Properties

Label 2646.2.a.bn.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: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2646.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} -1.41421 q^{10} -6.24264 q^{11} +3.00000 q^{13} +1.00000 q^{16} +1.58579 q^{17} +7.41421 q^{19} -1.41421 q^{20} -6.24264 q^{22} -1.00000 q^{23} -3.00000 q^{25} +3.00000 q^{26} +3.24264 q^{29} +7.24264 q^{31} +1.00000 q^{32} +1.58579 q^{34} +6.48528 q^{37} +7.41421 q^{38} -1.41421 q^{40} +2.82843 q^{41} -5.24264 q^{43} -6.24264 q^{44} -1.00000 q^{46} +7.07107 q^{47} -3.00000 q^{50} +3.00000 q^{52} +2.75736 q^{53} +8.82843 q^{55} +3.24264 q^{58} +5.82843 q^{59} +0.343146 q^{61} +7.24264 q^{62} +1.00000 q^{64} -4.24264 q^{65} +4.75736 q^{67} +1.58579 q^{68} +11.4853 q^{71} -9.89949 q^{73} +6.48528 q^{74} +7.41421 q^{76} +14.7279 q^{79} -1.41421 q^{80} +2.82843 q^{82} -5.31371 q^{83} -2.24264 q^{85} -5.24264 q^{86} -6.24264 q^{88} -16.0711 q^{89} -1.00000 q^{92} +7.07107 q^{94} -10.4853 q^{95} +9.17157 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 4 q^{11} + 6 q^{13} + 2 q^{16} + 6 q^{17} + 12 q^{19} - 4 q^{22} - 2 q^{23} - 6 q^{25} + 6 q^{26} - 2 q^{29} + 6 q^{31} + 2 q^{32} + 6 q^{34} - 4 q^{37} + 12 q^{38} - 2 q^{43} - 4 q^{44} - 2 q^{46} - 6 q^{50} + 6 q^{52} + 14 q^{53} + 12 q^{55} - 2 q^{58} + 6 q^{59} + 12 q^{61} + 6 q^{62} + 2 q^{64} + 18 q^{67} + 6 q^{68} + 6 q^{71} - 4 q^{74} + 12 q^{76} + 4 q^{79} + 12 q^{83} + 4 q^{85} - 2 q^{86} - 4 q^{88} - 18 q^{89} - 2 q^{92} - 4 q^{95} + 24 q^{97} + O(q^{100}) \)

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
\(5\) −1.41421 −0.632456 −0.316228 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.41421 −0.447214
\(11\) −6.24264 −1.88223 −0.941113 0.338091i \(-0.890219\pi\)
−0.941113 + 0.338091i \(0.890219\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.58579 0.384610 0.192305 0.981335i \(-0.438404\pi\)
0.192305 + 0.981335i \(0.438404\pi\)
\(18\) 0 0
\(19\) 7.41421 1.70094 0.850469 0.526026i \(-0.176318\pi\)
0.850469 + 0.526026i \(0.176318\pi\)
\(20\) −1.41421 −0.316228
\(21\) 0 0
\(22\) −6.24264 −1.33094
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 3.00000 0.588348
\(27\) 0 0
\(28\) 0 0
\(29\) 3.24264 0.602143 0.301072 0.953602i \(-0.402656\pi\)
0.301072 + 0.953602i \(0.402656\pi\)
\(30\) 0 0
\(31\) 7.24264 1.30082 0.650408 0.759585i \(-0.274598\pi\)
0.650408 + 0.759585i \(0.274598\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.58579 0.271960
\(35\) 0 0
\(36\) 0 0
\(37\) 6.48528 1.06617 0.533087 0.846061i \(-0.321032\pi\)
0.533087 + 0.846061i \(0.321032\pi\)
\(38\) 7.41421 1.20274
\(39\) 0 0
\(40\) −1.41421 −0.223607
\(41\) 2.82843 0.441726 0.220863 0.975305i \(-0.429113\pi\)
0.220863 + 0.975305i \(0.429113\pi\)
\(42\) 0 0
\(43\) −5.24264 −0.799495 −0.399748 0.916625i \(-0.630902\pi\)
−0.399748 + 0.916625i \(0.630902\pi\)
\(44\) −6.24264 −0.941113
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 7.07107 1.03142 0.515711 0.856763i \(-0.327528\pi\)
0.515711 + 0.856763i \(0.327528\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −3.00000 −0.424264
\(51\) 0 0
\(52\) 3.00000 0.416025
\(53\) 2.75736 0.378752 0.189376 0.981905i \(-0.439353\pi\)
0.189376 + 0.981905i \(0.439353\pi\)
\(54\) 0 0
\(55\) 8.82843 1.19042
\(56\) 0 0
\(57\) 0 0
\(58\) 3.24264 0.425780
\(59\) 5.82843 0.758797 0.379398 0.925233i \(-0.376131\pi\)
0.379398 + 0.925233i \(0.376131\pi\)
\(60\) 0 0
\(61\) 0.343146 0.0439353 0.0219677 0.999759i \(-0.493007\pi\)
0.0219677 + 0.999759i \(0.493007\pi\)
\(62\) 7.24264 0.919816
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.24264 −0.526235
\(66\) 0 0
\(67\) 4.75736 0.581204 0.290602 0.956844i \(-0.406144\pi\)
0.290602 + 0.956844i \(0.406144\pi\)
\(68\) 1.58579 0.192305
\(69\) 0 0
\(70\) 0 0
\(71\) 11.4853 1.36305 0.681526 0.731794i \(-0.261317\pi\)
0.681526 + 0.731794i \(0.261317\pi\)
\(72\) 0 0
\(73\) −9.89949 −1.15865 −0.579324 0.815097i \(-0.696683\pi\)
−0.579324 + 0.815097i \(0.696683\pi\)
\(74\) 6.48528 0.753899
\(75\) 0 0
\(76\) 7.41421 0.850469
\(77\) 0 0
\(78\) 0 0
\(79\) 14.7279 1.65702 0.828510 0.559974i \(-0.189189\pi\)
0.828510 + 0.559974i \(0.189189\pi\)
\(80\) −1.41421 −0.158114
\(81\) 0 0
\(82\) 2.82843 0.312348
