Properties

Label 4015.2.a.c.1.3
Level $4015$
Weight $2$
Character 4015.1
Self dual yes
Analytic conductor $32.060$
Analytic rank $1$
Dimension $23$
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: \(1\)
Dimension: \(23\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
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)
 
\(f(q)\) \(=\) \(q-2.26107 q^{2} -2.17834 q^{3} +3.11244 q^{4} +1.00000 q^{5} +4.92537 q^{6} +1.01936 q^{7} -2.51531 q^{8} +1.74515 q^{9} +O(q^{10})\) \(q-2.26107 q^{2} -2.17834 q^{3} +3.11244 q^{4} +1.00000 q^{5} +4.92537 q^{6} +1.01936 q^{7} -2.51531 q^{8} +1.74515 q^{9} -2.26107 q^{10} -1.00000 q^{11} -6.77995 q^{12} -1.53471 q^{13} -2.30485 q^{14} -2.17834 q^{15} -0.537594 q^{16} +4.77394 q^{17} -3.94591 q^{18} -3.44959 q^{19} +3.11244 q^{20} -2.22052 q^{21} +2.26107 q^{22} +0.771372 q^{23} +5.47919 q^{24} +1.00000 q^{25} +3.47008 q^{26} +2.73348 q^{27} +3.17271 q^{28} +2.66217 q^{29} +4.92537 q^{30} +0.126108 q^{31} +6.24615 q^{32} +2.17834 q^{33} -10.7942 q^{34} +1.01936 q^{35} +5.43169 q^{36} +7.53069 q^{37} +7.79977 q^{38} +3.34311 q^{39} -2.51531 q^{40} -10.6052 q^{41} +5.02075 q^{42} -0.100944 q^{43} -3.11244 q^{44} +1.74515 q^{45} -1.74413 q^{46} -3.79029 q^{47} +1.17106 q^{48} -5.96090 q^{49} -2.26107 q^{50} -10.3992 q^{51} -4.77668 q^{52} -12.5751 q^{53} -6.18059 q^{54} -1.00000 q^{55} -2.56401 q^{56} +7.51438 q^{57} -6.01935 q^{58} +3.22100 q^{59} -6.77995 q^{60} -8.65632 q^{61} -0.285140 q^{62} +1.77895 q^{63} -13.0478 q^{64} -1.53471 q^{65} -4.92537 q^{66} +0.447815 q^{67} +14.8586 q^{68} -1.68031 q^{69} -2.30485 q^{70} +1.85434 q^{71} -4.38960 q^{72} +1.00000 q^{73} -17.0274 q^{74} -2.17834 q^{75} -10.7367 q^{76} -1.01936 q^{77} -7.55900 q^{78} +7.51741 q^{79} -0.537594 q^{80} -11.1899 q^{81} +23.9791 q^{82} +9.63959 q^{83} -6.91123 q^{84} +4.77394 q^{85} +0.228241 q^{86} -5.79910 q^{87} +2.51531 q^{88} -7.08347 q^{89} -3.94591 q^{90} -1.56442 q^{91} +2.40085 q^{92} -0.274707 q^{93} +8.57011 q^{94} -3.44959 q^{95} -13.6062 q^{96} -2.72545 q^{97} +13.4780 q^{98} -1.74515 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 23 q^{11} - 18 q^{12} - 5 q^{13} - 17 q^{14} - 5 q^{15} - q^{16} - 36 q^{17} + q^{18} + 6 q^{19} + 15 q^{20} - 18 q^{21} + 3 q^{22} - 14 q^{23} - 7 q^{24} + 23 q^{25} - 21 q^{26} - 29 q^{27} + 28 q^{28} - 36 q^{29} - 5 q^{30} - 16 q^{31} - 5 q^{32} + 5 q^{33} - 28 q^{34} - 14 q^{36} - 24 q^{37} + q^{38} - 10 q^{39} - 6 q^{40} - 36 q^{41} - 5 q^{42} + 17 q^{43} - 15 q^{44} + 8 q^{45} - 25 q^{46} - 21 q^{47} - 17 q^{48} - 27 q^{49} - 3 q^{50} + 19 q^{51} - 21 q^{52} - 28 q^{53} - 15 q^{54} - 23 q^{55} - 46 q^{56} - 23 q^{57} - 16 q^{58} - 61 q^{59} - 18 q^{60} - 17 q^{61} - 22 q^{62} - 9 q^{63} - 18 q^{64} - 5 q^{65} + 5 q^{66} + 2 q^{67} - 39 q^{68} - 36 q^{69} - 17 q^{70} - 50 q^{71} + 15 q^{72} + 23 q^{73} + 17 q^{74} - 5 q^{75} - 21 q^{76} + 49 q^{78} - 18 q^{79} - q^{80} - 57 q^{81} + 14 q^{82} - 20 q^{83} - 38 q^{84} - 36 q^{85} - 45 q^{86} + 37 q^{87} + 6 q^{88} - 93 q^{89} + q^{90} - 42 q^{91} - 39 q^{92} - 18 q^{93} - 6 q^{94} + 6 q^{95} - 9 q^{96} - 31 q^{97} - 31 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.26107 −1.59882 −0.799409 0.600787i \(-0.794854\pi\)
−0.799409 + 0.600787i \(0.794854\pi\)
\(3\) −2.17834 −1.25766 −0.628832 0.777541i \(-0.716467\pi\)
−0.628832 + 0.777541i \(0.716467\pi\)
\(4\) 3.11244 1.55622
\(5\) 1.00000 0.447214
\(6\) 4.92537 2.01078
\(7\) 1.01936 0.385283 0.192642 0.981269i \(-0.438295\pi\)
0.192642 + 0.981269i \(0.438295\pi\)
\(8\) −2.51531 −0.889295
\(9\) 1.74515 0.581718
\(10\) −2.26107 −0.715013
\(11\) −1.00000 −0.301511
\(12\) −6.77995 −1.95720
\(13\) −1.53471 −0.425651 −0.212825 0.977090i \(-0.568267\pi\)
−0.212825 + 0.977090i \(0.568267\pi\)
\(14\) −2.30485 −0.615998
\(15\) −2.17834 −0.562444
\(16\) −0.537594 −0.134398
\(17\) 4.77394 1.15785 0.578925 0.815381i \(-0.303472\pi\)
0.578925 + 0.815381i \(0.303472\pi\)
\(18\) −3.94591 −0.930061
\(19\) −3.44959 −0.791391 −0.395695 0.918382i \(-0.629496\pi\)
−0.395695 + 0.918382i \(0.629496\pi\)
\(20\) 3.11244 0.695963
\(21\) −2.22052 −0.484557
\(22\) 2.26107 0.482062
\(23\) 0.771372 0.160842 0.0804211 0.996761i \(-0.474374\pi\)
0.0804211 + 0.996761i \(0.474374\pi\)
\(24\) 5.47919 1.11843
\(25\) 1.00000 0.200000
\(26\) 3.47008 0.680538
\(27\) 2.73348 0.526058
\(28\) 3.17271 0.599586
\(29\) 2.66217 0.494352 0.247176 0.968971i \(-0.420497\pi\)
0.247176 + 0.968971i \(0.420497\pi\)
\(30\) 4.92537 0.899246
\(31\) 0.126108 0.0226497 0.0113249 0.999936i \(-0.496395\pi\)
0.0113249 + 0.999936i \(0.496395\pi\)
\(32\) 6.24615 1.10417
\(33\) 2.17834 0.379200
\(34\) −10.7942 −1.85119
\(35\) 1.01936 0.172304
\(36\) 5.43169 0.905281
\(37\) 7.53069 1.23804 0.619019 0.785376i \(-0.287531\pi\)
0.619019 + 0.785376i \(0.287531\pi\)
\(38\) 7.79977 1.26529
\(39\) 3.34311 0.535325
\(40\) −2.51531 −0.397705
\(41\) −10.6052 −1.65625 −0.828127 0.560541i \(-0.810593\pi\)
−0.828127 + 0.560541i \(0.810593\pi\)
\(42\) 5.02075 0.774718
\(43\) −0.100944 −0.0153938 −0.00769689 0.999970i \(-0.502450\pi\)
−0.00769689 + 0.999970i \(0.502450\pi\)
\(44\) −3.11244 −0.469218
\(45\) 1.74515 0.260152
\(46\) −1.74413 −0.257157
\(47\) −3.79029 −0.552871 −0.276435 0.961033i \(-0.589153\pi\)
