Properties

Label 8047.2.a.c.1.10
Level $8047$
Weight $2$
Character 8047.1
Self dual yes
Analytic conductor $64.256$
Analytic rank $1$
Dimension $151$
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] = [8047,2,Mod(1,8047)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8047, 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("8047.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8047 = 13 \cdot 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8047.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: \(64.2556185065\)
Analytic rank: \(1\)
Dimension: \(151\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Character \(\chi\) \(=\) 8047.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.62707 q^{2} -2.08646 q^{3} +4.90149 q^{4} +1.65471 q^{5} +5.48127 q^{6} -3.37709 q^{7} -7.62241 q^{8} +1.35330 q^{9} +O(q^{10})\) \(q-2.62707 q^{2} -2.08646 q^{3} +4.90149 q^{4} +1.65471 q^{5} +5.48127 q^{6} -3.37709 q^{7} -7.62241 q^{8} +1.35330 q^{9} -4.34704 q^{10} +3.75982 q^{11} -10.2267 q^{12} -1.00000 q^{13} +8.87186 q^{14} -3.45248 q^{15} +10.2216 q^{16} +6.62891 q^{17} -3.55522 q^{18} -1.81610 q^{19} +8.11054 q^{20} +7.04616 q^{21} -9.87732 q^{22} +5.15156 q^{23} +15.9038 q^{24} -2.26194 q^{25} +2.62707 q^{26} +3.43576 q^{27} -16.5528 q^{28} +0.517013 q^{29} +9.06990 q^{30} +3.41712 q^{31} -11.6081 q^{32} -7.84471 q^{33} -17.4146 q^{34} -5.58811 q^{35} +6.63320 q^{36} -10.8938 q^{37} +4.77102 q^{38} +2.08646 q^{39} -12.6129 q^{40} +0.624662 q^{41} -18.5107 q^{42} -6.66643 q^{43} +18.4287 q^{44} +2.23932 q^{45} -13.5335 q^{46} -4.43666 q^{47} -21.3270 q^{48} +4.40476 q^{49} +5.94226 q^{50} -13.8309 q^{51} -4.90149 q^{52} +1.03442 q^{53} -9.02598 q^{54} +6.22142 q^{55} +25.7416 q^{56} +3.78922 q^{57} -1.35823 q^{58} +0.884692 q^{59} -16.9223 q^{60} -1.60178 q^{61} -8.97700 q^{62} -4.57023 q^{63} +10.0520 q^{64} -1.65471 q^{65} +20.6086 q^{66} +2.73694 q^{67} +32.4915 q^{68} -10.7485 q^{69} +14.6803 q^{70} +3.28850 q^{71} -10.3154 q^{72} -9.02450 q^{73} +28.6187 q^{74} +4.71943 q^{75} -8.90160 q^{76} -12.6973 q^{77} -5.48127 q^{78} +8.74165 q^{79} +16.9138 q^{80} -11.2285 q^{81} -1.64103 q^{82} -3.30673 q^{83} +34.5367 q^{84} +10.9689 q^{85} +17.5132 q^{86} -1.07873 q^{87} -28.6589 q^{88} -0.585740 q^{89} -5.88286 q^{90} +3.37709 q^{91} +25.2503 q^{92} -7.12967 q^{93} +11.6554 q^{94} -3.00512 q^{95} +24.2198 q^{96} -0.605988 q^{97} -11.5716 q^{98} +5.08818 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.62707 −1.85762 −0.928809 0.370559i \(-0.879166\pi\)
−0.928809 + 0.370559i \(0.879166\pi\)
\(3\) −2.08646 −1.20462 −0.602308 0.798264i \(-0.705752\pi\)
−0.602308 + 0.798264i \(0.705752\pi\)
\(4\) 4.90149 2.45075
\(5\) 1.65471 0.740009 0.370004 0.929030i \(-0.379356\pi\)
0.370004 + 0.929030i \(0.379356\pi\)
\(6\) 5.48127 2.23772
\(7\) −3.37709 −1.27642 −0.638211 0.769862i \(-0.720325\pi\)
−0.638211 + 0.769862i \(0.720325\pi\)
\(8\) −7.62241 −2.69493
\(9\) 1.35330 0.451101
\(10\) −4.34704 −1.37465
\(11\) 3.75982 1.13363 0.566815 0.823845i \(-0.308175\pi\)
0.566815 + 0.823845i \(0.308175\pi\)
\(12\) −10.2267 −2.95221
\(13\) −1.00000 −0.277350
\(14\) 8.87186 2.37110
\(15\) −3.45248 −0.891427
\(16\) 10.2216 2.55541
\(17\) 6.62891 1.60775 0.803873 0.594800i \(-0.202769\pi\)
0.803873 + 0.594800i \(0.202769\pi\)
\(18\) −3.55522 −0.837973
\(19\) −1.81610 −0.416642 −0.208321 0.978060i \(-0.566800\pi\)
−0.208321 + 0.978060i \(0.566800\pi\)
\(20\) 8.11054 1.81357
\(21\) 7.04616 1.53760
\(22\) −9.87732 −2.10585
\(23\) 5.15156 1.07417 0.537087 0.843527i \(-0.319525\pi\)
0.537087 + 0.843527i \(0.319525\pi\)
\(24\) 15.9038 3.24636
\(25\) −2.26194 −0.452387
\(26\) 2.62707 0.515211
\(27\) 3.43576 0.661213
\(28\) −16.5528 −3.12818
\(29\) 0.517013 0.0960070 0.0480035 0.998847i \(-0.484714\pi\)
0.0480035 + 0.998847i \(0.484714\pi\)
\(30\) 9.06990 1.65593
\(31\) 3.41712 0.613733 0.306866 0.951753i \(-0.400720\pi\)
0.306866 + 0.951753i \(0.400720\pi\)
\(32\) −11.6081 −2.05204
\(33\) −7.84471 −1.36559
\(34\) −17.4146 −2.98658
\(35\) −5.58811 −0.944563
\(36\) 6.63320 1.10553
\(37\) −10.8938 −1.79093 −0.895463 0.445137i \(-0.853155\pi\)
−0.895463 + 0.445137i \(0.853155\pi\)
\(38\) 4.77102 0.773962
\(39\) 2.08646 0.334101
\(40\) −12.6129 −1.99427
\(41\) 0.624662 0.0975558 0.0487779 0.998810i \(-0.484467\pi\)
0.0487779 + 0.998810i \(0.484467\pi\)
\(42\) −18.5107 −2.85627
\(43\) −6.66643 −1.01662 −0.508311 0.861174i \(-0.669730\pi\)
−0.508311 + 0.861174i \(0.669730\pi\)
\(44\) 18.4287 2.77824
\(45\) 2.23932 0.333819
\(46\) −13.5335 −1.99541
\(47\) −4.43666 −0.647153 −0.323577 0.946202i \(-0.604885\pi\)
−0.323577 + 0.946202i \(0.604885\pi\)
\(48\) −21.3270 −3.07829
\(49\) 4.40476 0.629252
\(50\) 5.94226 0.840362
\(51\) −13.8309 −1.93672
\(52\) −4.90149 −0.679714
\(53\) 1.03442 0.142088 0.0710441 0.997473i \(-0.477367\pi\)
0.0710441 + 0.997473i \(0.477367\pi\)
\(54\) −9.02598 −1.22828
\(55\) 6.22142 0.838896
\(56\) 25.7416 3.43987
\(57\) 3.78922 0.501894
\(58\) −1.35823 −0.178344
\(59\) 0.884692 0.115177 0.0575886 0.998340i \(-0.481659\pi\)
0.0575886 + 0.998340i \(0.481659\pi\)