\(83\) −5.31371 −0.583255 −0.291628 0.956532i \(-0.594197\pi\)
−0.291628 + 0.956532i \(0.594197\pi\)
\(84\) 0 0
\(85\) −2.24264 −0.243249
\(86\) −5.24264 −0.565328
\(87\) 0 0
\(88\) −6.24264 −0.665468
\(89\) −16.0711 −1.70353 −0.851765 0.523924i \(-0.824468\pi\)
−0.851765 + 0.523924i \(0.824468\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 7.07107 0.729325
\(95\) −10.4853 −1.07577
\(96\) 0 0
\(97\) 9.17157 0.931232 0.465616 0.884987i \(-0.345833\pi\)
0.465616 + 0.884987i \(0.345833\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) −1.75736 −0.174864 −0.0874319 0.996170i \(-0.527866\pi\)
−0.0874319 + 0.996170i \(0.527866\pi\)
\(102\) 0 0
\(103\) 16.0711 1.58353 0.791765 0.610826i \(-0.209163\pi\)
0.791765 + 0.610826i \(0.209163\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 2.75736 0.267818
\(107\) 16.4853 1.59369 0.796846 0.604182i \(-0.206500\pi\)
0.796846 + 0.604182i \(0.206500\pi\)
\(108\) 0 0
\(109\) −16.4853 −1.57900 −0.789502 0.613748i \(-0.789661\pi\)
−0.789502 + 0.613748i \(0.789661\pi\)
\(110\) 8.82843 0.841757
\(111\) 0 0
\(112\) 0 0
\(113\) 12.7279 1.19734 0.598671 0.800995i \(-0.295696\pi\)
0.598671 + 0.800995i \(0.295696\pi\)
\(114\) 0 0
\(115\) 1.41421 0.131876
\(116\) 3.24264 0.301072
\(117\) 0 0
\(118\) 5.82843 0.536550
\(119\) 0 0
\(120\) 0 0
\(121\) 27.9706 2.54278
\(122\) 0.343146 0.0310670
\(123\) 0 0
\(124\) 7.24264 0.650408
\(125\) 11.3137 1.01193
\(126\) 0 0
\(127\) −20.7279 −1.83931 −0.919653 0.392732i \(-0.871530\pi\)
−0.919653 + 0.392732i \(0.871530\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −4.24264 −0.372104
\(131\) −17.4853 −1.52770 −0.763848 0.645396i \(-0.776692\pi\)
−0.763848 + 0.645396i \(0.776692\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.75736 0.410973
\(135\) 0 0
\(136\) 1.58579 0.135980
\(137\) −22.2426 −1.90032 −0.950159 0.311767i \(-0.899079\pi\)
−0.950159 + 0.311767i \(0.899079\pi\)
\(138\) 0 0
\(139\) −5.31371 −0.450703 −0.225351 0.974278i \(-0.572353\pi\)
−0.225351 + 0.974278i \(0.572353\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 11.4853 0.963823
\(143\) −18.7279 −1.56611
\(144\) 0 0
\(145\) −4.58579 −0.380829
\(146\) −9.89949 −0.819288
\(147\) 0 0
\(148\) 6.48528 0.533087
\(149\) −0.757359 −0.0620453 −0.0310226 0.999519i \(-0.509876\pi\)
−0.0310226 + 0.999519i \(0.509876\pi\)
\(150\) 0 0
\(151\) −10.7279 −0.873026 −0.436513 0.899698i \(-0.643787\pi\)
−0.436513 + 0.899698i \(0.643787\pi\)
\(152\) 7.41421 0.601372
\(153\) 0 0
\(154\) 0 0
\(155\) −10.2426 −0.822709
\(156\) 0 0
\(157\) −3.34315 −0.266812 −0.133406 0.991061i \(-0.542591\pi\)
−0.133406 + 0.991061i \(0.542591\pi\)
\(158\) 14.7279 1.17169
\(159\) 0 0
\(160\) −1.41421 −0.111803
\(161\) 0 0
\(162\) 0 0
\(163\) −12.7574 −0.999233 −0.499617 0.866247i \(-0.666526\pi\)
−0.499617 + 0.866247i \(0.666526\pi\)
\(164\) 2.82843 0.220863
\(165\) 0 0
\(166\) −5.31371 −0.412424
\(167\) 15.8995 1.23034 0.615170 0.788395i \(-0.289087\pi\)
0.615170 + 0.788395i \(0.289087\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) −2.24264 −0.172003
\(171\) 0 0
\(172\) −5.24264 −0.399748
\(173\) −3.51472 −0.267219 −0.133610 0.991034i \(-0.542657\pi\)
−0.133610 + 0.991034i \(0.542657\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6.24264 −0.470557
\(177\) 0 0
\(178\) −16.0711 −1.20458
\(179\) 13.7574 1.02827 0.514137 0.857708i \(-0.328112\pi\)
0.514137 + 0.857708i \(0.328112\pi\)
\(180\) 0 0
\(181\) 23.4853 1.74565 0.872824 0.488036i \(-0.162286\pi\)
0.872824 + 0.488036i \(0.162286\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −9.17157 −0.674307
\(186\) 0 0
\(187\) −9.89949 −0.723923
\(188\) 7.07107 0.515711
\(189\) 0 0
\(190\) −10.4853 −0.760682
\(191\) 6.97056 0.504372 0.252186 0.967679i \(-0.418850\pi\)
0.252186 + 0.967679i \(0.418850\pi\)
\(192\) 0 0
\(193\) 5.00000 0.359908 0.179954 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 9.17157 0.658481
\(195\) 0 0
\(196\) 0 0
\(197\) −6.48528 −0.462057 −0.231029 0.972947i \(-0.574209\pi\)
−0.231029 + 0.972947i \(0.574209\pi\)
\(198\) 0 0
\(199\) 4.75736 0.337240 0.168620 0.985681i \(-0.446069\pi\)
0.168620 + 0.985681i \(0.446069\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −1.75736 −0.123647
\(203\) 0 0
