Properties

Label 8002.2.a.e.1.53
Level $8002$
Weight $2$
Character 8002.1
Self dual yes
Analytic conductor $63.896$
Analytic rank $0$
Dimension $77$
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] = [8002,2,Mod(1,8002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8002, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8002 = 2 \cdot 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8002.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: \(63.8962916974\)
Analytic rank: \(0\)
Dimension: \(77\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.53
Character \(\chi\) \(=\) 8002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.43148 q^{3} +1.00000 q^{4} +0.803871 q^{5} -1.43148 q^{6} -3.58216 q^{7} -1.00000 q^{8} -0.950869 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.43148 q^{3} +1.00000 q^{4} +0.803871 q^{5} -1.43148 q^{6} -3.58216 q^{7} -1.00000 q^{8} -0.950869 q^{9} -0.803871 q^{10} -1.17531 q^{11} +1.43148 q^{12} +4.60600 q^{13} +3.58216 q^{14} +1.15072 q^{15} +1.00000 q^{16} +1.42510 q^{17} +0.950869 q^{18} +7.85309 q^{19} +0.803871 q^{20} -5.12778 q^{21} +1.17531 q^{22} -1.12223 q^{23} -1.43148 q^{24} -4.35379 q^{25} -4.60600 q^{26} -5.65558 q^{27} -3.58216 q^{28} -5.21621 q^{29} -1.15072 q^{30} +5.09216 q^{31} -1.00000 q^{32} -1.68243 q^{33} -1.42510 q^{34} -2.87959 q^{35} -0.950869 q^{36} -3.48843 q^{37} -7.85309 q^{38} +6.59339 q^{39} -0.803871 q^{40} -4.00456 q^{41} +5.12778 q^{42} -1.27939 q^{43} -1.17531 q^{44} -0.764376 q^{45} +1.12223 q^{46} +6.62131 q^{47} +1.43148 q^{48} +5.83185 q^{49} +4.35379 q^{50} +2.04001 q^{51} +4.60600 q^{52} +12.3592 q^{53} +5.65558 q^{54} -0.944799 q^{55} +3.58216 q^{56} +11.2415 q^{57} +5.21621 q^{58} +4.07851 q^{59} +1.15072 q^{60} -5.46496 q^{61} -5.09216 q^{62} +3.40616 q^{63} +1.00000 q^{64} +3.70263 q^{65} +1.68243 q^{66} -5.10207 q^{67} +1.42510 q^{68} -1.60644 q^{69} +2.87959 q^{70} -6.40788 q^{71} +0.950869 q^{72} -0.792319 q^{73} +3.48843 q^{74} -6.23236 q^{75} +7.85309 q^{76} +4.21015 q^{77} -6.59339 q^{78} +7.21938 q^{79} +0.803871 q^{80} -5.24324 q^{81} +4.00456 q^{82} +6.15613 q^{83} -5.12778 q^{84} +1.14560 q^{85} +1.27939 q^{86} -7.46689 q^{87} +1.17531 q^{88} +8.37728 q^{89} +0.764376 q^{90} -16.4994 q^{91} -1.12223 q^{92} +7.28932 q^{93} -6.62131 q^{94} +6.31287 q^{95} -1.43148 q^{96} +6.80739 q^{97} -5.83185 q^{98} +1.11757 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.43148 0.826464 0.413232 0.910626i \(-0.364400\pi\)
0.413232 + 0.910626i \(0.364400\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.803871 0.359502 0.179751 0.983712i \(-0.442471\pi\)
0.179751 + 0.983712i \(0.442471\pi\)
\(6\) −1.43148 −0.584399
\(7\) −3.58216 −1.35393 −0.676964 0.736016i \(-0.736705\pi\)
−0.676964 + 0.736016i \(0.736705\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.950869 −0.316956
\(10\) −0.803871 −0.254206
\(11\) −1.17531 −0.354370 −0.177185 0.984178i \(-0.556699\pi\)
−0.177185 + 0.984178i \(0.556699\pi\)
\(12\) 1.43148 0.413232
\(13\) 4.60600 1.27748 0.638738 0.769425i \(-0.279457\pi\)
0.638738 + 0.769425i \(0.279457\pi\)
\(14\) 3.58216 0.957372
\(15\) 1.15072 0.297116
\(16\) 1.00000 0.250000
\(17\) 1.42510 0.345639 0.172819 0.984954i \(-0.444712\pi\)
0.172819 + 0.984954i \(0.444712\pi\)
\(18\) 0.950869 0.224122
\(19\) 7.85309 1.80162 0.900812 0.434210i \(-0.142972\pi\)
0.900812 + 0.434210i \(0.142972\pi\)
\(20\) 0.803871 0.179751
\(21\) −5.12778 −1.11897
\(22\) 1.17531 0.250577
\(23\) −1.12223 −0.234000 −0.117000 0.993132i \(-0.537328\pi\)
−0.117000 + 0.993132i \(0.537328\pi\)
\(24\) −1.43148 −0.292199
\(25\) −4.35379 −0.870758
\(26\) −4.60600 −0.903311
\(27\) −5.65558 −1.08842
\(28\) −3.58216 −0.676964
\(29\) −5.21621 −0.968626 −0.484313 0.874895i \(-0.660930\pi\)
−0.484313 + 0.874895i \(0.660930\pi\)
\(30\) −1.15072 −0.210092
\(31\) 5.09216 0.914579 0.457289 0.889318i \(-0.348820\pi\)
0.457289 + 0.889318i \(0.348820\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.68243 −0.292874
\(34\) −1.42510 −0.244403
\(35\) −2.87959 −0.486740
\(36\) −0.950869 −0.158478
\(37\) −3.48843 −0.573494 −0.286747 0.958006i \(-0.592574\pi\)
−0.286747 + 0.958006i \(0.592574\pi\)
\(38\) −7.85309 −1.27394
\(39\) 6.59339 1.05579
\(40\) −0.803871 −0.127103
\(41\) −4.00456 −0.625407 −0.312703 0.949851i \(-0.601235\pi\)
−0.312703 + 0.949851i \(0.601235\pi\)
\(42\) 5.12778 0.791234
\(43\) −1.27939 −0.195104 −0.0975522 0.995230i \(-0.531101\pi\)
−0.0975522 + 0.995230i \(0.531101\pi\)
\(44\) −1.17531 −0.177185
\(45\) −0.764376 −0.113946
\(46\) 1.12223 0.165463
\(47\) 6.62131 0.965818 0.482909 0.875671i \(-0.339580\pi\)
0.482909 + 0.875671i \(0.339580\pi\)
\(48\) 1.43148 0.206616
\(49\) 5.83185 0.833121
\(50\) 4.35379 0.615719
\(51\) 2.04001 0.285658
\(52\) 4.60600 0.638738
\(53\) 12.3592 1.69767 0.848834 0.528659i \(-0.177305\pi\)
0.848834 + 0.528659i \(0.177305\pi\)
\(54\) 5.65558 0.769628
\(55\) −0.944799 −0.127397
\(56\) 3.58216 0.478686
\(57\) 11.2415 1.48898
\(58\) 5.21621 0.684922
\(59\) 4.07851 0.530977 0.265489 0.964114i \(-0.414467\pi\)
0.265489 + 0.964114i \(0.414467\pi\)
\(60\) 1.15072 0.148558
