Properties

Label 8031.2.a.a.1.3
Level $8031$
Weight $2$
Character 8031.1
Self dual yes
Analytic conductor $64.128$
Analytic rank $1$
Dimension $92$
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: \(1\)
Dimension: \(92\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
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)
 
\(f(q)\) \(=\) \(q-2.66818 q^{2} +1.00000 q^{3} +5.11918 q^{4} +2.50193 q^{5} -2.66818 q^{6} +2.17887 q^{7} -8.32253 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.66818 q^{2} +1.00000 q^{3} +5.11918 q^{4} +2.50193 q^{5} -2.66818 q^{6} +2.17887 q^{7} -8.32253 q^{8} +1.00000 q^{9} -6.67559 q^{10} +3.36263 q^{11} +5.11918 q^{12} -0.728293 q^{13} -5.81361 q^{14} +2.50193 q^{15} +11.9676 q^{16} -1.01722 q^{17} -2.66818 q^{18} -8.14146 q^{19} +12.8078 q^{20} +2.17887 q^{21} -8.97211 q^{22} -8.21797 q^{23} -8.32253 q^{24} +1.25965 q^{25} +1.94322 q^{26} +1.00000 q^{27} +11.1540 q^{28} +1.36066 q^{29} -6.67559 q^{30} -0.119992 q^{31} -15.2868 q^{32} +3.36263 q^{33} +2.71414 q^{34} +5.45137 q^{35} +5.11918 q^{36} -6.71996 q^{37} +21.7229 q^{38} -0.728293 q^{39} -20.8224 q^{40} -1.65460 q^{41} -5.81361 q^{42} -8.77755 q^{43} +17.2139 q^{44} +2.50193 q^{45} +21.9270 q^{46} -8.23931 q^{47} +11.9676 q^{48} -2.25253 q^{49} -3.36096 q^{50} -1.01722 q^{51} -3.72826 q^{52} +9.36849 q^{53} -2.66818 q^{54} +8.41307 q^{55} -18.1337 q^{56} -8.14146 q^{57} -3.63050 q^{58} -6.58197 q^{59} +12.8078 q^{60} -11.5200 q^{61} +0.320160 q^{62} +2.17887 q^{63} +16.8525 q^{64} -1.82214 q^{65} -8.97211 q^{66} +0.965475 q^{67} -5.20735 q^{68} -8.21797 q^{69} -14.5452 q^{70} +0.0597271 q^{71} -8.32253 q^{72} +4.87655 q^{73} +17.9301 q^{74} +1.25965 q^{75} -41.6776 q^{76} +7.32674 q^{77} +1.94322 q^{78} -4.03279 q^{79} +29.9422 q^{80} +1.00000 q^{81} +4.41476 q^{82} +15.8058 q^{83} +11.1540 q^{84} -2.54502 q^{85} +23.4201 q^{86} +1.36066 q^{87} -27.9856 q^{88} -3.83206 q^{89} -6.67559 q^{90} -1.58685 q^{91} -42.0693 q^{92} -0.119992 q^{93} +21.9840 q^{94} -20.3694 q^{95} -15.2868 q^{96} -5.12282 q^{97} +6.01016 q^{98} +3.36263 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 92 q - 6 q^{2} + 92 q^{3} + 70 q^{4} - 18 q^{5} - 6 q^{6} - 42 q^{7} - 15 q^{8} + 92 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 92 q - 6 q^{2} + 92 q^{3} + 70 q^{4} - 18 q^{5} - 6 q^{6} - 42 q^{7} - 15 q^{8} + 92 q^{9} - 44 q^{10} - 24 q^{11} + 70 q^{12} - 48 q^{13} - 29 q^{14} - 18 q^{15} + 26 q^{16} - 69 q^{17} - 6 q^{18} - 74 q^{19} - 42 q^{20} - 42 q^{21} - 62 q^{22} - 19 q^{23} - 15 q^{24} + 16 q^{25} - 27 q^{26} + 92 q^{27} - 101 q^{28} - 54 q^{29} - 44 q^{30} - 67 q^{31} - 36 q^{32} - 24 q^{33} - 63 q^{34} - 31 q^{35} + 70 q^{36} - 70 q^{37} - 18 q^{38} - 48 q^{39} - 125 q^{40} - 98 q^{41} - 29 q^{42} - 159 q^{43} - 52 q^{44} - 18 q^{45} - 68 q^{46} - 15 q^{47} + 26 q^{48} - 28 q^{49} - 7 q^{50} - 69 q^{51} - 98 q^{52} - 23 q^{53} - 6 q^{54} - 93 q^{55} - 48 q^{56} - 74 q^{57} - 37 q^{58} - 36 q^{59} - 42 q^{60} - 172 q^{61} - 26 q^{62} - 42 q^{63} - 23 q^{64} - 66 q^{65} - 62 q^{66} - 143 q^{67} - 74 q^{68} - 19 q^{69} - 30 q^{70} - 9 q^{71} - 15 q^{72} - 134 q^{73} - 19 q^{74} + 16 q^{75} - 157 q^{76} - 25 q^{77} - 27 q^{78} - 138 q^{79} - 29 q^{80} + 92 q^{81} - 61 q^{82} - 24 q^{83} - 101 q^{84} - 84 q^{85} + 14 q^{86} - 54 q^{87} - 140 q^{88} - 148 q^{89} - 44 q^{90} - 115 q^{91} - 12 q^{92} - 67 q^{93} - 79 q^{94} - 10 q^{95} - 36 q^{96} - 165 q^{97} + 36 q^{98} - 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.66818 −1.88669 −0.943344 0.331817i \(-0.892338\pi\)
−0.943344 + 0.331817i \(0.892338\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.11918 2.55959
\(5\) 2.50193 1.11890 0.559448 0.828865i \(-0.311013\pi\)
0.559448 + 0.828865i \(0.311013\pi\)
\(6\) −2.66818 −1.08928
\(7\) 2.17887 0.823535 0.411767 0.911289i \(-0.364912\pi\)
0.411767 + 0.911289i \(0.364912\pi\)
\(8\) −8.32253 −2.94246
\(9\) 1.00000 0.333333
\(10\) −6.67559 −2.11101
\(11\) 3.36263 1.01387 0.506936 0.861984i \(-0.330778\pi\)
0.506936 + 0.861984i \(0.330778\pi\)
\(12\) 5.11918 1.47778
\(13\) −0.728293 −0.201992 −0.100996 0.994887i \(-0.532203\pi\)
−0.100996 + 0.994887i \(0.532203\pi\)
\(14\) −5.81361 −1.55375
\(15\) 2.50193 0.645995
\(16\) 11.9676 2.99191
\(17\) −1.01722 −0.246713 −0.123357 0.992362i \(-0.539366\pi\)
−0.123357 + 0.992362i \(0.539366\pi\)
\(18\) −2.66818 −0.628896
\(19\) −8.14146 −1.86778 −0.933890 0.357561i \(-0.883608\pi\)
−0.933890 + 0.357561i \(0.883608\pi\)
\(20\) 12.8078 2.86392
\(21\) 2.17887 0.475468
\(22\) −8.97211 −1.91286
\(23\) −8.21797 −1.71357 −0.856783 0.515677i \(-0.827540\pi\)
−0.856783 + 0.515677i \(0.827540\pi\)
\(24\) −8.32253 −1.69883
\(25\) 1.25965 0.251929
\(26\) 1.94322 0.381096
\(27\) 1.00000 0.192450
\(28\) 11.1540 2.10791
\(29\) 1.36066 0.252669 0.126335 0.991988i \(-0.459679\pi\)
0.126335 + 0.991988i \(0.459679\pi\)
\(30\) −6.67559 −1.21879
\(31\) −0.119992 −0.0215512 −0.0107756 0.999942i \(-0.503430\pi\)
−0.0107756 + 0.999942i \(0.503430\pi\)
\(32\) −15.2868 −2.70234
\(33\) 3.36263 0.585359
\(34\) 2.71414 0.465470
\(35\) 5.45137 0.921450
\(36\) 5.11918 0.853197
\(37\) −6.71996 −1.10476 −0.552378 0.833594i \(-0.686279\pi\)
−0.552378 + 0.833594i \(0.686279\pi\)
\(38\) 21.7229 3.52392
\(39\) −0.728293 −0.116620
