Properties

Label 245.6.a.i.1.2
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 1.2
Root \(-6.22733\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.22733 q^{2} -15.9209 q^{3} +20.2343 q^{4} -25.0000 q^{5} +115.066 q^{6} +85.0347 q^{8} +10.4759 q^{9} +O(q^{10})\) \(q-7.22733 q^{2} -15.9209 q^{3} +20.2343 q^{4} -25.0000 q^{5} +115.066 q^{6} +85.0347 q^{8} +10.4759 q^{9} +180.683 q^{10} -387.066 q^{11} -322.149 q^{12} -920.823 q^{13} +398.023 q^{15} -1262.07 q^{16} +678.473 q^{17} -75.7130 q^{18} +2764.19 q^{19} -505.857 q^{20} +2797.46 q^{22} +2065.73 q^{23} -1353.83 q^{24} +625.000 q^{25} +6655.09 q^{26} +3702.00 q^{27} -2921.75 q^{29} -2876.64 q^{30} +4255.06 q^{31} +6400.29 q^{32} +6162.46 q^{33} -4903.55 q^{34} +211.973 q^{36} -154.621 q^{37} -19977.7 q^{38} +14660.4 q^{39} -2125.87 q^{40} -7600.35 q^{41} +13533.2 q^{43} -7832.01 q^{44} -261.898 q^{45} -14929.7 q^{46} +15002.5 q^{47} +20093.3 q^{48} -4517.08 q^{50} -10801.9 q^{51} -18632.2 q^{52} +18078.9 q^{53} -26755.6 q^{54} +9676.66 q^{55} -44008.4 q^{57} +21116.4 q^{58} +6479.69 q^{59} +8053.71 q^{60} +4006.33 q^{61} -30752.7 q^{62} -5870.74 q^{64} +23020.6 q^{65} -44538.1 q^{66} -33620.3 q^{67} +13728.4 q^{68} -32888.3 q^{69} -61404.3 q^{71} +890.817 q^{72} -42664.3 q^{73} +1117.50 q^{74} -9950.58 q^{75} +55931.4 q^{76} -105955. q^{78} -107893. q^{79} +31551.8 q^{80} -61484.9 q^{81} +54930.2 q^{82} +99024.8 q^{83} -16961.8 q^{85} -97808.7 q^{86} +46516.9 q^{87} -32914.1 q^{88} +7661.84 q^{89} +1892.82 q^{90} +41798.5 q^{92} -67744.5 q^{93} -108428. q^{94} -69104.7 q^{95} -101899. q^{96} +87087.4 q^{97} -4054.88 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.22733 −1.27762 −0.638812 0.769363i \(-0.720574\pi\)
−0.638812 + 0.769363i \(0.720574\pi\)
\(3\) −15.9209 −1.02133 −0.510664 0.859780i \(-0.670600\pi\)
−0.510664 + 0.859780i \(0.670600\pi\)
\(4\) 20.2343 0.632321
\(5\) −25.0000 −0.447214
\(6\) 115.066 1.30487
\(7\) 0 0
\(8\) 85.0347 0.469755
\(9\) 10.4759 0.0431108
\(10\) 180.683 0.571371
\(11\) −387.066 −0.964504 −0.482252 0.876033i \(-0.660181\pi\)
−0.482252 + 0.876033i \(0.660181\pi\)
\(12\) −322.149 −0.645808
\(13\) −920.823 −1.51118 −0.755592 0.655042i \(-0.772651\pi\)
−0.755592 + 0.655042i \(0.772651\pi\)
\(14\) 0 0
\(15\) 398.023 0.456752
\(16\) −1262.07 −1.23249
\(17\) 678.473 0.569390 0.284695 0.958618i \(-0.408108\pi\)
0.284695 + 0.958618i \(0.408108\pi\)
\(18\) −75.7130 −0.0550794
\(19\) 2764.19 1.75664 0.878322 0.478070i \(-0.158663\pi\)
0.878322 + 0.478070i \(0.158663\pi\)
\(20\) −505.857 −0.282783
\(21\) 0 0
\(22\) 2797.46 1.23227
\(23\) 2065.73 0.814241 0.407121 0.913374i \(-0.366533\pi\)
0.407121 + 0.913374i \(0.366533\pi\)
\(24\) −1353.83 −0.479774
\(25\) 625.000 0.200000
\(26\) 6655.09 1.93073
\(27\) 3702.00 0.977298
\(28\) 0 0
\(29\) −2921.75 −0.645131 −0.322565 0.946547i \(-0.604545\pi\)
−0.322565 + 0.946547i \(0.604545\pi\)
\(30\) −2876.64 −0.583557
\(31\) 4255.06 0.795246 0.397623 0.917549i \(-0.369835\pi\)
0.397623 + 0.917549i \(0.369835\pi\)
\(32\) 6400.29 1.10490
\(33\) 6162.46 0.985074
\(34\) −4903.55 −0.727467
\(35\) 0 0
\(36\) 211.973 0.0272599
\(37\) −154.621 −0.0185680 −0.00928399 0.999957i \(-0.502955\pi\)
−0.00928399 + 0.999957i \(0.502955\pi\)
\(38\) −19977.7 −2.24433
\(39\) 14660.4 1.54342
\(40\) −2125.87 −0.210081
\(41\) −7600.35 −0.706112 −0.353056 0.935602i \(-0.614858\pi\)
−0.353056 + 0.935602i \(0.614858\pi\)
\(42\) 0 0
\(43\) 13533.2 1.11616 0.558082 0.829786i \(-0.311537\pi\)
0.558082 + 0.829786i \(0.311537\pi\)
\(44\) −7832.01 −0.609876
\(45\) −261.898 −0.0192797
\(46\) −14929.7 −1.04029
\(47\) 15002.5 0.990650 0.495325 0.868708i \(-0.335049\pi\)
0.495325 + 0.868708i \(0.335049\pi\)
\(48\) 20093.3 1.25878
\(49\) 0 0
\(50\) −4517.08 −0.255525
\(51\) −10801.9 −0.581534
\(52\) −18632.2 −0.955555
\(53\) 18078.9 0.884060 0.442030 0.897000i \(-0.354259\pi\)
0.442030 + 0.897000i \(0.354259\pi\)
\(54\) −26755.6 −1.24862
\(55\) 9676.66 0.431339
\(56\) 0 0
\(57\) −44008.4 −1.79411
\(58\) 21116.4 0.824234
\(59\) 6479.69 0.242339 0.121170 0.992632i \(-0.461335\pi\)
0.121170 + 0.992632i \(0.461335\pi\)
\(60\) 8053.71 0.288814
\(61\) 4006.33 0.137855 0.0689274 0.997622i \(-0.478042\pi\)
0.0689274 + 0.997622i \(0.478042\pi\)
\(62\) −30752.7 −1.01602
\(63\) 0 0
\(64\) −5870.74 −0.179161
\(65\) 23020.6 0.675822
\(66\) −44538.1 −1.25855
\(67\) −33620.3 −0.914986 −0.457493 0.889213i \(-0.651252\pi\)
−0.457493 + 0.889213i \(0.651252\pi\)
\(68\) 13728.4 0.360038
\(69\) −32888.3 −0.831608
\(70\) 0 0
\(71\) −61404.3 −1.44562 −0.722808 0.691049i \(-0.757149\pi\)
−0.722808 + 0.691049i \(0.757149\pi\)
\(72\) 890.817 0.0202515
\(73\) −42664.3 −0.937039 −0.468520 0.883453i \(-0.655212\pi\)
−0.468520 + 0.883453i \(0.655212\pi\)
\(74\) 1117.50 0.0237229
\(75\) −9950.58 −0.204266
\(76\) 55931.4 1.11076
\(77\) 0 0
\(78\) −105955. −1.97190
\(79\) −107893. −1.94502 −0.972510 0.232862i \(-0.925191\pi\)
