Properties

Label 1045.6.a.a.1.8
Level $1045$
Weight $6$
Character 1045.1
Self dual yes
Analytic conductor $167.601$
Analytic rank $1$
Dimension $35$
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: \(35\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
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)
 
\(f(q)\) \(=\) \(q-7.13459 q^{2} -23.4497 q^{3} +18.9024 q^{4} -25.0000 q^{5} +167.304 q^{6} -10.1832 q^{7} +93.4458 q^{8} +306.890 q^{9} +O(q^{10})\) \(q-7.13459 q^{2} -23.4497 q^{3} +18.9024 q^{4} -25.0000 q^{5} +167.304 q^{6} -10.1832 q^{7} +93.4458 q^{8} +306.890 q^{9} +178.365 q^{10} +121.000 q^{11} -443.257 q^{12} -724.498 q^{13} +72.6532 q^{14} +586.243 q^{15} -1271.58 q^{16} +1368.41 q^{17} -2189.53 q^{18} +361.000 q^{19} -472.561 q^{20} +238.794 q^{21} -863.286 q^{22} +2972.81 q^{23} -2191.28 q^{24} +625.000 q^{25} +5169.00 q^{26} -1498.20 q^{27} -192.488 q^{28} -1773.14 q^{29} -4182.61 q^{30} -6189.60 q^{31} +6081.91 q^{32} -2837.42 q^{33} -9763.06 q^{34} +254.581 q^{35} +5800.97 q^{36} -9314.71 q^{37} -2575.59 q^{38} +16989.3 q^{39} -2336.14 q^{40} +12420.1 q^{41} -1703.70 q^{42} -22399.2 q^{43} +2287.20 q^{44} -7672.25 q^{45} -21209.8 q^{46} -14672.3 q^{47} +29818.1 q^{48} -16703.3 q^{49} -4459.12 q^{50} -32088.9 q^{51} -13694.8 q^{52} +16934.8 q^{53} +10689.0 q^{54} -3025.00 q^{55} -951.579 q^{56} -8465.35 q^{57} +12650.6 q^{58} -14147.6 q^{59} +11081.4 q^{60} -49335.1 q^{61} +44160.3 q^{62} -3125.13 q^{63} -2701.56 q^{64} +18112.5 q^{65} +20243.8 q^{66} +49933.3 q^{67} +25866.3 q^{68} -69711.6 q^{69} -1816.33 q^{70} +3606.70 q^{71} +28677.6 q^{72} -37981.1 q^{73} +66456.7 q^{74} -14656.1 q^{75} +6823.78 q^{76} -1232.17 q^{77} -121212. q^{78} +51992.1 q^{79} +31789.4 q^{80} -39441.9 q^{81} -88612.3 q^{82} +110613. q^{83} +4513.79 q^{84} -34210.3 q^{85} +159809. q^{86} +41579.6 q^{87} +11306.9 q^{88} +72518.5 q^{89} +54738.4 q^{90} +7377.73 q^{91} +56193.4 q^{92} +145144. q^{93} +104681. q^{94} -9025.00 q^{95} -142619. q^{96} +29912.4 q^{97} +119171. q^{98} +37133.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 35 q - 4 q^{2} - 27 q^{3} + 520 q^{4} - 875 q^{5} - 291 q^{6} + 117 q^{7} - 498 q^{8} + 2046 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 35 q - 4 q^{2} - 27 q^{3} + 520 q^{4} - 875 q^{5} - 291 q^{6} + 117 q^{7} - 498 q^{8} + 2046 q^{9} + 100 q^{10} + 4235 q^{11} - 568 q^{12} - 717 q^{13} - 2585 q^{14} + 675 q^{15} + 3356 q^{16} - 3349 q^{17} - 5533 q^{18} + 12635 q^{19} - 13000 q^{20} + 289 q^{21} - 484 q^{22} - 820 q^{23} - 21748 q^{24} + 21875 q^{25} - 6267 q^{26} - 13650 q^{27} - 6487 q^{28} - 13357 q^{29} + 7275 q^{30} - 15341 q^{31} - 16405 q^{32} - 3267 q^{33} - 1255 q^{34} - 2925 q^{35} + 23487 q^{36} - 511 q^{37} - 1444 q^{38} - 33584 q^{39} + 12450 q^{40} - 36855 q^{41} + 16330 q^{42} + 10991 q^{43} + 62920 q^{44} - 51150 q^{45} - 20443 q^{46} - 33594 q^{47} + 36221 q^{48} + 23422 q^{49} - 2500 q^{50} - 53530 q^{51} + 89382 q^{52} + 13103 q^{53} + 65776 q^{54} - 105875 q^{55} + 130911 q^{56} - 9747 q^{57} + 127808 q^{58} - 161139 q^{59} + 14200 q^{60} - 91587 q^{61} + 131818 q^{62} + 16590 q^{63} - 23186 q^{64} + 17925 q^{65} - 35211 q^{66} + 39210 q^{67} + 26300 q^{68} - 23174 q^{69} + 64625 q^{70} - 167772 q^{71} + 135820 q^{72} - 5106 q^{73} - 256965 q^{74} - 16875 q^{75} + 187720 q^{76} + 14157 q^{77} + 492812 q^{78} - 156897 q^{79} - 83900 q^{80} + 31279 q^{81} + 46818 q^{82} - 185627 q^{83} + 165864 q^{84} + 83725 q^{85} - 159946 q^{86} - 112092 q^{87} - 60258 q^{88} - 144420 q^{89} + 138325 q^{90} - 442480 q^{91} - 205876 q^{92} + 125910 q^{93} - 110044 q^{94} - 315875 q^{95} - 554286 q^{96} + 41200 q^{97} + 41052 q^{98} + 247566 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.13459 −1.26123 −0.630615 0.776096i \(-0.717197\pi\)
−0.630615 + 0.776096i \(0.717197\pi\)
\(3\) −23.4497 −1.50430 −0.752150 0.658991i \(-0.770983\pi\)
−0.752150 + 0.658991i \(0.770983\pi\)
\(4\) 18.9024 0.590701
\(5\) −25.0000 −0.447214
\(6\) 167.304 1.89727
\(7\) −10.1832 −0.0785489 −0.0392745 0.999228i \(-0.512505\pi\)
−0.0392745 + 0.999228i \(0.512505\pi\)
\(8\) 93.4458 0.516220
\(9\) 306.890 1.26292
\(10\) 178.365 0.564039
\(11\) 121.000 0.301511
\(12\) −443.257 −0.888592
\(13\) −724.498 −1.18899 −0.594496 0.804099i \(-0.702648\pi\)
−0.594496 + 0.804099i \(0.702648\pi\)
\(14\) 72.6532 0.0990682
\(15\) 586.243 0.672744
\(16\) −1271.58 −1.24177
\(17\) 1368.41 1.14840 0.574201 0.818714i \(-0.305313\pi\)
0.574201 + 0.818714i \(0.305313\pi\)
\(18\) −2189.53 −1.59283
\(19\) 361.000 0.229416
\(20\) −472.561 −0.264170
\(21\) 238.794 0.118161
\(22\) −863.286 −0.380275
\(23\) 2972.81 1.17178 0.585892 0.810389i \(-0.300744\pi\)
0.585892 + 0.810389i \(0.300744\pi\)
\(24\) −2191.28 −0.776550
\(25\) 625.000 0.200000
\(26\) 5169.00 1.49959
\(27\) −1498.20 −0.395512
\(28\) −192.488 −0.0463989
\(29\) −1773.14 −0.391514 −0.195757 0.980652i \(-0.562716\pi\)
−0.195757 + 0.980652i \(0.562716\pi\)
\(30\) −4182.61 −0.848485
\(31\) −6189.60 −1.15680 −0.578400 0.815753i \(-0.696323\pi\)
−0.578400 + 0.815753i \(0.696323\pi\)
\(32\) 6081.91 1.04994
\(33\) −2837.42 −0.453564
\(34\) −9763.06 −1.44840
\(35\) 254.581 0.0351281
\(36\) 5800.97 0.746009
\(37\) −9314.71 −1.11858 −0.559288 0.828974i \(-0.688925\pi\)
−0.559288 + 0.828974i \(0.688925\pi\)
\(38\) −2575.59 −0.289346
\(39\) 16989.3 1.78860
\(40\) −2336.14 −0.230861
\(41\) 12420.1 1.15389 0.576946 0.816782i \(-0.304244\pi\)
