Properties

Label 5776.2.a.bi.1.2
Level $5776$
Weight $2$
Character 5776.1
Self dual yes
Analytic conductor $46.122$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5776,2,Mod(1,5776)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5776, 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("5776.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5776 = 2^{4} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5776.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: \(46.1215922075\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.347296\) of defining polynomial
Character \(\chi\) \(=\) 5776.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.652704 q^{3} -1.34730 q^{5} -1.53209 q^{7} -2.57398 q^{9} +O(q^{10})\) \(q-0.652704 q^{3} -1.34730 q^{5} -1.53209 q^{7} -2.57398 q^{9} +1.18479 q^{11} -2.71688 q^{13} +0.879385 q^{15} +3.87939 q^{17} +1.00000 q^{21} +5.06418 q^{23} -3.18479 q^{25} +3.63816 q^{27} +4.65270 q^{29} -3.83750 q^{31} -0.773318 q^{33} +2.06418 q^{35} +4.10607 q^{37} +1.77332 q^{39} +9.98545 q^{41} +8.70233 q^{43} +3.46791 q^{45} -0.573978 q^{47} -4.65270 q^{49} -2.53209 q^{51} -2.94356 q^{53} -1.59627 q^{55} -3.93582 q^{59} -4.51754 q^{61} +3.94356 q^{63} +3.66044 q^{65} -3.88713 q^{67} -3.30541 q^{69} -6.93582 q^{71} +6.12836 q^{73} +2.07873 q^{75} -1.81521 q^{77} +9.80840 q^{79} +5.34730 q^{81} -12.3182 q^{83} -5.22668 q^{85} -3.03684 q^{87} +2.42602 q^{89} +4.16250 q^{91} +2.50475 q^{93} -7.36959 q^{97} -3.04963 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{5} - 3 q^{15} + 6 q^{17} + 3 q^{21} + 6 q^{23} - 6 q^{25} - 6 q^{27} + 15 q^{29} - 9 q^{31} - 9 q^{33} - 3 q^{35} + 12 q^{39} + 12 q^{41} + 15 q^{45} + 6 q^{47} - 15 q^{49} - 3 q^{51} + 6 q^{53} + 9 q^{55} - 21 q^{59} + 9 q^{61} - 3 q^{63} - 12 q^{65} + 18 q^{67} - 12 q^{69} - 30 q^{71} + 15 q^{75} - 9 q^{77} - 9 q^{79} + 15 q^{81} - 9 q^{85} - 21 q^{87} + 15 q^{89} + 15 q^{91} + 24 q^{93} - 15 q^{97} + 18 q^{99}+O(q^{100}) \) 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
\(3\) −0.652704 −0.376839 −0.188419 0.982089i \(-0.560336\pi\)
−0.188419 + 0.982089i \(0.560336\pi\)
\(4\) 0 0
\(5\) −1.34730 −0.602529 −0.301265 0.953541i \(-0.597409\pi\)
−0.301265 + 0.953541i \(0.597409\pi\)
\(6\) 0 0
\(7\) −1.53209 −0.579075 −0.289538 0.957167i \(-0.593502\pi\)
−0.289538 + 0.957167i \(0.593502\pi\)
\(8\) 0 0
\(9\) −2.57398 −0.857993
\(10\) 0 0
\(11\) 1.18479 0.357228 0.178614 0.983919i \(-0.442839\pi\)
0.178614 + 0.983919i \(0.442839\pi\)
\(12\) 0 0
\(13\) −2.71688 −0.753527 −0.376764 0.926309i \(-0.622963\pi\)
−0.376764 + 0.926309i \(0.622963\pi\)
\(14\) 0 0
\(15\) 0.879385 0.227056
\(16\) 0 0
\(17\) 3.87939 0.940889 0.470445 0.882430i \(-0.344094\pi\)
0.470445 + 0.882430i \(0.344094\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 5.06418 1.05595 0.527977 0.849259i \(-0.322951\pi\)
0.527977 + 0.849259i \(0.322951\pi\)
\(24\) 0 0
\(25\) −3.18479 −0.636959
\(26\) 0 0
\(27\) 3.63816 0.700163
\(28\) 0 0
\(29\) 4.65270 0.863985 0.431993 0.901877i \(-0.357811\pi\)
0.431993 + 0.901877i \(0.357811\pi\)
\(30\) 0 0
\(31\) −3.83750 −0.689235 −0.344617 0.938743i \(-0.611991\pi\)
−0.344617 + 0.938743i \(0.611991\pi\)
\(32\) 0 0
\(33\) −0.773318 −0.134617
\(34\) 0 0
\(35\) 2.06418 0.348910
\(36\) 0 0
\(37\) 4.10607 0.675033 0.337517 0.941320i \(-0.390413\pi\)
0.337517 + 0.941320i \(0.390413\pi\)
\(38\) 0 0
\(39\) 1.77332 0.283958
\(40\) 0 0
\(41\) 9.98545 1.55947 0.779733 0.626112i \(-0.215355\pi\)
0.779733 + 0.626112i \(0.215355\pi\)
\(42\) 0 0
\(43\) 8.70233 1.32709 0.663547 0.748135i \(-0.269050\pi\)
0.663547 + 0.748135i \(0.269050\pi\)
\(44\) 0 0
\(45\) 3.46791 0.516966
\(46\) 0 0
\(47\) −0.573978 −0.0837233 −0.0418616 0.999123i \(-0.513329\pi\)
−0.0418616 + 0.999123i \(0.513329\pi\)
\(48\) 0 0
\(49\) −4.65270 −0.664672
\(50\) 0 0
\(51\) −2.53209 −0.354563
\(52\) 0 0
\(53\) −2.94356 −0.404329 −0.202165 0.979352i \(-0.564798\pi\)
−0.202165 + 0.979352i \(0.564798\pi\)
\(54\) 0 0
\(55\) −1.59627 −0.215241
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.93582 −0.512400 −0.256200 0.966624i \(-0.582471\pi\)
−0.256200 + 0.966624i \(0.582471\pi\)
\(60\) 0 0
\(61\) −4.51754 −0.578412 −0.289206 0.957267i \(-0.593391\pi\)
−0.289206 + 0.957267i \(0.593391\pi\)
\(62\) 0 0
\(63\) 3.94356 0.496842
\(64\) 0 0
\(65\) 3.66044 0.454022
\(66\) 0 0
\(67\) −3.88713 −0.474888 −0.237444 0.971401i \(-0.576310\pi\)
−0.237444 + 0.971401i \(0.576310\pi\)
\(68\) 0 0
\(69\) −3.30541 −0.397924
\(70\) 0 0
