Properties

Label 8015.2.a.m.1.8
Level $8015$
Weight $2$
Character 8015.1
Self dual yes
Analytic conductor $64.000$
Analytic rank $0$
Dimension $67$
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] = [8015,2,Mod(1,8015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8015, 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("8015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8015 = 5 \cdot 7 \cdot 229 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8015.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: \(64.0000972201\)
Analytic rank: \(0\)
Dimension: \(67\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 8015.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.30662 q^{2} +1.34001 q^{3} +3.32049 q^{4} +1.00000 q^{5} -3.09089 q^{6} -1.00000 q^{7} -3.04587 q^{8} -1.20437 q^{9} +O(q^{10})\) \(q-2.30662 q^{2} +1.34001 q^{3} +3.32049 q^{4} +1.00000 q^{5} -3.09089 q^{6} -1.00000 q^{7} -3.04587 q^{8} -1.20437 q^{9} -2.30662 q^{10} +2.13364 q^{11} +4.44949 q^{12} -5.11222 q^{13} +2.30662 q^{14} +1.34001 q^{15} +0.384674 q^{16} +6.69278 q^{17} +2.77803 q^{18} +1.81879 q^{19} +3.32049 q^{20} -1.34001 q^{21} -4.92149 q^{22} -3.98265 q^{23} -4.08149 q^{24} +1.00000 q^{25} +11.7919 q^{26} -5.63390 q^{27} -3.32049 q^{28} -1.75406 q^{29} -3.09089 q^{30} -2.92661 q^{31} +5.20444 q^{32} +2.85910 q^{33} -15.4377 q^{34} -1.00000 q^{35} -3.99910 q^{36} -9.15383 q^{37} -4.19525 q^{38} -6.85043 q^{39} -3.04587 q^{40} +7.82062 q^{41} +3.09089 q^{42} +2.14000 q^{43} +7.08473 q^{44} -1.20437 q^{45} +9.18646 q^{46} +8.57240 q^{47} +0.515467 q^{48} +1.00000 q^{49} -2.30662 q^{50} +8.96840 q^{51} -16.9751 q^{52} +1.87390 q^{53} +12.9953 q^{54} +2.13364 q^{55} +3.04587 q^{56} +2.43720 q^{57} +4.04595 q^{58} -2.65256 q^{59} +4.44949 q^{60} +9.50037 q^{61} +6.75058 q^{62} +1.20437 q^{63} -12.7740 q^{64} -5.11222 q^{65} -6.59485 q^{66} -6.97130 q^{67} +22.2233 q^{68} -5.33680 q^{69} +2.30662 q^{70} +0.142255 q^{71} +3.66836 q^{72} +10.1640 q^{73} +21.1144 q^{74} +1.34001 q^{75} +6.03927 q^{76} -2.13364 q^{77} +15.8013 q^{78} +4.48655 q^{79} +0.384674 q^{80} -3.93637 q^{81} -18.0392 q^{82} -15.2528 q^{83} -4.44949 q^{84} +6.69278 q^{85} -4.93617 q^{86} -2.35046 q^{87} -6.49878 q^{88} +11.1872 q^{89} +2.77803 q^{90} +5.11222 q^{91} -13.2244 q^{92} -3.92169 q^{93} -19.7733 q^{94} +1.81879 q^{95} +6.97400 q^{96} -2.15068 q^{97} -2.30662 q^{98} -2.56970 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 67 q + 3 q^{2} + 73 q^{4} + 67 q^{5} + 17 q^{6} - 67 q^{7} + 12 q^{8} + 97 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 67 q + 3 q^{2} + 73 q^{4} + 67 q^{5} + 17 q^{6} - 67 q^{7} + 12 q^{8} + 97 q^{9} + 3 q^{10} + 11 q^{11} + 9 q^{12} + 19 q^{13} - 3 q^{14} + 93 q^{16} + 7 q^{17} + 9 q^{18} + 36 q^{19} + 73 q^{20} - 8 q^{22} - 2 q^{23} + 37 q^{24} + 67 q^{25} + 39 q^{26} + 6 q^{27} - 73 q^{28} + 56 q^{29} + 17 q^{30} + 63 q^{31} + 27 q^{32} + 51 q^{33} + 55 q^{34} - 67 q^{35} + 148 q^{36} + 28 q^{37} + 10 q^{38} + 7 q^{39} + 12 q^{40} + 84 q^{41} - 17 q^{42} - 11 q^{43} + 41 q^{44} + 97 q^{45} + 17 q^{46} - 10 q^{47} + 10 q^{48} + 67 q^{49} + 3 q^{50} - q^{51} + 49 q^{52} + 3 q^{53} + 20 q^{54} + 11 q^{55} - 12 q^{56} + 45 q^{57} + 22 q^{58} + 80 q^{59} + 9 q^{60} + 64 q^{61} - 37 q^{62} - 97 q^{63} + 110 q^{64} + 19 q^{65} + 75 q^{66} + 22 q^{67} + 7 q^{68} + 107 q^{69} - 3 q^{70} + 24 q^{71} + 72 q^{72} + 83 q^{73} + 52 q^{74} + 115 q^{76} - 11 q^{77} - 70 q^{78} - 32 q^{79} + 93 q^{80} + 183 q^{81} + 56 q^{82} - 58 q^{83} - 9 q^{84} + 7 q^{85} + 51 q^{86} + 20 q^{87} - 5 q^{88} + 129 q^{89} + 9 q^{90} - 19 q^{91} - 37 q^{92} + 33 q^{93} + 89 q^{94} + 36 q^{95} + 129 q^{96} + 126 q^{97} + 3 q^{98} - 27 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.30662 −1.63103 −0.815513 0.578739i \(-0.803545\pi\)
−0.815513 + 0.578739i \(0.803545\pi\)
\(3\) 1.34001 0.773655 0.386828 0.922152i \(-0.373571\pi\)
0.386828 + 0.922152i \(0.373571\pi\)
\(4\) 3.32049 1.66025
\(5\) 1.00000 0.447214
\(6\) −3.09089 −1.26185
\(7\) −1.00000 −0.377964
\(8\) −3.04587 −1.07688
\(9\) −1.20437 −0.401457
\(10\) −2.30662 −0.729417
\(11\) 2.13364 0.643317 0.321658 0.946856i \(-0.395760\pi\)
0.321658 + 0.946856i \(0.395760\pi\)
\(12\) 4.44949 1.28446
\(13\) −5.11222 −1.41788 −0.708938 0.705271i \(-0.750825\pi\)
−0.708938 + 0.705271i \(0.750825\pi\)
\(14\) 2.30662 0.616470
\(15\) 1.34001 0.345989
\(16\) 0.384674 0.0961685
\(17\) 6.69278 1.62324 0.811619 0.584187i \(-0.198587\pi\)
0.811619 + 0.584187i \(0.198587\pi\)
\(18\) 2.77803 0.654787
\(19\) 1.81879 0.417259 0.208629 0.977995i \(-0.433100\pi\)
0.208629 + 0.977995i \(0.433100\pi\)
\(20\) 3.32049 0.742484
\(21\) −1.34001 −0.292414
\(22\) −4.92149 −1.04927
\(23\) −3.98265 −0.830441 −0.415220 0.909721i \(-0.636296\pi\)
−0.415220 + 0.909721i \(0.636296\pi\)
\(24\) −4.08149 −0.833131
\(25\) 1.00000 0.200000
\(26\) 11.7919 2.31259
\(27\) −5.63390 −1.08425
\(28\) −3.32049 −0.627514
\(29\) −1.75406 −0.325721 −0.162860 0.986649i \(-0.552072\pi\)
−0.162860 + 0.986649i \(0.552072\pi\)
\(30\) −3.09089 −0.564317
\(31\) −2.92661 −0.525635 −0.262818 0.964846i \(-0.584652\pi\)
−0.262818 + 0.964846i \(0.584652\pi\)
\(32\) 5.20444 0.920023
\(33\) 2.85910 0.497705
\(34\) −15.4377 −2.64754
\(35\) −1.00000 −0.169031
