Properties

Label 450.6.c.n.199.1
Level $450$
Weight $6$
Character 450.199
Analytic conductor $72.173$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 450.199
Dual form 450.6.c.n.199.2

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000i q^{2} -16.0000 q^{4} -233.000i q^{7} +64.0000i q^{8} +O(q^{10})\) \(q-4.00000i q^{2} -16.0000 q^{4} -233.000i q^{7} +64.0000i q^{8} +498.000 q^{11} +809.000i q^{13} -932.000 q^{14} +256.000 q^{16} -1002.00i q^{17} +1705.00 q^{19} -1992.00i q^{22} -1554.00i q^{23} +3236.00 q^{26} +3728.00i q^{28} +7830.00 q^{29} +977.000 q^{31} -1024.00i q^{32} -4008.00 q^{34} +4822.00i q^{37} -6820.00i q^{38} +8148.00 q^{41} +19469.0i q^{43} -7968.00 q^{44} -6216.00 q^{46} +8418.00i q^{47} -37482.0 q^{49} -12944.0i q^{52} -17664.0i q^{53} +14912.0 q^{56} -31320.0i q^{58} +35910.0 q^{59} +3527.00 q^{61} -3908.00i q^{62} -4096.00 q^{64} -57473.0i q^{67} +16032.0i q^{68} +7548.00 q^{71} -646.000i q^{73} +19288.0 q^{74} -27280.0 q^{76} -116034. i q^{77} +22720.0 q^{79} -32592.0i q^{82} -11574.0i q^{83} +77876.0 q^{86} +31872.0i q^{88} -78960.0 q^{89} +188497. q^{91} +24864.0i q^{92} +33672.0 q^{94} -54593.0i q^{97} +149928. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} + 996q^{11} - 1864q^{14} + 512q^{16} + 3410q^{19} + 6472q^{26} + 15660q^{29} + 1954q^{31} - 8016q^{34} + 16296q^{41} - 15936q^{44} - 12432q^{46} - 74964q^{49} + 29824q^{56} + 71820q^{59} + 7054q^{61} - 8192q^{64} + 15096q^{71} + 38576q^{74} - 54560q^{76} + 45440q^{79} + 155752q^{86} - 157920q^{89} + 376994q^{91} + 67344q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.00000i − 0.707107i
\(3\) 0 0
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 233.000i − 1.79726i −0.438708 0.898630i \(-0.644564\pi\)
0.438708 0.898630i \(-0.355436\pi\)
\(8\) 64.0000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 498.000 1.24093 0.620465 0.784234i \(-0.286944\pi\)
0.620465 + 0.784234i \(0.286944\pi\)
\(12\) 0 0
\(13\) 809.000i 1.32767i 0.747879 + 0.663835i \(0.231072\pi\)
−0.747879 + 0.663835i \(0.768928\pi\)
\(14\) −932.000 −1.27085
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) − 1002.00i − 0.840902i −0.907315 0.420451i \(-0.861872\pi\)
0.907315 0.420451i \(-0.138128\pi\)
\(18\) 0 0
\(19\) 1705.00 1.08353 0.541764 0.840530i \(-0.317757\pi\)
0.541764 + 0.840530i \(0.317757\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 1992.00i − 0.877471i
\(23\) − 1554.00i − 0.612536i −0.951945 0.306268i \(-0.900920\pi\)
0.951945 0.306268i \(-0.0990803\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3236.00 0.938804
\(27\) 0 0
\(28\) 3728.00i 0.898630i
\(29\) 7830.00 1.72889 0.864444 0.502729i \(-0.167671\pi\)
0.864444 + 0.502729i \(0.167671\pi\)
\(30\) 0 0
\(31\) 977.000 0.182596 0.0912978 0.995824i \(-0.470898\pi\)
0.0912978 + 0.995824i \(0.470898\pi\)
\(32\) − 1024.00i − 0.176777i
\(33\) 0 0
\(34\) −4008.00 −0.594608
\(35\) 0 0
\(36\) 0 0
\(37\) 4822.00i 0.579059i 0.957169 + 0.289530i \(0.0934989\pi\)
−0.957169 + 0.289530i \(0.906501\pi\)
\(38\) − 6820.00i − 0.766170i
\(39\) 0 0
\(40\) 0 0
\(41\) 8148.00 0.756992 0.378496 0.925603i \(-0.376441\pi\)
0.378496 + 0.925603i \(0.376441\pi\)
\(42\) 0 0
\(43\) 19469.0i 1.60573i 0.596161 + 0.802865i \(0.296692\pi\)
−0.596161 + 0.802865i \(0.703308\pi\)
\(44\) −7968.00 −0.620465
\(45\) 0 0
\(46\) −6216.00 −0.433128
\(47\) 8418.00i 0.555859i 0.960602 + 0.277929i \(0.0896481\pi\)
−0.960602 + 0.277929i \(0.910352\pi\)
\(48\) 0 0
\(49\) −37482.0 −2.23014
\(50\) 0 0
\(51\) 0 0
\(52\) − 12944.0i − 0.663835i
\(53\) − 17664.0i − 0.863773i −0.901928 0.431886i \(-0.857848\pi\)
0.901928 0.431886i \(-0.142152\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 14912.0 0.635427
\(57\) 0 0
\(58\) − 31320.0i − 1.22251i
\(59\) 35910.0 1.34303 0.671514 0.740991i \(-0.265644\pi\)
0.671514 + 0.740991i \(0.265644\pi\)
\(60\) 0 0
\(61\) 3527.00 0.121361 0.0606807 0.998157i \(-0.480673\pi\)
0.0606807 + 0.998157i \(0.480673\pi\)
\(62\) − 3908.00i − 0.129115i
\(63\) 0 0
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 57473.0i − 1.56414i −0.623188 0.782072i \(-0.714163\pi\)
0.623188 0.782072i \(-0.285837\pi\)
\(68\) 16032.0i 0.420451i
\(69\) 0 0
\(70\) 0 0
\(71\) 7548.00 0.177699 0.0888497 0.996045i \(-0.471681\pi\)
0.0888497 + 0.996045i \(0.471681\pi\)
\(72\) 0 0
\(73\) − 646.000i − 0.0141881i −0.999975 0.00709407i \(-0.997742\pi\)
0.999975 0.00709407i \(-0.00225813\pi\)
\(74\) 19288.0 0.409457
\(75\) 0 0
\(76\) −27280.0 −0.541764
\(77\) − 116034.i − 2.23028i
\(78\) 0 0