\(71\) −6.93582 −0.823131 −0.411565 0.911380i \(-0.635018\pi\)
−0.411565 + 0.911380i \(0.635018\pi\)
\(72\) 0 0
\(73\) 6.12836 0.717270 0.358635 0.933478i \(-0.383242\pi\)
0.358635 + 0.933478i \(0.383242\pi\)
\(74\) 0 0
\(75\) 2.07873 0.240031
\(76\) 0 0
\(77\) −1.81521 −0.206862
\(78\) 0 0
\(79\) 9.80840 1.10353 0.551766 0.833999i \(-0.313954\pi\)
0.551766 + 0.833999i \(0.313954\pi\)
\(80\) 0 0
\(81\) 5.34730 0.594144
\(82\) 0 0
\(83\) −12.3182 −1.35210 −0.676049 0.736857i \(-0.736309\pi\)
−0.676049 + 0.736857i \(0.736309\pi\)
\(84\) 0 0
\(85\) −5.22668 −0.566913
\(86\) 0 0
\(87\) −3.03684 −0.325583
\(88\) 0 0
\(89\) 2.42602 0.257158 0.128579 0.991699i \(-0.458958\pi\)
0.128579 + 0.991699i \(0.458958\pi\)
\(90\) 0 0
\(91\) 4.16250 0.436349
\(92\) 0 0
\(93\) 2.50475 0.259730
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7.36959 −0.748268 −0.374134 0.927375i \(-0.622060\pi\)
−0.374134 + 0.927375i \(0.622060\pi\)
\(98\) 0 0
\(99\) −3.04963 −0.306499
\(100\) 0 0
\(101\) 2.17024 0.215947 0.107974 0.994154i \(-0.465564\pi\)
0.107974 + 0.994154i \(0.465564\pi\)
\(102\) 0 0
\(103\) −12.4757 −1.22926 −0.614631 0.788815i \(-0.710695\pi\)
−0.614631 + 0.788815i \(0.710695\pi\)
\(104\) 0 0
\(105\) −1.34730 −0.131483
\(106\) 0 0
\(107\) −6.68004 −0.645784 −0.322892 0.946436i \(-0.604655\pi\)
−0.322892 + 0.946436i \(0.604655\pi\)
\(108\) 0 0
\(109\) −9.45336 −0.905468 −0.452734 0.891646i \(-0.649551\pi\)
−0.452734 + 0.891646i \(0.649551\pi\)
\(110\) 0 0
\(111\) −2.68004 −0.254379
\(112\) 0 0
\(113\) −1.31046 −0.123278 −0.0616388 0.998099i \(-0.519633\pi\)
−0.0616388 + 0.998099i \(0.519633\pi\)
\(114\) 0 0
\(115\) −6.82295 −0.636243
\(116\) 0 0
\(117\) 6.99319 0.646521
\(118\) 0 0
\(119\) −5.94356 −0.544846
\(120\) 0 0
\(121\) −9.59627 −0.872388
\(122\) 0 0
\(123\) −6.51754 −0.587667
\(124\) 0 0
\(125\) 11.0273 0.986315
\(126\) 0 0
\(127\) −14.5030 −1.28693 −0.643466 0.765474i \(-0.722504\pi\)
−0.643466 + 0.765474i \(0.722504\pi\)
\(128\) 0 0
\(129\) −5.68004 −0.500100
\(130\) 0 0
\(131\) 19.8084 1.73067 0.865334 0.501196i \(-0.167106\pi\)
0.865334 + 0.501196i \(0.167106\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −4.90167 −0.421869
\(136\) 0 0
\(137\) 10.2044 0.871820 0.435910 0.899990i \(-0.356427\pi\)
0.435910 + 0.899990i \(0.356427\pi\)
\(138\) 0 0
\(139\) 1.66044 0.140837 0.0704185 0.997518i \(-0.477567\pi\)
0.0704185 + 0.997518i \(0.477567\pi\)
\(140\) 0 0
\(141\) 0.374638 0.0315502
\(142\) 0 0
\(143\) −3.21894 −0.269181
\(144\) 0 0
\(145\) −6.26857 −0.520576
\(146\) 0 0
\(147\) 3.03684 0.250474
\(148\) 0 0
\(149\) 11.2071 0.918120 0.459060 0.888405i \(-0.348186\pi\)
0.459060 + 0.888405i \(0.348186\pi\)
\(150\) 0 0
\(151\) −11.0419 −0.898576 −0.449288 0.893387i \(-0.648322\pi\)
−0.449288 + 0.893387i \(0.648322\pi\)
\(152\) 0 0
\(153\) −9.98545 −0.807276
\(154\) 0 0
\(155\) 5.17024 0.415284
\(156\) 0 0
\(157\) 10.9932 0.877352 0.438676 0.898645i \(-0.355448\pi\)
0.438676 + 0.898645i \(0.355448\pi\)
\(158\) 0 0
\(159\) 1.92127 0.152367
\(160\) 0 0
\(161\) −7.75877 −0.611477
\(162\) 0 0
\(163\) 6.33275 0.496019 0.248010 0.968758i \(-0.420224\pi\)
0.248010 + 0.968758i \(0.420224\pi\)
\(164\) 0 0
\(165\) 1.04189 0.0811110
\(166\) 0 0
\(167\) −13.7784 −1.06620 −0.533101 0.846051i \(-0.678973\pi\)
−0.533101 + 0.846051i \(0.678973\pi\)
\(168\) 0 0
\(169\) −5.61856 −0.432197
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −25.2472 −1.91951 −0.959755 0.280838i \(-0.909388\pi\)
−0.959755 + 0.280838i \(0.909388\pi\)
\(174\) 0 0
\(175\) 4.87939 0.368847
\(176\) 0 0
\(177\) 2.56893 0.193092
\(178\) 0 0
\(179\) 5.83069 0.435806 0.217903 0.975970i \(-0.430078\pi\)
0.217903 + 0.975970i \(0.430078\pi\)
\(180\) 0 0
\(181\) −13.5621 −1.00806 −0.504032 0.863685i \(-0.668151\pi\)
−0.504032 + 0.863685i \(0.668151\pi\)
\(182\) 0 0
\(183\) 2.94862 0.217968
\(184\) 0 0
\(185\) −5.53209 −0.406727
\(186\) 0 0
\(187\) 4.59627 0.336112
\(188\) 0 0
\(189\) −5.57398 −0.405447
\(190\) 0 0
\(191\) 10.2841 0.744128 0.372064 0.928207i \(-0.378650\pi\)
0.372064 + 0.928207i \(0.378650\pi\)
\(192\) 0 0
\(193\) −13.8007 −0.993393 −0.496697 0.867924i \(-0.665454\pi\)
−0.496697 + 0.867924i \(0.665454\pi\)
\(194\) 0 0
\(195\) −2.38919 −0.171093
\(196\) 0 0
\(197\) −7.94087 −0.565764 −0.282882 0.959155i \(-0.591290\pi\)
−0.282882 + 0.959155i \(0.591290\pi\)
\(198\) 0 0
\(199\) −27.0351 −1.91647 −0.958233 0.285988i \(-0.907678\pi\)
−0.958233 + 0.285988i \(0.907678\pi\)
\(200\) 0 0
\(201\) 2.53714 0.178956
\(202\) 0 0
\(203\) −7.12836 −0.500312
\(204\) 0 0
