Properties

Label 1045.6.a.d.1.16
Level $1045$
Weight $6$
Character 1045.1
Self dual yes
Analytic conductor $167.601$
Analytic rank $0$
Dimension $37$
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] = [1045,6,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(167.601091705\)
Analytic rank: \(0\)
Dimension: \(37\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.37290 q^{2} -13.1216 q^{3} -26.3693 q^{4} -25.0000 q^{5} +31.1362 q^{6} +161.256 q^{7} +138.505 q^{8} -70.8249 q^{9} +O(q^{10})\) \(q-2.37290 q^{2} -13.1216 q^{3} -26.3693 q^{4} -25.0000 q^{5} +31.1362 q^{6} +161.256 q^{7} +138.505 q^{8} -70.8249 q^{9} +59.3226 q^{10} +121.000 q^{11} +346.007 q^{12} -854.616 q^{13} -382.644 q^{14} +328.039 q^{15} +515.160 q^{16} -773.382 q^{17} +168.061 q^{18} -361.000 q^{19} +659.233 q^{20} -2115.92 q^{21} -287.121 q^{22} +3015.46 q^{23} -1817.40 q^{24} +625.000 q^{25} +2027.92 q^{26} +4117.87 q^{27} -4252.20 q^{28} +1569.76 q^{29} -778.404 q^{30} +5621.67 q^{31} -5654.58 q^{32} -1587.71 q^{33} +1835.16 q^{34} -4031.39 q^{35} +1867.61 q^{36} -2170.05 q^{37} +856.618 q^{38} +11213.9 q^{39} -3462.62 q^{40} +8714.50 q^{41} +5020.88 q^{42} -2640.06 q^{43} -3190.69 q^{44} +1770.62 q^{45} -7155.40 q^{46} -7280.62 q^{47} -6759.70 q^{48} +9196.37 q^{49} -1483.06 q^{50} +10148.0 q^{51} +22535.6 q^{52} +4281.20 q^{53} -9771.30 q^{54} -3025.00 q^{55} +22334.7 q^{56} +4736.88 q^{57} -3724.88 q^{58} -15566.1 q^{59} -8650.16 q^{60} +1673.74 q^{61} -13339.7 q^{62} -11420.9 q^{63} -3067.37 q^{64} +21365.4 q^{65} +3767.48 q^{66} -5693.31 q^{67} +20393.6 q^{68} -39567.5 q^{69} +9566.10 q^{70} -66500.3 q^{71} -9809.58 q^{72} -27566.1 q^{73} +5149.32 q^{74} -8200.97 q^{75} +9519.33 q^{76} +19511.9 q^{77} -26609.5 q^{78} -38949.0 q^{79} -12879.0 q^{80} -36822.4 q^{81} -20678.7 q^{82} -17829.9 q^{83} +55795.5 q^{84} +19334.6 q^{85} +6264.60 q^{86} -20597.6 q^{87} +16759.1 q^{88} -107643. q^{89} -4201.51 q^{90} -137812. q^{91} -79515.7 q^{92} -73765.1 q^{93} +17276.2 q^{94} +9025.00 q^{95} +74196.8 q^{96} +17058.2 q^{97} -21822.1 q^{98} -8569.81 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 37 q + 4 q^{2} + 27 q^{3} + 616 q^{4} - 925 q^{5} + 141 q^{6} - 79 q^{7} + 72 q^{8} + 3140 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 37 q + 4 q^{2} + 27 q^{3} + 616 q^{4} - 925 q^{5} + 141 q^{6} - 79 q^{7} + 72 q^{8} + 3140 q^{9} - 100 q^{10} + 4477 q^{11} + 872 q^{12} + 719 q^{13} - 625 q^{14} - 675 q^{15} + 6940 q^{16} + 119 q^{17} - 4237 q^{18} - 13357 q^{19} - 15400 q^{20} + 2905 q^{21} + 484 q^{22} - 1252 q^{23} + 5884 q^{24} + 23125 q^{25} + 13201 q^{26} + 9918 q^{27} + 15461 q^{28} + 13221 q^{29} - 3525 q^{30} + 6419 q^{31} + 13173 q^{32} + 3267 q^{33} + 35415 q^{34} + 1975 q^{35} + 80543 q^{36} + 9037 q^{37} - 1444 q^{38} - 6184 q^{39} - 1800 q^{40} + 52577 q^{41} - 28578 q^{42} + 963 q^{43} + 74536 q^{44} - 78500 q^{45} - 10531 q^{46} + 49346 q^{47} + 80107 q^{48} + 70288 q^{49} + 2500 q^{50} + 140786 q^{51} + 165062 q^{52} - 34457 q^{53} + 34216 q^{54} - 111925 q^{55} - 64095 q^{56} - 9747 q^{57} - 126140 q^{58} + 56521 q^{59} - 21800 q^{60} + 6613 q^{61} + 494 q^{62} - 125618 q^{63} - 140426 q^{64} - 17975 q^{65} + 17061 q^{66} - 43534 q^{67} - 138520 q^{68} + 34618 q^{69} + 15625 q^{70} + 95986 q^{71} - 42192 q^{72} + 109218 q^{73} - 182005 q^{74} + 16875 q^{75} - 222376 q^{76} - 9559 q^{77} - 369624 q^{78} + 64943 q^{79} - 173500 q^{80} + 388941 q^{81} - 126926 q^{82} + 109741 q^{83} - 112886 q^{84} - 2975 q^{85} + 43866 q^{86} + 142492 q^{87} + 8712 q^{88} - 119092 q^{89} + 105925 q^{90} + 349320 q^{91} + 433396 q^{92} - 108630 q^{93} + 196160 q^{94} + 333925 q^{95} + 376630 q^{96} + 68774 q^{97} + 310926 q^{98} + 379940 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.37290 −0.419474 −0.209737 0.977758i \(-0.567261\pi\)
−0.209737 + 0.977758i \(0.567261\pi\)
\(3\) −13.1216 −0.841748 −0.420874 0.907119i \(-0.638277\pi\)
−0.420874 + 0.907119i \(0.638277\pi\)
\(4\) −26.3693 −0.824042
\(5\) −25.0000 −0.447214
\(6\) 31.1362 0.353091
\(7\) 161.256 1.24385 0.621927 0.783075i \(-0.286350\pi\)
0.621927 + 0.783075i \(0.286350\pi\)
\(8\) 138.505 0.765138
\(9\) −70.8249 −0.291460
\(10\) 59.3226 0.187594
\(11\) 121.000 0.301511
\(12\) 346.007 0.693635
\(13\) −854.616 −1.40253 −0.701265 0.712900i \(-0.747381\pi\)
−0.701265 + 0.712900i \(0.747381\pi\)
\(14\) −382.644 −0.521765
\(15\) 328.039 0.376441
\(16\) 515.160 0.503086
\(17\) −773.382 −0.649041 −0.324520 0.945879i \(-0.605203\pi\)
−0.324520 + 0.945879i \(0.605203\pi\)
\(18\) 168.061 0.122260
\(19\) −361.000 −0.229416
\(20\) 659.233 0.368523
\(21\) −2115.92 −1.04701
\(22\) −287.121 −0.126476
\(23\) 3015.46 1.18860 0.594298 0.804245i \(-0.297430\pi\)
0.594298 + 0.804245i \(0.297430\pi\)
\(24\) −1817.40 −0.644053
\(25\) 625.000 0.200000
\(26\) 2027.92 0.588325
\(27\) 4117.87 1.08708
\(28\) −4252.20 −1.02499
\(29\) 1569.76 0.346607 0.173303 0.984868i \(-0.444556\pi\)
0.173303 + 0.984868i \(0.444556\pi\)
\(30\) −778.404 −0.157907
\(31\) 5621.67 1.05066 0.525329 0.850899i \(-0.323942\pi\)
0.525329 + 0.850899i \(0.323942\pi\)
\(32\) −5654.58 −0.976169
\(33\) −1587.71 −0.253797
\(34\) 1835.16 0.272256
\(35\) −4031.39 −0.556269
\(36\) 1867.61 0.240176
\(37\) −2170.05 −0.260595 −0.130297 0.991475i \(-0.541593\pi\)
−0.130297 + 0.991475i \(0.541593\pi\)
\(38\) 856.618 0.0962339
\(39\) 11213.9 1.18058
\(40\) −3462.62 −0.342180
\(41\) 8714.50 0.809623 0.404811 0.914400i \(-0.367337\pi\)
