Properties

Label 2646.2.a.bn.1.2
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 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2646.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +1.41421 q^{10} +2.24264 q^{11} +3.00000 q^{13} +1.00000 q^{16} +4.41421 q^{17} +4.58579 q^{19} +1.41421 q^{20} +2.24264 q^{22} -1.00000 q^{23} -3.00000 q^{25} +3.00000 q^{26} -5.24264 q^{29} -1.24264 q^{31} +1.00000 q^{32} +4.41421 q^{34} -10.4853 q^{37} +4.58579 q^{38} +1.41421 q^{40} -2.82843 q^{41} +3.24264 q^{43} +2.24264 q^{44} -1.00000 q^{46} -7.07107 q^{47} -3.00000 q^{50} +3.00000 q^{52} +11.2426 q^{53} +3.17157 q^{55} -5.24264 q^{58} +0.171573 q^{59} +11.6569 q^{61} -1.24264 q^{62} +1.00000 q^{64} +4.24264 q^{65} +13.2426 q^{67} +4.41421 q^{68} -5.48528 q^{71} +9.89949 q^{73} -10.4853 q^{74} +4.58579 q^{76} -10.7279 q^{79} +1.41421 q^{80} -2.82843 q^{82} +17.3137 q^{83} +6.24264 q^{85} +3.24264 q^{86} +2.24264 q^{88} -1.92893 q^{89} -1.00000 q^{92} -7.07107 q^{94} +6.48528 q^{95} +14.8284 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 4 q^{11} + 6 q^{13} + 2 q^{16} + 6 q^{17} + 12 q^{19} - 4 q^{22} - 2 q^{23} - 6 q^{25} + 6 q^{26} - 2 q^{29} + 6 q^{31} + 2 q^{32} + 6 q^{34} - 4 q^{37} + 12 q^{38} - 2 q^{43} - 4 q^{44} - 2 q^{46} - 6 q^{50} + 6 q^{52} + 14 q^{53} + 12 q^{55} - 2 q^{58} + 6 q^{59} + 12 q^{61} + 6 q^{62} + 2 q^{64} + 18 q^{67} + 6 q^{68} + 6 q^{71} - 4 q^{74} + 12 q^{76} + 4 q^{79} + 12 q^{83} + 4 q^{85} - 2 q^{86} - 4 q^{88} - 18 q^{89} - 2 q^{92} - 4 q^{95} + 24 q^{97} + O(q^{100}) \)

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 \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.41421 0.447214
\(11\) 2.24264 0.676182 0.338091 0.941113i \(-0.390219\pi\)
0.338091 + 0.941113i \(0.390219\pi\)
\(12\) 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
\(17\) 4.41421 1.07060 0.535302 0.844661i \(-0.320198\pi\)
0.535302 + 0.844661i \(0.320198\pi\)
\(18\) 0 0
\(19\) 4.58579 1.05205 0.526026 0.850469i \(-0.323682\pi\)
0.526026 + 0.850469i \(0.323682\pi\)
\(20\) 1.41421 0.316228
\(21\) 0 0
\(22\) 2.24264 0.478133
\(23\) −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
\(29\) −5.24264 −0.973534 −0.486767 0.873532i \(-0.661824\pi\)
−0.486767 + 0.873532i \(0.661824\pi\)
\(30\) 0 0
\(31\) −1.24264 −0.223185 −0.111592 0.993754i \(-0.535595\pi\)
−0.111592 + 0.993754i \(0.535595\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.41421 0.757031
\(35\) 0 0
\(36\) 0 0
\(37\) −10.4853 −1.72377 −0.861885 0.507104i \(-0.830716\pi\)
−0.861885 + 0.507104i \(0.830716\pi\)
\(38\) 4.58579 0.743913
\(39\) 0 0
\(40\) 1.41421 0.223607
\(41\) −2.82843 −0.441726 −0.220863 0.975305i \(-0.570887\pi\)
−0.220863 + 0.975305i \(0.570887\pi\)
\(42\) 0 0
\(43\) 3.24264 0.494498 0.247249 0.968952i \(-0.420473\pi\)
0.247249 + 0.968952i \(0.420473\pi\)
\(44\) 2.24264 0.338091
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −7.07107 −1.03142 −0.515711 0.856763i \(-0.672472\pi\)
−0.515711 + 0.856763i \(0.672472\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −3.00000 −0.424264
\(51\) 0 0
\(52\) 3.00000 0.416025
\(53\) 11.2426 1.54430 0.772148 0.635443i \(-0.219183\pi\)
0.772148 + 0.635443i \(0.219183\pi\)
\(54\) 0 0
\(55\) 3.17157 0.427655
\(56\) 0 0
\(57\) 0 0
\(58\) −5.24264 −0.688392
\(59\) 0.171573 0.0223369 0.0111684 0.999938i \(-0.496445\pi\)
0.0111684 + 0.999938i \(0.496445\pi\)
\(60\) 0 0
\(61\) 11.6569 1.49251 0.746254 0.665662i \(-0.231851\pi\)
0.746254 + 0.665662i \(0.231851\pi\)
\(62\) −1.24264 −0.157816
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.24264 0.526235
\(66\) 0 0
\(67\) 13.2426 1.61785 0.808923 0.587915i \(-0.200051\pi\)
0.808923 + 0.587915i \(0.200051\pi\)
\(68\) 4.41421 0.535302
\(69\) 0 0
\(70\) 0 0
\(71\) −5.48528 −0.650983 −0.325492 0.945545i \(-0.605530\pi\)
−0.325492 + 0.945545i \(0.605530\pi\)
\(72\) 0 0
\(73\) 9.89949 1.15865 0.579324 0.815097i \(-0.303317\pi\)
0.579324 + 0.815097i \(0.303317\pi\)
\(74\) −10.4853 −1.21889
\(75\) 0 0
\(76\) 4.58579 0.526026
\(77\) 0 0
\(78\) 0 0
\(79\) −10.7279 −1.20699 −0.603493 0.797368i \(-0.706225\pi\)
−0.603493 + 0.797368i \(0.706225\pi\)
\(80\) 1.41421 0.158114
\(81\) 0 0
\(82\) −2.82843 −0.312348
