Properties

Label 245.6.a.i.1.6
Level $245$
Weight $6$
Character 245.1
Self dual yes
Analytic conductor $39.294$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.2940358542\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - x^{5} - 109x^{4} + 41x^{3} + 2208x^{2} - 3204x + 560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(9.72556\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.72556 q^{2} +4.73965 q^{3} +44.1354 q^{4} -25.0000 q^{5} +41.3561 q^{6} +105.888 q^{8} -220.536 q^{9} +O(q^{10})\) \(q+8.72556 q^{2} +4.73965 q^{3} +44.1354 q^{4} -25.0000 q^{5} +41.3561 q^{6} +105.888 q^{8} -220.536 q^{9} -218.139 q^{10} -679.156 q^{11} +209.186 q^{12} +58.8790 q^{13} -118.491 q^{15} -488.399 q^{16} +1427.56 q^{17} -1924.30 q^{18} -2197.34 q^{19} -1103.39 q^{20} -5926.02 q^{22} -2223.33 q^{23} +501.873 q^{24} +625.000 q^{25} +513.753 q^{26} -2197.00 q^{27} +2079.59 q^{29} -1033.90 q^{30} +3599.58 q^{31} -7649.98 q^{32} -3218.96 q^{33} +12456.2 q^{34} -9733.43 q^{36} +9181.95 q^{37} -19173.0 q^{38} +279.066 q^{39} -2647.21 q^{40} +10061.1 q^{41} -13956.6 q^{43} -29974.8 q^{44} +5513.39 q^{45} -19399.8 q^{46} -16272.5 q^{47} -2314.84 q^{48} +5453.48 q^{50} +6766.12 q^{51} +2598.65 q^{52} +1267.14 q^{53} -19170.0 q^{54} +16978.9 q^{55} -10414.6 q^{57} +18145.6 q^{58} -13465.7 q^{59} -5229.66 q^{60} +35031.5 q^{61} +31408.4 q^{62} -51121.6 q^{64} -1471.98 q^{65} -28087.2 q^{66} -36919.0 q^{67} +63005.9 q^{68} -10537.8 q^{69} +4118.92 q^{71} -23352.1 q^{72} -3517.37 q^{73} +80117.7 q^{74} +2962.28 q^{75} -96980.4 q^{76} +2435.01 q^{78} -72440.8 q^{79} +12210.0 q^{80} +43177.2 q^{81} +87789.0 q^{82} -42252.6 q^{83} -35688.9 q^{85} -121779. q^{86} +9856.51 q^{87} -71914.7 q^{88} +94517.0 q^{89} +48107.5 q^{90} -98127.7 q^{92} +17060.8 q^{93} -141987. q^{94} +54933.4 q^{95} -36258.2 q^{96} +85271.1 q^{97} +149778. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9} + 125 q^{10} - 924 q^{11} + 370 q^{12} - 150 q^{13} - 500 q^{15} + 435 q^{16} + 1540 q^{17} + 195 q^{18} + 92 q^{19} - 775 q^{20} - 6855 q^{22} - 3920 q^{23} - 7200 q^{24} + 3750 q^{25} + 2635 q^{26} + 2060 q^{27} + 1264 q^{29} - 2400 q^{30} - 7160 q^{31} + 9105 q^{32} + 4460 q^{33} + 2166 q^{34} - 26375 q^{36} - 14170 q^{37} - 46215 q^{38} - 15376 q^{39} + 3375 q^{40} + 4098 q^{41} - 24460 q^{43} - 27873 q^{44} - 9450 q^{45} + 6815 q^{46} + 42940 q^{47} + 11610 q^{48} - 3125 q^{50} - 42008 q^{51} - 36115 q^{52} - 2450 q^{53} + 19566 q^{54} + 23100 q^{55} - 97100 q^{57} - 36110 q^{58} - 64600 q^{59} - 9250 q^{60} + 73620 q^{61} + 111440 q^{62} - 157997 q^{64} + 3750 q^{65} - 139138 q^{66} - 142620 q^{67} + 124330 q^{68} + 17344 q^{69} - 154256 q^{71} - 117495 q^{72} - 5120 q^{73} + 2785 q^{74} + 12500 q^{75} + 7775 q^{76} - 214090 q^{78} - 222504 q^{79} - 10875 q^{80} - 43986 q^{81} + 31665 q^{82} + 179580 q^{83} - 38500 q^{85} - 207160 q^{86} - 209300 q^{87} - 45145 q^{88} + 41648 q^{89} - 4875 q^{90} - 292185 q^{92} - 198520 q^{93} - 333699 q^{94} - 2300 q^{95} + 61824 q^{96} - 73980 q^{97} - 190772 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.72556 1.54248 0.771238 0.636547i \(-0.219638\pi\)
0.771238 + 0.636547i \(0.219638\pi\)
\(3\) 4.73965 0.304048 0.152024 0.988377i \(-0.451421\pi\)
0.152024 + 0.988377i \(0.451421\pi\)
\(4\) 44.1354 1.37923
\(5\) −25.0000 −0.447214
\(6\) 41.3561 0.468987
\(7\) 0 0
\(8\) 105.888 0.584955
\(9\) −220.536 −0.907555
\(10\) −218.139 −0.689816
\(11\) −679.156 −1.69234 −0.846171 0.532912i \(-0.821098\pi\)
−0.846171 + 0.532912i \(0.821098\pi\)
\(12\) 209.186 0.419353
\(13\) 58.8790 0.0966279 0.0483139 0.998832i \(-0.484615\pi\)
0.0483139 + 0.998832i \(0.484615\pi\)
\(14\) 0 0
\(15\) −118.491 −0.135975
\(16\) −488.399 −0.476952
\(17\) 1427.56 1.19804 0.599020 0.800734i \(-0.295557\pi\)
0.599020 + 0.800734i \(0.295557\pi\)
\(18\) −1924.30 −1.39988
\(19\) −2197.34 −1.39641 −0.698205 0.715898i \(-0.746017\pi\)
−0.698205 + 0.715898i \(0.746017\pi\)
\(20\) −1103.39 −0.616811
\(21\) 0 0
\(22\) −5926.02 −2.61040
\(23\) −2223.33 −0.876365 −0.438182 0.898886i \(-0.644378\pi\)
−0.438182 + 0.898886i \(0.644378\pi\)
\(24\) 501.873 0.177855
\(25\) 625.000 0.200000
\(26\) 513.753 0.149046
\(27\) −2197.00 −0.579989
\(28\) 0 0
\(29\) 2079.59 0.459179 0.229590 0.973288i \(-0.426262\pi\)
0.229590 + 0.973288i \(0.426262\pi\)
\(30\) −1033.90 −0.209738
\(31\) 3599.58 0.672741 0.336371 0.941730i \(-0.390801\pi\)
0.336371 + 0.941730i \(0.390801\pi\)
\(32\) −7649.98 −1.32064
\(33\) −3218.96 −0.514554
\(34\) 12456.2 1.84795
\(35\) 0 0
\(36\) −9733.43 −1.25173
\(37\) 9181.95 1.10263 0.551316 0.834296i \(-0.314126\pi\)
0.551316 + 0.834296i \(0.314126\pi\)
\(38\) −19173.0 −2.15393
\(39\) 279.066 0.0293796
\(40\) −2647.21 −0.261600
\(41\) 10061.1 0.934732 0.467366 0.884064i \(-0.345203\pi\)
0.467366 + 0.884064i \(0.345203\pi\)
\(42\) 0 0
\(43\) −13956.6 −1.15108 −0.575542 0.817772i \(-0.695209\pi\)
−0.575542 + 0.817772i \(0.695209\pi\)
\(44\) −29974.8 −2.33413
\(45\) 5513.39 0.405871
\(46\) −19399.8 −1.35177
\(47\) −16272.5 −1.07451 −0.537254 0.843420i \(-0.680538\pi\)
−0.537254 + 0.843420i \(0.680538\pi\)
