Properties

Label 342.6.a.a.1.1
Level $342$
Weight $6$
Character 342.1
Self dual yes
Analytic conductor $54.851$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [342,6,Mod(1,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 342.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.8512663760\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 342.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +16.0000 q^{4} -21.0000 q^{5} -143.000 q^{7} -64.0000 q^{8} +O(q^{10})\) \(q-4.00000 q^{2} +16.0000 q^{4} -21.0000 q^{5} -143.000 q^{7} -64.0000 q^{8} +84.0000 q^{10} +205.000 q^{11} -78.0000 q^{13} +572.000 q^{14} +256.000 q^{16} +2125.00 q^{17} +361.000 q^{19} -336.000 q^{20} -820.000 q^{22} -20.0000 q^{23} -2684.00 q^{25} +312.000 q^{26} -2288.00 q^{28} +4866.00 q^{29} -1098.00 q^{31} -1024.00 q^{32} -8500.00 q^{34} +3003.00 q^{35} -15128.0 q^{37} -1444.00 q^{38} +1344.00 q^{40} +9400.00 q^{41} +20073.0 q^{43} +3280.00 q^{44} +80.0000 q^{46} -14105.0 q^{47} +3642.00 q^{49} +10736.0 q^{50} -1248.00 q^{52} -26386.0 q^{53} -4305.00 q^{55} +9152.00 q^{56} -19464.0 q^{58} +13216.0 q^{59} -2293.00 q^{61} +4392.00 q^{62} +4096.00 q^{64} +1638.00 q^{65} +35976.0 q^{67} +34000.0 q^{68} -12012.0 q^{70} -10180.0 q^{71} +33109.0 q^{73} +60512.0 q^{74} +5776.00 q^{76} -29315.0 q^{77} -53888.0 q^{79} -5376.00 q^{80} -37600.0 q^{82} -75196.0 q^{83} -44625.0 q^{85} -80292.0 q^{86} -13120.0 q^{88} -20618.0 q^{89} +11154.0 q^{91} -320.000 q^{92} +56420.0 q^{94} -7581.00 q^{95} -84130.0 q^{97} -14568.0 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) −21.0000 −0.375659 −0.187830 0.982202i \(-0.560145\pi\)
−0.187830 + 0.982202i \(0.560145\pi\)
\(6\) 0 0
\(7\) −143.000 −1.10304 −0.551520 0.834162i \(-0.685952\pi\)
−0.551520 + 0.834162i \(0.685952\pi\)
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) 84.0000 0.265631
\(11\) 205.000 0.510825 0.255413 0.966832i \(-0.417789\pi\)
0.255413 + 0.966832i \(0.417789\pi\)
\(12\) 0 0
\(13\) −78.0000 −0.128008 −0.0640039 0.997950i \(-0.520387\pi\)
−0.0640039 + 0.997950i \(0.520387\pi\)
\(14\) 572.000 0.779966
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 2125.00 1.78335 0.891675 0.452676i \(-0.149531\pi\)
0.891675 + 0.452676i \(0.149531\pi\)
\(18\) 0 0
\(19\) 361.000 0.229416
\(20\) −336.000 −0.187830
\(21\) 0 0
\(22\) −820.000 −0.361208
\(23\) −20.0000 −0.00788334 −0.00394167 0.999992i \(-0.501255\pi\)
−0.00394167 + 0.999992i \(0.501255\pi\)
\(24\) 0 0
\(25\) −2684.00 −0.858880
\(26\) 312.000 0.0905151
\(27\) 0 0
\(28\) −2288.00 −0.551520
\(29\) 4866.00 1.07443 0.537214 0.843446i \(-0.319477\pi\)
0.537214 + 0.843446i \(0.319477\pi\)
\(30\) 0 0
\(31\) −1098.00 −0.205210 −0.102605 0.994722i \(-0.532718\pi\)
−0.102605 + 0.994722i \(0.532718\pi\)
\(32\) −1024.00 −0.176777
\(33\) 0 0
\(34\) −8500.00 −1.26102
\(35\) 3003.00 0.414367
\(36\) 0 0
\(37\) −15128.0 −1.81667 −0.908337 0.418238i \(-0.862648\pi\)
−0.908337 + 0.418238i \(0.862648\pi\)
\(38\) −1444.00 −0.162221
\(39\) 0 0
\(40\) 1344.00 0.132816
\(41\) 9400.00 0.873310 0.436655 0.899629i \(-0.356163\pi\)
0.436655 + 0.899629i \(0.356163\pi\)
\(42\) 0 0
\(43\) 20073.0 1.65555 0.827773 0.561063i \(-0.189608\pi\)
0.827773 + 0.561063i \(0.189608\pi\)
\(44\) 3280.00 0.255413
\(45\) 0 0
\(46\) 80.0000 0.00557437
\(47\) −14105.0 −0.931383 −0.465692 0.884947i \(-0.654194\pi\)
−0.465692 + 0.884947i \(0.654194\pi\)
\(48\) 0 0
\(49\) 3642.00 0.216695
\(50\) 10736.0 0.607320
\(51\) 0 0
\(52\) −1248.00 −0.0640039
\(53\) −26386.0 −1.29028 −0.645140 0.764064i \(-0.723201\pi\)
−0.645140 + 0.764064i \(0.723201\pi\)
\(54\) 0 0
\(55\) −4305.00 −0.191896
\(56\) 9152.00 0.389983
\(57\) 0 0
\(58\) −19464.0 −0.759735
\(59\) 13216.0 0.494277 0.247138 0.968980i \(-0.420510\pi\)
0.247138 + 0.968980i \(0.420510\pi\)
\(60\) 0 0
\(61\) −2293.00 −0.0789004 −0.0394502 0.999222i \(-0.512561\pi\)
−0.0394502 + 0.999222i \(0.512561\pi\)
\(62\) 4392.00 0.145105
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 1638.00 0.0480873
\(66\) 0 0
\(67\) 35976.0 0.979097 0.489549 0.871976i \(-0.337162\pi\)
0.489549 + 0.871976i \(0.337162\pi\)
\(68\) 34000.0 0.891675
\(69\) 0 0
\(70\) −12012.0 −0.293002
\(71\) −10180.0 −0.239664 −0.119832 0.992794i \(-0.538236\pi\)
−0.119832 + 0.992794i \(0.538236\pi\)
\(72\) 0 0
\(73\) 33109.0 0.727175 0.363587 0.931560i \(-0.381552\pi\)
0.363587 + 0.931560i \(0.381552\pi\)
\(74\) 60512.0 1.28458
\(75\) 0 0
\(76\) 5776.00 0.114708
\(77\) −29315.0 −0.563460
\(78\) 0 0
\(79\) −53888.0 −0.971459 −0.485729 0.874109i \(-0.661446\pi\)
