Properties

Label 450.6.a.f.1.1
Level $450$
Weight $6$
Character 450.1
Self dual yes
Analytic conductor $72.173$
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] = [450,6,Mod(1,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, 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("450.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.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: \(72.1727189158\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +16.0000 q^{4} -4.00000 q^{7} -64.0000 q^{8} +O(q^{10})\) \(q-4.00000 q^{2} +16.0000 q^{4} -4.00000 q^{7} -64.0000 q^{8} +500.000 q^{11} -288.000 q^{13} +16.0000 q^{14} +256.000 q^{16} -1516.00 q^{17} -1344.00 q^{19} -2000.00 q^{22} +4100.00 q^{23} +1152.00 q^{26} -64.0000 q^{28} +2646.00 q^{29} -5612.00 q^{31} -1024.00 q^{32} +6064.00 q^{34} -7288.00 q^{37} +5376.00 q^{38} +18986.0 q^{41} -2404.00 q^{43} +8000.00 q^{44} -16400.0 q^{46} -8900.00 q^{47} -16791.0 q^{49} -4608.00 q^{52} -39804.0 q^{53} +256.000 q^{56} -10584.0 q^{58} +28300.0 q^{59} +18290.0 q^{61} +22448.0 q^{62} +4096.00 q^{64} +65956.0 q^{67} -24256.0 q^{68} +28800.0 q^{71} -30808.0 q^{73} +29152.0 q^{74} -21504.0 q^{76} -2000.00 q^{77} +60228.0 q^{79} -75944.0 q^{82} +2468.00 q^{83} +9616.00 q^{86} -32000.0 q^{88} -22678.0 q^{89} +1152.00 q^{91} +65600.0 q^{92} +35600.0 q^{94} -36968.0 q^{97} +67164.0 q^{98} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) −4.00000 −0.0308542 −0.0154271 0.999881i \(-0.504911\pi\)
−0.0154271 + 0.999881i \(0.504911\pi\)
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 500.000 1.24591 0.622957 0.782256i \(-0.285931\pi\)
0.622957 + 0.782256i \(0.285931\pi\)
\(12\) 0 0
\(13\) −288.000 −0.472644 −0.236322 0.971675i \(-0.575942\pi\)
−0.236322 + 0.971675i \(0.575942\pi\)
\(14\) 16.0000 0.0218172
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −1516.00 −1.27226 −0.636132 0.771581i \(-0.719466\pi\)
−0.636132 + 0.771581i \(0.719466\pi\)
\(18\) 0 0
\(19\) −1344.00 −0.854113 −0.427056 0.904225i \(-0.640449\pi\)
−0.427056 + 0.904225i \(0.640449\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2000.00 −0.880995
\(23\) 4100.00 1.61609 0.808043 0.589124i \(-0.200527\pi\)
0.808043 + 0.589124i \(0.200527\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1152.00 0.334210
\(27\) 0 0
\(28\) −64.0000 −0.0154271
\(29\) 2646.00 0.584245 0.292122 0.956381i \(-0.405639\pi\)
0.292122 + 0.956381i \(0.405639\pi\)
\(30\) 0 0
\(31\) −5612.00 −1.04885 −0.524425 0.851457i \(-0.675720\pi\)
−0.524425 + 0.851457i \(0.675720\pi\)
\(32\) −1024.00 −0.176777
\(33\) 0 0
\(34\) 6064.00 0.899626
\(35\) 0 0
\(36\) 0 0
\(37\) −7288.00 −0.875193 −0.437597 0.899171i \(-0.644170\pi\)
−0.437597 + 0.899171i \(0.644170\pi\)
\(38\) 5376.00 0.603949
\(39\) 0 0
\(40\) 0 0
\(41\) 18986.0 1.76390 0.881950 0.471343i \(-0.156231\pi\)
0.881950 + 0.471343i \(0.156231\pi\)
\(42\) 0 0
\(43\) −2404.00 −0.198273 −0.0991364 0.995074i \(-0.531608\pi\)
−0.0991364 + 0.995074i \(0.531608\pi\)
\(44\) 8000.00 0.622957
\(45\) 0 0
\(46\) −16400.0 −1.14274
\(47\) −8900.00 −0.587686 −0.293843 0.955854i \(-0.594934\pi\)
−0.293843 + 0.955854i \(0.594934\pi\)
\(48\) 0 0
\(49\) −16791.0 −0.999048
\(50\) 0 0
\(51\) 0 0
\(52\) −4608.00 −0.236322
\(53\) −39804.0 −1.94642 −0.973211 0.229913i \(-0.926156\pi\)
−0.973211 + 0.229913i \(0.926156\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 256.000 0.0109086
\(57\) 0 0
\(58\) −10584.0 −0.413123
\(59\) 28300.0 1.05842 0.529208 0.848492i \(-0.322489\pi\)
0.529208 + 0.848492i \(0.322489\pi\)
\(60\) 0 0
\(61\) 18290.0 0.629345 0.314673 0.949200i \(-0.398105\pi\)
0.314673 + 0.949200i \(0.398105\pi\)
\(62\) 22448.0 0.741649
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 65956.0 1.79501 0.897506 0.441002i \(-0.145377\pi\)
0.897506 + 0.441002i \(0.145377\pi\)
\(68\) −24256.0 −0.636132
\(69\) 0 0
\(70\) 0 0
\(71\) 28800.0 0.678026 0.339013 0.940782i \(-0.389907\pi\)
0.339013 + 0.940782i \(0.389907\pi\)
\(72\) 0 0
\(73\) −30808.0 −0.676638 −0.338319 0.941031i \(-0.609858\pi\)
−0.338319 + 0.941031i \(0.609858\pi\)
\(74\) 29152.0 0.618855
\(75\) 0 0
\(76\) −21504.0 −0.427056
\(77\) −2000.00 −0.0384418
\(78\) 0 0
\(79\) 60228.0 1.08575 0.542876 0.839813i \(-0.317335\pi\)
0.542876 + 0.839813i \(0.317335\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −75944.0 −1.24727
\(83\) 2468.00 0.0393233 0.0196616 0.999807i \(-0.493741\pi\)