\(48\) −2314.84 −0.145017
\(49\) 0 0
\(50\) 5453.48 0.308495
\(51\) 6766.12 0.364262
\(52\) 2598.65 0.133272
\(53\) 1267.14 0.0619634 0.0309817 0.999520i \(-0.490137\pi\)
0.0309817 + 0.999520i \(0.490137\pi\)
\(54\) −19170.0 −0.894619
\(55\) 16978.9 0.756838
\(56\) 0 0
\(57\) −10414.6 −0.424576
\(58\) 18145.6 0.708273
\(59\) −13465.7 −0.503615 −0.251807 0.967777i \(-0.581025\pi\)
−0.251807 + 0.967777i \(0.581025\pi\)
\(60\) −5229.66 −0.187540
\(61\) 35031.5 1.20541 0.602703 0.797965i \(-0.294090\pi\)
0.602703 + 0.797965i \(0.294090\pi\)
\(62\) 31408.4 1.03769
\(63\) 0 0
\(64\) −51121.6 −1.56011
\(65\) −1471.98 −0.0432133
\(66\) −28087.2 −0.793687
\(67\) −36919.0 −1.00476 −0.502380 0.864647i \(-0.667542\pi\)
−0.502380 + 0.864647i \(0.667542\pi\)
\(68\) 63005.9 1.65237
\(69\) −10537.8 −0.266457
\(70\) 0 0
\(71\) 4118.92 0.0969700 0.0484850 0.998824i \(-0.484561\pi\)
0.0484850 + 0.998824i \(0.484561\pi\)
\(72\) −23352.1 −0.530879
\(73\) −3517.37 −0.0772521 −0.0386261 0.999254i \(-0.512298\pi\)
−0.0386261 + 0.999254i \(0.512298\pi\)
\(74\) 80117.7 1.70078
\(75\) 2962.28 0.0608097
\(76\) −96980.4 −1.92597
\(77\) 0 0
\(78\) 2435.01 0.0453173
\(79\) −72440.8 −1.30592 −0.652959 0.757394i \(-0.726472\pi\)
−0.652959 + 0.757394i \(0.726472\pi\)
\(80\) 12210.0 0.213299
\(81\) 43177.2 0.731210
\(82\) 87789.0 1.44180
\(83\) −42252.6 −0.673223 −0.336611 0.941644i \(-0.609281\pi\)
−0.336611 + 0.941644i \(0.609281\pi\)
\(84\) 0 0
\(85\) −35688.9 −0.535780
\(86\) −121779. −1.77552
\(87\) 9856.51 0.139613
\(88\) −71914.7 −0.989944
\(89\) 94517.0 1.26484 0.632419 0.774627i \(-0.282062\pi\)
0.632419 + 0.774627i \(0.282062\pi\)
\(90\) 48107.5 0.626046
\(91\) 0 0
\(92\) −98127.7 −1.20871
\(93\) 17060.8 0.204546
\(94\) −141987. −1.65740
\(95\) 54933.4 0.624493
\(96\) −36258.2 −0.401539
\(97\) 85271.1 0.920180 0.460090 0.887872i \(-0.347817\pi\)
0.460090 + 0.887872i \(0.347817\pi\)
\(98\) 0 0
\(99\) 149778. 1.53589
\(100\) 27584.6 0.275846
\(101\) −180306. −1.75876 −0.879380 0.476121i \(-0.842042\pi\)
−0.879380 + 0.476121i \(0.842042\pi\)
\(102\) 59038.2 0.561866
\(103\) −173368. −1.61018 −0.805091 0.593152i \(-0.797883\pi\)
−0.805091 + 0.593152i \(0.797883\pi\)
\(104\) 6234.60 0.0565230
\(105\) 0 0
\(106\) 11056.5 0.0955770
\(107\) 155442. 1.31253 0.656264 0.754532i \(-0.272136\pi\)
0.656264 + 0.754532i \(0.272136\pi\)
\(108\) −96965.3 −0.799939
\(109\) 19088.6 0.153889 0.0769444 0.997035i \(-0.475484\pi\)
0.0769444 + 0.997035i \(0.475484\pi\)
\(110\) 148151. 1.16740
\(111\) 43519.2 0.335254
\(112\) 0 0
\(113\) 101150. 0.745197 0.372599 0.927993i \(-0.378467\pi\)
0.372599 + 0.927993i \(0.378467\pi\)
\(114\) −90873.3 −0.654899
\(115\) 55583.3 0.391922
\(116\) 91783.5 0.633315
\(117\) −12984.9 −0.0876951
\(118\) −117496. −0.776814
\(119\) 0 0
\(120\) −12546.8 −0.0795391
\(121\) 300202. 1.86402
\(122\) 305669. 1.85931
\(123\) 47686.2 0.284204
\(124\) 158869. 0.927866
\(125\) −15625.0 −0.0894427
\(126\) 0 0
\(127\) −112788. −0.620514 −0.310257 0.950653i \(-0.600415\pi\)
−0.310257 + 0.950653i \(0.600415\pi\)
\(128\) −201265. −1.08578
\(129\) −66149.1 −0.349985
\(130\) −12843.8 −0.0666555
\(131\) −183577. −0.934632 −0.467316 0.884090i \(-0.654779\pi\)
−0.467316 + 0.884090i \(0.654779\pi\)
\(132\) −142070. −0.709689
\(133\) 0 0
\(134\) −322139. −1.54982
\(135\) 54924.9 0.259379
\(136\) 151162. 0.700800
\(137\) −92212.1 −0.419746 −0.209873 0.977729i \(-0.567305\pi\)
−0.209873 + 0.977729i \(0.567305\pi\)
\(138\) −91948.3 −0.411004
\(139\) 424349. 1.86288 0.931442 0.363889i \(-0.118551\pi\)
0.931442 + 0.363889i \(0.118551\pi\)
\(140\) 0 0
\(141\) −77125.9 −0.326703
\(142\) 35939.9 0.149574
\(143\) −39988.1 −0.163527
\(144\) 107709. 0.432860
\(145\) −51989.7 −0.205351
\(146\) −30691.0 −0.119160
\(147\) 0 0
\(148\) 405249. 1.52079
\(149\) 137894. 0.508838 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(150\) 25847.5 0.0937975
\(151\) −308590. −1.10139 −0.550693 0.834708i \(-0.685636\pi\)
−0.550693 + 0.834708i \(0.685636\pi\)
\(152\) −232672. −0.816837
\(153\) −314828. −1.08729
\(154\) 0 0
\(155\) −89989.6 −0.300859
\(156\) 12316.7 0.0405212
\(157\) −209003. −0.676710 −0.338355 0.941018i \(-0.609871\pi\)
−0.338355 + 0.941018i \(0.609871\pi\)
\(158\) −632087. −2.01435
\(159\) 6005.80 0.0188399
\(160\) 191249. 0.590609
\(161\) 0 0
\(162\) 376745. 1.12787
\(163\) −249015. −0.734103 −0.367051 0.930201i \(-0.619633\pi\)
−0.367051 + 0.930201i \(0.619633\pi\)
\(164\) 444052. 1.28921
\(165\) 80474.0 0.230116
\(166\) −368678. −1.03843
\(167\) 550497. 1.52744 0.763720 0.645548i \(-0.223371\pi\)
0.763720 + 0.645548i \(0.223371\pi\)
\(168\) 0 0
\(169\) −367826. −0.990663
\(170\) −311406. −0.826428
\(171\) 484592. 1.26732
\(172\) −615978. −1.58761
\(173\) −103026. −0.261717 −0.130859 0.991401i \(-0.541773\pi\)
−0.130859 + 0.991401i \(0.541773\pi\)
\(174\) 86003.6 0.215349
\(175\) 0 0
\(176\) 331699. 0.807166
\(177\) −63822.6 −0.153123
\(178\) 824714. 1.95098
\(179\) 470596. 1.09778 0.548891 0.835894i \(-0.315050\pi\)
0.548891 + 0.835894i \(0.315050\pi\)
\(180\) 243336. 0.559790
