Properties

Label 3971.2.a.h.1.4
Level $3971$
Weight $2$
Character 3971.1
Self dual yes
Analytic conductor $31.709$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3971,2,Mod(1,3971)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3971, 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("3971.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3971 = 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3971.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: \(31.7085946427\)
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 \(\chi\) \(=\) 3971.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.779856 q^{2} +2.98063 q^{3} -1.39182 q^{4} -3.49235 q^{5} +2.32446 q^{6} +1.06736 q^{7} -2.64513 q^{8} +5.88418 q^{9} +O(q^{10})\) \(q+0.779856 q^{2} +2.98063 q^{3} -1.39182 q^{4} -3.49235 q^{5} +2.32446 q^{6} +1.06736 q^{7} -2.64513 q^{8} +5.88418 q^{9} -2.72353 q^{10} +1.00000 q^{11} -4.14852 q^{12} +0.0563258 q^{13} +0.832387 q^{14} -10.4094 q^{15} +0.720827 q^{16} -4.53628 q^{17} +4.58881 q^{18} +4.86074 q^{20} +3.18141 q^{21} +0.779856 q^{22} -1.07949 q^{23} -7.88418 q^{24} +7.19651 q^{25} +0.0439260 q^{26} +8.59667 q^{27} -1.48558 q^{28} -0.299905 q^{29} -8.11784 q^{30} -9.18548 q^{31} +5.85241 q^{32} +2.98063 q^{33} -3.53764 q^{34} -3.72760 q^{35} -8.18974 q^{36} -4.50448 q^{37} +0.167887 q^{39} +9.23774 q^{40} -12.0009 q^{41} +2.48104 q^{42} +10.7260 q^{43} -1.39182 q^{44} -20.5496 q^{45} -0.841844 q^{46} +2.89630 q^{47} +2.14852 q^{48} -5.86074 q^{49} +5.61224 q^{50} -13.5210 q^{51} -0.0783957 q^{52} +12.3213 q^{53} +6.70416 q^{54} -3.49235 q^{55} -2.82331 q^{56} -0.233882 q^{58} +1.14582 q^{59} +14.4881 q^{60} -8.09599 q^{61} -7.16335 q^{62} +6.28054 q^{63} +3.12238 q^{64} -0.196709 q^{65} +2.32446 q^{66} -11.2733 q^{67} +6.31370 q^{68} -3.21755 q^{69} -2.90699 q^{70} -13.4948 q^{71} -15.5644 q^{72} -11.1470 q^{73} -3.51284 q^{74} +21.4502 q^{75} +1.06736 q^{77} +0.130927 q^{78} -11.4250 q^{79} -2.51738 q^{80} +7.97100 q^{81} -9.35899 q^{82} -13.9802 q^{83} -4.42797 q^{84} +15.8423 q^{85} +8.36475 q^{86} -0.893906 q^{87} -2.64513 q^{88} +0.183185 q^{89} -16.0257 q^{90} +0.0601200 q^{91} +1.50246 q^{92} -27.3785 q^{93} +2.25870 q^{94} +17.4439 q^{96} -5.66263 q^{97} -4.57053 q^{98} +5.88418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} - 6 q^{8} + 4 q^{9} - 12 q^{10} + 5 q^{11} - 6 q^{12} - 4 q^{13} + 14 q^{14} - 3 q^{15} + 8 q^{16} - 4 q^{17} + 20 q^{18} - 8 q^{20} - 10 q^{21} - 2 q^{22} + 3 q^{23} - 14 q^{24} + 6 q^{25} - 6 q^{26} + 11 q^{27} - 10 q^{28} - 10 q^{29} + 6 q^{30} - 11 q^{31} - 14 q^{32} - q^{33} + 4 q^{34} - 8 q^{35} - 26 q^{36} - q^{37} + 2 q^{39} + 16 q^{40} - 2 q^{41} - 16 q^{42} + 20 q^{43} + 6 q^{44} - 28 q^{45} + 4 q^{46} - 20 q^{47} - 4 q^{48} + 3 q^{49} + 32 q^{50} - 24 q^{51} - 6 q^{52} + 14 q^{53} + 16 q^{54} - 5 q^{55} + 38 q^{56} - 6 q^{58} - 3 q^{59} + 40 q^{60} - 10 q^{61} - 6 q^{62} + 24 q^{63} - 2 q^{66} - 9 q^{67} + 24 q^{68} + 5 q^{69} - 50 q^{70} - 23 q^{71} + 12 q^{72} + 8 q^{74} + 18 q^{75} + 6 q^{77} + 22 q^{78} - 44 q^{79} - 18 q^{80} + q^{81} - 30 q^{82} - 14 q^{83} - 14 q^{84} - 12 q^{85} - 52 q^{86} + 28 q^{87} - 6 q^{88} + 27 q^{89} - 26 q^{90} - 24 q^{91} + 58 q^{92} - 27 q^{93} + 8 q^{94} + 50 q^{96} - 15 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100}) \) 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 \(-0.411084\pi\)
0.275721 + 0.961238i \(0.411084\pi\)