\(83\) 17.3137 1.90043 0.950213 0.311601i \(-0.100865\pi\)
0.950213 + 0.311601i \(0.100865\pi\)
\(84\) 0 0
\(85\) 6.24264 0.677109
\(86\) 3.24264 0.349663
\(87\) 0 0
\(88\) 2.24264 0.239066
\(89\) −1.92893 −0.204466 −0.102233 0.994760i \(-0.532599\pi\)
−0.102233 + 0.994760i \(0.532599\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −7.07107 −0.729325
\(95\) 6.48528 0.665376
\(96\) 0 0
\(97\) 14.8284 1.50560 0.752799 0.658250i \(-0.228703\pi\)
0.752799 + 0.658250i \(0.228703\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) −10.2426 −1.01918 −0.509590 0.860417i \(-0.670203\pi\)
−0.509590 + 0.860417i \(0.670203\pi\)
\(102\) 0 0
\(103\) 1.92893 0.190063 0.0950317 0.995474i \(-0.469705\pi\)
0.0950317 + 0.995474i \(0.469705\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 11.2426 1.09198
\(107\) −0.485281 −0.0469139 −0.0234570 0.999725i \(-0.507467\pi\)
−0.0234570 + 0.999725i \(0.507467\pi\)
\(108\) 0 0
\(109\) 0.485281 0.0464815 0.0232408 0.999730i \(-0.492602\pi\)
0.0232408 + 0.999730i \(0.492602\pi\)
\(110\) 3.17157 0.302398
\(111\) 0 0
\(112\) 0 0
\(113\) −12.7279 −1.19734 −0.598671 0.800995i \(-0.704304\pi\)
−0.598671 + 0.800995i \(0.704304\pi\)
\(114\) 0 0
\(115\) −1.41421 −0.131876
\(116\) −5.24264 −0.486767
\(117\) 0 0
\(118\) 0.171573 0.0157946
\(119\) 0 0
\(120\) 0 0
\(121\) −5.97056 −0.542778
\(122\) 11.6569 1.05536
\(123\) 0 0
\(124\) −1.24264 −0.111592
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) 4.72792 0.419535 0.209768 0.977751i \(-0.432729\pi\)
0.209768 + 0.977751i \(0.432729\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 4.24264 0.372104
\(131\) −0.514719 −0.0449712 −0.0224856 0.999747i \(-0.507158\pi\)
−0.0224856 + 0.999747i \(0.507158\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 13.2426 1.14399
\(135\) 0 0
\(136\) 4.41421 0.378516
\(137\) −13.7574 −1.17537 −0.587685 0.809090i \(-0.699961\pi\)
−0.587685 + 0.809090i \(0.699961\pi\)
\(138\) 0 0
\(139\) 17.3137 1.46853 0.734265 0.678863i \(-0.237527\pi\)
0.734265 + 0.678863i \(0.237527\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5.48528 −0.460315
\(143\) 6.72792 0.562617
\(144\) 0 0
\(145\) −7.41421 −0.615717
\(146\) 9.89949 0.819288
\(147\) 0 0
\(148\) −10.4853 −0.861885
\(149\) −9.24264 −0.757187 −0.378593 0.925563i \(-0.623592\pi\)
−0.378593 + 0.925563i \(0.623592\pi\)
\(150\) 0 0
\(151\) 14.7279 1.19854 0.599271 0.800546i \(-0.295457\pi\)
0.599271 + 0.800546i \(0.295457\pi\)
\(152\) 4.58579 0.371956
\(153\) 0 0
\(154\) 0 0
\(155\) −1.75736 −0.141154
\(156\) 0 0
\(157\) −14.6569 −1.16974 −0.584872 0.811125i \(-0.698855\pi\)
−0.584872 + 0.811125i \(0.698855\pi\)
\(158\) −10.7279 −0.853468
\(159\) 0 0
\(160\) 1.41421 0.111803
\(161\) 0 0
\(162\) 0 0
\(163\) −21.2426 −1.66385 −0.831926 0.554887i \(-0.812762\pi\)
−0.831926 + 0.554887i \(0.812762\pi\)
\(164\) −2.82843 −0.220863
\(165\) 0 0
\(166\) 17.3137 1.34380
\(167\) −3.89949 −0.301752 −0.150876 0.988553i \(-0.548209\pi\)
−0.150876 + 0.988553i \(0.548209\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 6.24264 0.478789
\(171\) 0 0
\(172\) 3.24264 0.247249
\(173\) −20.4853 −1.55747 −0.778734 0.627355i \(-0.784138\pi\)
−0.778734 + 0.627355i \(0.784138\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.24264 0.169045
\(177\) 0 0
\(178\) −1.92893 −0.144580
\(179\) 22.2426 1.66249 0.831247 0.555904i \(-0.187628\pi\)
0.831247 + 0.555904i \(0.187628\pi\)
\(180\) 0 0
\(181\) 6.51472 0.484235 0.242118 0.970247i \(-0.422158\pi\)
0.242118 + 0.970247i \(0.422158\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −14.8284 −1.09021
\(186\) 0 0
\(187\) 9.89949 0.723923
\(188\) −7.07107 −0.515711
\(189\) 0 0
\(190\) 6.48528 0.470492
\(191\) −26.9706 −1.95152 −0.975761 0.218840i \(-0.929773\pi\)
−0.975761 + 0.218840i \(0.929773\pi\)
\(192\) 0 0
\(193\) 5.00000 0.359908 0.179954 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 14.8284 1.06462
\(195\) 0 0
\(196\) 0 0
\(197\) 10.4853 0.747045 0.373523 0.927621i \(-0.378150\pi\)
0.373523 + 0.927621i \(0.378150\pi\)
\(198\) 0 0
\(199\) 13.2426 0.938746 0.469373 0.883000i \(-0.344480\pi\)
0.469373 + 0.883000i \(0.344480\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −10.2426 −0.720670