\(36\) −3.99910 −0.666517
\(37\) −9.15383 −1.50488 −0.752441 0.658660i \(-0.771124\pi\)
−0.752441 + 0.658660i \(0.771124\pi\)
\(38\) −4.19525 −0.680560
\(39\) −6.85043 −1.09695
\(40\) −3.04587 −0.481594
\(41\) 7.82062 1.22138 0.610688 0.791871i \(-0.290893\pi\)
0.610688 + 0.791871i \(0.290893\pi\)
\(42\) 3.09089 0.476935
\(43\) 2.14000 0.326347 0.163174 0.986597i \(-0.447827\pi\)
0.163174 + 0.986597i \(0.447827\pi\)
\(44\) 7.08473 1.06806
\(45\) −1.20437 −0.179537
\(46\) 9.18646 1.35447
\(47\) 8.57240 1.25041 0.625207 0.780459i \(-0.285015\pi\)
0.625207 + 0.780459i \(0.285015\pi\)
\(48\) 0.515467 0.0744013
\(49\) 1.00000 0.142857
\(50\) −2.30662 −0.326205
\(51\) 8.96840 1.25583
\(52\) −16.9751 −2.35402
\(53\) 1.87390 0.257400 0.128700 0.991684i \(-0.458920\pi\)
0.128700 + 0.991684i \(0.458920\pi\)
\(54\) 12.9953 1.76843
\(55\) 2.13364 0.287700
\(56\) 3.04587 0.407021
\(57\) 2.43720 0.322815
\(58\) 4.04595 0.531259
\(59\) −2.65256 −0.345334 −0.172667 0.984980i \(-0.555238\pi\)
−0.172667 + 0.984980i \(0.555238\pi\)
\(60\) 4.44949 0.574427
\(61\) 9.50037 1.21640 0.608199 0.793785i \(-0.291892\pi\)
0.608199 + 0.793785i \(0.291892\pi\)
\(62\) 6.75058 0.857324
\(63\) 1.20437 0.151737
\(64\) −12.7740 −1.59675
\(65\) −5.11222 −0.634093
\(66\) −6.59485 −0.811770
\(67\) −6.97130 −0.851680 −0.425840 0.904798i \(-0.640021\pi\)
−0.425840 + 0.904798i \(0.640021\pi\)
\(68\) 22.2233 2.69497
\(69\) −5.33680 −0.642475
\(70\) 2.30662 0.275694
\(71\) 0.142255 0.0168825 0.00844125 0.999964i \(-0.497313\pi\)
0.00844125 + 0.999964i \(0.497313\pi\)
\(72\) 3.66836 0.432320
\(73\) 10.1640 1.18960 0.594802 0.803872i \(-0.297230\pi\)
0.594802 + 0.803872i \(0.297230\pi\)
\(74\) 21.1144 2.45450
\(75\) 1.34001 0.154731
\(76\) 6.03927 0.692752
\(77\) −2.13364 −0.243151
\(78\) 15.8013 1.78915
\(79\) 4.48655 0.504777 0.252388 0.967626i \(-0.418784\pi\)
0.252388 + 0.967626i \(0.418784\pi\)
\(80\) 0.384674 0.0430079
\(81\) −3.93637 −0.437375
\(82\) −18.0392 −1.99210
\(83\) −15.2528 −1.67421 −0.837106 0.547041i \(-0.815754\pi\)
−0.837106 + 0.547041i \(0.815754\pi\)
\(84\) −4.44949 −0.485479
\(85\) 6.69278 0.725934
\(86\) −4.93617 −0.532281
\(87\) −2.35046 −0.251996
\(88\) −6.49878 −0.692773
\(89\) 11.1872 1.18584 0.592922 0.805260i \(-0.297974\pi\)
0.592922 + 0.805260i \(0.297974\pi\)
\(90\) 2.77803 0.292830
\(91\) 5.11222 0.535907
\(92\) −13.2244 −1.37874
\(93\) −3.92169 −0.406660
\(94\) −19.7733 −2.03946
\(95\) 1.81879 0.186604
\(96\) 6.97400 0.711781
\(97\) −2.15068 −0.218368 −0.109184 0.994022i \(-0.534824\pi\)
−0.109184 + 0.994022i \(0.534824\pi\)
\(98\) −2.30662 −0.233004
\(99\) −2.56970 −0.258264
\(100\) 3.32049 0.332049
\(101\) −7.98830 −0.794866 −0.397433 0.917631i \(-0.630099\pi\)
−0.397433 + 0.917631i \(0.630099\pi\)
\(102\) −20.6867 −2.04829
\(103\) −10.2062 −1.00564 −0.502821 0.864391i \(-0.667705\pi\)
−0.502821 + 0.864391i \(0.667705\pi\)
\(104\) 15.5712 1.52688
\(105\) −1.34001 −0.130772
\(106\) −4.32238 −0.419827
\(107\) 18.0542 1.74536 0.872681 0.488290i \(-0.162379\pi\)
0.872681 + 0.488290i \(0.162379\pi\)
\(108\) −18.7073 −1.80011
\(109\) 1.81976 0.174302 0.0871509 0.996195i \(-0.472224\pi\)
0.0871509 + 0.996195i \(0.472224\pi\)
\(110\) −4.92149 −0.469246
\(111\) −12.2662 −1.16426
\(112\) −0.384674 −0.0363483
\(113\) 5.19513 0.488716 0.244358 0.969685i \(-0.421423\pi\)
0.244358 + 0.969685i \(0.421423\pi\)
\(114\) −5.62168 −0.526519
\(115\) −3.98265 −0.371384
\(116\) −5.82434 −0.540776
\(117\) 6.15702 0.569216
\(118\) 6.11844 0.563248
\(119\) −6.69278 −0.613526
\(120\) −4.08149 −0.372588
\(121\) −6.44758 −0.586144
\(122\) −21.9137 −1.98398
\(123\) 10.4797 0.944924
\(124\) −9.71779 −0.872683
\(125\) 1.00000 0.0894427
\(126\) −2.77803 −0.247486
\(127\) −2.89657 −0.257029 −0.128514 0.991708i \(-0.541021\pi\)
−0.128514 + 0.991708i \(0.541021\pi\)
\(128\) 19.0559 1.68432
\(129\) 2.86763 0.252480
\(130\) 11.7919 1.03422
\(131\) −5.92357 −0.517545 −0.258773 0.965938i \(-0.583318\pi\)
−0.258773 + 0.965938i \(0.583318\pi\)
\(132\) 9.49361 0.826313
\(133\) −1.81879 −0.157709
\(134\) 16.0801 1.38911
\(135\) −5.63390 −0.484889
\(136\) −20.3853 −1.74803
\(137\) 12.8189 1.09519 0.547597 0.836742i \(-0.315543\pi\)
0.547597 + 0.836742i \(0.315543\pi\)
\(138\) 12.3100 1.04789
\(139\) 7.58899 0.643689 0.321845 0.946792i \(-0.395697\pi\)
0.321845 + 0.946792i \(0.395697\pi\)
\(140\) −3.32049 −0.280633
\(141\) 11.4871 0.967389
\(142\) −0.328127 −0.0275358
\(143\) −10.9076 −0.912143
\(144\) −0.463291 −0.0386075
\(145\) −1.75406 −0.145667
\(146\) −23.4444 −1.94028
\(147\) 1.34001 0.110522
\(148\) −30.3952 −2.49847
\(149\) 12.7889 1.04771 0.523856 0.851807i \(-0.324493\pi\)
0.523856 + 0.851807i \(0.324493\pi\)
\(150\) −3.09089 −0.252370
\(151\) −18.3320 −1.49184 −0.745919 0.666037i \(-0.767989\pi\)
−0.745919 + 0.666037i \(0.767989\pi\)
\(152\) −5.53979 −0.449336
\(153\) −8.06059 −0.651660
\(154\) 4.92149 0.396585
\(155\) −2.92661 −0.235071
\(156\) −22.7468 −1.82120
\(157\) 4.70588 0.375570 0.187785 0.982210i \(-0.439869\pi\)
0.187785 + 0.982210i \(0.439869\pi\)
\(158\) −10.3488 −0.823304
\(159\) 2.51105 0.199139
\(160\) 5.20444 0.411447
\(161\) 3.98265 0.313877
\(162\) 9.07971 0.713370
\(163\) 7.97144 0.624371 0.312186 0.950021i \(-0.398939\pi\)