\(79\) 22720.0 0.409582 0.204791 0.978806i \(-0.434349\pi\)
0.204791 + 0.978806i \(0.434349\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 32592.0i − 0.535274i
\(83\) − 11574.0i − 0.184412i −0.995740 0.0922058i \(-0.970608\pi\)
0.995740 0.0922058i \(-0.0293918\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 77876.0 1.13542
\(87\) 0 0
\(88\) 31872.0i 0.438735i
\(89\) −78960.0 −1.05665 −0.528326 0.849041i \(-0.677180\pi\)
−0.528326 + 0.849041i \(0.677180\pi\)
\(90\) 0 0
\(91\) 188497. 2.38617
\(92\) 24864.0i 0.306268i
\(93\) 0 0
\(94\) 33672.0 0.393051
\(95\) 0 0
\(96\) 0 0
\(97\) − 54593.0i − 0.589125i −0.955632 0.294563i \(-0.904826\pi\)
0.955632 0.294563i \(-0.0951740\pi\)
\(98\) 149928.i 1.57695i
\(99\) 0 0
\(100\) 0 0
\(101\) −105552. −1.02959 −0.514793 0.857314i \(-0.672131\pi\)
−0.514793 + 0.857314i \(0.672131\pi\)
\(102\) 0 0
\(103\) − 177436.i − 1.64797i −0.566613 0.823984i \(-0.691747\pi\)
0.566613 0.823984i \(-0.308253\pi\)
\(104\) −51776.0 −0.469402
\(105\) 0 0
\(106\) −70656.0 −0.610779
\(107\) − 183792.i − 1.55191i −0.630787 0.775956i \(-0.717268\pi\)
0.630787 0.775956i \(-0.282732\pi\)
\(108\) 0 0
\(109\) −169685. −1.36797 −0.683986 0.729495i \(-0.739755\pi\)
−0.683986 + 0.729495i \(0.739755\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 59648.0i − 0.449315i
\(113\) − 263484.i − 1.94115i −0.240808 0.970573i \(-0.577412\pi\)
0.240808 0.970573i \(-0.422588\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −125280. −0.864444
\(117\) 0 0
\(118\) − 143640.i − 0.949665i
\(119\) −233466. −1.51132
\(120\) 0 0
\(121\) 86953.0 0.539910
\(122\) − 14108.0i − 0.0858155i
\(123\) 0 0
\(124\) −15632.0 −0.0912978
\(125\) 0 0
\(126\) 0 0
\(127\) 256912.i 1.41343i 0.707497 + 0.706716i \(0.249824\pi\)
−0.707497 + 0.706716i \(0.750176\pi\)
\(128\) 16384.0i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 219048. 1.11522 0.557611 0.830103i \(-0.311718\pi\)
0.557611 + 0.830103i \(0.311718\pi\)
\(132\) 0 0
\(133\) − 397265.i − 1.94738i
\(134\) −229892. −1.10602
\(135\) 0 0
\(136\) 64128.0 0.297304
\(137\) 228678.i 1.04093i 0.853882 + 0.520467i \(0.174242\pi\)
−0.853882 + 0.520467i \(0.825758\pi\)
\(138\) 0 0
\(139\) −339740. −1.49145 −0.745727 0.666252i \(-0.767898\pi\)
−0.745727 + 0.666252i \(0.767898\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 30192.0i − 0.125652i
\(143\) 402882.i 1.64755i
\(144\) 0 0
\(145\) 0 0
\(146\) −2584.00 −0.0100325
\(147\) 0 0
\(148\) − 77152.0i − 0.289530i
\(149\) 306450. 1.13082 0.565411 0.824810i \(-0.308718\pi\)
0.565411 + 0.824810i \(0.308718\pi\)
\(150\) 0 0
\(151\) 198827. 0.709632 0.354816 0.934936i \(-0.384544\pi\)
0.354816 + 0.934936i \(0.384544\pi\)
\(152\) 109120.i 0.383085i
\(153\) 0 0
\(154\) −464136. −1.57704
\(155\) 0 0
\(156\) 0 0
\(157\) − 361283.i − 1.16976i −0.811118 0.584882i \(-0.801141\pi\)
0.811118 0.584882i \(-0.198859\pi\)
\(158\) − 90880.0i − 0.289618i
\(159\) 0 0
\(160\) 0 0
\(161\) −362082. −1.10089
\(162\) 0 0
\(163\) 338159.i 0.996901i 0.866918 + 0.498450i \(0.166097\pi\)
−0.866918 + 0.498450i \(0.833903\pi\)
\(164\) −130368. −0.378496
\(165\) 0 0
\(166\) −46296.0 −0.130399
\(167\) 430248.i 1.19379i 0.802320 + 0.596895i \(0.203599\pi\)
−0.802320 + 0.596895i \(0.796401\pi\)
\(168\) 0 0
\(169\) −283188. −0.762708
\(170\) 0 0
\(171\) 0 0
\(172\) − 311504.i − 0.802865i
\(173\) − 603354.i − 1.53270i −0.642424 0.766350i \(-0.722071\pi\)
0.642424 0.766350i \(-0.277929\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 127488. 0.310233
\(177\) 0 0
\(178\) 315840.i 0.747166i
\(179\) −374370. −0.873310 −0.436655 0.899629i \(-0.643837\pi\)
−0.436655 + 0.899629i \(0.643837\pi\)
\(180\) 0 0
\(181\) −232423. −0.527330 −0.263665 0.964614i \(-0.584931\pi\)
−0.263665 + 0.964614i \(0.584931\pi\)
\(182\) − 753988.i − 1.68728i
\(183\) 0 0
\(184\) 99456.0 0.216564
\(185\) 0 0
\(186\) 0 0
\(187\) − 498996.i − 1.04350i
\(188\) − 134688.i − 0.277929i
\(189\) 0 0
\(190\) 0 0
\(191\) 846198. 1.67837 0.839187 0.543843i \(-0.183031\pi\)
0.839187 + 0.543843i \(0.183031\pi\)
\(192\) 0 0
\(193\) − 155581.i − 0.300651i −0.988637 0.150326i \(-0.951968\pi\)
0.988637 0.150326i \(-0.0480322\pi\)
\(194\) −218372. −0.416574
\(195\) 0 0
\(196\) 599712. 1.11507
\(197\) − 103482.i − 0.189976i −0.995478 0.0949881i \(-0.969719\pi\)
0.995478 0.0949881i \(-0.0302813\pi\)
\(198\) 0 0
\(199\) 140425. 0.251369 0.125685 0.992070i \(-0.459887\pi\)
0.125685 + 0.992070i \(0.459887\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 422208.i 0.728028i
