Properties

Label 1143.2.a.e.1.1
Level $1143$
Weight $2$
Character 1143.1
Self dual yes
Analytic conductor $9.127$
Analytic rank $0$
Dimension $3$
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] = [1143,2,Mod(1,1143)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1143, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1143.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1143.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: \(9.12690095103\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 127)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 1143.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.879385 q^{2} -1.22668 q^{4} +2.34730 q^{5} -1.34730 q^{7} +2.83750 q^{8} +O(q^{10})\) \(q-0.879385 q^{2} -1.22668 q^{4} +2.34730 q^{5} -1.34730 q^{7} +2.83750 q^{8} -2.06418 q^{10} -5.29086 q^{11} -1.83750 q^{13} +1.18479 q^{14} -0.0418891 q^{16} +7.87939 q^{17} +0.652704 q^{19} -2.87939 q^{20} +4.65270 q^{22} +4.18479 q^{23} +0.509800 q^{25} +1.61587 q^{26} +1.65270 q^{28} +3.10607 q^{29} +1.28312 q^{31} -5.63816 q^{32} -6.92902 q^{34} -3.16250 q^{35} +5.14796 q^{37} -0.573978 q^{38} +6.66044 q^{40} +10.1284 q^{41} -5.08378 q^{43} +6.49020 q^{44} -3.68004 q^{46} -9.58172 q^{47} -5.18479 q^{49} -0.448311 q^{50} +2.25402 q^{52} -0.448311 q^{53} -12.4192 q^{55} -3.82295 q^{56} -2.73143 q^{58} +5.29086 q^{59} +11.9513 q^{61} -1.12836 q^{62} +5.04189 q^{64} -4.31315 q^{65} -0.652704 q^{67} -9.66550 q^{68} +2.78106 q^{70} +13.3405 q^{71} +2.32770 q^{73} -4.52704 q^{74} -0.800660 q^{76} +7.12836 q^{77} +16.1557 q^{79} -0.0983261 q^{80} -8.90673 q^{82} +14.9368 q^{83} +18.4953 q^{85} +4.47060 q^{86} -15.0128 q^{88} +17.3327 q^{89} +2.47565 q^{91} -5.13341 q^{92} +8.42602 q^{94} +1.53209 q^{95} -1.44562 q^{97} +4.55943 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8} + 3 q^{10} - 3 q^{13} + 3 q^{16} + 18 q^{17} + 3 q^{19} - 3 q^{20} + 15 q^{22} + 9 q^{23} + 3 q^{25} - 6 q^{26} + 6 q^{28} - 3 q^{29} + 12 q^{31} + 12 q^{34} - 12 q^{35} + 6 q^{38} - 3 q^{40} + 12 q^{41} - 9 q^{43} + 18 q^{44} + 9 q^{46} + 3 q^{47} - 12 q^{49} - 3 q^{50} - 21 q^{52} - 3 q^{53} - 3 q^{55} + 9 q^{56} - 18 q^{58} - 3 q^{61} + 15 q^{62} + 12 q^{64} + 9 q^{65} - 3 q^{67} + 9 q^{68} - 9 q^{70} - 3 q^{71} + 3 q^{73} - 24 q^{74} + 12 q^{76} + 3 q^{77} + 9 q^{79} - 12 q^{80} - 12 q^{83} + 39 q^{85} + 9 q^{86} - 6 q^{88} + 33 q^{89} - 12 q^{91} + 18 q^{92} + 33 q^{94} - 15 q^{97} - 12 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.879385 −0.621819 −0.310910 0.950439i \(-0.600634\pi\)
−0.310910 + 0.950439i \(0.600634\pi\)
\(3\) 0 0
\(4\) −1.22668 −0.613341
\(5\) 2.34730 1.04974 0.524871 0.851182i \(-0.324113\pi\)
0.524871 + 0.851182i \(0.324113\pi\)
\(6\) 0 0
\(7\) −1.34730 −0.509230 −0.254615 0.967042i \(-0.581949\pi\)
−0.254615 + 0.967042i \(0.581949\pi\)
\(8\) 2.83750 1.00321
\(9\) 0 0
\(10\) −2.06418 −0.652750
\(11\) −5.29086 −1.59525 −0.797627 0.603151i \(-0.793912\pi\)
−0.797627 + 0.603151i \(0.793912\pi\)
\(12\) 0 0
\(13\) −1.83750 −0.509630 −0.254815 0.966990i \(-0.582015\pi\)
−0.254815 + 0.966990i \(0.582015\pi\)
\(14\) 1.18479 0.316649
\(15\) 0 0
\(16\) −0.0418891 −0.0104723
\(17\) 7.87939 1.91103 0.955516 0.294940i \(-0.0952996\pi\)
0.955516 + 0.294940i \(0.0952996\pi\)
\(18\) 0 0
\(19\) 0.652704 0.149740 0.0748702 0.997193i \(-0.476146\pi\)
0.0748702 + 0.997193i \(0.476146\pi\)
\(20\) −2.87939 −0.643850
\(21\) 0 0
\(22\) 4.65270 0.991960
\(23\) 4.18479 0.872590 0.436295 0.899804i \(-0.356290\pi\)
0.436295 + 0.899804i \(0.356290\pi\)
\(24\) 0 0
\(25\) 0.509800 0.101960
\(26\) 1.61587 0.316898
\(27\) 0 0
\(28\) 1.65270 0.312332
\(29\) 3.10607 0.576782 0.288391 0.957513i \(-0.406880\pi\)
0.288391 + 0.957513i \(0.406880\pi\)
\(30\) 0 0
\(31\) 1.28312 0.230455 0.115227 0.993339i \(-0.463240\pi\)
0.115227 + 0.993339i \(0.463240\pi\)
\(32\) −5.63816 −0.996695
\(33\) 0 0
\(34\) −6.92902 −1.18832
\(35\) −3.16250 −0.534561
\(36\) 0 0
\(37\) 5.14796 0.846319 0.423159 0.906055i \(-0.360921\pi\)
0.423159 + 0.906055i \(0.360921\pi\)
\(38\) −0.573978 −0.0931115
\(39\) 0 0
\(40\) 6.66044 1.05311
\(41\) 10.1284 1.58178 0.790892 0.611956i \(-0.209617\pi\)
0.790892 + 0.611956i \(0.209617\pi\)
\(42\) 0 0
\(43\) −5.08378 −0.775269 −0.387634 0.921813i \(-0.626708\pi\)
−0.387634 + 0.921813i \(0.626708\pi\)
\(44\) 6.49020 0.978434
\(45\) 0 0
\(46\) −3.68004 −0.542593
\(47\) −9.58172 −1.39764 −0.698819 0.715299i \(-0.746291\pi\)
−0.698819 + 0.715299i \(0.746291\pi\)
\(48\) 0 0
\(49\) −5.18479 −0.740685
\(50\) −0.448311 −0.0634007
\(51\) 0 0
\(52\) 2.25402 0.312577
