Properties

Label 147.2.a.d.1.2
Level $147$
Weight $2$
Character 147.1
Self dual yes
Analytic conductor $1.174$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.17380090971\)
Analytic rank: \(1\)
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 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -3.41421 q^{5} -0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -3.41421 q^{5} -0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} -1.41421 q^{10} -2.00000 q^{11} +1.82843 q^{12} -2.58579 q^{13} +3.41421 q^{15} +3.00000 q^{16} +2.24264 q^{17} +0.414214 q^{18} +2.82843 q^{19} +6.24264 q^{20} -0.828427 q^{22} -7.65685 q^{23} +1.58579 q^{24} +6.65685 q^{25} -1.07107 q^{26} -1.00000 q^{27} -6.82843 q^{29} +1.41421 q^{30} +1.17157 q^{31} +4.41421 q^{32} +2.00000 q^{33} +0.928932 q^{34} -1.82843 q^{36} -4.00000 q^{37} +1.17157 q^{38} +2.58579 q^{39} +5.41421 q^{40} -6.24264 q^{41} +5.65685 q^{43} +3.65685 q^{44} -3.41421 q^{45} -3.17157 q^{46} +2.82843 q^{47} -3.00000 q^{48} +2.75736 q^{50} -2.24264 q^{51} +4.72792 q^{52} -2.00000 q^{53} -0.414214 q^{54} +6.82843 q^{55} -2.82843 q^{57} -2.82843 q^{58} +1.17157 q^{59} -6.24264 q^{60} -12.2426 q^{61} +0.485281 q^{62} -4.17157 q^{64} +8.82843 q^{65} +0.828427 q^{66} -5.65685 q^{67} -4.10051 q^{68} +7.65685 q^{69} +9.31371 q^{71} -1.58579 q^{72} -13.8995 q^{73} -1.65685 q^{74} -6.65685 q^{75} -5.17157 q^{76} +1.07107 q^{78} +13.6569 q^{79} -10.2426 q^{80} +1.00000 q^{81} -2.58579 q^{82} -7.31371 q^{83} -7.65685 q^{85} +2.34315 q^{86} +6.82843 q^{87} +3.17157 q^{88} +14.2426 q^{89} -1.41421 q^{90} +14.0000 q^{92} -1.17157 q^{93} +1.17157 q^{94} -9.65685 q^{95} -4.41421 q^{96} -2.58579 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 6q^{8} + 2q^{9} - 4q^{11} - 2q^{12} - 8q^{13} + 4q^{15} + 6q^{16} - 4q^{17} - 2q^{18} + 4q^{20} + 4q^{22} - 4q^{23} + 6q^{24} + 2q^{25} + 12q^{26} - 2q^{27} - 8q^{29} + 8q^{31} + 6q^{32} + 4q^{33} + 16q^{34} + 2q^{36} - 8q^{37} + 8q^{38} + 8q^{39} + 8q^{40} - 4q^{41} - 4q^{44} - 4q^{45} - 12q^{46} - 6q^{48} + 14q^{50} + 4q^{51} - 16q^{52} - 4q^{53} + 2q^{54} + 8q^{55} + 8q^{59} - 4q^{60} - 16q^{61} - 16q^{62} - 14q^{64} + 12q^{65} - 4q^{66} - 28q^{68} + 4q^{69} - 4q^{71} - 6q^{72} - 8q^{73} + 8q^{74} - 2q^{75} - 16q^{76} - 12q^{78} + 16q^{79} - 12q^{80} + 2q^{81} - 8q^{82} + 8q^{83} - 4q^{85} + 16q^{86} + 8q^{87} + 12q^{88} + 20q^{89} + 28q^{92} - 8q^{93} + 8q^{94} - 8q^{95} - 6q^{96} - 8q^{97} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.82843 −0.914214
\(5\) −3.41421 −1.52688 −0.763441 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) −0.414214 −0.169102
\(7\) 0 0
\(8\) −1.58579 −0.560660
\(9\) 1.00000 0.333333
\(10\) −1.41421 −0.447214
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.82843 0.527821
\(13\) −2.58579 −0.717168 −0.358584 0.933497i \(-0.616740\pi\)
−0.358584 + 0.933497i \(0.616740\pi\)
\(14\) 0 0
\(15\) 3.41421 0.881546
\(16\) 3.00000 0.750000
\(17\) 2.24264 0.543920 0.271960 0.962309i \(-0.412328\pi\)
0.271960 + 0.962309i \(0.412328\pi\)
\(18\) 0.414214 0.0976311
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 6.24264 1.39590
\(21\) 0 0
\(22\) −0.828427 −0.176621
