Properties

Label 4022.2.a.f.1.19
Level $4022$
Weight $2$
Character 4022.1
Self dual yes
Analytic conductor $32.116$
Analytic rank $0$
Dimension $50$
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] = [4022,2,Mod(1,4022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4022, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4022 = 2 \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4022.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: \(32.1158316930\)
Analytic rank: \(0\)
Dimension: \(50\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Character \(\chi\) \(=\) 4022.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.424610 q^{3} +1.00000 q^{4} -2.05502 q^{5} -0.424610 q^{6} +3.71032 q^{7} +1.00000 q^{8} -2.81971 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.424610 q^{3} +1.00000 q^{4} -2.05502 q^{5} -0.424610 q^{6} +3.71032 q^{7} +1.00000 q^{8} -2.81971 q^{9} -2.05502 q^{10} +0.844185 q^{11} -0.424610 q^{12} +6.36186 q^{13} +3.71032 q^{14} +0.872583 q^{15} +1.00000 q^{16} -1.24571 q^{17} -2.81971 q^{18} +4.46042 q^{19} -2.05502 q^{20} -1.57544 q^{21} +0.844185 q^{22} -1.59263 q^{23} -0.424610 q^{24} -0.776890 q^{25} +6.36186 q^{26} +2.47111 q^{27} +3.71032 q^{28} +1.82436 q^{29} +0.872583 q^{30} +6.01560 q^{31} +1.00000 q^{32} -0.358450 q^{33} -1.24571 q^{34} -7.62478 q^{35} -2.81971 q^{36} -2.03037 q^{37} +4.46042 q^{38} -2.70131 q^{39} -2.05502 q^{40} -9.59480 q^{41} -1.57544 q^{42} -3.91594 q^{43} +0.844185 q^{44} +5.79455 q^{45} -1.59263 q^{46} -10.4573 q^{47} -0.424610 q^{48} +6.76645 q^{49} -0.776890 q^{50} +0.528942 q^{51} +6.36186 q^{52} +11.0584 q^{53} +2.47111 q^{54} -1.73482 q^{55} +3.71032 q^{56} -1.89394 q^{57} +1.82436 q^{58} -9.42185 q^{59} +0.872583 q^{60} +4.44405 q^{61} +6.01560 q^{62} -10.4620 q^{63} +1.00000 q^{64} -13.0738 q^{65} -0.358450 q^{66} +5.39038 q^{67} -1.24571 q^{68} +0.676247 q^{69} -7.62478 q^{70} +11.7903 q^{71} -2.81971 q^{72} +15.1801 q^{73} -2.03037 q^{74} +0.329876 q^{75} +4.46042 q^{76} +3.13219 q^{77} -2.70131 q^{78} +8.93597 q^{79} -2.05502 q^{80} +7.40986 q^{81} -9.59480 q^{82} +6.99898 q^{83} -1.57544 q^{84} +2.55997 q^{85} -3.91594 q^{86} -0.774644 q^{87} +0.844185 q^{88} -6.60152 q^{89} +5.79455 q^{90} +23.6045 q^{91} -1.59263 q^{92} -2.55428 q^{93} -10.4573 q^{94} -9.16625 q^{95} -0.424610 q^{96} +12.0468 q^{97} +6.76645 q^{98} -2.38035 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50 q + 50 q^{2} + 18 q^{3} + 50 q^{4} + 11 q^{5} + 18 q^{6} + 30 q^{7} + 50 q^{8} + 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 50 q + 50 q^{2} + 18 q^{3} + 50 q^{4} + 11 q^{5} + 18 q^{6} + 30 q^{7} + 50 q^{8} + 66 q^{9} + 11 q^{10} + 21 q^{11} + 18 q^{12} + 26 q^{13} + 30 q^{14} + 17 q^{15} + 50 q^{16} + 24 q^{17} + 66 q^{18} + 39 q^{19} + 11 q^{20} - q^{21} + 21 q^{22} + 28 q^{23} + 18 q^{24} + 79 q^{25} + 26 q^{26} + 66 q^{27} + 30 q^{28} - 5 q^{29} + 17 q^{30} + 60 q^{31} + 50 q^{32} + 37 q^{33} + 24 q^{34} + 38 q^{35} + 66 q^{36} + 35 q^{37} + 39 q^{38} + 37 q^{39} + 11 q^{40} + 42 q^{41} - q^{42} + 44 q^{43} + 21 q^{44} + 31 q^{45} + 28 q^{46} + 60 q^{47} + 18 q^{48} + 92 q^{49} + 79 q^{50} + 26 q^{51} + 26 q^{52} - 2 q^{53} + 66 q^{54} + 33 q^{55} + 30 q^{56} + 15 q^{57} - 5 q^{58} + 65 q^{59} + 17 q^{60} + 15 q^{61} + 60 q^{62} + 56 q^{63} + 50 q^{64} + 6 q^{65} + 37 q^{66} + 48 q^{67} + 24 q^{68} - 9 q^{69} + 38 q^{70} + 34 q^{71} + 66 q^{72} + 91 q^{73} + 35 q^{74} + 54 q^{75} + 39 q^{76} - 6 q^{77} + 37 q^{78} + 29 q^{79} + 11 q^{80} + 66 q^{81} + 42 q^{82} + 43 q^{83} - q^{84} + 44 q^{86} + 32 q^{87} + 21 q^{88} + 38 q^{89} + 31 q^{90} + 55 q^{91} + 28 q^{92} - 15 q^{93} + 60 q^{94} + 9 q^{95} + 18 q^{96} + 80 q^{97} + 92 q^{98} + 20 q^{99}+O(q^{100}) \) 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
\(3\) −0.424610 −0.245149 −0.122574 0.992459i \(-0.539115\pi\)
−0.122574 + 0.992459i \(0.539115\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.05502 −0.919033 −0.459517 0.888169i \(-0.651977\pi\)
−0.459517 + 0.888169i \(0.651977\pi\)
\(6\) −0.424610 −0.173346
\(7\) 3.71032 1.40237 0.701184 0.712980i \(-0.252655\pi\)
0.701184 + 0.712980i \(0.252655\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.81971 −0.939902
\(10\) −2.05502 −0.649855
\(11\) 0.844185 0.254531 0.127266 0.991869i \(-0.459380\pi\)
0.127266 + 0.991869i \(0.459380\pi\)
\(12\) −0.424610 −0.122574
\(13\) 6.36186 1.76446 0.882232 0.470815i \(-0.156040\pi\)
0.882232 + 0.470815i \(0.156040\pi\)
\(14\) 3.71032 0.991624
\(15\) 0.872583 0.225300
\(16\) 1.00000 0.250000
\(17\) −1.24571 −0.302130 −0.151065 0.988524i \(-0.548270\pi\)
−0.151065 + 0.988524i \(0.548270\pi\)
\(18\) −2.81971 −0.664611
\(19\) 4.46042 1.02329 0.511645 0.859197i \(-0.329036\pi\)
0.511645 + 0.859197i \(0.329036\pi\)
\(20\) −2.05502 −0.459517
\(21\) −1.57544 −0.343789
\(22\) 0.844185 0.179981
\(23\) −1.59263 −0.332086 −0.166043 0.986118i \(-0.553099\pi\)
−0.166043 + 0.986118i \(0.553099\pi\)
\(24\) −0.424610 −0.0866732
\(25\) −0.776890 −0.155378
\(26\) 6.36186 1.24766
\(27\) 2.47111 0.475565
\(28\) 3.71032 0.701184
\(29\) 1.82436 0.338776 0.169388 0.985549i \(-0.445821\pi\)
0.169388 + 0.985549i \(0.445821\pi\)
\(30\) 0.872583 0.159311
\(31\) 6.01560 1.08043 0.540217 0.841526i \(-0.318342\pi\)
0.540217 + 0.841526i \(0.318342\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.358450 −0.0623981