−0.485729 + 0.874109i \(0.661446\pi\)
\(80\) −5376.00 −0.0939149
\(81\) 0 0
\(82\) −37600.0 −0.617523
\(83\) −75196.0 −1.19812 −0.599059 0.800705i \(-0.704458\pi\)
−0.599059 + 0.800705i \(0.704458\pi\)
\(84\) 0 0
\(85\) −44625.0 −0.669932
\(86\) −80292.0 −1.17065
\(87\) 0 0
\(88\) −13120.0 −0.180604
\(89\) −20618.0 −0.275913 −0.137956 0.990438i \(-0.544053\pi\)
−0.137956 + 0.990438i \(0.544053\pi\)
\(90\) 0 0
\(91\) 11154.0 0.141198
\(92\) −320.000 −0.00394167
\(93\) 0 0
\(94\) 56420.0 0.658587
\(95\) −7581.00 −0.0861822
\(96\) 0 0
\(97\) −84130.0 −0.907866 −0.453933 0.891036i \(-0.649979\pi\)
−0.453933 + 0.891036i \(0.649979\pi\)
\(98\) −14568.0 −0.153227
\(99\) 0 0
\(100\) −42944.0 −0.429440
\(101\) −163714. −1.59692 −0.798459 0.602050i \(-0.794351\pi\)
−0.798459 + 0.602050i \(0.794351\pi\)
\(102\) 0 0
\(103\) −139062. −1.29156 −0.645781 0.763523i \(-0.723468\pi\)
−0.645781 + 0.763523i \(0.723468\pi\)
\(104\) 4992.00 0.0452576
\(105\) 0 0
\(106\) 105544. 0.912366
\(107\) −124690. −1.05286 −0.526432 0.850217i \(-0.676470\pi\)
−0.526432 + 0.850217i \(0.676470\pi\)
\(108\) 0 0
\(109\) −11836.0 −0.0954198 −0.0477099 0.998861i \(-0.515192\pi\)
−0.0477099 + 0.998861i \(0.515192\pi\)
\(110\) 17220.0 0.135691
\(111\) 0 0
\(112\) −36608.0 −0.275760
\(113\) −57674.0 −0.424897 −0.212449 0.977172i \(-0.568144\pi\)
−0.212449 + 0.977172i \(0.568144\pi\)
\(114\) 0 0
\(115\) 420.000 0.00296145
\(116\) 77856.0 0.537214
\(117\) 0 0
\(118\) −52864.0 −0.349506
\(119\) −303875. −1.96711
\(120\) 0 0
\(121\) −119026. −0.739058
\(122\) 9172.00 0.0557910
\(123\) 0 0
\(124\) −17568.0 −0.102605
\(125\) 121989. 0.698306
\(126\) 0 0
\(127\) 134314. 0.738945 0.369472 0.929242i \(-0.379538\pi\)
0.369472 + 0.929242i \(0.379538\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 0 0
\(130\) −6552.00 −0.0340029
\(131\) 365955. 1.86316 0.931578 0.363540i \(-0.118432\pi\)
0.931578 + 0.363540i \(0.118432\pi\)
\(132\) 0 0
\(133\) −51623.0 −0.253055
\(134\) −143904. −0.692326
\(135\) 0 0
\(136\) −136000. −0.630510
\(137\) −44763.0 −0.203759 −0.101880 0.994797i \(-0.532486\pi\)
−0.101880 + 0.994797i \(0.532486\pi\)
\(138\) 0 0
\(139\) −422179. −1.85336 −0.926680 0.375852i \(-0.877350\pi\)
−0.926680 + 0.375852i \(0.877350\pi\)
\(140\) 48048.0 0.207184
\(141\) 0 0
\(142\) 40720.0 0.169468
\(143\) −15990.0 −0.0653896
\(144\) 0 0
\(145\) −102186. −0.403619
\(146\) −132436. −0.514190
\(147\) 0 0
\(148\) −242048. −0.908337
\(149\) 41741.0 0.154027 0.0770136 0.997030i \(-0.475462\pi\)
0.0770136 + 0.997030i \(0.475462\pi\)
\(150\) 0 0
\(151\) −41240.0 −0.147189 −0.0735947 0.997288i \(-0.523447\pi\)
−0.0735947 + 0.997288i \(0.523447\pi\)
\(152\) −23104.0 −0.0811107
\(153\) 0 0
\(154\) 117260. 0.398426
\(155\) 23058.0 0.0770890
\(156\) 0 0
\(157\) −345954. −1.12013 −0.560066 0.828448i \(-0.689224\pi\)
−0.560066 + 0.828448i \(0.689224\pi\)
\(158\) 215552. 0.686925
\(159\) 0 0
\(160\) 21504.0 0.0664078
\(161\) 2860.00 0.00869564
\(162\) 0 0
\(163\) −298144. −0.878936 −0.439468 0.898258i \(-0.644833\pi\)
−0.439468 + 0.898258i \(0.644833\pi\)
\(164\) 150400. 0.436655
\(165\) 0 0
\(166\) 300784. 0.847197
\(167\) 73290.0 0.203354 0.101677 0.994817i \(-0.467579\pi\)
0.101677 + 0.994817i \(0.467579\pi\)
\(168\) 0 0
\(169\) −365209. −0.983614
\(170\) 178500. 0.473714
\(171\) 0 0
\(172\) 321168. 0.827773
\(173\) −282102. −0.716623 −0.358312 0.933602i \(-0.616647\pi\)
−0.358312 + 0.933602i \(0.616647\pi\)
\(174\) 0 0
\(175\) 383812. 0.947378
\(176\) 52480.0 0.127706
\(177\) 0 0
\(178\) 82472.0 0.195100
\(179\) −193946. −0.452427 −0.226213 0.974078i \(-0.572635\pi\)
−0.226213 + 0.974078i \(0.572635\pi\)
\(180\) 0 0
\(181\) −283446. −0.643093 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(182\) −44616.0 −0.0998417
\(183\) 0 0
\(184\) 1280.00 0.00278718
\(185\) 317688. 0.682451
\(186\) 0 0
\(187\) 435625. 0.910980
\(188\) −225680. −0.465692
\(189\) 0 0
\(190\) 30324.0 0.0609400
\(191\) −50495.0 −0.100153 −0.0500766 0.998745i \(-0.515947\pi\)
−0.0500766 + 0.998745i \(0.515947\pi\)
\(192\) 0 0
\(193\) 231092. 0.446572 0.223286 0.974753i \(-0.428322\pi\)
0.223286 + 0.974753i \(0.428322\pi\)
\(194\) 336520. 0.641958
\(195\) 0 0
\(196\) 58272.0 0.108348
\(197\) 452482. 0.830684 0.415342 0.909665i \(-0.363662\pi\)
0.415342 + 0.909665i \(0.363662\pi\)
\(198\) 0 0
\(199\) −207199. −0.370898 −0.185449 0.982654i \(-0.559374\pi\)
−0.185449 + 0.982654i \(0.559374\pi\)
\(200\) 171776. 0.303660
\(201\) 0 0
\(202\) 654856. 1.12919
\(203\) −695838. −1.18514
\(204\) 0 0
\(205\) −197400. −0.328067