\(181\) −529125. −1.20050 −0.600250 0.799812i \(-0.704932\pi\)
−0.600250 + 0.799812i \(0.704932\pi\)
\(182\) 0 0
\(183\) 166037. 0.366502
\(184\) −235425. −0.512634
\(185\) −229549. −0.493112
\(186\) 148865. 0.315507
\(187\) −969535. −2.02749
\(188\) −718194. −1.48200
\(189\) 0 0
\(190\) 479325. 0.963266
\(191\) 180604. 0.358215 0.179108 0.983829i \(-0.442679\pi\)
0.179108 + 0.983829i \(0.442679\pi\)
\(192\) −242298. −0.474348
\(193\) −512865. −0.991083 −0.495541 0.868584i \(-0.665030\pi\)
−0.495541 + 0.868584i \(0.665030\pi\)
\(194\) 744038. 1.41935
\(195\) −6976.65 −0.0131389
\(196\) 0 0
\(197\) −28199.0 −0.0517687 −0.0258844 0.999665i \(-0.508240\pi\)
−0.0258844 + 0.999665i \(0.508240\pi\)
\(198\) 1.30690e6 2.36908
\(199\) 934000. 1.67192 0.835958 0.548794i \(-0.184913\pi\)
0.835958 + 0.548794i \(0.184913\pi\)
\(200\) 66180.1 0.116991
\(201\) −174983. −0.305496
\(202\) −1.57327e6 −2.71284
\(203\) 0 0
\(204\) 298625. 0.502402
\(205\) −251528. −0.418025
\(206\) −1.51273e6 −2.48367
\(207\) 490324. 0.795349
\(208\) −28756.5 −0.0460869
\(209\) 1.49234e6 2.36320
\(210\) 0 0
\(211\) −206876. −0.319893 −0.159946 0.987126i \(-0.551132\pi\)
−0.159946 + 0.987126i \(0.551132\pi\)
\(212\) 55925.8 0.0854619
\(213\) 19522.2 0.0294836
\(214\) 1.35632e6 2.02454
\(215\) 348914. 0.514780
\(216\) −232636. −0.339268
\(217\) 0 0
\(218\) 166559. 0.237370
\(219\) −16671.1 −0.0234884
\(220\) 749371. 1.04386
\(221\) 84053.3 0.115764
\(222\) 379730. 0.517121
\(223\) 347934. 0.468527 0.234263 0.972173i \(-0.424732\pi\)
0.234263 + 0.972173i \(0.424732\pi\)
\(224\) 0 0
\(225\) −137835. −0.181511
\(226\) 882594. 1.14945
\(227\) −792997. −1.02143 −0.510713 0.859751i \(-0.670619\pi\)
−0.510713 + 0.859751i \(0.670619\pi\)
\(228\) −459653. −0.585589
\(229\) −275845. −0.347598 −0.173799 0.984781i \(-0.555604\pi\)
−0.173799 + 0.984781i \(0.555604\pi\)
\(230\) 484996. 0.604531
\(231\) 0 0
\(232\) 220204. 0.268599
\(233\) 1.43590e6 1.73275 0.866373 0.499398i \(-0.166446\pi\)
0.866373 + 0.499398i \(0.166446\pi\)
\(234\) −113301. −0.135267
\(235\) 406813. 0.480535
\(236\) −594314. −0.694602
\(237\) −343344. −0.397062
\(238\) 0 0
\(239\) 127902. 0.144838 0.0724191 0.997374i \(-0.476928\pi\)
0.0724191 + 0.997374i \(0.476928\pi\)
\(240\) 57871.0 0.0648534
\(241\) 602677. 0.668409 0.334204 0.942501i \(-0.391532\pi\)
0.334204 + 0.942501i \(0.391532\pi\)
\(242\) 2.61943e6 2.87521
\(243\) 738515. 0.802312
\(244\) 1.54613e6 1.66253
\(245\) 0 0
\(246\) 416089. 0.438377
\(247\) −129377. −0.134932
\(248\) 381154. 0.393524
\(249\) −200263. −0.204692
\(250\) −136337. −0.137963
\(251\) −756978. −0.758401 −0.379201 0.925314i \(-0.623801\pi\)
−0.379201 + 0.925314i \(0.623801\pi\)
\(252\) 0 0
\(253\) 1.50999e6 1.48311
\(254\) −984135. −0.957128
\(255\) −169153. −0.162903
\(256\) −120261. −0.114689
\(257\) 606610. 0.572898 0.286449 0.958096i \(-0.407525\pi\)
0.286449 + 0.958096i \(0.407525\pi\)
\(258\) −577188. −0.539844
\(259\) 0 0
\(260\) −64966.3 −0.0596011
\(261\) −458624. −0.416730
\(262\) −1.60181e6 −1.44165
\(263\) −1.41962e6 −1.26556 −0.632780 0.774331i \(-0.718086\pi\)
−0.632780 + 0.774331i \(0.718086\pi\)
\(264\) −340850. −0.300991
\(265\) −31678.5 −0.0277109
\(266\) 0 0
\(267\) 447977. 0.384572
\(268\) −1.62943e6 −1.38580
\(269\) −496984. −0.418756 −0.209378 0.977835i \(-0.567144\pi\)
−0.209378 + 0.977835i \(0.567144\pi\)
\(270\) 479250. 0.400086
\(271\) 155568. 0.128676 0.0643378 0.997928i \(-0.479506\pi\)
0.0643378 + 0.997928i \(0.479506\pi\)
\(272\) −697218. −0.571408
\(273\) 0 0
\(274\) −804602. −0.647448
\(275\) −424473. −0.338468
\(276\) −465091. −0.367506
\(277\) −1.77069e6 −1.38657 −0.693285 0.720663i \(-0.743837\pi\)
−0.693285 + 0.720663i \(0.743837\pi\)
\(278\) 3.70268e6 2.87345
\(279\) −793837. −0.610549
\(280\) 0 0
\(281\) −2.38775e6 −1.80394 −0.901972 0.431794i \(-0.857881\pi\)
−0.901972 + 0.431794i \(0.857881\pi\)
\(282\) −672967. −0.503931
\(283\) −193515. −0.143631 −0.0718156 0.997418i \(-0.522879\pi\)
−0.0718156 + 0.997418i \(0.522879\pi\)
\(284\) 181790. 0.133744
\(285\) 260365. 0.189876
\(286\) −348918. −0.252237
\(287\) 0 0
\(288\) 1.68709e6 1.19855
\(289\) 618065. 0.435301
\(290\) −453639. −0.316749
\(291\) 404155. 0.279779
\(292\) −155240. −0.106549
\(293\) −66529.2 −0.0452734 −0.0226367 0.999744i \(-0.507206\pi\)
−0.0226367 + 0.999744i \(0.507206\pi\)
\(294\) 0 0
\(295\) 336642. 0.225223
\(296\) 972261. 0.644991
\(297\) 1.49210e6 0.981540
\(298\) 1.20320e6 0.784870
\(299\) −130908. −0.0846813
\(300\) 130741. 0.0838706
\(301\) 0 0
\(302\) −2.69262e6 −1.69886
\(303\) −854586. −0.534748
\(304\) 1.07318e6 0.666020
\(305\) −875786. −0.539074
\(306\) −2.74705e6 −1.67711
\(307\) 1.08295e6 0.655787 0.327893 0.944715i \(-0.393661\pi\)
0.327893 + 0.944715i \(0.393661\pi\)
\(308\) 0 0
\(309\) −821701. −0.489573
\(310\) −785210. −0.464068
\(311\) −1.48395e6 −0.869999 −0.435000 0.900431i \(-0.643251\pi\)
−0.435000 + 0.900431i \(0.643251\pi\)
\(312\) 29549.8 0.0171857
\(313\) −2.59381e6 −1.49650 −0.748251 0.663416i \(-0.769106\pi\)
−0.748251 + 0.663416i \(0.769106\pi\)
\(314\) −1.82367e6 −1.04381
\(315\) 0 0
\(316\) −3.19721e6 −1.80116
