Properties

Label 6039.2.a.c.1.5
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
Dimension $11$
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] = [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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 2x^{10} - 14x^{9} + 27x^{8} + 66x^{7} - 125x^{6} - 115x^{5} + 227x^{4} + 40x^{3} - 129x^{2} + 26x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.0512060\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+0.0512060 q^{2} -1.99738 q^{4} -3.74211 q^{5} -4.65643 q^{7} -0.204690 q^{8} +O(q^{10})\) \(q+0.0512060 q^{2} -1.99738 q^{4} -3.74211 q^{5} -4.65643 q^{7} -0.204690 q^{8} -0.191618 q^{10} -1.00000 q^{11} -0.845440 q^{13} -0.238437 q^{14} +3.98427 q^{16} -1.99161 q^{17} -0.676988 q^{19} +7.47440 q^{20} -0.0512060 q^{22} +3.54686 q^{23} +9.00337 q^{25} -0.0432916 q^{26} +9.30066 q^{28} +0.677171 q^{29} +7.69354 q^{31} +0.613399 q^{32} -0.101982 q^{34} +17.4249 q^{35} -5.80691 q^{37} -0.0346659 q^{38} +0.765971 q^{40} +3.31122 q^{41} -8.83130 q^{43} +1.99738 q^{44} +0.181620 q^{46} +5.37613 q^{47} +14.6824 q^{49} +0.461027 q^{50} +1.68866 q^{52} +7.32872 q^{53} +3.74211 q^{55} +0.953125 q^{56} +0.0346752 q^{58} -0.636788 q^{59} +1.00000 q^{61} +0.393955 q^{62} -7.93714 q^{64} +3.16373 q^{65} -6.23733 q^{67} +3.97799 q^{68} +0.892259 q^{70} +2.06467 q^{71} +7.66830 q^{73} -0.297349 q^{74} +1.35220 q^{76} +4.65643 q^{77} +2.09202 q^{79} -14.9096 q^{80} +0.169554 q^{82} +3.47537 q^{83} +7.45280 q^{85} -0.452216 q^{86} +0.204690 q^{88} -8.92973 q^{89} +3.93673 q^{91} -7.08441 q^{92} +0.275290 q^{94} +2.53336 q^{95} -16.1658 q^{97} +0.751826 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 2 q^{2} + 10 q^{4} + q^{5} - 11 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 2 q^{2} + 10 q^{4} + q^{5} - 11 q^{7} + 3 q^{8} - 8 q^{10} - 11 q^{11} - 13 q^{13} - 5 q^{14} + 4 q^{16} + 13 q^{17} - 12 q^{19} + 7 q^{20} - 2 q^{22} + 3 q^{23} + 12 q^{25} - 12 q^{26} - 13 q^{28} - 2 q^{29} + q^{31} + 23 q^{32} - 14 q^{34} + 4 q^{35} - 14 q^{37} + 8 q^{38} - 34 q^{40} - 3 q^{41} - 21 q^{43} - 10 q^{44} - 12 q^{46} + 16 q^{47} - 18 q^{49} + 13 q^{50} - 33 q^{52} - q^{55} - 16 q^{56} - 17 q^{58} - 3 q^{59} + 11 q^{61} + 21 q^{62} - 7 q^{64} + q^{65} - 24 q^{67} - 2 q^{68} + 4 q^{70} - 7 q^{71} - 42 q^{73} + 16 q^{74} - 13 q^{76} + 11 q^{77} - 11 q^{79} - 42 q^{80} - 38 q^{82} + 34 q^{83} - 14 q^{85} - 42 q^{86} - 3 q^{88} - 29 q^{89} + 9 q^{91} - 42 q^{92} - 33 q^{94} + 31 q^{95} - 45 q^{97} + 33 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0512060 0.0362081 0.0181041 0.999836i \(-0.494237\pi\)
0.0181041 + 0.999836i \(0.494237\pi\)
\(3\) 0 0
\(4\) −1.99738 −0.998689
\(5\) −3.74211 −1.67352 −0.836761 0.547569i \(-0.815553\pi\)
−0.836761 + 0.547569i \(0.815553\pi\)
\(6\) 0 0
\(7\) −4.65643 −1.75997 −0.879983 0.475005i \(-0.842446\pi\)
−0.879983 + 0.475005i \(0.842446\pi\)
\(8\) −0.204690 −0.0723688
\(9\) 0 0
\(10\) −0.191618 −0.0605951
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −0.845440 −0.234483 −0.117241 0.993103i \(-0.537405\pi\)
−0.117241 + 0.993103i \(0.537405\pi\)
\(14\) −0.238437 −0.0637251
\(15\) 0 0
\(16\) 3.98427 0.996069
\(17\) −1.99161 −0.483035 −0.241518 0.970396i \(-0.577645\pi\)
−0.241518 + 0.970396i \(0.577645\pi\)
\(18\) 0 0
\(19\) −0.676988 −0.155312 −0.0776559 0.996980i \(-0.524744\pi\)
−0.0776559 + 0.996980i \(0.524744\pi\)
\(20\) 7.47440 1.67133
\(21\) 0 0
\(22\) −0.0512060 −0.0109172
\(23\) 3.54686 0.739570 0.369785 0.929117i \(-0.379431\pi\)
0.369785 + 0.929117i \(0.379431\pi\)
\(24\) 0 0
\(25\) 9.00337 1.80067
\(26\) −0.0432916 −0.00849019
\(27\) 0 0
\(28\) 9.30066 1.75766
\(29\) 0.677171 0.125748 0.0628738 0.998021i \(-0.479973\pi\)
0.0628738 + 0.998021i \(0.479973\pi\)
\(30\) 0 0
\(31\) 7.69354 1.38180 0.690900 0.722950i \(-0.257214\pi\)
0.690900 + 0.722950i \(0.257214\pi\)
\(32\) 0.613399 0.108435
\(33\) 0 0
\(34\) −0.101982 −0.0174898
\(35\) 17.4249 2.94534
\(36\) 0 0
\(37\) −5.80691 −0.954650 −0.477325 0.878727i \(-0.658394\pi\)
−0.477325 + 0.878727i \(0.658394\pi\)
\(38\) −0.0346659 −0.00562355
\(39\) 0 0
\(40\) 0.765971 0.121111
\(41\) 3.31122 0.517126 0.258563 0.965994i \(-0.416751\pi\)
0.258563 + 0.965994i \(0.416751\pi\)
\(42\) 0 0
\(43\) −8.83130 −1.34676 −0.673380 0.739297i \(-0.735158\pi\)
−0.673380 + 0.739297i \(0.735158\pi\)
\(44\) 1.99738 0.301116
\(45\) 0 0
\(46\) 0.181620 0.0267785
\(47\) 5.37613 0.784189 0.392095 0.919925i \(-0.371751\pi\)
0.392095 + 0.919925i \(0.371751\pi\)
\(48\) 0 0
\(49\) 14.6824 2.09748
\(50\) 0.461027 0.0651990
\(51\) 0 0
\(52\) 1.68866 0.234175