\(40\) −20.8224 −3.29231
\(41\) −1.65460 −0.258405 −0.129202 0.991618i \(-0.541242\pi\)
−0.129202 + 0.991618i \(0.541242\pi\)
\(42\) −5.81361 −0.897060
\(43\) −8.77755 −1.33856 −0.669282 0.743009i \(-0.733398\pi\)
−0.669282 + 0.743009i \(0.733398\pi\)
\(44\) 17.2139 2.59510
\(45\) 2.50193 0.372965
\(46\) 21.9270 3.23296
\(47\) −8.23931 −1.20183 −0.600913 0.799314i \(-0.705196\pi\)
−0.600913 + 0.799314i \(0.705196\pi\)
\(48\) 11.9676 1.72738
\(49\) −2.25253 −0.321790
\(50\) −3.36096 −0.475312
\(51\) −1.01722 −0.142440
\(52\) −3.72826 −0.517017
\(53\) 9.36849 1.28686 0.643430 0.765505i \(-0.277511\pi\)
0.643430 + 0.765505i \(0.277511\pi\)
\(54\) −2.66818 −0.363093
\(55\) 8.41307 1.13442
\(56\) −18.1337 −2.42322
\(57\) −8.14146 −1.07836
\(58\) −3.63050 −0.476708
\(59\) −6.58197 −0.856900 −0.428450 0.903566i \(-0.640940\pi\)
−0.428450 + 0.903566i \(0.640940\pi\)
\(60\) 12.8078 1.65348
\(61\) −11.5200 −1.47498 −0.737490 0.675358i \(-0.763989\pi\)
−0.737490 + 0.675358i \(0.763989\pi\)
\(62\) 0.320160 0.0406604
\(63\) 2.17887 0.274512
\(64\) 16.8525 2.10657
\(65\) −1.82214 −0.226008
\(66\) −8.97211 −1.10439
\(67\) 0.965475 0.117951 0.0589757 0.998259i \(-0.481217\pi\)
0.0589757 + 0.998259i \(0.481217\pi\)
\(68\) −5.20735 −0.631484
\(69\) −8.21797 −0.989328
\(70\) −14.5452 −1.73849
\(71\) 0.0597271 0.00708830 0.00354415 0.999994i \(-0.498872\pi\)
0.00354415 + 0.999994i \(0.498872\pi\)
\(72\) −8.32253 −0.980820
\(73\) 4.87655 0.570756 0.285378 0.958415i \(-0.407881\pi\)
0.285378 + 0.958415i \(0.407881\pi\)
\(74\) 17.9301 2.08433
\(75\) 1.25965 0.145451
\(76\) −41.6776 −4.78075
\(77\) 7.32674 0.834959
\(78\) 1.94322 0.220026
\(79\) −4.03279 −0.453724 −0.226862 0.973927i \(-0.572847\pi\)
−0.226862 + 0.973927i \(0.572847\pi\)
\(80\) 29.9422 3.34764
\(81\) 1.00000 0.111111
\(82\) 4.41476 0.487529
\(83\) 15.8058 1.73491 0.867457 0.497512i \(-0.165753\pi\)
0.867457 + 0.497512i \(0.165753\pi\)
\(84\) 11.1540 1.21700
\(85\) −2.54502 −0.276046
\(86\) 23.4201 2.52545
\(87\) 1.36066 0.145879
\(88\) −27.9856 −2.98328
\(89\) −3.83206 −0.406197 −0.203099 0.979158i \(-0.565101\pi\)
−0.203099 + 0.979158i \(0.565101\pi\)
\(90\) −6.67559 −0.703669
\(91\) −1.58685 −0.166348
\(92\) −42.0693 −4.38603
\(93\) −0.119992 −0.0124426
\(94\) 21.9840 2.26747
\(95\) −20.3694 −2.08985
\(96\) −15.2868 −1.56020
\(97\) −5.12282 −0.520144 −0.260072 0.965589i \(-0.583746\pi\)
−0.260072 + 0.965589i \(0.583746\pi\)
\(98\) 6.01016 0.607118
\(99\) 3.36263 0.337957
\(100\) 6.44835 0.644835
\(101\) −1.55509 −0.154737 −0.0773684 0.997003i \(-0.524652\pi\)
−0.0773684 + 0.997003i \(0.524652\pi\)
\(102\) 2.71414 0.268740
\(103\) 7.41047 0.730175 0.365087 0.930973i \(-0.381039\pi\)
0.365087 + 0.930973i \(0.381039\pi\)
\(104\) 6.06124 0.594354
\(105\) 5.45137 0.532000
\(106\) −24.9968 −2.42790
\(107\) −8.52845 −0.824476 −0.412238 0.911076i \(-0.635253\pi\)
−0.412238 + 0.911076i \(0.635253\pi\)
\(108\) 5.11918 0.492593
\(109\) −18.9520 −1.81528 −0.907638 0.419755i \(-0.862116\pi\)
−0.907638 + 0.419755i \(0.862116\pi\)
\(110\) −22.4476 −2.14029
\(111\) −6.71996 −0.637831
\(112\) 26.0759 2.46394
\(113\) 0.530104 0.0498680 0.0249340 0.999689i \(-0.492062\pi\)
0.0249340 + 0.999689i \(0.492062\pi\)
\(114\) 21.7229 2.03453
\(115\) −20.5608 −1.91730
\(116\) 6.96549 0.646729
\(117\) −0.728293 −0.0673307
\(118\) 17.5619 1.61670
\(119\) −2.21640 −0.203177
\(120\) −20.8224 −1.90081
\(121\) 0.307304 0.0279367
\(122\) 30.7373 2.78283
\(123\) −1.65460 −0.149190
\(124\) −0.614260 −0.0551622
\(125\) −9.35810 −0.837014
\(126\) −5.81361 −0.517918
\(127\) 12.4958 1.10883 0.554413 0.832242i \(-0.312943\pi\)
0.554413 + 0.832242i \(0.312943\pi\)
\(128\) −14.3921 −1.27209
\(129\) −8.77755 −0.772820
\(130\) 4.86179 0.426407
\(131\) −9.15214 −0.799626 −0.399813 0.916597i \(-0.630925\pi\)
−0.399813 + 0.916597i \(0.630925\pi\)
\(132\) 17.2139 1.49828
\(133\) −17.7392 −1.53818
\(134\) −2.57606 −0.222538
\(135\) 2.50193 0.215332
\(136\) 8.46588 0.725943
\(137\) 5.72120 0.488795 0.244398 0.969675i \(-0.421410\pi\)
0.244398 + 0.969675i \(0.421410\pi\)
\(138\) 21.9270 1.86655
\(139\) 9.43803 0.800523 0.400261 0.916401i \(-0.368919\pi\)
0.400261 + 0.916401i \(0.368919\pi\)
\(140\) 27.9066 2.35854
\(141\) −8.23931 −0.693875
\(142\) −0.159363 −0.0133734
\(143\) −2.44898 −0.204794
\(144\) 11.9676 0.997304
\(145\) 3.40428 0.282710
\(146\) −13.0115 −1.07684
\(147\) −2.25253 −0.185786
\(148\) −34.4007 −2.82772
\(149\) 10.2856 0.842627 0.421313 0.906915i \(-0.361569\pi\)
0.421313 + 0.906915i \(0.361569\pi\)
\(150\) −3.36096 −0.274421
\(151\) −21.2747 −1.73131 −0.865657 0.500638i \(-0.833099\pi\)
−0.865657 + 0.500638i \(0.833099\pi\)
\(152\) 67.7576 5.49587
\(153\) −1.01722 −0.0822377
\(154\) −19.5490 −1.57531
\(155\) −0.300211 −0.0241135
\(156\) −3.72826 −0.298500
\(157\) −4.19220 −0.334574 −0.167287 0.985908i \(-0.553501\pi\)
−0.167287 + 0.985908i \(0.553501\pi\)
\(158\) 10.7602 0.856035
\(159\) 9.36849 0.742969
\(160\) −38.2464 −3.02364
\(161\) −17.9059 −1.41118
\(162\) −2.66818 −0.209632
\(163\) 13.3785 1.04788 0.523941 0.851755i \(-0.324461\pi\)
0.523941 + 0.851755i \(0.324461\pi\)
\(164\) −8.47018 −0.661410
\(165\) 8.41307 0.654956
\(166\) −42.1728 −3.27324
\(167\) 24.3499 1.88425 0.942125 0.335262i \(-0.108825\pi\)
