Properties

Label 4015.2.a.i.1.10
Level $4015$
Weight $2$
Character 4015.1
Self dual yes
Analytic conductor $32.060$
Analytic rank $0$
Dimension $38$
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] = [4015,2,Mod(1,4015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4015, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4015.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: \(32.0599364115\)
Analytic rank: \(0\)
Dimension: \(38\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Character \(\chi\) \(=\) 4015.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.61749 q^{2} +2.78278 q^{3} +0.616262 q^{4} +1.00000 q^{5} -4.50111 q^{6} -5.17171 q^{7} +2.23818 q^{8} +4.74386 q^{9} +O(q^{10})\) \(q-1.61749 q^{2} +2.78278 q^{3} +0.616262 q^{4} +1.00000 q^{5} -4.50111 q^{6} -5.17171 q^{7} +2.23818 q^{8} +4.74386 q^{9} -1.61749 q^{10} -1.00000 q^{11} +1.71492 q^{12} -1.96466 q^{13} +8.36518 q^{14} +2.78278 q^{15} -4.85275 q^{16} -1.20619 q^{17} -7.67313 q^{18} +1.46585 q^{19} +0.616262 q^{20} -14.3917 q^{21} +1.61749 q^{22} +5.25760 q^{23} +6.22835 q^{24} +1.00000 q^{25} +3.17781 q^{26} +4.85277 q^{27} -3.18713 q^{28} -3.23003 q^{29} -4.50111 q^{30} +4.38231 q^{31} +3.37289 q^{32} -2.78278 q^{33} +1.95100 q^{34} -5.17171 q^{35} +2.92346 q^{36} +1.24419 q^{37} -2.37100 q^{38} -5.46722 q^{39} +2.23818 q^{40} +0.338077 q^{41} +23.2784 q^{42} +0.413393 q^{43} -0.616262 q^{44} +4.74386 q^{45} -8.50409 q^{46} -3.40991 q^{47} -13.5041 q^{48} +19.7466 q^{49} -1.61749 q^{50} -3.35657 q^{51} -1.21075 q^{52} +5.96665 q^{53} -7.84930 q^{54} -1.00000 q^{55} -11.5752 q^{56} +4.07914 q^{57} +5.22453 q^{58} +7.57994 q^{59} +1.71492 q^{60} +11.2671 q^{61} -7.08832 q^{62} -24.5339 q^{63} +4.24988 q^{64} -1.96466 q^{65} +4.50111 q^{66} +2.61273 q^{67} -0.743332 q^{68} +14.6307 q^{69} +8.36518 q^{70} -12.8357 q^{71} +10.6176 q^{72} -1.00000 q^{73} -2.01245 q^{74} +2.78278 q^{75} +0.903349 q^{76} +5.17171 q^{77} +8.84315 q^{78} -4.74712 q^{79} -4.85275 q^{80} -0.727378 q^{81} -0.546834 q^{82} +9.19417 q^{83} -8.86908 q^{84} -1.20619 q^{85} -0.668658 q^{86} -8.98847 q^{87} -2.23818 q^{88} +10.4557 q^{89} -7.67313 q^{90} +10.1607 q^{91} +3.24006 q^{92} +12.1950 q^{93} +5.51548 q^{94} +1.46585 q^{95} +9.38602 q^{96} +14.7591 q^{97} -31.9399 q^{98} -4.74386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9} + 4 q^{10} - 38 q^{11} + 12 q^{12} - q^{13} + 23 q^{14} + 5 q^{15} + 74 q^{16} + 26 q^{17} + 16 q^{18} - 10 q^{19} + 50 q^{20} + 21 q^{21} - 4 q^{22} + 10 q^{23} + 41 q^{24} + 38 q^{25} + 25 q^{26} + 5 q^{27} + 2 q^{28} + 28 q^{29} + 11 q^{30} + 24 q^{31} + 39 q^{32} - 5 q^{33} + 38 q^{34} + 111 q^{36} + 12 q^{37} + 19 q^{38} - 18 q^{39} + 15 q^{40} + 62 q^{41} - 17 q^{42} - 32 q^{43} - 50 q^{44} + 63 q^{45} - 9 q^{46} + 31 q^{47} + 53 q^{48} + 88 q^{49} + 4 q^{50} - 3 q^{51} - 21 q^{52} + 30 q^{53} + 49 q^{54} - 38 q^{55} + 32 q^{56} + 49 q^{57} + 12 q^{58} + 31 q^{59} + 12 q^{60} + 25 q^{61} + 12 q^{62} + 15 q^{63} + 137 q^{64} - q^{65} - 11 q^{66} + 20 q^{67} + 75 q^{68} + 92 q^{69} + 23 q^{70} + 32 q^{71} + 6 q^{72} - 38 q^{73} + 55 q^{74} + 5 q^{75} - 57 q^{76} - 17 q^{78} - 2 q^{79} + 74 q^{80} + 118 q^{81} + 14 q^{82} + 4 q^{83} + 22 q^{84} + 26 q^{85} + 5 q^{86} + 24 q^{87} - 15 q^{88} + 143 q^{89} + 16 q^{90} + 66 q^{91} + 29 q^{92} - 8 q^{93} - 7 q^{94} - 10 q^{95} + 59 q^{96} + 41 q^{97} - 10 q^{98} - 63 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61749 −1.14374 −0.571868 0.820346i \(-0.693781\pi\)
−0.571868 + 0.820346i \(0.693781\pi\)
\(3\) 2.78278 1.60664 0.803319 0.595549i \(-0.203065\pi\)
0.803319 + 0.595549i \(0.203065\pi\)
\(4\) 0.616262 0.308131
\(5\) 1.00000 0.447214
\(6\) −4.50111 −1.83757
\(7\) −5.17171 −1.95472 −0.977362 0.211573i \(-0.932141\pi\)
−0.977362 + 0.211573i \(0.932141\pi\)
\(8\) 2.23818 0.791315
\(9\) 4.74386 1.58129
\(10\) −1.61749 −0.511494
\(11\) −1.00000 −0.301511
\(12\) 1.71492 0.495055
\(13\) −1.96466 −0.544899 −0.272450 0.962170i \(-0.587834\pi\)
−0.272450 + 0.962170i \(0.587834\pi\)
\(14\) 8.36518 2.23569
\(15\) 2.78278 0.718510
\(16\) −4.85275 −1.21319
\(17\) −1.20619 −0.292545 −0.146273 0.989244i \(-0.546728\pi\)
−0.146273 + 0.989244i \(0.546728\pi\)
\(18\) −7.67313 −1.80857
\(19\) 1.46585 0.336290 0.168145 0.985762i \(-0.446222\pi\)
0.168145 + 0.985762i \(0.446222\pi\)
\(20\) 0.616262 0.137800
\(21\) −14.3917 −3.14054
\(22\) 1.61749 0.344849
\(23\) 5.25760 1.09628 0.548142 0.836385i \(-0.315335\pi\)
0.548142 + 0.836385i \(0.315335\pi\)
\(24\) 6.22835 1.27136
\(25\) 1.00000 0.200000
\(26\) 3.17781 0.623220
\(27\) 4.85277 0.933917
\(28\) −3.18713 −0.602311
\(29\) −3.23003 −0.599802 −0.299901 0.953970i \(-0.596954\pi\)
−0.299901 + 0.953970i \(0.596954\pi\)
\(30\) −4.50111 −0.821786
\(31\) 4.38231 0.787085 0.393543 0.919306i \(-0.371249\pi\)
0.393543 + 0.919306i \(0.371249\pi\)
\(32\) 3.37289 0.596249
\(33\) −2.78278 −0.484420
\(34\) 1.95100 0.334594
\(35\) −5.17171 −0.874179
\(36\) 2.92346 0.487243
\(37\) 1.24419 0.204543 0.102272 0.994757i \(-0.467389\pi\)
0.102272 + 0.994757i \(0.467389\pi\)
\(38\) −2.37100 −0.384626
\(39\) −5.46722 −0.875456
\(40\) 2.23818 0.353887
\(41\) 0.338077 0.0527987 0.0263994 0.999651i \(-0.491596\pi\)
0.0263994 + 0.999651i \(0.491596\pi\)