0.576946 + 0.816782i \(0.304244\pi\)
\(42\) −1703.70 −0.149028
\(43\) −22399.2 −1.84740 −0.923702 0.383113i \(-0.874852\pi\)
−0.923702 + 0.383113i \(0.874852\pi\)
\(44\) 2287.20 0.178103
\(45\) −7672.25 −0.564795
\(46\) −21209.8 −1.47789
\(47\) −14672.3 −0.968842 −0.484421 0.874835i \(-0.660970\pi\)
−0.484421 + 0.874835i \(0.660970\pi\)
\(48\) 29818.1 1.86800
\(49\) −16703.3 −0.993830
\(50\) −4459.12 −0.252246
\(51\) −32088.9 −1.72754
\(52\) −13694.8 −0.702339
\(53\) 16934.8 0.828116 0.414058 0.910250i \(-0.364111\pi\)
0.414058 + 0.910250i \(0.364111\pi\)
\(54\) 10689.0 0.498832
\(55\) −3025.00 −0.134840
\(56\) −951.579 −0.0405485
\(57\) −8465.35 −0.345110
\(58\) 12650.6 0.493789
\(59\) −14147.6 −0.529119 −0.264560 0.964369i \(-0.585227\pi\)
−0.264560 + 0.964369i \(0.585227\pi\)
\(60\) 11081.4 0.397391
\(61\) −49335.1 −1.69759 −0.848793 0.528726i \(-0.822670\pi\)
−0.848793 + 0.528726i \(0.822670\pi\)
\(62\) 44160.3 1.45899
\(63\) −3125.13 −0.0992011
\(64\) −2701.56 −0.0824450
\(65\) 18112.5 0.531733
\(66\) 20243.8 0.572048
\(67\) 49933.3 1.35895 0.679475 0.733699i \(-0.262208\pi\)
0.679475 + 0.733699i \(0.262208\pi\)
\(68\) 25866.3 0.678363
\(69\) −69711.6 −1.76272
\(70\) −1816.33 −0.0443047
\(71\) 3606.70 0.0849111 0.0424556 0.999098i \(-0.486482\pi\)
0.0424556 + 0.999098i \(0.486482\pi\)
\(72\) 28677.6 0.651945
\(73\) −37981.1 −0.834181 −0.417090 0.908865i \(-0.636950\pi\)
−0.417090 + 0.908865i \(0.636950\pi\)
\(74\) 66456.7 1.41078
\(75\) −14656.1 −0.300860
\(76\) 6823.78 0.135516
\(77\) −1232.17 −0.0236834
\(78\) −121212. −2.25584
\(79\) 51992.1 0.937281 0.468641 0.883389i \(-0.344744\pi\)
0.468641 + 0.883389i \(0.344744\pi\)
\(80\) 31789.4 0.555338
\(81\) −39441.9 −0.667951
\(82\) −88612.3 −1.45532
\(83\) 110613. 1.76242 0.881209 0.472726i \(-0.156730\pi\)
0.881209 + 0.472726i \(0.156730\pi\)
\(84\) 4513.79 0.0697980
\(85\) −34210.3 −0.513581
\(86\) 159809. 2.33000
\(87\) 41579.6 0.588955
\(88\) 11306.9 0.155646
\(89\) 72518.5 0.970451 0.485226 0.874389i \(-0.338737\pi\)
0.485226 + 0.874389i \(0.338737\pi\)
\(90\) 54738.4 0.712337
\(91\) 7377.73 0.0933940
\(92\) 56193.4 0.692174
\(93\) 145144. 1.74018
\(94\) 104681. 1.22193
\(95\) −9025.00 −0.102598
\(96\) −142619. −1.57943
\(97\) 29912.4 0.322791 0.161395 0.986890i \(-0.448401\pi\)
0.161395 + 0.986890i \(0.448401\pi\)
\(98\) 119171. 1.25345
\(99\) 37133.7 0.380785
\(100\) 11814.0 0.118140
\(101\) −8365.38 −0.0815985 −0.0407992 0.999167i \(-0.512990\pi\)
−0.0407992 + 0.999167i \(0.512990\pi\)
\(102\) 228941. 2.17883
\(103\) 136335. 1.26623 0.633117 0.774056i \(-0.281775\pi\)
0.633117 + 0.774056i \(0.281775\pi\)
\(104\) −67701.3 −0.613781
\(105\) −5969.85 −0.0528433
\(106\) −120823. −1.04444
\(107\) 112954. 0.953767 0.476883 0.878967i \(-0.341766\pi\)
0.476883 + 0.878967i \(0.341766\pi\)
\(108\) −28319.6 −0.233630
\(109\) 135933. 1.09587 0.547936 0.836520i \(-0.315414\pi\)
0.547936 + 0.836520i \(0.315414\pi\)
\(110\) 21582.1 0.170064
\(111\) 218427. 1.68267
\(112\) 12948.7 0.0975399
\(113\) −186974. −1.37748 −0.688739 0.725009i \(-0.741835\pi\)
−0.688739 + 0.725009i \(0.741835\pi\)
\(114\) 60396.9 0.435263
\(115\) −74320.3 −0.524038
\(116\) −33516.6 −0.231268
\(117\) −222341. −1.50160
\(118\) 100938. 0.667341
\(119\) −13934.8 −0.0902058
\(120\) 54782.0 0.347284
\(121\) 14641.0 0.0909091
\(122\) 351986. 2.14105
\(123\) −291248. −1.73580
\(124\) −116999. −0.683323
\(125\) −15625.0 −0.0894427
\(126\) 22296.5 0.125115
\(127\) −28581.6 −0.157245 −0.0786224 0.996904i \(-0.525052\pi\)
−0.0786224 + 0.996904i \(0.525052\pi\)
\(128\) −175347. −0.945960
\(129\) 525256. 2.77905
\(130\) −129225. −0.670638
\(131\) 45167.0 0.229955 0.114978 0.993368i \(-0.463320\pi\)
0.114978 + 0.993368i \(0.463320\pi\)
\(132\) −53634.1 −0.267921
\(133\) −3676.14 −0.0180204
\(134\) −356254. −1.71395
\(135\) 37455.0 0.176879
\(136\) 127872. 0.592828
\(137\) 290883. 1.32409 0.662045 0.749465i \(-0.269689\pi\)
0.662045 + 0.749465i \(0.269689\pi\)
\(138\) 497364. 2.22319
\(139\) −356868. −1.56665 −0.783323 0.621615i \(-0.786477\pi\)
−0.783323 + 0.621615i \(0.786477\pi\)
\(140\) 4812.19 0.0207502
\(141\) 344061. 1.45743
\(142\) −25732.4 −0.107092
\(143\) −87664.3 −0.358495
\(144\) −390234. −1.56826
\(145\) 44328.4 0.175090
\(146\) 270980. 1.05209
\(147\) 391688. 1.49502
\(148\) −176071. −0.660744
\(149\) 8184.78 0.0302024 0.0151012 0.999886i \(-0.495193\pi\)
0.0151012 + 0.999886i \(0.495193\pi\)
\(150\) 104565. 0.379454
\(151\) −412405. −1.47191 −0.735956 0.677029i \(-0.763267\pi\)
−0.735956 + 0.677029i \(0.763267\pi\)
\(152\) 33733.9 0.118429
\(153\) 419951. 1.45034
\(154\) 8791.03 0.0298702
\(155\) 154740. 0.517337
\(156\) 321139. 1.05653
\(157\) 243063. 0.786991 0.393496 0.919326i \(-0.371266\pi\)
0.393496 + 0.919326i \(0.371266\pi\)
\(158\) −370943. −1.18213
\(159\) −397117. −1.24574
\(160\) −152048. −0.469548
\(161\) −30272.8 −0.0920424
\(162\) 281402. 0.842440
\(163\) 333447. 0.983010 0.491505 0.870875i \(-0.336447\pi\)
0.491505 + 0.870875i \(0.336447\pi\)
\(164\) 234770. 0.681605
\(165\) 70935.4 0.202840
\(166\) −789175. −2.22282
\(167\) 100210. 0.278049 0.139024 0.990289i \(-0.455603\pi\)
0.139024 + 0.990289i \(0.455603\pi\)
\(168\) 22314.3 0.0609972
\(169\) 153605. 0.413702
\(170\) 244076. 0.647744
\(171\) 110787. 0.289734
\(172\) −423400. −1.09126
\(173\) 712310. 1.80948 0.904740 0.425964i \(-0.140065\pi\)