\(53\) 7.32872 1.00668 0.503339 0.864089i \(-0.332105\pi\)
0.503339 + 0.864089i \(0.332105\pi\)
\(54\) 0 0
\(55\) 3.74211 0.504586
\(56\) 0.953125 0.127367
\(57\) 0 0
\(58\) 0.0346752 0.00455308
\(59\) −0.636788 −0.0829027 −0.0414513 0.999141i \(-0.513198\pi\)
−0.0414513 + 0.999141i \(0.513198\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037
\(62\) 0.393955 0.0500324
\(63\) 0 0
\(64\) −7.93714 −0.992142
\(65\) 3.16373 0.392412
\(66\) 0 0
\(67\) −6.23733 −0.762012 −0.381006 0.924573i \(-0.624422\pi\)
−0.381006 + 0.924573i \(0.624422\pi\)
\(68\) 3.97799 0.482402
\(69\) 0 0
\(70\) 0.892259 0.106645
\(71\) 2.06467 0.245032 0.122516 0.992467i \(-0.460904\pi\)
0.122516 + 0.992467i \(0.460904\pi\)
\(72\) 0 0
\(73\) 7.66830 0.897506 0.448753 0.893656i \(-0.351868\pi\)
0.448753 + 0.893656i \(0.351868\pi\)
\(74\) −0.297349 −0.0345661
\(75\) 0 0
\(76\) 1.35220 0.155108
\(77\) 4.65643 0.530650
\(78\) 0 0
\(79\) 2.09202 0.235371 0.117685 0.993051i \(-0.462453\pi\)
0.117685 + 0.993051i \(0.462453\pi\)
\(80\) −14.9096 −1.66694
\(81\) 0 0
\(82\) 0.169554 0.0187242
\(83\) 3.47537 0.381471 0.190736 0.981641i \(-0.438913\pi\)
0.190736 + 0.981641i \(0.438913\pi\)
\(84\) 0 0
\(85\) 7.45280 0.808370
\(86\) −0.452216 −0.0487637
\(87\) 0 0
\(88\) 0.204690 0.0218200
\(89\) −8.92973 −0.946550 −0.473275 0.880915i \(-0.656928\pi\)
−0.473275 + 0.880915i \(0.656928\pi\)
\(90\) 0 0
\(91\) 3.93673 0.412682
\(92\) −7.08441 −0.738601
\(93\) 0 0
\(94\) 0.275290 0.0283940
\(95\) 2.53336 0.259918
\(96\) 0 0
\(97\) −16.1658 −1.64138 −0.820692 0.571371i \(-0.806412\pi\)
−0.820692 + 0.571371i \(0.806412\pi\)
\(98\) 0.751826 0.0759459
\(99\) 0 0
\(100\) −17.9831 −1.79831
\(101\) 1.26799 0.126170 0.0630851 0.998008i \(-0.479906\pi\)
0.0630851 + 0.998008i \(0.479906\pi\)
\(102\) 0 0
\(103\) −1.31535 −0.129605 −0.0648026 0.997898i \(-0.520642\pi\)
−0.0648026 + 0.997898i \(0.520642\pi\)
\(104\) 0.173053 0.0169692
\(105\) 0 0
\(106\) 0.375275 0.0364499
\(107\) 20.2934 1.96183 0.980916 0.194430i \(-0.0622856\pi\)
0.980916 + 0.194430i \(0.0622856\pi\)
\(108\) 0 0
\(109\) −17.2096 −1.64838 −0.824189 0.566315i \(-0.808369\pi\)
−0.824189 + 0.566315i \(0.808369\pi\)
\(110\) 0.191618 0.0182701
\(111\) 0 0
\(112\) −18.5525 −1.75305
\(113\) 10.5949 0.996680 0.498340 0.866982i \(-0.333943\pi\)
0.498340 + 0.866982i \(0.333943\pi\)
\(114\) 0 0
\(115\) −13.2727 −1.23769
\(116\) −1.35257 −0.125583
\(117\) 0 0
\(118\) −0.0326074 −0.00300175
\(119\) 9.27378 0.850126
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0.0512060 0.00463598
\(123\) 0 0
\(124\) −15.3669 −1.37999
\(125\) −14.9810 −1.33994
\(126\) 0 0
\(127\) 0.743672 0.0659902 0.0329951 0.999456i \(-0.489495\pi\)
0.0329951 + 0.999456i \(0.489495\pi\)
\(128\) −1.63323 −0.144358
\(129\) 0 0
\(130\) 0.162002 0.0142085
\(131\) 9.42555 0.823514 0.411757 0.911294i \(-0.364915\pi\)
0.411757 + 0.911294i \(0.364915\pi\)
\(132\) 0 0
\(133\) 3.15235 0.273343
\(134\) −0.319389 −0.0275910
\(135\) 0 0
\(136\) 0.407662 0.0349567
\(137\) −14.3819 −1.22873 −0.614364 0.789022i \(-0.710588\pi\)
−0.614364 + 0.789022i \(0.710588\pi\)
\(138\) 0 0
\(139\) −15.0951 −1.28035 −0.640177 0.768227i \(-0.721139\pi\)
−0.640177 + 0.768227i \(0.721139\pi\)
\(140\) −34.8041 −2.94148
\(141\) 0 0
\(142\) 0.105724 0.00887215
\(143\) 0.845440 0.0706992
\(144\) 0 0
\(145\) −2.53405 −0.210441
\(146\) 0.392663 0.0324970
\(147\) 0 0
\(148\) 11.5986 0.953398
\(149\) 16.2231 1.32905 0.664523 0.747268i \(-0.268635\pi\)
0.664523 + 0.747268i \(0.268635\pi\)
\(150\) 0 0
\(151\) −9.18378 −0.747366 −0.373683 0.927557i \(-0.621905\pi\)
−0.373683 + 0.927557i \(0.621905\pi\)
\(152\) 0.138573 0.0112397
\(153\) 0 0
\(154\) 0.238437 0.0192138
\(155\) −28.7900 −2.31247
\(156\) 0 0
\(157\) 22.2002 1.77177 0.885886 0.463904i \(-0.153552\pi\)
0.885886 + 0.463904i \(0.153552\pi\)
\(158\) 0.107124 0.00852234
\(159\) 0 0
\(160\) −2.29540 −0.181468
\(161\) −16.5157 −1.30162
\(162\) 0 0
\(163\) −6.88448 −0.539234 −0.269617 0.962968i \(-0.586897\pi\)
−0.269617 + 0.962968i \(0.586897\pi\)
\(164\) −6.61376 −0.516448
\(165\) 0 0
\(166\) 0.177960 0.0138124
\(167\) −11.1697 −0.864340 −0.432170 0.901792i \(-0.642252\pi\)
−0.432170 + 0.901792i \(0.642252\pi\)
\(168\) 0 0
\(169\) −12.2852 −0.945018
\(170\) 0.381629 0.0292696
\(171\) 0 0
\(172\) 17.6394 1.34499
\(173\) 17.3542 1.31942 0.659708 0.751522i \(-0.270680\pi\)
0.659708 + 0.751522i \(0.270680\pi\)
\(174\) 0 0
\(175\) −41.9236 −3.16913
\(176\) −3.98427 −0.300326
\(177\) 0 0
\(178\) −0.457256 −0.0342728
\(179\) 10.7455 0.803160 0.401580 0.915824i \(-0.368461\pi\)
0.401580 + 0.915824i \(0.368461\pi\)