\(34\) −1.24571 −0.213638
\(35\) −7.62478 −1.28882
\(36\) −2.81971 −0.469951
\(37\) −2.03037 −0.333791 −0.166896 0.985975i \(-0.553374\pi\)
−0.166896 + 0.985975i \(0.553374\pi\)
\(38\) 4.46042 0.723575
\(39\) −2.70131 −0.432556
\(40\) −2.05502 −0.324927
\(41\) −9.59480 −1.49846 −0.749228 0.662312i \(-0.769575\pi\)
−0.749228 + 0.662312i \(0.769575\pi\)
\(42\) −1.57544 −0.243095
\(43\) −3.91594 −0.597175 −0.298587 0.954382i \(-0.596515\pi\)
−0.298587 + 0.954382i \(0.596515\pi\)
\(44\) 0.844185 0.127266
\(45\) 5.79455 0.863801
\(46\) −1.59263 −0.234820
\(47\) −10.4573 −1.52536 −0.762678 0.646778i \(-0.776116\pi\)
−0.762678 + 0.646778i \(0.776116\pi\)
\(48\) −0.424610 −0.0612872
\(49\) 6.76645 0.966636
\(50\) −0.776890 −0.109869
\(51\) 0.528942 0.0740667
\(52\) 6.36186 0.882232
\(53\) 11.0584 1.51899 0.759493 0.650516i \(-0.225447\pi\)
0.759493 + 0.650516i \(0.225447\pi\)
\(54\) 2.47111 0.336275
\(55\) −1.73482 −0.233923
\(56\) 3.71032 0.495812
\(57\) −1.89394 −0.250858
\(58\) 1.82436 0.239551
\(59\) −9.42185 −1.22662 −0.613310 0.789842i \(-0.710163\pi\)
−0.613310 + 0.789842i \(0.710163\pi\)
\(60\) 0.872583 0.112650
\(61\) 4.44405 0.569003 0.284501 0.958676i \(-0.408172\pi\)
0.284501 + 0.958676i \(0.408172\pi\)
\(62\) 6.01560 0.763982
\(63\) −10.4620 −1.31809
\(64\) 1.00000 0.125000
\(65\) −13.0738 −1.62160
\(66\) −0.358450 −0.0441221
\(67\) 5.39038 0.658540 0.329270 0.944236i \(-0.393197\pi\)
0.329270 + 0.944236i \(0.393197\pi\)
\(68\) −1.24571 −0.151065
\(69\) 0.676247 0.0814106
\(70\) −7.62478 −0.911335
\(71\) 11.7903 1.39925 0.699623 0.714512i \(-0.253351\pi\)
0.699623 + 0.714512i \(0.253351\pi\)
\(72\) −2.81971 −0.332306
\(73\) 15.1801 1.77669 0.888346 0.459175i \(-0.151855\pi\)
0.888346 + 0.459175i \(0.151855\pi\)
\(74\) −2.03037 −0.236026
\(75\) 0.329876 0.0380908
\(76\) 4.46042 0.511645
\(77\) 3.13219 0.356947
\(78\) −2.70131 −0.305863
\(79\) 8.93597 1.00538 0.502688 0.864468i \(-0.332345\pi\)
0.502688 + 0.864468i \(0.332345\pi\)
\(80\) −2.05502 −0.229758
\(81\) 7.40986 0.823318
\(82\) −9.59480 −1.05957
\(83\) 6.99898 0.768237 0.384119 0.923284i \(-0.374505\pi\)
0.384119 + 0.923284i \(0.374505\pi\)
\(84\) −1.57544 −0.171894
\(85\) 2.55997 0.277667
\(86\) −3.91594 −0.422266
\(87\) −0.774644 −0.0830505
\(88\) 0.844185 0.0899904
\(89\) −6.60152 −0.699760 −0.349880 0.936794i \(-0.613778\pi\)
−0.349880 + 0.936794i \(0.613778\pi\)
\(90\) 5.79455 0.610800
\(91\) 23.6045 2.47443
\(92\) −1.59263 −0.166043
\(93\) −2.55428 −0.264867
\(94\) −10.4573 −1.07859
\(95\) −9.16625 −0.940437
\(96\) −0.424610 −0.0433366
\(97\) 12.0468 1.22317 0.611584 0.791180i \(-0.290533\pi\)
0.611584 + 0.791180i \(0.290533\pi\)
\(98\) 6.76645 0.683515
\(99\) −2.38035 −0.239234
\(100\) −0.776890 −0.0776890
\(101\) 9.59434 0.954672 0.477336 0.878721i \(-0.341602\pi\)
0.477336 + 0.878721i \(0.341602\pi\)
\(102\) 0.528942 0.0523731
\(103\) −10.7695 −1.06115 −0.530576 0.847637i \(-0.678024\pi\)
−0.530576 + 0.847637i \(0.678024\pi\)
\(104\) 6.36186 0.623832
\(105\) 3.23756 0.315953
\(106\) 11.0584 1.07409
\(107\) 10.6939 1.03382 0.516910 0.856040i \(-0.327082\pi\)
0.516910 + 0.856040i \(0.327082\pi\)
\(108\) 2.47111 0.237782
\(109\) 0.791338 0.0757965 0.0378982 0.999282i \(-0.487934\pi\)
0.0378982 + 0.999282i \(0.487934\pi\)
\(110\) −1.73482 −0.165408
\(111\) 0.862118 0.0818286
\(112\) 3.71032 0.350592
\(113\) −1.79381 −0.168748 −0.0843738 0.996434i \(-0.526889\pi\)
−0.0843738 + 0.996434i \(0.526889\pi\)
\(114\) −1.89394 −0.177384
\(115\) 3.27289 0.305198
\(116\) 1.82436 0.169388
\(117\) −17.9386 −1.65842
\(118\) −9.42185 −0.867352
\(119\) −4.62199 −0.423697
\(120\) 0.872583 0.0796556
\(121\) −10.2874 −0.935214
\(122\) 4.44405 0.402346
\(123\) 4.07405 0.367345
\(124\) 6.01560 0.540217
\(125\) 11.8716 1.06183
\(126\) −10.4620 −0.932029
\(127\) 20.0409 1.77834 0.889171 0.457575i \(-0.151282\pi\)
0.889171 + 0.457575i \(0.151282\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.66275 0.146397
\(130\) −13.0738 −1.14664
\(131\) 0.607831 0.0531064 0.0265532 0.999647i \(-0.491547\pi\)
0.0265532 + 0.999647i \(0.491547\pi\)
\(132\) −0.358450 −0.0311990
\(133\) 16.5496 1.43503
\(134\) 5.39038 0.465658
\(135\) −5.07818 −0.437060
\(136\) −1.24571 −0.106819
\(137\) 10.6998 0.914147 0.457073 0.889429i \(-0.348898\pi\)
0.457073 + 0.889429i \(0.348898\pi\)
\(138\) 0.676247 0.0575660
\(139\) −15.1789 −1.28746 −0.643728 0.765254i \(-0.722613\pi\)
−0.643728 + 0.765254i \(0.722613\pi\)
\(140\) −7.62478 −0.644411
\(141\) 4.44029 0.373939
\(142\) 11.7903 0.989417
\(143\) 5.37059 0.449111
\(144\) −2.81971 −0.234976
\(145\) −3.74911 −0.311346
\(146\) 15.1801 1.25631
\(147\) −2.87311 −0.236970
\(148\) −2.03037 −0.166896
\(149\) 10.7807 0.883186 0.441593 0.897216i \(-0.354414\pi\)
0.441593 + 0.897216i \(0.354414\pi\)
\(150\) 0.329876 0.0269342
\(151\) −20.7420 −1.68796 −0.843981 0.536374i \(-0.819794\pi\)
−0.843981 + 0.536374i \(0.819794\pi\)
\(152\) 4.46042 0.361787
\(153\) 3.51254 0.283972
\(154\) 3.13219 0.252399
\(155\) −12.3622 −0.992954
\(156\) −2.70131 −0.216278
\(157\) −0.0174066 −0.00138920 −0.000694599 1.00000i \(-0.500221\pi\)
−0.000694599 1.00000i \(0.500221\pi\)
\(158\) 8.93597 0.710908
\(159\) −4.69550 −0.372378
\(160\) −2.05502 −0.162464
\(161\) −5.90916 −0.465707
\(162\) 7.40986 0.582174
