Properties

Label 6039.2.a.m.1.19
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
Dimension $25$
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] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, 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("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
Analytic rank: \(1\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Character \(\chi\) \(=\) 6039.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.23685 q^{2} -0.470198 q^{4} -0.403292 q^{5} -0.303505 q^{7} -3.05527 q^{8} +O(q^{10})\) \(q+1.23685 q^{2} -0.470198 q^{4} -0.403292 q^{5} -0.303505 q^{7} -3.05527 q^{8} -0.498813 q^{10} +1.00000 q^{11} -3.53056 q^{13} -0.375391 q^{14} -2.83852 q^{16} +7.19973 q^{17} +3.67072 q^{19} +0.189627 q^{20} +1.23685 q^{22} -3.45314 q^{23} -4.83736 q^{25} -4.36677 q^{26} +0.142707 q^{28} +2.49100 q^{29} +2.86904 q^{31} +2.59971 q^{32} +8.90500 q^{34} +0.122401 q^{35} +3.00574 q^{37} +4.54014 q^{38} +1.23217 q^{40} +4.84782 q^{41} -5.93177 q^{43} -0.470198 q^{44} -4.27103 q^{46} -8.13596 q^{47} -6.90788 q^{49} -5.98309 q^{50} +1.66006 q^{52} +1.91085 q^{53} -0.403292 q^{55} +0.927290 q^{56} +3.08100 q^{58} +1.68282 q^{59} +1.00000 q^{61} +3.54858 q^{62} +8.89249 q^{64} +1.42385 q^{65} -0.0898202 q^{67} -3.38530 q^{68} +0.151392 q^{70} -13.0923 q^{71} -14.0564 q^{73} +3.71766 q^{74} -1.72597 q^{76} -0.303505 q^{77} +3.46200 q^{79} +1.14475 q^{80} +5.99604 q^{82} -2.45660 q^{83} -2.90360 q^{85} -7.33671 q^{86} -3.05527 q^{88} -12.1773 q^{89} +1.07154 q^{91} +1.62366 q^{92} -10.0630 q^{94} -1.48037 q^{95} -1.70611 q^{97} -8.54403 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.23685 0.874586 0.437293 0.899319i \(-0.355937\pi\)
0.437293 + 0.899319i \(0.355937\pi\)
\(3\) 0 0
\(4\) −0.470198 −0.235099
\(5\) −0.403292 −0.180358 −0.0901789 0.995926i \(-0.528744\pi\)
−0.0901789 + 0.995926i \(0.528744\pi\)
\(6\) 0 0
\(7\) −0.303505 −0.114714 −0.0573571 0.998354i \(-0.518267\pi\)
−0.0573571 + 0.998354i \(0.518267\pi\)
\(8\) −3.05527 −1.08020
\(9\) 0 0
\(10\) −0.498813 −0.157738
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −3.53056 −0.979200 −0.489600 0.871947i \(-0.662857\pi\)
−0.489600 + 0.871947i \(0.662857\pi\)
\(14\) −0.375391 −0.100327
\(15\) 0 0
\(16\) −2.83852 −0.709630
\(17\) 7.19973 1.74619 0.873096 0.487549i \(-0.162109\pi\)
0.873096 + 0.487549i \(0.162109\pi\)
\(18\) 0 0
\(19\) 3.67072 0.842122 0.421061 0.907032i \(-0.361658\pi\)
0.421061 + 0.907032i \(0.361658\pi\)
\(20\) 0.189627 0.0424019
\(21\) 0 0
\(22\) 1.23685 0.263698
\(23\) −3.45314 −0.720030 −0.360015 0.932946i \(-0.617228\pi\)
−0.360015 + 0.932946i \(0.617228\pi\)
\(24\) 0 0
\(25\) −4.83736 −0.967471
\(26\) −4.36677 −0.856395
\(27\) 0 0
\(28\) 0.142707 0.0269692
\(29\) 2.49100 0.462567 0.231283 0.972886i \(-0.425708\pi\)
0.231283 + 0.972886i \(0.425708\pi\)
\(30\) 0 0
\(31\) 2.86904 0.515295 0.257648 0.966239i \(-0.417053\pi\)
0.257648 + 0.966239i \(0.417053\pi\)
\(32\) 2.59971 0.459568
\(33\) 0 0
\(34\) 8.90500 1.52720
\(35\) 0.122401 0.0206896
\(36\) 0 0
\(37\) 3.00574 0.494141 0.247071 0.968997i \(-0.420532\pi\)
0.247071 + 0.968997i \(0.420532\pi\)
\(38\) 4.54014 0.736508
\(39\) 0 0
\(40\) 1.23217 0.194822
\(41\) 4.84782 0.757103 0.378551 0.925580i \(-0.376422\pi\)
0.378551 + 0.925580i \(0.376422\pi\)
\(42\) 0 0
\(43\) −5.93177 −0.904586 −0.452293 0.891869i \(-0.649394\pi\)
−0.452293 + 0.891869i \(0.649394\pi\)
\(44\) −0.470198 −0.0708850
\(45\) 0 0
\(46\) −4.27103 −0.629729
\(47\) −8.13596 −1.18675 −0.593376 0.804925i \(-0.702205\pi\)
−0.593376 + 0.804925i \(0.702205\pi\)
\(48\) 0 0
\(49\) −6.90788 −0.986841
\(50\) −5.98309 −0.846137
\(51\) 0 0
\(52\) 1.66006 0.230209
\(53\) 1.91085 0.262475 0.131238 0.991351i \(-0.458105\pi\)
0.131238 + 0.991351i \(0.458105\pi\)
\(54\) 0 0
\(55\) −0.403292 −0.0543799
\(56\) 0.927290 0.123914
\(57\) 0 0
\(58\) 3.08100 0.404555
\(59\) 1.68282 0.219084 0.109542 0.993982i \(-0.465062\pi\)
0.109542 + 0.993982i \(0.465062\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037
\(62\) 3.54858 0.450670
\(63\) 0 0
\(64\) 8.89249 1.11156
\(65\) 1.42385 0.176606
\(66\) 0 0
\(67\) −0.0898202 −0.0109733 −0.00548664 0.999985i \(-0.501746\pi\)
−0.00548664 + 0.999985i \(0.501746\pi\)
\(68\) −3.38530 −0.410528
\(69\) 0 0
\(70\) 0.151392 0.0180948
\(71\) −13.0923 −1.55377 −0.776886 0.629642i \(-0.783202\pi\)
−0.776886 + 0.629642i \(0.783202\pi\)
\(72\) 0 0
\(73\) −14.0564 −1.64518 −0.822589 0.568637i \(-0.807471\pi\)
