Properties

Label 4001.2.a.a.1.58
Level $4001$
Weight $2$
Character 4001.1
Self dual yes
Analytic conductor $31.948$
Analytic rank $1$
Dimension $149$
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] = [4001,2,Mod(1,4001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4001, 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("4001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4001.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: \(31.9481458487\)
Analytic rank: \(1\)
Dimension: \(149\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.58
Character \(\chi\) \(=\) 4001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.769245 q^{2} -2.98873 q^{3} -1.40826 q^{4} -1.88149 q^{5} +2.29906 q^{6} +1.15926 q^{7} +2.62179 q^{8} +5.93250 q^{9} +O(q^{10})\) \(q-0.769245 q^{2} -2.98873 q^{3} -1.40826 q^{4} -1.88149 q^{5} +2.29906 q^{6} +1.15926 q^{7} +2.62179 q^{8} +5.93250 q^{9} +1.44733 q^{10} -1.23417 q^{11} +4.20892 q^{12} -1.79142 q^{13} -0.891755 q^{14} +5.62327 q^{15} +0.799730 q^{16} -2.20700 q^{17} -4.56355 q^{18} -5.66298 q^{19} +2.64963 q^{20} -3.46472 q^{21} +0.949379 q^{22} -3.44247 q^{23} -7.83581 q^{24} -1.45999 q^{25} +1.37804 q^{26} -8.76446 q^{27} -1.63254 q^{28} +8.45248 q^{29} -4.32567 q^{30} -4.76215 q^{31} -5.85876 q^{32} +3.68860 q^{33} +1.69772 q^{34} -2.18114 q^{35} -8.35453 q^{36} +6.89479 q^{37} +4.35622 q^{38} +5.35405 q^{39} -4.93287 q^{40} +6.86584 q^{41} +2.66521 q^{42} +7.89845 q^{43} +1.73804 q^{44} -11.1620 q^{45} +2.64810 q^{46} -8.34671 q^{47} -2.39018 q^{48} -5.65611 q^{49} +1.12309 q^{50} +6.59612 q^{51} +2.52278 q^{52} +7.02234 q^{53} +6.74201 q^{54} +2.32208 q^{55} +3.03934 q^{56} +16.9251 q^{57} -6.50203 q^{58} -12.6414 q^{59} -7.91904 q^{60} +1.75090 q^{61} +3.66326 q^{62} +6.87732 q^{63} +2.90736 q^{64} +3.37053 q^{65} -2.83744 q^{66} +7.30899 q^{67} +3.10803 q^{68} +10.2886 q^{69} +1.67783 q^{70} -1.18255 q^{71} +15.5538 q^{72} +1.03781 q^{73} -5.30378 q^{74} +4.36352 q^{75} +7.97497 q^{76} -1.43073 q^{77} -4.11858 q^{78} +15.9912 q^{79} -1.50469 q^{80} +8.39709 q^{81} -5.28151 q^{82} +1.65913 q^{83} +4.87923 q^{84} +4.15245 q^{85} -6.07584 q^{86} -25.2622 q^{87} -3.23573 q^{88} +12.1183 q^{89} +8.58627 q^{90} -2.07672 q^{91} +4.84790 q^{92} +14.2328 q^{93} +6.42066 q^{94} +10.6549 q^{95} +17.5103 q^{96} -4.20548 q^{97} +4.35093 q^{98} -7.32172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9} - 48 q^{10} - 31 q^{11} - 61 q^{12} - 54 q^{13} - 44 q^{14} - 65 q^{15} + 58 q^{16} - 26 q^{17} - 23 q^{18} - 86 q^{19} - 52 q^{20} - 30 q^{21} - 56 q^{22} - 63 q^{23} - 90 q^{24} + 92 q^{25} - 38 q^{26} - 103 q^{27} - 77 q^{28} - 51 q^{29} - 22 q^{30} - 256 q^{31} - 21 q^{32} - 36 q^{33} - 124 q^{34} - 50 q^{35} + 45 q^{36} - 42 q^{37} - 14 q^{38} - 119 q^{39} - 131 q^{40} - 55 q^{41} - 5 q^{42} - 55 q^{43} - 54 q^{44} - 68 q^{45} - 59 q^{46} - 82 q^{47} - 89 q^{48} + 30 q^{49} + 13 q^{50} - 83 q^{51} - 126 q^{52} - 23 q^{53} - 83 q^{54} - 244 q^{55} - 94 q^{56} - 14 q^{57} - 60 q^{58} - 93 q^{59} - 70 q^{60} - 139 q^{61} - 10 q^{62} - 120 q^{63} - 45 q^{64} - 37 q^{65} - 28 q^{66} - 110 q^{67} - 27 q^{68} - 79 q^{69} - 78 q^{70} - 123 q^{71} - 74 q^{73} - 25 q^{74} - 146 q^{75} - 192 q^{76} - q^{77} + 26 q^{78} - 273 q^{79} - 51 q^{80} + 41 q^{81} - 120 q^{82} - 27 q^{83} - 14 q^{84} - 71 q^{85} + 3 q^{86} - 98 q^{87} - 143 q^{88} - 70 q^{89} - 24 q^{90} - 256 q^{91} - 47 q^{92} + 16 q^{93} - 151 q^{94} - 83 q^{95} - 137 q^{96} - 108 q^{97} + 17 q^{98} - 131 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.769245 −0.543938 −0.271969 0.962306i \(-0.587675\pi\)
−0.271969 + 0.962306i \(0.587675\pi\)
\(3\) −2.98873 −1.72554 −0.862772 0.505593i \(-0.831274\pi\)
−0.862772 + 0.505593i \(0.831274\pi\)
\(4\) −1.40826 −0.704131
\(5\) −1.88149 −0.841428 −0.420714 0.907193i \(-0.638220\pi\)
−0.420714 + 0.907193i \(0.638220\pi\)
\(6\) 2.29906 0.938589
\(7\) 1.15926 0.438159 0.219080 0.975707i \(-0.429695\pi\)
0.219080 + 0.975707i \(0.429695\pi\)
\(8\) 2.62179 0.926942
\(9\) 5.93250 1.97750
\(10\) 1.44733 0.457685
\(11\) −1.23417 −0.372116 −0.186058 0.982539i \(-0.559571\pi\)
−0.186058 + 0.982539i \(0.559571\pi\)
\(12\) 4.20892 1.21501
\(13\) −1.79142 −0.496849 −0.248425 0.968651i \(-0.579913\pi\)
−0.248425 + 0.968651i \(0.579913\pi\)
\(14\) −0.891755 −0.238332
\(15\) 5.62327 1.45192
\(16\) 0.799730 0.199933
\(17\) −2.20700 −0.535276 −0.267638 0.963520i \(-0.586243\pi\)
−0.267638 + 0.963520i \(0.586243\pi\)
\(18\) −4.56355 −1.07564
\(19\) −5.66298 −1.29918 −0.649589 0.760286i \(-0.725059\pi\)
−0.649589 + 0.760286i \(0.725059\pi\)
\(20\) 2.64963 0.592476
\(21\) −3.46472 −0.756063
\(22\) 0.949379 0.202408
\(23\) −3.44247 −0.717804 −0.358902 0.933375i \(-0.616849\pi\)
−0.358902 + 0.933375i \(0.616849\pi\)
\(24\) −7.83581 −1.59948
\(25\) −1.45999 −0.291998
\(26\) 1.37804 0.270255
\(27\) −8.76446 −1.68672
\(28\) −1.63254 −0.308522
\(29\) 8.45248 1.56959 0.784793 0.619757i \(-0.212769\pi\)
0.784793 + 0.619757i \(0.212769\pi\)
\(30\) −4.32567 −0.789755
\(31\) −4.76215 −0.855308 −0.427654 0.903943i \(-0.640660\pi\)
−0.427654 + 0.903943i \(0.640660\pi\)
\(32\) −5.85876 −1.03569
\(33\) 3.68860 0.642103
\(34\) 1.69772 0.291157
\(35\) −2.18114 −0.368680
\(36\) −8.35453 −1.39242
