Properties

Label 825.6.a.v.1.8
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.88656\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.886559 q^{2} -9.00000 q^{3} -31.2140 q^{4} -7.97903 q^{6} -224.774 q^{7} -56.0430 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+0.886559 q^{2} -9.00000 q^{3} -31.2140 q^{4} -7.97903 q^{6} -224.774 q^{7} -56.0430 q^{8} +81.0000 q^{9} +121.000 q^{11} +280.926 q^{12} -1065.64 q^{13} -199.275 q^{14} +949.163 q^{16} +1347.75 q^{17} +71.8113 q^{18} +691.252 q^{19} +2022.96 q^{21} +107.274 q^{22} -3396.13 q^{23} +504.387 q^{24} -944.749 q^{26} -729.000 q^{27} +7016.09 q^{28} +8603.02 q^{29} -320.509 q^{31} +2634.86 q^{32} -1089.00 q^{33} +1194.86 q^{34} -2528.34 q^{36} +1906.23 q^{37} +612.836 q^{38} +9590.72 q^{39} +5821.85 q^{41} +1793.48 q^{42} -1421.42 q^{43} -3776.90 q^{44} -3010.87 q^{46} +6454.51 q^{47} -8542.47 q^{48} +33716.2 q^{49} -12129.7 q^{51} +33262.8 q^{52} -9109.26 q^{53} -646.302 q^{54} +12597.0 q^{56} -6221.27 q^{57} +7627.08 q^{58} +10213.6 q^{59} -34276.3 q^{61} -284.150 q^{62} -18206.7 q^{63} -28037.3 q^{64} -965.463 q^{66} +68436.0 q^{67} -42068.6 q^{68} +30565.2 q^{69} +1883.92 q^{71} -4539.48 q^{72} +87733.7 q^{73} +1689.99 q^{74} -21576.8 q^{76} -27197.6 q^{77} +8502.74 q^{78} -52408.8 q^{79} +6561.00 q^{81} +5161.42 q^{82} -58665.8 q^{83} -63144.8 q^{84} -1260.17 q^{86} -77427.1 q^{87} -6781.20 q^{88} -113843. q^{89} +239527. q^{91} +106007. q^{92} +2884.58 q^{93} +5722.30 q^{94} -23713.8 q^{96} +115329. q^{97} +29891.4 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.886559 0.156723 0.0783615 0.996925i \(-0.475031\pi\)
0.0783615 + 0.996925i \(0.475031\pi\)
\(3\) −9.00000 −0.577350
\(4\) −31.2140 −0.975438
\(5\) 0 0
\(6\) −7.97903 −0.0904841
\(7\) −224.774 −1.73380 −0.866902 0.498478i \(-0.833892\pi\)
−0.866902 + 0.498478i \(0.833892\pi\)
\(8\) −56.0430 −0.309597
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 280.926 0.563169
\(13\) −1065.64 −1.74884 −0.874420 0.485169i \(-0.838758\pi\)
−0.874420 + 0.485169i \(0.838758\pi\)
\(14\) −199.275 −0.271727
\(15\) 0 0
\(16\) 949.163 0.926917
\(17\) 1347.75 1.13106 0.565531 0.824727i \(-0.308671\pi\)
0.565531 + 0.824727i \(0.308671\pi\)
\(18\) 71.8113 0.0522410
\(19\) 691.252 0.439291 0.219646 0.975580i \(-0.429510\pi\)
0.219646 + 0.975580i \(0.429510\pi\)
\(20\) 0 0
\(21\) 2022.96 1.00101
\(22\) 107.274 0.0472538
\(23\) −3396.13 −1.33864 −0.669321 0.742973i \(-0.733415\pi\)
−0.669321 + 0.742973i \(0.733415\pi\)
\(24\) 504.387 0.178746
\(25\) 0 0
\(26\) −944.749 −0.274084
\(27\) −729.000 −0.192450
\(28\) 7016.09 1.69122
\(29\) 8603.02 1.89957 0.949786 0.312900i \(-0.101301\pi\)
0.949786 + 0.312900i \(0.101301\pi\)
\(30\) 0 0
\(31\) −320.509 −0.0599013 −0.0299507 0.999551i \(-0.509535\pi\)
−0.0299507 + 0.999551i \(0.509535\pi\)
\(32\) 2634.86 0.454866
\(33\) −1089.00 −0.174078
\(34\) 1194.86 0.177263
\(35\) 0 0
\(36\) −2528.34 −0.325146
\(37\) 1906.23 0.228913 0.114457 0.993428i \(-0.463487\pi\)
0.114457 + 0.993428i \(0.463487\pi\)
\(38\) 612.836 0.0688471
\(39\) 9590.72 1.00969
\(40\) 0 0
\(41\) 5821.85 0.540881 0.270440 0.962737i \(-0.412831\pi\)
0.270440 + 0.962737i \(0.412831\pi\)
\(42\) 1793.48 0.156882
\(43\) −1421.42 −0.117234 −0.0586168 0.998281i \(-0.518669\pi\)
−0.0586168 + 0.998281i \(0.518669\pi\)
\(44\) −3776.90 −0.294106
\(45\) 0 0
\(46\) −3010.87 −0.209796
\(47\) 6454.51 0.426205 0.213102 0.977030i \(-0.431643\pi\)
0.213102 + 0.977030i \(0.431643\pi\)
\(48\) −8542.47 −0.535156
\(49\) 33716.2 2.00608
\(50\) 0 0
\(51\) −12129.7 −0.653019
\(52\) 33262.8 1.70589
\(53\) −9109.26 −0.445444 −0.222722 0.974882i \(-0.571494\pi\)
−0.222722 + 0.974882i \(0.571494\pi\)
\(54\) −646.302 −0.0301614
\(55\) 0 0
\(56\) 12597.0 0.536780
\(57\) −6221.27 −0.253625
\(58\) 7627.08 0.297707
\(59\) 10213.6 0.381989 0.190995 0.981591i \(-0.438829\pi\)
0.190995 + 0.981591i \(0.438829\pi\)
\(60\) 0 0
\(61\) −34276.3 −1.17942 −0.589712 0.807614i \(-0.700759\pi\)
−0.589712 + 0.807614i \(0.700759\pi\)
\(62\) −284.150 −0.00938791
\(63\) −18206.7 −0.577935
\(64\) −28037.3 −0.855629
\(65\) 0 0
\(66\) −965.463 −0.0272820
\(67\) 68436.0 1.86251 0.931253 0.364374i \(-0.118717\pi\)
0.931253 + 0.364374i \(0.118717\pi\)
\(68\) −42068.6 −1.10328
\(69\) 30565.2 0.772866
\(70\) 0 0
\(71\) 1883.92 0.0443524 0.0221762 0.999754i \(-0.492941\pi\)
0.0221762 + 0.999754i \(0.492941\pi\)
\(72\) −4539.48 −0.103199
\(73\) 87733.7 1.92690 0.963451 0.267885i \(-0.0863248\pi\)
0.963451 + 0.267885i \(0.0863248\pi\)
\(74\) 1689.99 0.0358760
\(75\) 0 0
\(76\) −21576.8 −0.428501
\(77\) −27197.6 −0.522762
\(78\) 8502.74 0.158242
\(79\) −52408.8 −0.944793 −0.472397 0.881386i \(-0.656611\pi\)
−0.472397 + 0.881386i \(0.656611\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 5161.42 0.0847685