\(42\) 23.2784 3.59194
\(43\) 0.413393 0.0630418 0.0315209 0.999503i \(-0.489965\pi\)
0.0315209 + 0.999503i \(0.489965\pi\)
\(44\) −0.616262 −0.0929050
\(45\) 4.74386 0.707173
\(46\) −8.50409 −1.25386
\(47\) −3.40991 −0.497386 −0.248693 0.968582i \(-0.580001\pi\)
−0.248693 + 0.968582i \(0.580001\pi\)
\(48\) −13.5041 −1.94915
\(49\) 19.7466 2.82095
\(50\) −1.61749 −0.228747
\(51\) −3.35657 −0.470014
\(52\) −1.21075 −0.167900
\(53\) 5.96665 0.819582 0.409791 0.912179i \(-0.365602\pi\)
0.409791 + 0.912179i \(0.365602\pi\)
\(54\) −7.84930 −1.06815
\(55\) −1.00000 −0.134840
\(56\) −11.5752 −1.54680
\(57\) 4.07914 0.540296
\(58\) 5.22453 0.686015
\(59\) 7.57994 0.986824 0.493412 0.869796i \(-0.335749\pi\)
0.493412 + 0.869796i \(0.335749\pi\)
\(60\) 1.71492 0.221395
\(61\) 11.2671 1.44261 0.721305 0.692617i \(-0.243543\pi\)
0.721305 + 0.692617i \(0.243543\pi\)
\(62\) −7.08832 −0.900218
\(63\) −24.5339 −3.09098
\(64\) 4.24988 0.531235
\(65\) −1.96466 −0.243686
\(66\) 4.50111 0.554048
\(67\) 2.61273 0.319195 0.159598 0.987182i \(-0.448980\pi\)
0.159598 + 0.987182i \(0.448980\pi\)
\(68\) −0.743332 −0.0901422
\(69\) 14.6307 1.76133
\(70\) 8.36518 0.999830
\(71\) −12.8357 −1.52332 −0.761661 0.647975i \(-0.775616\pi\)
−0.761661 + 0.647975i \(0.775616\pi\)
\(72\) 10.6176 1.25130
\(73\) −1.00000 −0.117041
\(74\) −2.01245 −0.233943
\(75\) 2.78278 0.321328
\(76\) 0.903349 0.103621
\(77\) 5.17171 0.589372
\(78\) 8.84315 1.00129
\(79\) −4.74712 −0.534092 −0.267046 0.963684i \(-0.586048\pi\)
−0.267046 + 0.963684i \(0.586048\pi\)
\(80\) −4.85275 −0.542553
\(81\) −0.727378 −0.0808198
\(82\) −0.546834 −0.0603878
\(83\) 9.19417 1.00919 0.504596 0.863356i \(-0.331642\pi\)
0.504596 + 0.863356i \(0.331642\pi\)
\(84\) −8.86908 −0.967696
\(85\) −1.20619 −0.130830
\(86\) −0.668658 −0.0721032
\(87\) −8.98847 −0.963665
\(88\) −2.23818 −0.238591
\(89\) 10.4557 1.10831 0.554153 0.832415i \(-0.313042\pi\)
0.554153 + 0.832415i \(0.313042\pi\)
\(90\) −7.67313 −0.808819
\(91\) 10.1607 1.06513
\(92\) 3.24006 0.337799
\(93\) 12.1950 1.26456
\(94\) 5.51548 0.568878
\(95\) 1.46585 0.150393
\(96\) 9.38602 0.957957
\(97\) 14.7591 1.49856 0.749278 0.662256i \(-0.230401\pi\)
0.749278 + 0.662256i \(0.230401\pi\)
\(98\) −31.9399 −3.22642
\(99\) −4.74386 −0.476776
\(100\) 0.616262 0.0616262
\(101\) 3.81148 0.379256 0.189628 0.981856i \(-0.439272\pi\)
0.189628 + 0.981856i \(0.439272\pi\)
\(102\) 5.42921 0.537572
\(103\) −13.4112 −1.32144 −0.660721 0.750631i \(-0.729749\pi\)
−0.660721 + 0.750631i \(0.729749\pi\)
\(104\) −4.39726 −0.431187
\(105\) −14.3917 −1.40449
\(106\) −9.65098 −0.937386
\(107\) −4.48228 −0.433318 −0.216659 0.976247i \(-0.569516\pi\)
−0.216659 + 0.976247i \(0.569516\pi\)
\(108\) 2.99058 0.287769
\(109\) 5.50832 0.527601 0.263800 0.964577i \(-0.415024\pi\)
0.263800 + 0.964577i \(0.415024\pi\)
\(110\) 1.61749 0.154221
\(111\) 3.46230 0.328627
\(112\) 25.0970 2.37144
\(113\) −18.1196 −1.70455 −0.852276 0.523093i \(-0.824778\pi\)
−0.852276 + 0.523093i \(0.824778\pi\)
\(114\) −6.59796 −0.617955
\(115\) 5.25760 0.490273
\(116\) −1.99055 −0.184818
\(117\) −9.32008 −0.861642
\(118\) −12.2604 −1.12867
\(119\) 6.23809 0.571845
\(120\) 6.22835 0.568568
\(121\) 1.00000 0.0909091
\(122\) −18.2245 −1.64996
\(123\) 0.940793 0.0848284
\(124\) 2.70065 0.242525
\(125\) 1.00000 0.0894427
\(126\) 39.6832 3.53526
\(127\) 1.42881 0.126786 0.0633932 0.997989i \(-0.479808\pi\)
0.0633932 + 0.997989i \(0.479808\pi\)
\(128\) −13.6199 −1.20384
\(129\) 1.15038 0.101285
\(130\) 3.17781 0.278713
\(131\) 14.9359 1.30496 0.652478 0.757808i \(-0.273730\pi\)
0.652478 + 0.757808i \(0.273730\pi\)
\(132\) −1.71492 −0.149265
\(133\) −7.58097 −0.657353
\(134\) −4.22605 −0.365075
\(135\) 4.85277 0.417660
\(136\) −2.69968 −0.231495
\(137\) 19.4229 1.65941 0.829705 0.558202i \(-0.188508\pi\)
0.829705 + 0.558202i \(0.188508\pi\)
\(138\) −23.6650 −2.01450
\(139\) 15.6235 1.32517 0.662586 0.748986i \(-0.269459\pi\)
0.662586 + 0.748986i \(0.269459\pi\)
\(140\) −3.18713 −0.269362
\(141\) −9.48902 −0.799120
\(142\) 20.7616 1.74228
\(143\) 1.96466 0.164293
\(144\) −23.0207 −1.91839
\(145\) −3.23003 −0.268240
\(146\) 1.61749 0.133864
\(147\) 54.9505 4.53224
\(148\) 0.766745 0.0630260
\(149\) 16.4438 1.34713 0.673564 0.739129i \(-0.264763\pi\)
0.673564 + 0.739129i \(0.264763\pi\)
\(150\) −4.50111 −0.367514
\(151\) 20.1213 1.63744 0.818722 0.574190i \(-0.194683\pi\)
0.818722 + 0.574190i \(0.194683\pi\)
\(152\) 3.28084 0.266111
\(153\) −5.72202 −0.462598
\(154\) −8.36518 −0.674085
\(155\) 4.38231 0.351995
\(156\) −3.36924 −0.269755
\(157\) −3.60715 −0.287882 −0.143941 0.989586i \(-0.545978\pi\)
−0.143941 + 0.989586i \(0.545978\pi\)
\(158\) 7.67840 0.610860
\(159\) 16.6039 1.31677
\(160\) 3.37289 0.266651
\(161\) −27.1908 −2.14293
\(162\) 1.17652 0.0924365
\(163\) 24.8027 1.94270 0.971350 0.237654i \(-0.0763783\pi\)
0.971350 + 0.237654i \(0.0763783\pi\)
\(164\) 0.208344 0.0162689
\(165\) −2.78278 −0.216639
\(166\) −14.8715 −1.15425
\(167\) 6.22933 0.482040 0.241020 0.970520i \(-0.422518\pi\)
0.241020 + 0.970520i \(0.422518\pi\)
\(168\) −32.2113 −2.48515
\(169\) −9.14010 −0.703085
\(170\) 1.95100 0.149635
\(171\) 6.95380 0.531770
\(172\) 0.254758 0.0194251