\(23\) −7.65685 −1.59656 −0.798282 0.602284i \(-0.794258\pi\)
−0.798282 + 0.602284i \(0.794258\pi\)
\(24\) 1.58579 0.323697
\(25\) 6.65685 1.33137
\(26\) −1.07107 −0.210054
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.82843 −1.26801 −0.634004 0.773330i \(-0.718590\pi\)
−0.634004 + 0.773330i \(0.718590\pi\)
\(30\) 1.41421 0.258199
\(31\) 1.17157 0.210421 0.105210 0.994450i \(-0.466448\pi\)
0.105210 + 0.994450i \(0.466448\pi\)
\(32\) 4.41421 0.780330
\(33\) 2.00000 0.348155
\(34\) 0.928932 0.159311
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 1.17157 0.190054
\(39\) 2.58579 0.414057
\(40\) 5.41421 0.856062
\(41\) −6.24264 −0.974937 −0.487468 0.873141i \(-0.662080\pi\)
−0.487468 + 0.873141i \(0.662080\pi\)
\(42\) 0 0
\(43\) 5.65685 0.862662 0.431331 0.902194i \(-0.358044\pi\)
0.431331 + 0.902194i \(0.358044\pi\)
\(44\) 3.65685 0.551292
\(45\) −3.41421 −0.508961
\(46\) −3.17157 −0.467623
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) −3.00000 −0.433013
\(49\) 0 0
\(50\) 2.75736 0.389949
\(51\) −2.24264 −0.314033
\(52\) 4.72792 0.655645
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −0.414214 −0.0563673
\(55\) 6.82843 0.920745
\(56\) 0 0
\(57\) −2.82843 −0.374634
\(58\) −2.82843 −0.371391
\(59\) 1.17157 0.152526 0.0762629 0.997088i \(-0.475701\pi\)
0.0762629 + 0.997088i \(0.475701\pi\)
\(60\) −6.24264 −0.805921
\(61\) −12.2426 −1.56751 −0.783755 0.621070i \(-0.786698\pi\)
−0.783755 + 0.621070i \(0.786698\pi\)
\(62\) 0.485281 0.0616308
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) 8.82843 1.09503
\(66\) 0.828427 0.101972
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) −4.10051 −0.497259
\(69\) 7.65685 0.921777
\(70\) 0 0
\(71\) 9.31371 1.10533 0.552667 0.833402i \(-0.313610\pi\)
0.552667 + 0.833402i \(0.313610\pi\)
\(72\) −1.58579 −0.186887
\(73\) −13.8995 −1.62681 −0.813406 0.581696i \(-0.802389\pi\)
−0.813406 + 0.581696i \(0.802389\pi\)
\(74\) −1.65685 −0.192605
\(75\) −6.65685 −0.768667
\(76\) −5.17157 −0.593220
\(77\) 0 0
\(78\) 1.07107 0.121275
\(79\) 13.6569 1.53652 0.768258 0.640140i \(-0.221124\pi\)
0.768258 + 0.640140i \(0.221124\pi\)
\(80\) −10.2426 −1.14516
\(81\) 1.00000 0.111111
\(82\) −2.58579 −0.285552
\(83\) −7.31371 −0.802784 −0.401392 0.915906i \(-0.631473\pi\)
−0.401392 + 0.915906i \(0.631473\pi\)
\(84\) 0 0
\(85\) −7.65685 −0.830502
\(86\) 2.34315 0.252668
\(87\) 6.82843 0.732084
\(88\) 3.17157 0.338091
\(89\) 14.2426 1.50972 0.754858 0.655888i \(-0.227706\pi\)
0.754858 + 0.655888i \(0.227706\pi\)
\(90\) −1.41421 −0.149071
\(91\) 0 0
\(92\) 14.0000 1.45960
\(93\) −1.17157 −0.121486
\(94\) 1.17157 0.120839
\(95\) −9.65685 −0.990772
\(96\) −4.41421 −0.450524
\(97\) −2.58579 −0.262547 −0.131273 0.991346i \(-0.541907\pi\)
−0.131273 + 0.991346i \(0.541907\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −12.1716 −1.21716
\(101\) −2.92893 −0.291440 −0.145720 0.989326i \(-0.546550\pi\)
−0.145720 + 0.989326i \(0.546550\pi\)
\(102\) −0.928932 −0.0919780
\(103\) −4.48528 −0.441948 −0.220974 0.975280i \(-0.570924\pi\)
−0.220974 + 0.975280i \(0.570924\pi\)
\(104\) 4.10051 0.402088
\(105\) 0 0
\(106\) −0.828427 −0.0804640
