Properties

Label 8041.2.a.d.1.7
Level $8041$
Weight $2$
Character 8041.1
Self dual yes
Analytic conductor $64.208$
Analytic rank $1$
Dimension $62$
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] = [8041,2,Mod(1,8041)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, 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("8041.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8041 = 11 \cdot 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8041.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: \(64.2077082653\)
Analytic rank: \(1\)
Dimension: \(62\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 8041.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.36585 q^{2} +2.45510 q^{3} +3.59727 q^{4} +1.09940 q^{5} -5.80841 q^{6} +2.69770 q^{7} -3.77890 q^{8} +3.02752 q^{9} +O(q^{10})\) \(q-2.36585 q^{2} +2.45510 q^{3} +3.59727 q^{4} +1.09940 q^{5} -5.80841 q^{6} +2.69770 q^{7} -3.77890 q^{8} +3.02752 q^{9} -2.60102 q^{10} +1.00000 q^{11} +8.83165 q^{12} -0.515099 q^{13} -6.38237 q^{14} +2.69914 q^{15} +1.74579 q^{16} -1.00000 q^{17} -7.16267 q^{18} -4.85774 q^{19} +3.95483 q^{20} +6.62313 q^{21} -2.36585 q^{22} -7.58610 q^{23} -9.27757 q^{24} -3.79132 q^{25} +1.21865 q^{26} +0.0675680 q^{27} +9.70434 q^{28} -5.74572 q^{29} -6.38577 q^{30} -4.06124 q^{31} +3.42751 q^{32} +2.45510 q^{33} +2.36585 q^{34} +2.96585 q^{35} +10.8908 q^{36} -5.18985 q^{37} +11.4927 q^{38} -1.26462 q^{39} -4.15452 q^{40} +5.65407 q^{41} -15.6694 q^{42} +1.00000 q^{43} +3.59727 q^{44} +3.32846 q^{45} +17.9476 q^{46} +4.43739 q^{47} +4.28609 q^{48} +0.277587 q^{49} +8.96971 q^{50} -2.45510 q^{51} -1.85295 q^{52} -1.94057 q^{53} -0.159856 q^{54} +1.09940 q^{55} -10.1943 q^{56} -11.9262 q^{57} +13.5935 q^{58} +2.90607 q^{59} +9.70952 q^{60} -9.75564 q^{61} +9.60830 q^{62} +8.16735 q^{63} -11.6006 q^{64} -0.566300 q^{65} -5.80841 q^{66} -4.84702 q^{67} -3.59727 q^{68} -18.6246 q^{69} -7.01677 q^{70} -14.9495 q^{71} -11.4407 q^{72} +14.4159 q^{73} +12.2784 q^{74} -9.30807 q^{75} -17.4746 q^{76} +2.69770 q^{77} +2.99190 q^{78} +9.29512 q^{79} +1.91932 q^{80} -8.91668 q^{81} -13.3767 q^{82} -15.8395 q^{83} +23.8251 q^{84} -1.09940 q^{85} -2.36585 q^{86} -14.1063 q^{87} -3.77890 q^{88} +7.64563 q^{89} -7.87465 q^{90} -1.38958 q^{91} -27.2892 q^{92} -9.97076 q^{93} -10.4982 q^{94} -5.34060 q^{95} +8.41490 q^{96} +6.14325 q^{97} -0.656729 q^{98} +3.02752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 62 q - 7 q^{2} - 8 q^{3} + 49 q^{4} - 13 q^{5} - 2 q^{6} - 11 q^{7} - 9 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 62 q - 7 q^{2} - 8 q^{3} + 49 q^{4} - 13 q^{5} - 2 q^{6} - 11 q^{7} - 9 q^{8} + 40 q^{9} - 7 q^{10} + 62 q^{11} - 17 q^{12} - 31 q^{14} - 20 q^{15} + 27 q^{16} - 62 q^{17} + 3 q^{18} - 29 q^{20} - 18 q^{21} - 7 q^{22} - 50 q^{23} - 31 q^{24} + 35 q^{25} - 32 q^{26} - 14 q^{27} - 13 q^{28} - 26 q^{29} - 10 q^{30} - 58 q^{31} - 5 q^{32} - 8 q^{33} + 7 q^{34} - 32 q^{35} - 29 q^{36} - 41 q^{37} - 10 q^{38} - 53 q^{39} - 31 q^{40} - 55 q^{41} - 7 q^{42} + 62 q^{43} + 49 q^{44} - 34 q^{45} - 39 q^{46} - 31 q^{47} - 30 q^{48} + 35 q^{49} - 40 q^{50} + 8 q^{51} + 13 q^{52} - 74 q^{53} + 48 q^{54} - 13 q^{55} - 75 q^{56} - 43 q^{57} - 46 q^{58} - 65 q^{59} - 8 q^{60} - 14 q^{61} - 29 q^{62} - 23 q^{63} - 15 q^{64} - 9 q^{65} - 2 q^{66} - q^{67} - 49 q^{68} - 59 q^{69} - 31 q^{70} - 141 q^{71} + 9 q^{72} - 4 q^{73} - 94 q^{74} - 43 q^{75} + 34 q^{76} - 11 q^{77} - 11 q^{78} - 63 q^{79} - 41 q^{80} - 30 q^{81} + 38 q^{82} - 44 q^{83} - 16 q^{84} + 13 q^{85} - 7 q^{86} - 8 q^{87} - 9 q^{88} - 58 q^{89} - 55 q^{90} - 78 q^{91} - 104 q^{92} - 5 q^{94} - 99 q^{95} - 148 q^{96} - 26 q^{97} + 16 q^{98} + 40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.36585 −1.67291 −0.836456 0.548035i \(-0.815376\pi\)
−0.836456 + 0.548035i \(0.815376\pi\)
\(3\) 2.45510 1.41745 0.708727 0.705483i \(-0.249270\pi\)
0.708727 + 0.705483i \(0.249270\pi\)
\(4\) 3.59727 1.79863
\(5\) 1.09940 0.491667 0.245833 0.969312i \(-0.420938\pi\)
0.245833 + 0.969312i \(0.420938\pi\)
\(6\) −5.80841 −2.37127
\(7\) 2.69770 1.01963 0.509817 0.860283i \(-0.329713\pi\)
0.509817 + 0.860283i \(0.329713\pi\)
\(8\) −3.77890 −1.33604
\(9\) 3.02752 1.00917
\(10\) −2.60102 −0.822515
\(11\) 1.00000 0.301511
\(12\) 8.83165 2.54948
\(13\) −0.515099 −0.142863 −0.0714313 0.997446i \(-0.522757\pi\)
−0.0714313 + 0.997446i \(0.522757\pi\)
\(14\) −6.38237 −1.70576
\(15\) 2.69914 0.696915
\(16\) 1.74579 0.436447
\(17\) −1.00000 −0.242536
\(18\) −7.16267 −1.68826
\(19\) −4.85774 −1.11444 −0.557221 0.830364i \(-0.688132\pi\)
−0.557221 + 0.830364i \(0.688132\pi\)
\(20\) 3.95483 0.884328
\(21\) 6.62313 1.44528
\(22\) −2.36585 −0.504402
\(23\) −7.58610 −1.58181 −0.790906 0.611938i \(-0.790390\pi\)
−0.790906 + 0.611938i \(0.790390\pi\)
\(24\) −9.27757 −1.89378
\(25\) −3.79132 −0.758264
\(26\) 1.21865 0.238997
\(27\) 0.0675680 0.0130035
\(28\) 9.70434 1.83395
\(29\) −5.74572 −1.06695 −0.533477 0.845815i \(-0.679115\pi\)
−0.533477 + 0.845815i \(0.679115\pi\)
\(30\) −6.38577 −1.16588
\(31\) −4.06124 −0.729420 −0.364710 0.931121i \(-0.618832\pi\)
−0.364710 + 0.931121i \(0.618832\pi\)
\(32\) 3.42751 0.605905
\(33\) 2.45510 0.427378
\(34\) 2.36585 0.405741
\(35\) 2.96585 0.501321
\(36\) 10.8908 1.81513
