Properties

Label 2646.2.a.bo.1.1
Level $2646$
Weight $2$
Character 2646.1
Self dual yes
Analytic conductor $21.128$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.64575\) of defining polynomial
Character \(\chi\) \(=\) 2646.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.64575 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.64575 q^{5} +1.00000 q^{8} -1.64575 q^{10} -1.64575 q^{11} -0.645751 q^{13} +1.00000 q^{16} +1.64575 q^{17} -2.00000 q^{19} -1.64575 q^{20} -1.64575 q^{22} +9.29150 q^{23} -2.29150 q^{25} -0.645751 q^{26} +7.64575 q^{29} -0.645751 q^{31} +1.00000 q^{32} +1.64575 q^{34} +3.93725 q^{37} -2.00000 q^{38} -1.64575 q^{40} +4.93725 q^{41} +5.00000 q^{43} -1.64575 q^{44} +9.29150 q^{46} +10.9373 q^{47} -2.29150 q^{50} -0.645751 q^{52} +6.00000 q^{53} +2.70850 q^{55} +7.64575 q^{58} -13.6458 q^{59} -12.6458 q^{61} -0.645751 q^{62} +1.00000 q^{64} +1.06275 q^{65} +8.29150 q^{67} +1.64575 q^{68} +10.3542 q^{71} +10.5830 q^{73} +3.93725 q^{74} -2.00000 q^{76} -15.2288 q^{79} -1.64575 q^{80} +4.93725 q^{82} +2.70850 q^{83} -2.70850 q^{85} +5.00000 q^{86} -1.64575 q^{88} +10.9373 q^{89} +9.29150 q^{92} +10.9373 q^{94} +3.29150 q^{95} +7.58301 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{13} + 2 q^{16} - 2 q^{17} - 4 q^{19} + 2 q^{20} + 2 q^{22} + 8 q^{23} + 6 q^{25} + 4 q^{26} + 10 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{34} - 8 q^{37} - 4 q^{38} + 2 q^{40} - 6 q^{41} + 10 q^{43} + 2 q^{44} + 8 q^{46} + 6 q^{47} + 6 q^{50} + 4 q^{52} + 12 q^{53} + 16 q^{55} + 10 q^{58} - 22 q^{59} - 20 q^{61} + 4 q^{62} + 2 q^{64} + 18 q^{65} + 6 q^{67} - 2 q^{68} + 26 q^{71} - 8 q^{74} - 4 q^{76} - 4 q^{79} + 2 q^{80} - 6 q^{82} + 16 q^{83} - 16 q^{85} + 10 q^{86} + 2 q^{88} + 6 q^{89} + 8 q^{92} + 6 q^{94} - 4 q^{95} - 6 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.64575 −0.736002 −0.368001 0.929825i \(-0.619958\pi\)
−0.368001 + 0.929825i \(0.619958\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.64575 −0.520432
\(11\) −1.64575 −0.496213 −0.248106 0.968733i \(-0.579808\pi\)
−0.248106 + 0.968733i \(0.579808\pi\)
\(12\) 0 0
\(13\) −0.645751 −0.179099 −0.0895496 0.995982i \(-0.528543\pi\)
−0.0895496 + 0.995982i \(0.528543\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.64575 0.399153 0.199577 0.979882i \(-0.436043\pi\)
0.199577 + 0.979882i \(0.436043\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −1.64575 −0.368001
\(21\) 0 0
\(22\) −1.64575 −0.350875
\(23\) 9.29150 1.93741 0.968706 0.248211i \(-0.0798425\pi\)
0.968706 + 0.248211i \(0.0798425\pi\)
\(24\) 0 0
\(25\) −2.29150 −0.458301
\(26\) −0.645751 −0.126642
\(27\) 0 0
\(28\) 0 0
\(29\) 7.64575 1.41978 0.709890 0.704312i \(-0.248745\pi\)
0.709890 + 0.704312i \(0.248745\pi\)
\(30\) 0 0
\(31\) −0.645751 −0.115980 −0.0579902 0.998317i \(-0.518469\pi\)
−0.0579902 + 0.998317i \(0.518469\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.64575 0.282244
\(35\) 0 0
\(36\) 0 0
\(37\) 3.93725 0.647281 0.323640 0.946180i \(-0.395093\pi\)
0.323640 + 0.946180i \(0.395093\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −1.64575 −0.260216
\(41\) 4.93725 0.771070 0.385535 0.922693i \(-0.374017\pi\)
0.385535 + 0.922693i \(0.374017\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) −1.64575 −0.248106
\(45\) 0 0
\(46\) 9.29150 1.36996
\(47\) 10.9373 1.59536 0.797681 0.603079i \(-0.206060\pi\)
0.797681 + 0.603079i \(0.206060\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.29150 −0.324067
\(51\) 0 0
\(52\) −0.645751 −0.0895496
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 2.70850 0.365214
\(56\) 0 0
\(57\) 0 0
\(58\) 7.64575 1.00394
\(59\) −13.6458 −1.77653 −0.888263 0.459336i \(-0.848088\pi\)
−0.888263 + 0.459336i \(0.848088\pi\)
\(60\) 0 0
\(61\) −12.6458 −1.61912 −0.809561 0.587035i \(-0.800295\pi\)
−0.809561 + 0.587035i \(0.800295\pi\)
\(62\) −0.645751 −0.0820105
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.06275 0.131817
\(66\) 0 0
\(67\) 8.29150 1.01297 0.506484 0.862249i \(-0.330945\pi\)
0.506484 + 0.862249i \(0.330945\pi\)
\(68\) 1.64575 0.199577
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3542 1.22882 0.614412 0.788986i \(-0.289393\pi\)
0.614412 + 0.788986i \(0.289393\pi\)
\(72\) 0 0
\(73\) 10.5830 1.23865 0.619324 0.785136i \(-0.287407\pi\)
0.619324 + 0.785136i \(0.287407\pi\)
