Properties

Label 3920.2.a.bt.1.2
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.37228 q^{3} +1.00000 q^{5} +2.62772 q^{9} +O(q^{10})\) \(q+2.37228 q^{3} +1.00000 q^{5} +2.62772 q^{9} -6.37228 q^{11} -4.37228 q^{13} +2.37228 q^{15} +0.372281 q^{17} -4.74456 q^{19} +4.74456 q^{23} +1.00000 q^{25} -0.883156 q^{27} -4.37228 q^{29} -8.00000 q^{31} -15.1168 q^{33} -2.00000 q^{37} -10.3723 q^{39} -6.74456 q^{41} +8.74456 q^{43} +2.62772 q^{45} -7.11684 q^{47} +0.883156 q^{51} +10.7446 q^{53} -6.37228 q^{55} -11.2554 q^{57} +8.00000 q^{59} +2.74456 q^{61} -4.37228 q^{65} +4.00000 q^{67} +11.2554 q^{69} -8.00000 q^{71} +6.00000 q^{73} +2.37228 q^{75} -15.1168 q^{79} -9.97825 q^{81} -9.48913 q^{83} +0.372281 q^{85} -10.3723 q^{87} -14.7446 q^{89} -18.9783 q^{93} -4.74456 q^{95} +9.86141 q^{97} -16.7446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 2q^{5} + 11q^{9} + O(q^{10}) \) \( 2q - q^{3} + 2q^{5} + 11q^{9} - 7q^{11} - 3q^{13} - q^{15} - 5q^{17} + 2q^{19} - 2q^{23} + 2q^{25} - 19q^{27} - 3q^{29} - 16q^{31} - 13q^{33} - 4q^{37} - 15q^{39} - 2q^{41} + 6q^{43} + 11q^{45} + 3q^{47} + 19q^{51} + 10q^{53} - 7q^{55} - 34q^{57} + 16q^{59} - 6q^{61} - 3q^{65} + 8q^{67} + 34q^{69} - 16q^{71} + 12q^{73} - q^{75} - 13q^{79} + 26q^{81} + 4q^{83} - 5q^{85} - 15q^{87} - 18q^{89} + 8q^{93} + 2q^{95} - 9q^{97} - 22q^{99} + O(q^{100}) \)

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
\(3\) 2.37228 1.36964 0.684819 0.728714i \(-0.259881\pi\)
0.684819 + 0.728714i \(0.259881\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.62772 0.875906
\(10\) 0 0
\(11\) −6.37228 −1.92132 −0.960658 0.277736i \(-0.910416\pi\)
−0.960658 + 0.277736i \(0.910416\pi\)
\(12\) 0 0
\(13\) −4.37228 −1.21265 −0.606326 0.795216i \(-0.707357\pi\)
−0.606326 + 0.795216i \(0.707357\pi\)
\(14\) 0 0
\(15\) 2.37228 0.612520
\(16\) 0 0
\(17\) 0.372281 0.0902915 0.0451457 0.998980i \(-0.485625\pi\)
0.0451457 + 0.998980i \(0.485625\pi\)
\(18\) 0 0
\(19\) −4.74456 −1.08848 −0.544239 0.838930i \(-0.683181\pi\)
−0.544239 + 0.838930i \(0.683181\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.74456 0.989310 0.494655 0.869090i \(-0.335294\pi\)
0.494655 + 0.869090i \(0.335294\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −0.883156 −0.169963
\(28\) 0 0
\(29\) −4.37228 −0.811912 −0.405956 0.913893i \(-0.633061\pi\)
−0.405956 + 0.913893i \(0.633061\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) −15.1168 −2.63150
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −10.3723 −1.66089
\(40\) 0 0
\(41\) −6.74456 −1.05332 −0.526662 0.850075i \(-0.676557\pi\)
−0.526662 + 0.850075i \(0.676557\pi\)
\(42\) 0 0
\(43\) 8.74456 1.33353 0.666767 0.745267i \(-0.267678\pi\)
0.666767 + 0.745267i \(0.267678\pi\)
\(44\) 0 0
\(45\) 2.62772 0.391717
\(46\) 0 0
\(47\) −7.11684 −1.03810 −0.519049 0.854744i \(-0.673714\pi\)
−0.519049 + 0.854744i \(0.673714\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0.883156 0.123667
\(52\) 0 0
\(53\) 10.7446 1.47588 0.737940 0.674867i \(-0.235799\pi\)
0.737940 + 0.674867i \(0.235799\pi\)
\(54\) 0 0
\(55\) −6.37228 −0.859238
\(56\) 0 0
\(57\) −11.2554 −1.49082
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 2.74456 0.351405 0.175703 0.984443i \(-0.443780\pi\)
0.175703 + 0.984443i \(0.443780\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.37228 −0.542315
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 11.2554 1.35500
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 2.37228 0.273927
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −15.1168 −1.70078 −0.850389 0.526155i \(-0.823633\pi\)
−0.850389 + 0.526155i \(0.823633\pi\)
\(80\) 0 0
\(81\) −9.97825 −1.10869
\(82\) 0 0
\(83\) −9.48913 −1.04157 −0.520783 0.853689i \(-0.674360\pi\)
−0.520783 + 0.853689i \(0.674360\pi\)
\(84\) 0 0
\(85\) 0.372281 0.0403796
\(86\) 0 0
\(87\) −10.3723 −1.11203
