Properties

Label 1710.2.a.u.1.2
Level $1710$
Weight $2$
Character 1710.1
Self dual yes
Analytic conductor $13.654$
Analytic rank $0$
Dimension $2$
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] = [1710,2,Mod(1,1710)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1710, 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("1710.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.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: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1710.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +3.46410 q^{11} +5.46410 q^{13} -2.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} +1.00000 q^{19} -1.00000 q^{20} -3.46410 q^{22} -6.92820 q^{23} +1.00000 q^{25} -5.46410 q^{26} +2.00000 q^{28} -3.46410 q^{29} -1.46410 q^{31} -1.00000 q^{32} -3.46410 q^{34} -2.00000 q^{35} -1.46410 q^{37} -1.00000 q^{38} +1.00000 q^{40} +5.46410 q^{43} +3.46410 q^{44} +6.92820 q^{46} -6.92820 q^{47} -3.00000 q^{49} -1.00000 q^{50} +5.46410 q^{52} +12.9282 q^{53} -3.46410 q^{55} -2.00000 q^{56} +3.46410 q^{58} +3.46410 q^{59} +2.00000 q^{61} +1.46410 q^{62} +1.00000 q^{64} -5.46410 q^{65} +14.9282 q^{67} +3.46410 q^{68} +2.00000 q^{70} +6.92820 q^{71} -4.92820 q^{73} +1.46410 q^{74} +1.00000 q^{76} +6.92820 q^{77} -1.46410 q^{79} -1.00000 q^{80} +2.53590 q^{83} -3.46410 q^{85} -5.46410 q^{86} -3.46410 q^{88} -6.92820 q^{89} +10.9282 q^{91} -6.92820 q^{92} +6.92820 q^{94} -1.00000 q^{95} +18.3923 q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} + 4 q^{13} - 4 q^{14} + 2 q^{16} + 2 q^{19} - 2 q^{20} + 2 q^{25} - 4 q^{26} + 4 q^{28} + 4 q^{31} - 2 q^{32} - 4 q^{35} + 4 q^{37} - 2 q^{38} + 2 q^{40} + 4 q^{43} - 6 q^{49} - 2 q^{50} + 4 q^{52} + 12 q^{53} - 4 q^{56} + 4 q^{61} - 4 q^{62} + 2 q^{64} - 4 q^{65} + 16 q^{67} + 4 q^{70} + 4 q^{73} - 4 q^{74} + 2 q^{76} + 4 q^{79} - 2 q^{80} + 12 q^{83} - 4 q^{86} + 8 q^{91} - 2 q^{95} + 16 q^{97} + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) 5.46410 1.51547 0.757735 0.652563i \(-0.226306\pi\)
0.757735 + 0.652563i \(0.226306\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −3.46410 −0.738549
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −5.46410 −1.07160
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) −1.46410 −0.262960 −0.131480 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.46410 −0.594089
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −1.46410 −0.240697 −0.120348 0.992732i \(-0.538401\pi\)
−0.120348 + 0.992732i \(0.538401\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 5.46410 0.833268 0.416634 0.909074i \(-0.363210\pi\)
0.416634 + 0.909074i \(0.363210\pi\)
\(44\) 3.46410 0.522233
\(45\) 0 0
\(46\) 6.92820 1.02151
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 5.46410 0.757735
\(53\) 12.9282 1.77583 0.887913 0.460012i \(-0.152155\pi\)
0.887913 + 0.460012i \(0.152155\pi\)
\(54\) 0 0
\(55\) −3.46410 −0.467099
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 3.46410 0.454859
\(59\) 3.46410 0.450988 0.225494 0.974245i \(-0.427600\pi\)
0.225494 + 0.974245i \(0.427600\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 1.46410 0.185941
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −5.46410 −0.677738
\(66\) 0 0
\(67\) 14.9282 1.82377 0.911885 0.410445i \(-0.134627\pi\)
0.911885 + 0.410445i \(0.134627\pi\)
\(68\) 3.46410 0.420084
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 6.92820 0.822226 0.411113 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 0 0
\(73\) −4.92820 −0.576803 −0.288401 0.957510i \(-0.593124\pi\)
−0.288401 + 0.957510i \(0.593124\pi\)
\(74\) 1.46410 0.170198
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 6.92820 0.789542
\(78\) 0 0
\(79\) −1.46410 −0.164724 −0.0823622 0.996602i \(-0.526246\pi\)
−0.0823622 + 0.996602i \(0.526246\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 0 0
\(83\) 2.53590 0.278351 0.139176 0.990268i \(-0.455555\pi\)
0.139176 + 0.990268i \(0.455555\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) −5.46410 −0.589209
\(87\) 0 0
\(88\) −3.46410 −0.369274
\(89\) −6.92820 −0.734388 −0.367194 0.930144i \(-0.619682\pi\)
