Properties

Label 435.2.a.j.1.2
Level $435$
Weight $2$
Character 435.1
Self dual yes
Analytic conductor $3.473$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [435,2,Mod(1,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, 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("435.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.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: \(3.47349248793\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.820249\) of defining polynomial
Character \(\chi\) \(=\) 435.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.82025 q^{2} -1.00000 q^{3} +1.31331 q^{4} -1.00000 q^{5} +1.82025 q^{6} -0.729126 q^{7} +1.24995 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.82025 q^{2} -1.00000 q^{3} +1.31331 q^{4} -1.00000 q^{5} +1.82025 q^{6} -0.729126 q^{7} +1.24995 q^{8} +1.00000 q^{9} +1.82025 q^{10} +0.729126 q^{11} -1.31331 q^{12} +3.38351 q^{13} +1.32719 q^{14} +1.00000 q^{15} -4.90184 q^{16} -5.74301 q^{17} -1.82025 q^{18} +6.11263 q^{19} -1.31331 q^{20} +0.729126 q^{21} -1.32719 q^{22} -9.48602 q^{23} -1.24995 q^{24} +1.00000 q^{25} -6.15883 q^{26} -1.00000 q^{27} -0.957567 q^{28} -1.00000 q^{29} -1.82025 q^{30} +5.48602 q^{31} +6.42266 q^{32} -0.729126 q^{33} +10.4537 q^{34} +0.729126 q^{35} +1.31331 q^{36} -10.2949 q^{37} -11.1265 q^{38} -3.38351 q^{39} -1.24995 q^{40} -11.3088 q^{41} -1.32719 q^{42} -10.1404 q^{43} +0.957567 q^{44} -1.00000 q^{45} +17.2669 q^{46} -1.89749 q^{47} +4.90184 q^{48} -6.46838 q^{49} -1.82025 q^{50} +5.74301 q^{51} +4.44359 q^{52} +8.14040 q^{53} +1.82025 q^{54} -0.729126 q^{55} -0.911372 q^{56} -6.11263 q^{57} +1.82025 q^{58} +8.68215 q^{59} +1.31331 q^{60} -15.5709 q^{61} -9.98592 q^{62} -0.729126 q^{63} -1.88717 q^{64} -3.38351 q^{65} +1.32719 q^{66} -2.55187 q^{67} -7.54234 q^{68} +9.48602 q^{69} -1.32719 q^{70} -4.83164 q^{71} +1.24995 q^{72} +6.29488 q^{73} +18.7392 q^{74} -1.00000 q^{75} +8.02777 q^{76} -0.531625 q^{77} +6.15883 q^{78} -5.39363 q^{79} +4.90184 q^{80} +1.00000 q^{81} +20.5848 q^{82} +0.0848668 q^{83} +0.957567 q^{84} +5.74301 q^{85} +18.4581 q^{86} +1.00000 q^{87} +0.911372 q^{88} +4.63674 q^{89} +1.82025 q^{90} -2.46700 q^{91} -12.4581 q^{92} -5.48602 q^{93} +3.45390 q^{94} -6.11263 q^{95} -6.42266 q^{96} +1.30377 q^{97} +11.7741 q^{98} +0.729126 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 4 q^{3} + 5 q^{4} - 4 q^{5} + 3 q^{6} + 2 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} - 4 q^{3} + 5 q^{4} - 4 q^{5} + 3 q^{6} + 2 q^{7} - 12 q^{8} + 4 q^{9} + 3 q^{10} - 2 q^{11} - 5 q^{12} - 8 q^{13} - 3 q^{14} + 4 q^{15} + 11 q^{16} - 10 q^{17} - 3 q^{18} - 2 q^{19} - 5 q^{20} - 2 q^{21} + 3 q^{22} - 12 q^{23} + 12 q^{24} + 4 q^{25} - 7 q^{26} - 4 q^{27} - 9 q^{28} - 4 q^{29} - 3 q^{30} - 4 q^{31} - 17 q^{32} + 2 q^{33} - q^{34} - 2 q^{35} + 5 q^{36} - 16 q^{37} - 10 q^{38} + 8 q^{39} + 12 q^{40} - 12 q^{41} + 3 q^{42} + 2 q^{43} + 9 q^{44} - 4 q^{45} - 8 q^{46} - 12 q^{47} - 11 q^{48} + 6 q^{49} - 3 q^{50} + 10 q^{51} - 3 q^{52} - 10 q^{53} + 3 q^{54} + 2 q^{55} + 2 q^{57} + 3 q^{58} + 2 q^{59} + 5 q^{60} - 26 q^{61} + 20 q^{62} + 2 q^{63} + 34 q^{64} + 8 q^{65} - 3 q^{66} + 2 q^{67} + 9 q^{68} + 12 q^{69} + 3 q^{70} - 10 q^{71} - 12 q^{72} + 48 q^{74} - 4 q^{75} + 16 q^{76} - 34 q^{77} + 7 q^{78} + 22 q^{79} - 11 q^{80} + 4 q^{81} + 38 q^{82} - 10 q^{83} + 9 q^{84} + 10 q^{85} - 4 q^{86} + 4 q^{87} - 4 q^{89} + 3 q^{90} - 8 q^{91} + 28 q^{92} + 4 q^{93} + 39 q^{94} + 2 q^{95} + 17 q^{96} - 22 q^{97} + 34 q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.82025 −1.28711 −0.643555 0.765400i \(-0.722541\pi\)
−0.643555 + 0.765400i \(0.722541\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.31331 0.656654
\(5\) −1.00000 −0.447214
\(6\) 1.82025 0.743114
\(7\) −0.729126 −0.275584 −0.137792 0.990461i \(-0.544001\pi\)
−0.137792 + 0.990461i \(0.544001\pi\)
\(8\) 1.24995 0.441925
\(9\) 1.00000 0.333333
\(10\) 1.82025 0.575613
\(11\) 0.729126 0.219840 0.109920 0.993940i \(-0.464941\pi\)
0.109920 + 0.993940i \(0.464941\pi\)
\(12\) −1.31331 −0.379119
\(13\) 3.38351 0.938416 0.469208 0.883088i \(-0.344539\pi\)
0.469208 + 0.883088i \(0.344539\pi\)
\(14\) 1.32719 0.354707
\(15\) 1.00000 0.258199
\(16\) −4.90184 −1.22546
\(17\) −5.74301 −1.39288 −0.696442 0.717613i \(-0.745235\pi\)
−0.696442 + 0.717613i \(0.745235\pi\)
\(18\) −1.82025 −0.429037
\(19\) 6.11263 1.40233 0.701167 0.712997i \(-0.252663\pi\)
0.701167 + 0.712997i \(0.252663\pi\)
\(20\) −1.31331 −0.293664
\(21\) 0.729126 0.159108
\(22\) −1.32719 −0.282958
\(23\) −9.48602 −1.97797 −0.988986 0.148010i \(-0.952713\pi\)
−0.988986 + 0.148010i \(0.952713\pi\)
\(24\) −1.24995 −0.255145
\(25\) 1.00000 0.200000
\(26\) −6.15883 −1.20785
\(27\) −1.00000 −0.192450
\(28\) −0.957567 −0.180963
\(29\) −1.00000 −0.185695
\(30\) −1.82025 −0.332331
\(31\) 5.48602 0.985318 0.492659 0.870222i \(-0.336025\pi\)
0.492659 + 0.870222i \(0.336025\pi\)
\(32\) 6.42266 1.13538
\(33\) −0.729126 −0.126925
\(34\) 10.4537 1.79280
\(35\) 0.729126 0.123245
\(36\) 1.31331 0.218885
\(37\) −10.2949 −1.69247 −0.846234 0.532811i \(-0.821135\pi\)
−0.846234 + 0.532811i \(0.821135\pi\)
\(38\) −11.1265 −1.80496
\(39\) −3.38351 −0.541795
\(40\) −1.24995 −0.197635
\(41\) −11.3088 −1.76613 −0.883066 0.469249i \(-0.844525\pi\)