\(173\) −5.79410 −0.440517 −0.220258 0.975442i \(-0.570690\pi\)
−0.220258 + 0.975442i \(0.570690\pi\)
\(174\) 14.5387 1.10218
\(175\) −5.17171 −0.390945
\(176\) 4.85275 0.365789
\(177\) 21.0933 1.58547
\(178\) −16.9120 −1.26761
\(179\) 0.408017 0.0304966 0.0152483 0.999884i \(-0.495146\pi\)
0.0152483 + 0.999884i \(0.495146\pi\)
\(180\) 2.92346 0.217902
\(181\) 16.8460 1.25215 0.626076 0.779762i \(-0.284660\pi\)
0.626076 + 0.779762i \(0.284660\pi\)
\(182\) −16.4347 −1.21822
\(183\) 31.3540 2.31775
\(184\) 11.7674 0.867507
\(185\) 1.24419 0.0914744
\(186\) −19.7252 −1.44632
\(187\) 1.20619 0.0882057
\(188\) −2.10140 −0.153260
\(189\) −25.0972 −1.82555
\(190\) −2.37100 −0.172010
\(191\) −21.3403 −1.54413 −0.772064 0.635545i \(-0.780775\pi\)
−0.772064 + 0.635545i \(0.780775\pi\)
\(192\) 11.8265 0.853502
\(193\) 16.3527 1.17709 0.588547 0.808463i \(-0.299700\pi\)
0.588547 + 0.808463i \(0.299700\pi\)
\(194\) −23.8726 −1.71395
\(195\) −5.46722 −0.391516
\(196\) 12.1691 0.869221
\(197\) −2.50342 −0.178361 −0.0891806 0.996015i \(-0.528425\pi\)
−0.0891806 + 0.996015i \(0.528425\pi\)
\(198\) 7.67313 0.545305
\(199\) 1.96225 0.139100 0.0695499 0.997578i \(-0.477844\pi\)
0.0695499 + 0.997578i \(0.477844\pi\)
\(200\) 2.23818 0.158263
\(201\) 7.27064 0.512831
\(202\) −6.16501 −0.433769
\(203\) 16.7048 1.17245
\(204\) −2.06853 −0.144826
\(205\) 0.338077 0.0236123
\(206\) 21.6924 1.51138
\(207\) 24.9413 1.73354
\(208\) 9.53400 0.661064
\(209\) −1.46585 −0.101395
\(210\) 23.2784 1.60637
\(211\) 6.77598 0.466478 0.233239 0.972419i \(-0.425068\pi\)
0.233239 + 0.972419i \(0.425068\pi\)
\(212\) 3.67702 0.252539
\(213\) −35.7190 −2.44743
\(214\) 7.25002 0.495601
\(215\) 0.413393 0.0281932
\(216\) 10.8614 0.739023
\(217\) −22.6640 −1.53854
\(218\) −8.90963 −0.603436
\(219\) −2.78278 −0.188043
\(220\) −0.616262 −0.0415484
\(221\) 2.36976 0.159408
\(222\) −5.60022 −0.375862
\(223\) −8.18519 −0.548121 −0.274060 0.961712i \(-0.588367\pi\)
−0.274060 + 0.961712i \(0.588367\pi\)
\(224\) −17.4436 −1.16550
\(225\) 4.74386 0.316257
\(226\) 29.3082 1.94956
\(227\) −2.00813 −0.133284 −0.0666422 0.997777i \(-0.521229\pi\)
−0.0666422 + 0.997777i \(0.521229\pi\)
\(228\) 2.51382 0.166482
\(229\) 2.02799 0.134013 0.0670067 0.997753i \(-0.478655\pi\)
0.0670067 + 0.997753i \(0.478655\pi\)
\(230\) −8.50409 −0.560743
\(231\) 14.3917 0.946907
\(232\) −7.22939 −0.474632
\(233\) −3.39690 −0.222538 −0.111269 0.993790i \(-0.535492\pi\)
−0.111269 + 0.993790i \(0.535492\pi\)
\(234\) 15.0751 0.985490
\(235\) −3.40991 −0.222438
\(236\) 4.67123 0.304071
\(237\) −13.2102 −0.858093
\(238\) −10.0900 −0.654040
\(239\) −21.4343 −1.38647 −0.693234 0.720713i \(-0.743815\pi\)
−0.693234 + 0.720713i \(0.743815\pi\)
\(240\) −13.5041 −0.871687
\(241\) −22.4218 −1.44432 −0.722159 0.691727i \(-0.756850\pi\)
−0.722159 + 0.691727i \(0.756850\pi\)
\(242\) −1.61749 −0.103976
\(243\) −16.5825 −1.06377
\(244\) 6.94351 0.444513
\(245\) 19.7466 1.26157
\(246\) −1.52172 −0.0970213
\(247\) −2.87990 −0.183244
\(248\) 9.80838 0.622833
\(249\) 25.5854 1.62141
\(250\) −1.61749 −0.102299
\(251\) 12.4099 0.783308 0.391654 0.920113i \(-0.371903\pi\)
0.391654 + 0.920113i \(0.371903\pi\)
\(252\) −15.1193 −0.952426
\(253\) −5.25760 −0.330542
\(254\) −2.31108 −0.145010
\(255\) −3.35657 −0.210197
\(256\) 13.5303 0.845641
\(257\) −10.1366 −0.632301 −0.316150 0.948709i \(-0.602390\pi\)
−0.316150 + 0.948709i \(0.602390\pi\)
\(258\) −1.86073 −0.115844
\(259\) −6.43458 −0.399825
\(260\) −1.21075 −0.0750873
\(261\) −15.3228 −0.948459
\(262\) −24.1586 −1.49252
\(263\) 13.3700 0.824428 0.412214 0.911087i \(-0.364756\pi\)
0.412214 + 0.911087i \(0.364756\pi\)
\(264\) −6.22835 −0.383329
\(265\) 5.96665 0.366528
\(266\) 12.2621 0.751838
\(267\) 29.0960 1.78065
\(268\) 1.61012 0.0983540
\(269\) −8.84755 −0.539444 −0.269722 0.962938i \(-0.586932\pi\)
−0.269722 + 0.962938i \(0.586932\pi\)
\(270\) −7.84930 −0.477693
\(271\) −9.97716 −0.606069 −0.303035 0.952980i \(-0.598000\pi\)
−0.303035 + 0.952980i \(0.598000\pi\)
\(272\) 5.85335 0.354912
\(273\) 28.2749 1.71127
\(274\) −31.4163 −1.89793
\(275\) −1.00000 −0.0603023
\(276\) 9.01636 0.542721
\(277\) −0.0362213 −0.00217633 −0.00108816 0.999999i \(-0.500346\pi\)
−0.00108816 + 0.999999i \(0.500346\pi\)
\(278\) −25.2709 −1.51565
\(279\) 20.7890 1.24461
\(280\) −11.5752 −0.691751
\(281\) 5.16239 0.307963 0.153981 0.988074i \(-0.450790\pi\)
0.153981 + 0.988074i \(0.450790\pi\)
\(282\) 15.3484 0.913982
\(283\) 3.77015 0.224112 0.112056 0.993702i \(-0.464256\pi\)
0.112056 + 0.993702i \(0.464256\pi\)
\(284\) −7.91018 −0.469383
\(285\) 4.07914 0.241628
\(286\) −3.17781 −0.187908
\(287\) −1.74844 −0.103207
\(288\) 16.0005 0.942841
\(289\) −15.5451 −0.914417
\(290\) 5.22453 0.306795
\(291\) 41.0712 2.40764
\(292\) −0.616262 −0.0360640
\(293\) 22.4755 1.31303 0.656515 0.754313i \(-0.272030\pi\)
0.656515 + 0.754313i \(0.272030\pi\)
\(294\) −88.8817 −5.18369
\(295\) 7.57994 0.441321
\(296\) 2.78471 0.161858
\(297\) −4.85277 −0.281587
\(298\) −26.5976 −1.54076
\(299\) −10.3294 −0.597364
\(300\) 1.71492 0.0990110
\(301\) −2.13795 −0.123229
\(302\) −32.5459 −1.87280
\(303\) 10.6065 0.609328
\(304\) −7.11341 −0.407982
\(305\) 11.2671 0.645155
\(306\) 9.25528 0.529089
