Properties

Label 5040.2.a.by.1.2
Level $5040$
Weight $2$
Character 5040.1
Self dual yes
Analytic conductor $40.245$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(\chi\) \(=\) 5040.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +1.00000 q^{7} +6.37228 q^{11} +4.37228 q^{13} +0.372281 q^{17} +4.74456 q^{19} -4.74456 q^{23} +1.00000 q^{25} +4.37228 q^{29} +8.00000 q^{31} +1.00000 q^{35} -2.00000 q^{37} -6.74456 q^{41} +8.74456 q^{43} -7.11684 q^{47} +1.00000 q^{49} -10.7446 q^{53} +6.37228 q^{55} +8.00000 q^{59} -2.74456 q^{61} +4.37228 q^{65} +4.00000 q^{67} +8.00000 q^{71} -6.00000 q^{73} +6.37228 q^{77} -15.1168 q^{79} -9.48913 q^{83} +0.372281 q^{85} -14.7446 q^{89} +4.37228 q^{91} +4.74456 q^{95} -9.86141 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{5} + 2q^{7} + 7q^{11} + 3q^{13} - 5q^{17} - 2q^{19} + 2q^{23} + 2q^{25} + 3q^{29} + 16q^{31} + 2q^{35} - 4q^{37} - 2q^{41} + 6q^{43} + 3q^{47} + 2q^{49} - 10q^{53} + 7q^{55} + 16q^{59} + 6q^{61} + 3q^{65} + 8q^{67} + 16q^{71} - 12q^{73} + 7q^{77} - 13q^{79} + 4q^{83} - 5q^{85} - 18q^{89} + 3q^{91} - 2q^{95} + 9q^{97} + O(q^{100}) \)

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.37228 1.92132 0.960658 0.277736i \(-0.0895839\pi\)
0.960658 + 0.277736i \(0.0895839\pi\)
\(12\) 0 0
\(13\) 4.37228 1.21265 0.606326 0.795216i \(-0.292643\pi\)
0.606326 + 0.795216i \(0.292643\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 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 \(-0.316819\pi\)
0.544239 + 0.838930i \(0.316819\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.74456 −0.989310 −0.494655 0.869090i \(-0.664706\pi\)
−0.494655 + 0.869090i \(0.664706\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.37228 0.811912 0.405956 0.913893i \(-0.366939\pi\)
0.405956 + 0.913893i \(0.366939\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 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
\(45\) 0 0
\(46\) 0 0
\(47\) −7.11684 −1.03810 −0.519049 0.854744i \(-0.673714\pi\)
−0.519049 + 0.854744i \(0.673714\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.7446 −1.47588 −0.737940 0.674867i \(-0.764201\pi\)
−0.737940 + 0.674867i \(0.764201\pi\)
\(54\) 0 0
\(55\) 6.37228 0.859238
\(56\) 0 0
\(57\) 0 0
\(58\) 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 \(-0.556220\pi\)
−0.175703 + 0.984443i \(0.556220\pi\)
\(62\) 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
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.37228 0.726189
\(78\) 0 0
\(79\) −15.1168 −1.70078 −0.850389 0.526155i \(-0.823633\pi\)
−0.850389 + 0.526155i \(0.823633\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 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
\(87\) 0 0
\(88\) 0 0
\(89\) −14.7446 −1.56292 −0.781460 0.623955i \(-0.785525\pi\)
−0.781460 + 0.623955i \(0.785525\pi\)
\(90\) 0 0
\(91\) 4.37228 0.458340
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.74456 0.486782
\(96\) 0 0
\(97\) −9.86141 −1.00127 −0.500637 0.865657i \(-0.666901\pi\)
−0.500637 + 0.865657i \(0.666901\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 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 \(-0.410574\pi\)
0.277258 + 0.960796i \(0.410574\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.74456 0.845369 0.422684 0.906277i \(-0.361088\pi\)
0.422684 + 0.906277i \(0.361088\pi\)
\(108\) 0 0
\(109\) 0.372281 0.0356581 0.0178290 0.999841i \(-0.494325\pi\)
0.0178290 + 0.999841i \(0.494325\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −4.74456 −0.442433
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.372281 0.0341270
\(120\) 0 0
\(121\) 29.6060 2.69145
\(122\) 0 0
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) 0 0
\(131\) −4.74456 −0.414534 −0.207267 0.978284i \(-0.566457\pi\)
−0.207267 + 0.978284i \(0.566457\pi\)
\(132\) 0 0
\(133\) 4.74456 0.411406
