Properties

Label 2320.2.a.k.1.2
Level $2320$
Weight $2$
Character 2320.1
Self dual yes
Analytic conductor $18.525$
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] = [2320,2,Mod(1,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2320.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.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: \(18.5252932689\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2320.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{3} +1.00000 q^{5} +4.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} +1.00000 q^{5} +4.82843 q^{7} +1.00000 q^{9} -0.828427 q^{11} -2.00000 q^{13} +2.00000 q^{15} +2.82843 q^{17} +4.82843 q^{19} +9.65685 q^{21} +3.17157 q^{23} +1.00000 q^{25} -4.00000 q^{27} +1.00000 q^{29} -6.48528 q^{31} -1.65685 q^{33} +4.82843 q^{35} -8.48528 q^{37} -4.00000 q^{39} -6.00000 q^{41} +6.00000 q^{43} +1.00000 q^{45} +11.6569 q^{47} +16.3137 q^{49} +5.65685 q^{51} -3.65685 q^{53} -0.828427 q^{55} +9.65685 q^{57} -3.65685 q^{61} +4.82843 q^{63} -2.00000 q^{65} -6.48528 q^{67} +6.34315 q^{69} +15.3137 q^{71} +8.48528 q^{73} +2.00000 q^{75} -4.00000 q^{77} +2.48528 q^{79} -11.0000 q^{81} -7.17157 q^{83} +2.82843 q^{85} +2.00000 q^{87} -7.65685 q^{89} -9.65685 q^{91} -12.9706 q^{93} +4.82843 q^{95} -12.4853 q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{19} + 8 q^{21} + 12 q^{23} + 2 q^{25} - 8 q^{27} + 2 q^{29} + 4 q^{31} + 8 q^{33} + 4 q^{35} - 8 q^{39} - 12 q^{41} + 12 q^{43} + 2 q^{45} + 12 q^{47} + 10 q^{49} + 4 q^{53} + 4 q^{55} + 8 q^{57} + 4 q^{61} + 4 q^{63} - 4 q^{65} + 4 q^{67} + 24 q^{69} + 8 q^{71} + 4 q^{75} - 8 q^{77} - 12 q^{79} - 22 q^{81} - 20 q^{83} + 4 q^{87} - 4 q^{89} - 8 q^{91} + 8 q^{93} + 4 q^{95} - 8 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.82843 1.82497 0.912487 0.409106i \(-0.134159\pi\)
0.912487 + 0.409106i \(0.134159\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.828427 −0.249780 −0.124890 0.992171i \(-0.539858\pi\)
−0.124890 + 0.992171i \(0.539858\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) 4.82843 1.10772 0.553859 0.832611i \(-0.313155\pi\)
0.553859 + 0.832611i \(0.313155\pi\)
\(20\) 0 0
\(21\) 9.65685 2.10730
\(22\) 0 0
\(23\) 3.17157 0.661319 0.330659 0.943750i \(-0.392729\pi\)
0.330659 + 0.943750i \(0.392729\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −6.48528 −1.16479 −0.582395 0.812906i \(-0.697884\pi\)
−0.582395 + 0.812906i \(0.697884\pi\)
\(32\) 0 0
\(33\) −1.65685 −0.288421
\(34\) 0 0
\(35\) 4.82843 0.816153
\(36\) 0 0
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 11.6569 1.70033 0.850163 0.526519i \(-0.176503\pi\)
0.850163 + 0.526519i \(0.176503\pi\)
\(48\) 0 0
\(49\) 16.3137 2.33053
\(50\) 0 0
\(51\) 5.65685 0.792118
\(52\) 0 0
\(53\) −3.65685 −0.502308 −0.251154 0.967947i \(-0.580810\pi\)
−0.251154 + 0.967947i \(0.580810\pi\)
\(54\) 0 0
\(55\) −0.828427 −0.111705
\(56\) 0 0
\(57\) 9.65685 1.27908
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −3.65685 −0.468212 −0.234106 0.972211i \(-0.575216\pi\)
−0.234106 + 0.972211i \(0.575216\pi\)
\(62\) 0 0
\(63\) 4.82843 0.608325
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −6.48528 −0.792303 −0.396152 0.918185i \(-0.629655\pi\)
−0.396152 + 0.918185i \(0.629655\pi\)
\(68\) 0 0
\(69\) 6.34315 0.763625
\(70\) 0 0
\(71\) 15.3137 1.81740 0.908701 0.417447i \(-0.137075\pi\)
0.908701 + 0.417447i \(0.137075\pi\)
\(72\) 0 0
\(73\) 8.48528 0.993127 0.496564 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) 0 0
\(75\) 2.00000 0.230940
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 2.48528 0.279616 0.139808 0.990179i \(-0.455351\pi\)
0.139808 + 0.990179i \(0.455351\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −7.17157 −0.787182 −0.393591 0.919286i \(-0.628767\pi\)
−0.393591 + 0.919286i \(0.628767\pi\)
\(84\) 0 0
\(85\) 2.82843 0.306786