\(88\) 0 0
\(89\) −14.7446 −1.56292 −0.781460 0.623955i \(-0.785525\pi\)
−0.781460 + 0.623955i \(0.785525\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −18.9783 −1.96795
\(94\) 0 0
\(95\) −4.74456 −0.486782
\(96\) 0 0
\(97\) 9.86141 1.00127 0.500637 0.865657i \(-0.333099\pi\)
0.500637 + 0.865657i \(0.333099\pi\)
\(98\) 0 0
\(99\) −16.7446 −1.68289
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −5.62772 −0.554516 −0.277258 0.960796i \(-0.589426\pi\)
−0.277258 + 0.960796i \(0.589426\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.74456 −0.845369 −0.422684 0.906277i \(-0.638912\pi\)
−0.422684 + 0.906277i \(0.638912\pi\)
\(108\) 0 0
\(109\) 0.372281 0.0356581 0.0178290 0.999841i \(-0.494325\pi\)
0.0178290 + 0.999841i \(0.494325\pi\)
\(110\) 0 0
\(111\) −4.74456 −0.450334
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 4.74456 0.442433
\(116\) 0 0
\(117\) −11.4891 −1.06217
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 29.6060 2.69145
\(122\) 0 0
\(123\) −16.0000 −1.44267
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 20.7446 1.82646
\(130\) 0 0
\(131\) −4.74456 −0.414534 −0.207267 0.978284i \(-0.566457\pi\)
−0.207267 + 0.978284i \(0.566457\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.883156 −0.0760100
\(136\) 0 0
\(137\) 14.7446 1.25971 0.629857 0.776712i \(-0.283114\pi\)
0.629857 + 0.776712i \(0.283114\pi\)
\(138\) 0 0
\(139\) −4.74456 −0.402429 −0.201214 0.979547i \(-0.564489\pi\)
−0.201214 + 0.979547i \(0.564489\pi\)
\(140\) 0 0
\(141\) −16.8832 −1.42182
\(142\) 0 0
\(143\) 27.8614 2.32989
\(144\) 0 0
\(145\) −4.37228 −0.363098
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.4891 1.26892 0.634459 0.772956i \(-0.281223\pi\)
0.634459 + 0.772956i \(0.281223\pi\)
\(150\) 0 0
\(151\) −15.1168 −1.23019 −0.615096 0.788452i \(-0.710883\pi\)
−0.615096 + 0.788452i \(0.710883\pi\)
\(152\) 0 0
\(153\) 0.978251 0.0790869
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 15.4891 1.23617 0.618083 0.786113i \(-0.287909\pi\)
0.618083 + 0.786113i \(0.287909\pi\)
\(158\) 0 0
\(159\) 25.4891 2.02142
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.7446 1.31154 0.655768 0.754963i \(-0.272345\pi\)
0.655768 + 0.754963i \(0.272345\pi\)
\(164\) 0 0
\(165\) −15.1168 −1.17684
\(166\) 0 0
\(167\) −5.62772 −0.435486 −0.217743 0.976006i \(-0.569869\pi\)
−0.217743 + 0.976006i \(0.569869\pi\)
\(168\) 0 0
\(169\) 6.11684 0.470526
\(170\) 0 0
\(171\) −12.4674 −0.953404
\(172\) 0 0
\(173\) 0.372281 0.0283040 0.0141520 0.999900i \(-0.495495\pi\)
0.0141520 + 0.999900i \(0.495495\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 18.9783 1.42649
\(178\) 0 0
\(179\) 22.9783 1.71748 0.858738 0.512416i \(-0.171249\pi\)
0.858738 + 0.512416i \(0.171249\pi\)
\(180\) 0 0
\(181\) −16.2337 −1.20664 −0.603320 0.797499i \(-0.706156\pi\)
−0.603320 + 0.797499i \(0.706156\pi\)
\(182\) 0 0
\(183\) 6.51087 0.481298
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −2.37228 −0.173478
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.86141 −0.279402 −0.139701 0.990194i \(-0.544614\pi\)
−0.139701 + 0.990194i \(0.544614\pi\)
\(192\) 0 0
\(193\) 6.74456 0.485484 0.242742 0.970091i \(-0.421953\pi\)
0.242742 + 0.970091i \(0.421953\pi\)
\(194\) 0 0
\(195\) −10.3723 −0.742774
\(196\) 0 0
\(197\) −8.23369 −0.586626 −0.293313 0.956016i \(-0.594758\pi\)
−0.293313 + 0.956016i \(0.594758\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 9.48913 0.669311
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −6.74456 −0.471061
\(206\) 0 0
\(207\) 12.4674 0.866543
\(208\) 0 0
\(209\) 30.2337 2.09131
\(210\) 0 0
\(211\) 14.3723 0.989429 0.494714 0.869056i \(-0.335273\pi\)
0.494714 + 0.869056i \(0.335273\pi\)
\(212\) 0 0
\(213\) −18.9783 −1.30037
\(214\) 0 0
\(215\) 8.74456 0.596374
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 14.2337 0.961823