\(61\) −5.46496 −0.699717 −0.349858 0.936803i \(-0.613770\pi\)
−0.349858 + 0.936803i \(0.613770\pi\)
\(62\) −5.09216 −0.646705
\(63\) 3.40616 0.429136
\(64\) 1.00000 0.125000
\(65\) 3.70263 0.459255
\(66\) 1.68243 0.207093
\(67\) −5.10207 −0.623317 −0.311659 0.950194i \(-0.600885\pi\)
−0.311659 + 0.950194i \(0.600885\pi\)
\(68\) 1.42510 0.172819
\(69\) −1.60644 −0.193393
\(70\) 2.87959 0.344177
\(71\) −6.40788 −0.760475 −0.380238 0.924889i \(-0.624158\pi\)
−0.380238 + 0.924889i \(0.624158\pi\)
\(72\) 0.950869 0.112061
\(73\) −0.792319 −0.0927339 −0.0463670 0.998924i \(-0.514764\pi\)
−0.0463670 + 0.998924i \(0.514764\pi\)
\(74\) 3.48843 0.405522
\(75\) −6.23236 −0.719651
\(76\) 7.85309 0.900812
\(77\) 4.21015 0.479791
\(78\) −6.59339 −0.746555
\(79\) 7.21938 0.812244 0.406122 0.913819i \(-0.366881\pi\)
0.406122 + 0.913819i \(0.366881\pi\)
\(80\) 0.803871 0.0898755
\(81\) −5.24324 −0.582582
\(82\) 4.00456 0.442229
\(83\) 6.15613 0.675723 0.337862 0.941196i \(-0.390296\pi\)
0.337862 + 0.941196i \(0.390296\pi\)
\(84\) −5.12778 −0.559487
\(85\) 1.14560 0.124258
\(86\) 1.27939 0.137960
\(87\) −7.46689 −0.800535
\(88\) 1.17531 0.125289
\(89\) 8.37728 0.887990 0.443995 0.896029i \(-0.353561\pi\)
0.443995 + 0.896029i \(0.353561\pi\)
\(90\) 0.764376 0.0805723
\(91\) −16.4994 −1.72961
\(92\) −1.12223 −0.117000
\(93\) 7.28932 0.755867
\(94\) −6.62131 −0.682936
\(95\) 6.31287 0.647687
\(96\) −1.43148 −0.146100
\(97\) 6.80739 0.691186 0.345593 0.938384i \(-0.387678\pi\)
0.345593 + 0.938384i \(0.387678\pi\)
\(98\) −5.83185 −0.589106
\(99\) 1.11757 0.112320
\(100\) −4.35379 −0.435379
\(101\) 15.1085 1.50335 0.751676 0.659532i \(-0.229246\pi\)
0.751676 + 0.659532i \(0.229246\pi\)
\(102\) −2.04001 −0.201991
\(103\) 1.93373 0.190537 0.0952683 0.995452i \(-0.469629\pi\)
0.0952683 + 0.995452i \(0.469629\pi\)
\(104\) −4.60600 −0.451656
\(105\) −4.12207 −0.402273
\(106\) −12.3592 −1.20043
\(107\) 4.83216 0.467142 0.233571 0.972340i \(-0.424959\pi\)
0.233571 + 0.972340i \(0.424959\pi\)
\(108\) −5.65558 −0.544209
\(109\) −6.80875 −0.652160 −0.326080 0.945342i \(-0.605728\pi\)
−0.326080 + 0.945342i \(0.605728\pi\)
\(110\) 0.944799 0.0900830
\(111\) −4.99361 −0.473973
\(112\) −3.58216 −0.338482
\(113\) 13.9602 1.31327 0.656635 0.754209i \(-0.271979\pi\)
0.656635 + 0.754209i \(0.271979\pi\)
\(114\) −11.2415 −1.05287
\(115\) −0.902124 −0.0841235
\(116\) −5.21621 −0.484313
\(117\) −4.37971 −0.404904
\(118\) −4.07851 −0.375458
\(119\) −5.10495 −0.467970
\(120\) −1.15072 −0.105046
\(121\) −9.61864 −0.874422
\(122\) 5.46496 0.494775
\(123\) −5.73244 −0.516876
\(124\) 5.09216 0.457289
\(125\) −7.51924 −0.672541
\(126\) −3.40616 −0.303445
\(127\) −19.0630 −1.69157 −0.845784 0.533526i \(-0.820867\pi\)
−0.845784 + 0.533526i \(0.820867\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.83141 −0.161247
\(130\) −3.70263 −0.324742
\(131\) 1.62818 0.142255 0.0711274 0.997467i \(-0.477340\pi\)
0.0711274 + 0.997467i \(0.477340\pi\)
\(132\) −1.68243 −0.146437
\(133\) −28.1310 −2.43927
\(134\) 5.10207 0.440752
\(135\) −4.54636 −0.391288
\(136\) −1.42510 −0.122202
\(137\) 6.36086 0.543445 0.271723 0.962376i \(-0.412407\pi\)
0.271723 + 0.962376i \(0.412407\pi\)
\(138\) 1.60644 0.136749
\(139\) 16.7084 1.41719 0.708595 0.705616i \(-0.249329\pi\)
0.708595 + 0.705616i \(0.249329\pi\)
\(140\) −2.87959 −0.243370
\(141\) 9.47827 0.798214
\(142\) 6.40788 0.537737
\(143\) −5.41349 −0.452699
\(144\) −0.950869 −0.0792391
\(145\) −4.19316 −0.348223
\(146\) 0.792319 0.0655728
\(147\) 8.34817 0.688545
\(148\) −3.48843 −0.286747
\(149\) 9.78960 0.801995 0.400998 0.916079i \(-0.368664\pi\)
0.400998 + 0.916079i \(0.368664\pi\)
\(150\) 6.23236 0.508870
\(151\) −18.6569 −1.51828 −0.759138 0.650929i \(-0.774379\pi\)
−0.759138 + 0.650929i \(0.774379\pi\)
\(152\) −7.85309 −0.636970
\(153\) −1.35509 −0.109552
\(154\) −4.21015 −0.339264
\(155\) 4.09344 0.328793
\(156\) 6.59339 0.527894
\(157\) 8.84383 0.705815 0.352907 0.935658i \(-0.385193\pi\)
0.352907 + 0.935658i \(0.385193\pi\)
\(158\) −7.21938 −0.574343
\(159\) 17.6920 1.40306
\(160\) −0.803871 −0.0635516
\(161\) 4.01999 0.316819
\(162\) 5.24324 0.411948
\(163\) −25.4226 −1.99125 −0.995625 0.0934397i \(-0.970214\pi\)
−0.995625 + 0.0934397i \(0.970214\pi\)
\(164\) −4.00456 −0.312703
\(165\) −1.35246 −0.105289
\(166\) −6.15613 −0.477809
\(167\) 5.46543 0.422928 0.211464 0.977386i \(-0.432177\pi\)
0.211464 + 0.977386i \(0.432177\pi\)
\(168\) 5.12778 0.395617
\(169\) 8.21526 0.631943
\(170\) −1.14560 −0.0878635
\(171\) −7.46727 −0.571036
\(172\) −1.27939 −0.0975522
\(173\) −18.0771 −1.37437 −0.687186 0.726481i \(-0.741154\pi\)
−0.687186 + 0.726481i \(0.741154\pi\)
\(174\) 7.46689 0.566064
\(175\) 15.5960 1.17894
\(176\) −1.17531 −0.0885925
\(177\) 5.83831 0.438834
\(178\) −8.37728 −0.627903
\(179\) −4.58739 −0.342877 −0.171439 0.985195i \(-0.554842\pi\)
−0.171439 + 0.985195i \(0.554842\pi\)
\(180\) −0.764376 −0.0569732
\(181\) 19.1771 1.42542 0.712711 0.701458i \(-0.247467\pi\)
0.712711 + 0.701458i \(0.247467\pi\)
\(182\) 16.4994 1.22302
\(183\) −7.82298 −0.578291
\(184\) 1.12223 0.0827316
\(185\) −2.80425 −0.206172
\(186\) −7.28932 −0.534479