−0.367194 + 0.930144i \(0.619682\pi\)
\(90\) 0 0
\(91\) 10.9282 1.14559
\(92\) −6.92820 −0.722315
\(93\) 0 0
\(94\) 6.92820 0.714590
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 18.3923 1.86746 0.933728 0.357984i \(-0.116536\pi\)
0.933728 + 0.357984i \(0.116536\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −4.92820 −0.485590 −0.242795 0.970078i \(-0.578064\pi\)
−0.242795 + 0.970078i \(0.578064\pi\)
\(104\) −5.46410 −0.535799
\(105\) 0 0
\(106\) −12.9282 −1.25570
\(107\) −6.92820 −0.669775 −0.334887 0.942258i \(-0.608698\pi\)
−0.334887 + 0.942258i \(0.608698\pi\)
\(108\) 0 0
\(109\) −14.3923 −1.37853 −0.689266 0.724508i \(-0.742067\pi\)
−0.689266 + 0.724508i \(0.742067\pi\)
\(110\) 3.46410 0.330289
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 6.92820 0.646058
\(116\) −3.46410 −0.321634
\(117\) 0 0
\(118\) −3.46410 −0.318896
\(119\) 6.92820 0.635107
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) −1.46410 −0.131480
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 14.0000 1.24230 0.621150 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 5.46410 0.479233
\(131\) −3.46410 −0.302660 −0.151330 0.988483i \(-0.548356\pi\)
−0.151330 + 0.988483i \(0.548356\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) −14.9282 −1.28960
\(135\) 0 0
\(136\) −3.46410 −0.297044
\(137\) 15.4641 1.32119 0.660594 0.750744i \(-0.270305\pi\)
0.660594 + 0.750744i \(0.270305\pi\)
\(138\) 0 0
\(139\) 9.85641 0.836009 0.418005 0.908445i \(-0.362730\pi\)
0.418005 + 0.908445i \(0.362730\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) −6.92820 −0.581402
\(143\) 18.9282 1.58286
\(144\) 0 0
\(145\) 3.46410 0.287678
\(146\) 4.92820 0.407861
\(147\) 0 0
\(148\) −1.46410 −0.120348
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −8.39230 −0.682956 −0.341478 0.939890i \(-0.610927\pi\)
−0.341478 + 0.939890i \(0.610927\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −6.92820 −0.558291
\(155\) 1.46410 0.117599
\(156\) 0 0
\(157\) −15.3205 −1.22271 −0.611355 0.791357i \(-0.709375\pi\)
−0.611355 + 0.791357i \(0.709375\pi\)
\(158\) 1.46410 0.116478
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −13.8564 −1.09204
\(162\) 0 0
\(163\) −6.53590 −0.511931 −0.255966 0.966686i \(-0.582393\pi\)
−0.255966 + 0.966686i \(0.582393\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −2.53590 −0.196824
\(167\) 18.9282 1.46471 0.732354 0.680924i \(-0.238422\pi\)
0.732354 + 0.680924i \(0.238422\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 3.46410 0.265684
\(171\) 0 0
\(172\) 5.46410 0.416634
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 3.46410 0.261116
\(177\) 0 0
\(178\) 6.92820 0.519291
\(179\) −1.60770 −0.120165 −0.0600824 0.998193i \(-0.519136\pi\)
−0.0600824 + 0.998193i \(0.519136\pi\)
\(180\) 0 0
\(181\) 18.3923 1.36709 0.683545 0.729909i \(-0.260437\pi\)
0.683545 + 0.729909i \(0.260437\pi\)
\(182\) −10.9282 −0.810052
\(183\) 0 0
\(184\) 6.92820 0.510754
\(185\) 1.46410 0.107643
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) −6.92820 −0.505291
\(189\) 0 0
\(190\) 1.00000 0.0725476
\(191\) −11.3205 −0.819123 −0.409562 0.912282i \(-0.634318\pi\)
−0.409562 + 0.912282i \(0.634318\pi\)
\(192\) 0 0
\(193\) 6.39230 0.460128 0.230064 0.973175i \(-0.426106\pi\)
0.230064 + 0.973175i \(0.426106\pi\)
\(194\) −18.3923 −1.32049
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −0.928203 −0.0661317 −0.0330659 0.999453i \(-0.510527\pi\)
−0.0330659 + 0.999453i \(0.510527\pi\)
\(198\) 0 0
\(199\) 2.92820 0.207575 0.103787 0.994600i \(-0.466904\pi\)
0.103787 + 0.994600i \(0.466904\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) −6.92820 −0.486265
\(204\) 0 0
\(205\) 0 0
\(206\) 4.92820 0.343364
\(207\) 0 0
\(208\) 5.46410 0.378867
\(209\) 3.46410 0.239617
\(210\) 0 0
\(211\) −17.8564 −1.22929 −0.614643 0.788806i \(-0.710700\pi\)
−0.614643 + 0.788806i \(0.710700\pi\)
\(212\) 12.9282 0.887913
\(213\) 0 0
\(214\) 6.92820 0.473602
\(215\) −5.46410 −0.372649
\(216\) 0 0
\(217\) −2.92820 −0.198779
\(218\) 14.3923 0.974770
\(219\) 0 0
\(220\) −3.46410 −0.233550
\(221\) 18.9282 1.27325
\(222\) 0 0
