Properties

Label 91.2.a.c.1.1
Level $91$
Weight $2$
Character 91.1
Self dual yes
Analytic conductor $0.727$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{2} +1.41421 q^{3} +1.58579 q^{5} -2.00000 q^{6} +1.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{2} +1.41421 q^{3} +1.58579 q^{5} -2.00000 q^{6} +1.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} -2.24264 q^{10} +4.24264 q^{11} -1.00000 q^{13} -1.41421 q^{14} +2.24264 q^{15} -4.00000 q^{16} +1.41421 q^{17} +1.41421 q^{18} -7.24264 q^{19} +1.41421 q^{21} -6.00000 q^{22} -5.82843 q^{23} +4.00000 q^{24} -2.48528 q^{25} +1.41421 q^{26} -5.65685 q^{27} +0.171573 q^{29} -3.17157 q^{30} +3.24264 q^{31} +6.00000 q^{33} -2.00000 q^{34} +1.58579 q^{35} +2.24264 q^{37} +10.2426 q^{38} -1.41421 q^{39} +4.48528 q^{40} +8.82843 q^{41} -2.00000 q^{42} -5.00000 q^{43} -1.58579 q^{45} +8.24264 q^{46} +1.58579 q^{47} -5.65685 q^{48} +1.00000 q^{49} +3.51472 q^{50} +2.00000 q^{51} -0.171573 q^{53} +8.00000 q^{54} +6.72792 q^{55} +2.82843 q^{56} -10.2426 q^{57} -0.242641 q^{58} +0.343146 q^{59} +6.00000 q^{61} -4.58579 q^{62} -1.00000 q^{63} +8.00000 q^{64} -1.58579 q^{65} -8.48528 q^{66} -14.4853 q^{67} -8.24264 q^{69} -2.24264 q^{70} -13.0711 q^{71} -2.82843 q^{72} -9.24264 q^{73} -3.17157 q^{74} -3.51472 q^{75} +4.24264 q^{77} +2.00000 q^{78} +15.4853 q^{79} -6.34315 q^{80} -5.00000 q^{81} -12.4853 q^{82} +13.2426 q^{83} +2.24264 q^{85} +7.07107 q^{86} +0.242641 q^{87} +12.0000 q^{88} +1.58579 q^{89} +2.24264 q^{90} -1.00000 q^{91} +4.58579 q^{93} -2.24264 q^{94} -11.4853 q^{95} +11.7279 q^{97} -1.41421 q^{98} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} - 4 q^{6} + 2 q^{7} - 2 q^{9} + O(q^{10}) \) \( 2 q + 6 q^{5} - 4 q^{6} + 2 q^{7} - 2 q^{9} + 4 q^{10} - 2 q^{13} - 4 q^{15} - 8 q^{16} - 6 q^{19} - 12 q^{22} - 6 q^{23} + 8 q^{24} + 12 q^{25} + 6 q^{29} - 12 q^{30} - 2 q^{31} + 12 q^{33} - 4 q^{34} + 6 q^{35} - 4 q^{37} + 12 q^{38} - 8 q^{40} + 12 q^{41} - 4 q^{42} - 10 q^{43} - 6 q^{45} + 8 q^{46} + 6 q^{47} + 2 q^{49} + 24 q^{50} + 4 q^{51} - 6 q^{53} + 16 q^{54} - 12 q^{55} - 12 q^{57} + 8 q^{58} + 12 q^{59} + 12 q^{61} - 12 q^{62} - 2 q^{63} + 16 q^{64} - 6 q^{65} - 12 q^{67} - 8 q^{69} + 4 q^{70} - 12 q^{71} - 10 q^{73} - 12 q^{74} - 24 q^{75} + 4 q^{78} + 14 q^{79} - 24 q^{80} - 10 q^{81} - 8 q^{82} + 18 q^{83} - 4 q^{85} - 8 q^{87} + 24 q^{88} + 6 q^{89} - 4 q^{90} - 2 q^{91} + 12 q^{93} + 4 q^{94} - 6 q^{95} - 2 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −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 \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) 1.58579 0.709185 0.354593 0.935021i \(-0.384620\pi\)
0.354593 + 0.935021i \(0.384620\pi\)
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964
\(8\) 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
\(14\) −1.41421 −0.377964
\(15\) 2.24264 0.579047
\(16\) −4.00000 −1.00000
\(17\) 1.41421 0.342997 0.171499 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) 1.41421 0.333333
\(19\) −7.24264 −1.66158 −0.830788 0.556589i \(-0.812110\pi\)
−0.830788 + 0.556589i \(0.812110\pi\)
\(20\) 0 0
\(21\) 1.41421 0.308607
\(22\) −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 \(-0.405946\pi\)