\(37\) 6.89479 1.13350 0.566748 0.823891i \(-0.308201\pi\)
0.566748 + 0.823891i \(0.308201\pi\)
\(38\) 4.35622 0.706672
\(39\) 5.35405 0.857335
\(40\) −4.93287 −0.779955
\(41\) 6.86584 1.07226 0.536132 0.844134i \(-0.319885\pi\)
0.536132 + 0.844134i \(0.319885\pi\)
\(42\) 2.66521 0.411252
\(43\) 7.89845 1.20450 0.602251 0.798307i \(-0.294271\pi\)
0.602251 + 0.798307i \(0.294271\pi\)
\(44\) 1.73804 0.262019
\(45\) −11.1620 −1.66393
\(46\) 2.64810 0.390441
\(47\) −8.34671 −1.21749 −0.608747 0.793365i \(-0.708327\pi\)
−0.608747 + 0.793365i \(0.708327\pi\)
\(48\) −2.39018 −0.344992
\(49\) −5.65611 −0.808016
\(50\) 1.12309 0.158829
\(51\) 6.59612 0.923641
\(52\) 2.52278 0.349847
\(53\) 7.02234 0.964592 0.482296 0.876008i \(-0.339803\pi\)
0.482296 + 0.876008i \(0.339803\pi\)
\(54\) 6.74201 0.917472
\(55\) 2.32208 0.313109
\(56\) 3.03934 0.406148
\(57\) 16.9251 2.24179
\(58\) −6.50203 −0.853758
\(59\) −12.6414 −1.64578 −0.822888 0.568204i \(-0.807639\pi\)
−0.822888 + 0.568204i \(0.807639\pi\)
\(60\) −7.91904 −1.02234
\(61\) 1.75090 0.224179 0.112090 0.993698i \(-0.464246\pi\)
0.112090 + 0.993698i \(0.464246\pi\)
\(62\) 3.66326 0.465235
\(63\) 6.87732 0.866461
\(64\) 2.90736 0.363420
\(65\) 3.37053 0.418063
\(66\) −2.83744 −0.349264
\(67\) 7.30899 0.892935 0.446468 0.894800i \(-0.352682\pi\)
0.446468 + 0.894800i \(0.352682\pi\)
\(68\) 3.10803 0.376904
\(69\) 10.2886 1.23860
\(70\) 1.67783 0.200539
\(71\) −1.18255 −0.140343 −0.0701714 0.997535i \(-0.522355\pi\)
−0.0701714 + 0.997535i \(0.522355\pi\)
\(72\) 15.5538 1.83303
\(73\) 1.03781 0.121466 0.0607330 0.998154i \(-0.480656\pi\)
0.0607330 + 0.998154i \(0.480656\pi\)
\(74\) −5.30378 −0.616551
\(75\) 4.36352 0.503856
\(76\) 7.97497 0.914792
\(77\) −1.43073 −0.163046
\(78\) −4.11858 −0.466337
\(79\) 15.9912 1.79915 0.899575 0.436766i \(-0.143876\pi\)
0.899575 + 0.436766i \(0.143876\pi\)
\(80\) −1.50469 −0.168229
\(81\) 8.39709 0.933010
\(82\) −5.28151 −0.583245
\(83\) 1.65913 0.182113 0.0910565 0.995846i \(-0.470976\pi\)
0.0910565 + 0.995846i \(0.470976\pi\)
\(84\) 4.87923 0.532368
\(85\) 4.15245 0.450396
\(86\) −6.07584 −0.655175
\(87\) −25.2622 −2.70839
\(88\) −3.23573 −0.344930
\(89\) 12.1183 1.28454 0.642269 0.766479i \(-0.277993\pi\)
0.642269 + 0.766479i \(0.277993\pi\)
\(90\) 8.58627 0.905073
\(91\) −2.07672 −0.217699
\(92\) 4.84790 0.505428
\(93\) 14.2328 1.47587
\(94\) 6.42066 0.662241
\(95\) 10.6549 1.09316
\(96\) 17.5103 1.78713
\(97\) −4.20548 −0.427002 −0.213501 0.976943i \(-0.568487\pi\)
−0.213501 + 0.976943i \(0.568487\pi\)
\(98\) 4.35093 0.439511
\(99\) −7.32172 −0.735861
\(100\) 2.05605 0.205605
\(101\) 11.8276 1.17689 0.588447 0.808536i \(-0.299739\pi\)
0.588447 + 0.808536i \(0.299739\pi\)
\(102\) −5.07403 −0.502404
\(103\) 10.5426 1.03880 0.519398 0.854533i \(-0.326156\pi\)
0.519398 + 0.854533i \(0.326156\pi\)
\(104\) −4.69671 −0.460550
\(105\) 6.51883 0.636173
\(106\) −5.40189 −0.524678
\(107\) 10.7514 1.03938 0.519688 0.854356i \(-0.326048\pi\)
0.519688 + 0.854356i \(0.326048\pi\)
\(108\) 12.3427 1.18767
\(109\) 10.3332 0.989736 0.494868 0.868968i \(-0.335216\pi\)
0.494868 + 0.868968i \(0.335216\pi\)
\(110\) −1.78625 −0.170312
\(111\) −20.6067 −1.95590
\(112\) 0.927096 0.0876023
\(113\) −7.11858 −0.669660 −0.334830 0.942279i \(-0.608679\pi\)
−0.334830 + 0.942279i \(0.608679\pi\)
\(114\) −13.0196 −1.21939
\(115\) 6.47697 0.603980
\(116\) −11.9033 −1.10520
\(117\) −10.6276 −0.982520
\(118\) 9.72436 0.895200
\(119\) −2.55849 −0.234536
\(120\) 14.7430 1.34585
\(121\) −9.47682 −0.861529
\(122\) −1.34687 −0.121940
\(123\) −20.5201 −1.85024
\(124\) 6.70636 0.602249
\(125\) 12.1544 1.08712
\(126\) −5.29034 −0.471301
\(127\) 8.35066 0.741001 0.370501 0.928832i \(-0.379186\pi\)
0.370501 + 0.928832i \(0.379186\pi\)
\(128\) 9.48105 0.838015
\(129\) −23.6063 −2.07842
\(130\) −2.59276 −0.227400
\(131\) 7.34443 0.641686 0.320843 0.947132i \(-0.396034\pi\)
0.320843 + 0.947132i \(0.396034\pi\)
\(132\) −5.19452 −0.452125
\(133\) −6.56488 −0.569247
\(134\) −5.62240 −0.485701
\(135\) 16.4903 1.41926
\(136\) −5.78628 −0.496169
\(137\) −5.73568 −0.490032 −0.245016 0.969519i \(-0.578793\pi\)
−0.245016 + 0.969519i \(0.578793\pi\)
\(138\) −7.91445 −0.673722
\(139\) −15.2992 −1.29766 −0.648832 0.760932i \(-0.724742\pi\)
−0.648832 + 0.760932i \(0.724742\pi\)
\(140\) 3.07162 0.259599
\(141\) 24.9461 2.10084
\(142\) 0.909670 0.0763378
\(143\) 2.21091 0.184886
\(144\) 4.74440 0.395367
\(145\) −15.9033 −1.32069
\(146\) −0.798326 −0.0660699
\(147\) 16.9046 1.39427
\(148\) −9.70967 −0.798130
\(149\) 0.636582 0.0521508 0.0260754 0.999660i \(-0.491699\pi\)
0.0260754 + 0.999660i \(0.491699\pi\)
\(150\) −3.35661 −0.274066
\(151\) 6.86672 0.558806 0.279403 0.960174i \(-0.409863\pi\)
0.279403 + 0.960174i \(0.409863\pi\)
\(152\) −14.8471 −1.20426
\(153\) −13.0930 −1.05851
\(154\) 1.10058 0.0886871
\(155\) 8.95995 0.719680
\(156\) −7.53992 −0.603676
\(157\) −13.9086 −1.11003 −0.555014 0.831841i \(-0.687287\pi\)
−0.555014 + 0.831841i \(0.687287\pi\)
\(158\) −12.3011 −0.978626
\(159\) −20.9879 −1.66445
\(160\) 11.0232 0.871461
\(161\) −3.99072 −0.314512
\(162\) −6.45942 −0.507500
\(163\) −5.88216 −0.460726 −0.230363 0.973105i \(-0.573991\pi\)
−0.230363 + 0.973105i \(0.573991\pi\)
\(164\) −9.66891 −0.755015
\(165\) −6.94007 −0.540284