−0.276435 + 0.961033i \(0.589153\pi\)
\(48\) 1.17106 0.169028
\(49\) −5.96090 −0.851557
\(50\) −2.26107 −0.319764
\(51\) −10.3992 −1.45619
\(52\) −4.77668 −0.662406
\(53\) −12.5751 −1.72732 −0.863659 0.504076i \(-0.831833\pi\)
−0.863659 + 0.504076i \(0.831833\pi\)
\(54\) −6.18059 −0.841072
\(55\) −1.00000 −0.134840
\(56\) −2.56401 −0.342631
\(57\) 7.51438 0.995303
\(58\) −6.01935 −0.790379
\(59\) 3.22100 0.419338 0.209669 0.977772i \(-0.432761\pi\)
0.209669 + 0.977772i \(0.432761\pi\)
\(60\) −6.77995 −0.875287
\(61\) −8.65632 −1.10833 −0.554164 0.832407i \(-0.686962\pi\)
−0.554164 + 0.832407i \(0.686962\pi\)
\(62\) −0.285140 −0.0362128
\(63\) 1.77895 0.224126
\(64\) −13.0478 −1.63098
\(65\) −1.53471 −0.190357
\(66\) −4.92537 −0.606272
\(67\) 0.447815 0.0547093 0.0273547 0.999626i \(-0.491292\pi\)
0.0273547 + 0.999626i \(0.491292\pi\)
\(68\) 14.8586 1.80187
\(69\) −1.68031 −0.202285
\(70\) −2.30485 −0.275483
\(71\) 1.85434 0.220069 0.110035 0.993928i \(-0.464904\pi\)
0.110035 + 0.993928i \(0.464904\pi\)
\(72\) −4.38960 −0.517319
\(73\) 1.00000 0.117041
\(74\) −17.0274 −1.97940
\(75\) −2.17834 −0.251533
\(76\) −10.7367 −1.23158
\(77\) −1.01936 −0.116167
\(78\) −7.55900 −0.855888
\(79\) 7.51741 0.845775 0.422887 0.906182i \(-0.361017\pi\)
0.422887 + 0.906182i \(0.361017\pi\)
\(80\) −0.537594 −0.0601048
\(81\) −11.1899 −1.24332
\(82\) 23.9791 2.64805
\(83\) 9.63959 1.05808 0.529041 0.848596i \(-0.322552\pi\)
0.529041 + 0.848596i \(0.322552\pi\)
\(84\) −6.91123 −0.754077
\(85\) 4.77394 0.517806
\(86\) 0.228241 0.0246119
\(87\) −5.79910 −0.621728
\(88\) 2.51531 0.268133
\(89\) −7.08347 −0.750846 −0.375423 0.926854i \(-0.622503\pi\)
−0.375423 + 0.926854i \(0.622503\pi\)
\(90\) −3.94591 −0.415936
\(91\) −1.56442 −0.163996
\(92\) 2.40085 0.250306
\(93\) −0.274707 −0.0284858
\(94\) 8.57011 0.883940
\(95\) −3.44959 −0.353921
\(96\) −13.6062 −1.38868
\(97\) −2.72545 −0.276728 −0.138364 0.990381i \(-0.544184\pi\)
−0.138364 + 0.990381i \(0.544184\pi\)
\(98\) 13.4780 1.36148
\(99\) −1.74515 −0.175394
\(100\) 3.11244 0.311244
\(101\) −1.50808 −0.150060 −0.0750298 0.997181i \(-0.523905\pi\)
−0.0750298 + 0.997181i \(0.523905\pi\)
\(102\) 23.5134 2.32818
\(103\) 18.2298 1.79623 0.898116 0.439759i \(-0.144936\pi\)
0.898116 + 0.439759i \(0.144936\pi\)
\(104\) 3.86026 0.378529
\(105\) −2.22052 −0.216700
\(106\) 28.4331 2.76167
\(107\) 1.39080 0.134454 0.0672269 0.997738i \(-0.478585\pi\)
0.0672269 + 0.997738i \(0.478585\pi\)
\(108\) 8.50779 0.818663
\(109\) 12.9399 1.23942 0.619710 0.784831i \(-0.287250\pi\)
0.619710 + 0.784831i \(0.287250\pi\)
\(110\) 2.26107 0.215585
\(111\) −16.4044 −1.55703
\(112\) −0.548004 −0.0517815
\(113\) −8.97149 −0.843967 −0.421983 0.906604i \(-0.638666\pi\)
−0.421983 + 0.906604i \(0.638666\pi\)
\(114\) −16.9905 −1.59131
\(115\) 0.771372 0.0719308
\(116\) 8.28584 0.769321
\(117\) −2.67830 −0.247609
\(118\) −7.28291 −0.670446
\(119\) 4.86638 0.446100
\(120\) 5.47919 0.500179
\(121\) 1.00000 0.0909091
\(122\) 19.5725 1.77202
\(123\) 23.1017 2.08301
\(124\) 0.392505 0.0352480
\(125\) 1.00000 0.0894427
\(126\) −4.02232 −0.358337
\(127\) −7.68524 −0.681955 −0.340977 0.940071i \(-0.610758\pi\)
−0.340977 + 0.940071i \(0.610758\pi\)
\(128\) 17.0097 1.50346
\(129\) 0.219890 0.0193602
\(130\) 3.47008 0.304346
\(131\) 0.162327 0.0141826 0.00709131 0.999975i \(-0.497743\pi\)
0.00709131 + 0.999975i \(0.497743\pi\)
\(132\) 6.77995 0.590119
\(133\) −3.51639 −0.304910
\(134\) −1.01254 −0.0874703
\(135\) 2.73348 0.235260
\(136\) −12.0079 −1.02967
\(137\) 9.34442 0.798348 0.399174 0.916875i \(-0.369297\pi\)
0.399174 + 0.916875i \(0.369297\pi\)
\(138\) 3.79930 0.323417
\(139\) 7.79712 0.661343 0.330672 0.943746i \(-0.392725\pi\)
0.330672 + 0.943746i \(0.392725\pi\)
\(140\) 3.17271 0.268143
\(141\) 8.25653 0.695325
\(142\) −4.19278 −0.351851
\(143\) 1.53471 0.128339
\(144\) −0.938183 −0.0781820
\(145\) 2.66217 0.221081
\(146\) −2.26107 −0.187128
\(147\) 12.9848 1.07097
\(148\) 23.4388 1.92666
\(149\) −14.6093 −1.19684 −0.598421 0.801182i \(-0.704205\pi\)
−0.598421 + 0.801182i \(0.704205\pi\)
\(150\) 4.92537 0.402155
\(151\) 0.757490 0.0616437 0.0308218 0.999525i \(-0.490188\pi\)
0.0308218 + 0.999525i \(0.490188\pi\)
\(152\) 8.67678 0.703780
\(153\) 8.33125 0.673542
\(154\) 2.30485 0.185730
\(155\) 0.126108 0.0101293
\(156\) 10.4052 0.833084
\(157\) 3.32182 0.265110 0.132555 0.991176i \(-0.457682\pi\)
0.132555 + 0.991176i \(0.457682\pi\)
\(158\) −16.9974 −1.35224
\(159\) 27.3927 2.17239
\(160\) 6.24615 0.493802
\(161\) 0.786308 0.0619698
\(162\) 25.3012 1.98785
\(163\) 11.0194 0.863104 0.431552 0.902088i \(-0.357966\pi\)
0.431552 + 0.902088i \(0.357966\pi\)
\(164\) −33.0081 −2.57750
\(165\) 2.17834 0.169583
\(166\) −21.7958 −1.69168
\(167\) −19.4056 −1.50165 −0.750825 0.660502i \(-0.770344\pi\)
−0.750825 + 0.660502i \(0.770344\pi\)
\(168\) 5.58528 0.430914
\(169\) −10.6447 −0.818821
\(170\) −10.7942 −0.827878
\(171\) −6.02007 −0.460366
\(172\) −0.314182 −0.0239561
\(173\) −23.2231 −1.76562 −0.882808 0.469734i \(-0.844350\pi\)
−0.882808 + 0.469734i \(0.844350\pi\)
\(174\) 13.1122 0.994031
\(175\) 1.01936 0.0770566
\(176\) 0.537594 0.0405227
\(177\) −7.01642 −0.527387