0.291198 + 0.956663i \(0.405946\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) −2.00000 −0.342997
\(35\) 1.58579 0.268047
\(36\) 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 \(-0.257881\pi\)
0.689384 + 0.724396i \(0.257881\pi\)
\(42\) −2.00000 −0.308607
\(43\) −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 \(-0.463103\pi\)
0.115655 + 0.993289i \(0.463103\pi\)
\(48\) −5.65685 −0.816497
\(49\) 1.00000 0.142857
\(50\) 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
\(56\) 2.82843 0.377964
\(57\) −10.2426 −1.35667
\(58\) −0.242641 −0.0318603
\(59\) 0.343146 0.0446738 0.0223369 0.999751i \(-0.492889\pi\)
0.0223369 + 0.999751i \(0.492889\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −4.58579 −0.582395
\(63\) −1.00000 −0.125988
\(64\) 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
\(70\) −2.24264 −0.268047
\(71\) −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 \(-0.681910\pi\)
−0.540885 + 0.841097i \(0.681910\pi\)
\(74\) −3.17157 −0.368688
\(75\) −3.51472 −0.405845
\(76\) 0 0
\(77\) 4.24264 0.483494
\(78\) 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 \(-0.241014\pi\)
0.726784 + 0.686866i \(0.241014\pi\)
\(84\) 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 \(-0.473216\pi\)
0.0840465 + 0.996462i \(0.473216\pi\)
\(90\) 2.24264 0.236395
\(91\) −1.00000 −0.104828
\(92\) 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 \(-0.296996\pi\)
0.595395 + 0.803433i \(0.296996\pi\)
\(98\) −1.41421 −0.142857
\(99\) −4.24264 −0.426401
\(100\) 0 0
\(101\) 10.2426 1.01918 0.509590 0.860417i \(-0.329797\pi\)
0.509590 + 0.860417i \(0.329797\pi\)
\(102\) −2.82843 −0.280056
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −2.82843 −0.277350
\(105\) 2.24264 0.218859
\(106\) 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
\(112\) −4.00000 −0.377964
\(113\) 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
\(119\) 1.41421 0.129641
\(120\) 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
\(126\) 1.41421 0.125988
\(127\) 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 \(-0.539431\pi\)
−0.123560 + 0.992337i \(0.539431\pi\)
\(132\) 0 0
\(133\) −7.24264 −0.628017
\(134\) 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 \(-0.414712\pi\)
0.264747 + 0.964318i \(0.414712\pi\)
\(140\) 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
\(147\) 1.41421 0.116642
\(148\) 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
\(154\) −6.00000 −0.483494
\(155\) 5.14214 0.413026
\(156\) 0 0
\(157\) −3.75736 −0.299870 −0.149935 0.988696i \(-0.547906\pi\)
−0.149935 + 0.988696i \(0.547906\pi\)
\(158\) −21.8995 −1.74223
\(159\) −0.242641 −0.0192427
\(160\) 0 0
\(161\) −5.82843 −0.459344
\(162\) 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 \(-0.297051\pi\)
0.595255 + 0.803537i \(0.297051\pi\)
\(168\) 4.00000 0.308607
\(169\) 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 \(-0.110811\pi\)
0.940015 + 0.341134i \(0.110811\pi\)
\(174\) −0.343146 −0.0260138
\(175\) −2.48528 −0.187870
\(176\) −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 \(-0.745047\pi\)
−0.696018 + 0.718025i \(0.745047\pi\)
\(182\) 1.41421 0.104828
\(183\) 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
\(189\) −5.65685 −0.411476
