Properties

Label 441.6.a.bb.1.4
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(4.27213\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.09163 q^{2} -22.4418 q^{4} +13.7926 q^{5} -168.314 q^{8} +O(q^{10})\) \(q+3.09163 q^{2} -22.4418 q^{4} +13.7926 q^{5} -168.314 q^{8} +42.6416 q^{10} +3.66659 q^{11} +780.229 q^{13} +197.776 q^{16} +50.0832 q^{17} -1063.38 q^{19} -309.532 q^{20} +11.3357 q^{22} -4102.45 q^{23} -2934.76 q^{25} +2412.18 q^{26} +1487.11 q^{29} -5519.97 q^{31} +5997.49 q^{32} +154.838 q^{34} +6143.27 q^{37} -3287.59 q^{38} -2321.49 q^{40} +10757.9 q^{41} +17696.7 q^{43} -82.2850 q^{44} -12683.3 q^{46} +29468.5 q^{47} -9073.19 q^{50} -17509.8 q^{52} +19255.9 q^{53} +50.5718 q^{55} +4597.58 q^{58} +6619.18 q^{59} -36750.3 q^{61} -17065.7 q^{62} +12213.2 q^{64} +10761.4 q^{65} +46909.2 q^{67} -1123.96 q^{68} +41693.7 q^{71} -29451.1 q^{73} +18992.7 q^{74} +23864.3 q^{76} +22124.4 q^{79} +2727.84 q^{80} +33259.5 q^{82} -3896.35 q^{83} +690.778 q^{85} +54711.6 q^{86} -617.138 q^{88} +20530.8 q^{89} +92066.6 q^{92} +91105.4 q^{94} -14666.8 q^{95} -17742.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} - 864 q^{10} - 604 q^{11} - 1352 q^{13} + 4578 q^{16} + 3028 q^{17} - 1728 q^{19} + 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} + 14172 q^{26} + 5320 q^{29} - 3976 q^{31} + 37326 q^{32} + 16336 q^{34} + 22680 q^{37} + 52744 q^{38} - 100600 q^{40} + 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} + 51552 q^{47} + 40622 q^{50} - 119296 q^{52} - 80884 q^{53} - 11656 q^{55} - 70464 q^{58} + 8872 q^{59} - 50896 q^{61} + 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} + 37348 q^{68} + 110852 q^{71} - 64232 q^{73} + 27464 q^{74} + 194864 q^{76} + 111696 q^{79} - 308940 q^{80} + 189640 q^{82} + 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} - 35012 q^{89} + 449260 q^{92} + 121016 q^{94} + 119080 q^{95} - 70952 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.09163 0.546527 0.273264 0.961939i \(-0.411897\pi\)
0.273264 + 0.961939i \(0.411897\pi\)
\(3\) 0 0
\(4\) −22.4418 −0.701308
\(5\) 13.7926 0.246730 0.123365 0.992361i \(-0.460631\pi\)
0.123365 + 0.992361i \(0.460631\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −168.314 −0.929811
\(9\) 0 0
\(10\) 42.6416 0.134845
\(11\) 3.66659 0.00913651 0.00456826 0.999990i \(-0.498546\pi\)
0.00456826 + 0.999990i \(0.498546\pi\)
\(12\) 0 0
\(13\) 780.229 1.28045 0.640226 0.768186i \(-0.278841\pi\)
0.640226 + 0.768186i \(0.278841\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 197.776 0.193140
\(17\) 50.0832 0.0420310 0.0210155 0.999779i \(-0.493310\pi\)
0.0210155 + 0.999779i \(0.493310\pi\)
\(18\) 0 0
\(19\) −1063.38 −0.675781 −0.337891 0.941185i \(-0.609713\pi\)
−0.337891 + 0.941185i \(0.609713\pi\)
\(20\) −309.532 −0.173033
\(21\) 0 0
\(22\) 11.3357 0.00499336
\(23\) −4102.45 −1.61705 −0.808526 0.588460i \(-0.799734\pi\)
−0.808526 + 0.588460i \(0.799734\pi\)
\(24\) 0 0
\(25\) −2934.76 −0.939124
\(26\) 2412.18 0.699803
\(27\) 0 0
\(28\) 0 0
\(29\) 1487.11 0.328358 0.164179 0.986431i \(-0.447503\pi\)
0.164179 + 0.986431i \(0.447503\pi\)
\(30\) 0 0
\(31\) −5519.97 −1.03165 −0.515825 0.856694i \(-0.672515\pi\)
−0.515825 + 0.856694i \(0.672515\pi\)
\(32\) 5997.49 1.03537
\(33\) 0 0
\(34\) 154.838 0.0229711
\(35\) 0 0
\(36\) 0 0
\(37\) 6143.27 0.737726 0.368863 0.929484i \(-0.379747\pi\)
0.368863 + 0.929484i \(0.379747\pi\)
\(38\) −3287.59 −0.369333
\(39\) 0 0
\(40\) −2321.49 −0.229412
\(41\) 10757.9 0.999468 0.499734 0.866179i \(-0.333431\pi\)
0.499734 + 0.866179i \(0.333431\pi\)
\(42\) 0 0
\(43\) 17696.7 1.45956 0.729779 0.683683i \(-0.239623\pi\)
0.729779 + 0.683683i \(0.239623\pi\)
\(44\) −82.2850 −0.00640751
\(45\) 0 0
\(46\) −12683.3 −0.883763
\(47\) 29468.5 1.94587 0.972933 0.231089i \(-0.0742289\pi\)
0.972933 + 0.231089i \(0.0742289\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −9073.19 −0.513257
\(51\) 0 0
\(52\) −17509.8 −0.897991
\(53\) 19255.9 0.941616 0.470808 0.882236i \(-0.343962\pi\)
0.470808 + 0.882236i \(0.343962\pi\)
\(54\) 0 0
\(55\) 50.5718 0.00225425
\(56\) 0 0
\(57\) 0 0
\(58\) 4597.58 0.179457
\(59\) 6619.18 0.247557 0.123778 0.992310i \(-0.460499\pi\)
0.123778 + 0.992310i \(0.460499\pi\)
\(60\) 0 0
\(61\) −36750.3 −1.26455 −0.632275 0.774744i \(-0.717879\pi\)
−0.632275 + 0.774744i \(0.717879\pi\)
\(62\) −17065.7 −0.563825
\(63\) 0 0
\(64\) 12213.2 0.372717
\(65\) 10761.4 0.315926
\(66\) 0 0
\(67\) 46909.2 1.27665 0.638323 0.769768i \(-0.279628\pi\)
0.638323 + 0.769768i \(0.279628\pi\)