−0.822589 + 0.568637i \(0.807471\pi\)
\(74\) 3.71766 0.432169
\(75\) 0 0
\(76\) −1.72597 −0.197982
\(77\) −0.303505 −0.0345876
\(78\) 0 0
\(79\) 3.46200 0.389505 0.194753 0.980852i \(-0.437610\pi\)
0.194753 + 0.980852i \(0.437610\pi\)
\(80\) 1.14475 0.127987
\(81\) 0 0
\(82\) 5.99604 0.662152
\(83\) −2.45660 −0.269646 −0.134823 0.990870i \(-0.543047\pi\)
−0.134823 + 0.990870i \(0.543047\pi\)
\(84\) 0 0
\(85\) −2.90360 −0.314939
\(86\) −7.33671 −0.791138
\(87\) 0 0
\(88\) −3.05527 −0.325693
\(89\) −12.1773 −1.29079 −0.645397 0.763848i \(-0.723308\pi\)
−0.645397 + 0.763848i \(0.723308\pi\)
\(90\) 0 0
\(91\) 1.07154 0.112328
\(92\) 1.62366 0.169278
\(93\) 0 0
\(94\) −10.0630 −1.03792
\(95\) −1.48037 −0.151883
\(96\) 0 0
\(97\) −1.70611 −0.173229 −0.0866145 0.996242i \(-0.527605\pi\)
−0.0866145 + 0.996242i \(0.527605\pi\)
\(98\) −8.54403 −0.863077
\(99\) 0 0
\(100\) 2.27451 0.227451
\(101\) −3.09935 −0.308397 −0.154198 0.988040i \(-0.549279\pi\)
−0.154198 + 0.988040i \(0.549279\pi\)
\(102\) 0 0
\(103\) −1.70468 −0.167967 −0.0839834 0.996467i \(-0.526764\pi\)
−0.0839834 + 0.996467i \(0.526764\pi\)
\(104\) 10.7868 1.05773
\(105\) 0 0
\(106\) 2.36344 0.229557
\(107\) −11.9236 −1.15270 −0.576350 0.817203i \(-0.695523\pi\)
−0.576350 + 0.817203i \(0.695523\pi\)
\(108\) 0 0
\(109\) −3.88221 −0.371848 −0.185924 0.982564i \(-0.559528\pi\)
−0.185924 + 0.982564i \(0.559528\pi\)
\(110\) −0.498813 −0.0475599
\(111\) 0 0
\(112\) 0.861505 0.0814046
\(113\) −13.8764 −1.30538 −0.652692 0.757623i \(-0.726361\pi\)
−0.652692 + 0.757623i \(0.726361\pi\)
\(114\) 0 0
\(115\) 1.39263 0.129863
\(116\) −1.17126 −0.108749
\(117\) 0 0
\(118\) 2.08140 0.191608
\(119\) −2.18516 −0.200313
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.23685 0.111979
\(123\) 0 0
\(124\) −1.34902 −0.121145
\(125\) 3.96733 0.354849
\(126\) 0 0
\(127\) 12.9779 1.15160 0.575801 0.817590i \(-0.304690\pi\)
0.575801 + 0.817590i \(0.304690\pi\)
\(128\) 5.79927 0.512588
\(129\) 0 0
\(130\) 1.76109 0.154457
\(131\) 0.0196484 0.00171669 0.000858346 1.00000i \(-0.499727\pi\)
0.000858346 1.00000i \(0.499727\pi\)
\(132\) 0 0
\(133\) −1.11408 −0.0966033
\(134\) −0.111094 −0.00959708
\(135\) 0 0
\(136\) −21.9971 −1.88624
\(137\) −0.274897 −0.0234860 −0.0117430 0.999931i \(-0.503738\pi\)
−0.0117430 + 0.999931i \(0.503738\pi\)
\(138\) 0 0
\(139\) −14.3695 −1.21881 −0.609404 0.792860i \(-0.708591\pi\)
−0.609404 + 0.792860i \(0.708591\pi\)
\(140\) −0.0575528 −0.00486410
\(141\) 0 0
\(142\) −16.1932 −1.35891
\(143\) −3.53056 −0.295240
\(144\) 0 0
\(145\) −1.00460 −0.0834275
\(146\) −17.3857 −1.43885
\(147\) 0 0
\(148\) −1.41329 −0.116172
\(149\) −15.6059 −1.27848 −0.639242 0.769006i \(-0.720752\pi\)
−0.639242 + 0.769006i \(0.720752\pi\)
\(150\) 0 0
\(151\) 20.4498 1.66418 0.832091 0.554639i \(-0.187143\pi\)
0.832091 + 0.554639i \(0.187143\pi\)
\(152\) −11.2150 −0.909660
\(153\) 0 0
\(154\) −0.375391 −0.0302499
\(155\) −1.15706 −0.0929375
\(156\) 0 0
\(157\) −22.7960 −1.81932 −0.909659 0.415356i \(-0.863657\pi\)
−0.909659 + 0.415356i \(0.863657\pi\)
\(158\) 4.28198 0.340656
\(159\) 0 0
\(160\) −1.04844 −0.0828867
\(161\) 1.04805 0.0825977
\(162\) 0 0
\(163\) 5.52845 0.433022 0.216511 0.976280i \(-0.430532\pi\)
0.216511 + 0.976280i \(0.430532\pi\)
\(164\) −2.27944 −0.177994
\(165\) 0 0
\(166\) −3.03844 −0.235829
\(167\) 22.2926 1.72505 0.862526 0.506013i \(-0.168881\pi\)
0.862526 + 0.506013i \(0.168881\pi\)
\(168\) 0 0
\(169\) −0.535178 −0.0411675
\(170\) −3.59132 −0.275441
\(171\) 0 0
\(172\) 2.78910 0.212667
\(173\) 2.92197 0.222153 0.111077 0.993812i \(-0.464570\pi\)
0.111077 + 0.993812i \(0.464570\pi\)
\(174\) 0 0
\(175\) 1.46816 0.110983
\(176\) −2.83852 −0.213961
\(177\) 0 0
\(178\) −15.0615 −1.12891
\(179\) −8.74757 −0.653824 −0.326912 0.945055i \(-0.606008\pi\)
−0.326912 + 0.945055i \(0.606008\pi\)
\(180\) 0 0
\(181\) −21.9185 −1.62919 −0.814596 0.580028i \(-0.803041\pi\)
−0.814596 + 0.580028i \(0.803041\pi\)
\(182\) 1.32534 0.0982406
\(183\) 0 0
\(184\) 10.5503 0.777777
\(185\) −1.21219 −0.0891222
\(186\) 0 0
\(187\) 7.19973 0.526497
\(188\) 3.82551 0.279004
\(189\) 0 0
\(190\) −1.83100 −0.132835
\(191\) −1.79904 −0.130174 −0.0650869 0.997880i \(-0.520732\pi\)
−0.0650869 + 0.997880i \(0.520732\pi\)
\(192\) 0 0
\(193\) 6.99299 0.503366 0.251683 0.967810i \(-0.419016\pi\)
0.251683 + 0.967810i \(0.419016\pi\)
\(194\) −2.11020 −0.151504
\(195\) 0 0
\(196\) 3.24807 0.232005
\(197\) 19.7619 1.40798 0.703989 0.710211i \(-0.251400\pi\)