\(60\) −16.9223 −2.18466
\(61\) −1.60178 −0.205086 −0.102543 0.994729i \(-0.532698\pi\)
−0.102543 + 0.994729i \(0.532698\pi\)
\(62\) −8.97700 −1.14008
\(63\) −4.57023 −0.575795
\(64\) 10.0520 1.25650
\(65\) −1.65471 −0.205241
\(66\) 20.6086 2.53674
\(67\) 2.73694 0.334370 0.167185 0.985926i \(-0.446532\pi\)
0.167185 + 0.985926i \(0.446532\pi\)
\(68\) 32.4915 3.94018
\(69\) −10.7485 −1.29397
\(70\) 14.6803 1.75464
\(71\) 3.28850 0.390273 0.195136 0.980776i \(-0.437485\pi\)
0.195136 + 0.980776i \(0.437485\pi\)
\(72\) −10.3154 −1.21569
\(73\) −9.02450 −1.05624 −0.528119 0.849170i \(-0.677103\pi\)
−0.528119 + 0.849170i \(0.677103\pi\)
\(74\) 28.6187 3.32686
\(75\) 4.71943 0.544953
\(76\) −8.90160 −1.02108
\(77\) −12.6973 −1.44699
\(78\) −5.48127 −0.620631
\(79\) 8.74165 0.983513 0.491756 0.870733i \(-0.336355\pi\)
0.491756 + 0.870733i \(0.336355\pi\)
\(80\) 16.9138 1.89102
\(81\) −11.2285 −1.24761
\(82\) −1.64103 −0.181221
\(83\) −3.30673 −0.362961 −0.181481 0.983395i \(-0.558089\pi\)
−0.181481 + 0.983395i \(0.558089\pi\)
\(84\) 34.5367 3.76826
\(85\) 10.9689 1.18975
\(86\) 17.5132 1.88849
\(87\) −1.07873 −0.115652
\(88\) −28.6589 −3.05505
\(89\) −0.585740 −0.0620883 −0.0310441 0.999518i \(-0.509883\pi\)
−0.0310441 + 0.999518i \(0.509883\pi\)
\(90\) −5.88286 −0.620108
\(91\) 3.37709 0.354016
\(92\) 25.2503 2.63253
\(93\) −7.12967 −0.739312
\(94\) 11.6554 1.20216
\(95\) −3.00512 −0.308319
\(96\) 24.2198 2.47192
\(97\) −0.605988 −0.0615288 −0.0307644 0.999527i \(-0.509794\pi\)
−0.0307644 + 0.999527i \(0.509794\pi\)
\(98\) −11.5716 −1.16891
\(99\) 5.08818 0.511382
\(100\) −11.0869 −1.10869
\(101\) 15.1065 1.50315 0.751577 0.659645i \(-0.229294\pi\)
0.751577 + 0.659645i \(0.229294\pi\)
\(102\) 36.3348 3.59768
\(103\) −7.29579 −0.718875 −0.359438 0.933169i \(-0.617031\pi\)
−0.359438 + 0.933169i \(0.617031\pi\)
\(104\) 7.62241 0.747439
\(105\) 11.6594 1.13784
\(106\) −2.71749 −0.263946
\(107\) −10.8313 −1.04710 −0.523551 0.851994i \(-0.675393\pi\)
−0.523551 + 0.851994i \(0.675393\pi\)
\(108\) 16.8404 1.62046
\(109\) 1.90864 0.182814 0.0914071 0.995814i \(-0.470864\pi\)
0.0914071 + 0.995814i \(0.470864\pi\)
\(110\) −16.3441 −1.55835
\(111\) 22.7294 2.15738
\(112\) −34.5194 −3.26178
\(113\) 1.39133 0.130886 0.0654428 0.997856i \(-0.479154\pi\)
0.0654428 + 0.997856i \(0.479154\pi\)
\(114\) −9.95454 −0.932328
\(115\) 8.52433 0.794898
\(116\) 2.53414 0.235289
\(117\) −1.35330 −0.125113
\(118\) −2.32415 −0.213955
\(119\) −22.3865 −2.05216
\(120\) 26.3162 2.40233
\(121\) 3.13628 0.285117
\(122\) 4.20798 0.380972
\(123\) −1.30333 −0.117517
\(124\) 16.7490 1.50410
\(125\) −12.0164 −1.07478
\(126\) 12.0063 1.06961
\(127\) 11.9762 1.06272 0.531358 0.847148i \(-0.321682\pi\)
0.531358 + 0.847148i \(0.321682\pi\)
\(128\) −3.19110 −0.282056
\(129\) 13.9092 1.22464
\(130\) 4.34704 0.381260
\(131\) −1.33810 −0.116911 −0.0584553 0.998290i \(-0.518618\pi\)
−0.0584553 + 0.998290i \(0.518618\pi\)
\(132\) −38.4508 −3.34671
\(133\) 6.13314 0.531811
\(134\) −7.19012 −0.621132
\(135\) 5.68519 0.489303
\(136\) −50.5283 −4.33277
\(137\) −12.9635 −1.10755 −0.553775 0.832666i \(-0.686813\pi\)
−0.553775 + 0.832666i \(0.686813\pi\)
\(138\) 28.2371 2.40370
\(139\) 1.21252 0.102845 0.0514224 0.998677i \(-0.483625\pi\)
0.0514224 + 0.998677i \(0.483625\pi\)
\(140\) −27.3901 −2.31488
\(141\) 9.25690 0.779571
\(142\) −8.63911 −0.724978
\(143\) −3.75982 −0.314412
\(144\) 13.8330 1.15275
\(145\) 0.855507 0.0710460
\(146\) 23.7080 1.96209
\(147\) −9.19035 −0.758007
\(148\) −53.3957 −4.38910
\(149\) 6.02220 0.493358 0.246679 0.969097i \(-0.420661\pi\)
0.246679 + 0.969097i \(0.420661\pi\)
\(150\) −12.3983 −1.01231
\(151\) −9.13308 −0.743239 −0.371620 0.928385i \(-0.621197\pi\)
−0.371620 + 0.928385i \(0.621197\pi\)
\(152\) 13.8431 1.12282
\(153\) 8.97092 0.725256
\(154\) 33.3566 2.68795
\(155\) 5.65434 0.454167
\(156\) 10.2267 0.818795
\(157\) −3.82902 −0.305589 −0.152795 0.988258i \(-0.548827\pi\)
−0.152795 + 0.988258i \(0.548827\pi\)
\(158\) −22.9649 −1.82699
\(159\) −2.15827 −0.171162
\(160\) −19.2080 −1.51853
\(161\) −17.3973 −1.37110
\(162\) 29.4980 2.31758
\(163\) −8.16075 −0.639199 −0.319600 0.947553i \(-0.603548\pi\)
−0.319600 + 0.947553i \(0.603548\pi\)
\(164\) 3.06177 0.239084
\(165\) −12.9807 −1.01055
\(166\) 8.68702 0.674243
\(167\) −9.77275 −0.756238 −0.378119 0.925757i \(-0.623429\pi\)
−0.378119 + 0.925757i \(0.623429\pi\)
\(168\) −53.7088 −4.14372
\(169\) 1.00000 0.0769231
\(170\) −28.8161 −2.21010
\(171\) −2.45774 −0.187948
\(172\) −32.6755 −2.49148
\(173\) 2.73054 0.207599 0.103799 0.994598i \(-0.466900\pi\)
0.103799 + 0.994598i \(0.466900\pi\)
\(174\) 2.83389 0.214836
\(175\) 7.63877 0.577437
\(176\) 38.4315 2.89689
\(177\) −1.84587 −0.138744
\(178\) 1.53878 0.115336
\(179\) 8.99387 0.672233 0.336117 0.941820i \(-0.390886\pi\)
0.336117 + 0.941820i \(0.390886\pi\)
\(180\) 10.9760 0.818105
\(181\) −8.12912 −0.604233 −0.302116 0.953271i \(-0.597693\pi\)
−0.302116 + 0.953271i \(0.597693\pi\)
\(182\) −8.87186 −0.657626
\(183\) 3.34204 0.247051
\(184\) −39.2673 −2.89482
\(185\) −18.0260 −1.32530
