Properties

Label 1334.2.a.j.1.9
Level $1334$
Weight $2$
Character 1334.1
Self dual yes
Analytic conductor $10.652$
Analytic rank $0$
Dimension $9$
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] = [1334,2,Mod(1,1334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1334, 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("1334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1334 = 2 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1334.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: \(10.6520436296\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 16x^{7} + 44x^{6} + 87x^{5} - 209x^{4} - 160x^{3} + 348x^{2} + 12x - 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-2.50055\) of defining polynomial
Character \(\chi\) \(=\) 1334.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.50055 q^{3} +1.00000 q^{4} +1.67075 q^{5} -2.50055 q^{6} -2.56147 q^{7} -1.00000 q^{8} +3.25275 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.50055 q^{3} +1.00000 q^{4} +1.67075 q^{5} -2.50055 q^{6} -2.56147 q^{7} -1.00000 q^{8} +3.25275 q^{9} -1.67075 q^{10} +0.797665 q^{11} +2.50055 q^{12} +4.57607 q^{13} +2.56147 q^{14} +4.17781 q^{15} +1.00000 q^{16} +5.52539 q^{17} -3.25275 q^{18} -2.88478 q^{19} +1.67075 q^{20} -6.40507 q^{21} -0.797665 q^{22} -1.00000 q^{23} -2.50055 q^{24} -2.20858 q^{25} -4.57607 q^{26} +0.632025 q^{27} -2.56147 q^{28} +1.00000 q^{29} -4.17781 q^{30} +2.79767 q^{31} -1.00000 q^{32} +1.99460 q^{33} -5.52539 q^{34} -4.27958 q^{35} +3.25275 q^{36} +10.0324 q^{37} +2.88478 q^{38} +11.4427 q^{39} -1.67075 q^{40} +8.03244 q^{41} +6.40507 q^{42} +4.20344 q^{43} +0.797665 q^{44} +5.43455 q^{45} +1.00000 q^{46} -3.01372 q^{47} +2.50055 q^{48} -0.438895 q^{49} +2.20858 q^{50} +13.8165 q^{51} +4.57607 q^{52} +2.83475 q^{53} -0.632025 q^{54} +1.33270 q^{55} +2.56147 q^{56} -7.21354 q^{57} -1.00000 q^{58} -4.44699 q^{59} +4.17781 q^{60} +1.99094 q^{61} -2.79767 q^{62} -8.33182 q^{63} +1.00000 q^{64} +7.64549 q^{65} -1.99460 q^{66} -8.31278 q^{67} +5.52539 q^{68} -2.50055 q^{69} +4.27958 q^{70} -5.77544 q^{71} -3.25275 q^{72} +2.16312 q^{73} -10.0324 q^{74} -5.52266 q^{75} -2.88478 q^{76} -2.04319 q^{77} -11.4427 q^{78} -9.57229 q^{79} +1.67075 q^{80} -8.17785 q^{81} -8.03244 q^{82} +8.08767 q^{83} -6.40507 q^{84} +9.23157 q^{85} -4.20344 q^{86} +2.50055 q^{87} -0.797665 q^{88} +11.4861 q^{89} -5.43455 q^{90} -11.7214 q^{91} -1.00000 q^{92} +6.99570 q^{93} +3.01372 q^{94} -4.81976 q^{95} -2.50055 q^{96} -2.22607 q^{97} +0.438895 q^{98} +2.59461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} + 5 q^{5} + 3 q^{6} + 6 q^{7} - 9 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} + 5 q^{5} + 3 q^{6} + 6 q^{7} - 9 q^{8} + 14 q^{9} - 5 q^{10} - 3 q^{11} - 3 q^{12} + 13 q^{13} - 6 q^{14} - 3 q^{15} + 9 q^{16} + 2 q^{17} - 14 q^{18} + 16 q^{19} + 5 q^{20} + 8 q^{21} + 3 q^{22} - 9 q^{23} + 3 q^{24} + 20 q^{25} - 13 q^{26} - 21 q^{27} + 6 q^{28} + 9 q^{29} + 3 q^{30} + 15 q^{31} - 9 q^{32} + 13 q^{33} - 2 q^{34} + 14 q^{36} + 12 q^{37} - 16 q^{38} - 5 q^{39} - 5 q^{40} - 6 q^{41} - 8 q^{42} - 3 q^{43} - 3 q^{44} + 20 q^{45} + 9 q^{46} - 19 q^{47} - 3 q^{48} + 37 q^{49} - 20 q^{50} + 6 q^{51} + 13 q^{52} + 5 q^{53} + 21 q^{54} + q^{55} - 6 q^{56} - 20 q^{57} - 9 q^{58} + 12 q^{59} - 3 q^{60} + 12 q^{61} - 15 q^{62} + 6 q^{63} + 9 q^{64} + 19 q^{65} - 13 q^{66} - 6 q^{67} + 2 q^{68} + 3 q^{69} + 12 q^{71} - 14 q^{72} - 12 q^{74} + 16 q^{75} + 16 q^{76} + 34 q^{77} + 5 q^{78} + 29 q^{79} + 5 q^{80} + 5 q^{81} + 6 q^{82} + 24 q^{83} + 8 q^{84} + 12 q^{85} + 3 q^{86} - 3 q^{87} + 3 q^{88} - 2 q^{89} - 20 q^{90} + 58 q^{91} - 9 q^{92} + 7 q^{93} + 19 q^{94} + 6 q^{95} + 3 q^{96} + 12 q^{97} - 37 q^{98} - 20 q^{99}+O(q^{100}) \) 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
\(3\) 2.50055 1.44369 0.721847 0.692053i \(-0.243294\pi\)
0.721847 + 0.692053i \(0.243294\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.67075 0.747184 0.373592 0.927593i \(-0.378126\pi\)
0.373592 + 0.927593i \(0.378126\pi\)
\(6\) −2.50055 −1.02085
\(7\) −2.56147 −0.968143 −0.484071 0.875028i \(-0.660843\pi\)
−0.484071 + 0.875028i \(0.660843\pi\)
\(8\) −1.00000 −0.353553
\(9\) 3.25275 1.08425
\(10\) −1.67075 −0.528339
\(11\) 0.797665 0.240505 0.120253 0.992743i \(-0.461630\pi\)
0.120253 + 0.992743i \(0.461630\pi\)
\(12\) 2.50055 0.721847
\(13\) 4.57607 1.26917 0.634586 0.772852i \(-0.281170\pi\)
0.634586 + 0.772852i \(0.281170\pi\)
\(14\) 2.56147 0.684580
\(15\) 4.17781 1.07871
\(16\) 1.00000 0.250000
\(17\) 5.52539 1.34010 0.670052 0.742314i \(-0.266272\pi\)
0.670052 + 0.742314i \(0.266272\pi\)
\(18\) −3.25275 −0.766682
\(19\) −2.88478 −0.661814 −0.330907 0.943663i \(-0.607355\pi\)
−0.330907 + 0.943663i \(0.607355\pi\)
\(20\) 1.67075 0.373592
\(21\) −6.40507 −1.39770
\(22\) −0.797665 −0.170063
\(23\) −1.00000 −0.208514
\(24\) −2.50055 −0.510423
\(25\) −2.20858 −0.441716
\(26\) −4.57607 −0.897441
\(27\) 0.632025 0.121633
\(28\) −2.56147 −0.484071
\(29\) 1.00000 0.185695
\(30\) −4.17781 −0.762760
\(31\) 2.79767 0.502476 0.251238 0.967925i \(-0.419162\pi\)
0.251238 + 0.967925i \(0.419162\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.99460 0.347216
\(34\) −5.52539 −0.947597
\(35\) −4.27958 −0.723381
\(36\) 3.25275 0.542126