0.0196616 + 0.999807i \(0.493741\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9616.00 0.140200
\(87\) 0 0
\(88\) −32000.0 −0.440497
\(89\) −22678.0 −0.303480 −0.151740 0.988420i \(-0.548488\pi\)
−0.151740 + 0.988420i \(0.548488\pi\)
\(90\) 0 0
\(91\) 1152.00 0.0145831
\(92\) 65600.0 0.808043
\(93\) 0 0
\(94\) 35600.0 0.415557
\(95\) 0 0
\(96\) 0 0
\(97\) −36968.0 −0.398930 −0.199465 0.979905i \(-0.563920\pi\)
−0.199465 + 0.979905i \(0.563920\pi\)
\(98\) 67164.0 0.706434
\(99\) 0 0
\(100\) 0 0
\(101\) −167918. −1.63792 −0.818962 0.573848i \(-0.805450\pi\)
−0.818962 + 0.573848i \(0.805450\pi\)
\(102\) 0 0
\(103\) −154364. −1.43368 −0.716841 0.697236i \(-0.754413\pi\)
−0.716841 + 0.697236i \(0.754413\pi\)
\(104\) 18432.0 0.167105
\(105\) 0 0
\(106\) 159216. 1.37633
\(107\) −136788. −1.15502 −0.577509 0.816385i \(-0.695975\pi\)
−0.577509 + 0.816385i \(0.695975\pi\)
\(108\) 0 0
\(109\) −53810.0 −0.433807 −0.216904 0.976193i \(-0.569596\pi\)
−0.216904 + 0.976193i \(0.569596\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1024.00 −0.00771356
\(113\) −82692.0 −0.609211 −0.304605 0.952479i \(-0.598525\pi\)
−0.304605 + 0.952479i \(0.598525\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 42336.0 0.292122
\(117\) 0 0
\(118\) −113200. −0.748413
\(119\) 6064.00 0.0392547
\(120\) 0 0
\(121\) 88949.0 0.552303
\(122\) −73160.0 −0.445014
\(123\) 0 0
\(124\) −89792.0 −0.524425
\(125\) 0 0
\(126\) 0 0
\(127\) −211780. −1.16513 −0.582567 0.812783i \(-0.697952\pi\)
−0.582567 + 0.812783i \(0.697952\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −169500. −0.862962 −0.431481 0.902122i \(-0.642009\pi\)
−0.431481 + 0.902122i \(0.642009\pi\)
\(132\) 0 0
\(133\) 5376.00 0.0263530
\(134\) −263824. −1.26927
\(135\) 0 0
\(136\) 97024.0 0.449813
\(137\) −252036. −1.14726 −0.573629 0.819115i \(-0.694465\pi\)
−0.573629 + 0.819115i \(0.694465\pi\)
\(138\) 0 0
\(139\) 192016. 0.842947 0.421474 0.906841i \(-0.361513\pi\)
0.421474 + 0.906841i \(0.361513\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −115200. −0.479437
\(143\) −144000. −0.588874
\(144\) 0 0
\(145\) 0 0
\(146\) 123232. 0.478455
\(147\) 0 0
\(148\) −116608. −0.437597
\(149\) −235694. −0.869727 −0.434863 0.900496i \(-0.643203\pi\)
−0.434863 + 0.900496i \(0.643203\pi\)
\(150\) 0 0
\(151\) −371492. −1.32589 −0.662944 0.748669i \(-0.730693\pi\)
−0.662944 + 0.748669i \(0.730693\pi\)
\(152\) 86016.0 0.301975
\(153\) 0 0
\(154\) 8000.00 0.0271824
\(155\) 0 0
\(156\) 0 0
\(157\) 264952. 0.857863 0.428932 0.903337i \(-0.358890\pi\)
0.428932 + 0.903337i \(0.358890\pi\)
\(158\) −240912. −0.767743
\(159\) 0 0
\(160\) 0 0
\(161\) −16400.0 −0.0498631
\(162\) 0 0
\(163\) −403124. −1.18842 −0.594210 0.804310i \(-0.702535\pi\)
−0.594210 + 0.804310i \(0.702535\pi\)
\(164\) 303776. 0.881950
\(165\) 0 0
\(166\) −9872.00 −0.0278058
\(167\) 261900. 0.726682 0.363341 0.931656i \(-0.381636\pi\)
0.363341 + 0.931656i \(0.381636\pi\)
\(168\) 0 0
\(169\) −288349. −0.776608
\(170\) 0 0
\(171\) 0 0
\(172\) −38464.0 −0.0991364
\(173\) −326228. −0.828716 −0.414358 0.910114i \(-0.635994\pi\)
−0.414358 + 0.910114i \(0.635994\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 128000. 0.311479
\(177\) 0 0
\(178\) 90712.0 0.214593
\(179\) 109516. 0.255473 0.127736 0.991808i \(-0.459229\pi\)
0.127736 + 0.991808i \(0.459229\pi\)
\(180\) 0 0
\(181\) −53146.0 −0.120580 −0.0602898 0.998181i \(-0.519202\pi\)
−0.0602898 + 0.998181i \(0.519202\pi\)
\(182\) −4608.00 −0.0103118
\(183\) 0 0
\(184\) −262400. −0.571372
\(185\) 0 0
\(186\) 0 0
\(187\) −758000. −1.58513
\(188\) −142400. −0.293843
\(189\) 0 0
\(190\) 0 0
\(191\) −232056. −0.460267 −0.230133 0.973159i \(-0.573916\pi\)
−0.230133 + 0.973159i \(0.573916\pi\)
\(192\) 0 0
\(193\) −1.03067e6 −1.99172 −0.995858 0.0909274i \(-0.971017\pi\)
−0.995858 + 0.0909274i \(0.971017\pi\)
\(194\) 147872. 0.282086
\(195\) 0 0
\(196\) −268656. −0.499524
\(197\) 522796. 0.959769 0.479884 0.877332i \(-0.340679\pi\)
0.479884 + 0.877332i \(0.340679\pi\)
\(198\) 0 0
\(199\) −215292. −0.385385 −0.192693 0.981259i \(-0.561722\pi\)
−0.192693 + 0.981259i \(0.561722\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 671672. 1.15819
\(203\) −10584.0 −0.0180264
\(204\) 0 0
\(205\) 0 0
\(206\) 617456. 1.01377
\(207\) 0 0
\(208\) −73728.0 −0.118161
\(209\) −672000. −1.06415
\(210\) 0 0
\(211\) −1.03008e6 −1.59281 −0.796407 0.604762i \(-0.793268\pi\)