−0.883066 + 0.469249i \(0.844525\pi\)
\(42\) −1.32719 −0.204790
\(43\) −10.1404 −1.54640 −0.773198 0.634164i \(-0.781344\pi\)
−0.773198 + 0.634164i \(0.781344\pi\)
\(44\) 0.957567 0.144359
\(45\) −1.00000 −0.149071
\(46\) 17.2669 2.54587
\(47\) −1.89749 −0.276777 −0.138389 0.990378i \(-0.544192\pi\)
−0.138389 + 0.990378i \(0.544192\pi\)
\(48\) 4.90184 0.707519
\(49\) −6.46838 −0.924054
\(50\) −1.82025 −0.257422
\(51\) 5.74301 0.804182
\(52\) 4.44359 0.616215
\(53\) 8.14040 1.11817 0.559085 0.829110i \(-0.311152\pi\)
0.559085 + 0.829110i \(0.311152\pi\)
\(54\) 1.82025 0.247705
\(55\) −0.729126 −0.0983153
\(56\) −0.911372 −0.121787
\(57\) −6.11263 −0.809638
\(58\) 1.82025 0.239010
\(59\) 8.68215 1.13032 0.565160 0.824981i \(-0.308814\pi\)
0.565160 + 0.824981i \(0.308814\pi\)
\(60\) 1.31331 0.169547
\(61\) −15.5709 −1.99365 −0.996824 0.0796378i \(-0.974624\pi\)
−0.996824 + 0.0796378i \(0.974624\pi\)
\(62\) −9.98592 −1.26821
\(63\) −0.729126 −0.0918612
\(64\) −1.88717 −0.235897
\(65\) −3.38351 −0.419673
\(66\) 1.32719 0.163366
\(67\) −2.55187 −0.311761 −0.155880 0.987776i \(-0.549821\pi\)
−0.155880 + 0.987776i \(0.549821\pi\)
\(68\) −7.54234 −0.914643
\(69\) 9.48602 1.14198
\(70\) −1.32719 −0.158630
\(71\) −4.83164 −0.573410 −0.286705 0.958019i \(-0.592560\pi\)
−0.286705 + 0.958019i \(0.592560\pi\)
\(72\) 1.24995 0.147308
\(73\) 6.29488 0.736760 0.368380 0.929675i \(-0.379913\pi\)
0.368380 + 0.929675i \(0.379913\pi\)
\(74\) 18.7392 2.17839
\(75\) −1.00000 −0.115470
\(76\) 8.02777 0.920848
\(77\) −0.531625 −0.0605843
\(78\) 6.15883 0.697350
\(79\) −5.39363 −0.606831 −0.303415 0.952858i \(-0.598127\pi\)
−0.303415 + 0.952858i \(0.598127\pi\)
\(80\) 4.90184 0.548042
\(81\) 1.00000 0.111111
\(82\) 20.5848 2.27321
\(83\) 0.0848668 0.00931534 0.00465767 0.999989i \(-0.498517\pi\)
0.00465767 + 0.999989i \(0.498517\pi\)
\(84\) 0.957567 0.104479
\(85\) 5.74301 0.622917
\(86\) 18.4581 1.99038
\(87\) 1.00000 0.107211
\(88\) 0.911372 0.0971526
\(89\) 4.63674 0.491493 0.245747 0.969334i \(-0.420967\pi\)
0.245747 + 0.969334i \(0.420967\pi\)
\(90\) 1.82025 0.191871
\(91\) −2.46700 −0.258612
\(92\) −12.4581 −1.29884
\(93\) −5.48602 −0.568874
\(94\) 3.45390 0.356243
\(95\) −6.11263 −0.627143
\(96\) −6.42266 −0.655510
\(97\) 1.30377 0.132378 0.0661891 0.997807i \(-0.478916\pi\)
0.0661891 + 0.997807i \(0.478916\pi\)
\(98\) 11.7741 1.18936
\(99\) 0.729126 0.0732799
\(100\) 1.31331 0.131331
\(101\) −5.35574 −0.532916 −0.266458 0.963847i \(-0.585853\pi\)
−0.266458 + 0.963847i \(0.585853\pi\)
\(102\) −10.4537 −1.03507
\(103\) 16.7670 1.65210 0.826052 0.563594i \(-0.190582\pi\)
0.826052 + 0.563594i \(0.190582\pi\)
\(104\) 4.22922 0.414709
\(105\) −0.729126 −0.0711554
\(106\) −14.8176 −1.43921
\(107\) 7.30876 0.706565 0.353282 0.935517i \(-0.385065\pi\)
0.353282 + 0.935517i \(0.385065\pi\)
\(108\) −1.31331 −0.126373
\(109\) −8.21515 −0.786868 −0.393434 0.919353i \(-0.628713\pi\)
−0.393434 + 0.919353i \(0.628713\pi\)
\(110\) 1.32719 0.126543
\(111\) 10.2949 0.977147
\(112\) 3.57406 0.337717
\(113\) 8.51003 0.800556 0.400278 0.916394i \(-0.368914\pi\)
0.400278 + 0.916394i \(0.368914\pi\)
\(114\) 11.1265 1.04209
\(115\) 9.48602 0.884576
\(116\) −1.31331 −0.121938
\(117\) 3.38351 0.312805
\(118\) −15.8037 −1.45485
\(119\) 4.18738 0.383856
\(120\) 1.24995 0.114104
\(121\) −10.4684 −0.951670
\(122\) 28.3429 2.56605
\(123\) 11.3088 1.01968
\(124\) 7.20483 0.647013
\(125\) −1.00000 −0.0894427
\(126\) 1.32719 0.118236
\(127\) 4.51398 0.400551 0.200275 0.979740i \(-0.435816\pi\)
0.200275 + 0.979740i \(0.435816\pi\)
\(128\) −9.41020 −0.831752
\(129\) 10.1404 0.892813
\(130\) 6.15883 0.540165
\(131\) −14.7771 −1.29108 −0.645542 0.763724i \(-0.723369\pi\)
−0.645542 + 0.763724i \(0.723369\pi\)
\(132\) −0.957567 −0.0833455
\(133\) −4.45688 −0.386461
\(134\) 4.64504 0.401270
\(135\) 1.00000 0.0860663
\(136\) −7.17849 −0.615550
\(137\) −5.19114 −0.443509 −0.221754 0.975103i \(-0.571178\pi\)
−0.221754 + 0.975103i \(0.571178\pi\)
\(138\) −17.2669 −1.46986
\(139\) −1.15824 −0.0982406 −0.0491203 0.998793i \(-0.515642\pi\)
−0.0491203 + 0.998793i \(0.515642\pi\)
\(140\) 0.957567 0.0809291
\(141\) 1.89749 0.159797
\(142\) 8.79478 0.738042
\(143\) 2.46700 0.206301
\(144\) −4.90184 −0.408487
\(145\) 1.00000 0.0830455
\(146\) −11.4583 −0.948292
\(147\) 6.46838 0.533503
\(148\) −13.5203 −1.11137
\(149\) 6.68215 0.547423 0.273712 0.961812i \(-0.411749\pi\)
0.273712 + 0.961812i \(0.411749\pi\)
\(150\) 1.82025 0.148623
\(151\) 20.2253 1.64591 0.822955 0.568107i \(-0.192324\pi\)
0.822955 + 0.568107i \(0.192324\pi\)
\(152\) 7.64050 0.619726
\(153\) −5.74301 −0.464295
\(154\) 0.967690 0.0779787
\(155\) −5.48602 −0.440648
\(156\) −4.44359 −0.355772
\(157\) −23.8658 −1.90470 −0.952348 0.305014i \(-0.901339\pi\)
−0.952348 + 0.305014i \(0.901339\pi\)
\(158\) 9.81775 0.781059
\(159\) −8.14040 −0.645576
\(160\) −6.42266 −0.507756
\(161\) 6.91650 0.545097
\(162\) −1.82025 −0.143012
\(163\) −1.93538 −0.151591 −0.0757953 0.997123i \(-0.524150\pi\)
−0.0757953 + 0.997123i \(0.524150\pi\)
\(164\) −14.8519 −1.15974
\(165\) 0.729126 0.0567624
\(166\) −0.154479 −0.0119899
\(167\) −12.4025 −0.959736 −0.479868 0.877341i \(-0.659315\pi\)
−0.479868 + 0.877341i \(0.659315\pi\)
\(168\) 0.911372 0.0703139
\(169\) −1.55187 −0.119375
\(170\) −10.4537 −0.801763
