Properties

Label 6046.2.a.e.1.6
Level $6046$
Weight $2$
Character 6046.1
Self dual yes
Analytic conductor $48.278$
Analytic rank $1$
Dimension $56$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6046,2,Mod(1,6046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6046, 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("6046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6046 = 2 \cdot 3023 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6046.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: \(48.2775530621\)
Analytic rank: \(1\)
Dimension: \(56\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) \(=\) 6046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.89764 q^{3} +1.00000 q^{4} +3.79029 q^{5} -2.89764 q^{6} -4.56649 q^{7} +1.00000 q^{8} +5.39633 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.89764 q^{3} +1.00000 q^{4} +3.79029 q^{5} -2.89764 q^{6} -4.56649 q^{7} +1.00000 q^{8} +5.39633 q^{9} +3.79029 q^{10} +3.65865 q^{11} -2.89764 q^{12} +2.26961 q^{13} -4.56649 q^{14} -10.9829 q^{15} +1.00000 q^{16} +1.61568 q^{17} +5.39633 q^{18} -6.07444 q^{19} +3.79029 q^{20} +13.2321 q^{21} +3.65865 q^{22} -6.86880 q^{23} -2.89764 q^{24} +9.36631 q^{25} +2.26961 q^{26} -6.94371 q^{27} -4.56649 q^{28} -7.67395 q^{29} -10.9829 q^{30} +2.27430 q^{31} +1.00000 q^{32} -10.6015 q^{33} +1.61568 q^{34} -17.3083 q^{35} +5.39633 q^{36} -9.40992 q^{37} -6.07444 q^{38} -6.57652 q^{39} +3.79029 q^{40} -10.6980 q^{41} +13.2321 q^{42} +6.27023 q^{43} +3.65865 q^{44} +20.4537 q^{45} -6.86880 q^{46} -5.99379 q^{47} -2.89764 q^{48} +13.8529 q^{49} +9.36631 q^{50} -4.68165 q^{51} +2.26961 q^{52} +7.10368 q^{53} -6.94371 q^{54} +13.8674 q^{55} -4.56649 q^{56} +17.6016 q^{57} -7.67395 q^{58} -13.0409 q^{59} -10.9829 q^{60} +1.80554 q^{61} +2.27430 q^{62} -24.6423 q^{63} +1.00000 q^{64} +8.60249 q^{65} -10.6015 q^{66} +9.50832 q^{67} +1.61568 q^{68} +19.9033 q^{69} -17.3083 q^{70} +2.25981 q^{71} +5.39633 q^{72} +8.78795 q^{73} -9.40992 q^{74} -27.1402 q^{75} -6.07444 q^{76} -16.7072 q^{77} -6.57652 q^{78} -5.00610 q^{79} +3.79029 q^{80} +3.93139 q^{81} -10.6980 q^{82} +2.74130 q^{83} +13.2321 q^{84} +6.12388 q^{85} +6.27023 q^{86} +22.2364 q^{87} +3.65865 q^{88} -13.2225 q^{89} +20.4537 q^{90} -10.3642 q^{91} -6.86880 q^{92} -6.59012 q^{93} -5.99379 q^{94} -23.0239 q^{95} -2.89764 q^{96} +7.22829 q^{97} +13.8529 q^{98} +19.7433 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9} - 17 q^{10} - 53 q^{11} - 18 q^{12} - 21 q^{13} - 35 q^{14} - 36 q^{15} + 56 q^{16} - 22 q^{17} + 34 q^{18} - 31 q^{19} - 17 q^{20} - 23 q^{21} - 53 q^{22} - 59 q^{23} - 18 q^{24} + 41 q^{25} - 21 q^{26} - 63 q^{27} - 35 q^{28} - 88 q^{29} - 36 q^{30} - 44 q^{31} + 56 q^{32} + 4 q^{33} - 22 q^{34} - 51 q^{35} + 34 q^{36} - 60 q^{37} - 31 q^{38} - 62 q^{39} - 17 q^{40} - 39 q^{41} - 23 q^{42} - 66 q^{43} - 53 q^{44} - 34 q^{45} - 59 q^{46} - 51 q^{47} - 18 q^{48} + 41 q^{49} + 41 q^{50} - 48 q^{51} - 21 q^{52} - 75 q^{53} - 63 q^{54} - 41 q^{55} - 35 q^{56} - 12 q^{57} - 88 q^{58} - 77 q^{59} - 36 q^{60} - 43 q^{61} - 44 q^{62} - 88 q^{63} + 56 q^{64} - 54 q^{65} + 4 q^{66} - 62 q^{67} - 22 q^{68} - 48 q^{69} - 51 q^{70} - 122 q^{71} + 34 q^{72} - 7 q^{73} - 60 q^{74} - 63 q^{75} - 31 q^{76} - 39 q^{77} - 62 q^{78} - 91 q^{79} - 17 q^{80} + 8 q^{81} - 39 q^{82} - 51 q^{83} - 23 q^{84} - 72 q^{85} - 66 q^{86} - 19 q^{87} - 53 q^{88} - 62 q^{89} - 34 q^{90} - 48 q^{91} - 59 q^{92} - 41 q^{93} - 51 q^{94} - 120 q^{95} - 18 q^{96} + 6 q^{97} + 41 q^{98} - 128 q^{99}+O(q^{100}) \) 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
\(3\) −2.89764 −1.67295 −0.836477 0.548002i \(-0.815389\pi\)
−0.836477 + 0.548002i \(0.815389\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.79029 1.69507 0.847535 0.530740i \(-0.178086\pi\)
0.847535 + 0.530740i \(0.178086\pi\)
\(6\) −2.89764 −1.18296
\(7\) −4.56649 −1.72597 −0.862986 0.505227i \(-0.831409\pi\)
−0.862986 + 0.505227i \(0.831409\pi\)
\(8\) 1.00000 0.353553
\(9\) 5.39633 1.79878
\(10\) 3.79029 1.19860
\(11\) 3.65865 1.10313 0.551563 0.834133i \(-0.314032\pi\)
0.551563 + 0.834133i \(0.314032\pi\)
\(12\) −2.89764 −0.836477
\(13\) 2.26961 0.629477 0.314738 0.949178i \(-0.398083\pi\)
0.314738 + 0.949178i \(0.398083\pi\)
\(14\) −4.56649 −1.22045
\(15\) −10.9829 −2.83578
\(16\) 1.00000 0.250000
\(17\) 1.61568 0.391859 0.195929 0.980618i \(-0.437228\pi\)
0.195929 + 0.980618i \(0.437228\pi\)
\(18\) 5.39633 1.27193
\(19\) −6.07444 −1.39357 −0.696787 0.717279i \(-0.745387\pi\)
−0.696787 + 0.717279i \(0.745387\pi\)
\(20\) 3.79029 0.847535
\(21\) 13.2321 2.88747
\(22\) 3.65865 0.780027
\(23\) −6.86880 −1.43224 −0.716122 0.697975i \(-0.754084\pi\)
−0.716122 + 0.697975i \(0.754084\pi\)
\(24\) −2.89764 −0.591479
\(25\) 9.36631 1.87326
\(26\) 2.26961 0.445107
\(27\) −6.94371 −1.33632
\(28\) −4.56649 −0.862986
\(29\) −7.67395 −1.42502 −0.712509 0.701663i \(-0.752441\pi\)
−0.712509 + 0.701663i \(0.752441\pi\)
\(30\) −10.9829 −2.00520
\(31\) 2.27430 0.408477 0.204238 0.978921i \(-0.434528\pi\)
0.204238 + 0.978921i \(0.434528\pi\)
\(32\) 1.00000 0.176777
\(33\) −10.6015 −1.84548
\(34\) 1.61568 0.277086
\(35\) −17.3083 −2.92564
\(36\) 5.39633 0.899388
\(37\) −9.40992 −1.54698 −0.773491 0.633807i \(-0.781491\pi\)
