Properties

Label 18.6.a.c.1.1
Level $18$
Weight $6$
Character 18.1
Self dual yes
Analytic conductor $2.887$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.88690875663\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 18.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} +16.0000 q^{4} +96.0000 q^{5} -148.000 q^{7} +64.0000 q^{8} +O(q^{10})\) \(q+4.00000 q^{2} +16.0000 q^{4} +96.0000 q^{5} -148.000 q^{7} +64.0000 q^{8} +384.000 q^{10} -384.000 q^{11} -334.000 q^{13} -592.000 q^{14} +256.000 q^{16} -576.000 q^{17} -664.000 q^{19} +1536.00 q^{20} -1536.00 q^{22} +3840.00 q^{23} +6091.00 q^{25} -1336.00 q^{26} -2368.00 q^{28} -96.0000 q^{29} -4564.00 q^{31} +1024.00 q^{32} -2304.00 q^{34} -14208.0 q^{35} +5798.00 q^{37} -2656.00 q^{38} +6144.00 q^{40} +6720.00 q^{41} -14872.0 q^{43} -6144.00 q^{44} +15360.0 q^{46} +19200.0 q^{47} +5097.00 q^{49} +24364.0 q^{50} -5344.00 q^{52} -7776.00 q^{53} -36864.0 q^{55} -9472.00 q^{56} -384.000 q^{58} +13056.0 q^{59} +42782.0 q^{61} -18256.0 q^{62} +4096.00 q^{64} -32064.0 q^{65} +36656.0 q^{67} -9216.00 q^{68} -56832.0 q^{70} -64512.0 q^{71} -16810.0 q^{73} +23192.0 q^{74} -10624.0 q^{76} +56832.0 q^{77} +28076.0 q^{79} +24576.0 q^{80} +26880.0 q^{82} +66432.0 q^{83} -55296.0 q^{85} -59488.0 q^{86} -24576.0 q^{88} +81792.0 q^{89} +49432.0 q^{91} +61440.0 q^{92} +76800.0 q^{94} -63744.0 q^{95} -29938.0 q^{97} +20388.0 q^{98} +O(q^{100})\)

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
\(5\) 96.0000 1.71730 0.858650 0.512562i \(-0.171304\pi\)
0.858650 + 0.512562i \(0.171304\pi\)
\(6\) 0 0
\(7\) −148.000 −1.14161 −0.570803 0.821087i \(-0.693368\pi\)
−0.570803 + 0.821087i \(0.693368\pi\)
\(8\) 64.0000 0.353553
\(9\) 0 0
\(10\) 384.000 1.21431
\(11\) −384.000 −0.956862 −0.478431 0.878125i \(-0.658794\pi\)
−0.478431 + 0.878125i \(0.658794\pi\)
\(12\) 0 0
\(13\) −334.000 −0.548136 −0.274068 0.961710i \(-0.588369\pi\)
−0.274068 + 0.961710i \(0.588369\pi\)
\(14\) −592.000 −0.807238
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −576.000 −0.483393 −0.241696 0.970352i \(-0.577704\pi\)
−0.241696 + 0.970352i \(0.577704\pi\)
\(18\) 0 0
\(19\) −664.000 −0.421972 −0.210986 0.977489i \(-0.567668\pi\)
−0.210986 + 0.977489i \(0.567668\pi\)
\(20\) 1536.00 0.858650
\(21\) 0 0
\(22\) −1536.00 −0.676604
\(23\) 3840.00 1.51360 0.756801 0.653645i \(-0.226761\pi\)
0.756801 + 0.653645i \(0.226761\pi\)
\(24\) 0 0
\(25\) 6091.00 1.94912
\(26\) −1336.00 −0.387590
\(27\) 0 0
\(28\) −2368.00 −0.570803
\(29\) −96.0000 −0.0211971 −0.0105985 0.999944i \(-0.503374\pi\)
−0.0105985 + 0.999944i \(0.503374\pi\)
\(30\) 0 0
\(31\) −4564.00 −0.852985 −0.426493 0.904491i \(-0.640251\pi\)
−0.426493 + 0.904491i \(0.640251\pi\)
\(32\) 1024.00 0.176777
\(33\) 0 0
\(34\) −2304.00 −0.341810
\(35\) −14208.0 −1.96048
\(36\) 0 0
\(37\) 5798.00 0.696264 0.348132 0.937446i \(-0.386816\pi\)
0.348132 + 0.937446i \(0.386816\pi\)
\(38\) −2656.00 −0.298380
\(39\) 0 0
\(40\) 6144.00 0.607157
\(41\) 6720.00 0.624323 0.312162 0.950029i \(-0.398947\pi\)
0.312162 + 0.950029i \(0.398947\pi\)
\(42\) 0 0
\(43\) −14872.0 −1.22659 −0.613293 0.789855i \(-0.710156\pi\)
−0.613293 + 0.789855i \(0.710156\pi\)
\(44\) −6144.00 −0.478431
\(45\) 0 0
\(46\) 15360.0 1.07028
\(47\) 19200.0 1.26782 0.633909 0.773408i \(-0.281450\pi\)
0.633909 + 0.773408i \(0.281450\pi\)
\(48\) 0 0
\(49\) 5097.00 0.303266
\(50\) 24364.0 1.37824
\(51\) 0 0
\(52\) −5344.00 −0.274068
\(53\) −7776.00 −0.380248 −0.190124 0.981760i \(-0.560889\pi\)
−0.190124 + 0.981760i \(0.560889\pi\)
\(54\) 0 0
\(55\) −36864.0 −1.64322
\(56\) −9472.00 −0.403619
\(57\) 0 0
\(58\) −384.000 −0.0149886
\(59\) 13056.0 0.488293 0.244146 0.969738i \(-0.421492\pi\)
0.244146 + 0.969738i \(0.421492\pi\)
\(60\) 0 0
\(61\) 42782.0 1.47210 0.736049 0.676929i \(-0.236689\pi\)
0.736049 + 0.676929i \(0.236689\pi\)
\(62\) −18256.0 −0.603151
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) −32064.0 −0.941314
\(66\) 0 0
\(67\) 36656.0 0.997604 0.498802 0.866716i \(-0.333774\pi\)
0.498802 + 0.866716i \(0.333774\pi\)
\(68\) −9216.00 −0.241696
\(69\) 0 0
\(70\) −56832.0 −1.38627
\(71\) −64512.0 −1.51878 −0.759390 0.650636i \(-0.774502\pi\)
−0.759390 + 0.650636i \(0.774502\pi\)
\(72\) 0 0
\(73\) −16810.0 −0.369199 −0.184600 0.982814i \(-0.559099\pi\)
−0.184600 + 0.982814i \(0.559099\pi\)
\(74\) 23192.0 0.492333
\(75\) 0 0
\(76\) −10624.0 −0.210986
\(77\) 56832.0 1.09236
\(78\) 0 0
\(79\) 28076.0 0.506136 0.253068 0.967448i \(-0.418560\pi\)
0.253068 + 0.967448i \(0.418560\pi\)
\(80\) 24576.0 0.429325
\(81\) 0 0
\(82\) 26880.0 0.441463
\(83\) 66432.0 1.05848 0.529239 0.848473i \(-0.322477\pi\)
