Properties

Label 35.6.a.b.1.1
Level $35$
Weight $6$
Character 35.1
Self dual yes
Analytic conductor $5.613$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 35.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.61343369345\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.53113\) of defining polynomial
Character \(\chi\) \(=\) 35.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.53113 q^{2} +13.5934 q^{3} -19.5311 q^{4} -25.0000 q^{5} -48.0000 q^{6} -49.0000 q^{7} +181.963 q^{8} -58.2198 q^{9} +O(q^{10})\) \(q-3.53113 q^{2} +13.5934 q^{3} -19.5311 q^{4} -25.0000 q^{5} -48.0000 q^{6} -49.0000 q^{7} +181.963 q^{8} -58.2198 q^{9} +88.2782 q^{10} -691.520 q^{11} -265.494 q^{12} -502.150 q^{13} +173.025 q^{14} -339.835 q^{15} -17.5389 q^{16} -991.313 q^{17} +205.582 q^{18} +661.677 q^{19} +488.278 q^{20} -666.076 q^{21} +2441.84 q^{22} +3415.08 q^{23} +2473.49 q^{24} +625.000 q^{25} +1773.16 q^{26} -4094.60 q^{27} +957.025 q^{28} +6751.92 q^{29} +1200.00 q^{30} -3922.76 q^{31} -5760.89 q^{32} -9400.09 q^{33} +3500.46 q^{34} +1225.00 q^{35} +1137.10 q^{36} +627.222 q^{37} -2336.47 q^{38} -6825.92 q^{39} -4549.08 q^{40} +16277.9 q^{41} +2352.00 q^{42} -17277.7 q^{43} +13506.2 q^{44} +1455.50 q^{45} -12059.1 q^{46} -4295.47 q^{47} -238.413 q^{48} +2401.00 q^{49} -2206.96 q^{50} -13475.3 q^{51} +9807.55 q^{52} -25960.9 q^{53} +14458.6 q^{54} +17288.0 q^{55} -8916.19 q^{56} +8994.43 q^{57} -23841.9 q^{58} +8902.63 q^{59} +6637.35 q^{60} -48924.6 q^{61} +13851.8 q^{62} +2852.77 q^{63} +20903.7 q^{64} +12553.7 q^{65} +33192.9 q^{66} -4257.80 q^{67} +19361.5 q^{68} +46422.5 q^{69} -4325.63 q^{70} +18990.9 q^{71} -10593.9 q^{72} +10132.5 q^{73} -2214.80 q^{74} +8495.87 q^{75} -12923.3 q^{76} +33884.5 q^{77} +24103.2 q^{78} -96986.5 q^{79} +438.472 q^{80} -41512.0 q^{81} -57479.2 q^{82} +70732.1 q^{83} +13009.2 q^{84} +24782.8 q^{85} +61009.8 q^{86} +91781.4 q^{87} -125831. q^{88} +4241.12 q^{89} -5139.54 q^{90} +24605.3 q^{91} -66700.3 q^{92} -53323.6 q^{93} +15167.9 q^{94} -16541.9 q^{95} -78309.9 q^{96} -104376. q^{97} -8478.24 q^{98} +40260.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} - 31 q^{4} - 50 q^{5} - 96 q^{6} - 98 q^{7} - 15 q^{8} - 189 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 3 q^{3} - 31 q^{4} - 50 q^{5} - 96 q^{6} - 98 q^{7} - 15 q^{8} - 189 q^{9} - 25 q^{10} - 601 q^{11} - 144 q^{12} - 577 q^{13} - 49 q^{14} - 75 q^{15} - 543 q^{16} + 41 q^{17} - 387 q^{18} + 630 q^{19} + 775 q^{20} - 147 q^{21} + 2852 q^{22} - 442 q^{23} + 4560 q^{24} + 1250 q^{25} + 1434 q^{26} - 135 q^{27} + 1519 q^{28} + 5885 q^{29} + 2400 q^{30} - 396 q^{31} - 1839 q^{32} - 10359 q^{33} + 8178 q^{34} + 2450 q^{35} + 2637 q^{36} - 8904 q^{37} - 2480 q^{38} - 6033 q^{39} + 375 q^{40} + 1774 q^{41} + 4704 q^{42} - 27122 q^{43} + 12468 q^{44} + 4725 q^{45} - 29536 q^{46} - 21289 q^{47} + 5328 q^{48} + 4802 q^{49} + 625 q^{50} - 24411 q^{51} + 10666 q^{52} - 55582 q^{53} + 32400 q^{54} + 15025 q^{55} + 735 q^{56} + 9330 q^{57} - 27770 q^{58} + 59600 q^{59} + 3600 q^{60} - 51846 q^{61} + 29832 q^{62} + 9261 q^{63} + 55489 q^{64} + 14425 q^{65} + 28848 q^{66} - 45344 q^{67} + 7522 q^{68} + 87282 q^{69} + 1225 q^{70} + 80744 q^{71} + 15165 q^{72} - 13532 q^{73} - 45402 q^{74} + 1875 q^{75} - 12560 q^{76} + 29449 q^{77} + 27696 q^{78} - 51795 q^{79} + 13575 q^{80} - 51678 q^{81} - 123198 q^{82} + 109828 q^{83} + 7056 q^{84} - 1025 q^{85} + 16404 q^{86} + 100965 q^{87} - 143660 q^{88} - 37650 q^{89} + 9675 q^{90} + 28273 q^{91} - 22464 q^{92} - 90684 q^{93} - 61832 q^{94} - 15750 q^{95} - 119856 q^{96} - 96339 q^{97} + 2401 q^{98} + 28422 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.53113 −0.624221 −0.312111 0.950046i \(-0.601036\pi\)
−0.312111 + 0.950046i \(0.601036\pi\)
\(3\) 13.5934 0.872016 0.436008 0.899943i \(-0.356392\pi\)
0.436008 + 0.899943i \(0.356392\pi\)
\(4\) −19.5311 −0.610348
\(5\) −25.0000 −0.447214
\(6\) −48.0000 −0.544331
\(7\) −49.0000 −0.377964
\(8\) 181.963 1.00521
\(9\) −58.2198 −0.239588
\(10\) 88.2782 0.279160
\(11\) −691.520 −1.72315 −0.861574 0.507632i \(-0.830521\pi\)
−0.861574 + 0.507632i \(0.830521\pi\)
\(12\) −265.494 −0.532233
\(13\) −502.150 −0.824091 −0.412045 0.911163i \(-0.635185\pi\)
−0.412045 + 0.911163i \(0.635185\pi\)
\(14\) 173.025 0.235933
\(15\) −339.835 −0.389977
\(16\) −17.5389 −0.0171278
\(17\) −991.313 −0.831934 −0.415967 0.909380i \(-0.636557\pi\)
−0.415967 + 0.909380i \(0.636557\pi\)
\(18\) 205.582 0.149556
\(19\) 661.677 0.420496 0.210248 0.977648i \(-0.432573\pi\)
0.210248 + 0.977648i \(0.432573\pi\)
\(20\) 488.278 0.272956
\(21\) −666.076 −0.329591
\(22\) 2441.84 1.07563
\(23\) 3415.08 1.34611 0.673056 0.739592i \(-0.264981\pi\)
0.673056 + 0.739592i \(0.264981\pi\)
\(24\) 2473.49 0.876562
\(25\) 625.000 0.200000
\(26\) 1773.16 0.514415
\(27\) −4094.60 −1.08094
\(28\) 957.025 0.230690
\(29\) 6751.92 1.49084 0.745422 0.666593i \(-0.232248\pi\)
0.745422 + 0.666593i \(0.232248\pi\)
\(30\) 1200.00 0.243432
\(31\) −3922.76 −0.733142 −0.366571 0.930390i \(-0.619468\pi\)
−0.366571 + 0.930390i \(0.619468\pi\)
\(32\) −5760.89 −0.994522
\(33\) −9400.09 −1.50261
\(34\) 3500.46 0.519311
\(35\) 1225.00 0.169031
\(36\) 1137.10 0.146232
\(37\) 627.222 0.0753212 0.0376606 0.999291i \(-0.488009\pi\)
0.0376606 + 0.999291i \(0.488009\pi\)
\(38\) −2336.47 −0.262483
\(39\) −6825.92 −0.718620
\(40\) −4549.08 −0.449545
\(41\) 16277.9 1.51230 0.756149 0.654399i \(-0.227078\pi\)
