Properties

Label 108.6.a.b.1.1
Level 108
Weight 6
Character 108.1
Self dual yes
Analytic conductor 17.321
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 108.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3214525398\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 108.1

$q$-expansion

\(f(q)\) \(=\) \(q-57.6281 q^{5} +156.884 q^{7} +O(q^{10})\) \(q-57.6281 q^{5} +156.884 q^{7} +218.025 q^{11} -587.537 q^{13} -2124.56 q^{17} +29.7687 q^{19} -3171.49 q^{23} +196.000 q^{25} -3642.13 q^{29} +6331.88 q^{31} -9040.95 q^{35} -8435.69 q^{37} -10146.4 q^{41} -6855.76 q^{43} -24447.8 q^{47} +7805.70 q^{49} +25483.2 q^{53} -12564.4 q^{55} +20696.2 q^{59} +45580.2 q^{61} +33858.7 q^{65} +19099.3 q^{67} -23055.0 q^{71} -89651.6 q^{73} +34204.7 q^{77} -34023.6 q^{79} +108497. q^{83} +122435. q^{85} +110055. q^{89} -92175.4 q^{91} -1715.51 q^{95} -28832.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{7} + O(q^{10}) \) \( 2q - 32q^{7} - 486q^{11} + 208q^{13} - 1944q^{17} - 632q^{19} - 6804q^{23} + 392q^{25} - 11664q^{29} + 3328q^{31} - 19926q^{35} - 9956q^{37} - 13608q^{41} + 4960q^{43} - 18468q^{47} + 26676q^{49} + 11664q^{53} - 53136q^{55} - 1944q^{59} + 8176q^{61} + 79704q^{65} + 90064q^{67} + 44712q^{71} - 121214q^{73} + 167184q^{77} + 28768q^{79} + 15066q^{83} + 132840q^{85} + 178848q^{89} - 242440q^{91} - 39852q^{95} + 88942q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −57.6281 −1.03088 −0.515442 0.856925i \(-0.672372\pi\)
−0.515442 + 0.856925i \(0.672372\pi\)
\(6\) 0 0
\(7\) 156.884 1.21014 0.605068 0.796173i \(-0.293146\pi\)
0.605068 + 0.796173i \(0.293146\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 218.025 0.543281 0.271640 0.962399i \(-0.412434\pi\)
0.271640 + 0.962399i \(0.412434\pi\)
\(12\) 0 0
\(13\) −587.537 −0.964222 −0.482111 0.876110i \(-0.660130\pi\)
−0.482111 + 0.876110i \(0.660130\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2124.56 −1.78298 −0.891491 0.453038i \(-0.850340\pi\)
−0.891491 + 0.453038i \(0.850340\pi\)
\(18\) 0 0
\(19\) 29.7687 0.0189180 0.00945902 0.999955i \(-0.496989\pi\)
0.00945902 + 0.999955i \(0.496989\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3171.49 −1.25010 −0.625048 0.780586i \(-0.714921\pi\)
−0.625048 + 0.780586i \(0.714921\pi\)
\(24\) 0 0
\(25\) 196.000 0.0627200
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3642.13 −0.804194 −0.402097 0.915597i \(-0.631718\pi\)
−0.402097 + 0.915597i \(0.631718\pi\)
\(30\) 0 0
\(31\) 6331.88 1.18339 0.591696 0.806162i \(-0.298459\pi\)
0.591696 + 0.806162i \(0.298459\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9040.95 −1.24751
\(36\) 0 0
\(37\) −8435.69 −1.01302 −0.506508 0.862235i \(-0.669064\pi\)
−0.506508 + 0.862235i \(0.669064\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10146.4 −0.942657 −0.471328 0.881958i \(-0.656225\pi\)
−0.471328 + 0.881958i \(0.656225\pi\)
\(42\) 0 0
\(43\) −6855.76 −0.565437 −0.282718 0.959203i \(-0.591236\pi\)
−0.282718 + 0.959203i \(0.591236\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −24447.8 −1.61434 −0.807171 0.590318i \(-0.799002\pi\)
−0.807171 + 0.590318i \(0.799002\pi\)
\(48\) 0 0
\(49\) 7805.70 0.464432
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 25483.2 1.24613 0.623066 0.782169i \(-0.285887\pi\)
0.623066 + 0.782169i \(0.285887\pi\)
\(54\) 0 0
\(55\) −12564.4 −0.560059
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 20696.2 0.774034 0.387017 0.922073i \(-0.373505\pi\)
0.387017 + 0.922073i \(0.373505\pi\)
\(60\) 0 0
\(61\) 45580.2 1.56838 0.784191 0.620519i \(-0.213078\pi\)
0.784191 + 0.620519i \(0.213078\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 33858.7 0.994000
\(66\) 0 0
\(67\) 19099.3 0.519794 0.259897 0.965636i \(-0.416311\pi\)
0.259897 + 0.965636i \(0.416311\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −23055.0 −0.542773 −0.271387 0.962470i \(-0.587482\pi\)
−0.271387 + 0.962470i \(0.587482\pi\)
\(72\) 0 0
\(73\) −89651.6 −1.96902 −0.984511 0.175320i \(-0.943904\pi\)