\(53\) −0.448311 −0.0615802 −0.0307901 0.999526i \(-0.509802\pi\)
−0.0307901 + 0.999526i \(0.509802\pi\)
\(54\) 0 0
\(55\) −12.4192 −1.67461
\(56\) −3.82295 −0.510863
\(57\) 0 0
\(58\) −2.73143 −0.358654
\(59\) 5.29086 0.688811 0.344406 0.938821i \(-0.388080\pi\)
0.344406 + 0.938821i \(0.388080\pi\)
\(60\) 0 0
\(61\) 11.9513 1.53021 0.765104 0.643907i \(-0.222688\pi\)
0.765104 + 0.643907i \(0.222688\pi\)
\(62\) −1.12836 −0.143301
\(63\) 0 0
\(64\) 5.04189 0.630236
\(65\) −4.31315 −0.534980
\(66\) 0 0
\(67\) −0.652704 −0.0797404 −0.0398702 0.999205i \(-0.512694\pi\)
−0.0398702 + 0.999205i \(0.512694\pi\)
\(68\) −9.66550 −1.17211
\(69\) 0 0
\(70\) 2.78106 0.332400
\(71\) 13.3405 1.58322 0.791612 0.611024i \(-0.209242\pi\)
0.791612 + 0.611024i \(0.209242\pi\)
\(72\) 0 0
\(73\) 2.32770 0.272436 0.136218 0.990679i \(-0.456505\pi\)
0.136218 + 0.990679i \(0.456505\pi\)
\(74\) −4.52704 −0.526257
\(75\) 0 0
\(76\) −0.800660 −0.0918419
\(77\) 7.12836 0.812352
\(78\) 0 0
\(79\) 16.1557 1.81766 0.908829 0.417169i \(-0.136978\pi\)
0.908829 + 0.417169i \(0.136978\pi\)
\(80\) −0.0983261 −0.0109932
\(81\) 0 0
\(82\) −8.90673 −0.983583
\(83\) 14.9368 1.63952 0.819761 0.572706i \(-0.194106\pi\)
0.819761 + 0.572706i \(0.194106\pi\)
\(84\) 0 0
\(85\) 18.4953 2.00609
\(86\) 4.47060 0.482077
\(87\) 0 0
\(88\) −15.0128 −1.60037
\(89\) 17.3327 1.83727 0.918634 0.395110i \(-0.129294\pi\)
0.918634 + 0.395110i \(0.129294\pi\)
\(90\) 0 0
\(91\) 2.47565 0.259519
\(92\) −5.13341 −0.535195
\(93\) 0 0
\(94\) 8.42602 0.869078
\(95\) 1.53209 0.157189
\(96\) 0 0
\(97\) −1.44562 −0.146781 −0.0733904 0.997303i \(-0.523382\pi\)
−0.0733904 + 0.997303i \(0.523382\pi\)
\(98\) 4.55943 0.460572
\(99\) 0 0
\(100\) −0.625362 −0.0625362
\(101\) −7.15570 −0.712018 −0.356009 0.934482i \(-0.615863\pi\)
−0.356009 + 0.934482i \(0.615863\pi\)
\(102\) 0 0
\(103\) 5.94356 0.585637 0.292818 0.956168i \(-0.405407\pi\)
0.292818 + 0.956168i \(0.405407\pi\)
\(104\) −5.21389 −0.511264
\(105\) 0 0
\(106\) 0.394238 0.0382918
\(107\) −2.01455 −0.194754 −0.0973768 0.995248i \(-0.531045\pi\)
−0.0973768 + 0.995248i \(0.531045\pi\)
\(108\) 0 0
\(109\) 0.177052 0.0169585 0.00847924 0.999964i \(-0.497301\pi\)
0.00847924 + 0.999964i \(0.497301\pi\)
\(110\) 10.9213 1.04130
\(111\) 0 0
\(112\) 0.0564370 0.00533279
\(113\) −3.70233 −0.348286 −0.174143 0.984720i \(-0.555716\pi\)
−0.174143 + 0.984720i \(0.555716\pi\)
\(114\) 0 0
\(115\) 9.82295 0.915995
\(116\) −3.81016 −0.353764
\(117\) 0 0
\(118\) −4.65270 −0.428316
\(119\) −10.6159 −0.973155
\(120\) 0 0
\(121\) 16.9932 1.54484
\(122\) −10.5098 −0.951513
\(123\) 0 0
\(124\) −1.57398 −0.141347
\(125\) −10.5398 −0.942711
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 6.84255 0.604802
\(129\) 0 0
\(130\) 3.79292 0.332661
\(131\) −15.2395 −1.33148 −0.665740 0.746184i \(-0.731884\pi\)
−0.665740 + 0.746184i \(0.731884\pi\)
\(132\) 0 0
\(133\) −0.879385 −0.0762524
\(134\) 0.573978 0.0495841
\(135\) 0 0
\(136\) 22.3577 1.91716
\(137\) 11.0000 0.939793 0.469897 0.882721i \(-0.344291\pi\)
0.469897 + 0.882721i \(0.344291\pi\)
\(138\) 0 0
\(139\) −0.221629 −0.0187983 −0.00939917 0.999956i \(-0.502992\pi\)
−0.00939917 + 0.999956i \(0.502992\pi\)
\(140\) 3.87939 0.327868
\(141\) 0 0
\(142\) −11.7314 −0.984480
\(143\) 9.72193 0.812989
\(144\) 0 0
\(145\) 7.29086 0.605473
\(146\) −2.04694 −0.169406
\(147\) 0 0
\(148\) −6.31490 −0.519082
\(149\) −2.19160 −0.179543 −0.0897714 0.995962i \(-0.528614\pi\)
−0.0897714 + 0.995962i \(0.528614\pi\)
\(150\) 0 0
\(151\) −12.5621 −1.02229 −0.511145 0.859494i \(-0.670779\pi\)
−0.511145 + 0.859494i \(0.670779\pi\)
\(152\) 1.85204 0.150221
\(153\) 0 0
\(154\) −6.26857 −0.505136
\(155\) 3.01186 0.241918
\(156\) 0 0
\(157\) −9.32770 −0.744431 −0.372216 0.928146i \(-0.621402\pi\)
−0.372216 + 0.928146i \(0.621402\pi\)
\(158\) −14.2071 −1.13025
\(159\) 0 0
\(160\) −13.2344 −1.04627
\(161\) −5.63816 −0.444349
\(162\) 0 0
\(163\) −10.8229 −0.847719 −0.423859 0.905728i \(-0.639325\pi\)
−0.423859 + 0.905728i \(0.639325\pi\)
\(164\) −12.4243 −0.970172
\(165\) 0 0
\(166\) −13.1352 −1.01949
\(167\) −1.36959 −0.105982 −0.0529908 0.998595i \(-0.516875\pi\)
−0.0529908 + 0.998595i \(0.516875\pi\)
\(168\) 0 0
\(169\) −9.62361 −0.740278
\(170\) −16.2645 −1.24743
\(171\) 0 0
\(172\) 6.23618 0.475504
\(173\) 16.6263 1.26407 0.632037 0.774938i \(-0.282219\pi\)
0.632037 + 0.774938i \(0.282219\pi\)
\(174\) 0 0
\(175\) −0.686852 −0.0519211
\(176\) 0.221629 0.0167059
\(177\) 0 0
\(178\) −15.2422 −1.14245
\(179\) −6.96316 −0.520451 −0.260226 0.965548i \(-0.583797\pi\)
−0.260226 + 0.965548i \(0.583797\pi\)
\(180\) 0 0