\(206\) 556248. 0.913273
\(207\) 0 0
\(208\) −19968.0 −0.0320019
\(209\) 74005.0 0.117191
\(210\) 0 0
\(211\) 985948. 1.52457 0.762286 0.647240i \(-0.224077\pi\)
0.762286 + 0.647240i \(0.224077\pi\)
\(212\) −422176. −0.645140
\(213\) 0 0
\(214\) 498760. 0.744487
\(215\) −421533. −0.621921
\(216\) 0 0
\(217\) 157014. 0.226354
\(218\) 47344.0 0.0674720
\(219\) 0 0
\(220\) −68880.0 −0.0959481
\(221\) −165750. −0.228283
\(222\) 0 0
\(223\) 177756. 0.239366 0.119683 0.992812i \(-0.461812\pi\)
0.119683 + 0.992812i \(0.461812\pi\)
\(224\) 146432. 0.194992
\(225\) 0 0
\(226\) 230696. 0.300448
\(227\) −276382. −0.355996 −0.177998 0.984031i \(-0.556962\pi\)
−0.177998 + 0.984031i \(0.556962\pi\)
\(228\) 0 0
\(229\) 986125. 1.24263 0.621317 0.783559i \(-0.286598\pi\)
0.621317 + 0.783559i \(0.286598\pi\)
\(230\) −1680.00 −0.00209406
\(231\) 0 0
\(232\) −311424. −0.379867
\(233\) 116691. 0.140815 0.0704073 0.997518i \(-0.477570\pi\)
0.0704073 + 0.997518i \(0.477570\pi\)
\(234\) 0 0
\(235\) 296205. 0.349883
\(236\) 211456. 0.247138
\(237\) 0 0
\(238\) 1.21550e6 1.39095
\(239\) 1.25870e6 1.42537 0.712685 0.701484i \(-0.247479\pi\)
0.712685 + 0.701484i \(0.247479\pi\)
\(240\) 0 0
\(241\) −143492. −0.159142 −0.0795710 0.996829i \(-0.525355\pi\)
−0.0795710 + 0.996829i \(0.525355\pi\)
\(242\) 476104. 0.522593
\(243\) 0 0
\(244\) −36688.0 −0.0394502
\(245\) −76482.0 −0.0814037
\(246\) 0 0
\(247\) −28158.0 −0.0293670
\(248\) 70272.0 0.0725526
\(249\) 0 0
\(250\) −487956. −0.493777
\(251\) 884163. 0.885825 0.442913 0.896565i \(-0.353945\pi\)
0.442913 + 0.896565i \(0.353945\pi\)
\(252\) 0 0
\(253\) −4100.00 −0.00402701
\(254\) −537256. −0.522513
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) −1.04230e6 −0.984370 −0.492185 0.870491i \(-0.663802\pi\)
−0.492185 + 0.870491i \(0.663802\pi\)
\(258\) 0 0
\(259\) 2.16330e6 2.00386
\(260\) 26208.0 0.0240437
\(261\) 0 0
\(262\) −1.46382e6 −1.31745
\(263\) −998121. −0.889803 −0.444901 0.895580i \(-0.646761\pi\)
−0.444901 + 0.895580i \(0.646761\pi\)
\(264\) 0 0
\(265\) 554106. 0.484706
\(266\) 206492. 0.178937
\(267\) 0 0
\(268\) 575616. 0.489549
\(269\) −1.73550e6 −1.46232 −0.731162 0.682204i \(-0.761022\pi\)
−0.731162 + 0.682204i \(0.761022\pi\)
\(270\) 0 0
\(271\) −1.01674e6 −0.840982 −0.420491 0.907297i \(-0.638142\pi\)
−0.420491 + 0.907297i \(0.638142\pi\)
\(272\) 544000. 0.445838
\(273\) 0 0
\(274\) 179052. 0.144080
\(275\) −550220. −0.438737
\(276\) 0 0
\(277\) −1.37870e6 −1.07962 −0.539810 0.841787i \(-0.681504\pi\)
−0.539810 + 0.841787i \(0.681504\pi\)
\(278\) 1.68872e6 1.31052
\(279\) 0 0
\(280\) −192192. −0.146501
\(281\) 2.09789e6 1.58495 0.792476 0.609903i \(-0.208792\pi\)
0.792476 + 0.609903i \(0.208792\pi\)
\(282\) 0 0
\(283\) −215963. −0.160293 −0.0801463 0.996783i \(-0.525539\pi\)
−0.0801463 + 0.996783i \(0.525539\pi\)
\(284\) −162880. −0.119832
\(285\) 0 0
\(286\) 63960.0 0.0462374
\(287\) −1.34420e6 −0.963295
\(288\) 0 0
\(289\) 3.09577e6 2.18034
\(290\) 408744. 0.285402
\(291\) 0 0
\(292\) 529744. 0.363587
\(293\) 2.47872e6 1.68678 0.843389 0.537304i \(-0.180557\pi\)
0.843389 + 0.537304i \(0.180557\pi\)
\(294\) 0 0
\(295\) −277536. −0.185680
\(296\) 968192. 0.642292
\(297\) 0 0
\(298\) −166964. −0.108914
\(299\) 1560.00 0.00100913
\(300\) 0 0
\(301\) −2.87044e6 −1.82613
\(302\) 164960. 0.104079
\(303\) 0 0
\(304\) 92416.0 0.0573539
\(305\) 48153.0 0.0296397
\(306\) 0 0
\(307\) 495420. 0.300004 0.150002 0.988686i \(-0.452072\pi\)
0.150002 + 0.988686i \(0.452072\pi\)
\(308\) −469040. −0.281730
\(309\) 0 0
\(310\) −92232.0 −0.0545102
\(311\) −2.15964e6 −1.26614 −0.633070 0.774095i \(-0.718205\pi\)
−0.633070 + 0.774095i \(0.718205\pi\)
\(312\) 0 0
\(313\) −1.34757e6 −0.777482 −0.388741 0.921347i \(-0.627090\pi\)
−0.388741 + 0.921347i \(0.627090\pi\)
\(314\) 1.38382e6 0.792053
\(315\) 0 0
\(316\) −862208. −0.485729
\(317\) 1.01166e6 0.565440 0.282720 0.959203i \(-0.408763\pi\)
0.282720 + 0.959203i \(0.408763\pi\)
\(318\) 0 0
\(319\) 997530. 0.548844
\(320\) −86016.0 −0.0469574
\(321\) 0 0
\(322\) −11440.0 −0.00614874
\(323\) 767125. 0.409129
\(324\) 0 0
\(325\) 209352. 0.109943
\(326\) 1.19258e6 0.621501
\(327\) 0 0
\(328\) −601600. −0.308762
\(329\) 2.01702e6 1.02735
\(330\) 0 0
\(331\) −1.57062e6 −0.787954 −0.393977 0.919120i \(-0.628901\pi\)
−0.393977 + 0.919120i \(0.628901\pi\)
\(332\) −1.20314e6 −0.599059
\(333\) 0 0
\(334\) −293160. −0.143793
\(335\) −755496. −0.367807
\(336\) 0 0
\(337\) 2.16228e6 1.03714 0.518570 0.855035i \(-0.326464\pi\)
0.518570 + 0.855035i \(0.326464\pi\)
