Properties

Label 8034.2.a.bd.1.15
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} - 36 x^{14} + 196 x^{13} + 498 x^{12} - 3101 x^{11} - 3150 x^{10} + 25368 x^{9} + 6763 x^{8} - 113788 x^{7} + 19731 x^{6} + 270913 x^{5} - 122680 x^{4} + \cdots - 66432 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(3.75055\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.75055 q^{5} +1.00000 q^{6} -2.75149 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.75055 q^{5} +1.00000 q^{6} -2.75149 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.75055 q^{10} +1.40678 q^{11} +1.00000 q^{12} -1.00000 q^{13} -2.75149 q^{14} +3.75055 q^{15} +1.00000 q^{16} +4.85100 q^{17} +1.00000 q^{18} -4.50999 q^{19} +3.75055 q^{20} -2.75149 q^{21} +1.40678 q^{22} -4.42327 q^{23} +1.00000 q^{24} +9.06665 q^{25} -1.00000 q^{26} +1.00000 q^{27} -2.75149 q^{28} +6.19385 q^{29} +3.75055 q^{30} -8.77193 q^{31} +1.00000 q^{32} +1.40678 q^{33} +4.85100 q^{34} -10.3196 q^{35} +1.00000 q^{36} +4.12167 q^{37} -4.50999 q^{38} -1.00000 q^{39} +3.75055 q^{40} -1.99643 q^{41} -2.75149 q^{42} +11.5347 q^{43} +1.40678 q^{44} +3.75055 q^{45} -4.42327 q^{46} +11.3867 q^{47} +1.00000 q^{48} +0.570676 q^{49} +9.06665 q^{50} +4.85100 q^{51} -1.00000 q^{52} +6.40341 q^{53} +1.00000 q^{54} +5.27619 q^{55} -2.75149 q^{56} -4.50999 q^{57} +6.19385 q^{58} -9.19555 q^{59} +3.75055 q^{60} +14.5433 q^{61} -8.77193 q^{62} -2.75149 q^{63} +1.00000 q^{64} -3.75055 q^{65} +1.40678 q^{66} +6.93725 q^{67} +4.85100 q^{68} -4.42327 q^{69} -10.3196 q^{70} +14.4592 q^{71} +1.00000 q^{72} -5.76412 q^{73} +4.12167 q^{74} +9.06665 q^{75} -4.50999 q^{76} -3.87073 q^{77} -1.00000 q^{78} -3.87580 q^{79} +3.75055 q^{80} +1.00000 q^{81} -1.99643 q^{82} +10.5152 q^{83} -2.75149 q^{84} +18.1939 q^{85} +11.5347 q^{86} +6.19385 q^{87} +1.40678 q^{88} +7.07682 q^{89} +3.75055 q^{90} +2.75149 q^{91} -4.42327 q^{92} -8.77193 q^{93} +11.3867 q^{94} -16.9150 q^{95} +1.00000 q^{96} +17.8888 q^{97} +0.570676 q^{98} +1.40678 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9} + 5 q^{10} + 18 q^{11} + 16 q^{12} - 16 q^{13} + 4 q^{14} + 5 q^{15} + 16 q^{16} + 17 q^{17} + 16 q^{18} + 8 q^{19} + 5 q^{20} + 4 q^{21} + 18 q^{22} + 9 q^{23} + 16 q^{24} + 17 q^{25} - 16 q^{26} + 16 q^{27} + 4 q^{28} + 14 q^{29} + 5 q^{30} + 12 q^{31} + 16 q^{32} + 18 q^{33} + 17 q^{34} + 16 q^{35} + 16 q^{36} + 31 q^{37} + 8 q^{38} - 16 q^{39} + 5 q^{40} + 29 q^{41} + 4 q^{42} + 30 q^{43} + 18 q^{44} + 5 q^{45} + 9 q^{46} - q^{47} + 16 q^{48} + 36 q^{49} + 17 q^{50} + 17 q^{51} - 16 q^{52} + 12 q^{53} + 16 q^{54} + 30 q^{55} + 4 q^{56} + 8 q^{57} + 14 q^{58} + 38 q^{59} + 5 q^{60} + 12 q^{62} + 4 q^{63} + 16 q^{64} - 5 q^{65} + 18 q^{66} + 28 q^{67} + 17 q^{68} + 9 q^{69} + 16 q^{70} + 32 q^{71} + 16 q^{72} + 20 q^{73} + 31 q^{74} + 17 q^{75} + 8 q^{76} + 26 q^{77} - 16 q^{78} + 13 q^{79} + 5 q^{80} + 16 q^{81} + 29 q^{82} + 39 q^{83} + 4 q^{84} + 31 q^{85} + 30 q^{86} + 14 q^{87} + 18 q^{88} + 9 q^{89} + 5 q^{90} - 4 q^{91} + 9 q^{92} + 12 q^{93} - q^{94} - 20 q^{95} + 16 q^{96} + 35 q^{97} + 36 q^{98} + 18 q^{99}+O(q^{100}) \) 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
\(5\) 3.75055 1.67730 0.838649 0.544672i \(-0.183346\pi\)
0.838649 + 0.544672i \(0.183346\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.75149 −1.03996 −0.519982 0.854177i \(-0.674061\pi\)
−0.519982 + 0.854177i \(0.674061\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.75055 1.18603
\(11\) 1.40678 0.424159 0.212080 0.977252i \(-0.431976\pi\)
0.212080 + 0.977252i \(0.431976\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) −2.75149 −0.735366
\(15\) 3.75055 0.968389
\(16\) 1.00000 0.250000
\(17\) 4.85100 1.17654 0.588270 0.808664i \(-0.299809\pi\)
0.588270 + 0.808664i \(0.299809\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.50999 −1.03466 −0.517331 0.855785i \(-0.673074\pi\)
−0.517331 + 0.855785i \(0.673074\pi\)
\(20\) 3.75055 0.838649
\(21\) −2.75149 −0.600424
\(22\) 1.40678 0.299926
\(23\) −4.42327 −0.922315 −0.461157 0.887318i \(-0.652566\pi\)
−0.461157 + 0.887318i \(0.652566\pi\)
\(24\) 1.00000 0.204124
\(25\) 9.06665 1.81333
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −2.75149 −0.519982
\(29\) 6.19385 1.15017 0.575085 0.818094i \(-0.304969\pi\)
0.575085 + 0.818094i \(0.304969\pi\)
\(30\) 3.75055 0.684754
\(31\) −8.77193 −1.57549 −0.787743 0.616005i \(-0.788750\pi\)
−0.787743 + 0.616005i \(0.788750\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.40678 0.244889
\(34\) 4.85100 0.831940
\(35\) −10.3196 −1.74433
\(36\) 1.00000 0.166667
\(37\) 4.12167 0.677599 0.338800 0.940859i \(-0.389979\pi\)
0.338800 + 0.940859i \(0.389979\pi\)
\(38\) −4.50999 −0.731617
\(39\) −1.00000 −0.160128
\(40\) 3.75055 0.593015
\(41\) −1.99643 −0.311790 −0.155895 0.987774i \(-0.549826\pi\)
−0.155895 + 0.987774i \(0.549826\pi\)
\(42\) −2.75149 −0.424564
\(43\) 11.5347 1.75903 0.879515 0.475871i \(-0.157867\pi\)
0.879515 + 0.475871i \(0.157867\pi\)
\(44\) 1.40678 0.212080
\(45\) 3.75055 0.559099
\(46\) −4.42327 −0.652175
\(47\) 11.3867 1.66092 0.830462 0.557076i \(-0.188077\pi\)
0.830462 + 0.557076i \(0.188077\pi\)
\(48\) 1.00000 0.144338
\(49\) 0.570676 0.0815252
