Properties

Label 825.6.a.c.1.1
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.37228 q^{2} -9.00000 q^{3} +55.8397 q^{4} +84.3505 q^{6} +105.081 q^{7} -223.432 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-9.37228 q^{2} -9.00000 q^{3} +55.8397 q^{4} +84.3505 q^{6} +105.081 q^{7} -223.432 q^{8} +81.0000 q^{9} -121.000 q^{11} -502.557 q^{12} -147.549 q^{13} -984.853 q^{14} +307.198 q^{16} +1432.49 q^{17} -759.155 q^{18} +2033.18 q^{19} -945.733 q^{21} +1134.05 q^{22} -828.853 q^{23} +2010.89 q^{24} +1382.87 q^{26} -729.000 q^{27} +5867.71 q^{28} +4633.44 q^{29} -9835.65 q^{31} +4270.67 q^{32} +1089.00 q^{33} -13425.7 q^{34} +4523.01 q^{36} -7134.34 q^{37} -19055.6 q^{38} +1327.94 q^{39} +18265.0 q^{41} +8863.68 q^{42} -13822.5 q^{43} -6756.60 q^{44} +7768.24 q^{46} -22991.2 q^{47} -2764.78 q^{48} -5764.89 q^{49} -12892.4 q^{51} -8239.08 q^{52} -14311.9 q^{53} +6832.39 q^{54} -23478.6 q^{56} -18298.7 q^{57} -43425.9 q^{58} -7081.12 q^{59} -18470.2 q^{61} +92182.5 q^{62} +8511.60 q^{63} -49856.3 q^{64} -10206.4 q^{66} -16229.5 q^{67} +79989.7 q^{68} +7459.68 q^{69} +28198.9 q^{71} -18098.0 q^{72} +39382.9 q^{73} +66865.1 q^{74} +113532. q^{76} -12714.9 q^{77} -12445.8 q^{78} -41243.7 q^{79} +6561.00 q^{81} -171185. q^{82} +23355.3 q^{83} -52809.4 q^{84} +129549. q^{86} -41701.0 q^{87} +27035.3 q^{88} -103803. q^{89} -15504.6 q^{91} -46282.9 q^{92} +88520.9 q^{93} +215480. q^{94} -38436.1 q^{96} +149289. q^{97} +54030.2 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9} - 242 q^{11} - 333 q^{12} + 130 q^{13} - 74 q^{14} + 241 q^{16} + 728 q^{17} - 1053 q^{18} - 828 q^{19} + 1314 q^{21} + 1573 q^{22} + 238 q^{23} + 351 q^{24} + 376 q^{26} - 1458 q^{27} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} - 1391 q^{32} + 2178 q^{33} - 10870 q^{34} + 2997 q^{36} + 1908 q^{37} - 8676 q^{38} - 1170 q^{39} + 36484 q^{41} + 666 q^{42} - 9768 q^{43} - 4477 q^{44} + 3898 q^{46} - 43742 q^{47} - 2169 q^{48} + 40470 q^{49} - 6552 q^{51} - 13468 q^{52} + 12174 q^{53} + 9477 q^{54} - 69786 q^{56} + 7452 q^{57} - 29142 q^{58} - 2788 q^{59} - 25302 q^{61} + 94520 q^{62} - 11826 q^{63} - 27199 q^{64} - 14157 q^{66} + 40520 q^{67} + 93262 q^{68} - 2142 q^{69} + 31386 q^{71} - 3159 q^{72} + 46780 q^{73} + 34062 q^{74} + 167436 q^{76} + 17666 q^{77} - 3384 q^{78} - 16850 q^{79} + 13122 q^{81} - 237278 q^{82} - 79440 q^{83} - 95382 q^{84} + 114840 q^{86} - 6264 q^{87} + 4719 q^{88} - 54204 q^{89} - 85192 q^{91} - 66382 q^{92} + 94320 q^{93} + 290758 q^{94} + 12519 q^{96} + 241568 q^{97} - 113697 q^{98} - 19602 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.37228 −1.65680 −0.828400 0.560136i \(-0.810749\pi\)
−0.828400 + 0.560136i \(0.810749\pi\)
\(3\) −9.00000 −0.577350
\(4\) 55.8397 1.74499
\(5\) 0 0
\(6\) 84.3505 0.956554
\(7\) 105.081 0.810552 0.405276 0.914194i \(-0.367175\pi\)
0.405276 + 0.914194i \(0.367175\pi\)
\(8\) −223.432 −1.23430
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −502.557 −1.00747
\(13\) −147.549 −0.242146 −0.121073 0.992644i \(-0.538634\pi\)
−0.121073 + 0.992644i \(0.538634\pi\)
\(14\) −984.853 −1.34292
\(15\) 0 0
\(16\) 307.198 0.299998
\(17\) 1432.49 1.20218 0.601089 0.799182i \(-0.294734\pi\)
0.601089 + 0.799182i \(0.294734\pi\)
\(18\) −759.155 −0.552267
\(19\) 2033.18 1.29209 0.646045 0.763300i \(-0.276422\pi\)
0.646045 + 0.763300i \(0.276422\pi\)
\(20\) 0 0
\(21\) −945.733 −0.467972
\(22\) 1134.05 0.499544
\(23\) −828.853 −0.326707 −0.163353 0.986568i \(-0.552231\pi\)
−0.163353 + 0.986568i \(0.552231\pi\)
\(24\) 2010.89 0.712623
\(25\) 0 0
\(26\) 1382.87 0.401188
\(27\) −729.000 −0.192450
\(28\) 5867.71 1.41440
\(29\) 4633.44 1.02308 0.511539 0.859260i \(-0.329076\pi\)
0.511539 + 0.859260i \(0.329076\pi\)
\(30\) 0 0
\(31\) −9835.65 −1.83823 −0.919113 0.393994i \(-0.871093\pi\)
−0.919113 + 0.393994i \(0.871093\pi\)
\(32\) 4270.67 0.737261
\(33\) 1089.00 0.174078
\(34\) −13425.7 −1.99177
\(35\) 0 0
\(36\) 4523.01 0.581663
\(37\) −7134.34 −0.856741 −0.428371 0.903603i \(-0.640912\pi\)
−0.428371 + 0.903603i \(0.640912\pi\)
\(38\) −19055.6 −2.14074
\(39\) 1327.94 0.139803
\(40\) 0 0
\(41\) 18265.0 1.69691 0.848456 0.529265i \(-0.177532\pi\)
0.848456 + 0.529265i \(0.177532\pi\)
\(42\) 8863.68 0.775337
\(43\) −13822.5 −1.14003 −0.570016 0.821634i \(-0.693063\pi\)
−0.570016 + 0.821634i \(0.693063\pi\)
\(44\) −6756.60 −0.526134
\(45\) 0 0
\(46\) 7768.24 0.541288
\(47\) −22991.2 −1.51816 −0.759079 0.650999i \(-0.774350\pi\)
−0.759079 + 0.650999i \(0.774350\pi\)
\(48\) −2764.78 −0.173204
\(49\) −5764.89 −0.343005
\(50\) 0 0
\(51\) −12892.4 −0.694078
\(52\) −8239.08 −0.422542
\(53\) −14311.9 −0.699856 −0.349928 0.936777i \(-0.613794\pi\)
−0.349928 + 0.936777i \(0.613794\pi\)
\(54\) 6832.39 0.318851
\(55\) 0 0
\(56\) −23478.6 −1.00046
\(57\) −18298.7 −0.745988
\(58\) −43425.9 −1.69504
\(59\) −7081.12 −0.264833 −0.132416 0.991194i \(-0.542274\pi\)
−0.132416 + 0.991194i \(0.542274\pi\)
