Properties

Label 8043.2.a.s.1.18
Level $8043$
Weight $2$
Character 8043.1
Self dual yes
Analytic conductor $64.224$
Analytic rank $0$
Dimension $50$
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: \(50\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
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)
 
\(f(q)\) \(=\) \(q-1.15299 q^{2} -1.00000 q^{3} -0.670610 q^{4} +0.489993 q^{5} +1.15299 q^{6} -1.00000 q^{7} +3.07919 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.15299 q^{2} -1.00000 q^{3} -0.670610 q^{4} +0.489993 q^{5} +1.15299 q^{6} -1.00000 q^{7} +3.07919 q^{8} +1.00000 q^{9} -0.564958 q^{10} +1.66126 q^{11} +0.670610 q^{12} -2.35343 q^{13} +1.15299 q^{14} -0.489993 q^{15} -2.20906 q^{16} +3.53643 q^{17} -1.15299 q^{18} +4.21017 q^{19} -0.328594 q^{20} +1.00000 q^{21} -1.91542 q^{22} +7.68558 q^{23} -3.07919 q^{24} -4.75991 q^{25} +2.71348 q^{26} -1.00000 q^{27} +0.670610 q^{28} +5.40181 q^{29} +0.564958 q^{30} +6.30168 q^{31} -3.61135 q^{32} -1.66126 q^{33} -4.07747 q^{34} -0.489993 q^{35} -0.670610 q^{36} -9.09193 q^{37} -4.85429 q^{38} +2.35343 q^{39} +1.50878 q^{40} -3.34227 q^{41} -1.15299 q^{42} +8.34436 q^{43} -1.11406 q^{44} +0.489993 q^{45} -8.86141 q^{46} +2.83247 q^{47} +2.20906 q^{48} +1.00000 q^{49} +5.48813 q^{50} -3.53643 q^{51} +1.57823 q^{52} +3.98738 q^{53} +1.15299 q^{54} +0.814005 q^{55} -3.07919 q^{56} -4.21017 q^{57} -6.22824 q^{58} +12.4932 q^{59} +0.328594 q^{60} +11.1893 q^{61} -7.26579 q^{62} -1.00000 q^{63} +8.58198 q^{64} -1.15316 q^{65} +1.91542 q^{66} +2.79602 q^{67} -2.37156 q^{68} -7.68558 q^{69} +0.564958 q^{70} +10.9437 q^{71} +3.07919 q^{72} -10.2725 q^{73} +10.4829 q^{74} +4.75991 q^{75} -2.82338 q^{76} -1.66126 q^{77} -2.71348 q^{78} -16.3957 q^{79} -1.08242 q^{80} +1.00000 q^{81} +3.85361 q^{82} +13.2813 q^{83} -0.670610 q^{84} +1.73282 q^{85} -9.62098 q^{86} -5.40181 q^{87} +5.11533 q^{88} +2.37465 q^{89} -0.564958 q^{90} +2.35343 q^{91} -5.15403 q^{92} -6.30168 q^{93} -3.26581 q^{94} +2.06295 q^{95} +3.61135 q^{96} +0.810176 q^{97} -1.15299 q^{98} +1.66126 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50 q - q^{2} - 50 q^{3} + 53 q^{4} + 11 q^{5} + q^{6} - 50 q^{7} - 6 q^{8} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 50 q - q^{2} - 50 q^{3} + 53 q^{4} + 11 q^{5} + q^{6} - 50 q^{7} - 6 q^{8} + 50 q^{9} + 16 q^{10} - 31 q^{11} - 53 q^{12} + 42 q^{13} + q^{14} - 11 q^{15} + 59 q^{16} + 44 q^{17} - q^{18} + 11 q^{19} + 7 q^{20} + 50 q^{21} + 19 q^{22} - 16 q^{23} + 6 q^{24} + 71 q^{25} + q^{26} - 50 q^{27} - 53 q^{28} + 3 q^{29} - 16 q^{30} + 13 q^{31} - 23 q^{32} + 31 q^{33} + q^{34} - 11 q^{35} + 53 q^{36} + 53 q^{37} + 28 q^{38} - 42 q^{39} + 50 q^{40} + 23 q^{41} - q^{42} + 9 q^{43} - 78 q^{44} + 11 q^{45} - 8 q^{46} + 26 q^{47} - 59 q^{48} + 50 q^{49} - 38 q^{50} - 44 q^{51} + 86 q^{52} + 58 q^{53} + q^{54} + 28 q^{55} + 6 q^{56} - 11 q^{57} - 4 q^{58} + 7 q^{59} - 7 q^{60} + 51 q^{61} + 7 q^{62} - 50 q^{63} + 74 q^{64} - 14 q^{65} - 19 q^{66} + 23 q^{67} + 98 q^{68} + 16 q^{69} - 16 q^{70} - 75 q^{71} - 6 q^{72} + 34 q^{73} - 68 q^{74} - 71 q^{75} + 31 q^{76} + 31 q^{77} - q^{78} - 18 q^{79} - 21 q^{80} + 50 q^{81} + 31 q^{82} + 40 q^{83} + 53 q^{84} + 30 q^{85} - 15 q^{86} - 3 q^{87} + 70 q^{88} + 63 q^{89} + 16 q^{90} - 42 q^{91} - 38 q^{92} - 13 q^{93} + q^{94} - 77 q^{95} + 23 q^{96} + 77 q^{97} - q^{98} - 31 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.15299 −0.815288 −0.407644 0.913141i \(-0.633650\pi\)
−0.407644 + 0.913141i \(0.633650\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.670610 −0.335305
\(5\) 0.489993 0.219131 0.109566 0.993980i \(-0.465054\pi\)
0.109566 + 0.993980i \(0.465054\pi\)
\(6\) 1.15299 0.470707
\(7\) −1.00000 −0.377964
\(8\) 3.07919 1.08866
\(9\) 1.00000 0.333333
\(10\) −0.564958 −0.178655
\(11\) 1.66126 0.500888 0.250444 0.968131i \(-0.419423\pi\)
0.250444 + 0.968131i \(0.419423\pi\)
\(12\) 0.670610 0.193588
\(13\) −2.35343 −0.652723 −0.326362 0.945245i \(-0.605823\pi\)
−0.326362 + 0.945245i \(0.605823\pi\)
\(14\) 1.15299 0.308150
\(15\) −0.489993 −0.126516
\(16\) −2.20906 −0.552266
\(17\) 3.53643 0.857709 0.428855 0.903373i \(-0.358917\pi\)
0.428855 + 0.903373i \(0.358917\pi\)
\(18\) −1.15299 −0.271763
\(19\) 4.21017 0.965880 0.482940 0.875654i \(-0.339569\pi\)
0.482940 + 0.875654i \(0.339569\pi\)
\(20\) −0.328594 −0.0734758
\(21\) 1.00000 0.218218
\(22\) −1.91542 −0.408368
\(23\) 7.68558 1.60255 0.801277 0.598293i \(-0.204154\pi\)
0.801277 + 0.598293i \(0.204154\pi\)
\(24\) −3.07919 −0.628537
\(25\) −4.75991 −0.951981
\(26\) 2.71348 0.532158
\(27\) −1.00000 −0.192450
\(28\) 0.670610 0.126733
\(29\) 5.40181 1.00309 0.501545 0.865131i \(-0.332765\pi\)
0.501545 + 0.865131i \(0.332765\pi\)
\(30\) 0.564958 0.103147
\(31\) 6.30168 1.13182 0.565908 0.824469i \(-0.308526\pi\)
0.565908 + 0.824469i \(0.308526\pi\)
\(32\) −3.61135 −0.638403
\(33\) −1.66126 −0.289188
\(34\) −4.07747 −0.699281
\(35\) −0.489993 −0.0828239
\(36\) −0.670610 −0.111768
\(37\) −9.09193 −1.49470 −0.747352 0.664428i \(-0.768675\pi\)
−0.747352 + 0.664428i \(0.768675\pi\)
\(38\) −4.85429 −0.787470
\(39\) 2.35343 0.376850
\(40\) 1.50878 0.238559
\(41\) −3.34227 −0.521975 −0.260988 0.965342i \(-0.584048\pi\)