\(166\) −1.27627 −0.0990581
\(167\) 1.75646 0.135919 0.0679597 0.997688i \(-0.478351\pi\)
0.0679597 + 0.997688i \(0.478351\pi\)
\(168\) −9.08375 −0.700827
\(169\) −9.79083 −0.753141
\(170\) −3.19425 −0.244988
\(171\) −33.5957 −2.56913
\(172\) −11.1231 −0.848128
\(173\) 5.40460 0.410904 0.205452 0.978667i \(-0.434134\pi\)
0.205452 + 0.978667i \(0.434134\pi\)
\(174\) 19.4328 1.47320
\(175\) −1.69251 −0.127942
\(176\) −0.987003 −0.0743982
\(177\) 37.7819 2.83986
\(178\) −9.32194 −0.698709
\(179\) −17.8483 −1.33405 −0.667024 0.745036i \(-0.732432\pi\)
−0.667024 + 0.745036i \(0.732432\pi\)
\(180\) 15.7190 1.17162
\(181\) 7.39098 0.549368 0.274684 0.961535i \(-0.411427\pi\)
0.274684 + 0.961535i \(0.411427\pi\)
\(182\) 1.59750 0.118415
\(183\) −5.23296 −0.386831
\(184\) −9.02541 −0.665362
\(185\) −12.9725 −0.953756
\(186\) −10.9485 −0.802783
\(187\) 2.72381 0.199185
\(188\) 11.7544 0.857275
\(189\) −10.1603 −0.739053
\(190\) −8.19619 −0.594614
\(191\) −25.3418 −1.83367 −0.916835 0.399265i \(-0.869265\pi\)
−0.916835 + 0.399265i \(0.869265\pi\)
\(192\) −8.68932 −0.627097
\(193\) −3.08605 −0.222139 −0.111069 0.993813i \(-0.535428\pi\)
−0.111069 + 0.993813i \(0.535428\pi\)
\(194\) 3.23504 0.232263
\(195\) −10.0736 −0.721386
\(196\) 7.96530 0.568950
\(197\) 11.6044 0.826783 0.413391 0.910553i \(-0.364344\pi\)
0.413391 + 0.910553i \(0.364344\pi\)
\(198\) 5.63219 0.400263
\(199\) 14.8121 1.05000 0.525000 0.851102i \(-0.324065\pi\)
0.525000 + 0.851102i \(0.324065\pi\)
\(200\) −3.82779 −0.270665
\(201\) −21.8446 −1.54080
\(202\) −9.09835 −0.640158
\(203\) 9.79863 0.687729
\(204\) −9.28907 −0.650365
\(205\) −12.9180 −0.902233
\(206\) −8.10985 −0.565040
\(207\) −20.4224 −1.41946
\(208\) −1.43265 −0.0993363
\(209\) 6.98909 0.483445
\(210\) −5.01458 −0.346039
\(211\) 11.1856 0.770046 0.385023 0.922907i \(-0.374193\pi\)
0.385023 + 0.922907i \(0.374193\pi\)
\(212\) −9.88930 −0.679200
\(213\) 3.53432 0.242168
\(214\) −8.27044 −0.565356
\(215\) −14.8609 −1.01350
\(216\) −22.9786 −1.56349
\(217\) −5.52058 −0.374761
\(218\) −7.94872 −0.538355
\(219\) −3.10172 −0.209595
\(220\) −3.27010 −0.220470
\(221\) 3.95365 0.265951
\(222\) 15.8516 1.06389
\(223\) −1.30279 −0.0872410 −0.0436205 0.999048i \(-0.513889\pi\)
−0.0436205 + 0.999048i \(0.513889\pi\)
\(224\) −6.79183 −0.453799
\(225\) −8.66140 −0.577427
\(226\) 5.47593 0.364254
\(227\) 2.22961 0.147984 0.0739922 0.997259i \(-0.476426\pi\)
0.0739922 + 0.997259i \(0.476426\pi\)
\(228\) −23.8350 −1.57851
\(229\) −22.5889 −1.49272 −0.746359 0.665544i \(-0.768200\pi\)
−0.746359 + 0.665544i \(0.768200\pi\)
\(230\) −4.98237 −0.328528
\(231\) 4.27605 0.281343
\(232\) 22.1606 1.45492
\(233\) 13.5988 0.890884 0.445442 0.895311i \(-0.353047\pi\)
0.445442 + 0.895311i \(0.353047\pi\)
\(234\) 8.17521 0.534430
\(235\) 15.7043 1.02443
\(236\) 17.8025 1.15884
\(237\) −47.7934 −3.10451
\(238\) 1.96810 0.127573
\(239\) 1.21432 0.0785477 0.0392738 0.999228i \(-0.487496\pi\)
0.0392738 + 0.999228i \(0.487496\pi\)
\(240\) 4.49710 0.290286
\(241\) −23.4075 −1.50781 −0.753906 0.656983i \(-0.771832\pi\)
−0.753906 + 0.656983i \(0.771832\pi\)
\(242\) 7.28999 0.468619
\(243\) 1.19675 0.0767713
\(244\) −2.46572 −0.157852
\(245\) 10.6419 0.679888
\(246\) 15.7850 1.00641
\(247\) 10.1448 0.645495
\(248\) −12.4854 −0.792821
\(249\) −4.95868 −0.314244
\(250\) −9.34972 −0.591328
\(251\) 16.8290 1.06224 0.531118 0.847298i \(-0.321772\pi\)
0.531118 + 0.847298i \(0.321772\pi\)
\(252\) −9.68507 −0.610102
\(253\) 4.24859 0.267106
\(254\) −6.42370 −0.403059
\(255\) −12.4105 −0.777178
\(256\) −13.1080 −0.819248
\(257\) 2.49984 0.155936 0.0779679 0.996956i \(-0.475157\pi\)
0.0779679 + 0.996956i \(0.475157\pi\)
\(258\) 18.1590 1.13053
\(259\) 7.99286 0.496652
\(260\) −4.74659 −0.294371
\(261\) 50.1444 3.10386
\(262\) −5.64966 −0.349037
\(263\) 11.6908 0.720888 0.360444 0.932781i \(-0.382625\pi\)
0.360444 + 0.932781i \(0.382625\pi\)
\(264\) 9.67073 0.595192
\(265\) −13.2125 −0.811635
\(266\) 5.04999 0.309635
\(267\) −36.2183 −2.21653
\(268\) −10.2930 −0.628744
\(269\) 12.9245 0.788019 0.394009 0.919106i \(-0.371088\pi\)
0.394009 + 0.919106i \(0.371088\pi\)
\(270\) −12.6850 −0.771987
\(271\) −25.1070 −1.52514 −0.762570 0.646906i \(-0.776062\pi\)
−0.762570 + 0.646906i \(0.776062\pi\)
\(272\) −1.76500 −0.107019
\(273\) 6.20675 0.375649
\(274\) 4.41214 0.266547
\(275\) 1.80188 0.108657
\(276\) −14.4891 −0.872138
\(277\) 1.61680 0.0971440 0.0485720 0.998820i \(-0.484533\pi\)
0.0485720 + 0.998820i \(0.484533\pi\)
\(278\) 11.7688 0.705849
\(279\) −28.2515 −1.69137
\(280\) −5.71848 −0.341745
\(281\) −0.240678 −0.0143576 −0.00717882 0.999974i \(-0.502285\pi\)
−0.00717882 + 0.999974i \(0.502285\pi\)
\(282\) −19.1896 −1.14273
\(283\) −16.5928 −0.986337 −0.493169 0.869934i \(-0.664161\pi\)
−0.493169 + 0.869934i \(0.664161\pi\)
\(284\) 1.66534 0.0988198
\(285\) −31.8445 −1.88630
\(286\) −1.70073 −0.100566
\(287\) 7.95930 0.469823
\(288\) −34.7571 −2.04808
\(289\) −12.1292 −0.713480
\(290\) 12.2335 0.718376
\(291\) 12.5690 0.736811
\(292\) −1.46150 −0.0855280
\(293\) −25.4600 −1.48739 −0.743695 0.668520i \(-0.766928\pi\)
−0.743695 + 0.668520i \(0.766928\pi\)
\(294\) −13.0038 −0.758395
\(295\) 23.7848 1.38480
\(296\) 18.0767 1.05068
\(297\) 10.8168 0.627657