\(181\) −2.49020 −0.185095 −0.0925475 0.995708i \(-0.529501\pi\)
−0.0925475 + 0.995708i \(0.529501\pi\)
\(182\) −2.17705 −0.161374
\(183\) 0 0
\(184\) 11.8743 0.875387
\(185\) 12.0838 0.888417
\(186\) 0 0
\(187\) −41.6887 −3.04858
\(188\) 11.7537 0.857228
\(189\) 0 0
\(190\) −1.34730 −0.0977432
\(191\) 13.5398 0.979708 0.489854 0.871805i \(-0.337050\pi\)
0.489854 + 0.871805i \(0.337050\pi\)
\(192\) 0 0
\(193\) 21.7888 1.56839 0.784196 0.620513i \(-0.213076\pi\)
0.784196 + 0.620513i \(0.213076\pi\)
\(194\) 1.27126 0.0912711
\(195\) 0 0
\(196\) 6.36009 0.454292
\(197\) 5.07192 0.361359 0.180680 0.983542i \(-0.442170\pi\)
0.180680 + 0.983542i \(0.442170\pi\)
\(198\) 0 0
\(199\) −7.08378 −0.502156 −0.251078 0.967967i \(-0.580785\pi\)
−0.251078 + 0.967967i \(0.580785\pi\)
\(200\) 1.44656 0.102287
\(201\) 0 0
\(202\) 6.29261 0.442747
\(203\) −4.18479 −0.293715
\(204\) 0 0
\(205\) 23.7743 1.66047
\(206\) −5.22668 −0.364160
\(207\) 0 0
\(208\) 0.0769710 0.00533698
\(209\) −3.45336 −0.238874
\(210\) 0 0
\(211\) −3.67230 −0.252812 −0.126406 0.991979i \(-0.540344\pi\)
−0.126406 + 0.991979i \(0.540344\pi\)
\(212\) 0.549935 0.0377697
\(213\) 0 0
\(214\) 1.77156 0.121102
\(215\) −11.9331 −0.813833
\(216\) 0 0
\(217\) −1.72874 −0.117355
\(218\) −0.155697 −0.0105451
\(219\) 0 0
\(220\) 15.2344 1.02710
\(221\) −14.4783 −0.973919
\(222\) 0 0
\(223\) 24.8753 1.66577 0.832886 0.553445i \(-0.186687\pi\)
0.832886 + 0.553445i \(0.186687\pi\)
\(224\) 7.59627 0.507547
\(225\) 0 0
\(226\) 3.25578 0.216571
\(227\) 10.5321 0.699039 0.349520 0.936929i \(-0.386345\pi\)
0.349520 + 0.936929i \(0.386345\pi\)
\(228\) 0 0
\(229\) −7.12061 −0.470543 −0.235272 0.971930i \(-0.575598\pi\)
−0.235272 + 0.971930i \(0.575598\pi\)
\(230\) −8.63816 −0.569583
\(231\) 0 0
\(232\) 8.81345 0.578632
\(233\) −3.23442 −0.211894 −0.105947 0.994372i \(-0.533787\pi\)
−0.105947 + 0.994372i \(0.533787\pi\)
\(234\) 0 0
\(235\) −22.4911 −1.46716
\(236\) −6.49020 −0.422476
\(237\) 0 0
\(238\) 9.33544 0.605126
\(239\) 8.00505 0.517804 0.258902 0.965904i \(-0.416639\pi\)
0.258902 + 0.965904i \(0.416639\pi\)
\(240\) 0 0
\(241\) −28.5749 −1.84067 −0.920336 0.391129i \(-0.872085\pi\)
−0.920336 + 0.391129i \(0.872085\pi\)
\(242\) −14.9436 −0.960609
\(243\) 0 0
\(244\) −14.6604 −0.938539
\(245\) −12.1702 −0.777528
\(246\) 0 0
\(247\) −1.19934 −0.0763122
\(248\) 3.64084 0.231194
\(249\) 0 0
\(250\) 9.26857 0.586196
\(251\) −19.2540 −1.21530 −0.607652 0.794204i \(-0.707888\pi\)
−0.607652 + 0.794204i \(0.707888\pi\)
\(252\) 0 0
\(253\) −22.1411 −1.39200
\(254\) 0.879385 0.0551775
\(255\) 0 0
\(256\) −16.1010 −1.00631
\(257\) 20.1780 1.25867 0.629334 0.777135i \(-0.283328\pi\)
0.629334 + 0.777135i \(0.283328\pi\)
\(258\) 0 0
\(259\) −6.93582 −0.430971
\(260\) 5.29086 0.328125
\(261\) 0 0
\(262\) 13.4014 0.827939
\(263\) −7.52940 −0.464283 −0.232141 0.972682i \(-0.574573\pi\)
−0.232141 + 0.972682i \(0.574573\pi\)
\(264\) 0 0
\(265\) −1.05232 −0.0646434
\(266\) 0.773318 0.0474152
\(267\) 0 0
\(268\) 0.800660 0.0489081
\(269\) 13.3405 0.813384 0.406692 0.913565i \(-0.366682\pi\)
0.406692 + 0.913565i \(0.366682\pi\)
\(270\) 0 0
\(271\) −26.5226 −1.61113 −0.805566 0.592505i \(-0.798139\pi\)
−0.805566 + 0.592505i \(0.798139\pi\)
\(272\) −0.330060 −0.0200128
\(273\) 0 0
\(274\) −9.67324 −0.584382
\(275\) −2.69728 −0.162652
\(276\) 0 0
\(277\) −8.82800 −0.530423 −0.265212 0.964190i \(-0.585442\pi\)
−0.265212 + 0.964190i \(0.585442\pi\)
\(278\) 0.194897 0.0116892
\(279\) 0 0
\(280\) −8.97359 −0.536275
\(281\) −17.8922 −1.06736 −0.533679 0.845687i \(-0.679191\pi\)
−0.533679 + 0.845687i \(0.679191\pi\)
\(282\) 0 0
\(283\) 12.4483 0.739975 0.369988 0.929037i \(-0.379362\pi\)
0.369988 + 0.929037i \(0.379362\pi\)
\(284\) −16.3645 −0.971056
\(285\) 0 0
\(286\) −8.54933 −0.505532
\(287\) −13.6459 −0.805492
\(288\) 0 0
\(289\) 45.0847 2.65204
\(290\) −6.41147 −0.376495
\(291\) 0 0
\(292\) −2.85534 −0.167096
\(293\) −32.8316 −1.91804 −0.959022 0.283332i \(-0.908560\pi\)
−0.959022 + 0.283332i \(0.908560\pi\)
\(294\) 0 0
\(295\) 12.4192 0.723075
\(296\) 14.6073 0.849032
\(297\) 0 0
\(298\) 1.92726 0.111643
\(299\) −7.68954 −0.444698
\(300\) 0 0
\(301\) 6.84936 0.394790
\(302\) 11.0469 0.635680
\(303\) 0 0
\(304\) −0.0273411 −0.00156812
\(305\) 28.0533 1.60632
\(306\) 0 0
\(307\) −16.5125 −0.942417 −0.471209 0.882022i \(-0.656182\pi\)
−0.471209 + 0.882022i \(0.656182\pi\)
\(308\) −8.74422 −0.498248
\(309\) 0 0
\(310\) −2.64858 −0.150430
\(311\) −13.9290 −0.789842 −0.394921 0.918715i \(-0.629228\pi\)
−0.394921 + 0.918715i \(0.629228\pi\)
\(312\) 0 0
\(313\) −10.1797 −0.575393 −0.287696 0.957722i \(-0.592889\pi\)