0.703989 + 0.710211i \(0.251400\pi\)
\(198\) 0 0
\(199\) −4.95005 −0.350900 −0.175450 0.984488i \(-0.556138\pi\)
−0.175450 + 0.984488i \(0.556138\pi\)
\(200\) 14.7794 1.04506
\(201\) 0 0
\(202\) −3.83343 −0.269720
\(203\) −0.756031 −0.0530630
\(204\) 0 0
\(205\) −1.95509 −0.136549
\(206\) −2.10843 −0.146901
\(207\) 0 0
\(208\) 10.0215 0.694869
\(209\) 3.67072 0.253909
\(210\) 0 0
\(211\) −8.51717 −0.586346 −0.293173 0.956059i \(-0.594711\pi\)
−0.293173 + 0.956059i \(0.594711\pi\)
\(212\) −0.898477 −0.0617076
\(213\) 0 0
\(214\) −14.7477 −1.00813
\(215\) 2.39223 0.163149
\(216\) 0 0
\(217\) −0.870770 −0.0591117
\(218\) −4.80172 −0.325213
\(219\) 0 0
\(220\) 0.189627 0.0127847
\(221\) −25.4191 −1.70987
\(222\) 0 0
\(223\) −5.10025 −0.341538 −0.170769 0.985311i \(-0.554625\pi\)
−0.170769 + 0.985311i \(0.554625\pi\)
\(224\) −0.789026 −0.0527190
\(225\) 0 0
\(226\) −17.1631 −1.14167
\(227\) 13.1071 0.869951 0.434976 0.900442i \(-0.356757\pi\)
0.434976 + 0.900442i \(0.356757\pi\)
\(228\) 0 0
\(229\) 8.15429 0.538851 0.269425 0.963021i \(-0.413166\pi\)
0.269425 + 0.963021i \(0.413166\pi\)
\(230\) 1.72247 0.113576
\(231\) 0 0
\(232\) −7.61067 −0.499665
\(233\) −10.5083 −0.688419 −0.344209 0.938893i \(-0.611853\pi\)
−0.344209 + 0.938893i \(0.611853\pi\)
\(234\) 0 0
\(235\) 3.28117 0.214040
\(236\) −0.791258 −0.0515065
\(237\) 0 0
\(238\) −2.70271 −0.175191
\(239\) −21.1592 −1.36868 −0.684338 0.729165i \(-0.739909\pi\)
−0.684338 + 0.729165i \(0.739909\pi\)
\(240\) 0 0
\(241\) −29.1626 −1.87853 −0.939264 0.343194i \(-0.888491\pi\)
−0.939264 + 0.343194i \(0.888491\pi\)
\(242\) 1.23685 0.0795078
\(243\) 0 0
\(244\) −0.470198 −0.0301013
\(245\) 2.78590 0.177984
\(246\) 0 0
\(247\) −12.9597 −0.824606
\(248\) −8.76570 −0.556622
\(249\) 0 0
\(250\) 4.90700 0.310346
\(251\) −24.0553 −1.51835 −0.759177 0.650884i \(-0.774398\pi\)
−0.759177 + 0.650884i \(0.774398\pi\)
\(252\) 0 0
\(253\) −3.45314 −0.217097
\(254\) 16.0517 1.00718
\(255\) 0 0
\(256\) −10.6121 −0.663259
\(257\) 12.6806 0.790993 0.395497 0.918467i \(-0.370573\pi\)
0.395497 + 0.918467i \(0.370573\pi\)
\(258\) 0 0
\(259\) −0.912259 −0.0566850
\(260\) −0.669489 −0.0415199
\(261\) 0 0
\(262\) 0.0243022 0.00150140
\(263\) −25.0698 −1.54587 −0.772934 0.634487i \(-0.781211\pi\)
−0.772934 + 0.634487i \(0.781211\pi\)
\(264\) 0 0
\(265\) −0.770630 −0.0473394
\(266\) −1.37796 −0.0844879
\(267\) 0 0
\(268\) 0.0422332 0.00257981
\(269\) −4.34238 −0.264760 −0.132380 0.991199i \(-0.542262\pi\)
−0.132380 + 0.991199i \(0.542262\pi\)
\(270\) 0 0
\(271\) 17.9773 1.09204 0.546021 0.837771i \(-0.316142\pi\)
0.546021 + 0.837771i \(0.316142\pi\)
\(272\) −20.4366 −1.23915
\(273\) 0 0
\(274\) −0.340007 −0.0205406
\(275\) −4.83736 −0.291704
\(276\) 0 0
\(277\) 15.1868 0.912485 0.456242 0.889856i \(-0.349195\pi\)
0.456242 + 0.889856i \(0.349195\pi\)
\(278\) −17.7730 −1.06595
\(279\) 0 0
\(280\) −0.373969 −0.0223489
\(281\) 21.9272 1.30807 0.654034 0.756465i \(-0.273075\pi\)
0.654034 + 0.756465i \(0.273075\pi\)
\(282\) 0 0
\(283\) −21.5407 −1.28046 −0.640230 0.768183i \(-0.721161\pi\)
−0.640230 + 0.768183i \(0.721161\pi\)
\(284\) 6.15598 0.365290
\(285\) 0 0
\(286\) −4.36677 −0.258213
\(287\) −1.47134 −0.0868504
\(288\) 0 0
\(289\) 34.8362 2.04919
\(290\) −1.24254 −0.0729645
\(291\) 0 0
\(292\) 6.60929 0.386779
\(293\) −29.3795 −1.71637 −0.858183 0.513344i \(-0.828407\pi\)
−0.858183 + 0.513344i \(0.828407\pi\)
\(294\) 0 0
\(295\) −0.678668 −0.0395136
\(296\) −9.18335 −0.533771
\(297\) 0 0
\(298\) −19.3022 −1.11814
\(299\) 12.1915 0.705054
\(300\) 0 0
\(301\) 1.80032 0.103769
\(302\) 25.2934 1.45547
\(303\) 0 0
\(304\) −10.4194 −0.597595
\(305\) −0.403292 −0.0230924
\(306\) 0 0
\(307\) 27.0803 1.54555 0.772777 0.634678i \(-0.218867\pi\)
0.772777 + 0.634678i \(0.218867\pi\)
\(308\) 0.142707 0.00813151
\(309\) 0 0
\(310\) −1.43112 −0.0812819
\(311\) −29.9461 −1.69809 −0.849044 0.528321i \(-0.822822\pi\)
−0.849044 + 0.528321i \(0.822822\pi\)
\(312\) 0 0
\(313\) 5.01342 0.283376 0.141688 0.989911i \(-0.454747\pi\)
0.141688 + 0.989911i \(0.454747\pi\)
\(314\) −28.1953 −1.59115
\(315\) 0 0
\(316\) −1.62782 −0.0915723
\(317\) −4.61539 −0.259226 −0.129613 0.991565i \(-0.541374\pi\)
−0.129613 + 0.991565i \(0.541374\pi\)
\(318\) 0 0
\(319\) 2.49100 0.139469
\(320\) −3.58627 −0.200479
\(321\) 0 0
\(322\) 1.29628 0.0722388
\(323\) 26.4282 1.47051
\(324\) 0 0
\(325\) 17.0786 0.947348
\(326\) 6.83787 0.378715
\(327\) 0 0
\(328\) −14.8114 −0.817823