−0.984511 + 0.175320i \(0.943904\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 34204.7 0.657444
\(78\) 0 0
\(79\) −34023.6 −0.613356 −0.306678 0.951813i \(-0.599218\pi\)
−0.306678 + 0.951813i \(0.599218\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 108497. 1.72872 0.864359 0.502875i \(-0.167724\pi\)
0.864359 + 0.502875i \(0.167724\pi\)
\(84\) 0 0
\(85\) 122435. 1.83805
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 110055. 1.47277 0.736384 0.676564i \(-0.236532\pi\)
0.736384 + 0.676564i \(0.236532\pi\)
\(90\) 0 0
\(91\) −92175.4 −1.16684
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1715.51 −0.0195023
\(96\) 0 0
\(97\) −28832.0 −0.311132 −0.155566 0.987825i \(-0.549720\pi\)
−0.155566 + 0.987825i \(0.549720\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 40290.3 0.393004 0.196502 0.980503i \(-0.437042\pi\)
0.196502 + 0.980503i \(0.437042\pi\)
\(102\) 0 0
\(103\) 204547. 1.89977 0.949885 0.312600i \(-0.101200\pi\)
0.949885 + 0.312600i \(0.101200\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −164042. −1.38515 −0.692573 0.721348i \(-0.743523\pi\)
−0.692573 + 0.721348i \(0.743523\pi\)
\(108\) 0 0
\(109\) −131554. −1.06057 −0.530285 0.847819i \(-0.677915\pi\)
−0.530285 + 0.847819i \(0.677915\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 94322.0 0.694891 0.347446 0.937700i \(-0.387049\pi\)
0.347446 + 0.937700i \(0.387049\pi\)
\(114\) 0 0
\(115\) 182767. 1.28870
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −333311. −2.15765
\(120\) 0 0
\(121\) −113516. −0.704846
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 168793. 0.966226
\(126\) 0 0
\(127\) −303144. −1.66778 −0.833891 0.551929i \(-0.813892\pi\)
−0.833891 + 0.551929i \(0.813892\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −316573. −1.61174 −0.805871 0.592091i \(-0.798303\pi\)
−0.805871 + 0.592091i \(0.798303\pi\)
\(132\) 0 0
\(133\) 4670.24 0.0228934
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 982.957 0.00447438 0.00223719 0.999997i \(-0.499288\pi\)
0.00223719 + 0.999997i \(0.499288\pi\)
\(138\) 0 0
\(139\) 77533.8 0.340372 0.170186 0.985412i \(-0.445563\pi\)
0.170186 + 0.985412i \(0.445563\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −128098. −0.523844
\(144\) 0 0
\(145\) 209889. 0.829030
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −119209. −0.439890 −0.219945 0.975512i \(-0.570588\pi\)
−0.219945 + 0.975512i \(0.570588\pi\)
\(150\) 0 0
\(151\) −59568.9 −0.212607 −0.106303 0.994334i \(-0.533901\pi\)
−0.106303 + 0.994334i \(0.533901\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −364894. −1.21994
\(156\) 0 0
\(157\) 297561. 0.963446 0.481723 0.876324i \(-0.340011\pi\)
0.481723 + 0.876324i \(0.340011\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −497557. −1.51279
\(162\) 0 0
\(163\) −98681.9 −0.290917 −0.145458 0.989364i \(-0.546466\pi\)
−0.145458 + 0.989364i \(0.546466\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18553.0 0.0514782 0.0257391 0.999669i \(-0.491806\pi\)
0.0257391 + 0.999669i \(0.491806\pi\)
\(168\) 0 0
\(169\) −26092.8 −0.0702755
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 496652. 1.26164 0.630822 0.775927i \(-0.282718\pi\)
0.630822 + 0.775927i \(0.282718\pi\)
\(174\) 0 0
\(175\) 30749.3 0.0758998
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 150072. 0.350080 0.175040 0.984561i \(-0.443995\pi\)
0.175040 + 0.984561i \(0.443995\pi\)
\(180\) 0 0
\(181\) −73515.6 −0.166795 −0.0833976 0.996516i \(-0.526577\pi\)
−0.0833976 + 0.996516i \(0.526577\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 486133. 1.04430
\(186\) 0 0
\(187\) −463208. −0.968661
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 599319. 1.18871 0.594353 0.804204i \(-0.297408\pi\)
0.594353 + 0.804204i \(0.297408\pi\)
\(192\) 0 0
\(193\) −723746. −1.39860 −0.699299 0.714829i \(-0.746504\pi\)
−0.699299 + 0.714829i \(0.746504\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 416589. 0.764790 0.382395 0.923999i \(-0.375099\pi\)
0.382395 + 0.923999i \(0.375099\pi\)
\(198\) 0 0
\(199\) 688177. 1.23188 0.615938 0.787794i \(-0.288777\pi\)
