Properties

Label 5054.2.a.l.1.1
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
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] = [5054,2,Mod(1,5054)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5054, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5054.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.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: \(40.3563931816\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -2.61803 q^{5} -1.61803 q^{6} +1.00000 q^{7} +1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -2.61803 q^{5} -1.61803 q^{6} +1.00000 q^{7} +1.00000 q^{8} -0.381966 q^{9} -2.61803 q^{10} -4.85410 q^{11} -1.61803 q^{12} +4.47214 q^{13} +1.00000 q^{14} +4.23607 q^{15} +1.00000 q^{16} +0.763932 q^{17} -0.381966 q^{18} -2.61803 q^{20} -1.61803 q^{21} -4.85410 q^{22} +8.94427 q^{23} -1.61803 q^{24} +1.85410 q^{25} +4.47214 q^{26} +5.47214 q^{27} +1.00000 q^{28} -0.145898 q^{29} +4.23607 q^{30} -6.00000 q^{31} +1.00000 q^{32} +7.85410 q^{33} +0.763932 q^{34} -2.61803 q^{35} -0.381966 q^{36} -7.85410 q^{37} -7.23607 q^{39} -2.61803 q^{40} +9.56231 q^{41} -1.61803 q^{42} +3.85410 q^{43} -4.85410 q^{44} +1.00000 q^{45} +8.94427 q^{46} -10.8541 q^{47} -1.61803 q^{48} +1.00000 q^{49} +1.85410 q^{50} -1.23607 q^{51} +4.47214 q^{52} +5.14590 q^{53} +5.47214 q^{54} +12.7082 q^{55} +1.00000 q^{56} -0.145898 q^{58} +11.5623 q^{59} +4.23607 q^{60} -8.56231 q^{61} -6.00000 q^{62} -0.381966 q^{63} +1.00000 q^{64} -11.7082 q^{65} +7.85410 q^{66} -5.23607 q^{67} +0.763932 q^{68} -14.4721 q^{69} -2.61803 q^{70} -8.56231 q^{71} -0.381966 q^{72} -7.85410 q^{74} -3.00000 q^{75} -4.85410 q^{77} -7.23607 q^{78} -0.326238 q^{79} -2.61803 q^{80} -7.70820 q^{81} +9.56231 q^{82} +11.2361 q^{83} -1.61803 q^{84} -2.00000 q^{85} +3.85410 q^{86} +0.236068 q^{87} -4.85410 q^{88} +3.14590 q^{89} +1.00000 q^{90} +4.47214 q^{91} +8.94427 q^{92} +9.70820 q^{93} -10.8541 q^{94} -1.61803 q^{96} -7.14590 q^{97} +1.00000 q^{98} +1.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 3 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 3 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9} - 3 q^{10} - 3 q^{11} - q^{12} + 2 q^{14} + 4 q^{15} + 2 q^{16} + 6 q^{17} - 3 q^{18} - 3 q^{20} - q^{21} - 3 q^{22} - q^{24} - 3 q^{25} + 2 q^{27} + 2 q^{28} - 7 q^{29} + 4 q^{30} - 12 q^{31} + 2 q^{32} + 9 q^{33} + 6 q^{34} - 3 q^{35} - 3 q^{36} - 9 q^{37} - 10 q^{39} - 3 q^{40} - q^{41} - q^{42} + q^{43} - 3 q^{44} + 2 q^{45} - 15 q^{47} - q^{48} + 2 q^{49} - 3 q^{50} + 2 q^{51} + 17 q^{53} + 2 q^{54} + 12 q^{55} + 2 q^{56} - 7 q^{58} + 3 q^{59} + 4 q^{60} + 3 q^{61} - 12 q^{62} - 3 q^{63} + 2 q^{64} - 10 q^{65} + 9 q^{66} - 6 q^{67} + 6 q^{68} - 20 q^{69} - 3 q^{70} + 3 q^{71} - 3 q^{72} - 9 q^{74} - 6 q^{75} - 3 q^{77} - 10 q^{78} + 15 q^{79} - 3 q^{80} - 2 q^{81} - q^{82} + 18 q^{83} - q^{84} - 4 q^{85} + q^{86} - 4 q^{87} - 3 q^{88} + 13 q^{89} + 2 q^{90} + 6 q^{93} - 15 q^{94} - q^{96} - 21 q^{97} + 2 q^{98} - 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.61803 −0.934172 −0.467086 0.884212i \(-0.654696\pi\)
−0.467086 + 0.884212i \(0.654696\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.61803 −1.17082 −0.585410 0.810737i \(-0.699067\pi\)
−0.585410 + 0.810737i \(0.699067\pi\)
\(6\) −1.61803 −0.660560
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −0.381966 −0.127322
\(10\) −2.61803 −0.827895
\(11\) −4.85410 −1.46357 −0.731783 0.681537i \(-0.761312\pi\)
−0.731783 + 0.681537i \(0.761312\pi\)
\(12\) −1.61803 −0.467086
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 1.00000 0.267261
\(15\) 4.23607 1.09375
\(16\) 1.00000 0.250000
\(17\) 0.763932 0.185281 0.0926404 0.995700i \(-0.470469\pi\)
0.0926404 + 0.995700i \(0.470469\pi\)
\(18\) −0.381966 −0.0900303
\(19\) 0 0
\(20\) −2.61803 −0.585410
\(21\) −1.61803 −0.353084
\(22\) −4.85410 −1.03490
\(23\) 8.94427 1.86501 0.932505 0.361158i \(-0.117618\pi\)
0.932505 + 0.361158i \(0.117618\pi\)
\(24\) −1.61803 −0.330280
\(25\) 1.85410 0.370820
\(26\) 4.47214 0.877058
\(27\) 5.47214 1.05311
\(28\) 1.00000 0.188982
\(29\) −0.145898 −0.0270926 −0.0135463 0.999908i \(-0.504312\pi\)
−0.0135463 + 0.999908i \(0.504312\pi\)
\(30\) 4.23607 0.773397
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 1.00000 0.176777
\(33\) 7.85410 1.36722
\(34\) 0.763932 0.131013
\(35\) −2.61803 −0.442529
\(36\) −0.381966 −0.0636610
\(37\) −7.85410 −1.29121 −0.645603 0.763673i \(-0.723394\pi\)
−0.645603 + 0.763673i \(0.723394\pi\)
\(38\) 0 0
\(39\) −7.23607 −1.15870
\(40\) −2.61803 −0.413948
\(41\) 9.56231 1.49338 0.746691 0.665171i \(-0.231642\pi\)
0.746691 + 0.665171i \(0.231642\pi\)
\(42\) −1.61803 −0.249668
\(43\) 3.85410 0.587745 0.293873 0.955845i \(-0.405056\pi\)
0.293873 + 0.955845i \(0.405056\pi\)
\(44\) −4.85410 −0.731783
\(45\) 1.00000 0.149071
\(46\) 8.94427 1.31876
\(47\) −10.8541 −1.58323 −0.791617 0.611018i \(-0.790760\pi\)
−0.791617 + 0.611018i \(0.790760\pi\)
\(48\) −1.61803 −0.233543
\(49\) 1.00000 0.142857
\(50\) 1.85410 0.262210