\(74\) 3.93725 0.457696
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) −15.2288 −1.71337 −0.856684 0.515841i \(-0.827480\pi\)
−0.856684 + 0.515841i \(0.827480\pi\)
\(80\) −1.64575 −0.184001
\(81\) 0 0
\(82\) 4.93725 0.545228
\(83\) 2.70850 0.297296 0.148648 0.988890i \(-0.452508\pi\)
0.148648 + 0.988890i \(0.452508\pi\)
\(84\) 0 0
\(85\) −2.70850 −0.293778
\(86\) 5.00000 0.539164
\(87\) 0 0
\(88\) −1.64575 −0.175438
\(89\) 10.9373 1.15935 0.579673 0.814849i \(-0.303180\pi\)
0.579673 + 0.814849i \(0.303180\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.29150 0.968706
\(93\) 0 0
\(94\) 10.9373 1.12809
\(95\) 3.29150 0.337701
\(96\) 0 0
\(97\) 7.58301 0.769938 0.384969 0.922930i \(-0.374212\pi\)
0.384969 + 0.922930i \(0.374212\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.29150 −0.229150
\(101\) −13.6458 −1.35780 −0.678902 0.734229i \(-0.737544\pi\)
−0.678902 + 0.734229i \(0.737544\pi\)
\(102\) 0 0
\(103\) 11.9373 1.17621 0.588106 0.808784i \(-0.299874\pi\)
0.588106 + 0.808784i \(0.299874\pi\)
\(104\) −0.645751 −0.0633211
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) −8.64575 −0.828113 −0.414056 0.910251i \(-0.635888\pi\)
−0.414056 + 0.910251i \(0.635888\pi\)
\(110\) 2.70850 0.258245
\(111\) 0 0
\(112\) 0 0
\(113\) 7.64575 0.719252 0.359626 0.933097i \(-0.382904\pi\)
0.359626 + 0.933097i \(0.382904\pi\)
\(114\) 0 0
\(115\) −15.2915 −1.42594
\(116\) 7.64575 0.709890
\(117\) 0 0
\(118\) −13.6458 −1.25619
\(119\) 0 0
\(120\) 0 0
\(121\) −8.29150 −0.753773
\(122\) −12.6458 −1.14489
\(123\) 0 0
\(124\) −0.645751 −0.0579902
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 6.64575 0.589715 0.294858 0.955541i \(-0.404728\pi\)
0.294858 + 0.955541i \(0.404728\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 1.06275 0.0932090
\(131\) −6.58301 −0.575160 −0.287580 0.957757i \(-0.592851\pi\)
−0.287580 + 0.957757i \(0.592851\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 8.29150 0.716277
\(135\) 0 0
\(136\) 1.64575 0.141122
\(137\) −9.29150 −0.793827 −0.396913 0.917856i \(-0.629919\pi\)
−0.396913 + 0.917856i \(0.629919\pi\)
\(138\) 0 0
\(139\) 19.5830 1.66101 0.830504 0.557012i \(-0.188052\pi\)
0.830504 + 0.557012i \(0.188052\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.3542 0.868909
\(143\) 1.06275 0.0888713
\(144\) 0 0
\(145\) −12.5830 −1.04496
\(146\) 10.5830 0.875856
\(147\) 0 0
\(148\) 3.93725 0.323640
\(149\) 7.06275 0.578603 0.289301 0.957238i \(-0.406577\pi\)
0.289301 + 0.957238i \(0.406577\pi\)
\(150\) 0 0
\(151\) 7.22876 0.588268 0.294134 0.955764i \(-0.404969\pi\)
0.294134 + 0.955764i \(0.404969\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) 0 0
\(155\) 1.06275 0.0853618
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −15.2288 −1.21153
\(159\) 0 0
\(160\) −1.64575 −0.130108
\(161\) 0 0
\(162\) 0 0
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 4.93725 0.385535
\(165\) 0 0
\(166\) 2.70850 0.210220
\(167\) −12.5830 −0.973702 −0.486851 0.873485i \(-0.661855\pi\)
−0.486851 + 0.873485i \(0.661855\pi\)
\(168\) 0 0
\(169\) −12.5830 −0.967923
\(170\) −2.70850 −0.207732
\(171\) 0 0
\(172\) 5.00000 0.381246
\(173\) −6.58301 −0.500497 −0.250248 0.968182i \(-0.580512\pi\)
−0.250248 + 0.968182i \(0.580512\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.64575 −0.124053
\(177\) 0 0
\(178\) 10.9373 0.819782
\(179\) 1.06275 0.0794334 0.0397167 0.999211i \(-0.487354\pi\)
0.0397167 + 0.999211i \(0.487354\pi\)
\(180\) 0 0
\(181\) 13.2915 0.987950 0.493975 0.869476i \(-0.335543\pi\)
0.493975 + 0.869476i \(0.335543\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 9.29150 0.684979
\(185\) −6.47974 −0.476400
\(186\) 0 0
\(187\) −2.70850 −0.198065
\(188\) 10.9373 0.797681
\(189\) 0 0
\(190\) 3.29150 0.238791
\(191\) 26.8118 1.94003 0.970015 0.243043i \(-0.0781456\pi\)
0.970015 + 0.243043i \(0.0781456\pi\)
\(192\) 0 0
\(193\) −7.00000 −0.503871 −0.251936 0.967744i \(-0.581067\pi\)
−0.251936 + 0.967744i \(0.581067\pi\)
\(194\) 7.58301 0.544428
\(195\) 0 0
\(196\) 0 0
\(197\) 15.2915 1.08947 0.544737 0.838607i \(-0.316629\pi\)
0.544737 + 0.838607i \(0.316629\pi\)
\(198\) 0 0
\(199\) −21.9373 −1.55509 −0.777545 0.628827i \(-0.783535\pi\)
−0.777545 + 0.628827i \(0.783535\pi\)
\(200\) −2.29150 −0.162034
\(201\) 0 0
\(202\) −13.6458 −0.960112
\(203\) 0 0
\(204\) 0 0
\(205\) −8.12549 −0.567509