0.529239 + 0.848473i \(0.322477\pi\)
\(84\) 0 0
\(85\) −55296.0 −0.830131
\(86\) −59488.0 −0.867328
\(87\) 0 0
\(88\) −24576.0 −0.338302
\(89\) 81792.0 1.09455 0.547275 0.836953i \(-0.315665\pi\)
0.547275 + 0.836953i \(0.315665\pi\)
\(90\) 0 0
\(91\) 49432.0 0.625756
\(92\) 61440.0 0.756801
\(93\) 0 0
\(94\) 76800.0 0.896482
\(95\) −63744.0 −0.724653
\(96\) 0 0
\(97\) −29938.0 −0.323068 −0.161534 0.986867i \(-0.551644\pi\)
−0.161534 + 0.986867i \(0.551644\pi\)
\(98\) 20388.0 0.214442
\(99\) 0 0
\(100\) 97456.0 0.974560
\(101\) −178656. −1.74267 −0.871333 0.490692i \(-0.836744\pi\)
−0.871333 + 0.490692i \(0.836744\pi\)
\(102\) 0 0
\(103\) −115228. −1.07020 −0.535100 0.844789i \(-0.679726\pi\)
−0.535100 + 0.844789i \(0.679726\pi\)
\(104\) −21376.0 −0.193795
\(105\) 0 0
\(106\) −31104.0 −0.268876
\(107\) 76032.0 0.642003 0.321001 0.947079i \(-0.395981\pi\)
0.321001 + 0.947079i \(0.395981\pi\)
\(108\) 0 0
\(109\) −231118. −1.86323 −0.931617 0.363441i \(-0.881602\pi\)
−0.931617 + 0.363441i \(0.881602\pi\)
\(110\) −147456. −1.16193
\(111\) 0 0
\(112\) −37888.0 −0.285402
\(113\) −142464. −1.04956 −0.524782 0.851237i \(-0.675853\pi\)
−0.524782 + 0.851237i \(0.675853\pi\)
\(114\) 0 0
\(115\) 368640. 2.59931
\(116\) −1536.00 −0.0105985
\(117\) 0 0
\(118\) 52224.0 0.345275
\(119\) 85248.0 0.551845
\(120\) 0 0
\(121\) −13595.0 −0.0844143
\(122\) 171128. 1.04093
\(123\) 0 0
\(124\) −73024.0 −0.426493
\(125\) 284736. 1.62992
\(126\) 0 0
\(127\) −988.000 −0.00543560 −0.00271780 0.999996i \(-0.500865\pi\)
−0.00271780 + 0.999996i \(0.500865\pi\)
\(128\) 16384.0 0.0883883
\(129\) 0 0
\(130\) −128256. −0.665609
\(131\) −224256. −1.14174 −0.570868 0.821042i \(-0.693393\pi\)
−0.570868 + 0.821042i \(0.693393\pi\)
\(132\) 0 0
\(133\) 98272.0 0.481727
\(134\) 146624. 0.705412
\(135\) 0 0
\(136\) −36864.0 −0.170905
\(137\) 278976. 1.26989 0.634944 0.772558i \(-0.281023\pi\)
0.634944 + 0.772558i \(0.281023\pi\)
\(138\) 0 0
\(139\) 177200. 0.777905 0.388953 0.921258i \(-0.372837\pi\)
0.388953 + 0.921258i \(0.372837\pi\)
\(140\) −227328. −0.980241
\(141\) 0 0
\(142\) −258048. −1.07394
\(143\) 128256. 0.524490
\(144\) 0 0
\(145\) −9216.00 −0.0364018
\(146\) −67240.0 −0.261063
\(147\) 0 0
\(148\) 92768.0 0.348132
\(149\) −236064. −0.871092 −0.435546 0.900166i \(-0.643445\pi\)
−0.435546 + 0.900166i \(0.643445\pi\)
\(150\) 0 0
\(151\) −482836. −1.72329 −0.861643 0.507515i \(-0.830564\pi\)
−0.861643 + 0.507515i \(0.830564\pi\)
\(152\) −42496.0 −0.149190
\(153\) 0 0
\(154\) 227328. 0.772416
\(155\) −438144. −1.46483
\(156\) 0 0
\(157\) 381086. 1.23388 0.616941 0.787009i \(-0.288372\pi\)
0.616941 + 0.787009i \(0.288372\pi\)
\(158\) 112304. 0.357892
\(159\) 0 0
\(160\) 98304.0 0.303579
\(161\) −568320. −1.72794
\(162\) 0 0
\(163\) 162920. 0.480292 0.240146 0.970737i \(-0.422805\pi\)
0.240146 + 0.970737i \(0.422805\pi\)
\(164\) 107520. 0.312162
\(165\) 0 0
\(166\) 265728. 0.748457
\(167\) 566016. 1.57050 0.785249 0.619180i \(-0.212535\pi\)
0.785249 + 0.619180i \(0.212535\pi\)
\(168\) 0 0
\(169\) −259737. −0.699547
\(170\) −221184. −0.586991
\(171\) 0 0
\(172\) −237952. −0.613293
\(173\) 218208. 0.554313 0.277157 0.960825i \(-0.410608\pi\)
0.277157 + 0.960825i \(0.410608\pi\)
\(174\) 0 0
\(175\) −901468. −2.22513
\(176\) −98304.0 −0.239216
\(177\) 0 0
\(178\) 327168. 0.773964
\(179\) 412416. 0.962062 0.481031 0.876704i \(-0.340262\pi\)
0.481031 + 0.876704i \(0.340262\pi\)
\(180\) 0 0
\(181\) −25558.0 −0.0579870 −0.0289935 0.999580i \(-0.509230\pi\)
−0.0289935 + 0.999580i \(0.509230\pi\)
\(182\) 197728. 0.442476
\(183\) 0 0
\(184\) 245760. 0.535139
\(185\) 556608. 1.19569
\(186\) 0 0
\(187\) 221184. 0.462540
\(188\) 307200. 0.633909
\(189\) 0 0
\(190\) −254976. −0.512407
\(191\) −400128. −0.793625 −0.396813 0.917900i \(-0.629884\pi\)
−0.396813 + 0.917900i \(0.629884\pi\)
\(192\) 0 0
\(193\) 699650. 1.35203 0.676017 0.736886i \(-0.263705\pi\)
0.676017 + 0.736886i \(0.263705\pi\)
\(194\) −119752. −0.228443
\(195\) 0 0
\(196\) 81552.0 0.151633
\(197\) −406368. −0.746026 −0.373013 0.927826i \(-0.621675\pi\)
−0.373013 + 0.927826i \(0.621675\pi\)
\(198\) 0 0
\(199\) −361996. −0.647994 −0.323997 0.946058i \(-0.605027\pi\)
−0.323997 + 0.946058i \(0.605027\pi\)
\(200\) 389824. 0.689118
\(201\) 0 0
\(202\) −714624. −1.23225
\(203\) 14208.0 0.0241987
\(204\) 0 0
\(205\) 645120. 1.07215
\(206\) −460912. −0.756746
\(207\) 0 0
\(208\) −85504.0 −0.137034
\(209\) 254976. 0.403770
\(210\) 0 0
\(211\) 151856. 0.234815 0.117407 0.993084i \(-0.462542\pi\)
0.117407 + 0.993084i \(0.462542\pi\)
\(212\) −124416. −0.190124
\(213\) 0 0
\(214\) 304128. 0.453965
\(215\) −1.42771e6 −2.10642
\(216\) 0 0
