Properties

Label 225.12.a.b.1.1
Level $225$
Weight $12$
Character 225.1
Self dual yes
Analytic conductor $172.877$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(172.877215626\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 225.1

$q$-expansion

\(f(q)\) \(=\) \(q-24.0000 q^{2} -1472.00 q^{4} +16744.0 q^{7} +84480.0 q^{8} +O(q^{10})\) \(q-24.0000 q^{2} -1472.00 q^{4} +16744.0 q^{7} +84480.0 q^{8} -534612. q^{11} +577738. q^{13} -401856. q^{14} +987136. q^{16} -6.90593e6 q^{17} +1.06614e7 q^{19} +1.28307e7 q^{22} +1.86433e7 q^{23} -1.38657e7 q^{26} -2.46472e7 q^{28} -1.28407e8 q^{29} -5.28432e7 q^{31} -1.96706e8 q^{32} +1.65742e8 q^{34} +1.82213e8 q^{37} -2.55874e8 q^{38} -3.08120e8 q^{41} +1.71257e7 q^{43} +7.86949e8 q^{44} -4.47439e8 q^{46} +2.68735e9 q^{47} -1.69697e9 q^{49} -8.50430e8 q^{52} -1.59606e9 q^{53} +1.41453e9 q^{56} +3.08176e9 q^{58} +5.18920e9 q^{59} +6.95648e9 q^{61} +1.26824e9 q^{62} +2.69930e9 q^{64} +1.54818e10 q^{67} +1.01655e10 q^{68} -9.79149e9 q^{71} -1.46379e9 q^{73} -4.37312e9 q^{74} -1.56936e10 q^{76} -8.95154e9 q^{77} +3.81168e10 q^{79} +7.39489e9 q^{82} -2.93351e10 q^{83} -4.11017e8 q^{86} -4.51640e10 q^{88} +2.49929e10 q^{89} +9.67365e9 q^{91} -2.74429e10 q^{92} -6.44964e10 q^{94} -7.50136e10 q^{97} +4.07272e10 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −24.0000 −0.530330 −0.265165 0.964203i \(-0.585426\pi\)
−0.265165 + 0.964203i \(0.585426\pi\)
\(3\) 0 0
\(4\) −1472.00 −0.718750
\(5\) 0 0
\(6\) 0 0
\(7\) 16744.0 0.376548 0.188274 0.982117i \(-0.439711\pi\)
0.188274 + 0.982117i \(0.439711\pi\)
\(8\) 84480.0 0.911505
\(9\) 0 0
\(10\) 0 0
\(11\) −534612. −1.00087 −0.500436 0.865773i \(-0.666827\pi\)
−0.500436 + 0.865773i \(0.666827\pi\)
\(12\) 0 0
\(13\) 577738. 0.431561 0.215781 0.976442i \(-0.430770\pi\)
0.215781 + 0.976442i \(0.430770\pi\)
\(14\) −401856. −0.199695
\(15\) 0 0
\(16\) 987136. 0.235352
\(17\) −6.90593e6 −1.17965 −0.589825 0.807531i \(-0.700803\pi\)
−0.589825 + 0.807531i \(0.700803\pi\)
\(18\) 0 0
\(19\) 1.06614e7 0.987803 0.493901 0.869518i \(-0.335570\pi\)
0.493901 + 0.869518i \(0.335570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.28307e7 0.530793
\(23\) 1.86433e7 0.603975 0.301988 0.953312i \(-0.402350\pi\)
0.301988 + 0.953312i \(0.402350\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.38657e7 −0.228870
\(27\) 0 0
\(28\) −2.46472e7 −0.270644
\(29\) −1.28407e8 −1.16251 −0.581257 0.813720i \(-0.697439\pi\)
−0.581257 + 0.813720i \(0.697439\pi\)
\(30\) 0 0
\(31\) −5.28432e7 −0.331512 −0.165756 0.986167i \(-0.553006\pi\)
−0.165756 + 0.986167i \(0.553006\pi\)
\(32\) −1.96706e8 −1.03632
\(33\) 0 0
\(34\) 1.65742e8 0.625604
\(35\) 0 0
\(36\) 0 0
\(37\) 1.82213e8 0.431987 0.215993 0.976395i \(-0.430701\pi\)
0.215993 + 0.976395i \(0.430701\pi\)
\(38\) −2.55874e8 −0.523862
\(39\) 0 0
\(40\) 0 0
\(41\) −3.08120e8 −0.415345 −0.207673 0.978198i \(-0.566589\pi\)
−0.207673 + 0.978198i \(0.566589\pi\)
\(42\) 0 0
\(43\) 1.71257e7 0.0177653 0.00888264 0.999961i \(-0.497173\pi\)
0.00888264 + 0.999961i \(0.497173\pi\)
\(44\) 7.86949e8 0.719377
\(45\) 0 0
\(46\) −4.47439e8 −0.320306
\(47\) 2.68735e9 1.70917 0.854586 0.519310i \(-0.173811\pi\)
0.854586 + 0.519310i \(0.173811\pi\)
\(48\) 0 0
\(49\) −1.69697e9 −0.858212
\(50\) 0 0
\(51\) 0 0
\(52\) −8.50430e8 −0.310185
\(53\) −1.59606e9 −0.524241 −0.262120 0.965035i \(-0.584422\pi\)
−0.262120 + 0.965035i \(0.584422\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.41453e9 0.343225
\(57\) 0 0
\(58\) 3.08176e9 0.616517
\(59\) 5.18920e9 0.944963 0.472481 0.881341i \(-0.343358\pi\)
0.472481 + 0.881341i \(0.343358\pi\)
\(60\) 0 0
\(61\) 6.95648e9 1.05457 0.527285 0.849689i \(-0.323210\pi\)
0.527285 + 0.849689i \(0.323210\pi\)
\(62\) 1.26824e9 0.175811
\(63\) 0 0
\(64\) 2.69930e9 0.314240
\(65\) 0 0
\(66\) 0 0
\(67\) 1.54818e10 1.40091 0.700456 0.713696i \(-0.252980\pi\)
0.700456 + 0.713696i \(0.252980\pi\)
\(68\) 1.01655e10 0.847874
\(69\) 0 0
\(70\) 0 0
\(71\) −9.79149e9 −0.644062 −0.322031 0.946729i \(-0.604366\pi\)
−0.322031 + 0.946729i \(0.604366\pi\)
\(72\) 0 0
\(73\) −1.46379e9 −0.0826425 −0.0413212 0.999146i \(-0.513157\pi\)
−0.0413212 + 0.999146i \(0.513157\pi\)
\(74\) −4.37312e9 −0.229096
\(75\) 0 0
\(76\) −1.56936e10 −0.709983
\(77\) −8.95154e9 −0.376876
\(78\) 0 0
\(79\) 3.81168e10 1.39370 0.696848 0.717219i \(-0.254585\pi\)
0.696848 + 0.717219i \(0.254585\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 7.39489e9 0.220270
\(83\) −2.93351e10 −0.817444 −0.408722 0.912659i \(-0.634025\pi\)
−0.408722 + 0.912659i \(0.634025\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.11017e8 −0.00942146
