Properties

Label 5520.2.be.a.1471.3
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5520,2,Mod(1471,5520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5520.1471");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} - 45408 x^{7} + 62624 x^{6} - 18048 x^{5} + 2160 x^{4} - 1664 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.3
Root \(3.68710 - 3.68710i\) of defining polynomial
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.a.1471.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.00000i q^{5} -1.58474 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +1.00000i q^{5} -1.58474 q^{7} -1.00000 q^{9} +2.94482 q^{11} -1.23011 q^{13} +1.00000 q^{15} +1.25392i q^{17} -5.26714 q^{19} +1.58474i q^{21} +(-1.83865 - 4.42937i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +7.54806 q^{29} -0.690205i q^{31} -2.94482i q^{33} -1.58474i q^{35} -4.83340i q^{37} +1.23011i q^{39} +6.98783 q^{41} -4.06681 q^{43} -1.00000i q^{45} +8.95368i q^{47} -4.48861 q^{49} +1.25392 q^{51} +10.8852i q^{53} +2.94482i q^{55} +5.26714i q^{57} +2.33680i q^{59} +5.28658i q^{61} +1.58474 q^{63} -1.23011i q^{65} +9.49359 q^{67} +(-4.42937 + 1.83865i) q^{69} -13.5262i q^{71} -9.51828 q^{73} +1.00000i q^{75} -4.66677 q^{77} +3.71307 q^{79} +1.00000 q^{81} -16.5130 q^{83} -1.25392 q^{85} -7.54806i q^{87} +12.2406i q^{89} +1.94940 q^{91} -0.690205 q^{93} -5.26714i q^{95} +7.70591i q^{97} -2.94482 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{7} - 16 q^{9} + 8 q^{11} + 8 q^{13} + 16 q^{15} + 12 q^{23} - 16 q^{25} - 4 q^{29} + 4 q^{41} + 20 q^{49} - 4 q^{51} + 8 q^{63} - 16 q^{67} + 40 q^{73} + 24 q^{77} + 32 q^{79} + 16 q^{81} + 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1841\) \(4417\) \(4831\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) −1.58474 −0.598974 −0.299487 0.954100i \(-0.596816\pi\)
−0.299487 + 0.954100i \(0.596816\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.94482 0.887898 0.443949 0.896052i \(-0.353577\pi\)
0.443949 + 0.896052i \(0.353577\pi\)
\(12\) 0 0
\(13\) −1.23011 −0.341172 −0.170586 0.985343i \(-0.554566\pi\)
−0.170586 + 0.985343i \(0.554566\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 1.25392i 0.304120i 0.988371 + 0.152060i \(0.0485906\pi\)
−0.988371 + 0.152060i \(0.951409\pi\)
\(18\) 0 0
\(19\) −5.26714 −1.20836 −0.604182 0.796846i \(-0.706500\pi\)
−0.604182 + 0.796846i \(0.706500\pi\)
\(20\) 0 0
\(21\) 1.58474i 0.345818i
\(22\) 0 0
\(23\) −1.83865 4.42937i −0.383386 0.923588i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 7.54806 1.40164 0.700820 0.713338i \(-0.252818\pi\)
0.700820 + 0.713338i \(0.252818\pi\)
\(30\) 0 0
\(31\) 0.690205i 0.123965i −0.998077 0.0619823i \(-0.980258\pi\)
0.998077 0.0619823i \(-0.0197422\pi\)
\(32\) 0 0
\(33\) 2.94482i 0.512628i
\(34\) 0 0
\(35\) 1.58474i 0.267869i
\(36\) 0 0
\(37\) 4.83340i 0.794606i −0.917688 0.397303i \(-0.869946\pi\)
0.917688 0.397303i \(-0.130054\pi\)
\(38\) 0 0
\(39\) 1.23011i 0.196976i
\(40\) 0 0
\(41\) 6.98783 1.09132 0.545658 0.838008i \(-0.316280\pi\)
0.545658 + 0.838008i \(0.316280\pi\)
\(42\) 0 0
\(43\) −4.06681 −0.620183 −0.310091 0.950707i \(-0.600360\pi\)
−0.310091 + 0.950707i \(0.600360\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 8.95368i 1.30603i 0.757346 + 0.653014i \(0.226496\pi\)
−0.757346 + 0.653014i \(0.773504\pi\)
\(48\) 0 0
\(49\) −4.48861 −0.641230
\(50\) 0 0
\(51\) 1.25392 0.175583
\(52\) 0 0
\(53\) 10.8852i 1.49520i 0.664148 + 0.747601i \(0.268794\pi\)
−0.664148 + 0.747601i \(0.731206\pi\)
\(54\) 0 0
\(55\) 2.94482i 0.397080i
\(56\) 0 0
\(57\) 5.26714i 0.697649i
\(58\) 0 0
\(59\) 2.33680i 0.304226i 0.988363 + 0.152113i \(0.0486077\pi\)
−0.988363 + 0.152113i \(0.951392\pi\)
\(60\) 0 0
\(61\) 5.28658i 0.676877i 0.940989 + 0.338438i \(0.109899\pi\)
−0.940989 + 0.338438i \(0.890101\pi\)
\(62\) 0 0
\(63\) 1.58474 0.199658
\(64\) 0 0
\(65\) 1.23011i 0.152577i
\(66\) 0 0
\(67\) 9.49359 1.15983 0.579913 0.814678i \(-0.303087\pi\)
0.579913 + 0.814678i \(0.303087\pi\)
\(68\) 0 0
\(69\) −4.42937 + 1.83865i −0.533234 + 0.221348i
\(70\) 0 0
\(71\) 13.5262i 1.60527i −0.596473 0.802633i \(-0.703432\pi\)
0.596473 0.802633i \(-0.296568\pi\)
\(72\) 0 0
\(73\) −9.51828 −1.11403 −0.557015 0.830502i \(-0.688054\pi\)