0.756149 + 0.654399i \(0.227078\pi\)
\(42\) 2352.00 0.205738
\(43\) −17277.7 −1.42500 −0.712500 0.701672i \(-0.752437\pi\)
−0.712500 + 0.701672i \(0.752437\pi\)
\(44\) 13506.2 1.05172
\(45\) 1455.50 0.107147
\(46\) −12059.1 −0.840272
\(47\) −4295.47 −0.283639 −0.141820 0.989893i \(-0.545295\pi\)
−0.141820 + 0.989893i \(0.545295\pi\)
\(48\) −238.413 −0.0149357
\(49\) 2401.00 0.142857
\(50\) −2206.96 −0.124844
\(51\) −13475.3 −0.725460
\(52\) 9807.55 0.502982
\(53\) −25960.9 −1.26949 −0.634745 0.772721i \(-0.718895\pi\)
−0.634745 + 0.772721i \(0.718895\pi\)
\(54\) 14458.6 0.674746
\(55\) 17288.0 0.770615
\(56\) −8916.19 −0.379935
\(57\) 8994.43 0.366679
\(58\) −23841.9 −0.930616
\(59\) 8902.63 0.332957 0.166479 0.986045i \(-0.446760\pi\)
0.166479 + 0.986045i \(0.446760\pi\)
\(60\) 6637.35 0.238022
\(61\) −48924.6 −1.68346 −0.841730 0.539898i \(-0.818463\pi\)
−0.841730 + 0.539898i \(0.818463\pi\)
\(62\) 13851.8 0.457643
\(63\) 2852.77 0.0905557
\(64\) 20903.7 0.637930
\(65\) 12553.7 0.368545
\(66\) 33192.9 0.937963
\(67\) −4257.80 −0.115877 −0.0579387 0.998320i \(-0.518453\pi\)
−0.0579387 + 0.998320i \(0.518453\pi\)
\(68\) 19361.5 0.507769
\(69\) 46422.5 1.17383
\(70\) −4325.63 −0.105513
\(71\) 18990.9 0.447095 0.223547 0.974693i \(-0.428236\pi\)
0.223547 + 0.974693i \(0.428236\pi\)
\(72\) −10593.9 −0.240837
\(73\) 10132.5 0.222541 0.111270 0.993790i \(-0.464508\pi\)
0.111270 + 0.993790i \(0.464508\pi\)
\(74\) −2214.80 −0.0470171
\(75\) 8495.87 0.174403
\(76\) −12923.3 −0.256649
\(77\) 33884.5 0.651289
\(78\) 24103.2 0.448578
\(79\) −96986.5 −1.74841 −0.874205 0.485557i \(-0.838617\pi\)
−0.874205 + 0.485557i \(0.838617\pi\)
\(80\) 438.472 0.00765979
\(81\) −41512.0 −0.703010
\(82\) −57479.2 −0.944009
\(83\) 70732.1 1.12699 0.563497 0.826118i \(-0.309456\pi\)
0.563497 + 0.826118i \(0.309456\pi\)
\(84\) 13009.2 0.201165
\(85\) 24782.8 0.372052
\(86\) 61009.8 0.889515
\(87\) 91781.4 1.30004
\(88\) −125831. −1.73213
\(89\) 4241.12 0.0567552 0.0283776 0.999597i \(-0.490966\pi\)
0.0283776 + 0.999597i \(0.490966\pi\)
\(90\) −5139.54 −0.0668834
\(91\) 24605.3 0.311477
\(92\) −66700.3 −0.821596
\(93\) −53323.6 −0.639311
\(94\) 15167.9 0.177054
\(95\) −16541.9 −0.188052
\(96\) −78309.9 −0.867239
\(97\) −104376. −1.12634 −0.563170 0.826341i \(-0.690418\pi\)
−0.563170 + 0.826341i \(0.690418\pi\)
\(98\) −8478.24 −0.0891745
\(99\) 40260.2 0.412845
\(100\) −12207.0 −0.122070
\(101\) −45715.1 −0.445919 −0.222959 0.974828i \(-0.571572\pi\)
−0.222959 + 0.974828i \(0.571572\pi\)
\(102\) 47583.0 0.452847
\(103\) 89278.1 0.829186 0.414593 0.910007i \(-0.363924\pi\)
0.414593 + 0.910007i \(0.363924\pi\)
\(104\) −91372.7 −0.828387
\(105\) 16651.9 0.147398
\(106\) 91671.2 0.792443
\(107\) −106330. −0.897834 −0.448917 0.893573i \(-0.648190\pi\)
−0.448917 + 0.893573i \(0.648190\pi\)
\(108\) 79972.1 0.659750
\(109\) −49816.5 −0.401613 −0.200806 0.979631i \(-0.564356\pi\)
−0.200806 + 0.979631i \(0.564356\pi\)
\(110\) −61046.1 −0.481035
\(111\) 8526.08 0.0656813
\(112\) 859.405 0.00647370
\(113\) −37160.7 −0.273771 −0.136886 0.990587i \(-0.543709\pi\)
−0.136886 + 0.990587i \(0.543709\pi\)
\(114\) −31760.5 −0.228889
\(115\) −85377.0 −0.601999
\(116\) −131873. −0.909933
\(117\) 29235.1 0.197442
\(118\) −31436.3 −0.207839
\(119\) 48574.4 0.314441
\(120\) −61837.4 −0.392011
\(121\) 317148. 1.96924
\(122\) 172759. 1.05085
\(123\) 221271. 1.31875
\(124\) 76616.0 0.447471
\(125\) −15625.0 −0.0894427
\(126\) −10073.5 −0.0565268
\(127\) −46510.2 −0.255882 −0.127941 0.991782i \(-0.540837\pi\)
−0.127941 + 0.991782i \(0.540837\pi\)
\(128\) 110535. 0.596313
\(129\) −234862. −1.24262
\(130\) −44328.9 −0.230053
\(131\) 381771. 1.94368 0.971839 0.235646i \(-0.0757206\pi\)
0.971839 + 0.235646i \(0.0757206\pi\)
\(132\) 183594. 0.917117
\(133\) −32422.2 −0.158933
\(134\) 15034.9 0.0723331
\(135\) 102365. 0.483411
\(136\) −180382. −0.836271
\(137\) −1894.54 −0.00862389 −0.00431194 0.999991i \(-0.501373\pi\)
−0.00431194 + 0.999991i \(0.501373\pi\)
\(138\) −163924. −0.732730
\(139\) 201798. 0.885889 0.442944 0.896549i \(-0.353934\pi\)
0.442944 + 0.896549i \(0.353934\pi\)
\(140\) −23925.6 −0.103168
\(141\) −58390.0 −0.247338
\(142\) −67059.3 −0.279086
\(143\) 347246. 1.42003
\(144\) 1021.11 0.00410362
\(145\) −168798. −0.666726
\(146\) −35779.1 −0.138915
\(147\) 32637.7 0.124574
\(148\) −12250.4 −0.0459721
\(149\) −466237. −1.72045 −0.860224 0.509917i \(-0.829676\pi\)
−0.860224 + 0.509917i \(0.829676\pi\)
\(150\) −30000.0 −0.108866
\(151\) −122212. −0.436185 −0.218093 0.975928i \(-0.569983\pi\)
−0.218093 + 0.975928i \(0.569983\pi\)
\(152\) 120401. 0.422688
\(153\) 57714.1 0.199321
\(154\) −119650. −0.406548
\(155\) 98069.1 0.327871
\(156\) 133318. 0.438608
\(157\) −410638. −1.32957 −0.664784 0.747036i \(-0.731476\pi\)
−0.664784 + 0.747036i \(0.731476\pi\)
\(158\) 342472. 1.09139
\(159\) −352896. −1.10702
\(160\) 144022. 0.444764
\(161\) −167339. −0.508782
\(162\) 146584. 0.438834
\(163\) 78525.4 0.231495 0.115747 0.993279i \(-0.463074\pi\)
0.115747 + 0.993279i \(0.463074\pi\)
\(164\) −317925. −0.923028
\(165\) 235002. 0.671989
\(166\) −249764. −0.703494
\(167\) −597714. −1.65845 −0.829224 0.558916i \(-0.811217\pi\)
−0.829224 + 0.558916i \(0.811217\pi\)
\(168\) −121201. −0.331309
\(169\) −119139. −0.320875
\(170\) −87511.4 −0.232243
\(171\) −38522.7 −0.100746
\(172\) 337453. 0.869745
\(173\) −59874.0 −0.152098 −0.0760490 0.997104i \(-0.524231\pi\)
−0.0760490 + 0.997104i \(0.524231\pi\)