\(317\) −2.77640e6 −1.55180 −0.775898 0.630858i \(-0.782703\pi\)
−0.775898 + 0.630858i \(0.782703\pi\)
\(318\) 52404.0 0.0290601
\(319\) −1.41237e6 −0.777088
\(320\) 1.27804e6 0.697701
\(321\) 736739. 0.399072
\(322\) 0 0
\(323\) −3.13683e6 −1.67295
\(324\) 1.90564e6 1.00851
\(325\) 36799.4 0.0193256
\(326\) −2.17280e6 −1.13234
\(327\) 90473.1 0.0467897
\(328\) 1.06535e6 0.546776
\(329\) 0 0
\(330\) 702181. 0.354948
\(331\) −1.99398e6 −1.00035 −0.500174 0.865925i \(-0.666731\pi\)
−0.500174 + 0.865925i \(0.666731\pi\)
\(332\) −1.86484e6 −0.928530
\(333\) −2.02495e6 −1.00070
\(334\) 4.80340e6 2.35604
\(335\) 922974. 0.449343
\(336\) 0 0
\(337\) −19232.2 −0.00922473 −0.00461236 0.999989i \(-0.501468\pi\)
−0.00461236 + 0.999989i \(0.501468\pi\)
\(338\) −3.20949e6 −1.52807
\(339\) 479417. 0.226576
\(340\) −1.57515e6 −0.738965
\(341\) −2.44468e6 −1.13851
\(342\) 4.22833e6 1.95481
\(343\) 0 0
\(344\) −1.47783e6 −0.673333
\(345\) 263445. 0.119163
\(346\) −898961. −0.403692
\(347\) −2.23061e6 −0.994489 −0.497245 0.867610i \(-0.665655\pi\)
−0.497245 + 0.867610i \(0.665655\pi\)
\(348\) 435021. 0.192558
\(349\) −313791. −0.137904 −0.0689520 0.997620i \(-0.521966\pi\)
−0.0689520 + 0.997620i \(0.521966\pi\)
\(350\) 0 0
\(351\) −129357. −0.0560431
\(352\) 5.19553e6 2.23498
\(353\) 1.47348e6 0.629370 0.314685 0.949196i \(-0.398101\pi\)
0.314685 + 0.949196i \(0.398101\pi\)
\(354\) −556888. −0.236189
\(355\) −102973. −0.0433663
\(356\) 4.17155e6 1.74450
\(357\) 0 0
\(358\) 4.10622e6 1.69330
\(359\) −1.21115e6 −0.495977 −0.247988 0.968763i \(-0.579769\pi\)
−0.247988 + 0.968763i \(0.579769\pi\)
\(360\) 583803. 0.237416
\(361\) 2.35219e6 0.949960
\(362\) −4.61692e6 −1.85174
\(363\) 1.42285e6 0.566753
\(364\) 0 0
\(365\) 87934.1 0.0345482
\(366\) 1.44876e6 0.565321
\(367\) 2.43854e6 0.945073 0.472536 0.881311i \(-0.343339\pi\)
0.472536 + 0.881311i \(0.343339\pi\)
\(368\) 1.08587e6 0.417984
\(369\) −2.21884e6 −0.848320
\(370\) −2.00294e6 −0.760614
\(371\) 0 0
\(372\) 752983. 0.282116
\(373\) −3.48803e6 −1.29810 −0.649050 0.760746i \(-0.724833\pi\)
−0.649050 + 0.760746i \(0.724833\pi\)
\(374\) −8.45974e6 −3.12736
\(375\) −74057.0 −0.0271949
\(376\) −1.72307e6 −0.628539
\(377\) 122444. 0.0443695
\(378\) 0 0
\(379\) −964503. −0.344910 −0.172455 0.985017i \(-0.555170\pi\)
−0.172455 + 0.985017i \(0.555170\pi\)
\(380\) 2.42451e6 0.861321
\(381\) −534573. −0.188666
\(382\) 1.57587e6 0.552539
\(383\) 3.36433e6 1.17193 0.585965 0.810336i \(-0.300716\pi\)
0.585965 + 0.810336i \(0.300716\pi\)
\(384\) −953926. −0.330131
\(385\) 0 0
\(386\) −4.47504e6 −1.52872
\(387\) 3.07792e6 1.04467
\(388\) 3.76348e6 1.26914
\(389\) −1.18786e6 −0.398007 −0.199003 0.979999i \(-0.563770\pi\)
−0.199003 + 0.979999i \(0.563770\pi\)
\(390\) −60875.2 −0.0202665
\(391\) −3.17394e6 −1.04992
\(392\) 0 0
\(393\) −870091. −0.284173
\(394\) −246052. −0.0798520
\(395\) 1.81102e6 0.584024
\(396\) 6.61052e6 2.11835
\(397\) 1.63665e6 0.521171 0.260585 0.965451i \(-0.416084\pi\)
0.260585 + 0.965451i \(0.416084\pi\)
\(398\) 8.14967e6 2.57889
\(399\) 0 0
\(400\) −305249. −0.0953904
\(401\) −892182. −0.277072 −0.138536 0.990357i \(-0.544240\pi\)
−0.138536 + 0.990357i \(0.544240\pi\)
\(402\) −1.52682e6 −0.471220
\(403\) 211940. 0.0650056
\(404\) −7.95787e6 −2.42574
\(405\) −1.07943e6 −0.327007
\(406\) 0 0
\(407\) −6.23598e6 −1.86603
\(408\) 716453. 0.213077
\(409\) −3.38800e6 −1.00146 −0.500732 0.865603i \(-0.666936\pi\)
−0.500732 + 0.865603i \(0.666936\pi\)
\(410\) −2.19472e6 −0.644793
\(411\) −437053. −0.127623
\(412\) −7.65165e6 −2.22081
\(413\) 0 0
\(414\) 4.27836e6 1.22681
\(415\) 1.05632e6 0.301074
\(416\) −450423. −0.127611
\(417\) 2.01126e6 0.566407
\(418\) 1.30215e7 3.64518
\(419\) −5.29618e6 −1.47376 −0.736882 0.676022i \(-0.763703\pi\)
−0.736882 + 0.676022i \(0.763703\pi\)
\(420\) 0 0
\(421\) 5.38110e6 1.47967 0.739836 0.672787i \(-0.234903\pi\)
0.739836 + 0.672787i \(0.234903\pi\)
\(422\) −1.80511e6 −0.493427
\(423\) 3.58867e6 0.975175
\(424\) 134175. 0.0362458
\(425\) 892224. 0.239608
\(426\) 170342. 0.0454777
\(427\) 0 0
\(428\) 6.86049e6 1.81028
\(429\) −189529. −0.0497203
\(430\) 3.04447e6 0.794036
\(431\) −6.29797e6 −1.63308 −0.816540 0.577290i \(-0.804110\pi\)
−0.816540 + 0.577290i \(0.804110\pi\)
\(432\) 1.07301e6 0.276627
\(433\) −4.54572e6 −1.16515 −0.582577 0.812775i \(-0.697956\pi\)
−0.582577 + 0.812775i \(0.697956\pi\)
\(434\) 0 0
\(435\) −246413. −0.0624367
\(436\) 842482. 0.212248
\(437\) 4.88541e6 1.22376
\(438\) −145464. −0.0362303
\(439\) −2.69494e6 −0.667402 −0.333701 0.942679i \(-0.608298\pi\)
−0.333701 + 0.942679i \(0.608298\pi\)
\(440\) 1.79787e6 0.442717
\(441\) 0 0
\(442\) 733412. 0.178563
\(443\) −3.00924e6 −0.728530 −0.364265 0.931295i \(-0.618680\pi\)
−0.364265 + 0.931295i \(0.618680\pi\)
\(444\) 1.92074e6 0.462393
\(445\) −2.36292e6 −0.565653
\(446\) 3.03592e6 0.722691
\(447\) 653569. 0.154711
\(448\) 0 0
\(449\) 3.52092e6 0.824214 0.412107 0.911135i \(-0.364793\pi\)
0.412107 + 0.911135i \(0.364793\pi\)
\(450\) −1.20269e6 −0.279976
\(451\) −6.83308e6 −1.58189
\(452\) 4.46431e6 1.02780