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −7.65685 −0.811625 −0.405812 0.913956i \(-0.633011\pi\)
−0.405812 + 0.913956i \(0.633011\pi\)
\(90\) 0 0
\(91\) −9.65685 −1.01231
\(92\) 0 0
\(93\) −12.9706 −1.34498
\(94\) 0 0
\(95\) 4.82843 0.495386
\(96\) 0 0
\(97\) −12.4853 −1.26769 −0.633844 0.773461i \(-0.718524\pi\)
−0.633844 + 0.773461i \(0.718524\pi\)
\(98\) 0 0
\(99\) −0.828427 −0.0832601
\(100\) 0 0
\(101\) 15.6569 1.55792 0.778958 0.627077i \(-0.215749\pi\)
0.778958 + 0.627077i \(0.215749\pi\)
\(102\) 0 0
\(103\) −16.1421 −1.59053 −0.795266 0.606261i \(-0.792669\pi\)
−0.795266 + 0.606261i \(0.792669\pi\)
\(104\) 0 0
\(105\) 9.65685 0.942412
\(106\) 0 0
\(107\) −20.1421 −1.94721 −0.973607 0.228232i \(-0.926706\pi\)
−0.973607 + 0.228232i \(0.926706\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −16.9706 −1.61077
\(112\) 0 0
\(113\) −2.82843 −0.266076 −0.133038 0.991111i \(-0.542473\pi\)
−0.133038 + 0.991111i \(0.542473\pi\)
\(114\) 0 0
\(115\) 3.17157 0.295751
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 13.6569 1.25192
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 12.1421 1.06086 0.530432 0.847728i \(-0.322030\pi\)
0.530432 + 0.847728i \(0.322030\pi\)
\(132\) 0 0
\(133\) 23.3137 2.02155
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) −5.17157 −0.441837 −0.220919 0.975292i \(-0.570906\pi\)
−0.220919 + 0.975292i \(0.570906\pi\)
\(138\) 0 0
\(139\) −21.6569 −1.83691 −0.918455 0.395525i \(-0.870563\pi\)
−0.918455 + 0.395525i \(0.870563\pi\)
\(140\) 0 0
\(141\) 23.3137 1.96337
\(142\) 0 0
\(143\) 1.65685 0.138553
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) 32.6274 2.69106
\(148\) 0 0
\(149\) 9.31371 0.763009 0.381504 0.924367i \(-0.375406\pi\)
0.381504 + 0.924367i \(0.375406\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 2.82843 0.228665
\(154\) 0 0
\(155\) −6.48528 −0.520910
\(156\) 0 0
\(157\) 0.485281 0.0387297 0.0193648 0.999812i \(-0.493836\pi\)
0.0193648 + 0.999812i \(0.493836\pi\)
\(158\) 0 0
\(159\) −7.31371 −0.580015
\(160\) 0 0
\(161\) 15.3137 1.20689
\(162\) 0 0
\(163\) 8.34315 0.653486 0.326743 0.945113i \(-0.394049\pi\)
0.326743 + 0.945113i \(0.394049\pi\)
\(164\) 0 0
\(165\) −1.65685 −0.128986
\(166\) 0 0
\(167\) 2.48528 0.192317 0.0961584 0.995366i \(-0.469344\pi\)
0.0961584 + 0.995366i \(0.469344\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.82843 0.369239
\(172\) 0 0
\(173\) 17.3137 1.31634 0.658168 0.752871i \(-0.271331\pi\)
0.658168 + 0.752871i \(0.271331\pi\)
\(174\) 0 0
\(175\) 4.82843 0.364995
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 23.3137 1.74255 0.871274 0.490797i \(-0.163294\pi\)
0.871274 + 0.490797i \(0.163294\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) −7.31371 −0.540645
\(184\) 0 0
\(185\) −8.48528 −0.623850
\(186\) 0 0
\(187\) −2.34315 −0.171348
\(188\) 0 0
\(189\) −19.3137 −1.40487
\(190\) 0 0
\(191\) 20.8284 1.50709 0.753546 0.657395i \(-0.228342\pi\)
0.753546 + 0.657395i \(0.228342\pi\)
\(192\) 0 0
\(193\) 4.48528 0.322858 0.161429 0.986884i \(-0.448390\pi\)
0.161429 + 0.986884i \(0.448390\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) −19.6569 −1.40049 −0.700246 0.713901i \(-0.746927\pi\)
−0.700246 + 0.713901i \(0.746927\pi\)
\(198\) 0 0
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 0 0
\(201\) −12.9706 −0.914873
\(202\) 0 0
\(203\) 4.82843 0.338889
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 3.17157 0.220440
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 0.828427 0.0570313 0.0285156 0.999593i \(-0.490922\pi\)
0.0285156 + 0.999593i \(0.490922\pi\)
\(212\) 0 0
\(213\) 30.6274 2.09856
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) −31.3137 −2.12571
\(218\) 0 0
\(219\) 16.9706 1.14676
\(220\) 0 0
\(221\) −5.65685 −0.380521
\(222\) 0 0
\(223\) 17.7990 1.19191 0.595954 0.803018i \(-0.296774\pi\)
