Properties

Label 450.6.a.p.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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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

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 150)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

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

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
\(7\) −47.0000 −0.362537 −0.181269 0.983434i \(-0.558020\pi\)
−0.181269 + 0.983434i \(0.558020\pi\)
\(8\) 64.0000 0.353553
\(9\) 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
\(13\) −101.000 −0.165754 −0.0828768 0.996560i \(-0.526411\pi\)
−0.0828768 + 0.996560i \(0.526411\pi\)
\(14\) −188.000 −0.256353
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 162.000 0.135954 0.0679771 0.997687i \(-0.478346\pi\)
0.0679771 + 0.997687i \(0.478346\pi\)
\(18\) 0 0
\(19\) 1685.00 1.07082 0.535409 0.844593i \(-0.320157\pi\)
0.535409 + 0.844593i \(0.320157\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −888.000 −0.391162
\(23\) 306.000 0.120615 0.0603076 0.998180i \(-0.480792\pi\)
0.0603076 + 0.998180i \(0.480792\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −404.000 −0.117206
\(27\) 0 0
\(28\) −752.000 −0.181269
\(29\) −7890.00 −1.74214 −0.871068 0.491163i \(-0.836572\pi\)
−0.871068 + 0.491163i \(0.836572\pi\)
\(30\) 0 0
\(31\) −8593.00 −1.60598 −0.802991 0.595991i \(-0.796759\pi\)
−0.802991 + 0.595991i \(0.796759\pi\)
\(32\) 1024.00 0.176777
\(33\) 0 0
\(34\) 648.000 0.0961342
\(35\) 0 0
\(36\) 0 0
\(37\) −8642.00 −1.03779 −0.518896 0.854838i \(-0.673657\pi\)
−0.518896 + 0.854838i \(0.673657\pi\)
\(38\) 6740.00 0.757183
\(39\) 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
\(43\) −14351.0 −1.18362 −0.591808 0.806079i \(-0.701586\pi\)
−0.591808 + 0.806079i \(0.701586\pi\)
\(44\) −3552.00 −0.276593
\(45\) 0 0
\(46\) 1224.00 0.0852878
\(47\) −1098.00 −0.0725033 −0.0362516 0.999343i \(-0.511542\pi\)
−0.0362516 + 0.999343i \(0.511542\pi\)
\(48\) 0 0
\(49\) −14598.0 −0.868567
\(50\) 0 0
\(51\) 0 0
\(52\) −1616.00 −0.0828768
\(53\) 17916.0 0.876095 0.438048 0.898952i \(-0.355670\pi\)
0.438048 + 0.898952i \(0.355670\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3008.00 −0.128176
\(57\) 0 0
\(58\) −31560.0 −1.23188
\(59\) −17610.0 −0.658612 −0.329306 0.944223i \(-0.606815\pi\)
−0.329306 + 0.944223i \(0.606815\pi\)
\(60\) 0 0
\(61\) −21853.0 −0.751946 −0.375973 0.926631i \(-0.622691\pi\)
−0.375973 + 0.926631i \(0.622691\pi\)
\(62\) −34372.0 −1.13560
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −107.000 −0.00291204 −0.00145602 0.999999i \(-0.500463\pi\)
−0.00145602 + 0.999999i \(0.500463\pi\)
\(68\) 2592.00 0.0679771
\(69\) 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
\(73\) −34706.0 −0.762250 −0.381125 0.924524i \(-0.624463\pi\)
−0.381125 + 0.924524i \(0.624463\pi\)
\(74\) −34568.0 −0.733829
\(75\) 0 0
\(76\) 26960.0 0.535409
\(77\) 10434.0 0.200551
\(78\) 0 0
\(79\) −69160.0 −1.24677 −0.623386 0.781914i \(-0.714244\pi\)
−0.623386 + 0.781914i \(0.714244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 72672.0 1.19353
\(83\) −108534. −1.72930 −0.864650 0.502374i \(-0.832460\pi\)
−0.864650 + 0.502374i \(0.832460\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −57404.0 −0.836943
\(87\) 0 0
\(88\) −14208.0 −0.195581
\(89\) −35040.0 −0.468910 −0.234455 0.972127i \(-0.575330\pi\)
−0.234455 + 0.972127i \(0.575330\pi\)
\(90\) 0 0
\(91\) 4747.00 0.0600919
\(92\) 4896.00 0.0603076
\(93\) 0 0
\(94\) −4392.00 −0.0512676
\(95\) 0 0
\(96\) 0 0
\(97\) 823.000 0.00888118 0.00444059 0.999990i \(-0.498587\pi\)
0.00444059 + 0.999990i \(0.498587\pi\)
\(98\) −58392.0 −0.614169
\(99\) 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. 1.23938 0.619692 0.784845i \(-0.287257\pi\)
0.619692 + 0.784845i \(0.287257\pi\)
\(104\) −6464.00 −0.0586028
\(105\) 0 0
\(106\) 71664.0 0.619493
\(107\) 81252.0 0.686080 0.343040 0.939321i \(-0.388543\pi\)
0.343040 + 0.939321i \(0.388543\pi\)
\(108\) 0 0