\(307\) −21.1061 −1.20459 −0.602293 0.798275i \(-0.705746\pi\)
−0.602293 + 0.798275i \(0.705746\pi\)
\(308\) 3.18713 0.181604
\(309\) −37.3203 −2.12308
\(310\) −7.08832 −0.402590
\(311\) −7.31842 −0.414989 −0.207495 0.978236i \(-0.566531\pi\)
−0.207495 + 0.978236i \(0.566531\pi\)
\(312\) −12.2366 −0.692761
\(313\) −10.8806 −0.615010 −0.307505 0.951546i \(-0.599494\pi\)
−0.307505 + 0.951546i \(0.599494\pi\)
\(314\) 5.83451 0.329261
\(315\) −24.5339 −1.38233
\(316\) −2.92547 −0.164570
\(317\) 5.69923 0.320100 0.160050 0.987109i \(-0.448834\pi\)
0.160050 + 0.987109i \(0.448834\pi\)
\(318\) −26.8565 −1.50604
\(319\) 3.23003 0.180847
\(320\) 4.24988 0.237576
\(321\) −12.4732 −0.696185
\(322\) 43.9807 2.45095
\(323\) −1.76810 −0.0983799
\(324\) −0.448255 −0.0249031
\(325\) −1.96466 −0.108980
\(326\) −40.1181 −2.22193
\(327\) 15.3284 0.847664
\(328\) 0.756676 0.0417804
\(329\) 17.6351 0.972253
\(330\) 4.50111 0.247778
\(331\) 32.0185 1.75989 0.879947 0.475072i \(-0.157578\pi\)
0.879947 + 0.475072i \(0.157578\pi\)
\(332\) 5.66602 0.310963
\(333\) 5.90225 0.323441
\(334\) −10.0759 −0.551326
\(335\) 2.61273 0.142748
\(336\) 69.8394 3.81005
\(337\) −15.7934 −0.860319 −0.430159 0.902753i \(-0.641543\pi\)
−0.430159 + 0.902753i \(0.641543\pi\)
\(338\) 14.7840 0.804143
\(339\) −50.4229 −2.73860
\(340\) −0.743332 −0.0403128
\(341\) −4.38231 −0.237315
\(342\) −11.2477 −0.608204
\(343\) −65.9220 −3.55945
\(344\) 0.925247 0.0498860
\(345\) 14.6307 0.787692
\(346\) 9.37187 0.503835
\(347\) −0.903324 −0.0484930 −0.0242465 0.999706i \(-0.507719\pi\)
−0.0242465 + 0.999706i \(0.507719\pi\)
\(348\) −5.53925 −0.296935
\(349\) −11.4215 −0.611379 −0.305689 0.952131i \(-0.598887\pi\)
−0.305689 + 0.952131i \(0.598887\pi\)
\(350\) 8.36518 0.447138
\(351\) −9.53406 −0.508891
\(352\) −3.37289 −0.179776
\(353\) 19.1505 1.01928 0.509639 0.860388i \(-0.329779\pi\)
0.509639 + 0.860388i \(0.329779\pi\)
\(354\) −34.1181 −1.81336
\(355\) −12.8357 −0.681251
\(356\) 6.44347 0.341503
\(357\) 17.3592 0.918748
\(358\) −0.659961 −0.0348800
\(359\) −21.6269 −1.14142 −0.570712 0.821150i \(-0.693333\pi\)
−0.570712 + 0.821150i \(0.693333\pi\)
\(360\) 10.6176 0.559597
\(361\) −16.8513 −0.886909
\(362\) −27.2482 −1.43213
\(363\) 2.78278 0.146058
\(364\) 6.26163 0.328199
\(365\) −1.00000 −0.0523424
\(366\) −50.7146 −2.65090
\(367\) −10.0892 −0.526652 −0.263326 0.964707i \(-0.584819\pi\)
−0.263326 + 0.964707i \(0.584819\pi\)
\(368\) −25.5138 −1.33000
\(369\) 1.60379 0.0834899
\(370\) −2.01245 −0.104623
\(371\) −30.8578 −1.60206
\(372\) 7.51531 0.389651
\(373\) −8.23827 −0.426562 −0.213281 0.976991i \(-0.568415\pi\)
−0.213281 + 0.976991i \(0.568415\pi\)
\(374\) −1.95100 −0.100884
\(375\) 2.78278 0.143702
\(376\) −7.63198 −0.393589
\(377\) 6.34592 0.326832
\(378\) 40.5943 2.08795
\(379\) −12.9820 −0.666838 −0.333419 0.942779i \(-0.608202\pi\)
−0.333419 + 0.942779i \(0.608202\pi\)
\(380\) 0.903349 0.0463408
\(381\) 3.97606 0.203700
\(382\) 34.5176 1.76607
\(383\) 15.1968 0.776522 0.388261 0.921550i \(-0.373076\pi\)
0.388261 + 0.921550i \(0.373076\pi\)
\(384\) −37.9012 −1.93414
\(385\) 5.17171 0.263575
\(386\) −26.4503 −1.34628
\(387\) 1.96108 0.0996872
\(388\) 9.09544 0.461751
\(389\) 26.1275 1.32472 0.662359 0.749187i \(-0.269555\pi\)
0.662359 + 0.749187i \(0.269555\pi\)
\(390\) 8.84315 0.447790
\(391\) −6.34168 −0.320713
\(392\) 44.1965 2.23226
\(393\) 41.5633 2.09659
\(394\) 4.04925 0.203998
\(395\) −4.74712 −0.238853
\(396\) −2.92346 −0.146909
\(397\) 10.5780 0.530896 0.265448 0.964125i \(-0.414480\pi\)
0.265448 + 0.964125i \(0.414480\pi\)
\(398\) −3.17391 −0.159093
\(399\) −21.0962 −1.05613
\(400\) −4.85275 −0.242637
\(401\) 14.8349 0.740822 0.370411 0.928868i \(-0.379217\pi\)
0.370411 + 0.928868i \(0.379217\pi\)
\(402\) −11.7602 −0.586543
\(403\) −8.60975 −0.428882
\(404\) 2.34887 0.116861
\(405\) −0.727378 −0.0361437
\(406\) −27.0198 −1.34097
\(407\) −1.24419 −0.0616720
\(408\) −7.51260 −0.371929
\(409\) 39.4074 1.94857 0.974284 0.225322i \(-0.0723435\pi\)
0.974284 + 0.225322i \(0.0723435\pi\)
\(410\) −0.546834 −0.0270062
\(411\) 54.0497 2.66607
\(412\) −8.26480 −0.407177
\(413\) −39.2013 −1.92897
\(414\) −40.3422 −1.98271
\(415\) 9.19417 0.451324
\(416\) −6.62660 −0.324896
\(417\) 43.4768 2.12907
\(418\) 2.37100 0.115969
\(419\) 26.4856 1.29391 0.646954 0.762529i \(-0.276043\pi\)
0.646954 + 0.762529i \(0.276043\pi\)
\(420\) −8.86908 −0.432767
\(421\) −5.18173 −0.252542 −0.126271 0.991996i \(-0.540301\pi\)
−0.126271 + 0.991996i \(0.540301\pi\)
\(422\) −10.9601 −0.533527
\(423\) −16.1761 −0.786510
\(424\) 13.3544 0.648548
\(425\) −1.20619 −0.0585090
\(426\) 57.7751 2.79921
\(427\) −58.2705 −2.81991
\(428\) −2.76226 −0.133519
\(429\) 5.46722 0.263960
\(430\) −0.668658 −0.0322455
\(431\) −12.0400 −0.579946 −0.289973 0.957035i \(-0.593646\pi\)
−0.289973 + 0.957035i \(0.593646\pi\)
\(432\) −23.5493 −1.13302
\(433\) −6.02638 −0.289609 −0.144805 0.989460i \(-0.546255\pi\)
−0.144805 + 0.989460i \(0.546255\pi\)
\(434\) 36.6588 1.75968
\(435\) −8.98847 −0.430964
\(436\) 3.39457 0.162570
\(437\) 7.70686 0.368669
\(438\) 4.50111 0.215071
\(439\) −2.84212 −0.135647 −0.0678236 0.997697i \(-0.521606\pi\)
−0.0678236 + 0.997697i \(0.521606\pi\)
\(440\) −2.23818 −0.106701