0.595954 + 0.803018i \(0.296774\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −20.1421 −1.33688 −0.668440 0.743766i \(-0.733038\pi\)
−0.668440 + 0.743766i \(0.733038\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 11.6569 0.760409
\(236\) 0 0
\(237\) 4.97056 0.322873
\(238\) 0 0
\(239\) 0.686292 0.0443925 0.0221963 0.999754i \(-0.492934\pi\)
0.0221963 + 0.999754i \(0.492934\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) 16.3137 1.04224
\(246\) 0 0
\(247\) −9.65685 −0.614451
\(248\) 0 0
\(249\) −14.3431 −0.908960
\(250\) 0 0
\(251\) −8.82843 −0.557245 −0.278623 0.960401i \(-0.589878\pi\)
−0.278623 + 0.960401i \(0.589878\pi\)
\(252\) 0 0
\(253\) −2.62742 −0.165184
\(254\) 0 0
\(255\) 5.65685 0.354246
\(256\) 0 0
\(257\) 6.68629 0.417079 0.208540 0.978014i \(-0.433129\pi\)
0.208540 + 0.978014i \(0.433129\pi\)
\(258\) 0 0
\(259\) −40.9706 −2.54579
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 19.6569 1.21209 0.606047 0.795429i \(-0.292754\pi\)
0.606047 + 0.795429i \(0.292754\pi\)
\(264\) 0 0
\(265\) −3.65685 −0.224639
\(266\) 0 0
\(267\) −15.3137 −0.937184
\(268\) 0 0
\(269\) −21.3137 −1.29952 −0.649760 0.760140i \(-0.725131\pi\)
−0.649760 + 0.760140i \(0.725131\pi\)
\(270\) 0 0
\(271\) 9.79899 0.595246 0.297623 0.954683i \(-0.403806\pi\)
0.297623 + 0.954683i \(0.403806\pi\)
\(272\) 0 0
\(273\) −19.3137 −1.16892
\(274\) 0 0
\(275\) −0.828427 −0.0499560
\(276\) 0 0
\(277\) −3.65685 −0.219719 −0.109860 0.993947i \(-0.535040\pi\)
−0.109860 + 0.993947i \(0.535040\pi\)
\(278\) 0 0
\(279\) −6.48528 −0.388264
\(280\) 0 0
\(281\) −29.3137 −1.74871 −0.874355 0.485288i \(-0.838715\pi\)
−0.874355 + 0.485288i \(0.838715\pi\)
\(282\) 0 0
\(283\) −4.82843 −0.287020 −0.143510 0.989649i \(-0.545839\pi\)
−0.143510 + 0.989649i \(0.545839\pi\)
\(284\) 0 0
\(285\) 9.65685 0.572023
\(286\) 0 0
\(287\) −28.9706 −1.71008
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) −24.9706 −1.46380
\(292\) 0 0
\(293\) 8.48528 0.495715 0.247858 0.968796i \(-0.420273\pi\)
0.247858 + 0.968796i \(0.420273\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.31371 0.192281
\(298\) 0 0
\(299\) −6.34315 −0.366834
\(300\) 0 0
\(301\) 28.9706 1.66984
\(302\) 0 0
\(303\) 31.3137 1.79893
\(304\) 0 0
\(305\) −3.65685 −0.209391
\(306\) 0 0
\(307\) 22.9706 1.31100 0.655500 0.755195i \(-0.272458\pi\)
0.655500 + 0.755195i \(0.272458\pi\)
\(308\) 0 0
\(309\) −32.2843 −1.83659
\(310\) 0 0
\(311\) −14.4853 −0.821385 −0.410692 0.911774i \(-0.634713\pi\)
−0.410692 + 0.911774i \(0.634713\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 4.82843 0.272051
\(316\) 0 0
\(317\) 2.82843 0.158860 0.0794301 0.996840i \(-0.474690\pi\)
0.0794301 + 0.996840i \(0.474690\pi\)
\(318\) 0 0
\(319\) −0.828427 −0.0463830
\(320\) 0 0
\(321\) −40.2843 −2.24845
\(322\) 0 0
\(323\) 13.6569 0.759888
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 4.00000 0.221201
\(328\) 0 0
\(329\) 56.2843 3.10305
\(330\) 0 0
\(331\) −21.7990 −1.19818 −0.599090 0.800681i \(-0.704471\pi\)
−0.599090 + 0.800681i \(0.704471\pi\)
\(332\) 0 0
\(333\) −8.48528 −0.464991
\(334\) 0 0
\(335\) −6.48528 −0.354329
\(336\) 0 0
\(337\) −1.17157 −0.0638196 −0.0319098 0.999491i \(-0.510159\pi\)
−0.0319098 + 0.999491i \(0.510159\pi\)
\(338\) 0 0
\(339\) −5.65685 −0.307238
\(340\) 0 0
\(341\) 5.37258 0.290942
\(342\) 0 0
\(343\) 44.9706 2.42818
\(344\) 0 0
\(345\) 6.34315 0.341503
\(346\) 0 0
\(347\) 8.14214 0.437093 0.218546 0.975827i \(-0.429869\pi\)
0.218546 + 0.975827i \(0.429869\pi\)
\(348\) 0 0
\(349\) 20.6274 1.10416 0.552080 0.833791i \(-0.313834\pi\)
0.552080 + 0.833791i \(0.313834\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) −4.34315 −0.231162 −0.115581 0.993298i \(-0.536873\pi\)
−0.115581 + 0.993298i \(0.536873\pi\)
\(354\) 0 0
\(355\) 15.3137 0.812767
\(356\) 0 0
\(357\) 27.3137 1.44559
\(358\) 0 0