\(109\) −217015. −1.74954 −0.874769 0.484540i \(-0.838987\pi\)
−0.874769 + 0.484540i \(0.838987\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −12032.0 −0.0906343
\(113\) −138324. −1.01906 −0.509532 0.860452i \(-0.670181\pi\)
−0.509532 + 0.860452i \(0.670181\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −126240. −0.871068
\(117\) 0 0
\(118\) −70440.0 −0.465709
\(119\) −7614.00 −0.0492885
\(120\) 0 0
\(121\) −111767. −0.693985
\(122\) −87412.0 −0.531706
\(123\) 0 0
\(124\) −137488. −0.802991
\(125\) 0 0
\(126\) 0 0
\(127\) 256048. 1.40868 0.704340 0.709863i \(-0.251243\pi\)
0.704340 + 0.709863i \(0.251243\pi\)
\(128\) 16384.0 0.0883883
\(129\) 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
\(133\) −79195.0 −0.388212
\(134\) −428.000 −0.00205912
\(135\) 0 0
\(136\) 10368.0 0.0480671
\(137\) −13218.0 −0.0601678 −0.0300839 0.999547i \(-0.509577\pi\)
−0.0300839 + 0.999547i \(0.509577\pi\)
\(138\) 0 0
\(139\) −350740. −1.53974 −0.769872 0.638199i \(-0.779680\pi\)
−0.769872 + 0.638199i \(0.779680\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 162912. 0.678004
\(143\) 22422.0 0.0916926
\(144\) 0 0
\(145\) 0 0
\(146\) −138824. −0.538992
\(147\) 0 0
\(148\) −138272. −0.518896
\(149\) −109890. −0.405502 −0.202751 0.979230i \(-0.564988\pi\)
−0.202751 + 0.979230i \(0.564988\pi\)
\(150\) 0 0
\(151\) −172603. −0.616036 −0.308018 0.951381i \(-0.599666\pi\)
−0.308018 + 0.951381i \(0.599666\pi\)
\(152\) 107840. 0.378592
\(153\) 0 0
\(154\) 41736.0 0.141811
\(155\) 0 0
\(156\) 0 0
\(157\) 349993. 1.13321 0.566605 0.823990i \(-0.308257\pi\)
0.566605 + 0.823990i \(0.308257\pi\)
\(158\) −276640. −0.881601
\(159\) 0 0
\(160\) 0 0
\(161\) −14382.0 −0.0437275
\(162\) 0 0
\(163\) −192581. −0.567733 −0.283867 0.958864i \(-0.591617\pi\)
−0.283867 + 0.958864i \(0.591617\pi\)
\(164\) 290688. 0.843951
\(165\) 0 0
\(166\) −434136. −1.22280
\(167\) 580692. 1.61122 0.805610 0.592447i \(-0.201838\pi\)
0.805610 + 0.592447i \(0.201838\pi\)
\(168\) 0 0
\(169\) −361092. −0.972526
\(170\) 0 0
\(171\) 0 0
\(172\) −229616. −0.591808
\(173\) 738126. 1.87506 0.937530 0.347904i \(-0.113106\pi\)
0.937530 + 0.347904i \(0.113106\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −56832.0 −0.138297
\(177\) 0 0
\(178\) −140160. −0.331569
\(179\) −497370. −1.16024 −0.580119 0.814532i \(-0.696994\pi\)
−0.580119 + 0.814532i \(0.696994\pi\)
\(180\) 0 0
\(181\) −333163. −0.755893 −0.377947 0.925827i \(-0.623370\pi\)
−0.377947 + 0.925827i \(0.623370\pi\)
\(182\) 18988.0 0.0424914
\(183\) 0 0
\(184\) 19584.0 0.0426439
\(185\) 0 0
\(186\) 0 0
\(187\) −35964.0 −0.0752080
\(188\) −17568.0 −0.0362516
\(189\) 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. −0.955885 −0.477942 0.878391i \(-0.658617\pi\)
−0.477942 + 0.878391i \(0.658617\pi\)
\(194\) 3292.00 0.00627994
\(195\) 0 0
\(196\) −233568. −0.434283
\(197\) 552342. 1.01401 0.507005 0.861943i \(-0.330752\pi\)
0.507005 + 0.861943i \(0.330752\pi\)
\(198\) 0 0
\(199\) 685625. 1.22731 0.613655 0.789575i \(-0.289699\pi\)
0.613655 + 0.789575i \(0.289699\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 135312. 0.233323
\(203\) 370830. 0.631589
\(204\) 0 0
\(205\) 0 0
\(206\) 533776. 0.876377
\(207\) 0 0
\(208\) −25856.0 −0.0414384
\(209\) −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. 0.438048
\(213\) 0 0
\(214\) 325008. 0.485132
\(215\) 0 0
\(216\) 0 0
\(217\) 403871. 0.582228
\(218\) −868060. −1.23711
\(219\) 0 0
\(220\) 0 0
\(221\) −16362.0 −0.0225349
\(222\) 0 0
\(223\) −169271. −0.227940 −0.113970 0.993484i \(-0.536357\pi\)
−0.113970 + 0.993484i \(0.536357\pi\)
\(224\) −48128.0 −0.0640882
\(225\) 0 0
\(226\) −553296. −0.720587
\(227\) −46488.0 −0.0598792 −0.0299396 0.999552i \(-0.509532\pi\)
−0.0299396 + 0.999552i \(0.509532\pi\)
\(228\) 0 0
\(229\) −90115.0 −0.113556 −0.0567778 0.998387i \(-0.518083\pi\)
−0.0567778 + 0.998387i \(0.518083\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −504960. −0.615938
\(233\) 1.06414e6 1.28413 0.642063 0.766652i \(-0.278079\pi\)