\(51\) −1.23607 −0.173084
\(52\) 4.47214 0.620174
\(53\) 5.14590 0.706843 0.353422 0.935464i \(-0.385018\pi\)
0.353422 + 0.935464i \(0.385018\pi\)
\(54\) 5.47214 0.744663
\(55\) 12.7082 1.71357
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −0.145898 −0.0191574
\(59\) 11.5623 1.50528 0.752642 0.658430i \(-0.228779\pi\)
0.752642 + 0.658430i \(0.228779\pi\)
\(60\) 4.23607 0.546874
\(61\) −8.56231 −1.09629 −0.548145 0.836383i \(-0.684666\pi\)
−0.548145 + 0.836383i \(0.684666\pi\)
\(62\) −6.00000 −0.762001
\(63\) −0.381966 −0.0481232
\(64\) 1.00000 0.125000
\(65\) −11.7082 −1.45222
\(66\) 7.85410 0.966773
\(67\) −5.23607 −0.639688 −0.319844 0.947470i \(-0.603630\pi\)
−0.319844 + 0.947470i \(0.603630\pi\)
\(68\) 0.763932 0.0926404
\(69\) −14.4721 −1.74224
\(70\) −2.61803 −0.312915
\(71\) −8.56231 −1.01616 −0.508079 0.861310i \(-0.669644\pi\)
−0.508079 + 0.861310i \(0.669644\pi\)
\(72\) −0.381966 −0.0450151
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −7.85410 −0.913021
\(75\) −3.00000 −0.346410
\(76\) 0 0
\(77\) −4.85410 −0.553176
\(78\) −7.23607 −0.819323
\(79\) −0.326238 −0.0367046 −0.0183523 0.999832i \(-0.505842\pi\)
−0.0183523 + 0.999832i \(0.505842\pi\)
\(80\) −2.61803 −0.292705
\(81\) −7.70820 −0.856467
\(82\) 9.56231 1.05598
\(83\) 11.2361 1.23332 0.616659 0.787230i \(-0.288486\pi\)
0.616659 + 0.787230i \(0.288486\pi\)
\(84\) −1.61803 −0.176542
\(85\) −2.00000 −0.216930
\(86\) 3.85410 0.415599
\(87\) 0.236068 0.0253091
\(88\) −4.85410 −0.517449
\(89\) 3.14590 0.333465 0.166732 0.986002i \(-0.446678\pi\)
0.166732 + 0.986002i \(0.446678\pi\)
\(90\) 1.00000 0.105409
\(91\) 4.47214 0.468807
\(92\) 8.94427 0.932505
\(93\) 9.70820 1.00669
\(94\) −10.8541 −1.11952
\(95\) 0 0
\(96\) −1.61803 −0.165140
\(97\) −7.14590 −0.725556 −0.362778 0.931876i \(-0.618172\pi\)
−0.362778 + 0.931876i \(0.618172\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.85410 0.186344
\(100\) 1.85410 0.185410
\(101\) 2.94427 0.292966 0.146483 0.989213i \(-0.453205\pi\)
0.146483 + 0.989213i \(0.453205\pi\)
\(102\) −1.23607 −0.122389
\(103\) −8.94427 −0.881305 −0.440653 0.897678i \(-0.645253\pi\)
−0.440653 + 0.897678i \(0.645253\pi\)
\(104\) 4.47214 0.438529
\(105\) 4.23607 0.413398
\(106\) 5.14590 0.499814
\(107\) −19.7082 −1.90526 −0.952632 0.304125i \(-0.901636\pi\)
−0.952632 + 0.304125i \(0.901636\pi\)
\(108\) 5.47214 0.526557
\(109\) −10.8541 −1.03963 −0.519817 0.854278i \(-0.674000\pi\)
−0.519817 + 0.854278i \(0.674000\pi\)
\(110\) 12.7082 1.21168
\(111\) 12.7082 1.20621
\(112\) 1.00000 0.0944911
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) −23.4164 −2.18359
\(116\) −0.145898 −0.0135463
\(117\) −1.70820 −0.157924
\(118\) 11.5623 1.06440
\(119\) 0.763932 0.0700295
\(120\) 4.23607 0.386698
\(121\) 12.5623 1.14203
\(122\) −8.56231 −0.775195
\(123\) −15.4721 −1.39508
\(124\) −6.00000 −0.538816
\(125\) 8.23607 0.736656
\(126\) −0.381966 −0.0340282
\(127\) −22.0902 −1.96019 −0.980093 0.198540i \(-0.936380\pi\)
−0.980093 + 0.198540i \(0.936380\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.23607 −0.549055
\(130\) −11.7082 −1.02688
\(131\) −4.47214 −0.390732 −0.195366 0.980730i \(-0.562590\pi\)
−0.195366 + 0.980730i \(0.562590\pi\)
\(132\) 7.85410 0.683612
\(133\) 0 0
\(134\) −5.23607 −0.452327
\(135\) −14.3262 −1.23301
\(136\) 0.763932 0.0655066
\(137\) 3.38197 0.288941 0.144470 0.989509i \(-0.453852\pi\)
0.144470 + 0.989509i \(0.453852\pi\)
\(138\) −14.4721 −1.23195
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) −2.61803 −0.221264
\(141\) 17.5623 1.47901
\(142\) −8.56231 −0.718533
\(143\) −21.7082 −1.81533
\(144\) −0.381966 −0.0318305
\(145\) 0.381966 0.0317206
\(146\) 0 0
\(147\) −1.61803 −0.133453
\(148\) −7.85410 −0.645603
\(149\) −17.2361 −1.41203 −0.706017 0.708195i \(-0.749510\pi\)
−0.706017 + 0.708195i \(0.749510\pi\)
\(150\) −3.00000 −0.244949
\(151\) −3.05573 −0.248672 −0.124336 0.992240i \(-0.539680\pi\)
−0.124336 + 0.992240i \(0.539680\pi\)
\(152\) 0 0
\(153\) −0.291796 −0.0235903
\(154\) −4.85410 −0.391155
\(155\) 15.7082 1.26171
\(156\) −7.23607 −0.579349
\(157\) 2.85410 0.227782 0.113891 0.993493i \(-0.463669\pi\)
0.113891 + 0.993493i \(0.463669\pi\)
\(158\) −0.326238 −0.0259541
\(159\) −8.32624 −0.660314
\(160\) −2.61803 −0.206974
\(161\) 8.94427 0.704907
\(162\) −7.70820 −0.605614
\(163\) −20.2705 −1.58771 −0.793854 0.608108i \(-0.791929\pi\)
−0.793854 + 0.608108i \(0.791929\pi\)
\(164\) 9.56231 0.746691
\(165\) −20.5623 −1.60077
\(166\) 11.2361 0.872088
\(167\) 10.0000 0.773823 0.386912 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) −1.61803 −0.124834
\(169\) 7.00000 0.538462
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) 3.85410 0.293873
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0.236068 0.0178963
\(175\) 1.85410 0.140157
\(176\) −4.85410 −0.365892
\(177\) −18.7082 −1.40619
\(178\) 3.14590 0.235795
\(179\) −17.7082 −1.32357 −0.661787 0.749692i \(-0.730202\pi\)
−0.661787 + 0.749692i \(0.730202\pi\)
\(180\) 1.00000 0.0745356
\(181\) 7.41641 0.551257 0.275629 0.961264i \(-0.411114\pi\)