\(338\) 1.46084e6 0.695520
\(339\) 0 0
\(340\) −714000. −0.334966
\(341\) −225090. −0.104826
\(342\) 0 0
\(343\) 1.88259e6 0.864016
\(344\) −1.28467e6 −0.585324
\(345\) 0 0
\(346\) 1.12841e6 0.506729
\(347\) −4.16873e6 −1.85858 −0.929288 0.369356i \(-0.879578\pi\)
−0.929288 + 0.369356i \(0.879578\pi\)
\(348\) 0 0
\(349\) 722845. 0.317674 0.158837 0.987305i \(-0.449226\pi\)
0.158837 + 0.987305i \(0.449226\pi\)
\(350\) −1.53525e6 −0.669898
\(351\) 0 0
\(352\) −209920. −0.0903020
\(353\) −1.45102e6 −0.619780 −0.309890 0.950772i \(-0.600292\pi\)
−0.309890 + 0.950772i \(0.600292\pi\)
\(354\) 0 0
\(355\) 213780. 0.0900319
\(356\) −329888. −0.137956
\(357\) 0 0
\(358\) 775784. 0.319914
\(359\) −517179. −0.211790 −0.105895 0.994377i \(-0.533771\pi\)
−0.105895 + 0.994377i \(0.533771\pi\)
\(360\) 0 0
\(361\) 130321. 0.0526316
\(362\) 1.13378e6 0.454736
\(363\) 0 0
\(364\) 178464. 0.0705988
\(365\) −695289. −0.273170
\(366\) 0 0
\(367\) −3.59392e6 −1.39285 −0.696423 0.717631i \(-0.745226\pi\)
−0.696423 + 0.717631i \(0.745226\pi\)
\(368\) −5120.00 −0.00197084
\(369\) 0 0
\(370\) −1.27075e6 −0.482566
\(371\) 3.77320e6 1.42323
\(372\) 0 0
\(373\) −1.77578e6 −0.660870 −0.330435 0.943829i \(-0.607195\pi\)
−0.330435 + 0.943829i \(0.607195\pi\)
\(374\) −1.74250e6 −0.644160
\(375\) 0 0
\(376\) 902720. 0.329294
\(377\) −379548. −0.137535
\(378\) 0 0
\(379\) 2.48464e6 0.888516 0.444258 0.895899i \(-0.353467\pi\)
0.444258 + 0.895899i \(0.353467\pi\)
\(380\) −121296. −0.0430911
\(381\) 0 0
\(382\) 201980. 0.0708190
\(383\) −1.70987e6 −0.595614 −0.297807 0.954626i \(-0.596255\pi\)
−0.297807 + 0.954626i \(0.596255\pi\)
\(384\) 0 0
\(385\) 615615. 0.211669
\(386\) −924368. −0.315774
\(387\) 0 0
\(388\) −1.34608e6 −0.453933
\(389\) 244715. 0.0819949 0.0409974 0.999159i \(-0.486946\pi\)
0.0409974 + 0.999159i \(0.486946\pi\)
\(390\) 0 0
\(391\) −42500.0 −0.0140588
\(392\) −233088. −0.0766134
\(393\) 0 0
\(394\) −1.80993e6 −0.587382
\(395\) 1.13165e6 0.364938
\(396\) 0 0
\(397\) −4.11195e6 −1.30940 −0.654698 0.755890i \(-0.727204\pi\)
−0.654698 + 0.755890i \(0.727204\pi\)
\(398\) 828796. 0.262265
\(399\) 0 0
\(400\) −687104. −0.214720
\(401\) 1.87955e6 0.583704 0.291852 0.956464i \(-0.405729\pi\)
0.291852 + 0.956464i \(0.405729\pi\)
\(402\) 0 0
\(403\) 85644.0 0.0262684
\(404\) −2.61942e6 −0.798459
\(405\) 0 0
\(406\) 2.78335e6 0.838017
\(407\) −3.10124e6 −0.928003
\(408\) 0 0
\(409\) −2.68258e6 −0.792948 −0.396474 0.918046i \(-0.629766\pi\)
−0.396474 + 0.918046i \(0.629766\pi\)
\(410\) 789600. 0.231978
\(411\) 0 0
\(412\) −2.22499e6 −0.645781
\(413\) −1.88989e6 −0.545206
\(414\) 0 0
\(415\) 1.57912e6 0.450084
\(416\) 79872.0 0.0226288
\(417\) 0 0
\(418\) −296020. −0.0828668
\(419\) 5.70599e6 1.58780 0.793900 0.608048i \(-0.208047\pi\)
0.793900 + 0.608048i \(0.208047\pi\)
\(420\) 0 0
\(421\) −4.24578e6 −1.16749 −0.583744 0.811938i \(-0.698413\pi\)
−0.583744 + 0.811938i \(0.698413\pi\)
\(422\) −3.94379e6 −1.07804
\(423\) 0 0
\(424\) 1.68870e6 0.456183
\(425\) −5.70350e6 −1.53168
\(426\) 0 0
\(427\) 327899. 0.0870303
\(428\) −1.99504e6 −0.526432
\(429\) 0 0
\(430\) 1.68613e6 0.439765
\(431\) 1.28341e6 0.332793 0.166396 0.986059i \(-0.446787\pi\)
0.166396 + 0.986059i \(0.446787\pi\)
\(432\) 0 0
\(433\) −6.39182e6 −1.63834 −0.819171 0.573549i \(-0.805566\pi\)
−0.819171 + 0.573549i \(0.805566\pi\)
\(434\) −628056. −0.160057
\(435\) 0 0
\(436\) −189376. −0.0477099
\(437\) −7220.00 −0.00180856
\(438\) 0 0
\(439\) 1.27464e6 0.315664 0.157832 0.987466i \(-0.449549\pi\)
0.157832 + 0.987466i \(0.449549\pi\)
\(440\) 275520. 0.0678456
\(441\) 0 0
\(442\) 663000. 0.161420
\(443\) −2.35546e6 −0.570253 −0.285126 0.958490i \(-0.592036\pi\)
−0.285126 + 0.958490i \(0.592036\pi\)
\(444\) 0 0
\(445\) 432978. 0.103649
\(446\) −711024. −0.169257
\(447\) 0 0
\(448\) −585728. −0.137880
\(449\) −7.99412e6 −1.87135 −0.935675 0.352863i \(-0.885208\pi\)
−0.935675 + 0.352863i \(0.885208\pi\)
\(450\) 0 0
\(451\) 1.92700e6 0.446108
\(452\) −922784. −0.212449
\(453\) 0 0
\(454\) 1.10553e6 0.251727
\(455\) −234234. −0.0530422
\(456\) 0 0
\(457\) −3.03844e6 −0.680550 −0.340275 0.940326i \(-0.610520\pi\)
−0.340275 + 0.940326i \(0.610520\pi\)
\(458\) −3.94450e6 −0.878675
\(459\) 0 0
\(460\) 6720.00 0.00148073
\(461\) 5.98744e6 1.31217 0.656084 0.754688i \(-0.272212\pi\)
0.656084 + 0.754688i \(0.272212\pi\)
\(462\) 0 0
\(463\) 8.05385e6 1.74603 0.873014 0.487695i \(-0.162162\pi\)
0.873014 + 0.487695i \(0.162162\pi\)
\(464\) 1.24570e6 0.268607
\(465\) 0 0
\(466\) −466764. −0.0995709