\(171\) 6.11263 0.467445
\(172\) −13.3175 −1.01545
\(173\) −21.7113 −1.65068 −0.825339 0.564637i \(-0.809016\pi\)
−0.825339 + 0.564637i \(0.809016\pi\)
\(174\) −1.82025 −0.137993
\(175\) −0.729126 −0.0551167
\(176\) −3.57406 −0.269405
\(177\) −8.68215 −0.652590
\(178\) −8.44002 −0.632606
\(179\) 6.08487 0.454804 0.227402 0.973801i \(-0.426977\pi\)
0.227402 + 0.973801i \(0.426977\pi\)
\(180\) −1.31331 −0.0978881
\(181\) 6.77714 0.503741 0.251870 0.967761i \(-0.418954\pi\)
0.251870 + 0.967761i \(0.418954\pi\)
\(182\) 4.49056 0.332863
\(183\) 15.5709 1.15103
\(184\) −11.8571 −0.874115
\(185\) 10.2949 0.756895
\(186\) 9.98592 0.732203
\(187\) −4.18738 −0.306211
\(188\) −2.49199 −0.181747
\(189\) 0.729126 0.0530361
\(190\) 11.1265 0.807202
\(191\) −18.9645 −1.37222 −0.686112 0.727496i \(-0.740684\pi\)
−0.686112 + 0.727496i \(0.740684\pi\)
\(192\) 1.88717 0.136195
\(193\) 3.69760 0.266159 0.133079 0.991105i \(-0.457513\pi\)
0.133079 + 0.991105i \(0.457513\pi\)
\(194\) −2.37319 −0.170385
\(195\) 3.38351 0.242298
\(196\) −8.49496 −0.606783
\(197\) 20.4783 1.45902 0.729509 0.683971i \(-0.239748\pi\)
0.729509 + 0.683971i \(0.239748\pi\)
\(198\) −1.32719 −0.0943194
\(199\) −10.8418 −0.768552 −0.384276 0.923218i \(-0.625549\pi\)
−0.384276 + 0.923218i \(0.625549\pi\)
\(200\) 1.24995 0.0883849
\(201\) 2.55187 0.179995
\(202\) 9.74878 0.685922
\(203\) 0.729126 0.0511746
\(204\) 7.54234 0.528069
\(205\) 11.3088 0.789838
\(206\) −30.5201 −2.12644
\(207\) −9.48602 −0.659324
\(208\) −16.5854 −1.14999
\(209\) 4.45688 0.308289
\(210\) 1.32719 0.0915849
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 10.6908 0.734251
\(213\) 4.83164 0.331058
\(214\) −13.3038 −0.909427
\(215\) 10.1404 0.691570
\(216\) −1.24995 −0.0850484
\(217\) −4.00000 −0.271538
\(218\) 14.9536 1.01279
\(219\) −6.29488 −0.425369
\(220\) −0.957567 −0.0645591
\(221\) −19.4315 −1.30711
\(222\) −18.7392 −1.25770
\(223\) −4.21515 −0.282267 −0.141134 0.989991i \(-0.545075\pi\)
−0.141134 + 0.989991i \(0.545075\pi\)
\(224\) −4.68293 −0.312892
\(225\) 1.00000 0.0666667
\(226\) −15.4904 −1.03040
\(227\) −15.4214 −1.02355 −0.511777 0.859118i \(-0.671013\pi\)
−0.511777 + 0.859118i \(0.671013\pi\)
\(228\) −8.02777 −0.531652
\(229\) −24.2530 −1.60269 −0.801343 0.598205i \(-0.795881\pi\)
−0.801343 + 0.598205i \(0.795881\pi\)
\(230\) −17.2669 −1.13855
\(231\) 0.531625 0.0349783
\(232\) −1.24995 −0.0820634
\(233\) −5.33653 −0.349608 −0.174804 0.984603i \(-0.555929\pi\)
−0.174804 + 0.984603i \(0.555929\pi\)
\(234\) −6.15883 −0.402615
\(235\) 1.89749 0.123779
\(236\) 11.4023 0.742229
\(237\) 5.39363 0.350354
\(238\) −7.62207 −0.494066
\(239\) 16.0202 1.03626 0.518132 0.855301i \(-0.326628\pi\)
0.518132 + 0.855301i \(0.326628\pi\)
\(240\) −4.90184 −0.316412
\(241\) −6.60741 −0.425620 −0.212810 0.977094i \(-0.568262\pi\)
−0.212810 + 0.977094i \(0.568262\pi\)
\(242\) 19.0551 1.22491
\(243\) −1.00000 −0.0641500
\(244\) −20.4494 −1.30914
\(245\) 6.46838 0.413249
\(246\) −20.5848 −1.31244
\(247\) 20.6821 1.31597
\(248\) 6.85726 0.435436
\(249\) −0.0848668 −0.00537821
\(250\) 1.82025 0.115123
\(251\) 25.6834 1.62112 0.810560 0.585655i \(-0.199163\pi\)
0.810560 + 0.585655i \(0.199163\pi\)
\(252\) −0.957567 −0.0603210
\(253\) −6.91650 −0.434837
\(254\) −8.21657 −0.515553
\(255\) −5.74301 −0.359641
\(256\) 20.9033 1.30645
\(257\) −7.93538 −0.494995 −0.247498 0.968888i \(-0.579608\pi\)
−0.247498 + 0.968888i \(0.579608\pi\)
\(258\) −18.4581 −1.14915
\(259\) 7.50627 0.466417
\(260\) −4.44359 −0.275580
\(261\) −1.00000 −0.0618984
\(262\) 26.8981 1.66177
\(263\) 23.5618 1.45288 0.726441 0.687228i \(-0.241173\pi\)
0.726441 + 0.687228i \(0.241173\pi\)
\(264\) −0.911372 −0.0560911
\(265\) −8.14040 −0.500061
\(266\) 8.11263 0.497418
\(267\) −4.63674 −0.283764
\(268\) −3.35139 −0.204719
\(269\) −10.6645 −0.650226 −0.325113 0.945675i \(-0.605402\pi\)
−0.325113 + 0.945675i \(0.605402\pi\)
\(270\) −1.82025 −0.110777
\(271\) −3.30876 −0.200993 −0.100497 0.994937i \(-0.532043\pi\)
−0.100497 + 0.994937i \(0.532043\pi\)
\(272\) 28.1513 1.70692
\(273\) 2.46700 0.149310
\(274\) 9.44917 0.570845
\(275\) 0.729126 0.0439680
\(276\) 12.4581 0.749887
\(277\) −0.0469761 −0.00282252 −0.00141126 0.999999i \(-0.500449\pi\)
−0.00141126 + 0.999999i \(0.500449\pi\)
\(278\) 2.10828 0.126447
\(279\) 5.48602 0.328439
\(280\) 0.911372 0.0544649
\(281\) 29.8037 1.77794 0.888969 0.457967i \(-0.151422\pi\)
0.888969 + 0.457967i \(0.151422\pi\)
\(282\) −3.45390 −0.205677
\(283\) 5.79498 0.344476 0.172238 0.985055i \(-0.444900\pi\)
0.172238 + 0.985055i \(0.444900\pi\)
\(284\) −6.34542 −0.376532
\(285\) 6.11263 0.362081
\(286\) −4.49056 −0.265533
\(287\) 8.24552 0.486717
\(288\) 6.42266 0.378459
\(289\) 15.9822 0.940127
\(290\) −1.82025 −0.106889
\(291\) −1.30377 −0.0764286
\(292\) 8.26711 0.483796
\(293\) 2.99624 0.175042 0.0875211 0.996163i \(-0.472105\pi\)
0.0875211 + 0.996163i \(0.472105\pi\)
\(294\) −11.7741 −0.686677
\(295\) −8.68215 −0.505494
\(296\) −12.8681 −0.747943
\(297\) −0.729126 −0.0423082
\(298\) −12.1632 −0.704594
\(299\) −32.0960 −1.85616
\(300\) −1.31331 −0.0758238
\(301\) 7.39363 0.426162
\(302\) −36.8150 −2.11847
\(303\) 5.35574 0.307679
\(304\) −29.9631 −1.71850
\(305\) 15.5709 0.891586
\(306\) 10.4537 0.597599
\(307\) 4.05554 0.231462 0.115731 0.993281i \(-0.463079\pi\)
