Properties

Label 825.6.a.q.1.8
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(8.70245\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.70245 q^{2} -9.00000 q^{3} +62.1375 q^{4} -87.3220 q^{6} -79.4231 q^{7} +292.408 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+9.70245 q^{2} -9.00000 q^{3} +62.1375 q^{4} -87.3220 q^{6} -79.4231 q^{7} +292.408 q^{8} +81.0000 q^{9} -121.000 q^{11} -559.238 q^{12} -780.381 q^{13} -770.598 q^{14} +848.670 q^{16} +1974.55 q^{17} +785.898 q^{18} +2318.66 q^{19} +714.808 q^{21} -1174.00 q^{22} +1282.47 q^{23} -2631.67 q^{24} -7571.61 q^{26} -729.000 q^{27} -4935.15 q^{28} -1278.69 q^{29} -6820.62 q^{31} -1122.87 q^{32} +1089.00 q^{33} +19158.0 q^{34} +5033.14 q^{36} -14719.4 q^{37} +22496.6 q^{38} +7023.43 q^{39} -8353.18 q^{41} +6935.38 q^{42} +13380.3 q^{43} -7518.64 q^{44} +12443.1 q^{46} +1363.31 q^{47} -7638.03 q^{48} -10499.0 q^{49} -17770.9 q^{51} -48490.9 q^{52} -35100.9 q^{53} -7073.08 q^{54} -23223.9 q^{56} -20867.9 q^{57} -12406.4 q^{58} -34493.5 q^{59} -45097.0 q^{61} -66176.7 q^{62} -6433.27 q^{63} -38052.0 q^{64} +10566.0 q^{66} +24151.0 q^{67} +122694. q^{68} -11542.3 q^{69} -18253.3 q^{71} +23685.0 q^{72} +32455.4 q^{73} -142815. q^{74} +144075. q^{76} +9610.19 q^{77} +68144.5 q^{78} -28488.4 q^{79} +6561.00 q^{81} -81046.3 q^{82} -44345.4 q^{83} +44416.4 q^{84} +129821. q^{86} +11508.2 q^{87} -35381.3 q^{88} -82213.9 q^{89} +61980.3 q^{91} +79689.6 q^{92} +61385.6 q^{93} +13227.5 q^{94} +10105.8 q^{96} -20136.3 q^{97} -101866. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9} - 968 q^{11} - 963 q^{12} + 382 q^{13} + 1048 q^{14} - 325 q^{16} + 288 q^{17} + 729 q^{18} - 988 q^{19} + 594 q^{21} - 1089 q^{22} + 5972 q^{23} - 1863 q^{24} - 14579 q^{26} - 5832 q^{27} + 5942 q^{28} + 1032 q^{29} - 4682 q^{31} + 3863 q^{32} + 8712 q^{33} + 2206 q^{34} + 8667 q^{36} - 17200 q^{37} + 11011 q^{38} - 3438 q^{39} - 13220 q^{41} - 9432 q^{42} + 22872 q^{43} - 12947 q^{44} + 9101 q^{46} + 6700 q^{47} + 2925 q^{48} - 43466 q^{49} - 2592 q^{51} + 5009 q^{52} + 6224 q^{53} - 6561 q^{54} + 20992 q^{56} + 8892 q^{57} + 33015 q^{58} - 77556 q^{59} + 11554 q^{61} + 12135 q^{62} - 5346 q^{63} - 149917 q^{64} + 9801 q^{66} + 20894 q^{67} + 91776 q^{68} - 53748 q^{69} - 21648 q^{71} + 16767 q^{72} + 64660 q^{73} - 179522 q^{74} + 24401 q^{76} + 7986 q^{77} + 131211 q^{78} - 22660 q^{79} + 52488 q^{81} - 56080 q^{82} + 100390 q^{83} - 53478 q^{84} + 47271 q^{86} - 9288 q^{87} - 25047 q^{88} - 25578 q^{89} + 73250 q^{91} + 95311 q^{92} + 42138 q^{93} - 170120 q^{94} - 34767 q^{96} + 142828 q^{97} + 303397 q^{98} - 78408 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.70245 1.71517 0.857583 0.514345i \(-0.171965\pi\)
0.857583 + 0.514345i \(0.171965\pi\)
\(3\) −9.00000 −0.577350
\(4\) 62.1375 1.94180
\(5\) 0 0
\(6\) −87.3220 −0.990252
\(7\) −79.4231 −0.612635 −0.306317 0.951929i \(-0.599097\pi\)
−0.306317 + 0.951929i \(0.599097\pi\)
\(8\) 292.408 1.61534
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −559.238 −1.12110
\(13\) −780.381 −1.28070 −0.640352 0.768082i \(-0.721211\pi\)
−0.640352 + 0.768082i \(0.721211\pi\)
\(14\) −770.598 −1.05077
\(15\) 0 0
\(16\) 848.670 0.828779
\(17\) 1974.55 1.65709 0.828544 0.559924i \(-0.189170\pi\)
0.828544 + 0.559924i \(0.189170\pi\)
\(18\) 785.898 0.571722
\(19\) 2318.66 1.47351 0.736754 0.676161i \(-0.236358\pi\)
0.736754 + 0.676161i \(0.236358\pi\)
\(20\) 0 0
\(21\) 714.808 0.353705
\(22\) −1174.00 −0.517142
\(23\) 1282.47 0.505509 0.252754 0.967531i \(-0.418664\pi\)
0.252754 + 0.967531i \(0.418664\pi\)
\(24\) −2631.67 −0.932616
\(25\) 0 0
\(26\) −7571.61 −2.19662
\(27\) −729.000 −0.192450
\(28\) −4935.15 −1.18961
\(29\) −1278.69 −0.282338 −0.141169 0.989986i \(-0.545086\pi\)
−0.141169 + 0.989986i \(0.545086\pi\)
\(30\) 0 0
\(31\) −6820.62 −1.27473 −0.637367 0.770560i \(-0.719976\pi\)
−0.637367 + 0.770560i \(0.719976\pi\)
\(32\) −1122.87 −0.193845
\(33\) 1089.00 0.174078
\(34\) 19158.0 2.84218
\(35\) 0 0
\(36\) 5033.14 0.647266
\(37\) −14719.4 −1.76761 −0.883807 0.467853i \(-0.845028\pi\)
−0.883807 + 0.467853i \(0.845028\pi\)
\(38\) 22496.6 2.52731
\(39\) 7023.43 0.739414
\(40\) 0 0
\(41\) −8353.18 −0.776055 −0.388027 0.921648i \(-0.626843\pi\)
−0.388027 + 0.921648i \(0.626843\pi\)
\(42\) 6935.38 0.606663
\(43\) 13380.3 1.10355 0.551777 0.833991i \(-0.313950\pi\)
0.551777 + 0.833991i \(0.313950\pi\)
\(44\) −7518.64 −0.585474
\(45\) 0 0
\(46\) 12443.1 0.867031
\(47\) 1363.31 0.0900225 0.0450112 0.998986i \(-0.485668\pi\)
0.0450112 + 0.998986i \(0.485668\pi\)
\(48\) −7638.03 −0.478496
\(49\) −10499.0 −0.624679
\(50\) 0 0
\(51\) −17770.9 −0.956720
\(52\) −48490.9 −2.48687
\(53\) −35100.9 −1.71644 −0.858220 0.513282i \(-0.828430\pi\)
−0.858220 + 0.513282i \(0.828430\pi\)
\(54\) −7073.08 −0.330084
\(55\) 0 0
\(56\) −23223.9 −0.989613
\(57\) −20867.9 −0.850730
\(58\) −12406.4 −0.484256