0.312186 + 0.950021i \(0.398939\pi\)
\(164\) 25.9683 2.02778
\(165\) 2.85910 0.222581
\(166\) 35.1824 2.73068
\(167\) −4.51016 −0.349006 −0.174503 0.984657i \(-0.555832\pi\)
−0.174503 + 0.984657i \(0.555832\pi\)
\(168\) 4.08149 0.314894
\(169\) 13.1348 1.01037
\(170\) −15.4377 −1.18402
\(171\) −2.19050 −0.167512
\(172\) 7.10586 0.541817
\(173\) 2.43568 0.185182 0.0925908 0.995704i \(-0.470485\pi\)
0.0925908 + 0.995704i \(0.470485\pi\)
\(174\) 5.42161 0.411011
\(175\) −1.00000 −0.0755929
\(176\) 0.820756 0.0618668
\(177\) −3.55446 −0.267169
\(178\) −25.8047 −1.93414
\(179\) 21.0843 1.57592 0.787959 0.615728i \(-0.211138\pi\)
0.787959 + 0.615728i \(0.211138\pi\)
\(180\) −3.99910 −0.298076
\(181\) 16.3844 1.21785 0.608923 0.793229i \(-0.291602\pi\)
0.608923 + 0.793229i \(0.291602\pi\)
\(182\) −11.7919 −0.874077
\(183\) 12.7306 0.941073
\(184\) 12.1306 0.894282
\(185\) −9.15383 −0.673003
\(186\) 9.04585 0.663274
\(187\) 14.2800 1.04426
\(188\) 28.4646 2.07599
\(189\) 5.63390 0.409806
\(190\) −4.19525 −0.304356
\(191\) 19.3468 1.39989 0.699944 0.714198i \(-0.253208\pi\)
0.699944 + 0.714198i \(0.253208\pi\)
\(192\) −17.1173 −1.23533
\(193\) 8.93297 0.643009 0.321505 0.946908i \(-0.395811\pi\)
0.321505 + 0.946908i \(0.395811\pi\)
\(194\) 4.96080 0.356165
\(195\) −6.85043 −0.490570
\(196\) 3.32049 0.237178
\(197\) −14.8284 −1.05648 −0.528241 0.849095i \(-0.677148\pi\)
−0.528241 + 0.849095i \(0.677148\pi\)
\(198\) 5.92731 0.421235
\(199\) 0.193330 0.0137048 0.00685241 0.999977i \(-0.497819\pi\)
0.00685241 + 0.999977i \(0.497819\pi\)
\(200\) −3.04587 −0.215375
\(201\) −9.34162 −0.658907
\(202\) 18.4260 1.29645
\(203\) 1.75406 0.123111
\(204\) 29.7795 2.08498
\(205\) 7.82062 0.546216
\(206\) 23.5417 1.64023
\(207\) 4.79660 0.333386
\(208\) −1.96654 −0.136355
\(209\) 3.88064 0.268429
\(210\) 3.09089 0.213292
\(211\) 11.5861 0.797623 0.398811 0.917033i \(-0.369423\pi\)
0.398811 + 0.917033i \(0.369423\pi\)
\(212\) 6.22228 0.427348
\(213\) 0.190623 0.0130612
\(214\) −41.6441 −2.84673
\(215\) 2.14000 0.145947
\(216\) 17.1601 1.16760
\(217\) 2.92661 0.198671
\(218\) −4.19750 −0.284291
\(219\) 13.6199 0.920344
\(220\) 7.08473 0.477652
\(221\) −34.2150 −2.30155
\(222\) 28.2935 1.89894
\(223\) 8.31512 0.556822 0.278411 0.960462i \(-0.410192\pi\)
0.278411 + 0.960462i \(0.410192\pi\)
\(224\) −5.20444 −0.347736
\(225\) −1.20437 −0.0802914
\(226\) −11.9832 −0.797109
\(227\) 5.24230 0.347943 0.173972 0.984751i \(-0.444340\pi\)
0.173972 + 0.984751i \(0.444340\pi\)
\(228\) 8.09269 0.535951
\(229\) −1.00000 −0.0660819
\(230\) 9.18646 0.605738
\(231\) −2.85910 −0.188115
\(232\) 5.34263 0.350761
\(233\) −20.0824 −1.31564 −0.657820 0.753175i \(-0.728521\pi\)
−0.657820 + 0.753175i \(0.728521\pi\)
\(234\) −14.2019 −0.928407
\(235\) 8.57240 0.559202
\(236\) −8.80780 −0.573339
\(237\) 6.01203 0.390523
\(238\) 15.4377 1.00068
\(239\) 21.6093 1.39779 0.698894 0.715225i \(-0.253676\pi\)
0.698894 + 0.715225i \(0.253676\pi\)
\(240\) 0.515467 0.0332733
\(241\) 12.5819 0.810473 0.405237 0.914212i \(-0.367189\pi\)
0.405237 + 0.914212i \(0.367189\pi\)
\(242\) 14.8721 0.956016
\(243\) 11.6269 0.745868
\(244\) 31.5459 2.01952
\(245\) 1.00000 0.0638877
\(246\) −24.1727 −1.54120
\(247\) −9.29805 −0.591621
\(248\) 8.91407 0.566044
\(249\) −20.4389 −1.29526
\(250\) −2.30662 −0.145883
\(251\) −21.7549 −1.37316 −0.686578 0.727056i \(-0.740888\pi\)
−0.686578 + 0.727056i \(0.740888\pi\)
\(252\) 3.99910 0.251920
\(253\) −8.49755 −0.534236
\(254\) 6.68128 0.419221
\(255\) 8.96840 0.561623
\(256\) −18.4066 −1.15041
\(257\) −10.6124 −0.661984 −0.330992 0.943634i \(-0.607383\pi\)
−0.330992 + 0.943634i \(0.607383\pi\)
\(258\) −6.61452 −0.411802
\(259\) 9.15383 0.568792
\(260\) −16.9751 −1.05275
\(261\) 2.11254 0.130763
\(262\) 13.6634 0.844129
\(263\) 19.9740 1.23165 0.615824 0.787883i \(-0.288823\pi\)
0.615824 + 0.787883i \(0.288823\pi\)
\(264\) −8.70844 −0.535967
\(265\) 1.87390 0.115113
\(266\) 4.19525 0.257227
\(267\) 14.9910 0.917435
\(268\) −23.1481 −1.41400
\(269\) 18.1961 1.10943 0.554717 0.832039i \(-0.312827\pi\)
0.554717 + 0.832039i \(0.312827\pi\)
\(270\) 12.9953 0.790867
\(271\) 17.6524 1.07231 0.536155 0.844120i \(-0.319876\pi\)
0.536155 + 0.844120i \(0.319876\pi\)
\(272\) 2.57454 0.156104
\(273\) 6.85043 0.414607
\(274\) −29.5684 −1.78629
\(275\) 2.13364 0.128663
\(276\) −17.7208 −1.06667
\(277\) 21.8701 1.31405 0.657023 0.753870i \(-0.271815\pi\)
0.657023 + 0.753870i \(0.271815\pi\)
\(278\) −17.5049 −1.04987
\(279\) 3.52473 0.211020
\(280\) 3.04587 0.182025
\(281\) 11.8771 0.708530 0.354265 0.935145i \(-0.384731\pi\)
0.354265 + 0.935145i \(0.384731\pi\)
\(282\) −26.4964 −1.57784
\(283\) −23.7270 −1.41042 −0.705212 0.708997i \(-0.749148\pi\)
−0.705212 + 0.708997i \(0.749148\pi\)
\(284\) 0.472355 0.0280291
\(285\) 2.43720 0.144367
\(286\) 25.1598 1.48773
\(287\) −7.82062 −0.461637
\(288\) −6.26808 −0.369350
\(289\) 27.7933 1.63490
\(290\) 4.04595 0.237586
\(291\) −2.88193 −0.168942
\(292\) 33.7494 1.97504
\(293\) −31.7416 −1.85437 −0.927183 0.374609i \(-0.877777\pi\)
−0.927183 + 0.374609i \(0.877777\pi\)
\(294\) −3.09089 −0.180265
\(295\) −2.65256 −0.154438
\(296\) 27.8814 1.62057
\(297\) −12.0207 −0.697513
\(298\) −29.4992 −1.70884
\(299\) 20.3602 1.17746
\(300\) 4.44949 0.256892