\(217\) 675472. 0.973774
\(218\) −924472. −1.31751
\(219\) 0 0
\(220\) −589824. −0.821610
\(221\) 192384. 0.264965
\(222\) 0 0
\(223\) −1.09332e6 −1.47227 −0.736134 0.676836i \(-0.763351\pi\)
−0.736134 + 0.676836i \(0.763351\pi\)
\(224\) −151552. −0.201810
\(225\) 0 0
\(226\) −569856. −0.742154
\(227\) 566400. 0.729556 0.364778 0.931095i \(-0.381145\pi\)
0.364778 + 0.931095i \(0.381145\pi\)
\(228\) 0 0
\(229\) −587206. −0.739949 −0.369974 0.929042i \(-0.620634\pi\)
−0.369974 + 0.929042i \(0.620634\pi\)
\(230\) 1.47456e6 1.83799
\(231\) 0 0
\(232\) −6144.00 −0.00749430
\(233\) −579456. −0.699247 −0.349624 0.936890i \(-0.613691\pi\)
−0.349624 + 0.936890i \(0.613691\pi\)
\(234\) 0 0
\(235\) 1.84320e6 2.17722
\(236\) 208896. 0.244146
\(237\) 0 0
\(238\) 340992. 0.390213
\(239\) 584448. 0.661837 0.330919 0.943659i \(-0.392641\pi\)
0.330919 + 0.943659i \(0.392641\pi\)
\(240\) 0 0
\(241\) −414130. −0.459298 −0.229649 0.973274i \(-0.573758\pi\)
−0.229649 + 0.973274i \(0.573758\pi\)
\(242\) −54380.0 −0.0596899
\(243\) 0 0
\(244\) 684512. 0.736049
\(245\) 489312. 0.520800
\(246\) 0 0
\(247\) 221776. 0.231298
\(248\) −292096. −0.301576
\(249\) 0 0
\(250\) 1.13894e6 1.15253
\(251\) 1.89965e6 1.90322 0.951610 0.307309i \(-0.0994287\pi\)
0.951610 + 0.307309i \(0.0994287\pi\)
\(252\) 0 0
\(253\) −1.47456e6 −1.44831
\(254\) −3952.00 −0.00384355
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) −447744. −0.422860 −0.211430 0.977393i \(-0.567812\pi\)
−0.211430 + 0.977393i \(0.567812\pi\)
\(258\) 0 0
\(259\) −858104. −0.794860
\(260\) −513024. −0.470657
\(261\) 0 0
\(262\) −897024. −0.807330
\(263\) 67584.0 0.0602496 0.0301248 0.999546i \(-0.490410\pi\)
0.0301248 + 0.999546i \(0.490410\pi\)
\(264\) 0 0
\(265\) −746496. −0.652999
\(266\) 393088. 0.340632
\(267\) 0 0
\(268\) 586496. 0.498802
\(269\) −564192. −0.475386 −0.237693 0.971340i \(-0.576391\pi\)
−0.237693 + 0.971340i \(0.576391\pi\)
\(270\) 0 0
\(271\) 720308. 0.595792 0.297896 0.954598i \(-0.403715\pi\)
0.297896 + 0.954598i \(0.403715\pi\)
\(272\) −147456. −0.120848
\(273\) 0 0
\(274\) 1.11590e6 0.897946
\(275\) −2.33894e6 −1.86504
\(276\) 0 0
\(277\) −141142. −0.110524 −0.0552620 0.998472i \(-0.517599\pi\)
−0.0552620 + 0.998472i \(0.517599\pi\)
\(278\) 708800. 0.550062
\(279\) 0 0
\(280\) −909312. −0.693135
\(281\) −584448. −0.441550 −0.220775 0.975325i \(-0.570859\pi\)
−0.220775 + 0.975325i \(0.570859\pi\)
\(282\) 0 0
\(283\) 177056. 0.131415 0.0657074 0.997839i \(-0.479070\pi\)
0.0657074 + 0.997839i \(0.479070\pi\)
\(284\) −1.03219e6 −0.759390
\(285\) 0 0
\(286\) 513024. 0.370871
\(287\) −994560. −0.712732
\(288\) 0 0
\(289\) −1.08808e6 −0.766331
\(290\) −36864.0 −0.0257399
\(291\) 0 0
\(292\) −268960. −0.184600
\(293\) 956832. 0.651128 0.325564 0.945520i \(-0.394446\pi\)
0.325564 + 0.945520i \(0.394446\pi\)
\(294\) 0 0
\(295\) 1.25338e6 0.838545
\(296\) 371072. 0.246166
\(297\) 0 0
\(298\) −944256. −0.615955
\(299\) −1.28256e6 −0.829659
\(300\) 0 0
\(301\) 2.20106e6 1.40028
\(302\) −1.93134e6 −1.21855
\(303\) 0 0
\(304\) −169984. −0.105493
\(305\) 4.10707e6 2.52803
\(306\) 0 0
\(307\) 2.88286e6 1.74573 0.872867 0.487958i \(-0.162258\pi\)
0.872867 + 0.487958i \(0.162258\pi\)
\(308\) 909312. 0.546180
\(309\) 0 0
\(310\) −1.75258e6 −1.03579
\(311\) 2.60045e6 1.52457 0.762285 0.647242i \(-0.224078\pi\)
0.762285 + 0.647242i \(0.224078\pi\)
\(312\) 0 0
\(313\) −2.58079e6 −1.48899 −0.744495 0.667628i \(-0.767310\pi\)
−0.744495 + 0.667628i \(0.767310\pi\)
\(314\) 1.52434e6 0.872487
\(315\) 0 0
\(316\) 449216. 0.253068
\(317\) −2.31101e6 −1.29168 −0.645838 0.763475i \(-0.723492\pi\)
−0.645838 + 0.763475i \(0.723492\pi\)
\(318\) 0 0
\(319\) 36864.0 0.0202827
\(320\) 393216. 0.214663
\(321\) 0 0
\(322\) −2.27328e6 −1.22184
\(323\) 382464. 0.203978
\(324\) 0 0
\(325\) −2.03439e6 −1.06838
\(326\) 651680. 0.339618
\(327\) 0 0
\(328\) 430080. 0.220732
\(329\) −2.84160e6 −1.44735
\(330\) 0 0
\(331\) −637024. −0.319585 −0.159792 0.987151i \(-0.551082\pi\)
−0.159792 + 0.987151i \(0.551082\pi\)
\(332\) 1.06291e6 0.529239
\(333\) 0 0
\(334\) 2.26406e6 1.11051
\(335\) 3.51898e6 1.71319
\(336\) 0 0
\(337\) 3.38665e6 1.62441 0.812206 0.583371i \(-0.198267\pi\)
0.812206 + 0.583371i \(0.198267\pi\)
\(338\) −1.03895e6 −0.494655
\(339\) 0 0
\(340\) −884736. −0.415065
\(341\) 1.75258e6 0.816189
\(342\) 0 0
\(343\) 1.73308e6 0.795396
\(344\) −951808. −0.433664
\(345\) 0 0
\(346\) 872832. 0.391959
\(347\) −2.77824e6 −1.23864 −0.619321 0.785138i \(-0.712592\pi\)
−0.619321 + 0.785138i \(0.712592\pi\)
\(348\) 0 0
\(349\) 1.55536e6 0.683545 0.341772 0.939783i \(-0.388973\pi\)
0.341772 + 0.939783i \(0.388973\pi\)