\(220\) 0 0
\(221\) −1.62772 −0.109492
\(222\) 0 0
\(223\) −5.62772 −0.376860 −0.188430 0.982087i \(-0.560340\pi\)
−0.188430 + 0.982087i \(0.560340\pi\)
\(224\) 0 0
\(225\) 2.62772 0.175181
\(226\) 0 0
\(227\) 19.8614 1.31825 0.659124 0.752034i \(-0.270927\pi\)
0.659124 + 0.752034i \(0.270927\pi\)
\(228\) 0 0
\(229\) 12.2337 0.808425 0.404212 0.914665i \(-0.367546\pi\)
0.404212 + 0.914665i \(0.367546\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.25544 −0.0822464 −0.0411232 0.999154i \(-0.513094\pi\)
−0.0411232 + 0.999154i \(0.513094\pi\)
\(234\) 0 0
\(235\) −7.11684 −0.464252
\(236\) 0 0
\(237\) −35.8614 −2.32945
\(238\) 0 0
\(239\) −13.6277 −0.881504 −0.440752 0.897629i \(-0.645288\pi\)
−0.440752 + 0.897629i \(0.645288\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 0 0
\(243\) −21.0217 −1.34855
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 20.7446 1.31994
\(248\) 0 0
\(249\) −22.5109 −1.42657
\(250\) 0 0
\(251\) 4.74456 0.299474 0.149737 0.988726i \(-0.452157\pi\)
0.149737 + 0.988726i \(0.452157\pi\)
\(252\) 0 0
\(253\) −30.2337 −1.90078
\(254\) 0 0
\(255\) 0.883156 0.0553054
\(256\) 0 0
\(257\) 23.4891 1.46521 0.732606 0.680653i \(-0.238304\pi\)
0.732606 + 0.680653i \(0.238304\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −11.4891 −0.711159
\(262\) 0 0
\(263\) −22.2337 −1.37099 −0.685494 0.728078i \(-0.740414\pi\)
−0.685494 + 0.728078i \(0.740414\pi\)
\(264\) 0 0
\(265\) 10.7446 0.660033
\(266\) 0 0
\(267\) −34.9783 −2.14063
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 9.48913 0.576423 0.288212 0.957567i \(-0.406939\pi\)
0.288212 + 0.957567i \(0.406939\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.37228 −0.384263
\(276\) 0 0
\(277\) 24.9783 1.50080 0.750399 0.660985i \(-0.229861\pi\)
0.750399 + 0.660985i \(0.229861\pi\)
\(278\) 0 0
\(279\) −21.0217 −1.25854
\(280\) 0 0
\(281\) 18.6060 1.10994 0.554970 0.831871i \(-0.312730\pi\)
0.554970 + 0.831871i \(0.312730\pi\)
\(282\) 0 0
\(283\) −8.88316 −0.528049 −0.264024 0.964516i \(-0.585050\pi\)
−0.264024 + 0.964516i \(0.585050\pi\)
\(284\) 0 0
\(285\) −11.2554 −0.666715
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.8614 −0.991847
\(290\) 0 0
\(291\) 23.3940 1.37138
\(292\) 0 0
\(293\) −25.1168 −1.46734 −0.733671 0.679505i \(-0.762195\pi\)
−0.733671 + 0.679505i \(0.762195\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 5.62772 0.326553
\(298\) 0 0
\(299\) −20.7446 −1.19969
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 14.2337 0.817704
\(304\) 0 0
\(305\) 2.74456 0.157153
\(306\) 0 0
\(307\) 31.1168 1.77593 0.887966 0.459909i \(-0.152118\pi\)
0.887966 + 0.459909i \(0.152118\pi\)
\(308\) 0 0
\(309\) −13.3505 −0.759485
\(310\) 0 0
\(311\) −12.7446 −0.722678 −0.361339 0.932435i \(-0.617680\pi\)
−0.361339 + 0.932435i \(0.617680\pi\)
\(312\) 0 0
\(313\) −2.88316 −0.162966 −0.0814828 0.996675i \(-0.525966\pi\)
−0.0814828 + 0.996675i \(0.525966\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 0 0
\(319\) 27.8614 1.55994
\(320\) 0 0
\(321\) −20.7446 −1.15785
\(322\) 0 0
\(323\) −1.76631 −0.0982802
\(324\) 0 0
\(325\) −4.37228 −0.242531
\(326\) 0 0
\(327\) 0.883156 0.0488386
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) −5.25544 −0.287996
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −7.48913 −0.407959 −0.203979 0.978975i \(-0.565388\pi\)
−0.203979 + 0.978975i \(0.565388\pi\)
\(338\) 0 0
\(339\) 4.74456 0.257689
\(340\) 0 0
\(341\) 50.9783 2.76063
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 11.2554 0.605972
\(346\) 0 0
\(347\) 24.7446 1.32836 0.664179 0.747574i \(-0.268781\pi\)
0.664179 + 0.747574i \(0.268781\pi\)
\(348\) 0 0
\(349\) −19.4891 −1.04323 −0.521614 0.853181i \(-0.674670\pi\)
−0.521614 + 0.853181i \(0.674670\pi\)
\(350\) 0 0
\(351\) 3.86141 0.206107
\(352\) 0 0