\(107\) −0.343146 −0.0331732 −0.0165866 0.999862i \(-0.505280\pi\)
−0.0165866 + 0.999862i \(0.505280\pi\)
\(108\) 1.82843 0.175940
\(109\) −5.65685 −0.541828 −0.270914 0.962604i \(-0.587326\pi\)
−0.270914 + 0.962604i \(0.587326\pi\)
\(110\) 2.82843 0.269680
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −5.31371 −0.499872 −0.249936 0.968262i \(-0.580410\pi\)
−0.249936 + 0.968262i \(0.580410\pi\)
\(114\) −1.17157 −0.109728
\(115\) 26.1421 2.43777
\(116\) 12.4853 1.15923
\(117\) −2.58579 −0.239056
\(118\) 0.485281 0.0446738
\(119\) 0 0
\(120\) −5.41421 −0.494248
\(121\) −7.00000 −0.636364
\(122\) −5.07107 −0.459113
\(123\) 6.24264 0.562880
\(124\) −2.14214 −0.192369
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −1.65685 −0.147022 −0.0735110 0.997294i \(-0.523420\pi\)
−0.0735110 + 0.997294i \(0.523420\pi\)
\(128\) −10.5563 −0.933058
\(129\) −5.65685 −0.498058
\(130\) 3.65685 0.320727
\(131\) 15.3137 1.33796 0.668982 0.743278i \(-0.266730\pi\)
0.668982 + 0.743278i \(0.266730\pi\)
\(132\) −3.65685 −0.318288
\(133\) 0 0
\(134\) −2.34315 −0.202417
\(135\) 3.41421 0.293849
\(136\) −3.55635 −0.304954
\(137\) 14.1421 1.20824 0.604122 0.796892i \(-0.293524\pi\)
0.604122 + 0.796892i \(0.293524\pi\)
\(138\) 3.17157 0.269982
\(139\) 17.6569 1.49763 0.748817 0.662776i \(-0.230622\pi\)
0.748817 + 0.662776i \(0.230622\pi\)
\(140\) 0 0
\(141\) −2.82843 −0.238197
\(142\) 3.85786 0.323745
\(143\) 5.17157 0.432469
\(144\) 3.00000 0.250000
\(145\) 23.3137 1.93610
\(146\) −5.75736 −0.476482
\(147\) 0 0
\(148\) 7.31371 0.601183
\(149\) 17.3137 1.41839 0.709197 0.705010i \(-0.249058\pi\)
0.709197 + 0.705010i \(0.249058\pi\)
\(150\) −2.75736 −0.225137
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) −4.48528 −0.363804
\(153\) 2.24264 0.181307
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) −4.72792 −0.378537
\(157\) −11.7574 −0.938339 −0.469170 0.883108i \(-0.655447\pi\)
−0.469170 + 0.883108i \(0.655447\pi\)
\(158\) 5.65685 0.450035
\(159\) 2.00000 0.158610
\(160\) −15.0711 −1.19147
\(161\) 0 0
\(162\) 0.414214 0.0325437
\(163\) −11.3137 −0.886158 −0.443079 0.896483i \(-0.646114\pi\)
−0.443079 + 0.896483i \(0.646114\pi\)
\(164\) 11.4142 0.891300
\(165\) −6.82843 −0.531592
\(166\) −3.02944 −0.235130
\(167\) 19.7990 1.53209 0.766046 0.642786i \(-0.222221\pi\)
0.766046 + 0.642786i \(0.222221\pi\)
\(168\) 0 0
\(169\) −6.31371 −0.485670
\(170\) −3.17157 −0.243249
\(171\) 2.82843 0.216295
\(172\) −10.3431 −0.788657
\(173\) −21.0711 −1.60200 −0.801002 0.598662i \(-0.795699\pi\)
−0.801002 + 0.598662i \(0.795699\pi\)
\(174\) 2.82843 0.214423
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) −1.17157 −0.0880608
\(178\) 5.89949 0.442186
\(179\) −19.6569 −1.46922 −0.734611 0.678488i \(-0.762635\pi\)
−0.734611 + 0.678488i \(0.762635\pi\)
\(180\) 6.24264 0.465299
\(181\) 2.58579 0.192200 0.0961000 0.995372i \(-0.469363\pi\)
0.0961000 + 0.995372i \(0.469363\pi\)
\(182\) 0 0
\(183\) 12.2426 0.905002
\(184\) 12.1421 0.895130
\(185\) 13.6569 1.00407
\(186\) −0.485281 −0.0355826
\(187\) −4.48528 −0.327996
\(188\) −5.17157 −0.377176
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 4.17157 0.301057
\(193\) 5.31371 0.382489 0.191245 0.981542i \(-0.438748\pi\)
