Properties

Label 1045.6.a.e.1.4
Level $1045$
Weight $6$
Character 1045.1
Self dual yes
Analytic conductor $167.601$
Analytic rank $1$
Dimension $38$
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: \(1\)
Dimension: \(38\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
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)
 
\(f(q)\) \(=\) \(q-9.91265 q^{2} +15.6713 q^{3} +66.2607 q^{4} +25.0000 q^{5} -155.344 q^{6} -122.128 q^{7} -339.614 q^{8} +2.59022 q^{9} +O(q^{10})\) \(q-9.91265 q^{2} +15.6713 q^{3} +66.2607 q^{4} +25.0000 q^{5} -155.344 q^{6} -122.128 q^{7} -339.614 q^{8} +2.59022 q^{9} -247.816 q^{10} +121.000 q^{11} +1038.39 q^{12} +368.674 q^{13} +1210.61 q^{14} +391.783 q^{15} +1246.13 q^{16} -858.004 q^{17} -25.6759 q^{18} -361.000 q^{19} +1656.52 q^{20} -1913.91 q^{21} -1199.43 q^{22} +1759.78 q^{23} -5322.20 q^{24} +625.000 q^{25} -3654.53 q^{26} -3767.54 q^{27} -8092.28 q^{28} -7399.72 q^{29} -3883.61 q^{30} +1183.72 q^{31} -1484.84 q^{32} +1896.23 q^{33} +8505.09 q^{34} -3053.20 q^{35} +171.629 q^{36} +14337.7 q^{37} +3578.47 q^{38} +5777.60 q^{39} -8490.35 q^{40} +19689.2 q^{41} +18971.9 q^{42} +8923.31 q^{43} +8017.54 q^{44} +64.7554 q^{45} -17444.0 q^{46} -13532.7 q^{47} +19528.6 q^{48} -1891.76 q^{49} -6195.41 q^{50} -13446.0 q^{51} +24428.6 q^{52} -4747.41 q^{53} +37346.3 q^{54} +3025.00 q^{55} +41476.4 q^{56} -5657.35 q^{57} +73350.9 q^{58} +22266.8 q^{59} +25959.8 q^{60} +22138.2 q^{61} -11733.8 q^{62} -316.338 q^{63} -25157.6 q^{64} +9216.84 q^{65} -18796.7 q^{66} +14492.3 q^{67} -56851.9 q^{68} +27578.0 q^{69} +30265.3 q^{70} +41494.3 q^{71} -879.673 q^{72} -57928.9 q^{73} -142125. q^{74} +9794.57 q^{75} -23920.1 q^{76} -14777.5 q^{77} -57271.3 q^{78} -76968.9 q^{79} +31153.3 q^{80} -59671.7 q^{81} -195172. q^{82} -33284.7 q^{83} -126817. q^{84} -21450.1 q^{85} -88453.7 q^{86} -115963. q^{87} -41093.3 q^{88} +40140.4 q^{89} -641.898 q^{90} -45025.3 q^{91} +116604. q^{92} +18550.4 q^{93} +134145. q^{94} -9025.00 q^{95} -23269.4 q^{96} +72189.7 q^{97} +18752.4 q^{98} +313.416 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 38 q - 24 q^{2} - 63 q^{3} + 594 q^{4} + 950 q^{5} - 67 q^{6} - 729 q^{7} - 1272 q^{8} + 3029 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 38 q - 24 q^{2} - 63 q^{3} + 594 q^{4} + 950 q^{5} - 67 q^{6} - 729 q^{7} - 1272 q^{8} + 3029 q^{9} - 600 q^{10} + 4598 q^{11} - 2008 q^{12} - 2663 q^{13} - 1565 q^{14} - 1575 q^{15} + 12390 q^{16} - 3311 q^{17} - 6383 q^{18} - 13718 q^{19} + 14850 q^{20} - 8179 q^{21} - 2904 q^{22} - 3412 q^{23} - 4100 q^{24} + 23750 q^{25} - 1399 q^{26} - 31596 q^{27} - 43653 q^{28} - 13633 q^{29} - 1675 q^{30} - 13789 q^{31} - 58603 q^{32} - 7623 q^{33} - 29149 q^{34} - 18225 q^{35} + 50641 q^{36} - 12103 q^{37} + 8664 q^{38} - 50960 q^{39} - 31800 q^{40} - 37885 q^{41} + 51100 q^{42} - 56119 q^{43} + 71874 q^{44} + 75725 q^{45} - 56291 q^{46} - 37532 q^{47} - 113895 q^{48} + 153501 q^{49} - 15000 q^{50} + 32882 q^{51} - 169554 q^{52} - 51511 q^{53} - 175060 q^{54} + 114950 q^{55} - 84247 q^{56} + 22743 q^{57} - 256962 q^{58} - 154267 q^{59} - 50200 q^{60} - 47165 q^{61} + 143002 q^{62} - 358780 q^{63} + 142292 q^{64} - 66575 q^{65} - 8107 q^{66} - 161712 q^{67} - 210188 q^{68} - 124602 q^{69} - 39125 q^{70} + 6118 q^{71} - 327878 q^{72} - 152182 q^{73} - 167349 q^{74} - 39375 q^{75} - 214434 q^{76} - 88209 q^{77} - 216594 q^{78} - 140433 q^{79} + 309750 q^{80} + 382874 q^{81} - 29842 q^{82} - 515287 q^{83} + 29222 q^{84} - 82775 q^{85} + 204974 q^{86} - 106764 q^{87} - 153912 q^{88} - 271610 q^{89} - 159575 q^{90} - 44332 q^{91} + 236348 q^{92} + 25202 q^{93} - 496224 q^{94} - 342950 q^{95} - 275218 q^{96} - 126390 q^{97} - 285506 q^{98} + 366509 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.91265 −1.75233 −0.876163 0.482015i \(-0.839905\pi\)
−0.876163 + 0.482015i \(0.839905\pi\)
\(3\) 15.6713 1.00532 0.502658 0.864485i \(-0.332356\pi\)
0.502658 + 0.864485i \(0.332356\pi\)
\(4\) 66.2607 2.07065
\(5\) 25.0000 0.447214
\(6\) −155.344 −1.76164
\(7\) −122.128 −0.942041 −0.471021 0.882122i \(-0.656114\pi\)
−0.471021 + 0.882122i \(0.656114\pi\)
\(8\) −339.614 −1.87612
\(9\) 2.59022 0.0106593
\(10\) −247.816 −0.783664
\(11\) 121.000 0.301511
\(12\) 1038.39 2.08165
\(13\) 368.674 0.605039 0.302520 0.953143i \(-0.402172\pi\)
0.302520 + 0.953143i \(0.402172\pi\)
\(14\) 1210.61 1.65076
\(15\) 391.783 0.449591
\(16\) 1246.13 1.21693
\(17\) −858.004 −0.720057 −0.360028 0.932941i \(-0.617233\pi\)
−0.360028 + 0.932941i \(0.617233\pi\)
\(18\) −25.6759 −0.0186786
\(19\) −361.000 −0.229416
\(20\) 1656.52 0.926021
\(21\) −1913.91 −0.947049
\(22\) −1199.43 −0.528346
\(23\) 1759.78 0.693646 0.346823 0.937931i \(-0.387261\pi\)
0.346823 + 0.937931i \(0.387261\pi\)
\(24\) −5322.20 −1.88609
\(25\) 625.000 0.200000
\(26\) −3654.53 −1.06023
\(27\) −3767.54 −0.994600
\(28\) −8092.28 −1.95063
\(29\) −7399.72 −1.63388 −0.816941 0.576722i \(-0.804332\pi\)
−0.816941 + 0.576722i \(0.804332\pi\)
\(30\) −3883.61 −0.787829
\(31\) 1183.72 0.221230 0.110615 0.993863i \(-0.464718\pi\)
0.110615 + 0.993863i \(0.464718\pi\)
\(32\) −1484.84 −0.256333
\(33\) 1896.23 0.303114
\(34\) 8505.09 1.26177
\(35\) −3053.20 −0.421294
\(36\) 171.629 0.0220717
\(37\) 14337.7 1.72177 0.860887 0.508796i \(-0.169909\pi\)
0.860887 + 0.508796i \(0.169909\pi\)
\(38\) 3578.47 0.402011
\(39\) 5777.60 0.608255
\(40\) −8490.35 −0.839026