\(3\) 2.98063 1.72087 0.860435 0.509561i \(-0.170192\pi\)
0.860435 + 0.509561i \(0.170192\pi\)
\(4\) −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 \(-0.497514\pi\)
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 \(-0.685412\pi\)
−0.550104 + 0.835096i \(0.685412\pi\)
\(18\) 4.58881 1.08159
\(19\) 0 0
\(20\) 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 \(-0.508865\pi\)
−0.0278455 + 0.999612i \(0.508865\pi\)
\(30\) −8.11784 −1.48211
\(31\) −9.18548 −1.64976 −0.824880 0.565307i \(-0.808758\pi\)
−0.824880 + 0.565307i \(0.808758\pi\)
\(32\) 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 \(-0.620733\pi\)
−0.370266 + 0.928926i \(0.620733\pi\)
\(38\) 0 0
\(39\) 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 \(-0.195165\pi\)
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 \(-0.178867\pi\)
0.846230 + 0.532818i \(0.178867\pi\)
\(54\) 6.70416 0.912321
\(55\) −3.49235 −0.470908
\(56\) −2.82331 −0.377281
\(57\) 0 0
\(58\) −0.233882 −0.0307103
\(59\) 1.14582 0.149173 0.0745863 0.997215i \(-0.476236\pi\)
0.0745863 + 0.997215i \(0.476236\pi\)
\(60\) 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 \(-0.741787\pi\)
−0.688628 + 0.725115i \(0.741787\pi\)
\(68\) 6.31370 0.765649
\(69\) −3.21755 −0.387348
\(70\) −2.90699 −0.347452
\(71\) −13.4948 −1.60154 −0.800771 0.598970i \(-0.795577\pi\)
−0.800771 + 0.598970i \(0.795577\pi\)
\(72\) −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
\(76\) 0 0
\(77\) 1.06736 0.121637
\(78\) 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 \(-0.778381\pi\)
−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 \(-0.496910\pi\)
0.00970878 + 0.999953i \(0.496910\pi\)
\(90\) −16.0257 −1.68926
\(91\) 0.0601200 0.00630228
\(92\) 1.50246 0.156642
\(93\) −27.3785 −2.83902
\(94\) 2.25870 0.232967
\(95\) 0 0
\(96\) 17.4439 1.78036
\(97\) −5.66263 −0.574953 −0.287477 0.957788i \(-0.592816\pi\)
−0.287477 + 0.957788i \(0.592816\pi\)
\(98\) −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 \(-0.598598\pi\)
−0.304824 + 0.952409i \(0.598598\pi\)
\(104\) −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 \(0.621843\pi\)
\(110\) −2.72353 −0.259678
\(111\) −13.4262 −1.27436
\(112\) 0.769382 0.0726998
\(113\) 0.430558 0.0405035 0.0202517 0.999795i \(-0.493553\pi\)
0.0202517 + 0.999795i \(0.493553\pi\)
\(114\) 0 0
\(115\) 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 \(-0.455526\pi\)
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 \(-0.317260\pi\)
0.543075 + 0.839684i \(0.317260\pi\)
\(132\) −4.14852 −0.361082
\(133\) 0 0
\(134\) −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 \(-0.560768\pi\)
−0.189751 + 0.981832i \(0.560768\pi\)
\(152\) 0 0
\(153\) −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 \(-0.383184\pi\)
0.358807 + 0.933412i \(0.383184\pi\)
\(168\) −8.41526 −0.649251
\(169\) −12.9968 −0.999756
\(170\) 12.3547 0.947561
\(171\) 0 0
\(172\) −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
\(176\) 0.720827 0.0543344
\(177\) 3.41526 0.256707
\(178\) 0.142858 0.0107076
\(179\) 22.9070 1.71215 0.856073 0.516854i \(-0.172897\pi\)
0.856073 + 0.516854i \(0.172897\pi\)
\(180\) 28.6015 2.13183
\(181\) −5.90522 −0.438931 −0.219466 0.975620i \(-0.570431\pi\)
−0.219466 + 0.975620i \(0.570431\pi\)
\(182\) 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
\(190\) 0 0
\(191\) 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 \(-0.554168\pi\)