\(60\) 0 0
\(61\) −18470.2 −0.635547 −0.317774 0.948167i \(-0.602935\pi\)
−0.317774 + 0.948167i \(0.602935\pi\)
\(62\) 92182.5 3.04557
\(63\) 8511.60 0.270184
\(64\) −49856.3 −1.52149
\(65\) 0 0
\(66\) −10206.4 −0.288412
\(67\) −16229.5 −0.441690 −0.220845 0.975309i \(-0.570881\pi\)
−0.220845 + 0.975309i \(0.570881\pi\)
\(68\) 79989.7 2.09779
\(69\) 7459.68 0.188624
\(70\) 0 0
\(71\) 28198.9 0.663875 0.331938 0.943301i \(-0.392298\pi\)
0.331938 + 0.943301i \(0.392298\pi\)
\(72\) −18098.0 −0.411433
\(73\) 39382.9 0.864968 0.432484 0.901642i \(-0.357637\pi\)
0.432484 + 0.901642i \(0.357637\pi\)
\(74\) 66865.1 1.41945
\(75\) 0 0
\(76\) 113532. 2.25468
\(77\) −12714.9 −0.244391
\(78\) −12445.8 −0.231626
\(79\) −41243.7 −0.743515 −0.371758 0.928330i \(-0.621245\pi\)
−0.371758 + 0.928330i \(0.621245\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −171185. −2.81145
\(83\) 23355.3 0.372126 0.186063 0.982538i \(-0.440427\pi\)
0.186063 + 0.982538i \(0.440427\pi\)
\(84\) −52809.4 −0.816607
\(85\) 0 0
\(86\) 129549. 1.88880
\(87\) −41701.0 −0.590675
\(88\) 27035.3 0.372155
\(89\) −103803. −1.38911 −0.694555 0.719440i \(-0.744399\pi\)
−0.694555 + 0.719440i \(0.744399\pi\)
\(90\) 0 0
\(91\) −15504.6 −0.196272
\(92\) −46282.9 −0.570099
\(93\) 88520.9 1.06130
\(94\) 215480. 2.51528
\(95\) 0 0
\(96\) −38436.1 −0.425658
\(97\) 149289. 1.61101 0.805503 0.592592i \(-0.201895\pi\)
0.805503 + 0.592592i \(0.201895\pi\)
\(98\) 54030.2 0.568292
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 62410.1 0.608768 0.304384 0.952549i \(-0.401549\pi\)
0.304384 + 0.952549i \(0.401549\pi\)
\(102\) 120831. 1.14995
\(103\) 90047.2 0.836329 0.418165 0.908371i \(-0.362674\pi\)
0.418165 + 0.908371i \(0.362674\pi\)
\(104\) 32967.1 0.298881
\(105\) 0 0
\(106\) 134136. 1.15952
\(107\) −120458. −1.01713 −0.508566 0.861023i \(-0.669824\pi\)
−0.508566 + 0.861023i \(0.669824\pi\)
\(108\) −40707.1 −0.335823
\(109\) −91854.0 −0.740511 −0.370256 0.928930i \(-0.620730\pi\)
−0.370256 + 0.928930i \(0.620730\pi\)
\(110\) 0 0
\(111\) 64209.1 0.494640
\(112\) 32280.8 0.243164
\(113\) 29619.6 0.218214 0.109107 0.994030i \(-0.465201\pi\)
0.109107 + 0.994030i \(0.465201\pi\)
\(114\) 171500. 1.23595
\(115\) 0 0
\(116\) 258730. 1.78526
\(117\) −11951.5 −0.0807153
\(118\) 66366.2 0.438775
\(119\) 150528. 0.974428
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 173108. 1.05298
\(123\) −164385. −0.979713
\(124\) −549219. −3.20768
\(125\) 0 0
\(126\) −79773.1 −0.447641
\(127\) −271128. −1.49165 −0.745823 0.666145i \(-0.767943\pi\)
−0.745823 + 0.666145i \(0.767943\pi\)
\(128\) 330606. 1.78355
\(129\) 124403. 0.658197
\(130\) 0 0
\(131\) −223990. −1.14038 −0.570191 0.821512i \(-0.693131\pi\)
−0.570191 + 0.821512i \(0.693131\pi\)
\(132\) 60809.4 0.303764
\(133\) 213650. 1.04731
\(134\) 152107. 0.731792
\(135\) 0 0
\(136\) −320064. −1.48385
\(137\) 348229. 1.58513 0.792563 0.609789i \(-0.208746\pi\)
0.792563 + 0.609789i \(0.208746\pi\)
\(138\) −69914.2 −0.312513
\(139\) 333802. 1.46538 0.732692 0.680560i \(-0.238264\pi\)
0.732692 + 0.680560i \(0.238264\pi\)
\(140\) 0 0
\(141\) 206921. 0.876509
\(142\) −264288. −1.09991
\(143\) 17853.4 0.0730098
\(144\) 24883.1 0.0999994
\(145\) 0 0
\(146\) −369107. −1.43308
\(147\) 51884.0 0.198034
\(148\) −398379. −1.49500
\(149\) 310249. 1.14484 0.572419 0.819961i \(-0.306005\pi\)
0.572419 + 0.819961i \(0.306005\pi\)
\(150\) 0 0
\(151\) −54751.5 −0.195413 −0.0977066 0.995215i \(-0.531151\pi\)
−0.0977066 + 0.995215i \(0.531151\pi\)
\(152\) −454278. −1.59482
\(153\) 116032. 0.400726
\(154\) 119167. 0.404907
\(155\) 0 0
\(156\) 74151.7 0.243955
\(157\) −221556. −0.717354 −0.358677 0.933462i \(-0.616772\pi\)
−0.358677 + 0.933462i \(0.616772\pi\)
\(158\) 386547. 1.23186
\(159\) 128807. 0.404062
\(160\) 0 0
\(161\) −87097.1 −0.264813
\(162\) −61491.5 −0.184089
\(163\) 225239. 0.664010 0.332005 0.943278i \(-0.392275\pi\)
0.332005 + 0.943278i \(0.392275\pi\)
\(164\) 1.01991e6 2.96109
\(165\) 0 0
\(166\) −218892. −0.616539
\(167\) −186796. −0.518294 −0.259147 0.965838i \(-0.583441\pi\)
−0.259147 + 0.965838i \(0.583441\pi\)
\(168\) 211307. 0.577618
\(169\) −349522. −0.941365
\(170\) 0 0
\(171\) 164688. 0.430697
\(172\) −771846. −1.98934
\(173\) 713662. 1.81292 0.906458 0.422296i \(-0.138776\pi\)
0.906458 + 0.422296i \(0.138776\pi\)
\(174\) 390833. 0.978630
\(175\) 0 0
\(176\) −37171.0 −0.0904529
\(177\) 63730.1 0.152901
\(178\) 972875. 2.30148
\(179\) 789450. 1.84159 0.920793 0.390051i \(-0.127543\pi\)
0.920793 + 0.390051i \(0.127543\pi\)
\(180\) 0 0
\(181\) 40330.7 0.0915039 0.0457519 0.998953i \(-0.485432\pi\)
0.0457519 + 0.998953i \(0.485432\pi\)
\(182\) 145314. 0.325184
\(183\) 166232. 0.366934
\(184\) 185192. 0.403254
\(185\) 0 0
\(186\) −829642. −1.75836
\(187\) −173331. −0.362470
\(188\) −1.28382e6 −2.64917
\(189\) −76604.4 −0.155991
\(190\) 0 0
\(191\) 133736. 0.265256 0.132628 0.991166i \(-0.457658\pi\)
0.132628 + 0.991166i \(0.457658\pi\)