0.904740 + 0.425964i \(0.140065\pi\)
\(174\) −296654. −0.742808
\(175\) −6364.51 −0.0157098
\(176\) −153861. −0.374409
\(177\) 331758. 0.795955
\(178\) −517390. −1.22396
\(179\) −785407. −1.83215 −0.916077 0.401002i \(-0.868662\pi\)
−0.916077 + 0.401002i \(0.868662\pi\)
\(180\) −145024. −0.333625
\(181\) −181159. −0.411021 −0.205510 0.978655i \(-0.565885\pi\)
−0.205510 + 0.978655i \(0.565885\pi\)
\(182\) −52637.1 −0.117791
\(183\) 1.15690e6 2.55368
\(184\) 277797. 0.604898
\(185\) 232868. 0.500242
\(186\) −1.03555e6 −2.19476
\(187\) 165578. 0.346256
\(188\) −277342. −0.572296
\(189\) 15256.5 0.0310671
\(190\) 64389.7 0.129399
\(191\) 316299. 0.627356 0.313678 0.949529i \(-0.398439\pi\)
0.313678 + 0.949529i \(0.398439\pi\)
\(192\) 63350.8 0.124022
\(193\) 608517. 1.17592 0.587962 0.808888i \(-0.299930\pi\)
0.587962 + 0.808888i \(0.299930\pi\)
\(194\) −213413. −0.407114
\(195\) −424732. −0.799887
\(196\) −315733. −0.587057
\(197\) −461024. −0.846366 −0.423183 0.906044i \(-0.639087\pi\)
−0.423183 + 0.906044i \(0.639087\pi\)
\(198\) −264934. −0.480258
\(199\) −101657. −0.181972 −0.0909862 0.995852i \(-0.529002\pi\)
−0.0909862 + 0.995852i \(0.529002\pi\)
\(200\) 58403.6 0.103244
\(201\) −1.17092e6 −2.04427
\(202\) 59683.6 0.102914
\(203\) 18056.3 0.0307530
\(204\) −606558. −1.02046
\(205\) −310502. −0.516036
\(206\) −972694. −1.59701
\(207\) 912325. 1.47987
\(208\) 921255. 1.47646
\(209\) 43681.0 0.0691714
\(210\) 42592.4 0.0666475
\(211\) 104595. 0.161735 0.0808675 0.996725i \(-0.474231\pi\)
0.0808675 + 0.996725i \(0.474231\pi\)
\(212\) 320110. 0.489169
\(213\) −84576.2 −0.127732
\(214\) −805881. −1.20292
\(215\) 559980. 0.826184
\(216\) −140000. −0.204171
\(217\) 63030.1 0.0908654
\(218\) −969829. −1.38215
\(219\) 890646. 1.25486
\(220\) −57179.9 −0.0796501
\(221\) −991411. −1.36544
\(222\) −1.55839e6 −2.12224
\(223\) 408179. 0.549653 0.274827 0.961494i \(-0.411380\pi\)
0.274827 + 0.961494i \(0.411380\pi\)
\(224\) −61933.5 −0.0824718
\(225\) 191806. 0.252584
\(226\) 1.33398e6 1.73732
\(227\) 196271. 0.252808 0.126404 0.991979i \(-0.459656\pi\)
0.126404 + 0.991979i \(0.459656\pi\)
\(228\) −160016. −0.203857
\(229\) 570260. 0.718595 0.359297 0.933223i \(-0.383016\pi\)
0.359297 + 0.933223i \(0.383016\pi\)
\(230\) 530245. 0.660932
\(231\) 28894.1 0.0356269
\(232\) −165692. −0.202107
\(233\) 785003. 0.947287 0.473643 0.880717i \(-0.342939\pi\)
0.473643 + 0.880717i \(0.342939\pi\)
\(234\) 1.58631e6 1.89387
\(235\) 366807. 0.433279
\(236\) −267425. −0.312551
\(237\) −1.21920e6 −1.40995
\(238\) 99419.4 0.113770
\(239\) 802419. 0.908671 0.454336 0.890831i \(-0.349877\pi\)
0.454336 + 0.890831i \(0.349877\pi\)
\(240\) −745453. −0.835395
\(241\) −25488.4 −0.0282683 −0.0141341 0.999900i \(-0.504499\pi\)
−0.0141341 + 0.999900i \(0.504499\pi\)
\(242\) −104458. −0.114657
\(243\) 1.28896e6 1.40031
\(244\) −932554. −1.00277
\(245\) 417583. 0.444454
\(246\) 2.07793e6 2.18924
\(247\) −261544. −0.272774
\(248\) −578392. −0.597163
\(249\) −2.59383e6 −2.65121
\(250\) 111478. 0.112808
\(251\) −678835. −0.680111 −0.340055 0.940405i \(-0.610446\pi\)
−0.340055 + 0.940405i \(0.610446\pi\)
\(252\) −59072.5 −0.0585982
\(253\) 359710. 0.353306
\(254\) 203918. 0.198322
\(255\) 802222. 0.772581
\(256\) 1.33748e6 1.27552
\(257\) −929690. −0.878022 −0.439011 0.898482i \(-0.644671\pi\)
−0.439011 + 0.898482i \(0.644671\pi\)
\(258\) −3.74749e6 −3.50502
\(259\) 94853.8 0.0878629
\(260\) 342370. 0.314096
\(261\) −544158. −0.494451
\(262\) −322248. −0.290026
\(263\) 413965. 0.369041 0.184520 0.982829i \(-0.440927\pi\)
0.184520 + 0.982829i \(0.440927\pi\)
\(264\) −265145. −0.234139
\(265\) −423371. −0.370345
\(266\) 26227.8 0.0227278
\(267\) −1.70054e6 −1.45985
\(268\) 943861. 0.802733
\(269\) 994735. 0.838159 0.419080 0.907949i \(-0.362353\pi\)
0.419080 + 0.907949i \(0.362353\pi\)
\(270\) −267226. −0.223085
\(271\) −512272. −0.423719 −0.211859 0.977300i \(-0.567952\pi\)
−0.211859 + 0.977300i \(0.567952\pi\)
\(272\) −1.74004e6 −1.42606
\(273\) −173006. −0.140493
\(274\) −2.07533e6 −1.66998
\(275\) 75625.0 0.0603023
\(276\) −1.31772e6 −1.04124
\(277\) −708677. −0.554944 −0.277472 0.960734i \(-0.589496\pi\)
−0.277472 + 0.960734i \(0.589496\pi\)
\(278\) 2.54611e6 1.97590
\(279\) −1.89952e6 −1.46095
\(280\) 23789.5 0.0181338
\(281\) 1.28836e6 0.973356 0.486678 0.873582i \(-0.338209\pi\)
0.486678 + 0.873582i \(0.338209\pi\)
\(282\) −2.45474e6 −1.83815
\(283\) −1.35284e6 −1.00411 −0.502055 0.864836i \(-0.667423\pi\)
−0.502055 + 0.864836i \(0.667423\pi\)
\(284\) 68175.5 0.0501571
\(285\) 211634. 0.154338
\(286\) 625449. 0.452144
\(287\) −126477. −0.0906369
\(288\) 1.86648e6 1.32599
\(289\) 452692. 0.318829
\(290\) −316265. −0.220829
\(291\) −701437. −0.485575
\(292\) −717935. −0.492751
\(293\) −267652. −0.182139 −0.0910693 0.995845i \(-0.529028\pi\)
−0.0910693 + 0.995845i \(0.529028\pi\)
\(294\) −2.79453e6 −1.88556
\(295\) 353691. 0.236629
\(296\) −870421. −0.577431
\(297\) −181282. −0.119251
\(298\) −58395.1 −0.0380922
\(299\) −2.15380e6 −1.39324
\(300\) −277036. −0.177718
\(301\) 228096. 0.145111
\(302\) 2.94234e6 1.85642
\(303\) 196166. 0.122749
\(304\) −459039. −0.284882
\(305\) 1.23338e6 0.759183
\(306\) −2.99618e6 −1.82922
\(307\) 590360. 0.357496 0.178748 0.983895i \(-0.442795\pi\)
0.178748 + 0.983895i \(0.442795\pi\)
\(308\) −23291.0 −0.0139898
\(309\) −3.19702e6 −1.90480
\(310\) −1.10401e6 −0.652481