\(59\) −34493.5 −1.29005 −0.645025 0.764161i \(-0.723153\pi\)
−0.645025 + 0.764161i \(0.723153\pi\)
\(60\) 0 0
\(61\) −45097.0 −1.55176 −0.775878 0.630883i \(-0.782693\pi\)
−0.775878 + 0.630883i \(0.782693\pi\)
\(62\) −66176.7 −2.18638
\(63\) −6433.27 −0.204212
\(64\) −38052.0 −1.16126
\(65\) 0 0
\(66\) 10566.0 0.298572
\(67\) 24151.0 0.657275 0.328638 0.944456i \(-0.393411\pi\)
0.328638 + 0.944456i \(0.393411\pi\)
\(68\) 122694. 3.21773
\(69\) −11542.3 −0.291855
\(70\) 0 0
\(71\) −18253.3 −0.429731 −0.214865 0.976644i \(-0.568931\pi\)
−0.214865 + 0.976644i \(0.568931\pi\)
\(72\) 23685.0 0.538446
\(73\) 32455.4 0.712819 0.356410 0.934330i \(-0.384001\pi\)
0.356410 + 0.934330i \(0.384001\pi\)
\(74\) −142815. −3.03175
\(75\) 0 0
\(76\) 144075. 2.86125
\(77\) 9610.19 0.184716
\(78\) 68144.5 1.26822
\(79\) −28488.4 −0.513570 −0.256785 0.966469i \(-0.582663\pi\)
−0.256785 + 0.966469i \(0.582663\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −81046.3 −1.33106
\(83\) −44345.4 −0.706566 −0.353283 0.935516i \(-0.614935\pi\)
−0.353283 + 0.935516i \(0.614935\pi\)
\(84\) 44416.4 0.686823
\(85\) 0 0
\(86\) 129821. 1.89278
\(87\) 11508.2 0.163008
\(88\) −35381.3 −0.487043
\(89\) −82213.9 −1.10020 −0.550098 0.835100i \(-0.685410\pi\)
−0.550098 + 0.835100i \(0.685410\pi\)
\(90\) 0 0
\(91\) 61980.3 0.784603
\(92\) 79689.6 0.981595
\(93\) 61385.6 0.735968
\(94\) 13227.5 0.154404
\(95\) 0 0
\(96\) 10105.8 0.111917
\(97\) −20136.3 −0.217295 −0.108648 0.994080i \(-0.534652\pi\)
−0.108648 + 0.994080i \(0.534652\pi\)
\(98\) −101866. −1.07143
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) −39455.8 −0.384865 −0.192432 0.981310i \(-0.561638\pi\)
−0.192432 + 0.981310i \(0.561638\pi\)
\(102\) −172422. −1.64093
\(103\) 100304. 0.931587 0.465793 0.884893i \(-0.345769\pi\)
0.465793 + 0.884893i \(0.345769\pi\)
\(104\) −228189. −2.06877
\(105\) 0 0
\(106\) −340565. −2.94398
\(107\) 172037. 1.45265 0.726325 0.687351i \(-0.241227\pi\)
0.726325 + 0.687351i \(0.241227\pi\)
\(108\) −45298.2 −0.373699
\(109\) 217939. 1.75699 0.878495 0.477752i \(-0.158548\pi\)
0.878495 + 0.477752i \(0.158548\pi\)
\(110\) 0 0
\(111\) 132475. 1.02053
\(112\) −67403.9 −0.507739
\(113\) −49169.5 −0.362243 −0.181121 0.983461i \(-0.557973\pi\)
−0.181121 + 0.983461i \(0.557973\pi\)
\(114\) −202470. −1.45914
\(115\) 0 0
\(116\) −79454.3 −0.548242
\(117\) −63210.9 −0.426901
\(118\) −334671. −2.21265
\(119\) −156825. −1.01519
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −437552. −2.66152
\(123\) 75178.6 0.448055
\(124\) −423816. −2.47528
\(125\) 0 0
\(126\) −62418.5 −0.350257
\(127\) −55222.2 −0.303812 −0.151906 0.988395i \(-0.548541\pi\)
−0.151906 + 0.988395i \(0.548541\pi\)
\(128\) −333266. −1.79790
\(129\) −120422. −0.637138
\(130\) 0 0
\(131\) 56033.9 0.285281 0.142640 0.989775i \(-0.454441\pi\)
0.142640 + 0.989775i \(0.454441\pi\)
\(132\) 67667.7 0.338023
\(133\) −184155. −0.902722
\(134\) 234323. 1.12734
\(135\) 0 0
\(136\) 577373. 2.67676
\(137\) −370432. −1.68619 −0.843096 0.537763i \(-0.819269\pi\)
−0.843096 + 0.537763i \(0.819269\pi\)
\(138\) −111988. −0.500581
\(139\) −144540. −0.634527 −0.317264 0.948337i \(-0.602764\pi\)
−0.317264 + 0.948337i \(0.602764\pi\)
\(140\) 0 0
\(141\) −12269.8 −0.0519745
\(142\) −177102. −0.737060
\(143\) 94426.1 0.386147
\(144\) 68742.2 0.276260
\(145\) 0 0
\(146\) 314897. 1.22260
\(147\) 94490.8 0.360658
\(148\) −914630. −3.43235
\(149\) 63090.5 0.232808 0.116404 0.993202i \(-0.462863\pi\)
0.116404 + 0.993202i \(0.462863\pi\)
\(150\) 0 0
\(151\) 400569. 1.42967 0.714834 0.699294i \(-0.246502\pi\)
0.714834 + 0.699294i \(0.246502\pi\)
\(152\) 677993. 2.38021
\(153\) 159938. 0.552363
\(154\) 93242.4 0.316819
\(155\) 0 0
\(156\) 436419. 1.43579
\(157\) −119299. −0.386267 −0.193133 0.981173i \(-0.561865\pi\)
−0.193133 + 0.981173i \(0.561865\pi\)
\(158\) −276407. −0.880858
\(159\) 315908. 0.990987
\(160\) 0 0
\(161\) −101858. −0.309692
\(162\) 63657.8 0.190574
\(163\) −496732. −1.46438 −0.732189 0.681101i \(-0.761501\pi\)
−0.732189 + 0.681101i \(0.761501\pi\)
\(164\) −519046. −1.50694
\(165\) 0 0
\(166\) −430259. −1.21188
\(167\) −402617. −1.11712 −0.558562 0.829463i \(-0.688647\pi\)
−0.558562 + 0.829463i \(0.688647\pi\)
\(168\) 209015. 0.571353
\(169\) 237702. 0.640200
\(170\) 0 0
\(171\) 187811. 0.491169
\(172\) 831417. 2.14288
\(173\) 642176. 1.63132 0.815660 0.578532i \(-0.196374\pi\)
0.815660 + 0.578532i \(0.196374\pi\)
\(174\) 111657. 0.279585
\(175\) 0 0
\(176\) −102689. −0.249886
\(177\) 310441. 0.744811
\(178\) −797676. −1.88702
\(179\) −628315. −1.46570 −0.732850 0.680390i \(-0.761810\pi\)
−0.732850 + 0.680390i \(0.761810\pi\)
\(180\) 0 0
\(181\) 227359. 0.515842 0.257921 0.966166i \(-0.416963\pi\)
0.257921 + 0.966166i \(0.416963\pi\)
\(182\) 601360. 1.34573
\(183\) 405873. 0.895906
\(184\) 375005. 0.816568
\(185\) 0 0
\(186\) 595590. 1.26231
\(187\) −238920. −0.499631
\(188\) 84712.9 0.174805