\(441\) 93.6753 4.46073
\(442\) −3.83306 −0.182320
\(443\) 20.9707 0.996348 0.498174 0.867077i \(-0.334004\pi\)
0.498174 + 0.867077i \(0.334004\pi\)
\(444\) 2.13368 0.101260
\(445\) 10.4557 0.495649
\(446\) 13.2394 0.626905
\(447\) 45.7594 2.16435
\(448\) −21.9792 −1.03842
\(449\) 12.4509 0.587594 0.293797 0.955868i \(-0.405081\pi\)
0.293797 + 0.955868i \(0.405081\pi\)
\(450\) −7.67313 −0.361715
\(451\) −0.338077 −0.0159194
\(452\) −11.1664 −0.525225
\(453\) 55.9930 2.63078
\(454\) 3.24813 0.152442
\(455\) 10.1607 0.476340
\(456\) 9.12984 0.427544
\(457\) −41.7766 −1.95423 −0.977114 0.212715i \(-0.931770\pi\)
−0.977114 + 0.212715i \(0.931770\pi\)
\(458\) −3.28025 −0.153276
\(459\) −5.85339 −0.273213
\(460\) 3.24006 0.151068
\(461\) 38.3701 1.78708 0.893538 0.448988i \(-0.148215\pi\)
0.893538 + 0.448988i \(0.148215\pi\)
\(462\) −23.2784 −1.08301
\(463\) 33.5022 1.55698 0.778490 0.627657i \(-0.215986\pi\)
0.778490 + 0.627657i \(0.215986\pi\)
\(464\) 15.6745 0.727672
\(465\) 12.1950 0.565529
\(466\) 5.49444 0.254525
\(467\) −23.3483 −1.08043 −0.540216 0.841526i \(-0.681658\pi\)
−0.540216 + 0.841526i \(0.681658\pi\)
\(468\) −5.74361 −0.265498
\(469\) −13.5123 −0.623939
\(470\) 5.51548 0.254410
\(471\) −10.0379 −0.462522
\(472\) 16.9653 0.780889
\(473\) −0.413393 −0.0190078
\(474\) 21.3673 0.981432
\(475\) 1.46585 0.0672579
\(476\) 3.84430 0.176203
\(477\) 28.3049 1.29599
\(478\) 34.6696 1.58575
\(479\) 15.3855 0.702979 0.351490 0.936192i \(-0.385675\pi\)
0.351490 + 0.936192i \(0.385675\pi\)
\(480\) 9.38602 0.428411
\(481\) −2.44441 −0.111455
\(482\) 36.2670 1.65192
\(483\) −75.6659 −3.44292
\(484\) 0.616262 0.0280119
\(485\) 14.7591 0.670174
\(486\) 26.8219 1.21667
\(487\) −24.1495 −1.09432 −0.547159 0.837028i \(-0.684291\pi\)
−0.547159 + 0.837028i \(0.684291\pi\)
\(488\) 25.2179 1.14156
\(489\) 69.0205 3.12122
\(490\) −31.9399 −1.44290
\(491\) 12.7278 0.574398 0.287199 0.957871i \(-0.407276\pi\)
0.287199 + 0.957871i \(0.407276\pi\)
\(492\) 0.579775 0.0261383
\(493\) 3.89605 0.175469
\(494\) 4.65820 0.209583
\(495\) −4.74386 −0.213221
\(496\) −21.2662 −0.954881
\(497\) 66.3828 2.97768
\(498\) −41.3840 −1.85446
\(499\) 26.8541 1.20215 0.601077 0.799191i \(-0.294739\pi\)
0.601077 + 0.799191i \(0.294739\pi\)
\(500\) 0.616262 0.0275601
\(501\) 17.3349 0.774464
\(502\) −20.0729 −0.895897
\(503\) −3.42219 −0.152588 −0.0762940 0.997085i \(-0.524309\pi\)
−0.0762940 + 0.997085i \(0.524309\pi\)
\(504\) −54.9112 −2.44594
\(505\) 3.81148 0.169609
\(506\) 8.50409 0.378053
\(507\) −25.4349 −1.12960
\(508\) 0.880522 0.0390668
\(509\) −18.8514 −0.835575 −0.417787 0.908545i \(-0.637194\pi\)
−0.417787 + 0.908545i \(0.637194\pi\)
\(510\) 5.42921 0.240409
\(511\) 5.17171 0.228783
\(512\) 5.35481 0.236652
\(513\) 7.11345 0.314066
\(514\) 16.3957 0.723185
\(515\) −13.4112 −0.590967
\(516\) 0.708936 0.0312092
\(517\) 3.40991 0.149968
\(518\) 10.4078 0.457294
\(519\) −16.1237 −0.707751
\(520\) −4.39726 −0.192833
\(521\) −10.7512 −0.471017 −0.235509 0.971872i \(-0.575676\pi\)
−0.235509 + 0.971872i \(0.575676\pi\)
\(522\) 24.7845 1.08479
\(523\) −12.5296 −0.547879 −0.273940 0.961747i \(-0.588327\pi\)
−0.273940 + 0.961747i \(0.588327\pi\)
\(524\) 9.20442 0.402097
\(525\) −14.3917 −0.628107
\(526\) −21.6257 −0.942927
\(527\) −5.28591 −0.230258
\(528\) 13.5041 0.587691
\(529\) 4.64231 0.201840
\(530\) −9.65098 −0.419212
\(531\) 35.9582 1.56045
\(532\) −4.67186 −0.202551
\(533\) −0.664206 −0.0287700
\(534\) −47.0624 −2.03659
\(535\) −4.48228 −0.193786
\(536\) 5.84774 0.252584
\(537\) 1.13542 0.0489970
\(538\) 14.3108 0.616982
\(539\) −19.7466 −0.850548
\(540\) 2.99058 0.128694
\(541\) 21.8026 0.937368 0.468684 0.883366i \(-0.344728\pi\)
0.468684 + 0.883366i \(0.344728\pi\)
\(542\) 16.1379 0.693183
\(543\) 46.8787 2.01176
\(544\) −4.06837 −0.174430
\(545\) 5.50832 0.235950
\(546\) −45.7343 −1.95725
\(547\) −1.26711 −0.0541776 −0.0270888 0.999633i \(-0.508624\pi\)
−0.0270888 + 0.999633i \(0.508624\pi\)
\(548\) 11.9696 0.511316
\(549\) 53.4498 2.28118
\(550\) 1.61749 0.0689698
\(551\) −4.73475 −0.201707
\(552\) 32.7462 1.39377
\(553\) 24.5507 1.04400
\(554\) 0.0585874 0.00248914
\(555\) 3.46230 0.146966
\(556\) 9.62819 0.408326
\(557\) 35.0295 1.48425 0.742124 0.670263i \(-0.233819\pi\)
0.742124 + 0.670263i \(0.233819\pi\)
\(558\) −33.6260 −1.42350
\(559\) −0.812177 −0.0343514
\(560\) 25.0970 1.06054
\(561\) 3.35657 0.141715
\(562\) −8.35010 −0.352228
\(563\) 3.56704 0.150333 0.0751664 0.997171i \(-0.476051\pi\)
0.0751664 + 0.997171i \(0.476051\pi\)
\(564\) −5.84772 −0.246234
\(565\) −18.1196 −0.762298
\(566\) −6.09816 −0.256325
\(567\) 3.76179 0.157980
\(568\) −28.7287 −1.20543
\(569\) 44.4117 1.86183 0.930917 0.365230i \(-0.119010\pi\)
0.930917 + 0.365230i \(0.119010\pi\)
\(570\) −6.59796 −0.276358
\(571\) −25.4825 −1.06641 −0.533206 0.845986i \(-0.679013\pi\)
−0.533206 + 0.845986i \(0.679013\pi\)
\(572\) 1.21075 0.0506238
\(573\) −59.3852 −2.48085
\(574\) 2.82807 0.118041
\(575\) 5.25760 0.219257
\(576\) 20.1608 0.840035
\(577\) −11.2279 −0.467426 −0.233713 0.972306i \(-0.575088\pi\)
−0.233713 + 0.972306i \(0.575088\pi\)
\(578\) 25.1440 1.04585
\(579\) 45.5060 1.89116
\(580\) −1.99055 −0.0826529
\(581\) −47.5496 −1.97269
\(582\) −66.4321 −2.75370