\(41\) 19689.2 1.82923 0.914615 0.404326i \(-0.132494\pi\)
0.914615 + 0.404326i \(0.132494\pi\)
\(42\) 18971.9 1.65954
\(43\) 8923.31 0.735961 0.367981 0.929833i \(-0.380049\pi\)
0.367981 + 0.929833i \(0.380049\pi\)
\(44\) 8017.54 0.624323
\(45\) 64.7554 0.00476699
\(46\) −17444.0 −1.21549
\(47\) −13532.7 −0.893594 −0.446797 0.894635i \(-0.647435\pi\)
−0.446797 + 0.894635i \(0.647435\pi\)
\(48\) 19528.6 1.22340
\(49\) −1891.76 −0.112558
\(50\) −6195.41 −0.350465
\(51\) −13446.0 −0.723884
\(52\) 24428.6 1.25282
\(53\) −4747.41 −0.232149 −0.116074 0.993241i \(-0.537031\pi\)
−0.116074 + 0.993241i \(0.537031\pi\)
\(54\) 37346.3 1.74286
\(55\) 3025.00 0.134840
\(56\) 41476.4 1.76738
\(57\) −5657.35 −0.230635
\(58\) 73350.9 2.86309
\(59\) 22266.8 0.832776 0.416388 0.909187i \(-0.363296\pi\)
0.416388 + 0.909187i \(0.363296\pi\)
\(60\) 25959.8 0.930943
\(61\) 22138.2 0.761759 0.380879 0.924625i \(-0.375621\pi\)
0.380879 + 0.924625i \(0.375621\pi\)
\(62\) −11733.8 −0.387667
\(63\) −316.338 −0.0100415
\(64\) −25157.6 −0.767748
\(65\) 9216.84 0.270582
\(66\) −18796.7 −0.531155
\(67\) 14492.3 0.394413 0.197206 0.980362i \(-0.436813\pi\)
0.197206 + 0.980362i \(0.436813\pi\)
\(68\) −56851.9 −1.49098
\(69\) 27578.0 0.697333
\(70\) 30265.3 0.738244
\(71\) 41494.3 0.976884 0.488442 0.872596i \(-0.337565\pi\)
0.488442 + 0.872596i \(0.337565\pi\)
\(72\) −879.673 −0.0199982
\(73\) −57928.9 −1.27230 −0.636148 0.771567i \(-0.719473\pi\)
−0.636148 + 0.771567i \(0.719473\pi\)
\(74\) −142125. −3.01711
\(75\) 9794.57 0.201063
\(76\) −23920.1 −0.475039
\(77\) −14777.5 −0.284036
\(78\) −57271.3 −1.06586
\(79\) −76968.9 −1.38755 −0.693774 0.720193i \(-0.744053\pi\)
−0.693774 + 0.720193i \(0.744053\pi\)
\(80\) 31153.3 0.544226
\(81\) −59671.7 −1.01055
\(82\) −195172. −3.20541
\(83\) −33284.7 −0.530333 −0.265167 0.964203i \(-0.585427\pi\)
−0.265167 + 0.964203i \(0.585427\pi\)
\(84\) −126817. −1.96100
\(85\) −21450.1 −0.322019
\(86\) −88453.7 −1.28964
\(87\) −115963. −1.64257
\(88\) −41093.3 −0.565671
\(89\) 40140.4 0.537163 0.268582 0.963257i \(-0.413445\pi\)
0.268582 + 0.963257i \(0.413445\pi\)
\(90\) −641.898 −0.00835333
\(91\) −45025.3 −0.569972
\(92\) 116604. 1.43629
\(93\) 18550.4 0.222406
\(94\) 134145. 1.56587
\(95\) −9025.00 −0.102598
\(96\) −23269.4 −0.257696
\(97\) 72189.7 0.779015 0.389508 0.921023i \(-0.372645\pi\)
0.389508 + 0.921023i \(0.372645\pi\)
\(98\) 18752.4 0.197238
\(99\) 313.416 0.00321391
\(100\) 41412.9 0.414129
\(101\) −26226.8 −0.255824 −0.127912 0.991786i \(-0.540827\pi\)
−0.127912 + 0.991786i \(0.540827\pi\)
\(102\) 133286. 1.26848
\(103\) −97439.9 −0.904990 −0.452495 0.891767i \(-0.649466\pi\)
−0.452495 + 0.891767i \(0.649466\pi\)
\(104\) −125207. −1.13513
\(105\) −47847.7 −0.423533
\(106\) 47059.4 0.406801
\(107\) −161675. −1.36516 −0.682580 0.730811i \(-0.739142\pi\)
−0.682580 + 0.730811i \(0.739142\pi\)
\(108\) −249640. −2.05946
\(109\) −11322.3 −0.0912784 −0.0456392 0.998958i \(-0.514532\pi\)
−0.0456392 + 0.998958i \(0.514532\pi\)
\(110\) −29985.8 −0.236284
\(111\) 224691. 1.73093
\(112\) −152188. −1.14640
\(113\) 4809.56 0.0354331 0.0177166 0.999843i \(-0.494360\pi\)
0.0177166 + 0.999843i \(0.494360\pi\)
\(114\) 56079.3 0.404148
\(115\) 43994.4 0.310208
\(116\) −490311. −3.38319
\(117\) 954.944 0.00644931
\(118\) −220723. −1.45929
\(119\) 104786. 0.678323
\(120\) −133055. −0.843486
\(121\) 14641.0 0.0909091
\(122\) −219448. −1.33485
\(123\) 308556. 1.83895
\(124\) 78434.0 0.458089
\(125\) 15625.0 0.0894427
\(126\) 3135.75 0.0175960
\(127\) −254115. −1.39805 −0.699023 0.715099i \(-0.746382\pi\)
−0.699023 + 0.715099i \(0.746382\pi\)
\(128\) 296893. 1.60168
\(129\) 139840. 0.739873
\(130\) −91363.3 −0.474147
\(131\) −22841.9 −0.116293 −0.0581465 0.998308i \(-0.518519\pi\)
−0.0581465 + 0.998308i \(0.518519\pi\)
\(132\) 125645. 0.627642
\(133\) 44088.2 0.216119
\(134\) −143657. −0.691139
\(135\) −94188.5 −0.444798
\(136\) 291390. 1.35091
\(137\) −26462.3 −0.120456 −0.0602278 0.998185i \(-0.519183\pi\)
−0.0602278 + 0.998185i \(0.519183\pi\)
\(138\) −273371. −1.22195
\(139\) −101775. −0.446791 −0.223396 0.974728i \(-0.571714\pi\)
−0.223396 + 0.974728i \(0.571714\pi\)
\(140\) −202307. −0.872350
\(141\) −212075. −0.898344
\(142\) −411319. −1.71182
\(143\) 44609.5 0.182426
\(144\) 3227.75 0.0129716
\(145\) −184993. −0.730694
\(146\) 574229. 2.22948
\(147\) −29646.4 −0.113156
\(148\) 950028. 3.56518
\(149\) 39518.6 0.145826 0.0729132 0.997338i \(-0.476770\pi\)
0.0729132 + 0.997338i \(0.476770\pi\)
\(150\) −97090.2 −0.352328
\(151\) 14475.2 0.0516632 0.0258316 0.999666i \(-0.491777\pi\)
0.0258316 + 0.999666i \(0.491777\pi\)
\(152\) 122601. 0.430411
\(153\) −2222.41 −0.00767532
\(154\) 146484. 0.497724
\(155\) 29593.0 0.0989371
\(156\) 382828. 1.25948
\(157\) −470159. −1.52228 −0.761142 0.648585i \(-0.775361\pi\)
−0.761142 + 0.648585i \(0.775361\pi\)
\(158\) 762966. 2.43143
\(159\) −74398.1 −0.233383
\(160\) −37121.0 −0.114636
\(161\) −214918. −0.653443
\(162\) 591505. 1.77081
\(163\) 236406. 0.696929 0.348465 0.937322i \(-0.386703\pi\)
0.348465 + 0.937322i \(0.386703\pi\)
\(164\) 1.30462e6 3.78769
\(165\) 47405.7 0.135557
\(166\) 329939. 0.929317
\(167\) −295646. −0.820316 −0.410158 0.912015i \(-0.634526\pi\)
−0.410158 + 0.912015i \(0.634526\pi\)
\(168\) 649989. 1.77678
\(169\) −235373. −0.633927
\(170\) 212627. 0.564283
\(171\) −935.068 −0.00244542
\(172\) 591264. 1.52391
\(173\) 185379. 0.470918 0.235459 0.971884i \(-0.424341\pi\)