−0.557015 + 0.830502i \(0.688054\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) −4.66677 −0.531828
\(78\) 0 0
\(79\) 3.71307 0.417753 0.208876 0.977942i \(-0.433019\pi\)
0.208876 + 0.977942i \(0.433019\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.5130 −1.81253 −0.906267 0.422705i \(-0.861081\pi\)
−0.906267 + 0.422705i \(0.861081\pi\)
\(84\) 0 0
\(85\) −1.25392 −0.136006
\(86\) 0 0
\(87\) 7.54806i 0.809237i
\(88\) 0 0
\(89\) 12.2406i 1.29750i 0.761003 + 0.648748i \(0.224707\pi\)
−0.761003 + 0.648748i \(0.775293\pi\)
\(90\) 0 0
\(91\) 1.94940 0.204353
\(92\) 0 0
\(93\) −0.690205 −0.0715710
\(94\) 0 0
\(95\) 5.26714i 0.540397i
\(96\) 0 0
\(97\) 7.70591i 0.782417i 0.920302 + 0.391208i \(0.127943\pi\)
−0.920302 + 0.391208i \(0.872057\pi\)
\(98\) 0 0
\(99\) −2.94482 −0.295966
\(100\) 0 0
\(101\) −9.43609 −0.938926 −0.469463 0.882952i \(-0.655552\pi\)
−0.469463 + 0.882952i \(0.655552\pi\)
\(102\) 0 0
\(103\) −7.07692 −0.697310 −0.348655 0.937251i \(-0.613361\pi\)
−0.348655 + 0.937251i \(0.613361\pi\)
\(104\) 0 0
\(105\) −1.58474 −0.154654
\(106\) 0 0
\(107\) −6.20879 −0.600227 −0.300113 0.953904i \(-0.597024\pi\)
−0.300113 + 0.953904i \(0.597024\pi\)
\(108\) 0 0
\(109\) 5.46182i 0.523147i 0.965184 + 0.261574i \(0.0842414\pi\)
−0.965184 + 0.261574i \(0.915759\pi\)
\(110\) 0 0
\(111\) −4.83340 −0.458766
\(112\) 0 0
\(113\) 1.98548i 0.186778i 0.995630 + 0.0933891i \(0.0297700\pi\)
−0.995630 + 0.0933891i \(0.970230\pi\)
\(114\) 0 0
\(115\) 4.42937 1.83865i 0.413041 0.171455i
\(116\) 0 0
\(117\) 1.23011 0.113724
\(118\) 0 0
\(119\) 1.98713i 0.182160i
\(120\) 0 0
\(121\) −2.32801 −0.211638
\(122\) 0 0
\(123\) 6.98783i 0.630071i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 4.35495i 0.386439i 0.981156 + 0.193220i \(0.0618929\pi\)
−0.981156 + 0.193220i \(0.938107\pi\)
\(128\) 0 0
\(129\) 4.06681i 0.358063i
\(130\) 0 0
\(131\) 1.92541i 0.168224i 0.996456 + 0.0841118i \(0.0268053\pi\)
−0.996456 + 0.0841118i \(0.973195\pi\)
\(132\) 0 0
\(133\) 8.34702 0.723779
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 11.9282i 1.01910i 0.860442 + 0.509548i \(0.170187\pi\)
−0.860442 + 0.509548i \(0.829813\pi\)
\(138\) 0 0
\(139\) 1.31703i 0.111709i −0.998439 0.0558545i \(-0.982212\pi\)
0.998439 0.0558545i \(-0.0177883\pi\)
\(140\) 0 0
\(141\) 8.95368 0.754036
\(142\) 0 0
\(143\) −3.62246 −0.302925
\(144\) 0 0
\(145\) 7.54806i 0.626832i
\(146\) 0 0
\(147\) 4.48861i 0.370214i
\(148\) 0 0
\(149\) 7.50615i 0.614928i −0.951560 0.307464i \(-0.900520\pi\)
0.951560 0.307464i \(-0.0994804\pi\)
\(150\) 0 0
\(151\) 4.78698i 0.389559i −0.980847 0.194780i \(-0.937601\pi\)
0.980847 0.194780i \(-0.0623992\pi\)
\(152\) 0 0
\(153\) 1.25392i 0.101373i
\(154\) 0 0
\(155\) 0.690205 0.0554386
\(156\) 0 0
\(157\) 6.01182i 0.479795i 0.970798 + 0.239898i \(0.0771139\pi\)
−0.970798 + 0.239898i \(0.922886\pi\)
\(158\) 0 0
\(159\) 10.8852 0.863255
\(160\) 0 0
\(161\) 2.91378 + 7.01939i 0.229638 + 0.553206i
\(162\) 0 0
\(163\) 9.41003i 0.737051i 0.929618 + 0.368525i \(0.120137\pi\)
−0.929618 + 0.368525i \(0.879863\pi\)
\(164\) 0 0
\(165\) 2.94482 0.229254
\(166\) 0 0
\(167\) 2.39043i 0.184977i −0.995714 0.0924884i \(-0.970518\pi\)
0.995714 0.0924884i \(-0.0294821\pi\)
\(168\) 0 0
\(169\) −11.4868 −0.883602
\(170\) 0 0
\(171\) 5.26714 0.402788
\(172\) 0 0
\(173\) 7.24943 0.551164 0.275582 0.961278i \(-0.411129\pi\)
0.275582 + 0.961278i \(0.411129\pi\)
\(174\) 0 0
\(175\) 1.58474 0.119795
\(176\) 0 0
\(177\) 2.33680 0.175645
\(178\) 0 0
\(179\) 12.2088i 0.912532i −0.889843 0.456266i \(-0.849187\pi\)
0.889843 0.456266i \(-0.150813\pi\)
\(180\) 0 0
\(181\) 16.9328i 1.25860i 0.777161 + 0.629302i \(0.216659\pi\)
−0.777161 + 0.629302i \(0.783341\pi\)
\(182\) 0 0
\(183\) 5.28658 0.390795
\(184\) 0 0
\(185\) 4.83340 0.355359
\(186\) 0 0
\(187\) 3.69256i 0.270027i
\(188\) 0 0
\(189\) 1.58474i 0.115273i
\(190\) 0 0
\(191\) 6.46417 0.467731 0.233865 0.972269i \(-0.424862\pi\)
0.233865 + 0.972269i \(0.424862\pi\)
\(192\) 0 0
\(193\) −21.5952 −1.55445 −0.777227 0.629220i \(-0.783375\pi\)
−0.777227 + 0.629220i \(0.783375\pi\)
\(194\) 0 0
\(195\) −1.23011 −0.0880901
\(196\) 0 0
\(197\) −4.11055 −0.292865 −0.146432 0.989221i \(-0.546779\pi\)
−0.146432 + 0.989221i \(0.546779\pi\)
\(198\) 0 0