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.7446 −1.25971 −0.629857 0.776712i \(-0.716886\pi\)
−0.629857 + 0.776712i \(0.716886\pi\)
\(138\) 0 0
\(139\) 4.74456 0.402429 0.201214 0.979547i \(-0.435511\pi\)
0.201214 + 0.979547i \(0.435511\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 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 \(-0.718777\pi\)
−0.634459 + 0.772956i \(0.718777\pi\)
\(150\) 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 0
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −15.4891 −1.23617 −0.618083 0.786113i \(-0.712091\pi\)
−0.618083 + 0.786113i \(0.712091\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.74456 −0.373924
\(162\) 0 0
\(163\) 16.7446 1.31154 0.655768 0.754963i \(-0.272345\pi\)
0.655768 + 0.754963i \(0.272345\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) 0 0
\(173\) 0.372281 0.0283040 0.0141520 0.999900i \(-0.495495\pi\)
0.0141520 + 0.999900i \(0.495495\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −22.9783 −1.71748 −0.858738 0.512416i \(-0.828751\pi\)
−0.858738 + 0.512416i \(0.828751\pi\)
\(180\) 0 0
\(181\) 16.2337 1.20664 0.603320 0.797499i \(-0.293844\pi\)
0.603320 + 0.797499i \(0.293844\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 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 \(-0.455386\pi\)
0.139701 + 0.990194i \(0.455386\pi\)
\(192\) 0 0
\(193\) 6.74456 0.485484 0.242742 0.970091i \(-0.421953\pi\)
0.242742 + 0.970091i \(0.421953\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.23369 0.586626 0.293313 0.956016i \(-0.405242\pi\)
0.293313 + 0.956016i \(0.405242\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.37228 0.306874
\(204\) 0 0
\(205\) −6.74456 −0.471061
\(206\) 0 0
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 0 0
\(215\) 8.74456 0.596374
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.62772 0.109492
\(222\) 0 0
\(223\) 5.62772 0.376860 0.188430 0.982087i \(-0.439660\pi\)
0.188430 + 0.982087i \(0.439660\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 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 \(-0.632454\pi\)
−0.404212 + 0.914665i \(0.632454\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.25544 0.0822464 0.0411232 0.999154i \(-0.486906\pi\)
0.0411232 + 0.999154i \(0.486906\pi\)
\(234\) 0 0
\(235\) −7.11684 −0.464252
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 13.6277 0.881504 0.440752 0.897629i \(-0.354712\pi\)
0.440752 + 0.897629i \(0.354712\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 20.7446 1.31994
\(248\) 0 0
\(249\) 0 0
\(250\) 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 0
\(256\) 0 0
\(257\) 23.4891 1.46521 0.732606 0.680653i \(-0.238304\pi\)
0.732606 + 0.680653i \(0.238304\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 22.2337 1.37099 0.685494 0.728078i \(-0.259586\pi\)
0.685494 + 0.728078i \(0.259586\pi\)
\(264\) 0 0
\(265\) −10.7446 −0.660033
\(266\) 0 0
\(267\) 0 0
\(268\) 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 \(-0.593061\pi\)
−0.288212 + 0.957567i \(0.593061\pi\)
\(272\) 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
\(279\) 0 0
\(280\) 0 0
\(281\) −18.6060 −1.10994 −0.554970 0.831871i \(-0.687270\pi\)
−0.554970 + 0.831871i \(0.687270\pi\)
\(282\) 0 0
\(283\) 8.88316 0.528049 0.264024 0.964516i \(-0.414950\pi\)
0.264024 + 0.964516i \(0.414950\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.74456 −0.398119
\(288\) 0 0
\(289\) −16.8614 −0.991847
\(290\) 0 0
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) 0 0
\(299\) −20.7446 −1.19969
\(300\) 0 0
\(301\) 8.74456 0.504028
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.74456 −0.157153
\(306\) 0 0
\(307\) −31.1168 −1.77593 −0.887966 0.459909i \(-0.847882\pi\)
−0.887966 + 0.459909i \(0.847882\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 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 \(-0.474034\pi\)