−0.972510 + 0.232862i \(0.925191\pi\)
\(80\) 31551.8 0.551187
\(81\) −61484.9 −1.04125
\(82\) 54930.2 0.902145
\(83\) 99024.8 1.57779 0.788894 0.614529i \(-0.210654\pi\)
0.788894 + 0.614529i \(0.210654\pi\)
\(84\) 0 0
\(85\) −16961.8 −0.254639
\(86\) −97808.7 −1.42604
\(87\) 46516.9 0.658890
\(88\) −32914.1 −0.453080
\(89\) 7661.84 0.102532 0.0512659 0.998685i \(-0.483674\pi\)
0.0512659 + 0.998685i \(0.483674\pi\)
\(90\) 1892.82 0.0246323
\(91\) 0 0
\(92\) 41798.5 0.514862
\(93\) −67744.5 −0.812207
\(94\) −108428. −1.26568
\(95\) −69104.7 −0.785595
\(96\) −101899. −1.12847
\(97\) 87087.4 0.939779 0.469890 0.882725i \(-0.344294\pi\)
0.469890 + 0.882725i \(0.344294\pi\)
\(98\) 0 0
\(99\) −4054.88 −0.0415805
\(100\) 12646.4 0.126464
\(101\) 72757.1 0.709695 0.354848 0.934924i \(-0.384533\pi\)
0.354848 + 0.934924i \(0.384533\pi\)
\(102\) 78069.0 0.742982
\(103\) 49822.1 0.462732 0.231366 0.972867i \(-0.425681\pi\)
0.231366 + 0.972867i \(0.425681\pi\)
\(104\) −78301.9 −0.709886
\(105\) 0 0
\(106\) −130662. −1.12950
\(107\) −79308.8 −0.669672 −0.334836 0.942276i \(-0.608681\pi\)
−0.334836 + 0.942276i \(0.608681\pi\)
\(108\) 74907.3 0.617966
\(109\) −183107. −1.47618 −0.738089 0.674703i \(-0.764272\pi\)
−0.738089 + 0.674703i \(0.764272\pi\)
\(110\) −69936.4 −0.551089
\(111\) 2461.71 0.0189640
\(112\) 0 0
\(113\) 137042. 1.00962 0.504809 0.863231i \(-0.331563\pi\)
0.504809 + 0.863231i \(0.331563\pi\)
\(114\) 318064. 2.29220
\(115\) −51643.2 −0.364140
\(116\) −59119.5 −0.407930
\(117\) −9646.47 −0.0651484
\(118\) −46830.8 −0.309618
\(119\) 0 0
\(120\) 33845.8 0.214561
\(121\) −11230.6 −0.0697329
\(122\) −28955.0 −0.176126
\(123\) 121005. 0.721172
\(124\) 86098.1 0.502851
\(125\) −15625.0 −0.0894427
\(126\) 0 0
\(127\) −43001.7 −0.236579 −0.118290 0.992979i \(-0.537741\pi\)
−0.118290 + 0.992979i \(0.537741\pi\)
\(128\) −162380. −0.876005
\(129\) −215461. −1.13997
\(130\) −166377. −0.863447
\(131\) −21205.7 −0.107963 −0.0539815 0.998542i \(-0.517191\pi\)
−0.0539815 + 0.998542i \(0.517191\pi\)
\(132\) 124693. 0.622884
\(133\) 0 0
\(134\) 242985. 1.16901
\(135\) −92550.0 −0.437061
\(136\) 57693.7 0.267474
\(137\) 285197. 1.29820 0.649102 0.760701i \(-0.275145\pi\)
0.649102 + 0.760701i \(0.275145\pi\)
\(138\) 237694. 1.06248
\(139\) −282350. −1.23951 −0.619756 0.784794i \(-0.712769\pi\)
−0.619756 + 0.784794i \(0.712769\pi\)
\(140\) 0 0
\(141\) −238855. −1.01178
\(142\) 443789. 1.84695
\(143\) 356420. 1.45754
\(144\) −13221.4 −0.0531337
\(145\) 73043.7 0.288511
\(146\) 308349. 1.19718
\(147\) 0 0
\(148\) −3128.65 −0.0117409
\(149\) 69436.1 0.256224 0.128112 0.991760i \(-0.459108\pi\)
0.128112 + 0.991760i \(0.459108\pi\)
\(150\) 71916.1 0.260974
\(151\) 258978. 0.924315 0.462157 0.886798i \(-0.347075\pi\)
0.462157 + 0.886798i \(0.347075\pi\)
\(152\) 235052. 0.825192
\(153\) 7107.63 0.0245469
\(154\) 0 0
\(155\) −106376. −0.355645
\(156\) 296642. 0.975935
\(157\) −102995. −0.333477 −0.166738 0.986001i \(-0.553324\pi\)
−0.166738 + 0.986001i \(0.553324\pi\)
\(158\) 779775. 2.48500
\(159\) −287832. −0.902915
\(160\) −160007. −0.494128
\(161\) 0 0
\(162\) 444372. 1.33033
\(163\) −581424. −1.71405 −0.857026 0.515272i \(-0.827691\pi\)
−0.857026 + 0.515272i \(0.827691\pi\)
\(164\) −153788. −0.446490
\(165\) −154061. −0.440539
\(166\) −715685. −2.01582
\(167\) −475920. −1.32051 −0.660257 0.751040i \(-0.729553\pi\)
−0.660257 + 0.751040i \(0.729553\pi\)
\(168\) 0 0
\(169\) 476621. 1.28368
\(170\) 122589. 0.325333
\(171\) 28957.4 0.0757304
\(172\) 273834. 0.705775
\(173\) 285658. 0.725657 0.362828 0.931856i \(-0.381811\pi\)
0.362828 + 0.931856i \(0.381811\pi\)
\(174\) −336193. −0.841813
\(175\) 0 0
\(176\) 488505. 1.18874
\(177\) −103163. −0.247508
\(178\) −55374.7 −0.130997
\(179\) −450892. −1.05182 −0.525909 0.850541i \(-0.676275\pi\)
−0.525909 + 0.850541i \(0.676275\pi\)
\(180\) −5299.32 −0.0121910
\(181\) 151606. 0.343969 0.171985 0.985100i \(-0.444982\pi\)
0.171985 + 0.985100i \(0.444982\pi\)
\(182\) 0 0
\(183\) −63784.4 −0.140795
\(184\) 175658. 0.382494
\(185\) 3865.53 0.00830385
\(186\) 489612. 1.03769
\(187\) −262614. −0.549179
\(188\) 303566. 0.626409
\(189\) 0 0
\(190\) 499443. 1.00369
\(191\) 514986. 1.02144 0.510719 0.859748i \(-0.329379\pi\)
0.510719 + 0.859748i \(0.329379\pi\)
\(192\) 93467.7 0.182982
\(193\) −769968. −1.48792 −0.743959 0.668225i \(-0.767054\pi\)
−0.743959 + 0.668225i \(0.767054\pi\)
\(194\) −629409. −1.20068
\(195\) −366509. −0.690236
\(196\) 0 0
\(197\) −246901. −0.453269 −0.226635 0.973980i \(-0.572772\pi\)
−0.226635 + 0.973980i \(0.572772\pi\)
\(198\) 29306.0 0.0531243
\(199\) 890639. 1.59430 0.797148 0.603783i \(-0.206341\pi\)
0.797148 + 0.603783i \(0.206341\pi\)
\(200\) 53146.7 0.0939509
\(201\) 535266. 0.934501
\(202\) −525839. −0.906723
\(203\) 0 0
\(204\) −218569. −0.367717
\(205\) 190009. 0.315783
\(206\) −360081. −0.591197
\(207\) 21640.4 0.0351026
\(208\) 1.16214e6 1.86252
\(209\) −1.06992e6 −1.69429