\(199\) 3.25603 0.230814 0.115407 0.993318i \(-0.463183\pi\)
0.115407 + 0.993318i \(0.463183\pi\)
\(200\) 0 0
\(201\) 9.49359i 0.669626i
\(202\) 0 0
\(203\) −11.9617 −0.839546
\(204\) 0 0
\(205\) 6.98783i 0.488051i
\(206\) 0 0
\(207\) 1.83865 + 4.42937i 0.127795 + 0.307863i
\(208\) 0 0
\(209\) −15.5108 −1.07290
\(210\) 0 0
\(211\) 18.1011i 1.24613i 0.782169 + 0.623066i \(0.214113\pi\)
−0.782169 + 0.623066i \(0.785887\pi\)
\(212\) 0 0
\(213\) −13.5262 −0.926801
\(214\) 0 0
\(215\) 4.06681i 0.277354i
\(216\) 0 0
\(217\) 1.09379i 0.0742516i
\(218\) 0 0
\(219\) 9.51828i 0.643186i
\(220\) 0 0
\(221\) 1.54246i 0.103757i
\(222\) 0 0
\(223\) 2.72334i 0.182368i −0.995834 0.0911842i \(-0.970935\pi\)
0.995834 0.0911842i \(-0.0290652\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −21.7426 −1.44311 −0.721554 0.692358i \(-0.756572\pi\)
−0.721554 + 0.692358i \(0.756572\pi\)
\(228\) 0 0
\(229\) 1.81560i 0.119978i −0.998199 0.0599891i \(-0.980893\pi\)
0.998199 0.0599891i \(-0.0191066\pi\)
\(230\) 0 0
\(231\) 4.66677i 0.307051i
\(232\) 0 0
\(233\) −25.1093 −1.64496 −0.822482 0.568792i \(-0.807411\pi\)
−0.822482 + 0.568792i \(0.807411\pi\)
\(234\) 0 0
\(235\) −8.95368 −0.584074
\(236\) 0 0
\(237\) 3.71307i 0.241190i
\(238\) 0 0
\(239\) 25.5463i 1.65245i 0.563338 + 0.826226i \(0.309517\pi\)
−0.563338 + 0.826226i \(0.690483\pi\)
\(240\) 0 0
\(241\) 17.7500i 1.14338i 0.820471 + 0.571689i \(0.193711\pi\)
−0.820471 + 0.571689i \(0.806289\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 4.48861i 0.286767i
\(246\) 0 0
\(247\) 6.47916 0.412259
\(248\) 0 0
\(249\) 16.5130i 1.04647i
\(250\) 0 0
\(251\) −18.8514 −1.18989 −0.594946 0.803766i \(-0.702826\pi\)
−0.594946 + 0.803766i \(0.702826\pi\)
\(252\) 0 0
\(253\) −5.41451 13.0437i −0.340407 0.820052i
\(254\) 0 0
\(255\) 1.25392i 0.0785233i
\(256\) 0 0
\(257\) 18.1245 1.13058 0.565289 0.824893i \(-0.308765\pi\)
0.565289 + 0.824893i \(0.308765\pi\)
\(258\) 0 0
\(259\) 7.65967i 0.475948i
\(260\) 0 0
\(261\) −7.54806 −0.467213
\(262\) 0 0
\(263\) 29.0810 1.79321 0.896606 0.442829i \(-0.146025\pi\)
0.896606 + 0.442829i \(0.146025\pi\)
\(264\) 0 0
\(265\) −10.8852 −0.668675
\(266\) 0 0
\(267\) 12.2406 0.749110
\(268\) 0 0
\(269\) −21.2290 −1.29435 −0.647176 0.762340i \(-0.724050\pi\)
−0.647176 + 0.762340i \(0.724050\pi\)
\(270\) 0 0
\(271\) 9.18826i 0.558147i 0.960270 + 0.279073i \(0.0900273\pi\)
−0.960270 + 0.279073i \(0.909973\pi\)
\(272\) 0 0
\(273\) 1.94940i 0.117983i
\(274\) 0 0
\(275\) −2.94482 −0.177580
\(276\) 0 0
\(277\) 23.9814 1.44090 0.720451 0.693505i \(-0.243935\pi\)
0.720451 + 0.693505i \(0.243935\pi\)
\(278\) 0 0
\(279\) 0.690205i 0.0413215i
\(280\) 0 0
\(281\) 18.5638i 1.10742i 0.832709 + 0.553711i \(0.186789\pi\)
−0.832709 + 0.553711i \(0.813211\pi\)
\(282\) 0 0
\(283\) −12.8548 −0.764138 −0.382069 0.924134i \(-0.624788\pi\)
−0.382069 + 0.924134i \(0.624788\pi\)
\(284\) 0 0
\(285\) −5.26714 −0.311998
\(286\) 0 0
\(287\) −11.0739 −0.653670
\(288\) 0 0
\(289\) 15.4277 0.907511
\(290\) 0 0
\(291\) 7.70591 0.451728
\(292\) 0 0
\(293\) 25.3576i 1.48141i 0.671831 + 0.740705i \(0.265508\pi\)
−0.671831 + 0.740705i \(0.734492\pi\)
\(294\) 0 0
\(295\) −2.33680 −0.136054
\(296\) 0 0
\(297\) 2.94482i 0.170876i
\(298\) 0 0
\(299\) 2.26175 + 5.44862i 0.130800 + 0.315102i
\(300\) 0 0
\(301\) 6.44483 0.371474
\(302\) 0 0
\(303\) 9.43609i 0.542089i
\(304\) 0 0
\(305\) −5.28658 −0.302709
\(306\) 0 0
\(307\) 12.7630i 0.728422i 0.931316 + 0.364211i \(0.118661\pi\)
−0.931316 + 0.364211i \(0.881339\pi\)
\(308\) 0 0
\(309\) 7.07692i 0.402592i
\(310\) 0 0
\(311\) 14.7842i 0.838333i −0.907909 0.419166i \(-0.862322\pi\)
0.907909 0.419166i \(-0.137678\pi\)
\(312\) 0 0
\(313\) 1.48723i 0.0840632i −0.999116 0.0420316i \(-0.986617\pi\)
0.999116 0.0420316i \(-0.0133830\pi\)
\(314\) 0 0
\(315\) 1.58474i 0.0892898i
\(316\) 0 0
\(317\) 0.771575 0.0433360 0.0216680 0.999765i \(-0.493102\pi\)
0.0216680 + 0.999765i \(0.493102\pi\)
\(318\) 0 0
\(319\) 22.2277 1.24451
\(320\) 0 0
\(321\) 6.20879i 0.346541i
\(322\) 0 0
\(323\) 6.60455i 0.367487i
\(324\) 0 0
\(325\) 1.23011 0.0682343
\(326\) 0 0
\(327\) 5.46182 0.302039
\(328\) 0 0
\(329\) 14.1892i 0.782277i
\(330\) 0 0
\(331\) 33.8360i 1.85980i −0.367817 0.929898i \(-0.619895\pi\)