\(178\) 16.0162 1.20047
\(179\) −6.39292 −0.477829 −0.238915 0.971041i \(-0.576792\pi\)
−0.238915 + 0.971041i \(0.576792\pi\)
\(180\) 5.43169 0.404854
\(181\) 5.77136 0.428982 0.214491 0.976726i \(-0.431191\pi\)
0.214491 + 0.976726i \(0.431191\pi\)
\(182\) 3.53727 0.262200
\(183\) 18.8564 1.39390
\(184\) −1.94024 −0.143036
\(185\) 7.53069 0.553667
\(186\) 0.621131 0.0455436
\(187\) −4.77394 −0.349105
\(188\) −11.7971 −0.860389
\(189\) 2.78641 0.202681
\(190\) 7.79977 0.565855
\(191\) 8.99854 0.651112 0.325556 0.945523i \(-0.394449\pi\)
0.325556 + 0.945523i \(0.394449\pi\)
\(192\) 28.4225 2.05122
\(193\) 4.60153 0.331226 0.165613 0.986191i \(-0.447040\pi\)
0.165613 + 0.986191i \(0.447040\pi\)
\(194\) 6.16244 0.442437
\(195\) 3.34311 0.239405
\(196\) −18.5529 −1.32521
\(197\) 11.8571 0.844785 0.422393 0.906413i \(-0.361190\pi\)
0.422393 + 0.906413i \(0.361190\pi\)
\(198\) 3.94591 0.280424
\(199\) −5.74613 −0.407332 −0.203666 0.979040i \(-0.565286\pi\)
−0.203666 + 0.979040i \(0.565286\pi\)
\(200\) −2.51531 −0.177859
\(201\) −0.975493 −0.0688059
\(202\) 3.40988 0.239918
\(203\) 2.71372 0.190466
\(204\) −32.3670 −2.26615
\(205\) −10.6052 −0.740699
\(206\) −41.2188 −2.87185
\(207\) 1.34616 0.0935647
\(208\) 0.825048 0.0572068
\(209\) 3.44959 0.238613
\(210\) 5.02075 0.346464
\(211\) 5.71879 0.393698 0.196849 0.980434i \(-0.436929\pi\)
0.196849 + 0.980434i \(0.436929\pi\)
\(212\) −39.1391 −2.68809
\(213\) −4.03937 −0.276773
\(214\) −3.14470 −0.214967
\(215\) −0.100944 −0.00688431
\(216\) −6.87554 −0.467821
\(217\) 0.128550 0.00872657
\(218\) −29.2581 −1.98161
\(219\) −2.17834 −0.147198
\(220\) −3.11244 −0.209841
\(221\) −7.32659 −0.492840
\(222\) 37.0914 2.48941
\(223\) 14.3707 0.962330 0.481165 0.876630i \(-0.340214\pi\)
0.481165 + 0.876630i \(0.340214\pi\)
\(224\) 6.36710 0.425420
\(225\) 1.74515 0.116344
\(226\) 20.2852 1.34935
\(227\) 8.03332 0.533190 0.266595 0.963809i \(-0.414101\pi\)
0.266595 + 0.963809i \(0.414101\pi\)
\(228\) 23.3880 1.54891
\(229\) 18.7860 1.24141 0.620707 0.784042i \(-0.286846\pi\)
0.620707 + 0.784042i \(0.286846\pi\)
\(230\) −1.74413 −0.115004
\(231\) 2.22052 0.146099
\(232\) −6.69617 −0.439625
\(233\) 18.2338 1.19454 0.597268 0.802041i \(-0.296253\pi\)
0.597268 + 0.802041i \(0.296253\pi\)
\(234\) 6.05582 0.395881
\(235\) −3.79029 −0.247251
\(236\) 10.0252 0.652583
\(237\) −16.3755 −1.06370
\(238\) −11.0032 −0.713233
\(239\) 4.27458 0.276499 0.138250 0.990397i \(-0.455852\pi\)
0.138250 + 0.990397i \(0.455852\pi\)
\(240\) 1.17106 0.0755916
\(241\) 9.04162 0.582422 0.291211 0.956659i \(-0.405942\pi\)
0.291211 + 0.956659i \(0.405942\pi\)
\(242\) −2.26107 −0.145347
\(243\) 16.1749 1.03762
\(244\) −26.9423 −1.72480
\(245\) −5.96090 −0.380828
\(246\) −52.2346 −3.33036
\(247\) 5.29411 0.336856
\(248\) −0.317202 −0.0201423
\(249\) −20.9983 −1.33071
\(250\) −2.26107 −0.143003
\(251\) −5.28247 −0.333427 −0.166713 0.986005i \(-0.553315\pi\)
−0.166713 + 0.986005i \(0.553315\pi\)
\(252\) 5.53686 0.348790
\(253\) −0.771372 −0.0484957
\(254\) 17.3769 1.09032
\(255\) −10.3992 −0.651226
\(256\) −12.3645 −0.772783
\(257\) −11.6631 −0.727526 −0.363763 0.931491i \(-0.618508\pi\)
−0.363763 + 0.931491i \(0.618508\pi\)
\(258\) −0.497186 −0.0309535
\(259\) 7.67651 0.476995
\(260\) −4.77668 −0.296237
\(261\) 4.64589 0.287573
\(262\) −0.367034 −0.0226754
\(263\) −1.33569 −0.0823625 −0.0411812 0.999152i \(-0.513112\pi\)
−0.0411812 + 0.999152i \(0.513112\pi\)
\(264\) −5.47919 −0.337221
\(265\) −12.5751 −0.772480
\(266\) 7.95080 0.487495
\(267\) 15.4302 0.944312
\(268\) 1.39380 0.0851398
\(269\) −12.3338 −0.752004 −0.376002 0.926619i \(-0.622701\pi\)
−0.376002 + 0.926619i \(0.622701\pi\)
\(270\) −6.18059 −0.376139
\(271\) 6.89798 0.419023 0.209511 0.977806i \(-0.432813\pi\)
0.209511 + 0.977806i \(0.432813\pi\)
\(272\) −2.56644 −0.155613
\(273\) 3.40784 0.206252
\(274\) −21.1284 −1.27641
\(275\) −1.00000 −0.0603023
\(276\) −5.22986 −0.314801
\(277\) 6.91692 0.415597 0.207799 0.978172i \(-0.433370\pi\)
0.207799 + 0.978172i \(0.433370\pi\)
\(278\) −17.6298 −1.05737
\(279\) 0.220079 0.0131758
\(280\) −2.56401 −0.153229
\(281\) −27.4989 −1.64045 −0.820223 0.572044i \(-0.806151\pi\)
−0.820223 + 0.572044i \(0.806151\pi\)
\(282\) −18.6686 −1.11170
\(283\) −11.4651 −0.681531 −0.340765 0.940148i \(-0.610686\pi\)
−0.340765 + 0.940148i \(0.610686\pi\)
\(284\) 5.77151 0.342476
\(285\) 7.51438 0.445113
\(286\) −3.47008 −0.205190
\(287\) −10.8106 −0.638127
\(288\) 10.9005 0.642318
\(289\) 5.79049 0.340617
\(290\) −6.01935 −0.353468
\(291\) 5.93695 0.348030
\(292\) 3.11244 0.182142
\(293\) −21.5489 −1.25890 −0.629451 0.777040i \(-0.716720\pi\)
−0.629451 + 0.777040i \(0.716720\pi\)
\(294\) −29.3597 −1.71229
\(295\) 3.22100 0.187534
\(296\) −18.9420 −1.10098
\(297\) −2.73348 −0.158613
\(298\) 33.0327 1.91353
\(299\) −1.18383 −0.0684626
\(300\) −6.77995 −0.391440
\(301\) −0.102898 −0.00593097
\(302\) −1.71274 −0.0985570
\(303\) 3.28511 0.188725
\(304\) 1.85448 0.106362
\(305\) −8.65632 −0.495659
\(306\) −18.8376 −1.07687
\(307\) 2.33986 0.133543 0.0667713 0.997768i \(-0.478730\pi\)
0.0667713 + 0.997768i \(0.478730\pi\)
\(308\) −3.17271 −0.180782
\(309\) −39.7106 −2.25905
\(310\) −0.285140 −0.0161949