\(83\) −58665.8 −0.934737 −0.467369 0.884062i \(-0.654798\pi\)
−0.467369 + 0.884062i \(0.654798\pi\)
\(84\) −63144.8 −0.976426
\(85\) 0 0
\(86\) −1260.17 −0.0183732
\(87\) −77427.1 −1.09672
\(88\) −6781.20 −0.0933469
\(89\) −113843. −1.52346 −0.761728 0.647896i \(-0.775649\pi\)
−0.761728 + 0.647896i \(0.775649\pi\)
\(90\) 0 0
\(91\) 239527. 3.03215
\(92\) 106007. 1.30576
\(93\) 2884.58 0.0345840
\(94\) 5722.30 0.0667961
\(95\) 0 0
\(96\) −23713.8 −0.262617
\(97\) 115329. 1.24454 0.622271 0.782802i \(-0.286210\pi\)
0.622271 + 0.782802i \(0.286210\pi\)
\(98\) 29891.4 0.314399
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 39906.6 0.389261 0.194631 0.980877i \(-0.437649\pi\)
0.194631 + 0.980877i \(0.437649\pi\)
\(102\) −10753.7 −0.102343
\(103\) −69320.0 −0.643822 −0.321911 0.946770i \(-0.604325\pi\)
−0.321911 + 0.946770i \(0.604325\pi\)
\(104\) 59721.4 0.541435
\(105\) 0 0
\(106\) −8075.90 −0.0698114
\(107\) −131149. −1.10740 −0.553700 0.832717i \(-0.686784\pi\)
−0.553700 + 0.832717i \(0.686784\pi\)
\(108\) 22755.0 0.187723
\(109\) 146929. 1.18451 0.592257 0.805749i \(-0.298237\pi\)
0.592257 + 0.805749i \(0.298237\pi\)
\(110\) 0 0
\(111\) −17156.1 −0.132163
\(112\) −213347. −1.60709
\(113\) 57346.0 0.422481 0.211240 0.977434i \(-0.432250\pi\)
0.211240 + 0.977434i \(0.432250\pi\)
\(114\) −5515.52 −0.0397489
\(115\) 0 0
\(116\) −268535. −1.85291
\(117\) −86316.5 −0.582947
\(118\) 9055.00 0.0598665
\(119\) −302938. −1.96104
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −30388.0 −0.184843
\(123\) −52396.7 −0.312278
\(124\) 10004.4 0.0584300
\(125\) 0 0
\(126\) −16141.3 −0.0905757
\(127\) −172649. −0.949852 −0.474926 0.880026i \(-0.657525\pi\)
−0.474926 + 0.880026i \(0.657525\pi\)
\(128\) −109172. −0.588963
\(129\) 12792.8 0.0676848
\(130\) 0 0
\(131\) 297723. 1.51577 0.757886 0.652387i \(-0.226233\pi\)
0.757886 + 0.652387i \(0.226233\pi\)
\(132\) 33992.1 0.169802
\(133\) −155375. −0.761645
\(134\) 60672.6 0.291897
\(135\) 0 0
\(136\) −75531.8 −0.350173
\(137\) 45559.8 0.207387 0.103693 0.994609i \(-0.466934\pi\)
0.103693 + 0.994609i \(0.466934\pi\)
\(138\) 27097.8 0.121126
\(139\) −124306. −0.545701 −0.272850 0.962056i \(-0.587966\pi\)
−0.272850 + 0.962056i \(0.587966\pi\)
\(140\) 0 0
\(141\) −58090.6 −0.246070
\(142\) 1670.21 0.00695104
\(143\) −128942. −0.527295
\(144\) 76882.2 0.308972
\(145\) 0 0
\(146\) 77781.2 0.301990
\(147\) −303445. −1.15821
\(148\) −59501.1 −0.223291
\(149\) −107578. −0.396971 −0.198485 0.980104i \(-0.563602\pi\)
−0.198485 + 0.980104i \(0.563602\pi\)
\(150\) 0 0
\(151\) 147513. 0.526487 0.263244 0.964729i \(-0.415208\pi\)
0.263244 + 0.964729i \(0.415208\pi\)
\(152\) −38739.8 −0.136003
\(153\) 109168. 0.377021
\(154\) −24112.3 −0.0819288
\(155\) 0 0
\(156\) −299365. −0.984893
\(157\) −231901. −0.750849 −0.375425 0.926853i \(-0.622503\pi\)
−0.375425 + 0.926853i \(0.622503\pi\)
\(158\) −46463.5 −0.148071
\(159\) 81983.3 0.257177
\(160\) 0 0
\(161\) 763360. 2.32094
\(162\) 5816.72 0.0174137
\(163\) 6862.35 0.0202304 0.0101152 0.999949i \(-0.496780\pi\)
0.0101152 + 0.999949i \(0.496780\pi\)
\(164\) −181723. −0.527596
\(165\) 0 0
\(166\) −52010.7 −0.146495
\(167\) 10230.0 0.0283847 0.0141923 0.999899i \(-0.495482\pi\)
0.0141923 + 0.999899i \(0.495482\pi\)
\(168\) −113373. −0.309910
\(169\) 764285. 2.05844
\(170\) 0 0
\(171\) 55991.4 0.146430
\(172\) 44368.3 0.114354
\(173\) −421259. −1.07012 −0.535062 0.844813i \(-0.679712\pi\)
−0.535062 + 0.844813i \(0.679712\pi\)
\(174\) −68643.8 −0.171881
\(175\) 0 0
\(176\) 114849. 0.279476
\(177\) −91922.8 −0.220541
\(178\) −100928. −0.238761
\(179\) 330517. 0.771012 0.385506 0.922705i \(-0.374027\pi\)
0.385506 + 0.922705i \(0.374027\pi\)
\(180\) 0 0
\(181\) 668805. 1.51741 0.758705 0.651434i \(-0.225832\pi\)
0.758705 + 0.651434i \(0.225832\pi\)
\(182\) 212355. 0.475207
\(183\) 308487. 0.680940
\(184\) 190329. 0.414439
\(185\) 0 0
\(186\) 2557.35 0.00542012
\(187\) 163078. 0.341028
\(188\) −201471. −0.415736
\(189\) 163860. 0.333671
\(190\) 0 0
\(191\) −961866. −1.90779 −0.953896 0.300137i \(-0.902967\pi\)
−0.953896 + 0.300137i \(0.902967\pi\)
\(192\) 252335. 0.493998
\(193\) 87470.4 0.169032 0.0845158 0.996422i \(-0.473066\pi\)
0.0845158 + 0.996422i \(0.473066\pi\)
\(194\) 102246. 0.195048
\(195\) 0 0
\(196\) −1.05242e6 −1.95680
\(197\) −799656. −1.46804 −0.734019 0.679129i \(-0.762358\pi\)
−0.734019 + 0.679129i \(0.762358\pi\)
\(198\) 8689.17 0.0157513
\(199\) −635515. −1.13761 −0.568804 0.822473i \(-0.692594\pi\)
−0.568804 + 0.822473i \(0.692594\pi\)
\(200\) 0 0
\(201\) −615924. −1.07532
\(202\) 35379.6 0.0610062
\(203\) −1.93373e6 −3.29349
\(204\) 378618. 0.636980
\(205\) 0 0
\(206\) −61456.3 −0.100902
\(207\) −275086. −0.446214
\(208\) −1.01146e6 −1.62103
\(209\) 83641.5 0.132451
\(210\) 0 0