0.367817 0.929898i \(-0.380105\pi\)
\(332\) 0 0
\(333\) 4.83340i 0.264869i
\(334\) 0 0
\(335\) 9.49359i 0.518690i
\(336\) 0 0
\(337\) 25.3943i 1.38331i 0.722226 + 0.691657i \(0.243119\pi\)
−0.722226 + 0.691657i \(0.756881\pi\)
\(338\) 0 0
\(339\) 1.98548 0.107836
\(340\) 0 0
\(341\) 2.03253i 0.110068i
\(342\) 0 0
\(343\) 18.2064 0.983054
\(344\) 0 0
\(345\) −1.83865 4.42937i −0.0989898 0.238469i
\(346\) 0 0
\(347\) 7.66039i 0.411231i −0.978633 0.205616i \(-0.934080\pi\)
0.978633 0.205616i \(-0.0659197\pi\)
\(348\) 0 0
\(349\) 15.8624 0.849093 0.424546 0.905406i \(-0.360434\pi\)
0.424546 + 0.905406i \(0.360434\pi\)
\(350\) 0 0
\(351\) 1.23011i 0.0656585i
\(352\) 0 0
\(353\) −26.7036 −1.42129 −0.710643 0.703552i \(-0.751596\pi\)
−0.710643 + 0.703552i \(0.751596\pi\)
\(354\) 0 0
\(355\) 13.5262 0.717897
\(356\) 0 0
\(357\) −1.98713 −0.105170
\(358\) 0 0
\(359\) −17.3015 −0.913137 −0.456569 0.889688i \(-0.650922\pi\)
−0.456569 + 0.889688i \(0.650922\pi\)
\(360\) 0 0
\(361\) 8.74271 0.460143
\(362\) 0 0
\(363\) 2.32801i 0.122189i
\(364\) 0 0
\(365\) 9.51828i 0.498210i
\(366\) 0 0
\(367\) −19.1050 −0.997271 −0.498635 0.866812i \(-0.666165\pi\)
−0.498635 + 0.866812i \(0.666165\pi\)
\(368\) 0 0
\(369\) −6.98783 −0.363772
\(370\) 0 0
\(371\) 17.2502i 0.895587i
\(372\) 0 0
\(373\) 29.5261i 1.52880i −0.644740 0.764402i \(-0.723034\pi\)
0.644740 0.764402i \(-0.276966\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −9.28496 −0.478200
\(378\) 0 0
\(379\) 29.9649 1.53920 0.769598 0.638529i \(-0.220457\pi\)
0.769598 + 0.638529i \(0.220457\pi\)
\(380\) 0 0
\(381\) 4.35495 0.223111
\(382\) 0 0
\(383\) −6.78342 −0.346616 −0.173308 0.984868i \(-0.555446\pi\)
−0.173308 + 0.984868i \(0.555446\pi\)
\(384\) 0 0
\(385\) 4.66677i 0.237841i
\(386\) 0 0
\(387\) 4.06681 0.206728
\(388\) 0 0
\(389\) 8.20216i 0.415866i 0.978143 + 0.207933i \(0.0666736\pi\)
−0.978143 + 0.207933i \(0.933326\pi\)
\(390\) 0 0
\(391\) 5.55407 2.30552i 0.280881 0.116595i
\(392\) 0 0
\(393\) 1.92541 0.0971240
\(394\) 0 0
\(395\) 3.71307i 0.186825i
\(396\) 0 0
\(397\) −28.2572 −1.41819 −0.709095 0.705113i \(-0.750896\pi\)
−0.709095 + 0.705113i \(0.750896\pi\)
\(398\) 0 0
\(399\) 8.34702i 0.417874i
\(400\) 0 0
\(401\) 17.1193i 0.854899i 0.904039 + 0.427449i \(0.140588\pi\)
−0.904039 + 0.427449i \(0.859412\pi\)
\(402\) 0 0
\(403\) 0.849030i 0.0422932i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 14.2335i 0.705529i
\(408\) 0 0
\(409\) 22.8130 1.12803 0.564016 0.825764i \(-0.309256\pi\)
0.564016 + 0.825764i \(0.309256\pi\)
\(410\) 0 0
\(411\) 11.9282 0.588375
\(412\) 0 0
\(413\) 3.70322i 0.182223i
\(414\) 0 0
\(415\) 16.5130i 0.810590i
\(416\) 0 0
\(417\) −1.31703 −0.0644952
\(418\) 0 0
\(419\) −4.54342 −0.221960 −0.110980 0.993823i \(-0.535399\pi\)
−0.110980 + 0.993823i \(0.535399\pi\)
\(420\) 0 0
\(421\) 2.02280i 0.0985853i 0.998784 + 0.0492926i \(0.0156967\pi\)
−0.998784 + 0.0492926i \(0.984303\pi\)
\(422\) 0 0
\(423\) 8.95368i 0.435343i
\(424\) 0 0
\(425\) 1.25392i 0.0608239i
\(426\) 0 0
\(427\) 8.37783i 0.405432i
\(428\) 0 0
\(429\) 3.62246i 0.174894i
\(430\) 0 0
\(431\) −10.5534 −0.508339 −0.254169 0.967160i \(-0.581802\pi\)
−0.254169 + 0.967160i \(0.581802\pi\)
\(432\) 0 0
\(433\) 33.9550i 1.63177i 0.578211 + 0.815887i \(0.303751\pi\)
−0.578211 + 0.815887i \(0.696249\pi\)
\(434\) 0 0
\(435\) 7.54806 0.361902
\(436\) 0 0
\(437\) 9.68444 + 23.3301i 0.463270 + 1.11603i
\(438\) 0 0
\(439\) 2.99231i 0.142815i 0.997447 + 0.0714076i \(0.0227491\pi\)
−0.997447 + 0.0714076i \(0.977251\pi\)
\(440\) 0 0
\(441\) 4.48861 0.213743
\(442\) 0 0
\(443\) 20.9670i 0.996174i −0.867127 0.498087i \(-0.834036\pi\)
0.867127 0.498087i \(-0.165964\pi\)
\(444\) 0 0
\(445\) −12.2406 −0.580258
\(446\) 0 0
\(447\) −7.50615 −0.355029
\(448\) 0 0
\(449\) 11.8249 0.558053 0.279026 0.960283i \(-0.409988\pi\)
0.279026 + 0.960283i \(0.409988\pi\)
\(450\) 0 0
\(451\) 20.5779 0.968977
\(452\) 0 0
\(453\) −4.78698 −0.224912
\(454\) 0 0
\(455\) 1.94940i 0.0913894i
\(456\) 0 0
\(457\) 10.8771i 0.508810i −0.967098 0.254405i \(-0.918120\pi\)
0.967098 0.254405i \(-0.0818796\pi\)
\(458\) 0 0
\(459\) −1.25392 −0.0585278
\(460\) 0 0
\(461\) 15.2626 0.710849 0.355425 0.934705i \(-0.384336\pi\)
0.355425 + 0.934705i \(0.384336\pi\)