\(583\) −5.96665 −0.247113
\(584\) −2.23818 −0.0926164
\(585\) −9.32008 −0.385338
\(586\) −36.3537 −1.50176
\(587\) −28.5203 −1.17716 −0.588579 0.808440i \(-0.700312\pi\)
−0.588579 + 0.808440i \(0.700312\pi\)
\(588\) 33.8639 1.39652
\(589\) 6.42381 0.264689
\(590\) −12.2604 −0.504755
\(591\) −6.96646 −0.286562
\(592\) −6.03772 −0.248149
\(593\) −41.2072 −1.69218 −0.846088 0.533044i \(-0.821048\pi\)
−0.846088 + 0.533044i \(0.821048\pi\)
\(594\) 7.84930 0.322061
\(595\) 6.23809 0.255737
\(596\) 10.1337 0.415092
\(597\) 5.46050 0.223483
\(598\) 16.7077 0.683227
\(599\) 1.93477 0.0790525 0.0395263 0.999219i \(-0.487415\pi\)
0.0395263 + 0.999219i \(0.487415\pi\)
\(600\) 6.22835 0.254271
\(601\) −45.6035 −1.86021 −0.930104 0.367297i \(-0.880283\pi\)
−0.930104 + 0.367297i \(0.880283\pi\)
\(602\) 3.45811 0.140942
\(603\) 12.3944 0.504739
\(604\) 12.4000 0.504547
\(605\) 1.00000 0.0406558
\(606\) −17.1559 −0.696910
\(607\) 40.0947 1.62739 0.813697 0.581289i \(-0.197451\pi\)
0.813697 + 0.581289i \(0.197451\pi\)
\(608\) 4.94416 0.200512
\(609\) 46.4858 1.88370
\(610\) −18.2245 −0.737887
\(611\) 6.69932 0.271025
\(612\) −3.52626 −0.142541
\(613\) −25.4550 −1.02812 −0.514059 0.857755i \(-0.671859\pi\)
−0.514059 + 0.857755i \(0.671859\pi\)
\(614\) 34.1388 1.37773
\(615\) 0.940793 0.0379364
\(616\) 11.5752 0.466379
\(617\) 25.8050 1.03887 0.519434 0.854510i \(-0.326143\pi\)
0.519434 + 0.854510i \(0.326143\pi\)
\(618\) 60.3651 2.42824
\(619\) 16.9067 0.679536 0.339768 0.940509i \(-0.389651\pi\)
0.339768 + 0.940509i \(0.389651\pi\)
\(620\) 2.70065 0.108461
\(621\) 25.5139 1.02384
\(622\) 11.8374 0.474638
\(623\) −54.0741 −2.16643
\(624\) 26.5310 1.06209
\(625\) 1.00000 0.0400000
\(626\) 17.5993 0.703409
\(627\) −4.07914 −0.162905
\(628\) −2.22295 −0.0887053
\(629\) −1.50073 −0.0598381
\(630\) 39.6832 1.58102
\(631\) 31.9729 1.27282 0.636410 0.771351i \(-0.280419\pi\)
0.636410 + 0.771351i \(0.280419\pi\)
\(632\) −10.6249 −0.422635
\(633\) 18.8561 0.749461
\(634\) −9.21842 −0.366110
\(635\) 1.42881 0.0567006
\(636\) 10.2323 0.405738
\(637\) −38.7955 −1.53713
\(638\) −5.22453 −0.206841
\(639\) −60.8910 −2.40881
\(640\) −13.6199 −0.538374
\(641\) 36.4576 1.43999 0.719994 0.693980i \(-0.244145\pi\)
0.719994 + 0.693980i \(0.244145\pi\)
\(642\) 20.1752 0.796252
\(643\) −48.0124 −1.89342 −0.946712 0.322083i \(-0.895617\pi\)
−0.946712 + 0.322083i \(0.895617\pi\)
\(644\) −16.7566 −0.660304
\(645\) 1.15038 0.0452962
\(646\) 2.85988 0.112521
\(647\) −47.7463 −1.87710 −0.938550 0.345144i \(-0.887830\pi\)
−0.938550 + 0.345144i \(0.887830\pi\)
\(648\) −1.62800 −0.0639539
\(649\) −7.57994 −0.297539
\(650\) 3.17781 0.124644
\(651\) −63.0690 −2.47187
\(652\) 15.2850 0.598606
\(653\) −22.7666 −0.890928 −0.445464 0.895300i \(-0.646961\pi\)
−0.445464 + 0.895300i \(0.646961\pi\)
\(654\) −24.7935 −0.969503
\(655\) 14.9359 0.583594
\(656\) −1.64060 −0.0640547
\(657\) −4.74386 −0.185076
\(658\) −28.5245 −1.11200
\(659\) 24.3477 0.948452 0.474226 0.880403i \(-0.342728\pi\)
0.474226 + 0.880403i \(0.342728\pi\)
\(660\) −1.71492 −0.0667532
\(661\) −7.05610 −0.274450 −0.137225 0.990540i \(-0.543818\pi\)
−0.137225 + 0.990540i \(0.543818\pi\)
\(662\) −51.7894 −2.01285
\(663\) 6.59453 0.256110
\(664\) 20.5782 0.798589
\(665\) −7.58097 −0.293977
\(666\) −9.54680 −0.369931
\(667\) −16.9822 −0.657554
\(668\) 3.83890 0.148531
\(669\) −22.7776 −0.880632
\(670\) −4.22605 −0.163267
\(671\) −11.2671 −0.434963
\(672\) −48.5418 −1.87254
\(673\) 30.5927 1.17926 0.589630 0.807673i \(-0.299273\pi\)
0.589630 + 0.807673i \(0.299273\pi\)
\(674\) 25.5455 0.983977
\(675\) 4.85277 0.186783
\(676\) −5.63270 −0.216642
\(677\) −40.7328 −1.56549 −0.782744 0.622344i \(-0.786181\pi\)
−0.782744 + 0.622344i \(0.786181\pi\)
\(678\) 81.5584 3.13223
\(679\) −76.3296 −2.92926
\(680\) −2.69968 −0.103528
\(681\) −5.58819 −0.214140
\(682\) 7.08832 0.271426
\(683\) 19.5672 0.748720 0.374360 0.927283i \(-0.377862\pi\)
0.374360 + 0.927283i \(0.377862\pi\)
\(684\) 4.28536 0.163855
\(685\) 19.4229 0.742111
\(686\) 106.628 4.07107
\(687\) 5.64345 0.215311
\(688\) −2.00609 −0.0764815
\(689\) −11.7224 −0.446590
\(690\) −23.6650 −0.900911
\(691\) −7.76507 −0.295397 −0.147699 0.989032i \(-0.547187\pi\)
−0.147699 + 0.989032i \(0.547187\pi\)
\(692\) −3.57068 −0.135737
\(693\) 24.5339 0.931965
\(694\) 1.46111 0.0554631
\(695\) 15.6235 0.592634
\(696\) −20.1178 −0.762563
\(697\) −0.407786 −0.0154460
\(698\) 18.4741 0.699256
\(699\) −9.45283 −0.357539
\(700\) −3.18713 −0.120462
\(701\) −15.3991 −0.581615 −0.290807 0.956782i \(-0.593924\pi\)
−0.290807 + 0.956782i \(0.593924\pi\)
\(702\) 15.4212 0.582036
\(703\) 1.82379 0.0687857
\(704\) −4.24988 −0.160173
\(705\) −9.48902 −0.357377
\(706\) −30.9757 −1.16578
\(707\) −19.7119 −0.741341
\(708\) 12.9990 0.488532
\(709\) 19.4906 0.731984 0.365992 0.930618i \(-0.380730\pi\)
0.365992 + 0.930618i \(0.380730\pi\)
\(710\) 20.7616 0.779170
\(711\) −22.5197 −0.844553
\(712\) 23.4018 0.877019
\(713\) 23.0404 0.862870
\(714\) −28.0783 −1.05080
\(715\) 1.96466 0.0734742
\(716\) 0.251445 0.00939694
\(717\) −59.6469 −2.22755
\(718\) 34.9812 1.30549
\(719\) −20.8639 −0.778092 −0.389046 0.921218i \(-0.627195\pi\)
−0.389046 + 0.921218i \(0.627195\pi\)