\(211\) −349594. −0.540578 −0.270289 0.962779i \(-0.587119\pi\)
−0.270289 + 0.962779i \(0.587119\pi\)
\(212\) 284337. 0.434503
\(213\) −16955.3 −0.0256069
\(214\) −116271. −0.173555
\(215\) 0 0
\(216\) 40855.3 0.0595819
\(217\) 72042.0 0.103857
\(218\) 130261. 0.185641
\(219\) −789604. −1.11250
\(220\) 0 0
\(221\) −1.43621e6 −1.97805
\(222\) −15209.9 −0.0207130
\(223\) 309538. 0.416824 0.208412 0.978041i \(-0.433171\pi\)
0.208412 + 0.978041i \(0.433171\pi\)
\(224\) −592248. −0.788648
\(225\) 0 0
\(226\) 50840.6 0.0662124
\(227\) 509733. 0.656566 0.328283 0.944579i \(-0.393530\pi\)
0.328283 + 0.944579i \(0.393530\pi\)
\(228\) 194191. 0.247395
\(229\) −1.28351e6 −1.61737 −0.808684 0.588243i \(-0.799820\pi\)
−0.808684 + 0.588243i \(0.799820\pi\)
\(230\) 0 0
\(231\) 244778. 0.301817
\(232\) −482139. −0.588101
\(233\) −312877. −0.377558 −0.188779 0.982020i \(-0.560453\pi\)
−0.188779 + 0.982020i \(0.560453\pi\)
\(234\) −76524.7 −0.0913612
\(235\) 0 0
\(236\) −318809. −0.372607
\(237\) 471679. 0.545477
\(238\) −268573. −0.307340
\(239\) −357254. −0.404559 −0.202280 0.979328i \(-0.564835\pi\)
−0.202280 + 0.979328i \(0.564835\pi\)
\(240\) 0 0
\(241\) −680168. −0.754351 −0.377176 0.926142i \(-0.623105\pi\)
−0.377176 + 0.926142i \(0.623105\pi\)
\(242\) 12980.1 0.0142475
\(243\) −59049.0 −0.0641500
\(244\) 1.06990e6 1.15045
\(245\) 0 0
\(246\) −46452.8 −0.0489411
\(247\) −736623. −0.768250
\(248\) 17962.3 0.0185452
\(249\) 527992. 0.539671
\(250\) 0 0
\(251\) 157817. 0.158114 0.0790568 0.996870i \(-0.474809\pi\)
0.0790568 + 0.996870i \(0.474809\pi\)
\(252\) 568303. 0.563740
\(253\) −410932. −0.403616
\(254\) −153064. −0.148864
\(255\) 0 0
\(256\) 800404. 0.763325
\(257\) −557296. −0.526324 −0.263162 0.964752i \(-0.584765\pi\)
−0.263162 + 0.964752i \(0.584765\pi\)
\(258\) 11341.6 0.0106078
\(259\) −428470. −0.396891
\(260\) 0 0
\(261\) 696844. 0.633191
\(262\) 263949. 0.237556
\(263\) 645647. 0.575580 0.287790 0.957693i \(-0.407079\pi\)
0.287790 + 0.957693i \(0.407079\pi\)
\(264\) 61030.8 0.0538939
\(265\) 0 0
\(266\) −137749. −0.119367
\(267\) 1.02458e6 0.879568
\(268\) −2.13616e6 −1.81676
\(269\) −219755. −0.185165 −0.0925825 0.995705i \(-0.529512\pi\)
−0.0925825 + 0.995705i \(0.529512\pi\)
\(270\) 0 0
\(271\) 1.54792e6 1.28034 0.640171 0.768233i \(-0.278864\pi\)
0.640171 + 0.768233i \(0.278864\pi\)
\(272\) 1.27923e6 1.04840
\(273\) −2.15574e6 −1.75061
\(274\) 40391.5 0.0325023
\(275\) 0 0
\(276\) −954061. −0.753882
\(277\) 980724. 0.767976 0.383988 0.923338i \(-0.374550\pi\)
0.383988 + 0.923338i \(0.374550\pi\)
\(278\) −110205. −0.0855239
\(279\) −25961.2 −0.0199671
\(280\) 0 0
\(281\) 2.46283e6 1.86067 0.930334 0.366713i \(-0.119517\pi\)
0.930334 + 0.366713i \(0.119517\pi\)
\(282\) −51500.7 −0.0385648
\(283\) 838610. 0.622435 0.311217 0.950339i \(-0.399263\pi\)
0.311217 + 0.950339i \(0.399263\pi\)
\(284\) −58804.8 −0.0432630
\(285\) 0 0
\(286\) −114315. −0.0826393
\(287\) −1.30860e6 −0.937782
\(288\) 213424. 0.151622
\(289\) 396569. 0.279302
\(290\) 0 0
\(291\) −1.03796e6 −0.718536
\(292\) −2.73852e6 −1.87957
\(293\) −1.32689e6 −0.902955 −0.451478 0.892282i \(-0.649103\pi\)
−0.451478 + 0.892282i \(0.649103\pi\)
\(294\) −269022. −0.181518
\(295\) 0 0
\(296\) −106831. −0.0708708
\(297\) −88209.0 −0.0580259
\(298\) −95374.4 −0.0622144
\(299\) 3.61904e6 2.34107
\(300\) 0 0
\(301\) 319498. 0.203260
\(302\) 130779. 0.0825126
\(303\) −359159. −0.224740
\(304\) 656111. 0.407187
\(305\) 0 0
\(306\) 96783.6 0.0590878
\(307\) −1.04842e6 −0.634878 −0.317439 0.948279i \(-0.602823\pi\)
−0.317439 + 0.948279i \(0.602823\pi\)
\(308\) 848946. 0.509922
\(309\) 623880. 0.371711
\(310\) 0 0
\(311\) −1.53786e6 −0.901601 −0.450801 0.892625i \(-0.648861\pi\)
−0.450801 + 0.892625i \(0.648861\pi\)
\(312\) −537492. −0.312598
\(313\) 2.58473e6 1.49126 0.745631 0.666359i \(-0.232148\pi\)
0.745631 + 0.666359i \(0.232148\pi\)
\(314\) −205594. −0.117675
\(315\) 0 0
\(316\) 1.63589e6 0.921587
\(317\) 1.29146e6 0.721827 0.360914 0.932599i \(-0.382465\pi\)
0.360914 + 0.932599i \(0.382465\pi\)
\(318\) 72683.1 0.0403056
\(319\) 1.04096e6 0.572742
\(320\) 0 0
\(321\) 1.18034e6 0.639357
\(322\) 676764. 0.363745
\(323\) 931634. 0.496866
\(324\) −204795. −0.108382
\(325\) 0 0
\(326\) 6083.88 0.00317057
\(327\) −1.32236e6 −0.683880
\(328\) −326274. −0.167455
\(329\) −1.45080e6 −0.738956
\(330\) 0 0
\(331\) 1.77716e6 0.891573 0.445787 0.895139i \(-0.352924\pi\)
0.445787 + 0.895139i \(0.352924\pi\)
\(332\) 1.83119e6 0.911778
\(333\) 154405. 0.0763045
\(334\) 9069.48 0.00444853
\(335\) 0 0
\(336\) 1.92012e6 0.927856
\(337\) −1.99819e6 −0.958435 −0.479217 0.877696i \(-0.659079\pi\)
−0.479217 + 0.877696i \(0.659079\pi\)
\(338\) 677584. 0.322605
\(339\) −516114. −0.243919
\(340\) 0 0
\(341\) −38781.6 −0.0180609