\(462\) 0 0
\(463\) 38.1730i 1.77405i 0.461721 + 0.887025i \(0.347232\pi\)
−0.461721 + 0.887025i \(0.652768\pi\)
\(464\) 0 0
\(465\) 0.690205i 0.0320075i
\(466\) 0 0
\(467\) −4.21856 −0.195212 −0.0976059 0.995225i \(-0.531118\pi\)
−0.0976059 + 0.995225i \(0.531118\pi\)
\(468\) 0 0
\(469\) −15.0448 −0.694706
\(470\) 0 0
\(471\) 6.01182 0.277010
\(472\) 0 0
\(473\) −11.9760 −0.550659
\(474\) 0 0
\(475\) 5.26714 0.241673
\(476\) 0 0
\(477\) 10.8852i 0.498401i
\(478\) 0 0
\(479\) 0.347762 0.0158897 0.00794483 0.999968i \(-0.497471\pi\)
0.00794483 + 0.999968i \(0.497471\pi\)
\(480\) 0 0
\(481\) 5.94562i 0.271097i
\(482\) 0 0
\(483\) 7.01939 2.91378i 0.319393 0.132582i
\(484\) 0 0
\(485\) −7.70591 −0.349907
\(486\) 0 0
\(487\) 12.4417i 0.563789i −0.959445 0.281894i \(-0.909037\pi\)
0.959445 0.281894i \(-0.0909628\pi\)
\(488\) 0 0
\(489\) 9.41003 0.425536
\(490\) 0 0
\(491\) 4.12588i 0.186198i 0.995657 + 0.0930991i \(0.0296774\pi\)
−0.995657 + 0.0930991i \(0.970323\pi\)
\(492\) 0 0
\(493\) 9.46464i 0.426266i
\(494\) 0 0
\(495\) 2.94482i 0.132360i
\(496\) 0 0
\(497\) 21.4355i 0.961514i
\(498\) 0 0
\(499\) 24.3075i 1.08815i −0.839035 0.544077i \(-0.816880\pi\)
0.839035 0.544077i \(-0.183120\pi\)
\(500\) 0 0
\(501\) −2.39043 −0.106796
\(502\) 0 0
\(503\) 11.9762 0.533991 0.266995 0.963698i \(-0.413969\pi\)
0.266995 + 0.963698i \(0.413969\pi\)
\(504\) 0 0
\(505\) 9.43609i 0.419900i
\(506\) 0 0
\(507\) 11.4868i 0.510148i
\(508\) 0 0
\(509\) 43.4081 1.92403 0.962015 0.272997i \(-0.0880150\pi\)
0.962015 + 0.272997i \(0.0880150\pi\)
\(510\) 0 0
\(511\) 15.0840 0.667276
\(512\) 0 0
\(513\) 5.26714i 0.232550i
\(514\) 0 0
\(515\) 7.07692i 0.311846i
\(516\) 0 0
\(517\) 26.3670i 1.15962i
\(518\) 0 0
\(519\) 7.24943i 0.318215i
\(520\) 0 0
\(521\) 30.1561i 1.32116i −0.750754 0.660582i \(-0.770310\pi\)
0.750754 0.660582i \(-0.229690\pi\)
\(522\) 0 0
\(523\) 25.0503 1.09537 0.547686 0.836684i \(-0.315509\pi\)
0.547686 + 0.836684i \(0.315509\pi\)
\(524\) 0 0
\(525\) 1.58474i 0.0691636i
\(526\) 0 0
\(527\) 0.865460 0.0377000
\(528\) 0 0
\(529\) −16.2387 + 16.2882i −0.706031 + 0.708181i
\(530\) 0 0
\(531\) 2.33680i 0.101409i
\(532\) 0 0
\(533\) −8.59581 −0.372326
\(534\) 0 0
\(535\) 6.20879i 0.268430i
\(536\) 0 0
\(537\) −12.2088 −0.526851
\(538\) 0 0
\(539\) −13.2182 −0.569346
\(540\) 0 0
\(541\) 38.9949 1.67652 0.838260 0.545271i \(-0.183573\pi\)
0.838260 + 0.545271i \(0.183573\pi\)
\(542\) 0 0
\(543\) 16.9328 0.726656
\(544\) 0 0
\(545\) −5.46182 −0.233959
\(546\) 0 0
\(547\) 35.9994i 1.53922i 0.638512 + 0.769612i \(0.279550\pi\)
−0.638512 + 0.769612i \(0.720450\pi\)
\(548\) 0 0
\(549\) 5.28658i 0.225626i
\(550\) 0 0
\(551\) −39.7567 −1.69369
\(552\) 0 0
\(553\) −5.88424 −0.250223
\(554\) 0 0
\(555\) 4.83340i 0.205166i
\(556\) 0 0
\(557\) 22.1601i 0.938954i −0.882945 0.469477i \(-0.844442\pi\)
0.882945 0.469477i \(-0.155558\pi\)
\(558\) 0 0
\(559\) 5.00263 0.211589
\(560\) 0 0
\(561\) 3.69256 0.155900
\(562\) 0 0
\(563\) 2.91544 0.122871 0.0614356 0.998111i \(-0.480432\pi\)
0.0614356 + 0.998111i \(0.480432\pi\)
\(564\) 0 0
\(565\) −1.98548 −0.0835297
\(566\) 0 0
\(567\) −1.58474 −0.0665527
\(568\) 0 0
\(569\) 14.9836i 0.628144i 0.949399 + 0.314072i \(0.101693\pi\)
−0.949399 + 0.314072i \(0.898307\pi\)
\(570\) 0 0
\(571\) 3.15642 0.132092 0.0660461 0.997817i \(-0.478962\pi\)
0.0660461 + 0.997817i \(0.478962\pi\)
\(572\) 0 0
\(573\) 6.46417i 0.270044i
\(574\) 0 0
\(575\) 1.83865 + 4.42937i 0.0766772 + 0.184718i
\(576\) 0 0
\(577\) −6.25189 −0.260270 −0.130135 0.991496i \(-0.541541\pi\)
−0.130135 + 0.991496i \(0.541541\pi\)
\(578\) 0 0
\(579\) 21.5952i 0.897465i
\(580\) 0 0
\(581\) 26.1687 1.08566
\(582\) 0 0
\(583\) 32.0551i 1.32759i
\(584\) 0 0
\(585\) 1.23011i 0.0508589i
\(586\) 0 0
\(587\) 3.22099i 0.132945i −0.997788 0.0664723i \(-0.978826\pi\)
0.997788 0.0664723i \(-0.0211744\pi\)
\(588\) 0 0
\(589\) 3.63541i 0.149794i
\(590\) 0 0
\(591\) 4.11055i 0.169085i
\(592\) 0 0
\(593\) −18.3093 −0.751873 −0.375937 0.926645i \(-0.622679\pi\)
−0.375937 + 0.926645i \(0.622679\pi\)
\(594\) 0 0
\(595\) 1.98713 0.0814643
\(596\) 0 0
\(597\) 3.25603i 0.133261i
\(598\) 0 0
\(599\) 5.79992i 0.236978i −0.992955 0.118489i \(-0.962195\pi\)
0.992955 0.118489i \(-0.0378051\pi\)