\(210\) 0 0
\(211\) −538746. −0.833063 −0.416531 0.909121i \(-0.636754\pi\)
−0.416531 + 0.909121i \(0.636754\pi\)
\(212\) 365813. 0.559010
\(213\) 977614. 1.47645
\(214\) 573191. 0.855588
\(215\) −338329. −0.499164
\(216\) 314798. 0.459090
\(217\) 0 0
\(218\) 1.32338e6 1.88600
\(219\) 679256. 0.957025
\(220\) 195800. 0.272745
\(221\) −624753. −0.860454
\(222\) −17791.6 −0.0242288
\(223\) 1.34117e6 1.80602 0.903011 0.429618i \(-0.141352\pi\)
0.903011 + 0.429618i \(0.141352\pi\)
\(224\) 0 0
\(225\) 6547.46 0.00862216
\(226\) −990447. −1.28991
\(227\) −581025. −0.748394 −0.374197 0.927349i \(-0.622082\pi\)
−0.374197 + 0.927349i \(0.622082\pi\)
\(228\) −890479. −1.13445
\(229\) −998618. −1.25838 −0.629188 0.777253i \(-0.716613\pi\)
−0.629188 + 0.777253i \(0.716613\pi\)
\(230\) 373242. 0.465234
\(231\) 0 0
\(232\) −248450. −0.303053
\(233\) 1.37979e6 1.66503 0.832517 0.553999i \(-0.186899\pi\)
0.832517 + 0.553999i \(0.186899\pi\)
\(234\) 69718.2 0.0832351
\(235\) −375064. −0.443032
\(236\) 131112. 0.153236
\(237\) 1.71775e6 1.98650
\(238\) 0 0
\(239\) 508394. 0.575713 0.287856 0.957674i \(-0.407057\pi\)
0.287856 + 0.957674i \(0.407057\pi\)
\(240\) −502333. −0.562942
\(241\) 185871. 0.206143 0.103072 0.994674i \(-0.467133\pi\)
0.103072 + 0.994674i \(0.467133\pi\)
\(242\) 81166.9 0.0890924
\(243\) 79311.0 0.0861624
\(244\) 81065.1 0.0871685
\(245\) 0 0
\(246\) −874540. −0.921386
\(247\) −2.54533e6 −2.65461
\(248\) 361828. 0.373570
\(249\) −1.57657e6 −1.61144
\(250\) 112927. 0.114274
\(251\) −421837. −0.422630 −0.211315 0.977418i \(-0.567775\pi\)
−0.211315 + 0.977418i \(0.567775\pi\)
\(252\) 0 0
\(253\) −799573. −0.785339
\(254\) 310788. 0.302259
\(255\) 270048. 0.260070
\(256\) 1.36143e6 1.29836
\(257\) −1.94672e6 −1.83853 −0.919265 0.393639i \(-0.871216\pi\)
−0.919265 + 0.393639i \(0.871216\pi\)
\(258\) 1.55720e6 1.45645
\(259\) 0 0
\(260\) 465805. 0.427337
\(261\) −30608.0 −0.0278121
\(262\) 153261. 0.137936
\(263\) 1.63031e6 1.45339 0.726693 0.686962i \(-0.241056\pi\)
0.726693 + 0.686962i \(0.241056\pi\)
\(264\) 524023. 0.462743
\(265\) −451972. −0.395363
\(266\) 0 0
\(267\) −121984. −0.104719
\(268\) −680282. −0.578565
\(269\) 1.37145e6 1.15558 0.577788 0.816187i \(-0.303916\pi\)
0.577788 + 0.816187i \(0.303916\pi\)
\(270\) 668889. 0.558399
\(271\) −761737. −0.630060 −0.315030 0.949082i \(-0.602015\pi\)
−0.315030 + 0.949082i \(0.602015\pi\)
\(272\) −856281. −0.701769
\(273\) 0 0
\(274\) −2.06121e6 −1.65862
\(275\) −241917. −0.192901
\(276\) −665471. −0.525843
\(277\) −2.06875e6 −1.61997 −0.809987 0.586448i \(-0.800526\pi\)
−0.809987 + 0.586448i \(0.800526\pi\)
\(278\) 2.04064e6 1.58363
\(279\) 44575.7 0.0342837
\(280\) 0 0
\(281\) 431842. 0.326257 0.163128 0.986605i \(-0.447842\pi\)
0.163128 + 0.986605i \(0.447842\pi\)
\(282\) 1.72628e6 1.29267
\(283\) −1.43347e6 −1.06395 −0.531977 0.846759i \(-0.678551\pi\)
−0.531977 + 0.846759i \(0.678551\pi\)
\(284\) −1.24247e6 −0.914094
\(285\) 1.10021e6 0.802350
\(286\) −2.57596e6 −1.86219
\(287\) 0 0
\(288\) 67049.0 0.0476333
\(289\) −959531. −0.675794
\(290\) −527911. −0.368609
\(291\) −1.38651e6 −0.959823
\(292\) −863282. −0.592510
\(293\) −138921. −0.0945363 −0.0472682 0.998882i \(-0.515052\pi\)
−0.0472682 + 0.998882i \(0.515052\pi\)
\(294\) 0 0
\(295\) −161992. −0.108377
\(296\) −13148.2 −0.00872240
\(297\) −1.43292e6 −0.942607
\(298\) −501838. −0.327358
\(299\) −1.90217e6 −1.23047
\(300\) −201343. −0.129162
\(301\) 0 0
\(302\) −1.87172e6 −1.18093
\(303\) −1.15836e6 −0.724832
\(304\) −3.48860e6 −2.16505
\(305\) −100158. −0.0616505
\(306\) −51369.2 −0.0313617
\(307\) −631297. −0.382285 −0.191143 0.981562i \(-0.561219\pi\)
−0.191143 + 0.981562i \(0.561219\pi\)
\(308\) 0 0
\(309\) −793215. −0.472601
\(310\) 768818. 0.454380
\(311\) −2.20444e6 −1.29240 −0.646199 0.763169i \(-0.723642\pi\)
−0.646199 + 0.763169i \(0.723642\pi\)
\(312\) 1.24664e6 0.725027
\(313\) 441934. 0.254974 0.127487 0.991840i \(-0.459309\pi\)
0.127487 + 0.991840i \(0.459309\pi\)
\(314\) 744376. 0.426058
\(315\) 0 0
\(316\) −2.18313e6 −1.22988
\(317\) 1.33132e6 0.744103 0.372052 0.928212i \(-0.378654\pi\)
0.372052 + 0.928212i \(0.378654\pi\)
\(318\) 2.08026e6 1.15359
\(319\) 1.13091e6 0.622231
\(320\) 146769. 0.0801232
\(321\) 1.26267e6 0.683954
\(322\) 0 0
\(323\) 1.87543e6 1.00022
\(324\) −1.24410e6 −0.658406
\(325\) −575514. −0.302237
\(326\) 4.20215e6 2.18991
\(327\) 2.91523e6 1.50766
\(328\) −646293. −0.331700
\(329\) 0 0
\(330\) 1.11345e6 0.562843
\(331\) 1.50280e6 0.753931 0.376966 0.926227i \(-0.376967\pi\)
0.376966 + 0.926227i \(0.376967\pi\)
\(332\) 2.00370e6 0.997670
\(333\) −1619.80 −0.000800481 0
\(334\) 3.43963e6 1.68712
\(335\) 840507. 0.409194
\(336\) 0 0
\(337\) −2.34434e6 −1.12447 −0.562233 0.826979i \(-0.690058\pi\)
−0.562233 + 0.826979i \(0.690058\pi\)
\(338\) −3.44470e6 −1.64006
\(339\) −2.18183e6 −1.03115
\(340\) −343210. −0.161014
\(341\) −1.64699e6 −0.767017