\(359\) 3.85786 0.203610 0.101805 0.994804i \(-0.467538\pi\)
0.101805 + 0.994804i \(0.467538\pi\)
\(360\) 0 0
\(361\) 4.31371 0.227037
\(362\) 0 0
\(363\) −20.6274 −1.08266
\(364\) 0 0
\(365\) 8.48528 0.444140
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −17.6569 −0.916698
\(372\) 0 0
\(373\) −6.97056 −0.360922 −0.180461 0.983582i \(-0.557759\pi\)
−0.180461 + 0.983582i \(0.557759\pi\)
\(374\) 0 0
\(375\) 2.00000 0.103280
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −22.4853 −1.15499 −0.577496 0.816394i \(-0.695970\pi\)
−0.577496 + 0.816394i \(0.695970\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) −2.48528 −0.126992 −0.0634960 0.997982i \(-0.520225\pi\)
−0.0634960 + 0.997982i \(0.520225\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 0 0
\(387\) 6.00000 0.304997
\(388\) 0 0
\(389\) −29.3137 −1.48626 −0.743132 0.669145i \(-0.766661\pi\)
−0.743132 + 0.669145i \(0.766661\pi\)
\(390\) 0 0
\(391\) 8.97056 0.453661
\(392\) 0 0
\(393\) 24.2843 1.22498
\(394\) 0 0
\(395\) 2.48528 0.125048
\(396\) 0 0
\(397\) −19.6569 −0.986549 −0.493275 0.869874i \(-0.664200\pi\)
−0.493275 + 0.869874i \(0.664200\pi\)
\(398\) 0 0
\(399\) 46.6274 2.33429
\(400\) 0 0
\(401\) −6.68629 −0.333897 −0.166949 0.985966i \(-0.553391\pi\)
−0.166949 + 0.985966i \(0.553391\pi\)
\(402\) 0 0
\(403\) 12.9706 0.646110
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) 7.02944 0.348436
\(408\) 0 0
\(409\) −2.97056 −0.146885 −0.0734424 0.997299i \(-0.523399\pi\)
−0.0734424 + 0.997299i \(0.523399\pi\)
\(410\) 0 0
\(411\) −10.3431 −0.510190
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −7.17157 −0.352039
\(416\) 0 0
\(417\) −43.3137 −2.12108
\(418\) 0 0
\(419\) −28.9706 −1.41530 −0.707652 0.706561i \(-0.750246\pi\)
−0.707652 + 0.706561i \(0.750246\pi\)
\(420\) 0 0
\(421\) 18.9706 0.924569 0.462284 0.886732i \(-0.347030\pi\)
0.462284 + 0.886732i \(0.347030\pi\)
\(422\) 0 0
\(423\) 11.6569 0.566776
\(424\) 0 0
\(425\) 2.82843 0.137199
\(426\) 0 0
\(427\) −17.6569 −0.854475
\(428\) 0 0
\(429\) 3.31371 0.159987
\(430\) 0 0
\(431\) −3.31371 −0.159616 −0.0798079 0.996810i \(-0.525431\pi\)
−0.0798079 + 0.996810i \(0.525431\pi\)
\(432\) 0 0
\(433\) −29.1716 −1.40190 −0.700948 0.713212i \(-0.747240\pi\)
−0.700948 + 0.713212i \(0.747240\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) 0 0
\(437\) 15.3137 0.732554
\(438\) 0 0
\(439\) 10.3431 0.493651 0.246826 0.969060i \(-0.420612\pi\)
0.246826 + 0.969060i \(0.420612\pi\)
\(440\) 0 0
\(441\) 16.3137 0.776843
\(442\) 0 0
\(443\) 7.65685 0.363788 0.181894 0.983318i \(-0.441777\pi\)
0.181894 + 0.983318i \(0.441777\pi\)
\(444\) 0 0
\(445\) −7.65685 −0.362970
\(446\) 0 0
\(447\) 18.6274 0.881047
\(448\) 0 0
\(449\) 11.6569 0.550121 0.275060 0.961427i \(-0.411302\pi\)
0.275060 + 0.961427i \(0.411302\pi\)
\(450\) 0 0
\(451\) 4.97056 0.234055
\(452\) 0 0
\(453\) 24.0000 1.12762
\(454\) 0 0
\(455\) −9.65685 −0.452720
\(456\) 0 0
\(457\) 19.6569 0.919509 0.459754 0.888046i \(-0.347937\pi\)
0.459754 + 0.888046i \(0.347937\pi\)
\(458\) 0 0
\(459\) −11.3137 −0.528079
\(460\) 0 0
\(461\) −35.6569 −1.66071 −0.830353 0.557238i \(-0.811861\pi\)
−0.830353 + 0.557238i \(0.811861\pi\)
\(462\) 0 0
\(463\) −21.7990 −1.01308 −0.506542 0.862215i \(-0.669077\pi\)
−0.506542 + 0.862215i \(0.669077\pi\)
\(464\) 0 0
\(465\) −12.9706 −0.601495
\(466\) 0 0
\(467\) −10.9706 −0.507657 −0.253829 0.967249i \(-0.581690\pi\)
−0.253829 + 0.967249i \(0.581690\pi\)
\(468\) 0 0
\(469\) −31.3137 −1.44593
\(470\) 0 0
\(471\) 0.970563 0.0447212
\(472\) 0 0
\(473\) −4.97056 −0.228547
\(474\) 0 0
\(475\) 4.82843 0.221543
\(476\) 0 0
\(477\) −3.65685 −0.167436
\(478\) 0 0
\(479\) −7.17157 −0.327678 −0.163839 0.986487i \(-0.552388\pi\)
−0.163839 + 0.986487i \(0.552388\pi\)
\(480\) 0 0
\(481\) 16.9706 0.773791
\(482\) 0 0
\(483\) 30.6274 1.39360
\(484\) 0 0
\(485\) −12.4853 −0.566927
\(486\) 0 0