\(223\) −23.8564 −1.59754 −0.798772 0.601634i \(-0.794516\pi\)
−0.798772 + 0.601634i \(0.794516\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −28.9282 −1.91163 −0.955815 0.293970i \(-0.905024\pi\)
−0.955815 + 0.293970i \(0.905024\pi\)
\(230\) −6.92820 −0.456832
\(231\) 0 0
\(232\) 3.46410 0.227429
\(233\) 8.53590 0.559205 0.279603 0.960116i \(-0.409797\pi\)
0.279603 + 0.960116i \(0.409797\pi\)
\(234\) 0 0
\(235\) 6.92820 0.451946
\(236\) 3.46410 0.225494
\(237\) 0 0
\(238\) −6.92820 −0.449089
\(239\) −2.53590 −0.164034 −0.0820168 0.996631i \(-0.526136\pi\)
−0.0820168 + 0.996631i \(0.526136\pi\)
\(240\) 0 0
\(241\) 0.143594 0.00924967 0.00462484 0.999989i \(-0.498528\pi\)
0.00462484 + 0.999989i \(0.498528\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 5.46410 0.347672
\(248\) 1.46410 0.0929705
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 10.3923 0.655956 0.327978 0.944685i \(-0.393633\pi\)
0.327978 + 0.944685i \(0.393633\pi\)
\(252\) 0 0
\(253\) −24.0000 −1.50887
\(254\) −14.0000 −0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 24.9282 1.55498 0.777489 0.628896i \(-0.216493\pi\)
0.777489 + 0.628896i \(0.216493\pi\)
\(258\) 0 0
\(259\) −2.92820 −0.181950
\(260\) −5.46410 −0.338869
\(261\) 0 0
\(262\) 3.46410 0.214013
\(263\) 25.8564 1.59437 0.797187 0.603732i \(-0.206320\pi\)
0.797187 + 0.603732i \(0.206320\pi\)
\(264\) 0 0
\(265\) −12.9282 −0.794173
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) 14.9282 0.911885
\(269\) −22.3923 −1.36528 −0.682641 0.730753i \(-0.739169\pi\)
−0.682641 + 0.730753i \(0.739169\pi\)
\(270\) 0 0
\(271\) 16.7846 1.01959 0.509796 0.860295i \(-0.329721\pi\)
0.509796 + 0.860295i \(0.329721\pi\)
\(272\) 3.46410 0.210042
\(273\) 0 0
\(274\) −15.4641 −0.934221
\(275\) 3.46410 0.208893
\(276\) 0 0
\(277\) 0.392305 0.0235713 0.0117857 0.999931i \(-0.496248\pi\)
0.0117857 + 0.999931i \(0.496248\pi\)
\(278\) −9.85641 −0.591148
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) 17.0718 1.01842 0.509209 0.860643i \(-0.329938\pi\)
0.509209 + 0.860643i \(0.329938\pi\)
\(282\) 0 0
\(283\) −20.3923 −1.21220 −0.606098 0.795390i \(-0.707266\pi\)
−0.606098 + 0.795390i \(0.707266\pi\)
\(284\) 6.92820 0.411113
\(285\) 0 0
\(286\) −18.9282 −1.11925
\(287\) 0 0
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) −3.46410 −0.203419
\(291\) 0 0
\(292\) −4.92820 −0.288401
\(293\) −12.9282 −0.755274 −0.377637 0.925954i \(-0.623263\pi\)
−0.377637 + 0.925954i \(0.623263\pi\)
\(294\) 0 0
\(295\) −3.46410 −0.201688
\(296\) 1.46410 0.0850992
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −37.8564 −2.18929
\(300\) 0 0
\(301\) 10.9282 0.629891
\(302\) 8.39230 0.482923
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −24.7846 −1.41453 −0.707266 0.706947i \(-0.750072\pi\)
−0.707266 + 0.706947i \(0.750072\pi\)
\(308\) 6.92820 0.394771
\(309\) 0 0
\(310\) −1.46410 −0.0831554
\(311\) 16.3923 0.929522 0.464761 0.885436i \(-0.346140\pi\)
0.464761 + 0.885436i \(0.346140\pi\)
\(312\) 0 0
\(313\) 22.7846 1.28786 0.643931 0.765083i \(-0.277302\pi\)
0.643931 + 0.765083i \(0.277302\pi\)
\(314\) 15.3205 0.864586
\(315\) 0 0
\(316\) −1.46410 −0.0823622
\(317\) 19.8564 1.11525 0.557623 0.830094i \(-0.311713\pi\)
0.557623 + 0.830094i \(0.311713\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 13.8564 0.772187
\(323\) 3.46410 0.192748
\(324\) 0 0
\(325\) 5.46410 0.303094
\(326\) 6.53590 0.361990
\(327\) 0 0
\(328\) 0 0
\(329\) −13.8564 −0.763928
\(330\) 0 0
\(331\) 14.9282 0.820528 0.410264 0.911967i \(-0.365437\pi\)
0.410264 + 0.911967i \(0.365437\pi\)
\(332\) 2.53590 0.139176
\(333\) 0 0
\(334\) −18.9282 −1.03571
\(335\) −14.9282 −0.815615
\(336\) 0 0
\(337\) 1.32051 0.0719327 0.0359663 0.999353i \(-0.488549\pi\)
0.0359663 + 0.999353i \(0.488549\pi\)
\(338\) −16.8564 −0.916868
\(339\) 0 0
\(340\) −3.46410 −0.187867
\(341\) −5.07180 −0.274653
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) −5.46410 −0.294605
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −37.1769 −1.99576 −0.997881 0.0650705i \(-0.979273\pi\)
−0.997881 + 0.0650705i \(0.979273\pi\)
\(348\) 0 0