\(342\) −209285. −0.0967549
\(343\) 0 0
\(344\) 1.15079e6 0.524324
\(345\) 822207. 0.371906
\(346\) −2.06454e6 −0.927116
\(347\) 279935. 0.124805 0.0624026 0.998051i \(-0.480124\pi\)
0.0624026 + 0.998051i \(0.480124\pi\)
\(348\) 941237. 0.416630
\(349\) 1.16659e6 0.512691 0.256346 0.966585i \(-0.417481\pi\)
0.256346 + 0.966585i \(0.417481\pi\)
\(350\) 0 0
\(351\) −3.40888e6 −1.47688
\(352\) −2.47734e6 −1.06568
\(353\) 1.55443e6 0.663949 0.331975 0.943288i \(-0.392285\pi\)
0.331975 + 0.943288i \(0.392285\pi\)
\(354\) 745590. 0.316222
\(355\) 1.53511e6 0.646499
\(356\) 155032. 0.0648330
\(357\) 0 0
\(358\) 3.25875e6 1.34383
\(359\) 856494. 0.350742 0.175371 0.984502i \(-0.443887\pi\)
0.175371 + 0.984502i \(0.443887\pi\)
\(360\) −22270.4 −0.00905675
\(361\) 5.16464e6 2.08580
\(362\) −1.09571e6 −0.439463
\(363\) 178801. 0.0712202
\(364\) 0 0
\(365\) 1.06661e6 0.419057
\(366\) 460991. 0.179883
\(367\) −909272. −0.352394 −0.176197 0.984355i \(-0.556380\pi\)
−0.176197 + 0.984355i \(0.556380\pi\)
\(368\) −2.60709e6 −1.00355
\(369\) −79620.7 −0.0304411
\(370\) −27937.5 −0.0106092
\(371\) 0 0
\(372\) −1.37076e6 −0.513576
\(373\) −133835. −0.0498080 −0.0249040 0.999690i \(-0.507928\pi\)
−0.0249040 + 0.999690i \(0.507928\pi\)
\(374\) 1.89800e6 0.701644
\(375\) 248764. 0.0913504
\(376\) 1.27574e6 0.465363
\(377\) 2.69041e6 0.974912
\(378\) 0 0
\(379\) 1.07715e6 0.385192 0.192596 0.981278i \(-0.438309\pi\)
0.192596 + 0.981278i \(0.438309\pi\)
\(380\) −1.39828e6 −0.496749
\(381\) 684627. 0.241625
\(382\) −3.72198e6 −1.30501
\(383\) 461084. 0.160614 0.0803069 0.996770i \(-0.474410\pi\)
0.0803069 + 0.996770i \(0.474410\pi\)
\(384\) 2.58523e6 0.894688
\(385\) 0 0
\(386\) 5.56481e6 1.90100
\(387\) 141773. 0.0481188
\(388\) 1.76215e6 0.594243
\(389\) 235175. 0.0787983 0.0393992 0.999224i \(-0.487456\pi\)
0.0393992 + 0.999224i \(0.487456\pi\)
\(390\) 2.64888e6 0.881862
\(391\) 1.40154e6 0.463621
\(392\) 0 0
\(393\) 337615. 0.110266
\(394\) 1.78443e6 0.579108
\(395\) 2.69732e6 0.869839
\(396\) −82047.6 −0.0262923
\(397\) −3.60978e6 −1.14949 −0.574744 0.818334i \(-0.694898\pi\)
−0.574744 + 0.818334i \(0.694898\pi\)
\(398\) −6.43694e6 −2.03691
\(399\) 0 0
\(400\) −788794. −0.246498
\(401\) 4.78064e6 1.48465 0.742327 0.670038i \(-0.233722\pi\)
0.742327 + 0.670038i \(0.233722\pi\)
\(402\) −3.86854e6 −1.19394
\(403\) −3.91815e6 −1.20176
\(404\) 1.47219e6 0.448756
\(405\) 1.53712e6 0.465662
\(406\) 0 0
\(407\) 59848.7 0.0179089
\(408\) −918538. −0.273179
\(409\) 332478. 0.0982778 0.0491389 0.998792i \(-0.484352\pi\)
0.0491389 + 0.998792i \(0.484352\pi\)
\(410\) −1.37325e6 −0.403452
\(411\) −4.54060e6 −1.32589
\(412\) 1.00812e6 0.292595
\(413\) 0 0
\(414\) −156402. −0.0448479
\(415\) −2.47562e6 −0.705609
\(416\) −5.89353e6 −1.66972
\(417\) 4.49528e6 1.26595
\(418\) 7.73270e6 2.16466
\(419\) −3.01019e6 −0.837643 −0.418822 0.908068i \(-0.637557\pi\)
−0.418822 + 0.908068i \(0.637557\pi\)
\(420\) 0 0
\(421\) −2.57143e6 −0.707083 −0.353541 0.935419i \(-0.615023\pi\)
−0.353541 + 0.935419i \(0.615023\pi\)
\(422\) 3.89369e6 1.06434
\(423\) 157166. 0.0427078
\(424\) 1.53733e6 0.415291
\(425\) 424046. 0.113878
\(426\) −7.06554e6 −1.88634
\(427\) 0 0
\(428\) −1.60476e6 −0.423448
\(429\) −5.67453e6 −1.48863
\(430\) 2.44522e6 0.637744
\(431\) −5.76915e6 −1.49596 −0.747978 0.663723i \(-0.768975\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(432\) −4.67218e6 −1.20451
\(433\) −1.93748e6 −0.496611 −0.248306 0.968682i \(-0.579874\pi\)
−0.248306 + 0.968682i \(0.579874\pi\)
\(434\) 0 0
\(435\) −1.16292e6 −0.294665
\(436\) −3.70504e6 −0.933419
\(437\) 5.71006e6 1.43033
\(438\) −4.90921e6 −1.22272
\(439\) 4.48026e6 1.10954 0.554769 0.832004i \(-0.312807\pi\)
0.554769 + 0.832004i \(0.312807\pi\)
\(440\) 822852. 0.202624
\(441\) 0 0
\(442\) 4.51530e6 1.09934
\(443\) 2.62761e6 0.636139 0.318069 0.948067i \(-0.396965\pi\)
0.318069 + 0.948067i \(0.396965\pi\)
\(444\) 49811.0 0.0119913
\(445\) −191546. −0.0458536
\(446\) −9.69310e6 −2.30742
\(447\) −1.10549e6 −0.261689
\(448\) 0 0
\(449\) −6.11083e6 −1.43049 −0.715244 0.698875i \(-0.753684\pi\)
−0.715244 + 0.698875i \(0.753684\pi\)
\(450\) −47320.6 −0.0110159
\(451\) 2.94184e6 0.681048
\(452\) 2.77294e6 0.638403
\(453\) −4.12316e6 −0.944029
\(454\) 4.19926e6 0.956166
\(455\) 0 0
\(456\) −3.74224e6 −0.842791
\(457\) 3.93604e6 0.881595 0.440797 0.897607i \(-0.354696\pi\)
0.440797 + 0.897607i \(0.354696\pi\)
\(458\) 7.21734e6 1.60773
\(459\) 2.51171e6 0.556464
\(460\) −1.04496e6 −0.230253
\(461\) 3.61769e6 0.792827 0.396414 0.918072i \(-0.370255\pi\)
0.396414 + 0.918072i \(0.370255\pi\)
\(462\) 0 0
\(463\) 6.68308e6 1.44885 0.724426 0.689352i \(-0.242105\pi\)
0.724426 + 0.689352i \(0.242105\pi\)
\(464\) 3.68745e6 0.795118
\(465\) 1.69361e6 0.363230
\(466\) −9.97219e6 −2.12729
\(467\) −2.36830e6 −0.502510 −0.251255 0.967921i \(-0.580843\pi\)
−0.251255 + 0.967921i \(0.580843\pi\)
\(468\) −195189. −0.0411947