0.404811 + 0.914400i \(0.367337\pi\)
\(42\) 5020.88 0.439194
\(43\) −2640.06 −0.217742 −0.108871 0.994056i \(-0.534724\pi\)
−0.108871 + 0.994056i \(0.534724\pi\)
\(44\) −3190.69 −0.248458
\(45\) 1770.62 0.130345
\(46\) −7155.40 −0.498585
\(47\) −7280.62 −0.480755 −0.240378 0.970679i \(-0.577271\pi\)
−0.240378 + 0.970679i \(0.577271\pi\)
\(48\) −6759.70 −0.423472
\(49\) 9196.37 0.547175
\(50\) −1483.06 −0.0838948
\(51\) 10148.0 0.546329
\(52\) 22535.6 1.15574
\(53\) 4281.20 0.209351 0.104676 0.994506i \(-0.466620\pi\)
0.104676 + 0.994506i \(0.466620\pi\)
\(54\) −9771.30 −0.456003
\(55\) −3025.00 −0.134840
\(56\) 22334.7 0.951721
\(57\) 4736.88 0.193110
\(58\) −3724.88 −0.145393
\(59\) −15566.1 −0.582170 −0.291085 0.956697i \(-0.594016\pi\)
−0.291085 + 0.956697i \(0.594016\pi\)
\(60\) −8650.16 −0.310203
\(61\) 1673.74 0.0575922 0.0287961 0.999585i \(-0.490833\pi\)
0.0287961 + 0.999585i \(0.490833\pi\)
\(62\) −13339.7 −0.440723
\(63\) −11420.9 −0.362535
\(64\) −3067.37 −0.0936086
\(65\) 21365.4 0.627231
\(66\) 3767.48 0.106461
\(67\) −5693.31 −0.154945 −0.0774726 0.996994i \(-0.524685\pi\)
−0.0774726 + 0.996994i \(0.524685\pi\)
\(68\) 20393.6 0.534836
\(69\) −39567.5 −1.00050
\(70\) 9566.10 0.233340
\(71\) −66500.3 −1.56559 −0.782795 0.622280i \(-0.786207\pi\)
−0.782795 + 0.622280i \(0.786207\pi\)
\(72\) −9809.58 −0.223007
\(73\) −27566.1 −0.605435 −0.302718 0.953080i \(-0.597894\pi\)
−0.302718 + 0.953080i \(0.597894\pi\)
\(74\) 5149.32 0.109313
\(75\) −8200.97 −0.168350
\(76\) 9519.33 0.189048
\(77\) 19511.9 0.375036
\(78\) −26609.5 −0.495221
\(79\) −38949.0 −0.702148 −0.351074 0.936348i \(-0.614183\pi\)
−0.351074 + 0.936348i \(0.614183\pi\)
\(80\) −12879.0 −0.224987
\(81\) −36822.4 −0.623590
\(82\) −20678.7 −0.339616
\(83\) −17829.9 −0.284088 −0.142044 0.989860i \(-0.545367\pi\)
−0.142044 + 0.989860i \(0.545367\pi\)
\(84\) 55795.5 0.862782
\(85\) 19334.6 0.290260
\(86\) 6264.60 0.0913370
\(87\) −20597.6 −0.291756
\(88\) 16759.1 0.230698
\(89\) −107643. −1.44049 −0.720246 0.693719i \(-0.755971\pi\)
−0.720246 + 0.693719i \(0.755971\pi\)
\(90\) −4201.51 −0.0546764
\(91\) −137812. −1.74454
\(92\) −79515.7 −0.979453
\(93\) −73765.1 −0.884389
\(94\) 17276.2 0.201664
\(95\) 9025.00 0.102598
\(96\) 74196.8 0.821689
\(97\) 17058.2 0.184079 0.0920395 0.995755i \(-0.470661\pi\)
0.0920395 + 0.995755i \(0.470661\pi\)
\(98\) −21822.1 −0.229526
\(99\) −8569.81 −0.0878786
\(100\) −16480.8 −0.164808
\(101\) −60004.0 −0.585298 −0.292649 0.956220i \(-0.594537\pi\)
−0.292649 + 0.956220i \(0.594537\pi\)
\(102\) −24080.2 −0.229171
\(103\) 125218. 1.16298 0.581492 0.813552i \(-0.302469\pi\)
0.581492 + 0.813552i \(0.302469\pi\)
\(104\) −118368. −1.07313
\(105\) 52898.1 0.468238
\(106\) −10158.9 −0.0878174
\(107\) 48206.6 0.407050 0.203525 0.979070i \(-0.434760\pi\)
0.203525 + 0.979070i \(0.434760\pi\)
\(108\) −108585. −0.895803
\(109\) −218634. −1.76259 −0.881293 0.472570i \(-0.843327\pi\)
−0.881293 + 0.472570i \(0.843327\pi\)
\(110\) 7178.03 0.0565619
\(111\) 28474.4 0.219355
\(112\) 83072.5 0.625766
\(113\) −106273. −0.782935 −0.391467 0.920192i \(-0.628032\pi\)
−0.391467 + 0.920192i \(0.628032\pi\)
\(114\) −11240.2 −0.0810047
\(115\) −75386.6 −0.531556
\(116\) −41393.4 −0.285619
\(117\) 60528.1 0.408782
\(118\) 36936.9 0.244205
\(119\) −124712. −0.807312
\(120\) 45434.9 0.288029
\(121\) 14641.0 0.0909091
\(122\) −3971.63 −0.0241584
\(123\) −114348. −0.681498
\(124\) −148240. −0.865786
\(125\) −15625.0 −0.0894427
\(126\) 27100.7 0.152074
\(127\) 136934. 0.753358 0.376679 0.926344i \(-0.377066\pi\)
0.376679 + 0.926344i \(0.377066\pi\)
\(128\) 188225. 1.01544
\(129\) 34641.6 0.183284
\(130\) −50698.0 −0.263107
\(131\) 171446. 0.872868 0.436434 0.899736i \(-0.356241\pi\)
0.436434 + 0.899736i \(0.356241\pi\)
\(132\) 41866.8 0.209139
\(133\) −58213.3 −0.285360
\(134\) 13509.7 0.0649955
\(135\) −102947. −0.486159
\(136\) −107117. −0.496606
\(137\) −190342. −0.866430 −0.433215 0.901291i \(-0.642621\pi\)
−0.433215 + 0.901291i \(0.642621\pi\)
\(138\) 93889.9 0.419683
\(139\) −130774. −0.574098 −0.287049 0.957916i \(-0.592674\pi\)
−0.287049 + 0.957916i \(0.592674\pi\)
\(140\) 106305. 0.458389
\(141\) 95533.0 0.404675
\(142\) 157799. 0.656724
\(143\) −103408. −0.422879
\(144\) −36486.2 −0.146630
\(145\) −39243.9 −0.155007
\(146\) 65411.6 0.253964
\(147\) −120671. −0.460584
\(148\) 57222.8 0.214741
\(149\) 152156. 0.561466 0.280733 0.959786i \(-0.409422\pi\)
0.280733 + 0.959786i \(0.409422\pi\)
\(150\) 19460.1 0.0706183
\(151\) 163485. 0.583494 0.291747 0.956496i \(-0.405764\pi\)
0.291747 + 0.956496i \(0.405764\pi\)
\(152\) −50000.2 −0.175535
\(153\) 54774.7 0.189170
\(154\) −46299.9 −0.157318
\(155\) −140542. −0.469868
\(156\) −295703. −0.972845
\(157\) 57420.1 0.185915 0.0929576 0.995670i \(-0.470368\pi\)
0.0929576 + 0.995670i \(0.470368\pi\)
\(158\) 92422.2 0.294533
\(159\) −56175.9 −0.176221
\(160\) 141364. 0.436556
\(161\) 486260. 1.47844
\(162\) 87375.9 0.261580
\(163\) −69419.2 −0.204650 −0.102325 0.994751i \(-0.532628\pi\)
−0.102325 + 0.994751i \(0.532628\pi\)
\(164\) −229795. −0.667163
\(165\) 39692.7 0.113501
\(166\) 42308.6 0.119168
\(167\) −461436. −1.28032 −0.640162 0.768240i \(-0.721133\pi\)
−0.640162 + 0.768240i \(0.721133\pi\)
\(168\) −293066. −0.801109
\(169\) 359075. 0.967093
\(170\) −45879.0 −0.121756
\(171\) 25567.8 0.0668656
\(172\) 69616.5 0.179428