−0.260988 + 0.965342i \(0.584048\pi\)
\(42\) −1.15299 −0.177911
\(43\) 8.34436 1.27250 0.636251 0.771482i \(-0.280484\pi\)
0.636251 + 0.771482i \(0.280484\pi\)
\(44\) −1.11406 −0.167950
\(45\) 0.489993 0.0730438
\(46\) −8.86141 −1.30654
\(47\) 2.83247 0.413158 0.206579 0.978430i \(-0.433767\pi\)
0.206579 + 0.978430i \(0.433767\pi\)
\(48\) 2.20906 0.318851
\(49\) 1.00000 0.142857
\(50\) 5.48813 0.776139
\(51\) −3.53643 −0.495199
\(52\) 1.57823 0.218861
\(53\) 3.98738 0.547709 0.273854 0.961771i \(-0.411701\pi\)
0.273854 + 0.961771i \(0.411701\pi\)
\(54\) 1.15299 0.156902
\(55\) 0.814005 0.109760
\(56\) −3.07919 −0.411474
\(57\) −4.21017 −0.557651
\(58\) −6.22824 −0.817808
\(59\) 12.4932 1.62648 0.813241 0.581927i \(-0.197701\pi\)
0.813241 + 0.581927i \(0.197701\pi\)
\(60\) 0.328594 0.0424213
\(61\) 11.1893 1.43265 0.716323 0.697769i \(-0.245824\pi\)
0.716323 + 0.697769i \(0.245824\pi\)
\(62\) −7.26579 −0.922756
\(63\) −1.00000 −0.125988
\(64\) 8.58198 1.07275
\(65\) −1.15316 −0.143032
\(66\) 1.91542 0.235772
\(67\) 2.79602 0.341588 0.170794 0.985307i \(-0.445367\pi\)
0.170794 + 0.985307i \(0.445367\pi\)
\(68\) −2.37156 −0.287594
\(69\) −7.68558 −0.925235
\(70\) 0.564958 0.0675253
\(71\) 10.9437 1.29878 0.649392 0.760454i \(-0.275024\pi\)
0.649392 + 0.760454i \(0.275024\pi\)
\(72\) 3.07919 0.362886
\(73\) −10.2725 −1.20231 −0.601154 0.799133i \(-0.705292\pi\)
−0.601154 + 0.799133i \(0.705292\pi\)
\(74\) 10.4829 1.21861
\(75\) 4.75991 0.549627
\(76\) −2.82338 −0.323864
\(77\) −1.66126 −0.189318
\(78\) −2.71348 −0.307241
\(79\) −16.3957 −1.84466 −0.922331 0.386401i \(-0.873718\pi\)
−0.922331 + 0.386401i \(0.873718\pi\)
\(80\) −1.08242 −0.121019
\(81\) 1.00000 0.111111
\(82\) 3.85361 0.425560
\(83\) 13.2813 1.45781 0.728907 0.684613i \(-0.240029\pi\)
0.728907 + 0.684613i \(0.240029\pi\)
\(84\) −0.670610 −0.0731695
\(85\) 1.73282 0.187951
\(86\) −9.62098 −1.03746
\(87\) −5.40181 −0.579134
\(88\) 5.11533 0.545296
\(89\) 2.37465 0.251712 0.125856 0.992048i \(-0.459832\pi\)
0.125856 + 0.992048i \(0.459832\pi\)
\(90\) −0.564958 −0.0595518
\(91\) 2.35343 0.246706
\(92\) −5.15403 −0.537344
\(93\) −6.30168 −0.653454
\(94\) −3.26581 −0.336843
\(95\) 2.06295 0.211655
\(96\) 3.61135 0.368582
\(97\) 0.810176 0.0822609 0.0411305 0.999154i \(-0.486904\pi\)
0.0411305 + 0.999154i \(0.486904\pi\)
\(98\) −1.15299 −0.116470
\(99\) 1.66126 0.166963
\(100\) 3.19204 0.319204
\(101\) −17.1527 −1.70676 −0.853378 0.521292i \(-0.825450\pi\)
−0.853378 + 0.521292i \(0.825450\pi\)
\(102\) 4.07747 0.403730
\(103\) −18.2308 −1.79633 −0.898167 0.439655i \(-0.855101\pi\)
−0.898167 + 0.439655i \(0.855101\pi\)
\(104\) −7.24665 −0.710593
\(105\) 0.489993 0.0478184
\(106\) −4.59742 −0.446540
\(107\) 6.63153 0.641094 0.320547 0.947233i \(-0.396133\pi\)
0.320547 + 0.947233i \(0.396133\pi\)
\(108\) 0.670610 0.0645295
\(109\) −14.8320 −1.42065 −0.710324 0.703875i \(-0.751452\pi\)
−0.710324 + 0.703875i \(0.751452\pi\)
\(110\) −0.938541 −0.0894863
\(111\) 9.09193 0.862968
\(112\) 2.20906 0.208737
\(113\) 10.9562 1.03067 0.515337 0.856988i \(-0.327667\pi\)
0.515337 + 0.856988i \(0.327667\pi\)
\(114\) 4.85429 0.454646
\(115\) 3.76588 0.351170
\(116\) −3.62250 −0.336341
\(117\) −2.35343 −0.217574
\(118\) −14.4046 −1.32605
\(119\) −3.53643 −0.324184
\(120\) −1.50878 −0.137732
\(121\) −8.24022 −0.749111
\(122\) −12.9012 −1.16802
\(123\) 3.34227 0.301362
\(124\) −4.22597 −0.379503
\(125\) −4.78228 −0.427740
\(126\) 1.15299 0.102717
\(127\) 8.25401 0.732425 0.366213 0.930531i \(-0.380654\pi\)
0.366213 + 0.930531i \(0.380654\pi\)
\(128\) −2.67226 −0.236196
\(129\) −8.34436 −0.734680
\(130\) 1.32959 0.116612
\(131\) −21.0890 −1.84255 −0.921275 0.388911i \(-0.872851\pi\)
−0.921275 + 0.388911i \(0.872851\pi\)
\(132\) 1.11406 0.0969662
\(133\) −4.21017 −0.365068
\(134\) −3.22379 −0.278493
\(135\) −0.489993 −0.0421719
\(136\) 10.8893 0.933753
\(137\) 18.9167 1.61616 0.808081 0.589071i \(-0.200506\pi\)
0.808081 + 0.589071i \(0.200506\pi\)
\(138\) 8.86141 0.754333
\(139\) −9.55723 −0.810634 −0.405317 0.914176i \(-0.632839\pi\)
−0.405317 + 0.914176i \(0.632839\pi\)
\(140\) 0.328594 0.0277713
\(141\) −2.83247 −0.238537
\(142\) −12.6180 −1.05888
\(143\) −3.90965 −0.326941
\(144\) −2.20906 −0.184089
\(145\) 2.64685 0.219809
\(146\) 11.8441 0.980227
\(147\) −1.00000 −0.0824786
\(148\) 6.09714 0.501182
\(149\) 4.15137 0.340093 0.170047 0.985436i \(-0.445608\pi\)
0.170047 + 0.985436i \(0.445608\pi\)
\(150\) −5.48813 −0.448104
\(151\) 0.748490 0.0609112 0.0304556 0.999536i \(-0.490304\pi\)
0.0304556 + 0.999536i \(0.490304\pi\)
\(152\) 12.9639 1.05151
\(153\) 3.53643 0.285903
\(154\) 1.91542 0.154349
\(155\) 3.08778 0.248016
\(156\) −1.57823 −0.126360
\(157\) 18.2085 1.45320 0.726600 0.687061i \(-0.241100\pi\)
0.726600 + 0.687061i \(0.241100\pi\)
\(158\) 18.9041 1.50393
\(159\) −3.98738 −0.316220
\(160\) −1.76954 −0.139894
\(161\) −7.68558 −0.605709
\(162\) −1.15299 −0.0905876
\(163\) 19.3114 1.51259 0.756293 0.654233i \(-0.227009\pi\)
0.756293 + 0.654233i \(0.227009\pi\)
\(164\) 2.24136 0.175021
\(165\) −0.814005 −0.0633702
\(166\) −15.3132 −1.18854
\(167\) 6.14235 0.475309 0.237655 0.971350i \(-0.423621\pi\)
0.237655 + 0.971350i \(0.423621\pi\)
\(168\) 3.07919 0.237565
\(169\) −7.46138 −0.573953