−0.287696 + 0.957722i \(0.592889\pi\)
\(314\) 8.20264 0.462902
\(315\) 0 0
\(316\) −19.8179 −1.11484
\(317\) −19.9864 −1.12255 −0.561274 0.827630i \(-0.689688\pi\)
−0.561274 + 0.827630i \(0.689688\pi\)
\(318\) 0 0
\(319\) −16.4338 −0.920114
\(320\) 11.8348 0.661586
\(321\) 0 0
\(322\) 4.95811 0.276305
\(323\) 5.14290 0.286159
\(324\) 0 0
\(325\) −0.936756 −0.0519619
\(326\) 9.51754 0.527128
\(327\) 0 0
\(328\) 28.7392 1.58686
\(329\) 12.9094 0.711719
\(330\) 0 0
\(331\) 6.70140 0.368342 0.184171 0.982894i \(-0.441040\pi\)
0.184171 + 0.982894i \(0.441040\pi\)
\(332\) −18.3226 −1.00559
\(333\) 0 0
\(334\) 1.20439 0.0659014
\(335\) −1.53209 −0.0837070
\(336\) 0 0
\(337\) 0.524348 0.0285631 0.0142815 0.999898i \(-0.495454\pi\)
0.0142815 + 0.999898i \(0.495454\pi\)
\(338\) 8.46286 0.460319
\(339\) 0 0
\(340\) −22.6878 −1.23042
\(341\) −6.78880 −0.367634
\(342\) 0 0
\(343\) 16.4165 0.886409
\(344\) −14.4252 −0.777755
\(345\) 0 0
\(346\) −14.6209 −0.786025
\(347\) 19.4037 1.04165 0.520823 0.853664i \(-0.325625\pi\)
0.520823 + 0.853664i \(0.325625\pi\)
\(348\) 0 0
\(349\) −20.5645 −1.10079 −0.550396 0.834904i \(-0.685523\pi\)
−0.550396 + 0.834904i \(0.685523\pi\)
\(350\) 0.604007 0.0322856
\(351\) 0 0
\(352\) 29.8307 1.58998
\(353\) −28.0925 −1.49521 −0.747605 0.664143i \(-0.768796\pi\)
−0.747605 + 0.664143i \(0.768796\pi\)
\(354\) 0 0
\(355\) 31.3141 1.66198
\(356\) −21.2618 −1.12687
\(357\) 0 0
\(358\) 6.12330 0.323627
\(359\) −2.94356 −0.155355 −0.0776777 0.996979i \(-0.524750\pi\)
−0.0776777 + 0.996979i \(0.524750\pi\)
\(360\) 0 0
\(361\) −18.5740 −0.977578
\(362\) 2.18984 0.115096
\(363\) 0 0
\(364\) −3.03684 −0.159173
\(365\) 5.46379 0.285988
\(366\) 0 0
\(367\) 18.2668 0.953520 0.476760 0.879033i \(-0.341811\pi\)
0.476760 + 0.879033i \(0.341811\pi\)
\(368\) −0.175297 −0.00913799
\(369\) 0 0
\(370\) −10.6263 −0.552435
\(371\) 0.604007 0.0313585
\(372\) 0 0
\(373\) 6.05138 0.313329 0.156664 0.987652i \(-0.449926\pi\)
0.156664 + 0.987652i \(0.449926\pi\)
\(374\) 36.6604 1.89567
\(375\) 0 0
\(376\) −27.1881 −1.40212
\(377\) −5.70739 −0.293945
\(378\) 0 0
\(379\) 13.5449 0.695754 0.347877 0.937540i \(-0.386903\pi\)
0.347877 + 0.937540i \(0.386903\pi\)
\(380\) −1.87939 −0.0964104
\(381\) 0 0
\(382\) −11.9067 −0.609201
\(383\) 15.9504 0.815026 0.407513 0.913199i \(-0.366396\pi\)
0.407513 + 0.913199i \(0.366396\pi\)
\(384\) 0 0
\(385\) 16.7324 0.852760
\(386\) −19.1607 −0.975256
\(387\) 0 0
\(388\) 1.77332 0.0900266
\(389\) 28.5226 1.44615 0.723076 0.690768i \(-0.242727\pi\)
0.723076 + 0.690768i \(0.242727\pi\)
\(390\) 0 0
\(391\) 32.9736 1.66755
\(392\) −14.7118 −0.743060
\(393\) 0 0
\(394\) −4.46017 −0.224700
\(395\) 37.9222 1.90807
\(396\) 0 0
\(397\) 14.3182 0.718610 0.359305 0.933220i \(-0.383014\pi\)
0.359305 + 0.933220i \(0.383014\pi\)
\(398\) 6.22937 0.312250
\(399\) 0 0
\(400\) −0.0213551 −0.00106775
\(401\) −0.709141 −0.0354128 −0.0177064 0.999843i \(-0.505636\pi\)
−0.0177064 + 0.999843i \(0.505636\pi\)
\(402\) 0 0
\(403\) −2.35773 −0.117447
\(404\) 8.77776 0.436710
\(405\) 0 0
\(406\) 3.68004 0.182638
\(407\) −27.2371 −1.35009
\(408\) 0 0
\(409\) 23.2472 1.14950 0.574750 0.818329i \(-0.305099\pi\)
0.574750 + 0.818329i \(0.305099\pi\)
\(410\) −20.9067 −1.03251
\(411\) 0 0
\(412\) −7.29086 −0.359195
\(413\) −7.12836 −0.350763
\(414\) 0 0
\(415\) 35.0610 1.72108
\(416\) 10.3601 0.507945
\(417\) 0 0
\(418\) 3.03684 0.148537
\(419\) −2.32089 −0.113383 −0.0566914 0.998392i \(-0.518055\pi\)
−0.0566914 + 0.998392i \(0.518055\pi\)
\(420\) 0 0
\(421\) −23.1266 −1.12712 −0.563561 0.826075i \(-0.690569\pi\)
−0.563561 + 0.826075i \(0.690569\pi\)
\(422\) 3.22937 0.157203
\(423\) 0 0
\(424\) −1.27208 −0.0617777
\(425\) 4.01691 0.194849
\(426\) 0 0
\(427\) −16.1019 −0.779228
\(428\) 2.47121 0.119450
\(429\) 0 0
\(430\) 10.4938 0.506057
\(431\) −13.3105 −0.641142 −0.320571 0.947224i \(-0.603875\pi\)
−0.320571 + 0.947224i \(0.603875\pi\)
\(432\) 0 0
\(433\) −8.46522 −0.406813 −0.203406 0.979094i \(-0.565201\pi\)
−0.203406 + 0.979094i \(0.565201\pi\)
\(434\) 1.52023 0.0729733
\(435\) 0 0
\(436\) −0.217186 −0.0104013
\(437\) 2.73143 0.130662
\(438\) 0 0
\(439\) 17.1952 0.820683 0.410342 0.911932i \(-0.365409\pi\)
0.410342 + 0.911932i \(0.365409\pi\)
\(440\) −35.2395 −1.67998
\(441\) 0 0
\(442\) 12.7320 0.605601
\(443\) 28.8949 1.37284 0.686418 0.727207i \(-0.259182\pi\)
0.686418 + 0.727207i \(0.259182\pi\)
\(444\) 0 0
\(445\) 40.6851 1.92866
\(446\) −21.8749 −1.03581
\(447\) 0 0
\(448\) −6.79292 −0.320935
\(449\) 3.61081 0.170405 0.0852024 0.996364i \(-0.472846\pi\)
