Properties

Label 450.6.c.e.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.e.199.2

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000i q^{2} -16.0000 q^{4} +47.0000i q^{7} +64.0000i q^{8} +O(q^{10})\) \(q-4.00000i q^{2} -16.0000 q^{4} +47.0000i q^{7} +64.0000i q^{8} -222.000 q^{11} -101.000i q^{13} +188.000 q^{14} +256.000 q^{16} -162.000i q^{17} -1685.00 q^{19} +888.000i q^{22} +306.000i q^{23} -404.000 q^{26} -752.000i q^{28} +7890.00 q^{29} -8593.00 q^{31} -1024.00i q^{32} -648.000 q^{34} +8642.00i q^{37} +6740.00i q^{38} +18168.0 q^{41} -14351.0i q^{43} +3552.00 q^{44} +1224.00 q^{46} +1098.00i q^{47} +14598.0 q^{49} +1616.00i q^{52} +17916.0i q^{53} -3008.00 q^{56} -31560.0i q^{58} +17610.0 q^{59} -21853.0 q^{61} +34372.0i q^{62} -4096.00 q^{64} +107.000i q^{67} +2592.00i q^{68} +40728.0 q^{71} -34706.0i q^{73} +34568.0 q^{74} +26960.0 q^{76} -10434.0i q^{77} +69160.0 q^{79} -72672.0i q^{82} -108534. i q^{83} -57404.0 q^{86} -14208.0i q^{88} +35040.0 q^{89} +4747.00 q^{91} -4896.00i q^{92} +4392.00 q^{94} -823.000i q^{97} -58392.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} - 444q^{11} + 376q^{14} + 512q^{16} - 3370q^{19} - 808q^{26} + 15780q^{29} - 17186q^{31} - 1296q^{34} + 36336q^{41} + 7104q^{44} + 2448q^{46} + 29196q^{49} - 6016q^{56} + 35220q^{59} - 43706q^{61} - 8192q^{64} + 81456q^{71} + 69136q^{74} + 53920q^{76} + 138320q^{79} - 114808q^{86} + 70080q^{89} + 9494q^{91} + 8784q^{94} + O(q^{100}) \)

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
\(7\) 47.0000i 0.362537i 0.983434 + 0.181269i \(0.0580204\pi\)
−0.983434 + 0.181269i \(0.941980\pi\)
\(8\) 64.0000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −222.000 −0.553186 −0.276593 0.960987i \(-0.589205\pi\)
−0.276593 + 0.960987i \(0.589205\pi\)
\(12\) 0 0
\(13\) − 101.000i − 0.165754i −0.996560 0.0828768i \(-0.973589\pi\)
0.996560 0.0828768i \(-0.0264108\pi\)
\(14\) 188.000 0.256353
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) − 162.000i − 0.135954i −0.997687 0.0679771i \(-0.978346\pi\)
0.997687 0.0679771i \(-0.0216545\pi\)
\(18\) 0 0
\(19\) −1685.00 −1.07082 −0.535409 0.844593i \(-0.679843\pi\)
−0.535409 + 0.844593i \(0.679843\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 888.000i 0.391162i
\(23\) 306.000i 0.120615i 0.998180 + 0.0603076i \(0.0192082\pi\)
−0.998180 + 0.0603076i \(0.980792\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −404.000 −0.117206
\(27\) 0 0
\(28\) − 752.000i − 0.181269i
\(29\) 7890.00 1.74214 0.871068 0.491163i \(-0.163428\pi\)
0.871068 + 0.491163i \(0.163428\pi\)
\(30\) 0 0
\(31\) −8593.00 −1.60598 −0.802991 0.595991i \(-0.796759\pi\)
−0.802991 + 0.595991i \(0.796759\pi\)
\(32\) − 1024.00i − 0.176777i
\(33\) 0 0
\(34\) −648.000 −0.0961342
\(35\) 0 0
\(36\) 0 0
\(37\) 8642.00i 1.03779i 0.854838 + 0.518896i \(0.173657\pi\)
−0.854838 + 0.518896i \(0.826343\pi\)
\(38\) 6740.00i 0.757183i
\(39\) 0 0
\(40\) 0 0
\(41\) 18168.0 1.68790 0.843951 0.536420i \(-0.180223\pi\)
0.843951 + 0.536420i \(0.180223\pi\)
\(42\) 0 0
\(43\) − 14351.0i − 1.18362i −0.806079 0.591808i \(-0.798414\pi\)
0.806079 0.591808i \(-0.201586\pi\)
\(44\) 3552.00 0.276593
\(45\) 0 0
\(46\) 1224.00 0.0852878
\(47\) 1098.00i 0.0725033i 0.999343 + 0.0362516i \(0.0115418\pi\)
−0.999343 + 0.0362516i \(0.988458\pi\)
\(48\) 0 0
\(49\) 14598.0 0.868567
\(50\) 0 0
\(51\) 0 0
\(52\) 1616.00i 0.0828768i
\(53\) 17916.0i 0.876095i 0.898952 + 0.438048i \(0.144330\pi\)
−0.898952 + 0.438048i \(0.855670\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3008.00 −0.128176
\(57\) 0 0
\(58\) − 31560.0i − 1.23188i
\(59\) 17610.0 0.658612 0.329306 0.944223i \(-0.393185\pi\)
0.329306 + 0.944223i \(0.393185\pi\)
\(60\) 0 0
\(61\) −21853.0 −0.751946 −0.375973 0.926631i \(-0.622691\pi\)
−0.375973 + 0.926631i \(0.622691\pi\)
\(62\) 34372.0i 1.13560i
\(63\) 0 0
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 107.000i 0.00291204i 0.999999 + 0.00145602i \(0.000463465\pi\)
−0.999999 + 0.00145602i \(0.999537\pi\)
\(68\) 2592.00i 0.0679771i
\(69\) 0 0
\(70\) 0 0
\(71\) 40728.0 0.958842 0.479421 0.877585i \(-0.340847\pi\)
0.479421 + 0.877585i \(0.340847\pi\)
\(72\) 0 0
\(73\) − 34706.0i − 0.762250i −0.924524 0.381125i \(-0.875537\pi\)
0.924524 0.381125i \(-0.124463\pi\)
\(74\) 34568.0 0.733829
\(75\) 0 0
\(76\) 26960.0 0.535409
\(77\) − 10434.0i − 0.200551i
\(78\) 0 0
\(79\) 69160.0 1.24677 0.623386 0.781914i \(-0.285756\pi\)
0.623386 + 0.781914i \(0.285756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 72672.0i − 1.19353i