\(329\) 2.46931 0.136137
\(330\) 0 0
\(331\) 4.66498 0.256410 0.128205 0.991748i \(-0.459078\pi\)
0.128205 + 0.991748i \(0.459078\pi\)
\(332\) 1.15509 0.0633936
\(333\) 0 0
\(334\) 27.5726 1.50871
\(335\) 0.0362238 0.00197912
\(336\) 0 0
\(337\) 19.6742 1.07172 0.535861 0.844306i \(-0.319987\pi\)
0.535861 + 0.844306i \(0.319987\pi\)
\(338\) −0.661936 −0.0360046
\(339\) 0 0
\(340\) 1.36526 0.0740419
\(341\) 2.86904 0.155367
\(342\) 0 0
\(343\) 4.22112 0.227919
\(344\) 18.1231 0.977134
\(345\) 0 0
\(346\) 3.61404 0.194292
\(347\) −3.66629 −0.196817 −0.0984083 0.995146i \(-0.531375\pi\)
−0.0984083 + 0.995146i \(0.531375\pi\)
\(348\) 0 0
\(349\) 10.1270 0.542084 0.271042 0.962567i \(-0.412632\pi\)
0.271042 + 0.962567i \(0.412632\pi\)
\(350\) 1.81590 0.0970639
\(351\) 0 0
\(352\) 2.59971 0.138565
\(353\) 16.6514 0.886265 0.443133 0.896456i \(-0.353867\pi\)
0.443133 + 0.896456i \(0.353867\pi\)
\(354\) 0 0
\(355\) 5.28003 0.280235
\(356\) 5.72575 0.303464
\(357\) 0 0
\(358\) −10.8194 −0.571825
\(359\) −11.9942 −0.633027 −0.316514 0.948588i \(-0.602512\pi\)
−0.316514 + 0.948588i \(0.602512\pi\)
\(360\) 0 0
\(361\) −5.52579 −0.290831
\(362\) −27.1100 −1.42487
\(363\) 0 0
\(364\) −0.503837 −0.0264082
\(365\) 5.66884 0.296720
\(366\) 0 0
\(367\) 9.28210 0.484522 0.242261 0.970211i \(-0.422111\pi\)
0.242261 + 0.970211i \(0.422111\pi\)
\(368\) 9.80181 0.510955
\(369\) 0 0
\(370\) −1.49930 −0.0779450
\(371\) −0.579952 −0.0301096
\(372\) 0 0
\(373\) 10.6865 0.553327 0.276663 0.960967i \(-0.410771\pi\)
0.276663 + 0.960967i \(0.410771\pi\)
\(374\) 8.90500 0.460467
\(375\) 0 0
\(376\) 24.8575 1.28193
\(377\) −8.79461 −0.452945
\(378\) 0 0
\(379\) 32.1527 1.65157 0.825787 0.563982i \(-0.190731\pi\)
0.825787 + 0.563982i \(0.190731\pi\)
\(380\) 0.696069 0.0357076
\(381\) 0 0
\(382\) −2.22514 −0.113848
\(383\) −22.4294 −1.14609 −0.573046 0.819524i \(-0.694238\pi\)
−0.573046 + 0.819524i \(0.694238\pi\)
\(384\) 0 0
\(385\) 0.122401 0.00623815
\(386\) 8.64929 0.440237
\(387\) 0 0
\(388\) 0.802209 0.0407260
\(389\) −23.7359 −1.20346 −0.601729 0.798700i \(-0.705521\pi\)
−0.601729 + 0.798700i \(0.705521\pi\)
\(390\) 0 0
\(391\) −24.8617 −1.25731
\(392\) 21.1054 1.06599
\(393\) 0 0
\(394\) 24.4426 1.23140
\(395\) −1.39620 −0.0702503
\(396\) 0 0
\(397\) 13.5028 0.677686 0.338843 0.940843i \(-0.389964\pi\)
0.338843 + 0.940843i \(0.389964\pi\)
\(398\) −6.12248 −0.306892
\(399\) 0 0
\(400\) 13.7309 0.686546
\(401\) 28.6582 1.43112 0.715561 0.698550i \(-0.246171\pi\)
0.715561 + 0.698550i \(0.246171\pi\)
\(402\) 0 0
\(403\) −10.1293 −0.504577
\(404\) 1.45731 0.0725037
\(405\) 0 0
\(406\) −0.935098 −0.0464081
\(407\) 3.00574 0.148989
\(408\) 0 0
\(409\) 19.4603 0.962252 0.481126 0.876651i \(-0.340228\pi\)
0.481126 + 0.876651i \(0.340228\pi\)
\(410\) −2.41816 −0.119424
\(411\) 0 0
\(412\) 0.801536 0.0394888
\(413\) −0.510744 −0.0251321
\(414\) 0 0
\(415\) 0.990726 0.0486328
\(416\) −9.17842 −0.450009
\(417\) 0 0
\(418\) 4.54014 0.222066
\(419\) −17.3821 −0.849169 −0.424585 0.905388i \(-0.639580\pi\)
−0.424585 + 0.905388i \(0.639580\pi\)
\(420\) 0 0
\(421\) 10.5440 0.513885 0.256943 0.966427i \(-0.417285\pi\)
0.256943 + 0.966427i \(0.417285\pi\)
\(422\) −10.5345 −0.512810
\(423\) 0 0
\(424\) −5.83815 −0.283526
\(425\) −34.8277 −1.68939
\(426\) 0 0
\(427\) −0.303505 −0.0146876
\(428\) 5.60646 0.270998
\(429\) 0 0
\(430\) 2.95884 0.142688
\(431\) 32.8548 1.58256 0.791281 0.611453i \(-0.209415\pi\)
0.791281 + 0.611453i \(0.209415\pi\)
\(432\) 0 0
\(433\) 8.53171 0.410008 0.205004 0.978761i \(-0.434279\pi\)
0.205004 + 0.978761i \(0.434279\pi\)
\(434\) −1.07701 −0.0516983
\(435\) 0 0
\(436\) 1.82541 0.0874211
\(437\) −12.6755 −0.606353
\(438\) 0 0
\(439\) 14.2501 0.680122 0.340061 0.940403i \(-0.389552\pi\)
0.340061 + 0.940403i \(0.389552\pi\)
\(440\) 1.23217 0.0587412
\(441\) 0 0
\(442\) −31.4396 −1.49543
\(443\) 8.82606 0.419339 0.209669 0.977772i \(-0.432761\pi\)
0.209669 + 0.977772i \(0.432761\pi\)
\(444\) 0 0
\(445\) 4.91102 0.232805
\(446\) −6.30825 −0.298704
\(447\) 0 0
\(448\) −2.69892 −0.127512
\(449\) −10.0380 −0.473721 −0.236860 0.971544i \(-0.576118\pi\)
−0.236860 + 0.971544i \(0.576118\pi\)
\(450\) 0 0
\(451\) 4.84782 0.228275
\(452\) 6.52467 0.306895
\(453\) 0 0
\(454\) 16.2116 0.760848
\(455\) −0.432144 −0.0202592
\(456\) 0 0
\(457\) −20.9769 −0.981260 −0.490630 0.871368i \(-0.663233\pi\)
−0.490630 + 0.871368i \(0.663233\pi\)
\(458\) 10.0857 0.471272
\(459\) 0 0