0.235459 + 0.971884i \(0.424341\pi\)
\(174\) 1.14950e6 2.87831
\(175\) −76330.0 −0.188408
\(176\) 150782. 0.366917
\(177\) 348950. 0.837202
\(178\) −397897. −0.941285
\(179\) −295793. −0.690010 −0.345005 0.938601i \(-0.612123\pi\)
−0.345005 + 0.938601i \(0.612123\pi\)
\(180\) 4290.73 0.00987076
\(181\) 756527. 1.71644 0.858219 0.513284i \(-0.171571\pi\)
0.858219 + 0.513284i \(0.171571\pi\)
\(182\) 446321. 0.998777
\(183\) 346935. 0.765808
\(184\) −597644. −1.30136
\(185\) 358443. 0.770001
\(186\) −183884. −0.389728
\(187\) −103818. −0.217105
\(188\) −896686. −1.85032
\(189\) 460122. 0.936954
\(190\) 89461.7 0.179785
\(191\) 384626. 0.762879 0.381440 0.924394i \(-0.375428\pi\)
0.381440 + 0.924394i \(0.375428\pi\)
\(192\) −394252. −0.771829
\(193\) −390296. −0.754225 −0.377112 0.926167i \(-0.623083\pi\)
−0.377112 + 0.926167i \(0.623083\pi\)
\(194\) −715592. −1.36509
\(195\) 144440. 0.272020
\(196\) −125349. −0.233068
\(197\) −838013. −1.53846 −0.769229 0.638974i \(-0.779359\pi\)
−0.769229 + 0.638974i \(0.779359\pi\)
\(198\) −3106.78 −0.00563181
\(199\) 340911. 0.610252 0.305126 0.952312i \(-0.401301\pi\)
0.305126 + 0.952312i \(0.401301\pi\)
\(200\) −212259. −0.375224
\(201\) 227114. 0.396509
\(202\) 259977. 0.448287
\(203\) 903713. 1.53918
\(204\) −890944. −1.49891
\(205\) 492230. 0.818056
\(206\) 965888. 1.58584
\(207\) 4558.20 0.00739379
\(208\) 459416. 0.736289
\(209\) −43681.0 −0.0691714
\(210\) 474297. 0.742168
\(211\) −167344. −0.258765 −0.129382 0.991595i \(-0.541299\pi\)
−0.129382 + 0.991595i \(0.541299\pi\)
\(212\) −314566. −0.480698
\(213\) 650271. 0.982077
\(214\) 1.60263e6 2.39221
\(215\) 223083. 0.329132
\(216\) 1.27951e6 1.86599
\(217\) −144565. −0.208408
\(218\) 112234. 0.159949
\(219\) −907822. −1.27906
\(220\) 200438. 0.279206
\(221\) −316323. −0.435663
\(222\) −2.22729e6 −3.03315
\(223\) 958374. 1.29054 0.645272 0.763953i \(-0.276744\pi\)
0.645272 + 0.763953i \(0.276744\pi\)
\(224\) 181341. 0.241476
\(225\) 1618.88 0.00213186
\(226\) −47675.5 −0.0620904
\(227\) −1.14011e6 −1.46852 −0.734261 0.678867i \(-0.762471\pi\)
−0.734261 + 0.678867i \(0.762471\pi\)
\(228\) −374859. −0.477564
\(229\) −305837. −0.385390 −0.192695 0.981259i \(-0.561723\pi\)
−0.192695 + 0.981259i \(0.561723\pi\)
\(230\) −436101. −0.543585
\(231\) −231583. −0.285546
\(232\) 2.51305e6 3.06536
\(233\) 86798.3 0.104742 0.0523711 0.998628i \(-0.483322\pi\)
0.0523711 + 0.998628i \(0.483322\pi\)
\(234\) −9466.03 −0.0113013
\(235\) −338318. −0.399627
\(236\) 1.47541e6 1.72438
\(237\) −1.20620e6 −1.39492
\(238\) −1.03871e6 −1.18864
\(239\) 312504. 0.353884 0.176942 0.984221i \(-0.443379\pi\)
0.176942 + 0.984221i \(0.443379\pi\)
\(240\) 488214. 0.547119
\(241\) −433881. −0.481202 −0.240601 0.970624i \(-0.577345\pi\)
−0.240601 + 0.970624i \(0.577345\pi\)
\(242\) −145131. −0.159302
\(243\) −19622.6 −0.0213177
\(244\) 1.46689e6 1.57733
\(245\) −47294.0 −0.0503374
\(246\) −3.05860e6 −3.22244
\(247\) −133091. −0.138806
\(248\) −402007. −0.415054
\(249\) −521615. −0.533152
\(250\) −154885. −0.156733
\(251\) 174142. 0.174469 0.0872347 0.996188i \(-0.472197\pi\)
0.0872347 + 0.996188i \(0.472197\pi\)
\(252\) −20960.7 −0.0207924
\(253\) 212933. 0.209142
\(254\) 2.51896e6 2.44983
\(255\) −336151. −0.323731
\(256\) −2.13796e6 −2.03891
\(257\) −377892. −0.356891 −0.178445 0.983950i \(-0.557107\pi\)
−0.178445 + 0.983950i \(0.557107\pi\)
\(258\) −1.38619e6 −1.29650
\(259\) −1.75104e6 −1.62198
\(260\) 610714. 0.560279
\(261\) −19166.9 −0.0174161
\(262\) 226424. 0.203783
\(263\) −617400. −0.550399 −0.275199 0.961387i \(-0.588744\pi\)
−0.275199 + 0.961387i \(0.588744\pi\)
\(264\) −643986. −0.568678
\(265\) −118685. −0.103820
\(266\) −437031. −0.378711
\(267\) 629052. 0.540019
\(268\) 960270. 0.816688
\(269\) 105081. 0.0885404 0.0442702 0.999020i \(-0.485904\pi\)
0.0442702 + 0.999020i \(0.485904\pi\)
\(270\) 933657. 0.779432
\(271\) −831128. −0.687455 −0.343728 0.939069i \(-0.611690\pi\)
−0.343728 + 0.939069i \(0.611690\pi\)
\(272\) −1.06919e6 −0.876257
\(273\) −705607. −0.573002
\(274\) 262312. 0.211077
\(275\) 75625.0 0.0603023
\(276\) 1.82734e6 1.44393
\(277\) 664598. 0.520426 0.260213 0.965551i \(-0.416207\pi\)
0.260213 + 0.965551i \(0.416207\pi\)
\(278\) 1.00886e6 0.782924
\(279\) 3066.09 0.00235816
\(280\) 1.03691e6 0.790397
\(281\) −1.85375e6 −1.40051 −0.700253 0.713895i \(-0.746930\pi\)
−0.700253 + 0.713895i \(0.746930\pi\)
\(282\) 2.10223e6 1.57419
\(283\) 2.26829e6 1.68357 0.841787 0.539810i \(-0.181504\pi\)
0.841787 + 0.539810i \(0.181504\pi\)
\(284\) 2.74944e6 2.02278
\(285\) −141434. −0.103143
\(286\) −442198. −0.319670
\(287\) −2.40460e6 −1.72321
\(288\) −3846.06 −0.00273234
\(289\) −683687. −0.481518
\(290\) 1.83377e6 1.28041
\(291\) 1.13131e6 0.783156
\(292\) −3.83841e6 −2.63447
\(293\) 533780. 0.363239 0.181620 0.983369i \(-0.441866\pi\)
0.181620 + 0.983369i \(0.441866\pi\)
\(294\) 293874. 0.198287
\(295\) 556670. 0.372429
\(296\) −4.86929e6 −3.23025
\(297\) −455872. −0.299883
\(298\) −391734. −0.255535
\(299\) 648783. 0.419683
\(300\) 648995. 0.416330
\(301\) −1.08979e6 −0.693306
\(302\) −143487. −0.0905307
\(303\) −411008. −0.257184
\(304\) −449854. −0.279182
\(305\) 553455. 0.340669
\(306\) 22030.0 0.0134497
\(307\) 852984. 0.516529 0.258265 0.966074i \(-0.416849\pi\)
0.258265 + 0.966074i \(0.416849\pi\)
\(308\) −979166. −0.588138
\(309\) −1.52701e6 −0.909801
\(310\) −293345. −0.173370