\(350\) −3.60587e6 −1.57340
\(351\) 0 0
\(352\) −393216. −0.169151
\(353\) −2.11776e6 −0.904565 −0.452283 0.891875i \(-0.649390\pi\)
−0.452283 + 0.891875i \(0.649390\pi\)
\(354\) 0 0
\(355\) −6.19315e6 −2.60820
\(356\) 1.30867e6 0.547275
\(357\) 0 0
\(358\) 1.64966e6 0.680280
\(359\) −2.17498e6 −0.890673 −0.445337 0.895363i \(-0.646916\pi\)
−0.445337 + 0.895363i \(0.646916\pi\)
\(360\) 0 0
\(361\) −2.03520e6 −0.821939
\(362\) −102232. −0.0410030
\(363\) 0 0
\(364\) 790912. 0.312878
\(365\) −1.61376e6 −0.634026
\(366\) 0 0
\(367\) 1.05336e6 0.408235 0.204117 0.978946i \(-0.434568\pi\)
0.204117 + 0.978946i \(0.434568\pi\)
\(368\) 983040. 0.378400
\(369\) 0 0
\(370\) 2.22643e6 0.845483
\(371\) 1.15085e6 0.434093
\(372\) 0 0
\(373\) −677098. −0.251988 −0.125994 0.992031i \(-0.540212\pi\)
−0.125994 + 0.992031i \(0.540212\pi\)
\(374\) 884736. 0.327065
\(375\) 0 0
\(376\) 1.22880e6 0.448241
\(377\) 32064.0 0.0116189
\(378\) 0 0
\(379\) −5.10748e6 −1.82645 −0.913227 0.407452i \(-0.866418\pi\)
−0.913227 + 0.407452i \(0.866418\pi\)
\(380\) −1.01990e6 −0.362327
\(381\) 0 0
\(382\) −1.60051e6 −0.561178
\(383\) −1.63200e6 −0.568491 −0.284245 0.958752i \(-0.591743\pi\)
−0.284245 + 0.958752i \(0.591743\pi\)
\(384\) 0 0
\(385\) 5.45587e6 1.87591
\(386\) 2.79860e6 0.956032
\(387\) 0 0
\(388\) −479008. −0.161534
\(389\) 4.46563e6 1.49627 0.748133 0.663549i \(-0.230950\pi\)
0.748133 + 0.663549i \(0.230950\pi\)
\(390\) 0 0
\(391\) −2.21184e6 −0.731664
\(392\) 326208. 0.107221
\(393\) 0 0
\(394\) −1.62547e6 −0.527520
\(395\) 2.69530e6 0.869188
\(396\) 0 0
\(397\) −611026. −0.194573 −0.0972867 0.995256i \(-0.531016\pi\)
−0.0972867 + 0.995256i \(0.531016\pi\)
\(398\) −1.44798e6 −0.458201
\(399\) 0 0
\(400\) 1.55930e6 0.487280
\(401\) 6.09158e6 1.89177 0.945887 0.324496i \(-0.105195\pi\)
0.945887 + 0.324496i \(0.105195\pi\)
\(402\) 0 0
\(403\) 1.52438e6 0.467552
\(404\) −2.85850e6 −0.871333
\(405\) 0 0
\(406\) 56832.0 0.0171111
\(407\) −2.22643e6 −0.666229
\(408\) 0 0
\(409\) 2.89108e6 0.854578 0.427289 0.904115i \(-0.359469\pi\)
0.427289 + 0.904115i \(0.359469\pi\)
\(410\) 2.58048e6 0.758125
\(411\) 0 0
\(412\) −1.84365e6 −0.535100
\(413\) −1.93229e6 −0.557438
\(414\) 0 0
\(415\) 6.37747e6 1.81773
\(416\) −342016. −0.0968976
\(417\) 0 0
\(418\) 1.01990e6 0.285508
\(419\) 2.98406e6 0.830373 0.415186 0.909736i \(-0.363716\pi\)
0.415186 + 0.909736i \(0.363716\pi\)
\(420\) 0 0
\(421\) 822074. 0.226051 0.113025 0.993592i \(-0.463946\pi\)
0.113025 + 0.993592i \(0.463946\pi\)
\(422\) 607424. 0.166039
\(423\) 0 0
\(424\) −497664. −0.134438
\(425\) −3.50842e6 −0.942191
\(426\) 0 0
\(427\) −6.33174e6 −1.68056
\(428\) 1.21651e6 0.321001
\(429\) 0 0
\(430\) −5.71085e6 −1.48946
\(431\) −6.32448e6 −1.63995 −0.819977 0.572397i \(-0.806014\pi\)
−0.819977 + 0.572397i \(0.806014\pi\)
\(432\) 0 0
\(433\) −851902. −0.218358 −0.109179 0.994022i \(-0.534822\pi\)
−0.109179 + 0.994022i \(0.534822\pi\)
\(434\) 2.70189e6 0.688562
\(435\) 0 0
\(436\) −3.69789e6 −0.931617
\(437\) −2.54976e6 −0.638698
\(438\) 0 0
\(439\) −334732. −0.0828964 −0.0414482 0.999141i \(-0.513197\pi\)
−0.0414482 + 0.999141i \(0.513197\pi\)
\(440\) −2.35930e6 −0.580966
\(441\) 0 0
\(442\) 769536. 0.187358
\(443\) 1.76218e6 0.426619 0.213309 0.976985i \(-0.431576\pi\)
0.213309 + 0.976985i \(0.431576\pi\)
\(444\) 0 0
\(445\) 7.85203e6 1.87967
\(446\) −4.37330e6 −1.04105
\(447\) 0 0
\(448\) −606208. −0.142701
\(449\) −6.51859e6 −1.52594 −0.762971 0.646433i \(-0.776260\pi\)
−0.762971 + 0.646433i \(0.776260\pi\)
\(450\) 0 0
\(451\) −2.58048e6 −0.597392
\(452\) −2.27942e6 −0.524782
\(453\) 0 0
\(454\) 2.26560e6 0.515874
\(455\) 4.74547e6 1.07461
\(456\) 0 0
\(457\) −92074.0 −0.0206227 −0.0103114 0.999947i \(-0.503282\pi\)
−0.0103114 + 0.999947i \(0.503282\pi\)
\(458\) −2.34882e6 −0.523223
\(459\) 0 0
\(460\) 5.89824e6 1.29965
\(461\) −257568. −0.0564468 −0.0282234 0.999602i \(-0.508985\pi\)
−0.0282234 + 0.999602i \(0.508985\pi\)
\(462\) 0 0
\(463\) 3.96228e6 0.858998 0.429499 0.903067i \(-0.358690\pi\)
0.429499 + 0.903067i \(0.358690\pi\)
\(464\) −24576.0 −0.00529927
\(465\) 0 0
\(466\) −2.31782e6 −0.494442
\(467\) 3.48941e6 0.740388 0.370194 0.928954i \(-0.379291\pi\)
0.370194 + 0.928954i \(0.379291\pi\)
\(468\) 0 0
\(469\) −5.42509e6 −1.13887
\(470\) 7.37280e6 1.53953
\(471\) 0 0
\(472\) 835584. 0.172637
\(473\) 5.71085e6 1.17367
\(474\) 0 0
\(475\) −4.04442e6 −0.822475
\(476\) 1.36397e6 0.275922
\(477\) 0 0
\(478\) 2.33779e6 0.467990
\(479\) −513024. −0.102164 −0.0510821 0.998694i \(-0.516267\pi\)
−0.0510821 + 0.998694i \(0.516267\pi\)
\(480\) 0 0
\(481\) −1.93653e6 −0.381647
\(482\) −1.65652e6 −0.324772