\(349\) −11.8564 −0.634659 −0.317329 0.948315i \(-0.602786\pi\)
−0.317329 + 0.948315i \(0.602786\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) −3.46410 −0.184637
\(353\) 13.6077 0.724265 0.362132 0.932127i \(-0.382049\pi\)
0.362132 + 0.932127i \(0.382049\pi\)
\(354\) 0 0
\(355\) −6.92820 −0.367711
\(356\) −6.92820 −0.367194
\(357\) 0 0
\(358\) 1.60770 0.0849693
\(359\) −33.4641 −1.76617 −0.883084 0.469215i \(-0.844537\pi\)
−0.883084 + 0.469215i \(0.844537\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −18.3923 −0.966678
\(363\) 0 0
\(364\) 10.9282 0.572793
\(365\) 4.92820 0.257954
\(366\) 0 0
\(367\) −25.7128 −1.34220 −0.671099 0.741368i \(-0.734177\pi\)
−0.671099 + 0.741368i \(0.734177\pi\)
\(368\) −6.92820 −0.361158
\(369\) 0 0
\(370\) −1.46410 −0.0761150
\(371\) 25.8564 1.34240
\(372\) 0 0
\(373\) 0.392305 0.0203128 0.0101564 0.999948i \(-0.496767\pi\)
0.0101564 + 0.999948i \(0.496767\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) 6.92820 0.357295
\(377\) −18.9282 −0.974852
\(378\) 0 0
\(379\) −9.07180 −0.465987 −0.232993 0.972478i \(-0.574852\pi\)
−0.232993 + 0.972478i \(0.574852\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 0 0
\(382\) 11.3205 0.579208
\(383\) −18.9282 −0.967186 −0.483593 0.875293i \(-0.660669\pi\)
−0.483593 + 0.875293i \(0.660669\pi\)
\(384\) 0 0
\(385\) −6.92820 −0.353094
\(386\) −6.39230 −0.325360
\(387\) 0 0
\(388\) 18.3923 0.933728
\(389\) 26.7846 1.35803 0.679017 0.734123i \(-0.262406\pi\)
0.679017 + 0.734123i \(0.262406\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) 0.928203 0.0467622
\(395\) 1.46410 0.0736669
\(396\) 0 0
\(397\) 19.3205 0.969669 0.484834 0.874606i \(-0.338880\pi\)
0.484834 + 0.874606i \(0.338880\pi\)
\(398\) −2.92820 −0.146778
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 6.92820 0.343841
\(407\) −5.07180 −0.251400
\(408\) 0 0
\(409\) −30.7846 −1.52220 −0.761100 0.648634i \(-0.775341\pi\)
−0.761100 + 0.648634i \(0.775341\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.92820 −0.242795
\(413\) 6.92820 0.340915
\(414\) 0 0
\(415\) −2.53590 −0.124482
\(416\) −5.46410 −0.267900
\(417\) 0 0
\(418\) −3.46410 −0.169435
\(419\) −13.6077 −0.664779 −0.332390 0.943142i \(-0.607855\pi\)
−0.332390 + 0.943142i \(0.607855\pi\)
\(420\) 0 0
\(421\) 39.1769 1.90937 0.954683 0.297625i \(-0.0961944\pi\)
0.954683 + 0.297625i \(0.0961944\pi\)
\(422\) 17.8564 0.869236
\(423\) 0 0
\(424\) −12.9282 −0.627849
\(425\) 3.46410 0.168034
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) −6.92820 −0.334887
\(429\) 0 0
\(430\) 5.46410 0.263502
\(431\) −18.9282 −0.911739 −0.455870 0.890047i \(-0.650672\pi\)
−0.455870 + 0.890047i \(0.650672\pi\)
\(432\) 0 0
\(433\) −21.3205 −1.02460 −0.512299 0.858807i \(-0.671206\pi\)
−0.512299 + 0.858807i \(0.671206\pi\)
\(434\) 2.92820 0.140558
\(435\) 0 0
\(436\) −14.3923 −0.689266
\(437\) −6.92820 −0.331421
\(438\) 0 0
\(439\) −3.32051 −0.158479 −0.0792396 0.996856i \(-0.525249\pi\)
−0.0792396 + 0.996856i \(0.525249\pi\)
\(440\) 3.46410 0.165145
\(441\) 0 0
\(442\) −18.9282 −0.900323
\(443\) 4.39230 0.208685 0.104342 0.994541i \(-0.466726\pi\)
0.104342 + 0.994541i \(0.466726\pi\)
\(444\) 0 0
\(445\) 6.92820 0.328428
\(446\) 23.8564 1.12963
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) 1.85641 0.0876092 0.0438046 0.999040i \(-0.486052\pi\)
0.0438046 + 0.999040i \(0.486052\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 0 0
\(455\) −10.9282 −0.512322
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 28.9282 1.35173
\(459\) 0 0
\(460\) 6.92820 0.323029
\(461\) 36.9282 1.71992 0.859959 0.510363i \(-0.170489\pi\)
0.859959 + 0.510363i \(0.170489\pi\)
\(462\) 0 0
\(463\) −15.0718 −0.700446 −0.350223 0.936666i \(-0.613894\pi\)
−0.350223 + 0.936666i \(0.613894\pi\)
\(464\) −3.46410 −0.160817
\(465\) 0 0
\(466\) −8.53590 −0.395418
\(467\) −21.4641 −0.993240 −0.496620 0.867968i \(-0.665426\pi\)
−0.496620 + 0.867968i \(0.665426\pi\)
\(468\) 0 0
\(469\) 29.8564 1.37864
\(470\) −6.92820 −0.319574
\(471\) 0 0
\(472\) −3.46410 −0.159448
\(473\) 18.9282 0.870320
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 6.92820 0.317554