\(173\) 204310. 0.519007 0.259504 0.965742i \(-0.416441\pi\)
0.259504 + 0.965742i \(0.416441\pi\)
\(174\) 48876.2 0.122384
\(175\) 100785. 0.248771
\(176\) 62334.4 0.151686
\(177\) 204251. 0.490040
\(178\) 255426. 0.604248
\(179\) −58332.8 −0.136076 −0.0680378 0.997683i \(-0.521674\pi\)
−0.0680378 + 0.997683i \(0.521674\pi\)
\(180\) −46690.1 −0.107410
\(181\) 776342. 1.76140 0.880698 0.473679i \(-0.157074\pi\)
0.880698 + 0.473679i \(0.157074\pi\)
\(182\) 327013. 0.731791
\(183\) −21962.1 −0.0484781
\(184\) 417656. 0.909440
\(185\) 54251.3 0.116542
\(186\) 175037. 0.370978
\(187\) −93579.2 −0.195693
\(188\) 191985. 0.396162
\(189\) 664030. 1.35218
\(190\) −21415.4 −0.0430371
\(191\) 125267. 0.248459 0.124229 0.992254i \(-0.460354\pi\)
0.124229 + 0.992254i \(0.460354\pi\)
\(192\) 40248.6 0.0787949
\(193\) 682643. 1.31917 0.659585 0.751630i \(-0.270732\pi\)
0.659585 + 0.751630i \(0.270732\pi\)
\(194\) −40477.5 −0.0772163
\(195\) −280347. −0.527970
\(196\) −242502. −0.450895
\(197\) 576442. 1.05825 0.529127 0.848543i \(-0.322519\pi\)
0.529127 + 0.848543i \(0.322519\pi\)
\(198\) 20335.3 0.0368628
\(199\) −410738. −0.735245 −0.367622 0.929975i \(-0.619828\pi\)
−0.367622 + 0.929975i \(0.619828\pi\)
\(200\) 86565.5 0.153028
\(201\) 74705.1 0.130425
\(202\) 142384. 0.245517
\(203\) 253132. 0.431129
\(204\) −267595. −0.450197
\(205\) −217862. −0.362074
\(206\) −297130. −0.487842
\(207\) −213570. −0.346429
\(208\) −440264. −0.705594
\(209\) −43681.0 −0.0691714
\(210\) −125522. −0.196414
\(211\) 850360. 1.31491 0.657456 0.753493i \(-0.271633\pi\)
0.657456 + 0.753493i \(0.271633\pi\)
\(212\) −112892. −0.172514
\(213\) 872588. 1.31783
\(214\) −114390. −0.170747
\(215\) 66001.4 0.0973771
\(216\) 570344. 0.831769
\(217\) 906526. 1.30687
\(218\) 518796. 0.739359
\(219\) 361710. 0.509624
\(220\) 79767.2 0.111114
\(221\) 660944. 0.910299
\(222\) −67567.1 −0.0920138
\(223\) −1.18730e6 −1.59882 −0.799409 0.600787i \(-0.794854\pi\)
−0.799409 + 0.600787i \(0.794854\pi\)
\(224\) −911832. −1.21421
\(225\) −44265.6 −0.0582921
\(226\) 252175. 0.328421
\(227\) 91416.9 0.117750 0.0588751 0.998265i \(-0.481249\pi\)
0.0588751 + 0.998265i \(0.481249\pi\)
\(228\) −124908. −0.159131
\(229\) 274091. 0.345387 0.172694 0.984976i \(-0.444753\pi\)
0.172694 + 0.984976i \(0.444753\pi\)
\(230\) 178885. 0.222974
\(231\) −256027. −0.315686
\(232\) 217419. 0.265202
\(233\) −791866. −0.955568 −0.477784 0.878477i \(-0.658560\pi\)
−0.477784 + 0.878477i \(0.658560\pi\)
\(234\) −143627. −0.171473
\(235\) 182016. 0.215000
\(236\) 410468. 0.479732
\(237\) 511071. 0.591032
\(238\) 295930. 0.338646
\(239\) −1.19500e6 −1.35323 −0.676617 0.736335i \(-0.736555\pi\)
−0.676617 + 0.736335i \(0.736555\pi\)
\(240\) 168993. 0.189382
\(241\) −1.11921e6 −1.24128 −0.620639 0.784096i \(-0.713127\pi\)
−0.620639 + 0.784096i \(0.713127\pi\)
\(242\) −34741.7 −0.0381340
\(243\) −517475. −0.562178
\(244\) −44135.5 −0.0474584
\(245\) −229909. −0.244704
\(246\) 271336. 0.285871
\(247\) 308516. 0.321763
\(248\) 778628. 0.803898
\(249\) 233956. 0.239131
\(250\) 37076.6 0.0375189
\(251\) 480820. 0.481724 0.240862 0.970559i \(-0.422570\pi\)
0.240862 + 0.970559i \(0.422570\pi\)
\(252\) 301162. 0.298744
\(253\) 364871. 0.358375
\(254\) −324931. −0.316014
\(255\) −253699. −0.244326
\(256\) −348484. −0.332340
\(257\) 1.48286e6 1.40045 0.700227 0.713920i \(-0.253082\pi\)
0.700227 + 0.713920i \(0.253082\pi\)
\(258\) −82201.2 −0.0768828
\(259\) −349933. −0.324142
\(260\) −563391. −0.516864
\(261\) −111178. −0.101022
\(262\) −406824. −0.366145
\(263\) 26645.2 0.0237536 0.0118768 0.999929i \(-0.496219\pi\)
0.0118768 + 0.999929i \(0.496219\pi\)
\(264\) −219905. −0.194189
\(265\) −107030. −0.0936247
\(266\) 138134. 0.119701
\(267\) 1.41244e6 1.21253
\(268\) 150129. 0.127681
\(269\) 1.79237e6 1.51024 0.755120 0.655587i \(-0.227579\pi\)
0.755120 + 0.655587i \(0.227579\pi\)
\(270\) 244283. 0.203931
\(271\) 42086.2 0.0348110 0.0174055 0.999849i \(-0.494459\pi\)
0.0174055 + 0.999849i \(0.494459\pi\)
\(272\) −398416. −0.326523
\(273\) 1.80830e6 1.46847
\(274\) 451663. 0.363445
\(275\) 75625.0 0.0603023
\(276\) 1.04337e6 0.824452
\(277\) 192387. 0.150652 0.0753261 0.997159i \(-0.476000\pi\)
0.0753261 + 0.997159i \(0.476000\pi\)
\(278\) 310315. 0.240819
\(279\) −398154. −0.306225
\(280\) −558367. −0.425622
\(281\) −662659. −0.500639 −0.250319 0.968163i \(-0.580536\pi\)
−0.250319 + 0.968163i \(0.580536\pi\)
\(282\) −226691. −0.169750
\(283\) −896996. −0.665770 −0.332885 0.942967i \(-0.608022\pi\)
−0.332885 + 0.942967i \(0.608022\pi\)
\(284\) 1.75357e6 1.29011
\(285\) −118422. −0.0863615
\(286\) 245378. 0.177387
\(287\) 1.40526e6 1.00705
\(288\) 400485. 0.284515
\(289\) −821737. −0.578746
\(290\) 93122.0 0.0650215
\(291\) −223830. −0.154948
\(292\) 726899. 0.498904
\(293\) 2.17973e6 1.48332 0.741659 0.670777i \(-0.234039\pi\)
0.741659 + 0.670777i \(0.234039\pi\)
\(294\) 286340. 0.193203
\(295\) 389153. 0.260354
\(296\) −300563. −0.199391
\(297\) 498262. 0.327768
\(298\) −361052. −0.235521
\(299\) −2.57706e6 −1.66704
\(300\) 216254. 0.138727
\(301\) −425724. −0.270839
\(302\) −387935. −0.244760
\(303\) 787346. 0.492673
\(304\) −185973. −0.115416
\(305\) −41843.6 −0.0257560
\(306\) −129975. −0.0793517
\(307\) −1.50259e6 −0.909901 −0.454950 0.890517i \(-0.650343\pi\)
−0.454950 + 0.890517i \(0.650343\pi\)
\(308\) −514517. −0.309046
\(309\) −1.64306e6 −0.978940
\(310\) 333492. 0.197098
\(311\) −750226. −0.439837 −0.219918 0.975518i \(-0.570579\pi\)