0.642063 + 0.766652i \(0.278079\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −281760. −0.329306
\(237\) 0 0
\(238\) −30456.0 −0.0348522
\(239\) −1.15158e6 −1.30407 −0.652033 0.758191i \(-0.726084\pi\)
−0.652033 + 0.758191i \(0.726084\pi\)
\(240\) 0 0
\(241\) 856217. 0.949601 0.474801 0.880093i \(-0.342520\pi\)
0.474801 + 0.880093i \(0.342520\pi\)
\(242\) −447068. −0.490722
\(243\) 0 0
\(244\) −349648. −0.375973
\(245\) 0 0
\(246\) 0 0
\(247\) −170185. −0.177492
\(248\) −549952. −0.567800
\(249\) 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
\(253\) −67932.0 −0.0667226
\(254\) 1.02419e6 0.996087
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) −1.45319e6 −1.37243 −0.686213 0.727401i \(-0.740728\pi\)
−0.686213 + 0.727401i \(0.740728\pi\)
\(258\) 0 0
\(259\) 406174. 0.376238
\(260\) 0 0
\(261\) 0 0
\(262\) −473808. −0.426431
\(263\) 169296. 0.150924 0.0754618 0.997149i \(-0.475957\pi\)
0.0754618 + 0.997149i \(0.475957\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −316780. −0.274507
\(267\) 0 0
\(268\) −1712.00 −0.00145602
\(269\) 1.58109e6 1.33222 0.666110 0.745854i \(-0.267958\pi\)
0.666110 + 0.745854i \(0.267958\pi\)
\(270\) 0 0
\(271\) 822512. 0.680329 0.340165 0.940366i \(-0.389517\pi\)
0.340165 + 0.940366i \(0.389517\pi\)
\(272\) 41472.0 0.0339886
\(273\) 0 0
\(274\) −52872.0 −0.0425451
\(275\) 0 0
\(276\) 0 0
\(277\) 546823. 0.428201 0.214100 0.976812i \(-0.431318\pi\)
0.214100 + 0.976812i \(0.431318\pi\)
\(278\) −1.40296e6 −1.08876
\(279\) 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
\(283\) 2.48480e6 1.84427 0.922136 0.386865i \(-0.126442\pi\)
0.922136 + 0.386865i \(0.126442\pi\)
\(284\) 651648. 0.479421
\(285\) 0 0
\(286\) 89688.0 0.0648365
\(287\) −853896. −0.611928
\(288\) 0 0
\(289\) −1.39361e6 −0.981516
\(290\) 0 0
\(291\) 0 0
\(292\) −555296. −0.381125
\(293\) −341394. −0.232320 −0.116160 0.993231i \(-0.537059\pi\)
−0.116160 + 0.993231i \(0.537059\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −553088. −0.366915
\(297\) 0 0
\(298\) −439560. −0.286733
\(299\) −30906.0 −0.0199924
\(300\) 0 0
\(301\) 674497. 0.429105
\(302\) −690412. −0.435603
\(303\) 0 0
\(304\) 431360. 0.267705
\(305\) 0 0
\(306\) 0 0
\(307\) −2.02898e6 −1.22866 −0.614329 0.789050i \(-0.710573\pi\)
−0.614329 + 0.789050i \(0.710573\pi\)
\(308\) 166944. 0.100275
\(309\) 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
\(313\) 3.34223e6 1.92830 0.964152 0.265352i \(-0.0854881\pi\)
0.964152 + 0.265352i \(0.0854881\pi\)
\(314\) 1.39997e6 0.801300
\(315\) 0 0
\(316\) −1.10656e6 −0.623386
\(317\) −2.53289e6 −1.41569 −0.707844 0.706368i \(-0.750332\pi\)
−0.707844 + 0.706368i \(0.750332\pi\)
\(318\) 0 0
\(319\) 1.75158e6 0.963725
\(320\) 0 0
\(321\) 0 0
\(322\) −57528.0 −0.0309200
\(323\) 272970. 0.145582
\(324\) 0 0
\(325\) 0 0
\(326\) −770324. −0.401448
\(327\) 0 0
\(328\) 1.16275e6 0.596764
\(329\) 51606.0 0.0262851
\(330\) 0 0
\(331\) 602132. 0.302080 0.151040 0.988528i \(-0.451738\pi\)
0.151040 + 0.988528i \(0.451738\pi\)
\(332\) −1.73654e6 −0.864650
\(333\) 0 0
\(334\) 2.32277e6 1.13930
\(335\) 0 0
\(336\) 0 0
\(337\) −209777. −0.100620 −0.0503099 0.998734i \(-0.516021\pi\)
−0.0503099 + 0.998734i \(0.516021\pi\)
\(338\) −1.44437e6 −0.687680
\(339\) 0 0
\(340\) 0 0
\(341\) 1.90765e6 0.888407
\(342\) 0 0
\(343\) 1.47603e6 0.677425
\(344\) −918464. −0.418472
\(345\) 0 0
\(346\) 2.95250e6 1.32587
\(347\) 4.02166e6 1.79301 0.896503 0.443037i \(-0.146099\pi\)
0.896503 + 0.443037i \(0.146099\pi\)
\(348\) 0 0
\(349\) 8330.00 0.00366085 0.00183042 0.999998i \(-0.499417\pi\)
0.00183042 + 0.999998i \(0.499417\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −227328. −0.0977904
\(353\) 1.95001e6 0.832912 0.416456 0.909156i \(-0.363272\pi\)
0.416456 + 0.909156i \(0.363272\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −560640. −0.234455
\(357\) 0 0
\(358\) −1.98948e6 −0.820412
\(359\) −2.27088e6 −0.929947 −0.464973 0.885325i \(-0.653936\pi\)
−0.464973 + 0.885325i \(0.653936\pi\)
\(360\) 0 0