\(180\) 0 0
\(181\) 6.08954 0.452632 0.226316 0.974054i \(-0.427332\pi\)
0.226316 + 0.974054i \(0.427332\pi\)
\(182\) 0.201585 0.0149424
\(183\) 0 0
\(184\) −0.726005 −0.0535218
\(185\) 21.7301 1.59763
\(186\) 0 0
\(187\) 1.99161 0.145641
\(188\) −10.7382 −0.783161
\(189\) 0 0
\(190\) 0.129723 0.00941113
\(191\) −26.7454 −1.93523 −0.967616 0.252426i \(-0.918772\pi\)
−0.967616 + 0.252426i \(0.918772\pi\)
\(192\) 0 0
\(193\) 19.0367 1.37029 0.685147 0.728405i \(-0.259738\pi\)
0.685147 + 0.728405i \(0.259738\pi\)
\(194\) −0.827784 −0.0594314
\(195\) 0 0
\(196\) −29.3262 −2.09473
\(197\) 25.4318 1.81194 0.905970 0.423341i \(-0.139143\pi\)
0.905970 + 0.423341i \(0.139143\pi\)
\(198\) 0 0
\(199\) −27.4140 −1.94333 −0.971663 0.236371i \(-0.924042\pi\)
−0.971663 + 0.236371i \(0.924042\pi\)
\(200\) −1.84290 −0.130313
\(201\) 0 0
\(202\) 0.0649290 0.00456839
\(203\) −3.15320 −0.221311
\(204\) 0 0
\(205\) −12.3909 −0.865421
\(206\) −0.0673538 −0.00469276
\(207\) 0 0
\(208\) −3.36846 −0.233561
\(209\) 0.676988 0.0468283
\(210\) 0 0
\(211\) 15.3356 1.05575 0.527874 0.849323i \(-0.322989\pi\)
0.527874 + 0.849323i \(0.322989\pi\)
\(212\) −14.6382 −1.00536
\(213\) 0 0
\(214\) 1.03914 0.0710343
\(215\) 33.0477 2.25383
\(216\) 0 0
\(217\) −35.8244 −2.43192
\(218\) −0.881234 −0.0596847
\(219\) 0 0
\(220\) −7.47440 −0.503924
\(221\) 1.68378 0.113264
\(222\) 0 0
\(223\) 20.8414 1.39564 0.697821 0.716272i \(-0.254153\pi\)
0.697821 + 0.716272i \(0.254153\pi\)
\(224\) −2.85625 −0.190841
\(225\) 0 0
\(226\) 0.542520 0.0360879
\(227\) 12.4259 0.824735 0.412368 0.911018i \(-0.364702\pi\)
0.412368 + 0.911018i \(0.364702\pi\)
\(228\) 0 0
\(229\) −7.52577 −0.497317 −0.248658 0.968591i \(-0.579990\pi\)
−0.248658 + 0.968591i \(0.579990\pi\)
\(230\) −0.679643 −0.0448143
\(231\) 0 0
\(232\) −0.138610 −0.00910020
\(233\) −3.98849 −0.261295 −0.130647 0.991429i \(-0.541706\pi\)
−0.130647 + 0.991429i \(0.541706\pi\)
\(234\) 0 0
\(235\) −20.1181 −1.31236
\(236\) 1.27191 0.0827940
\(237\) 0 0
\(238\) 0.474873 0.0307815
\(239\) 7.60269 0.491777 0.245889 0.969298i \(-0.420920\pi\)
0.245889 + 0.969298i \(0.420920\pi\)
\(240\) 0 0
\(241\) 2.67663 0.172417 0.0862083 0.996277i \(-0.472525\pi\)
0.0862083 + 0.996277i \(0.472525\pi\)
\(242\) 0.0512060 0.00329165
\(243\) 0 0
\(244\) −1.99738 −0.127869
\(245\) −54.9430 −3.51018
\(246\) 0 0
\(247\) 0.572353 0.0364179
\(248\) −1.57479 −0.0999992
\(249\) 0 0
\(250\) −0.767119 −0.0485169
\(251\) −0.514360 −0.0324661 −0.0162331 0.999868i \(-0.505167\pi\)
−0.0162331 + 0.999868i \(0.505167\pi\)
\(252\) 0 0
\(253\) −3.54686 −0.222989
\(254\) 0.0380805 0.00238938
\(255\) 0 0
\(256\) 15.7906 0.986915
\(257\) 6.99608 0.436404 0.218202 0.975904i \(-0.429981\pi\)
0.218202 + 0.975904i \(0.429981\pi\)
\(258\) 0 0
\(259\) 27.0395 1.68015
\(260\) −6.31916 −0.391898
\(261\) 0 0
\(262\) 0.482645 0.0298179
\(263\) 15.1643 0.935072 0.467536 0.883974i \(-0.345142\pi\)
0.467536 + 0.883974i \(0.345142\pi\)
\(264\) 0 0
\(265\) −27.4249 −1.68470
\(266\) 0.161419 0.00989726
\(267\) 0 0
\(268\) 12.4583 0.761013
\(269\) −26.7507 −1.63102 −0.815509 0.578745i \(-0.803543\pi\)
−0.815509 + 0.578745i \(0.803543\pi\)
\(270\) 0 0
\(271\) 3.00706 0.182666 0.0913331 0.995820i \(-0.470887\pi\)
0.0913331 + 0.995820i \(0.470887\pi\)
\(272\) −7.93511 −0.481136
\(273\) 0 0
\(274\) −0.736440 −0.0444900
\(275\) −9.00337 −0.542924
\(276\) 0 0
\(277\) 0.761643 0.0457627 0.0228813 0.999738i \(-0.492716\pi\)
0.0228813 + 0.999738i \(0.492716\pi\)
\(278\) −0.772963 −0.0463592
\(279\) 0 0
\(280\) −3.56670 −0.213151
\(281\) −19.2471 −1.14819 −0.574094 0.818789i \(-0.694646\pi\)
−0.574094 + 0.818789i \(0.694646\pi\)
\(282\) 0 0
\(283\) −20.5209 −1.21984 −0.609921 0.792463i \(-0.708799\pi\)
−0.609921 + 0.792463i \(0.708799\pi\)
\(284\) −4.12394 −0.244711
\(285\) 0 0
\(286\) 0.0432916 0.00255989
\(287\) −15.4185 −0.910124
\(288\) 0 0
\(289\) −13.0335 −0.766677
\(290\) −0.129758 −0.00761968
\(291\) 0 0
\(292\) −15.3165 −0.896330
\(293\) 11.1033 0.648660 0.324330 0.945944i \(-0.394861\pi\)
0.324330 + 0.945944i \(0.394861\pi\)
\(294\) 0 0
\(295\) 2.38293 0.138739
\(296\) 1.18862 0.0690869
\(297\) 0 0
\(298\) 0.830720 0.0481223
\(299\) −2.99865 −0.173417
\(300\) 0 0
\(301\) 41.1223 2.37025
\(302\) −0.470265 −0.0270607
\(303\) 0 0
\(304\) −2.69731 −0.154701
\(305\) −3.74211 −0.214272
\(306\) 0 0
\(307\) 23.7237 1.35398 0.676991 0.735991i \(-0.263284\pi\)
0.676991 + 0.735991i \(0.263284\pi\)
\(308\) −9.30066 −0.529954
\(309\) 0 0
\(310\) −1.47422 −0.0837303
\(311\) −8.52580 −0.483454 −0.241727 0.970344i \(-0.577714\pi\)