0.115731 + 0.993281i \(0.463079\pi\)
\(308\) −0.698187 −0.0397829
\(309\) −16.7670 −0.953842
\(310\) 9.98592 0.567162
\(311\) 13.6734 0.775347 0.387674 0.921797i \(-0.373279\pi\)
0.387674 + 0.921797i \(0.373279\pi\)
\(312\) −4.22922 −0.239433
\(313\) −4.25200 −0.240337 −0.120169 0.992753i \(-0.538344\pi\)
−0.120169 + 0.992753i \(0.538344\pi\)
\(314\) 43.4416 2.45155
\(315\) 0.729126 0.0410816
\(316\) −7.08350 −0.398478
\(317\) −4.28476 −0.240656 −0.120328 0.992734i \(-0.538395\pi\)
−0.120328 + 0.992734i \(0.538395\pi\)
\(318\) 14.8176 0.830928
\(319\) −0.729126 −0.0408232
\(320\) 1.88717 0.105496
\(321\) −7.30876 −0.407935
\(322\) −12.5898 −0.701600
\(323\) −35.1049 −1.95329
\(324\) 1.31331 0.0729615
\(325\) 3.38351 0.187683
\(326\) 3.52287 0.195114
\(327\) 8.21515 0.454299
\(328\) −14.1354 −0.780497
\(329\) 1.38351 0.0762753
\(330\) −1.32719 −0.0730595
\(331\) 3.61772 0.198848 0.0994240 0.995045i \(-0.468300\pi\)
0.0994240 + 0.995045i \(0.468300\pi\)
\(332\) 0.111456 0.00611695
\(333\) −10.2949 −0.564156
\(334\) 22.5757 1.23529
\(335\) 2.55187 0.139424
\(336\) −3.57406 −0.194981
\(337\) 23.4442 1.27709 0.638543 0.769586i \(-0.279538\pi\)
0.638543 + 0.769586i \(0.279538\pi\)
\(338\) 2.82479 0.153648
\(339\) −8.51003 −0.462201
\(340\) 7.54234 0.409041
\(341\) 4.00000 0.216612
\(342\) −11.1265 −0.601653
\(343\) 9.82014 0.530238
\(344\) −12.6750 −0.683391
\(345\) −9.48602 −0.510710
\(346\) 39.5200 2.12461
\(347\) 20.8226 1.11781 0.558907 0.829231i \(-0.311221\pi\)
0.558907 + 0.829231i \(0.311221\pi\)
\(348\) 1.31331 0.0704007
\(349\) −18.0555 −0.966491 −0.483245 0.875485i \(-0.660542\pi\)
−0.483245 + 0.875485i \(0.660542\pi\)
\(350\) 1.32719 0.0709414
\(351\) −3.38351 −0.180598
\(352\) 4.68293 0.249601
\(353\) −14.3645 −0.764545 −0.382272 0.924050i \(-0.624858\pi\)
−0.382272 + 0.924050i \(0.624858\pi\)
\(354\) 15.8037 0.839956
\(355\) 4.83164 0.256437
\(356\) 6.08946 0.322741
\(357\) −4.18738 −0.221620
\(358\) −11.0760 −0.585383
\(359\) 6.20502 0.327489 0.163744 0.986503i \(-0.447643\pi\)
0.163744 + 0.986503i \(0.447643\pi\)
\(360\) −1.24995 −0.0658782
\(361\) 18.3643 0.966542
\(362\) −12.3361 −0.648370
\(363\) 10.4684 0.549447
\(364\) −3.23993 −0.169819
\(365\) −6.29488 −0.329489
\(366\) −28.3429 −1.48151
\(367\) 18.1975 0.949902 0.474951 0.880012i \(-0.342466\pi\)
0.474951 + 0.880012i \(0.342466\pi\)
\(368\) 46.4989 2.42392
\(369\) −11.3088 −0.588711
\(370\) −18.7392 −0.974207
\(371\) −5.93538 −0.308150
\(372\) −7.20483 −0.373553
\(373\) 34.8606 1.80501 0.902506 0.430677i \(-0.141725\pi\)
0.902506 + 0.430677i \(0.141725\pi\)
\(374\) 7.62207 0.394128
\(375\) 1.00000 0.0516398
\(376\) −2.37177 −0.122315
\(377\) −3.38351 −0.174260
\(378\) −1.32719 −0.0682633
\(379\) −6.41005 −0.329262 −0.164631 0.986355i \(-0.552643\pi\)
−0.164631 + 0.986355i \(0.552643\pi\)
\(380\) −8.02777 −0.411816
\(381\) −4.51398 −0.231258
\(382\) 34.5201 1.76620
\(383\) 20.7202 1.05875 0.529376 0.848387i \(-0.322426\pi\)
0.529376 + 0.848387i \(0.322426\pi\)
\(384\) 9.41020 0.480212
\(385\) 0.531625 0.0270941
\(386\) −6.73055 −0.342576
\(387\) −10.1404 −0.515466
\(388\) 1.71226 0.0869266
\(389\) −24.7581 −1.25528 −0.627642 0.778502i \(-0.715980\pi\)
−0.627642 + 0.778502i \(0.715980\pi\)
\(390\) −6.15883 −0.311864
\(391\) 54.4783 2.75509
\(392\) −8.08516 −0.408362
\(393\) 14.7771 0.745408
\(394\) −37.2756 −1.87792
\(395\) 5.39363 0.271383
\(396\) 0.957567 0.0481195
\(397\) −12.7468 −0.639742 −0.319871 0.947461i \(-0.603640\pi\)
−0.319871 + 0.947461i \(0.603640\pi\)
\(398\) 19.7347 0.989211
\(399\) 4.45688 0.223123
\(400\) −4.90184 −0.245092
\(401\) 28.1959 1.40804 0.704019 0.710181i \(-0.251387\pi\)
0.704019 + 0.710181i \(0.251387\pi\)
\(402\) −4.64504 −0.231674
\(403\) 18.5620 0.924639
\(404\) −7.03373 −0.349941
\(405\) −1.00000 −0.0496904
\(406\) −1.32719 −0.0658674
\(407\) −7.50627 −0.372072
\(408\) 7.17849 0.355388
\(409\) −24.0112 −1.18728 −0.593638 0.804732i \(-0.702309\pi\)
−0.593638 + 0.804732i \(0.702309\pi\)
\(410\) −20.5848 −1.01661
\(411\) 5.19114 0.256060
\(412\) 22.0202 1.08486
\(413\) −6.33038 −0.311498
\(414\) 17.2669 0.848623
\(415\) −0.0848668 −0.00416595
\(416\) 21.7311 1.06546
\(417\) 1.15824 0.0567192
\(418\) −8.11263 −0.396802
\(419\) 14.7670 0.721416 0.360708 0.932679i \(-0.382535\pi\)
0.360708 + 0.932679i \(0.382535\pi\)
\(420\) −0.957567 −0.0467245
\(421\) −12.7302 −0.620430 −0.310215 0.950666i \(-0.600401\pi\)
−0.310215 + 0.950666i \(0.600401\pi\)
\(422\) −3.64050 −0.177217
\(423\) −1.89749 −0.0922591
\(424\) 10.1751 0.494147
\(425\) −5.74301 −0.278577
\(426\) −8.79478 −0.426109
\(427\) 11.3531 0.549417
\(428\) 9.59865 0.463968
\(429\) −2.46700 −0.119108
\(430\) −18.4581 −0.890127
\(431\) 9.66327 0.465464 0.232732 0.972541i \(-0.425234\pi\)
0.232732 + 0.972541i \(0.425234\pi\)
\(432\) 4.90184 0.235840
\(433\) 13.1190 0.630459 0.315229 0.949016i \(-0.397919\pi\)
0.315229 + 0.949016i \(0.397919\pi\)
\(434\) 7.28100 0.349499
\(435\) −1.00000 −0.0479463
\(436\) −10.7890 −0.516700
\(437\) −57.9846 −2.77378
\(438\) 11.4583 0.547496
\(439\) −19.9253 −0.950981 −0.475490 0.879721i \(-0.657729\pi\)
−0.475490 + 0.879721i \(0.657729\pi\)
\(440\) −0.911372 −0.0434480
\(441\) −6.46838 −0.308018
\(442\) 35.3702 1.68239
\(443\) 1.94304 0.0923167 0.0461583 0.998934i \(-0.485302\pi\)