\(170\) −1.99793 −0.153234
\(171\) 4.21017 0.321960
\(172\) −5.59581 −0.426676
\(173\) 0.0491798 0.00373907 0.00186954 0.999998i \(-0.499405\pi\)
0.00186954 + 0.999998i \(0.499405\pi\)
\(174\) 6.22824 0.472162
\(175\) 4.75991 0.359815
\(176\) −3.66983 −0.276623
\(177\) −12.4932 −0.939050
\(178\) −2.73795 −0.205218
\(179\) −15.3656 −1.14848 −0.574241 0.818687i \(-0.694703\pi\)
−0.574241 + 0.818687i \(0.694703\pi\)
\(180\) −0.328594 −0.0244919
\(181\) 15.2957 1.13692 0.568460 0.822711i \(-0.307539\pi\)
0.568460 + 0.822711i \(0.307539\pi\)
\(182\) −2.71348 −0.201137
\(183\) −11.1893 −0.827138
\(184\) 23.6654 1.74463
\(185\) −4.45498 −0.327537
\(186\) 7.26579 0.532753
\(187\) 5.87492 0.429617
\(188\) −1.89948 −0.138534
\(189\) 1.00000 0.0727393
\(190\) −2.37857 −0.172559
\(191\) 11.2686 0.815368 0.407684 0.913123i \(-0.366337\pi\)
0.407684 + 0.913123i \(0.366337\pi\)
\(192\) −8.58198 −0.619351
\(193\) −2.49259 −0.179420 −0.0897102 0.995968i \(-0.528594\pi\)
−0.0897102 + 0.995968i \(0.528594\pi\)
\(194\) −0.934126 −0.0670664
\(195\) 1.15316 0.0825796
\(196\) −0.670610 −0.0479007
\(197\) −18.2325 −1.29901 −0.649506 0.760356i \(-0.725024\pi\)
−0.649506 + 0.760356i \(0.725024\pi\)
\(198\) −1.91542 −0.136123
\(199\) −7.36296 −0.521946 −0.260973 0.965346i \(-0.584043\pi\)
−0.260973 + 0.965346i \(0.584043\pi\)
\(200\) −14.6567 −1.03638
\(201\) −2.79602 −0.197216
\(202\) 19.7769 1.39150
\(203\) −5.40181 −0.379132
\(204\) 2.37156 0.166043
\(205\) −1.63769 −0.114381
\(206\) 21.0200 1.46453
\(207\) 7.68558 0.534185
\(208\) 5.19887 0.360477
\(209\) 6.99418 0.483798
\(210\) −0.564958 −0.0389858
\(211\) −20.3317 −1.39969 −0.699847 0.714293i \(-0.746749\pi\)
−0.699847 + 0.714293i \(0.746749\pi\)
\(212\) −2.67398 −0.183649
\(213\) −10.9437 −0.749853
\(214\) −7.64610 −0.522677
\(215\) 4.08867 0.278845
\(216\) −3.07919 −0.209512
\(217\) −6.30168 −0.427786
\(218\) 17.1012 1.15824
\(219\) 10.2725 0.694153
\(220\) −0.545879 −0.0368032
\(221\) −8.32272 −0.559847
\(222\) −10.4829 −0.703568
\(223\) 26.3274 1.76302 0.881508 0.472169i \(-0.156529\pi\)
0.881508 + 0.472169i \(0.156529\pi\)
\(224\) 3.61135 0.241294
\(225\) −4.75991 −0.317327
\(226\) −12.6324 −0.840296
\(227\) −27.1964 −1.80509 −0.902544 0.430599i \(-0.858302\pi\)
−0.902544 + 0.430599i \(0.858302\pi\)
\(228\) 2.82338 0.186983
\(229\) −1.78613 −0.118031 −0.0590153 0.998257i \(-0.518796\pi\)
−0.0590153 + 0.998257i \(0.518796\pi\)
\(230\) −4.34203 −0.286305
\(231\) 1.66126 0.109303
\(232\) 16.6332 1.09202
\(233\) −21.8254 −1.42983 −0.714915 0.699212i \(-0.753534\pi\)
−0.714915 + 0.699212i \(0.753534\pi\)
\(234\) 2.71348 0.177386
\(235\) 1.38789 0.0905358
\(236\) −8.37809 −0.545367
\(237\) 16.3957 1.06502
\(238\) 4.07747 0.264303
\(239\) 19.7241 1.27585 0.637923 0.770100i \(-0.279794\pi\)
0.637923 + 0.770100i \(0.279794\pi\)
\(240\) 1.08242 0.0698702
\(241\) 20.1026 1.29492 0.647461 0.762099i \(-0.275831\pi\)
0.647461 + 0.762099i \(0.275831\pi\)
\(242\) 9.50091 0.610741
\(243\) −1.00000 −0.0641500
\(244\) −7.50367 −0.480373
\(245\) 0.489993 0.0313045
\(246\) −3.85361 −0.245697
\(247\) −9.90833 −0.630452
\(248\) 19.4041 1.23216
\(249\) −13.2813 −0.841669
\(250\) 5.51393 0.348732
\(251\) −12.9654 −0.818369 −0.409185 0.912452i \(-0.634187\pi\)
−0.409185 + 0.912452i \(0.634187\pi\)
\(252\) 0.670610 0.0422444
\(253\) 12.7677 0.802701
\(254\) −9.51681 −0.597138
\(255\) −1.73282 −0.108514
\(256\) −14.0829 −0.880180
\(257\) 22.9838 1.43369 0.716844 0.697234i \(-0.245586\pi\)
0.716844 + 0.697234i \(0.245586\pi\)
\(258\) 9.62098 0.598976
\(259\) 9.09193 0.564945
\(260\) 0.773322 0.0479594
\(261\) 5.40181 0.334363
\(262\) 24.3154 1.50221
\(263\) −3.77883 −0.233013 −0.116506 0.993190i \(-0.537170\pi\)
−0.116506 + 0.993190i \(0.537170\pi\)
\(264\) −5.11533 −0.314827
\(265\) 1.95379 0.120020
\(266\) 4.85429 0.297636
\(267\) −2.37465 −0.145326
\(268\) −1.87504 −0.114536
\(269\) −4.37680 −0.266858 −0.133429 0.991058i \(-0.542599\pi\)
−0.133429 + 0.991058i \(0.542599\pi\)
\(270\) 0.564958 0.0343822
\(271\) −6.23702 −0.378872 −0.189436 0.981893i \(-0.560666\pi\)
−0.189436 + 0.981893i \(0.560666\pi\)
\(272\) −7.81219 −0.473684
\(273\) −2.35343 −0.142436
\(274\) −21.8108 −1.31764
\(275\) −7.90744 −0.476836
\(276\) 5.15403 0.310236
\(277\) −11.7889 −0.708326 −0.354163 0.935184i \(-0.615234\pi\)
−0.354163 + 0.935184i \(0.615234\pi\)
\(278\) 11.0194 0.660900
\(279\) 6.30168 0.377272
\(280\) −1.50878 −0.0901669
\(281\) −26.4366 −1.57708 −0.788538 0.614986i \(-0.789162\pi\)
−0.788538 + 0.614986i \(0.789162\pi\)
\(282\) 3.26581 0.194476
\(283\) 20.9061 1.24274 0.621370 0.783517i \(-0.286576\pi\)
0.621370 + 0.783517i \(0.286576\pi\)
\(284\) −7.33898 −0.435488
\(285\) −2.06295 −0.122199
\(286\) 4.50779 0.266552
\(287\) 3.34227 0.197288
\(288\) −3.61135 −0.212801
\(289\) −4.49369 −0.264334
\(290\) −3.05179 −0.179207
\(291\) −0.810176 −0.0474934
\(292\) 6.88885 0.403140
\(293\) −16.3510 −0.955238 −0.477619 0.878567i \(-0.658500\pi\)
−0.477619 + 0.878567i \(0.658500\pi\)
\(294\) 1.15299 0.0672439
\(295\) 6.12160 0.356413
\(296\) −27.9958 −1.62722
\(297\) −1.66126 −0.0963960
\(298\) −4.78649 −0.277274
\(299\) −18.0874 −1.04602
\(300\) −3.19204 −0.184293
\(301\) −8.34436 −0.480961
\(302\) −0.863002 −0.0496602
