Properties

Label 6034.2.a.m.1.2
Level $6034$
Weight $2$
Character 6034.1
Self dual yes
Analytic conductor $48.182$
Analytic rank $1$
Dimension $21$
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] = [6034,2,Mod(1,6034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6034, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6034.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: \(48.1817325796\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Character \(\chi\) \(=\) 6034.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.48749 q^{3} +1.00000 q^{4} +1.32783 q^{5} -2.48749 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.18763 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.48749 q^{3} +1.00000 q^{4} +1.32783 q^{5} -2.48749 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.18763 q^{9} +1.32783 q^{10} +1.36398 q^{11} -2.48749 q^{12} +0.281781 q^{13} +1.00000 q^{14} -3.30297 q^{15} +1.00000 q^{16} -0.670158 q^{17} +3.18763 q^{18} -2.87655 q^{19} +1.32783 q^{20} -2.48749 q^{21} +1.36398 q^{22} -1.59928 q^{23} -2.48749 q^{24} -3.23687 q^{25} +0.281781 q^{26} -0.466731 q^{27} +1.00000 q^{28} -1.25930 q^{29} -3.30297 q^{30} -1.39495 q^{31} +1.00000 q^{32} -3.39290 q^{33} -0.670158 q^{34} +1.32783 q^{35} +3.18763 q^{36} -9.31561 q^{37} -2.87655 q^{38} -0.700928 q^{39} +1.32783 q^{40} -6.42674 q^{41} -2.48749 q^{42} -12.7746 q^{43} +1.36398 q^{44} +4.23263 q^{45} -1.59928 q^{46} +8.45192 q^{47} -2.48749 q^{48} +1.00000 q^{49} -3.23687 q^{50} +1.66701 q^{51} +0.281781 q^{52} -3.58042 q^{53} -0.466731 q^{54} +1.81114 q^{55} +1.00000 q^{56} +7.15541 q^{57} -1.25930 q^{58} +1.26320 q^{59} -3.30297 q^{60} -2.22784 q^{61} -1.39495 q^{62} +3.18763 q^{63} +1.00000 q^{64} +0.374156 q^{65} -3.39290 q^{66} -2.81723 q^{67} -0.670158 q^{68} +3.97821 q^{69} +1.32783 q^{70} +11.6926 q^{71} +3.18763 q^{72} +2.26070 q^{73} -9.31561 q^{74} +8.05170 q^{75} -2.87655 q^{76} +1.36398 q^{77} -0.700928 q^{78} -7.76845 q^{79} +1.32783 q^{80} -8.40190 q^{81} -6.42674 q^{82} -0.662846 q^{83} -2.48749 q^{84} -0.889854 q^{85} -12.7746 q^{86} +3.13250 q^{87} +1.36398 q^{88} -2.92269 q^{89} +4.23263 q^{90} +0.281781 q^{91} -1.59928 q^{92} +3.46993 q^{93} +8.45192 q^{94} -3.81957 q^{95} -2.48749 q^{96} +14.5527 q^{97} +1.00000 q^{98} +4.34788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q + 21 q^{2} - 6 q^{3} + 21 q^{4} - 11 q^{5} - 6 q^{6} + 21 q^{7} + 21 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q + 21 q^{2} - 6 q^{3} + 21 q^{4} - 11 q^{5} - 6 q^{6} + 21 q^{7} + 21 q^{8} + 5 q^{9} - 11 q^{10} - 34 q^{11} - 6 q^{12} - 19 q^{13} + 21 q^{14} - 24 q^{15} + 21 q^{16} - 17 q^{17} + 5 q^{18} - 15 q^{19} - 11 q^{20} - 6 q^{21} - 34 q^{22} - 32 q^{23} - 6 q^{24} + 6 q^{25} - 19 q^{26} - 3 q^{27} + 21 q^{28} - 46 q^{29} - 24 q^{30} + 7 q^{31} + 21 q^{32} - 13 q^{33} - 17 q^{34} - 11 q^{35} + 5 q^{36} - 34 q^{37} - 15 q^{38} - 25 q^{39} - 11 q^{40} - 27 q^{41} - 6 q^{42} - 47 q^{43} - 34 q^{44} - 13 q^{45} - 32 q^{46} - 7 q^{47} - 6 q^{48} + 21 q^{49} + 6 q^{50} - 29 q^{51} - 19 q^{52} - 57 q^{53} - 3 q^{54} + 17 q^{55} + 21 q^{56} - 28 q^{57} - 46 q^{58} - 30 q^{59} - 24 q^{60} - 17 q^{61} + 7 q^{62} + 5 q^{63} + 21 q^{64} - 40 q^{65} - 13 q^{66} - 38 q^{67} - 17 q^{68} - 13 q^{69} - 11 q^{70} - 66 q^{71} + 5 q^{72} - 15 q^{73} - 34 q^{74} + 15 q^{75} - 15 q^{76} - 34 q^{77} - 25 q^{78} - 17 q^{79} - 11 q^{80} - 11 q^{81} - 27 q^{82} - 19 q^{83} - 6 q^{84} - 28 q^{85} - 47 q^{86} + 45 q^{87} - 34 q^{88} - 39 q^{89} - 13 q^{90} - 19 q^{91} - 32 q^{92} - 25 q^{93} - 7 q^{94} - 35 q^{95} - 6 q^{96} + 21 q^{98} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.48749 −1.43616 −0.718078 0.695963i \(-0.754978\pi\)
−0.718078 + 0.695963i \(0.754978\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.32783 0.593823 0.296911 0.954905i \(-0.404043\pi\)
0.296911 + 0.954905i \(0.404043\pi\)
\(6\) −2.48749 −1.01552
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 3.18763 1.06254
\(10\) 1.32783 0.419896
\(11\) 1.36398 0.411257 0.205628 0.978630i \(-0.434076\pi\)
0.205628 + 0.978630i \(0.434076\pi\)
\(12\) −2.48749 −0.718078
\(13\) 0.281781 0.0781519 0.0390759 0.999236i \(-0.487559\pi\)
0.0390759 + 0.999236i \(0.487559\pi\)
\(14\) 1.00000 0.267261
\(15\) −3.30297 −0.852822
\(16\) 1.00000 0.250000
\(17\) −0.670158 −0.162537 −0.0812685 0.996692i \(-0.525897\pi\)
−0.0812685 + 0.996692i \(0.525897\pi\)
\(18\) 3.18763 0.751332
\(19\) −2.87655 −0.659927 −0.329963 0.943994i \(-0.607036\pi\)
−0.329963 + 0.943994i \(0.607036\pi\)
\(20\) 1.32783 0.296911
\(21\) −2.48749 −0.542816
\(22\) 1.36398 0.290802
\(23\) −1.59928 −0.333474 −0.166737 0.986001i \(-0.553323\pi\)
−0.166737 + 0.986001i \(0.553323\pi\)
\(24\) −2.48749 −0.507758
\(25\) −3.23687 −0.647374
\(26\) 0.281781 0.0552617
\(27\) −0.466731 −0.0898224
\(28\) 1.00000 0.188982
\(29\) −1.25930 −0.233846 −0.116923 0.993141i \(-0.537303\pi\)
−0.116923 + 0.993141i \(0.537303\pi\)
\(30\) −3.30297 −0.603036
\(31\) −1.39495 −0.250541 −0.125270 0.992123i \(-0.539980\pi\)
−0.125270 + 0.992123i \(0.539980\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.39290 −0.590629
\(34\) −0.670158 −0.114931
\(35\) 1.32783 0.224444
\(36\) 3.18763 0.531272
\(37\) −9.31561 −1.53148 −0.765739 0.643152i \(-0.777626\pi\)
−0.765739 + 0.643152i \(0.777626\pi\)