\(37\) 10.0324 1.64932 0.824662 0.565626i \(-0.191366\pi\)
0.824662 + 0.565626i \(0.191366\pi\)
\(38\) 2.88478 0.467973
\(39\) 11.4427 1.83230
\(40\) −1.67075 −0.264170
\(41\) 8.03244 1.25446 0.627228 0.778835i \(-0.284189\pi\)
0.627228 + 0.778835i \(0.284189\pi\)
\(42\) 6.40507 0.988324
\(43\) 4.20344 0.641018 0.320509 0.947245i \(-0.396146\pi\)
0.320509 + 0.947245i \(0.396146\pi\)
\(44\) 0.797665 0.120253
\(45\) 5.43455 0.810136
\(46\) 1.00000 0.147442
\(47\) −3.01372 −0.439596 −0.219798 0.975545i \(-0.570540\pi\)
−0.219798 + 0.975545i \(0.570540\pi\)
\(48\) 2.50055 0.360923
\(49\) −0.438895 −0.0626993
\(50\) 2.20858 0.312340
\(51\) 13.8165 1.93470
\(52\) 4.57607 0.634586
\(53\) 2.83475 0.389383 0.194692 0.980864i \(-0.437629\pi\)
0.194692 + 0.980864i \(0.437629\pi\)
\(54\) −0.632025 −0.0860077
\(55\) 1.33270 0.179702
\(56\) 2.56147 0.342290
\(57\) −7.21354 −0.955456
\(58\) −1.00000 −0.131306
\(59\) −4.44699 −0.578948 −0.289474 0.957186i \(-0.593480\pi\)
−0.289474 + 0.957186i \(0.593480\pi\)
\(60\) 4.17781 0.539353
\(61\) 1.99094 0.254913 0.127457 0.991844i \(-0.459319\pi\)
0.127457 + 0.991844i \(0.459319\pi\)
\(62\) −2.79767 −0.355304
\(63\) −8.33182 −1.04971
\(64\) 1.00000 0.125000
\(65\) 7.64549 0.948306
\(66\) −1.99460 −0.245519
\(67\) −8.31278 −1.01557 −0.507784 0.861484i \(-0.669535\pi\)
−0.507784 + 0.861484i \(0.669535\pi\)
\(68\) 5.52539 0.670052
\(69\) −2.50055 −0.301031
\(70\) 4.27958 0.511508
\(71\) −5.77544 −0.685419 −0.342709 0.939441i \(-0.611345\pi\)
−0.342709 + 0.939441i \(0.611345\pi\)
\(72\) −3.25275 −0.383341
\(73\) 2.16312 0.253174 0.126587 0.991956i \(-0.459598\pi\)
0.126587 + 0.991956i \(0.459598\pi\)
\(74\) −10.0324 −1.16625
\(75\) −5.52266 −0.637702
\(76\) −2.88478 −0.330907
\(77\) −2.04319 −0.232843
\(78\) −11.4427 −1.29563
\(79\) −9.57229 −1.07697 −0.538484 0.842636i \(-0.681003\pi\)
−0.538484 + 0.842636i \(0.681003\pi\)
\(80\) 1.67075 0.186796
\(81\) −8.17785 −0.908650
\(82\) −8.03244 −0.887035
\(83\) 8.08767 0.887737 0.443869 0.896092i \(-0.353606\pi\)
0.443869 + 0.896092i \(0.353606\pi\)
\(84\) −6.40507 −0.698851
\(85\) 9.23157 1.00130
\(86\) −4.20344 −0.453268
\(87\) 2.50055 0.268087
\(88\) −0.797665 −0.0850314
\(89\) 11.4861 1.21752 0.608761 0.793354i \(-0.291667\pi\)
0.608761 + 0.793354i \(0.291667\pi\)
\(90\) −5.43455 −0.572852
\(91\) −11.7214 −1.22874
\(92\) −1.00000 −0.104257
\(93\) 6.99570 0.725421
\(94\) 3.01372 0.310841
\(95\) −4.81976 −0.494497
\(96\) −2.50055 −0.255211
\(97\) −2.22607 −0.226023 −0.113012 0.993594i \(-0.536050\pi\)
−0.113012 + 0.993594i \(0.536050\pi\)
\(98\) 0.438895 0.0443351
\(99\) 2.59461 0.260768
\(100\) −2.20858 −0.220858
\(101\) −13.7286 −1.36604 −0.683022 0.730398i \(-0.739335\pi\)
−0.683022 + 0.730398i \(0.739335\pi\)
\(102\) −13.8165 −1.36804
\(103\) −0.609882 −0.0600935 −0.0300467 0.999548i \(-0.509566\pi\)
−0.0300467 + 0.999548i \(0.509566\pi\)
\(104\) −4.57607 −0.448720
\(105\) −10.7013 −1.04434
\(106\) −2.83475 −0.275336
\(107\) 4.12882 0.399149 0.199574 0.979883i \(-0.436044\pi\)
0.199574 + 0.979883i \(0.436044\pi\)
\(108\) 0.632025 0.0608166
\(109\) 16.5257 1.58288 0.791438 0.611250i \(-0.209333\pi\)
0.791438 + 0.611250i \(0.209333\pi\)
\(110\) −1.33270 −0.127068
\(111\) 25.0866 2.38112
\(112\) −2.56147 −0.242036
\(113\) −7.88907 −0.742141 −0.371071 0.928605i \(-0.621009\pi\)
−0.371071 + 0.928605i \(0.621009\pi\)
\(114\) 7.21354 0.675610
\(115\) −1.67075 −0.155799
\(116\) 1.00000 0.0928477
\(117\) 14.8848 1.37610
\(118\) 4.44699 0.409378
\(119\) −14.1531 −1.29741
\(120\) −4.17781 −0.381380
\(121\) −10.3637 −0.942157
\(122\) −1.99094 −0.180251
\(123\) 20.0855 1.81105
\(124\) 2.79767 0.251238
\(125\) −12.0438 −1.07723
\(126\) 8.33182 0.742257
\(127\) 5.24015 0.464989 0.232494 0.972598i \(-0.425311\pi\)
0.232494 + 0.972598i \(0.425311\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 10.5109 0.925434
\(130\) −7.64549 −0.670554
\(131\) −13.2715 −1.15953 −0.579767 0.814782i \(-0.696856\pi\)
−0.579767 + 0.814782i \(0.696856\pi\)
\(132\) 1.99460 0.173608
\(133\) 7.38926 0.640730
\(134\) 8.31278 0.718115
\(135\) 1.05596 0.0908824
\(136\) −5.52539 −0.473798
\(137\) −14.2454 −1.21707 −0.608534 0.793528i \(-0.708242\pi\)
−0.608534 + 0.793528i \(0.708242\pi\)
\(138\) 2.50055 0.212861
\(139\) 7.66694 0.650301 0.325151 0.945662i \(-0.394585\pi\)
0.325151 + 0.945662i \(0.394585\pi\)
\(140\) −4.27958 −0.361691
\(141\) −7.53596 −0.634642
\(142\) 5.77544 0.484664
\(143\) 3.65017 0.305243
\(144\) 3.25275 0.271063
\(145\) 1.67075 0.138749
\(146\) −2.16312 −0.179021
\(147\) −1.09748 −0.0905186
\(148\) 10.0324 0.824662
\(149\) 7.10247 0.581857 0.290929 0.956745i \(-0.406036\pi\)
0.290929 + 0.956745i \(0.406036\pi\)
\(150\) 5.52266 0.450923
\(151\) 10.5988 0.862516 0.431258 0.902229i \(-0.358070\pi\)
0.431258 + 0.902229i \(0.358070\pi\)
\(152\) 2.88478 0.233986
\(153\) 17.9727 1.45301
\(154\) 2.04319 0.164645
\(155\) 4.67421 0.375442
\(156\) 11.4427 0.916148
\(157\) −2.81055 −0.224307 −0.112153 0.993691i \(-0.535775\pi\)
−0.112153 + 0.993691i \(0.535775\pi\)
\(158\) 9.57229 0.761531
\(159\) 7.08845 0.562150
\(160\) −1.67075 −0.132085
\(161\) 2.56147 0.201872
\(162\) 8.17785 0.642513
\(163\) 3.78302 0.296309 0.148154 0.988964i \(-0.452667\pi\)
0.148154 + 0.988964i \(0.452667\pi\)
\(164\) 8.03244 0.627228
\(165\) 3.33249 0.259434
