Properties

Label 35.6.a.b.1.2
Level $35$
Weight $6$
Character 35.1
Self dual yes
Analytic conductor $5.613$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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 1.2
Root \(4.53113\) of defining polynomial
Character \(\chi\) \(=\) 35.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.53113 q^{2} -10.5934 q^{3} -11.4689 q^{4} -25.0000 q^{5} -48.0000 q^{6} -49.0000 q^{7} -196.963 q^{8} -130.780 q^{9} +O(q^{10})\) \(q+4.53113 q^{2} -10.5934 q^{3} -11.4689 q^{4} -25.0000 q^{5} -48.0000 q^{6} -49.0000 q^{7} -196.963 q^{8} -130.780 q^{9} -113.278 q^{10} +90.5195 q^{11} +121.494 q^{12} -74.8502 q^{13} -222.025 q^{14} +264.835 q^{15} -525.461 q^{16} +1032.31 q^{17} -592.582 q^{18} -31.6771 q^{19} +286.722 q^{20} +519.076 q^{21} +410.156 q^{22} -3857.08 q^{23} +2086.51 q^{24} +625.000 q^{25} -339.156 q^{26} +3959.60 q^{27} +561.975 q^{28} -866.917 q^{29} +1200.00 q^{30} +3526.76 q^{31} +3921.89 q^{32} -958.908 q^{33} +4677.54 q^{34} +1225.00 q^{35} +1499.90 q^{36} -9531.22 q^{37} -143.533 q^{38} +792.917 q^{39} +4924.08 q^{40} -14503.9 q^{41} +2352.00 q^{42} -9844.30 q^{43} -1038.16 q^{44} +3269.50 q^{45} -17476.9 q^{46} -16993.5 q^{47} +5566.41 q^{48} +2401.00 q^{49} +2831.96 q^{50} -10935.7 q^{51} +858.447 q^{52} -29621.1 q^{53} +17941.4 q^{54} -2262.99 q^{55} +9651.19 q^{56} +335.568 q^{57} -3928.11 q^{58} +50697.4 q^{59} -3037.35 q^{60} -2921.38 q^{61} +15980.2 q^{62} +6408.23 q^{63} +34585.3 q^{64} +1871.25 q^{65} -4344.94 q^{66} -41086.2 q^{67} -11839.5 q^{68} +40859.5 q^{69} +5550.63 q^{70} +61753.1 q^{71} +25758.9 q^{72} -23664.5 q^{73} -43187.2 q^{74} -6620.87 q^{75} +363.300 q^{76} -4435.46 q^{77} +3592.81 q^{78} +45191.5 q^{79} +13136.5 q^{80} -10166.0 q^{81} -65718.8 q^{82} +39095.9 q^{83} -5953.22 q^{84} -25807.8 q^{85} -44605.8 q^{86} +9183.58 q^{87} -17829.0 q^{88} -41891.1 q^{89} +14814.5 q^{90} +3667.66 q^{91} +44236.3 q^{92} -37360.4 q^{93} -76999.9 q^{94} +791.927 q^{95} -41546.1 q^{96} +8036.53 q^{97} +10879.2 q^{98} -11838.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} - 31 q^{4} - 50 q^{5} - 96 q^{6} - 98 q^{7} - 15 q^{8} - 189 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.53113 0.800998 0.400499 0.916297i \(-0.368837\pi\)
0.400499 + 0.916297i \(0.368837\pi\)
\(3\) −10.5934 −0.679566 −0.339783 0.940504i \(-0.610354\pi\)
−0.339783 + 0.940504i \(0.610354\pi\)
\(4\) −11.4689 −0.358402
\(5\) −25.0000 −0.447214
\(6\) −48.0000 −0.544331
\(7\) −49.0000 −0.377964
\(8\) −196.963 −1.08808
\(9\) −130.780 −0.538190
\(10\) −113.278 −0.358217
\(11\) 90.5195 0.225559 0.112780 0.993620i \(-0.464025\pi\)
0.112780 + 0.993620i \(0.464025\pi\)
\(12\) 121.494 0.243558
\(13\) −74.8502 −0.122838 −0.0614192 0.998112i \(-0.519563\pi\)
−0.0614192 + 0.998112i \(0.519563\pi\)
\(14\) −222.025 −0.302749
\(15\) 264.835 0.303911
\(16\) −525.461 −0.513146
\(17\) 1032.31 0.866342 0.433171 0.901312i \(-0.357395\pi\)
0.433171 + 0.901312i \(0.357395\pi\)
\(18\) −592.582 −0.431089
\(19\) −31.6771 −0.0201308 −0.0100654 0.999949i \(-0.503204\pi\)
−0.0100654 + 0.999949i \(0.503204\pi\)
\(20\) 286.722 0.160282
\(21\) 519.076 0.256852
\(22\) 410.156 0.180672
\(23\) −3857.08 −1.52033 −0.760167 0.649728i \(-0.774883\pi\)
−0.760167 + 0.649728i \(0.774883\pi\)
\(24\) 2086.51 0.739421
\(25\) 625.000 0.200000
\(26\) −339.156 −0.0983934
\(27\) 3959.60 1.04530
\(28\) 561.975 0.135463
\(29\) −866.917 −0.191418 −0.0957089 0.995409i \(-0.530512\pi\)
−0.0957089 + 0.995409i \(0.530512\pi\)
\(30\) 1200.00 0.243432
\(31\) 3526.76 0.659131 0.329566 0.944133i \(-0.393098\pi\)
0.329566 + 0.944133i \(0.393098\pi\)
\(32\) 3921.89 0.677049
\(33\) −958.908 −0.153282
\(34\) 4677.54 0.693938
\(35\) 1225.00 0.169031
\(36\) 1499.90 0.192888
\(37\) −9531.22 −1.14458 −0.572288 0.820053i \(-0.693944\pi\)
−0.572288 + 0.820053i \(0.693944\pi\)
\(38\) −143.533 −0.0161247
\(39\) 792.917 0.0834769
\(40\) 4924.08 0.486603
\(41\) −14503.9 −1.34748 −0.673742 0.738967i \(-0.735314\pi\)
−0.673742 + 0.738967i \(0.735314\pi\)
\(42\) 2352.00 0.205738
\(43\) −9844.30 −0.811921 −0.405960 0.913891i \(-0.633063\pi\)
−0.405960 + 0.913891i \(0.633063\pi\)
\(44\) −1038.16 −0.0808409
\(45\) 3269.50 0.240686
\(46\) −17476.9 −1.21778
\(47\) −16993.5 −1.12212 −0.561059 0.827775i \(-0.689606\pi\)
−0.561059 + 0.827775i \(0.689606\pi\)
\(48\) 5566.41 0.348716
\(49\) 2401.00 0.142857
\(50\) 2831.96 0.160200
\(51\) −10935.7 −0.588736
\(52\) 858.447 0.0440256
\(53\) −29621.1 −1.44848 −0.724239 0.689549i \(-0.757809\pi\)
−0.724239 + 0.689549i \(0.757809\pi\)
\(54\) 17941.4 0.837285
\(55\) −2262.99 −0.100873
\(56\) 9651.19 0.411255
\(57\) 335.568 0.0136802
\(58\) −3928.11 −0.153325
\(59\) 50697.4 1.89607 0.948037 0.318159i \(-0.103065\pi\)
0.948037 + 0.318159i \(0.103065\pi\)
\(60\) −3037.35 −0.108922
\(61\) −2921.38 −0.100522 −0.0502612 0.998736i \(-0.516005\pi\)
−0.0502612 + 0.998736i \(0.516005\pi\)
\(62\) 15980.2 0.527963
\(63\) 6408.23 0.203417
\(64\) 34585.3 1.05546
\(65\) 1871.25 0.0549350
\(66\) −4344.94 −0.122779
\(67\) −41086.2 −1.11817 −0.559086 0.829109i \(-0.688848\pi\)
−0.559086 + 0.829109i \(0.688848\pi\)
\(68\) −11839.5 −0.310499
\(69\) 40859.5 1.03317
\(70\) 5550.63 0.135393
\(71\) 61753.1 1.45383 0.726914 0.686729i \(-0.240954\pi\)
0.726914 + 0.686729i \(0.240954\pi\)