\(186\) 18.7301 1.37336
\(187\) 24.9235 1.82259
\(188\) −21.7462 −1.58601
\(189\) −11.6029 −0.843986
\(190\) 7.89466 0.572739
\(191\) −23.7447 −1.71811 −0.859053 0.511886i \(-0.828947\pi\)
−0.859053 + 0.511886i \(0.828947\pi\)
\(192\) −20.9730 −1.51360
\(193\) 6.53531 0.470422 0.235211 0.971944i \(-0.424422\pi\)
0.235211 + 0.971944i \(0.424422\pi\)
\(194\) 1.59197 0.114297
\(195\) 3.45248 0.247237
\(196\) 21.5899 1.54214
\(197\) −1.38546 −0.0987102 −0.0493551 0.998781i \(-0.515717\pi\)
−0.0493551 + 0.998781i \(0.515717\pi\)
\(198\) −13.3670 −0.949952
\(199\) 18.9068 1.34027 0.670134 0.742240i \(-0.266237\pi\)
0.670134 + 0.742240i \(0.266237\pi\)
\(200\) 17.2414 1.21915
\(201\) −5.71050 −0.402788
\(202\) −39.6858 −2.79229
\(203\) −1.74600 −0.122545
\(204\) −67.7922 −4.74640
\(205\) 1.03363 0.0721922
\(206\) 19.1665 1.33540
\(207\) 6.97162 0.484561
\(208\) −10.2216 −0.708742
\(209\) −6.82822 −0.472318
\(210\) −30.6299 −2.11367
\(211\) 22.7008 1.56279 0.781395 0.624036i \(-0.214508\pi\)
0.781395 + 0.624036i \(0.214508\pi\)
\(212\) 5.07019 0.348222
\(213\) −6.86131 −0.470129
\(214\) 28.4546 1.94511
\(215\) −11.0310 −0.752309
\(216\) −26.1888 −1.78192
\(217\) −11.5399 −0.783381
\(218\) −5.01412 −0.339599
\(219\) 18.8292 1.27236
\(220\) 30.4942 2.05592
\(221\) −6.62891 −0.445909
\(222\) −59.7117 −4.00759
\(223\) 25.8965 1.73415 0.867077 0.498173i \(-0.165996\pi\)
0.867077 + 0.498173i \(0.165996\pi\)
\(224\) 39.2016 2.61927
\(225\) −3.06108 −0.204072
\(226\) −3.65513 −0.243136
\(227\) −4.63673 −0.307750 −0.153875 0.988090i \(-0.549175\pi\)
−0.153875 + 0.988090i \(0.549175\pi\)
\(228\) 18.5728 1.23001
\(229\) −26.8828 −1.77646 −0.888232 0.459395i \(-0.848066\pi\)
−0.888232 + 0.459395i \(0.848066\pi\)
\(230\) −22.3940 −1.47662
\(231\) 26.4923 1.74307
\(232\) −3.94089 −0.258732
\(233\) −27.6598 −1.81205 −0.906027 0.423219i \(-0.860900\pi\)
−0.906027 + 0.423219i \(0.860900\pi\)
\(234\) 3.55522 0.232412
\(235\) −7.34138 −0.478899
\(236\) 4.33631 0.282270
\(237\) −18.2391 −1.18476
\(238\) 58.8108 3.81213
\(239\) −5.79091 −0.374583 −0.187292 0.982304i \(-0.559971\pi\)
−0.187292 + 0.982304i \(0.559971\pi\)
\(240\) −35.2900 −2.27796
\(241\) 17.3411 1.11704 0.558520 0.829491i \(-0.311369\pi\)
0.558520 + 0.829491i \(0.311369\pi\)
\(242\) −8.23923 −0.529638
\(243\) 13.1205 0.841677
\(244\) −7.85109 −0.502615
\(245\) 7.28860 0.465652
\(246\) 3.42394 0.218302
\(247\) 1.81610 0.115556
\(248\) −26.0467 −1.65397
\(249\) 6.89936 0.437229
\(250\) 31.5679 1.99653
\(251\) 11.4533 0.722925 0.361463 0.932387i \(-0.382277\pi\)
0.361463 + 0.932387i \(0.382277\pi\)
\(252\) −22.4009 −1.41113
\(253\) 19.3690 1.21772
\(254\) −31.4623 −1.97412
\(255\) −22.8862 −1.43319
\(256\) −11.7208 −0.732547
\(257\) −1.62503 −0.101367 −0.0506834 0.998715i \(-0.516140\pi\)
−0.0506834 + 0.998715i \(0.516140\pi\)
\(258\) −36.5405 −2.27491
\(259\) 36.7893 2.28598
\(260\) −8.11054 −0.502995
\(261\) 0.699676 0.0433088
\(262\) 3.51529 0.217175
\(263\) 19.5544 1.20578 0.602888 0.797826i \(-0.294017\pi\)
0.602888 + 0.797826i \(0.294017\pi\)
\(264\) 59.7957 3.68017
\(265\) 1.71166 0.105147
\(266\) −16.1122 −0.987902
\(267\) 1.22212 0.0747925
\(268\) 13.4151 0.819456
\(269\) 13.5396 0.825525 0.412763 0.910839i \(-0.364564\pi\)
0.412763 + 0.910839i \(0.364564\pi\)
\(270\) −14.9354 −0.908939
\(271\) −11.4778 −0.697225 −0.348613 0.937267i \(-0.613347\pi\)
−0.348613 + 0.937267i \(0.613347\pi\)
\(272\) 67.7582 4.10845
\(273\) −7.04616 −0.426453
\(274\) 34.0561 2.05740
\(275\) −8.50448 −0.512840
\(276\) −52.6837 −3.17119
\(277\) −18.0658 −1.08547 −0.542735 0.839904i \(-0.682611\pi\)
−0.542735 + 0.839904i \(0.682611\pi\)
\(278\) −3.18538 −0.191046
\(279\) 4.62440 0.276855
\(280\) 42.5949 2.54553
\(281\) 5.66773 0.338109 0.169054 0.985607i \(-0.445929\pi\)
0.169054 + 0.985607i \(0.445929\pi\)
\(282\) −24.3185 −1.44815
\(283\) 8.40234 0.499467 0.249734 0.968315i \(-0.419657\pi\)
0.249734 + 0.968315i \(0.419657\pi\)
\(284\) 16.1185 0.956459
\(285\) 6.27006 0.371406
\(286\) 9.87732 0.584058
\(287\) −2.10954 −0.124522
\(288\) −15.7093 −0.925677
\(289\) 26.9425 1.58485
\(290\) −2.24748 −0.131976
\(291\) 1.26437 0.0741186
\(292\) −44.2335 −2.58857
\(293\) 22.5963 1.32009 0.660045 0.751226i \(-0.270537\pi\)
0.660045 + 0.751226i \(0.270537\pi\)
\(294\) 24.1437 1.40809
\(295\) 1.46391 0.0852321
\(296\) 83.0369 4.82642
\(297\) 12.9179 0.749571
\(298\) −15.8207 −0.916471
\(299\) −5.15156 −0.297922
\(300\) 23.1322 1.33554
\(301\) 22.5132 1.29764
\(302\) 23.9932 1.38066
\(303\) −31.5191 −1.81072
\(304\) −18.5635 −1.06469
\(305\) −2.65048 −0.151766
\(306\) −23.5672 −1.34725
\(307\) −0.759501 −0.0433470 −0.0216735 0.999765i \(-0.506899\pi\)
−0.0216735 + 0.999765i \(0.506899\pi\)
\(308\) −62.2356 −3.54620
\(309\) 15.2223 0.865969
\(310\) −14.8543 −0.843670
\(311\) −3.87253 −0.219591 −0.109796 0.993954i \(-0.535020\pi\)
−0.109796 + 0.993954i \(0.535020\pi\)
\(312\) −15.9038 −0.900378
\(313\) −9.19425 −0.519690 −0.259845 0.965650i \(-0.583671\pi\)
−0.259845 + 0.965650i \(0.583671\pi\)
\(314\) 10.0591 0.567668
\(315\) −7.56241 −0.426093
\(316\) 42.8471 2.41034