\(205\) −13.4534 −0.939624
\(206\) 0 0
\(207\) −13.0351 −0.906001
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.07192 −0.555694 −0.277847 0.960625i \(-0.589621\pi\)
−0.277847 + 0.960625i \(0.589621\pi\)
\(212\) 0 0
\(213\) 4.52704 0.310187
\(214\) 0 0
\(215\) −11.7246 −0.799613
\(216\) 0 0
\(217\) 5.87939 0.399119
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −10.5398 −0.708986
\(222\) 0 0
\(223\) −15.4757 −1.03633 −0.518163 0.855282i \(-0.673384\pi\)
−0.518163 + 0.855282i \(0.673384\pi\)
\(224\) 0 0
\(225\) 8.19759 0.546506
\(226\) 0 0
\(227\) 9.87258 0.655266 0.327633 0.944805i \(-0.393749\pi\)
0.327633 + 0.944805i \(0.393749\pi\)
\(228\) 0 0
\(229\) 20.1189 1.32949 0.664746 0.747070i \(-0.268540\pi\)
0.664746 + 0.747070i \(0.268540\pi\)
\(230\) 0 0
\(231\) 1.18479 0.0779536
\(232\) 0 0
\(233\) 3.53478 0.231571 0.115785 0.993274i \(-0.463061\pi\)
0.115785 + 0.993274i \(0.463061\pi\)
\(234\) 0 0
\(235\) 0.773318 0.0504457
\(236\) 0 0
\(237\) −6.40198 −0.415853
\(238\) 0 0
\(239\) −11.9736 −0.774507 −0.387254 0.921973i \(-0.626576\pi\)
−0.387254 + 0.921973i \(0.626576\pi\)
\(240\) 0 0
\(241\) −12.9017 −0.831070 −0.415535 0.909577i \(-0.636406\pi\)
−0.415535 + 0.909577i \(0.636406\pi\)
\(242\) 0 0
\(243\) −14.4047 −0.924060
\(244\) 0 0
\(245\) 6.26857 0.400484
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 8.04013 0.509523
\(250\) 0 0
\(251\) −14.3628 −0.906571 −0.453285 0.891366i \(-0.649748\pi\)
−0.453285 + 0.891366i \(0.649748\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 0 0
\(255\) 3.41147 0.213635
\(256\) 0 0
\(257\) −4.97771 −0.310501 −0.155251 0.987875i \(-0.549618\pi\)
−0.155251 + 0.987875i \(0.549618\pi\)
\(258\) 0 0
\(259\) −6.29086 −0.390895
\(260\) 0 0
\(261\) −11.9760 −0.741293
\(262\) 0 0
\(263\) 24.0428 1.48254 0.741272 0.671205i \(-0.234223\pi\)
0.741272 + 0.671205i \(0.234223\pi\)
\(264\) 0 0
\(265\) 3.96585 0.243620
\(266\) 0 0
\(267\) −1.58347 −0.0969070
\(268\) 0 0
\(269\) 13.1111 0.799399 0.399700 0.916646i \(-0.369114\pi\)
0.399700 + 0.916646i \(0.369114\pi\)
\(270\) 0 0
\(271\) 26.5699 1.61400 0.807002 0.590549i \(-0.201089\pi\)
0.807002 + 0.590549i \(0.201089\pi\)
\(272\) 0 0
\(273\) −2.71688 −0.164433
\(274\) 0 0
\(275\) −3.77332 −0.227540
\(276\) 0 0
\(277\) −16.5107 −0.992034 −0.496017 0.868313i \(-0.665205\pi\)
−0.496017 + 0.868313i \(0.665205\pi\)
\(278\) 0 0
\(279\) 9.87763 0.591358
\(280\) 0 0
\(281\) 19.3901 1.15672 0.578359 0.815783i \(-0.303693\pi\)
0.578359 + 0.815783i \(0.303693\pi\)
\(282\) 0 0
\(283\) 11.3105 0.672337 0.336169 0.941802i \(-0.390869\pi\)
0.336169 + 0.941802i \(0.390869\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −15.2986 −0.903048
\(288\) 0 0
\(289\) −1.95037 −0.114728
\(290\) 0 0
\(291\) 4.81016 0.281976
\(292\) 0 0
\(293\) −3.89899 −0.227781 −0.113891 0.993493i \(-0.536331\pi\)
−0.113891 + 0.993493i \(0.536331\pi\)
\(294\) 0 0
\(295\) 5.30272 0.308736
\(296\) 0 0
\(297\) 4.31046 0.250118
\(298\) 0 0
\(299\) −13.7588 −0.795690
\(300\) 0 0
\(301\) −13.3327 −0.768487
\(302\) 0 0
\(303\) −1.41653 −0.0813773
\(304\) 0 0
\(305\) 6.08647 0.348510
\(306\) 0 0
\(307\) 23.1753 1.32268 0.661342 0.750084i \(-0.269987\pi\)
0.661342 + 0.750084i \(0.269987\pi\)
\(308\) 0 0
\(309\) 8.14290 0.463234
\(310\) 0 0
\(311\) 3.46110 0.196261 0.0981306 0.995174i \(-0.468714\pi\)
0.0981306 + 0.995174i \(0.468714\pi\)
\(312\) 0 0
\(313\) −22.8898 −1.29381 −0.646904 0.762571i \(-0.723937\pi\)
−0.646904 + 0.762571i \(0.723937\pi\)
\(314\) 0 0
\(315\) −5.31315 −0.299362
\(316\) 0 0
\(317\) −26.1206 −1.46708 −0.733540 0.679646i \(-0.762133\pi\)
−0.733540 + 0.679646i \(0.762133\pi\)
\(318\) 0 0
\(319\) 5.51249 0.308640
\(320\) 0 0
\(321\) 4.36009 0.243356
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 8.65270 0.479966
\(326\) 0 0
\(327\) 6.17024 0.341215
\(328\) 0 0
\(329\) 0.879385 0.0484821
\(330\) 0 0
\(331\) 19.0446 1.04678 0.523392 0.852092i \(-0.324666\pi\)
0.523392 + 0.852092i \(0.324666\pi\)
\(332\) 0 0
\(333\) −10.5689 −0.579174
\(334\) 0 0
\(335\) 5.23711 0.286134
\(336\) 0 0
\(337\) −1.70140 −0.0926812 −0.0463406 0.998926i \(-0.514756\pi\)
−0.0463406 + 0.998926i \(0.514756\pi\)
\(338\) 0 0
\(339\) 0.855342 0.0464558
\(340\) 0 0
\(341\) −4.54664 −0.246214
\(342\) 0 0
\(343\) 17.8530 0.963970
\(344\) 0 0
\(345\) 4.45336 0.239761
\(346\) 0 0
\(347\) 4.90167 0.263136 0.131568 0.991307i \(-0.457999\pi\)
0.131568 + 0.991307i \(0.457999\pi\)
\(348\) 0 0
\(349\) −28.1293 −1.50573 −0.752863 0.658177i \(-0.771328\pi\)
