Properties

Label 1856.2.a.v.1.2
Level $1856$
Weight $2$
Character 1856.1
Self dual yes
Analytic conductor $14.820$
Analytic rank $1$
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] = [1856,2,Mod(1,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, 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("1856.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.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: \(14.8202346151\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 232)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.41421 q^{3} -1.82843 q^{5} -4.00000 q^{7} +2.82843 q^{9} +O(q^{10})\) \(q+2.41421 q^{3} -1.82843 q^{5} -4.00000 q^{7} +2.82843 q^{9} -0.414214 q^{11} +6.65685 q^{13} -4.41421 q^{15} -7.65685 q^{17} -2.00000 q^{19} -9.65685 q^{21} +0.828427 q^{23} -1.65685 q^{25} -0.414214 q^{27} -1.00000 q^{29} -5.58579 q^{31} -1.00000 q^{33} +7.31371 q^{35} -9.65685 q^{37} +16.0711 q^{39} +1.65685 q^{41} -4.41421 q^{43} -5.17157 q^{45} +2.07107 q^{47} +9.00000 q^{49} -18.4853 q^{51} +7.00000 q^{53} +0.757359 q^{55} -4.82843 q^{57} +6.48528 q^{59} -6.00000 q^{61} -11.3137 q^{63} -12.1716 q^{65} -5.65685 q^{67} +2.00000 q^{69} -10.4853 q^{71} +4.00000 q^{73} -4.00000 q^{75} +1.65685 q^{77} -15.7279 q^{79} -9.48528 q^{81} +3.17157 q^{83} +14.0000 q^{85} -2.41421 q^{87} +1.65685 q^{89} -26.6274 q^{91} -13.4853 q^{93} +3.65685 q^{95} +15.3137 q^{97} -1.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 8 q^{7} + 2 q^{11} + 2 q^{13} - 6 q^{15} - 4 q^{17} - 4 q^{19} - 8 q^{21} - 4 q^{23} + 8 q^{25} + 2 q^{27} - 2 q^{29} - 14 q^{31} - 2 q^{33} - 8 q^{35} - 8 q^{37} + 18 q^{39} - 8 q^{41} - 6 q^{43} - 16 q^{45} - 10 q^{47} + 18 q^{49} - 20 q^{51} + 14 q^{53} + 10 q^{55} - 4 q^{57} - 4 q^{59} - 12 q^{61} - 30 q^{65} + 4 q^{69} - 4 q^{71} + 8 q^{73} - 8 q^{75} - 8 q^{77} - 6 q^{79} - 2 q^{81} + 12 q^{83} + 28 q^{85} - 2 q^{87} - 8 q^{89} - 8 q^{91} - 10 q^{93} - 4 q^{95} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) 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
\(3\) 2.41421 1.39385 0.696923 0.717146i \(-0.254552\pi\)
0.696923 + 0.717146i \(0.254552\pi\)
\(4\) 0 0
\(5\) −1.82843 −0.817697 −0.408849 0.912602i \(-0.634070\pi\)
−0.408849 + 0.912602i \(0.634070\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) −0.414214 −0.124890 −0.0624450 0.998048i \(-0.519890\pi\)
−0.0624450 + 0.998048i \(0.519890\pi\)
\(12\) 0 0
\(13\) 6.65685 1.84628 0.923140 0.384465i \(-0.125614\pi\)
0.923140 + 0.384465i \(0.125614\pi\)
\(14\) 0 0
\(15\) −4.41421 −1.13975
\(16\) 0 0
\(17\) −7.65685 −1.85706 −0.928530 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −9.65685 −2.10730
\(22\) 0 0
\(23\) 0.828427 0.172739 0.0863695 0.996263i \(-0.472473\pi\)
0.0863695 + 0.996263i \(0.472473\pi\)
\(24\) 0 0
\(25\) −1.65685 −0.331371
\(26\) 0 0
\(27\) −0.414214 −0.0797154
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −5.58579 −1.00324 −0.501618 0.865089i \(-0.667262\pi\)
−0.501618 + 0.865089i \(0.667262\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 7.31371 1.23624
\(36\) 0 0
\(37\) −9.65685 −1.58758 −0.793789 0.608194i \(-0.791894\pi\)
−0.793789 + 0.608194i \(0.791894\pi\)
\(38\) 0 0
\(39\) 16.0711 2.57343
\(40\) 0 0
\(41\) 1.65685 0.258757 0.129379 0.991595i \(-0.458702\pi\)
0.129379 + 0.991595i \(0.458702\pi\)
\(42\) 0 0
\(43\) −4.41421 −0.673161 −0.336581 0.941655i \(-0.609270\pi\)
−0.336581 + 0.941655i \(0.609270\pi\)
\(44\) 0 0
\(45\) −5.17157 −0.770933
\(46\) 0 0
\(47\) 2.07107 0.302096 0.151048 0.988526i \(-0.451735\pi\)
0.151048 + 0.988526i \(0.451735\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −18.4853 −2.58846
\(52\) 0 0
\(53\) 7.00000 0.961524 0.480762 0.876851i \(-0.340360\pi\)
0.480762 + 0.876851i \(0.340360\pi\)
\(54\) 0 0
\(55\) 0.757359 0.102122
\(56\) 0 0
\(57\) −4.82843 −0.639541
\(58\) 0 0
\(59\) 6.48528 0.844312 0.422156 0.906523i \(-0.361273\pi\)
0.422156 + 0.906523i \(0.361273\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −11.3137 −1.42539
\(64\) 0 0
\(65\) −12.1716 −1.50970
