Properties

Label 8034.2.a.y.1.13
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 5 x^{12} - 25 x^{11} + 134 x^{10} + 196 x^{9} - 1151 x^{8} - 801 x^{7} + 4263 x^{6} + 2205 x^{5} - 6840 x^{4} - 3579 x^{3} + 3559 x^{2} + 1839 x - 180 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(0.914750\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.65794 q^{5} -1.00000 q^{6} +0.914750 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.65794 q^{5} -1.00000 q^{6} +0.914750 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.65794 q^{10} -4.42842 q^{11} -1.00000 q^{12} +1.00000 q^{13} +0.914750 q^{14} -3.65794 q^{15} +1.00000 q^{16} +5.23101 q^{17} +1.00000 q^{18} +1.71880 q^{19} +3.65794 q^{20} -0.914750 q^{21} -4.42842 q^{22} +1.96141 q^{23} -1.00000 q^{24} +8.38051 q^{25} +1.00000 q^{26} -1.00000 q^{27} +0.914750 q^{28} +0.973442 q^{29} -3.65794 q^{30} +1.71880 q^{31} +1.00000 q^{32} +4.42842 q^{33} +5.23101 q^{34} +3.34610 q^{35} +1.00000 q^{36} +7.31297 q^{37} +1.71880 q^{38} -1.00000 q^{39} +3.65794 q^{40} -6.72972 q^{41} -0.914750 q^{42} -1.19832 q^{43} -4.42842 q^{44} +3.65794 q^{45} +1.96141 q^{46} +0.170842 q^{47} -1.00000 q^{48} -6.16323 q^{49} +8.38051 q^{50} -5.23101 q^{51} +1.00000 q^{52} -4.21836 q^{53} -1.00000 q^{54} -16.1989 q^{55} +0.914750 q^{56} -1.71880 q^{57} +0.973442 q^{58} +0.893529 q^{59} -3.65794 q^{60} +1.22151 q^{61} +1.71880 q^{62} +0.914750 q^{63} +1.00000 q^{64} +3.65794 q^{65} +4.42842 q^{66} +6.26257 q^{67} +5.23101 q^{68} -1.96141 q^{69} +3.34610 q^{70} +8.89499 q^{71} +1.00000 q^{72} +10.4828 q^{73} +7.31297 q^{74} -8.38051 q^{75} +1.71880 q^{76} -4.05089 q^{77} -1.00000 q^{78} -15.7891 q^{79} +3.65794 q^{80} +1.00000 q^{81} -6.72972 q^{82} +6.55258 q^{83} -0.914750 q^{84} +19.1347 q^{85} -1.19832 q^{86} -0.973442 q^{87} -4.42842 q^{88} -5.81526 q^{89} +3.65794 q^{90} +0.914750 q^{91} +1.96141 q^{92} -1.71880 q^{93} +0.170842 q^{94} +6.28725 q^{95} -1.00000 q^{96} +14.4016 q^{97} -6.16323 q^{98} -4.42842 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{2} - 13 q^{3} + 13 q^{4} + 3 q^{5} - 13 q^{6} + 5 q^{7} + 13 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{2} - 13 q^{3} + 13 q^{4} + 3 q^{5} - 13 q^{6} + 5 q^{7} + 13 q^{8} + 13 q^{9} + 3 q^{10} + 8 q^{11} - 13 q^{12} + 13 q^{13} + 5 q^{14} - 3 q^{15} + 13 q^{16} - q^{17} + 13 q^{18} + 5 q^{19} + 3 q^{20} - 5 q^{21} + 8 q^{22} - 9 q^{23} - 13 q^{24} + 24 q^{25} + 13 q^{26} - 13 q^{27} + 5 q^{28} - q^{29} - 3 q^{30} + 5 q^{31} + 13 q^{32} - 8 q^{33} - q^{34} + 28 q^{35} + 13 q^{36} + 35 q^{37} + 5 q^{38} - 13 q^{39} + 3 q^{40} + 24 q^{41} - 5 q^{42} + 9 q^{43} + 8 q^{44} + 3 q^{45} - 9 q^{46} - 8 q^{47} - 13 q^{48} - 16 q^{49} + 24 q^{50} + q^{51} + 13 q^{52} + 32 q^{53} - 13 q^{54} - 4 q^{55} + 5 q^{56} - 5 q^{57} - q^{58} + 12 q^{59} - 3 q^{60} + 17 q^{61} + 5 q^{62} + 5 q^{63} + 13 q^{64} + 3 q^{65} - 8 q^{66} + 20 q^{67} - q^{68} + 9 q^{69} + 28 q^{70} + 16 q^{71} + 13 q^{72} + 46 q^{73} + 35 q^{74} - 24 q^{75} + 5 q^{76} - 7 q^{77} - 13 q^{78} + 17 q^{79} + 3 q^{80} + 13 q^{81} + 24 q^{82} + 13 q^{83} - 5 q^{84} + 11 q^{85} + 9 q^{86} + q^{87} + 8 q^{88} + 14 q^{89} + 3 q^{90} + 5 q^{91} - 9 q^{92} - 5 q^{93} - 8 q^{94} + 2 q^{95} - 13 q^{96} + 46 q^{97} - 16 q^{98} + 8 q^{99}+O(q^{100}) \) 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
\(5\) 3.65794 1.63588 0.817940 0.575304i \(-0.195116\pi\)
0.817940 + 0.575304i \(0.195116\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0.914750 0.345743 0.172872 0.984944i \(-0.444695\pi\)
0.172872 + 0.984944i \(0.444695\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.65794 1.15674
\(11\) −4.42842 −1.33522 −0.667609 0.744512i \(-0.732682\pi\)
−0.667609 + 0.744512i \(0.732682\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 0.914750 0.244477
\(15\) −3.65794 −0.944475
\(16\) 1.00000 0.250000
\(17\) 5.23101 1.26871 0.634354 0.773043i \(-0.281266\pi\)
0.634354 + 0.773043i \(0.281266\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.71880 0.394319 0.197159 0.980371i \(-0.436828\pi\)
0.197159 + 0.980371i \(0.436828\pi\)
\(20\) 3.65794 0.817940
\(21\) −0.914750 −0.199615
\(22\) −4.42842 −0.944141
\(23\) 1.96141 0.408981 0.204491 0.978869i \(-0.434446\pi\)
0.204491 + 0.978869i \(0.434446\pi\)
\(24\) −1.00000 −0.204124
\(25\) 8.38051 1.67610
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 0.914750 0.172872
\(29\) 0.973442 0.180764 0.0903818 0.995907i \(-0.471191\pi\)
0.0903818 + 0.995907i \(0.471191\pi\)
\(30\) −3.65794 −0.667845
\(31\) 1.71880 0.308705 0.154352 0.988016i \(-0.450671\pi\)
0.154352 + 0.988016i \(0.450671\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.42842 0.770888
\(34\) 5.23101 0.897112
\(35\) 3.34610 0.565594
\(36\) 1.00000 0.166667
\(37\) 7.31297 1.20224 0.601122 0.799157i \(-0.294720\pi\)
0.601122 + 0.799157i \(0.294720\pi\)
\(38\) 1.71880 0.278826
\(39\) −1.00000 −0.160128
\(40\) 3.65794 0.578371
\(41\) −6.72972 −1.05101 −0.525503 0.850792i \(-0.676123\pi\)
−0.525503 + 0.850792i \(0.676123\pi\)
\(42\) −0.914750 −0.141149
\(43\) −1.19832 −0.182742 −0.0913709 0.995817i \(-0.529125\pi\)
−0.0913709 + 0.995817i \(0.529125\pi\)
\(44\) −4.42842 −0.667609
\(45\) 3.65794 0.545293
\(46\) 1.96141 0.289194