\(311\) −1.29944e6 −0.761824 −0.380912 0.924611i \(-0.624390\pi\)
−0.380912 + 0.924611i \(0.624390\pi\)
\(312\) −1.96215e6 −1.14116
\(313\) −2.13933e6 −1.23429 −0.617144 0.786850i \(-0.711711\pi\)
−0.617144 + 0.786850i \(0.711711\pi\)
\(314\) 4.66052e6 2.66754
\(315\) −7908.44 −0.00449071
\(316\) −5.10001e6 −2.87312
\(317\) −2.97033e6 −1.66018 −0.830092 0.557627i \(-0.811712\pi\)
−0.830092 + 0.557627i \(0.811712\pi\)
\(318\) 737483. 0.408963
\(319\) −895367. −0.492634
\(320\) −628939. −0.343347
\(321\) −2.53366e6 −1.37242
\(322\) 2.13040e6 1.14504
\(323\) 309739. 0.165192
\(324\) −3.95389e6 −2.09248
\(325\) 230421. 0.121008
\(326\) −2.34341e6 −1.22125
\(327\) −177435. −0.0917636
\(328\) −6.68672e6 −3.43185
\(329\) 1.65272e6 0.841802
\(330\) −469917. −0.237540
\(331\) 1.92488e6 0.965683 0.482841 0.875708i \(-0.339605\pi\)
0.482841 + 0.875708i \(0.339605\pi\)
\(332\) −2.20546e6 −1.09813
\(333\) 37137.8 0.0183529
\(334\) 2.93064e6 1.43746
\(335\) 362308. 0.176387
\(336\) −2.38498e6 −1.15249
\(337\) 773927. 0.371215 0.185607 0.982624i \(-0.440575\pi\)
0.185607 + 0.982624i \(0.440575\pi\)
\(338\) 2.33317e6 1.11085
\(339\) 75372.2 0.0356215
\(340\) −1.42130e6 −0.666788
\(341\) 143230. 0.0667034
\(342\) 9269.00 0.00428517
\(343\) 2.28364e6 1.04808
\(344\) −3.03048e6 −1.38075
\(345\) 689450. 0.311857
\(346\) −1.83760e6 −0.825202
\(347\) −3.74600e6 −1.67011 −0.835053 0.550170i \(-0.814563\pi\)
−0.835053 + 0.550170i \(0.814563\pi\)
\(348\) −7.68381e6 −3.40117
\(349\) 3.31098e6 1.45510 0.727549 0.686055i \(-0.240659\pi\)
0.727549 + 0.686055i \(0.240659\pi\)
\(350\) 756632. 0.330153
\(351\) −1.38899e6 −0.601772
\(352\) −179666. −0.0772874
\(353\) −2.79778e6 −1.19502 −0.597512 0.801860i \(-0.703844\pi\)
−0.597512 + 0.801860i \(0.703844\pi\)
\(354\) −3.45902e6 −1.46705
\(355\) 1.03736e6 0.436876
\(356\) 2.65973e6 1.11227
\(357\) 1.64214e6 0.681929
\(358\) 2.93209e6 1.20912
\(359\) 26730.3 0.0109463 0.00547315 0.999985i \(-0.498258\pi\)
0.00547315 + 0.999985i \(0.498258\pi\)
\(360\) −21991.8 −0.00894345
\(361\) 130321. 0.0526316
\(362\) −7.49919e6 −3.00776
\(363\) 229444. 0.0913923
\(364\) −2.98341e6 −1.18021
\(365\) −1.44822e6 −0.568988
\(366\) −3.43904e6 −1.34194
\(367\) 4.12634e6 1.59919 0.799595 0.600540i \(-0.205048\pi\)
0.799595 + 0.600540i \(0.205048\pi\)
\(368\) 2.19291e6 0.844116
\(369\) 50999.2 0.0194983
\(370\) −3.55312e6 −1.34929
\(371\) 579791. 0.218694
\(372\) 1.22916e6 0.460524
\(373\) −928274. −0.345465 −0.172733 0.984969i \(-0.555260\pi\)
−0.172733 + 0.984969i \(0.555260\pi\)
\(374\) 1.02912e6 0.380439
\(375\) 244864. 0.0899182
\(376\) 4.59590e6 1.67649
\(377\) −2.72808e6 −0.988562
\(378\) −4.56103e6 −1.64185
\(379\) −2.52172e6 −0.901778 −0.450889 0.892580i \(-0.648893\pi\)
−0.450889 + 0.892580i \(0.648893\pi\)
\(380\) −598002. −0.212444
\(381\) −3.98232e6 −1.40548
\(382\) −3.81267e6 −1.33681
\(383\) −1.33224e6 −0.464072 −0.232036 0.972707i \(-0.574539\pi\)
−0.232036 + 0.972707i \(0.574539\pi\)
\(384\) 4.65271e6 1.61019
\(385\) −369437. −0.127025
\(386\) 3.86887e6 1.32165
\(387\) 23113.3 0.00784485
\(388\) 4.78334e6 1.61306
\(389\) −2.97937e6 −0.998275 −0.499137 0.866523i \(-0.666350\pi\)
−0.499137 + 0.866523i \(0.666350\pi\)
\(390\) −1.43178e6 −0.476668
\(391\) −1.50989e6 −0.499464
\(392\) 642469. 0.211172
\(393\) −357962. −0.116911
\(394\) 8.30693e6 2.69588
\(395\) −1.92422e6 −0.620530
\(396\) 20767.2 0.00665486
\(397\) −1.70619e6 −0.543316 −0.271658 0.962394i \(-0.587572\pi\)
−0.271658 + 0.962394i \(0.587572\pi\)
\(398\) −3.37934e6 −1.06936
\(399\) 690920. 0.217268
\(400\) 778833. 0.243385
\(401\) 3.83304e6 1.19037 0.595186 0.803588i \(-0.297078\pi\)
0.595186 + 0.803588i \(0.297078\pi\)
\(402\) −2.25130e6 −0.694813
\(403\) 436406. 0.133853
\(404\) −1.73780e6 −0.529721
\(405\) −1.49179e6 −0.451930
\(406\) −8.95819e6 −2.69715
\(407\) 1.73487e6 0.519134
\(408\) 4.56647e6 1.35809
\(409\) 6.67302e6 1.97249 0.986243 0.165303i \(-0.0528602\pi\)
0.986243 + 0.165303i \(0.0528602\pi\)
\(410\) −4.87930e6 −1.43350
\(411\) −414700. −0.121096
\(412\) −6.45643e6 −1.87391
\(413\) −2.71940e6 −0.784509
\(414\) −45183.8 −0.0129563
\(415\) −832117. −0.237172
\(416\) −547421. −0.155092
\(417\) −1.59495e6 −0.449166
\(418\) 432995. 0.121211
\(419\) 39601.7 0.0110199 0.00550996 0.999985i \(-0.498246\pi\)
0.00550996 + 0.999985i \(0.498246\pi\)
\(420\) −3.17042e6 −0.876987
\(421\) −3.25180e6 −0.894166 −0.447083 0.894493i \(-0.647537\pi\)
−0.447083 + 0.894493i \(0.647537\pi\)
\(422\) 1.65883e6 0.453440
\(423\) −35052.6 −0.00952511
\(424\) 1.61229e6 0.435539
\(425\) −536252. −0.144011
\(426\) −6.44591e6 −1.72092
\(427\) −2.70369e6 −0.717608
\(428\) −1.07127e7 −2.82676
\(429\) 699090. 0.183396
\(430\) −2.21134e6 −0.576746
\(431\) −3.53830e6 −0.917490 −0.458745 0.888568i \(-0.651701\pi\)
−0.458745 + 0.888568i \(0.651701\pi\)
\(432\) −4.69486e6 −1.21036
\(433\) 65925.2 0.0168978 0.00844892 0.999964i \(-0.497311\pi\)
0.00844892 + 0.999964i \(0.497311\pi\)
\(434\) 1.43302e6 0.365199
\(435\) −2.89909e6 −0.734578
\(436\) −750222. −0.189005
\(437\) −635279. −0.159133
\(438\) 8.99892e6 2.24133
\(439\) 4.77825e6 1.18334 0.591668 0.806182i \(-0.298470\pi\)
0.591668 + 0.806182i \(0.298470\pi\)
\(440\) −1.02733e6 −0.252976
\(441\) −4900.07 −0.00119979
\(442\) 3.13560e6 0.763423
\(443\) −3.56592e6 −0.863301 −0.431650 0.902041i \(-0.642069\pi\)
−0.431650 + 0.902041i \(0.642069\pi\)
\(444\) 1.48882e7 3.58413
\(445\) 1.00351e6 0.240227
