Properties

Label 8031.2.a.d.1.9
Level $8031$
Weight $2$
Character 8031.1
Self dual yes
Analytic conductor $64.128$
Analytic rank $0$
Dimension $132$
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] = [8031,2,Mod(1,8031)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8031, 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("8031.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8031 = 3 \cdot 2677 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8031.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: \(64.1278578633\)
Analytic rank: \(0\)
Dimension: \(132\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Character \(\chi\) \(=\) 8031.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.57206 q^{2} +1.00000 q^{3} +4.61548 q^{4} +3.83645 q^{5} -2.57206 q^{6} -2.68797 q^{7} -6.72717 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.57206 q^{2} +1.00000 q^{3} +4.61548 q^{4} +3.83645 q^{5} -2.57206 q^{6} -2.68797 q^{7} -6.72717 q^{8} +1.00000 q^{9} -9.86756 q^{10} -2.81629 q^{11} +4.61548 q^{12} +4.42524 q^{13} +6.91362 q^{14} +3.83645 q^{15} +8.07170 q^{16} +5.87877 q^{17} -2.57206 q^{18} -5.15734 q^{19} +17.7070 q^{20} -2.68797 q^{21} +7.24365 q^{22} -6.88942 q^{23} -6.72717 q^{24} +9.71832 q^{25} -11.3820 q^{26} +1.00000 q^{27} -12.4063 q^{28} -3.16316 q^{29} -9.86756 q^{30} -3.23509 q^{31} -7.30655 q^{32} -2.81629 q^{33} -15.1205 q^{34} -10.3123 q^{35} +4.61548 q^{36} +5.45431 q^{37} +13.2650 q^{38} +4.42524 q^{39} -25.8084 q^{40} +7.13649 q^{41} +6.91362 q^{42} +11.2124 q^{43} -12.9985 q^{44} +3.83645 q^{45} +17.7200 q^{46} +2.04252 q^{47} +8.07170 q^{48} +0.225198 q^{49} -24.9961 q^{50} +5.87877 q^{51} +20.4246 q^{52} +10.0537 q^{53} -2.57206 q^{54} -10.8045 q^{55} +18.0824 q^{56} -5.15734 q^{57} +8.13583 q^{58} -11.4678 q^{59} +17.7070 q^{60} +4.97599 q^{61} +8.32083 q^{62} -2.68797 q^{63} +2.64946 q^{64} +16.9772 q^{65} +7.24365 q^{66} -7.78664 q^{67} +27.1333 q^{68} -6.88942 q^{69} +26.5237 q^{70} +14.4119 q^{71} -6.72717 q^{72} -14.8241 q^{73} -14.0288 q^{74} +9.71832 q^{75} -23.8036 q^{76} +7.57010 q^{77} -11.3820 q^{78} +11.1854 q^{79} +30.9666 q^{80} +1.00000 q^{81} -18.3555 q^{82} -0.358839 q^{83} -12.4063 q^{84} +22.5536 q^{85} -28.8390 q^{86} -3.16316 q^{87} +18.9456 q^{88} -2.28043 q^{89} -9.86756 q^{90} -11.8949 q^{91} -31.7980 q^{92} -3.23509 q^{93} -5.25349 q^{94} -19.7858 q^{95} -7.30655 q^{96} -14.0986 q^{97} -0.579221 q^{98} -2.81629 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 4 q^{2} + 132 q^{3} + 156 q^{4} + 20 q^{5} + 4 q^{6} + 44 q^{7} + 9 q^{8} + 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 132 q + 4 q^{2} + 132 q^{3} + 156 q^{4} + 20 q^{5} + 4 q^{6} + 44 q^{7} + 9 q^{8} + 132 q^{9} + 40 q^{10} + 24 q^{11} + 156 q^{12} + 62 q^{13} + 25 q^{14} + 20 q^{15} + 192 q^{16} + 77 q^{17} + 4 q^{18} + 86 q^{19} + 26 q^{20} + 44 q^{21} + 52 q^{22} + 17 q^{23} + 9 q^{24} + 212 q^{25} + 13 q^{26} + 132 q^{27} + 95 q^{28} + 52 q^{29} + 40 q^{30} + 59 q^{31} - 8 q^{32} + 24 q^{33} + 41 q^{34} + 21 q^{35} + 156 q^{36} + 76 q^{37} + 2 q^{38} + 62 q^{39} + 91 q^{40} + 114 q^{41} + 25 q^{42} + 173 q^{43} + 44 q^{44} + 20 q^{45} + 48 q^{46} + 15 q^{47} + 192 q^{48} + 262 q^{49} - 9 q^{50} + 77 q^{51} + 144 q^{52} + 15 q^{53} + 4 q^{54} + 111 q^{55} + 66 q^{56} + 86 q^{57} + 33 q^{58} + 20 q^{59} + 26 q^{60} + 182 q^{61} + 16 q^{62} + 44 q^{63} + 255 q^{64} + 70 q^{65} + 52 q^{66} + 169 q^{67} + 128 q^{68} + 17 q^{69} + 2 q^{70} + 23 q^{71} + 9 q^{72} + 148 q^{73} + 57 q^{74} + 212 q^{75} + 143 q^{76} + 31 q^{77} + 13 q^{78} + 152 q^{79} + 27 q^{80} + 132 q^{81} + 67 q^{82} + 28 q^{83} + 95 q^{84} + 88 q^{85} - 10 q^{86} + 52 q^{87} + 130 q^{88} + 136 q^{89} + 40 q^{90} + 125 q^{91} + 59 q^{93} + 95 q^{94} + 2 q^{95} - 8 q^{96} + 147 q^{97} - 18 q^{98} + 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.57206 −1.81872 −0.909360 0.416011i \(-0.863428\pi\)
−0.909360 + 0.416011i \(0.863428\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.61548 2.30774
\(5\) 3.83645 1.71571 0.857855 0.513891i \(-0.171796\pi\)
0.857855 + 0.513891i \(0.171796\pi\)
\(6\) −2.57206 −1.05004
\(7\) −2.68797 −1.01596 −0.507979 0.861369i \(-0.669607\pi\)
−0.507979 + 0.861369i \(0.669607\pi\)
\(8\) −6.72717 −2.37841
\(9\) 1.00000 0.333333
\(10\) −9.86756 −3.12040
\(11\) −2.81629 −0.849143 −0.424571 0.905395i \(-0.639575\pi\)
−0.424571 + 0.905395i \(0.639575\pi\)
\(12\) 4.61548 1.33237
\(13\) 4.42524 1.22734 0.613670 0.789562i \(-0.289692\pi\)
0.613670 + 0.789562i \(0.289692\pi\)
\(14\) 6.91362 1.84774
\(15\) 3.83645 0.990566
\(16\) 8.07170 2.01793
\(17\) 5.87877 1.42581 0.712905 0.701260i \(-0.247379\pi\)
0.712905 + 0.701260i \(0.247379\pi\)
\(18\) −2.57206 −0.606240
\(19\) −5.15734 −1.18317 −0.591587 0.806241i \(-0.701498\pi\)
−0.591587 + 0.806241i \(0.701498\pi\)
\(20\) 17.7070 3.95942
\(21\) −2.68797 −0.586564
\(22\) 7.24365 1.54435
\(23\) −6.88942 −1.43654 −0.718272 0.695762i \(-0.755067\pi\)
−0.718272 + 0.695762i \(0.755067\pi\)
\(24\) −6.72717 −1.37318
\(25\) 9.71832 1.94366
\(26\) −11.3820 −2.23219
\(27\) 1.00000 0.192450
\(28\) −12.4063 −2.34457
\(29\) −3.16316 −0.587384 −0.293692 0.955900i \(-0.594884\pi\)
−0.293692 + 0.955900i \(0.594884\pi\)
\(30\) −9.86756 −1.80156