\(467\) 8.29332e6 1.75969 0.879845 0.475261i \(-0.157646\pi\)
0.879845 + 0.475261i \(0.157646\pi\)
\(468\) 0 0
\(469\) −5.14457e6 −1.07998
\(470\) −1.18482e6 −0.247405
\(471\) 0 0
\(472\) −845824. −0.174753
\(473\) 4.11496e6 0.845694
\(474\) 0 0
\(475\) −968924. −0.197041
\(476\) −4.86200e6 −0.983553
\(477\) 0 0
\(478\) −5.03480e6 −1.00789
\(479\) 4.58472e6 0.913007 0.456503 0.889722i \(-0.349102\pi\)
0.456503 + 0.889722i \(0.349102\pi\)
\(480\) 0 0
\(481\) 1.17998e6 0.232548
\(482\) 573968. 0.112530
\(483\) 0 0
\(484\) −1.90442e6 −0.369529
\(485\) 1.76673e6 0.341048
\(486\) 0 0
\(487\) 55304.0 0.0105666 0.00528329 0.999986i \(-0.498318\pi\)
0.00528329 + 0.999986i \(0.498318\pi\)
\(488\) 146752. 0.0278955
\(489\) 0 0
\(490\) 305928. 0.0575611
\(491\) 6.88932e6 1.28965 0.644826 0.764329i \(-0.276930\pi\)
0.644826 + 0.764329i \(0.276930\pi\)
\(492\) 0 0
\(493\) 1.03402e7 1.91608
\(494\) 112632. 0.0207656
\(495\) 0 0
\(496\) −281088. −0.0513025
\(497\) 1.45574e6 0.264358
\(498\) 0 0
\(499\) 5.61676e6 1.00980 0.504899 0.863179i \(-0.331530\pi\)
0.504899 + 0.863179i \(0.331530\pi\)
\(500\) 1.95182e6 0.349153
\(501\) 0 0
\(502\) −3.53665e6 −0.626373
\(503\) 5.16131e6 0.909578 0.454789 0.890599i \(-0.349715\pi\)
0.454789 + 0.890599i \(0.349715\pi\)
\(504\) 0 0
\(505\) 3.43799e6 0.599897
\(506\) 16400.0 0.00284753
\(507\) 0 0
\(508\) 2.14902e6 0.369472
\(509\) 1.81839e6 0.311094 0.155547 0.987828i \(-0.450286\pi\)
0.155547 + 0.987828i \(0.450286\pi\)
\(510\) 0 0
\(511\) −4.73459e6 −0.802102
\(512\) −262144. −0.0441942
\(513\) 0 0
\(514\) 4.16918e6 0.696055
\(515\) 2.92030e6 0.485188
\(516\) 0 0
\(517\) −2.89152e6 −0.475774
\(518\) −8.65322e6 −1.41695
\(519\) 0 0
\(520\) −104832. −0.0170014
\(521\) −1.10957e7 −1.79085 −0.895424 0.445215i \(-0.853127\pi\)
−0.895424 + 0.445215i \(0.853127\pi\)
\(522\) 0 0
\(523\) −7.51297e6 −1.20104 −0.600520 0.799610i \(-0.705040\pi\)
−0.600520 + 0.799610i \(0.705040\pi\)
\(524\) 5.85528e6 0.931578
\(525\) 0 0
\(526\) 3.99248e6 0.629186
\(527\) −2.33325e6 −0.365961
\(528\) 0 0
\(529\) −6.43594e6 −0.999938
\(530\) −2.21642e6 −0.342739
\(531\) 0 0
\(532\) −825968. −0.126527
\(533\) −733200. −0.111790
\(534\) 0 0
\(535\) 2.61849e6 0.395518
\(536\) −2.30246e6 −0.346163
\(537\) 0 0
\(538\) 6.94199e6 1.03402
\(539\) 746610. 0.110693
\(540\) 0 0
\(541\) 3.15622e6 0.463632 0.231816 0.972760i \(-0.425533\pi\)
0.231816 + 0.972760i \(0.425533\pi\)
\(542\) 4.06696e6 0.594664
\(543\) 0 0
\(544\) −2.17600e6 −0.315255
\(545\) 248556. 0.0358454
\(546\) 0 0
\(547\) −1.17063e7 −1.67283 −0.836417 0.548094i \(-0.815353\pi\)
−0.836417 + 0.548094i \(0.815353\pi\)
\(548\) −716208. −0.101880
\(549\) 0 0
\(550\) 2.20088e6 0.310234
\(551\) 1.75663e6 0.246491
\(552\) 0 0
\(553\) 7.70598e6 1.07156
\(554\) 5.51481e6 0.763407
\(555\) 0 0
\(556\) −6.75486e6 −0.926680
\(557\) −5.56111e6 −0.759493 −0.379746 0.925091i \(-0.623989\pi\)
−0.379746 + 0.925091i \(0.623989\pi\)
\(558\) 0 0
\(559\) −1.56569e6 −0.211923
\(560\) 768768. 0.103592
\(561\) 0 0
\(562\) −8.39154e6 −1.12073
\(563\) 1.17669e7 1.56456 0.782280 0.622928i \(-0.214057\pi\)
0.782280 + 0.622928i \(0.214057\pi\)
\(564\) 0 0
\(565\) 1.21115e6 0.159617
\(566\) 863852. 0.113344
\(567\) 0 0
\(568\) 651520. 0.0847338
\(569\) −6.49649e6 −0.841198 −0.420599 0.907247i \(-0.638180\pi\)
−0.420599 + 0.907247i \(0.638180\pi\)
\(570\) 0 0
\(571\) −7.15432e6 −0.918286 −0.459143 0.888362i \(-0.651843\pi\)
−0.459143 + 0.888362i \(0.651843\pi\)
\(572\) −255840. −0.0326948
\(573\) 0 0
\(574\) 5.37680e6 0.681152
\(575\) 53680.0 0.00677085
\(576\) 0 0
\(577\) −8.50781e6 −1.06385 −0.531923 0.846793i \(-0.678530\pi\)
−0.531923 + 0.846793i \(0.678530\pi\)
\(578\) −1.23831e7 −1.54173
\(579\) 0 0
\(580\) −1.63498e6 −0.201809
\(581\) 1.07530e7 1.32157
\(582\) 0 0
\(583\) −5.40913e6 −0.659107
\(584\) −2.11898e6 −0.257095
\(585\) 0 0
\(586\) −9.91486e6 −1.19273
\(587\) 854707. 0.102382 0.0511908 0.998689i \(-0.483698\pi\)
0.0511908 + 0.998689i \(0.483698\pi\)
\(588\) 0 0
\(589\) −396378. −0.0470784
\(590\) 1.11014e6 0.131295
\(591\) 0 0
\(592\) −3.87277e6 −0.454169
\(593\) 5.55213e6 0.648370 0.324185 0.945994i \(-0.394910\pi\)
0.324185 + 0.945994i \(0.394910\pi\)
\(594\) 0 0
\(595\) 6.38138e6 0.738962
\(596\) 667856. 0.0770136
\(597\) 0 0
\(598\) −6240.00 −0.000713562 0
\(599\) 1.30669e6 0.148801 0.0744003 0.997228i \(-0.476296\pi\)
0.0744003 + 0.997228i \(0.476296\pi\)
\(600\) 0 0
\(601\) −7.94858e6 −0.897643 −0.448821 0.893621i \(-0.648156\pi\)
−0.448821 + 0.893621i \(0.648156\pi\)