−0.219918 + 0.975518i \(0.570579\pi\)
\(312\) 1.55318e6 0.903304
\(313\) −1.35177e6 −0.779906 −0.389953 0.920835i \(-0.627509\pi\)
−0.389953 + 0.920835i \(0.627509\pi\)
\(314\) −136252. −0.0779866
\(315\) 285523. 0.162130
\(316\) 1.02706e6 0.578599
\(317\) −2.40047e6 −1.34168 −0.670838 0.741604i \(-0.734065\pi\)
−0.670838 + 0.741604i \(0.734065\pi\)
\(318\) 133300. 0.0739201
\(319\) 189941. 0.104506
\(320\) 76684.2 0.0418631
\(321\) −632546. −0.342633
\(322\) −1.15385e6 −0.620168
\(323\) 279191. 0.148900
\(324\) 970982. 0.513864
\(325\) −534135. −0.280506
\(326\) 164725. 0.0858451
\(327\) 2.86881e6 1.48365
\(328\) 1.20700e6 0.619473
\(329\) −1.17404e6 −0.597990
\(330\) −94186.9 −0.0476108
\(331\) −461188. −0.231371 −0.115685 0.993286i \(-0.536906\pi\)
−0.115685 + 0.993286i \(0.536906\pi\)
\(332\) 470162. 0.234101
\(333\) 153694. 0.0759531
\(334\) 1.09494e6 0.537063
\(335\) 142333. 0.0692936
\(336\) −1.09004e6 −0.526738
\(337\) 3.24545e6 1.55668 0.778342 0.627840i \(-0.216061\pi\)
0.778342 + 0.627840i \(0.216061\pi\)
\(338\) −852049. −0.405670
\(339\) 1.39446e6 0.659034
\(340\) −509839. −0.239186
\(341\) 680222. 0.316785
\(342\) −60669.9 −0.0280484
\(343\) −1.22726e6 −0.563248
\(344\) −365660. −0.166603
\(345\) 989189. 0.447437
\(346\) −484807. −0.217710
\(347\) −1.12740e6 −0.502636 −0.251318 0.967905i \(-0.580864\pi\)
−0.251318 + 0.967905i \(0.580864\pi\)
\(348\) 543146. 0.240419
\(349\) 1.75750e6 0.772380 0.386190 0.922419i \(-0.373791\pi\)
0.386190 + 0.922419i \(0.373791\pi\)
\(350\) −239152. −0.104353
\(351\) −3.51920e6 −1.52467
\(352\) −684204. −0.294326
\(353\) 4.16035e6 1.77702 0.888512 0.458854i \(-0.151740\pi\)
0.888512 + 0.458854i \(0.151740\pi\)
\(354\) −484669. −0.205559
\(355\) 1.66251e6 0.700153
\(356\) 2.83847e6 1.18702
\(357\) 1.63642e6 0.679553
\(358\) 138418. 0.0570802
\(359\) −3.27805e6 −1.34239 −0.671197 0.741279i \(-0.734219\pi\)
−0.671197 + 0.741279i \(0.734219\pi\)
\(360\) 245240. 0.0997320
\(361\) 130321. 0.0526316
\(362\) −1.84219e6 −0.738859
\(363\) −192113. −0.0765225
\(364\) 3.63400e6 1.43758
\(365\) 689152. 0.270759
\(366\) 52113.9 0.0203353
\(367\) 184003. 0.0713117 0.0356559 0.999364i \(-0.488648\pi\)
0.0356559 + 0.999364i \(0.488648\pi\)
\(368\) 1.55345e6 0.597966
\(369\) −617203. −0.235973
\(370\) −128733. −0.0488861
\(371\) 690367. 0.260403
\(372\) 1.94514e6 0.728773
\(373\) −2.27618e6 −0.847100 −0.423550 0.905873i \(-0.639216\pi\)
−0.423550 + 0.905873i \(0.639216\pi\)
\(374\) 222054. 0.0820881
\(375\) 205024. 0.0752882
\(376\) −1.00840e6 −0.367844
\(377\) −1.34154e6 −0.486127
\(378\) −1.57568e6 −0.567202
\(379\) 2.09363e6 0.748690 0.374345 0.927289i \(-0.377868\pi\)
0.374345 + 0.927289i \(0.377868\pi\)
\(380\) −237983. −0.0845449
\(381\) −1.79678e6 −0.634138
\(382\) −297247. −0.104222
\(383\) −1.49234e6 −0.519840 −0.259920 0.965630i \(-0.583696\pi\)
−0.259920 + 0.965630i \(0.583696\pi\)
\(384\) −2.46980e6 −0.854741
\(385\) −487798. −0.167721
\(386\) −1.61985e6 −0.553357
\(387\) 186982. 0.0634631
\(388\) −449814. −0.151689
\(389\) 4.53488e6 1.51947 0.759734 0.650234i \(-0.225329\pi\)
0.759734 + 0.650234i \(0.225329\pi\)
\(390\) 665236. 0.221470
\(391\) −2.33210e6 −0.771447
\(392\) 1.27374e6 0.418665
\(393\) −2.24963e6 −0.734735
\(394\) −1.36784e6 −0.443910
\(395\) 973725. 0.314010
\(396\) 225980. 0.0724157
\(397\) 1.08979e6 0.347030 0.173515 0.984831i \(-0.444487\pi\)
0.173515 + 0.984831i \(0.444487\pi\)
\(398\) 974640. 0.308416
\(399\) 763849. 0.240201
\(400\) 321975. 0.100617
\(401\) 4.34966e6 1.35081 0.675405 0.737447i \(-0.263969\pi\)
0.675405 + 0.737447i \(0.263969\pi\)
\(402\) −177268. −0.0547098
\(403\) −4.80437e6 −1.47358
\(404\) 1.58227e6 0.482310
\(405\) 920560. 0.278878
\(406\) −600658. −0.180847
\(407\) −262576. −0.0785723
\(408\) 1.40554e6 0.418017
\(409\) 3.77403e6 1.11557 0.557785 0.829986i \(-0.311651\pi\)
0.557785 + 0.829986i \(0.311651\pi\)
\(410\) 516966. 0.151881
\(411\) 2.49758e6 0.729315
\(412\) −3.30192e6 −0.958348
\(413\) −2.51012e6 −0.724135
\(414\) 506780. 0.145318
\(415\) 445747. 0.127048
\(416\) 4.83249e6 1.36911
\(417\) 1.71596e6 0.483246
\(418\) 103651. 0.0290156
\(419\) 1.88686e6 0.525055 0.262527 0.964925i \(-0.415444\pi\)
0.262527 + 0.964925i \(0.415444\pi\)
\(420\) −1.39489e6 −0.385848
\(421\) 2.56104e6 0.704224 0.352112 0.935958i \(-0.385464\pi\)
0.352112 + 0.935958i \(0.385464\pi\)
\(422\) −2.01782e6 −0.551571
\(423\) 515649. 0.140121
\(424\) 592966. 0.160183
\(425\) −483364. −0.129808
\(426\) −2.07057e6 −0.552796
\(427\) 269900. 0.0716364
\(428\) −1.27118e6 −0.335426
\(429\) 1.35688e6 0.355957
\(430\) −156615. −0.0408472
\(431\) 5.99609e6 1.55480 0.777400 0.629006i \(-0.216538\pi\)
0.777400 + 0.629006i \(0.216538\pi\)
\(432\) 2.12136e6 0.546897
\(433\) 693248. 0.177692 0.0888461 0.996045i \(-0.471682\pi\)
0.0888461 + 0.996045i \(0.471682\pi\)
\(434\) −2.15110e6 −0.548196
\(435\) 514941. 0.130477
\(436\) 5.76522e6 1.45244
\(437\) −1.08858e6 −0.272683
\(438\) −858302. −0.213774
\(439\) 4.18152e6 1.03555 0.517777 0.855515i \(-0.326760\pi\)
0.517777 + 0.855515i \(0.326760\pi\)
\(440\) −418977. −0.103171
\(441\) −651332. −0.159480
\(442\) −1.56836e6 −0.381847
\(443\) −959488. −0.232290 −0.116145 0.993232i \(-0.537054\pi\)
−0.116145 + 0.993232i \(0.537054\pi\)
\(444\) −750852. −0.180758
\(445\) 2.69107e6 0.644207
\(446\) 2.81735e6 0.670662
\(447\) −1.99652e6 −0.472613
\(448\) −494630. −0.116436
