Properties

Label 8034.2.a.v.1.4
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $11$
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] = [8034,2,Mod(1,8034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, 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("8034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.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: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5 x^{10} - 31 x^{9} + 168 x^{8} + 300 x^{7} - 1928 x^{6} - 736 x^{5} + 8532 x^{4} - 2065 x^{3} - 10494 x^{2} + 4024 x + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.31499\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.31499 q^{5} +1.00000 q^{6} +3.43679 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.31499 q^{5} +1.00000 q^{6} +3.43679 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.31499 q^{10} -3.11170 q^{11} +1.00000 q^{12} +1.00000 q^{13} +3.43679 q^{14} -1.31499 q^{15} +1.00000 q^{16} +6.00690 q^{17} +1.00000 q^{18} +2.43679 q^{19} -1.31499 q^{20} +3.43679 q^{21} -3.11170 q^{22} +5.59372 q^{23} +1.00000 q^{24} -3.27079 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.43679 q^{28} +6.46722 q^{29} -1.31499 q^{30} -2.43679 q^{31} +1.00000 q^{32} -3.11170 q^{33} +6.00690 q^{34} -4.51935 q^{35} +1.00000 q^{36} +2.81393 q^{37} +2.43679 q^{38} +1.00000 q^{39} -1.31499 q^{40} +0.544657 q^{41} +3.43679 q^{42} -12.8675 q^{43} -3.11170 q^{44} -1.31499 q^{45} +5.59372 q^{46} -5.36242 q^{47} +1.00000 q^{48} +4.81150 q^{49} -3.27079 q^{50} +6.00690 q^{51} +1.00000 q^{52} +10.1760 q^{53} +1.00000 q^{54} +4.09187 q^{55} +3.43679 q^{56} +2.43679 q^{57} +6.46722 q^{58} -4.51171 q^{59} -1.31499 q^{60} +13.8024 q^{61} -2.43679 q^{62} +3.43679 q^{63} +1.00000 q^{64} -1.31499 q^{65} -3.11170 q^{66} -14.5505 q^{67} +6.00690 q^{68} +5.59372 q^{69} -4.51935 q^{70} -9.93655 q^{71} +1.00000 q^{72} -2.55975 q^{73} +2.81393 q^{74} -3.27079 q^{75} +2.43679 q^{76} -10.6943 q^{77} +1.00000 q^{78} +12.3825 q^{79} -1.31499 q^{80} +1.00000 q^{81} +0.544657 q^{82} +8.21206 q^{83} +3.43679 q^{84} -7.89903 q^{85} -12.8675 q^{86} +6.46722 q^{87} -3.11170 q^{88} -1.59953 q^{89} -1.31499 q^{90} +3.43679 q^{91} +5.59372 q^{92} -2.43679 q^{93} -5.36242 q^{94} -3.20436 q^{95} +1.00000 q^{96} +9.40912 q^{97} +4.81150 q^{98} -3.11170 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 11 q^{2} + 11 q^{3} + 11 q^{4} + 5 q^{5} + 11 q^{6} + 4 q^{7} + 11 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 11 q^{2} + 11 q^{3} + 11 q^{4} + 5 q^{5} + 11 q^{6} + 4 q^{7} + 11 q^{8} + 11 q^{9} + 5 q^{10} - 7 q^{11} + 11 q^{12} + 11 q^{13} + 4 q^{14} + 5 q^{15} + 11 q^{16} + 10 q^{17} + 11 q^{18} - 7 q^{19} + 5 q^{20} + 4 q^{21} - 7 q^{22} + 18 q^{23} + 11 q^{24} + 32 q^{25} + 11 q^{26} + 11 q^{27} + 4 q^{28} + 29 q^{29} + 5 q^{30} + 7 q^{31} + 11 q^{32} - 7 q^{33} + 10 q^{34} + 31 q^{35} + 11 q^{36} + 21 q^{37} - 7 q^{38} + 11 q^{39} + 5 q^{40} - 3 q^{41} + 4 q^{42} - 17 q^{43} - 7 q^{44} + 5 q^{45} + 18 q^{46} + 12 q^{47} + 11 q^{48} + 21 q^{49} + 32 q^{50} + 10 q^{51} + 11 q^{52} + 11 q^{53} + 11 q^{54} + 4 q^{55} + 4 q^{56} - 7 q^{57} + 29 q^{58} - 48 q^{59} + 5 q^{60} - q^{61} + 7 q^{62} + 4 q^{63} + 11 q^{64} + 5 q^{65} - 7 q^{66} - 9 q^{67} + 10 q^{68} + 18 q^{69} + 31 q^{70} + 17 q^{71} + 11 q^{72} - 23 q^{73} + 21 q^{74} + 32 q^{75} - 7 q^{76} + 26 q^{77} + 11 q^{78} + 41 q^{79} + 5 q^{80} + 11 q^{81} - 3 q^{82} + 19 q^{83} + 4 q^{84} + 17 q^{85} - 17 q^{86} + 29 q^{87} - 7 q^{88} + 32 q^{89} + 5 q^{90} + 4 q^{91} + 18 q^{92} + 7 q^{93} + 12 q^{94} + 26 q^{95} + 11 q^{96} - 16 q^{97} + 21 q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.31499 −0.588083 −0.294042 0.955793i \(-0.595000\pi\)
−0.294042 + 0.955793i \(0.595000\pi\)
\(6\) 1.00000 0.408248
\(7\) 3.43679 1.29898 0.649492 0.760369i \(-0.274982\pi\)
0.649492 + 0.760369i \(0.274982\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.31499 −0.415837
\(11\) −3.11170 −0.938213 −0.469107 0.883142i \(-0.655424\pi\)
−0.469107 + 0.883142i \(0.655424\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 3.43679 0.918520
\(15\) −1.31499 −0.339530
\(16\) 1.00000 0.250000
\(17\) 6.00690 1.45689 0.728444 0.685106i \(-0.240244\pi\)
0.728444 + 0.685106i \(0.240244\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.43679 0.559037 0.279519 0.960140i \(-0.409825\pi\)
0.279519 + 0.960140i \(0.409825\pi\)
\(20\) −1.31499 −0.294042
\(21\) 3.43679 0.749968
\(22\) −3.11170 −0.663417
\(23\) 5.59372 1.16637 0.583186 0.812339i \(-0.301806\pi\)
0.583186 + 0.812339i \(0.301806\pi\)
\(24\) 1.00000 0.204124
\(25\) −3.27079 −0.654158
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 3.43679 0.649492
\(29\) 6.46722 1.20093 0.600466 0.799650i \(-0.294982\pi\)
0.600466 + 0.799650i \(0.294982\pi\)
\(30\) −1.31499 −0.240084
\(31\) −2.43679 −0.437660 −0.218830 0.975763i \(-0.570224\pi\)
−0.218830 + 0.975763i \(0.570224\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.11170 −0.541678
\(34\) 6.00690 1.03017
\(35\) −4.51935 −0.763910
\(36\) 1.00000 0.166667
\(37\) 2.81393 0.462608 0.231304 0.972882i \(-0.425701\pi\)
0.231304 + 0.972882i \(0.425701\pi\)
\(38\) 2.43679 0.395299
\(39\) 1.00000 0.160128
\(40\) −1.31499 −0.207919
\(41\) 0.544657 0.0850612 0.0425306 0.999095i \(-0.486458\pi\)
