Properties

Label 8043.2.a.p.1.17
Level $8043$
Weight $2$
Character 8043.1
Self dual yes
Analytic conductor $64.224$
Analytic rank $0$
Dimension $41$
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] = [8043,2,Mod(1,8043)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8043, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8043.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8043.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: \(64.2236783457\)
Analytic rank: \(0\)
Dimension: \(41\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Character \(\chi\) \(=\) 8043.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.306328 q^{2} +1.00000 q^{3} -1.90616 q^{4} -2.43700 q^{5} -0.306328 q^{6} -1.00000 q^{7} +1.19657 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.306328 q^{2} +1.00000 q^{3} -1.90616 q^{4} -2.43700 q^{5} -0.306328 q^{6} -1.00000 q^{7} +1.19657 q^{8} +1.00000 q^{9} +0.746523 q^{10} +2.36218 q^{11} -1.90616 q^{12} +2.69040 q^{13} +0.306328 q^{14} -2.43700 q^{15} +3.44578 q^{16} +0.171352 q^{17} -0.306328 q^{18} +2.51953 q^{19} +4.64533 q^{20} -1.00000 q^{21} -0.723603 q^{22} +2.11753 q^{23} +1.19657 q^{24} +0.938987 q^{25} -0.824146 q^{26} +1.00000 q^{27} +1.90616 q^{28} +7.92009 q^{29} +0.746523 q^{30} +3.29881 q^{31} -3.44868 q^{32} +2.36218 q^{33} -0.0524900 q^{34} +2.43700 q^{35} -1.90616 q^{36} -2.37894 q^{37} -0.771802 q^{38} +2.69040 q^{39} -2.91604 q^{40} +3.38192 q^{41} +0.306328 q^{42} +2.22643 q^{43} -4.50270 q^{44} -2.43700 q^{45} -0.648658 q^{46} -0.738129 q^{47} +3.44578 q^{48} +1.00000 q^{49} -0.287638 q^{50} +0.171352 q^{51} -5.12834 q^{52} -9.97324 q^{53} -0.306328 q^{54} -5.75664 q^{55} -1.19657 q^{56} +2.51953 q^{57} -2.42615 q^{58} -13.5075 q^{59} +4.64533 q^{60} -4.21286 q^{61} -1.01052 q^{62} -1.00000 q^{63} -5.83514 q^{64} -6.55652 q^{65} -0.723603 q^{66} -10.6261 q^{67} -0.326625 q^{68} +2.11753 q^{69} -0.746523 q^{70} +0.960879 q^{71} +1.19657 q^{72} -6.48583 q^{73} +0.728737 q^{74} +0.938987 q^{75} -4.80263 q^{76} -2.36218 q^{77} -0.824146 q^{78} -0.640875 q^{79} -8.39739 q^{80} +1.00000 q^{81} -1.03598 q^{82} -2.87557 q^{83} +1.90616 q^{84} -0.417585 q^{85} -0.682019 q^{86} +7.92009 q^{87} +2.82651 q^{88} +6.58516 q^{89} +0.746523 q^{90} -2.69040 q^{91} -4.03635 q^{92} +3.29881 q^{93} +0.226110 q^{94} -6.14010 q^{95} -3.44868 q^{96} +5.45203 q^{97} -0.306328 q^{98} +2.36218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9} + 18 q^{10} + 8 q^{11} + 45 q^{12} + 23 q^{13} - 7 q^{14} + 17 q^{15} + 37 q^{16} + 15 q^{17} + 7 q^{18} + 15 q^{19} + 53 q^{20} - 41 q^{21} + 13 q^{22} + 44 q^{23} + 12 q^{24} + 58 q^{25} + 9 q^{26} + 41 q^{27} - 45 q^{28} + 21 q^{29} + 18 q^{30} + 39 q^{31} + 61 q^{32} + 8 q^{33} + 9 q^{34} - 17 q^{35} + 45 q^{36} + 11 q^{37} + 44 q^{38} + 23 q^{39} + 24 q^{40} + 17 q^{41} - 7 q^{42} + 7 q^{43} + 30 q^{44} + 17 q^{45} - 12 q^{46} + 36 q^{47} + 37 q^{48} + 41 q^{49} + 28 q^{50} + 15 q^{51} + 58 q^{52} + 26 q^{53} + 7 q^{54} + 32 q^{55} - 12 q^{56} + 15 q^{57} - 4 q^{58} + 33 q^{59} + 53 q^{60} + 59 q^{61} - q^{62} - 41 q^{63} + 16 q^{64} + 72 q^{65} + 13 q^{66} + 12 q^{67} + 52 q^{68} + 44 q^{69} - 18 q^{70} + 33 q^{71} + 12 q^{72} + 18 q^{73} + 42 q^{74} + 58 q^{75} + 7 q^{76} - 8 q^{77} + 9 q^{78} + 22 q^{79} + 69 q^{80} + 41 q^{81} + 41 q^{82} + 32 q^{83} - 45 q^{84} - 44 q^{85} + 11 q^{86} + 21 q^{87} + 52 q^{88} + 63 q^{89} + 18 q^{90} - 23 q^{91} + 52 q^{92} + 39 q^{93} + 17 q^{94} + 37 q^{95} + 61 q^{96} + 8 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.306328 −0.216607 −0.108303 0.994118i \(-0.534542\pi\)
−0.108303 + 0.994118i \(0.534542\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.90616 −0.953081
\(5\) −2.43700 −1.08986 −0.544931 0.838481i \(-0.683444\pi\)
−0.544931 + 0.838481i \(0.683444\pi\)
\(6\) −0.306328 −0.125058
\(7\) −1.00000 −0.377964
\(8\) 1.19657 0.423051
\(9\) 1.00000 0.333333
\(10\) 0.746523 0.236071
\(11\) 2.36218 0.712224 0.356112 0.934443i \(-0.384102\pi\)
0.356112 + 0.934443i \(0.384102\pi\)
\(12\) −1.90616 −0.550262
\(13\) 2.69040 0.746183 0.373091 0.927795i \(-0.378298\pi\)
0.373091 + 0.927795i \(0.378298\pi\)
\(14\) 0.306328 0.0818697
\(15\) −2.43700 −0.629232
\(16\) 3.44578 0.861446
\(17\) 0.171352 0.0415590 0.0207795 0.999784i \(-0.493385\pi\)
0.0207795 + 0.999784i \(0.493385\pi\)
\(18\) −0.306328 −0.0722023
\(19\) 2.51953 0.578019 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(20\) 4.64533 1.03873
\(21\) −1.00000 −0.218218
\(22\) −0.723603 −0.154273
\(23\) 2.11753 0.441535 0.220767 0.975327i \(-0.429144\pi\)
0.220767 + 0.975327i \(0.429144\pi\)
\(24\) 1.19657 0.244248
\(25\) 0.938987 0.187797
\(26\) −0.824146 −0.161628
\(27\) 1.00000 0.192450
\(28\) 1.90616 0.360231
\(29\) 7.92009 1.47072 0.735362 0.677674i \(-0.237012\pi\)
0.735362 + 0.677674i \(0.237012\pi\)
\(30\) 0.746523 0.136296
\(31\) 3.29881 0.592484 0.296242 0.955113i \(-0.404266\pi\)
0.296242 + 0.955113i \(0.404266\pi\)
\(32\) −3.44868 −0.609646
\(33\) 2.36218 0.411203
\(34\) −0.0524900 −0.00900195
\(35\) 2.43700 0.411929
\(36\) −1.90616 −0.317694
\(37\) −2.37894 −0.391096 −0.195548 0.980694i \(-0.562648\pi\)
−0.195548 + 0.980694i \(0.562648\pi\)
\(38\) −0.771802 −0.125203
\(39\) 2.69040 0.430809
\(40\) −2.91604 −0.461067