\(87\) 0 0
\(88\) −4.51640e10 −0.912300
\(89\) 2.49929e10 0.474430 0.237215 0.971457i \(-0.423765\pi\)
0.237215 + 0.971457i \(0.423765\pi\)
\(90\) 0 0
\(91\) 9.67365e9 0.162503
\(92\) −2.74429e10 −0.434107
\(93\) 0 0
\(94\) −6.44964e10 −0.906425
\(95\) 0 0
\(96\) 0 0
\(97\) −7.50136e10 −0.886942 −0.443471 0.896289i \(-0.646253\pi\)
−0.443471 + 0.896289i \(0.646253\pi\)
\(98\) 4.07272e10 0.455136
\(99\) 0 0
\(100\) 0 0
\(101\) −8.17430e10 −0.773896 −0.386948 0.922101i \(-0.626471\pi\)
−0.386948 + 0.922101i \(0.626471\pi\)
\(102\) 0 0
\(103\) 2.25755e11 1.91881 0.959407 0.282025i \(-0.0910061\pi\)
0.959407 + 0.282025i \(0.0910061\pi\)
\(104\) 4.88073e10 0.393370
\(105\) 0 0
\(106\) 3.83053e10 0.278021
\(107\) 9.02413e10 0.622006 0.311003 0.950409i \(-0.399335\pi\)
0.311003 + 0.950409i \(0.399335\pi\)
\(108\) 0 0
\(109\) 7.34827e10 0.457445 0.228723 0.973492i \(-0.426545\pi\)
0.228723 + 0.973492i \(0.426545\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.65286e10 0.0886211
\(113\) −8.51469e10 −0.434748 −0.217374 0.976088i \(-0.569749\pi\)
−0.217374 + 0.976088i \(0.569749\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.89015e11 0.835557
\(117\) 0 0
\(118\) −1.24541e11 −0.501142
\(119\) −1.15633e11 −0.444195
\(120\) 0 0
\(121\) 4.98320e8 0.00174658
\(122\) −1.66955e11 −0.559270
\(123\) 0 0
\(124\) 7.77851e10 0.238274
\(125\) 0 0
\(126\) 0 0
\(127\) 2.62717e11 0.705615 0.352808 0.935696i \(-0.385227\pi\)
0.352808 + 0.935696i \(0.385227\pi\)
\(128\) 3.38071e11 0.869668
\(129\) 0 0
\(130\) 0 0
\(131\) −6.31529e11 −1.43021 −0.715107 0.699015i \(-0.753622\pi\)
−0.715107 + 0.699015i \(0.753622\pi\)
\(132\) 0 0
\(133\) 1.78515e11 0.371955
\(134\) −3.71564e11 −0.742946
\(135\) 0 0
\(136\) −5.83413e11 −1.07526
\(137\) −2.97199e11 −0.526119 −0.263059 0.964780i \(-0.584732\pi\)
−0.263059 + 0.964780i \(0.584732\pi\)
\(138\) 0 0
\(139\) 5.96794e11 0.975535 0.487767 0.872974i \(-0.337811\pi\)
0.487767 + 0.872974i \(0.337811\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.34996e11 0.341565
\(143\) −3.08866e11 −0.431938
\(144\) 0 0
\(145\) 0 0
\(146\) 3.51310e10 0.0438278
\(147\) 0 0
\(148\) −2.68218e11 −0.310491
\(149\) 1.11543e12 1.24428 0.622142 0.782905i \(-0.286263\pi\)
0.622142 + 0.782905i \(0.286263\pi\)
\(150\) 0 0
\(151\) −8.24447e11 −0.854653 −0.427326 0.904097i \(-0.640544\pi\)
−0.427326 + 0.904097i \(0.640544\pi\)
\(152\) 9.00677e11 0.900387
\(153\) 0 0
\(154\) 2.14837e11 0.199869
\(155\) 0 0
\(156\) 0 0
\(157\) −1.31512e12 −1.10031 −0.550156 0.835062i \(-0.685432\pi\)
−0.550156 + 0.835062i \(0.685432\pi\)
\(158\) −9.14804e11 −0.739119
\(159\) 0 0
\(160\) 0 0
\(161\) 3.12163e11 0.227425
\(162\) 0 0
\(163\) 3.57833e11 0.243584 0.121792 0.992556i \(-0.461136\pi\)
0.121792 + 0.992556i \(0.461136\pi\)
\(164\) 4.53553e11 0.298529
\(165\) 0 0
\(166\) 7.04042e11 0.433515
\(167\) 2.75483e12 1.64117 0.820587 0.571521i \(-0.193646\pi\)
0.820587 + 0.571521i \(0.193646\pi\)
\(168\) 0 0
\(169\) −1.45838e12 −0.813755
\(170\) 0 0
\(171\) 0 0
\(172\) −2.52090e10 −0.0127688
\(173\) −9.50387e11 −0.466280 −0.233140 0.972443i \(-0.574900\pi\)
−0.233140 + 0.972443i \(0.574900\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.27735e11 −0.235557
\(177\) 0 0
\(178\) −5.99830e11 −0.251604
\(179\) −1.68138e12 −0.683873 −0.341936 0.939723i \(-0.611083\pi\)
−0.341936 + 0.939723i \(0.611083\pi\)
\(180\) 0 0
\(181\) −9.96774e11 −0.381386 −0.190693 0.981650i \(-0.561073\pi\)
−0.190693 + 0.981650i \(0.561073\pi\)
\(182\) −2.32167e11 −0.0861804
\(183\) 0 0
\(184\) 1.57498e12 0.550526
\(185\) 0 0
\(186\) 0 0
\(187\) 3.69200e12 1.18068
\(188\) −3.95578e12 −1.22847
\(189\) 0 0
\(190\) 0 0
\(191\) −2.76240e12 −0.786328 −0.393164 0.919468i \(-0.628619\pi\)
−0.393164 + 0.919468i \(0.628619\pi\)
\(192\) 0 0
\(193\) −5.44239e12 −1.46293 −0.731466 0.681878i \(-0.761164\pi\)
−0.731466 + 0.681878i \(0.761164\pi\)
\(194\) 1.80033e12 0.470372
\(195\) 0 0
\(196\) 2.49793e12 0.616840
\(197\) −2.87609e12 −0.690619 −0.345309 0.938489i \(-0.612226\pi\)
−0.345309 + 0.938489i \(0.612226\pi\)
\(198\) 0 0
\(199\) 7.28391e11 0.165452 0.0827262 0.996572i \(-0.473637\pi\)
0.0827262 + 0.996572i \(0.473637\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1.96183e12 0.410421
\(203\) −2.15004e12 −0.437742
\(204\) 0 0
\(205\) 0 0
\(206\) −5.41812e12 −1.01760
\(207\) 0 0
\(208\) 5.70306e11 0.101569
\(209\) −5.69972e12 −0.988665
\(210\) 0 0
\(211\) −6.79317e12 −1.11820 −0.559099 0.829101i \(-0.688853\pi\)
−0.559099 + 0.829101i \(0.688853\pi\)
\(212\) 2.34939e12 0.376798
\(213\) 0 0
\(214\) −2.16579e12 −0.329868
\(215\) 0 0
\(216\) 0 0