\(47\) 0.170842 0.0249199 0.0124599 0.999922i \(-0.496034\pi\)
0.0124599 + 0.999922i \(0.496034\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.16323 −0.880462
\(50\) 8.38051 1.18518
\(51\) −5.23101 −0.732489
\(52\) 1.00000 0.138675
\(53\) −4.21836 −0.579436 −0.289718 0.957112i \(-0.593561\pi\)
−0.289718 + 0.957112i \(0.593561\pi\)
\(54\) −1.00000 −0.136083
\(55\) −16.1989 −2.18425
\(56\) 0.914750 0.122239
\(57\) −1.71880 −0.227660
\(58\) 0.973442 0.127819
\(59\) 0.893529 0.116328 0.0581638 0.998307i \(-0.481475\pi\)
0.0581638 + 0.998307i \(0.481475\pi\)
\(60\) −3.65794 −0.472238
\(61\) 1.22151 0.156398 0.0781990 0.996938i \(-0.475083\pi\)
0.0781990 + 0.996938i \(0.475083\pi\)
\(62\) 1.71880 0.218287
\(63\) 0.914750 0.115248
\(64\) 1.00000 0.125000
\(65\) 3.65794 0.453711
\(66\) 4.42842 0.545100
\(67\) 6.26257 0.765095 0.382548 0.923936i \(-0.375047\pi\)
0.382548 + 0.923936i \(0.375047\pi\)
\(68\) 5.23101 0.634354
\(69\) −1.96141 −0.236126
\(70\) 3.34610 0.399935
\(71\) 8.89499 1.05564 0.527821 0.849356i \(-0.323009\pi\)
0.527821 + 0.849356i \(0.323009\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.4828 1.22692 0.613459 0.789727i \(-0.289778\pi\)
0.613459 + 0.789727i \(0.289778\pi\)
\(74\) 7.31297 0.850116
\(75\) −8.38051 −0.967698
\(76\) 1.71880 0.197159
\(77\) −4.05089 −0.461642
\(78\) −1.00000 −0.113228
\(79\) −15.7891 −1.77641 −0.888205 0.459448i \(-0.848047\pi\)
−0.888205 + 0.459448i \(0.848047\pi\)
\(80\) 3.65794 0.408970
\(81\) 1.00000 0.111111
\(82\) −6.72972 −0.743173
\(83\) 6.55258 0.719240 0.359620 0.933099i \(-0.382906\pi\)
0.359620 + 0.933099i \(0.382906\pi\)
\(84\) −0.914750 −0.0998074
\(85\) 19.1347 2.07545
\(86\) −1.19832 −0.129218
\(87\) −0.973442 −0.104364
\(88\) −4.42842 −0.472071
\(89\) −5.81526 −0.616417 −0.308208 0.951319i \(-0.599729\pi\)
−0.308208 + 0.951319i \(0.599729\pi\)
\(90\) 3.65794 0.385580
\(91\) 0.914750 0.0958919
\(92\) 1.96141 0.204491
\(93\) −1.71880 −0.178231
\(94\) 0.170842 0.0176210
\(95\) 6.28725 0.645058
\(96\) −1.00000 −0.102062
\(97\) 14.4016 1.46226 0.731128 0.682240i \(-0.238994\pi\)
0.731128 + 0.682240i \(0.238994\pi\)
\(98\) −6.16323 −0.622580
\(99\) −4.42842 −0.445072
\(100\) 8.38051 0.838051
\(101\) −5.12273 −0.509731 −0.254865 0.966977i \(-0.582031\pi\)
−0.254865 + 0.966977i \(0.582031\pi\)
\(102\) −5.23101 −0.517948
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) −3.34610 −0.326546
\(106\) −4.21836 −0.409723
\(107\) −7.37919 −0.713373 −0.356687 0.934224i \(-0.616094\pi\)
−0.356687 + 0.934224i \(0.616094\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 9.79616 0.938302 0.469151 0.883118i \(-0.344560\pi\)
0.469151 + 0.883118i \(0.344560\pi\)
\(110\) −16.1989 −1.54450
\(111\) −7.31297 −0.694116
\(112\) 0.914750 0.0864358
\(113\) 0.0993838 0.00934924 0.00467462 0.999989i \(-0.498512\pi\)
0.00467462 + 0.999989i \(0.498512\pi\)
\(114\) −1.71880 −0.160980
\(115\) 7.17470 0.669044
\(116\) 0.973442 0.0903818
\(117\) 1.00000 0.0924500
\(118\) 0.893529 0.0822560
\(119\) 4.78507 0.438647
\(120\) −3.65794 −0.333923
\(121\) 8.61086 0.782806
\(122\) 1.22151 0.110590
\(123\) 6.72972 0.606798
\(124\) 1.71880 0.154352
\(125\) 12.3657 1.10602
\(126\) 0.914750 0.0814924
\(127\) 5.04372 0.447558 0.223779 0.974640i \(-0.428161\pi\)
0.223779 + 0.974640i \(0.428161\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.19832 0.105506
\(130\) 3.65794 0.320822
\(131\) −2.24919 −0.196513 −0.0982564 0.995161i \(-0.531327\pi\)
−0.0982564 + 0.995161i \(0.531327\pi\)
\(132\) 4.42842 0.385444
\(133\) 1.57227 0.136333
\(134\) 6.26257 0.541004
\(135\) −3.65794 −0.314825
\(136\) 5.23101 0.448556
\(137\) 4.58596 0.391805 0.195902 0.980623i \(-0.437236\pi\)
0.195902 + 0.980623i \(0.437236\pi\)
\(138\) −1.96141 −0.166966
\(139\) −16.5435 −1.40320 −0.701602 0.712569i \(-0.747532\pi\)
−0.701602 + 0.712569i \(0.747532\pi\)
\(140\) 3.34610 0.282797
\(141\) −0.170842 −0.0143875
\(142\) 8.89499 0.746451
\(143\) −4.42842 −0.370323
\(144\) 1.00000 0.0833333
\(145\) 3.56079 0.295707
\(146\) 10.4828 0.867562
\(147\) 6.16323 0.508335
\(148\) 7.31297 0.601122
\(149\) 17.8038 1.45855 0.729273 0.684222i \(-0.239858\pi\)
0.729273 + 0.684222i \(0.239858\pi\)
\(150\) −8.38051 −0.684266
\(151\) 18.2792 1.48754 0.743771 0.668434i \(-0.233035\pi\)
0.743771 + 0.668434i \(0.233035\pi\)
\(152\) 1.71880 0.139413
\(153\) 5.23101 0.422902
\(154\) −4.05089 −0.326430
\(155\) 6.28725 0.505004
\(156\) −1.00000 −0.0800641
\(157\) −7.83062 −0.624951 −0.312476 0.949926i \(-0.601158\pi\)
−0.312476 + 0.949926i \(0.601158\pi\)
\(158\) −15.7891 −1.25611
\(159\) 4.21836 0.334537
\(160\) 3.65794 0.289185
\(161\) 1.79420 0.141402
\(162\) 1.00000 0.0785674
\(163\) −4.97429 −0.389616 −0.194808 0.980841i \(-0.562408\pi\)
−0.194808 + 0.980841i \(0.562408\pi\)
\(164\) −6.72972 −0.525503
\(165\) 16.1989 1.26108
\(166\) 6.55258 0.508579
\(167\) 0.527598 0.0408268 0.0204134 0.999792i \(-0.493502\pi\)
0.0204134 + 0.999792i \(0.493502\pi\)
\(168\) −0.914750 −0.0705745
\(169\) 1.00000 0.0769231
\(170\) 19.1347 1.46757
\(171\) 1.71880 0.131440
\(172\) −1.19832 −0.0913709
\(173\) 1.10173 0.0837629 0.0418814 0.999123i \(-0.486665\pi\)
0.0418814 + 0.999123i \(0.486665\pi\)
\(174\) −0.973442 −0.0737964
\(175\) 7.66607 0.579500
\(176\) −4.42842 −0.333804