−0.773491 + 0.633807i \(0.781491\pi\)
\(38\) −6.07444 −0.985405
\(39\) −6.57652 −1.05309
\(40\) 3.79029 0.599298
\(41\) −10.6980 −1.67075 −0.835377 0.549677i \(-0.814751\pi\)
−0.835377 + 0.549677i \(0.814751\pi\)
\(42\) 13.2321 2.04175
\(43\) 6.27023 0.956200 0.478100 0.878305i \(-0.341326\pi\)
0.478100 + 0.878305i \(0.341326\pi\)
\(44\) 3.65865 0.551563
\(45\) 20.4537 3.04905
\(46\) −6.86880 −1.01275
\(47\) −5.99379 −0.874284 −0.437142 0.899393i \(-0.644009\pi\)
−0.437142 + 0.899393i \(0.644009\pi\)
\(48\) −2.89764 −0.418239
\(49\) 13.8529 1.97898
\(50\) 9.36631 1.32460
\(51\) −4.68165 −0.655562
\(52\) 2.26961 0.314738
\(53\) 7.10368 0.975766 0.487883 0.872909i \(-0.337769\pi\)
0.487883 + 0.872909i \(0.337769\pi\)
\(54\) −6.94371 −0.944919
\(55\) 13.8674 1.86987
\(56\) −4.56649 −0.610223
\(57\) 17.6016 2.33138
\(58\) −7.67395 −1.00764
\(59\) −13.0409 −1.69778 −0.848892 0.528567i \(-0.822730\pi\)
−0.848892 + 0.528567i \(0.822730\pi\)
\(60\) −10.9829 −1.41789
\(61\) 1.80554 0.231176 0.115588 0.993297i \(-0.463125\pi\)
0.115588 + 0.993297i \(0.463125\pi\)
\(62\) 2.27430 0.288837
\(63\) −24.6423 −3.10464
\(64\) 1.00000 0.125000
\(65\) 8.60249 1.06701
\(66\) −10.6015 −1.30495
\(67\) 9.50832 1.16163 0.580813 0.814037i \(-0.302735\pi\)
0.580813 + 0.814037i \(0.302735\pi\)
\(68\) 1.61568 0.195929
\(69\) 19.9033 2.39608
\(70\) −17.3083 −2.06874
\(71\) 2.25981 0.268190 0.134095 0.990968i \(-0.457187\pi\)
0.134095 + 0.990968i \(0.457187\pi\)
\(72\) 5.39633 0.635964
\(73\) 8.78795 1.02855 0.514276 0.857625i \(-0.328061\pi\)
0.514276 + 0.857625i \(0.328061\pi\)
\(74\) −9.40992 −1.09388
\(75\) −27.1402 −3.13388
\(76\) −6.07444 −0.696787
\(77\) −16.7072 −1.90396
\(78\) −6.57652 −0.744644
\(79\) −5.00610 −0.563231 −0.281615 0.959527i \(-0.590870\pi\)
−0.281615 + 0.959527i \(0.590870\pi\)
\(80\) 3.79029 0.423767
\(81\) 3.93139 0.436822
\(82\) −10.6980 −1.18140
\(83\) 2.74130 0.300897 0.150449 0.988618i \(-0.451928\pi\)
0.150449 + 0.988618i \(0.451928\pi\)
\(84\) 13.2321 1.44374
\(85\) 6.12388 0.664228
\(86\) 6.27023 0.676136
\(87\) 22.2364 2.38399
\(88\) 3.65865 0.390014
\(89\) −13.2225 −1.40158 −0.700792 0.713366i \(-0.747170\pi\)
−0.700792 + 0.713366i \(0.747170\pi\)
\(90\) 20.4537 2.15601
\(91\) −10.3642 −1.08646
\(92\) −6.86880 −0.716122
\(93\) −6.59012 −0.683363
\(94\) −5.99379 −0.618212
\(95\) −23.0239 −2.36220
\(96\) −2.89764 −0.295739
\(97\) 7.22829 0.733922 0.366961 0.930236i \(-0.380398\pi\)
0.366961 + 0.930236i \(0.380398\pi\)
\(98\) 13.8529 1.39935
\(99\) 19.7433 1.98428
\(100\) 9.36631 0.936631
\(101\) 2.14597 0.213532 0.106766 0.994284i \(-0.465950\pi\)
0.106766 + 0.994284i \(0.465950\pi\)
\(102\) −4.68165 −0.463552
\(103\) 18.3353 1.80663 0.903316 0.428975i \(-0.141125\pi\)
0.903316 + 0.428975i \(0.141125\pi\)
\(104\) 2.26961 0.222554
\(105\) 50.1534 4.89447
\(106\) 7.10368 0.689971
\(107\) −14.3251 −1.38486 −0.692432 0.721483i \(-0.743461\pi\)
−0.692432 + 0.721483i \(0.743461\pi\)
\(108\) −6.94371 −0.668159
\(109\) −19.8425 −1.90057 −0.950283 0.311386i \(-0.899207\pi\)
−0.950283 + 0.311386i \(0.899207\pi\)
\(110\) 13.8674 1.32220
\(111\) 27.2666 2.58803
\(112\) −4.56649 −0.431493
\(113\) −15.5873 −1.46633 −0.733166 0.680050i \(-0.761958\pi\)
−0.733166 + 0.680050i \(0.761958\pi\)
\(114\) 17.6016 1.64854
\(115\) −26.0348 −2.42775
\(116\) −7.67395 −0.712509
\(117\) 12.2476 1.13229
\(118\) −13.0409 −1.20051
\(119\) −7.37797 −0.676338
\(120\) −10.9829 −1.00260
\(121\) 2.38574 0.216886
\(122\) 1.80554 0.163466
\(123\) 30.9991 2.79510
\(124\) 2.27430 0.204238
\(125\) 16.5496 1.48024
\(126\) −24.6423 −2.19531
\(127\) 3.64159 0.323139 0.161569 0.986861i \(-0.448344\pi\)
0.161569 + 0.986861i \(0.448344\pi\)
\(128\) 1.00000 0.0883883
\(129\) −18.1689 −1.59968
\(130\) 8.60249 0.754488
\(131\) 10.1897 0.890276 0.445138 0.895462i \(-0.353155\pi\)
0.445138 + 0.895462i \(0.353155\pi\)
\(132\) −10.6015 −0.922739
\(133\) 27.7389 2.40527
\(134\) 9.50832 0.821394
\(135\) −26.3187 −2.26515
\(136\) 1.61568 0.138543
\(137\) 8.19163 0.699858 0.349929 0.936776i \(-0.386206\pi\)
0.349929 + 0.936776i \(0.386206\pi\)
\(138\) 19.9033 1.69428
\(139\) −0.273303 −0.0231813 −0.0115906 0.999933i \(-0.503689\pi\)
−0.0115906 + 0.999933i \(0.503689\pi\)
\(140\) −17.3083 −1.46282
\(141\) 17.3678 1.46264
\(142\) 2.25981 0.189639
\(143\) 8.30372 0.694392
\(144\) 5.39633 0.449694
\(145\) −29.0865 −2.41550
\(146\) 8.78795 0.727296
\(147\) −40.1407 −3.31075
\(148\) −9.40992 −0.773491
\(149\) 11.0896 0.908497 0.454248 0.890875i \(-0.349908\pi\)
0.454248 + 0.890875i \(0.349908\pi\)
\(150\) −27.1402 −2.21599
\(151\) −5.89412 −0.479657 −0.239828 0.970815i \(-0.577091\pi\)
−0.239828 + 0.970815i \(0.577091\pi\)
\(152\) −6.07444 −0.492702
\(153\) 8.71872 0.704867
\(154\) −16.7072 −1.34631
\(155\) 8.62027 0.692397
\(156\) −6.57652 −0.526543
\(157\) 13.8986 1.10923 0.554615 0.832107i \(-0.312865\pi\)
0.554615 + 0.832107i \(0.312865\pi\)
\(158\) −5.00610 −0.398264
\(159\) −20.5839 −1.63241
\(160\) 3.79029 0.299649
\(161\) 31.3663 2.47201
\(162\) 3.93139 0.308880
\(163\) −5.81128 −0.455175 −0.227587 0.973758i \(-0.573084\pi\)
−0.227587 + 0.973758i \(0.573084\pi\)
\(164\) −10.6980 −0.835377
\(165\) −40.1827 −3.12822