\(311\) −22.8336 −1.29478 −0.647389 0.762160i \(-0.724139\pi\)
−0.647389 + 0.762160i \(0.724139\pi\)
\(312\) −8.40894 −0.476062
\(313\) −10.2076 −0.576967 −0.288484 0.957485i \(-0.593151\pi\)
−0.288484 + 0.957485i \(0.593151\pi\)
\(314\) −7.51086 −0.423863
\(315\) 1.77895 0.100232
\(316\) 23.3975 1.31621
\(317\) −22.3192 −1.25357 −0.626785 0.779192i \(-0.715630\pi\)
−0.626785 + 0.779192i \(0.715630\pi\)
\(318\) −61.9369 −3.47325
\(319\) −2.66217 −0.149053
\(320\) −13.0478 −0.729394
\(321\) −3.02964 −0.169098
\(322\) −1.77790 −0.0990784
\(323\) −16.4681 −0.916312
\(324\) −34.8279 −1.93488
\(325\) −1.53471 −0.0851301
\(326\) −24.9156 −1.37995
\(327\) −28.1875 −1.55877
\(328\) 26.6753 1.47290
\(329\) −3.86368 −0.213012
\(330\) −4.92537 −0.271133
\(331\) −17.8332 −0.980204 −0.490102 0.871665i \(-0.663040\pi\)
−0.490102 + 0.871665i \(0.663040\pi\)
\(332\) 30.0026 1.64661
\(333\) 13.1422 0.720188
\(334\) 43.8774 2.40086
\(335\) 0.447815 0.0244668
\(336\) 1.19374 0.0651237
\(337\) −2.60832 −0.142084 −0.0710420 0.997473i \(-0.522632\pi\)
−0.0710420 + 0.997473i \(0.522632\pi\)
\(338\) 24.0684 1.30915
\(339\) 19.5429 1.06143
\(340\) 14.8586 0.805821
\(341\) −0.126108 −0.00682916
\(342\) 13.6118 0.736042
\(343\) −13.2119 −0.713374
\(344\) 0.253905 0.0136896
\(345\) −1.68031 −0.0904647
\(346\) 52.5090 2.82290
\(347\) 32.2905 1.73344 0.866721 0.498792i \(-0.166223\pi\)
0.866721 + 0.498792i \(0.166223\pi\)
\(348\) −18.0493 −0.967547
\(349\) 22.4713 1.20286 0.601430 0.798925i \(-0.294598\pi\)
0.601430 + 0.798925i \(0.294598\pi\)
\(350\) −2.30485 −0.123200
\(351\) −4.19509 −0.223917
\(352\) −6.24615 −0.332921
\(353\) 2.01085 0.107027 0.0535135 0.998567i \(-0.482958\pi\)
0.0535135 + 0.998567i \(0.482958\pi\)
\(354\) 15.8646 0.843195
\(355\) 1.85434 0.0984179
\(356\) −22.0469 −1.16848
\(357\) −10.6006 −0.561044
\(358\) 14.4548 0.763962
\(359\) 0.155016 0.00818144 0.00409072 0.999992i \(-0.498698\pi\)
0.00409072 + 0.999992i \(0.498698\pi\)
\(360\) −4.38960 −0.231352
\(361\) −7.10031 −0.373701
\(362\) −13.0495 −0.685864
\(363\) −2.17834 −0.114333
\(364\) −4.86917 −0.255214
\(365\) 1.00000 0.0523424
\(366\) −42.6356 −2.22860
\(367\) −32.5646 −1.69986 −0.849929 0.526898i \(-0.823355\pi\)
−0.849929 + 0.526898i \(0.823355\pi\)
\(368\) −0.414685 −0.0216169
\(369\) −18.5077 −0.963472
\(370\) −17.0274 −0.885213
\(371\) −12.8186 −0.665507
\(372\) −0.855009 −0.0443301
\(373\) −22.5314 −1.16663 −0.583317 0.812245i \(-0.698245\pi\)
−0.583317 + 0.812245i \(0.698245\pi\)
\(374\) 10.7942 0.558155
\(375\) −2.17834 −0.112489
\(376\) 9.53374 0.491665
\(377\) −4.08564 −0.210421
\(378\) −6.30027 −0.324051
\(379\) −37.6303 −1.93294 −0.966469 0.256782i \(-0.917338\pi\)
−0.966469 + 0.256782i \(0.917338\pi\)
\(380\) −10.7367 −0.550779
\(381\) 16.7410 0.857670
\(382\) −20.3463 −1.04101
\(383\) −20.5106 −1.04804 −0.524022 0.851704i \(-0.675569\pi\)
−0.524022 + 0.851704i \(0.675569\pi\)
\(384\) −37.0529 −1.89085
\(385\) −1.01936 −0.0519516
\(386\) −10.4044 −0.529570
\(387\) −0.176162 −0.00895484
\(388\) −8.48281 −0.430649
\(389\) 13.4390 0.681386 0.340693 0.940175i \(-0.389338\pi\)
0.340693 + 0.940175i \(0.389338\pi\)
\(390\) −7.55900 −0.382765
\(391\) 3.68248 0.186231
\(392\) 14.9935 0.757286
\(393\) −0.353604 −0.0178370
\(394\) −26.8098 −1.35066
\(395\) 7.51741 0.378242
\(396\) −5.43169 −0.272952
\(397\) −9.57942 −0.480777 −0.240389 0.970677i \(-0.577275\pi\)
−0.240389 + 0.970677i \(0.577275\pi\)
\(398\) 12.9924 0.651251
\(399\) 7.65988 0.383474
\(400\) −0.537594 −0.0268797
\(401\) −14.9797 −0.748048 −0.374024 0.927419i \(-0.622022\pi\)
−0.374024 + 0.927419i \(0.622022\pi\)
\(402\) 2.20566 0.110008
\(403\) −0.193539 −0.00964088
\(404\) −4.69381 −0.233526
\(405\) −11.1899 −0.556031
\(406\) −6.13590 −0.304520
\(407\) −7.53069 −0.373282
\(408\) 26.1573 1.29498
\(409\) 33.4925 1.65610 0.828050 0.560655i \(-0.189451\pi\)
0.828050 + 0.560655i \(0.189451\pi\)
\(410\) 23.9791 1.18424
\(411\) −20.3553 −1.00405
\(412\) 56.7390 2.79533
\(413\) 3.28337 0.161564
\(414\) −3.04377 −0.149593
\(415\) 9.63959 0.473189
\(416\) −9.58600 −0.469993
\(417\) −16.9848 −0.831747
\(418\) −7.79977 −0.381499
\(419\) −18.3882 −0.898322 −0.449161 0.893451i \(-0.648277\pi\)
−0.449161 + 0.893451i \(0.648277\pi\)
\(420\) −6.91123 −0.337233
\(421\) −12.4968 −0.609058 −0.304529 0.952503i \(-0.598499\pi\)
−0.304529 + 0.952503i \(0.598499\pi\)
\(422\) −12.9306 −0.629452
\(423\) −6.61464 −0.321615
\(424\) 31.6302 1.53610
\(425\) 4.77394 0.231570
\(426\) 9.13329 0.442510
\(427\) −8.82394 −0.427020
\(428\) 4.32879 0.209240
\(429\) −3.34311 −0.161407
\(430\) 0.228241 0.0110068
\(431\) 27.4430 1.32188 0.660941 0.750437i \(-0.270157\pi\)
0.660941 + 0.750437i \(0.270157\pi\)
\(432\) −1.46950 −0.0707014
\(433\) 3.25015 0.156192 0.0780960 0.996946i \(-0.475116\pi\)
0.0780960 + 0.996946i \(0.475116\pi\)
\(434\) −0.290661 −0.0139522
\(435\) −5.79910 −0.278045
\(436\) 40.2748 1.92881
\(437\) −2.66092 −0.127289
\(438\) 4.92537 0.235343
\(439\) 8.48640 0.405034 0.202517 0.979279i \(-0.435088\pi\)
0.202517 + 0.979279i \(0.435088\pi\)
\(440\) 2.51531 0.119913
\(441\) −10.4027 −0.495366
\(442\) 16.5659 0.787961
\(443\) 31.9581 1.51838 0.759188 0.650872i \(-0.225596\pi\)