\(602\) 1.14818e7 1.29127
\(603\) 0 0
\(604\) −659840. −0.0735947
\(605\) 2.49955e6 0.277634
\(606\) 0 0
\(607\) 2.83746e6 0.312578 0.156289 0.987711i \(-0.450047\pi\)
0.156289 + 0.987711i \(0.450047\pi\)
\(608\) −369664. −0.0405554
\(609\) 0 0
\(610\) −192612. −0.0209584
\(611\) 1.10019e6 0.119224
\(612\) 0 0
\(613\) 1.49674e7 1.60877 0.804387 0.594106i \(-0.202494\pi\)
0.804387 + 0.594106i \(0.202494\pi\)
\(614\) −1.98168e6 −0.212135
\(615\) 0 0
\(616\) 1.87616e6 0.199213
\(617\) 3.59751e6 0.380443 0.190221 0.981741i \(-0.439079\pi\)
0.190221 + 0.981741i \(0.439079\pi\)
\(618\) 0 0
\(619\) 6.74758e6 0.707818 0.353909 0.935280i \(-0.384852\pi\)
0.353909 + 0.935280i \(0.384852\pi\)
\(620\) 368928. 0.0385445
\(621\) 0 0
\(622\) 8.63858e6 0.895295
\(623\) 2.94837e6 0.304342
\(624\) 0 0
\(625\) 5.82573e6 0.596555
\(626\) 5.39028e6 0.549763
\(627\) 0 0
\(628\) −5.53526e6 −0.560066
\(629\) −3.21470e7 −3.23977
\(630\) 0 0
\(631\) 1.36044e7 1.36021 0.680103 0.733117i \(-0.261935\pi\)
0.680103 + 0.733117i \(0.261935\pi\)
\(632\) 3.44883e6 0.343463
\(633\) 0 0
\(634\) −4.04664e6 −0.399826
\(635\) −2.82059e6 −0.277592
\(636\) 0 0
\(637\) −284076. −0.0277387
\(638\) −3.99012e6 −0.388092
\(639\) 0 0
\(640\) 344064. 0.0332039
\(641\) −1.91221e7 −1.83819 −0.919097 0.394030i \(-0.871080\pi\)
−0.919097 + 0.394030i \(0.871080\pi\)
\(642\) 0 0
\(643\) 8.38738e6 0.800016 0.400008 0.916512i \(-0.369007\pi\)
0.400008 + 0.916512i \(0.369007\pi\)
\(644\) 45760.0 0.00434782
\(645\) 0 0
\(646\) −3.06850e6 −0.289298
\(647\) −7.90221e6 −0.742143 −0.371072 0.928604i \(-0.621010\pi\)
−0.371072 + 0.928604i \(0.621010\pi\)
\(648\) 0 0
\(649\) 2.70928e6 0.252489
\(650\) −837408. −0.0777416
\(651\) 0 0
\(652\) −4.77030e6 −0.439468
\(653\) 279213. 0.0256243 0.0128122 0.999918i \(-0.495922\pi\)
0.0128122 + 0.999918i \(0.495922\pi\)
\(654\) 0 0
\(655\) −7.68506e6 −0.699912
\(656\) 2.40640e6 0.218327
\(657\) 0 0
\(658\) −8.06806e6 −0.726448
\(659\) 6.82671e6 0.612348 0.306174 0.951976i \(-0.400951\pi\)
0.306174 + 0.951976i \(0.400951\pi\)
\(660\) 0 0
\(661\) 1.85059e7 1.64742 0.823712 0.567008i \(-0.191899\pi\)
0.823712 + 0.567008i \(0.191899\pi\)
\(662\) 6.28248e6 0.557168
\(663\) 0 0
\(664\) 4.81254e6 0.423599
\(665\) 1.08408e6 0.0950623
\(666\) 0 0
\(667\) −97320.0 −0.00847008
\(668\) 1.17264e6 0.101677
\(669\) 0 0
\(670\) 3.02198e6 0.260079
\(671\) −470065. −0.0403043
\(672\) 0 0
\(673\) 1.25059e7 1.06433 0.532166 0.846640i \(-0.321378\pi\)
0.532166 + 0.846640i \(0.321378\pi\)
\(674\) −8.64913e6 −0.733369
\(675\) 0 0
\(676\) −5.84334e6 −0.491807
\(677\) −5.47029e6 −0.458710 −0.229355 0.973343i \(-0.573662\pi\)
−0.229355 + 0.973343i \(0.573662\pi\)
\(678\) 0 0
\(679\) 1.20306e7 1.00141
\(680\) 2.85600e6 0.236857
\(681\) 0 0
\(682\) 900360. 0.0741234
\(683\) −1.02415e7 −0.840061 −0.420031 0.907510i \(-0.637981\pi\)
−0.420031 + 0.907510i \(0.637981\pi\)
\(684\) 0 0
\(685\) 940023. 0.0765442
\(686\) −7.53038e6 −0.610951
\(687\) 0 0
\(688\) 5.13869e6 0.413886
\(689\) 2.05811e6 0.165166
\(690\) 0 0
\(691\) 1.84945e6 0.147349 0.0736744 0.997282i \(-0.476527\pi\)
0.0736744 + 0.997282i \(0.476527\pi\)
\(692\) −4.51363e6 −0.358312
\(693\) 0 0
\(694\) 1.66749e7 1.31421
\(695\) 8.86576e6 0.696232
\(696\) 0 0
\(697\) 1.99750e7 1.55742
\(698\) −2.89138e6 −0.224629
\(699\) 0 0
\(700\) 6.14099e6 0.473689
\(701\) −1.65813e7 −1.27445 −0.637226 0.770677i \(-0.719918\pi\)
−0.637226 + 0.770677i \(0.719918\pi\)
\(702\) 0 0
\(703\) −5.46121e6 −0.416774
\(704\) 839680. 0.0638531
\(705\) 0 0
\(706\) 5.80409e6 0.438250
\(707\) 2.34111e7 1.76146
\(708\) 0 0
\(709\) 1.10501e7 0.825562 0.412781 0.910830i \(-0.364558\pi\)
0.412781 + 0.910830i \(0.364558\pi\)
\(710\) −855120. −0.0636621
\(711\) 0 0
\(712\) 1.31955e6 0.0975498
\(713\) 21960.0 0.00161774
\(714\) 0 0
\(715\) 335790. 0.0245642
\(716\) −3.10314e6 −0.226213
\(717\) 0 0
\(718\) 2.06872e6 0.149758
\(719\) −1.02255e7 −0.737669 −0.368835 0.929495i \(-0.620243\pi\)
−0.368835 + 0.929495i \(0.620243\pi\)
\(720\) 0 0
\(721\) 1.98859e7 1.42464
\(722\) −521284. −0.0372161
\(723\) 0 0
\(724\) −4.53514e6 −0.321547
\(725\) −1.30603e7 −0.922804
\(726\) 0 0
\(727\) 2.61985e7 1.83840 0.919202 0.393787i \(-0.128835\pi\)
0.919202 + 0.393787i \(0.128835\pi\)
\(728\) −713856. −0.0499209
\(729\) 0 0
\(730\) 2.78116e6 0.193160
\(731\) 4.26551e7 2.95242
\(732\) 0 0
\(733\) −2.56029e7 −1.76007 −0.880033 0.474913i \(-0.842480\pi\)
−0.880033 + 0.474913i \(0.842480\pi\)
\(734\) 1.43757e7 0.984891
\(735\) 0 0
\(736\) 20480.0 0.00139359
\(737\) 7.37508e6 0.500147