0.0852024 + 0.996364i \(0.472846\pi\)
\(450\) 0 0
\(451\) −53.5877 −2.52335
\(452\) 4.54158 0.213618
\(453\) 0 0
\(454\) −9.26176 −0.434676
\(455\) 5.81109 0.272428
\(456\) 0 0
\(457\) −6.61175 −0.309285 −0.154642 0.987971i \(-0.549422\pi\)
−0.154642 + 0.987971i \(0.549422\pi\)
\(458\) 6.26176 0.292593
\(459\) 0 0
\(460\) −12.0496 −0.561817
\(461\) −36.3141 −1.69132 −0.845658 0.533726i \(-0.820792\pi\)
−0.845658 + 0.533726i \(0.820792\pi\)
\(462\) 0 0
\(463\) −21.4688 −0.997742 −0.498871 0.866676i \(-0.666252\pi\)
−0.498871 + 0.866676i \(0.666252\pi\)
\(464\) −0.130110 −0.00604022
\(465\) 0 0
\(466\) 2.84430 0.131760
\(467\) −33.4861 −1.54955 −0.774776 0.632236i \(-0.782137\pi\)
−0.774776 + 0.632236i \(0.782137\pi\)
\(468\) 0 0
\(469\) 0.879385 0.0406062
\(470\) 19.7784 0.912308
\(471\) 0 0
\(472\) 15.0128 0.691020
\(473\) 26.8976 1.23675
\(474\) 0 0
\(475\) 0.332748 0.0152675
\(476\) 13.0223 0.596876
\(477\) 0 0
\(478\) −7.03952 −0.321980
\(479\) −31.1607 −1.42377 −0.711885 0.702296i \(-0.752159\pi\)
−0.711885 + 0.702296i \(0.752159\pi\)
\(480\) 0 0
\(481\) −9.45935 −0.431309
\(482\) 25.1284 1.14457
\(483\) 0 0
\(484\) −20.8452 −0.947511
\(485\) −3.39330 −0.154082
\(486\) 0 0
\(487\) 5.54158 0.251113 0.125557 0.992086i \(-0.459928\pi\)
0.125557 + 0.992086i \(0.459928\pi\)
\(488\) 33.9118 1.53511
\(489\) 0 0
\(490\) 10.7023 0.483482
\(491\) −13.2618 −0.598495 −0.299248 0.954176i \(-0.596736\pi\)
−0.299248 + 0.954176i \(0.596736\pi\)
\(492\) 0 0
\(493\) 24.4739 1.10225
\(494\) 1.05468 0.0474524
\(495\) 0 0
\(496\) −0.0537486 −0.00241339
\(497\) −17.9736 −0.806226
\(498\) 0 0
\(499\) 7.02465 0.314467 0.157233 0.987561i \(-0.449743\pi\)
0.157233 + 0.987561i \(0.449743\pi\)
\(500\) 12.9290 0.578203
\(501\) 0 0
\(502\) 16.9317 0.755699
\(503\) 12.5381 0.559045 0.279523 0.960139i \(-0.409824\pi\)
0.279523 + 0.960139i \(0.409824\pi\)
\(504\) 0 0
\(505\) −16.7965 −0.747436
\(506\) 19.4706 0.865574
\(507\) 0 0
\(508\) 1.22668 0.0544252
\(509\) 17.4320 0.772660 0.386330 0.922361i \(-0.373743\pi\)
0.386330 + 0.922361i \(0.373743\pi\)
\(510\) 0 0
\(511\) −3.13610 −0.138733
\(512\) 0.473897 0.0209435
\(513\) 0 0
\(514\) −17.7442 −0.782664
\(515\) 13.9513 0.614768
\(516\) 0 0
\(517\) 50.6955 2.22959
\(518\) 6.09926 0.267986
\(519\) 0 0
\(520\) −12.2385 −0.536696
\(521\) −13.8871 −0.608406 −0.304203 0.952607i \(-0.598390\pi\)
−0.304203 + 0.952607i \(0.598390\pi\)
\(522\) 0 0
\(523\) 7.05375 0.308439 0.154219 0.988037i \(-0.450714\pi\)
0.154219 + 0.988037i \(0.450714\pi\)
\(524\) 18.6940 0.816650
\(525\) 0 0
\(526\) 6.62124 0.288700
\(527\) 10.1102 0.440407
\(528\) 0 0
\(529\) −5.48751 −0.238587
\(530\) 0.925393 0.0401965
\(531\) 0 0
\(532\) 1.07873 0.0467687
\(533\) −18.6108 −0.806124
\(534\) 0 0
\(535\) −4.72874 −0.204441
\(536\) −1.85204 −0.0799961
\(537\) 0 0
\(538\) −11.7314 −0.505778
\(539\) 27.4320 1.18158
\(540\) 0 0
\(541\) 26.3969 1.13489 0.567446 0.823410i \(-0.307931\pi\)
0.567446 + 0.823410i \(0.307931\pi\)
\(542\) 23.3236 1.00183
\(543\) 0 0
\(544\) −44.4252 −1.90471
\(545\) 0.415593 0.0178020
\(546\) 0 0
\(547\) −33.4834 −1.43165 −0.715823 0.698282i \(-0.753948\pi\)
−0.715823 + 0.698282i \(0.753948\pi\)
\(548\) −13.4935 −0.576414
\(549\) 0 0
\(550\) 2.37195 0.101140
\(551\) 2.02734 0.0863676
\(552\) 0 0
\(553\) −21.7665 −0.925606
\(554\) 7.76321 0.329827
\(555\) 0 0
\(556\) 0.271868 0.0115298
\(557\) 27.6682 1.17234 0.586169 0.810189i \(-0.300635\pi\)
0.586169 + 0.810189i \(0.300635\pi\)
\(558\) 0 0
\(559\) 9.34142 0.395100
\(560\) 0.132474 0.00559806
\(561\) 0 0
\(562\) 15.7341 0.663704
\(563\) −2.64321 −0.111398 −0.0556990 0.998448i \(-0.517739\pi\)
−0.0556990 + 0.998448i \(0.517739\pi\)
\(564\) 0 0
\(565\) −8.69047 −0.365611
\(566\) −10.9469 −0.460131
\(567\) 0 0
\(568\) 37.8536 1.58830
\(569\) −11.8939 −0.498620 −0.249310 0.968424i \(-0.580204\pi\)
−0.249310 + 0.968424i \(0.580204\pi\)
\(570\) 0 0
\(571\) −19.1334 −0.800708 −0.400354 0.916361i \(-0.631113\pi\)
−0.400354 + 0.916361i \(0.631113\pi\)
\(572\) −11.9257 −0.498639
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 2.13341 0.0889693
\(576\) 0 0
\(577\) −18.7561 −0.780826 −0.390413 0.920640i \(-0.627668\pi\)
−0.390413 + 0.920640i \(0.627668\pi\)
\(578\) −39.6468 −1.64909
\(579\) 0 0
\(580\) −8.94356 −0.371361
\(581\) −20.1242 −0.834894
\(582\) 0 0
\(583\) 2.37195 0.0982361
\(584\) 6.60483 0.273310
\(585\) 0 0
\(586\) 28.8716 1.19268
\(587\) 28.2909 1.16769 0.583844 0.811866i \(-0.301548\pi\)
0.583844 + 0.811866i \(0.301548\pi\)
\(588\) 0 0
\(589\) 0.837496 0.0345084
\(590\) −10.9213 −0.449622
\(591\) 0 0
\(592\) −0.215643 −0.00886288