−0.752863 + 0.658177i \(0.771328\pi\)
\(350\) 0 0
\(351\) −9.88444 −0.527592
\(352\) 0 0
\(353\) −8.31996 −0.442827 −0.221413 0.975180i \(-0.571067\pi\)
−0.221413 + 0.975180i \(0.571067\pi\)
\(354\) 0 0
\(355\) 9.34461 0.495960
\(356\) 0 0
\(357\) 3.87939 0.205319
\(358\) 0 0
\(359\) −24.9290 −1.31570 −0.657852 0.753148i \(-0.728535\pi\)
−0.657852 + 0.753148i \(0.728535\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) 6.26352 0.328749
\(364\) 0 0
\(365\) −8.25671 −0.432176
\(366\) 0 0
\(367\) 2.58584 0.134980 0.0674898 0.997720i \(-0.478501\pi\)
0.0674898 + 0.997720i \(0.478501\pi\)
\(368\) 0 0
\(369\) −25.7023 −1.33801
\(370\) 0 0
\(371\) 4.50980 0.234137
\(372\) 0 0
\(373\) 23.3833 1.21074 0.605371 0.795943i \(-0.293025\pi\)
0.605371 + 0.795943i \(0.293025\pi\)
\(374\) 0 0
\(375\) −7.19759 −0.371682
\(376\) 0 0
\(377\) −12.6408 −0.651037
\(378\) 0 0
\(379\) 25.4388 1.30670 0.653352 0.757054i \(-0.273362\pi\)
0.653352 + 0.757054i \(0.273362\pi\)
\(380\) 0 0
\(381\) 9.46616 0.484966
\(382\) 0 0
\(383\) −27.4807 −1.40420 −0.702099 0.712079i \(-0.747754\pi\)
−0.702099 + 0.712079i \(0.747754\pi\)
\(384\) 0 0
\(385\) 2.44562 0.124640
\(386\) 0 0
\(387\) −22.3996 −1.13864
\(388\) 0 0
\(389\) −3.34224 −0.169458 −0.0847292 0.996404i \(-0.527003\pi\)
−0.0847292 + 0.996404i \(0.527003\pi\)
\(390\) 0 0
\(391\) 19.6459 0.993536
\(392\) 0 0
\(393\) −12.9290 −0.652183
\(394\) 0 0
\(395\) −13.2148 −0.664910
\(396\) 0 0
\(397\) −13.1233 −0.658640 −0.329320 0.944218i \(-0.606819\pi\)
−0.329320 + 0.944218i \(0.606819\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −17.1138 −0.854623 −0.427311 0.904105i \(-0.640539\pi\)
−0.427311 + 0.904105i \(0.640539\pi\)
\(402\) 0 0
\(403\) 10.4260 0.519357
\(404\) 0 0
\(405\) −7.20439 −0.357989
\(406\) 0 0
\(407\) 4.86484 0.241141
\(408\) 0 0
\(409\) 8.79797 0.435032 0.217516 0.976057i \(-0.430205\pi\)
0.217516 + 0.976057i \(0.430205\pi\)
\(410\) 0 0
\(411\) −6.66044 −0.328535
\(412\) 0 0
\(413\) 6.03003 0.296718
\(414\) 0 0
\(415\) 16.5963 0.814679
\(416\) 0 0
\(417\) −1.08378 −0.0530728
\(418\) 0 0
\(419\) −6.84018 −0.334165 −0.167082 0.985943i \(-0.553435\pi\)
−0.167082 + 0.985943i \(0.553435\pi\)
\(420\) 0 0
\(421\) 4.82295 0.235056 0.117528 0.993070i \(-0.462503\pi\)
0.117528 + 0.993070i \(0.462503\pi\)
\(422\) 0 0
\(423\) 1.47741 0.0718340
\(424\) 0 0
\(425\) −12.3550 −0.599307
\(426\) 0 0
\(427\) 6.92127 0.334944
\(428\) 0 0
\(429\) 2.10101 0.101438
\(430\) 0 0
\(431\) 1.30365 0.0627947 0.0313974 0.999507i \(-0.490004\pi\)
0.0313974 + 0.999507i \(0.490004\pi\)
\(432\) 0 0
\(433\) −19.8239 −0.952675 −0.476337 0.879263i \(-0.658036\pi\)
−0.476337 + 0.879263i \(0.658036\pi\)
\(434\) 0 0
\(435\) 4.09152 0.196173
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −34.5672 −1.64980 −0.824901 0.565278i \(-0.808769\pi\)
−0.824901 + 0.565278i \(0.808769\pi\)
\(440\) 0 0
\(441\) 11.9760 0.570284
\(442\) 0 0
\(443\) −17.0101 −0.808174 −0.404087 0.914720i \(-0.632411\pi\)
−0.404087 + 0.914720i \(0.632411\pi\)
\(444\) 0 0
\(445\) −3.26857 −0.154945
\(446\) 0 0
\(447\) −7.31490 −0.345983
\(448\) 0 0
\(449\) −37.4097 −1.76547 −0.882737 0.469868i \(-0.844302\pi\)
−0.882737 + 0.469868i \(0.844302\pi\)
\(450\) 0 0
\(451\) 11.8307 0.557085
\(452\) 0 0
\(453\) 7.20708 0.338618
\(454\) 0 0
\(455\) −5.60813 −0.262913
\(456\) 0 0
\(457\) 9.11112 0.426200 0.213100 0.977030i \(-0.431644\pi\)
0.213100 + 0.977030i \(0.431644\pi\)
\(458\) 0 0
\(459\) 14.1138 0.658776
\(460\) 0 0
\(461\) −24.4483 −1.13867 −0.569336 0.822105i \(-0.692800\pi\)
−0.569336 + 0.822105i \(0.692800\pi\)
\(462\) 0 0
\(463\) 0.250725 0.0116522 0.00582609 0.999983i \(-0.498145\pi\)
0.00582609 + 0.999983i \(0.498145\pi\)
\(464\) 0 0
\(465\) −3.37464 −0.156495
\(466\) 0 0
\(467\) −15.3618 −0.710861 −0.355431 0.934703i \(-0.615666\pi\)
−0.355431 + 0.934703i \(0.615666\pi\)
\(468\) 0 0
\(469\) 5.95542 0.274996
\(470\) 0 0
\(471\) −7.17530 −0.330620
\(472\) 0 0
\(473\) 10.3105 0.474075
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 7.57667 0.346912
\(478\) 0 0
\(479\) 0.719246 0.0328632 0.0164316 0.999865i \(-0.494769\pi\)
0.0164316 + 0.999865i \(0.494769\pi\)
\(480\) 0 0
\(481\) −11.1557 −0.508656
\(482\) 0 0
\(483\) 5.06418 0.230428
\(484\) 0 0
\(485\) 9.92902 0.450853
\(486\) 0 0
\(487\) 11.7469 0.532303 0.266152 0.963931i \(-0.414248\pi\)
0.266152 + 0.963931i \(0.414248\pi\)
\(488\) 0 0
\(489\) −4.13341 −0.186919
\(490\) 0 0
\(491\) −0.0888306 −0.00400887 −0.00200443 0.999998i \(-0.500638\pi\)