\(190\) 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 \(-0.542518\pi\)
−0.133176 + 0.991092i \(0.542518\pi\)
\(200\) −7.02944 −0.497056
\(201\) −20.4853 −1.44492
\(202\) −14.4853 −1.01918
\(203\) 0.171573 0.0120421
\(204\) 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
\(210\) −3.17157 −0.218859
\(211\) −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
\(217\) 3.24264 0.220125
\(218\) −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 \(-0.491927\pi\)
0.0253583 + 0.999678i \(0.491927\pi\)
\(224\) 0 0
\(225\) 2.48528 0.165685
\(226\) −3.27208 −0.217655
\(227\) 26.8284 1.78067 0.890333 0.455311i \(-0.150472\pi\)
0.890333 + 0.455311i \(0.150472\pi\)
\(228\) 0 0
\(229\) 29.4558 1.94650 0.973248 0.229755i \(-0.0737925\pi\)
0.973248 + 0.229755i \(0.0737925\pi\)
\(230\) 13.0711 0.861881
\(231\) 6.00000 0.394771
\(232\) 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
\(238\) −2.00000 −0.129641
\(239\) 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 \(-0.725660\pi\)
−0.651023 + 0.759058i \(0.725660\pi\)
\(242\) −9.89949 −0.636364
\(243\) 9.89949 0.635053
\(244\) 0 0
\(245\) 1.58579 0.101312
\(246\) −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 \(-0.324647\pi\)
0.523443 + 0.852061i \(0.324647\pi\)
\(252\) 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 \(-0.707032\pi\)
−0.605513 + 0.795836i \(0.707032\pi\)
\(258\) 10.0000 0.622573
\(259\) 2.24264 0.139351
\(260\) 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
\(266\) 10.2426 0.628017
\(267\) 2.24264 0.137247
\(268\) 0 0
\(269\) 9.17157 0.559201 0.279600 0.960116i \(-0.409798\pi\)
0.279600 + 0.960116i \(0.409798\pi\)
\(270\) 12.6863 0.772063
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −5.65685 −0.342997
\(273\) −1.41421 −0.0855921
\(274\) 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
\(280\) 4.48528 0.268047
\(281\) 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 \(-0.581154\pi\)
−0.252199 + 0.967675i \(0.581154\pi\)
\(284\) 0 0
\(285\) −16.2426 −0.962131
\(286\) 6.00000 0.354787
\(287\) 8.82843 0.521126
\(288\) 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 \(-0.351640\pi\)
0.449394 + 0.893334i \(0.351640\pi\)
\(294\) −2.00000 −0.116642
\(295\) 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
\(301\) −5.00000 −0.288195
\(302\) 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 \(-0.376647\pi\)
0.377899 + 0.925847i \(0.376647\pi\)
\(308\) 0 0
\(309\) −11.3137 −0.643614
\(310\) −7.27208 −0.413026
\(311\) −7.41421 −0.420421 −0.210211 0.977656i \(-0.567415\pi\)
−0.210211 + 0.977656i \(0.567415\pi\)
\(312\) −4.00000 −0.226455
\(313\) 23.2132 1.31209 0.656044 0.754723i \(-0.272229\pi\)
0.656044 + 0.754723i \(0.272229\pi\)
\(314\) 5.31371 0.299870
\(315\) −1.58579 −0.0893489
\(316\) 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
\(322\) 8.24264 0.459344
\(323\) −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
\(329\) 1.58579 0.0874272
\(330\) −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
\(336\) −5.65685 −0.308607
\(337\) −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
\(343\) 1.00000 0.0539949
\(344\) −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 \(-0.741771\pi\)
−0.688592 + 0.725149i \(0.741771\pi\)