\(738\) 0 0
\(739\) −2.10847e7 −1.42022 −0.710112 0.704089i \(-0.751356\pi\)
−0.710112 + 0.704089i \(0.751356\pi\)
\(740\) 5.08301e6 0.341226
\(741\) 0 0
\(742\) −1.50928e7 −1.00637
\(743\) −8.71807e6 −0.579360 −0.289680 0.957124i \(-0.593549\pi\)
−0.289680 + 0.957124i \(0.593549\pi\)
\(744\) 0 0
\(745\) −876561. −0.0578617
\(746\) 7.10310e6 0.467306
\(747\) 0 0
\(748\) 6.97000e6 0.455490
\(749\) 1.78307e7 1.16135
\(750\) 0 0
\(751\) −1.26501e7 −0.818452 −0.409226 0.912433i \(-0.634201\pi\)
−0.409226 + 0.912433i \(0.634201\pi\)
\(752\) −3.61088e6 −0.232846
\(753\) 0 0
\(754\) 1.51819e6 0.0972520
\(755\) 866040. 0.0552931
\(756\) 0 0
\(757\) −5.69653e6 −0.361302 −0.180651 0.983547i \(-0.557821\pi\)
−0.180651 + 0.983547i \(0.557821\pi\)
\(758\) −9.93855e6 −0.628275
\(759\) 0 0
\(760\) 485184. 0.0304700
\(761\) 2.51658e7 1.57525 0.787625 0.616155i \(-0.211310\pi\)
0.787625 + 0.616155i \(0.211310\pi\)
\(762\) 0 0
\(763\) 1.69255e6 0.105252
\(764\) −807920. −0.0500766
\(765\) 0 0
\(766\) 6.83946e6 0.421163
\(767\) −1.03085e6 −0.0632712
\(768\) 0 0
\(769\) −2.35673e7 −1.43713 −0.718563 0.695462i \(-0.755200\pi\)
−0.718563 + 0.695462i \(0.755200\pi\)
\(770\) −2.46246e6 −0.149673
\(771\) 0 0
\(772\) 3.69747e6 0.223286
\(773\) −1.61071e7 −0.969548 −0.484774 0.874639i \(-0.661098\pi\)
−0.484774 + 0.874639i \(0.661098\pi\)
\(774\) 0 0
\(775\) 2.94703e6 0.176251
\(776\) 5.38432e6 0.320979
\(777\) 0 0
\(778\) −978860. −0.0579791
\(779\) 3.39340e6 0.200351
\(780\) 0 0
\(781\) −2.08690e6 −0.122426
\(782\) 170000. 0.00994104
\(783\) 0 0
\(784\) 932352. 0.0541739
\(785\) 7.26503e6 0.420788
\(786\) 0 0
\(787\) −8.09622e6 −0.465956 −0.232978 0.972482i \(-0.574847\pi\)
−0.232978 + 0.972482i \(0.574847\pi\)
\(788\) 7.23971e6 0.415342
\(789\) 0 0
\(790\) −4.52659e6 −0.258050
\(791\) 8.24738e6 0.468678
\(792\) 0 0
\(793\) 178854. 0.0100999
\(794\) 1.64478e7 0.925883
\(795\) 0 0
\(796\) −3.31518e6 −0.185449
\(797\) −2.23729e7 −1.24760 −0.623800 0.781584i \(-0.714412\pi\)
−0.623800 + 0.781584i \(0.714412\pi\)
\(798\) 0 0
\(799\) −2.99731e7 −1.66098
\(800\) 2.74842e6 0.151830
\(801\) 0 0
\(802\) −7.51819e6 −0.412741
\(803\) 6.78735e6 0.371459
\(804\) 0 0
\(805\) −60060.0 −0.00326660
\(806\) −342576. −0.0185746
\(807\) 0 0
\(808\) 1.04777e7 0.564595
\(809\) −915669. −0.0491889 −0.0245945 0.999698i \(-0.507829\pi\)
−0.0245945 + 0.999698i \(0.507829\pi\)
\(810\) 0 0
\(811\) −3.00496e7 −1.60430 −0.802151 0.597121i \(-0.796311\pi\)
−0.802151 + 0.597121i \(0.796311\pi\)
\(812\) −1.11334e7 −0.592568
\(813\) 0 0
\(814\) 1.24050e7 0.656197
\(815\) 6.26102e6 0.330180
\(816\) 0 0
\(817\) 7.24635e6 0.379808
\(818\) 1.07303e7 0.560699
\(819\) 0 0
\(820\) −3.15840e6 −0.164033
\(821\) −3.25959e6 −0.168774 −0.0843868 0.996433i \(-0.526893\pi\)
−0.0843868 + 0.996433i \(0.526893\pi\)
\(822\) 0 0
\(823\) 2.63514e7 1.35614 0.678068 0.734999i \(-0.262817\pi\)
0.678068 + 0.734999i \(0.262817\pi\)
\(824\) 8.89997e6 0.456636
\(825\) 0 0
\(826\) 7.55955e6 0.385519
\(827\) −3.88895e6 −0.197728 −0.0988640 0.995101i \(-0.531521\pi\)
−0.0988640 + 0.995101i \(0.531521\pi\)
\(828\) 0 0
\(829\) −2.49909e7 −1.26298 −0.631488 0.775385i \(-0.717556\pi\)
−0.631488 + 0.775385i \(0.717556\pi\)
\(830\) −6.31646e6 −0.318258
\(831\) 0 0
\(832\) −319488. −0.0160010
\(833\) 7.73925e6 0.386444
\(834\) 0 0
\(835\) −1.53909e6 −0.0763920
\(836\) 1.18408e6 0.0585956
\(837\) 0 0
\(838\) −2.28240e7 −1.12274
\(839\) 2.77733e6 0.136214 0.0681072 0.997678i \(-0.478304\pi\)
0.0681072 + 0.997678i \(0.478304\pi\)
\(840\) 0 0
\(841\) 3.16681e6 0.154394
\(842\) 1.69831e7 0.825539
\(843\) 0 0
\(844\) 1.57752e7 0.762286
\(845\) 7.66939e6 0.369504
\(846\) 0 0
\(847\) 1.70207e7 0.815210
\(848\) −6.75482e6 −0.322570
\(849\) 0 0
\(850\) 2.28140e7 1.08306
\(851\) 302560. 0.0143215
\(852\) 0 0
\(853\) 1.63379e6 0.0768820 0.0384410 0.999261i \(-0.487761\pi\)
0.0384410 + 0.999261i \(0.487761\pi\)
\(854\) −1.31160e6 −0.0615397
\(855\) 0 0
\(856\) 7.98016e6 0.372244
\(857\) −2.45037e7 −1.13967 −0.569836 0.821758i \(-0.692993\pi\)
−0.569836 + 0.821758i \(0.692993\pi\)
\(858\) 0 0
\(859\) 1.45576e7 0.673143 0.336572 0.941658i \(-0.390733\pi\)
0.336572 + 0.941658i \(0.390733\pi\)
\(860\) −6.74453e6 −0.310961
\(861\) 0 0
\(862\) −5.13366e6 −0.235320
\(863\) −1.76706e7 −0.807651 −0.403826 0.914836i \(-0.632320\pi\)
−0.403826 + 0.914836i \(0.632320\pi\)
\(864\) 0 0
\(865\) 5.92414e6 0.269206
\(866\) 2.55673e7 1.15848
\(867\) 0 0
\(868\) 2.51222e6 0.113177
\(869\) −1.10470e7 −0.496245
\(870\) 0 0