0.0814828 + 0.996675i \(0.474034\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −14.0000 −0.786318 −0.393159 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) 0 0
\(319\) 27.8614 1.55994
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.76631 0.0982802
\(324\) 0 0
\(325\) 4.37228 0.242531
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −7.11684 −0.392364
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) 0 0
\(341\) 50.9783 2.76063
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −24.7446 −1.32836 −0.664179 0.747574i \(-0.731219\pi\)
−0.664179 + 0.747574i \(0.731219\pi\)
\(348\) 0 0
\(349\) 19.4891 1.04323 0.521614 0.853181i \(-0.325330\pi\)
0.521614 + 0.853181i \(0.325330\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 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
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −19.8614 −1.03676 −0.518378 0.855151i \(-0.673464\pi\)
−0.518378 + 0.855151i \(0.673464\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −10.7446 −0.557830
\(372\) 0 0
\(373\) −30.7446 −1.59189 −0.795947 0.605367i \(-0.793026\pi\)
−0.795947 + 0.605367i \(0.793026\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 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
\(381\) 0 0
\(382\) 0 0
\(383\) −17.4891 −0.893653 −0.446826 0.894621i \(-0.647446\pi\)
−0.446826 + 0.894621i \(0.647446\pi\)
\(384\) 0 0
\(385\) 6.37228 0.324762
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −17.8614 −0.905609 −0.452805 0.891610i \(-0.649576\pi\)
−0.452805 + 0.891610i \(0.649576\pi\)
\(390\) 0 0
\(391\) −1.76631 −0.0893262
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −15.1168 −0.760611
\(396\) 0 0
\(397\) 31.6277 1.58735 0.793675 0.608342i \(-0.208165\pi\)
0.793675 + 0.608342i \(0.208165\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 38.6060 1.92789 0.963945 0.266101i \(-0.0857356\pi\)
0.963945 + 0.266101i \(0.0857356\pi\)
\(402\) 0 0
\(403\) 34.9783 1.74239
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12.7446 −0.631725
\(408\) 0 0
\(409\) 11.4891 0.568101 0.284050 0.958809i \(-0.408322\pi\)
0.284050 + 0.958809i \(0.408322\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) −9.48913 −0.465803
\(416\) 0 0
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) 0 0
\(425\) 0.372281 0.0180583
\(426\) 0 0
\(427\) −2.74456 −0.132819
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.3723 0.884962 0.442481 0.896778i \(-0.354098\pi\)
0.442481 + 0.896778i \(0.354098\pi\)
\(432\) 0 0
\(433\) 28.9783 1.39261 0.696303 0.717748i \(-0.254827\pi\)
0.696303 + 0.717748i \(0.254827\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −22.5109 −1.07684
\(438\) 0 0
\(439\) 6.23369 0.297518 0.148759 0.988874i \(-0.452472\pi\)
0.148759 + 0.988874i \(0.452472\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.7446 −0.795558 −0.397779 0.917481i \(-0.630219\pi\)
−0.397779 + 0.917481i \(0.630219\pi\)
\(444\) 0 0
\(445\) −14.7446 −0.698959
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −17.1168 −0.807794 −0.403897 0.914805i \(-0.632345\pi\)
−0.403897 + 0.914805i \(0.632345\pi\)
\(450\) 0 0
\(451\) −42.9783 −2.02377
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.37228 0.204976
\(456\) 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 0
\(460\) 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
\(465\) 0 0
\(466\) 0 0
\(467\) 8.60597 0.398237 0.199118 0.979975i \(-0.436192\pi\)
0.199118 + 0.979975i \(0.436192\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 55.7228 2.56214
\(474\) 0 0
\(475\) 4.74456 0.217695
\(476\) 0 0
\(477\) 0 0
\(478\) 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
\(489\) 0 0
\(490\) 0 0
\(491\) 35.1168 1.58480 0.792400 0.610001i \(-0.208831\pi\)
0.792400 + 0.610001i \(0.208831\pi\)
\(492\) 0 0
\(493\) 1.62772 0.0733088