0.275629 + 0.961264i \(0.411114\pi\)
\(182\) 4.47214 0.331497
\(183\) 13.8541 1.02412
\(184\) 8.94427 0.659380
\(185\) 20.5623 1.51177
\(186\) 9.70820 0.711840
\(187\) −3.70820 −0.271171
\(188\) −10.8541 −0.791617
\(189\) 5.47214 0.398039
\(190\) 0 0
\(191\) 14.9443 1.08133 0.540665 0.841238i \(-0.318173\pi\)
0.540665 + 0.841238i \(0.318173\pi\)
\(192\) −1.61803 −0.116772
\(193\) 20.9443 1.50760 0.753801 0.657103i \(-0.228218\pi\)
0.753801 + 0.657103i \(0.228218\pi\)
\(194\) −7.14590 −0.513046
\(195\) 18.9443 1.35663
\(196\) 1.00000 0.0714286
\(197\) −5.23607 −0.373054 −0.186527 0.982450i \(-0.559723\pi\)
−0.186527 + 0.982450i \(0.559723\pi\)
\(198\) 1.85410 0.131765
\(199\) 1.56231 0.110749 0.0553745 0.998466i \(-0.482365\pi\)
0.0553745 + 0.998466i \(0.482365\pi\)
\(200\) 1.85410 0.131105
\(201\) 8.47214 0.597578
\(202\) 2.94427 0.207158
\(203\) −0.145898 −0.0102400
\(204\) −1.23607 −0.0865421
\(205\) −25.0344 −1.74848
\(206\) −8.94427 −0.623177
\(207\) −3.41641 −0.237457
\(208\) 4.47214 0.310087
\(209\) 0 0
\(210\) 4.23607 0.292316
\(211\) −17.2361 −1.18658 −0.593290 0.804989i \(-0.702171\pi\)
−0.593290 + 0.804989i \(0.702171\pi\)
\(212\) 5.14590 0.353422
\(213\) 13.8541 0.949267
\(214\) −19.7082 −1.34723
\(215\) −10.0902 −0.688144
\(216\) 5.47214 0.372332
\(217\) −6.00000 −0.407307
\(218\) −10.8541 −0.735133
\(219\) 0 0
\(220\) 12.7082 0.856787
\(221\) 3.41641 0.229812
\(222\) 12.7082 0.852919
\(223\) 5.23607 0.350633 0.175317 0.984512i \(-0.443905\pi\)
0.175317 + 0.984512i \(0.443905\pi\)
\(224\) 1.00000 0.0668153
\(225\) −0.708204 −0.0472136
\(226\) −10.0000 −0.665190
\(227\) −19.4164 −1.28871 −0.644356 0.764726i \(-0.722875\pi\)
−0.644356 + 0.764726i \(0.722875\pi\)
\(228\) 0 0
\(229\) 23.8541 1.57632 0.788162 0.615468i \(-0.211033\pi\)
0.788162 + 0.615468i \(0.211033\pi\)
\(230\) −23.4164 −1.54403
\(231\) 7.85410 0.516762
\(232\) −0.145898 −0.00957868
\(233\) 4.09017 0.267956 0.133978 0.990984i \(-0.457225\pi\)
0.133978 + 0.990984i \(0.457225\pi\)
\(234\) −1.70820 −0.111669
\(235\) 28.4164 1.85368
\(236\) 11.5623 0.752642
\(237\) 0.527864 0.0342885
\(238\) 0.763932 0.0495184
\(239\) 14.1803 0.917250 0.458625 0.888630i \(-0.348342\pi\)
0.458625 + 0.888630i \(0.348342\pi\)
\(240\) 4.23607 0.273437
\(241\) −17.5623 −1.13129 −0.565644 0.824650i \(-0.691372\pi\)
−0.565644 + 0.824650i \(0.691372\pi\)
\(242\) 12.5623 0.807536
\(243\) −3.94427 −0.253025
\(244\) −8.56231 −0.548145
\(245\) −2.61803 −0.167260
\(246\) −15.4721 −0.986467
\(247\) 0 0
\(248\) −6.00000 −0.381000
\(249\) −18.1803 −1.15213
\(250\) 8.23607 0.520895
\(251\) 9.05573 0.571592 0.285796 0.958290i \(-0.407742\pi\)
0.285796 + 0.958290i \(0.407742\pi\)
\(252\) −0.381966 −0.0240616
\(253\) −43.4164 −2.72957
\(254\) −22.0902 −1.38606
\(255\) 3.23607 0.202650
\(256\) 1.00000 0.0625000
\(257\) −9.85410 −0.614682 −0.307341 0.951599i \(-0.599439\pi\)
−0.307341 + 0.951599i \(0.599439\pi\)
\(258\) −6.23607 −0.388241
\(259\) −7.85410 −0.488030
\(260\) −11.7082 −0.726112
\(261\) 0.0557281 0.00344948
\(262\) −4.47214 −0.276289
\(263\) 12.6525 0.780185 0.390093 0.920776i \(-0.372443\pi\)
0.390093 + 0.920776i \(0.372443\pi\)
\(264\) 7.85410 0.483387
\(265\) −13.4721 −0.827587
\(266\) 0 0
\(267\) −5.09017 −0.311513
\(268\) −5.23607 −0.319844
\(269\) 4.58359 0.279467 0.139733 0.990189i \(-0.455375\pi\)
0.139733 + 0.990189i \(0.455375\pi\)
\(270\) −14.3262 −0.871867
\(271\) 2.43769 0.148079 0.0740397 0.997255i \(-0.476411\pi\)
0.0740397 + 0.997255i \(0.476411\pi\)
\(272\) 0.763932 0.0463202
\(273\) −7.23607 −0.437947
\(274\) 3.38197 0.204312
\(275\) −9.00000 −0.542720
\(276\) −14.4721 −0.871120
\(277\) −25.4164 −1.52712 −0.763562 0.645735i \(-0.776551\pi\)
−0.763562 + 0.645735i \(0.776551\pi\)
\(278\) 6.00000 0.359856
\(279\) 2.29180 0.137206
\(280\) −2.61803 −0.156457
\(281\) 23.4164 1.39691 0.698453 0.715656i \(-0.253872\pi\)
0.698453 + 0.715656i \(0.253872\pi\)
\(282\) 17.5623 1.04582
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) −8.56231 −0.508079
\(285\) 0 0
\(286\) −21.7082 −1.28363
\(287\) 9.56231 0.564445
\(288\) −0.381966 −0.0225076
\(289\) −16.4164 −0.965671
\(290\) 0.381966 0.0224298
\(291\) 11.5623 0.677794
\(292\) 0 0
\(293\) −31.4164 −1.83537 −0.917683 0.397313i \(-0.869943\pi\)
−0.917683 + 0.397313i \(0.869943\pi\)
\(294\) −1.61803 −0.0943657
\(295\) −30.2705 −1.76242
\(296\) −7.85410 −0.456510
\(297\) −26.5623 −1.54130
\(298\) −17.2361 −0.998459
\(299\) 40.0000 2.31326
\(300\) −3.00000 −0.173205
\(301\) 3.85410 0.222147
\(302\) −3.05573 −0.175837
\(303\) −4.76393 −0.273681
\(304\) 0 0
\(305\) 22.4164 1.28356
\(306\) −0.291796 −0.0166809
\(307\) −18.9787 −1.08317 −0.541586 0.840645i \(-0.682176\pi\)
−0.541586 + 0.840645i \(0.682176\pi\)
\(308\) −4.85410 −0.276588
\(309\) 14.4721 0.823291
\(310\) 15.7082 0.892166
\(311\) −21.3262 −1.20930 −0.604650 0.796491i \(-0.706687\pi\)
−0.604650 + 0.796491i \(0.706687\pi\)
\(312\) −7.23607 −0.409662
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 2.85410 0.161066