\(361\) 363126. 0.146652
\(362\) −1.33265e6 −0.534497
\(363\) 0 0
\(364\) 75952.0 0.0300459
\(365\) 0 0
\(366\) 0 0
\(367\) −2.86154e6 −1.10901 −0.554503 0.832181i \(-0.687092\pi\)
−0.554503 + 0.832181i \(0.687092\pi\)
\(368\) 78336.0 0.0301538
\(369\) 0 0
\(370\) 0 0
\(371\) −842052. −0.317617
\(372\) 0 0
\(373\) −615311. −0.228993 −0.114497 0.993424i \(-0.536526\pi\)
−0.114497 + 0.993424i \(0.536526\pi\)
\(374\) −143856. −0.0531801
\(375\) 0 0
\(376\) −70272.0 −0.0256338
\(377\) 796890. 0.288765
\(378\) 0 0
\(379\) 5.39878e6 1.93062 0.965311 0.261103i \(-0.0840863\pi\)
0.965311 + 0.261103i \(0.0840863\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 162552. 0.0569946
\(383\) 1.08688e6 0.378602 0.189301 0.981919i \(-0.439378\pi\)
0.189301 + 0.981919i \(0.439378\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.97860e6 −0.675913
\(387\) 0 0
\(388\) 13168.0 0.00444059
\(389\) 3.48432e6 1.16747 0.583733 0.811946i \(-0.301592\pi\)
0.583733 + 0.811946i \(0.301592\pi\)
\(390\) 0 0
\(391\) 49572.0 0.0163981
\(392\) −934272. −0.307085
\(393\) 0 0
\(394\) 2.20937e6 0.717014
\(395\) 0 0
\(396\) 0 0
\(397\) −3.26591e6 −1.03999 −0.519993 0.854170i \(-0.674065\pi\)
−0.519993 + 0.854170i \(0.674065\pi\)
\(398\) 2.74250e6 0.867839
\(399\) 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. 0.266197
\(404\) 541248. 0.164984
\(405\) 0 0
\(406\) 1.48332e6 0.446601
\(407\) 1.91852e6 0.574092
\(408\) 0 0
\(409\) −1.45188e6 −0.429162 −0.214581 0.976706i \(-0.568839\pi\)
−0.214581 + 0.976706i \(0.568839\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.13510e6 0.619692
\(413\) 827670. 0.238771
\(414\) 0 0
\(415\) 0 0
\(416\) −103424. −0.0293014
\(417\) 0 0
\(418\) −1.49628e6 −0.418863
\(419\) −559380. −0.155658 −0.0778291 0.996967i \(-0.524799\pi\)
−0.0778291 + 0.996967i \(0.524799\pi\)
\(420\) 0 0
\(421\) −3.91470e6 −1.07645 −0.538224 0.842802i \(-0.680905\pi\)
−0.538224 + 0.842802i \(0.680905\pi\)
\(422\) 2.99791e6 0.819478
\(423\) 0 0
\(424\) 1.14662e6 0.309746
\(425\) 0 0
\(426\) 0 0
\(427\) 1.02709e6 0.272608
\(428\) 1.30003e6 0.343040
\(429\) 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
\(433\) −7.15969e6 −1.83516 −0.917581 0.397548i \(-0.869861\pi\)
−0.917581 + 0.397548i \(0.869861\pi\)
\(434\) 1.61548e6 0.411698
\(435\) 0 0
\(436\) −3.47224e6 −0.874769
\(437\) 515610. 0.129157
\(438\) 0 0
\(439\) 1.71790e6 0.425437 0.212719 0.977114i \(-0.431768\pi\)
0.212719 + 0.977114i \(0.431768\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −65448.0 −0.0159346
\(443\) 3.39670e6 0.822332 0.411166 0.911560i \(-0.365122\pi\)
0.411166 + 0.911560i \(0.365122\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −677084. −0.161178
\(447\) 0 0
\(448\) −192512. −0.0453172
\(449\) 3.39606e6 0.794986 0.397493 0.917605i \(-0.369880\pi\)
0.397493 + 0.917605i \(0.369880\pi\)
\(450\) 0 0
\(451\) −4.03330e6 −0.933724
\(452\) −2.21318e6 −0.509532
\(453\) 0 0
\(454\) −185952. −0.0423410
\(455\) 0 0
\(456\) 0 0
\(457\) 4.52814e6 1.01421 0.507106 0.861883i \(-0.330715\pi\)
0.507106 + 0.861883i \(0.330715\pi\)
\(458\) −360460. −0.0802959
\(459\) 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
\(463\) 7.19862e6 1.56062 0.780310 0.625393i \(-0.215061\pi\)
0.780310 + 0.625393i \(0.215061\pi\)
\(464\) −2.01984e6 −0.435534
\(465\) 0 0
\(466\) 4.25654e6 0.908014
\(467\) 4.83034e6 1.02491 0.512455 0.858714i \(-0.328736\pi\)
0.512455 + 0.858714i \(0.328736\pi\)
\(468\) 0 0
\(469\) 5029.00 0.00105572
\(470\) 0 0
\(471\) 0 0
\(472\) −1.12704e6 −0.232854
\(473\) 3.18592e6 0.654760
\(474\) 0 0
\(475\) 0 0
\(476\) −121824. −0.0246442
\(477\) 0 0
\(478\) −4.60632e6 −0.922113
\(479\) −748650. −0.149087 −0.0745435 0.997218i \(-0.523750\pi\)
−0.0745435 + 0.997218i \(0.523750\pi\)
\(480\) 0 0
\(481\) 872842. 0.172018
\(482\) 3.42487e6 0.671469
\(483\) 0 0
\(484\) −1.78827e6 −0.346993
\(485\) 0 0
\(486\) 0 0
\(487\) 5.16394e6 0.986641 0.493320 0.869848i \(-0.335783\pi\)
0.493320 + 0.869848i \(0.335783\pi\)
\(488\) −1.39859e6 −0.265853