\(203\) 0 0
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 1.92893 0.134395
\(207\) 0 0
\(208\) 3.00000 0.208013
\(209\) 10.2843 0.711378
\(210\) 0 0
\(211\) 19.2426 1.32472 0.662359 0.749187i \(-0.269555\pi\)
0.662359 + 0.749187i \(0.269555\pi\)
\(212\) 11.2426 0.772148
\(213\) 0 0
\(214\) −0.485281 −0.0331732
\(215\) 4.58579 0.312748
\(216\) 0 0
\(217\) 0 0
\(218\) 0.485281 0.0328674
\(219\) 0 0
\(220\) 3.17157 0.213827
\(221\) 13.2426 0.890796
\(222\) 0 0
\(223\) −16.9706 −1.13643 −0.568216 0.822879i \(-0.692366\pi\)
−0.568216 + 0.822879i \(0.692366\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −12.7279 −0.846649
\(227\) −19.9706 −1.32549 −0.662746 0.748844i \(-0.730609\pi\)
−0.662746 + 0.748844i \(0.730609\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) −1.41421 −0.0932505
\(231\) 0 0
\(232\) −5.24264 −0.344196
\(233\) 8.48528 0.555889 0.277945 0.960597i \(-0.410347\pi\)
0.277945 + 0.960597i \(0.410347\pi\)
\(234\) 0 0
\(235\) −10.0000 −0.652328
\(236\) 0.171573 0.0111684
\(237\) 0 0
\(238\) 0 0
\(239\) 22.9706 1.48584 0.742921 0.669379i \(-0.233440\pi\)
0.742921 + 0.669379i \(0.233440\pi\)
\(240\) 0 0
\(241\) −0.727922 −0.0468896 −0.0234448 0.999725i \(-0.507463\pi\)
−0.0234448 + 0.999725i \(0.507463\pi\)
\(242\) −5.97056 −0.383802
\(243\) 0 0
\(244\) 11.6569 0.746254
\(245\) 0 0
\(246\) 0 0
\(247\) 13.7574 0.875360
\(248\) −1.24264 −0.0789078
\(249\) 0 0
\(250\) −11.3137 −0.715542
\(251\) −28.6274 −1.80695 −0.903473 0.428644i \(-0.858991\pi\)
−0.903473 + 0.428644i \(0.858991\pi\)
\(252\) 0 0
\(253\) −2.24264 −0.140994
\(254\) 4.72792 0.296656
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.9706 −1.43286 −0.716432 0.697657i \(-0.754226\pi\)
−0.716432 + 0.697657i \(0.754226\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 4.24264 0.263117
\(261\) 0 0
\(262\) −0.514719 −0.0317994
\(263\) −6.51472 −0.401715 −0.200857 0.979620i \(-0.564373\pi\)
−0.200857 + 0.979620i \(0.564373\pi\)
\(264\) 0 0
\(265\) 15.8995 0.976698
\(266\) 0 0
\(267\) 0 0
\(268\) 13.2426 0.808923
\(269\) 3.55635 0.216834 0.108417 0.994105i \(-0.465422\pi\)
0.108417 + 0.994105i \(0.465422\pi\)
\(270\) 0 0
\(271\) −6.89949 −0.419114 −0.209557 0.977796i \(-0.567202\pi\)
−0.209557 + 0.977796i \(0.567202\pi\)
\(272\) 4.41421 0.267651
\(273\) 0 0
\(274\) −13.7574 −0.831112
\(275\) −6.72792 −0.405709
\(276\) 0 0
\(277\) −4.48528 −0.269494 −0.134747 0.990880i \(-0.543022\pi\)
−0.134747 + 0.990880i \(0.543022\pi\)
\(278\) 17.3137 1.03841
\(279\) 0 0
\(280\) 0 0
\(281\) −22.4853 −1.34136 −0.670680 0.741747i \(-0.733997\pi\)
−0.670680 + 0.741747i \(0.733997\pi\)
\(282\) 0 0
\(283\) 17.6569 1.04959 0.524796 0.851228i \(-0.324142\pi\)
0.524796 + 0.851228i \(0.324142\pi\)
\(284\) −5.48528 −0.325492
\(285\) 0 0
\(286\) 6.72792 0.397830
\(287\) 0 0
\(288\) 0 0
\(289\) 2.48528 0.146193
\(290\) −7.41421 −0.435378
\(291\) 0 0
\(292\) 9.89949 0.579324
\(293\) −27.1716 −1.58738 −0.793690 0.608322i \(-0.791843\pi\)
−0.793690 + 0.608322i \(0.791843\pi\)
\(294\) 0 0
\(295\) 0.242641 0.0141271
\(296\) −10.4853 −0.609445
\(297\) 0 0
\(298\) −9.24264 −0.535412
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 0 0
\(302\) 14.7279 0.847497
\(303\) 0 0
\(304\) 4.58579 0.263013
\(305\) 16.4853 0.943944
\(306\) 0 0
\(307\) −12.3431 −0.704461 −0.352230 0.935913i \(-0.614577\pi\)
−0.352230 + 0.935913i \(0.614577\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.75736 −0.0998113
\(311\) −0.686292 −0.0389160 −0.0194580 0.999811i \(-0.506194\pi\)
−0.0194580 + 0.999811i \(0.506194\pi\)
\(312\) 0 0
\(313\) −4.24264 −0.239808 −0.119904 0.992785i \(-0.538259\pi\)
−0.119904 + 0.992785i \(0.538259\pi\)
\(314\) −14.6569 −0.827134
\(315\) 0 0
\(316\) −10.7279 −0.603493
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) −11.7574 −0.658286
\(320\) 1.41421 0.0790569
\(321\) 0 0
\(322\) 0 0
\(323\) 20.2426 1.12633
\(324\) 0 0
\(325\) −9.00000 −0.499230
\(326\) −21.2426 −1.17652
\(327\) 0 0
\(328\) −2.82843 −0.156174
\(329\) 0 0
\(330\) 0 0
\(331\) 1.72792 0.0949752 0.0474876 0.998872i \(-0.484879\pi\)
0.0474876 + 0.998872i \(0.484879\pi\)
\(332\) 17.3137 0.950213
\(333\) 0 0
\(334\) −3.89949 −0.213371
\(335\) 18.7279 1.02322