\(298\) −0.489687 −0.0283668
\(299\) 6.16688 0.356640
\(300\) −6.14498 −0.354781
\(301\) 9.15636 0.527764
\(302\) −5.28219 −0.303956
\(303\) −35.3496 −2.03078
\(304\) −4.52886 −0.259748
\(305\) −3.29430 −0.188631
\(306\) 10.0717 0.575763
\(307\) −24.3963 −1.39237 −0.696186 0.717861i \(-0.745121\pi\)
−0.696186 + 0.717861i \(0.745121\pi\)
\(308\) 2.01484 0.114806
\(309\) −31.5090 −1.79249
\(310\) −6.89239 −0.391462
\(311\) 11.9703 0.678774 0.339387 0.940647i \(-0.389780\pi\)
0.339387 + 0.940647i \(0.389780\pi\)
\(312\) 14.0372 0.794700
\(313\) −15.2036 −0.859361 −0.429680 0.902981i \(-0.641374\pi\)
−0.429680 + 0.902981i \(0.641374\pi\)
\(314\) 10.6991 0.603786
\(315\) −12.9396 −0.729065
\(316\) −22.5198 −1.26684
\(317\) 17.0036 0.955016 0.477508 0.878627i \(-0.341540\pi\)
0.477508 + 0.878627i \(0.341540\pi\)
\(318\) 16.1448 0.905356
\(319\) −10.4318 −0.584069
\(320\) −5.47017 −0.305792
\(321\) −32.1330 −1.79349
\(322\) 3.06984 0.171075
\(323\) 12.4982 0.695418
\(324\) −11.8253 −0.656962
\(325\) 2.61545 0.145079
\(326\) 4.52482 0.250607
\(327\) −30.8830 −1.70783
\(328\) 18.0008 0.993926
\(329\) −9.67602 −0.533456
\(330\) 5.33861 0.293881
\(331\) 15.1664 0.833618 0.416809 0.908994i \(-0.363148\pi\)
0.416809 + 0.908994i \(0.363148\pi\)
\(332\) −2.33649 −0.128231
\(333\) 40.9033 2.24149
\(334\) −1.35115 −0.0739317
\(335\) −13.7518 −0.751341
\(336\) −2.77084 −0.151162
\(337\) 8.01882 0.436813 0.218407 0.975858i \(-0.429914\pi\)
0.218407 + 0.975858i \(0.429914\pi\)
\(338\) 7.53154 0.409662
\(339\) 21.2755 1.15553
\(340\) −5.84774 −0.317138
\(341\) 5.87731 0.318274
\(342\) 25.8433 1.39744
\(343\) −14.6717 −0.792199
\(344\) 20.7081 1.11650
\(345\) −19.3579 −1.04219
\(346\) −4.15746 −0.223506
\(347\) −20.4071 −1.09551 −0.547755 0.836638i \(-0.684518\pi\)
−0.547755 + 0.836638i \(0.684518\pi\)
\(348\) 35.5758 1.90706
\(349\) 3.38606 0.181252 0.0906258 0.995885i \(-0.471113\pi\)
0.0906258 + 0.995885i \(0.471113\pi\)
\(350\) 1.30195 0.0695924
\(351\) 15.7008 0.838046
\(352\) 7.23071 0.385398
\(353\) 34.9227 1.85875 0.929373 0.369141i \(-0.120348\pi\)
0.929373 + 0.369141i \(0.120348\pi\)
\(354\) −29.0635 −1.54471
\(355\) 2.22496 0.118088
\(356\) −17.0658 −0.904483
\(357\) 7.64662 0.404702
\(358\) 13.7297 0.725639
\(359\) −32.1291 −1.69571 −0.847855 0.530228i \(-0.822106\pi\)
−0.847855 + 0.530228i \(0.822106\pi\)
\(360\) −29.2643 −1.54236
\(361\) 13.0694 0.687862
\(362\) −5.68547 −0.298822
\(363\) 28.3237 1.48661
\(364\) 2.92456 0.153289
\(365\) −1.95262 −0.102205
\(366\) 4.02542 0.210412
\(367\) 7.04206 0.367593 0.183796 0.982964i \(-0.441161\pi\)
0.183796 + 0.982964i \(0.441161\pi\)
\(368\) −2.75304 −0.143512
\(369\) 40.7316 2.12040
\(370\) 9.97901 0.518784
\(371\) 8.14072 0.422645
\(372\) −20.0435 −1.03921
\(373\) 4.61216 0.238809 0.119404 0.992846i \(-0.461902\pi\)
0.119404 + 0.992846i \(0.461902\pi\)
\(374\) −2.09528 −0.108344
\(375\) −36.3263 −1.87588
\(376\) −21.8833 −1.12855
\(377\) −15.1419 −0.779848
\(378\) 7.81575 0.401999
\(379\) 4.16167 0.213771 0.106885 0.994271i \(-0.465912\pi\)
0.106885 + 0.994271i \(0.465912\pi\)
\(380\) −15.0048 −0.769732
\(381\) −24.9579 −1.27863
\(382\) 19.4941 0.997403
\(383\) 13.6658 0.698291 0.349145 0.937069i \(-0.386472\pi\)
0.349145 + 0.937069i \(0.386472\pi\)
\(384\) −28.3363 −1.44603
\(385\) 2.69190 0.137192
\(386\) 2.37392 0.120830
\(387\) 46.8576 2.38190
\(388\) 5.92242 0.300665
\(389\) −11.8251 −0.599554 −0.299777 0.954009i \(-0.596912\pi\)
−0.299777 + 0.954009i \(0.596912\pi\)
\(390\) 7.74907 0.392389
\(391\) 7.59751 0.384223
\(392\) −14.8291 −0.748984
\(393\) −21.9505 −1.10726
\(394\) −8.92666 −0.449718
\(395\) −30.0873 −1.51386
\(396\) 10.3109 0.518143
\(397\) 11.6358 0.583985 0.291992 0.956421i \(-0.405682\pi\)
0.291992 + 0.956421i \(0.405682\pi\)
\(398\) −11.3941 −0.571135
\(399\) 19.6206 0.982260
\(400\) −1.16760 −0.0583799
\(401\) 19.9344 0.995475 0.497738 0.867328i \(-0.334164\pi\)
0.497738 + 0.867328i \(0.334164\pi\)
\(402\) 16.8038 0.838099
\(403\) 8.53099 0.424959
\(404\) −16.6564 −0.828689
\(405\) −15.7991 −0.785061
\(406\) −7.53755 −0.374082
\(407\) −8.50934 −0.421792
\(408\) 17.2936 0.856162
\(409\) −10.5743 −0.522867 −0.261434 0.965221i \(-0.584195\pi\)
−0.261434 + 0.965221i \(0.584195\pi\)
\(410\) 9.93711 0.490759
\(411\) 17.1424 0.845572
\(412\) −14.8468 −0.731448
\(413\) −14.6547 −0.721112
\(414\) 15.7098 0.772097
\(415\) −3.12163 −0.153235
\(416\) 10.4955 0.514583
\(417\) 45.7253 2.23918
\(418\) −5.37632 −0.262964
\(419\) −15.6866 −0.766341 −0.383171 0.923678i \(-0.625168\pi\)
−0.383171 + 0.923678i \(0.625168\pi\)
\(420\) −9.18023 −0.447949
\(421\) 22.5092 1.09703 0.548516 0.836140i \(-0.315193\pi\)
0.548516 + 0.836140i \(0.315193\pi\)
\(422\) −8.60444 −0.418857
\(423\) −49.5169 −2.40759
\(424\) 18.4111 0.894121
\(425\) 3.22220 0.156299
\(426\) −2.71876 −0.131724
\(427\) 2.02975 0.0982263
\(428\) −15.1408 −0.731857
\(429\) −6.60781 −0.319028
\(430\) 11.4316 0.551283
\(431\) 30.4291 1.46572 0.732860 0.680380i \(-0.238185\pi\)
0.732860 + 0.680380i \(0.238185\pi\)
\(432\) −7.00920 −0.337230
\(433\) −8.64527 −0.415465 −0.207733 0.978186i \(-0.566608\pi\)
−0.207733 + 0.978186i \(0.566608\pi\)
\(434\) 4.24667 0.203847
\(435\) 47.5306 2.27892
\(436\) −14.5518 −0.696904
\(437\) 19.4946 0.932554