\(315\) 1.00000 0.0563436
\(316\) −0.326238 −0.0183523
\(317\) 25.6869 1.44272 0.721361 0.692560i \(-0.243517\pi\)
0.721361 + 0.692560i \(0.243517\pi\)
\(318\) −8.32624 −0.466912
\(319\) 0.708204 0.0396518
\(320\) −2.61803 −0.146353
\(321\) 31.8885 1.77984
\(322\) 8.94427 0.498445
\(323\) 0 0
\(324\) −7.70820 −0.428234
\(325\) 8.29180 0.459946
\(326\) −20.2705 −1.12268
\(327\) 17.5623 0.971198
\(328\) 9.56231 0.527990
\(329\) −10.8541 −0.598406
\(330\) −20.5623 −1.13192
\(331\) 3.70820 0.203821 0.101911 0.994794i \(-0.467504\pi\)
0.101911 + 0.994794i \(0.467504\pi\)
\(332\) 11.2361 0.616659
\(333\) 3.00000 0.164399
\(334\) 10.0000 0.547176
\(335\) 13.7082 0.748959
\(336\) −1.61803 −0.0882710
\(337\) −9.70820 −0.528840 −0.264420 0.964408i \(-0.585180\pi\)
−0.264420 + 0.964408i \(0.585180\pi\)
\(338\) 7.00000 0.380750
\(339\) 16.1803 0.878795
\(340\) −2.00000 −0.108465
\(341\) 29.1246 1.57719
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 3.85410 0.207799
\(345\) 37.8885 2.03985
\(346\) −18.0000 −0.967686
\(347\) 8.94427 0.480154 0.240077 0.970754i \(-0.422827\pi\)
0.240077 + 0.970754i \(0.422827\pi\)
\(348\) 0.236068 0.0126546
\(349\) −16.8328 −0.901040 −0.450520 0.892766i \(-0.648761\pi\)
−0.450520 + 0.892766i \(0.648761\pi\)
\(350\) 1.85410 0.0991059
\(351\) 24.4721 1.30623
\(352\) −4.85410 −0.258725
\(353\) 13.4164 0.714083 0.357042 0.934088i \(-0.383785\pi\)
0.357042 + 0.934088i \(0.383785\pi\)
\(354\) −18.7082 −0.994330
\(355\) 22.4164 1.18974
\(356\) 3.14590 0.166732
\(357\) −1.23607 −0.0654197
\(358\) −17.7082 −0.935908
\(359\) 2.29180 0.120956 0.0604782 0.998170i \(-0.480737\pi\)
0.0604782 + 0.998170i \(0.480737\pi\)
\(360\) 1.00000 0.0527046
\(361\) 0 0
\(362\) 7.41641 0.389798
\(363\) −20.3262 −1.06685
\(364\) 4.47214 0.234404
\(365\) 0 0
\(366\) 13.8541 0.724166
\(367\) −6.27051 −0.327318 −0.163659 0.986517i \(-0.552330\pi\)
−0.163659 + 0.986517i \(0.552330\pi\)
\(368\) 8.94427 0.466252
\(369\) −3.65248 −0.190140
\(370\) 20.5623 1.06898
\(371\) 5.14590 0.267162
\(372\) 9.70820 0.503347
\(373\) −14.6180 −0.756893 −0.378447 0.925623i \(-0.623542\pi\)
−0.378447 + 0.925623i \(0.623542\pi\)
\(374\) −3.70820 −0.191747
\(375\) −13.3262 −0.688164
\(376\) −10.8541 −0.559758
\(377\) −0.652476 −0.0336042
\(378\) 5.47214 0.281456
\(379\) −32.1803 −1.65299 −0.826497 0.562942i \(-0.809669\pi\)
−0.826497 + 0.562942i \(0.809669\pi\)
\(380\) 0 0
\(381\) 35.7426 1.83115
\(382\) 14.9443 0.764615
\(383\) 13.1246 0.670636 0.335318 0.942105i \(-0.391156\pi\)
0.335318 + 0.942105i \(0.391156\pi\)
\(384\) −1.61803 −0.0825700
\(385\) 12.7082 0.647670
\(386\) 20.9443 1.06604
\(387\) −1.47214 −0.0748329
\(388\) −7.14590 −0.362778
\(389\) −37.3050 −1.89144 −0.945718 0.324988i \(-0.894640\pi\)
−0.945718 + 0.324988i \(0.894640\pi\)
\(390\) 18.9443 0.959280
\(391\) 6.83282 0.345550
\(392\) 1.00000 0.0505076
\(393\) 7.23607 0.365011
\(394\) −5.23607 −0.263789
\(395\) 0.854102 0.0429745
\(396\) 1.85410 0.0931721
\(397\) −28.5623 −1.43350 −0.716751 0.697330i \(-0.754371\pi\)
−0.716751 + 0.697330i \(0.754371\pi\)
\(398\) 1.56231 0.0783113
\(399\) 0 0
\(400\) 1.85410 0.0927051
\(401\) 25.4164 1.26923 0.634617 0.772826i \(-0.281158\pi\)
0.634617 + 0.772826i \(0.281158\pi\)
\(402\) 8.47214 0.422552
\(403\) −26.8328 −1.33664
\(404\) 2.94427 0.146483
\(405\) 20.1803 1.00277
\(406\) −0.145898 −0.00724080
\(407\) 38.1246 1.88977
\(408\) −1.23607 −0.0611945
\(409\) 1.79837 0.0889239 0.0444619 0.999011i \(-0.485843\pi\)
0.0444619 + 0.999011i \(0.485843\pi\)
\(410\) −25.0344 −1.23636
\(411\) −5.47214 −0.269921
\(412\) −8.94427 −0.440653
\(413\) 11.5623 0.568944
\(414\) −3.41641 −0.167907
\(415\) −29.4164 −1.44399
\(416\) 4.47214 0.219265
\(417\) −9.70820 −0.475413
\(418\) 0 0
\(419\) −12.6525 −0.618114 −0.309057 0.951044i \(-0.600013\pi\)
−0.309057 + 0.951044i \(0.600013\pi\)
\(420\) 4.23607 0.206699
\(421\) −19.5279 −0.951730 −0.475865 0.879518i \(-0.657865\pi\)
−0.475865 + 0.879518i \(0.657865\pi\)
\(422\) −17.2361 −0.839039
\(423\) 4.14590 0.201580
\(424\) 5.14590 0.249907
\(425\) 1.41641 0.0687059
\(426\) 13.8541 0.671233
\(427\) −8.56231 −0.414359
\(428\) −19.7082 −0.952632
\(429\) 35.1246 1.69583
\(430\) −10.0902 −0.486591
\(431\) −22.8541 −1.10084 −0.550422 0.834887i \(-0.685533\pi\)
−0.550422 + 0.834887i \(0.685533\pi\)
\(432\) 5.47214 0.263278
\(433\) 3.38197 0.162527 0.0812635 0.996693i \(-0.474104\pi\)
0.0812635 + 0.996693i \(0.474104\pi\)
\(434\) −6.00000 −0.288009
\(435\) −0.618034 −0.0296325
\(436\) −10.8541 −0.519817
\(437\) 0 0
\(438\) 0 0
\(439\) 2.18034 0.104062 0.0520310 0.998645i \(-0.483431\pi\)
0.0520310 + 0.998645i \(0.483431\pi\)
\(440\) 12.7082 0.605840
\(441\) −0.381966 −0.0181889
\(442\) 3.41641 0.162502
\(443\) 1.09017 0.0517955 0.0258978 0.999665i \(-0.491756\pi\)
0.0258978 + 0.999665i \(0.491756\pi\)
\(444\) 12.7082 0.603105
\(445\) −8.23607 −0.390427
\(446\) 5.23607 0.247935
\(447\) 27.8885 1.31908
\(448\) 1.00000 0.0472456
\(449\) 0.291796 0.0137707 0.00688535 0.999976i \(-0.497808\pi\)