\(203\) − 1.82439e6i − 3.10726i
\(204\) 0 0
\(205\) 0 0
\(206\) −709744. −1.16529
\(207\) 0 0
\(208\) 207104.i 0.331917i
\(209\) 849090. 1.34458
\(210\) 0 0
\(211\) −462673. −0.715431 −0.357716 0.933831i \(-0.616444\pi\)
−0.357716 + 0.933831i \(0.616444\pi\)
\(212\) 282624.i 0.431886i
\(213\) 0 0
\(214\) −735168. −1.09737
\(215\) 0 0
\(216\) 0 0
\(217\) − 227641.i − 0.328172i
\(218\) 678740.i 0.967302i
\(219\) 0 0
\(220\) 0 0
\(221\) 810618. 1.11644
\(222\) 0 0
\(223\) − 735271.i − 0.990114i −0.868860 0.495057i \(-0.835147\pi\)
0.868860 0.495057i \(-0.164853\pi\)
\(224\) −238592. −0.317714
\(225\) 0 0
\(226\) −1.05394e6 −1.37260
\(227\) 967188.i 1.24579i 0.782304 + 0.622897i \(0.214044\pi\)
−0.782304 + 0.622897i \(0.785956\pi\)
\(228\) 0 0
\(229\) 83695.0 0.105466 0.0527328 0.998609i \(-0.483207\pi\)
0.0527328 + 0.998609i \(0.483207\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 501120.i 0.611254i
\(233\) 873876.i 1.05453i 0.849700 + 0.527266i \(0.176783\pi\)
−0.849700 + 0.527266i \(0.823217\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −574560. −0.671514
\(237\) 0 0
\(238\) 933864.i 1.06866i
\(239\) −1.06056e6 −1.20099 −0.600497 0.799627i \(-0.705030\pi\)
−0.600497 + 0.799627i \(0.705030\pi\)
\(240\) 0 0
\(241\) −756823. −0.839367 −0.419683 0.907671i \(-0.637859\pi\)
−0.419683 + 0.907671i \(0.637859\pi\)
\(242\) − 347812.i − 0.381774i
\(243\) 0 0
\(244\) −56432.0 −0.0606807
\(245\) 0 0
\(246\) 0 0
\(247\) 1.37934e6i 1.43857i
\(248\) 62528.0i 0.0645573i
\(249\) 0 0
\(250\) 0 0
\(251\) 635148. 0.636342 0.318171 0.948033i \(-0.396931\pi\)
0.318171 + 0.948033i \(0.396931\pi\)
\(252\) 0 0
\(253\) − 773892.i − 0.760115i
\(254\) 1.02765e6 0.999448
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 30708.0i 0.0290014i 0.999895 + 0.0145007i \(0.00461588\pi\)
−0.999895 + 0.0145007i \(0.995384\pi\)
\(258\) 0 0
\(259\) 1.12353e6 1.04072
\(260\) 0 0
\(261\) 0 0
\(262\) − 876192.i − 0.788581i
\(263\) 189516.i 0.168949i 0.996426 + 0.0844747i \(0.0269212\pi\)
−0.996426 + 0.0844747i \(0.973079\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.58906e6 −1.37701
\(267\) 0 0
\(268\) 919568.i 0.782072i
\(269\) 1.10997e6 0.935256 0.467628 0.883925i \(-0.345109\pi\)
0.467628 + 0.883925i \(0.345109\pi\)
\(270\) 0 0
\(271\) 211952. 0.175313 0.0876565 0.996151i \(-0.472062\pi\)
0.0876565 + 0.996151i \(0.472062\pi\)
\(272\) − 256512.i − 0.210226i
\(273\) 0 0
\(274\) 914712. 0.736051
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.04741e6i − 0.820198i −0.912041 0.410099i \(-0.865494\pi\)
0.912041 0.410099i \(-0.134506\pi\)
\(278\) 1.35896e6i 1.05462i
\(279\) 0 0
\(280\) 0 0
\(281\) −34002.0 −0.0256885 −0.0128442 0.999918i \(-0.504089\pi\)
−0.0128442 + 0.999918i \(0.504089\pi\)
\(282\) 0 0
\(283\) − 281101.i − 0.208639i −0.994544 0.104320i \(-0.966733\pi\)
0.994544 0.104320i \(-0.0332665\pi\)
\(284\) −120768. −0.0888497
\(285\) 0 0
\(286\) 1.61153e6 1.16499
\(287\) − 1.89848e6i − 1.36051i
\(288\) 0 0
\(289\) 415853. 0.292884
\(290\) 0 0
\(291\) 0 0
\(292\) 10336.0i 0.00709407i
\(293\) 1.55851e6i 1.06057i 0.847819 + 0.530285i \(0.177915\pi\)
−0.847819 + 0.530285i \(0.822085\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −308608. −0.204728
\(297\) 0 0
\(298\) − 1.22580e6i − 0.799611i
\(299\) 1.25719e6 0.813245
\(300\) 0 0
\(301\) 4.53628e6 2.88591
\(302\) − 795308.i − 0.501785i
\(303\) 0 0
\(304\) 436480. 0.270882
\(305\) 0 0
\(306\) 0 0
\(307\) 839917.i 0.508616i 0.967123 + 0.254308i \(0.0818478\pi\)
−0.967123 + 0.254308i \(0.918152\pi\)
\(308\) 1.85654e6i 1.11514i
\(309\) 0 0
\(310\) 0 0
\(311\) 292698. 0.171601 0.0858003 0.996312i \(-0.472655\pi\)
0.0858003 + 0.996312i \(0.472655\pi\)
\(312\) 0 0
\(313\) 127859.i 0.0737684i 0.999320 + 0.0368842i \(0.0117433\pi\)
−0.999320 + 0.0368842i \(0.988257\pi\)
\(314\) −1.44513e6 −0.827148
\(315\) 0 0
\(316\) −363520. −0.204791
\(317\) 648048.i 0.362209i 0.983464 + 0.181104i \(0.0579672\pi\)
−0.983464 + 0.181104i \(0.942033\pi\)
\(318\) 0 0
\(319\) 3.89934e6 2.14543
\(320\) 0 0
\(321\) 0 0
\(322\) 1.44833e6i 0.778444i
\(323\) − 1.70841e6i − 0.911141i
\(324\) 0 0
\(325\) 0 0
\(326\) 1.35264e6 0.704915
\(327\) 0 0
\(328\) 521472.i 0.267637i
\(329\) 1.96139e6 0.999022
\(330\) 0 0
\(331\) −3.39315e6 −1.70229 −0.851144 0.524933i \(-0.824090\pi\)
−0.851144 + 0.524933i \(0.824090\pi\)
\(332\) 185184.i 0.0922058i
\(333\) 0 0
\(334\) 1.72099e6 0.844137
\(335\) 0 0
\(336\) 0 0
\(337\) 1.62085e6i 0.777441i 0.921356 + 0.388720i \(0.127083\pi\)
−0.921356 + 0.388720i \(0.872917\pi\)