\(72\) 25758.9 0.585592
\(73\) −23664.5 −0.519745 −0.259872 0.965643i \(-0.583680\pi\)
−0.259872 + 0.965643i \(0.583680\pi\)
\(74\) −43187.2 −0.916802
\(75\) −6620.87 −0.135913
\(76\) 363.300 0.00721493
\(77\) −4435.46 −0.0852533
\(78\) 3592.81 0.0668648
\(79\) 45191.5 0.814683 0.407341 0.913276i \(-0.366456\pi\)
0.407341 + 0.913276i \(0.366456\pi\)
\(80\) 13136.5 0.229486
\(81\) −10166.0 −0.172162
\(82\) −65718.8 −1.07933
\(83\) 39095.9 0.622925 0.311462 0.950259i \(-0.399181\pi\)
0.311462 + 0.950259i \(0.399181\pi\)
\(84\) −5953.22 −0.0920563
\(85\) −25807.8 −0.387440
\(86\) −44605.8 −0.650347
\(87\) 9183.58 0.130081
\(88\) −17829.0 −0.245426
\(89\) −41891.1 −0.560592 −0.280296 0.959914i \(-0.590433\pi\)
−0.280296 + 0.959914i \(0.590433\pi\)
\(90\) 14814.5 0.192789
\(91\) 3667.66 0.0464286
\(92\) 44236.3 0.544891
\(93\) −37360.4 −0.447923
\(94\) −76999.9 −0.898815
\(95\) 791.927 0.00900277
\(96\) −41546.1 −0.460099
\(97\) 8036.53 0.0867240 0.0433620 0.999059i \(-0.486193\pi\)
0.0433620 + 0.999059i \(0.486193\pi\)
\(98\) 10879.2 0.114428
\(99\) −11838.2 −0.121394
\(100\) −7168.04 −0.0716804
\(101\) 124979. 1.21908 0.609542 0.792754i \(-0.291353\pi\)
0.609542 + 0.792754i \(0.291353\pi\)
\(102\) −49551.0 −0.471577
\(103\) −76635.1 −0.711762 −0.355881 0.934531i \(-0.615819\pi\)
−0.355881 + 0.934531i \(0.615819\pi\)
\(104\) 14742.7 0.133658
\(105\) −12976.9 −0.114868
\(106\) −134217. −1.16023
\(107\) 135296. 1.14242 0.571209 0.820804i \(-0.306474\pi\)
0.571209 + 0.820804i \(0.306474\pi\)
\(108\) −45412.1 −0.374638
\(109\) −199508. −1.60840 −0.804202 0.594356i \(-0.797407\pi\)
−0.804202 + 0.594356i \(0.797407\pi\)
\(110\) −10253.9 −0.0807992
\(111\) 100968. 0.777814
\(112\) 25747.6 0.193951
\(113\) 122569. 0.902991 0.451496 0.892273i \(-0.350891\pi\)
0.451496 + 0.892273i \(0.350891\pi\)
\(114\) 1520.50 0.0109578
\(115\) 96427.0 0.679914
\(116\) 9942.56 0.0686046
\(117\) 9788.92 0.0661104
\(118\) 229716. 1.51875
\(119\) −50583.4 −0.327446
\(120\) −52162.6 −0.330679
\(121\) −152857. −0.949123
\(122\) −13237.1 −0.0805183
\(123\) 153645. 0.915704
\(124\) −40448.0 −0.236234
\(125\) −15625.0 −0.0894427
\(126\) 29036.5 0.162936
\(127\) 40046.2 0.220319 0.110160 0.993914i \(-0.464864\pi\)
0.110160 + 0.993914i \(0.464864\pi\)
\(128\) 31210.2 0.168373
\(129\) 104284. 0.551754
\(130\) 8478.89 0.0440029
\(131\) 211963. 1.07915 0.539576 0.841937i \(-0.318585\pi\)
0.539576 + 0.841937i \(0.318585\pi\)
\(132\) 10997.6 0.0549367
\(133\) 1552.18 0.00760873
\(134\) −186167. −0.895654
\(135\) −98989.9 −0.467473
\(136\) −203328. −0.942647
\(137\) 40980.5 0.186542 0.0932709 0.995641i \(-0.470268\pi\)
0.0932709 + 0.995641i \(0.470268\pi\)
\(138\) 185140. 0.827565
\(139\) −418948. −1.83917 −0.919587 0.392886i \(-0.871477\pi\)
−0.919587 + 0.392886i \(0.871477\pi\)
\(140\) −14049.4 −0.0605810
\(141\) 180019. 0.762554
\(142\) 279811. 1.16451
\(143\) −6775.40 −0.0277073
\(144\) 68719.9 0.276170
\(145\) 21672.9 0.0856047
\(146\) −107227. −0.416314
\(147\) −25434.7 −0.0970809
\(148\) 109312. 0.410218
\(149\) −64962.7 −0.239717 −0.119858 0.992791i \(-0.538244\pi\)
−0.119858 + 0.992791i \(0.538244\pi\)
\(150\) −30000.0 −0.108866
\(151\) 379801. 1.35554 0.677772 0.735272i \(-0.262946\pi\)
0.677772 + 0.735272i \(0.262946\pi\)
\(152\) 6239.22 0.0219039
\(153\) −135006. −0.466256
\(154\) −20097.6 −0.0682878
\(155\) −88169.1 −0.294773
\(156\) −9093.86 −0.0299183
\(157\) −281546. −0.911590 −0.455795 0.890085i \(-0.650645\pi\)
−0.455795 + 0.890085i \(0.650645\pi\)
\(158\) 204768. 0.652559
\(159\) 313788. 0.984337
\(160\) −98047.1 −0.302785
\(161\) 188997. 0.574632
\(162\) −46063.3 −0.137901
\(163\) −382587. −1.12788 −0.563938 0.825817i \(-0.690714\pi\)
−0.563938 + 0.825817i \(0.690714\pi\)
\(164\) 166343. 0.482941
\(165\) 23972.7 0.0685499
\(166\) 177148. 0.498961
\(167\) −388635. −1.07833 −0.539164 0.842201i \(-0.681260\pi\)
−0.539164 + 0.842201i \(0.681260\pi\)
\(168\) −102239. −0.279475
\(169\) −365690. −0.984911
\(170\) −116939. −0.310339
\(171\) 4142.73 0.0108342
\(172\) 112903. 0.290994
\(173\) −389693. −0.989936 −0.494968 0.868911i \(-0.664820\pi\)
−0.494968 + 0.868911i \(0.664820\pi\)
\(174\) 41612.0 0.104195
\(175\) −30625.0 −0.0755929
\(176\) −47564.5 −0.115745
\(177\) −537057. −1.28851
\(178\) −189814. −0.449033
\(179\) −149812. −0.349473 −0.174737 0.984615i \(-0.555907\pi\)
−0.174737 + 0.984615i \(0.555907\pi\)
\(180\) −37497.5 −0.0862623
\(181\) −821019. −1.86276 −0.931380 0.364049i \(-0.881394\pi\)
−0.931380 + 0.364049i \(0.881394\pi\)
\(182\) 16618.6 0.0371892
\(183\) 30947.3 0.0683117
\(184\) 759702. 1.65424
\(185\) 238281. 0.511870
\(186\) −169285. −0.358786
\(187\) 93444.5 0.195411
\(188\) 194897. 0.402170
\(189\) −194020. −0.395087
\(190\) 3588.32 0.00721120
\(191\) 286619. 0.568487 0.284244 0.958752i \(-0.408258\pi\)
0.284244 + 0.958752i \(0.408258\pi\)
\(192\) −366376. −0.717255
\(193\) 993573. 1.92002 0.960012 0.279959i \(-0.0903209\pi\)
0.960012 + 0.279959i \(0.0903209\pi\)
\(194\) 36414.6 0.0694657
\(195\) −19822.9 −0.0373320
\(196\) −27536.8 −0.0512003
\(197\) −38209.0 −0.0701456 −0.0350728 0.999385i \(-0.511166\pi\)
−0.0350728 + 0.999385i \(0.511166\pi\)
\(198\) −53640.2 −0.0972361
\(199\) 730487. 1.30761 0.653807 0.756661i \(-0.273171\pi\)