\(192\) 448707. 0.878435
\(193\) 462266. 0.893303 0.446652 0.894708i \(-0.352616\pi\)
0.446652 + 0.894708i \(0.352616\pi\)
\(194\) −1.39917e6 −2.66912
\(195\) 0 0
\(196\) −321910. −0.598541
\(197\) −96432.5 −0.177035 −0.0885173 0.996075i \(-0.528213\pi\)
−0.0885173 + 0.996075i \(0.528213\pi\)
\(198\) 91857.7 0.166515
\(199\) −1.08264e6 −1.93800 −0.968999 0.247066i \(-0.920533\pi\)
−0.968999 + 0.247066i \(0.920533\pi\)
\(200\) 0 0
\(201\) 146065. 0.255010
\(202\) −584925. −1.00861
\(203\) 486889. 0.829258
\(204\) −719907. −1.21116
\(205\) 0 0
\(206\) −843948. −1.38563
\(207\) −67137.1 −0.108902
\(208\) −45326.7 −0.0726434
\(209\) −246015. −0.389580
\(210\) 0 0
\(211\) 875588. 1.35392 0.676961 0.736019i \(-0.263297\pi\)
0.676961 + 0.736019i \(0.263297\pi\)
\(212\) −799174. −1.22124
\(213\) −253790. −0.383289
\(214\) 1.12897e6 1.68518
\(215\) 0 0
\(216\) 162882. 0.237541
\(217\) −1.03354e6 −1.48998
\(218\) 860881. 1.22688
\(219\) −354446. −0.499390
\(220\) 0 0
\(221\) −211362. −0.291103
\(222\) −601786. −0.819520
\(223\) −988042. −1.33049 −0.665247 0.746623i \(-0.731674\pi\)
−0.665247 + 0.746623i \(0.731674\pi\)
\(224\) 448769. 0.597589
\(225\) 0 0
\(226\) −277603. −0.361537
\(227\) −67066.0 −0.0863849 −0.0431925 0.999067i \(-0.513753\pi\)
−0.0431925 + 0.999067i \(0.513753\pi\)
\(228\) −1.02179e6 −1.30174
\(229\) −662131. −0.834363 −0.417182 0.908823i \(-0.636982\pi\)
−0.417182 + 0.908823i \(0.636982\pi\)
\(230\) 0 0
\(231\) 114434. 0.141099
\(232\) −1.03526e6 −1.26278
\(233\) −1.37961e6 −1.66482 −0.832411 0.554159i \(-0.813040\pi\)
−0.832411 + 0.554159i \(0.813040\pi\)
\(234\) 112012. 0.133729
\(235\) 0 0
\(236\) −395407. −0.462130
\(237\) 371193. 0.429269
\(238\) −1.41079e6 −1.61443
\(239\) 423791. 0.479907 0.239954 0.970784i \(-0.422868\pi\)
0.239954 + 0.970784i \(0.422868\pi\)
\(240\) 0 0
\(241\) −1.34802e6 −1.49504 −0.747521 0.664238i \(-0.768756\pi\)
−0.747521 + 0.664238i \(0.768756\pi\)
\(242\) −137220. −0.150618
\(243\) −59049.0 −0.0641500
\(244\) −1.03137e6 −1.10902
\(245\) 0 0
\(246\) 1.54066e6 1.62319
\(247\) −299994. −0.312874
\(248\) 2.19760e6 2.26892
\(249\) −210198. −0.214847
\(250\) 0 0
\(251\) −1.45914e6 −1.46188 −0.730940 0.682442i \(-0.760918\pi\)
−0.730940 + 0.682442i \(0.760918\pi\)
\(252\) 475285. 0.471468
\(253\) 100291. 0.0985057
\(254\) 2.54109e6 2.47136
\(255\) 0 0
\(256\) −1.50313e6 −1.43349
\(257\) −548503. −0.518020 −0.259010 0.965875i \(-0.583396\pi\)
−0.259010 + 0.965875i \(0.583396\pi\)
\(258\) −1.16594e6 −1.09050
\(259\) −749687. −0.694434
\(260\) 0 0
\(261\) 375309. 0.341026
\(262\) 2.09930e6 1.88939
\(263\) −905299. −0.807054 −0.403527 0.914968i \(-0.632216\pi\)
−0.403527 + 0.914968i \(0.632216\pi\)
\(264\) −243317. −0.214864
\(265\) 0 0
\(266\) −2.00239e6 −1.73518
\(267\) 934231. 0.802003
\(268\) −906248. −0.770744
\(269\) −164041. −0.138220 −0.0691100 0.997609i \(-0.522016\pi\)
−0.0691100 + 0.997609i \(0.522016\pi\)
\(270\) 0 0
\(271\) 989636. 0.818563 0.409282 0.912408i \(-0.365779\pi\)
0.409282 + 0.912408i \(0.365779\pi\)
\(272\) 440058. 0.360651
\(273\) 139542. 0.113318
\(274\) −3.26370e6 −2.62624
\(275\) 0 0
\(276\) 416546. 0.329147
\(277\) −727149. −0.569409 −0.284704 0.958615i \(-0.591895\pi\)
−0.284704 + 0.958615i \(0.591895\pi\)
\(278\) −3.12848e6 −2.42785
\(279\) −796688. −0.612742
\(280\) 0 0
\(281\) −379100. −0.286410 −0.143205 0.989693i \(-0.545741\pi\)
−0.143205 + 0.989693i \(0.545741\pi\)
\(282\) −1.93932e6 −1.45220
\(283\) 695371. 0.516120 0.258060 0.966129i \(-0.416917\pi\)
0.258060 + 0.966129i \(0.416917\pi\)
\(284\) 1.57462e6 1.15846
\(285\) 0 0
\(286\) −167327. −0.120963
\(287\) 1.91931e6 1.37544
\(288\) 345925. 0.245754
\(289\) 632167. 0.445233
\(290\) 0 0
\(291\) −1.34360e6 −0.930115
\(292\) 2.19913e6 1.50936
\(293\) −268373. −0.182629 −0.0913146 0.995822i \(-0.529107\pi\)
−0.0913146 + 0.995822i \(0.529107\pi\)
\(294\) −486272. −0.328103
\(295\) 0 0
\(296\) 1.59404e6 1.05747
\(297\) 88209.0 0.0580259
\(298\) −2.90774e6 −1.89677
\(299\) 122296. 0.0791107
\(300\) 0 0
\(301\) −1.45249e6 −0.924055
\(302\) 513147. 0.323761
\(303\) −561691. −0.351472
\(304\) 624591. 0.387625
\(305\) 0 0
\(306\) −1.08748e6 −0.663923
\(307\) −1.78841e6 −1.08298 −0.541491 0.840706i \(-0.682140\pi\)
−0.541491 + 0.840706i \(0.682140\pi\)
\(308\) −709993. −0.426459
\(309\) −810425. −0.482855
\(310\) 0 0
\(311\) 362727. 0.212657 0.106328 0.994331i \(-0.466091\pi\)
0.106328 + 0.994331i \(0.466091\pi\)
\(312\) −296704. −0.172559
\(313\) 1.14382e6 0.659926 0.329963 0.943994i \(-0.392964\pi\)
0.329963 + 0.943994i \(0.392964\pi\)
\(314\) 2.07648e6 1.18851
\(315\) 0 0
\(316\) −2.30303e6 −1.29743
\(317\) −3.43750e6 −1.92130 −0.960649 0.277766i \(-0.910406\pi\)
−0.960649 + 0.277766i \(0.910406\pi\)
\(318\) −1.20722e6 −0.669451
\(319\) −560647. −0.308470
\(320\) 0 0
\(321\) 1.08412e6 0.587241
\(322\) 816298. 0.438742
\(323\) 2.91251e6 1.55332
\(324\) 366364. 0.193888
\(325\) 0 0