\(487\) −9.79899 −0.444035 −0.222017 0.975043i \(-0.571264\pi\)
−0.222017 + 0.975043i \(0.571264\pi\)
\(488\) 0 0
\(489\) 16.6863 0.754580
\(490\) 0 0
\(491\) 7.45584 0.336478 0.168239 0.985746i \(-0.446192\pi\)
0.168239 + 0.985746i \(0.446192\pi\)
\(492\) 0 0
\(493\) 2.82843 0.127386
\(494\) 0 0
\(495\) −0.828427 −0.0372350
\(496\) 0 0
\(497\) 73.9411 3.31671
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) 4.97056 0.222068
\(502\) 0 0
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) 15.6569 0.696721
\(506\) 0 0
\(507\) −18.0000 −0.799408
\(508\) 0 0
\(509\) 0.627417 0.0278098 0.0139049 0.999903i \(-0.495574\pi\)
0.0139049 + 0.999903i \(0.495574\pi\)
\(510\) 0 0
\(511\) 40.9706 1.81243
\(512\) 0 0
\(513\) −19.3137 −0.852721
\(514\) 0 0
\(515\) −16.1421 −0.711307
\(516\) 0 0
\(517\) −9.65685 −0.424708
\(518\) 0 0
\(519\) 34.6274 1.51997
\(520\) 0 0
\(521\) −21.3137 −0.933771 −0.466885 0.884318i \(-0.654624\pi\)
−0.466885 + 0.884318i \(0.654624\pi\)
\(522\) 0 0
\(523\) −2.48528 −0.108674 −0.0543369 0.998523i \(-0.517304\pi\)
−0.0543369 + 0.998523i \(0.517304\pi\)
\(524\) 0 0
\(525\) 9.65685 0.421460
\(526\) 0 0
\(527\) −18.3431 −0.799040
\(528\) 0 0
\(529\) −12.9411 −0.562658
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) −20.1421 −0.870820
\(536\) 0 0
\(537\) 46.6274 2.01212
\(538\) 0 0
\(539\) −13.5147 −0.582120
\(540\) 0 0
\(541\) −5.02944 −0.216232 −0.108116 0.994138i \(-0.534482\pi\)
−0.108116 + 0.994138i \(0.534482\pi\)
\(542\) 0 0
\(543\) −12.0000 −0.514969
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 2.48528 0.106263 0.0531315 0.998588i \(-0.483080\pi\)
0.0531315 + 0.998588i \(0.483080\pi\)
\(548\) 0 0
\(549\) −3.65685 −0.156071
\(550\) 0 0
\(551\) 4.82843 0.205698
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 0 0
\(555\) −16.9706 −0.720360
\(556\) 0 0
\(557\) −27.9411 −1.18390 −0.591952 0.805973i \(-0.701642\pi\)
−0.591952 + 0.805973i \(0.701642\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −4.68629 −0.197855
\(562\) 0 0
\(563\) −7.65685 −0.322698 −0.161349 0.986897i \(-0.551584\pi\)
−0.161349 + 0.986897i \(0.551584\pi\)
\(564\) 0 0
\(565\) −2.82843 −0.118993
\(566\) 0 0
\(567\) −53.1127 −2.23052
\(568\) 0 0
\(569\) 27.6569 1.15944 0.579718 0.814817i \(-0.303163\pi\)
0.579718 + 0.814817i \(0.303163\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 41.6569 1.74024
\(574\) 0 0
\(575\) 3.17157 0.132264
\(576\) 0 0
\(577\) 23.7990 0.990765 0.495382 0.868675i \(-0.335028\pi\)
0.495382 + 0.868675i \(0.335028\pi\)
\(578\) 0 0
\(579\) 8.97056 0.372804
\(580\) 0 0
\(581\) −34.6274 −1.43659
\(582\) 0 0
\(583\) 3.02944 0.125466
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) 29.7990 1.22994 0.614968 0.788552i \(-0.289169\pi\)
0.614968 + 0.788552i \(0.289169\pi\)
\(588\) 0 0
\(589\) −31.3137 −1.29026
\(590\) 0 0
\(591\) −39.3137 −1.61715
\(592\) 0 0
\(593\) −7.65685 −0.314429 −0.157215 0.987564i \(-0.550251\pi\)
−0.157215 + 0.987564i \(0.550251\pi\)
\(594\) 0 0
\(595\) 13.6569 0.559876
\(596\) 0 0
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) 37.7990 1.54442 0.772212 0.635364i \(-0.219150\pi\)
0.772212 + 0.635364i \(0.219150\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) −6.48528 −0.264101
\(604\) 0 0
\(605\) −10.3137 −0.419312
\(606\) 0 0
\(607\) −9.02944 −0.366494 −0.183247 0.983067i \(-0.558661\pi\)
−0.183247 + 0.983067i \(0.558661\pi\)
\(608\) 0 0
\(609\) 9.65685 0.391315
\(610\) 0 0
\(611\) −23.3137 −0.943172
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 0 0
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −9.17157 −0.369234 −0.184617 0.982811i \(-0.559104\pi\)
−0.184617 + 0.982811i \(0.559104\pi\)
\(618\) 0 0
\(619\) 9.79899 0.393855 0.196927 0.980418i \(-0.436904\pi\)
0.196927 + 0.980418i \(0.436904\pi\)
\(620\) 0 0
\(621\) −12.6863 −0.509083
\(622\) 0 0
\(623\) −36.9706 −1.48119