0.0425306 + 0.999095i \(0.486458\pi\)
\(42\) 3.43679 0.530308
\(43\) −12.8675 −1.96227 −0.981135 0.193323i \(-0.938073\pi\)
−0.981135 + 0.193323i \(0.938073\pi\)
\(44\) −3.11170 −0.469107
\(45\) −1.31499 −0.196028
\(46\) 5.59372 0.824749
\(47\) −5.36242 −0.782189 −0.391095 0.920350i \(-0.627903\pi\)
−0.391095 + 0.920350i \(0.627903\pi\)
\(48\) 1.00000 0.144338
\(49\) 4.81150 0.687357
\(50\) −3.27079 −0.462560
\(51\) 6.00690 0.841134
\(52\) 1.00000 0.138675
\(53\) 10.1760 1.39778 0.698888 0.715231i \(-0.253679\pi\)
0.698888 + 0.715231i \(0.253679\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.09187 0.551747
\(56\) 3.43679 0.459260
\(57\) 2.43679 0.322760
\(58\) 6.46722 0.849188
\(59\) −4.51171 −0.587375 −0.293688 0.955901i \(-0.594883\pi\)
−0.293688 + 0.955901i \(0.594883\pi\)
\(60\) −1.31499 −0.169765
\(61\) 13.8024 1.76722 0.883610 0.468224i \(-0.155106\pi\)
0.883610 + 0.468224i \(0.155106\pi\)
\(62\) −2.43679 −0.309472
\(63\) 3.43679 0.432994
\(64\) 1.00000 0.125000
\(65\) −1.31499 −0.163105
\(66\) −3.11170 −0.383024
\(67\) −14.5505 −1.77763 −0.888813 0.458271i \(-0.848469\pi\)
−0.888813 + 0.458271i \(0.848469\pi\)
\(68\) 6.00690 0.728444
\(69\) 5.59372 0.673405
\(70\) −4.51935 −0.540166
\(71\) −9.93655 −1.17925 −0.589626 0.807676i \(-0.700725\pi\)
−0.589626 + 0.807676i \(0.700725\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.55975 −0.299596 −0.149798 0.988717i \(-0.547862\pi\)
−0.149798 + 0.988717i \(0.547862\pi\)
\(74\) 2.81393 0.327113
\(75\) −3.27079 −0.377679
\(76\) 2.43679 0.279519
\(77\) −10.6943 −1.21872
\(78\) 1.00000 0.113228
\(79\) 12.3825 1.39314 0.696570 0.717489i \(-0.254709\pi\)
0.696570 + 0.717489i \(0.254709\pi\)
\(80\) −1.31499 −0.147021
\(81\) 1.00000 0.111111
\(82\) 0.544657 0.0601474
\(83\) 8.21206 0.901391 0.450695 0.892678i \(-0.351176\pi\)
0.450695 + 0.892678i \(0.351176\pi\)
\(84\) 3.43679 0.374984
\(85\) −7.89903 −0.856771
\(86\) −12.8675 −1.38753
\(87\) 6.46722 0.693359
\(88\) −3.11170 −0.331708
\(89\) −1.59953 −0.169550 −0.0847749 0.996400i \(-0.527017\pi\)
−0.0847749 + 0.996400i \(0.527017\pi\)
\(90\) −1.31499 −0.138612
\(91\) 3.43679 0.360273
\(92\) 5.59372 0.583186
\(93\) −2.43679 −0.252683
\(94\) −5.36242 −0.553091
\(95\) −3.20436 −0.328760
\(96\) 1.00000 0.102062
\(97\) 9.40912 0.955351 0.477676 0.878536i \(-0.341479\pi\)
0.477676 + 0.878536i \(0.341479\pi\)
\(98\) 4.81150 0.486035
\(99\) −3.11170 −0.312738
\(100\) −3.27079 −0.327079
\(101\) 13.2699 1.32041 0.660203 0.751087i \(-0.270470\pi\)
0.660203 + 0.751087i \(0.270470\pi\)
\(102\) 6.00690 0.594772
\(103\) −1.00000 −0.0985329
\(104\) 1.00000 0.0980581
\(105\) −4.51935 −0.441044
\(106\) 10.1760 0.988377
\(107\) −0.861674 −0.0833012 −0.0416506 0.999132i \(-0.513262\pi\)
−0.0416506 + 0.999132i \(0.513262\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0.888682 0.0851203 0.0425602 0.999094i \(-0.486449\pi\)
0.0425602 + 0.999094i \(0.486449\pi\)
\(110\) 4.09187 0.390144
\(111\) 2.81393 0.267087
\(112\) 3.43679 0.324746
\(113\) −6.57570 −0.618590 −0.309295 0.950966i \(-0.600093\pi\)
−0.309295 + 0.950966i \(0.600093\pi\)
\(114\) 2.43679 0.228226
\(115\) −7.35571 −0.685923
\(116\) 6.46722 0.600466
\(117\) 1.00000 0.0924500
\(118\) −4.51171 −0.415337
\(119\) 20.6444 1.89247
\(120\) −1.31499 −0.120042
\(121\) −1.31732 −0.119756
\(122\) 13.8024 1.24961
\(123\) 0.544657 0.0491101
\(124\) −2.43679 −0.218830
\(125\) 10.8760 0.972782
\(126\) 3.43679 0.306173
\(127\) 16.4226 1.45727 0.728637 0.684901i \(-0.240154\pi\)
0.728637 + 0.684901i \(0.240154\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.8675 −1.13292
\(130\) −1.31499 −0.115333
\(131\) −10.8094 −0.944422 −0.472211 0.881485i \(-0.656544\pi\)
−0.472211 + 0.881485i \(0.656544\pi\)
\(132\) −3.11170 −0.270839
\(133\) 8.37472 0.726180
\(134\) −14.5505 −1.25697
\(135\) −1.31499 −0.113177
\(136\) 6.00690 0.515087
\(137\) −10.6942 −0.913669 −0.456834 0.889552i \(-0.651017\pi\)
−0.456834 + 0.889552i \(0.651017\pi\)
\(138\) 5.59372 0.476169
\(139\) −20.9071 −1.77331 −0.886657 0.462427i \(-0.846979\pi\)
−0.886657 + 0.462427i \(0.846979\pi\)
\(140\) −4.51935 −0.381955
\(141\) −5.36242 −0.451597
\(142\) −9.93655 −0.833857
\(143\) −3.11170 −0.260213
\(144\) 1.00000 0.0833333
\(145\) −8.50435 −0.706248
\(146\) −2.55975 −0.211846
\(147\) 4.81150 0.396846
\(148\) 2.81393 0.231304
\(149\) 5.57577 0.456785 0.228393 0.973569i \(-0.426653\pi\)
0.228393 + 0.973569i \(0.426653\pi\)
\(150\) −3.27079 −0.267059
\(151\) −14.9584 −1.21730 −0.608649 0.793440i \(-0.708288\pi\)
−0.608649 + 0.793440i \(0.708288\pi\)
\(152\) 2.43679 0.197649
\(153\) 6.00690 0.485629
\(154\) −10.6943 −0.861767
\(155\) 3.20436 0.257380
\(156\) 1.00000 0.0800641
\(157\) 22.7031 1.81190 0.905951 0.423384i \(-0.139158\pi\)
0.905951 + 0.423384i \(0.139158\pi\)
\(158\) 12.3825 0.985098
\(159\) 10.1760 0.807007
\(160\) −1.31499 −0.103959
\(161\) 19.2244 1.51510
\(162\) 1.00000 0.0785674
\(163\) 3.89555 0.305123 0.152562 0.988294i \(-0.451248\pi\)
0.152562 + 0.988294i \(0.451248\pi\)
\(164\) 0.544657 0.0425306
\(165\) 4.09187 0.318551
\(166\) 8.21206 0.637379
\(167\) −5.44190 −0.421107 −0.210553 0.977582i \(-0.567527\pi\)
−0.210553 + 0.977582i \(0.567527\pi\)
\(168\) 3.43679 0.265154
\(169\) 1.00000 0.0769231