\(326\) −2.11100e6 −1.10013
\(327\) 826686. 0.427534
\(328\) −4.08098e6 −2.09450
\(329\) −2.41595e6 −1.23055
\(330\) 0 0
\(331\) 1.11428e6 0.559015 0.279507 0.960144i \(-0.409829\pi\)
0.279507 + 0.960144i \(0.409829\pi\)
\(332\) 1.30415e6 0.649356
\(333\) −577882. −0.285580
\(334\) 1.75070e6 0.858710
\(335\) 0 0
\(336\) −290528. −0.140391
\(337\) 2.25804e6 1.08307 0.541535 0.840678i \(-0.317843\pi\)
0.541535 + 0.840678i \(0.317843\pi\)
\(338\) 3.27582e6 1.55965
\(339\) −266576. −0.125986
\(340\) 0 0
\(341\) 1.19011e6 0.554246
\(342\) −1.54350e6 −0.713578
\(343\) −2.37189e6 −1.08858
\(344\) 3.08840e6 1.40714
\(345\) 0 0
\(346\) −6.68865e6 −3.00364
\(347\) 1.17055e6 0.521873 0.260937 0.965356i \(-0.415969\pi\)
0.260937 + 0.965356i \(0.415969\pi\)
\(348\) −2.32857e6 −1.03072
\(349\) 1.21704e6 0.534863 0.267432 0.963577i \(-0.413825\pi\)
0.267432 + 0.963577i \(0.413825\pi\)
\(350\) 0 0
\(351\) 107563. 0.0466010
\(352\) −516752. −0.222293
\(353\) −286188. −0.122240 −0.0611201 0.998130i \(-0.519467\pi\)
−0.0611201 + 0.998130i \(0.519467\pi\)
\(354\) −597296. −0.253327
\(355\) 0 0
\(356\) −5.79635e6 −2.42398
\(357\) −1.35475e6 −0.562586
\(358\) −7.39895e6 −3.05114
\(359\) 1.19066e6 0.487588 0.243794 0.969827i \(-0.421608\pi\)
0.243794 + 0.969827i \(0.421608\pi\)
\(360\) 0 0
\(361\) 1.65774e6 0.669495
\(362\) −377991. −0.151604
\(363\) −131769. −0.0524864
\(364\) −865774. −0.342492
\(365\) 0 0
\(366\) −1.55797e6 −0.607936
\(367\) −3.11120e6 −1.20577 −0.602884 0.797829i \(-0.705982\pi\)
−0.602884 + 0.797829i \(0.705982\pi\)
\(368\) −254622. −0.0980114
\(369\) 1.47946e6 0.565638
\(370\) 0 0
\(371\) −1.50392e6 −0.567270
\(372\) 4.94297e6 1.85196
\(373\) −3.58536e6 −1.33432 −0.667160 0.744914i \(-0.732490\pi\)
−0.667160 + 0.744914i \(0.732490\pi\)
\(374\) 1.62451e6 0.600541
\(375\) 0 0
\(376\) 5.13697e6 1.87386
\(377\) −683659. −0.247734
\(378\) 717958. 0.258446
\(379\) 1.87822e6 0.671660 0.335830 0.941923i \(-0.390983\pi\)
0.335830 + 0.941923i \(0.390983\pi\)
\(380\) 0 0
\(381\) 2.44015e6 0.861202
\(382\) −1.25341e6 −0.439476
\(383\) 662654. 0.230829 0.115414 0.993317i \(-0.463180\pi\)
0.115414 + 0.993317i \(0.463180\pi\)
\(384\) −2.97545e6 −1.02973
\(385\) 0 0
\(386\) −4.33249e6 −1.48003
\(387\) −1.11963e6 −0.380010
\(388\) 8.33622e6 2.81119
\(389\) −5.08370e6 −1.70336 −0.851678 0.524065i \(-0.824415\pi\)
−0.851678 + 0.524065i \(0.824415\pi\)
\(390\) 0 0
\(391\) −1.18732e6 −0.392760
\(392\) 1.28806e6 0.423371
\(393\) 2.01591e6 0.658400
\(394\) 903793. 0.293311
\(395\) 0 0
\(396\) −547284. −0.175378
\(397\) 518385. 0.165073 0.0825366 0.996588i \(-0.473698\pi\)
0.0825366 + 0.996588i \(0.473698\pi\)
\(398\) 1.01468e7 3.21088
\(399\) −1.92285e6 −0.604662
\(400\) 0 0
\(401\) 1.12538e6 0.349493 0.174747 0.984613i \(-0.444089\pi\)
0.174747 + 0.984613i \(0.444089\pi\)
\(402\) −1.36896e6 −0.422500
\(403\) 1.45124e6 0.445119
\(404\) 3.48496e6 1.06229
\(405\) 0 0
\(406\) −4.56326e6 −1.37392
\(407\) 863256. 0.258317
\(408\) 2.88057e6 0.856700
\(409\) 3.15394e6 0.932279 0.466139 0.884711i \(-0.345645\pi\)
0.466139 + 0.884711i \(0.345645\pi\)
\(410\) 0 0
\(411\) −3.13406e6 −0.915173
\(412\) 5.02821e6 1.45939
\(413\) −744094. −0.214661
\(414\) 629228. 0.180429
\(415\) 0 0
\(416\) −630133. −0.178525
\(417\) −3.00421e6 −0.846040
\(418\) 2.30572e6 0.645456
\(419\) 801315. 0.222981 0.111491 0.993765i \(-0.464438\pi\)
0.111491 + 0.993765i \(0.464438\pi\)
\(420\) 0 0
\(421\) −1.85449e6 −0.509940 −0.254970 0.966949i \(-0.582066\pi\)
−0.254970 + 0.966949i \(0.582066\pi\)
\(422\) −8.20626e6 −2.24318
\(423\) −1.86229e6 −0.506052
\(424\) 3.19775e6 0.863832
\(425\) 0 0
\(426\) 2.37859e6 0.635033
\(427\) −1.94088e6 −0.515144
\(428\) −6.72635e6 −1.77488
\(429\) −160681. −0.0421522
\(430\) 0 0
\(431\) 2.46004e6 0.637895 0.318947 0.947772i \(-0.396671\pi\)
0.318947 + 0.947772i \(0.396671\pi\)
\(432\) −223948. −0.0577347
\(433\) −3.12223e6 −0.800287 −0.400144 0.916452i \(-0.631040\pi\)
−0.400144 + 0.916452i \(0.631040\pi\)
\(434\) 9.68667e6 2.46860
\(435\) 0 0
\(436\) −5.12910e6 −1.29218
\(437\) −1.68521e6 −0.422134
\(438\) 3.32197e6 0.827389
\(439\) −6.09275e6 −1.50887 −0.754436 0.656374i \(-0.772089\pi\)
−0.754436 + 0.656374i \(0.772089\pi\)
\(440\) 0 0
\(441\) −466956. −0.114335
\(442\) 1.98094e6 0.482299
\(443\) 7.93457e6 1.92094 0.960470 0.278382i \(-0.0897982\pi\)
0.960470 + 0.278382i \(0.0897982\pi\)
\(444\) 3.58541e6 0.863141
\(445\) 0 0
\(446\) 9.26021e6 2.20436
\(447\) −2.79224e6 −0.660973
\(448\) −5.23897e6 −1.23325
\(449\) 4.61011e6 1.07918 0.539592 0.841926i \(-0.318578\pi\)
0.539592 + 0.841926i \(0.318578\pi\)
\(450\) 0 0
\(451\) −2.21006e6 −0.511638
\(452\) 1.65395e6 0.380781
\(453\) 492764. 0.112822
\(454\) 628562. 0.143123
\(455\) 0 0
\(456\) 4.08850e6 0.920773
\(457\) 8.05458e6 1.80407 0.902033 0.431667i \(-0.142074\pi\)
0.902033 + 0.431667i \(0.142074\pi\)
\(458\) 6.20568e6 1.38237
\(459\) −1.04428e6 −0.231359
\(460\) 0 0