\(38\) −2.87655 −0.466639
\(39\) −0.700928 −0.112238
\(40\) 1.32783 0.209948
\(41\) −6.42674 −1.00369 −0.501844 0.864958i \(-0.667345\pi\)
−0.501844 + 0.864958i \(0.667345\pi\)
\(42\) −2.48749 −0.383829
\(43\) −12.7746 −1.94811 −0.974055 0.226313i \(-0.927333\pi\)
−0.974055 + 0.226313i \(0.927333\pi\)
\(44\) 1.36398 0.205628
\(45\) 4.23263 0.630963
\(46\) −1.59928 −0.235802
\(47\) 8.45192 1.23284 0.616420 0.787418i \(-0.288583\pi\)
0.616420 + 0.787418i \(0.288583\pi\)
\(48\) −2.48749 −0.359039
\(49\) 1.00000 0.142857
\(50\) −3.23687 −0.457763
\(51\) 1.66701 0.233429
\(52\) 0.281781 0.0390759
\(53\) −3.58042 −0.491809 −0.245905 0.969294i \(-0.579085\pi\)
−0.245905 + 0.969294i \(0.579085\pi\)
\(54\) −0.466731 −0.0635140
\(55\) 1.81114 0.244214
\(56\) 1.00000 0.133631
\(57\) 7.15541 0.947757
\(58\) −1.25930 −0.165354
\(59\) 1.26320 0.164455 0.0822274 0.996614i \(-0.473797\pi\)
0.0822274 + 0.996614i \(0.473797\pi\)
\(60\) −3.30297 −0.426411
\(61\) −2.22784 −0.285246 −0.142623 0.989777i \(-0.545554\pi\)
−0.142623 + 0.989777i \(0.545554\pi\)
\(62\) −1.39495 −0.177159
\(63\) 3.18763 0.401604
\(64\) 1.00000 0.125000
\(65\) 0.374156 0.0464084
\(66\) −3.39290 −0.417637
\(67\) −2.81723 −0.344180 −0.172090 0.985081i \(-0.555052\pi\)
−0.172090 + 0.985081i \(0.555052\pi\)
\(68\) −0.670158 −0.0812685
\(69\) 3.97821 0.478921
\(70\) 1.32783 0.158706
\(71\) 11.6926 1.38766 0.693829 0.720139i \(-0.255922\pi\)
0.693829 + 0.720139i \(0.255922\pi\)
\(72\) 3.18763 0.375666
\(73\) 2.26070 0.264595 0.132298 0.991210i \(-0.457765\pi\)
0.132298 + 0.991210i \(0.457765\pi\)
\(74\) −9.31561 −1.08292
\(75\) 8.05170 0.929730
\(76\) −2.87655 −0.329963
\(77\) 1.36398 0.155440
\(78\) −0.700928 −0.0793644
\(79\) −7.76845 −0.874019 −0.437010 0.899457i \(-0.643962\pi\)
−0.437010 + 0.899457i \(0.643962\pi\)
\(80\) 1.32783 0.148456
\(81\) −8.40190 −0.933545
\(82\) −6.42674 −0.709714
\(83\) −0.662846 −0.0727568 −0.0363784 0.999338i \(-0.511582\pi\)
−0.0363784 + 0.999338i \(0.511582\pi\)
\(84\) −2.48749 −0.271408
\(85\) −0.889854 −0.0965183
\(86\) −12.7746 −1.37752
\(87\) 3.13250 0.335839
\(88\) 1.36398 0.145401
\(89\) −2.92269 −0.309805 −0.154902 0.987930i \(-0.549506\pi\)
−0.154902 + 0.987930i \(0.549506\pi\)
\(90\) 4.23263 0.446158
\(91\) 0.281781 0.0295386
\(92\) −1.59928 −0.166737
\(93\) 3.46993 0.359815
\(94\) 8.45192 0.871749
\(95\) −3.81957 −0.391880
\(96\) −2.48749 −0.253879
\(97\) 14.5527 1.47761 0.738803 0.673921i \(-0.235391\pi\)
0.738803 + 0.673921i \(0.235391\pi\)
\(98\) 1.00000 0.101015
\(99\) 4.34788 0.436978
\(100\) −3.23687 −0.323687
\(101\) 14.1058 1.40358 0.701792 0.712382i \(-0.252384\pi\)
0.701792 + 0.712382i \(0.252384\pi\)
\(102\) 1.66701 0.165059
\(103\) 5.20884 0.513242 0.256621 0.966512i \(-0.417391\pi\)
0.256621 + 0.966512i \(0.417391\pi\)
\(104\) 0.281781 0.0276309
\(105\) −3.30297 −0.322337
\(106\) −3.58042 −0.347761
\(107\) −0.703567 −0.0680164 −0.0340082 0.999422i \(-0.510827\pi\)
−0.0340082 + 0.999422i \(0.510827\pi\)
\(108\) −0.466731 −0.0449112
\(109\) −9.64219 −0.923554 −0.461777 0.886996i \(-0.652788\pi\)
−0.461777 + 0.886996i \(0.652788\pi\)
\(110\) 1.81114 0.172685
\(111\) 23.1725 2.19944
\(112\) 1.00000 0.0944911
\(113\) −4.14750 −0.390164 −0.195082 0.980787i \(-0.562497\pi\)
−0.195082 + 0.980787i \(0.562497\pi\)
\(114\) 7.15541 0.670166
\(115\) −2.12358 −0.198024
\(116\) −1.25930 −0.116923
\(117\) 0.898213 0.0830398
\(118\) 1.26320 0.116287
\(119\) −0.670158 −0.0614332
\(120\) −3.30297 −0.301518
\(121\) −9.13955 −0.830868
\(122\) −2.22784 −0.201700
\(123\) 15.9865 1.44145
\(124\) −1.39495 −0.125270
\(125\) −10.9372 −0.978249
\(126\) 3.18763 0.283977
\(127\) −19.2644 −1.70944 −0.854721 0.519088i \(-0.826272\pi\)
−0.854721 + 0.519088i \(0.826272\pi\)
\(128\) 1.00000 0.0883883
\(129\) 31.7768 2.79779
\(130\) 0.374156 0.0328157
\(131\) −14.3216 −1.25129 −0.625644 0.780109i \(-0.715164\pi\)
−0.625644 + 0.780109i \(0.715164\pi\)
\(132\) −3.39290 −0.295314
\(133\) −2.87655 −0.249429
\(134\) −2.81723 −0.243372
\(135\) −0.619739 −0.0533386
\(136\) −0.670158 −0.0574655
\(137\) 18.3309 1.56611 0.783057 0.621950i \(-0.213659\pi\)
0.783057 + 0.621950i \(0.213659\pi\)
\(138\) 3.97821 0.338648
\(139\) −9.54684 −0.809752 −0.404876 0.914372i \(-0.632685\pi\)
−0.404876 + 0.914372i \(0.632685\pi\)
\(140\) 1.32783 0.112222
\(141\) −21.0241 −1.77055
\(142\) 11.6926 0.981223
\(143\) 0.384344 0.0321405
\(144\) 3.18763 0.265636
\(145\) −1.67213 −0.138863
\(146\) 2.26070 0.187097
\(147\) −2.48749 −0.205165
\(148\) −9.31561 −0.765739
\(149\) −17.0618 −1.39775 −0.698877 0.715242i \(-0.746316\pi\)
−0.698877 + 0.715242i \(0.746316\pi\)
\(150\) 8.05170 0.657419
\(151\) −10.8435 −0.882434 −0.441217 0.897400i \(-0.645453\pi\)
−0.441217 + 0.897400i \(0.645453\pi\)
\(152\) −2.87655 −0.233319
\(153\) −2.13621 −0.172703
\(154\) 1.36398 0.109913
\(155\) −1.85226 −0.148777
\(156\) −0.700928 −0.0561191
\(157\) 12.8469 1.02530 0.512648 0.858599i \(-0.328665\pi\)
0.512648 + 0.858599i \(0.328665\pi\)
\(158\) −7.76845 −0.618025
\(159\) 8.90629 0.706314
\(160\) 1.32783 0.104974
\(161\) −1.59928 −0.126041
\(162\) −8.40190 −0.660116
\(163\) 10.9064 0.854252 0.427126 0.904192i \(-0.359526\pi\)
0.427126 + 0.904192i \(0.359526\pi\)
\(164\) −6.42674 −0.501844
\(165\) −4.50519 −0.350729
\(166\) −0.662846 −0.0514468