\(449\) −867910. −0.203170 −0.101585 0.994827i \(-0.532391\pi\)
−0.101585 + 0.994827i \(0.532391\pi\)
\(450\) 105038. 0.0244520
\(451\) 1.05445e6 0.244110
\(452\) 2.80234e6 0.645171
\(453\) −2.14518e6 −0.491155
\(454\) −216923. −0.0493932
\(455\) 3.44529e6 0.780184
\(456\) 656080. 0.147756
\(457\) 4.28127e6 0.958919 0.479460 0.877564i \(-0.340833\pi\)
0.479460 + 0.877564i \(0.340833\pi\)
\(458\) −650391. −0.144881
\(459\) −3.18469e6 −0.705562
\(460\) 1.98789e6 0.438025
\(461\) 2.06835e6 0.453285 0.226642 0.973978i \(-0.427225\pi\)
0.226642 + 0.973978i \(0.427225\pi\)
\(462\) 607527. 0.132422
\(463\) 2.45612e6 0.532471 0.266236 0.963908i \(-0.414220\pi\)
0.266236 + 0.963908i \(0.414220\pi\)
\(464\) 808676. 0.174373
\(465\) 1.84413e6 0.395511
\(466\) 1.87902e6 0.400836
\(467\) 1.41736e6 0.300738 0.150369 0.988630i \(-0.451954\pi\)
0.150369 + 0.988630i \(0.451954\pi\)
\(468\) −1.59608e6 −0.336854
\(469\) −918079. −0.192729
\(470\) −431905. −0.0901870
\(471\) −753441. −0.156494
\(472\) −2.15598e6 −0.445440
\(473\) −319447. −0.0656516
\(474\) −1.21272e6 −0.247922
\(475\) −225625. −0.0458831
\(476\) 3.28858e6 0.665259
\(477\) −303215. −0.0610176
\(478\) 2.83562e6 0.567646
\(479\) 915681. 0.182350 0.0911749 0.995835i \(-0.470938\pi\)
0.0911749 + 0.995835i \(0.470938\pi\)
\(480\) −1.85492e6 −0.367470
\(481\) 1.85456e6 0.365492
\(482\) 2.65578e6 0.520684
\(483\) −6.38049e6 −1.24448
\(484\) −386073. −0.0749129
\(485\) −426455. −0.0823226
\(486\) 1.22792e6 0.235819
\(487\) −2.18548e6 −0.417566 −0.208783 0.977962i \(-0.566950\pi\)
−0.208783 + 0.977962i \(0.566950\pi\)
\(488\) 231821. 0.0440660
\(489\) 910888. 0.172263
\(490\) 545553. 0.102647
\(491\) 9.04192e6 1.69261 0.846305 0.532699i \(-0.178822\pi\)
0.846305 + 0.532699i \(0.178822\pi\)
\(492\) 3.01527e6 0.561583
\(493\) −1.21402e6 −0.224962
\(494\) −732079. −0.134971
\(495\) 214245. 0.0393005
\(496\) 2.89606e6 0.528571
\(497\) −1.07236e7 −1.94737
\(498\) −555154. −0.100309
\(499\) −4.20662e6 −0.756279 −0.378140 0.925749i \(-0.623436\pi\)
−0.378140 + 0.925749i \(0.623436\pi\)
\(500\) 412021. 0.0737045
\(501\) 6.05475e6 1.07771
\(502\) −1.14094e6 −0.202071
\(503\) −950827. −0.167564 −0.0837821 0.996484i \(-0.526700\pi\)
−0.0837821 + 0.996484i \(0.526700\pi\)
\(504\) −1.58185e6 −0.277389
\(505\) 1.50010e6 0.261753
\(506\) −865803. −0.150329
\(507\) −4.71162e6 −0.814048
\(508\) −3.61085e6 −0.620798
\(509\) −2.16476e6 −0.370352 −0.185176 0.982705i \(-0.559286\pi\)
−0.185176 + 0.982705i \(0.559286\pi\)
\(510\) 602004. 0.102488
\(511\) −4.44518e6 −0.753074
\(512\) −5.19628e6 −0.876028
\(513\) −1.48655e6 −0.249394
\(514\) −3.51869e6 −0.587454
\(515\) −3.13045e6 −0.520102
\(516\) −913477. −0.151033
\(517\) −880955. −0.144953
\(518\) 830357. 0.135969
\(519\) −2.68086e6 −0.436873
\(520\) 2.95921e6 0.479918
\(521\) 4.67888e6 0.755176 0.377588 0.925974i \(-0.376754\pi\)
0.377588 + 0.925974i \(0.376754\pi\)
\(522\) 263814. 0.0423762
\(523\) 8.60530e6 1.37566 0.687831 0.725871i \(-0.258563\pi\)
0.687831 + 0.725871i \(0.258563\pi\)
\(524\) −4.52091e6 −0.719279
\(525\) −1.32245e6 −0.209402
\(526\) −63226.4 −0.00996400
\(527\) −4.34770e6 −0.681919
\(528\) −817924. −0.127682
\(529\) 2.65667e6 0.412761
\(530\) 253972. 0.0392731
\(531\) 1.10247e6 0.169680
\(532\) 1.53505e6 0.235148
\(533\) −7.44754e6 −1.13552
\(534\) −3.35159e6 −0.508625
\(535\) −1.20517e6 −0.182038
\(536\) −788551. −0.118554
\(537\) 765417. 0.114541
\(538\) −4.25311e6 −0.633506
\(539\) 1.11276e6 0.164980
\(540\) 2.71464e6 0.400615
\(541\) 4.94858e6 0.726922 0.363461 0.931609i \(-0.381595\pi\)
0.363461 + 0.931609i \(0.381595\pi\)
\(542\) −99866.5 −0.0146023
\(543\) −1.01868e7 −1.48265
\(544\) 4.37315e6 0.633574
\(545\) 5.46584e6 0.788253
\(546\) −4.29092e6 −0.615984
\(547\) 5.10257e6 0.729156 0.364578 0.931173i \(-0.381213\pi\)
0.364578 + 0.931173i \(0.381213\pi\)
\(548\) 5.01919e6 0.713974
\(549\) −118543. −0.0167859
\(550\) −179451. −0.0252952
\(551\) −566682. −0.0795171
\(552\) −5.48029e6 −0.765519
\(553\) −6.28075e6 −0.873370
\(554\) −456515. −0.0631946
\(555\) −711861. −0.0980986
\(556\) 3.44843e6 0.473081
\(557\) 5.90274e6 0.806150 0.403075 0.915167i \(-0.367941\pi\)
0.403075 + 0.915167i \(0.367941\pi\)
\(558\) 944781. 0.128453
\(559\) 2.25623e6 0.305390
\(560\) −2.07681e6 −0.279851
\(561\) 1.22790e6 0.164724
\(562\) 1.57243e6 0.210005
\(563\) 2.20840e6 0.293635 0.146817 0.989164i \(-0.453097\pi\)
0.146817 + 0.989164i \(0.453097\pi\)
\(564\) −2.51914e6 −0.333469
\(565\) 2.65682e6 0.350139
\(566\) 2.12848e6 0.279273
\(567\) −5.93782e6 −0.775656
\(568\) −9.21061e6 −1.19789
\(569\) 455199. 0.0589414 0.0294707 0.999566i \(-0.490618\pi\)
0.0294707 + 0.999566i \(0.490618\pi\)
\(570\) 281004. 0.0362264
\(571\) −2.85732e6 −0.366748 −0.183374 0.983043i \(-0.558702\pi\)
−0.183374 + 0.983043i \(0.558702\pi\)
\(572\) 2.72681e6 0.348470
\(573\) −1.64370e6 −0.209140
\(574\) −3.33455e6 −0.422433
\(575\) 1.88466e6 0.237719
\(576\) 217246. 0.0272832
\(577\) −1.28731e7 −1.60969 −0.804846 0.593484i \(-0.797752\pi\)
−0.804846 + 0.593484i \(0.797752\pi\)
\(578\) 1.94990e6 0.242769
\(579\) −8.95734e6 −1.11041
\(580\) 1.03484e6 0.127732
\(581\) −2.87517e6 −0.353365
\(582\) 531127. 0.0649967
\(583\) 518025. 0.0631218
\(584\) −3.81803e6 −0.463241
\(585\) −1.51320e6 −0.182813
\(586\) −5.17229e6 −0.622213
\(587\) −7.94013e6 −0.951114 −0.475557 0.879685i \(-0.657753\pi\)
−0.475557 + 0.879685i \(0.657753\pi\)
\(588\) 3.18201e6 0.379540