0.0461583 + 0.998934i \(0.485302\pi\)
\(444\) 13.5203 0.641647
\(445\) −4.63674 −0.219802
\(446\) 7.67262 0.363309
\(447\) −6.68215 −0.316055
\(448\) 1.37599 0.0650093
\(449\) 17.0215 0.803293 0.401647 0.915795i \(-0.368438\pi\)
0.401647 + 0.915795i \(0.368438\pi\)
\(450\) −1.82025 −0.0858074
\(451\) −8.24552 −0.388266
\(452\) 11.1763 0.525688
\(453\) −20.2253 −0.950266
\(454\) 28.0708 1.31743
\(455\) 2.46700 0.115655
\(456\) −7.64050 −0.357799
\(457\) 15.9633 0.746731 0.373366 0.927684i \(-0.378204\pi\)
0.373366 + 0.927684i \(0.378204\pi\)
\(458\) 44.1466 2.06283
\(459\) 5.74301 0.268061
\(460\) 12.4581 0.580860
\(461\) −17.6910 −0.823954 −0.411977 0.911194i \(-0.635162\pi\)
−0.411977 + 0.911194i \(0.635162\pi\)
\(462\) −0.967690 −0.0450210
\(463\) −2.60741 −0.121176 −0.0605882 0.998163i \(-0.519298\pi\)
−0.0605882 + 0.998163i \(0.519298\pi\)
\(464\) 4.90184 0.227562
\(465\) 5.48602 0.254408
\(466\) 9.71382 0.449984
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 4.44359 0.205405
\(469\) 1.86064 0.0859162
\(470\) −3.45390 −0.159317
\(471\) 23.8658 1.09968
\(472\) 10.8523 0.499516
\(473\) −7.39363 −0.339960
\(474\) −9.81775 −0.450944
\(475\) 6.11263 0.280467
\(476\) 5.49931 0.252061
\(477\) 8.14040 0.372723
\(478\) −29.1608 −1.33379
\(479\) 20.6833 0.945045 0.472523 0.881319i \(-0.343343\pi\)
0.472523 + 0.881319i \(0.343343\pi\)
\(480\) 6.42266 0.293153
\(481\) −34.8328 −1.58824
\(482\) 12.0271 0.547821
\(483\) −6.91650 −0.314712
\(484\) −13.7482 −0.624918
\(485\) −1.30377 −0.0592013
\(486\) 1.82025 0.0825682
\(487\) −32.1035 −1.45475 −0.727375 0.686240i \(-0.759260\pi\)
−0.727375 + 0.686240i \(0.759260\pi\)
\(488\) −19.4629 −0.881042
\(489\) 1.93538 0.0875209
\(490\) −11.7741 −0.531898
\(491\) −40.6933 −1.83646 −0.918232 0.396044i \(-0.870383\pi\)
−0.918232 + 0.396044i \(0.870383\pi\)
\(492\) 14.8519 0.669575
\(493\) 5.74301 0.258652
\(494\) −37.6467 −1.69380
\(495\) −0.729126 −0.0327718
\(496\) −26.8916 −1.20747
\(497\) 3.52287 0.158022
\(498\) 0.154479 0.00692236
\(499\) 10.8418 0.485344 0.242672 0.970108i \(-0.421976\pi\)
0.242672 + 0.970108i \(0.421976\pi\)
\(500\) −1.31331 −0.0587329
\(501\) 12.4025 0.554104
\(502\) −46.7502 −2.08656
\(503\) −2.47727 −0.110456 −0.0552280 0.998474i \(-0.517589\pi\)
−0.0552280 + 0.998474i \(0.517589\pi\)
\(504\) −0.911372 −0.0405958
\(505\) 5.35574 0.238327
\(506\) 12.5898 0.559683
\(507\) 1.55187 0.0689210
\(508\) 5.92824 0.263023
\(509\) −24.3101 −1.07753 −0.538764 0.842457i \(-0.681109\pi\)
−0.538764 + 0.842457i \(0.681109\pi\)
\(510\) 10.4537 0.462898
\(511\) −4.58976 −0.203039
\(512\) −19.2287 −0.849798
\(513\) −6.11263 −0.269879
\(514\) 14.4444 0.637114
\(515\) −16.7670 −0.738843
\(516\) 13.3175 0.586269
\(517\) −1.38351 −0.0608466
\(518\) −13.6633 −0.600330
\(519\) 21.7113 0.953020
\(520\) −4.22922 −0.185464
\(521\) 25.2997 1.10840 0.554200 0.832384i \(-0.313024\pi\)
0.554200 + 0.832384i \(0.313024\pi\)
\(522\) 1.82025 0.0796701
\(523\) 39.2277 1.71531 0.857653 0.514228i \(-0.171922\pi\)
0.857653 + 0.514228i \(0.171922\pi\)
\(524\) −19.4069 −0.847796
\(525\) 0.729126 0.0318217
\(526\) −42.8884 −1.87002
\(527\) −31.5063 −1.37243
\(528\) 3.57406 0.155541
\(529\) 66.9846 2.91237
\(530\) 14.8176 0.643634
\(531\) 8.68215 0.376773
\(532\) −5.85325 −0.253771
\(533\) −38.2633 −1.65737
\(534\) 8.44002 0.365235
\(535\) −7.30876 −0.315985
\(536\) −3.18972 −0.137775
\(537\) −6.08487 −0.262581
\(538\) 19.4121 0.836913
\(539\) −4.71626 −0.203144
\(540\) 1.31331 0.0565157
\(541\) 6.28080 0.270033 0.135016 0.990843i \(-0.456891\pi\)
0.135016 + 0.990843i \(0.456891\pi\)
\(542\) 6.02278 0.258700
\(543\) −6.77714 −0.290835
\(544\) −36.8854 −1.58145
\(545\) 8.21515 0.351898
\(546\) −4.49056 −0.192178
\(547\) −20.2809 −0.867151 −0.433575 0.901117i \(-0.642748\pi\)
−0.433575 + 0.901117i \(0.642748\pi\)
\(548\) −6.81756 −0.291232
\(549\) −15.5709 −0.664549
\(550\) −1.32719 −0.0565916
\(551\) −6.11263 −0.260407
\(552\) 11.8571 0.504670
\(553\) 3.93264 0.167233
\(554\) 0.0855081 0.00363289
\(555\) −10.2949 −0.436993
\(556\) −1.52112 −0.0645100
\(557\) 16.3252 0.691720 0.345860 0.938286i \(-0.387587\pi\)
0.345860 + 0.938286i \(0.387587\pi\)
\(558\) −9.98592 −0.422738
\(559\) −34.3101 −1.45116
\(560\) −3.57406 −0.151032
\(561\) 4.18738 0.176791
\(562\) −54.2501 −2.28840
\(563\) −12.8037 −0.539613 −0.269806 0.962915i \(-0.586960\pi\)
−0.269806 + 0.962915i \(0.586960\pi\)
\(564\) 2.49199 0.104932
\(565\) −8.51003 −0.358020
\(566\) −10.5483 −0.443378
\(567\) −0.729126 −0.0306204
\(568\) −6.03931 −0.253404
\(569\) 9.35574 0.392213 0.196107 0.980583i \(-0.437170\pi\)
0.196107 + 0.980583i \(0.437170\pi\)
\(570\) −11.1265 −0.466039
\(571\) −27.0125 −1.13044 −0.565220 0.824940i \(-0.691209\pi\)
−0.565220 + 0.824940i \(0.691209\pi\)
\(572\) 3.23993 0.135468
\(573\) 18.9645 0.792254
\(574\) −15.0089 −0.626459
\(575\) −9.48602 −0.395594
\(576\) −1.88717 −0.0786322
\(577\) −25.9859 −1.08181 −0.540904 0.841084i \(-0.681918\pi\)
−0.540904 + 0.841084i \(0.681918\pi\)
\(578\) −29.0915 −1.21005
\(579\) −3.69760 −0.153667
\(580\) 1.31331 0.0545321
\(581\) −0.0618786 −0.00256716
\(582\) 2.37319 0.0983720
\(583\) 5.93538 0.245818
\(584\) 7.86830 0.325592
\(585\) −3.38351 −0.139891
\(586\) −5.45390 −0.225299
\(587\) 19.6467 0.810905 0.405452 0.914116i \(-0.367114\pi\)