\(37\) −5.18985 −0.853206 −0.426603 0.904439i \(-0.640290\pi\)
−0.426603 + 0.904439i \(0.640290\pi\)
\(38\) 11.4927 1.86436
\(39\) −1.26462 −0.202501
\(40\) −4.15452 −0.656887
\(41\) 5.65407 0.883017 0.441508 0.897257i \(-0.354444\pi\)
0.441508 + 0.897257i \(0.354444\pi\)
\(42\) −15.6694 −2.41783
\(43\) 1.00000 0.152499
\(44\) 3.59727 0.542308
\(45\) 3.32846 0.496177
\(46\) 17.9476 2.64623
\(47\) 4.43739 0.647259 0.323630 0.946184i \(-0.395097\pi\)
0.323630 + 0.946184i \(0.395097\pi\)
\(48\) 4.28609 0.618643
\(49\) 0.277587 0.0396552
\(50\) 8.96971 1.26851
\(51\) −2.45510 −0.343783
\(52\) −1.85295 −0.256957
\(53\) −1.94057 −0.266558 −0.133279 0.991079i \(-0.542551\pi\)
−0.133279 + 0.991079i \(0.542551\pi\)
\(54\) −0.159856 −0.0217536
\(55\) 1.09940 0.148243
\(56\) −10.1943 −1.36227
\(57\) −11.9262 −1.57967
\(58\) 13.5935 1.78492
\(59\) 2.90607 0.378338 0.189169 0.981945i \(-0.439421\pi\)
0.189169 + 0.981945i \(0.439421\pi\)
\(60\) 9.70952 1.25349
\(61\) −9.75564 −1.24908 −0.624541 0.780992i \(-0.714714\pi\)
−0.624541 + 0.780992i \(0.714714\pi\)
\(62\) 9.60830 1.22026
\(63\) 8.16735 1.02899
\(64\) −11.6006 −1.45007
\(65\) −0.566300 −0.0702408
\(66\) −5.80841 −0.714966
\(67\) −4.84702 −0.592158 −0.296079 0.955163i \(-0.595679\pi\)
−0.296079 + 0.955163i \(0.595679\pi\)
\(68\) −3.59727 −0.436233
\(69\) −18.6246 −2.24214
\(70\) −7.01677 −0.838665
\(71\) −14.9495 −1.77418 −0.887091 0.461594i \(-0.847278\pi\)
−0.887091 + 0.461594i \(0.847278\pi\)
\(72\) −11.4407 −1.34830
\(73\) 14.4159 1.68725 0.843627 0.536929i \(-0.180416\pi\)
0.843627 + 0.536929i \(0.180416\pi\)
\(74\) 12.2784 1.42734
\(75\) −9.30807 −1.07480
\(76\) −17.4746 −2.00447
\(77\) 2.69770 0.307431
\(78\) 2.99190 0.338766
\(79\) 9.29512 1.04578 0.522891 0.852399i \(-0.324853\pi\)
0.522891 + 0.852399i \(0.324853\pi\)
\(80\) 1.91932 0.214586
\(81\) −8.91668 −0.990742
\(82\) −13.3767 −1.47721
\(83\) −15.8395 −1.73861 −0.869303 0.494280i \(-0.835432\pi\)
−0.869303 + 0.494280i \(0.835432\pi\)
\(84\) 23.8251 2.59954
\(85\) −1.09940 −0.119247
\(86\) −2.36585 −0.255117
\(87\) −14.1063 −1.51236
\(88\) −3.77890 −0.402832
\(89\) 7.64563 0.810435 0.405217 0.914220i \(-0.367196\pi\)
0.405217 + 0.914220i \(0.367196\pi\)
\(90\) −7.87465 −0.830061
\(91\) −1.38958 −0.145668
\(92\) −27.2892 −2.84510
\(93\) −9.97076 −1.03392
\(94\) −10.4982 −1.08281
\(95\) −5.34060 −0.547934
\(96\) 8.41490 0.858842
\(97\) 6.14325 0.623753 0.311876 0.950123i \(-0.399043\pi\)
0.311876 + 0.950123i \(0.399043\pi\)
\(98\) −0.656729 −0.0663397
\(99\) 3.02752 0.304277
\(100\) −13.6384 −1.36384
\(101\) 2.68999 0.267664 0.133832 0.991004i \(-0.457272\pi\)
0.133832 + 0.991004i \(0.457272\pi\)
\(102\) 5.80841 0.575118
\(103\) −9.72591 −0.958322 −0.479161 0.877727i \(-0.659059\pi\)
−0.479161 + 0.877727i \(0.659059\pi\)
\(104\) 1.94650 0.190870
\(105\) 7.28147 0.710598
\(106\) 4.59111 0.445928
\(107\) −18.5163 −1.79004 −0.895019 0.446029i \(-0.852838\pi\)
−0.895019 + 0.446029i \(0.852838\pi\)
\(108\) 0.243060 0.0233885
\(109\) 8.69013 0.832364 0.416182 0.909281i \(-0.363368\pi\)
0.416182 + 0.909281i \(0.363368\pi\)
\(110\) −2.60102 −0.247998
\(111\) −12.7416 −1.20938
\(112\) 4.70961 0.445016
\(113\) 0.909959 0.0856018 0.0428009 0.999084i \(-0.486372\pi\)
0.0428009 + 0.999084i \(0.486372\pi\)
\(114\) 28.2157 2.64265
\(115\) −8.34016 −0.777724
\(116\) −20.6689 −1.91906
\(117\) −1.55947 −0.144173
\(118\) −6.87533 −0.632925
\(119\) −2.69770 −0.247298
\(120\) −10.1998 −0.931107
\(121\) 1.00000 0.0909091
\(122\) 23.0804 2.08960
\(123\) 13.8813 1.25163
\(124\) −14.6094 −1.31196
\(125\) −9.66518 −0.864480
\(126\) −19.3227 −1.72141
\(127\) −0.696723 −0.0618242 −0.0309121 0.999522i \(-0.509841\pi\)
−0.0309121 + 0.999522i \(0.509841\pi\)
\(128\) 20.5902 1.81994
\(129\) 2.45510 0.216160
\(130\) 1.33978 0.117507
\(131\) −1.80562 −0.157758 −0.0788789 0.996884i \(-0.525134\pi\)
−0.0788789 + 0.996884i \(0.525134\pi\)
\(132\) 8.83165 0.768696
\(133\) −13.1047 −1.13632
\(134\) 11.4673 0.990628
\(135\) 0.0742843 0.00639337
\(136\) 3.77890 0.324038
\(137\) −5.52702 −0.472205 −0.236103 0.971728i \(-0.575870\pi\)
−0.236103 + 0.971728i \(0.575870\pi\)
\(138\) 44.0632 3.75091
\(139\) 7.19577 0.610337 0.305169 0.952298i \(-0.401287\pi\)
0.305169 + 0.952298i \(0.401287\pi\)
\(140\) 10.6690 0.901692
\(141\) 10.8942 0.917460
\(142\) 35.3684 2.96805
\(143\) −0.515099 −0.0430747
\(144\) 5.28541 0.440451
\(145\) −6.31684 −0.524585
\(146\) −34.1059 −2.82263
\(147\) 0.681503 0.0562094
\(148\) −18.6693 −1.53460
\(149\) −11.5691 −0.947781 −0.473891 0.880584i \(-0.657151\pi\)
−0.473891 + 0.880584i \(0.657151\pi\)
\(150\) 22.0215 1.79805
\(151\) 8.41378 0.684704 0.342352 0.939572i \(-0.388776\pi\)
0.342352 + 0.939572i \(0.388776\pi\)
\(152\) 18.3569 1.48894
\(153\) −3.02752 −0.244761
\(154\) −6.38237 −0.514306
\(155\) −4.46493 −0.358632
\(156\) −4.54917 −0.364225
\(157\) 4.37885 0.349471 0.174735 0.984615i \(-0.444093\pi\)
0.174735 + 0.984615i \(0.444093\pi\)
\(158\) −21.9909 −1.74950
\(159\) −4.76430 −0.377833
\(160\) 3.76821 0.297903
\(161\) −20.4650 −1.61287
\(162\) 21.0956 1.65742
\(163\) 11.6055 0.909014 0.454507 0.890743i \(-0.349816\pi\)
0.454507 + 0.890743i \(0.349816\pi\)
\(164\) 20.3392 1.58822
\(165\) 2.69914 0.210128
\(166\) 37.4738 2.90853