\(469\) 0 0
\(470\) 2.71071e6 0.566028
\(471\) 1.63977e6 0.340589
\(472\) 550998. 0.113840
\(473\) −5.23824e6 −1.07654
\(474\) −1.24147e7 −2.53800
\(475\) 1.72762e6 0.351329
\(476\) 0 0
\(477\) 189393. 0.0381125
\(478\) −3.67433e6 −0.735544
\(479\) −2.26976e6 −0.452004 −0.226002 0.974127i \(-0.572566\pi\)
−0.226002 + 0.974127i \(0.572566\pi\)
\(480\) 2.54746e6 0.504667
\(481\) 142379. 0.0280597
\(482\) −1.34335e6 −0.263373
\(483\) 0 0
\(484\) −227242. −0.0440936
\(485\) −2.17718e6 −0.420282
\(486\) −573207. −0.110083
\(487\) −1.69003e6 −0.322904 −0.161452 0.986881i \(-0.551618\pi\)
−0.161452 + 0.986881i \(0.551618\pi\)
\(488\) 340677. 0.0647579
\(489\) 9.25681e6 1.75061
\(490\) 0 0
\(491\) 3.12583e6 0.585143 0.292572 0.956244i \(-0.405489\pi\)
0.292572 + 0.956244i \(0.405489\pi\)
\(492\) 2.44844e6 0.456013
\(493\) −1.98233e6 −0.367331
\(494\) 1.83959e7 3.39160
\(495\) 101372. 0.0185954
\(496\) −5.37018e6 −0.980133
\(497\) 0 0
\(498\) 1.13944e7 2.05881
\(499\) −9.40680e6 −1.69118 −0.845591 0.533831i \(-0.820752\pi\)
−0.845591 + 0.533831i \(0.820752\pi\)
\(500\) −316161. −0.0565565
\(501\) 7.57709e6 1.34868
\(502\) 3.04875e6 0.539961
\(503\) 7.23245e6 1.27458 0.637288 0.770626i \(-0.280056\pi\)
0.637288 + 0.770626i \(0.280056\pi\)
\(504\) 0 0
\(505\) −1.81893e6 −0.317385
\(506\) 5.77878e6 1.00337
\(507\) −7.58825e6 −1.31106
\(508\) −870109. −0.149594
\(509\) 4.10701e6 0.702638 0.351319 0.936256i \(-0.385733\pi\)
0.351319 + 0.936256i \(0.385733\pi\)
\(510\) −1.95173e6 −0.332272
\(511\) 0 0
\(512\) −4.64339e6 −0.782816
\(513\) 1.02330e7 1.71676
\(514\) 1.40696e7 2.34895
\(515\) −1.24555e6 −0.206940
\(516\) −4.35969e6 −0.720828
\(517\) −5.80698e6 −0.955486
\(518\) 0 0
\(519\) −4.54794e6 −0.741134
\(520\) 1.95755e6 0.317471
\(521\) 6.80125e6 1.09773 0.548863 0.835912i \(-0.315061\pi\)
0.548863 + 0.835912i \(0.315061\pi\)
\(522\) 221214. 0.0355334
\(523\) −3.31515e6 −0.529968 −0.264984 0.964253i \(-0.585367\pi\)
−0.264984 + 0.964253i \(0.585367\pi\)
\(524\) −429083. −0.0682674
\(525\) 0 0
\(526\) −1.17828e7 −1.85688
\(527\) 2.88694e6 0.452805
\(528\) −7.77746e6 −1.21410
\(529\) −2.16912e6 −0.337011
\(530\) 3.26655e6 0.505126
\(531\) 67880.7 0.0104475
\(532\) 0 0
\(533\) 6.99857e6 1.06707
\(534\) 881616. 0.133791
\(535\) 1.98272e6 0.299486
\(536\) −2.85889e6 −0.429819
\(537\) 7.17863e6 1.07425
\(538\) −9.91190e6 −1.47639
\(539\) 0 0
\(540\) −1.87268e6 −0.276363
\(541\) 3.47955e6 0.511129 0.255564 0.966792i \(-0.417739\pi\)
0.255564 + 0.966792i \(0.417739\pi\)
\(542\) 5.50532e6 0.804979
\(543\) −2.41371e6 −0.351305
\(544\) 4.34242e6 0.629122
\(545\) 4.57768e6 0.660167
\(546\) 0 0
\(547\) 2.55022e6 0.364426 0.182213 0.983259i \(-0.441674\pi\)
0.182213 + 0.983259i \(0.441674\pi\)
\(548\) 5.77075e6 0.820882
\(549\) 41970.0 0.00594303
\(550\) 1.74841e6 0.246454
\(551\) −8.07626e6 −1.13326
\(552\) −2.79664e6 −0.390652
\(553\) 0 0
\(554\) 1.49515e7 2.06972
\(555\) −61542.8 −0.00848096
\(556\) −5.71315e6 −0.783771
\(557\) 3.34227e6 0.456461 0.228230 0.973607i \(-0.426706\pi\)
0.228230 + 0.973607i \(0.426706\pi\)
\(558\) −322163. −0.0438016
\(559\) −1.24616e7 −1.68673
\(560\) 0 0
\(561\) 4.18106e6 0.560892
\(562\) −3.12107e6 −0.416833
\(563\) −6.30412e6 −0.838212 −0.419106 0.907937i \(-0.637656\pi\)
−0.419106 + 0.907937i \(0.637656\pi\)
\(564\) −4.83305e6 −0.639770
\(565\) −3.42605e6 −0.451515
\(566\) 1.03602e7 1.35933
\(567\) 0 0
\(568\) −5.22150e6 −0.679085
\(569\) 3.52056e6 0.455860 0.227930 0.973678i \(-0.426804\pi\)
0.227930 + 0.973678i \(0.426804\pi\)
\(570\) −7.95159e6 −1.02510
\(571\) −7.51318e6 −0.964348 −0.482174 0.876076i \(-0.660153\pi\)
−0.482174 + 0.876076i \(0.660153\pi\)
\(572\) 7.21190e6 0.921636
\(573\) −8.19906e6 −1.04322
\(574\) 0 0
\(575\) 1.29108e6 0.162848
\(576\) −61501.5 −0.00772377
\(577\) −6.67495e6 −0.834658 −0.417329 0.908756i \(-0.637034\pi\)
−0.417329 + 0.908756i \(0.637034\pi\)
\(578\) 6.93485e6 0.863411
\(579\) 1.22586e7 1.51965
\(580\) 1.47799e6 0.182432
\(581\) 0 0
\(582\) 1.00208e7 1.22629
\(583\) −6.99772e6 −0.852679
\(584\) −3.62795e6 −0.440179
\(585\) 241162. 0.0291353
\(586\) 1.00403e6 0.120782
\(587\) −1.44246e7 −1.72785 −0.863927 0.503617i \(-0.832002\pi\)
−0.863927 + 0.503617i \(0.832002\pi\)
\(588\) 0 0
\(589\) 1.17618e7 1.39696
\(590\) 1.17077e6 0.138466
\(591\) 3.93089e6 0.462937
\(592\) 195143. 0.0228849
\(593\) 8.63292e6 1.00814 0.504070 0.863663i \(-0.331835\pi\)
0.504070 + 0.863663i \(0.331835\pi\)
\(594\) 1.03562e7 1.20430
\(595\) 0 0
\(596\) 1.40499e6 0.162016
\(597\) −1.41798e7 −1.62830
\(598\) 1.37476e7 1.57208
\(599\) 5.75996e6 0.655922 0.327961 0.944691i \(-0.393638\pi\)
0.327961 + 0.944691i \(0.393638\pi\)
\(600\) −846144. −0.0959547
\(601\) −6.23001e6 −0.703562 −0.351781 0.936082i \(-0.614424\pi\)
−0.351781 + 0.936082i \(0.614424\pi\)
\(602\) 0 0
\(603\) −352204. −0.0394458
\(604\) 5.24023e6 0.584464
\(605\) 280764. 0.0311855
\(606\) 8.37185e6 0.926062