\(68\) −1123.96 −0.0294767
\(69\) 0 0
\(70\) 0 0
\(71\) 41693.7 0.981577 0.490789 0.871279i \(-0.336709\pi\)
0.490789 + 0.871279i \(0.336709\pi\)
\(72\) 0 0
\(73\) −29451.1 −0.646837 −0.323418 0.946256i \(-0.604832\pi\)
−0.323418 + 0.946256i \(0.604832\pi\)
\(74\) 18992.7 0.403188
\(75\) 0 0
\(76\) 23864.3 0.473931
\(77\) 0 0
\(78\) 0 0
\(79\) 22124.4 0.398844 0.199422 0.979914i \(-0.436094\pi\)
0.199422 + 0.979914i \(0.436094\pi\)
\(80\) 2727.84 0.0476535
\(81\) 0 0
\(82\) 33259.5 0.546237
\(83\) −3896.35 −0.0620816 −0.0310408 0.999518i \(-0.509882\pi\)
−0.0310408 + 0.999518i \(0.509882\pi\)
\(84\) 0 0
\(85\) 690.778 0.0103703
\(86\) 54711.6 0.797689
\(87\) 0 0
\(88\) −617.138 −0.00849523
\(89\) 20530.8 0.274746 0.137373 0.990519i \(-0.456134\pi\)
0.137373 + 0.990519i \(0.456134\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 92066.6 1.13405
\(93\) 0 0
\(94\) 91105.4 1.06347
\(95\) −14666.8 −0.166735
\(96\) 0 0
\(97\) −17742.9 −0.191468 −0.0957338 0.995407i \(-0.530520\pi\)
−0.0957338 + 0.995407i \(0.530520\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 65861.5 0.658615
\(101\) −142085. −1.38594 −0.692971 0.720965i \(-0.743699\pi\)
−0.692971 + 0.720965i \(0.743699\pi\)
\(102\) 0 0
\(103\) 210538. 1.95540 0.977702 0.209996i \(-0.0673450\pi\)
0.977702 + 0.209996i \(0.0673450\pi\)
\(104\) −131323. −1.19058
\(105\) 0 0
\(106\) 59532.0 0.514619
\(107\) 202671. 1.71132 0.855660 0.517538i \(-0.173151\pi\)
0.855660 + 0.517538i \(0.173151\pi\)
\(108\) 0 0
\(109\) 139538. 1.12493 0.562466 0.826821i \(-0.309853\pi\)
0.562466 + 0.826821i \(0.309853\pi\)
\(110\) 156.349 0.00123201
\(111\) 0 0
\(112\) 0 0
\(113\) 206316. 1.51998 0.759989 0.649936i \(-0.225204\pi\)
0.759989 + 0.649936i \(0.225204\pi\)
\(114\) 0 0
\(115\) −56583.5 −0.398975
\(116\) −33373.4 −0.230280
\(117\) 0 0
\(118\) 20464.0 0.135296
\(119\) 0 0
\(120\) 0 0
\(121\) −161038. −0.999917
\(122\) −113618. −0.691111
\(123\) 0 0
\(124\) 123878. 0.723504
\(125\) −83580.0 −0.478440
\(126\) 0 0
\(127\) 89874.2 0.494454 0.247227 0.968958i \(-0.420481\pi\)
0.247227 + 0.968958i \(0.420481\pi\)
\(128\) −154161. −0.831668
\(129\) 0 0
\(130\) 33270.2 0.172662
\(131\) 320115. 1.62978 0.814889 0.579617i \(-0.196798\pi\)
0.814889 + 0.579617i \(0.196798\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 145026. 0.697722
\(135\) 0 0
\(136\) −8429.69 −0.0390809
\(137\) −258971. −1.17883 −0.589414 0.807831i \(-0.700641\pi\)
−0.589414 + 0.807831i \(0.700641\pi\)
\(138\) 0 0
\(139\) −282366. −1.23958 −0.619791 0.784767i \(-0.712783\pi\)
−0.619791 + 0.784767i \(0.712783\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 128901. 0.536459
\(143\) 2860.78 0.0116989
\(144\) 0 0
\(145\) 20511.1 0.0810156
\(146\) −91051.9 −0.353514
\(147\) 0 0
\(148\) −137866. −0.517373
\(149\) 319669. 1.17960 0.589800 0.807549i \(-0.299207\pi\)
0.589800 + 0.807549i \(0.299207\pi\)
\(150\) 0 0
\(151\) −84046.3 −0.299969 −0.149985 0.988688i \(-0.547922\pi\)
−0.149985 + 0.988688i \(0.547922\pi\)
\(152\) 178982. 0.628349
\(153\) 0 0
\(154\) 0 0
\(155\) −76134.8 −0.254539
\(156\) 0 0
\(157\) −126195. −0.408596 −0.204298 0.978909i \(-0.565491\pi\)
−0.204298 + 0.978909i \(0.565491\pi\)
\(158\) 68400.2 0.217979
\(159\) 0 0
\(160\) 82721.1 0.255456
\(161\) 0 0
\(162\) 0 0
\(163\) −235790. −0.695114 −0.347557 0.937659i \(-0.612989\pi\)
−0.347557 + 0.937659i \(0.612989\pi\)
\(164\) −241428. −0.700935
\(165\) 0 0
\(166\) −12046.1 −0.0339293
\(167\) 149843. 0.415761 0.207881 0.978154i \(-0.433343\pi\)
0.207881 + 0.978154i \(0.433343\pi\)
\(168\) 0 0
\(169\) 237464. 0.639559
\(170\) 2135.63 0.00566765
\(171\) 0 0
\(172\) −397147. −1.02360
\(173\) 704460. 1.78954 0.894769 0.446530i \(-0.147340\pi\)
0.894769 + 0.446530i \(0.147340\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 725.162 0.00176463
\(177\) 0 0
\(178\) 63473.6 0.150156
\(179\) 557573. 1.30067 0.650337 0.759645i \(-0.274627\pi\)
0.650337 + 0.759645i \(0.274627\pi\)
\(180\) 0 0
\(181\) −443092. −1.00530 −0.502652 0.864489i \(-0.667642\pi\)
−0.502652 + 0.864489i \(0.667642\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 690500. 1.50355
\(185\) 84731.7 0.182019
\(186\) 0 0
\(187\) 183.634 0.000384017 0
\(188\) −661327. −1.36465
\(189\) 0 0
\(190\) −45344.4 −0.0911254
\(191\) −157481. −0.312353 −0.156177 0.987729i \(-0.549917\pi\)
−0.156177 + 0.987729i \(0.549917\pi\)
\(192\) 0 0
\(193\) −778040. −1.50352 −0.751759 0.659437i \(-0.770795\pi\)
−0.751759 + 0.659437i \(0.770795\pi\)
\(194\) −54854.4 −0.104642