\(167\) −5.80665 −0.449332 −0.224666 0.974436i \(-0.572129\pi\)
−0.224666 + 0.974436i \(0.572129\pi\)
\(168\) −25.0281 −1.93096
\(169\) −12.7347 −0.979590
\(170\) 2.60102 0.199489
\(171\) −14.7069 −1.12467
\(172\) 3.59727 0.274289
\(173\) −11.0016 −0.836436 −0.418218 0.908347i \(-0.637345\pi\)
−0.418218 + 0.908347i \(0.637345\pi\)
\(174\) 33.3735 2.53004
\(175\) −10.2278 −0.773152
\(176\) 1.74579 0.131594
\(177\) 7.13469 0.536276
\(178\) −18.0884 −1.35579
\(179\) −6.29037 −0.470165 −0.235082 0.971975i \(-0.575536\pi\)
−0.235082 + 0.971975i \(0.575536\pi\)
\(180\) 11.9733 0.892441
\(181\) 1.17111 0.0870479 0.0435239 0.999052i \(-0.486142\pi\)
0.0435239 + 0.999052i \(0.486142\pi\)
\(182\) 3.28755 0.243689
\(183\) −23.9511 −1.77052
\(184\) 28.6671 2.11337
\(185\) −5.70572 −0.419493
\(186\) 23.5894 1.72966
\(187\) −1.00000 −0.0731272
\(188\) 15.9625 1.16418
\(189\) 0.182278 0.0132588
\(190\) 12.6351 0.916645
\(191\) −11.8307 −0.856040 −0.428020 0.903769i \(-0.640789\pi\)
−0.428020 + 0.903769i \(0.640789\pi\)
\(192\) −28.4806 −2.05541
\(193\) 26.1116 1.87955 0.939777 0.341787i \(-0.111032\pi\)
0.939777 + 0.341787i \(0.111032\pi\)
\(194\) −14.5340 −1.04348
\(195\) −1.39032 −0.0995631
\(196\) 0.998552 0.0713252
\(197\) −2.34507 −0.167079 −0.0835395 0.996504i \(-0.526622\pi\)
−0.0835395 + 0.996504i \(0.526622\pi\)
\(198\) −7.16267 −0.509029
\(199\) 16.5562 1.17364 0.586819 0.809718i \(-0.300380\pi\)
0.586819 + 0.809718i \(0.300380\pi\)
\(200\) 14.3270 1.01307
\(201\) −11.8999 −0.839356
\(202\) −6.36412 −0.447778
\(203\) −15.5002 −1.08790
\(204\) −8.83165 −0.618339
\(205\) 6.21608 0.434150
\(206\) 23.0101 1.60319
\(207\) −22.9671 −1.59632
\(208\) −0.899253 −0.0623520
\(209\) −4.85774 −0.336017
\(210\) −17.2269 −1.18877
\(211\) 4.07452 0.280502 0.140251 0.990116i \(-0.455209\pi\)
0.140251 + 0.990116i \(0.455209\pi\)
\(212\) −6.98075 −0.479440
\(213\) −36.7026 −2.51482
\(214\) 43.8068 2.99457
\(215\) 1.09940 0.0749785
\(216\) −0.255332 −0.0173732
\(217\) −10.9560 −0.743742
\(218\) −20.5596 −1.39247
\(219\) 35.3925 2.39160
\(220\) 3.95483 0.266635
\(221\) 0.515099 0.0346493
\(222\) 30.1448 2.02318
\(223\) −15.3677 −1.02909 −0.514547 0.857462i \(-0.672040\pi\)
−0.514547 + 0.857462i \(0.672040\pi\)
\(224\) 9.24641 0.617802
\(225\) −11.4783 −0.765220
\(226\) −2.15283 −0.143204
\(227\) 17.6543 1.17176 0.585879 0.810399i \(-0.300749\pi\)
0.585879 + 0.810399i \(0.300749\pi\)
\(228\) −42.9018 −2.84124
\(229\) 11.6975 0.772995 0.386497 0.922291i \(-0.373685\pi\)
0.386497 + 0.922291i \(0.373685\pi\)
\(230\) 19.7316 1.30106
\(231\) 6.62313 0.435770
\(232\) 21.7125 1.42549
\(233\) −7.39715 −0.484603 −0.242302 0.970201i \(-0.577902\pi\)
−0.242302 + 0.970201i \(0.577902\pi\)
\(234\) 3.68948 0.241189
\(235\) 4.87846 0.318236
\(236\) 10.4539 0.680490
\(237\) 22.8205 1.48235
\(238\) 6.38237 0.413707
\(239\) −5.48879 −0.355040 −0.177520 0.984117i \(-0.556807\pi\)
−0.177520 + 0.984117i \(0.556807\pi\)
\(240\) 4.71212 0.304166
\(241\) 13.2782 0.855326 0.427663 0.903938i \(-0.359337\pi\)
0.427663 + 0.903938i \(0.359337\pi\)
\(242\) −2.36585 −0.152083
\(243\) −22.0941 −1.41733
\(244\) −35.0936 −2.24664
\(245\) 0.305179 0.0194972
\(246\) −32.8411 −2.09387
\(247\) 2.50221 0.159212
\(248\) 15.3470 0.974536
\(249\) −38.8875 −2.46439
\(250\) 22.8664 1.44620
\(251\) 6.76490 0.426997 0.213498 0.976943i \(-0.431514\pi\)
0.213498 + 0.976943i \(0.431514\pi\)
\(252\) 29.3801 1.85077
\(253\) −7.58610 −0.476934
\(254\) 1.64835 0.103426
\(255\) −2.69914 −0.169027
\(256\) −25.5123 −1.59452
\(257\) −15.1225 −0.943316 −0.471658 0.881782i \(-0.656344\pi\)
−0.471658 + 0.881782i \(0.656344\pi\)
\(258\) −5.80841 −0.361616
\(259\) −14.0007 −0.869958
\(260\) −2.03713 −0.126337
\(261\) −17.3953 −1.07674
\(262\) 4.27183 0.263915
\(263\) −5.51617 −0.340142 −0.170071 0.985432i \(-0.554400\pi\)
−0.170071 + 0.985432i \(0.554400\pi\)
\(264\) −9.27757 −0.570995
\(265\) −2.13346 −0.131058
\(266\) 31.0039 1.90097
\(267\) 18.7708 1.14875
\(268\) −17.4360 −1.06507
\(269\) −25.5878 −1.56011 −0.780056 0.625709i \(-0.784810\pi\)
−0.780056 + 0.625709i \(0.784810\pi\)
\(270\) −0.175746 −0.0106955
\(271\) 8.35581 0.507579 0.253790 0.967259i \(-0.418323\pi\)
0.253790 + 0.967259i \(0.418323\pi\)
\(272\) −1.74579 −0.105854
\(273\) −3.41156 −0.206477
\(274\) 13.0761 0.789957
\(275\) −3.79132 −0.228625
\(276\) −66.9978 −4.03279
\(277\) 6.54336 0.393153 0.196576 0.980489i \(-0.437018\pi\)
0.196576 + 0.980489i \(0.437018\pi\)
\(278\) −17.0241 −1.02104
\(279\) −12.2955 −0.736112
\(280\) −11.2077 −0.669785
\(281\) 4.85109 0.289392 0.144696 0.989476i \(-0.453780\pi\)
0.144696 + 0.989476i \(0.453780\pi\)
\(282\) −25.7742 −1.53483
\(283\) −12.9371 −0.769031 −0.384515 0.923119i \(-0.625631\pi\)
−0.384515 + 0.923119i \(0.625631\pi\)
\(284\) −53.7774 −3.19110
\(285\) −13.1117 −0.776671
\(286\) 1.21865 0.0720602
\(287\) 15.2530 0.900355
\(288\) 10.3769 0.611463
\(289\) 1.00000 0.0588235
\(290\) 14.9447 0.877585
\(291\) 15.0823 0.884140
\(292\) 51.8578 3.03475
\(293\) 24.5242 1.43272 0.716359 0.697732i \(-0.245807\pi\)
0.716359 + 0.697732i \(0.245807\pi\)
\(294\) −1.61234 −0.0940334
\(295\) 3.19493 0.186016
\(296\) 19.6119 1.13992
\(297\) 0.0675680 0.00392069
\(298\) 27.3709 1.58555
\(299\) 3.90759 0.225982