0.653807 + 0.756661i \(0.273171\pi\)
\(200\) −123102. −0.217615
\(201\) 435242. 0.759872
\(202\) 566296. 0.976484
\(203\) 42478.9 0.0723491
\(204\) 125420. 0.211004
\(205\) 362596. 0.602613
\(206\) −347244. −0.570120
\(207\) 504429. 0.818228
\(208\) 39330.9 0.0630340
\(209\) −2867.39 −0.00454069
\(210\) −58800.0 −0.0920087
\(211\) 194286. 0.300424 0.150212 0.988654i \(-0.452004\pi\)
0.150212 + 0.988654i \(0.452004\pi\)
\(212\) 339721. 0.519138
\(213\) −654175. −0.987972
\(214\) 613043. 0.915075
\(215\) 246107. 0.363102
\(216\) −779894. −1.13737
\(217\) −172811. −0.249128
\(218\) −903999. −1.28833
\(219\) 250687. 0.353201
\(220\) 25953.9 0.0361531
\(221\) −77268.8 −0.106420
\(222\) 457499. 0.623028
\(223\) 1.13569e6 1.52931 0.764656 0.644438i \(-0.222909\pi\)
0.764656 + 0.644438i \(0.222909\pi\)
\(224\) −192172. −0.255900
\(225\) −81737.6 −0.107638
\(226\) 555375. 0.723294
\(227\) −143806. −0.185231 −0.0926155 0.995702i \(-0.529523\pi\)
−0.0926155 + 0.995702i \(0.529523\pi\)
\(228\) −3848.58 −0.00490302
\(229\) −3832.50 −0.00482940 −0.00241470 0.999997i \(-0.500769\pi\)
−0.00241470 + 0.999997i \(0.500769\pi\)
\(230\) 436923. 0.544610
\(231\) 46986.5 0.0579353
\(232\) 170751. 0.208277
\(233\) −1.35599e6 −1.63631 −0.818157 0.574995i \(-0.805004\pi\)
−0.818157 + 0.574995i \(0.805004\pi\)
\(234\) 44354.8 0.0529543
\(235\) 424838. 0.501827
\(236\) −581442. −0.679557
\(237\) −478731. −0.553631
\(238\) −229200. −0.262284
\(239\) −478372. −0.541716 −0.270858 0.962619i \(-0.587307\pi\)
−0.270858 + 0.962619i \(0.587307\pi\)
\(240\) −139160. −0.155951
\(241\) −1.31082e6 −1.45379 −0.726894 0.686750i \(-0.759037\pi\)
−0.726894 + 0.686750i \(0.759037\pi\)
\(242\) −692616. −0.760246
\(243\) −854490. −0.928307
\(244\) 33504.9 0.0360275
\(245\) −60025.0 −0.0638877
\(246\) 696185. 0.733477
\(247\) 2371.04 0.00247284
\(248\) −694642. −0.717186
\(249\) −414157. −0.423318
\(250\) −70798.9 −0.0716434
\(251\) 340113. 0.340753 0.170376 0.985379i \(-0.445502\pi\)
0.170376 + 0.985379i \(0.445502\pi\)
\(252\) −73495.1 −0.0729050
\(253\) −349141. −0.342925
\(254\) 181454. 0.176475
\(255\) 273392. 0.263291
\(256\) −965313. −0.920594
\(257\) −58839.6 −0.0555696 −0.0277848 0.999614i \(-0.508845\pi\)
−0.0277848 + 0.999614i \(0.508845\pi\)
\(258\) 472526. 0.441954
\(259\) 467030. 0.432609
\(260\) −21461.2 −0.0196888
\(261\) 113376. 0.103019
\(262\) 960433. 0.864398
\(263\) −1380.79 −0.00123095 −0.000615473 1.00000i \(-0.500196\pi\)
−0.000615473 1.00000i \(0.500196\pi\)
\(264\) 188869. 0.166783
\(265\) 740528. 0.647779
\(266\) 7033.11 0.00609458
\(267\) 443769. 0.380959
\(268\) 471212. 0.400756
\(269\) 886839. 0.747247 0.373624 0.927580i \(-0.378115\pi\)
0.373624 + 0.927580i \(0.378115\pi\)
\(270\) −448536. −0.374445
\(271\) −376955. −0.311793 −0.155897 0.987773i \(-0.549827\pi\)
−0.155897 + 0.987773i \(0.549827\pi\)
\(272\) −542441. −0.444559
\(273\) −38852.9 −0.0315513
\(274\) 185688. 0.149420
\(275\) 56574.7 0.0451118
\(276\) −468613. −0.370289
\(277\) −880363. −0.689386 −0.344693 0.938716i \(-0.612017\pi\)
−0.344693 + 0.938716i \(0.612017\pi\)
\(278\) −1.89831e6 −1.47317
\(279\) −461231. −0.354738
\(280\) −241280. −0.183919
\(281\) 2.08492e6 1.57515 0.787577 0.616217i \(-0.211335\pi\)
0.787577 + 0.616217i \(0.211335\pi\)
\(282\) 815689. 0.610804
\(283\) 692401. 0.513915 0.256958 0.966423i \(-0.417280\pi\)
0.256958 + 0.966423i \(0.417280\pi\)
\(284\) −708238. −0.521055
\(285\) −8389.19 −0.00611798
\(286\) −30700.2 −0.0221935
\(287\) 710689. 0.509301
\(288\) −512905. −0.364381
\(289\) −354186. −0.249452
\(290\) 98202.8 0.0685692
\(291\) −85134.1 −0.0589347
\(292\) 271405. 0.186278
\(293\) 2.27624e6 1.54899 0.774495 0.632580i \(-0.218004\pi\)
0.774495 + 0.632580i \(0.218004\pi\)
\(294\) −115248. −0.0777616
\(295\) −1.26743e6 −0.847950
\(296\) 1.87730e6 1.24539
\(297\) 358421. 0.235777
\(298\) −294354. −0.192013
\(299\) 288703. 0.186755
\(300\) 75933.9 0.0487116
\(301\) 482371. 0.306877
\(302\) 1.72093e6 1.08579
\(303\) −1.32395e6 −0.828448
\(304\) 16645.1 0.0103300
\(305\) 73034.5 0.0449550
\(306\) −611730. −0.373470
\(307\) −2.65187e6 −1.60586 −0.802928 0.596076i \(-0.796726\pi\)
−0.802928 + 0.596076i \(0.796726\pi\)
\(308\) 50869.7 0.0305550
\(309\) 811825. 0.483689
\(310\) −399505. −0.236112
\(311\) 2.57626e6 1.51039 0.755195 0.655501i \(-0.227542\pi\)
0.755195 + 0.655501i \(0.227542\pi\)
\(312\) −156175. −0.0908293
\(313\) 2.51549e6 1.45131 0.725657 0.688057i \(-0.241536\pi\)
0.725657 + 0.688057i \(0.241536\pi\)
\(314\) −1.27572e6 −0.730182
\(315\) −160206. −0.0909707
\(316\) −518295. −0.291984
\(317\) −2.30397e6 −1.28774 −0.643870 0.765135i \(-0.722672\pi\)
−0.643870 + 0.765135i \(0.722672\pi\)
\(318\) 1.42181e6 0.788452
\(319\) −78472.9 −0.0431760
\(320\) −864633. −0.472016
\(321\) −1.43324e6 −0.776349
\(322\) 856369. 0.460279
\(323\) −32700.7 −0.0174402
\(324\) 116592. 0.0617031
\(325\) −46781.4 −0.0245677
\(326\) −1.73355e6 −0.903427
\(327\) 2.11347e6 1.09302
\(328\) 2.85672e6 1.46617
\(329\) 832683. 0.424121
\(330\) 108623. 0.0549084
\(331\) 697305. 0.349827 0.174913 0.984584i \(-0.444035\pi\)
0.174913 + 0.984584i \(0.444035\pi\)
\(332\) −448385. −0.223258
\(333\) 1.24649e6 0.615999
\(334\) −1.76096e6 −0.863739
\(335\) 1.02715e6 0.500062
\(336\) −272754. −0.131802