\(350\) 3.51472 0.187870
\(351\) 5.65685 0.301941
\(352\) 0 0
\(353\) −8.48528 −0.451626 −0.225813 0.974171i \(-0.572504\pi\)
−0.225813 + 0.974171i \(0.572504\pi\)
\(354\) −0.686292 −0.0364760
\(355\) −20.7279 −1.10012
\(356\) 0 0
\(357\) 2.00000 0.105851
\(358\) 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 \(-0.413858\pi\)
0.267331 + 0.963605i \(0.413858\pi\)
\(368\) 23.3137 1.21531
\(369\) −8.82843 −0.459590
\(370\) −5.02944 −0.261468
\(371\) −0.171573 −0.00890762
\(372\) 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
\(378\) 8.00000 0.411476
\(379\) 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 \(-0.675327\pi\)
−0.523374 + 0.852103i \(0.675327\pi\)
\(384\) −16.0000 −0.816497
\(385\) 6.72792 0.342887
\(386\) 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
\(392\) 2.82843 0.142857
\(393\) −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 \(-0.651093\pi\)
−0.457047 + 0.889442i \(0.651093\pi\)
\(398\) 5.31371 0.266352
\(399\) −10.2426 −0.512773
\(400\) 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
\(406\) −0.242641 −0.0120421
\(407\) 9.51472 0.471627
\(408\) 5.65685 0.280056
\(409\) 3.24264 0.160338 0.0801691 0.996781i \(-0.474454\pi\)
0.0801691 + 0.996781i \(0.474454\pi\)
\(410\) −19.7990 −0.977802
\(411\) −6.48528 −0.319895
\(412\) 0 0
\(413\) 0.343146 0.0168851
\(414\) −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 \(-0.670275\pi\)
−0.509785 + 0.860302i \(0.670275\pi\)
\(420\) 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
\(427\) 6.00000 0.290360
\(428\) 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 \(-0.704834\pi\)
−0.600004 + 0.799997i \(0.704834\pi\)
\(434\) −4.58579 −0.220125
\(435\) 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 \(-0.807671\pi\)
−0.822946 + 0.568119i \(0.807671\pi\)
\(440\) 19.0294 0.907193
\(441\) −1.00000 −0.0476190
\(442\) 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
\(448\) 8.00000 0.377964
\(449\) 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
\(455\) −1.58579 −0.0743428
\(456\) −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 \(-0.434089\pi\)
0.205590 + 0.978638i \(0.434089\pi\)
\(462\) −8.48528 −0.394771
\(463\) −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 \(-0.528758\pi\)
−0.0902236 + 0.995922i \(0.528758\pi\)
\(468\) 0 0
\(469\) −14.4853 −0.668868
\(470\) −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 \(-0.545335\pi\)
−0.141944 + 0.989875i \(0.545335\pi\)
\(480\) 0 0
\(481\) −2.24264 −0.102256
\(482\) 28.5858 1.30205
\(483\) −8.24264 −0.375053
\(484\) 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
\(490\) −2.24264 −0.101312
\(491\) 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
\(497\) −13.0711 −0.586318
\(498\) −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 \(-0.720328\pi\)
−0.638217 + 0.769857i \(0.720328\pi\)
\(504\) −2.82843 −0.125988
\(505\) 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 \(-0.536058\pi\)
−0.113038 + 0.993591i \(0.536058\pi\)
\(510\) −4.48528 −0.198612
\(511\) −9.24264 −0.408870
\(512\) 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
\(518\) −3.17157 −0.139351
\(519\) 34.9706 1.53504
\(520\) −4.48528 −0.196693
\(521\) −6.34315 −0.277898 −0.138949 0.990300i \(-0.544372\pi\)