\(338\) 1.13275e6i 0.539316i
\(339\) 0 0
\(340\) 0 0
\(341\) 486546. 0.226589
\(342\) 0 0
\(343\) 4.81728e6i 2.21088i
\(344\) −1.24602e6 −0.567711
\(345\) 0 0
\(346\) −2.41342e6 −1.08378
\(347\) − 1.28638e6i − 0.573517i −0.958003 0.286758i \(-0.907422\pi\)
0.958003 0.286758i \(-0.0925777\pi\)
\(348\) 0 0
\(349\) 2.73055e6 1.20001 0.600007 0.799994i \(-0.295164\pi\)
0.600007 + 0.799994i \(0.295164\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 509952.i − 0.219368i
\(353\) 2.13649e6i 0.912564i 0.889835 + 0.456282i \(0.150819\pi\)
−0.889835 + 0.456282i \(0.849181\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.26336e6 0.528326
\(357\) 0 0
\(358\) 1.49748e6i 0.617523i
\(359\) 3.26406e6 1.33666 0.668332 0.743863i \(-0.267009\pi\)
0.668332 + 0.743863i \(0.267009\pi\)
\(360\) 0 0
\(361\) 430926. 0.174034
\(362\) 929692.i 0.372879i
\(363\) 0 0
\(364\) −3.01595e6 −1.19308
\(365\) 0 0
\(366\) 0 0
\(367\) 4.15078e6i 1.60866i 0.594183 + 0.804330i \(0.297476\pi\)
−0.594183 + 0.804330i \(0.702524\pi\)
\(368\) − 397824.i − 0.153134i
\(369\) 0 0
\(370\) 0 0
\(371\) −4.11571e6 −1.55242
\(372\) 0 0
\(373\) − 2.14242e6i − 0.797320i −0.917099 0.398660i \(-0.869475\pi\)
0.917099 0.398660i \(-0.130525\pi\)
\(374\) −1.99598e6 −0.737867
\(375\) 0 0
\(376\) −538752. −0.196526
\(377\) 6.33447e6i 2.29539i
\(378\) 0 0
\(379\) 1.94699e6 0.696253 0.348126 0.937448i \(-0.386818\pi\)
0.348126 + 0.937448i \(0.386818\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 3.38479e6i − 1.18679i
\(383\) − 2.65052e6i − 0.923283i −0.887067 0.461641i \(-0.847261\pi\)
0.887067 0.461641i \(-0.152739\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −622324. −0.212593
\(387\) 0 0
\(388\) 873488.i 0.294563i
\(389\) 282540. 0.0946686 0.0473343 0.998879i \(-0.484927\pi\)
0.0473343 + 0.998879i \(0.484927\pi\)
\(390\) 0 0
\(391\) −1.55711e6 −0.515083
\(392\) − 2.39885e6i − 0.788474i
\(393\) 0 0
\(394\) −413928. −0.134333
\(395\) 0 0
\(396\) 0 0
\(397\) 2.62066e6i 0.834515i 0.908788 + 0.417257i \(0.137009\pi\)
−0.908788 + 0.417257i \(0.862991\pi\)
\(398\) − 561700.i − 0.177745i
\(399\) 0 0
\(400\) 0 0
\(401\) 286248. 0.0888959 0.0444479 0.999012i \(-0.485847\pi\)
0.0444479 + 0.999012i \(0.485847\pi\)
\(402\) 0 0
\(403\) 790393.i 0.242427i
\(404\) 1.68883e6 0.514793
\(405\) 0 0
\(406\) −7.29756e6 −2.19716
\(407\) 2.40136e6i 0.718572i
\(408\) 0 0
\(409\) −4.12069e6 −1.21804 −0.609019 0.793155i \(-0.708437\pi\)
−0.609019 + 0.793155i \(0.708437\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.83898e6i 0.823984i
\(413\) − 8.36703e6i − 2.41377i
\(414\) 0 0
\(415\) 0 0
\(416\) 828416. 0.234701
\(417\) 0 0
\(418\) − 3.39636e6i − 0.950765i
\(419\) −2.37948e6 −0.662136 −0.331068 0.943607i \(-0.607409\pi\)
−0.331068 + 0.943607i \(0.607409\pi\)
\(420\) 0 0
\(421\) −741298. −0.203839 −0.101920 0.994793i \(-0.532498\pi\)
−0.101920 + 0.994793i \(0.532498\pi\)
\(422\) 1.85069e6i 0.505886i
\(423\) 0 0
\(424\) 1.13050e6 0.305390
\(425\) 0 0
\(426\) 0 0
\(427\) − 821791.i − 0.218118i
\(428\) 2.94067e6i 0.775956i
\(429\) 0 0
\(430\) 0 0
\(431\) 187398. 0.0485928 0.0242964 0.999705i \(-0.492265\pi\)
0.0242964 + 0.999705i \(0.492265\pi\)
\(432\) 0 0
\(433\) 6.55110e6i 1.67917i 0.543229 + 0.839585i \(0.317202\pi\)
−0.543229 + 0.839585i \(0.682798\pi\)
\(434\) −910564. −0.232052
\(435\) 0 0
\(436\) 2.71496e6 0.683986
\(437\) − 2.64957e6i − 0.663700i
\(438\) 0 0
\(439\) −270065. −0.0668817 −0.0334408 0.999441i \(-0.510647\pi\)
−0.0334408 + 0.999441i \(0.510647\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 3.24247e6i − 0.789443i
\(443\) − 2.77934e6i − 0.672873i −0.941706 0.336436i \(-0.890778\pi\)
0.941706 0.336436i \(-0.109222\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2.94108e6 −0.700116
\(447\) 0 0
\(448\) 954368.i 0.224657i
\(449\) 7.86630e6 1.84143 0.920714 0.390238i \(-0.127607\pi\)
0.920714 + 0.390238i \(0.127607\pi\)
\(450\) 0 0
\(451\) 4.05770e6 0.939375
\(452\) 4.21574e6i 0.970573i
\(453\) 0 0
\(454\) 3.86875e6 0.880909
\(455\) 0 0
\(456\) 0 0
\(457\) − 6.23356e6i − 1.39619i −0.716004 0.698097i \(-0.754031\pi\)
0.716004 0.698097i \(-0.245969\pi\)
\(458\) − 334780.i − 0.0745754i
\(459\) 0 0
\(460\) 0 0
\(461\) 4.68305e6 1.02630 0.513152 0.858298i \(-0.328478\pi\)
0.513152 + 0.858298i \(0.328478\pi\)
\(462\) 0 0
\(463\) − 382816.i − 0.0829923i −0.999139 0.0414961i \(-0.986788\pi\)
0.999139 0.0414961i \(-0.0132124\pi\)
\(464\) 2.00448e6 0.432222
\(465\) 0 0
\(466\) 3.49550e6 0.745667
\(467\) − 1.93540e6i − 0.410657i −0.978693 0.205328i \(-0.934174\pi\)