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 16.0711 1.11972
\(207\) 0 0
\(208\) 3.00000 0.208013
\(209\) −46.2843 −3.20155
\(210\) 0 0
\(211\) 10.7574 0.740567 0.370284 0.928919i \(-0.379261\pi\)
0.370284 + 0.928919i \(0.379261\pi\)
\(212\) 2.75736 0.189376
\(213\) 0 0
\(214\) 16.4853 1.12691
\(215\) 7.41421 0.505645
\(216\) 0 0
\(217\) 0 0
\(218\) −16.4853 −1.11652
\(219\) 0 0
\(220\) 8.82843 0.595212
\(221\) 4.75736 0.320015
\(222\) 0 0
\(223\) 16.9706 1.13643 0.568216 0.822879i \(-0.307634\pi\)
0.568216 + 0.822879i \(0.307634\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 12.7279 0.846649
\(227\) 13.9706 0.927259 0.463629 0.886029i \(-0.346547\pi\)
0.463629 + 0.886029i \(0.346547\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 1.41421 0.0932505
\(231\) 0 0
\(232\) 3.24264 0.212890
\(233\) −8.48528 −0.555889 −0.277945 0.960597i \(-0.589653\pi\)
−0.277945 + 0.960597i \(0.589653\pi\)
\(234\) 0 0
\(235\) −10.0000 −0.652328
\(236\) 5.82843 0.379398
\(237\) 0 0
\(238\) 0 0
\(239\) −10.9706 −0.709627 −0.354813 0.934937i \(-0.615456\pi\)
−0.354813 + 0.934937i \(0.615456\pi\)
\(240\) 0 0
\(241\) 24.7279 1.59287 0.796433 0.604727i \(-0.206718\pi\)
0.796433 + 0.604727i \(0.206718\pi\)
\(242\) 27.9706 1.79802
\(243\) 0 0
\(244\) 0.343146 0.0219677
\(245\) 0 0
\(246\) 0 0
\(247\) 22.2426 1.41527
\(248\) 7.24264 0.459908
\(249\) 0 0
\(250\) 11.3137 0.715542
\(251\) 16.6274 1.04951 0.524757 0.851252i \(-0.324156\pi\)
0.524757 + 0.851252i \(0.324156\pi\)
\(252\) 0 0
\(253\) 6.24264 0.392471
\(254\) −20.7279 −1.30059
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.9706 0.684325 0.342162 0.939641i \(-0.388841\pi\)
0.342162 + 0.939641i \(0.388841\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −4.24264 −0.263117
\(261\) 0 0
\(262\) −17.4853 −1.08024
\(263\) −23.4853 −1.44816 −0.724082 0.689714i \(-0.757736\pi\)
−0.724082 + 0.689714i \(0.757736\pi\)
\(264\) 0 0
\(265\) −3.89949 −0.239544
\(266\) 0 0
\(267\) 0 0
\(268\) 4.75736 0.290602
\(269\) −27.5563 −1.68014 −0.840070 0.542478i \(-0.817486\pi\)
−0.840070 + 0.542478i \(0.817486\pi\)
\(270\) 0 0
\(271\) 12.8995 0.783589 0.391794 0.920053i \(-0.371855\pi\)
0.391794 + 0.920053i \(0.371855\pi\)
\(272\) 1.58579 0.0961524
\(273\) 0 0
\(274\) −22.2426 −1.34373
\(275\) 18.7279 1.12934
\(276\) 0 0
\(277\) 12.4853 0.750168 0.375084 0.926991i \(-0.377614\pi\)
0.375084 + 0.926991i \(0.377614\pi\)
\(278\) −5.31371 −0.318695
\(279\) 0 0
\(280\) 0 0
\(281\) −5.51472 −0.328981 −0.164490 0.986379i \(-0.552598\pi\)
−0.164490 + 0.986379i \(0.552598\pi\)
\(282\) 0 0
\(283\) 6.34315 0.377061 0.188530 0.982067i \(-0.439628\pi\)
0.188530 + 0.982067i \(0.439628\pi\)
\(284\) 11.4853 0.681526
\(285\) 0 0
\(286\) −18.7279 −1.10741
\(287\) 0 0
\(288\) 0 0
\(289\) −14.4853 −0.852075
\(290\) −4.58579 −0.269287
\(291\) 0 0
\(292\) −9.89949 −0.579324
\(293\) −32.8284 −1.91786 −0.958929 0.283648i \(-0.908455\pi\)
−0.958929 + 0.283648i \(0.908455\pi\)
\(294\) 0 0
\(295\) −8.24264 −0.479905
\(296\) 6.48528 0.376949
\(297\) 0 0
\(298\) −0.757359 −0.0438726
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 0 0
\(302\) −10.7279 −0.617323
\(303\) 0 0
\(304\) 7.41421 0.425234
\(305\) −0.485281 −0.0277871
\(306\) 0 0
\(307\) −23.6569 −1.35017 −0.675084 0.737741i \(-0.735893\pi\)
−0.675084 + 0.737741i \(0.735893\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −10.2426 −0.581743
\(311\) −23.3137 −1.32200 −0.661000 0.750386i \(-0.729867\pi\)
−0.661000 + 0.750386i \(0.729867\pi\)
\(312\) 0 0
\(313\) 4.24264 0.239808 0.119904 0.992785i \(-0.461741\pi\)
0.119904 + 0.992785i \(0.461741\pi\)
\(314\) −3.34315 −0.188665
\(315\) 0 0
\(316\) 14.7279 0.828510
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) −20.2426 −1.13337
\(320\) −1.41421 −0.0790569
\(321\) 0 0
\(322\) 0 0
\(323\) 11.7574 0.654197
\(324\) 0 0
\(325\) −9.00000 −0.499230
\(326\) −12.7574 −0.706565
\(327\) 0 0
\(328\) 2.82843 0.156174
\(329\) 0 0
\(330\) 0 0
\(331\) −23.7279 −1.30420 −0.652102 0.758131i \(-0.726113\pi\)
−0.652102 + 0.758131i \(0.726113\pi\)
\(332\) −5.31371 −0.291628
\(333\) 0 0
\(334\) 15.8995 0.869982
\(335\) −6.72792 −0.367586
\(336\) 0 0
\(337\) 8.51472 0.463826 0.231913 0.972736i \(-0.425501\pi\)
0.231913 + 0.972736i \(0.425501\pi\)