\(600\) 0 0
\(601\) −12.1653 −0.496233 −0.248116 0.968730i \(-0.579812\pi\)
−0.248116 + 0.968730i \(0.579812\pi\)
\(602\) 0 0
\(603\) −9.49359 −0.386609
\(604\) 0 0
\(605\) 2.32801i 0.0946472i
\(606\) 0 0
\(607\) 33.5799i 1.36296i 0.731834 + 0.681482i \(0.238664\pi\)
−0.731834 + 0.681482i \(0.761336\pi\)
\(608\) 0 0
\(609\) 11.9617i 0.484712i
\(610\) 0 0
\(611\) 11.0140i 0.445580i
\(612\) 0 0
\(613\) 24.5388i 0.991113i −0.868576 0.495556i \(-0.834964\pi\)
0.868576 0.495556i \(-0.165036\pi\)
\(614\) 0 0
\(615\) 6.98783 0.281776
\(616\) 0 0
\(617\) 10.9240i 0.439782i −0.975524 0.219891i \(-0.929430\pi\)
0.975524 0.219891i \(-0.0705701\pi\)
\(618\) 0 0
\(619\) −29.8447 −1.19956 −0.599780 0.800165i \(-0.704745\pi\)
−0.599780 + 0.800165i \(0.704745\pi\)
\(620\) 0 0
\(621\) 4.42937 1.83865i 0.177745 0.0737826i
\(622\) 0 0
\(623\) 19.3981i 0.777167i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 15.5108i 0.619441i
\(628\) 0 0
\(629\) 6.06068 0.241655
\(630\) 0 0
\(631\) 4.26198 0.169667 0.0848333 0.996395i \(-0.472964\pi\)
0.0848333 + 0.996395i \(0.472964\pi\)
\(632\) 0 0
\(633\) 18.1011 0.719455
\(634\) 0 0
\(635\) −4.35495 −0.172821
\(636\) 0 0
\(637\) 5.52149 0.218769
\(638\) 0 0
\(639\) 13.5262i 0.535089i
\(640\) 0 0
\(641\) 43.5638i 1.72067i 0.509731 + 0.860334i \(0.329745\pi\)
−0.509731 + 0.860334i \(0.670255\pi\)
\(642\) 0 0
\(643\) 11.2217 0.442542 0.221271 0.975212i \(-0.428979\pi\)
0.221271 + 0.975212i \(0.428979\pi\)
\(644\) 0 0
\(645\) −4.06681 −0.160131
\(646\) 0 0
\(647\) 36.7686i 1.44552i 0.691098 + 0.722761i \(0.257127\pi\)
−0.691098 + 0.722761i \(0.742873\pi\)
\(648\) 0 0
\(649\) 6.88147i 0.270121i
\(650\) 0 0
\(651\) 1.09379 0.0428692
\(652\) 0 0
\(653\) −31.9401 −1.24991 −0.624957 0.780659i \(-0.714883\pi\)
−0.624957 + 0.780659i \(0.714883\pi\)
\(654\) 0 0
\(655\) −1.92541 −0.0752319
\(656\) 0 0
\(657\) 9.51828 0.371344
\(658\) 0 0
\(659\) −43.5902 −1.69803 −0.849016 0.528367i \(-0.822804\pi\)
−0.849016 + 0.528367i \(0.822804\pi\)
\(660\) 0 0
\(661\) 10.4393i 0.406042i −0.979174 0.203021i \(-0.934924\pi\)
0.979174 0.203021i \(-0.0650759\pi\)
\(662\) 0 0
\(663\) −1.54246 −0.0599041
\(664\) 0 0
\(665\) 8.34702i 0.323684i
\(666\) 0 0
\(667\) −13.8783 33.4332i −0.537369 1.29454i
\(668\) 0 0
\(669\) −2.72334 −0.105290
\(670\) 0 0
\(671\) 15.5680i 0.600997i
\(672\) 0 0
\(673\) 2.83519 0.109288 0.0546442 0.998506i \(-0.482598\pi\)
0.0546442 + 0.998506i \(0.482598\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 36.3830i 1.39831i −0.714969 0.699156i \(-0.753559\pi\)
0.714969 0.699156i \(-0.246441\pi\)
\(678\) 0 0
\(679\) 12.2118i 0.468647i
\(680\) 0 0
\(681\) 21.7426i 0.833179i
\(682\) 0 0
\(683\) 2.81755i 0.107811i 0.998546 + 0.0539053i \(0.0171669\pi\)
−0.998546 + 0.0539053i \(0.982833\pi\)
\(684\) 0 0
\(685\) −11.9282 −0.455754
\(686\) 0 0
\(687\) −1.81560 −0.0692695
\(688\) 0 0
\(689\) 13.3901i 0.510120i
\(690\) 0 0
\(691\) 49.1990i 1.87162i −0.352506 0.935809i \(-0.614670\pi\)
0.352506 0.935809i \(-0.385330\pi\)
\(692\) 0 0
\(693\) 4.66677 0.177276
\(694\) 0 0
\(695\) 1.31703 0.0499578
\(696\) 0 0
\(697\) 8.76216i 0.331890i
\(698\) 0 0
\(699\) 25.1093i 0.949720i
\(700\) 0 0
\(701\) 15.5212i 0.586228i −0.956078 0.293114i \(-0.905308\pi\)
0.956078 0.293114i \(-0.0946915\pi\)
\(702\) 0 0
\(703\) 25.4582i 0.960173i
\(704\) 0 0
\(705\) 8.95368i 0.337215i
\(706\) 0 0
\(707\) 14.9537 0.562392
\(708\) 0 0
\(709\) 1.54150i 0.0578923i 0.999581 + 0.0289462i \(0.00921514\pi\)
−0.999581 + 0.0289462i \(0.990785\pi\)
\(710\) 0 0
\(711\) −3.71307 −0.139251
\(712\) 0 0
\(713\) −3.05718 + 1.26905i −0.114492 + 0.0475263i
\(714\) 0 0
\(715\) 3.62246i 0.135472i
\(716\) 0 0
\(717\) 25.5463 0.954044
\(718\) 0 0
\(719\) 2.00464i 0.0747606i 0.999301 + 0.0373803i \(0.0119013\pi\)
−0.999301 + 0.0373803i \(0.988099\pi\)
\(720\) 0 0
\(721\) 11.2151 0.417671
\(722\) 0 0
\(723\) 17.7500 0.660129
\(724\) 0 0
\(725\) −7.54806 −0.280328
\(726\) 0 0
\(727\) 17.3038 0.641761 0.320881 0.947120i \(-0.396021\pi\)
0.320881 + 0.947120i \(0.396021\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 5.09944i 0.188610i
\(732\) 0 0
\(733\) 26.4881i 0.978358i −0.872183 0.489179i \(-0.837296\pi\)
0.872183 0.489179i \(-0.162704\pi\)
\(734\) 0 0
\(735\) −4.48861 −0.165565