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −31.8614 −1.42631 −0.713156 0.701005i \(-0.752735\pi\)
−0.713156 + 0.701005i \(0.752735\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 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
\(507\) 0 0
\(508\) 0 0
\(509\) 40.9783 1.81633 0.908165 0.418613i \(-0.137484\pi\)
0.908165 + 0.418613i \(0.137484\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.62772 0.247987
\(516\) 0 0
\(517\) −45.3505 −1.99451
\(518\) 0 0
\(519\) 0 0
\(520\) 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 \(-0.206259\pi\)
0.797304 + 0.603579i \(0.206259\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.97825 0.129735
\(528\) 0 0
\(529\) −0.489125 −0.0212663
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −29.4891 −1.27732
\(534\) 0 0
\(535\) 8.74456 0.378060
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.37228 0.274474
\(540\) 0 0
\(541\) −31.3505 −1.34786 −0.673932 0.738793i \(-0.735396\pi\)
−0.673932 + 0.738793i \(0.735396\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) 0 0
\(551\) 20.7446 0.883748
\(552\) 0 0
\(553\) −15.1168 −0.642834
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.76631 0.159584 0.0797919 0.996812i \(-0.474574\pi\)
0.0797919 + 0.996812i \(0.474574\pi\)
\(558\) 0 0
\(559\) 38.2337 1.61711
\(560\) 0 0
\(561\) 0 0
\(562\) 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 \(-0.324599\pi\)
0.523571 + 0.851982i \(0.324599\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.74456 −0.197862
\(576\) 0 0
\(577\) −22.6060 −0.941099 −0.470549 0.882374i \(-0.655944\pi\)
−0.470549 + 0.882374i \(0.655944\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −9.48913 −0.393675
\(582\) 0 0
\(583\) −68.4674 −2.83563
\(584\) 0 0
\(585\) 0 0
\(586\) 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
\(591\) 0 0
\(592\) 0 0
\(593\) 19.6277 0.806014 0.403007 0.915197i \(-0.367965\pi\)
0.403007 + 0.915197i \(0.367965\pi\)
\(594\) 0 0
\(595\) 0.372281 0.0152620
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −32.6060 −1.33224 −0.666122 0.745843i \(-0.732047\pi\)
−0.666122 + 0.745843i \(0.732047\pi\)
\(600\) 0 0
\(601\) 16.5109 0.673493 0.336746 0.941595i \(-0.390674\pi\)
0.336746 + 0.941595i \(0.390674\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 29.6060 1.20365
\(606\) 0 0
\(607\) −24.6060 −0.998725 −0.499363 0.866393i \(-0.666432\pi\)
−0.499363 + 0.866393i \(0.666432\pi\)
\(608\) 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
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 12.7446 0.512247 0.256124 0.966644i \(-0.417555\pi\)
0.256124 + 0.966644i \(0.417555\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −14.7446 −0.590728
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 0 0
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 4.37228 0.173236
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) −18.3723 −0.724532 −0.362266 0.932075i \(-0.617997\pi\)
−0.362266 + 0.932075i \(0.617997\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 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 \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) −4.74456 −0.185385
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −11.1168 −0.433051 −0.216525 0.976277i \(-0.569472\pi\)
−0.216525 + 0.976277i \(0.569472\pi\)
\(660\) 0 0
\(661\) 14.7446 0.573497 0.286749 0.958006i \(-0.407426\pi\)
0.286749 + 0.958006i \(0.407426\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.74456 0.183986
\(666\) 0 0
\(667\) −20.7446 −0.803233
\(668\) 0 0
\(669\) 0 0
\(670\) 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 0
\(676\) 0 0
\(677\) −15.3505 −0.589969 −0.294984 0.955502i \(-0.595314\pi\)
−0.294984 + 0.955502i \(0.595314\pi\)
\(678\) 0 0
\(679\) −9.86141 −0.378446
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) −14.7446 −0.563361