\(166\) 2.74130 0.212766
\(167\) −13.9647 −1.08062 −0.540309 0.841467i \(-0.681693\pi\)
−0.540309 + 0.841467i \(0.681693\pi\)
\(168\) 13.2321 1.02088
\(169\) −7.84887 −0.603759
\(170\) 6.12388 0.469680
\(171\) −32.7797 −2.50673
\(172\) 6.27023 0.478100
\(173\) −21.7129 −1.65080 −0.825400 0.564548i \(-0.809050\pi\)
−0.825400 + 0.564548i \(0.809050\pi\)
\(174\) 22.2364 1.68574
\(175\) −42.7712 −3.23320
\(176\) 3.65865 0.275781
\(177\) 37.7879 2.84031
\(178\) −13.2225 −0.991070
\(179\) −19.7679 −1.47753 −0.738763 0.673966i \(-0.764589\pi\)
−0.738763 + 0.673966i \(0.764589\pi\)
\(180\) 20.4537 1.52453
\(181\) −6.86724 −0.510438 −0.255219 0.966883i \(-0.582148\pi\)
−0.255219 + 0.966883i \(0.582148\pi\)
\(182\) −10.3642 −0.768243
\(183\) −5.23182 −0.386747
\(184\) −6.86880 −0.506375
\(185\) −35.6664 −2.62224
\(186\) −6.59012 −0.483211
\(187\) 5.91120 0.432270
\(188\) −5.99379 −0.437142
\(189\) 31.7084 2.30645
\(190\) −23.0239 −1.67033
\(191\) −20.5068 −1.48382 −0.741911 0.670499i \(-0.766080\pi\)
−0.741911 + 0.670499i \(0.766080\pi\)
\(192\) −2.89764 −0.209119
\(193\) −6.04232 −0.434936 −0.217468 0.976067i \(-0.569780\pi\)
−0.217468 + 0.976067i \(0.569780\pi\)
\(194\) 7.22829 0.518961
\(195\) −24.9269 −1.78505
\(196\) 13.8529 0.989491
\(197\) 22.0543 1.57130 0.785652 0.618668i \(-0.212328\pi\)
0.785652 + 0.618668i \(0.212328\pi\)
\(198\) 19.7433 1.40310
\(199\) −25.3534 −1.79725 −0.898627 0.438714i \(-0.855434\pi\)
−0.898627 + 0.438714i \(0.855434\pi\)
\(200\) 9.36631 0.662298
\(201\) −27.5517 −1.94335
\(202\) 2.14597 0.150990
\(203\) 35.0431 2.45954
\(204\) −4.68165 −0.327781
\(205\) −40.5487 −2.83205
\(206\) 18.3353 1.27748
\(207\) −37.0663 −2.57629
\(208\) 2.26961 0.157369
\(209\) −22.2243 −1.53729
\(210\) 50.1534 3.46091
\(211\) 23.5618 1.62206 0.811032 0.585002i \(-0.198906\pi\)
0.811032 + 0.585002i \(0.198906\pi\)
\(212\) 7.10368 0.487883
\(213\) −6.54812 −0.448670
\(214\) −14.3251 −0.979247
\(215\) 23.7660 1.62083
\(216\) −6.94371 −0.472460
\(217\) −10.3856 −0.705020
\(218\) −19.8425 −1.34390
\(219\) −25.4643 −1.72072
\(220\) 13.8674 0.934937
\(221\) 3.66696 0.246666
\(222\) 27.2666 1.83001
\(223\) 18.3676 1.22998 0.614992 0.788533i \(-0.289159\pi\)
0.614992 + 0.788533i \(0.289159\pi\)
\(224\) −4.56649 −0.305112
\(225\) 50.5437 3.36958
\(226\) −15.5873 −1.03685
\(227\) 1.23503 0.0819718 0.0409859 0.999160i \(-0.486950\pi\)
0.0409859 + 0.999160i \(0.486950\pi\)
\(228\) 17.6016 1.16569
\(229\) −3.42062 −0.226041 −0.113020 0.993593i \(-0.536053\pi\)
−0.113020 + 0.993593i \(0.536053\pi\)
\(230\) −26.0348 −1.71668
\(231\) 48.4115 3.18525
\(232\) −7.67395 −0.503820
\(233\) −21.3961 −1.40170 −0.700852 0.713306i \(-0.747197\pi\)
−0.700852 + 0.713306i \(0.747197\pi\)
\(234\) 12.2476 0.800649
\(235\) −22.7182 −1.48197
\(236\) −13.0409 −0.848892
\(237\) 14.5059 0.942259
\(238\) −7.37797 −0.478243
\(239\) −12.8100 −0.828611 −0.414306 0.910138i \(-0.635976\pi\)
−0.414306 + 0.910138i \(0.635976\pi\)
\(240\) −10.9829 −0.708944
\(241\) 2.55865 0.164817 0.0824087 0.996599i \(-0.473739\pi\)
0.0824087 + 0.996599i \(0.473739\pi\)
\(242\) 2.38574 0.153361
\(243\) 9.43935 0.605535
\(244\) 1.80554 0.115588
\(245\) 52.5064 3.35451
\(246\) 30.9991 1.97643
\(247\) −13.7866 −0.877222
\(248\) 2.27430 0.144418
\(249\) −7.94331 −0.503387
\(250\) 16.5496 1.04669
\(251\) −13.2223 −0.834583 −0.417291 0.908773i \(-0.637021\pi\)
−0.417291 + 0.908773i \(0.637021\pi\)
\(252\) −24.6423 −1.55232
\(253\) −25.1306 −1.57994
\(254\) 3.64159 0.228494
\(255\) −17.7448 −1.11122
\(256\) 1.00000 0.0625000
\(257\) 6.88848 0.429692 0.214846 0.976648i \(-0.431075\pi\)
0.214846 + 0.976648i \(0.431075\pi\)
\(258\) −18.1689 −1.13114
\(259\) 42.9704 2.67005
\(260\) 8.60249 0.533504
\(261\) −41.4112 −2.56329
\(262\) 10.1897 0.629520
\(263\) 6.88858 0.424768 0.212384 0.977186i \(-0.431877\pi\)
0.212384 + 0.977186i \(0.431877\pi\)
\(264\) −10.6015 −0.652475
\(265\) 26.9250 1.65399
\(266\) 27.7389 1.70078
\(267\) 38.3141 2.34479
\(268\) 9.50832 0.580813
\(269\) 0.730619 0.0445467 0.0222733 0.999752i \(-0.492910\pi\)
0.0222733 + 0.999752i \(0.492910\pi\)
\(270\) −26.3187 −1.60170
\(271\) 29.9786 1.82107 0.910534 0.413435i \(-0.135671\pi\)
0.910534 + 0.413435i \(0.135671\pi\)
\(272\) 1.61568 0.0979647
\(273\) 30.0316 1.81760
\(274\) 8.19163 0.494874
\(275\) 34.2681 2.06644
\(276\) 19.9033 1.19804
\(277\) 14.5178 0.872288 0.436144 0.899877i \(-0.356344\pi\)
0.436144 + 0.899877i \(0.356344\pi\)
\(278\) −0.273303 −0.0163916
\(279\) 12.2729 0.734759
\(280\) −17.3083 −1.03437
\(281\) −5.12599 −0.305791 −0.152896 0.988242i \(-0.548860\pi\)
−0.152896 + 0.988242i \(0.548860\pi\)
\(282\) 17.3678 1.03424
\(283\) −17.9473 −1.06686 −0.533429 0.845845i \(-0.679097\pi\)
−0.533429 + 0.845845i \(0.679097\pi\)
\(284\) 2.25981 0.134095
\(285\) 66.7151 3.95186
\(286\) 8.30372 0.491009
\(287\) 48.8526 2.88368
\(288\) 5.39633 0.317982
\(289\) −14.3896 −0.846447
\(290\) −29.0865 −1.70802
\(291\) −20.9450 −1.22782
\(292\) 8.78795 0.514276
\(293\) 27.1006 1.58323 0.791616 0.611018i \(-0.209240\pi\)
0.791616 + 0.611018i \(0.209240\pi\)
\(294\) −40.1407 −2.34105
\(295\) −49.4289 −2.87786
\(296\) −9.40992 −0.546941
\(297\) −25.4046 −1.47413
\(298\) 11.0896 0.642404
\(299\) −15.5895 −0.901564