\(206\) 11.9373 0.831708
\(207\) 0 0
\(208\) −0.645751 −0.0447748
\(209\) 3.29150 0.227678
\(210\) 0 0
\(211\) −16.8745 −1.16169 −0.580845 0.814015i \(-0.697278\pi\)
−0.580845 + 0.814015i \(0.697278\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −6.00000 −0.410152
\(215\) −8.22876 −0.561197
\(216\) 0 0
\(217\) 0 0
\(218\) −8.64575 −0.585564
\(219\) 0 0
\(220\) 2.70850 0.182607
\(221\) −1.06275 −0.0714880
\(222\) 0 0
\(223\) −17.8745 −1.19697 −0.598483 0.801136i \(-0.704230\pi\)
−0.598483 + 0.801136i \(0.704230\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 7.64575 0.508588
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) 0 0
\(229\) 14.6458 0.967818 0.483909 0.875118i \(-0.339216\pi\)
0.483909 + 0.875118i \(0.339216\pi\)
\(230\) −15.2915 −1.00829
\(231\) 0 0
\(232\) 7.64575 0.501968
\(233\) −8.70850 −0.570513 −0.285256 0.958451i \(-0.592079\pi\)
−0.285256 + 0.958451i \(0.592079\pi\)
\(234\) 0 0
\(235\) −18.0000 −1.17419
\(236\) −13.6458 −0.888263
\(237\) 0 0
\(238\) 0 0
\(239\) 4.93725 0.319364 0.159682 0.987168i \(-0.448953\pi\)
0.159682 + 0.987168i \(0.448953\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) −8.29150 −0.532998
\(243\) 0 0
\(244\) −12.6458 −0.809561
\(245\) 0 0
\(246\) 0 0
\(247\) 1.29150 0.0821763
\(248\) −0.645751 −0.0410052
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) −15.2915 −0.961369
\(254\) 6.64575 0.416992
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.8745 0.990225 0.495112 0.868829i \(-0.335127\pi\)
0.495112 + 0.868829i \(0.335127\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.06275 0.0659087
\(261\) 0 0
\(262\) −6.58301 −0.406699
\(263\) 10.9373 0.674420 0.337210 0.941429i \(-0.390517\pi\)
0.337210 + 0.941429i \(0.390517\pi\)
\(264\) 0 0
\(265\) −9.87451 −0.606586
\(266\) 0 0
\(267\) 0 0
\(268\) 8.29150 0.506484
\(269\) 27.2915 1.66399 0.831996 0.554781i \(-0.187198\pi\)
0.831996 + 0.554781i \(0.187198\pi\)
\(270\) 0 0
\(271\) 21.2288 1.28956 0.644778 0.764370i \(-0.276950\pi\)
0.644778 + 0.764370i \(0.276950\pi\)
\(272\) 1.64575 0.0997883
\(273\) 0 0
\(274\) −9.29150 −0.561320
\(275\) 3.77124 0.227415
\(276\) 0 0
\(277\) −17.9373 −1.07775 −0.538873 0.842387i \(-0.681150\pi\)
−0.538873 + 0.842387i \(0.681150\pi\)
\(278\) 19.5830 1.17451
\(279\) 0 0
\(280\) 0 0
\(281\) −17.5203 −1.04517 −0.522586 0.852587i \(-0.675032\pi\)
−0.522586 + 0.852587i \(0.675032\pi\)
\(282\) 0 0
\(283\) −8.29150 −0.492879 −0.246439 0.969158i \(-0.579261\pi\)
−0.246439 + 0.969158i \(0.579261\pi\)
\(284\) 10.3542 0.614412
\(285\) 0 0
\(286\) 1.06275 0.0628415
\(287\) 0 0
\(288\) 0 0
\(289\) −14.2915 −0.840677
\(290\) −12.5830 −0.738900
\(291\) 0 0
\(292\) 10.5830 0.619324
\(293\) −22.9373 −1.34001 −0.670004 0.742357i \(-0.733708\pi\)
−0.670004 + 0.742357i \(0.733708\pi\)
\(294\) 0 0
\(295\) 22.4575 1.30753
\(296\) 3.93725 0.228848
\(297\) 0 0
\(298\) 7.06275 0.409134
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) 7.22876 0.415968
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 20.8118 1.19168
\(306\) 0 0
\(307\) 13.5830 0.775223 0.387612 0.921823i \(-0.373300\pi\)
0.387612 + 0.921823i \(0.373300\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.06275 0.0603599
\(311\) −23.5203 −1.33371 −0.666856 0.745187i \(-0.732360\pi\)
−0.666856 + 0.745187i \(0.732360\pi\)
\(312\) 0 0
\(313\) −23.2915 −1.31651 −0.658257 0.752793i \(-0.728706\pi\)
−0.658257 + 0.752793i \(0.728706\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −15.2288 −0.856684
\(317\) −20.8118 −1.16890 −0.584452 0.811428i \(-0.698691\pi\)
−0.584452 + 0.811428i \(0.698691\pi\)
\(318\) 0 0
\(319\) −12.5830 −0.704513
\(320\) −1.64575 −0.0920003
\(321\) 0 0
\(322\) 0 0
\(323\) −3.29150 −0.183144
\(324\) 0 0
\(325\) 1.47974 0.0820813
\(326\) −1.00000 −0.0553849
\(327\) 0 0
\(328\) 4.93725 0.272614
\(329\) 0 0
\(330\) 0 0
\(331\) −0.125492 −0.00689767 −0.00344884 0.999994i \(-0.501098\pi\)
−0.00344884 + 0.999994i \(0.501098\pi\)
\(332\) 2.70850 0.148648
\(333\) 0 0
\(334\) −12.5830 −0.688511
\(335\) −13.6458 −0.745547
\(336\) 0 0
\(337\) −9.41699 −0.512976 −0.256488 0.966547i \(-0.582565\pi\)
−0.256488 + 0.966547i \(0.582565\pi\)
\(338\) −12.5830 −0.684425
\(339\) 0 0
\(340\) −2.70850 −0.146889
\(341\) 1.06275 0.0575509
\(342\) 0 0
\(343\) 0 0
\(344\) 5.00000 0.269582
\(345\) 0 0
\(346\) −6.58301 −0.353905