\(453\) −1.46261e6 −0.334875
\(454\) −6.91935e6 −1.57553
\(455\) 0 0
\(456\) −1.10278e6 −0.248358
\(457\) 2.51620e6 0.563578 0.281789 0.959476i \(-0.409072\pi\)
0.281789 + 0.959476i \(0.409072\pi\)
\(458\) −2.40691e6 −0.536161
\(459\) −3.13634e6 −0.694850
\(460\) 2.45319e6 0.540552
\(461\) −200800. −0.0440060 −0.0220030 0.999758i \(-0.507004\pi\)
−0.0220030 + 0.999758i \(0.507004\pi\)
\(462\) 0 0
\(463\) 192190. 0.0416656 0.0208328 0.999783i \(-0.493368\pi\)
0.0208328 + 0.999783i \(0.493368\pi\)
\(464\) −1.01567e6 −0.219007
\(465\) −426519. −0.0914757
\(466\) 1.25290e7 2.67272
\(467\) 1.49891e6 0.318040 0.159020 0.987275i \(-0.449167\pi\)
0.159020 + 0.987275i \(0.449167\pi\)
\(468\) −573095. −0.120952
\(469\) 0 0
\(470\) 3.54967e6 0.741213
\(471\) −990599. −0.205753
\(472\) −1.42586e6 −0.294592
\(473\) 9.47868e6 1.94803
\(474\) −2.99587e6 −0.612459
\(475\) −1.37334e6 −0.279282
\(476\) 0 0
\(477\) −279450. −0.0562352
\(478\) 1.11602e6 0.223409
\(479\) 7.48400e6 1.49037 0.745186 0.666857i \(-0.232361\pi\)
0.745186 + 0.666857i \(0.232361\pi\)
\(480\) 906455. 0.179574
\(481\) 540625. 0.106545
\(482\) 5.25870e6 1.03100
\(483\) 0 0
\(484\) 1.32496e7 2.57092
\(485\) −2.13178e6 −0.411517
\(486\) 6.44395e6 1.23755
\(487\) −2.19160e6 −0.418734 −0.209367 0.977837i \(-0.567140\pi\)
−0.209367 + 0.977837i \(0.567140\pi\)
\(488\) 3.70942e6 0.705109
\(489\) −1.18024e6 −0.223203
\(490\) 0 0
\(491\) 3.71570e6 0.695564 0.347782 0.937575i \(-0.386935\pi\)
0.347782 + 0.937575i \(0.386935\pi\)
\(492\) 2.10465e6 0.391983
\(493\) 2.96873e6 0.550115
\(494\) −1.12889e6 −0.208129
\(495\) −3.74446e6 −0.686872
\(496\) −1.75803e6 −0.320865
\(497\) 0 0
\(498\) −1.74740e6 −0.315733
\(499\) 5.74147e6 1.03222 0.516109 0.856523i \(-0.327380\pi\)
0.516109 + 0.856523i \(0.327380\pi\)
\(500\) −689616. −0.123362
\(501\) 2.60916e6 0.464416
\(502\) −6.60506e6 −1.16982
\(503\) 8.85276e6 1.56012 0.780061 0.625703i \(-0.215188\pi\)
0.780061 + 0.625703i \(0.215188\pi\)
\(504\) 0 0
\(505\) 4.50765e6 0.786541
\(506\) 1.31755e7 2.28766
\(507\) −1.74337e6 −0.301210
\(508\) −4.97792e6 −0.855833
\(509\) 478571. 0.0818751 0.0409376 0.999162i \(-0.486966\pi\)
0.0409376 + 0.999162i \(0.486966\pi\)
\(510\) −1.47595e6 −0.251274
\(511\) 0 0
\(512\) 5.39114e6 0.908879
\(513\) 4.82754e6 0.809902
\(514\) 5.29302e6 0.883681
\(515\) 4.33419e6 0.720095
\(516\) −2.91952e6 −0.482711
\(517\) 1.10516e7 1.81844
\(518\) 0 0
\(519\) −488307. −0.0795747
\(520\) −155865. −0.0252778
\(521\) −8.96169e6 −1.44642 −0.723212 0.690626i \(-0.757335\pi\)
−0.723212 + 0.690626i \(0.757335\pi\)
\(522\) −4.00175e6 −0.642796
\(523\) 1.08551e7 1.73531 0.867656 0.497164i \(-0.165625\pi\)
0.867656 + 0.497164i \(0.165625\pi\)
\(524\) −8.10226e6 −1.28907
\(525\) 0 0
\(526\) −1.23870e7 −1.95210
\(527\) 5.13861e6 0.805971
\(528\) 1.57214e6 0.245418
\(529\) −1.49313e6 −0.231984
\(530\) −276413. −0.0427434
\(531\) 2.96967e6 0.457058
\(532\) 0 0
\(533\) 592389. 0.0903211
\(534\) 3.90885e6 0.593193
\(535\) −3.88604e6 −0.586980
\(536\) −3.90929e6 −0.587740
\(537\) 2.23046e6 0.333779
\(538\) −4.33646e6 −0.645922
\(539\) 0 0
\(540\) 2.42413e6 0.357744
\(541\) −1.15845e7 −1.70170 −0.850852 0.525405i \(-0.823914\pi\)
−0.850852 + 0.525405i \(0.823914\pi\)
\(542\) 1.35742e6 0.198479
\(543\) −2.50787e6 −0.365010
\(544\) −1.09208e7 −1.58218
\(545\) −477214. −0.0688212
\(546\) 0 0
\(547\) 594715. 0.0849846 0.0424923 0.999097i \(-0.486470\pi\)
0.0424923 + 0.999097i \(0.486470\pi\)
\(548\) −4.06982e6 −0.578927
\(549\) −7.72569e6 −1.09397
\(550\) −3.70376e6 −0.522079
\(551\) −4.56956e6 −0.641202
\(552\) −1.11583e6 −0.155866
\(553\) 0 0
\(554\) −1.54502e7 −2.13875
\(555\) −1.08798e6 −0.149930
\(556\) 1.87288e7 2.56935
\(557\) 4.57366e6 0.624634 0.312317 0.949978i \(-0.398895\pi\)
0.312317 + 0.949978i \(0.398895\pi\)
\(558\) −6.92667e6 −0.941758
\(559\) −821748. −0.111227
\(560\) 0 0
\(561\) −4.59525e6 −0.616456
\(562\) −2.08345e7 −2.78254
\(563\) 2.88071e6 0.383027 0.191513 0.981490i \(-0.438660\pi\)
0.191513 + 0.981490i \(0.438660\pi\)
\(564\) −3.40398e6 −0.450599
\(565\) −2.52876e6 −0.333262
\(566\) −1.68853e6 −0.221548
\(567\) 0 0
\(568\) 436145. 0.0567231
\(569\) 1.18609e7 1.53581 0.767903 0.640566i \(-0.221300\pi\)
0.767903 + 0.640566i \(0.221300\pi\)
\(570\) 2.27183e6 0.292880
\(571\) −3.94838e6 −0.506791 −0.253396 0.967363i \(-0.581547\pi\)
−0.253396 + 0.967363i \(0.581547\pi\)
\(572\) −1.76489e6 −0.225542
\(573\) 856000. 0.108915
\(574\) 0 0
\(575\) −1.38958e6 −0.175273
\(576\) 1.12741e7 1.41588
\(577\) 1.33953e6 0.167500 0.0837498 0.996487i \(-0.473310\pi\)
0.0837498 + 0.996487i \(0.473310\pi\)
\(578\) 5.39296e6 0.671441
\(579\) −2.43080e6 −0.301337
\(580\) −2.29459e6 −0.283227
\(581\) 0 0
\(582\) 3.52648e6 0.431553
\(583\) −860587. −0.104863
\(584\) −372448. −0.0451890
\(585\) 324623. 0.0392184
\(586\) −580504. −0.0698331
\(587\) −1.53600e7 −1.83990 −0.919951 0.392034i \(-0.871772\pi\)
−0.919951 + 0.392034i \(0.871772\pi\)
\(588\) 0 0
\(589\) −7.90950e6 −0.939422
\(590\) 2.93739e6 0.347402
\(591\) −133653. −0.0157402
\(592\) −4.48446e6 −0.525903