\(311\) −1.15358e6 −0.676309 −0.338155 0.941091i \(-0.609803\pi\)
−0.338155 + 0.941091i \(0.609803\pi\)
\(312\) 1.58758e6 0.923312
\(313\) −869231. −0.501504 −0.250752 0.968051i \(-0.580678\pi\)
−0.250752 + 0.968051i \(0.580678\pi\)
\(314\) −1.73416e6 −0.992577
\(315\) 78128.2 0.0443641
\(316\) 982778. 0.553653
\(317\) 1.38504e6 0.774128 0.387064 0.922053i \(-0.373489\pi\)
0.387064 + 0.922053i \(0.373489\pi\)
\(318\) 2.83327e6 1.57116
\(319\) −214550. −0.118046
\(320\) 67538.9 0.0368705
\(321\) −2.64874e6 −1.43475
\(322\) 215984. 0.116087
\(323\) 493996. 0.263462
\(324\) −745547. −0.394560
\(325\) −452811. −0.237798
\(326\) −2.37901e6 −1.23980
\(327\) −3.18760e6 −1.64852
\(328\) 1.16061e6 0.595662
\(329\) 149411. 0.0761015
\(330\) −506096. −0.255828
\(331\) −2.31804e6 −1.16292 −0.581462 0.813573i \(-0.697519\pi\)
−0.581462 + 0.813573i \(0.697519\pi\)
\(332\) 2.09085e6 1.04106
\(333\) −2.85859e6 −1.41267
\(334\) −714959. −0.350683
\(335\) −1.24833e6 −0.607741
\(336\) −303644. −0.146729
\(337\) −388169. −0.186186 −0.0930928 0.995657i \(-0.529675\pi\)
−0.0930928 + 0.995657i \(0.529675\pi\)
\(338\) −1.09591e6 −0.521774
\(339\) 4.38448e6 2.07214
\(340\) −646658. −0.303373
\(341\) −748941. −0.348788
\(342\) −790422. −0.365421
\(343\) 341243. 0.156613
\(344\) −2.09311e6 −0.953666
\(345\) 1.74279e6 0.788311
\(346\) −5.08204e6 −2.28217
\(347\) 1.84083e6 0.820710 0.410355 0.911926i \(-0.365405\pi\)
0.410355 + 0.911926i \(0.365405\pi\)
\(348\) 785956. 0.347896
\(349\) 426759. 0.187551 0.0937755 0.995593i \(-0.470106\pi\)
0.0937755 + 0.995593i \(0.470106\pi\)
\(350\) 45408.2 0.0198136
\(351\) 1.08544e6 0.470261
\(352\) 735911. 0.316569
\(353\) −3.30384e6 −1.41118 −0.705590 0.708621i \(-0.749318\pi\)
−0.705590 + 0.708621i \(0.749318\pi\)
\(354\) −2.36696e6 −1.00388
\(355\) −90167.6 −0.0379734
\(356\) 1.37078e6 0.573247
\(357\) 326768. 0.135697
\(358\) 5.60356e6 2.31077
\(359\) 3.47865e6 1.42454 0.712271 0.701905i \(-0.247667\pi\)
0.712271 + 0.701905i \(0.247667\pi\)
\(360\) −716939. −0.291559
\(361\) 130321. 0.0526316
\(362\) 1.29250e6 0.518392
\(363\) −343327. −0.136755
\(364\) 139457. 0.0551680
\(365\) 949527. 0.373057
\(366\) −8.25398e6 −3.22078
\(367\) −1.60279e6 −0.621170 −0.310585 0.950546i \(-0.600525\pi\)
−0.310585 + 0.950546i \(0.600525\pi\)
\(368\) −3.78015e6 −1.45509
\(369\) 3.81160e6 1.45727
\(370\) −1.66142e6 −0.630920
\(371\) −172451. −0.0650476
\(372\) 2.74358e6 1.02792
\(373\) −3.65780e6 −1.36128 −0.680640 0.732618i \(-0.738298\pi\)
−0.680640 + 0.732618i \(0.738298\pi\)
\(374\) −1.18133e6 −0.436709
\(375\) 366402. 0.134549
\(376\) −1.37106e6 −0.500136
\(377\) 1.28463e6 0.465507
\(378\) −108849. −0.0391827
\(379\) 4.71626e6 1.68655 0.843276 0.537481i \(-0.180624\pi\)
0.843276 + 0.537481i \(0.180624\pi\)
\(380\) −170595. −0.0606047
\(381\) 670230. 0.236544
\(382\) −2.25666e6 −0.791241
\(383\) 1.46343e6 0.509770 0.254885 0.966971i \(-0.417962\pi\)
0.254885 + 0.966971i \(0.417962\pi\)
\(384\) 4.11183e6 1.42301
\(385\) 30804.3 0.0105915
\(386\) −4.34152e6 −1.48311
\(387\) −6.87409e6 −2.33312
\(388\) 565417. 0.190673
\(389\) −5.25615e6 −1.76114 −0.880569 0.473918i \(-0.842839\pi\)
−0.880569 + 0.473918i \(0.842839\pi\)
\(390\) 3.03029e6 1.00884
\(391\) 4.06803e6 1.34568
\(392\) −1.56085e6 −0.513035
\(393\) −1.05915e6 −0.345922
\(394\) 3.28922e6 1.06746
\(395\) −1.29980e6 −0.419165
\(396\) 701917. 0.224930
\(397\) −1.54470e6 −0.491889 −0.245944 0.969284i \(-0.579098\pi\)
−0.245944 + 0.969284i \(0.579098\pi\)
\(398\) 725283. 0.229509
\(399\) 86204.6 0.0271080
\(400\) −794735. −0.248355
\(401\) 5.97875e6 1.85673 0.928366 0.371666i \(-0.121213\pi\)
0.928366 + 0.371666i \(0.121213\pi\)
\(402\) 8.35406e6 2.57829
\(403\) 4.48435e6 1.37543
\(404\) −158126. −0.0482003
\(405\) 986047. 0.298717
\(406\) −128824. −0.0387866
\(407\) −1.12708e6 −0.337263
\(408\) −2.99857e6 −0.891792
\(409\) 4.14833e6 1.22621 0.613105 0.790001i \(-0.289920\pi\)
0.613105 + 0.790001i \(0.289920\pi\)
\(410\) 2.21531e6 0.650840
\(411\) −6.82113e6 −1.99183
\(412\) 2.57706e6 0.747966
\(413\) 144068. 0.0415617
\(414\) −6.50907e6 −1.86646
\(415\) −2.76531e6 −0.788178
\(416\) −4.40634e6 −1.24837
\(417\) 8.36846e6 2.35671
\(418\) −311646. −0.0872411
\(419\) −3.20310e6 −0.891324 −0.445662 0.895201i \(-0.647032\pi\)
−0.445662 + 0.895201i \(0.647032\pi\)
\(420\) −112845. −0.0312146
\(421\) −1.09464e6 −0.301000 −0.150500 0.988610i \(-0.548088\pi\)
−0.150500 + 0.988610i \(0.548088\pi\)
\(422\) −746242. −0.203985
\(423\) −4.50277e6 −1.22357
\(424\) 1.58249e6 0.427490
\(425\) 855257. 0.229681
\(426\) 603417. 0.161099
\(427\) 502391. 0.133343
\(428\) 2.13511e6 0.563391
\(429\) 2.05570e6 0.539284
\(430\) −3.99523e6 −1.04201
\(431\) 4.49794e6 1.16633 0.583163 0.812355i \(-0.301815\pi\)
0.583163 + 0.812355i \(0.301815\pi\)
\(432\) 1.90507e6 0.491137
\(433\) 3.87599e6 0.993488 0.496744 0.867897i \(-0.334529\pi\)
0.496744 + 0.867897i \(0.334529\pi\)
\(434\) −449694. −0.114602
\(435\) −1.03949e6 −0.263389
\(436\) 2.56947e6 0.647333
\(437\) 1.07318e6 0.268826
\(438\) −6.35440e6 −1.58267
\(439\) −386368. −0.0956842 −0.0478421 0.998855i \(-0.515234\pi\)
−0.0478421 + 0.998855i \(0.515234\pi\)
\(440\) −282673. −0.0696071
\(441\) −5.12607e6 −1.25513
\(442\) 7.07332e6 1.72214
\(443\) −2.92841e6 −0.708961 −0.354481 0.935063i \(-0.615342\pi\)
−0.354481 + 0.935063i \(0.615342\pi\)
\(444\) 4.12881e6 0.993957
\(445\) −1.81296e6 −0.433999
\(446\) −2.91219e6 −0.693239