−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
\(209\) 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
\(220\) 4.86074 0.327711
\(221\) −0.255509 −0.0171874
\(222\) −10.4705 −0.702734
\(223\) −8.47427 −0.567479 −0.283740 0.958901i \(-0.591575\pi\)
−0.283740 + 0.958901i \(0.591575\pi\)
\(224\) 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 \(0.650818\pi\)
\(228\) 0 0
\(229\) −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 \(-0.568271\pi\)
−0.212838 + 0.977087i \(0.568271\pi\)
\(242\) 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 0
\(248\) 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 \(-0.443220\pi\)
0.177436 + 0.984132i \(0.443220\pi\)
\(258\) 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 0
\(267\) 0.546007 0.0334151
\(268\) 15.6905 0.958450
\(269\) −15.4020 −0.939075 −0.469538 0.882912i \(-0.655579\pi\)
−0.469538 + 0.882912i \(0.655579\pi\)
\(270\) −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 \(-0.559025\pi\)
−0.184373 + 0.982856i \(0.559025\pi\)
\(282\) 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
\(285\) 0 0
\(286\) 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 \(-0.688783\pi\)
−0.558916 + 0.829224i \(0.688783\pi\)
\(294\) −13.6231 −0.794514
\(295\) −4.00159 −0.232982
\(296\) 11.9149 0.692542
\(297\) 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 0
\(305\) 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 \(0.562400\pi\)
\(308\) −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 \(-0.666472\pi\)
−0.499471 + 0.866330i \(0.666472\pi\)
\(318\) 28.6404 1.60607
\(319\) −0.299905 −0.0167914
\(320\) −10.9045 −0.609577
\(321\) 21.7296 1.21283
\(322\) −0.898551 −0.0500743
\(323\) 0 0
\(324\) −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 \(-0.241770\pi\)
0.725152 + 0.688589i \(0.241770\pi\)
\(332\) 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
\(341\) −9.18548 −0.497422
\(342\) 0 0
\(343\) −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
\(361\) 0 0
\(362\) −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 \(-0.567022\pi\)
−0.209003 + 0.977915i \(0.567022\pi\)
\(374\) −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 \(-0.620020\pi\)
−0.368183 + 0.929754i \(0.620020\pi\)
\(380\) 0 0
\(381\) 9.35586 0.479315
\(382\) 5.02717 0.257212
\(383\) 9.79867 0.500689 0.250344 0.968157i \(-0.419456\pi\)
0.250344 + 0.968157i \(0.419456\pi\)
\(384\) −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
\(399\) 0 0
\(400\) 5.18744 0.259372
\(401\) 18.6293 0.930301 0.465151 0.885232i \(-0.346000\pi\)
0.465151 + 0.885232i \(0.346000\pi\)
\(402\) −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 \(-0.223952\pi\)
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 \(-0.351443\pi\)
0.449945 + 0.893056i \(0.351443\pi\)
\(422\) 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 \(-0.801084\pi\)
−0.811013 + 0.585027i \(0.801084\pi\)
\(432\) 6.19671 0.298139
\(433\) −11.4041 −0.548047 −0.274024 0.961723i \(-0.588355\pi\)
−0.274024 + 0.961723i \(0.588355\pi\)
\(434\) −7.64588 −0.367014
\(435\) 3.12183 0.149680
\(436\) 10.8548 0.519849
\(437\) 0 0
\(438\) −25.9108 −1.23807
\(439\) 3.76890 0.179880 0.0899399 0.995947i \(-0.471333\pi\)
0.0899399 + 0.995947i \(0.471333\pi\)
\(440\) 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 \(-0.156474\pi\)
0.881590 + 0.472016i \(0.156474\pi\)