\(177\) −0.893529 −0.0671617
\(178\) −5.81526 −0.435872
\(179\) 21.9135 1.63789 0.818945 0.573872i \(-0.194559\pi\)
0.818945 + 0.573872i \(0.194559\pi\)
\(180\) 3.65794 0.272647
\(181\) 8.70219 0.646829 0.323414 0.946257i \(-0.395169\pi\)
0.323414 + 0.946257i \(0.395169\pi\)
\(182\) 0.914750 0.0678058
\(183\) −1.22151 −0.0902964
\(184\) 1.96141 0.144597
\(185\) 26.7504 1.96673
\(186\) −1.71880 −0.126028
\(187\) −23.1651 −1.69400
\(188\) 0.170842 0.0124599
\(189\) −0.914750 −0.0665383
\(190\) 6.28725 0.456125
\(191\) −12.1667 −0.880354 −0.440177 0.897911i \(-0.645084\pi\)
−0.440177 + 0.897911i \(0.645084\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −7.71466 −0.555314 −0.277657 0.960680i \(-0.589558\pi\)
−0.277657 + 0.960680i \(0.589558\pi\)
\(194\) 14.4016 1.03397
\(195\) −3.65794 −0.261950
\(196\) −6.16323 −0.440231
\(197\) −4.95515 −0.353040 −0.176520 0.984297i \(-0.556484\pi\)
−0.176520 + 0.984297i \(0.556484\pi\)
\(198\) −4.42842 −0.314714
\(199\) −0.902701 −0.0639908 −0.0319954 0.999488i \(-0.510186\pi\)
−0.0319954 + 0.999488i \(0.510186\pi\)
\(200\) 8.38051 0.592591
\(201\) −6.26257 −0.441728
\(202\) −5.12273 −0.360434
\(203\) 0.890456 0.0624977
\(204\) −5.23101 −0.366244
\(205\) −24.6169 −1.71932
\(206\) 1.00000 0.0696733
\(207\) 1.96141 0.136327
\(208\) 1.00000 0.0693375
\(209\) −7.61154 −0.526501
\(210\) −3.34610 −0.230903
\(211\) −24.4435 −1.68276 −0.841380 0.540445i \(-0.818256\pi\)
−0.841380 + 0.540445i \(0.818256\pi\)
\(212\) −4.21836 −0.289718
\(213\) −8.89499 −0.609475
\(214\) −7.37919 −0.504431
\(215\) −4.38337 −0.298943
\(216\) −1.00000 −0.0680414
\(217\) 1.57227 0.106733
\(218\) 9.79616 0.663480
\(219\) −10.4828 −0.708361
\(220\) −16.1989 −1.09213
\(221\) 5.23101 0.351876
\(222\) −7.31297 −0.490814
\(223\) −1.85778 −0.124406 −0.0622031 0.998064i \(-0.519813\pi\)
−0.0622031 + 0.998064i \(0.519813\pi\)
\(224\) 0.914750 0.0611193
\(225\) 8.38051 0.558701
\(226\) 0.0993838 0.00661091
\(227\) 18.9743 1.25937 0.629684 0.776852i \(-0.283185\pi\)
0.629684 + 0.776852i \(0.283185\pi\)
\(228\) −1.71880 −0.113830
\(229\) 4.58595 0.303048 0.151524 0.988454i \(-0.451582\pi\)
0.151524 + 0.988454i \(0.451582\pi\)
\(230\) 7.17470 0.473086
\(231\) 4.05089 0.266529
\(232\) 0.973442 0.0639096
\(233\) 6.46254 0.423375 0.211687 0.977337i \(-0.432104\pi\)
0.211687 + 0.977337i \(0.432104\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0.624929 0.0407659
\(236\) 0.893529 0.0581638
\(237\) 15.7891 1.02561
\(238\) 4.78507 0.310170
\(239\) −27.2859 −1.76498 −0.882489 0.470333i \(-0.844134\pi\)
−0.882489 + 0.470333i \(0.844134\pi\)
\(240\) −3.65794 −0.236119
\(241\) 24.0349 1.54823 0.774113 0.633048i \(-0.218196\pi\)
0.774113 + 0.633048i \(0.218196\pi\)
\(242\) 8.61086 0.553527
\(243\) −1.00000 −0.0641500
\(244\) 1.22151 0.0781990
\(245\) −22.5447 −1.44033
\(246\) 6.72972 0.429071
\(247\) 1.71880 0.109364
\(248\) 1.71880 0.109144
\(249\) −6.55258 −0.415253
\(250\) 12.3657 0.782075
\(251\) −20.2541 −1.27843 −0.639215 0.769028i \(-0.720740\pi\)
−0.639215 + 0.769028i \(0.720740\pi\)
\(252\) 0.914750 0.0576238
\(253\) −8.68592 −0.546079
\(254\) 5.04372 0.316471
\(255\) −19.1347 −1.19826
\(256\) 1.00000 0.0625000
\(257\) −2.81242 −0.175434 −0.0877171 0.996145i \(-0.527957\pi\)
−0.0877171 + 0.996145i \(0.527957\pi\)
\(258\) 1.19832 0.0746040
\(259\) 6.68954 0.415668
\(260\) 3.65794 0.226856
\(261\) 0.973442 0.0602545
\(262\) −2.24919 −0.138955
\(263\) −4.00949 −0.247236 −0.123618 0.992330i \(-0.539450\pi\)
−0.123618 + 0.992330i \(0.539450\pi\)
\(264\) 4.42842 0.272550
\(265\) −15.4305 −0.947887
\(266\) 1.57227 0.0964020
\(267\) 5.81526 0.355888
\(268\) 6.26257 0.382548
\(269\) −18.8430 −1.14888 −0.574438 0.818548i \(-0.694779\pi\)
−0.574438 + 0.818548i \(0.694779\pi\)
\(270\) −3.65794 −0.222615
\(271\) −19.0394 −1.15656 −0.578281 0.815837i \(-0.696276\pi\)
−0.578281 + 0.815837i \(0.696276\pi\)
\(272\) 5.23101 0.317177
\(273\) −0.914750 −0.0553632
\(274\) 4.58596 0.277048
\(275\) −37.1124 −2.23796
\(276\) −1.96141 −0.118063
\(277\) 25.1167 1.50911 0.754557 0.656235i \(-0.227852\pi\)
0.754557 + 0.656235i \(0.227852\pi\)
\(278\) −16.5435 −0.992215
\(279\) 1.71880 0.102902
\(280\) 3.34610 0.199968
\(281\) −25.7761 −1.53767 −0.768837 0.639445i \(-0.779164\pi\)
−0.768837 + 0.639445i \(0.779164\pi\)
\(282\) −0.170842 −0.0101735
\(283\) 28.5628 1.69788 0.848940 0.528489i \(-0.177241\pi\)
0.848940 + 0.528489i \(0.177241\pi\)
\(284\) 8.89499 0.527821
\(285\) −6.28725 −0.372425
\(286\) −4.42842 −0.261858
\(287\) −6.15601 −0.363378
\(288\) 1.00000 0.0589256
\(289\) 10.3635 0.609619
\(290\) 3.56079 0.209097
\(291\) −14.4016 −0.844234
\(292\) 10.4828 0.613459
\(293\) 29.7590 1.73854 0.869270 0.494338i \(-0.164590\pi\)
0.869270 + 0.494338i \(0.164590\pi\)
\(294\) 6.16323 0.359447
\(295\) 3.26847 0.190298
\(296\) 7.31297 0.425058
\(297\) 4.42842 0.256963
\(298\) 17.8038 1.03135
\(299\) 1.96141 0.113431
\(300\) −8.38051 −0.483849
\(301\) −1.09616 −0.0631817
\(302\) 18.2792 1.05185
\(303\) 5.12273 0.294293
\(304\) 1.71880 0.0985797
\(305\) 4.46820 0.255848
\(306\) 5.23101 0.299037
\(307\) −11.9130 −0.679912 −0.339956 0.940441i \(-0.610412\pi\)
−0.339956 + 0.940441i \(0.610412\pi\)
\(308\) −4.05089 −0.230821
\(309\) −1.00000 −0.0568880