\(195\) 0 0
\(196\) 0 0
\(197\) 340283. 0.624704 0.312352 0.949966i \(-0.398883\pi\)
0.312352 + 0.949966i \(0.398883\pi\)
\(198\) 0 0
\(199\) −296509. −0.530768 −0.265384 0.964143i \(-0.585499\pi\)
−0.265384 + 0.964143i \(0.585499\pi\)
\(200\) 493961. 0.873209
\(201\) 0 0
\(202\) −439274. −0.757456
\(203\) 0 0
\(204\) 0 0
\(205\) 148380. 0.246599
\(206\) 650904. 1.06868
\(207\) 0 0
\(208\) 154310. 0.247307
\(209\) −3898.99 −0.00617429
\(210\) 0 0
\(211\) 506728. 0.783554 0.391777 0.920060i \(-0.371860\pi\)
0.391777 + 0.920060i \(0.371860\pi\)
\(212\) −432138. −0.660363
\(213\) 0 0
\(214\) 626582. 0.935283
\(215\) 244084. 0.360116
\(216\) 0 0
\(217\) 0 0
\(218\) 431399. 0.614806
\(219\) 0 0
\(220\) −1134.93 −0.00158092
\(221\) 39076.3 0.0538187
\(222\) 0 0
\(223\) 462362. 0.622615 0.311308 0.950309i \(-0.399233\pi\)
0.311308 + 0.950309i \(0.399233\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 637852. 0.830709
\(227\) −98226.6 −0.126521 −0.0632607 0.997997i \(-0.520150\pi\)
−0.0632607 + 0.997997i \(0.520150\pi\)
\(228\) 0 0
\(229\) −501403. −0.631827 −0.315914 0.948788i \(-0.602311\pi\)
−0.315914 + 0.948788i \(0.602311\pi\)
\(230\) −174935. −0.218051
\(231\) 0 0
\(232\) −250301. −0.305311
\(233\) 613916. 0.740831 0.370415 0.928866i \(-0.379215\pi\)
0.370415 + 0.928866i \(0.379215\pi\)
\(234\) 0 0
\(235\) 406447. 0.480103
\(236\) −148547. −0.173613
\(237\) 0 0
\(238\) 0 0
\(239\) −1.10020e6 −1.24588 −0.622940 0.782269i \(-0.714062\pi\)
−0.622940 + 0.782269i \(0.714062\pi\)
\(240\) 0 0
\(241\) −89883.5 −0.0996868 −0.0498434 0.998757i \(-0.515872\pi\)
−0.0498434 + 0.998757i \(0.515872\pi\)
\(242\) −497868. −0.546482
\(243\) 0 0
\(244\) 824744. 0.886839
\(245\) 0 0
\(246\) 0 0
\(247\) −829683. −0.865306
\(248\) 929087. 0.959240
\(249\) 0 0
\(250\) −258398. −0.261480
\(251\) 1.01050e6 1.01240 0.506202 0.862415i \(-0.331049\pi\)
0.506202 + 0.862415i \(0.331049\pi\)
\(252\) 0 0
\(253\) −15042.0 −0.0147742
\(254\) 277858. 0.270233
\(255\) 0 0
\(256\) −867430. −0.827246
\(257\) −1.21233e6 −1.14495 −0.572475 0.819922i \(-0.694017\pi\)
−0.572475 + 0.819922i \(0.694017\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −241505. −0.221561
\(261\) 0 0
\(262\) 989677. 0.890718
\(263\) −384901. −0.343131 −0.171566 0.985173i \(-0.554883\pi\)
−0.171566 + 0.985173i \(0.554883\pi\)
\(264\) 0 0
\(265\) 265589. 0.232325
\(266\) 0 0
\(267\) 0 0
\(268\) −1.05273e6 −0.895322
\(269\) −1.57455e6 −1.32671 −0.663355 0.748305i \(-0.730868\pi\)
−0.663355 + 0.748305i \(0.730868\pi\)
\(270\) 0 0
\(271\) −332270. −0.274832 −0.137416 0.990513i \(-0.543880\pi\)
−0.137416 + 0.990513i \(0.543880\pi\)
\(272\) 9905.24 0.00811788
\(273\) 0 0
\(274\) −800643. −0.644262
\(275\) −10760.6 −0.00858032
\(276\) 0 0
\(277\) 2.23543e6 1.75050 0.875251 0.483669i \(-0.160696\pi\)
0.875251 + 0.483669i \(0.160696\pi\)
\(278\) −872970. −0.677465
\(279\) 0 0
\(280\) 0 0
\(281\) −69723.6 −0.0526761 −0.0263381 0.999653i \(-0.508385\pi\)
−0.0263381 + 0.999653i \(0.508385\pi\)
\(282\) 0 0
\(283\) 476452. 0.353633 0.176817 0.984244i \(-0.443420\pi\)
0.176817 + 0.984244i \(0.443420\pi\)
\(284\) −935684. −0.688388
\(285\) 0 0
\(286\) 8844.46 0.00639376
\(287\) 0 0
\(288\) 0 0
\(289\) −1.41735e6 −0.998233
\(290\) 63412.6 0.0442773
\(291\) 0 0
\(292\) 660938. 0.453632
\(293\) 2.27589e6 1.54875 0.774377 0.632724i \(-0.218063\pi\)
0.774377 + 0.632724i \(0.218063\pi\)
\(294\) 0 0
\(295\) 91295.8 0.0610796
\(296\) −1.03400e6 −0.685946
\(297\) 0 0
\(298\) 988297. 0.644684
\(299\) −3.20085e6 −2.07056
\(300\) 0 0
\(301\) 0 0
\(302\) −259840. −0.163941
\(303\) 0 0
\(304\) −210312. −0.130521
\(305\) −506882. −0.312002
\(306\) 0 0
\(307\) 1.61790e6 0.979731 0.489866 0.871798i \(-0.337046\pi\)
0.489866 + 0.871798i \(0.337046\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −235380. −0.139112
\(311\) −1.02667e6 −0.601910 −0.300955 0.953638i \(-0.597305\pi\)
−0.300955 + 0.953638i \(0.597305\pi\)
\(312\) 0 0
\(313\) −317214. −0.183017 −0.0915085 0.995804i \(-0.529169\pi\)
−0.0915085 + 0.995804i \(0.529169\pi\)
\(314\) −390149. −0.223309
\(315\) 0 0
\(316\) −496512. −0.279712
\(317\) −1.57126e6 −0.878212 −0.439106 0.898435i \(-0.644705\pi\)
−0.439106 + 0.898435i \(0.644705\pi\)
\(318\) 0 0
\(319\) 5452.61 0.00300005
\(320\) 168452. 0.0919602
\(321\) 0 0
\(322\) 0 0
\(323\) −53257.7 −0.0284038
\(324\) 0 0
\(325\) −2.28979e6 −1.20250
\(326\) −728974. −0.379899
\(327\) 0 0
\(328\) −1.81071e6 −0.929317
\(329\) 0 0
\(330\) 0 0
\(331\) 1.53832e6 0.771751 0.385875 0.922551i \(-0.373899\pi\)