\(461\) 3.22431e6 0.706617 0.353309 0.935507i \(-0.385057\pi\)
0.353309 + 0.935507i \(0.385057\pi\)
\(462\) −1.07250e6 −0.233773
\(463\) −1.29607e6 −0.280979 −0.140490 0.990082i \(-0.544868\pi\)
−0.140490 + 0.990082i \(0.544868\pi\)
\(464\) 1.42339e6 0.306922
\(465\) 0 0
\(466\) 1.29301e7 2.75828
\(467\) 2.13902e6 0.453860 0.226930 0.973911i \(-0.427131\pi\)
0.226930 + 0.973911i \(0.427131\pi\)
\(468\) −667365. −0.140847
\(469\) −1.70542e6 −0.358012
\(470\) 0 0
\(471\) 1.99400e6 0.414165
\(472\) 1.58215e6 0.326883
\(473\) 1.67253e6 0.343732
\(474\) −3.47893e6 −0.711213
\(475\) 0 0
\(476\) 8.40543e6 1.70037
\(477\) −1.15927e6 −0.233285
\(478\) −3.97189e6 −0.795111
\(479\) −4.77343e6 −0.950586 −0.475293 0.879828i \(-0.657658\pi\)
−0.475293 + 0.879828i \(0.657658\pi\)
\(480\) 0 0
\(481\) 1.05266e6 0.207457
\(482\) 1.26340e7 2.47699
\(483\) 783873. 0.152890
\(484\) 817548. 0.158635
\(485\) 0 0
\(486\) 553424. 0.106284
\(487\) −7.98376e6 −1.52541 −0.762703 0.646749i \(-0.776128\pi\)
−0.762703 + 0.646749i \(0.776128\pi\)
\(488\) 4.12684e6 0.784456
\(489\) −2.02715e6 −0.383366
\(490\) 0 0
\(491\) −3.70031e6 −0.692682 −0.346341 0.938109i \(-0.612576\pi\)
−0.346341 + 0.938109i \(0.612576\pi\)
\(492\) −9.17919e6 −1.70959
\(493\) 6.63736e6 1.22992
\(494\) 2.81163e6 0.518371
\(495\) 0 0
\(496\) −3.02149e6 −0.551465
\(497\) 2.96318e6 0.538105
\(498\) 1.97003e6 0.355959
\(499\) −7.05137e6 −1.26772 −0.633858 0.773449i \(-0.718530\pi\)
−0.633858 + 0.773449i \(0.718530\pi\)
\(500\) 0 0
\(501\) 1.68116e6 0.299237
\(502\) 1.36754e7 2.42204
\(503\) 7.77325e6 1.36988 0.684940 0.728599i \(-0.259828\pi\)
0.684940 + 0.728599i \(0.259828\pi\)
\(504\) −1.90176e6 −0.333488
\(505\) 0 0
\(506\) −939957. −0.163204
\(507\) 3.14570e6 0.543498
\(508\) −1.51397e7 −2.60290
\(509\) 4.11598e6 0.704173 0.352086 0.935968i \(-0.385472\pi\)
0.352086 + 0.935968i \(0.385472\pi\)
\(510\) 0 0
\(511\) 4.13841e6 0.701102
\(512\) 3.50836e6 0.591465
\(513\) −1.48219e6 −0.248663
\(514\) 5.14073e6 0.858256
\(515\) 0 0
\(516\) 6.94661e6 1.14855
\(517\) 2.78193e6 0.457742
\(518\) 7.02628e6 1.15054
\(519\) −6.42296e6 −1.04669
\(520\) 0 0
\(521\) −2.56352e6 −0.413754 −0.206877 0.978367i \(-0.566330\pi\)
−0.206877 + 0.978367i \(0.566330\pi\)
\(522\) −3.51750e6 −0.565012
\(523\) 3.69240e6 0.590274 0.295137 0.955455i \(-0.404635\pi\)
0.295137 + 0.955455i \(0.404635\pi\)
\(524\) −1.25075e7 −1.98996
\(525\) 0 0
\(526\) 8.48471e6 1.33713
\(527\) −1.40895e7 −2.20988
\(528\) 334539. 0.0522230
\(529\) −5.74935e6 −0.893263
\(530\) 0 0
\(531\) −573570. −0.0882776
\(532\) 1.19301e7 1.82754
\(533\) −2.69498e6 −0.410901
\(534\) −8.75587e6 −1.32876
\(535\) 0 0
\(536\) 3.62618e6 0.545177
\(537\) −7.10505e6 −1.06324
\(538\) 1.53743e6 0.229003
\(539\) 697552. 0.103420
\(540\) 0 0
\(541\) −9.33636e6 −1.37146 −0.685732 0.727854i \(-0.740518\pi\)
−0.685732 + 0.727854i \(0.740518\pi\)
\(542\) −9.27515e6 −1.35620
\(543\) −362977. −0.0528298
\(544\) 6.11769e6 0.886320
\(545\) 0 0
\(546\) −1.30782e6 −0.187745
\(547\) −1.30814e6 −0.186933 −0.0934664 0.995622i \(-0.529795\pi\)
−0.0934664 + 0.995622i \(0.529795\pi\)
\(548\) 1.94450e7 2.76603
\(549\) −1.49609e6 −0.211849
\(550\) 0 0
\(551\) 9.42064e6 1.32191
\(552\) −1.66673e6 −0.232819
\(553\) −4.33395e6 −0.602658
\(554\) 6.81505e6 0.943397
\(555\) 0 0
\(556\) 1.86394e7 2.55708
\(557\) −6.95945e6 −0.950467 −0.475234 0.879860i \(-0.657636\pi\)
−0.475234 + 0.879860i \(0.657636\pi\)
\(558\) 7.46678e6 1.01519
\(559\) 2.03950e6 0.276054
\(560\) 0 0
\(561\) 1.55998e6 0.209272
\(562\) 3.55303e6 0.474524
\(563\) −6.90198e6 −0.917704 −0.458852 0.888513i \(-0.651739\pi\)
−0.458852 + 0.888513i \(0.651739\pi\)
\(564\) 1.15544e7 1.52950
\(565\) 0 0
\(566\) −6.51721e6 −0.855107
\(567\) 689439. 0.0900613
\(568\) −6.30054e6 −0.819421
\(569\) 3.31261e6 0.428933 0.214466 0.976731i \(-0.431199\pi\)
0.214466 + 0.976731i \(0.431199\pi\)
\(570\) 0 0
\(571\) 1.43930e7 1.84741 0.923703 0.383109i \(-0.125147\pi\)
0.923703 + 0.383109i \(0.125147\pi\)
\(572\) 996928. 0.127401
\(573\) −1.20362e6 −0.153146
\(574\) −1.79883e7 −2.27882
\(575\) 0 0
\(576\) −4.03836e6 −0.507165
\(577\) −3.31761e6 −0.414845 −0.207423 0.978251i \(-0.566508\pi\)
−0.207423 + 0.978251i \(0.566508\pi\)
\(578\) −5.92484e6 −0.737662
\(579\) −4.16040e6 −0.515749
\(580\) 0 0
\(581\) 2.45421e6 0.301628
\(582\) 1.25926e7 1.54101
\(583\) 1.73174e6 0.211015
\(584\) −8.79939e6 −1.06763
\(585\) 0 0
\(586\) 2.51527e6 0.302580
\(587\) −2.10735e6 −0.252431 −0.126215 0.992003i \(-0.540283\pi\)
−0.126215 + 0.992003i \(0.540283\pi\)
\(588\) 2.89719e6 0.345568
\(589\) −1.99977e7 −2.37515
\(590\) 0 0
\(591\) 867893. 0.102211
\(592\) −2.19166e6 −0.257021
\(593\) −525039. −0.0613133 −0.0306566 0.999530i \(-0.509760\pi\)
−0.0306566 + 0.999530i \(0.509760\pi\)
\(594\) −826720. −0.0961373
\(595\) 0 0
\(596\) 1.73242e7 1.99773
\(597\) 9.74380e6 1.11890
\(598\) −1.14619e6 −0.131071
\(599\) −7.02964e6 −0.800509 −0.400254 0.916404i \(-0.631078\pi\)