\(83\) − 108534.i − 1.72930i −0.502374 0.864650i \(-0.667540\pi\)
0.502374 0.864650i \(-0.332460\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −57404.0 −0.836943
\(87\) 0 0
\(88\) − 14208.0i − 0.195581i
\(89\) 35040.0 0.468910 0.234455 0.972127i \(-0.424670\pi\)
0.234455 + 0.972127i \(0.424670\pi\)
\(90\) 0 0
\(91\) 4747.00 0.0600919
\(92\) − 4896.00i − 0.0603076i
\(93\) 0 0
\(94\) 4392.00 0.0512676
\(95\) 0 0
\(96\) 0 0
\(97\) − 823.000i − 0.00888118i −0.999990 0.00444059i \(-0.998587\pi\)
0.999990 0.00444059i \(-0.00141349\pi\)
\(98\) − 58392.0i − 0.614169i
\(99\) 0 0
\(100\) 0 0
\(101\) 33828.0 0.329969 0.164984 0.986296i \(-0.447243\pi\)
0.164984 + 0.986296i \(0.447243\pi\)
\(102\) 0 0
\(103\) 133444.i 1.23938i 0.784845 + 0.619692i \(0.212743\pi\)
−0.784845 + 0.619692i \(0.787257\pi\)
\(104\) 6464.00 0.0586028
\(105\) 0 0
\(106\) 71664.0 0.619493
\(107\) − 81252.0i − 0.686080i −0.939321 0.343040i \(-0.888543\pi\)
0.939321 0.343040i \(-0.111457\pi\)
\(108\) 0 0
\(109\) 217015. 1.74954 0.874769 0.484540i \(-0.161013\pi\)
0.874769 + 0.484540i \(0.161013\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 12032.0i 0.0906343i
\(113\) − 138324.i − 1.01906i −0.860452 0.509532i \(-0.829819\pi\)
0.860452 0.509532i \(-0.170181\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −126240. −0.871068
\(117\) 0 0
\(118\) − 70440.0i − 0.465709i
\(119\) 7614.00 0.0492885
\(120\) 0 0
\(121\) −111767. −0.693985
\(122\) 87412.0i 0.531706i
\(123\) 0 0
\(124\) 137488. 0.802991
\(125\) 0 0
\(126\) 0 0
\(127\) − 256048.i − 1.40868i −0.709863 0.704340i \(-0.751243\pi\)
0.709863 0.704340i \(-0.248757\pi\)
\(128\) 16384.0i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −118452. −0.603065 −0.301533 0.953456i \(-0.597498\pi\)
−0.301533 + 0.953456i \(0.597498\pi\)
\(132\) 0 0
\(133\) − 79195.0i − 0.388212i
\(134\) 428.000 0.00205912
\(135\) 0 0
\(136\) 10368.0 0.0480671
\(137\) 13218.0i 0.0601678i 0.999547 + 0.0300839i \(0.00957745\pi\)
−0.999547 + 0.0300839i \(0.990423\pi\)
\(138\) 0 0
\(139\) 350740. 1.53974 0.769872 0.638199i \(-0.220320\pi\)
0.769872 + 0.638199i \(0.220320\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 162912.i − 0.678004i
\(143\) 22422.0i 0.0916926i
\(144\) 0 0
\(145\) 0 0
\(146\) −138824. −0.538992
\(147\) 0 0
\(148\) − 138272.i − 0.518896i
\(149\) 109890. 0.405502 0.202751 0.979230i \(-0.435012\pi\)
0.202751 + 0.979230i \(0.435012\pi\)
\(150\) 0 0
\(151\) −172603. −0.616036 −0.308018 0.951381i \(-0.599666\pi\)
−0.308018 + 0.951381i \(0.599666\pi\)
\(152\) − 107840.i − 0.378592i
\(153\) 0 0
\(154\) −41736.0 −0.141811
\(155\) 0 0
\(156\) 0 0
\(157\) − 349993.i − 1.13321i −0.823990 0.566605i \(-0.808257\pi\)
0.823990 0.566605i \(-0.191743\pi\)
\(158\) − 276640.i − 0.881601i
\(159\) 0 0
\(160\) 0 0
\(161\) −14382.0 −0.0437275
\(162\) 0 0
\(163\) − 192581.i − 0.567733i −0.958864 0.283867i \(-0.908383\pi\)
0.958864 0.283867i \(-0.0916173\pi\)
\(164\) −290688. −0.843951
\(165\) 0 0
\(166\) −434136. −1.22280
\(167\) − 580692.i − 1.61122i −0.592447 0.805610i \(-0.701838\pi\)
0.592447 0.805610i \(-0.298162\pi\)
\(168\) 0 0
\(169\) 361092. 0.972526
\(170\) 0 0
\(171\) 0 0
\(172\) 229616.i 0.591808i
\(173\) 738126.i 1.87506i 0.347904 + 0.937530i \(0.386894\pi\)
−0.347904 + 0.937530i \(0.613106\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −56832.0 −0.138297
\(177\) 0 0
\(178\) − 140160.i − 0.331569i
\(179\) 497370. 1.16024 0.580119 0.814532i \(-0.303006\pi\)
0.580119 + 0.814532i \(0.303006\pi\)
\(180\) 0 0
\(181\) −333163. −0.755893 −0.377947 0.925827i \(-0.623370\pi\)
−0.377947 + 0.925827i \(0.623370\pi\)
\(182\) − 18988.0i − 0.0424914i
\(183\) 0 0
\(184\) −19584.0 −0.0426439
\(185\) 0 0
\(186\) 0 0
\(187\) 35964.0i 0.0752080i
\(188\) − 17568.0i − 0.0362516i
\(189\) 0 0
\(190\) 0 0
\(191\) 40638.0 0.0806026 0.0403013 0.999188i \(-0.487168\pi\)
0.0403013 + 0.999188i \(0.487168\pi\)
\(192\) 0 0
\(193\) − 494651.i − 0.955885i −0.878391 0.477942i \(-0.841383\pi\)
0.878391 0.477942i \(-0.158617\pi\)
\(194\) −3292.00 −0.00627994
\(195\) 0 0
\(196\) −233568. −0.434283
\(197\) − 552342.i − 1.01401i −0.861943 0.507005i \(-0.830752\pi\)
0.861943 0.507005i \(-0.169248\pi\)
\(198\) 0 0
\(199\) −685625. −1.22731 −0.613655 0.789575i \(-0.710301\pi\)
−0.613655 + 0.789575i \(0.710301\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 135312.i − 0.233323i
\(203\) 370830.i 0.631589i
\(204\) 0 0
\(205\) 0 0
\(206\) 533776. 0.876377
\(207\) 0 0