\(489\) 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
\(493\) −1.27818e6 −0.236851
\(494\) −680740. −0.125506
\(495\) 0 0
\(496\) −2.19981e6 −0.401495
\(497\) −1.91422e6 −0.347616
\(498\) 0 0
\(499\) −4.20588e6 −0.756145 −0.378072 0.925776i \(-0.623413\pi\)
−0.378072 + 0.925776i \(0.623413\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 830832. 0.147148
\(503\) −8.18342e6 −1.44217 −0.721083 0.692849i \(-0.756355\pi\)
−0.721083 + 0.692849i \(0.756355\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −271728. −0.0471800
\(507\) 0 0
\(508\) 4.09677e6 0.704340
\(509\) −3.85923e6 −0.660247 −0.330123 0.943938i \(-0.607090\pi\)
−0.330123 + 0.943938i \(0.607090\pi\)
\(510\) 0 0
\(511\) 1.63118e6 0.276344
\(512\) 262144. 0.0441942
\(513\) 0 0
\(514\) −5.81275e6 −0.970452
\(515\) 0 0
\(516\) 0 0
\(517\) 243756. 0.0401078
\(518\) 1.62470e6 0.266040
\(519\) 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
\(523\) −4.82224e6 −0.770894 −0.385447 0.922730i \(-0.625953\pi\)
−0.385447 + 0.922730i \(0.625953\pi\)
\(524\) −1.89523e6 −0.301533
\(525\) 0 0
\(526\) 677184. 0.106719
\(527\) −1.39207e6 −0.218340
\(528\) 0 0
\(529\) −6.34271e6 −0.985452
\(530\) 0 0
\(531\) 0 0
\(532\) −1.26712e6 −0.194106
\(533\) −1.83497e6 −0.279776
\(534\) 0 0
\(535\) 0 0
\(536\) −6848.00 −0.00102956
\(537\) 0 0
\(538\) 6.32436e6 0.942022
\(539\) 3.24076e6 0.480479
\(540\) 0 0
\(541\) 362537. 0.0532549 0.0266274 0.999645i \(-0.491523\pi\)
0.0266274 + 0.999645i \(0.491523\pi\)
\(542\) 3.29005e6 0.481065
\(543\) 0 0
\(544\) 165888. 0.0240335
\(545\) 0 0
\(546\) 0 0
\(547\) −3.11439e6 −0.445046 −0.222523 0.974927i \(-0.571429\pi\)
−0.222523 + 0.974927i \(0.571429\pi\)
\(548\) −211488. −0.0300839
\(549\) 0 0
\(550\) 0 0
\(551\) −1.32947e7 −1.86551
\(552\) 0 0
\(553\) 3.25052e6 0.452002
\(554\) 2.18729e6 0.302784
\(555\) 0 0
\(556\) −5.61184e6 −0.769872
\(557\) −7.99304e6 −1.09163 −0.545813 0.837907i \(-0.683779\pi\)
−0.545813 + 0.837907i \(0.683779\pi\)
\(558\) 0 0
\(559\) 1.44945e6 0.196189
\(560\) 0 0
\(561\) 0 0
\(562\) 4.36999e6 0.583633
\(563\) −1.23236e7 −1.63857 −0.819286 0.573385i \(-0.805630\pi\)
−0.819286 + 0.573385i \(0.805630\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 9.93920e6 1.30410
\(567\) 0 0
\(568\) 2.60659e6 0.339002
\(569\) −1.01364e7 −1.31252 −0.656258 0.754537i \(-0.727862\pi\)
−0.656258 + 0.754537i \(0.727862\pi\)
\(570\) 0 0
\(571\) 6.53084e6 0.838260 0.419130 0.907926i \(-0.362335\pi\)
0.419130 + 0.907926i \(0.362335\pi\)
\(572\) 358752. 0.0458463
\(573\) 0 0
\(574\) −3.41558e6 −0.432698
\(575\) 0 0
\(576\) 0 0
\(577\) 1.24453e6 0.155621 0.0778103 0.996968i \(-0.475207\pi\)
0.0778103 + 0.996968i \(0.475207\pi\)
\(578\) −5.57445e6 −0.694037
\(579\) 0 0
\(580\) 0 0
\(581\) 5.10110e6 0.626936
\(582\) 0 0
\(583\) −3.97735e6 −0.484644
\(584\) −2.22118e6 −0.269496
\(585\) 0 0
\(586\) −1.36558e6 −0.164275
\(587\) −1.33403e6 −0.159797 −0.0798987 0.996803i \(-0.525460\pi\)
−0.0798987 + 0.996803i \(0.525460\pi\)
\(588\) 0 0
\(589\) −1.44792e7 −1.71972
\(590\) 0 0
\(591\) 0 0
\(592\) −2.21235e6 −0.259448
\(593\) −1.19401e7 −1.39435 −0.697177 0.716899i \(-0.745561\pi\)
−0.697177 + 0.716899i \(0.745561\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.75824e6 −0.202751
\(597\) 0 0
\(598\) −123624. −0.0141368
\(599\) 7.16430e6 0.815843 0.407922 0.913017i \(-0.366254\pi\)
0.407922 + 0.913017i \(0.366254\pi\)
\(600\) 0 0
\(601\) 1.15163e6 0.130055 0.0650273 0.997883i \(-0.479287\pi\)
0.0650273 + 0.997883i \(0.479287\pi\)
\(602\) 2.69799e6 0.303423
\(603\) 0 0
\(604\) −2.76165e6 −0.308018
\(605\) 0 0
\(606\) 0 0
\(607\) 1.34268e7 1.47911 0.739554 0.673097i \(-0.235037\pi\)
0.739554 + 0.673097i \(0.235037\pi\)
\(608\) 1.72544e6 0.189296
\(609\) 0 0
\(610\) 0 0
\(611\) 110898. 0.0120177
\(612\) 0 0
\(613\) −1.20184e7 −1.29180 −0.645900 0.763422i \(-0.723518\pi\)
−0.645900 + 0.763422i \(0.723518\pi\)
\(614\) −8.11591e6 −0.868793
\(615\) 0 0
\(616\) 667776. 0.0709054
\(617\) −6.98519e6 −0.738695 −0.369348 0.929291i \(-0.620419\pi\)