0.615938 + 0.787794i \(0.288777\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −571393. −0.973184
\(204\) 0 0
\(205\) 584720. 0.971769
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6490.32 0.0102778
\(210\) 0 0
\(211\) −321874. −0.497714 −0.248857 0.968540i \(-0.580055\pi\)
−0.248857 + 0.968540i \(0.580055\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 395084. 0.582899
\(216\) 0 0
\(217\) 993373. 1.43207
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.24826e6 1.71919
\(222\) 0 0
\(223\) −354474. −0.477334 −0.238667 0.971101i \(-0.576710\pi\)
−0.238667 + 0.971101i \(0.576710\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 226523. 0.291774 0.145887 0.989301i \(-0.453396\pi\)
0.145887 + 0.989301i \(0.453396\pi\)
\(228\) 0 0
\(229\) −1.45431e6 −1.83260 −0.916299 0.400494i \(-0.868839\pi\)
−0.916299 + 0.400494i \(0.868839\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 423642. 0.511221 0.255611 0.966780i \(-0.417723\pi\)
0.255611 + 0.966780i \(0.417723\pi\)
\(234\) 0 0
\(235\) 1.40888e6 1.66420
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 747052. 0.845972 0.422986 0.906136i \(-0.360982\pi\)
0.422986 + 0.906136i \(0.360982\pi\)
\(240\) 0 0
\(241\) 611874. 0.678609 0.339305 0.940677i \(-0.389808\pi\)
0.339305 + 0.940677i \(0.389808\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −449828. −0.478775
\(246\) 0 0
\(247\) −17490.2 −0.0182412
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 985152. 0.987004 0.493502 0.869745i \(-0.335717\pi\)
0.493502 + 0.869745i \(0.335717\pi\)
\(252\) 0 0
\(253\) −691463. −0.679153
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.38778e6 −1.31065 −0.655327 0.755345i \(-0.727469\pi\)
−0.655327 + 0.755345i \(0.727469\pi\)
\(258\) 0 0
\(259\) −1.32343e6 −1.22589
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −877536. −0.782304 −0.391152 0.920326i \(-0.627923\pi\)
−0.391152 + 0.920326i \(0.627923\pi\)
\(264\) 0 0
\(265\) −1.46855e6 −1.28462
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −911089. −0.767680 −0.383840 0.923400i \(-0.625399\pi\)
−0.383840 + 0.923400i \(0.625399\pi\)
\(270\) 0 0
\(271\) 620129. 0.512931 0.256465 0.966553i \(-0.417442\pi\)
0.256465 + 0.966553i \(0.417442\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 42732.9 0.0340746
\(276\) 0 0
\(277\) −236728. −0.185374 −0.0926871 0.995695i \(-0.529546\pi\)
−0.0926871 + 0.995695i \(0.529546\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.07691e6 −0.813609 −0.406804 0.913515i \(-0.633357\pi\)
−0.406804 + 0.913515i \(0.633357\pi\)
\(282\) 0 0
\(283\) 2.05317e6 1.52391 0.761953 0.647633i \(-0.224241\pi\)
0.761953 + 0.647633i \(0.224241\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.59182e6 −1.14074
\(288\) 0 0
\(289\) 3.09391e6 2.17903
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.12866e6 −1.44856 −0.724280 0.689506i \(-0.757828\pi\)
−0.724280 + 0.689506i \(0.757828\pi\)
\(294\) 0 0
\(295\) −1.19268e6 −0.797939
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.86337e6 1.20537
\(300\) 0 0
\(301\) −1.07556e6 −0.684256
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.62670e6 −1.61682
\(306\) 0 0
\(307\) −1.54154e6 −0.933489 −0.466744 0.884392i \(-0.654573\pi\)
−0.466744 + 0.884392i \(0.654573\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.73639e6 1.01800 0.508999 0.860767i \(-0.330016\pi\)
0.508999 + 0.860767i \(0.330016\pi\)
\(312\) 0 0
\(313\) 2.27414e6 1.31207 0.656034 0.754732i \(-0.272233\pi\)
0.656034 + 0.754732i \(0.272233\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 316836. 0.177087 0.0885435 0.996072i \(-0.471779\pi\)
0.0885435 + 0.996072i \(0.471779\pi\)
\(318\) 0 0
\(319\) −794076. −0.436903
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −63245.5 −0.0337305
\(324\) 0 0
\(325\) −115157. −0.0604760
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.83548e6 −1.95357
\(330\) 0 0
\(331\) −448791. −0.225151 −0.112576 0.993643i \(-0.535910\pi\)
−0.112576 + 0.993643i \(0.535910\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.10066e6 −0.535847