\(300\) −27.1402 −1.56694
\(301\) −28.6330 −1.65038
\(302\) −5.89412 −0.339169
\(303\) −6.21825 −0.357229
\(304\) −6.07444 −0.348393
\(305\) 6.84353 0.391859
\(306\) 8.71872 0.498416
\(307\) −23.5113 −1.34186 −0.670931 0.741520i \(-0.734105\pi\)
−0.670931 + 0.741520i \(0.734105\pi\)
\(308\) −16.7072 −0.951982
\(309\) −53.1292 −3.02241
\(310\) 8.62027 0.489599
\(311\) 4.10835 0.232963 0.116481 0.993193i \(-0.462838\pi\)
0.116481 + 0.993193i \(0.462838\pi\)
\(312\) −6.57652 −0.372322
\(313\) 18.3293 1.03603 0.518016 0.855371i \(-0.326671\pi\)
0.518016 + 0.855371i \(0.326671\pi\)
\(314\) 13.8986 0.784345
\(315\) −93.4016 −5.26258
\(316\) −5.00610 −0.281615
\(317\) −31.1070 −1.74714 −0.873572 0.486695i \(-0.838202\pi\)
−0.873572 + 0.486695i \(0.838202\pi\)
\(318\) −20.5839 −1.15429
\(319\) −28.0763 −1.57197
\(320\) 3.79029 0.211884
\(321\) 41.5092 2.31682
\(322\) 31.3663 1.74798
\(323\) −9.81433 −0.546084
\(324\) 3.93139 0.218411
\(325\) 21.2579 1.17918
\(326\) −5.81128 −0.321857
\(327\) 57.4965 3.17956
\(328\) −10.6980 −0.590701
\(329\) 27.3706 1.50899
\(330\) −40.1827 −2.21198
\(331\) −16.9778 −0.933183 −0.466591 0.884473i \(-0.654518\pi\)
−0.466591 + 0.884473i \(0.654518\pi\)
\(332\) 2.74130 0.150449
\(333\) −50.7791 −2.78268
\(334\) −13.9647 −0.764112
\(335\) 36.0393 1.96904
\(336\) 13.2321 0.721868
\(337\) −6.86104 −0.373745 −0.186872 0.982384i \(-0.559835\pi\)
−0.186872 + 0.982384i \(0.559835\pi\)
\(338\) −7.84887 −0.426922
\(339\) 45.1665 2.45311
\(340\) 6.12388 0.332114
\(341\) 8.32089 0.450601
\(342\) −32.7797 −1.77252
\(343\) −31.2936 −1.68970
\(344\) 6.27023 0.338068
\(345\) 75.4394 4.06152
\(346\) −21.7129 −1.16729
\(347\) 7.51465 0.403408 0.201704 0.979447i \(-0.435352\pi\)
0.201704 + 0.979447i \(0.435352\pi\)
\(348\) 22.2364 1.19199
\(349\) −23.9846 −1.28387 −0.641934 0.766760i \(-0.721868\pi\)
−0.641934 + 0.766760i \(0.721868\pi\)
\(350\) −42.7712 −2.28622
\(351\) −15.7595 −0.841181
\(352\) 3.65865 0.195007
\(353\) −24.4745 −1.30265 −0.651324 0.758800i \(-0.725786\pi\)
−0.651324 + 0.758800i \(0.725786\pi\)
\(354\) 37.7879 2.00841
\(355\) 8.56534 0.454601
\(356\) −13.2225 −0.700792
\(357\) 21.3787 1.13148
\(358\) −19.7679 −1.04477
\(359\) −2.53346 −0.133711 −0.0668553 0.997763i \(-0.521297\pi\)
−0.0668553 + 0.997763i \(0.521297\pi\)
\(360\) 20.4537 1.07800
\(361\) 17.8989 0.942046
\(362\) −6.86724 −0.360934
\(363\) −6.91303 −0.362840
\(364\) −10.3642 −0.543230
\(365\) 33.3089 1.74347
\(366\) −5.23182 −0.273471
\(367\) −11.8394 −0.618013 −0.309007 0.951060i \(-0.599997\pi\)
−0.309007 + 0.951060i \(0.599997\pi\)
\(368\) −6.86880 −0.358061
\(369\) −57.7302 −3.00531
\(370\) −35.6664 −1.85421
\(371\) −32.4389 −1.68415
\(372\) −6.59012 −0.341682
\(373\) 23.6275 1.22339 0.611694 0.791095i \(-0.290488\pi\)
0.611694 + 0.791095i \(0.290488\pi\)
\(374\) 5.91120 0.305661
\(375\) −47.9548 −2.47638
\(376\) −5.99379 −0.309106
\(377\) −17.4169 −0.897015
\(378\) 31.7084 1.63090
\(379\) −7.20684 −0.370190 −0.185095 0.982721i \(-0.559259\pi\)
−0.185095 + 0.982721i \(0.559259\pi\)
\(380\) −23.0239 −1.18110
\(381\) −10.5520 −0.540596
\(382\) −20.5068 −1.04922
\(383\) −18.1294 −0.926370 −0.463185 0.886262i \(-0.653294\pi\)
−0.463185 + 0.886262i \(0.653294\pi\)
\(384\) −2.89764 −0.147870
\(385\) −63.3252 −3.22735
\(386\) −6.04232 −0.307546
\(387\) 33.8362 1.71999
\(388\) 7.22829 0.366961
\(389\) 29.6967 1.50568 0.752841 0.658202i \(-0.228683\pi\)
0.752841 + 0.658202i \(0.228683\pi\)
\(390\) −24.9269 −1.26222
\(391\) −11.0978 −0.561237
\(392\) 13.8529 0.699676
\(393\) −29.5260 −1.48939
\(394\) 22.0543 1.11108
\(395\) −18.9746 −0.954715
\(396\) 19.7433 0.992138
\(397\) −21.4192 −1.07500 −0.537498 0.843265i \(-0.680631\pi\)
−0.537498 + 0.843265i \(0.680631\pi\)
\(398\) −25.3534 −1.27085
\(399\) −80.3775 −4.02391
\(400\) 9.36631 0.468316
\(401\) −24.5090 −1.22392 −0.611960 0.790889i \(-0.709619\pi\)
−0.611960 + 0.790889i \(0.709619\pi\)
\(402\) −27.5517 −1.37415
\(403\) 5.16178 0.257127
\(404\) 2.14597 0.106766
\(405\) 14.9011 0.740443
\(406\) 35.0431 1.73916
\(407\) −34.4276 −1.70651
\(408\) −4.68165 −0.231776
\(409\) 7.65341 0.378437 0.189218 0.981935i \(-0.439405\pi\)
0.189218 + 0.981935i \(0.439405\pi\)
\(410\) −40.5487 −2.00256
\(411\) −23.7364 −1.17083
\(412\) 18.3353 0.903316
\(413\) 59.5513 2.93033
\(414\) −37.0663 −1.82171
\(415\) 10.3903 0.510041
\(416\) 2.26961 0.111277
\(417\) 0.791934 0.0387812
\(418\) −22.2243 −1.08703
\(419\) 9.03421 0.441350 0.220675 0.975347i \(-0.429174\pi\)
0.220675 + 0.975347i \(0.429174\pi\)
\(420\) 50.1534 2.44724
\(421\) −19.0591 −0.928882 −0.464441 0.885604i \(-0.653745\pi\)
−0.464441 + 0.885604i \(0.653745\pi\)
\(422\) 23.5618 1.14697
\(423\) −32.3445 −1.57264
\(424\) 7.10368 0.344985
\(425\) 15.1329 0.734055
\(426\) −6.54812 −0.317257
\(427\) −8.24500 −0.399003
\(428\) −14.3251 −0.692432
\(429\) −24.0612 −1.16169
\(430\) 23.7660 1.14610
\(431\) −32.1407 −1.54816 −0.774080 0.633087i \(-0.781787\pi\)
−0.774080 + 0.633087i \(0.781787\pi\)
\(432\) −6.94371 −0.334079
\(433\) −3.96443 −0.190518 −0.0952592 0.995453i \(-0.530368\pi\)
−0.0952592 + 0.995453i \(0.530368\pi\)
\(434\) −10.3856 −0.498525
\(435\) 84.2824 4.04103
\(436\) −19.8425 −0.950283
\(437\) 41.7241 1.99594