0.942125 + 0.335262i \(0.108825\pi\)
\(168\) −18.1337 −1.39905
\(169\) −12.4696 −0.959199
\(170\) 6.79057 0.520813
\(171\) −8.14146 −0.622593
\(172\) −44.9339 −3.42617
\(173\) 10.6996 0.813474 0.406737 0.913545i \(-0.366667\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(174\) −3.63050 −0.275227
\(175\) 2.74460 0.207472
\(176\) 40.2428 3.03342
\(177\) −6.58197 −0.494731
\(178\) 10.2246 0.766368
\(179\) 15.2684 1.14121 0.570606 0.821224i \(-0.306708\pi\)
0.570606 + 0.821224i \(0.306708\pi\)
\(180\) 12.8078 0.954639
\(181\) −5.91757 −0.439850 −0.219925 0.975517i \(-0.570581\pi\)
−0.219925 + 0.975517i \(0.570581\pi\)
\(182\) 4.23401 0.313846
\(183\) −11.5200 −0.851581
\(184\) 68.3944 5.04210
\(185\) −16.8129 −1.23611
\(186\) 0.320160 0.0234753
\(187\) −3.42055 −0.250135
\(188\) −42.1785 −3.07618
\(189\) 2.17887 0.158489
\(190\) 54.3491 3.94290
\(191\) −14.3070 −1.03522 −0.517608 0.855618i \(-0.673177\pi\)
−0.517608 + 0.855618i \(0.673177\pi\)
\(192\) 16.8525 1.21623
\(193\) −26.2676 −1.89078 −0.945390 0.325940i \(-0.894319\pi\)
−0.945390 + 0.325940i \(0.894319\pi\)
\(194\) 13.6686 0.981349
\(195\) −1.82214 −0.130486
\(196\) −11.5311 −0.823651
\(197\) −23.8132 −1.69662 −0.848311 0.529498i \(-0.822380\pi\)
−0.848311 + 0.529498i \(0.822380\pi\)
\(198\) −8.97211 −0.637620
\(199\) −0.780516 −0.0553293 −0.0276647 0.999617i \(-0.508807\pi\)
−0.0276647 + 0.999617i \(0.508807\pi\)
\(200\) −10.4834 −0.741291
\(201\) 0.965475 0.0680993
\(202\) 4.14925 0.291940
\(203\) 2.96471 0.208082
\(204\) −5.20735 −0.364588
\(205\) −4.13968 −0.289128
\(206\) −19.7724 −1.37761
\(207\) −8.21797 −0.571189
\(208\) −8.71595 −0.604343
\(209\) −27.3768 −1.89369
\(210\) −14.5452 −1.00372
\(211\) −17.4775 −1.20320 −0.601599 0.798798i \(-0.705470\pi\)
−0.601599 + 0.798798i \(0.705470\pi\)
\(212\) 47.9590 3.29384
\(213\) 0.0597271 0.00409243
\(214\) 22.7554 1.55553
\(215\) −21.9608 −1.49771
\(216\) −8.32253 −0.566277
\(217\) −0.261447 −0.0177482
\(218\) 50.5674 3.42486
\(219\) 4.87655 0.329526
\(220\) 43.0680 2.90365
\(221\) 0.740837 0.0498341
\(222\) 17.9301 1.20339
\(223\) 23.8007 1.59381 0.796906 0.604103i \(-0.206469\pi\)
0.796906 + 0.604103i \(0.206469\pi\)
\(224\) −33.3079 −2.22547
\(225\) 1.25965 0.0839764
\(226\) −1.41441 −0.0940854
\(227\) −2.96344 −0.196690 −0.0983452 0.995152i \(-0.531355\pi\)
−0.0983452 + 0.995152i \(0.531355\pi\)
\(228\) −41.6776 −2.76017
\(229\) 18.7987 1.24225 0.621127 0.783710i \(-0.286675\pi\)
0.621127 + 0.783710i \(0.286675\pi\)
\(230\) 54.8599 3.61735
\(231\) 7.32674 0.482064
\(232\) −11.3242 −0.743468
\(233\) 11.2917 0.739745 0.369872 0.929083i \(-0.379401\pi\)
0.369872 + 0.929083i \(0.379401\pi\)
\(234\) 1.94322 0.127032
\(235\) −20.6142 −1.34472
\(236\) −33.6943 −2.19331
\(237\) −4.03279 −0.261958
\(238\) 5.91375 0.383331
\(239\) −16.1382 −1.04389 −0.521947 0.852978i \(-0.674794\pi\)
−0.521947 + 0.852978i \(0.674794\pi\)
\(240\) 29.9422 1.93276
\(241\) −15.1276 −0.974455 −0.487227 0.873275i \(-0.661992\pi\)
−0.487227 + 0.873275i \(0.661992\pi\)
\(242\) −0.819941 −0.0527078
\(243\) 1.00000 0.0641500
\(244\) −58.9728 −3.77535
\(245\) −5.63567 −0.360050
\(246\) 4.41476 0.281475
\(247\) 5.92937 0.377277
\(248\) 0.998637 0.0634135
\(249\) 15.8058 1.00165
\(250\) 24.9691 1.57918
\(251\) −15.4459 −0.974934 −0.487467 0.873141i \(-0.662079\pi\)
−0.487467 + 0.873141i \(0.662079\pi\)
\(252\) 11.1540 0.702637
\(253\) −27.6340 −1.73734
\(254\) −33.3411 −2.09201
\(255\) −2.54502 −0.159375
\(256\) 4.69552 0.293470
\(257\) −0.0223687 −0.00139532 −0.000697660 1.00000i \(-0.500222\pi\)
−0.000697660 1.00000i \(0.500222\pi\)
\(258\) 23.4201 1.45807
\(259\) −14.6419 −0.909804
\(260\) −9.32784 −0.578488
\(261\) 1.36066 0.0842230
\(262\) 24.4196 1.50865
\(263\) −1.11182 −0.0685577 −0.0342788 0.999412i \(-0.510913\pi\)
−0.0342788 + 0.999412i \(0.510913\pi\)
\(264\) −27.9856 −1.72240
\(265\) 23.4393 1.43986
\(266\) 47.3313 2.90207
\(267\) −3.83206 −0.234518
\(268\) 4.94244 0.301908
\(269\) 14.1411 0.862201 0.431100 0.902304i \(-0.358125\pi\)
0.431100 + 0.902304i \(0.358125\pi\)
\(270\) −6.67559 −0.406264
\(271\) 17.2975 1.05075 0.525373 0.850872i \(-0.323926\pi\)
0.525373 + 0.850872i \(0.323926\pi\)
\(272\) −12.1738 −0.738144
\(273\) −1.58685 −0.0960408
\(274\) −15.2652 −0.922204
\(275\) 4.23573 0.255424
\(276\) −42.0693 −2.53227
\(277\) −26.8386 −1.61257 −0.806287 0.591525i \(-0.798526\pi\)
−0.806287 + 0.591525i \(0.798526\pi\)
\(278\) −25.1823 −1.51034
\(279\) −0.119992 −0.00718373
\(280\) −45.3692 −2.71133
\(281\) −3.89982 −0.232644 −0.116322 0.993212i \(-0.537110\pi\)
−0.116322 + 0.993212i \(0.537110\pi\)
\(282\) 21.9840 1.30913
\(283\) 2.85301 0.169594 0.0847970 0.996398i \(-0.472976\pi\)
0.0847970 + 0.996398i \(0.472976\pi\)
\(284\) 0.305754 0.0181431
\(285\) −20.3694 −1.20658
\(286\) 6.53432 0.386383
\(287\) −3.60515 −0.212805
\(288\) −15.2868 −0.900781
\(289\) −15.9653 −0.939133
\(290\) −9.08324 −0.533386
\(291\) −5.12282 −0.300305
\(292\) 24.9639 1.46090
\(293\) −7.32108 −0.427702 −0.213851 0.976866i \(-0.568601\pi\)
−0.213851 + 0.976866i \(0.568601\pi\)
\(294\) 6.01016 0.350520
\(295\) −16.4676 −0.958782
\(296\) 55.9271 3.25070
\(297\) 3.36263 0.195120
\(298\) −27.4437 −1.58977
\(299\) 5.98509 0.346127
\(300\) 6.44835 0.372296