\(336\) 0 0
\(337\) 25.4853 1.38827 0.694136 0.719844i \(-0.255787\pi\)
0.694136 + 0.719844i \(0.255787\pi\)
\(338\) −4.00000 −0.217571
\(339\) 0 0
\(340\) 6.24264 0.338555
\(341\) −2.78680 −0.150913
\(342\) 0 0
\(343\) 0 0
\(344\) 3.24264 0.174831
\(345\) 0 0
\(346\) −20.4853 −1.10130
\(347\) 2.48528 0.133417 0.0667084 0.997773i \(-0.478750\pi\)
0.0667084 + 0.997773i \(0.478750\pi\)
\(348\) 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
\(352\) 2.24264 0.119533
\(353\) −12.5563 −0.668307 −0.334154 0.942519i \(-0.608450\pi\)
−0.334154 + 0.942519i \(0.608450\pi\)
\(354\) 0 0
\(355\) −7.75736 −0.411718
\(356\) −1.92893 −0.102233
\(357\) 0 0
\(358\) 22.2426 1.17556
\(359\) −21.9706 −1.15956 −0.579781 0.814772i \(-0.696862\pi\)
−0.579781 + 0.814772i \(0.696862\pi\)
\(360\) 0 0
\(361\) 2.02944 0.106812
\(362\) 6.51472 0.342406
\(363\) 0 0
\(364\) 0 0
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) −13.5858 −0.709172 −0.354586 0.935023i \(-0.615378\pi\)
−0.354586 + 0.935023i \(0.615378\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −14.8284 −0.770893
\(371\) 0 0
\(372\) 0 0
\(373\) 33.6985 1.74484 0.872421 0.488756i \(-0.162549\pi\)
0.872421 + 0.488756i \(0.162549\pi\)
\(374\) 9.89949 0.511891
\(375\) 0 0
\(376\) −7.07107 −0.364662
\(377\) −15.7279 −0.810029
\(378\) 0 0
\(379\) −18.9706 −0.974452 −0.487226 0.873276i \(-0.661991\pi\)
−0.487226 + 0.873276i \(0.661991\pi\)
\(380\) 6.48528 0.332688
\(381\) 0 0
\(382\) −26.9706 −1.37993
\(383\) 9.55635 0.488307 0.244153 0.969737i \(-0.421490\pi\)
0.244153 + 0.969737i \(0.421490\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5.00000 0.254493
\(387\) 0 0
\(388\) 14.8284 0.752799
\(389\) −38.9706 −1.97589 −0.987943 0.154818i \(-0.950521\pi\)
−0.987943 + 0.154818i \(0.950521\pi\)
\(390\) 0 0
\(391\) −4.41421 −0.223236
\(392\) 0 0
\(393\) 0 0
\(394\) 10.4853 0.528241
\(395\) −15.1716 −0.763365
\(396\) 0 0
\(397\) 14.8284 0.744217 0.372109 0.928189i \(-0.378635\pi\)
0.372109 + 0.928189i \(0.378635\pi\)
\(398\) 13.2426 0.663794
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) 14.4853 0.723360 0.361680 0.932302i \(-0.382203\pi\)
0.361680 + 0.932302i \(0.382203\pi\)
\(402\) 0 0
\(403\) −3.72792 −0.185701
\(404\) −10.2426 −0.509590
\(405\) 0 0
\(406\) 0 0
\(407\) −23.5147 −1.16558
\(408\) 0 0
\(409\) 14.4853 0.716251 0.358126 0.933673i \(-0.383416\pi\)
0.358126 + 0.933673i \(0.383416\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) 1.92893 0.0950317
\(413\) 0 0
\(414\) 0 0
\(415\) 24.4853 1.20194
\(416\) 3.00000 0.147087
\(417\) 0 0
\(418\) 10.2843 0.503020
\(419\) 26.6569 1.30227 0.651136 0.758961i \(-0.274293\pi\)
0.651136 + 0.758961i \(0.274293\pi\)
\(420\) 0 0
\(421\) −40.2426 −1.96131 −0.980653 0.195753i \(-0.937285\pi\)
−0.980653 + 0.195753i \(0.937285\pi\)
\(422\) 19.2426 0.936717
\(423\) 0 0
\(424\) 11.2426 0.545991
\(425\) −13.2426 −0.642362
\(426\) 0 0
\(427\) 0 0
\(428\) −0.485281 −0.0234570
\(429\) 0 0
\(430\) 4.58579 0.221146
\(431\) 34.9706 1.68447 0.842236 0.539108i \(-0.181239\pi\)
0.842236 + 0.539108i \(0.181239\pi\)
\(432\) 0 0
\(433\) −32.4853 −1.56114 −0.780571 0.625067i \(-0.785072\pi\)
−0.780571 + 0.625067i \(0.785072\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.485281 0.0232408
\(437\) −4.58579 −0.219368
\(438\) 0 0
\(439\) 12.2132 0.582904 0.291452 0.956585i \(-0.405862\pi\)
0.291452 + 0.956585i \(0.405862\pi\)
\(440\) 3.17157 0.151199
\(441\) 0 0
\(442\) 13.2426 0.629888
\(443\) −26.9706 −1.28141 −0.640705 0.767787i \(-0.721358\pi\)
−0.640705 + 0.767787i \(0.721358\pi\)
\(444\) 0 0
\(445\) −2.72792 −0.129316
\(446\) −16.9706 −0.803579
\(447\) 0 0
\(448\) 0 0
\(449\) −0.727922 −0.0343528 −0.0171764 0.999852i \(-0.505468\pi\)
−0.0171764 + 0.999852i \(0.505468\pi\)
\(450\) 0 0
\(451\) −6.34315 −0.298687
\(452\) −12.7279 −0.598671
\(453\) 0 0
\(454\) −19.9706 −0.937265
\(455\) 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
\(461\) 1.07107 0.0498846 0.0249423 0.999689i \(-0.492060\pi\)
0.0249423 + 0.999689i \(0.492060\pi\)
\(462\) 0 0
\(463\) −23.2132 −1.07881 −0.539405 0.842047i \(-0.681351\pi\)
−0.539405 + 0.842047i \(0.681351\pi\)
\(464\) −5.24264 −0.243383
\(465\) 0 0