\(217\) −8.84806e11 −0.124830
\(218\) −1.76358e12 −0.242597
\(219\) 0 0
\(220\) 0 0
\(221\) −3.98982e12 −0.509092
\(222\) 0 0
\(223\) −7.33486e12 −0.890667 −0.445333 0.895365i \(-0.646915\pi\)
−0.445333 + 0.895365i \(0.646915\pi\)
\(224\) −3.29365e12 −0.390223
\(225\) 0 0
\(226\) 2.04352e12 0.230560
\(227\) −1.35984e12 −0.149743 −0.0748713 0.997193i \(-0.523855\pi\)
−0.0748713 + 0.997193i \(0.523855\pi\)
\(228\) 0 0
\(229\) −1.18244e13 −1.24075 −0.620375 0.784305i \(-0.713020\pi\)
−0.620375 + 0.784305i \(0.713020\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.08478e13 −1.05964
\(233\) −1.75634e13 −1.67552 −0.837761 0.546038i \(-0.816135\pi\)
−0.837761 + 0.546038i \(0.816135\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.63851e12 −0.679192
\(237\) 0 0
\(238\) 2.77519e12 0.235570
\(239\) 7.13958e12 0.592221 0.296111 0.955154i \(-0.404310\pi\)
0.296111 + 0.955154i \(0.404310\pi\)
\(240\) 0 0
\(241\) −2.31307e11 −0.0183271 −0.00916357 0.999958i \(-0.502917\pi\)
−0.00916357 + 0.999958i \(0.502917\pi\)
\(242\) −1.19597e10 −0.000926264 0
\(243\) 0 0
\(244\) −1.02399e13 −0.757972
\(245\) 0 0
\(246\) 0 0
\(247\) 6.15951e12 0.426297
\(248\) −4.46419e12 −0.302175
\(249\) 0 0
\(250\) 0 0
\(251\) −1.29831e13 −0.822567 −0.411284 0.911507i \(-0.634919\pi\)
−0.411284 + 0.911507i \(0.634919\pi\)
\(252\) 0 0
\(253\) −9.96692e12 −0.604502
\(254\) −6.30521e12 −0.374209
\(255\) 0 0
\(256\) −1.36419e13 −0.775451
\(257\) 2.39612e13 1.33314 0.666571 0.745442i \(-0.267761\pi\)
0.666571 + 0.745442i \(0.267761\pi\)
\(258\) 0 0
\(259\) 3.05098e12 0.162664
\(260\) 0 0
\(261\) 0 0
\(262\) 1.51567e13 0.758485
\(263\) −2.42737e13 −1.18954 −0.594771 0.803895i \(-0.702757\pi\)
−0.594771 + 0.803895i \(0.702757\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.28436e12 −0.197259
\(267\) 0 0
\(268\) −2.27892e13 −1.00691
\(269\) −2.58377e13 −1.11845 −0.559225 0.829016i \(-0.688901\pi\)
−0.559225 + 0.829016i \(0.688901\pi\)
\(270\) 0 0
\(271\) −3.76793e12 −0.156593 −0.0782964 0.996930i \(-0.524948\pi\)
−0.0782964 + 0.996930i \(0.524948\pi\)
\(272\) −6.81710e12 −0.277633
\(273\) 0 0
\(274\) 7.13277e12 0.279017
\(275\) 0 0
\(276\) 0 0
\(277\) 1.64189e13 0.604931 0.302466 0.953160i \(-0.402190\pi\)
0.302466 + 0.953160i \(0.402190\pi\)
\(278\) −1.43230e13 −0.517355
\(279\) 0 0
\(280\) 0 0
\(281\) −2.10357e13 −0.716263 −0.358132 0.933671i \(-0.616586\pi\)
−0.358132 + 0.933671i \(0.616586\pi\)
\(282\) 0 0
\(283\) −1.67132e13 −0.547310 −0.273655 0.961828i \(-0.588233\pi\)
−0.273655 + 0.961828i \(0.588233\pi\)
\(284\) 1.44131e13 0.462920
\(285\) 0 0
\(286\) 7.41278e12 0.229070
\(287\) −5.15917e12 −0.156397
\(288\) 0 0
\(289\) 1.34200e13 0.391575
\(290\) 0 0
\(291\) 0 0
\(292\) 2.15470e12 0.0593993
\(293\) −2.39269e13 −0.647312 −0.323656 0.946175i \(-0.604912\pi\)
−0.323656 + 0.946175i \(0.604912\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.53934e13 0.393758
\(297\) 0 0
\(298\) −2.67704e13 −0.659881
\(299\) 1.07709e13 0.260652
\(300\) 0 0
\(301\) 2.86753e11 0.00668947
\(302\) 1.97867e13 0.453248
\(303\) 0 0
\(304\) 1.05243e13 0.232481
\(305\) 0 0
\(306\) 0 0
\(307\) −1.53111e13 −0.320439 −0.160219 0.987081i \(-0.551220\pi\)
−0.160219 + 0.987081i \(0.551220\pi\)
\(308\) 1.31767e13 0.270880
\(309\) 0 0
\(310\) 0 0
\(311\) −4.98752e13 −0.972080 −0.486040 0.873936i \(-0.661559\pi\)
−0.486040 + 0.873936i \(0.661559\pi\)
\(312\) 0 0
\(313\) 9.94808e13 1.87174 0.935870 0.352345i \(-0.114616\pi\)
0.935870 + 0.352345i \(0.114616\pi\)
\(314\) 3.15628e13 0.583529
\(315\) 0 0
\(316\) −5.61080e13 −1.00172
\(317\) 8.33692e13 1.46278 0.731392 0.681958i \(-0.238871\pi\)
0.731392 + 0.681958i \(0.238871\pi\)
\(318\) 0 0
\(319\) 6.86477e13 1.16353
\(320\) 0 0
\(321\) 0 0
\(322\) −7.49191e12 −0.120611
\(323\) −7.36271e13 −1.16526
\(324\) 0 0
\(325\) 0 0
\(326\) −8.58799e12 −0.129180
\(327\) 0 0
\(328\) −2.60300e13 −0.378589
\(329\) 4.49970e13 0.643585
\(330\) 0 0
\(331\) −6.35840e13 −0.879618 −0.439809 0.898091i \(-0.644954\pi\)
−0.439809 + 0.898091i \(0.644954\pi\)
\(332\) 4.31813e13 0.587538
\(333\) 0 0
\(334\) −6.61160e13 −0.870364
\(335\) 0 0
\(336\) 0 0
\(337\) −1.21001e14 −1.51644 −0.758221 0.651997i \(-0.773931\pi\)
−0.758221 + 0.651997i \(0.773931\pi\)
\(338\) 3.50011e13 0.431559
\(339\) 0 0
\(340\) 0 0
\(341\) 2.82506e13 0.331802
\(342\) 0 0
\(343\) −6.15223e13 −0.699705
\(344\) 1.44678e12 0.0161931
\(345\) 0 0
\(346\) 2.28093e13 0.247283
\(347\) −1.55662e14 −1.66100 −0.830499 0.557020i \(-0.811945\pi\)
−0.830499 + 0.557020i \(0.811945\pi\)
\(348\) 0 0
\(349\) −2.56430e13 −0.265112 −0.132556 0.991176i \(-0.542318\pi\)