0.405452 + 0.914116i \(0.367114\pi\)
\(588\) 8.49496 0.350326
\(589\) 33.5340 1.38175
\(590\) 15.8037 0.650627
\(591\) −20.4783 −0.842365
\(592\) 50.4638 2.07405
\(593\) 26.1682 1.07460 0.537299 0.843392i \(-0.319445\pi\)
0.537299 + 0.843392i \(0.319445\pi\)
\(594\) 1.32719 0.0544553
\(595\) −4.18738 −0.171666
\(596\) 8.77572 0.359467
\(597\) 10.8418 0.443724
\(598\) 58.4228 2.38908
\(599\) −3.45565 −0.141194 −0.0705970 0.997505i \(-0.522490\pi\)
−0.0705970 + 0.997505i \(0.522490\pi\)
\(600\) −1.24995 −0.0510291
\(601\) 13.5063 0.550932 0.275466 0.961311i \(-0.411168\pi\)
0.275466 + 0.961311i \(0.411168\pi\)
\(602\) −13.4583 −0.548517
\(603\) −2.55187 −0.103920
\(604\) 26.5620 1.08079
\(605\) 10.4684 0.425600
\(606\) −9.74878 −0.396017
\(607\) 12.0773 0.490204 0.245102 0.969497i \(-0.421178\pi\)
0.245102 + 0.969497i \(0.421178\pi\)
\(608\) 39.2594 1.59218
\(609\) −0.729126 −0.0295457
\(610\) −28.3429 −1.14757
\(611\) −6.42017 −0.259732
\(612\) −7.54234 −0.304881
\(613\) −19.9935 −0.807531 −0.403765 0.914863i \(-0.632299\pi\)
−0.403765 + 0.914863i \(0.632299\pi\)
\(614\) −7.38209 −0.297917
\(615\) −11.3088 −0.456013
\(616\) −0.664505 −0.0267737
\(617\) −31.9784 −1.28740 −0.643701 0.765277i \(-0.722602\pi\)
−0.643701 + 0.765277i \(0.722602\pi\)
\(618\) 30.5201 1.22770
\(619\) −33.0862 −1.32985 −0.664924 0.746911i \(-0.731536\pi\)
−0.664924 + 0.746911i \(0.731536\pi\)
\(620\) −7.20483 −0.289353
\(621\) 9.48602 0.380661
\(622\) −24.8890 −0.997958
\(623\) −3.38077 −0.135448
\(624\) 16.5854 0.663948
\(625\) 1.00000 0.0400000
\(626\) 7.73970 0.309341
\(627\) −4.45688 −0.177991
\(628\) −31.3431 −1.25073
\(629\) 59.1236 2.35741
\(630\) −1.32719 −0.0528766
\(631\) −3.02654 −0.120485 −0.0602423 0.998184i \(-0.519187\pi\)
−0.0602423 + 0.998184i \(0.519187\pi\)
\(632\) −6.74178 −0.268174
\(633\) −2.00000 −0.0794929
\(634\) 7.79933 0.309751
\(635\) −4.51398 −0.179132
\(636\) −10.6908 −0.423920
\(637\) −21.8858 −0.867147
\(638\) 1.32719 0.0525440
\(639\) −4.83164 −0.191137
\(640\) 9.41020 0.371971
\(641\) 14.0747 0.555919 0.277959 0.960593i \(-0.410342\pi\)
0.277959 + 0.960593i \(0.410342\pi\)
\(642\) 13.3038 0.525058
\(643\) 4.25044 0.167621 0.0838104 0.996482i \(-0.473291\pi\)
0.0838104 + 0.996482i \(0.473291\pi\)
\(644\) 9.08350 0.357940
\(645\) −10.1404 −0.399278
\(646\) 63.8997 2.51410
\(647\) −43.0185 −1.69123 −0.845616 0.533792i \(-0.820766\pi\)
−0.845616 + 0.533792i \(0.820766\pi\)
\(648\) 1.24995 0.0491027
\(649\) 6.33038 0.248489
\(650\) −6.15883 −0.241569
\(651\) 4.00000 0.156772
\(652\) −2.54175 −0.0995425
\(653\) 18.5175 0.724648 0.362324 0.932052i \(-0.381983\pi\)
0.362324 + 0.932052i \(0.381983\pi\)
\(654\) −14.9536 −0.584733
\(655\) 14.7771 0.577391
\(656\) 55.4337 2.16432
\(657\) 6.29488 0.245587
\(658\) −2.51833 −0.0981747
\(659\) 6.19736 0.241415 0.120707 0.992688i \(-0.461484\pi\)
0.120707 + 0.992688i \(0.461484\pi\)
\(660\) 0.957567 0.0372732
\(661\) −29.0936 −1.13161 −0.565805 0.824539i \(-0.691435\pi\)
−0.565805 + 0.824539i \(0.691435\pi\)
\(662\) −6.58516 −0.255939
\(663\) 19.4315 0.754658
\(664\) 0.106079 0.00411668
\(665\) 4.45688 0.172830
\(666\) 18.7392 0.726131
\(667\) 9.48602 0.367300
\(668\) −16.2883 −0.630214
\(669\) 4.21515 0.162967
\(670\) −4.64504 −0.179454
\(671\) −11.3531 −0.438283
\(672\) 4.68293 0.180648
\(673\) −36.6288 −1.41194 −0.705969 0.708243i \(-0.749488\pi\)
−0.705969 + 0.708243i \(0.749488\pi\)
\(674\) −42.6742 −1.64375
\(675\) −1.00000 −0.0384900
\(676\) −2.03808 −0.0783878
\(677\) 25.9330 0.996686 0.498343 0.866980i \(-0.333942\pi\)
0.498343 + 0.866980i \(0.333942\pi\)
\(678\) 15.4904 0.594904
\(679\) −0.950615 −0.0364813
\(680\) 7.17849 0.275282
\(681\) 15.4214 0.590949
\(682\) −7.28100 −0.278804
\(683\) 22.4403 0.858653 0.429327 0.903149i \(-0.358751\pi\)
0.429327 + 0.903149i \(0.358751\pi\)
\(684\) 8.02777 0.306949
\(685\) 5.19114 0.198343
\(686\) −17.8751 −0.682475
\(687\) 24.2530 0.925311
\(688\) 49.7066 1.89505
\(689\) 27.5431 1.04931
\(690\) 17.2669 0.657340
\(691\) 16.3000 0.620082 0.310041 0.950723i \(-0.399657\pi\)
0.310041 + 0.950723i \(0.399657\pi\)
\(692\) −28.5136 −1.08392
\(693\) −0.531625 −0.0201948
\(694\) −37.9022 −1.43875
\(695\) 1.15824 0.0439345
\(696\) 1.24995 0.0473793
\(697\) 64.9463 2.46002
\(698\) 32.8656 1.24398
\(699\) 5.33653 0.201846
\(700\) −0.957567 −0.0361926
\(701\) −10.9811 −0.414751 −0.207376 0.978261i \(-0.566492\pi\)
−0.207376 + 0.978261i \(0.566492\pi\)
\(702\) 6.15883 0.232450
\(703\) −62.9288 −2.37341
\(704\) −1.37599 −0.0518595
\(705\) −1.89749 −0.0714636
\(706\) 26.1470 0.984054
\(707\) 3.90501 0.146863
\(708\) −11.4023 −0.428526
\(709\) −33.0681 −1.24190 −0.620949 0.783851i \(-0.713252\pi\)
−0.620949 + 0.783851i \(0.713252\pi\)
\(710\) −8.79478 −0.330062
\(711\) −5.39363 −0.202277
\(712\) 5.79570 0.217203
\(713\) −52.0405 −1.94893
\(714\) 7.62207 0.285249
\(715\) −2.46700 −0.0922607
\(716\) 7.99130 0.298649
\(717\) −16.0202 −0.598287
\(718\) −11.2947 −0.421514
\(719\) −44.0289 −1.64200 −0.821001 0.570926i \(-0.806584\pi\)
−0.821001 + 0.570926i \(0.806584\pi\)
\(720\) 4.90184 0.182681
\(721\) −12.2253 −0.455293
\(722\) −33.4276 −1.24405
\(723\) 6.60741 0.245732
\(724\) 8.90047 0.330783
\(725\) −1.00000 −0.0371391
\(726\) −19.0551 −0.707199