\(301\) −2.14000 −0.123348
\(302\) 42.2849 2.43322
\(303\) −10.7044 −0.614952
\(304\) 0.699641 0.0401272
\(305\) 9.50037 0.543990
\(306\) 18.5927 1.06287
\(307\) 4.21455 0.240537 0.120269 0.992741i \(-0.461624\pi\)
0.120269 + 0.992741i \(0.461624\pi\)
\(308\) −7.08473 −0.403690
\(309\) −13.6764 −0.778021
\(310\) 6.75058 0.383407
\(311\) −12.3433 −0.699925 −0.349962 0.936764i \(-0.613806\pi\)
−0.349962 + 0.936764i \(0.613806\pi\)
\(312\) 20.8655 1.18128
\(313\) −4.35993 −0.246438 −0.123219 0.992380i \(-0.539322\pi\)
−0.123219 + 0.992380i \(0.539322\pi\)
\(314\) −10.8547 −0.612565
\(315\) 1.20437 0.0678587
\(316\) 14.8976 0.838053
\(317\) 11.3273 0.636204 0.318102 0.948056i \(-0.396954\pi\)
0.318102 + 0.948056i \(0.396954\pi\)
\(318\) −5.79203 −0.324801
\(319\) −3.74253 −0.209541
\(320\) −12.7740 −0.714088
\(321\) 24.1928 1.35031
\(322\) −9.18646 −0.511942
\(323\) 12.1728 0.677310
\(324\) −13.0707 −0.726149
\(325\) −5.11222 −0.283575
\(326\) −18.3871 −1.01837
\(327\) 2.43850 0.134850
\(328\) −23.8206 −1.31527
\(329\) −8.57240 −0.472612
\(330\) −6.59485 −0.363035
\(331\) 31.0058 1.70423 0.852117 0.523351i \(-0.175318\pi\)
0.852117 + 0.523351i \(0.175318\pi\)
\(332\) −50.6468 −2.77960
\(333\) 11.0246 0.604145
\(334\) 10.4032 0.569238
\(335\) −6.97130 −0.380883
\(336\) −0.515467 −0.0281210
\(337\) 21.4857 1.17040 0.585200 0.810889i \(-0.301016\pi\)
0.585200 + 0.810889i \(0.301016\pi\)
\(338\) −30.2970 −1.64794
\(339\) 6.96152 0.378098
\(340\) 22.2233 1.20523
\(341\) −6.24433 −0.338150
\(342\) 5.05264 0.273216
\(343\) −1.00000 −0.0539949
\(344\) −6.51816 −0.351436
\(345\) −5.33680 −0.287324
\(346\) −5.61819 −0.302036
\(347\) −30.3497 −1.62926 −0.814629 0.579982i \(-0.803059\pi\)
−0.814629 + 0.579982i \(0.803059\pi\)
\(348\) −7.80467 −0.418374
\(349\) −10.6029 −0.567558 −0.283779 0.958890i \(-0.591588\pi\)
−0.283779 + 0.958890i \(0.591588\pi\)
\(350\) 2.30662 0.123294
\(351\) 28.8018 1.53732
\(352\) 11.1044 0.591866
\(353\) 8.89537 0.473453 0.236726 0.971576i \(-0.423925\pi\)
0.236726 + 0.971576i \(0.423925\pi\)
\(354\) 8.19878 0.435760
\(355\) 0.142255 0.00755009
\(356\) 37.1471 1.96879
\(357\) −8.96840 −0.474658
\(358\) −48.6335 −2.57036
\(359\) 22.6885 1.19745 0.598725 0.800954i \(-0.295674\pi\)
0.598725 + 0.800954i \(0.295674\pi\)
\(360\) 3.66836 0.193339
\(361\) −15.6920 −0.825895
\(362\) −37.7927 −1.98634
\(363\) −8.63983 −0.453473
\(364\) 16.9751 0.889736
\(365\) 10.1640 0.532007
\(366\) −29.3646 −1.53491
\(367\) 4.27002 0.222893 0.111447 0.993770i \(-0.464452\pi\)
0.111447 + 0.993770i \(0.464452\pi\)
\(368\) −1.53202 −0.0798622
\(369\) −9.41894 −0.490330
\(370\) 21.1144 1.09769
\(371\) −1.87390 −0.0972882
\(372\) −13.0219 −0.675156
\(373\) −0.378616 −0.0196040 −0.00980199 0.999952i \(-0.503120\pi\)
−0.00980199 + 0.999952i \(0.503120\pi\)
\(374\) −32.9385 −1.70321
\(375\) 1.34001 0.0691978
\(376\) −26.1104 −1.34654
\(377\) 8.96714 0.461831
\(378\) −12.9953 −0.668404
\(379\) −6.03309 −0.309899 −0.154950 0.987922i \(-0.549522\pi\)
−0.154950 + 0.987922i \(0.549522\pi\)
\(380\) 6.03927 0.309808
\(381\) −3.88143 −0.198852
\(382\) −44.6257 −2.28325
\(383\) −24.4874 −1.25125 −0.625624 0.780124i \(-0.715156\pi\)
−0.625624 + 0.780124i \(0.715156\pi\)
\(384\) 25.5351 1.30308
\(385\) −2.13364 −0.108740
\(386\) −20.6050 −1.04876
\(387\) −2.57736 −0.131014
\(388\) −7.14131 −0.362545
\(389\) 9.71828 0.492736 0.246368 0.969176i \(-0.420763\pi\)
0.246368 + 0.969176i \(0.420763\pi\)
\(390\) 15.8013 0.800132
\(391\) −26.6550 −1.34800
\(392\) −3.04587 −0.153840
\(393\) −7.93765 −0.400402
\(394\) 34.2035 1.72315
\(395\) 4.48655 0.225743
\(396\) −8.53265 −0.428782
\(397\) −5.00564 −0.251226 −0.125613 0.992079i \(-0.540090\pi\)
−0.125613 + 0.992079i \(0.540090\pi\)
\(398\) −0.445939 −0.0223529
\(399\) −2.43720 −0.122012
\(400\) 0.384674 0.0192337
\(401\) −3.46530 −0.173049 −0.0865243 0.996250i \(-0.527576\pi\)
−0.0865243 + 0.996250i \(0.527576\pi\)
\(402\) 21.5476 1.07469
\(403\) 14.9615 0.745285
\(404\) −26.5251 −1.31967
\(405\) −3.93637 −0.195600
\(406\) −4.04595 −0.200797
\(407\) −19.5310 −0.968115
\(408\) −27.3165 −1.35237
\(409\) 25.1305 1.24262 0.621312 0.783563i \(-0.286600\pi\)
0.621312 + 0.783563i \(0.286600\pi\)
\(410\) −18.0392 −0.890892
\(411\) 17.1775 0.847303
\(412\) −33.8894 −1.66961
\(413\) 2.65256 0.130524
\(414\) −11.0639 −0.543762
\(415\) −15.2528 −0.748730
\(416\) −26.6062 −1.30448
\(417\) 10.1693 0.497994
\(418\) −8.95116 −0.437815
\(419\) −10.0820 −0.492537 −0.246268 0.969202i \(-0.579204\pi\)
−0.246268 + 0.969202i \(0.579204\pi\)
\(420\) −4.44949 −0.217113
\(421\) −32.3274 −1.57554 −0.787770 0.615969i \(-0.788764\pi\)
−0.787770 + 0.615969i \(0.788764\pi\)
\(422\) −26.7248 −1.30094
\(423\) −10.3244 −0.501988
\(424\) −5.70766 −0.277188
\(425\) 6.69278 0.324647
\(426\) −0.439694 −0.0213032
\(427\) −9.50037 −0.459755
\(428\) 59.9487 2.89773
\(429\) −14.6164 −0.705684
\(430\) −4.93617 −0.238043
\(431\) 2.37000 0.114159 0.0570794 0.998370i \(-0.481821\pi\)
0.0570794 + 0.998370i \(0.481821\pi\)
\(432\) −2.16722 −0.104270
\(433\) 37.6774 1.81066 0.905329 0.424710i \(-0.139624\pi\)
0.905329 + 0.424710i \(0.139624\pi\)
\(434\) −6.75058 −0.324038
\(435\) −2.35046 −0.112696
\(436\) 6.04251 0.289384
\(437\) −7.24361 −0.346509
\(438\) −31.4158 −1.50111