\(336\) 0 0
\(337\) −1.85869e6 −0.891524 −0.445762 0.895151i \(-0.647067\pi\)
−0.445762 + 0.895151i \(0.647067\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.38051e6 0.642914
\(342\) 0 0
\(343\) −1.41216e6 −0.648111
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 223643. 0.0997084 0.0498542 0.998757i \(-0.484124\pi\)
0.0498542 + 0.998757i \(0.484124\pi\)
\(348\) 0 0
\(349\) 1.33981e6 0.588815 0.294407 0.955680i \(-0.404878\pi\)
0.294407 + 0.955680i \(0.404878\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.39683e6 −1.45090 −0.725449 0.688276i \(-0.758368\pi\)
−0.725449 + 0.688276i \(0.758368\pi\)
\(354\) 0 0
\(355\) 1.32861e6 0.559536
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 761965. 0.312032 0.156016 0.987755i \(-0.450135\pi\)
0.156016 + 0.987755i \(0.450135\pi\)
\(360\) 0 0
\(361\) −2.47521e6 −0.999642
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.16645e6 2.02983
\(366\) 0 0
\(367\) 1.39712e6 0.541461 0.270730 0.962655i \(-0.412735\pi\)
0.270730 + 0.962655i \(0.412735\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.99791e6 1.50799
\(372\) 0 0
\(373\) 3.25243e6 1.21042 0.605209 0.796067i \(-0.293090\pi\)
0.605209 + 0.796067i \(0.293090\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.13989e6 0.775421
\(378\) 0 0
\(379\) −700188. −0.250390 −0.125195 0.992132i \(-0.539956\pi\)
−0.125195 + 0.992132i \(0.539956\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.30721e6 0.803693 0.401847 0.915707i \(-0.368369\pi\)
0.401847 + 0.915707i \(0.368369\pi\)
\(384\) 0 0
\(385\) −1.97115e6 −0.677748
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.01126e6 0.673899 0.336950 0.941523i \(-0.390605\pi\)
0.336950 + 0.941523i \(0.390605\pi\)
\(390\) 0 0
\(391\) 6.73802e6 2.22890
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.96072e6 0.632299
\(396\) 0 0
\(397\) −165619. −0.0527392 −0.0263696 0.999652i \(-0.508395\pi\)
−0.0263696 + 0.999652i \(0.508395\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.09454e6 −0.339915 −0.169958 0.985451i \(-0.554363\pi\)
−0.169958 + 0.985451i \(0.554363\pi\)
\(402\) 0 0
\(403\) −3.72021e6 −1.14105
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.83919e6 −0.550352
\(408\) 0 0
\(409\) −1.07293e6 −0.317149 −0.158574 0.987347i \(-0.550690\pi\)
−0.158574 + 0.987347i \(0.550690\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3.24691e6 0.936687
\(414\) 0 0
\(415\) −6.25250e6 −1.78211
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −852181. −0.237135 −0.118568 0.992946i \(-0.537830\pi\)
−0.118568 + 0.992946i \(0.537830\pi\)
\(420\) 0 0
\(421\) 3.18784e6 0.876579 0.438290 0.898834i \(-0.355584\pi\)
0.438290 + 0.898834i \(0.355584\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −416414. −0.111829
\(426\) 0 0
\(427\) 7.15083e6 1.89796
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.54409e6 1.17829 0.589147 0.808026i \(-0.299464\pi\)
0.589147 + 0.808026i \(0.299464\pi\)
\(432\) 0 0
\(433\) −6.16131e6 −1.57926 −0.789629 0.613585i \(-0.789727\pi\)
−0.789629 + 0.613585i \(0.789727\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −94411.1 −0.0236494
\(438\) 0 0
\(439\) 902834. 0.223587 0.111793 0.993731i \(-0.464340\pi\)
0.111793 + 0.993731i \(0.464340\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.68712e6 0.892642 0.446321 0.894873i \(-0.352734\pi\)
0.446321 + 0.894873i \(0.352734\pi\)
\(444\) 0 0
\(445\) −6.34225e6 −1.51825
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −7.71368e6 −1.80570 −0.902851 0.429954i \(-0.858530\pi\)
−0.902851 + 0.429954i \(0.858530\pi\)
\(450\) 0 0
\(451\) −2.21218e6 −0.512128
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.31190e6 1.20288
\(456\) 0 0
\(457\) 3.79267e6 0.849484 0.424742 0.905315i \(-0.360365\pi\)
0.424742 + 0.905315i \(0.360365\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.77898e6 1.04733 0.523664 0.851925i \(-0.324565\pi\)
0.523664 + 0.851925i \(0.324565\pi\)
\(462\) 0 0
\(463\) 2.55903e6 0.554782 0.277391 0.960757i \(-0.410530\pi\)