\(31\) −3.23509 −0.581038 −0.290519 0.956869i \(-0.593828\pi\)
−0.290519 + 0.956869i \(0.593828\pi\)
\(32\) −7.30655 −1.29163
\(33\) −2.81629 −0.490253
\(34\) −15.1205 −2.59315
\(35\) −10.3123 −1.74309
\(36\) 4.61548 0.769247
\(37\) 5.45431 0.896683 0.448341 0.893862i \(-0.352015\pi\)
0.448341 + 0.893862i \(0.352015\pi\)
\(38\) 13.2650 2.15186
\(39\) 4.42524 0.708606
\(40\) −25.8084 −4.08067
\(41\) 7.13649 1.11453 0.557266 0.830334i \(-0.311850\pi\)
0.557266 + 0.830334i \(0.311850\pi\)
\(42\) 6.91362 1.06679
\(43\) 11.2124 1.70988 0.854938 0.518731i \(-0.173595\pi\)
0.854938 + 0.518731i \(0.173595\pi\)
\(44\) −12.9985 −1.95960
\(45\) 3.83645 0.571904
\(46\) 17.7200 2.61267
\(47\) 2.04252 0.297933 0.148966 0.988842i \(-0.452405\pi\)
0.148966 + 0.988842i \(0.452405\pi\)
\(48\) 8.07170 1.16505
\(49\) 0.225198 0.0321711
\(50\) −24.9961 −3.53498
\(51\) 5.87877 0.823192
\(52\) 20.4246 2.83238
\(53\) 10.0537 1.38098 0.690489 0.723342i \(-0.257395\pi\)
0.690489 + 0.723342i \(0.257395\pi\)
\(54\) −2.57206 −0.350013
\(55\) −10.8045 −1.45688
\(56\) 18.0824 2.41637
\(57\) −5.15734 −0.683106
\(58\) 8.13583 1.06829
\(59\) −11.4678 −1.49298 −0.746490 0.665397i \(-0.768262\pi\)
−0.746490 + 0.665397i \(0.768262\pi\)
\(60\) 17.7070 2.28597
\(61\) 4.97599 0.637111 0.318555 0.947904i \(-0.396802\pi\)
0.318555 + 0.947904i \(0.396802\pi\)
\(62\) 8.32083 1.05675
\(63\) −2.68797 −0.338653
\(64\) 2.64946 0.331182
\(65\) 16.9772 2.10576
\(66\) 7.24365 0.891632
\(67\) −7.78664 −0.951290 −0.475645 0.879637i \(-0.657785\pi\)
−0.475645 + 0.879637i \(0.657785\pi\)
\(68\) 27.1333 3.29040
\(69\) −6.88942 −0.829389
\(70\) 26.5237 3.17019
\(71\) 14.4119 1.71038 0.855188 0.518317i \(-0.173441\pi\)
0.855188 + 0.518317i \(0.173441\pi\)
\(72\) −6.72717 −0.792804
\(73\) −14.8241 −1.73503 −0.867513 0.497415i \(-0.834283\pi\)
−0.867513 + 0.497415i \(0.834283\pi\)
\(74\) −14.0288 −1.63081
\(75\) 9.71832 1.12217
\(76\) −23.8036 −2.73046
\(77\) 7.57010 0.862693
\(78\) −11.3820 −1.28875
\(79\) 11.1854 1.25846 0.629228 0.777221i \(-0.283371\pi\)
0.629228 + 0.777221i \(0.283371\pi\)
\(80\) 30.9666 3.46218
\(81\) 1.00000 0.111111
\(82\) −18.3555 −2.02702
\(83\) −0.358839 −0.0393877 −0.0196939 0.999806i \(-0.506269\pi\)
−0.0196939 + 0.999806i \(0.506269\pi\)
\(84\) −12.4063 −1.35364
\(85\) 22.5536 2.44628
\(86\) −28.8390 −3.10978
\(87\) −3.16316 −0.339126
\(88\) 18.9456 2.01961
\(89\) −2.28043 −0.241725 −0.120862 0.992669i \(-0.538566\pi\)
−0.120862 + 0.992669i \(0.538566\pi\)
\(90\) −9.86756 −1.04013
\(91\) −11.8949 −1.24693
\(92\) −31.7980 −3.31517
\(93\) −3.23509 −0.335463
\(94\) −5.25349 −0.541856
\(95\) −19.7858 −2.02999
\(96\) −7.30655 −0.745721
\(97\) −14.0986 −1.43149 −0.715747 0.698359i \(-0.753914\pi\)
−0.715747 + 0.698359i \(0.753914\pi\)
\(98\) −0.579221 −0.0585102
\(99\) −2.81629 −0.283048
\(100\) 44.8547 4.48547
\(101\) 5.20985 0.518400 0.259200 0.965824i \(-0.416541\pi\)
0.259200 + 0.965824i \(0.416541\pi\)
\(102\) −15.1205 −1.49716
\(103\) 15.3371 1.51121 0.755606 0.655027i \(-0.227343\pi\)
0.755606 + 0.655027i \(0.227343\pi\)
\(104\) −29.7693 −2.91912
\(105\) −10.3123 −1.00637
\(106\) −25.8586 −2.51161
\(107\) 9.09892 0.879626 0.439813 0.898089i \(-0.355045\pi\)
0.439813 + 0.898089i \(0.355045\pi\)
\(108\) 4.61548 0.444125
\(109\) 18.0356 1.72750 0.863750 0.503920i \(-0.168109\pi\)
0.863750 + 0.503920i \(0.168109\pi\)
\(110\) 27.7899 2.64966
\(111\) 5.45431 0.517700
\(112\) −21.6965 −2.05013
\(113\) −15.4150 −1.45012 −0.725058 0.688687i \(-0.758187\pi\)
−0.725058 + 0.688687i \(0.758187\pi\)
\(114\) 13.2650 1.24238
\(115\) −26.4309 −2.46469
\(116\) −14.5995 −1.35553
\(117\) 4.42524 0.409114
\(118\) 29.4958 2.71531
\(119\) −15.8020 −1.44856
\(120\) −25.8084 −2.35598
\(121\) −3.06853 −0.278957
\(122\) −12.7985 −1.15873
\(123\) 7.13649 0.643475
\(124\) −14.9315 −1.34089
\(125\) 18.1016 1.61905
\(126\) 6.91362 0.615914
\(127\) 19.4907 1.72952 0.864762 0.502182i \(-0.167469\pi\)
0.864762 + 0.502182i \(0.167469\pi\)
\(128\) 7.79854 0.689300
\(129\) 11.2124 0.987197
\(130\) −43.6663 −3.82979
\(131\) −0.671792 −0.0586947 −0.0293474 0.999569i \(-0.509343\pi\)
−0.0293474 + 0.999569i \(0.509343\pi\)
\(132\) −12.9985 −1.13138
\(133\) 13.8628 1.20206
\(134\) 20.0277 1.73013
\(135\) 3.83645 0.330189
\(136\) −39.5475 −3.39117
\(137\) −17.7915 −1.52003 −0.760015 0.649905i \(-0.774808\pi\)
−0.760015 + 0.649905i \(0.774808\pi\)
\(138\) 17.7200 1.50843
\(139\) −12.2753 −1.04118 −0.520588 0.853808i \(-0.674287\pi\)
−0.520588 + 0.853808i \(0.674287\pi\)
\(140\) −47.5961 −4.02260
\(141\) 2.04252 0.172012
\(142\) −37.0682 −3.11070
\(143\) −12.4627 −1.04219
\(144\) 8.07170 0.672642
\(145\) −12.1353 −1.00778
\(146\) 38.1283 3.15552
\(147\) 0.225198 0.0185740
\(148\) 25.1743 2.06931
\(149\) −18.2342 −1.49380 −0.746900 0.664936i \(-0.768459\pi\)
−0.746900 + 0.664936i \(0.768459\pi\)
\(150\) −24.9961 −2.04092
\(151\) 9.27800 0.755033 0.377516 0.926003i \(-0.376778\pi\)
0.377516 + 0.926003i \(0.376778\pi\)
\(152\) 34.6943 2.81408
\(153\) 5.87877 0.475270
\(154\) −19.4707 −1.56900
\(155\) −12.4112 −0.996894
\(156\) 20.4246 1.63528
\(157\) 17.3396 1.38385 0.691926 0.721968i \(-0.256762\pi\)
0.691926 + 0.721968i \(0.256762\pi\)
\(158\) −28.7695 −2.28878
\(159\) 10.0537 0.797309
