Properties

Label 637.2.a.g.1.1
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{2} -1.41421 q^{3} -1.58579 q^{5} +2.00000 q^{6} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{2} -1.41421 q^{3} -1.58579 q^{5} +2.00000 q^{6} +2.82843 q^{8} -1.00000 q^{9} +2.24264 q^{10} +4.24264 q^{11} +1.00000 q^{13} +2.24264 q^{15} -4.00000 q^{16} -1.41421 q^{17} +1.41421 q^{18} +7.24264 q^{19} -6.00000 q^{22} -5.82843 q^{23} -4.00000 q^{24} -2.48528 q^{25} -1.41421 q^{26} +5.65685 q^{27} +0.171573 q^{29} -3.17157 q^{30} -3.24264 q^{31} -6.00000 q^{33} +2.00000 q^{34} +2.24264 q^{37} -10.2426 q^{38} -1.41421 q^{39} -4.48528 q^{40} -8.82843 q^{41} -5.00000 q^{43} +1.58579 q^{45} +8.24264 q^{46} -1.58579 q^{47} +5.65685 q^{48} +3.51472 q^{50} +2.00000 q^{51} -0.171573 q^{53} -8.00000 q^{54} -6.72792 q^{55} -10.2426 q^{57} -0.242641 q^{58} -0.343146 q^{59} -6.00000 q^{61} +4.58579 q^{62} +8.00000 q^{64} -1.58579 q^{65} +8.48528 q^{66} -14.4853 q^{67} +8.24264 q^{69} -13.0711 q^{71} -2.82843 q^{72} +9.24264 q^{73} -3.17157 q^{74} +3.51472 q^{75} +2.00000 q^{78} +15.4853 q^{79} +6.34315 q^{80} -5.00000 q^{81} +12.4853 q^{82} -13.2426 q^{83} +2.24264 q^{85} +7.07107 q^{86} -0.242641 q^{87} +12.0000 q^{88} -1.58579 q^{89} -2.24264 q^{90} +4.58579 q^{93} +2.24264 q^{94} -11.4853 q^{95} -11.7279 q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{5} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{5} + 4 q^{6} - 2 q^{9} - 4 q^{10} + 2 q^{13} - 4 q^{15} - 8 q^{16} + 6 q^{19} - 12 q^{22} - 6 q^{23} - 8 q^{24} + 12 q^{25} + 6 q^{29} - 12 q^{30} + 2 q^{31} - 12 q^{33} + 4 q^{34} - 4 q^{37} - 12 q^{38} + 8 q^{40} - 12 q^{41} - 10 q^{43} + 6 q^{45} + 8 q^{46} - 6 q^{47} + 24 q^{50} + 4 q^{51} - 6 q^{53} - 16 q^{54} + 12 q^{55} - 12 q^{57} + 8 q^{58} - 12 q^{59} - 12 q^{61} + 12 q^{62} + 16 q^{64} - 6 q^{65} - 12 q^{67} + 8 q^{69} - 12 q^{71} + 10 q^{73} - 12 q^{74} + 24 q^{75} + 4 q^{78} + 14 q^{79} + 24 q^{80} - 10 q^{81} + 8 q^{82} - 18 q^{83} - 4 q^{85} + 8 q^{87} + 24 q^{88} - 6 q^{89} + 4 q^{90} + 12 q^{93} - 4 q^{94} - 6 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) −1.58579 −0.709185 −0.354593 0.935021i \(-0.615380\pi\)
−0.354593 + 0.935021i \(0.615380\pi\)
\(6\) 2.00000 0.816497
\(7\) 0 0
\(8\) 2.82843 1.00000
\(9\) −1.00000 −0.333333
\(10\) 2.24264 0.709185
\(11\) 4.24264 1.27920 0.639602 0.768706i \(-0.279099\pi\)
0.639602 + 0.768706i \(0.279099\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 2.24264 0.579047
\(16\) −4.00000 −1.00000
\(17\) −1.41421 −0.342997 −0.171499 0.985184i \(-0.554861\pi\)
−0.171499 + 0.985184i \(0.554861\pi\)
\(18\) 1.41421 0.333333
\(19\) 7.24264 1.66158 0.830788 0.556589i \(-0.187890\pi\)
0.830788 + 0.556589i \(0.187890\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) −5.82843 −1.21531 −0.607656 0.794201i \(-0.707890\pi\)
−0.607656 + 0.794201i \(0.707890\pi\)
\(24\) −4.00000 −0.816497
\(25\) −2.48528 −0.497056
\(26\) −1.41421 −0.277350
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 0.171573 0.0318603 0.0159301 0.999873i \(-0.494929\pi\)
0.0159301 + 0.999873i \(0.494929\pi\)
\(30\) −3.17157 −0.579047
\(31\) −3.24264 −0.582395 −0.291198 0.956663i \(-0.594054\pi\)
−0.291198 + 0.956663i \(0.594054\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) 2.24264 0.368688 0.184344 0.982862i \(-0.440984\pi\)
0.184344 + 0.982862i \(0.440984\pi\)
\(38\) −10.2426 −1.66158
\(39\) −1.41421 −0.226455
\(40\) −4.48528 −0.709185
\(41\) −8.82843 −1.37877 −0.689384 0.724396i \(-0.742119\pi\)
−0.689384 + 0.724396i \(0.742119\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 1.58579 0.236395
\(46\) 8.24264 1.21531
\(47\) −1.58579 −0.231311 −0.115655 0.993289i \(-0.536897\pi\)
−0.115655 + 0.993289i \(0.536897\pi\)
\(48\) 5.65685 0.816497
\(49\) 0 0
\(50\) 3.51472 0.497056
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −0.171573 −0.0235673 −0.0117837 0.999931i \(-0.503751\pi\)
−0.0117837 + 0.999931i \(0.503751\pi\)
\(54\) −8.00000 −1.08866
\(55\) −6.72792 −0.907193
\(56\) 0 0
\(57\) −10.2426 −1.35667
\(58\) −0.242641 −0.0318603
\(59\) −0.343146 −0.0446738 −0.0223369 0.999751i \(-0.507111\pi\)
−0.0223369 + 0.999751i \(0.507111\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 4.58579 0.582395
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) −1.58579 −0.196693
\(66\) 8.48528 1.04447
\(67\) −14.4853 −1.76966 −0.884829 0.465915i \(-0.845725\pi\)
−0.884829 + 0.465915i \(0.845725\pi\)
\(68\) 0 0
\(69\) 8.24264 0.992297
\(70\) 0 0
\(71\) −13.0711 −1.55125 −0.775625 0.631194i \(-0.782565\pi\)
−0.775625 + 0.631194i \(0.782565\pi\)
\(72\) −2.82843 −0.333333