−0.00200443 + 0.999998i \(0.500638\pi\)
\(492\) 0 0
\(493\) 18.0496 0.812914
\(494\) 0 0
\(495\) 4.10876 0.184675
\(496\) 0 0
\(497\) 10.6263 0.476655
\(498\) 0 0
\(499\) −14.6905 −0.657636 −0.328818 0.944393i \(-0.606650\pi\)
−0.328818 + 0.944393i \(0.606650\pi\)
\(500\) 0 0
\(501\) 8.99319 0.401786
\(502\) 0 0
\(503\) 4.90404 0.218660 0.109330 0.994005i \(-0.465129\pi\)
0.109330 + 0.994005i \(0.465129\pi\)
\(504\) 0 0
\(505\) −2.92396 −0.130115
\(506\) 0 0
\(507\) 3.66725 0.162868
\(508\) 0 0
\(509\) −6.41384 −0.284288 −0.142144 0.989846i \(-0.545400\pi\)
−0.142144 + 0.989846i \(0.545400\pi\)
\(510\) 0 0
\(511\) −9.38919 −0.415353
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.8084 0.740667
\(516\) 0 0
\(517\) −0.680045 −0.0299083
\(518\) 0 0
\(519\) 16.4789 0.723346
\(520\) 0 0
\(521\) 35.8135 1.56902 0.784508 0.620119i \(-0.212916\pi\)
0.784508 + 0.620119i \(0.212916\pi\)
\(522\) 0 0
\(523\) 38.7725 1.69540 0.847701 0.530474i \(-0.177986\pi\)
0.847701 + 0.530474i \(0.177986\pi\)
\(524\) 0 0
\(525\) −3.18479 −0.138996
\(526\) 0 0
\(527\) −14.8871 −0.648493
\(528\) 0 0
\(529\) 2.64590 0.115039
\(530\) 0 0
\(531\) 10.1307 0.439636
\(532\) 0 0
\(533\) −27.1293 −1.17510
\(534\) 0 0
\(535\) 9.00000 0.389104
\(536\) 0 0
\(537\) −3.80571 −0.164229
\(538\) 0 0
\(539\) −5.51249 −0.237440
\(540\) 0 0
\(541\) 9.49020 0.408016 0.204008 0.978969i \(-0.434603\pi\)
0.204008 + 0.978969i \(0.434603\pi\)
\(542\) 0 0
\(543\) 8.85204 0.379878
\(544\) 0 0
\(545\) 12.7365 0.545571
\(546\) 0 0
\(547\) 14.2121 0.607667 0.303833 0.952725i \(-0.401733\pi\)
0.303833 + 0.952725i \(0.401733\pi\)
\(548\) 0 0
\(549\) 11.6281 0.496273
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −15.0273 −0.639028
\(554\) 0 0
\(555\) 3.61081 0.153271
\(556\) 0 0
\(557\) 22.5398 0.955043 0.477522 0.878620i \(-0.341535\pi\)
0.477522 + 0.878620i \(0.341535\pi\)
\(558\) 0 0
\(559\) −23.6432 −1.00000
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) 0 0
\(563\) −42.9718 −1.81105 −0.905524 0.424296i \(-0.860522\pi\)
−0.905524 + 0.424296i \(0.860522\pi\)
\(564\) 0 0
\(565\) 1.76558 0.0742784
\(566\) 0 0
\(567\) −8.19253 −0.344054
\(568\) 0 0
\(569\) −7.42696 −0.311354 −0.155677 0.987808i \(-0.549756\pi\)
−0.155677 + 0.987808i \(0.549756\pi\)
\(570\) 0 0
\(571\) −4.04458 −0.169260 −0.0846301 0.996412i \(-0.526971\pi\)
−0.0846301 + 0.996412i \(0.526971\pi\)
\(572\) 0 0
\(573\) −6.71244 −0.280416
\(574\) 0 0
\(575\) −16.1284 −0.672599
\(576\) 0 0
\(577\) 3.23442 0.134651 0.0673254 0.997731i \(-0.478553\pi\)
0.0673254 + 0.997731i \(0.478553\pi\)
\(578\) 0 0
\(579\) 9.00774 0.374349
\(580\) 0 0
\(581\) 18.8726 0.782966
\(582\) 0 0
\(583\) −3.48751 −0.144438
\(584\) 0 0
\(585\) −9.42190 −0.389548
\(586\) 0 0
\(587\) 40.8084 1.68434 0.842171 0.539210i \(-0.181277\pi\)
0.842171 + 0.539210i \(0.181277\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 5.18304 0.213202
\(592\) 0 0
\(593\) −11.0642 −0.454351 −0.227176 0.973854i \(-0.572949\pi\)
−0.227176 + 0.973854i \(0.572949\pi\)
\(594\) 0 0
\(595\) 8.00774 0.328285
\(596\) 0 0
\(597\) 17.6459 0.722198
\(598\) 0 0
\(599\) −44.5577 −1.82058 −0.910289 0.413974i \(-0.864140\pi\)
−0.910289 + 0.413974i \(0.864140\pi\)
\(600\) 0 0
\(601\) 4.99907 0.203916 0.101958 0.994789i \(-0.467489\pi\)
0.101958 + 0.994789i \(0.467489\pi\)
\(602\) 0 0
\(603\) 10.0054 0.407450
\(604\) 0 0
\(605\) 12.9290 0.525639
\(606\) 0 0
\(607\) −31.1881 −1.26589 −0.632943 0.774199i \(-0.718153\pi\)
−0.632943 + 0.774199i \(0.718153\pi\)
\(608\) 0 0
\(609\) 4.65270 0.188537
\(610\) 0 0
\(611\) 1.55943 0.0630878
\(612\) 0 0
\(613\) 16.3696 0.661161 0.330581 0.943778i \(-0.392755\pi\)
0.330581 + 0.943778i \(0.392755\pi\)
\(614\) 0 0
\(615\) 8.78106 0.354086
\(616\) 0 0
\(617\) 16.0583 0.646483 0.323241 0.946317i \(-0.395227\pi\)
0.323241 + 0.946317i \(0.395227\pi\)
\(618\) 0 0
\(619\) −23.8425 −0.958313 −0.479156 0.877730i \(-0.659057\pi\)
−0.479156 + 0.877730i \(0.659057\pi\)
\(620\) 0 0
\(621\) 18.4243 0.739340
\(622\) 0 0
\(623\) −3.71688 −0.148914
\(624\) 0 0
\(625\) 1.06687 0.0426746
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 15.9290 0.635131
\(630\) 0 0
\(631\) −21.4730 −0.854825 −0.427413 0.904057i \(-0.640575\pi\)
−0.427413 + 0.904057i \(0.640575\pi\)
\(632\) 0 0
\(633\) 5.26857 0.209407
\(634\) 0 0
\(635\) 19.5398 0.775414
\(636\) 0 0
\(637\) 12.6408 0.500848
\(638\) 0 0
\(639\) 17.8527 0.706240
\(640\) 0 0
\(641\) −12.7537 −0.503742 −0.251871 0.967761i \(-0.581046\pi\)
−0.251871 + 0.967761i \(0.581046\pi\)
