Properties

Label 450.6.a.k.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 18)
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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +16.0000 q^{4} +148.000 q^{7} -64.0000 q^{8} +O(q^{10})\) \(q-4.00000 q^{2} +16.0000 q^{4} +148.000 q^{7} -64.0000 q^{8} -384.000 q^{11} +334.000 q^{13} -592.000 q^{14} +256.000 q^{16} +576.000 q^{17} -664.000 q^{19} +1536.00 q^{22} -3840.00 q^{23} -1336.00 q^{26} +2368.00 q^{28} -96.0000 q^{29} -4564.00 q^{31} -1024.00 q^{32} -2304.00 q^{34} -5798.00 q^{37} +2656.00 q^{38} +6720.00 q^{41} +14872.0 q^{43} -6144.00 q^{44} +15360.0 q^{46} -19200.0 q^{47} +5097.00 q^{49} +5344.00 q^{52} +7776.00 q^{53} -9472.00 q^{56} +384.000 q^{58} +13056.0 q^{59} +42782.0 q^{61} +18256.0 q^{62} +4096.00 q^{64} -36656.0 q^{67} +9216.00 q^{68} -64512.0 q^{71} +16810.0 q^{73} +23192.0 q^{74} -10624.0 q^{76} -56832.0 q^{77} +28076.0 q^{79} -26880.0 q^{82} -66432.0 q^{83} -59488.0 q^{86} +24576.0 q^{88} +81792.0 q^{89} +49432.0 q^{91} -61440.0 q^{92} +76800.0 q^{94} +29938.0 q^{97} -20388.0 q^{98} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 148.000 1.14161 0.570803 0.821087i \(-0.306632\pi\)
0.570803 + 0.821087i \(0.306632\pi\)
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −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 \(-0.411631\pi\)
0.274068 + 0.961710i \(0.411631\pi\)
\(14\) −592.000 −0.807238
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 576.000 0.483393 0.241696 0.970352i \(-0.422296\pi\)
0.241696 + 0.970352i \(0.422296\pi\)
\(18\) 0 0
\(19\) −664.000 −0.421972 −0.210986 0.977489i \(-0.567668\pi\)
−0.210986 + 0.977489i \(0.567668\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1536.00 0.676604
\(23\) −3840.00 −1.51360 −0.756801 0.653645i \(-0.773239\pi\)
−0.756801 + 0.653645i \(0.773239\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −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
\(35\) 0 0
\(36\) 0 0
\(37\) −5798.00 −0.696264 −0.348132 0.937446i \(-0.613184\pi\)
−0.348132 + 0.937446i \(0.613184\pi\)
\(38\) 2656.00 0.298380
\(39\) 0 0
\(40\) 0 0
\(41\) 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 \(-0.289844\pi\)
0.613293 + 0.789855i \(0.289844\pi\)
\(44\) −6144.00 −0.478431
\(45\) 0 0
\(46\) 15360.0 1.07028
\(47\) −19200.0 −1.26782 −0.633909 0.773408i \(-0.718550\pi\)
−0.633909 + 0.773408i \(0.718550\pi\)
\(48\) 0 0
\(49\) 5097.00 0.303266
\(50\) 0 0
\(51\) 0 0
\(52\) 5344.00 0.274068
\(53\) 7776.00 0.380248 0.190124 0.981760i \(-0.439111\pi\)
0.190124 + 0.981760i \(0.439111\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −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
\(65\) 0 0
\(66\) 0 0
\(67\) −36656.0 −0.997604 −0.498802 0.866716i \(-0.666226\pi\)
−0.498802 + 0.866716i \(0.666226\pi\)
\(68\) 9216.00 0.241696
\(69\) 0 0
\(70\) 0 0
\(71\) −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 \(-0.440901\pi\)
0.184600 + 0.982814i \(0.440901\pi\)
\(74\) 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 \(0.418560\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −26880.0 −0.441463
\(83\) −66432.0 −1.05848 −0.529239 0.848473i \(-0.677523\pi\)
−0.529239 + 0.848473i \(0.677523\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −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