\(166\) −8.08767 −0.627725
\(167\) −3.69622 −0.286022 −0.143011 0.989721i \(-0.545678\pi\)
−0.143011 + 0.989721i \(0.545678\pi\)
\(168\) 6.40507 0.494162
\(169\) 7.94040 0.610800
\(170\) −9.23157 −0.708029
\(171\) −9.38348 −0.717572
\(172\) 4.20344 0.320509
\(173\) −19.2973 −1.46715 −0.733575 0.679609i \(-0.762150\pi\)
−0.733575 + 0.679609i \(0.762150\pi\)
\(174\) −2.50055 −0.189566
\(175\) 5.65720 0.427644
\(176\) 0.797665 0.0601263
\(177\) −11.1199 −0.835824
\(178\) −11.4861 −0.860918
\(179\) −15.3120 −1.14447 −0.572234 0.820090i \(-0.693923\pi\)
−0.572234 + 0.820090i \(0.693923\pi\)
\(180\) 5.43455 0.405068
\(181\) 9.51985 0.707605 0.353802 0.935320i \(-0.384889\pi\)
0.353802 + 0.935320i \(0.384889\pi\)
\(182\) 11.7214 0.868851
\(183\) 4.97844 0.368017
\(184\) 1.00000 0.0737210
\(185\) 16.7618 1.23235
\(186\) −6.99570 −0.512950
\(187\) 4.40741 0.322302
\(188\) −3.01372 −0.219798
\(189\) −1.61891 −0.117758
\(190\) 4.81976 0.349662
\(191\) −22.7186 −1.64386 −0.821931 0.569588i \(-0.807103\pi\)
−0.821931 + 0.569588i \(0.807103\pi\)
\(192\) 2.50055 0.180462
\(193\) −10.7819 −0.776098 −0.388049 0.921639i \(-0.626851\pi\)
−0.388049 + 0.921639i \(0.626851\pi\)
\(194\) 2.22607 0.159822
\(195\) 19.1179 1.36906
\(196\) −0.438895 −0.0313497
\(197\) −10.7522 −0.766063 −0.383031 0.923735i \(-0.625120\pi\)
−0.383031 + 0.923735i \(0.625120\pi\)
\(198\) −2.59461 −0.184391
\(199\) 13.6905 0.970493 0.485247 0.874377i \(-0.338730\pi\)
0.485247 + 0.874377i \(0.338730\pi\)
\(200\) 2.20858 0.156170
\(201\) −20.7865 −1.46617
\(202\) 13.7286 0.965939
\(203\) −2.56147 −0.179780
\(204\) 13.8165 0.967350
\(205\) 13.4202 0.937310
\(206\) 0.609882 0.0424925
\(207\) −3.25275 −0.226082
\(208\) 4.57607 0.317293
\(209\) −2.30109 −0.159170
\(210\) 10.7013 0.738460
\(211\) −19.9301 −1.37204 −0.686022 0.727581i \(-0.740645\pi\)
−0.686022 + 0.727581i \(0.740645\pi\)
\(212\) 2.83475 0.194692
\(213\) −14.4418 −0.989535
\(214\) −4.12882 −0.282241
\(215\) 7.02291 0.478959
\(216\) −0.632025 −0.0430038
\(217\) −7.16612 −0.486468
\(218\) −16.5257 −1.11926
\(219\) 5.40898 0.365505
\(220\) 1.33270 0.0898508
\(221\) 25.2846 1.70082
\(222\) −25.0866 −1.68370
\(223\) 0.826796 0.0553664 0.0276832 0.999617i \(-0.491187\pi\)
0.0276832 + 0.999617i \(0.491187\pi\)
\(224\) 2.56147 0.171145
\(225\) −7.18396 −0.478931
\(226\) 7.88907 0.524773
\(227\) −8.39006 −0.556868 −0.278434 0.960455i \(-0.589815\pi\)
−0.278434 + 0.960455i \(0.589815\pi\)
\(228\) −7.21354 −0.477728
\(229\) 23.8313 1.57482 0.787409 0.616431i \(-0.211422\pi\)
0.787409 + 0.616431i \(0.211422\pi\)
\(230\) 1.67075 0.110166
\(231\) −5.10911 −0.336155
\(232\) −1.00000 −0.0656532
\(233\) 12.4854 0.817945 0.408973 0.912547i \(-0.365887\pi\)
0.408973 + 0.912547i \(0.365887\pi\)
\(234\) −14.8848 −0.973051
\(235\) −5.03519 −0.328459
\(236\) −4.44699 −0.289474
\(237\) −23.9360 −1.55481
\(238\) 14.1531 0.917409
\(239\) 17.6596 1.14230 0.571152 0.820844i \(-0.306497\pi\)
0.571152 + 0.820844i \(0.306497\pi\)
\(240\) 4.17781 0.269676
\(241\) −13.0049 −0.837719 −0.418859 0.908051i \(-0.637570\pi\)
−0.418859 + 0.908051i \(0.637570\pi\)
\(242\) 10.3637 0.666206
\(243\) −22.3452 −1.43345
\(244\) 1.99094 0.127457
\(245\) −0.733286 −0.0468480
\(246\) −20.0855 −1.28061
\(247\) −13.2009 −0.839956
\(248\) −2.79767 −0.177652
\(249\) 20.2236 1.28162
\(250\) 12.0438 0.761715
\(251\) 6.59864 0.416502 0.208251 0.978075i \(-0.433223\pi\)
0.208251 + 0.978075i \(0.433223\pi\)
\(252\) −8.33182 −0.524855
\(253\) −0.797665 −0.0501488
\(254\) −5.24015 −0.328797
\(255\) 23.0840 1.44558
\(256\) 1.00000 0.0625000
\(257\) −0.405645 −0.0253035 −0.0126517 0.999920i \(-0.504027\pi\)
−0.0126517 + 0.999920i \(0.504027\pi\)
\(258\) −10.5109 −0.654380
\(259\) −25.6978 −1.59678
\(260\) 7.64549 0.474153
\(261\) 3.25275 0.201340
\(262\) 13.2715 0.819915
\(263\) 24.4097 1.50516 0.752582 0.658499i \(-0.228808\pi\)
0.752582 + 0.658499i \(0.228808\pi\)
\(264\) −1.99460 −0.122759
\(265\) 4.73618 0.290941
\(266\) −7.38926 −0.453065
\(267\) 28.7215 1.75773
\(268\) −8.31278 −0.507784
\(269\) −19.3943 −1.18249 −0.591245 0.806492i \(-0.701363\pi\)
−0.591245 + 0.806492i \(0.701363\pi\)
\(270\) −1.05596 −0.0642636
\(271\) 32.8754 1.99704 0.998520 0.0543852i \(-0.0173199\pi\)
0.998520 + 0.0543852i \(0.0173199\pi\)
\(272\) 5.52539 0.335026
\(273\) −29.3101 −1.77393
\(274\) 14.2454 0.860596
\(275\) −1.76171 −0.106235
\(276\) −2.50055 −0.150515
\(277\) −4.27774 −0.257024 −0.128512 0.991708i \(-0.541020\pi\)
−0.128512 + 0.991708i \(0.541020\pi\)
\(278\) −7.66694 −0.459833
\(279\) 9.10012 0.544810
\(280\) 4.27958 0.255754
\(281\) −25.1758 −1.50186 −0.750930 0.660381i \(-0.770395\pi\)
−0.750930 + 0.660381i \(0.770395\pi\)
\(282\) 7.53596 0.448760
\(283\) −1.12929 −0.0671295 −0.0335648 0.999437i \(-0.510686\pi\)
−0.0335648 + 0.999437i \(0.510686\pi\)
\(284\) −5.77544 −0.342709
\(285\) −12.0520 −0.713902
\(286\) −3.65017 −0.215839
\(287\) −20.5748 −1.21449
\(288\) −3.25275 −0.191670
\(289\) 13.5299 0.795879
\(290\) −1.67075 −0.0981101
\(291\) −5.56640 −0.326308
\(292\) 2.16312 0.126587
\(293\) −32.9587 −1.92547 −0.962735 0.270446i \(-0.912829\pi\)
−0.962735 + 0.270446i \(0.912829\pi\)
\(294\) 1.09748 0.0640063
\(295\) −7.42982 −0.432581
\(296\) −10.0324 −0.583124
\(297\) 0.504144 0.0292534