−0.369348 + 0.929291i \(0.620419\pi\)
\(618\) 0 0
\(619\) −8.20625e6 −0.860832 −0.430416 0.902631i \(-0.641633\pi\)
−0.430416 + 0.902631i \(0.641633\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 826392. 0.0856466
\(623\) 1.64688e6 0.169997
\(624\) 0 0
\(625\) 0 0
\(626\) 1.33689e7 1.36352
\(627\) 0 0
\(628\) 5.59989e6 0.566605
\(629\) −1.40000e6 −0.141092
\(630\) 0 0
\(631\) −1.07686e7 −1.07668 −0.538338 0.842729i \(-0.680947\pi\)
−0.538338 + 0.842729i \(0.680947\pi\)
\(632\) −4.42624e6 −0.440801
\(633\) 0 0
\(634\) −1.01316e7 −1.00104
\(635\) 0 0
\(636\) 0 0
\(637\) 1.47440e6 0.143968
\(638\) 7.00632e6 0.681457
\(639\) 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
\(643\) −1.00999e7 −0.963364 −0.481682 0.876346i \(-0.659974\pi\)
−0.481682 + 0.876346i \(0.659974\pi\)
\(644\) −230112. −0.0218637
\(645\) 0 0
\(646\) 1.09188e6 0.102942
\(647\) 7.52113e6 0.706354 0.353177 0.935556i \(-0.385101\pi\)
0.353177 + 0.935556i \(0.385101\pi\)
\(648\) 0 0
\(649\) 3.90942e6 0.364335
\(650\) 0 0
\(651\) 0 0
\(652\) −3.08130e6 −0.283867
\(653\) −2.67197e6 −0.245216 −0.122608 0.992455i \(-0.539126\pi\)
−0.122608 + 0.992455i \(0.539126\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.65101e6 0.421976
\(657\) 0 0
\(658\) 206424. 0.0185864
\(659\) −6.99948e6 −0.627845 −0.313922 0.949449i \(-0.601643\pi\)
−0.313922 + 0.949449i \(0.601643\pi\)
\(660\) 0 0
\(661\) 408122. 0.0363318 0.0181659 0.999835i \(-0.494217\pi\)
0.0181659 + 0.999835i \(0.494217\pi\)
\(662\) 2.40853e6 0.213603
\(663\) 0 0
\(664\) −6.94618e6 −0.611400
\(665\) 0 0
\(666\) 0 0
\(667\) −2.41434e6 −0.210128
\(668\) 9.29107e6 0.805610
\(669\) 0 0
\(670\) 0 0
\(671\) 4.85137e6 0.415966
\(672\) 0 0
\(673\) −1.74939e7 −1.48885 −0.744423 0.667709i \(-0.767275\pi\)
−0.744423 + 0.667709i \(0.767275\pi\)
\(674\) −839108. −0.0711489
\(675\) 0 0
\(676\) −5.77747e6 −0.486263
\(677\) −8.67440e6 −0.727391 −0.363695 0.931518i \(-0.618485\pi\)
−0.363695 + 0.931518i \(0.618485\pi\)
\(678\) 0 0
\(679\) −38681.0 −0.00321976
\(680\) 0 0
\(681\) 0 0
\(682\) 7.63058e6 0.628198
\(683\) 1.18478e7 0.971822 0.485911 0.874008i \(-0.338488\pi\)
0.485911 + 0.874008i \(0.338488\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 5.90414e6 0.479012
\(687\) 0 0
\(688\) −3.67386e6 −0.295904
\(689\) −1.80952e6 −0.145216
\(690\) 0 0
\(691\) 9.47775e6 0.755110 0.377555 0.925987i \(-0.376765\pi\)
0.377555 + 0.925987i \(0.376765\pi\)
\(692\) 1.18100e7 0.937530
\(693\) 0 0
\(694\) 1.60866e7 1.26785
\(695\) 0 0
\(696\) 0 0
\(697\) 2.94322e6 0.229478
\(698\) 33320.0 0.00258861
\(699\) 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
\(703\) −1.45618e7 −1.11129
\(704\) −909312. −0.0691483
\(705\) 0 0
\(706\) 7.80002e6 0.588958
\(707\) −1.58992e6 −0.119626
\(708\) 0 0
\(709\) 1.27436e7 0.952090 0.476045 0.879421i \(-0.342070\pi\)
0.476045 + 0.879421i \(0.342070\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −2.24256e6 −0.165785
\(713\) −2.62946e6 −0.193706
\(714\) 0 0
\(715\) 0 0
\(716\) −7.95792e6 −0.580119
\(717\) 0 0
\(718\) −9.08352e6 −0.657572
\(719\) −2.44929e6 −0.176692 −0.0883462 0.996090i \(-0.528158\pi\)
−0.0883462 + 0.996090i \(0.528158\pi\)
\(720\) 0 0
\(721\) −6.27187e6 −0.449323
\(722\) 1.45250e6 0.103699
\(723\) 0 0
\(724\) −5.33061e6 −0.377947
\(725\) 0 0
\(726\) 0 0
\(727\) 415033. 0.0291237 0.0145619 0.999894i \(-0.495365\pi\)
0.0145619 + 0.999894i \(0.495365\pi\)
\(728\) 303808. 0.0212457
\(729\) 0 0
\(730\) 0 0
\(731\) −2.32486e6 −0.160918
\(732\) 0 0
\(733\) 1.72877e7 1.18844 0.594221 0.804302i \(-0.297461\pi\)
0.594221 + 0.804302i \(0.297461\pi\)
\(734\) −1.14461e7 −0.784186
\(735\) 0 0
\(736\) 313344. 0.0213219
\(737\) 23754.0 0.00161090
\(738\) 0 0
\(739\) 5.18834e6 0.349476 0.174738 0.984615i \(-0.444092\pi\)
0.174738 + 0.984615i \(0.444092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −3.36821e6 −0.224589
\(743\) 4.79572e6 0.318700 0.159350 0.987222i \(-0.449060\pi\)
0.159350 + 0.987222i \(0.449060\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2.46124e6 −0.161923