\(50\) 9.06665 1.28222
\(51\) 4.85100 0.679276
\(52\) −1.00000 −0.138675
\(53\) 6.40341 0.879576 0.439788 0.898102i \(-0.355054\pi\)
0.439788 + 0.898102i \(0.355054\pi\)
\(54\) 1.00000 0.136083
\(55\) 5.27619 0.711442
\(56\) −2.75149 −0.367683
\(57\) −4.50999 −0.597363
\(58\) 6.19385 0.813293
\(59\) −9.19555 −1.19716 −0.598579 0.801064i \(-0.704268\pi\)
−0.598579 + 0.801064i \(0.704268\pi\)
\(60\) 3.75055 0.484194
\(61\) 14.5433 1.86208 0.931040 0.364918i \(-0.118903\pi\)
0.931040 + 0.364918i \(0.118903\pi\)
\(62\) −8.77193 −1.11404
\(63\) −2.75149 −0.346655
\(64\) 1.00000 0.125000
\(65\) −3.75055 −0.465199
\(66\) 1.40678 0.173162
\(67\) 6.93725 0.847520 0.423760 0.905775i \(-0.360710\pi\)
0.423760 + 0.905775i \(0.360710\pi\)
\(68\) 4.85100 0.588270
\(69\) −4.42327 −0.532499
\(70\) −10.3196 −1.23343
\(71\) 14.4592 1.71599 0.857996 0.513656i \(-0.171709\pi\)
0.857996 + 0.513656i \(0.171709\pi\)
\(72\) 1.00000 0.117851
\(73\) −5.76412 −0.674639 −0.337319 0.941390i \(-0.609520\pi\)
−0.337319 + 0.941390i \(0.609520\pi\)
\(74\) 4.12167 0.479135
\(75\) 9.06665 1.04693
\(76\) −4.50999 −0.517331
\(77\) −3.87073 −0.441111
\(78\) −1.00000 −0.113228
\(79\) −3.87580 −0.436061 −0.218031 0.975942i \(-0.569963\pi\)
−0.218031 + 0.975942i \(0.569963\pi\)
\(80\) 3.75055 0.419325
\(81\) 1.00000 0.111111
\(82\) −1.99643 −0.220469
\(83\) 10.5152 1.15420 0.577098 0.816675i \(-0.304185\pi\)
0.577098 + 0.816675i \(0.304185\pi\)
\(84\) −2.75149 −0.300212
\(85\) 18.1939 1.97341
\(86\) 11.5347 1.24382
\(87\) 6.19385 0.664051
\(88\) 1.40678 0.149963
\(89\) 7.07682 0.750142 0.375071 0.926996i \(-0.377618\pi\)
0.375071 + 0.926996i \(0.377618\pi\)
\(90\) 3.75055 0.395343
\(91\) 2.75149 0.288434
\(92\) −4.42327 −0.461157
\(93\) −8.77193 −0.909607
\(94\) 11.3867 1.17445
\(95\) −16.9150 −1.73544
\(96\) 1.00000 0.102062
\(97\) 17.8888 1.81633 0.908164 0.418614i \(-0.137484\pi\)
0.908164 + 0.418614i \(0.137484\pi\)
\(98\) 0.570676 0.0576470
\(99\) 1.40678 0.141386
\(100\) 9.06665 0.906665
\(101\) −7.86666 −0.782762 −0.391381 0.920229i \(-0.628003\pi\)
−0.391381 + 0.920229i \(0.628003\pi\)
\(102\) 4.85100 0.480321
\(103\) 1.00000 0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −10.3196 −1.00709
\(106\) 6.40341 0.621954
\(107\) −11.1912 −1.08189 −0.540946 0.841058i \(-0.681933\pi\)
−0.540946 + 0.841058i \(0.681933\pi\)
\(108\) 1.00000 0.0962250
\(109\) −9.76772 −0.935578 −0.467789 0.883840i \(-0.654949\pi\)
−0.467789 + 0.883840i \(0.654949\pi\)
\(110\) 5.27619 0.503065
\(111\) 4.12167 0.391212
\(112\) −2.75149 −0.259991
\(113\) −11.9388 −1.12311 −0.561555 0.827440i \(-0.689797\pi\)
−0.561555 + 0.827440i \(0.689797\pi\)
\(114\) −4.50999 −0.422399
\(115\) −16.5897 −1.54700
\(116\) 6.19385 0.575085
\(117\) −1.00000 −0.0924500
\(118\) −9.19555 −0.846519
\(119\) −13.3475 −1.22356
\(120\) 3.75055 0.342377
\(121\) −9.02098 −0.820089
\(122\) 14.5433 1.31669
\(123\) −1.99643 −0.180012
\(124\) −8.77193 −0.787743
\(125\) 15.2522 1.36420
\(126\) −2.75149 −0.245122
\(127\) 13.6249 1.20901 0.604506 0.796600i \(-0.293370\pi\)
0.604506 + 0.796600i \(0.293370\pi\)
\(128\) 1.00000 0.0883883
\(129\) 11.5347 1.01558
\(130\) −3.75055 −0.328945
\(131\) −2.27585 −0.198842 −0.0994209 0.995045i \(-0.531699\pi\)
−0.0994209 + 0.995045i \(0.531699\pi\)
\(132\) 1.40678 0.122444
\(133\) 12.4092 1.07601
\(134\) 6.93725 0.599287
\(135\) 3.75055 0.322796
\(136\) 4.85100 0.415970
\(137\) −11.7402 −1.00303 −0.501516 0.865149i \(-0.667224\pi\)
−0.501516 + 0.865149i \(0.667224\pi\)
\(138\) −4.42327 −0.376533
\(139\) 1.88819 0.160154 0.0800770 0.996789i \(-0.474483\pi\)
0.0800770 + 0.996789i \(0.474483\pi\)
\(140\) −10.3196 −0.872165
\(141\) 11.3867 0.958934
\(142\) 14.4592 1.21339
\(143\) −1.40678 −0.117641
\(144\) 1.00000 0.0833333
\(145\) 23.2304 1.92918
\(146\) −5.76412 −0.477042
\(147\) 0.570676 0.0470686
\(148\) 4.12167 0.338800
\(149\) 20.1646 1.65195 0.825973 0.563710i \(-0.190627\pi\)
0.825973 + 0.563710i \(0.190627\pi\)
\(150\) 9.06665 0.740289
\(151\) −4.02291 −0.327380 −0.163690 0.986512i \(-0.552340\pi\)
−0.163690 + 0.986512i \(0.552340\pi\)
\(152\) −4.50999 −0.365808
\(153\) 4.85100 0.392180
\(154\) −3.87073 −0.311912
\(155\) −32.8996 −2.64256
\(156\) −1.00000 −0.0800641
\(157\) −12.3376 −0.984645 −0.492322 0.870413i \(-0.663852\pi\)
−0.492322 + 0.870413i \(0.663852\pi\)
\(158\) −3.87580 −0.308342
\(159\) 6.40341 0.507823
\(160\) 3.75055 0.296507
\(161\) 12.1706 0.959174
\(162\) 1.00000 0.0785674
\(163\) 6.87000 0.538100 0.269050 0.963126i \(-0.413290\pi\)
0.269050 + 0.963126i \(0.413290\pi\)
\(164\) −1.99643 −0.155895
\(165\) 5.27619 0.410751
\(166\) 10.5152 0.816139
\(167\) −22.2739 −1.72361 −0.861803 0.507243i \(-0.830665\pi\)
−0.861803 + 0.507243i \(0.830665\pi\)
\(168\) −2.75149 −0.212282
\(169\) 1.00000 0.0769231
\(170\) 18.1939 1.39541
\(171\) −4.50999 −0.344887
\(172\) 11.5347 0.879515
\(173\) 4.21358 0.320353 0.160176 0.987088i \(-0.448794\pi\)
0.160176 + 0.987088i \(0.448794\pi\)
\(174\) 6.19385 0.469555
\(175\) −24.9468 −1.88580
\(176\) 1.40678 0.106040
\(177\) −9.19555 −0.691180
\(178\) 7.07682 0.530430
\(179\) −25.2297 −1.88576 −0.942878 0.333139i \(-0.891892\pi\)
−0.942878 + 0.333139i \(0.891892\pi\)