0.00688535 + 0.999976i \(0.497808\pi\)
\(450\) −0.708204 −0.0333851
\(451\) −46.4164 −2.18566
\(452\) −10.0000 −0.470360
\(453\) 4.94427 0.232302
\(454\) −19.4164 −0.911257
\(455\) −11.7082 −0.548889
\(456\) 0 0
\(457\) 31.1459 1.45694 0.728472 0.685076i \(-0.240231\pi\)
0.728472 + 0.685076i \(0.240231\pi\)
\(458\) 23.8541 1.11463
\(459\) 4.18034 0.195122
\(460\) −23.4164 −1.09180
\(461\) −41.5066 −1.93315 −0.966577 0.256376i \(-0.917471\pi\)
−0.966577 + 0.256376i \(0.917471\pi\)
\(462\) 7.85410 0.365406
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) −0.145898 −0.00677315
\(465\) −25.4164 −1.17866
\(466\) 4.09017 0.189473
\(467\) −40.3607 −1.86767 −0.933835 0.357705i \(-0.883559\pi\)
−0.933835 + 0.357705i \(0.883559\pi\)
\(468\) −1.70820 −0.0789618
\(469\) −5.23607 −0.241779
\(470\) 28.4164 1.31075
\(471\) −4.61803 −0.212788
\(472\) 11.5623 0.532198
\(473\) −18.7082 −0.860204
\(474\) 0.527864 0.0242456
\(475\) 0 0
\(476\) 0.763932 0.0350148
\(477\) −1.96556 −0.0899967
\(478\) 14.1803 0.648594
\(479\) 36.3262 1.65979 0.829894 0.557921i \(-0.188401\pi\)
0.829894 + 0.557921i \(0.188401\pi\)
\(480\) 4.23607 0.193349
\(481\) −35.1246 −1.60154
\(482\) −17.5623 −0.799941
\(483\) −14.4721 −0.658505
\(484\) 12.5623 0.571014
\(485\) 18.7082 0.849496
\(486\) −3.94427 −0.178916
\(487\) −16.9098 −0.766258 −0.383129 0.923695i \(-0.625153\pi\)
−0.383129 + 0.923695i \(0.625153\pi\)
\(488\) −8.56231 −0.387597
\(489\) 32.7984 1.48319
\(490\) −2.61803 −0.118271
\(491\) −40.3607 −1.82145 −0.910726 0.413011i \(-0.864477\pi\)
−0.910726 + 0.413011i \(0.864477\pi\)
\(492\) −15.4721 −0.697538
\(493\) −0.111456 −0.00501973
\(494\) 0 0
\(495\) −4.85410 −0.218176
\(496\) −6.00000 −0.269408
\(497\) −8.56231 −0.384072
\(498\) −18.1803 −0.814681
\(499\) −18.6869 −0.836541 −0.418271 0.908322i \(-0.637364\pi\)
−0.418271 + 0.908322i \(0.637364\pi\)
\(500\) 8.23607 0.368328
\(501\) −16.1803 −0.722884
\(502\) 9.05573 0.404177
\(503\) 30.2148 1.34721 0.673605 0.739091i \(-0.264745\pi\)
0.673605 + 0.739091i \(0.264745\pi\)
\(504\) −0.381966 −0.0170141
\(505\) −7.70820 −0.343011
\(506\) −43.4164 −1.93009
\(507\) −11.3262 −0.503016
\(508\) −22.0902 −0.980093
\(509\) 35.1246 1.55687 0.778436 0.627725i \(-0.216014\pi\)
0.778436 + 0.627725i \(0.216014\pi\)
\(510\) 3.23607 0.143295
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −9.85410 −0.434646
\(515\) 23.4164 1.03185
\(516\) −6.23607 −0.274528
\(517\) 52.6869 2.31717
\(518\) −7.85410 −0.345089
\(519\) 29.1246 1.27843
\(520\) −11.7082 −0.513439
\(521\) 21.4164 0.938270 0.469135 0.883127i \(-0.344566\pi\)
0.469135 + 0.883127i \(0.344566\pi\)
\(522\) 0.0557281 0.00243915
\(523\) −24.0000 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) −4.47214 −0.195366
\(525\) −3.00000 −0.130931
\(526\) 12.6525 0.551674
\(527\) −4.58359 −0.199664
\(528\) 7.85410 0.341806
\(529\) 57.0000 2.47826
\(530\) −13.4721 −0.585192
\(531\) −4.41641 −0.191656
\(532\) 0 0
\(533\) 42.7639 1.85231
\(534\) −5.09017 −0.220273
\(535\) 51.5967 2.23072
\(536\) −5.23607 −0.226164
\(537\) 28.6525 1.23645
\(538\) 4.58359 0.197613
\(539\) −4.85410 −0.209081
\(540\) −14.3262 −0.616503
\(541\) 19.1246 0.822231 0.411116 0.911583i \(-0.365139\pi\)
0.411116 + 0.911583i \(0.365139\pi\)
\(542\) 2.43769 0.104708
\(543\) −12.0000 −0.514969
\(544\) 0.763932 0.0327533
\(545\) 28.4164 1.21723
\(546\) −7.23607 −0.309675
\(547\) −12.6525 −0.540981 −0.270490 0.962723i \(-0.587186\pi\)
−0.270490 + 0.962723i \(0.587186\pi\)
\(548\) 3.38197 0.144470
\(549\) 3.27051 0.139582
\(550\) −9.00000 −0.383761
\(551\) 0 0
\(552\) −14.4721 −0.615975
\(553\) −0.326238 −0.0138730
\(554\) −25.4164 −1.07984
\(555\) −33.2705 −1.41225
\(556\) 6.00000 0.254457
\(557\) −32.0689 −1.35880 −0.679401 0.733767i \(-0.737760\pi\)
−0.679401 + 0.733767i \(0.737760\pi\)
\(558\) 2.29180 0.0970195
\(559\) 17.2361 0.729008
\(560\) −2.61803 −0.110632
\(561\) 6.00000 0.253320
\(562\) 23.4164 0.987762
\(563\) 6.56231 0.276568 0.138284 0.990393i \(-0.455841\pi\)
0.138284 + 0.990393i \(0.455841\pi\)
\(564\) 17.5623 0.739506
\(565\) 26.1803 1.10142
\(566\) 8.00000 0.336265
\(567\) −7.70820 −0.323714
\(568\) −8.56231 −0.359266
\(569\) 25.7082 1.07774 0.538872 0.842388i \(-0.318851\pi\)
0.538872 + 0.842388i \(0.318851\pi\)
\(570\) 0 0
\(571\) 15.6869 0.656477 0.328239 0.944595i \(-0.393545\pi\)
0.328239 + 0.944595i \(0.393545\pi\)
\(572\) −21.7082 −0.907666
\(573\) −24.1803 −1.01015
\(574\) 9.56231 0.399123
\(575\) 16.5836 0.691584
\(576\) −0.381966 −0.0159153
\(577\) −13.1246 −0.546385 −0.273192 0.961959i \(-0.588080\pi\)
−0.273192 + 0.961959i \(0.588080\pi\)
\(578\) −16.4164 −0.682833
\(579\) −33.8885 −1.40836
\(580\) 0.381966 0.0158603
\(581\) 11.2361 0.466151
\(582\) 11.5623 0.479273
\(583\) −24.9787 −1.03451
\(584\) 0 0
\(585\) 4.47214 0.184900
\(586\) −31.4164 −1.29780
\(587\) 19.5279 0.806001 0.403001 0.915200i \(-0.367967\pi\)
0.403001 + 0.915200i \(0.367967\pi\)
\(588\) −1.61803 −0.0667266
\(589\) 0 0
\(590\) −30.2705 −1.24622
\(591\) 8.47214 0.348497
\(592\) −7.85410 −0.322802