\(593\) −963371. −0.112501 −0.0562506 0.998417i \(-0.517915\pi\)
−0.0562506 + 0.998417i \(0.517915\pi\)
\(594\) 1.30194e7 1.51400
\(595\) 0 0
\(596\) 6.08600e6 0.701805
\(597\) 4.42683e6 0.508343
\(598\) −1.14224e6 −0.130619
\(599\) −1.39812e6 −0.159212 −0.0796062 0.996826i \(-0.525366\pi\)
−0.0796062 + 0.996826i \(0.525366\pi\)
\(600\) 313670. 0.0355710
\(601\) 9.79835e6 1.10654 0.553270 0.833002i \(-0.313380\pi\)
0.553270 + 0.833002i \(0.313380\pi\)
\(602\) 0 0
\(603\) 8.14195e6 0.911875
\(604\) −1.36197e7 −1.51907
\(605\) −7.50506e6 −0.833616
\(606\) −7.45674e6 −0.824836
\(607\) −1.05240e7 −1.15933 −0.579666 0.814854i \(-0.696817\pi\)
−0.579666 + 0.814854i \(0.696817\pi\)
\(608\) 1.68096e7 1.84416
\(609\) 0 0
\(610\) −7.64173e6 −0.831509
\(611\) −958110. −0.103827
\(612\) −1.38950e7 −1.49962
\(613\) −1.17388e7 −1.26175 −0.630874 0.775886i \(-0.717303\pi\)
−0.630874 + 0.775886i \(0.717303\pi\)
\(614\) 9.44935e6 1.01154
\(615\) −1.19215e6 −0.127100
\(616\) 0 0
\(617\) −6.83389e6 −0.722695 −0.361348 0.932431i \(-0.617683\pi\)
−0.361348 + 0.932431i \(0.617683\pi\)
\(618\) −7.16980e6 −0.755155
\(619\) 1.06954e7 1.12194 0.560969 0.827837i \(-0.310429\pi\)
0.560969 + 0.827837i \(0.310429\pi\)
\(620\) −3.97173e6 −0.414954
\(621\) 4.88465e6 0.508282
\(622\) −1.29483e7 −1.34195
\(623\) 0 0
\(624\) −136295. −0.0140126
\(625\) 390625. 0.0400000
\(626\) −2.26324e7 −2.30832
\(627\) 7.07315e6 0.718528
\(628\) −9.22442e6 −0.933340
\(629\) 1.31078e7 1.32100
\(630\) 0 0
\(631\) 5.17826e6 0.517739 0.258869 0.965912i \(-0.416650\pi\)
0.258869 + 0.965912i \(0.416650\pi\)
\(632\) −7.67063e6 −0.763903
\(633\) −980521. −0.0972630
\(634\) −2.42257e7 −2.39361
\(635\) 2.81969e6 0.277502
\(636\) 265068. 0.0259846
\(637\) 0 0
\(638\) −1.23237e7 −1.19864
\(639\) −908369. −0.0880055
\(640\) 5.03163e6 0.485578
\(641\) 2.00548e7 1.92785 0.963927 0.266165i \(-0.0857568\pi\)
0.963927 + 0.266165i \(0.0857568\pi\)
\(642\) 6.42846e6 0.615559
\(643\) −1.00964e6 −0.0963032 −0.0481516 0.998840i \(-0.515333\pi\)
−0.0481516 + 0.998840i \(0.515333\pi\)
\(644\) 0 0
\(645\) 1.65373e6 0.156518
\(646\) −2.73706e7 −2.58049
\(647\) 2.66309e6 0.250107 0.125053 0.992150i \(-0.460090\pi\)
0.125053 + 0.992150i \(0.460090\pi\)
\(648\) 4.57196e6 0.427725
\(649\) 9.14531e6 0.852289
\(650\) 321095. 0.0298092
\(651\) 0 0
\(652\) −1.09904e7 −1.01250
\(653\) 6.98872e6 0.641379 0.320689 0.947184i \(-0.396085\pi\)
0.320689 + 0.947184i \(0.396085\pi\)
\(654\) 789429. 0.0721720
\(655\) 4.58943e6 0.417980
\(656\) −4.91384e6 −0.445822
\(657\) 775705. 0.0701105
\(658\) 0 0
\(659\) 1.22406e7 1.09797 0.548985 0.835833i \(-0.315015\pi\)
0.548985 + 0.835833i \(0.315015\pi\)
\(660\) 3.55175e6 0.317383
\(661\) 8.94566e6 0.796359 0.398180 0.917307i \(-0.369642\pi\)
0.398180 + 0.917307i \(0.369642\pi\)
\(662\) −1.73986e7 −1.54301
\(663\) 398383. 0.0351979
\(664\) −4.47406e6 −0.393805
\(665\) 0 0
\(666\) −1.76688e7 −1.54355
\(667\) −4.62362e6 −0.402409
\(668\) 2.42964e7 2.10669
\(669\) 1.64908e6 0.142455
\(670\) 8.05347e6 0.693100
\(671\) −2.37918e7 −2.03996
\(672\) 0 0
\(673\) −8.40438e6 −0.715267 −0.357633 0.933862i \(-0.616416\pi\)
−0.357633 + 0.933862i \(0.616416\pi\)
\(674\) −167811. −0.0142289
\(675\) −1.37312e6 −0.115998
\(676\) −1.62342e7 −1.36635
\(677\) 2.67443e6 0.224264 0.112132 0.993693i \(-0.464232\pi\)
0.112132 + 0.993693i \(0.464232\pi\)
\(678\) 4.18318e6 0.349488
\(679\) 0 0
\(680\) −3.77904e6 −0.313407
\(681\) −3.75853e6 −0.310563
\(682\) −2.13312e7 −1.75612
\(683\) 8.95973e6 0.734925 0.367463 0.930038i \(-0.380227\pi\)
0.367463 + 0.930038i \(0.380227\pi\)
\(684\) 2.13876e7 1.74792
\(685\) 2.30530e6 0.187716
\(686\) 0 0
\(687\) −1.30741e6 −0.105687
\(688\) 6.81636e6 0.549012
\(689\) 74608.0 0.00598739
\(690\) 2.29871e6 0.183807
\(691\) −22616.0 −0.00180186 −0.000900928 1.00000i \(-0.500287\pi\)
−0.000900928 1.00000i \(0.500287\pi\)
\(692\) −4.54710e6 −0.360968
\(693\) 0 0
\(694\) −1.94633e7 −1.53398
\(695\) −1.06087e7 −0.833107
\(696\) 1.04369e6 0.0816672
\(697\) 1.43628e7 1.11985
\(698\) −2.73800e6 −0.212714
\(699\) 6.80566e6 0.526839
\(700\) 0 0
\(701\) 1.80427e7 1.38677 0.693386 0.720566i \(-0.256118\pi\)
0.693386 + 0.720566i \(0.256118\pi\)
\(702\) −1.12871e6 −0.0864451
\(703\) −2.01759e7 −1.53973
\(704\) 3.47195e7 2.64023
\(705\) 1.92815e6 0.146106
\(706\) 1.28569e7 0.970789
\(707\) 0 0
\(708\) −2.81684e6 −0.211193
\(709\) 1.52807e7 1.14164 0.570819 0.821076i \(-0.306626\pi\)
0.570819 + 0.821076i \(0.306626\pi\)
\(710\) −898497. −0.0668915
\(711\) 1.59758e7 1.18519
\(712\) 1.00082e7 0.739874
\(713\) −8.00308e6 −0.589567
\(714\) 0 0
\(715\) 999702. 0.0731317
\(716\) 2.07700e7 1.51409
\(717\) 606211. 0.0440378
\(718\) −1.05679e7 −0.765032
\(719\) 1.45192e7 1.04742 0.523710 0.851897i \(-0.324548\pi\)
0.523710 + 0.851897i \(0.324548\pi\)
\(720\) −2.69274e6 −0.193581
\(721\) 0 0
\(722\) 2.05242e7 1.46529
\(723\) 2.85648e6 0.203229
\(724\) −2.33532e7 −1.65577
\(725\) 1.29974e6 0.0918359
\(726\) 1.24152e7 0.874202
\(727\) 1.98706e7 1.39436 0.697179 0.716897i \(-0.254438\pi\)
0.697179 + 0.716897i \(0.254438\pi\)
\(728\) 0 0
\(729\) −6.99176e6 −0.487268