\(300\) −33.4836 −1.93318
\(301\) 2.69770 0.155493
\(302\) −19.9058 −1.14545
\(303\) 6.60420 0.379401
\(304\) −8.48058 −0.486395
\(305\) −10.7254 −0.614132
\(306\) 7.16267 0.409463
\(307\) 30.3334 1.73122 0.865610 0.500719i \(-0.166931\pi\)
0.865610 + 0.500719i \(0.166931\pi\)
\(308\) 9.70434 0.552956
\(309\) −23.8781 −1.35838
\(310\) 10.5634 0.599959
\(311\) −2.26764 −0.128586 −0.0642931 0.997931i \(-0.520479\pi\)
−0.0642931 + 0.997931i \(0.520479\pi\)
\(312\) 4.77887 0.270550
\(313\) 5.79665 0.327646 0.163823 0.986490i \(-0.447617\pi\)
0.163823 + 0.986490i \(0.447617\pi\)
\(314\) −10.3597 −0.584633
\(315\) 8.97918 0.505920
\(316\) 33.4370 1.88098
\(317\) −19.2754 −1.08261 −0.541307 0.840825i \(-0.682070\pi\)
−0.541307 + 0.840825i \(0.682070\pi\)
\(318\) 11.2716 0.632082
\(319\) −5.74572 −0.321698
\(320\) −12.7537 −0.712952
\(321\) −45.4594 −2.53729
\(322\) 48.4173 2.69819
\(323\) 4.85774 0.270292
\(324\) −32.0757 −1.78198
\(325\) 1.95290 0.108328
\(326\) −27.4569 −1.52070
\(327\) 21.3352 1.17984
\(328\) −21.3661 −1.17975
\(329\) 11.9707 0.659968
\(330\) −6.38577 −0.351525
\(331\) −10.0549 −0.552667 −0.276334 0.961062i \(-0.589119\pi\)
−0.276334 + 0.961062i \(0.589119\pi\)
\(332\) −56.9787 −3.12711
\(333\) −15.7124 −0.861033
\(334\) 13.7377 0.751692
\(335\) −5.32882 −0.291144
\(336\) 11.5626 0.630790
\(337\) −7.97201 −0.434263 −0.217132 0.976142i \(-0.569670\pi\)
−0.217132 + 0.976142i \(0.569670\pi\)
\(338\) 30.1284 1.63877
\(339\) 2.23404 0.121336
\(340\) −3.95483 −0.214481
\(341\) −4.06124 −0.219928
\(342\) 34.7944 1.88147
\(343\) −18.1351 −0.979201
\(344\) −3.77890 −0.203744
\(345\) −20.4759 −1.10239
\(346\) 26.0282 1.39928
\(347\) 3.16925 0.170134 0.0850671 0.996375i \(-0.472890\pi\)
0.0850671 + 0.996375i \(0.472890\pi\)
\(348\) −50.7442 −2.72017
\(349\) 7.57402 0.405428 0.202714 0.979238i \(-0.435024\pi\)
0.202714 + 0.979238i \(0.435024\pi\)
\(350\) 24.1976 1.29342
\(351\) −0.0348042 −0.00185771
\(352\) 3.42751 0.182687
\(353\) 30.6821 1.63305 0.816523 0.577313i \(-0.195899\pi\)
0.816523 + 0.577313i \(0.195899\pi\)
\(354\) −16.8796 −0.897142
\(355\) −16.4355 −0.872306
\(356\) 27.5034 1.45767
\(357\) −6.62313 −0.350533
\(358\) 14.8821 0.786544
\(359\) −3.07564 −0.162326 −0.0811631 0.996701i \(-0.525863\pi\)
−0.0811631 + 0.996701i \(0.525863\pi\)
\(360\) −12.5779 −0.662914
\(361\) 4.59763 0.241980
\(362\) −2.77067 −0.145623
\(363\) 2.45510 0.128859
\(364\) −4.99869 −0.262003
\(365\) 15.8489 0.829567
\(366\) 56.6648 2.96192
\(367\) −6.81938 −0.355969 −0.177984 0.984033i \(-0.556958\pi\)
−0.177984 + 0.984033i \(0.556958\pi\)
\(368\) −13.2437 −0.690377
\(369\) 17.1178 0.891117
\(370\) 13.4989 0.701774
\(371\) −5.23508 −0.271792
\(372\) −35.8675 −1.85964
\(373\) 14.6839 0.760303 0.380152 0.924924i \(-0.375872\pi\)
0.380152 + 0.924924i \(0.375872\pi\)
\(374\) 2.36585 0.122335
\(375\) −23.7290 −1.22536
\(376\) −16.7684 −0.864766
\(377\) 2.95961 0.152428
\(378\) −0.431243 −0.0221808
\(379\) −12.6962 −0.652161 −0.326081 0.945342i \(-0.605728\pi\)
−0.326081 + 0.945342i \(0.605728\pi\)
\(380\) −19.2116 −0.985532
\(381\) −1.71053 −0.0876329
\(382\) 27.9897 1.43208
\(383\) −28.9038 −1.47691 −0.738457 0.674300i \(-0.764445\pi\)
−0.738457 + 0.674300i \(0.764445\pi\)
\(384\) 50.5511 2.57968
\(385\) 2.96585 0.151154
\(386\) −61.7763 −3.14433
\(387\) 3.02752 0.153898
\(388\) 22.0989 1.12190
\(389\) 29.7565 1.50872 0.754358 0.656463i \(-0.227948\pi\)
0.754358 + 0.656463i \(0.227948\pi\)
\(390\) 3.28930 0.166560
\(391\) 7.58610 0.383646
\(392\) −1.04897 −0.0529810
\(393\) −4.43298 −0.223614
\(394\) 5.54809 0.279509
\(395\) 10.2191 0.514177
\(396\) 10.8908 0.547283
\(397\) −21.4070 −1.07439 −0.537193 0.843459i \(-0.680515\pi\)
−0.537193 + 0.843459i \(0.680515\pi\)
\(398\) −39.1695 −1.96339
\(399\) −32.1734 −1.61069
\(400\) −6.61884 −0.330942
\(401\) −10.2067 −0.509700 −0.254850 0.966981i \(-0.582026\pi\)
−0.254850 + 0.966981i \(0.582026\pi\)
\(402\) 28.1535 1.40417
\(403\) 2.09194 0.104207
\(404\) 9.67660 0.481429
\(405\) −9.80300 −0.487115
\(406\) 36.6713 1.81996
\(407\) −5.18985 −0.257251
\(408\) 9.27757 0.459308
\(409\) −29.9332 −1.48010 −0.740051 0.672551i \(-0.765199\pi\)
−0.740051 + 0.672551i \(0.765199\pi\)
\(410\) −14.7063 −0.726294
\(411\) −13.5694 −0.669329
\(412\) −34.9867 −1.72367
\(413\) 7.83969 0.385766
\(414\) 54.3368 2.67051
\(415\) −17.4139 −0.854815
\(416\) −1.76551 −0.0865612
\(417\) 17.6663 0.865124
\(418\) 11.4927 0.562126
\(419\) 23.1330 1.13012 0.565061 0.825049i \(-0.308853\pi\)
0.565061 + 0.825049i \(0.308853\pi\)
\(420\) 26.1934 1.27811
\(421\) 10.9290 0.532646 0.266323 0.963884i \(-0.414191\pi\)
0.266323 + 0.963884i \(0.414191\pi\)
\(422\) −9.63972 −0.469254
\(423\) 13.4343 0.653197
\(424\) 7.33322 0.356132
\(425\) 3.79132 0.183906
\(426\) 86.8330 4.20707
\(427\) −26.3178 −1.27361
\(428\) −66.6080 −3.21962
\(429\) −1.26462 −0.0610564
\(430\) −2.60102 −0.125432
\(431\) 29.7067 1.43092 0.715460 0.698654i \(-0.246217\pi\)
0.715460 + 0.698654i \(0.246217\pi\)
\(432\) 0.117959 0.00567532
\(433\) 29.7726 1.43078 0.715391 0.698724i \(-0.246249\pi\)
0.715391 + 0.698724i \(0.246249\pi\)
\(434\) 25.9203 1.24422
\(435\) −15.5085 −0.743575
\(436\) 31.2607 1.49712
\(437\) 36.8513 1.76284
\(438\) −83.7335 −4.00094