\(41\) 3.38192 0.528167 0.264083 0.964500i \(-0.414931\pi\)
0.264083 + 0.964500i \(0.414931\pi\)
\(42\) 0.306328 0.0472675
\(43\) 2.22643 0.339528 0.169764 0.985485i \(-0.445700\pi\)
0.169764 + 0.985485i \(0.445700\pi\)
\(44\) −4.50270 −0.678808
\(45\) −2.43700 −0.363287
\(46\) −0.648658 −0.0956394
\(47\) −0.738129 −0.107667 −0.0538336 0.998550i \(-0.517144\pi\)
−0.0538336 + 0.998550i \(0.517144\pi\)
\(48\) 3.44578 0.497356
\(49\) 1.00000 0.142857
\(50\) −0.287638 −0.0406782
\(51\) 0.171352 0.0239941
\(52\) −5.12834 −0.711173
\(53\) −9.97324 −1.36993 −0.684965 0.728576i \(-0.740183\pi\)
−0.684965 + 0.728576i \(0.740183\pi\)
\(54\) −0.306328 −0.0416860
\(55\) −5.75664 −0.776226
\(56\) −1.19657 −0.159898
\(57\) 2.51953 0.333720
\(58\) −2.42615 −0.318569
\(59\) −13.5075 −1.75852 −0.879261 0.476340i \(-0.841963\pi\)
−0.879261 + 0.476340i \(0.841963\pi\)
\(60\) 4.64533 0.599709
\(61\) −4.21286 −0.539401 −0.269700 0.962944i \(-0.586925\pi\)
−0.269700 + 0.962944i \(0.586925\pi\)
\(62\) −1.01052 −0.128336
\(63\) −1.00000 −0.125988
\(64\) −5.83514 −0.729392
\(65\) −6.55652 −0.813236
\(66\) −0.723603 −0.0890693
\(67\) −10.6261 −1.29819 −0.649093 0.760709i \(-0.724851\pi\)
−0.649093 + 0.760709i \(0.724851\pi\)
\(68\) −0.326625 −0.0396091
\(69\) 2.11753 0.254920
\(70\) −0.746523 −0.0892266
\(71\) 0.960879 0.114035 0.0570177 0.998373i \(-0.481841\pi\)
0.0570177 + 0.998373i \(0.481841\pi\)
\(72\) 1.19657 0.141017
\(73\) −6.48583 −0.759109 −0.379554 0.925169i \(-0.623923\pi\)
−0.379554 + 0.925169i \(0.623923\pi\)
\(74\) 0.728737 0.0847140
\(75\) 0.938987 0.108425
\(76\) −4.80263 −0.550899
\(77\) −2.36218 −0.269195
\(78\) −0.824146 −0.0933161
\(79\) −0.640875 −0.0721041 −0.0360520 0.999350i \(-0.511478\pi\)
−0.0360520 + 0.999350i \(0.511478\pi\)
\(80\) −8.39739 −0.938856
\(81\) 1.00000 0.111111
\(82\) −1.03598 −0.114405
\(83\) −2.87557 −0.315634 −0.157817 0.987468i \(-0.550446\pi\)
−0.157817 + 0.987468i \(0.550446\pi\)
\(84\) 1.90616 0.207979
\(85\) −0.417585 −0.0452935
\(86\) −0.682019 −0.0735440
\(87\) 7.92009 0.849123
\(88\) 2.82651 0.301307
\(89\) 6.58516 0.698026 0.349013 0.937118i \(-0.386517\pi\)
0.349013 + 0.937118i \(0.386517\pi\)
\(90\) 0.746523 0.0786905
\(91\) −2.69040 −0.282031
\(92\) −4.03635 −0.420818
\(93\) 3.29881 0.342071
\(94\) 0.226110 0.0233214
\(95\) −6.14010 −0.629961
\(96\) −3.44868 −0.351979
\(97\) 5.45203 0.553570 0.276785 0.960932i \(-0.410731\pi\)
0.276785 + 0.960932i \(0.410731\pi\)
\(98\) −0.306328 −0.0309438
\(99\) 2.36218 0.237408
\(100\) −1.78986 −0.178986
\(101\) −9.41257 −0.936586 −0.468293 0.883573i \(-0.655131\pi\)
−0.468293 + 0.883573i \(0.655131\pi\)
\(102\) −0.0524900 −0.00519728
\(103\) 17.4495 1.71935 0.859673 0.510844i \(-0.170667\pi\)
0.859673 + 0.510844i \(0.170667\pi\)
\(104\) 3.21925 0.315673
\(105\) 2.43700 0.237827
\(106\) 3.05509 0.296736
\(107\) 13.7634 1.33056 0.665278 0.746596i \(-0.268313\pi\)
0.665278 + 0.746596i \(0.268313\pi\)
\(108\) −1.90616 −0.183421
\(109\) 6.73939 0.645516 0.322758 0.946481i \(-0.395390\pi\)
0.322758 + 0.946481i \(0.395390\pi\)
\(110\) 1.76342 0.168136
\(111\) −2.37894 −0.225799
\(112\) −3.44578 −0.325596
\(113\) −9.37588 −0.882009 −0.441005 0.897505i \(-0.645378\pi\)
−0.441005 + 0.897505i \(0.645378\pi\)
\(114\) −0.771802 −0.0722859
\(115\) −5.16042 −0.481211
\(116\) −15.0970 −1.40172
\(117\) 2.69040 0.248728
\(118\) 4.13772 0.380908
\(119\) −0.171352 −0.0157078
\(120\) −2.91604 −0.266197
\(121\) −5.42010 −0.492737
\(122\) 1.29052 0.116838
\(123\) 3.38192 0.304937
\(124\) −6.28808 −0.564686
\(125\) 9.89670 0.885188
\(126\) 0.306328 0.0272899
\(127\) −8.58514 −0.761808 −0.380904 0.924615i \(-0.624387\pi\)
−0.380904 + 0.924615i \(0.624387\pi\)
\(128\) 8.68482 0.767637
\(129\) 2.22643 0.196026
\(130\) 2.00845 0.176152
\(131\) 22.3408 1.95192 0.975962 0.217942i \(-0.0699345\pi\)
0.975962 + 0.217942i \(0.0699345\pi\)
\(132\) −4.50270 −0.391910
\(133\) −2.51953 −0.218471
\(134\) 3.25508 0.281196
\(135\) −2.43700 −0.209744
\(136\) 0.205034 0.0175816
\(137\) 11.0019 0.939956 0.469978 0.882678i \(-0.344262\pi\)
0.469978 + 0.882678i \(0.344262\pi\)
\(138\) −0.648658 −0.0552174
\(139\) −2.37063 −0.201075 −0.100537 0.994933i \(-0.532056\pi\)
−0.100537 + 0.994933i \(0.532056\pi\)
\(140\) −4.64533 −0.392602
\(141\) −0.738129 −0.0621617
\(142\) −0.294344 −0.0247008
\(143\) 6.35521 0.531450
\(144\) 3.44578 0.287149
\(145\) −19.3013 −1.60289
\(146\) 1.98679 0.164428
\(147\) 1.00000 0.0824786
\(148\) 4.53465 0.372746
\(149\) 22.1546 1.81497 0.907487 0.420081i \(-0.137998\pi\)
0.907487 + 0.420081i \(0.137998\pi\)
\(150\) −0.287638 −0.0234856
\(151\) 19.9741 1.62547 0.812734 0.582635i \(-0.197978\pi\)
0.812734 + 0.582635i \(0.197978\pi\)
\(152\) 3.01479 0.244531
\(153\) 0.171352 0.0138530
\(154\) 0.723603 0.0583096
\(155\) −8.03922 −0.645726
\(156\) −5.12834 −0.410596
\(157\) 2.43145 0.194050 0.0970252 0.995282i \(-0.469067\pi\)
0.0970252 + 0.995282i \(0.469067\pi\)
\(158\) 0.196318 0.0156182
\(159\) −9.97324 −0.790929
\(160\) 8.40444 0.664429
\(161\) −2.11753 −0.166884
\(162\) −0.306328 −0.0240674
\(163\) 20.6882 1.62043 0.810214 0.586134i \(-0.199351\pi\)
0.810214 + 0.586134i \(0.199351\pi\)
\(164\) −6.44649 −0.503386
\(165\) −5.75664 −0.448154
\(166\) 0.880867 0.0683685
\(167\) −2.73919 −0.211965 −0.105982 0.994368i \(-0.533799\pi\)