\(298\) −7.10247 −0.411435
\(299\) −4.57607 −0.264641
\(300\) −5.52266 −0.318851
\(301\) −10.7670 −0.620597
\(302\) −10.5988 −0.609891
\(303\) −34.3290 −1.97215
\(304\) −2.88478 −0.165453
\(305\) 3.32637 0.190467
\(306\) −17.9727 −1.02743
\(307\) 9.51666 0.543145 0.271572 0.962418i \(-0.412456\pi\)
0.271572 + 0.962418i \(0.412456\pi\)
\(308\) −2.04319 −0.116422
\(309\) −1.52504 −0.0867566
\(310\) −4.67421 −0.265477
\(311\) −12.9151 −0.732347 −0.366174 0.930547i \(-0.619332\pi\)
−0.366174 + 0.930547i \(0.619332\pi\)
\(312\) −11.4427 −0.647815
\(313\) −12.8731 −0.727629 −0.363814 0.931471i \(-0.618526\pi\)
−0.363814 + 0.931471i \(0.618526\pi\)
\(314\) 2.81055 0.158609
\(315\) −13.9204 −0.784327
\(316\) −9.57229 −0.538484
\(317\) 0.422244 0.0237156 0.0118578 0.999930i \(-0.496225\pi\)
0.0118578 + 0.999930i \(0.496225\pi\)
\(318\) −7.08845 −0.397500
\(319\) 0.797665 0.0446607
\(320\) 1.67075 0.0933980
\(321\) 10.3243 0.576248
\(322\) −2.56147 −0.142745
\(323\) −15.9395 −0.886899
\(324\) −8.17785 −0.454325
\(325\) −10.1066 −0.560614
\(326\) −3.78302 −0.209522
\(327\) 41.3234 2.28519
\(328\) −8.03244 −0.443517
\(329\) 7.71954 0.425592
\(330\) −3.33249 −0.183448
\(331\) −3.74901 −0.206064 −0.103032 0.994678i \(-0.532854\pi\)
−0.103032 + 0.994678i \(0.532854\pi\)
\(332\) 8.08767 0.443869
\(333\) 32.6331 1.78828
\(334\) 3.69622 0.202248
\(335\) −13.8886 −0.758817
\(336\) −6.40507 −0.349425
\(337\) −7.09496 −0.386487 −0.193243 0.981151i \(-0.561901\pi\)
−0.193243 + 0.981151i \(0.561901\pi\)
\(338\) −7.94040 −0.431901
\(339\) −19.7270 −1.07142
\(340\) 9.23157 0.500652
\(341\) 2.23160 0.120848
\(342\) 9.38348 0.507400
\(343\) 19.0545 1.02884
\(344\) −4.20344 −0.226634
\(345\) −4.17781 −0.224926
\(346\) 19.2973 1.03743
\(347\) −13.3641 −0.717421 −0.358711 0.933449i \(-0.616784\pi\)
−0.358711 + 0.933449i \(0.616784\pi\)
\(348\) 2.50055 0.134044
\(349\) 2.15690 0.115456 0.0577280 0.998332i \(-0.481614\pi\)
0.0577280 + 0.998332i \(0.481614\pi\)
\(350\) −5.65720 −0.302390
\(351\) 2.89219 0.154374
\(352\) −0.797665 −0.0425157
\(353\) −30.3090 −1.61318 −0.806592 0.591109i \(-0.798690\pi\)
−0.806592 + 0.591109i \(0.798690\pi\)
\(354\) 11.1199 0.591017
\(355\) −9.64935 −0.512134
\(356\) 11.4861 0.608761
\(357\) −35.3905 −1.87307
\(358\) 15.3120 0.809262
\(359\) 36.0830 1.90439 0.952193 0.305496i \(-0.0988224\pi\)
0.952193 + 0.305496i \(0.0988224\pi\)
\(360\) −5.43455 −0.286426
\(361\) −10.6781 −0.562003
\(362\) −9.51985 −0.500352
\(363\) −25.9150 −1.36019
\(364\) −11.7214 −0.614370
\(365\) 3.61404 0.189167
\(366\) −4.97844 −0.260227
\(367\) −31.3302 −1.63542 −0.817712 0.575628i \(-0.804758\pi\)
−0.817712 + 0.575628i \(0.804758\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 26.1276 1.36015
\(370\) −16.7618 −0.871402
\(371\) −7.26112 −0.376979
\(372\) 6.99570 0.362710
\(373\) 15.6582 0.810752 0.405376 0.914150i \(-0.367141\pi\)
0.405376 + 0.914150i \(0.367141\pi\)
\(374\) −4.40741 −0.227902
\(375\) −30.1161 −1.55519
\(376\) 3.01372 0.155421
\(377\) 4.57607 0.235679
\(378\) 1.61891 0.0832677
\(379\) −16.3594 −0.840328 −0.420164 0.907448i \(-0.638027\pi\)
−0.420164 + 0.907448i \(0.638027\pi\)
\(380\) −4.81976 −0.247248
\(381\) 13.1033 0.671301
\(382\) 22.7186 1.16239
\(383\) −5.88216 −0.300564 −0.150282 0.988643i \(-0.548018\pi\)
−0.150282 + 0.988643i \(0.548018\pi\)
\(384\) −2.50055 −0.127606
\(385\) −3.41367 −0.173977
\(386\) 10.7819 0.548784
\(387\) 13.6727 0.695025
\(388\) −2.22607 −0.113012
\(389\) 23.6381 1.19850 0.599249 0.800563i \(-0.295466\pi\)
0.599249 + 0.800563i \(0.295466\pi\)
\(390\) −19.1179 −0.968074
\(391\) −5.52539 −0.279431
\(392\) 0.438895 0.0221676
\(393\) −33.1860 −1.67401
\(394\) 10.7522 0.541688
\(395\) −15.9930 −0.804693
\(396\) 2.59461 0.130384
\(397\) −11.5658 −0.580470 −0.290235 0.956955i \(-0.593734\pi\)
−0.290235 + 0.956955i \(0.593734\pi\)
\(398\) −13.6905 −0.686242
\(399\) 18.4772 0.925018
\(400\) −2.20858 −0.110429
\(401\) 19.1684 0.957222 0.478611 0.878027i \(-0.341140\pi\)
0.478611 + 0.878027i \(0.341140\pi\)
\(402\) 20.7865 1.03674
\(403\) 12.8023 0.637728
\(404\) −13.7286 −0.683022
\(405\) −13.6632 −0.678929
\(406\) 2.56147 0.127123
\(407\) 8.00253 0.396671
\(408\) −13.8165 −0.684020
\(409\) −21.3116 −1.05379 −0.526896 0.849930i \(-0.676644\pi\)
−0.526896 + 0.849930i \(0.676644\pi\)
\(410\) −13.4202 −0.662779
\(411\) −35.6214 −1.75707
\(412\) −0.609882 −0.0300467
\(413\) 11.3908 0.560505
\(414\) 3.25275 0.159864
\(415\) 13.5125 0.663303
\(416\) −4.57607 −0.224360
\(417\) 19.1716 0.938836
\(418\) 2.30109 0.112550
\(419\) −14.3124 −0.699208 −0.349604 0.936898i \(-0.613684\pi\)
−0.349604 + 0.936898i \(0.613684\pi\)
\(420\) −10.7013 −0.522170
\(421\) −9.65385 −0.470500 −0.235250 0.971935i \(-0.575591\pi\)
−0.235250 + 0.971935i \(0.575591\pi\)
\(422\) 19.9301 0.970182
\(423\) −9.80289 −0.476633
\(424\) −2.83475 −0.137668
\(425\) −12.2033 −0.591945
\(426\) 14.4418 0.699707
\(427\) −5.09971 −0.246792
\(428\) 4.12882 0.199574
\(429\) 9.12744 0.440677
\(430\) −7.02291 −0.338675
\(431\) 5.59657 0.269577 0.134789 0.990874i \(-0.456964\pi\)
0.134789 + 0.990874i \(0.456964\pi\)
\(432\) 0.632025 0.0304083
\(433\) 1.30326 0.0626305 0.0313152 0.999510i \(-0.490030\pi\)
0.0313152 + 0.999510i \(0.490030\pi\)
\(434\) 7.16612 0.343985
\(435\) 4.17781 0.200311