0.385875 + 0.922551i \(0.373899\pi\)
\(332\) 87441.4 0.0435383
\(333\) 0 0
\(334\) 463257. 0.227225
\(335\) 647000. 0.314987
\(336\) 0 0
\(337\) −2.86811e6 −1.37569 −0.687846 0.725857i \(-0.741444\pi\)
−0.687846 + 0.725857i \(0.741444\pi\)
\(338\) 734149. 0.349537
\(339\) 0 0
\(340\) −15502.3 −0.00727277
\(341\) −20239.5 −0.00942569
\(342\) 0 0
\(343\) 0 0
\(344\) −2.97860e6 −1.35711
\(345\) 0 0
\(346\) 2.17793e6 0.978031
\(347\) 111966. 0.0499185 0.0249593 0.999688i \(-0.492054\pi\)
0.0249593 + 0.999688i \(0.492054\pi\)
\(348\) 0 0
\(349\) 3.75314e6 1.64942 0.824711 0.565555i \(-0.191338\pi\)
0.824711 + 0.565555i \(0.191338\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 21990.3 0.00945965
\(353\) 3.22923e6 1.37931 0.689655 0.724138i \(-0.257762\pi\)
0.689655 + 0.724138i \(0.257762\pi\)
\(354\) 0 0
\(355\) 575065. 0.242184
\(356\) −460749. −0.192681
\(357\) 0 0
\(358\) 1.72381e6 0.710855
\(359\) −1.83659e6 −0.752100 −0.376050 0.926599i \(-0.622718\pi\)
−0.376050 + 0.926599i \(0.622718\pi\)
\(360\) 0 0
\(361\) −1.34531e6 −0.543319
\(362\) −1.36988e6 −0.549427
\(363\) 0 0
\(364\) 0 0
\(365\) −406208. −0.159594
\(366\) 0 0
\(367\) 339025. 0.131391 0.0656957 0.997840i \(-0.479073\pi\)
0.0656957 + 0.997840i \(0.479073\pi\)
\(368\) −811366. −0.312318
\(369\) 0 0
\(370\) 261959. 0.0994784
\(371\) 0 0
\(372\) 0 0
\(373\) −706622. −0.262975 −0.131488 0.991318i \(-0.541975\pi\)
−0.131488 + 0.991318i \(0.541975\pi\)
\(374\) 567.729 0.000209876 0
\(375\) 0 0
\(376\) −4.95995e6 −1.80929
\(377\) 1.16028e6 0.420447
\(378\) 0 0
\(379\) 648296. 0.231833 0.115917 0.993259i \(-0.463019\pi\)
0.115917 + 0.993259i \(0.463019\pi\)
\(380\) 329151. 0.116933
\(381\) 0 0
\(382\) −486873. −0.170709
\(383\) −3.07022e6 −1.06948 −0.534741 0.845016i \(-0.679591\pi\)
−0.534741 + 0.845016i \(0.679591\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.40541e6 −0.821714
\(387\) 0 0
\(388\) 398183. 0.134278
\(389\) 4.59998e6 1.54128 0.770641 0.637270i \(-0.219936\pi\)
0.770641 + 0.637270i \(0.219936\pi\)
\(390\) 0 0
\(391\) −205464. −0.0679663
\(392\) 0 0
\(393\) 0 0
\(394\) 1.05203e6 0.341418
\(395\) 305153. 0.0984066
\(396\) 0 0
\(397\) −2.64159e6 −0.841181 −0.420590 0.907251i \(-0.638177\pi\)
−0.420590 + 0.907251i \(0.638177\pi\)
\(398\) −916694. −0.290079
\(399\) 0 0
\(400\) −580425. −0.181383
\(401\) −4.89677e6 −1.52072 −0.760359 0.649503i \(-0.774977\pi\)
−0.760359 + 0.649503i \(0.774977\pi\)
\(402\) 0 0
\(403\) −4.30684e6 −1.32098
\(404\) 3.18865e6 0.971972
\(405\) 0 0
\(406\) 0 0
\(407\) 22524.8 0.00674025
\(408\) 0 0
\(409\) −96336.3 −0.0284762 −0.0142381 0.999899i \(-0.504532\pi\)
−0.0142381 + 0.999899i \(0.504532\pi\)
\(410\) 458735. 0.134773
\(411\) 0 0
\(412\) −4.72485e6 −1.37134
\(413\) 0 0
\(414\) 0 0
\(415\) −53740.9 −0.0153174
\(416\) 4.67941e6 1.32574
\(417\) 0 0
\(418\) −12054.2 −0.00337442
\(419\) 1.09657e6 0.305142 0.152571 0.988292i \(-0.451245\pi\)
0.152571 + 0.988292i \(0.451245\pi\)
\(420\) 0 0
\(421\) −1.93660e6 −0.532517 −0.266259 0.963902i \(-0.585788\pi\)
−0.266259 + 0.963902i \(0.585788\pi\)
\(422\) 1.56661e6 0.428234
\(423\) 0 0
\(424\) −3.24103e6 −0.875526
\(425\) −146982. −0.0394723
\(426\) 0 0
\(427\) 0 0
\(428\) −4.54830e6 −1.20016
\(429\) 0 0
\(430\) 754616. 0.196813
\(431\) 3.07330e6 0.796914 0.398457 0.917187i \(-0.369546\pi\)
0.398457 + 0.917187i \(0.369546\pi\)
\(432\) 0 0
\(433\) −3.80919e6 −0.976366 −0.488183 0.872741i \(-0.662340\pi\)
−0.488183 + 0.872741i \(0.662340\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.13149e6 −0.788923
\(437\) 4.36248e6 1.09277
\(438\) 0 0
\(439\) −533019. −0.132002 −0.0660011 0.997820i \(-0.521024\pi\)
−0.0660011 + 0.997820i \(0.521024\pi\)
\(440\) −8511.94 −0.00209603
\(441\) 0 0
\(442\) 120809. 0.0294134
\(443\) 4.66745e6 1.12998 0.564990 0.825098i \(-0.308880\pi\)
0.564990 + 0.825098i \(0.308880\pi\)
\(444\) 0 0
\(445\) 283173. 0.0677879
\(446\) 1.42945e6 0.340276
\(447\) 0 0
\(448\) 0 0
\(449\) 6.10134e6 1.42827 0.714133 0.700010i \(-0.246821\pi\)
0.714133 + 0.700010i \(0.246821\pi\)
\(450\) 0 0
\(451\) 39444.9 0.00913166
\(452\) −4.63012e6 −1.06597
\(453\) 0 0
\(454\) −303680. −0.0691475
\(455\) 0 0
\(456\) 0 0
\(457\) 5.94498e6 1.33156 0.665779 0.746149i \(-0.268099\pi\)
0.665779 + 0.746149i \(0.268099\pi\)
\(458\) −1.55015e6 −0.345311
\(459\) 0 0
\(460\) 1.26984e6 0.279804
\(461\) 2.09415e6 0.458940 0.229470 0.973316i \(-0.426301\pi\)
0.229470 + 0.973316i \(0.426301\pi\)
\(462\) 0 0