0.277391 + 0.960757i \(0.410530\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8.22563e6 −1.74533 −0.872664 0.488321i \(-0.837610\pi\)
−0.872664 + 0.488321i \(0.837610\pi\)
\(468\) 0 0
\(469\) 2.99639e6 0.629022
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.49473e6 −0.307191
\(474\) 0 0
\(475\) 5834.67 0.00118654
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.54369e6 −0.506554 −0.253277 0.967394i \(-0.581508\pi\)
−0.253277 + 0.967394i \(0.581508\pi\)
\(480\) 0 0
\(481\) 4.95628e6 0.976772
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.66153e6 0.320741
\(486\) 0 0
\(487\) 1.25686e6 0.240139 0.120070 0.992765i \(-0.461688\pi\)
0.120070 + 0.992765i \(0.461688\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −9.89966e6 −1.85318 −0.926588 0.376078i \(-0.877273\pi\)
−0.926588 + 0.376078i \(0.877273\pi\)
\(492\) 0 0
\(493\) 7.73794e6 1.43386
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.61696e6 −0.656830
\(498\) 0 0
\(499\) −3.04151e6 −0.546812 −0.273406 0.961899i \(-0.588150\pi\)
−0.273406 + 0.961899i \(0.588150\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2.75985e6 −0.486368 −0.243184 0.969980i \(-0.578192\pi\)
−0.243184 + 0.969980i \(0.578192\pi\)
\(504\) 0 0
\(505\) −2.32186e6 −0.405142
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.44508e6 0.247227 0.123614 0.992330i \(-0.460552\pi\)
0.123614 + 0.992330i \(0.460552\pi\)
\(510\) 0 0
\(511\) −1.40649e7 −2.38279
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.17877e7 −1.95844
\(516\) 0 0
\(517\) −5.33024e6 −0.877041
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.28492e6 −0.207388 −0.103694 0.994609i \(-0.533066\pi\)
−0.103694 + 0.994609i \(0.533066\pi\)
\(522\) 0 0
\(523\) −2.38275e6 −0.380912 −0.190456 0.981696i \(-0.560997\pi\)
−0.190456 + 0.981696i \(0.560997\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.34525e7 −2.10997
\(528\) 0 0
\(529\) 3.62199e6 0.562740
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.96141e6 0.908931
\(534\) 0 0
\(535\) 9.45342e6 1.42792
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.70184e6 0.252317
\(540\) 0 0
\(541\) 2.22158e6 0.326339 0.163170 0.986598i \(-0.447828\pi\)
0.163170 + 0.986598i \(0.447828\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.58124e6 1.09332
\(546\) 0 0
\(547\) −1.81689e6 −0.259634 −0.129817 0.991538i \(-0.541439\pi\)
−0.129817 + 0.991538i \(0.541439\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −108422. −0.0152138
\(552\) 0 0
\(553\) −5.33777e6 −0.742245
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.53109e6 0.482248 0.241124 0.970494i \(-0.422484\pi\)
0.241124 + 0.970494i \(0.422484\pi\)
\(558\) 0 0
\(559\) 4.02801e6 0.545207
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 327456. 0.0435393 0.0217697 0.999763i \(-0.493070\pi\)
0.0217697 + 0.999763i \(0.493070\pi\)
\(564\) 0 0
\(565\) −5.43560e6 −0.716351
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.06773e6 −1.04465 −0.522325 0.852747i \(-0.674935\pi\)
−0.522325 + 0.852747i \(0.674935\pi\)
\(570\) 0 0
\(571\) −211349. −0.0271275 −0.0135637 0.999908i \(-0.504318\pi\)
−0.0135637 + 0.999908i \(0.504318\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −621612. −0.0784060
\(576\) 0 0
\(577\) −1.60363e6 −0.200523 −0.100261 0.994961i \(-0.531968\pi\)
−0.100261 + 0.994961i \(0.531968\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.70216e7 2.09199
\(582\) 0 0
\(583\) 5.55597e6 0.677000
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9.92480e6 −1.18885 −0.594424 0.804151i \(-0.702620\pi\)
−0.594424 + 0.804151i \(0.702620\pi\)
\(588\) 0 0
\(589\) 188492. 0.0223874
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −5.74316e6 −0.670678 −0.335339 0.942098i \(-0.608851\pi\)
−0.335339 + 0.942098i \(0.608851\pi\)
\(594\) 0 0
\(595\) 1.92081e7 2.22429
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.27574e6 −0.600781 −0.300391 0.953816i \(-0.597117\pi\)
−0.300391 + 0.953816i \(0.597117\pi\)
\(600\) 0 0
\(601\) 4.14952e6 0.468610 0.234305 0.972163i \(-0.424719\pi\)