\(466\) 8.48528 0.393073
\(467\) 23.6569 1.09471 0.547354 0.836901i \(-0.315635\pi\)
0.547354 + 0.836901i \(0.315635\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −10.0000 −0.461266
\(471\) 0 0
\(472\) 0.171573 0.00789728
\(473\) 7.27208 0.334370
\(474\) 0 0
\(475\) −13.7574 −0.631231
\(476\) 0 0
\(477\) 0 0
\(478\) 22.9706 1.05065
\(479\) −18.0416 −0.824343 −0.412172 0.911106i \(-0.635230\pi\)
−0.412172 + 0.911106i \(0.635230\pi\)
\(480\) 0 0
\(481\) −31.4558 −1.43426
\(482\) −0.727922 −0.0331559
\(483\) 0 0
\(484\) −5.97056 −0.271389
\(485\) 20.9706 0.952224
\(486\) 0 0
\(487\) −22.2426 −1.00791 −0.503955 0.863730i \(-0.668122\pi\)
−0.503955 + 0.863730i \(0.668122\pi\)
\(488\) 11.6569 0.527681
\(489\) 0 0
\(490\) 0 0
\(491\) −10.2426 −0.462244 −0.231122 0.972925i \(-0.574240\pi\)
−0.231122 + 0.972925i \(0.574240\pi\)
\(492\) 0 0
\(493\) −23.1421 −1.04227
\(494\) 13.7574 0.618973
\(495\) 0 0
\(496\) −1.24264 −0.0557962
\(497\) 0 0
\(498\) 0 0
\(499\) 24.9706 1.11784 0.558918 0.829223i \(-0.311217\pi\)
0.558918 + 0.829223i \(0.311217\pi\)
\(500\) −11.3137 −0.505964
\(501\) 0 0
\(502\) −28.6274 −1.27770
\(503\) 8.10051 0.361184 0.180592 0.983558i \(-0.442199\pi\)
0.180592 + 0.983558i \(0.442199\pi\)
\(504\) 0 0
\(505\) −14.4853 −0.644587
\(506\) −2.24264 −0.0996975
\(507\) 0 0
\(508\) 4.72792 0.209768
\(509\) −8.14214 −0.360894 −0.180447 0.983585i \(-0.557754\pi\)
−0.180447 + 0.983585i \(0.557754\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −22.9706 −1.01319
\(515\) 2.72792 0.120207
\(516\) 0 0
\(517\) −15.8579 −0.697428
\(518\) 0 0
\(519\) 0 0
\(520\) 4.24264 0.186052
\(521\) 10.4142 0.456255 0.228127 0.973631i \(-0.426740\pi\)
0.228127 + 0.973631i \(0.426740\pi\)
\(522\) 0 0
\(523\) −11.2721 −0.492894 −0.246447 0.969156i \(-0.579263\pi\)
−0.246447 + 0.969156i \(0.579263\pi\)
\(524\) −0.514719 −0.0224856
\(525\) 0 0
\(526\) −6.51472 −0.284055
\(527\) −5.48528 −0.238943
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 15.8995 0.690630
\(531\) 0 0
\(532\) 0 0
\(533\) −8.48528 −0.367538
\(534\) 0 0
\(535\) −0.686292 −0.0296710
\(536\) 13.2426 0.571995
\(537\) 0 0
\(538\) 3.55635 0.153325
\(539\) 0 0
\(540\) 0 0
\(541\) 10.7279 0.461229 0.230615 0.973045i \(-0.425926\pi\)
0.230615 + 0.973045i \(0.425926\pi\)
\(542\) −6.89949 −0.296359
\(543\) 0 0
\(544\) 4.41421 0.189258
\(545\) 0.686292 0.0293975
\(546\) 0 0
\(547\) −4.48528 −0.191777 −0.0958884 0.995392i \(-0.530569\pi\)
−0.0958884 + 0.995392i \(0.530569\pi\)
\(548\) −13.7574 −0.587685
\(549\) 0 0
\(550\) −6.72792 −0.286880
\(551\) −24.0416 −1.02421
\(552\) 0 0
\(553\) 0 0
\(554\) −4.48528 −0.190561
\(555\) 0 0
\(556\) 17.3137 0.734265
\(557\) −14.6985 −0.622795 −0.311397 0.950280i \(-0.600797\pi\)
−0.311397 + 0.950280i \(0.600797\pi\)
\(558\) 0 0
\(559\) 9.72792 0.411447
\(560\) 0 0
\(561\) 0 0
\(562\) −22.4853 −0.948484
\(563\) 41.1421 1.73393 0.866967 0.498365i \(-0.166066\pi\)
0.866967 + 0.498365i \(0.166066\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) 17.6569 0.742173
\(567\) 0 0
\(568\) −5.48528 −0.230157
\(569\) 1.21320 0.0508601 0.0254301 0.999677i \(-0.491904\pi\)
0.0254301 + 0.999677i \(0.491904\pi\)
\(570\) 0 0
\(571\) −3.24264 −0.135700 −0.0678501 0.997696i \(-0.521614\pi\)
−0.0678501 + 0.997696i \(0.521614\pi\)
\(572\) 6.72792 0.281309
\(573\) 0 0
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) 0 0
\(577\) −2.48528 −0.103464 −0.0517318 0.998661i \(-0.516474\pi\)
−0.0517318 + 0.998661i \(0.516474\pi\)
\(578\) 2.48528 0.103374
\(579\) 0 0
\(580\) −7.41421 −0.307858
\(581\) 0 0
\(582\) 0 0
\(583\) 25.2132 1.04422
\(584\) 9.89949 0.409644
\(585\) 0 0
\(586\) −27.1716 −1.12245
\(587\) 40.4558 1.66979 0.834896 0.550408i \(-0.185528\pi\)
0.834896 + 0.550408i \(0.185528\pi\)
\(588\) 0 0
\(589\) −5.69848 −0.234802
\(590\) 0.242641 0.00998936
\(591\) 0 0
\(592\) −10.4853 −0.430942
\(593\) −20.1421 −0.827138 −0.413569 0.910473i \(-0.635718\pi\)
−0.413569 + 0.910473i \(0.635718\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −9.24264 −0.378593
\(597\) 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
\(601\) −34.2426 −1.39679 −0.698393 0.715714i \(-0.746101\pi\)