\(477\) 0 0
\(478\) 2.53590 0.115989
\(479\) −7.60770 −0.347604 −0.173802 0.984781i \(-0.555605\pi\)
−0.173802 + 0.984781i \(0.555605\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) −0.143594 −0.00654051
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −18.3923 −0.835152
\(486\) 0 0
\(487\) 34.7846 1.57624 0.788121 0.615521i \(-0.211054\pi\)
0.788121 + 0.615521i \(0.211054\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) −3.00000 −0.135526
\(491\) −41.3205 −1.86477 −0.932384 0.361469i \(-0.882275\pi\)
−0.932384 + 0.361469i \(0.882275\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) −5.46410 −0.245842
\(495\) 0 0
\(496\) −1.46410 −0.0657401
\(497\) 13.8564 0.621545
\(498\) 0 0
\(499\) −22.9282 −1.02641 −0.513204 0.858267i \(-0.671541\pi\)
−0.513204 + 0.858267i \(0.671541\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −10.3923 −0.463831
\(503\) 6.92820 0.308913 0.154457 0.988000i \(-0.450637\pi\)
0.154457 + 0.988000i \(0.450637\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 24.0000 1.06693
\(507\) 0 0
\(508\) 14.0000 0.621150
\(509\) −36.2487 −1.60670 −0.803348 0.595510i \(-0.796950\pi\)
−0.803348 + 0.595510i \(0.796950\pi\)
\(510\) 0 0
\(511\) −9.85641 −0.436022
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −24.9282 −1.09954
\(515\) 4.92820 0.217163
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) 2.92820 0.128658
\(519\) 0 0
\(520\) 5.46410 0.239617
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) −3.46410 −0.151330
\(525\) 0 0
\(526\) −25.8564 −1.12739
\(527\) −5.07180 −0.220931
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 12.9282 0.561565
\(531\) 0 0
\(532\) 2.00000 0.0867110
\(533\) 0 0
\(534\) 0 0
\(535\) 6.92820 0.299532
\(536\) −14.9282 −0.644800
\(537\) 0 0
\(538\) 22.3923 0.965401
\(539\) −10.3923 −0.447628
\(540\) 0 0
\(541\) −39.5692 −1.70121 −0.850607 0.525802i \(-0.823765\pi\)
−0.850607 + 0.525802i \(0.823765\pi\)
\(542\) −16.7846 −0.720961
\(543\) 0 0
\(544\) −3.46410 −0.148522
\(545\) 14.3923 0.616499
\(546\) 0 0
\(547\) 26.9282 1.15137 0.575683 0.817673i \(-0.304736\pi\)
0.575683 + 0.817673i \(0.304736\pi\)
\(548\) 15.4641 0.660594
\(549\) 0 0
\(550\) −3.46410 −0.147710
\(551\) −3.46410 −0.147576
\(552\) 0 0
\(553\) −2.92820 −0.124520
\(554\) −0.392305 −0.0166674
\(555\) 0 0
\(556\) 9.85641 0.418005
\(557\) −12.9282 −0.547786 −0.273893 0.961760i \(-0.588311\pi\)
−0.273893 + 0.961760i \(0.588311\pi\)
\(558\) 0 0
\(559\) 29.8564 1.26279
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −17.0718 −0.720130
\(563\) 8.78461 0.370227 0.185114 0.982717i \(-0.440735\pi\)
0.185114 + 0.982717i \(0.440735\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 20.3923 0.857153
\(567\) 0 0
\(568\) −6.92820 −0.290701
\(569\) −25.8564 −1.08396 −0.541978 0.840392i \(-0.682325\pi\)
−0.541978 + 0.840392i \(0.682325\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 18.9282 0.791428
\(573\) 0 0
\(574\) 0 0
\(575\) −6.92820 −0.288926
\(576\) 0 0
\(577\) −6.78461 −0.282447 −0.141223 0.989978i \(-0.545104\pi\)
−0.141223 + 0.989978i \(0.545104\pi\)
\(578\) 5.00000 0.207973
\(579\) 0 0
\(580\) 3.46410 0.143839
\(581\) 5.07180 0.210414
\(582\) 0 0
\(583\) 44.7846 1.85479
\(584\) 4.92820 0.203931
\(585\) 0 0
\(586\) 12.9282 0.534059
\(587\) −40.3923 −1.66717 −0.833584 0.552392i \(-0.813715\pi\)
−0.833584 + 0.552392i \(0.813715\pi\)
\(588\) 0 0
\(589\) −1.46410 −0.0603273
\(590\) 3.46410 0.142615
\(591\) 0 0
\(592\) −1.46410 −0.0601742
\(593\) −12.2487 −0.502994 −0.251497 0.967858i \(-0.580923\pi\)
−0.251497 + 0.967858i \(0.580923\pi\)
\(594\) 0 0
\(595\) −6.92820 −0.284029
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 37.8564 1.54806
\(599\) −8.78461 −0.358929 −0.179465 0.983764i \(-0.557437\pi\)
−0.179465 + 0.983764i \(0.557437\pi\)
\(600\) 0 0
\(601\) 19.0718 0.777955 0.388977 0.921247i \(-0.372828\pi\)
0.388977 + 0.921247i \(0.372828\pi\)
\(602\) −10.9282 −0.445400
\(603\) 0 0
\(604\) −8.39230 −0.341478
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 24.6410 1.00015 0.500074 0.865983i \(-0.333306\pi\)
0.500074 + 0.865983i \(0.333306\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) −37.8564 −1.53151