\(187\) −1.67494 −0.122484
\(188\) 6.62131 0.482909
\(189\) 20.2592 1.47364
\(190\) −6.31287 −0.457984
\(191\) −11.2058 −0.810826 −0.405413 0.914134i \(-0.632872\pi\)
−0.405413 + 0.914134i \(0.632872\pi\)
\(192\) 1.43148 0.103308
\(193\) 22.9269 1.65031 0.825156 0.564905i \(-0.191087\pi\)
0.825156 + 0.564905i \(0.191087\pi\)
\(194\) −6.80739 −0.488742
\(195\) 5.30024 0.379558
\(196\) 5.83185 0.416561
\(197\) 10.4987 0.748003 0.374001 0.927428i \(-0.377986\pi\)
0.374001 + 0.927428i \(0.377986\pi\)
\(198\) −1.11757 −0.0794221
\(199\) 20.8739 1.47971 0.739856 0.672765i \(-0.234894\pi\)
0.739856 + 0.672765i \(0.234894\pi\)
\(200\) 4.35379 0.307860
\(201\) −7.30350 −0.515149
\(202\) −15.1085 −1.06303
\(203\) 18.6853 1.31145
\(204\) 2.04001 0.142829
\(205\) −3.21915 −0.224835
\(206\) −1.93373 −0.134730
\(207\) 1.06709 0.0741679
\(208\) 4.60600 0.319369
\(209\) −9.22983 −0.638441
\(210\) 4.12207 0.284450
\(211\) 17.1402 1.17998 0.589989 0.807411i \(-0.299132\pi\)
0.589989 + 0.807411i \(0.299132\pi\)
\(212\) 12.3592 0.848834
\(213\) −9.17274 −0.628506
\(214\) −4.83216 −0.330320
\(215\) −1.02846 −0.0701404
\(216\) 5.65558 0.384814
\(217\) −18.2409 −1.23827
\(218\) 6.80875 0.461147
\(219\) −1.13419 −0.0766413
\(220\) −0.944799 −0.0636983
\(221\) 6.56404 0.441545
\(222\) 4.99361 0.335149
\(223\) 8.10757 0.542923 0.271461 0.962449i \(-0.412493\pi\)
0.271461 + 0.962449i \(0.412493\pi\)
\(224\) 3.58216 0.239343
\(225\) 4.13989 0.275992
\(226\) −13.9602 −0.928622
\(227\) 6.14699 0.407990 0.203995 0.978972i \(-0.434607\pi\)
0.203995 + 0.978972i \(0.434607\pi\)
\(228\) 11.2415 0.744489
\(229\) 13.2543 0.875870 0.437935 0.899007i \(-0.355710\pi\)
0.437935 + 0.899007i \(0.355710\pi\)
\(230\) 0.902124 0.0594843
\(231\) 6.02674 0.396530
\(232\) 5.21621 0.342461
\(233\) 29.9547 1.96240 0.981198 0.193003i \(-0.0618228\pi\)
0.981198 + 0.193003i \(0.0618228\pi\)
\(234\) 4.37971 0.286310
\(235\) 5.32268 0.347213
\(236\) 4.07851 0.265489
\(237\) 10.3344 0.671291
\(238\) 5.10495 0.330905
\(239\) 9.68475 0.626454 0.313227 0.949678i \(-0.398590\pi\)
0.313227 + 0.949678i \(0.398590\pi\)
\(240\) 1.15072 0.0742789
\(241\) 23.5432 1.51655 0.758275 0.651935i \(-0.226043\pi\)
0.758275 + 0.651935i \(0.226043\pi\)
\(242\) 9.61864 0.618310
\(243\) 9.46117 0.606934
\(244\) −5.46496 −0.349858
\(245\) 4.68805 0.299509
\(246\) 5.73244 0.365487
\(247\) 36.1714 2.30153
\(248\) −5.09216 −0.323352
\(249\) 8.81237 0.558461
\(250\) 7.51924 0.475558
\(251\) 15.0546 0.950240 0.475120 0.879921i \(-0.342405\pi\)
0.475120 + 0.879921i \(0.342405\pi\)
\(252\) 3.40616 0.214568
\(253\) 1.31897 0.0829226
\(254\) 19.0630 1.19612
\(255\) 1.63990 0.102695
\(256\) 1.00000 0.0625000
\(257\) −29.4035 −1.83414 −0.917069 0.398729i \(-0.869451\pi\)
−0.917069 + 0.398729i \(0.869451\pi\)
\(258\) 1.83141 0.114019
\(259\) 12.4961 0.776470
\(260\) 3.70263 0.229627
\(261\) 4.95993 0.307012
\(262\) −1.62818 −0.100589
\(263\) 20.3220 1.25311 0.626554 0.779378i \(-0.284465\pi\)
0.626554 + 0.779378i \(0.284465\pi\)
\(264\) 1.68243 0.103547
\(265\) 9.93521 0.610315
\(266\) 28.1310 1.72482
\(267\) 11.9919 0.733892
\(268\) −5.10207 −0.311659
\(269\) 7.76046 0.473164 0.236582 0.971612i \(-0.423973\pi\)
0.236582 + 0.971612i \(0.423973\pi\)
\(270\) 4.54636 0.276683
\(271\) 0.473929 0.0287891 0.0143946 0.999896i \(-0.495418\pi\)
0.0143946 + 0.999896i \(0.495418\pi\)
\(272\) 1.42510 0.0864097
\(273\) −23.6186 −1.42946
\(274\) −6.36086 −0.384274
\(275\) 5.11706 0.308571
\(276\) −1.60644 −0.0966964
\(277\) 6.44464 0.387221 0.193611 0.981078i \(-0.437980\pi\)
0.193611 + 0.981078i \(0.437980\pi\)
\(278\) −16.7084 −1.00210
\(279\) −4.84198 −0.289882
\(280\) 2.87959 0.172088
\(281\) −17.6878 −1.05517 −0.527583 0.849504i \(-0.676901\pi\)
−0.527583 + 0.849504i \(0.676901\pi\)
\(282\) −9.47827 −0.564423
\(283\) −10.8789 −0.646684 −0.323342 0.946282i \(-0.604806\pi\)
−0.323342 + 0.946282i \(0.604806\pi\)
\(284\) −6.40788 −0.380238
\(285\) 9.03674 0.535290
\(286\) 5.41349 0.320106
\(287\) 14.3450 0.846756
\(288\) 0.950869 0.0560305
\(289\) −14.9691 −0.880534
\(290\) 4.19316 0.246231
\(291\) 9.74464 0.571241
\(292\) −0.792319 −0.0463670
\(293\) −9.02055 −0.526986 −0.263493 0.964661i \(-0.584875\pi\)
−0.263493 + 0.964661i \(0.584875\pi\)
\(294\) −8.34817 −0.486875
\(295\) 3.27860 0.190887
\(296\) 3.48843 0.202761
\(297\) 6.64708 0.385702
\(298\) −9.78960 −0.567096
\(299\) −5.16897 −0.298929
\(300\) −6.23236 −0.359825
\(301\) 4.58296 0.264157
\(302\) 18.6569 1.07358
\(303\) 21.6275 1.24247
\(304\) 7.85309 0.450406
\(305\) −4.39312 −0.251550
\(306\) 1.35509 0.0774653
\(307\) −32.9966 −1.88321 −0.941607 0.336715i \(-0.890684\pi\)
−0.941607 + 0.336715i \(0.890684\pi\)
\(308\) 4.21015 0.239896
\(309\) 2.76810 0.157472
\(310\) −4.09344 −0.232492
\(311\) 7.35795 0.417231 0.208616 0.977998i \(-0.433104\pi\)
0.208616 + 0.977998i \(0.433104\pi\)
\(312\) −6.59339 −0.373277
\(313\) 11.9851 0.677435 0.338718 0.940888i \(-0.390007\pi\)
0.338718 + 0.940888i \(0.390007\pi\)
\(314\) −8.84383 −0.499086
\(315\) 2.73811 0.154275
\(316\) 7.21938 0.406122
\(317\) −12.4797 −0.700927 −0.350464 0.936576i \(-0.613976\pi\)