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −8.00000 −0.319489
\(628\) 0 0
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −36.9706 −1.47177 −0.735887 0.677104i \(-0.763235\pi\)
−0.735887 + 0.677104i \(0.763235\pi\)
\(632\) 0 0
\(633\) 1.65685 0.0658540
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) −32.6274 −1.29275
\(638\) 0 0
\(639\) 15.3137 0.605801
\(640\) 0 0
\(641\) 0.627417 0.0247815 0.0123907 0.999923i \(-0.496056\pi\)
0.0123907 + 0.999923i \(0.496056\pi\)
\(642\) 0 0
\(643\) −19.4558 −0.767264 −0.383632 0.923486i \(-0.625327\pi\)
−0.383632 + 0.923486i \(0.625327\pi\)
\(644\) 0 0
\(645\) 12.0000 0.472500
\(646\) 0 0
\(647\) 41.1127 1.61631 0.808153 0.588972i \(-0.200467\pi\)
0.808153 + 0.588972i \(0.200467\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −62.6274 −2.45456
\(652\) 0 0
\(653\) −17.1716 −0.671976 −0.335988 0.941866i \(-0.609070\pi\)
−0.335988 + 0.941866i \(0.609070\pi\)
\(654\) 0 0
\(655\) 12.1421 0.474432
\(656\) 0 0
\(657\) 8.48528 0.331042
\(658\) 0 0
\(659\) −1.79899 −0.0700787 −0.0350393 0.999386i \(-0.511156\pi\)
−0.0350393 + 0.999386i \(0.511156\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 0 0
\(663\) −11.3137 −0.439388
\(664\) 0 0
\(665\) 23.3137 0.904067
\(666\) 0 0
\(667\) 3.17157 0.122804
\(668\) 0 0
\(669\) 35.5980 1.37630
\(670\) 0 0
\(671\) 3.02944 0.116950
\(672\) 0 0
\(673\) 22.9706 0.885450 0.442725 0.896657i \(-0.354012\pi\)
0.442725 + 0.896657i \(0.354012\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −36.7696 −1.41317 −0.706584 0.707629i \(-0.749765\pi\)
−0.706584 + 0.707629i \(0.749765\pi\)
\(678\) 0 0
\(679\) −60.2843 −2.31350
\(680\) 0 0
\(681\) −40.2843 −1.54370
\(682\) 0 0
\(683\) −11.8579 −0.453729 −0.226864 0.973926i \(-0.572847\pi\)
−0.226864 + 0.973926i \(0.572847\pi\)
\(684\) 0 0
\(685\) −5.17157 −0.197596
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) 0 0
\(689\) 7.31371 0.278630
\(690\) 0 0
\(691\) 44.9706 1.71076 0.855380 0.518000i \(-0.173323\pi\)
0.855380 + 0.518000i \(0.173323\pi\)
\(692\) 0 0
\(693\) −4.00000 −0.151947
\(694\) 0 0
\(695\) −21.6569 −0.821491
\(696\) 0 0
\(697\) −16.9706 −0.642806
\(698\) 0 0
\(699\) 36.0000 1.36165
\(700\) 0 0
\(701\) 6.68629 0.252538 0.126269 0.991996i \(-0.459700\pi\)
0.126269 + 0.991996i \(0.459700\pi\)
\(702\) 0 0
\(703\) −40.9706 −1.54523
\(704\) 0 0
\(705\) 23.3137 0.878045
\(706\) 0 0
\(707\) 75.5980 2.84315
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 2.48528 0.0932053
\(712\) 0 0
\(713\) −20.5685 −0.770298
\(714\) 0 0
\(715\) 1.65685 0.0619628
\(716\) 0 0
\(717\) 1.37258 0.0512601
\(718\) 0 0
\(719\) 34.6274 1.29138 0.645692 0.763598i \(-0.276569\pi\)
0.645692 + 0.763598i \(0.276569\pi\)
\(720\) 0 0
\(721\) −77.9411 −2.90268
\(722\) 0 0
\(723\) 20.0000 0.743808
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 23.9411 0.887927 0.443964 0.896045i \(-0.353572\pi\)
0.443964 + 0.896045i \(0.353572\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 16.9706 0.627679
\(732\) 0 0
\(733\) −22.8284 −0.843187 −0.421594 0.906785i \(-0.638529\pi\)
−0.421594 + 0.906785i \(0.638529\pi\)
\(734\) 0 0
\(735\) 32.6274 1.20348
\(736\) 0 0
\(737\) 5.37258 0.197902
\(738\) 0 0
\(739\) −14.4853 −0.532850 −0.266425 0.963856i \(-0.585842\pi\)
−0.266425 + 0.963856i \(0.585842\pi\)
\(740\) 0 0
\(741\) −19.3137 −0.709507
\(742\) 0 0
\(743\) 52.6274 1.93071 0.965356 0.260935i \(-0.0840309\pi\)
0.965356 + 0.260935i \(0.0840309\pi\)
\(744\) 0 0
\(745\) 9.31371 0.341228
\(746\) 0 0
\(747\) −7.17157 −0.262394
\(748\) 0 0
\(749\) −97.2548 −3.55361
\(750\) 0 0
\(751\) −16.1421 −0.589035 −0.294517 0.955646i \(-0.595159\pi\)
−0.294517 + 0.955646i \(0.595159\pi\)
\(752\) 0 0
\(753\) −17.6569 −0.643452
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 19.5147 0.709275 0.354637 0.935004i \(-0.384604\pi\)