\(301\) −19.1251 −1.10235
\(302\) 56.7648 3.26645
\(303\) −1.55509 −0.0893374
\(304\) −97.4342 −5.58823
\(305\) −28.8221 −1.65035
\(306\) 2.71414 0.155157
\(307\) −28.1823 −1.60845 −0.804225 0.594324i \(-0.797420\pi\)
−0.804225 + 0.594324i \(0.797420\pi\)
\(308\) 37.5069 2.13715
\(309\) 7.41047 0.421567
\(310\) 0.801017 0.0454947
\(311\) 2.09340 0.118706 0.0593530 0.998237i \(-0.481096\pi\)
0.0593530 + 0.998237i \(0.481096\pi\)
\(312\) 6.06124 0.343150
\(313\) −11.9799 −0.677144 −0.338572 0.940940i \(-0.609944\pi\)
−0.338572 + 0.940940i \(0.609944\pi\)
\(314\) 11.1855 0.631237
\(315\) 5.45137 0.307150
\(316\) −20.6446 −1.16135
\(317\) 22.1655 1.24494 0.622470 0.782644i \(-0.286129\pi\)
0.622470 + 0.782644i \(0.286129\pi\)
\(318\) −24.9968 −1.40175
\(319\) 4.57542 0.256174
\(320\) 42.1638 2.35703
\(321\) −8.52845 −0.476011
\(322\) 47.7761 2.66246
\(323\) 8.28169 0.460806
\(324\) 5.11918 0.284399
\(325\) −0.917391 −0.0508877
\(326\) −35.6961 −1.97702
\(327\) −18.9520 −1.04805
\(328\) 13.7704 0.760345
\(329\) −17.9524 −0.989746
\(330\) −22.4476 −1.23570
\(331\) 17.7872 0.977675 0.488838 0.872375i \(-0.337421\pi\)
0.488838 + 0.872375i \(0.337421\pi\)
\(332\) 80.9128 4.44067
\(333\) −6.71996 −0.368252
\(334\) −64.9698 −3.55499
\(335\) 2.41555 0.131976
\(336\) 26.0759 1.42256
\(337\) 13.3148 0.725306 0.362653 0.931924i \(-0.381871\pi\)
0.362653 + 0.931924i \(0.381871\pi\)
\(338\) 33.2711 1.80971
\(339\) 0.530104 0.0287913
\(340\) −13.0284 −0.706566
\(341\) −0.403489 −0.0218502
\(342\) 21.7229 1.17464
\(343\) −20.1601 −1.08854
\(344\) 73.0514 3.93867
\(345\) −20.5608 −1.10696
\(346\) −28.5484 −1.53477
\(347\) 21.8646 1.17375 0.586876 0.809677i \(-0.300358\pi\)
0.586876 + 0.809677i \(0.300358\pi\)
\(348\) 6.96549 0.373389
\(349\) 4.10810 0.219902 0.109951 0.993937i \(-0.464931\pi\)
0.109951 + 0.993937i \(0.464931\pi\)
\(350\) −7.32309 −0.391436
\(351\) −0.728293 −0.0388734
\(352\) −51.4038 −2.73983
\(353\) 17.4549 0.929032 0.464516 0.885565i \(-0.346228\pi\)
0.464516 + 0.885565i \(0.346228\pi\)
\(354\) 17.5619 0.933403
\(355\) 0.149433 0.00793107
\(356\) −19.6170 −1.03970
\(357\) −2.21640 −0.117304
\(358\) −40.7387 −2.15311
\(359\) −32.5122 −1.71593 −0.857965 0.513708i \(-0.828271\pi\)
−0.857965 + 0.513708i \(0.828271\pi\)
\(360\) −20.8224 −1.09744
\(361\) 47.2834 2.48860
\(362\) 15.7891 0.829859
\(363\) 0.307304 0.0161293
\(364\) −8.12339 −0.425781
\(365\) 12.2008 0.638617
\(366\) 30.7373 1.60667
\(367\) 23.0488 1.20314 0.601569 0.798821i \(-0.294543\pi\)
0.601569 + 0.798821i \(0.294543\pi\)
\(368\) −98.3498 −5.12684
\(369\) −1.65460 −0.0861349
\(370\) 44.8597 2.33215
\(371\) 20.4127 1.05977
\(372\) −0.614260 −0.0318479
\(373\) 21.3192 1.10387 0.551934 0.833888i \(-0.313890\pi\)
0.551934 + 0.833888i \(0.313890\pi\)
\(374\) 9.12664 0.471928
\(375\) −9.35810 −0.483250
\(376\) 68.5719 3.53633
\(377\) −0.990962 −0.0510371
\(378\) −5.81361 −0.299020
\(379\) 35.3028 1.81338 0.906691 0.421796i \(-0.138600\pi\)
0.906691 + 0.421796i \(0.138600\pi\)
\(380\) −104.274 −5.34916
\(381\) 12.4958 0.640180
\(382\) 38.1735 1.95313
\(383\) 13.0399 0.666308 0.333154 0.942872i \(-0.391887\pi\)
0.333154 + 0.942872i \(0.391887\pi\)
\(384\) −14.3921 −0.734442
\(385\) 18.3310 0.934233
\(386\) 70.0866 3.56731
\(387\) −8.77755 −0.446188
\(388\) −26.2247 −1.33136
\(389\) 12.9689 0.657552 0.328776 0.944408i \(-0.393364\pi\)
0.328776 + 0.944408i \(0.393364\pi\)
\(390\) 4.86179 0.246186
\(391\) 8.35952 0.422759
\(392\) 18.7468 0.946855
\(393\) −9.15214 −0.461665
\(394\) 63.5380 3.20100
\(395\) −10.0897 −0.507670
\(396\) 17.2139 0.865032
\(397\) 27.6360 1.38701 0.693506 0.720451i \(-0.256065\pi\)
0.693506 + 0.720451i \(0.256065\pi\)
\(398\) 2.08256 0.104389
\(399\) −17.7392 −0.888070
\(400\) 15.0750 0.753750
\(401\) 21.8747 1.09237 0.546186 0.837664i \(-0.316079\pi\)
0.546186 + 0.837664i \(0.316079\pi\)
\(402\) −2.57606 −0.128482
\(403\) 0.0873893 0.00435317
\(404\) −7.96077 −0.396063
\(405\) 2.50193 0.124322
\(406\) −7.91037 −0.392585
\(407\) −22.5968 −1.12008
\(408\) 8.46588 0.419124
\(409\) 2.50714 0.123970 0.0619849 0.998077i \(-0.480257\pi\)
0.0619849 + 0.998077i \(0.480257\pi\)
\(410\) 11.0454 0.545494
\(411\) 5.72120 0.282206
\(412\) 37.9355 1.86895
\(413\) −14.3412 −0.705687
\(414\) 21.9270 1.07765
\(415\) 39.5450 1.94119
\(416\) 11.1332 0.545852
\(417\) 9.43803 0.462182
\(418\) 73.0461 3.57280
\(419\) 17.9077 0.874850 0.437425 0.899255i \(-0.355891\pi\)
0.437425 + 0.899255i \(0.355891\pi\)
\(420\) 27.9066 1.36170
\(421\) −36.3205 −1.77015 −0.885076 0.465446i \(-0.845894\pi\)
−0.885076 + 0.465446i \(0.845894\pi\)
\(422\) 46.6330 2.27006
\(423\) −8.23931 −0.400609
\(424\) −77.9696 −3.78654
\(425\) −1.28134 −0.0621542
\(426\) −0.159363 −0.00772114
\(427\) −25.1005 −1.21470
\(428\) −43.6587 −2.11032
\(429\) −2.44898 −0.118238
\(430\) 58.5953 2.82572
\(431\) 20.2500 0.975408 0.487704 0.873009i \(-0.337835\pi\)
0.487704 + 0.873009i \(0.337835\pi\)
\(432\) 11.9676 0.575794
\(433\) 15.8477 0.761592 0.380796 0.924659i \(-0.375650\pi\)
0.380796 + 0.924659i \(0.375650\pi\)
\(434\) 0.697587 0.0334852
\(435\) 3.40428 0.163223
\(436\) −97.0189 −4.64636
\(437\) 66.9063 3.20056
\(438\) −13.0115 −0.621713
\(439\) 36.0011 1.71824 0.859119 0.511777i \(-0.171012\pi\)