\(170\) −7.89903 −0.605828
\(171\) 2.43679 0.186346
\(172\) −12.8675 −0.981135
\(173\) 20.1657 1.53317 0.766583 0.642145i \(-0.221955\pi\)
0.766583 + 0.642145i \(0.221955\pi\)
\(174\) 6.46722 0.490279
\(175\) −11.2410 −0.849741
\(176\) −3.11170 −0.234553
\(177\) −4.51171 −0.339121
\(178\) −1.59953 −0.119890
\(179\) 11.0633 0.826912 0.413456 0.910524i \(-0.364322\pi\)
0.413456 + 0.910524i \(0.364322\pi\)
\(180\) −1.31499 −0.0980138
\(181\) 0.606839 0.0451060 0.0225530 0.999746i \(-0.492821\pi\)
0.0225530 + 0.999746i \(0.492821\pi\)
\(182\) 3.43679 0.254752
\(183\) 13.8024 1.02030
\(184\) 5.59372 0.412374
\(185\) −3.70030 −0.272052
\(186\) −2.43679 −0.178674
\(187\) −18.6917 −1.36687
\(188\) −5.36242 −0.391095
\(189\) 3.43679 0.249989
\(190\) −3.20436 −0.232469
\(191\) 6.43044 0.465291 0.232645 0.972562i \(-0.425262\pi\)
0.232645 + 0.972562i \(0.425262\pi\)
\(192\) 1.00000 0.0721688
\(193\) −21.8446 −1.57241 −0.786205 0.617966i \(-0.787957\pi\)
−0.786205 + 0.617966i \(0.787957\pi\)
\(194\) 9.40912 0.675535
\(195\) −1.31499 −0.0941686
\(196\) 4.81150 0.343679
\(197\) 16.8084 1.19755 0.598775 0.800917i \(-0.295654\pi\)
0.598775 + 0.800917i \(0.295654\pi\)
\(198\) −3.11170 −0.221139
\(199\) 27.6236 1.95819 0.979094 0.203410i \(-0.0652023\pi\)
0.979094 + 0.203410i \(0.0652023\pi\)
\(200\) −3.27079 −0.231280
\(201\) −14.5505 −1.02631
\(202\) 13.2699 0.933668
\(203\) 22.2265 1.55999
\(204\) 6.00690 0.420567
\(205\) −0.716221 −0.0500231
\(206\) −1.00000 −0.0696733
\(207\) 5.59372 0.388790
\(208\) 1.00000 0.0693375
\(209\) −7.58255 −0.524496
\(210\) −4.51935 −0.311865
\(211\) −0.511831 −0.0352359 −0.0176180 0.999845i \(-0.505608\pi\)
−0.0176180 + 0.999845i \(0.505608\pi\)
\(212\) 10.1760 0.698888
\(213\) −9.93655 −0.680841
\(214\) −0.861674 −0.0589028
\(215\) 16.9206 1.15398
\(216\) 1.00000 0.0680414
\(217\) −8.37472 −0.568513
\(218\) 0.888682 0.0601892
\(219\) −2.55975 −0.172972
\(220\) 4.09187 0.275874
\(221\) 6.00690 0.404068
\(222\) 2.81393 0.188859
\(223\) −14.3697 −0.962264 −0.481132 0.876648i \(-0.659774\pi\)
−0.481132 + 0.876648i \(0.659774\pi\)
\(224\) 3.43679 0.229630
\(225\) −3.27079 −0.218053
\(226\) −6.57570 −0.437409
\(227\) 26.2398 1.74160 0.870799 0.491639i \(-0.163602\pi\)
0.870799 + 0.491639i \(0.163602\pi\)
\(228\) 2.43679 0.161380
\(229\) 19.1941 1.26838 0.634192 0.773175i \(-0.281333\pi\)
0.634192 + 0.773175i \(0.281333\pi\)
\(230\) −7.35571 −0.485021
\(231\) −10.6943 −0.703630
\(232\) 6.46722 0.424594
\(233\) 7.61515 0.498885 0.249443 0.968390i \(-0.419753\pi\)
0.249443 + 0.968390i \(0.419753\pi\)
\(234\) 1.00000 0.0653720
\(235\) 7.05155 0.459992
\(236\) −4.51171 −0.293688
\(237\) 12.3825 0.804329
\(238\) 20.6444 1.33818
\(239\) 22.9290 1.48316 0.741578 0.670867i \(-0.234078\pi\)
0.741578 + 0.670867i \(0.234078\pi\)
\(240\) −1.31499 −0.0848825
\(241\) 13.1026 0.844014 0.422007 0.906593i \(-0.361326\pi\)
0.422007 + 0.906593i \(0.361326\pi\)
\(242\) −1.31732 −0.0846804
\(243\) 1.00000 0.0641500
\(244\) 13.8024 0.883610
\(245\) −6.32709 −0.404223
\(246\) 0.544657 0.0347261
\(247\) 2.43679 0.155049
\(248\) −2.43679 −0.154736
\(249\) 8.21206 0.520418
\(250\) 10.8760 0.687861
\(251\) 15.6751 0.989404 0.494702 0.869063i \(-0.335277\pi\)
0.494702 + 0.869063i \(0.335277\pi\)
\(252\) 3.43679 0.216497
\(253\) −17.4060 −1.09430
\(254\) 16.4226 1.03045
\(255\) −7.89903 −0.494657
\(256\) 1.00000 0.0625000
\(257\) −5.32596 −0.332224 −0.166112 0.986107i \(-0.553121\pi\)
−0.166112 + 0.986107i \(0.553121\pi\)
\(258\) −12.8675 −0.801093
\(259\) 9.67088 0.600919
\(260\) −1.31499 −0.0815524
\(261\) 6.46722 0.400311
\(262\) −10.8094 −0.667807
\(263\) −19.1914 −1.18339 −0.591697 0.806160i \(-0.701542\pi\)
−0.591697 + 0.806160i \(0.701542\pi\)
\(264\) −3.11170 −0.191512
\(265\) −13.3813 −0.822008
\(266\) 8.37472 0.513487
\(267\) −1.59953 −0.0978896
\(268\) −14.5505 −0.888813
\(269\) −20.8026 −1.26836 −0.634179 0.773187i \(-0.718662\pi\)
−0.634179 + 0.773187i \(0.718662\pi\)
\(270\) −1.31499 −0.0800280
\(271\) −18.4540 −1.12100 −0.560501 0.828153i \(-0.689392\pi\)
−0.560501 + 0.828153i \(0.689392\pi\)
\(272\) 6.00690 0.364222
\(273\) 3.43679 0.208004
\(274\) −10.6942 −0.646061
\(275\) 10.1777 0.613740
\(276\) 5.59372 0.336702
\(277\) −16.1807 −0.972205 −0.486103 0.873902i \(-0.661582\pi\)
−0.486103 + 0.873902i \(0.661582\pi\)
\(278\) −20.9071 −1.25392
\(279\) −2.43679 −0.145887
\(280\) −4.51935 −0.270083
\(281\) 20.1897 1.20441 0.602207 0.798340i \(-0.294288\pi\)
0.602207 + 0.798340i \(0.294288\pi\)
\(282\) −5.36242 −0.319327
\(283\) −21.2857 −1.26530 −0.632652 0.774436i \(-0.718034\pi\)
−0.632652 + 0.774436i \(0.718034\pi\)
\(284\) −9.93655 −0.589626
\(285\) −3.20436 −0.189810
\(286\) −3.11170 −0.183999
\(287\) 1.87187 0.110493
\(288\) 1.00000 0.0589256
\(289\) 19.0828 1.12252
\(290\) −8.50435 −0.499393
\(291\) 9.40912 0.551572
\(292\) −2.55975 −0.149798
\(293\) 23.1886 1.35469 0.677345 0.735666i \(-0.263131\pi\)
0.677345 + 0.735666i \(0.263131\pi\)
\(294\) 4.81150 0.280613
\(295\) 5.93287 0.345425
\(296\) 2.81393 0.163556
\(297\) −3.11170 −0.180559
\(298\) 5.57577 0.322996
\(299\) 5.59372 0.323493
\(300\) −3.27079 −0.188839
\(301\) −44.2227 −2.54896
\(302\) −14.9584 −0.860760
\(303\) 13.2699 0.762337
\(304\) 2.43679 0.139759
\(305\) −18.1501 −1.03927