\(353\) 1.86141 0.0990727 0.0495363 0.998772i \(-0.484226\pi\)
0.0495363 + 0.998772i \(0.484226\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 3.51087 0.184783
\(362\) 0 0
\(363\) 70.2337 3.68631
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 19.8614 1.03676 0.518378 0.855151i \(-0.326536\pi\)
0.518378 + 0.855151i \(0.326536\pi\)
\(368\) 0 0
\(369\) −17.7228 −0.922613
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −30.7446 −1.59189 −0.795947 0.605367i \(-0.793026\pi\)
−0.795947 + 0.605367i \(0.793026\pi\)
\(374\) 0 0
\(375\) 2.37228 0.122504
\(376\) 0 0
\(377\) 19.1168 0.984568
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) −18.9783 −0.972285
\(382\) 0 0
\(383\) −17.4891 −0.893653 −0.446826 0.894621i \(-0.647446\pi\)
−0.446826 + 0.894621i \(0.647446\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 22.9783 1.16805
\(388\) 0 0
\(389\) 17.8614 0.905609 0.452805 0.891610i \(-0.350424\pi\)
0.452805 + 0.891610i \(0.350424\pi\)
\(390\) 0 0
\(391\) 1.76631 0.0893262
\(392\) 0 0
\(393\) −11.2554 −0.567762
\(394\) 0 0
\(395\) −15.1168 −0.760611
\(396\) 0 0
\(397\) −31.6277 −1.58735 −0.793675 0.608342i \(-0.791835\pi\)
−0.793675 + 0.608342i \(0.791835\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −38.6060 −1.92789 −0.963945 0.266101i \(-0.914264\pi\)
−0.963945 + 0.266101i \(0.914264\pi\)
\(402\) 0 0
\(403\) 34.9783 1.74239
\(404\) 0 0
\(405\) −9.97825 −0.495823
\(406\) 0 0
\(407\) 12.7446 0.631725
\(408\) 0 0
\(409\) −11.4891 −0.568101 −0.284050 0.958809i \(-0.591678\pi\)
−0.284050 + 0.958809i \(0.591678\pi\)
\(410\) 0 0
\(411\) 34.9783 1.72535
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −9.48913 −0.465803
\(416\) 0 0
\(417\) −11.2554 −0.551181
\(418\) 0 0
\(419\) −14.5109 −0.708903 −0.354451 0.935074i \(-0.615332\pi\)
−0.354451 + 0.935074i \(0.615332\pi\)
\(420\) 0 0
\(421\) −18.6060 −0.906799 −0.453400 0.891307i \(-0.649789\pi\)
−0.453400 + 0.891307i \(0.649789\pi\)
\(422\) 0 0
\(423\) −18.7011 −0.909277
\(424\) 0 0
\(425\) 0.372281 0.0180583
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 66.0951 3.19110
\(430\) 0 0
\(431\) −18.3723 −0.884962 −0.442481 0.896778i \(-0.645902\pi\)
−0.442481 + 0.896778i \(0.645902\pi\)
\(432\) 0 0
\(433\) −28.9783 −1.39261 −0.696303 0.717748i \(-0.745173\pi\)
−0.696303 + 0.717748i \(0.745173\pi\)
\(434\) 0 0
\(435\) −10.3723 −0.497313
\(436\) 0 0
\(437\) −22.5109 −1.07684
\(438\) 0 0
\(439\) −6.23369 −0.297518 −0.148759 0.988874i \(-0.547528\pi\)
−0.148759 + 0.988874i \(0.547528\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16.7446 0.795558 0.397779 0.917481i \(-0.369781\pi\)
0.397779 + 0.917481i \(0.369781\pi\)
\(444\) 0 0
\(445\) −14.7446 −0.698959
\(446\) 0 0
\(447\) 36.7446 1.73796
\(448\) 0 0
\(449\) 17.1168 0.807794 0.403897 0.914805i \(-0.367655\pi\)
0.403897 + 0.914805i \(0.367655\pi\)
\(450\) 0 0
\(451\) 42.9783 2.02377
\(452\) 0 0
\(453\) −35.8614 −1.68492
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.25544 0.245839 0.122919 0.992417i \(-0.460774\pi\)
0.122919 + 0.992417i \(0.460774\pi\)
\(458\) 0 0
\(459\) −0.328782 −0.0153463
\(460\) 0 0
\(461\) 1.25544 0.0584715 0.0292358 0.999573i \(-0.490693\pi\)
0.0292358 + 0.999573i \(0.490693\pi\)
\(462\) 0 0
\(463\) 6.51087 0.302586 0.151293 0.988489i \(-0.451656\pi\)
0.151293 + 0.988489i \(0.451656\pi\)
\(464\) 0 0
\(465\) −18.9783 −0.880095
\(466\) 0 0
\(467\) 8.60597 0.398237 0.199118 0.979975i \(-0.436192\pi\)
0.199118 + 0.979975i \(0.436192\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 36.7446 1.69310
\(472\) 0 0
\(473\) −55.7228 −2.56214
\(474\) 0 0
\(475\) −4.74456 −0.217695
\(476\) 0 0
\(477\) 28.2337 1.29273
\(478\) 0 0
\(479\) 22.2337 1.01588 0.507942 0.861392i \(-0.330407\pi\)
0.507942 + 0.861392i \(0.330407\pi\)
\(480\) 0 0
\(481\) 8.74456 0.398718
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.86141 0.447783