\(163\) −1.34937 −0.105691 −0.0528455 0.998603i \(-0.516829\pi\)
−0.0528455 + 0.998603i \(0.516829\pi\)
\(164\) −9.59480 −0.749228
\(165\) 0.736621 0.0573459
\(166\) 6.99898 0.543226
\(167\) −6.95361 −0.538086 −0.269043 0.963128i \(-0.586707\pi\)
−0.269043 + 0.963128i \(0.586707\pi\)
\(168\) −1.57544 −0.121548
\(169\) 27.4733 2.11333
\(170\) 2.55997 0.196340
\(171\) −12.5771 −0.961792
\(172\) −3.91594 −0.298587
\(173\) −2.24992 −0.171058 −0.0855291 0.996336i \(-0.527258\pi\)
−0.0855291 + 0.996336i \(0.527258\pi\)
\(174\) −0.774644 −0.0587256
\(175\) −2.88251 −0.217897
\(176\) 0.844185 0.0636328
\(177\) 4.00062 0.300705
\(178\) −6.60152 −0.494805
\(179\) −0.0932950 −0.00697319 −0.00348660 0.999994i \(-0.501110\pi\)
−0.00348660 + 0.999994i \(0.501110\pi\)
\(180\) 5.79455 0.431901
\(181\) 21.7581 1.61727 0.808635 0.588311i \(-0.200207\pi\)
0.808635 + 0.588311i \(0.200207\pi\)
\(182\) 23.6045 1.74968
\(183\) −1.88699 −0.139490
\(184\) −1.59263 −0.117410
\(185\) 4.17246 0.306765
\(186\) −2.55428 −0.187289
\(187\) −1.05161 −0.0769015
\(188\) −10.4573 −0.762678
\(189\) 9.16859 0.666917
\(190\) −9.16625 −0.664989
\(191\) −6.28030 −0.454426 −0.227213 0.973845i \(-0.572961\pi\)
−0.227213 + 0.973845i \(0.572961\pi\)
\(192\) −0.424610 −0.0306436
\(193\) −14.0894 −1.01418 −0.507088 0.861895i \(-0.669278\pi\)
−0.507088 + 0.861895i \(0.669278\pi\)
\(194\) 12.0468 0.864910
\(195\) 5.55125 0.397534
\(196\) 6.76645 0.483318
\(197\) −4.04257 −0.288021 −0.144011 0.989576i \(-0.546000\pi\)
−0.144011 + 0.989576i \(0.546000\pi\)
\(198\) −2.38035 −0.169164
\(199\) −2.51251 −0.178107 −0.0890536 0.996027i \(-0.528384\pi\)
−0.0890536 + 0.996027i \(0.528384\pi\)
\(200\) −0.776890 −0.0549344
\(201\) −2.28881 −0.161440
\(202\) 9.59434 0.675055
\(203\) 6.76897 0.475089
\(204\) 0.528942 0.0370334
\(205\) 19.7175 1.37713
\(206\) −10.7695 −0.750348
\(207\) 4.49075 0.312128
\(208\) 6.36186 0.441116
\(209\) 3.76542 0.260459
\(210\) 3.23756 0.223413
\(211\) 5.25931 0.362066 0.181033 0.983477i \(-0.442056\pi\)
0.181033 + 0.983477i \(0.442056\pi\)
\(212\) 11.0584 0.759493
\(213\) −5.00627 −0.343024
\(214\) 10.6939 0.731021
\(215\) 8.04733 0.548824
\(216\) 2.47111 0.168138
\(217\) 22.3198 1.51517
\(218\) 0.791338 0.0535962
\(219\) −6.44561 −0.435554
\(220\) −1.73482 −0.116961
\(221\) −7.92505 −0.533097
\(222\) 0.862118 0.0578615
\(223\) 19.9512 1.33603 0.668016 0.744147i \(-0.267144\pi\)
0.668016 + 0.744147i \(0.267144\pi\)
\(224\) 3.71032 0.247906
\(225\) 2.19060 0.146040
\(226\) −1.79381 −0.119323
\(227\) −28.2375 −1.87419 −0.937094 0.349078i \(-0.886495\pi\)
−0.937094 + 0.349078i \(0.886495\pi\)
\(228\) −1.89394 −0.125429
\(229\) 23.7303 1.56814 0.784070 0.620672i \(-0.213140\pi\)
0.784070 + 0.620672i \(0.213140\pi\)
\(230\) 3.27289 0.215808
\(231\) −1.32996 −0.0875051
\(232\) 1.82436 0.119775
\(233\) −1.53854 −0.100793 −0.0503966 0.998729i \(-0.516049\pi\)
−0.0503966 + 0.998729i \(0.516049\pi\)
\(234\) −17.9386 −1.17268
\(235\) 21.4900 1.40185
\(236\) −9.42185 −0.613310
\(237\) −3.79431 −0.246467
\(238\) −4.62199 −0.299599
\(239\) 27.6458 1.78826 0.894129 0.447810i \(-0.147796\pi\)
0.894129 + 0.447810i \(0.147796\pi\)
\(240\) 0.872583 0.0563250
\(241\) −22.7447 −1.46512 −0.732558 0.680705i \(-0.761674\pi\)
−0.732558 + 0.680705i \(0.761674\pi\)
\(242\) −10.2874 −0.661296
\(243\) −10.5596 −0.677400
\(244\) 4.44405 0.284501
\(245\) −13.9052 −0.888371
\(246\) 4.07405 0.259752
\(247\) 28.3766 1.80556
\(248\) 6.01560 0.381991
\(249\) −2.97184 −0.188333
\(250\) 11.8716 0.750828
\(251\) −15.3512 −0.968960 −0.484480 0.874802i \(-0.660991\pi\)
−0.484480 + 0.874802i \(0.660991\pi\)
\(252\) −10.4620 −0.659044
\(253\) −1.34447 −0.0845263
\(254\) 20.0409 1.25748
\(255\) −1.08699 −0.0680698
\(256\) 1.00000 0.0625000
\(257\) −12.0406 −0.751070 −0.375535 0.926808i \(-0.622541\pi\)
−0.375535 + 0.926808i \(0.622541\pi\)
\(258\) 1.66275 0.103518
\(259\) −7.53333 −0.468098
\(260\) −13.0738 −0.810800
\(261\) −5.14417 −0.318416
\(262\) 0.607831 0.0375519
\(263\) 20.5239 1.26556 0.632780 0.774332i \(-0.281914\pi\)
0.632780 + 0.774332i \(0.281914\pi\)
\(264\) −0.358450 −0.0220610
\(265\) −22.7252 −1.39600
\(266\) 16.5496 1.01472
\(267\) 2.80308 0.171545
\(268\) 5.39038 0.329270
\(269\) 8.83023 0.538389 0.269194 0.963086i \(-0.413243\pi\)
0.269194 + 0.963086i \(0.413243\pi\)
\(270\) −5.07818 −0.309048
\(271\) 19.0015 1.15426 0.577129 0.816653i \(-0.304173\pi\)
0.577129 + 0.816653i \(0.304173\pi\)
\(272\) −1.24571 −0.0755324
\(273\) −10.0227 −0.606603
\(274\) 10.6998 0.646399
\(275\) −0.655839 −0.0395486
\(276\) 0.676247 0.0407053
\(277\) −28.2096 −1.69495 −0.847475 0.530835i \(-0.821878\pi\)
−0.847475 + 0.530835i \(0.821878\pi\)
\(278\) −15.1789 −0.910369
\(279\) −16.9622 −1.01550
\(280\) −7.62478 −0.455668
\(281\) −16.1237 −0.961860 −0.480930 0.876759i \(-0.659701\pi\)
−0.480930 + 0.876759i \(0.659701\pi\)
\(282\) 4.44029 0.264415
\(283\) −26.7967 −1.59290 −0.796450 0.604705i \(-0.793291\pi\)
−0.796450 + 0.604705i \(0.793291\pi\)
\(284\) 11.7903 0.699623
\(285\) 3.89208 0.230547
\(286\) 5.37059 0.317570
\(287\) −35.5997 −2.10139
\(288\) −2.81971 −0.166153
\(289\) −15.4482 −0.908718
\(290\) −3.74911 −0.220155
\(291\) −5.11519 −0.299858
\(292\) 15.1801 0.888346
\(293\) 7.13880 0.417053 0.208527 0.978017i \(-0.433133\pi\)
0.208527 + 0.978017i \(0.433133\pi\)
\(294\) −2.87311 −0.167563