−0.138949 + 0.990300i \(0.544372\pi\)
\(522\) 0.242641 0.0106201
\(523\) 30.9706 1.35425 0.677124 0.735869i \(-0.263226\pi\)
0.677124 + 0.735869i \(0.263226\pi\)
\(524\) 0 0
\(525\) −3.51472 −0.153395
\(526\) −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
\(539\) 4.24264 0.182743
\(540\) 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
\(546\) 2.00000 0.0855921
\(547\) 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
\(553\) 15.4853 0.658501
\(554\) 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
\(560\) −6.34315 −0.268047
\(561\) 8.48528 0.358249
\(562\) −22.0000 −0.928014
\(563\) −6.34315 −0.267332 −0.133666 0.991026i \(-0.542675\pi\)
−0.133666 + 0.991026i \(0.542675\pi\)
\(564\) 0 0
\(565\) 3.66905 0.154358
\(566\) 12.0000 0.504398
\(567\) −5.00000 −0.209980
\(568\) −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
\(574\) −12.4853 −0.521126
\(575\) 14.4853 0.604078
\(576\) −8.00000 −0.333333
\(577\) −6.97056 −0.290188 −0.145094 0.989418i \(-0.546349\pi\)
−0.145094 + 0.989418i \(0.546349\pi\)
\(578\) 21.2132 0.882353
\(579\) −20.4853 −0.851339
\(580\) 0 0
\(581\) 13.2426 0.549397
\(582\) −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 \(-0.543201\pi\)
−0.135305 + 0.990804i \(0.543201\pi\)
\(588\) 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 \(-0.503636\pi\)
−0.0114233 + 0.999935i \(0.503636\pi\)
\(594\) 33.9411 1.39262
\(595\) 2.24264 0.0919393
\(596\) 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 \(-0.792435\pi\)
−0.794821 + 0.606844i \(0.792435\pi\)
\(602\) 7.07107 0.288195
\(603\) 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 \(-0.682494\pi\)
−0.542426 + 0.840103i \(0.682494\pi\)
\(608\) 0 0
\(609\) 0.242641 0.00983230
\(610\) −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
\(616\) 12.0000 0.483494
\(617\) 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 \(-0.493791\pi\)
0.0195051 + 0.999810i \(0.493791\pi\)
\(620\) 0 0
\(621\) 32.9706 1.32306
\(622\) 10.4853 0.420421
\(623\) 1.58579 0.0635332
\(624\) 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
\(630\) 2.24264 0.0893489
\(631\) 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
\(637\) −1.00000 −0.0396214
\(638\) −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 \(-0.579177\pi\)
−0.246186 + 0.969223i \(0.579177\pi\)
\(644\) 0 0
\(645\) −11.2132 −0.441519
\(646\) 14.4853 0.569916
\(647\) 14.5269 0.571112 0.285556 0.958362i \(-0.407822\pi\)
0.285556 + 0.958362i \(0.407822\pi\)
\(648\) −14.1421 −0.555556
\(649\) 1.45584 0.0571469
\(650\) −3.51472 −0.137859
\(651\) 4.58579 0.179731
\(652\) 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
\(658\) −2.24264 −0.0874272
\(659\) −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 \(-0.567088\pi\)
−0.209206 + 0.977872i \(0.567088\pi\)
\(662\) 25.4558 0.989369
\(663\) −2.00000 −0.0776736
\(664\) 37.4558 1.45357
\(665\) −11.4853 −0.445380
\(666\) 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 \(-0.792376\pi\)
−0.794707 + 0.606993i \(0.792376\pi\)
\(678\) −4.62742 −0.177715
\(679\) 11.7279 0.450076
\(680\) 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
\(686\) −1.41421 −0.0539949
\(687\) 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 \(-0.316199\pi\)