\(439\) −3.54480 −0.169184 −0.0845921 0.996416i \(-0.526959\pi\)
−0.0845921 + 0.996416i \(0.526959\pi\)
\(440\) −4.15452 −0.198059
\(441\) 0.840399 0.0400190
\(442\) −1.21865 −0.0579652
\(443\) 9.39914 0.446567 0.223283 0.974754i \(-0.428322\pi\)
0.223283 + 0.974754i \(0.428322\pi\)
\(444\) −45.8349 −2.17523
\(445\) 8.40561 0.398464
\(446\) 36.3576 1.72158
\(447\) −28.4034 −1.34344
\(448\) −31.2949 −1.47854
\(449\) −2.29059 −0.108100 −0.0540498 0.998538i \(-0.517213\pi\)
−0.0540498 + 0.998538i \(0.517213\pi\)
\(450\) 27.1560 1.28015
\(451\) 5.65407 0.266240
\(452\) 3.27336 0.153966
\(453\) 20.6567 0.970536
\(454\) −41.7675 −1.96025
\(455\) −1.52771 −0.0716200
\(456\) 45.0680 2.11050
\(457\) −5.69855 −0.266567 −0.133283 0.991078i \(-0.542552\pi\)
−0.133283 + 0.991078i \(0.542552\pi\)
\(458\) −27.6747 −1.29315
\(459\) −0.0675680 −0.00315380
\(460\) −30.0018 −1.39884
\(461\) 37.7082 1.75624 0.878122 0.478436i \(-0.158796\pi\)
0.878122 + 0.478436i \(0.158796\pi\)
\(462\) −15.6694 −0.729004
\(463\) 2.73444 0.127080 0.0635401 0.997979i \(-0.479761\pi\)
0.0635401 + 0.997979i \(0.479761\pi\)
\(464\) −10.0308 −0.465668
\(465\) −10.9619 −0.508344
\(466\) 17.5006 0.810699
\(467\) −0.612711 −0.0283529 −0.0141764 0.999900i \(-0.504513\pi\)
−0.0141764 + 0.999900i \(0.504513\pi\)
\(468\) −5.60984 −0.259315
\(469\) −13.0758 −0.603785
\(470\) −11.5417 −0.532381
\(471\) 10.7505 0.495358
\(472\) −10.9817 −0.505475
\(473\) 1.00000 0.0459800
\(474\) −53.9899 −2.47984
\(475\) 18.4172 0.845041
\(476\) −9.70434 −0.444798
\(477\) −5.87512 −0.269003
\(478\) 12.9857 0.593951
\(479\) −37.3057 −1.70454 −0.852270 0.523102i \(-0.824774\pi\)
−0.852270 + 0.523102i \(0.824774\pi\)
\(480\) 9.25134 0.422264
\(481\) 2.67328 0.121891
\(482\) −31.4143 −1.43088
\(483\) −50.2437 −2.28617
\(484\) 3.59727 0.163512
\(485\) 6.75389 0.306678
\(486\) 52.2713 2.37107
\(487\) 20.5030 0.929080 0.464540 0.885552i \(-0.346220\pi\)
0.464540 + 0.885552i \(0.346220\pi\)
\(488\) 36.8656 1.66883
\(489\) 28.4927 1.28848
\(490\) −0.722008 −0.0326170
\(491\) 12.8644 0.580564 0.290282 0.956941i \(-0.406251\pi\)
0.290282 + 0.956941i \(0.406251\pi\)
\(492\) 49.9347 2.25123
\(493\) 5.74572 0.258774
\(494\) −5.91987 −0.266348
\(495\) 3.32846 0.149603
\(496\) −7.09006 −0.318353
\(497\) −40.3293 −1.80902
\(498\) 92.0021 4.12271
\(499\) 5.46925 0.244837 0.122419 0.992479i \(-0.460935\pi\)
0.122419 + 0.992479i \(0.460935\pi\)
\(500\) −34.7682 −1.55488
\(501\) −14.2559 −0.636907
\(502\) −16.0048 −0.714328
\(503\) −20.7481 −0.925114 −0.462557 0.886590i \(-0.653068\pi\)
−0.462557 + 0.886590i \(0.653068\pi\)
\(504\) −30.8636 −1.37477
\(505\) 2.95737 0.131601
\(506\) 17.9476 0.797868
\(507\) −31.2649 −1.38852
\(508\) −2.50630 −0.111199
\(509\) 22.8313 1.01198 0.505989 0.862540i \(-0.331128\pi\)
0.505989 + 0.862540i \(0.331128\pi\)
\(510\) 6.38577 0.282767
\(511\) 38.8898 1.72038
\(512\) 19.1780 0.847557
\(513\) −0.328228 −0.0144916
\(514\) 35.7776 1.57808
\(515\) −10.6927 −0.471175
\(516\) 8.83165 0.388792
\(517\) 4.43739 0.195156
\(518\) 33.1235 1.45536
\(519\) −27.0100 −1.18561
\(520\) 2.13999 0.0938447
\(521\) −3.59424 −0.157467 −0.0787333 0.996896i \(-0.525088\pi\)
−0.0787333 + 0.996896i \(0.525088\pi\)
\(522\) 41.1547 1.80129
\(523\) −17.0696 −0.746400 −0.373200 0.927751i \(-0.621740\pi\)
−0.373200 + 0.927751i \(0.621740\pi\)
\(524\) −6.49530 −0.283748
\(525\) −25.1104 −1.09591
\(526\) 13.0505 0.569027
\(527\) 4.06124 0.176910
\(528\) 4.28609 0.186528
\(529\) 34.5489 1.50213
\(530\) 5.04746 0.219248
\(531\) 8.79818 0.381808
\(532\) −47.1412 −2.04383
\(533\) −2.91240 −0.126150
\(534\) −44.4089 −1.92176
\(535\) −20.3568 −0.880102
\(536\) 18.3164 0.791148
\(537\) −15.4435 −0.666436
\(538\) 60.5369 2.60993
\(539\) 0.277587 0.0119565
\(540\) 0.267220 0.0114993
\(541\) 15.8809 0.682772 0.341386 0.939923i \(-0.389104\pi\)
0.341386 + 0.939923i \(0.389104\pi\)
\(542\) −19.7686 −0.849135
\(543\) 2.87519 0.123386
\(544\) −3.42751 −0.146953
\(545\) 9.55393 0.409246
\(546\) 8.07126 0.345418
\(547\) 30.7601 1.31521 0.657604 0.753363i \(-0.271570\pi\)
0.657604 + 0.753363i \(0.271570\pi\)
\(548\) −19.8822 −0.849323
\(549\) −29.5354 −1.26054
\(550\) 8.96971 0.382470
\(551\) 27.9112 1.18906
\(552\) 70.3806 2.99560
\(553\) 25.0755 1.06632
\(554\) −15.4806 −0.657710
\(555\) −14.0081 −0.594612
\(556\) 25.8851 1.09777
\(557\) 37.1957 1.57603 0.788016 0.615655i \(-0.211108\pi\)
0.788016 + 0.615655i \(0.211108\pi\)
\(558\) 29.0893 1.23145
\(559\) −0.515099 −0.0217864
\(560\) 5.17775 0.218800
\(561\) −2.45510 −0.103654
\(562\) −11.4770 −0.484127
\(563\) 32.3657 1.36405 0.682026 0.731328i \(-0.261099\pi\)
0.682026 + 0.731328i \(0.261099\pi\)
\(564\) 39.1895 1.65017
\(565\) 1.00041 0.0420875
\(566\) 30.6073 1.28652
\(567\) −24.0545 −1.01020
\(568\) 56.4927 2.37038
\(569\) −47.1163 −1.97522 −0.987609 0.156932i \(-0.949840\pi\)
−0.987609 + 0.156932i \(0.949840\pi\)
\(570\) 31.0204 1.29930
\(571\) 0.292248 0.0122302 0.00611511 0.999981i \(-0.498053\pi\)
0.00611511 + 0.999981i \(0.498053\pi\)
\(572\) −1.85295 −0.0774756
\(573\) −29.0456 −1.21340
\(574\) −36.0863 −1.50621
\(575\) 28.7613 1.19943
\(576\) −35.1210 −1.46337
\(577\) 25.4245 1.05844 0.529219 0.848485i \(-0.322485\pi\)
0.529219 + 0.848485i \(0.322485\pi\)
\(578\) −2.36585 −0.0984066