\(310\) 6.28725 0.357092
\(311\) −17.0556 −0.967132 −0.483566 0.875308i \(-0.660659\pi\)
−0.483566 + 0.875308i \(0.660659\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 29.1954 1.65022 0.825110 0.564972i \(-0.191113\pi\)
0.825110 + 0.564972i \(0.191113\pi\)
\(314\) −7.83062 −0.441907
\(315\) 3.34610 0.188531
\(316\) −15.7891 −0.888205
\(317\) −11.2339 −0.630957 −0.315478 0.948933i \(-0.602165\pi\)
−0.315478 + 0.948933i \(0.602165\pi\)
\(318\) 4.21836 0.236554
\(319\) −4.31080 −0.241359
\(320\) 3.65794 0.204485
\(321\) 7.37919 0.411866
\(322\) 1.79420 0.0999867
\(323\) 8.99105 0.500275
\(324\) 1.00000 0.0555556
\(325\) 8.38051 0.464867
\(326\) −4.97429 −0.275500
\(327\) −9.79616 −0.541729
\(328\) −6.72972 −0.371587
\(329\) 0.156278 0.00861587
\(330\) 16.1989 0.891718
\(331\) −36.0555 −1.98179 −0.990895 0.134639i \(-0.957012\pi\)
−0.990895 + 0.134639i \(0.957012\pi\)
\(332\) 6.55258 0.359620
\(333\) 7.31297 0.400748
\(334\) 0.527598 0.0288689
\(335\) 22.9081 1.25160
\(336\) −0.914750 −0.0499037
\(337\) −19.4420 −1.05907 −0.529536 0.848287i \(-0.677634\pi\)
−0.529536 + 0.848287i \(0.677634\pi\)
\(338\) 1.00000 0.0543928
\(339\) −0.0993838 −0.00539779
\(340\) 19.1347 1.03773
\(341\) −7.61154 −0.412188
\(342\) 1.71880 0.0929418
\(343\) −12.0411 −0.650157
\(344\) −1.19832 −0.0646090
\(345\) −7.17470 −0.386273
\(346\) 1.10173 0.0592293
\(347\) 4.24821 0.228056 0.114028 0.993478i \(-0.463625\pi\)
0.114028 + 0.993478i \(0.463625\pi\)
\(348\) −0.973442 −0.0521819
\(349\) 13.2219 0.707751 0.353875 0.935293i \(-0.384864\pi\)
0.353875 + 0.935293i \(0.384864\pi\)
\(350\) 7.66607 0.409769
\(351\) −1.00000 −0.0533761
\(352\) −4.42842 −0.236035
\(353\) 28.2177 1.50188 0.750938 0.660373i \(-0.229602\pi\)
0.750938 + 0.660373i \(0.229602\pi\)
\(354\) −0.893529 −0.0474905
\(355\) 32.5373 1.72690
\(356\) −5.81526 −0.308208
\(357\) −4.78507 −0.253253
\(358\) 21.9135 1.15816
\(359\) 19.9685 1.05389 0.526947 0.849898i \(-0.323336\pi\)
0.526947 + 0.849898i \(0.323336\pi\)
\(360\) 3.65794 0.192790
\(361\) −16.0457 −0.844513
\(362\) 8.70219 0.457377
\(363\) −8.61086 −0.451953
\(364\) 0.914750 0.0479459
\(365\) 38.3454 2.00709
\(366\) −1.22151 −0.0638492
\(367\) −9.37152 −0.489189 −0.244595 0.969625i \(-0.578655\pi\)
−0.244595 + 0.969625i \(0.578655\pi\)
\(368\) 1.96141 0.102245
\(369\) −6.72972 −0.350335
\(370\) 26.7504 1.39069
\(371\) −3.85874 −0.200336
\(372\) −1.71880 −0.0891154
\(373\) 10.9714 0.568076 0.284038 0.958813i \(-0.408326\pi\)
0.284038 + 0.958813i \(0.408326\pi\)
\(374\) −23.1651 −1.19784
\(375\) −12.3657 −0.638561
\(376\) 0.170842 0.00881050
\(377\) 0.973442 0.0501348
\(378\) −0.914750 −0.0470497
\(379\) −4.03145 −0.207081 −0.103541 0.994625i \(-0.533017\pi\)
−0.103541 + 0.994625i \(0.533017\pi\)
\(380\) 6.28725 0.322529
\(381\) −5.04372 −0.258397
\(382\) −12.1667 −0.622505
\(383\) 12.2886 0.627919 0.313959 0.949436i \(-0.398344\pi\)
0.313959 + 0.949436i \(0.398344\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −14.8179 −0.755191
\(386\) −7.71466 −0.392666
\(387\) −1.19832 −0.0609139
\(388\) 14.4016 0.731128
\(389\) −7.96737 −0.403962 −0.201981 0.979389i \(-0.564738\pi\)
−0.201981 + 0.979389i \(0.564738\pi\)
\(390\) −3.65794 −0.185227
\(391\) 10.2601 0.518878
\(392\) −6.16323 −0.311290
\(393\) 2.24919 0.113457
\(394\) −4.95515 −0.249637
\(395\) −57.7555 −2.90599
\(396\) −4.42842 −0.222536
\(397\) −8.42629 −0.422904 −0.211452 0.977388i \(-0.567819\pi\)
−0.211452 + 0.977388i \(0.567819\pi\)
\(398\) −0.902701 −0.0452483
\(399\) −1.57227 −0.0787119
\(400\) 8.38051 0.419025
\(401\) 14.5861 0.728396 0.364198 0.931322i \(-0.381343\pi\)
0.364198 + 0.931322i \(0.381343\pi\)
\(402\) −6.26257 −0.312349
\(403\) 1.71880 0.0856193
\(404\) −5.12273 −0.254865
\(405\) 3.65794 0.181764
\(406\) 0.890456 0.0441926
\(407\) −32.3849 −1.60526
\(408\) −5.23101 −0.258974
\(409\) −8.12621 −0.401815 −0.200908 0.979610i \(-0.564389\pi\)
−0.200908 + 0.979610i \(0.564389\pi\)
\(410\) −24.6169 −1.21574
\(411\) −4.58596 −0.226209
\(412\) 1.00000 0.0492665
\(413\) 0.817355 0.0402194
\(414\) 1.96141 0.0963978
\(415\) 23.9689 1.17659
\(416\) 1.00000 0.0490290
\(417\) 16.5435 0.810141
\(418\) −7.61154 −0.372293
\(419\) 35.1857 1.71893 0.859467 0.511191i \(-0.170796\pi\)
0.859467 + 0.511191i \(0.170796\pi\)
\(420\) −3.34610 −0.163273
\(421\) −18.3372 −0.893701 −0.446850 0.894609i \(-0.647454\pi\)
−0.446850 + 0.894609i \(0.647454\pi\)
\(422\) −24.4435 −1.18989
\(423\) 0.170842 0.00830662
\(424\) −4.21836 −0.204861
\(425\) 43.8386 2.12648
\(426\) −8.89499 −0.430964
\(427\) 1.11737 0.0540735
\(428\) −7.37919 −0.356687
\(429\) 4.42842 0.213806
\(430\) −4.38337 −0.211385
\(431\) 10.2701 0.494693 0.247346 0.968927i \(-0.420441\pi\)
0.247346 + 0.968927i \(0.420441\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 19.5716 0.940553 0.470277 0.882519i \(-0.344154\pi\)
0.470277 + 0.882519i \(0.344154\pi\)
\(434\) 1.57227 0.0754713
\(435\) −3.56079 −0.170727
\(436\) 9.79616 0.469151
\(437\) 3.37126 0.161269
\(438\) −10.4828 −0.500887
\(439\) −36.2242 −1.72889 −0.864443 0.502731i \(-0.832328\pi\)
−0.864443 + 0.502731i \(0.832328\pi\)
\(440\) −16.1989 −0.772251
\(441\) −6.16323 −0.293487
\(442\) 5.23101 0.248814
\(443\) −19.4044 −0.921930 −0.460965 0.887418i \(-0.652497\pi\)