−0.698393 + 0.715714i \(0.746101\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 14.7279 0.599271
\(605\) −8.44365 −0.343283
\(606\) 0 0
\(607\) −32.6985 −1.32719 −0.663595 0.748092i \(-0.730970\pi\)
−0.663595 + 0.748092i \(0.730970\pi\)
\(608\) 4.58579 0.185978
\(609\) 0 0
\(610\) 16.4853 0.667470
\(611\) −21.2132 −0.858194
\(612\) 0 0
\(613\) 28.2426 1.14071 0.570355 0.821398i \(-0.306806\pi\)
0.570355 + 0.821398i \(0.306806\pi\)
\(614\) −12.3431 −0.498129
\(615\) 0 0
\(616\) 0 0
\(617\) −35.9411 −1.44694 −0.723468 0.690358i \(-0.757453\pi\)
−0.723468 + 0.690358i \(0.757453\pi\)
\(618\) 0 0
\(619\) 8.82843 0.354844 0.177422 0.984135i \(-0.443224\pi\)
0.177422 + 0.984135i \(0.443224\pi\)
\(620\) −1.75736 −0.0705772
\(621\) 0 0
\(622\) −0.686292 −0.0275178
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) −4.24264 −0.169570
\(627\) 0 0
\(628\) −14.6569 −0.584872
\(629\) −46.2843 −1.84547
\(630\) 0 0
\(631\) 3.75736 0.149578 0.0747891 0.997199i \(-0.476172\pi\)
0.0747891 + 0.997199i \(0.476172\pi\)
\(632\) −10.7279 −0.426734
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) 6.68629 0.265337
\(636\) 0 0
\(637\) 0 0
\(638\) −11.7574 −0.465478
\(639\) 0 0
\(640\) 1.41421 0.0559017
\(641\) 9.45584 0.373483 0.186742 0.982409i \(-0.440207\pi\)
0.186742 + 0.982409i \(0.440207\pi\)
\(642\) 0 0
\(643\) −16.9289 −0.667612 −0.333806 0.942642i \(-0.608333\pi\)
−0.333806 + 0.942642i \(0.608333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 20.2426 0.796436
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 0 0
\(649\) 0.384776 0.0151038
\(650\) −9.00000 −0.353009
\(651\) 0 0
\(652\) −21.2426 −0.831926
\(653\) 32.6985 1.27959 0.639795 0.768545i \(-0.279019\pi\)
0.639795 + 0.768545i \(0.279019\pi\)
\(654\) 0 0
\(655\) −0.727922 −0.0284423
\(656\) −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
\(661\) 21.9411 0.853411 0.426705 0.904391i \(-0.359674\pi\)
0.426705 + 0.904391i \(0.359674\pi\)
\(662\) 1.72792 0.0671576
\(663\) 0 0
\(664\) 17.3137 0.671902
\(665\) 0 0
\(666\) 0 0
\(667\) 5.24264 0.202996
\(668\) −3.89949 −0.150876
\(669\) 0 0
\(670\) 18.7279 0.723523
\(671\) 26.1421 1.00921
\(672\) 0 0
\(673\) −28.4558 −1.09689 −0.548446 0.836186i \(-0.684780\pi\)
−0.548446 + 0.836186i \(0.684780\pi\)
\(674\) 25.4853 0.981656
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) 29.6985 1.14141 0.570703 0.821157i \(-0.306671\pi\)
0.570703 + 0.821157i \(0.306671\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 6.24264 0.239394
\(681\) 0 0
\(682\) −2.78680 −0.106712
\(683\) −13.7574 −0.526411 −0.263205 0.964740i \(-0.584780\pi\)
−0.263205 + 0.964740i \(0.584780\pi\)
\(684\) 0 0
\(685\) −19.4558 −0.743370
\(686\) 0 0
\(687\) 0 0
\(688\) 3.24264 0.123625
\(689\) 33.7279 1.28493
\(690\) 0 0
\(691\) 33.2132 1.26349 0.631745 0.775176i \(-0.282339\pi\)
0.631745 + 0.775176i \(0.282339\pi\)
\(692\) −20.4853 −0.778734
\(693\) 0 0
\(694\) 2.48528 0.0943400
\(695\) 24.4853 0.928780
\(696\) 0 0
\(697\) −12.4853 −0.472914
\(698\) −3.00000 −0.113552
\(699\) 0 0
\(700\) 0 0
\(701\) 1.02944 0.0388813 0.0194407 0.999811i \(-0.493811\pi\)
0.0194407 + 0.999811i \(0.493811\pi\)
\(702\) 0 0
\(703\) −48.0833 −1.81349
\(704\) 2.24264 0.0845227
\(705\) 0 0
\(706\) −12.5563 −0.472564
\(707\) 0 0
\(708\) 0 0
\(709\) 19.2132 0.721567 0.360784 0.932650i \(-0.382509\pi\)
0.360784 + 0.932650i \(0.382509\pi\)
\(710\) −7.75736 −0.291129
\(711\) 0 0
\(712\) −1.92893 −0.0722898
\(713\) 1.24264 0.0465373
\(714\) 0 0
\(715\) 9.51472 0.355830
\(716\) 22.2426 0.831247
\(717\) 0 0
\(718\) −21.9706 −0.819934
\(719\) 34.6274 1.29138 0.645692 0.763598i \(-0.276569\pi\)
0.645692 + 0.763598i \(0.276569\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2.02944 0.0755278
\(723\) 0 0
\(724\) 6.51472 0.242118
\(725\) 15.7279 0.584120
\(726\) 0 0
\(727\) −46.8406 −1.73722 −0.868611 0.495494i \(-0.834987\pi\)
−0.868611 + 0.495494i \(0.834987\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 14.0000 0.518163
\(731\) 14.3137 0.529412
\(732\) 0 0
\(733\) −36.9411 −1.36445 −0.682226 0.731142i \(-0.738988\pi\)
−0.682226 + 0.731142i \(0.738988\pi\)
\(734\) −13.5858 −0.501461
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 29.6985 1.09396
\(738\) 0 0