\(450\) 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
\(456\) 0 0
\(457\) 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
\(475\) 0 0
\(476\) 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 \(-0.683532\pi\)
−0.545161 + 0.838331i \(0.683532\pi\)
\(488\) 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 0
\(495\) −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 \(-0.425408\pi\)
0.232197 + 0.972669i \(0.425408\pi\)
\(510\) 36.8248 1.63063
\(511\) −11.8979 −0.526332
\(512\) 8.03194 0.354965
\(513\) 0 0
\(514\) 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 \(-0.317532\pi\)
0.542357 + 0.840148i \(0.317532\pi\)
\(522\) −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
\(528\) 2.14852 0.0935023
\(529\) −21.8347 −0.949335
\(530\) −33.5574 −1.45764
\(531\) 6.74219 0.292586
\(532\) 0 0
\(533\) −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 \(-0.622188\pi\)
−0.374507 + 0.927224i \(0.622188\pi\)
\(548\) −19.2491 −0.822283
\(549\) −47.6382 −2.03315
\(550\) 5.61224 0.239307
\(551\) 0 0
\(552\) 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 \(-0.714486\pi\)
−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 \(0.472891\pi\)
\(570\) 0 0
\(571\) 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
\(589\) 0 0
\(590\) −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 \(-0.537346\pi\)
−0.117058 + 0.993125i \(0.537346\pi\)
\(600\) −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
\(605\) −3.49235 −0.141984
\(606\) 18.6134 0.756116
\(607\) −0.156175 −0.00633897 −0.00316948 0.999995i \(-0.501009\pi\)
−0.00316948 + 0.999995i \(0.501009\pi\)
\(608\) 0 0
\(609\) −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
\(627\) 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
\(638\) −0.233882 −0.00925950
\(639\) −79.4060 −3.14125
\(640\) 32.3734 1.27967
\(641\) −2.84360 −0.112315 −0.0561576 0.998422i \(-0.517885\pi\)
−0.0561576 + 0.998422i \(0.517885\pi\)
\(642\) 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
\(646\) 0 0
\(647\) −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 \(-0.429775\pi\)
0.218832 + 0.975763i \(0.429775\pi\)
\(660\) 14.4881 0.563948
\(661\) 22.9273 0.891768 0.445884 0.895091i \(-0.352889\pi\)
0.445884 + 0.895091i \(0.352889\pi\)
\(662\) 20.5772 0.799757
\(663\) −0.761580 −0.0295773
\(664\) 36.9794 1.43508
\(665\) 0 0
\(666\) −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 \(-0.480981\pi\)
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
\(682\) −7.16335 −0.274299
\(683\) −6.01634 −0.230209 −0.115104 0.993353i \(-0.536720\pi\)
−0.115104 + 0.993353i \(0.536720\pi\)
\(684\) 0 0
\(685\) −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
\(703\) 0 0
\(704\) 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 0
\(723\) −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 0
\(742\) 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
\(748\) 6.31370 0.230852
\(749\) 7.78135 0.284324
\(750\) −17.8310 −0.651095
\(751\) −9.55633 −0.348715 −0.174358 0.984682i \(-0.555785\pi\)
−0.174358 + 0.984682i \(0.555785\pi\)
\(752\) 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
\(760\) 0 0
\(761\) 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 \(-0.553590\pi\)
−0.167563 + 0.985861i \(0.553590\pi\)
\(774\) 49.2196 1.76916
\(775\) −66.1034 −2.37451
\(776\) 14.9784 0.537694
\(777\) −14.3306 −0.514107
\(778\) −15.5375 −0.557047
\(779\) 0 0