−0.460965 + 0.887418i \(0.652497\pi\)
\(444\) −7.31297 −0.347058
\(445\) −21.2719 −1.00838
\(446\) −1.85778 −0.0879685
\(447\) −17.8038 −0.842092
\(448\) 0.914750 0.0432179
\(449\) −15.8186 −0.746527 −0.373263 0.927725i \(-0.621761\pi\)
−0.373263 + 0.927725i \(0.621761\pi\)
\(450\) 8.38051 0.395061
\(451\) 29.8020 1.40332
\(452\) 0.0993838 0.00467462
\(453\) −18.2792 −0.858833
\(454\) 18.9743 0.890507
\(455\) 3.34610 0.156868
\(456\) −1.71880 −0.0804900
\(457\) −4.13162 −0.193269 −0.0966346 0.995320i \(-0.530808\pi\)
−0.0966346 + 0.995320i \(0.530808\pi\)
\(458\) 4.58595 0.214287
\(459\) −5.23101 −0.244163
\(460\) 7.17470 0.334522
\(461\) −7.23501 −0.336968 −0.168484 0.985704i \(-0.553887\pi\)
−0.168484 + 0.985704i \(0.553887\pi\)
\(462\) 4.05089 0.188465
\(463\) 2.76102 0.128315 0.0641577 0.997940i \(-0.479564\pi\)
0.0641577 + 0.997940i \(0.479564\pi\)
\(464\) 0.973442 0.0451909
\(465\) −6.28725 −0.291564
\(466\) 6.46254 0.299371
\(467\) 25.9191 1.19939 0.599697 0.800227i \(-0.295288\pi\)
0.599697 + 0.800227i \(0.295288\pi\)
\(468\) 1.00000 0.0462250
\(469\) 5.72869 0.264526
\(470\) 0.624929 0.0288258
\(471\) 7.83062 0.360816
\(472\) 0.893529 0.0411280
\(473\) 5.30665 0.244000
\(474\) 15.7891 0.725216
\(475\) 14.4044 0.660919
\(476\) 4.78507 0.219323
\(477\) −4.21836 −0.193145
\(478\) −27.2859 −1.24803
\(479\) −18.8215 −0.859977 −0.429988 0.902834i \(-0.641482\pi\)
−0.429988 + 0.902834i \(0.641482\pi\)
\(480\) −3.65794 −0.166961
\(481\) 7.31297 0.333443
\(482\) 24.0349 1.09476
\(483\) −1.79420 −0.0816388
\(484\) 8.61086 0.391403
\(485\) 52.6800 2.39207
\(486\) −1.00000 −0.0453609
\(487\) 29.4004 1.33226 0.666129 0.745837i \(-0.267950\pi\)
0.666129 + 0.745837i \(0.267950\pi\)
\(488\) 1.22151 0.0552950
\(489\) 4.97429 0.224945
\(490\) −22.5447 −1.01847
\(491\) 7.82617 0.353190 0.176595 0.984284i \(-0.443492\pi\)
0.176595 + 0.984284i \(0.443492\pi\)
\(492\) 6.72972 0.303399
\(493\) 5.09209 0.229336
\(494\) 1.71880 0.0773323
\(495\) −16.1989 −0.728085
\(496\) 1.71880 0.0771762
\(497\) 8.13669 0.364981
\(498\) −6.55258 −0.293628
\(499\) −26.4365 −1.18346 −0.591730 0.806137i \(-0.701555\pi\)
−0.591730 + 0.806137i \(0.701555\pi\)
\(500\) 12.3657 0.553010
\(501\) −0.527598 −0.0235713
\(502\) −20.2541 −0.903986
\(503\) −17.1437 −0.764401 −0.382201 0.924079i \(-0.624834\pi\)
−0.382201 + 0.924079i \(0.624834\pi\)
\(504\) 0.914750 0.0407462
\(505\) −18.7386 −0.833858
\(506\) −8.68592 −0.386136
\(507\) −1.00000 −0.0444116
\(508\) 5.04372 0.223779
\(509\) 29.4586 1.30573 0.652866 0.757474i \(-0.273567\pi\)
0.652866 + 0.757474i \(0.273567\pi\)
\(510\) −19.1347 −0.847300
\(511\) 9.58913 0.424198
\(512\) 1.00000 0.0441942
\(513\) −1.71880 −0.0758867
\(514\) −2.81242 −0.124051
\(515\) 3.65794 0.161188
\(516\) 1.19832 0.0527530
\(517\) −0.756559 −0.0332734
\(518\) 6.68954 0.293922
\(519\) −1.10173 −0.0483605
\(520\) 3.65794 0.160411
\(521\) −15.2884 −0.669799 −0.334899 0.942254i \(-0.608702\pi\)
−0.334899 + 0.942254i \(0.608702\pi\)
\(522\) 0.973442 0.0426064
\(523\) 22.7942 0.996720 0.498360 0.866970i \(-0.333936\pi\)
0.498360 + 0.866970i \(0.333936\pi\)
\(524\) −2.24919 −0.0982564
\(525\) −7.66607 −0.334575
\(526\) −4.00949 −0.174822
\(527\) 8.99105 0.391656
\(528\) 4.42842 0.192722
\(529\) −19.1529 −0.832734
\(530\) −15.4305 −0.670257
\(531\) 0.893529 0.0387758
\(532\) 1.57227 0.0681665
\(533\) −6.72972 −0.291497
\(534\) 5.81526 0.251651
\(535\) −26.9926 −1.16699
\(536\) 6.26257 0.270502
\(537\) −21.9135 −0.945636
\(538\) −18.8430 −0.812378
\(539\) 27.2934 1.17561
\(540\) −3.65794 −0.157413
\(541\) 12.4956 0.537228 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(542\) −19.0394 −0.817814
\(543\) −8.70219 −0.373447
\(544\) 5.23101 0.224278
\(545\) 35.8338 1.53495
\(546\) −0.914750 −0.0391477
\(547\) −12.6299 −0.540016 −0.270008 0.962858i \(-0.587026\pi\)
−0.270008 + 0.962858i \(0.587026\pi\)
\(548\) 4.58596 0.195902
\(549\) 1.22151 0.0521327
\(550\) −37.1124 −1.58248
\(551\) 1.67315 0.0712785
\(552\) −1.96141 −0.0834830
\(553\) −14.4431 −0.614181
\(554\) 25.1167 1.06710
\(555\) −26.7504 −1.13549
\(556\) −16.5435 −0.701602
\(557\) 12.9474 0.548600 0.274300 0.961644i \(-0.411554\pi\)
0.274300 + 0.961644i \(0.411554\pi\)
\(558\) 1.71880 0.0727624
\(559\) −1.19832 −0.0506834
\(560\) 3.34610 0.141398
\(561\) 23.1651 0.978032
\(562\) −25.7761 −1.08730
\(563\) −5.91159 −0.249144 −0.124572 0.992211i \(-0.539756\pi\)
−0.124572 + 0.992211i \(0.539756\pi\)
\(564\) −0.170842 −0.00719375
\(565\) 0.363540 0.0152942
\(566\) 28.5628 1.20058
\(567\) 0.914750 0.0384159
\(568\) 8.89499 0.373226
\(569\) −2.74793 −0.115199 −0.0575995 0.998340i \(-0.518345\pi\)
−0.0575995 + 0.998340i \(0.518345\pi\)
\(570\) −6.28725 −0.263344
\(571\) 15.9042 0.665569 0.332784 0.943003i \(-0.392012\pi\)
0.332784 + 0.943003i \(0.392012\pi\)
\(572\) −4.42842 −0.185161
\(573\) 12.1667 0.508273
\(574\) −6.15601 −0.256947
\(575\) 16.4376 0.685494
\(576\) 1.00000 0.0416667
\(577\) −29.3737 −1.22285 −0.611423 0.791304i \(-0.709402\pi\)
−0.611423 + 0.791304i \(0.709402\pi\)
\(578\) 10.3635 0.431065
\(579\) 7.71466 0.320610
\(580\) 3.56079 0.147854
\(581\) 5.99398 0.248672
\(582\) −14.4016 −0.596963
\(583\) 18.6806 0.773673
\(584\) 10.4828 0.433781
\(585\) 3.65794 0.151237