\(174\) −324092. −0.811512
\(175\) −30625.0 −0.0755929
\(176\) 12128.5 0.0295138
\(177\) 121017. 0.290344
\(178\) −14975.9 −0.0354278
\(179\) 616812. 1.43887 0.719433 0.694562i \(-0.244402\pi\)
0.719433 + 0.694562i \(0.244402\pi\)
\(180\) −28427.5 −0.0653969
\(181\) −37287.0 −0.0845981 −0.0422990 0.999105i \(-0.513468\pi\)
−0.0422990 + 0.999105i \(0.513468\pi\)
\(182\) −86884.6 −0.194431
\(183\) −665051. −1.46800
\(184\) 621418. 1.35313
\(185\) −15680.6 −0.0336847
\(186\) 188293. 0.399072
\(187\) 685513. 1.43355
\(188\) 83895.4 0.173119
\(189\) 200635. 0.408557
\(190\) 58411.7 0.117386
\(191\) 326760. 0.648106 0.324053 0.946039i \(-0.394954\pi\)
0.324053 + 0.946039i \(0.394954\pi\)
\(192\) 284152. 0.556285
\(193\) −265735. −0.513518 −0.256759 0.966475i \(-0.582655\pi\)
−0.256759 + 0.966475i \(0.582655\pi\)
\(194\) 368563. 0.703085
\(195\) 170648. 0.321377
\(196\) −46894.2 −0.0871925
\(197\) −517865. −0.950716 −0.475358 0.879792i \(-0.657682\pi\)
−0.475358 + 0.879792i \(0.657682\pi\)
\(198\) −142164. −0.257707
\(199\) −148687. −0.266158 −0.133079 0.991105i \(-0.542486\pi\)
−0.133079 + 0.991105i \(0.542486\pi\)
\(200\) 113727. 0.201043
\(201\) −57878.0 −0.101047
\(202\) 161426. 0.278352
\(203\) −330844. −0.563486
\(204\) 263188. 0.442783
\(205\) −406946. −0.676320
\(206\) −315252. −0.517595
\(207\) −198825. −0.322512
\(208\) 8807.15 0.0141149
\(209\) −457563. −0.724577
\(210\) −58800.0 −0.0920087
\(211\) 7443.09 0.0115093 0.00575463 0.999983i \(-0.498168\pi\)
0.00575463 + 0.999983i \(0.498168\pi\)
\(212\) 507045. 0.774831
\(213\) 258151. 0.389874
\(214\) 375465. 0.560447
\(215\) 431943. 0.637279
\(216\) −745066. −1.08658
\(217\) 192215. 0.277101
\(218\) 175909. 0.250695
\(219\) 137735. 0.194059
\(220\) −337654. −0.470343
\(221\) 497788. 0.685589
\(222\) −30106.7 −0.0409997
\(223\) 119157. 0.160456 0.0802280 0.996777i \(-0.474435\pi\)
0.0802280 + 0.996777i \(0.474435\pi\)
\(224\) 282283. 0.375894
\(225\) −36387.4 −0.0479176
\(226\) 131219. 0.170894
\(227\) −388843. −0.500852 −0.250426 0.968136i \(-0.580571\pi\)
−0.250426 + 0.968136i \(0.580571\pi\)
\(228\) −175671. −0.223802
\(229\) 732622. 0.923191 0.461595 0.887091i \(-0.347277\pi\)
0.461595 + 0.887091i \(0.347277\pi\)
\(230\) 301477. 0.375781
\(231\) 460605. 0.567934
\(232\) 1.22860e6 1.49862
\(233\) −1.12639e6 −1.35925 −0.679626 0.733559i \(-0.737858\pi\)
−0.679626 + 0.733559i \(0.737858\pi\)
\(234\) −103233. −0.123248
\(235\) 107387. 0.126847
\(236\) −173878. −0.203220
\(237\) −1.31837e6 −1.52464
\(238\) −171522. −0.196281
\(239\) 772317. 0.874583 0.437292 0.899320i \(-0.355938\pi\)
0.437292 + 0.899320i \(0.355938\pi\)
\(240\) 5960.32 0.00667946
\(241\) 1.40297e6 1.55598 0.777991 0.628275i \(-0.216239\pi\)
0.777991 + 0.628275i \(0.216239\pi\)
\(242\) −1.11989e6 −1.22924
\(243\) 430698. 0.467905
\(244\) 955553. 1.02750
\(245\) −60025.0 −0.0638877
\(246\) −781337. −0.823191
\(247\) −332261. −0.346527
\(248\) −713798. −0.736964
\(249\) 961489. 0.982757
\(250\) 55173.9 0.0558320
\(251\) −1.63922e6 −1.64230 −0.821151 0.570712i \(-0.806667\pi\)
−0.821151 + 0.570712i \(0.806667\pi\)
\(252\) −55717.9 −0.0552705
\(253\) −2.36159e6 −2.31955
\(254\) 164234. 0.159727
\(255\) 336883. 0.324435
\(256\) −1.05923e6 −1.01016
\(257\) −223664. −0.211234 −0.105617 0.994407i \(-0.533682\pi\)
−0.105617 + 0.994407i \(0.533682\pi\)
\(258\) 829330. 0.775672
\(259\) −30733.9 −0.0284687
\(260\) −245189. −0.224940
\(261\) −393096. −0.357188
\(262\) −1.34808e6 −1.21329
\(263\) 299519. 0.267014 0.133507 0.991048i \(-0.457376\pi\)
0.133507 + 0.991048i \(0.457376\pi\)
\(264\) −1.71047e6 −1.51045
\(265\) 649022. 0.567734
\(266\) 114487. 0.0992091
\(267\) 57651.2 0.0494914
\(268\) 83159.7 0.0707255
\(269\) 134341. 0.113195 0.0565974 0.998397i \(-0.481975\pi\)
0.0565974 + 0.998397i \(0.481975\pi\)
\(270\) −361464. −0.301756
\(271\) 1.93414e6 1.59980 0.799898 0.600136i \(-0.204887\pi\)
0.799898 + 0.600136i \(0.204887\pi\)
\(272\) 17386.5 0.0142492
\(273\) 334470. 0.271613
\(274\) 6689.88 0.00538321
\(275\) −432200. −0.344630
\(276\) −906683. −0.716445
\(277\) 177599. 0.139072 0.0695362 0.997579i \(-0.477848\pi\)
0.0695362 + 0.997579i \(0.477848\pi\)
\(278\) −712574. −0.552991
\(279\) 228383. 0.175652
\(280\) 222905. 0.169912
\(281\) 1.85131e6 1.39867 0.699333 0.714796i \(-0.253481\pi\)
0.699333 + 0.714796i \(0.253481\pi\)
\(282\) 206183. 0.154394
\(283\) 2.39851e6 1.78023 0.890114 0.455737i \(-0.150624\pi\)
0.890114 + 0.455737i \(0.150624\pi\)
\(284\) −370914. −0.272883
\(285\) −224861. −0.163984
\(286\) −1.22617e6 −0.886413
\(287\) −797615. −0.571595
\(288\) 335398. 0.238275
\(289\) −437155. −0.307887
\(290\) 596047. 0.416184
\(291\) −1.41882e6 −0.982186
\(292\) −197899. −0.135827
\(293\) 2.49922e6 1.70073 0.850364 0.526196i \(-0.176382\pi\)
0.850364 + 0.526196i \(0.176382\pi\)
\(294\) −115248. −0.0777616
\(295\) −222566. −0.148903
\(296\) 114131. 0.0757139
\(297\) 2.83149e6 1.86262
\(298\) 1.64634e6 1.07394
\(299\) −1.71488e6 −1.10932
\(300\) −165934. −0.106447
\(301\) 846607. 0.538599
\(302\) 431546. 0.272276
\(303\) −621422. −0.388848
\(304\) −11605.1 −0.00720218
\(305\) 1.22312e6 0.752866
\(306\) −203796. −0.124421
\(307\) 3.07195e6 1.86024 0.930119 0.367258i \(-0.119703\pi\)
0.930119 + 0.367258i \(0.119703\pi\)
\(308\) −661802. −0.397513
\(309\) 1.21359e6 0.723063
\(310\) −346295. −0.204664
\(311\) 661233. 0.387662 0.193831 0.981035i \(-0.437909\pi\)
0.193831 + 0.981035i \(0.437909\pi\)
\(312\) −1.24206e6 −0.722367