−0.132556 + 0.991176i \(0.542318\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.05162e14 1.03722
\(353\) 2.49098e13 0.241885 0.120943 0.992659i \(-0.461408\pi\)
0.120943 + 0.992659i \(0.461408\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3.67896e13 −0.340996
\(357\) 0 0
\(358\) 4.03532e13 0.362678
\(359\) −1.57584e14 −1.39474 −0.697370 0.716712i \(-0.745646\pi\)
−0.697370 + 0.716712i \(0.745646\pi\)
\(360\) 0 0
\(361\) −2.82438e12 −0.0242457
\(362\) 2.39226e13 0.202260
\(363\) 0 0
\(364\) −1.42396e13 −0.116799
\(365\) 0 0
\(366\) 0 0
\(367\) 1.77901e14 1.39481 0.697406 0.716676i \(-0.254338\pi\)
0.697406 + 0.716676i \(0.254338\pi\)
\(368\) 1.84034e13 0.142146
\(369\) 0 0
\(370\) 0 0
\(371\) −2.67244e13 −0.197402
\(372\) 0 0
\(373\) 5.51617e13 0.395585 0.197792 0.980244i \(-0.436623\pi\)
0.197792 + 0.980244i \(0.436623\pi\)
\(374\) −8.86079e13 −0.626150
\(375\) 0 0
\(376\) 2.27027e14 1.55792
\(377\) −7.41854e13 −0.501696
\(378\) 0 0
\(379\) 1.46463e14 0.962083 0.481042 0.876698i \(-0.340259\pi\)
0.481042 + 0.876698i \(0.340259\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 6.62977e13 0.417013
\(383\) 2.31450e14 1.43504 0.717519 0.696539i \(-0.245278\pi\)
0.717519 + 0.696539i \(0.245278\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.30617e14 0.775837
\(387\) 0 0
\(388\) 1.10420e14 0.637490
\(389\) 1.49872e14 0.853093 0.426547 0.904466i \(-0.359730\pi\)
0.426547 + 0.904466i \(0.359730\pi\)
\(390\) 0 0
\(391\) −1.28749e14 −0.712480
\(392\) −1.43360e14 −0.782264
\(393\) 0 0
\(394\) 6.90262e13 0.366256
\(395\) 0 0
\(396\) 0 0
\(397\) −2.08111e14 −1.05912 −0.529562 0.848271i \(-0.677644\pi\)
−0.529562 + 0.848271i \(0.677644\pi\)
\(398\) −1.74814e13 −0.0877443
\(399\) 0 0
\(400\) 0 0
\(401\) 1.33408e14 0.642521 0.321261 0.946991i \(-0.395893\pi\)
0.321261 + 0.946991i \(0.395893\pi\)
\(402\) 0 0
\(403\) −3.05295e13 −0.143068
\(404\) 1.20326e14 0.556238
\(405\) 0 0
\(406\) 5.16010e13 0.232148
\(407\) −9.74134e13 −0.432364
\(408\) 0 0
\(409\) −2.06168e14 −0.890722 −0.445361 0.895351i \(-0.646925\pi\)
−0.445361 + 0.895351i \(0.646925\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3.32312e14 −1.37915
\(413\) 8.68880e13 0.355824
\(414\) 0 0
\(415\) 0 0
\(416\) −1.13645e14 −0.447235
\(417\) 0 0
\(418\) 1.36793e14 0.524319
\(419\) −7.34035e13 −0.277677 −0.138838 0.990315i \(-0.544337\pi\)
−0.138838 + 0.990315i \(0.544337\pi\)
\(420\) 0 0
\(421\) 1.71112e14 0.630563 0.315282 0.948998i \(-0.397901\pi\)
0.315282 + 0.948998i \(0.397901\pi\)
\(422\) 1.63036e14 0.593014
\(423\) 0 0
\(424\) −1.34835e14 −0.477848
\(425\) 0 0
\(426\) 0 0
\(427\) 1.16479e14 0.397096
\(428\) −1.32835e14 −0.447067
\(429\) 0 0
\(430\) 0 0
\(431\) 7.17758e13 0.232463 0.116231 0.993222i \(-0.462919\pi\)
0.116231 + 0.993222i \(0.462919\pi\)
\(432\) 0 0
\(433\) −9.98812e13 −0.315356 −0.157678 0.987491i \(-0.550401\pi\)
−0.157678 + 0.987491i \(0.550401\pi\)
\(434\) 2.12353e13 0.0662012
\(435\) 0 0
\(436\) −1.08166e14 −0.328789
\(437\) 1.98764e14 0.596608
\(438\) 0 0
\(439\) −2.90312e13 −0.0849788 −0.0424894 0.999097i \(-0.513529\pi\)
−0.0424894 + 0.999097i \(0.513529\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 9.57557e13 0.269987
\(443\) 3.28370e14 0.914414 0.457207 0.889360i \(-0.348850\pi\)
0.457207 + 0.889360i \(0.348850\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.76037e14 0.472347
\(447\) 0 0
\(448\) 4.51970e13 0.118326
\(449\) 6.12368e14 1.58364 0.791822 0.610752i \(-0.209133\pi\)
0.791822 + 0.610752i \(0.209133\pi\)
\(450\) 0 0
\(451\) 1.64725e14 0.415708
\(452\) 1.25336e14 0.312475
\(453\) 0 0
\(454\) 3.26361e13 0.0794130
\(455\) 0 0
\(456\) 0 0
\(457\) −3.03483e14 −0.712189 −0.356095 0.934450i \(-0.615892\pi\)
−0.356095 + 0.934450i \(0.615892\pi\)
\(458\) 2.83786e14 0.658007
\(459\) 0 0
\(460\) 0 0
\(461\) 7.29308e14 1.63138 0.815691 0.578487i \(-0.196357\pi\)
0.815691 + 0.578487i \(0.196357\pi\)
\(462\) 0 0
\(463\) −1.22188e14 −0.266891 −0.133445 0.991056i \(-0.542604\pi\)
−0.133445 + 0.991056i \(0.542604\pi\)
\(464\) −1.26755e14 −0.273600
\(465\) 0 0
\(466\) 4.21520e14 0.888579
\(467\) −6.17381e14 −1.28621 −0.643103 0.765780i \(-0.722353\pi\)
−0.643103 + 0.765780i \(0.722353\pi\)
\(468\) 0 0
\(469\) 2.59228e14 0.527510
\(470\) 0 0
\(471\) 0 0
\(472\) 4.38384e14 0.861338
\(473\) −9.15561e12 −0.0177808
\(474\) 0 0
\(475\) 0 0
\(476\) 1.70212e14 0.319265
\(477\) 0 0
\(478\) −1.71350e14 −0.314073
\(479\) −1.05084e15 −1.90410 −0.952052 0.305938i \(-0.901030\pi\)
−0.952052 + 0.305938i \(0.901030\pi\)
\(480\) 0 0
\(481\) 1.05272e14 0.186429
\(482\) 5.55137e12 0.00971944
\(483\) 0 0
\(484\) −7.33527e11 −0.00125536
\(485\) 0 0