−0.105982 + 0.994368i \(0.533799\pi\)
\(168\) −1.19657 −0.0923172
\(169\) −5.76174 −0.443211
\(170\) 0.127918 0.00981088
\(171\) 2.51953 0.192673
\(172\) −4.24394 −0.323597
\(173\) 9.56588 0.727281 0.363640 0.931539i \(-0.381534\pi\)
0.363640 + 0.931539i \(0.381534\pi\)
\(174\) −2.42615 −0.183926
\(175\) −0.938987 −0.0709808
\(176\) 8.13956 0.613543
\(177\) −13.5075 −1.01528
\(178\) −2.01722 −0.151197
\(179\) 18.4310 1.37760 0.688800 0.724952i \(-0.258138\pi\)
0.688800 + 0.724952i \(0.258138\pi\)
\(180\) 4.64533 0.346242
\(181\) 15.9260 1.18377 0.591885 0.806023i \(-0.298384\pi\)
0.591885 + 0.806023i \(0.298384\pi\)
\(182\) 0.824146 0.0610898
\(183\) −4.21286 −0.311423
\(184\) 2.53376 0.186792
\(185\) 5.79749 0.426240
\(186\) −1.01052 −0.0740949
\(187\) 0.404764 0.0295993
\(188\) 1.40699 0.102616
\(189\) −1.00000 −0.0727393
\(190\) 1.88089 0.136454
\(191\) −18.5006 −1.33866 −0.669329 0.742966i \(-0.733419\pi\)
−0.669329 + 0.742966i \(0.733419\pi\)
\(192\) −5.83514 −0.421115
\(193\) 5.00974 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(194\) −1.67011 −0.119907
\(195\) −6.55652 −0.469522
\(196\) −1.90616 −0.136154
\(197\) −16.2211 −1.15570 −0.577852 0.816142i \(-0.696109\pi\)
−0.577852 + 0.816142i \(0.696109\pi\)
\(198\) −0.723603 −0.0514242
\(199\) −20.7295 −1.46948 −0.734738 0.678352i \(-0.762695\pi\)
−0.734738 + 0.678352i \(0.762695\pi\)
\(200\) 1.12356 0.0794479
\(201\) −10.6261 −0.749508
\(202\) 2.88334 0.202871
\(203\) −7.92009 −0.555881
\(204\) −0.326625 −0.0228683
\(205\) −8.24175 −0.575629
\(206\) −5.34526 −0.372422
\(207\) 2.11753 0.147178
\(208\) 9.27054 0.642796
\(209\) 5.95158 0.411679
\(210\) −0.746523 −0.0515150
\(211\) −19.6700 −1.35414 −0.677068 0.735921i \(-0.736750\pi\)
−0.677068 + 0.735921i \(0.736750\pi\)
\(212\) 19.0106 1.30565
\(213\) 0.960879 0.0658383
\(214\) −4.21611 −0.288207
\(215\) −5.42582 −0.370038
\(216\) 1.19657 0.0814162
\(217\) −3.29881 −0.223938
\(218\) −2.06447 −0.139823
\(219\) −6.48583 −0.438271
\(220\) 10.9731 0.739806
\(221\) 0.461006 0.0310106
\(222\) 0.728737 0.0489096
\(223\) 13.1389 0.879844 0.439922 0.898036i \(-0.355006\pi\)
0.439922 + 0.898036i \(0.355006\pi\)
\(224\) 3.44868 0.230424
\(225\) 0.938987 0.0625992
\(226\) 2.87210 0.191049
\(227\) 14.1801 0.941164 0.470582 0.882356i \(-0.344044\pi\)
0.470582 + 0.882356i \(0.344044\pi\)
\(228\) −4.80263 −0.318062
\(229\) −1.32939 −0.0878486 −0.0439243 0.999035i \(-0.513986\pi\)
−0.0439243 + 0.999035i \(0.513986\pi\)
\(230\) 1.58078 0.104234
\(231\) −2.36218 −0.155420
\(232\) 9.47693 0.622191
\(233\) 22.9228 1.50172 0.750860 0.660462i \(-0.229639\pi\)
0.750860 + 0.660462i \(0.229639\pi\)
\(234\) −0.824146 −0.0538761
\(235\) 1.79882 0.117342
\(236\) 25.7474 1.67601
\(237\) −0.640875 −0.0416293
\(238\) 0.0524900 0.00340242
\(239\) −10.9920 −0.711016 −0.355508 0.934673i \(-0.615692\pi\)
−0.355508 + 0.934673i \(0.615692\pi\)
\(240\) −8.39739 −0.542049
\(241\) 19.7582 1.27274 0.636370 0.771384i \(-0.280435\pi\)
0.636370 + 0.771384i \(0.280435\pi\)
\(242\) 1.66033 0.106730
\(243\) 1.00000 0.0641500
\(244\) 8.03039 0.514093
\(245\) −2.43700 −0.155694
\(246\) −1.03598 −0.0660515
\(247\) 6.77854 0.431308
\(248\) 3.94725 0.250651
\(249\) −2.87557 −0.182232
\(250\) −3.03164 −0.191738
\(251\) −20.1895 −1.27435 −0.637176 0.770718i \(-0.719898\pi\)
−0.637176 + 0.770718i \(0.719898\pi\)
\(252\) 1.90616 0.120077
\(253\) 5.00198 0.314472
\(254\) 2.62987 0.165013
\(255\) −0.417585 −0.0261502
\(256\) 9.00987 0.563117
\(257\) 14.1397 0.882013 0.441007 0.897504i \(-0.354622\pi\)
0.441007 + 0.897504i \(0.354622\pi\)
\(258\) −0.682019 −0.0424606
\(259\) 2.37894 0.147820
\(260\) 12.4978 0.775080
\(261\) 7.92009 0.490241
\(262\) −6.84361 −0.422800
\(263\) 18.6300 1.14878 0.574388 0.818583i \(-0.305240\pi\)
0.574388 + 0.818583i \(0.305240\pi\)
\(264\) 2.82651 0.173960
\(265\) 24.3048 1.49303
\(266\) 0.771802 0.0473222
\(267\) 6.58516 0.403005
\(268\) 20.2551 1.23728
\(269\) −13.2895 −0.810275 −0.405138 0.914256i \(-0.632776\pi\)
−0.405138 + 0.914256i \(0.632776\pi\)
\(270\) 0.746523 0.0454320
\(271\) −5.23080 −0.317749 −0.158874 0.987299i \(-0.550786\pi\)
−0.158874 + 0.987299i \(0.550786\pi\)
\(272\) 0.590442 0.0358008
\(273\) −2.69040 −0.162830
\(274\) −3.37020 −0.203601
\(275\) 2.21806 0.133754
\(276\) −4.03635 −0.242960
\(277\) 2.73078 0.164077 0.0820385 0.996629i \(-0.473857\pi\)
0.0820385 + 0.996629i \(0.473857\pi\)
\(278\) 0.726192 0.0435541
\(279\) 3.29881 0.197495
\(280\) 2.91604 0.174267
\(281\) −16.1088 −0.960968 −0.480484 0.877003i \(-0.659539\pi\)
−0.480484 + 0.877003i \(0.659539\pi\)
\(282\) 0.226110 0.0134646
\(283\) −19.9453 −1.18562 −0.592812 0.805341i \(-0.701982\pi\)
−0.592812 + 0.805341i \(0.701982\pi\)
\(284\) −1.83159 −0.108685
\(285\) −6.14010 −0.363708
\(286\) −1.94678 −0.115116
\(287\) −3.38192 −0.199628
\(288\) −3.44868 −0.203215
\(289\) −16.9706 −0.998273
\(290\) 5.91253 0.347196
\(291\) 5.45203 0.319604
\(292\) 12.3630 0.723492
\(293\) −2.08843 −0.122007 −0.0610037 0.998138i \(-0.519430\pi\)
−0.0610037 + 0.998138i \(0.519430\pi\)
\(294\) −0.306328 −0.0178654
\(295\) 32.9177 1.91654
\(296\) −2.84657 −0.165453
\(297\) 2.36218 0.137068
\(298\) −6.78657 −0.393136
\(299\) 5.69699 0.329466
\(300\) −1.78986 −0.103338
\(301\) −2.22643 −0.128329
\(302\) −6.11863 −0.352088