0.234305 + 0.972163i \(0.424719\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 6.54172e6 0.726614
\(606\) 0 0
\(607\) −1.58501e7 −1.74606 −0.873031 0.487665i \(-0.837849\pi\)
−0.873031 + 0.487665i \(0.837849\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.43640e7 1.55658
\(612\) 0 0
\(613\) 1.48963e7 1.60113 0.800565 0.599246i \(-0.204533\pi\)
0.800565 + 0.599246i \(0.204533\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.44790e7 −1.53118 −0.765591 0.643328i \(-0.777553\pi\)
−0.765591 + 0.643328i \(0.777553\pi\)
\(618\) 0 0
\(619\) −1.05892e7 −1.11080 −0.555402 0.831582i \(-0.687435\pi\)
−0.555402 + 0.831582i \(0.687435\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.72659e7 1.78225
\(624\) 0 0
\(625\) −1.03397e7 −1.05879
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.79221e7 1.80619
\(630\) 0 0
\(631\) −4.93118e6 −0.493035 −0.246517 0.969138i \(-0.579286\pi\)
−0.246517 + 0.969138i \(0.579286\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.74696e7 1.71929
\(636\) 0 0
\(637\) −4.58614e6 −0.447815
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.90060e7 1.82703 0.913514 0.406807i \(-0.133358\pi\)
0.913514 + 0.406807i \(0.133358\pi\)
\(642\) 0 0
\(643\) −6.69738e6 −0.638818 −0.319409 0.947617i \(-0.603484\pi\)
−0.319409 + 0.947617i \(0.603484\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.62821e7 −1.52915 −0.764574 0.644536i \(-0.777051\pi\)
−0.764574 + 0.644536i \(0.777051\pi\)
\(648\) 0 0
\(649\) 4.51228e6 0.420518
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.52086e6 0.506668 0.253334 0.967379i \(-0.418473\pi\)
0.253334 + 0.967379i \(0.418473\pi\)
\(654\) 0 0
\(655\) 1.82435e7 1.66152
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −139176. −0.0124839 −0.00624197 0.999981i \(-0.501987\pi\)
−0.00624197 + 0.999981i \(0.501987\pi\)
\(660\) 0 0
\(661\) −1.94550e7 −1.73192 −0.865960 0.500113i \(-0.833292\pi\)
−0.865960 + 0.500113i \(0.833292\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −269137. −0.0236004
\(666\) 0 0
\(667\) 1.15510e7 1.00532
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9.93763e6 0.852072
\(672\) 0 0
\(673\) −9.15012e6 −0.778734 −0.389367 0.921083i \(-0.627306\pi\)
−0.389367 + 0.921083i \(0.627306\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.92716e6 0.832441 0.416220 0.909264i \(-0.363354\pi\)
0.416220 + 0.909264i \(0.363354\pi\)
\(678\) 0 0
\(679\) −4.52328e6 −0.376513
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.27969e7 −1.04967 −0.524835 0.851204i \(-0.675873\pi\)
−0.524835 + 0.851204i \(0.675873\pi\)
\(684\) 0 0
\(685\) −56646.0 −0.00461257
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.49723e7 −1.20155
\(690\) 0 0
\(691\) 8.05692e6 0.641909 0.320955 0.947095i \(-0.395996\pi\)
0.320955 + 0.947095i \(0.395996\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.46813e6 −0.350884
\(696\) 0 0
\(697\) 2.15567e7 1.68074
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.05658e7 0.812100 0.406050 0.913851i \(-0.366906\pi\)
0.406050 + 0.913851i \(0.366906\pi\)
\(702\) 0 0
\(703\) −251120. −0.0191643
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.32092e6 0.475589
\(708\) 0 0
\(709\) −1.51822e7 −1.13428 −0.567138 0.823623i \(-0.691949\pi\)
−0.567138 + 0.823623i \(0.691949\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.00815e7 −1.47935
\(714\) 0 0
\(715\) 7.38204e6 0.540021
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2.93525e6 −0.211750 −0.105875 0.994379i \(-0.533764\pi\)
−0.105875 + 0.994379i \(0.533764\pi\)
\(720\) 0 0
\(721\) 3.20903e7 2.29898
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −713858. −0.0504390
\(726\) 0 0
\(727\) −9.33363e6 −0.654960 −0.327480 0.944858i \(-0.606199\pi\)
−0.327480 + 0.944858i \(0.606199\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.45655e7 1.00816
\(732\) 0 0
\(733\) −4.73250e6 −0.325335 −0.162667 0.986681i \(-0.552010\pi\)
−0.162667 + 0.986681i \(0.552010\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.16413e6 0.282394
\(738\) 0 0
\(739\) −2.42162e7 −1.63115 −0.815575 0.578651i \(-0.803579\pi\)