\(337\) −1.14848e6 −0.550868 −0.275434 0.961320i \(-0.588822\pi\)
−0.275434 + 0.961320i \(0.588822\pi\)
\(338\) −1.65699e6 −0.788911
\(339\) −1.29842e6 −0.613642
\(340\) 295987. 0.138859
\(341\) 319241. 0.148673
\(342\) 18771.3 0.00867817
\(343\) −117649. −0.0539949
\(344\) 1.93896e6 0.883433
\(345\) −1.02149e6 −0.462046
\(346\) −1.76575e6 −0.792937
\(347\) −2.43716e6 −1.08658 −0.543289 0.839546i \(-0.682821\pi\)
−0.543289 + 0.839546i \(0.682821\pi\)
\(348\) −105325. −0.0466213
\(349\) 896801. 0.394124 0.197062 0.980391i \(-0.436860\pi\)
0.197062 + 0.980391i \(0.436860\pi\)
\(350\) −138766. −0.0605498
\(351\) −296377. −0.128403
\(352\) 355007. 0.152715
\(353\) 4.33422e6 1.85129 0.925644 0.378396i \(-0.123524\pi\)
0.925644 + 0.378396i \(0.123524\pi\)
\(354\) −2.43347e6 −1.03209
\(355\) −1.54383e6 −0.650172
\(356\) 480444. 0.200917
\(357\) 535849. 0.222521
\(358\) −678817. −0.279927
\(359\) −3.59167e6 −1.47082 −0.735411 0.677621i \(-0.763011\pi\)
−0.735411 + 0.677621i \(0.763011\pi\)
\(360\) −643972. −0.261885
\(361\) −2.47510e6 −0.999595
\(362\) −3.72014e6 −1.49207
\(363\) 1.61928e6 0.644992
\(364\) −42063.9 −0.0166401
\(365\) 591612. 0.232437
\(366\) 140226. 0.0547175
\(367\) 361791. 0.140215 0.0701073 0.997539i \(-0.477666\pi\)
0.0701073 + 0.997539i \(0.477666\pi\)
\(368\) 2.02674e6 0.780152
\(369\) 1.89682e6 0.725202
\(370\) 1.07968e6 0.410006
\(371\) 1.45144e6 0.547473
\(372\) 428481. 0.160537
\(373\) −3.08325e6 −1.14746 −0.573729 0.819045i \(-0.694504\pi\)
−0.573729 + 0.819045i \(0.694504\pi\)
\(374\) 423409. 0.156524
\(375\) 165522. 0.0607822
\(376\) 3.34710e6 1.22095
\(377\) 64888.9 0.0235135
\(378\) −879131. −0.316464
\(379\) −5.04227e6 −1.80313 −0.901567 0.432639i \(-0.857582\pi\)
−0.901567 + 0.432639i \(0.857582\pi\)
\(380\) −9082.51 −0.00322661
\(381\) −424225. −0.149721
\(382\) 1.29871e6 0.455357
\(383\) −3.11928e6 −1.08657 −0.543285 0.839548i \(-0.682820\pi\)
−0.543285 + 0.839548i \(0.682820\pi\)
\(384\) −330622. −0.114420
\(385\) 110886. 0.0381265
\(386\) 4.50201e6 1.53794
\(387\) 1.28744e6 0.436968
\(388\) −92169.9 −0.0310821
\(389\) −1.64560e6 −0.551378 −0.275689 0.961247i \(-0.588906\pi\)
−0.275689 + 0.961247i \(0.588906\pi\)
\(390\) −89820.2 −0.0299028
\(391\) −3.98171e6 −1.31713
\(392\) −472908. −0.155440
\(393\) −2.24541e6 −0.733355
\(394\) −173130. −0.0561865
\(395\) −1.12979e6 −0.364337
\(396\) 135770. 0.0435078
\(397\) 3.72727e6 1.18690 0.593451 0.804870i \(-0.297765\pi\)
0.593451 + 0.804870i \(0.297765\pi\)
\(398\) 3.30993e6 1.04740
\(399\) −16442.8 −0.00517063
\(400\) −328413. −0.102629
\(401\) 4.84820e6 1.50564 0.752818 0.658229i \(-0.228694\pi\)
0.752818 + 0.658229i \(0.228694\pi\)
\(402\) 1.97214e6 0.608656
\(403\) −263979. −0.0809667
\(404\) −1.43337e6 −0.436923
\(405\) 254149. 0.0769930
\(406\) 192477. 0.0579515
\(407\) −862761. −0.258169
\(408\) 2.15393e6 0.640591
\(409\) −3.03126e6 −0.896014 −0.448007 0.894030i \(-0.647866\pi\)
−0.448007 + 0.894030i \(0.647866\pi\)
\(410\) 1.64297e6 0.482692
\(411\) −434123. −0.126768
\(412\) 878918. 0.255097
\(413\) −2.48417e6 −0.716649
\(414\) 2.28563e6 0.655399
\(415\) −977396. −0.278580
\(416\) −293554. −0.0831676
\(417\) 4.43808e6 1.24984
\(418\) −12992.5 −0.00363708
\(419\) 1.66905e6 0.464444 0.232222 0.972663i \(-0.425400\pi\)
0.232222 + 0.972663i \(0.425400\pi\)
\(420\) 148830. 0.0411688
\(421\) −1.76031e6 −0.484043 −0.242022 0.970271i \(-0.577810\pi\)
−0.242022 + 0.970271i \(0.577810\pi\)
\(422\) 880335. 0.240639
\(423\) 2.22242e6 0.603913
\(424\) 5.83427e6 1.57606
\(425\) 645196. 0.173268
\(426\) −2.96415e6 −0.791364
\(427\) 143148. 0.0379939
\(428\) −1.55169e6 −0.409445
\(429\) 71774.4 0.0188290
\(430\) 1.11514e6 0.290844
\(431\) −648832. −0.168244 −0.0841219 0.996455i \(-0.526809\pi\)
−0.0841219 + 0.996455i \(0.526809\pi\)
\(432\) −2.08061e6 −0.536392
\(433\) −360993. −0.0925293 −0.0462646 0.998929i \(-0.514732\pi\)
−0.0462646 + 0.998929i \(0.514732\pi\)
\(434\) −783031. −0.199551
\(435\) −229590. −0.0581740
\(436\) 2.28814e6 0.576455
\(437\) 122181. 0.0306055
\(438\) 1.13590e6 0.282913
\(439\) 1.07021e6 0.265038 0.132519 0.991180i \(-0.457693\pi\)
0.132519 + 0.991180i \(0.457693\pi\)
\(440\) 445725. 0.109758
\(441\) −314003. −0.0768843
\(442\) −350115. −0.0852423
\(443\) −797145. −0.192987 −0.0964935 0.995334i \(-0.530763\pi\)
−0.0964935 + 0.995334i \(0.530763\pi\)
\(444\) −1.15799e6 −0.278770
\(445\) 1.04728e6 0.250704
\(446\) 5.14594e6 1.22498
\(447\) 688175. 0.162903
\(448\) −1.69468e6 −0.398927
\(449\) 6.40103e6 1.49842 0.749211 0.662331i \(-0.230433\pi\)
0.749211 + 0.662331i \(0.230433\pi\)
\(450\) −370364. −0.0862178
\(451\) −1.31288e6 −0.303937
\(452\) −1.40572e6 −0.323634
\(453\) −4.02338e6 −0.921181
\(454\) −651605. −0.148370
\(455\) −91691.5 −0.0207635
\(456\) −66094.4 −0.0148851
\(457\) 5.91843e6 1.32561 0.662806 0.748791i \(-0.269366\pi\)
0.662806 + 0.748791i \(0.269366\pi\)
\(458\) −17365.5 −0.00386834
\(459\) 4.08755e6 0.905588
\(460\) −1.10591e6 −0.243683
\(461\) −8.84337e6 −1.93805 −0.969026 0.246960i \(-0.920569\pi\)
−0.969026 + 0.246960i \(0.920569\pi\)
\(462\) 212902. 0.0464060
\(463\) −6.76858e6 −1.46739 −0.733695 0.679479i \(-0.762206\pi\)
−0.733695 + 0.679479i \(0.762206\pi\)
\(464\) 455531. 0.0982252
\(465\) 934009. 0.200317
\(466\) −6.14416e6 −1.31068
