Properties

Label 2299.2.a.n.1.4
Level $2299$
Weight $2$
Character 2299.1
Self dual yes
Analytic conductor $18.358$
Analytic rank $1$
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] = [2299,2,Mod(1,2299)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2299, 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("2299.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2299 = 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2299.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: \(18.3576074247\)
Analytic rank: \(1\)
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: no (minimal twist has level 209)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.71250\) of defining polynomial
Character \(\chi\) \(=\) 2299.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.779856 q^{2} -2.98063 q^{3} -1.39182 q^{4} -3.49235 q^{5} -2.32446 q^{6} -1.06736 q^{7} -2.64513 q^{8} +5.88418 q^{9} +O(q^{10})\) \(q+0.779856 q^{2} -2.98063 q^{3} -1.39182 q^{4} -3.49235 q^{5} -2.32446 q^{6} -1.06736 q^{7} -2.64513 q^{8} +5.88418 q^{9} -2.72353 q^{10} +4.14852 q^{12} +0.0563258 q^{13} -0.832387 q^{14} +10.4094 q^{15} +0.720827 q^{16} +4.53628 q^{17} +4.58881 q^{18} +1.00000 q^{19} +4.86074 q^{20} +3.18141 q^{21} -1.07949 q^{23} +7.88418 q^{24} +7.19651 q^{25} +0.0439260 q^{26} -8.59667 q^{27} +1.48558 q^{28} -0.299905 q^{29} +8.11784 q^{30} +9.18548 q^{31} +5.85241 q^{32} +3.53764 q^{34} +3.72760 q^{35} -8.18974 q^{36} +4.50448 q^{37} +0.779856 q^{38} -0.167887 q^{39} +9.23774 q^{40} -12.0009 q^{41} +2.48104 q^{42} -10.7260 q^{43} -20.5496 q^{45} -0.841844 q^{46} +2.89630 q^{47} -2.14852 q^{48} -5.86074 q^{49} +5.61224 q^{50} -13.5210 q^{51} -0.0783957 q^{52} -12.3213 q^{53} -6.70416 q^{54} +2.82331 q^{56} -2.98063 q^{57} -0.233882 q^{58} -1.14582 q^{59} -14.4881 q^{60} +8.09599 q^{61} +7.16335 q^{62} -6.28054 q^{63} +3.12238 q^{64} -0.196709 q^{65} +11.2733 q^{67} -6.31370 q^{68} +3.21755 q^{69} +2.90699 q^{70} +13.4948 q^{71} -15.5644 q^{72} +11.1470 q^{73} +3.51284 q^{74} -21.4502 q^{75} -1.39182 q^{76} -0.130927 q^{78} -11.4250 q^{79} -2.51738 q^{80} +7.97100 q^{81} -9.35899 q^{82} +13.9802 q^{83} -4.42797 q^{84} -15.8423 q^{85} -8.36475 q^{86} +0.893906 q^{87} -0.183185 q^{89} -16.0257 q^{90} -0.0601200 q^{91} +1.50246 q^{92} -27.3785 q^{93} +2.25870 q^{94} -3.49235 q^{95} -17.4439 q^{96} +5.66263 q^{97} -4.57053 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} + 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} + 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} - 12 q^{10} + 6 q^{12} - 4 q^{13} - 14 q^{14} + 3 q^{15} + 8 q^{16} + 4 q^{17} + 20 q^{18} + 5 q^{19} - 8 q^{20} - 10 q^{21} + 3 q^{23} + 14 q^{24} + 6 q^{25} - 6 q^{26} - 11 q^{27} + 10 q^{28} - 10 q^{29} - 6 q^{30} + 11 q^{31} - 14 q^{32} - 4 q^{34} + 8 q^{35} - 26 q^{36} + q^{37} - 2 q^{38} - 2 q^{39} + 16 q^{40} - 2 q^{41} - 16 q^{42} - 20 q^{43} - 28 q^{45} + 4 q^{46} - 20 q^{47} + 4 q^{48} + 3 q^{49} + 32 q^{50} - 24 q^{51} - 6 q^{52} - 14 q^{53} - 16 q^{54} - 38 q^{56} + q^{57} - 6 q^{58} + 3 q^{59} - 40 q^{60} + 10 q^{61} + 6 q^{62} - 24 q^{63} + 9 q^{67} - 24 q^{68} - 5 q^{69} + 50 q^{70} + 23 q^{71} + 12 q^{72} - 8 q^{74} - 18 q^{75} + 6 q^{76} - 22 q^{78} - 44 q^{79} - 18 q^{80} + q^{81} - 30 q^{82} + 14 q^{83} - 14 q^{84} + 12 q^{85} + 52 q^{86} - 28 q^{87} - 27 q^{89} - 26 q^{90} + 24 q^{91} + 58 q^{92} - 27 q^{93} + 8 q^{94} - 5 q^{95} - 50 q^{96} + 15 q^{97} + 10 q^{98}+O(q^{100}) \) 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 \(-0.411084\pi\)