−0.796407 + 0.604762i \(0.793268\pi\)
\(212\) −636864. −0.973211
\(213\) 0 0
\(214\) 547152. 0.816721
\(215\) 0 0
\(216\) 0 0
\(217\) 22448.0 0.0323615
\(218\) 215240. 0.306748
\(219\) 0 0
\(220\) 0 0
\(221\) 436608. 0.601327
\(222\) 0 0
\(223\) 456020. 0.614075 0.307038 0.951697i \(-0.400662\pi\)
0.307038 + 0.951697i \(0.400662\pi\)
\(224\) 4096.00 0.00545431
\(225\) 0 0
\(226\) 330768. 0.430777
\(227\) 434252. 0.559342 0.279671 0.960096i \(-0.409775\pi\)
0.279671 + 0.960096i \(0.409775\pi\)
\(228\) 0 0
\(229\) −722710. −0.910700 −0.455350 0.890313i \(-0.650486\pi\)
−0.455350 + 0.890313i \(0.650486\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −169344. −0.206562
\(233\) 565348. 0.682223 0.341111 0.940023i \(-0.389197\pi\)
0.341111 + 0.940023i \(0.389197\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 452800. 0.529208
\(237\) 0 0
\(238\) −24256.0 −0.0277573
\(239\) −324904. −0.367926 −0.183963 0.982933i \(-0.558893\pi\)
−0.183963 + 0.982933i \(0.558893\pi\)
\(240\) 0 0
\(241\) 915262. 1.01509 0.507543 0.861626i \(-0.330554\pi\)
0.507543 + 0.861626i \(0.330554\pi\)
\(242\) −355796. −0.390537
\(243\) 0 0
\(244\) 292640. 0.314673
\(245\) 0 0
\(246\) 0 0
\(247\) 387072. 0.403691
\(248\) 359168. 0.370825
\(249\) 0 0
\(250\) 0 0
\(251\) −1.36708e6 −1.36965 −0.684823 0.728709i \(-0.740121\pi\)
−0.684823 + 0.728709i \(0.740121\pi\)
\(252\) 0 0
\(253\) 2.05000e6 2.01350
\(254\) 847120. 0.823874
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) −892932. −0.843307 −0.421653 0.906757i \(-0.638550\pi\)
−0.421653 + 0.906757i \(0.638550\pi\)
\(258\) 0 0
\(259\) 29152.0 0.0270034
\(260\) 0 0
\(261\) 0 0
\(262\) 678000. 0.610206
\(263\) 1.86650e6 1.66394 0.831972 0.554818i \(-0.187212\pi\)
0.831972 + 0.554818i \(0.187212\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −21504.0 −0.0186344
\(267\) 0 0
\(268\) 1.05530e6 0.897506
\(269\) 1.37227e6 1.15627 0.578133 0.815943i \(-0.303782\pi\)
0.578133 + 0.815943i \(0.303782\pi\)
\(270\) 0 0
\(271\) 458644. 0.379361 0.189680 0.981846i \(-0.439255\pi\)
0.189680 + 0.981846i \(0.439255\pi\)
\(272\) −388096. −0.318066
\(273\) 0 0
\(274\) 1.00814e6 0.811234
\(275\) 0 0
\(276\) 0 0
\(277\) −985408. −0.771643 −0.385822 0.922573i \(-0.626082\pi\)
−0.385822 + 0.922573i \(0.626082\pi\)
\(278\) −768064. −0.596054
\(279\) 0 0
\(280\) 0 0
\(281\) −165798. −0.125260 −0.0626302 0.998037i \(-0.519949\pi\)
−0.0626302 + 0.998037i \(0.519949\pi\)
\(282\) 0 0
\(283\) −1.66471e6 −1.23558 −0.617792 0.786342i \(-0.711972\pi\)
−0.617792 + 0.786342i \(0.711972\pi\)
\(284\) 460800. 0.339013
\(285\) 0 0
\(286\) 576000. 0.416397
\(287\) −75944.0 −0.0544238
\(288\) 0 0
\(289\) 878399. 0.618653
\(290\) 0 0
\(291\) 0 0
\(292\) −492928. −0.338319
\(293\) −2.55104e6 −1.73600 −0.867998 0.496567i \(-0.834594\pi\)
−0.867998 + 0.496567i \(0.834594\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 466432. 0.309428
\(297\) 0 0
\(298\) 942776. 0.614990
\(299\) −1.18080e6 −0.763833
\(300\) 0 0
\(301\) 9616.00 0.00611756
\(302\) 1.48597e6 0.937545
\(303\) 0 0
\(304\) −344064. −0.213528
\(305\) 0 0
\(306\) 0 0
\(307\) 736020. 0.445701 0.222851 0.974853i \(-0.428464\pi\)
0.222851 + 0.974853i \(0.428464\pi\)
\(308\) −32000.0 −0.0192209
\(309\) 0 0
\(310\) 0 0
\(311\) −1.71660e6 −1.00639 −0.503197 0.864172i \(-0.667843\pi\)
−0.503197 + 0.864172i \(0.667843\pi\)
\(312\) 0 0
\(313\) 2.83851e6 1.63768 0.818842 0.574020i \(-0.194617\pi\)
0.818842 + 0.574020i \(0.194617\pi\)
\(314\) −1.05981e6 −0.606601
\(315\) 0 0
\(316\) 963648. 0.542876
\(317\) 1.27605e6 0.713215 0.356607 0.934254i \(-0.383933\pi\)
0.356607 + 0.934254i \(0.383933\pi\)
\(318\) 0 0
\(319\) 1.32300e6 0.727919
\(320\) 0 0
\(321\) 0 0
\(322\) 65600.0 0.0352585
\(323\) 2.03750e6 1.08666
\(324\) 0 0
\(325\) 0 0
\(326\) 1.61250e6 0.840339
\(327\) 0 0
\(328\) −1.21510e6 −0.623633
\(329\) 35600.0 0.0181326
\(330\) 0 0
\(331\) 443992. 0.222744 0.111372 0.993779i \(-0.464476\pi\)
0.111372 + 0.993779i \(0.464476\pi\)
\(332\) 39488.0 0.0196616
\(333\) 0 0
\(334\) −1.04760e6 −0.513842
\(335\) 0 0
\(336\) 0 0
\(337\) 2.71326e6 1.30142 0.650708 0.759328i \(-0.274472\pi\)
0.650708 + 0.759328i \(0.274472\pi\)
\(338\) 1.15340e6 0.549145
\(339\) 0 0
\(340\) 0 0
\(341\) −2.80600e6 −1.30678
\(342\) 0 0
\(343\) 134392. 0.0616791
\(344\) 153856. 0.0701001
\(345\) 0 0
\(346\) 1.30491e6 0.585991
\(347\) 1.31051e6 0.584273 0.292137 0.956377i \(-0.405634\pi\)