\(642\) 0 0
\(643\) −28.6081 −1.12819 −0.564097 0.825708i \(-0.690776\pi\)
−0.564097 + 0.825708i \(0.690776\pi\)
\(644\) 0 0
\(645\) 7.65270 0.301325
\(646\) 0 0
\(647\) −16.7128 −0.657046 −0.328523 0.944496i \(-0.606551\pi\)
−0.328523 + 0.944496i \(0.606551\pi\)
\(648\) 0 0
\(649\) −4.66313 −0.183044
\(650\) 0 0
\(651\) −3.83750 −0.150403
\(652\) 0 0
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) 0 0
\(655\) −26.6878 −1.04278
\(656\) 0 0
\(657\) −15.7743 −0.615412
\(658\) 0 0
\(659\) −43.9009 −1.71013 −0.855067 0.518517i \(-0.826484\pi\)
−0.855067 + 0.518517i \(0.826484\pi\)
\(660\) 0 0
\(661\) 10.7561 0.418363 0.209182 0.977877i \(-0.432920\pi\)
0.209182 + 0.977877i \(0.432920\pi\)
\(662\) 0 0
\(663\) 6.87939 0.267173
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 23.5621 0.912329
\(668\) 0 0
\(669\) 10.1010 0.390528
\(670\) 0 0
\(671\) −5.35235 −0.206625
\(672\) 0 0
\(673\) −4.65776 −0.179543 −0.0897717 0.995962i \(-0.528614\pi\)
−0.0897717 + 0.995962i \(0.528614\pi\)
\(674\) 0 0
\(675\) −11.5868 −0.445975
\(676\) 0 0
\(677\) −3.26857 −0.125621 −0.0628107 0.998025i \(-0.520006\pi\)
−0.0628107 + 0.998025i \(0.520006\pi\)
\(678\) 0 0
\(679\) 11.2909 0.433303
\(680\) 0 0
\(681\) −6.44387 −0.246930
\(682\) 0 0
\(683\) 6.21894 0.237961 0.118981 0.992897i \(-0.462037\pi\)
0.118981 + 0.992897i \(0.462037\pi\)
\(684\) 0 0
\(685\) −13.7483 −0.525297
\(686\) 0 0
\(687\) −13.1317 −0.501004
\(688\) 0 0
\(689\) 7.99731 0.304673
\(690\) 0 0
\(691\) −22.2175 −0.845194 −0.422597 0.906318i \(-0.638881\pi\)
−0.422597 + 0.906318i \(0.638881\pi\)
\(692\) 0 0
\(693\) 4.67230 0.177486
\(694\) 0 0
\(695\) −2.23711 −0.0848584
\(696\) 0 0
\(697\) 38.7374 1.46728
\(698\) 0 0
\(699\) −2.30716 −0.0872649
\(700\) 0 0
\(701\) −27.7725 −1.04895 −0.524476 0.851425i \(-0.675739\pi\)
−0.524476 + 0.851425i \(0.675739\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −0.504748 −0.0190099
\(706\) 0 0
\(707\) −3.32501 −0.125050
\(708\) 0 0
\(709\) −6.10782 −0.229384 −0.114692 0.993401i \(-0.536588\pi\)
−0.114692 + 0.993401i \(0.536588\pi\)
\(710\) 0 0
\(711\) −25.2466 −0.946822
\(712\) 0 0
\(713\) −19.4338 −0.727800
\(714\) 0 0
\(715\) 4.33687 0.162190
\(716\) 0 0
\(717\) 7.81521 0.291864
\(718\) 0 0
\(719\) −38.7238 −1.44415 −0.722077 0.691813i \(-0.756812\pi\)
−0.722077 + 0.691813i \(0.756812\pi\)
\(720\) 0 0
\(721\) 19.1138 0.711835
\(722\) 0 0
\(723\) 8.42097 0.313179
\(724\) 0 0
\(725\) −14.8179 −0.550323
\(726\) 0 0
\(727\) 11.0779 0.410857 0.205428 0.978672i \(-0.434141\pi\)
0.205428 + 0.978672i \(0.434141\pi\)
\(728\) 0 0
\(729\) −6.63991 −0.245923
\(730\) 0 0
\(731\) 33.7597 1.24865
\(732\) 0 0
\(733\) −15.8075 −0.583862 −0.291931 0.956439i \(-0.594298\pi\)
−0.291931 + 0.956439i \(0.594298\pi\)
\(734\) 0 0
\(735\) −4.09152 −0.150918
\(736\) 0 0
\(737\) −4.60544 −0.169643
\(738\) 0 0
\(739\) −1.54933 −0.0569928 −0.0284964 0.999594i \(-0.509072\pi\)
−0.0284964 + 0.999594i \(0.509072\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 38.1634 1.40008 0.700040 0.714103i \(-0.253165\pi\)
0.700040 + 0.714103i \(0.253165\pi\)
\(744\) 0 0
\(745\) −15.0993 −0.553194
\(746\) 0 0
\(747\) 31.7068 1.16009
\(748\) 0 0
\(749\) 10.2344 0.373958
\(750\) 0 0
\(751\) −25.3482 −0.924970 −0.462485 0.886627i \(-0.653042\pi\)
−0.462485 + 0.886627i \(0.653042\pi\)
\(752\) 0 0
\(753\) 9.37464 0.341631
\(754\) 0 0
\(755\) 14.8767 0.541418
\(756\) 0 0
\(757\) 42.3705 1.53998 0.769991 0.638054i \(-0.220260\pi\)
0.769991 + 0.638054i \(0.220260\pi\)
\(758\) 0 0
\(759\) −3.91622 −0.142150
\(760\) 0 0
\(761\) −2.85710 −0.103570 −0.0517848 0.998658i \(-0.516491\pi\)
−0.0517848 + 0.998658i \(0.516491\pi\)
\(762\) 0 0
\(763\) 14.4834 0.524334
\(764\) 0 0
\(765\) 13.4534 0.486407
\(766\) 0 0
\(767\) 10.6932 0.386108
\(768\) 0 0
\(769\) 19.1206 0.689507 0.344754 0.938693i \(-0.387963\pi\)
0.344754 + 0.938693i \(0.387963\pi\)
\(770\) 0 0
\(771\) 3.24897 0.117009
\(772\) 0 0
\(773\) −2.51485 −0.0904530 −0.0452265 0.998977i \(-0.514401\pi\)
−0.0452265 + 0.998977i \(0.514401\pi\)
\(774\) 0 0
\(775\) 12.2216 0.439014
\(776\) 0 0
\(777\) 4.10607 0.147304
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −8.21751 −0.294046
\(782\) 0 0
\(783\) 16.9273 0.604931
\(784\) 0 0
\(785\) −14.8111 −0.528630
\(786\) 0 0
\(787\) 2.72605 0.0971733 0.0485866 0.998819i \(-0.484528\pi\)
0.0485866 + 0.998819i \(0.484528\pi\)
\(788\) 0 0
\(789\) −15.6928 −0.558680
\(790\) 0 0
\(791\) 2.00774 0.0713870
\(792\) 0 0
\(793\) 12.2736 0.435849
\(794\) 0 0