\(167\) −5.03627 −0.389719 −0.194859 0.980831i \(-0.562425\pi\)
−0.194859 + 0.980831i \(0.562425\pi\)
\(168\) −2.48749 −0.191914
\(169\) −12.9206 −0.993892
\(170\) −0.889854 −0.0682487
\(171\) −9.16939 −0.701201
\(172\) −12.7746 −0.974055
\(173\) −17.8313 −1.35569 −0.677846 0.735204i \(-0.737086\pi\)
−0.677846 + 0.735204i \(0.737086\pi\)
\(174\) 3.13250 0.237474
\(175\) −3.23687 −0.244684
\(176\) 1.36398 0.102814
\(177\) −3.14221 −0.236183
\(178\) −2.92269 −0.219065
\(179\) 19.2812 1.44115 0.720573 0.693379i \(-0.243879\pi\)
0.720573 + 0.693379i \(0.243879\pi\)
\(180\) 4.23263 0.315481
\(181\) −0.943125 −0.0701019 −0.0350510 0.999386i \(-0.511159\pi\)
−0.0350510 + 0.999386i \(0.511159\pi\)
\(182\) 0.281781 0.0208870
\(183\) 5.54175 0.409658
\(184\) −1.59928 −0.117901
\(185\) −12.3695 −0.909426
\(186\) 3.46993 0.254428
\(187\) −0.914084 −0.0668444
\(188\) 8.45192 0.616420
\(189\) −0.466731 −0.0339497
\(190\) −3.81957 −0.277101
\(191\) 20.2936 1.46840 0.734198 0.678936i \(-0.237558\pi\)
0.734198 + 0.678936i \(0.237558\pi\)
\(192\) −2.48749 −0.179519
\(193\) −18.4467 −1.32782 −0.663910 0.747813i \(-0.731104\pi\)
−0.663910 + 0.747813i \(0.731104\pi\)
\(194\) 14.5527 1.04483
\(195\) −0.930712 −0.0666497
\(196\) 1.00000 0.0714286
\(197\) 8.74474 0.623037 0.311519 0.950240i \(-0.399162\pi\)
0.311519 + 0.950240i \(0.399162\pi\)
\(198\) 4.34788 0.308990
\(199\) 25.9588 1.84017 0.920085 0.391719i \(-0.128119\pi\)
0.920085 + 0.391719i \(0.128119\pi\)
\(200\) −3.23687 −0.228881
\(201\) 7.00785 0.494296
\(202\) 14.1058 0.992484
\(203\) −1.25930 −0.0883855
\(204\) 1.66701 0.116714
\(205\) −8.53360 −0.596013
\(206\) 5.20884 0.362917
\(207\) −5.09793 −0.354331
\(208\) 0.281781 0.0195380
\(209\) −3.92357 −0.271399
\(210\) −3.30297 −0.227926
\(211\) 18.3688 1.26456 0.632280 0.774740i \(-0.282119\pi\)
0.632280 + 0.774740i \(0.282119\pi\)
\(212\) −3.58042 −0.245905
\(213\) −29.0853 −1.99289
\(214\) −0.703567 −0.0480948
\(215\) −16.9625 −1.15683
\(216\) −0.466731 −0.0317570
\(217\) −1.39495 −0.0946954
\(218\) −9.64219 −0.653051
\(219\) −5.62349 −0.380000
\(220\) 1.81114 0.122107
\(221\) −0.188837 −0.0127026
\(222\) 23.1725 1.55524
\(223\) 2.02562 0.135645 0.0678227 0.997697i \(-0.478395\pi\)
0.0678227 + 0.997697i \(0.478395\pi\)
\(224\) 1.00000 0.0668153
\(225\) −10.3180 −0.687863
\(226\) −4.14750 −0.275888
\(227\) −4.94194 −0.328008 −0.164004 0.986460i \(-0.552441\pi\)
−0.164004 + 0.986460i \(0.552441\pi\)
\(228\) 7.15541 0.473879
\(229\) −21.2745 −1.40586 −0.702931 0.711258i \(-0.748126\pi\)
−0.702931 + 0.711258i \(0.748126\pi\)
\(230\) −2.12358 −0.140024
\(231\) −3.39290 −0.223237
\(232\) −1.25930 −0.0826771
\(233\) 9.01899 0.590854 0.295427 0.955365i \(-0.404538\pi\)
0.295427 + 0.955365i \(0.404538\pi\)
\(234\) 0.898213 0.0587180
\(235\) 11.2227 0.732088
\(236\) 1.26320 0.0822274
\(237\) 19.3240 1.25523
\(238\) −0.670158 −0.0434399
\(239\) −5.91684 −0.382729 −0.191364 0.981519i \(-0.561291\pi\)
−0.191364 + 0.981519i \(0.561291\pi\)
\(240\) −3.30297 −0.213206
\(241\) −8.37298 −0.539351 −0.269675 0.962951i \(-0.586916\pi\)
−0.269675 + 0.962951i \(0.586916\pi\)
\(242\) −9.13955 −0.587512
\(243\) 22.2999 1.43054
\(244\) −2.22784 −0.142623
\(245\) 1.32783 0.0848319
\(246\) 15.9865 1.01926
\(247\) −0.810557 −0.0515745
\(248\) −1.39495 −0.0885795
\(249\) 1.64883 0.104490
\(250\) −10.9372 −0.691726
\(251\) −25.7847 −1.62752 −0.813759 0.581202i \(-0.802583\pi\)
−0.813759 + 0.581202i \(0.802583\pi\)
\(252\) 3.18763 0.200802
\(253\) −2.18140 −0.137143
\(254\) −19.2644 −1.20876
\(255\) 2.21351 0.138615
\(256\) 1.00000 0.0625000
\(257\) −18.2012 −1.13536 −0.567681 0.823248i \(-0.692159\pi\)
−0.567681 + 0.823248i \(0.692159\pi\)
\(258\) 31.7768 1.97834
\(259\) −9.31561 −0.578844
\(260\) 0.374156 0.0232042
\(261\) −4.01418 −0.248472
\(262\) −14.3216 −0.884794
\(263\) −8.76899 −0.540719 −0.270359 0.962759i \(-0.587143\pi\)
−0.270359 + 0.962759i \(0.587143\pi\)
\(264\) −3.39290 −0.208819
\(265\) −4.75419 −0.292047
\(266\) −2.87655 −0.176373
\(267\) 7.27018 0.444928
\(268\) −2.81723 −0.172090
\(269\) −2.09481 −0.127723 −0.0638613 0.997959i \(-0.520342\pi\)
−0.0638613 + 0.997959i \(0.520342\pi\)
\(270\) −0.619739 −0.0377161
\(271\) 28.2295 1.71482 0.857411 0.514632i \(-0.172072\pi\)
0.857411 + 0.514632i \(0.172072\pi\)
\(272\) −0.670158 −0.0406343
\(273\) −0.700928 −0.0424221
\(274\) 18.3309 1.10741
\(275\) −4.41504 −0.266237
\(276\) 3.97821 0.239460
\(277\) −21.9882 −1.32114 −0.660571 0.750764i \(-0.729686\pi\)
−0.660571 + 0.750764i \(0.729686\pi\)
\(278\) −9.54684 −0.572581
\(279\) −4.44659 −0.266210
\(280\) 1.32783 0.0793529
\(281\) 1.37003 0.0817290 0.0408645 0.999165i \(-0.486989\pi\)
0.0408645 + 0.999165i \(0.486989\pi\)
\(282\) −21.0241 −1.25197
\(283\) 22.6165 1.34441 0.672205 0.740365i \(-0.265347\pi\)
0.672205 + 0.740365i \(0.265347\pi\)
\(284\) 11.6926 0.693829
\(285\) 9.50116 0.562800
\(286\) 0.384344 0.0227267
\(287\) −6.42674 −0.379358
\(288\) 3.18763 0.187833
\(289\) −16.5509 −0.973582
\(290\) −1.67213 −0.0981911
\(291\) −36.1999 −2.12207
\(292\) 2.26070 0.132298
\(293\) 12.8871 0.752872 0.376436 0.926443i \(-0.377149\pi\)
0.376436 + 0.926443i \(0.377149\pi\)
\(294\) −2.48749 −0.145074
\(295\) 1.67731 0.0976570
\(296\) −9.31561 −0.541459
\(297\) −0.636614 −0.0369401
\(298\) −17.0618 −0.988361