\(303\) −9.41257 −0.540738
\(304\) 8.68174 0.497932
\(305\) 10.2667 0.587872
\(306\) −0.0524900 −0.00300065
\(307\) −23.3529 −1.33282 −0.666409 0.745586i \(-0.732170\pi\)
−0.666409 + 0.745586i \(0.732170\pi\)
\(308\) 4.50270 0.256565
\(309\) 17.4495 0.992665
\(310\) 2.46264 0.139869
\(311\) 7.02184 0.398172 0.199086 0.979982i \(-0.436203\pi\)
0.199086 + 0.979982i \(0.436203\pi\)
\(312\) 3.21925 0.182254
\(313\) −8.12440 −0.459218 −0.229609 0.973283i \(-0.573745\pi\)
−0.229609 + 0.973283i \(0.573745\pi\)
\(314\) −0.744821 −0.0420327
\(315\) 2.43700 0.137310
\(316\) 1.22161 0.0687210
\(317\) −8.49255 −0.476989 −0.238495 0.971144i \(-0.576654\pi\)
−0.238495 + 0.971144i \(0.576654\pi\)
\(318\) 3.05509 0.171321
\(319\) 18.7087 1.04749
\(320\) 14.2203 0.794936
\(321\) 13.7634 0.768196
\(322\) 0.648658 0.0361483
\(323\) 0.431726 0.0240219
\(324\) −1.90616 −0.105898
\(325\) 2.52625 0.140131
\(326\) −6.33739 −0.350996
\(327\) 6.73939 0.372689
\(328\) 4.04670 0.223441
\(329\) 0.738129 0.0406944
\(330\) 1.76342 0.0970732
\(331\) −18.5559 −1.01992 −0.509962 0.860197i \(-0.670340\pi\)
−0.509962 + 0.860197i \(0.670340\pi\)
\(332\) 5.48130 0.300825
\(333\) −2.37894 −0.130365
\(334\) 0.839091 0.0459130
\(335\) 25.8959 1.41484
\(336\) −3.44578 −0.187983
\(337\) −7.71312 −0.420161 −0.210080 0.977684i \(-0.567373\pi\)
−0.210080 + 0.977684i \(0.567373\pi\)
\(338\) 1.76498 0.0960025
\(339\) −9.37588 −0.509228
\(340\) 0.795986 0.0431684
\(341\) 7.79239 0.421982
\(342\) −0.771802 −0.0417343
\(343\) −1.00000 −0.0539949
\(344\) 2.66408 0.143637
\(345\) −5.16042 −0.277828
\(346\) −2.93030 −0.157534
\(347\) 14.0342 0.753397 0.376699 0.926336i \(-0.377059\pi\)
0.376699 + 0.926336i \(0.377059\pi\)
\(348\) −15.0970 −0.809283
\(349\) 25.5369 1.36696 0.683479 0.729971i \(-0.260466\pi\)
0.683479 + 0.729971i \(0.260466\pi\)
\(350\) 0.287638 0.0153749
\(351\) 2.69040 0.143603
\(352\) −8.14640 −0.434205
\(353\) 22.3356 1.18881 0.594403 0.804167i \(-0.297388\pi\)
0.594403 + 0.804167i \(0.297388\pi\)
\(354\) 4.13772 0.219917
\(355\) −2.34167 −0.124283
\(356\) −12.5524 −0.665275
\(357\) −0.171352 −0.00906891
\(358\) −5.64594 −0.298397
\(359\) −11.3514 −0.599106 −0.299553 0.954080i \(-0.596837\pi\)
−0.299553 + 0.954080i \(0.596837\pi\)
\(360\) −2.91604 −0.153689
\(361\) −12.6520 −0.665894
\(362\) −4.87858 −0.256412
\(363\) −5.42010 −0.284482
\(364\) 5.12834 0.268798
\(365\) 15.8060 0.827323
\(366\) 1.29052 0.0674564
\(367\) 11.2450 0.586985 0.293492 0.955961i \(-0.405183\pi\)
0.293492 + 0.955961i \(0.405183\pi\)
\(368\) 7.29653 0.380358
\(369\) 3.38192 0.176056
\(370\) −1.77594 −0.0923265
\(371\) 9.97324 0.517785
\(372\) −6.28808 −0.326022
\(373\) 31.7766 1.64533 0.822665 0.568527i \(-0.192486\pi\)
0.822665 + 0.568527i \(0.192486\pi\)
\(374\) −0.123991 −0.00641141
\(375\) 9.89670 0.511064
\(376\) −0.883222 −0.0455487
\(377\) 21.3082 1.09743
\(378\) 0.306328 0.0157558
\(379\) 1.52851 0.0785143 0.0392572 0.999229i \(-0.487501\pi\)
0.0392572 + 0.999229i \(0.487501\pi\)
\(380\) 11.7040 0.600404
\(381\) −8.58514 −0.439830
\(382\) 5.66727 0.289963
\(383\) −1.00000 −0.0510976
\(384\) 8.68482 0.443195
\(385\) 5.75664 0.293386
\(386\) −1.53463 −0.0781104
\(387\) 2.22643 0.113176
\(388\) −10.3925 −0.527597
\(389\) −28.0586 −1.42263 −0.711315 0.702874i \(-0.751900\pi\)
−0.711315 + 0.702874i \(0.751900\pi\)
\(390\) 2.00845 0.101702
\(391\) 0.362842 0.0183497
\(392\) 1.19657 0.0604358
\(393\) 22.3408 1.12694
\(394\) 4.96897 0.250333
\(395\) 1.56181 0.0785834
\(396\) −4.50270 −0.226269
\(397\) −4.94249 −0.248057 −0.124028 0.992279i \(-0.539581\pi\)
−0.124028 + 0.992279i \(0.539581\pi\)
\(398\) 6.35003 0.318298
\(399\) −2.51953 −0.126134
\(400\) 3.23555 0.161777
\(401\) −27.2911 −1.36285 −0.681425 0.731888i \(-0.738639\pi\)
−0.681425 + 0.731888i \(0.738639\pi\)
\(402\) 3.25508 0.162348
\(403\) 8.87513 0.442102
\(404\) 17.9419 0.892643
\(405\) −2.43700 −0.121096
\(406\) 2.42615 0.120408
\(407\) −5.61949 −0.278548
\(408\) 0.205034 0.0101507
\(409\) −23.3549 −1.15483 −0.577413 0.816452i \(-0.695938\pi\)
−0.577413 + 0.816452i \(0.695938\pi\)
\(410\) 2.52468 0.124685
\(411\) 11.0019 0.542684
\(412\) −33.2615 −1.63868
\(413\) 13.5075 0.664659
\(414\) −0.648658 −0.0318798
\(415\) 7.00776 0.343998
\(416\) −9.27832 −0.454907
\(417\) −2.37063 −0.116090
\(418\) −1.82314 −0.0891725
\(419\) 33.3672 1.63009 0.815047 0.579394i \(-0.196711\pi\)
0.815047 + 0.579394i \(0.196711\pi\)
\(420\) −4.64533 −0.226669
\(421\) 13.4422 0.655135 0.327568 0.944828i \(-0.393771\pi\)
0.327568 + 0.944828i \(0.393771\pi\)
\(422\) 6.02546 0.293315
\(423\) −0.738129 −0.0358891
\(424\) −11.9337 −0.579550
\(425\) 0.160897 0.00780467
\(426\) −0.294344 −0.0142610
\(427\) 4.21286 0.203874
\(428\) −26.2352 −1.26813
\(429\) 6.35521 0.306833
\(430\) 1.66208 0.0801527
\(431\) 25.4599 1.22636 0.613180 0.789943i \(-0.289890\pi\)
0.613180 + 0.789943i \(0.289890\pi\)
\(432\) 3.44578 0.165785
\(433\) −5.92183 −0.284585 −0.142293 0.989825i \(-0.545447\pi\)
−0.142293 + 0.989825i \(0.545447\pi\)
\(434\) 1.01052 0.0485065
\(435\) −19.3013 −0.925426
\(436\) −12.8464 −0.615230
\(437\) 5.33516 0.255215
\(438\) 1.98679 0.0949326
\(439\) 30.7000 1.46523 0.732615 0.680643i \(-0.238300\pi\)
0.732615 + 0.680643i \(0.238300\pi\)
\(440\) −6.88822 −0.328383