0.275721 + 0.961238i \(0.411084\pi\)
\(3\) −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 \(-0.564651\pi\)
−0.201712 + 0.979445i \(0.564651\pi\)
\(8\) −2.64513 −0.935196
\(9\) 5.88418 1.96139
\(10\) −2.72353 −0.861256
\(11\) 0 0
\(12\) 4.14852 1.19757
\(13\) 0.0563258 0.0156220 0.00781098 0.999969i \(-0.497514\pi\)
0.00781098 + 0.999969i \(0.497514\pi\)
\(14\) −0.832387 −0.222465
\(15\) 10.4094 2.68770
\(16\) 0.720827 0.180207
\(17\) 4.53628 1.10021 0.550104 0.835096i \(-0.314588\pi\)
0.550104 + 0.835096i \(0.314588\pi\)
\(18\) 4.58881 1.08159
\(19\) 1.00000 0.229416
\(20\) 4.86074 1.08689
\(21\) 3.18141 0.694241
\(22\) 0 0
\(23\) −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 \(-0.508865\pi\)
−0.0278455 + 0.999612i \(0.508865\pi\)
\(30\) 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
\(33\) 0 0
\(34\) 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 \(-0.886514\pi\)
−0.937115 + 0.349020i \(0.886514\pi\)
\(42\) 2.48104 0.382833
\(43\) −10.7260 −1.63570 −0.817851 0.575430i \(-0.804835\pi\)
−0.817851 + 0.575430i \(0.804835\pi\)
\(44\) 0 0
\(45\) −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
\(55\) 0 0
\(56\) 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 \(-0.326568\pi\)
0.518293 + 0.855203i \(0.326568\pi\)
\(62\) 7.16335 0.909746
\(63\) −6.28054 −0.791273
\(64\) 3.12238 0.390298
\(65\) −0.196709 −0.0243988
\(66\) 0 0
\(67\) 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 \(-0.273792\pi\)
0.652330 + 0.757935i \(0.273792\pi\)
\(74\) 3.51284 0.408360
\(75\) −21.4502 −2.47685
\(76\) −1.39182 −0.159653
\(77\) 0 0
\(78\) −0.130927 −0.0148246
\(79\) −11.4250 −1.28541 −0.642706 0.766113i \(-0.722188\pi\)
−0.642706 + 0.766113i \(0.722188\pi\)
\(80\) −2.51738 −0.281452
\(81\) 7.97100 0.885666
\(82\) −9.35899 −1.03353
\(83\) 13.9802 1.53452 0.767261 0.641335i \(-0.221619\pi\)
0.767261 + 0.641335i \(0.221619\pi\)
\(84\) −4.42797 −0.483131
\(85\) −15.8423 −1.71834
\(86\) −8.36475 −0.901994
\(87\) 0.893906 0.0958368
\(88\) 0 0
\(89\) −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
\(99\) 0 0
\(100\) −10.0163 −1.00163
\(101\) −8.00759 −0.796785 −0.398392 0.917215i \(-0.630432\pi\)
−0.398392 + 0.917215i \(0.630432\pi\)
\(102\) −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 \(-0.385370\pi\)
0.352388 + 0.935854i \(0.385370\pi\)
\(108\) 11.9651 1.15134
\(109\) −7.79895 −0.747004 −0.373502 0.927629i \(-0.621843\pi\)
−0.373502 + 0.927629i \(0.621843\pi\)
\(110\) 0 0
\(111\) −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
\(121\) 0 0
\(122\) 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 \(-0.455526\pi\)
0.139265 + 0.990255i \(0.455526\pi\)
\(128\) −9.26981 −0.819343
\(129\) 31.9703 2.81483
\(130\) −0.153405 −0.0134545
\(131\) −12.4315 −1.08615 −0.543075 0.839684i \(-0.682740\pi\)
−0.543075 + 0.839684i \(0.682740\pi\)
\(132\) 0 0
\(133\) −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 \(-0.737334\pi\)
−0.678417 + 0.734677i \(0.737334\pi\)
\(140\) −5.18816 −0.438480
\(141\) −8.63281 −0.727014
\(142\) 10.5240 0.883157
\(143\) 0 0
\(144\) 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 \(-0.654092\pi\)
−0.465406 + 0.885097i \(0.654092\pi\)
\(150\) −16.7280 −1.36584
\(151\) −4.66341 −0.379503 −0.189751 0.981832i \(-0.560768\pi\)
−0.189751 + 0.981832i \(0.560768\pi\)
\(152\) −2.64513 −0.214549
\(153\) 26.6923 2.15794