\(446\) −9.50003e6 −2.26145
\(447\) 619309. 0.146602
\(448\) 3.07244e6 0.723250
\(449\) 971747. 0.227477 0.113738 0.993511i \(-0.463717\pi\)
0.113738 + 0.993511i \(0.463717\pi\)
\(450\) −16047.4 −0.00373572
\(451\) 2.38239e6 0.551533
\(452\) 318685. 0.0733694
\(453\) 226845. 0.0519378
\(454\) 1.13015e7 2.57333
\(455\) −1.12563e6 −0.254899
\(456\) 1.92131e6 0.432699
\(457\) 524697. 0.117522 0.0587609 0.998272i \(-0.481285\pi\)
0.0587609 + 0.998272i \(0.481285\pi\)
\(458\) 3.03165e6 0.675329
\(459\) 3.23256e6 0.716168
\(460\) 2.91510e6 0.642330
\(461\) −2.78946e6 −0.611319 −0.305659 0.952141i \(-0.598877\pi\)
−0.305659 + 0.952141i \(0.598877\pi\)
\(462\) 2.29560e6 0.500370
\(463\) −6.76123e6 −1.46580 −0.732898 0.680339i \(-0.761833\pi\)
−0.732898 + 0.680339i \(0.761833\pi\)
\(464\) −9.22104e6 −1.98831
\(465\) 463761. 0.0994630
\(466\) −860401. −0.183542
\(467\) −6.79585e6 −1.44195 −0.720977 0.692959i \(-0.756307\pi\)
−0.720977 + 0.692959i \(0.756307\pi\)
\(468\) 63275.2 0.0133542
\(469\) −1.76992e6 −0.371553
\(470\) 3.35363e6 0.700277
\(471\) −7.36801e6 −1.53038
\(472\) −7.56212e6 −1.56239
\(473\) 1.07972e6 0.221901
\(474\) 1.19567e7 2.44436
\(475\) −225625. −0.0458831
\(476\) 6.94320e6 1.40457
\(477\) −12296.8 −0.00247455
\(478\) −3.09775e6 −0.620121
\(479\) −7.83424e6 −1.56012 −0.780061 0.625704i \(-0.784812\pi\)
−0.780061 + 0.625704i \(0.784812\pi\)
\(480\) −581735. −0.115245
\(481\) 5.28594e6 1.04174
\(482\) 4.30091e6 0.843223
\(483\) −3.36804e6 −0.656916
\(484\) 970122. 0.188240
\(485\) 1.80474e6 0.348386
\(486\) 194512. 0.0373556
\(487\) −1.90142e6 −0.363291 −0.181646 0.983364i \(-0.558142\pi\)
−0.181646 + 0.983364i \(0.558142\pi\)
\(488\) −7.51844e6 −1.42915
\(489\) 3.70479e6 0.700634
\(490\) 468809. 0.0882076
\(491\) 4.88670e6 0.914770 0.457385 0.889269i \(-0.348786\pi\)
0.457385 + 0.889269i \(0.348786\pi\)
\(492\) 2.04451e7 3.80782
\(493\) 6.34899e6 1.17649
\(494\) 1.31929e6 0.243233
\(495\) 7835.40 0.00143730
\(496\) 1.47507e6 0.269221
\(497\) −5.06762e6 −0.920265
\(498\) 5.17058e6 0.934257
\(499\) −8.97412e6 −1.61339 −0.806697 0.590965i \(-0.798747\pi\)
−0.806697 + 0.590965i \(0.798747\pi\)
\(500\) 1.03532e6 0.185204
\(501\) −4.63317e6 −0.824676
\(502\) −1.72621e6 −0.305727
\(503\) −9.18627e6 −1.61890 −0.809449 0.587190i \(-0.800234\pi\)
−0.809449 + 0.587190i \(0.800234\pi\)
\(504\) 107433. 0.0188391
\(505\) −655669. −0.114408
\(506\) −2.11073e6 −0.366485
\(507\) −3.68860e6 −0.637297
\(508\) −1.68378e7 −2.89486
\(509\) 4.47325e6 0.765294 0.382647 0.923895i \(-0.375013\pi\)
0.382647 + 0.923895i \(0.375013\pi\)
\(510\) 3.33215e6 0.567282
\(511\) 7.07474e6 1.19856
\(512\) 1.16922e7 1.97116
\(513\) 1.36008e6 0.228177
\(514\) 3.74591e6 0.625389
\(515\) −2.43600e6 −0.404724
\(516\) 9.26589e6 1.53201
\(517\) −1.63746e6 −0.269429
\(518\) 1.73574e7 2.84224
\(519\) 2.90513e6 0.473421
\(520\) −3.13017e6 −0.507644
\(521\) 4.21769e6 0.680739 0.340369 0.940292i \(-0.389448\pi\)
0.340369 + 0.940292i \(0.389448\pi\)
\(522\) 189995. 0.0305186
\(523\) −8.62086e6 −1.37815 −0.689075 0.724690i \(-0.741983\pi\)
−0.689075 + 0.724690i \(0.741983\pi\)
\(524\) −1.51352e6 −0.240802
\(525\) −1.19619e6 −0.189410
\(526\) 6.12007e6 0.964478
\(527\) −1.01564e6 −0.159298
\(528\) 2.36295e6 0.368868
\(529\) −3.33953e6 −0.518856
\(530\) 1.17648e6 0.181927
\(531\) 57675.8 0.00887682
\(532\) 2.92131e6 0.447506
\(533\) 7.25888e6 1.10676
\(534\) −6.23558e6 −0.946288
\(535\) −4.04188e6 −0.610518
\(536\) −4.92179e6 −0.739965
\(537\) −4.63547e6 −0.693678
\(538\) −1.04163e6 −0.155152
\(539\) −228903. −0.0339375
\(540\) −6.24099e6 −0.921020
\(541\) −4.35985e6 −0.640440 −0.320220 0.947343i \(-0.603757\pi\)
−0.320220 + 0.947343i \(0.603757\pi\)
\(542\) 8.23868e6 1.20465
\(543\) 1.18558e7 1.72556
\(544\) 1.27400e6 0.184574
\(545\) −283057. −0.0408209
\(546\) 6.99443e6 1.00409
\(547\) −1.08483e6 −0.155022 −0.0775111 0.996991i \(-0.524697\pi\)
−0.0775111 + 0.996991i \(0.524697\pi\)
\(548\) −1.75341e6 −0.249421
\(549\) 57342.7 0.00811983
\(550\) −749644. −0.105669
\(551\) 2.67130e6 0.374838
\(552\) −9.36587e6 −1.30828
\(553\) 9.40006e6 1.30713
\(554\) −6.58792e6 −0.911957
\(555\) 5.61728e6 0.774094
\(556\) −6.74369e6 −0.925146
\(557\) −6.27051e6 −0.856377 −0.428188 0.903689i \(-0.640848\pi\)
−0.428188 + 0.903689i \(0.640848\pi\)
\(558\) −30393.1 −0.00413227
\(559\) 3.28979e6 0.445285
\(560\) −3.80469e6 −0.512684
\(561\) −1.62697e6 −0.218259
\(562\) 1.83756e7 2.45414
\(563\) 4.93153e6 0.655708 0.327854 0.944728i \(-0.393675\pi\)
0.327854 + 0.944728i \(0.393675\pi\)
\(564\) −1.40523e7 −1.86015
\(565\) 120239. 0.0158462
\(566\) −2.24848e7 −2.95017
\(567\) 7.28758e6 0.951976
\(568\) −1.40921e7 −1.83275
\(569\) 7.96172e6 1.03092 0.515461 0.856913i \(-0.327621\pi\)
0.515461 + 0.856913i \(0.327621\pi\)
\(570\) 1.40198e6 0.180740
\(571\) 1.19086e6 0.152852 0.0764259 0.997075i \(-0.475649\pi\)
0.0764259 + 0.997075i \(0.475649\pi\)
\(572\) 2.95585e6 0.377740
\(573\) 6.02760e6 0.766934
\(574\) 2.38360e7 3.01963
\(575\) 1.09986e6 0.138729
\(576\) −65163.5 −0.00818367
\(577\) −2.85119e6 −0.356523 −0.178261 0.983983i \(-0.557047\pi\)
−0.178261 + 0.983983i \(0.557047\pi\)
\(578\) 6.77715e6 0.843776
\(579\) −6.11645e6 −0.758234
\(580\) −1.22578e7 −1.51301
\(581\) 4.06499e6 0.499596
\(582\) −1.12143e7 −1.37235
\(583\) −574436. −0.0699956
\(584\) 1.96735e7 2.38698
\(585\) 23873.6 0.00288422
\(586\) −5.29117e6 −0.636514