\(317\) 29.2747 1.64423 0.822116 0.569320i \(-0.192793\pi\)
0.822116 + 0.569320i \(0.192793\pi\)
\(318\) 5.66992 0.317953
\(319\) 1.94388 0.108836
\(320\) 16.6331 0.929820
\(321\) 22.5990 1.26136
\(322\) 45.7039 2.54698
\(323\) −12.0388 −0.669855
\(324\) −55.0363 −3.05757
\(325\) 2.26194 0.125470
\(326\) 21.4389 1.18739
\(327\) −3.98229 −0.220221
\(328\) −4.76143 −0.262906
\(329\) 14.9830 0.826040
\(330\) 34.1013 1.87721
\(331\) −24.0559 −1.32223 −0.661116 0.750284i \(-0.729917\pi\)
−0.661116 + 0.750284i \(0.729917\pi\)
\(332\) −16.2079 −0.889525
\(333\) −14.7426 −0.807888
\(334\) 25.6737 1.40480
\(335\) 4.52884 0.247437
\(336\) 72.0232 3.92919
\(337\) −14.6077 −0.795733 −0.397866 0.917443i \(-0.630249\pi\)
−0.397866 + 0.917443i \(0.630249\pi\)
\(338\) −2.62707 −0.142894
\(339\) −2.90296 −0.157667
\(340\) 53.7641 2.91577
\(341\) 12.8478 0.695746
\(342\) 6.45664 0.349135
\(343\) 8.76436 0.473231
\(344\) 50.8143 2.73972
\(345\) −17.7857 −0.957548
\(346\) −7.17331 −0.385639
\(347\) 6.43170 0.345272 0.172636 0.984986i \(-0.444772\pi\)
0.172636 + 0.984986i \(0.444772\pi\)
\(348\) −5.28736 −0.283433
\(349\) −7.31729 −0.391685 −0.195843 0.980635i \(-0.562744\pi\)
−0.195843 + 0.980635i \(0.562744\pi\)
\(350\) −20.0676 −1.07266
\(351\) −3.43576 −0.183387
\(352\) −43.6444 −2.32625
\(353\) 6.80465 0.362175 0.181087 0.983467i \(-0.442038\pi\)
0.181087 + 0.983467i \(0.442038\pi\)
\(354\) 4.84923 0.257734
\(355\) 5.44151 0.288805
\(356\) −2.87100 −0.152163
\(357\) 46.7084 2.47207
\(358\) −23.6275 −1.24875
\(359\) −31.0689 −1.63975 −0.819876 0.572542i \(-0.805958\pi\)
−0.819876 + 0.572542i \(0.805958\pi\)
\(360\) −17.0691 −0.899618
\(361\) −15.7018 −0.826409
\(362\) 21.3558 1.12243
\(363\) −6.54372 −0.343456
\(364\) 16.5528 0.867602
\(365\) −14.9329 −0.781626
\(366\) −8.77976 −0.458926
\(367\) −13.5108 −0.705260 −0.352630 0.935763i \(-0.614713\pi\)
−0.352630 + 0.935763i \(0.614713\pi\)
\(368\) 52.6573 2.74495
\(369\) 0.845357 0.0440075
\(370\) 47.3556 2.46190
\(371\) −3.49333 −0.181364
\(372\) −34.9460 −1.81187
\(373\) 22.6497 1.17276 0.586379 0.810037i \(-0.300553\pi\)
0.586379 + 0.810037i \(0.300553\pi\)
\(374\) −65.4759 −3.38568
\(375\) 25.0717 1.29470
\(376\) 33.8181 1.74403
\(377\) −0.517013 −0.0266275
\(378\) 30.4816 1.56780
\(379\) 24.6538 1.26638 0.633191 0.773996i \(-0.281745\pi\)
0.633191 + 0.773996i \(0.281745\pi\)
\(380\) −14.7296 −0.755611
\(381\) −24.9878 −1.28016
\(382\) 62.3790 3.19159
\(383\) 10.6370 0.543526 0.271763 0.962364i \(-0.412393\pi\)
0.271763 + 0.962364i \(0.412393\pi\)
\(384\) 6.65808 0.339769
\(385\) −21.0103 −1.07078
\(386\) −17.1687 −0.873864
\(387\) −9.02171 −0.458599
\(388\) −2.97025 −0.150791
\(389\) 25.0086 1.26799 0.633994 0.773338i \(-0.281414\pi\)
0.633994 + 0.773338i \(0.281414\pi\)
\(390\) −9.06990 −0.459272
\(391\) 34.1492 1.72700
\(392\) −33.5749 −1.69579
\(393\) 2.79190 0.140833
\(394\) 3.63971 0.183366
\(395\) 14.4649 0.727808
\(396\) 24.9397 1.25327
\(397\) −13.4520 −0.675139 −0.337569 0.941301i \(-0.609605\pi\)
−0.337569 + 0.941301i \(0.609605\pi\)
\(398\) −49.6695 −2.48971
\(399\) −12.7965 −0.640628
\(400\) −23.1207 −1.15603
\(401\) −20.6387 −1.03065 −0.515324 0.856995i \(-0.672328\pi\)
−0.515324 + 0.856995i \(0.672328\pi\)
\(402\) 15.0019 0.748226
\(403\) −3.41712 −0.170219
\(404\) 74.0444 3.68385
\(405\) −18.5799 −0.923241
\(406\) 4.58687 0.227642
\(407\) −40.9587 −2.03025
\(408\) 105.425 5.21932
\(409\) 21.0978 1.04322 0.521610 0.853184i \(-0.325332\pi\)
0.521610 + 0.853184i \(0.325332\pi\)
\(410\) −2.71543 −0.134105
\(411\) 27.0479 1.33417
\(412\) −35.7602 −1.76178
\(413\) −2.98769 −0.147015
\(414\) −18.3149 −0.900129
\(415\) −5.47169 −0.268594
\(416\) 11.6081 0.569133
\(417\) −2.52987 −0.123888
\(418\) 17.9382 0.877387
\(419\) 25.4740 1.24449 0.622244 0.782823i \(-0.286221\pi\)
0.622244 + 0.782823i \(0.286221\pi\)
\(420\) 57.1482 2.78855
\(421\) −20.7905 −1.01327 −0.506635 0.862161i \(-0.669111\pi\)
−0.506635 + 0.862161i \(0.669111\pi\)
\(422\) −59.6367 −2.90307
\(423\) −6.00414 −0.291931
\(424\) −7.88476 −0.382918
\(425\) −14.9942 −0.727324
\(426\) 18.0251 0.873320
\(427\) 5.40935 0.261777
\(428\) −53.0895 −2.56618
\(429\) 7.84471 0.378746
\(430\) 28.9792 1.39750
\(431\) 40.8120 1.96585 0.982923 0.184018i \(-0.0589105\pi\)
0.982923 + 0.184018i \(0.0589105\pi\)
\(432\) 35.1191 1.68967
\(433\) −8.84111 −0.424877 −0.212438 0.977174i \(-0.568140\pi\)
−0.212438 + 0.977174i \(0.568140\pi\)
\(434\) 30.3162 1.45522
\(435\) −1.78498 −0.0855832
\(436\) 9.35516 0.448031
\(437\) −9.35575 −0.447546
\(438\) −49.4657 −2.36356
\(439\) −34.2338 −1.63389 −0.816945 0.576716i \(-0.804334\pi\)
−0.816945 + 0.576716i \(0.804334\pi\)
\(440\) −47.4222 −2.26077
\(441\) 5.96098 0.283856
\(442\) 17.4146 0.828328
\(443\) 0.0486450 0.00231120 0.00115560 0.999999i \(-0.499632\pi\)
0.00115560 + 0.999999i \(0.499632\pi\)
\(444\) 111.408 5.28718
\(445\) −0.969229 −0.0459459
\(446\) −68.0318 −3.22140
\(447\) −12.5651 −0.594307
\(448\) −33.9465 −1.60382
\(449\) −1.90081 −0.0897046 −0.0448523 0.998994i \(-0.514282\pi\)
−0.0448523 + 0.998994i \(0.514282\pi\)