\(154\) 0 0
\(155\) −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
\(165\) 0 0
\(166\) 10.9025 0.846199
\(167\) 9.27361 0.717613 0.358807 0.933412i \(-0.383184\pi\)
0.358807 + 0.933412i \(0.383184\pi\)
\(168\) −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 \(-0.633567\pi\)
−0.407406 + 0.913247i \(0.633567\pi\)
\(174\) 0.697118 0.0528484
\(175\) −7.68128 −0.580650
\(176\) 0 0
\(177\) 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
\(187\) 0 0
\(188\) −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 \(-0.516459\pi\)
−0.0516849 + 0.998663i \(0.516459\pi\)
\(194\) 4.41604 0.317053
\(195\) 0.586319 0.0419872
\(196\) 8.15713 0.582652
\(197\) 4.75399 0.338708 0.169354 0.985555i \(-0.445832\pi\)
0.169354 + 0.985555i \(0.445832\pi\)
\(198\) 0 0
\(199\) −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
\(209\) 0 0
\(210\) −8.66467 −0.597919
\(211\) 24.5133 1.68756 0.843781 0.536687i \(-0.180324\pi\)
0.843781 + 0.536687i \(0.180324\pi\)
\(212\) 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
\(220\) 0 0
\(221\) 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 \(-0.650818\pi\)
−0.456279 + 0.889837i \(0.650818\pi\)
\(228\) 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
\(231\) 0 0
\(232\) 0.793288 0.0520819
\(233\) −11.3481 −0.743437 −0.371718 0.928346i \(-0.621231\pi\)
−0.371718 + 0.928346i \(0.621231\pi\)
\(234\) 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 \(-0.553898\pi\)
−0.168518 + 0.985699i \(0.553898\pi\)
\(240\) 7.50339 0.484341
\(241\) −6.60827 −0.425676 −0.212838 0.977087i \(-0.568271\pi\)
−0.212838 + 0.977087i \(0.568271\pi\)
\(242\) 0 0
\(243\) 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
\(253\) 0 0
\(254\) 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 \(-0.641909\pi\)
−0.431197 + 0.902258i \(0.641909\pi\)
\(264\) 0 0
\(265\) 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 \(-0.219658\pi\)
0.771199 + 0.636595i \(0.219658\pi\)
\(272\) 3.26987 0.198265
\(273\) 0.179196 0.0108454
\(274\) 10.7855 0.651577
\(275\) 0 0
\(276\) −4.47827 −0.269560
\(277\) −19.9798 −1.20047 −0.600235 0.799824i \(-0.704926\pi\)
−0.600235 + 0.799824i \(0.704926\pi\)
\(278\) −12.4752 −0.748214
\(279\) 54.0490 3.23583
\(280\) −9.86000 −0.589248
\(281\) −6.18130 −0.368745 −0.184373 0.982856i \(-0.559025\pi\)
−0.184373 + 0.982856i \(0.559025\pi\)
\(282\) −6.73235 −0.400906
\(283\) 7.12127 0.423316 0.211658 0.977344i \(-0.432114\pi\)
0.211658 + 0.977344i \(0.432114\pi\)
\(284\) −18.7825 −1.11453
\(285\) 10.4094 0.616601
\(286\) 0 0
\(287\) 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 \(-0.688783\pi\)
−0.558916 + 0.829224i \(0.688783\pi\)
\(294\) 13.6231 0.794514
\(295\) 4.00159 0.232982
\(296\) −11.9149 −0.692542
\(297\) 0 0
\(298\) −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 \(-0.562400\pi\)
−0.194782 + 0.980846i \(0.562400\pi\)
\(308\) 0 0
\(309\) −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 0
\(320\) −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
\(330\) 0 0
\(331\) −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 \(-0.382916\pi\)
0.359590 + 0.933110i \(0.382916\pi\)
\(338\) −10.1357 −0.551307
\(339\) 1.28334 0.0697012
\(340\) 22.0497 1.19581
\(341\) 0 0
\(342\) 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 \(-0.645862\pi\)
−0.442370 + 0.896833i \(0.645862\pi\)
\(348\) −1.24416 −0.0666940
\(349\) 19.0165 1.01793 0.508966 0.860787i \(-0.330028\pi\)