\(486\) 0 0
\(487\) −30.2337 −1.37002 −0.685010 0.728534i \(-0.740202\pi\)
−0.685010 + 0.728534i \(0.740202\pi\)
\(488\) 0 0
\(489\) 39.7228 1.79633
\(490\) 0 0
\(491\) −35.1168 −1.58480 −0.792400 0.610001i \(-0.791169\pi\)
−0.792400 + 0.610001i \(0.791169\pi\)
\(492\) 0 0
\(493\) −1.62772 −0.0733088
\(494\) 0 0
\(495\) −16.7446 −0.752612
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −31.8614 −1.42631 −0.713156 0.701005i \(-0.752735\pi\)
−0.713156 + 0.701005i \(0.752735\pi\)
\(500\) 0 0
\(501\) −13.3505 −0.596458
\(502\) 0 0
\(503\) −18.3723 −0.819180 −0.409590 0.912270i \(-0.634328\pi\)
−0.409590 + 0.912270i \(0.634328\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 14.5109 0.644451
\(508\) 0 0
\(509\) 40.9783 1.81633 0.908165 0.418613i \(-0.137484\pi\)
0.908165 + 0.418613i \(0.137484\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 4.19019 0.185001
\(514\) 0 0
\(515\) −5.62772 −0.247987
\(516\) 0 0
\(517\) 45.3505 1.99451
\(518\) 0 0
\(519\) 0.883156 0.0387662
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) −36.4674 −1.59461 −0.797304 0.603579i \(-0.793741\pi\)
−0.797304 + 0.603579i \(0.793741\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.97825 −0.129735
\(528\) 0 0
\(529\) −0.489125 −0.0212663
\(530\) 0 0
\(531\) 21.0217 0.912266
\(532\) 0 0
\(533\) 29.4891 1.27732
\(534\) 0 0
\(535\) −8.74456 −0.378060
\(536\) 0 0
\(537\) 54.5109 2.35232
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −31.3505 −1.34786 −0.673932 0.738793i \(-0.735396\pi\)
−0.673932 + 0.738793i \(0.735396\pi\)
\(542\) 0 0
\(543\) −38.5109 −1.65266
\(544\) 0 0
\(545\) 0.372281 0.0159468
\(546\) 0 0
\(547\) 30.9783 1.32453 0.662267 0.749268i \(-0.269594\pi\)
0.662267 + 0.749268i \(0.269594\pi\)
\(548\) 0 0
\(549\) 7.21194 0.307798
\(550\) 0 0
\(551\) 20.7446 0.883748
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −4.74456 −0.201395
\(556\) 0 0
\(557\) −3.76631 −0.159584 −0.0797919 0.996812i \(-0.525426\pi\)
−0.0797919 + 0.996812i \(0.525426\pi\)
\(558\) 0 0
\(559\) −38.2337 −1.61711
\(560\) 0 0
\(561\) −5.62772 −0.237602
\(562\) 0 0
\(563\) 17.4891 0.737079 0.368539 0.929612i \(-0.379858\pi\)
0.368539 + 0.929612i \(0.379858\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.9783 −1.04714 −0.523571 0.851982i \(-0.675401\pi\)
−0.523571 + 0.851982i \(0.675401\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) −9.16034 −0.382679
\(574\) 0 0
\(575\) 4.74456 0.197862
\(576\) 0 0
\(577\) 22.6060 0.941099 0.470549 0.882374i \(-0.344056\pi\)
0.470549 + 0.882374i \(0.344056\pi\)
\(578\) 0 0
\(579\) 16.0000 0.664937
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −68.4674 −2.83563
\(584\) 0 0
\(585\) −11.4891 −0.475017
\(586\) 0 0
\(587\) −34.9783 −1.44371 −0.721853 0.692046i \(-0.756710\pi\)
−0.721853 + 0.692046i \(0.756710\pi\)
\(588\) 0 0
\(589\) 37.9565 1.56397
\(590\) 0 0
\(591\) −19.5326 −0.803465
\(592\) 0 0
\(593\) 19.6277 0.806014 0.403007 0.915197i \(-0.367965\pi\)
0.403007 + 0.915197i \(0.367965\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −37.9565 −1.55346
\(598\) 0 0
\(599\) 32.6060 1.33224 0.666122 0.745843i \(-0.267953\pi\)
0.666122 + 0.745843i \(0.267953\pi\)
\(600\) 0 0
\(601\) −16.5109 −0.673493 −0.336746 0.941595i \(-0.609326\pi\)
−0.336746 + 0.941595i \(0.609326\pi\)
\(602\) 0 0
\(603\) 10.5109 0.428036
\(604\) 0 0
\(605\) 29.6060 1.20365
\(606\) 0 0
\(607\) 24.6060 0.998725 0.499363 0.866393i \(-0.333568\pi\)
0.499363 + 0.866393i \(0.333568\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 31.1168 1.25885
\(612\) 0 0
\(613\) −8.51087 −0.343751 −0.171875 0.985119i \(-0.554983\pi\)
−0.171875 + 0.985119i \(0.554983\pi\)
\(614\) 0 0
\(615\) −16.0000 −0.645182
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −12.7446 −0.512247 −0.256124 0.966644i \(-0.582445\pi\)
−0.256124 + 0.966644i \(0.582445\pi\)