\(180\) 3.75055 0.279550
\(181\) −1.22302 −0.0909060 −0.0454530 0.998966i \(-0.514473\pi\)
−0.0454530 + 0.998966i \(0.514473\pi\)
\(182\) 2.75149 0.203954
\(183\) 14.5433 1.07507
\(184\) −4.42327 −0.326087
\(185\) 15.4586 1.13654
\(186\) −8.77193 −0.643189
\(187\) 6.82428 0.499041
\(188\) 11.3867 0.830462
\(189\) −2.75149 −0.200141
\(190\) −16.9150 −1.22714
\(191\) −16.9267 −1.22477 −0.612385 0.790560i \(-0.709790\pi\)
−0.612385 + 0.790560i \(0.709790\pi\)
\(192\) 1.00000 0.0721688
\(193\) 22.1717 1.59595 0.797977 0.602688i \(-0.205904\pi\)
0.797977 + 0.602688i \(0.205904\pi\)
\(194\) 17.8888 1.28434
\(195\) −3.75055 −0.268583
\(196\) 0.570676 0.0407626
\(197\) −4.41932 −0.314864 −0.157432 0.987530i \(-0.550321\pi\)
−0.157432 + 0.987530i \(0.550321\pi\)
\(198\) 1.40678 0.0999753
\(199\) −15.6553 −1.10977 −0.554887 0.831926i \(-0.687238\pi\)
−0.554887 + 0.831926i \(0.687238\pi\)
\(200\) 9.06665 0.641109
\(201\) 6.93725 0.489316
\(202\) −7.86666 −0.553496
\(203\) −17.0423 −1.19614
\(204\) 4.85100 0.339638
\(205\) −7.48771 −0.522965
\(206\) 1.00000 0.0696733
\(207\) −4.42327 −0.307438
\(208\) −1.00000 −0.0693375
\(209\) −6.34455 −0.438862
\(210\) −10.3196 −0.712120
\(211\) −11.3565 −0.781814 −0.390907 0.920430i \(-0.627839\pi\)
−0.390907 + 0.920430i \(0.627839\pi\)
\(212\) 6.40341 0.439788
\(213\) 14.4592 0.990729
\(214\) −11.1912 −0.765012
\(215\) 43.2616 2.95042
\(216\) 1.00000 0.0680414
\(217\) 24.1358 1.63845
\(218\) −9.76772 −0.661553
\(219\) −5.76412 −0.389503
\(220\) 5.27619 0.355721
\(221\) −4.85100 −0.326314
\(222\) 4.12167 0.276629
\(223\) −18.6322 −1.24770 −0.623852 0.781543i \(-0.714433\pi\)
−0.623852 + 0.781543i \(0.714433\pi\)
\(224\) −2.75149 −0.183841
\(225\) 9.06665 0.604443
\(226\) −11.9388 −0.794158
\(227\) −16.1448 −1.07157 −0.535785 0.844355i \(-0.679984\pi\)
−0.535785 + 0.844355i \(0.679984\pi\)
\(228\) −4.50999 −0.298681
\(229\) 13.4124 0.886318 0.443159 0.896443i \(-0.353858\pi\)
0.443159 + 0.896443i \(0.353858\pi\)
\(230\) −16.5897 −1.09389
\(231\) −3.87073 −0.254675
\(232\) 6.19385 0.406646
\(233\) 6.75929 0.442816 0.221408 0.975181i \(-0.428935\pi\)
0.221408 + 0.975181i \(0.428935\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 42.7065 2.78586
\(236\) −9.19555 −0.598579
\(237\) −3.87580 −0.251760
\(238\) −13.3475 −0.865187
\(239\) 12.9876 0.840099 0.420049 0.907501i \(-0.362013\pi\)
0.420049 + 0.907501i \(0.362013\pi\)
\(240\) 3.75055 0.242097
\(241\) −3.25984 −0.209985 −0.104992 0.994473i \(-0.533482\pi\)
−0.104992 + 0.994473i \(0.533482\pi\)
\(242\) −9.02098 −0.579890
\(243\) 1.00000 0.0641500
\(244\) 14.5433 0.931040
\(245\) 2.14035 0.136742
\(246\) −1.99643 −0.127288
\(247\) 4.50999 0.286964
\(248\) −8.77193 −0.557018
\(249\) 10.5152 0.666375
\(250\) 15.2522 0.964633
\(251\) −12.0501 −0.760597 −0.380299 0.924864i \(-0.624179\pi\)
−0.380299 + 0.924864i \(0.624179\pi\)
\(252\) −2.75149 −0.173327
\(253\) −6.22255 −0.391209
\(254\) 13.6249 0.854901
\(255\) 18.1939 1.13935
\(256\) 1.00000 0.0625000
\(257\) −13.5708 −0.846525 −0.423263 0.906007i \(-0.639115\pi\)
−0.423263 + 0.906007i \(0.639115\pi\)
\(258\) 11.5347 0.718121
\(259\) −11.3407 −0.704679
\(260\) −3.75055 −0.232599
\(261\) 6.19385 0.383390
\(262\) −2.27585 −0.140602
\(263\) 13.2870 0.819310 0.409655 0.912241i \(-0.365649\pi\)
0.409655 + 0.912241i \(0.365649\pi\)
\(264\) 1.40678 0.0865812
\(265\) 24.0163 1.47531
\(266\) 12.4092 0.760855
\(267\) 7.07682 0.433094
\(268\) 6.93725 0.423760
\(269\) −22.3684 −1.36383 −0.681913 0.731434i \(-0.738852\pi\)
−0.681913 + 0.731434i \(0.738852\pi\)
\(270\) 3.75055 0.228251
\(271\) 27.7261 1.68424 0.842121 0.539288i \(-0.181307\pi\)
0.842121 + 0.539288i \(0.181307\pi\)
\(272\) 4.85100 0.294135
\(273\) 2.75149 0.166528
\(274\) −11.7402 −0.709250
\(275\) 12.7548 0.769141
\(276\) −4.42327 −0.266249
\(277\) 19.8500 1.19267 0.596335 0.802735i \(-0.296623\pi\)
0.596335 + 0.802735i \(0.296623\pi\)
\(278\) 1.88819 0.113246
\(279\) −8.77193 −0.525162
\(280\) −10.3196 −0.616714
\(281\) 7.52692 0.449018 0.224509 0.974472i \(-0.427922\pi\)
0.224509 + 0.974472i \(0.427922\pi\)
\(282\) 11.3867 0.678069
\(283\) −23.8501 −1.41774 −0.708871 0.705338i \(-0.750795\pi\)
−0.708871 + 0.705338i \(0.750795\pi\)
\(284\) 14.4592 0.857996
\(285\) −16.9150 −1.00196
\(286\) −1.40678 −0.0831845
\(287\) 5.49315 0.324250
\(288\) 1.00000 0.0589256
\(289\) 6.53220 0.384247
\(290\) 23.2304 1.36413
\(291\) 17.8888 1.04866
\(292\) −5.76412 −0.337319
\(293\) −2.38503 −0.139335 −0.0696676 0.997570i \(-0.522194\pi\)
−0.0696676 + 0.997570i \(0.522194\pi\)
\(294\) 0.570676 0.0332825
\(295\) −34.4884 −2.00799
\(296\) 4.12167 0.239567
\(297\) 1.40678 0.0816295
\(298\) 20.1646 1.16810
\(299\) 4.42327 0.255804
\(300\) 9.06665 0.523463
\(301\) −31.7377 −1.82933
\(302\) −4.02291 −0.231493
\(303\) −7.86666 −0.451928
\(304\) −4.50999 −0.258666
\(305\) 54.5454 3.12326
\(306\) 4.85100 0.277313
\(307\) −12.7915 −0.730050 −0.365025 0.930998i \(-0.618940\pi\)
−0.365025 + 0.930998i \(0.618940\pi\)
\(308\) −3.87073 −0.220555
\(309\) 1.00000 0.0568880
\(310\) −32.8996 −1.86857
\(311\) −10.9137 −0.618857 −0.309428 0.950923i \(-0.600138\pi\)