\(303\) 17.1527 0.985396
\(304\) −9.30053 −0.533422
\(305\) 5.48268 0.313938
\(306\) −4.07747 −0.233094
\(307\) −30.0983 −1.71780 −0.858899 0.512145i \(-0.828851\pi\)
−0.858899 + 0.512145i \(0.828851\pi\)
\(308\) 1.11406 0.0634792
\(309\) 18.2308 1.03711
\(310\) −3.56018 −0.202205
\(311\) −14.1735 −0.803708 −0.401854 0.915704i \(-0.631634\pi\)
−0.401854 + 0.915704i \(0.631634\pi\)
\(312\) 7.24665 0.410261
\(313\) 22.9571 1.29761 0.648806 0.760954i \(-0.275269\pi\)
0.648806 + 0.760954i \(0.275269\pi\)
\(314\) −20.9943 −1.18478
\(315\) −0.489993 −0.0276080
\(316\) 10.9951 0.618524
\(317\) 5.17947 0.290908 0.145454 0.989365i \(-0.453536\pi\)
0.145454 + 0.989365i \(0.453536\pi\)
\(318\) 4.59742 0.257810
\(319\) 8.97380 0.502436
\(320\) 4.20511 0.235073
\(321\) −6.63153 −0.370136
\(322\) 8.86141 0.493827
\(323\) 14.8890 0.828444
\(324\) −0.670610 −0.0372561
\(325\) 11.2021 0.621380
\(326\) −22.2659 −1.23319
\(327\) 14.8320 0.820211
\(328\) −10.2915 −0.568253
\(329\) −2.83247 −0.156159
\(330\) 0.938541 0.0516650
\(331\) 22.6512 1.24502 0.622511 0.782611i \(-0.286112\pi\)
0.622511 + 0.782611i \(0.286112\pi\)
\(332\) −8.90658 −0.488812
\(333\) −9.09193 −0.498235
\(334\) −7.08208 −0.387514
\(335\) 1.37003 0.0748526
\(336\) −2.20906 −0.120514
\(337\) −12.5057 −0.681231 −0.340616 0.940203i \(-0.610635\pi\)
−0.340616 + 0.940203i \(0.610635\pi\)
\(338\) 8.60291 0.467937
\(339\) −10.9562 −0.595059
\(340\) −1.16205 −0.0630209
\(341\) 10.4687 0.566913
\(342\) −4.85429 −0.262490
\(343\) −1.00000 −0.0539949
\(344\) 25.6939 1.38532
\(345\) −3.76588 −0.202748
\(346\) −0.0567040 −0.00304842
\(347\) −32.4067 −1.73968 −0.869840 0.493334i \(-0.835778\pi\)
−0.869840 + 0.493334i \(0.835778\pi\)
\(348\) 3.62250 0.194187
\(349\) 4.01916 0.215141 0.107570 0.994197i \(-0.465693\pi\)
0.107570 + 0.994197i \(0.465693\pi\)
\(350\) −5.48813 −0.293353
\(351\) 2.35343 0.125617
\(352\) −5.99939 −0.319768
\(353\) 25.0605 1.33384 0.666918 0.745131i \(-0.267613\pi\)
0.666918 + 0.745131i \(0.267613\pi\)
\(354\) 14.4046 0.765596
\(355\) 5.36235 0.284604
\(356\) −1.59246 −0.0844004
\(357\) 3.53643 0.187168
\(358\) 17.7164 0.936343
\(359\) 20.4261 1.07805 0.539024 0.842291i \(-0.318793\pi\)
0.539024 + 0.842291i \(0.318793\pi\)
\(360\) 1.50878 0.0795198
\(361\) −1.27446 −0.0670767
\(362\) −17.6358 −0.926918
\(363\) 8.24022 0.432499
\(364\) −1.57823 −0.0827218
\(365\) −5.03346 −0.263463
\(366\) 12.9012 0.674356
\(367\) −27.7671 −1.44943 −0.724715 0.689049i \(-0.758029\pi\)
−0.724715 + 0.689049i \(0.758029\pi\)
\(368\) −16.9779 −0.885036
\(369\) −3.34227 −0.173992
\(370\) 5.13656 0.267037
\(371\) −3.98738 −0.207014
\(372\) 4.22597 0.219106
\(373\) 26.3795 1.36588 0.682940 0.730474i \(-0.260701\pi\)
0.682940 + 0.730474i \(0.260701\pi\)
\(374\) −6.77373 −0.350261
\(375\) 4.78228 0.246956
\(376\) 8.72171 0.449788
\(377\) −12.7128 −0.654740
\(378\) −1.15299 −0.0593035
\(379\) −11.6844 −0.600185 −0.300093 0.953910i \(-0.597018\pi\)
−0.300093 + 0.953910i \(0.597018\pi\)
\(380\) −1.38344 −0.0709688
\(381\) −8.25401 −0.422866
\(382\) −12.9926 −0.664760
\(383\) 1.00000 0.0510976
\(384\) 2.67226 0.136368
\(385\) −0.814005 −0.0414855
\(386\) 2.87393 0.146279
\(387\) 8.34436 0.424168
\(388\) −0.543312 −0.0275825
\(389\) −10.7525 −0.545174 −0.272587 0.962131i \(-0.587879\pi\)
−0.272587 + 0.962131i \(0.587879\pi\)
\(390\) −1.32959 −0.0673262
\(391\) 27.1795 1.37453
\(392\) 3.07919 0.155523
\(393\) 21.0890 1.06380
\(394\) 21.0219 1.05907
\(395\) −8.03378 −0.404223
\(396\) −1.11406 −0.0559834
\(397\) 16.6700 0.836644 0.418322 0.908299i \(-0.362618\pi\)
0.418322 + 0.908299i \(0.362618\pi\)
\(398\) 8.48943 0.425537
\(399\) 4.21017 0.210772
\(400\) 10.5149 0.525747
\(401\) 13.8324 0.690756 0.345378 0.938464i \(-0.387751\pi\)
0.345378 + 0.938464i \(0.387751\pi\)
\(402\) 3.22379 0.160788
\(403\) −14.8305 −0.738762
\(404\) 11.5028 0.572284
\(405\) 0.489993 0.0243479
\(406\) 6.22824 0.309102
\(407\) −15.1040 −0.748680
\(408\) −10.8893 −0.539102
\(409\) 2.71892 0.134442 0.0672210 0.997738i \(-0.478587\pi\)
0.0672210 + 0.997738i \(0.478587\pi\)
\(410\) 1.88824 0.0932536
\(411\) −18.9167 −0.933092
\(412\) 12.2257 0.602319
\(413\) −12.4932 −0.614752
\(414\) −8.86141 −0.435515
\(415\) 6.50775 0.319453
\(416\) 8.49905 0.416700
\(417\) 9.55723 0.468020
\(418\) −8.06424 −0.394435
\(419\) 2.92821 0.143052 0.0715262 0.997439i \(-0.477213\pi\)
0.0715262 + 0.997439i \(0.477213\pi\)
\(420\) −0.328594 −0.0160337
\(421\) 30.6924 1.49586 0.747929 0.663779i \(-0.231048\pi\)
0.747929 + 0.663779i \(0.231048\pi\)
\(422\) 23.4423 1.14115
\(423\) 2.83247 0.137719
\(424\) 12.2779 0.596268
\(425\) −16.8331 −0.816523
\(426\) 12.6180 0.611346
\(427\) −11.1893 −0.541489
\(428\) −4.44717 −0.214962
\(429\) 3.90965 0.188760
\(430\) −4.71421 −0.227339
\(431\) −3.45996 −0.166660 −0.0833301 0.996522i \(-0.526556\pi\)
−0.0833301 + 0.996522i \(0.526556\pi\)
\(432\) 2.20906 0.106284
\(433\) 0.986867 0.0474258 0.0237129 0.999719i \(-0.492451\pi\)
0.0237129 + 0.999719i \(0.492451\pi\)
\(434\) 7.26579 0.348769
\(435\) −2.64685 −0.126907
\(436\) 9.94648 0.476350
\(437\) 32.3576 1.54787
\(438\) −11.8441 −0.565935
\(439\) 29.5399 1.40986 0.704930 0.709277i \(-0.250978\pi\)
0.704930 + 0.709277i \(0.250978\pi\)
\(440\) 2.50648 0.119492