\(299\) −0.450647 −0.0260616
\(300\) 8.05170 0.464865
\(301\) −12.7746 −0.736316
\(302\) −10.8435 −0.623975
\(303\) −35.0882 −2.01577
\(304\) −2.87655 −0.164982
\(305\) −2.95820 −0.169386
\(306\) −2.13621 −0.122119
\(307\) 18.5453 1.05843 0.529217 0.848486i \(-0.322486\pi\)
0.529217 + 0.848486i \(0.322486\pi\)
\(308\) 1.36398 0.0777202
\(309\) −12.9570 −0.737095
\(310\) −1.85226 −0.105201
\(311\) −26.7289 −1.51566 −0.757829 0.652453i \(-0.773740\pi\)
−0.757829 + 0.652453i \(0.773740\pi\)
\(312\) −0.700928 −0.0396822
\(313\) −26.7284 −1.51078 −0.755388 0.655277i \(-0.772552\pi\)
−0.755388 + 0.655277i \(0.772552\pi\)
\(314\) 12.8469 0.724993
\(315\) 4.23263 0.238482
\(316\) −7.76845 −0.437010
\(317\) −3.61201 −0.202871 −0.101435 0.994842i \(-0.532344\pi\)
−0.101435 + 0.994842i \(0.532344\pi\)
\(318\) 8.90629 0.499440
\(319\) −1.71766 −0.0961707
\(320\) 1.32783 0.0742279
\(321\) 1.75012 0.0976821
\(322\) −1.59928 −0.0891247
\(323\) 1.92774 0.107263
\(324\) −8.40190 −0.466772
\(325\) −0.912088 −0.0505935
\(326\) 10.9064 0.604047
\(327\) 23.9849 1.32637
\(328\) −6.42674 −0.354857
\(329\) 8.45192 0.465969
\(330\) −4.50519 −0.248003
\(331\) 5.82013 0.319903 0.159952 0.987125i \(-0.448866\pi\)
0.159952 + 0.987125i \(0.448866\pi\)
\(332\) −0.662846 −0.0363784
\(333\) −29.6947 −1.62726
\(334\) −5.03627 −0.275573
\(335\) −3.74080 −0.204382
\(336\) −2.48749 −0.135704
\(337\) 6.61801 0.360506 0.180253 0.983620i \(-0.442308\pi\)
0.180253 + 0.983620i \(0.442308\pi\)
\(338\) −12.9206 −0.702788
\(339\) 10.3169 0.560337
\(340\) −0.889854 −0.0482591
\(341\) −1.90269 −0.103036
\(342\) −9.16939 −0.495824
\(343\) 1.00000 0.0539949
\(344\) −12.7746 −0.688761
\(345\) 5.28238 0.284394
\(346\) −17.8313 −0.958619
\(347\) 15.5183 0.833064 0.416532 0.909121i \(-0.363245\pi\)
0.416532 + 0.909121i \(0.363245\pi\)
\(348\) 3.13250 0.167920
\(349\) −7.42938 −0.397685 −0.198843 0.980031i \(-0.563718\pi\)
−0.198843 + 0.980031i \(0.563718\pi\)
\(350\) −3.23687 −0.173018
\(351\) −0.131516 −0.00701979
\(352\) 1.36398 0.0727006
\(353\) 13.6875 0.728510 0.364255 0.931299i \(-0.381324\pi\)
0.364255 + 0.931299i \(0.381324\pi\)
\(354\) −3.14221 −0.167006
\(355\) 15.5258 0.824024
\(356\) −2.92269 −0.154902
\(357\) 1.66701 0.0882277
\(358\) 19.2812 1.01904
\(359\) −35.3180 −1.86401 −0.932005 0.362445i \(-0.881942\pi\)
−0.932005 + 0.362445i \(0.881942\pi\)
\(360\) 4.23263 0.223079
\(361\) −10.7254 −0.564497
\(362\) −0.943125 −0.0495696
\(363\) 22.7346 1.19326
\(364\) 0.281781 0.0147693
\(365\) 3.00183 0.157123
\(366\) 5.54175 0.289672
\(367\) −5.49026 −0.286589 −0.143295 0.989680i \(-0.545770\pi\)
−0.143295 + 0.989680i \(0.545770\pi\)
\(368\) −1.59928 −0.0833685
\(369\) −20.4861 −1.06646
\(370\) −12.3695 −0.643062
\(371\) −3.58042 −0.185886
\(372\) 3.46993 0.179908
\(373\) 0.659604 0.0341530 0.0170765 0.999854i \(-0.494564\pi\)
0.0170765 + 0.999854i \(0.494564\pi\)
\(374\) −0.914084 −0.0472662
\(375\) 27.2061 1.40492
\(376\) 8.45192 0.435875
\(377\) −0.354846 −0.0182755
\(378\) −0.466731 −0.0240061
\(379\) 10.5619 0.542528 0.271264 0.962505i \(-0.412558\pi\)
0.271264 + 0.962505i \(0.412558\pi\)
\(380\) −3.81957 −0.195940
\(381\) 47.9202 2.45502
\(382\) 20.2936 1.03831
\(383\) −16.2108 −0.828332 −0.414166 0.910201i \(-0.635927\pi\)
−0.414166 + 0.910201i \(0.635927\pi\)
\(384\) −2.48749 −0.126939
\(385\) 1.81114 0.0923041
\(386\) −18.4467 −0.938910
\(387\) −40.7207 −2.06995
\(388\) 14.5527 0.738803
\(389\) −9.87917 −0.500894 −0.250447 0.968130i \(-0.580578\pi\)
−0.250447 + 0.968130i \(0.580578\pi\)
\(390\) −0.930712 −0.0471284
\(391\) 1.07177 0.0542019
\(392\) 1.00000 0.0505076
\(393\) 35.6250 1.79704
\(394\) 8.74474 0.440554
\(395\) −10.3152 −0.519013
\(396\) 4.34788 0.218489
\(397\) −32.9334 −1.65288 −0.826439 0.563026i \(-0.809637\pi\)
−0.826439 + 0.563026i \(0.809637\pi\)
\(398\) 25.9588 1.30120
\(399\) 7.15541 0.358219
\(400\) −3.23687 −0.161844
\(401\) −33.2223 −1.65904 −0.829520 0.558477i \(-0.811386\pi\)
−0.829520 + 0.558477i \(0.811386\pi\)
\(402\) 7.00785 0.349520
\(403\) −0.393070 −0.0195802
\(404\) 14.1058 0.701792
\(405\) −11.1563 −0.554360
\(406\) −1.25930 −0.0624980
\(407\) −12.7063 −0.629830
\(408\) 1.66701 0.0825295
\(409\) −29.6001 −1.46363 −0.731815 0.681503i \(-0.761327\pi\)
−0.731815 + 0.681503i \(0.761327\pi\)
\(410\) −8.53360 −0.421445
\(411\) −45.5980 −2.24918
\(412\) 5.20884 0.256621
\(413\) 1.26320 0.0621581
\(414\) −5.09793 −0.250550
\(415\) −0.880146 −0.0432047
\(416\) 0.281781 0.0138154
\(417\) 23.7477 1.16293
\(418\) −3.92357 −0.191908
\(419\) 0.686821 0.0335534 0.0167767 0.999859i \(-0.494660\pi\)
0.0167767 + 0.999859i \(0.494660\pi\)
\(420\) −3.30297 −0.161168
\(421\) −6.61491 −0.322391 −0.161196 0.986922i \(-0.551535\pi\)
−0.161196 + 0.986922i \(0.551535\pi\)
\(422\) 18.3688 0.894179
\(423\) 26.9416 1.30995
\(424\) −3.58042 −0.173881
\(425\) 2.16921 0.105222
\(426\) −29.0853 −1.40919
\(427\) −2.22784 −0.107813
\(428\) −0.703567 −0.0340082
\(429\) −0.956054 −0.0461587
\(430\) −16.9625 −0.818004
\(431\) −1.00000 −0.0481683
\(432\) −0.466731 −0.0224556
\(433\) −35.6021 −1.71093 −0.855463 0.517864i \(-0.826727\pi\)
−0.855463 + 0.517864i \(0.826727\pi\)
\(434\) −1.39495 −0.0669598
\(435\) 4.15942 0.199429
\(436\) −9.64219 −0.461777
\(437\) 4.60043 0.220068
\(438\) −5.62349 −0.268701