\(208\) − 25856.0i − 0.0414384i
\(209\) 374070. 0.592362
\(210\) 0 0
\(211\) 749477. 1.15892 0.579458 0.815002i \(-0.303264\pi\)
0.579458 + 0.815002i \(0.303264\pi\)
\(212\) − 286656.i − 0.438048i
\(213\) 0 0
\(214\) −325008. −0.485132
\(215\) 0 0
\(216\) 0 0
\(217\) − 403871.i − 0.582228i
\(218\) − 868060.i − 1.23711i
\(219\) 0 0
\(220\) 0 0
\(221\) −16362.0 −0.0225349
\(222\) 0 0
\(223\) − 169271.i − 0.227940i −0.993484 0.113970i \(-0.963643\pi\)
0.993484 0.113970i \(-0.0363568\pi\)
\(224\) 48128.0 0.0640882
\(225\) 0 0
\(226\) −553296. −0.720587
\(227\) 46488.0i 0.0598792i 0.999552 + 0.0299396i \(0.00953150\pi\)
−0.999552 + 0.0299396i \(0.990468\pi\)
\(228\) 0 0
\(229\) 90115.0 0.113556 0.0567778 0.998387i \(-0.481917\pi\)
0.0567778 + 0.998387i \(0.481917\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 504960.i 0.615938i
\(233\) 1.06414e6i 1.28413i 0.766652 + 0.642063i \(0.221921\pi\)
−0.766652 + 0.642063i \(0.778079\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −281760. −0.329306
\(237\) 0 0
\(238\) − 30456.0i − 0.0348522i
\(239\) 1.15158e6 1.30407 0.652033 0.758191i \(-0.273916\pi\)
0.652033 + 0.758191i \(0.273916\pi\)
\(240\) 0 0
\(241\) 856217. 0.949601 0.474801 0.880093i \(-0.342520\pi\)
0.474801 + 0.880093i \(0.342520\pi\)
\(242\) 447068.i 0.490722i
\(243\) 0 0
\(244\) 349648. 0.375973
\(245\) 0 0
\(246\) 0 0
\(247\) 170185.i 0.177492i
\(248\) − 549952.i − 0.567800i
\(249\) 0 0
\(250\) 0 0
\(251\) 207708. 0.208098 0.104049 0.994572i \(-0.466820\pi\)
0.104049 + 0.994572i \(0.466820\pi\)
\(252\) 0 0
\(253\) − 67932.0i − 0.0667226i
\(254\) −1.02419e6 −0.996087
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 1.45319e6i 1.37243i 0.727401 + 0.686213i \(0.240728\pi\)
−0.727401 + 0.686213i \(0.759272\pi\)
\(258\) 0 0
\(259\) −406174. −0.376238
\(260\) 0 0
\(261\) 0 0
\(262\) 473808.i 0.426431i
\(263\) 169296.i 0.150924i 0.997149 + 0.0754618i \(0.0240431\pi\)
−0.997149 + 0.0754618i \(0.975957\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −316780. −0.274507
\(267\) 0 0
\(268\) − 1712.00i − 0.00145602i
\(269\) −1.58109e6 −1.33222 −0.666110 0.745854i \(-0.732042\pi\)
−0.666110 + 0.745854i \(0.732042\pi\)
\(270\) 0 0
\(271\) 822512. 0.680329 0.340165 0.940366i \(-0.389517\pi\)
0.340165 + 0.940366i \(0.389517\pi\)
\(272\) − 41472.0i − 0.0339886i
\(273\) 0 0
\(274\) 52872.0 0.0425451
\(275\) 0 0
\(276\) 0 0
\(277\) − 546823.i − 0.428201i −0.976812 0.214100i \(-0.931318\pi\)
0.976812 0.214100i \(-0.0686820\pi\)
\(278\) − 1.40296e6i − 1.08876i
\(279\) 0 0
\(280\) 0 0
\(281\) 1.09250e6 0.825382 0.412691 0.910871i \(-0.364589\pi\)
0.412691 + 0.910871i \(0.364589\pi\)
\(282\) 0 0
\(283\) 2.48480e6i 1.84427i 0.386865 + 0.922136i \(0.373558\pi\)
−0.386865 + 0.922136i \(0.626442\pi\)
\(284\) −651648. −0.479421
\(285\) 0 0
\(286\) 89688.0 0.0648365
\(287\) 853896.i 0.611928i
\(288\) 0 0
\(289\) 1.39361e6 0.981516
\(290\) 0 0
\(291\) 0 0
\(292\) 555296.i 0.381125i
\(293\) − 341394.i − 0.232320i −0.993231 0.116160i \(-0.962941\pi\)
0.993231 0.116160i \(-0.0370586\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −553088. −0.366915
\(297\) 0 0
\(298\) − 439560.i − 0.286733i
\(299\) 30906.0 0.0199924
\(300\) 0 0
\(301\) 674497. 0.429105
\(302\) 690412.i 0.435603i
\(303\) 0 0
\(304\) −431360. −0.267705
\(305\) 0 0
\(306\) 0 0
\(307\) 2.02898e6i 1.22866i 0.789050 + 0.614329i \(0.210573\pi\)
−0.789050 + 0.614329i \(0.789427\pi\)
\(308\) 166944.i 0.100275i
\(309\) 0 0
\(310\) 0 0
\(311\) 206598. 0.121123 0.0605613 0.998164i \(-0.480711\pi\)
0.0605613 + 0.998164i \(0.480711\pi\)
\(312\) 0 0
\(313\) 3.34223e6i 1.92830i 0.265352 + 0.964152i \(0.414512\pi\)
−0.265352 + 0.964152i \(0.585488\pi\)
\(314\) −1.39997e6 −0.801300
\(315\) 0 0
\(316\) −1.10656e6 −0.623386
\(317\) 2.53289e6i 1.41569i 0.706368 + 0.707844i \(0.250332\pi\)
−0.706368 + 0.707844i \(0.749668\pi\)
\(318\) 0 0
\(319\) −1.75158e6 −0.963725
\(320\) 0 0
\(321\) 0 0
\(322\) 57528.0i 0.0309200i
\(323\) 272970.i 0.145582i
\(324\) 0 0
\(325\) 0 0
\(326\) −770324. −0.401448
\(327\) 0 0
\(328\) 1.16275e6i 0.596764i
\(329\) −51606.0 −0.0262851
\(330\) 0 0
\(331\) 602132. 0.302080 0.151040 0.988528i \(-0.451738\pi\)
0.151040 + 0.988528i \(0.451738\pi\)
\(332\) 1.73654e6i 0.864650i
\(333\) 0 0
\(334\) −2.32277e6 −1.13930
\(335\) 0 0
\(336\) 0 0
\(337\) 209777.i 0.100620i 0.998734 + 0.0503099i \(0.0160209\pi\)
−0.998734 + 0.0503099i \(0.983979\pi\)
\(338\) − 1.44437e6i − 0.687680i
\(339\) 0 0
\(340\) 0 0
\(341\) 1.90765e6 0.888407