0.545871 + 0.837869i \(0.316199\pi\)
\(692\) 0 0
\(693\) −4.24264 −0.161165
\(694\) 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
\(707\) 10.2426 0.385214
\(708\) 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
\(714\) −2.82843 −0.105851
\(715\) −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 \(-0.510433\pi\)
−0.0327692 + 0.999463i \(0.510433\pi\)
\(720\) 6.34315 0.236395
\(721\) −8.00000 −0.297936
\(722\) −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 \(-0.215805\pi\)
0.778847 + 0.627214i \(0.215805\pi\)
\(728\) −2.82843 −0.104828
\(729\) 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 \(-0.599789\pi\)
−0.308386 + 0.951261i \(0.599789\pi\)
\(734\) −14.4853 −0.534661
\(735\) 2.24264 0.0827210
\(736\) 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
\(742\) 0.242641 0.00890762
\(743\) −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
\(749\) −20.1421 −0.735978
\(750\) 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 \(-0.283064\pi\)
0.629977 + 0.776614i \(0.283064\pi\)
\(762\) −4.00000 −0.144905
\(763\) 16.7279 0.605591
\(764\) 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 \(-0.109484\pi\)
0.941428 + 0.337214i \(0.109484\pi\)
\(770\) −9.51472 −0.342887
\(771\) −27.4558 −0.988798
\(772\) 0 0
\(773\) −32.8284 −1.18076 −0.590378 0.807127i \(-0.701021\pi\)
−0.590378 + 0.807127i \(0.701021\pi\)
\(774\) −7.07107 −0.254164
\(775\) −8.05887 −0.289483
\(776\) 33.1716 1.19079
\(777\) 3.17157 0.113780
\(778\) −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
\(784\) −4.00000 −0.142857
\(785\) −5.95837 −0.212663
\(786\) 5.65685 0.201773
\(787\) −24.7574 −0.882505 −0.441252 0.897383i \(-0.645466\pi\)
−0.441252 + 0.897383i \(0.645466\pi\)
\(788\) 0 0
\(789\) 2.78680 0.0992126
\(790\) −34.7279 −1.23556
\(791\) 2.31371 0.0822660
\(792\) −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 \(-0.641888\pi\)
−0.431139 + 0.902285i \(0.641888\pi\)
\(798\) 14.4853 0.512773
\(799\) 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
\(805\) −9.24264 −0.325760
\(806\) 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 \(-0.625877\pi\)
−0.385229 + 0.922821i \(0.625877\pi\)
\(812\) 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
\(819\) 1.00000 0.0349428
\(820\) 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
\(826\) −0.485281 −0.0168851
\(827\) −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 \(-0.156920\pi\)
0.880927 + 0.473253i \(0.156920\pi\)
\(830\) −29.6985 −1.03085
\(831\) −10.5858 −0.367217
\(832\) −8.00000 −0.277350
\(833\) 1.41421 0.0489996
\(834\) −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 \(-0.694067\pi\)
−0.572605 + 0.819831i \(0.694067\pi\)
\(840\) 6.34315 0.218859
\(841\) −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
\(847\) 7.00000 0.240523
\(848\) 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 \(-0.261924\pi\)
0.680129 + 0.733092i \(0.261924\pi\)
\(854\) −8.48528 −0.290360
\(855\) 11.4853 0.392788
\(856\) −56.9706 −1.94721
\(857\) −54.7696 −1.87089 −0.935446 0.353469i \(-0.885002\pi\)
−0.935446 + 0.353469i \(0.885002\pi\)
\(858\) 8.48528 0.289683
\(859\) 30.9706 1.05670 0.528351 0.849026i \(-0.322811\pi\)
0.528351 + 0.849026i \(0.322811\pi\)