−0.309428 + 0.950923i \(0.600138\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −5.14958 −0.291072 −0.145536 0.989353i \(-0.546491\pi\)
−0.145536 + 0.989353i \(0.546491\pi\)
\(314\) −12.3376 −0.696249
\(315\) −10.3196 −0.581443
\(316\) −3.87580 −0.218031
\(317\) 34.4704 1.93605 0.968025 0.250855i \(-0.0807118\pi\)
0.968025 + 0.250855i \(0.0807118\pi\)
\(318\) 6.40341 0.359085
\(319\) 8.71337 0.487855
\(320\) 3.75055 0.209662
\(321\) −11.1912 −0.624630
\(322\) 12.1706 0.678239
\(323\) −21.8780 −1.21732
\(324\) 1.00000 0.0555556
\(325\) −9.06665 −0.502927
\(326\) 6.87000 0.380494
\(327\) −9.76772 −0.540156
\(328\) −1.99643 −0.110234
\(329\) −31.3304 −1.72730
\(330\) 5.27619 0.290445
\(331\) 14.9647 0.822537 0.411268 0.911514i \(-0.365086\pi\)
0.411268 + 0.911514i \(0.365086\pi\)
\(332\) 10.5152 0.577098
\(333\) 4.12167 0.225866
\(334\) −22.2739 −1.21877
\(335\) 26.0185 1.42154
\(336\) −2.75149 −0.150106
\(337\) −25.1159 −1.36815 −0.684074 0.729413i \(-0.739793\pi\)
−0.684074 + 0.729413i \(0.739793\pi\)
\(338\) 1.00000 0.0543928
\(339\) −11.9388 −0.648427
\(340\) 18.1939 0.986705
\(341\) −12.3402 −0.668257
\(342\) −4.50999 −0.243872
\(343\) 17.6902 0.955181
\(344\) 11.5347 0.621911
\(345\) −16.5897 −0.893159
\(346\) 4.21358 0.226524
\(347\) −17.6919 −0.949751 −0.474875 0.880053i \(-0.657507\pi\)
−0.474875 + 0.880053i \(0.657507\pi\)
\(348\) 6.19385 0.332025
\(349\) −30.3608 −1.62518 −0.812589 0.582837i \(-0.801943\pi\)
−0.812589 + 0.582837i \(0.801943\pi\)
\(350\) −24.9468 −1.33346
\(351\) −1.00000 −0.0533761
\(352\) 1.40678 0.0749815
\(353\) −28.0436 −1.49261 −0.746306 0.665603i \(-0.768174\pi\)
−0.746306 + 0.665603i \(0.768174\pi\)
\(354\) −9.19555 −0.488738
\(355\) 54.2300 2.87823
\(356\) 7.07682 0.375071
\(357\) −13.3475 −0.706422
\(358\) −25.2297 −1.33343
\(359\) −15.6900 −0.828084 −0.414042 0.910258i \(-0.635883\pi\)
−0.414042 + 0.910258i \(0.635883\pi\)
\(360\) 3.75055 0.197672
\(361\) 1.34000 0.0705263
\(362\) −1.22302 −0.0642803
\(363\) −9.02098 −0.473478
\(364\) 2.75149 0.144217
\(365\) −21.6186 −1.13157
\(366\) 14.5433 0.760191
\(367\) −33.6308 −1.75552 −0.877758 0.479105i \(-0.840961\pi\)
−0.877758 + 0.479105i \(0.840961\pi\)
\(368\) −4.42327 −0.230579
\(369\) −1.99643 −0.103930
\(370\) 15.4586 0.803652
\(371\) −17.6189 −0.914727
\(372\) −8.77193 −0.454803
\(373\) −17.0354 −0.882059 −0.441030 0.897493i \(-0.645387\pi\)
−0.441030 + 0.897493i \(0.645387\pi\)
\(374\) 6.82428 0.352875
\(375\) 15.2522 0.787619
\(376\) 11.3867 0.587225
\(377\) −6.19385 −0.319000
\(378\) −2.75149 −0.141521
\(379\) 24.5883 1.26301 0.631507 0.775370i \(-0.282437\pi\)
0.631507 + 0.775370i \(0.282437\pi\)
\(380\) −16.9150 −0.867719
\(381\) 13.6249 0.698024
\(382\) −16.9267 −0.866043
\(383\) −7.75115 −0.396065 −0.198033 0.980195i \(-0.563455\pi\)
−0.198033 + 0.980195i \(0.563455\pi\)
\(384\) 1.00000 0.0510310
\(385\) −14.5174 −0.739874
\(386\) 22.1717 1.12851
\(387\) 11.5347 0.586343
\(388\) 17.8888 0.908164
\(389\) −31.5554 −1.59992 −0.799961 0.600052i \(-0.795147\pi\)
−0.799961 + 0.600052i \(0.795147\pi\)
\(390\) −3.75055 −0.189917
\(391\) −21.4573 −1.08514
\(392\) 0.570676 0.0288235
\(393\) −2.27585 −0.114801
\(394\) −4.41932 −0.222642
\(395\) −14.5364 −0.731405
\(396\) 1.40678 0.0706932
\(397\) 18.1186 0.909346 0.454673 0.890658i \(-0.349756\pi\)
0.454673 + 0.890658i \(0.349756\pi\)
\(398\) −15.6553 −0.784728
\(399\) 12.4092 0.621236
\(400\) 9.06665 0.453332
\(401\) −13.1398 −0.656172 −0.328086 0.944648i \(-0.606404\pi\)
−0.328086 + 0.944648i \(0.606404\pi\)
\(402\) 6.93725 0.345999
\(403\) 8.77193 0.436961
\(404\) −7.86666 −0.391381
\(405\) 3.75055 0.186366
\(406\) −17.0423 −0.845795
\(407\) 5.79828 0.287410
\(408\) 4.85100 0.240160
\(409\) −2.70569 −0.133788 −0.0668939 0.997760i \(-0.521309\pi\)
−0.0668939 + 0.997760i \(0.521309\pi\)
\(410\) −7.48771 −0.369792
\(411\) −11.7402 −0.579100
\(412\) 1.00000 0.0492665
\(413\) 25.3014 1.24500
\(414\) −4.42327 −0.217392
\(415\) 39.4379 1.93593
\(416\) −1.00000 −0.0490290
\(417\) 1.88819 0.0924650
\(418\) −6.34455 −0.310322
\(419\) 20.1362 0.983718 0.491859 0.870675i \(-0.336318\pi\)
0.491859 + 0.870675i \(0.336318\pi\)
\(420\) −10.3196 −0.503545
\(421\) 8.01317 0.390538 0.195269 0.980750i \(-0.437442\pi\)
0.195269 + 0.980750i \(0.437442\pi\)
\(422\) −11.3565 −0.552826
\(423\) 11.3867 0.553641
\(424\) 6.40341 0.310977
\(425\) 43.9823 2.13346
\(426\) 14.4592 0.700551
\(427\) −40.0157 −1.93650
\(428\) −11.1912 −0.540946
\(429\) −1.40678 −0.0679199
\(430\) 43.2616 2.08626
\(431\) 26.7201 1.28706 0.643530 0.765421i \(-0.277469\pi\)
0.643530 + 0.765421i \(0.277469\pi\)
\(432\) 1.00000 0.0481125
\(433\) 24.8075 1.19217 0.596087 0.802920i \(-0.296721\pi\)
0.596087 + 0.802920i \(0.296721\pi\)
\(434\) 24.1358 1.15856
\(435\) 23.2304 1.11381
\(436\) −9.76772 −0.467789
\(437\) 19.9489 0.954284
\(438\) −5.76412 −0.275420
\(439\) −25.4088 −1.21270 −0.606348 0.795199i \(-0.707366\pi\)
−0.606348 + 0.795199i \(0.707366\pi\)
\(440\) 5.27619 0.251533
\(441\) 0.570676 0.0271751
\(442\) −4.85100 −0.230739
\(443\) 30.5178 1.44995 0.724973 0.688778i \(-0.241852\pi\)
0.724973 + 0.688778i \(0.241852\pi\)
\(444\) 4.12167 0.195606
\(445\) 26.5420 1.25821
\(446\) −18.6322 −0.882260