\(73\) 9.24264 1.08177 0.540885 0.841097i \(-0.318090\pi\)
0.540885 + 0.841097i \(0.318090\pi\)
\(74\) −3.17157 −0.368688
\(75\) 3.51472 0.405845
\(76\) 0 0
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 15.4853 1.74223 0.871115 0.491079i \(-0.163397\pi\)
0.871115 + 0.491079i \(0.163397\pi\)
\(80\) 6.34315 0.709185
\(81\) −5.00000 −0.555556
\(82\) 12.4853 1.37877
\(83\) −13.2426 −1.45357 −0.726784 0.686866i \(-0.758986\pi\)
−0.726784 + 0.686866i \(0.758986\pi\)
\(84\) 0 0
\(85\) 2.24264 0.243249
\(86\) 7.07107 0.762493
\(87\) −0.242641 −0.0260138
\(88\) 12.0000 1.27920
\(89\) −1.58579 −0.168093 −0.0840465 0.996462i \(-0.526784\pi\)
−0.0840465 + 0.996462i \(0.526784\pi\)
\(90\) −2.24264 −0.236395
\(91\) 0 0
\(92\) 0 0
\(93\) 4.58579 0.475524
\(94\) 2.24264 0.231311
\(95\) −11.4853 −1.17837
\(96\) 0 0
\(97\) −11.7279 −1.19079 −0.595395 0.803433i \(-0.703004\pi\)
−0.595395 + 0.803433i \(0.703004\pi\)
\(98\) 0 0
\(99\) −4.24264 −0.426401
\(100\) 0 0
\(101\) −10.2426 −1.01918 −0.509590 0.860417i \(-0.670203\pi\)
−0.509590 + 0.860417i \(0.670203\pi\)
\(102\) −2.82843 −0.280056
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 2.82843 0.277350
\(105\) 0 0
\(106\) 0.242641 0.0235673
\(107\) −20.1421 −1.94721 −0.973607 0.228232i \(-0.926706\pi\)
−0.973607 + 0.228232i \(0.926706\pi\)
\(108\) 0 0
\(109\) 16.7279 1.60224 0.801122 0.598501i \(-0.204237\pi\)
0.801122 + 0.598501i \(0.204237\pi\)
\(110\) 9.51472 0.907193
\(111\) −3.17157 −0.301032
\(112\) 0 0
\(113\) 2.31371 0.217655 0.108828 0.994061i \(-0.465290\pi\)
0.108828 + 0.994061i \(0.465290\pi\)
\(114\) 14.4853 1.35667
\(115\) 9.24264 0.861881
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0.485281 0.0446738
\(119\) 0 0
\(120\) 6.34315 0.579047
\(121\) 7.00000 0.636364
\(122\) 8.48528 0.768221
\(123\) 12.4853 1.12576
\(124\) 0 0
\(125\) 11.8701 1.06169
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −11.3137 −1.00000
\(129\) 7.07107 0.622573
\(130\) 2.24264 0.196693
\(131\) 2.82843 0.247121 0.123560 0.992337i \(-0.460569\pi\)
0.123560 + 0.992337i \(0.460569\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 20.4853 1.76966
\(135\) −8.97056 −0.772063
\(136\) −4.00000 −0.342997
\(137\) −4.58579 −0.391790 −0.195895 0.980625i \(-0.562761\pi\)
−0.195895 + 0.980625i \(0.562761\pi\)
\(138\) −11.6569 −0.992297
\(139\) −6.24264 −0.529494 −0.264747 0.964318i \(-0.585288\pi\)
−0.264747 + 0.964318i \(0.585288\pi\)
\(140\) 0 0
\(141\) 2.24264 0.188864
\(142\) 18.4853 1.55125
\(143\) 4.24264 0.354787
\(144\) 4.00000 0.333333
\(145\) −0.272078 −0.0225948
\(146\) −13.0711 −1.08177
\(147\) 0 0
\(148\) 0 0
\(149\) 16.2426 1.33065 0.665324 0.746554i \(-0.268293\pi\)
0.665324 + 0.746554i \(0.268293\pi\)
\(150\) −4.97056 −0.405845
\(151\) −9.75736 −0.794043 −0.397021 0.917809i \(-0.629956\pi\)
−0.397021 + 0.917809i \(0.629956\pi\)
\(152\) 20.4853 1.66158
\(153\) 1.41421 0.114332
\(154\) 0 0
\(155\) 5.14214 0.413026
\(156\) 0 0
\(157\) 3.75736 0.299870 0.149935 0.988696i \(-0.452094\pi\)
0.149935 + 0.988696i \(0.452094\pi\)
\(158\) −21.8995 −1.74223
\(159\) 0.242641 0.0192427
\(160\) 0 0
\(161\) 0 0
\(162\) 7.07107 0.555556
\(163\) 8.48528 0.664619 0.332309 0.943170i \(-0.392172\pi\)
0.332309 + 0.943170i \(0.392172\pi\)
\(164\) 0 0
\(165\) 9.51472 0.740720
\(166\) 18.7279 1.45357
\(167\) −15.3848 −1.19051 −0.595255 0.803537i \(-0.702949\pi\)
−0.595255 + 0.803537i \(0.702949\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −3.17157 −0.243249
\(171\) −7.24264 −0.553859
\(172\) 0 0
\(173\) −24.7279 −1.88003 −0.940015 0.341134i \(-0.889189\pi\)
−0.940015 + 0.341134i \(0.889189\pi\)
\(174\) 0.343146 0.0260138
\(175\) 0 0
\(176\) −16.9706 −1.27920
\(177\) 0.485281 0.0364760
\(178\) 2.24264 0.168093
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 18.7279 1.39204 0.696018 0.718025i \(-0.254953\pi\)
0.696018 + 0.718025i \(0.254953\pi\)
\(182\) 0 0
\(183\) 8.48528 0.627250
\(184\) −16.4853 −1.21531
\(185\) −3.55635 −0.261468
\(186\) −6.48528 −0.475524
\(187\) −6.00000 −0.438763
\(188\) 0 0
\(189\) 0 0
\(190\) 16.2426 1.17837
\(191\) 15.1716 1.09778 0.548888 0.835896i \(-0.315051\pi\)
0.548888 + 0.835896i \(0.315051\pi\)
\(192\) −11.3137 −0.816497
\(193\) −14.4853 −1.04267 −0.521337 0.853351i \(-0.674566\pi\)
−0.521337 + 0.853351i \(0.674566\pi\)
\(194\) 16.5858 1.19079
\(195\) 2.24264 0.160599
\(196\) 0 0
\(197\) 12.3431 0.879413 0.439706 0.898142i \(-0.355083\pi\)
0.439706 + 0.898142i \(0.355083\pi\)
\(198\) 6.00000 0.426401
\(199\) 3.75736 0.266352 0.133176 0.991092i \(-0.457482\pi\)
0.133176 + 0.991092i \(0.457482\pi\)
\(200\) −7.02944 −0.497056
\(201\) 20.4853 1.44492
\(202\) 14.4853 1.01918