\(736\) 0 0
\(737\) 27.9569 1.02981
\(738\) 0 0
\(739\) 7.11302i 0.261657i −0.991405 0.130828i \(-0.958236\pi\)
0.991405 0.130828i \(-0.0417637\pi\)
\(740\) 0 0
\(741\) 6.47916i 0.238018i
\(742\) 0 0
\(743\) −38.7611 −1.42201 −0.711003 0.703189i \(-0.751759\pi\)
−0.711003 + 0.703189i \(0.751759\pi\)
\(744\) 0 0
\(745\) 7.50615 0.275004
\(746\) 0 0
\(747\) 16.5130 0.604178
\(748\) 0 0
\(749\) 9.83930 0.359520
\(750\) 0 0
\(751\) −26.4017 −0.963412 −0.481706 0.876333i \(-0.659983\pi\)
−0.481706 + 0.876333i \(0.659983\pi\)
\(752\) 0 0
\(753\) 18.8514i 0.686984i
\(754\) 0 0
\(755\) 4.78698 0.174216
\(756\) 0 0
\(757\) 19.6012i 0.712419i −0.934406 0.356210i \(-0.884069\pi\)
0.934406 0.356210i \(-0.115931\pi\)
\(758\) 0 0
\(759\) −13.0437 + 5.41451i −0.473457 + 0.196534i
\(760\) 0 0
\(761\) −1.90461 −0.0690421 −0.0345211 0.999404i \(-0.510991\pi\)
−0.0345211 + 0.999404i \(0.510991\pi\)
\(762\) 0 0
\(763\) 8.65554i 0.313352i
\(764\) 0 0
\(765\) 1.25392 0.0453355
\(766\) 0 0
\(767\) 2.87453i 0.103793i
\(768\) 0 0
\(769\) 24.1496i 0.870858i −0.900223 0.435429i \(-0.856597\pi\)
0.900223 0.435429i \(-0.143403\pi\)
\(770\) 0 0
\(771\) 18.1245i 0.652739i
\(772\) 0 0
\(773\) 35.9153i 1.29179i −0.763428 0.645893i \(-0.776485\pi\)
0.763428 0.645893i \(-0.223515\pi\)
\(774\) 0 0
\(775\) 0.690205i 0.0247929i
\(776\) 0 0
\(777\) 7.65967 0.274789
\(778\) 0 0
\(779\) −36.8058 −1.31871
\(780\) 0 0
\(781\) 39.8323i 1.42531i
\(782\) 0 0
\(783\) 7.54806i 0.269746i
\(784\) 0 0
\(785\) −6.01182 −0.214571
\(786\) 0 0
\(787\) 25.6536 0.914452 0.457226 0.889351i \(-0.348843\pi\)
0.457226 + 0.889351i \(0.348843\pi\)
\(788\) 0 0
\(789\) 29.0810i 1.03531i
\(790\) 0 0
\(791\) 3.14646i 0.111875i
\(792\) 0 0
\(793\) 6.50308i 0.230931i
\(794\) 0 0
\(795\) 10.8852i 0.386059i
\(796\) 0 0
\(797\) 11.6082i 0.411185i −0.978638 0.205592i \(-0.934088\pi\)
0.978638 0.205592i \(-0.0659121\pi\)
\(798\) 0 0
\(799\) −11.2272 −0.397189
\(800\) 0 0
\(801\) 12.2406i 0.432499i
\(802\) 0 0
\(803\) −28.0297 −0.989145
\(804\) 0 0
\(805\) −7.01939 + 2.91378i −0.247401 + 0.102697i
\(806\) 0 0
\(807\) 21.2290i 0.747295i
\(808\) 0 0
\(809\) −33.1262 −1.16465 −0.582327 0.812955i \(-0.697858\pi\)
−0.582327 + 0.812955i \(0.697858\pi\)
\(810\) 0 0
\(811\) 45.0060i 1.58037i 0.612867 + 0.790186i \(0.290016\pi\)
−0.612867 + 0.790186i \(0.709984\pi\)
\(812\) 0 0
\(813\) 9.18826 0.322246
\(814\) 0 0
\(815\) −9.41003 −0.329619
\(816\) 0 0
\(817\) 21.4204 0.749407
\(818\) 0 0
\(819\) −1.94940 −0.0681177
\(820\) 0 0
\(821\) 26.5182 0.925493 0.462746 0.886491i \(-0.346864\pi\)
0.462746 + 0.886491i \(0.346864\pi\)
\(822\) 0 0
\(823\) 10.3626i 0.361216i 0.983555 + 0.180608i \(0.0578066\pi\)
−0.983555 + 0.180608i \(0.942193\pi\)
\(824\) 0 0
\(825\) 2.94482i 0.102526i
\(826\) 0 0
\(827\) −47.8592 −1.66423 −0.832114 0.554605i \(-0.812870\pi\)
−0.832114 + 0.554605i \(0.812870\pi\)
\(828\) 0 0
\(829\) −10.3355 −0.358968 −0.179484 0.983761i \(-0.557443\pi\)
−0.179484 + 0.983761i \(0.557443\pi\)
\(830\) 0 0
\(831\) 23.9814i 0.831906i
\(832\) 0 0
\(833\) 5.62834i 0.195011i
\(834\) 0 0
\(835\) 2.39043 0.0827241
\(836\) 0 0
\(837\) 0.690205 0.0238570
\(838\) 0 0
\(839\) 6.62580 0.228748 0.114374 0.993438i \(-0.463514\pi\)
0.114374 + 0.993438i \(0.463514\pi\)
\(840\) 0 0
\(841\) 27.9732 0.964594
\(842\) 0 0
\(843\) 18.5638 0.639370
\(844\) 0 0
\(845\) 11.4868i 0.395159i
\(846\) 0 0
\(847\) 3.68929 0.126766
\(848\) 0 0
\(849\) 12.8548i 0.441175i
\(850\) 0 0
\(851\) −21.4089 + 8.88695i −0.733889 + 0.304641i
\(852\) 0 0
\(853\) −44.4991 −1.52362 −0.761810 0.647800i \(-0.775689\pi\)
−0.761810 + 0.647800i \(0.775689\pi\)
\(854\) 0 0
\(855\) 5.26714i 0.180132i
\(856\) 0 0
\(857\) 9.12506 0.311706 0.155853 0.987780i \(-0.450187\pi\)
0.155853 + 0.987780i \(0.450187\pi\)
\(858\) 0 0
\(859\) 30.9281i 1.05525i −0.849476 0.527627i \(-0.823082\pi\)
0.849476 0.527627i \(-0.176918\pi\)
\(860\) 0 0
\(861\) 11.0739i 0.377397i
\(862\) 0 0
\(863\) 8.46438i 0.288131i −0.989568 0.144065i \(-0.953982\pi\)
0.989568 0.144065i \(-0.0460176\pi\)
\(864\) 0 0
\(865\) 7.24943i 0.246488i
\(866\) 0 0
\(867\) 15.4277i 0.523952i
\(868\) 0 0
\(869\) 10.9343 0.370922
\(870\) 0 0
\(871\) −11.6782 −0.395700
\(872\) 0 0
\(873\) 7.70591i 0.260806i
\(874\) 0 0