\(730\) 767275. 0.0532898
\(731\) −1.99238e7 −1.37904
\(732\) 7.32810e6 0.505491
\(733\) −1.58619e7 −1.09042 −0.545211 0.838299i \(-0.683550\pi\)
−0.545211 + 0.838299i \(0.683550\pi\)
\(734\) 2.12777e7 1.45775
\(735\) 0 0
\(736\) 1.70085e7 1.15736
\(737\) 2.50738e7 1.70040
\(738\) −1.93606e7 −1.30851
\(739\) 2.51524e7 1.69421 0.847107 0.531422i \(-0.178342\pi\)
0.847107 + 0.531422i \(0.178342\pi\)
\(740\) −1.01312e7 −0.680116
\(741\) −613202. −0.0410259
\(742\) 0 0
\(743\) 7.30561e6 0.485495 0.242747 0.970090i \(-0.421951\pi\)
0.242747 + 0.970090i \(0.421951\pi\)
\(744\) 1.80653e6 0.119650
\(745\) −3.44735e6 −0.227559
\(746\) −3.04350e7 −2.00229
\(747\) 9.31822e6 0.610986
\(748\) −4.27908e7 −2.79638
\(749\) 0 0
\(750\) −646189. −0.0419475
\(751\) −826829. −0.0534953 −0.0267477 0.999642i \(-0.508515\pi\)
−0.0267477 + 0.999642i \(0.508515\pi\)
\(752\) 7.94747e6 0.512489
\(753\) −3.58781e6 −0.230591
\(754\) 1.06839e6 0.0684389
\(755\) 7.71475e6 0.492554
\(756\) 0 0
\(757\) 4.70431e6 0.298371 0.149185 0.988809i \(-0.452335\pi\)
0.149185 + 0.988809i \(0.452335\pi\)
\(758\) −8.41583e6 −0.532015
\(759\) 7.15683e6 0.450937
\(760\) 5.81680e6 0.365301
\(761\) −1.42681e6 −0.0893107 −0.0446553 0.999002i \(-0.514219\pi\)
−0.0446553 + 0.999002i \(0.514219\pi\)
\(762\) −4.66445e6 −0.291013
\(763\) 0 0
\(764\) 7.97104e6 0.494062
\(765\) 7.87069e6 0.486250
\(766\) 2.93557e7 1.80767
\(767\) −792847. −0.0486632
\(768\) −569993. −0.0348712
\(769\) −9.32444e6 −0.568600 −0.284300 0.958735i \(-0.591761\pi\)
−0.284300 + 0.958735i \(0.591761\pi\)
\(770\) 0 0
\(771\) 2.87512e6 0.174189
\(772\) −2.26355e7 −1.36693
\(773\) −1.52132e7 −0.915741 −0.457870 0.889019i \(-0.651388\pi\)
−0.457870 + 0.889019i \(0.651388\pi\)
\(774\) 2.68566e7 1.61138
\(775\) 2.24974e6 0.134548
\(776\) 9.02921e6 0.538264
\(777\) 0 0
\(778\) −1.03647e7 −0.613916
\(779\) −2.21077e7 −1.30527
\(780\) −307917. −0.0181216
\(781\) −2.79739e6 −0.164106
\(782\) −2.76944e7 −1.61948
\(783\) −4.56885e6 −0.266319
\(784\) 0 0
\(785\) 5.22507e6 0.302634
\(786\) −7.59203e6 −0.438331
\(787\) 2.20016e7 1.26624 0.633122 0.774052i \(-0.281773\pi\)
0.633122 + 0.774052i \(0.281773\pi\)
\(788\) −1.24457e6 −0.0714011
\(789\) −6.72850e6 −0.384792
\(790\) 1.58022e7 0.900843
\(791\) 0 0
\(792\) 1.58598e7 0.898428
\(793\) 2.06262e6 0.116476
\(794\) 1.42807e7 0.803893
\(795\) −150145. −0.00842545
\(796\) 4.12225e7 2.30596
\(797\) 3.02905e7 1.68912 0.844559 0.535462i \(-0.179862\pi\)
0.844559 + 0.535462i \(0.179862\pi\)
\(798\) 0 0
\(799\) −2.32299e7 −1.28730
\(800\) −4.78124e6 −0.264128
\(801\) −2.08444e7 −1.14791
\(802\) −7.78479e6 −0.427377
\(803\) 2.38884e6 0.130737
\(804\) −7.72294e6 −0.421350
\(805\) 0 0
\(806\) 1.84930e6 0.100269
\(807\) −2.35553e6 −0.127322
\(808\) −1.90923e7 −1.02880
\(809\) −1.29572e7 −0.696050 −0.348025 0.937485i \(-0.613148\pi\)
−0.348025 + 0.937485i \(0.613148\pi\)
\(810\) −9.41863e6 −0.504400
\(811\) −1.87248e7 −0.999688 −0.499844 0.866115i \(-0.666609\pi\)
−0.499844 + 0.866115i \(0.666609\pi\)
\(812\) 0 0
\(813\) 737336. 0.0391236
\(814\) −5.44124e7 −2.87831
\(815\) 6.22538e6 0.328301
\(816\) −3.30457e6 −0.173736
\(817\) 3.06673e7 1.60738
\(818\) −2.95622e7 −1.54473
\(819\) 0 0
\(820\) −1.11013e7 −0.576553
\(821\) 2.07863e7 1.07626 0.538132 0.842861i \(-0.319130\pi\)
0.538132 + 0.842861i \(0.319130\pi\)
\(822\) −3.81353e6 −0.196856
\(823\) −2.40815e7 −1.23932 −0.619660 0.784870i \(-0.712730\pi\)
−0.619660 + 0.784870i \(0.712730\pi\)
\(824\) −1.83576e7 −0.941884
\(825\) −2.01185e6 −0.102911
\(826\) 0 0
\(827\) −1.53428e6 −0.0780081 −0.0390041 0.999239i \(-0.512419\pi\)
−0.0390041 + 0.999239i \(0.512419\pi\)
\(828\) 2.16407e7 1.09697
\(829\) −1.47802e7 −0.746952 −0.373476 0.927640i \(-0.621834\pi\)
−0.373476 + 0.927640i \(0.621834\pi\)
\(830\) 9.21695e6 0.464400
\(831\) −8.39242e6 −0.421585
\(832\) −3.00999e6 −0.150750
\(833\) 0 0
\(834\) 1.75494e7 0.873669
\(835\) −1.37624e7 −0.683092
\(836\) 6.58649e7 3.25940
\(837\) −7.90827e6 −0.390183
\(838\) −4.62121e7 −2.27324
\(839\) 2.41688e6 0.118536 0.0592681 0.998242i \(-0.481123\pi\)
0.0592681 + 0.998242i \(0.481123\pi\)
\(840\) 0 0
\(841\) −1.61865e7 −0.789154
\(842\) 4.69531e7 2.28236
\(843\) −1.13171e7 −0.548486
\(844\) −9.13057e6 −0.441206
\(845\) 9.19566e6 0.443038
\(846\) 3.13132e7 1.50418
\(847\) 0 0
\(848\) −618870. −0.0295536
\(849\) −917194. −0.0436709
\(850\) 7.78515e6 0.369590
\(851\) −2.04145e7 −0.966309
\(852\) 861621. 0.0406647
\(853\) −3.98657e7 −1.87598 −0.937988 0.346669i \(-0.887313\pi\)
−0.937988 + 0.346669i \(0.887313\pi\)
\(854\) 0 0
\(855\) −1.21148e7 −0.566762
\(856\) 1.64595e7 0.767770
\(857\) 3.36474e7 1.56495 0.782473 0.622684i \(-0.213958\pi\)
0.782473 + 0.622684i \(0.213958\pi\)
\(858\) −1.65375e6 −0.0766923
\(859\) 8.27879e6 0.382811 0.191405 0.981511i \(-0.438695\pi\)
0.191405 + 0.981511i \(0.438695\pi\)
\(860\) 1.53995e7 0.710001
\(861\) 0 0
\(862\) −5.49533e7 −2.51898
\(863\) 1.06111e7 0.484990 0.242495 0.970153i \(-0.422034\pi\)
0.242495 + 0.970153i \(0.422034\pi\)
\(864\) 1.68070e7 0.765958
\(865\) 2.57565e6 0.117043
\(866\) −3.96640e7 −1.79722