\(593\) −7.34730 −0.301717 −0.150859 0.988555i \(-0.548204\pi\)
−0.150859 + 0.988555i \(0.548204\pi\)
\(594\) 0 0
\(595\) −24.9186 −1.02156
\(596\) 2.68839 0.110121
\(597\) 0 0
\(598\) 6.76207 0.276522
\(599\) 28.8922 1.18050 0.590251 0.807220i \(-0.299029\pi\)
0.590251 + 0.807220i \(0.299029\pi\)
\(600\) 0 0
\(601\) 20.2249 0.824992 0.412496 0.910959i \(-0.364657\pi\)
0.412496 + 0.910959i \(0.364657\pi\)
\(602\) −6.02322 −0.245488
\(603\) 0 0
\(604\) 15.4097 0.627013
\(605\) 39.8881 1.62168
\(606\) 0 0
\(607\) −19.7128 −0.800116 −0.400058 0.916490i \(-0.631010\pi\)
−0.400058 + 0.916490i \(0.631010\pi\)
\(608\) −3.68004 −0.149246
\(609\) 0 0
\(610\) −24.6696 −0.998844
\(611\) 17.6064 0.712278
\(612\) 0 0
\(613\) 4.69728 0.189721 0.0948607 0.995491i \(-0.469759\pi\)
0.0948607 + 0.995491i \(0.469759\pi\)
\(614\) 14.5208 0.586013
\(615\) 0 0
\(616\) 20.2267 0.814956
\(617\) 41.9368 1.68831 0.844155 0.536099i \(-0.180103\pi\)
0.844155 + 0.536099i \(0.180103\pi\)
\(618\) 0 0
\(619\) 20.0814 0.807140 0.403570 0.914949i \(-0.367769\pi\)
0.403570 + 0.914949i \(0.367769\pi\)
\(620\) −3.69459 −0.148378
\(621\) 0 0
\(622\) 12.2490 0.491139
\(623\) −23.3523 −0.935592
\(624\) 0 0
\(625\) −27.2891 −1.09156
\(626\) 8.95191 0.357790
\(627\) 0 0
\(628\) 11.4421 0.456590
\(629\) 40.5627 1.61734
\(630\) 0 0
\(631\) −9.97090 −0.396935 −0.198468 0.980107i \(-0.563596\pi\)
−0.198468 + 0.980107i \(0.563596\pi\)
\(632\) 45.8417 1.82349
\(633\) 0 0
\(634\) 17.5757 0.698022
\(635\) −2.34730 −0.0931496
\(636\) 0 0
\(637\) 9.52704 0.377475
\(638\) 14.4516 0.572145
\(639\) 0 0
\(640\) 16.0615 0.634886
\(641\) 11.9513 0.472048 0.236024 0.971747i \(-0.424156\pi\)
0.236024 + 0.971747i \(0.424156\pi\)
\(642\) 0 0
\(643\) −22.2668 −0.878118 −0.439059 0.898458i \(-0.644688\pi\)
−0.439059 + 0.898458i \(0.644688\pi\)
\(644\) 6.91622 0.272537
\(645\) 0 0
\(646\) −4.52259 −0.177939
\(647\) 2.62773 0.103307 0.0516533 0.998665i \(-0.483551\pi\)
0.0516533 + 0.998665i \(0.483551\pi\)
\(648\) 0 0
\(649\) −27.9932 −1.09883
\(650\) 0.823769 0.0323109
\(651\) 0 0
\(652\) 13.2763 0.519940
\(653\) 39.9873 1.56482 0.782412 0.622761i \(-0.213989\pi\)
0.782412 + 0.622761i \(0.213989\pi\)
\(654\) 0 0
\(655\) −35.7716 −1.39771
\(656\) −0.424267 −0.0165649
\(657\) 0 0
\(658\) −11.3523 −0.442561
\(659\) 20.4388 0.796183 0.398092 0.917346i \(-0.369673\pi\)
0.398092 + 0.917346i \(0.369673\pi\)
\(660\) 0 0
\(661\) −20.5767 −0.800340 −0.400170 0.916441i \(-0.631049\pi\)
−0.400170 + 0.916441i \(0.631049\pi\)
\(662\) −5.89311 −0.229042
\(663\) 0 0
\(664\) 42.3830 1.64478
\(665\) −2.06418 −0.0800454
\(666\) 0 0
\(667\) 12.9982 0.503294
\(668\) 1.68004 0.0650029
\(669\) 0 0
\(670\) 1.34730 0.0520506
\(671\) −63.2327 −2.44107
\(672\) 0 0
\(673\) 27.6064 1.06415 0.532074 0.846698i \(-0.321413\pi\)
0.532074 + 0.846698i \(0.321413\pi\)
\(674\) −0.461104 −0.0177611
\(675\) 0 0
\(676\) 11.8051 0.454042
\(677\) −21.5107 −0.826725 −0.413362 0.910567i \(-0.635646\pi\)
−0.413362 + 0.910567i \(0.635646\pi\)
\(678\) 0 0
\(679\) 1.94768 0.0747452
\(680\) 52.4802 2.01252
\(681\) 0 0
\(682\) 5.96997 0.228602
\(683\) 27.7452 1.06164 0.530819 0.847485i \(-0.321884\pi\)
0.530819 + 0.847485i \(0.321884\pi\)
\(684\) 0 0
\(685\) 25.8203 0.986541
\(686\) −14.4365 −0.551186
\(687\) 0 0
\(688\) 0.212955 0.00811882
\(689\) 0.823769 0.0313831
\(690\) 0 0
\(691\) 1.86989 0.0711340 0.0355670 0.999367i \(-0.488676\pi\)
0.0355670 + 0.999367i \(0.488676\pi\)
\(692\) −20.3952 −0.775308
\(693\) 0 0
\(694\) −17.0634 −0.647716
\(695\) −0.520229 −0.0197334
\(696\) 0 0
\(697\) 79.8052 3.02284
\(698\) 18.0841 0.684493
\(699\) 0 0
\(700\) 0.842549 0.0318453
\(701\) 36.0036 1.35984 0.679919 0.733287i \(-0.262015\pi\)
0.679919 + 0.733287i \(0.262015\pi\)
\(702\) 0 0
\(703\) 3.36009 0.126728
\(704\) −26.6759 −1.00539
\(705\) 0 0
\(706\) 24.7041 0.929751
\(707\) 9.64084 0.362581
\(708\) 0 0
\(709\) 11.6723 0.438363 0.219181 0.975684i \(-0.429661\pi\)
0.219181 + 0.975684i \(0.429661\pi\)
\(710\) −27.5371 −1.03345
\(711\) 0 0
\(712\) 49.1816 1.84316
\(713\) 5.36959 0.201093
\(714\) 0 0
\(715\) 22.8203 0.853429
\(716\) 8.54158 0.319214
\(717\) 0 0
\(718\) 2.58853 0.0966029
\(719\) 45.8539 1.71006 0.855031 0.518577i \(-0.173538\pi\)
0.855031 + 0.518577i \(0.173538\pi\)
\(720\) 0 0
\(721\) −8.00774 −0.298224
\(722\) 16.3337 0.607877
\(723\) 0 0
\(724\) 3.05468 0.113526
\(725\) 1.58347 0.0588087
\(726\) 0 0
\(727\) 0.111444 0.00413321 0.00206661 0.999998i \(-0.499342\pi\)
0.00206661 + 0.999998i \(0.499342\pi\)
\(728\) 7.02465 0.260351
\(729\) 0 0
\(730\) −4.80478 −0.177833