0.354637 + 0.935004i \(0.384604\pi\)
\(758\) 0 0
\(759\) −5.25483 −0.190738
\(760\) 0 0
\(761\) 8.62742 0.312744 0.156372 0.987698i \(-0.450020\pi\)
0.156372 + 0.987698i \(0.450020\pi\)
\(762\) 0 0
\(763\) 9.65685 0.349602
\(764\) 0 0
\(765\) 2.82843 0.102262
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −15.6569 −0.564601 −0.282300 0.959326i \(-0.591097\pi\)
−0.282300 + 0.959326i \(0.591097\pi\)
\(770\) 0 0
\(771\) 13.3726 0.481602
\(772\) 0 0
\(773\) −8.48528 −0.305194 −0.152597 0.988288i \(-0.548764\pi\)
−0.152597 + 0.988288i \(0.548764\pi\)
\(774\) 0 0
\(775\) −6.48528 −0.232958
\(776\) 0 0
\(777\) −81.9411 −2.93962
\(778\) 0 0
\(779\) −28.9706 −1.03798
\(780\) 0 0
\(781\) −12.6863 −0.453951
\(782\) 0 0
\(783\) −4.00000 −0.142948
\(784\) 0 0
\(785\) 0.485281 0.0173204
\(786\) 0 0
\(787\) −17.7990 −0.634465 −0.317233 0.948348i \(-0.602754\pi\)
−0.317233 + 0.948348i \(0.602754\pi\)
\(788\) 0 0
\(789\) 39.3137 1.39961
\(790\) 0 0
\(791\) −13.6569 −0.485582
\(792\) 0 0
\(793\) 7.31371 0.259717
\(794\) 0 0
\(795\) −7.31371 −0.259391
\(796\) 0 0
\(797\) −5.85786 −0.207496 −0.103748 0.994604i \(-0.533084\pi\)
−0.103748 + 0.994604i \(0.533084\pi\)
\(798\) 0 0
\(799\) 32.9706 1.16641
\(800\) 0 0
\(801\) −7.65685 −0.270542
\(802\) 0 0
\(803\) −7.02944 −0.248063
\(804\) 0 0
\(805\) 15.3137 0.539737
\(806\) 0 0
\(807\) −42.6274 −1.50056
\(808\) 0 0
\(809\) 42.2843 1.48664 0.743318 0.668938i \(-0.233251\pi\)
0.743318 + 0.668938i \(0.233251\pi\)
\(810\) 0 0
\(811\) −37.6569 −1.32231 −0.661155 0.750249i \(-0.729934\pi\)
−0.661155 + 0.750249i \(0.729934\pi\)
\(812\) 0 0
\(813\) 19.5980 0.687331
\(814\) 0 0
\(815\) 8.34315 0.292248
\(816\) 0 0
\(817\) 28.9706 1.01355
\(818\) 0 0
\(819\) −9.65685 −0.337438
\(820\) 0 0
\(821\) −22.6863 −0.791757 −0.395879 0.918303i \(-0.629560\pi\)
−0.395879 + 0.918303i \(0.629560\pi\)
\(822\) 0 0
\(823\) 30.9706 1.07957 0.539783 0.841804i \(-0.318506\pi\)
0.539783 + 0.841804i \(0.318506\pi\)
\(824\) 0 0
\(825\) −1.65685 −0.0576843
\(826\) 0 0
\(827\) −17.3137 −0.602057 −0.301028 0.953615i \(-0.597330\pi\)
−0.301028 + 0.953615i \(0.597330\pi\)
\(828\) 0 0
\(829\) 20.6274 0.716420 0.358210 0.933641i \(-0.383387\pi\)
0.358210 + 0.933641i \(0.383387\pi\)
\(830\) 0 0
\(831\) −7.31371 −0.253710
\(832\) 0 0
\(833\) 46.1421 1.59873
\(834\) 0 0
\(835\) 2.48528 0.0860067
\(836\) 0 0
\(837\) 25.9411 0.896656
\(838\) 0 0
\(839\) −2.48528 −0.0858014 −0.0429007 0.999079i \(-0.513660\pi\)
−0.0429007 + 0.999079i \(0.513660\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −58.6274 −2.01924
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −49.7990 −1.71111
\(848\) 0 0
\(849\) −9.65685 −0.331422
\(850\) 0 0
\(851\) −26.9117 −0.922521
\(852\) 0 0
\(853\) 51.1127 1.75007 0.875033 0.484064i \(-0.160840\pi\)
0.875033 + 0.484064i \(0.160840\pi\)
\(854\) 0 0
\(855\) 4.82843 0.165129
\(856\) 0 0
\(857\) 3.37258 0.115205 0.0576026 0.998340i \(-0.481654\pi\)
0.0576026 + 0.998340i \(0.481654\pi\)
\(858\) 0 0
\(859\) 56.4264 1.92524 0.962622 0.270848i \(-0.0873041\pi\)
0.962622 + 0.270848i \(0.0873041\pi\)
\(860\) 0 0
\(861\) −57.9411 −1.97463
\(862\) 0 0
\(863\) 36.1421 1.23029 0.615146 0.788413i \(-0.289097\pi\)
0.615146 + 0.788413i \(0.289097\pi\)
\(864\) 0 0
\(865\) 17.3137 0.588684
\(866\) 0 0
\(867\) −18.0000 −0.611312
\(868\) 0 0
\(869\) −2.05887 −0.0698425
\(870\) 0 0
\(871\) 12.9706 0.439491
\(872\) 0 0
\(873\) −12.4853 −0.422563
\(874\) 0 0
\(875\) 4.82843 0.163231
\(876\) 0 0
\(877\) 38.2843 1.29277 0.646384 0.763012i \(-0.276280\pi\)
0.646384 + 0.763012i \(0.276280\pi\)
\(878\) 0 0
\(879\) 16.9706 0.572403
\(880\) 0 0
\(881\) 29.3137 0.987604 0.493802 0.869574i \(-0.335607\pi\)
0.493802 + 0.869574i \(0.335607\pi\)
\(882\) 0 0
\(883\) 14.4853 0.487469 0.243734 0.969842i \(-0.421628\pi\)
0.243734 + 0.969842i \(0.421628\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.68629 0.224504 0.112252 0.993680i \(-0.464194\pi\)