\(593\) 16.4721 0.676430 0.338215 0.941069i \(-0.390177\pi\)
0.338215 + 0.941069i \(0.390177\pi\)
\(594\) −26.5623 −1.08986
\(595\) −2.00000 −0.0819920
\(596\) −17.2361 −0.706017
\(597\) −2.52786 −0.103459
\(598\) 40.0000 1.63572
\(599\) −11.9787 −0.489437 −0.244718 0.969594i \(-0.578696\pi\)
−0.244718 + 0.969594i \(0.578696\pi\)
\(600\) −3.00000 −0.122474
\(601\) −10.3607 −0.422621 −0.211310 0.977419i \(-0.567773\pi\)
−0.211310 + 0.977419i \(0.567773\pi\)
\(602\) 3.85410 0.157081
\(603\) 2.00000 0.0814463
\(604\) −3.05573 −0.124336
\(605\) −32.8885 −1.33711
\(606\) −4.76393 −0.193522
\(607\) −16.3607 −0.664060 −0.332030 0.943269i \(-0.607733\pi\)
−0.332030 + 0.943269i \(0.607733\pi\)
\(608\) 0 0
\(609\) 0.236068 0.00956596
\(610\) 22.4164 0.907614
\(611\) −48.5410 −1.96376
\(612\) −0.291796 −0.0117952
\(613\) 12.2918 0.496461 0.248230 0.968701i \(-0.420151\pi\)
0.248230 + 0.968701i \(0.420151\pi\)
\(614\) −18.9787 −0.765919
\(615\) 40.5066 1.63338
\(616\) −4.85410 −0.195577
\(617\) 33.9230 1.36569 0.682844 0.730564i \(-0.260743\pi\)
0.682844 + 0.730564i \(0.260743\pi\)
\(618\) 14.4721 0.582155
\(619\) −35.4164 −1.42351 −0.711753 0.702430i \(-0.752098\pi\)
−0.711753 + 0.702430i \(0.752098\pi\)
\(620\) 15.7082 0.630857
\(621\) 48.9443 1.96407
\(622\) −21.3262 −0.855104
\(623\) 3.14590 0.126038
\(624\) −7.23607 −0.289675
\(625\) −30.8328 −1.23331
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 2.85410 0.113891
\(629\) −6.00000 −0.239236
\(630\) 1.00000 0.0398410
\(631\) 22.8328 0.908960 0.454480 0.890757i \(-0.349825\pi\)
0.454480 + 0.890757i \(0.349825\pi\)
\(632\) −0.326238 −0.0129770
\(633\) 27.8885 1.10847
\(634\) 25.6869 1.02016
\(635\) 57.8328 2.29503
\(636\) −8.32624 −0.330157
\(637\) 4.47214 0.177192
\(638\) 0.708204 0.0280381
\(639\) 3.27051 0.129379
\(640\) −2.61803 −0.103487
\(641\) 7.41641 0.292930 0.146465 0.989216i \(-0.453210\pi\)
0.146465 + 0.989216i \(0.453210\pi\)
\(642\) 31.8885 1.25854
\(643\) 21.4164 0.844581 0.422290 0.906461i \(-0.361226\pi\)
0.422290 + 0.906461i \(0.361226\pi\)
\(644\) 8.94427 0.352454
\(645\) 16.3262 0.642845
\(646\) 0 0
\(647\) −0.326238 −0.0128257 −0.00641287 0.999979i \(-0.502041\pi\)
−0.00641287 + 0.999979i \(0.502041\pi\)
\(648\) −7.70820 −0.302807
\(649\) −56.1246 −2.20308
\(650\) 8.29180 0.325231
\(651\) 9.70820 0.380495
\(652\) −20.2705 −0.793854
\(653\) −13.5279 −0.529386 −0.264693 0.964333i \(-0.585271\pi\)
−0.264693 + 0.964333i \(0.585271\pi\)
\(654\) 17.5623 0.686741
\(655\) 11.7082 0.457477
\(656\) 9.56231 0.373345
\(657\) 0 0
\(658\) −10.8541 −0.423137
\(659\) 0.875388 0.0341003 0.0170501 0.999855i \(-0.494573\pi\)
0.0170501 + 0.999855i \(0.494573\pi\)
\(660\) −20.5623 −0.800387
\(661\) 26.8328 1.04368 0.521838 0.853045i \(-0.325247\pi\)
0.521838 + 0.853045i \(0.325247\pi\)
\(662\) 3.70820 0.144123
\(663\) −5.52786 −0.214684
\(664\) 11.2361 0.436044
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) −1.30495 −0.0505279
\(668\) 10.0000 0.386912
\(669\) −8.47214 −0.327552
\(670\) 13.7082 0.529594
\(671\) 41.5623 1.60450
\(672\) −1.61803 −0.0624170
\(673\) 3.05573 0.117790 0.0588948 0.998264i \(-0.481242\pi\)
0.0588948 + 0.998264i \(0.481242\pi\)
\(674\) −9.70820 −0.373946
\(675\) 10.1459 0.390516
\(676\) 7.00000 0.269231
\(677\) 41.7082 1.60298 0.801488 0.598011i \(-0.204042\pi\)
0.801488 + 0.598011i \(0.204042\pi\)
\(678\) 16.1803 0.621402
\(679\) −7.14590 −0.274234
\(680\) −2.00000 −0.0766965
\(681\) 31.4164 1.20388
\(682\) 29.1246 1.11524
\(683\) 2.29180 0.0876931 0.0438466 0.999038i \(-0.486039\pi\)
0.0438466 + 0.999038i \(0.486039\pi\)
\(684\) 0 0
\(685\) −8.85410 −0.338298
\(686\) 1.00000 0.0381802
\(687\) −38.5967 −1.47256
\(688\) 3.85410 0.146936
\(689\) 23.0132 0.876731
\(690\) 37.8885 1.44239
\(691\) 20.5410 0.781417 0.390709 0.920514i \(-0.372230\pi\)
0.390709 + 0.920514i \(0.372230\pi\)
\(692\) −18.0000 −0.684257
\(693\) 1.85410 0.0704315
\(694\) 8.94427 0.339520
\(695\) −15.7082 −0.595846
\(696\) 0.236068 0.00894813
\(697\) 7.30495 0.276695
\(698\) −16.8328 −0.637131
\(699\) −6.61803 −0.250317
\(700\) 1.85410 0.0700785
\(701\) −5.23607 −0.197764 −0.0988818 0.995099i \(-0.531527\pi\)
−0.0988818 + 0.995099i \(0.531527\pi\)
\(702\) 24.4721 0.923641
\(703\) 0 0
\(704\) −4.85410 −0.182946
\(705\) −45.9787 −1.73166
\(706\) 13.4164 0.504933
\(707\) 2.94427 0.110731
\(708\) −18.7082 −0.703097
\(709\) 9.12461 0.342682 0.171341 0.985212i \(-0.445190\pi\)
0.171341 + 0.985212i \(0.445190\pi\)
\(710\) 22.4164 0.841273
\(711\) 0.124612 0.00467331
\(712\) 3.14590 0.117898
\(713\) −53.6656 −2.00979
\(714\) −1.23607 −0.0462587
\(715\) 56.8328 2.12543
\(716\) −17.7082 −0.661787
\(717\) −22.9443 −0.856870
\(718\) 2.29180 0.0855291
\(719\) 16.3607 0.610150 0.305075 0.952328i \(-0.401318\pi\)
0.305075 + 0.952328i \(0.401318\pi\)
\(720\) 1.00000 0.0372678
\(721\) −8.94427 −0.333102
\(722\) 0 0
\(723\) 28.4164 1.05682
\(724\) 7.41641 0.275629
\(725\) −0.270510 −0.0100465
\(726\) −20.3262 −0.754377
\(727\) 28.5623 1.05932 0.529659 0.848211i \(-0.322320\pi\)