\(441\) 1.00000 0.0476190
\(442\) 9.59603 0.456437
\(443\) 10.9121 0.518448 0.259224 0.965817i \(-0.416533\pi\)
0.259224 + 0.965817i \(0.416533\pi\)
\(444\) −6.09714 −0.289357
\(445\) 1.16356 0.0551581
\(446\) −30.3553 −1.43737
\(447\) −4.15137 −0.196353
\(448\) −8.58198 −0.405461
\(449\) 30.0120 1.41635 0.708176 0.706036i \(-0.249518\pi\)
0.708176 + 0.706036i \(0.249518\pi\)
\(450\) 5.48813 0.258713
\(451\) −5.55238 −0.261451
\(452\) −7.34734 −0.345590
\(453\) −0.748490 −0.0351671
\(454\) 31.3572 1.47167
\(455\) 1.15316 0.0540611
\(456\) −12.9639 −0.607091
\(457\) −22.5872 −1.05658 −0.528291 0.849063i \(-0.677167\pi\)
−0.528291 + 0.849063i \(0.677167\pi\)
\(458\) 2.05939 0.0962290
\(459\) −3.53643 −0.165066
\(460\) −2.52543 −0.117749
\(461\) 27.8924 1.29908 0.649538 0.760329i \(-0.274962\pi\)
0.649538 + 0.760329i \(0.274962\pi\)
\(462\) −1.91542 −0.0891133
\(463\) 6.73171 0.312849 0.156425 0.987690i \(-0.450003\pi\)
0.156425 + 0.987690i \(0.450003\pi\)
\(464\) −11.9329 −0.553972
\(465\) −3.08778 −0.143192
\(466\) 25.1645 1.16572
\(467\) 9.29858 0.430287 0.215144 0.976582i \(-0.430978\pi\)
0.215144 + 0.976582i \(0.430978\pi\)
\(468\) 1.57823 0.0729537
\(469\) −2.79602 −0.129108
\(470\) −1.60022 −0.0738128
\(471\) −18.2085 −0.839005
\(472\) 38.4691 1.77068
\(473\) 13.8621 0.637382
\(474\) −18.9041 −0.868295
\(475\) −20.0400 −0.919499
\(476\) 2.37156 0.108700
\(477\) 3.98738 0.182570
\(478\) −22.7417 −1.04018
\(479\) 4.75609 0.217311 0.108656 0.994079i \(-0.465345\pi\)
0.108656 + 0.994079i \(0.465345\pi\)
\(480\) 1.76954 0.0807679
\(481\) 21.3972 0.975628
\(482\) −23.1781 −1.05573
\(483\) 7.68558 0.349706
\(484\) 5.52597 0.251181
\(485\) 0.396980 0.0180259
\(486\) 1.15299 0.0523008
\(487\) −18.8401 −0.853727 −0.426863 0.904316i \(-0.640381\pi\)
−0.426863 + 0.904316i \(0.640381\pi\)
\(488\) 34.4541 1.55966
\(489\) −19.3114 −0.873292
\(490\) −0.564958 −0.0255222
\(491\) 6.02096 0.271722 0.135861 0.990728i \(-0.456620\pi\)
0.135861 + 0.990728i \(0.456620\pi\)
\(492\) −2.24136 −0.101048
\(493\) 19.1031 0.860360
\(494\) 11.4242 0.514000
\(495\) 0.814005 0.0365868
\(496\) −13.9208 −0.625063
\(497\) −10.9437 −0.490894
\(498\) 15.3132 0.686203
\(499\) −8.75520 −0.391936 −0.195968 0.980610i \(-0.562785\pi\)
−0.195968 + 0.980610i \(0.562785\pi\)
\(500\) 3.20705 0.143423
\(501\) −6.14235 −0.274420
\(502\) 14.9490 0.667207
\(503\) 4.53083 0.202020 0.101010 0.994885i \(-0.467793\pi\)
0.101010 + 0.994885i \(0.467793\pi\)
\(504\) −3.07919 −0.137158
\(505\) −8.40469 −0.374004
\(506\) −14.7211 −0.654433
\(507\) 7.46138 0.331372
\(508\) −5.53522 −0.245586
\(509\) 4.28491 0.189925 0.0949627 0.995481i \(-0.469727\pi\)
0.0949627 + 0.995481i \(0.469727\pi\)
\(510\) 1.99793 0.0884699
\(511\) 10.2725 0.454430
\(512\) 21.5820 0.953797
\(513\) −4.21017 −0.185884
\(514\) −26.5001 −1.16887
\(515\) −8.93295 −0.393633
\(516\) 5.59581 0.246342
\(517\) 4.70546 0.206946
\(518\) −10.4829 −0.460593
\(519\) −0.0491798 −0.00215876
\(520\) −3.55081 −0.155713
\(521\) −0.784693 −0.0343780 −0.0171890 0.999852i \(-0.505472\pi\)
−0.0171890 + 0.999852i \(0.505472\pi\)
\(522\) −6.22824 −0.272603
\(523\) 39.3317 1.71986 0.859928 0.510415i \(-0.170508\pi\)
0.859928 + 0.510415i \(0.170508\pi\)
\(524\) 14.1425 0.617816
\(525\) −4.75991 −0.207739
\(526\) 4.35696 0.189972
\(527\) 22.2854 0.970769
\(528\) 3.66983 0.159709
\(529\) 36.0681 1.56818
\(530\) −2.25270 −0.0978510
\(531\) 12.4932 0.542161
\(532\) 2.82338 0.122409
\(533\) 7.86579 0.340705
\(534\) 2.73795 0.118483
\(535\) 3.24940 0.140484
\(536\) 8.60947 0.371873
\(537\) 15.3656 0.663076
\(538\) 5.04641 0.217566
\(539\) 1.66126 0.0715555
\(540\) 0.328594 0.0141404
\(541\) 25.8582 1.11173 0.555866 0.831272i \(-0.312387\pi\)
0.555866 + 0.831272i \(0.312387\pi\)
\(542\) 7.19123 0.308890
\(543\) −15.2957 −0.656401
\(544\) −12.7713 −0.547564
\(545\) −7.26757 −0.311309
\(546\) 2.71348 0.116126
\(547\) −15.6118 −0.667513 −0.333757 0.942659i \(-0.608316\pi\)
−0.333757 + 0.942659i \(0.608316\pi\)
\(548\) −12.6857 −0.541907
\(549\) 11.1893 0.477548
\(550\) 9.11721 0.388759
\(551\) 22.7425 0.968864
\(552\) −23.6654 −1.00727
\(553\) 16.3957 0.697217
\(554\) 13.5925 0.577490
\(555\) 4.45498 0.189103
\(556\) 6.40917 0.271810
\(557\) 17.7852 0.753581 0.376791 0.926298i \(-0.377028\pi\)
0.376791 + 0.926298i \(0.377028\pi\)
\(558\) −7.26579 −0.307585
\(559\) −19.6378 −0.830592
\(560\) 1.08242 0.0457408
\(561\) −5.87492 −0.248039
\(562\) 30.4812 1.28577
\(563\) −8.24347 −0.347421 −0.173710 0.984797i \(-0.555576\pi\)
−0.173710 + 0.984797i \(0.555576\pi\)
\(564\) 1.89948 0.0799825
\(565\) 5.36846 0.225853
\(566\) −24.1046 −1.01319
\(567\) −1.00000 −0.0419961
\(568\) 33.6979 1.41393
\(569\) 34.4633 1.44478 0.722389 0.691487i \(-0.243044\pi\)
0.722389 + 0.691487i \(0.243044\pi\)
\(570\) 2.37857 0.0996273
\(571\) −25.0582 −1.04865 −0.524327 0.851517i \(-0.675683\pi\)
−0.524327 + 0.851517i \(0.675683\pi\)
\(572\) 2.62185 0.109625
\(573\) −11.2686 −0.470753
\(574\) −3.85361 −0.160847
\(575\) −36.5826 −1.52560
\(576\) 8.58198 0.357583
\(577\) −16.2254 −0.675470 −0.337735 0.941241i \(-0.609661\pi\)
−0.337735 + 0.941241i \(0.609661\pi\)
\(578\) 5.18118 0.215509
\(579\) 2.49259 0.103588
\(580\) −1.77500 −0.0737029
\(581\) −13.2813 −0.551002