\(189\) 57899.4 0.117902
\(190\) 0 0
\(191\) 232254. 0.460659 0.230329 0.973113i \(-0.426020\pi\)
0.230329 + 0.973113i \(0.426020\pi\)
\(192\) 342468. 0.670451
\(193\) 199404. 0.385336 0.192668 0.981264i \(-0.438286\pi\)
0.192668 + 0.981264i \(0.438286\pi\)
\(194\) −195371. −0.372697
\(195\) 0 0
\(196\) −652380. −1.21300
\(197\) −66319.8 −0.121752 −0.0608762 0.998145i \(-0.519389\pi\)
−0.0608762 + 0.998145i \(0.519389\pi\)
\(198\) −95093.7 −0.172381
\(199\) −472509. −0.845819 −0.422909 0.906172i \(-0.638991\pi\)
−0.422909 + 0.906172i \(0.638991\pi\)
\(200\) 0 0
\(201\) −217359. −0.379478
\(202\) −382818. −0.660107
\(203\) 101557. 0.172970
\(204\) −1.10424e6 −1.85776
\(205\) 0 0
\(206\) 973190. 1.59783
\(207\) 103880. 0.168503
\(208\) −662286. −1.06142
\(209\) −280557. −0.444279
\(210\) 0 0
\(211\) −287964. −0.445280 −0.222640 0.974901i \(-0.571467\pi\)
−0.222640 + 0.974901i \(0.571467\pi\)
\(212\) −2.18108e6 −3.33298
\(213\) 164280. 0.248105
\(214\) 1.66918e6 2.49154
\(215\) 0 0
\(216\) −213165. −0.310872
\(217\) 541715. 0.780946
\(218\) 2.11454e6 3.01353
\(219\) −292098. −0.411546
\(220\) 0 0
\(221\) −1.54090e6 −2.12224
\(222\) 1.28533e6 1.75038
\(223\) −1.05057e6 −1.41469 −0.707347 0.706867i \(-0.750108\pi\)
−0.707347 + 0.706867i \(0.750108\pi\)
\(224\) 89181.8 0.118756
\(225\) 0 0
\(226\) −477065. −0.621307
\(227\) 304726. 0.392505 0.196253 0.980553i \(-0.437123\pi\)
0.196253 + 0.980553i \(0.437123\pi\)
\(228\) −1.29668e6 −1.65194
\(229\) −612466. −0.771780 −0.385890 0.922545i \(-0.626106\pi\)
−0.385890 + 0.922545i \(0.626106\pi\)
\(230\) 0 0
\(231\) −86491.7 −0.106646
\(232\) −373897. −0.456071
\(233\) 1.51321e6 1.82604 0.913021 0.407914i \(-0.133744\pi\)
0.913021 + 0.407914i \(0.133744\pi\)
\(234\) −613300. −0.732206
\(235\) 0 0
\(236\) −2.14334e6 −2.50502
\(237\) 256395. 0.296510
\(238\) −1.52158e6 −1.74122
\(239\) −749979. −0.849287 −0.424644 0.905361i \(-0.639601\pi\)
−0.424644 + 0.905361i \(0.639601\pi\)
\(240\) 0 0
\(241\) 251133. 0.278523 0.139261 0.990256i \(-0.455527\pi\)
0.139261 + 0.990256i \(0.455527\pi\)
\(242\) 142054. 0.155924
\(243\) −59049.0 −0.0641500
\(244\) −2.80222e6 −3.01319
\(245\) 0 0
\(246\) 729417. 0.768490
\(247\) −1.80944e6 −1.88713
\(248\) −1.99440e6 −2.05913
\(249\) 399108. 0.407936
\(250\) 0 0
\(251\) 901997. 0.903692 0.451846 0.892096i \(-0.350766\pi\)
0.451846 + 0.892096i \(0.350766\pi\)
\(252\) −399747. −0.396537
\(253\) −155179. −0.152417
\(254\) −535791. −0.521088
\(255\) 0 0
\(256\) −2.01583e6 −1.92245
\(257\) 1.08538e6 1.02506 0.512528 0.858671i \(-0.328709\pi\)
0.512528 + 0.858671i \(0.328709\pi\)
\(258\) −1.16839e6 −1.09280
\(259\) 1.16906e6 1.08290
\(260\) 0 0
\(261\) −103574. −0.0941126
\(262\) 543666. 0.489304
\(263\) 522831. 0.466092 0.233046 0.972466i \(-0.425131\pi\)
0.233046 + 0.972466i \(0.425131\pi\)
\(264\) 318432. 0.281194
\(265\) 0 0
\(266\) −1.78675e6 −1.54832
\(267\) 739925. 0.635199
\(268\) 1.50068e6 1.27630
\(269\) 1.48280e6 1.24940 0.624699 0.780866i \(-0.285222\pi\)
0.624699 + 0.780866i \(0.285222\pi\)
\(270\) 0 0
\(271\) −1.16789e6 −0.966002 −0.483001 0.875620i \(-0.660453\pi\)
−0.483001 + 0.875620i \(0.660453\pi\)
\(272\) 1.67574e6 1.37336
\(273\) −557822. −0.452991
\(274\) −3.59410e6 −2.89210
\(275\) 0 0
\(276\) −717207. −0.566724
\(277\) 793314. 0.621220 0.310610 0.950537i \(-0.399467\pi\)
0.310610 + 0.950537i \(0.399467\pi\)
\(278\) −1.40239e6 −1.08832
\(279\) −552470. −0.424911
\(280\) 0 0
\(281\) −1.79487e6 −1.35602 −0.678010 0.735053i \(-0.737157\pi\)
−0.678010 + 0.735053i \(0.737157\pi\)
\(282\) −119047. −0.0891449
\(283\) 1.36160e6 1.01061 0.505306 0.862940i \(-0.331379\pi\)
0.505306 + 0.862940i \(0.331379\pi\)
\(284\) −1.13422e6 −0.834450
\(285\) 0 0
\(286\) 916165. 0.662306
\(287\) 663435. 0.475438
\(288\) −90952.5 −0.0646150
\(289\) 2.47899e6 1.74594
\(290\) 0 0
\(291\) 181227. 0.125455
\(292\) 2.01670e6 1.38415
\(293\) 353801. 0.240763 0.120382 0.992728i \(-0.461588\pi\)
0.120382 + 0.992728i \(0.461588\pi\)
\(294\) 916792. 0.618589
\(295\) 0 0
\(296\) −4.30408e6 −2.85529
\(297\) 88209.0 0.0580259
\(298\) 612132. 0.399305
\(299\) −1.00082e6 −0.647406
\(300\) 0 0
\(301\) −1.06270e6 −0.676076
\(302\) 3.88650e6 2.45212
\(303\) 355103. 0.222202
\(304\) 1.96777e6 1.22121
\(305\) 0 0
\(306\) 1.55179e6 0.947394
\(307\) 2.53723e6 1.53644 0.768218 0.640189i \(-0.221144\pi\)
0.768218 + 0.640189i \(0.221144\pi\)
\(308\) 597153. 0.358682
\(309\) −902732. −0.537852
\(310\) 0 0
\(311\) 1.13460e6 0.665183 0.332592 0.943071i \(-0.392077\pi\)
0.332592 + 0.943071i \(0.392077\pi\)
\(312\) 2.05370e6 1.19440
\(313\) −1.91840e6 −1.10682 −0.553412 0.832908i \(-0.686675\pi\)
−0.553412 + 0.832908i \(0.686675\pi\)
\(314\) −1.15749e6 −0.662511
\(315\) 0 0
\(316\) −1.77020e6 −0.997249
\(317\) −1.47779e6 −0.825970 −0.412985 0.910738i \(-0.635514\pi\)
−0.412985 + 0.910738i \(0.635514\pi\)
\(318\) 3.06508e6 1.69971