\(203\) 0 0
\(204\) 0 0
\(205\) 14.0000 0.977802
\(206\) −11.3137 −0.788263
\(207\) 5.82843 0.405104
\(208\) −4.00000 −0.277350
\(209\) 30.7279 2.12549
\(210\) 0 0
\(211\) −15.9706 −1.09946 −0.549729 0.835343i \(-0.685269\pi\)
−0.549729 + 0.835343i \(0.685269\pi\)
\(212\) 0 0
\(213\) 18.4853 1.26659
\(214\) 28.4853 1.94721
\(215\) 7.92893 0.540749
\(216\) 16.0000 1.08866
\(217\) 0 0
\(218\) −23.6569 −1.60224
\(219\) −13.0711 −0.883261
\(220\) 0 0
\(221\) −1.41421 −0.0951303
\(222\) 4.48528 0.301032
\(223\) −0.757359 −0.0507165 −0.0253583 0.999678i \(-0.508073\pi\)
−0.0253583 + 0.999678i \(0.508073\pi\)
\(224\) 0 0
\(225\) 2.48528 0.165685
\(226\) −3.27208 −0.217655
\(227\) −26.8284 −1.78067 −0.890333 0.455311i \(-0.849528\pi\)
−0.890333 + 0.455311i \(0.849528\pi\)
\(228\) 0 0
\(229\) −29.4558 −1.94650 −0.973248 0.229755i \(-0.926207\pi\)
−0.973248 + 0.229755i \(0.926207\pi\)
\(230\) −13.0711 −0.861881
\(231\) 0 0
\(232\) 0.485281 0.0318603
\(233\) −14.6569 −0.960202 −0.480101 0.877213i \(-0.659400\pi\)
−0.480101 + 0.877213i \(0.659400\pi\)
\(234\) 1.41421 0.0924500
\(235\) 2.51472 0.164042
\(236\) 0 0
\(237\) −21.8995 −1.42253
\(238\) 0 0
\(239\) 3.51472 0.227348 0.113674 0.993518i \(-0.463738\pi\)
0.113674 + 0.993518i \(0.463738\pi\)
\(240\) −8.97056 −0.579047
\(241\) 20.2132 1.30205 0.651023 0.759058i \(-0.274340\pi\)
0.651023 + 0.759058i \(0.274340\pi\)
\(242\) −9.89949 −0.636364
\(243\) −9.89949 −0.635053
\(244\) 0 0
\(245\) 0 0
\(246\) −17.6569 −1.12576
\(247\) 7.24264 0.460838
\(248\) −9.17157 −0.582395
\(249\) 18.7279 1.18683
\(250\) −16.7868 −1.06169
\(251\) −16.5858 −1.04689 −0.523443 0.852061i \(-0.675353\pi\)
−0.523443 + 0.852061i \(0.675353\pi\)
\(252\) 0 0
\(253\) −24.7279 −1.55463
\(254\) −2.82843 −0.177471
\(255\) −3.17157 −0.198612
\(256\) 0 0
\(257\) 19.4142 1.21103 0.605513 0.795836i \(-0.292968\pi\)
0.605513 + 0.795836i \(0.292968\pi\)
\(258\) −10.0000 −0.622573
\(259\) 0 0
\(260\) 0 0
\(261\) −0.171573 −0.0106201
\(262\) −4.00000 −0.247121
\(263\) 1.97056 0.121510 0.0607551 0.998153i \(-0.480649\pi\)
0.0607551 + 0.998153i \(0.480649\pi\)
\(264\) −16.9706 −1.04447
\(265\) 0.272078 0.0167136
\(266\) 0 0
\(267\) 2.24264 0.137247
\(268\) 0 0
\(269\) −9.17157 −0.559201 −0.279600 0.960116i \(-0.590202\pi\)
−0.279600 + 0.960116i \(0.590202\pi\)
\(270\) 12.6863 0.772063
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 5.65685 0.342997
\(273\) 0 0
\(274\) 6.48528 0.391790
\(275\) −10.5442 −0.635837
\(276\) 0 0
\(277\) −7.48528 −0.449747 −0.224873 0.974388i \(-0.572197\pi\)
−0.224873 + 0.974388i \(0.572197\pi\)
\(278\) 8.82843 0.529494
\(279\) 3.24264 0.194132
\(280\) 0 0
\(281\) 15.5563 0.928014 0.464007 0.885832i \(-0.346411\pi\)
0.464007 + 0.885832i \(0.346411\pi\)
\(282\) −3.17157 −0.188864
\(283\) 8.48528 0.504398 0.252199 0.967675i \(-0.418846\pi\)
0.252199 + 0.967675i \(0.418846\pi\)
\(284\) 0 0
\(285\) 16.2426 0.962131
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 0.384776 0.0225948
\(291\) 16.5858 0.972276
\(292\) 0 0
\(293\) −15.3848 −0.898788 −0.449394 0.893334i \(-0.648360\pi\)
−0.449394 + 0.893334i \(0.648360\pi\)
\(294\) 0 0
\(295\) 0.544156 0.0316820
\(296\) 6.34315 0.368688
\(297\) 24.0000 1.39262
\(298\) −22.9706 −1.33065
\(299\) −5.82843 −0.337067
\(300\) 0 0
\(301\) 0 0
\(302\) 13.7990 0.794043
\(303\) 14.4853 0.832158
\(304\) −28.9706 −1.66158
\(305\) 9.51472 0.544811
\(306\) −2.00000 −0.114332
\(307\) −13.2426 −0.755797 −0.377899 0.925847i \(-0.623353\pi\)
−0.377899 + 0.925847i \(0.623353\pi\)
\(308\) 0 0
\(309\) −11.3137 −0.643614
\(310\) −7.27208 −0.413026
\(311\) 7.41421 0.420421 0.210211 0.977656i \(-0.432585\pi\)
0.210211 + 0.977656i \(0.432585\pi\)
\(312\) −4.00000 −0.226455
\(313\) −23.2132 −1.31209 −0.656044 0.754723i \(-0.727771\pi\)
−0.656044 + 0.754723i \(0.727771\pi\)
\(314\) −5.31371 −0.299870
\(315\) 0 0
\(316\) 0 0
\(317\) −11.3137 −0.635441 −0.317721 0.948184i \(-0.602917\pi\)
−0.317721 + 0.948184i \(0.602917\pi\)
\(318\) −0.343146 −0.0192427
\(319\) 0.727922 0.0407558
\(320\) −12.6863 −0.709185
\(321\) 28.4853 1.58989
\(322\) 0 0
\(323\) −10.2426 −0.569916
\(324\) 0 0
\(325\) −2.48528 −0.137859
\(326\) −12.0000 −0.664619
\(327\) −23.6569 −1.30823
\(328\) −24.9706 −1.37877
\(329\) 0 0
\(330\) −13.4558 −0.740720
\(331\) −18.0000 −0.989369 −0.494685 0.869072i \(-0.664716\pi\)
−0.494685 + 0.869072i \(0.664716\pi\)
\(332\) 0 0
\(333\) −2.24264 −0.122896
\(334\) 21.7574 1.19051
\(335\) 22.9706 1.25502
\(336\) 0 0
\(337\) −33.0000 −1.79762 −0.898812 0.438334i \(-0.855569\pi\)
−0.898812 + 0.438334i \(0.855569\pi\)
\(338\) −1.41421 −0.0769231
\(339\) −3.27208 −0.177715