0.759188 + 0.650872i \(0.225596\pi\)
\(444\) −51.0576 −2.42309
\(445\) −7.08347 −0.335789
\(446\) −32.4931 −1.53859
\(447\) 31.8240 1.50522
\(448\) −13.3005 −0.628387
\(449\) 4.16081 0.196361 0.0981804 0.995169i \(-0.468698\pi\)
0.0981804 + 0.995169i \(0.468698\pi\)
\(450\) −3.94591 −0.186012
\(451\) 10.6052 0.499379
\(452\) −27.9232 −1.31340
\(453\) −1.65007 −0.0775270
\(454\) −18.1639 −0.852474
\(455\) −1.56442 −0.0733413
\(456\) −18.9010 −0.885119
\(457\) −36.0169 −1.68480 −0.842401 0.538852i \(-0.818858\pi\)
−0.842401 + 0.538852i \(0.818858\pi\)
\(458\) −42.4765 −1.98480
\(459\) 13.0495 0.609097
\(460\) 2.40085 0.111940
\(461\) 24.9416 1.16165 0.580823 0.814030i \(-0.302731\pi\)
0.580823 + 0.814030i \(0.302731\pi\)
\(462\) −5.02075 −0.233586
\(463\) 7.51867 0.349422 0.174711 0.984620i \(-0.444101\pi\)
0.174711 + 0.984620i \(0.444101\pi\)
\(464\) −1.43116 −0.0664401
\(465\) −0.274707 −0.0127392
\(466\) −41.2279 −1.90985
\(467\) 2.89112 0.133785 0.0668926 0.997760i \(-0.478691\pi\)
0.0668926 + 0.997760i \(0.478691\pi\)
\(468\) −8.33604 −0.385333
\(469\) 0.456487 0.0210786
\(470\) 8.57011 0.395310
\(471\) −7.23604 −0.333419
\(472\) −8.10180 −0.372916
\(473\) 0.100944 0.00464140
\(474\) 37.0261 1.70066
\(475\) −3.44959 −0.158278
\(476\) 15.1463 0.694230
\(477\) −21.9454 −1.00481
\(478\) −9.66512 −0.442072
\(479\) −43.1382 −1.97104 −0.985518 0.169573i \(-0.945761\pi\)
−0.985518 + 0.169573i \(0.945761\pi\)
\(480\) −13.6062 −0.621036
\(481\) −11.5574 −0.526971
\(482\) −20.4437 −0.931187
\(483\) −1.71284 −0.0779371
\(484\) 3.11244 0.141475
\(485\) −2.72545 −0.123756
\(486\) −36.5727 −1.65897
\(487\) −5.67414 −0.257120 −0.128560 0.991702i \(-0.541035\pi\)
−0.128560 + 0.991702i \(0.541035\pi\)
\(488\) 21.7733 0.985631
\(489\) −24.0039 −1.08549
\(490\) 13.4780 0.608875
\(491\) −33.5416 −1.51371 −0.756855 0.653582i \(-0.773265\pi\)
−0.756855 + 0.653582i \(0.773265\pi\)
\(492\) 71.9027 3.24162
\(493\) 12.7090 0.572386
\(494\) −11.9704 −0.538572
\(495\) −1.74515 −0.0784388
\(496\) −0.0677951 −0.00304409
\(497\) 1.89024 0.0847889
\(498\) 47.4786 2.12757
\(499\) 11.3451 0.507877 0.253939 0.967220i \(-0.418274\pi\)
0.253939 + 0.967220i \(0.418274\pi\)
\(500\) 3.11244 0.139193
\(501\) 42.2719 1.88857
\(502\) 11.9440 0.533089
\(503\) −21.1105 −0.941272 −0.470636 0.882327i \(-0.655976\pi\)
−0.470636 + 0.882327i \(0.655976\pi\)
\(504\) −4.47459 −0.199314
\(505\) −1.50808 −0.0671087
\(506\) 1.74413 0.0775359
\(507\) 23.1877 1.02980
\(508\) −23.9199 −1.06127
\(509\) −16.3206 −0.723399 −0.361700 0.932295i \(-0.617803\pi\)
−0.361700 + 0.932295i \(0.617803\pi\)
\(510\) 23.5134 1.04119
\(511\) 1.01936 0.0450940
\(512\) −6.06232 −0.267919
\(513\) −9.42939 −0.416318
\(514\) 26.3712 1.16318
\(515\) 18.2298 0.803299
\(516\) 0.684394 0.0301287
\(517\) 3.79029 0.166697
\(518\) −17.3571 −0.762628
\(519\) 50.5876 2.22055
\(520\) 3.86026 0.169283
\(521\) −40.6600 −1.78135 −0.890673 0.454644i \(-0.849767\pi\)
−0.890673 + 0.454644i \(0.849767\pi\)
\(522\) −10.5047 −0.459777
\(523\) −6.33991 −0.277225 −0.138612 0.990347i \(-0.544264\pi\)
−0.138612 + 0.990347i \(0.544264\pi\)
\(524\) 0.505234 0.0220713
\(525\) −2.22052 −0.0969113
\(526\) 3.02010 0.131683
\(527\) 0.602034 0.0262250
\(528\) −1.17106 −0.0509639
\(529\) −22.4050 −0.974130
\(530\) 28.4331 1.23506
\(531\) 5.62114 0.243937
\(532\) −10.9446 −0.474506
\(533\) 16.2759 0.704986
\(534\) −34.8887 −1.50978
\(535\) 1.39080 0.0601296
\(536\) −1.12639 −0.0486528
\(537\) 13.9259 0.600949
\(538\) 27.8876 1.20232
\(539\) 5.96090 0.256754
\(540\) 8.50779 0.366117
\(541\) 7.24782 0.311608 0.155804 0.987788i \(-0.450203\pi\)
0.155804 + 0.987788i \(0.450203\pi\)
\(542\) −15.5968 −0.669941
\(543\) −12.5720 −0.539515
\(544\) 29.8187 1.27847
\(545\) 12.9399 0.554286
\(546\) −7.70537 −0.329759
\(547\) 12.9456 0.553513 0.276756 0.960940i \(-0.410740\pi\)
0.276756 + 0.960940i \(0.410740\pi\)
\(548\) 29.0840 1.24241
\(549\) −15.1066 −0.644734
\(550\) 2.26107 0.0964124
\(551\) −9.18339 −0.391226
\(552\) 4.22649 0.179891
\(553\) 7.66297 0.325863
\(554\) −15.6396 −0.664465
\(555\) −16.4044 −0.696327
\(556\) 24.2681 1.02920
\(557\) −35.0209 −1.48388 −0.741942 0.670464i \(-0.766095\pi\)
−0.741942 + 0.670464i \(0.766095\pi\)
\(558\) −0.497613 −0.0210656
\(559\) 0.154919 0.00655238
\(560\) −0.548004 −0.0231574
\(561\) 10.3992 0.439057
\(562\) 62.1769 2.62278
\(563\) 1.43449 0.0604564 0.0302282 0.999543i \(-0.490377\pi\)
0.0302282 + 0.999543i \(0.490377\pi\)
\(564\) 25.6980 1.08208
\(565\) −8.97149 −0.377433
\(566\) 25.9234 1.08964
\(567\) −11.4066 −0.479031
\(568\) −4.66422 −0.195706
\(569\) −15.8061 −0.662628 −0.331314 0.943521i \(-0.607492\pi\)
−0.331314 + 0.943521i \(0.607492\pi\)
\(570\) −16.9905 −0.711655
\(571\) 15.4889 0.648190 0.324095 0.946025i \(-0.394940\pi\)
0.324095 + 0.946025i \(0.394940\pi\)
\(572\) 4.77668 0.199723
\(573\) −19.6019 −0.818879
\(574\) 24.4434 1.02025
\(575\) 0.771372 0.0321684
\(576\) −22.7704 −0.948767
\(577\) 1.69633 0.0706193 0.0353097 0.999376i \(-0.488758\pi\)
0.0353097 + 0.999376i \(0.488758\pi\)
\(578\) −13.0927 −0.544585
\(579\) −10.0237 −0.416570
\(580\) 8.28584 0.344051
\(581\) 9.82624 0.407661
\(582\) −13.4239 −0.556437
\(583\) 12.5751 0.520806