\(586\) 29.7590 1.22933
\(587\) −27.8661 −1.15016 −0.575079 0.818098i \(-0.695029\pi\)
−0.575079 + 0.818098i \(0.695029\pi\)
\(588\) 6.16323 0.254167
\(589\) 2.95426 0.121728
\(590\) 3.26847 0.134561
\(591\) 4.95515 0.203828
\(592\) 7.31297 0.300561
\(593\) 35.4100 1.45412 0.727058 0.686576i \(-0.240887\pi\)
0.727058 + 0.686576i \(0.240887\pi\)
\(594\) 4.42842 0.181700
\(595\) 17.5035 0.717573
\(596\) 17.8038 0.729273
\(597\) 0.902701 0.0369451
\(598\) 1.96141 0.0802079
\(599\) −10.7617 −0.439713 −0.219856 0.975532i \(-0.570559\pi\)
−0.219856 + 0.975532i \(0.570559\pi\)
\(600\) −8.38051 −0.342133
\(601\) 21.4956 0.876824 0.438412 0.898774i \(-0.355541\pi\)
0.438412 + 0.898774i \(0.355541\pi\)
\(602\) −1.09616 −0.0446762
\(603\) 6.26257 0.255032
\(604\) 18.2792 0.743771
\(605\) 31.4980 1.28058
\(606\) 5.12273 0.208097
\(607\) 27.6113 1.12071 0.560354 0.828253i \(-0.310665\pi\)
0.560354 + 0.828253i \(0.310665\pi\)
\(608\) 1.71880 0.0697064
\(609\) −0.890456 −0.0360831
\(610\) 4.46820 0.180912
\(611\) 0.170842 0.00691153
\(612\) 5.23101 0.211451
\(613\) 7.24980 0.292817 0.146408 0.989224i \(-0.453229\pi\)
0.146408 + 0.989224i \(0.453229\pi\)
\(614\) −11.9130 −0.480770
\(615\) 24.6169 0.992649
\(616\) −4.05089 −0.163215
\(617\) 15.1433 0.609646 0.304823 0.952409i \(-0.401403\pi\)
0.304823 + 0.952409i \(0.401403\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 34.2081 1.37494 0.687471 0.726212i \(-0.258721\pi\)
0.687471 + 0.726212i \(0.258721\pi\)
\(620\) 6.28725 0.252502
\(621\) −1.96141 −0.0787085
\(622\) −17.0556 −0.683866
\(623\) −5.31951 −0.213122
\(624\) −1.00000 −0.0400320
\(625\) 3.33038 0.133215
\(626\) 29.1954 1.16688
\(627\) 7.61154 0.303976
\(628\) −7.83062 −0.312476
\(629\) 38.2543 1.52530
\(630\) 3.34610 0.133312
\(631\) 37.1474 1.47881 0.739407 0.673259i \(-0.235106\pi\)
0.739407 + 0.673259i \(0.235106\pi\)
\(632\) −15.7891 −0.628056
\(633\) 24.4435 0.971542
\(634\) −11.2339 −0.446154
\(635\) 18.4496 0.732150
\(636\) 4.21836 0.167269
\(637\) −6.16323 −0.244196
\(638\) −4.31080 −0.170666
\(639\) 8.89499 0.351880
\(640\) 3.65794 0.144593
\(641\) −3.59215 −0.141881 −0.0709407 0.997481i \(-0.522600\pi\)
−0.0709407 + 0.997481i \(0.522600\pi\)
\(642\) 7.37919 0.291233
\(643\) −4.96575 −0.195830 −0.0979151 0.995195i \(-0.531217\pi\)
−0.0979151 + 0.995195i \(0.531217\pi\)
\(644\) 1.79420 0.0707012
\(645\) 4.38337 0.172595
\(646\) 8.99105 0.353748
\(647\) −31.2408 −1.22820 −0.614101 0.789227i \(-0.710481\pi\)
−0.614101 + 0.789227i \(0.710481\pi\)
\(648\) 1.00000 0.0392837
\(649\) −3.95692 −0.155323
\(650\) 8.38051 0.328711
\(651\) −1.57227 −0.0616221
\(652\) −4.97429 −0.194808
\(653\) −7.76229 −0.303762 −0.151881 0.988399i \(-0.548533\pi\)
−0.151881 + 0.988399i \(0.548533\pi\)
\(654\) −9.79616 −0.383060
\(655\) −8.22740 −0.321471
\(656\) −6.72972 −0.262751
\(657\) 10.4828 0.408973
\(658\) 0.156278 0.00609234
\(659\) 38.8977 1.51524 0.757620 0.652696i \(-0.226362\pi\)
0.757620 + 0.652696i \(0.226362\pi\)
\(660\) 16.1989 0.630540
\(661\) −23.0813 −0.897760 −0.448880 0.893592i \(-0.648177\pi\)
−0.448880 + 0.893592i \(0.648177\pi\)
\(662\) −36.0555 −1.40134
\(663\) −5.23101 −0.203156
\(664\) 6.55258 0.254290
\(665\) 5.75126 0.223024
\(666\) 7.31297 0.283372
\(667\) 1.90931 0.0739289
\(668\) 0.527598 0.0204134
\(669\) 1.85778 0.0718260
\(670\) 22.9081 0.885017
\(671\) −5.40934 −0.208825
\(672\) −0.914750 −0.0352873
\(673\) −22.0362 −0.849433 −0.424717 0.905326i \(-0.639626\pi\)
−0.424717 + 0.905326i \(0.639626\pi\)
\(674\) −19.4420 −0.748877
\(675\) −8.38051 −0.322566
\(676\) 1.00000 0.0384615
\(677\) −3.39656 −0.130540 −0.0652702 0.997868i \(-0.520791\pi\)
−0.0652702 + 0.997868i \(0.520791\pi\)
\(678\) −0.0993838 −0.00381681
\(679\) 13.1738 0.505565
\(680\) 19.1347 0.733783
\(681\) −18.9743 −0.727096
\(682\) −7.61154 −0.291461
\(683\) −19.6276 −0.751029 −0.375515 0.926817i \(-0.622534\pi\)
−0.375515 + 0.926817i \(0.622534\pi\)
\(684\) 1.71880 0.0657198
\(685\) 16.7751 0.640945
\(686\) −12.0411 −0.459730
\(687\) −4.58595 −0.174965
\(688\) −1.19832 −0.0456854
\(689\) −4.21836 −0.160707
\(690\) −7.17470 −0.273136
\(691\) −12.1157 −0.460904 −0.230452 0.973084i \(-0.574020\pi\)
−0.230452 + 0.973084i \(0.574020\pi\)
\(692\) 1.10173 0.0418814
\(693\) −4.05089 −0.153881
\(694\) 4.24821 0.161260
\(695\) −60.5152 −2.29547
\(696\) −0.973442 −0.0368982
\(697\) −35.2033 −1.33342
\(698\) 13.2219 0.500455
\(699\) −6.46254 −0.244436
\(700\) 7.66607 0.289750
\(701\) −6.28294 −0.237303 −0.118652 0.992936i \(-0.537857\pi\)
−0.118652 + 0.992936i \(0.537857\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 12.5695 0.474068
\(704\) −4.42842 −0.166902
\(705\) −0.624929 −0.0235362
\(706\) 28.2177 1.06199
\(707\) −4.68602 −0.176236
\(708\) −0.893529 −0.0335809
\(709\) −30.3734 −1.14070 −0.570349 0.821403i \(-0.693192\pi\)
−0.570349 + 0.821403i \(0.693192\pi\)
\(710\) 32.5373 1.22110
\(711\) −15.7891 −0.592137
\(712\) −5.81526 −0.217936
\(713\) 3.37126 0.126255
\(714\) −4.78507 −0.179077
\(715\) −16.1989 −0.605803
\(716\) 21.9135 0.818945
\(717\) 27.2859 1.01901
\(718\) 19.9685 0.745216
\(719\) −41.3809 −1.54325 −0.771623 0.636080i \(-0.780555\pi\)
−0.771623 + 0.636080i \(0.780555\pi\)
\(720\) 3.65794 0.136323
\(721\) 0.914750 0.0340671