\(338\) −4.00000 −0.217571
\(339\) 0 0
\(340\) −2.24264 −0.121624
\(341\) −45.2132 −2.44843
\(342\) 0 0
\(343\) 0 0
\(344\) −5.24264 −0.282664
\(345\) 0 0
\(346\) −3.51472 −0.188952
\(347\) −14.4853 −0.777611 −0.388805 0.921320i \(-0.627112\pi\)
−0.388805 + 0.921320i \(0.627112\pi\)
\(348\) 0 0
\(349\) −3.00000 −0.160586 −0.0802932 0.996771i \(-0.525586\pi\)
−0.0802932 + 0.996771i \(0.525586\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6.24264 −0.332734
\(353\) 18.5563 0.987655 0.493827 0.869560i \(-0.335597\pi\)
0.493827 + 0.869560i \(0.335597\pi\)
\(354\) 0 0
\(355\) −16.2426 −0.862070
\(356\) −16.0711 −0.851765
\(357\) 0 0
\(358\) 13.7574 0.727099
\(359\) 11.9706 0.631782 0.315891 0.948795i \(-0.397697\pi\)
0.315891 + 0.948795i \(0.397697\pi\)
\(360\) 0 0
\(361\) 35.9706 1.89319
\(362\) 23.4853 1.23436
\(363\) 0 0
\(364\) 0 0
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) −16.4142 −0.856815 −0.428407 0.903586i \(-0.640925\pi\)
−0.428407 + 0.903586i \(0.640925\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −9.17157 −0.476807
\(371\) 0 0
\(372\) 0 0
\(373\) −25.6985 −1.33062 −0.665309 0.746569i \(-0.731700\pi\)
−0.665309 + 0.746569i \(0.731700\pi\)
\(374\) −9.89949 −0.511891
\(375\) 0 0
\(376\) 7.07107 0.364662
\(377\) 9.72792 0.501013
\(378\) 0 0
\(379\) 14.9706 0.768986 0.384493 0.923128i \(-0.374376\pi\)
0.384493 + 0.923128i \(0.374376\pi\)
\(380\) −10.4853 −0.537884
\(381\) 0 0
\(382\) 6.97056 0.356645
\(383\) −21.5563 −1.10148 −0.550739 0.834678i \(-0.685654\pi\)
−0.550739 + 0.834678i \(0.685654\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5.00000 0.254493
\(387\) 0 0
\(388\) 9.17157 0.465616
\(389\) −5.02944 −0.255003 −0.127501 0.991838i \(-0.540696\pi\)
−0.127501 + 0.991838i \(0.540696\pi\)
\(390\) 0 0
\(391\) −1.58579 −0.0801967
\(392\) 0 0
\(393\) 0 0
\(394\) −6.48528 −0.326724
\(395\) −20.8284 −1.04799
\(396\) 0 0
\(397\) 9.17157 0.460308 0.230154 0.973154i \(-0.426077\pi\)
0.230154 + 0.973154i \(0.426077\pi\)
\(398\) 4.75736 0.238465
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) −2.48528 −0.124109 −0.0620545 0.998073i \(-0.519765\pi\)
−0.0620545 + 0.998073i \(0.519765\pi\)
\(402\) 0 0
\(403\) 21.7279 1.08234
\(404\) −1.75736 −0.0874319
\(405\) 0 0
\(406\) 0 0
\(407\) −40.4853 −2.00678
\(408\) 0 0
\(409\) −2.48528 −0.122889 −0.0614446 0.998110i \(-0.519571\pi\)
−0.0614446 + 0.998110i \(0.519571\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) 16.0711 0.791765
\(413\) 0 0
\(414\) 0 0
\(415\) 7.51472 0.368883
\(416\) 3.00000 0.147087
\(417\) 0 0
\(418\) −46.2843 −2.26384
\(419\) 15.3431 0.749562 0.374781 0.927113i \(-0.377718\pi\)
0.374781 + 0.927113i \(0.377718\pi\)
\(420\) 0 0
\(421\) −31.7574 −1.54776 −0.773879 0.633333i \(-0.781686\pi\)
−0.773879 + 0.633333i \(0.781686\pi\)
\(422\) 10.7574 0.523660
\(423\) 0 0
\(424\) 2.75736 0.133909
\(425\) −4.75736 −0.230766
\(426\) 0 0
\(427\) 0 0
\(428\) 16.4853 0.796846
\(429\) 0 0
\(430\) 7.41421 0.357545
\(431\) 1.02944 0.0495862 0.0247931 0.999693i \(-0.492107\pi\)
0.0247931 + 0.999693i \(0.492107\pi\)
\(432\) 0 0
\(433\) −15.5147 −0.745590 −0.372795 0.927914i \(-0.621600\pi\)
−0.372795 + 0.927914i \(0.621600\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −16.4853 −0.789502
\(437\) −7.41421 −0.354670
\(438\) 0 0
\(439\) −30.2132 −1.44200 −0.720999 0.692936i \(-0.756317\pi\)
−0.720999 + 0.692936i \(0.756317\pi\)
\(440\) 8.82843 0.420879
\(441\) 0 0
\(442\) 4.75736 0.226285
\(443\) 6.97056 0.331181 0.165591 0.986195i \(-0.447047\pi\)
0.165591 + 0.986195i \(0.447047\pi\)
\(444\) 0 0
\(445\) 22.7279 1.07741
\(446\) 16.9706 0.803579
\(447\) 0 0
\(448\) 0 0
\(449\) 24.7279 1.16698 0.583491 0.812119i \(-0.301686\pi\)
0.583491 + 0.812119i \(0.301686\pi\)
\(450\) 0 0
\(451\) −17.6569 −0.831429
\(452\) 12.7279 0.598671
\(453\) 0 0
\(454\) 13.9706 0.655671
\(455\) 0 0
\(456\) 0 0
\(457\) −15.0000 −0.701670 −0.350835 0.936437i \(-0.614102\pi\)
−0.350835 + 0.936437i \(0.614102\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 1.41421 0.0659380
\(461\) −13.0711 −0.608780 −0.304390 0.952547i \(-0.598453\pi\)
−0.304390 + 0.952547i \(0.598453\pi\)
\(462\) 0 0
\(463\) 19.2132 0.892913 0.446457 0.894805i \(-0.352686\pi\)