\(584\) −2.51531 −0.104084
\(585\) −2.67830 −0.110734
\(586\) 48.7236 2.01276
\(587\) −16.8643 −0.696066 −0.348033 0.937482i \(-0.613150\pi\)
−0.348033 + 0.937482i \(0.613150\pi\)
\(588\) 40.4146 1.66667
\(589\) −0.435023 −0.0179248
\(590\) −7.28291 −0.299833
\(591\) −25.8288 −1.06246
\(592\) −4.04845 −0.166390
\(593\) 6.48817 0.266437 0.133219 0.991087i \(-0.457469\pi\)
0.133219 + 0.991087i \(0.457469\pi\)
\(594\) 6.18059 0.253593
\(595\) 4.86638 0.199502
\(596\) −45.4706 −1.86255
\(597\) 12.5170 0.512287
\(598\) 2.67672 0.109459
\(599\) −38.7831 −1.58463 −0.792317 0.610110i \(-0.791125\pi\)
−0.792317 + 0.610110i \(0.791125\pi\)
\(600\) 5.47919 0.223687
\(601\) −22.2458 −0.907424 −0.453712 0.891148i \(-0.649901\pi\)
−0.453712 + 0.891148i \(0.649901\pi\)
\(602\) 0.232661 0.00948254
\(603\) 0.781506 0.0318254
\(604\) 2.35764 0.0959311
\(605\) 1.00000 0.0406558
\(606\) −7.42786 −0.301736
\(607\) 28.4788 1.15592 0.577959 0.816066i \(-0.303850\pi\)
0.577959 + 0.816066i \(0.303850\pi\)
\(608\) −21.5467 −0.873833
\(609\) −5.91139 −0.239542
\(610\) 19.5725 0.792469
\(611\) 5.81698 0.235330
\(612\) 25.9305 1.04818
\(613\) 18.0424 0.728726 0.364363 0.931257i \(-0.381287\pi\)
0.364363 + 0.931257i \(0.381287\pi\)
\(614\) −5.29058 −0.213510
\(615\) 23.1017 0.931550
\(616\) 2.56401 0.103307
\(617\) −33.9415 −1.36643 −0.683217 0.730215i \(-0.739420\pi\)
−0.683217 + 0.730215i \(0.739420\pi\)
\(618\) 89.7884 3.61182
\(619\) 8.71706 0.350368 0.175184 0.984536i \(-0.443948\pi\)
0.175184 + 0.984536i \(0.443948\pi\)
\(620\) 0.392505 0.0157634
\(621\) 2.10853 0.0846124
\(622\) 51.6285 2.07011
\(623\) −7.22063 −0.289288
\(624\) −1.79723 −0.0719469
\(625\) 1.00000 0.0400000
\(626\) 23.0801 0.922466
\(627\) −7.51438 −0.300095
\(628\) 10.3390 0.412569
\(629\) 35.9510 1.43346
\(630\) −4.02232 −0.160253
\(631\) −39.2382 −1.56205 −0.781024 0.624501i \(-0.785302\pi\)
−0.781024 + 0.624501i \(0.785302\pi\)
\(632\) −18.9086 −0.752143
\(633\) −12.4575 −0.495140
\(634\) 50.4653 2.00423
\(635\) −7.68524 −0.304980
\(636\) 85.2583 3.38071
\(637\) 9.14822 0.362466
\(638\) 6.01935 0.238308
\(639\) 3.23610 0.128018
\(640\) 17.0097 0.672368
\(641\) 16.9106 0.667927 0.333963 0.942586i \(-0.391614\pi\)
0.333963 + 0.942586i \(0.391614\pi\)
\(642\) 6.85022 0.270357
\(643\) −23.8305 −0.939782 −0.469891 0.882725i \(-0.655707\pi\)
−0.469891 + 0.882725i \(0.655707\pi\)
\(644\) 2.44734 0.0964386
\(645\) 0.219890 0.00865815
\(646\) 37.2356 1.46502
\(647\) −22.7529 −0.894510 −0.447255 0.894407i \(-0.647598\pi\)
−0.447255 + 0.894407i \(0.647598\pi\)
\(648\) 28.1460 1.10568
\(649\) −3.22100 −0.126435
\(650\) 3.47008 0.136108
\(651\) −0.280026 −0.0109751
\(652\) 34.2972 1.34318
\(653\) 38.2483 1.49677 0.748386 0.663264i \(-0.230829\pi\)
0.748386 + 0.663264i \(0.230829\pi\)
\(654\) 63.7340 2.49220
\(655\) 0.162327 0.00634266
\(656\) 5.70129 0.222598
\(657\) 1.74515 0.0680849
\(658\) 8.73606 0.340567
\(659\) 15.0056 0.584537 0.292268 0.956336i \(-0.405590\pi\)
0.292268 + 0.956336i \(0.405590\pi\)
\(660\) 6.77995 0.263909
\(661\) −38.0233 −1.47893 −0.739467 0.673192i \(-0.764923\pi\)
−0.739467 + 0.673192i \(0.764923\pi\)
\(662\) 40.3222 1.56717
\(663\) 15.9598 0.619827
\(664\) −24.2465 −0.940948
\(665\) −3.51639 −0.136360
\(666\) −29.7154 −1.15145
\(667\) 2.05352 0.0795126
\(668\) −60.3987 −2.33690
\(669\) −31.3041 −1.21029
\(670\) −1.01254 −0.0391179
\(671\) 8.65632 0.334173
\(672\) −13.8697 −0.535035
\(673\) 12.8520 0.495409 0.247705 0.968836i \(-0.420324\pi\)
0.247705 + 0.968836i \(0.420324\pi\)
\(674\) 5.89759 0.227167
\(675\) 2.73348 0.105212
\(676\) −33.1309 −1.27427
\(677\) −14.4613 −0.555792 −0.277896 0.960611i \(-0.589637\pi\)
−0.277896 + 0.960611i \(0.589637\pi\)
\(678\) −44.1879 −1.69703
\(679\) −2.77823 −0.106619
\(680\) −12.0079 −0.460483
\(681\) −17.4993 −0.670574
\(682\) 0.285140 0.0109186
\(683\) 26.3011 1.00638 0.503192 0.864175i \(-0.332159\pi\)
0.503192 + 0.864175i \(0.332159\pi\)
\(684\) −18.7371 −0.716431
\(685\) 9.34442 0.357032
\(686\) 29.8730 1.14056
\(687\) −40.9223 −1.56128
\(688\) 0.0542668 0.00206890
\(689\) 19.2990 0.735234
\(690\) 3.79930 0.144637
\(691\) −16.6751 −0.634350 −0.317175 0.948367i \(-0.602734\pi\)
−0.317175 + 0.948367i \(0.602734\pi\)
\(692\) −72.2804 −2.74769
\(693\) −1.77895 −0.0675765
\(694\) −73.0110 −2.77146
\(695\) 7.79712 0.295762
\(696\) 14.5865 0.552900
\(697\) −50.6286 −1.91769
\(698\) −50.8092 −1.92316
\(699\) −39.7194 −1.50233
\(700\) 3.17271 0.119917
\(701\) 3.17852 0.120051 0.0600255 0.998197i \(-0.480882\pi\)
0.0600255 + 0.998197i \(0.480882\pi\)
\(702\) 9.48539 0.358003
\(703\) −25.9778 −0.979771
\(704\) 13.0478 0.491758
\(705\) 8.25653 0.310959
\(706\) −4.54668 −0.171117
\(707\) −1.53728 −0.0578155
\(708\) −21.8382 −0.820730
\(709\) 33.0245 1.24026 0.620132 0.784498i \(-0.287079\pi\)
0.620132 + 0.784498i \(0.287079\pi\)
\(710\) −4.19278 −0.157352
\(711\) 13.1190 0.492002
\(712\) 17.8171 0.667724
\(713\) 0.0972765 0.00364303
\(714\) 23.9687 0.897008
\(715\) 1.53471 0.0573947
\(716\) −19.8976 −0.743608
\(717\) −9.31147 −0.347743
\(718\) −0.350502 −0.0130806
\(719\) −3.22205 −0.120162 −0.0600810 0.998194i \(-0.519136\pi\)
−0.0600810 + 0.998194i \(0.519136\pi\)