\(342\) 49639.7 0.0229490
\(343\) −3.80073e6 −1.74434
\(344\) 79660.7 0.0362951
\(345\) 0 0
\(346\) −373471. −0.167713
\(347\) −2.94954e6 −1.31501 −0.657507 0.753449i \(-0.728389\pi\)
−0.657507 + 0.753449i \(0.728389\pi\)
\(348\) 2.41681e6 1.06978
\(349\) −1.15473e6 −0.507476 −0.253738 0.967273i \(-0.581660\pi\)
−0.253738 + 0.967273i \(0.581660\pi\)
\(350\) 0 0
\(351\) 776848. 0.336565
\(352\) 318819. 0.137147
\(353\) 829780. 0.354427 0.177213 0.984172i \(-0.443292\pi\)
0.177213 + 0.984172i \(0.443292\pi\)
\(354\) −81495.0 −0.0345639
\(355\) 0 0
\(356\) 3.55349e6 1.48604
\(357\) 2.72644e6 1.13221
\(358\) 293023. 0.120835
\(359\) 3.60644e6 1.47687 0.738435 0.674325i \(-0.235565\pi\)
0.738435 + 0.674325i \(0.235565\pi\)
\(360\) 0 0
\(361\) −1.99827e6 −0.807023
\(362\) 592935. 0.237813
\(363\) −131769. −0.0524864
\(364\) −7.47659e6 −2.95767
\(365\) 0 0
\(366\) 273492. 0.106719
\(367\) 403294. 0.156299 0.0781496 0.996942i \(-0.475099\pi\)
0.0781496 + 0.996942i \(0.475099\pi\)
\(368\) −3.22348e6 −1.24081
\(369\) 471570. 0.180294
\(370\) 0 0
\(371\) 2.04752e6 0.772313
\(372\) −90039.4 −0.0337346
\(373\) −3.48177e6 −1.29577 −0.647884 0.761739i \(-0.724346\pi\)
−0.647884 + 0.761739i \(0.724346\pi\)
\(374\) 144578. 0.0534470
\(375\) 0 0
\(376\) −361730. −0.131952
\(377\) −9.16768e6 −3.32205
\(378\) 145272. 0.0522939
\(379\) −704863. −0.252062 −0.126031 0.992026i \(-0.540224\pi\)
−0.126031 + 0.992026i \(0.540224\pi\)
\(380\) 0 0
\(381\) 1.55384e6 0.548397
\(382\) −852751. −0.298995
\(383\) 2.16463e6 0.754028 0.377014 0.926208i \(-0.376951\pi\)
0.377014 + 0.926208i \(0.376951\pi\)
\(384\) 982551. 0.340038
\(385\) 0 0
\(386\) 77547.7 0.0264911
\(387\) −115135. −0.0390778
\(388\) −3.59988e6 −1.21397
\(389\) 2.50750e6 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(390\) 0 0
\(391\) −4.57713e6 −1.51409
\(392\) −1.88955e6 −0.621075
\(393\) −2.67950e6 −0.875131
\(394\) −708942. −0.230075
\(395\) 0 0
\(396\) −305929. −0.0980352
\(397\) 2.75852e6 0.878417 0.439208 0.898385i \(-0.355259\pi\)
0.439208 + 0.898385i \(0.355259\pi\)
\(398\) −563422. −0.178289
\(399\) 1.39838e6 0.439736
\(400\) 0 0
\(401\) −6.11243e6 −1.89825 −0.949124 0.314901i \(-0.898029\pi\)
−0.949124 + 0.314901i \(0.898029\pi\)
\(402\) −546053. −0.168527
\(403\) 341546. 0.104758
\(404\) −1.24565e6 −0.379700
\(405\) 0 0
\(406\) −1.71437e6 −0.516165
\(407\) 230654. 0.0690200
\(408\) 679786. 0.202172
\(409\) 4.96574e6 1.46783 0.733915 0.679241i \(-0.237691\pi\)
0.733915 + 0.679241i \(0.237691\pi\)
\(410\) 0 0
\(411\) −410039. −0.119735
\(412\) 2.16376e6 0.628008
\(413\) −2.29576e6 −0.662294
\(414\) −243880. −0.0699320
\(415\) 0 0
\(416\) −2.80780e6 −0.795488
\(417\) 1.11875e6 0.315061
\(418\) 74153.2 0.0207582
\(419\) −3.30467e6 −0.919589 −0.459794 0.888025i \(-0.652077\pi\)
−0.459794 + 0.888025i \(0.652077\pi\)
\(420\) 0 0
\(421\) 5.04577e6 1.38747 0.693733 0.720232i \(-0.255965\pi\)
0.693733 + 0.720232i \(0.255965\pi\)
\(422\) −309936. −0.0847209
\(423\) 522815. 0.142068
\(424\) 510510. 0.137908
\(425\) 0 0
\(426\) −15031.9 −0.00401318
\(427\) 7.70441e6 2.04489
\(428\) 4.09367e6 1.08020
\(429\) 1.16048e6 0.304434
\(430\) 0 0
\(431\) 889002. 0.230520 0.115260 0.993335i \(-0.463230\pi\)
0.115260 + 0.993335i \(0.463230\pi\)
\(432\) −691940. −0.178385
\(433\) −406627. −0.104226 −0.0521131 0.998641i \(-0.516596\pi\)
−0.0521131 + 0.998641i \(0.516596\pi\)
\(434\) 63869.5 0.0162768
\(435\) 0 0
\(436\) −4.58623e6 −1.15542
\(437\) −2.34758e6 −0.588054
\(438\) −700031. −0.174354
\(439\) −5.04798e6 −1.25013 −0.625067 0.780571i \(-0.714928\pi\)
−0.625067 + 0.780571i \(0.714928\pi\)
\(440\) 0 0
\(441\) 2.73101e6 0.668693
\(442\) −1.27328e6 −0.310006
\(443\) 888258. 0.215045 0.107523 0.994203i \(-0.465708\pi\)
0.107523 + 0.994203i \(0.465708\pi\)
\(444\) 535510. 0.128917
\(445\) 0 0
\(446\) 274424. 0.0653259
\(447\) 968203. 0.229191
\(448\) 6.30203e6 1.48349
\(449\) 3.82230e6 0.894765 0.447383 0.894343i \(-0.352356\pi\)
0.447383 + 0.894343i \(0.352356\pi\)
\(450\) 0 0
\(451\) 704444. 0.163082
\(452\) −1.79000e6 −0.412104
\(453\) −1.32762e6 −0.303967
\(454\) 451909. 0.102899
\(455\) 0 0
\(456\) 348658. 0.0785214
\(457\) −2.10600e6 −0.471703 −0.235852 0.971789i \(-0.575788\pi\)
−0.235852 + 0.971789i \(0.575788\pi\)
\(458\) −1.13790e6 −0.253479
\(459\) −982509. −0.217673
\(460\) 0 0
\(461\) −2.43217e6 −0.533017 −0.266508 0.963833i \(-0.585870\pi\)
−0.266508 + 0.963833i \(0.585870\pi\)
\(462\) 217011. 0.0473016
\(463\) −7.53506e6 −1.63356 −0.816779 0.576951i \(-0.804242\pi\)
−0.816779 + 0.576951i \(0.804242\pi\)
\(464\) 8.16566e6 1.76075
\(465\) 0 0
\(466\) −277384. −0.0591720
\(467\) 3.17287e6 0.673225 0.336612 0.941643i \(-0.390719\pi\)
0.336612 + 0.941643i \(0.390719\pi\)
\(468\) 2.69428e6 0.568628
\(469\) −1.53826e7 −3.22922
\(470\) 0 0