\(313\) −3.29393e6 −1.90043 −0.950217 0.311588i \(-0.899139\pi\)
−0.950217 + 0.311588i \(0.899139\pi\)
\(314\) 1.45002e6 0.829944
\(315\) −71319.3 −0.0404977
\(316\) 1.89426e6 1.06714
\(317\) 639724. 0.357556 0.178778 0.983889i \(-0.442786\pi\)
0.178778 + 0.983889i \(0.442786\pi\)
\(318\) 1.24612e6 0.691023
\(319\) −4.66908e6 −2.56894
\(320\) −522592. −0.285291
\(321\) −1.44538e6 −0.782926
\(322\) 590895. 0.317593
\(323\) −655929. −0.349825
\(324\) 810777. 0.429081
\(325\) −313844. −0.164818
\(326\) −277283. −0.144504
\(327\) −677175. −0.350213
\(328\) 2.96197e6 1.52018
\(329\) 210478. 0.107206
\(330\) −829823. −0.419470
\(331\) −1.13876e6 −0.571298 −0.285649 0.958334i \(-0.592209\pi\)
−0.285649 + 0.958334i \(0.592209\pi\)
\(332\) −1.38148e6 −0.687858
\(333\) −36516.8 −0.0180460
\(334\) 2.11060e6 1.03524
\(335\) 106445. 0.0518219
\(336\) 11682.2 0.00564518
\(337\) 685493. 0.328797 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(338\) 420694. 0.200297
\(339\) −505140. −0.238733
\(340\) −484037. −0.227081
\(341\) 2.71267e6 1.26331
\(342\) 136029. 0.0628877
\(343\) −117649. −0.0539949
\(344\) −3.14390e6 −1.43243
\(345\) −1.16056e6 −0.524953
\(346\) 211423. 0.0949428
\(347\) −1.25151e6 −0.557970 −0.278985 0.960295i \(-0.589998\pi\)
−0.278985 + 0.960295i \(0.589998\pi\)
\(348\) −1.79259e6 −0.793476
\(349\) −3.16606e6 −1.39141 −0.695706 0.718327i \(-0.744908\pi\)
−0.695706 + 0.718327i \(0.744908\pi\)
\(350\) 108141. 0.0471867
\(351\) 2.05610e6 0.890793
\(352\) 3.98376e6 1.71371
\(353\) −2.43368e6 −1.03951 −0.519754 0.854316i \(-0.673976\pi\)
−0.519754 + 0.854316i \(0.673976\pi\)
\(354\) −427326. −0.181239
\(355\) −474772. −0.199947
\(356\) −82833.8 −0.0346404
\(357\) 660290. 0.274198
\(358\) −2.17804e6 −0.898171
\(359\) −2.13021e6 −0.872341 −0.436170 0.899864i \(-0.643665\pi\)
−0.436170 + 0.899864i \(0.643665\pi\)
\(360\) 264847. 0.107706
\(361\) −2.03828e6 −0.823183
\(362\) 131665. 0.0528079
\(363\) 4.31112e6 1.71721
\(364\) −480570. −0.190109
\(365\) −253312. −0.0995232
\(366\) 2.34838e6 0.916360
\(367\) −3.10976e6 −1.20521 −0.602604 0.798041i \(-0.705870\pi\)
−0.602604 + 0.798041i \(0.705870\pi\)
\(368\) −59896.7 −0.0230559
\(369\) −947694. −0.362328
\(370\) 55370.1 0.0210267
\(371\) 1.27208e6 0.479822
\(372\) 1.04147e6 0.390202
\(373\) −3.15189e6 −1.17300 −0.586502 0.809948i \(-0.699495\pi\)
−0.586502 + 0.809948i \(0.699495\pi\)
\(374\) −2.42063e6 −0.894849
\(375\) −212397. −0.0779955
\(376\) −781617. −0.285118
\(377\) −3.39047e6 −1.22859
\(378\) −708469. −0.255030
\(379\) 342350. 0.122426 0.0612129 0.998125i \(-0.480503\pi\)
0.0612129 + 0.998125i \(0.480503\pi\)
\(380\) 323083. 0.114777
\(381\) −632231. −0.223133
\(382\) −1.15383e6 −0.404562
\(383\) 3.69387e6 1.28672 0.643361 0.765563i \(-0.277539\pi\)
0.643361 + 0.765563i \(0.277539\pi\)
\(384\) 1.50254e6 0.519994
\(385\) −847111. −0.291265
\(386\) 938345. 0.320549
\(387\) 1.00590e6 0.341413
\(388\) 2.03857e6 0.687459
\(389\) 2.05313e6 0.687928 0.343964 0.938983i \(-0.388230\pi\)
0.343964 + 0.938983i \(0.388230\pi\)
\(390\) −602580. −0.200610
\(391\) −3.38541e6 −1.11988
\(392\) 436893. 0.143602
\(393\) 5.18956e6 1.69492
\(394\) 1.82865e6 0.593457
\(395\) 2.42466e6 0.781913
\(396\) −786326. −0.251979
\(397\) −2.28107e6 −0.726377 −0.363189 0.931716i \(-0.618312\pi\)
−0.363189 + 0.931716i \(0.618312\pi\)
\(398\) 525031. 0.166141
\(399\) −440727. −0.138592
\(400\) −10961.8 −0.00342556
\(401\) 3.32082e6 1.03130 0.515649 0.856800i \(-0.327551\pi\)
0.515649 + 0.856800i \(0.327551\pi\)
\(402\) 204375. 0.0630756
\(403\) 1.96981e6 0.604175
\(404\) 892867. 0.272166
\(405\) 1.03780e6 0.314396
\(406\) 1.16825e6 0.351740
\(407\) −433737. −0.129790
\(408\) −2.45201e6 −0.729242
\(409\) 5.15938e6 1.52507 0.762534 0.646948i \(-0.223955\pi\)
0.762534 + 0.646948i \(0.223955\pi\)
\(410\) 1.43698e6 0.422174
\(411\) −25753.3 −0.00752017
\(412\) −1.74370e6 −0.506092
\(413\) −436229. −0.125846
\(414\) 702078. 0.201319
\(415\) −1.76830e6 −0.504007
\(416\) 2.89283e6 0.819576
\(417\) 2.74311e6 0.772509
\(418\) 1.61571e6 0.452297
\(419\) −4.85187e6 −1.35012 −0.675062 0.737761i \(-0.735883\pi\)
−0.675062 + 0.737761i \(0.735883\pi\)
\(420\) −325230. −0.0899638
\(421\) −6.14767e6 −1.69046 −0.845231 0.534401i \(-0.820537\pi\)
−0.845231 + 0.534401i \(0.820537\pi\)
\(422\) −26282.5 −0.00718432
\(423\) 250082. 0.0679565
\(424\) −4.72392e6 −1.27611
\(425\) −619571. −0.166387
\(426\) −911563. −0.243368
\(427\) 2.39731e6 0.636288
\(428\) 2.07674e6 0.547991
\(429\) 4.72025e6 1.23829
\(430\) −1.52524e6 −0.397803
\(431\) 3.55411e6 0.921590 0.460795 0.887507i \(-0.347564\pi\)
0.460795 + 0.887507i \(0.347564\pi\)
\(432\) 71814.7 0.0185141
\(433\) 2.82650e6 0.724485 0.362243 0.932084i \(-0.382011\pi\)
0.362243 + 0.932084i \(0.382011\pi\)
\(434\) −678737. −0.172973
\(435\) −2.29454e6 −0.581395
\(436\) 972973. 0.245123
\(437\) 2.25968e6 0.566035
\(438\) −486360. −0.121136
\(439\) 4.64410e6 1.15011 0.575056 0.818114i \(-0.304980\pi\)
0.575056 + 0.818114i \(0.304980\pi\)
\(440\) 3.14578e6 0.774633
\(441\) −139786. −0.0342268
\(442\) −1.75775e6 −0.427959
\(443\) −6.15534e6 −1.49019 −0.745097 0.666957i \(-0.767597\pi\)
−0.745097 + 0.666957i \(0.767597\pi\)
\(444\) −166524. −0.0400884
\(445\) −106028. −0.0253817
\(446\) −420757. −0.100160
\(447\) −6.33774e6 −1.50026
\(448\) −1.02428e6 −0.241115
\(449\) 3.67035e6 0.859196 0.429598 0.903020i \(-0.358655\pi\)
0.429598 + 0.903020i \(0.358655\pi\)
\(450\) 128489. 0.0299112