\(295\) 19.3621 1.12731
\(296\) −2.03037 −0.118013
\(297\) 2.08607 0.121046
\(298\) 10.7807 0.624507
\(299\) −10.1321 −0.585954
\(300\) 0.329876 0.0190454
\(301\) −14.5294 −0.837459
\(302\) −20.7420 −1.19357
\(303\) −4.07386 −0.234037
\(304\) 4.46042 0.255822
\(305\) −9.13262 −0.522932
\(306\) 3.51254 0.200799
\(307\) 5.40001 0.308195 0.154097 0.988056i \(-0.450753\pi\)
0.154097 + 0.988056i \(0.450753\pi\)
\(308\) 3.13219 0.178473
\(309\) 4.57285 0.260140
\(310\) −12.3622 −0.702125
\(311\) 10.8282 0.614013 0.307006 0.951707i \(-0.400673\pi\)
0.307006 + 0.951707i \(0.400673\pi\)
\(312\) −2.70131 −0.152932
\(313\) −7.97695 −0.450884 −0.225442 0.974257i \(-0.572383\pi\)
−0.225442 + 0.974257i \(0.572383\pi\)
\(314\) −0.0174066 −0.000982311 0
\(315\) 21.4996 1.21137
\(316\) 8.93597 0.502688
\(317\) −13.2265 −0.742874 −0.371437 0.928458i \(-0.621135\pi\)
−0.371437 + 0.928458i \(0.621135\pi\)
\(318\) −4.69550 −0.263311
\(319\) 1.54010 0.0862291
\(320\) −2.05502 −0.114879
\(321\) −4.54075 −0.253440
\(322\) −5.90916 −0.329305
\(323\) −5.55640 −0.309166
\(324\) 7.40986 0.411659
\(325\) −4.94247 −0.274159
\(326\) −1.34937 −0.0747348
\(327\) −0.336010 −0.0185814
\(328\) −9.59480 −0.529784
\(329\) −38.8000 −2.13911
\(330\) 0.736621 0.0405497
\(331\) −15.2737 −0.839518 −0.419759 0.907636i \(-0.637885\pi\)
−0.419759 + 0.907636i \(0.637885\pi\)
\(332\) 6.99898 0.384119
\(333\) 5.72506 0.313731
\(334\) −6.95361 −0.380484
\(335\) −11.0773 −0.605220
\(336\) −1.57544 −0.0859472
\(337\) −25.7109 −1.40056 −0.700281 0.713867i \(-0.746942\pi\)
−0.700281 + 0.713867i \(0.746942\pi\)
\(338\) 27.4733 1.49435
\(339\) 0.761671 0.0413683
\(340\) 2.55997 0.138834
\(341\) 5.07828 0.275004
\(342\) −12.5771 −0.680090
\(343\) −0.866531 −0.0467883
\(344\) −3.91594 −0.211133
\(345\) −1.38970 −0.0748190
\(346\) −2.24992 −0.120956
\(347\) −11.3063 −0.606952 −0.303476 0.952839i \(-0.598147\pi\)
−0.303476 + 0.952839i \(0.598147\pi\)
\(348\) −0.774644 −0.0415253
\(349\) 16.2197 0.868219 0.434110 0.900860i \(-0.357063\pi\)
0.434110 + 0.900860i \(0.357063\pi\)
\(350\) −2.88251 −0.154077
\(351\) 15.7208 0.839117
\(352\) 0.844185 0.0449952
\(353\) 29.6415 1.57766 0.788828 0.614614i \(-0.210688\pi\)
0.788828 + 0.614614i \(0.210688\pi\)
\(354\) 4.00062 0.212630
\(355\) −24.2292 −1.28595
\(356\) −6.60152 −0.349880
\(357\) 1.96254 0.103869
\(358\) −0.0932950 −0.00493079
\(359\) 20.3512 1.07409 0.537046 0.843553i \(-0.319540\pi\)
0.537046 + 0.843553i \(0.319540\pi\)
\(360\) 5.79455 0.305400
\(361\) 0.895308 0.0471215
\(362\) 21.7581 1.14358
\(363\) 4.36812 0.229267
\(364\) 23.6045 1.23721
\(365\) −31.1953 −1.63284
\(366\) −1.88699 −0.0986346
\(367\) 21.3358 1.11372 0.556860 0.830606i \(-0.312006\pi\)
0.556860 + 0.830606i \(0.312006\pi\)
\(368\) −1.59263 −0.0830215
\(369\) 27.0545 1.40840
\(370\) 4.17246 0.216916
\(371\) 41.0301 2.13018
\(372\) −2.55428 −0.132434
\(373\) −31.0301 −1.60668 −0.803339 0.595523i \(-0.796945\pi\)
−0.803339 + 0.595523i \(0.796945\pi\)
\(374\) −1.05161 −0.0543775
\(375\) −5.04082 −0.260307
\(376\) −10.4573 −0.539295
\(377\) 11.6064 0.597758
\(378\) 9.16859 0.471581
\(379\) −4.29950 −0.220850 −0.110425 0.993884i \(-0.535221\pi\)
−0.110425 + 0.993884i \(0.535221\pi\)
\(380\) −9.16625 −0.470219
\(381\) −8.50957 −0.435958
\(382\) −6.28030 −0.321328
\(383\) −1.37802 −0.0704134 −0.0352067 0.999380i \(-0.511209\pi\)
−0.0352067 + 0.999380i \(0.511209\pi\)
\(384\) −0.424610 −0.0216683
\(385\) −6.43672 −0.328046
\(386\) −14.0894 −0.717130
\(387\) 11.0418 0.561286
\(388\) 12.0468 0.611584
\(389\) 30.2447 1.53347 0.766734 0.641964i \(-0.221880\pi\)
0.766734 + 0.641964i \(0.221880\pi\)
\(390\) 5.55125 0.281099
\(391\) 1.98396 0.100333
\(392\) 6.76645 0.341758
\(393\) −0.258091 −0.0130190
\(394\) −4.04257 −0.203662
\(395\) −18.3636 −0.923974
\(396\) −2.38035 −0.119617
\(397\) 36.3903 1.82638 0.913188 0.407538i \(-0.133612\pi\)
0.913188 + 0.407538i \(0.133612\pi\)
\(398\) −2.51251 −0.125941
\(399\) −7.02711 −0.351796
\(400\) −0.776890 −0.0388445
\(401\) 34.3590 1.71581 0.857904 0.513810i \(-0.171766\pi\)
0.857904 + 0.513810i \(0.171766\pi\)
\(402\) −2.28881 −0.114156
\(403\) 38.2704 1.90639
\(404\) 9.59434 0.477336
\(405\) −15.2274 −0.756656
\(406\) 6.76897 0.335938
\(407\) −1.71401 −0.0849604
\(408\) 0.528942 0.0261866
\(409\) 18.4837 0.913961 0.456981 0.889477i \(-0.348931\pi\)
0.456981 + 0.889477i \(0.348931\pi\)
\(410\) 19.7175 0.973778
\(411\) −4.54325 −0.224102
\(412\) −10.7695 −0.530576
\(413\) −34.9581 −1.72017
\(414\) 4.49075 0.220708
\(415\) −14.3830 −0.706036
\(416\) 6.36186 0.311916
\(417\) 6.44511 0.315619
\(418\) 3.76542 0.184172
\(419\) 25.5378 1.24760 0.623801 0.781583i \(-0.285588\pi\)
0.623801 + 0.781583i \(0.285588\pi\)
\(420\) 3.23756 0.157977
\(421\) 30.4856 1.48578 0.742888 0.669416i \(-0.233455\pi\)
0.742888 + 0.669416i \(0.233455\pi\)
\(422\) 5.25931 0.256019
\(423\) 29.4866 1.43369
\(424\) 11.0584 0.537043
\(425\) 0.967782 0.0469443
\(426\) −5.00627 −0.242554
\(427\) 16.4888 0.797951
\(428\) 10.6939 0.516910
\(429\) −2.28041 −0.110099
\(430\) 8.04733 0.388077
\(431\) 3.38191 0.162901 0.0814505 0.996677i \(-0.474045\pi\)
0.0814505 + 0.996677i \(0.474045\pi\)
\(432\) 2.47111 0.118891
\(433\) 19.2457 0.924891 0.462445 0.886648i \(-0.346972\pi\)
0.462445 + 0.886648i \(0.346972\pi\)
\(434\) 22.3198 1.07138