\(486\) 0 0
\(487\) 2.19910e14 0.363777 0.181889 0.983319i \(-0.441779\pi\)
0.181889 + 0.983319i \(0.441779\pi\)
\(488\) 5.87683e14 0.961246
\(489\) 0 0
\(490\) 0 0
\(491\) 4.83863e14 0.765199 0.382599 0.923914i \(-0.375029\pi\)
0.382599 + 0.923914i \(0.375029\pi\)
\(492\) 0 0
\(493\) 8.86768e14 1.37136
\(494\) −1.47828e14 −0.226078
\(495\) 0 0
\(496\) −5.21634e13 −0.0780219
\(497\) −1.63949e14 −0.242520
\(498\) 0 0
\(499\) −1.08878e14 −0.157538 −0.0787691 0.996893i \(-0.525099\pi\)
−0.0787691 + 0.996893i \(0.525099\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3.11593e14 0.436232
\(503\) 5.06588e14 0.701506 0.350753 0.936468i \(-0.385926\pi\)
0.350753 + 0.936468i \(0.385926\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.39206e14 0.320586
\(507\) 0 0
\(508\) −3.86720e14 −0.507161
\(509\) −8.57534e13 −0.111251 −0.0556254 0.998452i \(-0.517715\pi\)
−0.0556254 + 0.998452i \(0.517715\pi\)
\(510\) 0 0
\(511\) −2.45097e13 −0.0311188
\(512\) −3.64965e14 −0.458423
\(513\) 0 0
\(514\) −5.75069e14 −0.707005
\(515\) 0 0
\(516\) 0 0
\(517\) −1.43669e15 −1.71066
\(518\) −7.32235e13 −0.0862654
\(519\) 0 0
\(520\) 0 0
\(521\) −9.27575e14 −1.05862 −0.529312 0.848428i \(-0.677550\pi\)
−0.529312 + 0.848428i \(0.677550\pi\)
\(522\) 0 0
\(523\) 2.18187e13 0.0243820 0.0121910 0.999926i \(-0.496119\pi\)
0.0121910 + 0.999926i \(0.496119\pi\)
\(524\) 9.29610e14 1.02797
\(525\) 0 0
\(526\) 5.82569e14 0.630850
\(527\) 3.64931e14 0.391069
\(528\) 0 0
\(529\) −6.05238e14 −0.635214
\(530\) 0 0
\(531\) 0 0
\(532\) −2.62774e14 −0.267343
\(533\) −1.78013e14 −0.179247
\(534\) 0 0
\(535\) 0 0
\(536\) 1.30790e15 1.27694
\(537\) 0 0
\(538\) 6.20105e14 0.593147
\(539\) 9.07218e14 0.858961
\(540\) 0 0
\(541\) −1.69527e15 −1.57273 −0.786363 0.617765i \(-0.788038\pi\)
−0.786363 + 0.617765i \(0.788038\pi\)
\(542\) 9.04304e13 0.0830459
\(543\) 0 0
\(544\) 1.35844e15 1.22249
\(545\) 0 0
\(546\) 0 0
\(547\) −7.52145e14 −0.656706 −0.328353 0.944555i \(-0.606494\pi\)
−0.328353 + 0.944555i \(0.606494\pi\)
\(548\) 4.37477e14 0.378148
\(549\) 0 0
\(550\) 0 0
\(551\) −1.36900e15 −1.14834
\(552\) 0 0
\(553\) 6.38228e14 0.524793
\(554\) −3.94054e14 −0.320813
\(555\) 0 0
\(556\) −8.78480e14 −0.701166
\(557\) 1.87489e14 0.148174 0.0740870 0.997252i \(-0.476396\pi\)
0.0740870 + 0.997252i \(0.476396\pi\)
\(558\) 0 0
\(559\) 9.89417e12 0.00766681
\(560\) 0 0
\(561\) 0 0
\(562\) 5.04857e14 0.379856
\(563\) 2.44971e14 0.182524 0.0912618 0.995827i \(-0.470910\pi\)
0.0912618 + 0.995827i \(0.470910\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 4.01116e14 0.290255
\(567\) 0 0
\(568\) −8.27185e14 −0.587066
\(569\) −1.35243e15 −0.950596 −0.475298 0.879825i \(-0.657660\pi\)
−0.475298 + 0.879825i \(0.657660\pi\)
\(570\) 0 0
\(571\) 1.43223e15 0.987447 0.493723 0.869619i \(-0.335636\pi\)
0.493723 + 0.869619i \(0.335636\pi\)
\(572\) 4.54650e14 0.310455
\(573\) 0 0
\(574\) 1.23820e14 0.0829422
\(575\) 0 0
\(576\) 0 0
\(577\) 8.77659e14 0.571293 0.285647 0.958335i \(-0.407792\pi\)
0.285647 + 0.958335i \(0.407792\pi\)
\(578\) −3.22081e14 −0.207664
\(579\) 0 0
\(580\) 0 0
\(581\) −4.91187e14 −0.307807
\(582\) 0 0
\(583\) 8.53271e14 0.524698
\(584\) −1.23661e14 −0.0753290
\(585\) 0 0
\(586\) 5.74245e14 0.343289
\(587\) −2.43425e15 −1.44164 −0.720818 0.693124i \(-0.756234\pi\)
−0.720818 + 0.693124i \(0.756234\pi\)
\(588\) 0 0
\(589\) −5.63383e14 −0.327469
\(590\) 0 0
\(591\) 0 0
\(592\) 1.79869e14 0.101669
\(593\) −3.03318e14 −0.169863 −0.0849313 0.996387i \(-0.527067\pi\)
−0.0849313 + 0.996387i \(0.527067\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.64192e15 −0.894329
\(597\) 0 0
\(598\) −2.58502e14 −0.138232
\(599\) 1.70198e15 0.901795 0.450898 0.892576i \(-0.351104\pi\)
0.450898 + 0.892576i \(0.351104\pi\)
\(600\) 0 0
\(601\) 2.33922e15 1.21692 0.608458 0.793586i \(-0.291788\pi\)
0.608458 + 0.793586i \(0.291788\pi\)
\(602\) −6.88207e12 −0.00354763
\(603\) 0 0
\(604\) 1.21359e15 0.614282
\(605\) 0 0
\(606\) 0 0
\(607\) 2.49607e15 1.22947 0.614737 0.788732i \(-0.289262\pi\)
0.614737 + 0.788732i \(0.289262\pi\)
\(608\) −2.09717e15 −1.02368
\(609\) 0 0
\(610\) 0 0
\(611\) 1.55258e15 0.737612
\(612\) 0 0
\(613\) −2.47301e15 −1.15397 −0.576983 0.816756i \(-0.695770\pi\)
−0.576983 + 0.816756i \(0.695770\pi\)
\(614\) 3.67466e14 0.169938
\(615\) 0 0
\(616\) −7.56226e14 −0.343525
\(617\) 2.43368e13 0.0109571 0.00547854 0.999985i \(-0.498256\pi\)
0.00547854 + 0.999985i \(0.498256\pi\)
\(618\) 0 0
\(619\) 4.22545e15 1.86885 0.934425 0.356160i \(-0.115914\pi\)
0.934425 + 0.356160i \(0.115914\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 1.19700e15 0.515523