\(589\) −2.02942e6 −0.241037
\(590\) −923421. −0.109212
\(591\) −7.56381e6 −0.890783
\(592\) −1.11792e6 −0.131102
\(593\) 4.48562e6 0.523825 0.261912 0.965092i \(-0.415647\pi\)
0.261912 + 0.965092i \(0.415647\pi\)
\(594\) −1.18233e6 −0.137490
\(595\) 3.11781e6 0.361041
\(596\) −4.01226e6 −0.462672
\(597\) 5.38951e6 0.618891
\(598\) 6.11512e6 0.699281
\(599\) −5.08340e6 −0.578878 −0.289439 0.957196i \(-0.593469\pi\)
−0.289439 + 0.957196i \(0.593469\pi\)
\(600\) −1.13587e6 −0.128811
\(601\) −7.95366e6 −0.898216 −0.449108 0.893477i \(-0.648258\pi\)
−0.449108 + 0.893477i \(0.648258\pi\)
\(602\) 1.01020e6 0.113610
\(603\) 403228. 0.0451604
\(604\) −4.31100e6 −0.480823
\(605\) −366025. −0.0406558
\(606\) −1.86830e6 −0.206664
\(607\) 1.07655e6 0.118594 0.0592969 0.998240i \(-0.481114\pi\)
0.0592969 + 0.998240i \(0.481114\pi\)
\(608\) 2.04130e6 0.223949
\(609\) −3.32148e6 −0.362902
\(610\) 99290.7 0.0108040
\(611\) 6.22213e6 0.674274
\(612\) −1.44437e6 −0.155884
\(613\) −1.63843e6 −0.176107 −0.0880534 0.996116i \(-0.528065\pi\)
−0.0880534 + 0.996116i \(0.528065\pi\)
\(614\) 3.56550e6 0.381680
\(615\) 2.85869e6 0.304775
\(616\) 2.70250e6 0.286955
\(617\) −1.01236e6 −0.107058 −0.0535292 0.998566i \(-0.517047\pi\)
−0.0535292 + 0.998566i \(0.517047\pi\)
\(618\) 3.89881e6 0.410640
\(619\) 1.59040e7 1.66832 0.834161 0.551521i \(-0.185952\pi\)
0.834161 + 0.551521i \(0.185952\pi\)
\(620\) 3.70599e6 0.387191
\(621\) 1.24173e7 1.29210
\(622\) 1.78021e6 0.184500
\(623\) −1.73580e7 −1.79176
\(624\) 5.77695e6 0.593932
\(625\) 390625. 0.0400000
\(626\) 3.20762e6 0.327150
\(627\) 573162. 0.0582249
\(628\) −1.51413e6 −0.153202
\(629\) 1.67828e6 0.169137
\(630\) −677518. −0.0680095
\(631\) 1.80933e7 1.80902 0.904512 0.426448i \(-0.140235\pi\)
0.904512 + 0.426448i \(0.140235\pi\)
\(632\) −5.39462e6 −0.537240
\(633\) −1.11580e7 −1.10682
\(634\) 5.69608e6 0.562798
\(635\) −3.42335e6 −0.336912
\(636\) 1.48132e6 0.145213
\(637\) −7.85936e6 −0.767430
\(638\) −450710. −0.0438375
\(639\) 4.70988e6 0.456308
\(640\) −4.70563e6 −0.454117
\(641\) −1.44097e7 −1.38519 −0.692596 0.721326i \(-0.743533\pi\)
−0.692596 + 0.721326i \(0.743533\pi\)
\(642\) 1.50097e6 0.143726
\(643\) 2.24743e6 0.214367 0.107184 0.994239i \(-0.465817\pi\)
0.107184 + 0.994239i \(0.465817\pi\)
\(644\) −1.28224e7 −1.21830
\(645\) −866041. −0.0819670
\(646\) −662493. −0.0624597
\(647\) −2.00218e6 −0.188036 −0.0940181 0.995570i \(-0.529971\pi\)
−0.0940181 + 0.995570i \(0.529971\pi\)
\(648\) −5.10008e6 −0.477133
\(649\) −1.88350e6 −0.175531
\(650\) 1.26745e6 0.117665
\(651\) −1.18950e7 −1.10005
\(652\) 1.83054e6 0.168640
\(653\) 1.28042e7 1.17508 0.587541 0.809194i \(-0.300096\pi\)
0.587541 + 0.809194i \(0.300096\pi\)
\(654\) −6.80741e6 −0.622354
\(655\) −4.28614e6 −0.390358
\(656\) 4.48936e6 0.407310
\(657\) 1.95236e6 0.176460
\(658\) 2.78589e6 0.250841
\(659\) 3.02033e6 0.270919 0.135460 0.990783i \(-0.456749\pi\)
0.135460 + 0.990783i \(0.456749\pi\)
\(660\) −1.04667e6 −0.0935298
\(661\) 3.07171e6 0.273450 0.136725 0.990609i \(-0.456342\pi\)
0.136725 + 0.990609i \(0.456342\pi\)
\(662\) 1.09435e6 0.0970540
\(663\) −8.67262e6 −0.766243
\(664\) −2.46952e6 −0.217367
\(665\) 1.45533e6 0.127617
\(666\) −364700. −0.0318603
\(667\) 4.73354e6 0.411976
\(668\) 1.21678e7 1.05504
\(669\) 1.55792e7 1.34580
\(670\) −337742. −0.0290669
\(671\) 202523. 0.0173647
\(672\) 1.19647e7 1.02206
\(673\) −1.86853e7 −1.59024 −0.795119 0.606453i \(-0.792592\pi\)
−0.795119 + 0.606453i \(0.792592\pi\)
\(674\) −7.70114e6 −0.652988
\(675\) 2.57367e6 0.217417
\(676\) −9.46856e6 −0.796925
\(677\) 2.29336e7 1.92309 0.961546 0.274643i \(-0.0885596\pi\)
0.961546 + 0.274643i \(0.0885596\pi\)
\(678\) −3.30892e6 −0.276447
\(679\) 2.75073e6 0.228968
\(680\) 2.67793e6 0.222089
\(681\) −1.19953e6 −0.0991161
\(682\) −1.61410e6 −0.132883
\(683\) 1.11687e7 0.916113 0.458057 0.888923i \(-0.348546\pi\)
0.458057 + 0.888923i \(0.348546\pi\)
\(684\) −674205. −0.0551001
\(685\) 4.75855e6 0.387479
\(686\) 2.91216e6 0.236268
\(687\) −3.59650e6 −0.290729
\(688\) −1.36005e6 −0.109543
\(689\) −3.65878e6 −0.293621
\(690\) −2.34725e6 −0.187688
\(691\) −1.19214e7 −0.949801 −0.474901 0.880039i \(-0.657516\pi\)
−0.474901 + 0.880039i \(0.657516\pi\)
\(692\) −5.38751e6 −0.427684
\(693\) −1.38193e6 −0.109308
\(694\) 2.67520e6 0.210843
\(695\) 3.26936e6 0.256744
\(696\) −2.85287e6 −0.223233
\(697\) −6.73964e6 −0.525478
\(698\) −4.17037e6 −0.323993
\(699\) 1.03905e7 0.804348
\(700\) −2.65763e6 −0.204998
\(701\) −2.85037e6 −0.219082 −0.109541 0.993982i \(-0.534938\pi\)
−0.109541 + 0.993982i \(0.534938\pi\)
\(702\) 8.35071e6 0.639559
\(703\) 783389. 0.0597846
\(704\) −371151. −0.0282241
\(705\) −2.38833e6 −0.180976
\(706\) −9.87211e6 −0.745415
\(707\) −9.67599e6 −0.728026
\(708\) −5.38597e6 −0.403814
\(709\) 1.01938e7 0.761590 0.380795 0.924659i \(-0.375650\pi\)
0.380795 + 0.924659i \(0.375650\pi\)
\(710\) −3.94497e6 −0.293696
\(711\) 2.75856e6 0.204648
\(712\) −1.49091e7 −1.10217
\(713\) 1.69519e7 1.24881
\(714\) −3.88306e6 −0.285055
\(715\) 2.58521e6 0.189117
\(716\) 1.53820e6 0.112132
\(717\) 1.56802e7 1.13908
\(718\) 7.77850e6 0.563099
\(719\) 1.44249e7 1.04061 0.520307 0.853979i \(-0.325818\pi\)
0.520307 + 0.853979i \(0.325818\pi\)
\(720\) 912154. 0.0655748
\(721\) 2.01921e7 1.44658
\(722\) −309239. −0.0220776
\(723\) 1.46858e7 1.04484
\(724\) −2.04716e7 −1.45146
\(725\) 981098. 0.0693214
\(726\) 455865. 0.0320992