\(867\) 2.92941e6 0.132353
\(868\) 0 0
\(869\) 4.91987e7 2.21006
\(870\) −2.15009e6 −0.0963072
\(871\) −2.17375e6 −0.0970879
\(872\) 2.02126e6 0.0900181
\(873\) −1.88053e7 −0.835113
\(874\) 4.26280e7 1.88763
\(875\) 0 0
\(876\) −735784. −0.0323959
\(877\) −1.80051e6 −0.0790492 −0.0395246 0.999219i \(-0.512584\pi\)
−0.0395246 + 0.999219i \(0.512584\pi\)
\(878\) −2.35148e7 −1.02945
\(879\) −315325. −0.0137653
\(880\) −8.29248e6 −0.360976
\(881\) −3.86161e7 −1.67621 −0.838105 0.545508i \(-0.816337\pi\)
−0.838105 + 0.545508i \(0.816337\pi\)
\(882\) 0 0
\(883\) 8.80155e6 0.379890 0.189945 0.981795i \(-0.439169\pi\)
0.189945 + 0.981795i \(0.439169\pi\)
\(884\) 3.70972e6 0.159665
\(885\) 1.59557e6 0.0684789
\(886\) −2.62573e7 −1.12374
\(887\) 2.76706e7 1.18089 0.590444 0.807078i \(-0.298953\pi\)
0.590444 + 0.807078i \(0.298953\pi\)
\(888\) 4.60817e6 0.196108
\(889\) 0 0
\(890\) −2.06178e7 −0.872505
\(891\) −2.93241e7 −1.23746
\(892\) 1.53562e7 0.646207
\(893\) 3.57562e7 1.50045
\(894\) 5.70275e6 0.238639
\(895\) −1.17649e7 −0.490943
\(896\) 0 0
\(897\) −620457. −0.0257472
\(898\) 3.07220e7 1.27133
\(899\) 7.48565e6 0.308909
\(900\) −6.08340e6 −0.250346
\(901\) 1.80892e6 0.0742347
\(902\) −5.96224e7 −2.44002
\(903\) 0 0
\(904\) 1.07106e7 0.435907
\(905\) 1.32281e7 0.536880
\(906\) −1.27621e7 −0.516536
\(907\) −3.47961e7 −1.40447 −0.702235 0.711945i \(-0.747815\pi\)
−0.702235 + 0.711945i \(0.747815\pi\)
\(908\) −3.49993e7 −1.40878
\(909\) 3.97639e7 1.59617
\(910\) 0 0
\(911\) 2.48752e7 0.993049 0.496524 0.868023i \(-0.334609\pi\)
0.496524 + 0.868023i \(0.334609\pi\)
\(912\) 5.08648e6 0.202502
\(913\) 2.86962e7 1.13932
\(914\) 2.19552e7 0.869306
\(915\) −4.15092e6 −0.163905
\(916\) −1.21745e7 −0.479418
\(917\) 0 0
\(918\) −2.73663e7 −1.07179
\(919\) −1.88971e7 −0.738086 −0.369043 0.929412i \(-0.620314\pi\)
−0.369043 + 0.929412i \(0.620314\pi\)
\(920\) 5.88562e6 0.229257
\(921\) 5.13280e6 0.199391
\(922\) −1.75209e6 −0.0678782
\(923\) 242518. 0.00937000
\(924\) 0 0
\(925\) 5.73872e6 0.220527
\(926\) 1.67696e6 0.0642681
\(927\) 3.82337e7 1.46133
\(928\) −1.59088e7 −0.606412
\(929\) −2.29202e7 −0.871324 −0.435662 0.900110i \(-0.643486\pi\)
−0.435662 + 0.900110i \(0.643486\pi\)
\(930\) −3.72162e6 −0.141099
\(931\) 0 0
\(932\) 6.33741e7 2.38986
\(933\) −7.03341e6 −0.264522
\(934\) 1.30788e7 0.490570
\(935\) 2.42384e7 0.906723
\(936\) −1.37495e6 −0.0512977
\(937\) 3.70307e7 1.37789 0.688943 0.724815i \(-0.258075\pi\)
0.688943 + 0.724815i \(0.258075\pi\)
\(938\) 0 0
\(939\) −1.22937e7 −0.455009
\(940\) 1.79548e7 0.662769
\(941\) −1.25128e7 −0.460660 −0.230330 0.973113i \(-0.573981\pi\)
−0.230330 + 0.973113i \(0.573981\pi\)
\(942\) −8.64353e6 −0.317369
\(943\) −2.23692e7 −0.819166
\(944\) 6.57663e6 0.240200
\(945\) 0 0
\(946\) 8.27068e7 3.00479
\(947\) −1.22987e7 −0.445640 −0.222820 0.974860i \(-0.571526\pi\)
−0.222820 + 0.974860i \(0.571526\pi\)
\(948\) −1.51536e7 −0.547641
\(949\) −207099. −0.00746471
\(950\) −1.19831e7 −0.430786
\(951\) −1.31592e7 −0.471821
\(952\) 0 0
\(953\) 3.53421e6 0.126055 0.0630275 0.998012i \(-0.479924\pi\)
0.0630275 + 0.998012i \(0.479924\pi\)
\(954\) −2.43836e6 −0.0867414
\(955\) −4.51510e6 −0.160199
\(956\) 5.64501e6 0.199765
\(957\) −6.69411e6 −0.236273
\(958\) 6.53021e7 2.29886
\(959\) 0 0
\(960\) 6.05746e6 0.212135
\(961\) −1.56721e7 −0.547419
\(962\) 4.71725e6 0.164343
\(963\) −3.42805e7 −1.19119
\(964\) 2.65994e7 0.921890
\(965\) 1.28216e7 0.443226
\(966\) 0 0
\(967\) −9.47189e6 −0.325740 −0.162870 0.986648i \(-0.552075\pi\)
−0.162870 + 0.986648i \(0.552075\pi\)
\(968\) 3.17879e7 1.09037
\(969\) −1.48675e7 −0.508659
\(970\) −1.86010e7 −0.634755
\(971\) 1.18431e7 0.403106 0.201553 0.979478i \(-0.435401\pi\)
0.201553 + 0.979478i \(0.435401\pi\)
\(972\) 3.25946e7 1.10657
\(973\) 0 0
\(974\) −1.91229e7 −0.645887
\(975\) 174416. 0.00587591
\(976\) −1.71093e7 −0.574921
\(977\) −7.17342e6 −0.240431 −0.120215 0.992748i \(-0.538358\pi\)
−0.120215 + 0.992748i \(0.538358\pi\)
\(978\) −1.02983e7 −0.344285
\(979\) −6.41918e7 −2.14054
\(980\) 0 0
\(981\) −4.20971e6 −0.139663
\(982\) 3.24216e7 1.07289
\(983\) 3.00611e7 0.992251 0.496126 0.868251i \(-0.334755\pi\)
0.496126 + 0.868251i \(0.334755\pi\)
\(984\) 5.04940e6 0.166246
\(985\) 704974. 0.0231517
\(986\) 2.59039e7 0.848540
\(987\) 0 0
\(988\) −5.71011e6 −0.186103
\(989\) 3.10301e7 1.00877
\(990\) −3.26725e7 −1.05948
\(991\) −2.55378e7 −0.826035 −0.413018 0.910723i \(-0.635525\pi\)
−0.413018 + 0.910723i \(0.635525\pi\)
\(992\) −2.75367e7 −0.888451
\(993\) −9.45077e6 −0.304154
\(994\) 0 0
\(995\) −2.33500e7 −0.747703
\(996\) −8.83867e6 −0.282318
\(997\) −2.42489e7 −0.772598 −0.386299 0.922374i \(-0.626247\pi\)
−0.386299 + 0.922374i \(0.626247\pi\)
\(998\) 5.00975e7 1.59217
\(999\) −2.01727e7 −0.639515
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 245.6.a.i.1.6 6
7.2 even 3 35.6.e.a.11.1 12
7.4 even 3 35.6.e.a.16.1 yes 12
7.6 odd 2 245.6.a.h.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.6.e.a.11.1 12 7.2 even 3
35.6.e.a.16.1 yes 12 7.4 even 3
245.6.a.h.1.6 6 7.6 odd 2
245.6.a.i.1.6 6 1.1 even 1 trivial