\(160\) −28.0312 −2.21606
\(161\) 18.5186 1.45947
\(162\) −2.57206 −0.202080
\(163\) −0.588856 −0.0461228 −0.0230614 0.999734i \(-0.507341\pi\)
−0.0230614 + 0.999734i \(0.507341\pi\)
\(164\) 32.9383 2.57205
\(165\) −10.8045 −0.841132
\(166\) 0.922955 0.0716352
\(167\) −12.5907 −0.974301 −0.487150 0.873318i \(-0.661964\pi\)
−0.487150 + 0.873318i \(0.661964\pi\)
\(168\) 18.0824 1.39509
\(169\) 6.58275 0.506366
\(170\) −58.0091 −4.44909
\(171\) −5.15734 −0.394391
\(172\) 51.7506 3.94595
\(173\) −0.933654 −0.0709844 −0.0354922 0.999370i \(-0.511300\pi\)
−0.0354922 + 0.999370i \(0.511300\pi\)
\(174\) 8.13583 0.616775
\(175\) −26.1226 −1.97468
\(176\) −22.7322 −1.71351
\(177\) −11.4678 −0.861972
\(178\) 5.86539 0.439629
\(179\) 3.33908 0.249574 0.124787 0.992184i \(-0.460175\pi\)
0.124787 + 0.992184i \(0.460175\pi\)
\(180\) 17.7070 1.31981
\(181\) 20.6096 1.53190 0.765949 0.642901i \(-0.222269\pi\)
0.765949 + 0.642901i \(0.222269\pi\)
\(182\) 30.5944 2.26781
\(183\) 4.97599 0.367836
\(184\) 46.3463 3.41670
\(185\) 20.9252 1.53845
\(186\) 8.32083 0.610113
\(187\) −16.5563 −1.21072
\(188\) 9.42723 0.687551
\(189\) −2.68797 −0.195521
\(190\) 50.8903 3.69197
\(191\) −4.86708 −0.352169 −0.176085 0.984375i \(-0.556343\pi\)
−0.176085 + 0.984375i \(0.556343\pi\)
\(192\) 2.64946 0.191208
\(193\) 18.1019 1.30300 0.651502 0.758647i \(-0.274139\pi\)
0.651502 + 0.758647i \(0.274139\pi\)
\(194\) 36.2624 2.60349
\(195\) 16.9772 1.21576
\(196\) 1.03940 0.0742425
\(197\) 8.14430 0.580257 0.290129 0.956988i \(-0.406302\pi\)
0.290129 + 0.956988i \(0.406302\pi\)
\(198\) 7.24365 0.514784
\(199\) 23.2617 1.64898 0.824489 0.565878i \(-0.191463\pi\)
0.824489 + 0.565878i \(0.191463\pi\)
\(200\) −65.3768 −4.62284
\(201\) −7.78664 −0.549228
\(202\) −13.4000 −0.942824
\(203\) 8.50248 0.596757
\(204\) 27.1333 1.89971
\(205\) 27.3787 1.91221
\(206\) −39.4480 −2.74847
\(207\) −6.88942 −0.478848
\(208\) 35.7192 2.47668
\(209\) 14.5245 1.00468
\(210\) 26.5237 1.83031
\(211\) 22.2384 1.53095 0.765477 0.643463i \(-0.222503\pi\)
0.765477 + 0.643463i \(0.222503\pi\)
\(212\) 46.4026 3.18694
\(213\) 14.4119 0.987486
\(214\) −23.4030 −1.59979
\(215\) 43.0158 2.93365
\(216\) −6.72717 −0.457726
\(217\) 8.69582 0.590311
\(218\) −46.3887 −3.14184
\(219\) −14.8241 −1.00172
\(220\) −49.8681 −3.36211
\(221\) 26.0150 1.74996
\(222\) −14.0288 −0.941551
\(223\) 3.42042 0.229048 0.114524 0.993420i \(-0.463466\pi\)
0.114524 + 0.993420i \(0.463466\pi\)
\(224\) 19.6398 1.31224
\(225\) 9.71832 0.647888
\(226\) 39.6482 2.63736
\(227\) 20.6053 1.36762 0.683810 0.729661i \(-0.260322\pi\)
0.683810 + 0.729661i \(0.260322\pi\)
\(228\) −23.8036 −1.57643
\(229\) 18.4194 1.21719 0.608596 0.793480i \(-0.291733\pi\)
0.608596 + 0.793480i \(0.291733\pi\)
\(230\) 67.9818 4.48259
\(231\) 7.57010 0.498076
\(232\) 21.2791 1.39704
\(233\) 4.44717 0.291344 0.145672 0.989333i \(-0.453466\pi\)
0.145672 + 0.989333i \(0.453466\pi\)
\(234\) −11.3820 −0.744063
\(235\) 7.83603 0.511166
\(236\) −52.9294 −3.44541
\(237\) 11.1854 0.726570
\(238\) 40.6436 2.63453
\(239\) 9.55835 0.618278 0.309139 0.951017i \(-0.399959\pi\)
0.309139 + 0.951017i \(0.399959\pi\)
\(240\) 30.9666 1.99889
\(241\) −9.35677 −0.602723 −0.301361 0.953510i \(-0.597441\pi\)
−0.301361 + 0.953510i \(0.597441\pi\)
\(242\) 7.89242 0.507344
\(243\) 1.00000 0.0641500
\(244\) 22.9666 1.47029
\(245\) 0.863958 0.0551963
\(246\) −18.3555 −1.17030
\(247\) −22.8225 −1.45216
\(248\) 21.7630 1.38195
\(249\) −0.358839 −0.0227405
\(250\) −46.5583 −2.94461
\(251\) 2.01942 0.127465 0.0637324 0.997967i \(-0.479700\pi\)
0.0637324 + 0.997967i \(0.479700\pi\)
\(252\) −12.4063 −0.781523
\(253\) 19.4026 1.21983
\(254\) −50.1313 −3.14552
\(255\) 22.5536 1.41236
\(256\) −25.3572 −1.58483
\(257\) −22.1534 −1.38189 −0.690946 0.722906i \(-0.742806\pi\)
−0.690946 + 0.722906i \(0.742806\pi\)
\(258\) −28.8390 −1.79543
\(259\) −14.6610 −0.910992
\(260\) 78.3579 4.85955
\(261\) −3.16316 −0.195795
\(262\) 1.72789 0.106749
\(263\) 12.0481 0.742917 0.371458 0.928450i \(-0.378858\pi\)
0.371458 + 0.928450i \(0.378858\pi\)
\(264\) 18.9456 1.16602
\(265\) 38.5704 2.36936
\(266\) −35.6559 −2.18620
\(267\) −2.28043 −0.139560
\(268\) −35.9391 −2.19533
\(269\) 1.91786 0.116934 0.0584670 0.998289i \(-0.481379\pi\)
0.0584670 + 0.998289i \(0.481379\pi\)
\(270\) −9.86756 −0.600521
\(271\) −19.8197 −1.20396 −0.601980 0.798511i \(-0.705621\pi\)
−0.601980 + 0.798511i \(0.705621\pi\)
\(272\) 47.4517 2.87718
\(273\) −11.8949 −0.719914
\(274\) 45.7608 2.76451
\(275\) −27.3696 −1.65045
\(276\) −31.7980 −1.91401
\(277\) 12.5533 0.754253 0.377126 0.926162i \(-0.376912\pi\)
0.377126 + 0.926162i \(0.376912\pi\)
\(278\) 31.5728 1.89361
\(279\) −3.23509 −0.193679
\(280\) 69.3723 4.14579
\(281\) 6.34273 0.378375 0.189188 0.981941i \(-0.439415\pi\)
0.189188 + 0.981941i \(0.439415\pi\)
\(282\) −5.25349 −0.312841
\(283\) 2.05838 0.122358 0.0611789 0.998127i \(-0.480514\pi\)
0.0611789 + 0.998127i \(0.480514\pi\)
\(284\) 66.5178 3.94711
\(285\) −19.7858 −1.17201
\(286\) 32.0549 1.89545
\(287\) −19.1827 −1.13232
\(288\) −7.30655 −0.430542
\(289\) 17.5599 1.03294
\(290\) 31.2127 1.83287
\(291\) −14.0986 −0.826474
\(292\) −68.4202 −4.00399
\(293\) −25.9128 −1.51384 −0.756922 0.653506i \(-0.773298\pi\)