\(579\) 64.1066 2.66418
\(580\) −22.7234 −0.943536
\(581\) −42.7301 −1.77274
\(582\) −35.6825 −1.47909
\(583\) −1.94057 −0.0803702
\(584\) −54.4762 −2.25424
\(585\) −1.71448 −0.0708852
\(586\) −58.0206 −2.39681
\(587\) 12.4695 0.514671 0.257335 0.966322i \(-0.417155\pi\)
0.257335 + 0.966322i \(0.417155\pi\)
\(588\) 2.45155 0.101100
\(589\) 19.7284 0.812896
\(590\) −7.55874 −0.311188
\(591\) −5.75738 −0.236827
\(592\) −9.06037 −0.372379
\(593\) 3.15522 0.129569 0.0647847 0.997899i \(-0.479364\pi\)
0.0647847 + 0.997899i \(0.479364\pi\)
\(594\) −0.159856 −0.00655897
\(595\) −2.96585 −0.121588
\(596\) −41.6173 −1.70471
\(597\) 40.6471 1.66358
\(598\) −9.24479 −0.378047
\(599\) 15.9386 0.651232 0.325616 0.945502i \(-0.394428\pi\)
0.325616 + 0.945502i \(0.394428\pi\)
\(600\) 35.1742 1.43598
\(601\) 27.8894 1.13763 0.568816 0.822465i \(-0.307402\pi\)
0.568816 + 0.822465i \(0.307402\pi\)
\(602\) −6.38237 −0.260126
\(603\) −14.6745 −0.597590
\(604\) 30.2666 1.23153
\(605\) 1.09940 0.0446970
\(606\) −15.6246 −0.634704
\(607\) −26.6709 −1.08254 −0.541270 0.840849i \(-0.682056\pi\)
−0.541270 + 0.840849i \(0.682056\pi\)
\(608\) −16.6500 −0.675246
\(609\) −38.0546 −1.54205
\(610\) 25.3746 1.02739
\(611\) −2.28569 −0.0924692
\(612\) −10.8908 −0.440234
\(613\) −23.7466 −0.959115 −0.479557 0.877511i \(-0.659203\pi\)
−0.479557 + 0.877511i \(0.659203\pi\)
\(614\) −71.7644 −2.89618
\(615\) 15.2611 0.615387
\(616\) −10.1943 −0.410741
\(617\) −18.8314 −0.758122 −0.379061 0.925372i \(-0.623753\pi\)
−0.379061 + 0.925372i \(0.623753\pi\)
\(618\) 56.4921 2.27244
\(619\) 1.36437 0.0548388 0.0274194 0.999624i \(-0.491271\pi\)
0.0274194 + 0.999624i \(0.491271\pi\)
\(620\) −16.0615 −0.645047
\(621\) −0.512577 −0.0205690
\(622\) 5.36491 0.215113
\(623\) 20.6256 0.826348
\(624\) −2.20776 −0.0883810
\(625\) 8.33069 0.333228
\(626\) −13.7140 −0.548123
\(627\) −11.9262 −0.476288
\(628\) 15.7519 0.628569
\(629\) 5.18985 0.206933
\(630\) −21.2434 −0.846359
\(631\) 6.14796 0.244746 0.122373 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(632\) −35.1253 −1.39721
\(633\) 10.0034 0.397598
\(634\) 45.6028 1.81112
\(635\) −0.765978 −0.0303969
\(636\) −17.1384 −0.679583
\(637\) −0.142984 −0.00566525
\(638\) 13.5935 0.538173
\(639\) −45.2600 −1.79046
\(640\) 22.6369 0.894803
\(641\) −25.9152 −1.02359 −0.511795 0.859108i \(-0.671019\pi\)
−0.511795 + 0.859108i \(0.671019\pi\)
\(642\) 107.550 4.24467
\(643\) −0.140942 −0.00555819 −0.00277910 0.999996i \(-0.500885\pi\)
−0.00277910 + 0.999996i \(0.500885\pi\)
\(644\) −73.6181 −2.90096
\(645\) 2.69914 0.106278
\(646\) −11.4927 −0.452174
\(647\) −37.9116 −1.49046 −0.745228 0.666809i \(-0.767660\pi\)
−0.745228 + 0.666809i \(0.767660\pi\)
\(648\) 33.6952 1.32367
\(649\) 2.90607 0.114073
\(650\) −4.62028 −0.181222
\(651\) −26.8981 −1.05422
\(652\) 41.7481 1.63498
\(653\) 15.4659 0.605228 0.302614 0.953113i \(-0.402141\pi\)
0.302614 + 0.953113i \(0.402141\pi\)
\(654\) −50.4759 −1.97376
\(655\) −1.98510 −0.0775643
\(656\) 9.87080 0.385390
\(657\) 43.6445 1.70273
\(658\) −28.3210 −1.10407
\(659\) 35.5619 1.38530 0.692648 0.721276i \(-0.256444\pi\)
0.692648 + 0.721276i \(0.256444\pi\)
\(660\) 9.70952 0.377943
\(661\) 26.3928 1.02656 0.513281 0.858221i \(-0.328430\pi\)
0.513281 + 0.858221i \(0.328430\pi\)
\(662\) 23.7884 0.924563
\(663\) 1.26462 0.0491137
\(664\) 59.8557 2.32285
\(665\) −14.4073 −0.558693
\(666\) 37.1732 1.44043
\(667\) 43.5876 1.68772
\(668\) −20.8880 −0.808183
\(669\) −37.7291 −1.45869
\(670\) 12.6072 0.487059
\(671\) −9.75564 −0.376612
\(672\) 22.7009 0.875705
\(673\) −0.823571 −0.0317463 −0.0158732 0.999874i \(-0.505053\pi\)
−0.0158732 + 0.999874i \(0.505053\pi\)
\(674\) 18.8606 0.726484
\(675\) −0.256172 −0.00986006
\(676\) −45.8100 −1.76192
\(677\) 28.1551 1.08209 0.541045 0.840994i \(-0.318029\pi\)
0.541045 + 0.840994i \(0.318029\pi\)
\(678\) −5.28542 −0.202985
\(679\) 16.5726 0.636000
\(680\) 4.15452 0.159319
\(681\) 43.3431 1.66091
\(682\) 9.60830 0.367921
\(683\) 4.59032 0.175644 0.0878219 0.996136i \(-0.472009\pi\)
0.0878219 + 0.996136i \(0.472009\pi\)
\(684\) −52.9047 −2.02286
\(685\) −6.07641 −0.232168
\(686\) 42.9049 1.63812
\(687\) 28.7186 1.09568
\(688\) 1.74579 0.0665575
\(689\) 0.999585 0.0380812
\(690\) 48.4431 1.84420
\(691\) −15.5542 −0.591709 −0.295854 0.955233i \(-0.595604\pi\)
−0.295854 + 0.955233i \(0.595604\pi\)
\(692\) −39.5756 −1.50444
\(693\) 8.16735 0.310252
\(694\) −7.49798 −0.284620
\(695\) 7.91103 0.300082
\(696\) 53.3063 2.02057
\(697\) −5.65407 −0.214163
\(698\) −17.9190 −0.678245
\(699\) −18.1608 −0.686903
\(700\) −36.7923 −1.39062
\(701\) −19.3849 −0.732156 −0.366078 0.930584i \(-0.619300\pi\)
−0.366078 + 0.930584i \(0.619300\pi\)
\(702\) 0.0823416 0.00310778
\(703\) 25.2109 0.950848
\(704\) −11.6006 −0.437213
\(705\) 11.9771 0.451085
\(706\) −72.5895 −2.73194
\(707\) 7.25678 0.272919
\(708\) 25.6654 0.964563
\(709\) 12.8044 0.480879 0.240440 0.970664i \(-0.422708\pi\)
0.240440 + 0.970664i \(0.422708\pi\)
\(710\) 38.8840 1.45929
\(711\) 28.1412 1.05538
\(712\) −28.8920 −1.08277
\(713\) 30.8090 1.15381
\(714\) 15.6694 0.586411
\(715\) −0.566300 −0.0211784
\(716\) −22.6281 −0.845654
\(717\) −13.4755 −0.503253
\(718\) 7.27652 0.271557
\(719\) −11.3657 −0.423868 −0.211934 0.977284i \(-0.567976\pi\)