\(342\) 0 0
\(343\) 1.47603e6i 0.677425i
\(344\) 918464. 0.418472
\(345\) 0 0
\(346\) 2.95250e6 1.32587
\(347\) − 4.02166e6i − 1.79301i −0.443037 0.896503i \(-0.646099\pi\)
0.443037 0.896503i \(-0.353901\pi\)
\(348\) 0 0
\(349\) −8330.00 −0.00366085 −0.00183042 0.999998i \(-0.500583\pi\)
−0.00183042 + 0.999998i \(0.500583\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 227328.i 0.0977904i
\(353\) 1.95001e6i 0.832912i 0.909156 + 0.416456i \(0.136728\pi\)
−0.909156 + 0.416456i \(0.863272\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −560640. −0.234455
\(357\) 0 0
\(358\) − 1.98948e6i − 0.820412i
\(359\) 2.27088e6 0.929947 0.464973 0.885325i \(-0.346064\pi\)
0.464973 + 0.885325i \(0.346064\pi\)
\(360\) 0 0
\(361\) 363126. 0.146652
\(362\) 1.33265e6i 0.534497i
\(363\) 0 0
\(364\) −75952.0 −0.0300459
\(365\) 0 0
\(366\) 0 0
\(367\) 2.86154e6i 1.10901i 0.832181 + 0.554503i \(0.187092\pi\)
−0.832181 + 0.554503i \(0.812908\pi\)
\(368\) 78336.0i 0.0301538i
\(369\) 0 0
\(370\) 0 0
\(371\) −842052. −0.317617
\(372\) 0 0
\(373\) − 615311.i − 0.228993i −0.993424 0.114497i \(-0.963474\pi\)
0.993424 0.114497i \(-0.0365255\pi\)
\(374\) 143856. 0.0531801
\(375\) 0 0
\(376\) −70272.0 −0.0256338
\(377\) − 796890.i − 0.288765i
\(378\) 0 0
\(379\) −5.39878e6 −1.93062 −0.965311 0.261103i \(-0.915914\pi\)
−0.965311 + 0.261103i \(0.915914\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 162552.i − 0.0569946i
\(383\) 1.08688e6i 0.378602i 0.981919 + 0.189301i \(0.0606222\pi\)
−0.981919 + 0.189301i \(0.939378\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.97860e6 −0.675913
\(387\) 0 0
\(388\) 13168.0i 0.00444059i
\(389\) −3.48432e6 −1.16747 −0.583733 0.811946i \(-0.698408\pi\)
−0.583733 + 0.811946i \(0.698408\pi\)
\(390\) 0 0
\(391\) 49572.0 0.0163981
\(392\) 934272.i 0.307085i
\(393\) 0 0
\(394\) −2.20937e6 −0.717014
\(395\) 0 0
\(396\) 0 0
\(397\) 3.26591e6i 1.03999i 0.854170 + 0.519993i \(0.174065\pi\)
−0.854170 + 0.519993i \(0.825935\pi\)
\(398\) 2.74250e6i 0.867839i
\(399\) 0 0
\(400\) 0 0
\(401\) 4.27319e6 1.32706 0.663531 0.748149i \(-0.269057\pi\)
0.663531 + 0.748149i \(0.269057\pi\)
\(402\) 0 0
\(403\) 867893.i 0.266197i
\(404\) −541248. −0.164984
\(405\) 0 0
\(406\) 1.48332e6 0.446601
\(407\) − 1.91852e6i − 0.574092i
\(408\) 0 0
\(409\) 1.45188e6 0.429162 0.214581 0.976706i \(-0.431161\pi\)
0.214581 + 0.976706i \(0.431161\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 2.13510e6i − 0.619692i
\(413\) 827670.i 0.238771i
\(414\) 0 0
\(415\) 0 0
\(416\) −103424. −0.0293014
\(417\) 0 0
\(418\) − 1.49628e6i − 0.418863i
\(419\) 559380. 0.155658 0.0778291 0.996967i \(-0.475201\pi\)
0.0778291 + 0.996967i \(0.475201\pi\)
\(420\) 0 0
\(421\) −3.91470e6 −1.07645 −0.538224 0.842802i \(-0.680905\pi\)
−0.538224 + 0.842802i \(0.680905\pi\)
\(422\) − 2.99791e6i − 0.819478i
\(423\) 0 0
\(424\) −1.14662e6 −0.309746
\(425\) 0 0
\(426\) 0 0
\(427\) − 1.02709e6i − 0.272608i
\(428\) 1.30003e6i 0.343040i
\(429\) 0 0
\(430\) 0 0
\(431\) 3.57500e6 0.927006 0.463503 0.886095i \(-0.346592\pi\)
0.463503 + 0.886095i \(0.346592\pi\)
\(432\) 0 0
\(433\) − 7.15969e6i − 1.83516i −0.397548 0.917581i \(-0.630139\pi\)
0.397548 0.917581i \(-0.369861\pi\)
\(434\) −1.61548e6 −0.411698
\(435\) 0 0
\(436\) −3.47224e6 −0.874769
\(437\) − 515610.i − 0.129157i
\(438\) 0 0
\(439\) −1.71790e6 −0.425437 −0.212719 0.977114i \(-0.568232\pi\)
−0.212719 + 0.977114i \(0.568232\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 65448.0i 0.0159346i
\(443\) 3.39670e6i 0.822332i 0.911560 + 0.411166i \(0.134878\pi\)
−0.911560 + 0.411166i \(0.865122\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −677084. −0.161178
\(447\) 0 0
\(448\) − 192512.i − 0.0453172i
\(449\) −3.39606e6 −0.794986 −0.397493 0.917605i \(-0.630120\pi\)
−0.397493 + 0.917605i \(0.630120\pi\)
\(450\) 0 0
\(451\) −4.03330e6 −0.933724
\(452\) 2.21318e6i 0.509532i
\(453\) 0 0
\(454\) 185952. 0.0423410
\(455\) 0 0
\(456\) 0 0
\(457\) − 4.52814e6i − 1.01421i −0.861883 0.507106i \(-0.830715\pi\)
0.861883 0.507106i \(-0.169285\pi\)
\(458\) − 360460.i − 0.0802959i
\(459\) 0 0
\(460\) 0 0
\(461\) 1.27895e6 0.280285 0.140143 0.990131i \(-0.455244\pi\)
0.140143 + 0.990131i \(0.455244\pi\)
\(462\) 0 0
\(463\) 7.19862e6i 1.56062i 0.625393 + 0.780310i \(0.284939\pi\)
−0.625393 + 0.780310i \(0.715061\pi\)
\(464\) 2.01984e6 0.435534
\(465\) 0 0
\(466\) 4.25654e6 0.908014
\(467\) − 4.83034e6i − 1.02491i −0.858714 0.512455i \(-0.828736\pi\)