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) −10.4853 −1.24437 −0.622187 0.782869i \(-0.713756\pi\)
−0.622187 + 0.782869i \(0.713756\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 1.65685 0.188816
\(78\) 0 0
\(79\) −15.7279 −1.76953 −0.884765 0.466038i \(-0.845681\pi\)
−0.884765 + 0.466038i \(0.845681\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) 3.17157 0.348125 0.174063 0.984735i \(-0.444310\pi\)
0.174063 + 0.984735i \(0.444310\pi\)
\(84\) 0 0
\(85\) 14.0000 1.51851
\(86\) 0 0
\(87\) −2.41421 −0.258831
\(88\) 0 0
\(89\) 1.65685 0.175626 0.0878131 0.996137i \(-0.472012\pi\)
0.0878131 + 0.996137i \(0.472012\pi\)
\(90\) 0 0
\(91\) −26.6274 −2.79131
\(92\) 0 0
\(93\) −13.4853 −1.39836
\(94\) 0 0
\(95\) 3.65685 0.375185
\(96\) 0 0
\(97\) 15.3137 1.55487 0.777436 0.628962i \(-0.216520\pi\)
0.777436 + 0.628962i \(0.216520\pi\)
\(98\) 0 0
\(99\) −1.17157 −0.117748
\(100\) 0 0
\(101\) 11.3137 1.12576 0.562878 0.826540i \(-0.309694\pi\)
0.562878 + 0.826540i \(0.309694\pi\)
\(102\) 0 0
\(103\) −11.1716 −1.10077 −0.550384 0.834912i \(-0.685519\pi\)
−0.550384 + 0.834912i \(0.685519\pi\)
\(104\) 0 0
\(105\) 17.6569 1.72313
\(106\) 0 0
\(107\) −13.6569 −1.32026 −0.660129 0.751152i \(-0.729498\pi\)
−0.660129 + 0.751152i \(0.729498\pi\)
\(108\) 0 0
\(109\) 3.82843 0.366697 0.183348 0.983048i \(-0.441306\pi\)
0.183348 + 0.983048i \(0.441306\pi\)
\(110\) 0 0
\(111\) −23.3137 −2.21284
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) −1.51472 −0.141248
\(116\) 0 0
\(117\) 18.8284 1.74069
\(118\) 0 0
\(119\) 30.6274 2.80761
\(120\) 0 0
\(121\) −10.8284 −0.984402
\(122\) 0 0
\(123\) 4.00000 0.360668
\(124\) 0 0
\(125\) 12.1716 1.08866
\(126\) 0 0
\(127\) 4.34315 0.385392 0.192696 0.981259i \(-0.438277\pi\)
0.192696 + 0.981259i \(0.438277\pi\)
\(128\) 0 0
\(129\) −10.6569 −0.938284
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 0 0
\(135\) 0.757359 0.0651831
\(136\) 0 0
\(137\) 17.6569 1.50853 0.754263 0.656572i \(-0.227994\pi\)
0.754263 + 0.656572i \(0.227994\pi\)
\(138\) 0 0
\(139\) −3.17157 −0.269009 −0.134505 0.990913i \(-0.542944\pi\)
−0.134505 + 0.990913i \(0.542944\pi\)
\(140\) 0 0
\(141\) 5.00000 0.421076
\(142\) 0 0
\(143\) −2.75736 −0.230582
\(144\) 0 0
\(145\) 1.82843 0.151843
\(146\) 0 0
\(147\) 21.7279 1.79209
\(148\) 0 0
\(149\) 2.65685 0.217658 0.108829 0.994060i \(-0.465290\pi\)
0.108829 + 0.994060i \(0.465290\pi\)
\(150\) 0 0
\(151\) 17.6569 1.43689 0.718447 0.695581i \(-0.244853\pi\)
0.718447 + 0.695581i \(0.244853\pi\)
\(152\) 0 0
\(153\) −21.6569 −1.75085
\(154\) 0 0
\(155\) 10.2132 0.820344
\(156\) 0 0
\(157\) −11.3137 −0.902932 −0.451466 0.892288i \(-0.649099\pi\)
−0.451466 + 0.892288i \(0.649099\pi\)
\(158\) 0 0
\(159\) 16.8995 1.34022
\(160\) 0 0
\(161\) −3.31371 −0.261157
\(162\) 0 0
\(163\) 12.8995 1.01037 0.505183 0.863012i \(-0.331425\pi\)
0.505183 + 0.863012i \(0.331425\pi\)
\(164\) 0 0
\(165\) 1.82843 0.142343
\(166\) 0 0
\(167\) −18.4853 −1.43043 −0.715217 0.698902i \(-0.753672\pi\)
−0.715217 + 0.698902i \(0.753672\pi\)
\(168\) 0 0
\(169\) 31.3137 2.40875
\(170\) 0 0
\(171\) −5.65685 −0.432590
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 6.62742 0.500986
\(176\) 0 0
\(177\) 15.6569 1.17684
\(178\) 0 0
\(179\) −22.4853 −1.68063 −0.840314 0.542099i \(-0.817630\pi\)
−0.840314 + 0.542099i \(0.817630\pi\)
\(180\) 0 0
\(181\) 11.4853 0.853694 0.426847 0.904324i \(-0.359624\pi\)
0.426847 + 0.904324i \(0.359624\pi\)
\(182\) 0 0
\(183\) −14.4853 −1.07078
\(184\) 0 0
\(185\) 17.6569 1.29816
\(186\) 0 0
\(187\) 3.17157 0.231928
\(188\) 0 0
\(189\) 1.65685 0.120518
\(190\) 0 0
\(191\) 7.65685 0.554031 0.277015 0.960866i \(-0.410655\pi\)
0.277015 + 0.960866i \(0.410655\pi\)
\(192\) 0 0
\(193\) −5.65685 −0.407189 −0.203595 0.979055i \(-0.565262\pi\)
−0.203595 + 0.979055i \(0.565262\pi\)
\(194\) 0 0
\(195\) −29.3848 −2.10429
\(196\) 0 0
\(197\) 9.31371 0.663574 0.331787 0.943354i \(-0.392348\pi\)
0.331787 + 0.943354i \(0.392348\pi\)
\(198\) 0 0
\(199\) −14.3431 −1.01676 −0.508379 0.861133i \(-0.669755\pi\)
−0.508379 + 0.861133i \(0.669755\pi\)