\(463\) 2.41123e6 0.522741 0.261371 0.965239i \(-0.415826\pi\)
0.261371 + 0.965239i \(0.415826\pi\)
\(464\) 294114. 0.0634191
\(465\) 0 0
\(466\) 1.89800e6 0.404884
\(467\) 4.51833e6 0.958706 0.479353 0.877622i \(-0.340871\pi\)
0.479353 + 0.877622i \(0.340871\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1.25658e6 0.262389
\(471\) 0 0
\(472\) −1.11410e6 −0.230181
\(473\) 64886.6 0.0133353
\(474\) 0 0
\(475\) 3.12078e6 0.634643
\(476\) 0 0
\(477\) 0 0
\(478\) −3.40140e6 −0.680908
\(479\) −8.35928e6 −1.66468 −0.832339 0.554267i \(-0.812999\pi\)
−0.832339 + 0.554267i \(0.812999\pi\)
\(480\) 0 0
\(481\) 4.79315e6 0.944624
\(482\) −277886. −0.0544815
\(483\) 0 0
\(484\) 3.61398e6 0.701249
\(485\) −244721. −0.0472407
\(486\) 0 0
\(487\) 598304. 0.114314 0.0571570 0.998365i \(-0.481796\pi\)
0.0571570 + 0.998365i \(0.481796\pi\)
\(488\) 6.18558e6 1.17579
\(489\) 0 0
\(490\) 0 0
\(491\) −2.76760e6 −0.518084 −0.259042 0.965866i \(-0.583407\pi\)
−0.259042 + 0.965866i \(0.583407\pi\)
\(492\) 0 0
\(493\) 74479.1 0.0138012
\(494\) −2.56507e6 −0.472914
\(495\) 0 0
\(496\) −1.09172e6 −0.199253
\(497\) 0 0
\(498\) 0 0
\(499\) 3.03921e6 0.546398 0.273199 0.961958i \(-0.411918\pi\)
0.273199 + 0.961958i \(0.411918\pi\)
\(500\) 1.87569e6 0.335533
\(501\) 0 0
\(502\) 3.12410e6 0.553307
\(503\) −9.37896e6 −1.65286 −0.826428 0.563043i \(-0.809631\pi\)
−0.826428 + 0.563043i \(0.809631\pi\)
\(504\) 0 0
\(505\) −1.95973e6 −0.341953
\(506\) −46504.3 −0.00807452
\(507\) 0 0
\(508\) −2.01694e6 −0.346764
\(509\) −5.90139e6 −1.00962 −0.504812 0.863229i \(-0.668438\pi\)
−0.504812 + 0.863229i \(0.668438\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 2.25139e6 0.379555
\(513\) 0 0
\(514\) −3.74806e6 −0.625747
\(515\) 2.90386e6 0.482456
\(516\) 0 0
\(517\) 108049. 0.0177784
\(518\) 0 0
\(519\) 0 0
\(520\) −1.81129e6 −0.293751
\(521\) −9.37426e6 −1.51301 −0.756506 0.653986i \(-0.773095\pi\)
−0.756506 + 0.653986i \(0.773095\pi\)
\(522\) 0 0
\(523\) 6.46830e6 1.03404 0.517018 0.855975i \(-0.327042\pi\)
0.517018 + 0.855975i \(0.327042\pi\)
\(524\) −7.18398e6 −1.14298
\(525\) 0 0
\(526\) −1.18997e6 −0.187531
\(527\) −276458. −0.0433613
\(528\) 0 0
\(529\) 1.03938e7 1.61486
\(530\) 821102. 0.126972
\(531\) 0 0
\(532\) 0 0
\(533\) 8.39364e6 1.27977
\(534\) 0 0
\(535\) 2.79536e6 0.422234
\(536\) −7.89546e6 −1.18704
\(537\) 0 0
\(538\) −4.86792e6 −0.725083
\(539\) 0 0
\(540\) 0 0
\(541\) −2.46089e6 −0.361492 −0.180746 0.983530i \(-0.557851\pi\)
−0.180746 + 0.983530i \(0.557851\pi\)
\(542\) −1.02725e6 −0.150203
\(543\) 0 0
\(544\) 300373. 0.0435175
\(545\) 1.92459e6 0.277554
\(546\) 0 0
\(547\) 503726. 0.0719824 0.0359912 0.999352i \(-0.488541\pi\)
0.0359912 + 0.999352i \(0.488541\pi\)
\(548\) 5.81180e6 0.826721
\(549\) 0 0
\(550\) −33267.7 −0.00468938
\(551\) −1.58137e6 −0.221898
\(552\) 0 0
\(553\) 0 0
\(554\) 6.91113e6 0.956697
\(555\) 0 0
\(556\) 6.33681e6 0.869328
\(557\) −632648. −0.0864021 −0.0432011 0.999066i \(-0.513756\pi\)
−0.0432011 + 0.999066i \(0.513756\pi\)
\(558\) 0 0
\(559\) 1.38075e7 1.86890
\(560\) 0 0
\(561\) 0 0
\(562\) −215559. −0.0287889
\(563\) −1.00780e7 −1.33999 −0.669995 0.742366i \(-0.733704\pi\)
−0.669995 + 0.742366i \(0.733704\pi\)
\(564\) 0 0
\(565\) 2.84564e6 0.375024
\(566\) 1.47301e6 0.193270
\(567\) 0 0
\(568\) −7.01763e6 −0.912682
\(569\) −2.01685e6 −0.261151 −0.130576 0.991438i \(-0.541683\pi\)
−0.130576 + 0.991438i \(0.541683\pi\)
\(570\) 0 0
\(571\) 4.09727e6 0.525902 0.262951 0.964809i \(-0.415304\pi\)
0.262951 + 0.964809i \(0.415304\pi\)
\(572\) −64201.1 −0.00820451
\(573\) 0 0
\(574\) 0 0
\(575\) 1.20397e7 1.51861
\(576\) 0 0
\(577\) −8.14322e6 −1.01826 −0.509128 0.860691i \(-0.670032\pi\)
−0.509128 + 0.860691i \(0.670032\pi\)
\(578\) −4.38191e6 −0.545562
\(579\) 0 0
\(580\) −460307. −0.0568169
\(581\) 0 0
\(582\) 0 0
\(583\) 70603.4 0.00860309
\(584\) 4.95703e6 0.601436
\(585\) 0 0
\(586\) 7.03621e6 0.846437
\(587\) 8.81465e6 1.05587 0.527934 0.849285i \(-0.322967\pi\)
0.527934 + 0.849285i \(0.322967\pi\)
\(588\) 0 0
\(589\) 5.86985e6 0.697170
\(590\) 282253. 0.0333816
\(591\) 0 0
\(592\) 1.21499e6 0.142485
\(593\) 1.15739e7 1.35158 0.675792 0.737093i \(-0.263802\pi\)
0.675792 + 0.737093i \(0.263802\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −7.17396e6 −0.827263
\(597\) 0 0
\(598\) −9.89584e6 −1.13162
\(599\) 1.23551e7 1.40695 0.703474 0.710721i \(-0.251631\pi\)
0.703474 + 0.710721i \(0.251631\pi\)
\(600\) 0 0
\(601\) −8.33752e6 −0.941566 −0.470783 0.882249i \(-0.656029\pi\)