−0.756922 + 0.653506i \(0.773298\pi\)
\(294\) −0.579221 −0.0337809
\(295\) −43.9956 −2.56152
\(296\) −36.6920 −2.13268
\(297\) −2.81629 −0.163418
\(298\) 46.8993 2.71680
\(299\) −30.4874 −1.76313
\(300\) 44.8547 2.58969
\(301\) −30.1386 −1.73716
\(302\) −23.8636 −1.37319
\(303\) 5.20985 0.299298
\(304\) −41.6285 −2.38756
\(305\) 19.0901 1.09310
\(306\) −15.1205 −0.864383
\(307\) 19.5895 1.11803 0.559017 0.829156i \(-0.311179\pi\)
0.559017 + 0.829156i \(0.311179\pi\)
\(308\) 34.9397 1.99087
\(309\) 15.3371 0.872498
\(310\) 31.9224 1.81307
\(311\) −29.2485 −1.65853 −0.829265 0.558855i \(-0.811241\pi\)
−0.829265 + 0.558855i \(0.811241\pi\)
\(312\) −29.7693 −1.68536
\(313\) −5.66840 −0.320397 −0.160199 0.987085i \(-0.551213\pi\)
−0.160199 + 0.987085i \(0.551213\pi\)
\(314\) −44.5985 −2.51684
\(315\) −10.3123 −0.581030
\(316\) 51.6260 2.90419
\(317\) 13.2127 0.742098 0.371049 0.928613i \(-0.378998\pi\)
0.371049 + 0.928613i \(0.378998\pi\)
\(318\) −25.8586 −1.45008
\(319\) 8.90836 0.498773
\(320\) 10.1645 0.568213
\(321\) 9.09892 0.507852
\(322\) −47.6309 −2.65436
\(323\) −30.3188 −1.68698
\(324\) 4.61548 0.256416
\(325\) 43.0059 2.38554
\(326\) 1.51457 0.0838844
\(327\) 18.0356 0.997373
\(328\) −48.0083 −2.65082
\(329\) −5.49025 −0.302687
\(330\) 27.7899 1.52978
\(331\) 8.88215 0.488207 0.244104 0.969749i \(-0.421506\pi\)
0.244104 + 0.969749i \(0.421506\pi\)
\(332\) −1.65622 −0.0908966
\(333\) 5.45431 0.298894
\(334\) 32.3841 1.77198
\(335\) −29.8730 −1.63214
\(336\) −21.6965 −1.18364
\(337\) 17.3095 0.942908 0.471454 0.881891i \(-0.343729\pi\)
0.471454 + 0.881891i \(0.343729\pi\)
\(338\) −16.9312 −0.920937
\(339\) −15.4150 −0.837225
\(340\) 104.096 5.64538
\(341\) 9.11093 0.493384
\(342\) 13.2650 0.717287
\(343\) 18.2105 0.983274
\(344\) −75.4277 −4.06679
\(345\) −26.4309 −1.42299
\(346\) 2.40141 0.129101
\(347\) 16.0953 0.864043 0.432021 0.901863i \(-0.357801\pi\)
0.432021 + 0.901863i \(0.357801\pi\)
\(348\) −14.5995 −0.782615
\(349\) 16.6352 0.890462 0.445231 0.895416i \(-0.353121\pi\)
0.445231 + 0.895416i \(0.353121\pi\)
\(350\) 67.1888 3.59139
\(351\) 4.42524 0.236202
\(352\) 20.5773 1.09678
\(353\) −17.4445 −0.928479 −0.464239 0.885710i \(-0.653672\pi\)
−0.464239 + 0.885710i \(0.653672\pi\)
\(354\) 29.4958 1.56769
\(355\) 55.2904 2.93451
\(356\) −10.5253 −0.557838
\(357\) −15.8020 −0.836329
\(358\) −8.58830 −0.453906
\(359\) −5.19012 −0.273924 −0.136962 0.990576i \(-0.543734\pi\)
−0.136962 + 0.990576i \(0.543734\pi\)
\(360\) −25.8084 −1.36022
\(361\) 7.59813 0.399902
\(362\) −53.0091 −2.78609
\(363\) −3.06853 −0.161056
\(364\) −54.9008 −2.87758
\(365\) −56.8717 −2.97680
\(366\) −12.7985 −0.668990
\(367\) 21.2962 1.11165 0.555827 0.831298i \(-0.312402\pi\)
0.555827 + 0.831298i \(0.312402\pi\)
\(368\) −55.6094 −2.89884
\(369\) 7.13649 0.371511
\(370\) −53.8207 −2.79801
\(371\) −27.0240 −1.40302
\(372\) −14.9315 −0.774161
\(373\) −18.0070 −0.932368 −0.466184 0.884688i \(-0.654371\pi\)
−0.466184 + 0.884688i \(0.654371\pi\)
\(374\) 42.5838 2.20195
\(375\) 18.1016 0.934762
\(376\) −13.7404 −0.708607
\(377\) −13.9977 −0.720920
\(378\) 6.91362 0.355598
\(379\) −10.9441 −0.562163 −0.281081 0.959684i \(-0.590693\pi\)
−0.281081 + 0.959684i \(0.590693\pi\)
\(380\) −91.3212 −4.68468
\(381\) 19.4907 0.998541
\(382\) 12.5184 0.640497
\(383\) −4.73611 −0.242004 −0.121002 0.992652i \(-0.538611\pi\)
−0.121002 + 0.992652i \(0.538611\pi\)
\(384\) 7.79854 0.397967
\(385\) 29.0423 1.48013
\(386\) −46.5592 −2.36980
\(387\) 11.2124 0.569959
\(388\) −65.0718 −3.30352
\(389\) −2.36277 −0.119797 −0.0598986 0.998204i \(-0.519078\pi\)
−0.0598986 + 0.998204i \(0.519078\pi\)
\(390\) −43.6663 −2.21113
\(391\) −40.5013 −2.04824
\(392\) −1.51494 −0.0765161
\(393\) −0.671792 −0.0338874
\(394\) −20.9476 −1.05533
\(395\) 42.9122 2.15915
\(396\) −12.9985 −0.653200
\(397\) 6.23818 0.313085 0.156543 0.987671i \(-0.449965\pi\)
0.156543 + 0.987671i \(0.449965\pi\)
\(398\) −59.8304 −2.99903
\(399\) 13.8628 0.694007
\(400\) 78.4434 3.92217
\(401\) −18.3745 −0.917581 −0.458790 0.888545i \(-0.651717\pi\)
−0.458790 + 0.888545i \(0.651717\pi\)
\(402\) 20.0277 0.998891
\(403\) −14.3160 −0.713132
\(404\) 24.0460 1.19633
\(405\) 3.83645 0.190635
\(406\) −21.8689 −1.08533
\(407\) −15.3609 −0.761411
\(408\) −39.5475 −1.95789
\(409\) 33.0442 1.63393 0.816966 0.576685i \(-0.195654\pi\)
0.816966 + 0.576685i \(0.195654\pi\)
\(410\) −70.4197 −3.47778
\(411\) −17.7915 −0.877590
\(412\) 70.7882 3.48748
\(413\) 30.8251 1.51680
\(414\) 17.7200 0.870890
\(415\) −1.37667 −0.0675779
\(416\) −32.3332 −1.58527
\(417\) −12.2753 −0.601124
\(418\) −37.3580 −1.82724
\(419\) −0.0276870 −0.00135260 −0.000676299 1.00000i \(-0.500215\pi\)
−0.000676299 1.00000i \(0.500215\pi\)
\(420\) −47.5961 −2.32245
\(421\) 33.3026 1.62307 0.811534 0.584306i \(-0.198633\pi\)
0.811534 + 0.584306i \(0.198633\pi\)
\(422\) −57.1984 −2.78438
\(423\) 2.04252 0.0993109
\(424\) −67.6328 −3.28454
\(425\) 57.1317 2.77130
\(426\) −37.0682 −1.79596
\(427\) −13.3753 −0.647278
\(428\) 41.9959 2.02995
\(429\) −12.4627 −0.601707
\(430\) −110.639 −5.33549
\(431\) 11.7691 0.566898 0.283449 0.958987i \(-0.408521\pi\)
0.283449 + 0.958987i \(0.408521\pi\)
\(432\) 8.07170 0.388350
\(433\) −34.7155 −1.66832 −0.834159 0.551524i \(-0.814047\pi\)