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −25.2288 −1.35046 −0.675232 0.737605i \(-0.735957\pi\)
−0.675232 + 0.737605i \(0.735957\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.64575 −0.0877188
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 0 0
\(355\) −17.0405 −0.904417
\(356\) 10.9373 0.579673
\(357\) 0 0
\(358\) 1.06275 0.0561679
\(359\) −31.1660 −1.64488 −0.822440 0.568853i \(-0.807388\pi\)
−0.822440 + 0.568853i \(0.807388\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 13.2915 0.698586
\(363\) 0 0
\(364\) 0 0
\(365\) −17.4170 −0.911647
\(366\) 0 0
\(367\) 1.87451 0.0978485 0.0489243 0.998802i \(-0.484421\pi\)
0.0489243 + 0.998802i \(0.484421\pi\)
\(368\) 9.29150 0.484353
\(369\) 0 0
\(370\) −6.47974 −0.336866
\(371\) 0 0
\(372\) 0 0
\(373\) −16.5830 −0.858635 −0.429318 0.903154i \(-0.641246\pi\)
−0.429318 + 0.903154i \(0.641246\pi\)
\(374\) −2.70850 −0.140053
\(375\) 0 0
\(376\) 10.9373 0.564046
\(377\) −4.93725 −0.254282
\(378\) 0 0
\(379\) 4.41699 0.226886 0.113443 0.993545i \(-0.463812\pi\)
0.113443 + 0.993545i \(0.463812\pi\)
\(380\) 3.29150 0.168851
\(381\) 0 0
\(382\) 26.8118 1.37181
\(383\) 0.583005 0.0297902 0.0148951 0.999889i \(-0.495259\pi\)
0.0148951 + 0.999889i \(0.495259\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −7.00000 −0.356291
\(387\) 0 0
\(388\) 7.58301 0.384969
\(389\) 8.70850 0.441538 0.220769 0.975326i \(-0.429143\pi\)
0.220769 + 0.975326i \(0.429143\pi\)
\(390\) 0 0
\(391\) 15.2915 0.773325
\(392\) 0 0
\(393\) 0 0
\(394\) 15.2915 0.770375
\(395\) 25.0627 1.26104
\(396\) 0 0
\(397\) 11.3542 0.569853 0.284927 0.958549i \(-0.408031\pi\)
0.284927 + 0.958549i \(0.408031\pi\)
\(398\) −21.9373 −1.09962
\(399\) 0 0
\(400\) −2.29150 −0.114575
\(401\) 26.8118 1.33892 0.669458 0.742850i \(-0.266527\pi\)
0.669458 + 0.742850i \(0.266527\pi\)
\(402\) 0 0
\(403\) 0.416995 0.0207720
\(404\) −13.6458 −0.678902
\(405\) 0 0
\(406\) 0 0
\(407\) −6.47974 −0.321189
\(408\) 0 0
\(409\) 16.8745 0.834391 0.417195 0.908817i \(-0.363013\pi\)
0.417195 + 0.908817i \(0.363013\pi\)
\(410\) −8.12549 −0.401289
\(411\) 0 0
\(412\) 11.9373 0.588106
\(413\) 0 0
\(414\) 0 0
\(415\) −4.45751 −0.218811
\(416\) −0.645751 −0.0316606
\(417\) 0 0
\(418\) 3.29150 0.160993
\(419\) −13.0627 −0.638157 −0.319078 0.947728i \(-0.603373\pi\)
−0.319078 + 0.947728i \(0.603373\pi\)
\(420\) 0 0
\(421\) −25.2915 −1.23263 −0.616316 0.787499i \(-0.711376\pi\)
−0.616316 + 0.787499i \(0.711376\pi\)
\(422\) −16.8745 −0.821438
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −3.77124 −0.182932
\(426\) 0 0
\(427\) 0 0
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) −8.22876 −0.396826
\(431\) −8.81176 −0.424448 −0.212224 0.977221i \(-0.568071\pi\)
−0.212224 + 0.977221i \(0.568071\pi\)
\(432\) 0 0
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −8.64575 −0.414056
\(437\) −18.5830 −0.888946
\(438\) 0 0
\(439\) 17.1660 0.819289 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(440\) 2.70850 0.129123
\(441\) 0 0
\(442\) −1.06275 −0.0505497
\(443\) −8.70850 −0.413753 −0.206877 0.978367i \(-0.566330\pi\)
−0.206877 + 0.978367i \(0.566330\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) −17.8745 −0.846382
\(447\) 0 0
\(448\) 0 0
\(449\) 7.64575 0.360825 0.180413 0.983591i \(-0.442257\pi\)
0.180413 + 0.983591i \(0.442257\pi\)
\(450\) 0 0
\(451\) −8.12549 −0.382614
\(452\) 7.64575 0.359626
\(453\) 0 0
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) 2.29150 0.107192 0.0535960 0.998563i \(-0.482932\pi\)
0.0535960 + 0.998563i \(0.482932\pi\)
\(458\) 14.6458 0.684351
\(459\) 0 0
\(460\) −15.2915 −0.712970
\(461\) 2.22876 0.103804 0.0519018 0.998652i \(-0.483472\pi\)
0.0519018 + 0.998652i \(0.483472\pi\)
\(462\) 0 0
\(463\) 29.2915 1.36129 0.680646 0.732613i \(-0.261699\pi\)
0.680646 + 0.732613i \(0.261699\pi\)
\(464\) 7.64575 0.354945
\(465\) 0 0
\(466\) −8.70850 −0.403413
\(467\) 26.2288 1.21372 0.606861 0.794808i \(-0.292428\pi\)
0.606861 + 0.794808i \(0.292428\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −18.0000 −0.830278
\(471\) 0 0
\(472\) −13.6458 −0.628097
\(473\) −8.22876 −0.378359
\(474\) 0 0
\(475\) 4.58301 0.210283
\(476\) 0 0
\(477\) 0 0
\(478\) 4.93725 0.225825
\(479\) 1.64575 0.0751963 0.0375981 0.999293i \(-0.488029\pi\)
0.0375981 + 0.999293i \(0.488029\pi\)
\(480\) 0 0