\(95\) 0 0
\(96\) 0 0
\(97\) 29938.0 0.323068 0.161534 0.986867i \(-0.448356\pi\)
0.161534 + 0.986867i \(0.448356\pi\)
\(98\) −20388.0 −0.214442
\(99\) 0 0
\(100\) 0 0
\(101\) −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 \(-0.320274\pi\)
0.535100 + 0.844789i \(0.320274\pi\)
\(104\) −21376.0 −0.193795
\(105\) 0 0
\(106\) −31104.0 −0.268876
\(107\) −76032.0 −0.642003 −0.321001 0.947079i \(-0.604019\pi\)
−0.321001 + 0.947079i \(0.604019\pi\)
\(108\) 0 0
\(109\) −231118. −1.86323 −0.931617 0.363441i \(-0.881602\pi\)
−0.931617 + 0.363441i \(0.881602\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 37888.0 0.285402
\(113\) 142464. 1.04956 0.524782 0.851237i \(-0.324147\pi\)
0.524782 + 0.851237i \(0.324147\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −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
\(125\) 0 0
\(126\) 0 0
\(127\) 988.000 0.00543560 0.00271780 0.999996i \(-0.499135\pi\)
0.00271780 + 0.999996i \(0.499135\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −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 \(-0.718977\pi\)
−0.634944 + 0.772558i \(0.718977\pi\)
\(138\) 0 0
\(139\) 177200. 0.777905 0.388953 0.921258i \(-0.372837\pi\)
0.388953 + 0.921258i \(0.372837\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 258048. 1.07394
\(143\) −128256. −0.524490
\(144\) 0 0
\(145\) 0 0
\(146\) −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
\(155\) 0 0
\(156\) 0 0
\(157\) −381086. −1.23388 −0.616941 0.787009i \(-0.711628\pi\)
−0.616941 + 0.787009i \(0.711628\pi\)
\(158\) −112304. −0.357892
\(159\) 0 0
\(160\) 0 0
\(161\) −568320. −1.72794
\(162\) 0 0
\(163\) −162920. −0.480292 −0.240146 0.970737i \(-0.577195\pi\)
−0.240146 + 0.970737i \(0.577195\pi\)
\(164\) 107520. 0.312162
\(165\) 0 0
\(166\) 265728. 0.748457
\(167\) −566016. −1.57050 −0.785249 0.619180i \(-0.787465\pi\)
−0.785249 + 0.619180i \(0.787465\pi\)
\(168\) 0 0
\(169\) −259737. −0.699547
\(170\) 0 0
\(171\) 0 0
\(172\) 237952. 0.613293
\(173\) −218208. −0.554313 −0.277157 0.960825i \(-0.589392\pi\)
−0.277157 + 0.960825i \(0.589392\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −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
\(185\) 0 0
\(186\) 0 0
\(187\) −221184. −0.462540
\(188\) −307200. −0.633909
\(189\) 0 0
\(190\) 0 0
\(191\) −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 \(-0.736295\pi\)
−0.676017 + 0.736886i \(0.736295\pi\)
\(194\) −119752. −0.228443
\(195\) 0 0
\(196\) 81552.0 0.151633
\(197\) 406368. 0.746026 0.373013 0.927826i \(-0.378325\pi\)
0.373013 + 0.927826i \(0.378325\pi\)
\(198\) 0 0
\(199\) −361996. −0.647994 −0.323997 0.946058i \(-0.605027\pi\)
−0.323997 + 0.946058i \(0.605027\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 714624. 1.23225
\(203\) −14208.0 −0.0241987
\(204\) 0 0
\(205\) 0 0
\(206\) −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
\(215\) 0 0
\(216\) 0 0
\(217\) −675472. −0.973774
\(218\) 924472. 1.31751
\(219\) 0 0
\(220\) 0 0
\(221\) 192384. 0.264965
\(222\) 0 0
\(223\) 1.09332e6 1.47227 0.736134 0.676836i \(-0.236649\pi\)
0.736134 + 0.676836i \(0.236649\pi\)
\(224\) −151552. −0.201810
\(225\) 0 0
\(226\) −569856. −0.742154
\(227\) −566400. −0.729556 −0.364778 0.931095i \(-0.618855\pi\)
−0.364778 + 0.931095i \(0.618855\pi\)
\(228\) 0 0