\(447\) −191931. −0.0454335
\(448\) 27510.6 0.00647596
\(449\) −5.59637e6 −1.31006 −0.655029 0.755604i \(-0.727343\pi\)
−0.655029 + 0.755604i \(0.727343\pi\)
\(450\) −1.36846e6 −0.318567
\(451\) 1.50283e6 0.347911
\(452\) −3.53426e6 −0.813678
\(453\) 9.67079e6 2.21420
\(454\) −1.40031e6 −0.318850
\(455\) −184443. −0.0417671
\(456\) −791051. −0.178153
\(457\) −179779. −0.0402669 −0.0201335 0.999797i \(-0.506409\pi\)
−0.0201335 + 0.999797i \(0.506409\pi\)
\(458\) −4.06857e6 −0.906313
\(459\) −2.05015e6 −0.454208
\(460\) −1.40483e6 −0.309550
\(461\) −2.89395e6 −0.634217 −0.317109 0.948389i \(-0.602712\pi\)
−0.317109 + 0.948389i \(0.602712\pi\)
\(462\) −206147. −0.0449338
\(463\) 4.62210e6 1.00205 0.501023 0.865434i \(-0.332957\pi\)
0.501023 + 0.865434i \(0.332957\pi\)
\(464\) 2.25468e6 0.486172
\(465\) −3.62861e6 −0.778230
\(466\) −5.60068e6 −1.19475
\(467\) −548881. −0.116463 −0.0582313 0.998303i \(-0.518546\pi\)
−0.0582313 + 0.998303i \(0.518546\pi\)
\(468\) −4.20279e6 −0.886999
\(469\) −508482. −0.106744
\(470\) −2.61702e6 −0.546465
\(471\) −5.69976e6 −1.18387
\(472\) −1.32204e6 −0.273142
\(473\) −2.71031e6 −0.557013
\(474\) 8.69851e6 1.77827
\(475\) 225625. 0.0458831
\(476\) −263402. −0.0532847
\(477\) 5.19713e6 1.04585
\(478\) −5.72494e6 −1.14604
\(479\) 5.56572e6 1.10836 0.554182 0.832396i \(-0.313031\pi\)
0.554182 + 0.832396i \(0.313031\pi\)
\(480\) 3.56548e6 0.706342
\(481\) 6.74849e6 1.32998
\(482\) 181849. 0.0356528
\(483\) 709889. 0.138459
\(484\) 276751. 0.0537001
\(485\) −747809. −0.144357
\(486\) −9.19623e6 −1.76612
\(487\) 4.22188e6 0.806648 0.403324 0.915057i \(-0.367855\pi\)
0.403324 + 0.915057i \(0.367855\pi\)
\(488\) −4.61016e6 −0.876327
\(489\) −7.81924e6 −1.47874
\(490\) −2.97928e6 −0.560559
\(491\) 6.35541e6 1.18971 0.594853 0.803835i \(-0.297210\pi\)
0.594853 + 0.803835i \(0.297210\pi\)
\(492\) −5.50529e6 −1.02534
\(493\) −2.42638e6 −0.449616
\(494\) 1.86601e6 0.344030
\(495\) −928342. −0.170292
\(496\) 7.87054e6 1.43648
\(497\) −36727.9 −0.00666968
\(498\) 1.85060e7 3.34378
\(499\) −8.86729e6 −1.59419 −0.797094 0.603856i \(-0.793630\pi\)
−0.797094 + 0.603856i \(0.793630\pi\)
\(500\) −295351. −0.0528339
\(501\) −2.34990e6 −0.418269
\(502\) 4.84321e6 0.857776
\(503\) 7.66117e6 1.35013 0.675064 0.737759i \(-0.264116\pi\)
0.675064 + 0.737759i \(0.264116\pi\)
\(504\) −292030. −0.0512096
\(505\) 209134. 0.0364919
\(506\) −2.56639e6 −0.445600
\(507\) −3.60199e6 −0.622333
\(508\) −540261. −0.0928848
\(509\) −5.46044e6 −0.934185 −0.467093 0.884208i \(-0.654699\pi\)
−0.467093 + 0.884208i \(0.654699\pi\)
\(510\) −5.72353e6 −0.974402
\(511\) 386770. 0.0655240
\(512\) −3.93126e6 −0.662762
\(513\) −540850. −0.0907368
\(514\) 6.63296e6 1.10739
\(515\) −3.40837e6 −0.566277
\(516\) 9.92861e6 1.64159
\(517\) −1.77535e6 −0.292117
\(518\) −676743. −0.110815
\(519\) −1.67035e7 −2.72200
\(520\) 1.69253e6 0.274491
\(521\) −6.66984e6 −1.07652 −0.538259 0.842779i \(-0.680918\pi\)
−0.538259 + 0.842779i \(0.680918\pi\)
\(522\) 3.88235e6 0.623617
\(523\) −7.64901e6 −1.22279 −0.611394 0.791326i \(-0.709391\pi\)
−0.611394 + 0.791326i \(0.709391\pi\)
\(524\) 853767. 0.135835
\(525\) 149246. 0.0236322
\(526\) −2.95347e6 −0.465446
\(527\) −8.46991e6 −1.32847
\(528\) 3.60799e6 0.563223
\(529\) 2.40126e6 0.373078
\(530\) 3.02058e6 0.467090
\(531\) −4.34176e6 −0.668236
\(532\) −69488.1 −0.0106446
\(533\) −8.99833e6 −1.37197
\(534\) 1.21327e7 1.84121
\(535\) −2.82385e6 −0.426537
\(536\) 4.66606e6 0.701517
\(537\) 1.84176e7 2.75611
\(538\) −7.09703e6 −1.05711
\(539\) −2.02110e6 −0.299651
\(540\) 707990. 0.104482
\(541\) 7.74562e6 1.13779 0.568896 0.822409i \(-0.307371\pi\)
0.568896 + 0.822409i \(0.307371\pi\)
\(542\) 3.65486e6 0.534407
\(543\) 4.24813e6 0.618299
\(544\) 8.32256e6 1.20576
\(545\) −3.39833e6 −0.490089
\(546\) 1.23433e6 0.177194
\(547\) −4.59792e6 −0.657042 −0.328521 0.944497i \(-0.606550\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(548\) 5.49840e6 0.782141
\(549\) −1.51404e7 −2.14392
\(550\) −539554. −0.0760550
\(551\) −640103. −0.0898195
\(552\) −6.51426e6 −0.909949
\(553\) −529447. −0.0736224
\(554\) 5.05612e6 0.699911
\(555\) −5.46069e6 −0.752514
\(556\) −6.74568e6 −0.925420
\(557\) 1.01353e7 1.38420 0.692099 0.721803i \(-0.256686\pi\)
0.692099 + 0.721803i \(0.256686\pi\)
\(558\) 1.35523e7 1.84259
\(559\) 1.62282e7 2.19655
\(560\) −323719. −0.0436212
\(561\) −3.88275e6 −0.520874
\(562\) −9.19193e6 −1.22763
\(563\) −5.95706e6 −0.792065 −0.396032 0.918236i \(-0.629613\pi\)
−0.396032 + 0.918236i \(0.629613\pi\)
\(564\) 6.50359e6 0.860906
\(565\) 4.67434e6 0.616027
\(566\) 9.65199e6 1.26641
\(567\) 401645. 0.0524669
\(568\) 337031. 0.0438328
\(569\) 1.37133e6 0.177567 0.0887836 0.996051i \(-0.471702\pi\)
0.0887836 + 0.996051i \(0.471702\pi\)
\(570\) −1.50992e6 −0.194656
\(571\) −1.40265e7 −1.80035 −0.900176 0.435526i \(-0.856563\pi\)
−0.900176 + 0.435526i \(0.856563\pi\)
\(572\) −1.65707e6 −0.211763
\(573\) −7.41712e6 −0.943733
\(574\) 902359. 0.114314
\(575\) 1.85801e6 0.234357
\(576\) −829080. −0.104121
\(577\) 7.32574e6 0.916035 0.458018 0.888943i \(-0.348560\pi\)
0.458018 + 0.888943i \(0.348560\pi\)
\(578\) −3.22977e6 −0.402117
\(579\) −1.42696e7 −1.76894
\(580\) 837915. 0.103426
\(581\) −1.12639e6 −0.138436
\(582\) 5.00447e6 0.612422
\(583\) 2.04911e6 0.249686
\(584\) −3.54917e6 −0.430621
\(585\) 5.55853e6 0.671537
\(586\) 1.90959e6 0.229719
\(587\) 6.69340e6 0.801773 0.400887 0.916128i \(-0.368702\pi\)