\(306\) 6.00690 0.343392
\(307\) −24.3962 −1.39236 −0.696182 0.717865i \(-0.745119\pi\)
−0.696182 + 0.717865i \(0.745119\pi\)
\(308\) −10.6943 −0.609362
\(309\) −1.00000 −0.0568880
\(310\) 3.20436 0.181995
\(311\) −27.1959 −1.54214 −0.771068 0.636753i \(-0.780277\pi\)
−0.771068 + 0.636753i \(0.780277\pi\)
\(312\) 1.00000 0.0566139
\(313\) 2.70927 0.153137 0.0765684 0.997064i \(-0.475604\pi\)
0.0765684 + 0.997064i \(0.475604\pi\)
\(314\) 22.7031 1.28121
\(315\) −4.51935 −0.254637
\(316\) 12.3825 0.696570
\(317\) −27.6179 −1.55117 −0.775587 0.631241i \(-0.782546\pi\)
−0.775587 + 0.631241i \(0.782546\pi\)
\(318\) 10.1760 0.570640
\(319\) −20.1241 −1.12673
\(320\) −1.31499 −0.0735104
\(321\) −0.861674 −0.0480940
\(322\) 19.2244 1.07134
\(323\) 14.6375 0.814454
\(324\) 1.00000 0.0555556
\(325\) −3.27079 −0.181431
\(326\) 3.89555 0.215755
\(327\) 0.888682 0.0491442
\(328\) 0.544657 0.0300737
\(329\) −18.4295 −1.01605
\(330\) 4.09187 0.225250
\(331\) −2.45599 −0.134993 −0.0674966 0.997720i \(-0.521501\pi\)
−0.0674966 + 0.997720i \(0.521501\pi\)
\(332\) 8.21206 0.450695
\(333\) 2.81393 0.154203
\(334\) −5.44190 −0.297767
\(335\) 19.1338 1.04539
\(336\) 3.43679 0.187492
\(337\) −12.6506 −0.689122 −0.344561 0.938764i \(-0.611972\pi\)
−0.344561 + 0.938764i \(0.611972\pi\)
\(338\) 1.00000 0.0543928
\(339\) −6.57570 −0.357143
\(340\) −7.89903 −0.428385
\(341\) 7.58255 0.410618
\(342\) 2.43679 0.131766
\(343\) −7.52140 −0.406117
\(344\) −12.8675 −0.693767
\(345\) −7.35571 −0.396018
\(346\) 20.1657 1.08411
\(347\) −7.26365 −0.389933 −0.194967 0.980810i \(-0.562460\pi\)
−0.194967 + 0.980810i \(0.562460\pi\)
\(348\) 6.46722 0.346679
\(349\) 1.47059 0.0787191 0.0393595 0.999225i \(-0.487468\pi\)
0.0393595 + 0.999225i \(0.487468\pi\)
\(350\) −11.2410 −0.600857
\(351\) 1.00000 0.0533761
\(352\) −3.11170 −0.165854
\(353\) 31.5648 1.68003 0.840014 0.542565i \(-0.182547\pi\)
0.840014 + 0.542565i \(0.182547\pi\)
\(354\) −4.51171 −0.239795
\(355\) 13.0665 0.693498
\(356\) −1.59953 −0.0847749
\(357\) 20.6444 1.09262
\(358\) 11.0633 0.584715
\(359\) 3.69648 0.195092 0.0975462 0.995231i \(-0.468901\pi\)
0.0975462 + 0.995231i \(0.468901\pi\)
\(360\) −1.31499 −0.0693062
\(361\) −13.0621 −0.687477
\(362\) 0.606839 0.0318947
\(363\) −1.31732 −0.0691413
\(364\) 3.43679 0.180137
\(365\) 3.36605 0.176187
\(366\) 13.8024 0.721464
\(367\) −7.08638 −0.369906 −0.184953 0.982747i \(-0.559213\pi\)
−0.184953 + 0.982747i \(0.559213\pi\)
\(368\) 5.59372 0.291593
\(369\) 0.544657 0.0283537
\(370\) −3.70030 −0.192370
\(371\) 34.9726 1.81569
\(372\) −2.43679 −0.126342
\(373\) 9.77642 0.506204 0.253102 0.967440i \(-0.418549\pi\)
0.253102 + 0.967440i \(0.418549\pi\)
\(374\) −18.6917 −0.966523
\(375\) 10.8760 0.561636
\(376\) −5.36242 −0.276546
\(377\) 6.46722 0.333079
\(378\) 3.43679 0.176769
\(379\) 18.2285 0.936333 0.468167 0.883640i \(-0.344915\pi\)
0.468167 + 0.883640i \(0.344915\pi\)
\(380\) −3.20436 −0.164380
\(381\) 16.4226 0.841357
\(382\) 6.43044 0.329010
\(383\) −3.84084 −0.196258 −0.0981288 0.995174i \(-0.531286\pi\)
−0.0981288 + 0.995174i \(0.531286\pi\)
\(384\) 1.00000 0.0510310
\(385\) 14.0629 0.716710
\(386\) −21.8446 −1.11186
\(387\) −12.8675 −0.654090
\(388\) 9.40912 0.477676
\(389\) −27.5604 −1.39737 −0.698685 0.715429i \(-0.746231\pi\)
−0.698685 + 0.715429i \(0.746231\pi\)
\(390\) −1.31499 −0.0665873
\(391\) 33.6009 1.69927
\(392\) 4.81150 0.243018
\(393\) −10.8094 −0.545262
\(394\) 16.8084 0.846796
\(395\) −16.2829 −0.819282
\(396\) −3.11170 −0.156369
\(397\) −37.5736 −1.88576 −0.942882 0.333128i \(-0.891896\pi\)
−0.942882 + 0.333128i \(0.891896\pi\)
\(398\) 27.6236 1.38465
\(399\) 8.37472 0.419260
\(400\) −3.27079 −0.163540
\(401\) −30.7512 −1.53564 −0.767820 0.640666i \(-0.778658\pi\)
−0.767820 + 0.640666i \(0.778658\pi\)
\(402\) −14.5505 −0.725713
\(403\) −2.43679 −0.121385
\(404\) 13.2699 0.660203
\(405\) −1.31499 −0.0653426
\(406\) 22.2265 1.10308
\(407\) −8.75611 −0.434024
\(408\) 6.00690 0.297386
\(409\) 25.0321 1.23776 0.618879 0.785486i \(-0.287587\pi\)
0.618879 + 0.785486i \(0.287587\pi\)
\(410\) −0.716221 −0.0353716
\(411\) −10.6942 −0.527507
\(412\) −1.00000 −0.0492665
\(413\) −15.5058 −0.762990
\(414\) 5.59372 0.274916
\(415\) −10.7988 −0.530092
\(416\) 1.00000 0.0490290
\(417\) −20.9071 −1.02382
\(418\) −7.58255 −0.370875
\(419\) 1.04667 0.0511332 0.0255666 0.999673i \(-0.491861\pi\)
0.0255666 + 0.999673i \(0.491861\pi\)
\(420\) −4.51935 −0.220522
\(421\) −12.7139 −0.619638 −0.309819 0.950796i \(-0.600268\pi\)
−0.309819 + 0.950796i \(0.600268\pi\)
\(422\) −0.511831 −0.0249156
\(423\) −5.36242 −0.260730
\(424\) 10.1760 0.494189
\(425\) −19.6473 −0.953035
\(426\) −9.93655 −0.481428
\(427\) 47.4360 2.29559
\(428\) −0.861674 −0.0416506
\(429\) −3.11170 −0.150234
\(430\) 16.9206 0.815985
\(431\) 0.741467 0.0357152 0.0178576 0.999841i \(-0.494315\pi\)
0.0178576 + 0.999841i \(0.494315\pi\)
\(432\) 1.00000 0.0481125
\(433\) −31.1023 −1.49468 −0.747341 0.664440i \(-0.768670\pi\)
−0.747341 + 0.664440i \(0.768670\pi\)
\(434\) −8.37472 −0.401999
\(435\) −8.50435 −0.407753
\(436\) 0.888682 0.0425602
\(437\) 13.6307 0.652045
\(438\) −2.55975 −0.122309
\(439\) −38.0269 −1.81492 −0.907462 0.420134i \(-0.861983\pi\)
−0.907462 + 0.420134i \(0.861983\pi\)