\(607\) 2.79863e6 0.308301 0.154150 0.988047i \(-0.450736\pi\)
0.154150 + 0.988047i \(0.450736\pi\)
\(608\) 1.76916e7 1.94092
\(609\) 0 0
\(610\) 723876. 0.0787661
\(611\) −1.38147e7 −1.49706
\(612\) 143818. 0.0155215
\(613\) 1.24192e7 1.33488 0.667438 0.744665i \(-0.267391\pi\)
0.667438 + 0.744665i \(0.267391\pi\)
\(614\) 4.56259e6 0.488417
\(615\) −3.02511e6 −0.322518
\(616\) 0 0
\(617\) 5.63980e6 0.596418 0.298209 0.954501i \(-0.403611\pi\)
0.298209 + 0.954501i \(0.403611\pi\)
\(618\) 5.73282e6 0.603806
\(619\) 5.79714e6 0.608117 0.304059 0.952653i \(-0.401658\pi\)
0.304059 + 0.952653i \(0.401658\pi\)
\(620\) −2.15245e6 −0.224882
\(621\) 7.64732e6 0.795756
\(622\) 1.59322e7 1.65120
\(623\) 0 0
\(624\) −1.85024e7 −1.90225
\(625\) 390625. 0.0400000
\(626\) −3.19400e6 −0.325761
\(627\) 1.70342e7 1.73042
\(628\) −2.08402e6 −0.210864
\(629\) −104906. −0.0105724
\(630\) 0 0
\(631\) −3.41197e6 −0.341140 −0.170570 0.985346i \(-0.554561\pi\)
−0.170570 + 0.985346i \(0.554561\pi\)
\(632\) −9.17461e6 −0.913682
\(633\) 8.57733e6 0.850830
\(634\) −9.62186e6 −0.950684
\(635\) 1.07504e6 0.105801
\(636\) −5.82408e6 −0.570932
\(637\) 0 0
\(638\) −8.17346e6 −0.794977
\(639\) −643267. −0.0623217
\(640\) 4.05949e6 0.391761
\(641\) −1.18486e7 −1.13900 −0.569499 0.821992i \(-0.692863\pi\)
−0.569499 + 0.821992i \(0.692863\pi\)
\(642\) −9.12573e6 −0.873836
\(643\) −6.05156e6 −0.577218 −0.288609 0.957447i \(-0.593193\pi\)
−0.288609 + 0.957447i \(0.593193\pi\)
\(644\) 0 0
\(645\) 5.38651e6 0.509810
\(646\) −1.35543e7 −1.27790
\(647\) −9.82251e6 −0.922490 −0.461245 0.887273i \(-0.652597\pi\)
−0.461245 + 0.887273i \(0.652597\pi\)
\(648\) −5.22835e6 −0.489133
\(649\) −2.50807e6 −0.233737
\(650\) 4.15943e6 0.386145
\(651\) 0 0
\(652\) −1.17647e7 −1.08383
\(653\) −957185. −0.0878442 −0.0439221 0.999035i \(-0.513985\pi\)
−0.0439221 + 0.999035i \(0.513985\pi\)
\(654\) −2.10694e7 −1.92622
\(655\) 530144. 0.0482826
\(656\) 9.59217e6 0.870277
\(657\) −446949. −0.0403965
\(658\) 0 0
\(659\) −1.54679e7 −1.38745 −0.693725 0.720240i \(-0.744032\pi\)
−0.693725 + 0.720240i \(0.744032\pi\)
\(660\) −3.11732e6 −0.278562
\(661\) −3.95036e6 −0.351668 −0.175834 0.984420i \(-0.556262\pi\)
−0.175834 + 0.984420i \(0.556262\pi\)
\(662\) −1.08612e7 −0.963240
\(663\) 9.94665e6 0.878806
\(664\) 8.42055e6 0.741174
\(665\) 0 0
\(666\) 11706.8 0.00102271
\(667\) −6.03553e6 −0.525292
\(668\) −9.62990e6 −0.834989
\(669\) −2.13527e7 −1.84454
\(670\) −6.07462e6 −0.522796
\(671\) −1.55071e6 −0.132961
\(672\) 0 0
\(673\) −1.64772e7 −1.40231 −0.701157 0.713007i \(-0.747333\pi\)
−0.701157 + 0.713007i \(0.747333\pi\)
\(674\) 1.69433e7 1.43664
\(675\) 2.31375e6 0.195460
\(676\) 9.64409e6 0.811698
\(677\) −9.00324e6 −0.754966 −0.377483 0.926017i \(-0.623210\pi\)
−0.377483 + 0.926017i \(0.623210\pi\)
\(678\) 1.57688e7 1.31742
\(679\) 0 0
\(680\) −1.44234e6 −0.119618
\(681\) 9.25046e6 0.764356
\(682\) 1.19033e7 0.979959
\(683\) 9.38382e6 0.769711 0.384856 0.922977i \(-0.374251\pi\)
0.384856 + 0.922977i \(0.374251\pi\)
\(684\) 585933. 0.0478859
\(685\) −7.12992e6 −0.580575
\(686\) 0 0
\(687\) 1.58989e7 1.28522
\(688\) −1.70798e7 −1.37566
\(689\) −1.66474e7 −1.33598
\(690\) −5.94236e6 −0.475156
\(691\) −5.05803e6 −0.402983 −0.201491 0.979490i \(-0.564579\pi\)
−0.201491 + 0.979490i \(0.564579\pi\)
\(692\) 5.78009e6 0.458848
\(693\) 0 0
\(694\) −2.02318e6 −0.159454
\(695\) 7.05875e6 0.554327
\(696\) 3.95555e6 0.309517
\(697\) −5.15663e6 −0.402054
\(698\) −8.43135e6 −0.655026
\(699\) −2.19675e7 −1.70055
\(700\) 0 0
\(701\) −2.25374e7 −1.73224 −0.866121 0.499835i \(-0.833394\pi\)
−0.866121 + 0.499835i \(0.833394\pi\)
\(702\) 2.46371e7 1.88689
\(703\) −427402. −0.0326173
\(704\) 2.27237e6 0.172801
\(705\) 5.97136e6 0.452481
\(706\) −1.12344e7 −0.848277
\(707\) 0 0
\(708\) −2.08742e6 −0.156505
\(709\) −2.37037e6 −0.177093 −0.0885463 0.996072i \(-0.528222\pi\)
−0.0885463 + 0.996072i \(0.528222\pi\)
\(710\) −1.10947e7 −0.825982
\(711\) −1.13028e6 −0.0838514
\(712\) 651523. 0.0481648
\(713\) 8.78979e6 0.647522
\(714\) 0 0
\(715\) −8.91049e6 −0.651833
\(716\) −9.12349e6 −0.665087
\(717\) −8.09411e6 −0.587992
\(718\) −6.19016e6 −0.448117
\(719\) −1.25461e7 −0.905077 −0.452539 0.891745i \(-0.649482\pi\)
−0.452539 + 0.891745i \(0.649482\pi\)
\(720\) 330534. 0.0237621
\(721\) 0 0
\(722\) −3.73266e7 −2.66486
\(723\) −2.95924e6 −0.210540
\(724\) 3.06764e6 0.217499
\(725\) −1.82609e6 −0.129026
\(726\) −1.29225e6 −0.0909926
\(727\) 5.18009e6 0.363498 0.181749 0.983345i \(-0.441824\pi\)
0.181749 + 0.983345i \(0.441824\pi\)
\(728\) 0 0
\(729\) 1.36781e7 0.953252
\(730\) −7.70873e6 −0.535397
\(731\) 9.18189e6 0.635534
\(732\) −1.29063e6 −0.0890276
\(733\) −1.54379e7 −1.06127 −0.530637 0.847599i \(-0.678047\pi\)
−0.530637 + 0.847599i \(0.678047\pi\)
\(734\) 6.57161e6 0.450227
\(735\) 0 0
\(736\) 1.32213e7 0.899659
\(737\) 1.30133e7 0.882507
\(738\) 575445. 0.0388922