−0.241727 + 0.970344i \(0.577714\pi\)
\(312\) 0 0
\(313\) 2.02754 0.114603 0.0573016 0.998357i \(-0.481750\pi\)
0.0573016 + 0.998357i \(0.481750\pi\)
\(314\) 1.13679 0.0641525
\(315\) 0 0
\(316\) −4.17856 −0.235062
\(317\) −31.9495 −1.79446 −0.897231 0.441561i \(-0.854425\pi\)
−0.897231 + 0.441561i \(0.854425\pi\)
\(318\) 0 0
\(319\) −0.677171 −0.0379143
\(320\) 29.7016 1.66037
\(321\) 0 0
\(322\) −0.845703 −0.0471292
\(323\) 1.34829 0.0750211
\(324\) 0 0
\(325\) −7.61181 −0.422227
\(326\) −0.352527 −0.0195247
\(327\) 0 0
\(328\) −0.677773 −0.0374238
\(329\) −25.0336 −1.38015
\(330\) 0 0
\(331\) −18.3691 −1.00966 −0.504828 0.863220i \(-0.668444\pi\)
−0.504828 + 0.863220i \(0.668444\pi\)
\(332\) −6.94163 −0.380971
\(333\) 0 0
\(334\) −0.571958 −0.0312961
\(335\) 23.3408 1.27524
\(336\) 0 0
\(337\) −8.49840 −0.462937 −0.231469 0.972842i \(-0.574353\pi\)
−0.231469 + 0.972842i \(0.574353\pi\)
\(338\) −0.629078 −0.0342173
\(339\) 0 0
\(340\) −14.8861 −0.807310
\(341\) −7.69354 −0.416628
\(342\) 0 0
\(343\) −35.7724 −1.93153
\(344\) 1.80768 0.0974634
\(345\) 0 0
\(346\) 0.888640 0.0477736
\(347\) 17.1538 0.920865 0.460433 0.887695i \(-0.347694\pi\)
0.460433 + 0.887695i \(0.347694\pi\)
\(348\) 0 0
\(349\) −8.06840 −0.431892 −0.215946 0.976405i \(-0.569283\pi\)
−0.215946 + 0.976405i \(0.569283\pi\)
\(350\) −2.14674 −0.114748
\(351\) 0 0
\(352\) −0.613399 −0.0326943
\(353\) −17.6885 −0.941462 −0.470731 0.882277i \(-0.656010\pi\)
−0.470731 + 0.882277i \(0.656010\pi\)
\(354\) 0 0
\(355\) −7.72624 −0.410066
\(356\) 17.8361 0.945309
\(357\) 0 0
\(358\) 0.550237 0.0290809
\(359\) −19.1909 −1.01286 −0.506428 0.862282i \(-0.669034\pi\)
−0.506428 + 0.862282i \(0.669034\pi\)
\(360\) 0 0
\(361\) −18.5417 −0.975878
\(362\) 0.311821 0.0163890
\(363\) 0 0
\(364\) −7.86315 −0.412141
\(365\) −28.6956 −1.50200
\(366\) 0 0
\(367\) 6.55481 0.342158 0.171079 0.985257i \(-0.445275\pi\)
0.171079 + 0.985257i \(0.445275\pi\)
\(368\) 14.1316 0.736663
\(369\) 0 0
\(370\) 1.11271 0.0578471
\(371\) −34.1257 −1.77172
\(372\) 0 0
\(373\) 35.5579 1.84112 0.920559 0.390602i \(-0.127733\pi\)
0.920559 + 0.390602i \(0.127733\pi\)
\(374\) 0.101982 0.00527338
\(375\) 0 0
\(376\) −1.10044 −0.0567508
\(377\) −0.572507 −0.0294856
\(378\) 0 0
\(379\) 28.4271 1.46020 0.730102 0.683338i \(-0.239473\pi\)
0.730102 + 0.683338i \(0.239473\pi\)
\(380\) −5.06008 −0.259577
\(381\) 0 0
\(382\) −1.36953 −0.0700712
\(383\) −21.6032 −1.10387 −0.551936 0.833887i \(-0.686111\pi\)
−0.551936 + 0.833887i \(0.686111\pi\)
\(384\) 0 0
\(385\) −17.4249 −0.888054
\(386\) 0.974795 0.0496158
\(387\) 0 0
\(388\) 32.2891 1.63923
\(389\) 20.4103 1.03485 0.517423 0.855730i \(-0.326891\pi\)
0.517423 + 0.855730i \(0.326891\pi\)
\(390\) 0 0
\(391\) −7.06394 −0.357239
\(392\) −3.00533 −0.151792
\(393\) 0 0
\(394\) 1.30226 0.0656070
\(395\) −7.82857 −0.393898
\(396\) 0 0
\(397\) 17.7419 0.890439 0.445220 0.895421i \(-0.353126\pi\)
0.445220 + 0.895421i \(0.353126\pi\)
\(398\) −1.40376 −0.0703642
\(399\) 0 0
\(400\) 35.8719 1.79359
\(401\) −9.35139 −0.466986 −0.233493 0.972358i \(-0.575016\pi\)
−0.233493 + 0.972358i \(0.575016\pi\)
\(402\) 0 0
\(403\) −6.50442 −0.324008
\(404\) −2.53267 −0.126005
\(405\) 0 0
\(406\) −0.161463 −0.00801327
\(407\) 5.80691 0.287838
\(408\) 0 0
\(409\) 7.73559 0.382500 0.191250 0.981541i \(-0.438746\pi\)
0.191250 + 0.981541i \(0.438746\pi\)
\(410\) −0.634491 −0.0313353
\(411\) 0 0
\(412\) 2.62725 0.129435
\(413\) 2.96516 0.145906
\(414\) 0 0
\(415\) −13.0052 −0.638401
\(416\) −0.518592 −0.0254260
\(417\) 0 0
\(418\) 0.0346659 0.00169556
\(419\) −28.4807 −1.39137 −0.695686 0.718346i \(-0.744899\pi\)
−0.695686 + 0.718346i \(0.744899\pi\)
\(420\) 0 0
\(421\) 15.6217 0.761357 0.380679 0.924707i \(-0.375690\pi\)
0.380679 + 0.924707i \(0.375690\pi\)
\(422\) 0.785277 0.0382267
\(423\) 0 0
\(424\) −1.50012 −0.0728520
\(425\) −17.9312 −0.869789
\(426\) 0 0
\(427\) −4.65643 −0.225341
\(428\) −40.5335 −1.95926
\(429\) 0 0
\(430\) 1.69224 0.0816070
\(431\) 16.7493 0.806783 0.403392 0.915027i \(-0.367831\pi\)
0.403392 + 0.915027i \(0.367831\pi\)
\(432\) 0 0
\(433\) −7.73632 −0.371784 −0.185892 0.982570i \(-0.559517\pi\)
−0.185892 + 0.982570i \(0.559517\pi\)
\(434\) −1.83443 −0.0880553
\(435\) 0 0
\(436\) 34.3740 1.64622
\(437\) −2.40118 −0.114864
\(438\) 0 0
\(439\) 31.8535 1.52028 0.760142 0.649757i \(-0.225130\pi\)
0.760142 + 0.649757i \(0.225130\pi\)
\(440\) −0.765971 −0.0365163
\(441\) 0 0
\(442\) 0.0862199 0.00410106
\(443\) 5.53681 0.263062 0.131531 0.991312i \(-0.458011\pi\)
0.131531 + 0.991312i \(0.458011\pi\)
\(444\) 0 0
\(445\) 33.4160 1.58407