0.292137 + 0.956377i \(0.405634\pi\)
\(348\) 0 0
\(349\) −298910. −0.131364 −0.0656821 0.997841i \(-0.520922\pi\)
−0.0656821 + 0.997841i \(0.520922\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −512000. −0.220249
\(353\) −737996. −0.315223 −0.157611 0.987501i \(-0.550379\pi\)
−0.157611 + 0.987501i \(0.550379\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −362848. −0.151740
\(357\) 0 0
\(358\) −438064. −0.180647
\(359\) 2.34074e6 0.958557 0.479278 0.877663i \(-0.340898\pi\)
0.479278 + 0.877663i \(0.340898\pi\)
\(360\) 0 0
\(361\) −669763. −0.270491
\(362\) 212584. 0.0852627
\(363\) 0 0
\(364\) 18432.0 0.00729154
\(365\) 0 0
\(366\) 0 0
\(367\) 127292. 0.0493328 0.0246664 0.999696i \(-0.492148\pi\)
0.0246664 + 0.999696i \(0.492148\pi\)
\(368\) 1.04960e6 0.404021
\(369\) 0 0
\(370\) 0 0
\(371\) 159216. 0.0600554
\(372\) 0 0
\(373\) 4.03870e6 1.50303 0.751517 0.659713i \(-0.229322\pi\)
0.751517 + 0.659713i \(0.229322\pi\)
\(374\) 3.03200e6 1.12086
\(375\) 0 0
\(376\) 569600. 0.207778
\(377\) −762048. −0.276140
\(378\) 0 0
\(379\) 1.01214e6 0.361944 0.180972 0.983488i \(-0.442076\pi\)
0.180972 + 0.983488i \(0.442076\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 928224. 0.325458
\(383\) 2.37610e6 0.827690 0.413845 0.910347i \(-0.364186\pi\)
0.413845 + 0.910347i \(0.364186\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.12269e6 1.40836
\(387\) 0 0
\(388\) −591488. −0.199465
\(389\) −1.42497e6 −0.477456 −0.238728 0.971087i \(-0.576730\pi\)
−0.238728 + 0.971087i \(0.576730\pi\)
\(390\) 0 0
\(391\) −6.21560e6 −2.05609
\(392\) 1.07462e6 0.353217
\(393\) 0 0
\(394\) −2.09118e6 −0.678659
\(395\) 0 0
\(396\) 0 0
\(397\) 1.69345e6 0.539257 0.269628 0.962964i \(-0.413099\pi\)
0.269628 + 0.962964i \(0.413099\pi\)
\(398\) 861168. 0.272509
\(399\) 0 0
\(400\) 0 0
\(401\) 2.84501e6 0.883532 0.441766 0.897130i \(-0.354352\pi\)
0.441766 + 0.897130i \(0.354352\pi\)
\(402\) 0 0
\(403\) 1.61626e6 0.495733
\(404\) −2.68669e6 −0.818962
\(405\) 0 0
\(406\) 42336.0 0.0127466
\(407\) −3.64400e6 −1.09042
\(408\) 0 0
\(409\) −1.89069e6 −0.558873 −0.279436 0.960164i \(-0.590148\pi\)
−0.279436 + 0.960164i \(0.590148\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.46982e6 −0.716841
\(413\) −113200. −0.0326566
\(414\) 0 0
\(415\) 0 0
\(416\) 294912. 0.0835524
\(417\) 0 0
\(418\) 2.68800e6 0.752469
\(419\) −4.60930e6 −1.28263 −0.641313 0.767280i \(-0.721610\pi\)
−0.641313 + 0.767280i \(0.721610\pi\)
\(420\) 0 0
\(421\) −6.04151e6 −1.66127 −0.830635 0.556817i \(-0.812022\pi\)
−0.830635 + 0.556817i \(0.812022\pi\)
\(422\) 4.12032e6 1.12629
\(423\) 0 0
\(424\) 2.54746e6 0.688164
\(425\) 0 0
\(426\) 0 0
\(427\) −73160.0 −0.0194180
\(428\) −2.18861e6 −0.577509
\(429\) 0 0
\(430\) 0 0
\(431\) 3800.00 0.000985350 0 0.000492675 1.00000i \(-0.499843\pi\)
0.000492675 1.00000i \(0.499843\pi\)
\(432\) 0 0
\(433\) 250736. 0.0642683 0.0321342 0.999484i \(-0.489770\pi\)
0.0321342 + 0.999484i \(0.489770\pi\)
\(434\) −89792.0 −0.0228830
\(435\) 0 0
\(436\) −860960. −0.216904
\(437\) −5.51040e6 −1.38032
\(438\) 0 0
\(439\) −3.58873e6 −0.888750 −0.444375 0.895841i \(-0.646574\pi\)
−0.444375 + 0.895841i \(0.646574\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.74643e6 −0.425203
\(443\) 1.41479e6 0.342517 0.171258 0.985226i \(-0.445217\pi\)
0.171258 + 0.985226i \(0.445217\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1.82408e6 −0.434217
\(447\) 0 0
\(448\) −16384.0 −0.00385678
\(449\) 829806. 0.194250 0.0971249 0.995272i \(-0.469035\pi\)
0.0971249 + 0.995272i \(0.469035\pi\)
\(450\) 0 0
\(451\) 9.49300e6 2.19767
\(452\) −1.32307e6 −0.304605
\(453\) 0 0
\(454\) −1.73701e6 −0.395514
\(455\) 0 0
\(456\) 0 0
\(457\) 4.68198e6 1.04867 0.524335 0.851512i \(-0.324314\pi\)
0.524335 + 0.851512i \(0.324314\pi\)
\(458\) 2.89084e6 0.643962
\(459\) 0 0
\(460\) 0 0
\(461\) 141930. 0.0311044 0.0155522 0.999879i \(-0.495049\pi\)
0.0155522 + 0.999879i \(0.495049\pi\)
\(462\) 0 0
\(463\) 727476. 0.157713 0.0788563 0.996886i \(-0.474873\pi\)
0.0788563 + 0.996886i \(0.474873\pi\)
\(464\) 677376. 0.146061
\(465\) 0 0
\(466\) −2.26139e6 −0.482404
\(467\) 4.47640e6 0.949809 0.474905 0.880037i \(-0.342483\pi\)
0.474905 + 0.880037i \(0.342483\pi\)
\(468\) 0 0
\(469\) −263824. −0.0553837
\(470\) 0 0
\(471\) 0 0
\(472\) −1.81120e6 −0.374207
\(473\) −1.20200e6 −0.247031
\(474\) 0 0
\(475\) 0 0