\(871\) −2.80613e6 −0.125332
\(872\) 757504. 0.0337360
\(873\) 0 0
\(874\) 28880.0 0.00127885
\(875\) −1.74444e7 −0.770259
\(876\) 0 0
\(877\) −4.46464e7 −1.96014 −0.980070 0.198650i \(-0.936344\pi\)
−0.980070 + 0.198650i \(0.936344\pi\)
\(878\) −5.09855e6 −0.223208
\(879\) 0 0
\(880\) −1.10208e6 −0.0479741
\(881\) 1.97316e7 0.856490 0.428245 0.903663i \(-0.359132\pi\)
0.428245 + 0.903663i \(0.359132\pi\)
\(882\) 0 0
\(883\) −2.12686e7 −0.917989 −0.458994 0.888439i \(-0.651790\pi\)
−0.458994 + 0.888439i \(0.651790\pi\)
\(884\) −2.65200e6 −0.114141
\(885\) 0 0
\(886\) 9.42186e6 0.403230
\(887\) 3.40623e7 1.45367 0.726835 0.686813i \(-0.240991\pi\)
0.726835 + 0.686813i \(0.240991\pi\)
\(888\) 0 0
\(889\) −1.92069e7 −0.815085
\(890\) −1.73191e6 −0.0732910
\(891\) 0 0
\(892\) 2.84410e6 0.119683
\(893\) −5.09190e6 −0.213674
\(894\) 0 0
\(895\) 4.07287e6 0.169958
\(896\) 2.34291e6 0.0974958
\(897\) 0 0
\(898\) 3.19765e7 1.32324
\(899\) −5.34287e6 −0.220483
\(900\) 0 0
\(901\) −5.60702e7 −2.30102
\(902\) −7.70800e6 −0.315446
\(903\) 0 0
\(904\) 3.69114e6 0.150224
\(905\) 5.95237e6 0.241584
\(906\) 0 0
\(907\) −1.59880e7 −0.645323 −0.322662 0.946514i \(-0.604578\pi\)
−0.322662 + 0.946514i \(0.604578\pi\)
\(908\) −4.42211e6 −0.177998
\(909\) 0 0
\(910\) 936936. 0.0375065
\(911\) −2.70045e7 −1.07805 −0.539026 0.842289i \(-0.681208\pi\)
−0.539026 + 0.842289i \(0.681208\pi\)
\(912\) 0 0
\(913\) −1.54152e7 −0.612029
\(914\) 1.21537e7 0.481221
\(915\) 0 0
\(916\) 1.57780e7 0.621317
\(917\) −5.23316e7 −2.05514
\(918\) 0 0
\(919\) −6.71840e6 −0.262408 −0.131204 0.991355i \(-0.541884\pi\)
−0.131204 + 0.991355i \(0.541884\pi\)
\(920\) −26880.0 −0.00104703
\(921\) 0 0
\(922\) −2.39498e7 −0.927842
\(923\) 794040. 0.0306788
\(924\) 0 0
\(925\) 4.06036e7 1.56031
\(926\) −3.22154e7 −1.23463
\(927\) 0 0
\(928\) −4.98278e6 −0.189934
\(929\) 1.29929e7 0.493933 0.246967 0.969024i \(-0.420566\pi\)
0.246967 + 0.969024i \(0.420566\pi\)
\(930\) 0 0
\(931\) 1.31476e6 0.0497133
\(932\) 1.86706e6 0.0704073
\(933\) 0 0
\(934\) −3.31733e7 −1.24429
\(935\) −9.14812e6 −0.342218
\(936\) 0 0
\(937\) 2.87830e6 0.107099 0.0535497 0.998565i \(-0.482946\pi\)
0.0535497 + 0.998565i \(0.482946\pi\)
\(938\) 2.05783e7 0.763663
\(939\) 0 0
\(940\) 4.73928e6 0.174941
\(941\) 1.11189e7 0.409345 0.204672 0.978831i \(-0.434387\pi\)
0.204672 + 0.978831i \(0.434387\pi\)
\(942\) 0 0
\(943\) −188000. −0.00688460
\(944\) 3.38330e6 0.123569
\(945\) 0 0
\(946\) −1.64599e7 −0.597996
\(947\) 1.59967e7 0.579638 0.289819 0.957081i \(-0.406405\pi\)
0.289819 + 0.957081i \(0.406405\pi\)
\(948\) 0 0
\(949\) −2.58250e6 −0.0930840
\(950\) 3.87570e6 0.139329
\(951\) 0 0
\(952\) 1.94480e7 0.695477
\(953\) −3.80244e7 −1.35622 −0.678110 0.734960i \(-0.737201\pi\)
−0.678110 + 0.734960i \(0.737201\pi\)
\(954\) 0 0
\(955\) 1.06040e6 0.0376235
\(956\) 2.01392e7 0.712685
\(957\) 0 0
\(958\) −1.83389e7 −0.645593
\(959\) 6.40111e6 0.224755
\(960\) 0 0
\(961\) −2.74235e7 −0.957889
\(962\) −4.71994e6 −0.164437
\(963\) 0 0
\(964\) −2.29587e6 −0.0795710
\(965\) −4.85293e6 −0.167759
\(966\) 0 0
\(967\) 2.60950e6 0.0897409 0.0448705 0.998993i \(-0.485712\pi\)
0.0448705 + 0.998993i \(0.485712\pi\)
\(968\) 7.61766e6 0.261296
\(969\) 0 0
\(970\) −7.06692e6 −0.241158
\(971\) −1.40618e7 −0.478623 −0.239312 0.970943i \(-0.576922\pi\)
−0.239312 + 0.970943i \(0.576922\pi\)
\(972\) 0 0
\(973\) 6.03716e7 2.04433
\(974\) −221216. −0.00747170
\(975\) 0 0
\(976\) −587008. −0.0197251
\(977\) −4.66346e7 −1.56305 −0.781523 0.623876i \(-0.785557\pi\)
−0.781523 + 0.623876i \(0.785557\pi\)
\(978\) 0 0
\(979\) −4.22669e6 −0.140943
\(980\) −1.22371e6 −0.0407018
\(981\) 0 0
\(982\) −2.75573e7 −0.911922
\(983\) 2.96980e6 0.0980263 0.0490132 0.998798i \(-0.484392\pi\)
0.0490132 + 0.998798i \(0.484392\pi\)
\(984\) 0 0
\(985\) −9.50212e6 −0.312054
\(986\) −4.13610e7 −1.35487
\(987\) 0 0
\(988\) −450528. −0.0146835
\(989\) −401460. −0.0130512
\(990\) 0 0
\(991\) 3.88690e7 1.25724 0.628621 0.777712i \(-0.283620\pi\)
0.628621 + 0.777712i \(0.283620\pi\)
\(992\) 1.12435e6 0.0362763
\(993\) 0 0
\(994\) −5.82296e6 −0.186930
\(995\) 4.35118e6 0.139331
\(996\) 0 0
\(997\) −4.92115e7 −1.56794 −0.783969 0.620801i \(-0.786808\pi\)
−0.783969 + 0.620801i \(0.786808\pi\)
\(998\) −2.24670e7 −0.714034
\(999\) 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 342.6.a.a.1.1 1
3.2 odd 2 114.6.a.c.1.1 1
12.11 even 2 912.6.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.6.a.c.1.1 1 3.2 odd 2
342.6.a.a.1.1 1 1.1 even 1 trivial
912.6.a.f.1.1 1 12.11 even 2