0.400887 + 0.916128i \(0.368702\pi\)
\(588\) 7.40386e6 0.883110
\(589\) −2.23445e6 −0.265388
\(590\) −2.52344e6 −0.298444
\(591\) 1.08109e7 1.27319
\(592\) 1.18444e7 1.38902
\(593\) 4.85520e6 0.566983 0.283492 0.958975i \(-0.408507\pi\)
0.283492 + 0.958975i \(0.408507\pi\)
\(594\) 1.29337e6 0.150404
\(595\) 348371. 0.0403413
\(596\) 154712. 0.0178406
\(597\) 2.38383e6 0.273741
\(598\) 1.53665e7 1.75720
\(599\) −1.30661e7 −1.48791 −0.743957 0.668227i \(-0.767053\pi\)
−0.743957 + 0.668227i \(0.767053\pi\)
\(600\) −1.36955e6 −0.155310
\(601\) −8.60041e6 −0.971255 −0.485627 0.874166i \(-0.661409\pi\)
−0.485627 + 0.874166i \(0.661409\pi\)
\(602\) −1.62737e6 −0.183019
\(603\) 1.53240e7 1.71625
\(604\) −7.79547e6 −0.869460
\(605\) −366025. −0.0406558
\(606\) −1.39956e6 −0.154814
\(607\) 960299. 0.105788 0.0528938 0.998600i \(-0.483156\pi\)
0.0528938 + 0.998600i \(0.483156\pi\)
\(608\) 2.19557e6 0.240873
\(609\) −423414. −0.0462618
\(610\) −8.79965e6 −0.957505
\(611\) 1.06300e7 1.15195
\(612\) 7.93811e6 0.856719
\(613\) 1.37066e7 1.47325 0.736626 0.676300i \(-0.236418\pi\)
0.736626 + 0.676300i \(0.236418\pi\)
\(614\) −4.21198e6 −0.450885
\(615\) 7.28119e6 0.776274
\(616\) −115141. −0.0122258
\(617\) −1.05006e7 −1.11046 −0.555229 0.831697i \(-0.687369\pi\)
−0.555229 + 0.831697i \(0.687369\pi\)
\(618\) 2.28094e7 2.40239
\(619\) 1.16029e7 1.21713 0.608567 0.793503i \(-0.291745\pi\)
0.608567 + 0.793503i \(0.291745\pi\)
\(620\) 2.92496e6 0.305591
\(621\) −4.45386e6 −0.463455
\(622\) 8.23030e6 0.852982
\(623\) −738472. −0.0762279
\(624\) −2.16032e7 −2.22104
\(625\) 390625. 0.0400000
\(626\) 6.20161e6 0.632512
\(627\) −1.02431e6 −0.104055
\(628\) 4.59449e6 0.464877
\(629\) −1.27464e7 −1.28457
\(630\) −557413. −0.0559533
\(631\) −6.57254e6 −0.657143 −0.328572 0.944479i \(-0.606567\pi\)
−0.328572 + 0.944479i \(0.606567\pi\)
\(632\) 4.85845e6 0.483843
\(633\) −2.45272e6 −0.243298
\(634\) −9.88167e6 −0.976354
\(635\) 714539. 0.0703221
\(636\) −7.50648e6 −0.735858
\(637\) 1.21015e7 1.18166
\(638\) 1.53072e6 0.148883
\(639\) 1.10686e6 0.107236
\(640\) 4.38367e6 0.423046
\(641\) 8.79244e6 0.845210 0.422605 0.906314i \(-0.361116\pi\)
0.422605 + 0.906314i \(0.361116\pi\)
\(642\) 1.88977e7 1.80955
\(643\) 1.57144e7 1.49890 0.749448 0.662063i \(-0.230319\pi\)
0.749448 + 0.662063i \(0.230319\pi\)
\(644\) −572230. −0.0543695
\(645\) −1.31314e7 −1.24283
\(646\) −3.52446e6 −0.332286
\(647\) −1.88098e6 −0.176654 −0.0883272 0.996092i \(-0.528152\pi\)
−0.0883272 + 0.996092i \(0.528152\pi\)
\(648\) −3.68568e6 −0.344810
\(649\) −1.71186e6 −0.159535
\(650\) 3.23063e6 0.299919
\(651\) −1.47804e6 −0.136689
\(652\) 6.30296e6 0.580665
\(653\) 6.22718e6 0.571490 0.285745 0.958306i \(-0.407759\pi\)
0.285745 + 0.958306i \(0.407759\pi\)
\(654\) 2.27422e7 2.07916
\(655\) −1.12918e6 −0.102839
\(656\) −1.57931e7 −1.43287
\(657\) −1.16560e7 −1.05350
\(658\) −1.06599e6 −0.0959815
\(659\) 1.32051e7 1.18448 0.592241 0.805761i \(-0.298243\pi\)
0.592241 + 0.805761i \(0.298243\pi\)
\(660\) 1.34085e6 0.119818
\(661\) 1.56980e7 1.39746 0.698731 0.715384i \(-0.253748\pi\)
0.698731 + 0.715384i \(0.253748\pi\)
\(662\) 1.65383e7 1.46671
\(663\) 2.32483e7 2.05404
\(664\) 1.03363e7 0.909795
\(665\) 91903.6 0.00805895
\(666\) 2.03949e7 1.78170
\(667\) −5.27120e6 −0.458770
\(668\) 1.89422e6 0.164244
\(669\) −9.57170e6 −0.826844
\(670\) 8.90635e6 0.766501
\(671\) −5.96955e6 −0.511841
\(672\) 1.45232e6 0.124062
\(673\) 1.98667e7 1.69078 0.845392 0.534146i \(-0.179367\pi\)
0.845392 + 0.534146i \(0.179367\pi\)
\(674\) 2.76943e6 0.234823
\(675\) −936374. −0.0791025
\(676\) 2.90350e6 0.244374
\(677\) −1.07136e7 −0.898384 −0.449192 0.893435i \(-0.648288\pi\)
−0.449192 + 0.893435i \(0.648288\pi\)
\(678\) −3.12815e7 −2.61345
\(679\) −304604. −0.0253549
\(680\) −3.19681e6 −0.265121
\(681\) −4.60250e6 −0.380300
\(682\) 5.34339e6 0.439902
\(683\) −7.82645e6 −0.641967 −0.320984 0.947085i \(-0.604013\pi\)
−0.320984 + 0.947085i \(0.604013\pi\)
\(684\) 2.09415e6 0.171146
\(685\) −7.27208e6 −0.592151
\(686\) −2.43463e6 −0.197525
\(687\) −1.33724e7 −1.08098
\(688\) 2.84823e7 2.29406
\(689\) −1.22693e7 −0.984623
\(690\) −1.24341e7 −0.994241
\(691\) −1.76929e7 −1.40963 −0.704814 0.709392i \(-0.748970\pi\)
−0.704814 + 0.709392i \(0.748970\pi\)
\(692\) 1.34644e7 1.06886
\(693\) −378140. −0.0299102
\(694\) −1.31336e7 −1.03510
\(695\) 8.92171e6 0.700625
\(696\) 3.88544e6 0.304030
\(697\) 1.69958e7 1.32513
\(698\) −3.04475e6 −0.236545
\(699\) −1.84081e7 −1.42500
\(700\) −120305. −0.00927979
\(701\) 1.48548e7 1.14175 0.570874 0.821037i \(-0.306604\pi\)
0.570874 + 0.821037i \(0.306604\pi\)
\(702\) −7.74419e6 −0.593108
\(703\) −3.36261e6 −0.256619
\(704\) −326888. −0.0248581
\(705\) −8.60153e6 −0.651782
\(706\) 2.35716e7 1.77982
\(707\) 85186.5 0.00640947
\(708\) 6.27104e6 0.470171
\(709\) −1.65746e7 −1.23831 −0.619153 0.785271i \(-0.712524\pi\)
−0.619153 + 0.785271i \(0.712524\pi\)
\(710\) 643309. 0.0478932
\(711\) 1.59559e7 1.18371
\(712\) 6.77655e6 0.500966
\(713\) −1.84005e7 −1.35552
\(714\) −2.33136e6 −0.171145
\(715\) 2.19161e6 0.160324
\(716\) −1.48461e7 −1.08226
\(717\) −1.88165e7 −1.36691
\(718\) −2.48188e7 −1.79667
\(719\) −6.21856e6 −0.448609 −0.224304 0.974519i \(-0.572011\pi\)
−0.224304 + 0.974519i \(0.572011\pi\)
\(720\) 9.75584e6 0.701348
\(721\) −1.38833e6 −0.0994613
\(722\) −929787. −0.0663805
\(723\) 597695. 0.0425240