\(438\) −25.4643 −1.21673
\(439\) 23.9432 1.14275 0.571374 0.820690i \(-0.306410\pi\)
0.571374 + 0.820690i \(0.306410\pi\)
\(440\) 13.8674 0.661101
\(441\) 74.7547 3.55975
\(442\) 3.66696 0.174419
\(443\) 8.73126 0.414834 0.207417 0.978253i \(-0.433494\pi\)
0.207417 + 0.978253i \(0.433494\pi\)
\(444\) 27.2666 1.29402
\(445\) −50.1172 −2.37578
\(446\) 18.3676 0.869730
\(447\) −32.1337 −1.51987
\(448\) −4.56649 −0.215747
\(449\) −12.3418 −0.582446 −0.291223 0.956655i \(-0.594062\pi\)
−0.291223 + 0.956655i \(0.594062\pi\)
\(450\) 50.5437 2.38265
\(451\) −39.1404 −1.84305
\(452\) −15.5873 −0.733166
\(453\) 17.0791 0.802444
\(454\) 1.23503 0.0579628
\(455\) −39.2832 −1.84163
\(456\) 17.6016 0.824269
\(457\) 8.92101 0.417307 0.208654 0.977990i \(-0.433092\pi\)
0.208654 + 0.977990i \(0.433092\pi\)
\(458\) −3.42062 −0.159835
\(459\) −11.2188 −0.523648
\(460\) −26.0348 −1.21388
\(461\) −6.74591 −0.314188 −0.157094 0.987584i \(-0.550213\pi\)
−0.157094 + 0.987584i \(0.550213\pi\)
\(462\) 48.4115 2.25231
\(463\) 9.99033 0.464290 0.232145 0.972681i \(-0.425426\pi\)
0.232145 + 0.972681i \(0.425426\pi\)
\(464\) −7.67395 −0.356254
\(465\) −24.9785 −1.15835
\(466\) −21.3961 −0.991155
\(467\) 20.1174 0.930922 0.465461 0.885068i \(-0.345889\pi\)
0.465461 + 0.885068i \(0.345889\pi\)
\(468\) 12.2476 0.566144
\(469\) −43.4197 −2.00494
\(470\) −22.7182 −1.04791
\(471\) −40.2732 −1.85569
\(472\) −13.0409 −0.600257
\(473\) 22.9406 1.05481
\(474\) 14.5059 0.666278
\(475\) −56.8951 −2.61053
\(476\) −7.37797 −0.338169
\(477\) 38.3338 1.75519
\(478\) −12.8100 −0.585917
\(479\) −4.31252 −0.197044 −0.0985221 0.995135i \(-0.531412\pi\)
−0.0985221 + 0.995135i \(0.531412\pi\)
\(480\) −10.9829 −0.501299
\(481\) −21.3569 −0.973789
\(482\) 2.55865 0.116544
\(483\) −90.8884 −4.13557
\(484\) 2.38574 0.108443
\(485\) 27.3973 1.24405
\(486\) 9.43935 0.428178
\(487\) 9.56529 0.433445 0.216722 0.976233i \(-0.430463\pi\)
0.216722 + 0.976233i \(0.430463\pi\)
\(488\) 1.80554 0.0817331
\(489\) 16.8390 0.761487
\(490\) 52.5064 2.37200
\(491\) −1.40407 −0.0633648 −0.0316824 0.999498i \(-0.510087\pi\)
−0.0316824 + 0.999498i \(0.510087\pi\)
\(492\) 30.9991 1.39755
\(493\) −12.3986 −0.558406
\(494\) −13.7866 −0.620289
\(495\) 74.8329 3.36349
\(496\) 2.27430 0.102119
\(497\) −10.3194 −0.462889
\(498\) −7.94331 −0.355948
\(499\) 30.8884 1.38275 0.691376 0.722495i \(-0.257005\pi\)
0.691376 + 0.722495i \(0.257005\pi\)
\(500\) 16.5496 0.740120
\(501\) 40.4646 1.80782
\(502\) −13.2223 −0.590139
\(503\) −10.8726 −0.484786 −0.242393 0.970178i \(-0.577932\pi\)
−0.242393 + 0.970178i \(0.577932\pi\)
\(504\) −24.6423 −1.09766
\(505\) 8.13385 0.361952
\(506\) −25.1306 −1.11719
\(507\) 22.7432 1.01006
\(508\) 3.64159 0.161569
\(509\) −18.3173 −0.811902 −0.405951 0.913895i \(-0.633060\pi\)
−0.405951 + 0.913895i \(0.633060\pi\)
\(510\) −17.7448 −0.785754
\(511\) −40.1301 −1.77525
\(512\) 1.00000 0.0441942
\(513\) 42.1792 1.86226
\(514\) 6.88848 0.303838
\(515\) 69.4962 3.06237
\(516\) −18.1689 −0.799840
\(517\) −21.9292 −0.964444
\(518\) 42.9704 1.88801
\(519\) 62.9162 2.76171
\(520\) 8.60249 0.377244
\(521\) −19.2444 −0.843113 −0.421557 0.906802i \(-0.638516\pi\)
−0.421557 + 0.906802i \(0.638516\pi\)
\(522\) −41.4112 −1.81252
\(523\) −13.9335 −0.609271 −0.304636 0.952469i \(-0.598535\pi\)
−0.304636 + 0.952469i \(0.598535\pi\)
\(524\) 10.1897 0.445138
\(525\) 123.936 5.40900
\(526\) 6.88858 0.300356
\(527\) 3.67454 0.160065
\(528\) −10.6015 −0.461370
\(529\) 24.1804 1.05132
\(530\) 26.9250 1.16955
\(531\) −70.3731 −3.05393
\(532\) 27.7389 1.20263
\(533\) −24.2804 −1.05170
\(534\) 38.3141 1.65801
\(535\) −54.2965 −2.34744
\(536\) 9.50832 0.410697
\(537\) 57.2804 2.47183
\(538\) 0.730619 0.0314992
\(539\) 50.6829 2.18307
\(540\) −26.3187 −1.13258
\(541\) 13.2662 0.570358 0.285179 0.958474i \(-0.407947\pi\)
0.285179 + 0.958474i \(0.407947\pi\)
\(542\) 29.9786 1.28769
\(543\) 19.8988 0.853939
\(544\) 1.61568 0.0692715
\(545\) −75.2089 −3.22159
\(546\) 30.0316 1.28524
\(547\) −13.3252 −0.569745 −0.284872 0.958565i \(-0.591951\pi\)
−0.284872 + 0.958565i \(0.591951\pi\)
\(548\) 8.19163 0.349929
\(549\) 9.74330 0.415834
\(550\) 34.2681 1.46120
\(551\) 46.6150 1.98587
\(552\) 19.9033 0.847142
\(553\) 22.8603 0.972121
\(554\) 14.5178 0.616801
\(555\) 103.348 4.38689
\(556\) −0.273303 −0.0115906
\(557\) −12.3910 −0.525025 −0.262512 0.964929i \(-0.584551\pi\)
−0.262512 + 0.964929i \(0.584551\pi\)
\(558\) 12.2729 0.519553
\(559\) 14.2310 0.601906
\(560\) −17.3083 −0.731411
\(561\) −17.1285 −0.723167
\(562\) −5.12599 −0.216227
\(563\) −3.97975 −0.167726 −0.0838632 0.996477i \(-0.526726\pi\)
−0.0838632 + 0.996477i \(0.526726\pi\)
\(564\) 17.3678 0.731318
\(565\) −59.0805 −2.48554
\(566\) −17.9473 −0.754382
\(567\) −17.9527 −0.753942
\(568\) 2.25981 0.0948195
\(569\) −21.1758 −0.887737 −0.443869 0.896092i \(-0.646394\pi\)
−0.443869 + 0.896092i \(0.646394\pi\)
\(570\) 66.7151 2.79439
\(571\) 27.4574 1.14906 0.574528 0.818485i \(-0.305186\pi\)
0.574528 + 0.818485i \(0.305186\pi\)
\(572\) 8.30372 0.347196
\(573\) 59.4214 2.48237
\(574\) 48.8526 2.03907
\(575\) −64.3353 −2.68297
\(576\) 5.39633 0.224847
\(577\) −25.3723 −1.05626 −0.528131 0.849163i \(-0.677107\pi\)
−0.528131 + 0.849163i \(0.677107\pi\)