\(582\) 0.934126 0.0387208
\(583\) 6.62407 0.274341
\(584\) −31.6311 −1.30890
\(585\) −1.15316 −0.0476774
\(586\) 18.8526 0.778795
\(587\) 26.4202 1.09048 0.545239 0.838281i \(-0.316439\pi\)
0.545239 + 0.838281i \(0.316439\pi\)
\(588\) 0.670610 0.0276555
\(589\) 26.5312 1.09320
\(590\) −7.05815 −0.290580
\(591\) 18.2325 0.749985
\(592\) 20.0846 0.825474
\(593\) 5.78924 0.237736 0.118868 0.992910i \(-0.462074\pi\)
0.118868 + 0.992910i \(0.462074\pi\)
\(594\) 1.91542 0.0785905
\(595\) −1.73282 −0.0710388
\(596\) −2.78395 −0.114035
\(597\) 7.36296 0.301346
\(598\) 20.8547 0.852811
\(599\) −13.3764 −0.546543 −0.273271 0.961937i \(-0.588106\pi\)
−0.273271 + 0.961937i \(0.588106\pi\)
\(600\) 14.6567 0.598356
\(601\) −20.4029 −0.832253 −0.416126 0.909307i \(-0.636613\pi\)
−0.416126 + 0.909307i \(0.636613\pi\)
\(602\) 9.62098 0.392122
\(603\) 2.79602 0.113863
\(604\) −0.501944 −0.0204238
\(605\) −4.03765 −0.164154
\(606\) −19.7769 −0.803382
\(607\) 33.9762 1.37905 0.689525 0.724262i \(-0.257819\pi\)
0.689525 + 0.724262i \(0.257819\pi\)
\(608\) −15.2044 −0.616620
\(609\) 5.40181 0.218892
\(610\) −6.32149 −0.255950
\(611\) −6.66600 −0.269678
\(612\) −2.37156 −0.0958647
\(613\) −11.4377 −0.461963 −0.230981 0.972958i \(-0.574194\pi\)
−0.230981 + 0.972958i \(0.574194\pi\)
\(614\) 34.7030 1.40050
\(615\) 1.63769 0.0660380
\(616\) −5.11533 −0.206103
\(617\) 24.6581 0.992697 0.496348 0.868123i \(-0.334674\pi\)
0.496348 + 0.868123i \(0.334674\pi\)
\(618\) −21.0200 −0.845547
\(619\) −14.7312 −0.592098 −0.296049 0.955173i \(-0.595669\pi\)
−0.296049 + 0.955173i \(0.595669\pi\)
\(620\) −2.07069 −0.0831611
\(621\) −7.68558 −0.308412
\(622\) 16.3420 0.655254
\(623\) −2.37465 −0.0951383
\(624\) −5.19887 −0.208121
\(625\) 21.4563 0.858250
\(626\) −26.4693 −1.05793
\(627\) −6.99418 −0.279321
\(628\) −12.2108 −0.487265
\(629\) −32.1529 −1.28202
\(630\) 0.564958 0.0225084
\(631\) 1.66358 0.0662261 0.0331131 0.999452i \(-0.489458\pi\)
0.0331131 + 0.999452i \(0.489458\pi\)
\(632\) −50.4855 −2.00821
\(633\) 20.3317 0.808114
\(634\) −5.97188 −0.237174
\(635\) 4.04441 0.160497
\(636\) 2.67398 0.106030
\(637\) −2.35343 −0.0932462
\(638\) −10.3467 −0.409630
\(639\) 10.9437 0.432928
\(640\) −1.30939 −0.0517580
\(641\) 21.0005 0.829470 0.414735 0.909942i \(-0.363874\pi\)
0.414735 + 0.909942i \(0.363874\pi\)
\(642\) 7.64610 0.301767
\(643\) −33.1526 −1.30741 −0.653706 0.756749i \(-0.726787\pi\)
−0.653706 + 0.756749i \(0.726787\pi\)
\(644\) 5.15403 0.203097
\(645\) −4.08867 −0.160991
\(646\) −17.1669 −0.675421
\(647\) −14.0292 −0.551545 −0.275772 0.961223i \(-0.588934\pi\)
−0.275772 + 0.961223i \(0.588934\pi\)
\(648\) 3.07919 0.120962
\(649\) 20.7545 0.814686
\(650\) −12.9159 −0.506604
\(651\) 6.30168 0.246982
\(652\) −12.9504 −0.507177
\(653\) −33.2643 −1.30173 −0.650866 0.759193i \(-0.725594\pi\)
−0.650866 + 0.759193i \(0.725594\pi\)
\(654\) −17.1012 −0.668709
\(655\) −10.3334 −0.403761
\(656\) 7.38329 0.288269
\(657\) −10.2725 −0.400769
\(658\) 3.26581 0.127315
\(659\) 6.54041 0.254778 0.127389 0.991853i \(-0.459340\pi\)
0.127389 + 0.991853i \(0.459340\pi\)
\(660\) 0.545879 0.0212483
\(661\) 6.14150 0.238877 0.119438 0.992842i \(-0.461891\pi\)
0.119438 + 0.992842i \(0.461891\pi\)
\(662\) −26.1167 −1.01505
\(663\) 8.32272 0.323228
\(664\) 40.8957 1.58706
\(665\) −2.06295 −0.0799979
\(666\) 10.4829 0.406205
\(667\) 41.5160 1.60751
\(668\) −4.11912 −0.159373
\(669\) −26.3274 −1.01788
\(670\) −1.57963 −0.0610265
\(671\) 18.5884 0.717595
\(672\) −3.61135 −0.139311
\(673\) 35.9115 1.38429 0.692143 0.721761i \(-0.256667\pi\)
0.692143 + 0.721761i \(0.256667\pi\)
\(674\) 14.4190 0.555400
\(675\) 4.75991 0.183209
\(676\) 5.00368 0.192449
\(677\) 9.06359 0.348342 0.174171 0.984715i \(-0.444275\pi\)
0.174171 + 0.984715i \(0.444275\pi\)
\(678\) 12.6324 0.485145
\(679\) −0.810176 −0.0310917
\(680\) 5.33569 0.204615
\(681\) 27.1964 1.04217
\(682\) −12.0704 −0.462198
\(683\) 12.8911 0.493266 0.246633 0.969109i \(-0.420676\pi\)
0.246633 + 0.969109i \(0.420676\pi\)
\(684\) −2.82338 −0.107955
\(685\) 9.26904 0.354152
\(686\) 1.15299 0.0440214
\(687\) 1.78613 0.0681450
\(688\) −18.4332 −0.702760
\(689\) −9.38400 −0.357502
\(690\) 4.34203 0.165298
\(691\) 18.4567 0.702126 0.351063 0.936352i \(-0.385820\pi\)
0.351063 + 0.936352i \(0.385820\pi\)
\(692\) −0.0329805 −0.00125373
\(693\) −1.66126 −0.0631060
\(694\) 37.3646 1.41834
\(695\) −4.68297 −0.177635
\(696\) −16.6332 −0.630480
\(697\) −11.8197 −0.447703
\(698\) −4.63406 −0.175402
\(699\) 21.8254 0.825512
\(700\) −3.19204 −0.120648
\(701\) −31.2914 −1.18186 −0.590929 0.806723i \(-0.701239\pi\)
−0.590929 + 0.806723i \(0.701239\pi\)
\(702\) −2.71348 −0.102414
\(703\) −38.2786 −1.44370
\(704\) 14.2569 0.537327
\(705\) −1.38789 −0.0522709
\(706\) −28.8945 −1.08746
\(707\) 17.1527 0.645093
\(708\) 8.37809 0.314868
\(709\) 27.9348 1.04911 0.524556 0.851376i \(-0.324231\pi\)
0.524556 + 0.851376i \(0.324231\pi\)
\(710\) −6.18275 −0.232035
\(711\) −16.3957 −0.614887
\(712\) 7.31200 0.274029
\(713\) 48.4321 1.81380
\(714\) −4.07747 −0.152596
\(715\) −1.91570 −0.0716431
\(716\) 10.3043 0.385091
\(717\) −19.7241 −0.736610
\(718\) −23.5511 −0.878919
\(719\) −12.5904 −0.469543 −0.234771 0.972051i \(-0.575434\pi\)
−0.234771 + 0.972051i \(0.575434\pi\)