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −46.9783 −1.78973
\(690\) 0 0
\(691\) 9.48913 0.360983 0.180492 0.983577i \(-0.442231\pi\)
0.180492 + 0.983577i \(0.442231\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.74456 0.179972
\(696\) 0 0
\(697\) −2.51087 −0.0951062
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.13859 −0.0807736 −0.0403868 0.999184i \(-0.512859\pi\)
−0.0403868 + 0.999184i \(0.512859\pi\)
\(702\) 0 0
\(703\) −9.48913 −0.357889
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 6.60597 0.248092 0.124046 0.992276i \(-0.460413\pi\)
0.124046 + 0.992276i \(0.460413\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −37.9565 −1.42148
\(714\) 0 0
\(715\) 27.8614 1.04196
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.25544 0.121407 0.0607037 0.998156i \(-0.480666\pi\)
0.0607037 + 0.998156i \(0.480666\pi\)
\(720\) 0 0
\(721\) 5.62772 0.209587
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.37228 0.162382
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.25544 0.120407
\(732\) 0 0
\(733\) −10.1386 −0.374477 −0.187239 0.982314i \(-0.559954\pi\)
−0.187239 + 0.982314i \(0.559954\pi\)
\(734\) 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
\(741\) 0 0
\(742\) 0 0
\(743\) −6.51087 −0.238861 −0.119430 0.992843i \(-0.538107\pi\)
−0.119430 + 0.992843i \(0.538107\pi\)
\(744\) 0 0
\(745\) −15.4891 −0.567478
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.74456 0.319519
\(750\) 0 0
\(751\) −20.1386 −0.734868 −0.367434 0.930050i \(-0.619764\pi\)
−0.367434 + 0.930050i \(0.619764\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 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
\(759\) 0 0
\(760\) 0 0
\(761\) −4.97825 −0.180461 −0.0902307 0.995921i \(-0.528760\pi\)
−0.0902307 + 0.995921i \(0.528760\pi\)
\(762\) 0 0
\(763\) 0.372281 0.0134775
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 34.9783 1.26299
\(768\) 0 0
\(769\) 3.48913 0.125821 0.0629105 0.998019i \(-0.479962\pi\)
0.0629105 + 0.998019i \(0.479962\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 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
\(783\) 0 0
\(784\) 0 0
\(785\) −15.4891 −0.552831
\(786\) 0 0
\(787\) 31.1168 1.10920 0.554598 0.832119i \(-0.312872\pi\)
0.554598 + 0.832119i \(0.312872\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 0 0
\(795\) 0 0
\(796\) 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
\(801\) 0 0
\(802\) 0 0
\(803\) −38.2337 −1.34924
\(804\) 0 0
\(805\) −4.74456 −0.167224
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −20.3723 −0.716251 −0.358126 0.933673i \(-0.616584\pi\)
−0.358126 + 0.933673i \(0.616584\pi\)
\(810\) 0 0
\(811\) −12.7446 −0.447522 −0.223761 0.974644i \(-0.571834\pi\)
−0.223761 + 0.974644i \(0.571834\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 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 \(-0.355584\pi\)
0.438292 + 0.898833i \(0.355584\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 24.7446 0.860453 0.430226 0.902721i \(-0.358434\pi\)
0.430226 + 0.902721i \(0.358434\pi\)
\(828\) 0 0
\(829\) −12.2337 −0.424894 −0.212447 0.977173i \(-0.568143\pi\)
−0.212447 + 0.977173i \(0.568143\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.372281 0.0128988
\(834\) 0 0
\(835\) −5.62772 −0.194755
\(836\) 0 0
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) 0 0
\(845\) 6.11684 0.210426
\(846\) 0 0
\(847\) 29.6060 1.01727
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 9.48913 0.325283
\(852\) 0 0
\(853\) −39.4891 −1.35208 −0.676041 0.736864i \(-0.736306\pi\)
−0.676041 + 0.736864i \(0.736306\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 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 \(-0.774118\pi\)
−0.758604 + 0.651552i \(0.774118\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.4891 0.867660 0.433830 0.900995i \(-0.357162\pi\)