\(795\) −2.58853 −0.0918056
\(796\) 0 0
\(797\) −22.0327 −0.780439 −0.390219 0.920722i \(-0.627601\pi\)
−0.390219 + 0.920722i \(0.627601\pi\)
\(798\) 0 0
\(799\) −2.22668 −0.0787743
\(800\) 0 0
\(801\) −6.24453 −0.220640
\(802\) 0 0
\(803\) 7.26083 0.256229
\(804\) 0 0
\(805\) 10.4534 0.368433
\(806\) 0 0
\(807\) −8.55768 −0.301244
\(808\) 0 0
\(809\) 54.7205 1.92387 0.961935 0.273278i \(-0.0881077\pi\)
0.961935 + 0.273278i \(0.0881077\pi\)
\(810\) 0 0
\(811\) −2.31046 −0.0811312 −0.0405656 0.999177i \(-0.512916\pi\)
−0.0405656 + 0.999177i \(0.512916\pi\)
\(812\) 0 0
\(813\) −17.3422 −0.608219
\(814\) 0 0
\(815\) −8.53209 −0.298866
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −10.7142 −0.374384
\(820\) 0 0
\(821\) 1.11112 0.0387783 0.0193892 0.999812i \(-0.493828\pi\)
0.0193892 + 0.999812i \(0.493828\pi\)
\(822\) 0 0
\(823\) −20.6477 −0.719732 −0.359866 0.933004i \(-0.617178\pi\)
−0.359866 + 0.933004i \(0.617178\pi\)
\(824\) 0 0
\(825\) 2.46286 0.0857457
\(826\) 0 0
\(827\) 36.3054 1.26246 0.631231 0.775595i \(-0.282550\pi\)
0.631231 + 0.775595i \(0.282550\pi\)
\(828\) 0 0
\(829\) 7.14971 0.248320 0.124160 0.992262i \(-0.460376\pi\)
0.124160 + 0.992262i \(0.460376\pi\)
\(830\) 0 0
\(831\) 10.7766 0.373837
\(832\) 0 0
\(833\) −18.0496 −0.625383
\(834\) 0 0
\(835\) 18.5635 0.642418
\(836\) 0 0
\(837\) −13.9614 −0.482577
\(838\) 0 0
\(839\) −34.6067 −1.19476 −0.597378 0.801960i \(-0.703791\pi\)
−0.597378 + 0.801960i \(0.703791\pi\)
\(840\) 0 0
\(841\) −7.35235 −0.253529
\(842\) 0 0
\(843\) −12.6560 −0.435896
\(844\) 0 0
\(845\) 7.56986 0.260411
\(846\) 0 0
\(847\) 14.7023 0.505178
\(848\) 0 0
\(849\) −7.38238 −0.253363
\(850\) 0 0
\(851\) 20.7939 0.712804
\(852\) 0 0
\(853\) 33.2508 1.13849 0.569243 0.822169i \(-0.307236\pi\)
0.569243 + 0.822169i \(0.307236\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.88619 0.132750 0.0663749 0.997795i \(-0.478857\pi\)
0.0663749 + 0.997795i \(0.478857\pi\)
\(858\) 0 0
\(859\) −1.65776 −0.0565619 −0.0282810 0.999600i \(-0.509003\pi\)
−0.0282810 + 0.999600i \(0.509003\pi\)
\(860\) 0 0
\(861\) 9.98545 0.340303
\(862\) 0 0
\(863\) −52.7187 −1.79457 −0.897284 0.441455i \(-0.854463\pi\)
−0.897284 + 0.441455i \(0.854463\pi\)
\(864\) 0 0
\(865\) 34.0155 1.15656
\(866\) 0 0
\(867\) 1.27301 0.0432338
\(868\) 0 0
\(869\) 11.6209 0.394213
\(870\) 0 0
\(871\) 10.5609 0.357841
\(872\) 0 0
\(873\) 18.9691 0.642008
\(874\) 0 0
\(875\) −16.8949 −0.571151
\(876\) 0 0
\(877\) 21.1898 0.715530 0.357765 0.933812i \(-0.383539\pi\)
0.357765 + 0.933812i \(0.383539\pi\)
\(878\) 0 0
\(879\) 2.54488 0.0858367
\(880\) 0 0
\(881\) 32.1010 1.08151 0.540755 0.841180i \(-0.318138\pi\)
0.540755 + 0.841180i \(0.318138\pi\)
\(882\) 0 0
\(883\) 47.2968 1.59167 0.795833 0.605516i \(-0.207033\pi\)
0.795833 + 0.605516i \(0.207033\pi\)
\(884\) 0 0
\(885\) −3.46110 −0.116344
\(886\) 0 0
\(887\) 10.5631 0.354673 0.177336 0.984150i \(-0.443252\pi\)
0.177336 + 0.984150i \(0.443252\pi\)
\(888\) 0 0
\(889\) 22.2199 0.745231
\(890\) 0 0
\(891\) 6.33544 0.212245
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −7.85567 −0.262586
\(896\) 0 0
\(897\) 8.98040 0.299847
\(898\) 0 0
\(899\) −17.8547 −0.595489
\(900\) 0 0
\(901\) −11.4192 −0.380429
\(902\) 0 0
\(903\) 8.70233 0.289596
\(904\) 0 0
\(905\) 18.2722 0.607388
\(906\) 0 0
\(907\) 42.9205 1.42515 0.712575 0.701596i \(-0.247529\pi\)
0.712575 + 0.701596i \(0.247529\pi\)
\(908\) 0 0
\(909\) −5.58616 −0.185281
\(910\) 0 0
\(911\) −55.1411 −1.82691 −0.913454 0.406942i \(-0.866595\pi\)
−0.913454 + 0.406942i \(0.866595\pi\)
\(912\) 0 0
\(913\) −14.5945 −0.483008
\(914\) 0 0
\(915\) −3.97266 −0.131332
\(916\) 0 0
\(917\) −30.3482 −1.00219
\(918\) 0 0
\(919\) 24.5577 0.810083 0.405041 0.914298i \(-0.367257\pi\)
0.405041 + 0.914298i \(0.367257\pi\)
\(920\) 0 0
\(921\) −15.1266 −0.498438
\(922\) 0 0
\(923\) 18.8438 0.620251
\(924\) 0 0
\(925\) −13.0770 −0.429968
\(926\) 0 0
\(927\) 32.1121 1.05470
\(928\) 0 0
\(929\) −22.2772 −0.730893 −0.365446 0.930832i \(-0.619084\pi\)
−0.365446 + 0.930832i \(0.619084\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −2.25908 −0.0739588
\(934\) 0 0
\(935\) −6.19253 −0.202517
\(936\) 0 0
\(937\) −9.55169 −0.312040 −0.156020 0.987754i \(-0.549866\pi\)
−0.156020 + 0.987754i \(0.549866\pi\)
\(938\) 0 0
\(939\) 14.9403 0.487557
\(940\) 0 0
\(941\) 55.7256 1.81660 0.908301 0.418318i \(-0.137380\pi\)
0.908301 + 0.418318i \(0.137380\pi\)
\(942\) 0 0
\(943\) 50.5681 1.64672
\(944\) 0 0
\(945\) 7.50980 0.244294
\(946\) 0 0