\(467\) −1.04740e6 −0.222239 −0.111120 0.993807i \(-0.535444\pi\)
−0.111120 + 0.993807i \(0.535444\pi\)
\(468\) −112268. −0.0236941
\(469\) 2.01322e6 0.422630
\(470\) 1.92500e6 0.401962
\(471\) 2.98252e6 0.619486
\(472\) −9.98551e6 −2.06308
\(473\) −891101. −0.183136
\(474\) −2.16919e6 −0.443457
\(475\) −19798.2 −0.00402616
\(476\) 580134. 0.117358
\(477\) 3.87386e6 0.779556
\(478\) −2.16757e6 −0.433913
\(479\) −9.37725e6 −1.86740 −0.933699 0.358060i \(-0.883438\pi\)
−0.933699 + 0.358060i \(0.883438\pi\)
\(480\) 1.03865e6 0.205763
\(481\) 713414. 0.140598
\(482\) −5.93950e6 −1.16448
\(483\) −2.00212e6 −0.390500
\(484\) 1.75310e6 0.340168
\(485\) −200913. −0.0387841
\(486\) −3.87181e6 −0.743572
\(487\) −2.88960e6 −0.552098 −0.276049 0.961144i \(-0.589025\pi\)
−0.276049 + 0.961144i \(0.589025\pi\)
\(488\) 575404. 0.109376
\(489\) 4.05290e6 0.766467
\(490\) −271981. −0.0511739
\(491\) −1.04845e6 −0.196265 −0.0981324 0.995173i \(-0.531287\pi\)
−0.0981324 + 0.995173i \(0.531287\pi\)
\(492\) −1.76213e6 −0.328191
\(493\) −894930. −0.165833
\(494\) 10743.5 0.00198074
\(495\) 295954. 0.0542889
\(496\) −1.85318e6 −0.338230
\(497\) −3.02590e6 −0.549495
\(498\) −1.87660e6 −0.339077
\(499\) 6.38524e6 1.14796 0.573979 0.818870i \(-0.305399\pi\)
0.573979 + 0.818870i \(0.305399\pi\)
\(500\) 179201. 0.0320565
\(501\) 4.11696e6 0.732795
\(502\) 1.54110e6 0.272942
\(503\) 1.35497e6 0.238787 0.119394 0.992847i \(-0.461905\pi\)
0.119394 + 0.992847i \(0.461905\pi\)
\(504\) −1.26218e6 −0.221333
\(505\) −3.12448e6 −0.545191
\(506\) −1.58200e6 −0.274682
\(507\) 3.87390e6 0.669312
\(508\) −459285. −0.0789628
\(509\) 6.43410e6 1.10076 0.550381 0.834914i \(-0.314482\pi\)
0.550381 + 0.834914i \(0.314482\pi\)
\(510\) 1.23878e6 0.210896
\(511\) 1.15956e6 0.196445
\(512\) −5.37268e6 −0.905767
\(513\) −125429. −0.0210428
\(514\) −266610. −0.0445111
\(515\) 1.91588e6 0.318310
\(516\) −1.19603e6 −0.197750
\(517\) −1.53825e6 −0.253104
\(518\) 2.11617e6 0.346519
\(519\) 4.12817e6 0.672727
\(520\) −368568. −0.0597736
\(521\) 6.16552e6 0.995120 0.497560 0.867430i \(-0.334229\pi\)
0.497560 + 0.867430i \(0.334229\pi\)
\(522\) 513719. 0.0825181
\(523\) 7.12400e6 1.13886 0.569429 0.822040i \(-0.307164\pi\)
0.569429 + 0.822040i \(0.307164\pi\)
\(524\) −2.43098e6 −0.386770
\(525\) 324422. 0.0513704
\(526\) −6256.55 −0.000985985 0
\(527\) 3.64072e6 0.571033
\(528\) 503869. 0.0786562
\(529\) 8.44071e6 1.31141
\(530\) 3.35543e6 0.518870
\(531\) −6.63021e6 −1.02045
\(532\) −17801.7 −0.00272699
\(533\) 1.08562e6 0.165523
\(534\) 2.01077e6 0.305148
\(535\) −3.38240e6 −0.510905
\(536\) 8.09246e6 1.21666
\(537\) 1.58702e6 0.237490
\(538\) 4.01838e6 0.598544
\(539\) 217337. 0.0322227
\(540\) 1.13530e6 0.167543
\(541\) −9.51129e6 −1.39716 −0.698580 0.715532i \(-0.746184\pi\)
−0.698580 + 0.715532i \(0.746184\pi\)
\(542\) −1.70803e6 −0.249746
\(543\) 8.69737e6 1.26587
\(544\) 4.04862e6 0.586556
\(545\) 4.98771e6 0.719300
\(546\) −176048. −0.0252725
\(547\) 1.48406e6 0.212071 0.106036 0.994362i \(-0.466184\pi\)
0.106036 + 0.994362i \(0.466184\pi\)
\(548\) −470001. −0.0668570
\(549\) 382058. 0.0541002
\(550\) 256347. 0.0361345
\(551\) 27461.4 0.00385340
\(552\) −8.04782e6 −1.12417
\(553\) −2.21438e6 −0.307921
\(554\) −3.98904e6 −0.552197
\(555\) −2.52420e6 −0.347849
\(556\) 4.80486e6 0.659164
\(557\) −7.72192e6 −1.05460 −0.527299 0.849680i \(-0.676795\pi\)
−0.527299 + 0.849680i \(0.676795\pi\)
\(558\) −2.08990e6 −0.284144
\(559\) 736847. 0.0997351
\(560\) −643690. −0.0867374
\(561\) −989894. −0.132795
\(562\) 9.44703e6 1.26169
\(563\) 3.41747e6 0.454395 0.227198 0.973849i \(-0.427044\pi\)
0.227198 + 0.973849i \(0.427044\pi\)
\(564\) −2.06461e6 −0.273301
\(565\) −3.06422e6 −0.403830
\(566\) 3.13736e6 0.411645
\(567\) 498133. 0.0650710
\(568\) −1.21631e7 −1.58188
\(569\) −2.42117e6 −0.313505 −0.156752 0.987638i \(-0.550102\pi\)
−0.156752 + 0.987638i \(0.550102\pi\)
\(570\) −38012.5 −0.00490049
\(571\) 2.31941e6 0.297705 0.148853 0.988859i \(-0.452442\pi\)
0.148853 + 0.988859i \(0.452442\pi\)
\(572\) 77706.2 0.00993037
\(573\) −3.03626e6 −0.386325
\(574\) 3.22022e6 0.407949
\(575\) −2.41067e6 −0.304067
\(576\) −4.52307e6 −0.568038
\(577\) 2.56645e6 0.320917 0.160459 0.987043i \(-0.448703\pi\)
0.160459 + 0.987043i \(0.448703\pi\)
\(578\) −1.60486e6 −0.199811
\(579\) −1.05253e7 −1.30478
\(580\) −248564. −0.0306809
\(581\) −1.91570e6 −0.235443
\(582\) −385753. −0.0472066
\(583\) −2.68129e6 −0.326718
\(584\) 4.66103e6 0.565522
\(585\) −244723. −0.0295655
\(586\) 1.03139e7 1.24074
\(587\) 2.09851e6 0.251371 0.125685 0.992070i \(-0.459887\pi\)
0.125685 + 0.992070i \(0.459887\pi\)
\(588\) 291708. 0.0347940
\(589\) −111718. −0.0132688
\(590\) −5.74291e6 −0.679207
\(591\) 404763. 0.0476686
\(592\) 5.00829e6 0.587334
\(593\) −5.11831e6 −0.597709 −0.298854 0.954299i \(-0.596604\pi\)
−0.298854 + 0.954299i \(0.596604\pi\)
\(594\) 1.62405e6 0.188857
\(595\) 1.26458e6 0.146438
\(596\) 745048. 0.0859150
\(597\) −7.73833e6 −0.888610
\(598\) 1.30815e6 0.149591
\(599\) −2.61775e6 −0.298099 −0.149049 0.988830i \(-0.547621\pi\)
−0.149049 + 0.988830i \(0.547621\pi\)
\(600\) 1.30407e6 0.147884
\(601\) 9.49925e6 1.07276 0.536381 0.843976i \(-0.319791\pi\)
0.536381 + 0.843976i \(0.319791\pi\)
\(602\) 2.18568e6 0.245808
\(603\) 5.37326e6 0.601789
\(604\) −4.35589e6 −0.485830