0.508966 + 0.860787i \(0.330028\pi\)
\(350\) −5.99029 −0.320194
\(351\) −0.484214 −0.0258455
\(352\) 0 0
\(353\) 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 \(-0.475721\pi\)
0.0761998 + 0.997093i \(0.475721\pi\)
\(360\) 54.3565 2.86484
\(361\) 1.00000 0.0526316
\(362\) 4.60522 0.242045
\(363\) 0 0
\(364\) 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 \(-0.567022\pi\)
−0.209003 + 0.977915i \(0.567022\pi\)
\(374\) 0 0
\(375\) 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
\(385\) 0 0
\(386\) −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
\(396\) 0 0
\(397\) 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
\(407\) 0 0
\(408\) 35.7648 1.77062
\(409\) 30.8428 1.52508 0.762540 0.646941i \(-0.223952\pi\)
0.762540 + 0.646941i \(0.223952\pi\)
\(410\) 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 0
\(419\) 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 0
\(430\) 29.2126 1.40876
\(431\) −33.6742 −1.62203 −0.811013 0.585027i \(-0.801084\pi\)
−0.811013 + 0.585027i \(0.801084\pi\)
\(432\) −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 \(-0.471333\pi\)
0.0899399 + 0.995947i \(0.471333\pi\)
\(440\) 0 0
\(441\) −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
\(451\) 0 0
\(452\) 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 \(-0.570389\pi\)
−0.219336 + 0.975649i \(0.570389\pi\)
\(458\) −1.55690 −0.0727494
\(459\) −38.9969 −1.82022
\(460\) −5.24710 −0.244648
\(461\) −31.6785 −1.47542 −0.737708 0.675120i \(-0.764092\pi\)
−0.737708 + 0.675120i \(0.764092\pi\)
\(462\) 0 0
\(463\) −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
\(473\) 0 0
\(474\) 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 \(-0.722571\pi\)
−0.643627 + 0.765340i \(0.722571\pi\)
\(480\) 60.9202 2.78061
\(481\) 0.253718 0.0115686
\(482\) −5.15350 −0.234735
\(483\) −3.43429 −0.156266
\(484\) 0 0
\(485\) −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 \(-0.272744\pi\)
0.654821 + 0.755784i \(0.272744\pi\)
\(492\) −49.7861 −2.24453
\(493\) −1.36045 −0.0612716
\(494\) 0.0439260 0.00197632
\(495\) 0 0
\(496\) 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 \(-0.278277\pi\)
0.641585 + 0.767052i \(0.278277\pi\)
\(504\) 16.6129 0.739996
\(505\) 27.9653 1.24444
\(506\) 0 0
\(507\) 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
\(517\) 0 0
\(518\) −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 \(-0.605303\pi\)
−0.324818 + 0.945777i \(0.605303\pi\)
\(524\) 17.3025 0.755865
\(525\) 22.8951 0.999223
\(526\) −10.9068 −0.475560
\(527\) 41.6679 1.81508
\(528\) 0 0
\(529\) −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
\(539\) 0 0
\(540\) −41.7862 −1.79819
\(541\) −3.88960 −0.167227 −0.0836134 0.996498i \(-0.526646\pi\)
−0.0836134 + 0.996498i \(0.526646\pi\)
\(542\) 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 \(-0.622188\pi\)
−0.374507 + 0.927224i \(0.622188\pi\)
\(548\) −19.2491 −0.822283
\(549\) 47.6382 2.03315
\(550\) 0 0
\(551\) −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 \(-0.710476\pi\)
−0.614088 + 0.789237i \(0.710476\pi\)
\(558\) 42.1504 1.78437
\(559\) −0.604152 −0.0255529
\(560\) 2.68695 0.113544
\(561\) 0 0
\(562\) −4.82052 −0.203341
\(563\) −29.6112 −1.24796 −0.623982 0.781438i \(-0.714486\pi\)
−0.623982 + 0.781438i \(0.714486\pi\)
\(564\) 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 \(-0.472891\pi\)
0.0850640 + 0.996375i \(0.472891\pi\)
\(570\) 8.11784 0.340019