\(947\) 27.0428 0.878774 0.439387 0.898298i \(-0.355196\pi\)
0.439387 + 0.898298i \(0.355196\pi\)
\(948\) 0 0
\(949\) −16.6500 −0.540482
\(950\) 0 0
\(951\) 17.0490 0.552852
\(952\) 0 0
\(953\) −23.1310 −0.749288 −0.374644 0.927169i \(-0.622235\pi\)
−0.374644 + 0.927169i \(0.622235\pi\)
\(954\) 0 0
\(955\) −13.8557 −0.448359
\(956\) 0 0
\(957\) −3.59802 −0.116308
\(958\) 0 0
\(959\) −15.6340 −0.504849
\(960\) 0 0
\(961\) −16.2736 −0.524956
\(962\) 0 0
\(963\) 17.1943 0.554078
\(964\) 0 0
\(965\) 18.5936 0.598548
\(966\) 0 0
\(967\) −39.0351 −1.25528 −0.627642 0.778502i \(-0.715980\pi\)
−0.627642 + 0.778502i \(0.715980\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −41.2026 −1.32226 −0.661128 0.750273i \(-0.729922\pi\)
−0.661128 + 0.750273i \(0.729922\pi\)
\(972\) 0 0
\(973\) −2.54395 −0.0815552
\(974\) 0 0
\(975\) −5.64765 −0.180870
\(976\) 0 0
\(977\) −22.4938 −0.719641 −0.359821 0.933022i \(-0.617162\pi\)
−0.359821 + 0.933022i \(0.617162\pi\)
\(978\) 0 0
\(979\) 2.87433 0.0918641
\(980\) 0 0
\(981\) 24.3327 0.776885
\(982\) 0 0
\(983\) −44.5461 −1.42080 −0.710401 0.703798i \(-0.751486\pi\)
−0.710401 + 0.703798i \(0.751486\pi\)
\(984\) 0 0
\(985\) 10.6987 0.340889
\(986\) 0 0
\(987\) −0.573978 −0.0182699
\(988\) 0 0
\(989\) 44.0702 1.40135
\(990\) 0 0
\(991\) −45.3296 −1.43994 −0.719971 0.694005i \(-0.755845\pi\)
−0.719971 + 0.694005i \(0.755845\pi\)
\(992\) 0 0
\(993\) −12.4305 −0.394469
\(994\) 0 0
\(995\) 36.4243 1.15473
\(996\) 0 0
\(997\) −10.4911 −0.332258 −0.166129 0.986104i \(-0.553127\pi\)
−0.166129 + 0.986104i \(0.553127\pi\)
\(998\) 0 0
\(999\) 14.9385 0.472634
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 5776.2.a.bi.1.2 3
4.3 odd 2 361.2.a.h.1.3 3
12.11 even 2 3249.2.a.s.1.1 3
19.14 odd 18 304.2.u.b.177.1 6
19.15 odd 18 304.2.u.b.225.1 6
19.18 odd 2 5776.2.a.br.1.2 3
20.19 odd 2 9025.2.a.x.1.1 3
76.3 even 18 361.2.e.f.28.1 6
76.7 odd 6 361.2.c.h.68.1 6
76.11 odd 6 361.2.c.h.292.1 6
76.15 even 18 19.2.e.a.16.1 yes 6
76.23 odd 18 361.2.e.h.54.1 6
76.27 even 6 361.2.c.i.292.3 6
76.31 even 6 361.2.c.i.68.3 6
76.35 odd 18 361.2.e.b.28.1 6
76.43 odd 18 361.2.e.h.234.1 6
76.47 odd 18 361.2.e.a.62.1 6
76.51 even 18 361.2.e.f.245.1 6
76.55 odd 18 361.2.e.a.99.1 6
76.59 even 18 361.2.e.g.99.1 6
76.63 odd 18 361.2.e.b.245.1 6
76.67 even 18 361.2.e.g.62.1 6
76.71 even 18 19.2.e.a.6.1 6
76.75 even 2 361.2.a.g.1.1 3
228.71 odd 18 171.2.u.c.82.1 6
228.167 odd 18 171.2.u.c.73.1 6
228.227 odd 2 3249.2.a.z.1.3 3
380.147 odd 36 475.2.u.a.424.1 12
380.167 odd 36 475.2.u.a.149.2 12
380.223 odd 36 475.2.u.a.424.2 12
380.243 odd 36 475.2.u.a.149.1 12
380.299 even 18 475.2.l.a.101.1 6
380.319 even 18 475.2.l.a.301.1 6
380.379 even 2 9025.2.a.bd.1.3 3
532.167 odd 18 931.2.w.a.491.1 6
532.223 odd 18 931.2.w.a.785.1 6
532.243 odd 18 931.2.v.a.263.1 6
532.299 odd 18 931.2.x.b.557.1 6
532.319 even 18 931.2.x.a.814.1 6
532.375 even 18 931.2.v.b.177.1 6
532.395 odd 18 931.2.x.b.814.1 6
532.451 odd 18 931.2.v.a.177.1 6
532.471 even 18 931.2.v.b.263.1 6
532.527 even 18 931.2.x.a.557.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
19.2.e.a.6.1 6 76.71 even 18
19.2.e.a.16.1 yes 6 76.15 even 18
171.2.u.c.73.1 6 228.167 odd 18
171.2.u.c.82.1 6 228.71 odd 18
304.2.u.b.177.1 6 19.14 odd 18
304.2.u.b.225.1 6 19.15 odd 18
361.2.a.g.1.1 3 76.75 even 2
361.2.a.h.1.3 3 4.3 odd 2
361.2.c.h.68.1 6 76.7 odd 6
361.2.c.h.292.1 6 76.11 odd 6
361.2.c.i.68.3 6 76.31 even 6
361.2.c.i.292.3 6 76.27 even 6
361.2.e.a.62.1 6 76.47 odd 18
361.2.e.a.99.1 6 76.55 odd 18
361.2.e.b.28.1 6 76.35 odd 18
361.2.e.b.245.1 6 76.63 odd 18
361.2.e.f.28.1 6 76.3 even 18
361.2.e.f.245.1 6 76.51 even 18
361.2.e.g.62.1 6 76.67 even 18
361.2.e.g.99.1 6 76.59 even 18
361.2.e.h.54.1 6 76.23 odd 18
361.2.e.h.234.1 6 76.43 odd 18
475.2.l.a.101.1 6 380.299 even 18
475.2.l.a.301.1 6 380.319 even 18
475.2.u.a.149.1 12 380.243 odd 36
475.2.u.a.149.2 12 380.167 odd 36
475.2.u.a.424.1 12 380.147 odd 36
475.2.u.a.424.2 12 380.223 odd 36
931.2.v.a.177.1 6 532.451 odd 18
931.2.v.a.263.1 6 532.243 odd 18
931.2.v.b.177.1 6 532.375 even 18
931.2.v.b.263.1 6 532.471 even 18
931.2.w.a.491.1 6 532.167 odd 18
931.2.w.a.785.1 6 532.223 odd 18
931.2.x.a.557.1 6 532.527 even 18
931.2.x.a.814.1 6 532.319 even 18
931.2.x.b.557.1 6 532.299 odd 18
931.2.x.b.814.1 6 532.395 odd 18
3249.2.a.s.1.1 3 12.11 even 2
3249.2.a.z.1.3 3 228.227 odd 2
5776.2.a.bi.1.2 3 1.1 even 1 trivial
5776.2.a.br.1.2 3 19.18 odd 2
9025.2.a.x.1.1 3 20.19 odd 2
9025.2.a.bd.1.3 3 380.379 even 2