\(471\) 2.08711e6 0.433503
\(472\) −572403. −0.118262
\(473\) −171992. −0.0353472
\(474\) 418172. 0.0854887
\(475\) 0 0
\(476\) 9.45592e6 1.91287
\(477\) −737850. −0.148481
\(478\) −316727. −0.0634038
\(479\) 3.20281e6 0.637811 0.318905 0.947787i \(-0.396685\pi\)
0.318905 + 0.947787i \(0.396685\pi\)
\(480\) 0 0
\(481\) −2.03135e6 −0.400333
\(482\) −603009. −0.118224
\(483\) −6.87024e6 −1.34000
\(484\) −457004. −0.0886762
\(485\) 0 0
\(486\) −52350.4 −0.0100538
\(487\) 4.45541e6 0.851266 0.425633 0.904896i \(-0.360051\pi\)
0.425633 + 0.904896i \(0.360051\pi\)
\(488\) 1.92095e6 0.365145
\(489\) −61761.2 −0.0116800
\(490\) 0 0
\(491\) 150957. 0.0282586 0.0141293 0.999900i \(-0.495502\pi\)
0.0141293 + 0.999900i \(0.495502\pi\)
\(492\) 1.63551e6 0.304608
\(493\) 1.15947e7 2.14853
\(494\) −653060. −0.120403
\(495\) 0 0
\(496\) −304216. −0.0555235
\(497\) −423456. −0.0768984
\(498\) 468096. 0.0845789
\(499\) −4.56196e6 −0.820163 −0.410081 0.912049i \(-0.634500\pi\)
−0.410081 + 0.912049i \(0.634500\pi\)
\(500\) 0 0
\(501\) −92069.8 −0.0163879
\(502\) 139914. 0.0247800
\(503\) 5.18773e6 0.914234 0.457117 0.889407i \(-0.348882\pi\)
0.457117 + 0.889407i \(0.348882\pi\)
\(504\) 1.02036e6 0.178927
\(505\) 0 0
\(506\) −364315. −0.0632559
\(507\) −6.87857e6 −1.18844
\(508\) 5.38908e6 0.926521
\(509\) 6.85927e6 1.17350 0.586750 0.809768i \(-0.300407\pi\)
0.586750 + 0.809768i \(0.300407\pi\)
\(510\) 0 0
\(511\) −1.97202e7 −3.34087
\(512\) 4.20312e6 0.708593
\(513\) −503923. −0.0845416
\(514\) −494076. −0.0824871
\(515\) 0 0
\(516\) −399314. −0.0660223
\(517\) 780995. 0.128506
\(518\) −379864. −0.0622020
\(519\) 3.79133e6 0.617836
\(520\) 0 0
\(521\) −3.91860e6 −0.632466 −0.316233 0.948682i \(-0.602418\pi\)
−0.316233 + 0.948682i \(0.602418\pi\)
\(522\) 617794. 0.0992356
\(523\) 3.64804e6 0.583183 0.291592 0.956543i \(-0.405815\pi\)
0.291592 + 0.956543i \(0.405815\pi\)
\(524\) −9.29312e6 −1.47854
\(525\) 0 0
\(526\) 572405. 0.0902067
\(527\) −431966. −0.0677521
\(528\) −1.03364e6 −0.161356
\(529\) 5.09735e6 0.791964
\(530\) 0 0
\(531\) 827305. 0.127330
\(532\) 4.84988e6 0.742938
\(533\) −6.20397e6 −0.945914
\(534\) 908355. 0.137849
\(535\) 0 0
\(536\) −3.83536e6 −0.576625
\(537\) −2.97465e6 −0.445144
\(538\) −194826. −0.0290196
\(539\) 4.07966e6 0.604855
\(540\) 0 0
\(541\) −4.23861e6 −0.622630 −0.311315 0.950307i \(-0.600769\pi\)
−0.311315 + 0.950307i \(0.600769\pi\)
\(542\) 1.37232e6 0.200659
\(543\) −6.01924e6 −0.876077
\(544\) 3.55113e6 0.514482
\(545\) 0 0
\(546\) −1.91119e6 −0.274361
\(547\) 513913. 0.0734380 0.0367190 0.999326i \(-0.488309\pi\)
0.0367190 + 0.999326i \(0.488309\pi\)
\(548\) −1.42211e6 −0.202293
\(549\) −2.77638e6 −0.393141
\(550\) 0 0
\(551\) 5.94685e6 0.834465
\(552\) −1.71296e6 −0.239277
\(553\) 1.17801e7 1.63809
\(554\) 869470. 0.120359
\(555\) 0 0
\(556\) 3.88009e6 0.532297
\(557\) 1.04431e7 1.42624 0.713121 0.701041i \(-0.247281\pi\)
0.713121 + 0.701041i \(0.247281\pi\)
\(558\) −23016.2 −0.00312930
\(559\) 1.51472e6 0.205023
\(560\) 0 0
\(561\) −1.46770e6 −0.196893
\(562\) 2.18345e6 0.291610
\(563\) −5.42113e6 −0.720806 −0.360403 0.932797i \(-0.617361\pi\)
−0.360403 + 0.932797i \(0.617361\pi\)
\(564\) 1.81324e6 0.240026
\(565\) 0 0
\(566\) 743477. 0.0975499
\(567\) −1.47474e6 −0.192645
\(568\) −105581. −0.0137313
\(569\) −4.65152e6 −0.602301 −0.301151 0.953577i \(-0.597371\pi\)
−0.301151 + 0.953577i \(0.597371\pi\)
\(570\) 0 0
\(571\) 836920. 0.107422 0.0537111 0.998557i \(-0.482895\pi\)
0.0537111 + 0.998557i \(0.482895\pi\)
\(572\) 4.02479e6 0.514344
\(573\) 8.65679e6 1.10146
\(574\) −1.16015e6 −0.146972
\(575\) 0 0
\(576\) −2.27102e6 −0.285210
\(577\) 1.22020e7 1.52578 0.762892 0.646526i \(-0.223779\pi\)
0.762892 + 0.646526i \(0.223779\pi\)
\(578\) 351582. 0.0437730
\(579\) −787233. −0.0975904
\(580\) 0 0
\(581\) 1.31865e7 1.62065
\(582\) −920214. −0.112611
\(583\) −1.10222e6 −0.134307
\(584\) −4.91686e6 −0.596562
\(585\) 0 0
\(586\) −1.17637e6 −0.141514
\(587\) 4.64629e6 0.556559 0.278279 0.960500i \(-0.410236\pi\)
0.278279 + 0.960500i \(0.410236\pi\)
\(588\) 9.47175e6 1.12976
\(589\) −221553. −0.0263141
\(590\) 0 0
\(591\) 7.19690e6 0.847572
\(592\) 1.80932e6 0.212184
\(593\) 3.76104e6 0.439209 0.219605 0.975589i \(-0.429523\pi\)
0.219605 + 0.975589i \(0.429523\pi\)
\(594\) −78202.5 −0.00909399
\(595\) 0 0
\(596\) 3.35794e6 0.387220
\(597\) 5.71963e6 0.656799
\(598\) 3.20849e6 0.366900
\(599\) −2.65769e6 −0.302648 −0.151324 0.988484i \(-0.548354\pi\)
−0.151324 + 0.988484i \(0.548354\pi\)
\(600\) 0 0
\(601\) 5.84003e6 0.659521 0.329761 0.944065i \(-0.393032\pi\)
0.329761 + 0.944065i \(0.393032\pi\)
\(602\) 283254. 0.0318555
\(603\) 5.54331e6 0.620835
\(604\) −4.60447e6 −0.513555
\(605\) 0 0
\(606\) −318416. −0.0352220
\(607\) 7.10119e6 0.782274 0.391137 0.920332i \(-0.372082\pi\)