\(476\) 97024.0 0.0196274
\(477\) 0 0
\(478\) 1.29962e6 0.260163
\(479\) 1.32718e6 0.264297 0.132149 0.991230i \(-0.457812\pi\)
0.132149 + 0.991230i \(0.457812\pi\)
\(480\) 0 0
\(481\) 2.09894e6 0.413655
\(482\) −3.66105e6 −0.717774
\(483\) 0 0
\(484\) 1.42318e6 0.276152
\(485\) 0 0
\(486\) 0 0
\(487\) −4.11647e6 −0.786507 −0.393253 0.919430i \(-0.628650\pi\)
−0.393253 + 0.919430i \(0.628650\pi\)
\(488\) −1.17056e6 −0.222507
\(489\) 0 0
\(490\) 0 0
\(491\) 6.12316e6 1.14623 0.573115 0.819475i \(-0.305735\pi\)
0.573115 + 0.819475i \(0.305735\pi\)
\(492\) 0 0
\(493\) −4.01134e6 −0.743313
\(494\) −1.54829e6 −0.285453
\(495\) 0 0
\(496\) −1.43667e6 −0.262213
\(497\) −115200. −0.0209200
\(498\) 0 0
\(499\) −7.90490e6 −1.42117 −0.710584 0.703613i \(-0.751569\pi\)
−0.710584 + 0.703613i \(0.751569\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 5.46830e6 0.968486
\(503\) −3.97628e6 −0.700741 −0.350370 0.936611i \(-0.613944\pi\)
−0.350370 + 0.936611i \(0.613944\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −8.20000e6 −1.42376
\(507\) 0 0
\(508\) −3.38848e6 −0.582567
\(509\) −781914. −0.133772 −0.0668859 0.997761i \(-0.521306\pi\)
−0.0668859 + 0.997761i \(0.521306\pi\)
\(510\) 0 0
\(511\) 123232. 0.0208772
\(512\) −262144. −0.0441942
\(513\) 0 0
\(514\) 3.57173e6 0.596308
\(515\) 0 0
\(516\) 0 0
\(517\) −4.45000e6 −0.732207
\(518\) −116608. −0.0190943
\(519\) 0 0
\(520\) 0 0
\(521\) −5.82694e6 −0.940472 −0.470236 0.882541i \(-0.655831\pi\)
−0.470236 + 0.882541i \(0.655831\pi\)
\(522\) 0 0
\(523\) −9.78938e6 −1.56495 −0.782476 0.622681i \(-0.786043\pi\)
−0.782476 + 0.622681i \(0.786043\pi\)
\(524\) −2.71200e6 −0.431481
\(525\) 0 0
\(526\) −7.46600e6 −1.17659
\(527\) 8.50779e6 1.33441
\(528\) 0 0
\(529\) 1.03737e7 1.61173
\(530\) 0 0
\(531\) 0 0
\(532\) 86016.0 0.0131765
\(533\) −5.46797e6 −0.833696
\(534\) 0 0
\(535\) 0 0
\(536\) −4.22118e6 −0.634633
\(537\) 0 0
\(538\) −5.48906e6 −0.817603
\(539\) −8.39550e6 −1.24473
\(540\) 0 0
\(541\) 4.76059e6 0.699307 0.349653 0.936879i \(-0.386299\pi\)
0.349653 + 0.936879i \(0.386299\pi\)
\(542\) −1.83458e6 −0.268249
\(543\) 0 0
\(544\) 1.55238e6 0.224906
\(545\) 0 0
\(546\) 0 0
\(547\) −1.16595e6 −0.166614 −0.0833069 0.996524i \(-0.526548\pi\)
−0.0833069 + 0.996524i \(0.526548\pi\)
\(548\) −4.03258e6 −0.573629
\(549\) 0 0
\(550\) 0 0
\(551\) −3.55622e6 −0.499011
\(552\) 0 0
\(553\) −240912. −0.0335001
\(554\) 3.94163e6 0.545634
\(555\) 0 0
\(556\) 3.07226e6 0.421474
\(557\) 1.61293e6 0.220282 0.110141 0.993916i \(-0.464870\pi\)
0.110141 + 0.993916i \(0.464870\pi\)
\(558\) 0 0
\(559\) 692352. 0.0937125
\(560\) 0 0
\(561\) 0 0
\(562\) 663192. 0.0885724
\(563\) −3.40603e6 −0.452874 −0.226437 0.974026i \(-0.572708\pi\)
−0.226437 + 0.974026i \(0.572708\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6.65883e6 0.873689
\(567\) 0 0
\(568\) −1.84320e6 −0.239719
\(569\) 1.44009e7 1.86470 0.932350 0.361557i \(-0.117755\pi\)
0.932350 + 0.361557i \(0.117755\pi\)
\(570\) 0 0
\(571\) 4.74772e6 0.609389 0.304695 0.952450i \(-0.401446\pi\)
0.304695 + 0.952450i \(0.401446\pi\)
\(572\) −2.30400e6 −0.294437
\(573\) 0 0
\(574\) 303776. 0.0384834
\(575\) 0 0
\(576\) 0 0
\(577\) −1.09094e7 −1.36415 −0.682074 0.731283i \(-0.738922\pi\)
−0.682074 + 0.731283i \(0.738922\pi\)
\(578\) −3.51360e6 −0.437454
\(579\) 0 0
\(580\) 0 0
\(581\) −9872.00 −0.00121329
\(582\) 0 0
\(583\) −1.99020e7 −2.42508
\(584\) 1.97171e6 0.239228
\(585\) 0 0
\(586\) 1.02042e7 1.22754
\(587\) −8.53223e6 −1.02204 −0.511019 0.859569i \(-0.670732\pi\)
−0.511019 + 0.859569i \(0.670732\pi\)
\(588\) 0 0
\(589\) 7.54253e6 0.895836
\(590\) 0 0
\(591\) 0 0
\(592\) −1.86573e6 −0.218798
\(593\) 4.63182e6 0.540897 0.270449 0.962734i \(-0.412828\pi\)
0.270449 + 0.962734i \(0.412828\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.77110e6 −0.434863
\(597\) 0 0
\(598\) 4.72320e6 0.540111
\(599\) −6.27598e6 −0.714684 −0.357342 0.933974i \(-0.616317\pi\)
−0.357342 + 0.933974i \(0.616317\pi\)
\(600\) 0 0
\(601\) 7.71988e6 0.871815 0.435907 0.899992i \(-0.356428\pi\)
0.435907 + 0.899992i \(0.356428\pi\)
\(602\) −38464.0 −0.00432577
\(603\) 0 0
\(604\) −5.94387e6 −0.662944
\(605\) 0 0
\(606\) 0 0
\(607\) −6.06160e6 −0.667753 −0.333876 0.942617i \(-0.608357\pi\)
−0.333876 + 0.942617i \(0.608357\pi\)
\(608\) 1.37626e6 0.150987
\(609\) 0 0
\(610\) 0 0
\(611\) 2.56320e6 0.277766
\(612\) 0 0