\(200\) 0 0
\(201\) −13.6569 −0.963280
\(202\) 0 0
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) −3.02944 −0.211585
\(206\) 0 0
\(207\) 2.34315 0.162860
\(208\) 0 0
\(209\) 0.828427 0.0573035
\(210\) 0 0
\(211\) 27.2426 1.87546 0.937730 0.347364i \(-0.112923\pi\)
0.937730 + 0.347364i \(0.112923\pi\)
\(212\) 0 0
\(213\) −25.3137 −1.73447
\(214\) 0 0
\(215\) 8.07107 0.550442
\(216\) 0 0
\(217\) 22.3431 1.51675
\(218\) 0 0
\(219\) 9.65685 0.652550
\(220\) 0 0
\(221\) −50.9706 −3.42865
\(222\) 0 0
\(223\) −0.142136 −0.00951811 −0.00475905 0.999989i \(-0.501515\pi\)
−0.00475905 + 0.999989i \(0.501515\pi\)
\(224\) 0 0
\(225\) −4.68629 −0.312419
\(226\) 0 0
\(227\) 16.8284 1.11694 0.558471 0.829524i \(-0.311388\pi\)
0.558471 + 0.829524i \(0.311388\pi\)
\(228\) 0 0
\(229\) −6.34315 −0.419167 −0.209583 0.977791i \(-0.567211\pi\)
−0.209583 + 0.977791i \(0.567211\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) −6.65685 −0.436105 −0.218053 0.975937i \(-0.569970\pi\)
−0.218053 + 0.975937i \(0.569970\pi\)
\(234\) 0 0
\(235\) −3.78680 −0.247023
\(236\) 0 0
\(237\) −37.9706 −2.46645
\(238\) 0 0
\(239\) −20.1421 −1.30289 −0.651443 0.758697i \(-0.725836\pi\)
−0.651443 + 0.758697i \(0.725836\pi\)
\(240\) 0 0
\(241\) 4.31371 0.277870 0.138935 0.990301i \(-0.455632\pi\)
0.138935 + 0.990301i \(0.455632\pi\)
\(242\) 0 0
\(243\) −21.6569 −1.38929
\(244\) 0 0
\(245\) −16.4558 −1.05133
\(246\) 0 0
\(247\) −13.3137 −0.847131
\(248\) 0 0
\(249\) 7.65685 0.485233
\(250\) 0 0
\(251\) −24.2132 −1.52832 −0.764162 0.645024i \(-0.776847\pi\)
−0.764162 + 0.645024i \(0.776847\pi\)
\(252\) 0 0
\(253\) −0.343146 −0.0215734
\(254\) 0 0
\(255\) 33.7990 2.11657
\(256\) 0 0
\(257\) −2.17157 −0.135459 −0.0677295 0.997704i \(-0.521575\pi\)
−0.0677295 + 0.997704i \(0.521575\pi\)
\(258\) 0 0
\(259\) 38.6274 2.40019
\(260\) 0 0
\(261\) −2.82843 −0.175075
\(262\) 0 0
\(263\) 23.3848 1.44197 0.720984 0.692952i \(-0.243690\pi\)
0.720984 + 0.692952i \(0.243690\pi\)
\(264\) 0 0
\(265\) −12.7990 −0.786236
\(266\) 0 0
\(267\) 4.00000 0.244796
\(268\) 0 0
\(269\) 10.9706 0.668887 0.334444 0.942416i \(-0.391452\pi\)
0.334444 + 0.942416i \(0.391452\pi\)
\(270\) 0 0
\(271\) −6.41421 −0.389636 −0.194818 0.980839i \(-0.562412\pi\)
−0.194818 + 0.980839i \(0.562412\pi\)
\(272\) 0 0
\(273\) −64.2843 −3.89066
\(274\) 0 0
\(275\) 0.686292 0.0413849
\(276\) 0 0
\(277\) −16.6274 −0.999045 −0.499522 0.866301i \(-0.666491\pi\)
−0.499522 + 0.866301i \(0.666491\pi\)
\(278\) 0 0
\(279\) −15.7990 −0.945861
\(280\) 0 0
\(281\) −1.00000 −0.0596550 −0.0298275 0.999555i \(-0.509496\pi\)
−0.0298275 + 0.999555i \(0.509496\pi\)
\(282\) 0 0
\(283\) −8.82843 −0.524796 −0.262398 0.964960i \(-0.584513\pi\)
−0.262398 + 0.964960i \(0.584513\pi\)
\(284\) 0 0
\(285\) 8.82843 0.522951
\(286\) 0 0
\(287\) −6.62742 −0.391204
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) 0 0
\(291\) 36.9706 2.16725
\(292\) 0 0
\(293\) 33.3137 1.94621 0.973104 0.230367i \(-0.0739927\pi\)
0.973104 + 0.230367i \(0.0739927\pi\)
\(294\) 0 0
\(295\) −11.8579 −0.690392
\(296\) 0 0
\(297\) 0.171573 0.00995567
\(298\) 0 0
\(299\) 5.51472 0.318924
\(300\) 0 0
\(301\) 17.6569 1.01772
\(302\) 0 0
\(303\) 27.3137 1.56913
\(304\) 0 0
\(305\) 10.9706 0.628173
\(306\) 0 0
\(307\) 4.27208 0.243820 0.121910 0.992541i \(-0.461098\pi\)
0.121910 + 0.992541i \(0.461098\pi\)
\(308\) 0 0
\(309\) −26.9706 −1.53430
\(310\) 0 0
\(311\) −14.9706 −0.848903 −0.424451 0.905451i \(-0.639533\pi\)
−0.424451 + 0.905451i \(0.639533\pi\)
\(312\) 0 0
\(313\) −8.51472 −0.481280 −0.240640 0.970614i \(-0.577357\pi\)
−0.240640 + 0.970614i \(0.577357\pi\)
\(314\) 0 0
\(315\) 20.6863 1.16554
\(316\) 0 0
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 0 0
\(319\) 0.414214 0.0231915
\(320\) 0 0
\(321\) −32.9706 −1.84024
\(322\) 0 0
\(323\) 15.3137 0.852078
\(324\) 0 0
\(325\) −11.0294 −0.611803
\(326\) 0 0
\(327\) 9.24264 0.511119
\(328\) 0 0
\(329\) −8.28427 −0.456727
\(330\) 0 0
\(331\) −8.07107 −0.443626 −0.221813 0.975089i \(-0.571197\pi\)
−0.221813 + 0.975089i \(0.571197\pi\)