\(460\) −0.654810 −0.0305307
\(461\) 24.8875 1.15913 0.579564 0.814927i \(-0.303223\pi\)
0.579564 + 0.814927i \(0.303223\pi\)
\(462\) 0 0
\(463\) 21.9458 1.01991 0.509954 0.860202i \(-0.329663\pi\)
0.509954 + 0.860202i \(0.329663\pi\)
\(464\) −7.07074 −0.328251
\(465\) 0 0
\(466\) −12.9972 −0.602082
\(467\) −11.5803 −0.535871 −0.267936 0.963437i \(-0.586341\pi\)
−0.267936 + 0.963437i \(0.586341\pi\)
\(468\) 0 0
\(469\) 0.0272609 0.00125879
\(470\) 4.05832 0.187196
\(471\) 0 0
\(472\) −5.14146 −0.236655
\(473\) −5.93177 −0.272743
\(474\) 0 0
\(475\) −17.7566 −0.814728
\(476\) 1.02746 0.0470934
\(477\) 0 0
\(478\) −26.1708 −1.19702
\(479\) −21.6662 −0.989952 −0.494976 0.868907i \(-0.664823\pi\)
−0.494976 + 0.868907i \(0.664823\pi\)
\(480\) 0 0
\(481\) −10.6119 −0.483863
\(482\) −36.0698 −1.64294
\(483\) 0 0
\(484\) −0.470198 −0.0213726
\(485\) 0.688060 0.0312432
\(486\) 0 0
\(487\) 11.2365 0.509173 0.254587 0.967050i \(-0.418061\pi\)
0.254587 + 0.967050i \(0.418061\pi\)
\(488\) −3.05527 −0.138306
\(489\) 0 0
\(490\) 3.44574 0.155663
\(491\) −40.3512 −1.82102 −0.910512 0.413482i \(-0.864313\pi\)
−0.910512 + 0.413482i \(0.864313\pi\)
\(492\) 0 0
\(493\) 17.9345 0.807730
\(494\) −16.0292 −0.721189
\(495\) 0 0
\(496\) −8.14383 −0.365669
\(497\) 3.97358 0.178240
\(498\) 0 0
\(499\) −36.3298 −1.62635 −0.813173 0.582022i \(-0.802262\pi\)
−0.813173 + 0.582022i \(0.802262\pi\)
\(500\) −1.86543 −0.0834245
\(501\) 0 0
\(502\) −29.7528 −1.32793
\(503\) 5.74480 0.256148 0.128074 0.991765i \(-0.459121\pi\)
0.128074 + 0.991765i \(0.459121\pi\)
\(504\) 0 0
\(505\) 1.24994 0.0556217
\(506\) −4.27103 −0.189870
\(507\) 0 0
\(508\) −6.10218 −0.270740
\(509\) −40.4047 −1.79091 −0.895453 0.445156i \(-0.853148\pi\)
−0.895453 + 0.445156i \(0.853148\pi\)
\(510\) 0 0
\(511\) 4.26619 0.188725
\(512\) −24.7242 −1.09267
\(513\) 0 0
\(514\) 15.6840 0.691792
\(515\) 0.687483 0.0302941
\(516\) 0 0
\(517\) −8.13596 −0.357819
\(518\) −1.12833 −0.0495759
\(519\) 0 0
\(520\) −4.35023 −0.190770
\(521\) −26.7769 −1.17312 −0.586559 0.809907i \(-0.699518\pi\)
−0.586559 + 0.809907i \(0.699518\pi\)
\(522\) 0 0
\(523\) −7.37927 −0.322673 −0.161336 0.986899i \(-0.551580\pi\)
−0.161336 + 0.986899i \(0.551580\pi\)
\(524\) −0.00923865 −0.000403592 0
\(525\) 0 0
\(526\) −31.0076 −1.35199
\(527\) 20.6563 0.899805
\(528\) 0 0
\(529\) −11.0758 −0.481556
\(530\) −0.953155 −0.0414024
\(531\) 0 0
\(532\) 0.523840 0.0227113
\(533\) −17.1155 −0.741355
\(534\) 0 0
\(535\) 4.80870 0.207898
\(536\) 0.274425 0.0118533
\(537\) 0 0
\(538\) −5.37088 −0.231555
\(539\) −6.90788 −0.297544
\(540\) 0 0
\(541\) 26.7137 1.14851 0.574256 0.818676i \(-0.305291\pi\)
0.574256 + 0.818676i \(0.305291\pi\)
\(542\) 22.2352 0.955085
\(543\) 0 0
\(544\) 18.7172 0.802494
\(545\) 1.56567 0.0670657
\(546\) 0 0
\(547\) −12.2664 −0.524472 −0.262236 0.965004i \(-0.584460\pi\)
−0.262236 + 0.965004i \(0.584460\pi\)
\(548\) 0.129256 0.00552154
\(549\) 0 0
\(550\) −5.98309 −0.255120
\(551\) 9.14377 0.389538
\(552\) 0 0
\(553\) −1.05073 −0.0446818
\(554\) 18.7838 0.798047
\(555\) 0 0
\(556\) 6.75653 0.286541
\(557\) −18.9900 −0.804632 −0.402316 0.915501i \(-0.631795\pi\)
−0.402316 + 0.915501i \(0.631795\pi\)
\(558\) 0 0
\(559\) 20.9424 0.885770
\(560\) −0.347438 −0.0146819
\(561\) 0 0
\(562\) 27.1207 1.14402
\(563\) −12.4311 −0.523907 −0.261953 0.965081i \(-0.584367\pi\)
−0.261953 + 0.965081i \(0.584367\pi\)
\(564\) 0 0
\(565\) 5.59626 0.235436
\(566\) −26.6426 −1.11987
\(567\) 0 0
\(568\) 40.0005 1.67838
\(569\) −24.3719 −1.02172 −0.510861 0.859663i \(-0.670673\pi\)
−0.510861 + 0.859663i \(0.670673\pi\)
\(570\) 0 0
\(571\) −22.3684 −0.936088 −0.468044 0.883705i \(-0.655041\pi\)
−0.468044 + 0.883705i \(0.655041\pi\)
\(572\) 1.66006 0.0694106
\(573\) 0 0
\(574\) −1.81983 −0.0759582
\(575\) 16.7041 0.696609
\(576\) 0 0
\(577\) −0.888594 −0.0369927 −0.0184963 0.999829i \(-0.505888\pi\)
−0.0184963 + 0.999829i \(0.505888\pi\)
\(578\) 43.0872 1.79219
\(579\) 0 0
\(580\) 0.472361 0.0196137
\(581\) 0.745590 0.0309323
\(582\) 0 0
\(583\) 1.91085 0.0791392
\(584\) 42.9461 1.77712
\(585\) 0 0
\(586\) −36.3380 −1.50111
\(587\) −38.8288 −1.60264 −0.801319 0.598238i \(-0.795868\pi\)
−0.801319 + 0.598238i \(0.795868\pi\)
\(588\) 0 0
\(589\) 10.5315 0.433942
\(590\) −0.839411 −0.0345580
\(591\) 0 0
\(592\) −8.53186 −0.350657
\(593\) 27.1156 1.11350 0.556751 0.830679i \(-0.312048\pi\)
0.556751 + 0.830679i \(0.312048\pi\)
\(594\) 0 0
\(595\) 0.881256 0.0361280
\(596\) 7.33785 0.300570