\(446\) 1.06720 0.0505336
\(447\) 0 0
\(448\) 36.9588 1.74614
\(449\) 22.8076 1.07636 0.538179 0.842830i \(-0.319112\pi\)
0.538179 + 0.842830i \(0.319112\pi\)
\(450\) 0 0
\(451\) −3.31122 −0.155919
\(452\) −21.1619 −0.995373
\(453\) 0 0
\(454\) 0.636280 0.0298621
\(455\) −14.7317 −0.690632
\(456\) 0 0
\(457\) −25.1371 −1.17587 −0.587933 0.808910i \(-0.700058\pi\)
−0.587933 + 0.808910i \(0.700058\pi\)
\(458\) −0.385365 −0.0180069
\(459\) 0 0
\(460\) 26.5106 1.23606
\(461\) 1.84514 0.0859366 0.0429683 0.999076i \(-0.486319\pi\)
0.0429683 + 0.999076i \(0.486319\pi\)
\(462\) 0 0
\(463\) 27.1385 1.26123 0.630617 0.776094i \(-0.282802\pi\)
0.630617 + 0.776094i \(0.282802\pi\)
\(464\) 2.69804 0.125253
\(465\) 0 0
\(466\) −0.204235 −0.00946099
\(467\) 40.1833 1.85946 0.929732 0.368238i \(-0.120039\pi\)
0.929732 + 0.368238i \(0.120039\pi\)
\(468\) 0 0
\(469\) 29.0437 1.34111
\(470\) −1.03017 −0.0475180
\(471\) 0 0
\(472\) 0.130344 0.00599957
\(473\) 8.83130 0.406063
\(474\) 0 0
\(475\) −6.09518 −0.279666
\(476\) −18.5232 −0.849011
\(477\) 0 0
\(478\) 0.389304 0.0178063
\(479\) −35.3352 −1.61451 −0.807253 0.590206i \(-0.799046\pi\)
−0.807253 + 0.590206i \(0.799046\pi\)
\(480\) 0 0
\(481\) 4.90939 0.223849
\(482\) 0.137059 0.00624289
\(483\) 0 0
\(484\) −1.99738 −0.0907899
\(485\) 60.4940 2.74689
\(486\) 0 0
\(487\) −23.3147 −1.05649 −0.528244 0.849092i \(-0.677149\pi\)
−0.528244 + 0.849092i \(0.677149\pi\)
\(488\) −0.204690 −0.00926587
\(489\) 0 0
\(490\) −2.81341 −0.127097
\(491\) 21.8539 0.986253 0.493127 0.869957i \(-0.335854\pi\)
0.493127 + 0.869957i \(0.335854\pi\)
\(492\) 0 0
\(493\) −1.34866 −0.0607405
\(494\) 0.0293079 0.00131863
\(495\) 0 0
\(496\) 30.6532 1.37637
\(497\) −9.61402 −0.431248
\(498\) 0 0
\(499\) 3.66654 0.164137 0.0820685 0.996627i \(-0.473847\pi\)
0.0820685 + 0.996627i \(0.473847\pi\)
\(500\) 29.9228 1.33819
\(501\) 0 0
\(502\) −0.0263384 −0.00117554
\(503\) −1.25493 −0.0559546 −0.0279773 0.999609i \(-0.508907\pi\)
−0.0279773 + 0.999609i \(0.508907\pi\)
\(504\) 0 0
\(505\) −4.74497 −0.211149
\(506\) −0.181620 −0.00807401
\(507\) 0 0
\(508\) −1.48539 −0.0659037
\(509\) −6.93766 −0.307507 −0.153753 0.988109i \(-0.549136\pi\)
−0.153753 + 0.988109i \(0.549136\pi\)
\(510\) 0 0
\(511\) −35.7069 −1.57958
\(512\) 4.07503 0.180093
\(513\) 0 0
\(514\) 0.358242 0.0158014
\(515\) 4.92218 0.216897
\(516\) 0 0
\(517\) −5.37613 −0.236442
\(518\) 1.38458 0.0608352
\(519\) 0 0
\(520\) −0.647583 −0.0283984
\(521\) 18.4786 0.809561 0.404781 0.914414i \(-0.367348\pi\)
0.404781 + 0.914414i \(0.367348\pi\)
\(522\) 0 0
\(523\) 12.3942 0.541961 0.270980 0.962585i \(-0.412652\pi\)
0.270980 + 0.962585i \(0.412652\pi\)
\(524\) −18.8264 −0.822434
\(525\) 0 0
\(526\) 0.776505 0.0338572
\(527\) −15.3225 −0.667458
\(528\) 0 0
\(529\) −10.4198 −0.453036
\(530\) −1.40432 −0.0609997
\(531\) 0 0
\(532\) −6.29644 −0.272985
\(533\) −2.79944 −0.121257
\(534\) 0 0
\(535\) −75.9399 −3.28317
\(536\) 1.27672 0.0551459
\(537\) 0 0
\(538\) −1.36980 −0.0590561
\(539\) −14.6824 −0.632414
\(540\) 0 0
\(541\) 8.95379 0.384953 0.192477 0.981302i \(-0.438348\pi\)
0.192477 + 0.981302i \(0.438348\pi\)
\(542\) 0.153980 0.00661400
\(543\) 0 0
\(544\) −1.22165 −0.0523777
\(545\) 64.4001 2.75860
\(546\) 0 0
\(547\) 1.30109 0.0556308 0.0278154 0.999613i \(-0.491145\pi\)
0.0278154 + 0.999613i \(0.491145\pi\)
\(548\) 28.7261 1.22712
\(549\) 0 0
\(550\) −0.461027 −0.0196582
\(551\) −0.458437 −0.0195301
\(552\) 0 0
\(553\) −9.74136 −0.414245
\(554\) 0.0390007 0.00165698
\(555\) 0 0
\(556\) 30.1507 1.27868
\(557\) 7.18106 0.304271 0.152136 0.988360i \(-0.451385\pi\)
0.152136 + 0.988360i \(0.451385\pi\)
\(558\) 0 0
\(559\) 7.46633 0.315792
\(560\) 69.4255 2.93376
\(561\) 0 0
\(562\) −0.985570 −0.0415738
\(563\) −39.3214 −1.65720 −0.828599 0.559842i \(-0.810862\pi\)
−0.828599 + 0.559842i \(0.810862\pi\)
\(564\) 0 0
\(565\) −39.6471 −1.66797
\(566\) −1.05079 −0.0441682
\(567\) 0 0
\(568\) −0.422618 −0.0177327
\(569\) −31.0189 −1.30038 −0.650189 0.759772i \(-0.725310\pi\)
−0.650189 + 0.759772i \(0.725310\pi\)
\(570\) 0 0
\(571\) −4.12927 −0.172805 −0.0864023 0.996260i \(-0.527537\pi\)
−0.0864023 + 0.996260i \(0.527537\pi\)
\(572\) −1.68866 −0.0706065
\(573\) 0 0
\(574\) −0.789519 −0.0329539
\(575\) 31.9336 1.33173
\(576\) 0 0
\(577\) −12.2320 −0.509226 −0.254613 0.967043i \(-0.581948\pi\)
−0.254613 + 0.967043i \(0.581948\pi\)
\(578\) −0.667394 −0.0277599
\(579\) 0 0
\(580\) 5.06145 0.210165
\(581\) −16.1828 −0.671377
\(582\) 0 0
\(583\) −7.32872 −0.303525
\(584\) −1.56962 −0.0649515
\(585\) 0 0
\(586\) 0.568554 0.0234868