\(332\) 0 0
\(333\) −27.3137 −1.49678
\(334\) 0 0
\(335\) 10.3431 0.565106
\(336\) 0 0
\(337\) −9.31371 −0.507350 −0.253675 0.967290i \(-0.581639\pi\)
−0.253675 + 0.967290i \(0.581639\pi\)
\(338\) 0 0
\(339\) −24.1421 −1.31122
\(340\) 0 0
\(341\) 2.31371 0.125294
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) −3.65685 −0.196878
\(346\) 0 0
\(347\) 10.4853 0.562879 0.281440 0.959579i \(-0.409188\pi\)
0.281440 + 0.959579i \(0.409188\pi\)
\(348\) 0 0
\(349\) −4.31371 −0.230908 −0.115454 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(350\) 0 0
\(351\) −2.75736 −0.147177
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 19.1716 1.01752
\(356\) 0 0
\(357\) 73.9411 3.91338
\(358\) 0 0
\(359\) 12.2721 0.647696 0.323848 0.946109i \(-0.395023\pi\)
0.323848 + 0.946109i \(0.395023\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −26.1421 −1.37211
\(364\) 0 0
\(365\) −7.31371 −0.382817
\(366\) 0 0
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 0 0
\(369\) 4.68629 0.243959
\(370\) 0 0
\(371\) −28.0000 −1.45369
\(372\) 0 0
\(373\) −1.48528 −0.0769050 −0.0384525 0.999260i \(-0.512243\pi\)
−0.0384525 + 0.999260i \(0.512243\pi\)
\(374\) 0 0
\(375\) 29.3848 1.51742
\(376\) 0 0
\(377\) −6.65685 −0.342845
\(378\) 0 0
\(379\) −8.34315 −0.428559 −0.214279 0.976772i \(-0.568740\pi\)
−0.214279 + 0.976772i \(0.568740\pi\)
\(380\) 0 0
\(381\) 10.4853 0.537177
\(382\) 0 0
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) −3.02944 −0.154394
\(386\) 0 0
\(387\) −12.4853 −0.634663
\(388\) 0 0
\(389\) 17.6569 0.895238 0.447619 0.894224i \(-0.352272\pi\)
0.447619 + 0.894224i \(0.352272\pi\)
\(390\) 0 0
\(391\) −6.34315 −0.320787
\(392\) 0 0
\(393\) 24.1421 1.21781
\(394\) 0 0
\(395\) 28.7574 1.44694
\(396\) 0 0
\(397\) −18.4558 −0.926272 −0.463136 0.886287i \(-0.653276\pi\)
−0.463136 + 0.886287i \(0.653276\pi\)
\(398\) 0 0
\(399\) 19.3137 0.966895
\(400\) 0 0
\(401\) 8.65685 0.432303 0.216151 0.976360i \(-0.430650\pi\)
0.216151 + 0.976360i \(0.430650\pi\)
\(402\) 0 0
\(403\) −37.1838 −1.85226
\(404\) 0 0
\(405\) 17.3431 0.861788
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 0.343146 0.0169675 0.00848373 0.999964i \(-0.497300\pi\)
0.00848373 + 0.999964i \(0.497300\pi\)
\(410\) 0 0
\(411\) 42.6274 2.10266
\(412\) 0 0
\(413\) −25.9411 −1.27648
\(414\) 0 0
\(415\) −5.79899 −0.284661
\(416\) 0 0
\(417\) −7.65685 −0.374958
\(418\) 0 0
\(419\) −26.4853 −1.29389 −0.646945 0.762536i \(-0.723954\pi\)
−0.646945 + 0.762536i \(0.723954\pi\)
\(420\) 0 0
\(421\) 1.02944 0.0501717 0.0250859 0.999685i \(-0.492014\pi\)
0.0250859 + 0.999685i \(0.492014\pi\)
\(422\) 0 0
\(423\) 5.85786 0.284819
\(424\) 0 0
\(425\) 12.6863 0.615376
\(426\) 0 0
\(427\) 24.0000 1.16144
\(428\) 0 0
\(429\) −6.65685 −0.321396
\(430\) 0 0
\(431\) −13.7990 −0.664674 −0.332337 0.943161i \(-0.607837\pi\)
−0.332337 + 0.943161i \(0.607837\pi\)
\(432\) 0 0
\(433\) −8.00000 −0.384455 −0.192228 0.981350i \(-0.561571\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(434\) 0 0
\(435\) 4.41421 0.211645
\(436\) 0 0
\(437\) −1.65685 −0.0792581
\(438\) 0 0
\(439\) −20.1421 −0.961332 −0.480666 0.876904i \(-0.659605\pi\)
−0.480666 + 0.876904i \(0.659605\pi\)
\(440\) 0 0
\(441\) 25.4558 1.21218
\(442\) 0 0
\(443\) 8.62742 0.409901 0.204950 0.978772i \(-0.434297\pi\)
0.204950 + 0.978772i \(0.434297\pi\)
\(444\) 0 0
\(445\) −3.02944 −0.143609
\(446\) 0 0
\(447\) 6.41421 0.303382
\(448\) 0 0
\(449\) 28.6274 1.35101 0.675506 0.737355i \(-0.263925\pi\)
0.675506 + 0.737355i \(0.263925\pi\)
\(450\) 0 0
\(451\) −0.686292 −0.0323162
\(452\) 0 0
\(453\) 42.6274 2.00281
\(454\) 0 0
\(455\) 48.6863 2.28245
\(456\) 0 0
\(457\) −28.6274 −1.33913 −0.669567 0.742752i \(-0.733520\pi\)
−0.669567 + 0.742752i \(0.733520\pi\)
\(458\) 0 0
\(459\) 3.17157 0.148036
\(460\) 0 0
\(461\) −18.6863 −0.870307 −0.435154 0.900356i \(-0.643306\pi\)
−0.435154 + 0.900356i \(0.643306\pi\)
\(462\) 0 0
\(463\) 31.4558 1.46188 0.730939 0.682443i \(-0.239083\pi\)
0.730939 + 0.682443i \(0.239083\pi\)
\(464\) 0 0
\(465\) 24.6569 1.14343