\(620\) 0 0
\(621\) −4.19019 −0.168146
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 71.7228 2.86433
\(628\) 0 0
\(629\) −0.744563 −0.0296877
\(630\) 0 0
\(631\) −2.37228 −0.0944390 −0.0472195 0.998885i \(-0.515036\pi\)
−0.0472195 + 0.998885i \(0.515036\pi\)
\(632\) 0 0
\(633\) 34.0951 1.35516
\(634\) 0 0
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −21.0217 −0.831608
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 18.3723 0.724532 0.362266 0.932075i \(-0.382003\pi\)
0.362266 + 0.932075i \(0.382003\pi\)
\(644\) 0 0
\(645\) 20.7446 0.816816
\(646\) 0 0
\(647\) −16.0000 −0.629025 −0.314512 0.949253i \(-0.601841\pi\)
−0.314512 + 0.949253i \(0.601841\pi\)
\(648\) 0 0
\(649\) −50.9783 −2.00107
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) −4.74456 −0.185385
\(656\) 0 0
\(657\) 15.7663 0.615102
\(658\) 0 0
\(659\) 11.1168 0.433051 0.216525 0.976277i \(-0.430528\pi\)
0.216525 + 0.976277i \(0.430528\pi\)
\(660\) 0 0
\(661\) −14.7446 −0.573497 −0.286749 0.958006i \(-0.592574\pi\)
−0.286749 + 0.958006i \(0.592574\pi\)
\(662\) 0 0
\(663\) −3.86141 −0.149965
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −20.7446 −0.803233
\(668\) 0 0
\(669\) −13.3505 −0.516161
\(670\) 0 0
\(671\) −17.4891 −0.675160
\(672\) 0 0
\(673\) 25.7228 0.991542 0.495771 0.868453i \(-0.334886\pi\)
0.495771 + 0.868453i \(0.334886\pi\)
\(674\) 0 0
\(675\) −0.883156 −0.0339927
\(676\) 0 0
\(677\) −15.3505 −0.589969 −0.294984 0.955502i \(-0.595314\pi\)
−0.294984 + 0.955502i \(0.595314\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 47.1168 1.80552
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 14.7446 0.563361
\(686\) 0 0
\(687\) 29.0217 1.10725
\(688\) 0 0
\(689\) −46.9783 −1.78973
\(690\) 0 0
\(691\) −9.48913 −0.360983 −0.180492 0.983577i \(-0.557769\pi\)
−0.180492 + 0.983577i \(0.557769\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.74456 −0.179972
\(696\) 0 0
\(697\) −2.51087 −0.0951062
\(698\) 0 0
\(699\) −2.97825 −0.112648
\(700\) 0 0
\(701\) 2.13859 0.0807736 0.0403868 0.999184i \(-0.487141\pi\)
0.0403868 + 0.999184i \(0.487141\pi\)
\(702\) 0 0
\(703\) 9.48913 0.357889
\(704\) 0 0
\(705\) −16.8832 −0.635856
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.60597 0.248092 0.124046 0.992276i \(-0.460413\pi\)
0.124046 + 0.992276i \(0.460413\pi\)
\(710\) 0 0
\(711\) −39.7228 −1.48972
\(712\) 0 0
\(713\) −37.9565 −1.42148
\(714\) 0 0
\(715\) 27.8614 1.04196
\(716\) 0 0
\(717\) −32.3288 −1.20734
\(718\) 0 0
\(719\) 3.25544 0.121407 0.0607037 0.998156i \(-0.480666\pi\)
0.0607037 + 0.998156i \(0.480666\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −61.6793 −2.29388
\(724\) 0 0
\(725\) −4.37228 −0.162382
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) −19.9348 −0.738324
\(730\) 0 0
\(731\) 3.25544 0.120407
\(732\) 0 0
\(733\) 10.1386 0.374477 0.187239 0.982314i \(-0.440046\pi\)
0.187239 + 0.982314i \(0.440046\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −25.4891 −0.938904
\(738\) 0 0
\(739\) 20.6060 0.758003 0.379001 0.925396i \(-0.376268\pi\)
0.379001 + 0.925396i \(0.376268\pi\)
\(740\) 0 0
\(741\) 49.2119 1.80785
\(742\) 0 0
\(743\) 6.51087 0.238861 0.119430 0.992843i \(-0.461893\pi\)
0.119430 + 0.992843i \(0.461893\pi\)
\(744\) 0 0
\(745\) 15.4891 0.567478
\(746\) 0 0
\(747\) −24.9348 −0.912315
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −20.1386 −0.734868 −0.367434 0.930050i \(-0.619764\pi\)
−0.367434 + 0.930050i \(0.619764\pi\)
\(752\) 0 0
\(753\) 11.2554 0.410171
\(754\) 0 0
\(755\) −15.1168 −0.550158
\(756\) 0 0
\(757\) −3.76631 −0.136889 −0.0684445 0.997655i \(-0.521804\pi\)
−0.0684445 + 0.997655i \(0.521804\pi\)
\(758\) 0 0
\(759\) −71.7228 −2.60337
\(760\) 0 0
\(761\) −4.97825 −0.180461 −0.0902307 0.995921i \(-0.528760\pi\)
−0.0902307 + 0.995921i \(0.528760\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.978251 0.0353687