−0.815575 + 0.578651i \(0.803579\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −2.47448e7 −1.64441 −0.822207 0.569189i \(-0.807257\pi\)
−0.822207 + 0.569189i \(0.807257\pi\)
\(744\) 0 0
\(745\) 6.86981e6 0.453476
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2.57356e7 −1.67622
\(750\) 0 0
\(751\) −5.57065e6 −0.360418 −0.180209 0.983628i \(-0.557677\pi\)
−0.180209 + 0.983628i \(0.557677\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.43284e6 0.219173
\(756\) 0 0
\(757\) −2.72203e6 −0.172645 −0.0863224 0.996267i \(-0.527511\pi\)
−0.0863224 + 0.996267i \(0.527511\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.50617e7 0.942787 0.471393 0.881923i \(-0.343751\pi\)
0.471393 + 0.881923i \(0.343751\pi\)
\(762\) 0 0
\(763\) −2.06388e7 −1.28344
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.21598e7 −0.746341
\(768\) 0 0
\(769\) 9.13893e6 0.557288 0.278644 0.960395i \(-0.410115\pi\)
0.278644 + 0.960395i \(0.410115\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.35738e7 −0.817060 −0.408530 0.912745i \(-0.633959\pi\)
−0.408530 + 0.912745i \(0.633959\pi\)
\(774\) 0 0
\(775\) 1.24105e6 0.0742223
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −302046. −0.0178332
\(780\) 0 0
\(781\) −5.02656e6 −0.294878
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.71479e7 −0.993200
\(786\) 0 0
\(787\) −2.04292e7 −1.17575 −0.587875 0.808952i \(-0.700035\pi\)
−0.587875 + 0.808952i \(0.700035\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.47976e7 0.840913
\(792\) 0 0
\(793\) −2.67801e7 −1.51227
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.03966e7 −1.69504 −0.847518 0.530766i \(-0.821904\pi\)
−0.847518 + 0.530766i \(0.821904\pi\)
\(798\) 0 0
\(799\) 5.19409e7 2.87834
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.95463e7 −1.06973
\(804\) 0 0
\(805\) 2.86733e7 1.55951
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −3.50428e7 −1.88247 −0.941234 0.337755i \(-0.890333\pi\)
−0.941234 + 0.337755i \(0.890333\pi\)
\(810\) 0 0
\(811\) 1.55069e7 0.827893 0.413946 0.910301i \(-0.364150\pi\)
0.413946 + 0.910301i \(0.364150\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.68685e6 0.299901
\(816\) 0 0
\(817\) −204087. −0.0106970
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.51617e7 0.785038 0.392519 0.919744i \(-0.371604\pi\)
0.392519 + 0.919744i \(0.371604\pi\)
\(822\) 0 0
\(823\) −1.26532e7 −0.651180 −0.325590 0.945511i \(-0.605563\pi\)
−0.325590 + 0.945511i \(0.605563\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.21240e7 −0.616427 −0.308213 0.951317i \(-0.599731\pi\)
−0.308213 + 0.951317i \(0.599731\pi\)
\(828\) 0 0
\(829\) −1.79003e7 −0.904634 −0.452317 0.891857i \(-0.649402\pi\)
−0.452317 + 0.891857i \(0.649402\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.65837e7 −0.828073
\(834\) 0 0
\(835\) −1.06918e6 −0.0530680
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.34365e7 1.14945 0.574723 0.818348i \(-0.305110\pi\)
0.574723 + 0.818348i \(0.305110\pi\)
\(840\) 0 0
\(841\) −7.24603e6 −0.353273
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.50368e6 0.0724458
\(846\) 0 0
\(847\) −1.78089e7 −0.852960
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.67537e7 1.26637
\(852\) 0 0
\(853\) 2.89092e7 1.36039 0.680196 0.733030i \(-0.261895\pi\)
0.680196 + 0.733030i \(0.261895\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.06961e7 −0.497476 −0.248738 0.968571i \(-0.580016\pi\)
−0.248738 + 0.968571i \(0.580016\pi\)
\(858\) 0 0
\(859\) 3.69703e7 1.70950 0.854752 0.519037i \(-0.173709\pi\)
0.854752 + 0.519037i \(0.173709\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.83720e7 1.75383 0.876914 0.480646i \(-0.159598\pi\)
0.876914 + 0.480646i \(0.159598\pi\)
\(864\) 0 0
\(865\) −2.86211e7 −1.30061
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −7.41800e6 −0.333225
\(870\) 0 0
\(871\) −1.12216e7 −0.501197
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.64809e7 1.16927
\(876\) 0 0
\(877\) 3.66208e7 1.60779 0.803895 0.594771i \(-0.202757\pi\)