\(466\) 0 0
\(467\) 14.8995 0.689466 0.344733 0.938701i \(-0.387969\pi\)
0.344733 + 0.938701i \(0.387969\pi\)
\(468\) 0 0
\(469\) 22.6274 1.04484
\(470\) 0 0
\(471\) −27.3137 −1.25855
\(472\) 0 0
\(473\) 1.82843 0.0840712
\(474\) 0 0
\(475\) 3.31371 0.152043
\(476\) 0 0
\(477\) 19.7990 0.906533
\(478\) 0 0
\(479\) −4.75736 −0.217369 −0.108685 0.994076i \(-0.534664\pi\)
−0.108685 + 0.994076i \(0.534664\pi\)
\(480\) 0 0
\(481\) −64.2843 −2.93111
\(482\) 0 0
\(483\) −8.00000 −0.364013
\(484\) 0 0
\(485\) −28.0000 −1.27141
\(486\) 0 0
\(487\) −16.9706 −0.769010 −0.384505 0.923123i \(-0.625628\pi\)
−0.384505 + 0.923123i \(0.625628\pi\)
\(488\) 0 0
\(489\) 31.1421 1.40830
\(490\) 0 0
\(491\) 2.75736 0.124438 0.0622189 0.998063i \(-0.480182\pi\)
0.0622189 + 0.998063i \(0.480182\pi\)
\(492\) 0 0
\(493\) 7.65685 0.344847
\(494\) 0 0
\(495\) 2.14214 0.0962818
\(496\) 0 0
\(497\) 41.9411 1.88132
\(498\) 0 0
\(499\) −40.1421 −1.79701 −0.898504 0.438965i \(-0.855345\pi\)
−0.898504 + 0.438965i \(0.855345\pi\)
\(500\) 0 0
\(501\) −44.6274 −1.99381
\(502\) 0 0
\(503\) 11.2426 0.501285 0.250642 0.968080i \(-0.419358\pi\)
0.250642 + 0.968080i \(0.419358\pi\)
\(504\) 0 0
\(505\) −20.6863 −0.920528
\(506\) 0 0
\(507\) 75.5980 3.35742
\(508\) 0 0
\(509\) −7.34315 −0.325479 −0.162740 0.986669i \(-0.552033\pi\)
−0.162740 + 0.986669i \(0.552033\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 0 0
\(513\) 0.828427 0.0365760
\(514\) 0 0
\(515\) 20.4264 0.900095
\(516\) 0 0
\(517\) −0.857864 −0.0377288
\(518\) 0 0
\(519\) 33.7990 1.48361
\(520\) 0 0
\(521\) 38.1716 1.67233 0.836163 0.548480i \(-0.184793\pi\)
0.836163 + 0.548480i \(0.184793\pi\)
\(522\) 0 0
\(523\) −29.6569 −1.29680 −0.648402 0.761298i \(-0.724562\pi\)
−0.648402 + 0.761298i \(0.724562\pi\)
\(524\) 0 0
\(525\) 16.0000 0.698297
\(526\) 0 0
\(527\) 42.7696 1.86307
\(528\) 0 0
\(529\) −22.3137 −0.970161
\(530\) 0 0
\(531\) 18.3431 0.796025
\(532\) 0 0
\(533\) 11.0294 0.477738
\(534\) 0 0
\(535\) 24.9706 1.07957
\(536\) 0 0
\(537\) −54.2843 −2.34254
\(538\) 0 0
\(539\) −3.72792 −0.160573
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 0 0
\(543\) 27.7279 1.18992
\(544\) 0 0
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) −31.3137 −1.33888 −0.669439 0.742867i \(-0.733465\pi\)
−0.669439 + 0.742867i \(0.733465\pi\)
\(548\) 0 0
\(549\) −16.9706 −0.724286
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) 62.9117 2.67528
\(554\) 0 0
\(555\) 42.6274 1.80943
\(556\) 0 0
\(557\) −13.3137 −0.564120 −0.282060 0.959397i \(-0.591018\pi\)
−0.282060 + 0.959397i \(0.591018\pi\)
\(558\) 0 0
\(559\) −29.3848 −1.24284
\(560\) 0 0
\(561\) 7.65685 0.323273
\(562\) 0 0
\(563\) 20.8995 0.880809 0.440404 0.897800i \(-0.354835\pi\)
0.440404 + 0.897800i \(0.354835\pi\)
\(564\) 0 0
\(565\) 18.2843 0.769225
\(566\) 0 0
\(567\) 37.9411 1.59338
\(568\) 0 0
\(569\) 8.34315 0.349763 0.174881 0.984590i \(-0.444046\pi\)
0.174881 + 0.984590i \(0.444046\pi\)
\(570\) 0 0
\(571\) 8.97056 0.375406 0.187703 0.982226i \(-0.439896\pi\)
0.187703 + 0.982226i \(0.439896\pi\)
\(572\) 0 0
\(573\) 18.4853 0.772234
\(574\) 0 0
\(575\) −1.37258 −0.0572407
\(576\) 0 0
\(577\) 33.3137 1.38687 0.693434 0.720520i \(-0.256097\pi\)
0.693434 + 0.720520i \(0.256097\pi\)
\(578\) 0 0
\(579\) −13.6569 −0.567559
\(580\) 0 0
\(581\) −12.6863 −0.526316
\(582\) 0 0
\(583\) −2.89949 −0.120085
\(584\) 0 0
\(585\) −34.4264 −1.42336
\(586\) 0 0
\(587\) −33.1127 −1.36671 −0.683354 0.730088i \(-0.739479\pi\)
−0.683354 + 0.730088i \(0.739479\pi\)
\(588\) 0 0
\(589\) 11.1716 0.460317
\(590\) 0 0
\(591\) 22.4853 0.924921
\(592\) 0 0
\(593\) −0.171573 −0.00704565 −0.00352283 0.999994i \(-0.501121\pi\)
−0.00352283 + 0.999994i \(0.501121\pi\)
\(594\) 0 0
\(595\) −56.0000 −2.29578
\(596\) 0 0
\(597\) −34.6274 −1.41721
\(598\) 0 0
\(599\) −29.3848 −1.20063 −0.600315 0.799764i \(-0.704958\pi\)
−0.600315 + 0.799764i \(0.704958\pi\)
\(600\) 0 0
\(601\) −12.0000 −0.489490 −0.244745 0.969587i \(-0.578704\pi\)
−0.244745 + 0.969587i \(0.578704\pi\)
\(602\) 0 0
\(603\) −16.0000 −0.651570