\(229\) −587206. −0.739949 −0.369974 0.929042i \(-0.620634\pi\)
−0.369974 + 0.929042i \(0.620634\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6144.00 0.00749430
\(233\) 579456. 0.699247 0.349624 0.936890i \(-0.386309\pi\)
0.349624 + 0.936890i \(0.386309\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(245\) 0 0
\(246\) 0 0
\(247\) −221776. −0.231298
\(248\) 292096. 0.301576
\(249\) 0 0
\(250\) 0 0
\(251\) 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 \(-0.432188\pi\)
0.211430 + 0.977393i \(0.432188\pi\)
\(258\) 0 0
\(259\) −858104. −0.794860
\(260\) 0 0
\(261\) 0 0
\(262\) 897024. 0.807330
\(263\) −67584.0 −0.0602496 −0.0301248 0.999546i \(-0.509590\pi\)
−0.0301248 + 0.999546i \(0.509590\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 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
\(275\) 0 0
\(276\) 0 0
\(277\) 141142. 0.110524 0.0552620 0.998472i \(-0.482401\pi\)
0.0552620 + 0.998472i \(0.482401\pi\)
\(278\) −708800. −0.550062
\(279\) 0 0
\(280\) 0 0
\(281\) −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 \(-0.520930\pi\)
−0.0657074 + 0.997839i \(0.520930\pi\)
\(284\) −1.03219e6 −0.759390
\(285\) 0 0
\(286\) 513024. 0.370871
\(287\) 994560. 0.712732
\(288\) 0 0
\(289\) −1.08808e6 −0.766331
\(290\) 0 0
\(291\) 0 0
\(292\) 268960. 0.184600
\(293\) −956832. −0.651128 −0.325564 0.945520i \(-0.605554\pi\)
−0.325564 + 0.945520i \(0.605554\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) −2.88286e6 −1.74573 −0.872867 0.487958i \(-0.837742\pi\)
−0.872867 + 0.487958i \(0.837742\pi\)
\(308\) −909312. −0.546180
\(309\) 0 0
\(310\) 0 0
\(311\) 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 \(-0.232690\pi\)
0.744495 + 0.667628i \(0.232690\pi\)
\(314\) 1.52434e6 0.872487
\(315\) 0 0
\(316\) 449216. 0.253068
\(317\) 2.31101e6 1.29168 0.645838 0.763475i \(-0.276508\pi\)
0.645838 + 0.763475i \(0.276508\pi\)
\(318\) 0 0
\(319\) 36864.0 0.0202827
\(320\) 0 0
\(321\) 0 0
\(322\) 2.27328e6 1.22184
\(323\) −382464. −0.203978
\(324\) 0 0
\(325\) 0 0
\(326\) 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
\(335\) 0 0
\(336\) 0 0
\(337\) −3.38665e6 −1.62441 −0.812206 0.583371i \(-0.801733\pi\)
−0.812206 + 0.583371i \(0.801733\pi\)
\(338\) 1.03895e6 0.494655
\(339\) 0 0
\(340\) 0 0
\(341\) 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 \(-0.287408\pi\)
0.619321 + 0.785138i \(0.287408\pi\)
\(348\) 0 0
\(349\) 1.55536e6 0.683545 0.341772 0.939783i \(-0.388973\pi\)
0.341772 + 0.939783i \(0.388973\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 393216. 0.169151
\(353\) 2.11776e6 0.904565 0.452283 0.891875i \(-0.350610\pi\)
0.452283 + 0.891875i \(0.350610\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) −1.05336e6 −0.408235 −0.204117 0.978946i \(-0.565432\pi\)
−0.204117 + 0.978946i \(0.565432\pi\)
\(368\) −983040. −0.378400
\(369\) 0 0
\(370\) 0 0
\(371\) 1.15085e6 0.434093
\(372\) 0 0
\(373\) 677098. 0.251988 0.125994 0.992031i \(-0.459788\pi\)
0.125994 + 0.992031i \(0.459788\pi\)
\(374\) 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 \(0.866418\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.60051e6 0.561178
\(383\) 1.63200e6 0.568491 0.284245 0.958752i \(-0.408257\pi\)
0.284245 + 0.958752i \(0.408257\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 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