−0.400254 + 0.916404i \(0.631078\pi\)
\(600\) 0 0
\(601\) −1.11158e6 −0.125532 −0.0627660 0.998028i \(-0.519992\pi\)
−0.0627660 + 0.998028i \(0.519992\pi\)
\(602\) 1.36132e7 1.53097
\(603\) −1.31459e6 −0.147230
\(604\) −3.05731e6 −0.340994
\(605\) 0 0
\(606\) 5.26433e6 0.582320
\(607\) −1.53428e7 −1.69018 −0.845088 0.534627i \(-0.820452\pi\)
−0.845088 + 0.534627i \(0.820452\pi\)
\(608\) 8.68307e6 0.952608
\(609\) −4.38200e6 −0.478772
\(610\) 0 0
\(611\) 3.39232e6 0.367616
\(612\) 6.47916e6 0.699263
\(613\) −4.78522e6 −0.514341 −0.257170 0.966366i \(-0.582790\pi\)
−0.257170 + 0.966366i \(0.582790\pi\)
\(614\) 1.67615e7 1.79429
\(615\) 0 0
\(616\) 2.84091e6 0.301651
\(617\) −1.69116e7 −1.78843 −0.894215 0.447639i \(-0.852265\pi\)
−0.894215 + 0.447639i \(0.852265\pi\)
\(618\) 7.59553e6 0.799994
\(619\) −8.65985e6 −0.908414 −0.454207 0.890896i \(-0.650077\pi\)
−0.454207 + 0.890896i \(0.650077\pi\)
\(620\) 0 0
\(621\) 604234. 0.0628747
\(622\) −3.39958e6 −0.352330
\(623\) −1.09078e7 −1.12595
\(624\) 407941. 0.0419407
\(625\) 0 0
\(626\) −1.07202e7 −1.09337
\(627\) 2.21414e6 0.224924
\(628\) −1.23716e7 −1.25178
\(629\) −1.02199e7 −1.02996
\(630\) 0 0
\(631\) 9.04083e6 0.903930 0.451965 0.892036i \(-0.350723\pi\)
0.451965 + 0.892036i \(0.350723\pi\)
\(632\) 9.21516e6 0.917720
\(633\) −7.88029e6 −0.781687
\(634\) 3.22172e7 3.18321
\(635\) 0 0
\(636\) 7.19257e6 0.705084
\(637\) 850603. 0.0830574
\(638\) 5.25454e6 0.511073
\(639\) 2.28411e6 0.221292
\(640\) 0 0
\(641\) −1.21113e7 −1.16424 −0.582122 0.813101i \(-0.697778\pi\)
−0.582122 + 0.813101i \(0.697778\pi\)
\(642\) −1.01607e7 −0.972942
\(643\) −2.24955e6 −0.214570 −0.107285 0.994228i \(-0.534216\pi\)
−0.107285 + 0.994228i \(0.534216\pi\)
\(644\) −4.86347e6 −0.462095
\(645\) 0 0
\(646\) −2.72969e7 −2.57355
\(647\) −8.21206e6 −0.771244 −0.385622 0.922657i \(-0.626013\pi\)
−0.385622 + 0.922657i \(0.626013\pi\)
\(648\) −1.46594e6 −0.137144
\(649\) 856815. 0.0798501
\(650\) 0 0
\(651\) 9.30190e6 0.860239
\(652\) 1.25773e7 1.15869
\(653\) 1.15110e7 1.05641 0.528203 0.849118i \(-0.322866\pi\)
0.528203 + 0.849118i \(0.322866\pi\)
\(654\) −7.74793e6 −0.708340
\(655\) 0 0
\(656\) 5.61097e6 0.509071
\(657\) 3.19001e6 0.288323
\(658\) 2.26429e7 2.03877
\(659\) −1.79339e7 −1.60865 −0.804323 0.594192i \(-0.797472\pi\)
−0.804323 + 0.594192i \(0.797472\pi\)
\(660\) 0 0
\(661\) −8.63046e6 −0.768299 −0.384150 0.923271i \(-0.625505\pi\)
−0.384150 + 0.923271i \(0.625505\pi\)
\(662\) −1.04433e7 −0.926176
\(663\) 1.90226e6 0.168068
\(664\) −5.21832e6 −0.459315
\(665\) 0 0
\(666\) 5.41607e6 0.473150
\(667\) −3.84044e6 −0.334246
\(668\) −1.04306e7 −0.904418
\(669\) 8.89238e6 0.768162
\(670\) 0 0
\(671\) 2.23490e6 0.191625
\(672\) −4.03892e6 −0.345018
\(673\) 1.92878e7 1.64152 0.820758 0.571276i \(-0.193551\pi\)
0.820758 + 0.571276i \(0.193551\pi\)
\(674\) −2.11630e7 −1.79443
\(675\) 0 0
\(676\) −1.95172e7 −1.64267
\(677\) −6.67400e6 −0.559647 −0.279824 0.960051i \(-0.590276\pi\)
−0.279824 + 0.960051i \(0.590276\pi\)
\(678\) 2.49843e6 0.208734
\(679\) 1.56875e7 1.30580
\(680\) 0 0
\(681\) 603594. 0.0498744
\(682\) −1.11541e7 −0.918275
\(683\) 7.83722e6 0.642851 0.321426 0.946935i \(-0.395838\pi\)
0.321426 + 0.946935i \(0.395838\pi\)
\(684\) 9.19611e6 0.751561
\(685\) 0 0
\(686\) 2.22300e7 1.80355
\(687\) 5.95918e6 0.481720
\(688\) −4.24626e6 −0.342007
\(689\) 2.11171e6 0.169467
\(690\) 0 0
\(691\) −2.05838e7 −1.63995 −0.819974 0.572400i \(-0.806012\pi\)
−0.819974 + 0.572400i \(0.806012\pi\)
\(692\) 3.98507e7 3.16352
\(693\) −1.02990e6 −0.0814635
\(694\) −1.09707e7 −0.864640
\(695\) 0 0
\(696\) 9.31734e6 0.729069
\(697\) 2.61644e7 2.03999
\(698\) −1.14065e7 −0.886162
\(699\) 1.24165e7 0.961185
\(700\) 0 0
\(701\) −8.88356e6 −0.682797 −0.341399 0.939919i \(-0.610901\pi\)
−0.341399 + 0.939919i \(0.610901\pi\)
\(702\) −1.00811e6 −0.0772086
\(703\) −1.45054e7 −1.10699
\(704\) 6.03261e6 0.458748
\(705\) 0 0
\(706\) 2.68223e6 0.202528
\(707\) 6.55815e6 0.493438
\(708\) 3.55866e6 0.266811
\(709\) 6.33167e6 0.473045 0.236523 0.971626i \(-0.423992\pi\)
0.236523 + 0.971626i \(0.423992\pi\)
\(710\) 0 0
\(711\) −3.34074e6 −0.247838
\(712\) 2.31930e7 1.71458
\(713\) 8.15231e6 0.600560
\(714\) 1.26971e7 0.932094
\(715\) 0 0
\(716\) 4.40826e7 3.21355
\(717\) −3.81412e6 −0.277075
\(718\) −1.11592e7 −0.807836
\(719\) 3.88936e6 0.280579 0.140290 0.990111i \(-0.455197\pi\)
0.140290 + 0.990111i \(0.455197\pi\)
\(720\) 0 0
\(721\) 9.46229e6 0.677888
\(722\) −1.55368e7 −1.10922
\(723\) 1.21322e7 0.863163
\(724\) 2.25205e6 0.159673
\(725\) 0 0
\(726\) 1.23498e6 0.0869595
\(727\) −2.34319e7 −1.64426 −0.822131 0.569299i \(-0.807215\pi\)
−0.822131 + 0.569299i \(0.807215\pi\)
\(728\) 3.46423e6 0.242258
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.98006e7 −1.37052
\(732\) 9.28235e6 0.640295
\(733\) −1.77978e7 −1.22351 −0.611753 0.791049i \(-0.709535\pi\)
−0.611753 + 0.791049i \(0.709535\pi\)