\(766\) 0 0
\(767\) −34.9783 −1.26299
\(768\) 0 0
\(769\) −3.48913 −0.125821 −0.0629105 0.998019i \(-0.520038\pi\)
−0.0629105 + 0.998019i \(0.520038\pi\)
\(770\) 0 0
\(771\) 55.7228 2.00681
\(772\) 0 0
\(773\) −4.37228 −0.157260 −0.0786300 0.996904i \(-0.525055\pi\)
−0.0786300 + 0.996904i \(0.525055\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 32.0000 1.14652
\(780\) 0 0
\(781\) 50.9783 1.82415
\(782\) 0 0
\(783\) 3.86141 0.137995
\(784\) 0 0
\(785\) 15.4891 0.552831
\(786\) 0 0
\(787\) −31.1168 −1.10920 −0.554598 0.832119i \(-0.687128\pi\)
−0.554598 + 0.832119i \(0.687128\pi\)
\(788\) 0 0
\(789\) −52.7446 −1.87776
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 0 0
\(795\) 25.4891 0.904006
\(796\) 0 0
\(797\) −15.6277 −0.553562 −0.276781 0.960933i \(-0.589268\pi\)
−0.276781 + 0.960933i \(0.589268\pi\)
\(798\) 0 0
\(799\) −2.64947 −0.0937314
\(800\) 0 0
\(801\) −38.7446 −1.36897
\(802\) 0 0
\(803\) −38.2337 −1.34924
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.2337 0.501050
\(808\) 0 0
\(809\) 20.3723 0.716251 0.358126 0.933673i \(-0.383416\pi\)
0.358126 + 0.933673i \(0.383416\pi\)
\(810\) 0 0
\(811\) 12.7446 0.447522 0.223761 0.974644i \(-0.428166\pi\)
0.223761 + 0.974644i \(0.428166\pi\)
\(812\) 0 0
\(813\) 22.5109 0.789491
\(814\) 0 0
\(815\) 16.7446 0.586536
\(816\) 0 0
\(817\) −41.4891 −1.45152
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25.1168 −0.876584 −0.438292 0.898833i \(-0.644416\pi\)
−0.438292 + 0.898833i \(0.644416\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 0 0
\(825\) −15.1168 −0.526301
\(826\) 0 0
\(827\) −24.7446 −0.860453 −0.430226 0.902721i \(-0.641566\pi\)
−0.430226 + 0.902721i \(0.641566\pi\)
\(828\) 0 0
\(829\) 12.2337 0.424894 0.212447 0.977173i \(-0.431857\pi\)
0.212447 + 0.977173i \(0.431857\pi\)
\(830\) 0 0
\(831\) 59.2554 2.05555
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −5.62772 −0.194755
\(836\) 0 0
\(837\) 7.06525 0.244211
\(838\) 0 0
\(839\) −11.2554 −0.388581 −0.194290 0.980944i \(-0.562240\pi\)
−0.194290 + 0.980944i \(0.562240\pi\)
\(840\) 0 0
\(841\) −9.88316 −0.340798
\(842\) 0 0
\(843\) 44.1386 1.52021
\(844\) 0 0
\(845\) 6.11684 0.210426
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −21.0733 −0.723235
\(850\) 0 0
\(851\) −9.48913 −0.325283
\(852\) 0 0
\(853\) 39.4891 1.35208 0.676041 0.736864i \(-0.263694\pi\)
0.676041 + 0.736864i \(0.263694\pi\)
\(854\) 0 0
\(855\) −12.4674 −0.426375
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 44.4674 1.51721 0.758604 0.651552i \(-0.225882\pi\)
0.758604 + 0.651552i \(0.225882\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25.4891 −0.867660 −0.433830 0.900995i \(-0.642838\pi\)
−0.433830 + 0.900995i \(0.642838\pi\)
\(864\) 0 0
\(865\) 0.372281 0.0126579
\(866\) 0 0
\(867\) −40.0000 −1.35847
\(868\) 0 0
\(869\) 96.3288 3.26773
\(870\) 0 0
\(871\) −17.4891 −0.592596
\(872\) 0 0
\(873\) 25.9130 0.877022
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −7.02175 −0.237108 −0.118554 0.992948i \(-0.537826\pi\)
−0.118554 + 0.992948i \(0.537826\pi\)
\(878\) 0 0
\(879\) −59.5842 −2.00973
\(880\) 0 0
\(881\) 25.2554 0.850877 0.425439 0.904987i \(-0.360120\pi\)
0.425439 + 0.904987i \(0.360120\pi\)
\(882\) 0 0
\(883\) 10.5109 0.353719 0.176860 0.984236i \(-0.443406\pi\)
0.176860 + 0.984236i \(0.443406\pi\)
\(884\) 0 0
\(885\) 18.9783 0.637947
\(886\) 0 0
\(887\) −13.0217 −0.437228 −0.218614 0.975811i \(-0.570153\pi\)
−0.218614 + 0.975811i \(0.570153\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 63.5842 2.13015
\(892\) 0 0
\(893\) 33.7663 1.12995
\(894\) 0 0
\(895\) 22.9783 0.768078
\(896\) 0 0
\(897\) −49.2119 −1.64314
\(898\) 0 0
\(899\) 34.9783 1.16659
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −16.2337 −0.539626
\(906\) 0 0