\(439\) 20.5945 0.982924 0.491462 0.870899i \(-0.336463\pi\)
0.491462 + 0.870899i \(0.336463\pi\)
\(440\) 1.81114 0.0863426
\(441\) 3.18763 0.151792
\(442\) −0.188837 −0.00898208
\(443\) −38.1747 −1.81373 −0.906867 0.421416i \(-0.861533\pi\)
−0.906867 + 0.421416i \(0.861533\pi\)
\(444\) 23.1725 1.09972
\(445\) −3.88083 −0.183969
\(446\) 2.02562 0.0959158
\(447\) 42.4410 2.00739
\(448\) 1.00000 0.0472456
\(449\) −31.4244 −1.48301 −0.741504 0.670948i \(-0.765887\pi\)
−0.741504 + 0.670948i \(0.765887\pi\)
\(450\) −10.3180 −0.486393
\(451\) −8.76596 −0.412773
\(452\) −4.14750 −0.195082
\(453\) 26.9732 1.26731
\(454\) −4.94194 −0.231937
\(455\) 0.374156 0.0175407
\(456\) 7.15541 0.335083
\(457\) 22.7966 1.06638 0.533191 0.845995i \(-0.320993\pi\)
0.533191 + 0.845995i \(0.320993\pi\)
\(458\) −21.2745 −0.994094
\(459\) 0.312783 0.0145995
\(460\) −2.12358 −0.0990122
\(461\) −22.5292 −1.04929 −0.524645 0.851321i \(-0.675802\pi\)
−0.524645 + 0.851321i \(0.675802\pi\)
\(462\) −3.39290 −0.157852
\(463\) −16.3115 −0.758059 −0.379030 0.925385i \(-0.623742\pi\)
−0.379030 + 0.925385i \(0.623742\pi\)
\(464\) −1.25930 −0.0584615
\(465\) 4.60748 0.213667
\(466\) 9.01899 0.417797
\(467\) −3.99673 −0.184947 −0.0924734 0.995715i \(-0.529477\pi\)
−0.0924734 + 0.995715i \(0.529477\pi\)
\(468\) 0.898213 0.0415199
\(469\) −2.81723 −0.130088
\(470\) 11.2227 0.517665
\(471\) −31.9566 −1.47248
\(472\) 1.26320 0.0581435
\(473\) −17.4244 −0.801173
\(474\) 19.3240 0.887580
\(475\) 9.31103 0.427219
\(476\) −0.670158 −0.0307166
\(477\) −11.4131 −0.522569
\(478\) −5.91684 −0.270630
\(479\) 33.3604 1.52428 0.762139 0.647414i \(-0.224149\pi\)
0.762139 + 0.647414i \(0.224149\pi\)
\(480\) −3.30297 −0.150759
\(481\) −2.62496 −0.119688
\(482\) −8.37298 −0.381379
\(483\) 3.97821 0.181015
\(484\) −9.13955 −0.415434
\(485\) 19.3235 0.877437
\(486\) 22.2999 1.01154
\(487\) 9.68933 0.439065 0.219533 0.975605i \(-0.429547\pi\)
0.219533 + 0.975605i \(0.429547\pi\)
\(488\) −2.22784 −0.100850
\(489\) −27.1295 −1.22684
\(490\) 1.32783 0.0599852
\(491\) 23.5275 1.06178 0.530891 0.847440i \(-0.321857\pi\)
0.530891 + 0.847440i \(0.321857\pi\)
\(492\) 15.9865 0.720726
\(493\) 0.843929 0.0380087
\(494\) −0.810557 −0.0364687
\(495\) 5.77324 0.259488
\(496\) −1.39495 −0.0626351
\(497\) 11.6926 0.524486
\(498\) 1.64883 0.0738857
\(499\) −7.42227 −0.332266 −0.166133 0.986103i \(-0.553128\pi\)
−0.166133 + 0.986103i \(0.553128\pi\)
\(500\) −10.9372 −0.489124
\(501\) 12.5277 0.559697
\(502\) −25.7847 −1.15083
\(503\) 3.74884 0.167153 0.0835763 0.996501i \(-0.473366\pi\)
0.0835763 + 0.996501i \(0.473366\pi\)
\(504\) 3.18763 0.141988
\(505\) 18.7301 0.833481
\(506\) −2.18140 −0.0969750
\(507\) 32.1399 1.42738
\(508\) −19.2644 −0.854721
\(509\) −8.81025 −0.390507 −0.195254 0.980753i \(-0.562553\pi\)
−0.195254 + 0.980753i \(0.562553\pi\)
\(510\) 2.21351 0.0980158
\(511\) 2.26070 0.100008
\(512\) 1.00000 0.0441942
\(513\) 1.34258 0.0592762
\(514\) −18.2012 −0.802823
\(515\) 6.91644 0.304775
\(516\) 31.7768 1.39889
\(517\) 11.5283 0.507013
\(518\) −9.31561 −0.409305
\(519\) 44.3554 1.94699
\(520\) 0.374156 0.0164078
\(521\) 41.3020 1.80947 0.904737 0.425971i \(-0.140067\pi\)
0.904737 + 0.425971i \(0.140067\pi\)
\(522\) −4.01418 −0.175696
\(523\) −23.0466 −1.00776 −0.503878 0.863775i \(-0.668094\pi\)
−0.503878 + 0.863775i \(0.668094\pi\)
\(524\) −14.3216 −0.625644
\(525\) 8.05170 0.351405
\(526\) −8.76899 −0.382346
\(527\) 0.934837 0.0407221
\(528\) −3.39290 −0.147657
\(529\) −20.4423 −0.888795
\(530\) −4.75419 −0.206509
\(531\) 4.02662 0.174740
\(532\) −2.87655 −0.124714
\(533\) −1.81093 −0.0784401
\(534\) 7.27018 0.314611
\(535\) −0.934216 −0.0403897
\(536\) −2.81723 −0.121686
\(537\) −47.9619 −2.06971
\(538\) −2.09481 −0.0903135
\(539\) 1.36398 0.0587509
\(540\) −0.619739 −0.0266693
\(541\) 9.69628 0.416876 0.208438 0.978036i \(-0.433162\pi\)
0.208438 + 0.978036i \(0.433162\pi\)
\(542\) 28.2295 1.21256
\(543\) 2.34602 0.100677
\(544\) −0.670158 −0.0287328
\(545\) −12.8032 −0.548428
\(546\) −0.700928 −0.0299969
\(547\) −10.0725 −0.430667 −0.215334 0.976541i \(-0.569084\pi\)
−0.215334 + 0.976541i \(0.569084\pi\)
\(548\) 18.3309 0.783057
\(549\) −7.10155 −0.303087
\(550\) −4.41504 −0.188258
\(551\) 3.62244 0.154321
\(552\) 3.97821 0.169324
\(553\) −7.76845 −0.330348
\(554\) −21.9882 −0.934188
\(555\) 30.7692 1.30608
\(556\) −9.54684 −0.404876
\(557\) −3.24225 −0.137379 −0.0686894 0.997638i \(-0.521882\pi\)
−0.0686894 + 0.997638i \(0.521882\pi\)
\(558\) −4.44659 −0.188239
\(559\) −3.59964 −0.152248
\(560\) 1.32783 0.0561110
\(561\) 2.27378 0.0959990
\(562\) 1.37003 0.0577911
\(563\) 27.5227 1.15994 0.579971 0.814637i \(-0.303064\pi\)
0.579971 + 0.814637i \(0.303064\pi\)
\(564\) −21.0241 −0.885275
\(565\) −5.50718 −0.231689
\(566\) 22.6165 0.950641
\(567\) −8.40190 −0.352847
\(568\) 11.6926 0.490611
\(569\) −22.6831 −0.950924 −0.475462 0.879736i \(-0.657719\pi\)
−0.475462 + 0.879736i \(0.657719\pi\)
\(570\) 9.50116 0.397960
\(571\) 25.9223 1.08481 0.542407 0.840116i \(-0.317513\pi\)
0.542407 + 0.840116i \(0.317513\pi\)
\(572\) 0.384344 0.0160702
\(573\) −50.4803 −2.10884
\(574\) −6.42674 −0.268247
\(575\) 5.17668 0.215882
\(576\) 3.18763 0.132818
\(577\) 35.5454 1.47977 0.739887 0.672731i \(-0.234879\pi\)
0.739887 + 0.672731i \(0.234879\pi\)
\(578\) −16.5509 −0.688426