\(438\) 2.38598 0.114007
\(439\) −21.9272 −1.04653 −0.523264 0.852171i \(-0.675286\pi\)
−0.523264 + 0.852171i \(0.675286\pi\)
\(440\) 6.08800 0.290234
\(441\) −33.5549 −1.59785
\(442\) −3.04132 −0.144661
\(443\) −26.9727 −1.28151 −0.640757 0.767744i \(-0.721379\pi\)
−0.640757 + 0.767744i \(0.721379\pi\)
\(444\) 29.0196 1.37721
\(445\) −22.8005 −1.08085
\(446\) 1.00216 0.0474537
\(447\) −1.90257 −0.0899885
\(448\) 3.37039 0.159236
\(449\) 31.0658 1.46608 0.733042 0.680184i \(-0.238100\pi\)
0.733042 + 0.680184i \(0.238100\pi\)
\(450\) 6.66274 0.314084
\(451\) −8.47361 −0.399007
\(452\) 10.0248 0.471529
\(453\) −20.5228 −0.964244
\(454\) −1.71512 −0.0804944
\(455\) 3.90733 0.183178
\(456\) 44.3741 2.07801
\(457\) 12.8893 0.602935 0.301467 0.953477i \(-0.402524\pi\)
0.301467 + 0.953477i \(0.402524\pi\)
\(458\) 17.3764 0.811946
\(459\) 19.3431 0.902861
\(460\) −9.12127 −0.425282
\(461\) 30.0423 1.39921 0.699604 0.714531i \(-0.253360\pi\)
0.699604 + 0.714531i \(0.253360\pi\)
\(462\) −3.28933 −0.153033
\(463\) −4.78800 −0.222517 −0.111259 0.993791i \(-0.535488\pi\)
−0.111259 + 0.993791i \(0.535488\pi\)
\(464\) 6.75971 0.313811
\(465\) −26.7789 −1.24184
\(466\) −10.4608 −0.484586
\(467\) 18.8014 0.870025 0.435013 0.900424i \(-0.356744\pi\)
0.435013 + 0.900424i \(0.356744\pi\)
\(468\) 14.9664 0.691823
\(469\) 8.47302 0.391248
\(470\) −12.0804 −0.557228
\(471\) 41.5691 1.91540
\(472\) −33.1432 −1.52554
\(473\) −9.74803 −0.448215
\(474\) 36.7648 1.68866
\(475\) 8.26791 0.379358
\(476\) 3.60302 0.165144
\(477\) 41.6600 1.90748
\(478\) −0.934107 −0.0427251
\(479\) −27.6614 −1.26388 −0.631941 0.775017i \(-0.717741\pi\)
−0.631941 + 0.775017i \(0.717741\pi\)
\(480\) −32.9454 −1.50374
\(481\) −12.3514 −0.563176
\(482\) 18.0061 0.820156
\(483\) 11.9272 0.542705
\(484\) 13.3459 0.606630
\(485\) 7.91258 0.359292
\(486\) −0.920590 −0.0417588
\(487\) −2.99237 −0.135597 −0.0677986 0.997699i \(-0.521598\pi\)
−0.0677986 + 0.997699i \(0.521598\pi\)
\(488\) 4.59048 0.207801
\(489\) 17.5802 0.795004
\(490\) −8.18625 −0.369817
\(491\) 11.9556 0.539549 0.269774 0.962924i \(-0.413051\pi\)
0.269774 + 0.962924i \(0.413051\pi\)
\(492\) 28.8977 1.30281
\(493\) −18.6546 −0.840161
\(494\) −7.80380 −0.351109
\(495\) 13.7758 0.619174
\(496\) −3.80844 −0.171004
\(497\) −1.37088 −0.0614925
\(498\) 3.81444 0.170929
\(499\) −44.4358 −1.98922 −0.994610 0.103684i \(-0.966937\pi\)
−0.994610 + 0.103684i \(0.966937\pi\)
\(500\) −17.1166 −0.765478
\(501\) −5.24960 −0.234535
\(502\) −12.9456 −0.577790
\(503\) 31.1841 1.39043 0.695215 0.718801i \(-0.255309\pi\)
0.695215 + 0.718801i \(0.255309\pi\)
\(504\) 18.0309 0.803159
\(505\) −22.2536 −0.990273
\(506\) −3.26820 −0.145289
\(507\) 29.2621 1.29958
\(508\) −11.7599 −0.521762
\(509\) 28.6058 1.26793 0.633966 0.773361i \(-0.281426\pi\)
0.633966 + 0.773361i \(0.281426\pi\)
\(510\) 9.54674 0.422737
\(511\) 1.20309 0.0532214
\(512\) −8.87887 −0.392394
\(513\) 49.6330 2.19135
\(514\) −1.92299 −0.0848194
\(515\) −19.8358 −0.874072
\(516\) 33.2439 1.46348
\(517\) 10.3013 0.453049
\(518\) −6.14846 −0.270148
\(519\) −16.1529 −0.709033
\(520\) 8.83682 0.387520
\(521\) −45.2656 −1.98312 −0.991560 0.129646i \(-0.958616\pi\)
−0.991560 + 0.129646i \(0.958616\pi\)
\(522\) −38.5733 −1.68831
\(523\) 37.1552 1.62468 0.812341 0.583182i \(-0.198193\pi\)
0.812341 + 0.583182i \(0.198193\pi\)
\(524\) −10.3429 −0.451831
\(525\) 5.05846 0.220769
\(526\) −8.99312 −0.392119
\(527\) 10.5101 0.457825
\(528\) 2.94989 0.128377
\(529\) −11.1494 −0.484758
\(530\) 10.1636 0.441479
\(531\) −74.9954 −3.25452
\(532\) 9.24507 0.400825
\(533\) −12.2996 −0.532753
\(534\) 27.8608 1.20565
\(535\) −20.2286 −0.874560
\(536\) 19.1626 0.827699
\(537\) 53.3439 2.30196
\(538\) −9.94208 −0.428633
\(539\) 6.98061 0.300676
\(540\) −23.2226 −0.999342
\(541\) −0.284165 −0.0122172 −0.00610861 0.999981i \(-0.501944\pi\)
−0.00610861 + 0.999981i \(0.501944\pi\)
\(542\) 19.3134 0.829581
\(543\) −22.0897 −0.947958
\(544\) 12.9303 0.554381
\(545\) −19.4417 −0.832792
\(546\) −4.77451 −0.204330
\(547\) 30.9706 1.32421 0.662104 0.749412i \(-0.269664\pi\)
0.662104 + 0.749412i \(0.269664\pi\)
\(548\) 8.07735 0.345047
\(549\) 10.3872 0.443315
\(550\) −1.38608 −0.0591028
\(551\) −47.8663 −2.03917
\(552\) 26.9745 1.14811
\(553\) 18.5380 0.788315
\(554\) −1.24371 −0.0528403
\(555\) 38.7712 1.64575
\(556\) 21.5453 0.913726
\(557\) −16.3774 −0.693931 −0.346965 0.937878i \(-0.612788\pi\)
−0.346965 + 0.937878i \(0.612788\pi\)
\(558\) 21.7323 0.920002
\(559\) −14.1494 −0.598456
\(560\) −1.74432 −0.0737111
\(561\) −8.14073 −0.343702
\(562\) 0.185140 0.00780966
\(563\) −13.7536 −0.579644 −0.289822 0.957081i \(-0.593596\pi\)
−0.289822 + 0.957081i \(0.593596\pi\)
\(564\) −35.1306 −1.47927
\(565\) 13.3936 0.563471
\(566\) 12.7639 0.536506
\(567\) 9.73442 0.408807
\(568\) −3.10039 −0.130090
\(569\) −22.1204 −0.927333 −0.463667 0.886010i \(-0.653467\pi\)
−0.463667 + 0.886010i \(0.653467\pi\)
\(570\) 24.4962 1.02603
\(571\) −38.0461 −1.59218 −0.796089 0.605179i \(-0.793102\pi\)
−0.796089 + 0.605179i \(0.793102\pi\)
\(572\) −3.11354 −0.130184
\(573\) 75.7399 3.16408
\(574\) −6.12265 −0.255554
\(575\) 5.02597 0.209597
\(576\) 17.2479 0.718664
\(577\) 5.66941 0.236020 0.118010 0.993012i \(-0.462348\pi\)
0.118010 + 0.993012i \(0.462348\pi\)