\(340\) 0 0
\(341\) −13.7574 −0.745003
\(342\) 10.2426 0.553859
\(343\) 0 0
\(344\) −14.1421 −0.762493
\(345\) −13.0711 −0.703723
\(346\) 34.9706 1.88003
\(347\) −5.65685 −0.303676 −0.151838 0.988405i \(-0.548519\pi\)
−0.151838 + 0.988405i \(0.548519\pi\)
\(348\) 0 0
\(349\) 25.7279 1.37718 0.688592 0.725149i \(-0.258229\pi\)
0.688592 + 0.725149i \(0.258229\pi\)
\(350\) 0 0
\(351\) 5.65685 0.301941
\(352\) 0 0
\(353\) 8.48528 0.451626 0.225813 0.974171i \(-0.427496\pi\)
0.225813 + 0.974171i \(0.427496\pi\)
\(354\) −0.686292 −0.0364760
\(355\) 20.7279 1.10012
\(356\) 0 0
\(357\) 0 0
\(358\) 12.7279 0.672692
\(359\) −27.8995 −1.47248 −0.736240 0.676721i \(-0.763400\pi\)
−0.736240 + 0.676721i \(0.763400\pi\)
\(360\) 4.48528 0.236395
\(361\) 33.4558 1.76083
\(362\) −26.4853 −1.39204
\(363\) −9.89949 −0.519589
\(364\) 0 0
\(365\) −14.6569 −0.767175
\(366\) −12.0000 −0.627250
\(367\) −10.2426 −0.534661 −0.267331 0.963605i \(-0.586142\pi\)
−0.267331 + 0.963605i \(0.586142\pi\)
\(368\) 23.3137 1.21531
\(369\) 8.82843 0.459590
\(370\) 5.02944 0.261468
\(371\) 0 0
\(372\) 0 0
\(373\) 8.48528 0.439351 0.219676 0.975573i \(-0.429500\pi\)
0.219676 + 0.975573i \(0.429500\pi\)
\(374\) 8.48528 0.438763
\(375\) −16.7868 −0.866866
\(376\) −4.48528 −0.231311
\(377\) 0.171573 0.00883645
\(378\) 0 0
\(379\) 23.7574 1.22033 0.610167 0.792273i \(-0.291102\pi\)
0.610167 + 0.792273i \(0.291102\pi\)
\(380\) 0 0
\(381\) −2.82843 −0.144905
\(382\) −21.4558 −1.09778
\(383\) 20.4853 1.04675 0.523374 0.852103i \(-0.324673\pi\)
0.523374 + 0.852103i \(0.324673\pi\)
\(384\) 16.0000 0.816497
\(385\) 0 0
\(386\) 20.4853 1.04267
\(387\) 5.00000 0.254164
\(388\) 0 0
\(389\) 29.6569 1.50366 0.751831 0.659356i \(-0.229171\pi\)
0.751831 + 0.659356i \(0.229171\pi\)
\(390\) −3.17157 −0.160599
\(391\) 8.24264 0.416848
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) −17.4558 −0.879413
\(395\) −24.5563 −1.23556
\(396\) 0 0
\(397\) 18.2132 0.914094 0.457047 0.889442i \(-0.348907\pi\)
0.457047 + 0.889442i \(0.348907\pi\)
\(398\) −5.31371 −0.266352
\(399\) 0 0
\(400\) 9.94113 0.497056
\(401\) 6.34315 0.316762 0.158381 0.987378i \(-0.449373\pi\)
0.158381 + 0.987378i \(0.449373\pi\)
\(402\) −28.9706 −1.44492
\(403\) −3.24264 −0.161527
\(404\) 0 0
\(405\) 7.92893 0.393992
\(406\) 0 0
\(407\) 9.51472 0.471627
\(408\) 5.65685 0.280056
\(409\) −3.24264 −0.160338 −0.0801691 0.996781i \(-0.525546\pi\)
−0.0801691 + 0.996781i \(0.525546\pi\)
\(410\) −19.7990 −0.977802
\(411\) 6.48528 0.319895
\(412\) 0 0
\(413\) 0 0
\(414\) −8.24264 −0.405104
\(415\) 21.0000 1.03085
\(416\) 0 0
\(417\) 8.82843 0.432330
\(418\) −43.4558 −2.12549
\(419\) 20.8701 1.01957 0.509785 0.860302i \(-0.329725\pi\)
0.509785 + 0.860302i \(0.329725\pi\)
\(420\) 0 0
\(421\) 6.72792 0.327899 0.163949 0.986469i \(-0.447577\pi\)
0.163949 + 0.986469i \(0.447577\pi\)
\(422\) 22.5858 1.09946
\(423\) 1.58579 0.0771036
\(424\) −0.485281 −0.0235673
\(425\) 3.51472 0.170489
\(426\) −26.1421 −1.26659
\(427\) 0 0
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) −11.2132 −0.540749
\(431\) −12.3431 −0.594548 −0.297274 0.954792i \(-0.596078\pi\)
−0.297274 + 0.954792i \(0.596078\pi\)
\(432\) −22.6274 −1.08866
\(433\) 24.9706 1.20001 0.600004 0.799997i \(-0.295166\pi\)
0.600004 + 0.799997i \(0.295166\pi\)
\(434\) 0 0
\(435\) 0.384776 0.0184486
\(436\) 0 0
\(437\) −42.2132 −2.01933
\(438\) 18.4853 0.883261
\(439\) 34.4853 1.64589 0.822946 0.568119i \(-0.192329\pi\)
0.822946 + 0.568119i \(0.192329\pi\)
\(440\) −19.0294 −0.907193
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) −3.68629 −0.175141 −0.0875705 0.996158i \(-0.527910\pi\)
−0.0875705 + 0.996158i \(0.527910\pi\)
\(444\) 0 0
\(445\) 2.51472 0.119209
\(446\) 1.07107 0.0507165
\(447\) −22.9706 −1.08647
\(448\) 0 0
\(449\) 21.1716 0.999148 0.499574 0.866271i \(-0.333490\pi\)
0.499574 + 0.866271i \(0.333490\pi\)
\(450\) −3.51472 −0.165685
\(451\) −37.4558 −1.76373
\(452\) 0 0
\(453\) 13.7990 0.648333
\(454\) 37.9411 1.78067
\(455\) 0 0
\(456\) −28.9706 −1.35667
\(457\) −7.21320 −0.337419 −0.168710 0.985666i \(-0.553960\pi\)
−0.168710 + 0.985666i \(0.553960\pi\)
\(458\) 41.6569 1.94650
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) −8.82843 −0.411181 −0.205590 0.978638i \(-0.565911\pi\)
−0.205590 + 0.978638i \(0.565911\pi\)
\(462\) 0 0
\(463\) −4.24264 −0.197172 −0.0985861 0.995129i \(-0.531432\pi\)
−0.0985861 + 0.995129i \(0.531432\pi\)
\(464\) −0.686292 −0.0318603
\(465\) −7.27208 −0.337235
\(466\) 20.7279 0.960202
\(467\) 3.89949 0.180447 0.0902236 0.995922i \(-0.471242\pi\)
0.0902236 + 0.995922i \(0.471242\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −3.55635 −0.164042