0.858714 0.512455i \(-0.171264\pi\)
\(468\) 0 0
\(469\) −5029.00 −0.00105572
\(470\) 0 0
\(471\) 0 0
\(472\) 1.12704e6i 0.232854i
\(473\) 3.18592e6i 0.654760i
\(474\) 0 0
\(475\) 0 0
\(476\) −121824. −0.0246442
\(477\) 0 0
\(478\) − 4.60632e6i − 0.922113i
\(479\) 748650. 0.149087 0.0745435 0.997218i \(-0.476250\pi\)
0.0745435 + 0.997218i \(0.476250\pi\)
\(480\) 0 0
\(481\) 872842. 0.172018
\(482\) − 3.42487e6i − 0.671469i
\(483\) 0 0
\(484\) 1.78827e6 0.346993
\(485\) 0 0
\(486\) 0 0
\(487\) − 5.16394e6i − 0.986641i −0.869848 0.493320i \(-0.835783\pi\)
0.869848 0.493320i \(-0.164217\pi\)
\(488\) − 1.39859e6i − 0.265853i
\(489\) 0 0
\(490\) 0 0
\(491\) −8.54287e6 −1.59919 −0.799595 0.600539i \(-0.794953\pi\)
−0.799595 + 0.600539i \(0.794953\pi\)
\(492\) 0 0
\(493\) − 1.27818e6i − 0.236851i
\(494\) 680740. 0.125506
\(495\) 0 0
\(496\) −2.19981e6 −0.401495
\(497\) 1.91422e6i 0.347616i
\(498\) 0 0
\(499\) 4.20588e6 0.756145 0.378072 0.925776i \(-0.376587\pi\)
0.378072 + 0.925776i \(0.376587\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 830832.i − 0.147148i
\(503\) − 8.18342e6i − 1.44217i −0.692849 0.721083i \(-0.743645\pi\)
0.692849 0.721083i \(-0.256355\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −271728. −0.0471800
\(507\) 0 0
\(508\) 4.09677e6i 0.704340i
\(509\) 3.85923e6 0.660247 0.330123 0.943938i \(-0.392910\pi\)
0.330123 + 0.943938i \(0.392910\pi\)
\(510\) 0 0
\(511\) 1.63118e6 0.276344
\(512\) − 262144.i − 0.0441942i
\(513\) 0 0
\(514\) 5.81275e6 0.970452
\(515\) 0 0
\(516\) 0 0
\(517\) − 243756.i − 0.0401078i
\(518\) 1.62470e6i 0.266040i
\(519\) 0 0
\(520\) 0 0
\(521\) −4.55410e6 −0.735036 −0.367518 0.930016i \(-0.619792\pi\)
−0.367518 + 0.930016i \(0.619792\pi\)
\(522\) 0 0
\(523\) − 4.82224e6i − 0.770894i −0.922730 0.385447i \(-0.874047\pi\)
0.922730 0.385447i \(-0.125953\pi\)
\(524\) 1.89523e6 0.301533
\(525\) 0 0
\(526\) 677184. 0.106719
\(527\) 1.39207e6i 0.218340i
\(528\) 0 0
\(529\) 6.34271e6 0.985452
\(530\) 0 0
\(531\) 0 0
\(532\) 1.26712e6i 0.194106i
\(533\) − 1.83497e6i − 0.279776i
\(534\) 0 0
\(535\) 0 0
\(536\) −6848.00 −0.00102956
\(537\) 0 0
\(538\) 6.32436e6i 0.942022i
\(539\) −3.24076e6 −0.480479
\(540\) 0 0
\(541\) 362537. 0.0532549 0.0266274 0.999645i \(-0.491523\pi\)
0.0266274 + 0.999645i \(0.491523\pi\)
\(542\) − 3.29005e6i − 0.481065i
\(543\) 0 0
\(544\) −165888. −0.0240335
\(545\) 0 0
\(546\) 0 0
\(547\) 3.11439e6i 0.445046i 0.974927 + 0.222523i \(0.0714293\pi\)
−0.974927 + 0.222523i \(0.928571\pi\)
\(548\) − 211488.i − 0.0300839i
\(549\) 0 0
\(550\) 0 0
\(551\) −1.32947e7 −1.86551
\(552\) 0 0
\(553\) 3.25052e6i 0.452002i
\(554\) −2.18729e6 −0.302784
\(555\) 0 0
\(556\) −5.61184e6 −0.769872
\(557\) 7.99304e6i 1.09163i 0.837907 + 0.545813i \(0.183779\pi\)
−0.837907 + 0.545813i \(0.816221\pi\)
\(558\) 0 0
\(559\) −1.44945e6 −0.196189
\(560\) 0 0
\(561\) 0 0
\(562\) − 4.36999e6i − 0.583633i
\(563\) − 1.23236e7i − 1.63857i −0.573385 0.819286i \(-0.694370\pi\)
0.573385 0.819286i \(-0.305630\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 9.93920e6 1.30410
\(567\) 0 0
\(568\) 2.60659e6i 0.339002i
\(569\) 1.01364e7 1.31252 0.656258 0.754537i \(-0.272138\pi\)
0.656258 + 0.754537i \(0.272138\pi\)
\(570\) 0 0
\(571\) 6.53084e6 0.838260 0.419130 0.907926i \(-0.362335\pi\)
0.419130 + 0.907926i \(0.362335\pi\)
\(572\) − 358752.i − 0.0458463i
\(573\) 0 0
\(574\) 3.41558e6 0.432698
\(575\) 0 0
\(576\) 0 0
\(577\) − 1.24453e6i − 0.155621i −0.996968 0.0778103i \(-0.975207\pi\)
0.996968 0.0778103i \(-0.0247928\pi\)
\(578\) − 5.57445e6i − 0.694037i
\(579\) 0 0
\(580\) 0 0
\(581\) 5.10110e6 0.626936
\(582\) 0 0
\(583\) − 3.97735e6i − 0.484644i
\(584\) 2.22118e6 0.269496
\(585\) 0 0
\(586\) −1.36558e6 −0.164275
\(587\) 1.33403e6i 0.159797i 0.996803 + 0.0798987i \(0.0254597\pi\)
−0.996803 + 0.0798987i \(0.974540\pi\)
\(588\) 0 0
\(589\) 1.44792e7 1.71972
\(590\) 0 0
\(591\) 0 0
\(592\) 2.21235e6i 0.259448i
\(593\) − 1.19401e7i − 1.39435i −0.716899 0.697177i \(-0.754439\pi\)
0.716899 0.697177i \(-0.245561\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.75824e6 −0.202751
\(597\) 0 0
\(598\) − 123624.i − 0.0141368i
\(599\) −7.16430e6 −0.815843 −0.407922 0.913017i \(-0.633746\pi\)
−0.407922 + 0.913017i \(0.633746\pi\)
\(600\) 0 0
\(601\) 1.15163e6 0.130055 0.0650273 0.997883i \(-0.479287\pi\)
0.0650273 + 0.997883i \(0.479287\pi\)
\(602\) − 2.69799e6i − 0.303423i
\(603\) 0 0
\(604\) 2.76165e6 0.308018
\(605\) 0 0
\(606\) 0 0
\(607\) − 1.34268e7i − 1.47911i −0.673097 0.739554i \(-0.735037\pi\)