0.446457 + 0.894805i \(0.352686\pi\)
\(464\) 3.24264 0.150536
\(465\) 0 0
\(466\) −8.48528 −0.393073
\(467\) 12.3431 0.571173 0.285586 0.958353i \(-0.407812\pi\)
0.285586 + 0.958353i \(0.407812\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −10.0000 −0.461266
\(471\) 0 0
\(472\) 5.82843 0.268275
\(473\) 32.7279 1.50483
\(474\) 0 0
\(475\) −22.2426 −1.02056
\(476\) 0 0
\(477\) 0 0
\(478\) −10.9706 −0.501782
\(479\) 30.0416 1.37264 0.686319 0.727301i \(-0.259226\pi\)
0.686319 + 0.727301i \(0.259226\pi\)
\(480\) 0 0
\(481\) 19.4558 0.887110
\(482\) 24.7279 1.12633
\(483\) 0 0
\(484\) 27.9706 1.27139
\(485\) −12.9706 −0.588963
\(486\) 0 0
\(487\) −13.7574 −0.623405 −0.311703 0.950180i \(-0.600899\pi\)
−0.311703 + 0.950180i \(0.600899\pi\)
\(488\) 0.343146 0.0155335
\(489\) 0 0
\(490\) 0 0
\(491\) −1.75736 −0.0793085 −0.0396543 0.999213i \(-0.512626\pi\)
−0.0396543 + 0.999213i \(0.512626\pi\)
\(492\) 0 0
\(493\) 5.14214 0.231590
\(494\) 22.2426 1.00074
\(495\) 0 0
\(496\) 7.24264 0.325204
\(497\) 0 0
\(498\) 0 0
\(499\) −8.97056 −0.401578 −0.200789 0.979635i \(-0.564350\pi\)
−0.200789 + 0.979635i \(0.564350\pi\)
\(500\) 11.3137 0.505964
\(501\) 0 0
\(502\) 16.6274 0.742118
\(503\) 27.8995 1.24398 0.621988 0.783026i \(-0.286325\pi\)
0.621988 + 0.783026i \(0.286325\pi\)
\(504\) 0 0
\(505\) 2.48528 0.110594
\(506\) 6.24264 0.277519
\(507\) 0 0
\(508\) −20.7279 −0.919653
\(509\) 20.1421 0.892784 0.446392 0.894837i \(-0.352709\pi\)
0.446392 + 0.894837i \(0.352709\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 10.9706 0.483891
\(515\) −22.7279 −1.00151
\(516\) 0 0
\(517\) −44.1421 −1.94137
\(518\) 0 0
\(519\) 0 0
\(520\) −4.24264 −0.186052
\(521\) 7.58579 0.332339 0.166170 0.986097i \(-0.446860\pi\)
0.166170 + 0.986097i \(0.446860\pi\)
\(522\) 0 0
\(523\) −36.7279 −1.60600 −0.803000 0.595979i \(-0.796764\pi\)
−0.803000 + 0.595979i \(0.796764\pi\)
\(524\) −17.4853 −0.763848
\(525\) 0 0
\(526\) −23.4853 −1.02401
\(527\) 11.4853 0.500307
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) −3.89949 −0.169383
\(531\) 0 0
\(532\) 0 0
\(533\) 8.48528 0.367538
\(534\) 0 0
\(535\) −23.3137 −1.00794
\(536\) 4.75736 0.205487
\(537\) 0 0
\(538\) −27.5563 −1.18804
\(539\) 0 0
\(540\) 0 0
\(541\) −14.7279 −0.633203 −0.316601 0.948559i \(-0.602542\pi\)
−0.316601 + 0.948559i \(0.602542\pi\)
\(542\) 12.8995 0.554081
\(543\) 0 0
\(544\) 1.58579 0.0679900
\(545\) 23.3137 0.998650
\(546\) 0 0
\(547\) 12.4853 0.533832 0.266916 0.963720i \(-0.413995\pi\)
0.266916 + 0.963720i \(0.413995\pi\)
\(548\) −22.2426 −0.950159
\(549\) 0 0
\(550\) 18.7279 0.798561
\(551\) 24.0416 1.02421
\(552\) 0 0
\(553\) 0 0
\(554\) 12.4853 0.530449
\(555\) 0 0
\(556\) −5.31371 −0.225351
\(557\) 44.6985 1.89394 0.946968 0.321328i \(-0.104129\pi\)
0.946968 + 0.321328i \(0.104129\pi\)
\(558\) 0 0
\(559\) −15.7279 −0.665220
\(560\) 0 0
\(561\) 0 0
\(562\) −5.51472 −0.232624
\(563\) 12.8579 0.541894 0.270947 0.962594i \(-0.412663\pi\)
0.270947 + 0.962594i \(0.412663\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) 6.34315 0.266622
\(567\) 0 0
\(568\) 11.4853 0.481912
\(569\) −41.2132 −1.72775 −0.863874 0.503709i \(-0.831969\pi\)
−0.863874 + 0.503709i \(0.831969\pi\)
\(570\) 0 0
\(571\) 5.24264 0.219398 0.109699 0.993965i \(-0.465011\pi\)
0.109699 + 0.993965i \(0.465011\pi\)
\(572\) −18.7279 −0.783054
\(573\) 0 0
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) 0 0
\(577\) 14.4853 0.603030 0.301515 0.953461i \(-0.402508\pi\)
0.301515 + 0.953461i \(0.402508\pi\)
\(578\) −14.4853 −0.602508
\(579\) 0 0
\(580\) −4.58579 −0.190414
\(581\) 0 0
\(582\) 0 0
\(583\) −17.2132 −0.712898
\(584\) −9.89949 −0.409644
\(585\) 0 0
\(586\) −32.8284 −1.35613
\(587\) −10.4558 −0.431559 −0.215779 0.976442i \(-0.569229\pi\)
−0.215779 + 0.976442i \(0.569229\pi\)
\(588\) 0 0
\(589\) 53.6985 2.21261
\(590\) −8.24264 −0.339344
\(591\) 0 0
\(592\) 6.48528 0.266543
\(593\) 8.14214 0.334357 0.167179 0.985927i \(-0.446534\pi\)
0.167179 + 0.985927i \(0.446534\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.757359 −0.0310226
\(597\) 0 0
\(598\) −3.00000 −0.122679
\(599\) −7.00000 −0.286012 −0.143006 0.989722i \(-0.545677\pi\)
−0.143006 + 0.989722i \(0.545677\pi\)
\(600\) 0 0