\(440\) 4.09187 0.195072
\(441\) 4.81150 0.229119
\(442\) 6.00690 0.285719
\(443\) 38.2558 1.81759 0.908793 0.417248i \(-0.137005\pi\)
0.908793 + 0.417248i \(0.137005\pi\)
\(444\) 2.81393 0.133543
\(445\) 2.10337 0.0997094
\(446\) −14.3697 −0.680423
\(447\) 5.57577 0.263725
\(448\) 3.43679 0.162373
\(449\) 23.8234 1.12429 0.562147 0.827037i \(-0.309976\pi\)
0.562147 + 0.827037i \(0.309976\pi\)
\(450\) −3.27079 −0.154187
\(451\) −1.69481 −0.0798055
\(452\) −6.57570 −0.309295
\(453\) −14.9584 −0.702807
\(454\) 26.2398 1.23150
\(455\) −4.51935 −0.211871
\(456\) 2.43679 0.114113
\(457\) −19.3293 −0.904185 −0.452092 0.891971i \(-0.649322\pi\)
−0.452092 + 0.891971i \(0.649322\pi\)
\(458\) 19.1941 0.896884
\(459\) 6.00690 0.280378
\(460\) −7.35571 −0.342962
\(461\) 0.682323 0.0317789 0.0158895 0.999874i \(-0.494942\pi\)
0.0158895 + 0.999874i \(0.494942\pi\)
\(462\) −10.6943 −0.497542
\(463\) −21.9563 −1.02040 −0.510198 0.860057i \(-0.670428\pi\)
−0.510198 + 0.860057i \(0.670428\pi\)
\(464\) 6.46722 0.300233
\(465\) 3.20436 0.148599
\(466\) 7.61515 0.352765
\(467\) −16.6653 −0.771180 −0.385590 0.922670i \(-0.626002\pi\)
−0.385590 + 0.922670i \(0.626002\pi\)
\(468\) 1.00000 0.0462250
\(469\) −50.0069 −2.30911
\(470\) 7.05155 0.325264
\(471\) 22.7031 1.04610
\(472\) −4.51171 −0.207668
\(473\) 40.0397 1.84103
\(474\) 12.3825 0.568747
\(475\) −7.97022 −0.365699
\(476\) 20.6444 0.946236
\(477\) 10.1760 0.465925
\(478\) 22.9290 1.04875
\(479\) −16.7942 −0.767344 −0.383672 0.923469i \(-0.625341\pi\)
−0.383672 + 0.923469i \(0.625341\pi\)
\(480\) −1.31499 −0.0600210
\(481\) 2.81393 0.128304
\(482\) 13.1026 0.596808
\(483\) 19.2244 0.874741
\(484\) −1.31732 −0.0598781
\(485\) −12.3729 −0.561826
\(486\) 1.00000 0.0453609
\(487\) −16.0914 −0.729171 −0.364585 0.931170i \(-0.618789\pi\)
−0.364585 + 0.931170i \(0.618789\pi\)
\(488\) 13.8024 0.624806
\(489\) 3.89555 0.176163
\(490\) −6.32709 −0.285829
\(491\) 18.1187 0.817684 0.408842 0.912605i \(-0.365933\pi\)
0.408842 + 0.912605i \(0.365933\pi\)
\(492\) 0.544657 0.0245551
\(493\) 38.8479 1.74962
\(494\) 2.43679 0.109636
\(495\) 4.09187 0.183916
\(496\) −2.43679 −0.109415
\(497\) −34.1498 −1.53183
\(498\) 8.21206 0.367991
\(499\) 19.5924 0.877075 0.438537 0.898713i \(-0.355497\pi\)
0.438537 + 0.898713i \(0.355497\pi\)
\(500\) 10.8760 0.486391
\(501\) −5.44190 −0.243126
\(502\) 15.6751 0.699615
\(503\) −7.20486 −0.321249 −0.160624 0.987016i \(-0.551351\pi\)
−0.160624 + 0.987016i \(0.551351\pi\)
\(504\) 3.43679 0.153087
\(505\) −17.4499 −0.776508
\(506\) −17.4060 −0.773790
\(507\) 1.00000 0.0444116
\(508\) 16.4226 0.728637
\(509\) −25.8878 −1.14746 −0.573728 0.819046i \(-0.694503\pi\)
−0.573728 + 0.819046i \(0.694503\pi\)
\(510\) −7.89903 −0.349775
\(511\) −8.79730 −0.389170
\(512\) 1.00000 0.0441942
\(513\) 2.43679 0.107587
\(514\) −5.32596 −0.234918
\(515\) 1.31499 0.0579455
\(516\) −12.8675 −0.566459
\(517\) 16.6862 0.733860
\(518\) 9.67088 0.424914
\(519\) 20.1657 0.885174
\(520\) −1.31499 −0.0576663
\(521\) 14.5840 0.638935 0.319467 0.947597i \(-0.396496\pi\)
0.319467 + 0.947597i \(0.396496\pi\)
\(522\) 6.46722 0.283063
\(523\) −21.4457 −0.937757 −0.468879 0.883263i \(-0.655342\pi\)
−0.468879 + 0.883263i \(0.655342\pi\)
\(524\) −10.8094 −0.472211
\(525\) −11.2410 −0.490598
\(526\) −19.1914 −0.836786
\(527\) −14.6375 −0.637621
\(528\) −3.11170 −0.135419
\(529\) 8.28970 0.360422
\(530\) −13.3813 −0.581248
\(531\) −4.51171 −0.195792
\(532\) 8.37472 0.363090
\(533\) 0.544657 0.0235917
\(534\) −1.59953 −0.0692184
\(535\) 1.13310 0.0489880
\(536\) −14.5505 −0.628486
\(537\) 11.0633 0.477418
\(538\) −20.8026 −0.896864
\(539\) −14.9720 −0.644888
\(540\) −1.31499 −0.0565883
\(541\) 41.4939 1.78396 0.891982 0.452072i \(-0.149315\pi\)
0.891982 + 0.452072i \(0.149315\pi\)
\(542\) −18.4540 −0.792669
\(543\) 0.606839 0.0260419
\(544\) 6.00690 0.257544
\(545\) −1.16861 −0.0500578
\(546\) 3.43679 0.147081
\(547\) −28.3172 −1.21076 −0.605379 0.795937i \(-0.706978\pi\)
−0.605379 + 0.795937i \(0.706978\pi\)
\(548\) −10.6942 −0.456834
\(549\) 13.8024 0.589073
\(550\) 10.1777 0.433980
\(551\) 15.7592 0.671366
\(552\) 5.59372 0.238085
\(553\) 42.5560 1.80966
\(554\) −16.1807 −0.687453
\(555\) −3.70030 −0.157069
\(556\) −20.9071 −0.886657
\(557\) −35.7072 −1.51296 −0.756481 0.654015i \(-0.773083\pi\)
−0.756481 + 0.654015i \(0.773083\pi\)
\(558\) −2.43679 −0.103157
\(559\) −12.8675 −0.544236
\(560\) −4.51935 −0.190977
\(561\) −18.6917 −0.789163
\(562\) 20.1897 0.851649
\(563\) 39.2661 1.65487 0.827434 0.561564i \(-0.189800\pi\)
0.827434 + 0.561564i \(0.189800\pi\)
\(564\) −5.36242 −0.225799
\(565\) 8.64700 0.363782
\(566\) −21.2857 −0.894706
\(567\) 3.43679 0.144331
\(568\) −9.93655 −0.416928
\(569\) −1.78837 −0.0749722 −0.0374861 0.999297i \(-0.511935\pi\)
−0.0374861 + 0.999297i \(0.511935\pi\)
\(570\) −3.20436 −0.134216
\(571\) −7.03549 −0.294426 −0.147213 0.989105i \(-0.547030\pi\)
−0.147213 + 0.989105i \(0.547030\pi\)
\(572\) −3.11170 −0.130107
\(573\) 6.43044 0.268636
\(574\) 1.87187 0.0781304
\(575\) −18.2959 −0.762991
\(576\) 1.00000 0.0416667
\(577\) −7.86416 −0.327389 −0.163695 0.986511i \(-0.552341\pi\)
−0.163695 + 0.986511i \(0.552341\pi\)
\(578\) 19.0828 0.793742
\(579\) −21.8446 −0.907831
\(580\) −8.50435 −0.353124