0.673097 0.739554i \(-0.264963\pi\)
\(608\) 1.72544e6i 0.189296i
\(609\) 0 0
\(610\) 0 0
\(611\) 110898. 0.0120177
\(612\) 0 0
\(613\) − 1.20184e7i − 1.29180i −0.763422 0.645900i \(-0.776482\pi\)
0.763422 0.645900i \(-0.223518\pi\)
\(614\) 8.11591e6 0.868793
\(615\) 0 0
\(616\) 667776. 0.0709054
\(617\) 6.98519e6i 0.738695i 0.929291 + 0.369348i \(0.120419\pi\)
−0.929291 + 0.369348i \(0.879581\pi\)
\(618\) 0 0
\(619\) 8.20625e6 0.860832 0.430416 0.902631i \(-0.358367\pi\)
0.430416 + 0.902631i \(0.358367\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 826392.i − 0.0856466i
\(623\) 1.64688e6i 0.169997i
\(624\) 0 0
\(625\) 0 0
\(626\) 1.33689e7 1.36352
\(627\) 0 0
\(628\) 5.59989e6i 0.566605i
\(629\) 1.40000e6 0.141092
\(630\) 0 0
\(631\) −1.07686e7 −1.07668 −0.538338 0.842729i \(-0.680947\pi\)
−0.538338 + 0.842729i \(0.680947\pi\)
\(632\) 4.42624e6i 0.440801i
\(633\) 0 0
\(634\) 1.01316e7 1.00104
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.47440e6i − 0.143968i
\(638\) 7.00632e6i 0.681457i
\(639\) 0 0
\(640\) 0 0
\(641\) −1.92571e7 −1.85117 −0.925585 0.378539i \(-0.876427\pi\)
−0.925585 + 0.378539i \(0.876427\pi\)
\(642\) 0 0
\(643\) − 1.00999e7i − 0.963364i −0.876346 0.481682i \(-0.840026\pi\)
0.876346 0.481682i \(-0.159974\pi\)
\(644\) 230112. 0.0218637
\(645\) 0 0
\(646\) 1.09188e6 0.102942
\(647\) − 7.52113e6i − 0.706354i −0.935556 0.353177i \(-0.885101\pi\)
0.935556 0.353177i \(-0.114899\pi\)
\(648\) 0 0
\(649\) −3.90942e6 −0.364335
\(650\) 0 0
\(651\) 0 0
\(652\) 3.08130e6i 0.283867i
\(653\) − 2.67197e6i − 0.245216i −0.992455 0.122608i \(-0.960874\pi\)
0.992455 0.122608i \(-0.0391258\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.65101e6 0.421976
\(657\) 0 0
\(658\) 206424.i 0.0185864i
\(659\) 6.99948e6 0.627845 0.313922 0.949449i \(-0.398357\pi\)
0.313922 + 0.949449i \(0.398357\pi\)
\(660\) 0 0
\(661\) 408122. 0.0363318 0.0181659 0.999835i \(-0.494217\pi\)
0.0181659 + 0.999835i \(0.494217\pi\)
\(662\) − 2.40853e6i − 0.213603i
\(663\) 0 0
\(664\) 6.94618e6 0.611400
\(665\) 0 0
\(666\) 0 0
\(667\) 2.41434e6i 0.210128i
\(668\) 9.29107e6i 0.805610i
\(669\) 0 0
\(670\) 0 0
\(671\) 4.85137e6 0.415966
\(672\) 0 0
\(673\) − 1.74939e7i − 1.48885i −0.667709 0.744423i \(-0.732725\pi\)
0.667709 0.744423i \(-0.267275\pi\)
\(674\) 839108. 0.0711489
\(675\) 0 0
\(676\) −5.77747e6 −0.486263
\(677\) 8.67440e6i 0.727391i 0.931518 + 0.363695i \(0.118485\pi\)
−0.931518 + 0.363695i \(0.881515\pi\)
\(678\) 0 0
\(679\) 38681.0 0.00321976
\(680\) 0 0
\(681\) 0 0
\(682\) − 7.63058e6i − 0.628198i
\(683\) 1.18478e7i 0.971822i 0.874008 + 0.485911i \(0.161512\pi\)
−0.874008 + 0.485911i \(0.838488\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 5.90414e6 0.479012
\(687\) 0 0
\(688\) − 3.67386e6i − 0.295904i
\(689\) 1.80952e6 0.145216
\(690\) 0 0
\(691\) 9.47775e6 0.755110 0.377555 0.925987i \(-0.376765\pi\)
0.377555 + 0.925987i \(0.376765\pi\)
\(692\) − 1.18100e7i − 0.937530i
\(693\) 0 0
\(694\) −1.60866e7 −1.26785
\(695\) 0 0
\(696\) 0 0
\(697\) − 2.94322e6i − 0.229478i
\(698\) 33320.0i 0.00258861i
\(699\) 0 0
\(700\) 0 0
\(701\) 2.28147e7 1.75355 0.876777 0.480898i \(-0.159689\pi\)
0.876777 + 0.480898i \(0.159689\pi\)
\(702\) 0 0
\(703\) − 1.45618e7i − 1.11129i
\(704\) 909312. 0.0691483
\(705\) 0 0
\(706\) 7.80002e6 0.588958
\(707\) 1.58992e6i 0.119626i
\(708\) 0 0
\(709\) −1.27436e7 −0.952090 −0.476045 0.879421i \(-0.657930\pi\)
−0.476045 + 0.879421i \(0.657930\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.24256e6i 0.165785i
\(713\) − 2.62946e6i − 0.193706i
\(714\) 0 0
\(715\) 0 0
\(716\) −7.95792e6 −0.580119
\(717\) 0 0
\(718\) − 9.08352e6i − 0.657572i
\(719\) 2.44929e6 0.176692 0.0883462 0.996090i \(-0.471842\pi\)
0.0883462 + 0.996090i \(0.471842\pi\)
\(720\) 0 0
\(721\) −6.27187e6 −0.449323
\(722\) − 1.45250e6i − 0.103699i
\(723\) 0 0
\(724\) 5.33061e6 0.377947
\(725\) 0 0
\(726\) 0 0
\(727\) − 415033.i − 0.0291237i −0.999894 0.0145619i \(-0.995365\pi\)
0.999894 0.0145619i \(-0.00463535\pi\)
\(728\) 303808.i 0.0212457i
\(729\) 0 0
\(730\) 0 0
\(731\) −2.32486e6 −0.160918
\(732\) 0 0
\(733\) 1.72877e7i 1.18844i 0.804302 + 0.594221i \(0.202539\pi\)
−0.804302 + 0.594221i \(0.797461\pi\)
\(734\) 1.14461e7 0.784186
\(735\) 0 0
\(736\) 313344. 0.0213219
\(737\) − 23754.0i − 0.00161090i
\(738\) 0 0
\(739\) −5.18834e6 −0.349476 −0.174738 0.984615i \(-0.555908\pi\)
−0.174738 + 0.984615i \(0.555908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 3.36821e6i 0.224589i