\(727\) 20.2708 0.751803 0.375902 0.926660i \(-0.377333\pi\)
0.375902 + 0.926660i \(0.377333\pi\)
\(728\) −3.08364 −0.114287
\(729\) 1.00000 0.0370370
\(730\) 11.4583 0.424089
\(731\) 58.2364 2.15395
\(732\) 20.4494 0.755830
\(733\) 45.9138 1.69586 0.847932 0.530105i \(-0.177847\pi\)
0.847932 + 0.530105i \(0.177847\pi\)
\(734\) −33.1240 −1.22263
\(735\) −6.46838 −0.238590
\(736\) −60.9255 −2.24574
\(737\) −1.86064 −0.0685374
\(738\) 20.5848 0.757736
\(739\) 38.1884 1.40478 0.702392 0.711791i \(-0.252115\pi\)
0.702392 + 0.711791i \(0.252115\pi\)
\(740\) 13.5203 0.497018
\(741\) −20.6821 −0.759778
\(742\) 10.8039 0.396623
\(743\) −34.4773 −1.26485 −0.632424 0.774622i \(-0.717940\pi\)
−0.632424 + 0.774622i \(0.717940\pi\)
\(744\) −6.85726 −0.251399
\(745\) −6.68215 −0.244815
\(746\) −63.4550 −2.32325
\(747\) 0.0848668 0.00310511
\(748\) −5.49931 −0.201075
\(749\) −5.32901 −0.194718
\(750\) −1.82025 −0.0664661
\(751\) 41.7113 1.52207 0.761033 0.648713i \(-0.224692\pi\)
0.761033 + 0.648713i \(0.224692\pi\)
\(752\) 9.30118 0.339179
\(753\) −25.6834 −0.935954
\(754\) 6.15883 0.224291
\(755\) −20.2253 −0.736073
\(756\) 0.957567 0.0348264
\(757\) −13.5659 −0.493061 −0.246530 0.969135i \(-0.579291\pi\)
−0.246530 + 0.969135i \(0.579291\pi\)
\(758\) 11.6679 0.423797
\(759\) 6.91650 0.251053
\(760\) −7.64050 −0.277150
\(761\) 3.27982 0.118893 0.0594467 0.998231i \(-0.481066\pi\)
0.0594467 + 0.998231i \(0.481066\pi\)
\(762\) 8.21657 0.297655
\(763\) 5.98988 0.216848
\(764\) −24.9062 −0.901076
\(765\) 5.74301 0.207639
\(766\) −37.7159 −1.36273
\(767\) 29.3761 1.06071
\(768\) −20.9033 −0.754281
\(769\) −35.5527 −1.28206 −0.641032 0.767514i \(-0.721493\pi\)
−0.641032 + 0.767514i \(0.721493\pi\)
\(770\) −0.967690 −0.0348731
\(771\) 7.93538 0.285786
\(772\) 4.85608 0.174774
\(773\) −0.965870 −0.0347399 −0.0173700 0.999849i \(-0.505529\pi\)
−0.0173700 + 0.999849i \(0.505529\pi\)
\(774\) 18.4581 0.663461
\(775\) 5.48602 0.197064
\(776\) 1.62965 0.0585012
\(777\) −7.50627 −0.269286
\(778\) 45.0659 1.61569
\(779\) −69.1263 −2.47671
\(780\) 4.44359 0.159106
\(781\) −3.52287 −0.126058
\(782\) −99.1641 −3.54610
\(783\) 1.00000 0.0357371
\(784\) 31.7069 1.13239
\(785\) 23.8658 0.851806
\(786\) −26.8981 −0.959423
\(787\) 15.0756 0.537387 0.268693 0.963226i \(-0.413408\pi\)
0.268693 + 0.963226i \(0.413408\pi\)
\(788\) 26.8943 0.958070
\(789\) −23.5618 −0.838822
\(790\) −9.81775 −0.349300
\(791\) −6.20488 −0.220620
\(792\) 0.911372 0.0323842
\(793\) −52.6842 −1.87087
\(794\) 23.2023 0.823419
\(795\) 8.14040 0.288710
\(796\) −14.2386 −0.504673
\(797\) 27.9884 0.991399 0.495700 0.868494i \(-0.334912\pi\)
0.495700 + 0.868494i \(0.334912\pi\)
\(798\) −8.11263 −0.287184
\(799\) 10.8973 0.385519
\(800\) 6.42266 0.227075
\(801\) 4.63674 0.163831
\(802\) −51.3236 −1.81230
\(803\) 4.58976 0.161969
\(804\) 3.35139 0.118194
\(805\) −6.91650 −0.243775
\(806\) −33.7875 −1.19011
\(807\) 10.6645 0.375408
\(808\) −6.69442 −0.235509
\(809\) 10.0950 0.354921 0.177460 0.984128i \(-0.443212\pi\)
0.177460 + 0.984128i \(0.443212\pi\)
\(810\) 1.82025 0.0639570
\(811\) 16.3555 0.574321 0.287160 0.957882i \(-0.407289\pi\)
0.287160 + 0.957882i \(0.407289\pi\)
\(812\) 0.957567 0.0336040
\(813\) 3.30876 0.116043
\(814\) 13.6633 0.478898
\(815\) 1.93538 0.0677934
\(816\) −28.1513 −0.985493
\(817\) −61.9846 −2.16857
\(818\) 43.7063 1.52815
\(819\) −2.46700 −0.0862041
\(820\) 14.8519 0.518650
\(821\) 20.7468 0.724067 0.362034 0.932165i \(-0.382083\pi\)
0.362034 + 0.932165i \(0.382083\pi\)
\(822\) −9.44917 −0.329578
\(823\) 33.4619 1.16641 0.583204 0.812326i \(-0.301799\pi\)
0.583204 + 0.812326i \(0.301799\pi\)
\(824\) 20.9580 0.730105
\(825\) −0.729126 −0.0253849
\(826\) 11.5229 0.400932
\(827\) −20.3363 −0.707164 −0.353582 0.935404i \(-0.615036\pi\)
−0.353582 + 0.935404i \(0.615036\pi\)
\(828\) −12.4581 −0.432947
\(829\) 18.4858 0.642039 0.321020 0.947073i \(-0.395974\pi\)
0.321020 + 0.947073i \(0.395974\pi\)
\(830\) 0.154479 0.00536203
\(831\) 0.0469761 0.00162958
\(832\) −6.38526 −0.221369
\(833\) 37.1479 1.28710
\(834\) −2.10828 −0.0730039
\(835\) 12.4025 0.429207
\(836\) 5.85325 0.202439
\(837\) −5.48602 −0.189625
\(838\) −26.8797 −0.928542
\(839\) −22.3571 −0.771853 −0.385927 0.922529i \(-0.626118\pi\)
−0.385927 + 0.922529i \(0.626118\pi\)
\(840\) −0.911372 −0.0314453
\(841\) 1.00000 0.0344828
\(842\) 23.1721 0.798562
\(843\) −29.8037 −1.02649
\(844\) 2.62661 0.0904118
\(845\) 1.55187 0.0533860
\(846\) 3.45390 0.118748
\(847\) 7.63277 0.262265
\(848\) −39.9029 −1.37027
\(849\) −5.79498 −0.198883
\(850\) 10.4537 0.358559
\(851\) 97.6574 3.34765
\(852\) 6.34542 0.217391
\(853\) 16.1950 0.554505 0.277253 0.960797i \(-0.410576\pi\)
0.277253 + 0.960797i \(0.410576\pi\)
\(854\) −20.6655 −0.707160
\(855\) −6.11263 −0.209048
\(856\) 9.13560 0.312249
\(857\) 11.4785 0.392098 0.196049 0.980594i \(-0.437189\pi\)
0.196049 + 0.980594i \(0.437189\pi\)
\(858\) 4.49056 0.153305
\(859\) −21.6744 −0.739522 −0.369761 0.929127i \(-0.620560\pi\)
−0.369761 + 0.929127i \(0.620560\pi\)
\(860\) 13.3175 0.454122
\(861\) −8.24552 −0.281006
\(862\) −17.5896 −0.599103
\(863\) −30.7226 −1.04581 −0.522905 0.852391i \(-0.675152\pi\)
−0.522905 + 0.852391i \(0.675152\pi\)
\(864\) −6.42266 −0.218503
\(865\) 21.7113 0.738206
\(866\) −23.8798 −0.811470
\(867\) −15.9822 −0.542783