\(447\) 20.1646 0.953751
\(448\) −2.75149 −0.129996
\(449\) −7.30121 −0.344565 −0.172283 0.985048i \(-0.555114\pi\)
−0.172283 + 0.985048i \(0.555114\pi\)
\(450\) 9.06665 0.427406
\(451\) −2.80853 −0.132249
\(452\) −11.9388 −0.561555
\(453\) −4.02291 −0.189013
\(454\) −16.1448 −0.757714
\(455\) 10.3196 0.483790
\(456\) −4.50999 −0.211200
\(457\) −2.46773 −0.115435 −0.0577177 0.998333i \(-0.518382\pi\)
−0.0577177 + 0.998333i \(0.518382\pi\)
\(458\) 13.4124 0.626721
\(459\) 4.85100 0.226425
\(460\) −16.5897 −0.773498
\(461\) −20.1050 −0.936384 −0.468192 0.883627i \(-0.655095\pi\)
−0.468192 + 0.883627i \(0.655095\pi\)
\(462\) −3.87073 −0.180083
\(463\) −4.99320 −0.232054 −0.116027 0.993246i \(-0.537016\pi\)
−0.116027 + 0.993246i \(0.537016\pi\)
\(464\) 6.19385 0.287542
\(465\) −32.8996 −1.52568
\(466\) 6.75929 0.313118
\(467\) 12.2694 0.567760 0.283880 0.958860i \(-0.408378\pi\)
0.283880 + 0.958860i \(0.408378\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −19.0877 −0.881390
\(470\) 42.7065 1.96990
\(471\) −12.3376 −0.568485
\(472\) −9.19555 −0.423259
\(473\) 16.2268 0.746109
\(474\) −3.87580 −0.178021
\(475\) −40.8905 −1.87618
\(476\) −13.3475 −0.611780
\(477\) 6.40341 0.293192
\(478\) 12.9876 0.594039
\(479\) −26.9745 −1.23250 −0.616248 0.787552i \(-0.711348\pi\)
−0.616248 + 0.787552i \(0.711348\pi\)
\(480\) 3.75055 0.171189
\(481\) −4.12167 −0.187932
\(482\) −3.25984 −0.148482
\(483\) 12.1706 0.553779
\(484\) −9.02098 −0.410044
\(485\) 67.0927 3.04652
\(486\) 1.00000 0.0453609
\(487\) −13.0202 −0.590002 −0.295001 0.955497i \(-0.595320\pi\)
−0.295001 + 0.955497i \(0.595320\pi\)
\(488\) 14.5433 0.658344
\(489\) 6.87000 0.310672
\(490\) 2.14035 0.0966913
\(491\) 9.57244 0.431998 0.215999 0.976394i \(-0.430699\pi\)
0.215999 + 0.976394i \(0.430699\pi\)
\(492\) −1.99643 −0.0900060
\(493\) 30.0464 1.35322
\(494\) 4.50999 0.202914
\(495\) 5.27619 0.237147
\(496\) −8.77193 −0.393871
\(497\) −39.7843 −1.78457
\(498\) 10.5152 0.471198
\(499\) 39.4882 1.76773 0.883866 0.467740i \(-0.154931\pi\)
0.883866 + 0.467740i \(0.154931\pi\)
\(500\) 15.2522 0.682098
\(501\) −22.2739 −0.995125
\(502\) −12.0501 −0.537824
\(503\) −6.33659 −0.282534 −0.141267 0.989972i \(-0.545118\pi\)
−0.141267 + 0.989972i \(0.545118\pi\)
\(504\) −2.75149 −0.122561
\(505\) −29.5043 −1.31293
\(506\) −6.22255 −0.276626
\(507\) 1.00000 0.0444116
\(508\) 13.6249 0.604506
\(509\) −23.6417 −1.04790 −0.523950 0.851749i \(-0.675542\pi\)
−0.523950 + 0.851749i \(0.675542\pi\)
\(510\) 18.1939 0.805641
\(511\) 15.8599 0.701600
\(512\) 1.00000 0.0441942
\(513\) −4.50999 −0.199121
\(514\) −13.5708 −0.598584
\(515\) 3.75055 0.165269
\(516\) 11.5347 0.507788
\(517\) 16.0186 0.704496
\(518\) −11.3407 −0.498283
\(519\) 4.21358 0.184956
\(520\) −3.75055 −0.164473
\(521\) −27.3043 −1.19622 −0.598111 0.801413i \(-0.704082\pi\)
−0.598111 + 0.801413i \(0.704082\pi\)
\(522\) 6.19385 0.271098
\(523\) 17.4180 0.761634 0.380817 0.924650i \(-0.375643\pi\)
0.380817 + 0.924650i \(0.375643\pi\)
\(524\) −2.27585 −0.0994209
\(525\) −24.9468 −1.08877
\(526\) 13.2870 0.579339
\(527\) −42.5526 −1.85362
\(528\) 1.40678 0.0612221
\(529\) −3.43472 −0.149336
\(530\) 24.0163 1.04320
\(531\) −9.19555 −0.399053
\(532\) 12.4092 0.538006
\(533\) 1.99643 0.0864749
\(534\) 7.07682 0.306244
\(535\) −41.9731 −1.81465
\(536\) 6.93725 0.299644
\(537\) −25.2297 −1.08874
\(538\) −22.3684 −0.964370
\(539\) 0.802815 0.0345797
\(540\) 3.75055 0.161398
\(541\) −10.6788 −0.459119 −0.229560 0.973295i \(-0.573729\pi\)
−0.229560 + 0.973295i \(0.573729\pi\)
\(542\) 27.7261 1.19094
\(543\) −1.22302 −0.0524846
\(544\) 4.85100 0.207985
\(545\) −36.6343 −1.56924
\(546\) 2.75149 0.117753
\(547\) −2.35594 −0.100733 −0.0503665 0.998731i \(-0.516039\pi\)
−0.0503665 + 0.998731i \(0.516039\pi\)
\(548\) −11.7402 −0.501516
\(549\) 14.5433 0.620693
\(550\) 12.7548 0.543865
\(551\) −27.9342 −1.19004
\(552\) −4.42327 −0.188267
\(553\) 10.6642 0.453488
\(554\) 19.8500 0.843345
\(555\) 15.4586 0.656179
\(556\) 1.88819 0.0800770
\(557\) −15.2342 −0.645492 −0.322746 0.946486i \(-0.604606\pi\)
−0.322746 + 0.946486i \(0.604606\pi\)
\(558\) −8.77193 −0.371345
\(559\) −11.5347 −0.487867
\(560\) −10.3196 −0.436083
\(561\) 6.82428 0.288121
\(562\) 7.52692 0.317504
\(563\) 28.8572 1.21619 0.608093 0.793866i \(-0.291935\pi\)
0.608093 + 0.793866i \(0.291935\pi\)
\(564\) 11.3867 0.479467
\(565\) −44.7772 −1.88379
\(566\) −23.8501 −1.00249
\(567\) −2.75149 −0.115552
\(568\) 14.4592 0.606695
\(569\) 26.4704 1.10970 0.554849 0.831951i \(-0.312776\pi\)
0.554849 + 0.831951i \(0.312776\pi\)
\(570\) −16.9150 −0.708489
\(571\) −13.0509 −0.546163 −0.273082 0.961991i \(-0.588043\pi\)
−0.273082 + 0.961991i \(0.588043\pi\)
\(572\) −1.40678 −0.0588203
\(573\) −16.9267 −0.707122
\(574\) 5.49315 0.229279
\(575\) −40.1042 −1.67246
\(576\) 1.00000 0.0416667
\(577\) 31.9196 1.32883 0.664415 0.747364i \(-0.268681\pi\)
0.664415 + 0.747364i \(0.268681\pi\)
\(578\) 6.53220 0.271704
\(579\) 22.1717 0.921424
\(580\) 23.2304 0.964589
\(581\) −28.9325 −1.20032
\(582\) 17.8888 0.741513
\(583\) 9.00817 0.373080
\(584\) −5.76412 −0.238521
\(585\) −3.75055 −0.155066
\(586\) −2.38503 −0.0985248