\(743\) 4.79572e6i 0.318700i 0.987222 + 0.159350i \(0.0509398\pi\)
−0.987222 + 0.159350i \(0.949060\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2.46124e6 −0.161923
\(747\) 0 0
\(748\) − 575424.i − 0.0376040i
\(749\) 3.81884e6 0.248730
\(750\) 0 0
\(751\) −1.85654e7 −1.20117 −0.600585 0.799561i \(-0.705066\pi\)
−0.600585 + 0.799561i \(0.705066\pi\)
\(752\) 281088.i 0.0181258i
\(753\) 0 0
\(754\) −3.18756e6 −0.204188
\(755\) 0 0
\(756\) 0 0
\(757\) 2.82068e7i 1.78902i 0.447053 + 0.894508i \(0.352474\pi\)
−0.447053 + 0.894508i \(0.647526\pi\)
\(758\) 2.15951e7i 1.36516i
\(759\) 0 0
\(760\) 0 0
\(761\) −6.56161e6 −0.410723 −0.205361 0.978686i \(-0.565837\pi\)
−0.205361 + 0.978686i \(0.565837\pi\)
\(762\) 0 0
\(763\) 1.01997e7i 0.634273i
\(764\) −650208. −0.0403013
\(765\) 0 0
\(766\) 4.34750e6 0.267712
\(767\) − 1.77861e6i − 0.109167i
\(768\) 0 0
\(769\) −2.20930e7 −1.34722 −0.673610 0.739087i \(-0.735257\pi\)
−0.673610 + 0.739087i \(0.735257\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.91442e6i 0.477942i
\(773\) − 3.00787e7i − 1.81055i −0.424824 0.905276i \(-0.639664\pi\)
0.424824 0.905276i \(-0.360336\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 52672.0 0.00313997
\(777\) 0 0
\(778\) 1.39373e7i 0.825523i
\(779\) −3.06131e7 −1.80744
\(780\) 0 0
\(781\) −9.04162e6 −0.530418
\(782\) − 198288.i − 0.0115952i
\(783\) 0 0
\(784\) 3.73709e6 0.217142
\(785\) 0 0
\(786\) 0 0
\(787\) − 3.28954e6i − 0.189321i −0.995510 0.0946605i \(-0.969823\pi\)
0.995510 0.0946605i \(-0.0301766\pi\)
\(788\) 8.83747e6i 0.507005i
\(789\) 0 0
\(790\) 0 0
\(791\) 6.50123e6 0.369449
\(792\) 0 0
\(793\) 2.20715e6i 0.124638i
\(794\) 1.30636e7 0.735381
\(795\) 0 0
\(796\) 1.09700e7 0.613655
\(797\) − 6.71053e6i − 0.374206i −0.982340 0.187103i \(-0.940090\pi\)
0.982340 0.187103i \(-0.0599099\pi\)
\(798\) 0 0
\(799\) 177876. 0.00985713
\(800\) 0 0
\(801\) 0 0
\(802\) − 1.70928e7i − 0.938374i
\(803\) 7.70473e6i 0.421666i
\(804\) 0 0
\(805\) 0 0
\(806\) 3.47157e6 0.188230
\(807\) 0 0
\(808\) 2.16499e6i 0.116662i
\(809\) 8.74254e6 0.469641 0.234821 0.972039i \(-0.424550\pi\)
0.234821 + 0.972039i \(0.424550\pi\)
\(810\) 0 0
\(811\) −2.48410e7 −1.32622 −0.663112 0.748520i \(-0.730765\pi\)
−0.663112 + 0.748520i \(0.730765\pi\)
\(812\) − 5.93328e6i − 0.315795i
\(813\) 0 0
\(814\) −7.67410e6 −0.405944
\(815\) 0 0
\(816\) 0 0
\(817\) 2.41814e7i 1.26744i
\(818\) − 5.80750e6i − 0.303463i
\(819\) 0 0
\(820\) 0 0
\(821\) −2.12219e7 −1.09882 −0.549409 0.835554i \(-0.685147\pi\)
−0.549409 + 0.835554i \(0.685147\pi\)
\(822\) 0 0
\(823\) 8.70659e6i 0.448073i 0.974581 + 0.224036i \(0.0719234\pi\)
−0.974581 + 0.224036i \(0.928077\pi\)
\(824\) −8.54042e6 −0.438189
\(825\) 0 0
\(826\) 3.31068e6 0.168837
\(827\) − 3.71184e7i − 1.88723i −0.331040 0.943617i \(-0.607400\pi\)
0.331040 0.943617i \(-0.392600\pi\)
\(828\) 0 0
\(829\) −1.01765e6 −0.0514295 −0.0257147 0.999669i \(-0.508186\pi\)
−0.0257147 + 0.999669i \(0.508186\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 413696.i 0.0207192i
\(833\) − 2.36488e6i − 0.118085i
\(834\) 0 0
\(835\) 0 0
\(836\) −5.98512e6 −0.296181
\(837\) 0 0
\(838\) − 2.23752e6i − 0.110067i
\(839\) −3.36194e7 −1.64887 −0.824433 0.565960i \(-0.808506\pi\)
−0.824433 + 0.565960i \(0.808506\pi\)
\(840\) 0 0
\(841\) 4.17410e7 2.03504
\(842\) 1.56588e7i 0.761164i
\(843\) 0 0
\(844\) −1.19916e7 −0.579458
\(845\) 0 0
\(846\) 0 0
\(847\) − 5.25305e6i − 0.251596i
\(848\) 4.58650e6i 0.219024i
\(849\) 0 0
\(850\) 0 0
\(851\) −2.64445e6 −0.125173
\(852\) 0 0
\(853\) 3.52574e7i 1.65912i 0.558419 + 0.829559i \(0.311408\pi\)
−0.558419 + 0.829559i \(0.688592\pi\)
\(854\) −4.10836e6 −0.192763
\(855\) 0 0
\(856\) 5.20013e6 0.242566
\(857\) 3.14941e7i 1.46480i 0.680877 + 0.732398i \(0.261599\pi\)
−0.680877 + 0.732398i \(0.738401\pi\)
\(858\) 0 0
\(859\) −1.19344e7 −0.551848 −0.275924 0.961180i \(-0.588984\pi\)
−0.275924 + 0.961180i \(0.588984\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 1.43000e7i − 0.655492i
\(863\) 8.70442e6i 0.397844i 0.980015 + 0.198922i \(0.0637440\pi\)
−0.980015 + 0.198922i \(0.936256\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2.86388e7 −1.29766
\(867\) 0 0
\(868\) 6.46194e6i 0.291114i
\(869\) −1.53535e7 −0.689697
\(870\) 0 0
\(871\) 10807.0 0.000482681 0
\(872\) 1.38890e7i 0.618555i
\(873\) 0 0
\(874\) −2.06244e6 −0.0913277
\(875\) 0 0
\(876\) 0 0
\(877\) 1.17999e7i 0.518059i 0.965869 + 0.259029i \(0.0834026\pi\)
−0.965869 + 0.259029i \(0.916597\pi\)