−0.834159 + 0.551524i \(0.814047\pi\)
\(434\) −22.3662 −1.07361
\(435\) −12.1353 −0.581843
\(436\) 83.2431 3.98662
\(437\) 35.5311 1.69968
\(438\) 38.1283 1.82184
\(439\) 21.7387 1.03753 0.518765 0.854917i \(-0.326392\pi\)
0.518765 + 0.854917i \(0.326392\pi\)
\(440\) 72.6839 3.46507
\(441\) 0.225198 0.0107237
\(442\) −66.9120 −3.18268
\(443\) 1.23837 0.0588367 0.0294183 0.999567i \(-0.490634\pi\)
0.0294183 + 0.999567i \(0.490634\pi\)
\(444\) 25.1743 1.19472
\(445\) −8.74873 −0.414730
\(446\) −8.79752 −0.416575
\(447\) −18.2342 −0.862446
\(448\) −7.12167 −0.336467
\(449\) −27.9237 −1.31780 −0.658901 0.752230i \(-0.728978\pi\)
−0.658901 + 0.752230i \(0.728978\pi\)
\(450\) −24.9961 −1.17833
\(451\) −20.0984 −0.946397
\(452\) −71.1474 −3.34649
\(453\) 9.27800 0.435918
\(454\) −52.9979 −2.48732
\(455\) −45.6342 −2.13937
\(456\) 34.6943 1.62471
\(457\) 13.0679 0.611292 0.305646 0.952145i \(-0.401128\pi\)
0.305646 + 0.952145i \(0.401128\pi\)
\(458\) −47.3759 −2.21373
\(459\) 5.87877 0.274397
\(460\) −121.991 −5.68788
\(461\) 9.95387 0.463598 0.231799 0.972764i \(-0.425539\pi\)
0.231799 + 0.972764i \(0.425539\pi\)
\(462\) −19.4707 −0.905861
\(463\) −11.0167 −0.511989 −0.255994 0.966678i \(-0.582403\pi\)
−0.255994 + 0.966678i \(0.582403\pi\)
\(464\) −25.5321 −1.18530
\(465\) −12.4112 −0.575557
\(466\) −11.4384 −0.529873
\(467\) 6.45694 0.298791 0.149396 0.988777i \(-0.452267\pi\)
0.149396 + 0.988777i \(0.452267\pi\)
\(468\) 20.4246 0.944128
\(469\) 20.9303 0.966471
\(470\) −20.1547 −0.929668
\(471\) 17.3396 0.798967
\(472\) 77.1458 3.55092
\(473\) −31.5774 −1.45193
\(474\) −28.7695 −1.32143
\(475\) −50.1207 −2.29969
\(476\) −72.9337 −3.34291
\(477\) 10.0537 0.460326
\(478\) −24.5846 −1.12447
\(479\) 17.0126 0.777325 0.388663 0.921380i \(-0.372937\pi\)
0.388663 + 0.921380i \(0.372937\pi\)
\(480\) −28.0312 −1.27944
\(481\) 24.1366 1.10054
\(482\) 24.0662 1.09618
\(483\) 18.5186 0.842625
\(484\) −14.1627 −0.643760
\(485\) −54.0885 −2.45603
\(486\) −2.57206 −0.116671
\(487\) −26.4641 −1.19920 −0.599601 0.800299i \(-0.704674\pi\)
−0.599601 + 0.800299i \(0.704674\pi\)
\(488\) −33.4743 −1.51531
\(489\) −0.588856 −0.0266290
\(490\) −2.22215 −0.100387
\(491\) 7.13267 0.321893 0.160946 0.986963i \(-0.448545\pi\)
0.160946 + 0.986963i \(0.448545\pi\)
\(492\) 32.9383 1.48497
\(493\) −18.5955 −0.837498
\(494\) 58.7007 2.64107
\(495\) −10.8045 −0.485628
\(496\) −26.1126 −1.17249
\(497\) −38.7388 −1.73767
\(498\) 0.922955 0.0413586
\(499\) −33.8760 −1.51650 −0.758249 0.651965i \(-0.773945\pi\)
−0.758249 + 0.651965i \(0.773945\pi\)
\(500\) 83.5475 3.73636
\(501\) −12.5907 −0.562513
\(502\) −5.19407 −0.231823
\(503\) 10.4035 0.463870 0.231935 0.972731i \(-0.425494\pi\)
0.231935 + 0.972731i \(0.425494\pi\)
\(504\) 18.0824 0.805456
\(505\) 19.9873 0.889424
\(506\) −49.9046 −2.21853
\(507\) 6.58275 0.292350
\(508\) 89.9592 3.99129
\(509\) 43.7967 1.94125 0.970627 0.240590i \(-0.0773408\pi\)
0.970627 + 0.240590i \(0.0773408\pi\)
\(510\) −58.0091 −2.56869
\(511\) 39.8467 1.76271
\(512\) 49.6231 2.19305
\(513\) −5.15734 −0.227702
\(514\) 56.9799 2.51327
\(515\) 58.8400 2.59280
\(516\) 51.7506 2.27819
\(517\) −5.75233 −0.252987
\(518\) 37.7090 1.65684
\(519\) −0.933654 −0.0409828
\(520\) −114.208 −5.00837
\(521\) 2.19046 0.0959657 0.0479828 0.998848i \(-0.484721\pi\)
0.0479828 + 0.998848i \(0.484721\pi\)
\(522\) 8.13583 0.356095
\(523\) −27.5000 −1.20249 −0.601246 0.799064i \(-0.705329\pi\)
−0.601246 + 0.799064i \(0.705329\pi\)
\(524\) −3.10064 −0.135452
\(525\) −26.1226 −1.14008
\(526\) −30.9884 −1.35116
\(527\) −19.0183 −0.828451
\(528\) −22.7322 −0.989293
\(529\) 24.4642 1.06366
\(530\) −99.2053 −4.30920
\(531\) −11.4678 −0.497660
\(532\) 63.9834 2.77403
\(533\) 31.5807 1.36791
\(534\) 5.86539 0.253820
\(535\) 34.9075 1.50918
\(536\) 52.3821 2.26256
\(537\) 3.33908 0.144092
\(538\) −4.93285 −0.212670
\(539\) −0.634221 −0.0273178
\(540\) 17.7070 0.761990
\(541\) −39.5768 −1.70154 −0.850770 0.525538i \(-0.823864\pi\)
−0.850770 + 0.525538i \(0.823864\pi\)
\(542\) 50.9774 2.18967
\(543\) 20.6096 0.884442
\(544\) −42.9535 −1.84162
\(545\) 69.1928 2.96389
\(546\) 30.5944 1.30932
\(547\) 6.59572 0.282013 0.141006 0.990009i \(-0.454966\pi\)
0.141006 + 0.990009i \(0.454966\pi\)
\(548\) −82.1163 −3.50784
\(549\) 4.97599 0.212370
\(550\) 70.3961 3.00170
\(551\) 16.3135 0.694977
\(552\) 46.3463 1.97263
\(553\) −30.0661 −1.27854
\(554\) −32.2877 −1.37177
\(555\) 20.9252 0.888223
\(556\) −56.6564 −2.40277
\(557\) 29.9669 1.26974 0.634869 0.772619i \(-0.281054\pi\)
0.634869 + 0.772619i \(0.281054\pi\)
\(558\) 8.32083 0.352249
\(559\) 49.6176 2.09860
\(560\) −83.2375 −3.51743
\(561\) −16.5563 −0.699007
\(562\) −16.3139 −0.688159
\(563\) −42.2715 −1.78153 −0.890765 0.454465i \(-0.849831\pi\)
−0.890765 + 0.454465i \(0.849831\pi\)
\(564\) 9.42723 0.396958
\(565\) −59.1386 −2.48798
\(566\) −5.29426 −0.222534
\(567\) −2.68797 −0.112884
\(568\) −96.9512 −4.06798
\(569\) 0.826136 0.0346334 0.0173167 0.999850i \(-0.494488\pi\)
0.0173167 + 0.999850i \(0.494488\pi\)
\(570\) 50.8903 2.13156
\(571\) 13.1838 0.551726 0.275863 0.961197i \(-0.411036\pi\)
0.275863 + 0.961197i \(0.411036\pi\)
\(572\) −57.5216 −2.40510
\(573\) −4.86708 −0.203325
\(574\) 49.3390 2.05937