\(731\) −40.0570 −1.48156
\(732\) 0 0
\(733\) −24.0983 −0.890092 −0.445046 0.895508i \(-0.646813\pi\)
−0.445046 + 0.895508i \(0.646813\pi\)
\(734\) −16.0636 −0.592917
\(735\) 0 0
\(736\) −23.5945 −0.869705
\(737\) 3.45336 0.127206
\(738\) 0 0
\(739\) −8.54570 −0.314359 −0.157179 0.987570i \(-0.550240\pi\)
−0.157179 + 0.987570i \(0.550240\pi\)
\(740\) −14.8229 −0.544902
\(741\) 0 0
\(742\) −0.531155 −0.0194993
\(743\) −34.0401 −1.24881 −0.624406 0.781100i \(-0.714659\pi\)
−0.624406 + 0.781100i \(0.714659\pi\)
\(744\) 0 0
\(745\) −5.14433 −0.188474
\(746\) −5.32150 −0.194834
\(747\) 0 0
\(748\) 51.1388 1.86982
\(749\) 2.71419 0.0991745
\(750\) 0 0
\(751\) 9.29591 0.339213 0.169606 0.985512i \(-0.445750\pi\)
0.169606 + 0.985512i \(0.445750\pi\)
\(752\) 0.401369 0.0146364
\(753\) 0 0
\(754\) 5.01899 0.182781
\(755\) −29.4870 −1.07314
\(756\) 0 0
\(757\) 29.9436 1.08832 0.544159 0.838982i \(-0.316849\pi\)
0.544159 + 0.838982i \(0.316849\pi\)
\(758\) −11.9112 −0.432633
\(759\) 0 0
\(760\) 4.34730 0.157693
\(761\) 6.83656 0.247825 0.123913 0.992293i \(-0.460456\pi\)
0.123913 + 0.992293i \(0.460456\pi\)
\(762\) 0 0
\(763\) −0.238541 −0.00863577
\(764\) −16.6091 −0.600895
\(765\) 0 0
\(766\) −14.0265 −0.506799
\(767\) −9.72193 −0.351039
\(768\) 0 0
\(769\) −15.5253 −0.559856 −0.279928 0.960021i \(-0.590311\pi\)
−0.279928 + 0.960021i \(0.590311\pi\)
\(770\) −14.7142 −0.530263
\(771\) 0 0
\(772\) −26.7279 −0.961959
\(773\) −30.1857 −1.08571 −0.542853 0.839828i \(-0.682656\pi\)
−0.542853 + 0.839828i \(0.682656\pi\)
\(774\) 0 0
\(775\) 0.654134 0.0234972
\(776\) −4.10195 −0.147251
\(777\) 0 0
\(778\) −25.0823 −0.899246
\(779\) 6.61081 0.236857
\(780\) 0 0
\(781\) −70.5827 −2.52565
\(782\) −28.9965 −1.03691
\(783\) 0 0
\(784\) 0.217186 0.00775665
\(785\) −21.8949 −0.781461
\(786\) 0 0
\(787\) 19.0906 0.680506 0.340253 0.940334i \(-0.389487\pi\)
0.340253 + 0.940334i \(0.389487\pi\)
\(788\) −6.22163 −0.221636
\(789\) 0 0
\(790\) −33.3482 −1.18648
\(791\) 4.98814 0.177358
\(792\) 0 0
\(793\) −21.9605 −0.779839
\(794\) −12.5912 −0.446845
\(795\) 0 0
\(796\) 8.68954 0.307993
\(797\) −27.0452 −0.957990 −0.478995 0.877818i \(-0.658999\pi\)
−0.478995 + 0.877818i \(0.658999\pi\)
\(798\) 0 0
\(799\) −75.4981 −2.67093
\(800\) −2.87433 −0.101623
\(801\) 0 0
\(802\) 0.623608 0.0220204
\(803\) −12.3155 −0.434605
\(804\) 0 0
\(805\) −13.2344 −0.466452
\(806\) 2.07335 0.0730306
\(807\) 0 0
\(808\) −20.3043 −0.714301
\(809\) −2.24628 −0.0789751 −0.0394875 0.999220i \(-0.512573\pi\)
−0.0394875 + 0.999220i \(0.512573\pi\)
\(810\) 0 0
\(811\) 42.5090 1.49269 0.746346 0.665558i \(-0.231806\pi\)
0.746346 + 0.665558i \(0.231806\pi\)
\(812\) 5.13341 0.180147
\(813\) 0 0
\(814\) 23.9519 0.839514
\(815\) −25.4047 −0.889887
\(816\) 0 0
\(817\) −3.31820 −0.116089
\(818\) −20.4433 −0.714782
\(819\) 0 0
\(820\) −29.1634 −1.01843
\(821\) 43.2850 1.51066 0.755328 0.655347i \(-0.227477\pi\)
0.755328 + 0.655347i \(0.227477\pi\)
\(822\) 0 0
\(823\) 26.6340 0.928404 0.464202 0.885729i \(-0.346341\pi\)
0.464202 + 0.885729i \(0.346341\pi\)
\(824\) 16.8648 0.587514
\(825\) 0 0
\(826\) 6.26857 0.218111
\(827\) −44.9053 −1.56151 −0.780755 0.624837i \(-0.785165\pi\)
−0.780755 + 0.624837i \(0.785165\pi\)
\(828\) 0 0
\(829\) −25.1361 −0.873013 −0.436506 0.899701i \(-0.643784\pi\)
−0.436506 + 0.899701i \(0.643784\pi\)
\(830\) −30.8321 −1.07020
\(831\) 0 0
\(832\) −9.26445 −0.321187
\(833\) −40.8530 −1.41547
\(834\) 0 0
\(835\) −3.21482 −0.111254
\(836\) 4.23618 0.146511
\(837\) 0 0
\(838\) 2.04096 0.0705036
\(839\) −17.1530 −0.592188 −0.296094 0.955159i \(-0.595684\pi\)
−0.296094 + 0.955159i \(0.595684\pi\)
\(840\) 0 0
\(841\) −19.3523 −0.667322
\(842\) 20.3372 0.700866
\(843\) 0 0
\(844\) 4.50475 0.155060
\(845\) −22.5895 −0.777101
\(846\) 0 0
\(847\) −22.8949 −0.786677
\(848\) 0.0187793 0.000644884 0
\(849\) 0 0
\(850\) −3.53241 −0.121161
\(851\) 21.5431 0.738489
\(852\) 0 0
\(853\) 15.0327 0.514710 0.257355 0.966317i \(-0.417149\pi\)
0.257355 + 0.966317i \(0.417149\pi\)
\(854\) 14.1598 0.484539
\(855\) 0 0
\(856\) −5.71627 −0.195378
\(857\) 19.1916 0.655573 0.327786 0.944752i \(-0.393697\pi\)
0.327786 + 0.944752i \(0.393697\pi\)
\(858\) 0 0
\(859\) 13.3527 0.455587 0.227794 0.973709i \(-0.426849\pi\)
0.227794 + 0.973709i \(0.426849\pi\)
\(860\) 14.6382 0.499157
\(861\) 0 0
\(862\) 11.7050 0.398675
\(863\) −25.3577 −0.863187 −0.431594 0.902068i \(-0.642049\pi\)
−0.431594 + 0.902068i \(0.642049\pi\)
\(864\) 0 0
\(865\) 39.0268 1.32695
\(866\) 7.44419 0.252964
\(867\) 0 0
\(868\) 2.12061 0.0719784
\(869\) −85.4775 −2.89963
\(870\) 0 0
\(871\) 1.19934 0.0406381