\(587\) −7.83586e6 −0.938623 −0.469312 0.883033i \(-0.655498\pi\)
−0.469312 + 0.883033i \(0.655498\pi\)
\(588\) −1.96439e6 −0.234306
\(589\) −427323. −0.0507537
\(590\) −5.51808e6 −0.652616
\(591\) −1.31328e7 −1.54663
\(592\) 1.78667e7 2.09527
\(593\) −4.54441e6 −0.530690 −0.265345 0.964154i \(-0.585486\pi\)
−0.265345 + 0.964154i \(0.585486\pi\)
\(594\) 4.51890e6 0.525493
\(595\) 2.61966e6 0.303355
\(596\) 2.61853e6 0.301955
\(597\) 5.34253e6 0.613495
\(598\) −6.43116e6 −0.735421
\(599\) −8.76221e6 −0.997807 −0.498904 0.866657i \(-0.666264\pi\)
−0.498904 + 0.866657i \(0.666264\pi\)
\(600\) −3.32637e6 −0.377218
\(601\) 131994. 0.0149062 0.00745312 0.999972i \(-0.497628\pi\)
0.00745312 + 0.999972i \(0.497628\pi\)
\(602\) 1.08027e7 1.21490
\(603\) 37538.2 0.00420417
\(604\) 959134. 0.106976
\(605\) 366025. 0.0406558
\(606\) 4.07418e6 0.450670
\(607\) −1.16973e7 −1.28859 −0.644295 0.764777i \(-0.722849\pi\)
−0.644295 + 0.764777i \(0.722849\pi\)
\(608\) 536027. 0.0588069
\(609\) 1.41624e7 1.54737
\(610\) −5.48620e6 −0.596963
\(611\) −4.98915e6 −0.540659
\(612\) −147259. −0.0158929
\(613\) 1.41538e7 1.52133 0.760665 0.649145i \(-0.224873\pi\)
0.760665 + 0.649145i \(0.224873\pi\)
\(614\) −8.45533e6 −0.905128
\(615\) 7.71389e6 0.822405
\(616\) 5.01864e6 0.532886
\(617\) −4.99655e6 −0.528393 −0.264197 0.964469i \(-0.585107\pi\)
−0.264197 + 0.964469i \(0.585107\pi\)
\(618\) 1.51367e7 1.59427
\(619\) 8.20573e6 0.860777 0.430389 0.902644i \(-0.358377\pi\)
0.430389 + 0.902644i \(0.358377\pi\)
\(620\) 1.96085e6 0.204864
\(621\) −6.63002e6 −0.689900
\(622\) 1.28809e7 1.33496
\(623\) −4.90226e6 −0.506030
\(624\) 7.19966e6 0.740203
\(625\) 390625. 0.0400000
\(626\) 2.12064e7 2.16288
\(627\) −684539. −0.0695391
\(628\) −3.11530e7 −3.15211
\(629\) −1.23018e7 −1.23978
\(630\) 78393.6 0.00786918
\(631\) 1.47441e7 1.47416 0.737082 0.675803i \(-0.236203\pi\)
0.737082 + 0.675803i \(0.236203\pi\)
\(632\) 2.61397e7 2.60320
\(633\) −2.62251e6 −0.260140
\(634\) 2.94438e7 2.90918
\(635\) −6.35288e6 −0.625225
\(636\) −4.92967e6 −0.483253
\(637\) −697442. −0.0681020
\(638\) 8.87546e6 0.863255
\(639\) 107479. 0.0104129
\(640\) 7.42233e6 0.716292
\(641\) −6.94288e6 −0.667413 −0.333707 0.942677i \(-0.608299\pi\)
−0.333707 + 0.942677i \(0.608299\pi\)
\(642\) 2.51153e7 2.40492
\(643\) −1.32874e7 −1.26740 −0.633700 0.773579i \(-0.718465\pi\)
−0.633700 + 0.773579i \(0.718465\pi\)
\(644\) −1.42406e7 −1.35305
\(645\) 3.49600e6 0.330881
\(646\) −3.07034e6 −0.289471
\(647\) 9.12095e6 0.856603 0.428301 0.903636i \(-0.359112\pi\)
0.428301 + 0.903636i \(0.359112\pi\)
\(648\) 2.02653e7 1.89590
\(649\) 2.69428e6 0.251091
\(650\) −2.28408e6 −0.212045
\(651\) −2.26553e6 −0.209516
\(652\) 1.56644e7 1.44309
\(653\) −1.21176e7 −1.11208 −0.556039 0.831156i \(-0.687679\pi\)
−0.556039 + 0.831156i \(0.687679\pi\)
\(654\) 1.75885e6 0.160800
\(655\) −571047. −0.0520078
\(656\) 2.45354e7 2.22604
\(657\) −150048. −0.0135618
\(658\) −1.63829e7 −1.47511
\(659\) −1.13385e7 −1.01705 −0.508525 0.861047i \(-0.669809\pi\)
−0.508525 + 0.861047i \(0.669809\pi\)
\(660\) 3.14114e6 0.280690
\(661\) −476277. −0.0423990 −0.0211995 0.999775i \(-0.506749\pi\)
−0.0211995 + 0.999775i \(0.506749\pi\)
\(662\) −1.90807e7 −1.69219
\(663\) −4.95720e6 −0.437979
\(664\) 1.13039e7 0.994969
\(665\) 1.10220e6 0.0966514
\(666\) −368134. −0.0321603
\(667\) −1.30218e7 −1.13333
\(668\) −1.95897e7 −1.69858
\(669\) 1.50190e7 1.29740
\(670\) −3.59143e6 −0.309087
\(671\) 2.67872e6 0.229679
\(672\) 2.84184e6 0.242760
\(673\) 4.96425e6 0.422489 0.211245 0.977433i \(-0.432248\pi\)
0.211245 + 0.977433i \(0.432248\pi\)
\(674\) −7.67167e6 −0.650489
\(675\) −2.35471e6 −0.198920
\(676\) −1.55960e7 −1.31264
\(677\) 1.87249e7 1.57017 0.785086 0.619387i \(-0.212619\pi\)
0.785086 + 0.619387i \(0.212619\pi\)
\(678\) −747138. −0.0624204
\(679\) −8.81638e6 −0.733865
\(680\) 7.28475e6 0.604147
\(681\) −1.78670e7 −1.47633
\(682\) −1.41979e6 −0.116886
\(683\) 7.77676e6 0.637892 0.318946 0.947773i \(-0.396671\pi\)
0.318946 + 0.947773i \(0.396671\pi\)
\(684\) −61958.2 −0.00506359
\(685\) −661559. −0.0538694
\(686\) −2.26369e7 −1.83657
\(687\) −4.79286e6 −0.387439
\(688\) 1.11196e7 0.895611
\(689\) −1.75024e6 −0.140459
\(690\) −6.83428e6 −0.546474
\(691\) 2.34197e7 1.86589 0.932944 0.360022i \(-0.117231\pi\)
0.932944 + 0.360022i \(0.117231\pi\)
\(692\) 1.22833e7 0.975104
\(693\) −38276.9 −0.00302763
\(694\) 3.71328e7 2.92657
\(695\) −2.54438e6 −0.199811
\(696\) 3.93828e7 3.08165
\(697\) −1.68934e7 −1.31715
\(698\) −3.28205e7 −2.54981
\(699\) 1.36024e6 0.105299
\(700\) −5.05767e6 −0.390127
\(701\) −1.29951e7 −0.998811 −0.499406 0.866368i \(-0.666448\pi\)
−0.499406 + 0.866368i \(0.666448\pi\)
\(702\) 1.37686e7 1.05450
\(703\) −5.17592e6 −0.395002
\(704\) −3.04407e6 −0.231485
\(705\) −5.30189e6 −0.401752
\(706\) 2.77334e7 2.09407
\(707\) 3.20302e6 0.240997
\(708\) 2.31217e7 1.73355
\(709\) −2.42577e7 −1.81232 −0.906160 0.422935i \(-0.861000\pi\)
−0.906160 + 0.422935i \(0.861000\pi\)
\(710\) −1.02830e7 −0.765549
\(711\) −199366. −0.0147903
\(712\) −1.36322e7 −1.00778
\(713\) 2.08308e6 0.153455
\(714\) −1.62779e7 −1.19496
\(715\) 1.11524e6 0.0815835
\(716\) −1.95994e7 −1.42877
\(717\) 4.89736e6 0.355765
\(718\) −264968. −0.0191815
\(719\) 4.81309e6 0.347218 0.173609 0.984815i \(-0.444457\pi\)
0.173609 + 0.984815i \(0.444457\pi\)
\(720\) 80693.9 0.00580109
\(721\) 1.19001e7 0.852538
\(722\) −1.29183e6 −0.0922277
\(723\) −6.79948e6 −0.483760