\(471\) −5.31371 −0.244843
\(472\) −0.970563 −0.0446738
\(473\) −21.2132 −0.975384
\(474\) 30.9706 1.42253
\(475\) −18.0000 −0.825897
\(476\) 0 0
\(477\) 0.171573 0.00785578
\(478\) −4.97056 −0.227348
\(479\) 6.21320 0.283889 0.141944 0.989875i \(-0.454665\pi\)
0.141944 + 0.989875i \(0.454665\pi\)
\(480\) 0 0
\(481\) 2.24264 0.102256
\(482\) −28.5858 −1.30205
\(483\) 0 0
\(484\) 0 0
\(485\) 18.5980 0.844491
\(486\) 14.0000 0.635053
\(487\) −11.4558 −0.519114 −0.259557 0.965728i \(-0.583577\pi\)
−0.259557 + 0.965728i \(0.583577\pi\)
\(488\) −16.9706 −0.768221
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 16.6274 0.750385 0.375192 0.926947i \(-0.377577\pi\)
0.375192 + 0.926947i \(0.377577\pi\)
\(492\) 0 0
\(493\) −0.242641 −0.0109280
\(494\) −10.2426 −0.460838
\(495\) 6.72792 0.302398
\(496\) 12.9706 0.582395
\(497\) 0 0
\(498\) −26.4853 −1.18683
\(499\) −13.2721 −0.594140 −0.297070 0.954856i \(-0.596009\pi\)
−0.297070 + 0.954856i \(0.596009\pi\)
\(500\) 0 0
\(501\) 21.7574 0.972047
\(502\) 23.4558 1.04689
\(503\) 28.6274 1.27643 0.638217 0.769857i \(-0.279672\pi\)
0.638217 + 0.769857i \(0.279672\pi\)
\(504\) 0 0
\(505\) 16.2426 0.722788
\(506\) 34.9706 1.55463
\(507\) −1.41421 −0.0628074
\(508\) 0 0
\(509\) 5.10051 0.226076 0.113038 0.993591i \(-0.463942\pi\)
0.113038 + 0.993591i \(0.463942\pi\)
\(510\) 4.48528 0.198612
\(511\) 0 0
\(512\) 22.6274 1.00000
\(513\) 40.9706 1.80889
\(514\) −27.4558 −1.21103
\(515\) −12.6863 −0.559025
\(516\) 0 0
\(517\) −6.72792 −0.295894
\(518\) 0 0
\(519\) 34.9706 1.53504
\(520\) −4.48528 −0.196693
\(521\) 6.34315 0.277898 0.138949 0.990300i \(-0.455628\pi\)
0.138949 + 0.990300i \(0.455628\pi\)
\(522\) 0.242641 0.0106201
\(523\) −30.9706 −1.35425 −0.677124 0.735869i \(-0.736774\pi\)
−0.677124 + 0.735869i \(0.736774\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −2.78680 −0.121510
\(527\) 4.58579 0.199760
\(528\) 24.0000 1.04447
\(529\) 10.9706 0.476981
\(530\) −0.384776 −0.0167136
\(531\) 0.343146 0.0148913
\(532\) 0 0
\(533\) −8.82843 −0.382402
\(534\) −3.17157 −0.137247
\(535\) 31.9411 1.38094
\(536\) −40.9706 −1.76966
\(537\) 12.7279 0.549250
\(538\) 12.9706 0.559201
\(539\) 0 0
\(540\) 0 0
\(541\) −7.21320 −0.310120 −0.155060 0.987905i \(-0.549557\pi\)
−0.155060 + 0.987905i \(0.549557\pi\)
\(542\) 28.2843 1.21491
\(543\) −26.4853 −1.13659
\(544\) 0 0
\(545\) −26.5269 −1.13629
\(546\) 0 0
\(547\) 35.4853 1.51724 0.758621 0.651533i \(-0.225874\pi\)
0.758621 + 0.651533i \(0.225874\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 14.9117 0.635837
\(551\) 1.24264 0.0529383
\(552\) 23.3137 0.992297
\(553\) 0 0
\(554\) 10.5858 0.449747
\(555\) 5.02944 0.213488
\(556\) 0 0
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) −4.58579 −0.194132
\(559\) −5.00000 −0.211477
\(560\) 0 0
\(561\) 8.48528 0.358249
\(562\) −22.0000 −0.928014
\(563\) 6.34315 0.267332 0.133666 0.991026i \(-0.457325\pi\)
0.133666 + 0.991026i \(0.457325\pi\)
\(564\) 0 0
\(565\) −3.66905 −0.154358
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) −36.9706 −1.55125
\(569\) 5.14214 0.215570 0.107785 0.994174i \(-0.465624\pi\)
0.107785 + 0.994174i \(0.465624\pi\)
\(570\) −22.9706 −0.962131
\(571\) 42.4558 1.77672 0.888361 0.459146i \(-0.151844\pi\)
0.888361 + 0.459146i \(0.151844\pi\)
\(572\) 0 0
\(573\) −21.4558 −0.896331
\(574\) 0 0
\(575\) 14.4853 0.604078
\(576\) −8.00000 −0.333333
\(577\) 6.97056 0.290188 0.145094 0.989418i \(-0.453651\pi\)
0.145094 + 0.989418i \(0.453651\pi\)
\(578\) 21.2132 0.882353
\(579\) 20.4853 0.851339
\(580\) 0 0
\(581\) 0 0
\(582\) −23.4558 −0.972276
\(583\) −0.727922 −0.0301475
\(584\) 26.1421 1.08177
\(585\) 1.58579 0.0655642
\(586\) 21.7574 0.898788
\(587\) 6.55635 0.270609 0.135305 0.990804i \(-0.456799\pi\)
0.135305 + 0.990804i \(0.456799\pi\)
\(588\) 0 0
\(589\) −23.4853 −0.967694
\(590\) −0.769553 −0.0316820
\(591\) −17.4558 −0.718037
\(592\) −8.97056 −0.368688
\(593\) 0.556349 0.0228465 0.0114233 0.999935i \(-0.496364\pi\)
0.0114233 + 0.999935i \(0.496364\pi\)
\(594\) −33.9411 −1.39262
\(595\) 0 0
\(596\) 0 0
\(597\) −5.31371 −0.217476
\(598\) 8.24264 0.337067
\(599\) −28.7990 −1.17669 −0.588347 0.808608i \(-0.700221\pi\)
−0.588347 + 0.808608i \(0.700221\pi\)
\(600\) 9.94113 0.405845
\(601\) 38.9706 1.58964 0.794821 0.606844i \(-0.207565\pi\)
0.794821 + 0.606844i \(0.207565\pi\)
\(602\) 0 0
\(603\) 14.4853 0.589886
\(604\) 0 0
\(605\) −11.1005 −0.451300
\(606\) −20.4853 −0.832158
\(607\) 26.7279 1.08485 0.542426 0.840103i \(-0.317506\pi\)
0.542426 + 0.840103i \(0.317506\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −13.4558 −0.544811