\(436\) 16.5257 0.791438
\(437\) 2.88478 0.137998
\(438\) −5.40898 −0.258451
\(439\) 40.7442 1.94462 0.972308 0.233702i \(-0.0750841\pi\)
0.972308 + 0.233702i \(0.0750841\pi\)
\(440\) −1.33270 −0.0635341
\(441\) −1.42762 −0.0679818
\(442\) −25.2846 −1.20266
\(443\) −34.5364 −1.64087 −0.820436 0.571739i \(-0.806269\pi\)
−0.820436 + 0.571739i \(0.806269\pi\)
\(444\) 25.0866 1.19056
\(445\) 19.1904 0.909713
\(446\) −0.826796 −0.0391499
\(447\) 17.7601 0.840024
\(448\) −2.56147 −0.121018
\(449\) 29.2908 1.38232 0.691159 0.722703i \(-0.257101\pi\)
0.691159 + 0.722703i \(0.257101\pi\)
\(450\) 7.18396 0.338655
\(451\) 6.40720 0.301703
\(452\) −7.88907 −0.371071
\(453\) 26.5028 1.24521
\(454\) 8.39006 0.393765
\(455\) −19.5837 −0.918096
\(456\) 7.21354 0.337805
\(457\) 14.8536 0.694822 0.347411 0.937713i \(-0.387061\pi\)
0.347411 + 0.937713i \(0.387061\pi\)
\(458\) −23.8313 −1.11356
\(459\) 3.49218 0.163001
\(460\) −1.67075 −0.0778993
\(461\) −4.31663 −0.201046 −0.100523 0.994935i \(-0.532052\pi\)
−0.100523 + 0.994935i \(0.532052\pi\)
\(462\) 5.10911 0.237697
\(463\) 29.3881 1.36578 0.682891 0.730520i \(-0.260722\pi\)
0.682891 + 0.730520i \(0.260722\pi\)
\(464\) 1.00000 0.0464238
\(465\) 11.6881 0.542023
\(466\) −12.4854 −0.578375
\(467\) −18.9264 −0.875810 −0.437905 0.899021i \(-0.644279\pi\)
−0.437905 + 0.899021i \(0.644279\pi\)
\(468\) 14.8848 0.688051
\(469\) 21.2929 0.983215
\(470\) 5.03519 0.232256
\(471\) −7.02793 −0.323830
\(472\) 4.44699 0.204689
\(473\) 3.35294 0.154168
\(474\) 23.9360 1.09942
\(475\) 6.37126 0.292333
\(476\) −14.1531 −0.648706
\(477\) 9.22076 0.422189
\(478\) −17.6596 −0.807731
\(479\) −16.1906 −0.739768 −0.369884 0.929078i \(-0.620603\pi\)
−0.369884 + 0.929078i \(0.620603\pi\)
\(480\) −4.17781 −0.190690
\(481\) 45.9091 2.09328
\(482\) 13.0049 0.592357
\(483\) 6.40507 0.291441
\(484\) −10.3637 −0.471079
\(485\) −3.71922 −0.168881
\(486\) 22.3452 1.01360
\(487\) −31.6066 −1.43223 −0.716116 0.697981i \(-0.754082\pi\)
−0.716116 + 0.697981i \(0.754082\pi\)
\(488\) −1.99094 −0.0901255
\(489\) 9.45963 0.427779
\(490\) 0.733286 0.0331265
\(491\) −24.1949 −1.09190 −0.545950 0.837818i \(-0.683831\pi\)
−0.545950 + 0.837818i \(0.683831\pi\)
\(492\) 20.0855 0.905526
\(493\) 5.52539 0.248851
\(494\) 13.2009 0.593939
\(495\) 4.33496 0.194842
\(496\) 2.79767 0.125619
\(497\) 14.7936 0.663583
\(498\) −20.2236 −0.906242
\(499\) 4.23893 0.189761 0.0948803 0.995489i \(-0.469753\pi\)
0.0948803 + 0.995489i \(0.469753\pi\)
\(500\) −12.0438 −0.538614
\(501\) −9.24258 −0.412928
\(502\) −6.59864 −0.294512
\(503\) −9.94415 −0.443388 −0.221694 0.975116i \(-0.571159\pi\)
−0.221694 + 0.975116i \(0.571159\pi\)
\(504\) 8.33182 0.371129
\(505\) −22.9371 −1.02069
\(506\) 0.797665 0.0354606
\(507\) 19.8554 0.881808
\(508\) 5.24015 0.232494
\(509\) −20.8154 −0.922627 −0.461314 0.887237i \(-0.652622\pi\)
−0.461314 + 0.887237i \(0.652622\pi\)
\(510\) −23.0840 −1.02218
\(511\) −5.54075 −0.245108
\(512\) −1.00000 −0.0441942
\(513\) −1.82325 −0.0804985
\(514\) 0.405645 0.0178922
\(515\) −1.01896 −0.0449009
\(516\) 10.5109 0.462717
\(517\) −2.40394 −0.105725
\(518\) 25.6978 1.12909
\(519\) −48.2540 −2.11811
\(520\) −7.64549 −0.335277
\(521\) 11.9399 0.523097 0.261549 0.965190i \(-0.415767\pi\)
0.261549 + 0.965190i \(0.415767\pi\)
\(522\) −3.25275 −0.142369
\(523\) 31.7743 1.38939 0.694696 0.719303i \(-0.255539\pi\)
0.694696 + 0.719303i \(0.255539\pi\)
\(524\) −13.2715 −0.579767
\(525\) 14.1461 0.617387
\(526\) −24.4097 −1.06431
\(527\) 15.4582 0.673369
\(528\) 1.99460 0.0868040
\(529\) 1.00000 0.0434783
\(530\) −4.73618 −0.205726
\(531\) −14.4649 −0.627725
\(532\) 7.38926 0.320365
\(533\) 36.7570 1.59212
\(534\) −28.7215 −1.24290
\(535\) 6.89825 0.298238
\(536\) 8.31278 0.359058
\(537\) −38.2883 −1.65226
\(538\) 19.3943 0.836147
\(539\) −0.350092 −0.0150795
\(540\) 1.05596 0.0454412
\(541\) −32.9076 −1.41481 −0.707403 0.706810i \(-0.750134\pi\)
−0.707403 + 0.706810i \(0.750134\pi\)
\(542\) −32.8754 −1.41212
\(543\) 23.8049 1.02156
\(544\) −5.52539 −0.236899
\(545\) 27.6104 1.18270
\(546\) 29.3101 1.25435
\(547\) 12.7196 0.543850 0.271925 0.962319i \(-0.412340\pi\)
0.271925 + 0.962319i \(0.412340\pi\)
\(548\) −14.2454 −0.608534
\(549\) 6.47603 0.276390
\(550\) 1.76171 0.0751194
\(551\) −2.88478 −0.122896
\(552\) 2.50055 0.106431
\(553\) 24.5191 1.04266
\(554\) 4.27774 0.181744
\(555\) 41.9136 1.77913
\(556\) 7.66694 0.325151
\(557\) −37.1466 −1.57395 −0.786975 0.616985i \(-0.788354\pi\)
−0.786975 + 0.616985i \(0.788354\pi\)
\(558\) −9.10012 −0.385239
\(559\) 19.2352 0.813563
\(560\) −4.27958 −0.180845
\(561\) 11.0210 0.465305
\(562\) 25.1758 1.06198
\(563\) −16.8374 −0.709613 −0.354807 0.934940i \(-0.615453\pi\)
−0.354807 + 0.934940i \(0.615453\pi\)
\(564\) −7.53596 −0.317321
\(565\) −13.1807 −0.554516
\(566\) 1.12929 0.0474678
\(567\) 20.9473 0.879703
\(568\) 5.77544 0.242332
\(569\) 11.7904 0.494279 0.247139 0.968980i \(-0.420509\pi\)
0.247139 + 0.968980i \(0.420509\pi\)
\(570\) 12.0520 0.504805
\(571\) −20.8665 −0.873235 −0.436618 0.899647i \(-0.643824\pi\)
−0.436618 + 0.899647i \(0.643824\pi\)
\(572\) 3.65017 0.152621
\(573\) −56.8090 −2.37323
\(574\) 20.5748 0.858777
\(575\) 2.20858 0.0921041
\(576\) 3.25275 0.135531
\(577\) −5.34570 −0.222544 −0.111272 0.993790i \(-0.535493\pi\)