\(720\) −1.08242 −0.0403396
\(721\) 18.2308 0.678950
\(722\) 1.46944 0.0546869
\(723\) −20.1026 −0.747623
\(724\) −10.2574 −0.381215
\(725\) −25.7121 −0.954923
\(726\) −9.50091 −0.352612
\(727\) 29.5096 1.09445 0.547226 0.836985i \(-0.315684\pi\)
0.547226 + 0.836985i \(0.315684\pi\)
\(728\) 7.24665 0.268579
\(729\) 1.00000 0.0370370
\(730\) 5.80354 0.214799
\(731\) 29.5092 1.09144
\(732\) 7.50367 0.277343
\(733\) 22.8987 0.845782 0.422891 0.906181i \(-0.361015\pi\)
0.422891 + 0.906181i \(0.361015\pi\)
\(734\) 32.0152 1.18170
\(735\) −0.489993 −0.0180737
\(736\) −27.7553 −1.02307
\(737\) 4.64491 0.171097
\(738\) 3.85361 0.141853
\(739\) 34.1718 1.25703 0.628515 0.777798i \(-0.283663\pi\)
0.628515 + 0.777798i \(0.283663\pi\)
\(740\) 2.98755 0.109825
\(741\) 9.90833 0.363992
\(742\) 4.59742 0.168776
\(743\) 20.2979 0.744657 0.372329 0.928101i \(-0.378559\pi\)
0.372329 + 0.928101i \(0.378559\pi\)
\(744\) −19.4041 −0.711388
\(745\) 2.03414 0.0745251
\(746\) −30.4154 −1.11359
\(747\) 13.2813 0.485938
\(748\) −3.93978 −0.144053
\(749\) −6.63153 −0.242311
\(750\) −5.51393 −0.201340
\(751\) −23.1211 −0.843699 −0.421850 0.906666i \(-0.638619\pi\)
−0.421850 + 0.906666i \(0.638619\pi\)
\(752\) −6.25710 −0.228173
\(753\) 12.9654 0.472486
\(754\) 14.6577 0.533802
\(755\) 0.366754 0.0133476
\(756\) −0.670610 −0.0243898
\(757\) 32.1578 1.16880 0.584398 0.811467i \(-0.301331\pi\)
0.584398 + 0.811467i \(0.301331\pi\)
\(758\) 13.4720 0.489324
\(759\) −12.7677 −0.463440
\(760\) 6.35223 0.230420
\(761\) −15.4714 −0.560838 −0.280419 0.959878i \(-0.590473\pi\)
−0.280419 + 0.959878i \(0.590473\pi\)
\(762\) 9.51681 0.344758
\(763\) 14.8320 0.536954
\(764\) −7.55684 −0.273397
\(765\) 1.73282 0.0626504
\(766\) −1.15299 −0.0416593
\(767\) −29.4019 −1.06164
\(768\) 14.0829 0.508172
\(769\) 12.3400 0.444990 0.222495 0.974934i \(-0.428580\pi\)
0.222495 + 0.974934i \(0.428580\pi\)
\(770\) 0.938541 0.0338227
\(771\) −22.9838 −0.827740
\(772\) 1.67155 0.0601606
\(773\) 12.3636 0.444687 0.222343 0.974968i \(-0.428629\pi\)
0.222343 + 0.974968i \(0.428629\pi\)
\(774\) −9.62098 −0.345819
\(775\) −29.9954 −1.07747
\(776\) 2.49469 0.0895540
\(777\) −9.09193 −0.326171
\(778\) 12.3975 0.444474
\(779\) −14.0715 −0.504165
\(780\) −0.773322 −0.0276894
\(781\) 18.1804 0.650546
\(782\) −31.3377 −1.12064
\(783\) −5.40181 −0.193045
\(784\) −2.20906 −0.0788951
\(785\) 8.92205 0.318442
\(786\) −24.3154 −0.867301
\(787\) 8.56339 0.305252 0.152626 0.988284i \(-0.451227\pi\)
0.152626 + 0.988284i \(0.451227\pi\)
\(788\) 12.2269 0.435565
\(789\) 3.77883 0.134530
\(790\) 9.26288 0.329559
\(791\) −10.9562 −0.389558
\(792\) 5.11533 0.181765
\(793\) −26.3332 −0.935121
\(794\) −19.2204 −0.682106
\(795\) −1.95379 −0.0692937
\(796\) 4.93767 0.175011
\(797\) 30.2894 1.07290 0.536452 0.843931i \(-0.319764\pi\)
0.536452 + 0.843931i \(0.319764\pi\)
\(798\) −4.85429 −0.171840
\(799\) 10.0168 0.354369
\(800\) 17.1897 0.607747
\(801\) 2.37465 0.0839041
\(802\) −15.9486 −0.563166
\(803\) −17.0653 −0.602222
\(804\) 1.87504 0.0661275
\(805\) −3.76588 −0.132730
\(806\) 17.0995 0.602304
\(807\) 4.37680 0.154071
\(808\) −52.8164 −1.85807
\(809\) −8.23364 −0.289479 −0.144740 0.989470i \(-0.546234\pi\)
−0.144740 + 0.989470i \(0.546234\pi\)
\(810\) −0.564958 −0.0198506
\(811\) 30.5891 1.07413 0.537064 0.843541i \(-0.319533\pi\)
0.537064 + 0.843541i \(0.319533\pi\)
\(812\) 3.62250 0.127125
\(813\) 6.23702 0.218742
\(814\) 17.4148 0.610390
\(815\) 9.46244 0.331455
\(816\) 7.81219 0.273481
\(817\) 35.1312 1.22908
\(818\) −3.13489 −0.109609
\(819\) 2.35343 0.0822354
\(820\) 1.09825 0.0383525
\(821\) 36.0310 1.25749 0.628745 0.777611i \(-0.283569\pi\)
0.628745 + 0.777611i \(0.283569\pi\)
\(822\) 21.8108 0.760739
\(823\) −34.3488 −1.19733 −0.598663 0.801001i \(-0.704301\pi\)
−0.598663 + 0.801001i \(0.704301\pi\)
\(824\) −56.1361 −1.95559
\(825\) 7.90744 0.275302
\(826\) 14.4046 0.501200
\(827\) 34.7799 1.20942 0.604708 0.796447i \(-0.293290\pi\)
0.604708 + 0.796447i \(0.293290\pi\)
\(828\) −5.15403 −0.179115
\(829\) −0.441399 −0.0153304 −0.00766521 0.999971i \(-0.502440\pi\)
−0.00766521 + 0.999971i \(0.502440\pi\)
\(830\) −7.50338 −0.260446
\(831\) 11.7889 0.408952
\(832\) −20.1971 −0.700207
\(833\) 3.53643 0.122530
\(834\) −11.0194 −0.381571
\(835\) 3.00971 0.104155
\(836\) −4.69037 −0.162220
\(837\) −6.30168 −0.217818
\(838\) −3.37620 −0.116629
\(839\) 12.7763 0.441088 0.220544 0.975377i \(-0.429217\pi\)
0.220544 + 0.975377i \(0.429217\pi\)
\(840\) 1.50878 0.0520579
\(841\) 0.179514 0.00619013
\(842\) −35.3881 −1.21956
\(843\) 26.4366 0.910525
\(844\) 13.6347 0.469324
\(845\) −3.65602 −0.125771
\(846\) −3.26581 −0.112281
\(847\) 8.24022 0.283137
\(848\) −8.80837 −0.302481
\(849\) −20.9061 −0.717497
\(850\) 19.4084 0.665702
\(851\) −69.8768 −2.39534
\(852\) 7.33898 0.251429
\(853\) 8.42088 0.288325 0.144163 0.989554i \(-0.453951\pi\)
0.144163 + 0.989554i \(0.453951\pi\)
\(854\) 12.9012 0.441470
\(855\) 2.06295 0.0705515
\(856\) 20.4197 0.697933
\(857\) 0.224477 0.00766800 0.00383400 0.999993i \(-0.498780\pi\)
0.00383400 + 0.999993i \(0.498780\pi\)
\(858\) −4.50779 −0.153894
\(859\) 8.15203 0.278144 0.139072 0.990282i \(-0.455588\pi\)
0.139072 + 0.990282i \(0.455588\pi\)
\(860\) −2.74191 −0.0934982