\(878\) 6.87158e6i 0.300829i
\(879\) 0 0
\(880\) 0 0
\(881\) 2.73840e7 1.18866 0.594330 0.804221i \(-0.297417\pi\)
0.594330 + 0.804221i \(0.297417\pi\)
\(882\) 0 0
\(883\) 8.80577e6i 0.380072i 0.981777 + 0.190036i \(0.0608604\pi\)
−0.981777 + 0.190036i \(0.939140\pi\)
\(884\) 261792. 0.0112675
\(885\) 0 0
\(886\) 1.35868e7 0.581477
\(887\) − 250122.i − 0.0106744i −0.999986 0.00533719i \(-0.998301\pi\)
0.999986 0.00533719i \(-0.00169889\pi\)
\(888\) 0 0
\(889\) 1.20343e7 0.510699
\(890\) 0 0
\(891\) 0 0
\(892\) 2.70834e6i 0.113970i
\(893\) − 1.85013e6i − 0.0776379i
\(894\) 0 0
\(895\) 0 0
\(896\) −770048. −0.0320441
\(897\) 0 0
\(898\) 1.35842e7i 0.562140i
\(899\) −6.77988e7 −2.79784
\(900\) 0 0
\(901\) 2.90239e6 0.119109
\(902\) 1.61332e7i 0.660243i
\(903\) 0 0
\(904\) 8.85274e6 0.360294
\(905\) 0 0
\(906\) 0 0
\(907\) − 3.24955e7i − 1.31161i −0.754929 0.655806i \(-0.772329\pi\)
0.754929 0.655806i \(-0.227671\pi\)
\(908\) − 743808.i − 0.0299396i
\(909\) 0 0
\(910\) 0 0
\(911\) −4.24595e7 −1.69504 −0.847518 0.530766i \(-0.821904\pi\)
−0.847518 + 0.530766i \(0.821904\pi\)
\(912\) 0 0
\(913\) 2.40945e7i 0.956625i
\(914\) −1.81126e7 −0.717157
\(915\) 0 0
\(916\) −1.44184e6 −0.0567778
\(917\) − 5.56724e6i − 0.218634i
\(918\) 0 0
\(919\) −1.41629e7 −0.553176 −0.276588 0.960989i \(-0.589204\pi\)
−0.276588 + 0.960989i \(0.589204\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 5.11579e6i − 0.198192i
\(923\) − 4.11353e6i − 0.158932i
\(924\) 0 0
\(925\) 0 0
\(926\) 2.87945e7 1.10352
\(927\) 0 0
\(928\) − 8.07936e6i − 0.307969i
\(929\) 4.37292e7 1.66239 0.831194 0.555982i \(-0.187658\pi\)
0.831194 + 0.555982i \(0.187658\pi\)
\(930\) 0 0
\(931\) −2.45976e7 −0.930077
\(932\) − 1.70262e7i − 0.642063i
\(933\) 0 0
\(934\) −1.93214e7 −0.724721
\(935\) 0 0
\(936\) 0 0
\(937\) 5.73509e6i 0.213398i 0.994291 + 0.106699i \(0.0340282\pi\)
−0.994291 + 0.106699i \(0.965972\pi\)
\(938\) 20116.0i 0 0.000746508i
\(939\) 0 0
\(940\) 0 0
\(941\) 3.37395e7 1.24212 0.621061 0.783762i \(-0.286702\pi\)
0.621061 + 0.783762i \(0.286702\pi\)
\(942\) 0 0
\(943\) 5.55941e6i 0.203587i
\(944\) 4.50816e6 0.164653
\(945\) 0 0
\(946\) 1.27437e7 0.462985
\(947\) 3.07342e7i 1.11365i 0.830631 + 0.556823i \(0.187980\pi\)
−0.830631 + 0.556823i \(0.812020\pi\)
\(948\) 0 0
\(949\) −3.50531e6 −0.126346
\(950\) 0 0
\(951\) 0 0
\(952\) 487296.i 0.0174261i
\(953\) − 2.51847e7i − 0.898264i −0.893465 0.449132i \(-0.851733\pi\)
0.893465 0.449132i \(-0.148267\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.84253e7 −0.652033
\(957\) 0 0
\(958\) − 2.99460e6i − 0.105420i
\(959\) −621246. −0.0218131
\(960\) 0 0
\(961\) 4.52105e7 1.57918
\(962\) − 3.49137e6i − 0.121635i
\(963\) 0 0
\(964\) −1.36995e7 −0.474801
\(965\) 0 0
\(966\) 0 0
\(967\) 1.44556e7i 0.497130i 0.968615 + 0.248565i \(0.0799589\pi\)
−0.968615 + 0.248565i \(0.920041\pi\)
\(968\) − 7.15309e6i − 0.245361i
\(969\) 0 0
\(970\) 0 0
\(971\) 1.06974e7 0.364109 0.182054 0.983288i \(-0.441725\pi\)
0.182054 + 0.983288i \(0.441725\pi\)
\(972\) 0 0
\(973\) 1.64848e7i 0.558214i
\(974\) −2.06558e7 −0.697660
\(975\) 0 0
\(976\) −5.59437e6 −0.187986
\(977\) 8.41568e6i 0.282067i 0.990005 + 0.141034i \(0.0450426\pi\)
−0.990005 + 0.141034i \(0.954957\pi\)
\(978\) 0 0
\(979\) −7.77888e6 −0.259394
\(980\) 0 0
\(981\) 0 0
\(982\) 3.41715e7i 1.13080i
\(983\) 3.89409e7i 1.28535i 0.766138 + 0.642676i \(0.222176\pi\)
−0.766138 + 0.642676i \(0.777824\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −5.11272e6 −0.167479
\(987\) 0 0
\(988\) − 2.72296e6i − 0.0887460i
\(989\) 4.39141e6 0.142762
\(990\) 0 0
\(991\) 4.84592e7 1.56745 0.783723 0.621111i \(-0.213318\pi\)
0.783723 + 0.621111i \(0.213318\pi\)
\(992\) 8.79923e6i 0.283900i
\(993\) 0 0
\(994\) 7.65686e6 0.245802
\(995\) 0 0
\(996\) 0 0
\(997\) 3.84733e7i 1.22581i 0.790158 + 0.612903i \(0.209998\pi\)
−0.790158 + 0.612903i \(0.790002\pi\)
\(998\) − 1.68235e7i − 0.534675i
\(999\) 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.e.199.1 2
3.2 odd 2 150.6.c.g.49.2 2
5.2 odd 4 450.6.a.p.1.1 1
5.3 odd 4 450.6.a.i.1.1 1
5.4 even 2 inner 450.6.c.e.199.2 2
15.2 even 4 150.6.a.a.1.1 1
15.8 even 4 150.6.a.m.1.1 yes 1
15.14 odd 2 150.6.c.g.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.6.a.a.1.1 1 15.2 even 4
150.6.a.m.1.1 yes 1 15.8 even 4
150.6.c.g.49.1 2 15.14 odd 2
150.6.c.g.49.2 2 3.2 odd 2
450.6.a.i.1.1 1 5.3 odd 4
450.6.a.p.1.1 1 5.2 odd 4
450.6.c.e.199.1 2 1.1 even 1 trivial
450.6.c.e.199.2 2 5.4 even 2 inner