\(450\) 8.04168 0.379088
\(451\) 2.34862 0.110592
\(452\) 6.81961 0.320767
\(453\) 19.0558 0.895319
\(454\) 12.1810 0.571683
\(455\) 5.58811 0.261975
\(456\) −28.8830 −1.35257
\(457\) −12.8247 −0.599913 −0.299956 0.953953i \(-0.596972\pi\)
−0.299956 + 0.953953i \(0.596972\pi\)
\(458\) 70.6229 3.29999
\(459\) 22.7754 1.06306
\(460\) 41.7819 1.94809
\(461\) −36.9887 −1.72273 −0.861367 0.507983i \(-0.830391\pi\)
−0.861367 + 0.507983i \(0.830391\pi\)
\(462\) −69.5972 −3.23795
\(463\) 6.57372 0.305507 0.152753 0.988264i \(-0.451186\pi\)
0.152753 + 0.988264i \(0.451186\pi\)
\(464\) 5.28472 0.245337
\(465\) −11.7975 −0.547098
\(466\) 72.6642 3.36611
\(467\) 30.1017 1.39294 0.696470 0.717586i \(-0.254753\pi\)
0.696470 + 0.717586i \(0.254753\pi\)
\(468\) −6.63320 −0.306620
\(469\) −9.24290 −0.426797
\(470\) 19.2863 0.889611
\(471\) 7.98909 0.368118
\(472\) −6.74349 −0.310394
\(473\) −25.0646 −1.15247
\(474\) 47.9153 2.20082
\(475\) 4.10790 0.188484
\(476\) −109.727 −5.02933
\(477\) 1.39988 0.0640961
\(478\) 15.2131 0.695832
\(479\) 32.2322 1.47273 0.736363 0.676587i \(-0.236542\pi\)
0.736363 + 0.676587i \(0.236542\pi\)
\(480\) 40.0767 1.82924
\(481\) 10.8938 0.496713
\(482\) −45.5563 −2.07503
\(483\) 36.2987 1.65165
\(484\) 15.3725 0.698748
\(485\) −1.00274 −0.0455319
\(486\) −34.4683 −1.56352
\(487\) 30.5388 1.38385 0.691923 0.721971i \(-0.256764\pi\)
0.691923 + 0.721971i \(0.256764\pi\)
\(488\) 12.2094 0.552694
\(489\) 17.0271 0.769990
\(490\) −19.1477 −0.865003
\(491\) −16.8384 −0.759904 −0.379952 0.925006i \(-0.624060\pi\)
−0.379952 + 0.925006i \(0.624060\pi\)
\(492\) −6.38826 −0.288005
\(493\) 3.42723 0.154355
\(494\) −4.77102 −0.214658
\(495\) 8.41947 0.378427
\(496\) 34.9285 1.56834
\(497\) −11.1056 −0.498153
\(498\) −18.1251 −0.812205
\(499\) −27.7556 −1.24251 −0.621255 0.783608i \(-0.713377\pi\)
−0.621255 + 0.783608i \(0.713377\pi\)
\(500\) −58.8982 −2.63401
\(501\) 20.3904 0.910977
\(502\) −30.0886 −1.34292
\(503\) −35.7069 −1.59209 −0.796045 0.605238i \(-0.793078\pi\)
−0.796045 + 0.605238i \(0.793078\pi\)
\(504\) 34.8362 1.55173
\(505\) 24.9969 1.11235
\(506\) −50.8836 −2.26205
\(507\) −2.08646 −0.0926628
\(508\) 58.7012 2.60444
\(509\) −20.8992 −0.926339 −0.463170 0.886270i \(-0.653288\pi\)
−0.463170 + 0.886270i \(0.653288\pi\)
\(510\) 60.1236 2.66232
\(511\) 30.4766 1.34821
\(512\) 37.1734 1.64285
\(513\) −6.23969 −0.275489
\(514\) 4.26908 0.188301
\(515\) −12.0724 −0.531974
\(516\) 68.1760 3.00128
\(517\) −16.6811 −0.733632
\(518\) −96.6480 −4.24647
\(519\) −5.69715 −0.250077
\(520\) 12.6129 0.553112
\(521\) −11.1389 −0.488004 −0.244002 0.969775i \(-0.578460\pi\)
−0.244002 + 0.969775i \(0.578460\pi\)
\(522\) −1.83810 −0.0804513
\(523\) −1.61469 −0.0706053 −0.0353027 0.999377i \(-0.511240\pi\)
−0.0353027 + 0.999377i \(0.511240\pi\)
\(524\) −6.55870 −0.286518
\(525\) −15.9380 −0.695590
\(526\) −51.3707 −2.23987
\(527\) 22.6518 0.986727
\(528\) −80.1857 −3.48964
\(529\) 3.53855 0.153850
\(530\) −4.49665 −0.195322
\(531\) 1.19726 0.0519565
\(532\) 30.0616 1.30333
\(533\) −0.624662 −0.0270571
\(534\) −3.21059 −0.138936
\(535\) −17.9227 −0.774864
\(536\) −20.8621 −0.901104
\(537\) −18.7653 −0.809783
\(538\) −35.5695 −1.53351
\(539\) 16.5611 0.713339
\(540\) 27.8659 1.19916
\(541\) −1.46182 −0.0628483 −0.0314242 0.999506i \(-0.510004\pi\)
−0.0314242 + 0.999506i \(0.510004\pi\)
\(542\) 30.1529 1.29518
\(543\) 16.9611 0.727869
\(544\) −76.9490 −3.29916
\(545\) 3.15824 0.135284
\(546\) 18.5107 0.792187
\(547\) 40.2961 1.72294 0.861468 0.507811i \(-0.169545\pi\)
0.861468 + 0.507811i \(0.169545\pi\)
\(548\) −63.5407 −2.71432
\(549\) −2.16769 −0.0925147
\(550\) 22.3419 0.952660
\(551\) −0.938949 −0.0400006
\(552\) 81.9296 3.48715
\(553\) −29.5214 −1.25538
\(554\) 47.4602 2.01639
\(555\) 37.6105 1.59648
\(556\) 5.94316 0.252046
\(557\) −23.2093 −0.983411 −0.491706 0.870762i \(-0.663626\pi\)
−0.491706 + 0.870762i \(0.663626\pi\)
\(558\) −12.1486 −0.514292
\(559\) 6.66643 0.281960
\(560\) −57.1196 −2.41374
\(561\) −52.0019 −2.19552
\(562\) −14.8895 −0.628077
\(563\) −21.5878 −0.909817 −0.454909 0.890538i \(-0.650328\pi\)
−0.454909 + 0.890538i \(0.650328\pi\)
\(564\) 45.3726 1.91053
\(565\) 2.30225 0.0968566
\(566\) −22.0735 −0.927819
\(567\) 37.9196 1.59247
\(568\) −25.0663 −1.05176
\(569\) 30.9138 1.29597 0.647987 0.761651i \(-0.275611\pi\)
0.647987 + 0.761651i \(0.275611\pi\)
\(570\) −16.4719 −0.689931
\(571\) −11.9781 −0.501267 −0.250634 0.968082i \(-0.580639\pi\)
−0.250634 + 0.968082i \(0.580639\pi\)
\(572\) −18.4287 −0.770545
\(573\) 49.5423 2.06966
\(574\) 5.54191 0.231315
\(575\) −11.6525 −0.485942
\(576\) 13.6034 0.566808
\(577\) −38.9687 −1.62229 −0.811144 0.584846i \(-0.801155\pi\)
−0.811144 + 0.584846i \(0.801155\pi\)
\(578\) −70.7797 −2.94405
\(579\) −13.6356 −0.566678
\(580\) 4.19326 0.174116
\(581\) 11.1672 0.463292
\(582\) −3.32158 −0.137684
\(583\) 3.88923 0.161075
\(584\) 68.7885 2.84649
\(585\) −2.23932 −0.0925846
\(586\) −59.3620 −2.45222
\(587\) 44.1506 1.82229 0.911144 0.412087i \(-0.135200\pi\)
0.911144 + 0.412087i \(0.135200\pi\)
\(588\) −45.0464 −1.85768