−0.111272 + 0.993790i \(0.535493\pi\)
\(578\) −13.5299 −0.562771
\(579\) −26.9607 −1.12045
\(580\) 1.67075 0.0693743
\(581\) −20.7163 −0.859456
\(582\) 5.56640 0.230735
\(583\) 2.26118 0.0936487
\(584\) −2.16312 −0.0895104
\(585\) 24.8689 1.02820
\(586\) 32.9587 1.36151
\(587\) −18.7393 −0.773452 −0.386726 0.922195i \(-0.626394\pi\)
−0.386726 + 0.922195i \(0.626394\pi\)
\(588\) −1.09748 −0.0452593
\(589\) −8.07065 −0.332545
\(590\) 7.42982 0.305881
\(591\) −26.8864 −1.10596
\(592\) 10.0324 0.412331
\(593\) 27.5959 1.13323 0.566614 0.823983i \(-0.308253\pi\)
0.566614 + 0.823983i \(0.308253\pi\)
\(594\) −0.504144 −0.0206853
\(595\) −23.6464 −0.969406
\(596\) 7.10247 0.290929
\(597\) 34.2338 1.40109
\(598\) 4.57607 0.187129
\(599\) −11.2382 −0.459181 −0.229590 0.973287i \(-0.573739\pi\)
−0.229590 + 0.973287i \(0.573739\pi\)
\(600\) 5.52266 0.225462
\(601\) 11.2614 0.459361 0.229680 0.973266i \(-0.426232\pi\)
0.229680 + 0.973266i \(0.426232\pi\)
\(602\) 10.7670 0.438828
\(603\) −27.0394 −1.10113
\(604\) 10.5988 0.431258
\(605\) −17.3153 −0.703965
\(606\) 34.3290 1.39452
\(607\) 0.676999 0.0274785 0.0137393 0.999906i \(-0.495627\pi\)
0.0137393 + 0.999906i \(0.495627\pi\)
\(608\) 2.88478 0.116993
\(609\) −6.40507 −0.259547
\(610\) −3.32637 −0.134681
\(611\) −13.7910 −0.557924
\(612\) 17.9727 0.726505
\(613\) 24.4238 0.986468 0.493234 0.869897i \(-0.335815\pi\)
0.493234 + 0.869897i \(0.335815\pi\)
\(614\) −9.51666 −0.384061
\(615\) 33.5580 1.35319
\(616\) 2.04319 0.0823226
\(617\) 27.8613 1.12166 0.560828 0.827933i \(-0.310483\pi\)
0.560828 + 0.827933i \(0.310483\pi\)
\(618\) 1.52504 0.0613462
\(619\) 35.3185 1.41957 0.709785 0.704419i \(-0.248792\pi\)
0.709785 + 0.704419i \(0.248792\pi\)
\(620\) 4.67421 0.187721
\(621\) −0.632025 −0.0253623
\(622\) 12.9151 0.517848
\(623\) −29.4212 −1.17873
\(624\) 11.4427 0.458074
\(625\) −9.07929 −0.363172
\(626\) 12.8731 0.514511
\(627\) −5.75399 −0.229792
\(628\) −2.81055 −0.112153
\(629\) 55.4332 2.21026
\(630\) 13.9204 0.554603
\(631\) 3.46959 0.138122 0.0690611 0.997612i \(-0.478000\pi\)
0.0690611 + 0.997612i \(0.478000\pi\)
\(632\) 9.57229 0.380765
\(633\) −49.8362 −1.98081
\(634\) −0.422244 −0.0167694
\(635\) 8.75501 0.347432
\(636\) 7.08845 0.281075
\(637\) −2.00841 −0.0795763
\(638\) −0.797665 −0.0315799
\(639\) −18.7861 −0.743166
\(640\) −1.67075 −0.0660424
\(641\) −22.8078 −0.900855 −0.450427 0.892813i \(-0.648728\pi\)
−0.450427 + 0.892813i \(0.648728\pi\)
\(642\) −10.3243 −0.407469
\(643\) 3.36423 0.132672 0.0663362 0.997797i \(-0.478869\pi\)
0.0663362 + 0.997797i \(0.478869\pi\)
\(644\) 2.56147 0.100936
\(645\) 17.5611 0.691469
\(646\) 15.9395 0.627132
\(647\) −18.1759 −0.714570 −0.357285 0.933995i \(-0.616298\pi\)
−0.357285 + 0.933995i \(0.616298\pi\)
\(648\) 8.17785 0.321256
\(649\) −3.54721 −0.139240
\(650\) 10.1066 0.396414
\(651\) −17.9193 −0.702311
\(652\) 3.78302 0.148154
\(653\) 21.2301 0.830798 0.415399 0.909639i \(-0.363642\pi\)
0.415399 + 0.909639i \(0.363642\pi\)
\(654\) −41.3234 −1.61587
\(655\) −22.1734 −0.866386
\(656\) 8.03244 0.313614
\(657\) 7.03609 0.274504
\(658\) −7.71954 −0.300939
\(659\) −32.6757 −1.27287 −0.636433 0.771332i \(-0.719591\pi\)
−0.636433 + 0.771332i \(0.719591\pi\)
\(660\) 3.33249 0.129717
\(661\) −34.5493 −1.34381 −0.671907 0.740636i \(-0.734524\pi\)
−0.671907 + 0.740636i \(0.734524\pi\)
\(662\) 3.74901 0.145709
\(663\) 63.2253 2.45547
\(664\) −8.08767 −0.313862
\(665\) 12.3456 0.478744
\(666\) −32.6331 −1.26451
\(667\) −1.00000 −0.0387202
\(668\) −3.69622 −0.143011
\(669\) 2.06745 0.0799321
\(670\) 13.8886 0.536564
\(671\) 1.58810 0.0613080
\(672\) 6.40507 0.247081
\(673\) 18.5932 0.716716 0.358358 0.933584i \(-0.383337\pi\)
0.358358 + 0.933584i \(0.383337\pi\)
\(674\) 7.09496 0.273288
\(675\) −1.39588 −0.0537273
\(676\) 7.94040 0.305400
\(677\) 10.3135 0.396381 0.198190 0.980164i \(-0.436494\pi\)
0.198190 + 0.980164i \(0.436494\pi\)
\(678\) 19.7270 0.757612
\(679\) 5.70200 0.218823
\(680\) −9.23157 −0.354015
\(681\) −20.9798 −0.803946
\(682\) −2.23160 −0.0854524
\(683\) 4.79758 0.183574 0.0917872 0.995779i \(-0.470742\pi\)
0.0917872 + 0.995779i \(0.470742\pi\)
\(684\) −9.38348 −0.358786
\(685\) −23.8006 −0.909373
\(686\) −19.0545 −0.727503
\(687\) 59.5914 2.27355
\(688\) 4.20344 0.160254
\(689\) 12.9720 0.494195
\(690\) 4.17781 0.159046
\(691\) −15.7460 −0.599008 −0.299504 0.954095i \(-0.596821\pi\)
−0.299504 + 0.954095i \(0.596821\pi\)
\(692\) −19.2973 −0.733575
\(693\) −6.64600 −0.252461
\(694\) 13.3641 0.507293
\(695\) 12.8096 0.485895
\(696\) −2.50055 −0.0947831
\(697\) 44.3824 1.68110
\(698\) −2.15690 −0.0816398
\(699\) 31.2204 1.18086
\(700\) 5.65720 0.213822
\(701\) 1.37538 0.0519474 0.0259737 0.999663i \(-0.491731\pi\)
0.0259737 + 0.999663i \(0.491731\pi\)
\(702\) −2.89219 −0.109159
\(703\) −28.9414 −1.09154
\(704\) 0.797665 0.0300631
\(705\) −12.5907 −0.474195
\(706\) 30.3090 1.14069
\(707\) 35.1653 1.32253
\(708\) −11.1199 −0.417912
\(709\) 28.2420 1.06065 0.530325 0.847795i \(-0.322070\pi\)
0.530325 + 0.847795i \(0.322070\pi\)
\(710\) 9.64935 0.362133
\(711\) −31.1363 −1.16770
\(712\) −11.4861 −0.430459
\(713\) −2.79767 −0.104773
\(714\) 35.3905 1.32446
\(715\) 6.09854 0.228072
\(716\) −15.3120 −0.572234
\(717\) 44.1587 1.64914
\(718\) −36.0830 −1.34660