\(441\) 1.00000 0.0476190
\(442\) −0.141219 −0.00671710
\(443\) 4.78673 0.227424 0.113712 0.993514i \(-0.463726\pi\)
0.113712 + 0.993514i \(0.463726\pi\)
\(444\) 4.53465 0.215205
\(445\) −16.0481 −0.760751
\(446\) −4.02481 −0.190580
\(447\) 22.1546 1.04788
\(448\) 5.83514 0.275684
\(449\) 3.21623 0.151783 0.0758917 0.997116i \(-0.475820\pi\)
0.0758917 + 0.997116i \(0.475820\pi\)
\(450\) −0.287638 −0.0135594
\(451\) 7.98870 0.376173
\(452\) 17.8720 0.840626
\(453\) 19.9741 0.938465
\(454\) −4.34375 −0.203863
\(455\) 6.55652 0.307374
\(456\) 3.01479 0.141180
\(457\) 24.4198 1.14231 0.571155 0.820843i \(-0.306496\pi\)
0.571155 + 0.820843i \(0.306496\pi\)
\(458\) 0.407230 0.0190286
\(459\) 0.171352 0.00799803
\(460\) 9.83660 0.458634
\(461\) −12.4350 −0.579154 −0.289577 0.957155i \(-0.593515\pi\)
−0.289577 + 0.957155i \(0.593515\pi\)
\(462\) 0.723603 0.0336650
\(463\) −5.14321 −0.239025 −0.119513 0.992833i \(-0.538133\pi\)
−0.119513 + 0.992833i \(0.538133\pi\)
\(464\) 27.2909 1.26695
\(465\) −8.03922 −0.372810
\(466\) −7.02189 −0.325283
\(467\) 4.02430 0.186222 0.0931111 0.995656i \(-0.470319\pi\)
0.0931111 + 0.995656i \(0.470319\pi\)
\(468\) −5.12834 −0.237058
\(469\) 10.6261 0.490668
\(470\) −0.551030 −0.0254171
\(471\) 2.43145 0.112035
\(472\) −16.1626 −0.743944
\(473\) 5.25923 0.241820
\(474\) 0.196318 0.00901719
\(475\) 2.36580 0.108551
\(476\) 0.326625 0.0149708
\(477\) −9.97324 −0.456643
\(478\) 3.36717 0.154011
\(479\) 31.8259 1.45416 0.727081 0.686552i \(-0.240876\pi\)
0.727081 + 0.686552i \(0.240876\pi\)
\(480\) 8.40444 0.383608
\(481\) −6.40031 −0.291829
\(482\) −6.05251 −0.275684
\(483\) −2.11753 −0.0963507
\(484\) 10.3316 0.469618
\(485\) −13.2866 −0.603315
\(486\) −0.306328 −0.0138953
\(487\) −32.7544 −1.48424 −0.742122 0.670265i \(-0.766180\pi\)
−0.742122 + 0.670265i \(0.766180\pi\)
\(488\) −5.04097 −0.228194
\(489\) 20.6882 0.935555
\(490\) 0.746523 0.0337245
\(491\) 8.80770 0.397486 0.198743 0.980052i \(-0.436314\pi\)
0.198743 + 0.980052i \(0.436314\pi\)
\(492\) −6.44649 −0.290630
\(493\) 1.35712 0.0611218
\(494\) −2.07646 −0.0934243
\(495\) −5.75664 −0.258742
\(496\) 11.3670 0.510393
\(497\) −0.960879 −0.0431013
\(498\) 0.880867 0.0394726
\(499\) 18.5080 0.828530 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(500\) −18.8647 −0.843656
\(501\) −2.73919 −0.122378
\(502\) 6.18463 0.276033
\(503\) 4.74817 0.211711 0.105855 0.994382i \(-0.466242\pi\)
0.105855 + 0.994382i \(0.466242\pi\)
\(504\) −1.19657 −0.0532994
\(505\) 22.9385 1.02075
\(506\) −1.53225 −0.0681167
\(507\) −5.76174 −0.255888
\(508\) 16.3647 0.726065
\(509\) 43.0093 1.90635 0.953177 0.302412i \(-0.0977919\pi\)
0.953177 + 0.302412i \(0.0977919\pi\)
\(510\) 0.127918 0.00566431
\(511\) 6.48583 0.286916
\(512\) −20.1296 −0.889612
\(513\) 2.51953 0.111240
\(514\) −4.33140 −0.191050
\(515\) −42.5244 −1.87385
\(516\) −4.24394 −0.186829
\(517\) −1.74359 −0.0766832
\(518\) −0.728737 −0.0320189
\(519\) 9.56588 0.419896
\(520\) −7.84532 −0.344040
\(521\) −17.0535 −0.747129 −0.373564 0.927604i \(-0.621864\pi\)
−0.373564 + 0.927604i \(0.621864\pi\)
\(522\) −2.42615 −0.106190
\(523\) 20.7197 0.906008 0.453004 0.891508i \(-0.350352\pi\)
0.453004 + 0.891508i \(0.350352\pi\)
\(524\) −42.5852 −1.86034
\(525\) −0.938987 −0.0409808
\(526\) −5.70690 −0.248833
\(527\) 0.565258 0.0246230
\(528\) 8.13956 0.354229
\(529\) −18.5161 −0.805047
\(530\) −7.44526 −0.323401
\(531\) −13.5075 −0.586174
\(532\) 4.80263 0.208220
\(533\) 9.09872 0.394109
\(534\) −2.01722 −0.0872937
\(535\) −33.5414 −1.45012
\(536\) −12.7149 −0.549198
\(537\) 18.4310 0.795357
\(538\) 4.07095 0.175511
\(539\) 2.36218 0.101746
\(540\) 4.64533 0.199903
\(541\) 9.36879 0.402796 0.201398 0.979510i \(-0.435452\pi\)
0.201398 + 0.979510i \(0.435452\pi\)
\(542\) 1.60234 0.0688265
\(543\) 15.9260 0.683449
\(544\) −0.590938 −0.0253362
\(545\) −16.4239 −0.703523
\(546\) 0.824146 0.0352702
\(547\) 24.4702 1.04627 0.523135 0.852250i \(-0.324762\pi\)
0.523135 + 0.852250i \(0.324762\pi\)
\(548\) −20.9714 −0.895855
\(549\) −4.21286 −0.179800
\(550\) −0.679454 −0.0289720
\(551\) 19.9549 0.850107
\(552\) 2.53376 0.107844
\(553\) 0.640875 0.0272528
\(554\) −0.836516 −0.0355402
\(555\) 5.79749 0.246090
\(556\) 4.51882 0.191641
\(557\) −29.6662 −1.25700 −0.628499 0.777811i \(-0.716330\pi\)
−0.628499 + 0.777811i \(0.716330\pi\)
\(558\) −1.01052 −0.0427787
\(559\) 5.98999 0.253350
\(560\) 8.39739 0.354854
\(561\) 0.404764 0.0170892
\(562\) 4.93457 0.208152
\(563\) 28.1376 1.18586 0.592929 0.805255i \(-0.297972\pi\)
0.592929 + 0.805255i \(0.297972\pi\)
\(564\) 1.40699 0.0592451
\(565\) 22.8491 0.961267
\(566\) 6.10980 0.256814
\(567\) −1.00000 −0.0419961
\(568\) 1.14976 0.0482427
\(569\) −1.12434 −0.0471349 −0.0235675 0.999722i \(-0.507502\pi\)
−0.0235675 + 0.999722i \(0.507502\pi\)
\(570\) 1.88089 0.0787816
\(571\) 2.47463 0.103560 0.0517800 0.998659i \(-0.483511\pi\)
0.0517800 + 0.998659i \(0.483511\pi\)
\(572\) −12.1141 −0.506515
\(573\) −18.5006 −0.772875
\(574\) 1.03598 0.0432409
\(575\) 1.98833 0.0829191
\(576\) −5.83514 −0.243131
\(577\) −19.8352 −0.825752 −0.412876 0.910787i \(-0.635476\pi\)
−0.412876 + 0.910787i \(0.635476\pi\)
\(578\) 5.19859 0.216233
\(579\) 5.00974 0.208198
\(580\) 36.7914 1.52768
\(581\) 2.87557 0.119299
\(582\) −1.67011 −0.0692284