\(720\) −23.0207 −0.857932
\(721\) 69.3588 2.58306
\(722\) 27.2567 1.01439
\(723\) −62.3950 −2.32050
\(724\) 10.3815 0.385827
\(725\) −3.23003 −0.119960
\(726\) −4.50111 −0.167052
\(727\) 7.67699 0.284724 0.142362 0.989815i \(-0.454530\pi\)
0.142362 + 0.989815i \(0.454530\pi\)
\(728\) 22.7414 0.842852
\(729\) −43.9632 −1.62827
\(730\) 1.61749 0.0598659
\(731\) −0.498632 −0.0184426
\(732\) 19.3223 0.714171
\(733\) −1.71918 −0.0634994 −0.0317497 0.999496i \(-0.510108\pi\)
−0.0317497 + 0.999496i \(0.510108\pi\)
\(734\) 16.3191 0.602350
\(735\) 54.9505 2.02688
\(736\) 17.7333 0.653659
\(737\) −2.61273 −0.0962410
\(738\) −2.59411 −0.0954903
\(739\) 26.1527 0.962042 0.481021 0.876709i \(-0.340266\pi\)
0.481021 + 0.876709i \(0.340266\pi\)
\(740\) 0.766745 0.0281861
\(741\) −8.01413 −0.294407
\(742\) 49.9121 1.83233
\(743\) 11.6118 0.425994 0.212997 0.977053i \(-0.431677\pi\)
0.212997 + 0.977053i \(0.431677\pi\)
\(744\) 27.2945 1.00067
\(745\) 16.4438 0.602454
\(746\) 13.3253 0.487874
\(747\) 43.6159 1.59582
\(748\) 0.743332 0.0271789
\(749\) 23.1811 0.847017
\(750\) −4.50111 −0.164357
\(751\) 38.7626 1.41447 0.707234 0.706980i \(-0.249943\pi\)
0.707234 + 0.706980i \(0.249943\pi\)
\(752\) 16.5474 0.603422
\(753\) 34.5341 1.25849
\(754\) −10.2644 −0.373809
\(755\) 20.1213 0.732287
\(756\) −15.4664 −0.562509
\(757\) −19.5137 −0.709237 −0.354619 0.935011i \(-0.615389\pi\)
−0.354619 + 0.935011i \(0.615389\pi\)
\(758\) 20.9981 0.762687
\(759\) −14.6307 −0.531062
\(760\) 3.28084 0.119008
\(761\) 5.44925 0.197535 0.0987676 0.995111i \(-0.468510\pi\)
0.0987676 + 0.995111i \(0.468510\pi\)
\(762\) −6.43123 −0.232979
\(763\) −28.4874 −1.03131
\(764\) −13.1512 −0.475793
\(765\) −5.72202 −0.206880
\(766\) −24.5807 −0.888135
\(767\) −14.8920 −0.537720
\(768\) 37.6517 1.35864
\(769\) 5.77063 0.208094 0.104047 0.994572i \(-0.466821\pi\)
0.104047 + 0.994572i \(0.466821\pi\)
\(770\) −8.36518 −0.301460
\(771\) −28.2078 −1.01588
\(772\) 10.0776 0.362699
\(773\) 19.9927 0.719087 0.359544 0.933128i \(-0.382932\pi\)
0.359544 + 0.933128i \(0.382932\pi\)
\(774\) −3.17202 −0.114016
\(775\) 4.38231 0.157417
\(776\) 33.0334 1.18583
\(777\) −17.9060 −0.642374
\(778\) −42.2609 −1.51513
\(779\) 0.495570 0.0177557
\(780\) −3.36924 −0.120638
\(781\) 12.8357 0.459299
\(782\) 10.2576 0.366810
\(783\) −15.6746 −0.560165
\(784\) −95.8254 −3.42234
\(785\) −3.60715 −0.128745
\(786\) −67.2281 −2.39795
\(787\) −16.0333 −0.571527 −0.285763 0.958300i \(-0.592247\pi\)
−0.285763 + 0.958300i \(0.592247\pi\)
\(788\) −1.54276 −0.0549586
\(789\) 37.2057 1.32456
\(790\) 7.67840 0.273185
\(791\) 93.7095 3.33193
\(792\) −10.6176 −0.377280
\(793\) −22.1361 −0.786077
\(794\) −17.1098 −0.607205
\(795\) 16.6039 0.588879
\(796\) 1.20926 0.0428610
\(797\) −24.1224 −0.854459 −0.427229 0.904143i \(-0.640510\pi\)
−0.427229 + 0.904143i \(0.640510\pi\)
\(798\) 34.1228 1.20793
\(799\) 4.11301 0.145508
\(800\) 3.37289 0.119250
\(801\) 49.6005 1.75255
\(802\) −23.9953 −0.847304
\(803\) 1.00000 0.0352892
\(804\) 4.48062 0.158019
\(805\) −27.1908 −0.958349
\(806\) 13.9262 0.490528
\(807\) −24.6208 −0.866692
\(808\) 8.53076 0.300111
\(809\) 12.4127 0.436406 0.218203 0.975903i \(-0.429981\pi\)
0.218203 + 0.975903i \(0.429981\pi\)
\(810\) 1.17652 0.0413388
\(811\) −11.2548 −0.395211 −0.197606 0.980282i \(-0.563317\pi\)
−0.197606 + 0.980282i \(0.563317\pi\)
\(812\) 10.2945 0.361267
\(813\) −27.7642 −0.973734
\(814\) 2.01245 0.0705365
\(815\) 24.8027 0.868802
\(816\) 16.2886 0.570215
\(817\) 0.605973 0.0212003
\(818\) −63.7409 −2.22865
\(819\) 48.2008 1.68427
\(820\) 0.208344 0.00727568
\(821\) 16.6170 0.579937 0.289968 0.957036i \(-0.406355\pi\)
0.289968 + 0.957036i \(0.406355\pi\)
\(822\) −87.4246 −3.04928
\(823\) 5.60946 0.195533 0.0977667 0.995209i \(-0.468830\pi\)
0.0977667 + 0.995209i \(0.468830\pi\)
\(824\) −30.0166 −1.04568
\(825\) −2.78278 −0.0968839
\(826\) 63.4075 2.20623
\(827\) −47.4977 −1.65166 −0.825829 0.563921i \(-0.809292\pi\)
−0.825829 + 0.563921i \(0.809292\pi\)
\(828\) 15.3704 0.534157
\(829\) 40.2663 1.39851 0.699254 0.714874i \(-0.253516\pi\)
0.699254 + 0.714874i \(0.253516\pi\)
\(830\) −14.8715 −0.516196
\(831\) −0.100796 −0.00349657
\(832\) −8.34958 −0.289469
\(833\) −23.8183 −0.825254
\(834\) −70.3232 −2.43509
\(835\) 6.22933 0.215575
\(836\) −0.903349 −0.0312430
\(837\) 21.2663 0.735072
\(838\) −42.8401 −1.47989
\(839\) −40.0117 −1.38136 −0.690679 0.723161i \(-0.742688\pi\)
−0.690679 + 0.723161i \(0.742688\pi\)
\(840\) −32.2113 −1.11139
\(841\) −18.5669 −0.640237
\(842\) 8.38138 0.288842
\(843\) 14.3658 0.494785
\(844\) 4.17578 0.143736
\(845\) −9.14010 −0.314429
\(846\) 26.1647 0.899560
\(847\) −5.17171 −0.177702
\(848\) −28.9546 −0.994306
\(849\) 10.4915 0.360067
\(850\) 1.95100 0.0669188
\(851\) 6.54143 0.224237
\(852\) −22.0123 −0.754128
\(853\) −36.2334 −1.24061 −0.620303 0.784362i \(-0.712990\pi\)
−0.620303 + 0.784362i \(0.712990\pi\)
\(854\) 94.2517 3.22523
\(855\) 6.95380 0.237815
\(856\) −10.0321 −0.342891
\(857\) −16.7745 −0.573007 −0.286503 0.958079i \(-0.592493\pi\)
−0.286503 + 0.958079i \(0.592493\pi\)
\(858\) −8.84315 −0.301900
\(859\) −40.6752 −1.38782 −0.693911 0.720061i \(-0.744114\pi\)
−0.693911 + 0.720061i \(0.744114\pi\)