\(395\) 0 0
\(396\) 0 0
\(397\) 611026. 0.194573 0.0972867 0.995256i \(-0.468984\pi\)
0.0972867 + 0.995256i \(0.468984\pi\)
\(398\) 1.44798e6 0.458201
\(399\) 0 0
\(400\) 0 0
\(401\) 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 \(0.359469\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.84365e6 0.535100
\(413\) 1.93229e6 0.557438
\(414\) 0 0
\(415\) 0 0
\(416\) −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
\(425\) 0 0
\(426\) 0 0
\(427\) 6.33174e6 1.68056
\(428\) −1.21651e6 −0.321001
\(429\) 0 0
\(430\) 0 0
\(431\) −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 \(-0.465178\pi\)
0.109179 + 0.994022i \(0.465178\pi\)
\(434\) 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 \(0.513197\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −769536. −0.187358
\(443\) −1.76218e6 −0.426619 −0.213309 0.976985i \(-0.568424\pi\)
−0.213309 + 0.976985i \(0.568424\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −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
\(455\) 0 0
\(456\) 0 0
\(457\) 92074.0 0.0206227 0.0103114 0.999947i \(-0.496718\pi\)
0.0103114 + 0.999947i \(0.496718\pi\)
\(458\) 2.34882e6 0.523223
\(459\) 0 0
\(460\) 0 0
\(461\) −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 \(-0.641310\pi\)
−0.429499 + 0.903067i \(0.641310\pi\)
\(464\) −24576.0 −0.00529927
\(465\) 0 0
\(466\) −2.31782e6 −0.494442
\(467\) −3.48941e6 −0.740388 −0.370194 0.928954i \(-0.620709\pi\)
−0.370194 + 0.928954i \(0.620709\pi\)
\(468\) 0 0
\(469\) −5.42509e6 −1.13887
\(470\) 0 0
\(471\) 0 0
\(472\) −835584. −0.172637
\(473\) −5.71085e6 −1.17367
\(474\) 0 0
\(475\) 0 0
\(476\) 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
\(485\) 0 0
\(486\) 0 0
\(487\) −4.14499e6 −0.791956 −0.395978 0.918260i \(-0.629594\pi\)
−0.395978 + 0.918260i \(0.629594\pi\)
\(488\) −2.73805e6 −0.520465
\(489\) 0 0
\(490\) 0 0
\(491\) −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 \(0.481078\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −7.59859e6 −1.34578
\(503\) −944640. −0.166474 −0.0832370 0.996530i \(-0.526526\pi\)
−0.0832370 + 0.996530i \(0.526526\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −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
\(515\) 0 0
\(516\) 0 0
\(517\) 7.37280e6 1.21313
\(518\) 3.43242e6 0.562051
\(519\) 0 0
\(520\) 0 0
\(521\) −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 \(-0.673972\pi\)
−0.519742 + 0.854323i \(0.673972\pi\)
\(524\) −3.58810e6 −0.570868
\(525\) 0 0
\(526\) 270336. 0.0426029
\(527\) −2.62886e6 −0.412327
\(528\) 0 0
\(529\) 8.30926e6 1.29099
\(530\) 0 0
\(531\) 0 0
\(532\) −1.57235e6 −0.240863
\(533\) 2.24448e6 0.342214
\(534\) 0 0
\(535\) 0 0
\(536\) 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
\(545\) 0 0
\(546\) 0 0
\(547\) 3.42580e6 0.489546 0.244773 0.969580i \(-0.421287\pi\)
0.244773 + 0.969580i \(0.421287\pi\)
\(548\) −4.46362e6 −0.634944
\(549\) 0 0
\(550\) 0 0
\(551\) 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 \(-0.644783\pi\)
−0.439327 + 0.898327i \(0.644783\pi\)
\(558\) 0 0
\(559\) 4.96725e6 0.672336
\(560\) 0 0
\(561\) 0 0
\(562\) 2.33779e6 0.312223
\(563\) 753024. 0.100124 0.0500620 0.998746i \(-0.484058\pi\)