\(861\) −3.34227 −0.113904
\(862\) 3.98930 0.135876
\(863\) −6.14825 −0.209289 −0.104644 0.994510i \(-0.533370\pi\)
−0.104644 + 0.994510i \(0.533370\pi\)
\(864\) 3.61135 0.122861
\(865\) 0.0240978 0.000819349 0
\(866\) −1.13785 −0.0386657
\(867\) 4.49369 0.152614
\(868\) 4.22597 0.143439
\(869\) −27.2375 −0.923970
\(870\) 3.05179 0.103465
\(871\) −6.58022 −0.222962
\(872\) −45.6706 −1.54660
\(873\) 0.810176 0.0274203
\(874\) −37.3081 −1.26196
\(875\) 4.78228 0.161671
\(876\) −6.88885 −0.232753
\(877\) −22.2410 −0.751026 −0.375513 0.926817i \(-0.622533\pi\)
−0.375513 + 0.926817i \(0.622533\pi\)
\(878\) −34.0592 −1.14944
\(879\) 16.3510 0.551507
\(880\) −1.79819 −0.0606169
\(881\) 11.4881 0.387043 0.193522 0.981096i \(-0.438009\pi\)
0.193522 + 0.981096i \(0.438009\pi\)
\(882\) −1.15299 −0.0388233
\(883\) 6.37351 0.214486 0.107243 0.994233i \(-0.465798\pi\)
0.107243 + 0.994233i \(0.465798\pi\)
\(884\) 5.58130 0.187719
\(885\) −6.12160 −0.205775
\(886\) −12.5815 −0.422684
\(887\) −18.2567 −0.613001 −0.306500 0.951871i \(-0.599158\pi\)
−0.306500 + 0.951871i \(0.599158\pi\)
\(888\) 27.9958 0.939477
\(889\) −8.25401 −0.276831
\(890\) −1.34158 −0.0449697
\(891\) 1.66126 0.0556543
\(892\) −17.6554 −0.591148
\(893\) 11.9252 0.399061
\(894\) 4.78649 0.160084
\(895\) −7.52905 −0.251668
\(896\) 2.67226 0.0892738
\(897\) 18.0874 0.603922
\(898\) −34.6036 −1.15474
\(899\) 34.0405 1.13531
\(900\) 3.19204 0.106401
\(901\) 14.1011 0.469775
\(902\) 6.40185 0.213158
\(903\) 8.34436 0.277683
\(904\) 33.7363 1.12205
\(905\) 7.49478 0.249135
\(906\) 0.863002 0.0286713
\(907\) 8.75359 0.290658 0.145329 0.989383i \(-0.453576\pi\)
0.145329 + 0.989383i \(0.453576\pi\)
\(908\) 18.2382 0.605254
\(909\) −17.1527 −0.568919
\(910\) −1.32959 −0.0440754
\(911\) 27.1446 0.899340 0.449670 0.893195i \(-0.351542\pi\)
0.449670 + 0.893195i \(0.351542\pi\)
\(912\) 9.30053 0.307971
\(913\) 22.0637 0.730202
\(914\) 26.0428 0.861420
\(915\) −5.48268 −0.181252
\(916\) 1.19779 0.0395762
\(917\) 21.0890 0.696419
\(918\) 4.07747 0.134577
\(919\) 52.6646 1.73724 0.868622 0.495474i \(-0.165006\pi\)
0.868622 + 0.495474i \(0.165006\pi\)
\(920\) 11.5959 0.382304
\(921\) 30.0983 0.991771
\(922\) −32.1597 −1.05912
\(923\) −25.7553 −0.847746
\(924\) −1.11406 −0.0366498
\(925\) 43.2767 1.42293
\(926\) −7.76161 −0.255062
\(927\) −18.2308 −0.598778
\(928\) −19.5078 −0.640376
\(929\) −0.872316 −0.0286198 −0.0143099 0.999898i \(-0.504555\pi\)
−0.0143099 + 0.999898i \(0.504555\pi\)
\(930\) 3.56018 0.116743
\(931\) 4.21017 0.137983
\(932\) 14.6363 0.479429
\(933\) 14.1735 0.464021
\(934\) −10.7212 −0.350808
\(935\) 2.87867 0.0941425
\(936\) −7.24665 −0.236864
\(937\) 52.7623 1.72367 0.861834 0.507190i \(-0.169316\pi\)
0.861834 + 0.507190i \(0.169316\pi\)
\(938\) 3.22379 0.105260
\(939\) −22.9571 −0.749177
\(940\) −0.930731 −0.0303571
\(941\) −46.4145 −1.51307 −0.756535 0.653954i \(-0.773109\pi\)
−0.756535 + 0.653954i \(0.773109\pi\)
\(942\) 20.9943 0.684031
\(943\) −25.6873 −0.836493
\(944\) −27.5984 −0.898250
\(945\) 0.489993 0.0159395
\(946\) −15.9829 −0.519650
\(947\) −13.6636 −0.444009 −0.222004 0.975046i \(-0.571260\pi\)
−0.222004 + 0.975046i \(0.571260\pi\)
\(948\) −10.9951 −0.357105
\(949\) 24.1756 0.784774
\(950\) 23.1060 0.749657
\(951\) −5.17947 −0.167956
\(952\) −10.8893 −0.352925
\(953\) −60.3180 −1.95389 −0.976945 0.213490i \(-0.931517\pi\)
−0.976945 + 0.213490i \(0.931517\pi\)
\(954\) −4.59742 −0.148847
\(955\) 5.52153 0.178673
\(956\) −13.2272 −0.427797
\(957\) −8.97380 −0.290082
\(958\) −5.48373 −0.177171
\(959\) −18.9167 −0.610852
\(960\) −4.20511 −0.135719
\(961\) 8.71120 0.281006
\(962\) −24.6708 −0.795418
\(963\) 6.63153 0.213698
\(964\) −13.4810 −0.434194
\(965\) −1.22135 −0.0393167
\(966\) −8.86141 −0.285111
\(967\) −34.4948 −1.10928 −0.554638 0.832092i \(-0.687143\pi\)
−0.554638 + 0.832092i \(0.687143\pi\)
\(968\) −25.3732 −0.815526
\(969\) −14.8890 −0.478302
\(970\) −0.457715 −0.0146963
\(971\) −6.29857 −0.202131 −0.101065 0.994880i \(-0.532225\pi\)
−0.101065 + 0.994880i \(0.532225\pi\)
\(972\) 0.670610 0.0215098
\(973\) 9.55723 0.306391
\(974\) 21.7225 0.696033
\(975\) −11.2021 −0.358754
\(976\) −24.7179 −0.791201
\(977\) −11.5571 −0.369743 −0.184872 0.982763i \(-0.559187\pi\)
−0.184872 + 0.982763i \(0.559187\pi\)
\(978\) 22.2659 0.711984
\(979\) 3.94491 0.126080
\(980\) −0.328594 −0.0104965
\(981\) −14.8320 −0.473549
\(982\) −6.94211 −0.221532
\(983\) 50.0335 1.59582 0.797910 0.602777i \(-0.205939\pi\)
0.797910 + 0.602777i \(0.205939\pi\)
\(984\) 10.2915 0.328081
\(985\) −8.93379 −0.284654
\(986\) −22.0257 −0.701442
\(987\) 2.83247 0.0901584
\(988\) 6.64462 0.211394
\(989\) 64.1312 2.03925
\(990\) −0.938541 −0.0298288
\(991\) 5.03742 0.160019 0.0800094 0.996794i \(-0.474505\pi\)
0.0800094 + 0.996794i \(0.474505\pi\)
\(992\) −22.7576 −0.722554
\(993\) −22.6512 −0.718814
\(994\) 12.6180 0.400220
\(995\) −3.60780 −0.114375
\(996\) 8.90658 0.282216
\(997\) 21.7107 0.687586 0.343793 0.939045i \(-0.388288\pi\)
0.343793 + 0.939045i \(0.388288\pi\)
\(998\) 10.0947 0.319541
\(999\) 9.09193 0.287656
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.s.1.18 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8043.2.a.s.1.18 50 1.1 even 1 trivial