\(435\) 1.59191 0.0763262
\(436\) 0.791338 0.0378982
\(437\) −7.10379 −0.339820
\(438\) −6.44561 −0.307983
\(439\) −38.1157 −1.81916 −0.909582 0.415525i \(-0.863598\pi\)
−0.909582 + 0.415525i \(0.863598\pi\)
\(440\) −1.73482 −0.0827042
\(441\) −19.0794 −0.908543
\(442\) −7.92505 −0.376956
\(443\) −39.9123 −1.89629 −0.948145 0.317838i \(-0.897043\pi\)
−0.948145 + 0.317838i \(0.897043\pi\)
\(444\) 0.862118 0.0409143
\(445\) 13.5663 0.643103
\(446\) 19.9512 0.944717
\(447\) −4.57758 −0.216512
\(448\) 3.71032 0.175296
\(449\) −20.8390 −0.983455 −0.491728 0.870749i \(-0.663634\pi\)
−0.491728 + 0.870749i \(0.663634\pi\)
\(450\) 2.19060 0.103266
\(451\) −8.09978 −0.381404
\(452\) −1.79381 −0.0843738
\(453\) 8.80727 0.413802
\(454\) −28.2375 −1.32525
\(455\) −48.5078 −2.27408
\(456\) −1.89394 −0.0886918
\(457\) −28.9031 −1.35203 −0.676014 0.736889i \(-0.736294\pi\)
−0.676014 + 0.736889i \(0.736294\pi\)
\(458\) 23.7303 1.10884
\(459\) −3.07829 −0.143682
\(460\) 3.27289 0.152599
\(461\) −27.8908 −1.29900 −0.649501 0.760360i \(-0.725022\pi\)
−0.649501 + 0.760360i \(0.725022\pi\)
\(462\) −1.32996 −0.0618754
\(463\) −14.6630 −0.681448 −0.340724 0.940163i \(-0.610672\pi\)
−0.340724 + 0.940163i \(0.610672\pi\)
\(464\) 1.82436 0.0846940
\(465\) 5.24911 0.243422
\(466\) −1.53854 −0.0712715
\(467\) −15.9437 −0.737785 −0.368892 0.929472i \(-0.620263\pi\)
−0.368892 + 0.929472i \(0.620263\pi\)
\(468\) −17.9386 −0.829211
\(469\) 20.0000 0.923516
\(470\) 21.4900 0.991260
\(471\) 0.00739102 0.000340560 0
\(472\) −9.42185 −0.433676
\(473\) −3.30578 −0.152000
\(474\) −3.79431 −0.174278
\(475\) −3.46525 −0.158997
\(476\) −4.62199 −0.211849
\(477\) −31.1814 −1.42770
\(478\) 27.6458 1.26449
\(479\) −26.9483 −1.23130 −0.615650 0.788020i \(-0.711107\pi\)
−0.615650 + 0.788020i \(0.711107\pi\)
\(480\) 0.872583 0.0398278
\(481\) −12.9170 −0.588963
\(482\) −22.7447 −1.03599
\(483\) 2.50909 0.114168
\(484\) −10.2874 −0.467607
\(485\) −24.7564 −1.12413
\(486\) −10.5596 −0.478994
\(487\) −19.0499 −0.863235 −0.431617 0.902057i \(-0.642057\pi\)
−0.431617 + 0.902057i \(0.642057\pi\)
\(488\) 4.44405 0.201173
\(489\) 0.572957 0.0259100
\(490\) −13.9052 −0.628173
\(491\) −7.88816 −0.355988 −0.177994 0.984032i \(-0.556961\pi\)
−0.177994 + 0.984032i \(0.556961\pi\)
\(492\) 4.07405 0.183672
\(493\) −2.27263 −0.102354
\(494\) 28.3766 1.27672
\(495\) 4.89167 0.219864
\(496\) 6.01560 0.270108
\(497\) 43.7456 1.96226
\(498\) −2.97184 −0.133171
\(499\) −18.1973 −0.814621 −0.407310 0.913290i \(-0.633533\pi\)
−0.407310 + 0.913290i \(0.633533\pi\)
\(500\) 11.8716 0.530915
\(501\) 2.95257 0.131911
\(502\) −15.3512 −0.685158
\(503\) −5.19012 −0.231416 −0.115708 0.993283i \(-0.536914\pi\)
−0.115708 + 0.993283i \(0.536914\pi\)
\(504\) −10.4620 −0.466015
\(505\) −19.7166 −0.877376
\(506\) −1.34447 −0.0597691
\(507\) −11.6655 −0.518081
\(508\) 20.0409 0.889171
\(509\) −2.49385 −0.110538 −0.0552689 0.998472i \(-0.517602\pi\)
−0.0552689 + 0.998472i \(0.517602\pi\)
\(510\) −1.08699 −0.0481326
\(511\) 56.3228 2.49158
\(512\) 1.00000 0.0441942
\(513\) 11.0222 0.486640
\(514\) −12.0406 −0.531086
\(515\) 22.1316 0.975234
\(516\) 1.66275 0.0731984
\(517\) −8.82791 −0.388251
\(518\) −7.53333 −0.330996
\(519\) 0.955339 0.0419347
\(520\) −13.0738 −0.573322
\(521\) 16.8407 0.737802 0.368901 0.929469i \(-0.379734\pi\)
0.368901 + 0.929469i \(0.379734\pi\)
\(522\) −5.14417 −0.225154
\(523\) 1.12989 0.0494065 0.0247033 0.999695i \(-0.492136\pi\)
0.0247033 + 0.999695i \(0.492136\pi\)
\(524\) 0.607831 0.0265532
\(525\) 1.22394 0.0534173
\(526\) 20.5239 0.894886
\(527\) −7.49371 −0.326431
\(528\) −0.358450 −0.0155995
\(529\) −20.4635 −0.889719
\(530\) −22.7252 −0.987120
\(531\) 26.5669 1.15290
\(532\) 16.5496 0.717514
\(533\) −61.0408 −2.64397
\(534\) 2.80308 0.121301
\(535\) −21.9762 −0.950115
\(536\) 5.39038 0.232829
\(537\) 0.0396140 0.00170947
\(538\) 8.83023 0.380698
\(539\) 5.71214 0.246039
\(540\) −5.07818 −0.218530
\(541\) −8.88136 −0.381839 −0.190920 0.981606i \(-0.561147\pi\)
−0.190920 + 0.981606i \(0.561147\pi\)
\(542\) 19.0015 0.816183
\(543\) −9.23873 −0.396472
\(544\) −1.24571 −0.0534095
\(545\) −1.62622 −0.0696595
\(546\) −10.0227 −0.428933
\(547\) −16.4258 −0.702315 −0.351157 0.936316i \(-0.614212\pi\)
−0.351157 + 0.936316i \(0.614212\pi\)
\(548\) 10.6998 0.457073
\(549\) −12.5309 −0.534807
\(550\) −0.655839 −0.0279651
\(551\) 8.13742 0.346666
\(552\) 0.676247 0.0287830
\(553\) 33.1553 1.40991
\(554\) −28.2096 −1.19851
\(555\) −1.77167 −0.0752032
\(556\) −15.1789 −0.643728
\(557\) −33.2856 −1.41036 −0.705178 0.709030i \(-0.749133\pi\)
−0.705178 + 0.709030i \(0.749133\pi\)
\(558\) −16.9622 −0.718068
\(559\) −24.9127 −1.05369
\(560\) −7.62478 −0.322206
\(561\) 0.446525 0.0188523
\(562\) −16.1237 −0.680138
\(563\) 11.8854 0.500910 0.250455 0.968128i \(-0.419420\pi\)
0.250455 + 0.968128i \(0.419420\pi\)
\(564\) 4.44029 0.186970
\(565\) 3.68632 0.155085
\(566\) −26.7967 −1.12635
\(567\) 27.4929 1.15459
\(568\) 11.7903 0.494708
\(569\) −45.9701 −1.92717 −0.963584 0.267405i \(-0.913834\pi\)
−0.963584 + 0.267405i \(0.913834\pi\)
\(570\) 3.89208 0.163021
\(571\) 40.9461 1.71354 0.856770 0.515699i \(-0.172468\pi\)
0.856770 + 0.515699i \(0.172468\pi\)
\(572\) 5.37059 0.224556
\(573\) 2.66668 0.111402
\(574\) −35.5997 −1.48590
\(575\) 1.23730 0.0515989
\(576\) −2.81971 −0.117488