\(578\) 9.33029 0.388089
\(579\) 9.22336 0.383310
\(580\) 22.3960 0.929943
\(581\) 1.92336 0.0797945
\(582\) −9.66867 −0.400779
\(583\) −8.66676 −0.358941
\(584\) 2.72090 0.112592
\(585\) 19.9957 0.826720
\(586\) 19.5850 0.809047
\(587\) −29.2969 −1.20921 −0.604606 0.796524i \(-0.706670\pi\)
−0.604606 + 0.796524i \(0.706670\pi\)
\(588\) −23.8061 −0.981748
\(589\) 26.9680 1.11120
\(590\) −18.2963 −0.753247
\(591\) −34.6826 −1.42665
\(592\) 5.51397 0.226623
\(593\) −10.4679 −0.429867 −0.214933 0.976629i \(-0.568953\pi\)
−0.214933 + 0.976629i \(0.568953\pi\)
\(594\) −8.32079 −0.341406
\(595\) 4.81377 0.197345
\(596\) −0.896475 −0.0367210
\(597\) −44.2693 −1.81182
\(598\) −4.74384 −0.193990
\(599\) −43.1571 −1.76335 −0.881676 0.471855i \(-0.843584\pi\)
−0.881676 + 0.471855i \(0.843584\pi\)
\(600\) 11.4402 0.467045
\(601\) −30.5145 −1.24471 −0.622356 0.782734i \(-0.713824\pi\)
−0.622356 + 0.782734i \(0.713824\pi\)
\(602\) −7.04348 −0.287071
\(603\) 43.3606 1.76578
\(604\) −9.67015 −0.393473
\(605\) 17.8306 0.724915
\(606\) 27.1925 1.10462
\(607\) −16.6646 −0.676396 −0.338198 0.941075i \(-0.609817\pi\)
−0.338198 + 0.941075i \(0.609817\pi\)
\(608\) 33.1781 1.34555
\(609\) −29.2855 −1.18671
\(610\) 2.53412 0.102604
\(611\) 14.9524 0.604910
\(612\) 18.4384 0.745329
\(613\) 19.5027 0.787705 0.393853 0.919174i \(-0.371142\pi\)
0.393853 + 0.919174i \(0.371142\pi\)
\(614\) 18.7667 0.757364
\(615\) 38.6085 1.55684
\(616\) −3.75106 −0.151134
\(617\) −14.0955 −0.567464 −0.283732 0.958904i \(-0.591573\pi\)
−0.283732 + 0.958904i \(0.591573\pi\)
\(618\) 24.2382 0.975002
\(619\) −19.3521 −0.777829 −0.388914 0.921274i \(-0.627150\pi\)
−0.388914 + 0.921274i \(0.627150\pi\)
\(620\) −12.6180 −0.506750
\(621\) 30.1714 1.21073
\(622\) −9.20810 −0.369211
\(623\) 14.0483 0.562832
\(624\) 4.28180 0.171409
\(625\) −15.5685 −0.622739
\(626\) 11.6953 0.467439
\(627\) −20.8885 −0.834206
\(628\) 19.5870 0.781605
\(629\) −15.2168 −0.606733
\(630\) 9.95373 0.396566
\(631\) 31.8185 1.26667 0.633337 0.773876i \(-0.281685\pi\)
0.633337 + 0.773876i \(0.281685\pi\)
\(632\) 41.9255 1.66771
\(633\) −33.4306 −1.32875
\(634\) −13.0799 −0.519469
\(635\) −15.7117 −0.623500
\(636\) 29.5564 1.17199
\(637\) 10.1324 0.401462
\(638\) 8.02461 0.317697
\(639\) −7.01548 −0.277528
\(640\) −17.8385 −0.705129
\(641\) 22.6910 0.896239 0.448120 0.893974i \(-0.352094\pi\)
0.448120 + 0.893974i \(0.352094\pi\)
\(642\) 24.7181 0.975546
\(643\) 7.91528 0.312148 0.156074 0.987745i \(-0.450116\pi\)
0.156074 + 0.987745i \(0.450116\pi\)
\(644\) 5.61998 0.221458
\(645\) 44.4151 1.74884
\(646\) −9.61417 −0.378264
\(647\) 34.1361 1.34203 0.671015 0.741444i \(-0.265859\pi\)
0.671015 + 0.741444i \(0.265859\pi\)
\(648\) 22.0154 0.864846
\(649\) 15.6017 0.612420
\(650\) −2.01192 −0.0789140
\(651\) 16.4995 0.646667
\(652\) 8.28363 0.324412
\(653\) −9.96921 −0.390125 −0.195063 0.980791i \(-0.562491\pi\)
−0.195063 + 0.980791i \(0.562491\pi\)
\(654\) 23.7566 0.928955
\(655\) −13.8185 −0.539933
\(656\) 5.49082 0.214380
\(657\) 6.15678 0.240199
\(658\) 7.44322 0.290167
\(659\) −6.67772 −0.260127 −0.130063 0.991506i \(-0.541518\pi\)
−0.130063 + 0.991506i \(0.541518\pi\)
\(660\) 9.77344 0.380431
\(661\) 2.68068 0.104267 0.0521333 0.998640i \(-0.483398\pi\)
0.0521333 + 0.998640i \(0.483398\pi\)
\(662\) −11.6666 −0.453437
\(663\) −11.8164 −0.458910
\(664\) 4.34988 0.168808
\(665\) 12.3518 0.478981
\(666\) −31.4647 −1.21923
\(667\) −29.0974 −1.12666
\(668\) −2.47356 −0.0957051
\(669\) 3.89368 0.150538
\(670\) 10.5785 0.408683
\(671\) −2.16090 −0.0834208
\(672\) 20.2990 0.783049
\(673\) −12.2858 −0.473582 −0.236791 0.971561i \(-0.576096\pi\)
−0.236791 + 0.971561i \(0.576096\pi\)
\(674\) −6.16844 −0.237599
\(675\) 12.7960 0.492520
\(676\) 13.7881 0.530310
\(677\) −23.8413 −0.916297 −0.458149 0.888876i \(-0.651487\pi\)
−0.458149 + 0.888876i \(0.651487\pi\)
\(678\) −16.3661 −0.628535
\(679\) −4.87525 −0.187095
\(680\) 10.8868 0.417491
\(681\) −6.66370 −0.255354
\(682\) −4.52109 −0.173121
\(683\) 36.8014 1.40817 0.704083 0.710118i \(-0.251358\pi\)
0.704083 + 0.710118i \(0.251358\pi\)
\(684\) 47.3115 1.80900
\(685\) 10.7916 0.412327
\(686\) 11.2862 0.430907
\(687\) 67.5122 2.57575
\(688\) 6.31663 0.240819
\(689\) −12.5799 −0.479257
\(690\) 14.8910 0.566889
\(691\) 21.2987 0.810239 0.405120 0.914264i \(-0.367230\pi\)
0.405120 + 0.914264i \(0.367230\pi\)
\(692\) −7.61109 −0.289330
\(693\) −8.48778 −0.322424
\(694\) 15.6981 0.595890
\(695\) 28.7854 1.09189
\(696\) −66.2321 −2.51052
\(697\) −15.1529 −0.573957
\(698\) −2.60471 −0.0985896
\(699\) −40.6430 −1.53726
\(700\) 2.38350 0.0900878
\(701\) −18.5541 −0.700779 −0.350389 0.936604i \(-0.613951\pi\)
−0.350389 + 0.936604i \(0.613951\pi\)
\(702\) −12.0777 −0.455845
\(703\) −39.0451 −1.47261
\(704\) −3.58818 −0.135235
\(705\) −46.9358 −1.76770
\(706\) −26.8641 −1.01104
\(707\) 13.7113 0.515667
\(708\) −53.2068 −1.99963
\(709\) −27.7909 −1.04371 −0.521854 0.853035i \(-0.674759\pi\)
−0.521854 + 0.853035i \(0.674759\pi\)
\(710\) −1.71154 −0.0642328
\(711\) 94.8679 3.55782
\(712\) 31.7716 1.19069
\(713\) 16.3935 0.613943
\(714\) −5.88212 −0.220133
\(715\) −4.15981 −0.155568
\(716\) 25.1352 0.939345
\(717\) −3.62927 −0.135537
\(718\) 24.7151 0.922361