\(483\) 0 0
\(484\) −217520. −0.0422071
\(485\) −2.87405e6 −0.554804
\(486\) 0 0
\(487\) 4.14499e6 0.791956 0.395978 0.918260i \(-0.370406\pi\)
0.395978 + 0.918260i \(0.370406\pi\)
\(488\) 2.73805e6 0.520465
\(489\) 0 0
\(490\) 1.95725e6 0.368261
\(491\) −2.75866e6 −0.516409 −0.258205 0.966090i \(-0.583131\pi\)
−0.258205 + 0.966090i \(0.583131\pi\)
\(492\) 0 0
\(493\) 55296.0 0.0102465
\(494\) 887104. 0.163552
\(495\) 0 0
\(496\) −1.16838e6 −0.213246
\(497\) 9.54778e6 1.73385
\(498\) 0 0
\(499\) 660896. 0.118818 0.0594089 0.998234i \(-0.481078\pi\)
0.0594089 + 0.998234i \(0.481078\pi\)
\(500\) 4.55578e6 0.814962
\(501\) 0 0
\(502\) 7.59859e6 1.34578
\(503\) 944640. 0.166474 0.0832370 0.996530i \(-0.473474\pi\)
0.0832370 + 0.996530i \(0.473474\pi\)
\(504\) 0 0
\(505\) −1.71510e7 −2.99268
\(506\) −5.89824e6 −1.02411
\(507\) 0 0
\(508\) −15808.0 −0.00271780
\(509\) 7.83773e6 1.34090 0.670449 0.741956i \(-0.266102\pi\)
0.670449 + 0.741956i \(0.266102\pi\)
\(510\) 0 0
\(511\) 2.48788e6 0.421480
\(512\) 262144. 0.0441942
\(513\) 0 0
\(514\) −1.79098e6 −0.299007
\(515\) −1.10619e7 −1.83785
\(516\) 0 0
\(517\) −7.37280e6 −1.21313
\(518\) −3.43242e6 −0.562051
\(519\) 0 0
\(520\) −2.05210e6 −0.332805
\(521\) −3.29645e6 −0.532049 −0.266025 0.963966i \(-0.585710\pi\)
−0.266025 + 0.963966i \(0.585710\pi\)
\(522\) 0 0
\(523\) 6.50238e6 1.03948 0.519742 0.854323i \(-0.326028\pi\)
0.519742 + 0.854323i \(0.326028\pi\)
\(524\) −3.58810e6 −0.570868
\(525\) 0 0
\(526\) 270336. 0.0426029
\(527\) 2.62886e6 0.412327
\(528\) 0 0
\(529\) 8.30926e6 1.29099
\(530\) −2.98598e6 −0.461740
\(531\) 0 0
\(532\) 1.57235e6 0.240863
\(533\) −2.24448e6 −0.342214
\(534\) 0 0
\(535\) 7.29907e6 1.10251
\(536\) 2.34598e6 0.352706
\(537\) 0 0
\(538\) −2.25677e6 −0.336149
\(539\) −1.95725e6 −0.290184
\(540\) 0 0
\(541\) 9.82714e6 1.44356 0.721778 0.692124i \(-0.243325\pi\)
0.721778 + 0.692124i \(0.243325\pi\)
\(542\) 2.88123e6 0.421289
\(543\) 0 0
\(544\) −589824. −0.0854526
\(545\) −2.21873e7 −3.19973
\(546\) 0 0
\(547\) −3.42580e6 −0.489546 −0.244773 0.969580i \(-0.578713\pi\)
−0.244773 + 0.969580i \(0.578713\pi\)
\(548\) 4.46362e6 0.634944
\(549\) 0 0
\(550\) −9.35578e6 −1.31878
\(551\) 63744.0 0.00894459
\(552\) 0 0
\(553\) −4.15525e6 −0.577809
\(554\) −564568. −0.0781523
\(555\) 0 0
\(556\) 2.83520e6 0.388953
\(557\) 6.43363e6 0.878655 0.439327 0.898327i \(-0.355217\pi\)
0.439327 + 0.898327i \(0.355217\pi\)
\(558\) 0 0
\(559\) 4.96725e6 0.672336
\(560\) −3.63725e6 −0.490120
\(561\) 0 0
\(562\) −2.33779e6 −0.312223
\(563\) −753024. −0.100124 −0.0500620 0.998746i \(-0.515942\pi\)
−0.0500620 + 0.998746i \(0.515942\pi\)
\(564\) 0 0
\(565\) −1.36765e7 −1.80242
\(566\) 708224. 0.0929244
\(567\) 0 0
\(568\) −4.12877e6 −0.536970
\(569\) 1.11481e7 1.44351 0.721755 0.692148i \(-0.243336\pi\)
0.721755 + 0.692148i \(0.243336\pi\)
\(570\) 0 0
\(571\) 191024. 0.0245187 0.0122594 0.999925i \(-0.496098\pi\)
0.0122594 + 0.999925i \(0.496098\pi\)
\(572\) 2.05210e6 0.262245
\(573\) 0 0
\(574\) −3.97824e6 −0.503978
\(575\) 2.33894e7 2.95019
\(576\) 0 0
\(577\) 1.03722e7 1.29697 0.648486 0.761227i \(-0.275403\pi\)
0.648486 + 0.761227i \(0.275403\pi\)
\(578\) −4.35232e6 −0.541878
\(579\) 0 0
\(580\) −147456. −0.0182009
\(581\) −9.83194e6 −1.20837
\(582\) 0 0
\(583\) 2.98598e6 0.363845
\(584\) −1.07584e6 −0.130532
\(585\) 0 0
\(586\) 3.82733e6 0.460417
\(587\) 2.97062e6 0.355838 0.177919 0.984045i \(-0.443063\pi\)
0.177919 + 0.984045i \(0.443063\pi\)
\(588\) 0 0
\(589\) 3.03050e6 0.359936
\(590\) 5.01350e6 0.592941
\(591\) 0 0
\(592\) 1.48429e6 0.174066
\(593\) 7.55827e6 0.882644 0.441322 0.897349i \(-0.354510\pi\)
0.441322 + 0.897349i \(0.354510\pi\)
\(594\) 0 0
\(595\) 8.18381e6 0.947683
\(596\) −3.77702e6 −0.435546
\(597\) 0 0
\(598\) −5.13024e6 −0.586658
\(599\) −6.69158e6 −0.762012 −0.381006 0.924573i \(-0.624422\pi\)
−0.381006 + 0.924573i \(0.624422\pi\)
\(600\) 0 0
\(601\) −3.20359e6 −0.361785 −0.180893 0.983503i \(-0.557899\pi\)
−0.180893 + 0.983503i \(0.557899\pi\)
\(602\) 8.80422e6 0.990147
\(603\) 0 0
\(604\) −7.72538e6 −0.861643
\(605\) −1.30512e6 −0.144965
\(606\) 0 0
\(607\) −1.35585e7 −1.49362 −0.746809 0.665038i \(-0.768415\pi\)
−0.746809 + 0.665038i \(0.768415\pi\)
\(608\) −679936. −0.0745949
\(609\) 0 0
\(610\) 1.64283e7 1.78759
\(611\) −6.41280e6 −0.694936
\(612\) 0 0
\(613\) −1.07654e7 −1.15712 −0.578561 0.815639i \(-0.696385\pi\)
−0.578561 + 0.815639i \(0.696385\pi\)
\(614\) 1.15315e7 1.23442
\(615\) 0 0
\(616\) 3.63725e6 0.386208
\(617\) 9.33504e6 0.987196 0.493598 0.869690i \(-0.335681\pi\)
0.493598 + 0.869690i \(0.335681\pi\)
\(618\) 0 0