\(581\) 28.2231 1.17089
\(582\) 9.40912 0.390021
\(583\) −31.6646 −1.31141
\(584\) −2.55975 −0.105923
\(585\) −1.31499 −0.0543683
\(586\) 23.1886 0.957910
\(587\) −12.7866 −0.527761 −0.263881 0.964555i \(-0.585002\pi\)
−0.263881 + 0.964555i \(0.585002\pi\)
\(588\) 4.81150 0.198423
\(589\) −5.93793 −0.244668
\(590\) 5.93287 0.244253
\(591\) 16.8084 0.691406
\(592\) 2.81393 0.115652
\(593\) −1.44125 −0.0591851 −0.0295925 0.999562i \(-0.509421\pi\)
−0.0295925 + 0.999562i \(0.509421\pi\)
\(594\) −3.11170 −0.127675
\(595\) −27.1473 −1.11293
\(596\) 5.57577 0.228393
\(597\) 27.6236 1.13056
\(598\) 5.59372 0.228744
\(599\) −14.1708 −0.579001 −0.289501 0.957178i \(-0.593489\pi\)
−0.289501 + 0.957178i \(0.593489\pi\)
\(600\) −3.27079 −0.133530
\(601\) −24.0693 −0.981809 −0.490904 0.871214i \(-0.663334\pi\)
−0.490904 + 0.871214i \(0.663334\pi\)
\(602\) −44.2227 −1.80238
\(603\) −14.5505 −0.592542
\(604\) −14.9584 −0.608649
\(605\) 1.73227 0.0704266
\(606\) 13.2699 0.539053
\(607\) 39.3859 1.59862 0.799312 0.600916i \(-0.205197\pi\)
0.799312 + 0.600916i \(0.205197\pi\)
\(608\) 2.43679 0.0988247
\(609\) 22.2265 0.900661
\(610\) −18.1501 −0.734876
\(611\) −5.36242 −0.216940
\(612\) 6.00690 0.242815
\(613\) 13.1418 0.530792 0.265396 0.964140i \(-0.414497\pi\)
0.265396 + 0.964140i \(0.414497\pi\)
\(614\) −24.3962 −0.984550
\(615\) −0.716221 −0.0288808
\(616\) −10.6943 −0.430884
\(617\) 15.6549 0.630244 0.315122 0.949051i \(-0.397955\pi\)
0.315122 + 0.949051i \(0.397955\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 4.74807 0.190841 0.0954204 0.995437i \(-0.469580\pi\)
0.0954204 + 0.995437i \(0.469580\pi\)
\(620\) 3.20436 0.128690
\(621\) 5.59372 0.224468
\(622\) −27.1959 −1.09045
\(623\) −5.49724 −0.220242
\(624\) 1.00000 0.0400320
\(625\) 2.05204 0.0820815
\(626\) 2.70927 0.108284
\(627\) −7.58255 −0.302818
\(628\) 22.7031 0.905951
\(629\) 16.9030 0.673967
\(630\) −4.51935 −0.180055
\(631\) 5.40691 0.215246 0.107623 0.994192i \(-0.465676\pi\)
0.107623 + 0.994192i \(0.465676\pi\)
\(632\) 12.3825 0.492549
\(633\) −0.511831 −0.0203435
\(634\) −27.6179 −1.09685
\(635\) −21.5957 −0.856998
\(636\) 10.1760 0.403503
\(637\) 4.81150 0.190639
\(638\) −20.1241 −0.796719
\(639\) −9.93655 −0.393084
\(640\) −1.31499 −0.0519797
\(641\) 12.3293 0.486978 0.243489 0.969904i \(-0.421708\pi\)
0.243489 + 0.969904i \(0.421708\pi\)
\(642\) −0.861674 −0.0340076
\(643\) −7.50601 −0.296008 −0.148004 0.988987i \(-0.547285\pi\)
−0.148004 + 0.988987i \(0.547285\pi\)
\(644\) 19.2244 0.757548
\(645\) 16.9206 0.666249
\(646\) 14.6375 0.575906
\(647\) 2.18457 0.0858844 0.0429422 0.999078i \(-0.486327\pi\)
0.0429422 + 0.999078i \(0.486327\pi\)
\(648\) 1.00000 0.0392837
\(649\) 14.0391 0.551083
\(650\) −3.27079 −0.128291
\(651\) −8.37472 −0.328231
\(652\) 3.89555 0.152562
\(653\) 27.3469 1.07017 0.535084 0.844799i \(-0.320280\pi\)
0.535084 + 0.844799i \(0.320280\pi\)
\(654\) 0.888682 0.0347502
\(655\) 14.2143 0.555399
\(656\) 0.544657 0.0212653
\(657\) −2.55975 −0.0998652
\(658\) −18.4295 −0.718456
\(659\) 40.7841 1.58872 0.794361 0.607447i \(-0.207806\pi\)
0.794361 + 0.607447i \(0.207806\pi\)
\(660\) 4.09187 0.159276
\(661\) −33.3624 −1.29765 −0.648823 0.760939i \(-0.724738\pi\)
−0.648823 + 0.760939i \(0.724738\pi\)
\(662\) −2.45599 −0.0954546
\(663\) 6.00690 0.233289
\(664\) 8.21206 0.318690
\(665\) −11.0127 −0.427054
\(666\) 2.81393 0.109038
\(667\) 36.1758 1.40073
\(668\) −5.44190 −0.210553
\(669\) −14.3697 −0.555563
\(670\) 19.1338 0.739203
\(671\) −42.9490 −1.65803
\(672\) 3.43679 0.132577
\(673\) 32.9787 1.27124 0.635618 0.772003i \(-0.280745\pi\)
0.635618 + 0.772003i \(0.280745\pi\)
\(674\) −12.6506 −0.487283
\(675\) −3.27079 −0.125893
\(676\) 1.00000 0.0384615
\(677\) −18.7249 −0.719656 −0.359828 0.933019i \(-0.617165\pi\)
−0.359828 + 0.933019i \(0.617165\pi\)
\(678\) −6.57570 −0.252538
\(679\) 32.3371 1.24099
\(680\) −7.89903 −0.302914
\(681\) 26.2398 1.00551
\(682\) 7.58255 0.290351
\(683\) −34.5159 −1.32072 −0.660358 0.750951i \(-0.729595\pi\)
−0.660358 + 0.750951i \(0.729595\pi\)
\(684\) 2.43679 0.0931729
\(685\) 14.0628 0.537313
\(686\) −7.52140 −0.287168
\(687\) 19.1941 0.732302
\(688\) −12.8675 −0.490568
\(689\) 10.1760 0.387673
\(690\) −7.35571 −0.280027
\(691\) −5.75055 −0.218761 −0.109381 0.994000i \(-0.534887\pi\)
−0.109381 + 0.994000i \(0.534887\pi\)
\(692\) 20.1657 0.766583
\(693\) −10.6943 −0.406241
\(694\) −7.26365 −0.275725
\(695\) 27.4927 1.04286
\(696\) 6.46722 0.245139
\(697\) 3.27170 0.123925
\(698\) 1.47059 0.0556628
\(699\) 7.61515 0.288031
\(700\) −11.2410 −0.424870
\(701\) 12.9196 0.487966 0.243983 0.969779i \(-0.421546\pi\)
0.243983 + 0.969779i \(0.421546\pi\)
\(702\) 1.00000 0.0377426
\(703\) 6.85695 0.258615
\(704\) −3.11170 −0.117277
\(705\) 7.05155 0.265577
\(706\) 31.5648 1.18796
\(707\) 45.6059 1.71518
\(708\) −4.51171 −0.169561
\(709\) 21.8796 0.821706 0.410853 0.911702i \(-0.365231\pi\)
0.410853 + 0.911702i \(0.365231\pi\)
\(710\) 13.0665 0.490377
\(711\) 12.3825 0.464380
\(712\) −1.59953 −0.0599449
\(713\) −13.6307 −0.510474
\(714\) 20.6444 0.772598
\(715\) 4.09187 0.153027
\(716\) 11.0633 0.413456
\(717\) 22.9290 0.856301
\(718\) 3.69648 0.137951
\(719\) 24.1809 0.901796 0.450898 0.892575i \(-0.351104\pi\)
0.450898 + 0.892575i \(0.351104\pi\)