\(587\) −10.0217 −0.413642 −0.206821 0.978379i \(-0.566312\pi\)
−0.206821 + 0.978379i \(0.566312\pi\)
\(588\) 0 0
\(589\) −5.20843 −0.214610
\(590\) 0.122020 0.00502350
\(591\) 0 0
\(592\) −23.1363 −0.950897
\(593\) 7.55520 0.310255 0.155127 0.987894i \(-0.450421\pi\)
0.155127 + 0.987894i \(0.450421\pi\)
\(594\) 0 0
\(595\) −34.7035 −1.42270
\(596\) −32.4036 −1.32730
\(597\) 0 0
\(598\) −0.153549 −0.00627909
\(599\) 35.1654 1.43682 0.718410 0.695620i \(-0.244870\pi\)
0.718410 + 0.695620i \(0.244870\pi\)
\(600\) 0 0
\(601\) 16.4203 0.669799 0.334899 0.942254i \(-0.391298\pi\)
0.334899 + 0.942254i \(0.391298\pi\)
\(602\) 2.10571 0.0858224
\(603\) 0 0
\(604\) 18.3435 0.746386
\(605\) −3.74211 −0.152138
\(606\) 0 0
\(607\) 21.2135 0.861029 0.430515 0.902584i \(-0.358332\pi\)
0.430515 + 0.902584i \(0.358332\pi\)
\(608\) −0.415264 −0.0168412
\(609\) 0 0
\(610\) −0.191618 −0.00775841
\(611\) −4.54520 −0.183879
\(612\) 0 0
\(613\) 3.39536 0.137137 0.0685686 0.997646i \(-0.478157\pi\)
0.0685686 + 0.997646i \(0.478157\pi\)
\(614\) 1.21480 0.0490252
\(615\) 0 0
\(616\) −0.953125 −0.0384025
\(617\) −12.9192 −0.520109 −0.260055 0.965594i \(-0.583741\pi\)
−0.260055 + 0.965594i \(0.583741\pi\)
\(618\) 0 0
\(619\) −42.8280 −1.72140 −0.860700 0.509112i \(-0.829974\pi\)
−0.860700 + 0.509112i \(0.829974\pi\)
\(620\) 57.5046 2.30944
\(621\) 0 0
\(622\) −0.436573 −0.0175050
\(623\) 41.5807 1.66590
\(624\) 0 0
\(625\) 11.0438 0.441752
\(626\) 0.103822 0.00414957
\(627\) 0 0
\(628\) −44.3423 −1.76945
\(629\) 11.5651 0.461130
\(630\) 0 0
\(631\) −35.0931 −1.39703 −0.698516 0.715594i \(-0.746156\pi\)
−0.698516 + 0.715594i \(0.746156\pi\)
\(632\) −0.428216 −0.0170335
\(633\) 0 0
\(634\) −1.63601 −0.0649741
\(635\) −2.78290 −0.110436
\(636\) 0 0
\(637\) −12.4131 −0.491823
\(638\) −0.0346752 −0.00137281
\(639\) 0 0
\(640\) 6.11171 0.241587
\(641\) 13.2012 0.521418 0.260709 0.965417i \(-0.416044\pi\)
0.260709 + 0.965417i \(0.416044\pi\)
\(642\) 0 0
\(643\) 38.7731 1.52906 0.764532 0.644586i \(-0.222970\pi\)
0.764532 + 0.644586i \(0.222970\pi\)
\(644\) 32.9881 1.29991
\(645\) 0 0
\(646\) 0.0690408 0.00271637
\(647\) −8.06386 −0.317023 −0.158512 0.987357i \(-0.550670\pi\)
−0.158512 + 0.987357i \(0.550670\pi\)
\(648\) 0 0
\(649\) 0.636788 0.0249961
\(650\) −0.389770 −0.0152881
\(651\) 0 0
\(652\) 13.7509 0.538527
\(653\) −20.3231 −0.795304 −0.397652 0.917536i \(-0.630175\pi\)
−0.397652 + 0.917536i \(0.630175\pi\)
\(654\) 0 0
\(655\) −35.2714 −1.37817
\(656\) 13.1928 0.515093
\(657\) 0 0
\(658\) −1.28187 −0.0499725
\(659\) 12.8616 0.501015 0.250508 0.968115i \(-0.419403\pi\)
0.250508 + 0.968115i \(0.419403\pi\)
\(660\) 0 0
\(661\) 9.50897 0.369856 0.184928 0.982752i \(-0.440795\pi\)
0.184928 + 0.982752i \(0.440795\pi\)
\(662\) −0.940609 −0.0365578
\(663\) 0 0
\(664\) −0.711373 −0.0276066
\(665\) −11.7964 −0.457446
\(666\) 0 0
\(667\) 2.40183 0.0929991
\(668\) 22.3102 0.863207
\(669\) 0 0
\(670\) 1.19519 0.0461742
\(671\) −1.00000 −0.0386046
\(672\) 0 0
\(673\) −14.0528 −0.541695 −0.270848 0.962622i \(-0.587304\pi\)
−0.270848 + 0.962622i \(0.587304\pi\)
\(674\) −0.435169 −0.0167621
\(675\) 0 0
\(676\) 24.5383 0.943779
\(677\) 21.7382 0.835467 0.417733 0.908570i \(-0.362825\pi\)
0.417733 + 0.908570i \(0.362825\pi\)
\(678\) 0 0
\(679\) 75.2747 2.88878
\(680\) −1.52551 −0.0585008
\(681\) 0 0
\(682\) −0.393955 −0.0150853
\(683\) 31.1910 1.19349 0.596745 0.802431i \(-0.296461\pi\)
0.596745 + 0.802431i \(0.296461\pi\)
\(684\) 0 0
\(685\) 53.8186 2.05630
\(686\) −1.83176 −0.0699371
\(687\) 0 0
\(688\) −35.1863 −1.34147
\(689\) −6.19600 −0.236049
\(690\) 0 0
\(691\) −21.4676 −0.816666 −0.408333 0.912833i \(-0.633890\pi\)
−0.408333 + 0.912833i \(0.633890\pi\)
\(692\) −34.6629 −1.31769
\(693\) 0 0
\(694\) 0.878379 0.0333428
\(695\) 56.4877 2.14270
\(696\) 0 0
\(697\) −6.59465 −0.249790
\(698\) −0.413151 −0.0156380
\(699\) 0 0
\(700\) 83.7372 3.16497
\(701\) −21.0539 −0.795195 −0.397597 0.917560i \(-0.630156\pi\)
−0.397597 + 0.917560i \(0.630156\pi\)
\(702\) 0 0
\(703\) 3.93121 0.148268
\(704\) 7.93714 0.299142
\(705\) 0 0
\(706\) −0.905756 −0.0340886
\(707\) −5.90433 −0.222055
\(708\) 0 0
\(709\) −31.1132 −1.16848 −0.584241 0.811581i \(-0.698608\pi\)
−0.584241 + 0.811581i \(0.698608\pi\)
\(710\) −0.395630 −0.0148477
\(711\) 0 0
\(712\) 1.82783 0.0685007
\(713\) 27.2879 1.02194
\(714\) 0 0
\(715\) −3.16373 −0.118317
\(716\) −21.4629 −0.802107
\(717\) 0 0
\(718\) −0.982689 −0.0366736
\(719\) 5.72958 0.213677 0.106839 0.994276i \(-0.465927\pi\)
0.106839 + 0.994276i \(0.465927\pi\)
\(720\) 0 0
\(721\) 6.12484 0.228101
\(722\) −0.949446 −0.0353347