\(611\) −1.58579 −0.0641541
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 18.7279 0.755797
\(615\) −19.7990 −0.798372
\(616\) 0 0
\(617\) 12.3431 0.496916 0.248458 0.968643i \(-0.420076\pi\)
0.248458 + 0.968643i \(0.420076\pi\)
\(618\) 16.0000 0.643614
\(619\) −0.970563 −0.0390102 −0.0195051 0.999810i \(-0.506209\pi\)
−0.0195051 + 0.999810i \(0.506209\pi\)
\(620\) 0 0
\(621\) −32.9706 −1.32306
\(622\) −10.4853 −0.420421
\(623\) 0 0
\(624\) 5.65685 0.226455
\(625\) −6.39697 −0.255879
\(626\) 32.8284 1.31209
\(627\) −43.4558 −1.73546
\(628\) 0 0
\(629\) −3.17157 −0.126459
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) 43.7990 1.74223
\(633\) 22.5858 0.897704
\(634\) 16.0000 0.635441
\(635\) −3.17157 −0.125860
\(636\) 0 0
\(637\) 0 0
\(638\) −1.02944 −0.0407558
\(639\) 13.0711 0.517083
\(640\) 17.9411 0.709185
\(641\) 15.3431 0.606018 0.303009 0.952988i \(-0.402009\pi\)
0.303009 + 0.952988i \(0.402009\pi\)
\(642\) −40.2843 −1.58989
\(643\) 12.4853 0.492371 0.246186 0.969223i \(-0.420823\pi\)
0.246186 + 0.969223i \(0.420823\pi\)
\(644\) 0 0
\(645\) −11.2132 −0.441519
\(646\) 14.4853 0.569916
\(647\) −14.5269 −0.571112 −0.285556 0.958362i \(-0.592178\pi\)
−0.285556 + 0.958362i \(0.592178\pi\)
\(648\) −14.1421 −0.555556
\(649\) −1.45584 −0.0571469
\(650\) 3.51472 0.137859
\(651\) 0 0
\(652\) 0 0
\(653\) 17.3137 0.677538 0.338769 0.940870i \(-0.389990\pi\)
0.338769 + 0.940870i \(0.389990\pi\)
\(654\) 33.4558 1.30823
\(655\) −4.48528 −0.175254
\(656\) 35.3137 1.37877
\(657\) −9.24264 −0.360590
\(658\) 0 0
\(659\) −15.3431 −0.597684 −0.298842 0.954303i \(-0.596600\pi\)
−0.298842 + 0.954303i \(0.596600\pi\)
\(660\) 0 0
\(661\) 10.7574 0.418413 0.209206 0.977872i \(-0.432912\pi\)
0.209206 + 0.977872i \(0.432912\pi\)
\(662\) 25.4558 0.989369
\(663\) 2.00000 0.0776736
\(664\) −37.4558 −1.45357
\(665\) 0 0
\(666\) 3.17157 0.122896
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) 1.07107 0.0414099
\(670\) −32.4853 −1.25502
\(671\) −25.4558 −0.982712
\(672\) 0 0
\(673\) −26.9411 −1.03850 −0.519252 0.854621i \(-0.673789\pi\)
−0.519252 + 0.854621i \(0.673789\pi\)
\(674\) 46.6690 1.79762
\(675\) −14.0589 −0.541126
\(676\) 0 0
\(677\) 41.3553 1.58941 0.794707 0.606993i \(-0.207624\pi\)
0.794707 + 0.606993i \(0.207624\pi\)
\(678\) 4.62742 0.177715
\(679\) 0 0
\(680\) 6.34315 0.243249
\(681\) 37.9411 1.45391
\(682\) 19.4558 0.745003
\(683\) −21.1716 −0.810108 −0.405054 0.914293i \(-0.632747\pi\)
−0.405054 + 0.914293i \(0.632747\pi\)
\(684\) 0 0
\(685\) 7.27208 0.277852
\(686\) 0 0
\(687\) 41.6569 1.58931
\(688\) 20.0000 0.762493
\(689\) −0.171573 −0.00653641
\(690\) 18.4853 0.703723
\(691\) −28.6985 −1.09174 −0.545871 0.837869i \(-0.683801\pi\)
−0.545871 + 0.837869i \(0.683801\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 8.00000 0.303676
\(695\) 9.89949 0.375509
\(696\) −0.686292 −0.0260138
\(697\) 12.4853 0.472914
\(698\) −36.3848 −1.37718
\(699\) 20.7279 0.784002
\(700\) 0 0
\(701\) −10.7990 −0.407872 −0.203936 0.978984i \(-0.565373\pi\)
−0.203936 + 0.978984i \(0.565373\pi\)
\(702\) −8.00000 −0.301941
\(703\) 16.2426 0.612603
\(704\) 33.9411 1.27920
\(705\) −3.55635 −0.133940
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) 0 0
\(709\) 0.727922 0.0273377 0.0136688 0.999907i \(-0.495649\pi\)
0.0136688 + 0.999907i \(0.495649\pi\)
\(710\) −29.3137 −1.10012
\(711\) −15.4853 −0.580743
\(712\) −4.48528 −0.168093
\(713\) 18.8995 0.707792
\(714\) 0 0
\(715\) −6.72792 −0.251610
\(716\) 0 0
\(717\) −4.97056 −0.185629
\(718\) 39.4558 1.47248
\(719\) 1.75736 0.0655384 0.0327692 0.999463i \(-0.489567\pi\)
0.0327692 + 0.999463i \(0.489567\pi\)
\(720\) −6.34315 −0.236395
\(721\) 0 0
\(722\) −47.3137 −1.76083
\(723\) −28.5858 −1.06312
\(724\) 0 0
\(725\) −0.426407 −0.0158364
\(726\) 14.0000 0.519589
\(727\) −42.0000 −1.55769 −0.778847 0.627214i \(-0.784195\pi\)
−0.778847 + 0.627214i \(0.784195\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 20.7279 0.767175
\(731\) 7.07107 0.261533
\(732\) 0 0
\(733\) 16.6985 0.616773 0.308386 0.951261i \(-0.400211\pi\)
0.308386 + 0.951261i \(0.400211\pi\)
\(734\) 14.4853 0.534661
\(735\) 0 0
\(736\) 0 0
\(737\) −61.4558 −2.26376
\(738\) −12.4853 −0.459590
\(739\) −17.6985 −0.651049 −0.325525 0.945534i \(-0.605541\pi\)
−0.325525 + 0.945534i \(0.605541\pi\)
\(740\) 0 0
\(741\) −10.2426 −0.376273
\(742\) 0 0
\(743\) −29.6569 −1.08800 −0.544002 0.839084i \(-0.683092\pi\)
−0.544002 + 0.839084i \(0.683092\pi\)
\(744\) 12.9706 0.475524
\(745\) −25.7574 −0.943677
\(746\) −12.0000 −0.439351
\(747\) 13.2426 0.484523
\(748\) 0 0
\(749\) 0 0
\(750\) 23.7401 0.866866