\(605\) 3.82143e6 0.424461
\(606\) −5.99899e6 −0.663586
\(607\) 6.07366e6 0.669081 0.334541 0.942381i \(-0.391419\pi\)
0.334541 + 0.942381i \(0.391419\pi\)
\(608\) −124234. −0.0136295
\(609\) −449996. −0.0491660
\(610\) 330929. 0.0360089
\(611\) 1.27197e6 0.137839
\(612\) 1.54837e6 0.167107
\(613\) −9.20907e6 −0.989839 −0.494920 0.868939i \(-0.664802\pi\)
−0.494920 + 0.868939i \(0.664802\pi\)
\(614\) −1.20160e7 −1.28629
\(615\) −3.84112e6 −0.409515
\(616\) 873621. 0.0927622
\(617\) 1.37224e7 1.45117 0.725585 0.688133i \(-0.241569\pi\)
0.725585 + 0.688133i \(0.241569\pi\)
\(618\) 3.67848e6 0.387434
\(619\) −6.80356e6 −0.713690 −0.356845 0.934164i \(-0.616148\pi\)
−0.356845 + 0.934164i \(0.616148\pi\)
\(620\) 1.01120e6 0.105647
\(621\) −1.52725e7 −1.58921
\(622\) 1.16734e7 1.20982
\(623\) 2.05266e6 0.211884
\(624\) −416647. −0.0428358
\(625\) 390625. 0.0400000
\(626\) 1.13980e7 1.16250
\(627\) 30375.4 0.00308570
\(628\) 3.22901e6 0.326716
\(629\) −9.83921e6 −0.991593
\(630\) −725913. −0.0728674
\(631\) −2.80897e6 −0.280850 −0.140425 0.990091i \(-0.544847\pi\)
−0.140425 + 0.990091i \(0.544847\pi\)
\(632\) −8.90105e6 −0.886438
\(633\) −2.05815e6 −0.204158
\(634\) −1.04396e7 −1.03148
\(635\) −1.00115e6 −0.0985297
\(636\) −3.59880e6 −0.352788
\(637\) −179715. −0.0175484
\(638\) −355571. −0.0345839
\(639\) −8.07608e6 −0.782435
\(640\) −780256. −0.0752986
\(641\) −3.83494e6 −0.368649 −0.184325 0.982865i \(-0.559010\pi\)
−0.184325 + 0.982865i \(0.559010\pi\)
\(642\) −6.49420e6 −0.621854
\(643\) 1.59562e7 1.52195 0.760977 0.648779i \(-0.224720\pi\)
0.760977 + 0.648779i \(0.224720\pi\)
\(644\) −2.16758e6 −0.205949
\(645\) −2.60711e6 −0.246752
\(646\) −148171. −0.0139695
\(647\) −7.48025e6 −0.702514 −0.351257 0.936279i \(-0.614246\pi\)
−0.351257 + 0.936279i \(0.614246\pi\)
\(648\) 2.00232e6 0.187325
\(649\) 4.58910e6 0.427677
\(650\) −211972. −0.0196787
\(651\) 1.83066e6 0.169299
\(652\) 4.38785e6 0.404234
\(653\) 2.62102e6 0.240540 0.120270 0.992741i \(-0.461624\pi\)
0.120270 + 0.992741i \(0.461624\pi\)
\(654\) 9.57641e6 0.875504
\(655\) −5.29908e6 −0.482611
\(656\) 7.62121e6 0.691456
\(657\) 3.09485e6 0.279721
\(658\) 3.77299e6 0.339720
\(659\) −8.01276e6 −0.718734 −0.359367 0.933196i \(-0.617007\pi\)
−0.359367 + 0.933196i \(0.617007\pi\)
\(660\) −274940. −0.0245685
\(661\) −7.21439e6 −0.642238 −0.321119 0.947039i \(-0.604059\pi\)
−0.321119 + 0.947039i \(0.604059\pi\)
\(662\) 3.15958e6 0.280210
\(663\) 818539. 0.0723195
\(664\) −7.70044e6 −0.677790
\(665\) −38804.4 −0.00340273
\(666\) 5.64803e6 0.493414
\(667\) 3.34377e6 0.291019
\(668\) 4.45721e6 0.386475
\(669\) −1.20308e7 −1.03927
\(670\) 4.65417e6 0.400549
\(671\) −264442. −0.0226738
\(672\) 2.03576e6 0.173901
\(673\) 1.43323e7 1.21977 0.609887 0.792488i \(-0.291215\pi\)
0.609887 + 0.792488i \(0.291215\pi\)
\(674\) −5.20390e6 −0.441244
\(675\) 2.47475e6 0.209060
\(676\) 4.19406e6 0.352994
\(677\) −4.94781e6 −0.414898 −0.207449 0.978246i \(-0.566516\pi\)
−0.207449 + 0.978246i \(0.566516\pi\)
\(678\) −5.88330e6 −0.491526
\(679\) −393790. −0.0327786
\(680\) 5.08319e6 0.421565
\(681\) 1.52340e6 0.125877
\(682\) 1.44652e6 0.119087
\(683\) 7.77331e6 0.637608 0.318804 0.947821i \(-0.396719\pi\)
0.318804 + 0.947821i \(0.396719\pi\)
\(684\) −47512.5 −0.00388300
\(685\) −1.02451e6 −0.0834241
\(686\) −533083. −0.0432498
\(687\) 40599.1 0.00328190
\(688\) 5.17280e6 0.416634
\(689\) 2.21715e6 0.177929
\(690\) −4.62849e6 −0.370098
\(691\) −7.69151e6 −0.612797 −0.306398 0.951903i \(-0.599124\pi\)
−0.306398 + 0.951903i \(0.599124\pi\)
\(692\) 4.46934e6 0.354795
\(693\) 580070. 0.0458825
\(694\) −1.10431e7 −0.870347
\(695\) 1.04737e7 0.822504
\(696\) −1.80883e6 −0.141538
\(697\) −1.49725e7 −1.16738
\(698\) 4.06352e6 0.315692
\(699\) 1.43645e7 1.11198
\(700\) 351234. 0.0270927
\(701\) 1.53139e7 1.17704 0.588521 0.808482i \(-0.299711\pi\)
0.588521 + 0.808482i \(0.299711\pi\)
\(702\) −1.34292e6 −0.102851
\(703\) 301921. 0.0230412
\(704\) 3.13065e6 0.238069
\(705\) −4.50048e6 −0.341025
\(706\) 1.96389e7 1.48288
\(707\) −6.12397e6 −0.460771
\(708\) 6.15944e6 0.461804
\(709\) 1.84381e7 1.37753 0.688764 0.724986i \(-0.258154\pi\)
0.688764 + 0.724986i \(0.258154\pi\)
\(710\) −6.99528e6 −0.520786
\(711\) −5.91015e6 −0.438454
\(712\) 8.25100e6 0.609968
\(713\) −1.36030e7 −1.00210
\(714\) 2.42800e6 0.178239
\(715\) 169385. 0.0123911
\(716\) 1.71817e6 0.125252
\(717\) 5.06758e6 0.368132
\(718\) −1.62743e7 −1.17813
\(719\) −99951.2 −0.00721051 −0.00360525 0.999994i \(-0.501148\pi\)
−0.00360525 + 0.999994i \(0.501148\pi\)
\(720\) −1.71800e6 −0.123507
\(721\) 3.75512e6 0.269021
\(722\) −1.12150e7 −0.800673
\(723\) 1.38860e7 0.987945
\(724\) 9.41616e6 0.667617
\(725\) −541823. −0.0382836
\(726\) 7.33715e6 0.516637
\(727\) 1.25198e7 0.878538 0.439269 0.898356i \(-0.355238\pi\)
0.439269 + 0.898356i \(0.355238\pi\)
\(728\) −722393. −0.0505179
\(729\) 1.15223e7 0.803007
\(730\) 2.68067e6 0.186181
\(731\) −1.01624e7 −0.703401
\(732\) −354931. −0.0244831
\(733\) −1.72341e7 −1.18476 −0.592378 0.805660i \(-0.701811\pi\)
−0.592378 + 0.805660i \(0.701811\pi\)
\(734\) 1.63932e6 0.112312
\(735\) 635868. 0.0434159
\(736\) −1.51270e7 −1.02934
\(737\) −3.71910e6 −0.252214
\(738\) 8.59472e6 0.580886
\(739\) −1.83842e7 −1.23832 −0.619160 0.785265i \(-0.712527\pi\)