\(613\) −3.66489e6 −0.393921 −0.196961 0.980411i \(-0.563107\pi\)
−0.196961 + 0.980411i \(0.563107\pi\)
\(614\) −2.94408e6 −0.315158
\(615\) 0 0
\(616\) 128000. 0.0135912
\(617\) 9.32522e6 0.986157 0.493079 0.869985i \(-0.335871\pi\)
0.493079 + 0.869985i \(0.335871\pi\)
\(618\) 0 0
\(619\) −7.40162e6 −0.776426 −0.388213 0.921570i \(-0.626907\pi\)
−0.388213 + 0.921570i \(0.626907\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.86640e6 0.711628
\(623\) 90712.0 0.00936364
\(624\) 0 0
\(625\) 0 0
\(626\) −1.13540e7 −1.15802
\(627\) 0 0
\(628\) 4.23923e6 0.428932
\(629\) 1.10486e7 1.11348
\(630\) 0 0
\(631\) 160052. 0.0160025 0.00800125 0.999968i \(-0.497453\pi\)
0.00800125 + 0.999968i \(0.497453\pi\)
\(632\) −3.85459e6 −0.383871
\(633\) 0 0
\(634\) −5.10421e6 −0.504319
\(635\) 0 0
\(636\) 0 0
\(637\) 4.83581e6 0.472194
\(638\) −5.29200e6 −0.514717
\(639\) 0 0
\(640\) 0 0
\(641\) 1.69565e7 1.63002 0.815008 0.579450i \(-0.196732\pi\)
0.815008 + 0.579450i \(0.196732\pi\)
\(642\) 0 0
\(643\) 1.10128e7 1.05044 0.525219 0.850967i \(-0.323984\pi\)
0.525219 + 0.850967i \(0.323984\pi\)
\(644\) −262400. −0.0249315
\(645\) 0 0
\(646\) −8.15002e6 −0.768382
\(647\) −3.33848e6 −0.313537 −0.156768 0.987635i \(-0.550108\pi\)
−0.156768 + 0.987635i \(0.550108\pi\)
\(648\) 0 0
\(649\) 1.41500e7 1.31870
\(650\) 0 0
\(651\) 0 0
\(652\) −6.44998e6 −0.594210
\(653\) 4.76181e6 0.437008 0.218504 0.975836i \(-0.429882\pi\)
0.218504 + 0.975836i \(0.429882\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.86042e6 0.440975
\(657\) 0 0
\(658\) −142400. −0.0128217
\(659\) −798188. −0.0715965 −0.0357982 0.999359i \(-0.511397\pi\)
−0.0357982 + 0.999359i \(0.511397\pi\)
\(660\) 0 0
\(661\) −1.54048e7 −1.37136 −0.685682 0.727901i \(-0.740496\pi\)
−0.685682 + 0.727901i \(0.740496\pi\)
\(662\) −1.77597e6 −0.157503
\(663\) 0 0
\(664\) −157952. −0.0139029
\(665\) 0 0
\(666\) 0 0
\(667\) 1.08486e7 0.944189
\(668\) 4.19040e6 0.363341
\(669\) 0 0
\(670\) 0 0
\(671\) 9.14500e6 0.784111
\(672\) 0 0
\(673\) −976704. −0.0831238 −0.0415619 0.999136i \(-0.513233\pi\)
−0.0415619 + 0.999136i \(0.513233\pi\)
\(674\) −1.08530e7 −0.920240
\(675\) 0 0
\(676\) −4.61358e6 −0.388304
\(677\) −1.93885e7 −1.62582 −0.812911 0.582388i \(-0.802119\pi\)
−0.812911 + 0.582388i \(0.802119\pi\)
\(678\) 0 0
\(679\) 147872. 0.0123087
\(680\) 0 0
\(681\) 0 0
\(682\) 1.12240e7 0.924031
\(683\) 5.25573e6 0.431103 0.215552 0.976492i \(-0.430845\pi\)
0.215552 + 0.976492i \(0.430845\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −537568. −0.0436137
\(687\) 0 0
\(688\) −615424. −0.0495682
\(689\) 1.14636e7 0.919965
\(690\) 0 0
\(691\) −5.45034e6 −0.434238 −0.217119 0.976145i \(-0.569666\pi\)
−0.217119 + 0.976145i \(0.569666\pi\)
\(692\) −5.21965e6 −0.414358
\(693\) 0 0
\(694\) −5.24203e6 −0.413144
\(695\) 0 0
\(696\) 0 0
\(697\) −2.87828e7 −2.24414
\(698\) 1.19564e6 0.0928885
\(699\) 0 0
\(700\) 0 0
\(701\) 4.43961e6 0.341232 0.170616 0.985338i \(-0.445424\pi\)
0.170616 + 0.985338i \(0.445424\pi\)
\(702\) 0 0
\(703\) 9.79507e6 0.747514
\(704\) 2.04800e6 0.155739
\(705\) 0 0
\(706\) 2.95198e6 0.222896
\(707\) 671672. 0.0505369
\(708\) 0 0
\(709\) 4.55918e6 0.340621 0.170310 0.985390i \(-0.445523\pi\)
0.170310 + 0.985390i \(0.445523\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.45139e6 0.107296
\(713\) −2.30092e7 −1.69503
\(714\) 0 0
\(715\) 0 0
\(716\) 1.75226e6 0.127736
\(717\) 0 0
\(718\) −9.36298e6 −0.677802
\(719\) 2.06630e7 1.49063 0.745317 0.666710i \(-0.232298\pi\)
0.745317 + 0.666710i \(0.232298\pi\)
\(720\) 0 0
\(721\) 617456. 0.0442352
\(722\) 2.67905e6 0.191266
\(723\) 0 0
\(724\) −850336. −0.0602898
\(725\) 0 0
\(726\) 0 0
\(727\) 5.48161e6 0.384656 0.192328 0.981331i \(-0.438396\pi\)
0.192328 + 0.981331i \(0.438396\pi\)
\(728\) −73728.0 −0.00515589
\(729\) 0 0
\(730\) 0 0
\(731\) 3.64446e6 0.252255
\(732\) 0 0
\(733\) −8.55579e6 −0.588166 −0.294083 0.955780i \(-0.595014\pi\)
−0.294083 + 0.955780i \(0.595014\pi\)
\(734\) −509168. −0.0348836
\(735\) 0 0
\(736\) −4.19840e6 −0.285686
\(737\) 3.29780e7 2.23643
\(738\) 0 0
\(739\) −5.29119e6 −0.356404 −0.178202 0.983994i \(-0.557028\pi\)
−0.178202 + 0.983994i \(0.557028\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −636864. −0.0424656
\(743\) 2.36432e6 0.157121 0.0785606 0.996909i \(-0.474968\pi\)
0.0785606 + 0.996909i \(0.474968\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.61548e7 −1.06281