−0.470783 + 0.882249i \(0.656029\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.88616e6 0.210371
\(605\) −2.22113e6 −0.246709
\(606\) 0 0
\(607\) −8.67189e6 −0.955305 −0.477653 0.878549i \(-0.658512\pi\)
−0.477653 + 0.878549i \(0.658512\pi\)
\(608\) −6.37764e6 −0.699682
\(609\) 0 0
\(610\) −1.56709e6 −0.170518
\(611\) 2.29921e7 2.49159
\(612\) 0 0
\(613\) 5.24031e6 0.563256 0.281628 0.959524i \(-0.409126\pi\)
0.281628 + 0.959524i \(0.409126\pi\)
\(614\) 5.00196e6 0.535450
\(615\) 0 0
\(616\) 0 0
\(617\) 6.95644e6 0.735655 0.367828 0.929894i \(-0.380102\pi\)
0.367828 + 0.929894i \(0.380102\pi\)
\(618\) 0 0
\(619\) 1.82763e7 1.91717 0.958586 0.284802i \(-0.0919278\pi\)
0.958586 + 0.284802i \(0.0919278\pi\)
\(620\) 1.70861e6 0.178510
\(621\) 0 0
\(622\) −3.17409e6 −0.328960
\(623\) 0 0
\(624\) 0 0
\(625\) 8.01835e6 0.821079
\(626\) −980707. −0.100024
\(627\) 0 0
\(628\) 2.83205e6 0.286551
\(629\) 307674. 0.0310074
\(630\) 0 0
\(631\) 1.90708e7 1.90676 0.953379 0.301777i \(-0.0975798\pi\)
0.953379 + 0.301777i \(0.0975798\pi\)
\(632\) −3.72384e6 −0.370850
\(633\) 0 0
\(634\) −4.85774e6 −0.479967
\(635\) 1.23960e6 0.121997
\(636\) 0 0
\(637\) 0 0
\(638\) 16857.4 0.00163961
\(639\) 0 0
\(640\) −2.12628e6 −0.205197
\(641\) −3.20624e6 −0.308213 −0.154106 0.988054i \(-0.549250\pi\)
−0.154106 + 0.988054i \(0.549250\pi\)
\(642\) 0 0
\(643\) 4.35153e6 0.415063 0.207532 0.978228i \(-0.433457\pi\)
0.207532 + 0.978228i \(0.433457\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −164653. −0.0155234
\(647\) 5.59976e6 0.525907 0.262953 0.964809i \(-0.415303\pi\)
0.262953 + 0.964809i \(0.415303\pi\)
\(648\) 0 0
\(649\) 24269.8 0.00226180
\(650\) −7.07917e6 −0.657202
\(651\) 0 0
\(652\) 5.29156e6 0.487489
\(653\) 6.74495e6 0.619007 0.309504 0.950898i \(-0.399837\pi\)
0.309504 + 0.950898i \(0.399837\pi\)
\(654\) 0 0
\(655\) 4.41523e6 0.402115
\(656\) 2.12766e6 0.193038
\(657\) 0 0
\(658\) 0 0
\(659\) −2.09622e6 −0.188029 −0.0940143 0.995571i \(-0.529970\pi\)
−0.0940143 + 0.995571i \(0.529970\pi\)
\(660\) 0 0
\(661\) −1.94185e7 −1.72867 −0.864335 0.502917i \(-0.832260\pi\)
−0.864335 + 0.502917i \(0.832260\pi\)
\(662\) 4.75591e6 0.421783
\(663\) 0 0
\(664\) 655810. 0.0577242
\(665\) 0 0
\(666\) 0 0
\(667\) −6.10079e6 −0.530972
\(668\) −3.36274e6 −0.291577
\(669\) 0 0
\(670\) 2.00028e6 0.172149
\(671\) −134748. −0.0115536
\(672\) 0 0
\(673\) 1.27627e7 1.08619 0.543094 0.839672i \(-0.317253\pi\)
0.543094 + 0.839672i \(0.317253\pi\)
\(674\) −8.86713e6 −0.751853
\(675\) 0 0
\(676\) −5.32913e6 −0.448528
\(677\) 3.86502e6 0.324101 0.162050 0.986782i \(-0.448189\pi\)
0.162050 + 0.986782i \(0.448189\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −116267. −0.00964241
\(681\) 0 0
\(682\) −62572.8 −0.00515140
\(683\) 2.06418e7 1.69316 0.846578 0.532265i \(-0.178659\pi\)
0.846578 + 0.532265i \(0.178659\pi\)
\(684\) 0 0
\(685\) −3.57189e6 −0.290852
\(686\) 0 0
\(687\) 0 0
\(688\) 3.49998e6 0.281900
\(689\) 1.50240e7 1.20570
\(690\) 0 0
\(691\) −1.62329e7 −1.29330 −0.646651 0.762786i \(-0.723831\pi\)
−0.646651 + 0.762786i \(0.723831\pi\)
\(692\) −1.58094e7 −1.25502
\(693\) 0 0
\(694\) 346156. 0.0272818
\(695\) −3.89456e6 −0.305842
\(696\) 0 0
\(697\) 538791. 0.0420086
\(698\) 1.16033e7 0.901454
\(699\) 0 0
\(700\) 0 0
\(701\) 8.27590e6 0.636092 0.318046 0.948075i \(-0.396973\pi\)
0.318046 + 0.948075i \(0.396973\pi\)
\(702\) 0 0
\(703\) −6.53266e6 −0.498542
\(704\) 44780.7 0.00340533
\(705\) 0 0
\(706\) 9.98356e6 0.753830
\(707\) 0 0
\(708\) 0 0
\(709\) 2.38973e7 1.78539 0.892696 0.450660i \(-0.148811\pi\)
0.892696 + 0.450660i \(0.148811\pi\)
\(710\) 1.77789e6 0.132360
\(711\) 0 0
\(712\) −3.45562e6 −0.255462
\(713\) 2.26454e7 1.66823
\(714\) 0 0
\(715\) 39457.6 0.00288646
\(716\) −1.25130e7 −0.912173
\(717\) 0 0
\(718\) −5.67804e6 −0.411043
\(719\) 784349. 0.0565832 0.0282916 0.999600i \(-0.490993\pi\)
0.0282916 + 0.999600i \(0.490993\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −4.15920e6 −0.296939
\(723\) 0 0
\(724\) 9.94381e6 0.705028
\(725\) −4.36431e6 −0.308369
\(726\) 0 0
\(727\) −1.61766e7 −1.13514 −0.567571 0.823324i \(-0.692117\pi\)
−0.567571 + 0.823324i \(0.692117\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1.25584e6 −0.0872224
\(731\) 886307. 0.0613467
\(732\) 0 0
\(733\) −1.36978e7 −0.941654 −0.470827 0.882226i \(-0.656044\pi\)
−0.470827 + 0.882226i \(0.656044\pi\)
\(734\) 1.04814e6 0.0718090
\(735\) 0 0
\(736\) −2.46044e7 −1.67424
\(737\) 171997. 0.0116641
\(738\) 0 0
\(739\) −2.53712e7 −1.70895 −0.854477 0.519490i \(-0.826122\pi\)