\(727\) −2.94371e6 −0.206566 −0.103283 0.994652i \(-0.532935\pi\)
−0.103283 + 0.994652i \(0.532935\pi\)
\(728\) −1.90876e7 −1.33482
\(729\) 1.57379e7 1.09680
\(730\) −1.63529e6 −0.113576
\(731\) 2.04177e6 0.141323
\(732\) 579126. 0.0399480
\(733\) 5.37045e6 0.369191 0.184595 0.982815i \(-0.440903\pi\)
0.184595 + 0.982815i \(0.440903\pi\)
\(734\) −436622. −0.0299134
\(735\) 3.01677e6 0.205979
\(736\) −1.70512e7 −1.16027
\(737\) −688891. −0.0467177
\(738\) 1.46456e6 0.0989845
\(739\) 2.58078e7 1.73836 0.869180 0.494495i \(-0.164647\pi\)
0.869180 + 0.494495i \(0.164647\pi\)
\(740\) −1.43057e6 −0.0960351
\(741\) −4.04821e6 −0.270843
\(742\) −1.63817e6 −0.109232
\(743\) −1.23720e7 −0.822183 −0.411091 0.911594i \(-0.634852\pi\)
−0.411091 + 0.911594i \(0.634852\pi\)
\(744\) −1.02168e7 −0.676679
\(745\) −3.80390e6 −0.251095
\(746\) 5.40115e6 0.355336
\(747\) 1.26280e6 0.0828005
\(748\) 2.46762e6 0.161259
\(749\) 7.77359e6 0.506311
\(750\) −486503. −0.0315814
\(751\) −6.64194e6 −0.429730 −0.214865 0.976644i \(-0.568931\pi\)
−0.214865 + 0.976644i \(0.568931\pi\)
\(752\) −3.75069e6 −0.241861
\(753\) −6.30910e6 −0.405490
\(754\) 3.18334e6 0.203918
\(755\) −4.08713e6 −0.260946
\(756\) −1.75100e7 −1.11425
\(757\) −1.52835e7 −0.969355 −0.484678 0.874693i \(-0.661063\pi\)
−0.484678 + 0.874693i \(0.661063\pi\)
\(758\) −4.96798e6 −0.314056
\(759\) −4.78767e6 −0.301662
\(760\) 1.25001e6 0.0785015
\(761\) −1.33069e7 −0.832941 −0.416471 0.909149i \(-0.636733\pi\)
−0.416471 + 0.909149i \(0.636733\pi\)
\(762\) 4.26359e6 0.266004
\(763\) −3.52559e7 −2.19240
\(764\) −3.30322e6 −0.204740
\(765\) −1.36937e6 −0.0845992
\(766\) 3.54117e6 0.218059
\(767\) 1.33030e7 0.816511
\(768\) 4.57265e6 0.279747
\(769\) 7.47042e6 0.455543 0.227771 0.973715i \(-0.426856\pi\)
0.227771 + 0.973715i \(0.426856\pi\)
\(770\) 1.15750e6 0.0703547
\(771\) −1.94575e7 −1.17883
\(772\) −1.80008e7 −1.08705
\(773\) 7.29605e6 0.439177 0.219588 0.975593i \(-0.429529\pi\)
0.219588 + 0.975593i \(0.429529\pi\)
\(774\) −443689. −0.0266211
\(775\) 3.51354e6 0.210132
\(776\) 2.36264e6 0.140846
\(777\) 4.59166e6 0.272846
\(778\) −1.07608e7 −0.637377
\(779\) −3.14593e6 −0.185740
\(780\) 7.39256e6 0.435069
\(781\) −8.04654e6 −0.472043
\(782\) 5.53386e6 0.323602
\(783\) 6.46405e6 0.376791
\(784\) 4.73761e6 0.275276
\(785\) −1.43550e6 −0.0831438
\(786\) 5.33816e6 0.308202
\(787\) −2.75695e7 −1.58669 −0.793344 0.608774i \(-0.791662\pi\)
−0.793344 + 0.608774i \(0.791662\pi\)
\(788\) −1.52004e7 −0.872046
\(789\) −349626. −0.0199945
\(790\) −2.31055e6 −0.131719
\(791\) −1.71371e7 −0.973857
\(792\) −1.18696e6 −0.0672393
\(793\) −1.43041e6 −0.0807749
\(794\) −2.58597e6 −0.145570
\(795\) 1.40440e6 0.0788084
\(796\) 1.08309e7 0.605872
\(797\) 3.54357e6 0.197604 0.0988020 0.995107i \(-0.468499\pi\)
0.0988020 + 0.995107i \(0.468499\pi\)
\(798\) −1.81254e6 −0.100758
\(799\) 5.63070e6 0.312029
\(800\) −3.53411e6 −0.195234
\(801\) 7.62380e6 0.419846
\(802\) −1.03213e7 −0.566630
\(803\) −3.33549e6 −0.182546
\(804\) −1.96992e6 −0.107475
\(805\) −1.21565e7 −0.661179
\(806\) 1.14003e7 0.618128
\(807\) −2.35186e7 −1.27124
\(808\) −8.31084e6 −0.447834
\(809\) −3.25812e7 −1.75023 −0.875117 0.483911i \(-0.839216\pi\)
−0.875117 + 0.483911i \(0.839216\pi\)
\(810\) −2.18440e6 −0.116982
\(811\) −2.08727e7 −1.11436 −0.557181 0.830391i \(-0.688117\pi\)
−0.557181 + 0.830391i \(0.688117\pi\)
\(812\) −6.67492e6 −0.355268
\(813\) −552236. −0.0293021
\(814\) 623068. 0.0329590
\(815\) 1.73548e6 0.0915221
\(816\) 5.22783e6 0.274850
\(817\) 953060. 0.0499534
\(818\) −8.95540e6 −0.467952
\(819\) 9.76049e6 0.508466
\(820\) 5.74489e6 0.298364
\(821\) 3.66310e6 0.189667 0.0948334 0.995493i \(-0.469768\pi\)
0.0948334 + 0.995493i \(0.469768\pi\)
\(822\) −5.92652e6 −0.305929
\(823\) −78151.3 −0.00402195 −0.00201098 0.999998i \(-0.500640\pi\)
−0.00201098 + 0.999998i \(0.500640\pi\)
\(824\) 1.73433e7 0.889843
\(825\) −992317. −0.0507593
\(826\) 5.95627e6 0.303756
\(827\) 2.65484e7 1.34981 0.674907 0.737903i \(-0.264184\pi\)
0.674907 + 0.737903i \(0.264184\pi\)
\(828\) 5.63169e6 0.285472
\(829\) −947687. −0.0478937 −0.0239469 0.999713i \(-0.507623\pi\)
−0.0239469 + 0.999713i \(0.507623\pi\)
\(830\) −1.05771e6 −0.0532934
\(831\) −2.52441e6 −0.126811
\(832\) 2.62142e6 0.131289
\(833\) −7.11231e6 −0.355139
\(834\) −4.07181e6 −0.202709
\(835\) 1.15359e7 0.572578
\(836\) 1.15184e6 0.0570002
\(837\) 2.31493e7 1.14215
\(838\) −4.47733e6 −0.220247
\(839\) 2.73512e7 1.34144 0.670721 0.741709i \(-0.265985\pi\)
0.670721 + 0.741709i \(0.265985\pi\)
\(840\) 7.32664e6 0.358267
\(841\) −1.80470e7 −0.879864
\(842\) −6.07709e6 −0.295404
\(843\) 8.69511e6 0.421411
\(844\) −2.24234e7 −1.08354
\(845\) −8.97687e6 −0.432497
\(846\) −1.22359e6 −0.0587771
\(847\) 2.36094e6 0.113078
\(848\) 2.20550e6 0.105322
\(849\) 1.17700e7 0.560410
\(850\) 1.14698e6 0.0544511
\(851\) −6.54371e6 −0.309742
\(852\) −2.30096e7 −1.08595
\(853\) −2.25067e7 −1.05911 −0.529553 0.848277i \(-0.677640\pi\)
−0.529553 + 0.848277i \(0.677640\pi\)
\(854\) −640447. −0.0300496
\(855\) −639195. −0.0299032
\(856\) 6.67685e6 0.311449
\(857\) −807631. −0.0375630 −0.0187815 0.999824i \(-0.505979\pi\)
−0.0187815 + 0.999824i \(0.505979\pi\)
\(858\) −3.21974e6 −0.149315
\(859\) 2.14256e7 0.990719 0.495360 0.868688i \(-0.335036\pi\)
0.495360 + 0.868688i \(0.335036\pi\)
\(860\) −1.74041e6 −0.0802428
\(861\) −1.84392e7 −0.847685
\(862\) −1.42281e7 −0.652198