\(720\) −1.31499 −0.0490069
\(721\) −3.43679 −0.127993
\(722\) −13.0621 −0.486120
\(723\) 13.1026 0.487292
\(724\) 0.606839 0.0225530
\(725\) −21.1529 −0.785600
\(726\) −1.31732 −0.0488903
\(727\) 39.9266 1.48080 0.740398 0.672169i \(-0.234637\pi\)
0.740398 + 0.672169i \(0.234637\pi\)
\(728\) 3.43679 0.127376
\(729\) 1.00000 0.0370370
\(730\) 3.36605 0.124583
\(731\) −77.2936 −2.85881
\(732\) 13.8024 0.510152
\(733\) 52.3822 1.93478 0.967391 0.253289i \(-0.0815123\pi\)
0.967391 + 0.253289i \(0.0815123\pi\)
\(734\) −7.08638 −0.261563
\(735\) −6.32709 −0.233378
\(736\) 5.59372 0.206187
\(737\) 45.2768 1.66779
\(738\) 0.544657 0.0200491
\(739\) −30.5613 −1.12422 −0.562108 0.827064i \(-0.690009\pi\)
−0.562108 + 0.827064i \(0.690009\pi\)
\(740\) −3.70030 −0.136026
\(741\) 2.43679 0.0895176
\(742\) 34.9726 1.28389
\(743\) 35.2118 1.29179 0.645897 0.763424i \(-0.276483\pi\)
0.645897 + 0.763424i \(0.276483\pi\)
\(744\) −2.43679 −0.0893369
\(745\) −7.33211 −0.268628
\(746\) 9.77642 0.357940
\(747\) 8.21206 0.300464
\(748\) −18.6917 −0.683435
\(749\) −2.96139 −0.108207
\(750\) 10.8760 0.397137
\(751\) −22.5006 −0.821060 −0.410530 0.911847i \(-0.634656\pi\)
−0.410530 + 0.911847i \(0.634656\pi\)
\(752\) −5.36242 −0.195547
\(753\) 15.6751 0.571233
\(754\) 6.46722 0.235522
\(755\) 19.6702 0.715872
\(756\) 3.43679 0.124995
\(757\) −21.4959 −0.781282 −0.390641 0.920543i \(-0.627747\pi\)
−0.390641 + 0.920543i \(0.627747\pi\)
\(758\) 18.2285 0.662088
\(759\) −17.4060 −0.631797
\(760\) −3.20436 −0.116234
\(761\) −49.4797 −1.79364 −0.896818 0.442399i \(-0.854128\pi\)
−0.896818 + 0.442399i \(0.854128\pi\)
\(762\) 16.4226 0.594929
\(763\) 3.05421 0.110570
\(764\) 6.43044 0.232645
\(765\) −7.89903 −0.285590
\(766\) −3.84084 −0.138775
\(767\) −4.51171 −0.162909
\(768\) 1.00000 0.0360844
\(769\) −36.1691 −1.30429 −0.652145 0.758094i \(-0.726131\pi\)
−0.652145 + 0.758094i \(0.726131\pi\)
\(770\) 14.0629 0.506791
\(771\) −5.32596 −0.191810
\(772\) −21.8446 −0.786205
\(773\) −1.64867 −0.0592987 −0.0296493 0.999560i \(-0.509439\pi\)
−0.0296493 + 0.999560i \(0.509439\pi\)
\(774\) −12.8675 −0.462512
\(775\) 7.97022 0.286299
\(776\) 9.40912 0.337768
\(777\) 9.67088 0.346941
\(778\) −27.5604 −0.988090
\(779\) 1.32721 0.0475524
\(780\) −1.31499 −0.0470843
\(781\) 30.9196 1.10639
\(782\) 33.6009 1.20157
\(783\) 6.46722 0.231120
\(784\) 4.81150 0.171839
\(785\) −29.8544 −1.06555
\(786\) −10.8094 −0.385559
\(787\) 8.08810 0.288310 0.144155 0.989555i \(-0.453954\pi\)
0.144155 + 0.989555i \(0.453954\pi\)
\(788\) 16.8084 0.598775
\(789\) −19.1914 −0.683233
\(790\) −16.2829 −0.579320
\(791\) −22.5993 −0.803538
\(792\) −3.11170 −0.110569
\(793\) 13.8024 0.490139
\(794\) −37.5736 −1.33344
\(795\) −13.3813 −0.474587
\(796\) 27.6236 0.979094
\(797\) 15.3447 0.543538 0.271769 0.962363i \(-0.412391\pi\)
0.271769 + 0.962363i \(0.412391\pi\)
\(798\) 8.37472 0.296462
\(799\) −32.2115 −1.13956
\(800\) −3.27079 −0.115640
\(801\) −1.59953 −0.0565166
\(802\) −30.7512 −1.08586
\(803\) 7.96516 0.281084
\(804\) −14.5505 −0.513156
\(805\) −25.2800 −0.891003
\(806\) −2.43679 −0.0858321
\(807\) −20.8026 −0.732286
\(808\) 13.2699 0.466834
\(809\) −52.9166 −1.86045 −0.930224 0.366992i \(-0.880388\pi\)
−0.930224 + 0.366992i \(0.880388\pi\)
\(810\) −1.31499 −0.0462042
\(811\) −13.6572 −0.479569 −0.239785 0.970826i \(-0.577077\pi\)
−0.239785 + 0.970826i \(0.577077\pi\)
\(812\) 22.2265 0.779996
\(813\) −18.4540 −0.647211
\(814\) −8.75611 −0.306902
\(815\) −5.12262 −0.179438
\(816\) 6.00690 0.210284
\(817\) −31.3553 −1.09698
\(818\) 25.0321 0.875227
\(819\) 3.43679 0.120091
\(820\) −0.716221 −0.0250115
\(821\) −23.2542 −0.811579 −0.405789 0.913967i \(-0.633003\pi\)
−0.405789 + 0.913967i \(0.633003\pi\)
\(822\) −10.6942 −0.373004
\(823\) −27.5833 −0.961494 −0.480747 0.876859i \(-0.659635\pi\)
−0.480747 + 0.876859i \(0.659635\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 10.1777 0.354343
\(826\) −15.5058 −0.539516
\(827\) −15.7934 −0.549191 −0.274595 0.961560i \(-0.588544\pi\)
−0.274595 + 0.961560i \(0.588544\pi\)
\(828\) 5.59372 0.194395
\(829\) −36.5604 −1.26979 −0.634897 0.772597i \(-0.718958\pi\)
−0.634897 + 0.772597i \(0.718958\pi\)
\(830\) −10.7988 −0.374832
\(831\) −16.1807 −0.561303
\(832\) 1.00000 0.0346688
\(833\) 28.9022 1.00140
\(834\) −20.9071 −0.723953
\(835\) 7.15606 0.247646
\(836\) −7.58255 −0.262248
\(837\) −2.43679 −0.0842277
\(838\) 1.04667 0.0361566
\(839\) 7.79528 0.269123 0.134562 0.990905i \(-0.457037\pi\)
0.134562 + 0.990905i \(0.457037\pi\)
\(840\) −4.51935 −0.155932
\(841\) 12.8249 0.442239
\(842\) −12.7139 −0.438150
\(843\) 20.1897 0.695369
\(844\) −0.511831 −0.0176180
\(845\) −1.31499 −0.0452372
\(846\) −5.36242 −0.184364
\(847\) −4.52734 −0.155561
\(848\) 10.1760 0.349444
\(849\) −21.2857 −0.730524
\(850\) −19.6473 −0.673897
\(851\) 15.7403 0.539572
\(852\) −9.93655 −0.340421
\(853\) 33.3821 1.14298 0.571491 0.820609i \(-0.306365\pi\)
0.571491 + 0.820609i \(0.306365\pi\)
\(854\) 47.4360 1.62323
\(855\) −3.20436 −0.109587
\(856\) −0.861674 −0.0294514
\(857\) 10.9659 0.374587 0.187294 0.982304i \(-0.440028\pi\)
0.187294 + 0.982304i \(0.440028\pi\)
\(858\) −3.11170 −0.106232
\(859\) −22.9068 −0.781570 −0.390785 0.920482i \(-0.627796\pi\)
−0.390785 + 0.920482i \(0.627796\pi\)