\(722\) −16.0457 −0.597161
\(723\) −24.0349 −0.893868
\(724\) 8.70219 0.323414
\(725\) 8.15794 0.302978
\(726\) −8.61086 −0.319579
\(727\) 40.7346 1.51076 0.755381 0.655285i \(-0.227452\pi\)
0.755381 + 0.655285i \(0.227452\pi\)
\(728\) 0.914750 0.0339029
\(729\) 1.00000 0.0370370
\(730\) 38.3454 1.41923
\(731\) −6.26842 −0.231846
\(732\) −1.22151 −0.0451482
\(733\) −1.14253 −0.0422003 −0.0211001 0.999777i \(-0.506717\pi\)
−0.0211001 + 0.999777i \(0.506717\pi\)
\(734\) −9.37152 −0.345909
\(735\) 22.5447 0.831575
\(736\) 1.96141 0.0722984
\(737\) −27.7333 −1.02157
\(738\) −6.72972 −0.247724
\(739\) −6.08169 −0.223719 −0.111859 0.993724i \(-0.535681\pi\)
−0.111859 + 0.993724i \(0.535681\pi\)
\(740\) 26.7504 0.983364
\(741\) −1.71880 −0.0631416
\(742\) −3.85874 −0.141659
\(743\) −2.05578 −0.0754191 −0.0377096 0.999289i \(-0.512006\pi\)
−0.0377096 + 0.999289i \(0.512006\pi\)
\(744\) −1.71880 −0.0630141
\(745\) 65.1253 2.38601
\(746\) 10.9714 0.401690
\(747\) 6.55258 0.239747
\(748\) −23.1651 −0.847000
\(749\) −6.75012 −0.246644
\(750\) −12.3657 −0.451531
\(751\) −50.1829 −1.83120 −0.915600 0.402090i \(-0.868284\pi\)
−0.915600 + 0.402090i \(0.868284\pi\)
\(752\) 0.170842 0.00622997
\(753\) 20.2541 0.738101
\(754\) 0.973442 0.0354507
\(755\) 66.8643 2.43344
\(756\) −0.914750 −0.0332691
\(757\) 17.5089 0.636371 0.318185 0.948029i \(-0.396927\pi\)
0.318185 + 0.948029i \(0.396927\pi\)
\(758\) −4.03145 −0.146429
\(759\) 8.68592 0.315279
\(760\) 6.28725 0.228062
\(761\) −13.7120 −0.497058 −0.248529 0.968624i \(-0.579947\pi\)
−0.248529 + 0.968624i \(0.579947\pi\)
\(762\) −5.04372 −0.182715
\(763\) 8.96104 0.324412
\(764\) −12.1667 −0.440177
\(765\) 19.1347 0.691818
\(766\) 12.2886 0.444006
\(767\) 0.893529 0.0322635
\(768\) −1.00000 −0.0360844
\(769\) −15.2454 −0.549761 −0.274881 0.961478i \(-0.588638\pi\)
−0.274881 + 0.961478i \(0.588638\pi\)
\(770\) −14.8179 −0.534001
\(771\) 2.81242 0.101287
\(772\) −7.71466 −0.277657
\(773\) 18.1461 0.652670 0.326335 0.945254i \(-0.394186\pi\)
0.326335 + 0.945254i \(0.394186\pi\)
\(774\) −1.19832 −0.0430726
\(775\) 14.4044 0.517421
\(776\) 14.4016 0.516986
\(777\) −6.68954 −0.239986
\(778\) −7.96737 −0.285644
\(779\) −11.5670 −0.414431
\(780\) −3.65794 −0.130975
\(781\) −39.3907 −1.40951
\(782\) 10.2601 0.366902
\(783\) −0.973442 −0.0347880
\(784\) −6.16323 −0.220115
\(785\) −28.6439 −1.02234
\(786\) 2.24919 0.0802260
\(787\) −28.3674 −1.01119 −0.505595 0.862771i \(-0.668727\pi\)
−0.505595 + 0.862771i \(0.668727\pi\)
\(788\) −4.95515 −0.176520
\(789\) 4.00949 0.142742
\(790\) −57.7555 −2.05485
\(791\) 0.0909113 0.00323243
\(792\) −4.42842 −0.157357
\(793\) 1.22151 0.0433770
\(794\) −8.42629 −0.299038
\(795\) 15.4305 0.547263
\(796\) −0.902701 −0.0319954
\(797\) −14.8175 −0.524862 −0.262431 0.964951i \(-0.584524\pi\)
−0.262431 + 0.964951i \(0.584524\pi\)
\(798\) −1.57227 −0.0556577
\(799\) 0.893677 0.0316160
\(800\) 8.38051 0.296296
\(801\) −5.81526 −0.205472
\(802\) 14.5861 0.515054
\(803\) −46.4222 −1.63820
\(804\) −6.26257 −0.220864
\(805\) 6.56306 0.231317
\(806\) 1.71880 0.0605420
\(807\) 18.8430 0.663304
\(808\) −5.12273 −0.180217
\(809\) 23.8544 0.838674 0.419337 0.907831i \(-0.362263\pi\)
0.419337 + 0.907831i \(0.362263\pi\)
\(810\) 3.65794 0.128527
\(811\) −2.70581 −0.0950140 −0.0475070 0.998871i \(-0.515128\pi\)
−0.0475070 + 0.998871i \(0.515128\pi\)
\(812\) 0.890456 0.0312489
\(813\) 19.0394 0.667742
\(814\) −32.3849 −1.13509
\(815\) −18.1956 −0.637366
\(816\) −5.23101 −0.183122
\(817\) −2.05966 −0.0720585
\(818\) −8.12621 −0.284126
\(819\) 0.914750 0.0319640
\(820\) −24.6169 −0.859659
\(821\) −31.8825 −1.11271 −0.556354 0.830946i \(-0.687800\pi\)
−0.556354 + 0.830946i \(0.687800\pi\)
\(822\) −4.58596 −0.159954
\(823\) 36.0459 1.25648 0.628241 0.778019i \(-0.283775\pi\)
0.628241 + 0.778019i \(0.283775\pi\)
\(824\) 1.00000 0.0348367
\(825\) 37.1124 1.29209
\(826\) 0.817355 0.0284394
\(827\) 9.92174 0.345013 0.172506 0.985008i \(-0.444813\pi\)
0.172506 + 0.985008i \(0.444813\pi\)
\(828\) 1.96141 0.0681636
\(829\) 14.8089 0.514336 0.257168 0.966367i \(-0.417211\pi\)
0.257168 + 0.966367i \(0.417211\pi\)
\(830\) 23.9689 0.831974
\(831\) −25.1167 −0.871287
\(832\) 1.00000 0.0346688
\(833\) −32.2400 −1.11705
\(834\) 16.5435 0.572856
\(835\) 1.92992 0.0667877
\(836\) −7.61154 −0.263251
\(837\) −1.71880 −0.0594103
\(838\) 35.1857 1.21547
\(839\) −36.3475 −1.25486 −0.627428 0.778675i \(-0.715892\pi\)
−0.627428 + 0.778675i \(0.715892\pi\)
\(840\) −3.34610 −0.115451
\(841\) −28.0524 −0.967325
\(842\) −18.3372 −0.631942
\(843\) 25.7761 0.887776
\(844\) −24.4435 −0.841380
\(845\) 3.65794 0.125837
\(846\) 0.170842 0.00587367
\(847\) 7.87679 0.270650
\(848\) −4.21836 −0.144859
\(849\) −28.5628 −0.980271
\(850\) 43.8386 1.50365
\(851\) 14.3437 0.491696
\(852\) −8.89499 −0.304737
\(853\) −0.600488 −0.0205603 −0.0102802 0.999947i \(-0.503272\pi\)
−0.0102802 + 0.999947i \(0.503272\pi\)
\(854\) 1.11737 0.0382357
\(855\) 6.28725 0.215019
\(856\) −7.37919 −0.252216
\(857\) −53.8982 −1.84113 −0.920565 0.390591i \(-0.872271\pi\)
−0.920565 + 0.390591i \(0.872271\pi\)
\(858\) 4.42842 0.151184
\(859\) 25.4380 0.867934 0.433967 0.900929i \(-0.357113\pi\)
0.433967 + 0.900929i \(0.357113\pi\)
\(860\) −4.38337 −0.149472