−0.350464 + 0.936576i \(0.613976\pi\)
\(318\) −17.6920 −0.992115
\(319\) 6.13067 0.343252
\(320\) 0.803871 0.0449377
\(321\) 6.91713 0.386077
\(322\) −4.01999 −0.224025
\(323\) 11.1915 0.622711
\(324\) −5.24324 −0.291291
\(325\) −20.0536 −1.11237
\(326\) 25.4226 1.40803
\(327\) −9.74658 −0.538987
\(328\) 4.00456 0.221115
\(329\) −23.7186 −1.30765
\(330\) 1.35246 0.0744504
\(331\) −19.8116 −1.08894 −0.544471 0.838780i \(-0.683270\pi\)
−0.544471 + 0.838780i \(0.683270\pi\)
\(332\) 6.15613 0.337862
\(333\) 3.31704 0.181773
\(334\) −5.46543 −0.299055
\(335\) −4.10140 −0.224084
\(336\) −5.12778 −0.279743
\(337\) 21.8413 1.18977 0.594885 0.803811i \(-0.297198\pi\)
0.594885 + 0.803811i \(0.297198\pi\)
\(338\) −8.21526 −0.446851
\(339\) 19.9838 1.08537
\(340\) 1.14560 0.0621289
\(341\) −5.98488 −0.324099
\(342\) 7.46727 0.403783
\(343\) 4.18450 0.225942
\(344\) 1.27939 0.0689798
\(345\) −1.29137 −0.0695251
\(346\) 18.0771 0.971828
\(347\) 0.811855 0.0435827 0.0217913 0.999763i \(-0.493063\pi\)
0.0217913 + 0.999763i \(0.493063\pi\)
\(348\) −7.46689 −0.400267
\(349\) −11.5066 −0.615932 −0.307966 0.951397i \(-0.599648\pi\)
−0.307966 + 0.951397i \(0.599648\pi\)
\(350\) −15.5960 −0.833639
\(351\) −26.0496 −1.39043
\(352\) 1.17531 0.0626443
\(353\) −10.3775 −0.552341 −0.276171 0.961109i \(-0.589065\pi\)
−0.276171 + 0.961109i \(0.589065\pi\)
\(354\) −5.83831 −0.310302
\(355\) −5.15111 −0.273392
\(356\) 8.37728 0.443995
\(357\) −7.30763 −0.386761
\(358\) 4.58739 0.242451
\(359\) 1.98848 0.104948 0.0524740 0.998622i \(-0.483289\pi\)
0.0524740 + 0.998622i \(0.483289\pi\)
\(360\) 0.764376 0.0402861
\(361\) 42.6711 2.24585
\(362\) −19.1771 −1.00793
\(363\) −13.7689 −0.722679
\(364\) −16.4994 −0.864805
\(365\) −0.636922 −0.0333380
\(366\) 7.82298 0.408914
\(367\) 33.8501 1.76696 0.883481 0.468467i \(-0.155193\pi\)
0.883481 + 0.468467i \(0.155193\pi\)
\(368\) −1.12223 −0.0585000
\(369\) 3.80781 0.198227
\(370\) 2.80425 0.145786
\(371\) −44.2726 −2.29852
\(372\) 7.28932 0.377933
\(373\) −17.3730 −0.899542 −0.449771 0.893144i \(-0.648494\pi\)
−0.449771 + 0.893144i \(0.648494\pi\)
\(374\) 1.67494 0.0866092
\(375\) −10.7636 −0.555831
\(376\) −6.62131 −0.341468
\(377\) −24.0259 −1.23740
\(378\) −20.2592 −1.04202
\(379\) 14.8045 0.760457 0.380229 0.924893i \(-0.375845\pi\)
0.380229 + 0.924893i \(0.375845\pi\)
\(380\) 6.31287 0.323843
\(381\) −27.2883 −1.39802
\(382\) 11.2058 0.573340
\(383\) 17.7812 0.908578 0.454289 0.890854i \(-0.349893\pi\)
0.454289 + 0.890854i \(0.349893\pi\)
\(384\) −1.43148 −0.0730498
\(385\) 3.38442 0.172486
\(386\) −22.9269 −1.16695
\(387\) 1.21653 0.0618396
\(388\) 6.80739 0.345593
\(389\) −21.2040 −1.07508 −0.537542 0.843237i \(-0.680647\pi\)
−0.537542 + 0.843237i \(0.680647\pi\)
\(390\) −5.30024 −0.268388
\(391\) −1.59929 −0.0808795
\(392\) −5.83185 −0.294553
\(393\) 2.33071 0.117569
\(394\) −10.4987 −0.528918
\(395\) 5.80345 0.292003
\(396\) 1.11757 0.0561599
\(397\) 26.7934 1.34472 0.672362 0.740223i \(-0.265280\pi\)
0.672362 + 0.740223i \(0.265280\pi\)
\(398\) −20.8739 −1.04631
\(399\) −40.2689 −2.01597
\(400\) −4.35379 −0.217690
\(401\) 19.3584 0.966710 0.483355 0.875424i \(-0.339418\pi\)
0.483355 + 0.875424i \(0.339418\pi\)
\(402\) 7.30350 0.364266
\(403\) 23.4545 1.16835
\(404\) 15.1085 0.751676
\(405\) −4.21489 −0.209439
\(406\) −18.6853 −0.927335
\(407\) 4.09999 0.203229
\(408\) −2.04001 −0.100995
\(409\) 5.17694 0.255983 0.127992 0.991775i \(-0.459147\pi\)
0.127992 + 0.991775i \(0.459147\pi\)
\(410\) 3.21915 0.158982
\(411\) 9.10544 0.449138
\(412\) 1.93373 0.0952683
\(413\) −14.6099 −0.718905
\(414\) −1.06709 −0.0524446
\(415\) 4.94874 0.242924
\(416\) −4.60600 −0.225828
\(417\) 23.9177 1.17126
\(418\) 9.22983 0.451446
\(419\) 13.5001 0.659523 0.329761 0.944064i \(-0.393032\pi\)
0.329761 + 0.944064i \(0.393032\pi\)
\(420\) −4.12207 −0.201137
\(421\) −20.3494 −0.991767 −0.495884 0.868389i \(-0.665156\pi\)
−0.495884 + 0.868389i \(0.665156\pi\)
\(422\) −17.1402 −0.834371
\(423\) −6.29600 −0.306122
\(424\) −12.3592 −0.600217
\(425\) −6.20461 −0.300968
\(426\) 9.17274 0.444421
\(427\) 19.5764 0.947366
\(428\) 4.83216 0.233571
\(429\) −7.74929 −0.374139
\(430\) 1.02846 0.0495968
\(431\) 0.00245142 0.000118081 0 5.90404e−5 1.00000i \(-0.499981\pi\)
5.90404e−5 1.00000i \(0.499981\pi\)
\(432\) −5.65558 −0.272104
\(433\) 13.4348 0.645638 0.322819 0.946461i \(-0.395370\pi\)
0.322819 + 0.946461i \(0.395370\pi\)
\(434\) 18.2409 0.875592
\(435\) −6.00242 −0.287794
\(436\) −6.80875 −0.326080
\(437\) −8.81294 −0.421580
\(438\) 1.13419 0.0541936
\(439\) 29.4126 1.40379 0.701894 0.712281i \(-0.252338\pi\)
0.701894 + 0.712281i \(0.252338\pi\)
\(440\) 0.944799 0.0450415
\(441\) −5.54533 −0.264063
\(442\) −6.56404 −0.312219
\(443\) 25.2124 1.19788 0.598940 0.800794i \(-0.295589\pi\)
0.598940 + 0.800794i \(0.295589\pi\)
\(444\) −4.99361 −0.236986
\(445\) 6.73425 0.319234
\(446\) −8.10757 −0.383904
\(447\) 14.0136 0.662821
\(448\) −3.58216 −0.169241
\(449\) −6.24356 −0.294652 −0.147326 0.989088i \(-0.547067\pi\)
−0.147326 + 0.989088i \(0.547067\pi\)
\(450\) −4.13989 −0.195156
\(451\) 4.70660 0.221625
\(452\) 13.9602 0.656635