0.859119 + 0.511777i \(0.171012\pi\)
\(440\) −70.0180 −3.33798
\(441\) −2.25253 −0.107263
\(442\) −1.97669 −0.0940213
\(443\) −30.2718 −1.43826 −0.719129 0.694876i \(-0.755459\pi\)
−0.719129 + 0.694876i \(0.755459\pi\)
\(444\) −34.4007 −1.63259
\(445\) −9.58754 −0.454493
\(446\) −63.5045 −3.00703
\(447\) 10.2856 0.486491
\(448\) 36.7195 1.73483
\(449\) 5.21495 0.246109 0.123054 0.992400i \(-0.460731\pi\)
0.123054 + 0.992400i \(0.460731\pi\)
\(450\) −3.36096 −0.158437
\(451\) −5.56380 −0.261989
\(452\) 2.71370 0.127642
\(453\) −21.2747 −0.999574
\(454\) 7.90699 0.371093
\(455\) −3.97020 −0.186126
\(456\) 67.7576 3.17304
\(457\) −6.89838 −0.322693 −0.161346 0.986898i \(-0.551584\pi\)
−0.161346 + 0.986898i \(0.551584\pi\)
\(458\) −50.1583 −2.34374
\(459\) −1.01722 −0.0474800
\(460\) −105.254 −4.90751
\(461\) 20.2748 0.944290 0.472145 0.881521i \(-0.343480\pi\)
0.472145 + 0.881521i \(0.343480\pi\)
\(462\) −19.5490 −0.909504
\(463\) −2.87152 −0.133451 −0.0667255 0.997771i \(-0.521255\pi\)
−0.0667255 + 0.997771i \(0.521255\pi\)
\(464\) 16.2840 0.755964
\(465\) −0.300211 −0.0139220
\(466\) −30.1283 −1.39567
\(467\) 31.8861 1.47551 0.737756 0.675068i \(-0.235886\pi\)
0.737756 + 0.675068i \(0.235886\pi\)
\(468\) −3.72826 −0.172339
\(469\) 2.10364 0.0971372
\(470\) 55.0023 2.53707
\(471\) −4.19220 −0.193166
\(472\) 54.7787 2.52139
\(473\) −29.5157 −1.35713
\(474\) 10.7602 0.494232
\(475\) −10.2554 −0.470548
\(476\) −11.3461 −0.520049
\(477\) 9.36849 0.428954
\(478\) 43.0597 1.96950
\(479\) 36.0218 1.64588 0.822940 0.568128i \(-0.192332\pi\)
0.822940 + 0.568128i \(0.192332\pi\)
\(480\) −38.2464 −1.74570
\(481\) 4.89410 0.223152
\(482\) 40.3632 1.83849
\(483\) −17.9059 −0.814746
\(484\) 1.57314 0.0715065
\(485\) −12.8169 −0.581987
\(486\) −2.66818 −0.121031
\(487\) 3.61926 0.164004 0.0820022 0.996632i \(-0.473869\pi\)
0.0820022 + 0.996632i \(0.473869\pi\)
\(488\) 95.8753 4.34007
\(489\) 13.3785 0.604994
\(490\) 15.0370 0.679302
\(491\) 4.61584 0.208310 0.104155 0.994561i \(-0.466786\pi\)
0.104155 + 0.994561i \(0.466786\pi\)
\(492\) −8.47018 −0.381865
\(493\) −1.38410 −0.0623368
\(494\) −15.8206 −0.711803
\(495\) 8.41307 0.378139
\(496\) −1.43602 −0.0644793
\(497\) 0.130137 0.00583746
\(498\) −42.1728 −1.88981
\(499\) 32.1791 1.44054 0.720268 0.693696i \(-0.244019\pi\)
0.720268 + 0.693696i \(0.244019\pi\)
\(500\) −47.9058 −2.14241
\(501\) 24.3499 1.08787
\(502\) 41.2123 1.83940
\(503\) −10.9719 −0.489215 −0.244607 0.969622i \(-0.578659\pi\)
−0.244607 + 0.969622i \(0.578659\pi\)
\(504\) −18.1337 −0.807739
\(505\) −3.89071 −0.173134
\(506\) 73.7326 3.27781
\(507\) −12.4696 −0.553794
\(508\) 63.9684 2.83814
\(509\) −24.6477 −1.09249 −0.546245 0.837626i \(-0.683943\pi\)
−0.546245 + 0.837626i \(0.683943\pi\)
\(510\) 6.79057 0.300692
\(511\) 10.6254 0.470038
\(512\) 16.2556 0.718405
\(513\) −8.14146 −0.359454
\(514\) 0.0596837 0.00263253
\(515\) 18.5405 0.816990
\(516\) −44.9339 −1.97810
\(517\) −27.7058 −1.21850
\(518\) 39.0673 1.71652
\(519\) 10.6996 0.469659
\(520\) 15.1648 0.665020
\(521\) −2.87732 −0.126058 −0.0630288 0.998012i \(-0.520076\pi\)
−0.0630288 + 0.998012i \(0.520076\pi\)
\(522\) −3.63050 −0.158903
\(523\) −2.26922 −0.0992262 −0.0496131 0.998769i \(-0.515799\pi\)
−0.0496131 + 0.998769i \(0.515799\pi\)
\(524\) −46.8515 −2.04672
\(525\) 2.74460 0.119784
\(526\) 2.96653 0.129347
\(527\) 0.122059 0.00531696
\(528\) 40.2428 1.75134
\(529\) 44.5351 1.93631
\(530\) −62.5402 −2.71657
\(531\) −6.58197 −0.285633
\(532\) −90.8100 −3.93711
\(533\) 1.20503 0.0521957
\(534\) 10.2246 0.442463
\(535\) −21.3376 −0.922503
\(536\) −8.03519 −0.347068
\(537\) 15.2684 0.658879
\(538\) −37.7311 −1.62670
\(539\) −7.57444 −0.326254
\(540\) 12.8078 0.551161
\(541\) −14.8440 −0.638193 −0.319096 0.947722i \(-0.603379\pi\)
−0.319096 + 0.947722i \(0.603379\pi\)
\(542\) −46.1527 −1.98243
\(543\) −5.91757 −0.253947
\(544\) 15.5501 0.666704
\(545\) −47.4166 −2.03111
\(546\) 4.23401 0.181199
\(547\) −0.245704 −0.0105056 −0.00525278 0.999986i \(-0.501672\pi\)
−0.00525278 + 0.999986i \(0.501672\pi\)
\(548\) 29.2879 1.25112
\(549\) −11.5200 −0.491660
\(550\) −11.3017 −0.481905
\(551\) −11.0778 −0.471930
\(552\) 68.3944 2.91106
\(553\) −8.78691 −0.373657
\(554\) 71.6101 3.04242
\(555\) −16.8129 −0.713667
\(556\) 48.3150 2.04901
\(557\) −7.85104 −0.332659 −0.166330 0.986070i \(-0.553192\pi\)
−0.166330 + 0.986070i \(0.553192\pi\)
\(558\) 0.320160 0.0135535
\(559\) 6.39262 0.270379
\(560\) 65.2401 2.75690
\(561\) −3.42055 −0.144416
\(562\) 10.4054 0.438927
\(563\) 3.67086 0.154708 0.0773542 0.997004i \(-0.475353\pi\)
0.0773542 + 0.997004i \(0.475353\pi\)
\(564\) −42.1785 −1.77604
\(565\) 1.32628 0.0557971
\(566\) −7.61235 −0.319971
\(567\) 2.17887 0.0915039
\(568\) −0.497081 −0.0208570
\(569\) 14.7247 0.617290 0.308645 0.951177i \(-0.400125\pi\)
0.308645 + 0.951177i \(0.400125\pi\)
\(570\) 54.3491 2.27643
\(571\) 15.2220 0.637022 0.318511 0.947919i \(-0.396817\pi\)
0.318511 + 0.947919i \(0.396817\pi\)
\(572\) −12.5368 −0.524189
\(573\) −14.3070 −0.597682
\(574\) 9.61918 0.401497
\(575\) −10.3517 −0.431697
\(576\) 16.8525 0.702189
\(577\) −4.54156 −0.189068 −0.0945338 0.995522i \(-0.530136\pi\)
−0.0945338 + 0.995522i \(0.530136\pi\)
\(578\) 42.5982 1.77185
\(579\) −26.2676 −1.09164