\(612\) 0 0
\(613\) −8.39230 −0.338962 −0.169481 0.985533i \(-0.554209\pi\)
−0.169481 + 0.985533i \(0.554209\pi\)
\(614\) 24.7846 1.00023
\(615\) 0 0
\(616\) −6.92820 −0.279145
\(617\) 22.3923 0.901480 0.450740 0.892655i \(-0.351160\pi\)
0.450740 + 0.892655i \(0.351160\pi\)
\(618\) 0 0
\(619\) 28.7846 1.15695 0.578476 0.815700i \(-0.303648\pi\)
0.578476 + 0.815700i \(0.303648\pi\)
\(620\) 1.46410 0.0587997
\(621\) 0 0
\(622\) −16.3923 −0.657272
\(623\) −13.8564 −0.555145
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −22.7846 −0.910656
\(627\) 0 0
\(628\) −15.3205 −0.611355
\(629\) −5.07180 −0.202226
\(630\) 0 0
\(631\) −22.9282 −0.912757 −0.456379 0.889786i \(-0.650854\pi\)
−0.456379 + 0.889786i \(0.650854\pi\)
\(632\) 1.46410 0.0582388
\(633\) 0 0
\(634\) −19.8564 −0.788599
\(635\) −14.0000 −0.555573
\(636\) 0 0
\(637\) −16.3923 −0.649487
\(638\) 12.0000 0.475085
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 10.1436 0.400648 0.200324 0.979730i \(-0.435801\pi\)
0.200324 + 0.979730i \(0.435801\pi\)
\(642\) 0 0
\(643\) 45.1769 1.78160 0.890802 0.454392i \(-0.150143\pi\)
0.890802 + 0.454392i \(0.150143\pi\)
\(644\) −13.8564 −0.546019
\(645\) 0 0
\(646\) −3.46410 −0.136293
\(647\) −17.0718 −0.671162 −0.335581 0.942011i \(-0.608933\pi\)
−0.335581 + 0.942011i \(0.608933\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) −5.46410 −0.214320
\(651\) 0 0
\(652\) −6.53590 −0.255966
\(653\) 45.7128 1.78888 0.894440 0.447187i \(-0.147574\pi\)
0.894440 + 0.447187i \(0.147574\pi\)
\(654\) 0 0
\(655\) 3.46410 0.135354
\(656\) 0 0
\(657\) 0 0
\(658\) 13.8564 0.540179
\(659\) −22.3923 −0.872280 −0.436140 0.899879i \(-0.643655\pi\)
−0.436140 + 0.899879i \(0.643655\pi\)
\(660\) 0 0
\(661\) 42.3923 1.64887 0.824435 0.565957i \(-0.191493\pi\)
0.824435 + 0.565957i \(0.191493\pi\)
\(662\) −14.9282 −0.580201
\(663\) 0 0
\(664\) −2.53590 −0.0984119
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 18.9282 0.732354
\(669\) 0 0
\(670\) 14.9282 0.576727
\(671\) 6.92820 0.267460
\(672\) 0 0
\(673\) 2.67949 0.103287 0.0516434 0.998666i \(-0.483554\pi\)
0.0516434 + 0.998666i \(0.483554\pi\)
\(674\) −1.32051 −0.0508641
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 36.7846 1.41166
\(680\) 3.46410 0.132842
\(681\) 0 0
\(682\) 5.07180 0.194209
\(683\) 25.8564 0.989368 0.494684 0.869073i \(-0.335284\pi\)
0.494684 + 0.869073i \(0.335284\pi\)
\(684\) 0 0
\(685\) −15.4641 −0.590853
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) 5.46410 0.208317
\(689\) 70.6410 2.69121
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 37.1769 1.41122
\(695\) −9.85641 −0.373875
\(696\) 0 0
\(697\) 0 0
\(698\) 11.8564 0.448772
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) −23.0718 −0.871410 −0.435705 0.900090i \(-0.643501\pi\)
−0.435705 + 0.900090i \(0.643501\pi\)
\(702\) 0 0
\(703\) −1.46410 −0.0552196
\(704\) 3.46410 0.130558
\(705\) 0 0
\(706\) −13.6077 −0.512132
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) 46.7846 1.75703 0.878516 0.477712i \(-0.158534\pi\)
0.878516 + 0.477712i \(0.158534\pi\)
\(710\) 6.92820 0.260011
\(711\) 0 0
\(712\) 6.92820 0.259645
\(713\) 10.1436 0.379881
\(714\) 0 0
\(715\) −18.9282 −0.707875
\(716\) −1.60770 −0.0600824
\(717\) 0 0
\(718\) 33.4641 1.24887
\(719\) −14.5359 −0.542098 −0.271049 0.962566i \(-0.587370\pi\)
−0.271049 + 0.962566i \(0.587370\pi\)
\(720\) 0 0
\(721\) −9.85641 −0.367072
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 18.3923 0.683545
\(725\) −3.46410 −0.128654
\(726\) 0 0
\(727\) 0.143594 0.00532559 0.00266279 0.999996i \(-0.499152\pi\)
0.00266279 + 0.999996i \(0.499152\pi\)
\(728\) −10.9282 −0.405026
\(729\) 0 0
\(730\) −4.92820 −0.182401
\(731\) 18.9282 0.700085
\(732\) 0 0
\(733\) −41.1769 −1.52090 −0.760452 0.649394i \(-0.775023\pi\)
−0.760452 + 0.649394i \(0.775023\pi\)
\(734\) 25.7128 0.949077
\(735\) 0 0
\(736\) 6.92820 0.255377
\(737\) 51.7128 1.90487
\(738\) 0 0
\(739\) −12.7846 −0.470289 −0.235145 0.971960i \(-0.575556\pi\)
−0.235145 + 0.971960i \(0.575556\pi\)
\(740\) 1.46410 0.0538214
\(741\) 0 0