\(747\) 0 0
\(748\) −575424. −0.0376040
\(749\) −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. −0.0181258
\(753\) 0 0
\(754\) 3.18756e6 0.204188
\(755\) 0 0
\(756\) 0 0
\(757\) −2.82068e7 −1.78902 −0.894508 0.447053i \(-0.852474\pi\)
−0.894508 + 0.447053i \(0.852474\pi\)
\(758\) 2.15951e7 1.36516
\(759\) 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
\(763\) 1.01997e7 0.634273
\(764\) 650208. 0.0403013
\(765\) 0 0
\(766\) 4.34750e6 0.267712
\(767\) 1.77861e6 0.109167
\(768\) 0 0
\(769\) 2.20930e7 1.34722 0.673610 0.739087i \(-0.264743\pi\)
0.673610 + 0.739087i \(0.264743\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.91442e6 −0.477942
\(773\) −3.00787e7 −1.81055 −0.905276 0.424824i \(-0.860336\pi\)
−0.905276 + 0.424824i \(0.860336\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 52672.0 0.00313997
\(777\) 0 0
\(778\) 1.39373e7 0.825523
\(779\) 3.06131e7 1.80744
\(780\) 0 0
\(781\) −9.04162e6 −0.530418
\(782\) 198288. 0.0115952
\(783\) 0 0
\(784\) −3.73709e6 −0.217142
\(785\) 0 0
\(786\) 0 0
\(787\) 3.28954e6 0.189321 0.0946605 0.995510i \(-0.469823\pi\)
0.0946605 + 0.995510i \(0.469823\pi\)
\(788\) 8.83747e6 0.507005
\(789\) 0 0
\(790\) 0 0
\(791\) 6.50123e6 0.369449
\(792\) 0 0
\(793\) 2.20715e6 0.124638
\(794\) −1.30636e7 −0.735381
\(795\) 0 0
\(796\) 1.09700e7 0.613655
\(797\) 6.71053e6 0.374206 0.187103 0.982340i \(-0.440090\pi\)
0.187103 + 0.982340i \(0.440090\pi\)
\(798\) 0 0
\(799\) −177876. −0.00985713
\(800\) 0 0
\(801\) 0 0
\(802\) 1.70928e7 0.938374
\(803\) 7.70473e6 0.421666
\(804\) 0 0
\(805\) 0 0
\(806\) 3.47157e6 0.188230
\(807\) 0 0
\(808\) 2.16499e6 0.116662
\(809\) −8.74254e6 −0.469641 −0.234821 0.972039i \(-0.575450\pi\)
−0.234821 + 0.972039i \(0.575450\pi\)
\(810\) 0 0
\(811\) −2.48410e7 −1.32622 −0.663112 0.748520i \(-0.730765\pi\)
−0.663112 + 0.748520i \(0.730765\pi\)
\(812\) 5.93328e6 0.315795
\(813\) 0 0
\(814\) 7.67410e6 0.405944
\(815\) 0 0
\(816\) 0 0
\(817\) −2.41814e7 −1.26744
\(818\) −5.80750e6 −0.303463
\(819\) 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
\(823\) 8.70659e6 0.448073 0.224036 0.974581i \(-0.428077\pi\)
0.224036 + 0.974581i \(0.428077\pi\)
\(824\) 8.54042e6 0.438189
\(825\) 0 0
\(826\) 3.31068e6 0.168837
\(827\) 3.71184e7 1.88723 0.943617 0.331040i \(-0.107400\pi\)
0.943617 + 0.331040i \(0.107400\pi\)
\(828\) 0 0
\(829\) 1.01765e6 0.0514295 0.0257147 0.999669i \(-0.491814\pi\)
0.0257147 + 0.999669i \(0.491814\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −413696. −0.0207192
\(833\) −2.36488e6 −0.118085
\(834\) 0 0
\(835\) 0 0
\(836\) −5.98512e6 −0.296181
\(837\) 0 0
\(838\) −2.23752e6 −0.110067
\(839\) 3.36194e7 1.64887 0.824433 0.565960i \(-0.191494\pi\)
0.824433 + 0.565960i \(0.191494\pi\)
\(840\) 0 0
\(841\) 4.17410e7 2.03504
\(842\) −1.56588e7 −0.761164
\(843\) 0 0
\(844\) 1.19916e7 0.579458
\(845\) 0 0
\(846\) 0 0
\(847\) 5.25305e6 0.251596
\(848\) 4.58650e6 0.219024
\(849\) 0 0
\(850\) 0 0
\(851\) −2.64445e6 −0.125173
\(852\) 0 0
\(853\) 3.52574e7 1.65912 0.829559 0.558419i \(-0.188592\pi\)
0.829559 + 0.558419i \(0.188592\pi\)
\(854\) 4.10836e6 0.192763
\(855\) 0 0
\(856\) 5.20013e6 0.242566
\(857\) −3.14941e7 −1.46480 −0.732398 0.680877i \(-0.761599\pi\)
−0.732398 + 0.680877i \(0.761599\pi\)
\(858\) 0 0
\(859\) 1.19344e7 0.551848 0.275924 0.961180i \(-0.411016\pi\)
0.275924 + 0.961180i \(0.411016\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.43000e7 0.655492
\(863\) 8.70442e6 0.397844 0.198922 0.980015i \(-0.436256\pi\)
0.198922 + 0.980015i \(0.436256\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2.86388e7 −1.29766
\(867\) 0 0
\(868\) 6.46194e6 0.291114
\(869\) 1.53535e7 0.689697
\(870\) 0 0
\(871\) 10807.0 0.000482681 0
\(872\) −1.38890e7 −0.618555
\(873\) 0 0
\(874\) 2.06244e6 0.0913277
\(875\) 0 0
\(876\) 0 0
\(877\) −1.17999e7 −0.518059 −0.259029 0.965869i \(-0.583403\pi\)
−0.259029 + 0.965869i \(0.583403\pi\)
\(878\) 6.87158e6 0.300829