\(861\) 6.15601 0.209796
\(862\) 10.2701 0.349800
\(863\) 5.04305 0.171667 0.0858337 0.996309i \(-0.472645\pi\)
0.0858337 + 0.996309i \(0.472645\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 4.03006 0.137026
\(866\) 19.5716 0.665072
\(867\) −10.3635 −0.351963
\(868\) 1.57227 0.0533663
\(869\) 69.9206 2.37189
\(870\) −3.56079 −0.120722
\(871\) 6.26257 0.212199
\(872\) 9.79616 0.331740
\(873\) 14.4016 0.487419
\(874\) 3.37126 0.114034
\(875\) 11.3115 0.382399
\(876\) −10.4828 −0.354181
\(877\) 20.2628 0.684226 0.342113 0.939659i \(-0.388857\pi\)
0.342113 + 0.939659i \(0.388857\pi\)
\(878\) −36.2242 −1.22251
\(879\) −29.7590 −1.00375
\(880\) −16.1989 −0.546064
\(881\) 41.2377 1.38933 0.694667 0.719331i \(-0.255552\pi\)
0.694667 + 0.719331i \(0.255552\pi\)
\(882\) −6.16323 −0.207527
\(883\) −3.22454 −0.108514 −0.0542571 0.998527i \(-0.517279\pi\)
−0.0542571 + 0.998527i \(0.517279\pi\)
\(884\) 5.23101 0.175938
\(885\) −3.26847 −0.109869
\(886\) −19.4044 −0.651903
\(887\) −9.07062 −0.304562 −0.152281 0.988337i \(-0.548662\pi\)
−0.152281 + 0.988337i \(0.548662\pi\)
\(888\) −7.31297 −0.245407
\(889\) 4.61374 0.154740
\(890\) −21.2719 −0.713035
\(891\) −4.42842 −0.148357
\(892\) −1.85778 −0.0622031
\(893\) 0.293643 0.00982637
\(894\) −17.8038 −0.595449
\(895\) 80.1581 2.67939
\(896\) 0.914750 0.0305597
\(897\) −1.96141 −0.0654894
\(898\) −15.8186 −0.527874
\(899\) 1.67315 0.0558026
\(900\) 8.38051 0.279350
\(901\) −22.0663 −0.735135
\(902\) 29.8020 0.992298
\(903\) 1.09616 0.0364780
\(904\) 0.0993838 0.00330545
\(905\) 31.8321 1.05813
\(906\) −18.2792 −0.607287
\(907\) 10.8622 0.360675 0.180337 0.983605i \(-0.442281\pi\)
0.180337 + 0.983605i \(0.442281\pi\)
\(908\) 18.9743 0.629684
\(909\) −5.12273 −0.169910
\(910\) 3.34610 0.110922
\(911\) −13.9020 −0.460594 −0.230297 0.973120i \(-0.573970\pi\)
−0.230297 + 0.973120i \(0.573970\pi\)
\(912\) −1.71880 −0.0569150
\(913\) −29.0176 −0.960341
\(914\) −4.13162 −0.136662
\(915\) −4.46820 −0.147714
\(916\) 4.58595 0.151524
\(917\) −2.05745 −0.0679429
\(918\) −5.23101 −0.172649
\(919\) 23.4193 0.772533 0.386267 0.922387i \(-0.373765\pi\)
0.386267 + 0.922387i \(0.373765\pi\)
\(920\) 7.17470 0.236543
\(921\) 11.9130 0.392547
\(922\) −7.23501 −0.238272
\(923\) 8.89499 0.292782
\(924\) 4.05089 0.133265
\(925\) 61.2864 2.01508
\(926\) 2.76102 0.0907327
\(927\) 1.00000 0.0328443
\(928\) 0.973442 0.0319548
\(929\) 23.4469 0.769269 0.384634 0.923069i \(-0.374328\pi\)
0.384634 + 0.923069i \(0.374328\pi\)
\(930\) −6.28725 −0.206167
\(931\) −10.5933 −0.347183
\(932\) 6.46254 0.211687
\(933\) 17.0556 0.558374
\(934\) 25.9191 0.848099
\(935\) −84.7365 −2.77118
\(936\) 1.00000 0.0326860
\(937\) 8.69746 0.284134 0.142067 0.989857i \(-0.454625\pi\)
0.142067 + 0.989857i \(0.454625\pi\)
\(938\) 5.72869 0.187048
\(939\) −29.1954 −0.952755
\(940\) 0.624929 0.0203829
\(941\) −52.2988 −1.70489 −0.852446 0.522815i \(-0.824882\pi\)
−0.852446 + 0.522815i \(0.824882\pi\)
\(942\) 7.83062 0.255135
\(943\) −13.1997 −0.429842
\(944\) 0.893529 0.0290819
\(945\) −3.34610 −0.108849
\(946\) 5.30665 0.172534
\(947\) −45.7838 −1.48777 −0.743887 0.668306i \(-0.767020\pi\)
−0.743887 + 0.668306i \(0.767020\pi\)
\(948\) 15.7891 0.512805
\(949\) 10.4828 0.340286
\(950\) 14.4044 0.467340
\(951\) 11.2339 0.364283
\(952\) 4.78507 0.155085
\(953\) −38.2532 −1.23914 −0.619571 0.784941i \(-0.712693\pi\)
−0.619571 + 0.784941i \(0.712693\pi\)
\(954\) −4.21836 −0.136574
\(955\) −44.5052 −1.44015
\(956\) −27.2859 −0.882489
\(957\) 4.31080 0.139348
\(958\) −18.8215 −0.608096
\(959\) 4.19500 0.135464
\(960\) −3.65794 −0.118059
\(961\) −28.0457 −0.904701
\(962\) 7.31297 0.235780
\(963\) −7.37919 −0.237791
\(964\) 24.0349 0.774113
\(965\) −28.2198 −0.908426
\(966\) −1.79420 −0.0577273
\(967\) −18.0188 −0.579446 −0.289723 0.957111i \(-0.593563\pi\)
−0.289723 + 0.957111i \(0.593563\pi\)
\(968\) 8.61086 0.276764
\(969\) −8.99105 −0.288834
\(970\) 52.6800 1.69145
\(971\) −3.26308 −0.104717 −0.0523586 0.998628i \(-0.516674\pi\)
−0.0523586 + 0.998628i \(0.516674\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −15.1332 −0.485148
\(974\) 29.4004 0.942048
\(975\) −8.38051 −0.268391
\(976\) 1.22151 0.0390995
\(977\) −58.3105 −1.86552 −0.932759 0.360500i \(-0.882606\pi\)
−0.932759 + 0.360500i \(0.882606\pi\)
\(978\) 4.97429 0.159060
\(979\) 25.7524 0.823050
\(980\) −22.5447 −0.720165
\(981\) 9.79616 0.312767
\(982\) 7.82617 0.249743
\(983\) 42.3991 1.35232 0.676161 0.736754i \(-0.263642\pi\)
0.676161 + 0.736754i \(0.263642\pi\)
\(984\) 6.72972 0.214536
\(985\) −18.1256 −0.577531
\(986\) 5.09209 0.162165
\(987\) −0.156278 −0.00497437
\(988\) 1.71880 0.0546822
\(989\) −2.35039 −0.0747380
\(990\) −16.1989 −0.514834
\(991\) −25.7734 −0.818719 −0.409360 0.912373i \(-0.634248\pi\)
−0.409360 + 0.912373i \(0.634248\pi\)
\(992\) 1.71880 0.0545718
\(993\) 36.0555 1.14419
\(994\) 8.13669 0.258080
\(995\) −3.30202 −0.104681
\(996\) −6.55258 −0.207627
\(997\) 30.7216 0.972962 0.486481 0.873691i \(-0.338280\pi\)
0.486481 + 0.873691i \(0.338280\pi\)
\(998\) −26.4365 −0.836832
\(999\) −7.31297 −0.231372
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.y.1.13 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.y.1.13 13 1.1 even 1 trivial