\(451\) −1.12565e7 −2.60591
\(452\) 725791. 0.167096
\(453\) −1.66127e6 −0.380360
\(454\) 1.37305e6 0.312642
\(455\) −615134. −0.139297
\(456\) 1.63665e6 0.368591
\(457\) −866327. −0.194040 −0.0970201 0.995282i \(-0.530931\pi\)
−0.0970201 + 0.995282i \(0.530931\pi\)
\(458\) −2.58698e6 −0.576275
\(459\) 4.05903e6 0.899271
\(460\) 1.66751e6 0.367429
\(461\) −1.88572e6 −0.413261 −0.206631 0.978419i \(-0.566250\pi\)
−0.206631 + 0.978419i \(0.566250\pi\)
\(462\) −1.62645e6 −0.354517
\(463\) −8.49017e6 −1.84062 −0.920309 0.391192i \(-0.872063\pi\)
−0.920309 + 0.391192i \(0.872063\pi\)
\(464\) −118421. −0.0255349
\(465\) 1.33309e6 0.285909
\(466\) 3.97744e6 0.848474
\(467\) 1.82738e6 0.387737 0.193868 0.981028i \(-0.437896\pi\)
0.193868 + 0.981028i \(0.437896\pi\)
\(468\) −570994. −0.120508
\(469\) 208632. 0.0437975
\(470\) −379197. −0.0791808
\(471\) −5.58197e6 −1.15940
\(472\) 1.61995e6 0.334693
\(473\) 1.19479e7 2.45549
\(474\) 4.65535e6 0.951714
\(475\) 413548. 0.0840992
\(476\) −948712. −0.191919
\(477\) 1.51144e6 0.304155
\(478\) −2.72715e6 −0.545933
\(479\) 4.68744e6 0.933463 0.466731 0.884399i \(-0.345432\pi\)
0.466731 + 0.884399i \(0.345432\pi\)
\(480\) 1.95775e6 0.387841
\(481\) −314960. −0.0620715
\(482\) −4.95405e6 −0.971277
\(483\) −2.27470e6 −0.443666
\(484\) −6.19426e6 −1.20192
\(485\) 2.60939e6 0.503714
\(486\) −1.52085e6 −0.292076
\(487\) 4.59651e6 0.878225 0.439112 0.898432i \(-0.355293\pi\)
0.439112 + 0.898432i \(0.355293\pi\)
\(488\) −8.90247e6 −1.69224
\(489\) 1.06743e6 0.201867
\(490\) 211956. 0.0398800
\(491\) 6.62099e6 1.23942 0.619711 0.784830i \(-0.287250\pi\)
0.619711 + 0.784830i \(0.287250\pi\)
\(492\) −4.32167e6 −0.804895
\(493\) −6.69327e6 −1.24028
\(494\) 1.17326e6 0.216310
\(495\) −1.00650e6 −0.184630
\(496\) 68800.9 0.0125571
\(497\) −930554. −0.168986
\(498\) −3.39514e6 −0.613458
\(499\) −4.52632e6 −0.813756 −0.406878 0.913482i \(-0.633383\pi\)
−0.406878 + 0.913482i \(0.633383\pi\)
\(500\) 305174. 0.0545912
\(501\) −8.12495e6 −1.44619
\(502\) 5.78829e6 1.02516
\(503\) 3.83316e6 0.675518 0.337759 0.941233i \(-0.390331\pi\)
0.337759 + 0.941233i \(0.390331\pi\)
\(504\) 519099. 0.0910278
\(505\) 1.14288e6 0.199421
\(506\) 8.33909e6 1.44791
\(507\) −1.61950e6 −0.279808
\(508\) 908397. 0.156177
\(509\) −3.25460e6 −0.556806 −0.278403 0.960464i \(-0.589805\pi\)
−0.278403 + 0.960464i \(0.589805\pi\)
\(510\) −1.18958e6 −0.202519
\(511\) −496492. −0.0841124
\(512\) 203165. 0.0342511
\(513\) −2.70930e6 −0.454531
\(514\) 789788. 0.131857
\(515\) −2.23195e6 −0.370823
\(516\) 4.58713e6 0.758432
\(517\) 2.97040e6 0.488752
\(518\) 108525. 0.0177708
\(519\) −813891. −0.132632
\(520\) 2.28432e6 0.370466
\(521\) −1.07842e6 −0.174057 −0.0870287 0.996206i \(-0.527737\pi\)
−0.0870287 + 0.996206i \(0.527737\pi\)
\(522\) 1.38807e6 0.222964
\(523\) 408626. 0.0653238 0.0326619 0.999466i \(-0.489602\pi\)
0.0326619 + 0.999466i \(0.489602\pi\)
\(524\) −7.45641e6 −1.18632
\(525\) −416297. −0.0659182
\(526\) −1.05764e6 −0.166676
\(527\) 3.88869e6 0.609925
\(528\) 164867. 0.0257365
\(529\) 5.22642e6 0.812016
\(530\) −2.29178e6 −0.354391
\(531\) −518310. −0.0797724
\(532\) 633242. 0.0970042
\(533\) −8.17392e6 −1.24627
\(534\) −203574. −0.0308936
\(535\) 2.65825e6 0.401524
\(536\) −774763. −0.116481
\(537\) 8.38456e6 1.25471
\(538\) −474374. −0.0706586
\(539\) −1.66034e6 −0.246164
\(540\) −1.99930e6 −0.295049
\(541\) 1.13918e7 1.67340 0.836701 0.547659i \(-0.184481\pi\)
0.836701 + 0.547659i \(0.184481\pi\)
\(542\) −6.82970e6 −0.998627
\(543\) −506856. −0.0737709
\(544\) 5.71084e6 0.827376
\(545\) 1.24541e6 0.179607
\(546\) −1.18106e6 −0.169547
\(547\) −3.44866e6 −0.492813 −0.246406 0.969167i \(-0.579250\pi\)
−0.246406 + 0.969167i \(0.579250\pi\)
\(548\) 37002.6 0.00526357
\(549\) 2.84838e6 0.403337
\(550\) 1.52615e6 0.215125
\(551\) 4.46759e6 0.626894
\(552\) 8.44718e6 1.17995
\(553\) 4.75234e6 0.660837
\(554\) −627124. −0.0868119
\(555\) −213152. −0.0293736
\(556\) −3.94134e6 −0.540700
\(557\) −4.69772e6 −0.641577 −0.320789 0.947151i \(-0.603948\pi\)
−0.320789 + 0.947151i \(0.603948\pi\)
\(558\) −806449. −0.109646
\(559\) 8.67599e6 1.17433
\(560\) −21485.1 −0.00289513
\(561\) 9.31844e6 1.25007
\(562\) −6.53722e6 −0.873077
\(563\) −561642. −0.0746772 −0.0373386 0.999303i \(-0.511888\pi\)
−0.0373386 + 0.999303i \(0.511888\pi\)
\(564\) 1.14042e6 0.150962
\(565\) 929018. 0.122434
\(566\) −8.46945e6 −1.11126
\(567\) 2.03409e6 0.265713
\(568\) 3.45564e6 0.449426
\(569\) 5.19019e6 0.672051 0.336026 0.941853i \(-0.390917\pi\)
0.336026 + 0.941853i \(0.390917\pi\)
\(570\) 794012. 0.102362
\(571\) −5.20274e6 −0.667793 −0.333897 0.942610i \(-0.608364\pi\)
−0.333897 + 0.942610i \(0.608364\pi\)
\(572\) −6.78211e6 −0.866712
\(573\) 4.44178e6 0.565159
\(574\) 2.81648e6 0.356802
\(575\) 2.13442e6 0.269222
\(576\) −1.21701e6 −0.152840
\(577\) 8.70662e6 1.08870 0.544352 0.838857i \(-0.316775\pi\)
0.544352 + 0.838857i \(0.316775\pi\)
\(578\) 1.54365e6 0.192189
\(579\) −3.61224e6 −0.447796
\(580\) 3.29681e6 0.406934
\(581\) −3.46588e6 −0.425964
\(582\) 5.01003e6 0.613102
\(583\) 1.79524e7 2.18752
\(584\) 1.84374e6 0.223701
\(585\) −730877. −0.0882988
\(586\) −8.82505e6 −1.06163
\(587\) −4.35717e6 −0.521926 −0.260963 0.965349i \(-0.584040\pi\)
−0.260963 + 0.965349i \(0.584040\pi\)
\(588\) −637452. −0.0760333
\(589\) −2.59560e6 −0.308283
\(590\) 785908. 0.0929484
\(591\) −7.03954e6 −0.829040
\(592\) −11000.8 −0.00129009
