Properties

Label 4334.2.a.f.1.14
Level $4334$
Weight $2$
Character 4334.1
Self dual yes
Analytic conductor $34.607$
Analytic rank $0$
Dimension $24$
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] = [4334,2,Mod(1,4334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4334, 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("4334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4334 = 2 \cdot 11 \cdot 197 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4334.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: \(34.6071642360\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Character \(\chi\) \(=\) 4334.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.844390 q^{3} +1.00000 q^{4} +4.22697 q^{5} +0.844390 q^{6} -3.69998 q^{7} +1.00000 q^{8} -2.28700 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.844390 q^{3} +1.00000 q^{4} +4.22697 q^{5} +0.844390 q^{6} -3.69998 q^{7} +1.00000 q^{8} -2.28700 q^{9} +4.22697 q^{10} -1.00000 q^{11} +0.844390 q^{12} -2.61015 q^{13} -3.69998 q^{14} +3.56921 q^{15} +1.00000 q^{16} +3.89660 q^{17} -2.28700 q^{18} +6.44454 q^{19} +4.22697 q^{20} -3.12423 q^{21} -1.00000 q^{22} +3.21484 q^{23} +0.844390 q^{24} +12.8672 q^{25} -2.61015 q^{26} -4.46430 q^{27} -3.69998 q^{28} +6.10309 q^{29} +3.56921 q^{30} +8.08890 q^{31} +1.00000 q^{32} -0.844390 q^{33} +3.89660 q^{34} -15.6397 q^{35} -2.28700 q^{36} -9.72230 q^{37} +6.44454 q^{38} -2.20399 q^{39} +4.22697 q^{40} +5.54084 q^{41} -3.12423 q^{42} +1.26532 q^{43} -1.00000 q^{44} -9.66709 q^{45} +3.21484 q^{46} +4.11895 q^{47} +0.844390 q^{48} +6.68987 q^{49} +12.8672 q^{50} +3.29025 q^{51} -2.61015 q^{52} -10.5440 q^{53} -4.46430 q^{54} -4.22697 q^{55} -3.69998 q^{56} +5.44170 q^{57} +6.10309 q^{58} -0.892335 q^{59} +3.56921 q^{60} +8.76256 q^{61} +8.08890 q^{62} +8.46188 q^{63} +1.00000 q^{64} -11.0330 q^{65} -0.844390 q^{66} +4.58290 q^{67} +3.89660 q^{68} +2.71458 q^{69} -15.6397 q^{70} +4.06852 q^{71} -2.28700 q^{72} -8.29482 q^{73} -9.72230 q^{74} +10.8650 q^{75} +6.44454 q^{76} +3.69998 q^{77} -2.20399 q^{78} +13.7390 q^{79} +4.22697 q^{80} +3.09141 q^{81} +5.54084 q^{82} +8.88633 q^{83} -3.12423 q^{84} +16.4708 q^{85} +1.26532 q^{86} +5.15339 q^{87} -1.00000 q^{88} -16.2682 q^{89} -9.66709 q^{90} +9.65751 q^{91} +3.21484 q^{92} +6.83019 q^{93} +4.11895 q^{94} +27.2408 q^{95} +0.844390 q^{96} +3.99233 q^{97} +6.68987 q^{98} +2.28700 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 24 q^{2} + 8 q^{3} + 24 q^{4} + 8 q^{6} + 11 q^{7} + 24 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 24 q^{2} + 8 q^{3} + 24 q^{4} + 8 q^{6} + 11 q^{7} + 24 q^{8} + 28 q^{9} - 24 q^{11} + 8 q^{12} + 9 q^{13} + 11 q^{14} + 6 q^{15} + 24 q^{16} - q^{17} + 28 q^{18} + 35 q^{19} + 21 q^{21} - 24 q^{22} + 13 q^{23} + 8 q^{24} + 38 q^{25} + 9 q^{26} + 23 q^{27} + 11 q^{28} + 15 q^{29} + 6 q^{30} + 31 q^{31} + 24 q^{32} - 8 q^{33} - q^{34} + 28 q^{35} + 28 q^{36} + 13 q^{37} + 35 q^{38} + 31 q^{39} + 16 q^{41} + 21 q^{42} + 35 q^{43} - 24 q^{44} + 13 q^{45} + 13 q^{46} + 46 q^{47} + 8 q^{48} + 55 q^{49} + 38 q^{50} + 18 q^{51} + 9 q^{52} + 6 q^{53} + 23 q^{54} + 11 q^{56} + 18 q^{57} + 15 q^{58} + 13 q^{59} + 6 q^{60} + 60 q^{61} + 31 q^{62} + 52 q^{63} + 24 q^{64} - 9 q^{65} - 8 q^{66} + 28 q^{67} - q^{68} + 21 q^{69} + 28 q^{70} + 10 q^{71} + 28 q^{72} + 3 q^{73} + 13 q^{74} + 48 q^{75} + 35 q^{76} - 11 q^{77} + 31 q^{78} + 47 q^{79} + 16 q^{81} + 16 q^{82} + 66 q^{83} + 21 q^{84} + 37 q^{85} + 35 q^{86} + 34 q^{87} - 24 q^{88} - 5 q^{89} + 13 q^{90} + q^{91} + 13 q^{92} - 16 q^{93} + 46 q^{94} + 9 q^{95} + 8 q^{96} + 24 q^{97} + 55 q^{98} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0.844390 0.487509 0.243754 0.969837i \(-0.421621\pi\)
0.243754 + 0.969837i \(0.421621\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.22697 1.89036 0.945178 0.326555i \(-0.105888\pi\)
0.945178 + 0.326555i \(0.105888\pi\)
\(6\) 0.844390 0.344721
\(7\) −3.69998 −1.39846 −0.699231 0.714896i \(-0.746474\pi\)
−0.699231 + 0.714896i \(0.746474\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.28700 −0.762335
\(10\) 4.22697 1.33668
\(11\) −1.00000 −0.301511
\(12\) 0.844390 0.243754
\(13\) −2.61015 −0.723926 −0.361963 0.932193i \(-0.617893\pi\)
−0.361963 + 0.932193i \(0.617893\pi\)
\(14\) −3.69998 −0.988862
\(15\) 3.56921 0.921566
\(16\) 1.00000 0.250000
\(17\) 3.89660 0.945065 0.472532 0.881313i \(-0.343340\pi\)
0.472532 + 0.881313i \(0.343340\pi\)
\(18\) −2.28700 −0.539052
\(19\) 6.44454 1.47848 0.739239 0.673443i \(-0.235185\pi\)
0.739239 + 0.673443i \(0.235185\pi\)
\(20\) 4.22697 0.945178
\(21\) −3.12423 −0.681763
\(22\) −1.00000 −0.213201
\(23\) 3.21484 0.670340 0.335170 0.942158i \(-0.391206\pi\)
0.335170 + 0.942158i \(0.391206\pi\)
\(24\) 0.844390 0.172360
\(25\) 12.8672 2.57345
\(26\) −2.61015 −0.511893
\(27\) −4.46430 −0.859154
\(28\) −3.69998 −0.699231
\(29\) 6.10309 1.13332 0.566658 0.823953i \(-0.308236\pi\)
0.566658 + 0.823953i \(0.308236\pi\)
\(30\) 3.56921 0.651645
\(31\) 8.08890 1.45281 0.726404 0.687267i \(-0.241190\pi\)
0.726404 + 0.687267i \(0.241190\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.844390 −0.146989
\(34\) 3.89660 0.668262
\(35\) −15.6397 −2.64359
\(36\) −2.28700 −0.381167
\(37\) −9.72230 −1.59834 −0.799168 0.601108i \(-0.794726\pi\)