\(319\) 154721. 0.0851280
\(320\) 0 0
\(321\) −1.54833e6 −0.838688
\(322\) −988271. −0.531173
\(323\) 4.57830e6 2.44173
\(324\) 407684. 0.215755
\(325\) 0 0
\(326\) −4.81952e6 −2.51165
\(327\) −1.96145e6 −1.01440
\(328\) −2.44253e6 −1.25359
\(329\) −108278. −0.0551509
\(330\) 0 0
\(331\) −1.74625e6 −0.876066 −0.438033 0.898959i \(-0.644325\pi\)
−0.438033 + 0.898959i \(0.644325\pi\)
\(332\) −2.75551e6 −1.37201
\(333\) −1.19228e6 −0.589204
\(334\) −3.90637e6 −1.91605
\(335\) 0 0
\(336\) 606635. 0.293143
\(337\) 3.76831e6 1.80747 0.903736 0.428090i \(-0.140813\pi\)
0.903736 + 0.428090i \(0.140813\pi\)
\(338\) 2.30629e6 1.09805
\(339\) 442526. 0.209141
\(340\) 0 0
\(341\) 825295. 0.384347
\(342\) 1.82223e6 0.842437
\(343\) 2.16872e6 0.995335
\(344\) 3.91249e6 1.78261
\(345\) 0 0
\(346\) 6.23068e6 2.79799
\(347\) −1.08132e6 −0.482094 −0.241047 0.970513i \(-0.577491\pi\)
−0.241047 + 0.970513i \(0.577491\pi\)
\(348\) 715089. 0.316528
\(349\) 2.13960e6 0.940304 0.470152 0.882586i \(-0.344199\pi\)
0.470152 + 0.882586i \(0.344199\pi\)
\(350\) 0 0
\(351\) 568898. 0.246471
\(352\) 135867. 0.0584465
\(353\) −1.07749e6 −0.460234 −0.230117 0.973163i \(-0.573911\pi\)
−0.230117 + 0.973163i \(0.573911\pi\)
\(354\) 3.01204e6 1.27748
\(355\) 0 0
\(356\) −5.10857e6 −2.13636
\(357\) 1.41142e6 0.586120
\(358\) −6.09620e6 −2.51392
\(359\) −3.88382e6 −1.59046 −0.795230 0.606308i \(-0.792650\pi\)
−0.795230 + 0.606308i \(0.792650\pi\)
\(360\) 0 0
\(361\) 2.90007e6 1.17122
\(362\) 2.20594e6 0.884755
\(363\) −131769. −0.0524864
\(364\) 3.85130e6 1.52354
\(365\) 0 0
\(366\) 3.93796e6 1.53663
\(367\) 1.46017e6 0.565899 0.282949 0.959135i \(-0.408687\pi\)
0.282949 + 0.959135i \(0.408687\pi\)
\(368\) 1.08840e6 0.418955
\(369\) −676608. −0.258685
\(370\) 0 0
\(371\) 2.78782e6 1.05155
\(372\) 3.81435e6 1.42910
\(373\) 4.78602e6 1.78116 0.890579 0.454829i \(-0.150300\pi\)
0.890579 + 0.454829i \(0.150300\pi\)
\(374\) −2.31811e6 −0.856950
\(375\) 0 0
\(376\) 398643. 0.145417
\(377\) 997862. 0.361591
\(378\) 561766. 0.202221
\(379\) −1.63705e6 −0.585414 −0.292707 0.956202i \(-0.594556\pi\)
−0.292707 + 0.956202i \(0.594556\pi\)
\(380\) 0 0
\(381\) 497000. 0.175406
\(382\) 2.25343e6 0.790107
\(383\) −4.09383e6 −1.42604 −0.713022 0.701142i \(-0.752674\pi\)
−0.713022 + 0.701142i \(0.752674\pi\)
\(384\) 2.99939e6 1.03802
\(385\) 0 0
\(386\) 1.93470e6 0.660916
\(387\) 1.08380e6 0.367852
\(388\) −1.25122e6 −0.421943
\(389\) −181312. −0.0607509 −0.0303755 0.999539i \(-0.509670\pi\)
−0.0303755 + 0.999539i \(0.509670\pi\)
\(390\) 0 0
\(391\) 2.53230e6 0.837672
\(392\) −3.06998e6 −1.00907
\(393\) −504305. −0.164707
\(394\) −643464. −0.208826
\(395\) 0 0
\(396\) −609010. −0.195158
\(397\) 907.679 0.000289039 0 0.000144519 1.00000i \(-0.499954\pi\)
0.000144519 1.00000i \(0.499954\pi\)
\(398\) −4.58449e6 −1.45072
\(399\) 1.65739e6 0.521187
\(400\) 0 0
\(401\) −4.23119e6 −1.31402 −0.657009 0.753883i \(-0.728179\pi\)
−0.657009 + 0.753883i \(0.728179\pi\)
\(402\) −2.10891e6 −0.650868
\(403\) 5.32268e6 1.63256
\(404\) −2.45169e6 −0.747329
\(405\) 0 0
\(406\) 985353. 0.296672
\(407\) 1.78105e6 0.532955
\(408\) −5.19636e6 −1.54543
\(409\) −2.62102e6 −0.774750 −0.387375 0.921922i \(-0.626618\pi\)
−0.387375 + 0.921922i \(0.626618\pi\)
\(410\) 0 0
\(411\) 3.33389e6 0.973523
\(412\) 6.23261e6 1.80895
\(413\) 2.73958e6 0.790330
\(414\) 1.00789e6 0.289010
\(415\) 0 0
\(416\) 876267. 0.248258
\(417\) 1.30086e6 0.366345
\(418\) −2.72209e6 −0.762013
\(419\) 2.02046e6 0.562233 0.281116 0.959674i \(-0.409295\pi\)
0.281116 + 0.959674i \(0.409295\pi\)
\(420\) 0 0
\(421\) 5.46228e6 1.50200 0.750998 0.660304i \(-0.229573\pi\)
0.750998 + 0.660304i \(0.229573\pi\)
\(422\) −2.79396e6 −0.763729
\(423\) 110428. 0.0300075
\(424\) −1.02638e7 −2.77263
\(425\) 0 0
\(426\) 1.59392e6 0.425542
\(427\) 3.58174e6 0.950659
\(428\) 1.06899e7 2.82075
\(429\) −849835. −0.222942
\(430\) 0 0
\(431\) −640161. −0.165995 −0.0829977 0.996550i \(-0.526449\pi\)
−0.0829977 + 0.996550i \(0.526449\pi\)
\(432\) −618680. −0.159499
\(433\) 5.52690e6 1.41665 0.708324 0.705888i \(-0.249452\pi\)
0.708324 + 0.705888i \(0.249452\pi\)
\(434\) 5.25596e6 1.33945
\(435\) 0 0
\(436\) 1.35422e7 3.41172
\(437\) 2.97361e6 0.744870
\(438\) −2.83407e6 −0.705871
\(439\) 3.50611e6 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(440\) 0 0
\(441\) −850417. −0.208226
\(442\) −1.49505e7 −3.63999
\(443\) 3.37380e6 0.816790 0.408395 0.912805i \(-0.366089\pi\)
0.408395 + 0.912805i \(0.366089\pi\)
\(444\) 8.23167e6 1.98167
\(445\) 0 0
\(446\) −1.01931e7 −2.42644
\(447\) −567814. −0.134412
\(448\) 3.02221e6 0.711425
\(449\) −1.55483e6 −0.363970 −0.181985 0.983301i \(-0.558252\pi\)
−0.181985 + 0.983301i \(0.558252\pi\)
\(450\) 0 0
\(451\) 1.01073e6 0.233989
\(452\) −3.05527e6 −0.703402
\(453\) −3.60512e6 −0.825419
\(454\) 2.95659e6 0.673212
\(455\) 0 0
\(456\) −6.10193e6 −1.37422