\(860\) 16.9206 0.576989
\(861\) 1.87187 0.0637932
\(862\) 0.741467 0.0252545
\(863\) 42.8975 1.46025 0.730125 0.683314i \(-0.239462\pi\)
0.730125 + 0.683314i \(0.239462\pi\)
\(864\) 1.00000 0.0340207
\(865\) −26.5177 −0.901629
\(866\) −31.1023 −1.05690
\(867\) 19.0828 0.648087
\(868\) −8.37472 −0.284256
\(869\) −38.5306 −1.30706
\(870\) −8.50435 −0.288325
\(871\) −14.5505 −0.493025
\(872\) 0.888682 0.0300946
\(873\) 9.40912 0.318450
\(874\) 13.6307 0.461065
\(875\) 37.3786 1.26363
\(876\) −2.55975 −0.0864858
\(877\) −36.1793 −1.22169 −0.610844 0.791751i \(-0.709170\pi\)
−0.610844 + 0.791751i \(0.709170\pi\)
\(878\) −38.0269 −1.28335
\(879\) 23.1886 0.782130
\(880\) 4.09187 0.137937
\(881\) −34.7769 −1.17166 −0.585832 0.810433i \(-0.699232\pi\)
−0.585832 + 0.810433i \(0.699232\pi\)
\(882\) 4.81150 0.162012
\(883\) 0.809098 0.0272283 0.0136142 0.999907i \(-0.495666\pi\)
0.0136142 + 0.999907i \(0.495666\pi\)
\(884\) 6.00690 0.202034
\(885\) 5.93287 0.199431
\(886\) 38.2558 1.28523
\(887\) 3.06854 0.103031 0.0515157 0.998672i \(-0.483595\pi\)
0.0515157 + 0.998672i \(0.483595\pi\)
\(888\) 2.81393 0.0944294
\(889\) 56.4411 1.89297
\(890\) 2.10337 0.0705052
\(891\) −3.11170 −0.104246
\(892\) −14.3697 −0.481132
\(893\) −13.0671 −0.437273
\(894\) 5.57577 0.186482
\(895\) −14.5482 −0.486293
\(896\) 3.43679 0.114815
\(897\) 5.59372 0.186769
\(898\) 23.8234 0.794996
\(899\) −15.7592 −0.525600
\(900\) −3.27079 −0.109026
\(901\) 61.1260 2.03640
\(902\) −1.69481 −0.0564310
\(903\) −44.2227 −1.47164
\(904\) −6.57570 −0.218705
\(905\) −0.797989 −0.0265261
\(906\) −14.9584 −0.496960
\(907\) 0.553149 0.0183670 0.00918350 0.999958i \(-0.497077\pi\)
0.00918350 + 0.999958i \(0.497077\pi\)
\(908\) 26.2398 0.870799
\(909\) 13.2699 0.440135
\(910\) −4.51935 −0.149815
\(911\) 32.7891 1.08635 0.543176 0.839619i \(-0.317222\pi\)
0.543176 + 0.839619i \(0.317222\pi\)
\(912\) 2.43679 0.0806901
\(913\) −25.5535 −0.845696
\(914\) −19.3293 −0.639355
\(915\) −18.1501 −0.600024
\(916\) 19.1941 0.634192
\(917\) −37.1496 −1.22679
\(918\) 6.00690 0.198257
\(919\) 14.7305 0.485913 0.242957 0.970037i \(-0.421883\pi\)
0.242957 + 0.970037i \(0.421883\pi\)
\(920\) −7.35571 −0.242510
\(921\) −24.3962 −0.803882
\(922\) 0.682323 0.0224711
\(923\) −9.93655 −0.327066
\(924\) −10.6943 −0.351815
\(925\) −9.20379 −0.302619
\(926\) −21.9563 −0.721529
\(927\) −1.00000 −0.0328443
\(928\) 6.46722 0.212297
\(929\) −40.2478 −1.32049 −0.660244 0.751051i \(-0.729547\pi\)
−0.660244 + 0.751051i \(0.729547\pi\)
\(930\) 3.20436 0.105075
\(931\) 11.7246 0.384258
\(932\) 7.61515 0.249443
\(933\) −27.1959 −0.890353
\(934\) −16.6653 −0.545307
\(935\) 24.5794 0.803833
\(936\) 1.00000 0.0326860
\(937\) −18.4649 −0.603223 −0.301611 0.953431i \(-0.597525\pi\)
−0.301611 + 0.953431i \(0.597525\pi\)
\(938\) −50.0069 −1.63278
\(939\) 2.70927 0.0884135
\(940\) 7.05155 0.229996
\(941\) −3.86966 −0.126147 −0.0630736 0.998009i \(-0.520090\pi\)
−0.0630736 + 0.998009i \(0.520090\pi\)
\(942\) 22.7031 0.739705
\(943\) 3.04666 0.0992129
\(944\) −4.51171 −0.146844
\(945\) −4.51935 −0.147015
\(946\) 40.0397 1.30180
\(947\) 48.6165 1.57983 0.789913 0.613219i \(-0.210126\pi\)
0.789913 + 0.613219i \(0.210126\pi\)
\(948\) 12.3825 0.402165
\(949\) −2.55975 −0.0830929
\(950\) −7.97022 −0.258588
\(951\) −27.6179 −0.895571
\(952\) 20.6444 0.669090
\(953\) 7.16603 0.232131 0.116065 0.993242i \(-0.462972\pi\)
0.116065 + 0.993242i \(0.462972\pi\)
\(954\) 10.1760 0.329459
\(955\) −8.45599 −0.273630
\(956\) 22.9290 0.741578
\(957\) −20.1241 −0.650518
\(958\) −16.7942 −0.542594
\(959\) −36.7537 −1.18684
\(960\) −1.31499 −0.0424412
\(961\) −25.0621 −0.808454
\(962\) 2.81393 0.0907248
\(963\) −0.861674 −0.0277671
\(964\) 13.1026 0.422007
\(965\) 28.7255 0.924708
\(966\) 19.2244 0.618536
\(967\) −34.3488 −1.10458 −0.552292 0.833651i \(-0.686247\pi\)
−0.552292 + 0.833651i \(0.686247\pi\)
\(968\) −1.31732 −0.0423402
\(969\) 14.6375 0.470225
\(970\) −12.3729 −0.397271
\(971\) −8.27328 −0.265502 −0.132751 0.991149i \(-0.542381\pi\)
−0.132751 + 0.991149i \(0.542381\pi\)
\(972\) 1.00000 0.0320750
\(973\) −71.8531 −2.30351
\(974\) −16.0914 −0.515602
\(975\) −3.27079 −0.104749
\(976\) 13.8024 0.441805
\(977\) −3.73445 −0.119476 −0.0597379 0.998214i \(-0.519026\pi\)
−0.0597379 + 0.998214i \(0.519026\pi\)
\(978\) 3.89555 0.124566
\(979\) 4.97726 0.159074
\(980\) −6.32709 −0.202112
\(981\) 0.888682 0.0283734
\(982\) 18.1187 0.578190
\(983\) −41.0686 −1.30989 −0.654943 0.755679i \(-0.727307\pi\)
−0.654943 + 0.755679i \(0.727307\pi\)
\(984\) 0.544657 0.0173630
\(985\) −22.1030 −0.704259
\(986\) 38.8479 1.23717
\(987\) −18.4295 −0.586617
\(988\) 2.43679 0.0775245
\(989\) −71.9770 −2.28874
\(990\) 4.09187 0.130048
\(991\) 56.0647 1.78095 0.890477 0.455029i \(-0.150371\pi\)
0.890477 + 0.455029i \(0.150371\pi\)
\(992\) −2.43679 −0.0773681
\(993\) −2.45599 −0.0779384
\(994\) −34.1498 −1.08317
\(995\) −36.3249 −1.15158
\(996\) 8.21206 0.260209
\(997\) −46.8961 −1.48521 −0.742607 0.669727i \(-0.766411\pi\)
−0.742607 + 0.669727i \(0.766411\pi\)
\(998\) 19.5924 0.620185
\(999\) 2.81393 0.0890289
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 8034.2.a.v.1.4 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.v.1.4 11 1.1 even 1 trivial