−0.799168 + 0.601108i \(0.794726\pi\)
\(38\) 6.44454 1.04544
\(39\) −2.20399 −0.352920
\(40\) 4.22697 0.668342
\(41\) 5.54084 0.865334 0.432667 0.901554i \(-0.357573\pi\)
0.432667 + 0.901554i \(0.357573\pi\)
\(42\) −3.12423 −0.482079
\(43\) 1.26532 0.192959 0.0964795 0.995335i \(-0.469242\pi\)
0.0964795 + 0.995335i \(0.469242\pi\)
\(44\) −1.00000 −0.150756
\(45\) −9.66709 −1.44109
\(46\) 3.21484 0.474002
\(47\) 4.11895 0.600810 0.300405 0.953812i \(-0.402878\pi\)
0.300405 + 0.953812i \(0.402878\pi\)
\(48\) 0.844390 0.121877
\(49\) 6.68987 0.955696
\(50\) 12.8672 1.81970
\(51\) 3.29025 0.460728
\(52\) −2.61015 −0.361963
\(53\) −10.5440 −1.44832 −0.724162 0.689630i \(-0.757773\pi\)
−0.724162 + 0.689630i \(0.757773\pi\)
\(54\) −4.46430 −0.607514
\(55\) −4.22697 −0.569964
\(56\) −3.69998 −0.494431
\(57\) 5.44170 0.720771
\(58\) 6.10309 0.801375
\(59\) −0.892335 −0.116172 −0.0580860 0.998312i \(-0.518500\pi\)
−0.0580860 + 0.998312i \(0.518500\pi\)
\(60\) 3.56921 0.460783
\(61\) 8.76256 1.12193 0.560966 0.827839i \(-0.310430\pi\)
0.560966 + 0.827839i \(0.310430\pi\)
\(62\) 8.08890 1.02729
\(63\) 8.46188 1.06610
\(64\) 1.00000 0.125000
\(65\) −11.0330 −1.36848
\(66\) −0.844390 −0.103937
\(67\) 4.58290 0.559891 0.279945 0.960016i \(-0.409684\pi\)
0.279945 + 0.960016i \(0.409684\pi\)
\(68\) 3.89660 0.472532
\(69\) 2.71458 0.326797
\(70\) −15.6397 −1.86930
\(71\) 4.06852 0.482844 0.241422 0.970420i \(-0.422386\pi\)
0.241422 + 0.970420i \(0.422386\pi\)
\(72\) −2.28700 −0.269526
\(73\) −8.29482 −0.970836 −0.485418 0.874282i \(-0.661333\pi\)
−0.485418 + 0.874282i \(0.661333\pi\)
\(74\) −9.72230 −1.13019
\(75\) 10.8650 1.25458
\(76\) 6.44454 0.739239
\(77\) 3.69998 0.421652
\(78\) −2.20399 −0.249552
\(79\) 13.7390 1.54576 0.772879 0.634554i \(-0.218816\pi\)
0.772879 + 0.634554i \(0.218816\pi\)
\(80\) 4.22697 0.472589
\(81\) 3.09141 0.343490
\(82\) 5.54084 0.611884
\(83\) 8.88633 0.975401 0.487701 0.873011i \(-0.337836\pi\)
0.487701 + 0.873011i \(0.337836\pi\)
\(84\) −3.12423 −0.340881
\(85\) 16.4708 1.78651
\(86\) 1.26532 0.136443
\(87\) 5.15339 0.552501
\(88\) −1.00000 −0.106600
\(89\) −16.2682 −1.72442 −0.862212 0.506547i \(-0.830922\pi\)
−0.862212 + 0.506547i \(0.830922\pi\)
\(90\) −9.66709 −1.01900
\(91\) 9.65751 1.01238
\(92\) 3.21484 0.335170
\(93\) 6.83019 0.708257
\(94\) 4.11895 0.424837
\(95\) 27.2408 2.79485
\(96\) 0.844390 0.0861802
\(97\) 3.99233 0.405359 0.202680 0.979245i \(-0.435035\pi\)
0.202680 + 0.979245i \(0.435035\pi\)
\(98\) 6.68987 0.675779
\(99\) 2.28700 0.229853
\(100\) 12.8672 1.28672
\(101\) 6.93369 0.689928 0.344964 0.938616i \(-0.387891\pi\)
0.344964 + 0.938616i \(0.387891\pi\)
\(102\) 3.29025 0.325784
\(103\) −13.7159 −1.35147 −0.675733 0.737146i \(-0.736173\pi\)
−0.675733 + 0.737146i \(0.736173\pi\)
\(104\) −2.61015 −0.255946
\(105\) −13.2060 −1.28877
\(106\) −10.5440 −1.02412
\(107\) 2.82180 0.272794 0.136397 0.990654i \(-0.456448\pi\)
0.136397 + 0.990654i \(0.456448\pi\)
\(108\) −4.46430 −0.429577
\(109\) −10.1850 −0.975546 −0.487773 0.872970i \(-0.662191\pi\)
−0.487773 + 0.872970i \(0.662191\pi\)
\(110\) −4.22697 −0.403025
\(111\) −8.20941 −0.779203
\(112\) −3.69998 −0.349615
\(113\) −0.545928 −0.0513566 −0.0256783 0.999670i \(-0.508175\pi\)
−0.0256783 + 0.999670i \(0.508175\pi\)
\(114\) 5.44170 0.509662
\(115\) 13.5890 1.26718
\(116\) 6.10309 0.566658
\(117\) 5.96943 0.551874
\(118\) −0.892335 −0.0821461
\(119\) −14.4174 −1.32164
\(120\) 3.56921 0.325823
\(121\) 1.00000 0.0909091
\(122\) 8.76256 0.793325
\(123\) 4.67863 0.421858
\(124\) 8.08890 0.726404
\(125\) 33.2546 2.97438
\(126\) 8.46188 0.753844
\(127\) 7.73124 0.686036 0.343018 0.939329i \(-0.388551\pi\)
0.343018 + 0.939329i \(0.388551\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.06842 0.0940692
\(130\) −11.0330 −0.967660
\(131\) 5.94221 0.519174 0.259587 0.965720i \(-0.416414\pi\)
0.259587 + 0.965720i \(0.416414\pi\)
\(132\) −0.844390 −0.0734947
\(133\) −23.8447 −2.06760
\(134\) 4.58290 0.395903
\(135\) −18.8704 −1.62411
\(136\) 3.89660 0.334131
\(137\) −21.4264 −1.83058 −0.915292 0.402792i \(-0.868040\pi\)
−0.915292 + 0.402792i \(0.868040\pi\)
\(138\) 2.71458 0.231080
\(139\) 11.9196 1.01101 0.505504 0.862824i \(-0.331306\pi\)
0.505504 + 0.862824i \(0.331306\pi\)
\(140\) −15.6397 −1.32180
\(141\) 3.47800 0.292900
\(142\) 4.06852 0.341422
\(143\) 2.61015 0.218272
\(144\) −2.28700 −0.190584
\(145\) 25.7976 2.14237
\(146\) −8.29482 −0.686484
\(147\) 5.64886 0.465910
\(148\) −9.72230 −0.799168
\(149\) 10.4390 0.855199 0.427600 0.903968i \(-0.359359\pi\)
0.427600 + 0.903968i \(0.359359\pi\)
\(150\) 10.8650 0.887121
\(151\) −11.5544 −0.940283 −0.470142 0.882591i \(-0.655797\pi\)
−0.470142 + 0.882591i \(0.655797\pi\)
\(152\) 6.44454 0.522721
\(153\) −8.91155 −0.720456
\(154\) 3.69998 0.298153
\(155\) 34.1915 2.74633
\(156\) −2.20399 −0.176460
\(157\) −22.4382 −1.79076 −0.895381 0.445302i \(-0.853096\pi\)
−0.895381 + 0.445302i \(0.853096\pi\)
\(158\) 13.7390 1.09302
\(159\) −8.90321 −0.706071
\(160\) 4.22697 0.334171
\(161\) −11.8948 −0.937445
\(162\) 3.09141 0.242884
\(163\) −24.3335 −1.90595 −0.952973 0.303056i \(-0.901993\pi\)
−0.952973 + 0.303056i \(0.901993\pi\)
\(164\) 5.54084 0.432667
\(165\) −3.56921 −0.277863