\(577\) −37.8734 −1.57669 −0.788345 0.615233i \(-0.789062\pi\)
−0.788345 + 0.615233i \(0.789062\pi\)
\(578\) −15.4482 −0.642560
\(579\) 5.98249 0.248624
\(580\) −3.74911 −0.155673
\(581\) 25.9684 1.07735
\(582\) −5.11519 −0.212032
\(583\) 9.33532 0.386629
\(584\) 15.1801 0.628155
\(585\) 36.8642 1.52415
\(586\) 7.13880 0.294901
\(587\) −23.4310 −0.967099 −0.483550 0.875317i \(-0.660653\pi\)
−0.483550 + 0.875317i \(0.660653\pi\)
\(588\) −2.87311 −0.118485
\(589\) 26.8321 1.10560
\(590\) 19.3621 0.797125
\(591\) 1.71652 0.0706081
\(592\) −2.03037 −0.0834478
\(593\) 7.64818 0.314073 0.157036 0.987593i \(-0.449806\pi\)
0.157036 + 0.987593i \(0.449806\pi\)
\(594\) 2.08607 0.0855925
\(595\) 9.49828 0.389392
\(596\) 10.7807 0.441593
\(597\) 1.06684 0.0436628
\(598\) −10.1321 −0.414332
\(599\) −23.7044 −0.968535 −0.484268 0.874920i \(-0.660914\pi\)
−0.484268 + 0.874920i \(0.660914\pi\)
\(600\) 0.329876 0.0134671
\(601\) −29.9557 −1.22192 −0.610959 0.791662i \(-0.709216\pi\)
−0.610959 + 0.791662i \(0.709216\pi\)
\(602\) −14.5294 −0.592173
\(603\) −15.1993 −0.618963
\(604\) −20.7420 −0.843981
\(605\) 21.1407 0.859492
\(606\) −4.07386 −0.165489
\(607\) 25.9893 1.05487 0.527437 0.849594i \(-0.323153\pi\)
0.527437 + 0.849594i \(0.323153\pi\)
\(608\) 4.46042 0.180894
\(609\) −2.87417 −0.116467
\(610\) −9.13262 −0.369769
\(611\) −66.5280 −2.69144
\(612\) 3.51254 0.141986
\(613\) −12.3380 −0.498326 −0.249163 0.968462i \(-0.580155\pi\)
−0.249163 + 0.968462i \(0.580155\pi\)
\(614\) 5.40001 0.217927
\(615\) −8.37226 −0.337602
\(616\) 3.13219 0.126200
\(617\) −1.04812 −0.0421958 −0.0210979 0.999777i \(-0.506716\pi\)
−0.0210979 + 0.999777i \(0.506716\pi\)
\(618\) 4.57285 0.183947
\(619\) −38.7536 −1.55764 −0.778819 0.627249i \(-0.784181\pi\)
−0.778819 + 0.627249i \(0.784181\pi\)
\(620\) −12.3622 −0.496477
\(621\) −3.93556 −0.157929
\(622\) 10.8282 0.434173
\(623\) −24.4937 −0.981321
\(624\) −2.70131 −0.108139
\(625\) −20.5120 −0.820480
\(626\) −7.97695 −0.318823
\(627\) −1.59883 −0.0638513
\(628\) −0.0174066 −0.000694599 0
\(629\) 2.52926 0.100848
\(630\) 21.4996 0.856566
\(631\) −14.2614 −0.567738 −0.283869 0.958863i \(-0.591618\pi\)
−0.283869 + 0.958863i \(0.591618\pi\)
\(632\) 8.93597 0.355454
\(633\) −2.23316 −0.0887599
\(634\) −13.2265 −0.525291
\(635\) −41.1844 −1.63435
\(636\) −4.69550 −0.186189
\(637\) 43.0473 1.70559
\(638\) 1.54010 0.0609732
\(639\) −33.2451 −1.31515
\(640\) −2.05502 −0.0812318
\(641\) 21.9342 0.866350 0.433175 0.901310i \(-0.357393\pi\)
0.433175 + 0.901310i \(0.357393\pi\)
\(642\) −4.54075 −0.179209
\(643\) 26.5916 1.04867 0.524336 0.851511i \(-0.324314\pi\)
0.524336 + 0.851511i \(0.324314\pi\)
\(644\) −5.90916 −0.232854
\(645\) −3.41698 −0.134543
\(646\) −5.55640 −0.218613
\(647\) 25.1098 0.987169 0.493584 0.869698i \(-0.335686\pi\)
0.493584 + 0.869698i \(0.335686\pi\)
\(648\) 7.40986 0.291087
\(649\) −7.95379 −0.312213
\(650\) −4.94247 −0.193860
\(651\) −9.47721 −0.371441
\(652\) −1.34937 −0.0528455
\(653\) 21.4459 0.839244 0.419622 0.907699i \(-0.362163\pi\)
0.419622 + 0.907699i \(0.362163\pi\)
\(654\) −0.336010 −0.0131390
\(655\) −1.24911 −0.0488066
\(656\) −9.59480 −0.374614
\(657\) −42.8033 −1.66992
\(658\) −38.8000 −1.51258
\(659\) 27.7830 1.08227 0.541136 0.840935i \(-0.317994\pi\)
0.541136 + 0.840935i \(0.317994\pi\)
\(660\) 0.736621 0.0286729
\(661\) 16.5641 0.644270 0.322135 0.946694i \(-0.395599\pi\)
0.322135 + 0.946694i \(0.395599\pi\)
\(662\) −15.2737 −0.593629
\(663\) 3.36506 0.130688
\(664\) 6.99898 0.271613
\(665\) −34.0097 −1.31884
\(666\) 5.72506 0.221841
\(667\) −2.90554 −0.112503
\(668\) −6.95361 −0.269043
\(669\) −8.47149 −0.327527
\(670\) −11.0773 −0.427955
\(671\) 3.75160 0.144829
\(672\) −1.57544 −0.0607739
\(673\) 4.93101 0.190076 0.0950382 0.995474i \(-0.469703\pi\)
0.0950382 + 0.995474i \(0.469703\pi\)
\(674\) −25.7109 −0.990347
\(675\) −1.91978 −0.0738923
\(676\) 27.4733 1.05667
\(677\) −21.5899 −0.829766 −0.414883 0.909875i \(-0.636177\pi\)
−0.414883 + 0.909875i \(0.636177\pi\)
\(678\) 0.761671 0.0292518
\(679\) 44.6974 1.71533
\(680\) 2.55997 0.0981702
\(681\) 11.9899 0.459455
\(682\) 5.07828 0.194457
\(683\) −18.7222 −0.716385 −0.358192 0.933648i \(-0.616607\pi\)
−0.358192 + 0.933648i \(0.616607\pi\)
\(684\) −12.5771 −0.480896
\(685\) −21.9883 −0.840131
\(686\) −0.866531 −0.0330843
\(687\) −10.0761 −0.384428
\(688\) −3.91594 −0.149294
\(689\) 70.3519 2.68020
\(690\) −1.38970 −0.0529050
\(691\) −20.8934 −0.794822 −0.397411 0.917641i \(-0.630091\pi\)
−0.397411 + 0.917641i \(0.630091\pi\)
\(692\) −2.24992 −0.0855291
\(693\) −8.83187 −0.335495
\(694\) −11.3063 −0.429180
\(695\) 31.1929 1.18322
\(696\) −0.774644 −0.0293628
\(697\) 11.9524 0.452728
\(698\) 16.2197 0.613924
\(699\) 0.653280 0.0247093
\(700\) −2.88251 −0.108949
\(701\) −36.4697 −1.37744 −0.688721 0.725026i \(-0.741828\pi\)
−0.688721 + 0.725026i \(0.741828\pi\)
\(702\) 15.7208 0.593345
\(703\) −9.05631 −0.341565
\(704\) 0.844185 0.0318164
\(705\) −9.12488 −0.343663
\(706\) 29.6415 1.11557
\(707\) 35.5980 1.33880
\(708\) 4.00062 0.150352
\(709\) 24.8416 0.932944 0.466472 0.884536i \(-0.345525\pi\)
0.466472 + 0.884536i \(0.345525\pi\)
\(710\) −24.2292 −0.909307
\(711\) −25.1968 −0.944955
\(712\) −6.60152 −0.247403
\(713\) −9.58062 −0.358797
\(714\) 1.96254 0.0734464
\(715\) −11.0367 −0.412748