\(575\) −66.9536 −2.79216
\(576\) 2.64946 0.110394
\(577\) 16.1784 0.673516 0.336758 0.941591i \(-0.390670\pi\)
0.336758 + 0.941591i \(0.390670\pi\)
\(578\) −45.1651 −1.87862
\(579\) 18.1019 0.752290
\(580\) −56.0102 −2.32570
\(581\) 0.964550 0.0400163
\(582\) 36.2624 1.50312
\(583\) −28.3140 −1.17265
\(584\) 99.7240 4.12661
\(585\) 16.9772 0.701921
\(586\) 66.6493 2.75326
\(587\) 32.6181 1.34629 0.673146 0.739510i \(-0.264943\pi\)
0.673146 + 0.739510i \(0.264943\pi\)
\(588\) 1.03940 0.0428639
\(589\) 16.6844 0.687470
\(590\) 113.159 4.65869
\(591\) 8.14430 0.335012
\(592\) 44.0255 1.80944
\(593\) −6.49879 −0.266873 −0.133437 0.991057i \(-0.542601\pi\)
−0.133437 + 0.991057i \(0.542601\pi\)
\(594\) 7.24365 0.297211
\(595\) −60.6234 −2.48532
\(596\) −84.1594 −3.44730
\(597\) 23.2617 0.952038
\(598\) 78.4152 3.20664
\(599\) −20.2303 −0.826587 −0.413293 0.910598i \(-0.635622\pi\)
−0.413293 + 0.910598i \(0.635622\pi\)
\(600\) −65.3768 −2.66900
\(601\) 45.6710 1.86296 0.931479 0.363796i \(-0.118520\pi\)
0.931479 + 0.363796i \(0.118520\pi\)
\(602\) 77.5183 3.15941
\(603\) −7.78664 −0.317097
\(604\) 42.8224 1.74242
\(605\) −11.7722 −0.478609
\(606\) −13.4000 −0.544340
\(607\) −25.2483 −1.02480 −0.512399 0.858747i \(-0.671243\pi\)
−0.512399 + 0.858747i \(0.671243\pi\)
\(608\) 37.6823 1.52822
\(609\) 8.50248 0.344538
\(610\) −49.1009 −1.98804
\(611\) 9.03866 0.365665
\(612\) 27.1333 1.09680
\(613\) −43.2229 −1.74576 −0.872878 0.487939i \(-0.837749\pi\)
−0.872878 + 0.487939i \(0.837749\pi\)
\(614\) −50.3854 −2.03339
\(615\) 27.3787 1.10402
\(616\) −50.9254 −2.05184
\(617\) −0.423375 −0.0170444 −0.00852222 0.999964i \(-0.502713\pi\)
−0.00852222 + 0.999964i \(0.502713\pi\)
\(618\) −39.4480 −1.58683
\(619\) −34.3392 −1.38021 −0.690104 0.723710i \(-0.742435\pi\)
−0.690104 + 0.723710i \(0.742435\pi\)
\(620\) −57.2838 −2.30057
\(621\) −6.88942 −0.276463
\(622\) 75.2288 3.01640
\(623\) 6.12972 0.245582
\(624\) 35.7192 1.42991
\(625\) 20.8541 0.834166
\(626\) 14.5795 0.582713
\(627\) 14.5245 0.580054
\(628\) 80.0307 3.19357
\(629\) 32.0646 1.27850
\(630\) 26.5237 1.05673
\(631\) −18.8473 −0.750298 −0.375149 0.926965i \(-0.622408\pi\)
−0.375149 + 0.926965i \(0.622408\pi\)
\(632\) −75.2461 −2.99313
\(633\) 22.2384 0.883897
\(634\) −33.9838 −1.34967
\(635\) 74.7752 2.96736
\(636\) 46.4026 1.83998
\(637\) 0.996553 0.0394849
\(638\) −22.9128 −0.907127
\(639\) 14.4119 0.570126
\(640\) 29.9187 1.18264
\(641\) −4.87013 −0.192358 −0.0961792 0.995364i \(-0.530662\pi\)
−0.0961792 + 0.995364i \(0.530662\pi\)
\(642\) −23.4030 −0.923641
\(643\) 3.32173 0.130996 0.0654980 0.997853i \(-0.479136\pi\)
0.0654980 + 0.997853i \(0.479136\pi\)
\(644\) 85.4722 3.36808
\(645\) 43.0158 1.69374
\(646\) 77.9817 3.06815
\(647\) −37.0096 −1.45500 −0.727499 0.686109i \(-0.759317\pi\)
−0.727499 + 0.686109i \(0.759317\pi\)
\(648\) −6.72717 −0.264268
\(649\) 32.2966 1.26775
\(650\) −110.614 −4.33862
\(651\) 8.69582 0.340816
\(652\) −2.71786 −0.106439
\(653\) 35.9507 1.40686 0.703429 0.710766i \(-0.251651\pi\)
0.703429 + 0.710766i \(0.251651\pi\)
\(654\) −46.3887 −1.81394
\(655\) −2.57729 −0.100703
\(656\) 57.6036 2.24904
\(657\) −14.8241 −0.578342
\(658\) 14.1212 0.550503
\(659\) −31.0617 −1.20999 −0.604997 0.796228i \(-0.706826\pi\)
−0.604997 + 0.796228i \(0.706826\pi\)
\(660\) −49.8681 −1.94111
\(661\) 36.3795 1.41500 0.707500 0.706714i \(-0.249823\pi\)
0.707500 + 0.706714i \(0.249823\pi\)
\(662\) −22.8454 −0.887912
\(663\) 26.0150 1.01034
\(664\) 2.41397 0.0936803
\(665\) 53.1838 2.06238
\(666\) −14.0288 −0.543605
\(667\) 21.7923 0.843803
\(668\) −58.1123 −2.24843
\(669\) 3.42042 0.132241
\(670\) 76.8352 2.96840
\(671\) −14.0138 −0.540998
\(672\) 19.6398 0.757622
\(673\) −32.9850 −1.27148 −0.635739 0.771904i \(-0.719305\pi\)
−0.635739 + 0.771904i \(0.719305\pi\)
\(674\) −44.5210 −1.71488
\(675\) 9.71832 0.374058
\(676\) 30.3826 1.16856
\(677\) −48.7074 −1.87198 −0.935989 0.352028i \(-0.885492\pi\)
−0.935989 + 0.352028i \(0.885492\pi\)
\(678\) 39.6482 1.52268
\(679\) 37.8966 1.45434
\(680\) −151.722 −5.81826
\(681\) 20.6053 0.789595
\(682\) −23.4338 −0.897328
\(683\) 18.0998 0.692570 0.346285 0.938129i \(-0.387443\pi\)
0.346285 + 0.938129i \(0.387443\pi\)
\(684\) −23.8036 −0.910153
\(685\) −68.2561 −2.60793
\(686\) −46.8384 −1.78830
\(687\) 18.4194 0.702746
\(688\) 90.5032 3.45040
\(689\) 44.4899 1.69493
\(690\) 67.9818 2.58802
\(691\) 42.0561 1.59989 0.799945 0.600073i \(-0.204862\pi\)
0.799945 + 0.600073i \(0.204862\pi\)
\(692\) −4.30926 −0.163813
\(693\) 7.57010 0.287564
\(694\) −41.3981 −1.57145
\(695\) −47.0935 −1.78636
\(696\) 21.2791 0.806582
\(697\) 41.9537 1.58911
\(698\) −42.7867 −1.61950
\(699\) 4.44717 0.168208
\(700\) −120.568 −4.55705
\(701\) 24.4343 0.922869 0.461435 0.887174i \(-0.347335\pi\)
0.461435 + 0.887174i \(0.347335\pi\)
\(702\) −11.3820 −0.429585
\(703\) −28.1297 −1.06093
\(704\) −7.46163 −0.281221
\(705\) 7.83603 0.295122
\(706\) 44.8683 1.68864
\(707\) −14.0039 −0.526673
\(708\) −52.9294 −1.98921
\(709\) −28.4948 −1.07014 −0.535072 0.844806i \(-0.679716\pi\)
−0.535072 + 0.844806i \(0.679716\pi\)
\(710\) −142.210 −5.33705
\(711\) 11.1854 0.419485
\(712\) 15.3408 0.574921
\(713\) 22.2879 0.834687
\(714\) 40.6436 1.52105