\(580\) 17.4271 0.723623
\(581\) 34.4388 1.42876
\(582\) 13.6686 0.566582
\(583\) 31.5028 1.30471
\(584\) −40.5852 −1.67943
\(585\) −1.82214 −0.0753361
\(586\) 19.5339 0.806940
\(587\) −6.42545 −0.265207 −0.132603 0.991169i \(-0.542334\pi\)
−0.132603 + 0.991169i \(0.542334\pi\)
\(588\) −11.5311 −0.475535
\(589\) 0.976910 0.0402529
\(590\) 43.9386 1.80892
\(591\) −23.8132 −0.979545
\(592\) −80.4222 −3.30533
\(593\) −7.90780 −0.324734 −0.162367 0.986730i \(-0.551913\pi\)
−0.162367 + 0.986730i \(0.551913\pi\)
\(594\) −8.97211 −0.368130
\(595\) −5.54527 −0.227334
\(596\) 52.6537 2.15678
\(597\) −0.780516 −0.0319444
\(598\) −15.9693 −0.653033
\(599\) 26.9505 1.10117 0.550584 0.834780i \(-0.314405\pi\)
0.550584 + 0.834780i \(0.314405\pi\)
\(600\) −10.4834 −0.427985
\(601\) −5.73354 −0.233876 −0.116938 0.993139i \(-0.537308\pi\)
−0.116938 + 0.993139i \(0.537308\pi\)
\(602\) 51.0293 2.07980
\(603\) 0.965475 0.0393172
\(604\) −108.909 −4.43145
\(605\) 0.768851 0.0312583
\(606\) 4.14925 0.168552
\(607\) −23.0467 −0.935436 −0.467718 0.883878i \(-0.654924\pi\)
−0.467718 + 0.883878i \(0.654924\pi\)
\(608\) 124.457 5.04738
\(609\) 2.96471 0.120136
\(610\) 76.9026 3.11370
\(611\) 6.00063 0.242759
\(612\) −5.20735 −0.210495
\(613\) −35.8613 −1.44843 −0.724213 0.689577i \(-0.757797\pi\)
−0.724213 + 0.689577i \(0.757797\pi\)
\(614\) 75.1955 3.03464
\(615\) −4.13968 −0.166928
\(616\) −60.9770 −2.45683
\(617\) −6.29245 −0.253325 −0.126662 0.991946i \(-0.540426\pi\)
−0.126662 + 0.991946i \(0.540426\pi\)
\(618\) −19.7724 −0.795365
\(619\) −1.90810 −0.0766929 −0.0383465 0.999265i \(-0.512209\pi\)
−0.0383465 + 0.999265i \(0.512209\pi\)
\(620\) −1.53684 −0.0617208
\(621\) −8.21797 −0.329776
\(622\) −5.58557 −0.223961
\(623\) −8.34955 −0.334518
\(624\) −8.71595 −0.348917
\(625\) −29.7115 −1.18846
\(626\) 31.9645 1.27756
\(627\) −27.3768 −1.09332
\(628\) −21.4606 −0.856373
\(629\) 6.83571 0.272558
\(630\) −14.5452 −0.579496
\(631\) −33.7143 −1.34214 −0.671072 0.741392i \(-0.734166\pi\)
−0.671072 + 0.741392i \(0.734166\pi\)
\(632\) 33.5630 1.33506
\(633\) −17.4775 −0.694667
\(634\) −59.1416 −2.34881
\(635\) 31.2637 1.24066
\(636\) 47.9590 1.90170
\(637\) 1.64050 0.0649991
\(638\) −12.2080 −0.483320
\(639\) 0.0597271 0.00236277
\(640\) −36.0079 −1.42334
\(641\) 17.8889 0.706570 0.353285 0.935516i \(-0.385065\pi\)
0.353285 + 0.935516i \(0.385065\pi\)
\(642\) 22.7554 0.898085
\(643\) 27.6338 1.08977 0.544885 0.838511i \(-0.316573\pi\)
0.544885 + 0.838511i \(0.316573\pi\)
\(644\) −91.6635 −3.61205
\(645\) −21.9608 −0.864706
\(646\) −22.0970 −0.869396
\(647\) 3.53161 0.138842 0.0694210 0.997587i \(-0.477885\pi\)
0.0694210 + 0.997587i \(0.477885\pi\)
\(648\) −8.32253 −0.326940
\(649\) −22.1328 −0.868787
\(650\) 2.44776 0.0960092
\(651\) −0.261447 −0.0102469
\(652\) 68.4867 2.68215
\(653\) 19.5680 0.765756 0.382878 0.923799i \(-0.374933\pi\)
0.382878 + 0.923799i \(0.374933\pi\)
\(654\) 50.5674 1.97734
\(655\) −22.8980 −0.894699
\(656\) −19.8016 −0.773124
\(657\) 4.87655 0.190252
\(658\) 47.9001 1.86734
\(659\) −30.2620 −1.17884 −0.589420 0.807827i \(-0.700644\pi\)
−0.589420 + 0.807827i \(0.700644\pi\)
\(660\) 43.0680 1.67642
\(661\) 29.2882 1.13918 0.569590 0.821929i \(-0.307102\pi\)
0.569590 + 0.821929i \(0.307102\pi\)
\(662\) −47.4596 −1.84457
\(663\) 0.740837 0.0287717
\(664\) −131.544 −5.10491
\(665\) −44.3821 −1.72107
\(666\) 17.9301 0.694776
\(667\) −11.1819 −0.432965
\(668\) 124.651 4.82291
\(669\) 23.8007 0.920188
\(670\) −6.44512 −0.248997
\(671\) −38.7374 −1.49544
\(672\) −33.3079 −1.28488
\(673\) 43.4808 1.67606 0.838031 0.545623i \(-0.183707\pi\)
0.838031 + 0.545623i \(0.183707\pi\)
\(674\) −35.5264 −1.36843
\(675\) 1.25965 0.0484838
\(676\) −63.8341 −2.45516
\(677\) −33.9151 −1.30346 −0.651731 0.758450i \(-0.725957\pi\)
−0.651731 + 0.758450i \(0.725957\pi\)
\(678\) −1.41441 −0.0543202
\(679\) −11.1620 −0.428357
\(680\) 21.1810 0.812255
\(681\) −2.96344 −0.113559
\(682\) 1.07658 0.0412244
\(683\) −29.8612 −1.14261 −0.571304 0.820738i \(-0.693562\pi\)
−0.571304 + 0.820738i \(0.693562\pi\)
\(684\) −41.6776 −1.59358
\(685\) 14.3140 0.546911
\(686\) 53.7906 2.05374
\(687\) 18.7987 0.717216
\(688\) −105.047 −4.00487
\(689\) −6.82300 −0.259936
\(690\) 54.8599 2.08848
\(691\) 5.56571 0.211730 0.105865 0.994381i \(-0.466239\pi\)
0.105865 + 0.994381i \(0.466239\pi\)
\(692\) 54.7731 2.08216
\(693\) 7.32674 0.278320
\(694\) −58.3386 −2.21450
\(695\) 23.6133 0.895702
\(696\) −11.3242 −0.429242
\(697\) 1.68310 0.0637518
\(698\) −10.9611 −0.414886
\(699\) 11.2917 0.427092
\(700\) 14.0501 0.531044
\(701\) 13.5895 0.513270 0.256635 0.966508i \(-0.417386\pi\)
0.256635 + 0.966508i \(0.417386\pi\)
\(702\) 1.94322 0.0733419
\(703\) 54.7103 2.06344
\(704\) 56.6689 2.13579
\(705\) −20.6142 −0.776374
\(706\) −46.5729 −1.75279
\(707\) −3.38833 −0.127431
\(708\) −33.6943 −1.26631
\(709\) −8.68500 −0.326172 −0.163086 0.986612i \(-0.552145\pi\)
−0.163086 + 0.986612i \(0.552145\pi\)
\(710\) −0.398714 −0.0149635
\(711\) −4.03279 −0.151241
\(712\) 31.8924 1.19522
\(713\) 0.986091 0.0369294
\(714\) 5.91375 0.221316
\(715\) −6.12718 −0.229143
\(716\) 78.1615 2.92103
\(717\) −16.1382 −0.602693
\(718\) 86.7485 3.23742
\(719\) 0.633393 0.0236216 0.0118108 0.999930i \(-0.496240\pi\)