\(747\) 0 0
\(748\) −1.21280e7 −0.792566
\(749\) 547152. 0.0356372
\(750\) 0 0
\(751\) −8.79694e6 −0.569157 −0.284578 0.958653i \(-0.591854\pi\)
−0.284578 + 0.958653i \(0.591854\pi\)
\(752\) −2.27840e6 −0.146922
\(753\) 0 0
\(754\) 3.04819e6 0.195260
\(755\) 0 0
\(756\) 0 0
\(757\) 2.95808e7 1.87616 0.938079 0.346421i \(-0.112603\pi\)
0.938079 + 0.346421i \(0.112603\pi\)
\(758\) −4.04854e6 −0.255933
\(759\) 0 0
\(760\) 0 0
\(761\) 1.26296e7 0.790549 0.395274 0.918563i \(-0.370649\pi\)
0.395274 + 0.918563i \(0.370649\pi\)
\(762\) 0 0
\(763\) 215240. 0.0133848
\(764\) −3.71290e6 −0.230133
\(765\) 0 0
\(766\) −9.50440e6 −0.585265
\(767\) −8.15040e6 −0.500254
\(768\) 0 0
\(769\) −2.32186e7 −1.41586 −0.707929 0.706283i \(-0.750370\pi\)
−0.707929 + 0.706283i \(0.750370\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.64908e7 −0.995858
\(773\) 1.73201e7 1.04256 0.521280 0.853386i \(-0.325455\pi\)
0.521280 + 0.853386i \(0.325455\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2.36595e6 0.141043
\(777\) 0 0
\(778\) 5.69990e6 0.337612
\(779\) −2.55172e7 −1.50657
\(780\) 0 0
\(781\) 1.44000e7 0.844763
\(782\) 2.48624e7 1.45387
\(783\) 0 0
\(784\) −4.29850e6 −0.249762
\(785\) 0 0
\(786\) 0 0
\(787\) 556676. 0.0320380 0.0160190 0.999872i \(-0.494901\pi\)
0.0160190 + 0.999872i \(0.494901\pi\)
\(788\) 8.36474e6 0.479884
\(789\) 0 0
\(790\) 0 0
\(791\) 330768. 0.0187967
\(792\) 0 0
\(793\) −5.26752e6 −0.297456
\(794\) −6.77379e6 −0.381312
\(795\) 0 0
\(796\) −3.44467e6 −0.192693
\(797\) −3.00562e6 −0.167606 −0.0838028 0.996482i \(-0.526707\pi\)
−0.0838028 + 0.996482i \(0.526707\pi\)
\(798\) 0 0
\(799\) 1.34924e7 0.747691
\(800\) 0 0
\(801\) 0 0
\(802\) −1.13800e7 −0.624751
\(803\) −1.54040e7 −0.843033
\(804\) 0 0
\(805\) 0 0
\(806\) −6.46502e6 −0.350536
\(807\) 0 0
\(808\) 1.07468e7 0.579094
\(809\) −2.23153e6 −0.119876 −0.0599378 0.998202i \(-0.519090\pi\)
−0.0599378 + 0.998202i \(0.519090\pi\)
\(810\) 0 0
\(811\) 2.24862e7 1.20051 0.600253 0.799810i \(-0.295067\pi\)
0.600253 + 0.799810i \(0.295067\pi\)
\(812\) −169344. −0.00901322
\(813\) 0 0
\(814\) 1.45760e7 0.771041
\(815\) 0 0
\(816\) 0 0
\(817\) 3.23098e6 0.169347
\(818\) 7.56278e6 0.395183
\(819\) 0 0
\(820\) 0 0
\(821\) 1.65921e7 0.859098 0.429549 0.903044i \(-0.358673\pi\)
0.429549 + 0.903044i \(0.358673\pi\)
\(822\) 0 0
\(823\) 1.47544e7 0.759316 0.379658 0.925127i \(-0.376042\pi\)
0.379658 + 0.925127i \(0.376042\pi\)
\(824\) 9.87930e6 0.506883
\(825\) 0 0
\(826\) 452800. 0.0230917
\(827\) 3.39475e6 0.172601 0.0863006 0.996269i \(-0.472495\pi\)
0.0863006 + 0.996269i \(0.472495\pi\)
\(828\) 0 0
\(829\) −509442. −0.0257459 −0.0128730 0.999917i \(-0.504098\pi\)
−0.0128730 + 0.999917i \(0.504098\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.17965e6 −0.0590805
\(833\) 2.54552e7 1.27105
\(834\) 0 0
\(835\) 0 0
\(836\) −1.07520e7 −0.532076
\(837\) 0 0
\(838\) 1.84372e7 0.906953
\(839\) −4.00609e7 −1.96479 −0.982394 0.186819i \(-0.940182\pi\)
−0.982394 + 0.186819i \(0.940182\pi\)
\(840\) 0 0
\(841\) −1.35098e7 −0.658658
\(842\) 2.41660e7 1.17470
\(843\) 0 0
\(844\) −1.64813e7 −0.796407
\(845\) 0 0
\(846\) 0 0
\(847\) −355796. −0.0170409
\(848\) −1.01898e7 −0.486606
\(849\) 0 0
\(850\) 0 0
\(851\) −2.98808e7 −1.41439
\(852\) 0 0
\(853\) 9.67506e6 0.455283 0.227641 0.973745i \(-0.426899\pi\)
0.227641 + 0.973745i \(0.426899\pi\)
\(854\) 292640. 0.0137306
\(855\) 0 0
\(856\) 8.75443e6 0.408360
\(857\) −3.27535e7 −1.52337 −0.761686 0.647946i \(-0.775628\pi\)
−0.761686 + 0.647946i \(0.775628\pi\)
\(858\) 0 0
\(859\) −2.17420e7 −1.00535 −0.502675 0.864476i \(-0.667651\pi\)
−0.502675 + 0.864476i \(0.667651\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −15200.0 −0.000696748 0
\(863\) −2.08744e7 −0.954087 −0.477043 0.878880i \(-0.658292\pi\)
−0.477043 + 0.878880i \(0.658292\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −1.00294e6 −0.0454446
\(867\) 0 0
\(868\) 359168. 0.0161807
\(869\) 3.01140e7 1.35275
\(870\) 0 0
\(871\) −1.89953e7 −0.848401
\(872\) 3.44384e6 0.153374
\(873\) 0 0
\(874\) 2.20416e7 0.976033
\(875\) 0 0
\(876\) 0 0
\(877\) 3.96804e7 1.74212 0.871058 0.491181i \(-0.163434\pi\)
0.871058 + 0.491181i \(0.163434\pi\)
\(878\) 1.43549e7 0.628441
\(879\) 0 0
\(880\) 0 0
\(881\) −2.60742e7 −1.13180 −0.565902 0.824472i \(-0.691472\pi\)
−0.565902 + 0.824472i \(0.691472\pi\)
\(882\) 0 0