\(619\) 9.07664e6 0.952135 0.476067 0.879409i \(-0.342062\pi\)
0.476067 + 0.879409i \(0.342062\pi\)
\(620\) −7.01030e6 −0.732416
\(621\) 0 0
\(622\) 1.04018e7 1.07803
\(623\) −1.21052e7 −1.24955
\(624\) 0 0
\(625\) 8.30028e6 0.849949
\(626\) −1.03232e7 −1.05288
\(627\) 0 0
\(628\) 6.09738e6 0.616941
\(629\) −3.33965e6 −0.336569
\(630\) 0 0
\(631\) −1.13367e7 −1.13348 −0.566741 0.823896i \(-0.691796\pi\)
−0.566741 + 0.823896i \(0.691796\pi\)
\(632\) 1.79686e6 0.178946
\(633\) 0 0
\(634\) −9.24403e6 −0.913352
\(635\) −94848.0 −0.00933456
\(636\) 0 0
\(637\) −1.70240e6 −0.166231
\(638\) 147456. 0.0143420
\(639\) 0 0
\(640\) 1.57286e6 0.151789
\(641\) −1.55449e7 −1.49432 −0.747159 0.664646i \(-0.768582\pi\)
−0.747159 + 0.664646i \(0.768582\pi\)
\(642\) 0 0
\(643\) −8.80026e6 −0.839399 −0.419699 0.907663i \(-0.637864\pi\)
−0.419699 + 0.907663i \(0.637864\pi\)
\(644\) −9.09312e6 −0.863969
\(645\) 0 0
\(646\) 1.52986e6 0.144235
\(647\) −1.08449e7 −1.01851 −0.509256 0.860615i \(-0.670079\pi\)
−0.509256 + 0.860615i \(0.670079\pi\)
\(648\) 0 0
\(649\) −5.01350e6 −0.467229
\(650\) −8.13758e6 −0.755460
\(651\) 0 0
\(652\) 2.60672e6 0.240146
\(653\) 9.88771e6 0.907429 0.453715 0.891147i \(-0.350099\pi\)
0.453715 + 0.891147i \(0.350099\pi\)
\(654\) 0 0
\(655\) −2.15286e7 −1.96070
\(656\) 1.72032e6 0.156081
\(657\) 0 0
\(658\) −1.13664e7 −1.02343
\(659\) 1.46150e7 1.31095 0.655476 0.755216i \(-0.272468\pi\)
0.655476 + 0.755216i \(0.272468\pi\)
\(660\) 0 0
\(661\) 1.57792e7 1.40469 0.702347 0.711834i \(-0.252135\pi\)
0.702347 + 0.711834i \(0.252135\pi\)
\(662\) −2.54810e6 −0.225980
\(663\) 0 0
\(664\) 4.25165e6 0.374229
\(665\) 9.43411e6 0.827269
\(666\) 0 0
\(667\) −368640. −0.0320840
\(668\) 9.05626e6 0.785249
\(669\) 0 0
\(670\) 1.40759e7 1.21140
\(671\) −1.64283e7 −1.40859
\(672\) 0 0
\(673\) 6.64939e6 0.565906 0.282953 0.959134i \(-0.408686\pi\)
0.282953 + 0.959134i \(0.408686\pi\)
\(674\) 1.35466e7 1.14863
\(675\) 0 0
\(676\) −4.15579e6 −0.349774
\(677\) −4.42550e6 −0.371100 −0.185550 0.982635i \(-0.559407\pi\)
−0.185550 + 0.982635i \(0.559407\pi\)
\(678\) 0 0
\(679\) 4.43082e6 0.368816
\(680\) −3.53894e6 −0.293495
\(681\) 0 0
\(682\) 7.01030e6 0.577133
\(683\) 4.67827e6 0.383737 0.191869 0.981421i \(-0.438545\pi\)
0.191869 + 0.981421i \(0.438545\pi\)
\(684\) 0 0
\(685\) 2.67817e7 2.18078
\(686\) 6.93232e6 0.562430
\(687\) 0 0
\(688\) −3.80723e6 −0.306647
\(689\) 2.59718e6 0.208427
\(690\) 0 0
\(691\) −5.54102e6 −0.441463 −0.220731 0.975335i \(-0.570844\pi\)
−0.220731 + 0.975335i \(0.570844\pi\)
\(692\) 3.49133e6 0.277157
\(693\) 0 0
\(694\) −1.11130e7 −0.875853
\(695\) 1.70112e7 1.33590
\(696\) 0 0
\(697\) −3.87072e6 −0.301793
\(698\) 6.22143e6 0.483339
\(699\) 0 0
\(700\) −1.44235e7 −1.11256
\(701\) 6.53443e6 0.502242 0.251121 0.967956i \(-0.419201\pi\)
0.251121 + 0.967956i \(0.419201\pi\)
\(702\) 0 0
\(703\) −3.84987e6 −0.293804
\(704\) −1.57286e6 −0.119608
\(705\) 0 0
\(706\) −8.47104e6 −0.639624
\(707\) 2.64411e7 1.98944
\(708\) 0 0
\(709\) −3.86541e6 −0.288789 −0.144394 0.989520i \(-0.546123\pi\)
−0.144394 + 0.989520i \(0.546123\pi\)
\(710\) −2.47726e7 −1.84428
\(711\) 0 0
\(712\) 5.23469e6 0.386982
\(713\) −1.75258e7 −1.29108
\(714\) 0 0
\(715\) 1.23126e7 0.900708
\(716\) 6.59866e6 0.481031
\(717\) 0 0
\(718\) −8.69990e6 −0.629801
\(719\) −4.80614e6 −0.346717 −0.173358 0.984859i \(-0.555462\pi\)
−0.173358 + 0.984859i \(0.555462\pi\)
\(720\) 0 0
\(721\) 1.70537e7 1.22175
\(722\) −8.14081e6 −0.581199
\(723\) 0 0
\(724\) −408928. −0.0289935
\(725\) −584736. −0.0413157
\(726\) 0 0
\(727\) −1.90590e7 −1.33741 −0.668704 0.743529i \(-0.733150\pi\)
−0.668704 + 0.743529i \(0.733150\pi\)
\(728\) 3.16365e6 0.221238
\(729\) 0 0
\(730\) −6.45504e6 −0.448324
\(731\) 8.56627e6 0.592923
\(732\) 0 0
\(733\) 5.69616e6 0.391582 0.195791 0.980646i \(-0.437273\pi\)
0.195791 + 0.980646i \(0.437273\pi\)
\(734\) 4.21342e6 0.288666
\(735\) 0 0
\(736\) 3.93216e6 0.267570
\(737\) −1.40759e7 −0.954570
\(738\) 0 0
\(739\) 1.84902e7 1.24546 0.622730 0.782437i \(-0.286024\pi\)
0.622730 + 0.782437i \(0.286024\pi\)
\(740\) 8.90573e6 0.597847
\(741\) 0 0
\(742\) 4.60339e6 0.306950
\(743\) −9.90336e6 −0.658128 −0.329064 0.944308i \(-0.606733\pi\)
−0.329064 + 0.944308i \(0.606733\pi\)
\(744\) 0 0
\(745\) −2.26621e7 −1.49593
\(746\) −2.70839e6 −0.178182
\(747\) 0 0
\(748\) 3.53894e6 0.231270
\(749\) −1.12527e7 −0.732915
\(750\) 0 0
\(751\) −9.05914e6 −0.586121 −0.293060 0.956094i \(-0.594674\pi\)
−0.293060 + 0.956094i \(0.594674\pi\)
\(752\) 4.91520e6 0.316954
\(753\) 0 0
\(754\) 128256. 0.00821579
\(755\) −4.63523e7 −2.95940
\(756\) 0 0
\(757\) −1.16677e7 −0.740022 −0.370011 0.929027i \(-0.620646\pi\)