\(723\) 0 0
\(724\) −12.1631 −0.452039
\(725\) 6.09682 0.226430
\(726\) 0 0
\(727\) 6.70140 0.248541 0.124271 0.992248i \(-0.460341\pi\)
0.124271 + 0.992248i \(0.460341\pi\)
\(728\) −0.805810 −0.0298653
\(729\) 0 0
\(730\) −1.46939 −0.0543845
\(731\) 17.5885 0.650533
\(732\) 0 0
\(733\) 24.5158 0.905513 0.452756 0.891634i \(-0.350441\pi\)
0.452756 + 0.891634i \(0.350441\pi\)
\(734\) 0.335646 0.0123889
\(735\) 0 0
\(736\) 2.17564 0.0801950
\(737\) 6.23733 0.229755
\(738\) 0 0
\(739\) −21.1631 −0.778497 −0.389249 0.921133i \(-0.627265\pi\)
−0.389249 + 0.921133i \(0.627265\pi\)
\(740\) −43.4032 −1.59553
\(741\) 0 0
\(742\) −1.74744 −0.0641506
\(743\) 48.0278 1.76197 0.880985 0.473144i \(-0.156881\pi\)
0.880985 + 0.473144i \(0.156881\pi\)
\(744\) 0 0
\(745\) −60.7085 −2.22419
\(746\) 1.82078 0.0666635
\(747\) 0 0
\(748\) −3.97799 −0.145450
\(749\) −94.4947 −3.45276
\(750\) 0 0
\(751\) 4.94046 0.180280 0.0901400 0.995929i \(-0.471269\pi\)
0.0901400 + 0.995929i \(0.471269\pi\)
\(752\) 21.4200 0.781106
\(753\) 0 0
\(754\) −0.0293158 −0.00106762
\(755\) 34.3667 1.25073
\(756\) 0 0
\(757\) 35.6690 1.29641 0.648207 0.761464i \(-0.275519\pi\)
0.648207 + 0.761464i \(0.275519\pi\)
\(758\) 1.45564 0.0528712
\(759\) 0 0
\(760\) −0.518554 −0.0188099
\(761\) −25.9419 −0.940393 −0.470197 0.882562i \(-0.655817\pi\)
−0.470197 + 0.882562i \(0.655817\pi\)
\(762\) 0 0
\(763\) 80.1352 2.90109
\(764\) 53.4208 1.93270
\(765\) 0 0
\(766\) −1.10621 −0.0399691
\(767\) 0.538366 0.0194393
\(768\) 0 0
\(769\) −53.2246 −1.91933 −0.959664 0.281150i \(-0.909284\pi\)
−0.959664 + 0.281150i \(0.909284\pi\)
\(770\) −0.892259 −0.0321548
\(771\) 0 0
\(772\) −38.0235 −1.36850
\(773\) 20.0661 0.721729 0.360865 0.932618i \(-0.382482\pi\)
0.360865 + 0.932618i \(0.382482\pi\)
\(774\) 0 0
\(775\) 69.2677 2.48817
\(776\) 3.30897 0.118785
\(777\) 0 0
\(778\) 1.04513 0.0374698
\(779\) −2.24166 −0.0803157
\(780\) 0 0
\(781\) −2.06467 −0.0738799
\(782\) −0.361716 −0.0129349
\(783\) 0 0
\(784\) 58.4986 2.08924
\(785\) −83.0756 −2.96510
\(786\) 0 0
\(787\) −20.9934 −0.748336 −0.374168 0.927361i \(-0.622072\pi\)
−0.374168 + 0.927361i \(0.622072\pi\)
\(788\) −50.7969 −1.80957
\(789\) 0 0
\(790\) −0.400870 −0.0142623
\(791\) −49.3342 −1.75412
\(792\) 0 0
\(793\) −0.845440 −0.0300225
\(794\) 0.908491 0.0322411
\(795\) 0 0
\(796\) 54.7561 1.94078
\(797\) −15.0047 −0.531494 −0.265747 0.964043i \(-0.585619\pi\)
−0.265747 + 0.964043i \(0.585619\pi\)
\(798\) 0 0
\(799\) −10.7071 −0.378791
\(800\) 5.52265 0.195255
\(801\) 0 0
\(802\) −0.478847 −0.0169087
\(803\) −7.66830 −0.270608
\(804\) 0 0
\(805\) 61.8035 2.17829
\(806\) −0.333066 −0.0117317
\(807\) 0 0
\(808\) −0.259546 −0.00913079
\(809\) 11.9417 0.419848 0.209924 0.977718i \(-0.432678\pi\)
0.209924 + 0.977718i \(0.432678\pi\)
\(810\) 0 0
\(811\) 28.6992 1.00777 0.503883 0.863772i \(-0.331904\pi\)
0.503883 + 0.863772i \(0.331904\pi\)
\(812\) 6.29814 0.221021
\(813\) 0 0
\(814\) 0.297349 0.0104221
\(815\) 25.7625 0.902420
\(816\) 0 0
\(817\) 5.97868 0.209168
\(818\) 0.396109 0.0138496
\(819\) 0 0
\(820\) 24.7494 0.864287
\(821\) 16.2117 0.565791 0.282896 0.959151i \(-0.408705\pi\)
0.282896 + 0.959151i \(0.408705\pi\)
\(822\) 0 0
\(823\) 9.55261 0.332983 0.166491 0.986043i \(-0.446756\pi\)
0.166491 + 0.986043i \(0.446756\pi\)
\(824\) 0.269239 0.00937937
\(825\) 0 0
\(826\) 0.151834 0.00528298
\(827\) −44.0421 −1.53149 −0.765747 0.643142i \(-0.777631\pi\)
−0.765747 + 0.643142i \(0.777631\pi\)
\(828\) 0 0
\(829\) −36.8291 −1.27913 −0.639564 0.768738i \(-0.720885\pi\)
−0.639564 + 0.768738i \(0.720885\pi\)
\(830\) −0.665945 −0.0231153
\(831\) 0 0
\(832\) 6.71037 0.232640
\(833\) −29.2415 −1.01316
\(834\) 0 0
\(835\) 41.7983 1.44649
\(836\) −1.35220 −0.0467669
\(837\) 0 0
\(838\) −1.45838 −0.0503790
\(839\) 0.737017 0.0254447 0.0127223 0.999919i \(-0.495950\pi\)
0.0127223 + 0.999919i \(0.495950\pi\)
\(840\) 0 0
\(841\) −28.5414 −0.984188
\(842\) 0.799927 0.0275673
\(843\) 0 0
\(844\) −30.6311 −1.05436
\(845\) 45.9727 1.58151
\(846\) 0 0
\(847\) −4.65643 −0.159997
\(848\) 29.1996 1.00272
\(849\) 0 0
\(850\) −0.918184 −0.0314934
\(851\) −20.5963 −0.706031
\(852\) 0 0
\(853\) −5.92173 −0.202756 −0.101378 0.994848i \(-0.532325\pi\)
−0.101378 + 0.994848i \(0.532325\pi\)
\(854\) −0.238437 −0.00815916
\(855\) 0 0
\(856\) −4.15384 −0.141975
\(857\) −27.3082 −0.932831 −0.466415 0.884566i \(-0.654455\pi\)
−0.466415 + 0.884566i \(0.654455\pi\)
\(858\) 0 0
\(859\) −9.24564 −0.315457 −0.157729 0.987483i \(-0.550417\pi\)
−0.157729 + 0.987483i \(0.550417\pi\)
\(860\) −66.0087 −2.25088
\(861\) 0 0