\(578\) −14.3896 −0.598528
\(579\) 17.5085 0.727628
\(580\) −29.0865 −1.20775
\(581\) −12.5181 −0.519340
\(582\) −20.9450 −0.868198
\(583\) 25.9899 1.07639
\(584\) 8.78795 0.363648
\(585\) 46.4219 1.91931
\(586\) 27.1006 1.11951
\(587\) 45.3591 1.87217 0.936086 0.351772i \(-0.114421\pi\)
0.936086 + 0.351772i \(0.114421\pi\)
\(588\) −40.1407 −1.65537
\(589\) −13.8151 −0.569243
\(590\) −49.4289 −2.03496
\(591\) −63.9055 −2.62872
\(592\) −9.40992 −0.386745
\(593\) −29.5729 −1.21442 −0.607208 0.794543i \(-0.707710\pi\)
−0.607208 + 0.794543i \(0.707710\pi\)
\(594\) −25.4046 −1.04236
\(595\) −27.9647 −1.14644
\(596\) 11.0896 0.454248
\(597\) 73.4650 3.00672
\(598\) −15.5895 −0.637502
\(599\) 24.1337 0.986076 0.493038 0.870008i \(-0.335886\pi\)
0.493038 + 0.870008i \(0.335886\pi\)
\(600\) −27.1402 −1.10799
\(601\) 24.3385 0.992788 0.496394 0.868097i \(-0.334657\pi\)
0.496394 + 0.868097i \(0.334657\pi\)
\(602\) −28.6330 −1.16699
\(603\) 51.3101 2.08951
\(604\) −5.89412 −0.239828
\(605\) 9.04266 0.367636
\(606\) −6.21825 −0.252599
\(607\) −24.4757 −0.993438 −0.496719 0.867911i \(-0.665462\pi\)
−0.496719 + 0.867911i \(0.665462\pi\)
\(608\) −6.07444 −0.246351
\(609\) −101.542 −4.11470
\(610\) 6.84353 0.277086
\(611\) −13.6036 −0.550341
\(612\) 8.71872 0.352433
\(613\) −27.2523 −1.10071 −0.550356 0.834930i \(-0.685508\pi\)
−0.550356 + 0.834930i \(0.685508\pi\)
\(614\) −23.5113 −0.948840
\(615\) 117.496 4.73788
\(616\) −16.7072 −0.673153
\(617\) 17.8664 0.719272 0.359636 0.933093i \(-0.382901\pi\)
0.359636 + 0.933093i \(0.382901\pi\)
\(618\) −53.1292 −2.13717
\(619\) 16.5922 0.666898 0.333449 0.942768i \(-0.391787\pi\)
0.333449 + 0.942768i \(0.391787\pi\)
\(620\) 8.62027 0.346199
\(621\) 47.6949 1.91393
\(622\) 4.10835 0.164730
\(623\) 60.3806 2.41910
\(624\) −6.57652 −0.263271
\(625\) 15.8962 0.635849
\(626\) 18.3293 0.732585
\(627\) 64.3980 2.57181
\(628\) 13.8986 0.554615
\(629\) −15.2034 −0.606199
\(630\) −93.4016 −3.72121
\(631\) −23.5990 −0.939460 −0.469730 0.882810i \(-0.655649\pi\)
−0.469730 + 0.882810i \(0.655649\pi\)
\(632\) −5.00610 −0.199132
\(633\) −68.2738 −2.71364
\(634\) −31.1070 −1.23542
\(635\) 13.8027 0.547743
\(636\) −20.5839 −0.816206
\(637\) 31.4406 1.24572
\(638\) −28.0763 −1.11155
\(639\) 12.1947 0.482414
\(640\) 3.79029 0.149824
\(641\) −26.0328 −1.02823 −0.514116 0.857721i \(-0.671880\pi\)
−0.514116 + 0.857721i \(0.671880\pi\)
\(642\) 41.5092 1.63824
\(643\) 11.5997 0.457448 0.228724 0.973491i \(-0.426545\pi\)
0.228724 + 0.973491i \(0.426545\pi\)
\(644\) 31.3663 1.23601
\(645\) −68.8653 −2.71157
\(646\) −9.81433 −0.386140
\(647\) 28.0427 1.10247 0.551236 0.834350i \(-0.314157\pi\)
0.551236 + 0.834350i \(0.314157\pi\)
\(648\) 3.93139 0.154440
\(649\) −47.7122 −1.87287
\(650\) 21.2579 0.833803
\(651\) 30.0937 1.17947
\(652\) −5.81128 −0.227587
\(653\) 10.4246 0.407946 0.203973 0.978977i \(-0.434615\pi\)
0.203973 + 0.978977i \(0.434615\pi\)
\(654\) 57.4965 2.24829
\(655\) 38.6218 1.50908
\(656\) −10.6980 −0.417689
\(657\) 47.4227 1.85014
\(658\) 27.3706 1.06702
\(659\) −0.724819 −0.0282349 −0.0141175 0.999900i \(-0.504494\pi\)
−0.0141175 + 0.999900i \(0.504494\pi\)
\(660\) −40.1827 −1.56411
\(661\) 3.96086 0.154060 0.0770298 0.997029i \(-0.475456\pi\)
0.0770298 + 0.997029i \(0.475456\pi\)
\(662\) −16.9778 −0.659860
\(663\) −10.6255 −0.412661
\(664\) 2.74130 0.106383
\(665\) 105.139 4.07710
\(666\) −50.7791 −1.96765
\(667\) 52.7109 2.04097
\(668\) −13.9647 −0.540309
\(669\) −53.2227 −2.05771
\(670\) 36.0393 1.39232
\(671\) 6.60585 0.255016
\(672\) 13.2321 0.510438
\(673\) 12.5805 0.484944 0.242472 0.970158i \(-0.422042\pi\)
0.242472 + 0.970158i \(0.422042\pi\)
\(674\) −6.86104 −0.264277
\(675\) −65.0369 −2.50327
\(676\) −7.84887 −0.301880
\(677\) −22.7523 −0.874442 −0.437221 0.899354i \(-0.644037\pi\)
−0.437221 + 0.899354i \(0.644037\pi\)
\(678\) 45.1665 1.73461
\(679\) −33.0080 −1.26673
\(680\) 6.12388 0.234840
\(681\) −3.57868 −0.137135
\(682\) 8.32089 0.318623
\(683\) −30.5002 −1.16706 −0.583529 0.812092i \(-0.698329\pi\)
−0.583529 + 0.812092i \(0.698329\pi\)
\(684\) −32.7797 −1.25336
\(685\) 31.0487 1.18631
\(686\) −31.2936 −1.19480
\(687\) 9.91173 0.378156
\(688\) 6.27023 0.239050
\(689\) 16.1226 0.614222
\(690\) 75.4394 2.87193
\(691\) −6.05370 −0.230293 −0.115147 0.993348i \(-0.536734\pi\)
−0.115147 + 0.993348i \(0.536734\pi\)
\(692\) −21.7129 −0.825400
\(693\) −90.1577 −3.42481
\(694\) 7.51465 0.285252
\(695\) −1.03590 −0.0392938
\(696\) 22.2364 0.842868
\(697\) −17.2846 −0.654700
\(698\) −23.9846 −0.907831
\(699\) 61.9982 2.34499
\(700\) −42.7712 −1.61660
\(701\) 14.1688 0.535149 0.267575 0.963537i \(-0.413778\pi\)
0.267575 + 0.963537i \(0.413778\pi\)
\(702\) −15.7595 −0.594805
\(703\) 57.1600 2.15583
\(704\) 3.65865 0.137891
\(705\) 65.8292 2.47927
\(706\) −24.4745 −0.921111
\(707\) −9.79956 −0.368550
\(708\) 37.7879 1.42016
\(709\) 25.8582 0.971124 0.485562 0.874202i \(-0.338615\pi\)
0.485562 + 0.874202i \(0.338615\pi\)
\(710\) 8.56534 0.321451
\(711\) −27.0146 −1.01313
\(712\) −13.2225 −0.495535
\(713\) −15.6217 −0.585039
\(714\) 21.3787 0.800079
\(715\) 31.4735 1.17704
\(716\) −19.7679 −0.738763
\(717\) 37.1189 1.38623
\(718\) −2.53346 −0.0945477