0.0500620 + 0.998746i \(0.484058\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 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
\(575\) 0 0
\(576\) 0 0
\(577\) −1.03722e7 −1.29697 −0.648486 0.761227i \(-0.724597\pi\)
−0.648486 + 0.761227i \(0.724597\pi\)
\(578\) 4.35232e6 0.541878
\(579\) 0 0
\(580\) 0 0
\(581\) −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 \(-0.556937\pi\)
−0.177919 + 0.984045i \(0.556937\pi\)
\(588\) 0 0
\(589\) 3.03050e6 0.359936
\(590\) 0 0
\(591\) 0 0
\(592\) −1.48429e6 −0.174066
\(593\) −7.55827e6 −0.882644 −0.441322 0.897349i \(-0.645490\pi\)
−0.441322 + 0.897349i \(0.645490\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −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
\(605\) 0 0
\(606\) 0 0
\(607\) 1.35585e7 1.49362 0.746809 0.665038i \(-0.231585\pi\)
0.746809 + 0.665038i \(0.231585\pi\)
\(608\) 679936. 0.0745949
\(609\) 0 0
\(610\) 0 0
\(611\) −6.41280e6 −0.694936
\(612\) 0 0
\(613\) 1.07654e7 1.15712 0.578561 0.815639i \(-0.303615\pi\)
0.578561 + 0.815639i \(0.303615\pi\)
\(614\) 1.15315e7 1.23442
\(615\) 0 0
\(616\) 3.63725e6 0.386208
\(617\) −9.33504e6 −0.987196 −0.493598 0.869690i \(-0.664319\pi\)
−0.493598 + 0.869690i \(0.664319\pi\)
\(618\) 0 0
\(619\) 9.07664e6 0.952135 0.476067 0.879409i \(-0.342062\pi\)
0.476067 + 0.879409i \(0.342062\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −1.04018e7 −1.07803
\(623\) 1.21052e7 1.24955
\(624\) 0 0
\(625\) 0 0
\(626\) −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
\(635\) 0 0
\(636\) 0 0
\(637\) 1.70240e6 0.166231
\(638\) −147456. −0.0143420
\(639\) 0 0
\(640\) 0 0
\(641\) −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 \(-0.362136\pi\)
0.419699 + 0.907663i \(0.362136\pi\)
\(644\) −9.09312e6 −0.863969
\(645\) 0 0
\(646\) 1.52986e6 0.144235
\(647\) 1.08449e7 1.01851 0.509256 0.860615i \(-0.329921\pi\)
0.509256 + 0.860615i \(0.329921\pi\)
\(648\) 0 0
\(649\) −5.01350e6 −0.467229
\(650\) 0 0
\(651\) 0 0
\(652\) −2.60672e6 −0.240146
\(653\) −9.88771e6 −0.907429 −0.453715 0.891147i \(-0.649901\pi\)
−0.453715 + 0.891147i \(0.649901\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 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
\(665\) 0 0
\(666\) 0 0
\(667\) 368640. 0.0320840
\(668\) −9.05626e6 −0.785249
\(669\) 0 0
\(670\) 0 0
\(671\) −1.64283e7 −1.40859
\(672\) 0 0
\(673\) −6.64939e6 −0.565906 −0.282953 0.959134i \(-0.591314\pi\)
−0.282953 + 0.959134i \(0.591314\pi\)
\(674\) 1.35466e7 1.14863
\(675\) 0 0
\(676\) −4.15579e6 −0.349774
\(677\) 4.42550e6 0.371100 0.185550 0.982635i \(-0.440593\pi\)
0.185550 + 0.982635i \(0.440593\pi\)
\(678\) 0 0
\(679\) 4.43082e6 0.368816
\(680\) 0 0
\(681\) 0 0
\(682\) −7.01030e6 −0.577133
\(683\) −4.67827e6 −0.383737 −0.191869 0.981421i \(-0.561455\pi\)
−0.191869 + 0.981421i \(0.561455\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 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
\(695\) 0 0
\(696\) 0 0
\(697\) 3.87072e6 0.301793
\(698\) −6.22143e6 −0.483339
\(699\) 0 0
\(700\) 0 0
\(701\) 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 \(0.546123\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −5.23469e6 −0.386982
\(713\) 1.75258e7 1.29108
\(714\) 0 0
\(715\) 0 0
\(716\) 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
\(725\) 0 0
\(726\) 0 0
\(727\) 1.90590e7 1.33741 0.668704 0.743529i \(-0.266850\pi\)