\(601\) −25.7574 −1.05066 −0.525332 0.850897i \(-0.676059\pi\)
−0.525332 + 0.850897i \(0.676059\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −10.7279 −0.436513
\(605\) −39.5563 −1.60819
\(606\) 0 0
\(607\) 26.6985 1.08366 0.541829 0.840489i \(-0.317732\pi\)
0.541829 + 0.840489i \(0.317732\pi\)
\(608\) 7.41421 0.300686
\(609\) 0 0
\(610\) −0.485281 −0.0196485
\(611\) 21.2132 0.858194
\(612\) 0 0
\(613\) 19.7574 0.797992 0.398996 0.916953i \(-0.369359\pi\)
0.398996 + 0.916953i \(0.369359\pi\)
\(614\) −23.6569 −0.954713
\(615\) 0 0
\(616\) 0 0
\(617\) 31.9411 1.28590 0.642951 0.765908i \(-0.277710\pi\)
0.642951 + 0.765908i \(0.277710\pi\)
\(618\) 0 0
\(619\) 3.17157 0.127476 0.0637381 0.997967i \(-0.479698\pi\)
0.0637381 + 0.997967i \(0.479698\pi\)
\(620\) −10.2426 −0.411354
\(621\) 0 0
\(622\) −23.3137 −0.934795
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 4.24264 0.169570
\(627\) 0 0
\(628\) −3.34315 −0.133406
\(629\) 10.2843 0.410061
\(630\) 0 0
\(631\) 12.2426 0.487372 0.243686 0.969854i \(-0.421643\pi\)
0.243686 + 0.969854i \(0.421643\pi\)
\(632\) 14.7279 0.585845
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) 29.3137 1.16328
\(636\) 0 0
\(637\) 0 0
\(638\) −20.2426 −0.801414
\(639\) 0 0
\(640\) −1.41421 −0.0559017
\(641\) −41.4558 −1.63741 −0.818704 0.574216i \(-0.805307\pi\)
−0.818704 + 0.574216i \(0.805307\pi\)
\(642\) 0 0
\(643\) −31.0711 −1.22532 −0.612662 0.790345i \(-0.709901\pi\)
−0.612662 + 0.790345i \(0.709901\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 11.7574 0.462587
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 0 0
\(649\) −36.3848 −1.42823
\(650\) −9.00000 −0.353009
\(651\) 0 0
\(652\) −12.7574 −0.499617
\(653\) −26.6985 −1.04479 −0.522396 0.852703i \(-0.674962\pi\)
−0.522396 + 0.852703i \(0.674962\pi\)
\(654\) 0 0
\(655\) 24.7279 0.966200
\(656\) 2.82843 0.110432
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −45.9411 −1.78690 −0.893451 0.449160i \(-0.851723\pi\)
−0.893451 + 0.449160i \(0.851723\pi\)
\(662\) −23.7279 −0.922212
\(663\) 0 0
\(664\) −5.31371 −0.206212
\(665\) 0 0
\(666\) 0 0
\(667\) −3.24264 −0.125556
\(668\) 15.8995 0.615170
\(669\) 0 0
\(670\) −6.72792 −0.259922
\(671\) −2.14214 −0.0826962
\(672\) 0 0
\(673\) 22.4558 0.865609 0.432805 0.901488i \(-0.357524\pi\)
0.432805 + 0.901488i \(0.357524\pi\)
\(674\) 8.51472 0.327975
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) −29.6985 −1.14141 −0.570703 0.821157i \(-0.693329\pi\)
−0.570703 + 0.821157i \(0.693329\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.24264 −0.0860013
\(681\) 0 0
\(682\) −45.2132 −1.73130
\(683\) −22.2426 −0.851091 −0.425545 0.904937i \(-0.639918\pi\)
−0.425545 + 0.904937i \(0.639918\pi\)
\(684\) 0 0
\(685\) 31.4558 1.20187
\(686\) 0 0
\(687\) 0 0
\(688\) −5.24264 −0.199874
\(689\) 8.27208 0.315141
\(690\) 0 0
\(691\) −9.21320 −0.350487 −0.175243 0.984525i \(-0.556071\pi\)
−0.175243 + 0.984525i \(0.556071\pi\)
\(692\) −3.51472 −0.133610
\(693\) 0 0
\(694\) −14.4853 −0.549854
\(695\) 7.51472 0.285050
\(696\) 0 0
\(697\) 4.48528 0.169892
\(698\) −3.00000 −0.113552
\(699\) 0 0
\(700\) 0 0
\(701\) 34.9706 1.32082 0.660410 0.750905i \(-0.270383\pi\)
0.660410 + 0.750905i \(0.270383\pi\)
\(702\) 0 0
\(703\) 48.0833 1.81349
\(704\) −6.24264 −0.235278
\(705\) 0 0
\(706\) 18.5563 0.698377
\(707\) 0 0
\(708\) 0 0
\(709\) −23.2132 −0.871790 −0.435895 0.899997i \(-0.643568\pi\)
−0.435895 + 0.899997i \(0.643568\pi\)
\(710\) −16.2426 −0.609575
\(711\) 0 0
\(712\) −16.0711 −0.602289
\(713\) −7.24264 −0.271239
\(714\) 0 0
\(715\) 26.4853 0.990493
\(716\) 13.7574 0.514137
\(717\) 0 0
\(718\) 11.9706 0.446737
\(719\) −10.6274 −0.396336 −0.198168 0.980168i \(-0.563499\pi\)
−0.198168 + 0.980168i \(0.563499\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 35.9706 1.33869
\(723\) 0 0
\(724\) 23.4853 0.872824
\(725\) −9.72792 −0.361286
\(726\) 0 0
\(727\) 40.8406 1.51469 0.757347 0.653012i \(-0.226495\pi\)
0.757347 + 0.653012i \(0.226495\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 14.0000 0.518163
\(731\) −8.31371 −0.307494
\(732\) 0 0
\(733\) 30.9411 1.14284 0.571418 0.820659i \(-0.306393\pi\)
0.571418 + 0.820659i \(0.306393\pi\)
\(734\) −16.4142 −0.605860
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −29.6985 −1.09396