\(453\) −26.7070 −1.25480
\(454\) −6.14699 −0.288493
\(455\) −13.2634 −0.621798
\(456\) −11.2415 −0.526433
\(457\) 22.7691 1.06509 0.532546 0.846401i \(-0.321235\pi\)
0.532546 + 0.846401i \(0.321235\pi\)
\(458\) −13.2543 −0.619334
\(459\) −8.05980 −0.376199
\(460\) −0.902124 −0.0420618
\(461\) −23.9176 −1.11395 −0.556976 0.830529i \(-0.688038\pi\)
−0.556976 + 0.830529i \(0.688038\pi\)
\(462\) −6.02674 −0.280389
\(463\) 30.7084 1.42714 0.713570 0.700584i \(-0.247077\pi\)
0.713570 + 0.700584i \(0.247077\pi\)
\(464\) −5.21621 −0.242156
\(465\) 5.85967 0.271736
\(466\) −29.9547 −1.38762
\(467\) 23.7542 1.09921 0.549606 0.835424i \(-0.314778\pi\)
0.549606 + 0.835424i \(0.314778\pi\)
\(468\) −4.37971 −0.202452
\(469\) 18.2764 0.843926
\(470\) −5.32268 −0.245517
\(471\) 12.6598 0.583331
\(472\) −4.07851 −0.187729
\(473\) 1.50368 0.0691391
\(474\) −10.3344 −0.474674
\(475\) −34.1907 −1.56878
\(476\) −5.10495 −0.233985
\(477\) −11.7520 −0.538087
\(478\) −9.68475 −0.442970
\(479\) 39.1600 1.78926 0.894632 0.446803i \(-0.147438\pi\)
0.894632 + 0.446803i \(0.147438\pi\)
\(480\) −1.15072 −0.0525231
\(481\) −16.0677 −0.732625
\(482\) −23.5432 −1.07236
\(483\) 5.75453 0.261840
\(484\) −9.61864 −0.437211
\(485\) 5.47226 0.248483
\(486\) −9.46117 −0.429167
\(487\) −29.7377 −1.34754 −0.673771 0.738940i \(-0.735327\pi\)
−0.673771 + 0.738940i \(0.735327\pi\)
\(488\) 5.46496 0.247387
\(489\) −36.3919 −1.64570
\(490\) −4.68805 −0.211785
\(491\) 29.0228 1.30978 0.654891 0.755723i \(-0.272714\pi\)
0.654891 + 0.755723i \(0.272714\pi\)
\(492\) −5.73244 −0.258438
\(493\) −7.43365 −0.334795
\(494\) −36.1714 −1.62743
\(495\) 0.898380 0.0403792
\(496\) 5.09216 0.228645
\(497\) 22.9540 1.02963
\(498\) −8.81237 −0.394892
\(499\) −34.3533 −1.53786 −0.768932 0.639330i \(-0.779212\pi\)
−0.768932 + 0.639330i \(0.779212\pi\)
\(500\) −7.51924 −0.336271
\(501\) 7.82365 0.349535
\(502\) −15.0546 −0.671921
\(503\) 13.4589 0.600103 0.300052 0.953923i \(-0.402996\pi\)
0.300052 + 0.953923i \(0.402996\pi\)
\(504\) −3.40616 −0.151723
\(505\) 12.1453 0.540458
\(506\) −1.31897 −0.0586351
\(507\) 11.7600 0.522278
\(508\) −19.0630 −0.845784
\(509\) 27.1714 1.20435 0.602176 0.798363i \(-0.294301\pi\)
0.602176 + 0.798363i \(0.294301\pi\)
\(510\) −1.63990 −0.0726161
\(511\) 2.83821 0.125555
\(512\) −1.00000 −0.0441942
\(513\) −44.4138 −1.96092
\(514\) 29.4035 1.29693
\(515\) 1.55447 0.0684982
\(516\) −1.83141 −0.0806234
\(517\) −7.78211 −0.342257
\(518\) −12.4961 −0.549047
\(519\) −25.8769 −1.13587
\(520\) −3.70263 −0.162371
\(521\) −10.6612 −0.467077 −0.233539 0.972348i \(-0.575031\pi\)
−0.233539 + 0.972348i \(0.575031\pi\)
\(522\) −4.95993 −0.217090
\(523\) 35.0878 1.53428 0.767140 0.641480i \(-0.221679\pi\)
0.767140 + 0.641480i \(0.221679\pi\)
\(524\) 1.62818 0.0711274
\(525\) 22.3253 0.974356
\(526\) −20.3220 −0.886081
\(527\) 7.25686 0.316114
\(528\) −1.68243 −0.0732185
\(529\) −21.7406 −0.945244
\(530\) −9.93521 −0.431558
\(531\) −3.87813 −0.168297
\(532\) −28.1310 −1.21963
\(533\) −18.4450 −0.798942
\(534\) −11.9919 −0.518940
\(535\) 3.88443 0.167939
\(536\) 5.10207 0.220376
\(537\) −6.56674 −0.283376
\(538\) −7.76046 −0.334577
\(539\) −6.85424 −0.295233
\(540\) −4.54636 −0.195644
\(541\) −43.6364 −1.87607 −0.938037 0.346535i \(-0.887358\pi\)
−0.938037 + 0.346535i \(0.887358\pi\)
\(542\) −0.473929 −0.0203570
\(543\) 27.4516 1.17806
\(544\) −1.42510 −0.0611009
\(545\) −5.47335 −0.234453
\(546\) 23.6186 1.01078
\(547\) 17.6908 0.756403 0.378202 0.925723i \(-0.376543\pi\)
0.378202 + 0.925723i \(0.376543\pi\)
\(548\) 6.36086 0.271723
\(549\) 5.19647 0.221780
\(550\) −5.11706 −0.218192
\(551\) −40.9634 −1.74510
\(552\) 1.60644 0.0683747
\(553\) −25.8610 −1.09972
\(554\) −6.44464 −0.273807
\(555\) −4.01422 −0.170394
\(556\) 16.7084 0.708595
\(557\) 28.2587 1.19736 0.598679 0.800989i \(-0.295693\pi\)
0.598679 + 0.800989i \(0.295693\pi\)
\(558\) 4.84198 0.204977
\(559\) −5.89285 −0.249241
\(560\) −2.87959 −0.121685
\(561\) −2.39764 −0.101229
\(562\) 17.6878 0.746114
\(563\) 32.2465 1.35903 0.679514 0.733663i \(-0.262191\pi\)
0.679514 + 0.733663i \(0.262191\pi\)
\(564\) 9.47827 0.399107
\(565\) 11.2222 0.472123
\(566\) 10.8789 0.457274
\(567\) 18.7821 0.788774
\(568\) 6.40788 0.268869
\(569\) −32.0928 −1.34540 −0.672699 0.739916i \(-0.734865\pi\)
−0.672699 + 0.739916i \(0.734865\pi\)
\(570\) −9.03674 −0.378507
\(571\) −4.02161 −0.168299 −0.0841496 0.996453i \(-0.526817\pi\)
−0.0841496 + 0.996453i \(0.526817\pi\)
\(572\) −5.41349 −0.226349
\(573\) −16.0409 −0.670119
\(574\) −14.3450 −0.598747
\(575\) 4.88594 0.203758
\(576\) −0.950869 −0.0396196
\(577\) −15.4175 −0.641839 −0.320920 0.947106i \(-0.603992\pi\)
−0.320920 + 0.947106i \(0.603992\pi\)
\(578\) 14.9691 0.622631
\(579\) 32.8193 1.36392
\(580\) −4.19316 −0.174111
\(581\) −22.0522 −0.914881
\(582\) −9.74464 −0.403928
\(583\) −14.5259 −0.601603
\(584\) 0.792319 0.0327864
\(585\) −3.52072 −0.145564
\(586\) 9.02055 0.372635
\(587\) 34.4441 1.42166 0.710831 0.703363i \(-0.248319\pi\)
0.710831 + 0.703363i \(0.248319\pi\)
\(588\) 8.34817 0.344273
\(589\) 39.9892 1.64773
\(590\) −3.27860 −0.134978
\(591\) 15.0287 0.618198