0.391137 + 0.920332i \(0.372082\pi\)
\(608\) 1.82136e6 0.199819
\(609\) 1.74036e7 1.90150
\(610\) 0 0
\(611\) −6.87815e6 −0.745364
\(612\) −3.40756e6 −0.367760
\(613\) 1.10978e7 1.19285 0.596423 0.802670i \(-0.296588\pi\)
0.596423 + 0.802670i \(0.296588\pi\)
\(614\) −929488. −0.0995000
\(615\) 0 0
\(616\) 1.52423e6 0.161845
\(617\) −1.34866e7 −1.42623 −0.713114 0.701049i \(-0.752716\pi\)
−0.713114 + 0.701049i \(0.752716\pi\)
\(618\) 553107. 0.0582556
\(619\) −1.19228e7 −1.25069 −0.625346 0.780347i \(-0.715042\pi\)
−0.625346 + 0.780347i \(0.715042\pi\)
\(620\) 0 0
\(621\) 2.47578e6 0.257622
\(622\) −1.36340e6 −0.141302
\(623\) 2.55888e7 2.64138
\(624\) 9.10315e6 0.935902
\(625\) 0 0
\(626\) 2.29151e6 0.233715
\(627\) −752774. −0.0764708
\(628\) 7.23855e6 0.732407
\(629\) 2.56912e6 0.258915
\(630\) 0 0
\(631\) −5.56411e6 −0.556317 −0.278159 0.960535i \(-0.589724\pi\)
−0.278159 + 0.960535i \(0.589724\pi\)
\(632\) 2.93715e6 0.292505
\(633\) 3.14635e6 0.312103
\(634\) 1.14496e6 0.113127
\(635\) 0 0
\(636\) −2.55903e6 −0.250861
\(637\) −3.59291e7 −3.50831
\(638\) 922877. 0.0897619
\(639\) 152598. 0.0147841
\(640\) 0 0
\(641\) 7.43898e6 0.715103 0.357551 0.933894i \(-0.383612\pi\)
0.357551 + 0.933894i \(0.383612\pi\)
\(642\) 1.04644e6 0.100202
\(643\) −8.56732e6 −0.817180 −0.408590 0.912718i \(-0.633979\pi\)
−0.408590 + 0.912718i \(0.633979\pi\)
\(644\) −2.38275e7 −2.26394
\(645\) 0 0
\(646\) 825949. 0.0778703
\(647\) 1.65852e7 1.55762 0.778808 0.627263i \(-0.215825\pi\)
0.778808 + 0.627263i \(0.215825\pi\)
\(648\) −367698. −0.0343996
\(649\) 1.23585e6 0.115174
\(650\) 0 0
\(651\) −648378. −0.0599620
\(652\) −214202. −0.0197335
\(653\) 7.12943e6 0.654292 0.327146 0.944974i \(-0.393913\pi\)
0.327146 + 0.944974i \(0.393913\pi\)
\(654\) −1.17235e6 −0.107180
\(655\) 0 0
\(656\) 5.52589e6 0.501352
\(657\) 7.10643e6 0.642301
\(658\) −1.28622e6 −0.115811
\(659\) 1.00795e7 0.904116 0.452058 0.891988i \(-0.350690\pi\)
0.452058 + 0.891988i \(0.350690\pi\)
\(660\) 0 0
\(661\) 1.12712e7 1.00338 0.501689 0.865048i \(-0.332712\pi\)
0.501689 + 0.865048i \(0.332712\pi\)
\(662\) 1.57556e6 0.139730
\(663\) 1.29259e7 1.14203
\(664\) 3.28780e6 0.289392
\(665\) 0 0
\(666\) 136889. 0.0119587
\(667\) −2.92170e7 −2.54285
\(668\) −319319. −0.0276875
\(669\) −2.78585e6 −0.240653
\(670\) 0 0
\(671\) −4.14743e6 −0.355609
\(672\) 5.33023e6 0.455326
\(673\) 2.71329e6 0.230918 0.115459 0.993312i \(-0.463166\pi\)
0.115459 + 0.993312i \(0.463166\pi\)
\(674\) −1.77152e6 −0.150209
\(675\) 0 0
\(676\) −2.38564e7 −2.00788
\(677\) 1.70935e6 0.143337 0.0716686 0.997429i \(-0.477168\pi\)
0.0716686 + 0.997429i \(0.477168\pi\)
\(678\) −457565. −0.0382278
\(679\) −2.59229e7 −2.15779
\(680\) 0 0
\(681\) −4.58760e6 −0.379069
\(682\) −34382.2 −0.00283056
\(683\) −2.06905e7 −1.69715 −0.848573 0.529079i \(-0.822538\pi\)
−0.848573 + 0.529079i \(0.822538\pi\)
\(684\) −1.74772e6 −0.142834
\(685\) 0 0
\(686\) −3.36957e6 −0.273379
\(687\) 1.15516e7 0.933788
\(688\) −1.34916e6 −0.108666
\(689\) 9.70715e6 0.779011
\(690\) 0 0
\(691\) 3.42722e6 0.273053 0.136526 0.990636i \(-0.456406\pi\)
0.136526 + 0.990636i \(0.456406\pi\)
\(692\) 1.31492e7 1.04384
\(693\) −2.20301e6 −0.174254
\(694\) −2.61494e6 −0.206093
\(695\) 0 0
\(696\) 4.33925e6 0.339540
\(697\) 7.84639e6 0.611770
\(698\) −1.02373e6 −0.0795331
\(699\) 2.81589e6 0.217983
\(700\) 0 0
\(701\) 1.05204e7 0.808608 0.404304 0.914625i \(-0.367514\pi\)
0.404304 + 0.914625i \(0.367514\pi\)
\(702\) 688722. 0.0527474
\(703\) 1.31769e6 0.100560
\(704\) −3.39251e6 −0.257982
\(705\) 0 0
\(706\) 735650. 0.0555468
\(707\) −8.96995e6 −0.674903
\(708\) 2.86928e6 0.215124
\(709\) −5.27905e6 −0.394403 −0.197202 0.980363i \(-0.563185\pi\)
−0.197202 + 0.980363i \(0.563185\pi\)
\(710\) 0 0
\(711\) −4.24512e6 −0.314931
\(712\) 6.38008e6 0.471657
\(713\) 1.08849e6 0.0801864
\(714\) 2.41715e6 0.177443
\(715\) 0 0
\(716\) −1.03168e7 −0.752074
\(717\) 3.21529e6 0.233572
\(718\) 3.19732e6 0.231459
\(719\) −5.38506e6 −0.388480 −0.194240 0.980954i \(-0.562224\pi\)
−0.194240 + 0.980954i \(0.562224\pi\)
\(720\) 0 0
\(721\) 1.55813e7 1.11626
\(722\) −1.77158e6 −0.126479
\(723\) 6.12151e6 0.435525
\(724\) −2.08761e7 −1.48014
\(725\) 0 0
\(726\) −116821. −0.00822583
\(727\) 1.05731e6 0.0741934 0.0370967 0.999312i \(-0.488189\pi\)
0.0370967 + 0.999312i \(0.488189\pi\)
\(728\) −1.34238e7 −0.938743
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.91572e6 −0.132598
\(732\) −9.62911e6 −0.664215
\(733\) −1.02639e6 −0.0705589 −0.0352794 0.999377i \(-0.511232\pi\)
−0.0352794 + 0.999377i \(0.511232\pi\)
\(734\) 357544. 0.0244957
\(735\) 0 0
\(736\) −8.94834e6 −0.608903
\(737\) 8.28075e6 0.561566
\(738\) 418075. 0.0282562
\(739\) −3.33815e6 −0.224851 −0.112426 0.993660i \(-0.535862\pi\)