\(457\) −6.22946e6 −1.39527 −0.697637 0.716451i \(-0.745765\pi\)
−0.697637 + 0.716451i \(0.745765\pi\)
\(458\) −5.94242e6 −1.32373
\(459\) −1.43945e6 −0.318907
\(460\) 0 0
\(461\) 458075. 0.100389 0.0501943 0.998739i \(-0.484016\pi\)
0.0501943 + 0.998739i \(0.484016\pi\)
\(462\) −839181. −0.182916
\(463\) −5.20799e6 −1.12906 −0.564531 0.825412i \(-0.690943\pi\)
−0.564531 + 0.825412i \(0.690943\pi\)
\(464\) −1.08518e6 −0.233996
\(465\) 0 0
\(466\) 1.46819e7 3.13197
\(467\) −1.67532e6 −0.355473 −0.177736 0.984078i \(-0.556877\pi\)
−0.177736 + 0.984078i \(0.556877\pi\)
\(468\) −3.92777e6 −0.828955
\(469\) −1.91814e6 −0.402670
\(470\) 0 0
\(471\) 1.07369e6 0.223011
\(472\) −1.00862e7 −2.08387
\(473\) −1.61901e6 −0.332734
\(474\) 2.48766e6 0.508564
\(475\) 0 0
\(476\) −9.74470e6 −1.97129
\(477\) −2.84317e6 −0.572147
\(478\) −7.27664e6 −1.45667
\(479\) 2.61462e6 0.520679 0.260340 0.965517i \(-0.416166\pi\)
0.260340 + 0.965517i \(0.416166\pi\)
\(480\) 0 0
\(481\) 1.14868e7 2.26379
\(482\) 2.43660e6 0.477713
\(483\) 916721. 0.178801
\(484\) 909755. 0.176527
\(485\) 0 0
\(486\) −572920. −0.110028
\(487\) 6.20282e6 1.18513 0.592566 0.805522i \(-0.298115\pi\)
0.592566 + 0.805522i \(0.298115\pi\)
\(488\) −1.31867e7 −2.50661
\(489\) 4.47059e6 0.845459
\(490\) 0 0
\(491\) −6.40953e6 −1.19984 −0.599919 0.800061i \(-0.704801\pi\)
−0.599919 + 0.800061i \(0.704801\pi\)
\(492\) 4.67141e6 0.870032
\(493\) −2.52483e6 −0.467858
\(494\) −1.75560e7 −3.23673
\(495\) 0 0
\(496\) −5.78845e6 −1.05647
\(497\) 1.44974e6 0.263268
\(498\) 3.87233e6 0.699679
\(499\) −8.63006e6 −1.55154 −0.775769 0.631017i \(-0.782638\pi\)
−0.775769 + 0.631017i \(0.782638\pi\)
\(500\) 0 0
\(501\) 3.62355e6 0.644971
\(502\) 8.75158e6 1.54998
\(503\) −3.86443e6 −0.681028 −0.340514 0.940239i \(-0.610601\pi\)
−0.340514 + 0.940239i \(0.610601\pi\)
\(504\) −1.88114e6 −0.329871
\(505\) 0 0
\(506\) −1.50562e6 −0.261420
\(507\) −2.13932e6 −0.369620
\(508\) −3.43137e6 −0.589941
\(509\) 3.14562e6 0.538160 0.269080 0.963118i \(-0.413280\pi\)
0.269080 + 0.963118i \(0.413280\pi\)
\(510\) 0 0
\(511\) −2.57771e6 −0.436698
\(512\) −8.89398e6 −1.49941
\(513\) −1.69030e6 −0.283577
\(514\) 1.05308e7 1.75814
\(515\) 0 0
\(516\) −7.48275e6 −1.23719
\(517\) −164961. −0.0271428
\(518\) 1.13428e7 1.85736
\(519\) −5.77959e6 −0.941843
\(520\) 0 0
\(521\) −870807. −0.140549 −0.0702745 0.997528i \(-0.522388\pi\)
−0.0702745 + 0.997528i \(0.522388\pi\)
\(522\) −1.00492e6 −0.161419
\(523\) 3.73764e6 0.597508 0.298754 0.954330i \(-0.403429\pi\)
0.298754 + 0.954330i \(0.403429\pi\)
\(524\) 3.48181e6 0.553958
\(525\) 0 0
\(526\) 5.07274e6 0.799426
\(527\) −1.34676e7 −2.11235
\(528\) 924201. 0.144272
\(529\) −4.79161e6 −0.744461
\(530\) 0 0
\(531\) −2.79397e6 −0.430017
\(532\) −1.14429e7 −1.75290
\(533\) 6.51867e6 0.993895
\(534\) 7.17909e6 1.08947
\(535\) 0 0
\(536\) 7.06192e6 1.06172
\(537\) 5.65484e6 0.846222
\(538\) 1.43868e7 2.14293
\(539\) 1.27038e6 0.188348
\(540\) 0 0
\(541\) −3.18691e6 −0.468142 −0.234071 0.972220i \(-0.575205\pi\)
−0.234071 + 0.972220i \(0.575205\pi\)
\(542\) −1.13314e7 −1.65685
\(543\) −2.04623e6 −0.297821
\(544\) −2.21716e6 −0.321218
\(545\) 0 0
\(546\) −5.41224e6 −0.776955
\(547\) −3.82413e6 −0.546467 −0.273234 0.961948i \(-0.588093\pi\)
−0.273234 + 0.961948i \(0.588093\pi\)
\(548\) −2.30177e7 −3.27424
\(549\) −3.65286e6 −0.517252
\(550\) 0 0
\(551\) −2.96483e6 −0.416027
\(552\) −3.37504e6 −0.471446
\(553\) 2.26263e6 0.314631
\(554\) 7.69709e6 1.06550
\(555\) 0 0
\(556\) −8.98134e6 −1.23212
\(557\) −7.46292e6 −1.01923 −0.509613 0.860404i \(-0.670211\pi\)
−0.509613 + 0.860404i \(0.670211\pi\)
\(558\) −5.36031e6 −0.728794
\(559\) −1.04417e7 −1.41333
\(560\) 0 0
\(561\) 2.15028e6 0.288462
\(562\) −1.74146e7 −2.32580
\(563\) 5.50547e6 0.732021 0.366011 0.930611i \(-0.380723\pi\)
0.366011 + 0.930611i \(0.380723\pi\)
\(564\) −762416. −0.100924
\(565\) 0 0
\(566\) 1.32109e7 1.73337
\(567\) −521095. −0.0680705
\(568\) −5.33742e6 −0.694161
\(569\) −1.28692e6 −0.166637 −0.0833183 0.996523i \(-0.526552\pi\)
−0.0833183 + 0.996523i \(0.526552\pi\)
\(570\) 0 0
\(571\) −1.08430e7 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(572\) 5.86740e6 0.749818
\(573\) −2.09028e6 −0.265961
\(574\) 6.43695e6 0.815455
\(575\) 0 0
\(576\) −3.08221e6 −0.387085
\(577\) −2.33126e6 −0.291508 −0.145754 0.989321i \(-0.546561\pi\)
−0.145754 + 0.989321i \(0.546561\pi\)
\(578\) 2.40522e7 2.99458
\(579\) −1.79463e6 −0.222474
\(580\) 0 0
\(581\) 3.52205e6 0.432867
\(582\) 1.75834e6 0.215177
\(583\) 4.24721e6 0.517526
\(584\) 9.49020e6 1.15144
\(585\) 0 0
\(586\) 3.43274e6 0.412949
\(587\) 2.84062e6 0.340266 0.170133 0.985421i \(-0.445580\pi\)
0.170133 + 0.985421i \(0.445580\pi\)
\(588\) 5.87142e6 0.700326
\(589\) −1.58147e7 −1.87833
\(590\) 0 0
\(591\) 596878. 0.0702938
\(592\) −1.24919e7 −1.46496
\(593\) 1.53492e7 1.79246 0.896231 0.443587i \(-0.146294\pi\)
0.896231 + 0.443587i \(0.146294\pi\)