\(597\) 0 0
\(598\) 15.0791 0.616630
\(599\) 13.0804 0.534451 0.267225 0.963634i \(-0.413893\pi\)
0.267225 + 0.963634i \(0.413893\pi\)
\(600\) 0 0
\(601\) −9.30686 −0.379635 −0.189817 0.981819i \(-0.560790\pi\)
−0.189817 + 0.981819i \(0.560790\pi\)
\(602\) 2.22673 0.0907548
\(603\) 0 0
\(604\) −9.61546 −0.391248
\(605\) −0.403292 −0.0163962
\(606\) 0 0
\(607\) 32.2161 1.30761 0.653805 0.756663i \(-0.273172\pi\)
0.653805 + 0.756663i \(0.273172\pi\)
\(608\) 9.54282 0.387012
\(609\) 0 0
\(610\) −0.498813 −0.0201963
\(611\) 28.7245 1.16207
\(612\) 0 0
\(613\) 7.00638 0.282985 0.141492 0.989939i \(-0.454810\pi\)
0.141492 + 0.989939i \(0.454810\pi\)
\(614\) 33.4943 1.35172
\(615\) 0 0
\(616\) 0.927290 0.0373616
\(617\) 33.8218 1.36161 0.680807 0.732463i \(-0.261629\pi\)
0.680807 + 0.732463i \(0.261629\pi\)
\(618\) 0 0
\(619\) 17.7967 0.715309 0.357655 0.933854i \(-0.383577\pi\)
0.357655 + 0.933854i \(0.383577\pi\)
\(620\) 0.544048 0.0218495
\(621\) 0 0
\(622\) −37.0389 −1.48513
\(623\) 3.69588 0.148072
\(624\) 0 0
\(625\) 22.5868 0.903471
\(626\) 6.20086 0.247836
\(627\) 0 0
\(628\) 10.7186 0.427720
\(629\) 21.6405 0.862865
\(630\) 0 0
\(631\) 40.5090 1.61264 0.806319 0.591480i \(-0.201456\pi\)
0.806319 + 0.591480i \(0.201456\pi\)
\(632\) −10.5773 −0.420744
\(633\) 0 0
\(634\) −5.70856 −0.226716
\(635\) −5.23388 −0.207700
\(636\) 0 0
\(637\) 24.3887 0.966314
\(638\) 3.08100 0.121978
\(639\) 0 0
\(640\) −2.33880 −0.0924493
\(641\) 10.7585 0.424936 0.212468 0.977168i \(-0.431850\pi\)
0.212468 + 0.977168i \(0.431850\pi\)
\(642\) 0 0
\(643\) −35.6131 −1.40445 −0.702223 0.711957i \(-0.747809\pi\)
−0.702223 + 0.711957i \(0.747809\pi\)
\(644\) −0.492790 −0.0194186
\(645\) 0 0
\(646\) 32.6878 1.28608
\(647\) 28.1304 1.10592 0.552961 0.833207i \(-0.313498\pi\)
0.552961 + 0.833207i \(0.313498\pi\)
\(648\) 0 0
\(649\) 1.68282 0.0660564
\(650\) 21.1236 0.828537
\(651\) 0 0
\(652\) −2.59946 −0.101803
\(653\) −13.3332 −0.521769 −0.260884 0.965370i \(-0.584014\pi\)
−0.260884 + 0.965370i \(0.584014\pi\)
\(654\) 0 0
\(655\) −0.00792406 −0.000309619 0
\(656\) −13.7606 −0.537263
\(657\) 0 0
\(658\) 3.05417 0.119064
\(659\) 19.3913 0.755378 0.377689 0.925932i \(-0.376719\pi\)
0.377689 + 0.925932i \(0.376719\pi\)
\(660\) 0 0
\(661\) −26.7209 −1.03932 −0.519661 0.854373i \(-0.673942\pi\)
−0.519661 + 0.854373i \(0.673942\pi\)
\(662\) 5.76989 0.224253
\(663\) 0 0
\(664\) 7.50556 0.291272
\(665\) 0.449301 0.0174232
\(666\) 0 0
\(667\) −8.60178 −0.333062
\(668\) −10.4819 −0.405558
\(669\) 0 0
\(670\) 0.0448034 0.00173091
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) −41.3889 −1.59543 −0.797713 0.603038i \(-0.793957\pi\)
−0.797713 + 0.603038i \(0.793957\pi\)
\(674\) 24.3341 0.937314
\(675\) 0 0
\(676\) 0.251640 0.00967844
\(677\) 26.0151 0.999840 0.499920 0.866072i \(-0.333363\pi\)
0.499920 + 0.866072i \(0.333363\pi\)
\(678\) 0 0
\(679\) 0.517813 0.0198718
\(680\) 8.87126 0.340197
\(681\) 0 0
\(682\) 3.54858 0.135882
\(683\) 14.8239 0.567220 0.283610 0.958940i \(-0.408468\pi\)
0.283610 + 0.958940i \(0.408468\pi\)
\(684\) 0 0
\(685\) 0.110864 0.00423589
\(686\) 5.22089 0.199335
\(687\) 0 0
\(688\) 16.8374 0.641921
\(689\) −6.74635 −0.257016
\(690\) 0 0
\(691\) −48.7801 −1.85568 −0.927841 0.372975i \(-0.878338\pi\)
−0.927841 + 0.372975i \(0.878338\pi\)
\(692\) −1.37390 −0.0522280
\(693\) 0 0
\(694\) −4.53465 −0.172133
\(695\) 5.79512 0.219821
\(696\) 0 0
\(697\) 34.9030 1.32205
\(698\) 12.5256 0.474099
\(699\) 0 0
\(700\) −0.690327 −0.0260919
\(701\) −51.2249 −1.93474 −0.967369 0.253371i \(-0.918461\pi\)
−0.967369 + 0.253371i \(0.918461\pi\)
\(702\) 0 0
\(703\) 11.0333 0.416127
\(704\) 8.89249 0.335148
\(705\) 0 0
\(706\) 20.5953 0.775115
\(707\) 0.940668 0.0353775
\(708\) 0 0
\(709\) 25.1294 0.943753 0.471876 0.881665i \(-0.343577\pi\)
0.471876 + 0.881665i \(0.343577\pi\)
\(710\) 6.53061 0.245089
\(711\) 0 0
\(712\) 37.2050 1.39432
\(713\) −9.90722 −0.371028
\(714\) 0 0
\(715\) 1.42385 0.0532488
\(716\) 4.11309 0.153713
\(717\) 0 0
\(718\) −14.8350 −0.553637
\(719\) −3.40870 −0.127123 −0.0635615 0.997978i \(-0.520246\pi\)
−0.0635615 + 0.997978i \(0.520246\pi\)
\(720\) 0 0
\(721\) 0.517378 0.0192682
\(722\) −6.83458 −0.254357
\(723\) 0 0
\(724\) 10.3061 0.383021
\(725\) −12.0498 −0.447520
\(726\) 0 0
\(727\) 37.1314 1.37713 0.688563 0.725176i \(-0.258242\pi\)
0.688563 + 0.725176i \(0.258242\pi\)
\(728\) −3.27385 −0.121337
\(729\) 0 0
\(730\) 7.01151 0.259508
\(731\) −42.7071 −1.57958