\(592\) −3.48843 −0.143374
\(593\) −33.7207 −1.38474 −0.692371 0.721542i \(-0.743434\pi\)
−0.692371 + 0.721542i \(0.743434\pi\)
\(594\) −6.64708 −0.272733
\(595\) −4.10372 −0.168236
\(596\) 9.78960 0.400998
\(597\) 29.8806 1.22293
\(598\) 5.16897 0.211375
\(599\) −27.4060 −1.11978 −0.559888 0.828568i \(-0.689156\pi\)
−0.559888 + 0.828568i \(0.689156\pi\)
\(600\) 6.23236 0.254435
\(601\) −31.4801 −1.28410 −0.642049 0.766663i \(-0.721915\pi\)
−0.642049 + 0.766663i \(0.721915\pi\)
\(602\) −4.58296 −0.186787
\(603\) 4.85140 0.197564
\(604\) −18.6569 −0.759138
\(605\) −7.73214 −0.314356
\(606\) −21.6275 −0.878557
\(607\) −45.0275 −1.82761 −0.913806 0.406151i \(-0.866871\pi\)
−0.913806 + 0.406151i \(0.866871\pi\)
\(608\) −7.85309 −0.318485
\(609\) 26.7476 1.08387
\(610\) 4.39312 0.177872
\(611\) 30.4978 1.23381
\(612\) −1.35509 −0.0547762
\(613\) −15.0761 −0.608919 −0.304459 0.952525i \(-0.598476\pi\)
−0.304459 + 0.952525i \(0.598476\pi\)
\(614\) 32.9966 1.33163
\(615\) −4.60814 −0.185818
\(616\) −4.21015 −0.169632
\(617\) 33.3062 1.34086 0.670429 0.741974i \(-0.266110\pi\)
0.670429 + 0.741974i \(0.266110\pi\)
\(618\) −2.76810 −0.111349
\(619\) 32.7845 1.31772 0.658860 0.752266i \(-0.271039\pi\)
0.658860 + 0.752266i \(0.271039\pi\)
\(620\) 4.09344 0.164396
\(621\) 6.34684 0.254690
\(622\) −7.35795 −0.295027
\(623\) −30.0087 −1.20227
\(624\) 6.59339 0.263947
\(625\) 15.7245 0.628979
\(626\) −11.9851 −0.479019
\(627\) −13.2123 −0.527649
\(628\) 8.84383 0.352907
\(629\) −4.97138 −0.198222
\(630\) −2.73811 −0.109089
\(631\) −22.4125 −0.892227 −0.446114 0.894976i \(-0.647192\pi\)
−0.446114 + 0.894976i \(0.647192\pi\)
\(632\) −7.21938 −0.287172
\(633\) 24.5358 0.975211
\(634\) 12.4797 0.495630
\(635\) −15.3242 −0.608122
\(636\) 17.6920 0.701532
\(637\) 26.8615 1.06429
\(638\) −6.13067 −0.242716
\(639\) 6.09306 0.241038
\(640\) −0.803871 −0.0317758
\(641\) −34.4552 −1.36090 −0.680450 0.732795i \(-0.738216\pi\)
−0.680450 + 0.732795i \(0.738216\pi\)
\(642\) −6.91713 −0.272997
\(643\) −41.9310 −1.65360 −0.826798 0.562499i \(-0.809840\pi\)
−0.826798 + 0.562499i \(0.809840\pi\)
\(644\) 4.01999 0.158410
\(645\) −1.47222 −0.0579686
\(646\) −11.1915 −0.440323
\(647\) −31.9677 −1.25678 −0.628390 0.777899i \(-0.716286\pi\)
−0.628390 + 0.777899i \(0.716286\pi\)
\(648\) 5.24324 0.205974
\(649\) −4.79353 −0.188162
\(650\) 20.0536 0.786566
\(651\) −26.1115 −1.02339
\(652\) −25.4226 −0.995625
\(653\) 43.3264 1.69549 0.847747 0.530400i \(-0.177958\pi\)
0.847747 + 0.530400i \(0.177958\pi\)
\(654\) 9.74658 0.381121
\(655\) 1.30885 0.0511409
\(656\) −4.00456 −0.156352
\(657\) 0.753392 0.0293926
\(658\) 23.7186 0.924647
\(659\) −4.10123 −0.159761 −0.0798806 0.996804i \(-0.525454\pi\)
−0.0798806 + 0.996804i \(0.525454\pi\)
\(660\) −1.35246 −0.0526444
\(661\) −33.8905 −1.31819 −0.659094 0.752061i \(-0.729060\pi\)
−0.659094 + 0.752061i \(0.729060\pi\)
\(662\) 19.8116 0.769998
\(663\) 9.39628 0.364921
\(664\) −6.15613 −0.238904
\(665\) −22.6137 −0.876922
\(666\) −3.31704 −0.128533
\(667\) 5.85376 0.226659
\(668\) 5.46543 0.211464
\(669\) 11.6058 0.448707
\(670\) 4.10140 0.158451
\(671\) 6.42304 0.247959
\(672\) 5.12778 0.197808
\(673\) −33.8252 −1.30387 −0.651933 0.758276i \(-0.726042\pi\)
−0.651933 + 0.758276i \(0.726042\pi\)
\(674\) −21.8413 −0.841294
\(675\) 24.6232 0.947749
\(676\) 8.21526 0.315972
\(677\) −19.0158 −0.730838 −0.365419 0.930843i \(-0.619074\pi\)
−0.365419 + 0.930843i \(0.619074\pi\)
\(678\) −19.9838 −0.767473
\(679\) −24.3852 −0.935816
\(680\) −1.14560 −0.0439318
\(681\) 8.79929 0.337189
\(682\) 5.98488 0.229173
\(683\) −21.2328 −0.812452 −0.406226 0.913773i \(-0.633155\pi\)
−0.406226 + 0.913773i \(0.633155\pi\)
\(684\) −7.46727 −0.285518
\(685\) 5.11331 0.195370
\(686\) −4.18450 −0.159765
\(687\) 18.9733 0.723876
\(688\) −1.27939 −0.0487761
\(689\) 56.9266 2.16873
\(690\) 1.29137 0.0491617
\(691\) 51.5610 1.96147 0.980736 0.195339i \(-0.0625806\pi\)
0.980736 + 0.195339i \(0.0625806\pi\)
\(692\) −18.0771 −0.687186
\(693\) −4.00330 −0.152073
\(694\) −0.811855 −0.0308176
\(695\) 13.4314 0.509482
\(696\) 7.46689 0.283032
\(697\) −5.70691 −0.216165
\(698\) 11.5066 0.435530
\(699\) 42.8795 1.62185
\(700\) 15.5960 0.589472
\(701\) 24.8228 0.937546 0.468773 0.883319i \(-0.344696\pi\)
0.468773 + 0.883319i \(0.344696\pi\)
\(702\) 26.0496 0.983180
\(703\) −27.3950 −1.03322
\(704\) −1.17531 −0.0442962
\(705\) 7.61930 0.286959
\(706\) 10.3775 0.390564
\(707\) −54.1210 −2.03543
\(708\) 5.83831 0.219417
\(709\) −0.620603 −0.0233072 −0.0116536 0.999932i \(-0.503710\pi\)
−0.0116536 + 0.999932i \(0.503710\pi\)
\(710\) 5.15111 0.193318
\(711\) −6.86469 −0.257446
\(712\) −8.37728 −0.313952
\(713\) −5.71455 −0.214012
\(714\) 7.30763 0.273481
\(715\) −4.35175 −0.162746
\(716\) −4.58739 −0.171439
\(717\) 13.8635 0.517742
\(718\) −1.98848 −0.0742094
\(719\) 34.9883 1.30484 0.652422 0.757856i \(-0.273753\pi\)
0.652422 + 0.757856i \(0.273753\pi\)
\(720\) −0.764376 −0.0284866
\(721\) −6.92694 −0.257973
\(722\) −42.6711 −1.58805
\(723\) 33.7016 1.25337
\(724\) 19.1771 0.712711
\(725\) 22.7103 0.843439
\(726\) 13.7689 0.511011