\(439\) 10.0246 0.478448 0.239224 0.970964i \(-0.423107\pi\)
0.239224 + 0.970964i \(0.423107\pi\)
\(440\) −6.49878 −0.309817
\(441\) −1.20437 −0.0573510
\(442\) 78.9209 3.75389
\(443\) −1.84452 −0.0876360 −0.0438180 0.999040i \(-0.513952\pi\)
−0.0438180 + 0.999040i \(0.513952\pi\)
\(444\) −40.7299 −1.93296
\(445\) 11.1872 0.530326
\(446\) −19.1798 −0.908190
\(447\) 17.1373 0.810568
\(448\) 12.7740 0.603515
\(449\) 6.87189 0.324304 0.162152 0.986766i \(-0.448156\pi\)
0.162152 + 0.986766i \(0.448156\pi\)
\(450\) 2.77803 0.130957
\(451\) 16.6864 0.785732
\(452\) 17.2504 0.811389
\(453\) −24.5651 −1.15417
\(454\) −12.0920 −0.567505
\(455\) 5.11222 0.239665
\(456\) −7.42338 −0.347631
\(457\) −3.39644 −0.158879 −0.0794395 0.996840i \(-0.525313\pi\)
−0.0794395 + 0.996840i \(0.525313\pi\)
\(458\) 2.30662 0.107781
\(459\) −37.7065 −1.75999
\(460\) −13.2244 −0.616589
\(461\) −1.36813 −0.0637203 −0.0318601 0.999492i \(-0.510143\pi\)
−0.0318601 + 0.999492i \(0.510143\pi\)
\(462\) 6.59485 0.306820
\(463\) 11.7058 0.544016 0.272008 0.962295i \(-0.412312\pi\)
0.272008 + 0.962295i \(0.412312\pi\)
\(464\) −0.674741 −0.0313241
\(465\) −3.92169 −0.181864
\(466\) 46.3224 2.14584
\(467\) 20.3654 0.942396 0.471198 0.882027i \(-0.343822\pi\)
0.471198 + 0.882027i \(0.343822\pi\)
\(468\) 20.4443 0.945039
\(469\) 6.97130 0.321905
\(470\) −19.7733 −0.912073
\(471\) 6.30593 0.290562
\(472\) 8.07934 0.371882
\(473\) 4.56599 0.209945
\(474\) −13.8675 −0.636953
\(475\) 1.81879 0.0834518
\(476\) −22.2233 −1.01860
\(477\) −2.25688 −0.103335
\(478\) −49.8444 −2.27983
\(479\) 6.59641 0.301398 0.150699 0.988580i \(-0.451848\pi\)
0.150699 + 0.988580i \(0.451848\pi\)
\(480\) 6.97400 0.318318
\(481\) 46.7964 2.13373
\(482\) −29.0217 −1.32190
\(483\) 5.33680 0.242833
\(484\) −21.4091 −0.973142
\(485\) −2.15068 −0.0976573
\(486\) −26.8189 −1.21653
\(487\) 32.0801 1.45369 0.726843 0.686803i \(-0.240987\pi\)
0.726843 + 0.686803i \(0.240987\pi\)
\(488\) −28.9369 −1.30991
\(489\) 10.6818 0.483048
\(490\) −2.30662 −0.104202
\(491\) −12.4351 −0.561187 −0.280594 0.959827i \(-0.590531\pi\)
−0.280594 + 0.959827i \(0.590531\pi\)
\(492\) 34.7978 1.56881
\(493\) −11.7395 −0.528722
\(494\) 21.4471 0.964949
\(495\) −2.56970 −0.115499
\(496\) −1.12579 −0.0505495
\(497\) −0.142255 −0.00638099
\(498\) 47.1448 2.11261
\(499\) −8.41527 −0.376719 −0.188360 0.982100i \(-0.560317\pi\)
−0.188360 + 0.982100i \(0.560317\pi\)
\(500\) 3.32049 0.148497
\(501\) −6.04366 −0.270011
\(502\) 50.1802 2.23965
\(503\) −15.1360 −0.674882 −0.337441 0.941347i \(-0.609561\pi\)
−0.337441 + 0.941347i \(0.609561\pi\)
\(504\) −3.66836 −0.163402
\(505\) −7.98830 −0.355475
\(506\) 19.6006 0.871353
\(507\) 17.6008 0.781679
\(508\) −9.61803 −0.426731
\(509\) 41.8204 1.85366 0.926828 0.375486i \(-0.122524\pi\)
0.926828 + 0.375486i \(0.122524\pi\)
\(510\) −20.6867 −0.916021
\(511\) −10.1640 −0.449628
\(512\) 4.34534 0.192039
\(513\) −10.2469 −0.452411
\(514\) 24.4788 1.07971
\(515\) −10.2062 −0.449737
\(516\) 9.52192 0.419179
\(517\) 18.2904 0.804412
\(518\) −21.1144 −0.927714
\(519\) 3.26384 0.143267
\(520\) 15.5712 0.682840
\(521\) 8.61494 0.377427 0.188714 0.982032i \(-0.439568\pi\)
0.188714 + 0.982032i \(0.439568\pi\)
\(522\) −4.87282 −0.213278
\(523\) 24.9923 1.09284 0.546418 0.837512i \(-0.315991\pi\)
0.546418 + 0.837512i \(0.315991\pi\)
\(524\) −19.6692 −0.859252
\(525\) −1.34001 −0.0584829
\(526\) −46.0724 −2.00885
\(527\) −19.5872 −0.853230
\(528\) 1.09982 0.0478636
\(529\) −7.13847 −0.310368
\(530\) −4.32238 −0.187752
\(531\) 3.19467 0.138637
\(532\) −6.03927 −0.261836
\(533\) −39.9808 −1.73176
\(534\) −34.5785 −1.49636
\(535\) 18.0542 0.780550
\(536\) 21.2337 0.917154
\(537\) 28.2532 1.21922
\(538\) −41.9714 −1.80952
\(539\) 2.13364 0.0919024
\(540\) −18.7073 −0.805035
\(541\) −15.3379 −0.659428 −0.329714 0.944081i \(-0.606952\pi\)
−0.329714 + 0.944081i \(0.606952\pi\)
\(542\) −40.7175 −1.74897
\(543\) 21.9553 0.942194
\(544\) 34.8322 1.49342
\(545\) 1.81976 0.0779501
\(546\) −15.8013 −0.676235
\(547\) 7.37892 0.315500 0.157750 0.987479i \(-0.449576\pi\)
0.157750 + 0.987479i \(0.449576\pi\)
\(548\) 42.5651 1.81829
\(549\) −11.4420 −0.488332
\(550\) −4.92149 −0.209853
\(551\) −3.19026 −0.135910
\(552\) 16.2552 0.691866
\(553\) −4.48655 −0.190788
\(554\) −50.4460 −2.14324
\(555\) −12.2662 −0.520673
\(556\) 25.1992 1.06868
\(557\) −30.1600 −1.27792 −0.638959 0.769241i \(-0.720635\pi\)
−0.638959 + 0.769241i \(0.720635\pi\)
\(558\) −8.13021 −0.344179
\(559\) −10.9402 −0.462720
\(560\) −0.384674 −0.0162554
\(561\) 19.1353 0.807894
\(562\) −27.3960 −1.15563
\(563\) −34.7175 −1.46317 −0.731584 0.681752i \(-0.761219\pi\)
−0.731584 + 0.681752i \(0.761219\pi\)
\(564\) 38.1428 1.60610
\(565\) 5.19513 0.218561
\(566\) 54.7292 2.30044
\(567\) 3.93637 0.165312
\(568\) −0.433288 −0.0181804
\(569\) 23.4408 0.982687 0.491344 0.870966i \(-0.336506\pi\)
0.491344 + 0.870966i \(0.336506\pi\)
\(570\) −5.62168 −0.235466
\(571\) −20.8255 −0.871520 −0.435760 0.900063i \(-0.643520\pi\)
−0.435760 + 0.900063i \(0.643520\pi\)
\(572\) −36.2187 −1.51438
\(573\) 25.9249 1.08303
\(574\) 18.0392 0.752942
\(575\) −3.98265 −0.166088
\(576\) 15.3846 0.641027
\(577\) 1.69444 0.0705406 0.0352703 0.999378i \(-0.488771\pi\)