\(583\) −23.5586 −0.975697
\(584\) −7.76073 −0.321141
\(585\) −6.55652 −0.271079
\(586\) 0.639746 0.0264276
\(587\) 40.3637 1.66599 0.832994 0.553283i \(-0.186625\pi\)
0.832994 + 0.553283i \(0.186625\pi\)
\(588\) −1.90616 −0.0786088
\(589\) 8.31145 0.342467
\(590\) −10.0836 −0.415137
\(591\) −16.2211 −0.667246
\(592\) −8.19732 −0.336908
\(593\) 26.3896 1.08369 0.541845 0.840478i \(-0.317726\pi\)
0.541845 + 0.840478i \(0.317726\pi\)
\(594\) −0.723603 −0.0296898
\(595\) 0.417585 0.0171193
\(596\) −42.2302 −1.72982
\(597\) −20.7295 −0.848402
\(598\) −1.74515 −0.0713645
\(599\) 22.0752 0.901968 0.450984 0.892532i \(-0.351073\pi\)
0.450984 + 0.892532i \(0.351073\pi\)
\(600\) 1.12356 0.0458692
\(601\) 6.36684 0.259709 0.129854 0.991533i \(-0.458549\pi\)
0.129854 + 0.991533i \(0.458549\pi\)
\(602\) 0.682019 0.0277970
\(603\) −10.6261 −0.432728
\(604\) −38.0739 −1.54920
\(605\) 13.2088 0.537014
\(606\) 2.88334 0.117128
\(607\) 25.7019 1.04321 0.521603 0.853188i \(-0.325334\pi\)
0.521603 + 0.853188i \(0.325334\pi\)
\(608\) −8.68904 −0.352387
\(609\) −7.92009 −0.320938
\(610\) −3.14499 −0.127337
\(611\) −1.98586 −0.0803394
\(612\) −0.326625 −0.0132030
\(613\) 28.5562 1.15337 0.576687 0.816965i \(-0.304345\pi\)
0.576687 + 0.816965i \(0.304345\pi\)
\(614\) 7.15364 0.288697
\(615\) −8.24175 −0.332339
\(616\) −2.82651 −0.113883
\(617\) 44.2281 1.78056 0.890279 0.455416i \(-0.150509\pi\)
0.890279 + 0.455416i \(0.150509\pi\)
\(618\) −5.34526 −0.215018
\(619\) 36.6561 1.47333 0.736667 0.676256i \(-0.236399\pi\)
0.736667 + 0.676256i \(0.236399\pi\)
\(620\) 15.3241 0.615429
\(621\) 2.11753 0.0849734
\(622\) −2.15099 −0.0862467
\(623\) −6.58516 −0.263829
\(624\) 9.27054 0.371119
\(625\) −28.8132 −1.15253
\(626\) 2.48873 0.0994698
\(627\) 5.95158 0.237683
\(628\) −4.63473 −0.184946
\(629\) −0.407637 −0.0162535
\(630\) −0.746523 −0.0297422
\(631\) −46.2267 −1.84025 −0.920127 0.391620i \(-0.871915\pi\)
−0.920127 + 0.391620i \(0.871915\pi\)
\(632\) −0.766851 −0.0305037
\(633\) −19.6700 −0.781811
\(634\) 2.60151 0.103319
\(635\) 20.9220 0.830265
\(636\) 19.0106 0.753820
\(637\) 2.69040 0.106598
\(638\) −5.73100 −0.226892
\(639\) 0.960879 0.0380118
\(640\) −21.1649 −0.836618
\(641\) −0.547650 −0.0216309 −0.0108154 0.999942i \(-0.503443\pi\)
−0.0108154 + 0.999942i \(0.503443\pi\)
\(642\) −4.21611 −0.166397
\(643\) −2.70137 −0.106532 −0.0532659 0.998580i \(-0.516963\pi\)
−0.0532659 + 0.998580i \(0.516963\pi\)
\(644\) 4.03635 0.159054
\(645\) −5.42582 −0.213641
\(646\) −0.132250 −0.00520330
\(647\) −5.21463 −0.205008 −0.102504 0.994733i \(-0.532686\pi\)
−0.102504 + 0.994733i \(0.532686\pi\)
\(648\) 1.19657 0.0470056
\(649\) −31.9071 −1.25246
\(650\) −0.773863 −0.0303534
\(651\) −3.29881 −0.129291
\(652\) −39.4352 −1.54440
\(653\) 46.1150 1.80462 0.902309 0.431090i \(-0.141871\pi\)
0.902309 + 0.431090i \(0.141871\pi\)
\(654\) −2.06447 −0.0807270
\(655\) −54.4446 −2.12733
\(656\) 11.6534 0.454987
\(657\) −6.48583 −0.253036
\(658\) −0.226110 −0.00881468
\(659\) −12.6598 −0.493156 −0.246578 0.969123i \(-0.579306\pi\)
−0.246578 + 0.969123i \(0.579306\pi\)
\(660\) 10.9731 0.427127
\(661\) −7.95024 −0.309229 −0.154614 0.987975i \(-0.549413\pi\)
−0.154614 + 0.987975i \(0.549413\pi\)
\(662\) 5.68419 0.220922
\(663\) 0.461006 0.0179040
\(664\) −3.44081 −0.133529
\(665\) 6.14010 0.238103
\(666\) 0.728737 0.0282380
\(667\) 16.7710 0.649376
\(668\) 5.22134 0.202020
\(669\) 13.1389 0.507978
\(670\) −7.93263 −0.306464
\(671\) −9.95153 −0.384174
\(672\) 3.44868 0.133036
\(673\) 7.93560 0.305895 0.152947 0.988234i \(-0.451123\pi\)
0.152947 + 0.988234i \(0.451123\pi\)
\(674\) 2.36275 0.0910096
\(675\) 0.938987 0.0361416
\(676\) 10.9828 0.422416
\(677\) −27.9758 −1.07520 −0.537599 0.843201i \(-0.680668\pi\)
−0.537599 + 0.843201i \(0.680668\pi\)
\(678\) 2.87210 0.110302
\(679\) −5.45203 −0.209230
\(680\) −0.499669 −0.0191614
\(681\) 14.1801 0.543381
\(682\) −2.38703 −0.0914041
\(683\) 17.0364 0.651878 0.325939 0.945391i \(-0.394320\pi\)
0.325939 + 0.945391i \(0.394320\pi\)
\(684\) −4.80263 −0.183633
\(685\) −26.8117 −1.02442
\(686\) 0.306328 0.0116957
\(687\) −1.32939 −0.0507194
\(688\) 7.67180 0.292485
\(689\) −26.8320 −1.02222
\(690\) 1.58078 0.0601793
\(691\) −36.5982 −1.39226 −0.696131 0.717914i \(-0.745097\pi\)
−0.696131 + 0.717914i \(0.745097\pi\)
\(692\) −18.2341 −0.693158
\(693\) −2.36218 −0.0897318
\(694\) −4.29908 −0.163191
\(695\) 5.77725 0.219143
\(696\) 9.47693 0.359222
\(697\) 0.579498 0.0219501
\(698\) −7.82266 −0.296092
\(699\) 22.9228 0.867018
\(700\) 1.78986 0.0676505
\(701\) 21.2960 0.804338 0.402169 0.915565i \(-0.368256\pi\)
0.402169 + 0.915565i \(0.368256\pi\)
\(702\) −0.824146 −0.0311054
\(703\) −5.99381 −0.226061
\(704\) −13.7837 −0.519491
\(705\) 1.79882 0.0677476
\(706\) −6.84204 −0.257504
\(707\) 9.41257 0.353996
\(708\) 25.7474 0.967648
\(709\) −25.2962 −0.950018 −0.475009 0.879981i \(-0.657555\pi\)
−0.475009 + 0.879981i \(0.657555\pi\)
\(710\) 0.717318 0.0269205
\(711\) −0.640875 −0.0240347
\(712\) 7.87959 0.295300
\(713\) 6.98532 0.261602
\(714\) 0.0524900 0.00196439
\(715\) −15.4877 −0.579206
\(716\) −35.1325 −1.31296
\(717\) −10.9920 −0.410505
\(718\) 3.47726 0.129770
\(719\) 23.8157 0.888174 0.444087 0.895984i \(-0.353528\pi\)
0.444087 + 0.895984i \(0.353528\pi\)