0.978693 0.205328i \(-0.0658262\pi\)
\(468\) 0 0
\(469\) −1.33912e7 −2.81117
\(470\) 0 0
\(471\) 0 0
\(472\) 2.29824e6i 0.474832i
\(473\) 9.69556e6i 1.99260i
\(474\) 0 0
\(475\) 0 0
\(476\) 3.73546e6 0.755660
\(477\) 0 0
\(478\) 4.24224e6i 0.849230i
\(479\) −5.56917e6 −1.10905 −0.554526 0.832167i \(-0.687100\pi\)
−0.554526 + 0.832167i \(0.687100\pi\)
\(480\) 0 0
\(481\) −3.90100e6 −0.768799
\(482\) 3.02729e6i 0.593522i
\(483\) 0 0
\(484\) −1.39125e6 −0.269955
\(485\) 0 0
\(486\) 0 0
\(487\) − 4.22450e6i − 0.807148i −0.914947 0.403574i \(-0.867768\pi\)
0.914947 0.403574i \(-0.132232\pi\)
\(488\) 225728.i 0.0429078i
\(489\) 0 0
\(490\) 0 0
\(491\) −6.96295e6 −1.30344 −0.651718 0.758461i \(-0.725951\pi\)
−0.651718 + 0.758461i \(0.725951\pi\)
\(492\) 0 0
\(493\) − 7.84566e6i − 1.45383i
\(494\) 5.51738e6 1.01722
\(495\) 0 0
\(496\) 250112. 0.0456489
\(497\) − 1.75868e6i − 0.319372i
\(498\) 0 0
\(499\) 9.29582e6 1.67123 0.835616 0.549315i \(-0.185111\pi\)
0.835616 + 0.549315i \(0.185111\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 2.54059e6i − 0.449962i
\(503\) − 6.34136e6i − 1.11754i −0.829323 0.558770i \(-0.811274\pi\)
0.829323 0.558770i \(-0.188726\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −3.09557e6 −0.537482
\(507\) 0 0
\(508\) − 4.11059e6i − 0.706716i
\(509\) −7.38309e6 −1.26312 −0.631559 0.775328i \(-0.717585\pi\)
−0.631559 + 0.775328i \(0.717585\pi\)
\(510\) 0 0
\(511\) −150518. −0.0254998
\(512\) − 262144.i − 0.0441942i
\(513\) 0 0
\(514\) 122832. 0.0205071
\(515\) 0 0
\(516\) 0 0
\(517\) 4.19216e6i 0.689782i
\(518\) − 4.49410e6i − 0.735900i
\(519\) 0 0
\(520\) 0 0
\(521\) −4.19620e6 −0.677270 −0.338635 0.940918i \(-0.609965\pi\)
−0.338635 + 0.940918i \(0.609965\pi\)
\(522\) 0 0
\(523\) − 3.57942e6i − 0.572214i −0.958198 0.286107i \(-0.907639\pi\)
0.958198 0.286107i \(-0.0923613\pi\)
\(524\) −3.50477e6 −0.557611
\(525\) 0 0
\(526\) 758064. 0.119465
\(527\) − 978954.i − 0.153545i
\(528\) 0 0
\(529\) 4.02143e6 0.624800
\(530\) 0 0
\(531\) 0 0
\(532\) 6.35624e6i 0.973691i
\(533\) 6.59173e6i 1.00504i
\(534\) 0 0
\(535\) 0 0
\(536\) 3.67827e6 0.553009
\(537\) 0 0
\(538\) − 4.43988e6i − 0.661326i
\(539\) −1.86660e7 −2.76745
\(540\) 0 0
\(541\) 4.95548e6 0.727934 0.363967 0.931412i \(-0.381422\pi\)
0.363967 + 0.931412i \(0.381422\pi\)
\(542\) − 847808.i − 0.123965i
\(543\) 0 0
\(544\) −1.02605e6 −0.148652
\(545\) 0 0
\(546\) 0 0
\(547\) 5.31803e6i 0.759946i 0.924998 + 0.379973i \(0.124067\pi\)
−0.924998 + 0.379973i \(0.875933\pi\)
\(548\) − 3.65885e6i − 0.520467i
\(549\) 0 0
\(550\) 0 0
\(551\) 1.33502e7 1.87330
\(552\) 0 0
\(553\) − 5.29376e6i − 0.736125i
\(554\) −4.18965e6 −0.579967
\(555\) 0 0
\(556\) 5.43584e6 0.745727
\(557\) − 4.25794e6i − 0.581516i −0.956797 0.290758i \(-0.906093\pi\)
0.956797 0.290758i \(-0.0939075\pi\)
\(558\) 0 0
\(559\) −1.57504e7 −2.13188
\(560\) 0 0
\(561\) 0 0
\(562\) 136008.i 0.0181645i
\(563\) 4.37617e6i 0.581866i 0.956743 + 0.290933i \(0.0939656\pi\)
−0.956743 + 0.290933i \(0.906034\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.12440e6 −0.147530
\(567\) 0 0
\(568\) 483072.i 0.0628262i
\(569\) 3.57717e6 0.463190 0.231595 0.972812i \(-0.425606\pi\)
0.231595 + 0.972812i \(0.425606\pi\)
\(570\) 0 0
\(571\) 1.94693e6 0.249896 0.124948 0.992163i \(-0.460124\pi\)
0.124948 + 0.992163i \(0.460124\pi\)
\(572\) − 6.44611e6i − 0.823773i
\(573\) 0 0
\(574\) −7.59394e6 −0.962027
\(575\) 0 0
\(576\) 0 0
\(577\) − 6.95576e6i − 0.869772i −0.900486 0.434886i \(-0.856789\pi\)
0.900486 0.434886i \(-0.143211\pi\)
\(578\) − 1.66341e6i − 0.207100i
\(579\) 0 0
\(580\) 0 0
\(581\) −2.69674e6 −0.331436
\(582\) 0 0
\(583\) − 8.79667e6i − 1.07188i
\(584\) 41344.0 0.00501626
\(585\) 0 0
\(586\) 6.23402e6 0.749936
\(587\) 2.41853e6i 0.289705i 0.989453 + 0.144852i \(0.0462708\pi\)
−0.989453 + 0.144852i \(0.953729\pi\)
\(588\) 0 0
\(589\) 1.66578e6 0.197848
\(590\) 0 0
\(591\) 0 0
\(592\) 1.23443e6i 0.144765i
\(593\) 9.58396e6i 1.11920i 0.828763 + 0.559600i \(0.189045\pi\)
−0.828763 + 0.559600i \(0.810955\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.90320e6 −0.565411
\(597\) 0 0
\(598\) − 5.02874e6i − 0.575051i
\(599\) −1.52070e6 −0.173172 −0.0865858 0.996244i \(-0.527596\pi\)
−0.0865858 + 0.996244i \(0.527596\pi\)
\(600\) 0 0
\(601\) 3.88283e6 0.438492 0.219246 0.975670i \(-0.429640\pi\)
0.219246 + 0.975670i \(0.429640\pi\)
\(602\) − 1.81451e7i − 2.04065i
\(603\) 0 0
\(604\) −3.18123e6 −0.354816
\(605\) 0 0
\(606\) 0 0