\(734\) 2.91591e7 1.99772
\(735\) 0 0
\(736\) −3.53976e6 −0.240868
\(737\) 1.96376e6 0.133174
\(738\) −1.38659e7 −0.937149
\(739\) 3.15950e6 0.212817 0.106409 0.994322i \(-0.466065\pi\)
0.106409 + 0.994322i \(0.466065\pi\)
\(740\) 0 0
\(741\) 2.69994e6 0.180638
\(742\) 1.40952e7 0.939853
\(743\) 4.37086e6 0.290465 0.145233 0.989398i \(-0.453607\pi\)
0.145233 + 0.989398i \(0.453607\pi\)
\(744\) −1.97784e7 −1.30996
\(745\) 0 0
\(746\) 3.36030e7 2.21070
\(747\) 1.89178e6 0.124042
\(748\) −9.67875e6 −0.632507
\(749\) −1.26579e7 −0.824438
\(750\) 0 0
\(751\) 1.72544e7 1.11635 0.558173 0.829724i \(-0.311502\pi\)
0.558173 + 0.829724i \(0.311502\pi\)
\(752\) −7.06285e6 −0.455445
\(753\) 1.31322e7 0.844017
\(754\) 6.40745e6 0.410447
\(755\) 0 0
\(756\) −4.27756e6 −0.272202
\(757\) 1.57084e7 0.996307 0.498153 0.867089i \(-0.334012\pi\)
0.498153 + 0.867089i \(0.334012\pi\)
\(758\) −1.76033e7 −1.11281
\(759\) −902621. −0.0568723
\(760\) 0 0
\(761\) 5.01006e6 0.313604 0.156802 0.987630i \(-0.449882\pi\)
0.156802 + 0.987630i \(0.449882\pi\)
\(762\) −2.28698e7 −1.42684
\(763\) −9.65215e6 −0.600223
\(764\) 7.46778e6 0.462869
\(765\) 0 0
\(766\) −6.21058e6 −0.382437
\(767\) 1.04481e6 0.0641282
\(768\) 1.35282e7 0.827629
\(769\) 3.47489e6 0.211897 0.105949 0.994372i \(-0.466212\pi\)
0.105949 + 0.994372i \(0.466212\pi\)
\(770\) 0 0
\(771\) 4.93653e6 0.299079
\(772\) 2.58128e7 1.55880
\(773\) −2.29786e7 −1.38317 −0.691584 0.722296i \(-0.743087\pi\)
−0.691584 + 0.722296i \(0.743087\pi\)
\(774\) 1.04934e7 0.629602
\(775\) 0 0
\(776\) −3.33558e7 −1.98846
\(777\) 6.74718e6 0.400931
\(778\) 4.76458e7 2.82212
\(779\) 3.71361e7 2.19256
\(780\) 0 0
\(781\) −3.41207e6 −0.200166
\(782\) 1.11279e7 0.650724
\(783\) −3.37778e6 −0.196892
\(784\) −1.77096e6 −0.102901
\(785\) 0 0
\(786\) −1.88937e7 −1.09084
\(787\) 1.40930e7 0.811087 0.405544 0.914076i \(-0.367082\pi\)
0.405544 + 0.914076i \(0.367082\pi\)
\(788\) −5.38476e6 −0.308923
\(789\) 8.14769e6 0.465953
\(790\) 0 0
\(791\) 3.11247e6 0.176874
\(792\) 2.18986e6 0.124052
\(793\) 2.72526e6 0.153895
\(794\) −4.85845e6 −0.273493
\(795\) 0 0
\(796\) −6.04545e7 −3.38178
\(797\) 1.08281e7 0.603819 0.301909 0.953337i \(-0.402376\pi\)
0.301909 + 0.953337i \(0.402376\pi\)
\(798\) 1.80215e7 1.00181
\(799\) −3.29346e7 −1.82510
\(800\) 0 0
\(801\) −8.40808e6 −0.463037
\(802\) −1.05474e7 −0.579041
\(803\) −4.76533e6 −0.260798
\(804\) 8.15623e6 0.444989
\(805\) 0 0
\(806\) −1.36014e7 −0.737474
\(807\) 1.47637e6 0.0798013
\(808\) −1.39444e7 −0.751402
\(809\) −2.27494e7 −1.22208 −0.611040 0.791600i \(-0.709248\pi\)
−0.611040 + 0.791600i \(0.709248\pi\)
\(810\) 0 0
\(811\) −1.33958e7 −0.715179 −0.357590 0.933879i \(-0.616401\pi\)
−0.357590 + 0.933879i \(0.616401\pi\)
\(812\) 2.71877e7 1.44705
\(813\) −8.90673e6 −0.472598
\(814\) −8.09067e6 −0.427980
\(815\) 0 0
\(816\) −3.96052e6 −0.208222
\(817\) −2.81038e7 −1.47302
\(818\) −2.95596e7 −1.54460
\(819\) −1.25588e6 −0.0654240
\(820\) 0 0
\(821\) −1.77359e7 −0.918323 −0.459161 0.888353i \(-0.651850\pi\)
−0.459161 + 0.888353i \(0.651850\pi\)
\(822\) 2.93733e7 1.51626
\(823\) 2.24560e7 1.15567 0.577834 0.816154i \(-0.303898\pi\)
0.577834 + 0.816154i \(0.303898\pi\)
\(824\) −2.01194e7 −1.03228
\(825\) 0 0
\(826\) 6.97386e6 0.355650
\(827\) 1.97457e7 1.00394 0.501972 0.864884i \(-0.332608\pi\)
0.501972 + 0.864884i \(0.332608\pi\)
\(828\) −3.74891e6 −0.190033
\(829\) 2.61754e7 1.32284 0.661418 0.750017i \(-0.269955\pi\)
0.661418 + 0.750017i \(0.269955\pi\)
\(830\) 0 0
\(831\) 6.54434e6 0.328748
\(832\) 7.35624e6 0.368424
\(833\) −8.25814e6 −0.412354
\(834\) 2.81563e7 1.40172
\(835\) 0 0
\(836\) −1.37374e7 −0.679812
\(837\) 7.17019e6 0.353767
\(838\) −7.51015e6 −0.369435
\(839\) −3.69992e6 −0.181463 −0.0907313 0.995875i \(-0.528920\pi\)
−0.0907313 + 0.995875i \(0.528920\pi\)
\(840\) 0 0
\(841\) 957652. 0.0466893
\(842\) 1.73808e7 0.844869
\(843\) 3.41190e6 0.165359
\(844\) 4.88925e7 2.36258
\(845\) 0 0
\(846\) 1.74539e7 0.838428
\(847\) 1.53850e6 0.0736866
\(848\) −4.39660e6 −0.209956
\(849\) −6.25834e6 −0.297982
\(850\) 0 0
\(851\) 5.91332e6 0.279903
\(852\) −1.41716e7 −0.668834
\(853\) −2.90121e7 −1.36523 −0.682617 0.730776i \(-0.739158\pi\)
−0.682617 + 0.730776i \(0.739158\pi\)
\(854\) 1.81905e7 0.853492
\(855\) 0 0
\(856\) 2.69142e7 1.25544
\(857\) 3.60120e7 1.67493 0.837463 0.546494i \(-0.184038\pi\)
0.837463 + 0.546494i \(0.184038\pi\)
\(858\) 1.50594e6 0.0698378
\(859\) −2.14715e7 −0.992841 −0.496421 0.868082i \(-0.665353\pi\)
−0.496421 + 0.868082i \(0.665353\pi\)
\(860\) 0 0
\(861\) −1.72738e7 −0.794108
\(862\) −2.30562e7 −1.05686
\(863\) −1.43154e7 −0.654298 −0.327149 0.944973i \(-0.606088\pi\)
−0.327149 + 0.944973i \(0.606088\pi\)
\(864\) −3.11332e6 −0.141886
\(865\) 0 0
\(866\) 2.92625e7 1.32592
\(867\) −5.68950e6 −0.257055
\(868\) −5.77128e7 −2.60000
\(869\) 4.99049e6 0.224178
\(870\) 0 0
\(871\) 2.39464e6 0.106953
\(872\) 2.05231e7 0.914013
\(873\) 1.20924e7 0.537002