\(166\) 8.88633 0.689713
\(167\) −7.53899 −0.583385 −0.291692 0.956512i \(-0.594218\pi\)
−0.291692 + 0.956512i \(0.594218\pi\)
\(168\) −3.12423 −0.241040
\(169\) −6.18711 −0.475932
\(170\) 16.4708 1.26325
\(171\) −14.7387 −1.12710
\(172\) 1.26532 0.0964795
\(173\) 5.63897 0.428723 0.214361 0.976754i \(-0.431233\pi\)
0.214361 + 0.976754i \(0.431233\pi\)
\(174\) 5.15339 0.390677
\(175\) −47.6086 −3.59887
\(176\) −1.00000 −0.0753778
\(177\) −0.753479 −0.0566349
\(178\) −16.2682 −1.21935
\(179\) 9.90420 0.740275 0.370137 0.928977i \(-0.379311\pi\)
0.370137 + 0.928977i \(0.379311\pi\)
\(180\) −9.66709 −0.720543
\(181\) −19.7201 −1.46578 −0.732892 0.680345i \(-0.761830\pi\)
−0.732892 + 0.680345i \(0.761830\pi\)
\(182\) 9.65751 0.715863
\(183\) 7.39902 0.546952
\(184\) 3.21484 0.237001
\(185\) −41.0958 −3.02142
\(186\) 6.83019 0.500814
\(187\) −3.89660 −0.284948
\(188\) 4.11895 0.300405
\(189\) 16.5178 1.20149
\(190\) 27.2408 1.97626
\(191\) −12.0678 −0.873195 −0.436598 0.899657i \(-0.643817\pi\)
−0.436598 + 0.899657i \(0.643817\pi\)
\(192\) 0.844390 0.0609386
\(193\) −7.02489 −0.505663 −0.252831 0.967510i \(-0.581362\pi\)
−0.252831 + 0.967510i \(0.581362\pi\)
\(194\) 3.99233 0.286632
\(195\) −9.31618 −0.667145
\(196\) 6.68987 0.477848
\(197\) −1.00000 −0.0712470
\(198\) 2.28700 0.162530
\(199\) −3.09241 −0.219215 −0.109608 0.993975i \(-0.534959\pi\)
−0.109608 + 0.993975i \(0.534959\pi\)
\(200\) 12.8672 0.909851
\(201\) 3.86976 0.272952
\(202\) 6.93369 0.487852
\(203\) −22.5813 −1.58490
\(204\) 3.29025 0.230364
\(205\) 23.4210 1.63579
\(206\) −13.7159 −0.955631
\(207\) −7.35235 −0.511023
\(208\) −2.61015 −0.180981
\(209\) −6.44454 −0.445778
\(210\) −13.2060 −0.911301
\(211\) −1.18835 −0.0818097 −0.0409049 0.999163i \(-0.513024\pi\)
−0.0409049 + 0.999163i \(0.513024\pi\)
\(212\) −10.5440 −0.724162
\(213\) 3.43541 0.235391
\(214\) 2.82180 0.192895
\(215\) 5.34845 0.364761
\(216\) −4.46430 −0.303757
\(217\) −29.9288 −2.03170
\(218\) −10.1850 −0.689815
\(219\) −7.00407 −0.473291
\(220\) −4.22697 −0.284982
\(221\) −10.1707 −0.684157
\(222\) −8.20941 −0.550980
\(223\) 21.4276 1.43490 0.717450 0.696610i \(-0.245309\pi\)
0.717450 + 0.696610i \(0.245309\pi\)
\(224\) −3.69998 −0.247215
\(225\) −29.4274 −1.96183
\(226\) −0.545928 −0.0363146
\(227\) −0.337056 −0.0223712 −0.0111856 0.999937i \(-0.503561\pi\)
−0.0111856 + 0.999937i \(0.503561\pi\)
\(228\) 5.44170 0.360386
\(229\) 19.0636 1.25976 0.629881 0.776692i \(-0.283104\pi\)
0.629881 + 0.776692i \(0.283104\pi\)
\(230\) 13.5890 0.896032
\(231\) 3.12423 0.205559
\(232\) 6.10309 0.400687
\(233\) 6.56520 0.430101 0.215050 0.976603i \(-0.431008\pi\)
0.215050 + 0.976603i \(0.431008\pi\)
\(234\) 5.96943 0.390234
\(235\) 17.4106 1.13575
\(236\) −0.892335 −0.0580860
\(237\) 11.6011 0.753570
\(238\) −14.4174 −0.934539
\(239\) 0.818578 0.0529494 0.0264747 0.999649i \(-0.491572\pi\)
0.0264747 + 0.999649i \(0.491572\pi\)
\(240\) 3.56921 0.230391
\(241\) 7.99971 0.515307 0.257653 0.966237i \(-0.417051\pi\)
0.257653 + 0.966237i \(0.417051\pi\)
\(242\) 1.00000 0.0642824
\(243\) 16.0032 1.02661
\(244\) 8.76256 0.560966
\(245\) 28.2779 1.80661
\(246\) 4.67863 0.298299
\(247\) −16.8212 −1.07031
\(248\) 8.08890 0.513645
\(249\) 7.50353 0.475517
\(250\) 33.2546 2.10320
\(251\) −11.2651 −0.711047 −0.355524 0.934667i \(-0.615697\pi\)
−0.355524 + 0.934667i \(0.615697\pi\)
\(252\) 8.46188 0.533048
\(253\) −3.21484 −0.202115
\(254\) 7.73124 0.485101
\(255\) 13.9078 0.870939
\(256\) 1.00000 0.0625000
\(257\) −5.11751 −0.319221 −0.159611 0.987180i \(-0.551024\pi\)
−0.159611 + 0.987180i \(0.551024\pi\)
\(258\) 1.06842 0.0665170
\(259\) 35.9723 2.23521
\(260\) −11.0330 −0.684239
\(261\) −13.9578 −0.863966
\(262\) 5.94221 0.367111
\(263\) 1.99141 0.122795 0.0613977 0.998113i \(-0.480444\pi\)
0.0613977 + 0.998113i \(0.480444\pi\)
\(264\) −0.844390 −0.0519686
\(265\) −44.5689 −2.73785
\(266\) −23.8447 −1.46201
\(267\) −13.7367 −0.840673
\(268\) 4.58290 0.279945
\(269\) 3.83213 0.233649 0.116825 0.993153i \(-0.462728\pi\)
0.116825 + 0.993153i \(0.462728\pi\)
\(270\) −18.8704 −1.14842
\(271\) −12.4814 −0.758192 −0.379096 0.925357i \(-0.623765\pi\)
−0.379096 + 0.925357i \(0.623765\pi\)
\(272\) 3.89660 0.236266
\(273\) 8.15471 0.493546
\(274\) −21.4264 −1.29442
\(275\) −12.8672 −0.775924
\(276\) 2.71458 0.163398
\(277\) −4.92043 −0.295640 −0.147820 0.989014i \(-0.547226\pi\)
−0.147820 + 0.989014i \(0.547226\pi\)
\(278\) 11.9196 0.714891
\(279\) −18.4993 −1.10753
\(280\) −15.6397 −0.934651
\(281\) 31.2854 1.86633 0.933167 0.359444i \(-0.117034\pi\)
0.933167 + 0.359444i \(0.117034\pi\)
\(282\) 3.47800 0.207112
\(283\) −22.4086 −1.33205 −0.666027 0.745928i \(-0.732006\pi\)
−0.666027 + 0.745928i \(0.732006\pi\)
\(284\) 4.06852 0.241422
\(285\) 23.0019 1.36251
\(286\) 2.61015 0.154341
\(287\) −20.5010 −1.21014
\(288\) −2.28700 −0.134763
\(289\) −1.81649 −0.106852
\(290\) 25.7976 1.51488
\(291\) 3.37108 0.197616
\(292\) −8.29482 −0.485418
\(293\) −20.5399 −1.19995 −0.599976 0.800018i \(-0.704823\pi\)
−0.599976 + 0.800018i \(0.704823\pi\)
\(294\) 5.64886 0.329448
\(295\) −3.77187 −0.219607
\(296\) −9.72230 −0.565097
\(297\) 4.46430 0.259045
\(298\) 10.4390 0.604717
\(299\) −8.39121 −0.485276
\(300\) 10.8650 0.627290
\(301\) −4.68165 −0.269846