\(862\) 0.857663 0.0292121
\(863\) 14.7556 0.502288 0.251144 0.967950i \(-0.419193\pi\)
0.251144 + 0.967950i \(0.419193\pi\)
\(864\) 0 0
\(865\) −64.9413 −2.20807
\(866\) −0.396146 −0.0134616
\(867\) 0 0
\(868\) 71.5549 2.42873
\(869\) −2.09202 −0.0709670
\(870\) 0 0
\(871\) 5.27329 0.178679
\(872\) 3.52263 0.119291
\(873\) 0 0
\(874\) −0.122955 −0.00415901
\(875\) 69.7582 2.35826
\(876\) 0 0
\(877\) −52.2504 −1.76437 −0.882185 0.470903i \(-0.843928\pi\)
−0.882185 + 0.470903i \(0.843928\pi\)
\(878\) 1.63109 0.0550466
\(879\) 0 0
\(880\) 14.9096 0.502602
\(881\) −26.3428 −0.887510 −0.443755 0.896148i \(-0.646354\pi\)
−0.443755 + 0.896148i \(0.646354\pi\)
\(882\) 0 0
\(883\) 45.2865 1.52401 0.762006 0.647570i \(-0.224215\pi\)
0.762006 + 0.647570i \(0.224215\pi\)
\(884\) −3.36315 −0.113115
\(885\) 0 0
\(886\) 0.283518 0.00952498
\(887\) 10.1077 0.339384 0.169692 0.985497i \(-0.445723\pi\)
0.169692 + 0.985497i \(0.445723\pi\)
\(888\) 0 0
\(889\) −3.46286 −0.116141
\(890\) 1.71110 0.0573563
\(891\) 0 0
\(892\) −41.6281 −1.39381
\(893\) −3.63958 −0.121794
\(894\) 0 0
\(895\) −40.2110 −1.34411
\(896\) 7.60501 0.254066
\(897\) 0 0
\(898\) 1.16789 0.0389729
\(899\) 5.20984 0.173758
\(900\) 0 0
\(901\) −14.5959 −0.486261
\(902\) −0.169554 −0.00564555
\(903\) 0 0
\(904\) −2.16866 −0.0721285
\(905\) −22.7877 −0.757490
\(906\) 0 0
\(907\) −29.5028 −0.979625 −0.489813 0.871828i \(-0.662935\pi\)
−0.489813 + 0.871828i \(0.662935\pi\)
\(908\) −24.8192 −0.823654
\(909\) 0 0
\(910\) −0.754351 −0.0250065
\(911\) −54.3951 −1.80219 −0.901095 0.433623i \(-0.857235\pi\)
−0.901095 + 0.433623i \(0.857235\pi\)
\(912\) 0 0
\(913\) −3.47537 −0.115018
\(914\) −1.28717 −0.0425759
\(915\) 0 0
\(916\) 15.0318 0.496665
\(917\) −43.8894 −1.44936
\(918\) 0 0
\(919\) −41.6870 −1.37513 −0.687563 0.726125i \(-0.741319\pi\)
−0.687563 + 0.726125i \(0.741319\pi\)
\(920\) 2.71679 0.0895699
\(921\) 0 0
\(922\) 0.0944822 0.00311161
\(923\) −1.74556 −0.0574558
\(924\) 0 0
\(925\) −52.2817 −1.71901
\(926\) 1.38966 0.0456669
\(927\) 0 0
\(928\) 0.415376 0.0136354
\(929\) −12.8663 −0.422130 −0.211065 0.977472i \(-0.567693\pi\)
−0.211065 + 0.977472i \(0.567693\pi\)
\(930\) 0 0
\(931\) −9.93979 −0.325764
\(932\) 7.96652 0.260952
\(933\) 0 0
\(934\) 2.05763 0.0673277
\(935\) −7.45280 −0.243733
\(936\) 0 0
\(937\) −43.3290 −1.41550 −0.707748 0.706465i \(-0.750289\pi\)
−0.707748 + 0.706465i \(0.750289\pi\)
\(938\) 1.48721 0.0485593
\(939\) 0 0
\(940\) 40.1834 1.31064
\(941\) 53.7199 1.75122 0.875609 0.483020i \(-0.160460\pi\)
0.875609 + 0.483020i \(0.160460\pi\)
\(942\) 0 0
\(943\) 11.7444 0.382451
\(944\) −2.53714 −0.0825768
\(945\) 0 0
\(946\) 0.452216 0.0147028
\(947\) −6.94150 −0.225568 −0.112784 0.993620i \(-0.535977\pi\)
−0.112784 + 0.993620i \(0.535977\pi\)
\(948\) 0 0
\(949\) −6.48309 −0.210450
\(950\) −0.312110 −0.0101262
\(951\) 0 0
\(952\) −1.89825 −0.0615226
\(953\) 45.7414 1.48171 0.740855 0.671665i \(-0.234421\pi\)
0.740855 + 0.671665i \(0.234421\pi\)
\(954\) 0 0
\(955\) 100.084 3.23865
\(956\) −15.1854 −0.491132
\(957\) 0 0
\(958\) −1.80937 −0.0584582
\(959\) 66.9684 2.16252
\(960\) 0 0
\(961\) 28.1905 0.909371
\(962\) 0.251390 0.00810515
\(963\) 0 0
\(964\) −5.34624 −0.172191
\(965\) −71.2375 −2.29322
\(966\) 0 0
\(967\) 40.3862 1.29873 0.649366 0.760476i \(-0.275034\pi\)
0.649366 + 0.760476i \(0.275034\pi\)
\(968\) −0.204690 −0.00657898
\(969\) 0 0
\(970\) 3.09766 0.0994598
\(971\) 17.8315 0.572240 0.286120 0.958194i \(-0.407634\pi\)
0.286120 + 0.958194i \(0.407634\pi\)
\(972\) 0 0
\(973\) 70.2896 2.25338
\(974\) −1.19385 −0.0382535
\(975\) 0 0
\(976\) 3.98427 0.127534
\(977\) 14.1563 0.452899 0.226449 0.974023i \(-0.427288\pi\)
0.226449 + 0.974023i \(0.427288\pi\)
\(978\) 0 0
\(979\) 8.92973 0.285396
\(980\) 109.742 3.50558
\(981\) 0 0
\(982\) 1.11905 0.0357104
\(983\) −35.7906 −1.14154 −0.570772 0.821109i \(-0.693356\pi\)
−0.570772 + 0.821109i \(0.693356\pi\)
\(984\) 0 0
\(985\) −95.1685 −3.03232
\(986\) −0.0690594 −0.00219930
\(987\) 0 0
\(988\) −1.14321 −0.0363702
\(989\) −31.3233 −0.996024
\(990\) 0 0
\(991\) −9.02047 −0.286545 −0.143272 0.989683i \(-0.545762\pi\)
−0.143272 + 0.989683i \(0.545762\pi\)
\(992\) 4.71920 0.149835
\(993\) 0 0
\(994\) −0.492296 −0.0156147
\(995\) 102.586 3.25220
\(996\) 0 0
\(997\) 49.2128 1.55858 0.779292 0.626661i \(-0.215579\pi\)
0.779292 + 0.626661i \(0.215579\pi\)
\(998\) 0.187749 0.00594309
\(999\) 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.c.1.5 11
3.2 odd 2 2013.2.a.b.1.7 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.b.1.7 11 3.2 odd 2
6039.2.a.c.1.5 11 1.1 even 1 trivial