\(593\) −3.22991e6 −0.377184 −0.188592 0.982055i \(-0.560392\pi\)
−0.188592 + 0.982055i \(0.560392\pi\)
\(594\) −9.99837e6 −1.16269
\(595\) −1.21436e6 −0.140622
\(596\) 9.10614e6 1.05007
\(597\) −2.02115e6 −0.232094
\(598\) 6.05547e6 0.692460
\(599\) −7.23988e6 −0.824450 −0.412225 0.911082i \(-0.635248\pi\)
−0.412225 + 0.911082i \(0.635248\pi\)
\(600\) 1.54593e6 0.175312
\(601\) −1.06837e7 −1.20652 −0.603262 0.797543i \(-0.706133\pi\)
−0.603262 + 0.797543i \(0.706133\pi\)
\(602\) −2.98948e6 −0.336205
\(603\) 247889. 0.0277628
\(604\) 2.38693e6 0.266225
\(605\) −7.92871e6 −0.880671
\(606\) 2.19432e6 0.242727
\(607\) 2.51528e6 0.277086 0.138543 0.990356i \(-0.455758\pi\)
0.138543 + 0.990356i \(0.455758\pi\)
\(608\) −3.81185e6 −0.418193
\(609\) −4.49729e6 −0.491369
\(610\) −4.31898e6 −0.469955
\(611\) 2.15697e6 0.233744
\(612\) −1.12722e6 −0.121655
\(613\) 213999. 0.0230017 0.0115009 0.999934i \(-0.496339\pi\)
0.0115009 + 0.999934i \(0.496339\pi\)
\(614\) −1.08475e7 −1.16120
\(615\) −5.53178e6 −0.589762
\(616\) 6.16572e6 0.654684
\(617\) −127951. −0.0135310 −0.00676552 0.999977i \(-0.502154\pi\)
−0.00676552 + 0.999977i \(0.502154\pi\)
\(618\) −4.28535e6 −0.451352
\(619\) −1.23980e7 −1.30054 −0.650272 0.759701i \(-0.725345\pi\)
−0.650272 + 0.759701i \(0.725345\pi\)
\(620\) −1.91540e6 −0.200115
\(621\) −1.39834e7 −1.45507
\(622\) −2.33490e6 −0.241987
\(623\) −207815. −0.0214514
\(624\) 119719. 0.0123084
\(625\) 390625. 0.0400000
\(626\) 1.16313e7 1.18629
\(627\) −6.21983e6 −0.631843
\(628\) 8.02023e6 0.811499
\(629\) −621774. −0.0626622
\(630\) 251838. 0.0252795
\(631\) −5.87683e6 −0.587584 −0.293792 0.955869i \(-0.594917\pi\)
−0.293792 + 0.955869i \(0.594917\pi\)
\(632\) −1.76480e7 −1.75753
\(633\) 101177. 0.0100363
\(634\) −2.25895e6 −0.223194
\(635\) 1.16275e6 0.114434
\(636\) 6.89246e6 0.675665
\(637\) −1.20566e6 −0.117727
\(638\) 1.64871e7 1.60359
\(639\) −1.10565e6 −0.107118
\(640\) −2.76337e6 −0.266679
\(641\) −2.60122e6 −0.250053 −0.125026 0.992153i \(-0.539902\pi\)
−0.125026 + 0.992153i \(0.539902\pi\)
\(642\) 5.10384e6 0.488719
\(643\) 1.31345e7 1.25282 0.626408 0.779495i \(-0.284524\pi\)
0.626408 + 0.779495i \(0.284524\pi\)
\(644\) 3.26832e6 0.310534
\(645\) 5.87156e6 0.555718
\(646\) 2.31617e6 0.218368
\(647\) 5.54662e6 0.520916 0.260458 0.965485i \(-0.416126\pi\)
0.260458 + 0.965485i \(0.416126\pi\)
\(648\) −7.55366e6 −0.706675
\(649\) −6.15634e6 −0.573734
\(650\) 1.10822e6 0.102883
\(651\) 2.61286e6 0.241637
\(652\) −1.53369e6 −0.141292
\(653\) −1.54993e7 −1.42242 −0.711212 0.702978i \(-0.751853\pi\)
−0.711212 + 0.702978i \(0.751853\pi\)
\(654\) 2.39119e6 0.218610
\(655\) −9.54427e6 −0.869239
\(656\) −285495. −0.0259024
\(657\) −589912. −0.0533180
\(658\) −743225. −0.0669200
\(659\) −4.30145e6 −0.385835 −0.192917 0.981215i \(-0.561795\pi\)
−0.192917 + 0.981215i \(0.561795\pi\)
\(660\) −4.58986e6 −0.410147
\(661\) 861980. 0.0767350 0.0383675 0.999264i \(-0.487784\pi\)
0.0383675 + 0.999264i \(0.487784\pi\)
\(662\) 4.02111e6 0.356616
\(663\) 6.76662e6 0.597844
\(664\) 1.28706e7 1.13287
\(665\) 810554. 0.0710768
\(666\) 128945. 0.0112647
\(667\) 2.30583e7 2.00684
\(668\) 1.16740e7 1.01223
\(669\) 1.61974e6 0.139920
\(670\) −375871. −0.0323484
\(671\) 3.38323e7 2.90085
\(672\) 3.83719e6 0.327786
\(673\) −953818. −0.0811760 −0.0405880 0.999176i \(-0.512923\pi\)
−0.0405880 + 0.999176i \(0.512923\pi\)
\(674\) −2.42056e6 −0.205242
\(675\) −2.55912e6 −0.216188
\(676\) 2.32691e6 0.195845
\(677\) −1.48606e7 −1.24613 −0.623065 0.782170i \(-0.714113\pi\)
−0.623065 + 0.782170i \(0.714113\pi\)
\(678\) 1.78371e6 0.149022
\(679\) 5.11440e6 0.425716
\(680\) 4.50956e6 0.373992
\(681\) −5.28569e6 −0.436751
\(682\) −9.57878e6 −0.788586
\(683\) −1.57605e7 −1.29276 −0.646381 0.763015i \(-0.723718\pi\)
−0.646381 + 0.763015i \(0.723718\pi\)
\(684\) 752392. 0.0614900
\(685\) 47363.6 0.00385672
\(686\) 415434. 0.0337048
\(687\) 9.95882e6 0.805037
\(688\) 303032. 0.0244071
\(689\) 1.30362e7 1.04618
\(690\) 4.09809e6 0.327687
\(691\) 1.59740e7 1.27267 0.636337 0.771411i \(-0.280449\pi\)
0.636337 + 0.771411i \(0.280449\pi\)
\(692\) 1.16941e6 0.0928326
\(693\) −1.97275e6 −0.156041
\(694\) 4.41925e6 0.348297
\(695\) −5.04494e6 −0.396182
\(696\) 1.67008e7 1.30682
\(697\) −1.61365e7 −1.25813
\(698\) 1.11798e7 0.868549
\(699\) −1.53115e7 −1.18529
\(700\) 598141. 0.0461380
\(701\) −1.85736e7 −1.42758 −0.713790 0.700360i \(-0.753023\pi\)
−0.713790 + 0.700360i \(0.753023\pi\)
\(702\) −7.26036e6 −0.556052
\(703\) 415019. 0.0316723
\(704\) −1.44553e7 −1.09925
\(705\) 1.45975e6 0.110613
\(706\) 8.59365e6 0.648883
\(707\) 2.24004e6 0.168541
\(708\) −2.36360e6 −0.177211
\(709\) −1.71029e7 −1.27778 −0.638888 0.769300i \(-0.720605\pi\)
−0.638888 + 0.769300i \(0.720605\pi\)
\(710\) 1.67648e6 0.124811
\(711\) 5.64654e6 0.418898
\(712\) 771727. 0.0570511
\(713\) −1.33965e7 −0.986890
\(714\) −2.33157e6 −0.171160
\(715\) −8.68116e6 −0.635057
\(716\) −1.20470e7 −0.878208
\(717\) 1.04984e7 0.762651
\(718\) 7.52204e6 0.544534
\(719\) −1.40945e7 −1.01678 −0.508390 0.861127i \(-0.669759\pi\)
−0.508390 + 0.861127i \(0.669759\pi\)
\(720\) −25527.8 −0.00183519
\(721\) −4.37463e6 −0.313403
\(722\) 7.19744e6 0.513848
\(723\) 1.90711e7 1.35684
\(724\) 728256. 0.0516343
\(725\) 4.21995e6 0.298169
\(726\) −1.52231e7 −1.07192
\(727\) 2.24196e7 1.57323 0.786616 0.617443i \(-0.211831\pi\)
0.786616 + 0.617443i \(0.211831\pi\)
\(728\) 4.47726e6 0.313101
\(729\) 1.59421e7 1.11103
\(730\) 894478. 0.0621245