\(302\) −11.5544 −0.664881
\(303\) 5.85474 0.336346
\(304\) 6.44454 0.369619
\(305\) 37.0391 2.12085
\(306\) −8.91155 −0.509439
\(307\) −24.0190 −1.37084 −0.685418 0.728149i \(-0.740381\pi\)
−0.685418 + 0.728149i \(0.740381\pi\)
\(308\) 3.69998 0.210826
\(309\) −11.5816 −0.658852
\(310\) 34.1915 1.94195
\(311\) 14.8044 0.839481 0.419740 0.907644i \(-0.362121\pi\)
0.419740 + 0.907644i \(0.362121\pi\)
\(312\) −2.20399 −0.124776
\(313\) −29.6240 −1.67445 −0.837224 0.546860i \(-0.815823\pi\)
−0.837224 + 0.546860i \(0.815823\pi\)
\(314\) −22.4382 −1.26626
\(315\) 35.7681 2.01530
\(316\) 13.7390 0.772879
\(317\) 6.88637 0.386777 0.193389 0.981122i \(-0.438052\pi\)
0.193389 + 0.981122i \(0.438052\pi\)
\(318\) −8.90321 −0.499267
\(319\) −6.10309 −0.341707
\(320\) 4.22697 0.236295
\(321\) 2.38270 0.132990
\(322\) −11.8948 −0.662873
\(323\) 25.1118 1.39726
\(324\) 3.09141 0.171745
\(325\) −33.5855 −1.86299
\(326\) −24.3335 −1.34771
\(327\) −8.60012 −0.475587
\(328\) 5.54084 0.305942
\(329\) −15.2400 −0.840210
\(330\) −3.56921 −0.196479
\(331\) −4.84184 −0.266132 −0.133066 0.991107i \(-0.542482\pi\)
−0.133066 + 0.991107i \(0.542482\pi\)
\(332\) 8.88633 0.487701
\(333\) 22.2349 1.21847
\(334\) −7.53899 −0.412515
\(335\) 19.3718 1.05839
\(336\) −3.12423 −0.170441
\(337\) −27.4794 −1.49690 −0.748450 0.663191i \(-0.769202\pi\)
−0.748450 + 0.663191i \(0.769202\pi\)
\(338\) −6.18711 −0.336534
\(339\) −0.460976 −0.0250368
\(340\) 16.4708 0.893255
\(341\) −8.08890 −0.438038
\(342\) −14.7387 −0.796977
\(343\) 1.14747 0.0619575
\(344\) 1.26532 0.0682213
\(345\) 11.4744 0.617762
\(346\) 5.63897 0.303153
\(347\) 7.78099 0.417705 0.208853 0.977947i \(-0.433027\pi\)
0.208853 + 0.977947i \(0.433027\pi\)
\(348\) 5.15339 0.276251
\(349\) 20.7381 1.11008 0.555042 0.831822i \(-0.312702\pi\)
0.555042 + 0.831822i \(0.312702\pi\)
\(350\) −47.6086 −2.54479
\(351\) 11.6525 0.621964
\(352\) −1.00000 −0.0533002
\(353\) −2.88777 −0.153701 −0.0768503 0.997043i \(-0.524486\pi\)
−0.0768503 + 0.997043i \(0.524486\pi\)
\(354\) −0.753479 −0.0400469
\(355\) 17.1975 0.912747
\(356\) −16.2682 −0.862212
\(357\) −12.1739 −0.644310
\(358\) 9.90420 0.523453
\(359\) 15.2377 0.804212 0.402106 0.915593i \(-0.368278\pi\)
0.402106 + 0.915593i \(0.368278\pi\)
\(360\) −9.66709 −0.509501
\(361\) 22.5320 1.18590
\(362\) −19.7201 −1.03647
\(363\) 0.844390 0.0443190
\(364\) 9.65751 0.506191
\(365\) −35.0619 −1.83523
\(366\) 7.39902 0.386753
\(367\) 16.7576 0.874738 0.437369 0.899282i \(-0.355910\pi\)
0.437369 + 0.899282i \(0.355910\pi\)
\(368\) 3.21484 0.167585
\(369\) −12.6719 −0.659675
\(370\) −41.0958 −2.13647
\(371\) 39.0124 2.02542
\(372\) 6.83019 0.354129
\(373\) −35.4758 −1.83687 −0.918433 0.395577i \(-0.870545\pi\)
−0.918433 + 0.395577i \(0.870545\pi\)
\(374\) −3.89660 −0.201488
\(375\) 28.0798 1.45004
\(376\) 4.11895 0.212418
\(377\) −15.9300 −0.820436
\(378\) 16.5178 0.849585
\(379\) 2.08275 0.106984 0.0534919 0.998568i \(-0.482965\pi\)
0.0534919 + 0.998568i \(0.482965\pi\)
\(380\) 27.2408 1.39743
\(381\) 6.52818 0.334449
\(382\) −12.0678 −0.617442
\(383\) −3.98661 −0.203706 −0.101853 0.994799i \(-0.532477\pi\)
−0.101853 + 0.994799i \(0.532477\pi\)
\(384\) 0.844390 0.0430901
\(385\) 15.6397 0.797073
\(386\) −7.02489 −0.357557
\(387\) −2.89378 −0.147099
\(388\) 3.99233 0.202680
\(389\) 30.6751 1.55529 0.777644 0.628705i \(-0.216415\pi\)
0.777644 + 0.628705i \(0.216415\pi\)
\(390\) −9.31618 −0.471743
\(391\) 12.5269 0.633515
\(392\) 6.68987 0.337890
\(393\) 5.01755 0.253102
\(394\) −1.00000 −0.0503793
\(395\) 58.0742 2.92203
\(396\) 2.28700 0.114926
\(397\) 28.2667 1.41866 0.709332 0.704875i \(-0.248997\pi\)
0.709332 + 0.704875i \(0.248997\pi\)
\(398\) −3.09241 −0.155008
\(399\) −20.1342 −1.00797
\(400\) 12.8672 0.643362
\(401\) −7.18828 −0.358966 −0.179483 0.983761i \(-0.557442\pi\)
−0.179483 + 0.983761i \(0.557442\pi\)
\(402\) 3.86976 0.193006
\(403\) −21.1132 −1.05173
\(404\) 6.93369 0.344964
\(405\) 13.0673 0.649318
\(406\) −22.5813 −1.12069
\(407\) 9.72230 0.481916
\(408\) 3.29025 0.162892
\(409\) −27.2450 −1.34718 −0.673590 0.739106i \(-0.735248\pi\)
−0.673590 + 0.739106i \(0.735248\pi\)
\(410\) 23.4210 1.15668
\(411\) −18.0923 −0.892426
\(412\) −13.7159 −0.675733
\(413\) 3.30162 0.162462
\(414\) −7.35235 −0.361348
\(415\) 37.5622 1.84386
\(416\) −2.61015 −0.127973
\(417\) 10.0648 0.492876
\(418\) −6.44454 −0.315213
\(419\) −14.4906 −0.707915 −0.353957 0.935262i \(-0.615164\pi\)
−0.353957 + 0.935262i \(0.615164\pi\)
\(420\) −13.2060 −0.644387
\(421\) −34.0129 −1.65769 −0.828844 0.559480i \(-0.811001\pi\)
−0.828844 + 0.559480i \(0.811001\pi\)
\(422\) −1.18835 −0.0578482
\(423\) −9.42005 −0.458018
\(424\) −10.5440 −0.512060
\(425\) 50.1385 2.43208
\(426\) 3.43541 0.166446
\(427\) −32.4213 −1.56898
\(428\) 2.82180 0.136397
\(429\) 2.20399 0.106409
\(430\) 5.34845 0.257925
\(431\) −6.65099 −0.320367 −0.160183 0.987087i \(-0.551209\pi\)
−0.160183 + 0.987087i \(0.551209\pi\)
\(432\) −4.46430 −0.214789
\(433\) −39.0712 −1.87764 −0.938821 0.344405i \(-0.888081\pi\)
−0.938821 + 0.344405i \(0.888081\pi\)
\(434\) −29.9288 −1.43663
\(435\) 21.7832 1.04442
\(436\) −10.1850 −0.487773
\(437\) 20.7181 0.991083
\(438\) −7.00407 −0.334667
\(439\) −12.9506 −0.618097 −0.309048 0.951046i \(-0.600010\pi\)