\(607\) 107992.i 0.0118965i 0.999982 + 0.00594826i \(0.00189340\pi\)
−0.999982 + 0.00594826i \(0.998107\pi\)
\(608\) − 1.74592e6i − 0.191543i
\(609\) 0 0
\(610\) 0 0
\(611\) −6.81016e6 −0.737997
\(612\) 0 0
\(613\) 7.49923e6i 0.806057i 0.915187 + 0.403028i \(0.132042\pi\)
−0.915187 + 0.403028i \(0.867958\pi\)
\(614\) 3.35967e6 0.359646
\(615\) 0 0
\(616\) 7.42618e6 0.788521
\(617\) − 1.22695e6i − 0.129752i −0.997893 0.0648761i \(-0.979335\pi\)
0.997893 0.0648761i \(-0.0206652\pi\)
\(618\) 0 0
\(619\) −9.51340e6 −0.997950 −0.498975 0.866616i \(-0.666290\pi\)
−0.498975 + 0.866616i \(0.666290\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 1.17079e6i − 0.121340i
\(623\) 1.83977e7i 1.89908i
\(624\) 0 0
\(625\) 0 0
\(626\) 511436. 0.0521621
\(627\) 0 0
\(628\) 5.78053e6i 0.584882i
\(629\) 4.83164e6 0.486932
\(630\) 0 0
\(631\) −1.56449e7 −1.56423 −0.782114 0.623135i \(-0.785859\pi\)
−0.782114 + 0.623135i \(0.785859\pi\)
\(632\) 1.45408e6i 0.144809i
\(633\) 0 0
\(634\) 2.59219e6 0.256120
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.03229e7i − 2.96089i
\(638\) − 1.55974e7i − 1.51705i
\(639\) 0 0
\(640\) 0 0
\(641\) 9.60395e6 0.923219 0.461609 0.887083i \(-0.347272\pi\)
0.461609 + 0.887083i \(0.347272\pi\)
\(642\) 0 0
\(643\) − 8.24396e6i − 0.786336i −0.919467 0.393168i \(-0.871379\pi\)
0.919467 0.393168i \(-0.128621\pi\)
\(644\) 5.79331e6 0.550443
\(645\) 0 0
\(646\) −6.83364e6 −0.644274
\(647\) − 4.07353e6i − 0.382570i −0.981535 0.191285i \(-0.938735\pi\)
0.981535 0.191285i \(-0.0612654\pi\)
\(648\) 0 0
\(649\) 1.78832e7 1.66661
\(650\) 0 0
\(651\) 0 0
\(652\) − 5.41054e6i − 0.498450i
\(653\) − 1.68193e7i − 1.54357i −0.635886 0.771783i \(-0.719365\pi\)
0.635886 0.771783i \(-0.280635\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.08589e6 0.189248
\(657\) 0 0
\(658\) − 7.84558e6i − 0.706415i
\(659\) 2.87826e6 0.258176 0.129088 0.991633i \(-0.458795\pi\)
0.129088 + 0.991633i \(0.458795\pi\)
\(660\) 0 0
\(661\) 1.33386e7 1.18743 0.593713 0.804677i \(-0.297661\pi\)
0.593713 + 0.804677i \(0.297661\pi\)
\(662\) 1.35726e7i 1.20370i
\(663\) 0 0
\(664\) 740736. 0.0651993
\(665\) 0 0
\(666\) 0 0
\(667\) − 1.21678e7i − 1.05901i
\(668\) − 6.88397e6i − 0.596895i
\(669\) 0 0
\(670\) 0 0
\(671\) 1.75645e6 0.150601
\(672\) 0 0
\(673\) 6.37345e6i 0.542422i 0.962520 + 0.271211i \(0.0874241\pi\)
−0.962520 + 0.271211i \(0.912576\pi\)
\(674\) 6.48339e6 0.549734
\(675\) 0 0
\(676\) 4.53101e6 0.381354
\(677\) − 2.27210e7i − 1.90526i −0.304126 0.952632i \(-0.598364\pi\)
0.304126 0.952632i \(-0.401636\pi\)
\(678\) 0 0
\(679\) −1.27202e7 −1.05881
\(680\) 0 0
\(681\) 0 0
\(682\) − 1.94618e6i − 0.160222i
\(683\) 1.53612e7i 1.26001i 0.776593 + 0.630003i \(0.216946\pi\)
−0.776593 + 0.630003i \(0.783054\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.92691e7 1.56333
\(687\) 0 0
\(688\) 4.98406e6i 0.401432i
\(689\) 1.42902e7 1.14680
\(690\) 0 0
\(691\) 8.05035e6 0.641386 0.320693 0.947183i \(-0.396084\pi\)
0.320693 + 0.947183i \(0.396084\pi\)
\(692\) 9.65366e6i 0.766350i
\(693\) 0 0
\(694\) −5.14553e6 −0.405538
\(695\) 0 0
\(696\) 0 0
\(697\) − 8.16430e6i − 0.636556i
\(698\) − 1.09222e7i − 0.848539i
\(699\) 0 0
\(700\) 0 0
\(701\) −7.27840e6 −0.559424 −0.279712 0.960084i \(-0.590239\pi\)
−0.279712 + 0.960084i \(0.590239\pi\)
\(702\) 0 0
\(703\) 8.22151e6i 0.627427i
\(704\) −2.03981e6 −0.155116
\(705\) 0 0
\(706\) 8.54594e6 0.645280
\(707\) 2.45936e7i 1.85044i
\(708\) 0 0
\(709\) −1.37498e7 −1.02726 −0.513630 0.858012i \(-0.671700\pi\)
−0.513630 + 0.858012i \(0.671700\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 5.05344e6i − 0.373583i
\(713\) − 1.51826e6i − 0.111846i
\(714\) 0 0
\(715\) 0 0
\(716\) 5.98992e6 0.436655
\(717\) 0 0
\(718\) − 1.30562e7i − 0.945164i
\(719\) −9.05583e6 −0.653290 −0.326645 0.945147i \(-0.605918\pi\)
−0.326645 + 0.945147i \(0.605918\pi\)
\(720\) 0 0
\(721\) −4.13426e7 −2.96183
\(722\) − 1.72370e6i − 0.123061i
\(723\) 0 0
\(724\) 3.71877e6 0.263665
\(725\) 0 0
\(726\) 0 0
\(727\) 2.20979e7i 1.55065i 0.631560 + 0.775327i \(0.282415\pi\)
−0.631560 + 0.775327i \(0.717585\pi\)
\(728\) 1.20638e7i 0.843638i
\(729\) 0 0
\(730\) 0 0
\(731\) 1.95079e7 1.35026
\(732\) 0 0
\(733\) 2.07337e6i 0.142534i 0.997457 + 0.0712669i \(0.0227042\pi\)
−0.997457 + 0.0712669i \(0.977296\pi\)
\(734\) 1.66031e7 1.13749
\(735\) 0 0
\(736\) −1.59130e6 −0.108282
\(737\) − 2.86216e7i − 1.94100i
\(738\) 0 0
\(739\) −1.48849e7 −1.00262 −0.501310 0.865268i \(-0.667148\pi\)