\(623\) 4.18481e14 0.178645
\(624\) 0 0
\(625\) 0 0
\(626\) −2.38754e15 −0.992640
\(627\) 0 0
\(628\) 1.93585e15 0.790850
\(629\) −1.25835e15 −0.509594
\(630\) 0 0
\(631\) −4.26326e15 −1.69660 −0.848302 0.529513i \(-0.822375\pi\)
−0.848302 + 0.529513i \(0.822375\pi\)
\(632\) 3.22011e15 1.27036
\(633\) 0 0
\(634\) −2.00086e15 −0.775758
\(635\) 0 0
\(636\) 0 0
\(637\) −9.80401e14 −0.370371
\(638\) −1.64755e15 −0.617055
\(639\) 0 0
\(640\) 0 0
\(641\) −1.00830e15 −0.368018 −0.184009 0.982925i \(-0.558908\pi\)
−0.184009 + 0.982925i \(0.558908\pi\)
\(642\) 0 0
\(643\) −3.03982e14 −0.109066 −0.0545328 0.998512i \(-0.517367\pi\)
−0.0545328 + 0.998512i \(0.517367\pi\)
\(644\) −4.59504e14 −0.163462
\(645\) 0 0
\(646\) 1.76705e15 0.617974
\(647\) 3.43583e15 1.19140 0.595700 0.803207i \(-0.296875\pi\)
0.595700 + 0.803207i \(0.296875\pi\)
\(648\) 0 0
\(649\) −2.77421e15 −0.945788
\(650\) 0 0
\(651\) 0 0
\(652\) −5.26730e14 −0.175076
\(653\) −1.18539e15 −0.390695 −0.195347 0.980734i \(-0.562583\pi\)
−0.195347 + 0.980734i \(0.562583\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.04157e14 −0.0977522
\(657\) 0 0
\(658\) −1.07993e15 −0.341312
\(659\) 2.26510e15 0.709934 0.354967 0.934879i \(-0.384492\pi\)
0.354967 + 0.934879i \(0.384492\pi\)
\(660\) 0 0
\(661\) −5.33012e15 −1.64297 −0.821484 0.570232i \(-0.806853\pi\)
−0.821484 + 0.570232i \(0.806853\pi\)
\(662\) 1.52602e15 0.466488
\(663\) 0 0
\(664\) −2.47823e15 −0.745104
\(665\) 0 0
\(666\) 0 0
\(667\) −2.39392e15 −0.702130
\(668\) −4.05512e15 −1.17959
\(669\) 0 0
\(670\) 0 0
\(671\) −3.71902e15 −1.05549
\(672\) 0 0
\(673\) −4.74120e15 −1.32375 −0.661874 0.749615i \(-0.730239\pi\)
−0.661874 + 0.749615i \(0.730239\pi\)
\(674\) 2.90403e15 0.804215
\(675\) 0 0
\(676\) 2.14673e15 0.584886
\(677\) −1.41307e15 −0.381880 −0.190940 0.981602i \(-0.561154\pi\)
−0.190940 + 0.981602i \(0.561154\pi\)
\(678\) 0 0
\(679\) −1.25603e15 −0.333976
\(680\) 0 0
\(681\) 0 0
\(682\) −6.78014e14 −0.175964
\(683\) −3.03116e15 −0.780359 −0.390180 0.920739i \(-0.627587\pi\)
−0.390180 + 0.920739i \(0.627587\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.47654e15 0.371075
\(687\) 0 0
\(688\) 1.69054e13 0.00418109
\(689\) −9.22102e14 −0.226242
\(690\) 0 0
\(691\) −2.74731e15 −0.663405 −0.331703 0.943384i \(-0.607623\pi\)
−0.331703 + 0.943384i \(0.607623\pi\)
\(692\) 1.39897e15 0.335139
\(693\) 0 0
\(694\) 3.73588e15 0.880878
\(695\) 0 0
\(696\) 0 0
\(697\) 2.12786e15 0.489962
\(698\) 6.15433e14 0.140597
\(699\) 0 0
\(700\) 0 0
\(701\) −5.72747e15 −1.27795 −0.638974 0.769228i \(-0.720641\pi\)
−0.638974 + 0.769228i \(0.720641\pi\)
\(702\) 0 0
\(703\) 1.94265e15 0.426718
\(704\) −1.44308e15 −0.314514
\(705\) 0 0
\(706\) −5.97836e14 −0.128279
\(707\) −1.36870e15 −0.291409
\(708\) 0 0
\(709\) 6.98326e14 0.146388 0.0731938 0.997318i \(-0.476681\pi\)
0.0731938 + 0.997318i \(0.476681\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.11140e15 0.432445
\(713\) −9.85170e14 −0.200225
\(714\) 0 0
\(715\) 0 0
\(716\) 2.47500e15 0.491534
\(717\) 0 0
\(718\) 3.78202e15 0.739672
\(719\) −9.70979e15 −1.88452 −0.942260 0.334882i \(-0.891304\pi\)
−0.942260 + 0.334882i \(0.891304\pi\)
\(720\) 0 0
\(721\) 3.78004e15 0.722525
\(722\) 6.77852e13 0.0128582
\(723\) 0 0
\(724\) 1.46725e15 0.274121
\(725\) 0 0
\(726\) 0 0
\(727\) −2.46469e15 −0.450114 −0.225057 0.974346i \(-0.572257\pi\)
−0.225057 + 0.974346i \(0.572257\pi\)
\(728\) 8.17230e14 0.148123
\(729\) 0 0
\(730\) 0 0
\(731\) −1.18269e14 −0.0209568
\(732\) 0 0
\(733\) −7.91285e15 −1.38121 −0.690607 0.723230i \(-0.742657\pi\)
−0.690607 + 0.723230i \(0.742657\pi\)
\(734\) −4.26963e15 −0.739711
\(735\) 0 0
\(736\) −3.66725e15 −0.625911
\(737\) −8.27677e15 −1.40213
\(738\) 0 0
\(739\) −8.40694e15 −1.40312 −0.701558 0.712613i \(-0.747512\pi\)
−0.701558 + 0.712613i \(0.747512\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 6.41385e14 0.104688
\(743\) 1.36287e15 0.220809 0.110404 0.993887i \(-0.464785\pi\)
0.110404 + 0.993887i \(0.464785\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.32388e15 −0.209790
\(747\) 0 0
\(748\) −5.43462e15 −0.848614
\(749\) 1.51100e15 0.234215
\(750\) 0 0
\(751\) 6.81722e15 1.04133 0.520664 0.853762i \(-0.325684\pi\)
0.520664 + 0.853762i \(0.325684\pi\)
\(752\) 2.65278e15 0.402256
\(753\) 0 0
\(754\) 1.78045e15 0.266065
\(755\) 0 0
\(756\) 0 0
\(757\) 6.67049e14 0.0975282 0.0487641 0.998810i \(-0.484472\pi\)
0.0487641 + 0.998810i \(0.484472\pi\)
\(758\) −3.51511e15 −0.510222
\(759\) 0 0
\(760\) 0 0
\(761\) 7.74408e15 1.09990 0.549951 0.835197i \(-0.314646\pi\)
0.549951 + 0.835197i \(0.314646\pi\)
\(762\) 0 0