−0.619160 + 0.785265i \(0.712527\pi\)
\(740\) −2.73281e6 −0.183455
\(741\) −25117.3 −0.00168046
\(742\) 6.57664e6 0.438525
\(743\) 2.79507e7 1.85746 0.928732 0.370752i \(-0.120900\pi\)
0.928732 + 0.370752i \(0.120900\pi\)
\(744\) 7.35861e6 0.487375
\(745\) 1.62407e6 0.107205
\(746\) −1.39706e7 −0.919111
\(747\) −5.11296e6 −0.335252
\(748\) −1.07170e6 −0.0700358
\(749\) −6.62950e6 −0.431794
\(750\) 750000. 0.0486864
\(751\) 1.70136e7 1.10077 0.550383 0.834912i \(-0.314482\pi\)
0.550383 + 0.834912i \(0.314482\pi\)
\(752\) 8.92944e6 0.575810
\(753\) −3.60295e6 −0.231564
\(754\) 294020. 0.0188342
\(755\) −9.49502e6 −0.606218
\(756\) 2.22519e6 0.141600
\(757\) −1.91600e7 −1.21522 −0.607611 0.794234i \(-0.707872\pi\)
−0.607611 + 0.794234i \(0.707872\pi\)
\(758\) −2.28472e7 −1.44431
\(759\) 3.69858e6 0.233040
\(760\) −155980. −0.00979571
\(761\) −2.06404e7 −1.29198 −0.645992 0.763344i \(-0.723556\pi\)
−0.645992 + 0.763344i \(0.723556\pi\)
\(762\) −1.92222e6 −0.119926
\(763\) 9.77592e6 0.607919
\(764\) −3.28719e6 −0.203747
\(765\) 3.37515e6 0.208516
\(766\) −1.41339e7 −0.870341
\(767\) −3.79471e6 −0.232911
\(768\) 1.02259e7 0.625605
\(769\) 1.11748e7 0.681432 0.340716 0.940166i \(-0.389331\pi\)
0.340716 + 0.940166i \(0.389331\pi\)
\(770\) 502441. 0.0305392
\(771\) 623311. 0.0377632
\(772\) −1.13952e7 −0.688141
\(773\) −3.07555e7 −1.85129 −0.925645 0.378394i \(-0.876477\pi\)
−0.925645 + 0.378394i \(0.876477\pi\)
\(774\) 5.83355e6 0.350010
\(775\) 2.20423e6 0.131826
\(776\) −1.58290e6 −0.0943624
\(777\) −4.94743e6 −0.293986
\(778\) −7.45641e6 −0.441653
\(779\) 459440. 0.0271259
\(780\) 227347. 0.0133799
\(781\) 5.58986e6 0.327924
\(782\) −1.80417e7 −1.05502
\(783\) −3.43264e6 −0.200089
\(784\) −1.26163e6 −0.0733065
\(785\) 7.03864e6 0.407675
\(786\) −1.01742e7 −0.587416
\(787\) 3.52357e6 0.202790 0.101395 0.994846i \(-0.467669\pi\)
0.101395 + 0.994846i \(0.467669\pi\)
\(788\) 438214. 0.0251403
\(789\) 14627.3 0.000836509 0
\(790\) −5.11921e6 −0.291833
\(791\) −6.00587e6 −0.341299
\(792\) 2.33168e6 0.132086
\(793\) 218666. 0.0123480
\(794\) 1.68887e7 0.950706
\(795\) −7.84470e6 −0.440209
\(796\) −8.37786e6 −0.468652
\(797\) −3.57469e6 −0.199339 −0.0996695 0.995021i \(-0.531779\pi\)
−0.0996695 + 0.995021i \(0.531779\pi\)
\(798\) −74504.5 −0.00414167
\(799\) −1.75426e7 −0.972139
\(800\) 2.45118e6 0.135410
\(801\) 5.47853e6 0.301705
\(802\) 2.19678e7 1.20601
\(803\) −2.14210e6 −0.117233
\(804\) −4.99173e6 −0.272340
\(805\) −4.72492e6 −0.256983
\(806\) −1.19612e6 −0.0648542
\(807\) −9.39463e6 −0.507804
\(808\) −2.46163e7 −1.32646
\(809\) 3.38466e7 1.81821 0.909104 0.416569i \(-0.136767\pi\)
0.909104 + 0.416569i \(0.136767\pi\)
\(810\) 1.15158e6 0.0616712
\(811\) 3.21058e7 1.71408 0.857042 0.515247i \(-0.172300\pi\)
0.857042 + 0.515247i \(0.172300\pi\)
\(812\) −487185. −0.0259301
\(813\) 3.99323e6 0.211884
\(814\) −3.90928e6 −0.206793
\(815\) 9.56468e6 0.504402
\(816\) 5.74628e6 0.302108
\(817\) 311839. 0.0163446
\(818\) −1.37350e7 −0.717706
\(819\) −479657. −0.0249874
\(820\) −4.15857e6 −0.215978
\(821\) 3.04076e6 0.157443 0.0787216 0.996897i \(-0.474916\pi\)
0.0787216 + 0.996897i \(0.474916\pi\)
\(822\) −1.96707e6 −0.101541
\(823\) 3.72147e6 0.191520 0.0957601 0.995404i \(-0.469472\pi\)
0.0957601 + 0.995404i \(0.469472\pi\)
\(824\) 1.50943e7 0.774452
\(825\) −599318. −0.0306565
\(826\) −1.12561e7 −0.574034
\(827\) −8.77895e6 −0.446354 −0.223177 0.974778i \(-0.571643\pi\)
−0.223177 + 0.974778i \(0.571643\pi\)
\(828\) −5.78523e6 −0.293255
\(829\) 6.61553e6 0.334332 0.167166 0.985929i \(-0.446538\pi\)
0.167166 + 0.985929i \(0.446538\pi\)
\(830\) −4.42871e6 −0.223142
\(831\) 9.32602e6 0.468483
\(832\) −2.58872e6 −0.129651
\(833\) 2.47858e6 0.123763
\(834\) 2.01095e7 1.00112
\(835\) 9.71588e6 0.482243
\(836\) 32885.8 0.00162739
\(837\) 1.39646e7 0.688991
\(838\) 7.56267e6 0.372019
\(839\) −4.67110e6 −0.229094 −0.114547 0.993418i \(-0.536542\pi\)
−0.114547 + 0.993418i \(0.536542\pi\)
\(840\) 2.55597e6 0.124985
\(841\) −1.97596e7 −0.963359
\(842\) −7.97619e6 −0.387718
\(843\) −2.20863e7 −1.07042
\(844\) −2.22824e6 −0.107673
\(845\) 9.14226e6 0.440465
\(846\) 1.00701e7 0.483733
\(847\) 7.49000e6 0.358735
\(848\) 1.55648e7 0.743280
\(849\) −7.33487e6 −0.349239
\(850\) 2.92347e6 0.138788
\(851\) 3.67627e7 1.74014
\(852\) 7.50264e6 0.354091
\(853\) −3.66146e7 −1.72299 −0.861494 0.507768i \(-0.830471\pi\)
−0.861494 + 0.507768i \(0.830471\pi\)
\(854\) 648620. 0.0304331
\(855\) −103568. −0.00484520
\(856\) −2.66483e7 −1.24304
\(857\) 1.15520e7 0.537288 0.268644 0.963240i \(-0.413425\pi\)
0.268644 + 0.963240i \(0.413425\pi\)
\(858\) 325219. 0.0150820
\(859\) 3.53878e7 1.63633 0.818165 0.574983i \(-0.194991\pi\)
0.818165 + 0.574983i \(0.194991\pi\)
\(860\) −2.82257e6 −0.130137
\(861\) −7.52860e6 −0.346104
\(862\) −2.93994e6 −0.134763
\(863\) 1.07433e7 0.491034 0.245517 0.969392i \(-0.421042\pi\)
0.245517 + 0.969392i \(0.421042\pi\)
\(864\) 1.55291e7 0.707720
\(865\) 9.74232e6 0.442713
\(866\) −1.63571e6 −0.0741158
\(867\) 3.75203e6 0.169519
\(868\) 1.98195e6 0.0892881
\(869\) 4.09071e6 0.183759
\(870\) −1.04030e6 −0.0465973
\(871\) 3.07531e6 0.137355
\(872\) 3.92958e7 1.75007
\(873\) −1.05102e6 −0.0466740
\(874\) 553618. 0.0245150
\(875\) 765625. 0.0338062