\(719\) −43.6512 −1.62792 −0.813958 0.580923i \(-0.802692\pi\)
−0.813958 + 0.580923i \(0.802692\pi\)
\(720\) 20.4537 0.762263
\(721\) −83.7281 −3.11820
\(722\) 17.8989 0.666127
\(723\) −7.41407 −0.275732
\(724\) −6.86724 −0.255219
\(725\) −71.8766 −2.66943
\(726\) −6.91303 −0.256567
\(727\) −29.2645 −1.08536 −0.542679 0.839940i \(-0.682590\pi\)
−0.542679 + 0.839940i \(0.682590\pi\)
\(728\) −10.3642 −0.384121
\(729\) −39.1461 −1.44985
\(730\) 33.3089 1.23282
\(731\) 10.1307 0.374696
\(732\) −5.23182 −0.193373
\(733\) 34.1905 1.26285 0.631427 0.775436i \(-0.282470\pi\)
0.631427 + 0.775436i \(0.282470\pi\)
\(734\) −11.8394 −0.437001
\(735\) −152.145 −5.61195
\(736\) −6.86880 −0.253187
\(737\) 34.7877 1.28142
\(738\) −57.7302 −2.12508
\(739\) 38.2664 1.40765 0.703826 0.710372i \(-0.251473\pi\)
0.703826 + 0.710372i \(0.251473\pi\)
\(740\) −35.6664 −1.31112
\(741\) 39.9487 1.46755
\(742\) −32.4389 −1.19087
\(743\) −6.53713 −0.239824 −0.119912 0.992785i \(-0.538261\pi\)
−0.119912 + 0.992785i \(0.538261\pi\)
\(744\) −6.59012 −0.241605
\(745\) 42.0329 1.53997
\(746\) 23.6275 0.865066
\(747\) 14.7930 0.541247
\(748\) 5.91120 0.216135
\(749\) 65.4157 2.39024
\(750\) −47.9548 −1.75106
\(751\) 17.0455 0.622000 0.311000 0.950410i \(-0.399336\pi\)
0.311000 + 0.950410i \(0.399336\pi\)
\(752\) −5.99379 −0.218571
\(753\) 38.3134 1.39622
\(754\) −17.4169 −0.634286
\(755\) −22.3404 −0.813052
\(756\) 31.7084 1.15322
\(757\) 28.5345 1.03710 0.518552 0.855046i \(-0.326471\pi\)
0.518552 + 0.855046i \(0.326471\pi\)
\(758\) −7.20684 −0.261764
\(759\) 72.8194 2.64318
\(760\) −23.0239 −0.835165
\(761\) 24.3824 0.883861 0.441930 0.897049i \(-0.354294\pi\)
0.441930 + 0.897049i \(0.354294\pi\)
\(762\) −10.5520 −0.382259
\(763\) 90.6107 3.28033
\(764\) −20.5068 −0.741911
\(765\) 33.0465 1.19480
\(766\) −18.1294 −0.655043
\(767\) −29.5978 −1.06872
\(768\) −2.89764 −0.104560
\(769\) 29.2580 1.05507 0.527535 0.849534i \(-0.323117\pi\)
0.527535 + 0.849534i \(0.323117\pi\)
\(770\) −63.3252 −2.28208
\(771\) −19.9604 −0.718854
\(772\) −6.04232 −0.217468
\(773\) −4.01873 −0.144544 −0.0722718 0.997385i \(-0.523025\pi\)
−0.0722718 + 0.997385i \(0.523025\pi\)
\(774\) 33.8362 1.21622
\(775\) 21.3018 0.765185
\(776\) 7.22829 0.259481
\(777\) −124.513 −4.46687
\(778\) 29.6967 1.06468
\(779\) 64.9847 2.32832
\(780\) −24.9269 −0.892527
\(781\) 8.26786 0.295847
\(782\) −11.0978 −0.396855
\(783\) 53.2857 1.90428
\(784\) 13.8529 0.494745
\(785\) 52.6798 1.88022
\(786\) −29.5260 −1.05316
\(787\) 14.5095 0.517207 0.258604 0.965984i \(-0.416738\pi\)
0.258604 + 0.965984i \(0.416738\pi\)
\(788\) 22.0543 0.785652
\(789\) −19.9606 −0.710618
\(790\) −18.9746 −0.675086
\(791\) 71.1794 2.53085
\(792\) 19.7433 0.701548
\(793\) 4.09788 0.145520
\(794\) −21.4192 −0.760138
\(795\) −78.0191 −2.76705
\(796\) −25.3534 −0.898627
\(797\) −40.6870 −1.44121 −0.720604 0.693347i \(-0.756135\pi\)
−0.720604 + 0.693347i \(0.756135\pi\)
\(798\) −80.3775 −2.84533
\(799\) −9.68401 −0.342596
\(800\) 9.36631 0.331149
\(801\) −71.3531 −2.52114
\(802\) −24.5090 −0.865442
\(803\) 32.1521 1.13462
\(804\) −27.5517 −0.971674
\(805\) 118.888 4.19024
\(806\) 5.16178 0.181816
\(807\) −2.11707 −0.0745245
\(808\) 2.14597 0.0754949
\(809\) −7.60488 −0.267373 −0.133687 0.991024i \(-0.542682\pi\)
−0.133687 + 0.991024i \(0.542682\pi\)
\(810\) 14.9011 0.523572
\(811\) 47.6003 1.67147 0.835736 0.549131i \(-0.185041\pi\)
0.835736 + 0.549131i \(0.185041\pi\)
\(812\) 35.0431 1.22977
\(813\) −86.8671 −3.04656
\(814\) −34.4276 −1.20669
\(815\) −22.0265 −0.771553
\(816\) −4.68165 −0.163891
\(817\) −38.0881 −1.33254
\(818\) 7.65341 0.267595
\(819\) −55.9285 −1.95430
\(820\) −40.5487 −1.41602
\(821\) 34.3313 1.19817 0.599086 0.800685i \(-0.295531\pi\)
0.599086 + 0.800685i \(0.295531\pi\)
\(822\) −23.7364 −0.827902
\(823\) −28.5005 −0.993463 −0.496732 0.867904i \(-0.665467\pi\)
−0.496732 + 0.867904i \(0.665467\pi\)
\(824\) 18.3353 0.638741
\(825\) −99.2967 −3.45707
\(826\) 59.5513 2.07205
\(827\) −4.08419 −0.142021 −0.0710107 0.997476i \(-0.522622\pi\)
−0.0710107 + 0.997476i \(0.522622\pi\)
\(828\) −37.0663 −1.28814
\(829\) −37.6237 −1.30673 −0.653363 0.757044i \(-0.726643\pi\)
−0.653363 + 0.757044i \(0.726643\pi\)
\(830\) 10.3903 0.360654
\(831\) −42.0673 −1.45930
\(832\) 2.26961 0.0786846
\(833\) 22.3818 0.775482
\(834\) 0.791934 0.0274224
\(835\) −52.9301 −1.83172
\(836\) −22.2243 −0.768643
\(837\) −15.7921 −0.545855
\(838\) 9.03421 0.312081
\(839\) 12.1751 0.420330 0.210165 0.977666i \(-0.432600\pi\)
0.210165 + 0.977666i \(0.432600\pi\)
\(840\) 50.1534 1.73046
\(841\) 29.8896 1.03068
\(842\) −19.0591 −0.656818
\(843\) 14.8533 0.511575
\(844\) 23.5618 0.811032
\(845\) −29.7495 −1.02341
\(846\) −32.3445 −1.11203
\(847\) −10.8945 −0.374339
\(848\) 7.10368 0.243941
\(849\) 52.0049 1.78480
\(850\) 15.1329 0.519055
\(851\) 64.6349 2.21565
\(852\) −6.54812 −0.224335
\(853\) 18.8656 0.645945 0.322972 0.946408i \(-0.395318\pi\)
0.322972 + 0.946408i \(0.395318\pi\)
\(854\) −8.24500 −0.282138
\(855\) −124.245 −4.24908
\(856\) −14.3251 −0.489624
\(857\) 42.6515 1.45695 0.728473 0.685074i \(-0.240230\pi\)
0.728473 + 0.685074i \(0.240230\pi\)
\(858\) −24.0612 −0.821436
\(859\) 41.6830 1.42221 0.711103 0.703088i \(-0.248196\pi\)