\(604\) 0 0
\(605\) 19.7990 0.804943
\(606\) 0 0
\(607\) 25.8701 1.05003 0.525017 0.851092i \(-0.324059\pi\)
0.525017 + 0.851092i \(0.324059\pi\)
\(608\) 0 0
\(609\) 9.65685 0.391315
\(610\) 0 0
\(611\) 13.7868 0.557754
\(612\) 0 0
\(613\) −18.7990 −0.759284 −0.379642 0.925133i \(-0.623953\pi\)
−0.379642 + 0.925133i \(0.623953\pi\)
\(614\) 0 0
\(615\) −7.31371 −0.294917
\(616\) 0 0
\(617\) −36.9706 −1.48838 −0.744189 0.667969i \(-0.767164\pi\)
−0.744189 + 0.667969i \(0.767164\pi\)
\(618\) 0 0
\(619\) −26.6985 −1.07310 −0.536551 0.843868i \(-0.680273\pi\)
−0.536551 + 0.843868i \(0.680273\pi\)
\(620\) 0 0
\(621\) −0.343146 −0.0137700
\(622\) 0 0
\(623\) −6.62742 −0.265522
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) 2.00000 0.0798723
\(628\) 0 0
\(629\) 73.9411 2.94823
\(630\) 0 0
\(631\) 30.7696 1.22492 0.612458 0.790503i \(-0.290181\pi\)
0.612458 + 0.790503i \(0.290181\pi\)
\(632\) 0 0
\(633\) 65.7696 2.61411
\(634\) 0 0
\(635\) −7.94113 −0.315134
\(636\) 0 0
\(637\) 59.9117 2.37379
\(638\) 0 0
\(639\) −29.6569 −1.17321
\(640\) 0 0
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) 0 0
\(643\) 36.9706 1.45798 0.728988 0.684526i \(-0.239991\pi\)
0.728988 + 0.684526i \(0.239991\pi\)
\(644\) 0 0
\(645\) 19.4853 0.767232
\(646\) 0 0
\(647\) 4.82843 0.189825 0.0949125 0.995486i \(-0.469743\pi\)
0.0949125 + 0.995486i \(0.469743\pi\)
\(648\) 0 0
\(649\) −2.68629 −0.105446
\(650\) 0 0
\(651\) 53.9411 2.11412
\(652\) 0 0
\(653\) −1.37258 −0.0537133 −0.0268567 0.999639i \(-0.508550\pi\)
−0.0268567 + 0.999639i \(0.508550\pi\)
\(654\) 0 0
\(655\) −18.2843 −0.714426
\(656\) 0 0
\(657\) 11.3137 0.441390
\(658\) 0 0
\(659\) 12.0711 0.470222 0.235111 0.971969i \(-0.424455\pi\)
0.235111 + 0.971969i \(0.424455\pi\)
\(660\) 0 0
\(661\) 6.68629 0.260067 0.130033 0.991510i \(-0.458492\pi\)
0.130033 + 0.991510i \(0.458492\pi\)
\(662\) 0 0
\(663\) −123.054 −4.77901
\(664\) 0 0
\(665\) −14.6274 −0.567227
\(666\) 0 0
\(667\) −0.828427 −0.0320768
\(668\) 0 0
\(669\) −0.343146 −0.0132668
\(670\) 0 0
\(671\) 2.48528 0.0959432
\(672\) 0 0
\(673\) −35.2843 −1.36011 −0.680054 0.733162i \(-0.738044\pi\)
−0.680054 + 0.733162i \(0.738044\pi\)
\(674\) 0 0
\(675\) 0.686292 0.0264154
\(676\) 0 0
\(677\) −10.9706 −0.421633 −0.210816 0.977526i \(-0.567612\pi\)
−0.210816 + 0.977526i \(0.567612\pi\)
\(678\) 0 0
\(679\) −61.2548 −2.35074
\(680\) 0 0
\(681\) 40.6274 1.55685
\(682\) 0 0
\(683\) 28.9706 1.10853 0.554264 0.832341i \(-0.313000\pi\)
0.554264 + 0.832341i \(0.313000\pi\)
\(684\) 0 0
\(685\) −32.2843 −1.23352
\(686\) 0 0
\(687\) −15.3137 −0.584254
\(688\) 0 0
\(689\) 46.5980 1.77524
\(690\) 0 0
\(691\) 19.3137 0.734728 0.367364 0.930077i \(-0.380260\pi\)
0.367364 + 0.930077i \(0.380260\pi\)
\(692\) 0 0
\(693\) 4.68629 0.178017
\(694\) 0 0
\(695\) 5.79899 0.219968
\(696\) 0 0
\(697\) −12.6863 −0.480528
\(698\) 0 0
\(699\) −16.0711 −0.607864
\(700\) 0 0
\(701\) −37.6274 −1.42117 −0.710584 0.703612i \(-0.751569\pi\)
−0.710584 + 0.703612i \(0.751569\pi\)
\(702\) 0 0
\(703\) 19.3137 0.728430
\(704\) 0 0
\(705\) −9.14214 −0.344313
\(706\) 0 0
\(707\) −45.2548 −1.70198
\(708\) 0 0
\(709\) −32.9411 −1.23713 −0.618565 0.785734i \(-0.712286\pi\)
−0.618565 + 0.785734i \(0.712286\pi\)
\(710\) 0 0
\(711\) −44.4853 −1.66833
\(712\) 0 0
\(713\) −4.62742 −0.173298
\(714\) 0 0
\(715\) 5.04163 0.188546
\(716\) 0 0
\(717\) −48.6274 −1.81602
\(718\) 0 0
\(719\) −34.4853 −1.28608 −0.643042 0.765831i \(-0.722328\pi\)
−0.643042 + 0.765831i \(0.722328\pi\)
\(720\) 0 0
\(721\) 44.6863 1.66420
\(722\) 0 0
\(723\) 10.4142 0.387309
\(724\) 0 0
\(725\) 1.65685 0.0615340
\(726\) 0 0
\(727\) −18.2843 −0.678126 −0.339063 0.940764i \(-0.610110\pi\)
−0.339063 + 0.940764i \(0.610110\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) 33.7990 1.25010
\(732\) 0 0
\(733\) −2.62742 −0.0970459 −0.0485229 0.998822i \(-0.515451\pi\)
−0.0485229 + 0.998822i \(0.515451\pi\)
\(734\) 0 0
\(735\) −39.7279 −1.46539
\(736\) 0 0
\(737\) 2.34315 0.0863109
\(738\) 0 0
\(739\) 8.75736 0.322145 0.161072 0.986943i \(-0.448505\pi\)