\(719\) 41.5078 1.54798 0.773990 0.633198i \(-0.218258\pi\)
0.773990 + 0.633198i \(0.218258\pi\)
\(720\) 5.43455 0.202534
\(721\) 1.56219 0.0581791
\(722\) 10.6781 0.397396
\(723\) −32.5194 −1.20941
\(724\) 9.51985 0.353802
\(725\) −2.20858 −0.0820245
\(726\) 25.9150 0.961797
\(727\) −5.52586 −0.204943 −0.102471 0.994736i \(-0.532675\pi\)
−0.102471 + 0.994736i \(0.532675\pi\)
\(728\) 11.7214 0.434425
\(729\) −31.3418 −1.16081
\(730\) −3.61404 −0.133762
\(731\) 23.2256 0.859031
\(732\) 4.97844 0.184008
\(733\) 25.6062 0.945785 0.472892 0.881120i \(-0.343210\pi\)
0.472892 + 0.881120i \(0.343210\pi\)
\(734\) 31.3302 1.15642
\(735\) −1.83362 −0.0676341
\(736\) 1.00000 0.0368605
\(737\) −6.63082 −0.244249
\(738\) −26.1276 −0.961769
\(739\) 33.8805 1.24631 0.623157 0.782097i \(-0.285850\pi\)
0.623157 + 0.782097i \(0.285850\pi\)
\(740\) 16.7618 0.616174
\(741\) −33.0096 −1.21264
\(742\) 7.26112 0.266564
\(743\) −50.4493 −1.85081 −0.925404 0.378983i \(-0.876274\pi\)
−0.925404 + 0.378983i \(0.876274\pi\)
\(744\) −6.99570 −0.256475
\(745\) 11.8665 0.434755
\(746\) −15.6582 −0.573288
\(747\) 26.3072 0.962530
\(748\) 4.40741 0.161151
\(749\) −10.5758 −0.386433
\(750\) 30.1161 1.09968
\(751\) 4.77959 0.174410 0.0872049 0.996190i \(-0.472207\pi\)
0.0872049 + 0.996190i \(0.472207\pi\)
\(752\) −3.01372 −0.109899
\(753\) 16.5002 0.601302
\(754\) −4.57607 −0.166651
\(755\) 17.7080 0.644459
\(756\) −1.61891 −0.0588792
\(757\) 46.3627 1.68508 0.842541 0.538633i \(-0.181059\pi\)
0.842541 + 0.538633i \(0.181059\pi\)
\(758\) 16.3594 0.594202
\(759\) −1.99460 −0.0723995
\(760\) 4.81976 0.174831
\(761\) 45.1079 1.63516 0.817580 0.575815i \(-0.195315\pi\)
0.817580 + 0.575815i \(0.195315\pi\)
\(762\) −13.1033 −0.474681
\(763\) −42.3300 −1.53245
\(764\) −22.7186 −0.821931
\(765\) 30.0280 1.08567
\(766\) 5.88216 0.212531
\(767\) −20.3497 −0.734785
\(768\) 2.50055 0.0902309
\(769\) 13.2403 0.477456 0.238728 0.971087i \(-0.423270\pi\)
0.238728 + 0.971087i \(0.423270\pi\)
\(770\) 3.41367 0.123020
\(771\) −1.01434 −0.0365304
\(772\) −10.7819 −0.388049
\(773\) 23.0111 0.827651 0.413826 0.910356i \(-0.364192\pi\)
0.413826 + 0.910356i \(0.364192\pi\)
\(774\) −13.6727 −0.491457
\(775\) −6.17886 −0.221951
\(776\) 2.22607 0.0799112
\(777\) −64.2585 −2.30526
\(778\) −23.6381 −0.847466
\(779\) −23.1718 −0.830217
\(780\) 19.1179 0.684532
\(781\) −4.60687 −0.164847
\(782\) 5.52539 0.197588
\(783\) 0.632025 0.0225867
\(784\) −0.438895 −0.0156748
\(785\) −4.69574 −0.167598
\(786\) 33.1860 1.18371
\(787\) −44.0639 −1.57071 −0.785354 0.619047i \(-0.787519\pi\)
−0.785354 + 0.619047i \(0.787519\pi\)
\(788\) −10.7522 −0.383031
\(789\) 61.0376 2.17300
\(790\) 15.9930 0.569004
\(791\) 20.2076 0.718499
\(792\) −2.59461 −0.0921954
\(793\) 9.11066 0.323529
\(794\) 11.5658 0.410454
\(795\) 11.8431 0.420030
\(796\) 13.6905 0.485247
\(797\) 22.9165 0.811746 0.405873 0.913930i \(-0.366968\pi\)
0.405873 + 0.913930i \(0.366968\pi\)
\(798\) −18.4772 −0.654087
\(799\) −16.6520 −0.589105
\(800\) 2.20858 0.0780850
\(801\) 37.3614 1.32010
\(802\) −19.1684 −0.676858
\(803\) 1.72544 0.0608896
\(804\) −20.7865 −0.733085
\(805\) 4.27958 0.150835
\(806\) −12.8023 −0.450942
\(807\) −48.4964 −1.70715
\(808\) 13.7286 0.482969
\(809\) −31.9211 −1.12229 −0.561144 0.827718i \(-0.689639\pi\)
−0.561144 + 0.827718i \(0.689639\pi\)
\(810\) 13.6632 0.480075
\(811\) 32.2097 1.13103 0.565517 0.824737i \(-0.308677\pi\)
0.565517 + 0.824737i \(0.308677\pi\)
\(812\) −2.56147 −0.0898898
\(813\) 82.2067 2.88311
\(814\) −8.00253 −0.280489
\(815\) 6.32050 0.221397
\(816\) 13.8165 0.483675
\(817\) −12.1260 −0.424234
\(818\) 21.3116 0.745144
\(819\) −38.1270 −1.33226
\(820\) 13.4202 0.468655
\(821\) 31.5158 1.09991 0.549954 0.835195i \(-0.314645\pi\)
0.549954 + 0.835195i \(0.314645\pi\)
\(822\) 35.6214 1.24244
\(823\) 37.8969 1.32100 0.660501 0.750825i \(-0.270344\pi\)
0.660501 + 0.750825i \(0.270344\pi\)
\(824\) 0.609882 0.0212463
\(825\) −4.40524 −0.153371
\(826\) −11.3908 −0.396337
\(827\) 23.5001 0.817178 0.408589 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(828\) −3.25275 −0.113041
\(829\) 12.1249 0.421114 0.210557 0.977582i \(-0.432472\pi\)
0.210557 + 0.977582i \(0.432472\pi\)
\(830\) −13.5125 −0.469026
\(831\) −10.6967 −0.371064
\(832\) 4.57607 0.158647
\(833\) −2.42507 −0.0840236
\(834\) −19.1716 −0.663857
\(835\) −6.17547 −0.213711
\(836\) −2.30109 −0.0795848
\(837\) 1.76819 0.0611177
\(838\) 14.3124 0.494414
\(839\) 26.4482 0.913094 0.456547 0.889699i \(-0.349086\pi\)
0.456547 + 0.889699i \(0.349086\pi\)
\(840\) 10.7013 0.369230
\(841\) 1.00000 0.0344828
\(842\) 9.65385 0.332693
\(843\) −62.9533 −2.16823
\(844\) −19.9301 −0.686022
\(845\) 13.2665 0.456380
\(846\) 9.80289 0.337030
\(847\) 26.5463 0.912143
\(848\) 2.83475 0.0973458
\(849\) −2.82386 −0.0969145
\(850\) 12.2033 0.418568
\(851\) −10.0324 −0.343908
\(852\) −14.4418 −0.494767
\(853\) 14.0206 0.480057 0.240028 0.970766i \(-0.422843\pi\)
0.240028 + 0.970766i \(0.422843\pi\)
\(854\) 5.09971 0.174509
\(855\) −15.6775 −0.536159
\(856\) −4.12882 −0.141120
\(857\) 11.0292 0.376751 0.188375 0.982097i \(-0.439678\pi\)
0.188375 + 0.982097i \(0.439678\pi\)
\(858\) −9.12744 −0.311606
\(859\) −42.6614 −1.45559 −0.727793 0.685796i \(-0.759454\pi\)
−0.727793 + 0.685796i \(0.759454\pi\)