\(715\) −47.8127 −1.78809
\(716\) 15.4114 0.575953
\(717\) 9.55835 0.356963
\(718\) 13.3493 0.498191
\(719\) 41.6482 1.55322 0.776608 0.629984i \(-0.216939\pi\)
0.776608 + 0.629984i \(0.216939\pi\)
\(720\) 30.9666 1.15406
\(721\) −41.2258 −1.53533
\(722\) −19.5428 −0.727309
\(723\) −9.35677 −0.347982
\(724\) 95.1232 3.53523
\(725\) −30.7406 −1.14168
\(726\) 7.89242 0.292915
\(727\) 4.25729 0.157894 0.0789470 0.996879i \(-0.474844\pi\)
0.0789470 + 0.996879i \(0.474844\pi\)
\(728\) 80.0192 2.96571
\(729\) 1.00000 0.0370370
\(730\) 146.277 5.41397
\(731\) 65.9151 2.43796
\(732\) 22.9666 0.848870
\(733\) −37.7594 −1.39468 −0.697338 0.716743i \(-0.745632\pi\)
−0.697338 + 0.716743i \(0.745632\pi\)
\(734\) −54.7752 −2.02179
\(735\) 0.863958 0.0318676
\(736\) 50.3379 1.85548
\(737\) 21.9294 0.807781
\(738\) −18.3555 −0.675674
\(739\) 4.98373 0.183330 0.0916648 0.995790i \(-0.470781\pi\)
0.0916648 + 0.995790i \(0.470781\pi\)
\(740\) 96.5797 3.55034
\(741\) −22.8225 −0.838404
\(742\) 69.5073 2.55169
\(743\) −15.6328 −0.573511 −0.286755 0.958004i \(-0.592577\pi\)
−0.286755 + 0.958004i \(0.592577\pi\)
\(744\) 21.7630 0.797869
\(745\) −69.9544 −2.56293
\(746\) 46.3151 1.69572
\(747\) −0.358839 −0.0131292
\(748\) −76.4153 −2.79402
\(749\) −24.4577 −0.893663
\(750\) −46.5583 −1.70007
\(751\) −10.1825 −0.371564 −0.185782 0.982591i \(-0.559482\pi\)
−0.185782 + 0.982591i \(0.559482\pi\)
\(752\) 16.4866 0.601206
\(753\) 2.01942 0.0735918
\(754\) 36.0030 1.31115
\(755\) 35.5945 1.29542
\(756\) −12.4063 −0.451212
\(757\) 25.1100 0.912637 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(758\) 28.1490 1.02242
\(759\) 19.4026 0.704270
\(760\) 133.103 4.82814
\(761\) −20.1061 −0.728845 −0.364423 0.931234i \(-0.618734\pi\)
−0.364423 + 0.931234i \(0.618734\pi\)
\(762\) −50.1313 −1.81607
\(763\) −48.4793 −1.75507
\(764\) −22.4639 −0.812715
\(765\) 22.5536 0.815426
\(766\) 12.1815 0.440137
\(767\) −50.7477 −1.83239
\(768\) −25.3572 −0.914999
\(769\) 34.6635 1.25000 0.624999 0.780625i \(-0.285099\pi\)
0.624999 + 0.780625i \(0.285099\pi\)
\(770\) −74.6985 −2.69195
\(771\) −22.1534 −0.797836
\(772\) 83.5491 3.00700
\(773\) 1.83788 0.0661041 0.0330520 0.999454i \(-0.489477\pi\)
0.0330520 + 0.999454i \(0.489477\pi\)
\(774\) −28.8390 −1.03659
\(775\) −31.4396 −1.12934
\(776\) 94.8436 3.40469
\(777\) −14.6610 −0.525961
\(778\) 6.07718 0.217877
\(779\) −36.8053 −1.31869
\(780\) 78.3579 2.80566
\(781\) −40.5880 −1.45235
\(782\) 104.172 3.72517
\(783\) −3.16316 −0.113042
\(784\) 1.81773 0.0649188
\(785\) 66.5225 2.37429
\(786\) 1.72789 0.0616317
\(787\) 12.9160 0.460406 0.230203 0.973143i \(-0.426061\pi\)
0.230203 + 0.973143i \(0.426061\pi\)
\(788\) 37.5899 1.33908
\(789\) 12.0481 0.428923
\(790\) −110.373 −3.92688
\(791\) 41.4350 1.47326
\(792\) 18.9456 0.673204
\(793\) 22.0200 0.781952
\(794\) −16.0450 −0.569414
\(795\) 38.5704 1.36795
\(796\) 107.364 3.80541
\(797\) −16.1326 −0.571446 −0.285723 0.958312i \(-0.592234\pi\)
−0.285723 + 0.958312i \(0.592234\pi\)
\(798\) −35.6559 −1.26220
\(799\) 12.0075 0.424796
\(800\) −71.0074 −2.51049
\(801\) −2.28043 −0.0805749
\(802\) 47.2604 1.66882
\(803\) 41.7488 1.47328
\(804\) −35.9391 −1.26747
\(805\) 71.0456 2.50403
\(806\) 36.8217 1.29699
\(807\) 1.91786 0.0675119
\(808\) −35.0476 −1.23297
\(809\) 16.8840 0.593610 0.296805 0.954938i \(-0.404079\pi\)
0.296805 + 0.954938i \(0.404079\pi\)
\(810\) −9.86756 −0.346711
\(811\) −1.58705 −0.0557289 −0.0278645 0.999612i \(-0.508871\pi\)
−0.0278645 + 0.999612i \(0.508871\pi\)
\(812\) 39.2431 1.37716
\(813\) −19.8197 −0.695107
\(814\) 39.5091 1.38479
\(815\) −2.25912 −0.0791334
\(816\) 47.4517 1.66114
\(817\) −57.8262 −2.02308
\(818\) −84.9917 −2.97167
\(819\) −11.8949 −0.415642
\(820\) 126.366 4.41290
\(821\) −17.5015 −0.610807 −0.305403 0.952223i \(-0.598791\pi\)
−0.305403 + 0.952223i \(0.598791\pi\)
\(822\) 45.7608 1.59609
\(823\) 8.53337 0.297455 0.148727 0.988878i \(-0.452482\pi\)
0.148727 + 0.988878i \(0.452482\pi\)
\(824\) −103.175 −3.59428
\(825\) −27.3696 −0.952887
\(826\) −79.2840 −2.75864
\(827\) −36.2648 −1.26105 −0.630526 0.776168i \(-0.717161\pi\)
−0.630526 + 0.776168i \(0.717161\pi\)
\(828\) −31.7980 −1.10506
\(829\) 17.6559 0.613216 0.306608 0.951836i \(-0.400806\pi\)
0.306608 + 0.951836i \(0.400806\pi\)
\(830\) 3.54087 0.122905
\(831\) 12.5533 0.435468
\(832\) 11.7245 0.406473
\(833\) 1.32388 0.0458699
\(834\) 31.5728 1.09328
\(835\) −48.3037 −1.67162
\(836\) 67.0378 2.31855
\(837\) −3.23509 −0.111821
\(838\) 0.0712125 0.00246000
\(839\) −11.5870 −0.400027 −0.200013 0.979793i \(-0.564099\pi\)
−0.200013 + 0.979793i \(0.564099\pi\)
\(840\) 69.3723 2.39357
\(841\) −18.9944 −0.654980
\(842\) −85.6561 −2.95190
\(843\) 6.34273 0.218455
\(844\) 102.641 3.53304
\(845\) 25.2544 0.868777
\(846\) −5.25349 −0.180619
\(847\) 8.24811 0.283408
\(848\) 81.1503 2.78671
\(849\) 2.05838 0.0706433
\(850\) −146.946 −5.04021
\(851\) −37.5770 −1.28812
\(852\) 66.5178 2.27886
\(853\) 0.449358 0.0153857 0.00769286 0.999970i \(-0.497551\pi\)
0.00769286 + 0.999970i \(0.497551\pi\)
\(854\) 34.4021 1.17722
\(855\) −19.7858 −0.676662
\(856\) −61.2100 −2.09211
\(857\) −56.5401 −1.93137 −0.965687 0.259710i \(-0.916373\pi\)
−0.965687 + 0.259710i \(0.916373\pi\)
\(858\) 32.0549 1.09434