\(571\) −20.3380 −0.851117 −0.425559 0.904931i \(-0.639922\pi\)
−0.425559 + 0.904931i \(0.639922\pi\)
\(572\) 0 0
\(573\) −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
\(583\) 0 0
\(584\) −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 \(-0.202162\pi\)
0.805006 + 0.593267i \(0.202162\pi\)
\(594\) 0 0
\(595\) 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 \(-0.591605\pi\)
−0.283828 + 0.958875i \(0.591605\pi\)
\(602\) 8.92820 0.363886
\(603\) 66.3343 2.70134
\(604\) 6.49065 0.264101
\(605\) 0 0
\(606\) 18.6134 0.756116
\(607\) −0.156175 −0.00633897 −0.00316948 0.999995i \(-0.501009\pi\)
−0.00316948 + 0.999995i \(0.501009\pi\)
\(608\) 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 \(-0.801422\pi\)
−0.811634 + 0.584166i \(0.801422\pi\)
\(614\) −5.32308 −0.214822
\(615\) −124.923 −5.03737
\(616\) 0 0
\(617\) 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
\(627\) 0 0
\(628\) 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 0
\(639\) 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
\(649\) 0 0
\(650\) 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 \(-0.429775\pi\)
0.218832 + 0.975763i \(0.429775\pi\)
\(660\) 0 0
\(661\) −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
\(671\) 0 0
\(672\) 18.6189 0.718240
\(673\) 3.09828 0.119430 0.0597150 0.998215i \(-0.480981\pi\)
0.0597150 + 0.998215i \(0.480981\pi\)
\(674\) 10.2960 0.396586
\(675\) −61.8661 −2.38123
\(676\) 18.0893 0.695743
\(677\) −33.3985 −1.28361 −0.641804 0.766868i \(-0.721814\pi\)
−0.641804 + 0.766868i \(0.721814\pi\)
\(678\) 1.00082 0.0384361
\(679\) −6.04407 −0.231950
\(680\) 41.9049 1.60698
\(681\) 40.9810 1.57039
\(682\) 0 0
\(683\) 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
\(693\) 0 0
\(694\) −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 \(-0.610764\pi\)
−0.340996 + 0.940065i \(0.610764\pi\)
\(702\) −0.377617 −0.0142523
\(703\) 4.50448 0.169890
\(704\) 0 0
\(705\) 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 0
\(716\) 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
\(726\) 0 0
\(727\) −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 \(-0.811560\pi\)
−0.829826 + 0.558023i \(0.811560\pi\)
\(734\) −1.64419 −0.0606882
\(735\) −61.0069 −2.25027
\(736\) −6.31760 −0.232870
\(737\) 0 0
\(738\) −55.0700 −2.02715
\(739\) 3.19172 0.117409 0.0587047 0.998275i \(-0.481303\pi\)
0.0587047 + 0.998275i \(0.481303\pi\)
\(740\) 21.8951 0.804880
\(741\) −0.167887 −0.00616747
\(742\) 10.2561 0.376513
\(743\) −17.4348 −0.639619 −0.319810 0.947482i \(-0.603619\pi\)
−0.319810 + 0.947482i \(0.603619\pi\)
\(744\) 72.4199 2.65504
\(745\) 39.6801 1.45377
\(746\) −6.29581 −0.230506
\(747\) 82.2617 3.00980
\(748\) 0 0
\(749\) −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
\(759\) 0 0
\(760\) 9.23774 0.335088
\(761\) −38.6035 −1.39937 −0.699687 0.714449i \(-0.746677\pi\)
−0.699687 + 0.714449i \(0.746677\pi\)
\(762\) −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 \(-0.910129\pi\)
−0.960406 + 0.278603i \(0.910129\pi\)
\(770\) 0 0
\(771\) 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
\(781\) 0 0
\(782\) −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 \(-0.413490\pi\)
0.268444 + 0.963295i \(0.413490\pi\)
\(788\) −6.61672 −0.235711
\(789\) 41.6862 1.48407
\(790\) 31.1163 1.10707
\(791\) 0.459561 0.0163401
\(792\) 0 0
\(793\) 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
\(803\) 0 0