\(876\) −2.87510e6 −0.126588
\(877\) 4.58293e6 0.201208 0.100604 0.994927i \(-0.467923\pi\)
0.100604 + 0.994927i \(0.467923\pi\)
\(878\) 4.84927e6 0.212295
\(879\) −2.41131e7 −1.05264
\(880\) 1.18911e6 0.0517626
\(881\) 1.68070e7 0.729543 0.364772 0.931097i \(-0.381147\pi\)
0.364772 + 0.931097i \(0.381147\pi\)
\(882\) −1.42279e6 −0.0615842
\(883\) −5.28260e6 −0.228006 −0.114003 0.993480i \(-0.536367\pi\)
−0.114003 + 0.993480i \(0.536367\pi\)
\(884\) 886186. 0.0381412
\(885\) 1.34264e7 0.576238
\(886\) −3.61197e6 −0.154582
\(887\) 4.43107e7 1.89104 0.945518 0.325569i \(-0.105556\pi\)
0.945518 + 0.325569i \(0.105556\pi\)
\(888\) −1.98870e7 −0.846322
\(889\) −1.96226e6 −0.0832728
\(890\) 4.74535e6 0.200814
\(891\) −920219. −0.0388326
\(892\) −1.30250e7 −0.548109
\(893\) 538305. 0.0225892
\(894\) 3.11821e6 0.130485
\(895\) 3.74530e6 0.156289
\(896\) −1.52930e6 −0.0636389
\(897\) −3.05834e6 −0.126913
\(898\) 2.90039e7 1.20023
\(899\) −3.05741e6 −0.126170
\(900\) 937438. 0.0385777
\(901\) −3.05783e7 −1.25488
\(902\) −5.94883e6 −0.243453
\(903\) −5.10994e6 −0.208543
\(904\) −2.41415e7 −0.982524
\(905\) 2.05255e7 0.833051
\(906\) −1.82304e7 −0.737865
\(907\) −3.95608e7 −1.59679 −0.798394 0.602135i \(-0.794317\pi\)
−0.798394 + 0.602135i \(0.794317\pi\)
\(908\) 1.64930e6 0.0663872
\(909\) −1.63448e7 −0.656099
\(910\) −415466. −0.0166315
\(911\) −2.40982e7 −0.962030 −0.481015 0.876712i \(-0.659732\pi\)
−0.481015 + 0.876712i \(0.659732\pi\)
\(912\) −176328. −0.00701994
\(913\) 3.53894e6 0.140506
\(914\) 2.68172e7 1.06181
\(915\) −773682. −0.0305499
\(916\) 43954.4 0.00173087
\(917\) −1.03862e7 −0.407881
\(918\) 1.85212e7 0.725375
\(919\) −2.24660e7 −0.877480 −0.438740 0.898614i \(-0.644575\pi\)
−0.438740 + 0.898614i \(0.644575\pi\)
\(920\) −1.89925e7 −0.739799
\(921\) 2.80923e7 1.09129
\(922\) −4.00704e7 −1.55238
\(923\) −4.62223e6 −0.178586
\(924\) −538882. −0.0207641
\(925\) −5.95701e6 −0.228915
\(926\) −3.06693e7 −1.17538
\(927\) 1.00224e7 0.383063
\(928\) −3.39995e6 −0.129599
\(929\) −1.16280e7 −0.442043 −0.221021 0.975269i \(-0.570939\pi\)
−0.221021 + 0.975269i \(0.570939\pi\)
\(930\) 4.23212e6 0.160454
\(931\) −76056.7 −0.00287583
\(932\) 1.55517e7 0.586458
\(933\) −2.72913e7 −1.02641
\(934\) −4.74591e6 −0.178013
\(935\) −2.33611e6 −0.0873906
\(936\) −1.92806e6 −0.0719333
\(937\) −2.40956e7 −0.896580 −0.448290 0.893888i \(-0.647967\pi\)
−0.448290 + 0.893888i \(0.647967\pi\)
\(938\) 9.12218e6 0.338525
\(939\) −2.66475e7 −0.986264
\(940\) −4.87241e6 −0.179856
\(941\) −567590. −0.0208959 −0.0104479 0.999945i \(-0.503326\pi\)
−0.0104479 + 0.999945i \(0.503326\pi\)
\(942\) 1.35142e7 0.496207
\(943\) 5.59425e7 2.04863
\(944\) −2.66395e7 −0.972962
\(945\) 4.85051e6 0.176688
\(946\) −4.03769e6 −0.146692
\(947\) 9.04501e6 0.327743 0.163872 0.986482i \(-0.447602\pi\)
0.163872 + 0.986482i \(0.447602\pi\)
\(948\) 5.49050e6 0.198423
\(949\) 1.77129e6 0.0638446
\(950\) −89708.1 −0.00322495
\(951\) 2.44068e7 0.875105
\(952\) 9.96305e6 0.356287
\(953\) −1.48053e7 −0.528063 −0.264032 0.964514i \(-0.585052\pi\)
−0.264032 + 0.964514i \(0.585052\pi\)
\(954\) 1.75529e7 0.624423
\(955\) −7.16546e6 −0.254235
\(956\) 5.48639e6 0.194152
\(957\) 831294. 0.0293410
\(958\) −4.24895e7 −1.49578
\(959\) −2.00805e6 −0.0705062
\(960\) 9.15939e6 0.320766
\(961\) −1.61911e7 −0.565546
\(962\) 3.23257e6 0.112619
\(963\) −1.76940e7 −0.614838
\(964\) 1.50336e7 0.521041
\(965\) −2.48393e7 −0.858661
\(966\) −9.07185e6 −0.312790
\(967\) −2.02304e7 −0.695726 −0.347863 0.937545i \(-0.613093\pi\)
−0.347863 + 0.937545i \(0.613093\pi\)
\(968\) 3.01072e7 1.03272
\(969\) 346411. 0.0118517
\(970\) −910364. −0.0310660
\(971\) −255216. −0.00868679 −0.00434340 0.999991i \(-0.501383\pi\)
−0.00434340 + 0.999991i \(0.501383\pi\)
\(972\) 9.80004e6 0.332707
\(973\) 2.05284e7 0.695142
\(974\) −1.30932e7 −0.442229
\(975\) 495573. 0.0166954
\(976\) 1.53507e6 0.0515827
\(977\) 3.15610e7 1.05783 0.528914 0.848675i \(-0.322599\pi\)
0.528914 + 0.848675i \(0.322599\pi\)
\(978\) 1.83642e7 0.613938
\(979\) −3.79196e6 −0.126447
\(980\) 688419. 0.0228975
\(981\) 2.60917e7 0.865627
\(982\) −4.75064e6 −0.157208
\(983\) −5.24054e7 −1.72978 −0.864892 0.501958i \(-0.832613\pi\)
−0.864892 + 0.501958i \(0.832613\pi\)
\(984\) −3.02624e7 −0.996357
\(985\) 955225. 0.0313701
\(986\) −4.05504e6 −0.132832
\(987\) −8.82093e6 −0.288218
\(988\) −27193.1 −0.000886270 0
\(989\) 3.79702e7 1.23439
\(990\) 1.34101e6 0.0434853
\(991\) 3.73103e7 1.20683 0.603413 0.797429i \(-0.293807\pi\)
0.603413 + 0.797429i \(0.293807\pi\)
\(992\) 1.38316e7 0.446264
\(993\) −7.38682e6 −0.237730
\(994\) −1.37108e7 −0.440145
\(995\) −1.82622e7 −0.584783
\(996\) 4.74992e6 0.151718
\(997\) −2.27918e7 −0.726174 −0.363087 0.931755i \(-0.618277\pi\)
−0.363087 + 0.931755i \(0.618277\pi\)
\(998\) 2.89323e7 0.919512
\(999\) −3.77398e7 −1.19643
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.1.2 2
3.2 odd 2 315.6.a.c.1.1 2
4.3 odd 2 560.6.a.l.1.2 2
5.2 odd 4 175.6.b.d.99.4 4
5.3 odd 4 175.6.b.d.99.1 4
5.4 even 2 175.6.a.d.1.1 2
7.6 odd 2 245.6.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.6.a.b.1.2 2 1.1 even 1 trivial
175.6.a.d.1.1 2 5.4 even 2
175.6.b.d.99.1 4 5.3 odd 4
175.6.b.d.99.4 4 5.2 odd 4
245.6.a.c.1.2 2 7.6 odd 2
315.6.a.c.1.1 2 3.2 odd 2
560.6.a.l.1.2 2 4.3 odd 2