\(589\) −6.20583 −0.255707
\(590\) −3.84579 −0.158329
\(591\) 2.89071 0.118908
\(592\) −111.352 −4.57654
\(593\) 17.0372 0.699635 0.349818 0.936818i \(-0.386244\pi\)
0.349818 + 0.936818i \(0.386244\pi\)
\(594\) −33.9361 −1.39242
\(595\) −37.0431 −1.51862
\(596\) 29.5178 1.20909
\(597\) −39.4482 −1.61451
\(598\) 13.5335 0.553426
\(599\) 14.1582 0.578489 0.289245 0.957255i \(-0.406596\pi\)
0.289245 + 0.957255i \(0.406596\pi\)
\(600\) −35.9735 −1.46861
\(601\) 38.8754 1.58576 0.792881 0.609377i \(-0.208580\pi\)
0.792881 + 0.609377i \(0.208580\pi\)
\(602\) −59.1437 −2.41052
\(603\) 3.70391 0.150835
\(604\) −44.7657 −1.82149
\(605\) 5.18964 0.210989
\(606\) 82.8028 3.36363
\(607\) −6.39023 −0.259371 −0.129686 0.991555i \(-0.541397\pi\)
−0.129686 + 0.991555i \(0.541397\pi\)
\(608\) 21.0815 0.854966
\(609\) 3.64296 0.147620
\(610\) 6.96298 0.281923
\(611\) 4.43666 0.179488
\(612\) 43.9709 1.77742
\(613\) −18.1835 −0.734424 −0.367212 0.930137i \(-0.619688\pi\)
−0.367212 + 0.930137i \(0.619688\pi\)
\(614\) 1.99526 0.0805221
\(615\) −2.15663 −0.0869639
\(616\) 96.7839 3.89954
\(617\) −21.3145 −0.858091 −0.429045 0.903283i \(-0.641150\pi\)
−0.429045 + 0.903283i \(0.641150\pi\)
\(618\) −39.9902 −1.60864
\(619\) −1.00000 −0.0401934
\(620\) 27.7147 1.11305
\(621\) 17.6995 0.710258
\(622\) 10.1734 0.407916
\(623\) 1.97810 0.0792508
\(624\) 21.3270 0.853763
\(625\) −8.57397 −0.342959
\(626\) 24.1539 0.965385
\(627\) 14.2468 0.568962
\(628\) −18.7679 −0.748921
\(629\) −72.2139 −2.87935
\(630\) 19.8670 0.791519
\(631\) −24.9335 −0.992588 −0.496294 0.868155i \(-0.665306\pi\)
−0.496294 + 0.868155i \(0.665306\pi\)
\(632\) −66.6325 −2.65050
\(633\) −47.3643 −1.88256
\(634\) −76.9067 −3.05436
\(635\) 19.8171 0.786419
\(636\) −10.5787 −0.419474
\(637\) −4.40476 −0.174523
\(638\) −5.10670 −0.202176
\(639\) 4.45033 0.176052
\(640\) −5.28034 −0.208724
\(641\) −34.9738 −1.38138 −0.690690 0.723151i \(-0.742693\pi\)
−0.690690 + 0.723151i \(0.742693\pi\)
\(642\) −59.3693 −2.34312
\(643\) 35.1186 1.38494 0.692471 0.721446i \(-0.256522\pi\)
0.692471 + 0.721446i \(0.256522\pi\)
\(644\) −85.2727 −3.36021
\(645\) 23.0157 0.906244
\(646\) 31.6267 1.24434
\(647\) −44.5409 −1.75108 −0.875542 0.483142i \(-0.839495\pi\)
−0.875542 + 0.483142i \(0.839495\pi\)
\(648\) 85.5881 3.36222
\(649\) 3.32629 0.130568
\(650\) −5.94226 −0.233075
\(651\) 24.0776 0.943674
\(652\) −39.9998 −1.56651
\(653\) 8.66687 0.339161 0.169580 0.985516i \(-0.445759\pi\)
0.169580 + 0.985516i \(0.445759\pi\)
\(654\) 10.4617 0.409087
\(655\) −2.21417 −0.0865149
\(656\) 6.38506 0.249295
\(657\) −12.2129 −0.476470
\(658\) −39.3614 −1.53447
\(659\) 26.5626 1.03473 0.517367 0.855764i \(-0.326912\pi\)
0.517367 + 0.855764i \(0.326912\pi\)
\(660\) −63.6249 −2.47660
\(661\) −27.4564 −1.06793 −0.533965 0.845506i \(-0.679299\pi\)
−0.533965 + 0.845506i \(0.679299\pi\)
\(662\) 63.1965 2.45620
\(663\) 13.8309 0.537149
\(664\) 25.2053 0.978155
\(665\) 10.1486 0.393545
\(666\) 38.7298 1.50075
\(667\) 2.66342 0.103128
\(668\) −47.9010 −1.85335
\(669\) −54.0318 −2.08899
\(670\) −11.8976 −0.459643
\(671\) −6.02240 −0.232492
\(672\) −81.7925 −3.15521
\(673\) 1.88361 0.0726079 0.0363039 0.999341i \(-0.488442\pi\)
0.0363039 + 0.999341i \(0.488442\pi\)
\(674\) 38.3755 1.47817
\(675\) −7.77147 −0.299124
\(676\) 4.90149 0.188519
\(677\) −1.75643 −0.0675052 −0.0337526 0.999430i \(-0.510746\pi\)
−0.0337526 + 0.999430i \(0.510746\pi\)
\(678\) 7.62627 0.292885
\(679\) 2.04648 0.0785367
\(680\) −83.6097 −3.20628
\(681\) 9.67433 0.370721
\(682\) −33.7520 −1.29243
\(683\) −21.4990 −0.822635 −0.411318 0.911492i \(-0.634931\pi\)
−0.411318 + 0.911492i \(0.634931\pi\)
\(684\) −12.0466 −0.460612
\(685\) −21.4509 −0.819597
\(686\) −23.0246 −0.879082
\(687\) 56.0898 2.13996
\(688\) −68.1418 −2.59788
\(689\) −1.03442 −0.0394082
\(690\) 46.7241 1.77876
\(691\) 37.6199 1.43113 0.715564 0.698547i \(-0.246170\pi\)
0.715564 + 0.698547i \(0.246170\pi\)
\(692\) 13.3837 0.508772
\(693\) −17.1833 −0.652738
\(694\) −16.8965 −0.641383
\(695\) 2.00637 0.0761060
\(696\) 8.22250 0.311673
\(697\) 4.14083 0.156845
\(698\) 19.2230 0.727602
\(699\) 57.7110 2.18283
\(700\) 37.4413 1.41515
\(701\) −14.4320 −0.545088 −0.272544 0.962143i \(-0.587865\pi\)
−0.272544 + 0.962143i \(0.587865\pi\)
\(702\) 9.02598 0.340664
\(703\) 19.7842 0.746175
\(704\) 37.7937 1.42440
\(705\) 15.3175 0.576890
\(706\) −17.8763 −0.672783
\(707\) −51.0161 −1.91866
\(708\) −9.04753 −0.340027
\(709\) −34.7602 −1.30545 −0.652723 0.757596i \(-0.726374\pi\)
−0.652723 + 0.757596i \(0.726374\pi\)
\(710\) −14.2952 −0.536490
\(711\) 11.8301 0.443664
\(712\) 4.46475 0.167324
\(713\) 17.6035 0.659256
\(714\) −122.706 −4.59216
\(715\) −6.22142 −0.232668
\(716\) 44.0834 1.64747
\(717\) 12.0825 0.451229
\(718\) 81.6200 3.04603
\(719\) −32.7986 −1.22318 −0.611590 0.791175i \(-0.709470\pi\)
−0.611590 + 0.791175i \(0.709470\pi\)
\(720\) 22.8895 0.853042
\(721\) 24.6386 0.917588
\(722\) 41.2496 1.53515
\(723\) −36.1815 −1.34560
\(724\) −39.8448 −1.48082
\(725\) −1.16945 −0.0434323
\(726\) 17.1908 0.638011
\(727\) 21.6122 0.801552 0.400776 0.916176i \(-0.368741\pi\)