\(481\) −2.54249 −0.115927
\(482\) −5.00000 −0.227744
\(483\) 0 0
\(484\) −8.29150 −0.376886
\(485\) −12.4797 −0.566676
\(486\) 0 0
\(487\) −7.87451 −0.356828 −0.178414 0.983956i \(-0.557097\pi\)
−0.178414 + 0.983956i \(0.557097\pi\)
\(488\) −12.6458 −0.572446
\(489\) 0 0
\(490\) 0 0
\(491\) −37.7490 −1.70359 −0.851795 0.523876i \(-0.824486\pi\)
−0.851795 + 0.523876i \(0.824486\pi\)
\(492\) 0 0
\(493\) 12.5830 0.566710
\(494\) 1.29150 0.0581075
\(495\) 0 0
\(496\) −0.645751 −0.0289951
\(497\) 0 0
\(498\) 0 0
\(499\) 36.1660 1.61901 0.809506 0.587111i \(-0.199735\pi\)
0.809506 + 0.587111i \(0.199735\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 18.0000 0.803379
\(503\) 27.8745 1.24286 0.621431 0.783469i \(-0.286551\pi\)
0.621431 + 0.783469i \(0.286551\pi\)
\(504\) 0 0
\(505\) 22.4575 0.999346
\(506\) −15.2915 −0.679790
\(507\) 0 0
\(508\) 6.64575 0.294858
\(509\) −39.8745 −1.76741 −0.883703 0.468048i \(-0.844958\pi\)
−0.883703 + 0.468048i \(0.844958\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 15.8745 0.700195
\(515\) −19.6458 −0.865695
\(516\) 0 0
\(517\) −18.0000 −0.791639
\(518\) 0 0
\(519\) 0 0
\(520\) 1.06275 0.0466045
\(521\) 39.8745 1.74693 0.873467 0.486883i \(-0.161866\pi\)
0.873467 + 0.486883i \(0.161866\pi\)
\(522\) 0 0
\(523\) 1.00000 0.0437269 0.0218635 0.999761i \(-0.493040\pi\)
0.0218635 + 0.999761i \(0.493040\pi\)
\(524\) −6.58301 −0.287580
\(525\) 0 0
\(526\) 10.9373 0.476887
\(527\) −1.06275 −0.0462939
\(528\) 0 0
\(529\) 63.3320 2.75357
\(530\) −9.87451 −0.428921
\(531\) 0 0
\(532\) 0 0
\(533\) −3.18824 −0.138098
\(534\) 0 0
\(535\) 9.87451 0.426912
\(536\) 8.29150 0.358138
\(537\) 0 0
\(538\) 27.2915 1.17662
\(539\) 0 0
\(540\) 0 0
\(541\) −41.1660 −1.76987 −0.884933 0.465719i \(-0.845796\pi\)
−0.884933 + 0.465719i \(0.845796\pi\)
\(542\) 21.2288 0.911853
\(543\) 0 0
\(544\) 1.64575 0.0705610
\(545\) 14.2288 0.609493
\(546\) 0 0
\(547\) 7.70850 0.329592 0.164796 0.986328i \(-0.447303\pi\)
0.164796 + 0.986328i \(0.447303\pi\)
\(548\) −9.29150 −0.396913
\(549\) 0 0
\(550\) 3.77124 0.160806
\(551\) −15.2915 −0.651440
\(552\) 0 0
\(553\) 0 0
\(554\) −17.9373 −0.762081
\(555\) 0 0
\(556\) 19.5830 0.830504
\(557\) −32.2288 −1.36558 −0.682788 0.730616i \(-0.739233\pi\)
−0.682788 + 0.730616i \(0.739233\pi\)
\(558\) 0 0
\(559\) −3.22876 −0.136562
\(560\) 0 0
\(561\) 0 0
\(562\) −17.5203 −0.739048
\(563\) 34.4575 1.45221 0.726106 0.687583i \(-0.241328\pi\)
0.726106 + 0.687583i \(0.241328\pi\)
\(564\) 0 0
\(565\) −12.5830 −0.529371
\(566\) −8.29150 −0.348518
\(567\) 0 0
\(568\) 10.3542 0.434455
\(569\) −1.06275 −0.0445526 −0.0222763 0.999752i \(-0.507091\pi\)
−0.0222763 + 0.999752i \(0.507091\pi\)
\(570\) 0 0
\(571\) −1.29150 −0.0540477 −0.0270239 0.999635i \(-0.508603\pi\)
−0.0270239 + 0.999635i \(0.508603\pi\)
\(572\) 1.06275 0.0444356
\(573\) 0 0
\(574\) 0 0
\(575\) −21.2915 −0.887917
\(576\) 0 0
\(577\) 21.7085 0.903737 0.451868 0.892085i \(-0.350758\pi\)
0.451868 + 0.892085i \(0.350758\pi\)
\(578\) −14.2915 −0.594448
\(579\) 0 0
\(580\) −12.5830 −0.522481
\(581\) 0 0
\(582\) 0 0
\(583\) −9.87451 −0.408960
\(584\) 10.5830 0.437928
\(585\) 0 0
\(586\) −22.9373 −0.947529
\(587\) −38.2288 −1.57787 −0.788935 0.614477i \(-0.789367\pi\)
−0.788935 + 0.614477i \(0.789367\pi\)
\(588\) 0 0
\(589\) 1.29150 0.0532154
\(590\) 22.4575 0.924561
\(591\) 0 0
\(592\) 3.93725 0.161820
\(593\) −25.0627 −1.02920 −0.514602 0.857429i \(-0.672060\pi\)
−0.514602 + 0.857429i \(0.672060\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.06275 0.289301
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) −21.8745 −0.893768 −0.446884 0.894592i \(-0.647466\pi\)
−0.446884 + 0.894592i \(0.647466\pi\)
\(600\) 0 0
\(601\) 4.87451 0.198835 0.0994177 0.995046i \(-0.468302\pi\)
0.0994177 + 0.995046i \(0.468302\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 7.22876 0.294134
\(605\) 13.6458 0.554779
\(606\) 0 0
\(607\) −26.5830 −1.07897 −0.539485 0.841995i \(-0.681381\pi\)
−0.539485 + 0.841995i \(0.681381\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 20.8118 0.842644
\(611\) −7.06275 −0.285728
\(612\) 0 0
\(613\) 26.3948 1.06607 0.533037 0.846092i \(-0.321051\pi\)
0.533037 + 0.846092i \(0.321051\pi\)
\(614\) 13.5830 0.548165
\(615\) 0 0
\(616\) 0 0
\(617\) −35.5203 −1.42999 −0.714996 0.699129i \(-0.753571\pi\)