0.433830 + 0.900995i \(0.357162\pi\)
\(864\) 0 0
\(865\) 0.372281 0.0126579
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −96.3288 −3.26773
\(870\) 0 0
\(871\) 17.4891 0.592596
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −7.02175 −0.237108 −0.118554 0.992948i \(-0.537826\pi\)
−0.118554 + 0.992948i \(0.537826\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 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
\(885\) 0 0
\(886\) 0 0
\(887\) −13.0217 −0.437228 −0.218614 0.975811i \(-0.570153\pi\)
−0.218614 + 0.975811i \(0.570153\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −33.7663 −1.12995
\(894\) 0 0
\(895\) −22.9783 −0.768078
\(896\) 0 0
\(897\) 0 0
\(898\) 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
\(909\) 0 0
\(910\) 0 0
\(911\) −45.9565 −1.52261 −0.761303 0.648396i \(-0.775440\pi\)
−0.761303 + 0.648396i \(0.775440\pi\)
\(912\) 0 0
\(913\) −60.4674 −2.00118
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.74456 −0.156679
\(918\) 0 0
\(919\) −13.6277 −0.449537 −0.224768 0.974412i \(-0.572163\pi\)
−0.224768 + 0.974412i \(0.572163\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 34.9783 1.15132
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.76631 0.254804 0.127402 0.991851i \(-0.459336\pi\)
0.127402 + 0.991851i \(0.459336\pi\)
\(930\) 0 0
\(931\) 4.74456 0.155497
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.37228 0.0775819
\(936\) 0 0
\(937\) 28.0951 0.917827 0.458913 0.888481i \(-0.348239\pi\)
0.458913 + 0.888481i \(0.348239\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 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 \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 0 0
\(949\) −26.2337 −0.851582
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −37.2554 −1.20682 −0.603411 0.797430i \(-0.706192\pi\)
−0.603411 + 0.797430i \(0.706192\pi\)
\(954\) 0 0
\(955\) 3.86141 0.124952
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14.7446 −0.476127
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) 0 0
\(971\) 33.4891 1.07472 0.537359 0.843354i \(-0.319422\pi\)
0.537359 + 0.843354i \(0.319422\pi\)
\(972\) 0 0
\(973\) 4.74456 0.152104
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −52.9783 −1.69492 −0.847462 0.530856i \(-0.821871\pi\)
−0.847462 + 0.530856i \(0.821871\pi\)
\(978\) 0 0
\(979\) −93.9565 −3.00286
\(980\) 0 0
\(981\) 0 0
\(982\) 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
\(993\) 0 0
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) −6.88316 −0.217992 −0.108996 0.994042i \(-0.534764\pi\)
−0.108996 + 0.994042i \(0.534764\pi\)
\(998\) 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 5040.2.a.by.1.2 2
3.2 odd 2 560.2.a.h.1.1 2
4.3 odd 2 2520.2.a.x.1.1 2
12.11 even 2 280.2.a.c.1.2 2
15.2 even 4 2800.2.g.r.449.3 4
15.8 even 4 2800.2.g.r.449.2 4
15.14 odd 2 2800.2.a.bk.1.2 2
21.20 even 2 3920.2.a.bt.1.2 2
24.5 odd 2 2240.2.a.bg.1.2 2
24.11 even 2 2240.2.a.bk.1.1 2
60.23 odd 4 1400.2.g.i.449.3 4
60.47 odd 4 1400.2.g.i.449.2 4
60.59 even 2 1400.2.a.r.1.1 2
84.11 even 6 1960.2.q.t.961.1 4
84.23 even 6 1960.2.q.t.361.1 4
84.47 odd 6 1960.2.q.r.361.2 4
84.59 odd 6 1960.2.q.r.961.2 4
84.83 odd 2 1960.2.a.s.1.1 2
420.419 odd 2 9800.2.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.c.1.2 2 12.11 even 2
560.2.a.h.1.1 2 3.2 odd 2
1400.2.a.r.1.1 2 60.59 even 2
1400.2.g.i.449.2 4 60.47 odd 4
1400.2.g.i.449.3 4 60.23 odd 4
1960.2.a.s.1.1 2 84.83 odd 2
1960.2.q.r.361.2 4 84.47 odd 6
1960.2.q.r.961.2 4 84.59 odd 6
1960.2.q.t.361.1 4 84.23 even 6
1960.2.q.t.961.1 4 84.11 even 6
2240.2.a.bg.1.2 2 24.5 odd 2
2240.2.a.bk.1.1 2 24.11 even 2
2520.2.a.x.1.1 2 4.3 odd 2
2800.2.a.bk.1.2 2 15.14 odd 2
2800.2.g.r.449.2 4 15.8 even 4
2800.2.g.r.449.3 4 15.2 even 4
3920.2.a.bt.1.2 2 21.20 even 2
5040.2.a.by.1.2 2 1.1 even 1 trivial
9800.2.a.bu.1.2 2 420.419 odd 2