−0.854477 + 0.519490i \(0.826122\pi\)
\(740\) −1.90154e6 −0.127651
\(741\) 0 0
\(742\) 0 0
\(743\) −2.45336e7 −1.63038 −0.815192 0.579191i \(-0.803368\pi\)
−0.815192 + 0.579191i \(0.803368\pi\)
\(744\) 0 0
\(745\) 4.40907e6 0.291042
\(746\) −2.18461e6 −0.143723
\(747\) 0 0
\(748\) −4121.10 −0.000269314 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.84907e7 −1.19634 −0.598169 0.801370i \(-0.704105\pi\)
−0.598169 + 0.801370i \(0.704105\pi\)
\(752\) 5.82815e6 0.375825
\(753\) 0 0
\(754\) 3.58716e6 0.229786
\(755\) −1.15922e6 −0.0740113
\(756\) 0 0
\(757\) −2.94413e7 −1.86732 −0.933658 0.358166i \(-0.883402\pi\)
−0.933658 + 0.358166i \(0.883402\pi\)
\(758\) 2.00429e6 0.126703
\(759\) 0 0
\(760\) 2.46863e6 0.155032
\(761\) −1.86236e7 −1.16574 −0.582870 0.812565i \(-0.698070\pi\)
−0.582870 + 0.812565i \(0.698070\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 3.53417e6 0.219056
\(765\) 0 0
\(766\) −9.49199e6 −0.584501
\(767\) 5.16448e6 0.316984
\(768\) 0 0
\(769\) 1.13326e7 0.691056 0.345528 0.938408i \(-0.387700\pi\)
0.345528 + 0.938408i \(0.387700\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.74607e7 1.05443
\(773\) −8.22020e6 −0.494804 −0.247402 0.968913i \(-0.579577\pi\)
−0.247402 + 0.968913i \(0.579577\pi\)
\(774\) 0 0
\(775\) 1.61998e7 0.968848
\(776\) 2.98637e6 0.178029
\(777\) 0 0
\(778\) 1.42214e7 0.842352
\(779\) −1.14398e7 −0.675422
\(780\) 0 0
\(781\) 152874. 0.00896819
\(782\) −635217. −0.0371454
\(783\) 0 0
\(784\) 0 0
\(785\) −1.74056e6 −0.100813
\(786\) 0 0
\(787\) 5.43081e6 0.312556 0.156278 0.987713i \(-0.450050\pi\)
0.156278 + 0.987713i \(0.450050\pi\)
\(788\) −7.63657e6 −0.438110
\(789\) 0 0
\(790\) 943418. 0.0537819
\(791\) 0 0
\(792\) 0 0
\(793\) −2.86736e7 −1.61920
\(794\) −8.16681e6 −0.459728
\(795\) 0 0
\(796\) 6.65421e6 0.372232
\(797\) 1.78388e7 0.994765 0.497382 0.867531i \(-0.334295\pi\)
0.497382 + 0.867531i \(0.334295\pi\)
\(798\) 0 0
\(799\) 1.47587e6 0.0817866
\(800\) −1.76012e7 −0.972339
\(801\) 0 0
\(802\) −1.51390e7 −0.831115
\(803\) −107985. −0.00590983
\(804\) 0 0
\(805\) 0 0
\(806\) −1.33151e7 −0.721951
\(807\) 0 0
\(808\) 2.39149e7 1.28867
\(809\) −1.18431e7 −0.636199 −0.318099 0.948057i \(-0.603045\pi\)
−0.318099 + 0.948057i \(0.603045\pi\)
\(810\) 0 0
\(811\) −618053. −0.0329969 −0.0164985 0.999864i \(-0.505252\pi\)
−0.0164985 + 0.999864i \(0.505252\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 69638.4 0.00368373
\(815\) −3.25216e6 −0.171505
\(816\) 0 0
\(817\) −1.88184e7 −0.986342
\(818\) −297836. −0.0155630
\(819\) 0 0
\(820\) −3.32992e6 −0.172941
\(821\) −4.31256e6 −0.223294 −0.111647 0.993748i \(-0.535613\pi\)
−0.111647 + 0.993748i \(0.535613\pi\)
\(822\) 0 0
\(823\) 2.86620e7 1.47505 0.737524 0.675321i \(-0.235995\pi\)
0.737524 + 0.675321i \(0.235995\pi\)
\(824\) −3.54364e7 −1.81816
\(825\) 0 0
\(826\) 0 0
\(827\) −1.86894e7 −0.950235 −0.475117 0.879922i \(-0.657594\pi\)
−0.475117 + 0.879922i \(0.657594\pi\)
\(828\) 0 0
\(829\) −1.32486e7 −0.669550 −0.334775 0.942298i \(-0.608660\pi\)
−0.334775 + 0.942298i \(0.608660\pi\)
\(830\) −166147. −0.00837137
\(831\) 0 0
\(832\) 9.52907e6 0.477246
\(833\) 0 0
\(834\) 0 0
\(835\) 2.06672e6 0.102581
\(836\) 87500.6 0.00433007
\(837\) 0 0
\(838\) 3.39019e6 0.166769
\(839\) −2.52930e7 −1.24049 −0.620247 0.784406i \(-0.712968\pi\)
−0.620247 + 0.784406i \(0.712968\pi\)
\(840\) 0 0
\(841\) −1.82997e7 −0.892181
\(842\) −5.98723e6 −0.291035
\(843\) 0 0
\(844\) −1.13719e7 −0.549512
\(845\) 3.27525e6 0.157798
\(846\) 0 0
\(847\) 0 0
\(848\) 3.80835e6 0.181864
\(849\) 0 0
\(850\) −454414. −0.0215727
\(851\) −2.52025e7 −1.19294
\(852\) 0 0
\(853\) −4.05806e7 −1.90962 −0.954808 0.297223i \(-0.903939\pi\)
−0.954808 + 0.297223i \(0.903939\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3.41123e7 −1.59120
\(857\) 1.94469e7 0.904480 0.452240 0.891896i \(-0.350625\pi\)
0.452240 + 0.891896i \(0.350625\pi\)
\(858\) 0 0
\(859\) 2.03242e7 0.939788 0.469894 0.882723i \(-0.344292\pi\)
0.469894 + 0.882723i \(0.344292\pi\)
\(860\) −5.47769e6 −0.252552
\(861\) 0 0
\(862\) 9.50149e6 0.435536
\(863\) 2.61202e7 1.19385 0.596926 0.802297i \(-0.296389\pi\)
0.596926 + 0.802297i \(0.296389\pi\)
\(864\) 0 0
\(865\) 9.71634e6 0.441532
\(866\) −1.17766e7 −0.533611
\(867\) 0 0
\(868\) 0 0
\(869\) 81120.9 0.00364404
\(870\) 0 0
\(871\) 3.65999e7 1.63469
\(872\) −2.34862e7 −1.04597
\(873\) 0 0
\(874\) 1.34872e7 0.597231
\(875\) 0 0
\(876\) 0 0
\(877\) −1.62956e7 −0.715436 −0.357718 0.933830i \(-0.616445\pi\)