0.161072 + 0.986943i \(0.448505\pi\)
\(740\) 0 0
\(741\) −32.1421 −1.18077
\(742\) 0 0
\(743\) 29.5980 1.08584 0.542922 0.839783i \(-0.317318\pi\)
0.542922 + 0.839783i \(0.317318\pi\)
\(744\) 0 0
\(745\) −4.85786 −0.177978
\(746\) 0 0
\(747\) 8.97056 0.328216
\(748\) 0 0
\(749\) 54.6274 1.99604
\(750\) 0 0
\(751\) −12.6274 −0.460781 −0.230390 0.973098i \(-0.574000\pi\)
−0.230390 + 0.973098i \(0.574000\pi\)
\(752\) 0 0
\(753\) −58.4558 −2.13025
\(754\) 0 0
\(755\) −32.2843 −1.17494
\(756\) 0 0
\(757\) −9.02944 −0.328180 −0.164090 0.986445i \(-0.552469\pi\)
−0.164090 + 0.986445i \(0.552469\pi\)
\(758\) 0 0
\(759\) −0.828427 −0.0300700
\(760\) 0 0
\(761\) 15.9411 0.577865 0.288933 0.957349i \(-0.406700\pi\)
0.288933 + 0.957349i \(0.406700\pi\)
\(762\) 0 0
\(763\) −15.3137 −0.554393
\(764\) 0 0
\(765\) 39.5980 1.43167
\(766\) 0 0
\(767\) 43.1716 1.55884
\(768\) 0 0
\(769\) 28.6274 1.03233 0.516166 0.856489i \(-0.327359\pi\)
0.516166 + 0.856489i \(0.327359\pi\)
\(770\) 0 0
\(771\) −5.24264 −0.188809
\(772\) 0 0
\(773\) −14.3431 −0.515887 −0.257944 0.966160i \(-0.583045\pi\)
−0.257944 + 0.966160i \(0.583045\pi\)
\(774\) 0 0
\(775\) 9.25483 0.332443
\(776\) 0 0
\(777\) 93.2548 3.34550
\(778\) 0 0
\(779\) −3.31371 −0.118726
\(780\) 0 0
\(781\) 4.34315 0.155410
\(782\) 0 0
\(783\) 0.414214 0.0148028
\(784\) 0 0
\(785\) 20.6863 0.738325
\(786\) 0 0
\(787\) −20.1421 −0.717990 −0.358995 0.933340i \(-0.616880\pi\)
−0.358995 + 0.933340i \(0.616880\pi\)
\(788\) 0 0
\(789\) 56.4558 2.00988
\(790\) 0 0
\(791\) 40.0000 1.42224
\(792\) 0 0
\(793\) −39.9411 −1.41835
\(794\) 0 0
\(795\) −30.8995 −1.09589
\(796\) 0 0
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) −15.8579 −0.561011
\(800\) 0 0
\(801\) 4.68629 0.165582
\(802\) 0 0
\(803\) −1.65685 −0.0584691
\(804\) 0 0
\(805\) 6.05887 0.213547
\(806\) 0 0
\(807\) 26.4853 0.932326
\(808\) 0 0
\(809\) 5.65685 0.198884 0.0994422 0.995043i \(-0.468294\pi\)
0.0994422 + 0.995043i \(0.468294\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) −15.4853 −0.543093
\(814\) 0 0
\(815\) −23.5858 −0.826174
\(816\) 0 0
\(817\) 8.82843 0.308868
\(818\) 0 0
\(819\) −75.3137 −2.63167
\(820\) 0 0
\(821\) 6.65685 0.232326 0.116163 0.993230i \(-0.462941\pi\)
0.116163 + 0.993230i \(0.462941\pi\)
\(822\) 0 0
\(823\) 10.9706 0.382410 0.191205 0.981550i \(-0.438761\pi\)
0.191205 + 0.981550i \(0.438761\pi\)
\(824\) 0 0
\(825\) 1.65685 0.0576843
\(826\) 0 0
\(827\) 40.8995 1.42221 0.711107 0.703083i \(-0.248194\pi\)
0.711107 + 0.703083i \(0.248194\pi\)
\(828\) 0 0
\(829\) −22.0000 −0.764092 −0.382046 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) 0 0
\(831\) −40.1421 −1.39252
\(832\) 0 0
\(833\) −68.9117 −2.38765
\(834\) 0 0
\(835\) 33.7990 1.16966
\(836\) 0 0
\(837\) 2.31371 0.0799735
\(838\) 0 0
\(839\) −48.8406 −1.68617 −0.843083 0.537784i \(-0.819262\pi\)
−0.843083 + 0.537784i \(0.819262\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −2.41421 −0.0831499
\(844\) 0 0
\(845\) −57.2548 −1.96963
\(846\) 0 0
\(847\) 43.3137 1.48828
\(848\) 0 0
\(849\) −21.3137 −0.731485
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 15.9411 0.545814 0.272907 0.962040i \(-0.412015\pi\)
0.272907 + 0.962040i \(0.412015\pi\)
\(854\) 0 0
\(855\) 10.3431 0.353728
\(856\) 0 0
\(857\) −23.1421 −0.790520 −0.395260 0.918569i \(-0.629346\pi\)
−0.395260 + 0.918569i \(0.629346\pi\)
\(858\) 0 0
\(859\) −15.5858 −0.531780 −0.265890 0.964003i \(-0.585666\pi\)
−0.265890 + 0.964003i \(0.585666\pi\)
\(860\) 0 0
\(861\) −16.0000 −0.545279
\(862\) 0 0
\(863\) −20.8284 −0.709008 −0.354504 0.935055i \(-0.615350\pi\)
−0.354504 + 0.935055i \(0.615350\pi\)
\(864\) 0 0
\(865\) −25.5980 −0.870357
\(866\) 0 0
\(867\) 100.497 3.41307
\(868\) 0 0
\(869\) 6.51472 0.220997
\(870\) 0 0
\(871\) −37.6569 −1.27595
\(872\) 0 0
\(873\) 43.3137 1.46595
\(874\) 0 0
\(875\) −48.6863 −1.64590
\(876\) 0 0
\(877\) 5.62742 0.190024 0.0950122 0.995476i \(-0.469711\pi\)
0.0950122 + 0.995476i \(0.469711\pi\)
\(878\) 0 0
\(879\) 80.4264 2.71272
\(880\) 0 0
\(881\) 27.6569 0.931783 0.465892 0.884842i \(-0.345734\pi\)