−0.309048 + 0.951046i \(0.600010\pi\)
\(440\) −4.22697 −0.201513
\(441\) −15.2998 −0.728560
\(442\) −10.1707 −0.483772
\(443\) −13.3359 −0.633607 −0.316804 0.948491i \(-0.602610\pi\)
−0.316804 + 0.948491i \(0.602610\pi\)
\(444\) −8.20941 −0.389602
\(445\) −68.7651 −3.25978
\(446\) 21.4276 1.01463
\(447\) 8.81462 0.416917
\(448\) −3.69998 −0.174808
\(449\) 3.42835 0.161794 0.0808969 0.996722i \(-0.474222\pi\)
0.0808969 + 0.996722i \(0.474222\pi\)
\(450\) −29.4274 −1.38722
\(451\) −5.54084 −0.260908
\(452\) −0.545928 −0.0256783
\(453\) −9.75642 −0.458396
\(454\) −0.337056 −0.0158188
\(455\) 40.8220 1.91376
\(456\) 5.44170 0.254831
\(457\) −4.61725 −0.215986 −0.107993 0.994152i \(-0.534442\pi\)
−0.107993 + 0.994152i \(0.534442\pi\)
\(458\) 19.0636 0.890786
\(459\) −17.3956 −0.811956
\(460\) 13.5890 0.633591
\(461\) −13.8104 −0.643215 −0.321608 0.946873i \(-0.604223\pi\)
−0.321608 + 0.946873i \(0.604223\pi\)
\(462\) 3.12423 0.145352
\(463\) 29.0450 1.34984 0.674918 0.737892i \(-0.264179\pi\)
0.674918 + 0.737892i \(0.264179\pi\)
\(464\) 6.10309 0.283329
\(465\) 28.8710 1.33886
\(466\) 6.56520 0.304127
\(467\) −1.59622 −0.0738643 −0.0369321 0.999318i \(-0.511759\pi\)
−0.0369321 + 0.999318i \(0.511759\pi\)
\(468\) 5.96943 0.275937
\(469\) −16.9567 −0.782986
\(470\) 17.4106 0.803093
\(471\) −18.9466 −0.873012
\(472\) −0.892335 −0.0410730
\(473\) −1.26532 −0.0581793
\(474\) 11.6011 0.532855
\(475\) 82.9234 3.80479
\(476\) −14.4174 −0.660819
\(477\) 24.1141 1.10411
\(478\) 0.818578 0.0374409
\(479\) −24.1500 −1.10344 −0.551721 0.834029i \(-0.686029\pi\)
−0.551721 + 0.834029i \(0.686029\pi\)
\(480\) 3.56921 0.162911
\(481\) 25.3767 1.15708
\(482\) 7.99971 0.364377
\(483\) −10.0439 −0.457013
\(484\) 1.00000 0.0454545
\(485\) 16.8754 0.766274
\(486\) 16.0032 0.725922
\(487\) 40.0418 1.81447 0.907234 0.420625i \(-0.138189\pi\)
0.907234 + 0.420625i \(0.138189\pi\)
\(488\) 8.76256 0.396663
\(489\) −20.5470 −0.929166
\(490\) 28.2779 1.27746
\(491\) −38.6205 −1.74292 −0.871459 0.490469i \(-0.836826\pi\)
−0.871459 + 0.490469i \(0.836826\pi\)
\(492\) 4.67863 0.210929
\(493\) 23.7813 1.07106
\(494\) −16.8212 −0.756822
\(495\) 9.66709 0.434503
\(496\) 8.08890 0.363202
\(497\) −15.0534 −0.675239
\(498\) 7.50353 0.336241
\(499\) 35.1589 1.57393 0.786964 0.616998i \(-0.211652\pi\)
0.786964 + 0.616998i \(0.211652\pi\)
\(500\) 33.2546 1.48719
\(501\) −6.36585 −0.284405
\(502\) −11.2651 −0.502786
\(503\) −8.99226 −0.400945 −0.200473 0.979699i \(-0.564248\pi\)
−0.200473 + 0.979699i \(0.564248\pi\)
\(504\) 8.46188 0.376922
\(505\) 29.3085 1.30421
\(506\) −3.21484 −0.142917
\(507\) −5.22434 −0.232021
\(508\) 7.73124 0.343018
\(509\) 41.3096 1.83102 0.915508 0.402300i \(-0.131789\pi\)
0.915508 + 0.402300i \(0.131789\pi\)
\(510\) 13.9078 0.615847
\(511\) 30.6907 1.35768
\(512\) 1.00000 0.0441942
\(513\) −28.7703 −1.27024
\(514\) −5.11751 −0.225723
\(515\) −57.9766 −2.55475
\(516\) 1.06842 0.0470346
\(517\) −4.11895 −0.181151
\(518\) 35.9723 1.58053
\(519\) 4.76149 0.209006
\(520\) −11.0330 −0.483830
\(521\) 26.3354 1.15378 0.576888 0.816823i \(-0.304267\pi\)
0.576888 + 0.816823i \(0.304267\pi\)
\(522\) −13.9578 −0.610916
\(523\) −22.6110 −0.988711 −0.494355 0.869260i \(-0.664596\pi\)
−0.494355 + 0.869260i \(0.664596\pi\)
\(524\) 5.94221 0.259587
\(525\) −40.2002 −1.75448
\(526\) 1.99141 0.0868295
\(527\) 31.5192 1.37300
\(528\) −0.844390 −0.0367474
\(529\) −12.6648 −0.550645
\(530\) −44.5689 −1.93595
\(531\) 2.04077 0.0885620
\(532\) −23.8447 −1.03380
\(533\) −14.4624 −0.626438
\(534\) −13.7367 −0.594445
\(535\) 11.9277 0.515678
\(536\) 4.58290 0.197951
\(537\) 8.36301 0.360891
\(538\) 3.83213 0.165215
\(539\) −6.68987 −0.288153
\(540\) −18.8704 −0.812054
\(541\) −22.0328 −0.947264 −0.473632 0.880723i \(-0.657057\pi\)
−0.473632 + 0.880723i \(0.657057\pi\)
\(542\) −12.4814 −0.536123
\(543\) −16.6515 −0.714583
\(544\) 3.89660 0.167065
\(545\) −43.0517 −1.84413
\(546\) 8.15471 0.348989
\(547\) 38.2504 1.63547 0.817735 0.575595i \(-0.195229\pi\)
0.817735 + 0.575595i \(0.195229\pi\)
\(548\) −21.4264 −0.915292
\(549\) −20.0400 −0.855287
\(550\) −12.8672 −0.548661
\(551\) 39.3316 1.67558
\(552\) 2.71458 0.115540
\(553\) −50.8340 −2.16168
\(554\) −4.92043 −0.209049
\(555\) −34.7009 −1.47297
\(556\) 11.9196 0.505504
\(557\) 32.6703 1.38428 0.692142 0.721761i \(-0.256667\pi\)
0.692142 + 0.721761i \(0.256667\pi\)
\(558\) −18.4993 −0.783140
\(559\) −3.30267 −0.139688
\(560\) −15.6397 −0.660898
\(561\) −3.29025 −0.138915
\(562\) 31.2854 1.31970
\(563\) −33.7102 −1.42071 −0.710357 0.703842i \(-0.751466\pi\)
−0.710357 + 0.703842i \(0.751466\pi\)
\(564\) 3.47800 0.146450
\(565\) −2.30762 −0.0970822
\(566\) −22.4086 −0.941904
\(567\) −11.4382 −0.480357
\(568\) 4.06852 0.170711
\(569\) −26.7251 −1.12037 −0.560187 0.828366i \(-0.689271\pi\)
−0.560187 + 0.828366i \(0.689271\pi\)
\(570\) 23.0019 0.963443
\(571\) 45.0811 1.88659 0.943293 0.331962i \(-0.107711\pi\)
0.943293 + 0.331962i \(0.107711\pi\)
\(572\) 2.61015 0.109136
\(573\) −10.1899 −0.425691
\(574\) −20.5010 −0.855696
\(575\) 41.3661 1.72508
\(576\) −2.28700 −0.0952919
\(577\) 1.46900 0.0611554 0.0305777 0.999532i \(-0.490265\pi\)
0.0305777 + 0.999532i \(0.490265\pi\)
\(578\) −1.81649 −0.0755561