−0.501310 + 0.865268i \(0.667148\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.64628e7i 1.09773i
\(743\) − 2.58856e7i − 1.72023i −0.510099 0.860116i \(-0.670391\pi\)
0.510099 0.860116i \(-0.329609\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −8.56968e6 −0.563790
\(747\) 0 0
\(748\) 7.98394e6i 0.521751i
\(749\) −4.28235e7 −2.78919
\(750\) 0 0
\(751\) −7.98645e6 −0.516718 −0.258359 0.966049i \(-0.583182\pi\)
−0.258359 + 0.966049i \(0.583182\pi\)
\(752\) 2.15501e6i 0.138965i
\(753\) 0 0
\(754\) 2.53379e7 1.62309
\(755\) 0 0
\(756\) 0 0
\(757\) 1.15570e7i 0.733000i 0.930418 + 0.366500i \(0.119444\pi\)
−0.930418 + 0.366500i \(0.880556\pi\)
\(758\) − 7.78798e6i − 0.492325i
\(759\) 0 0
\(760\) 0 0
\(761\) −3.16705e6 −0.198241 −0.0991205 0.995075i \(-0.531603\pi\)
−0.0991205 + 0.995075i \(0.531603\pi\)
\(762\) 0 0
\(763\) 3.95366e7i 2.45860i
\(764\) −1.35392e7 −0.839187
\(765\) 0 0
\(766\) −1.06021e7 −0.652860
\(767\) 2.90512e7i 1.78310i
\(768\) 0 0
\(769\) −8.21560e6 −0.500983 −0.250492 0.968119i \(-0.580592\pi\)
−0.250492 + 0.968119i \(0.580592\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.48930e6i 0.150326i
\(773\) − 1.19708e7i − 0.720567i −0.932843 0.360284i \(-0.882680\pi\)
0.932843 0.360284i \(-0.117320\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 3.49395e6 0.208287
\(777\) 0 0
\(778\) − 1.13016e6i − 0.0669408i
\(779\) 1.38923e7 0.820223
\(780\) 0 0
\(781\) 3.75890e6 0.220513
\(782\) 6.22843e6i 0.364218i
\(783\) 0 0
\(784\) −9.59539e6 −0.557536
\(785\) 0 0
\(786\) 0 0
\(787\) 1.71154e7i 0.985032i 0.870304 + 0.492516i \(0.163923\pi\)
−0.870304 + 0.492516i \(0.836077\pi\)
\(788\) 1.65571e6i 0.0949881i
\(789\) 0 0
\(790\) 0 0
\(791\) −6.13918e7 −3.48874
\(792\) 0 0
\(793\) 2.85334e6i 0.161128i
\(794\) 1.04826e7 0.590091
\(795\) 0 0
\(796\) −2.24680e6 −0.125685
\(797\) − 2.80753e7i − 1.56559i −0.622277 0.782797i \(-0.713792\pi\)
0.622277 0.782797i \(-0.286208\pi\)
\(798\) 0 0
\(799\) 8.43484e6 0.467423
\(800\) 0 0
\(801\) 0 0
\(802\) − 1.14499e6i − 0.0628589i
\(803\) − 321708.i − 0.0176065i
\(804\) 0 0
\(805\) 0 0
\(806\) 3.16157e6 0.171422
\(807\) 0 0
\(808\) − 6.75533e6i − 0.364014i
\(809\) 9.90816e6 0.532257 0.266129 0.963938i \(-0.414255\pi\)
0.266129 + 0.963938i \(0.414255\pi\)
\(810\) 0 0
\(811\) 5.72573e6 0.305688 0.152844 0.988250i \(-0.451157\pi\)
0.152844 + 0.988250i \(0.451157\pi\)
\(812\) 2.91902e7i 1.55363i
\(813\) 0 0
\(814\) 9.60542e6 0.508107
\(815\) 0 0
\(816\) 0 0
\(817\) 3.31946e7i 1.73985i
\(818\) 1.64827e7i 0.861284i
\(819\) 0 0
\(820\) 0 0
\(821\) 5.96570e6 0.308890 0.154445 0.988001i \(-0.450641\pi\)
0.154445 + 0.988001i \(0.450641\pi\)
\(822\) 0 0
\(823\) − 1.97778e7i − 1.01784i −0.860815 0.508918i \(-0.830046\pi\)
0.860815 0.508918i \(-0.169954\pi\)
\(824\) 1.13559e7 0.582645
\(825\) 0 0
\(826\) −3.34681e7 −1.70679
\(827\) 2.04045e7i 1.03744i 0.854945 + 0.518719i \(0.173591\pi\)
−0.854945 + 0.518719i \(0.826409\pi\)
\(828\) 0 0
\(829\) 2.77183e7 1.40081 0.700406 0.713745i \(-0.253002\pi\)
0.700406 + 0.713745i \(0.253002\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 3.31366e6i − 0.165959i
\(833\) 3.75570e7i 1.87533i
\(834\) 0 0
\(835\) 0 0
\(836\) −1.35854e7 −0.672292
\(837\) 0 0
\(838\) 9.51792e6i 0.468201i
\(839\) −7.79841e6 −0.382473 −0.191237 0.981544i \(-0.561250\pi\)
−0.191237 + 0.981544i \(0.561250\pi\)
\(840\) 0 0
\(841\) 4.07978e7 1.98905
\(842\) 2.96519e6i 0.144136i
\(843\) 0 0
\(844\) 7.40277e6 0.357716
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.02600e7i − 0.970358i
\(848\) − 4.52198e6i − 0.215943i
\(849\) 0 0
\(850\) 0 0
\(851\) 7.49339e6 0.354694
\(852\) 0 0
\(853\) 6.22554e6i 0.292957i 0.989214 + 0.146479i \(0.0467940\pi\)
−0.989214 + 0.146479i \(0.953206\pi\)
\(854\) −3.28716e6 −0.154233
\(855\) 0 0
\(856\) 1.17627e7 0.548684
\(857\) 3.39757e7i 1.58022i 0.612968 + 0.790108i \(0.289976\pi\)
−0.612968 + 0.790108i \(0.710024\pi\)
\(858\) 0 0
\(859\) 1.47282e7 0.681033 0.340516 0.940239i \(-0.389398\pi\)
0.340516 + 0.940239i \(0.389398\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 749592.i − 0.0343603i
\(863\) 1.88594e7i 0.861988i 0.902355 + 0.430994i \(0.141837\pi\)
−0.902355 + 0.430994i \(0.858163\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.62044e7 1.18735
\(867\) 0 0
\(868\) 3.64226e6i 0.164086i
\(869\) 1.13146e7 0.508263
\(870\) 0 0
\(871\) 4.64957e7 2.07667
\(872\) − 1.08598e7i − 0.483651i
\(873\) 0 0
\(874\) −1.05983e7 −0.469307
\(875\) 0 0
\(876\) 0 0
\(877\) 1.82112e7i 0.799540i 0.916615 + 0.399770i \(0.130910\pi\)
−0.916615 + 0.399770i \(0.869090\pi\)