0.465892 + 0.884842i \(0.345734\pi\)
\(882\) 0 0
\(883\) 23.5980 0.794135 0.397068 0.917789i \(-0.370028\pi\)
0.397068 + 0.917789i \(0.370028\pi\)
\(884\) 0 0
\(885\) −28.6274 −0.962300
\(886\) 0 0
\(887\) −33.8701 −1.13725 −0.568623 0.822599i \(-0.692524\pi\)
−0.568623 + 0.822599i \(0.692524\pi\)
\(888\) 0 0
\(889\) −17.3726 −0.582658
\(890\) 0 0
\(891\) 3.92893 0.131624
\(892\) 0 0
\(893\) −4.14214 −0.138611
\(894\) 0 0
\(895\) 41.1127 1.37425
\(896\) 0 0
\(897\) 13.3137 0.444532
\(898\) 0 0
\(899\) 5.58579 0.186296
\(900\) 0 0
\(901\) −53.5980 −1.78561
\(902\) 0 0
\(903\) 42.6274 1.41855
\(904\) 0 0
\(905\) −21.0000 −0.698064
\(906\) 0 0
\(907\) −51.2548 −1.70189 −0.850944 0.525256i \(-0.823970\pi\)
−0.850944 + 0.525256i \(0.823970\pi\)
\(908\) 0 0
\(909\) 32.0000 1.06137
\(910\) 0 0
\(911\) 59.5269 1.97221 0.986107 0.166110i \(-0.0531206\pi\)
0.986107 + 0.166110i \(0.0531206\pi\)
\(912\) 0 0
\(913\) −1.31371 −0.0434774
\(914\) 0 0
\(915\) 26.4853 0.875576
\(916\) 0 0
\(917\) −40.0000 −1.32092
\(918\) 0 0
\(919\) 31.4558 1.03763 0.518816 0.854886i \(-0.326373\pi\)
0.518816 + 0.854886i \(0.326373\pi\)
\(920\) 0 0
\(921\) 10.3137 0.339848
\(922\) 0 0
\(923\) −69.7990 −2.29746
\(924\) 0 0
\(925\) 16.0000 0.526077
\(926\) 0 0
\(927\) −31.5980 −1.03781
\(928\) 0 0
\(929\) −51.9411 −1.70413 −0.852067 0.523434i \(-0.824651\pi\)
−0.852067 + 0.523434i \(0.824651\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 0 0
\(933\) −36.1421 −1.18324
\(934\) 0 0
\(935\) −5.79899 −0.189647
\(936\) 0 0
\(937\) 20.6274 0.673868 0.336934 0.941528i \(-0.390610\pi\)
0.336934 + 0.941528i \(0.390610\pi\)
\(938\) 0 0
\(939\) −20.5563 −0.670831
\(940\) 0 0
\(941\) 41.0833 1.33928 0.669638 0.742688i \(-0.266449\pi\)
0.669638 + 0.742688i \(0.266449\pi\)
\(942\) 0 0
\(943\) 1.37258 0.0446975
\(944\) 0 0
\(945\) −3.02944 −0.0985476
\(946\) 0 0
\(947\) 8.27208 0.268806 0.134403 0.990927i \(-0.457088\pi\)
0.134403 + 0.990927i \(0.457088\pi\)
\(948\) 0 0
\(949\) 26.6274 0.864363
\(950\) 0 0
\(951\) 33.7990 1.09601
\(952\) 0 0
\(953\) −37.0000 −1.19855 −0.599274 0.800544i \(-0.704544\pi\)
−0.599274 + 0.800544i \(0.704544\pi\)
\(954\) 0 0
\(955\) −14.0000 −0.453029
\(956\) 0 0
\(957\) 1.00000 0.0323254
\(958\) 0 0
\(959\) −70.6274 −2.28068
\(960\) 0 0
\(961\) 0.201010 0.00648420
\(962\) 0 0
\(963\) −38.6274 −1.24475
\(964\) 0 0
\(965\) 10.3431 0.332958
\(966\) 0 0
\(967\) −36.5563 −1.17557 −0.587786 0.809016i \(-0.700000\pi\)
−0.587786 + 0.809016i \(0.700000\pi\)
\(968\) 0 0
\(969\) 36.9706 1.18767
\(970\) 0 0
\(971\) 10.6863 0.342939 0.171470 0.985189i \(-0.445148\pi\)
0.171470 + 0.985189i \(0.445148\pi\)
\(972\) 0 0
\(973\) 12.6863 0.406704
\(974\) 0 0
\(975\) −26.6274 −0.852760
\(976\) 0 0
\(977\) −31.1421 −0.996325 −0.498163 0.867084i \(-0.665992\pi\)
−0.498163 + 0.867084i \(0.665992\pi\)
\(978\) 0 0
\(979\) −0.686292 −0.0219340
\(980\) 0 0
\(981\) 10.8284 0.345725
\(982\) 0 0
\(983\) −51.8701 −1.65440 −0.827199 0.561909i \(-0.810067\pi\)
−0.827199 + 0.561909i \(0.810067\pi\)
\(984\) 0 0
\(985\) −17.0294 −0.542603
\(986\) 0 0
\(987\) −20.0000 −0.636607
\(988\) 0 0
\(989\) −3.65685 −0.116281
\(990\) 0 0
\(991\) −21.5147 −0.683438 −0.341719 0.939802i \(-0.611009\pi\)
−0.341719 + 0.939802i \(0.611009\pi\)
\(992\) 0 0
\(993\) −19.4853 −0.618347
\(994\) 0 0
\(995\) 26.2254 0.831401
\(996\) 0 0
\(997\) −8.97056 −0.284101 −0.142050 0.989859i \(-0.545369\pi\)
−0.142050 + 0.989859i \(0.545369\pi\)
\(998\) 0 0
\(999\) 4.00000 0.126554
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 1856.2.a.v.1.2 2
4.3 odd 2 1856.2.a.s.1.1 2
8.3 odd 2 464.2.a.i.1.2 2
8.5 even 2 232.2.a.c.1.1 2
24.5 odd 2 2088.2.a.q.1.1 2
24.11 even 2 4176.2.a.br.1.1 2
40.29 even 2 5800.2.a.o.1.2 2
232.173 even 2 6728.2.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.2.a.c.1.1 2 8.5 even 2
464.2.a.i.1.2 2 8.3 odd 2
1856.2.a.s.1.1 2 4.3 odd 2
1856.2.a.v.1.2 2 1.1 even 1 trivial
2088.2.a.q.1.1 2 24.5 odd 2
4176.2.a.br.1.1 2 24.11 even 2
5800.2.a.o.1.2 2 40.29 even 2
6728.2.a.f.1.2 2 232.173 even 2