0.803895 + 0.594771i \(0.202757\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.70223e7 0.738889 0.369445 0.929253i \(-0.379548\pi\)
0.369445 + 0.929253i \(0.379548\pi\)
\(882\) 0 0
\(883\) 3.54409e7 1.52969 0.764844 0.644215i \(-0.222816\pi\)
0.764844 + 0.644215i \(0.222816\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.20072e7 0.512428 0.256214 0.966620i \(-0.417525\pi\)
0.256214 + 0.966620i \(0.417525\pi\)
\(888\) 0 0
\(889\) −4.75585e7 −2.01824
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −727780. −0.0305402
\(894\) 0 0
\(895\) −8.64837e6 −0.360892
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.30615e7 −0.951676
\(900\) 0 0
\(901\) −5.41406e7 −2.22183
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.23657e6 0.171946
\(906\) 0 0
\(907\) −8.42904e6 −0.340220 −0.170110 0.985425i \(-0.554412\pi\)
−0.170110 + 0.985425i \(0.554412\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 2.48847e7 0.993430 0.496715 0.867914i \(-0.334539\pi\)
0.496715 + 0.867914i \(0.334539\pi\)
\(912\) 0 0
\(913\) 2.36552e7 0.939180
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.96653e7 −1.95043
\(918\) 0 0
\(919\) −3.68789e7 −1.44042 −0.720210 0.693757i \(-0.755954\pi\)
−0.720210 + 0.693757i \(0.755954\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.35456e7 0.523354
\(924\) 0 0
\(925\) −1.65339e6 −0.0635363
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.60910e7 0.991863 0.495932 0.868362i \(-0.334827\pi\)
0.495932 + 0.868362i \(0.334827\pi\)
\(930\) 0 0
\(931\) 232366. 0.00878613
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.66938e7 0.998576
\(936\) 0 0
\(937\) 7.84259e6 0.291817 0.145908 0.989298i \(-0.453390\pi\)
0.145908 + 0.989298i \(0.453390\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 3.36164e7 1.23759 0.618796 0.785552i \(-0.287621\pi\)
0.618796 + 0.785552i \(0.287621\pi\)
\(942\) 0 0
\(943\) 3.21793e7 1.17841
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.57753e7 −0.571615 −0.285808 0.958287i \(-0.592262\pi\)
−0.285808 + 0.958287i \(0.592262\pi\)
\(948\) 0 0
\(949\) 5.26737e7 1.89858
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2.32737e7 0.830104 0.415052 0.909798i \(-0.363763\pi\)
0.415052 + 0.909798i \(0.363763\pi\)
\(954\) 0 0
\(955\) −3.45376e7 −1.22542
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 154211. 0.00541462
\(960\) 0 0
\(961\) 1.14635e7 0.400414
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.17081e7 1.44179
\(966\) 0 0
\(967\) 1.95430e7 0.672085 0.336043 0.941847i \(-0.390911\pi\)
0.336043 + 0.941847i \(0.390911\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5.59842e7 1.90554 0.952768 0.303700i \(-0.0982220\pi\)
0.952768 + 0.303700i \(0.0982220\pi\)
\(972\) 0 0
\(973\) 1.21638e7 0.411897
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.20225e6 −0.140846 −0.0704231 0.997517i \(-0.522435\pi\)
−0.0704231 + 0.997517i \(0.522435\pi\)
\(978\) 0 0
\(979\) 2.39947e7 0.800127
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.81575e7 −0.599340 −0.299670 0.954043i \(-0.596877\pi\)
−0.299670 + 0.954043i \(0.596877\pi\)
\(984\) 0 0
\(985\) −2.40072e7 −0.788409
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.17429e7 0.706851
\(990\) 0 0
\(991\) 6.51450e6 0.210716 0.105358 0.994434i \(-0.466401\pi\)
0.105358 + 0.994434i \(0.466401\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3.96583e7 −1.26992
\(996\) 0 0
\(997\) −2.89170e7 −0.921329 −0.460664 0.887574i \(-0.652389\pi\)
−0.460664 + 0.887574i \(0.652389\pi\)
\(998\) 0 0
\(999\) 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 108.6.a.b.1.1 2
3.2 odd 2 108.6.a.c.1.2 yes 2
4.3 odd 2 432.6.a.r.1.1 2
9.2 odd 6 324.6.e.f.109.1 4
9.4 even 3 324.6.e.g.217.2 4
9.5 odd 6 324.6.e.f.217.1 4
9.7 even 3 324.6.e.g.109.2 4
12.11 even 2 432.6.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.6.a.b.1.1 2 1.1 even 1 trivial
108.6.a.c.1.2 yes 2 3.2 odd 2
324.6.e.f.109.1 4 9.2 odd 6
324.6.e.f.217.1 4 9.5 odd 6
324.6.e.g.109.2 4 9.7 even 3
324.6.e.g.217.2 4 9.4 even 3
432.6.a.q.1.2 2 12.11 even 2
432.6.a.r.1.1 2 4.3 odd 2