\(731\) 1.71276e7 1.18551
\(732\) 1.29892e7 0.895993
\(733\) −8.02464e6 −0.551653 −0.275826 0.961208i \(-0.588951\pi\)
−0.275826 + 0.961208i \(0.588951\pi\)
\(734\) 1.09810e7 0.752316
\(735\) −815943. −0.0557111
\(736\) −1.96739e7 −1.33874
\(737\) 2.94435e6 0.199674
\(738\) 3.34643e6 0.226173
\(739\) 1.15678e7 0.779181 0.389591 0.920988i \(-0.372616\pi\)
0.389591 + 0.920988i \(0.372616\pi\)
\(740\) 306259. 0.0205594
\(741\) −4.51655e6 −0.302177
\(742\) −4.49189e6 −0.299515
\(743\) 3.79387e6 0.252122 0.126061 0.992023i \(-0.459767\pi\)
0.126061 + 0.992023i \(0.459767\pi\)
\(744\) −9.70293e6 −0.642644
\(745\) 1.16559e7 0.769407
\(746\) 1.11297e7 0.732214
\(747\) −4.11801e6 −0.270014
\(748\) −1.33888e7 −0.874961
\(749\) 5.21017e6 0.339349
\(750\) 750000. 0.0486864
\(751\) −9.07884e6 −0.587395 −0.293698 0.955898i \(-0.594886\pi\)
−0.293698 + 0.955898i \(0.594886\pi\)
\(752\) 75337.8 0.00485812
\(753\) −2.22825e7 −1.43211
\(754\) 1.19722e7 0.766912
\(755\) 3.05530e6 0.195068
\(756\) −3.91863e6 −0.249362
\(757\) −2.33210e7 −1.47913 −0.739565 0.673085i \(-0.764969\pi\)
−0.739565 + 0.673085i \(0.764969\pi\)
\(758\) −1.20888e6 −0.0764208
\(759\) −3.21020e7 −2.02269
\(760\) −3.01002e6 −0.189032
\(761\) 1.44859e7 0.906745 0.453373 0.891321i \(-0.350221\pi\)
0.453373 + 0.891321i \(0.350221\pi\)
\(762\) 2.23249e6 0.139284
\(763\) 2.44101e6 0.151795
\(764\) −6.38200e6 −0.395570
\(765\) −1.44285e6 −0.0891391
\(766\) −1.30435e7 −0.803200
\(767\) −4.47045e6 −0.274387
\(768\) −1.43985e7 −0.880876
\(769\) 1.59424e7 0.972162 0.486081 0.873914i \(-0.338426\pi\)
0.486081 + 0.873914i \(0.338426\pi\)
\(770\) 2.99126e6 0.181814
\(771\) −3.04036e6 −0.184200
\(772\) 5.19011e6 0.313425
\(773\) −3.36066e6 −0.202290 −0.101145 0.994872i \(-0.532251\pi\)
−0.101145 + 0.994872i \(0.532251\pi\)
\(774\) −3.55198e6 −0.213117
\(775\) −2.45173e6 −0.146628
\(776\) −1.89925e7 −1.13221
\(777\) −417778. −0.0248252
\(778\) −7.24988e6 −0.429419
\(779\) 1.07707e7 0.635916
\(780\) −3.33295e6 −0.196152
\(781\) −1.31326e7 −0.770411
\(782\) 1.19543e7 0.699050
\(783\) −2.76464e7 −1.61151
\(784\) −42110.9 −0.00244683
\(785\) 1.02660e7 0.594601
\(786\) −1.83250e7 −1.05800
\(787\) −3.28918e6 −0.189300 −0.0946500 0.995511i \(-0.530173\pi\)
−0.0946500 + 0.995511i \(0.530173\pi\)
\(788\) 1.01145e7 0.580268
\(789\) 4.07147e6 0.232841
\(790\) −8.56179e6 −0.488087
\(791\) 1.82088e6 0.103476
\(792\) 7.32586e6 0.414998
\(793\) 2.45675e7 1.38732
\(794\) 8.05475e6 0.453420
\(795\) 8.82240e6 0.495073
\(796\) 2.90402e6 0.162449
\(797\) −2.51939e6 −0.140492 −0.0702458 0.997530i \(-0.522378\pi\)
−0.0702458 + 0.997530i \(0.522378\pi\)
\(798\) 1.55626e6 0.0865120
\(799\) 4.25816e6 0.235969
\(800\) −3.60055e6 −0.198904
\(801\) −246917. −0.0135978
\(802\) −1.17262e7 −0.643758
\(803\) −7.00682e6 −0.383470
\(804\) 1.13042e6 0.0616738
\(805\) 4.18347e6 0.227534
\(806\) −6.95567e6 −0.377139
\(807\) 1.82614e6 0.0987077
\(808\) −8.31845e6 −0.448244
\(809\) 8.93808e6 0.480146 0.240073 0.970755i \(-0.422829\pi\)
0.240073 + 0.970755i \(0.422829\pi\)
\(810\) −3.66461e6 −0.196252
\(811\) 3.01341e7 1.60881 0.804406 0.594080i \(-0.202484\pi\)
0.804406 + 0.594080i \(0.202484\pi\)
\(812\) 6.46176e6 0.343922
\(813\) 2.62915e7 1.39505
\(814\) 1.53158e6 0.0810174
\(815\) −1.96313e6 −0.103528
\(816\) 236342. 0.0124255
\(817\) −1.14323e7 −0.599207
\(818\) −1.82184e7 −0.951980
\(819\) −1.43252e6 −0.0746261
\(820\) 7.94812e6 0.412791
\(821\) −3.25440e7 −1.68505 −0.842524 0.538658i \(-0.818931\pi\)
−0.842524 + 0.538658i \(0.818931\pi\)
\(822\) 90938.1 0.00469425
\(823\) −499640. −0.0257133 −0.0128566 0.999917i \(-0.504093\pi\)
−0.0128566 + 0.999917i \(0.504093\pi\)
\(824\) 1.62453e7 0.833509
\(825\) −5.87506e6 −0.300523
\(826\) 1.54038e6 0.0785557
\(827\) −4.64084e6 −0.235957 −0.117978 0.993016i \(-0.537641\pi\)
−0.117978 + 0.993016i \(0.537641\pi\)
\(828\) 3.88328e6 0.196844
\(829\) −3.08794e7 −1.56057 −0.780283 0.625427i \(-0.784925\pi\)
−0.780283 + 0.625427i \(0.784925\pi\)
\(830\) 6.24411e6 0.314612
\(831\) 2.41417e6 0.121273
\(832\) −1.04968e7 −0.525712
\(833\) −2.38014e6 −0.118848
\(834\) −9.68629e6 −0.482217
\(835\) 1.49428e7 0.741681
\(836\) 8.93671e6 0.442244
\(837\) 1.60621e7 0.792482
\(838\) 1.71326e7 0.842777
\(839\) 1.93485e7 0.948948 0.474474 0.880269i \(-0.342638\pi\)
0.474474 + 0.880269i \(0.342638\pi\)
\(840\) 3.03003e6 0.148166
\(841\) 2.50772e7 1.22261
\(842\) 2.17082e7 1.05522
\(843\) 2.51656e7 1.21966
\(844\) −145372. −0.00702465
\(845\) 2.97846e6 0.143500
\(846\) −883071. −0.0424199
\(847\) −1.55403e7 −0.744303
\(848\) 455325. 0.0217436
\(849\) 3.26039e7 1.55239
\(850\) 2.18778e6 0.103862
\(851\) 2.14201e6 0.101391
\(852\) −5.04197e6 −0.237959
\(853\) −3.05998e7 −1.43994 −0.719972 0.694003i \(-0.755845\pi\)
−0.719972 + 0.694003i \(0.755845\pi\)
\(854\) −8.46520e6 −0.397185
\(855\) 963068. 0.0450549
\(856\) −1.93481e7 −0.902515
\(857\) −1.33039e7 −0.618766 −0.309383 0.950937i \(-0.600123\pi\)
−0.309383 + 0.950937i \(0.600123\pi\)
\(858\) −1.66678e7 −0.772967
\(859\) −1.27708e7 −0.590520 −0.295260 0.955417i \(-0.595406\pi\)
−0.295260 + 0.955417i \(0.595406\pi\)
\(860\) −8.43633e6 −0.388962
\(861\) −1.08423e7 −0.498440
\(862\) −1.25500e7 −0.575276
\(863\) 1.37987e7 0.630684 0.315342 0.948978i \(-0.397881\pi\)
0.315342 + 0.948978i \(0.397881\pi\)
\(864\) 2.35885e7 1.07502
\(865\) 1.49685e6 0.0680203
\(866\) −9.98074e6 −0.452239
\(867\) −5.94241e6 −0.268482
\(868\) −3.75418e6 −0.169128
\(869\) 6.70680e7 3.01277