\(879\) 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
\(883\) 8.80577e6 0.380072 0.190036 0.981777i \(-0.439140\pi\)
0.190036 + 0.981777i \(0.439140\pi\)
\(884\) −261792. −0.0112675
\(885\) 0 0
\(886\) 1.35868e7 0.581477
\(887\) 250122. 0.0106744 0.00533719 0.999986i \(-0.498301\pi\)
0.00533719 + 0.999986i \(0.498301\pi\)
\(888\) 0 0
\(889\) −1.20343e7 −0.510699
\(890\) 0 0
\(891\) 0 0
\(892\) −2.70834e6 −0.113970
\(893\) −1.85013e6 −0.0776379
\(894\) 0 0
\(895\) 0 0
\(896\) −770048. −0.0320441
\(897\) 0 0
\(898\) 1.35842e7 0.562140
\(899\) 6.77988e7 2.79784
\(900\) 0 0
\(901\) 2.90239e6 0.119109
\(902\) −1.61332e7 −0.660243
\(903\) 0 0
\(904\) −8.85274e6 −0.360294
\(905\) 0 0
\(906\) 0 0
\(907\) 3.24955e7 1.31161 0.655806 0.754929i \(-0.272329\pi\)
0.655806 + 0.754929i \(0.272329\pi\)
\(908\) −743808. −0.0299396
\(909\) 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
\(913\) 2.40945e7 0.956625
\(914\) 1.81126e7 0.717157
\(915\) 0 0
\(916\) −1.44184e6 −0.0567778
\(917\) 5.56724e6 0.218634
\(918\) 0 0
\(919\) 1.41629e7 0.553176 0.276588 0.960989i \(-0.410796\pi\)
0.276588 + 0.960989i \(0.410796\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 5.11579e6 0.198192
\(923\) −4.11353e6 −0.158932
\(924\) 0 0
\(925\) 0 0
\(926\) 2.87945e7 1.10352
\(927\) 0 0
\(928\) −8.07936e6 −0.307969
\(929\) −4.37292e7 −1.66239 −0.831194 0.555982i \(-0.812342\pi\)
−0.831194 + 0.555982i \(0.812342\pi\)
\(930\) 0 0
\(931\) −2.45976e7 −0.930077
\(932\) 1.70262e7 0.642063
\(933\) 0 0
\(934\) 1.93214e7 0.724721
\(935\) 0 0
\(936\) 0 0
\(937\) −5.73509e6 −0.213398 −0.106699 0.994291i \(-0.534028\pi\)
−0.106699 + 0.994291i \(0.534028\pi\)
\(938\) 20116.0 0.000746508 0
\(939\) 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
\(943\) 5.55941e6 0.203587
\(944\) −4.50816e6 −0.164653
\(945\) 0 0
\(946\) 1.27437e7 0.462985
\(947\) −3.07342e7 −1.11365 −0.556823 0.830631i \(-0.687980\pi\)
−0.556823 + 0.830631i \(0.687980\pi\)
\(948\) 0 0
\(949\) 3.50531e6 0.126346
\(950\) 0 0
\(951\) 0 0
\(952\) −487296. −0.0174261
\(953\) −2.51847e7 −0.898264 −0.449132 0.893465i \(-0.648267\pi\)
−0.449132 + 0.893465i \(0.648267\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.84253e7 −0.652033
\(957\) 0 0
\(958\) −2.99460e6 −0.105420
\(959\) 621246. 0.0218131
\(960\) 0 0
\(961\) 4.52105e7 1.57918
\(962\) 3.49137e6 0.121635
\(963\) 0 0
\(964\) 1.36995e7 0.474801
\(965\) 0 0
\(966\) 0 0
\(967\) −1.44556e7 −0.497130 −0.248565 0.968615i \(-0.579959\pi\)
−0.248565 + 0.968615i \(0.579959\pi\)
\(968\) −7.15309e6 −0.245361
\(969\) 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
\(973\) 1.64848e7 0.558214
\(974\) 2.06558e7 0.697660
\(975\) 0 0
\(976\) −5.59437e6 −0.187986
\(977\) −8.41568e6 −0.282067 −0.141034 0.990005i \(-0.545043\pi\)
−0.141034 + 0.990005i \(0.545043\pi\)
\(978\) 0 0
\(979\) 7.77888e6 0.259394
\(980\) 0 0
\(981\) 0 0
\(982\) −3.41715e7 −1.13080
\(983\) 3.89409e7 1.28535 0.642676 0.766138i \(-0.277824\pi\)
0.642676 + 0.766138i \(0.277824\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −5.11272e6 −0.167479
\(987\) 0 0
\(988\) −2.72296e6 −0.0887460
\(989\) −4.39141e6 −0.142762
\(990\) 0 0
\(991\) 4.84592e7 1.56745 0.783723 0.621111i \(-0.213318\pi\)
0.783723 + 0.621111i \(0.213318\pi\)
\(992\) −8.79923e6 −0.283900
\(993\) 0 0
\(994\) −7.65686e6 −0.245802
\(995\) 0 0
\(996\) 0 0
\(997\) −3.84733e7 −1.22581 −0.612903 0.790158i \(-0.709998\pi\)
−0.612903 + 0.790158i \(0.709998\pi\)
\(998\) −1.68235e7 −0.534675
\(999\) 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.p.1.1 1
3.2 odd 2 150.6.a.a.1.1 1
5.2 odd 4 450.6.c.e.199.2 2
5.3 odd 4 450.6.c.e.199.1 2
5.4 even 2 450.6.a.i.1.1 1
15.2 even 4 150.6.c.g.49.1 2
15.8 even 4 150.6.c.g.49.2 2
15.14 odd 2 150.6.a.m.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.6.a.a.1.1 1 3.2 odd 2
150.6.a.m.1.1 yes 1 15.14 odd 2
150.6.c.g.49.1 2 15.2 even 4
150.6.c.g.49.2 2 15.8 even 4
450.6.a.i.1.1 1 5.4 even 2
450.6.a.p.1.1 1 1.1 even 1 trivial
450.6.c.e.199.1 2 5.3 odd 4
450.6.c.e.199.2 2 5.2 odd 4