\(763\) 1.23039e15 0.172250
\(764\) 4.06626e15 0.565173
\(765\) 0 0
\(766\) −5.55479e15 −0.761043
\(767\) 2.99800e15 0.407809
\(768\) 0 0
\(769\) 2.52411e15 0.338465 0.169232 0.985576i \(-0.445871\pi\)
0.169232 + 0.985576i \(0.445871\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.01119e15 1.05148
\(773\) −1.11453e16 −1.45246 −0.726229 0.687453i \(-0.758729\pi\)
−0.726229 + 0.687453i \(0.758729\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −6.33715e15 −0.808452
\(777\) 0 0
\(778\) −3.59692e15 −0.452421
\(779\) −3.28500e15 −0.410279
\(780\) 0 0
\(781\) 5.23465e15 0.644624
\(782\) 3.08998e15 0.377849
\(783\) 0 0
\(784\) −1.67514e15 −0.201981
\(785\) 0 0
\(786\) 0 0
\(787\) −1.32271e16 −1.56172 −0.780861 0.624705i \(-0.785219\pi\)
−0.780861 + 0.624705i \(0.785219\pi\)
\(788\) 4.23361e15 0.496382
\(789\) 0 0
\(790\) 0 0
\(791\) −1.42570e15 −0.163703
\(792\) 0 0
\(793\) 4.01902e15 0.455112
\(794\) 4.99466e15 0.561685
\(795\) 0 0
\(796\) −1.07219e15 −0.118919
\(797\) 2.30248e15 0.253615 0.126807 0.991927i \(-0.459527\pi\)
0.126807 + 0.991927i \(0.459527\pi\)
\(798\) 0 0
\(799\) −1.85587e16 −2.01623
\(800\) 0 0
\(801\) 0 0
\(802\) −3.20179e15 −0.340748
\(803\) 7.82560e14 0.0827146
\(804\) 0 0
\(805\) 0 0
\(806\) 7.32708e14 0.0758732
\(807\) 0 0
\(808\) −6.90565e15 −0.705410
\(809\) −5.60472e15 −0.568639 −0.284320 0.958730i \(-0.591768\pi\)
−0.284320 + 0.958730i \(0.591768\pi\)
\(810\) 0 0
\(811\) −5.08516e15 −0.508968 −0.254484 0.967077i \(-0.581906\pi\)
−0.254484 + 0.967077i \(0.581906\pi\)
\(812\) 3.16486e15 0.314627
\(813\) 0 0
\(814\) 2.33792e15 0.229296
\(815\) 0 0
\(816\) 0 0
\(817\) 1.82584e14 0.0175486
\(818\) 4.94802e15 0.472377
\(819\) 0 0
\(820\) 0 0
\(821\) −2.79111e14 −0.0261150 −0.0130575 0.999915i \(-0.504156\pi\)
−0.0130575 + 0.999915i \(0.504156\pi\)
\(822\) 0 0
\(823\) 1.35265e16 1.24878 0.624391 0.781112i \(-0.285347\pi\)
0.624391 + 0.781112i \(0.285347\pi\)
\(824\) 1.90718e16 1.74901
\(825\) 0 0
\(826\) −2.08531e15 −0.188704
\(827\) 2.72544e14 0.0244994 0.0122497 0.999925i \(-0.496101\pi\)
0.0122497 + 0.999925i \(0.496101\pi\)
\(828\) 0 0
\(829\) 1.80459e16 1.60077 0.800385 0.599486i \(-0.204628\pi\)
0.800385 + 0.599486i \(0.204628\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.55949e15 0.135614
\(833\) 1.17191e16 1.01239
\(834\) 0 0
\(835\) 0 0
\(836\) 8.38999e15 0.710603
\(837\) 0 0
\(838\) 1.76168e15 0.147260
\(839\) 7.96183e15 0.661184 0.330592 0.943774i \(-0.392752\pi\)
0.330592 + 0.943774i \(0.392752\pi\)
\(840\) 0 0
\(841\) 4.28775e15 0.351440
\(842\) −4.10669e15 −0.334407
\(843\) 0 0
\(844\) 9.99954e15 0.803705
\(845\) 0 0
\(846\) 0 0
\(847\) 8.34387e12 0.000657671 0
\(848\) −1.57552e15 −0.123381
\(849\) 0 0
\(850\) 0 0
\(851\) 3.39705e15 0.260909
\(852\) 0 0
\(853\) 1.49826e16 1.13598 0.567988 0.823037i \(-0.307722\pi\)
0.567988 + 0.823037i \(0.307722\pi\)
\(854\) −2.79550e15 −0.210592
\(855\) 0 0
\(856\) 7.62358e15 0.566961
\(857\) −2.22561e16 −1.64458 −0.822290 0.569068i \(-0.807304\pi\)
−0.822290 + 0.569068i \(0.807304\pi\)
\(858\) 0 0
\(859\) 5.44237e15 0.397032 0.198516 0.980098i \(-0.436388\pi\)
0.198516 + 0.980098i \(0.436388\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.72262e15 −0.123282
\(863\) 1.08110e16 0.768787 0.384393 0.923169i \(-0.374411\pi\)
0.384393 + 0.923169i \(0.374411\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.39715e15 0.167243
\(867\) 0 0
\(868\) 1.30243e15 0.0897217
\(869\) −2.03777e16 −1.39491
\(870\) 0 0
\(871\) 8.94444e15 0.604579
\(872\) 6.20782e15 0.416964
\(873\) 0 0
\(874\) −4.77033e15 −0.316399
\(875\) 0 0
\(876\) 0 0
\(877\) 2.81024e16 1.82914 0.914568 0.404431i \(-0.132530\pi\)
0.914568 + 0.404431i \(0.132530\pi\)
\(878\) 6.96749e14 0.0450668
\(879\) 0 0
\(880\) 0 0
\(881\) −4.22209e15 −0.268016 −0.134008 0.990980i \(-0.542785\pi\)
−0.134008 + 0.990980i \(0.542785\pi\)
\(882\) 0 0
\(883\) −5.16092e14 −0.0323551 −0.0161776 0.999869i \(-0.505150\pi\)
−0.0161776 + 0.999869i \(0.505150\pi\)
\(884\) 5.87302e15 0.365910
\(885\) 0 0
\(886\) −7.88088e15 −0.484941
\(887\) 5.71906e15 0.349740 0.174870 0.984592i \(-0.444050\pi\)
0.174870 + 0.984592i \(0.444050\pi\)
\(888\) 0 0
\(889\) 4.39894e15 0.265698
\(890\) 0 0
\(891\) 0 0
\(892\) 1.07969e16 0.640167
\(893\) 2.86510e16 1.68832
\(894\) 0 0
\(895\) 0 0
\(896\) 5.66067e15 0.327472
\(897\) 0 0
\(898\) −1.46968e16 −0.839854
\(899\) 6.78541e15 0.385388
\(900\) 0 0
\(901\) 1.10223e16 0.618421
\(902\) −3.95340e15 −0.220462
\(903\) 0 0
\(904\) −7.19321e15 −0.396275
\(905\) 0 0
\(906\) 0 0
\(907\) 8.43778e13 0.00456445 0.00228222 0.999997i \(-0.499274\pi\)
0.00228222 + 0.999997i \(0.499274\pi\)