\(720\) −8.39739 −0.312952
\(721\) −17.4495 −0.649852
\(722\) 3.87566 0.144237
\(723\) 19.7582 0.734817
\(724\) −30.3575 −1.12823
\(725\) 7.43687 0.276198
\(726\) 1.66033 0.0616206
\(727\) −19.4656 −0.721940 −0.360970 0.932577i \(-0.617554\pi\)
−0.360970 + 0.932577i \(0.617554\pi\)
\(728\) −3.21925 −0.119313
\(729\) 1.00000 0.0370370
\(730\) −4.84182 −0.179204
\(731\) 0.381503 0.0141104
\(732\) 8.03039 0.296812
\(733\) 6.14765 0.227068 0.113534 0.993534i \(-0.463783\pi\)
0.113534 + 0.993534i \(0.463783\pi\)
\(734\) −3.44466 −0.127145
\(735\) −2.43700 −0.0898902
\(736\) −7.30266 −0.269180
\(737\) −25.1008 −0.924599
\(738\) −1.03598 −0.0381348
\(739\) −25.9676 −0.955235 −0.477617 0.878568i \(-0.658500\pi\)
−0.477617 + 0.878568i \(0.658500\pi\)
\(740\) −11.0510 −0.406242
\(741\) 6.77854 0.249016
\(742\) −3.05509 −0.112156
\(743\) 42.2560 1.55022 0.775111 0.631825i \(-0.217694\pi\)
0.775111 + 0.631825i \(0.217694\pi\)
\(744\) 3.94725 0.144713
\(745\) −53.9908 −1.97807
\(746\) −9.73407 −0.356390
\(747\) −2.87557 −0.105211
\(748\) −0.771547 −0.0282105
\(749\) −13.7634 −0.502903
\(750\) −3.03164 −0.110700
\(751\) −16.8502 −0.614872 −0.307436 0.951569i \(-0.599471\pi\)
−0.307436 + 0.951569i \(0.599471\pi\)
\(752\) −2.54343 −0.0927494
\(753\) −20.1895 −0.735748
\(754\) −6.52731 −0.237711
\(755\) −48.6769 −1.77154
\(756\) 1.90616 0.0693265
\(757\) 39.2076 1.42502 0.712511 0.701661i \(-0.247558\pi\)
0.712511 + 0.701661i \(0.247558\pi\)
\(758\) −0.468226 −0.0170067
\(759\) 5.00198 0.181560
\(760\) −7.34704 −0.266505
\(761\) 30.4699 1.10453 0.552267 0.833667i \(-0.313763\pi\)
0.552267 + 0.833667i \(0.313763\pi\)
\(762\) 2.62987 0.0952702
\(763\) −6.73939 −0.243982
\(764\) 35.2652 1.27585
\(765\) −0.417585 −0.0150978
\(766\) 0.306328 0.0110681
\(767\) −36.3405 −1.31218
\(768\) 9.00987 0.325116
\(769\) −10.1240 −0.365080 −0.182540 0.983198i \(-0.558432\pi\)
−0.182540 + 0.983198i \(0.558432\pi\)
\(770\) −1.76342 −0.0635493
\(771\) 14.1397 0.509231
\(772\) −9.54938 −0.343690
\(773\) 8.48170 0.305066 0.152533 0.988298i \(-0.451257\pi\)
0.152533 + 0.988298i \(0.451257\pi\)
\(774\) −0.682019 −0.0245147
\(775\) 3.09754 0.111267
\(776\) 6.52373 0.234188
\(777\) 2.37894 0.0853441
\(778\) 8.59515 0.308151
\(779\) 8.52083 0.305291
\(780\) 12.4978 0.447493
\(781\) 2.26977 0.0812187
\(782\) −0.111149 −0.00397467
\(783\) 7.92009 0.283041
\(784\) 3.44578 0.123064
\(785\) −5.92544 −0.211488
\(786\) −6.84361 −0.244104
\(787\) −31.6332 −1.12760 −0.563801 0.825911i \(-0.690662\pi\)
−0.563801 + 0.825911i \(0.690662\pi\)
\(788\) 30.9200 1.10148
\(789\) 18.6300 0.663246
\(790\) −0.478428 −0.0170217
\(791\) 9.37588 0.333368
\(792\) 2.82651 0.100436
\(793\) −11.3343 −0.402492
\(794\) 1.51403 0.0537308
\(795\) 24.3048 0.862003
\(796\) 39.5138 1.40053
\(797\) 19.9684 0.707317 0.353659 0.935375i \(-0.384937\pi\)
0.353659 + 0.935375i \(0.384937\pi\)
\(798\) 0.771802 0.0273215
\(799\) −0.126480 −0.00447454
\(800\) −3.23826 −0.114490
\(801\) 6.58516 0.232675
\(802\) 8.36002 0.295203
\(803\) −15.3207 −0.540656
\(804\) 20.2551 0.714342
\(805\) 5.16042 0.181881
\(806\) −2.71870 −0.0957622
\(807\) −13.2895 −0.467813
\(808\) −11.2628 −0.396223
\(809\) −11.4147 −0.401319 −0.200660 0.979661i \(-0.564309\pi\)
−0.200660 + 0.979661i \(0.564309\pi\)
\(810\) 0.746523 0.0262302
\(811\) −39.4376 −1.38484 −0.692420 0.721495i \(-0.743455\pi\)
−0.692420 + 0.721495i \(0.743455\pi\)
\(812\) 15.0970 0.529800
\(813\) −5.23080 −0.183452
\(814\) 1.72141 0.0603354
\(815\) −50.4173 −1.76604
\(816\) 0.590442 0.0206696
\(817\) 5.60955 0.196253
\(818\) 7.15427 0.250143
\(819\) −2.69040 −0.0940102
\(820\) 15.7101 0.548621
\(821\) 9.09700 0.317487 0.158744 0.987320i \(-0.449256\pi\)
0.158744 + 0.987320i \(0.449256\pi\)
\(822\) −3.37020 −0.117549
\(823\) 35.0617 1.22217 0.611087 0.791564i \(-0.290733\pi\)
0.611087 + 0.791564i \(0.290733\pi\)
\(824\) 20.8795 0.727371
\(825\) 2.21806 0.0772229
\(826\) −4.13772 −0.143970
\(827\) −10.5157 −0.365668 −0.182834 0.983144i \(-0.558527\pi\)
−0.182834 + 0.983144i \(0.558527\pi\)
\(828\) −4.03635 −0.140273
\(829\) −25.7538 −0.894467 −0.447233 0.894417i \(-0.647591\pi\)
−0.447233 + 0.894417i \(0.647591\pi\)
\(830\) −2.14668 −0.0745122
\(831\) 2.73078 0.0947299
\(832\) −15.6989 −0.544260
\(833\) 0.171352 0.00593699
\(834\) 0.726192 0.0251460
\(835\) 6.67541 0.231012
\(836\) −11.3447 −0.392364
\(837\) 3.29881 0.114024
\(838\) −10.2213 −0.353090
\(839\) 38.3857 1.32522 0.662611 0.748963i \(-0.269448\pi\)
0.662611 + 0.748963i \(0.269448\pi\)
\(840\) 2.91604 0.100613
\(841\) 33.7279 1.16303
\(842\) −4.11774 −0.141907
\(843\) −16.1088 −0.554815
\(844\) 37.4941 1.29060
\(845\) 14.0414 0.483038
\(846\) 0.226110 0.00777381
\(847\) 5.42010 0.186237
\(848\) −34.3656 −1.18012
\(849\) −19.9453 −0.684520
\(850\) −0.0492874 −0.00169054
\(851\) −5.03747 −0.172682
\(852\) −1.83159 −0.0627493
\(853\) −7.13436 −0.244276 −0.122138 0.992513i \(-0.538975\pi\)
−0.122138 + 0.992513i \(0.538975\pi\)
\(854\) −1.29052 −0.0441606
\(855\) −6.14010 −0.209987
\(856\) 16.4688 0.562892
\(857\) 29.6253 1.01198 0.505990 0.862539i \(-0.331127\pi\)
0.505990 + 0.862539i \(0.331127\pi\)
\(858\) −1.94678 −0.0664620
\(859\) 31.0671 1.05999 0.529997 0.847999i \(-0.322193\pi\)
0.529997 + 0.847999i \(0.322193\pi\)
\(860\) 10.3425 0.352676