0.400776 + 0.916176i \(0.368741\pi\)
\(728\) −25.7416 −0.954048
\(729\) 6.31018 0.233710
\(730\) 39.2298 1.45196
\(731\) −44.1912 −1.63447
\(732\) 16.3810 0.605458
\(733\) 2.48808 0.0918993 0.0459497 0.998944i \(-0.485369\pi\)
0.0459497 + 0.998944i \(0.485369\pi\)
\(734\) 35.4939 1.31010
\(735\) −15.2074 −0.560932
\(736\) −59.7997 −2.20425
\(737\) 10.2904 0.379052
\(738\) −2.22081 −0.0817492
\(739\) 43.0241 1.58267 0.791334 0.611384i \(-0.209387\pi\)
0.791334 + 0.611384i \(0.209387\pi\)
\(740\) −88.3544 −3.24797
\(741\) −3.78922 −0.139200
\(742\) 9.17721 0.336906
\(743\) 2.48717 0.0912455 0.0456227 0.998959i \(-0.485473\pi\)
0.0456227 + 0.998959i \(0.485473\pi\)
\(744\) 54.3453 1.99240
\(745\) 9.96500 0.365089
\(746\) −59.5024 −2.17854
\(747\) −4.47501 −0.163732
\(748\) 122.163 4.46670
\(749\) 36.5783 1.33654
\(750\) −65.8651 −2.40505
\(751\) −35.5407 −1.29690 −0.648448 0.761259i \(-0.724582\pi\)
−0.648448 + 0.761259i \(0.724582\pi\)
\(752\) −45.3499 −1.65374
\(753\) −23.8968 −0.870848
\(754\) 1.35823 0.0494638
\(755\) −15.1126 −0.550004
\(756\) −56.8715 −2.06840
\(757\) −24.7209 −0.898495 −0.449248 0.893407i \(-0.648308\pi\)
−0.449248 + 0.893407i \(0.648308\pi\)
\(758\) −64.7673 −2.35245
\(759\) −40.4125 −1.46688
\(760\) 22.9063 0.830898
\(761\) −3.23611 −0.117309 −0.0586544 0.998278i \(-0.518681\pi\)
−0.0586544 + 0.998278i \(0.518681\pi\)
\(762\) 65.6447 2.37806
\(763\) −6.44564 −0.233348
\(764\) −116.384 −4.21064
\(765\) 14.8443 0.536696
\(766\) −27.9442 −1.00966
\(767\) −0.884692 −0.0319444
\(768\) 24.4549 0.882438
\(769\) 37.7883 1.36268 0.681340 0.731967i \(-0.261398\pi\)
0.681340 + 0.731967i \(0.261398\pi\)
\(770\) 55.1955 1.98911
\(771\) 3.39056 0.122108
\(772\) 32.0328 1.15288
\(773\) −26.7897 −0.963557 −0.481779 0.876293i \(-0.660009\pi\)
−0.481779 + 0.876293i \(0.660009\pi\)
\(774\) 23.7006 0.851902
\(775\) −7.72930 −0.277645
\(776\) 4.61910 0.165816
\(777\) −76.7593 −2.75372
\(778\) −65.6994 −2.35544
\(779\) −1.13445 −0.0406459
\(780\) 16.9223 0.605916
\(781\) 12.3642 0.442425
\(782\) −89.7123 −3.20811
\(783\) 1.77633 0.0634810
\(784\) 45.0238 1.60799
\(785\) −6.33592 −0.226139
\(786\) −7.33450 −0.261613
\(787\) −36.3767 −1.29669 −0.648345 0.761347i \(-0.724538\pi\)
−0.648345 + 0.761347i \(0.724538\pi\)
\(788\) −6.79084 −0.241914
\(789\) −40.7994 −1.45250
\(790\) −38.0003 −1.35199
\(791\) −4.69867 −0.167065
\(792\) −38.7842 −1.37814
\(793\) 1.60178 0.0568808
\(794\) 35.3395 1.25415
\(795\) −3.57131 −0.126661
\(796\) 92.6715 3.28465
\(797\) −27.3149 −0.967542 −0.483771 0.875195i \(-0.660733\pi\)
−0.483771 + 0.875195i \(0.660733\pi\)
\(798\) 33.6174 1.19004
\(799\) −29.4102 −1.04046
\(800\) 26.2567 0.928316
\(801\) −0.792683 −0.0280081
\(802\) 54.2193 1.91455
\(803\) −33.9306 −1.19738
\(804\) −27.9900 −0.987130
\(805\) −28.7875 −1.01463
\(806\) 8.97700 0.316202
\(807\) −28.2498 −0.994441
\(808\) −115.148 −4.05090
\(809\) 4.73446 0.166455 0.0832273 0.996531i \(-0.473477\pi\)
0.0832273 + 0.996531i \(0.473477\pi\)
\(810\) 48.8106 1.71503
\(811\) −12.6856 −0.445451 −0.222725 0.974881i \(-0.571495\pi\)
−0.222725 + 0.974881i \(0.571495\pi\)
\(812\) −8.55801 −0.300327
\(813\) 23.9479 0.839889
\(814\) 107.601 3.77142
\(815\) −13.5037 −0.473013
\(816\) −141.375 −4.94910
\(817\) 12.1069 0.423568
\(818\) −55.4254 −1.93790
\(819\) 4.57023 0.159697
\(820\) 5.06635 0.176925
\(821\) 29.6546 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(822\) −71.0566 −2.47838
\(823\) 49.1076 1.71178 0.855892 0.517155i \(-0.173009\pi\)
0.855892 + 0.517155i \(0.173009\pi\)
\(824\) 55.6115 1.93732
\(825\) 17.7442 0.617775
\(826\) 7.84886 0.273097
\(827\) 10.5392 0.366484 0.183242 0.983068i \(-0.441341\pi\)
0.183242 + 0.983068i \(0.441341\pi\)
\(828\) 34.1713 1.18754
\(829\) −13.2329 −0.459599 −0.229799 0.973238i \(-0.573807\pi\)
−0.229799 + 0.973238i \(0.573807\pi\)
\(830\) 14.3745 0.498946
\(831\) 37.6936 1.30758
\(832\) −10.0520 −0.348490
\(833\) 29.1988 1.01168
\(834\) 6.64615 0.230137
\(835\) −16.1711 −0.559623
\(836\) −33.4685 −1.15753
\(837\) 11.7404 0.405808
\(838\) −66.9220 −2.31178
\(839\) −2.75841 −0.0952309 −0.0476155 0.998866i \(-0.515162\pi\)
−0.0476155 + 0.998866i \(0.515162\pi\)
\(840\) −88.8724 −3.06639
\(841\) −28.7327 −0.990783
\(842\) 54.6182 1.88227
\(843\) −11.8255 −0.407291
\(844\) 111.268 3.83000
\(845\) 1.65471 0.0569237
\(846\) 15.7733 0.542297
\(847\) −10.5915 −0.363929
\(848\) 10.5734 0.363093
\(849\) −17.5311 −0.601667
\(850\) 39.3907 1.35109
\(851\) −56.1199 −1.92377
\(852\) −33.6306 −1.15217
\(853\) −40.7699 −1.39594 −0.697968 0.716129i \(-0.745912\pi\)
−0.697968 + 0.716129i \(0.745912\pi\)
\(854\) −14.2107 −0.486281
\(855\) −4.06684 −0.139083
\(856\) 82.5607 2.82187
\(857\) −38.8404 −1.32676 −0.663381 0.748282i \(-0.730879\pi\)
−0.663381 + 0.748282i \(0.730879\pi\)
\(858\) −20.6086 −0.703566
\(859\) −14.1524 −0.482874 −0.241437 0.970416i \(-0.577619\pi\)
−0.241437 + 0.970416i \(0.577619\pi\)
\(860\) −54.0684 −1.84372
\(861\) 4.40147 0.150002
\(862\) −107.216 −3.65179
\(863\) −41.6433 −1.41755 −0.708777 0.705433i \(-0.750753\pi\)
