) 12.4526 0.407461
\(935\) 15.8423 0.518098
\(936\) −0.876679 −0.0286552
\(937\) −54.1000 −1.76737 −0.883684 0.468083i \(-0.844945\pi\)
−0.883684 + 0.468083i \(0.844945\pi\)
\(938\) −9.38378 −0.306391
\(939\) 8.99277 0.293468
\(940\) 14.0782 0.459179
\(941\) 26.6955 0.870248 0.435124 0.900371i \(-0.356705\pi\)
0.435124 + 0.900371i \(0.356705\pi\)
\(942\) −21.0412 −0.685559
\(943\) −12.9548 −0.421868
\(944\) −0.825935 −0.0268819
\(945\) 32.0449 1.04242
\(946\) −8.36475 −0.271961
\(947\) −39.4463 −1.28183 −0.640916 0.767611i \(-0.721445\pi\)
−0.640916 + 0.767611i \(0.721445\pi\)
\(948\) 47.3968 1.53938
\(949\) 0.627865 0.0203814
\(950\) 5.61224 0.182085
\(951\) −53.0126 −1.71905
\(952\) −12.8073 −0.415088
\(953\) −48.7523 −1.57924 −0.789621 0.613595i \(-0.789723\pi\)
−0.789621 + 0.613595i \(0.789723\pi\)
\(954\) 56.5400 1.83055
\(955\) −22.5127 −0.728493
\(956\) −7.25203 −0.234547
\(957\) −0.893906 −0.0288959
\(958\) −21.9708 −0.709845
\(959\) 14.7618 0.476682
\(960\) 32.5022 1.04900
\(961\) 53.3730 1.72171
\(962\) 0.197864 0.00637938
\(963\) −42.8972 −1.38234
\(964\) −9.19755 −0.296233
\(965\) −5.01522 −0.161446
\(966\) −2.67825 −0.0861713
\(967\) −1.39478 −0.0448532 −0.0224266 0.999748i \(-0.507139\pi\)
−0.0224266 + 0.999748i \(0.507139\pi\)
\(968\) 2.64513 0.0850178
\(969\) −13.5210 −0.434356
\(970\) 15.4224 0.495182
\(971\) −3.11733 −0.100040 −0.0500200 0.998748i \(-0.515929\pi\)
−0.0500200 + 0.998748i \(0.515929\pi\)
\(972\) −2.82734 −0.0906870
\(973\) 17.0744 0.547380
\(974\) −18.7644 −0.601249
\(975\) 1.20820 0.0386933
\(976\) −5.83580 −0.186800
\(977\) −22.3690 −0.715647 −0.357823 0.933789i \(-0.616481\pi\)
−0.357823 + 0.933789i \(0.616481\pi\)
\(978\) 5.50343 0.175980
\(979\) −0.183185 −0.00585462
\(980\) −28.4875 −0.910001
\(981\) 45.8904 1.46517
\(982\) 22.6312 0.722191
\(983\) 42.2421 1.34731 0.673657 0.739044i \(-0.264723\pi\)
0.673657 + 0.739044i \(0.264723\pi\)
\(984\) −94.6174 −3.01629
\(985\) 16.6026 0.529003
\(986\) 1.06096 0.0337877
\(987\) −9.21433 −0.293295
\(988\) −0.0783957 −0.00249410
\(989\) −11.5786 −0.368178
\(990\) 16.0257 0.509331
\(991\) −1.69828 −0.0539477 −0.0269738 0.999636i \(-0.508587\pi\)
−0.0269738 + 0.999636i \(0.508587\pi\)
\(992\) −53.7572 −1.70679
\(993\) 78.6469 2.49578
\(994\) −11.2329 −0.356287
\(995\) 8.24203 0.261290
\(996\) −57.9970 −1.83770
\(997\) 18.9376 0.599759 0.299879 0.953977i \(-0.403054\pi\)
0.299879 + 0.953977i \(0.403054\pi\)
\(998\) 11.7179 0.370923
\(999\) −38.7235 −1.22516
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 209.2.a.c.1.2 5
3.2 odd 2 1881.2.a.k.1.4 5
4.3 odd 2 3344.2.a.t.1.5 5
5.4 even 2 5225.2.a.h.1.4 5
11.10 odd 2 2299.2.a.n.1.4 5
19.18 odd 2 3971.2.a.h.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.2 5 1.1 even 1 trivial
1881.2.a.k.1.4 5 3.2 odd 2
2299.2.a.n.1.4 5 11.10 odd 2
3344.2.a.t.1.5 5 4.3 odd 2
3971.2.a.h.1.4 5 19.18 odd 2
5225.2.a.h.1.4 5 5.4 even 2