\(719\) −23.3909 −0.872334 −0.436167 0.899866i \(-0.643664\pi\)
−0.436167 + 0.899866i \(0.643664\pi\)
\(720\) −8.92655 −0.332673
\(721\) 12.2216 0.455158
\(722\) −10.0536 −0.374155
\(723\) 69.9588 2.60179
\(724\) −10.4084 −0.386827
\(725\) −12.3405 −0.458317
\(726\) −21.7878 −0.808622
\(727\) −25.4118 −0.942471 −0.471236 0.882007i \(-0.656192\pi\)
−0.471236 + 0.882007i \(0.656192\pi\)
\(728\) −5.44471 −0.201794
\(729\) −28.7680 −1.06548
\(730\) 1.50204 0.0555931
\(731\) −17.4319 −0.644741
\(732\) 7.36938 0.272380
\(733\) −16.6366 −0.614486 −0.307243 0.951631i \(-0.599406\pi\)
−0.307243 + 0.951631i \(0.599406\pi\)
\(734\) −5.41707 −0.199948
\(735\) −31.8058 −1.17318
\(736\) 20.1686 0.743424
\(737\) −9.02054 −0.332276
\(738\) −31.3326 −1.15337
\(739\) −47.4815 −1.74664 −0.873318 0.487150i \(-0.838037\pi\)
−0.873318 + 0.487150i \(0.838037\pi\)
\(740\) 18.2687 0.671569
\(741\) −30.3199 −1.11383
\(742\) −6.26220 −0.229893
\(743\) −39.6440 −1.45440 −0.727198 0.686427i \(-0.759178\pi\)
−0.727198 + 0.686427i \(0.759178\pi\)
\(744\) 37.3153 1.36805
\(745\) −1.19772 −0.0438812
\(746\) −3.54788 −0.129897
\(747\) 9.84278 0.360129
\(748\) −3.83584 −0.140252
\(749\) 12.4637 0.455412
\(750\) 27.9438 1.02036
\(751\) −20.6767 −0.754505 −0.377252 0.926111i \(-0.623131\pi\)
−0.377252 + 0.926111i \(0.623131\pi\)
\(752\) −6.67512 −0.243416
\(753\) −50.2972 −1.83293
\(754\) 11.6478 0.424189
\(755\) −12.9197 −0.470195
\(756\) 14.3084 0.520390
\(757\) −29.4407 −1.07004 −0.535020 0.844839i \(-0.679696\pi\)
−0.535020 + 0.844839i \(0.679696\pi\)
\(758\) −3.20134 −0.116278
\(759\) −12.6979 −0.460904
\(760\) 27.9348 1.01330
\(761\) −13.5406 −0.490846 −0.245423 0.969416i \(-0.578927\pi\)
−0.245423 + 0.969416i \(0.578927\pi\)
\(762\) 19.1987 0.695496
\(763\) 11.9788 0.433662
\(764\) 35.6880 1.29115
\(765\) 24.6344 0.890659
\(766\) −10.5124 −0.379827
\(767\) 22.6461 0.817702
\(768\) 39.1762 1.41365
\(769\) −47.8047 −1.72388 −0.861941 0.507009i \(-0.830751\pi\)
−0.861941 + 0.507009i \(0.830751\pi\)
\(770\) −2.07073 −0.0746238
\(771\) −7.47134 −0.269074
\(772\) 4.34597 0.156415
\(773\) 22.6571 0.814921 0.407460 0.913223i \(-0.366414\pi\)
0.407460 + 0.913223i \(0.366414\pi\)
\(774\) −36.0449 −1.29561
\(775\) 6.95270 0.249748
\(776\) −11.0259 −0.395806
\(777\) −23.8885 −0.856995
\(778\) 9.09636 0.326120
\(779\) −38.8811 −1.39306
\(780\) 14.1863 0.507951
\(781\) 1.45947 0.0522239
\(782\) −5.84435 −0.208993
\(783\) −74.0815 −2.64746
\(784\) −4.52336 −0.161549
\(785\) 26.1689 0.934009
\(786\) 16.8853 0.602279
\(787\) 20.8385 0.742813 0.371406 0.928470i \(-0.378876\pi\)
0.371406 + 0.928470i \(0.378876\pi\)
\(788\) −16.3421 −0.582164
\(789\) −34.9408 −1.24392
\(790\) 23.1445 0.823444
\(791\) −8.25230 −0.293418
\(792\) −19.1960 −0.682100
\(793\) −3.13658 −0.111383
\(794\) −8.95078 −0.317651
\(795\) 39.4885 1.40051
\(796\) −20.8593 −0.739338
\(797\) 2.32140 0.0822281 0.0411140 0.999154i \(-0.486909\pi\)
0.0411140 + 0.999154i \(0.486909\pi\)
\(798\) −15.0931 −0.534289
\(799\) 18.4212 0.651694
\(800\) 8.55374 0.302420
\(801\) 71.8919 2.54017
\(802\) −15.3344 −0.541477
\(803\) −1.28083 −0.0451994
\(804\) 30.7629 1.08492
\(805\) 7.50850 0.264640
\(806\) −6.56242 −0.231151
\(807\) −38.6277 −1.35976
\(808\) 31.0096 1.09091
\(809\) −45.2074 −1.58941 −0.794704 0.606997i \(-0.792374\pi\)
−0.794704 + 0.606997i \(0.792374\pi\)
\(810\) 12.1533 0.427025
\(811\) −45.4734 −1.59679 −0.798394 0.602135i \(-0.794317\pi\)
−0.798394 + 0.602135i \(0.794317\pi\)
\(812\) −13.7991 −0.484252
\(813\) 75.0379 2.63169
\(814\) 6.54576 0.229429
\(815\) 11.0672 0.387668
\(816\) 5.27511 0.184666
\(817\) −44.7288 −1.56486
\(818\) 8.13425 0.284407
\(819\) −12.3201 −0.430500
\(820\) 18.1920 0.635291
\(821\) 2.35462 0.0821767 0.0410883 0.999156i \(-0.486917\pi\)
0.0410883 + 0.999156i \(0.486917\pi\)
\(822\) −13.1867 −0.459939
\(823\) −39.0138 −1.35994 −0.679968 0.733242i \(-0.738006\pi\)
−0.679968 + 0.733242i \(0.738006\pi\)
\(824\) 27.6405 0.962903
\(825\) −5.38532 −0.187493
\(826\) 11.2731 0.392240
\(827\) −18.8571 −0.655727 −0.327863 0.944725i \(-0.606329\pi\)
−0.327863 + 0.944725i \(0.606329\pi\)
\(828\) 28.7602 0.999485
\(829\) 44.9946 1.56273 0.781364 0.624076i \(-0.214524\pi\)
0.781364 + 0.624076i \(0.214524\pi\)
\(830\) 2.40130 0.0833503
\(831\) −4.83217 −0.167626
\(832\) −5.20829 −0.180565
\(833\) 12.4830 0.432511
\(834\) −35.1739 −1.21797
\(835\) −3.30477 −0.114366
\(836\) −9.84247 −0.340409
\(837\) 41.7377 1.44267
\(838\) 12.0668 0.416842
\(839\) 9.28809 0.320660 0.160330 0.987063i \(-0.448744\pi\)
0.160330 + 0.987063i \(0.448744\pi\)
\(840\) 17.0910 0.589695
\(841\) 42.4445 1.46360
\(842\) −17.3151 −0.596718
\(843\) 0.719321 0.0247747
\(844\) −15.7522 −0.542214
\(845\) 18.4214 0.633714
\(846\) 38.0906 1.30958
\(847\) −10.9861 −0.377487
\(848\) 5.61597 0.192853
\(849\) 49.5913 1.70197
\(850\) −2.47866 −0.0850172
\(851\) −23.7351 −0.813627
\(852\) −4.97725 −0.170518
\(853\) 19.6823 0.673909 0.336954 0.941521i \(-0.390603\pi\)
0.336954 + 0.941521i \(0.390603\pi\)
\(854\) −1.56137 −0.0534290
\(855\) 63.2100 2.16174
\(856\) 28.1878 0.963441
\(857\) 12.8317 0.438322 0.219161 0.975689i \(-0.429668\pi\)
0.219161 + 0.975689i \(0.429668\pi\)
\(858\) 5.08303 0.173532
\(859\) −17.6648 −0.602715 −0.301357 0.953511i \(-0.597440\pi\)