0.529659 + 0.848211i \(0.322320\pi\)
\(728\) 4.47214 0.165748
\(729\) 29.5066 1.09284
\(730\) 0 0
\(731\) 2.94427 0.108898
\(732\) 13.8541 0.512062
\(733\) −19.5623 −0.722550 −0.361275 0.932459i \(-0.617658\pi\)
−0.361275 + 0.932459i \(0.617658\pi\)
\(734\) −6.27051 −0.231449
\(735\) 4.23607 0.156250
\(736\) 8.94427 0.329690
\(737\) 25.4164 0.936225
\(738\) −3.65248 −0.134449
\(739\) 35.2705 1.29745 0.648723 0.761024i \(-0.275303\pi\)
0.648723 + 0.761024i \(0.275303\pi\)
\(740\) 20.5623 0.755885
\(741\) 0 0
\(742\) 5.14590 0.188912
\(743\) −24.9787 −0.916380 −0.458190 0.888854i \(-0.651502\pi\)
−0.458190 + 0.888854i \(0.651502\pi\)
\(744\) 9.70820 0.355920
\(745\) 45.1246 1.65324
\(746\) −14.6180 −0.535204
\(747\) −4.29180 −0.157029
\(748\) −3.70820 −0.135585
\(749\) −19.7082 −0.720122
\(750\) −13.3262 −0.486605
\(751\) 0.326238 0.0119046 0.00595230 0.999982i \(-0.498105\pi\)
0.00595230 + 0.999982i \(0.498105\pi\)
\(752\) −10.8541 −0.395808
\(753\) −14.6525 −0.533966
\(754\) −0.652476 −0.0237618
\(755\) 8.00000 0.291150
\(756\) 5.47214 0.199020
\(757\) 46.2492 1.68096 0.840478 0.541845i \(-0.182274\pi\)
0.840478 + 0.541845i \(0.182274\pi\)
\(758\) −32.1803 −1.16884
\(759\) 70.2492 2.54989
\(760\) 0 0
\(761\) −17.7771 −0.644419 −0.322209 0.946668i \(-0.604426\pi\)
−0.322209 + 0.946668i \(0.604426\pi\)
\(762\) 35.7426 1.29482
\(763\) −10.8541 −0.392945
\(764\) 14.9443 0.540665
\(765\) 0.763932 0.0276200
\(766\) 13.1246 0.474212
\(767\) 51.7082 1.86708
\(768\) −1.61803 −0.0583858
\(769\) 12.2918 0.443254 0.221627 0.975132i \(-0.428863\pi\)
0.221627 + 0.975132i \(0.428863\pi\)
\(770\) 12.7082 0.457972
\(771\) 15.9443 0.574219
\(772\) 20.9443 0.753801
\(773\) −1.70820 −0.0614398 −0.0307199 0.999528i \(-0.509780\pi\)
−0.0307199 + 0.999528i \(0.509780\pi\)
\(774\) −1.47214 −0.0529148
\(775\) −11.1246 −0.399608
\(776\) −7.14590 −0.256523
\(777\) 12.7082 0.455904
\(778\) −37.3050 −1.33745
\(779\) 0 0
\(780\) 18.9443 0.678314
\(781\) 41.5623 1.48722
\(782\) 6.83282 0.244341
\(783\) −0.798374 −0.0285316
\(784\) 1.00000 0.0357143
\(785\) −7.47214 −0.266692
\(786\) 7.23607 0.258102
\(787\) 27.2705 0.972089 0.486044 0.873934i \(-0.338439\pi\)
0.486044 + 0.873934i \(0.338439\pi\)
\(788\) −5.23607 −0.186527
\(789\) −20.4721 −0.728827
\(790\) 0.854102 0.0303876
\(791\) −10.0000 −0.355559
\(792\) 1.85410 0.0658826
\(793\) −38.2918 −1.35978
\(794\) −28.5623 −1.01364
\(795\) 21.7984 0.773109
\(796\) 1.56231 0.0553745
\(797\) 44.8328 1.58806 0.794030 0.607879i \(-0.207979\pi\)
0.794030 + 0.607879i \(0.207979\pi\)
\(798\) 0 0
\(799\) −8.29180 −0.293343
\(800\) 1.85410 0.0655524
\(801\) −1.20163 −0.0424574
\(802\) 25.4164 0.897485
\(803\) 0 0
\(804\) 8.47214 0.298789
\(805\) −23.4164 −0.825320
\(806\) −26.8328 −0.945146
\(807\) −7.41641 −0.261070
\(808\) 2.94427 0.103579
\(809\) −16.0344 −0.563741 −0.281870 0.959452i \(-0.590955\pi\)
−0.281870 + 0.959452i \(0.590955\pi\)
\(810\) 20.1803 0.709065
\(811\) −2.02129 −0.0709770 −0.0354885 0.999370i \(-0.511299\pi\)
−0.0354885 + 0.999370i \(0.511299\pi\)
\(812\) −0.145898 −0.00512002
\(813\) −3.94427 −0.138332
\(814\) 38.1246 1.33627
\(815\) 53.0689 1.85892
\(816\) −1.23607 −0.0432710
\(817\) 0 0
\(818\) 1.79837 0.0628787
\(819\) −1.70820 −0.0596895
\(820\) −25.0344 −0.874241
\(821\) 6.87539 0.239953 0.119976 0.992777i \(-0.461718\pi\)
0.119976 + 0.992777i \(0.461718\pi\)
\(822\) −5.47214 −0.190863
\(823\) −2.29180 −0.0798870 −0.0399435 0.999202i \(-0.512718\pi\)
−0.0399435 + 0.999202i \(0.512718\pi\)
\(824\) −8.94427 −0.311588
\(825\) 14.5623 0.506994
\(826\) 11.5623 0.402304
\(827\) 43.9574 1.52855 0.764275 0.644891i \(-0.223097\pi\)
0.764275 + 0.644891i \(0.223097\pi\)
\(828\) −3.41641 −0.118728
\(829\) 35.1246 1.21993 0.609964 0.792429i \(-0.291184\pi\)
0.609964 + 0.792429i \(0.291184\pi\)
\(830\) −29.4164 −1.02106
\(831\) 41.1246 1.42660
\(832\) 4.47214 0.155043
\(833\) 0.763932 0.0264687
\(834\) −9.70820 −0.336168
\(835\) −26.1803 −0.906008
\(836\) 0 0
\(837\) −32.8328 −1.13487
\(838\) −12.6525 −0.437073
\(839\) 4.83282 0.166847 0.0834237 0.996514i \(-0.473415\pi\)
0.0834237 + 0.996514i \(0.473415\pi\)
\(840\) 4.23607 0.146158
\(841\) −28.9787 −0.999266
\(842\) −19.5279 −0.672975
\(843\) −37.8885 −1.30495
\(844\) −17.2361 −0.593290
\(845\) −18.3262 −0.630442
\(846\) 4.14590 0.142539
\(847\) 12.5623 0.431646
\(848\) 5.14590 0.176711
\(849\) −12.9443 −0.444246
\(850\) 1.41641 0.0485824
\(851\) −70.2492 −2.40811
\(852\) 13.8541 0.474634
\(853\) −23.2705 −0.796767 −0.398384 0.917219i \(-0.630429\pi\)
−0.398384 + 0.917219i \(0.630429\pi\)
\(854\) −8.56231 −0.292996
\(855\) 0 0
\(856\) −19.7082 −0.673613
\(857\) −32.2492 −1.10161 −0.550806 0.834633i \(-0.685680\pi\)
−0.550806 + 0.834633i \(0.685680\pi\)
\(858\) 35.1246 1.19913
\(859\) −34.0000 −1.16007 −0.580033 0.814593i \(-0.696960\pi\)
−0.580033 + 0.814593i \(0.696960\pi\)
\(860\) −10.0902 −0.344072
\(861\) −15.4721 −0.527289
\(862\) −22.8541 −0.778414
\(863\) −48.9787 −1.66725 −0.833627 0.552327i \(-0.813740\pi\)