\(907\) −11.7228 −0.389250 −0.194625 0.980878i \(-0.562349\pi\)
−0.194625 + 0.980878i \(0.562349\pi\)
\(908\) 0 0
\(909\) 15.7663 0.522936
\(910\) 0 0
\(911\) 45.9565 1.52261 0.761303 0.648396i \(-0.224560\pi\)
0.761303 + 0.648396i \(0.224560\pi\)
\(912\) 0 0
\(913\) 60.4674 2.00118
\(914\) 0 0
\(915\) 6.51087 0.215243
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −13.6277 −0.449537 −0.224768 0.974412i \(-0.572163\pi\)
−0.224768 + 0.974412i \(0.572163\pi\)
\(920\) 0 0
\(921\) 73.8179 2.43238
\(922\) 0 0
\(923\) 34.9783 1.15132
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) −14.7881 −0.485704
\(928\) 0 0
\(929\) 7.76631 0.254804 0.127402 0.991851i \(-0.459336\pi\)
0.127402 + 0.991851i \(0.459336\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −30.2337 −0.989807
\(934\) 0 0
\(935\) −2.37228 −0.0775819
\(936\) 0 0
\(937\) −28.0951 −0.917827 −0.458913 0.888481i \(-0.651761\pi\)
−0.458913 + 0.888481i \(0.651761\pi\)
\(938\) 0 0
\(939\) −6.83966 −0.223204
\(940\) 0 0
\(941\) −32.2337 −1.05079 −0.525394 0.850859i \(-0.676082\pi\)
−0.525394 + 0.850859i \(0.676082\pi\)
\(942\) 0 0
\(943\) −32.0000 −1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 0 0
\(949\) −26.2337 −0.851582
\(950\) 0 0
\(951\) 33.2119 1.07697
\(952\) 0 0
\(953\) 37.2554 1.20682 0.603411 0.797430i \(-0.293808\pi\)
0.603411 + 0.797430i \(0.293808\pi\)
\(954\) 0 0
\(955\) −3.86141 −0.124952
\(956\) 0 0
\(957\) 66.0951 2.13655
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −22.9783 −0.740464
\(964\) 0 0
\(965\) 6.74456 0.217115
\(966\) 0 0
\(967\) −1.76631 −0.0568008 −0.0284004 0.999597i \(-0.509041\pi\)
−0.0284004 + 0.999597i \(0.509041\pi\)
\(968\) 0 0
\(969\) −4.19019 −0.134608
\(970\) 0 0
\(971\) 33.4891 1.07472 0.537359 0.843354i \(-0.319422\pi\)
0.537359 + 0.843354i \(0.319422\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −10.3723 −0.332179
\(976\) 0 0
\(977\) 52.9783 1.69492 0.847462 0.530856i \(-0.178129\pi\)
0.847462 + 0.530856i \(0.178129\pi\)
\(978\) 0 0
\(979\) 93.9565 3.00286
\(980\) 0 0
\(981\) 0.978251 0.0312331
\(982\) 0 0
\(983\) −10.3723 −0.330824 −0.165412 0.986225i \(-0.552895\pi\)
−0.165412 + 0.986225i \(0.552895\pi\)
\(984\) 0 0
\(985\) −8.23369 −0.262347
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 41.4891 1.31928
\(990\) 0 0
\(991\) 37.9565 1.20573 0.602864 0.797844i \(-0.294026\pi\)
0.602864 + 0.797844i \(0.294026\pi\)
\(992\) 0 0
\(993\) −28.4674 −0.903385
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 6.88316 0.217992 0.108996 0.994042i \(-0.465236\pi\)
0.108996 + 0.994042i \(0.465236\pi\)
\(998\) 0 0
\(999\) 1.76631 0.0558836
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 3920.2.a.bt.1.2 2
4.3 odd 2 1960.2.a.s.1.1 2
7.6 odd 2 560.2.a.h.1.1 2
20.19 odd 2 9800.2.a.bu.1.2 2
21.20 even 2 5040.2.a.by.1.2 2
28.3 even 6 1960.2.q.t.961.1 4
28.11 odd 6 1960.2.q.r.961.2 4
28.19 even 6 1960.2.q.t.361.1 4
28.23 odd 6 1960.2.q.r.361.2 4
28.27 even 2 280.2.a.c.1.2 2
35.13 even 4 2800.2.g.r.449.2 4
35.27 even 4 2800.2.g.r.449.3 4
35.34 odd 2 2800.2.a.bk.1.2 2
56.13 odd 2 2240.2.a.bg.1.2 2
56.27 even 2 2240.2.a.bk.1.1 2
84.83 odd 2 2520.2.a.x.1.1 2
140.27 odd 4 1400.2.g.i.449.2 4
140.83 odd 4 1400.2.g.i.449.3 4
140.139 even 2 1400.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.c.1.2 2 28.27 even 2
560.2.a.h.1.1 2 7.6 odd 2
1400.2.a.r.1.1 2 140.139 even 2
1400.2.g.i.449.2 4 140.27 odd 4
1400.2.g.i.449.3 4 140.83 odd 4
1960.2.a.s.1.1 2 4.3 odd 2
1960.2.q.r.361.2 4 28.23 odd 6
1960.2.q.r.961.2 4 28.11 odd 6
1960.2.q.t.361.1 4 28.19 even 6
1960.2.q.t.961.1 4 28.3 even 6
2240.2.a.bg.1.2 2 56.13 odd 2
2240.2.a.bk.1.1 2 56.27 even 2
2520.2.a.x.1.1 2 84.83 odd 2
2800.2.a.bk.1.2 2 35.34 odd 2
2800.2.g.r.449.2 4 35.13 even 4
2800.2.g.r.449.3 4 35.27 even 4
3920.2.a.bt.1.2 2 1.1 even 1 trivial
5040.2.a.by.1.2 2 21.20 even 2
9800.2.a.bu.1.2 2 20.19 odd 2