\(732\) 0 0
\(733\) 14.2953 0.528009 0.264004 0.964521i \(-0.414957\pi\)
0.264004 + 0.964521i \(0.414957\pi\)
\(734\) 11.4806 0.423756
\(735\) 0 0
\(736\) −8.97718 −0.330903
\(737\) −0.0898202 −0.00330857
\(738\) 0 0
\(739\) 34.9653 1.28622 0.643110 0.765774i \(-0.277644\pi\)
0.643110 + 0.765774i \(0.277644\pi\)
\(740\) 0.569970 0.0209525
\(741\) 0 0
\(742\) −0.717315 −0.0263335
\(743\) 23.7805 0.872422 0.436211 0.899844i \(-0.356320\pi\)
0.436211 + 0.899844i \(0.356320\pi\)
\(744\) 0 0
\(745\) 6.29373 0.230584
\(746\) 13.2176 0.483932
\(747\) 0 0
\(748\) −3.38530 −0.123779
\(749\) 3.61888 0.132231
\(750\) 0 0
\(751\) 0.735996 0.0268569 0.0134284 0.999910i \(-0.495725\pi\)
0.0134284 + 0.999910i \(0.495725\pi\)
\(752\) 23.0941 0.842154
\(753\) 0 0
\(754\) −10.8776 −0.396140
\(755\) −8.24725 −0.300148
\(756\) 0 0
\(757\) 46.9323 1.70578 0.852892 0.522087i \(-0.174846\pi\)
0.852892 + 0.522087i \(0.174846\pi\)
\(758\) 39.7681 1.44444
\(759\) 0 0
\(760\) 4.52294 0.164064
\(761\) 7.29456 0.264427 0.132214 0.991221i \(-0.457791\pi\)
0.132214 + 0.991221i \(0.457791\pi\)
\(762\) 0 0
\(763\) 1.17827 0.0426563
\(764\) 0.845904 0.0306037
\(765\) 0 0
\(766\) −27.7419 −1.00236
\(767\) −5.94129 −0.214527
\(768\) 0 0
\(769\) −34.6811 −1.25063 −0.625316 0.780372i \(-0.715030\pi\)
−0.625316 + 0.780372i \(0.715030\pi\)
\(770\) 0.151392 0.00545580
\(771\) 0 0
\(772\) −3.28809 −0.118341
\(773\) 15.5742 0.560165 0.280083 0.959976i \(-0.409638\pi\)
0.280083 + 0.959976i \(0.409638\pi\)
\(774\) 0 0
\(775\) −13.8786 −0.498533
\(776\) 5.21262 0.187122
\(777\) 0 0
\(778\) −29.3578 −1.05253
\(779\) 17.7950 0.637573
\(780\) 0 0
\(781\) −13.0923 −0.468480
\(782\) −30.7503 −1.09963
\(783\) 0 0
\(784\) 19.6082 0.700291
\(785\) 9.19344 0.328128
\(786\) 0 0
\(787\) 2.43019 0.0866271 0.0433135 0.999062i \(-0.486209\pi\)
0.0433135 + 0.999062i \(0.486209\pi\)
\(788\) −9.29201 −0.331014
\(789\) 0 0
\(790\) −1.72689 −0.0614399
\(791\) 4.21157 0.149746
\(792\) 0 0
\(793\) −3.53056 −0.125374
\(794\) 16.7010 0.592695
\(795\) 0 0
\(796\) 2.32750 0.0824962
\(797\) 16.5071 0.584712 0.292356 0.956310i \(-0.405561\pi\)
0.292356 + 0.956310i \(0.405561\pi\)
\(798\) 0 0
\(799\) −58.5768 −2.07230
\(800\) −12.5757 −0.444619
\(801\) 0 0
\(802\) 35.4459 1.25164
\(803\) −14.0564 −0.496040
\(804\) 0 0
\(805\) −0.422669 −0.0148971
\(806\) −12.5285 −0.441296
\(807\) 0 0
\(808\) 9.46934 0.333130
\(809\) −26.1046 −0.917788 −0.458894 0.888491i \(-0.651754\pi\)
−0.458894 + 0.888491i \(0.651754\pi\)
\(810\) 0 0
\(811\) −28.3961 −0.997122 −0.498561 0.866855i \(-0.666138\pi\)
−0.498561 + 0.866855i \(0.666138\pi\)
\(812\) 0.355484 0.0124750
\(813\) 0 0
\(814\) 3.71766 0.130304
\(815\) −2.22958 −0.0780988
\(816\) 0 0
\(817\) −21.7739 −0.761771
\(818\) 24.0696 0.841573
\(819\) 0 0
\(820\) 0.919279 0.0321026
\(821\) 7.36854 0.257164 0.128582 0.991699i \(-0.458957\pi\)
0.128582 + 0.991699i \(0.458957\pi\)
\(822\) 0 0
\(823\) −24.5592 −0.856080 −0.428040 0.903760i \(-0.640796\pi\)
−0.428040 + 0.903760i \(0.640796\pi\)
\(824\) 5.20825 0.181438
\(825\) 0 0
\(826\) −0.631715 −0.0219802
\(827\) −31.8506 −1.10755 −0.553776 0.832666i \(-0.686814\pi\)
−0.553776 + 0.832666i \(0.686814\pi\)
\(828\) 0 0
\(829\) 28.0544 0.974370 0.487185 0.873299i \(-0.338024\pi\)
0.487185 + 0.873299i \(0.338024\pi\)
\(830\) 1.22538 0.0425336
\(831\) 0 0
\(832\) −31.3954 −1.08844
\(833\) −49.7349 −1.72321
\(834\) 0 0
\(835\) −8.99042 −0.311126
\(836\) −1.72597 −0.0596938
\(837\) 0 0
\(838\) −21.4990 −0.742672
\(839\) −8.53067 −0.294511 −0.147256 0.989098i \(-0.547044\pi\)
−0.147256 + 0.989098i \(0.547044\pi\)
\(840\) 0 0
\(841\) −22.7949 −0.786032
\(842\) 13.0414 0.449437
\(843\) 0 0
\(844\) 4.00475 0.137849
\(845\) 0.215833 0.00742488
\(846\) 0 0
\(847\) −0.303505 −0.0104286
\(848\) −5.42398 −0.186260
\(849\) 0 0
\(850\) −43.0767 −1.47752
\(851\) −10.3793 −0.355797
\(852\) 0 0
\(853\) 47.7707 1.63564 0.817818 0.575477i \(-0.195183\pi\)
0.817818 + 0.575477i \(0.195183\pi\)
\(854\) −0.375391 −0.0128456
\(855\) 0 0
\(856\) 36.4298 1.24515
\(857\) −20.2990 −0.693399 −0.346700 0.937976i \(-0.612698\pi\)
−0.346700 + 0.937976i \(0.612698\pi\)
\(858\) 0 0
\(859\) 40.8031 1.39218 0.696091 0.717953i \(-0.254921\pi\)
0.696091 + 0.717953i \(0.254921\pi\)
\(860\) −1.12482 −0.0383562
\(861\) 0 0
\(862\) 40.6366 1.38409
\(863\) 13.1014 0.445979 0.222989 0.974821i \(-0.428419\pi\)
0.222989 + 0.974821i \(0.428419\pi\)
\(864\) 0 0
\(865\) −1.17841 −0.0400671