\(724\) 5.01280e7 3.55413
\(725\) −4.62483e6 −0.326776
\(726\) −2.27440e6 −0.160149
\(727\) 2.76680e7 1.94152 0.970759 0.240057i \(-0.0771662\pi\)
0.970759 + 0.240057i \(0.0771662\pi\)
\(728\) 1.52912e7 1.06934
\(729\) 1.41927e7 0.989115
\(730\) 1.43557e7 0.997052
\(731\) −7.65623e6 −0.529934
\(732\) 2.29881e7 1.58572
\(733\) −3.55639e6 −0.244483 −0.122242 0.992500i \(-0.539008\pi\)
−0.122242 + 0.992500i \(0.539008\pi\)
\(734\) −4.09030e7 −2.80230
\(735\) −741160. −0.0506050
\(736\) −2.61298e6 −0.177804
\(737\) 1.75357e6 0.118920
\(738\) −505538. −0.0341675
\(739\) −1.47974e7 −0.996724 −0.498362 0.866969i \(-0.666065\pi\)
−0.498362 + 0.866969i \(0.666065\pi\)
\(740\) 2.37507e7 1.59440
\(741\) −2.08571e6 −0.139543
\(742\) −5.74727e6 −0.383223
\(743\) −1.64371e7 −1.09233 −0.546165 0.837678i \(-0.683913\pi\)
−0.546165 + 0.837678i \(0.683913\pi\)
\(744\) −6.29999e6 −0.417260
\(745\) 987966. 0.0652156
\(746\) 9.20166e6 0.605367
\(747\) −86214.5 −0.00565299
\(748\) −6.87908e6 −0.449548
\(749\) 1.97450e7 1.28604
\(750\) −2.42725e6 −0.157566
\(751\) −5.47017e6 −0.353917 −0.176958 0.984218i \(-0.556626\pi\)
−0.176958 + 0.984218i \(0.556626\pi\)
\(752\) −1.68636e7 −1.08744
\(753\) 2.72903e6 0.175397
\(754\) 2.70425e7 1.73228
\(755\) 361879. 0.0231045
\(756\) 3.04880e7 1.94010
\(757\) 2.48028e6 0.157312 0.0786559 0.996902i \(-0.474937\pi\)
0.0786559 + 0.996902i \(0.474937\pi\)
\(758\) 2.49970e7 1.58021
\(759\) 3.33694e6 0.210254
\(760\) 3.06502e6 0.192486
\(761\) 7.15291e6 0.447735 0.223868 0.974620i \(-0.428132\pi\)
0.223868 + 0.974620i \(0.428132\pi\)
\(762\) 3.94754e7 2.46285
\(763\) 1.38277e6 0.0859880
\(764\) 2.54856e7 1.57965
\(765\) −55560.4 −0.00343251
\(766\) 1.32060e7 0.813205
\(767\) 8.20918e6 0.503862
\(768\) −3.35046e7 −2.04975
\(769\) 767976. 0.0468308 0.0234154 0.999726i \(-0.492546\pi\)
0.0234154 + 0.999726i \(0.492546\pi\)
\(770\) 3.66210e6 0.222589
\(771\) −5.92207e6 −0.358788
\(772\) −2.58613e7 −1.56173
\(773\) −2.63575e7 −1.58656 −0.793280 0.608858i \(-0.791628\pi\)
−0.793280 + 0.608858i \(0.791628\pi\)
\(774\) −229114. −0.0137467
\(775\) 739824. 0.0442460
\(776\) −2.45166e7 −1.46153
\(777\) −2.74411e7 −1.63060
\(778\) 2.95334e7 1.74930
\(779\) −7.10780e6 −0.419654
\(780\) 9.57069e6 0.563257
\(781\) 5.02081e6 0.294542
\(782\) 1.49670e7 0.875224
\(783\) 2.78787e7 1.62506
\(784\) −2.35739e6 −0.136975
\(785\) −1.17540e7 −0.680786
\(786\) 3.54836e6 0.204866
\(787\) −1.18887e7 −0.684221 −0.342110 0.939660i \(-0.611142\pi\)
−0.342110 + 0.939660i \(0.611142\pi\)
\(788\) −5.55273e7 −3.18560
\(789\) −9.67547e6 −0.553324
\(790\) 1.90742e7 1.08737
\(791\) −587382. −0.0333795
\(792\) −106440. −0.00602967
\(793\) 8.16176e6 0.460894
\(794\) 1.69129e7 0.952066
\(795\) −1.85995e6 −0.104372
\(796\) 2.25890e7 1.26361
\(797\) −2.89357e7 −1.61357 −0.806787 0.590843i \(-0.798795\pi\)
−0.806787 + 0.590843i \(0.798795\pi\)
\(798\) −6.84885e6 −0.380724
\(799\) 1.16111e7 0.643438
\(800\) −928025. −0.0512666
\(801\) 103972. 0.00572580
\(802\) −3.79956e7 −2.08592
\(803\) −7.00940e6 −0.383612
\(804\) 1.50487e7 0.821030
\(805\) −5.37294e6 −0.292229
\(806\) −4.32594e6 −0.234554
\(807\) 1.64675e6 0.0890110
\(808\) 8.90697e6 0.479956
\(809\) 3.57036e7 1.91797 0.958983 0.283464i \(-0.0914837\pi\)
0.958983 + 0.283464i \(0.0914837\pi\)
\(810\) 1.47876e7 0.791928
\(811\) −2.11113e7 −1.12710 −0.563550 0.826082i \(-0.690565\pi\)
−0.563550 + 0.826082i \(0.690565\pi\)
\(812\) 5.98806e7 3.18710
\(813\) −1.30249e7 −0.691110
\(814\) −1.71971e7 −0.909693
\(815\) 5.91014e6 0.311676
\(816\) −1.67556e7 −0.880915
\(817\) −3.22131e6 −0.168841
\(818\) −6.61473e7 −3.45644
\(819\) −116625. −0.00607552
\(820\) 3.26155e7 1.69390
\(821\) 7.63918e6 0.395539 0.197769 0.980249i \(-0.436630\pi\)
0.197769 + 0.980249i \(0.436630\pi\)
\(822\) 4.11077e6 0.212199
\(823\) 2.42232e7 1.24661 0.623306 0.781978i \(-0.285789\pi\)
0.623306 + 0.781978i \(0.285789\pi\)
\(824\) 3.30920e7 1.69787
\(825\) 1.18514e6 0.0606228
\(826\) 2.69565e7 1.37472
\(827\) −8.31960e6 −0.422998 −0.211499 0.977378i \(-0.567835\pi\)
−0.211499 + 0.977378i \(0.567835\pi\)
\(828\) 302029. 0.0153099
\(829\) −3.62770e6 −0.183335 −0.0916675 0.995790i \(-0.529220\pi\)
−0.0916675 + 0.995790i \(0.529220\pi\)
\(830\) 8.24848e6 0.415603
\(831\) 1.04151e7 0.523193
\(832\) −9.27493e6 −0.464518
\(833\) 1.62314e6 0.0810481
\(834\) 1.58102e7 0.787085
\(835\) −7.39115e6 −0.366856
\(836\) −2.89433e6 −0.143230
\(837\) −4.45971e6 −0.220035
\(838\) −392558. −0.0193105
\(839\) 5.97711e6 0.293148 0.146574 0.989200i \(-0.453175\pi\)
0.146574 + 0.989200i \(0.453175\pi\)
\(840\) 1.62497e7 0.794599
\(841\) 3.42448e7 1.66957
\(842\) 3.22339e7 1.56687
\(843\) −2.90507e7 −1.40795
\(844\) −1.10884e7 −0.535810
\(845\) −5.88432e6 −0.283501
\(846\) 347465. 0.0166911
\(847\) −1.78808e6 −0.0856401
\(848\) −5.91590e6 −0.282508
\(849\) 3.55471e7 1.69252
\(850\) 5.31568e6 0.252355
\(851\) 2.52312e7 1.19430
\(852\) 4.30874e7 2.03353
\(853\) −3.57598e7 −1.68276 −0.841380 0.540445i \(-0.818256\pi\)
−0.841380 + 0.540445i \(0.818256\pi\)
\(854\) 2.68007e7 1.25748
\(855\) −23376.7 −0.00109362
\(856\) 5.49071e7 2.56120
\(857\) −2.07247e7 −0.963911 −0.481956 0.876196i \(-0.660073\pi\)
−0.481956 + 0.876196i \(0.660073\pi\)
\(858\) −6.92983e6 −0.321369
\(859\) −3.11456e7 −1.44017 −0.720084 0.693887i \(-0.755897\pi\)
−0.720084 + 0.693887i \(0.755897\pi\)
\(860\) 1.47816e7 0.681515
\(861\) −3.76833e7 −1.73237