\(716\) −0.0932950 −0.00348660
\(717\) −11.7387 −0.438389
\(718\) 20.3512 0.759498
\(719\) 4.48805 0.167376 0.0836879 0.996492i \(-0.473330\pi\)
0.0836879 + 0.996492i \(0.473330\pi\)
\(720\) 5.79455 0.215950
\(721\) −39.9583 −1.48813
\(722\) 0.895308 0.0333199
\(723\) 9.65764 0.359171
\(724\) 21.7581 0.808635
\(725\) −1.41733 −0.0526383
\(726\) 4.36812 0.162116
\(727\) 6.63570 0.246105 0.123052 0.992400i \(-0.460732\pi\)
0.123052 + 0.992400i \(0.460732\pi\)
\(728\) 23.6045 0.874842
\(729\) −17.7459 −0.657254
\(730\) −31.1953 −1.15459
\(731\) 4.87813 0.180424
\(732\) −1.88699 −0.0697452
\(733\) −48.1406 −1.77811 −0.889056 0.457799i \(-0.848638\pi\)
−0.889056 + 0.457799i \(0.848638\pi\)
\(734\) 21.3358 0.787519
\(735\) 5.90429 0.217783
\(736\) −1.59263 −0.0587051
\(737\) 4.55048 0.167619
\(738\) 27.0545 0.995890
\(739\) 51.5187 1.89515 0.947574 0.319537i \(-0.103527\pi\)
0.947574 + 0.319537i \(0.103527\pi\)
\(740\) 4.17246 0.153383
\(741\) −12.0490 −0.442630
\(742\) 41.0301 1.50626
\(743\) 53.1507 1.94991 0.974956 0.222398i \(-0.0713885\pi\)
0.974956 + 0.222398i \(0.0713885\pi\)
\(744\) −2.55428 −0.0936446
\(745\) −22.1545 −0.811677
\(746\) −31.0301 −1.13609
\(747\) −19.7351 −0.722068
\(748\) −1.05161 −0.0384507
\(749\) 39.6778 1.44980
\(750\) −5.04082 −0.184065
\(751\) −13.0622 −0.476647 −0.238324 0.971186i \(-0.576598\pi\)
−0.238324 + 0.971186i \(0.576598\pi\)
\(752\) −10.4573 −0.381339
\(753\) 6.51828 0.237539
\(754\) 11.6064 0.422679
\(755\) 42.6253 1.55129
\(756\) 9.16859 0.333458
\(757\) −22.6331 −0.822615 −0.411308 0.911497i \(-0.634928\pi\)
−0.411308 + 0.911497i \(0.634928\pi\)
\(758\) −4.29950 −0.156165
\(759\) 0.570877 0.0207215
\(760\) −9.16625 −0.332495
\(761\) 44.9768 1.63041 0.815203 0.579175i \(-0.196625\pi\)
0.815203 + 0.579175i \(0.196625\pi\)
\(762\) −8.50957 −0.308269
\(763\) 2.93612 0.106295
\(764\) −6.28030 −0.227213
\(765\) −7.21835 −0.260980
\(766\) −1.37802 −0.0497898
\(767\) −59.9405 −2.16433
\(768\) −0.424610 −0.0153218
\(769\) 23.3725 0.842833 0.421416 0.906867i \(-0.361533\pi\)
0.421416 + 0.906867i \(0.361533\pi\)
\(770\) −6.43672 −0.231963
\(771\) 5.11255 0.184124
\(772\) −14.0894 −0.507088
\(773\) −31.9137 −1.14786 −0.573929 0.818905i \(-0.694581\pi\)
−0.573929 + 0.818905i \(0.694581\pi\)
\(774\) 11.0418 0.396889
\(775\) −4.67346 −0.167876
\(776\) 12.0468 0.432455
\(777\) 3.19873 0.114754
\(778\) 30.2447 1.08433
\(779\) −42.7968 −1.53335
\(780\) 5.55125 0.198767
\(781\) 9.95316 0.356152
\(782\) 1.98396 0.0709462
\(783\) 4.50820 0.161110
\(784\) 6.76645 0.241659
\(785\) 0.0357709 0.00127672
\(786\) −0.258091 −0.00920581
\(787\) 0.755053 0.0269147 0.0134574 0.999909i \(-0.495716\pi\)
0.0134574 + 0.999909i \(0.495716\pi\)
\(788\) −4.04257 −0.144011
\(789\) −8.71467 −0.310250
\(790\) −18.3636 −0.653348
\(791\) −6.65561 −0.236646
\(792\) −2.38035 −0.0845822
\(793\) 28.2725 1.00398
\(794\) 36.3903 1.29144
\(795\) 9.64936 0.342227
\(796\) −2.51251 −0.0890536
\(797\) −8.51913 −0.301763 −0.150881 0.988552i \(-0.548211\pi\)
−0.150881 + 0.988552i \(0.548211\pi\)
\(798\) −7.02711 −0.248757
\(799\) 13.0268 0.460856
\(800\) −0.776890 −0.0274672
\(801\) 18.6144 0.657706
\(802\) 34.3590 1.21326
\(803\) 12.8148 0.452224
\(804\) −2.28881 −0.0807202
\(805\) 12.1434 0.428000
\(806\) 38.2704 1.34802
\(807\) −3.74941 −0.131985
\(808\) 9.59434 0.337528
\(809\) 5.18639 0.182344 0.0911718 0.995835i \(-0.470939\pi\)
0.0911718 + 0.995835i \(0.470939\pi\)
\(810\) −15.2274 −0.535037
\(811\) 13.4138 0.471022 0.235511 0.971872i \(-0.424324\pi\)
0.235511 + 0.971872i \(0.424324\pi\)
\(812\) 6.76897 0.237544
\(813\) −8.06822 −0.282965
\(814\) −1.71401 −0.0600760
\(815\) 2.77299 0.0971334
\(816\) 0.528942 0.0185167
\(817\) −17.4667 −0.611083
\(818\) 18.4837 0.646268
\(819\) −66.5578 −2.32572
\(820\) 19.7175 0.688565
\(821\) −22.8863 −0.798736 −0.399368 0.916791i \(-0.630770\pi\)
−0.399368 + 0.916791i \(0.630770\pi\)
\(822\) −4.54325 −0.158464
\(823\) 19.7213 0.687443 0.343721 0.939072i \(-0.388312\pi\)
0.343721 + 0.939072i \(0.388312\pi\)
\(824\) −10.7695 −0.375174
\(825\) 0.278476 0.00969529
\(826\) −34.9581 −1.21635
\(827\) −6.65999 −0.231590 −0.115795 0.993273i \(-0.536942\pi\)
−0.115795 + 0.993273i \(0.536942\pi\)
\(828\) 4.49075 0.156064
\(829\) 6.07068 0.210843 0.105422 0.994428i \(-0.466381\pi\)
0.105422 + 0.994428i \(0.466381\pi\)
\(830\) −14.3830 −0.499243
\(831\) 11.9781 0.415515
\(832\) 6.36186 0.220558
\(833\) −8.42906 −0.292050
\(834\) 6.44511 0.223176
\(835\) 14.2898 0.494519
\(836\) 3.76542 0.130230
\(837\) 14.8652 0.513816
\(838\) 25.5378 0.882187
\(839\) −34.3310 −1.18524 −0.592620 0.805483i \(-0.701906\pi\)
−0.592620 + 0.805483i \(0.701906\pi\)
\(840\) 3.23756 0.111706
\(841\) −25.6717 −0.885231
\(842\) 30.4856 1.05060
\(843\) 6.84630 0.235799
\(844\) 5.25931 0.181033
\(845\) −56.4582 −1.94222
\(846\) 29.4866 1.01377
\(847\) −38.1693 −1.31151
\(848\) 11.0584 0.379746
\(849\) 11.3782 0.390497
\(850\) 0.967782 0.0331947
\(851\) 3.23363 0.110848
\(852\) −5.00627 −0.171512
\(853\) −8.79085 −0.300993 −0.150496 0.988611i \(-0.548087\pi\)
−0.150496 + 0.988611i \(0.548087\pi\)
\(854\) 16.4888 0.564237
\(855\) 25.8461 0.883919
\(856\) 10.6939 0.365511
\(857\) −33.7434 −1.15265 −0.576327 0.817219i \(-0.695514\pi\)
−0.576327 + 0.817219i \(0.695514\pi\)
\(858\) −2.28041 −0.0778518