−0.211934 + 0.977284i \(0.567976\pi\)
\(720\) 5.81078 0.216555
\(721\) −26.2376 −0.977139
\(722\) −10.8773 −0.404812
\(723\) 32.5994 1.21238
\(724\) 4.21279 0.156567
\(725\) 21.7838 0.809032
\(726\) −5.80841 −0.215570
\(727\) 11.1880 0.414939 0.207469 0.978242i \(-0.433477\pi\)
0.207469 + 0.978242i \(0.433477\pi\)
\(728\) 5.25109 0.194618
\(729\) −27.4931 −1.01826
\(730\) −37.4961 −1.38779
\(731\) −1.00000 −0.0369863
\(732\) −86.1584 −3.18451
\(733\) −38.4543 −1.42034 −0.710171 0.704030i \(-0.751382\pi\)
−0.710171 + 0.704030i \(0.751382\pi\)
\(734\) 16.1337 0.595505
\(735\) 0.749245 0.0276363
\(736\) −26.0015 −0.958427
\(737\) −4.84702 −0.178542
\(738\) −40.4982 −1.49076
\(739\) −45.8392 −1.68622 −0.843111 0.537739i \(-0.819279\pi\)
−0.843111 + 0.537739i \(0.819279\pi\)
\(740\) −20.5250 −0.754514
\(741\) 6.14319 0.225676
\(742\) 12.3854 0.454683
\(743\) −26.9014 −0.986917 −0.493458 0.869769i \(-0.664267\pi\)
−0.493458 + 0.869769i \(0.664267\pi\)
\(744\) 37.6785 1.38136
\(745\) −12.7191 −0.465993
\(746\) −34.7400 −1.27192
\(747\) −47.9543 −1.75456
\(748\) −3.59727 −0.131529
\(749\) −49.9514 −1.82518
\(750\) 56.1393 2.04992
\(751\) −7.27244 −0.265375 −0.132688 0.991158i \(-0.542361\pi\)
−0.132688 + 0.991158i \(0.542361\pi\)
\(752\) 7.74674 0.282494
\(753\) 16.6085 0.605248
\(754\) −7.00201 −0.254998
\(755\) 9.25012 0.336646
\(756\) 0.655703 0.0238477
\(757\) 27.8741 1.01310 0.506550 0.862210i \(-0.330921\pi\)
0.506550 + 0.862210i \(0.330921\pi\)
\(758\) 30.0374 1.09101
\(759\) −18.6246 −0.676032
\(760\) 20.1816 0.732063
\(761\) −14.2615 −0.516980 −0.258490 0.966014i \(-0.583225\pi\)
−0.258490 + 0.966014i \(0.583225\pi\)
\(762\) 4.04685 0.146602
\(763\) 23.4434 0.848707
\(764\) −42.5582 −1.53970
\(765\) −3.32846 −0.120341
\(766\) 68.3822 2.47075
\(767\) −1.49691 −0.0540503
\(768\) −62.6354 −2.26016
\(769\) −16.5906 −0.598272 −0.299136 0.954210i \(-0.596698\pi\)
−0.299136 + 0.954210i \(0.596698\pi\)
\(770\) −7.01677 −0.252867
\(771\) −37.1273 −1.33711
\(772\) 93.9304 3.38063
\(773\) 15.9947 0.575290 0.287645 0.957737i \(-0.407128\pi\)
0.287645 + 0.957737i \(0.407128\pi\)
\(774\) −7.16267 −0.257457
\(775\) 15.3975 0.553093
\(776\) −23.2147 −0.833359
\(777\) −34.3730 −1.23313
\(778\) −70.3996 −2.52395
\(779\) −27.4660 −0.984071
\(780\) −5.00136 −0.179077
\(781\) −14.9495 −0.534936
\(782\) −17.9476 −0.641805
\(783\) −0.388227 −0.0138741
\(784\) 0.484607 0.0173074
\(785\) 4.81411 0.171823
\(786\) 10.4878 0.374087
\(787\) −24.2651 −0.864958 −0.432479 0.901644i \(-0.642361\pi\)
−0.432479 + 0.901644i \(0.642361\pi\)
\(788\) −8.43583 −0.300514
\(789\) −13.5428 −0.482135
\(790\) −24.1768 −0.860172
\(791\) 2.45480 0.0872825
\(792\) −11.4407 −0.406527
\(793\) 5.02512 0.178447
\(794\) 50.6458 1.79735
\(795\) −5.23787 −0.185768
\(796\) 59.5570 2.11094
\(797\) −45.6833 −1.61819 −0.809093 0.587680i \(-0.800041\pi\)
−0.809093 + 0.587680i \(0.800041\pi\)
\(798\) 76.1176 2.69453
\(799\) −4.43739 −0.156983
\(800\) −12.9948 −0.459436
\(801\) 23.1473 0.817870
\(802\) 24.1476 0.852683
\(803\) 14.4159 0.508726
\(804\) −42.8072 −1.50969
\(805\) −22.4993 −0.792994
\(806\) −4.94922 −0.174329
\(807\) −62.8205 −2.21139
\(808\) −10.1652 −0.357610
\(809\) 45.0028 1.58221 0.791107 0.611677i \(-0.209505\pi\)
0.791107 + 0.611677i \(0.209505\pi\)
\(810\) 23.1925 0.814900
\(811\) 5.42318 0.190434 0.0952169 0.995457i \(-0.469646\pi\)
0.0952169 + 0.995457i \(0.469646\pi\)
\(812\) −55.7584 −1.95674
\(813\) 20.5144 0.719470
\(814\) 12.2784 0.430358
\(815\) 12.7591 0.446932
\(816\) −4.28609 −0.150043
\(817\) −4.85774 −0.169951
\(818\) 70.8176 2.47608
\(819\) −4.20699 −0.147004
\(820\) 22.3609 0.780876
\(821\) −29.3995 −1.02605 −0.513024 0.858374i \(-0.671475\pi\)
−0.513024 + 0.858374i \(0.671475\pi\)
\(822\) 32.1032 1.11973
\(823\) 28.8594 1.00597 0.502987 0.864294i \(-0.332234\pi\)
0.502987 + 0.864294i \(0.332234\pi\)
\(824\) 36.7532 1.28036
\(825\) −9.30807 −0.324065
\(826\) −18.5476 −0.645353
\(827\) 5.44279 0.189264 0.0946322 0.995512i \(-0.469832\pi\)
0.0946322 + 0.995512i \(0.469832\pi\)
\(828\) −82.6187 −2.87120
\(829\) 10.9220 0.379336 0.189668 0.981848i \(-0.439259\pi\)
0.189668 + 0.981848i \(0.439259\pi\)
\(830\) 41.1987 1.43003
\(831\) 16.0646 0.557276
\(832\) 5.97544 0.207161
\(833\) −0.277587 −0.00961780
\(834\) −41.7960 −1.44728
\(835\) −6.38383 −0.220921
\(836\) −17.4746 −0.604371
\(837\) −0.274410 −0.00948499
\(838\) −54.7293 −1.89059
\(839\) 1.23733 0.0427175 0.0213588 0.999772i \(-0.493201\pi\)
0.0213588 + 0.999772i \(0.493201\pi\)
\(840\) −27.5159 −0.949389
\(841\) 4.01327 0.138389
\(842\) −25.8564 −0.891069
\(843\) 11.9099 0.410200
\(844\) 14.6571 0.504519
\(845\) −14.0005 −0.481632
\(846\) −31.7836 −1.09274
\(847\) 2.69770 0.0926941
\(848\) −3.38782 −0.116338
\(849\) −31.7619 −1.09007
\(850\) −8.96971 −0.307658
\(851\) 39.3707 1.34961
\(852\) −132.029 −4.52324
\(853\) 21.2865 0.728837 0.364419 0.931235i \(-0.381268\pi\)
0.364419 + 0.931235i \(0.381268\pi\)
\(854\) 62.2641 2.13063
\(855\) −16.1688 −0.552961
\(856\) 69.9711 2.39156
\(857\) −38.0727 −1.30054 −0.650269 0.759704i \(-0.725344\pi\)
−0.650269 + 0.759704i \(0.725344\pi\)
\(858\) 2.99190 0.102142
\(859\) −36.9258 −1.25989 −0.629946 0.776639i \(-0.716923\pi\)
−0.629946 + 0.776639i \(0.716923\pi\)