−0.708777 + 0.705433i \(0.750753\pi\)
\(864\) −39.8826 −1.35683
\(865\) 4.51825 0.153625
\(866\) 23.2262 0.789258
\(867\) −56.2143 −1.90914
\(868\) −56.5629 −1.91987
\(869\) 32.8671 1.11494
\(870\) 4.68926 0.158981
\(871\) −2.73694 −0.0927376
\(872\) −14.5484 −0.492672
\(873\) −0.820086 −0.0277557
\(874\) 24.5782 0.831370
\(875\) 40.5805 1.37187
\(876\) 92.2913 3.11824
\(877\) −8.89547 −0.300379 −0.150189 0.988657i \(-0.547988\pi\)
−0.150189 + 0.988657i \(0.547988\pi\)
\(878\) 89.9345 3.03514
\(879\) −47.1462 −1.59020
\(880\) 63.5930 2.14372
\(881\) 19.2090 0.647168 0.323584 0.946200i \(-0.395112\pi\)
0.323584 + 0.946200i \(0.395112\pi\)
\(882\) −15.6599 −0.527296
\(883\) −26.1960 −0.881567 −0.440783 0.897613i \(-0.645299\pi\)
−0.440783 + 0.897613i \(0.645299\pi\)
\(884\) −32.4915 −1.09281
\(885\) −3.05438 −0.102672
\(886\) −0.127794 −0.00429332
\(887\) −16.2576 −0.545876 −0.272938 0.962032i \(-0.587995\pi\)
−0.272938 + 0.962032i \(0.587995\pi\)
\(888\) −173.253 −5.81398
\(889\) −40.4447 −1.35647
\(890\) 2.54623 0.0853499
\(891\) −42.2171 −1.41433
\(892\) 126.931 4.24997
\(893\) 8.05742 0.269631
\(894\) 33.0093 1.10400
\(895\) 14.8822 0.497459
\(896\) 10.7766 0.360022
\(897\) 10.7485 0.358882
\(898\) 4.99355 0.166637
\(899\) 1.76670 0.0589226
\(900\) −15.0039 −0.500129
\(901\) 6.85706 0.228442
\(902\) −6.16999 −0.205438
\(903\) −46.9728 −1.56316
\(904\) −10.6053 −0.352728
\(905\) −13.4513 −0.447137
\(906\) −50.0608 −1.66316
\(907\) −22.1068 −0.734043 −0.367022 0.930212i \(-0.619623\pi\)
−0.367022 + 0.930212i \(0.619623\pi\)
\(908\) −22.7269 −0.754218
\(909\) 20.4437 0.678074
\(910\) −14.6803 −0.486649
\(911\) −4.77016 −0.158043 −0.0790213 0.996873i \(-0.525180\pi\)
−0.0790213 + 0.996873i \(0.525180\pi\)
\(912\) 38.7320 1.28254
\(913\) −12.4327 −0.411464
\(914\) 33.6913 1.11441
\(915\) 5.53010 0.182820
\(916\) −131.766 −4.35366
\(917\) 4.51890 0.149227
\(918\) −59.8324 −1.97476
\(919\) −32.1812 −1.06156 −0.530781 0.847509i \(-0.678101\pi\)
−0.530781 + 0.847509i \(0.678101\pi\)
\(920\) −64.9760 −2.14220
\(921\) 1.58467 0.0522165
\(922\) 97.1718 3.20018
\(923\) −3.28850 −0.108242
\(924\) 129.852 4.27181
\(925\) 24.6410 0.810192
\(926\) −17.2696 −0.567515
\(927\) −9.87341 −0.324285
\(928\) −6.00154 −0.197010
\(929\) −38.1206 −1.25070 −0.625348 0.780346i \(-0.715043\pi\)
−0.625348 + 0.780346i \(0.715043\pi\)
\(930\) 30.9929 1.01630
\(931\) −7.99950 −0.262173
\(932\) −135.574 −4.44088
\(933\) 8.07987 0.264523
\(934\) −79.0792 −2.58755
\(935\) 41.2412 1.34873
\(936\) 10.3154 0.337171
\(937\) 0.514006 0.0167918 0.00839592 0.999965i \(-0.497327\pi\)
0.00839592 + 0.999965i \(0.497327\pi\)
\(938\) 24.2817 0.792826
\(939\) 19.1834 0.626027
\(940\) −35.9837 −1.17366
\(941\) −34.1679 −1.11384 −0.556921 0.830565i \(-0.688017\pi\)
−0.556921 + 0.830565i \(0.688017\pi\)
\(942\) −20.9879 −0.683822
\(943\) 3.21798 0.104792
\(944\) 9.04299 0.294324
\(945\) −19.1994 −0.624557
\(946\) 65.8465 2.14085
\(947\) −24.7445 −0.804087 −0.402043 0.915621i \(-0.631700\pi\)
−0.402043 + 0.915621i \(0.631700\pi\)
\(948\) −89.3987 −2.90353
\(949\) 9.02450 0.292948
\(950\) −10.7917 −0.350130
\(951\) −61.0805 −1.98067
\(952\) 170.639 5.53044
\(953\) −35.4480 −1.14827 −0.574136 0.818760i \(-0.694662\pi\)
−0.574136 + 0.818760i \(0.694662\pi\)
\(954\) −3.67758 −0.119066
\(955\) −39.2906 −1.27141
\(956\) −28.3841 −0.918008
\(957\) −4.05582 −0.131106
\(958\) −84.6762 −2.73576
\(959\) 43.7791 1.41370
\(960\) −34.7043 −1.12008
\(961\) −19.3233 −0.623332
\(962\) −28.6187 −0.922704
\(963\) −14.6580 −0.472349
\(964\) 84.9974 2.73758
\(965\) 10.8140 0.348116
\(966\) −95.3592 −3.06813
\(967\) −31.1278 −1.00100 −0.500502 0.865736i \(-0.666851\pi\)
−0.500502 + 0.865736i \(0.666851\pi\)
\(968\) −23.9061 −0.768370
\(969\) 25.1184 0.806919
\(970\) 2.63425 0.0845808
\(971\) −13.3557 −0.428605 −0.214303 0.976767i \(-0.568748\pi\)
−0.214303 + 0.976767i \(0.568748\pi\)
\(972\) 64.3098 2.06274
\(973\) −4.09480 −0.131273
\(974\) −80.2276 −2.57066
\(975\) −4.71943 −0.151143
\(976\) −16.3728 −0.524079
\(977\) 30.8552 0.987145 0.493572 0.869705i \(-0.335691\pi\)
0.493572 + 0.869705i \(0.335691\pi\)
\(978\) −44.7312 −1.43035
\(979\) −2.20228 −0.0703851
\(980\) 35.7250 1.14119
\(981\) 2.58296 0.0824677
\(982\) 44.2355 1.41161
\(983\) 8.65416 0.276025 0.138012 0.990430i \(-0.455929\pi\)
0.138012 + 0.990430i \(0.455929\pi\)
\(984\) 9.93453 0.316701
\(985\) −2.29254 −0.0730464
\(986\) −9.00358 −0.286732
\(987\) −31.2614 −0.995062
\(988\) 8.90160 0.283198
\(989\) −34.3425 −1.09203
\(990\) −22.1185 −0.702973
\(991\) 18.1441 0.576367 0.288184 0.957575i \(-0.406949\pi\)
0.288184 + 0.957575i \(0.406949\pi\)
\(992\) −39.6662 −1.25940
\(993\) 50.1916 1.59278
\(994\) 29.1751 0.925377
\(995\) 31.2853 0.991810
\(996\) 33.8171 1.07154
\(997\) 39.7778 1.25977 0.629887 0.776686i \(-0.283101\pi\)
0.629887 + 0.776686i \(0.283101\pi\)
\(998\) 72.9158 2.30811
\(999\) −37.4284 −1.18418
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 8047.2.a.c.1.10 151
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8047.2.a.c.1.10 151 1.1 even 1 trivial