−0.714996 + 0.699129i \(0.753571\pi\)
\(618\) 0 0
\(619\) −2.29150 −0.0921033 −0.0460516 0.998939i \(-0.514664\pi\)
−0.0460516 + 0.998939i \(0.514664\pi\)
\(620\) 1.06275 0.0426809
\(621\) 0 0
\(622\) −23.5203 −0.943076
\(623\) 0 0
\(624\) 0 0
\(625\) −8.29150 −0.331660
\(626\) −23.2915 −0.930916
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) 6.47974 0.258364
\(630\) 0 0
\(631\) 21.9373 0.873308 0.436654 0.899629i \(-0.356163\pi\)
0.436654 + 0.899629i \(0.356163\pi\)
\(632\) −15.2288 −0.605767
\(633\) 0 0
\(634\) −20.8118 −0.826541
\(635\) −10.9373 −0.434032
\(636\) 0 0
\(637\) 0 0
\(638\) −12.5830 −0.498166
\(639\) 0 0
\(640\) −1.64575 −0.0650540
\(641\) 8.22876 0.325016 0.162508 0.986707i \(-0.448042\pi\)
0.162508 + 0.986707i \(0.448042\pi\)
\(642\) 0 0
\(643\) 34.8745 1.37532 0.687658 0.726035i \(-0.258639\pi\)
0.687658 + 0.726035i \(0.258639\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3.29150 −0.129502
\(647\) 21.8745 0.859976 0.429988 0.902835i \(-0.358518\pi\)
0.429988 + 0.902835i \(0.358518\pi\)
\(648\) 0 0
\(649\) 22.4575 0.881534
\(650\) 1.47974 0.0580402
\(651\) 0 0
\(652\) −1.00000 −0.0391630
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 10.8340 0.423319
\(656\) 4.93725 0.192767
\(657\) 0 0
\(658\) 0 0
\(659\) 2.22876 0.0868200 0.0434100 0.999057i \(-0.486178\pi\)
0.0434100 + 0.999057i \(0.486178\pi\)
\(660\) 0 0
\(661\) −39.1660 −1.52338 −0.761691 0.647941i \(-0.775630\pi\)
−0.761691 + 0.647941i \(0.775630\pi\)
\(662\) −0.125492 −0.00487739
\(663\) 0 0
\(664\) 2.70850 0.105110
\(665\) 0 0
\(666\) 0 0
\(667\) 71.0405 2.75070
\(668\) −12.5830 −0.486851
\(669\) 0 0
\(670\) −13.6458 −0.527181
\(671\) 20.8118 0.803429
\(672\) 0 0
\(673\) −47.7490 −1.84059 −0.920295 0.391226i \(-0.872051\pi\)
−0.920295 + 0.391226i \(0.872051\pi\)
\(674\) −9.41699 −0.362729
\(675\) 0 0
\(676\) −12.5830 −0.483962
\(677\) 14.1255 0.542887 0.271443 0.962454i \(-0.412499\pi\)
0.271443 + 0.962454i \(0.412499\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.70850 −0.103866
\(681\) 0 0
\(682\) 1.06275 0.0406947
\(683\) 20.8118 0.796340 0.398170 0.917312i \(-0.369645\pi\)
0.398170 + 0.917312i \(0.369645\pi\)
\(684\) 0 0
\(685\) 15.2915 0.584258
\(686\) 0 0
\(687\) 0 0
\(688\) 5.00000 0.190623
\(689\) −3.87451 −0.147607
\(690\) 0 0
\(691\) −24.7490 −0.941497 −0.470748 0.882267i \(-0.656016\pi\)
−0.470748 + 0.882267i \(0.656016\pi\)
\(692\) −6.58301 −0.250248
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −32.2288 −1.22251
\(696\) 0 0
\(697\) 8.12549 0.307775
\(698\) −25.2288 −0.954923
\(699\) 0 0
\(700\) 0 0
\(701\) −5.41699 −0.204597 −0.102299 0.994754i \(-0.532620\pi\)
−0.102299 + 0.994754i \(0.532620\pi\)
\(702\) 0 0
\(703\) −7.87451 −0.296993
\(704\) −1.64575 −0.0620266
\(705\) 0 0
\(706\) 12.0000 0.451626
\(707\) 0 0
\(708\) 0 0
\(709\) −9.81176 −0.368488 −0.184244 0.982880i \(-0.558984\pi\)
−0.184244 + 0.982880i \(0.558984\pi\)
\(710\) −17.0405 −0.639519
\(711\) 0 0
\(712\) 10.9373 0.409891
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) −1.74902 −0.0654095
\(716\) 1.06275 0.0397167
\(717\) 0 0
\(718\) −31.1660 −1.16311
\(719\) −7.06275 −0.263396 −0.131698 0.991290i \(-0.542043\pi\)
−0.131698 + 0.991290i \(0.542043\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) 13.2915 0.493975
\(725\) −17.5203 −0.650686
\(726\) 0 0
\(727\) −25.2288 −0.935683 −0.467841 0.883812i \(-0.654968\pi\)
−0.467841 + 0.883812i \(0.654968\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −17.4170 −0.644632
\(731\) 8.22876 0.304352
\(732\) 0 0
\(733\) −8.77124 −0.323973 −0.161987 0.986793i \(-0.551790\pi\)
−0.161987 + 0.986793i \(0.551790\pi\)
\(734\) 1.87451 0.0691893
\(735\) 0 0
\(736\) 9.29150 0.342489
\(737\) −13.6458 −0.502648
\(738\) 0 0
\(739\) 21.4575 0.789327 0.394664 0.918826i \(-0.370861\pi\)
0.394664 + 0.918826i \(0.370861\pi\)
\(740\) −6.47974 −0.238200
\(741\) 0 0
\(742\) 0 0
\(743\) −13.0627 −0.479226 −0.239613 0.970869i \(-0.577021\pi\)
−0.239613 + 0.970869i \(0.577021\pi\)
\(744\) 0 0
\(745\) −11.6235 −0.425853
\(746\) −16.5830 −0.607147
\(747\) 0 0
\(748\) −2.70850 −0.0990325
\(749\) 0 0
\(750\) 0 0
\(751\) 0.457513 0.0166949 0.00834745 0.999965i \(-0.497343\pi\)
0.00834745 + 0.999965i \(0.497343\pi\)
\(752\) 10.9373 0.398841
\(753\) 0 0
\(754\) −4.93725 −0.179804
\(755\) −11.8967 −0.432967
\(756\) 0 0