\(579\) 45.8860 1.90696
\(580\) −1.67213 −0.0694316
\(581\) −0.662846 −0.0274995
\(582\) −36.1999 −1.50053
\(583\) −4.88364 −0.202260
\(584\) 2.26070 0.0935486
\(585\) 1.19267 0.0493109
\(586\) 12.8871 0.532361
\(587\) 38.0738 1.57148 0.785738 0.618560i \(-0.212284\pi\)
0.785738 + 0.618560i \(0.212284\pi\)
\(588\) −2.48749 −0.102583
\(589\) 4.01265 0.165338
\(590\) 1.67731 0.0690539
\(591\) −21.7525 −0.894778
\(592\) −9.31561 −0.382869
\(593\) 16.3486 0.671358 0.335679 0.941976i \(-0.391034\pi\)
0.335679 + 0.941976i \(0.391034\pi\)
\(594\) −0.636614 −0.0261206
\(595\) −0.889854 −0.0364805
\(596\) −17.0618 −0.698877
\(597\) −64.5724 −2.64277
\(598\) −0.450647 −0.0184283
\(599\) −15.2724 −0.624015 −0.312008 0.950080i \(-0.601001\pi\)
−0.312008 + 0.950080i \(0.601001\pi\)
\(600\) 8.05170 0.328709
\(601\) 46.7939 1.90876 0.954381 0.298591i \(-0.0965166\pi\)
0.954381 + 0.298591i \(0.0965166\pi\)
\(602\) −12.7746 −0.520654
\(603\) −8.98029 −0.365706
\(604\) −10.8435 −0.441217
\(605\) −12.1358 −0.493389
\(606\) −35.0882 −1.42536
\(607\) −0.257101 −0.0104354 −0.00521770 0.999986i \(-0.501661\pi\)
−0.00521770 + 0.999986i \(0.501661\pi\)
\(608\) −2.87655 −0.116660
\(609\) 3.13250 0.126935
\(610\) −2.95820 −0.119774
\(611\) 2.38159 0.0963487
\(612\) −2.13621 −0.0863514
\(613\) 46.4767 1.87718 0.938588 0.345041i \(-0.112135\pi\)
0.938588 + 0.345041i \(0.112135\pi\)
\(614\) 18.5453 0.748426
\(615\) 21.2273 0.855967
\(616\) 1.36398 0.0549565
\(617\) 43.4704 1.75005 0.875025 0.484077i \(-0.160845\pi\)
0.875025 + 0.484077i \(0.160845\pi\)
\(618\) −12.9570 −0.521205
\(619\) −31.4847 −1.26548 −0.632738 0.774366i \(-0.718069\pi\)
−0.632738 + 0.774366i \(0.718069\pi\)
\(620\) −1.85226 −0.0743884
\(621\) 0.746436 0.0299534
\(622\) −26.7289 −1.07173
\(623\) −2.92269 −0.117095
\(624\) −0.700928 −0.0280596
\(625\) 1.66169 0.0664677
\(626\) −26.7284 −1.06828
\(627\) 9.75987 0.389771
\(628\) 12.8469 0.512648
\(629\) 6.24293 0.248922
\(630\) 4.23263 0.168632
\(631\) 40.5403 1.61389 0.806943 0.590630i \(-0.201121\pi\)
0.806943 + 0.590630i \(0.201121\pi\)
\(632\) −7.76845 −0.309012
\(633\) −45.6923 −1.81611
\(634\) −3.61201 −0.143451
\(635\) −25.5799 −1.01511
\(636\) 8.90629 0.353157
\(637\) 0.281781 0.0111646
\(638\) −1.71766 −0.0680030
\(639\) 37.2718 1.47445
\(640\) 1.32783 0.0524870
\(641\) 6.42650 0.253831 0.126916 0.991914i \(-0.459492\pi\)
0.126916 + 0.991914i \(0.459492\pi\)
\(642\) 1.75012 0.0690717
\(643\) −7.75535 −0.305841 −0.152921 0.988238i \(-0.548868\pi\)
−0.152921 + 0.988238i \(0.548868\pi\)
\(644\) −1.59928 −0.0630207
\(645\) 42.1941 1.66139
\(646\) 1.92774 0.0758461
\(647\) −14.9289 −0.586914 −0.293457 0.955972i \(-0.594806\pi\)
−0.293457 + 0.955972i \(0.594806\pi\)
\(648\) −8.40190 −0.330058
\(649\) 1.72299 0.0676331
\(650\) −0.912088 −0.0357750
\(651\) 3.46993 0.135997
\(652\) 10.9064 0.427126
\(653\) 11.6313 0.455167 0.227584 0.973759i \(-0.426917\pi\)
0.227584 + 0.973759i \(0.426917\pi\)
\(654\) 23.9849 0.937884
\(655\) −19.0167 −0.743043
\(656\) −6.42674 −0.250922
\(657\) 7.20629 0.281144
\(658\) 8.45192 0.329490
\(659\) 34.2263 1.33327 0.666634 0.745385i \(-0.267734\pi\)
0.666634 + 0.745385i \(0.267734\pi\)
\(660\) −4.50519 −0.175364
\(661\) 16.3856 0.637327 0.318663 0.947868i \(-0.396766\pi\)
0.318663 + 0.947868i \(0.396766\pi\)
\(662\) 5.82013 0.226206
\(663\) 0.469732 0.0182429
\(664\) −0.662846 −0.0257234
\(665\) −3.81957 −0.148117
\(666\) −29.6947 −1.15065
\(667\) 2.01398 0.0779816
\(668\) −5.03627 −0.194859
\(669\) −5.03871 −0.194808
\(670\) −3.74080 −0.144520
\(671\) −3.03874 −0.117309
\(672\) −2.48749 −0.0959572
\(673\) 23.1554 0.892575 0.446288 0.894890i \(-0.352746\pi\)
0.446288 + 0.894890i \(0.352746\pi\)
\(674\) 6.61801 0.254916
\(675\) 1.51075 0.0581487
\(676\) −12.9206 −0.496946
\(677\) −18.1784 −0.698653 −0.349326 0.937001i \(-0.613590\pi\)
−0.349326 + 0.937001i \(0.613590\pi\)
\(678\) 10.3169 0.396218
\(679\) 14.5527 0.558483
\(680\) −0.889854 −0.0341244
\(681\) 12.2930 0.471070
\(682\) −1.90269 −0.0728578
\(683\) −36.6838 −1.40367 −0.701833 0.712342i \(-0.747635\pi\)
−0.701833 + 0.712342i \(0.747635\pi\)
\(684\) −9.16939 −0.350600
\(685\) 24.3403 0.929995
\(686\) 1.00000 0.0381802
\(687\) 52.9203 2.01904
\(688\) −12.7746 −0.487027
\(689\) −1.00889 −0.0384358
\(690\) 5.28238 0.201097
\(691\) −22.1102 −0.841113 −0.420556 0.907266i \(-0.638165\pi\)
−0.420556 + 0.907266i \(0.638165\pi\)
\(692\) −17.8313 −0.677846
\(693\) 4.34788 0.165162
\(694\) 15.5183 0.589065
\(695\) −12.6766 −0.480849
\(696\) 3.13250 0.118737
\(697\) 4.30693 0.163136
\(698\) −7.42938 −0.281206
\(699\) −22.4347 −0.848558
\(700\) −3.23687 −0.122342
\(701\) 16.1641 0.610511 0.305255 0.952271i \(-0.401258\pi\)
0.305255 + 0.952271i \(0.401258\pi\)
\(702\) −0.131516 −0.00496374
\(703\) 26.7969 1.01066
\(704\) 1.36398 0.0514071
\(705\) −27.9164 −1.05139
\(706\) 13.6875 0.515134
\(707\) 14.1058 0.530505
\(708\) −3.14221 −0.118091
\(709\) −29.5183 −1.10858 −0.554291 0.832323i \(-0.687011\pi\)
−0.554291 + 0.832323i \(0.687011\pi\)
\(710\) 15.5258 0.582673
\(711\) −24.7630 −0.928684
\(712\) −2.92269 −0.109532
\(713\) 2.23092 0.0835488
\(714\) 1.66701 0.0623864
\(715\) 0.510343 0.0190858
\(716\) 19.2812 0.720573
\(717\) 14.7181 0.549658
\(718\) −35.3180 −1.31805
\(719\) 43.8476 1.63524 0.817620 0.575759i \(-0.195293\pi\)