\(859\) −1.90843 −0.0651147 −0.0325574 0.999470i \(-0.510365\pi\)
−0.0325574 + 0.999470i \(0.510365\pi\)
\(860\) 8.04733 0.274412
\(861\) 15.1160 0.515153
\(862\) 3.38191 0.115188
\(863\) 48.0888 1.63696 0.818481 0.574533i \(-0.194816\pi\)
0.818481 + 0.574533i \(0.194816\pi\)
\(864\) 2.47111 0.0840688
\(865\) 4.62363 0.157208
\(866\) 19.2457 0.653997
\(867\) 6.55946 0.222771
\(868\) 22.3198 0.757583
\(869\) 7.54361 0.255900
\(870\) 1.59191 0.0539708
\(871\) 34.2929 1.16197
\(872\) 0.791338 0.0267981
\(873\) −33.9684 −1.14966
\(874\) −7.10379 −0.240289
\(875\) 44.0475 1.48908
\(876\) −6.44561 −0.217777
\(877\) 8.36376 0.282424 0.141212 0.989979i \(-0.454900\pi\)
0.141212 + 0.989979i \(0.454900\pi\)
\(878\) −38.1157 −1.28634
\(879\) −3.03121 −0.102240
\(880\) −1.73482 −0.0584807
\(881\) 27.3248 0.920596 0.460298 0.887764i \(-0.347743\pi\)
0.460298 + 0.887764i \(0.347743\pi\)
\(882\) −19.0794 −0.642437
\(883\) −8.46958 −0.285024 −0.142512 0.989793i \(-0.545518\pi\)
−0.142512 + 0.989793i \(0.545518\pi\)
\(884\) −7.92505 −0.266548
\(885\) −8.22135 −0.276358
\(886\) −39.9123 −1.34088
\(887\) 25.2990 0.849459 0.424729 0.905320i \(-0.360369\pi\)
0.424729 + 0.905320i \(0.360369\pi\)
\(888\) 0.862118 0.0289308
\(889\) 74.3581 2.49389
\(890\) 13.5663 0.454742
\(891\) 6.25529 0.209560
\(892\) 19.9512 0.668016
\(893\) −46.6440 −1.56088
\(894\) −4.57758 −0.153097
\(895\) 0.191723 0.00640860
\(896\) 3.71032 0.123953
\(897\) 4.30219 0.143646
\(898\) −20.8390 −0.695408
\(899\) 10.9746 0.366025
\(900\) 2.19060 0.0730201
\(901\) −13.7756 −0.458931
\(902\) −8.09978 −0.269693
\(903\) 6.16932 0.205302
\(904\) −1.79381 −0.0596613
\(905\) −44.7134 −1.48632
\(906\) 8.80727 0.292602
\(907\) −30.4682 −1.01168 −0.505841 0.862627i \(-0.668818\pi\)
−0.505841 + 0.862627i \(0.668818\pi\)
\(908\) −28.2375 −0.937094
\(909\) −27.0532 −0.897299
\(910\) −48.5078 −1.60802
\(911\) 37.6830 1.24849 0.624247 0.781227i \(-0.285406\pi\)
0.624247 + 0.781227i \(0.285406\pi\)
\(912\) −1.89394 −0.0627146
\(913\) 5.90843 0.195540
\(914\) −28.9031 −0.956029
\(915\) 3.87781 0.128196
\(916\) 23.7303 0.784070
\(917\) 2.25525 0.0744748
\(918\) −3.07829 −0.101599
\(919\) −14.6225 −0.482351 −0.241176 0.970481i \(-0.577533\pi\)
−0.241176 + 0.970481i \(0.577533\pi\)
\(920\) 3.27289 0.107904
\(921\) −2.29290 −0.0755537
\(922\) −27.8908 −0.918534
\(923\) 75.0080 2.46892
\(924\) −1.32996 −0.0437525
\(925\) 1.57738 0.0518639
\(926\) −14.6630 −0.481857
\(927\) 30.3669 0.997379
\(928\) 1.82436 0.0598877
\(929\) −46.7438 −1.53361 −0.766807 0.641878i \(-0.778156\pi\)
−0.766807 + 0.641878i \(0.778156\pi\)
\(930\) 5.24911 0.172125
\(931\) 30.1812 0.989149
\(932\) −1.53854 −0.0503966
\(933\) −4.59778 −0.150525
\(934\) −15.9437 −0.521693
\(935\) 2.16108 0.0706750
\(936\) −17.9386 −0.586341
\(937\) −14.9122 −0.487160 −0.243580 0.969881i \(-0.578322\pi\)
−0.243580 + 0.969881i \(0.578322\pi\)
\(938\) 20.0000 0.653024
\(939\) 3.38709 0.110534
\(940\) 21.4900 0.700927
\(941\) −44.5540 −1.45242 −0.726210 0.687473i \(-0.758720\pi\)
−0.726210 + 0.687473i \(0.758720\pi\)
\(942\) 0.00739102 0.000240812 0
\(943\) 15.2810 0.497617
\(944\) −9.42185 −0.306655
\(945\) −18.8416 −0.612919
\(946\) −3.30578 −0.107480
\(947\) 14.5801 0.473788 0.236894 0.971535i \(-0.423871\pi\)
0.236894 + 0.971535i \(0.423871\pi\)
\(948\) −3.79431 −0.123233
\(949\) 96.5735 3.13491
\(950\) −3.46525 −0.112428
\(951\) 5.61611 0.182115
\(952\) −4.62199 −0.149800
\(953\) −13.6884 −0.443412 −0.221706 0.975114i \(-0.571163\pi\)
−0.221706 + 0.975114i \(0.571163\pi\)
\(954\) −31.1814 −1.00953
\(955\) 12.9061 0.417633
\(956\) 27.6458 0.894129
\(957\) −0.653943 −0.0211390
\(958\) −26.9483 −0.870661
\(959\) 39.6997 1.28197
\(960\) 0.872583 0.0281625
\(961\) 5.18742 0.167336
\(962\) −12.9170 −0.416460
\(963\) −30.1537 −0.971690
\(964\) −22.7447 −0.732558
\(965\) 28.9539 0.932061
\(966\) 2.50909 0.0807287
\(967\) −24.6304 −0.792062 −0.396031 0.918237i \(-0.629613\pi\)
−0.396031 + 0.918237i \(0.629613\pi\)
\(968\) −10.2874 −0.330648
\(969\) 2.35930 0.0757917
\(970\) −24.7564 −0.794881
\(971\) 19.4244 0.623358 0.311679 0.950187i \(-0.399109\pi\)
0.311679 + 0.950187i \(0.399109\pi\)
\(972\) −10.5596 −0.338700
\(973\) −56.3185 −1.80549
\(974\) −19.0499 −0.610399
\(975\) 2.09862 0.0672098
\(976\) 4.44405 0.142251
\(977\) 21.7544 0.695985 0.347992 0.937497i \(-0.386863\pi\)
0.347992 + 0.937497i \(0.386863\pi\)
\(978\) 0.572957 0.0183211
\(979\) −5.57291 −0.178111
\(980\) −13.9052 −0.444185
\(981\) −2.23134 −0.0712412
\(982\) −7.88816 −0.251721
\(983\) 2.43517 0.0776697 0.0388349 0.999246i \(-0.487635\pi\)
0.0388349 + 0.999246i \(0.487635\pi\)
\(984\) 4.07405 0.129876
\(985\) 8.30757 0.264701
\(986\) −2.27263 −0.0723754
\(987\) 16.4749 0.524401
\(988\) 28.3766 0.902779
\(989\) 6.23664 0.198314
\(990\) 4.89167 0.155468
\(991\) 56.7974 1.80423 0.902115 0.431496i \(-0.142014\pi\)
0.902115 + 0.431496i \(0.142014\pi\)
\(992\) 6.01560 0.190995
\(993\) 6.48536 0.205807
\(994\) 43.7456 1.38753
\(995\) 5.16327 0.163687
\(996\) −2.97184 −0.0941663
\(997\) −7.35595 −0.232965 −0.116483 0.993193i \(-0.537162\pi\)
−0.116483 + 0.993193i \(0.537162\pi\)
\(998\) −18.1973 −0.576024
\(999\) −5.01727 −0.158739
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 4022.2.a.f.1.19 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4022.2.a.f.1.19 50 1.1 even 1 trivial