\(742\) −25.8564 −0.949219
\(743\) 10.1436 0.372132 0.186066 0.982537i \(-0.440426\pi\)
0.186066 + 0.982537i \(0.440426\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) −0.392305 −0.0143633
\(747\) 0 0
\(748\) 12.0000 0.438763
\(749\) −13.8564 −0.506302
\(750\) 0 0
\(751\) −1.46410 −0.0534258 −0.0267129 0.999643i \(-0.508504\pi\)
−0.0267129 + 0.999643i \(0.508504\pi\)
\(752\) −6.92820 −0.252646
\(753\) 0 0
\(754\) 18.9282 0.689325
\(755\) 8.39230 0.305427
\(756\) 0 0
\(757\) −41.1769 −1.49660 −0.748300 0.663360i \(-0.769130\pi\)
−0.748300 + 0.663360i \(0.769130\pi\)
\(758\) 9.07180 0.329502
\(759\) 0 0
\(760\) 1.00000 0.0362738
\(761\) 53.5692 1.94188 0.970941 0.239318i \(-0.0769237\pi\)
0.970941 + 0.239318i \(0.0769237\pi\)
\(762\) 0 0
\(763\) −28.7846 −1.04207
\(764\) −11.3205 −0.409562
\(765\) 0 0
\(766\) 18.9282 0.683904
\(767\) 18.9282 0.683458
\(768\) 0 0
\(769\) 27.8564 1.00453 0.502264 0.864714i \(-0.332501\pi\)
0.502264 + 0.864714i \(0.332501\pi\)
\(770\) 6.92820 0.249675
\(771\) 0 0
\(772\) 6.39230 0.230064
\(773\) −7.85641 −0.282575 −0.141288 0.989969i \(-0.545124\pi\)
−0.141288 + 0.989969i \(0.545124\pi\)
\(774\) 0 0
\(775\) −1.46410 −0.0525921
\(776\) −18.3923 −0.660245
\(777\) 0 0
\(778\) −26.7846 −0.960275
\(779\) 0 0
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 24.0000 0.858238
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 15.3205 0.546812
\(786\) 0 0
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −0.928203 −0.0330659
\(789\) 0 0
\(790\) −1.46410 −0.0520904
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 10.9282 0.388072
\(794\) −19.3205 −0.685659
\(795\) 0 0
\(796\) 2.92820 0.103787
\(797\) −33.7128 −1.19417 −0.597085 0.802178i \(-0.703674\pi\)
−0.597085 + 0.802178i \(0.703674\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −12.0000 −0.423735
\(803\) −17.0718 −0.602451
\(804\) 0 0
\(805\) 13.8564 0.488374
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 46.6410 1.63981 0.819905 0.572499i \(-0.194026\pi\)
0.819905 + 0.572499i \(0.194026\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) −6.92820 −0.243132
\(813\) 0 0
\(814\) 5.07180 0.177766
\(815\) 6.53590 0.228943
\(816\) 0 0
\(817\) 5.46410 0.191165
\(818\) 30.7846 1.07636
\(819\) 0 0
\(820\) 0 0
\(821\) −7.85641 −0.274190 −0.137095 0.990558i \(-0.543777\pi\)
−0.137095 + 0.990558i \(0.543777\pi\)
\(822\) 0 0
\(823\) −42.7846 −1.49138 −0.745689 0.666294i \(-0.767879\pi\)
−0.745689 + 0.666294i \(0.767879\pi\)
\(824\) 4.92820 0.171682
\(825\) 0 0
\(826\) −6.92820 −0.241063
\(827\) 5.07180 0.176364 0.0881818 0.996104i \(-0.471894\pi\)
0.0881818 + 0.996104i \(0.471894\pi\)
\(828\) 0 0
\(829\) 47.4641 1.64850 0.824248 0.566229i \(-0.191598\pi\)
0.824248 + 0.566229i \(0.191598\pi\)
\(830\) 2.53590 0.0880223
\(831\) 0 0
\(832\) 5.46410 0.189434
\(833\) −10.3923 −0.360072
\(834\) 0 0
\(835\) −18.9282 −0.655037
\(836\) 3.46410 0.119808
\(837\) 0 0
\(838\) 13.6077 0.470070
\(839\) 13.8564 0.478376 0.239188 0.970973i \(-0.423119\pi\)
0.239188 + 0.970973i \(0.423119\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) −39.1769 −1.35013
\(843\) 0 0
\(844\) −17.8564 −0.614643
\(845\) −16.8564 −0.579878
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 12.9282 0.443956
\(849\) 0 0
\(850\) −3.46410 −0.118818
\(851\) 10.1436 0.347718
\(852\) 0 0
\(853\) 19.3205 0.661522 0.330761 0.943715i \(-0.392695\pi\)
0.330761 + 0.943715i \(0.392695\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 6.92820 0.236801
\(857\) −14.7846 −0.505033 −0.252516 0.967593i \(-0.581258\pi\)
−0.252516 + 0.967593i \(0.581258\pi\)
\(858\) 0 0
\(859\) −41.8564 −1.42812 −0.714061 0.700083i \(-0.753146\pi\)
−0.714061 + 0.700083i \(0.753146\pi\)
\(860\) −5.46410 −0.186324
\(861\) 0 0
\(862\) 18.9282 0.644697
\(863\) −42.9282 −1.46129 −0.730647 0.682756i \(-0.760781\pi\)
−0.730647 + 0.682756i \(0.760781\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 21.3205 0.724500
\(867\) 0 0
\(868\) −2.92820 −0.0993897
\(869\) −5.07180 −0.172049
\(870\) 0 0
\(871\) 81.5692 2.76387
\(872\) 14.3923 0.487385
\(873\) 0 0
\(874\) 6.92820 0.234350