\(724\) −3.42435e6 −0.242790
\(725\) −1.10821e6 −0.0783028
\(726\) 2.44950e6 0.172479
\(727\) −1.25697e7 −0.882042 −0.441021 0.897497i \(-0.645384\pi\)
−0.441021 + 0.897497i \(0.645384\pi\)
\(728\) 689418. 0.0482119
\(729\) −2.06415e7 −1.43854
\(730\) −6.77449e6 −0.470511
\(731\) −3.06513e7 −2.12156
\(732\) 2.18681e7 1.50846
\(733\) −2.66588e7 −1.83265 −0.916327 0.400430i \(-0.868861\pi\)
−0.916327 + 0.400430i \(0.868861\pi\)
\(734\) 1.14352e7 0.783438
\(735\) −9.79220e6 −0.668593
\(736\) 1.80804e7 1.23031
\(737\) 6.04193e6 0.409739
\(738\) −2.71942e7 −1.83796
\(739\) −5.21068e6 −0.350981 −0.175490 0.984481i \(-0.556151\pi\)
−0.175490 + 0.984481i \(0.556151\pi\)
\(740\) 4.40177e6 0.295494
\(741\) 6.13313e6 0.410333
\(742\) 1.23037e6 0.0820400
\(743\) 2.04875e7 1.36149 0.680747 0.732518i \(-0.261655\pi\)
0.680747 + 0.732518i \(0.261655\pi\)
\(744\) 1.35631e7 0.898313
\(745\) −204620. −0.0135069
\(746\) 2.60969e7 1.71689
\(747\) 3.39459e7 2.22580
\(748\) 3.12982e6 0.204534
\(749\) −1.15024e6 −0.0749173
\(750\) −2.61413e6 −0.169697
\(751\) −7.21719e6 −0.466948 −0.233474 0.972363i \(-0.575009\pi\)
−0.233474 + 0.972363i \(0.575009\pi\)
\(752\) 1.86569e7 1.20308
\(753\) 1.59185e7 1.02309
\(754\) −9.16535e6 −0.587112
\(755\) 1.03101e7 0.658259
\(756\) 288385. 0.0183514
\(757\) −2.34446e7 −1.48697 −0.743487 0.668751i \(-0.766829\pi\)
−0.743487 + 0.668751i \(0.766829\pi\)
\(758\) −3.36486e7 −2.12713
\(759\) −8.43510e6 −0.531479
\(760\) −843348. −0.0529630
\(761\) −1.03281e6 −0.0646484 −0.0323242 0.999477i \(-0.510291\pi\)
−0.0323242 + 0.999477i \(0.510291\pi\)
\(762\) −4.78182e6 −0.298336
\(763\) −1.38424e6 −0.0860795
\(764\) 5.97882e6 0.370580
\(765\) −1.04988e7 −0.648613
\(766\) −1.04410e7 −0.642937
\(767\) 1.02499e7 0.629119
\(768\) −3.13635e7 −1.91876
\(769\) 2.44745e7 1.49245 0.746223 0.665696i \(-0.231865\pi\)
0.746223 + 0.665696i \(0.231865\pi\)
\(770\) −219776. −0.0133584
\(771\) 2.18010e7 1.32081
\(772\) 1.15025e7 0.694620
\(773\) 1.08231e7 0.651484 0.325742 0.945459i \(-0.394386\pi\)
0.325742 + 0.945459i \(0.394386\pi\)
\(774\) 4.90439e7 2.94261
\(775\) −3.86850e6 −0.231360
\(776\) 2.79518e6 0.166631
\(777\) −2.22430e6 −0.132172
\(778\) 3.75005e7 2.22120
\(779\) 4.48365e6 0.264721
\(780\) −8.02848e6 −0.472494
\(781\) 436411. 0.0256017
\(782\) −2.90237e7 −1.69721
\(783\) 2.65651e6 0.154849
\(784\) 2.12395e7 1.23411
\(785\) −6.07658e6 −0.351953
\(786\) 7.55664e6 0.436287
\(787\) −1.65559e7 −0.952833 −0.476417 0.879220i \(-0.658065\pi\)
−0.476417 + 0.879220i \(0.658065\pi\)
\(788\) −8.71448e6 −0.499949
\(789\) −9.70737e6 −0.555149
\(790\) 9.27357e6 0.528663
\(791\) 1.90400e6 0.108199
\(792\) 3.46998e6 0.196569
\(793\) 3.57432e7 2.01842
\(794\) 1.10208e7 0.620385
\(795\) 9.92793e6 0.557110
\(796\) −1.92157e6 −0.107491
\(797\) 2.25936e7 1.25991 0.629956 0.776631i \(-0.283073\pi\)
0.629956 + 0.776631i \(0.283073\pi\)
\(798\) −615035. −0.0341895
\(799\) −2.00777e7 −1.11262
\(800\) 3.80120e6 0.209988
\(801\) 2.22552e7 1.22560
\(802\) −4.26560e7 −2.34177
\(803\) −4.59571e6 −0.251515
\(804\) −2.21333e7 −1.20755
\(805\) 756820. 0.0411626
\(806\) −3.19940e7 −1.73473
\(807\) −2.33263e7 −1.26084
\(808\) −781709. −0.0421228
\(809\) 1.66890e7 0.896520 0.448260 0.893903i \(-0.352044\pi\)
0.448260 + 0.893903i \(0.352044\pi\)
\(810\) −7.03504e6 −0.376751
\(811\) −3.27766e7 −1.74990 −0.874948 0.484217i \(-0.839105\pi\)
−0.874948 + 0.484217i \(0.839105\pi\)
\(812\) 341307. 0.0181658
\(813\) 1.20126e7 0.637400
\(814\) 8.04126e6 0.425366
\(815\) −8.33618e6 −0.439615
\(816\) 4.08034e7 2.14522
\(817\) −8.08612e6 −0.423823
\(818\) −2.95967e7 −1.54653
\(819\) 2.26415e6 0.117949
\(820\) −5.86925e6 −0.304823
\(821\) −1.60822e7 −0.832697 −0.416348 0.909205i \(-0.636690\pi\)
−0.416348 + 0.909205i \(0.636690\pi\)
\(822\) 4.86660e7 2.51215
\(823\) 2.03558e7 1.04758 0.523791 0.851847i \(-0.324517\pi\)
0.523791 + 0.851847i \(0.324517\pi\)
\(824\) 1.27399e7 0.653655
\(825\) −1.77339e6 −0.0907128
\(826\) −1.02787e6 −0.0524189
\(827\) −2.47509e7 −1.25842 −0.629211 0.777234i \(-0.716622\pi\)
−0.629211 + 0.777234i \(0.716622\pi\)
\(828\) 1.72452e7 0.874162
\(829\) 2.98798e7 1.51005 0.755026 0.655695i \(-0.227624\pi\)
0.755026 + 0.655695i \(0.227624\pi\)
\(830\) 1.97294e7 0.994073
\(831\) 1.66183e7 0.834802
\(832\) 1.95727e6 0.0980264
\(833\) −2.28570e7 −1.14132
\(834\) −5.97056e7 −2.97235
\(835\) −2.50526e6 −0.124347
\(836\) 825677. 0.0408597
\(837\) 9.27325e6 0.457529
\(838\) 2.28528e7 1.12416
\(839\) −4.58350e6 −0.224798 −0.112399 0.993663i \(-0.535853\pi\)
−0.112399 + 0.993663i \(0.535853\pi\)
\(840\) −557857. −0.0272788
\(841\) −1.73671e7 −0.846717
\(842\) 7.80982e6 0.379630
\(843\) −3.02117e7 −1.46422
\(844\) 1.97710e6 0.0955371
\(845\) −3.84012e6 −0.185013
\(846\) 3.21255e7 1.54320
\(847\) −149093. −0.00714081
\(848\) −2.15339e7 −1.02833
\(849\) 3.17238e7 1.51048
\(850\) −6.10191e6 −0.289680
\(851\) −2.76909e7 −1.31073
\(852\) −1.59870e6 −0.0754514
\(853\) −2.81267e7 −1.32357 −0.661784 0.749695i \(-0.730200\pi\)
−0.661784 + 0.749695i \(0.730200\pi\)
\(854\) −3.58435e6 −0.168177
\(855\) −2.76968e6 −0.129573
\(856\) 1.05551e7 0.492353
\(857\) −1.32333e7 −0.615485 −0.307742 0.951470i \(-0.599574\pi\)
−0.307742 + 0.951470i \(0.599574\pi\)
\(858\) −1.46666e7 −0.680161
\(859\) −1.99257e7 −0.921365 −0.460682 0.887565i \(-0.652395\pi\)
−0.460682 + 0.887565i \(0.652395\pi\)
\(860\) 1.05850e7 0.488028