\(780\) 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 \(-0.413490\pi\)
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
\(792\) −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 \(-0.390399\pi\)
0.337557 + 0.941305i \(0.390399\pi\)
\(798\) 0 0
\(799\) −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 \(-0.224779\pi\)
0.760857 + 0.648920i \(0.224779\pi\)
\(812\) 0.445532 0.0156351
\(813\) −75.6814 −2.65426
\(814\) −3.51284 −0.123125
\(815\) −8.26854 −0.289634
\(816\) −9.74628 −0.341188
\(817\) 0 0
\(818\) 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 \(-0.326549\pi\)
0.518344 + 0.855172i \(0.326549\pi\)
\(828\) 8.84072 0.307236
\(829\) −52.6510 −1.82864 −0.914322 0.404988i \(-0.867276\pi\)
−0.914322 + 0.404988i \(0.867276\pi\)
\(830\) 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 \(-0.671918\pi\)
−0.514218 + 0.857660i \(0.671918\pi\)
\(840\) 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
\(855\) 0 0
\(856\) −19.2837 −0.659105
\(857\) 21.5877 0.737421 0.368711 0.929544i \(-0.379799\pi\)
0.368711 + 0.929544i \(0.379799\pi\)
\(858\) 0.130927 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 \(-0.538657\pi\)
−0.121145 + 0.992635i \(0.538657\pi\)
\(864\) 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 0
\(875\) −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
\(880\) −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 \(-0.560620\pi\)
−0.189295 + 0.981920i \(0.560620\pi\)
\(888\) 35.5141 1.19177
\(889\) 3.35032 0.112366
\(890\) −0.498910 −0.0167235
\(891\) 7.97100 0.267038
\(892\) 11.7947 0.394916
\(893\) 0 0
\(894\) 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 \(-0.630563\pi\)
−0.398770 + 0.917051i \(0.630563\pi\)
\(908\) 19.1363 0.635061
\(909\) 47.1181 1.56281
\(910\) −0.163739 −0.00542788
\(911\) −7.48540 −0.248002 −0.124001 0.992282i \(-0.539573\pi\)
−0.124001 + 0.992282i \(0.539573\pi\)
\(912\) 0 0
\(913\) −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
\(931\) 0 0
\(932\) −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 \(-0.643295\pi\)
−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
\(946\) 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
\(950\) 0 0
\(951\) −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
\(957\) −0.893906 −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
\(969\) 0 0
\(970\) 15.4224 0.495182
\(971\) 3.11733 0.100040 0.0500200 0.998748i \(-0.484071\pi\)
0.0500200 + 0.998748i \(0.484071\pi\)
\(972\) 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 \(-0.383519\pi\)
0.357823 + 0.933789i \(0.383519\pi\)
\(978\) 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 \(-0.735277\pi\)
−0.673657 + 0.739044i \(0.735277\pi\)
\(984\) 94.6174 3.01629
\(985\) 16.6026 0.529003
\(986\) 1.06096 0.0337877
\(987\) 9.21433 0.293295
\(988\) 0 0
\(989\) −11.5786 −0.368178
\(990\) −16.0257 −0.509331
\(991\) 1.69828 0.0539477 0.0269738 0.999636i \(-0.491413\pi\)
0.0269738 + 0.999636i \(0.491413\pi\)
\(992\) −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 3971.2.a.h.1.4 5
19.18 odd 2 209.2.a.c.1.2 5
57.56 even 2 1881.2.a.k.1.4 5
76.75 even 2 3344.2.a.t.1.5 5
95.94 odd 2 5225.2.a.h.1.4 5
209.208 even 2 2299.2.a.n.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.2 5 19.18 odd 2
1881.2.a.k.1.4 5 57.56 even 2
2299.2.a.n.1.4 5 209.208 even 2
3344.2.a.t.1.5 5 76.75 even 2
3971.2.a.h.1.4 5 1.1 even 1 trivial
5225.2.a.h.1.4 5 95.94 odd 2