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) −23.6077 −0.797175 −0.398588 0.917130i \(-0.630500\pi\)
−0.398588 + 0.917130i \(0.630500\pi\)
\(878\) 3.32051 0.112062
\(879\) 0 0
\(880\) −3.46410 −0.116775
\(881\) −6.92820 −0.233417 −0.116709 0.993166i \(-0.537234\pi\)
−0.116709 + 0.993166i \(0.537234\pi\)
\(882\) 0 0
\(883\) 3.60770 0.121409 0.0607043 0.998156i \(-0.480665\pi\)
0.0607043 + 0.998156i \(0.480665\pi\)
\(884\) 18.9282 0.636624
\(885\) 0 0
\(886\) −4.39230 −0.147562
\(887\) 41.5692 1.39576 0.697879 0.716216i \(-0.254127\pi\)
0.697879 + 0.716216i \(0.254127\pi\)
\(888\) 0 0
\(889\) 28.0000 0.939090
\(890\) −6.92820 −0.232234
\(891\) 0 0
\(892\) −23.8564 −0.798772
\(893\) −6.92820 −0.231843
\(894\) 0 0
\(895\) 1.60770 0.0537393
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −1.85641 −0.0619491
\(899\) 5.07180 0.169154
\(900\) 0 0
\(901\) 44.7846 1.49199
\(902\) 0 0
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −18.3923 −0.611381
\(906\) 0 0
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 10.9282 0.362266
\(911\) −10.1436 −0.336072 −0.168036 0.985781i \(-0.553743\pi\)
−0.168036 + 0.985781i \(0.553743\pi\)
\(912\) 0 0
\(913\) 8.78461 0.290728
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −28.9282 −0.955815
\(917\) −6.92820 −0.228789
\(918\) 0 0
\(919\) 38.9282 1.28412 0.642061 0.766653i \(-0.278079\pi\)
0.642061 + 0.766653i \(0.278079\pi\)
\(920\) −6.92820 −0.228416
\(921\) 0 0
\(922\) −36.9282 −1.21617
\(923\) 37.8564 1.24606
\(924\) 0 0
\(925\) −1.46410 −0.0481394
\(926\) 15.0718 0.495290
\(927\) 0 0
\(928\) 3.46410 0.113715
\(929\) −12.0000 −0.393707 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 8.53590 0.279603
\(933\) 0 0
\(934\) 21.4641 0.702327
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) 43.5692 1.42334 0.711672 0.702512i \(-0.247938\pi\)
0.711672 + 0.702512i \(0.247938\pi\)
\(938\) −29.8564 −0.974846
\(939\) 0 0
\(940\) 6.92820 0.225973
\(941\) −48.2487 −1.57286 −0.786432 0.617677i \(-0.788074\pi\)
−0.786432 + 0.617677i \(0.788074\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 3.46410 0.112747
\(945\) 0 0
\(946\) −18.9282 −0.615409
\(947\) −0.679492 −0.0220805 −0.0110403 0.999939i \(-0.503514\pi\)
−0.0110403 + 0.999939i \(0.503514\pi\)
\(948\) 0 0
\(949\) −26.9282 −0.874126
\(950\) −1.00000 −0.0324443
\(951\) 0 0
\(952\) −6.92820 −0.224544
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 11.3205 0.366323
\(956\) −2.53590 −0.0820168
\(957\) 0 0
\(958\) 7.60770 0.245793
\(959\) 30.9282 0.998724
\(960\) 0 0
\(961\) −28.8564 −0.930852
\(962\) 8.00000 0.257930
\(963\) 0 0
\(964\) 0.143594 0.00462484
\(965\) −6.39230 −0.205776
\(966\) 0 0
\(967\) 7.07180 0.227414 0.113707 0.993514i \(-0.463728\pi\)
0.113707 + 0.993514i \(0.463728\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 18.3923 0.590541
\(971\) −18.6795 −0.599453 −0.299727 0.954025i \(-0.596895\pi\)
−0.299727 + 0.954025i \(0.596895\pi\)
\(972\) 0 0
\(973\) 19.7128 0.631964
\(974\) −34.7846 −1.11457
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −23.5692 −0.754046 −0.377023 0.926204i \(-0.623052\pi\)
−0.377023 + 0.926204i \(0.623052\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) 41.3205 1.31859
\(983\) −60.4974 −1.92957 −0.964784 0.263043i \(-0.915274\pi\)
−0.964784 + 0.263043i \(0.915274\pi\)
\(984\) 0 0
\(985\) 0.928203 0.0295750
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 5.46410 0.173836
\(989\) −37.8564 −1.20376
\(990\) 0 0
\(991\) −8.39230 −0.266590 −0.133295 0.991076i \(-0.542556\pi\)
−0.133295 + 0.991076i \(0.542556\pi\)
\(992\) 1.46410 0.0464853
\(993\) 0 0
\(994\) −13.8564 −0.439499
\(995\) −2.92820 −0.0928303
\(996\) 0 0
\(997\) 12.3923 0.392468 0.196234 0.980557i \(-0.437129\pi\)
0.196234 + 0.980557i \(0.437129\pi\)
\(998\) 22.9282 0.725780
\(999\) 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 1710.2.a.u.1.2 2
3.2 odd 2 1710.2.a.y.1.1 yes 2
5.4 even 2 8550.2.a.bw.1.2 2
15.14 odd 2 8550.2.a.bo.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1710.2.a.u.1.2 2 1.1 even 1 trivial
1710.2.a.y.1.1 yes 2 3.2 odd 2
8550.2.a.bo.1.1 2 15.14 odd 2
8550.2.a.bw.1.2 2 5.4 even 2