−0.370011 + 0.929027i \(0.620646\pi\)
\(758\) −2.04299e7 −1.29150
\(759\) 0 0
\(760\) −4.07962e6 −0.256204
\(761\) 1.27398e7 0.797444 0.398722 0.917072i \(-0.369454\pi\)
0.398722 + 0.917072i \(0.369454\pi\)
\(762\) 0 0
\(763\) 3.42055e7 2.12708
\(764\) −6.40205e6 −0.396813
\(765\) 0 0
\(766\) −6.52800e6 −0.401983
\(767\) −4.36070e6 −0.267651
\(768\) 0 0
\(769\) 1.06783e7 0.651156 0.325578 0.945515i \(-0.394441\pi\)
0.325578 + 0.945515i \(0.394441\pi\)
\(770\) 2.18235e7 1.32647
\(771\) 0 0
\(772\) 1.11944e7 0.676017
\(773\) 9.18634e6 0.552960 0.276480 0.961020i \(-0.410832\pi\)
0.276480 + 0.961020i \(0.410832\pi\)
\(774\) 0 0
\(775\) −2.77993e7 −1.66257
\(776\) −1.91603e6 −0.114222
\(777\) 0 0
\(778\) 1.78625e7 1.05802
\(779\) −4.46208e6 −0.263447
\(780\) 0 0
\(781\) 2.47726e7 1.45326
\(782\) −8.84736e6 −0.517365
\(783\) 0 0
\(784\) 1.30483e6 0.0758166
\(785\) 3.65843e7 2.11895
\(786\) 0 0
\(787\) −9.28209e6 −0.534206 −0.267103 0.963668i \(-0.586066\pi\)
−0.267103 + 0.963668i \(0.586066\pi\)
\(788\) −6.50189e6 −0.373013
\(789\) 0 0
\(790\) 1.07812e7 0.614609
\(791\) 2.10847e7 1.19819
\(792\) 0 0
\(793\) −1.42892e7 −0.806909
\(794\) −2.44410e6 −0.137584
\(795\) 0 0
\(796\) −5.79194e6 −0.323997
\(797\) −21792.0 −0.00121521 −0.000607605 1.00000i \(-0.500193\pi\)
−0.000607605 1.00000i \(0.500193\pi\)
\(798\) 0 0
\(799\) −1.10592e7 −0.612854
\(800\) 6.23718e6 0.344559
\(801\) 0 0
\(802\) 2.43663e7 1.33769
\(803\) 6.45504e6 0.353273
\(804\) 0 0
\(805\) −5.45587e7 −2.96739
\(806\) 6.09750e6 0.330609
\(807\) 0 0
\(808\) −1.14340e7 −0.616125
\(809\) −1.23085e7 −0.661204 −0.330602 0.943770i \(-0.607252\pi\)
−0.330602 + 0.943770i \(0.607252\pi\)
\(810\) 0 0
\(811\) −2.34636e7 −1.25269 −0.626343 0.779547i \(-0.715449\pi\)
−0.626343 + 0.779547i \(0.715449\pi\)
\(812\) 227328. 0.0120994
\(813\) 0 0
\(814\) −8.90573e6 −0.471095
\(815\) 1.56403e7 0.824806
\(816\) 0 0
\(817\) 9.87501e6 0.517586
\(818\) 1.15643e7 0.604278
\(819\) 0 0
\(820\) 1.03219e7 0.536075
\(821\) 1.44206e7 0.746666 0.373333 0.927697i \(-0.378215\pi\)
0.373333 + 0.927697i \(0.378215\pi\)
\(822\) 0 0
\(823\) 3.43419e7 1.76736 0.883679 0.468093i \(-0.155059\pi\)
0.883679 + 0.468093i \(0.155059\pi\)
\(824\) −7.37459e6 −0.378373
\(825\) 0 0
\(826\) −7.72915e6 −0.394168
\(827\) −2.13327e7 −1.08463 −0.542316 0.840174i \(-0.682453\pi\)
−0.542316 + 0.840174i \(0.682453\pi\)
\(828\) 0 0
\(829\) 2.63751e6 0.133293 0.0666465 0.997777i \(-0.478770\pi\)
0.0666465 + 0.997777i \(0.478770\pi\)
\(830\) 2.55099e7 1.28533
\(831\) 0 0
\(832\) −1.36806e6 −0.0685170
\(833\) −2.93587e6 −0.146597
\(834\) 0 0
\(835\) 5.43375e7 2.69702
\(836\) 4.07962e6 0.201885
\(837\) 0 0
\(838\) 1.19363e7 0.587162
\(839\) 1.00577e7 0.493282 0.246641 0.969107i \(-0.420673\pi\)
0.246641 + 0.969107i \(0.420673\pi\)
\(840\) 0 0
\(841\) −2.05019e7 −0.999551
\(842\) 3.28830e6 0.159842
\(843\) 0 0
\(844\) 2.42970e6 0.117407
\(845\) −2.49348e7 −1.20133
\(846\) 0 0
\(847\) 2.01206e6 0.0963679
\(848\) −1.99066e6 −0.0950619
\(849\) 0 0
\(850\) −1.40337e7 −0.666229
\(851\) 2.22643e7 1.05387
\(852\) 0 0
\(853\) −2.30748e7 −1.08584 −0.542919 0.839785i \(-0.682681\pi\)
−0.542919 + 0.839785i \(0.682681\pi\)
\(854\) −2.53269e7 −1.18833
\(855\) 0 0
\(856\) 4.86605e6 0.226982
\(857\) −3.51646e7 −1.63551 −0.817756 0.575565i \(-0.804782\pi\)
−0.817756 + 0.575565i \(0.804782\pi\)
\(858\) 0 0
\(859\) 1.66022e7 0.767684 0.383842 0.923399i \(-0.374601\pi\)
0.383842 + 0.923399i \(0.374601\pi\)
\(860\) −2.28434e7 −1.05321
\(861\) 0 0
\(862\) −2.52979e7 −1.15962
\(863\) 2.97009e7 1.35751 0.678754 0.734366i \(-0.262520\pi\)
0.678754 + 0.734366i \(0.262520\pi\)
\(864\) 0 0
\(865\) 2.09480e7 0.951923
\(866\) −3.40761e6 −0.154403
\(867\) 0 0
\(868\) 1.08076e7 0.486887
\(869\) −1.07812e7 −0.484303
\(870\) 0 0
\(871\) −1.22431e7 −0.546822
\(872\) −1.47916e7 −0.658753
\(873\) 0 0
\(874\) −1.01990e7 −0.451628
\(875\) −4.21409e7 −1.86073
\(876\) 0 0
\(877\) −1.17943e7 −0.517811 −0.258906 0.965903i \(-0.583362\pi\)
−0.258906 + 0.965903i \(0.583362\pi\)
\(878\) −1.33893e6 −0.0586166
\(879\) 0 0
\(880\) −9.43718e6 −0.410805
\(881\) 2.10378e7 0.913190 0.456595 0.889675i \(-0.349069\pi\)
0.456595 + 0.889675i \(0.349069\pi\)
\(882\) 0 0
\(883\) 2.12192e7 0.915855 0.457928 0.888990i \(-0.348592\pi\)
0.457928 + 0.888990i \(0.348592\pi\)
\(884\) 3.07814e6 0.132482
\(885\) 0 0
\(886\) 7.04870e6 0.301665
\(887\) −2.28818e7 −0.976520 −0.488260 0.872698i \(-0.662368\pi\)
−0.488260 + 0.872698i \(0.662368\pi\)
\(888\) 0 0
\(889\) 146224. 0.00620532
\(890\) 3.14081e7 1.32913
\(891\) 0 0
\(892\) −1.74932e7 −0.736134
\(893\) −1.27488e7 −0.534984
\(894\) 0 0