\(739\) −1.64732e7 −1.10960 −0.554801 0.831983i \(-0.687206\pi\)
−0.554801 + 0.831983i \(0.687206\pi\)
\(740\) 78216.2 0.00525070
\(741\) 4.05240e7 2.71123
\(742\) 0 0
\(743\) −2.59403e7 −1.72386 −0.861931 0.507026i \(-0.830745\pi\)
−0.861931 + 0.507026i \(0.830745\pi\)
\(744\) −5.76063e6 −0.381538
\(745\) −1.73590e6 −0.114587
\(746\) 967273. 0.0636358
\(747\) 1.03738e6 0.0680198
\(748\) −5.31381e6 −0.347258
\(749\) 0 0
\(750\) −1.79790e6 −0.116711
\(751\) 1.18608e7 0.767386 0.383693 0.923461i \(-0.374652\pi\)
0.383693 + 0.923461i \(0.374652\pi\)
\(752\) −1.89343e7 −1.22097
\(753\) 6.71603e6 0.431643
\(754\) −1.94445e7 −1.24557
\(755\) −6.47444e6 −0.413366
\(756\) 0 0
\(757\) −6.25104e6 −0.396472 −0.198236 0.980154i \(-0.563521\pi\)
−0.198236 + 0.980154i \(0.563521\pi\)
\(758\) −7.78491e6 −0.492131
\(759\) 1.27300e7 0.802088
\(760\) −5.87630e6 −0.369037
\(761\) 2.20531e7 1.38041 0.690206 0.723613i \(-0.257520\pi\)
0.690206 + 0.723613i \(0.257520\pi\)
\(762\) −4.94803e6 −0.308706
\(763\) 0 0
\(764\) 1.04204e7 0.645877
\(765\) −177691. −0.0109777
\(766\) −3.33240e6 −0.205204
\(767\) −5.96664e6 −0.366220
\(768\) −2.16753e7 −1.32606
\(769\) −1.37289e7 −0.837181 −0.418591 0.908175i \(-0.637476\pi\)
−0.418591 + 0.908175i \(0.637476\pi\)
\(770\) 0 0
\(771\) 3.09936e7 1.87774
\(772\) −1.55797e7 −0.940843
\(773\) −1.50716e7 −0.907213 −0.453606 0.891202i \(-0.649863\pi\)
−0.453606 + 0.891202i \(0.649863\pi\)
\(774\) −1.02464e6 −0.0614777
\(775\) 2.65941e6 0.159049
\(776\) 7.40545e6 0.441466
\(777\) 0 0
\(778\) −1.69969e6 −0.100675
\(779\) −2.10088e7 −1.24039
\(780\) −7.41604e6 −0.436451
\(781\) 2.37675e7 1.39430
\(782\) −1.01294e7 −0.592333
\(783\) −1.08163e7 −0.630485
\(784\) 0 0
\(785\) 2.57487e6 0.149135
\(786\) −2.44006e6 −0.140878
\(787\) 1.93056e7 1.11108 0.555541 0.831489i \(-0.312511\pi\)
0.555541 + 0.831489i \(0.312511\pi\)
\(788\) −4.99586e6 −0.286612
\(789\) −2.59561e7 −1.48438
\(790\) −1.94944e7 −1.11133
\(791\) 0 0
\(792\) −344806. −0.0195327
\(793\) −3.68911e6 −0.208324
\(794\) 2.60891e7 1.46861
\(795\) 7.19581e6 0.403796
\(796\) 1.80215e7 1.00811
\(797\) 2.05347e7 1.14510 0.572549 0.819871i \(-0.305955\pi\)
0.572549 + 0.819871i \(0.305955\pi\)
\(798\) 0 0
\(799\) 1.01788e7 0.564067
\(800\) 4.00018e6 0.220981
\(801\) 80264.9 0.00442023
\(802\) −3.45513e7 −1.89683
\(803\) 1.65139e7 0.903778
\(804\) 1.08307e7 0.590905
\(805\) 0 0
\(806\) 2.83178e7 1.53540
\(807\) −2.18347e7 −1.18022
\(808\) 6.18688e6 0.333383
\(809\) 7.34708e6 0.394678 0.197339 0.980335i \(-0.436770\pi\)
0.197339 + 0.980335i \(0.436770\pi\)
\(810\) −1.11093e7 −0.594941
\(811\) 2.36201e7 1.26104 0.630522 0.776171i \(-0.282841\pi\)
0.630522 + 0.776171i \(0.282841\pi\)
\(812\) 0 0
\(813\) 1.21276e7 0.643498
\(814\) −432546. −0.0228808
\(815\) 1.45356e7 0.766548
\(816\) 1.36328e7 0.716736
\(817\) 3.74082e7 1.96070
\(818\) −2.40293e6 −0.125562
\(819\) 0 0
\(820\) 3.84469e6 0.199676
\(821\) 1.84495e7 0.955272 0.477636 0.878558i \(-0.341494\pi\)
0.477636 + 0.878558i \(0.341494\pi\)
\(822\) 3.28164e7 1.69399
\(823\) −1.01309e7 −0.521374 −0.260687 0.965423i \(-0.583949\pi\)
−0.260687 + 0.965423i \(0.583949\pi\)
\(824\) 4.23661e6 0.217370
\(825\) 3.85154e6 0.197015
\(826\) 0 0
\(827\) −9.10559e6 −0.462961 −0.231480 0.972840i \(-0.574357\pi\)
−0.231480 + 0.972840i \(0.574357\pi\)
\(828\) 437878. 0.0221961
\(829\) −2.40432e7 −1.21508 −0.607540 0.794289i \(-0.707844\pi\)
−0.607540 + 0.794289i \(0.707844\pi\)
\(830\) 1.78921e7 0.901502
\(831\) 3.29364e7 1.65452
\(832\) 5.40591e6 0.270745
\(833\) 0 0
\(834\) −3.24888e7 −1.61741
\(835\) 1.18980e7 0.590552
\(836\) −2.16492e7 −1.07134
\(837\) 1.57522e7 0.777192
\(838\) 2.17557e7 1.07019
\(839\) −3.70147e7 −1.81539 −0.907694 0.419633i \(-0.862159\pi\)
−0.907694 + 0.419633i \(0.862159\pi\)
\(840\) 0 0
\(841\) −1.19745e7 −0.583806
\(842\) 1.85846e7 0.903385
\(843\) −6.87533e6 −0.333215
\(844\) −1.09011e7 −0.526763
\(845\) −1.19155e7 −0.574079
\(846\) −1.13589e6 −0.0545644
\(847\) 0 0
\(848\) −2.28168e7 −1.08960
\(849\) 2.28222e7 1.08665
\(850\) −3.06472e6 −0.145493
\(851\) −319405. −0.0151188
\(852\) 1.97813e7 0.933590
\(853\) −79427.6 −0.00373765 −0.00186883 0.999998i \(-0.500595\pi\)
−0.00186883 + 0.999998i \(0.500595\pi\)
\(854\) 0 0
\(855\) −723936. −0.0338676
\(856\) −6.74400e6 −0.314581
\(857\) 1.28759e7 0.598859 0.299430 0.954118i \(-0.403204\pi\)
0.299430 + 0.954118i \(0.403204\pi\)
\(858\) 4.10117e7 1.90191
\(859\) 3.82897e6 0.177051 0.0885256 0.996074i \(-0.471784\pi\)
0.0885256 + 0.996074i \(0.471784\pi\)
\(860\) −6.84585e6 −0.315632
\(861\) 0 0
\(862\) 4.16956e7 1.91127
\(863\) −1.27055e7 −0.580718 −0.290359 0.956918i \(-0.593775\pi\)
−0.290359 + 0.956918i \(0.593775\pi\)
\(864\) 2.36939e7 1.07982
\(865\) −7.14145e6 −0.324524
\(866\) 1.40028e7 0.634482
\(867\) 1.52766e7 0.690208
\(868\) 0 0
\(869\) 4.17616e7 1.87598
\(870\) 8.40483e6 0.376470
\(871\) 3.09583e7 1.38271
\(872\) −1.55705e7 −0.693442