0.0352703 + 0.999378i \(0.488771\pi\)
\(578\) −64.1085 −2.66656
\(579\) 11.9703 0.497468
\(580\) −5.82434 −0.241842
\(581\) 15.2528 0.632793
\(582\) 6.64752 0.275549
\(583\) 3.99823 0.165590
\(584\) −30.9582 −1.28106
\(585\) 6.15702 0.254561
\(586\) 73.2158 3.02452
\(587\) −31.8268 −1.31363 −0.656816 0.754051i \(-0.728097\pi\)
−0.656816 + 0.754051i \(0.728097\pi\)
\(588\) 4.44949 0.183494
\(589\) −5.32289 −0.219326
\(590\) 6.11844 0.251892
\(591\) −19.8703 −0.817353
\(592\) −3.52124 −0.144722
\(593\) −25.8737 −1.06251 −0.531253 0.847213i \(-0.678279\pi\)
−0.531253 + 0.847213i \(0.678279\pi\)
\(594\) 27.7272 1.13766
\(595\) −6.69278 −0.274377
\(596\) 42.4656 1.73946
\(597\) 0.259065 0.0106028
\(598\) −46.9633 −1.92047
\(599\) −2.32211 −0.0948790 −0.0474395 0.998874i \(-0.515106\pi\)
−0.0474395 + 0.998874i \(0.515106\pi\)
\(600\) −4.08149 −0.166626
\(601\) 41.7928 1.70476 0.852382 0.522919i \(-0.175157\pi\)
0.852382 + 0.522919i \(0.175157\pi\)
\(602\) 4.93617 0.201183
\(603\) 8.39604 0.341913
\(604\) −60.8712 −2.47682
\(605\) −6.44758 −0.262131
\(606\) 24.6910 1.00300
\(607\) 7.93106 0.321912 0.160956 0.986962i \(-0.448542\pi\)
0.160956 + 0.986962i \(0.448542\pi\)
\(608\) 9.46577 0.383888
\(609\) 2.35046 0.0952454
\(610\) −21.9137 −0.887261
\(611\) −43.8240 −1.77293
\(612\) −26.7651 −1.08192
\(613\) 1.54597 0.0624411 0.0312206 0.999513i \(-0.490061\pi\)
0.0312206 + 0.999513i \(0.490061\pi\)
\(614\) −9.72136 −0.392322
\(615\) 10.4797 0.422583
\(616\) 6.49878 0.261843
\(617\) 44.2377 1.78094 0.890471 0.455040i \(-0.150375\pi\)
0.890471 + 0.455040i \(0.150375\pi\)
\(618\) 31.5461 1.26897
\(619\) −16.0738 −0.646059 −0.323029 0.946389i \(-0.604701\pi\)
−0.323029 + 0.946389i \(0.604701\pi\)
\(620\) −9.71779 −0.390276
\(621\) 22.4379 0.900401
\(622\) 28.4713 1.14160
\(623\) −11.1872 −0.448207
\(624\) −2.63518 −0.105492
\(625\) 1.00000 0.0400000
\(626\) 10.0567 0.401946
\(627\) 5.20010 0.207672
\(628\) 15.6258 0.623539
\(629\) −61.2646 −2.44278
\(630\) −2.77803 −0.110679
\(631\) 15.6480 0.622938 0.311469 0.950256i \(-0.399179\pi\)
0.311469 + 0.950256i \(0.399179\pi\)
\(632\) −13.6654 −0.543582
\(633\) 15.5255 0.617085
\(634\) −26.1278 −1.03767
\(635\) −2.89657 −0.114947
\(636\) 8.33792 0.330620
\(637\) −5.11222 −0.202554
\(638\) 8.63259 0.341768
\(639\) −0.171327 −0.00677761
\(640\) 19.0559 0.753250
\(641\) 28.5834 1.12898 0.564488 0.825442i \(-0.309074\pi\)
0.564488 + 0.825442i \(0.309074\pi\)
\(642\) −55.8035 −2.20239
\(643\) −19.9142 −0.785338 −0.392669 0.919680i \(-0.628448\pi\)
−0.392669 + 0.919680i \(0.628448\pi\)
\(644\) 13.2244 0.521113
\(645\) 2.86763 0.112913
\(646\) −28.0779 −1.10471
\(647\) 32.5545 1.27985 0.639924 0.768438i \(-0.278966\pi\)
0.639924 + 0.768438i \(0.278966\pi\)
\(648\) 11.9897 0.470999
\(649\) −5.65960 −0.222159
\(650\) 11.7919 0.462518
\(651\) 3.92169 0.153703
\(652\) 26.4691 1.03661
\(653\) −13.3877 −0.523901 −0.261951 0.965081i \(-0.584366\pi\)
−0.261951 + 0.965081i \(0.584366\pi\)
\(654\) −5.62470 −0.219943
\(655\) −5.92357 −0.231453
\(656\) 3.00839 0.117458
\(657\) −12.2412 −0.477575
\(658\) 19.7733 0.770842
\(659\) 25.6259 0.998244 0.499122 0.866532i \(-0.333656\pi\)
0.499122 + 0.866532i \(0.333656\pi\)
\(660\) 9.49361 0.369538
\(661\) −3.24804 −0.126334 −0.0631671 0.998003i \(-0.520120\pi\)
−0.0631671 + 0.998003i \(0.520120\pi\)
\(662\) −71.5186 −2.77965
\(663\) −45.8484 −1.78061
\(664\) 46.4580 1.80292
\(665\) −1.81879 −0.0705296
\(666\) −25.4296 −0.985377
\(667\) 6.98581 0.270492
\(668\) −14.9759 −0.579436
\(669\) 11.1423 0.430788
\(670\) 16.0801 0.621230
\(671\) 20.2704 0.782529
\(672\) −6.97400 −0.269028
\(673\) −22.3938 −0.863218 −0.431609 0.902061i \(-0.642054\pi\)
−0.431609 + 0.902061i \(0.642054\pi\)
\(674\) −49.5593 −1.90895
\(675\) −5.63390 −0.216849
\(676\) 43.6141 1.67746
\(677\) −40.6462 −1.56216 −0.781081 0.624430i \(-0.785331\pi\)
−0.781081 + 0.624430i \(0.785331\pi\)
\(678\) −16.0576 −0.616688
\(679\) 2.15068 0.0825355
\(680\) −20.3853 −0.781741
\(681\) 7.02473 0.269188
\(682\) 14.4033 0.551531
\(683\) −25.6084 −0.979878 −0.489939 0.871757i \(-0.662981\pi\)
−0.489939 + 0.871757i \(0.662981\pi\)
\(684\) −7.27353 −0.278110
\(685\) 12.8189 0.489786
\(686\) 2.30662 0.0880671
\(687\) −1.34001 −0.0511246
\(688\) 0.823203 0.0313843
\(689\) −9.57981 −0.364962
\(690\) 12.3100 0.468632
\(691\) 20.6425 0.785278 0.392639 0.919693i \(-0.371562\pi\)
0.392639 + 0.919693i \(0.371562\pi\)
\(692\) 8.08767 0.307447
\(693\) 2.56970 0.0976146
\(694\) 70.0052 2.65736
\(695\) 7.58899 0.287867
\(696\) 7.15918 0.271368
\(697\) 52.3417 1.98258
\(698\) 24.4567 0.925701
\(699\) −26.9106 −1.01785
\(700\) −3.32049 −0.125503
\(701\) −30.3761 −1.14729 −0.573644 0.819104i \(-0.694471\pi\)
−0.573644 + 0.819104i \(0.694471\pi\)
\(702\) −66.4347 −2.50742
\(703\) −16.6489 −0.627925
\(704\) −27.2551 −1.02722
\(705\) 11.4871 0.432630
\(706\) −20.5182 −0.772214
\(707\) 7.98830 0.300431
\(708\) −11.8025 −0.443567
\(709\) −29.0714 −1.09180 −0.545900 0.837850i \(-0.683812\pi\)
−0.545900 + 0.837850i \(0.683812\pi\)
\(710\) −0.328127 −0.0123144
\(711\) −5.40348 −0.202646
\(712\) −34.0748 −1.27701
\(713\) 11.6557 0.436509
\(714\) 20.6867 0.774179
\(715\) −10.9076 −0.407923
\(716\) 70.0104 2.61641
\(717\) 28.9567 1.08141
\(718\) −52.3336 −1.95307