0.668704 + 0.743529i \(0.266850\pi\)
\(728\) −3.16365e6 −0.221238
\(729\) 0 0
\(730\) 0 0
\(731\) 8.56627e6 0.592923
\(732\) 0 0
\(733\) −5.69616e6 −0.391582 −0.195791 0.980646i \(-0.562727\pi\)
−0.195791 + 0.980646i \(0.562727\pi\)
\(734\) 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 \(0.286024\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4.60339e6 −0.306950
\(743\) 9.90336e6 0.658128 0.329064 0.944308i \(-0.393267\pi\)
0.329064 + 0.944308i \(0.393267\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −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
\(755\) 0 0
\(756\) 0 0
\(757\) 1.16677e7 0.740022 0.370011 0.929027i \(-0.379354\pi\)
0.370011 + 0.929027i \(0.379354\pi\)
\(758\) 2.04299e7 1.29150
\(759\) 0 0
\(760\) 0 0
\(761\) 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 \(0.394441\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.11944e7 −0.676017
\(773\) −9.18634e6 −0.552960 −0.276480 0.961020i \(-0.589168\pi\)
−0.276480 + 0.961020i \(0.589168\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −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
\(785\) 0 0
\(786\) 0 0
\(787\) 9.28209e6 0.534206 0.267103 0.963668i \(-0.413934\pi\)
0.267103 + 0.963668i \(0.413934\pi\)
\(788\) 6.50189e6 0.373013
\(789\) 0 0
\(790\) 0 0
\(791\) 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 \(-0.499807\pi\)
0.000607605 1.00000i \(0.499807\pi\)
\(798\) 0 0
\(799\) −1.10592e7 −0.612854
\(800\) 0 0
\(801\) 0 0
\(802\) −2.43663e7 −1.33769
\(803\) −6.45504e6 −0.353273
\(804\) 0 0
\(805\) 0 0
\(806\) 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
\(815\) 0 0
\(816\) 0 0
\(817\) −9.87501e6 −0.517586
\(818\) −1.15643e7 −0.604278
\(819\) 0 0
\(820\) 0 0
\(821\) 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 \(-0.844941\pi\)
−0.883679 + 0.468093i \(0.844941\pi\)
\(824\) −7.37459e6 −0.378373
\(825\) 0 0
\(826\) −7.72915e6 −0.394168
\(827\) 2.13327e7 1.08463 0.542316 0.840174i \(-0.317547\pi\)
0.542316 + 0.840174i \(0.317547\pi\)
\(828\) 0 0
\(829\) 2.63751e6 0.133293 0.0666465 0.997777i \(-0.478770\pi\)
0.0666465 + 0.997777i \(0.478770\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.36806e6 0.0685170
\(833\) 2.93587e6 0.146597
\(834\) 0 0
\(835\) 0 0
\(836\) 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
\(845\) 0 0
\(846\) 0 0
\(847\) −2.01206e6 −0.0963679
\(848\) 1.99066e6 0.0950619
\(849\) 0 0
\(850\) 0 0
\(851\) 2.22643e7 1.05387
\(852\) 0 0
\(853\) 2.30748e7 1.08584 0.542919 0.839785i \(-0.317319\pi\)
0.542919 + 0.839785i \(0.317319\pi\)
\(854\) −2.53269e7 −1.18833
\(855\) 0 0
\(856\) 4.86605e6 0.226982
\(857\) 3.51646e7 1.63551 0.817756 0.575565i \(-0.195218\pi\)
0.817756 + 0.575565i \(0.195218\pi\)
\(858\) 0 0
\(859\) 1.66022e7 0.767684 0.383842 0.923399i \(-0.374601\pi\)
0.383842 + 0.923399i \(0.374601\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.52979e7 1.15962
\(863\) −2.97009e7 −1.35751 −0.678754 0.734366i \(-0.737480\pi\)
−0.678754 + 0.734366i \(0.737480\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −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
\(875\) 0 0
\(876\) 0 0
\(877\) 1.17943e7 0.517811 0.258906 0.965903i \(-0.416638\pi\)
0.258906 + 0.965903i \(0.416638\pi\)
\(878\) 1.33893e6 0.0586166
\(879\) 0 0
\(880\) 0 0