\(866\) 10.5525 0.358587
\(867\) 0 0
\(868\) 0.409434 0.0138971
\(869\) 3.46200 0.117440
\(870\) 0 0
\(871\) 0.317115 0.0107450
\(872\) 11.8612 0.401671
\(873\) 0 0
\(874\) −15.6778 −0.530308
\(875\) −1.20410 −0.0407062
\(876\) 0 0
\(877\) −17.8525 −0.602836 −0.301418 0.953492i \(-0.597460\pi\)
−0.301418 + 0.953492i \(0.597460\pi\)
\(878\) 17.6253 0.594825
\(879\) 0 0
\(880\) 1.14475 0.0385896
\(881\) −23.6982 −0.798412 −0.399206 0.916861i \(-0.630714\pi\)
−0.399206 + 0.916861i \(0.630714\pi\)
\(882\) 0 0
\(883\) 45.6569 1.53648 0.768239 0.640163i \(-0.221133\pi\)
0.768239 + 0.640163i \(0.221133\pi\)
\(884\) 11.9520 0.401989
\(885\) 0 0
\(886\) 10.9165 0.366748
\(887\) −23.1922 −0.778716 −0.389358 0.921086i \(-0.627303\pi\)
−0.389358 + 0.921086i \(0.627303\pi\)
\(888\) 0 0
\(889\) −3.93886 −0.132105
\(890\) 6.07420 0.203608
\(891\) 0 0
\(892\) 2.39813 0.0802952
\(893\) −29.8649 −0.999390
\(894\) 0 0
\(895\) 3.52783 0.117922
\(896\) −1.76011 −0.0588011
\(897\) 0 0
\(898\) −12.4155 −0.414310
\(899\) 7.14678 0.238359
\(900\) 0 0
\(901\) 13.7576 0.458332
\(902\) 5.99604 0.199646
\(903\) 0 0
\(904\) 42.3962 1.41008
\(905\) 8.83958 0.293837
\(906\) 0 0
\(907\) −0.587437 −0.0195055 −0.00975277 0.999952i \(-0.503104\pi\)
−0.00975277 + 0.999952i \(0.503104\pi\)
\(908\) −6.16295 −0.204525
\(909\) 0 0
\(910\) −0.534499 −0.0177185
\(911\) −0.0732945 −0.00242836 −0.00121418 0.999999i \(-0.500386\pi\)
−0.00121418 + 0.999999i \(0.500386\pi\)
\(912\) 0 0
\(913\) −2.45660 −0.0813015
\(914\) −25.9454 −0.858196
\(915\) 0 0
\(916\) −3.83413 −0.126683
\(917\) −0.00596340 −0.000196929 0
\(918\) 0 0
\(919\) 18.5785 0.612849 0.306424 0.951895i \(-0.400867\pi\)
0.306424 + 0.951895i \(0.400867\pi\)
\(920\) −4.25485 −0.140278
\(921\) 0 0
\(922\) 30.7822 1.01376
\(923\) 46.2231 1.52145
\(924\) 0 0
\(925\) −14.5398 −0.478067
\(926\) 27.1437 0.891997
\(927\) 0 0
\(928\) 6.47587 0.212581
\(929\) −8.28707 −0.271890 −0.135945 0.990716i \(-0.543407\pi\)
−0.135945 + 0.990716i \(0.543407\pi\)
\(930\) 0 0
\(931\) −25.3569 −0.831040
\(932\) 4.94096 0.161847
\(933\) 0 0
\(934\) −14.3231 −0.468666
\(935\) −2.90360 −0.0949577
\(936\) 0 0
\(937\) 38.3030 1.25131 0.625653 0.780102i \(-0.284833\pi\)
0.625653 + 0.780102i \(0.284833\pi\)
\(938\) 0.0337177 0.00110092
\(939\) 0 0
\(940\) −1.54280 −0.0503205
\(941\) 8.69924 0.283587 0.141793 0.989896i \(-0.454713\pi\)
0.141793 + 0.989896i \(0.454713\pi\)
\(942\) 0 0
\(943\) −16.7402 −0.545137
\(944\) −4.77671 −0.155469
\(945\) 0 0
\(946\) −7.33671 −0.238537
\(947\) 16.1962 0.526306 0.263153 0.964754i \(-0.415238\pi\)
0.263153 + 0.964754i \(0.415238\pi\)
\(948\) 0 0
\(949\) 49.6269 1.61096
\(950\) −21.9623 −0.712550
\(951\) 0 0
\(952\) 6.67624 0.216378
\(953\) 13.7098 0.444104 0.222052 0.975035i \(-0.428724\pi\)
0.222052 + 0.975035i \(0.428724\pi\)
\(954\) 0 0
\(955\) 0.725538 0.0234779
\(956\) 9.94901 0.321774
\(957\) 0 0
\(958\) −26.7978 −0.865799
\(959\) 0.0834327 0.00269418
\(960\) 0 0
\(961\) −22.7686 −0.734471
\(962\) −13.1254 −0.423180
\(963\) 0 0
\(964\) 13.7122 0.441640
\(965\) −2.82022 −0.0907860
\(966\) 0 0
\(967\) 29.8355 0.959444 0.479722 0.877420i \(-0.340737\pi\)
0.479722 + 0.877420i \(0.340737\pi\)
\(968\) −3.05527 −0.0982000
\(969\) 0 0
\(970\) 0.851028 0.0273249
\(971\) 45.1242 1.44811 0.724053 0.689745i \(-0.242277\pi\)
0.724053 + 0.689745i \(0.242277\pi\)
\(972\) 0 0
\(973\) 4.36123 0.139815
\(974\) 13.8979 0.445316
\(975\) 0 0
\(976\) −2.83852 −0.0908588
\(977\) −60.1493 −1.92435 −0.962173 0.272439i \(-0.912170\pi\)
−0.962173 + 0.272439i \(0.912170\pi\)
\(978\) 0 0
\(979\) −12.1773 −0.389189
\(980\) −1.30992 −0.0418439
\(981\) 0 0
\(982\) −49.9085 −1.59264
\(983\) 13.5937 0.433573 0.216786 0.976219i \(-0.430442\pi\)
0.216786 + 0.976219i \(0.430442\pi\)
\(984\) 0 0
\(985\) −7.96983 −0.253940
\(986\) 22.1823 0.706430
\(987\) 0 0
\(988\) 6.09362 0.193864
\(989\) 20.4832 0.651329
\(990\) 0 0
\(991\) −20.1950 −0.641517 −0.320758 0.947161i \(-0.603938\pi\)
−0.320758 + 0.947161i \(0.603938\pi\)
\(992\) 7.45868 0.236813
\(993\) 0 0
\(994\) 4.91473 0.155886
\(995\) 1.99632 0.0632875
\(996\) 0 0
\(997\) −39.4389 −1.24904 −0.624522 0.781007i \(-0.714706\pi\)
−0.624522 + 0.781007i \(0.714706\pi\)
\(998\) −44.9346 −1.42238
\(999\) 0 0
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 6039.2.a.m.1.19 25
3.2 odd 2 6039.2.a.p.1.7 yes 25
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6039.2.a.m.1.19 25 1.1 even 1 trivial
6039.2.a.p.1.7 yes 25 3.2 odd 2