\(872\) 0.502384 0.0170129
\(873\) 0 0
\(874\) −2.40198 −0.0812481
\(875\) 14.2003 0.480057
\(876\) 0 0
\(877\) −24.1257 −0.814666 −0.407333 0.913280i \(-0.633541\pi\)
−0.407333 + 0.913280i \(0.633541\pi\)
\(878\) −15.1212 −0.510317
\(879\) 0 0
\(880\) 0.520229 0.0175369
\(881\) 42.7606 1.44064 0.720321 0.693641i \(-0.243994\pi\)
0.720321 + 0.693641i \(0.243994\pi\)
\(882\) 0 0
\(883\) 21.2983 0.716744 0.358372 0.933579i \(-0.383332\pi\)
0.358372 + 0.933579i \(0.383332\pi\)
\(884\) 17.7603 0.597344
\(885\) 0 0
\(886\) −25.4097 −0.853656
\(887\) 27.9956 0.939999 0.469999 0.882667i \(-0.344254\pi\)
0.469999 + 0.882667i \(0.344254\pi\)
\(888\) 0 0
\(889\) 1.34730 0.0451869
\(890\) −35.7779 −1.19928
\(891\) 0 0
\(892\) −30.5140 −1.02169
\(893\) −6.25402 −0.209283
\(894\) 0 0
\(895\) −16.3446 −0.546340
\(896\) −9.21894 −0.307983
\(897\) 0 0
\(898\) −3.17530 −0.105961
\(899\) 3.98545 0.132922
\(900\) 0 0
\(901\) −3.53241 −0.117682
\(902\) 47.1242 1.56907
\(903\) 0 0
\(904\) −10.5054 −0.349403
\(905\) −5.84524 −0.194302
\(906\) 0 0
\(907\) 18.3750 0.610131 0.305065 0.952331i \(-0.401322\pi\)
0.305065 + 0.952331i \(0.401322\pi\)
\(908\) −12.9195 −0.428749
\(909\) 0 0
\(910\) −5.11019 −0.169401
\(911\) 16.3625 0.542112 0.271056 0.962564i \(-0.412627\pi\)
0.271056 + 0.962564i \(0.412627\pi\)
\(912\) 0 0
\(913\) −79.0283 −2.61545
\(914\) 5.81427 0.192319
\(915\) 0 0
\(916\) 8.73473 0.288604
\(917\) 20.5321 0.678029
\(918\) 0 0
\(919\) −43.5340 −1.43605 −0.718026 0.696016i \(-0.754954\pi\)
−0.718026 + 0.696016i \(0.754954\pi\)
\(920\) 27.8726 0.918932
\(921\) 0 0
\(922\) 31.9341 1.05169
\(923\) −24.5131 −0.806858
\(924\) 0 0
\(925\) 2.62443 0.0862907
\(926\) 18.8794 0.620415
\(927\) 0 0
\(928\) −17.5125 −0.574876
\(929\) −10.0368 −0.329298 −0.164649 0.986352i \(-0.552649\pi\)
−0.164649 + 0.986352i \(0.552649\pi\)
\(930\) 0 0
\(931\) −3.38413 −0.110910
\(932\) 3.96761 0.129963
\(933\) 0 0
\(934\) 29.4472 0.963541
\(935\) −97.8558 −3.20023
\(936\) 0 0
\(937\) −31.0865 −1.01555 −0.507775 0.861490i \(-0.669532\pi\)
−0.507775 + 0.861490i \(0.669532\pi\)
\(938\) −0.773318 −0.0252497
\(939\) 0 0
\(940\) 27.5895 0.899869
\(941\) 14.1976 0.462828 0.231414 0.972855i \(-0.425665\pi\)
0.231414 + 0.972855i \(0.425665\pi\)
\(942\) 0 0
\(943\) 42.3851 1.38025
\(944\) −0.221629 −0.00721341
\(945\) 0 0
\(946\) −23.6533 −0.769036
\(947\) −50.3236 −1.63530 −0.817648 0.575718i \(-0.804723\pi\)
−0.817648 + 0.575718i \(0.804723\pi\)
\(948\) 0 0
\(949\) −4.27713 −0.138842
\(950\) −0.292614 −0.00949365
\(951\) 0 0
\(952\) −30.1225 −0.976275
\(953\) 34.1539 1.10635 0.553177 0.833064i \(-0.313415\pi\)
0.553177 + 0.833064i \(0.313415\pi\)
\(954\) 0 0
\(955\) 31.7820 1.02844
\(956\) −9.81965 −0.317590
\(957\) 0 0
\(958\) 27.4023 0.885328
\(959\) −14.8203 −0.478571
\(960\) 0 0
\(961\) −29.3536 −0.946891
\(962\) 8.31841 0.268196
\(963\) 0 0
\(964\) 35.0523 1.12896
\(965\) 51.1448 1.64641
\(966\) 0 0
\(967\) 2.47472 0.0795816 0.0397908 0.999208i \(-0.487331\pi\)
0.0397908 + 0.999208i \(0.487331\pi\)
\(968\) 48.2181 1.54979
\(969\) 0 0
\(970\) 2.98402 0.0958112
\(971\) −35.0770 −1.12567 −0.562837 0.826568i \(-0.690290\pi\)
−0.562837 + 0.826568i \(0.690290\pi\)
\(972\) 0 0
\(973\) 0.298600 0.00957268
\(974\) −4.87319 −0.156147
\(975\) 0 0
\(976\) −0.500629 −0.0160247
\(977\) 16.6581 0.532939 0.266470 0.963843i \(-0.414143\pi\)
0.266470 + 0.963843i \(0.414143\pi\)
\(978\) 0 0
\(979\) −91.7051 −2.93091
\(980\) 14.9290 0.476890
\(981\) 0 0
\(982\) 11.6622 0.372156
\(983\) 42.9009 1.36832 0.684162 0.729330i \(-0.260168\pi\)
0.684162 + 0.729330i \(0.260168\pi\)
\(984\) 0 0
\(985\) 11.9053 0.379334
\(986\) −21.5220 −0.685400
\(987\) 0 0
\(988\) 1.47121 0.0468054
\(989\) −21.2746 −0.676492
\(990\) 0 0
\(991\) 24.9213 0.791650 0.395825 0.918326i \(-0.370459\pi\)
0.395825 + 0.918326i \(0.370459\pi\)
\(992\) −7.23442 −0.229693
\(993\) 0 0
\(994\) 15.8057 0.501327
\(995\) −16.6277 −0.527134
\(996\) 0 0
\(997\) 16.6141 0.526174 0.263087 0.964772i \(-0.415259\pi\)
0.263087 + 0.964772i \(0.415259\pi\)
\(998\) −6.17738 −0.195541
\(999\) 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 1143.2.a.e.1.1 3
3.2 odd 2 127.2.a.a.1.3 3
12.11 even 2 2032.2.a.k.1.3 3
15.14 odd 2 3175.2.a.h.1.1 3
21.20 even 2 6223.2.a.e.1.3 3
24.5 odd 2 8128.2.a.bd.1.3 3
24.11 even 2 8128.2.a.w.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
127.2.a.a.1.3 3 3.2 odd 2
1143.2.a.e.1.1 3 1.1 even 1 trivial
2032.2.a.k.1.3 3 12.11 even 2
3175.2.a.h.1.1 3 15.14 odd 2
6223.2.a.e.1.3 3 21.20 even 2
8128.2.a.w.1.1 3 24.11 even 2
8128.2.a.bd.1.3 3 24.5 odd 2