\(881\) 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 \(-0.651408\pi\)
−0.457928 + 0.888990i \(0.651408\pi\)
\(884\) 3.07814e6 0.132482
\(885\) 0 0
\(886\) 7.04870e6 0.301665
\(887\) 2.28818e7 0.976520 0.488260 0.872698i \(-0.337632\pi\)
0.488260 + 0.872698i \(0.337632\pi\)
\(888\) 0 0
\(889\) 146224. 0.00620532
\(890\) 0 0
\(891\) 0 0
\(892\) 1.74932e7 0.736134
\(893\) 1.27488e7 0.534984
\(894\) 0 0
\(895\) 0 0
\(896\) −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
\(905\) 0 0
\(906\) 0 0
\(907\) 9.68692e6 0.390992 0.195496 0.980705i \(-0.437368\pi\)
0.195496 + 0.980705i \(0.437368\pi\)
\(908\) −9.06240e6 −0.364778
\(909\) 0 0
\(910\) 0 0
\(911\) −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 \(0.808191\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.03027e6 0.0399139
\(923\) −2.15470e7 −0.832497
\(924\) 0 0
\(925\) 0 0
\(926\) 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
\(935\) 0 0
\(936\) 0 0
\(937\) −1.47847e7 −0.550128 −0.275064 0.961426i \(-0.588699\pi\)
−0.275064 + 0.961426i \(0.588699\pi\)
\(938\) 2.17004e7 0.805304
\(939\) 0 0
\(940\) 0 0
\(941\) 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 \(-0.483175\pi\)
0.0528320 + 0.998603i \(0.483175\pi\)
\(948\) 0 0
\(949\) 5.61454e6 0.202371
\(950\) 0 0
\(951\) 0 0
\(952\) −5.45587e6 −0.195107
\(953\) 1.40861e7 0.502410 0.251205 0.967934i \(-0.419173\pi\)
0.251205 + 0.967934i \(0.419173\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) −1.51949e7 −0.522553 −0.261276 0.965264i \(-0.584143\pi\)
−0.261276 + 0.965264i \(0.584143\pi\)
\(968\) 870080. 0.0298449
\(969\) 0 0
\(970\) 0 0
\(971\) 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 \(-0.303360\pi\)
0.579214 + 0.815176i \(0.303360\pi\)
\(978\) 0 0
\(979\) −3.14081e7 −1.04733
\(980\) 0 0
\(981\) 0 0
\(982\) 1.10346e7 0.365156
\(983\) 5.56385e7 1.83650 0.918252 0.395997i \(-0.129601\pi\)
0.918252 + 0.395997i \(0.129601\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) 0 0
\(997\) −2.35811e7 −0.751322 −0.375661 0.926757i \(-0.622584\pi\)
−0.375661 + 0.926757i \(0.622584\pi\)
\(998\) −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 450.6.a.k.1.1 1
3.2 odd 2 450.6.a.v.1.1 1
5.2 odd 4 450.6.c.c.199.1 2
5.3 odd 4 450.6.c.c.199.2 2
5.4 even 2 18.6.a.c.1.1 yes 1
15.2 even 4 450.6.c.m.199.2 2
15.8 even 4 450.6.c.m.199.1 2
15.14 odd 2 18.6.a.a.1.1 1
20.19 odd 2 144.6.a.l.1.1 1
35.34 odd 2 882.6.a.l.1.1 1
40.19 odd 2 576.6.a.b.1.1 1
40.29 even 2 576.6.a.a.1.1 1
45.4 even 6 162.6.c.a.55.1 2
45.14 odd 6 162.6.c.l.55.1 2
45.29 odd 6 162.6.c.l.109.1 2
45.34 even 6 162.6.c.a.109.1 2
60.59 even 2 144.6.a.a.1.1 1
105.104 even 2 882.6.a.k.1.1 1
120.29 odd 2 576.6.a.bh.1.1 1
120.59 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 15.14 odd 2
18.6.a.c.1.1 yes 1 5.4 even 2
144.6.a.a.1.1 1 60.59 even 2
144.6.a.l.1.1 1 20.19 odd 2
162.6.c.a.55.1 2 45.4 even 6
162.6.c.a.109.1 2 45.34 even 6
162.6.c.l.55.1 2 45.14 odd 6
162.6.c.l.109.1 2 45.29 odd 6
450.6.a.k.1.1 1 1.1 even 1 trivial
450.6.a.v.1.1 1 3.2 odd 2
450.6.c.c.199.1 2 5.2 odd 4
450.6.c.c.199.2 2 5.3 odd 4
450.6.c.m.199.1 2 15.8 even 4
450.6.c.m.199.2 2 15.2 even 4
576.6.a.a.1.1 1 40.29 even 2
576.6.a.b.1.1 1 40.19 odd 2
576.6.a.bh.1.1 1 120.29 odd 2
576.6.a.bi.1.1 1 120.59 even 2
882.6.a.k.1.1 1 105.104 even 2
882.6.a.l.1.1 1 35.34 odd 2