\(739\) 22.9706 0.844986 0.422493 0.906366i \(-0.361155\pi\)
0.422493 + 0.906366i \(0.361155\pi\)
\(740\) −14.8284 −0.545104
\(741\) 0 0
\(742\) 0 0
\(743\) 21.4853 0.788219 0.394109 0.919064i \(-0.371053\pi\)
0.394109 + 0.919064i \(0.371053\pi\)
\(744\) 0 0
\(745\) −13.0711 −0.478887
\(746\) 33.6985 1.23379
\(747\) 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
\(754\) −15.7279 −0.572777
\(755\) 20.8284 0.758024
\(756\) 0 0
\(757\) 31.6985 1.15210 0.576051 0.817414i \(-0.304593\pi\)
0.576051 + 0.817414i \(0.304593\pi\)
\(758\) −18.9706 −0.689042
\(759\) 0 0
\(760\) 6.48528 0.235246
\(761\) 11.5269 0.417850 0.208925 0.977932i \(-0.433004\pi\)
0.208925 + 0.977932i \(0.433004\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −26.9706 −0.975761
\(765\) 0 0
\(766\) 9.55635 0.345285
\(767\) 0.514719 0.0185854
\(768\) 0 0
\(769\) −27.2132 −0.981333 −0.490667 0.871347i \(-0.663247\pi\)
−0.490667 + 0.871347i \(0.663247\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.00000 0.179954
\(773\) −18.3848 −0.661254 −0.330627 0.943761i \(-0.607260\pi\)
−0.330627 + 0.943761i \(0.607260\pi\)
\(774\) 0 0
\(775\) 3.72792 0.133911
\(776\) 14.8284 0.532310
\(777\) 0 0
\(778\) −38.9706 −1.39716
\(779\) −12.9706 −0.464719
\(780\) 0 0
\(781\) −12.3015 −0.440183
\(782\) −4.41421 −0.157852
\(783\) 0 0
\(784\) 0 0
\(785\) −20.7279 −0.739811
\(786\) 0 0
\(787\) −32.8284 −1.17021 −0.585104 0.810959i \(-0.698946\pi\)
−0.585104 + 0.810959i \(0.698946\pi\)
\(788\) 10.4853 0.373523
\(789\) 0 0
\(790\) −15.1716 −0.539780
\(791\) 0 0
\(792\) 0 0
\(793\) 34.9706 1.24184
\(794\) 14.8284 0.526241
\(795\) 0 0
\(796\) 13.2426 0.469373
\(797\) −33.5147 −1.18715 −0.593576 0.804778i \(-0.702284\pi\)
−0.593576 + 0.804778i \(0.702284\pi\)
\(798\) 0 0
\(799\) −31.2132 −1.10424
\(800\) −3.00000 −0.106066
\(801\) 0 0
\(802\) 14.4853 0.511493
\(803\) 22.2010 0.783457
\(804\) 0 0
\(805\) 0 0
\(806\) −3.72792 −0.131310
\(807\) 0 0
\(808\) −10.2426 −0.360335
\(809\) −4.00000 −0.140633 −0.0703163 0.997525i \(-0.522401\pi\)
−0.0703163 + 0.997525i \(0.522401\pi\)
\(810\) 0 0
\(811\) 51.9411 1.82390 0.911950 0.410302i \(-0.134577\pi\)
0.911950 + 0.410302i \(0.134577\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −23.5147 −0.824190
\(815\) −30.0416 −1.05231
\(816\) 0 0
\(817\) 14.8701 0.520237
\(818\) 14.4853 0.506466
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 18.2132 0.635645 0.317823 0.948150i \(-0.397048\pi\)
0.317823 + 0.948150i \(0.397048\pi\)
\(822\) 0 0
\(823\) −6.97056 −0.242979 −0.121489 0.992593i \(-0.538767\pi\)
−0.121489 + 0.992593i \(0.538767\pi\)
\(824\) 1.92893 0.0671975
\(825\) 0 0
\(826\) 0 0
\(827\) 29.6985 1.03272 0.516359 0.856372i \(-0.327287\pi\)
0.516359 + 0.856372i \(0.327287\pi\)
\(828\) 0 0
\(829\) 21.1716 0.735319 0.367660 0.929960i \(-0.380159\pi\)
0.367660 + 0.929960i \(0.380159\pi\)
\(830\) 24.4853 0.849897
\(831\) 0 0
\(832\) 3.00000 0.104006
\(833\) 0 0
\(834\) 0 0
\(835\) −5.51472 −0.190845
\(836\) 10.2843 0.355689
\(837\) 0 0
\(838\) 26.6569 0.920846
\(839\) −1.79899 −0.0621080 −0.0310540 0.999518i \(-0.509886\pi\)
−0.0310540 + 0.999518i \(0.509886\pi\)
\(840\) 0 0
\(841\) −1.51472 −0.0522317
\(842\) −40.2426 −1.38685
\(843\) 0 0
\(844\) 19.2426 0.662359
\(845\) −5.65685 −0.194602
\(846\) 0 0
\(847\) 0 0
\(848\) 11.2426 0.386074
\(849\) 0 0
\(850\) −13.2426 −0.454219
\(851\) 10.4853 0.359431
\(852\) 0 0
\(853\) 45.4264 1.55537 0.777685 0.628654i \(-0.216394\pi\)
0.777685 + 0.628654i \(0.216394\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.485281 −0.0165866
\(857\) 43.5858 1.48886 0.744431 0.667699i \(-0.232721\pi\)
0.744431 + 0.667699i \(0.232721\pi\)
\(858\) 0 0
\(859\) −42.7696 −1.45928 −0.729639 0.683832i \(-0.760312\pi\)
−0.729639 + 0.683832i \(0.760312\pi\)
\(860\) 4.58579 0.156374
\(861\) 0 0
\(862\) 34.9706 1.19110
\(863\) 2.51472 0.0856020 0.0428010 0.999084i \(-0.486372\pi\)
0.0428010 + 0.999084i \(0.486372\pi\)
\(864\) 0 0
\(865\) −28.9706 −0.985029
\(866\) −32.4853 −1.10389
\(867\) 0 0
\(868\) 0 0
\(869\) −24.0589 −0.816141
\(870\) 0 0
\(871\) 39.7279 1.34613
\(872\) 0.485281 0.0164337
\(873\) 0 0
\(874\) −4.58579 −0.155117
\(875\) 0 0
\(876\) 0 0