−0.112426 + 0.993660i \(0.535862\pi\)
\(740\) 0 0
\(741\) 6.62960e6 0.443550
\(742\) 1.81525e6 0.121039
\(743\) 2.15478e7 1.43196 0.715981 0.698120i \(-0.245980\pi\)
0.715981 + 0.698120i \(0.245980\pi\)
\(744\) −161661. −0.0107071
\(745\) 0 0
\(746\) −3.08679e6 −0.203077
\(747\) −4.75193e6 −0.311579
\(748\) −5.09030e6 −0.332652
\(749\) 2.94787e7 1.92001
\(750\) 0 0
\(751\) −2.02621e7 −1.31095 −0.655474 0.755218i \(-0.727531\pi\)
−0.655474 + 0.755218i \(0.727531\pi\)
\(752\) 6.12638e6 0.395057
\(753\) −1.42035e6 −0.0912869
\(754\) −8.12769e6 −0.520641
\(755\) 0 0
\(756\) −5.11473e6 −0.325475
\(757\) −1.23455e7 −0.783012 −0.391506 0.920176i \(-0.628046\pi\)
−0.391506 + 0.920176i \(0.628046\pi\)
\(758\) −624903. −0.0395039
\(759\) 3.69838e6 0.233028
\(760\) 0 0
\(761\) 9.77671e6 0.611971 0.305986 0.952036i \(-0.401014\pi\)
0.305986 + 0.952036i \(0.401014\pi\)
\(762\) 1.37758e6 0.0859465
\(763\) −3.30257e7 −2.05372
\(764\) 3.00237e7 1.86093
\(765\) 0 0
\(766\) 1.91908e6 0.118174
\(767\) −1.08840e7 −0.668038
\(768\) −7.20364e6 −0.440706
\(769\) −4.31458e6 −0.263101 −0.131551 0.991309i \(-0.541996\pi\)
−0.131551 + 0.991309i \(0.541996\pi\)
\(770\) 0 0
\(771\) 5.01567e6 0.303873
\(772\) −2.73030e6 −0.164880
\(773\) −3.28419e7 −1.97688 −0.988439 0.151620i \(-0.951551\pi\)
−0.988439 + 0.151620i \(0.951551\pi\)
\(774\) −102074. −0.00612440
\(775\) 0 0
\(776\) −6.46338e6 −0.385306
\(777\) 3.85623e6 0.229145
\(778\) 2.22304e6 0.131674
\(779\) 4.02437e6 0.237604
\(780\) 0 0
\(781\) 227955. 0.0133727
\(782\) −4.05790e6 −0.237292
\(783\) −6.27160e6 −0.365573
\(784\) 3.20021e7 1.85947
\(785\) 0 0
\(786\) −2.37554e6 −0.137153
\(787\) −1.45886e7 −0.839608 −0.419804 0.907615i \(-0.637901\pi\)
−0.419804 + 0.907615i \(0.637901\pi\)
\(788\) 2.49605e7 1.43198
\(789\) −5.81083e6 −0.332312
\(790\) 0 0
\(791\) −1.28899e7 −0.732499
\(792\) −549277. −0.0311156
\(793\) 3.65261e7 2.06262
\(794\) 2.44560e6 0.137668
\(795\) 0 0
\(796\) 1.98370e7 1.10967
\(797\) −4.63451e6 −0.258439 −0.129220 0.991616i \(-0.541247\pi\)
−0.129220 + 0.991616i \(0.541247\pi\)
\(798\) 1.23974e6 0.0689168
\(799\) 8.69905e6 0.482064
\(800\) 0 0
\(801\) −9.22126e6 −0.507819
\(802\) −5.41903e6 −0.297499
\(803\) 1.06158e7 0.580983
\(804\) 1.92255e7 1.04891
\(805\) 0 0
\(806\) 302801. 0.0164180
\(807\) 1.97780e6 0.106905
\(808\) −2.23648e6 −0.120514
\(809\) 1.28933e6 0.0692618 0.0346309 0.999400i \(-0.488974\pi\)
0.0346309 + 0.999400i \(0.488974\pi\)
\(810\) 0 0
\(811\) 6.47343e6 0.345607 0.172803 0.984956i \(-0.444718\pi\)
0.172803 + 0.984956i \(0.444718\pi\)
\(812\) 6.03595e7 3.21259
\(813\) −1.39313e7 −0.739205
\(814\) 204488. 0.0108170
\(815\) 0 0
\(816\) −1.15131e7 −0.605294
\(817\) −982561. −0.0514997
\(818\) 4.40242e6 0.230043
\(819\) 1.94017e7 1.01072
\(820\) 0 0
\(821\) 2.48796e7 1.28820 0.644102 0.764939i \(-0.277231\pi\)
0.644102 + 0.764939i \(0.277231\pi\)
\(822\) −363524. −0.0187652
\(823\) −2.73317e7 −1.40659 −0.703295 0.710898i \(-0.748289\pi\)
−0.703295 + 0.710898i \(0.748289\pi\)
\(824\) 3.88490e6 0.199325
\(825\) 0 0
\(826\) −2.03533e6 −0.103797
\(827\) −3.20369e7 −1.62887 −0.814437 0.580252i \(-0.802954\pi\)
−0.814437 + 0.580252i \(0.802954\pi\)
\(828\) 8.58655e6 0.435254
\(829\) −2.54756e7 −1.28747 −0.643736 0.765247i \(-0.722617\pi\)
−0.643736 + 0.765247i \(0.722617\pi\)
\(830\) 0 0
\(831\) −8.82652e6 −0.443391
\(832\) 2.98775e7 1.49636
\(833\) 4.54409e7 2.26900
\(834\) 991841. 0.0493772
\(835\) 0 0
\(836\) −2.61079e6 −0.129198
\(837\) 233651. 0.0115280
\(838\) −2.92979e6 −0.144121
\(839\) −2.56179e7 −1.25643 −0.628216 0.778039i \(-0.716215\pi\)
−0.628216 + 0.778039i \(0.716215\pi\)
\(840\) 0 0
\(841\) 5.35007e7 2.60837
\(842\) 4.47338e6 0.217448
\(843\) −2.21655e7 −1.07426
\(844\) 1.09122e7 0.527300
\(845\) 0 0
\(846\) 463507. 0.0222654
\(847\) −3.29091e6 −0.157619
\(848\) −8.64617e6 −0.412890
\(849\) −7.54749e6 −0.359363
\(850\) 0 0
\(851\) −6.47381e6 −0.306433
\(852\) 529243. 0.0249779
\(853\) −8.49389e6 −0.399700 −0.199850 0.979827i \(-0.564045\pi\)
−0.199850 + 0.979827i \(0.564045\pi\)
\(854\) 6.83042e6 0.320481
\(855\) 0 0
\(856\) 7.34996e6 0.342847
\(857\) −2.29100e7 −1.06555 −0.532773 0.846258i \(-0.678850\pi\)
−0.532773 + 0.846258i \(0.678850\pi\)
\(858\) 1.02883e6 0.0477118
\(859\) 1.05172e7 0.486315 0.243158 0.969987i \(-0.421817\pi\)
0.243158 + 0.969987i \(0.421817\pi\)
\(860\) 0 0
\(861\) 1.17774e7 0.541429
\(862\) 788153. 0.0361279
\(863\) −2.93900e7 −1.34330 −0.671649 0.740870i \(-0.734413\pi\)
−0.671649 + 0.740870i \(0.734413\pi\)
\(864\) −1.92082e6 −0.0875390
\(865\) 0 0
\(866\) −360499. −0.0163346
\(867\) −3.56912e6 −0.161255
\(868\) −2.24872e6 −0.101306
\(869\) −6.34147e6 −0.284866
\(870\) 0 0
\(871\) −7.29278e7 −3.25722
\(872\) −8.23432e6 −0.366722
\(873\) 9.34165e6 0.414847
\(874\) −2.08127e6 −0.0921616