0.112252 + 0.993680i \(0.464194\pi\)
\(888\) 0 0
\(889\) −28.9706 −0.971641
\(890\) 0 0
\(891\) 9.11270 0.305287
\(892\) 0 0
\(893\) 56.2843 1.88348
\(894\) 0 0
\(895\) 23.3137 0.779291
\(896\) 0 0
\(897\) −12.6863 −0.423583
\(898\) 0 0
\(899\) −6.48528 −0.216296
\(900\) 0 0
\(901\) −10.3431 −0.344580
\(902\) 0 0
\(903\) 57.9411 1.92816
\(904\) 0 0
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) 0 0
\(909\) 15.6569 0.519305
\(910\) 0 0
\(911\) −32.1421 −1.06492 −0.532458 0.846456i \(-0.678732\pi\)
−0.532458 + 0.846456i \(0.678732\pi\)
\(912\) 0 0
\(913\) 5.94113 0.196623
\(914\) 0 0
\(915\) −7.31371 −0.241784
\(916\) 0 0
\(917\) 58.6274 1.93605
\(918\) 0 0
\(919\) 36.0000 1.18753 0.593765 0.804638i \(-0.297641\pi\)
0.593765 + 0.804638i \(0.297641\pi\)
\(920\) 0 0
\(921\) 45.9411 1.51381
\(922\) 0 0
\(923\) −30.6274 −1.00811
\(924\) 0 0
\(925\) −8.48528 −0.278994
\(926\) 0 0
\(927\) −16.1421 −0.530177
\(928\) 0 0
\(929\) −4.62742 −0.151821 −0.0759103 0.997115i \(-0.524186\pi\)
−0.0759103 + 0.997115i \(0.524186\pi\)
\(930\) 0 0
\(931\) 78.7696 2.58157
\(932\) 0 0
\(933\) −28.9706 −0.948454
\(934\) 0 0
\(935\) −2.34315 −0.0766291
\(936\) 0 0
\(937\) 19.6569 0.642161 0.321081 0.947052i \(-0.395954\pi\)
0.321081 + 0.947052i \(0.395954\pi\)
\(938\) 0 0
\(939\) −12.0000 −0.391605
\(940\) 0 0
\(941\) −27.9411 −0.910855 −0.455427 0.890273i \(-0.650514\pi\)
−0.455427 + 0.890273i \(0.650514\pi\)
\(942\) 0 0
\(943\) −19.0294 −0.619684
\(944\) 0 0
\(945\) −19.3137 −0.628275
\(946\) 0 0
\(947\) −44.9117 −1.45943 −0.729717 0.683749i \(-0.760348\pi\)
−0.729717 + 0.683749i \(0.760348\pi\)
\(948\) 0 0
\(949\) −16.9706 −0.550888
\(950\) 0 0
\(951\) 5.65685 0.183436
\(952\) 0 0
\(953\) 29.3137 0.949564 0.474782 0.880103i \(-0.342527\pi\)
0.474782 + 0.880103i \(0.342527\pi\)
\(954\) 0 0
\(955\) 20.8284 0.673992
\(956\) 0 0
\(957\) −1.65685 −0.0535585
\(958\) 0 0
\(959\) −24.9706 −0.806342
\(960\) 0 0
\(961\) 11.0589 0.356738
\(962\) 0 0
\(963\) −20.1421 −0.649071
\(964\) 0 0
\(965\) 4.48528 0.144386
\(966\) 0 0
\(967\) −14.9706 −0.481421 −0.240710 0.970597i \(-0.577380\pi\)
−0.240710 + 0.970597i \(0.577380\pi\)
\(968\) 0 0
\(969\) 27.3137 0.877443
\(970\) 0 0
\(971\) −28.1421 −0.903124 −0.451562 0.892240i \(-0.649133\pi\)
−0.451562 + 0.892240i \(0.649133\pi\)
\(972\) 0 0
\(973\) −104.569 −3.35231
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) −2.68629 −0.0859421 −0.0429710 0.999076i \(-0.513682\pi\)
−0.0429710 + 0.999076i \(0.513682\pi\)
\(978\) 0 0
\(979\) 6.34315 0.202728
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 9.31371 0.297061 0.148531 0.988908i \(-0.452546\pi\)
0.148531 + 0.988908i \(0.452546\pi\)
\(984\) 0 0
\(985\) −19.6569 −0.626319
\(986\) 0 0
\(987\) 112.569 3.58310
\(988\) 0 0
\(989\) 19.0294 0.605101
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 0 0
\(993\) −43.5980 −1.38354
\(994\) 0 0
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) 6.82843 0.216258 0.108129 0.994137i \(-0.465514\pi\)
0.108129 + 0.994137i \(0.465514\pi\)
\(998\) 0 0
\(999\) 33.9411 1.07385
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 2320.2.a.k.1.2 2
4.3 odd 2 145.2.a.b.1.2 2
8.3 odd 2 9280.2.a.be.1.1 2
8.5 even 2 9280.2.a.w.1.2 2
12.11 even 2 1305.2.a.n.1.1 2
20.3 even 4 725.2.b.c.349.2 4
20.7 even 4 725.2.b.c.349.3 4
20.19 odd 2 725.2.a.c.1.1 2
28.27 even 2 7105.2.a.e.1.2 2
60.59 even 2 6525.2.a.p.1.2 2
116.115 odd 2 4205.2.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.2 2 4.3 odd 2
725.2.a.c.1.1 2 20.19 odd 2
725.2.b.c.349.2 4 20.3 even 4
725.2.b.c.349.3 4 20.7 even 4
1305.2.a.n.1.1 2 12.11 even 2
2320.2.a.k.1.2 2 1.1 even 1 trivial
4205.2.a.d.1.1 2 116.115 odd 2
6525.2.a.p.1.2 2 60.59 even 2
7105.2.a.e.1.2 2 28.27 even 2
9280.2.a.w.1.2 2 8.5 even 2
9280.2.a.be.1.1 2 8.3 odd 2