\(594\) 855843. 0.0995241
\(595\) 0 0
\(596\) 3.92029e6 0.452066
\(597\) 4.25258e6 0.488334
\(598\) −9.71038e6 −1.11041
\(599\) 1.21107e7 1.37912 0.689560 0.724228i \(-0.257804\pi\)
0.689560 + 0.724228i \(0.257804\pi\)
\(600\) 0 0
\(601\) 1.27801e7 1.44327 0.721635 0.692273i \(-0.243391\pi\)
0.721635 + 0.692273i \(0.243391\pi\)
\(602\) −1.03108e7 −1.15958
\(603\) 1.95623e6 0.219092
\(604\) 2.48904e7 2.77612
\(605\) 0 0
\(606\) 3.44536e6 0.381113
\(607\) −1.21041e7 −1.33341 −0.666703 0.745323i \(-0.732295\pi\)
−0.666703 + 0.745323i \(0.732295\pi\)
\(608\) −2.60355e6 −0.285632
\(609\) −914014. −0.0998642
\(610\) 0 0
\(611\) −1.06390e6 −0.115292
\(612\) 9.93818e6 1.07258
\(613\) 3.32675e6 0.357577 0.178788 0.983888i \(-0.442782\pi\)
0.178788 + 0.983888i \(0.442782\pi\)
\(614\) 2.46174e7 2.63524
\(615\) 0 0
\(616\) 2.81009e6 0.298379
\(617\) −6.60697e6 −0.698698 −0.349349 0.936993i \(-0.613597\pi\)
−0.349349 + 0.936993i \(0.613597\pi\)
\(618\) −8.75871e6 −0.922506
\(619\) −6.09619e6 −0.639487 −0.319743 0.947504i \(-0.603597\pi\)
−0.319743 + 0.947504i \(0.603597\pi\)
\(620\) 0 0
\(621\) −934922. −0.0972852
\(622\) 1.10084e7 1.14090
\(623\) 6.52968e6 0.674018
\(624\) 5.96057e6 0.612811
\(625\) 0 0
\(626\) −1.86132e7 −1.89839
\(627\) 2.52502e6 0.256505
\(628\) −7.41293e6 −0.750051
\(629\) −2.90643e7 −2.92909
\(630\) 0 0
\(631\) −1.61232e7 −1.61205 −0.806026 0.591881i \(-0.798386\pi\)
−0.806026 + 0.591881i \(0.798386\pi\)
\(632\) −8.33021e6 −0.829590
\(633\) 2.59168e6 0.257082
\(634\) −1.43382e7 −1.41668
\(635\) 0 0
\(636\) 1.96297e7 1.92430
\(637\) 8.19320e6 0.800028
\(638\) 1.50117e6 0.146009
\(639\) −1.47852e6 −0.143244
\(640\) 0 0
\(641\) 8.95786e6 0.861111 0.430555 0.902564i \(-0.358318\pi\)
0.430555 + 0.902564i \(0.358318\pi\)
\(642\) −1.50226e7 −1.43849
\(643\) 1.99681e7 1.90462 0.952310 0.305131i \(-0.0987003\pi\)
0.952310 + 0.305131i \(0.0987003\pi\)
\(644\) −6.32920e6 −0.601359
\(645\) 0 0
\(646\) 4.44207e7 4.18798
\(647\) 7.36055e6 0.691273 0.345636 0.938369i \(-0.387663\pi\)
0.345636 + 0.938369i \(0.387663\pi\)
\(648\) 1.91849e6 0.179482
\(649\) 4.17371e6 0.388965
\(650\) 0 0
\(651\) −4.87543e6 −0.450880
\(652\) −3.08657e7 −2.84353
\(653\) 1.03621e7 0.950961 0.475481 0.879726i \(-0.342274\pi\)
0.475481 + 0.879726i \(0.342274\pi\)
\(654\) −1.90309e7 −1.73986
\(655\) 0 0
\(656\) −7.08909e6 −0.643178
\(657\) 2.62889e6 0.237606
\(658\) −1.05057e6 −0.0945930
\(659\) −1.14611e7 −1.02805 −0.514023 0.857777i \(-0.671845\pi\)
−0.514023 + 0.857777i \(0.671845\pi\)
\(660\) 0 0
\(661\) −1.23869e7 −1.10271 −0.551353 0.834272i \(-0.685888\pi\)
−0.551353 + 0.834272i \(0.685888\pi\)
\(662\) −1.69429e7 −1.50260
\(663\) 1.38681e7 1.22527
\(664\) −1.29669e7 −1.14134
\(665\) 0 0
\(666\) −1.15680e7 −1.01058
\(667\) −1.63988e6 −0.142724
\(668\) −2.50176e7 −2.16923
\(669\) 9.45512e6 0.816774
\(670\) 0 0
\(671\) 5.45674e6 0.467872
\(672\) −802637. −0.0685639
\(673\) 1.92657e7 1.63964 0.819818 0.572624i \(-0.194075\pi\)
0.819818 + 0.572624i \(0.194075\pi\)
\(674\) 3.65618e7 3.10012
\(675\) 0 0
\(676\) 1.47702e7 1.24314
\(677\) −1.13886e6 −0.0954988 −0.0477494 0.998859i \(-0.515205\pi\)
−0.0477494 + 0.998859i \(0.515205\pi\)
\(678\) 4.29358e6 0.358712
\(679\) 1.59929e6 0.133123
\(680\) 0 0
\(681\) −2.74254e6 −0.226613
\(682\) 8.00738e6 0.659219
\(683\) 1.17487e7 0.963691 0.481845 0.876256i \(-0.339967\pi\)
0.481845 + 0.876256i \(0.339967\pi\)
\(684\) 1.16701e7 0.953751
\(685\) 0 0
\(686\) 2.10419e7 1.70716
\(687\) 5.51220e6 0.445587
\(688\) 1.13554e7 0.914603
\(689\) 2.73921e7 2.19825
\(690\) 0 0
\(691\) 1.25506e7 0.999932 0.499966 0.866045i \(-0.333346\pi\)
0.499966 + 0.866045i \(0.333346\pi\)
\(692\) 3.99032e7 3.16769
\(693\) 778425. 0.0615721
\(694\) −1.04915e7 −0.826872
\(695\) 0 0
\(696\) 3.36508e6 0.263313
\(697\) −1.64938e7 −1.28599
\(698\) 2.07593e7 1.61278
\(699\) −1.36189e7 −1.05427
\(700\) 0 0
\(701\) −2.34000e7 −1.79854 −0.899272 0.437391i \(-0.855903\pi\)
−0.899272 + 0.437391i \(0.855903\pi\)
\(702\) 5.51970e6 0.422740
\(703\) −3.41293e7 −2.60459
\(704\) 4.60429e6 0.350132
\(705\) 0 0
\(706\) −1.04543e7 −0.789377
\(707\) 3.13370e6 0.235781
\(708\) 1.92900e7 1.44627
\(709\) 1.66095e7 1.24091 0.620455 0.784242i \(-0.286948\pi\)
0.620455 + 0.784242i \(0.286948\pi\)
\(710\) 0 0
\(711\) −2.30756e6 −0.171190
\(712\) −2.40400e7 −1.77719
\(713\) −8.74726e6 −0.644389
\(714\) 1.36943e7 1.00529
\(715\) 0 0
\(716\) −3.90420e7 −2.84609
\(717\) 6.74981e6 0.490336
\(718\) −3.76825e7 −2.72790
\(719\) 1.98028e7 1.42858 0.714291 0.699848i \(-0.246749\pi\)
0.714291 + 0.699848i \(0.246749\pi\)
\(720\) 0 0
\(721\) −7.96642e6 −0.570722
\(722\) 2.81377e7 2.00884
\(723\) −2.26020e6 −0.160805
\(724\) 1.41275e7 1.00166
\(725\) 0 0
\(726\) −1.27848e6 −0.0900229
\(727\) −2.73066e7 −1.91616 −0.958079 0.286504i \(-0.907507\pi\)
−0.958079 + 0.286504i \(0.907507\pi\)
\(728\) 1.81235e7 1.26740