−0.301357 + 0.953511i \(0.597440\pi\)
\(860\) 20.9280 0.713639
\(861\) −23.7882 −0.810699
\(862\) −23.4074 −0.797261
\(863\) −49.7153 −1.69233 −0.846164 0.532923i \(-0.821093\pi\)
−0.846164 + 0.532923i \(0.821093\pi\)
\(864\) 51.3489 1.74693
\(865\) −10.1687 −0.345746
\(866\) 6.65033 0.225987
\(867\) 36.2508 1.23114
\(868\) 7.77442 0.263881
\(869\) −19.7359 −0.669493
\(870\) −36.5626 −1.23959
\(871\) −13.0934 −0.443654
\(872\) 27.0913 0.917428
\(873\) −24.9490 −0.844397
\(874\) −14.9961 −0.507252
\(875\) 14.0901 0.476334
\(876\) 4.36804 0.147582
\(877\) −4.70793 −0.158976 −0.0794878 0.996836i \(-0.525328\pi\)
−0.0794878 + 0.996836i \(0.525328\pi\)
\(878\) 16.8674 0.569246
\(879\) 76.0931 2.56655
\(880\) 1.85704 0.0626007
\(881\) 15.4886 0.521824 0.260912 0.965363i \(-0.415977\pi\)
0.260912 + 0.965363i \(0.415977\pi\)
\(882\) 25.8119 0.869133
\(883\) −39.8260 −1.34025 −0.670125 0.742248i \(-0.733760\pi\)
−0.670125 + 0.742248i \(0.733760\pi\)
\(884\) −5.56778 −0.187265
\(885\) −71.0862 −2.38954
\(886\) 20.7486 0.697064
\(887\) 3.09616 0.103959 0.0519795 0.998648i \(-0.483447\pi\)
0.0519795 + 0.998648i \(0.483447\pi\)
\(888\) −54.0263 −1.81300
\(889\) 9.68059 0.324677
\(890\) 17.5391 0.587913
\(891\) −10.3634 −0.347188
\(892\) 1.83467 0.0614292
\(893\) 47.2673 1.58174
\(894\) 1.46354 0.0489482
\(895\) 33.5815 1.12251
\(896\) 10.9910 0.367184
\(897\) −18.4311 −0.615398
\(898\) −23.8972 −0.797459
\(899\) −40.2520 −1.34248
\(900\) 12.1975 0.406584
\(901\) −15.4983 −0.516323
\(902\) 6.51828 0.217035
\(903\) −27.3659 −0.910680
\(904\) −18.6634 −0.620736
\(905\) −13.9061 −0.462253
\(906\) 15.7870 0.524489
\(907\) −52.5988 −1.74652 −0.873258 0.487259i \(-0.837997\pi\)
−0.873258 + 0.487259i \(0.837997\pi\)
\(908\) −3.13988 −0.104201
\(909\) 70.1676 2.32731
\(910\) −3.00569 −0.0996376
\(911\) −48.7874 −1.61640 −0.808199 0.588910i \(-0.799557\pi\)
−0.808199 + 0.588910i \(0.799557\pi\)
\(912\) 13.5355 0.448206
\(913\) −2.04765 −0.0677672
\(914\) −9.91500 −0.327959
\(915\) 9.84576 0.325491
\(916\) 31.8111 1.05107
\(917\) 8.51411 0.281161
\(918\) −14.8796 −0.491100
\(919\) 12.5481 0.413924 0.206962 0.978349i \(-0.433642\pi\)
0.206962 + 0.978349i \(0.433642\pi\)
\(920\) 16.9812 0.559855
\(921\) 72.9141 2.40260
\(922\) −23.1098 −0.761082
\(923\) 2.11844 0.0697292
\(924\) −6.02180 −0.198103
\(925\) −10.0663 −0.330979
\(926\) 3.68314 0.121036
\(927\) 62.5441 2.05422
\(928\) −49.5211 −1.62561
\(929\) 28.2201 0.925873 0.462936 0.886391i \(-0.346796\pi\)
0.462936 + 0.886391i \(0.346796\pi\)
\(930\) 20.5995 0.675484
\(931\) 32.0305 1.04976
\(932\) −19.1506 −0.627299
\(933\) −35.7760 −1.17125
\(934\) −14.4629 −0.473240
\(935\) −5.12483 −0.167600
\(936\) −27.8632 −0.910739
\(937\) −15.5735 −0.508763 −0.254381 0.967104i \(-0.581872\pi\)
−0.254381 + 0.967104i \(0.581872\pi\)
\(938\) −6.51783 −0.212815
\(939\) 45.4396 1.48286
\(940\) −22.1157 −0.721336
\(941\) 49.8449 1.62490 0.812449 0.583032i \(-0.198134\pi\)
0.812449 + 0.583032i \(0.198134\pi\)
\(942\) −31.9768 −1.04186
\(943\) −23.6354 −0.769675
\(944\) −10.1097 −0.329044
\(945\) 19.1165 0.621860
\(946\) 7.49862 0.243801
\(947\) 16.8703 0.548211 0.274106 0.961700i \(-0.411618\pi\)
0.274106 + 0.961700i \(0.411618\pi\)
\(948\) 67.3056 2.18599
\(949\) −1.85914 −0.0603502
\(950\) −6.36004 −0.206347
\(951\) −50.8191 −1.64792
\(952\) −6.70781 −0.217401
\(953\) 40.3392 1.30671 0.653357 0.757050i \(-0.273360\pi\)
0.653357 + 0.757050i \(0.273360\pi\)
\(954\) −32.0468 −1.03755
\(955\) 47.6804 1.54290
\(956\) −1.71008 −0.0553079
\(957\) 31.1778 1.00784
\(958\) 21.2784 0.687473
\(959\) −6.64915 −0.214712
\(960\) 16.3489 0.527658
\(961\) −8.32190 −0.268448
\(962\) 9.50126 0.306333
\(963\) 63.7826 2.05537
\(964\) 32.9640 1.06170
\(965\) 5.80637 0.186914
\(966\) −9.17491 −0.295198
\(967\) 52.6815 1.69412 0.847062 0.531494i \(-0.178369\pi\)
0.847062 + 0.531494i \(0.178369\pi\)
\(968\) −24.8462 −0.798588
\(969\) −37.3537 −1.19997
\(970\) −6.08671 −0.195432
\(971\) 57.6289 1.84940 0.924700 0.380697i \(-0.124316\pi\)
0.924700 + 0.380697i \(0.124316\pi\)
\(972\) −1.68533 −0.0540571
\(973\) −17.7358 −0.568584
\(974\) 2.30186 0.0737565
\(975\) −7.81687 −0.250340
\(976\) 1.40024 0.0448207
\(977\) 7.22940 0.231289 0.115644 0.993291i \(-0.463107\pi\)
0.115644 + 0.993291i \(0.463107\pi\)
\(978\) −13.5235 −0.432433
\(979\) −14.9560 −0.477997
\(980\) −14.9866 −0.478730
\(981\) 61.3015 1.95720
\(982\) −9.19678 −0.293481
\(983\) −0.0912068 −0.00290904 −0.00145452 0.999999i \(-0.500463\pi\)
−0.00145452 + 0.999999i \(0.500463\pi\)
\(984\) −53.7994 −1.71506
\(985\) −21.8337 −0.695678
\(986\) 14.3500 0.456996
\(987\) 28.9190 0.920502
\(988\) −14.2865 −0.454514
\(989\) −27.1901 −0.864596
\(990\) −10.5969 −0.336792
\(991\) −31.7069 −1.00720 −0.503602 0.863936i \(-0.667992\pi\)
−0.503602 + 0.863936i \(0.667992\pi\)
\(992\) 27.9003 0.885836
\(993\) −45.3281 −1.43844
\(994\) 1.05454 0.0334481
\(995\) −27.8688 −0.883500
\(996\) 6.98313 0.221269
\(997\) 11.0006 0.348392 0.174196 0.984711i \(-0.444267\pi\)
0.174196 + 0.984711i \(0.444267\pi\)
\(998\) 34.1820 1.08201
\(999\) −60.4291 −1.91189
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 4001.2.a.a.1.58 149
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.a.1.58 149 1.1 even 1 trivial