−0.357718 + 0.933830i \(0.616445\pi\)
\(878\) −1.64789e6 −0.0721428
\(879\) 0 0
\(880\) 10001.9 0.000435387 0
\(881\) 2.93722e7 1.27496 0.637479 0.770467i \(-0.279977\pi\)
0.637479 + 0.770467i \(0.279977\pi\)
\(882\) 0 0
\(883\) 1.21821e7 0.525800 0.262900 0.964823i \(-0.415321\pi\)
0.262900 + 0.964823i \(0.415321\pi\)
\(884\) −876945. −0.0377435
\(885\) 0 0
\(886\) 1.44300e7 0.617565
\(887\) −2.38021e7 −1.01579 −0.507897 0.861418i \(-0.669577\pi\)
−0.507897 + 0.861418i \(0.669577\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 875466. 0.0370480
\(891\) 0 0
\(892\) −1.03763e7 −0.436645
\(893\) −3.13363e7 −1.31498
\(894\) 0 0
\(895\) 7.69038e6 0.320915
\(896\) 0 0
\(897\) 0 0
\(898\) 1.88631e7 0.780587
\(899\) −8.20879e6 −0.338750
\(900\) 0 0
\(901\) 964396. 0.0395771
\(902\) 121949. 0.00499070
\(903\) 0 0
\(904\) −3.47259e7 −1.41329
\(905\) −6.11140e6 −0.248039
\(906\) 0 0
\(907\) 2.10870e7 0.851133 0.425566 0.904927i \(-0.360075\pi\)
0.425566 + 0.904927i \(0.360075\pi\)
\(908\) 2.20439e6 0.0887305
\(909\) 0 0
\(910\) 0 0
\(911\) −1.71528e7 −0.684761 −0.342381 0.939561i \(-0.611233\pi\)
−0.342381 + 0.939561i \(0.611233\pi\)
\(912\) 0 0
\(913\) −14286.3 −0.000567210 0
\(914\) 1.83797e7 0.727733
\(915\) 0 0
\(916\) 1.12524e7 0.443105
\(917\) 0 0
\(918\) 0 0
\(919\) 7.35541e6 0.287289 0.143644 0.989629i \(-0.454118\pi\)
0.143644 + 0.989629i \(0.454118\pi\)
\(920\) 9.52379e6 0.370971
\(921\) 0 0
\(922\) 6.47433e6 0.250823
\(923\) 3.25306e7 1.25686
\(924\) 0 0
\(925\) −1.80290e7 −0.692817
\(926\) 7.45463e6 0.285692
\(927\) 0 0
\(928\) 8.91891e6 0.339971
\(929\) −3.34199e7 −1.27047 −0.635236 0.772318i \(-0.719097\pi\)
−0.635236 + 0.772318i \(0.719097\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.37774e7 −0.519550
\(933\) 0 0
\(934\) 1.39690e7 0.523959
\(935\) 2532.80 9.47483e−5 0
\(936\) 0 0
\(937\) 2.87696e7 1.07049 0.535247 0.844696i \(-0.320219\pi\)
0.535247 + 0.844696i \(0.320219\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −9.12142e6 −0.336700
\(941\) 2.16379e7 0.796601 0.398300 0.917255i \(-0.369600\pi\)
0.398300 + 0.917255i \(0.369600\pi\)
\(942\) 0 0
\(943\) −4.41339e7 −1.61619
\(944\) 1.30911e6 0.0478132
\(945\) 0 0
\(946\) 200605. 0.00728809
\(947\) 6.31254e6 0.228733 0.114367 0.993439i \(-0.463516\pi\)
0.114367 + 0.993439i \(0.463516\pi\)
\(948\) 0 0
\(949\) −2.29786e7 −0.828244
\(950\) 9.64829e6 0.346850
\(951\) 0 0
\(952\) 0 0
\(953\) 5.59599e6 0.199593 0.0997963 0.995008i \(-0.468181\pi\)
0.0997963 + 0.995008i \(0.468181\pi\)
\(954\) 0 0
\(955\) −2.17208e6 −0.0770668
\(956\) 2.46905e7 0.873746
\(957\) 0 0
\(958\) −2.58438e7 −0.909792
\(959\) 0 0
\(960\) 0 0
\(961\) 1.84091e6 0.0643021
\(962\) 1.48186e7 0.516263
\(963\) 0 0
\(964\) 2.01715e6 0.0699111
\(965\) −1.07312e7 −0.370963
\(966\) 0 0
\(967\) 1.33277e7 0.458340 0.229170 0.973386i \(-0.426399\pi\)
0.229170 + 0.973386i \(0.426399\pi\)
\(968\) 2.71048e7 0.929734
\(969\) 0 0
\(970\) −756585. −0.0258183
\(971\) −3.63257e7 −1.23642 −0.618210 0.786013i \(-0.712142\pi\)
−0.618210 + 0.786013i \(0.712142\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1.84973e6 0.0624758
\(975\) 0 0
\(976\) −7.26831e6 −0.244236
\(977\) 9.03739e6 0.302905 0.151453 0.988465i \(-0.451605\pi\)
0.151453 + 0.988465i \(0.451605\pi\)
\(978\) 0 0
\(979\) 75278.0 0.00251022
\(980\) 0 0
\(981\) 0 0
\(982\) −8.55639e6 −0.283147
\(983\) 1.98539e7 0.655332 0.327666 0.944794i \(-0.393738\pi\)
0.327666 + 0.944794i \(0.393738\pi\)
\(984\) 0 0
\(985\) 4.69339e6 0.154133
\(986\) 230261. 0.00754273
\(987\) 0 0
\(988\) 1.86196e7 0.606846
\(989\) −7.25999e7 −2.36018
\(990\) 0 0
\(991\) −4.55557e7 −1.47353 −0.736764 0.676150i \(-0.763647\pi\)
−0.736764 + 0.676150i \(0.763647\pi\)
\(992\) −3.31060e7 −1.06814
\(993\) 0 0
\(994\) 0 0
\(995\) −4.08963e6 −0.130956
\(996\) 0 0
\(997\) 4.51665e7 1.43906 0.719529 0.694462i \(-0.244358\pi\)
0.719529 + 0.694462i \(0.244358\pi\)
\(998\) 9.39610e6 0.298622
\(999\) 0 0
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 441.6.a.bb.1.4 6
3.2 odd 2 147.6.a.n.1.3 6
7.6 odd 2 441.6.a.ba.1.4 6
21.2 odd 6 147.6.e.q.67.4 12
21.5 even 6 147.6.e.p.67.4 12
21.11 odd 6 147.6.e.q.79.4 12
21.17 even 6 147.6.e.p.79.4 12
21.20 even 2 147.6.a.o.1.3 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.n.1.3 6 3.2 odd 2
147.6.a.o.1.3 yes 6 21.20 even 2
147.6.e.p.67.4 12 21.5 even 6
147.6.e.p.79.4 12 21.17 even 6
147.6.e.q.67.4 12 21.2 odd 6
147.6.e.q.79.4 12 21.11 odd 6
441.6.a.ba.1.4 6 7.6 odd 2
441.6.a.bb.1.4 6 1.1 even 1 trivial