\(895\) 3.95919e7 1.65215
\(896\) −2.42483e6 −0.100905
\(897\) 0 0
\(898\) −2.60744e7 −1.07900
\(899\) 438144. 0.0180808
\(900\) 0 0
\(901\) 4.47898e6 0.183809
\(902\) −1.03219e7 −0.422420
\(903\) 0 0
\(904\) −9.11770e6 −0.371077
\(905\) −2.45357e6 −0.0995810
\(906\) 0 0
\(907\) −9.68692e6 −0.390992 −0.195496 0.980705i \(-0.562632\pi\)
−0.195496 + 0.980705i \(0.562632\pi\)
\(908\) 9.06240e6 0.364778
\(909\) 0 0
\(910\) 1.89819e7 0.759864
\(911\) −9.38112e6 −0.374506 −0.187253 0.982312i \(-0.559958\pi\)
−0.187253 + 0.982312i \(0.559958\pi\)
\(912\) 0 0
\(913\) −2.55099e7 −1.01282
\(914\) −368296. −0.0145825
\(915\) 0 0
\(916\) −9.39530e6 −0.369974
\(917\) 3.31899e7 1.30341
\(918\) 0 0
\(919\) −4.21870e7 −1.64774 −0.823872 0.566775i \(-0.808191\pi\)
−0.823872 + 0.566775i \(0.808191\pi\)
\(920\) 2.35930e7 0.918994
\(921\) 0 0
\(922\) −1.03027e6 −0.0399139
\(923\) 2.15470e7 0.832497
\(924\) 0 0
\(925\) 3.53156e7 1.35710
\(926\) 1.58491e7 0.607403
\(927\) 0 0
\(928\) −98304.0 −0.00374715
\(929\) 3.04556e7 1.15779 0.578893 0.815404i \(-0.303485\pi\)
0.578893 + 0.815404i \(0.303485\pi\)
\(930\) 0 0
\(931\) −3.38441e6 −0.127970
\(932\) −9.27130e6 −0.349624
\(933\) 0 0
\(934\) 1.39576e7 0.523534
\(935\) 2.12337e7 0.794321
\(936\) 0 0
\(937\) 1.47847e7 0.550128 0.275064 0.961426i \(-0.411301\pi\)
0.275064 + 0.961426i \(0.411301\pi\)
\(938\) −2.17004e7 −0.805304
\(939\) 0 0
\(940\) 2.94912e7 1.08861
\(941\) 9.91997e6 0.365205 0.182602 0.983187i \(-0.441548\pi\)
0.182602 + 0.983187i \(0.441548\pi\)
\(942\) 0 0
\(943\) 2.58048e7 0.944977
\(944\) 3.34234e6 0.122073
\(945\) 0 0
\(946\) 2.28434e7 0.829913
\(947\) −2.91610e6 −0.105664 −0.0528320 0.998603i \(-0.516825\pi\)
−0.0528320 + 0.998603i \(0.516825\pi\)
\(948\) 0 0
\(949\) 5.61454e6 0.202371
\(950\) −1.61777e7 −0.581578
\(951\) 0 0
\(952\) 5.45587e6 0.195107
\(953\) −1.40861e7 −0.502410 −0.251205 0.967934i \(-0.580827\pi\)
−0.251205 + 0.967934i \(0.580827\pi\)
\(954\) 0 0
\(955\) −3.84123e7 −1.36289
\(956\) 9.35117e6 0.330919
\(957\) 0 0
\(958\) −2.05210e6 −0.0722410
\(959\) −4.12884e7 −1.44971
\(960\) 0 0
\(961\) −7.79906e6 −0.272417
\(962\) −7.74613e6 −0.269865
\(963\) 0 0
\(964\) −6.62608e6 −0.229649
\(965\) 6.71664e7 2.32185
\(966\) 0 0
\(967\) 1.51949e7 0.522553 0.261276 0.965264i \(-0.415857\pi\)
0.261276 + 0.965264i \(0.415857\pi\)
\(968\) −870080. −0.0298449
\(969\) 0 0
\(970\) −1.14962e7 −0.392306
\(971\) 5.61220e7 1.91023 0.955113 0.296240i \(-0.0957329\pi\)
0.955113 + 0.296240i \(0.0957329\pi\)
\(972\) 0 0
\(973\) −2.62256e7 −0.888062
\(974\) 1.65800e7 0.559997
\(975\) 0 0
\(976\) 1.09522e7 0.368024
\(977\) −3.45625e7 −1.15843 −0.579214 0.815176i \(-0.696640\pi\)
−0.579214 + 0.815176i \(0.696640\pi\)
\(978\) 0 0
\(979\) −3.14081e7 −1.04733
\(980\) 7.82899e6 0.260400
\(981\) 0 0
\(982\) −1.10346e7 −0.365156
\(983\) −5.56385e7 −1.83650 −0.918252 0.395997i \(-0.870399\pi\)
−0.918252 + 0.395997i \(0.870399\pi\)
\(984\) 0 0
\(985\) −3.90113e7 −1.28115
\(986\) 221184. 0.00724538
\(987\) 0 0
\(988\) 3.54842e6 0.115649
\(989\) −5.71085e7 −1.85656
\(990\) 0 0
\(991\) −3.60028e7 −1.16453 −0.582267 0.812998i \(-0.697834\pi\)
−0.582267 + 0.812998i \(0.697834\pi\)
\(992\) −4.67354e6 −0.150788
\(993\) 0 0
\(994\) 3.81911e7 1.22602
\(995\) −3.47516e7 −1.11280
\(996\) 0 0
\(997\) 2.35811e7 0.751322 0.375661 0.926757i \(-0.377416\pi\)
0.375661 + 0.926757i \(0.377416\pi\)
\(998\) 2.64358e6 0.0840169
\(999\) 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 18.6.a.c.1.1 yes 1
3.2 odd 2 18.6.a.a.1.1 1
4.3 odd 2 144.6.a.l.1.1 1
5.2 odd 4 450.6.c.c.199.2 2
5.3 odd 4 450.6.c.c.199.1 2
5.4 even 2 450.6.a.k.1.1 1
7.6 odd 2 882.6.a.l.1.1 1
8.3 odd 2 576.6.a.b.1.1 1
8.5 even 2 576.6.a.a.1.1 1
9.2 odd 6 162.6.c.l.109.1 2
9.4 even 3 162.6.c.a.55.1 2
9.5 odd 6 162.6.c.l.55.1 2
9.7 even 3 162.6.c.a.109.1 2
12.11 even 2 144.6.a.a.1.1 1
15.2 even 4 450.6.c.m.199.1 2
15.8 even 4 450.6.c.m.199.2 2
15.14 odd 2 450.6.a.v.1.1 1
21.20 even 2 882.6.a.k.1.1 1
24.5 odd 2 576.6.a.bh.1.1 1
24.11 even 2 576.6.a.bi.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.a.a.1.1 1 3.2 odd 2
18.6.a.c.1.1 yes 1 1.1 even 1 trivial
144.6.a.a.1.1 1 12.11 even 2
144.6.a.l.1.1 1 4.3 odd 2
162.6.c.a.55.1 2 9.4 even 3
162.6.c.a.109.1 2 9.7 even 3
162.6.c.l.55.1 2 9.5 odd 6
162.6.c.l.109.1 2 9.2 odd 6
450.6.a.k.1.1 1 5.4 even 2
450.6.a.v.1.1 1 15.14 odd 2
450.6.c.c.199.1 2 5.3 odd 4
450.6.c.c.199.2 2 5.2 odd 4
450.6.c.m.199.1 2 15.2 even 4
450.6.c.m.199.2 2 15.8 even 4
576.6.a.a.1.1 1 8.5 even 2
576.6.a.b.1.1 1 8.3 odd 2
576.6.a.bh.1.1 1 24.5 odd 2
576.6.a.bi.1.1 1 24.11 even 2
882.6.a.k.1.1 1 21.20 even 2
882.6.a.l.1.1 1 7.6 odd 2