\(587\) −0.597564 −0.0246641 −0.0123321 0.999924i \(-0.503926\pi\)
−0.0123321 + 0.999924i \(0.503926\pi\)
\(588\) 0.570676 0.0235343
\(589\) 39.5613 1.63010
\(590\) −34.4884 −1.41986
\(591\) −4.41932 −0.181787
\(592\) 4.12167 0.169400
\(593\) 12.0593 0.495217 0.247608 0.968860i \(-0.420355\pi\)
0.247608 + 0.968860i \(0.420355\pi\)
\(594\) 1.40678 0.0577208
\(595\) −50.0604 −2.05227
\(596\) 20.1646 0.825973
\(597\) −15.6553 −0.640728
\(598\) 4.42327 0.180881
\(599\) −17.3205 −0.707695 −0.353847 0.935303i \(-0.615127\pi\)
−0.353847 + 0.935303i \(0.615127\pi\)
\(600\) 9.06665 0.370144
\(601\) −2.24987 −0.0917740 −0.0458870 0.998947i \(-0.514611\pi\)
−0.0458870 + 0.998947i \(0.514611\pi\)
\(602\) −31.7377 −1.29353
\(603\) 6.93725 0.282507
\(604\) −4.02291 −0.163690
\(605\) −33.8337 −1.37553
\(606\) −7.86666 −0.319561
\(607\) −15.6789 −0.636386 −0.318193 0.948026i \(-0.603076\pi\)
−0.318193 + 0.948026i \(0.603076\pi\)
\(608\) −4.50999 −0.182904
\(609\) −17.0423 −0.690589
\(610\) 54.5454 2.20848
\(611\) −11.3867 −0.460657
\(612\) 4.85100 0.196090
\(613\) −2.57938 −0.104180 −0.0520900 0.998642i \(-0.516588\pi\)
−0.0520900 + 0.998642i \(0.516588\pi\)
\(614\) −12.7915 −0.516223
\(615\) −7.48771 −0.301934
\(616\) −3.87073 −0.155956
\(617\) 45.9164 1.84853 0.924263 0.381756i \(-0.124681\pi\)
0.924263 + 0.381756i \(0.124681\pi\)
\(618\) 1.00000 0.0402259
\(619\) 22.6301 0.909579 0.454790 0.890599i \(-0.349714\pi\)
0.454790 + 0.890599i \(0.349714\pi\)
\(620\) −32.8996 −1.32128
\(621\) −4.42327 −0.177500
\(622\) −10.9137 −0.437598
\(623\) −19.4718 −0.780120
\(624\) −1.00000 −0.0400320
\(625\) 11.8709 0.474835
\(626\) −5.14958 −0.205819
\(627\) −6.34455 −0.253377
\(628\) −12.3376 −0.492322
\(629\) 19.9942 0.797223
\(630\) −10.3196 −0.411143
\(631\) 37.5836 1.49618 0.748089 0.663598i \(-0.230972\pi\)
0.748089 + 0.663598i \(0.230972\pi\)
\(632\) −3.87580 −0.154171
\(633\) −11.3565 −0.451380
\(634\) 34.4704 1.36899
\(635\) 51.1008 2.02787
\(636\) 6.40341 0.253912
\(637\) −0.570676 −0.0226110
\(638\) 8.71337 0.344966
\(639\) 14.4592 0.571998
\(640\) 3.75055 0.148254
\(641\) 18.8532 0.744658 0.372329 0.928101i \(-0.378559\pi\)
0.372329 + 0.928101i \(0.378559\pi\)
\(642\) −11.1912 −0.441680
\(643\) −17.7100 −0.698413 −0.349207 0.937046i \(-0.613549\pi\)
−0.349207 + 0.937046i \(0.613549\pi\)
\(644\) 12.1706 0.479587
\(645\) 43.2616 1.70342
\(646\) −21.8780 −0.860777
\(647\) −5.52460 −0.217194 −0.108597 0.994086i \(-0.534636\pi\)
−0.108597 + 0.994086i \(0.534636\pi\)
\(648\) 1.00000 0.0392837
\(649\) −12.9361 −0.507786
\(650\) −9.06665 −0.355623
\(651\) 24.1358 0.945958
\(652\) 6.87000 0.269050
\(653\) −37.9594 −1.48546 −0.742732 0.669589i \(-0.766470\pi\)
−0.742732 + 0.669589i \(0.766470\pi\)
\(654\) −9.76772 −0.381948
\(655\) −8.53569 −0.333517
\(656\) −1.99643 −0.0779474
\(657\) −5.76412 −0.224880
\(658\) −31.3304 −1.22139
\(659\) −12.5990 −0.490788 −0.245394 0.969423i \(-0.578917\pi\)
−0.245394 + 0.969423i \(0.578917\pi\)
\(660\) 5.27619 0.205376
\(661\) −27.3101 −1.06224 −0.531120 0.847297i \(-0.678228\pi\)
−0.531120 + 0.847297i \(0.678228\pi\)
\(662\) 14.9647 0.581621
\(663\) −4.85100 −0.188397
\(664\) 10.5152 0.408070
\(665\) 46.5413 1.80479
\(666\) 4.12167 0.159712
\(667\) −27.3971 −1.06082
\(668\) −22.2739 −0.861803
\(669\) −18.6322 −0.720362
\(670\) 26.0185 1.00518
\(671\) 20.4592 0.789819
\(672\) −2.75149 −0.106141
\(673\) 10.1922 0.392880 0.196440 0.980516i \(-0.437062\pi\)
0.196440 + 0.980516i \(0.437062\pi\)
\(674\) −25.1159 −0.967427
\(675\) 9.06665 0.348975
\(676\) 1.00000 0.0384615
\(677\) 31.7391 1.21983 0.609917 0.792466i \(-0.291203\pi\)
0.609917 + 0.792466i \(0.291203\pi\)
\(678\) −11.9388 −0.458507
\(679\) −49.2207 −1.88892
\(680\) 18.1939 0.697705
\(681\) −16.1448 −0.618671
\(682\) −12.3402 −0.472529
\(683\) 9.40345 0.359813 0.179906 0.983684i \(-0.442421\pi\)
0.179906 + 0.983684i \(0.442421\pi\)
\(684\) −4.50999 −0.172444
\(685\) −44.0322 −1.68238
\(686\) 17.6902 0.675415
\(687\) 13.4124 0.511716
\(688\) 11.5347 0.439758
\(689\) −6.40341 −0.243950
\(690\) −16.5897 −0.631559
\(691\) −29.7503 −1.13175 −0.565877 0.824490i \(-0.691462\pi\)
−0.565877 + 0.824490i \(0.691462\pi\)
\(692\) 4.21358 0.160176
\(693\) −3.87073 −0.147037
\(694\) −17.6919 −0.671575
\(695\) 7.08175 0.268626
\(696\) 6.19385 0.234777
\(697\) −9.68468 −0.366833
\(698\) −30.3608 −1.14917
\(699\) 6.75929 0.255660
\(700\) −24.9468 −0.942899
\(701\) −27.3803 −1.03414 −0.517070 0.855943i \(-0.672977\pi\)
−0.517070 + 0.855943i \(0.672977\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −18.5887 −0.701086
\(704\) 1.40678 0.0530199
\(705\) 42.7065 1.60842
\(706\) −28.0436 −1.05544
\(707\) 21.6450 0.814044
\(708\) −9.19555 −0.345590
\(709\) 16.0203 0.601655 0.300827 0.953679i \(-0.402737\pi\)
0.300827 + 0.953679i \(0.402737\pi\)
\(710\) 54.2300 2.03522
\(711\) −3.87580 −0.145354
\(712\) 7.07682 0.265215
\(713\) 38.8006 1.45309
\(714\) −13.3475 −0.499516
\(715\) −5.27619 −0.197319
\(716\) −25.2297 −0.942878
\(717\) 12.9876 0.485031
\(718\) −15.6900 −0.585544
\(719\) −21.7279 −0.810314 −0.405157 0.914247i \(-0.632783\pi\)
−0.405157 + 0.914247i \(0.632783\pi\)
\(720\) 3.75055 0.139775
\(721\) −2.75149 −0.102471