\(860\) 0 0
\(861\) 12.4853 0.425497
\(862\) 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
\(875\) −11.8701 −0.401281
\(876\) 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 \(-0.429104\pi\)
0.220889 + 0.975299i \(0.429104\pi\)
\(882\) 1.41421 0.0476190
\(883\) 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 \(-0.396024\pi\)
0.320871 + 0.947123i \(0.396024\pi\)
\(888\) 8.97056 0.301032
\(889\) 2.00000 0.0670778
\(890\) −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
\(896\) −11.3137 −0.377964
\(897\) 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
\(903\) −7.07107 −0.235310
\(904\) 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
\(910\) 2.24264 0.0743428
\(911\) −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
\(917\) −2.82843 −0.0934029
\(918\) 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 \(-0.293047\pi\)
0.605317 + 0.795985i \(0.293047\pi\)
\(930\) −10.2843 −0.337235
\(931\) −7.24264 −0.237368
\(932\) 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 \(-0.235522\pi\)
0.738525 + 0.674226i \(0.235522\pi\)
\(938\) 20.4853 0.668868
\(939\) 32.8284 1.07132
\(940\) 0 0
\(941\) 40.0711 1.30628 0.653140 0.757237i \(-0.273451\pi\)
0.653140 + 0.757237i \(0.273451\pi\)
\(942\) 7.51472 0.244843
\(943\) −51.4558 −1.67563
\(944\) −1.37258 −0.0446738
\(945\) −8.97056 −0.291812
\(946\) 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
\(952\) 4.00000 0.129641
\(953\) 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
\(959\) −4.58579 −0.148083
\(960\) 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
\(966\) 11.6569 0.375053
\(967\) 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 \(-0.254392\pi\)
0.697282 + 0.716797i \(0.254392\pi\)
\(972\) 0 0
\(973\) 6.24264 0.200130
\(974\) 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 \(-0.577107\pi\)
−0.239877 + 0.970803i \(0.577107\pi\)
\(984\) 35.3137 1.12576
\(985\) 19.5736 0.623667
\(986\) −0.343146 −0.0109280
\(987\) 2.24264 0.0713840
\(988\) 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
\(994\) 18.4853 0.586318
\(995\) −5.95837 −0.188893
\(996\) 0 0
\(997\) −11.5147 −0.364675 −0.182337 0.983236i \(-0.558366\pi\)
−0.182337 + 0.983236i \(0.558366\pi\)
\(998\) 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 91.2.a.c.1.1 2
3.2 odd 2 819.2.a.h.1.2 2
4.3 odd 2 1456.2.a.q.1.1 2
5.4 even 2 2275.2.a.j.1.2 2
7.2 even 3 637.2.e.f.508.2 4
7.3 odd 6 637.2.e.g.79.2 4
7.4 even 3 637.2.e.f.79.2 4
7.5 odd 6 637.2.e.g.508.2 4
7.6 odd 2 637.2.a.g.1.1 2
8.3 odd 2 5824.2.a.bk.1.2 2
8.5 even 2 5824.2.a.bl.1.1 2
13.5 odd 4 1183.2.c.d.337.4 4
13.8 odd 4 1183.2.c.d.337.2 4
13.12 even 2 1183.2.a.d.1.2 2
21.20 even 2 5733.2.a.s.1.2 2
91.90 odd 2 8281.2.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.c.1.1 2 1.1 even 1 trivial
637.2.a.g.1.1 2 7.6 odd 2
637.2.e.f.79.2 4 7.4 even 3
637.2.e.f.508.2 4 7.2 even 3
637.2.e.g.79.2 4 7.3 odd 6
637.2.e.g.508.2 4 7.5 odd 6
819.2.a.h.1.2 2 3.2 odd 2
1183.2.a.d.1.2 2 13.12 even 2
1183.2.c.d.337.2 4 13.8 odd 4
1183.2.c.d.337.4 4 13.5 odd 4
1456.2.a.q.1.1 2 4.3 odd 2
2275.2.a.j.1.2 2 5.4 even 2
5733.2.a.s.1.2 2 21.20 even 2
5824.2.a.bk.1.2 2 8.3 odd 2
5824.2.a.bl.1.1 2 8.5 even 2
8281.2.a.v.1.2 2 91.90 odd 2