\(860\) 3.95483 0.134859
\(861\) 37.4476 1.27621
\(862\) −70.2816 −2.39380
\(863\) 0.0207226 0.000705406 0 0.000352703 1.00000i \(-0.499888\pi\)
0.000352703 1.00000i \(0.499888\pi\)
\(864\) 0.231590 0.00787886
\(865\) −12.0952 −0.411248
\(866\) −70.4377 −2.39357
\(867\) 2.45510 0.0833796
\(868\) −39.4117 −1.33772
\(869\) 9.29512 0.315315
\(870\) 36.6908 1.24394
\(871\) 2.49669 0.0845973
\(872\) −32.8391 −1.11207
\(873\) 18.5988 0.629475
\(874\) −87.1848 −2.94907
\(875\) −26.0738 −0.881454
\(876\) 127.316 4.30162
\(877\) 23.2900 0.786446 0.393223 0.919443i \(-0.371360\pi\)
0.393223 + 0.919443i \(0.371360\pi\)
\(878\) 8.38649 0.283030
\(879\) 60.2093 2.03081
\(880\) 1.91932 0.0647002
\(881\) 6.96971 0.234815 0.117408 0.993084i \(-0.462542\pi\)
0.117408 + 0.993084i \(0.462542\pi\)
\(882\) −1.98826 −0.0669483
\(883\) 1.91163 0.0643315 0.0321657 0.999483i \(-0.489760\pi\)
0.0321657 + 0.999483i \(0.489760\pi\)
\(884\) 1.85295 0.0623213
\(885\) 7.84388 0.263669
\(886\) −22.2370 −0.747067
\(887\) 2.84615 0.0955643 0.0477822 0.998858i \(-0.484785\pi\)
0.0477822 + 0.998858i \(0.484785\pi\)
\(888\) 48.1492 1.61578
\(889\) −1.87955 −0.0630381
\(890\) −19.8864 −0.666595
\(891\) −8.91668 −0.298720
\(892\) −55.2815 −1.85096
\(893\) −21.5557 −0.721333
\(894\) 67.1983 2.24745
\(895\) −6.91564 −0.231164
\(896\) 55.5463 1.85567
\(897\) 9.59353 0.320319
\(898\) 5.41920 0.180841
\(899\) 23.3347 0.778257
\(900\) −41.2905 −1.37635
\(901\) 1.94057 0.0646498
\(902\) −13.3767 −0.445395
\(903\) 6.62313 0.220404
\(904\) −3.43864 −0.114368
\(905\) 1.28752 0.0427986
\(906\) −48.8707 −1.62362
\(907\) 34.1374 1.13351 0.566757 0.823885i \(-0.308198\pi\)
0.566757 + 0.823885i \(0.308198\pi\)
\(908\) 63.5072 2.10756
\(909\) 8.14400 0.270119
\(910\) 3.61433 0.119814
\(911\) 8.49534 0.281463 0.140732 0.990048i \(-0.455055\pi\)
0.140732 + 0.990048i \(0.455055\pi\)
\(912\) −20.8207 −0.689442
\(913\) −15.8395 −0.524209
\(914\) 13.4819 0.445943
\(915\) −26.3318 −0.870503
\(916\) 42.0791 1.39033
\(917\) −4.87102 −0.160855
\(918\) 0.159856 0.00527603
\(919\) 31.5826 1.04181 0.520906 0.853614i \(-0.325594\pi\)
0.520906 + 0.853614i \(0.325594\pi\)
\(920\) 31.5166 1.03907
\(921\) 74.4716 2.45392
\(922\) −89.2120 −2.93804
\(923\) 7.70048 0.253464
\(924\) 23.8251 0.783790
\(925\) 19.6764 0.646955
\(926\) −6.46929 −0.212594
\(927\) −29.4454 −0.967114
\(928\) −19.6935 −0.646472
\(929\) −0.422315 −0.0138557 −0.00692786 0.999976i \(-0.502205\pi\)
−0.00692786 + 0.999976i \(0.502205\pi\)
\(930\) 25.9341 0.850414
\(931\) −1.34844 −0.0441934
\(932\) −26.6095 −0.871624
\(933\) −5.56729 −0.182265
\(934\) 1.44959 0.0474319
\(935\) −1.09940 −0.0359542
\(936\) 5.89308 0.192621
\(937\) −36.9710 −1.20779 −0.603896 0.797063i \(-0.706386\pi\)
−0.603896 + 0.797063i \(0.706386\pi\)
\(938\) 30.9355 1.01008
\(939\) 14.2314 0.464423
\(940\) 17.5491 0.572390
\(941\) 24.2905 0.791848 0.395924 0.918283i \(-0.370424\pi\)
0.395924 + 0.918283i \(0.370424\pi\)
\(942\) −25.4342 −0.828691
\(943\) −42.8923 −1.39677
\(944\) 5.07337 0.165124
\(945\) 0.200397 0.00651890
\(946\) −2.36585 −0.0769206
\(947\) −34.3135 −1.11504 −0.557519 0.830164i \(-0.688247\pi\)
−0.557519 + 0.830164i \(0.688247\pi\)
\(948\) 82.0913 2.66620
\(949\) −7.42561 −0.241046
\(950\) −43.5725 −1.41368
\(951\) −47.3230 −1.53455
\(952\) 10.1943 0.330400
\(953\) −31.5417 −1.02174 −0.510869 0.859659i \(-0.670676\pi\)
−0.510869 + 0.859659i \(0.670676\pi\)
\(954\) 13.8997 0.450019
\(955\) −13.0067 −0.420886
\(956\) −19.7446 −0.638587
\(957\) −14.1063 −0.455993
\(958\) 88.2598 2.85154
\(959\) −14.9102 −0.481477
\(960\) −31.3116 −1.01058
\(961\) −14.5063 −0.467946
\(962\) −6.32460 −0.203913
\(963\) −56.0585 −1.80646
\(964\) 47.7653 1.53842
\(965\) 28.7071 0.924115
\(966\) 118.869 3.82456
\(967\) −9.67987 −0.311283 −0.155642 0.987814i \(-0.549745\pi\)
−0.155642 + 0.987814i \(0.549745\pi\)
\(968\) −3.77890 −0.121458
\(969\) 11.9262 0.383126
\(970\) −15.9787 −0.513046
\(971\) −52.8570 −1.69626 −0.848130 0.529787i \(-0.822272\pi\)
−0.848130 + 0.529787i \(0.822272\pi\)
\(972\) −79.4782 −2.54926
\(973\) 19.4120 0.622321
\(974\) −48.5072 −1.55427
\(975\) 4.79457 0.153549
\(976\) −17.0313 −0.545158
\(977\) −22.3812 −0.716039 −0.358020 0.933714i \(-0.616548\pi\)
−0.358020 + 0.933714i \(0.616548\pi\)
\(978\) −67.4096 −2.15552
\(979\) 7.64563 0.244355
\(980\) 1.09781 0.0350682
\(981\) 26.3096 0.840000
\(982\) −30.4354 −0.971232
\(983\) −4.22132 −0.134639 −0.0673195 0.997731i \(-0.521445\pi\)
−0.0673195 + 0.997731i \(0.521445\pi\)
\(984\) −52.4560 −1.67224
\(985\) −2.57817 −0.0821472
\(986\) −13.5935 −0.432906
\(987\) 29.3894 0.935474
\(988\) 9.00113 0.286364
\(989\) −7.58610 −0.241224
\(990\) −7.87465 −0.250273
\(991\) −5.38609 −0.171095 −0.0855473 0.996334i \(-0.527264\pi\)
−0.0855473 + 0.996334i \(0.527264\pi\)
\(992\) −13.9200 −0.441959
\(993\) −24.6858 −0.783380
\(994\) 95.4133 3.02633
\(995\) 18.2019 0.577038
\(996\) −139.889 −4.43254
\(997\) −43.1298 −1.36594 −0.682968 0.730449i \(-0.739311\pi\)
−0.682968 + 0.730449i \(0.739311\pi\)
\(998\) −12.9395 −0.409591
\(999\) −0.350667 −0.0110946
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 8041.2.a.d.1.7 62
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8041.2.a.d.1.7 62 1.1 even 1 trivial