\(863\) −1.90437e7 −0.870410 −0.435205 0.900331i \(-0.643324\pi\)
−0.435205 + 0.900331i \(0.643324\pi\)
\(864\) −2.32848e7 −1.06118
\(865\) −5.10774e6 −0.232107
\(866\) −1.64501e6 −0.0745373
\(867\) 1.07825e7 0.487159
\(868\) −2.39045e7 −1.07691
\(869\) −4.71283e6 −0.211706
\(870\) −1.22190e6 −0.0547317
\(871\) 4.86559e6 0.217315
\(872\) −3.02818e7 −1.34862
\(873\) −1.20815e6 −0.0536517
\(874\) 2.58310e6 0.114383
\(875\) −2.51962e6 −0.111254
\(876\) −9.53804e6 −0.419951
\(877\) 266272. 0.0116903 0.00584516 0.999983i \(-0.498139\pi\)
0.00584516 + 0.999983i \(0.498139\pi\)
\(878\) −9.92234e6 −0.434388
\(879\) −2.86015e7 −1.24858
\(880\) −1.55836e6 −0.0678361
\(881\) 3.54570e7 1.53908 0.769541 0.638597i \(-0.220485\pi\)
0.769541 + 0.638597i \(0.220485\pi\)
\(882\) 1.54555e6 0.0668977
\(883\) −3.84411e7 −1.65918 −0.829591 0.558371i \(-0.811426\pi\)
−0.829591 + 0.558371i \(0.811426\pi\)
\(884\) −1.74287e7 −0.750125
\(885\) −5.10629e6 −0.219153
\(886\) 2.27677e6 0.0974395
\(887\) −4.44852e7 −1.89848 −0.949241 0.314550i \(-0.898146\pi\)
−0.949241 + 0.314550i \(0.898146\pi\)
\(888\) 3.94385e6 0.167837
\(889\) 2.20814e7 0.937068
\(890\) −6.38566e6 −0.270228
\(891\) −4.45551e6 −0.188020
\(892\) 3.13084e7 1.31749
\(893\) 2.62830e6 0.110293
\(894\) 4.73756e6 0.198249
\(895\) 1.45832e6 0.0608548
\(896\) 3.03523e7 1.26305
\(897\) 3.38150e7 1.40323
\(898\) 2.05947e6 0.0852244
\(899\) 8.82465e6 0.364165
\(900\) 1.16725e6 0.0480351
\(901\) −3.31100e6 −0.135877
\(902\) −2.50212e6 −0.102398
\(903\) 5.58616e6 0.227978
\(904\) −1.47193e7 −0.599053
\(905\) −1.94086e7 −0.787720
\(906\) 5.09030e6 0.206027
\(907\) −211109. −0.00852096 −0.00426048 0.999991i \(-0.501356\pi\)
−0.00426048 + 0.999991i \(0.501356\pi\)
\(908\) −2.41060e6 −0.0970311
\(909\) 4.24978e6 0.170591
\(910\) −8.17534e6 −0.327267
\(911\) 2.18697e7 0.873066 0.436533 0.899688i \(-0.356206\pi\)
0.436533 + 0.899688i \(0.356206\pi\)
\(912\) 2.44025e6 0.0971511
\(913\) −2.15742e6 −0.0856558
\(914\) −1.01590e7 −0.402242
\(915\) 549052. 0.0216801
\(916\) −7.22760e6 −0.284613
\(917\) 2.76466e7 1.08572
\(918\) 7.55695e6 0.295965
\(919\) 1.88742e7 0.737192 0.368596 0.929590i \(-0.379839\pi\)
0.368596 + 0.929590i \(0.379839\pi\)
\(920\) −1.04414e7 −0.406714
\(921\) 1.97163e7 0.765907
\(922\) −4.90799e6 −0.190141
\(923\) 5.68322e7 2.19579
\(924\) 6.75126e6 0.260138
\(925\) −1.35628e6 −0.0521190
\(926\) −5.82812e6 −0.223358
\(927\) −8.86856e6 −0.338964
\(928\) −8.87631e6 −0.338347
\(929\) −5.37016e6 −0.204149 −0.102075 0.994777i \(-0.532548\pi\)
−0.102075 + 0.994777i \(0.532548\pi\)
\(930\) −4.37593e6 −0.165906
\(931\) −3.31989e6 −0.125531
\(932\) 2.08810e7 0.787428
\(933\) 9.84413e6 0.370232
\(934\) −3.36326e6 −0.126152
\(935\) 2.33948e6 0.0875166
\(936\) 8.38342e6 0.312775
\(937\) 2.80796e7 1.04482 0.522411 0.852694i \(-0.325033\pi\)
0.522411 + 0.852694i \(0.325033\pi\)
\(938\) 2.17851e6 0.0808449
\(939\) 1.77373e7 0.656484
\(940\) −4.79963e6 −0.177169
\(941\) 1.33996e7 0.493307 0.246654 0.969104i \(-0.420669\pi\)
0.246654 + 0.969104i \(0.420669\pi\)
\(942\) 1.78784e6 0.0656450
\(943\) 2.62782e7 0.962314
\(944\) −8.01904e6 −0.292882
\(945\) −1.66007e7 −0.604711
\(946\) 758016. 0.0275392
\(947\) 1.81387e6 0.0657251 0.0328626 0.999460i \(-0.489538\pi\)
0.0328626 + 0.999460i \(0.489538\pi\)
\(948\) −1.34766e7 −0.487035
\(949\) 2.35584e7 0.849142
\(950\) 535386. 0.0192468
\(951\) 3.14979e7 1.12935
\(952\) −1.72732e7 −0.617705
\(953\) −4.14362e7 −1.47791 −0.738954 0.673756i \(-0.764680\pi\)
−0.738954 + 0.673756i \(0.764680\pi\)
\(954\) 719500. 0.0255953
\(955\) −3.13168e6 −0.111114
\(956\) 3.15113e7 1.11512
\(957\) −2.49231e6 −0.0879676
\(958\) −2.17282e6 −0.0764910
\(959\) −3.06937e7 −1.07771
\(960\) −1.00622e6 −0.0352381
\(961\) 2.97404e6 0.103882
\(962\) −4.40069e6 −0.153314
\(963\) −3.41423e6 −0.118639
\(964\) 2.95128e7 1.02286
\(965\) −1.70661e7 −0.589951
\(966\) 1.51403e7 0.522025
\(967\) 3.09184e7 1.06329 0.531644 0.846968i \(-0.321575\pi\)
0.531644 + 0.846968i \(0.321575\pi\)
\(968\) 2.02785e6 0.0695580
\(969\) −3.66342e6 −0.125336
\(970\) 1.01194e6 0.0345322
\(971\) −1.72970e7 −0.588738 −0.294369 0.955692i \(-0.595110\pi\)
−0.294369 + 0.955692i \(0.595110\pi\)
\(972\) 1.36455e7 0.463258
\(973\) −2.10881e7 −0.714094
\(974\) 5.18594e6 0.175158
\(975\) 7.00868e6 0.236115
\(976\) 862245. 0.0289739
\(977\) −3.25532e7 −1.09108 −0.545541 0.838084i \(-0.683676\pi\)
−0.545541 + 0.838084i \(0.683676\pi\)
\(978\) −2.16145e6 −0.0722600
\(979\) −1.30248e7 −0.434324
\(980\) 6.06256e6 0.201646
\(981\) 1.54847e7 0.513724
\(982\) −2.14556e7 −0.710006
\(983\) −5.93839e6 −0.196013 −0.0980066 0.995186i \(-0.531247\pi\)
−0.0980066 + 0.995186i \(0.531247\pi\)
\(984\) −1.58377e7 −0.521440
\(985\) −1.44111e7 −0.473266
\(986\) 2.88075e6 0.0943657
\(987\) 1.54052e7 0.503356
\(988\) −8.13537e6 −0.265146
\(989\) −7.96099e6 −0.258807
\(990\) −508383. −0.0164855
\(991\) −3.47991e7 −1.12560 −0.562800 0.826593i \(-0.690276\pi\)
−0.562800 + 0.826593i \(0.690276\pi\)
\(992\) −3.17882e7 −1.02562
\(993\) 6.05151e6 0.194756
\(994\) 2.54460e7 0.816870
\(995\) 1.02684e7 0.328811
\(996\) −6.16926e6 −0.197054
\(997\) 2.57562e7 0.820623 0.410312 0.911945i \(-0.365420\pi\)
0.410312 + 0.911945i \(0.365420\pi\)
\(998\) 9.98191e6 0.317239
\(999\) −8.93599e6 −0.283289
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 1045.6.a.d.1.16 37
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.6.a.d.1.16 37 1.1 even 1 trivial