\(862\) 3.50739e7 1.60774
\(863\) −3.12886e7 −1.43008 −0.715039 0.699085i \(-0.753591\pi\)
−0.715039 + 0.699085i \(0.753591\pi\)
\(864\) 5.59419e6 0.254949
\(865\) 4.63447e6 0.210601
\(866\) −653493. −0.0296105
\(867\) −1.07143e7 −0.484078
\(868\) −9.57898e6 −0.431539
\(869\) −9.31324e6 −0.418361
\(870\) 2.87376e7 1.28722
\(871\) 5.34293e6 0.238635
\(872\) 3.84521e6 0.171249
\(873\) 186987. 0.00830378
\(874\) 6.29730e6 0.278853
\(875\) −1.90825e6 −0.0842587
\(876\) −6.01529e7 −2.64848
\(877\) −1.96807e7 −0.864053 −0.432027 0.901861i \(-0.642201\pi\)
−0.432027 + 0.901861i \(0.642201\pi\)
\(878\) −4.73652e7 −2.07359
\(879\) 8.36503e6 0.365170
\(880\) 3.76955e6 0.164090
\(881\) 2.26413e7 0.982791 0.491395 0.870937i \(-0.336487\pi\)
0.491395 + 0.870937i \(0.336487\pi\)
\(882\) 48572.7 0.00210243
\(883\) 1.58038e7 0.682116 0.341058 0.940042i \(-0.389215\pi\)
0.341058 + 0.940042i \(0.389215\pi\)
\(884\) −2.09598e7 −0.902103
\(885\) 8.72376e6 0.374408
\(886\) 3.53477e7 1.51278
\(887\) −4.26767e7 −1.82130 −0.910652 0.413175i \(-0.864420\pi\)
−0.910652 + 0.413175i \(0.864420\pi\)
\(888\) −7.63083e7 −3.24742
\(889\) 3.10346e7 1.31702
\(890\) −9.94744e6 −0.420955
\(891\) −7.22028e6 −0.304691
\(892\) 6.35025e7 2.67226
\(893\) 4.88531e6 0.205004
\(894\) −6.13899e6 −0.256894
\(895\) −7.39483e6 −0.308582
\(896\) −3.62589e7 −1.50885
\(897\) 1.01673e7 0.421914
\(898\) −9.63259e6 −0.398614
\(899\) −8.75919e6 −0.361464
\(900\) 107268. 0.00441434
\(901\) 4.07329e6 0.167160
\(902\) −2.36158e7 −0.966466
\(903\) −1.70784e7 −0.696991
\(904\) −1.63339e6 −0.0664768
\(905\) 1.89132e7 0.767614
\(906\) −2.24863e6 −0.0910120
\(907\) 1.75475e7 0.708265 0.354133 0.935195i \(-0.384776\pi\)
0.354133 + 0.935195i \(0.384776\pi\)
\(908\) −7.55441e7 −3.04079
\(909\) −67933.0 −0.00272691
\(910\) 1.11580e7 0.446667
\(911\) −1.97498e7 −0.788438 −0.394219 0.919017i \(-0.628985\pi\)
−0.394219 + 0.919017i \(0.628985\pi\)
\(912\) −7.04981e6 −0.280666
\(913\) −4.02744e6 −0.159902
\(914\) −5.20114e6 −0.205936
\(915\) 8.67336e6 0.342480
\(916\) −2.02649e7 −0.798007
\(917\) 2.78963e6 0.109553
\(918\) −3.20433e7 −1.25496
\(919\) −1.06858e7 −0.417366 −0.208683 0.977983i \(-0.566918\pi\)
−0.208683 + 0.977983i \(0.566918\pi\)
\(920\) −1.49411e7 −0.581987
\(921\) 1.33674e7 0.519275
\(922\) 2.76509e7 1.07123
\(923\) 1.52979e7 0.591053
\(924\) −1.53448e7 −0.591264
\(925\) 8.96108e6 0.344355
\(926\) 6.70218e7 2.56855
\(927\) −252390. −0.00964659
\(928\) 1.09874e7 0.418818
\(929\) 2.27699e7 0.865607 0.432804 0.901488i \(-0.357524\pi\)
0.432804 + 0.901488i \(0.357524\pi\)
\(930\) −4.59710e6 −0.174292
\(931\) 682926. 0.0258226
\(932\) 5.75131e6 0.216884
\(933\) −2.03639e7 −0.765873
\(934\) 6.73648e7 2.52677
\(935\) −2.59546e6 −0.0970925
\(936\) −324312. −0.0120997
\(937\) −2.05541e7 −0.764803 −0.382402 0.923996i \(-0.624903\pi\)
−0.382402 + 0.923996i \(0.624903\pi\)
\(938\) 1.75446e7 0.651082
\(939\) −3.35261e7 −1.24085
\(940\) −2.24172e7 −0.827487
\(941\) 1.76886e7 0.651206 0.325603 0.945507i \(-0.394433\pi\)
0.325603 + 0.945507i \(0.394433\pi\)
\(942\) 7.30365e7 2.68172
\(943\) 3.46485e7 1.26884
\(944\) 2.77474e7 1.01343
\(945\) 1.15030e7 0.419019
\(946\) −1.07029e7 −0.388842
\(947\) 4.53179e7 1.64208 0.821040 0.570871i \(-0.193394\pi\)
0.821040 + 0.570871i \(0.193394\pi\)
\(948\) −7.99239e7 −2.88839
\(949\) −2.13569e7 −0.769789
\(950\) 2.23654e6 0.0804022
\(951\) −4.65489e7 −1.66901
\(952\) −3.55869e7 −1.27262
\(953\) −5.04177e7 −1.79825 −0.899126 0.437689i \(-0.855797\pi\)
−0.899126 + 0.437689i \(0.855797\pi\)
\(954\) 121894. 0.00433622
\(955\) 9.61566e6 0.341170
\(956\) 2.07067e7 0.732769
\(957\) −1.40316e7 −0.495252
\(958\) 7.76581e7 2.73384
\(959\) 3.23179e6 0.113474
\(960\) −9.85631e6 −0.345172
\(961\) −2.72280e7 −0.951057
\(962\) −5.23977e7 −1.82547
\(963\) −418773. −0.0145517
\(964\) −2.87492e7 −0.996399
\(965\) −9.75740e6 −0.337300
\(966\) 3.33863e7 1.15113
\(967\) −2.61680e7 −0.899921 −0.449960 0.893049i \(-0.648562\pi\)
−0.449960 + 0.893049i \(0.648562\pi\)
\(968\) −4.97229e6 −0.170556
\(969\) 4.85402e6 0.166070
\(970\) −1.78898e7 −0.610486
\(971\) −2.18018e7 −0.742068 −0.371034 0.928619i \(-0.620997\pi\)
−0.371034 + 0.928619i \(0.620997\pi\)
\(972\) −1.30021e6 −0.0441415
\(973\) 1.24296e7 0.420896
\(974\) 1.88481e7 0.636604
\(975\) 3.61100e6 0.121651
\(976\) 2.75871e7 0.927005
\(977\) −2.98192e7 −0.999445 −0.499723 0.866186i \(-0.666565\pi\)
−0.499723 + 0.866186i \(0.666565\pi\)
\(978\) −3.67243e7 −1.22774
\(979\) 4.85698e6 0.161961
\(980\) −3.13373e6 −0.104231
\(981\) −29327.2 −0.000972966 0
\(982\) −4.84401e7 −1.60297
\(983\) 3.50223e6 0.115601 0.0578004 0.998328i \(-0.481591\pi\)
0.0578004 + 0.998328i \(0.481591\pi\)
\(984\) −1.04790e8 −3.45010
\(985\) −2.09503e7 −0.688019
\(986\) −6.29353e7 −2.06159
\(987\) 2.59003e7 0.846277
\(988\) −8.81871e6 −0.287417
\(989\) 1.57030e7 0.510496
\(990\) −77669.6 −0.00251862
\(991\) 3.23436e7 1.04617 0.523087 0.852279i \(-0.324780\pi\)
0.523087 + 0.852279i \(0.324780\pi\)
\(992\) −1.75763e6 −0.0567086
\(993\) 3.01655e7 0.970816
\(994\) 5.02335e7 1.61260
\(995\) 8.52279e6 0.272913
\(996\) −3.45625e7 −1.10397
\(997\) 5.98804e7 1.90786 0.953930 0.300029i \(-0.0969963\pi\)
0.953930 + 0.300029i \(0.0969963\pi\)
\(998\) 8.89573e7 2.82719
\(999\) −5.40180e7 −1.71248
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.e.1.4 38
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.6.a.e.1.4 38 1.1 even 1 trivial