\(861\) −3.38192 −0.115255
\(862\) −7.79909 −0.265638
\(863\) −10.6748 −0.363374 −0.181687 0.983356i \(-0.558156\pi\)
−0.181687 + 0.983356i \(0.558156\pi\)
\(864\) −3.44868 −0.117326
\(865\) −23.3121 −0.792635
\(866\) 1.81402 0.0616431
\(867\) −16.9706 −0.576353
\(868\) 6.28808 0.213431
\(869\) −1.51386 −0.0513543
\(870\) 5.91253 0.200454
\(871\) −28.5885 −0.968684
\(872\) 8.06414 0.273086
\(873\) 5.45203 0.184523
\(874\) −1.63431 −0.0552814
\(875\) −9.89670 −0.334570
\(876\) 12.3630 0.417708
\(877\) 22.7772 0.769130 0.384565 0.923098i \(-0.374351\pi\)
0.384565 + 0.923098i \(0.374351\pi\)
\(878\) −9.40427 −0.317379
\(879\) −2.08843 −0.0704410
\(880\) −19.8361 −0.668676
\(881\) −26.4125 −0.889860 −0.444930 0.895565i \(-0.646771\pi\)
−0.444930 + 0.895565i \(0.646771\pi\)
\(882\) −0.306328 −0.0103146
\(883\) −41.7787 −1.40597 −0.702983 0.711206i \(-0.748149\pi\)
−0.702983 + 0.711206i \(0.748149\pi\)
\(884\) −0.878752 −0.0295556
\(885\) 32.9177 1.10652
\(886\) −1.46631 −0.0492617
\(887\) −14.8523 −0.498691 −0.249346 0.968415i \(-0.580216\pi\)
−0.249346 + 0.968415i \(0.580216\pi\)
\(888\) −2.84657 −0.0955245
\(889\) 8.58514 0.287936
\(890\) 4.91598 0.164784
\(891\) 2.36218 0.0791360
\(892\) −25.0448 −0.838563
\(893\) −1.85974 −0.0622337
\(894\) −6.78657 −0.226977
\(895\) −44.9165 −1.50139
\(896\) −8.68482 −0.290140
\(897\) 5.69699 0.190217
\(898\) −0.985223 −0.0328773
\(899\) 26.1269 0.871381
\(900\) −1.78986 −0.0596621
\(901\) −1.70893 −0.0569329
\(902\) −2.44717 −0.0814817
\(903\) −2.22643 −0.0740910
\(904\) −11.2189 −0.373135
\(905\) −38.8117 −1.29014
\(906\) −6.11863 −0.203278
\(907\) −22.8304 −0.758071 −0.379035 0.925382i \(-0.623744\pi\)
−0.379035 + 0.925382i \(0.623744\pi\)
\(908\) −27.0295 −0.897006
\(909\) −9.41257 −0.312195
\(910\) −2.00845 −0.0665794
\(911\) 0.990631 0.0328211 0.0164105 0.999865i \(-0.494776\pi\)
0.0164105 + 0.999865i \(0.494776\pi\)
\(912\) 8.68174 0.287481
\(913\) −6.79260 −0.224802
\(914\) −7.48047 −0.247432
\(915\) 10.2667 0.339408
\(916\) 2.53404 0.0837269
\(917\) −22.3408 −0.737758
\(918\) −0.0524900 −0.00173243
\(919\) 33.7570 1.11354 0.556771 0.830666i \(-0.312040\pi\)
0.556771 + 0.830666i \(0.312040\pi\)
\(920\) −6.17479 −0.203577
\(921\) −23.3529 −0.769503
\(922\) 3.80918 0.125449
\(923\) 2.58515 0.0850912
\(924\) 4.50270 0.148128
\(925\) −2.23380 −0.0734468
\(926\) 1.57551 0.0517745
\(927\) 17.4495 0.573115
\(928\) −27.3138 −0.896621
\(929\) −18.1225 −0.594579 −0.297289 0.954787i \(-0.596083\pi\)
−0.297289 + 0.954787i \(0.596083\pi\)
\(930\) 2.46264 0.0807532
\(931\) 2.51953 0.0825742
\(932\) −43.6945 −1.43126
\(933\) 7.02184 0.229885
\(934\) −1.23276 −0.0403370
\(935\) −0.986412 −0.0322591
\(936\) 3.21925 0.105224
\(937\) 35.2334 1.15102 0.575512 0.817793i \(-0.304803\pi\)
0.575512 + 0.817793i \(0.304803\pi\)
\(938\) −3.25508 −0.106282
\(939\) −8.12440 −0.265130
\(940\) −3.42885 −0.111837
\(941\) −31.3577 −1.02223 −0.511115 0.859512i \(-0.670768\pi\)
−0.511115 + 0.859512i \(0.670768\pi\)
\(942\) −0.744821 −0.0242676
\(943\) 7.16130 0.233204
\(944\) −46.5438 −1.51487
\(945\) 2.43700 0.0792757
\(946\) −1.61105 −0.0523798
\(947\) −25.8681 −0.840600 −0.420300 0.907385i \(-0.638075\pi\)
−0.420300 + 0.907385i \(0.638075\pi\)
\(948\) 1.22161 0.0396761
\(949\) −17.4495 −0.566434
\(950\) −0.724713 −0.0235128
\(951\) −8.49255 −0.275390
\(952\) −0.205034 −0.00664520
\(953\) 48.1449 1.55957 0.779784 0.626049i \(-0.215329\pi\)
0.779784 + 0.626049i \(0.215329\pi\)
\(954\) 3.05509 0.0989121
\(955\) 45.0861 1.45895
\(956\) 20.9526 0.677656
\(957\) 18.7087 0.604766
\(958\) −9.74917 −0.314981
\(959\) −11.0019 −0.355270
\(960\) 14.2203 0.458957
\(961\) −20.1178 −0.648962
\(962\) 1.96060 0.0632121
\(963\) 13.7634 0.443518
\(964\) −37.6624 −1.21302
\(965\) −12.2088 −0.393014
\(966\) 0.648658 0.0208702
\(967\) 40.9603 1.31719 0.658597 0.752495i \(-0.271150\pi\)
0.658597 + 0.752495i \(0.271150\pi\)
\(968\) −6.48552 −0.208453
\(969\) 0.431726 0.0138690
\(970\) 4.07007 0.130682
\(971\) 44.3932 1.42464 0.712322 0.701852i \(-0.247643\pi\)
0.712322 + 0.701852i \(0.247643\pi\)
\(972\) −1.90616 −0.0611402
\(973\) 2.37063 0.0759991
\(974\) 10.0336 0.321497
\(975\) 2.52625 0.0809048
\(976\) −14.5166 −0.464665
\(977\) 29.3721 0.939697 0.469849 0.882747i \(-0.344308\pi\)
0.469849 + 0.882747i \(0.344308\pi\)
\(978\) −6.33739 −0.202648
\(979\) 15.5553 0.497151
\(980\) 4.64533 0.148390
\(981\) 6.73939 0.215172
\(982\) −2.69805 −0.0860982
\(983\) 13.0541 0.416362 0.208181 0.978090i \(-0.433246\pi\)
0.208181 + 0.978090i \(0.433246\pi\)
\(984\) 4.04670 0.129004
\(985\) 39.5308 1.25956
\(986\) −0.415725 −0.0132394
\(987\) 0.738129 0.0234949
\(988\) −12.9210 −0.411072
\(989\) 4.71452 0.149913
\(990\) 1.76342 0.0560453
\(991\) −51.3168 −1.63013 −0.815066 0.579369i \(-0.803299\pi\)
−0.815066 + 0.579369i \(0.803299\pi\)
\(992\) −11.3765 −0.361206
\(993\) −18.5559 −0.588853
\(994\) 0.294344 0.00933603
\(995\) 50.5179 1.60152
\(996\) 5.48130 0.173681
\(997\) −39.8128 −1.26088 −0.630442 0.776237i \(-0.717126\pi\)
−0.630442 + 0.776237i \(0.717126\pi\)
\(998\) −5.66951 −0.179465
\(999\) −2.37894 −0.0752664
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 8043.2.a.p.1.17 41
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8043.2.a.p.1.17 41 1.1 even 1 trivial