\(875\) 1.58474i 0.0535739i
\(876\) 0 0
\(877\) 20.0279 0.676295 0.338147 0.941093i \(-0.390200\pi\)
0.338147 + 0.941093i \(0.390200\pi\)
\(878\) 0 0
\(879\) 25.3576 0.855292
\(880\) 0 0
\(881\) 38.0341i 1.28140i 0.767790 + 0.640701i \(0.221356\pi\)
−0.767790 + 0.640701i \(0.778644\pi\)
\(882\) 0 0
\(883\) 14.9644i 0.503592i −0.967780 0.251796i \(-0.918979\pi\)
0.967780 0.251796i \(-0.0810211\pi\)
\(884\) 0 0
\(885\) 2.33680i 0.0785508i
\(886\) 0 0
\(887\) 0.728725i 0.0244682i −0.999925 0.0122341i \(-0.996106\pi\)
0.999925 0.0122341i \(-0.00389433\pi\)
\(888\) 0 0
\(889\) 6.90144i 0.231467i
\(890\) 0 0
\(891\) 2.94482 0.0986553
\(892\) 0 0
\(893\) 47.1602i 1.57816i
\(894\) 0 0
\(895\) 12.2088 0.408097
\(896\) 0 0
\(897\) 5.44862 2.26175i 0.181924 0.0755176i
\(898\) 0 0
\(899\) 5.20971i 0.173754i
\(900\) 0 0
\(901\) −13.6492 −0.454720
\(902\) 0 0
\(903\) 6.44483i 0.214470i
\(904\) 0 0
\(905\) −16.9328 −0.562865
\(906\) 0 0
\(907\) 3.84373 0.127629 0.0638145 0.997962i \(-0.479673\pi\)
0.0638145 + 0.997962i \(0.479673\pi\)
\(908\) 0 0
\(909\) 9.43609 0.312975
\(910\) 0 0
\(911\) 55.8963 1.85193 0.925964 0.377611i \(-0.123254\pi\)
0.925964 + 0.377611i \(0.123254\pi\)
\(912\) 0 0
\(913\) −48.6278 −1.60935
\(914\) 0 0
\(915\) 5.28658i 0.174769i
\(916\) 0 0
\(917\) 3.05126i 0.100762i
\(918\) 0 0
\(919\) 0.643465 0.0212260 0.0106130 0.999944i \(-0.496622\pi\)
0.0106130 + 0.999944i \(0.496622\pi\)
\(920\) 0 0
\(921\) 12.7630 0.420555
\(922\) 0 0
\(923\) 16.6388i 0.547672i
\(924\) 0 0
\(925\) 4.83340i 0.158921i
\(926\) 0 0
\(927\) 7.07692 0.232437
\(928\) 0 0
\(929\) −1.92764 −0.0632440 −0.0316220 0.999500i \(-0.510067\pi\)
−0.0316220 + 0.999500i \(0.510067\pi\)
\(930\) 0 0
\(931\) 23.6421 0.774839
\(932\) 0 0
\(933\) −14.7842 −0.484012
\(934\) 0 0
\(935\) −3.69256 −0.120760
\(936\) 0 0
\(937\) 18.6541i 0.609404i 0.952448 + 0.304702i \(0.0985568\pi\)
−0.952448 + 0.304702i \(0.901443\pi\)
\(938\) 0 0
\(939\) −1.48723 −0.0485339
\(940\) 0 0
\(941\) 20.7327i 0.675868i 0.941170 + 0.337934i \(0.109728\pi\)
−0.941170 + 0.337934i \(0.890272\pi\)
\(942\) 0 0
\(943\) −12.8482 30.9517i −0.418395 1.00793i
\(944\) 0 0
\(945\) 1.58474 0.0515515
\(946\) 0 0
\(947\) 34.9977i 1.13727i −0.822589 0.568636i \(-0.807471\pi\)
0.822589 0.568636i \(-0.192529\pi\)
\(948\) 0 0
\(949\) 11.7086 0.380076
\(950\) 0 0
\(951\) 0.771575i 0.0250200i
\(952\) 0 0
\(953\) 42.1023i 1.36383i −0.731433 0.681914i \(-0.761148\pi\)
0.731433 0.681914i \(-0.238852\pi\)
\(954\) 0 0
\(955\) 6.46417i 0.209176i
\(956\) 0 0
\(957\) 22.2277i 0.718520i
\(958\) 0 0
\(959\) 18.9031i 0.610412i
\(960\) 0 0
\(961\) 30.5236 0.984633
\(962\) 0 0
\(963\) 6.20879 0.200076
\(964\) 0 0
\(965\) 21.5952i 0.695173i
\(966\) 0 0
\(967\) 21.6032i 0.694711i 0.937734 + 0.347355i \(0.112920\pi\)
−0.937734 + 0.347355i \(0.887080\pi\)
\(968\) 0 0
\(969\) −6.60455 −0.212169
\(970\) 0 0
\(971\) 38.8071 1.24538 0.622690 0.782469i \(-0.286040\pi\)
0.622690 + 0.782469i \(0.286040\pi\)
\(972\) 0 0
\(973\) 2.08715i 0.0669108i
\(974\) 0 0
\(975\) 1.23011i 0.0393951i
\(976\) 0 0
\(977\) 33.0657i 1.05787i 0.848664 + 0.528933i \(0.177408\pi\)
−0.848664 + 0.528933i \(0.822592\pi\)
\(978\) 0 0
\(979\) 36.0463i 1.15204i
\(980\) 0 0
\(981\) 5.46182i 0.174382i
\(982\) 0 0
\(983\) −19.4381 −0.619980 −0.309990 0.950740i \(-0.600326\pi\)
−0.309990 + 0.950740i \(0.600326\pi\)
\(984\) 0 0
\(985\) 4.11055i 0.130973i
\(986\) 0 0
\(987\) −14.1892 −0.451648
\(988\) 0 0
\(989\) 7.47746 + 18.0134i 0.237769 + 0.572794i
\(990\) 0 0
\(991\) 1.60244i 0.0509033i 0.999676 + 0.0254517i \(0.00810239\pi\)
−0.999676 + 0.0254517i \(0.991898\pi\)
\(992\) 0 0
\(993\) −33.8360 −1.07375
\(994\) 0 0
\(995\) 3.25603i 0.103223i
\(996\) 0 0
\(997\) 10.4207 0.330026 0.165013 0.986291i \(-0.447233\pi\)
0.165013 + 0.986291i \(0.447233\pi\)
\(998\) 0 0
\(999\) 4.83340 0.152922
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 5520.2.be.a.1471.3 16
4.3 odd 2 5520.2.be.b.1471.14 yes 16
23.22 odd 2 5520.2.be.b.1471.6 yes 16
92.91 even 2 inner 5520.2.be.a.1471.11 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5520.2.be.a.1471.3 16 1.1 even 1 trivial
5520.2.be.a.1471.11 yes 16 92.91 even 2 inner
5520.2.be.b.1471.6 yes 16 23.22 odd 2
5520.2.be.b.1471.14 yes 16 4.3 odd 2