\(727\) 42.4243 1.57343 0.786715 0.617316i \(-0.211780\pi\)
0.786715 + 0.617316i \(0.211780\pi\)
\(728\) 16.4994 0.611509
\(729\) 29.2732 1.08419
\(730\) 0.636922 0.0235735
\(731\) −1.82326 −0.0674356
\(732\) −7.82298 −0.289146
\(733\) −27.3968 −1.01192 −0.505962 0.862555i \(-0.668863\pi\)
−0.505962 + 0.862555i \(0.668863\pi\)
\(734\) −33.8501 −1.24943
\(735\) 6.71085 0.247533
\(736\) 1.12223 0.0413658
\(737\) 5.99652 0.220885
\(738\) −3.80781 −0.140167
\(739\) 40.9798 1.50747 0.753733 0.657180i \(-0.228251\pi\)
0.753733 + 0.657180i \(0.228251\pi\)
\(740\) −2.80425 −0.103086
\(741\) 51.7785 1.90213
\(742\) 44.2726 1.62530
\(743\) 1.95166 0.0715995 0.0357998 0.999359i \(-0.488602\pi\)
0.0357998 + 0.999359i \(0.488602\pi\)
\(744\) −7.28932 −0.267239
\(745\) 7.86957 0.288319
\(746\) 17.3730 0.636072
\(747\) −5.85368 −0.214175
\(748\) −1.67494 −0.0612420
\(749\) −17.3096 −0.632477
\(750\) 10.7636 0.393032
\(751\) 26.8155 0.978511 0.489256 0.872140i \(-0.337268\pi\)
0.489256 + 0.872140i \(0.337268\pi\)
\(752\) 6.62131 0.241454
\(753\) 21.5504 0.785340
\(754\) 24.0259 0.874971
\(755\) −14.9977 −0.545823
\(756\) 20.2592 0.736820
\(757\) −6.13139 −0.222849 −0.111425 0.993773i \(-0.535541\pi\)
−0.111425 + 0.993773i \(0.535541\pi\)
\(758\) −14.8045 −0.537725
\(759\) 1.88807 0.0685326
\(760\) −6.31287 −0.228992
\(761\) 5.93684 0.215210 0.107605 0.994194i \(-0.465682\pi\)
0.107605 + 0.994194i \(0.465682\pi\)
\(762\) 27.2883 0.988550
\(763\) 24.3900 0.882978
\(764\) −11.2058 −0.405413
\(765\) −1.08932 −0.0393843
\(766\) −17.7812 −0.642462
\(767\) 18.7856 0.678310
\(768\) 1.43148 0.0516540
\(769\) −13.3517 −0.481473 −0.240737 0.970591i \(-0.577389\pi\)
−0.240737 + 0.970591i \(0.577389\pi\)
\(770\) −3.38442 −0.121966
\(771\) −42.0904 −1.51585
\(772\) 22.9269 0.825156
\(773\) 39.3531 1.41543 0.707716 0.706497i \(-0.249726\pi\)
0.707716 + 0.706497i \(0.249726\pi\)
\(774\) −1.21653 −0.0437272
\(775\) −22.1702 −0.796377
\(776\) −6.80739 −0.244371
\(777\) 17.8879 0.641725
\(778\) 21.2040 0.760200
\(779\) −31.4482 −1.12675
\(780\) 5.30024 0.189779
\(781\) 7.53126 0.269490
\(782\) 1.59929 0.0571905
\(783\) 29.5007 1.05427
\(784\) 5.83185 0.208280
\(785\) 7.10930 0.253742
\(786\) −2.33071 −0.0831336
\(787\) 3.95473 0.140971 0.0704854 0.997513i \(-0.477545\pi\)
0.0704854 + 0.997513i \(0.477545\pi\)
\(788\) 10.4987 0.374001
\(789\) 29.0905 1.03565
\(790\) −5.80345 −0.206477
\(791\) −50.0078 −1.77807
\(792\) −1.11757 −0.0397111
\(793\) −25.1716 −0.893871
\(794\) −26.7934 −0.950863
\(795\) 14.2220 0.504404
\(796\) 20.8739 0.739856
\(797\) 22.4915 0.796691 0.398346 0.917235i \(-0.369584\pi\)
0.398346 + 0.917235i \(0.369584\pi\)
\(798\) 40.2689 1.42550
\(799\) 9.43607 0.333824
\(800\) 4.35379 0.153930
\(801\) −7.96570 −0.281454
\(802\) −19.3584 −0.683567
\(803\) 0.931222 0.0328621
\(804\) −7.30350 −0.257575
\(805\) 3.23155 0.113897
\(806\) −23.4545 −0.826149
\(807\) 11.1089 0.391053
\(808\) −15.1085 −0.531515
\(809\) −44.7344 −1.57278 −0.786388 0.617733i \(-0.788051\pi\)
−0.786388 + 0.617733i \(0.788051\pi\)
\(810\) 4.21489 0.148096
\(811\) 1.74111 0.0611387 0.0305694 0.999533i \(-0.490268\pi\)
0.0305694 + 0.999533i \(0.490268\pi\)
\(812\) 18.6853 0.655725
\(813\) 0.678419 0.0237932
\(814\) −4.09999 −0.143705
\(815\) −20.4365 −0.715858
\(816\) 2.04001 0.0714145
\(817\) −10.0471 −0.351505
\(818\) −5.17694 −0.181007
\(819\) 15.6888 0.548211
\(820\) −3.21915 −0.112417
\(821\) 27.5669 0.962091 0.481045 0.876696i \(-0.340257\pi\)
0.481045 + 0.876696i \(0.340257\pi\)
\(822\) −9.10544 −0.317589
\(823\) −22.7791 −0.794031 −0.397015 0.917812i \(-0.629954\pi\)
−0.397015 + 0.917812i \(0.629954\pi\)
\(824\) −1.93373 −0.0673648
\(825\) 7.32497 0.255023
\(826\) 14.6099 0.508343
\(827\) 40.8914 1.42193 0.710966 0.703226i \(-0.248258\pi\)
0.710966 + 0.703226i \(0.248258\pi\)
\(828\) 1.06709 0.0370839
\(829\) 0.833934 0.0289637 0.0144819 0.999895i \(-0.495390\pi\)
0.0144819 + 0.999895i \(0.495390\pi\)
\(830\) −4.94874 −0.171773
\(831\) 9.22537 0.320024
\(832\) 4.60600 0.159684
\(833\) 8.31100 0.287959
\(834\) −23.9177 −0.828204
\(835\) 4.39350 0.152043
\(836\) −9.22983 −0.319220
\(837\) −28.7991 −0.995444
\(838\) −13.5001 −0.466353
\(839\) 28.1092 0.970439 0.485219 0.874393i \(-0.338740\pi\)
0.485219 + 0.874393i \(0.338740\pi\)
\(840\) 4.12207 0.142225
\(841\) −1.79116 −0.0617640
\(842\) 20.3494 0.701285
\(843\) −25.3197 −0.872057
\(844\) 17.1402 0.589989
\(845\) 6.60401 0.227185
\(846\) 6.29600 0.216461
\(847\) 34.4555 1.18390
\(848\) 12.3592 0.424417
\(849\) −15.5729 −0.534461
\(850\) 6.20461 0.212816
\(851\) 3.91480 0.134198
\(852\) −9.17274 −0.314253
\(853\) 32.2317 1.10359 0.551796 0.833979i \(-0.313943\pi\)
0.551796 + 0.833979i \(0.313943\pi\)
\(854\) −19.5764 −0.669889
\(855\) −6.00272 −0.205289
\(856\) −4.83216 −0.165160
\(857\) 16.3053 0.556977 0.278489 0.960440i \(-0.410167\pi\)
0.278489 + 0.960440i \(0.410167\pi\)
\(858\) 7.74929 0.264557
\(859\) −46.6831 −1.59281 −0.796404 0.604766i \(-0.793267\pi\)
−0.796404 + 0.604766i \(0.793267\pi\)
\(860\) −1.02846 −0.0350702
\(861\) 20.5345 0.699813
\(862\) −0.00245142 −8.34958e−5 0