0.817620 + 0.575759i \(0.195293\pi\)
\(720\) 4.23263 0.157741
\(721\) 5.20884 0.193987
\(722\) −10.7254 −0.399160
\(723\) 20.8277 0.774592
\(724\) −0.943125 −0.0350510
\(725\) 4.07619 0.151386
\(726\) 22.7346 0.843759
\(727\) 28.6539 1.06271 0.531357 0.847148i \(-0.321682\pi\)
0.531357 + 0.847148i \(0.321682\pi\)
\(728\) 0.281781 0.0104435
\(729\) −30.2651 −1.12093
\(730\) 3.00183 0.111103
\(731\) 8.56100 0.316640
\(732\) 5.54175 0.204829
\(733\) 33.9673 1.25461 0.627306 0.778773i \(-0.284158\pi\)
0.627306 + 0.778773i \(0.284158\pi\)
\(734\) −5.49026 −0.202649
\(735\) −3.30297 −0.121832
\(736\) −1.59928 −0.0589504
\(737\) −3.84266 −0.141546
\(738\) −20.4861 −0.754102
\(739\) −36.0497 −1.32611 −0.663055 0.748571i \(-0.730740\pi\)
−0.663055 + 0.748571i \(0.730740\pi\)
\(740\) −12.3695 −0.454713
\(741\) 2.01626 0.0740690
\(742\) −3.58042 −0.131441
\(743\) 40.9311 1.50162 0.750808 0.660521i \(-0.229665\pi\)
0.750808 + 0.660521i \(0.229665\pi\)
\(744\) 3.46993 0.127214
\(745\) −22.6551 −0.830018
\(746\) 0.659604 0.0241498
\(747\) −2.11291 −0.0773073
\(748\) −0.914084 −0.0334222
\(749\) −0.703567 −0.0257078
\(750\) 27.2061 0.993427
\(751\) 3.96763 0.144781 0.0723904 0.997376i \(-0.476937\pi\)
0.0723904 + 0.997376i \(0.476937\pi\)
\(752\) 8.45192 0.308210
\(753\) 64.1394 2.33737
\(754\) −0.354846 −0.0129227
\(755\) −14.3983 −0.524009
\(756\) −0.466731 −0.0169748
\(757\) 41.9368 1.52422 0.762109 0.647449i \(-0.224164\pi\)
0.762109 + 0.647449i \(0.224164\pi\)
\(758\) 10.5619 0.383625
\(759\) 5.42622 0.196959
\(760\) −3.81957 −0.138550
\(761\) 39.8908 1.44604 0.723021 0.690826i \(-0.242753\pi\)
0.723021 + 0.690826i \(0.242753\pi\)
\(762\) 47.9202 1.73596
\(763\) −9.64219 −0.349071
\(764\) 20.2936 0.734198
\(765\) −2.83653 −0.102555
\(766\) −16.2108 −0.585719
\(767\) 0.355946 0.0128524
\(768\) −2.48749 −0.0897597
\(769\) 49.7943 1.79563 0.897814 0.440375i \(-0.145155\pi\)
0.897814 + 0.440375i \(0.145155\pi\)
\(770\) 1.81114 0.0652688
\(771\) 45.2755 1.63056
\(772\) −18.4467 −0.663910
\(773\) 38.1563 1.37238 0.686192 0.727420i \(-0.259281\pi\)
0.686192 + 0.727420i \(0.259281\pi\)
\(774\) −40.7207 −1.46368
\(775\) 4.51528 0.162194
\(776\) 14.5527 0.522413
\(777\) 23.1725 0.831310
\(778\) −9.87917 −0.354186
\(779\) 18.4868 0.662360
\(780\) −0.930712 −0.0333248
\(781\) 15.9485 0.570684
\(782\) 1.07177 0.0383265
\(783\) 0.587754 0.0210046
\(784\) 1.00000 0.0357143
\(785\) 17.0585 0.608844
\(786\) 35.6250 1.27070
\(787\) −19.8666 −0.708167 −0.354083 0.935214i \(-0.615207\pi\)
−0.354083 + 0.935214i \(0.615207\pi\)
\(788\) 8.74474 0.311519
\(789\) 21.8128 0.776556
\(790\) −10.3152 −0.366997
\(791\) −4.14750 −0.147468
\(792\) 4.34788 0.154495
\(793\) −0.627763 −0.0222925
\(794\) −32.9334 −1.16876
\(795\) 11.8260 0.419426
\(796\) 25.9588 0.920085
\(797\) −31.9491 −1.13170 −0.565848 0.824510i \(-0.691451\pi\)
−0.565848 + 0.824510i \(0.691451\pi\)
\(798\) 7.15541 0.253299
\(799\) −5.66412 −0.200382
\(800\) −3.23687 −0.114441
\(801\) −9.31646 −0.329181
\(802\) −33.2223 −1.17312
\(803\) 3.08356 0.108817
\(804\) 7.00785 0.247148
\(805\) −2.12358 −0.0748462
\(806\) −0.393070 −0.0138453
\(807\) 5.21082 0.183430
\(808\) 14.1058 0.496242
\(809\) −9.62596 −0.338431 −0.169215 0.985579i \(-0.554123\pi\)
−0.169215 + 0.985579i \(0.554123\pi\)
\(810\) −11.1563 −0.391992
\(811\) −21.0781 −0.740154 −0.370077 0.929001i \(-0.620669\pi\)
−0.370077 + 0.929001i \(0.620669\pi\)
\(812\) −1.25930 −0.0441928
\(813\) −70.2208 −2.46275
\(814\) −12.7063 −0.445357
\(815\) 14.4818 0.507274
\(816\) 1.66701 0.0583571
\(817\) 36.7468 1.28561
\(818\) −29.6001 −1.03494
\(819\) 0.898213 0.0313861
\(820\) −8.53360 −0.298006
\(821\) −35.4055 −1.23566 −0.617830 0.786312i \(-0.711988\pi\)
−0.617830 + 0.786312i \(0.711988\pi\)
\(822\) −45.5980 −1.59041
\(823\) −25.4055 −0.885579 −0.442789 0.896626i \(-0.646011\pi\)
−0.442789 + 0.896626i \(0.646011\pi\)
\(824\) 5.20884 0.181458
\(825\) 10.9824 0.382358
\(826\) 1.26320 0.0439524
\(827\) −18.7198 −0.650953 −0.325476 0.945550i \(-0.605525\pi\)
−0.325476 + 0.945550i \(0.605525\pi\)
\(828\) −5.09793 −0.177165
\(829\) −16.1210 −0.559905 −0.279953 0.960014i \(-0.590319\pi\)
−0.279953 + 0.960014i \(0.590319\pi\)
\(830\) −0.880146 −0.0305503
\(831\) 54.6955 1.89736
\(832\) 0.281781 0.00976898
\(833\) −0.670158 −0.0232196
\(834\) 23.7477 0.822316
\(835\) −6.68731 −0.231424
\(836\) −3.92357 −0.135700
\(837\) 0.651067 0.0225042
\(838\) 0.686821 0.0237258
\(839\) −7.33478 −0.253225 −0.126612 0.991952i \(-0.540410\pi\)
−0.126612 + 0.991952i \(0.540410\pi\)
\(840\) −3.30297 −0.113963
\(841\) −27.4142 −0.945316
\(842\) −6.61491 −0.227965
\(843\) −3.40794 −0.117376
\(844\) 18.3688 0.632280
\(845\) −17.1563 −0.590196
\(846\) 26.9416 0.926271
\(847\) −9.13955 −0.314039
\(848\) −3.58042 −0.122952
\(849\) −56.2584 −1.93078
\(850\) 2.16921 0.0744034
\(851\) 14.8983 0.510708
\(852\) −29.0853 −0.996447
\(853\) −3.53623 −0.121078 −0.0605392 0.998166i \(-0.519282\pi\)
−0.0605392 + 0.998166i \(0.519282\pi\)
\(854\) −2.22784 −0.0762353
\(855\) −12.1754 −0.416389
\(856\) −0.703567 −0.0240474
\(857\) 16.3278 0.557747 0.278873 0.960328i \(-0.410039\pi\)
0.278873 + 0.960328i \(0.410039\pi\)
\(858\) −0.956054 −0.0326392
\(859\) 1.20390 0.0410766 0.0205383 0.999789i \(-0.493462\pi\)