\(908\) 2.00168e15 0.107628
\(909\) 0 0
\(910\) 0 0
\(911\) 1.10091e16 0.581298 0.290649 0.956830i \(-0.406129\pi\)
0.290649 + 0.956830i \(0.406129\pi\)
\(912\) 0 0
\(913\) 1.56829e16 0.818158
\(914\) 7.28359e15 0.377695
\(915\) 0 0
\(916\) 1.74055e16 0.891789
\(917\) −1.05743e16 −0.538544
\(918\) 0 0
\(919\) −4.86351e15 −0.244746 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.75034e16 −0.865171
\(923\) −5.65691e15 −0.277952
\(924\) 0 0
\(925\) 0 0
\(926\) 2.93252e15 0.141540
\(927\) 0 0
\(928\) 2.52584e16 1.20474
\(929\) −3.57534e15 −0.169524 −0.0847620 0.996401i \(-0.527013\pi\)
−0.0847620 + 0.996401i \(0.527013\pi\)
\(930\) 0 0
\(931\) −1.80921e16 −0.847744
\(932\) 2.58533e16 1.20428
\(933\) 0 0
\(934\) 1.48171e16 0.682113
\(935\) 0 0
\(936\) 0 0
\(937\) −3.86373e16 −1.74759 −0.873795 0.486295i \(-0.838348\pi\)
−0.873795 + 0.486295i \(0.838348\pi\)
\(938\) −6.22147e15 −0.279754
\(939\) 0 0
\(940\) 0 0
\(941\) 3.48997e16 1.54198 0.770991 0.636846i \(-0.219761\pi\)
0.770991 + 0.636846i \(0.219761\pi\)
\(942\) 0 0
\(943\) −5.74437e15 −0.250858
\(944\) 5.12245e15 0.222398
\(945\) 0 0
\(946\) 2.19735e14 0.00942969
\(947\) −2.85123e16 −1.21649 −0.608243 0.793751i \(-0.708125\pi\)
−0.608243 + 0.793751i \(0.708125\pi\)
\(948\) 0 0
\(949\) −8.45688e14 −0.0356653
\(950\) 0 0
\(951\) 0 0
\(952\) −9.76867e15 −0.404886
\(953\) 4.00334e16 1.64973 0.824863 0.565332i \(-0.191252\pi\)
0.824863 + 0.565332i \(0.191252\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.05095e16 −0.425659
\(957\) 0 0
\(958\) 2.52201e16 1.00980
\(959\) −4.97630e15 −0.198109
\(960\) 0 0
\(961\) −2.26161e16 −0.890100
\(962\) −2.52652e15 −0.0988688
\(963\) 0 0
\(964\) 3.40484e14 0.0131726
\(965\) 0 0
\(966\) 0 0
\(967\) −1.84953e16 −0.703422 −0.351711 0.936109i \(-0.614400\pi\)
−0.351711 + 0.936109i \(0.614400\pi\)
\(968\) 4.20981e13 0.00159202
\(969\) 0 0
\(970\) 0 0
\(971\) 2.14877e16 0.798884 0.399442 0.916759i \(-0.369204\pi\)
0.399442 + 0.916759i \(0.369204\pi\)
\(972\) 0 0
\(973\) 9.99271e15 0.367335
\(974\) −5.27784e15 −0.192922
\(975\) 0 0
\(976\) 6.86699e15 0.248195
\(977\) −8.73880e15 −0.314074 −0.157037 0.987593i \(-0.550194\pi\)
−0.157037 + 0.987593i \(0.550194\pi\)
\(978\) 0 0
\(979\) −1.33615e16 −0.474844
\(980\) 0 0
\(981\) 0 0
\(982\) −1.16127e16 −0.405808
\(983\) 1.18924e16 0.413263 0.206631 0.978419i \(-0.433750\pi\)
0.206631 + 0.978419i \(0.433750\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.12824e16 −0.727274
\(987\) 0 0
\(988\) −9.06679e15 −0.306401
\(989\) 3.19279e14 0.0107298
\(990\) 0 0
\(991\) 2.34409e16 0.779056 0.389528 0.921015i \(-0.372638\pi\)
0.389528 + 0.921015i \(0.372638\pi\)
\(992\) 1.03946e16 0.343552
\(993\) 0 0
\(994\) 3.93477e15 0.128616
\(995\) 0 0
\(996\) 0 0
\(997\) 2.14004e16 0.688016 0.344008 0.938967i \(-0.388215\pi\)
0.344008 + 0.938967i \(0.388215\pi\)
\(998\) 2.61307e15 0.0835473
\(999\) 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 225.12.a.b.1.1 1
3.2 odd 2 25.12.a.b.1.1 1
5.2 odd 4 225.12.b.d.199.1 2
5.3 odd 4 225.12.b.d.199.2 2
5.4 even 2 9.12.a.b.1.1 1
15.2 even 4 25.12.b.b.24.2 2
15.8 even 4 25.12.b.b.24.1 2
15.14 odd 2 1.12.a.a.1.1 1
20.19 odd 2 144.12.a.d.1.1 1
45.4 even 6 81.12.c.b.55.1 2
45.14 odd 6 81.12.c.d.55.1 2
45.29 odd 6 81.12.c.d.28.1 2
45.34 even 6 81.12.c.b.28.1 2
60.59 even 2 16.12.a.a.1.1 1
105.44 odd 6 49.12.c.b.18.1 2
105.59 even 6 49.12.c.c.30.1 2
105.74 odd 6 49.12.c.b.30.1 2
105.89 even 6 49.12.c.c.18.1 2
105.104 even 2 49.12.a.a.1.1 1
120.29 odd 2 64.12.a.b.1.1 1
120.59 even 2 64.12.a.f.1.1 1
165.164 even 2 121.12.a.b.1.1 1
195.194 odd 2 169.12.a.a.1.1 1
240.29 odd 4 256.12.b.e.129.2 2
240.59 even 4 256.12.b.c.129.2 2
240.149 odd 4 256.12.b.e.129.1 2
240.179 even 4 256.12.b.c.129.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.12.a.a.1.1 1 15.14 odd 2
9.12.a.b.1.1 1 5.4 even 2
16.12.a.a.1.1 1 60.59 even 2
25.12.a.b.1.1 1 3.2 odd 2
25.12.b.b.24.1 2 15.8 even 4
25.12.b.b.24.2 2 15.2 even 4
49.12.a.a.1.1 1 105.104 even 2
49.12.c.b.18.1 2 105.44 odd 6
49.12.c.b.30.1 2 105.74 odd 6
49.12.c.c.18.1 2 105.89 even 6
49.12.c.c.30.1 2 105.59 even 6
64.12.a.b.1.1 1 120.29 odd 2
64.12.a.f.1.1 1 120.59 even 2
81.12.c.b.28.1 2 45.34 even 6
81.12.c.b.55.1 2 45.4 even 6
81.12.c.d.28.1 2 45.29 odd 6
81.12.c.d.55.1 2 45.14 odd 6
121.12.a.b.1.1 1 165.164 even 2
144.12.a.d.1.1 1 20.19 odd 2
169.12.a.a.1.1 1 195.194 odd 2
225.12.a.b.1.1 1 1.1 even 1 trivial
225.12.b.d.199.1 2 5.2 odd 4
225.12.b.d.199.2 2 5.3 odd 4
256.12.b.c.129.1 2 240.179 even 4
256.12.b.c.129.2 2 240.59 even 4
256.12.b.e.129.1 2 240.149 odd 4
256.12.b.e.129.2 2 240.29 odd 4