\(722\) 1.34000 0.0498696
\(723\) −3.25984 −0.121235
\(724\) −1.22302 −0.0454530
\(725\) 56.1575 2.08564
\(726\) −9.02098 −0.334800
\(727\) −15.2089 −0.564068 −0.282034 0.959404i \(-0.591009\pi\)
−0.282034 + 0.959404i \(0.591009\pi\)
\(728\) 2.75149 0.101977
\(729\) 1.00000 0.0370370
\(730\) −21.6186 −0.800141
\(731\) 55.9550 2.06957
\(732\) 14.5433 0.537536
\(733\) −12.2944 −0.454104 −0.227052 0.973883i \(-0.572909\pi\)
−0.227052 + 0.973883i \(0.572909\pi\)
\(734\) −33.6308 −1.24134
\(735\) 2.14035 0.0789481
\(736\) −4.42327 −0.163044
\(737\) 9.75917 0.359484
\(738\) −1.99643 −0.0734896
\(739\) −41.3497 −1.52107 −0.760537 0.649294i \(-0.775064\pi\)
−0.760537 + 0.649294i \(0.775064\pi\)
\(740\) 15.4586 0.568268
\(741\) 4.50999 0.165679
\(742\) −17.6189 −0.646810
\(743\) −26.0556 −0.955888 −0.477944 0.878390i \(-0.658618\pi\)
−0.477944 + 0.878390i \(0.658618\pi\)
\(744\) −8.77193 −0.321595
\(745\) 75.6283 2.77080
\(746\) −17.0354 −0.623710
\(747\) 10.5152 0.384732
\(748\) 6.82428 0.249520
\(749\) 30.7923 1.12513
\(750\) 15.2522 0.556931
\(751\) 32.2491 1.17679 0.588394 0.808575i \(-0.299761\pi\)
0.588394 + 0.808575i \(0.299761\pi\)
\(752\) 11.3867 0.415231
\(753\) −12.0501 −0.439131
\(754\) −6.19385 −0.225567
\(755\) −15.0882 −0.549114
\(756\) −2.75149 −0.100071
\(757\) −42.6448 −1.54995 −0.774977 0.631990i \(-0.782238\pi\)
−0.774977 + 0.631990i \(0.782238\pi\)
\(758\) 24.5883 0.893086
\(759\) −6.22255 −0.225864
\(760\) −16.9150 −0.613570
\(761\) −33.4844 −1.21381 −0.606904 0.794775i \(-0.707589\pi\)
−0.606904 + 0.794775i \(0.707589\pi\)
\(762\) 13.6249 0.493577
\(763\) 26.8757 0.972967
\(764\) −16.9267 −0.612385
\(765\) 18.1939 0.657803
\(766\) −7.75115 −0.280060
\(767\) 9.19555 0.332032
\(768\) 1.00000 0.0360844
\(769\) −4.78372 −0.172505 −0.0862527 0.996273i \(-0.527489\pi\)
−0.0862527 + 0.996273i \(0.527489\pi\)
\(770\) −14.5174 −0.523170
\(771\) −13.5708 −0.488741
\(772\) 22.1717 0.797977
\(773\) −9.82103 −0.353238 −0.176619 0.984279i \(-0.556516\pi\)
−0.176619 + 0.984279i \(0.556516\pi\)
\(774\) 11.5347 0.414607
\(775\) −79.5320 −2.85687
\(776\) 17.8888 0.642169
\(777\) −11.3407 −0.406846
\(778\) −31.5554 −1.13132
\(779\) 9.00387 0.322597
\(780\) −3.75055 −0.134291
\(781\) 20.3409 0.727855
\(782\) −21.4573 −0.767310
\(783\) 6.19385 0.221350
\(784\) 0.570676 0.0203813
\(785\) −46.2727 −1.65154
\(786\) −2.27585 −0.0811769
\(787\) 26.0077 0.927073 0.463536 0.886078i \(-0.346580\pi\)
0.463536 + 0.886078i \(0.346580\pi\)
\(788\) −4.41932 −0.157432
\(789\) 13.2870 0.473029
\(790\) −14.5364 −0.517181
\(791\) 32.8495 1.16799
\(792\) 1.40678 0.0499877
\(793\) −14.5433 −0.516448
\(794\) 18.1186 0.643005
\(795\) 24.0163 0.851771
\(796\) −15.6553 −0.554887
\(797\) −1.21383 −0.0429961 −0.0214980 0.999769i \(-0.506844\pi\)
−0.0214980 + 0.999769i \(0.506844\pi\)
\(798\) 12.4092 0.439280
\(799\) 55.2370 1.95414
\(800\) 9.06665 0.320554
\(801\) 7.07682 0.250047
\(802\) −13.1398 −0.463984
\(803\) −8.10883 −0.286154
\(804\) 6.93725 0.244658
\(805\) 45.6463 1.60882
\(806\) 8.77193 0.308978
\(807\) −22.3684 −0.787405
\(808\) −7.86666 −0.276748
\(809\) −36.6451 −1.28837 −0.644186 0.764869i \(-0.722804\pi\)
−0.644186 + 0.764869i \(0.722804\pi\)
\(810\) 3.75055 0.131781
\(811\) 39.2571 1.37850 0.689252 0.724522i \(-0.257939\pi\)
0.689252 + 0.724522i \(0.257939\pi\)
\(812\) −17.0423 −0.598068
\(813\) 27.7261 0.972398
\(814\) 5.79828 0.203230
\(815\) 25.7663 0.902554
\(816\) 4.85100 0.169819
\(817\) −52.0215 −1.82000
\(818\) −2.70569 −0.0946023
\(819\) 2.75149 0.0961447
\(820\) −7.48771 −0.261482
\(821\) −13.8433 −0.483136 −0.241568 0.970384i \(-0.577662\pi\)
−0.241568 + 0.970384i \(0.577662\pi\)
\(822\) −11.7402 −0.409486
\(823\) 14.6944 0.512215 0.256107 0.966648i \(-0.417560\pi\)
0.256107 + 0.966648i \(0.417560\pi\)
\(824\) 1.00000 0.0348367
\(825\) 12.7548 0.444064
\(826\) 25.3014 0.880349
\(827\) 22.2035 0.772092 0.386046 0.922480i \(-0.373841\pi\)
0.386046 + 0.922480i \(0.373841\pi\)
\(828\) −4.42327 −0.153719
\(829\) 36.1214 1.25455 0.627273 0.778799i \(-0.284171\pi\)
0.627273 + 0.778799i \(0.284171\pi\)
\(830\) 39.4379 1.36891
\(831\) 19.8500 0.688589
\(832\) −1.00000 −0.0346688
\(833\) 2.76835 0.0959177
\(834\) 1.88819 0.0653826
\(835\) −83.5395 −2.89100
\(836\) −6.34455 −0.219431
\(837\) −8.77193 −0.303202
\(838\) 20.1362 0.695594
\(839\) 49.0119 1.69208 0.846040 0.533120i \(-0.178981\pi\)
0.846040 + 0.533120i \(0.178981\pi\)
\(840\) −10.3196 −0.356060
\(841\) 9.36381 0.322890
\(842\) 8.01317 0.276152
\(843\) 7.52692 0.259241
\(844\) −11.3565 −0.390907
\(845\) 3.75055 0.129023
\(846\) 11.3867 0.391483
\(847\) 24.8211 0.852863
\(848\) 6.40341 0.219894
\(849\) −23.8501 −0.818534
\(850\) 43.9823 1.50858
\(851\) −18.2313 −0.624960
\(852\) 14.4592 0.495364
\(853\) −17.5841 −0.602068 −0.301034 0.953613i \(-0.597332\pi\)
−0.301034 + 0.953613i \(0.597332\pi\)
\(854\) −40.0157 −1.36931
\(855\) −16.9150 −0.578479
\(856\) −11.1912 −0.382506
\(857\) 49.6481 1.69595 0.847973 0.530040i \(-0.177823\pi\)
0.847973 + 0.530040i \(0.177823\pi\)
\(858\) −1.40678 −0.0480266
\(859\) −32.3009 −1.10209 −0.551046 0.834475i \(-0.685771\pi\)
−0.551046 + 0.834475i \(0.685771\pi\)