\(868\) −5.25323 −0.178306
\(869\) −3.93264 −0.133406
\(870\) 1.82025 0.0617122
\(871\) −8.63428 −0.292561
\(872\) −10.2685 −0.347737
\(873\) 1.30377 0.0441260
\(874\) 105.546 3.57016
\(875\) 0.729126 0.0246490
\(876\) −8.26711 −0.279320
\(877\) 34.2908 1.15792 0.578959 0.815357i \(-0.303459\pi\)
0.578959 + 0.815357i \(0.303459\pi\)
\(878\) 36.2689 1.22402
\(879\) −2.99624 −0.101061
\(880\) 3.57406 0.120481
\(881\) 12.8620 0.433332 0.216666 0.976246i \(-0.430482\pi\)
0.216666 + 0.976246i \(0.430482\pi\)
\(882\) 11.7741 0.396453
\(883\) 1.37495 0.0462707 0.0231354 0.999732i \(-0.492635\pi\)
0.0231354 + 0.999732i \(0.492635\pi\)
\(884\) −25.5196 −0.858316
\(885\) 8.68215 0.291847
\(886\) −3.53682 −0.118822
\(887\) −27.1710 −0.912312 −0.456156 0.889900i \(-0.650774\pi\)
−0.456156 + 0.889900i \(0.650774\pi\)
\(888\) 12.8681 0.431825
\(889\) −3.29126 −0.110385
\(890\) 8.44002 0.282910
\(891\) 0.729126 0.0244266
\(892\) −5.53578 −0.185352
\(893\) −11.5987 −0.388134
\(894\) 12.1632 0.406798
\(895\) −6.08487 −0.203395
\(896\) 6.86122 0.229217
\(897\) 32.0960 1.07166
\(898\) −30.9833 −1.03393
\(899\) −5.48602 −0.182969
\(900\) 1.31331 0.0437769
\(901\) −46.7504 −1.55748
\(902\) 15.0089 0.499741
\(903\) −7.39363 −0.246045
\(904\) 10.6371 0.353785
\(905\) −6.77714 −0.225280
\(906\) 36.8150 1.22310
\(907\) 0.327640 0.0108791 0.00543955 0.999985i \(-0.498269\pi\)
0.00543955 + 0.999985i \(0.498269\pi\)
\(908\) −20.2530 −0.672121
\(909\) −5.35574 −0.177639
\(910\) −4.49056 −0.148861
\(911\) 43.0377 1.42590 0.712951 0.701214i \(-0.247358\pi\)
0.712951 + 0.701214i \(0.247358\pi\)
\(912\) 29.9631 0.992179
\(913\) 0.0618786 0.00204788
\(914\) −29.0572 −0.961125
\(915\) −15.5709 −0.514758
\(916\) −31.8517 −1.05241
\(917\) 10.7744 0.355802
\(918\) −10.4537 −0.345024
\(919\) −28.2061 −0.930432 −0.465216 0.885197i \(-0.654023\pi\)
−0.465216 + 0.885197i \(0.654023\pi\)
\(920\) 11.8571 0.390916
\(921\) −4.05554 −0.133634
\(922\) 32.2021 1.06052
\(923\) −16.3479 −0.538097
\(924\) 0.698187 0.0229687
\(925\) −10.2949 −0.338494
\(926\) 4.74613 0.155967
\(927\) 16.7670 0.550701
\(928\) −6.42266 −0.210834
\(929\) −14.4505 −0.474107 −0.237053 0.971497i \(-0.576182\pi\)
−0.237053 + 0.971497i \(0.576182\pi\)
\(930\) −9.98592 −0.327451
\(931\) −39.5388 −1.29583
\(932\) −7.00851 −0.229571
\(933\) −13.6734 −0.447647
\(934\) 14.5620 0.476483
\(935\) 4.18738 0.136942
\(936\) 4.22922 0.138236
\(937\) 13.9657 0.456241 0.228121 0.973633i \(-0.426742\pi\)
0.228121 + 0.973633i \(0.426742\pi\)
\(938\) −3.38682 −0.110584
\(939\) 4.25200 0.138759
\(940\) 2.49199 0.0812796
\(941\) −40.6262 −1.32438 −0.662189 0.749337i \(-0.730372\pi\)
−0.662189 + 0.749337i \(0.730372\pi\)
\(942\) −43.4416 −1.41541
\(943\) 107.275 3.49336
\(944\) −42.5585 −1.38516
\(945\) −0.729126 −0.0237185
\(946\) 13.4583 0.437566
\(947\) −20.4495 −0.664519 −0.332260 0.943188i \(-0.607811\pi\)
−0.332260 + 0.943188i \(0.607811\pi\)
\(948\) 7.08350 0.230061
\(949\) 21.2988 0.691388
\(950\) −11.1265 −0.360992
\(951\) 4.28476 0.138943
\(952\) 5.23402 0.169636
\(953\) −51.5527 −1.66996 −0.834978 0.550283i \(-0.814520\pi\)
−0.834978 + 0.550283i \(0.814520\pi\)
\(954\) −14.8176 −0.479736
\(955\) 18.9645 0.613677
\(956\) 21.0395 0.680466
\(957\) 0.729126 0.0235693
\(958\) −37.6488 −1.21638
\(959\) 3.78499 0.122224
\(960\) −1.88717 −0.0609082
\(961\) −0.903587 −0.0291480
\(962\) 63.4044 2.04424
\(963\) 7.30876 0.235522
\(964\) −8.67756 −0.279485
\(965\) −3.69760 −0.119030
\(966\) 12.5898 0.405069
\(967\) 16.5429 0.531985 0.265992 0.963975i \(-0.414300\pi\)
0.265992 + 0.963975i \(0.414300\pi\)
\(968\) −13.0850 −0.420567
\(969\) 35.1049 1.12773
\(970\) 2.37319 0.0761986
\(971\) 32.5417 1.04431 0.522157 0.852849i \(-0.325127\pi\)
0.522157 + 0.852849i \(0.325127\pi\)
\(972\) −1.31331 −0.0421244
\(973\) 0.844503 0.0270735
\(974\) 58.4365 1.87242
\(975\) −3.38351 −0.108359
\(976\) 76.3260 2.44313
\(977\) −38.7111 −1.23848 −0.619239 0.785203i \(-0.712559\pi\)
−0.619239 + 0.785203i \(0.712559\pi\)
\(978\) −3.52287 −0.112649
\(979\) 3.38077 0.108050
\(980\) 8.49496 0.271362
\(981\) −8.21515 −0.262289
\(982\) 74.0720 2.36373
\(983\) −31.7315 −1.01208 −0.506039 0.862510i \(-0.668891\pi\)
−0.506039 + 0.862510i \(0.668891\pi\)
\(984\) 14.1354 0.450620
\(985\) −20.4783 −0.652493
\(986\) −10.4537 −0.332914
\(987\) −1.38351 −0.0440376
\(988\) 27.1620 0.864139
\(989\) 96.1921 3.05873
\(990\) 1.32719 0.0421809
\(991\) −22.8620 −0.726236 −0.363118 0.931743i \(-0.618288\pi\)
−0.363118 + 0.931743i \(0.618288\pi\)
\(992\) 35.2349 1.11871
\(993\) −3.61772 −0.114805
\(994\) −6.41251 −0.203392
\(995\) 10.8418 0.343707
\(996\) −0.111456 −0.00353162
\(997\) 40.9024 1.29539 0.647696 0.761898i \(-0.275733\pi\)
0.647696 + 0.761898i \(0.275733\pi\)
\(998\) −19.7347 −0.624691
\(999\) 10.2949 0.325716
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 435.2.a.j.1.2 4
3.2 odd 2 1305.2.a.r.1.3 4
4.3 odd 2 6960.2.a.co.1.3 4
5.2 odd 4 2175.2.c.n.349.2 8
5.3 odd 4 2175.2.c.n.349.7 8
5.4 even 2 2175.2.a.v.1.3 4
15.14 odd 2 6525.2.a.bi.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.2 4 1.1 even 1 trivial
1305.2.a.r.1.3 4 3.2 odd 2
2175.2.a.v.1.3 4 5.4 even 2
2175.2.c.n.349.2 8 5.2 odd 4
2175.2.c.n.349.7 8 5.3 odd 4
6525.2.a.bi.1.2 4 15.14 odd 2
6960.2.a.co.1.3 4 4.3 odd 2