\(757\) −40.9778 −1.48936 −0.744681 0.667420i \(-0.767398\pi\)
−0.744681 + 0.667420i \(0.767398\pi\)
\(758\) 4.41699 0.160432
\(759\) 0 0
\(760\) 3.29150 0.119395
\(761\) 18.5830 0.673633 0.336817 0.941570i \(-0.390650\pi\)
0.336817 + 0.941570i \(0.390650\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 26.8118 0.970015
\(765\) 0 0
\(766\) 0.583005 0.0210648
\(767\) 8.81176 0.318174
\(768\) 0 0
\(769\) −19.4170 −0.700195 −0.350097 0.936713i \(-0.613852\pi\)
−0.350097 + 0.936713i \(0.613852\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.00000 −0.251936
\(773\) −38.3320 −1.37871 −0.689353 0.724425i \(-0.742105\pi\)
−0.689353 + 0.724425i \(0.742105\pi\)
\(774\) 0 0
\(775\) 1.47974 0.0531539
\(776\) 7.58301 0.272214
\(777\) 0 0
\(778\) 8.70850 0.312215
\(779\) −9.87451 −0.353791
\(780\) 0 0
\(781\) −17.0405 −0.609758
\(782\) 15.2915 0.546823
\(783\) 0 0
\(784\) 0 0
\(785\) −6.58301 −0.234958
\(786\) 0 0
\(787\) 22.2915 0.794606 0.397303 0.917687i \(-0.369946\pi\)
0.397303 + 0.917687i \(0.369946\pi\)
\(788\) 15.2915 0.544737
\(789\) 0 0
\(790\) 25.0627 0.891692
\(791\) 0 0
\(792\) 0 0
\(793\) 8.16601 0.289984
\(794\) 11.3542 0.402947
\(795\) 0 0
\(796\) −21.9373 −0.777545
\(797\) −32.8118 −1.16225 −0.581126 0.813814i \(-0.697388\pi\)
−0.581126 + 0.813814i \(0.697388\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) −2.29150 −0.0810169
\(801\) 0 0
\(802\) 26.8118 0.946756
\(803\) −17.4170 −0.614632
\(804\) 0 0
\(805\) 0 0
\(806\) 0.416995 0.0146880
\(807\) 0 0
\(808\) −13.6458 −0.480056
\(809\) 36.5830 1.28619 0.643095 0.765786i \(-0.277650\pi\)
0.643095 + 0.765786i \(0.277650\pi\)
\(810\) 0 0
\(811\) 18.7085 0.656944 0.328472 0.944514i \(-0.393466\pi\)
0.328472 + 0.944514i \(0.393466\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −6.47974 −0.227115
\(815\) 1.64575 0.0576482
\(816\) 0 0
\(817\) −10.0000 −0.349856
\(818\) 16.8745 0.590003
\(819\) 0 0
\(820\) −8.12549 −0.283754
\(821\) −0.583005 −0.0203470 −0.0101735 0.999948i \(-0.503238\pi\)
−0.0101735 + 0.999948i \(0.503238\pi\)
\(822\) 0 0
\(823\) 23.1033 0.805329 0.402665 0.915348i \(-0.368084\pi\)
0.402665 + 0.915348i \(0.368084\pi\)
\(824\) 11.9373 0.415854
\(825\) 0 0
\(826\) 0 0
\(827\) 19.7490 0.686741 0.343370 0.939200i \(-0.388431\pi\)
0.343370 + 0.939200i \(0.388431\pi\)
\(828\) 0 0
\(829\) 18.7085 0.649773 0.324886 0.945753i \(-0.394674\pi\)
0.324886 + 0.945753i \(0.394674\pi\)
\(830\) −4.45751 −0.154723
\(831\) 0 0
\(832\) −0.645751 −0.0223874
\(833\) 0 0
\(834\) 0 0
\(835\) 20.7085 0.716647
\(836\) 3.29150 0.113839
\(837\) 0 0
\(838\) −13.0627 −0.451245
\(839\) 38.7085 1.33637 0.668183 0.743997i \(-0.267072\pi\)
0.668183 + 0.743997i \(0.267072\pi\)
\(840\) 0 0
\(841\) 29.4575 1.01578
\(842\) −25.2915 −0.871603
\(843\) 0 0
\(844\) −16.8745 −0.580845
\(845\) 20.7085 0.712394
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) −3.77124 −0.129353
\(851\) 36.5830 1.25405
\(852\) 0 0
\(853\) −16.1255 −0.552126 −0.276063 0.961139i \(-0.589030\pi\)
−0.276063 + 0.961139i \(0.589030\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −6.00000 −0.205076
\(857\) −20.8118 −0.710916 −0.355458 0.934692i \(-0.615675\pi\)
−0.355458 + 0.934692i \(0.615675\pi\)
\(858\) 0 0
\(859\) 7.00000 0.238837 0.119418 0.992844i \(-0.461897\pi\)
0.119418 + 0.992844i \(0.461897\pi\)
\(860\) −8.22876 −0.280598
\(861\) 0 0
\(862\) −8.81176 −0.300130
\(863\) −13.1660 −0.448176 −0.224088 0.974569i \(-0.571940\pi\)
−0.224088 + 0.974569i \(0.571940\pi\)
\(864\) 0 0
\(865\) 10.8340 0.368367
\(866\) 19.0000 0.645646
\(867\) 0 0
\(868\) 0 0
\(869\) 25.0627 0.850195
\(870\) 0 0
\(871\) −5.35425 −0.181422
\(872\) −8.64575 −0.292782
\(873\) 0 0
\(874\) −18.5830 −0.628580
\(875\) 0 0
\(876\) 0 0
\(877\) 21.3542 0.721082 0.360541 0.932743i \(-0.382592\pi\)
0.360541 + 0.932743i \(0.382592\pi\)
\(878\) 17.1660 0.579325
\(879\) 0 0
\(880\) 2.70850 0.0913034
\(881\) −27.8745 −0.939116 −0.469558 0.882902i \(-0.655587\pi\)
−0.469558 + 0.882902i \(0.655587\pi\)
\(882\) 0 0
\(883\) 11.8745 0.399609 0.199805 0.979836i \(-0.435969\pi\)
0.199805 + 0.979836i \(0.435969\pi\)
\(884\) −1.06275 −0.0357440
\(885\) 0 0
\(886\) −8.70850 −0.292568
\(887\) 15.8745 0.533014 0.266507 0.963833i \(-0.414130\pi\)
0.266507 + 0.963833i \(0.414130\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −18.0000 −0.603361
\(891\) 0 0