\(579\) −5.93175 −0.246515
\(580\) 25.7976 1.07119
\(581\) −32.8793 −1.36406
\(582\) 3.37108 0.139736
\(583\) 10.5440 0.436686
\(584\) −8.29482 −0.343242
\(585\) 25.2326 1.04324
\(586\) −20.5399 −0.848494
\(587\) −1.12752 −0.0465377 −0.0232688 0.999729i \(-0.507407\pi\)
−0.0232688 + 0.999729i \(0.507407\pi\)
\(588\) 5.64886 0.232955
\(589\) 52.1292 2.14795
\(590\) −3.77187 −0.155285
\(591\) −0.844390 −0.0347336
\(592\) −9.72230 −0.399584
\(593\) 37.5248 1.54096 0.770479 0.637465i \(-0.220017\pi\)
0.770479 + 0.637465i \(0.220017\pi\)
\(594\) 4.46430 0.183172
\(595\) −60.9417 −2.49837
\(596\) 10.4390 0.427600
\(597\) −2.61120 −0.106869
\(598\) −8.39121 −0.343142
\(599\) 24.8692 1.01613 0.508065 0.861319i \(-0.330361\pi\)
0.508065 + 0.861319i \(0.330361\pi\)
\(600\) 10.8650 0.443561
\(601\) −17.5955 −0.717734 −0.358867 0.933389i \(-0.616837\pi\)
−0.358867 + 0.933389i \(0.616837\pi\)
\(602\) −4.68165 −0.190810
\(603\) −10.4811 −0.426824
\(604\) −11.5544 −0.470142
\(605\) 4.22697 0.171851
\(606\) 5.85474 0.237832
\(607\) −39.8949 −1.61929 −0.809643 0.586923i \(-0.800339\pi\)
−0.809643 + 0.586923i \(0.800339\pi\)
\(608\) 6.44454 0.261360
\(609\) −19.0675 −0.772652
\(610\) 37.0391 1.49967
\(611\) −10.7511 −0.434942
\(612\) −8.91155 −0.360228
\(613\) −44.3957 −1.79313 −0.896563 0.442916i \(-0.853944\pi\)
−0.896563 + 0.442916i \(0.853944\pi\)
\(614\) −24.0190 −0.969328
\(615\) 19.7764 0.797462
\(616\) 3.69998 0.149077
\(617\) −20.3606 −0.819689 −0.409844 0.912155i \(-0.634417\pi\)
−0.409844 + 0.912155i \(0.634417\pi\)
\(618\) −11.5816 −0.465879
\(619\) −2.32871 −0.0935987 −0.0467994 0.998904i \(-0.514902\pi\)
−0.0467994 + 0.998904i \(0.514902\pi\)
\(620\) 34.1915 1.37316
\(621\) −14.3520 −0.575925
\(622\) 14.8044 0.593602
\(623\) 60.1920 2.41154
\(624\) −2.20399 −0.0882301
\(625\) 76.2297 3.04919
\(626\) −29.6240 −1.18401
\(627\) −5.44170 −0.217321
\(628\) −22.4382 −0.895381
\(629\) −37.8839 −1.51053
\(630\) 35.7681 1.42503
\(631\) −6.46527 −0.257378 −0.128689 0.991685i \(-0.541077\pi\)
−0.128689 + 0.991685i \(0.541077\pi\)
\(632\) 13.7390 0.546508
\(633\) −1.00344 −0.0398830
\(634\) 6.88637 0.273493
\(635\) 32.6797 1.29685
\(636\) −8.90321 −0.353035
\(637\) −17.4616 −0.691853
\(638\) −6.10309 −0.241624
\(639\) −9.30471 −0.368089
\(640\) 4.22697 0.167086
\(641\) 25.9044 1.02316 0.511581 0.859235i \(-0.329060\pi\)
0.511581 + 0.859235i \(0.329060\pi\)
\(642\) 2.38270 0.0940378
\(643\) 37.4755 1.47789 0.738946 0.673765i \(-0.235324\pi\)
0.738946 + 0.673765i \(0.235324\pi\)
\(644\) −11.8948 −0.468722
\(645\) 4.51618 0.177824
\(646\) 25.1118 0.988010
\(647\) 42.4037 1.66706 0.833531 0.552472i \(-0.186315\pi\)
0.833531 + 0.552472i \(0.186315\pi\)
\(648\) 3.09141 0.121442
\(649\) 0.892335 0.0350272
\(650\) −33.5855 −1.31733
\(651\) −25.2716 −0.990471
\(652\) −24.3335 −0.952973
\(653\) 27.7564 1.08619 0.543096 0.839671i \(-0.317252\pi\)
0.543096 + 0.839671i \(0.317252\pi\)
\(654\) −8.60012 −0.336291
\(655\) 25.1175 0.981423
\(656\) 5.54084 0.216334
\(657\) 18.9703 0.740102
\(658\) −15.2400 −0.594118
\(659\) −5.38453 −0.209752 −0.104876 0.994485i \(-0.533445\pi\)
−0.104876 + 0.994485i \(0.533445\pi\)
\(660\) −3.56921 −0.138931
\(661\) −20.2828 −0.788909 −0.394454 0.918916i \(-0.629066\pi\)
−0.394454 + 0.918916i \(0.629066\pi\)
\(662\) −4.84184 −0.188183
\(663\) −8.58806 −0.333533
\(664\) 8.88633 0.344856
\(665\) −100.791 −3.90849
\(666\) 22.2349 0.861586
\(667\) 19.6204 0.759706
\(668\) −7.53899 −0.291692
\(669\) 18.0933 0.699527
\(670\) 19.3718 0.748397
\(671\) −8.76256 −0.338275
\(672\) −3.12423 −0.120520
\(673\) 26.7211 1.03002 0.515011 0.857184i \(-0.327788\pi\)
0.515011 + 0.857184i \(0.327788\pi\)
\(674\) −27.4794 −1.05847
\(675\) −57.4432 −2.21099
\(676\) −6.18711 −0.237966
\(677\) 44.5073 1.71056 0.855278 0.518170i \(-0.173386\pi\)
0.855278 + 0.518170i \(0.173386\pi\)
\(678\) −0.460976 −0.0177037
\(679\) −14.7715 −0.566879
\(680\) 16.4708 0.631627
\(681\) −0.284607 −0.0109062
\(682\) −8.08890 −0.309740
\(683\) 37.6162 1.43934 0.719672 0.694314i \(-0.244292\pi\)
0.719672 + 0.694314i \(0.244292\pi\)
\(684\) −14.7387 −0.563548
\(685\) −90.5688 −3.46046
\(686\) 1.14747 0.0438106
\(687\) 16.0972 0.614145
\(688\) 1.26532 0.0482397
\(689\) 27.5213 1.04848
\(690\) 11.4744 0.436824
\(691\) −17.4575 −0.664113 −0.332057 0.943259i \(-0.607743\pi\)
−0.332057 + 0.943259i \(0.607743\pi\)
\(692\) 5.63897 0.214361
\(693\) −8.46188 −0.321440
\(694\) 7.78099 0.295362
\(695\) 50.3838 1.91117
\(696\) 5.15339 0.195339
\(697\) 21.5905 0.817797
\(698\) 20.7381 0.784948
\(699\) 5.54359 0.209678
\(700\) −47.6086 −1.79943
\(701\) 5.49542 0.207559 0.103780 0.994600i \(-0.466906\pi\)
0.103780 + 0.994600i \(0.466906\pi\)
\(702\) 11.6525 0.439795
\(703\) −62.6557 −2.36310
\(704\) −1.00000 −0.0376889
\(705\) 14.7014 0.553686
\(706\) −2.88777 −0.108683
\(707\) −25.6545 −0.964838
\(708\) −0.753479 −0.0283175
\(709\) −24.6020 −0.923947 −0.461974 0.886894i \(-0.652859\pi\)
−0.461974 + 0.886894i \(0.652859\pi\)
\(710\) 17.1975 0.645410
\(711\) −31.4211 −1.17838
\(712\) −16.2682 −0.609676
\(713\) 26.0045 0.973876
\(714\) −12.1739 −0.455596
\(715\) 11.0330 0.412612
\(716\) 9.90420 0.370137
\(717\) 0.691199 0.0258133
\(718\) 15.2377 0.568664
\(719\) −38.5150 −1.43637 −0.718184 0.695853i \(-0.755026\pi\)