\(875\) 0 0
\(876\) 2.46467e7 1.08517
\(877\) 3.81404e7 1.67451 0.837253 0.546816i \(-0.184160\pi\)
0.837253 + 0.546816i \(0.184160\pi\)
\(878\) −4.47533e6 −0.195925
\(879\) 1.19420e7 0.521321
\(880\) 0 0
\(881\) 2.39208e7 1.03833 0.519165 0.854674i \(-0.326243\pi\)
0.519165 + 0.854674i \(0.326243\pi\)
\(882\) 2.42120e6 0.104800
\(883\) 1.95650e6 0.0844457 0.0422229 0.999108i \(-0.486556\pi\)
0.0422229 + 0.999108i \(0.486556\pi\)
\(884\) 4.48298e7 1.92946
\(885\) 0 0
\(886\) 787493. 0.0337025
\(887\) 2.80192e7 1.19577 0.597884 0.801583i \(-0.296008\pi\)
0.597884 + 0.801583i \(0.296008\pi\)
\(888\) 961478. 0.0409173
\(889\) 3.88070e7 1.64686
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −9.66194e6 −0.406586
\(893\) 4.46169e6 0.187228
\(894\) 858369. 0.0359195
\(895\) 0 0
\(896\) 2.45391e7 1.02115
\(897\) −3.25713e7 −1.35162
\(898\) 3.38870e6 0.140230
\(899\) −2.75735e6 −0.113787
\(900\) 0 0
\(901\) −1.22770e7 −0.503825
\(902\) 624532. 0.0255587
\(903\) −2.87548e6 −0.117352
\(904\) −3.21384e6 −0.130799
\(905\) 0 0
\(906\) −1.17701e6 −0.0476387
\(907\) 5.99454e6 0.241957 0.120978 0.992655i \(-0.461397\pi\)
0.120978 + 0.992655i \(0.461397\pi\)
\(908\) −1.59108e7 −0.640439
\(909\) 3.23243e6 0.129754
\(910\) 0 0
\(911\) −3.83047e7 −1.52917 −0.764585 0.644523i \(-0.777056\pi\)
−0.764585 + 0.644523i \(0.777056\pi\)
\(912\) −5.90500e6 −0.235089
\(913\) −7.09856e6 −0.281834
\(914\) −1.86710e6 −0.0739267
\(915\) 0 0
\(916\) 4.00634e7 1.57764
\(917\) −6.69202e7 −2.62805
\(918\) −871052. −0.0341144
\(919\) −4.41558e7 −1.72464 −0.862321 0.506362i \(-0.830990\pi\)
−0.862321 + 0.506362i \(0.830990\pi\)
\(920\) 0 0
\(921\) 9.43580e6 0.366547
\(922\) −2.15626e6 −0.0835360
\(923\) −2.00757e6 −0.0775652
\(924\) −7.64052e6 −0.294403
\(925\) 0 0
\(926\) −6.68028e6 −0.256016
\(927\) −5.61492e6 −0.214607
\(928\) 2.26678e7 0.864050
\(929\) 1.00353e7 0.381499 0.190749 0.981639i \(-0.438908\pi\)
0.190749 + 0.981639i \(0.438908\pi\)
\(930\) 0 0
\(931\) 2.33064e7 0.881253
\(932\) 9.76613e6 0.368284
\(933\) 1.38407e7 0.520540
\(934\) 2.81294e6 0.105510
\(935\) 0 0
\(936\) 4.83743e6 0.180478
\(937\) −3.04913e7 −1.13456 −0.567279 0.823526i \(-0.692004\pi\)
−0.567279 + 0.823526i \(0.692004\pi\)
\(938\) −1.36376e7 −0.506093
\(939\) −2.32626e7 −0.860981
\(940\) 0 0
\(941\) −4.13865e7 −1.52365 −0.761824 0.647785i \(-0.775696\pi\)
−0.761824 + 0.647785i \(0.775696\pi\)
\(942\) 1.85034e6 0.0679399
\(943\) −1.97718e7 −0.724046
\(944\) 9.69442e6 0.354072
\(945\) 0 0
\(946\) −152481. −0.00553973
\(947\) 9.38834e6 0.340184 0.170092 0.985428i \(-0.445594\pi\)
0.170092 + 0.985428i \(0.445594\pi\)
\(948\) −1.47230e7 −0.532079
\(949\) −9.34922e7 −3.36984
\(950\) 0 0
\(951\) −1.16232e7 −0.416747
\(952\) 1.69776e7 0.607132
\(953\) −3.35369e7 −1.19616 −0.598081 0.801435i \(-0.704070\pi\)
−0.598081 + 0.801435i \(0.704070\pi\)
\(954\) −654148. −0.0232705
\(955\) 0 0
\(956\) 1.11513e7 0.394623
\(957\) −9.36868e6 −0.330673
\(958\) 2.83948e6 0.0999597
\(959\) −1.02406e7 −0.359568
\(960\) 0 0
\(961\) −2.85264e7 −0.996412
\(962\) −1.80091e6 −0.0627414
\(963\) −1.06230e7 −0.369133
\(964\) 2.12308e7 0.735823
\(965\) 0 0
\(966\) −6.09088e6 −0.210009
\(967\) 5.35398e7 1.84124 0.920620 0.390459i \(-0.127684\pi\)
0.920620 + 0.390459i \(0.127684\pi\)
\(968\) −820525. −0.0281451
\(969\) −8.38471e6 −0.286866
\(970\) 0 0
\(971\) −3.17286e7 −1.07995 −0.539974 0.841682i \(-0.681566\pi\)
−0.539974 + 0.841682i \(0.681566\pi\)
\(972\) 1.84316e6 0.0625744
\(973\) 2.79407e7 0.946139
\(974\) 3.94999e6 0.133413
\(975\) 0 0
\(976\) −3.25338e7 −1.09323
\(977\) −5.28990e6 −0.177301 −0.0886505 0.996063i \(-0.528255\pi\)
−0.0886505 + 0.996063i \(0.528255\pi\)
\(978\) −54754.9 −0.00183053
\(979\) −1.37750e7 −0.459340
\(980\) 0 0
\(981\) 1.19012e7 0.394838
\(982\) 133832. 0.00442877
\(983\) 2.55796e7 0.844327 0.422164 0.906520i \(-0.361271\pi\)
0.422164 + 0.906520i \(0.361271\pi\)
\(984\) 2.93647e6 0.0966801
\(985\) 0 0
\(986\) 1.02794e7 0.336725
\(987\) 1.30572e7 0.426637
\(988\) 2.29930e7 0.749380
\(989\) 4.82733e6 0.156934
\(990\) 0 0
\(991\) 4.14167e7 1.33965 0.669825 0.742519i \(-0.266369\pi\)
0.669825 + 0.742519i \(0.266369\pi\)
\(992\) −844498. −0.0272471
\(993\) −1.59945e7 −0.514750
\(994\) −375419. −0.0120517
\(995\) 0 0
\(996\) −1.64808e7 −0.526415
\(997\) −3.02210e7 −0.962877 −0.481438 0.876480i \(-0.659885\pi\)
−0.481438 + 0.876480i \(0.659885\pi\)
\(998\) −4.04445e6 −0.128538
\(999\) −1.38964e6 −0.0440544
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.v.1.8 13
5.2 odd 4 165.6.c.b.34.14 yes 26
5.3 odd 4 165.6.c.b.34.13 26
5.4 even 2 825.6.a.y.1.6 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.13 26 5.3 odd 4
165.6.c.b.34.14 yes 26 5.2 odd 4
825.6.a.v.1.8 13 1.1 even 1 trivial
825.6.a.y.1.6 13 5.4 even 2