\(738\) 0 0
\(739\) −10.9706 −0.403559 −0.201779 0.979431i \(-0.564672\pi\)
−0.201779 + 0.979431i \(0.564672\pi\)
\(740\) −9.17157 −0.337154
\(741\) 0 0
\(742\) 0 0
\(743\) 4.51472 0.165629 0.0828145 0.996565i \(-0.473609\pi\)
0.0828145 + 0.996565i \(0.473609\pi\)
\(744\) 0 0
\(745\) 1.07107 0.0392409
\(746\) −25.6985 −0.940888
\(747\) 0 0
\(748\) −9.89949 −0.361961
\(749\) 0 0
\(750\) 0 0
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) 7.07107 0.257855
\(753\) 0 0
\(754\) 9.72792 0.354270
\(755\) 15.1716 0.552150
\(756\) 0 0
\(757\) −27.6985 −1.00672 −0.503359 0.864077i \(-0.667903\pi\)
−0.503359 + 0.864077i \(0.667903\pi\)
\(758\) 14.9706 0.543755
\(759\) 0 0
\(760\) −10.4853 −0.380341
\(761\) −53.5269 −1.94035 −0.970175 0.242408i \(-0.922063\pi\)
−0.970175 + 0.242408i \(0.922063\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 6.97056 0.252186
\(765\) 0 0
\(766\) −21.5563 −0.778863
\(767\) 17.4853 0.631357
\(768\) 0 0
\(769\) 15.2132 0.548602 0.274301 0.961644i \(-0.411554\pi\)
0.274301 + 0.961644i \(0.411554\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.00000 0.179954
\(773\) 18.3848 0.661254 0.330627 0.943761i \(-0.392740\pi\)
0.330627 + 0.943761i \(0.392740\pi\)
\(774\) 0 0
\(775\) −21.7279 −0.780490
\(776\) 9.17157 0.329240
\(777\) 0 0
\(778\) −5.02944 −0.180314
\(779\) 20.9706 0.751348
\(780\) 0 0
\(781\) −71.6985 −2.56557
\(782\) −1.58579 −0.0567076
\(783\) 0 0
\(784\) 0 0
\(785\) 4.72792 0.168747
\(786\) 0 0
\(787\) −27.1716 −0.968562 −0.484281 0.874913i \(-0.660919\pi\)
−0.484281 + 0.874913i \(0.660919\pi\)
\(788\) −6.48528 −0.231029
\(789\) 0 0
\(790\) −20.8284 −0.741042
\(791\) 0 0
\(792\) 0 0
\(793\) 1.02944 0.0365564
\(794\) 9.17157 0.325487
\(795\) 0 0
\(796\) 4.75736 0.168620
\(797\) −50.4853 −1.78828 −0.894140 0.447787i \(-0.852212\pi\)
−0.894140 + 0.447787i \(0.852212\pi\)
\(798\) 0 0
\(799\) 11.2132 0.396695
\(800\) −3.00000 −0.106066
\(801\) 0 0
\(802\) −2.48528 −0.0877583
\(803\) 61.7990 2.18084
\(804\) 0 0
\(805\) 0 0
\(806\) 21.7279 0.765333
\(807\) 0 0
\(808\) −1.75736 −0.0618237
\(809\) −4.00000 −0.140633 −0.0703163 0.997525i \(-0.522401\pi\)
−0.0703163 + 0.997525i \(0.522401\pi\)
\(810\) 0 0
\(811\) −15.9411 −0.559769 −0.279884 0.960034i \(-0.590296\pi\)
−0.279884 + 0.960034i \(0.590296\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −40.4853 −1.41901
\(815\) 18.0416 0.631971
\(816\) 0 0
\(817\) −38.8701 −1.35989
\(818\) −2.48528 −0.0868958
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) −24.2132 −0.845047 −0.422523 0.906352i \(-0.638856\pi\)
−0.422523 + 0.906352i \(0.638856\pi\)
\(822\) 0 0
\(823\) 26.9706 0.940135 0.470067 0.882631i \(-0.344230\pi\)
0.470067 + 0.882631i \(0.344230\pi\)
\(824\) 16.0711 0.559862
\(825\) 0 0
\(826\) 0 0
\(827\) −29.6985 −1.03272 −0.516359 0.856372i \(-0.672713\pi\)
−0.516359 + 0.856372i \(0.672713\pi\)
\(828\) 0 0
\(829\) 26.8284 0.931790 0.465895 0.884840i \(-0.345732\pi\)
0.465895 + 0.884840i \(0.345732\pi\)
\(830\) 7.51472 0.260840
\(831\) 0 0
\(832\) 3.00000 0.104006
\(833\) 0 0
\(834\) 0 0
\(835\) −22.4853 −0.778135
\(836\) −46.2843 −1.60077
\(837\) 0 0
\(838\) 15.3431 0.530020
\(839\) 37.7990 1.30497 0.652483 0.757803i \(-0.273727\pi\)
0.652483 + 0.757803i \(0.273727\pi\)
\(840\) 0 0
\(841\) −18.4853 −0.637423
\(842\) −31.7574 −1.09443
\(843\) 0 0
\(844\) 10.7574 0.370284
\(845\) 5.65685 0.194602
\(846\) 0 0
\(847\) 0 0
\(848\) 2.75736 0.0946881
\(849\) 0 0
\(850\) −4.75736 −0.163176
\(851\) −6.48528 −0.222313
\(852\) 0 0
\(853\) −39.4264 −1.34993 −0.674967 0.737848i \(-0.735842\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 16.4853 0.563455
\(857\) 46.4142 1.58548 0.792740 0.609560i \(-0.208654\pi\)
0.792740 + 0.609560i \(0.208654\pi\)
\(858\) 0 0
\(859\) 30.7696 1.04984 0.524922 0.851150i \(-0.324095\pi\)
0.524922 + 0.851150i \(0.324095\pi\)
\(860\) 7.41421 0.252823
\(861\) 0 0
\(862\) 1.02944 0.0350628
\(863\) 19.4853 0.663287 0.331643 0.943405i \(-0.392397\pi\)
0.331643 + 0.943405i \(0.392397\pi\)
\(864\) 0 0
\(865\) 4.97056 0.169004
\(866\) −15.5147 −0.527212
\(867\) 0 0
\(868\) 0 0
\(869\) −91.9411 −3.11889
\(870\) 0 0
\(871\) 14.2721 0.483591
\(872\) −16.4853 −0.558262
\(873\) 0 0