\(804\) 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 \(-0.413743\pi\)
0.267681 + 0.963507i \(0.413743\pi\)
\(810\) −21.7093 −0.762785
\(811\) 43.3354 1.52171 0.760857 0.648920i \(-0.224779\pi\)
0.760857 + 0.648920i \(0.224779\pi\)
\(812\) −0.445532 −0.0156351
\(813\) −75.6814 −2.65426
\(814\) 0 0
\(815\) −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 \(-0.610095\pi\)
−0.339020 + 0.940779i \(0.610095\pi\)
\(822\) −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
\(825\) 0 0
\(826\) 0.953763 0.0331857
\(827\) 29.8127 1.03669 0.518344 0.855172i \(-0.326549\pi\)
0.518344 + 0.855172i \(0.326549\pi\)
\(828\) 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
\(836\) 0 0
\(837\) −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
\(847\) 0 0
\(848\) −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 \(-0.293858\pi\)
0.603285 + 0.797526i \(0.293858\pi\)
\(854\) −6.73900 −0.230604
\(855\) −20.5496 −0.702782
\(856\) −19.2837 −0.659105
\(857\) 21.5877 0.737421 0.368711 0.929544i \(-0.379799\pi\)
0.368711 + 0.929544i \(0.379799\pi\)
\(858\) 0 0
\(859\) 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
\(869\) 0 0
\(870\) −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 \(-0.232627\pi\)
0.744627 + 0.667481i \(0.232627\pi\)
\(878\) 2.93920 0.0991931
\(879\) 57.0321 1.92364
\(880\) 0 0
\(881\) −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 \(-0.560620\pi\)
−0.189295 + 0.981920i \(0.560620\pi\)
\(888\) 35.5141 1.19177
\(889\) −3.35032 −0.112366
\(890\) 0.498910 0.0167235
\(891\) 0 0
\(892\) −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
\(902\) 0 0
\(903\) −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
\(913\) 0 0
\(914\) −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 \(-0.729464\pi\)
−0.660047 + 0.751224i \(0.729464\pi\)
\(920\) −9.97201 −0.328768
\(921\) 20.3450 0.670390
\(922\) −24.7047 −0.813605
\(923\) 0.760108 0.0250193
\(924\) 0 0
\(925\) 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
\(935\) 0 0
\(936\) −0.876679 −0.0286552
\(937\) 54.1000 1.76737 0.883684 0.468083i \(-0.155055\pi\)
0.883684 + 0.468083i \(0.155055\pi\)
\(938\) −9.38378 −0.306391
\(939\) 8.99277 0.293468
\(940\) 14.0782 0.459179
\(941\) −26.6955 −0.870248 −0.435124 0.900371i \(-0.643295\pi\)
−0.435124 + 0.900371i \(0.643295\pi\)
\(942\) 21.0412 0.685559
\(943\) 12.9548 0.421868
\(944\) −0.825935 −0.0268819
\(945\) −32.0449 −1.04242
\(946\) 0 0
\(947\) −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 \(-0.210277\pi\)
0.789621 + 0.613595i \(0.210277\pi\)
\(954\) −56.5400 −1.83055
\(955\) −22.5127 −0.728493
\(956\) 7.25203 0.234547
\(957\) 0 0
\(958\) −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 \(-0.492861\pi\)
0.0224266 + 0.999748i \(0.492861\pi\)
\(968\) 0 0
\(969\) −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 0
\(980\) −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
\(990\) 0 0
\(991\) −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 \(-0.596946\pi\)
−0.299879 + 0.953977i \(0.596946\pi\)
\(998\) −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 2299.2.a.n.1.4 5
11.10 odd 2 209.2.a.c.1.2 5
33.32 even 2 1881.2.a.k.1.4 5
44.43 even 2 3344.2.a.t.1.5 5
55.54 odd 2 5225.2.a.h.1.4 5
209.208 even 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 11.10 odd 2
1881.2.a.k.1.4 5 33.32 even 2
2299.2.a.n.1.4 5 1.1 even 1 trivial
3344.2.a.t.1.5 5 44.43 even 2
3971.2.a.h.1.4 5 209.208 even 2
5225.2.a.h.1.4 5 55.54 odd 2