\(720\) −0.938183 −0.0349640
\(721\) 18.5827 0.692058
\(722\) 16.0543 0.597480
\(723\) −19.6957 −0.732491
\(724\) 17.9630 0.667591
\(725\) 2.66217 0.0988704
\(726\) 4.92537 0.182798
\(727\) −35.4486 −1.31471 −0.657357 0.753579i \(-0.728326\pi\)
−0.657357 + 0.753579i \(0.728326\pi\)
\(728\) 3.93500 0.145841
\(729\) −1.66477 −0.0616580
\(730\) −2.26107 −0.0836860
\(731\) −0.481900 −0.0178237
\(732\) 58.6894 2.16922
\(733\) −5.50396 −0.203293 −0.101647 0.994821i \(-0.532411\pi\)
−0.101647 + 0.994821i \(0.532411\pi\)
\(734\) 73.6308 2.71776
\(735\) 12.9848 0.478953
\(736\) 4.81811 0.177598
\(737\) −0.447815 −0.0164955
\(738\) 41.8472 1.54042
\(739\) 49.2340 1.81110 0.905551 0.424238i \(-0.139458\pi\)
0.905551 + 0.424238i \(0.139458\pi\)
\(740\) 23.4388 0.861628
\(741\) −11.5324 −0.423652
\(742\) 28.9837 1.06402
\(743\) 35.5739 1.30508 0.652540 0.757754i \(-0.273703\pi\)
0.652540 + 0.757754i \(0.273703\pi\)
\(744\) 0.690972 0.0253323
\(745\) −14.6093 −0.535244
\(746\) 50.9452 1.86524
\(747\) 16.8226 0.615505
\(748\) −14.8586 −0.543284
\(749\) 1.41773 0.0518028
\(750\) 4.92537 0.179849
\(751\) −24.3136 −0.887216 −0.443608 0.896221i \(-0.646302\pi\)
−0.443608 + 0.896221i \(0.646302\pi\)
\(752\) 2.03764 0.0743050
\(753\) 11.5070 0.419338
\(754\) 9.23793 0.336425
\(755\) 0.757490 0.0275679
\(756\) 8.67254 0.315417
\(757\) −16.0822 −0.584519 −0.292259 0.956339i \(-0.594407\pi\)
−0.292259 + 0.956339i \(0.594407\pi\)
\(758\) 85.0848 3.09042
\(759\) 1.68031 0.0609913
\(760\) 8.67678 0.314740
\(761\) −28.6409 −1.03823 −0.519115 0.854704i \(-0.673739\pi\)
−0.519115 + 0.854704i \(0.673739\pi\)
\(762\) −37.8527 −1.37126
\(763\) 13.1905 0.477528
\(764\) 28.0074 1.01327
\(765\) 8.33125 0.301217
\(766\) 46.3760 1.67563
\(767\) −4.94328 −0.178492
\(768\) 26.9341 0.971902
\(769\) −9.70738 −0.350057 −0.175028 0.984563i \(-0.556002\pi\)
−0.175028 + 0.984563i \(0.556002\pi\)
\(770\) 2.30485 0.0830611
\(771\) 25.4062 0.914983
\(772\) 14.3220 0.515460
\(773\) −38.9553 −1.40113 −0.700563 0.713591i \(-0.747068\pi\)
−0.700563 + 0.713591i \(0.747068\pi\)
\(774\) 0.398316 0.0143172
\(775\) 0.126108 0.00452995
\(776\) 6.85535 0.246093
\(777\) −16.7220 −0.599899
\(778\) −30.3866 −1.08941
\(779\) 36.5836 1.31074
\(780\) 10.4052 0.372567
\(781\) −1.85434 −0.0663533
\(782\) −8.32635 −0.297750
\(783\) 7.27698 0.260058
\(784\) 3.20454 0.114448
\(785\) 3.32182 0.118561
\(786\) 0.799523 0.0285181
\(787\) −27.2032 −0.969689 −0.484844 0.874600i \(-0.661124\pi\)
−0.484844 + 0.874600i \(0.661124\pi\)
\(788\) 36.9046 1.31467
\(789\) 2.90959 0.103584
\(790\) −16.9974 −0.604740
\(791\) −9.14521 −0.325166
\(792\) 4.38960 0.155978
\(793\) 13.2849 0.471761
\(794\) 21.6597 0.768676
\(795\) 27.3927 0.971520
\(796\) −17.8845 −0.633899
\(797\) −19.6032 −0.694380 −0.347190 0.937795i \(-0.612864\pi\)
−0.347190 + 0.937795i \(0.612864\pi\)
\(798\) −17.3195 −0.613105
\(799\) −18.0946 −0.640141
\(800\) 6.24615 0.220835
\(801\) −12.3617 −0.436780
\(802\) 33.8701 1.19599
\(803\) −1.00000 −0.0352892
\(804\) −3.03616 −0.107077
\(805\) 0.786308 0.0277137
\(806\) 0.437606 0.0154140
\(807\) 26.8671 0.945768
\(808\) 3.79329 0.133447
\(809\) 7.97624 0.280430 0.140215 0.990121i \(-0.455221\pi\)
0.140215 + 0.990121i \(0.455221\pi\)
\(810\) 25.3012 0.888992
\(811\) −35.0160 −1.22958 −0.614790 0.788691i \(-0.710759\pi\)
−0.614790 + 0.788691i \(0.710759\pi\)
\(812\) 8.44628 0.296406
\(813\) −15.0261 −0.526990
\(814\) 17.0274 0.596810
\(815\) 11.0194 0.385992
\(816\) 5.59057 0.195709
\(817\) 0.348215 0.0121825
\(818\) −75.7290 −2.64780
\(819\) −2.73016 −0.0953994
\(820\) −33.0081 −1.15269
\(821\) −16.0075 −0.558664 −0.279332 0.960195i \(-0.590113\pi\)
−0.279332 + 0.960195i \(0.590113\pi\)
\(822\) 46.0248 1.60530
\(823\) 15.1373 0.527654 0.263827 0.964570i \(-0.415015\pi\)
0.263827 + 0.964570i \(0.415015\pi\)
\(824\) −45.8534 −1.59738
\(825\) 2.17834 0.0758400
\(826\) −7.42393 −0.258312
\(827\) −31.1695 −1.08387 −0.541935 0.840420i \(-0.682308\pi\)
−0.541935 + 0.840420i \(0.682308\pi\)
\(828\) 4.18985 0.145607
\(829\) 34.1509 1.18611 0.593054 0.805163i \(-0.297922\pi\)
0.593054 + 0.805163i \(0.297922\pi\)
\(830\) −21.7958 −0.756543
\(831\) −15.0674 −0.522682
\(832\) 20.0245 0.694226
\(833\) −28.4570 −0.985975
\(834\) 38.4037 1.32981
\(835\) −19.4056 −0.671558
\(836\) 10.7367 0.371335
\(837\) 0.344715 0.0119151
\(838\) 41.5770 1.43625
\(839\) −22.7893 −0.786774 −0.393387 0.919373i \(-0.628697\pi\)
−0.393387 + 0.919373i \(0.628697\pi\)
\(840\) 5.58528 0.192711
\(841\) −21.9129 −0.755616
\(842\) 28.2562 0.973774
\(843\) 59.9019 2.06313
\(844\) 17.7994 0.612681
\(845\) −10.6447 −0.366188
\(846\) 14.9562 0.514203
\(847\) 1.01936 0.0350257
\(848\) 6.76028 0.232149
\(849\) 24.9749 0.857136
\(850\) −10.7942 −0.370238
\(851\) 5.80896 0.199129
\(852\) −12.5723 −0.430720
\(853\) 43.8953 1.50295 0.751474 0.659763i \(-0.229343\pi\)
0.751474 + 0.659763i \(0.229343\pi\)
\(854\) 19.9515 0.682728
\(855\) −6.02007 −0.205882
\(856\) −3.49829 −0.119569
\(857\) −29.4695 −1.00666 −0.503330 0.864094i \(-0.667892\pi\)
−0.503330 + 0.864094i \(0.667892\pi\)
\(858\) 7.55900 0.258060
\(859\) 44.8915 1.53168 0.765840 0.643031i \(-0.222324\pi\)
0.765840 + 0.643031i \(0.222324\pi\)