\(861\) 2.96584e6 0.136345
\(862\) −3.20909e7 −1.47101
\(863\) 3.60952e7 1.64976 0.824882 0.565304i \(-0.191241\pi\)
0.824882 + 0.565304i \(0.191241\pi\)
\(864\) −9.11192e6 −0.415265
\(865\) −1.78077e7 −0.809224
\(866\) −2.76536e7 −1.25302
\(867\) −1.06155e7 −0.479615
\(868\) 1.19142e6 0.0536743
\(869\) 6.29105e6 0.282601
\(870\) 7.41634e6 0.332194
\(871\) −3.61766e7 −1.61578
\(872\) 1.27024e7 0.565711
\(873\) 9.17980e6 0.407660
\(874\) −7.65674e6 −0.339051
\(875\) 159113. 0.00702563
\(876\) 1.68354e7 0.741246
\(877\) 2.63968e7 1.15892 0.579458 0.815002i \(-0.303264\pi\)
0.579458 + 0.815002i \(0.303264\pi\)
\(878\) 2.75658e6 0.120680
\(879\) 6.27637e6 0.273991
\(880\) 3.84652e6 0.167441
\(881\) 2.89523e7 1.25673 0.628367 0.777917i \(-0.283724\pi\)
0.628367 + 0.777917i \(0.283724\pi\)
\(882\) 3.65725e7 1.58301
\(883\) −2.41824e7 −1.04375 −0.521876 0.853022i \(-0.674767\pi\)
−0.521876 + 0.853022i \(0.674767\pi\)
\(884\) −1.87401e7 −0.806568
\(885\) −8.29395e6 −0.355962
\(886\) 2.08930e7 0.894163
\(887\) −2.52582e7 −1.07794 −0.538968 0.842326i \(-0.681186\pi\)
−0.538968 + 0.842326i \(0.681186\pi\)
\(888\) 2.04111e7 0.868629
\(889\) 291052. 0.0123514
\(890\) 1.29347e7 0.547373
\(891\) −4.77247e6 −0.201395
\(892\) 7.71559e6 0.324681
\(893\) −5.29669e6 −0.222268
\(894\) 1.36935e6 0.0573021
\(895\) 1.96352e7 0.819364
\(896\) 1.78559e6 0.0743041
\(897\) 5.05059e7 2.09586
\(898\) 3.99278e7 1.65228
\(899\) 1.09750e7 0.452903
\(900\) 3.62560e6 0.149202
\(901\) 2.31738e7 0.951011
\(902\) −1.07221e7 −0.438796
\(903\) −5.34879e6 −0.218291
\(904\) −1.74719e7 −0.711081
\(905\) 4.52898e6 0.183814
\(906\) −6.89972e7 −2.79261
\(907\) 3.28410e7 1.32556 0.662778 0.748816i \(-0.269377\pi\)
0.662778 + 0.748816i \(0.269377\pi\)
\(908\) 3.71000e6 0.149334
\(909\) −2.56725e6 −0.103052
\(910\) 1.31593e6 0.0526779
\(911\) −1.45892e7 −0.582418 −0.291209 0.956659i \(-0.594057\pi\)
−0.291209 + 0.956659i \(0.594057\pi\)
\(912\) 1.07643e7 0.428549
\(913\) 1.33841e7 0.531389
\(914\) 1.28265e6 0.0507859
\(915\) −2.89224e7 −1.14204
\(916\) 1.07793e7 0.424475
\(917\) −459946. −0.0180627
\(918\) 1.46270e7 0.572860
\(919\) −1.90083e7 −0.742428 −0.371214 0.928547i \(-0.621058\pi\)
−0.371214 + 0.928547i \(0.621058\pi\)
\(920\) −6.94492e6 −0.270519
\(921\) −1.38438e7 −0.537782
\(922\) 2.06471e7 0.799894
\(923\) −2.61305e6 −0.100959
\(924\) 546168. 0.0210449
\(925\) −5.82170e6 −0.223715
\(926\) −3.29768e7 −1.26381
\(927\) 4.18398e7 1.59915
\(928\) −1.07841e7 −0.411067
\(929\) 2.97742e7 1.13188 0.565941 0.824446i \(-0.308513\pi\)
0.565941 + 0.824446i \(0.308513\pi\)
\(930\) 2.58887e7 0.981527
\(931\) −6.02989e6 −0.228000
\(932\) 1.48385e7 0.559564
\(933\) 2.70511e7 1.01737
\(934\) 3.91605e6 0.146886
\(935\) −4.13944e6 −0.154851
\(936\) −2.07768e7 −0.775157
\(937\) 3.10909e7 1.15687 0.578434 0.815729i \(-0.303664\pi\)
0.578434 + 0.815729i \(0.303664\pi\)
\(938\) 3.62781e6 0.134629
\(939\) 2.03832e7 0.754413
\(940\) 6.93355e6 0.255939
\(941\) 1.24141e7 0.457025 0.228513 0.973541i \(-0.426614\pi\)
0.228513 + 0.973541i \(0.426614\pi\)
\(942\) 4.06655e7 1.49313
\(943\) 3.69226e7 1.35211
\(944\) 1.79898e7 0.657046
\(945\) −381412. −0.0138936
\(946\) 1.93369e7 0.702522
\(947\) −3.55400e7 −1.28778 −0.643891 0.765117i \(-0.722681\pi\)
−0.643891 + 0.765117i \(0.722681\pi\)
\(948\) −2.30459e7 −0.832861
\(949\) 2.75172e7 0.991834
\(950\) −1.60974e6 −0.0578692
\(951\) −3.24787e7 −1.16452
\(952\) −1.30215e6 −0.0465660
\(953\) 2.63749e7 0.940716 0.470358 0.882476i \(-0.344125\pi\)
0.470358 + 0.882476i \(0.344125\pi\)
\(954\) −3.70794e7 −1.31905
\(955\) −7.90747e6 −0.280562
\(956\) 1.51677e7 0.536753
\(957\) 5.03113e6 0.177577
\(958\) −3.97091e7 −1.39790
\(959\) −2.96213e6 −0.104006
\(960\) −1.58377e6 −0.0554643
\(961\) 9.68198e6 0.338186
\(962\) −4.81478e7 −1.67741
\(963\) 3.46644e7 1.20453
\(964\) −481792. −0.0166981
\(965\) −1.52129e7 −0.525889
\(966\) −5.06477e6 −0.174629
\(967\) −5.28241e7 −1.81663 −0.908314 0.418289i \(-0.862630\pi\)
−0.908314 + 0.418289i \(0.862630\pi\)
\(968\) 1.36814e6 0.0469291
\(969\) −1.15841e7 −0.396326
\(970\) 5.33531e6 0.182067
\(971\) 2.42410e7 0.825092 0.412546 0.910937i \(-0.364640\pi\)
0.412546 + 0.910937i \(0.364640\pi\)
\(972\) 2.43646e7 0.827166
\(973\) 3.63407e6 0.123058
\(974\) −3.01214e7 −1.01737
\(975\) 1.06183e7 0.357720
\(976\) 6.27334e7 2.10802
\(977\) −4.50760e7 −1.51081 −0.755404 0.655260i \(-0.772559\pi\)
−0.755404 + 0.655260i \(0.772559\pi\)
\(978\) 5.57871e7 1.86503
\(979\) 8.77474e6 0.292602
\(980\) 7.89333e6 0.262540
\(981\) 4.17166e7 1.38400
\(982\) −4.53433e7 −1.50049
\(983\) 1.19204e7 0.393465 0.196733 0.980457i \(-0.436967\pi\)
0.196733 + 0.980457i \(0.436967\pi\)
\(984\) −2.72159e7 −0.896055
\(985\) 1.15256e7 0.378506
\(986\) 1.73112e7 0.567069
\(987\) −3.50365e6 −0.114480
\(988\) −4.94382e6 −0.161128
\(989\) −6.65886e7 −2.16476
\(990\) 6.62334e6 0.214778
\(991\) 2.72613e7 0.881783 0.440892 0.897560i \(-0.354662\pi\)
0.440892 + 0.897560i \(0.354662\pi\)
\(992\) −3.76446e7 −1.21457
\(993\) 5.43575e7 1.74939
\(994\) 262039. 0.00841200
\(995\) 2.54143e6 0.0813805
\(996\) −4.90298e7 −1.56607
\(997\) −1.82305e7 −0.580846 −0.290423 0.956898i \(-0.593796\pi\)
−0.290423 + 0.956898i \(0.593796\pi\)
\(998\) 6.32645e7 2.01064
\(999\) 1.39553e7 0.442410
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.a.1.8 35
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.6.a.a.1.8 35 1.1 even 1 trivial