\(719\) −11.2470 −0.419442 −0.209721 0.977761i \(-0.567255\pi\)
−0.209721 + 0.977761i \(0.567255\pi\)
\(720\) −0.463291 −0.0172658
\(721\) 10.2062 0.380097
\(722\) 36.1955 1.34706
\(723\) 16.8599 0.627027
\(724\) 54.4044 2.02192
\(725\) −1.75406 −0.0651441
\(726\) 19.9288 0.739627
\(727\) −3.02930 −0.112350 −0.0561752 0.998421i \(-0.517891\pi\)
−0.0561752 + 0.998421i \(0.517891\pi\)
\(728\) −15.5712 −0.577105
\(729\) 27.3893 1.01442
\(730\) −23.4444 −0.867718
\(731\) 14.3226 0.529739
\(732\) 42.2718 1.56241
\(733\) −40.3539 −1.49050 −0.745252 0.666783i \(-0.767671\pi\)
−0.745252 + 0.666783i \(0.767671\pi\)
\(734\) −9.84931 −0.363545
\(735\) 1.34001 0.0494270
\(736\) −20.7275 −0.764025
\(737\) −14.8742 −0.547900
\(738\) 21.7259 0.799741
\(739\) 34.9877 1.28704 0.643521 0.765429i \(-0.277473\pi\)
0.643521 + 0.765429i \(0.277473\pi\)
\(740\) −30.3952 −1.11735
\(741\) −12.4595 −0.457711
\(742\) 4.32238 0.158680
\(743\) −8.40431 −0.308324 −0.154162 0.988046i \(-0.549268\pi\)
−0.154162 + 0.988046i \(0.549268\pi\)
\(744\) 11.9449 0.437923
\(745\) 12.7889 0.468551
\(746\) 0.873322 0.0319746
\(747\) 18.3700 0.672125
\(748\) 47.4165 1.73372
\(749\) −18.0542 −0.659685
\(750\) −3.09089 −0.112863
\(751\) 15.7806 0.575842 0.287921 0.957654i \(-0.407036\pi\)
0.287921 + 0.957654i \(0.407036\pi\)
\(752\) 3.29758 0.120250
\(753\) −29.1518 −1.06235
\(754\) −20.6838 −0.753259
\(755\) −18.3320 −0.667170
\(756\) 18.7073 0.680379
\(757\) −23.4198 −0.851206 −0.425603 0.904910i \(-0.639938\pi\)
−0.425603 + 0.904910i \(0.639938\pi\)
\(758\) 13.9160 0.505454
\(759\) −11.3868 −0.413315
\(760\) −5.53979 −0.200949
\(761\) −27.1790 −0.985237 −0.492619 0.870245i \(-0.663960\pi\)
−0.492619 + 0.870245i \(0.663960\pi\)
\(762\) 8.95299 0.324333
\(763\) −1.81976 −0.0658799
\(764\) 64.2409 2.32416
\(765\) −8.06059 −0.291431
\(766\) 56.4831 2.04082
\(767\) 13.5605 0.489640
\(768\) −24.6651 −0.890025
\(769\) −11.4436 −0.412667 −0.206334 0.978482i \(-0.566153\pi\)
−0.206334 + 0.978482i \(0.566153\pi\)
\(770\) 4.92149 0.177358
\(771\) −14.2207 −0.512148
\(772\) 29.6618 1.06755
\(773\) −18.6610 −0.671188 −0.335594 0.942007i \(-0.608937\pi\)
−0.335594 + 0.942007i \(0.608937\pi\)
\(774\) 5.94498 0.213688
\(775\) −2.92661 −0.105127
\(776\) 6.55069 0.235156
\(777\) 12.2662 0.440049
\(778\) −22.4164 −0.803665
\(779\) 14.2241 0.509630
\(780\) −22.7468 −0.814466
\(781\) 0.303520 0.0108608
\(782\) 61.4830 2.19863
\(783\) 9.88220 0.353161
\(784\) 0.384674 0.0137384
\(785\) 4.70588 0.167960
\(786\) 18.3091 0.653065
\(787\) 37.2693 1.32851 0.664254 0.747507i \(-0.268749\pi\)
0.664254 + 0.747507i \(0.268749\pi\)
\(788\) −49.2377 −1.75402
\(789\) 26.7654 0.952872
\(790\) −10.3488 −0.368193
\(791\) −5.19513 −0.184717
\(792\) 7.82695 0.278119
\(793\) −48.5680 −1.72470
\(794\) 11.5461 0.409756
\(795\) 2.51105 0.0890578
\(796\) 0.641951 0.0227533
\(797\) 36.4354 1.29061 0.645304 0.763926i \(-0.276731\pi\)
0.645304 + 0.763926i \(0.276731\pi\)
\(798\) 5.62168 0.199005
\(799\) 57.3732 2.02972
\(800\) 5.20444 0.184005
\(801\) −13.4736 −0.476066
\(802\) 7.99312 0.282247
\(803\) 21.6863 0.765292
\(804\) −31.0188 −1.09395
\(805\) 3.98265 0.140370
\(806\) −34.5105 −1.21558
\(807\) 24.3829 0.858320
\(808\) 24.3313 0.855972
\(809\) 2.43358 0.0855602 0.0427801 0.999085i \(-0.486379\pi\)
0.0427801 + 0.999085i \(0.486379\pi\)
\(810\) 9.07971 0.319029
\(811\) 39.3768 1.38271 0.691353 0.722517i \(-0.257015\pi\)
0.691353 + 0.722517i \(0.257015\pi\)
\(812\) 5.82434 0.204394
\(813\) 23.6545 0.829598
\(814\) 45.0505 1.57902
\(815\) 7.97144 0.279227
\(816\) 3.44991 0.120771
\(817\) 3.89221 0.136171
\(818\) −57.9665 −2.02675
\(819\) −6.15702 −0.215144
\(820\) 25.9683 0.906852
\(821\) −16.2534 −0.567249 −0.283625 0.958935i \(-0.591537\pi\)
−0.283625 + 0.958935i \(0.591537\pi\)
\(822\) −39.6219 −1.38197
\(823\) 20.1585 0.702680 0.351340 0.936248i \(-0.385726\pi\)
0.351340 + 0.936248i \(0.385726\pi\)
\(824\) 31.0866 1.08295
\(825\) 2.85910 0.0995411
\(826\) −6.11844 −0.212888
\(827\) 41.8891 1.45663 0.728314 0.685244i \(-0.240304\pi\)
0.728314 + 0.685244i \(0.240304\pi\)
\(828\) 15.9270 0.553503
\(829\) −11.9814 −0.416132 −0.208066 0.978115i \(-0.566717\pi\)
−0.208066 + 0.978115i \(0.566717\pi\)
\(830\) 35.1824 1.22120
\(831\) 29.3062 1.01662
\(832\) 65.3035 2.26399
\(833\) 6.69278 0.231891
\(834\) −23.4567 −0.812241
\(835\) −4.51016 −0.156080
\(836\) 12.8856 0.445659
\(837\) 16.4882 0.569917
\(838\) 23.2553 0.803340
\(839\) 27.0619 0.934282 0.467141 0.884183i \(-0.345284\pi\)
0.467141 + 0.884183i \(0.345284\pi\)
\(840\) 4.08149 0.140825
\(841\) −25.9233 −0.893906
\(842\) 74.5669 2.56975
\(843\) 15.9155 0.548158
\(844\) 38.4717 1.32425
\(845\) 13.1348 0.451852
\(846\) 23.8144 0.818755
\(847\) 6.44758 0.221542
\(848\) 0.720842 0.0247538
\(849\) −31.7944 −1.09118
\(850\) −15.4377 −0.529508
\(851\) 36.4566 1.24971
\(852\) 0.632960 0.0216849
\(853\) 55.5730 1.90278 0.951391 0.307984i \(-0.0996545\pi\)
0.951391 + 0.307984i \(0.0996545\pi\)
\(854\) 21.9137 0.749872
\(855\) −2.19050 −0.0749134
\(856\) −54.9906 −1.87954
\(857\) 2.79983 0.0956402 0.0478201 0.998856i \(-0.484773\pi\)
0.0478201 + 0.998856i \(0.484773\pi\)
\(858\) 33.7144 1.15099
\(859\) 14.7394 0.502902 0.251451 0.967870i \(-0.419092\pi\)