\(883\) 4.10486e7 1.77172 0.885862 0.463949i \(-0.153568\pi\)
0.885862 + 0.463949i \(0.153568\pi\)
\(884\) 6.98573e6 0.300664
\(885\) 0 0
\(886\) −5.65915e6 −0.242196
\(887\) −1.37553e7 −0.587031 −0.293515 0.955954i \(-0.594825\pi\)
−0.293515 + 0.955954i \(0.594825\pi\)
\(888\) 0 0
\(889\) 847120. 0.0359493
\(890\) 0 0
\(891\) 0 0
\(892\) 7.29632e6 0.307038
\(893\) 1.19616e7 0.501950
\(894\) 0 0
\(895\) 0 0
\(896\) 65536.0 0.00272716
\(897\) 0 0
\(898\) −3.31922e6 −0.137355
\(899\) −1.48494e7 −0.612785
\(900\) 0 0
\(901\) 6.03429e7 2.47636
\(902\) −3.79720e7 −1.55399
\(903\) 0 0
\(904\) 5.29229e6 0.215388
\(905\) 0 0
\(906\) 0 0
\(907\) −5.86936e6 −0.236904 −0.118452 0.992960i \(-0.537793\pi\)
−0.118452 + 0.992960i \(0.537793\pi\)
\(908\) 6.94803e6 0.279671
\(909\) 0 0
\(910\) 0 0
\(911\) −4.63982e7 −1.85227 −0.926137 0.377188i \(-0.876891\pi\)
−0.926137 + 0.377188i \(0.876891\pi\)
\(912\) 0 0
\(913\) 1.23400e6 0.0489935
\(914\) −1.87279e7 −0.741521
\(915\) 0 0
\(916\) −1.15634e7 −0.455350
\(917\) 678000. 0.0266260
\(918\) 0 0
\(919\) 2.27859e7 0.889975 0.444988 0.895537i \(-0.353208\pi\)
0.444988 + 0.895537i \(0.353208\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −567720. −0.0219941
\(923\) −8.29440e6 −0.320465
\(924\) 0 0
\(925\) 0 0
\(926\) −2.90990e6 −0.111520
\(927\) 0 0
\(928\) −2.70950e6 −0.103281
\(929\) −2.70352e7 −1.02775 −0.513877 0.857864i \(-0.671791\pi\)
−0.513877 + 0.857864i \(0.671791\pi\)
\(930\) 0 0
\(931\) 2.25671e7 0.853300
\(932\) 9.04557e6 0.341111
\(933\) 0 0
\(934\) −1.79056e7 −0.671616
\(935\) 0 0
\(936\) 0 0
\(937\) −2.86149e7 −1.06474 −0.532370 0.846512i \(-0.678699\pi\)
−0.532370 + 0.846512i \(0.678699\pi\)
\(938\) 1.05530e6 0.0391622
\(939\) 0 0
\(940\) 0 0
\(941\) 3.67892e7 1.35440 0.677200 0.735799i \(-0.263193\pi\)
0.677200 + 0.735799i \(0.263193\pi\)
\(942\) 0 0
\(943\) 7.78426e7 2.85061
\(944\) 7.24480e6 0.264604
\(945\) 0 0
\(946\) 4.80800e6 0.174677
\(947\) −7.96828e6 −0.288728 −0.144364 0.989525i \(-0.546114\pi\)
−0.144364 + 0.989525i \(0.546114\pi\)
\(948\) 0 0
\(949\) 8.87270e6 0.319809
\(950\) 0 0
\(951\) 0 0
\(952\) −388096. −0.0138786
\(953\) 4.82202e7 1.71987 0.859937 0.510400i \(-0.170503\pi\)
0.859937 + 0.510400i \(0.170503\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −5.19846e6 −0.183963
\(957\) 0 0
\(958\) −5.30874e6 −0.186886
\(959\) 1.00814e6 0.0353978
\(960\) 0 0
\(961\) 2.86539e6 0.100087
\(962\) −8.39578e6 −0.292498
\(963\) 0 0
\(964\) 1.46442e7 0.507543
\(965\) 0 0
\(966\) 0 0
\(967\) 4.83510e7 1.66280 0.831398 0.555678i \(-0.187541\pi\)
0.831398 + 0.555678i \(0.187541\pi\)
\(968\) −5.69274e6 −0.195269
\(969\) 0 0
\(970\) 0 0
\(971\) 4.05515e7 1.38025 0.690127 0.723688i \(-0.257555\pi\)
0.690127 + 0.723688i \(0.257555\pi\)
\(972\) 0 0
\(973\) −768064. −0.0260085
\(974\) 1.64659e7 0.556144
\(975\) 0 0
\(976\) 4.68224e6 0.157336
\(977\) 4.34929e7 1.45775 0.728874 0.684648i \(-0.240044\pi\)
0.728874 + 0.684648i \(0.240044\pi\)
\(978\) 0 0
\(979\) −1.13390e7 −0.378110
\(980\) 0 0
\(981\) 0 0
\(982\) −2.44926e7 −0.810507
\(983\) 3.34896e6 0.110542 0.0552709 0.998471i \(-0.482398\pi\)
0.0552709 + 0.998471i \(0.482398\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 1.60453e7 0.525602
\(987\) 0 0
\(988\) 6.19315e6 0.201846
\(989\) −9.85640e6 −0.320426
\(990\) 0 0
\(991\) −5.55726e7 −1.79753 −0.898766 0.438429i \(-0.855535\pi\)
−0.898766 + 0.438429i \(0.855535\pi\)
\(992\) 5.74669e6 0.185412
\(993\) 0 0
\(994\) 460800. 0.0147927
\(995\) 0 0
\(996\) 0 0
\(997\) −1.27342e7 −0.405726 −0.202863 0.979207i \(-0.565025\pi\)
−0.202863 + 0.979207i \(0.565025\pi\)
\(998\) 3.16196e7 1.00492
\(999\) 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 450.6.a.f.1.1 1
3.2 odd 2 150.6.a.j.1.1 1
5.2 odd 4 90.6.c.b.19.1 2
5.3 odd 4 90.6.c.b.19.2 2
5.4 even 2 450.6.a.s.1.1 1
15.2 even 4 30.6.c.a.19.2 yes 2
15.8 even 4 30.6.c.a.19.1 2
15.14 odd 2 150.6.a.f.1.1 1
20.3 even 4 720.6.f.g.289.2 2
20.7 even 4 720.6.f.g.289.1 2
60.23 odd 4 240.6.f.a.49.2 2
60.47 odd 4 240.6.f.a.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.6.c.a.19.1 2 15.8 even 4
30.6.c.a.19.2 yes 2 15.2 even 4
90.6.c.b.19.1 2 5.2 odd 4
90.6.c.b.19.2 2 5.3 odd 4
150.6.a.f.1.1 1 15.14 odd 2
150.6.a.j.1.1 1 3.2 odd 2
240.6.f.a.49.1 2 60.47 odd 4
240.6.f.a.49.2 2 60.23 odd 4
450.6.a.f.1.1 1 1.1 even 1 trivial
450.6.a.s.1.1 1 5.4 even 2
720.6.f.g.289.1 2 20.7 even 4
720.6.f.g.289.2 2 20.3 even 4