0.0118108 + 0.999930i \(0.496240\pi\)
\(720\) 29.9422 1.11588
\(721\) 16.1464 0.601324
\(722\) −126.161 −4.69521
\(723\) −15.1276 −0.562602
\(724\) −30.2931 −1.12583
\(725\) 1.71396 0.0636547
\(726\) −0.819941 −0.0304309
\(727\) −11.4686 −0.425348 −0.212674 0.977123i \(-0.568217\pi\)
−0.212674 + 0.977123i \(0.568217\pi\)
\(728\) 13.2066 0.489471
\(729\) 1.00000 0.0370370
\(730\) −32.5538 −1.20487
\(731\) 8.92873 0.330241
\(732\) −58.9728 −2.17970
\(733\) 5.46114 0.201712 0.100856 0.994901i \(-0.467842\pi\)
0.100856 + 0.994901i \(0.467842\pi\)
\(734\) −61.4983 −2.26994
\(735\) −5.63567 −0.207875
\(736\) 125.626 4.63065
\(737\) 3.24654 0.119588
\(738\) 4.41476 0.162510
\(739\) −25.5997 −0.941700 −0.470850 0.882213i \(-0.656053\pi\)
−0.470850 + 0.882213i \(0.656053\pi\)
\(740\) −86.0681 −3.16393
\(741\) 5.92937 0.217821
\(742\) −54.4647 −1.99946
\(743\) 0.163146 0.00598523 0.00299262 0.999996i \(-0.499047\pi\)
0.00299262 + 0.999996i \(0.499047\pi\)
\(744\) 0.998637 0.0366118
\(745\) 25.7338 0.942812
\(746\) −56.8836 −2.08266
\(747\) 15.8058 0.578305
\(748\) −17.5104 −0.640244
\(749\) −18.5824 −0.678985
\(750\) 24.9691 0.911742
\(751\) −15.6592 −0.571412 −0.285706 0.958317i \(-0.592228\pi\)
−0.285706 + 0.958317i \(0.592228\pi\)
\(752\) −98.6052 −3.59576
\(753\) −15.4459 −0.562879
\(754\) 2.64406 0.0962911
\(755\) −53.2279 −1.93716
\(756\) 11.1540 0.405668
\(757\) 39.3740 1.43107 0.715536 0.698576i \(-0.246182\pi\)
0.715536 + 0.698576i \(0.246182\pi\)
\(758\) −94.1941 −3.42129
\(759\) −27.6340 −1.00305
\(760\) 169.525 6.14931
\(761\) −16.6844 −0.604809 −0.302405 0.953180i \(-0.597789\pi\)
−0.302405 + 0.953180i \(0.597789\pi\)
\(762\) −33.3411 −1.20782
\(763\) −41.2940 −1.49494
\(764\) −73.2399 −2.64973
\(765\) −2.54502 −0.0920155
\(766\) −34.7928 −1.25711
\(767\) 4.79360 0.173087
\(768\) 4.69552 0.169435
\(769\) −22.4474 −0.809473 −0.404736 0.914433i \(-0.632637\pi\)
−0.404736 + 0.914433i \(0.632637\pi\)
\(770\) −48.9103 −1.76261
\(771\) −0.0223687 −0.000805589 0
\(772\) −134.468 −4.83962
\(773\) 48.3274 1.73821 0.869107 0.494623i \(-0.164694\pi\)
0.869107 + 0.494623i \(0.164694\pi\)
\(774\) 23.4201 0.841817
\(775\) −0.151147 −0.00542937
\(776\) 42.6349 1.53050
\(777\) −14.6419 −0.525276
\(778\) −34.6035 −1.24059
\(779\) 13.4708 0.482643
\(780\) −9.32784 −0.333990
\(781\) 0.200840 0.00718663
\(782\) −22.3047 −0.797614
\(783\) 1.36066 0.0486262
\(784\) −26.9575 −0.962769
\(785\) −10.4886 −0.374354
\(786\) 24.4196 0.871017
\(787\) 51.4259 1.83314 0.916568 0.399880i \(-0.130948\pi\)
0.916568 + 0.399880i \(0.130948\pi\)
\(788\) −121.904 −4.34266
\(789\) −1.11182 −0.0395818
\(790\) 26.9212 0.957815
\(791\) 1.15503 0.0410680
\(792\) −27.9856 −0.994426
\(793\) 8.38991 0.297934
\(794\) −73.7378 −2.61686
\(795\) 23.4393 0.831306
\(796\) −3.99560 −0.141620
\(797\) −29.4307 −1.04249 −0.521245 0.853407i \(-0.674532\pi\)
−0.521245 + 0.853407i \(0.674532\pi\)
\(798\) 47.3313 1.67551
\(799\) 8.38122 0.296506
\(800\) −19.2559 −0.680799
\(801\) −3.83206 −0.135399
\(802\) −58.3657 −2.06096
\(803\) 16.3980 0.578674
\(804\) 4.94244 0.174306
\(805\) −44.7992 −1.57897
\(806\) −0.233170 −0.00821307
\(807\) 14.1411 0.497792
\(808\) 12.9423 0.455307
\(809\) 12.6464 0.444625 0.222312 0.974976i \(-0.428640\pi\)
0.222312 + 0.974976i \(0.428640\pi\)
\(810\) −6.67559 −0.234556
\(811\) −44.3397 −1.55698 −0.778489 0.627658i \(-0.784014\pi\)
−0.778489 + 0.627658i \(0.784014\pi\)
\(812\) 15.1769 0.532604
\(813\) 17.2975 0.606648
\(814\) 60.2922 2.11324
\(815\) 33.4719 1.17247
\(816\) −12.1738 −0.426168
\(817\) 71.4621 2.50014
\(818\) −6.68949 −0.233892
\(819\) −1.58685 −0.0554492
\(820\) −21.1918 −0.740049
\(821\) −25.3121 −0.883399 −0.441699 0.897163i \(-0.645624\pi\)
−0.441699 + 0.897163i \(0.645624\pi\)
\(822\) −15.2652 −0.532435
\(823\) −56.6988 −1.97639 −0.988197 0.153185i \(-0.951047\pi\)
−0.988197 + 0.153185i \(0.951047\pi\)
\(824\) −61.6738 −2.14851
\(825\) 4.23573 0.147469
\(826\) 38.2650 1.33141
\(827\) 17.3079 0.601856 0.300928 0.953647i \(-0.402704\pi\)
0.300928 + 0.953647i \(0.402704\pi\)
\(828\) −42.0693 −1.46201
\(829\) 0.913024 0.0317106 0.0158553 0.999874i \(-0.494953\pi\)
0.0158553 + 0.999874i \(0.494953\pi\)
\(830\) −105.513 −3.66242
\(831\) −26.8386 −0.931020
\(832\) −12.2736 −0.425510
\(833\) 2.29133 0.0793899
\(834\) −25.1823 −0.871993
\(835\) 60.9216 2.10828
\(836\) −140.147 −4.84707
\(837\) −0.119992 −0.00414753
\(838\) −47.7810 −1.65057
\(839\) 32.9728 1.13835 0.569173 0.822218i \(-0.307263\pi\)
0.569173 + 0.822218i \(0.307263\pi\)
\(840\) −45.3692 −1.56539
\(841\) −27.1486 −0.936158
\(842\) 96.9096 3.33972
\(843\) −3.89982 −0.134317
\(844\) −89.4703 −3.07969
\(845\) −31.1980 −1.07324
\(846\) 21.9840 0.755824
\(847\) 0.669574 0.0230068
\(848\) 112.119 3.85017
\(849\) 2.85301 0.0979152
\(850\) 3.41885 0.117266
\(851\) 55.2245 1.89307
\(852\) 0.305754 0.0104749
\(853\) 6.78292 0.232243 0.116121 0.993235i \(-0.462954\pi\)
0.116121 + 0.993235i \(0.462954\pi\)
\(854\) 66.9726 2.29176
\(855\) −20.3694 −0.696617
\(856\) 70.9783 2.42599
\(857\) −20.9229 −0.714712 −0.357356 0.933968i \(-0.616322\pi\)
−0.357356 + 0.933968i \(0.616322\pi\)
\(858\) 6.53432 0.223078
\(859\) −10.4406 −0.356229 −0.178115 0.984010i \(-0.557000\pi\)