\(860\) −0.314182 −0.0107135
\(861\) 23.5490 0.802549
\(862\) −62.0506 −2.11345
\(863\) 21.8951 0.745317 0.372659 0.927969i \(-0.378446\pi\)
0.372659 + 0.927969i \(0.378446\pi\)
\(864\) 17.0737 0.580860
\(865\) −23.2231 −0.789607
\(866\) −7.34881 −0.249723
\(867\) −12.6136 −0.428382
\(868\) 0.400105 0.0135805
\(869\) −7.51741 −0.255011
\(870\) 13.1122 0.444544
\(871\) −0.687265 −0.0232871
\(872\) −32.5479 −1.10221
\(873\) −4.75633 −0.160977
\(874\) 6.01652 0.203512
\(875\) 1.01936 0.0344608
\(876\) −6.77995 −0.229073
\(877\) 12.6311 0.426523 0.213261 0.976995i \(-0.431591\pi\)
0.213261 + 0.976995i \(0.431591\pi\)
\(878\) −19.1884 −0.647576
\(879\) 46.9408 1.58327
\(880\) 0.537594 0.0181223
\(881\) 46.6844 1.57284 0.786419 0.617694i \(-0.211933\pi\)
0.786419 + 0.617694i \(0.211933\pi\)
\(882\) 23.5212 0.792000
\(883\) −38.4042 −1.29240 −0.646202 0.763166i \(-0.723644\pi\)
−0.646202 + 0.763166i \(0.723644\pi\)
\(884\) −22.8036 −0.766967
\(885\) −7.01642 −0.235854
\(886\) −72.2595 −2.42761
\(887\) 17.4443 0.585722 0.292861 0.956155i \(-0.405393\pi\)
0.292861 + 0.956155i \(0.405393\pi\)
\(888\) 41.2620 1.38466
\(889\) −7.83406 −0.262746
\(890\) 16.0162 0.536865
\(891\) 11.1899 0.374876
\(892\) 44.7278 1.49760
\(893\) 13.0750 0.437537
\(894\) −71.9563 −2.40658
\(895\) −6.39292 −0.213692
\(896\) 17.3391 0.579258
\(897\) 2.57878 0.0861029
\(898\) −9.40789 −0.313945
\(899\) 0.335722 0.0111969
\(900\) 5.43169 0.181056
\(901\) −60.0326 −1.99998
\(902\) −23.9791 −0.798417
\(903\) 0.224148 0.00745916
\(904\) 22.5660 0.750536
\(905\) 5.77136 0.191847
\(906\) 3.73092 0.123952
\(907\) −13.5165 −0.448807 −0.224404 0.974496i \(-0.572043\pi\)
−0.224404 + 0.974496i \(0.572043\pi\)
\(908\) 25.0032 0.829761
\(909\) −2.63183 −0.0872924
\(910\) 3.53727 0.117259
\(911\) −36.2561 −1.20122 −0.600610 0.799542i \(-0.705075\pi\)
−0.600610 + 0.799542i \(0.705075\pi\)
\(912\) −4.03968 −0.133767
\(913\) −9.63959 −0.319024
\(914\) 81.4368 2.69369
\(915\) 18.8564 0.623373
\(916\) 58.4704 1.93191
\(917\) 0.165471 0.00546432
\(918\) −29.5058 −0.973835
\(919\) 36.3161 1.19796 0.598979 0.800765i \(-0.295573\pi\)
0.598979 + 0.800765i \(0.295573\pi\)
\(920\) −1.94024 −0.0639677
\(921\) −5.09700 −0.167952
\(922\) −56.3948 −1.85726
\(923\) −2.84586 −0.0936726
\(924\) 6.91123 0.227363
\(925\) 7.53069 0.247607
\(926\) −17.0002 −0.558663
\(927\) 31.8137 1.04490
\(928\) 16.6283 0.545851
\(929\) 28.0949 0.921763 0.460881 0.887462i \(-0.347533\pi\)
0.460881 + 0.887462i \(0.347533\pi\)
\(930\) 0.621131 0.0203677
\(931\) 20.5627 0.673914
\(932\) 56.7517 1.85896
\(933\) 49.7394 1.62839
\(934\) −6.53704 −0.213898
\(935\) −4.77394 −0.156124
\(936\) 6.73674 0.220197
\(937\) 5.82865 0.190414 0.0952068 0.995458i \(-0.469649\pi\)
0.0952068 + 0.995458i \(0.469649\pi\)
\(938\) −1.03215 −0.0337008
\(939\) 22.2356 0.725631
\(940\) −11.7971 −0.384777
\(941\) −26.6635 −0.869205 −0.434602 0.900622i \(-0.643111\pi\)
−0.434602 + 0.900622i \(0.643111\pi\)
\(942\) 16.3612 0.533076
\(943\) −8.18055 −0.266395
\(944\) −1.73159 −0.0563584
\(945\) 2.78641 0.0906419
\(946\) −0.228241 −0.00742076
\(947\) 48.5378 1.57727 0.788634 0.614863i \(-0.210789\pi\)
0.788634 + 0.614863i \(0.210789\pi\)
\(948\) −50.9676 −1.65535
\(949\) −1.53471 −0.0498186
\(950\) 7.79977 0.253058
\(951\) 48.6187 1.57657
\(952\) −12.2404 −0.396715
\(953\) −22.4157 −0.726117 −0.363059 0.931766i \(-0.618268\pi\)
−0.363059 + 0.931766i \(0.618268\pi\)
\(954\) 49.6201 1.60651
\(955\) 8.99854 0.291186
\(956\) 13.3044 0.430294
\(957\) 5.79910 0.187458
\(958\) 97.5386 3.15133
\(959\) 9.52536 0.307590
\(960\) 28.4225 0.917333
\(961\) −30.9841 −0.999487
\(962\) 26.1321 0.842532
\(963\) 2.42716 0.0782142
\(964\) 28.1415 0.906377
\(965\) 4.60153 0.148129
\(966\) 3.87286 0.124607
\(967\) −35.1671 −1.13090 −0.565449 0.824783i \(-0.691297\pi\)
−0.565449 + 0.824783i \(0.691297\pi\)
\(968\) −2.51531 −0.0808450
\(969\) 35.8732 1.15241
\(970\) 6.16244 0.197864
\(971\) 0.120404 0.00386395 0.00193197 0.999998i \(-0.499385\pi\)
0.00193197 + 0.999998i \(0.499385\pi\)
\(972\) 50.3435 1.61477
\(973\) 7.94810 0.254804
\(974\) 12.8296 0.411088
\(975\) 3.34311 0.107065
\(976\) 4.65358 0.148958
\(977\) −19.5697 −0.626089 −0.313044 0.949738i \(-0.601349\pi\)
−0.313044 + 0.949738i \(0.601349\pi\)
\(978\) 54.2746 1.73551
\(979\) 7.08347 0.226389
\(980\) −18.5529 −0.592652
\(981\) 22.5822 0.720993
\(982\) 75.8399 2.42015
\(983\) 5.58741 0.178211 0.0891054 0.996022i \(-0.471599\pi\)
0.0891054 + 0.996022i \(0.471599\pi\)
\(984\) −58.1079 −1.85241
\(985\) 11.8571 0.377799
\(986\) −28.7360 −0.915141
\(987\) 8.41641 0.267897
\(988\) 16.4776 0.524222
\(989\) −0.0778652 −0.00247597
\(990\) 3.94591 0.125409
\(991\) −19.7298 −0.626739 −0.313370 0.949631i \(-0.601458\pi\)
−0.313370 + 0.949631i \(0.601458\pi\)
\(992\) 0.787693 0.0250093
\(993\) 38.8468 1.23277
\(994\) −4.27397 −0.135562
\(995\) −5.74613 −0.182165
\(996\) −65.3559 −2.07088
\(997\) 14.3506 0.454489 0.227245 0.973838i \(-0.427028\pi\)
0.227245 + 0.973838i \(0.427028\pi\)
\(998\) −25.6521 −0.812004
\(999\) 20.5850 0.651280
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.c.1.3 23
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4015.2.a.c.1.3 23 1.1 even 1 trivial