\(878\) 1.08026e6i 0.0472925i
\(879\) 0 0
\(880\) 0 0
\(881\) 7.65425e6 0.332248 0.166124 0.986105i \(-0.446875\pi\)
0.166124 + 0.986105i \(0.446875\pi\)
\(882\) 0 0
\(883\) − 597451.i − 0.0257870i −0.999917 0.0128935i \(-0.995896\pi\)
0.999917 0.0128935i \(-0.00410424\pi\)
\(884\) −1.29699e7 −0.558220
\(885\) 0 0
\(886\) −1.11174e7 −0.475793
\(887\) 2.06888e6i 0.0882929i 0.999025 + 0.0441465i \(0.0140568\pi\)
−0.999025 + 0.0441465i \(0.985943\pi\)
\(888\) 0 0
\(889\) 5.98605e7 2.54031
\(890\) 0 0
\(891\) 0 0
\(892\) 1.17643e7i 0.495057i
\(893\) 1.43527e7i 0.602289i
\(894\) 0 0
\(895\) 0 0
\(896\) 3.81747e6 0.158857
\(897\) 0 0
\(898\) − 3.14652e7i − 1.30209i
\(899\) 7.64991e6 0.315687
\(900\) 0 0
\(901\) −1.76993e7 −0.726348
\(902\) − 1.62308e7i − 0.664238i
\(903\) 0 0
\(904\) 1.68630e7 0.686299
\(905\) 0 0
\(906\) 0 0
\(907\) − 7.83331e6i − 0.316175i −0.987425 0.158087i \(-0.949467\pi\)
0.987425 0.158087i \(-0.0505327\pi\)
\(908\) − 1.54750e7i − 0.622897i
\(909\) 0 0
\(910\) 0 0
\(911\) −4.08133e7 −1.62932 −0.814659 0.579940i \(-0.803076\pi\)
−0.814659 + 0.579940i \(0.803076\pi\)
\(912\) 0 0
\(913\) − 5.76385e6i − 0.228842i
\(914\) −2.49342e7 −0.987258
\(915\) 0 0
\(916\) −1.33912e6 −0.0527328
\(917\) − 5.10382e7i − 2.00434i
\(918\) 0 0
\(919\) −6.21579e6 −0.242777 −0.121389 0.992605i \(-0.538735\pi\)
−0.121389 + 0.992605i \(0.538735\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 1.87322e7i − 0.725707i
\(923\) 6.10633e6i 0.235926i
\(924\) 0 0
\(925\) 0 0
\(926\) −1.53126e6 −0.0586844
\(927\) 0 0
\(928\) − 8.01792e6i − 0.305627i
\(929\) −4.64847e6 −0.176714 −0.0883570 0.996089i \(-0.528162\pi\)
−0.0883570 + 0.996089i \(0.528162\pi\)
\(930\) 0 0
\(931\) −6.39068e7 −2.41642
\(932\) − 1.39820e7i − 0.527266i
\(933\) 0 0
\(934\) −7.74161e6 −0.290378
\(935\) 0 0
\(936\) 0 0
\(937\) 1.33603e7i 0.497127i 0.968616 + 0.248563i \(0.0799584\pi\)
−0.968616 + 0.248563i \(0.920042\pi\)
\(938\) 5.35648e7i 1.98780i
\(939\) 0 0
\(940\) 0 0
\(941\) −3.32569e7 −1.22436 −0.612178 0.790720i \(-0.709706\pi\)
−0.612178 + 0.790720i \(0.709706\pi\)
\(942\) 0 0
\(943\) − 1.26620e7i − 0.463685i
\(944\) 9.19296e6 0.335757
\(945\) 0 0
\(946\) 3.87822e7 1.40898
\(947\) 4.21530e7i 1.52740i 0.645569 + 0.763702i \(0.276620\pi\)
−0.645569 + 0.763702i \(0.723380\pi\)
\(948\) 0 0
\(949\) 522614. 0.0188372
\(950\) 0 0
\(951\) 0 0
\(952\) − 1.49418e7i − 0.534332i
\(953\) − 2.24691e7i − 0.801406i −0.916208 0.400703i \(-0.868766\pi\)
0.916208 0.400703i \(-0.131234\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.69690e7 0.600497
\(957\) 0 0
\(958\) 2.22767e7i 0.784218i
\(959\) 5.32820e7 1.87083
\(960\) 0 0
\(961\) −2.76746e7 −0.966659
\(962\) 1.56040e7i 0.543623i
\(963\) 0 0
\(964\) 1.21092e7 0.419683
\(965\) 0 0
\(966\) 0 0
\(967\) 1.75000e7i 0.601826i 0.953652 + 0.300913i \(0.0972913\pi\)
−0.953652 + 0.300913i \(0.902709\pi\)
\(968\) 5.56499e6i 0.190887i
\(969\) 0 0
\(970\) 0 0
\(971\) −5.42920e7 −1.84794 −0.923970 0.382465i \(-0.875075\pi\)
−0.923970 + 0.382465i \(0.875075\pi\)
\(972\) 0 0
\(973\) 7.91594e7i 2.68053i
\(974\) −1.68980e7 −0.570740
\(975\) 0 0
\(976\) 902912. 0.0303404
\(977\) 2.55925e7i 0.857782i 0.903356 + 0.428891i \(0.141096\pi\)
−0.903356 + 0.428891i \(0.858904\pi\)
\(978\) 0 0
\(979\) −3.93221e7 −1.31123
\(980\) 0 0
\(981\) 0 0
\(982\) 2.78518e7i 0.921668i
\(983\) 4.82488e6i 0.159258i 0.996825 + 0.0796292i \(0.0253736\pi\)
−0.996825 + 0.0796292i \(0.974626\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −3.13826e7 −1.02801
\(987\) 0 0
\(988\) − 2.20695e7i − 0.719284i
\(989\) 3.02548e7 0.983567
\(990\) 0 0
\(991\) 1.18448e6 0.0383127 0.0191563 0.999817i \(-0.493902\pi\)
0.0191563 + 0.999817i \(0.493902\pi\)
\(992\) − 1.00045e6i − 0.0322786i
\(993\) 0 0
\(994\) −7.03474e6 −0.225830
\(995\) 0 0
\(996\) 0 0
\(997\) 2.58178e7i 0.822585i 0.911503 + 0.411293i \(0.134923\pi\)
−0.911503 + 0.411293i \(0.865077\pi\)
\(998\) − 3.71833e7i − 1.18174i
\(999\) 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.c.n.199.1 2
3.2 odd 2 150.6.c.e.49.2 2
5.2 odd 4 450.6.a.x.1.1 1
5.3 odd 4 450.6.a.a.1.1 1
5.4 even 2 inner 450.6.c.n.199.2 2
15.2 even 4 150.6.a.c.1.1 1
15.8 even 4 150.6.a.l.1.1 yes 1
15.14 odd 2 150.6.c.e.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.6.a.c.1.1 1 15.2 even 4
150.6.a.l.1.1 yes 1 15.8 even 4
150.6.c.e.49.1 2 15.14 odd 2
150.6.c.e.49.2 2 3.2 odd 2
450.6.a.a.1.1 1 5.3 odd 4
450.6.a.x.1.1 1 5.2 odd 4
450.6.c.n.199.1 2 1.1 even 1 trivial
450.6.c.n.199.2 2 5.4 even 2 inner