\(751\) 15.4853 0.565066 0.282533 0.959258i \(-0.408825\pi\)
0.282533 + 0.959258i \(0.408825\pi\)
\(752\) 6.34315 0.231311
\(753\) 23.4558 0.854778
\(754\) −0.242641 −0.00883645
\(755\) 15.4731 0.563123
\(756\) 0 0
\(757\) 21.4853 0.780896 0.390448 0.920625i \(-0.372320\pi\)
0.390448 + 0.920625i \(0.372320\pi\)
\(758\) −33.5980 −1.22033
\(759\) 34.9706 1.26935
\(760\) −32.4853 −1.17837
\(761\) −34.7574 −1.25995 −0.629977 0.776614i \(-0.716936\pi\)
−0.629977 + 0.776614i \(0.716936\pi\)
\(762\) 4.00000 0.144905
\(763\) 0 0
\(764\) 0 0
\(765\) −2.24264 −0.0810828
\(766\) −28.9706 −1.04675
\(767\) −0.343146 −0.0123903
\(768\) 0 0
\(769\) −52.2132 −1.88286 −0.941428 0.337214i \(-0.890516\pi\)
−0.941428 + 0.337214i \(0.890516\pi\)
\(770\) 0 0
\(771\) −27.4558 −0.988798
\(772\) 0 0
\(773\) 32.8284 1.18076 0.590378 0.807127i \(-0.298979\pi\)
0.590378 + 0.807127i \(0.298979\pi\)
\(774\) −7.07107 −0.254164
\(775\) 8.05887 0.289483
\(776\) −33.1716 −1.19079
\(777\) 0 0
\(778\) −41.9411 −1.50366
\(779\) −63.9411 −2.29093
\(780\) 0 0
\(781\) −55.4558 −1.98437
\(782\) −11.6569 −0.416848
\(783\) 0.970563 0.0346851
\(784\) 0 0
\(785\) −5.95837 −0.212663
\(786\) 5.65685 0.201773
\(787\) 24.7574 0.882505 0.441252 0.897383i \(-0.354534\pi\)
0.441252 + 0.897383i \(0.354534\pi\)
\(788\) 0 0
\(789\) −2.78680 −0.0992126
\(790\) 34.7279 1.23556
\(791\) 0 0
\(792\) −12.0000 −0.426401
\(793\) −6.00000 −0.213066
\(794\) −25.7574 −0.914094
\(795\) −0.384776 −0.0136466
\(796\) 0 0
\(797\) 24.3431 0.862278 0.431139 0.902285i \(-0.358112\pi\)
0.431139 + 0.902285i \(0.358112\pi\)
\(798\) 0 0
\(799\) 2.24264 0.0793389
\(800\) 0 0
\(801\) 1.58579 0.0560310
\(802\) −8.97056 −0.316762
\(803\) 39.2132 1.38380
\(804\) 0 0
\(805\) 0 0
\(806\) 4.58579 0.161527
\(807\) 12.9706 0.456585
\(808\) −28.9706 −1.01918
\(809\) 17.4853 0.614750 0.307375 0.951589i \(-0.400549\pi\)
0.307375 + 0.951589i \(0.400549\pi\)
\(810\) −11.2132 −0.393992
\(811\) 21.9411 0.770457 0.385229 0.922821i \(-0.374123\pi\)
0.385229 + 0.922821i \(0.374123\pi\)
\(812\) 0 0
\(813\) 28.2843 0.991973
\(814\) −13.4558 −0.471627
\(815\) −13.4558 −0.471338
\(816\) −8.00000 −0.280056
\(817\) −36.2132 −1.26694
\(818\) 4.58579 0.160338
\(819\) 0 0
\(820\) 0 0
\(821\) −20.1421 −0.702965 −0.351483 0.936194i \(-0.614322\pi\)
−0.351483 + 0.936194i \(0.614322\pi\)
\(822\) −9.17157 −0.319895
\(823\) 10.4853 0.365494 0.182747 0.983160i \(-0.441501\pi\)
0.182747 + 0.983160i \(0.441501\pi\)
\(824\) 22.6274 0.788263
\(825\) 14.9117 0.519158
\(826\) 0 0
\(827\) −49.4558 −1.71975 −0.859874 0.510506i \(-0.829458\pi\)
−0.859874 + 0.510506i \(0.829458\pi\)
\(828\) 0 0
\(829\) −50.7279 −1.76185 −0.880927 0.473253i \(-0.843080\pi\)
−0.880927 + 0.473253i \(0.843080\pi\)
\(830\) −29.6985 −1.03085
\(831\) 10.5858 0.367217
\(832\) 8.00000 0.277350
\(833\) 0 0
\(834\) −12.4853 −0.432330
\(835\) 24.3970 0.844292
\(836\) 0 0
\(837\) −18.3431 −0.634032
\(838\) −29.5147 −1.01957
\(839\) 33.1716 1.14521 0.572605 0.819831i \(-0.305933\pi\)
0.572605 + 0.819831i \(0.305933\pi\)
\(840\) 0 0
\(841\) −28.9706 −0.998985
\(842\) −9.51472 −0.327899
\(843\) −22.0000 −0.757720
\(844\) 0 0
\(845\) −1.58579 −0.0545527
\(846\) −2.24264 −0.0771036
\(847\) 0 0
\(848\) 0.686292 0.0235673
\(849\) −12.0000 −0.411839
\(850\) −4.97056 −0.170489
\(851\) −13.0711 −0.448070
\(852\) 0 0
\(853\) −39.7279 −1.36026 −0.680129 0.733092i \(-0.738076\pi\)
−0.680129 + 0.733092i \(0.738076\pi\)
\(854\) 0 0
\(855\) 11.4853 0.392788
\(856\) −56.9706 −1.94721
\(857\) 54.7696 1.87089 0.935446 0.353469i \(-0.114998\pi\)
0.935446 + 0.353469i \(0.114998\pi\)
\(858\) 8.48528 0.289683
\(859\) −30.9706 −1.05670 −0.528351 0.849026i \(-0.677189\pi\)
−0.528351 + 0.849026i \(0.677189\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 17.4558 0.594548
\(863\) 28.2843 0.962808 0.481404 0.876499i \(-0.340127\pi\)
0.481404 + 0.876499i \(0.340127\pi\)
\(864\) 0 0
\(865\) 39.2132 1.33329
\(866\) −35.3137 −1.20001
\(867\) 21.2132 0.720438
\(868\) 0 0
\(869\) 65.6985 2.22867
\(870\) −0.544156 −0.0184486
\(871\) −14.4853 −0.490815
\(872\) 47.3137 1.60224
\(873\) 11.7279 0.396930
\(874\) 59.6985 2.01933
\(875\) 0 0
\(876\) 0 0
\(877\) −10.2426 −0.345869 −0.172935 0.984933i \(-0.555325\pi\)
−0.172935 + 0.984933i \(0.555325\pi\)
\(878\) −48.7696 −1.64589
\(879\) 21.7574 0.733858
\(880\) 26.9117 0.907193
\(881\) −13.1127 −0.441778 −0.220889 0.975299i \(-0.570896\pi\)
−0.220889 + 0.975299i \(0.570896\pi\)
\(882\) 0 0
\(883\) 2.00000 0.0673054 0.0336527 0.999434i \(-0.489286\pi\)
0.0336527 + 0.999434i \(0.489286\pi\)
\(884\) 0 0
\(885\) −0.769553 −0.0258682
\(886\) 5.21320 0.175141