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 2.64200e7 1.82869
\(732\) 2.52199e7 1.73967
\(733\) −1.08525e7 −0.746052 −0.373026 0.927821i \(-0.621680\pi\)
−0.373026 + 0.927821i \(0.621680\pi\)
\(734\) 1.41672e7 0.970610
\(735\) 0 0
\(736\) −1.44005e6 −0.0979903
\(737\) −2.92227e6 −0.198176
\(738\) −6.56475e6 −0.443688
\(739\) 687400. 0.0463019 0.0231509 0.999732i \(-0.492630\pi\)
0.0231509 + 0.999732i \(0.492630\pi\)
\(740\) 0 0
\(741\) 1.62849e7 1.08953
\(742\) 2.70487e7 1.80359
\(743\) −4.30995e6 −0.286418 −0.143209 0.989692i \(-0.545742\pi\)
−0.143209 + 0.989692i \(0.545742\pi\)
\(744\) 1.79496e7 1.18884
\(745\) 0 0
\(746\) 4.64361e7 3.05498
\(747\) −3.59197e6 −0.235522
\(748\) −1.48459e7 −0.970182
\(749\) −1.36637e7 −0.889944
\(750\) 0 0
\(751\) −9.07828e6 −0.587359 −0.293680 0.955904i \(-0.594880\pi\)
−0.293680 + 0.955904i \(0.594880\pi\)
\(752\) 1.15700e6 0.0746087
\(753\) −8.11797e6 −0.521747
\(754\) 9.68171e6 0.620188
\(755\) 0 0
\(756\) 3.59773e6 0.228941
\(757\) 1.01884e7 0.646197 0.323098 0.946365i \(-0.395276\pi\)
0.323098 + 0.946365i \(0.395276\pi\)
\(758\) −1.58834e7 −1.00408
\(759\) 1.39661e6 0.0879977
\(760\) 0 0
\(761\) 1.93077e7 1.20856 0.604282 0.796771i \(-0.293460\pi\)
0.604282 + 0.796771i \(0.293460\pi\)
\(762\) 4.82212e6 0.300850
\(763\) −1.73094e7 −1.07639
\(764\) 1.44317e7 0.894506
\(765\) 0 0
\(766\) −3.97202e7 −2.44590
\(767\) 2.69181e7 1.65217
\(768\) 1.81425e7 1.10992
\(769\) −4.54597e6 −0.277211 −0.138605 0.990348i \(-0.544262\pi\)
−0.138605 + 0.990348i \(0.544262\pi\)
\(770\) 0 0
\(771\) −9.76838e6 −0.591816
\(772\) 1.23904e7 0.748245
\(773\) 2.23550e7 1.34563 0.672814 0.739811i \(-0.265085\pi\)
0.672814 + 0.739811i \(0.265085\pi\)
\(774\) 1.05155e7 0.630927
\(775\) 0 0
\(776\) −5.88800e6 −0.351005
\(777\) −1.05216e7 −0.625213
\(778\) −1.75917e6 −0.104198
\(779\) −1.93682e7 −1.14352
\(780\) 0 0
\(781\) 2.20866e6 0.129569
\(782\) 2.45696e7 1.43675
\(783\) 932162. 0.0543359
\(784\) −8.91016e6 −0.517721
\(785\) 0 0
\(786\) −4.89299e6 −0.282500
\(787\) −7.13003e6 −0.410350 −0.205175 0.978725i \(-0.565776\pi\)
−0.205175 + 0.978725i \(0.565776\pi\)
\(788\) −4.12094e6 −0.236418
\(789\) −4.70548e6 −0.269099
\(790\) 0 0
\(791\) 3.90519e6 0.221922
\(792\) −2.86589e6 −0.162348
\(793\) 3.51929e7 1.98734
\(794\) 8806.71 0.000495750 0
\(795\) 0 0
\(796\) −2.93605e7 −1.64241
\(797\) 2.47446e7 1.37986 0.689931 0.723875i \(-0.257641\pi\)
0.689931 + 0.723875i \(0.257641\pi\)
\(798\) 1.60808e7 0.893922
\(799\) 2.69193e6 0.149175
\(800\) 0 0
\(801\) −6.65933e6 −0.366732
\(802\) −4.10529e7 −2.25376
\(803\) −3.92710e6 −0.214923
\(804\) −1.35061e7 −0.736869
\(805\) 0 0
\(806\) 5.16431e7 2.80011
\(807\) −1.33452e7 −0.721340
\(808\) −1.15372e7 −0.621687
\(809\) −7.53550e6 −0.404800 −0.202400 0.979303i \(-0.564874\pi\)
−0.202400 + 0.979303i \(0.564874\pi\)
\(810\) 0 0
\(811\) 1.62999e7 0.870229 0.435114 0.900375i \(-0.356708\pi\)
0.435114 + 0.900375i \(0.356708\pi\)
\(812\) 6.31051e6 0.335872
\(813\) 1.05110e7 0.557721
\(814\) 1.72806e7 0.914107
\(815\) 0 0
\(816\) −1.50817e7 −0.792909
\(817\) 3.10242e7 1.62610
\(818\) −2.54303e7 −1.32882
\(819\) 5.02040e6 0.261534
\(820\) 0 0
\(821\) −2.27539e7 −1.17814 −0.589071 0.808081i \(-0.700506\pi\)
−0.589071 + 0.808081i \(0.700506\pi\)
\(822\) 3.23469e7 1.66975
\(823\) −3.55290e7 −1.82845 −0.914226 0.405204i \(-0.867200\pi\)
−0.914226 + 0.405204i \(0.867200\pi\)
\(824\) 2.93295e7 1.50483
\(825\) 0 0
\(826\) 2.65806e7 1.35555
\(827\) −4.75143e6 −0.241580 −0.120790 0.992678i \(-0.538543\pi\)
−0.120790 + 0.992678i \(0.538543\pi\)
\(828\) 6.45486e6 0.327198
\(829\) −3.32259e6 −0.167915 −0.0839577 0.996469i \(-0.526756\pi\)
−0.0839577 + 0.996469i \(0.526756\pi\)
\(830\) 0 0
\(831\) −7.13983e6 −0.358662
\(832\) 2.96951e7 1.48722
\(833\) −2.07307e7 −1.03515
\(834\) 1.26215e7 0.628342
\(835\) 0 0
\(836\) −1.74331e7 −0.862700
\(837\) 4.97223e6 0.245323
\(838\) 1.96034e7 0.964323
\(839\) −2.00974e7 −0.985679 −0.492840 0.870120i \(-0.664041\pi\)
−0.492840 + 0.870120i \(0.664041\pi\)
\(840\) 0 0
\(841\) −1.88761e7 −0.920285
\(842\) 5.29975e7 2.57617
\(843\) 1.61538e7 0.782899
\(844\) −1.78934e7 −0.864643
\(845\) 0 0
\(846\) 1.07143e6 0.0514678
\(847\) −1.16283e6 −0.0556941
\(848\) −2.97891e7 −1.42255
\(849\) −1.22544e7 −0.583478
\(850\) 0 0
\(851\) −1.88773e7 −0.893543
\(852\) 1.02080e7 0.481770
\(853\) 1.62177e7 0.763161 0.381581 0.924336i \(-0.375380\pi\)
0.381581 + 0.924336i \(0.375380\pi\)
\(854\) 3.47517e7 1.63054
\(855\) 0 0
\(856\) 5.03048e7 2.34652
\(857\) −2.68814e7 −1.25026 −0.625129 0.780521i \(-0.714954\pi\)
−0.625129 + 0.780521i \(0.714954\pi\)
\(858\) −8.24548e6 −0.382382
\(859\) 3.23807e7 1.49728 0.748640 0.662977i \(-0.230707\pi\)
0.748640 + 0.662977i \(0.230707\pi\)
\(860\) 0 0
\(861\) −5.97092e6 −0.274494
\(862\) −6.21113e6 −0.284710
\(863\) −6.65793e6 −0.304307 −0.152154 0.988357i \(-0.548621\pi\)
−0.152154 + 0.988357i \(0.548621\pi\)