\(860\) 0.254758 0.00868719
\(861\) −4.86551 −0.165816
\(862\) 19.4745 0.663305
\(863\) 2.64635 0.0900828 0.0450414 0.998985i \(-0.485658\pi\)
0.0450414 + 0.998985i \(0.485658\pi\)
\(864\) 16.3679 0.556847
\(865\) −5.79410 −0.197005
\(866\) 9.74759 0.331237
\(867\) −43.2586 −1.46914
\(868\) −13.9670 −0.474070
\(869\) 4.74712 0.161035
\(870\) 14.5387 0.492909
\(871\) −5.13312 −0.173929
\(872\) 12.3286 0.417499
\(873\) 70.0149 2.36964
\(874\) −12.4657 −0.421660
\(875\) −5.17171 −0.174836
\(876\) −1.71492 −0.0579418
\(877\) 29.6451 1.00104 0.500522 0.865724i \(-0.333142\pi\)
0.500522 + 0.865724i \(0.333142\pi\)
\(878\) 4.59710 0.155144
\(879\) 62.5442 2.10956
\(880\) 4.85275 0.163586
\(881\) 38.2096 1.28732 0.643658 0.765313i \(-0.277416\pi\)
0.643658 + 0.765313i \(0.277416\pi\)
\(882\) −151.518 −5.10189
\(883\) −25.9917 −0.874689 −0.437345 0.899294i \(-0.644081\pi\)
−0.437345 + 0.899294i \(0.644081\pi\)
\(884\) 1.46040 0.0491184
\(885\) 21.0933 0.709043
\(886\) −33.9198 −1.13956
\(887\) 11.4734 0.385238 0.192619 0.981274i \(-0.438302\pi\)
0.192619 + 0.981274i \(0.438302\pi\)
\(888\) 7.74923 0.260047
\(889\) −7.38940 −0.247833
\(890\) −16.9120 −0.566892
\(891\) 0.727378 0.0243681
\(892\) −5.04422 −0.168893
\(893\) −4.99842 −0.167266
\(894\) −74.0153 −2.47544
\(895\) 0.408017 0.0136385
\(896\) 70.4383 2.35318
\(897\) −28.7444 −0.959749
\(898\) −20.1391 −0.672052
\(899\) −14.1550 −0.472095
\(900\) 2.92346 0.0974487
\(901\) −7.19694 −0.239765
\(902\) 0.546834 0.0182076
\(903\) −5.94944 −0.197985
\(904\) −40.5549 −1.34884
\(905\) 16.8460 0.559979
\(906\) −90.5679 −3.00892
\(907\) 21.9329 0.728271 0.364135 0.931346i \(-0.381365\pi\)
0.364135 + 0.931346i \(0.381365\pi\)
\(908\) −1.23754 −0.0410691
\(909\) 18.0811 0.599713
\(910\) −16.4347 −0.544806
\(911\) −26.8498 −0.889574 −0.444787 0.895636i \(-0.646721\pi\)
−0.444787 + 0.895636i \(0.646721\pi\)
\(912\) −19.7950 −0.655479
\(913\) −9.19417 −0.304283
\(914\) 67.5731 2.23512
\(915\) 31.3540 1.03653
\(916\) 1.24977 0.0412937
\(917\) −77.2442 −2.55083
\(918\) 9.46778 0.312483
\(919\) 4.99711 0.164840 0.0824198 0.996598i \(-0.473735\pi\)
0.0824198 + 0.996598i \(0.473735\pi\)
\(920\) 11.7674 0.387961
\(921\) −58.7335 −1.93534
\(922\) −62.0632 −2.04394
\(923\) 25.2179 0.830057
\(924\) 8.86908 0.291771
\(925\) 1.24419 0.0409086
\(926\) −54.1894 −1.78077
\(927\) −63.6207 −2.08958
\(928\) −10.8946 −0.357631
\(929\) 46.6250 1.52972 0.764859 0.644198i \(-0.222809\pi\)
0.764859 + 0.644198i \(0.222809\pi\)
\(930\) −19.7252 −0.646816
\(931\) 28.9456 0.948655
\(932\) −2.09338 −0.0685710
\(933\) −20.3655 −0.666738
\(934\) 37.7656 1.23573
\(935\) 1.20619 0.0394468
\(936\) −20.8600 −0.681830
\(937\) −10.3738 −0.338899 −0.169449 0.985539i \(-0.554199\pi\)
−0.169449 + 0.985539i \(0.554199\pi\)
\(938\) 21.8559 0.713621
\(939\) −30.2784 −0.988099
\(940\) −2.10140 −0.0685400
\(941\) 45.4065 1.48021 0.740106 0.672491i \(-0.234776\pi\)
0.740106 + 0.672491i \(0.234776\pi\)
\(942\) 16.2362 0.529003
\(943\) 1.77747 0.0578824
\(944\) −36.7835 −1.19720
\(945\) −25.0972 −0.816411
\(946\) 0.668658 0.0217399
\(947\) 31.9501 1.03824 0.519120 0.854701i \(-0.326260\pi\)
0.519120 + 0.854701i \(0.326260\pi\)
\(948\) −8.14093 −0.264405
\(949\) 1.96466 0.0637756
\(950\) −2.37100 −0.0769253
\(951\) 15.8597 0.514286
\(952\) 13.9620 0.452510
\(953\) 50.1110 1.62325 0.811627 0.584176i \(-0.198582\pi\)
0.811627 + 0.584176i \(0.198582\pi\)
\(954\) −45.7829 −1.48228
\(955\) −21.3403 −0.690555
\(956\) −13.2091 −0.427214
\(957\) 8.98847 0.290556
\(958\) −24.8858 −0.804022
\(959\) −100.450 −3.24369
\(960\) 11.8265 0.381698
\(961\) −11.7954 −0.380497
\(962\) 3.95379 0.127475
\(963\) −21.2633 −0.685200
\(964\) −13.8177 −0.445039
\(965\) 16.3527 0.526413
\(966\) 122.389 3.93779
\(967\) −39.7754 −1.27909 −0.639545 0.768754i \(-0.720877\pi\)
−0.639545 + 0.768754i \(0.720877\pi\)
\(968\) 2.23818 0.0719377
\(969\) −4.92024 −0.158061
\(970\) −23.8726 −0.766502
\(971\) 26.0383 0.835608 0.417804 0.908537i \(-0.362800\pi\)
0.417804 + 0.908537i \(0.362800\pi\)
\(972\) −10.2191 −0.327779
\(973\) −80.8005 −2.59034
\(974\) 39.0615 1.25161
\(975\) −5.46722 −0.175091
\(976\) −54.6766 −1.75015
\(977\) −32.6256 −1.04379 −0.521893 0.853011i \(-0.674774\pi\)
−0.521893 + 0.853011i \(0.674774\pi\)
\(978\) −111.640 −3.56985
\(979\) −10.4557 −0.334167
\(980\) 12.1691 0.388728
\(981\) 26.1307 0.834288
\(982\) −20.5871 −0.656959
\(983\) −29.9795 −0.956199 −0.478099 0.878306i \(-0.658674\pi\)
−0.478099 + 0.878306i \(0.658674\pi\)
\(984\) 2.10566 0.0671260
\(985\) −2.50342 −0.0797656
\(986\) −6.30180 −0.200690
\(987\) 49.0745 1.56206
\(988\) −1.77477 −0.0564631
\(989\) 2.17345 0.0691118
\(990\) 7.67313 0.243868
\(991\) −21.9158 −0.696177 −0.348089 0.937462i \(-0.613169\pi\)
−0.348089 + 0.937462i \(0.613169\pi\)
\(992\) 14.7811 0.469299
\(993\) 89.1003 2.82751
\(994\) −107.373 −3.40567
\(995\) 1.96225 0.0622074
\(996\) 15.7673 0.499605
\(997\) 3.82514 0.121143 0.0605717 0.998164i \(-0.480708\pi\)
0.0605717 + 0.998164i \(0.480708\pi\)
\(998\) −43.4361 −1.37495
\(999\) 6.03776 0.191026
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 4015.2.a.i.1.10 38
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4015.2.a.i.1.10 38 1.1 even 1 trivial