\(863\) 11.4098 0.388395 0.194198 0.980962i \(-0.437790\pi\)
0.194198 + 0.980962i \(0.437790\pi\)
\(864\) 5.65558 0.192407
\(865\) −14.5316 −0.494090
\(866\) −13.4348 −0.456535
\(867\) −21.4279 −0.727730
\(868\) −18.2409 −0.619137
\(869\) −8.48502 −0.287835
\(870\) 6.00242 0.203501
\(871\) −23.5001 −0.796272
\(872\) 6.80875 0.230573
\(873\) −6.47294 −0.219076
\(874\) 8.81294 0.298102
\(875\) 26.9351 0.910572
\(876\) −1.13419 −0.0383206
\(877\) −33.4478 −1.12945 −0.564725 0.825279i \(-0.691018\pi\)
−0.564725 + 0.825279i \(0.691018\pi\)
\(878\) −29.4126 −0.992628
\(879\) −12.9127 −0.435535
\(880\) −0.944799 −0.0318492
\(881\) −34.5471 −1.16392 −0.581961 0.813217i \(-0.697714\pi\)
−0.581961 + 0.813217i \(0.697714\pi\)
\(882\) 5.54533 0.186721
\(883\) −26.6240 −0.895968 −0.447984 0.894041i \(-0.647858\pi\)
−0.447984 + 0.894041i \(0.647858\pi\)
\(884\) 6.56404 0.220772
\(885\) 4.69324 0.157762
\(886\) −25.2124 −0.847028
\(887\) 17.5073 0.587839 0.293920 0.955830i \(-0.405040\pi\)
0.293920 + 0.955830i \(0.405040\pi\)
\(888\) 4.99361 0.167575
\(889\) 68.2867 2.29026
\(890\) −6.73425 −0.225732
\(891\) 6.16244 0.206450
\(892\) 8.10757 0.271461
\(893\) 51.9978 1.74004
\(894\) −14.0136 −0.468685
\(895\) −3.68767 −0.123265
\(896\) 3.58216 0.119671
\(897\) −7.39927 −0.247055
\(898\) 6.24356 0.208350
\(899\) −26.5618 −0.885885
\(900\) 4.13989 0.137996
\(901\) 17.6132 0.586780
\(902\) −4.70660 −0.156713
\(903\) 6.56041 0.218317
\(904\) −13.9602 −0.464311
\(905\) 15.4159 0.512442
\(906\) 26.7070 0.887279
\(907\) −19.3253 −0.641687 −0.320843 0.947132i \(-0.603966\pi\)
−0.320843 + 0.947132i \(0.603966\pi\)
\(908\) 6.14699 0.203995
\(909\) −14.3662 −0.476497
\(910\) 13.2634 0.439678
\(911\) 17.4940 0.579601 0.289800 0.957087i \(-0.406411\pi\)
0.289800 + 0.957087i \(0.406411\pi\)
\(912\) 11.2415 0.372244
\(913\) −7.23538 −0.239456
\(914\) −22.7691 −0.753134
\(915\) −6.28866 −0.207897
\(916\) 13.2543 0.437935
\(917\) −5.83240 −0.192603
\(918\) 8.05980 0.266013
\(919\) −26.9194 −0.887990 −0.443995 0.896029i \(-0.646439\pi\)
−0.443995 + 0.896029i \(0.646439\pi\)
\(920\) 0.902124 0.0297422
\(921\) −47.2339 −1.55641
\(922\) 23.9176 0.787683
\(923\) −29.5147 −0.971488
\(924\) 6.02674 0.198265
\(925\) 15.1879 0.499375
\(926\) −30.7084 −1.00914
\(927\) −1.83873 −0.0603918
\(928\) 5.21621 0.171230
\(929\) −21.0988 −0.692230 −0.346115 0.938192i \(-0.612499\pi\)
−0.346115 + 0.938192i \(0.612499\pi\)
\(930\) −5.85967 −0.192146
\(931\) 45.7980 1.50097
\(932\) 29.9547 0.981198
\(933\) 10.5328 0.344827
\(934\) −23.7542 −0.777261
\(935\) −1.34644 −0.0440332
\(936\) 4.37971 0.143155
\(937\) −24.9317 −0.814483 −0.407242 0.913320i \(-0.633509\pi\)
−0.407242 + 0.913320i \(0.633509\pi\)
\(938\) −18.2764 −0.596746
\(939\) 17.1563 0.559876
\(940\) 5.32268 0.173607
\(941\) 10.5116 0.342668 0.171334 0.985213i \(-0.445192\pi\)
0.171334 + 0.985213i \(0.445192\pi\)
\(942\) −12.6598 −0.412477
\(943\) 4.49402 0.146345
\(944\) 4.07851 0.132744
\(945\) 16.2858 0.529776
\(946\) −1.50368 −0.0488888
\(947\) −17.0994 −0.555655 −0.277828 0.960631i \(-0.589614\pi\)
−0.277828 + 0.960631i \(0.589614\pi\)
\(948\) 10.3344 0.335645
\(949\) −3.64942 −0.118465
\(950\) 34.1907 1.10929
\(951\) −17.8644 −0.579291
\(952\) 5.10495 0.165452
\(953\) −30.2850 −0.981027 −0.490514 0.871434i \(-0.663191\pi\)
−0.490514 + 0.871434i \(0.663191\pi\)
\(954\) 11.7520 0.380485
\(955\) −9.00804 −0.291493
\(956\) 9.68475 0.313227
\(957\) 8.77593 0.283685
\(958\) −39.1600 −1.26520
\(959\) −22.7856 −0.735786
\(960\) 1.15072 0.0371394
\(961\) −5.06992 −0.163546
\(962\) 16.0677 0.518044
\(963\) −4.59475 −0.148064
\(964\) 23.5432 0.758275
\(965\) 18.4302 0.593290
\(966\) −5.75453 −0.185149
\(967\) 18.8905 0.607476 0.303738 0.952756i \(-0.401765\pi\)
0.303738 + 0.952756i \(0.401765\pi\)
\(968\) 9.61864 0.309155
\(969\) 16.0204 0.514648
\(970\) −5.47226 −0.175704
\(971\) −49.0867 −1.57527 −0.787633 0.616144i \(-0.788694\pi\)
−0.787633 + 0.616144i \(0.788694\pi\)
\(972\) 9.46117 0.303467
\(973\) −59.8522 −1.91877
\(974\) 29.7377 0.952856
\(975\) −28.7063 −0.919336
\(976\) −5.46496 −0.174929
\(977\) −17.6020 −0.563137 −0.281568 0.959541i \(-0.590855\pi\)
−0.281568 + 0.959541i \(0.590855\pi\)
\(978\) 36.3919 1.16368
\(979\) −9.84591 −0.314677
\(980\) 4.68805 0.149754
\(981\) 6.47423 0.206706
\(982\) −29.0228 −0.926156
\(983\) −3.69837 −0.117960 −0.0589799 0.998259i \(-0.518785\pi\)
−0.0589799 + 0.998259i \(0.518785\pi\)
\(984\) 5.73244 0.182743
\(985\) 8.43961 0.268908
\(986\) 7.43365 0.236736
\(987\) −33.9526 −1.08072
\(988\) 36.1714 1.15076
\(989\) 1.43576 0.0456545
\(990\) −0.898380 −0.0285524
\(991\) 22.2776 0.707671 0.353836 0.935308i \(-0.384877\pi\)
0.353836 + 0.935308i \(0.384877\pi\)
\(992\) −5.09216 −0.161676
\(993\) −28.3598 −0.899972
\(994\) −22.9540 −0.728058
\(995\) 16.7799 0.531959
\(996\) 8.81237 0.279231
\(997\) −42.8107 −1.35583 −0.677915 0.735141i \(-0.737116\pi\)
−0.677915 + 0.735141i \(0.737116\pi\)
\(998\) 34.3533 1.08743
\(999\) 19.7291 0.624201
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 8002.2.a.e.1.53 77
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8002.2.a.e.1.53 77 1.1 even 1 trivial