−0.833627 + 0.552327i \(0.813740\pi\)
\(864\) 5.47214 0.186166
\(865\) 47.1246 1.60228
\(866\) 3.38197 0.114924
\(867\) 26.5623 0.902103
\(868\) −6.00000 −0.203653
\(869\) 1.58359 0.0537197
\(870\) −0.618034 −0.0209533
\(871\) −23.4164 −0.793435
\(872\) −10.8541 −0.367566
\(873\) 2.72949 0.0923792
\(874\) 0 0
\(875\) 8.23607 0.278430
\(876\) 0 0
\(877\) 47.3951 1.60042 0.800210 0.599720i \(-0.204721\pi\)
0.800210 + 0.599720i \(0.204721\pi\)
\(878\) 2.18034 0.0735829
\(879\) 50.8328 1.71455
\(880\) 12.7082 0.428393
\(881\) 9.81966 0.330833 0.165416 0.986224i \(-0.447103\pi\)
0.165416 + 0.986224i \(0.447103\pi\)
\(882\) −0.381966 −0.0128615
\(883\) −27.6869 −0.931739 −0.465869 0.884853i \(-0.654258\pi\)
−0.465869 + 0.884853i \(0.654258\pi\)
\(884\) 3.41641 0.114906
\(885\) 48.9787 1.64640
\(886\) 1.09017 0.0366250
\(887\) −2.83282 −0.0951166 −0.0475583 0.998868i \(-0.515144\pi\)
−0.0475583 + 0.998868i \(0.515144\pi\)
\(888\) 12.7082 0.426459
\(889\) −22.0902 −0.740881
\(890\) −8.23607 −0.276074
\(891\) 37.4164 1.25350
\(892\) 5.23607 0.175317
\(893\) 0 0
\(894\) 27.8885 0.932732
\(895\) 46.3607 1.54967
\(896\) 1.00000 0.0334077
\(897\) −64.7214 −2.16098
\(898\) 0.291796 0.00973736
\(899\) 0.875388 0.0291958
\(900\) −0.708204 −0.0236068
\(901\) 3.93112 0.130964
\(902\) −46.4164 −1.54550
\(903\) −6.23607 −0.207523
\(904\) −10.0000 −0.332595
\(905\) −19.4164 −0.645423
\(906\) 4.94427 0.164262
\(907\) −48.0000 −1.59381 −0.796907 0.604102i \(-0.793532\pi\)
−0.796907 + 0.604102i \(0.793532\pi\)
\(908\) −19.4164 −0.644356
\(909\) −1.12461 −0.0373010
\(910\) −11.7082 −0.388123
\(911\) 49.8541 1.65174 0.825870 0.563861i \(-0.190684\pi\)
0.825870 + 0.563861i \(0.190684\pi\)
\(912\) 0 0
\(913\) −54.5410 −1.80504
\(914\) 31.1459 1.03021
\(915\) −36.2705 −1.19907
\(916\) 23.8541 0.788162
\(917\) −4.47214 −0.147683
\(918\) 4.18034 0.137972
\(919\) −12.2918 −0.405469 −0.202734 0.979234i \(-0.564983\pi\)
−0.202734 + 0.979234i \(0.564983\pi\)
\(920\) −23.4164 −0.772016
\(921\) 30.7082 1.01187
\(922\) −41.5066 −1.36695
\(923\) −38.2918 −1.26039
\(924\) 7.85410 0.258381
\(925\) −14.5623 −0.478806
\(926\) −12.0000 −0.394344
\(927\) 3.41641 0.112210
\(928\) −0.145898 −0.00478934
\(929\) 42.7639 1.40304 0.701520 0.712650i \(-0.252505\pi\)
0.701520 + 0.712650i \(0.252505\pi\)
\(930\) −25.4164 −0.833437
\(931\) 0 0
\(932\) 4.09017 0.133978
\(933\) 34.5066 1.12969
\(934\) −40.3607 −1.32064
\(935\) 9.70820 0.317492
\(936\) −1.70820 −0.0558344
\(937\) 13.7082 0.447828 0.223914 0.974609i \(-0.428117\pi\)
0.223914 + 0.974609i \(0.428117\pi\)
\(938\) −5.23607 −0.170964
\(939\) −16.1803 −0.528025
\(940\) 28.4164 0.926841
\(941\) 33.7082 1.09886 0.549428 0.835541i \(-0.314846\pi\)
0.549428 + 0.835541i \(0.314846\pi\)
\(942\) −4.61803 −0.150464
\(943\) 85.5279 2.78517
\(944\) 11.5623 0.376321
\(945\) −14.3262 −0.466033
\(946\) −18.7082 −0.608256
\(947\) −43.1459 −1.40205 −0.701027 0.713135i \(-0.747275\pi\)
−0.701027 + 0.713135i \(0.747275\pi\)
\(948\) 0.527864 0.0171442
\(949\) 0 0
\(950\) 0 0
\(951\) −41.5623 −1.34775
\(952\) 0.763932 0.0247592
\(953\) −2.29180 −0.0742386 −0.0371193 0.999311i \(-0.511818\pi\)
−0.0371193 + 0.999311i \(0.511818\pi\)
\(954\) −1.96556 −0.0636373
\(955\) −39.1246 −1.26604
\(956\) 14.1803 0.458625
\(957\) −1.14590 −0.0370416
\(958\) 36.3262 1.17365
\(959\) 3.38197 0.109209
\(960\) 4.23607 0.136719
\(961\) 5.00000 0.161290
\(962\) −35.1246 −1.13246
\(963\) 7.52786 0.242582
\(964\) −17.5623 −0.565644
\(965\) −54.8328 −1.76513
\(966\) −14.4721 −0.465633
\(967\) 54.2492 1.74454 0.872269 0.489027i \(-0.162648\pi\)
0.872269 + 0.489027i \(0.162648\pi\)
\(968\) 12.5623 0.403768
\(969\) 0 0
\(970\) 18.7082 0.600684
\(971\) −11.7295 −0.376417 −0.188209 0.982129i \(-0.560268\pi\)
−0.188209 + 0.982129i \(0.560268\pi\)
\(972\) −3.94427 −0.126513
\(973\) 6.00000 0.192351
\(974\) −16.9098 −0.541826
\(975\) −13.4164 −0.429669
\(976\) −8.56231 −0.274073
\(977\) −35.7082 −1.14241 −0.571203 0.820809i \(-0.693523\pi\)
−0.571203 + 0.820809i \(0.693523\pi\)
\(978\) 32.7984 1.04878
\(979\) −15.2705 −0.488048
\(980\) −2.61803 −0.0836300
\(981\) 4.14590 0.132368
\(982\) −40.3607 −1.28796
\(983\) −19.1246 −0.609980 −0.304990 0.952355i \(-0.598653\pi\)
−0.304990 + 0.952355i \(0.598653\pi\)
\(984\) −15.4721 −0.493234
\(985\) 13.7082 0.436780
\(986\) −0.111456 −0.00354949
\(987\) 17.5623 0.559014
\(988\) 0 0
\(989\) 34.4721 1.09615
\(990\) −4.85410 −0.154273
\(991\) −47.3951 −1.50556 −0.752778 0.658275i \(-0.771287\pi\)
−0.752778 + 0.658275i \(0.771287\pi\)
\(992\) −6.00000 −0.190500
\(993\) −6.00000 −0.190404
\(994\) −8.56231 −0.271580
\(995\) −4.09017 −0.129667
\(996\) −18.1803 −0.576066
\(997\) −51.8541 −1.64224 −0.821118 0.570759i \(-0.806649\pi\)
−0.821118 + 0.570759i \(0.806649\pi\)
\(998\) −18.6869 −0.591524
\(999\) −42.9787 −1.35979
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 5054.2.a.l.1.1 yes 2
19.18 odd 2 5054.2.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.g.1.2 2 19.18 odd 2
5054.2.a.l.1.1 yes 2 1.1 even 1 trivial