\(892\) −17.8745 −0.598483
\(893\) −21.8745 −0.732002
\(894\) 0 0
\(895\) −1.74902 −0.0584631
\(896\) 0 0
\(897\) 0 0
\(898\) 7.64575 0.255142
\(899\) −4.93725 −0.164667
\(900\) 0 0
\(901\) 9.87451 0.328968
\(902\) −8.12549 −0.270549
\(903\) 0 0
\(904\) 7.64575 0.254294
\(905\) −21.8745 −0.727133
\(906\) 0 0
\(907\) 25.7085 0.853637 0.426818 0.904337i \(-0.359634\pi\)
0.426818 + 0.904337i \(0.359634\pi\)
\(908\) −6.00000 −0.199117
\(909\) 0 0
\(910\) 0 0
\(911\) 13.0627 0.432788 0.216394 0.976306i \(-0.430570\pi\)
0.216394 + 0.976306i \(0.430570\pi\)
\(912\) 0 0
\(913\) −4.45751 −0.147522
\(914\) 2.29150 0.0757962
\(915\) 0 0
\(916\) 14.6458 0.483909
\(917\) 0 0
\(918\) 0 0
\(919\) 55.2288 1.82183 0.910914 0.412596i \(-0.135378\pi\)
0.910914 + 0.412596i \(0.135378\pi\)
\(920\) −15.2915 −0.504146
\(921\) 0 0
\(922\) 2.22876 0.0734002
\(923\) −6.68627 −0.220081
\(924\) 0 0
\(925\) −9.02223 −0.296649
\(926\) 29.2915 0.962579
\(927\) 0 0
\(928\) 7.64575 0.250984
\(929\) 1.06275 0.0348676 0.0174338 0.999848i \(-0.494450\pi\)
0.0174338 + 0.999848i \(0.494450\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −8.70850 −0.285256
\(933\) 0 0
\(934\) 26.2288 0.858231
\(935\) 4.45751 0.145776
\(936\) 0 0
\(937\) −18.7490 −0.612504 −0.306252 0.951951i \(-0.599075\pi\)
−0.306252 + 0.951951i \(0.599075\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −18.0000 −0.587095
\(941\) −15.3948 −0.501855 −0.250928 0.968006i \(-0.580736\pi\)
−0.250928 + 0.968006i \(0.580736\pi\)
\(942\) 0 0
\(943\) 45.8745 1.49388
\(944\) −13.6458 −0.444131
\(945\) 0 0
\(946\) −8.22876 −0.267540
\(947\) 47.5203 1.54420 0.772100 0.635500i \(-0.219206\pi\)
0.772100 + 0.635500i \(0.219206\pi\)
\(948\) 0 0
\(949\) −6.83399 −0.221841
\(950\) 4.58301 0.148692
\(951\) 0 0
\(952\) 0 0
\(953\) −32.3320 −1.04734 −0.523668 0.851922i \(-0.675437\pi\)
−0.523668 + 0.851922i \(0.675437\pi\)
\(954\) 0 0
\(955\) −44.1255 −1.42787
\(956\) 4.93725 0.159682
\(957\) 0 0
\(958\) 1.64575 0.0531718
\(959\) 0 0
\(960\) 0 0
\(961\) −30.5830 −0.986549
\(962\) −2.54249 −0.0819731
\(963\) 0 0
\(964\) −5.00000 −0.161039
\(965\) 11.5203 0.370850
\(966\) 0 0
\(967\) 51.9373 1.67019 0.835095 0.550106i \(-0.185413\pi\)
0.835095 + 0.550106i \(0.185413\pi\)
\(968\) −8.29150 −0.266499
\(969\) 0 0
\(970\) −12.4797 −0.400700
\(971\) −21.8745 −0.701986 −0.350993 0.936378i \(-0.614156\pi\)
−0.350993 + 0.936378i \(0.614156\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −7.87451 −0.252316
\(975\) 0 0
\(976\) −12.6458 −0.404781
\(977\) −20.1255 −0.643872 −0.321936 0.946762i \(-0.604334\pi\)
−0.321936 + 0.946762i \(0.604334\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) −37.7490 −1.20462
\(983\) 26.8118 0.855162 0.427581 0.903977i \(-0.359366\pi\)
0.427581 + 0.903977i \(0.359366\pi\)
\(984\) 0 0
\(985\) −25.1660 −0.801856
\(986\) 12.5830 0.400725
\(987\) 0 0
\(988\) 1.29150 0.0410882
\(989\) 46.4575 1.47726
\(990\) 0 0
\(991\) −41.9373 −1.33218 −0.666090 0.745871i \(-0.732034\pi\)
−0.666090 + 0.745871i \(0.732034\pi\)
\(992\) −0.645751 −0.0205026
\(993\) 0 0
\(994\) 0 0
\(995\) 36.1033 1.14455
\(996\) 0 0
\(997\) 55.6863 1.76360 0.881801 0.471622i \(-0.156331\pi\)
0.881801 + 0.471622i \(0.156331\pi\)
\(998\) 36.1660 1.14482
\(999\) 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 2646.2.a.bo.1.1 2
3.2 odd 2 2646.2.a.bf.1.2 2
7.3 odd 6 378.2.g.g.163.1 yes 4
7.5 odd 6 378.2.g.g.109.1 4
7.6 odd 2 2646.2.a.bl.1.2 2
21.5 even 6 378.2.g.h.109.2 yes 4
21.17 even 6 378.2.g.h.163.2 yes 4
21.20 even 2 2646.2.a.bi.1.1 2
63.5 even 6 1134.2.e.q.865.2 4
63.31 odd 6 1134.2.h.q.541.2 4
63.38 even 6 1134.2.e.q.919.2 4
63.40 odd 6 1134.2.e.t.865.1 4
63.47 even 6 1134.2.h.t.109.1 4
63.52 odd 6 1134.2.e.t.919.1 4
63.59 even 6 1134.2.h.t.541.1 4
63.61 odd 6 1134.2.h.q.109.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.g.g.109.1 4 7.5 odd 6
378.2.g.g.163.1 yes 4 7.3 odd 6
378.2.g.h.109.2 yes 4 21.5 even 6
378.2.g.h.163.2 yes 4 21.17 even 6
1134.2.e.q.865.2 4 63.5 even 6
1134.2.e.q.919.2 4 63.38 even 6
1134.2.e.t.865.1 4 63.40 odd 6
1134.2.e.t.919.1 4 63.52 odd 6
1134.2.h.q.109.2 4 63.61 odd 6
1134.2.h.q.541.2 4 63.31 odd 6
1134.2.h.t.109.1 4 63.47 even 6
1134.2.h.t.541.1 4 63.59 even 6
2646.2.a.bf.1.2 2 3.2 odd 2
2646.2.a.bi.1.1 2 21.20 even 2
2646.2.a.bl.1.2 2 7.6 odd 2
2646.2.a.bo.1.1 2 1.1 even 1 trivial