−0.718184 + 0.695853i \(0.755026\pi\)
\(720\) −9.66709 −0.360271
\(721\) 50.7485 1.88997
\(722\) 22.5320 0.838556
\(723\) 6.75488 0.251217
\(724\) −19.7201 −0.732892
\(725\) 78.5299 2.91653
\(726\) 0.844390 0.0313383
\(727\) −14.0857 −0.522411 −0.261205 0.965283i \(-0.584120\pi\)
−0.261205 + 0.965283i \(0.584120\pi\)
\(728\) 9.65751 0.357931
\(729\) 4.23876 0.156991
\(730\) −35.0619 −1.29770
\(731\) 4.93043 0.182359
\(732\) 7.39902 0.273476
\(733\) −13.4321 −0.496126 −0.248063 0.968744i \(-0.579794\pi\)
−0.248063 + 0.968744i \(0.579794\pi\)
\(734\) 16.7576 0.618533
\(735\) 23.8776 0.880737
\(736\) 3.21484 0.118500
\(737\) −4.58290 −0.168813
\(738\) −12.6719 −0.466460
\(739\) −32.1784 −1.18370 −0.591850 0.806048i \(-0.701602\pi\)
−0.591850 + 0.806048i \(0.701602\pi\)
\(740\) −41.0958 −1.51071
\(741\) −14.2037 −0.521785
\(742\) 39.0124 1.43219
\(743\) 7.65039 0.280666 0.140333 0.990104i \(-0.455183\pi\)
0.140333 + 0.990104i \(0.455183\pi\)
\(744\) 6.83019 0.250407
\(745\) 44.1254 1.61663
\(746\) −35.4758 −1.29886
\(747\) −20.3231 −0.743583
\(748\) −3.89660 −0.142474
\(749\) −10.4406 −0.381492
\(750\) 28.0798 1.02533
\(751\) −35.3070 −1.28837 −0.644186 0.764869i \(-0.722804\pi\)
−0.644186 + 0.764869i \(0.722804\pi\)
\(752\) 4.11895 0.150202
\(753\) −9.51215 −0.346642
\(754\) −15.9300 −0.580136
\(755\) −48.8400 −1.77747
\(756\) 16.5178 0.600747
\(757\) 29.0049 1.05420 0.527100 0.849803i \(-0.323279\pi\)
0.527100 + 0.849803i \(0.323279\pi\)
\(758\) 2.08275 0.0756489
\(759\) −2.71458 −0.0985329
\(760\) 27.2408 0.988129
\(761\) 14.1566 0.513176 0.256588 0.966521i \(-0.417402\pi\)
0.256588 + 0.966521i \(0.417402\pi\)
\(762\) 6.52818 0.236491
\(763\) 37.6843 1.36426
\(764\) −12.0678 −0.436598
\(765\) −37.6688 −1.36192
\(766\) −3.98661 −0.144042
\(767\) 2.32913 0.0841000
\(768\) 0.844390 0.0304693
\(769\) 19.1400 0.690207 0.345103 0.938565i \(-0.387844\pi\)
0.345103 + 0.938565i \(0.387844\pi\)
\(770\) 15.6397 0.563616
\(771\) −4.32117 −0.155623
\(772\) −7.02489 −0.252831
\(773\) −26.7827 −0.963307 −0.481654 0.876362i \(-0.659964\pi\)
−0.481654 + 0.876362i \(0.659964\pi\)
\(774\) −2.89378 −0.104015
\(775\) 104.082 3.73873
\(776\) 3.99233 0.143316
\(777\) 30.3747 1.08969
\(778\) 30.6751 1.09975
\(779\) 35.7082 1.27938
\(780\) −9.31618 −0.333573
\(781\) −4.06852 −0.145583
\(782\) 12.5269 0.447962
\(783\) −27.2460 −0.973693
\(784\) 6.68987 0.238924
\(785\) −94.8454 −3.38518
\(786\) 5.01755 0.178970
\(787\) 14.7535 0.525906 0.262953 0.964809i \(-0.415304\pi\)
0.262953 + 0.964809i \(0.415304\pi\)
\(788\) −1.00000 −0.0356235
\(789\) 1.68153 0.0598639
\(790\) 58.0742 2.06619
\(791\) 2.01992 0.0718202
\(792\) 2.28700 0.0812652
\(793\) −22.8716 −0.812195
\(794\) 28.2667 1.00315
\(795\) −37.6336 −1.33473
\(796\) −3.09241 −0.109608
\(797\) −33.8080 −1.19754 −0.598771 0.800920i \(-0.704344\pi\)
−0.598771 + 0.800920i \(0.704344\pi\)
\(798\) −20.1342 −0.712743
\(799\) 16.0499 0.567804
\(800\) 12.8672 0.454926
\(801\) 37.2054 1.31459
\(802\) −7.18828 −0.253827
\(803\) 8.29482 0.292718
\(804\) 3.86976 0.136476
\(805\) −50.2791 −1.77210
\(806\) −21.1132 −0.743682
\(807\) 3.23581 0.113906
\(808\) 6.93369 0.243926
\(809\) 15.5360 0.546218 0.273109 0.961983i \(-0.411948\pi\)
0.273109 + 0.961983i \(0.411948\pi\)
\(810\) 13.0673 0.459137
\(811\) 5.00745 0.175835 0.0879177 0.996128i \(-0.471979\pi\)
0.0879177 + 0.996128i \(0.471979\pi\)
\(812\) −22.5813 −0.792449
\(813\) −10.5392 −0.369626
\(814\) 9.72230 0.340766
\(815\) −102.857 −3.60292
\(816\) 3.29025 0.115182
\(817\) 8.15438 0.285286
\(818\) −27.2450 −0.952599
\(819\) −22.0868 −0.771775
\(820\) 23.4210 0.817895
\(821\) −7.50470 −0.261916 −0.130958 0.991388i \(-0.541805\pi\)
−0.130958 + 0.991388i \(0.541805\pi\)
\(822\) −18.0923 −0.631040
\(823\) 20.6529 0.719913 0.359957 0.932969i \(-0.382791\pi\)
0.359957 + 0.932969i \(0.382791\pi\)
\(824\) −13.7159 −0.477816
\(825\) −10.8650 −0.378270
\(826\) 3.30162 0.114878
\(827\) 34.3811 1.19555 0.597775 0.801664i \(-0.296052\pi\)
0.597775 + 0.801664i \(0.296052\pi\)
\(828\) −7.35235 −0.255512
\(829\) −48.8203 −1.69560 −0.847800 0.530316i \(-0.822073\pi\)
−0.847800 + 0.530316i \(0.822073\pi\)
\(830\) 37.5622 1.30380
\(831\) −4.15476 −0.144127
\(832\) −2.61015 −0.0904907
\(833\) 26.0678 0.903195
\(834\) 10.0648 0.348516
\(835\) −31.8671 −1.10281
\(836\) −6.44454 −0.222889
\(837\) −36.1112 −1.24819
\(838\) −14.4906 −0.500571
\(839\) −31.4609 −1.08615 −0.543076 0.839684i \(-0.682740\pi\)
−0.543076 + 0.839684i \(0.682740\pi\)
\(840\) −13.2060 −0.455651
\(841\) 8.24771 0.284404
\(842\) −34.0129 −1.17216
\(843\) 26.4171 0.909854
\(844\) −1.18835 −0.0409049
\(845\) −26.1527 −0.899680
\(846\) −9.42005 −0.323868
\(847\) −3.69998 −0.127133
\(848\) −10.5440 −0.362081
\(849\) −18.9216 −0.649388
\(850\) 50.1385 1.71974
\(851\) −31.2556 −1.07143
\(852\) 3.43541 0.117695
\(853\) −26.9406 −0.922430 −0.461215 0.887288i \(-0.652586\pi\)
−0.461215 + 0.887288i \(0.652586\pi\)
\(854\) −32.4213 −1.10944
\(855\) −62.2999 −2.13061
\(856\) 2.82180 0.0964473
\(857\) 14.8118 0.505963 0.252981 0.967471i \(-0.418589\pi\)
0.252981 + 0.967471i \(0.418589\pi\)
\(858\) 2.20399 0.0752429
\(859\) 27.8216 0.949259 0.474630 0.880186i \(-0.342582\pi\)
0.474630 + 0.880186i \(0.342582\pi\)