\(874\) −7.41421 −0.250790
\(875\) 0 0
\(876\) 0 0
\(877\) −12.4853 −0.421598 −0.210799 0.977529i \(-0.567607\pi\)
−0.210799 + 0.977529i \(0.567607\pi\)
\(878\) −30.2132 −1.01965
\(879\) 0 0
\(880\) 8.82843 0.297606
\(881\) −33.7279 −1.13632 −0.568161 0.822917i \(-0.692345\pi\)
−0.568161 + 0.822917i \(0.692345\pi\)
\(882\) 0 0
\(883\) −12.2721 −0.412988 −0.206494 0.978448i \(-0.566205\pi\)
−0.206494 + 0.978448i \(0.566205\pi\)
\(884\) 4.75736 0.160007
\(885\) 0 0
\(886\) 6.97056 0.234181
\(887\) 33.1716 1.11379 0.556896 0.830582i \(-0.311992\pi\)
0.556896 + 0.830582i \(0.311992\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 22.7279 0.761842
\(891\) 0 0
\(892\) 16.9706 0.568216
\(893\) 52.4264 1.75438
\(894\) 0 0
\(895\) −19.4558 −0.650337
\(896\) 0 0
\(897\) 0 0
\(898\) 24.7279 0.825181
\(899\) 23.4853 0.783278
\(900\) 0 0
\(901\) 4.37258 0.145672
\(902\) −17.6569 −0.587909
\(903\) 0 0
\(904\) 12.7279 0.423324
\(905\) −33.2132 −1.10404
\(906\) 0 0
\(907\) 43.9411 1.45904 0.729521 0.683959i \(-0.239743\pi\)
0.729521 + 0.683959i \(0.239743\pi\)
\(908\) 13.9706 0.463629
\(909\) 0 0
\(910\) 0 0
\(911\) 17.5147 0.580289 0.290144 0.956983i \(-0.406297\pi\)
0.290144 + 0.956983i \(0.406297\pi\)
\(912\) 0 0
\(913\) 33.1716 1.09782
\(914\) −15.0000 −0.496156
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 5.75736 0.189918 0.0949589 0.995481i \(-0.469728\pi\)
0.0949589 + 0.995481i \(0.469728\pi\)
\(920\) 1.41421 0.0466252
\(921\) 0 0
\(922\) −13.0711 −0.430473
\(923\) 34.4558 1.13413
\(924\) 0 0
\(925\) −19.4558 −0.639704
\(926\) 19.2132 0.631385
\(927\) 0 0
\(928\) 3.24264 0.106445
\(929\) 41.3137 1.35546 0.677729 0.735311i \(-0.262964\pi\)
0.677729 + 0.735311i \(0.262964\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −8.48528 −0.277945
\(933\) 0 0
\(934\) 12.3431 0.403880
\(935\) 14.0000 0.457849
\(936\) 0 0
\(937\) −37.0711 −1.21106 −0.605529 0.795823i \(-0.707039\pi\)
−0.605529 + 0.795823i \(0.707039\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −10.0000 −0.326164
\(941\) 22.2843 0.726446 0.363223 0.931702i \(-0.381676\pi\)
0.363223 + 0.931702i \(0.381676\pi\)
\(942\) 0 0
\(943\) −2.82843 −0.0921063
\(944\) 5.82843 0.189699
\(945\) 0 0
\(946\) 32.7279 1.06408
\(947\) −36.7279 −1.19350 −0.596749 0.802428i \(-0.703541\pi\)
−0.596749 + 0.802428i \(0.703541\pi\)
\(948\) 0 0
\(949\) −29.6985 −0.964054
\(950\) −22.2426 −0.721647
\(951\) 0 0
\(952\) 0 0
\(953\) 9.69848 0.314165 0.157082 0.987586i \(-0.449791\pi\)
0.157082 + 0.987586i \(0.449791\pi\)
\(954\) 0 0
\(955\) −9.85786 −0.318993
\(956\) −10.9706 −0.354813
\(957\) 0 0
\(958\) 30.0416 0.970601
\(959\) 0 0
\(960\) 0 0
\(961\) 21.4558 0.692124
\(962\) 19.4558 0.627282
\(963\) 0 0
\(964\) 24.7279 0.796433
\(965\) −7.07107 −0.227626
\(966\) 0 0
\(967\) −55.1543 −1.77364 −0.886822 0.462112i \(-0.847092\pi\)
−0.886822 + 0.462112i \(0.847092\pi\)
\(968\) 27.9706 0.899008
\(969\) 0 0
\(970\) −12.9706 −0.416460
\(971\) 27.7696 0.891167 0.445584 0.895240i \(-0.352996\pi\)
0.445584 + 0.895240i \(0.352996\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −13.7574 −0.440814
\(975\) 0 0
\(976\) 0.343146 0.0109838
\(977\) 52.6690 1.68503 0.842516 0.538671i \(-0.181073\pi\)
0.842516 + 0.538671i \(0.181073\pi\)
\(978\) 0 0
\(979\) 100.326 3.20643
\(980\) 0 0
\(981\) 0 0
\(982\) −1.75736 −0.0560796
\(983\) −9.55635 −0.304800 −0.152400 0.988319i \(-0.548700\pi\)
−0.152400 + 0.988319i \(0.548700\pi\)
\(984\) 0 0
\(985\) 9.17157 0.292231
\(986\) 5.14214 0.163759
\(987\) 0 0
\(988\) 22.2426 0.707633
\(989\) 5.24264 0.166706
\(990\) 0 0
\(991\) 39.4558 1.25336 0.626678 0.779278i \(-0.284414\pi\)
0.626678 + 0.779278i \(0.284414\pi\)
\(992\) 7.24264 0.229954
\(993\) 0 0
\(994\) 0 0
\(995\) −6.72792 −0.213289
\(996\) 0 0
\(997\) −47.1421 −1.49301 −0.746503 0.665382i \(-0.768269\pi\)
−0.746503 + 0.665382i \(0.768269\pi\)
\(998\) −8.97056 −0.283958
\(999\) 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.bn.1.1 yes 2
3.2 odd 2 2646.2.a.bh.1.2 yes 2
7.6 odd 2 2646.2.a.bm.1.2 yes 2
21.20 even 2 2646.2.a.bg.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2646.2.a.bg.1.1 2 21.20 even 2
2646.2.a.bh.1.2 yes 2 3.2 odd 2
2646.2.a.bm.1.2 yes 2 7.6 odd 2
2646.2.a.bn.1.1 yes 2 1.1 even 1 trivial