\(877\) 4.48528 0.151457 0.0757286 0.997128i \(-0.475872\pi\)
0.0757286 + 0.997128i \(0.475872\pi\)
\(878\) 12.2132 0.412176
\(879\) 0 0
\(880\) 3.17157 0.106914
\(881\) −8.27208 −0.278693 −0.139347 0.990244i \(-0.544500\pi\)
−0.139347 + 0.990244i \(0.544500\pi\)
\(882\) 0 0
\(883\) −37.7279 −1.26965 −0.634823 0.772658i \(-0.718927\pi\)
−0.634823 + 0.772658i \(0.718927\pi\)
\(884\) 13.2426 0.445398
\(885\) 0 0
\(886\) −26.9706 −0.906094
\(887\) 38.8284 1.30373 0.651865 0.758335i \(-0.273987\pi\)
0.651865 + 0.758335i \(0.273987\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −2.72792 −0.0914402
\(891\) 0 0
\(892\) −16.9706 −0.568216
\(893\) −32.4264 −1.08511
\(894\) 0 0
\(895\) 31.4558 1.05145
\(896\) 0 0
\(897\) 0 0
\(898\) −0.727922 −0.0242911
\(899\) 6.51472 0.217278
\(900\) 0 0
\(901\) 49.6274 1.65333
\(902\) −6.34315 −0.211204
\(903\) 0 0
\(904\) −12.7279 −0.423324
\(905\) 9.21320 0.306257
\(906\) 0 0
\(907\) −23.9411 −0.794952 −0.397476 0.917613i \(-0.630114\pi\)
−0.397476 + 0.917613i \(0.630114\pi\)
\(908\) −19.9706 −0.662746
\(909\) 0 0
\(910\) 0 0
\(911\) 34.4853 1.14255 0.571274 0.820759i \(-0.306449\pi\)
0.571274 + 0.820759i \(0.306449\pi\)
\(912\) 0 0
\(913\) 38.8284 1.28503
\(914\) −15.0000 −0.496156
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 14.2426 0.469821 0.234911 0.972017i \(-0.424520\pi\)
0.234911 + 0.972017i \(0.424520\pi\)
\(920\) −1.41421 −0.0466252
\(921\) 0 0
\(922\) 1.07107 0.0352737
\(923\) −16.4558 −0.541651
\(924\) 0 0
\(925\) 31.4558 1.03426
\(926\) −23.2132 −0.762833
\(927\) 0 0
\(928\) −5.24264 −0.172098
\(929\) 18.6863 0.613077 0.306539 0.951858i \(-0.400829\pi\)
0.306539 + 0.951858i \(0.400829\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 8.48528 0.277945
\(933\) 0 0
\(934\) 23.6569 0.774076
\(935\) 14.0000 0.457849
\(936\) 0 0
\(937\) −22.9289 −0.749056 −0.374528 0.927216i \(-0.622195\pi\)
−0.374528 + 0.927216i \(0.622195\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −10.0000 −0.326164
\(941\) −34.2843 −1.11764 −0.558818 0.829291i \(-0.688745\pi\)
−0.558818 + 0.829291i \(0.688745\pi\)
\(942\) 0 0
\(943\) 2.82843 0.0921063
\(944\) 0.171573 0.00558422
\(945\) 0 0
\(946\) 7.27208 0.236436
\(947\) −11.2721 −0.366293 −0.183147 0.983086i \(-0.558628\pi\)
−0.183147 + 0.983086i \(0.558628\pi\)
\(948\) 0 0
\(949\) 29.6985 0.964054
\(950\) −13.7574 −0.446348
\(951\) 0 0
\(952\) 0 0
\(953\) −49.6985 −1.60989 −0.804946 0.593348i \(-0.797806\pi\)
−0.804946 + 0.593348i \(0.797806\pi\)
\(954\) 0 0
\(955\) −38.1421 −1.23425
\(956\) 22.9706 0.742921
\(957\) 0 0
\(958\) −18.0416 −0.582899
\(959\) 0 0
\(960\) 0 0
\(961\) −29.4558 −0.950189
\(962\) −31.4558 −1.01418
\(963\) 0 0
\(964\) −0.727922 −0.0234448
\(965\) 7.07107 0.227626
\(966\) 0 0
\(967\) 55.1543 1.77364 0.886822 0.462112i \(-0.152908\pi\)
0.886822 + 0.462112i \(0.152908\pi\)
\(968\) −5.97056 −0.191901
\(969\) 0 0
\(970\) 20.9706 0.673324
\(971\) −45.7696 −1.46881 −0.734407 0.678709i \(-0.762540\pi\)
−0.734407 + 0.678709i \(0.762540\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −22.2426 −0.712700
\(975\) 0 0
\(976\) 11.6569 0.373127
\(977\) −40.6690 −1.30112 −0.650559 0.759456i \(-0.725465\pi\)
−0.650559 + 0.759456i \(0.725465\pi\)
\(978\) 0 0
\(979\) −4.32590 −0.138256
\(980\) 0 0
\(981\) 0 0
\(982\) −10.2426 −0.326856
\(983\) 21.5563 0.687541 0.343770 0.939054i \(-0.388296\pi\)
0.343770 + 0.939054i \(0.388296\pi\)
\(984\) 0 0
\(985\) 14.8284 0.472473
\(986\) −23.1421 −0.736996
\(987\) 0 0
\(988\) 13.7574 0.437680
\(989\) −3.24264 −0.103110
\(990\) 0 0
\(991\) −11.4558 −0.363907 −0.181953 0.983307i \(-0.558242\pi\)
−0.181953 + 0.983307i \(0.558242\pi\)
\(992\) −1.24264 −0.0394539
\(993\) 0 0
\(994\) 0 0
\(995\) 18.7279 0.593715
\(996\) 0 0
\(997\) −18.8579 −0.597235 −0.298617 0.954373i \(-0.596525\pi\)
−0.298617 + 0.954373i \(0.596525\pi\)
\(998\) 24.9706 0.790429
\(999\) 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.1.2 yes 2
3.2 odd 2 2646.2.a.bh.1.1 yes 2
7.6 odd 2 2646.2.a.bm.1.1 yes 2
21.20 even 2 2646.2.a.bg.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2646.2.a.bg.1.2 2 21.20 even 2
2646.2.a.bh.1.1 yes 2 3.2 odd 2
2646.2.a.bm.1.1 yes 2 7.6 odd 2
2646.2.a.bn.1.2 yes 2 1.1 even 1 trivial