\(874\) 1.57943e7 0.699392
\(875\) 0 0
\(876\) −1.97921e7 −0.871430
\(877\) −2.25088e7 −0.988221 −0.494110 0.869399i \(-0.664506\pi\)
−0.494110 + 0.869399i \(0.664506\pi\)
\(878\) 5.71030e7 2.49990
\(879\) 2.41536e6 0.105441
\(880\) 0 0
\(881\) 8.39629e6 0.364458 0.182229 0.983256i \(-0.441669\pi\)
0.182229 + 0.983256i \(0.441669\pi\)
\(882\) 4.37644e6 0.189431
\(883\) −1.08666e7 −0.469021 −0.234510 0.972114i \(-0.575349\pi\)
−0.234510 + 0.972114i \(0.575349\pi\)
\(884\) −1.18024e7 −0.507971
\(885\) 0 0
\(886\) −7.43650e7 −3.18262
\(887\) −1.53488e7 −0.655038 −0.327519 0.944845i \(-0.606212\pi\)
−0.327519 + 0.944845i \(0.606212\pi\)
\(888\) −1.43464e7 −0.610533
\(889\) −2.84905e7 −1.20906
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −5.51719e7 −2.32170
\(893\) −4.67453e7 −1.96160
\(894\) 2.61696e7 1.09510
\(895\) 0 0
\(896\) 3.47405e7 1.44566
\(897\) −1.10067e6 −0.0456746
\(898\) −4.32073e7 −1.78799
\(899\) −4.55729e7 −1.88065
\(900\) 0 0
\(901\) −2.05017e7 −0.841352
\(902\) 2.07133e7 0.847683
\(903\) 1.30724e7 0.533503
\(904\) −6.61796e6 −0.269341
\(905\) 0 0
\(906\) −4.61832e6 −0.186923
\(907\) −7.93009e6 −0.320081 −0.160041 0.987110i \(-0.551162\pi\)
−0.160041 + 0.987110i \(0.551162\pi\)
\(908\) −3.74494e6 −0.150741
\(909\) 5.05522e6 0.202923
\(910\) 0 0
\(911\) −4.31907e7 −1.72423 −0.862114 0.506715i \(-0.830860\pi\)
−0.862114 + 0.506715i \(0.830860\pi\)
\(912\) −5.62131e6 −0.223795
\(913\) −2.82599e6 −0.112200
\(914\) −7.54898e7 −2.98898
\(915\) 0 0
\(916\) −3.69732e7 −1.45595
\(917\) −2.35372e7 −0.924340
\(918\) 9.78733e6 0.383316
\(919\) −2.17987e7 −0.851414 −0.425707 0.904861i \(-0.639975\pi\)
−0.425707 + 0.904861i \(0.639975\pi\)
\(920\) 0 0
\(921\) 1.60957e7 0.625260
\(922\) −3.02191e7 −1.17072
\(923\) −4.16072e6 −0.160755
\(924\) 6.38994e6 0.246216
\(925\) 0 0
\(926\) 1.21471e7 0.465527
\(927\) 7.29382e6 0.278776
\(928\) 1.97879e7 0.754276
\(929\) −1.33390e7 −0.507089 −0.253545 0.967324i \(-0.581596\pi\)
−0.253545 + 0.967324i \(0.581596\pi\)
\(930\) 0 0
\(931\) −1.17211e7 −0.443194
\(932\) −7.70371e7 −2.90510
\(933\) −3.26454e6 −0.122777
\(934\) −2.00475e7 −0.751956
\(935\) 0 0
\(936\) 2.67034e6 0.0996269
\(937\) −1.33697e7 −0.497476 −0.248738 0.968571i \(-0.580016\pi\)
−0.248738 + 0.968571i \(0.580016\pi\)
\(938\) 1.59836e7 0.593155
\(939\) −1.02943e7 −0.381008
\(940\) 0 0
\(941\) 2.28926e7 0.842793 0.421397 0.906876i \(-0.361540\pi\)
0.421397 + 0.906876i \(0.361540\pi\)
\(942\) −1.86883e7 −0.686189
\(943\) −1.51390e7 −0.554393
\(944\) −2.17531e6 −0.0794494
\(945\) 0 0
\(946\) −1.56754e7 −0.569496
\(947\) −3.33079e6 −0.120690 −0.0603451 0.998178i \(-0.519220\pi\)
−0.0603451 + 0.998178i \(0.519220\pi\)
\(948\) 2.07273e7 0.749069
\(949\) −5.81089e6 −0.209449
\(950\) 0 0
\(951\) 3.09375e7 1.10926
\(952\) −3.36328e7 −1.20274
\(953\) −4.68461e7 −1.67087 −0.835433 0.549593i \(-0.814783\pi\)
−0.835433 + 0.549593i \(0.814783\pi\)
\(954\) 1.08650e7 0.386508
\(955\) 0 0
\(956\) 2.36644e7 0.837433
\(957\) 5.04582e6 0.178095
\(958\) 4.47379e7 1.57493
\(959\) 3.65924e7 1.28483
\(960\) 0 0
\(961\) 6.81109e7 2.37907
\(962\) −9.86586e6 −0.343714
\(963\) −9.75712e6 −0.339044
\(964\) −7.52730e7 −2.60883
\(965\) 0 0
\(966\) −7.34668e6 −0.253308
\(967\) 3.65887e7 1.25829 0.629146 0.777287i \(-0.283405\pi\)
0.629146 + 0.777287i \(0.283405\pi\)
\(968\) −3.27127e6 −0.112209
\(969\) −2.62126e7 −0.896811
\(970\) 0 0
\(971\) 1.62147e7 0.551901 0.275951 0.961172i \(-0.411007\pi\)
0.275951 + 0.961172i \(0.411007\pi\)
\(972\) −3.29728e6 −0.111941
\(973\) 3.50764e7 1.18777
\(974\) 7.48261e7 2.52729
\(975\) 0 0
\(976\) −5.67403e6 −0.190663
\(977\) 1.50181e6 0.0503360 0.0251680 0.999683i \(-0.491988\pi\)
0.0251680 + 0.999683i \(0.491988\pi\)
\(978\) 1.89990e7 0.635161
\(979\) 1.25602e7 0.418832
\(980\) 0 0
\(981\) −7.44017e6 −0.246837
\(982\) 3.46803e7 1.14764
\(983\) −3.72690e7 −1.23017 −0.615083 0.788462i \(-0.710878\pi\)
−0.615083 + 0.788462i \(0.710878\pi\)
\(984\) 3.67288e7 1.20926
\(985\) 0 0
\(986\) −6.22072e7 −2.03774
\(987\) 2.17435e7 0.710456
\(988\) −1.67516e7 −0.545962
\(989\) 1.14569e7 0.372456
\(990\) 0 0
\(991\) −3.20015e7 −1.03511 −0.517554 0.855651i \(-0.673157\pi\)
−0.517554 + 0.855651i \(0.673157\pi\)
\(992\) −4.20049e7 −1.35525
\(993\) −1.00285e7 −0.322747
\(994\) −2.77718e7 −0.891534
\(995\) 0 0
\(996\) −1.17374e7 −0.374906
\(997\) 5.59172e7 1.78159 0.890795 0.454406i \(-0.150148\pi\)
0.890795 + 0.454406i \(0.150148\pi\)
\(998\) 6.60874e7 2.10035
\(999\) 5.20094e6 0.164880
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 825.6.a.c.1.1 2
5.4 even 2 33.6.a.e.1.2 2
15.14 odd 2 99.6.a.d.1.1 2
20.19 odd 2 528.6.a.o.1.1 2
55.54 odd 2 363.6.a.f.1.1 2
165.164 even 2 1089.6.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.e.1.2 2 5.4 even 2
99.6.a.d.1.1 2 15.14 odd 2
363.6.a.f.1.1 2 55.54 odd 2
528.6.a.o.1.1 2 20.19 odd 2
825.6.a.c.1.1 2 1.1 even 1 trivial
1089.6.a.p.1.2 2 165.164 even 2