\(887\) −19.1127 −0.641742 −0.320871 0.947123i \(-0.603976\pi\)
−0.320871 + 0.947123i \(0.603976\pi\)
\(888\) −8.97056 −0.301032
\(889\) 0 0
\(890\) −3.55635 −0.119209
\(891\) −21.2132 −0.710669
\(892\) 0 0
\(893\) −11.4853 −0.384340
\(894\) 32.4853 1.08647
\(895\) 14.2721 0.477063
\(896\) 0 0
\(897\) 8.24264 0.275214
\(898\) −29.9411 −0.999148
\(899\) −0.556349 −0.0185553
\(900\) 0 0
\(901\) 0.242641 0.00808353
\(902\) 52.9706 1.76373
\(903\) 0 0
\(904\) 6.54416 0.217655
\(905\) −29.6985 −0.987211
\(906\) −19.5147 −0.648333
\(907\) 1.97056 0.0654315 0.0327157 0.999465i \(-0.489584\pi\)
0.0327157 + 0.999465i \(0.489584\pi\)
\(908\) 0 0
\(909\) 10.2426 0.339727
\(910\) 0 0
\(911\) −43.9706 −1.45681 −0.728405 0.685147i \(-0.759738\pi\)
−0.728405 + 0.685147i \(0.759738\pi\)
\(912\) 40.9706 1.35667
\(913\) −56.1838 −1.85941
\(914\) 10.2010 0.337419
\(915\) −13.4558 −0.444836
\(916\) 0 0
\(917\) 0 0
\(918\) 11.3137 0.373408
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 26.1421 0.861881
\(921\) 18.7279 0.617106
\(922\) 12.4853 0.411181
\(923\) −13.0711 −0.430239
\(924\) 0 0
\(925\) −5.57359 −0.183259
\(926\) 6.00000 0.197172
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −36.8995 −1.21063 −0.605317 0.795985i \(-0.706953\pi\)
−0.605317 + 0.795985i \(0.706953\pi\)
\(930\) 10.2843 0.337235
\(931\) 0 0
\(932\) 0 0
\(933\) −10.4853 −0.343273
\(934\) −5.51472 −0.180447
\(935\) 9.51472 0.311165
\(936\) −2.82843 −0.0924500
\(937\) −45.2132 −1.47705 −0.738525 0.674226i \(-0.764478\pi\)
−0.738525 + 0.674226i \(0.764478\pi\)
\(938\) 0 0
\(939\) 32.8284 1.07132
\(940\) 0 0
\(941\) −40.0711 −1.30628 −0.653140 0.757237i \(-0.726549\pi\)
−0.653140 + 0.757237i \(0.726549\pi\)
\(942\) 7.51472 0.244843
\(943\) 51.4558 1.67563
\(944\) 1.37258 0.0446738
\(945\) 0 0
\(946\) 30.0000 0.975384
\(947\) 15.1716 0.493010 0.246505 0.969142i \(-0.420718\pi\)
0.246505 + 0.969142i \(0.420718\pi\)
\(948\) 0 0
\(949\) 9.24264 0.300029
\(950\) 25.4558 0.825897
\(951\) 16.0000 0.518836
\(952\) 0 0
\(953\) 11.1421 0.360929 0.180465 0.983581i \(-0.442240\pi\)
0.180465 + 0.983581i \(0.442240\pi\)
\(954\) −0.242641 −0.00785578
\(955\) −24.0589 −0.778527
\(956\) 0 0
\(957\) −1.02944 −0.0332770
\(958\) −8.78680 −0.283889
\(959\) 0 0
\(960\) 17.9411 0.579047
\(961\) −20.4853 −0.660816
\(962\) −3.17157 −0.102256
\(963\) 20.1421 0.649071
\(964\) 0 0
\(965\) 22.9706 0.739449
\(966\) 0 0
\(967\) 17.6985 0.569145 0.284572 0.958655i \(-0.408148\pi\)
0.284572 + 0.958655i \(0.408148\pi\)
\(968\) 19.7990 0.636364
\(969\) 14.4853 0.465334
\(970\) −26.3015 −0.844491
\(971\) −43.4558 −1.39456 −0.697282 0.716797i \(-0.745608\pi\)
−0.697282 + 0.716797i \(0.745608\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 16.2010 0.519114
\(975\) 3.51472 0.112561
\(976\) 24.0000 0.768221
\(977\) 24.0416 0.769160 0.384580 0.923092i \(-0.374346\pi\)
0.384580 + 0.923092i \(0.374346\pi\)
\(978\) 16.9706 0.542659
\(979\) −6.72792 −0.215025
\(980\) 0 0
\(981\) −16.7279 −0.534081
\(982\) −23.5147 −0.750385
\(983\) 15.0416 0.479754 0.239877 0.970803i \(-0.422893\pi\)
0.239877 + 0.970803i \(0.422893\pi\)
\(984\) 35.3137 1.12576
\(985\) −19.5736 −0.623667
\(986\) 0.343146 0.0109280
\(987\) 0 0
\(988\) 0 0
\(989\) 29.1421 0.926666
\(990\) −9.51472 −0.302398
\(991\) 1.02944 0.0327012 0.0163506 0.999866i \(-0.494795\pi\)
0.0163506 + 0.999866i \(0.494795\pi\)
\(992\) 0 0
\(993\) 25.4558 0.807817
\(994\) 0 0
\(995\) −5.95837 −0.188893
\(996\) 0 0
\(997\) 11.5147 0.364675 0.182337 0.983236i \(-0.441634\pi\)
0.182337 + 0.983236i \(0.441634\pi\)
\(998\) 18.7696 0.594140
\(999\) 12.6863 0.401377
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 637.2.a.g.1.1 2
3.2 odd 2 5733.2.a.s.1.2 2
7.2 even 3 637.2.e.g.508.2 4
7.3 odd 6 637.2.e.f.79.2 4
7.4 even 3 637.2.e.g.79.2 4
7.5 odd 6 637.2.e.f.508.2 4
7.6 odd 2 91.2.a.c.1.1 2
13.12 even 2 8281.2.a.v.1.2 2
21.20 even 2 819.2.a.h.1.2 2
28.27 even 2 1456.2.a.q.1.1 2
35.34 odd 2 2275.2.a.j.1.2 2
56.13 odd 2 5824.2.a.bl.1.1 2
56.27 even 2 5824.2.a.bk.1.2 2
91.34 even 4 1183.2.c.d.337.2 4
91.83 even 4 1183.2.c.d.337.4 4
91.90 odd 2 1183.2.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.c.1.1 2 7.6 odd 2
637.2.a.g.1.1 2 1.1 even 1 trivial
637.2.e.f.79.2 4 7.3 odd 6
637.2.e.f.508.2 4 7.5 odd 6
637.2.e.g.79.2 4 7.4 even 3
637.2.e.g.508.2 4 7.2 even 3
819.2.a.h.1.2 2 21.20 even 2
1183.2.a.d.1.2 2 91.90 odd 2
1183.2.c.d.337.2 4 91.34 even 4
1183.2.c.d.337.4 4 91.83 even 4
1456.2.a.q.1.1 2 28.27 even 2
2275.2.a.j.1.2 2 35.34 odd 2
5733.2.a.s.1.2 2 3.2 odd 2
5824.2.a.bk.1.2 2 56.27 even 2
5824.2.a.bl.1.1 2 56.13 odd 2
8281.2.a.v.1.2 2 13.12 even 2