0.0205383 + 0.999789i \(0.493462\pi\)
\(860\) −16.9625 −0.578416
\(861\) 15.9865 0.544818
\(862\) −1.00000 −0.0340601
\(863\) 2.40294 0.0817971 0.0408985 0.999163i \(-0.486978\pi\)
0.0408985 + 0.999163i \(0.486978\pi\)
\(864\) −0.466731 −0.0158785
\(865\) −23.6770 −0.805041
\(866\) −35.6021 −1.20981
\(867\) 41.1703 1.39822
\(868\) −1.39495 −0.0473477
\(869\) −10.5960 −0.359446
\(870\) 4.15942 0.141018
\(871\) −0.793841 −0.0268983
\(872\) −9.64219 −0.326526
\(873\) 46.3888 1.57002
\(874\) 4.60043 0.155612
\(875\) −10.9372 −0.369743
\(876\) −5.62349 −0.190000
\(877\) 36.4502 1.23084 0.615418 0.788201i \(-0.288987\pi\)
0.615418 + 0.788201i \(0.288987\pi\)
\(878\) 20.5945 0.695032
\(879\) −32.0566 −1.08124
\(880\) 1.81114 0.0610534
\(881\) −26.2306 −0.883732 −0.441866 0.897081i \(-0.645683\pi\)
−0.441866 + 0.897081i \(0.645683\pi\)
\(882\) 3.18763 0.107333
\(883\) −31.1098 −1.04693 −0.523464 0.852048i \(-0.675361\pi\)
−0.523464 + 0.852048i \(0.675361\pi\)
\(884\) −0.188837 −0.00635129
\(885\) −4.17231 −0.140251
\(886\) −38.1747 −1.28250
\(887\) −11.0909 −0.372397 −0.186199 0.982512i \(-0.559617\pi\)
−0.186199 + 0.982512i \(0.559617\pi\)
\(888\) 23.1725 0.777619
\(889\) −19.2644 −0.646108
\(890\) −3.88083 −0.130086
\(891\) −11.4601 −0.383926
\(892\) 2.02562 0.0678227
\(893\) −24.3124 −0.813583
\(894\) 42.4410 1.41944
\(895\) 25.6021 0.855785
\(896\) 1.00000 0.0334077
\(897\) 1.12098 0.0374285
\(898\) −31.4244 −1.04864
\(899\) 1.75666 0.0585879
\(900\) −10.3180 −0.343932
\(901\) 2.39945 0.0799372
\(902\) −8.76596 −0.291875
\(903\) 31.7768 1.05746
\(904\) −4.14750 −0.137944
\(905\) −1.25231 −0.0416281
\(906\) 26.9732 0.896125
\(907\) 45.9040 1.52422 0.762109 0.647449i \(-0.224164\pi\)
0.762109 + 0.647449i \(0.224164\pi\)
\(908\) −4.94194 −0.164004
\(909\) 44.9642 1.49137
\(910\) 0.374156 0.0124032
\(911\) −0.905177 −0.0299898 −0.0149949 0.999888i \(-0.504773\pi\)
−0.0149949 + 0.999888i \(0.504773\pi\)
\(912\) 7.15541 0.236939
\(913\) −0.904112 −0.0299217
\(914\) 22.7966 0.754046
\(915\) 7.35850 0.243264
\(916\) −21.2745 −0.702931
\(917\) −14.3216 −0.472942
\(918\) 0.312783 0.0103234
\(919\) −14.8043 −0.488350 −0.244175 0.969731i \(-0.578517\pi\)
−0.244175 + 0.969731i \(0.578517\pi\)
\(920\) −2.12358 −0.0700122
\(921\) −46.1313 −1.52008
\(922\) −22.5292 −0.741960
\(923\) 3.29475 0.108448
\(924\) −3.39290 −0.111618
\(925\) 30.1534 0.991439
\(926\) −16.3115 −0.536029
\(927\) 16.6038 0.545342
\(928\) −1.25930 −0.0413385
\(929\) −26.8808 −0.881930 −0.440965 0.897524i \(-0.645364\pi\)
−0.440965 + 0.897524i \(0.645364\pi\)
\(930\) 4.60748 0.151085
\(931\) −2.87655 −0.0942752
\(932\) 9.01899 0.295427
\(933\) 66.4880 2.17672
\(934\) −3.99673 −0.130777
\(935\) −1.21375 −0.0396938
\(936\) 0.898213 0.0293590
\(937\) −38.8813 −1.27020 −0.635098 0.772432i \(-0.719040\pi\)
−0.635098 + 0.772432i \(0.719040\pi\)
\(938\) −2.81723 −0.0919859
\(939\) 66.4867 2.16971
\(940\) 11.2227 0.366044
\(941\) 40.3202 1.31440 0.657200 0.753716i \(-0.271741\pi\)
0.657200 + 0.753716i \(0.271741\pi\)
\(942\) −31.9566 −1.04120
\(943\) 10.2782 0.334704
\(944\) 1.26320 0.0411137
\(945\) −0.619739 −0.0201601
\(946\) −17.4244 −0.566515
\(947\) −58.4984 −1.90094 −0.950471 0.310814i \(-0.899398\pi\)
−0.950471 + 0.310814i \(0.899398\pi\)
\(948\) 19.3240 0.627614
\(949\) 0.637022 0.0206786
\(950\) 9.31103 0.302090
\(951\) 8.98486 0.291354
\(952\) −0.670158 −0.0217199
\(953\) 17.9630 0.581878 0.290939 0.956742i \(-0.406032\pi\)
0.290939 + 0.956742i \(0.406032\pi\)
\(954\) −11.4131 −0.369512
\(955\) 26.9465 0.871967
\(956\) −5.91684 −0.191364
\(957\) 4.27268 0.138116
\(958\) 33.3604 1.07783
\(959\) 18.3309 0.591936
\(960\) −3.30297 −0.106603
\(961\) −29.0541 −0.937229
\(962\) −2.62496 −0.0846321
\(963\) −2.24271 −0.0722704
\(964\) −8.37298 −0.269675
\(965\) −24.4940 −0.788490
\(966\) 3.97821 0.127997
\(967\) 7.26690 0.233688 0.116844 0.993150i \(-0.462722\pi\)
0.116844 + 0.993150i \(0.462722\pi\)
\(968\) −9.13955 −0.293756
\(969\) −4.79525 −0.154046
\(970\) 19.3235 0.620442
\(971\) −30.5015 −0.978839 −0.489420 0.872048i \(-0.662791\pi\)
−0.489420 + 0.872048i \(0.662791\pi\)
\(972\) 22.2999 0.715269
\(973\) −9.54684 −0.306058
\(974\) 9.68933 0.310466
\(975\) 2.26881 0.0726602
\(976\) −2.22784 −0.0713116
\(977\) −48.6210 −1.55552 −0.777761 0.628560i \(-0.783645\pi\)
−0.777761 + 0.628560i \(0.783645\pi\)
\(978\) −27.1295 −0.867506
\(979\) −3.98650 −0.127409
\(980\) 1.32783 0.0424159
\(981\) −30.7357 −0.981317
\(982\) 23.5275 0.750793
\(983\) 8.05361 0.256870 0.128435 0.991718i \(-0.459005\pi\)
0.128435 + 0.991718i \(0.459005\pi\)
\(984\) 15.9865 0.509630
\(985\) 11.6115 0.369974
\(986\) 0.843929 0.0268762
\(987\) −21.0241 −0.669205
\(988\) −0.810557 −0.0257872
\(989\) 20.4302 0.649644
\(990\) 5.77324 0.183485
\(991\) −25.0665 −0.796262 −0.398131 0.917329i \(-0.630341\pi\)
−0.398131 + 0.917329i \(0.630341\pi\)
\(992\) −1.39495 −0.0442897
\(993\) −14.4776 −0.459431
\(994\) 11.6926 0.370867
\(995\) 34.4688 1.09274
\(996\) 1.64883 0.0522451
\(997\) −39.8864 −1.26321 −0.631607 0.775289i \(-0.717604\pi\)
−0.631607 + 0.775289i \(0.717604\pi\)
\(998\) −7.42227 −0.234948
\(999\) 4.34788 0.137561
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 6034.2.a.m.1.2 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6034.2.a.m.1.2 21 1.1 even 1 trivial