\(870\) 8.10230e6 0.362919
\(871\) 2.13806e6 0.0954934
\(872\) −9.06477e6 −0.403706
\(873\) 6.07673e6 0.269857
\(874\) −7.97922e6 −0.353331
\(875\) 765625. 0.0338062
\(876\) −2.69012e6 −0.118443
\(877\) 6.68992e6 0.293712 0.146856 0.989158i \(-0.453085\pi\)
0.146856 + 0.989158i \(0.453085\pi\)
\(878\) −1.63989e7 −0.717924
\(879\) 3.39728e7 1.48306
\(880\) −303212. −0.0131990
\(881\) −7.25051e6 −0.314723 −0.157362 0.987541i \(-0.550299\pi\)
−0.157362 + 0.987541i \(0.550299\pi\)
\(882\) 493602. 0.0213651
\(883\) −2.55944e7 −1.10470 −0.552348 0.833613i \(-0.686268\pi\)
−0.552348 + 0.833613i \(0.686268\pi\)
\(884\) −9.72236e6 −0.418447
\(885\) −3.02542e6 −0.129846
\(886\) 2.17353e7 0.930210
\(887\) −2.33204e7 −0.995240 −0.497620 0.867395i \(-0.665793\pi\)
−0.497620 + 0.867395i \(0.665793\pi\)
\(888\) 1.55143e6 0.0660237
\(889\) 2.27900e6 0.0967141
\(890\) 374398. 0.0158438
\(891\) 2.87064e7 1.21139
\(892\) −2.32726e6 −0.0979340
\(893\) −2.84222e6 −0.119269
\(894\) 2.23794e7 0.936493
\(895\) −1.54203e7 −0.643480
\(896\) −5.41620e6 −0.225385
\(897\) −2.33110e7 −0.967343
\(898\) −1.29605e7 −0.536328
\(899\) −2.64862e7 −1.09300
\(900\) 710687. 0.0292464
\(901\) 2.57354e7 1.05613
\(902\) 3.97480e7 1.62667
\(903\) 1.15083e7 0.469667
\(904\) −6.76188e6 −0.275199
\(905\) 932174. 0.0378334
\(906\) 5.86617e6 0.237429
\(907\) 3.71721e7 1.50037 0.750187 0.661226i \(-0.229964\pi\)
0.750187 + 0.661226i \(0.229964\pi\)
\(908\) 7.59453e6 0.305694
\(909\) 2.66152e6 0.106837
\(910\) 2.17212e6 0.0869520
\(911\) −2.89923e7 −1.15741 −0.578704 0.815537i \(-0.696441\pi\)
−0.578704 + 0.815537i \(0.696441\pi\)
\(912\) −157752. −0.00628042
\(913\) −4.89127e7 −1.94198
\(914\) 3.05911e6 0.121124
\(915\) 1.66263e7 0.656512
\(916\) −1.43089e7 −0.563468
\(917\) −1.87068e7 −0.734641
\(918\) −1.43330e7 −0.561344
\(919\) −1.25020e6 −0.0488306 −0.0244153 0.999702i \(-0.507772\pi\)
−0.0244153 + 0.999702i \(0.507772\pi\)
\(920\) −1.55355e7 −0.605138
\(921\) 4.17582e7 1.62216
\(922\) 6.65872e6 0.257966
\(923\) −9.53627e6 −0.368447
\(924\) −8.99613e6 −0.346638
\(925\) 392014. 0.0150642
\(926\) 2.99799e7 1.14895
\(927\) −5.19776e6 −0.198663
\(928\) −3.88970e7 −1.48268
\(929\) 6.87875e6 0.261499 0.130749 0.991415i \(-0.458262\pi\)
0.130749 + 0.991415i \(0.458262\pi\)
\(930\) −4.70732e6 −0.178470
\(931\) 1.58869e6 0.0600709
\(932\) 2.19997e7 0.829616
\(933\) 8.98840e6 0.338048
\(934\) −6.45272e6 −0.242034
\(935\) −1.71378e7 −0.641101
\(936\) 5.31971e6 0.198471
\(937\) 2.18600e7 0.813396 0.406698 0.913563i \(-0.366680\pi\)
0.406698 + 0.913563i \(0.366680\pi\)
\(938\) −736708. −0.0273393
\(939\) −4.47756e7 −1.65721
\(940\) −2.09739e6 −0.0774210
\(941\) 2.26450e6 0.0833679 0.0416840 0.999131i \(-0.486728\pi\)
0.0416840 + 0.999131i \(0.486728\pi\)
\(942\) 1.97106e7 0.723725
\(943\) 5.55901e7 2.03572
\(944\) −156142. −0.00570283
\(945\) −5.01588e6 −0.182712
\(946\) −4.21895e7 −1.53277
\(947\) −2.45846e7 −0.890815 −0.445408 0.895328i \(-0.646941\pi\)
−0.445408 + 0.895328i \(0.646941\pi\)
\(948\) 2.57493e7 0.930562
\(949\) −5.08803e6 −0.183394
\(950\) −1.46029e6 −0.0524965
\(951\) 8.69601e6 0.311795
\(952\) 8.83874e6 0.316081
\(953\) 4.59681e7 1.63955 0.819774 0.572687i \(-0.194099\pi\)
0.819774 + 0.572687i \(0.194099\pi\)
\(954\) −5.33708e6 −0.189860
\(955\) −8.16901e6 −0.289842
\(956\) −1.50842e7 −0.533800
\(957\) −6.34686e7 −2.24016
\(958\) −1.65520e7 −0.582687
\(959\) 92832.6 0.00325952
\(960\) −7.10379e6 −0.248778
\(961\) −1.32411e7 −0.462503
\(962\) 1.11216e6 0.0387463
\(963\) 6.19051e6 0.215110
\(964\) −2.74015e7 −0.949690
\(965\) 6.64338e6 0.229652
\(966\) 8.03226e6 0.276946
\(967\) −5.30202e7 −1.82337 −0.911686 0.410888i \(-0.865219\pi\)
−0.911686 + 0.410888i \(0.865219\pi\)
\(968\) 5.77093e7 1.97951
\(969\) −8.91630e6 −0.305053
\(970\) −9.21409e6 −0.314429
\(971\) −5.43523e7 −1.84999 −0.924996 0.379976i \(-0.875932\pi\)
−0.924996 + 0.379976i \(0.875932\pi\)
\(972\) −8.41202e6 −0.285585
\(973\) −9.88809e6 −0.334835
\(974\) −1.62309e7 −0.548207
\(975\) −4.26620e6 −0.143724
\(976\) 858083. 0.0288340
\(977\) 1.08929e7 0.365097 0.182549 0.983197i \(-0.441565\pi\)
0.182549 + 0.983197i \(0.441565\pi\)
\(978\) −3.76922e6 −0.126010
\(979\) −2.93282e6 −0.0977976
\(980\) 1.17236e6 0.0389937
\(981\) 2.90031e6 0.0962215
\(982\) −2.33796e7 −0.773674
\(983\) −3.42136e7 −1.12932 −0.564658 0.825325i \(-0.690992\pi\)
−0.564658 + 0.825325i \(0.690992\pi\)
\(984\) 4.02632e7 1.32562
\(985\) 1.29466e7 0.425173
\(986\) 2.36348e7 0.774211
\(987\) 2.86111e6 0.0934850
\(988\) 6.48943e6 0.211502
\(989\) −5.90047e7 −1.91821
\(990\) 3.55409e6 0.115250
\(991\) 4.22117e6 0.136537 0.0682683 0.997667i \(-0.478253\pi\)
0.0682683 + 0.997667i \(0.478253\pi\)
\(992\) 2.25986e7 0.729125
\(993\) −1.54796e7 −0.498181
\(994\) 3.28591e6 0.105485
\(995\) 3.71716e6 0.119029
\(996\) −1.87790e7 −0.599824
\(997\) 4.30323e7 1.37106 0.685530 0.728044i \(-0.259571\pi\)
0.685530 + 0.728044i \(0.259571\pi\)
\(998\) 1.59830e7 0.507964
\(999\) −2.56822e6 −0.0814177
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 35.6.a.b.1.1 2
3.2 odd 2 315.6.a.c.1.2 2
4.3 odd 2 560.6.a.l.1.1 2
5.2 odd 4 175.6.b.d.99.2 4
5.3 odd 4 175.6.b.d.99.3 4
5.4 even 2 175.6.a.d.1.2 2
7.6 odd 2 245.6.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.6.a.b.1.1 2 1.1 even 1 trivial
175.6.a.d.1.2 2 5.4 even 2
175.6.b.d.99.2 4 5.2 odd 4
175.6.b.d.99.3 4 5.3 odd 4
245.6.a.c.1.1 2 7.6 odd 2
315.6.a.c.1.2 2 3.2 odd 2
560.6.a.l.1.1 2 4.3 odd 2