\(860\) 7.02291 0.239479
\(861\) −51.4484 −1.75336
\(862\) −5.59657 −0.190620
\(863\) −0.518121 −0.0176370 −0.00881851 0.999961i \(-0.502807\pi\)
−0.00881851 + 0.999961i \(0.502807\pi\)
\(864\) −0.632025 −0.0215019
\(865\) −32.2411 −1.09623
\(866\) −1.30326 −0.0442864
\(867\) 33.8323 1.14900
\(868\) −7.16612 −0.243234
\(869\) −7.63549 −0.259016
\(870\) −4.17781 −0.141641
\(871\) −38.0399 −1.28893
\(872\) −16.5257 −0.559631
\(873\) −7.24085 −0.245066
\(874\) −2.88478 −0.0975791
\(875\) 30.8497 1.04291
\(876\) 5.40898 0.182753
\(877\) 19.3340 0.652862 0.326431 0.945221i \(-0.394154\pi\)
0.326431 + 0.945221i \(0.394154\pi\)
\(878\) −40.7442 −1.37505
\(879\) −82.4150 −2.77979
\(880\) 1.33270 0.0449254
\(881\) 17.5513 0.591319 0.295660 0.955293i \(-0.404461\pi\)
0.295660 + 0.955293i \(0.404461\pi\)
\(882\) 1.42762 0.0480704
\(883\) −1.01735 −0.0342366 −0.0171183 0.999853i \(-0.505449\pi\)
−0.0171183 + 0.999853i \(0.505449\pi\)
\(884\) 25.2846 0.850412
\(885\) −18.5786 −0.624514
\(886\) 34.5364 1.16027
\(887\) 20.0120 0.671937 0.335969 0.941873i \(-0.390936\pi\)
0.335969 + 0.941873i \(0.390936\pi\)
\(888\) −25.0866 −0.841852
\(889\) −13.4225 −0.450175
\(890\) −19.1904 −0.643264
\(891\) −6.52319 −0.218535
\(892\) 0.826796 0.0276832
\(893\) 8.69392 0.290931
\(894\) −17.7601 −0.593986
\(895\) −25.5825 −0.855129
\(896\) 2.56147 0.0855726
\(897\) −11.4427 −0.382060
\(898\) −29.2908 −0.977446
\(899\) 2.79767 0.0933074
\(900\) −7.18396 −0.239465
\(901\) 15.6631 0.521814
\(902\) −6.40720 −0.213336
\(903\) −26.9233 −0.895952
\(904\) 7.88907 0.262387
\(905\) 15.9053 0.528711
\(906\) −26.5028 −0.880496
\(907\) −14.1315 −0.469228 −0.234614 0.972089i \(-0.575383\pi\)
−0.234614 + 0.972089i \(0.575383\pi\)
\(908\) −8.39006 −0.278434
\(909\) −44.6557 −1.48113
\(910\) 19.5837 0.649192
\(911\) 0.128010 0.00424118 0.00212059 0.999998i \(-0.499325\pi\)
0.00212059 + 0.999998i \(0.499325\pi\)
\(912\) −7.21354 −0.238864
\(913\) 6.45125 0.213505
\(914\) −14.8536 −0.491314
\(915\) 8.31775 0.274976
\(916\) 23.8313 0.787409
\(917\) 33.9944 1.12259
\(918\) −3.49218 −0.115259
\(919\) 4.54799 0.150024 0.0750121 0.997183i \(-0.476100\pi\)
0.0750121 + 0.997183i \(0.476100\pi\)
\(920\) 1.67075 0.0550832
\(921\) 23.7969 0.784134
\(922\) 4.31663 0.142161
\(923\) −26.4288 −0.869915
\(924\) −5.10911 −0.168077
\(925\) −22.1574 −0.728532
\(926\) −29.3881 −0.965754
\(927\) −1.98380 −0.0651564
\(928\) −1.00000 −0.0328266
\(929\) 43.6631 1.43254 0.716271 0.697823i \(-0.245848\pi\)
0.716271 + 0.697823i \(0.245848\pi\)
\(930\) −11.6881 −0.383268
\(931\) 1.26612 0.0414953
\(932\) 12.4854 0.408973
\(933\) −32.2948 −1.05729
\(934\) 18.9264 0.619291
\(935\) 7.36371 0.240819
\(936\) −14.8848 −0.486526
\(937\) 2.64749 0.0864896 0.0432448 0.999065i \(-0.486230\pi\)
0.0432448 + 0.999065i \(0.486230\pi\)
\(938\) −21.2929 −0.695238
\(939\) −32.1898 −1.05047
\(940\) −5.03519 −0.164230
\(941\) 1.71934 0.0560488 0.0280244 0.999607i \(-0.491078\pi\)
0.0280244 + 0.999607i \(0.491078\pi\)
\(942\) 7.02793 0.228982
\(943\) −8.03244 −0.261572
\(944\) −4.44699 −0.144737
\(945\) −2.70480 −0.0879872
\(946\) −3.35294 −0.109013
\(947\) 3.71518 0.120727 0.0603635 0.998176i \(-0.480774\pi\)
0.0603635 + 0.998176i \(0.480774\pi\)
\(948\) −23.9360 −0.777405
\(949\) 9.89857 0.321321
\(950\) −6.37126 −0.206711
\(951\) 1.05584 0.0342380
\(952\) 14.1531 0.458704
\(953\) −30.0759 −0.974255 −0.487127 0.873331i \(-0.661955\pi\)
−0.487127 + 0.873331i \(0.661955\pi\)
\(954\) −9.22076 −0.298533
\(955\) −37.9572 −1.22827
\(956\) 17.6596 0.571152
\(957\) 1.99460 0.0644764
\(958\) 16.1906 0.523095
\(959\) 36.4891 1.17829
\(960\) 4.17781 0.134838
\(961\) −23.1731 −0.747518
\(962\) −45.9091 −1.48017
\(963\) 13.4301 0.432777
\(964\) −13.0049 −0.418859
\(965\) −18.0139 −0.579888
\(966\) −6.40507 −0.206080
\(967\) −55.4004 −1.78156 −0.890778 0.454439i \(-0.849840\pi\)
−0.890778 + 0.454439i \(0.849840\pi\)
\(968\) 10.3637 0.333103
\(969\) −39.8576 −1.28041
\(970\) 3.71922 0.119417
\(971\) −36.5221 −1.17205 −0.586025 0.810293i \(-0.699308\pi\)
−0.586025 + 0.810293i \(0.699308\pi\)
\(972\) −22.3452 −0.716723
\(973\) −19.6386 −0.629585
\(974\) 31.6066 1.01274
\(975\) −25.2721 −0.809354
\(976\) 1.99094 0.0637283
\(977\) 18.6925 0.598025 0.299013 0.954249i \(-0.403343\pi\)
0.299013 + 0.954249i \(0.403343\pi\)
\(978\) −9.45963 −0.302486
\(979\) 9.16205 0.292820
\(980\) −0.733286 −0.0234240
\(981\) 53.7541 1.71623
\(982\) 24.1949 0.772090
\(983\) −20.0322 −0.638928 −0.319464 0.947598i \(-0.603503\pi\)
−0.319464 + 0.947598i \(0.603503\pi\)
\(984\) −20.0855 −0.640303
\(985\) −17.9643 −0.572390
\(986\) −5.52539 −0.175964
\(987\) 19.3031 0.614424
\(988\) −13.2009 −0.419978
\(989\) −4.20344 −0.133661
\(990\) −4.33496 −0.137774
\(991\) 51.9608 1.65059 0.825295 0.564703i \(-0.191009\pi\)
0.825295 + 0.564703i \(0.191009\pi\)
\(992\) −2.79767 −0.0888260
\(993\) −9.37459 −0.297494
\(994\) −14.7936 −0.469224
\(995\) 22.8735 0.725137
\(996\) 20.2236 0.640810
\(997\) −17.3787 −0.550388 −0.275194 0.961389i \(-0.588742\pi\)
−0.275194 + 0.961389i \(0.588742\pi\)
\(998\) −4.23893 −0.134181
\(999\) 6.34075 0.200612
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 1334.2.a.j.1.9 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1334.2.a.j.1.9 9 1.1 even 1 trivial