0.711103 + 0.703088i \(0.248196\pi\)
\(860\) 23.7660 0.810413
\(861\) −141.557 −4.82426
\(862\) −32.1407 −1.09471
\(863\) −10.3409 −0.352010 −0.176005 0.984389i \(-0.556317\pi\)
−0.176005 + 0.984389i \(0.556317\pi\)
\(864\) −6.94371 −0.236230
\(865\) −82.2982 −2.79822
\(866\) −3.96443 −0.134717
\(867\) 41.6959 1.41607
\(868\) −10.3856 −0.352510
\(869\) −18.3156 −0.621314
\(870\) 84.2824 2.85744
\(871\) 21.5802 0.731217
\(872\) −19.8425 −0.671952
\(873\) 39.0063 1.32016
\(874\) 41.7241 1.41134
\(875\) −75.5736 −2.55485
\(876\) −25.4643 −0.860360
\(877\) 22.9723 0.775721 0.387860 0.921718i \(-0.373214\pi\)
0.387860 + 0.921718i \(0.373214\pi\)
\(878\) 23.9432 0.808045
\(879\) −78.5278 −2.64868
\(880\) 13.8674 0.467469
\(881\) −11.8291 −0.398531 −0.199266 0.979946i \(-0.563856\pi\)
−0.199266 + 0.979946i \(0.563856\pi\)
\(882\) 74.7547 2.51712
\(883\) 7.91811 0.266466 0.133233 0.991085i \(-0.457464\pi\)
0.133233 + 0.991085i \(0.457464\pi\)
\(884\) 3.66696 0.123333
\(885\) 143.227 4.81453
\(886\) 8.73126 0.293332
\(887\) 37.6103 1.26283 0.631416 0.775445i \(-0.282474\pi\)
0.631416 + 0.775445i \(0.282474\pi\)
\(888\) 27.2666 0.915007
\(889\) −16.6293 −0.557729
\(890\) −50.1172 −1.67993
\(891\) 14.3836 0.481869
\(892\) 18.3676 0.614992
\(893\) 36.4089 1.21838
\(894\) −32.1337 −1.07471
\(895\) −74.9262 −2.50451
\(896\) −4.56649 −0.152556
\(897\) 45.1728 1.50828
\(898\) −12.3418 −0.411851
\(899\) −17.4529 −0.582087
\(900\) 50.5437 1.68479
\(901\) 11.4772 0.382363
\(902\) −39.1404 −1.30323
\(903\) 82.9681 2.76100
\(904\) −15.5873 −0.518427
\(905\) −26.0288 −0.865228
\(906\) 17.0791 0.567414
\(907\) 1.49096 0.0495064 0.0247532 0.999694i \(-0.492120\pi\)
0.0247532 + 0.999694i \(0.492120\pi\)
\(908\) 1.23503 0.0409859
\(909\) 11.5804 0.384096
\(910\) −39.2832 −1.30223
\(911\) −38.8643 −1.28763 −0.643816 0.765181i \(-0.722650\pi\)
−0.643816 + 0.765181i \(0.722650\pi\)
\(912\) 17.6016 0.582846
\(913\) 10.0295 0.331927
\(914\) 8.92101 0.295081
\(915\) −19.8301 −0.655563
\(916\) −3.42062 −0.113020
\(917\) −46.5311 −1.53659
\(918\) −11.2188 −0.370275
\(919\) −57.9899 −1.91291 −0.956456 0.291877i \(-0.905720\pi\)
−0.956456 + 0.291877i \(0.905720\pi\)
\(920\) −26.0348 −0.858340
\(921\) 68.1274 2.24487
\(922\) −6.74591 −0.222165
\(923\) 5.12889 0.168819
\(924\) 48.4115 1.59262
\(925\) −88.1363 −2.89790
\(926\) 9.99033 0.328303
\(927\) 98.9434 3.24973
\(928\) −7.67395 −0.251910
\(929\) −7.73785 −0.253871 −0.126935 0.991911i \(-0.540514\pi\)
−0.126935 + 0.991911i \(0.540514\pi\)
\(930\) −24.9785 −0.819076
\(931\) −84.1485 −2.75786
\(932\) −21.3961 −0.700852
\(933\) −11.9045 −0.389736
\(934\) 20.1174 0.658261
\(935\) 22.4052 0.732727
\(936\) 12.2476 0.400324
\(937\) 14.4494 0.472043 0.236021 0.971748i \(-0.424156\pi\)
0.236021 + 0.971748i \(0.424156\pi\)
\(938\) −43.4197 −1.41770
\(939\) −53.1117 −1.73323
\(940\) −22.7182 −0.740986
\(941\) 6.43072 0.209635 0.104818 0.994491i \(-0.466574\pi\)
0.104818 + 0.994491i \(0.466574\pi\)
\(942\) −40.2732 −1.31217
\(943\) 73.4827 2.39293
\(944\) −13.0409 −0.424446
\(945\) 120.184 3.90959
\(946\) 22.9406 0.745863
\(947\) 2.16728 0.0704272 0.0352136 0.999380i \(-0.488789\pi\)
0.0352136 + 0.999380i \(0.488789\pi\)
\(948\) 14.5059 0.471130
\(949\) 19.9452 0.647449
\(950\) −56.8951 −1.84592
\(951\) 90.1370 2.92289
\(952\) −7.37797 −0.239122
\(953\) −17.9623 −0.581857 −0.290928 0.956745i \(-0.593964\pi\)
−0.290928 + 0.956745i \(0.593964\pi\)
\(954\) 38.3338 1.24110
\(955\) −77.7268 −2.51518
\(956\) −12.8100 −0.414306
\(957\) 81.3552 2.62984
\(958\) −4.31252 −0.139331
\(959\) −37.4070 −1.20794
\(960\) −10.9829 −0.354472
\(961\) −25.8275 −0.833147
\(962\) −21.3569 −0.688573
\(963\) −77.3032 −2.49106
\(964\) 2.55865 0.0824087
\(965\) −22.9022 −0.737246
\(966\) −90.8884 −2.92429
\(967\) −3.64174 −0.117110 −0.0585552 0.998284i \(-0.518649\pi\)
−0.0585552 + 0.998284i \(0.518649\pi\)
\(968\) 2.38574 0.0766807
\(969\) 28.4384 0.913574
\(970\) 27.3973 0.879675
\(971\) 41.9662 1.34676 0.673380 0.739297i \(-0.264842\pi\)
0.673380 + 0.739297i \(0.264842\pi\)
\(972\) 9.43935 0.302767
\(973\) 1.24804 0.0400102
\(974\) 9.56529 0.306492
\(975\) −61.5977 −1.97271
\(976\) 1.80554 0.0577940
\(977\) 47.9869 1.53524 0.767619 0.640906i \(-0.221441\pi\)
0.767619 + 0.640906i \(0.221441\pi\)
\(978\) 16.8390 0.538452
\(979\) −48.3766 −1.54612
\(980\) 52.5064 1.67726
\(981\) −107.077 −3.41870
\(982\) −1.40407 −0.0448057
\(983\) 6.16410 0.196604 0.0983021 0.995157i \(-0.468659\pi\)
0.0983021 + 0.995157i \(0.468659\pi\)
\(984\) 30.9991 0.988216
\(985\) 83.5923 2.66347
\(986\) −12.3986 −0.394853
\(987\) −79.3102 −2.52447
\(988\) −13.7866 −0.438611
\(989\) −43.0689 −1.36951
\(990\) 74.8329 2.37835
\(991\) −21.9291 −0.696600 −0.348300 0.937383i \(-0.613241\pi\)
−0.348300 + 0.937383i \(0.613241\pi\)
\(992\) 2.27430 0.0722092
\(993\) 49.1955 1.56117
\(994\) −10.3194 −0.327312
\(995\) −96.0967 −3.04647
\(996\) −7.94331 −0.251694
\(997\) −22.7246 −0.719695 −0.359848 0.933011i \(-0.617171\pi\)
−0.359848 + 0.933011i \(0.617171\pi\)
\(998\) 30.8884 0.977754
\(999\) 65.3398 2.06726
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 6046.2.a.e.1.6 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6046.2.a.e.1.6 56 1.1 even 1 trivial