−0.178115 + 0.984010i \(0.557000\pi\)
\(860\) −112.421 −3.83353
\(861\) −3.60515 −0.122863
\(862\) −54.0306 −1.84029
\(863\) −6.60187 −0.224730 −0.112365 0.993667i \(-0.535843\pi\)
−0.112365 + 0.993667i \(0.535843\pi\)
\(864\) −15.2868 −0.520066
\(865\) 26.7696 0.910193
\(866\) −42.2845 −1.43689
\(867\) −15.9653 −0.542208
\(868\) −1.33839 −0.0454280
\(869\) −13.5608 −0.460018
\(870\) −9.08324 −0.307951
\(871\) −0.703148 −0.0238253
\(872\) 157.729 5.34137
\(873\) −5.12282 −0.173381
\(874\) −178.518 −6.03846
\(875\) −20.3901 −0.689310
\(876\) 24.9639 0.843453
\(877\) 13.9169 0.469940 0.234970 0.972003i \(-0.424501\pi\)
0.234970 + 0.972003i \(0.424501\pi\)
\(878\) −96.0573 −3.24178
\(879\) −7.32108 −0.246934
\(880\) 100.685 3.39408
\(881\) 9.87846 0.332814 0.166407 0.986057i \(-0.446783\pi\)
0.166407 + 0.986057i \(0.446783\pi\)
\(882\) 6.01016 0.202373
\(883\) 18.8888 0.635659 0.317829 0.948148i \(-0.397046\pi\)
0.317829 + 0.948148i \(0.397046\pi\)
\(884\) 3.79248 0.127555
\(885\) −16.4676 −0.553553
\(886\) 80.7707 2.71354
\(887\) −23.2220 −0.779719 −0.389859 0.920874i \(-0.627476\pi\)
−0.389859 + 0.920874i \(0.627476\pi\)
\(888\) 55.9271 1.87679
\(889\) 27.2268 0.913156
\(890\) 25.5813 0.857486
\(891\) 3.36263 0.112652
\(892\) 121.840 4.07951
\(893\) 67.0800 2.24475
\(894\) −27.4437 −0.917856
\(895\) 38.2004 1.27690
\(896\) −31.3584 −1.04761
\(897\) 5.98509 0.199836
\(898\) −13.9144 −0.464330
\(899\) −0.163269 −0.00544532
\(900\) 6.44835 0.214945
\(901\) −9.52985 −0.317485
\(902\) 14.8452 0.494292
\(903\) −19.1251 −0.636444
\(904\) −4.41181 −0.146735
\(905\) −14.8053 −0.492146
\(906\) 56.7648 1.88588
\(907\) −29.1180 −0.966848 −0.483424 0.875386i \(-0.660607\pi\)
−0.483424 + 0.875386i \(0.660607\pi\)
\(908\) −15.1704 −0.503447
\(909\) −1.55509 −0.0515789
\(910\) 10.5932 0.351161
\(911\) −23.7607 −0.787227 −0.393613 0.919276i \(-0.628775\pi\)
−0.393613 + 0.919276i \(0.628775\pi\)
\(912\) −97.4342 −3.22637
\(913\) 53.1492 1.75898
\(914\) 18.4061 0.608820
\(915\) −28.8221 −0.952831
\(916\) 96.2340 3.17966
\(917\) −19.9413 −0.658520
\(918\) 2.71414 0.0895798
\(919\) −36.7641 −1.21274 −0.606368 0.795184i \(-0.707374\pi\)
−0.606368 + 0.795184i \(0.707374\pi\)
\(920\) 171.118 5.64159
\(921\) −28.1823 −0.928640
\(922\) −54.0967 −1.78158
\(923\) −0.0434988 −0.00143178
\(924\) 37.5069 1.23389
\(925\) −8.46477 −0.278320
\(926\) 7.66174 0.251780
\(927\) 7.41047 0.243392
\(928\) −20.8002 −0.682799
\(929\) −54.6011 −1.79140 −0.895701 0.444656i \(-0.853326\pi\)
−0.895701 + 0.444656i \(0.853326\pi\)
\(930\) 0.801017 0.0262664
\(931\) 18.3389 0.601033
\(932\) 57.8043 1.89344
\(933\) 2.09340 0.0685349
\(934\) −85.0778 −2.78383
\(935\) −8.55797 −0.279876
\(936\) 6.06124 0.198118
\(937\) −18.3872 −0.600685 −0.300343 0.953831i \(-0.597101\pi\)
−0.300343 + 0.953831i \(0.597101\pi\)
\(938\) −5.61289 −0.183267
\(939\) −11.9799 −0.390949
\(940\) −105.528 −3.44193
\(941\) −32.5337 −1.06057 −0.530285 0.847820i \(-0.677915\pi\)
−0.530285 + 0.847820i \(0.677915\pi\)
\(942\) 11.1855 0.364445
\(943\) 13.5974 0.442793
\(944\) −78.7707 −2.56377
\(945\) 5.45137 0.177333
\(946\) 78.7531 2.56048
\(947\) −46.7308 −1.51855 −0.759273 0.650772i \(-0.774445\pi\)
−0.759273 + 0.650772i \(0.774445\pi\)
\(948\) −20.6446 −0.670504
\(949\) −3.55155 −0.115288
\(950\) 27.3631 0.887777
\(951\) 22.1655 0.718766
\(952\) 18.4460 0.597840
\(953\) −10.0674 −0.326114 −0.163057 0.986617i \(-0.552135\pi\)
−0.163057 + 0.986617i \(0.552135\pi\)
\(954\) −24.9968 −0.809301
\(955\) −35.7950 −1.15830
\(956\) −82.6145 −2.67194
\(957\) 4.57542 0.147902
\(958\) −96.1127 −3.10526
\(959\) 12.4657 0.402540
\(960\) 42.1638 1.36083
\(961\) −30.9856 −0.999536
\(962\) −13.0583 −0.421018
\(963\) −8.52845 −0.274825
\(964\) −77.4410 −2.49421
\(965\) −65.7196 −2.11559
\(966\) 47.7761 1.53717
\(967\) −2.61734 −0.0841681 −0.0420840 0.999114i \(-0.513400\pi\)
−0.0420840 + 0.999114i \(0.513400\pi\)
\(968\) −2.55754 −0.0822026
\(969\) 8.28169 0.266046
\(970\) 34.1979 1.09803
\(971\) −33.5130 −1.07548 −0.537742 0.843109i \(-0.680723\pi\)
−0.537742 + 0.843109i \(0.680723\pi\)
\(972\) 5.11918 0.164198
\(973\) 20.5642 0.659258
\(974\) −9.65684 −0.309425
\(975\) −0.917391 −0.0293800
\(976\) −137.867 −4.41301
\(977\) −9.85589 −0.315318 −0.157659 0.987494i \(-0.550395\pi\)
−0.157659 + 0.987494i \(0.550395\pi\)
\(978\) −35.6961 −1.14144
\(979\) −12.8858 −0.411832
\(980\) −28.8500 −0.921581
\(981\) −18.9520 −0.605092
\(982\) −12.3159 −0.393016
\(983\) 11.8088 0.376641 0.188320 0.982108i \(-0.439696\pi\)
0.188320 + 0.982108i \(0.439696\pi\)
\(984\) 13.7704 0.438986
\(985\) −59.5790 −1.89834
\(986\) 3.69303 0.117610
\(987\) −17.9524 −0.571430
\(988\) 30.3535 0.965674
\(989\) 72.1337 2.29372
\(990\) −22.4476 −0.713431
\(991\) 51.5702 1.63818 0.819091 0.573664i \(-0.194478\pi\)
0.819091 + 0.573664i \(0.194478\pi\)
\(992\) 1.83429 0.0582387
\(993\) 17.7872 0.564461
\(994\) −0.347230 −0.0110135
\(995\) −1.95280 −0.0619078
\(996\) 80.9128 2.56382
\(997\) 35.5520 1.12594 0.562972 0.826476i \(-0.309658\pi\)
0.562972 + 0.826476i \(0.309658\pi\)
\(998\) −85.8597 −2.71784
\(999\) −6.71996 −0.212610
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.a.1.3 92
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8031.2.a.a.1.3 92 1.1 even 1 trivial