\(873\) 912321. 0.0405147
\(874\) −4.12685e7 −1.82743
\(875\) 0 0
\(876\) 1.37443e7 0.605147
\(877\) −1.88584e7 −0.827953 −0.413977 0.910288i \(-0.635860\pi\)
−0.413977 + 0.910288i \(0.635860\pi\)
\(878\) −3.23803e7 −1.41757
\(879\) 2.21175e6 0.0965526
\(880\) −1.22126e7 −0.531622
\(881\) −3.82921e7 −1.66215 −0.831073 0.556163i \(-0.812273\pi\)
−0.831073 + 0.556163i \(0.812273\pi\)
\(882\) 0 0
\(883\) 3.02825e7 1.30704 0.653521 0.756909i \(-0.273291\pi\)
0.653521 + 0.756909i \(0.273291\pi\)
\(884\) −1.26414e7 −0.544084
\(885\) 2.57907e6 0.110689
\(886\) −1.89906e7 −0.812746
\(887\) −2.80881e7 −1.19871 −0.599354 0.800484i \(-0.704576\pi\)
−0.599354 + 0.800484i \(0.704576\pi\)
\(888\) 209331. 0.00890843
\(889\) 0 0
\(890\) 1.38437e6 0.0585836
\(891\) 2.37987e7 1.00429
\(892\) 2.71377e7 1.14199
\(893\) 4.14699e7 1.74022
\(894\) 7.98972e6 0.334340
\(895\) 1.12723e7 0.470387
\(896\) 0 0
\(897\) 3.02843e7 1.25671
\(898\) 4.41650e7 1.82762
\(899\) −1.24322e7 −0.513037
\(900\) 132483. 0.00545198
\(901\) 1.22660e7 0.503375
\(902\) −2.12616e7 −0.870122
\(903\) 0 0
\(904\) 1.16533e7 0.474273
\(905\) −3.79015e6 −0.153828
\(906\) 2.97995e7 1.20611
\(907\) 1.50416e7 0.607121 0.303561 0.952812i \(-0.401824\pi\)
0.303561 + 0.952812i \(0.401824\pi\)
\(908\) −1.17566e7 −0.473226
\(909\) 762198. 0.0305955
\(910\) 0 0
\(911\) 1.53071e7 0.611078 0.305539 0.952179i \(-0.401163\pi\)
0.305539 + 0.952179i \(0.401163\pi\)
\(912\) 5.55418e7 2.21122
\(913\) −3.83292e7 −1.52178
\(914\) −2.84471e7 −1.12635
\(915\) 1.59461e6 0.0629654
\(916\) −2.02063e7 −0.795698
\(917\) 0 0
\(918\) −1.81529e7 −0.710951
\(919\) 2.42413e7 0.946820 0.473410 0.880842i \(-0.343023\pi\)
0.473410 + 0.880842i \(0.343023\pi\)
\(920\) −4.39146e6 −0.171056
\(921\) 1.00508e7 0.390439
\(922\) −2.61462e7 −1.01293
\(923\) 5.65425e7 2.18459
\(924\) 0 0
\(925\) −96638.2 −0.00371360
\(926\) −4.83008e7 −1.85109
\(927\) 521933. 0.0199488
\(928\) −1.87000e7 −0.712808
\(929\) −2.10562e7 −0.800461 −0.400231 0.916414i \(-0.631070\pi\)
−0.400231 + 0.916414i \(0.631070\pi\)
\(930\) −1.22403e7 −0.464071
\(931\) 0 0
\(932\) 2.79191e7 1.05284
\(933\) 3.50967e7 1.31996
\(934\) 1.71165e7 0.642019
\(935\) 6.56535e6 0.245600
\(936\) −820285. −0.0306038
\(937\) −1.43242e7 −0.532993 −0.266497 0.963836i \(-0.585866\pi\)
−0.266497 + 0.963836i \(0.585866\pi\)
\(938\) 0 0
\(939\) −7.03600e6 −0.260413
\(940\) −7.58915e6 −0.280139
\(941\) −1.12722e7 −0.414988 −0.207494 0.978236i \(-0.566531\pi\)
−0.207494 + 0.978236i \(0.566531\pi\)
\(942\) −1.18512e7 −0.435145
\(943\) −1.57002e7 −0.574946
\(944\) −8.17782e6 −0.298681
\(945\) 0 0
\(946\) 3.78584e7 1.37542
\(947\) −2.86149e7 −1.03685 −0.518427 0.855122i \(-0.673482\pi\)
−0.518427 + 0.855122i \(0.673482\pi\)
\(948\) 3.47574e7 1.25611
\(949\) 3.92863e7 1.41604
\(950\) −1.24861e7 −0.448866
\(951\) −2.11958e7 −0.759974
\(952\) 0 0
\(953\) −2.67535e7 −0.954219 −0.477109 0.878844i \(-0.658315\pi\)
−0.477109 + 0.878844i \(0.658315\pi\)
\(954\) −1.36881e6 −0.0486935
\(955\) −1.28747e7 −0.456801
\(956\) 1.02870e7 0.364036
\(957\) −1.80051e7 −0.635502
\(958\) 1.64043e7 0.577490
\(959\) 0 0
\(960\) −2.33669e6 −0.0818320
\(961\) −1.05236e7 −0.367585
\(962\) −1.02902e6 −0.0358497
\(963\) −830833. −0.0288701
\(964\) 3.76096e6 0.130349
\(965\) 1.92492e7 0.665417
\(966\) 0 0
\(967\) −1.79212e7 −0.616311 −0.308155 0.951336i \(-0.599712\pi\)
−0.308155 + 0.951336i \(0.599712\pi\)
\(968\) −954987. −0.0327574
\(969\) −2.98585e7 −1.02155
\(970\) 1.57352e7 0.536962
\(971\) 3.38258e7 1.15133 0.575666 0.817685i \(-0.304743\pi\)
0.575666 + 0.817685i \(0.304743\pi\)
\(972\) 1.60480e6 0.0544823
\(973\) 0 0
\(974\) 1.22144e7 0.412549
\(975\) 9.16272e6 0.308683
\(976\) −5.05627e6 −0.169905
\(977\) 3.62165e7 1.21386 0.606931 0.794754i \(-0.292400\pi\)
0.606931 + 0.794754i \(0.292400\pi\)
\(978\) −6.69020e7 −2.23662
\(979\) −2.96564e6 −0.0988922
\(980\) 0 0
\(981\) −1.91822e6 −0.0636393
\(982\) −2.25914e7 −0.747593
\(983\) 1.07386e7 0.354456 0.177228 0.984170i \(-0.443287\pi\)
0.177228 + 0.984170i \(0.443287\pi\)
\(984\) 1.02896e7 0.338774
\(985\) 6.17251e6 0.202708
\(986\) 1.43269e7 0.469311
\(987\) 0 0
\(988\) −5.15029e7 −1.67857
\(989\) 2.79558e7 0.908828
\(990\) −732649. −0.0237579
\(991\) 5.29006e6 0.171110 0.0855552 0.996333i \(-0.472734\pi\)
0.0855552 + 0.996333i \(0.472734\pi\)
\(992\) 2.72336e7 0.878670
\(993\) −2.39260e7 −0.770011
\(994\) 0 0
\(995\) −2.22660e7 −0.712991
\(996\) −3.19007e7 −1.01895
\(997\) −1.70145e7 −0.542103 −0.271051 0.962565i \(-0.587371\pi\)
−0.271051 + 0.962565i \(0.587371\pi\)
\(998\) 6.79860e7 2.16069
\(999\) −572407. −0.0181464
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.1.2 6
7.2 even 3 35.6.e.a.11.5 12
7.4 even 3 35.6.e.a.16.5 yes 12
7.6 odd 2 245.6.a.h.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.6.e.a.11.5 12 7.2 even 3
35.6.e.a.16.5 yes 12 7.4 even 3
245.6.a.h.1.2 6 7.6 odd 2
245.6.a.i.1.2 6 1.1 even 1 trivial