\(859\) 29.4197 1.00379 0.501893 0.864929i \(-0.332637\pi\)
0.501893 + 0.864929i \(0.332637\pi\)
\(860\) 198.539 6.77011
\(861\) −19.1827 −0.653744
\(862\) −30.2708 −1.03103
\(863\) 45.7070 1.55588 0.777942 0.628336i \(-0.216264\pi\)
0.777942 + 0.628336i \(0.216264\pi\)
\(864\) −7.30655 −0.248574
\(865\) −3.58191 −0.121789
\(866\) 89.2902 3.03420
\(867\) 17.5599 0.596366
\(868\) 40.1354 1.36228
\(869\) −31.5013 −1.06861
\(870\) 31.2127 1.05821
\(871\) −34.4578 −1.16756
\(872\) −121.329 −4.10871
\(873\) −14.0986 −0.477165
\(874\) −91.3880 −3.09125
\(875\) −48.6566 −1.64489
\(876\) −68.4202 −2.31170
\(877\) 21.2195 0.716531 0.358266 0.933620i \(-0.383368\pi\)
0.358266 + 0.933620i \(0.383368\pi\)
\(878\) −55.9131 −1.88698
\(879\) −25.9128 −0.874018
\(880\) −87.2110 −2.93988
\(881\) 35.1817 1.18530 0.592651 0.805460i \(-0.298082\pi\)
0.592651 + 0.805460i \(0.298082\pi\)
\(882\) −0.579221 −0.0195034
\(883\) −32.8862 −1.10671 −0.553355 0.832946i \(-0.686653\pi\)
−0.553355 + 0.832946i \(0.686653\pi\)
\(884\) 120.072 4.03844
\(885\) −43.9956 −1.47890
\(886\) −3.18516 −0.107007
\(887\) 13.8102 0.463702 0.231851 0.972751i \(-0.425522\pi\)
0.231851 + 0.972751i \(0.425522\pi\)
\(888\) −36.6920 −1.23130
\(889\) −52.3906 −1.75712
\(890\) 22.5022 0.754277
\(891\) −2.81629 −0.0943492
\(892\) 15.7869 0.528584
\(893\) −10.5340 −0.352506
\(894\) 46.8993 1.56855
\(895\) 12.8102 0.428197
\(896\) −20.9623 −0.700300
\(897\) −30.4874 −1.01794
\(898\) 71.8214 2.39671
\(899\) 10.2331 0.341293
\(900\) 44.8547 1.49516
\(901\) 59.1032 1.96901
\(902\) 51.6942 1.72123
\(903\) −30.1386 −1.00295
\(904\) 103.699 3.44898
\(905\) 79.0676 2.62830
\(906\) −23.8636 −0.792813
\(907\) −18.6535 −0.619379 −0.309690 0.950838i \(-0.600225\pi\)
−0.309690 + 0.950838i \(0.600225\pi\)
\(908\) 95.1032 3.15611
\(909\) 5.20985 0.172800
\(910\) 117.374 3.89091
\(911\) 14.6054 0.483899 0.241949 0.970289i \(-0.422213\pi\)
0.241949 + 0.970289i \(0.422213\pi\)
\(912\) −41.6285 −1.37846
\(913\) 1.01059 0.0334458
\(914\) −33.6115 −1.11177
\(915\) 19.0901 0.631100
\(916\) 85.0146 2.80896
\(917\) 1.80576 0.0596314
\(918\) −15.1205 −0.499052
\(919\) −31.4267 −1.03667 −0.518335 0.855177i \(-0.673448\pi\)
−0.518335 + 0.855177i \(0.673448\pi\)
\(920\) 177.805 5.86206
\(921\) 19.5895 0.645497
\(922\) −25.6019 −0.843155
\(923\) 63.7761 2.09922
\(924\) 34.9397 1.14943
\(925\) 53.0067 1.74285
\(926\) 28.3356 0.931164
\(927\) 15.3371 0.503737
\(928\) 23.1118 0.758681
\(929\) 43.4149 1.42440 0.712198 0.701979i \(-0.247700\pi\)
0.712198 + 0.701979i \(0.247700\pi\)
\(930\) 31.9224 1.04678
\(931\) −1.16142 −0.0380640
\(932\) 20.5258 0.672346
\(933\) −29.2485 −0.957553
\(934\) −16.6076 −0.543418
\(935\) −63.5174 −2.07724
\(936\) −29.7693 −0.973041
\(937\) 35.8374 1.17076 0.585378 0.810760i \(-0.300946\pi\)
0.585378 + 0.810760i \(0.300946\pi\)
\(938\) −53.8339 −1.75774
\(939\) −5.66840 −0.184981
\(940\) 36.1671 1.17964
\(941\) 46.1799 1.50542 0.752710 0.658352i \(-0.228746\pi\)
0.752710 + 0.658352i \(0.228746\pi\)
\(942\) −44.5985 −1.45310
\(943\) −49.1663 −1.60107
\(944\) −92.5646 −3.01272
\(945\) −10.3123 −0.335458
\(946\) 81.2188 2.64065
\(947\) 17.1774 0.558190 0.279095 0.960263i \(-0.409966\pi\)
0.279095 + 0.960263i \(0.409966\pi\)
\(948\) 51.6260 1.67673
\(949\) −65.6000 −2.12947
\(950\) 128.913 4.18250
\(951\) 13.2127 0.428451
\(952\) 106.302 3.44528
\(953\) −55.1301 −1.78584 −0.892920 0.450215i \(-0.851347\pi\)
−0.892920 + 0.450215i \(0.851347\pi\)
\(954\) −25.8586 −0.837204
\(955\) −18.6723 −0.604221
\(956\) 44.1164 1.42683
\(957\) 8.90836 0.287967
\(958\) −43.7574 −1.41374
\(959\) 47.8231 1.54429
\(960\) 10.1645 0.328058
\(961\) −20.5342 −0.662394
\(962\) −62.0808 −2.00156
\(963\) 9.09892 0.293209
\(964\) −43.1860 −1.39093
\(965\) 69.4471 2.23558
\(966\) −47.6309 −1.53250
\(967\) 40.8535 1.31376 0.656881 0.753995i \(-0.271876\pi\)
0.656881 + 0.753995i \(0.271876\pi\)
\(968\) 20.6425 0.663475
\(969\) −30.3188 −0.973980
\(970\) 139.119 4.46683
\(971\) −11.2321 −0.360456 −0.180228 0.983625i \(-0.557684\pi\)
−0.180228 + 0.983625i \(0.557684\pi\)
\(972\) 4.61548 0.148042
\(973\) 32.9957 1.05779
\(974\) 68.0672 2.18101
\(975\) 43.0059 1.37729
\(976\) 40.1647 1.28564
\(977\) −39.5761 −1.26615 −0.633076 0.774090i \(-0.718208\pi\)
−0.633076 + 0.774090i \(0.718208\pi\)
\(978\) 1.51457 0.0484307
\(979\) 6.42234 0.205259
\(980\) 3.98758 0.127379
\(981\) 18.0356 0.575834
\(982\) −18.3456 −0.585433
\(983\) −51.6348 −1.64690 −0.823448 0.567392i \(-0.807953\pi\)
−0.823448 + 0.567392i \(0.807953\pi\)
\(984\) −48.0083 −1.53045
\(985\) 31.2452 0.995554
\(986\) 47.8286 1.52317
\(987\) −5.49025 −0.174757
\(988\) −105.337 −3.35120
\(989\) −77.2470 −2.45631
\(990\) 27.7899 0.883221
\(991\) 26.3986 0.838580 0.419290 0.907852i \(-0.362279\pi\)
0.419290 + 0.907852i \(0.362279\pi\)
\(992\) 23.6373 0.750485
\(993\) 8.88215 0.281867
\(994\) 99.6383 3.16034
\(995\) 89.2422 2.82917
\(996\) −1.65622 −0.0524792
\(997\) −23.9106 −0.757258 −0.378629 0.925549i \(-0.623604\pi\)
−0.378629 + 0.925549i \(0.623604\pi\)
\(998\) 87.1310 2.75808
\(999\) 5.45431 0.172567
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 8031.2.a.d.1.9 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8031.2.a.d.1.9 132 1.1 even 1 trivial