\(860\) 5.34845 0.182381
\(861\) −17.3109 −0.589953
\(862\) −6.65099 −0.226533
\(863\) −20.8280 −0.708992 −0.354496 0.935058i \(-0.615348\pi\)
−0.354496 + 0.935058i \(0.615348\pi\)
\(864\) −4.46430 −0.151878
\(865\) 23.8357 0.810439
\(866\) −39.0712 −1.32769
\(867\) −1.53383 −0.0520915
\(868\) −29.9288 −1.01585
\(869\) −13.7390 −0.466063
\(870\) 21.7832 0.738520
\(871\) −11.9621 −0.405319
\(872\) −10.1850 −0.344908
\(873\) −9.13047 −0.309020
\(874\) 20.7181 0.700801
\(875\) −123.041 −4.15956
\(876\) −7.00407 −0.236646
\(877\) −33.8906 −1.14440 −0.572202 0.820113i \(-0.693911\pi\)
−0.572202 + 0.820113i \(0.693911\pi\)
\(878\) −12.9506 −0.437060
\(879\) −17.3437 −0.584988
\(880\) −4.22697 −0.142491
\(881\) 5.48780 0.184889 0.0924443 0.995718i \(-0.470532\pi\)
0.0924443 + 0.995718i \(0.470532\pi\)
\(882\) −15.2998 −0.515170
\(883\) −39.5901 −1.33231 −0.666157 0.745812i \(-0.732062\pi\)
−0.666157 + 0.745812i \(0.732062\pi\)
\(884\) −10.1707 −0.342078
\(885\) −3.18493 −0.107060
\(886\) −13.3359 −0.448028
\(887\) 32.9813 1.10740 0.553701 0.832715i \(-0.313215\pi\)
0.553701 + 0.832715i \(0.313215\pi\)
\(888\) −8.20941 −0.275490
\(889\) −28.6054 −0.959396
\(890\) −68.7651 −2.30501
\(891\) −3.09141 −0.103566
\(892\) 21.4276 0.717450
\(893\) 26.5447 0.888284
\(894\) 8.81462 0.294805
\(895\) 41.8647 1.39938
\(896\) −3.69998 −0.123608
\(897\) −7.08546 −0.236577
\(898\) 3.42835 0.114405
\(899\) 49.3673 1.64649
\(900\) −29.4274 −0.980915
\(901\) −41.0856 −1.36876
\(902\) −5.54084 −0.184490
\(903\) −3.95314 −0.131552
\(904\) −0.545928 −0.0181573
\(905\) −83.3562 −2.77086
\(906\) −9.75642 −0.324135
\(907\) −3.67368 −0.121982 −0.0609912 0.998138i \(-0.519426\pi\)
−0.0609912 + 0.998138i \(0.519426\pi\)
\(908\) −0.337056 −0.0111856
\(909\) −15.8574 −0.525956
\(910\) 40.8220 1.35324
\(911\) −6.56180 −0.217402 −0.108701 0.994074i \(-0.534669\pi\)
−0.108701 + 0.994074i \(0.534669\pi\)
\(912\) 5.44170 0.180193
\(913\) −8.88633 −0.294095
\(914\) −4.61725 −0.152725
\(915\) 31.2754 1.03393
\(916\) 19.0636 0.629881
\(917\) −21.9861 −0.726044
\(918\) −17.3956 −0.574140
\(919\) −18.4078 −0.607217 −0.303608 0.952797i \(-0.598191\pi\)
−0.303608 + 0.952797i \(0.598191\pi\)
\(920\) 13.5890 0.448016
\(921\) −20.2814 −0.668295
\(922\) −13.8104 −0.454822
\(923\) −10.6194 −0.349543
\(924\) 3.12423 0.102780
\(925\) −125.099 −4.11323
\(926\) 29.0450 0.954479
\(927\) 31.3683 1.03027
\(928\) 6.10309 0.200344
\(929\) −27.4128 −0.899385 −0.449692 0.893183i \(-0.648466\pi\)
−0.449692 + 0.893183i \(0.648466\pi\)
\(930\) 28.8710 0.946716
\(931\) 43.1131 1.41298
\(932\) 6.56520 0.215050
\(933\) 12.5007 0.409254
\(934\) −1.59622 −0.0522299
\(935\) −16.4708 −0.538653
\(936\) 5.96943 0.195117
\(937\) 35.3582 1.15510 0.577551 0.816355i \(-0.304009\pi\)
0.577551 + 0.816355i \(0.304009\pi\)
\(938\) −16.9567 −0.553655
\(939\) −25.0142 −0.816309
\(940\) 17.4106 0.567873
\(941\) 15.7031 0.511907 0.255954 0.966689i \(-0.417611\pi\)
0.255954 + 0.966689i \(0.417611\pi\)
\(942\) −18.9466 −0.617313
\(943\) 17.8129 0.580068
\(944\) −0.892335 −0.0290430
\(945\) 69.8203 2.27125
\(946\) −1.26532 −0.0411390
\(947\) −25.7117 −0.835518 −0.417759 0.908558i \(-0.637184\pi\)
−0.417759 + 0.908558i \(0.637184\pi\)
\(948\) 11.6011 0.376785
\(949\) 21.6507 0.702813
\(950\) 82.9234 2.69039
\(951\) 5.81478 0.188557
\(952\) −14.4174 −0.467269
\(953\) −4.10747 −0.133054 −0.0665270 0.997785i \(-0.521192\pi\)
−0.0665270 + 0.997785i \(0.521192\pi\)
\(954\) 24.1141 0.780722
\(955\) −51.0102 −1.65065
\(956\) 0.818578 0.0264747
\(957\) −5.15339 −0.166585
\(958\) −24.1500 −0.780251
\(959\) 79.2774 2.56000
\(960\) 3.56921 0.115196
\(961\) 34.4303 1.11065
\(962\) 25.3767 0.818177
\(963\) −6.45348 −0.207960
\(964\) 7.99971 0.257653
\(965\) −29.6940 −0.955883
\(966\) −10.0439 −0.323157
\(967\) −2.07183 −0.0666257 −0.0333128 0.999445i \(-0.510606\pi\)
−0.0333128 + 0.999445i \(0.510606\pi\)
\(968\) 1.00000 0.0321412
\(969\) 21.2042 0.681176
\(970\) 16.8754 0.541837
\(971\) −16.2399 −0.521162 −0.260581 0.965452i \(-0.583914\pi\)
−0.260581 + 0.965452i \(0.583914\pi\)
\(972\) 16.0032 0.513304
\(973\) −44.1024 −1.41386
\(974\) 40.0418 1.28302
\(975\) −28.3592 −0.908222
\(976\) 8.76256 0.280483
\(977\) 55.5862 1.77836 0.889181 0.457556i \(-0.151275\pi\)
0.889181 + 0.457556i \(0.151275\pi\)
\(978\) −20.5470 −0.657019
\(979\) 16.2682 0.519934
\(980\) 28.2779 0.903303
\(981\) 23.2931 0.743693
\(982\) −38.6205 −1.23243
\(983\) −12.0362 −0.383896 −0.191948 0.981405i \(-0.561480\pi\)
−0.191948 + 0.981405i \(0.561480\pi\)
\(984\) 4.67863 0.149149
\(985\) −4.22697 −0.134682
\(986\) 23.7813 0.757351
\(987\) −12.8685 −0.409610
\(988\) −16.8212 −0.535154
\(989\) 4.06779 0.129348
\(990\) 9.66709 0.307240
\(991\) 51.6586 1.64099 0.820495 0.571654i \(-0.193698\pi\)
0.820495 + 0.571654i \(0.193698\pi\)
\(992\) 8.08890 0.256823
\(993\) −4.08840 −0.129742
\(994\) −15.0534 −0.477466
\(995\) −13.0715 −0.414395
\(996\) 7.50353 0.237758
\(997\) −28.4177 −0.899998 −0.449999 0.893029i \(-0.648576\pi\)
−0.449999 + 0.893029i \(0.648576\pi\)
\(998\) 35.1589 1.11294
\(999\) 43.4032 1.37322
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 4334.2.a.f.1.14 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4334.2.a.f.1.14 24 1.1 even 1 trivial