0.251451 + 0.967870i \(0.419092\pi\)
\(860\) 7.10586 0.242308
\(861\) −10.4797 −0.357148
\(862\) −5.46668 −0.186196
\(863\) 13.7773 0.468984 0.234492 0.972118i \(-0.424657\pi\)
0.234492 + 0.972118i \(0.424657\pi\)
\(864\) −29.3213 −0.997531
\(865\) 2.43568 0.0828157
\(866\) −86.9073 −2.95323
\(867\) 37.2433 1.26485
\(868\) 9.71779 0.329843
\(869\) 9.57269 0.324731
\(870\) 5.42161 0.183810
\(871\) 35.6389 1.20758
\(872\) −5.54276 −0.187701
\(873\) 2.59022 0.0876656
\(874\) 16.7082 0.565165
\(875\) −1.00000 −0.0338062
\(876\) 45.2246 1.52800
\(877\) 34.1222 1.15223 0.576113 0.817370i \(-0.304569\pi\)
0.576113 + 0.817370i \(0.304569\pi\)
\(878\) −23.1229 −0.780361
\(879\) −42.5341 −1.43464
\(880\) 0.820756 0.0276677
\(881\) 52.7646 1.77768 0.888842 0.458215i \(-0.151511\pi\)
0.888842 + 0.458215i \(0.151511\pi\)
\(882\) 2.77803 0.0935410
\(883\) 1.67074 0.0562250 0.0281125 0.999605i \(-0.491050\pi\)
0.0281125 + 0.999605i \(0.491050\pi\)
\(884\) −113.611 −3.82113
\(885\) −3.55446 −0.119482
\(886\) 4.25461 0.142937
\(887\) −8.15940 −0.273966 −0.136983 0.990573i \(-0.543741\pi\)
−0.136983 + 0.990573i \(0.543741\pi\)
\(888\) 37.3613 1.25376
\(889\) 2.89657 0.0971478
\(890\) −25.8047 −0.864975
\(891\) −8.39880 −0.281370
\(892\) 27.6103 0.924460
\(893\) 15.5914 0.521746
\(894\) −39.5293 −1.32206
\(895\) 21.0843 0.704772
\(896\) −19.0559 −0.636612
\(897\) 27.2829 0.910950
\(898\) −15.8508 −0.528949
\(899\) 5.13345 0.171210
\(900\) −3.99910 −0.133303
\(901\) 12.5416 0.417822
\(902\) −38.4891 −1.28155
\(903\) −2.86763 −0.0954286
\(904\) −15.8237 −0.526287
\(905\) 16.3844 0.544638
\(906\) 56.6623 1.88248
\(907\) 43.3624 1.43982 0.719912 0.694065i \(-0.244182\pi\)
0.719912 + 0.694065i \(0.244182\pi\)
\(908\) 17.4070 0.577671
\(909\) 9.62088 0.319105
\(910\) −11.7919 −0.390899
\(911\) 49.1852 1.62958 0.814789 0.579758i \(-0.196853\pi\)
0.814789 + 0.579758i \(0.196853\pi\)
\(912\) 0.937526 0.0310446
\(913\) −32.5440 −1.07705
\(914\) 7.83430 0.259136
\(915\) 12.7306 0.420861
\(916\) −3.32049 −0.109712
\(917\) 5.92357 0.195614
\(918\) 86.9744 2.87058
\(919\) −39.9475 −1.31775 −0.658873 0.752254i \(-0.728967\pi\)
−0.658873 + 0.752254i \(0.728967\pi\)
\(920\) 12.1306 0.399935
\(921\) 5.64754 0.186093
\(922\) 3.15576 0.103929
\(923\) −0.727237 −0.0239373
\(924\) −9.49361 −0.312317
\(925\) −9.15383 −0.300976
\(926\) −27.0009 −0.887304
\(927\) 12.2920 0.403722
\(928\) −9.12889 −0.299671
\(929\) 31.4181 1.03080 0.515398 0.856951i \(-0.327644\pi\)
0.515398 + 0.856951i \(0.327644\pi\)
\(930\) 9.04585 0.296625
\(931\) 1.81879 0.0596084
\(932\) −66.6833 −2.18429
\(933\) −16.5402 −0.541501
\(934\) −46.9751 −1.53707
\(935\) 14.2800 0.467005
\(936\) −18.7535 −0.612976
\(937\) 1.23159 0.0402342 0.0201171 0.999798i \(-0.493596\pi\)
0.0201171 + 0.999798i \(0.493596\pi\)
\(938\) −16.0801 −0.525035
\(939\) −5.84235 −0.190658
\(940\) 28.4646 0.928412
\(941\) 21.6748 0.706579 0.353289 0.935514i \(-0.385063\pi\)
0.353289 + 0.935514i \(0.385063\pi\)
\(942\) −14.5454 −0.473914
\(943\) −31.1468 −1.01428
\(944\) −1.02037 −0.0332102
\(945\) 5.63390 0.183271
\(946\) −10.5320 −0.342425
\(947\) −6.97102 −0.226528 −0.113264 0.993565i \(-0.536131\pi\)
−0.113264 + 0.993565i \(0.536131\pi\)
\(948\) 19.9629 0.648364
\(949\) −51.9606 −1.68671
\(950\) −4.19525 −0.136112
\(951\) 15.1787 0.492203
\(952\) 20.3853 0.660692
\(953\) −32.6424 −1.05739 −0.528696 0.848811i \(-0.677319\pi\)
−0.528696 + 0.848811i \(0.677319\pi\)
\(954\) 5.20575 0.168542
\(955\) 19.3468 0.626049
\(956\) 71.7534 2.32067
\(957\) −5.01503 −0.162113
\(958\) −15.2154 −0.491587
\(959\) −12.8189 −0.413944
\(960\) −17.1173 −0.552458
\(961\) −22.4349 −0.723708
\(962\) −107.942 −3.48018
\(963\) −21.7439 −0.700688
\(964\) 41.7782 1.34558
\(965\) 8.93297 0.287563
\(966\) −12.3100 −0.396066
\(967\) −25.2873 −0.813183 −0.406592 0.913610i \(-0.633283\pi\)
−0.406592 + 0.913610i \(0.633283\pi\)
\(968\) 19.6385 0.631205
\(969\) 16.3116 0.524005
\(970\) 4.96080 0.159282
\(971\) 9.07379 0.291192 0.145596 0.989344i \(-0.453490\pi\)
0.145596 + 0.989344i \(0.453490\pi\)
\(972\) 38.6071 1.23832
\(973\) −7.58899 −0.243292
\(974\) −73.9965 −2.37100
\(975\) −6.85043 −0.219389
\(976\) 3.65455 0.116979
\(977\) 52.2123 1.67042 0.835210 0.549932i \(-0.185346\pi\)
0.835210 + 0.549932i \(0.185346\pi\)
\(978\) −24.6389 −0.787864
\(979\) 23.8695 0.762873
\(980\) 3.32049 0.106069
\(981\) −2.19167 −0.0699747
\(982\) 28.6830 0.915311
\(983\) −54.7691 −1.74686 −0.873432 0.486946i \(-0.838111\pi\)
−0.873432 + 0.486946i \(0.838111\pi\)
\(984\) −31.9198 −1.01757
\(985\) −14.8284 −0.472473
\(986\) 27.0786 0.862359
\(987\) −11.4871 −0.365639
\(988\) −30.8741 −0.982236
\(989\) −8.52289 −0.271012
\(990\) 5.92731 0.188382
\(991\) 10.5226 0.334261 0.167130 0.985935i \(-0.446550\pi\)
0.167130 + 0.985935i \(0.446550\pi\)
\(992\) −15.2314 −0.483596
\(993\) 41.5481 1.31849
\(994\) 0.328127 0.0104076
\(995\) 0.193330 0.00612898
\(996\) −67.8672 −2.15045
\(997\) −28.9360 −0.916412 −0.458206 0.888846i \(-0.651508\pi\)
−0.458206 + 0.888846i \(0.651508\pi\)
\(998\) 19.4108 0.614438
\(999\) 51.5718 1.63166
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 8015.2.a.m.1.8 67
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8015.2.a.m.1.8 67 1.1 even 1 trivial