\(864\) 818573. 0.0373055
\(865\) 0 0
\(866\) 5.36245e7 2.42979
\(867\) −2.23109e7 −1.00802
\(868\) 3.36608e7 1.51644
\(869\) 3.44709e6 0.154847
\(870\) 0 0
\(871\) −1.88469e7 −0.841774
\(872\) 6.37271e7 2.83813
\(873\) −1.63104e6 −0.0724317
\(874\) 2.88513e7 1.27758
\(875\) 0 0
\(876\) −1.81503e7 −0.799140
\(877\) −1.18215e6 −0.0519006 −0.0259503 0.999663i \(-0.508261\pi\)
−0.0259503 + 0.999663i \(0.508261\pi\)
\(878\) 3.40179e7 1.48926
\(879\) −3.18421e6 −0.139005
\(880\) 0 0
\(881\) 3.03739e7 1.31844 0.659222 0.751949i \(-0.270886\pi\)
0.659222 + 0.751949i \(0.270886\pi\)
\(882\) −8.25113e6 −0.357143
\(883\) 2.06314e7 0.890485 0.445243 0.895410i \(-0.353117\pi\)
0.445243 + 0.895410i \(0.353117\pi\)
\(884\) −9.57477e7 −4.12095
\(885\) 0 0
\(886\) 3.27342e7 1.40093
\(887\) 3.19552e7 1.36374 0.681871 0.731473i \(-0.261167\pi\)
0.681871 + 0.731473i \(0.261167\pi\)
\(888\) 3.87367e7 1.64851
\(889\) 4.38592e6 0.186126
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −6.52797e7 −2.74705
\(893\) 3.16105e6 0.132649
\(894\) −5.50919e6 −0.230539
\(895\) 0 0
\(896\) 2.64690e7 1.10146
\(897\) 9.00736e6 0.373780
\(898\) −1.50856e7 −0.624270
\(899\) 8.72143e6 0.359905
\(900\) 0 0
\(901\) −6.93085e7 −2.84429
\(902\) 9.80660e6 0.401331
\(903\) 9.56432e6 0.390333
\(904\) −1.43775e7 −0.585145
\(905\) 0 0
\(906\) −3.49785e7 −1.41573
\(907\) 3.27668e7 1.32256 0.661280 0.750139i \(-0.270014\pi\)
0.661280 + 0.750139i \(0.270014\pi\)
\(908\) 1.89349e7 0.762165
\(909\) −3.19592e6 −0.128288
\(910\) 0 0
\(911\) 1.89497e7 0.756495 0.378248 0.925704i \(-0.376527\pi\)
0.378248 + 0.925704i \(0.376527\pi\)
\(912\) −1.77100e7 −0.705067
\(913\) 5.36579e6 0.213038
\(914\) −6.04410e7 −2.39313
\(915\) 0 0
\(916\) −3.80571e7 −1.49864
\(917\) −4.45038e6 −0.174773
\(918\) −1.39662e7 −0.546978
\(919\) −5.37819e6 −0.210062 −0.105031 0.994469i \(-0.533494\pi\)
−0.105031 + 0.994469i \(0.533494\pi\)
\(920\) 0 0
\(921\) −2.28351e7 −0.887061
\(922\) 4.44445e6 0.172183
\(923\) 1.42446e7 0.550358
\(924\) −5.37438e6 −0.207085
\(925\) 0 0
\(926\) −5.05303e7 −1.93653
\(927\) 8.12459e6 0.310529
\(928\) 1.43580e6 0.0547298
\(929\) −1.66380e7 −0.632503 −0.316251 0.948675i \(-0.602424\pi\)
−0.316251 + 0.948675i \(0.602424\pi\)
\(930\) 0 0
\(931\) −2.43435e7 −0.920469
\(932\) 9.40273e7 3.54580
\(933\) −1.02114e7 −0.384044
\(934\) −1.62547e7 −0.609695
\(935\) 0 0
\(936\) −1.84833e7 −0.689590
\(937\) −3.18707e7 −1.18588 −0.592942 0.805245i \(-0.702034\pi\)
−0.592942 + 0.805245i \(0.702034\pi\)
\(938\) −1.86107e7 −0.690645
\(939\) 1.72656e7 0.639025
\(940\) 0 0
\(941\) −2.37797e7 −0.875452 −0.437726 0.899108i \(-0.644216\pi\)
−0.437726 + 0.899108i \(0.644216\pi\)
\(942\) 1.04174e7 0.382501
\(943\) −1.07127e7 −0.392302
\(944\) −2.92736e7 −1.06917
\(945\) 0 0
\(946\) −1.57084e7 −0.570695
\(947\) −5.44978e7 −1.97471 −0.987357 0.158510i \(-0.949331\pi\)
−0.987357 + 0.158510i \(0.949331\pi\)
\(948\) 1.59318e7 0.575762
\(949\) −2.53276e7 −0.912910
\(950\) 0 0
\(951\) 1.33001e7 0.476874
\(952\) −4.58567e7 −1.63988
\(953\) 2.08795e7 0.744710 0.372355 0.928090i \(-0.378550\pi\)
0.372355 + 0.928090i \(0.378550\pi\)
\(954\) −2.75857e7 −0.981327
\(955\) 0 0
\(956\) −4.66019e7 −1.64914
\(957\) −1.39249e6 −0.0491487
\(958\) 2.53682e7 0.893051
\(959\) 2.94208e7 1.03302
\(960\) 0 0
\(961\) 1.78917e7 0.624947
\(962\) 1.11450e8 3.88277
\(963\) 1.39350e7 0.484217
\(964\) 1.56048e7 0.540835
\(965\) 0 0
\(966\) 8.89444e6 0.306673
\(967\) −4.72474e7 −1.62484 −0.812422 0.583070i \(-0.801851\pi\)
−0.812422 + 0.583070i \(0.801851\pi\)
\(968\) 4.28114e6 0.146849
\(969\) −4.12047e7 −1.40973
\(970\) 0 0
\(971\) −1.81878e7 −0.619059 −0.309530 0.950890i \(-0.600172\pi\)
−0.309530 + 0.950890i \(0.600172\pi\)
\(972\) −3.66916e6 −0.124566
\(973\) 1.14798e7 0.388734
\(974\) 6.01825e7 2.03270
\(975\) 0 0
\(976\) −3.82725e7 −1.28606
\(977\) −4.40237e7 −1.47554 −0.737769 0.675054i \(-0.764121\pi\)
−0.737769 + 0.675054i \(0.764121\pi\)
\(978\) 4.33757e7 1.45010
\(979\) 9.94788e6 0.331722
\(980\) 0 0
\(981\) 1.76531e7 0.585663
\(982\) −6.21882e7 −2.05792
\(983\) 6.68153e6 0.220542 0.110271 0.993902i \(-0.464828\pi\)
0.110271 + 0.993902i \(0.464828\pi\)
\(984\) 2.19828e7 0.723761
\(985\) 0 0
\(986\) −2.44970e7 −0.802455
\(987\) 974506. 0.0318414
\(988\) −1.12434e8 −3.66441
\(989\) 1.71598e7 0.557856
\(990\) 0 0
\(991\) −3.06716e7 −0.992092 −0.496046 0.868296i \(-0.665215\pi\)
−0.496046 + 0.868296i \(0.665215\pi\)
\(992\) 7.65867e6 0.247101
\(993\) 1.57163e7 0.505797
\(994\) 1.40660e7 0.451549
\(995\) 0 0
\(996\) 2.47996e7 0.792130
\(997\) 2.18670e7 0.696707 0.348354 0.937363i \(-0.386741\pi\)
0.348354 + 0.937363i \(0.386741\pi\)
\(998\) −8.37327e7 −2.66115
\(999\) 1.07305e7 0.340177
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.q.1.8 yes 8
5.4 even 2 825.6.a.p.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.p.1.1 8 5.4 even 2
825.6.a.q.1.8 yes 8 1.1 even 1 trivial