\(860\) 43.2616 1.47521
\(861\) 5.49315 0.187206
\(862\) 26.7201 0.910089
\(863\) 36.0519 1.22722 0.613610 0.789609i \(-0.289717\pi\)
0.613610 + 0.789609i \(0.289717\pi\)
\(864\) 1.00000 0.0340207
\(865\) 15.8033 0.537327
\(866\) 24.8075 0.842994
\(867\) 6.53220 0.221845
\(868\) 24.1358 0.819224
\(869\) −5.45239 −0.184960
\(870\) 23.2304 0.787583
\(871\) −6.93725 −0.235060
\(872\) −9.76772 −0.330777
\(873\) 17.8888 0.605443
\(874\) 19.9489 0.674781
\(875\) −41.9662 −1.41872
\(876\) −5.76412 −0.194751
\(877\) −7.30397 −0.246638 −0.123319 0.992367i \(-0.539354\pi\)
−0.123319 + 0.992367i \(0.539354\pi\)
\(878\) −25.4088 −0.857506
\(879\) −2.38503 −0.0804452
\(880\) 5.27619 0.177860
\(881\) 31.8389 1.07268 0.536340 0.844002i \(-0.319806\pi\)
0.536340 + 0.844002i \(0.319806\pi\)
\(882\) 0.570676 0.0192157
\(883\) 38.2054 1.28571 0.642857 0.765986i \(-0.277749\pi\)
0.642857 + 0.765986i \(0.277749\pi\)
\(884\) −4.85100 −0.163157
\(885\) −34.4884 −1.15931
\(886\) 30.5178 1.02527
\(887\) 16.8560 0.565969 0.282985 0.959124i \(-0.408675\pi\)
0.282985 + 0.959124i \(0.408675\pi\)
\(888\) 4.12167 0.138314
\(889\) −37.4887 −1.25733
\(890\) 26.5420 0.889690
\(891\) 1.40678 0.0471288
\(892\) −18.6322 −0.623852
\(893\) −51.3540 −1.71849
\(894\) 20.1646 0.674404
\(895\) −94.6253 −3.16297
\(896\) −2.75149 −0.0919207
\(897\) 4.42327 0.147689
\(898\) −7.30121 −0.243645
\(899\) −54.3320 −1.81208
\(900\) 9.06665 0.302222
\(901\) 31.0629 1.03486
\(902\) −2.80853 −0.0935139
\(903\) −31.7377 −1.05616
\(904\) −11.9388 −0.397079
\(905\) −4.58698 −0.152477
\(906\) −4.02291 −0.133652
\(907\) −27.5492 −0.914756 −0.457378 0.889272i \(-0.651211\pi\)
−0.457378 + 0.889272i \(0.651211\pi\)
\(908\) −16.1448 −0.535785
\(909\) −7.86666 −0.260921
\(910\) 10.3196 0.342091
\(911\) −23.2797 −0.771291 −0.385646 0.922647i \(-0.626021\pi\)
−0.385646 + 0.922647i \(0.626021\pi\)
\(912\) −4.50999 −0.149341
\(913\) 14.7926 0.489563
\(914\) −2.46773 −0.0816252
\(915\) 54.5454 1.80322
\(916\) 13.4124 0.443159
\(917\) 6.26197 0.206788
\(918\) 4.85100 0.160107
\(919\) 41.8555 1.38068 0.690342 0.723483i \(-0.257460\pi\)
0.690342 + 0.723483i \(0.257460\pi\)
\(920\) −16.5897 −0.546946
\(921\) −12.7915 −0.421495
\(922\) −20.1050 −0.662124
\(923\) −14.4592 −0.475931
\(924\) −3.87073 −0.127338
\(925\) 37.3698 1.22871
\(926\) −4.99320 −0.164087
\(927\) 1.00000 0.0328443
\(928\) 6.19385 0.203323
\(929\) 3.17728 0.104243 0.0521216 0.998641i \(-0.483402\pi\)
0.0521216 + 0.998641i \(0.483402\pi\)
\(930\) −32.8996 −1.07882
\(931\) −2.57374 −0.0843511
\(932\) 6.75929 0.221408
\(933\) −10.9137 −0.357297
\(934\) 12.2694 0.401467
\(935\) 25.5948 0.837040
\(936\) −1.00000 −0.0326860
\(937\) 14.2743 0.466322 0.233161 0.972438i \(-0.425093\pi\)
0.233161 + 0.972438i \(0.425093\pi\)
\(938\) −19.0877 −0.623237
\(939\) −5.14958 −0.168050
\(940\) 42.7065 1.39293
\(941\) 17.1305 0.558439 0.279220 0.960227i \(-0.409924\pi\)
0.279220 + 0.960227i \(0.409924\pi\)
\(942\) −12.3376 −0.401980
\(943\) 8.83074 0.287568
\(944\) −9.19555 −0.299290
\(945\) −10.3196 −0.335696
\(946\) 16.2268 0.527579
\(947\) 3.68806 0.119846 0.0599229 0.998203i \(-0.480915\pi\)
0.0599229 + 0.998203i \(0.480915\pi\)
\(948\) −3.87580 −0.125880
\(949\) 5.76412 0.187111
\(950\) −40.8905 −1.32666
\(951\) 34.4704 1.11778
\(952\) −13.3475 −0.432594
\(953\) −26.0185 −0.842820 −0.421410 0.906870i \(-0.638465\pi\)
−0.421410 + 0.906870i \(0.638465\pi\)
\(954\) 6.40341 0.207318
\(955\) −63.4844 −2.05431
\(956\) 12.9876 0.420049
\(957\) 8.71337 0.281663
\(958\) −26.9745 −0.871506
\(959\) 32.3029 1.04312
\(960\) 3.75055 0.121049
\(961\) 45.9468 1.48215
\(962\) −4.12167 −0.132888
\(963\) −11.1912 −0.360630
\(964\) −3.25984 −0.104992
\(965\) 83.1561 2.67689
\(966\) 12.1706 0.391581
\(967\) 9.62259 0.309442 0.154721 0.987958i \(-0.450552\pi\)
0.154721 + 0.987958i \(0.450552\pi\)
\(968\) −9.02098 −0.289945
\(969\) −21.8780 −0.702821
\(970\) 67.0927 2.15422
\(971\) 41.4954 1.33165 0.665825 0.746108i \(-0.268080\pi\)
0.665825 + 0.746108i \(0.268080\pi\)
\(972\) 1.00000 0.0320750
\(973\) −5.19532 −0.166554
\(974\) −13.0202 −0.417194
\(975\) −9.06665 −0.290365
\(976\) 14.5433 0.465520
\(977\) 61.1349 1.95588 0.977939 0.208893i \(-0.0669859\pi\)
0.977939 + 0.208893i \(0.0669859\pi\)
\(978\) 6.87000 0.219678
\(979\) 9.95552 0.318180
\(980\) 2.14035 0.0683711
\(981\) −9.76772 −0.311859
\(982\) 9.57244 0.305469
\(983\) −51.6042 −1.64592 −0.822960 0.568100i \(-0.807679\pi\)
−0.822960 + 0.568100i \(0.807679\pi\)
\(984\) −1.99643 −0.0636438
\(985\) −16.5749 −0.528120
\(986\) 30.0464 0.956872
\(987\) −31.3304 −0.997257
\(988\) 4.50999 0.143482
\(989\) −51.0212 −1.62238
\(990\) 5.27619 0.167688
\(991\) 36.0436 1.14496 0.572482 0.819918i \(-0.305981\pi\)
0.572482 + 0.819918i \(0.305981\pi\)
\(992\) −8.77193 −0.278509
\(993\) 14.9647 0.474892
\(994\) −39.7843 −1.26188
\(995\) −58.7160 −1.86142
\(996\) 10.5152 0.333188
\(997\) −53.4859 −1.69391 −0.846957 0.531661i \(-0.821568\pi\)
−0.846957 + 0.531661i \(0.821568\pi\)
\(998\) 39.4882 1.24998
\(999\) 4.12167 0.130404
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.bd.1.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bd.1.15 16 1.1 even 1 trivial