0.191245 + 0.981542i \(0.438748\pi\)
\(194\) −1.07107 −0.0768982
\(195\) −8.82843 −0.632217
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −0.828427 −0.0588738
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) −10.5563 −0.746447
\(201\) 5.65685 0.399004
\(202\) −1.21320 −0.0853607
\(203\) 0 0
\(204\) 4.10051 0.287093
\(205\) 21.3137 1.48861
\(206\) −1.85786 −0.129444
\(207\) −7.65685 −0.532188
\(208\) −7.75736 −0.537876
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) 12.9706 0.892930 0.446465 0.894801i \(-0.352683\pi\)
0.446465 + 0.894801i \(0.352683\pi\)
\(212\) 3.65685 0.251154
\(213\) −9.31371 −0.638165
\(214\) −0.142136 −0.00971619
\(215\) −19.3137 −1.31718
\(216\) 1.58579 0.107899
\(217\) 0 0
\(218\) −2.34315 −0.158698
\(219\) 13.8995 0.939241
\(220\) −12.4853 −0.841757
\(221\) −5.79899 −0.390082
\(222\) 1.65685 0.111201
\(223\) −24.9706 −1.67215 −0.836076 0.548613i \(-0.815156\pi\)
−0.836076 + 0.548613i \(0.815156\pi\)
\(224\) 0 0
\(225\) 6.65685 0.443790
\(226\) −2.20101 −0.146409
\(227\) −23.7990 −1.57959 −0.789797 0.613368i \(-0.789814\pi\)
−0.789797 + 0.613368i \(0.789814\pi\)
\(228\) 5.17157 0.342496
\(229\) 0.242641 0.0160341 0.00801707 0.999968i \(-0.497448\pi\)
0.00801707 + 0.999968i \(0.497448\pi\)
\(230\) 10.8284 0.714005
\(231\) 0 0
\(232\) 10.8284 0.710921
\(233\) −6.14214 −0.402385 −0.201192 0.979552i \(-0.564482\pi\)
−0.201192 + 0.979552i \(0.564482\pi\)
\(234\) −1.07107 −0.0700179
\(235\) −9.65685 −0.629944
\(236\) −2.14214 −0.139441
\(237\) −13.6569 −0.887108
\(238\) 0 0
\(239\) −15.6569 −1.01276 −0.506379 0.862311i \(-0.669016\pi\)
−0.506379 + 0.862311i \(0.669016\pi\)
\(240\) 10.2426 0.661160
\(241\) 16.2426 1.04628 0.523140 0.852247i \(-0.324760\pi\)
0.523140 + 0.852247i \(0.324760\pi\)
\(242\) −2.89949 −0.186387
\(243\) −1.00000 −0.0641500
\(244\) 22.3848 1.43304
\(245\) 0 0
\(246\) 2.58579 0.164864
\(247\) −7.31371 −0.465360
\(248\) −1.85786 −0.117975
\(249\) 7.31371 0.463487
\(250\) −2.34315 −0.148194
\(251\) 12.4853 0.788064 0.394032 0.919097i \(-0.371080\pi\)
0.394032 + 0.919097i \(0.371080\pi\)
\(252\) 0 0
\(253\) 15.3137 0.962765
\(254\) −0.686292 −0.0430618
\(255\) 7.65685 0.479491
\(256\) 3.97056 0.248160
\(257\) −23.2132 −1.44800 −0.724000 0.689800i \(-0.757698\pi\)
−0.724000 + 0.689800i \(0.757698\pi\)
\(258\) −2.34315 −0.145878
\(259\) 0 0
\(260\) −16.1421 −1.00109
\(261\) −6.82843 −0.422669
\(262\) 6.34315 0.391881
\(263\) 5.31371 0.327657 0.163829 0.986489i \(-0.447616\pi\)
0.163829 + 0.986489i \(0.447616\pi\)
\(264\) −3.17157 −0.195197
\(265\) 6.82843 0.419467
\(266\) 0 0
\(267\) −14.2426 −0.871635
\(268\) 10.3431 0.631808
\(269\) 14.7279 0.897977 0.448989 0.893537i \(-0.351784\pi\)
0.448989 + 0.893537i \(0.351784\pi\)
\(270\) 1.41421 0.0860663
\(271\) 10.1421 0.616091 0.308045 0.951372i \(-0.400325\pi\)
0.308045 + 0.951372i \(0.400325\pi\)
\(272\) 6.72792 0.407940
\(273\) 0 0
\(274\) 5.85786 0.353887
\(275\) −13.3137 −0.802847
\(276\) −14.0000 −0.842701
\(277\) −9.31371 −0.559607 −0.279803 0.960057i \(-0.590269\pi\)
−0.279803 + 0.960057i \(0.590269\pi\)
\(278\) 7.31371 0.438647
\(279\) 1.17157 0.0701402
\(280\) 0 0
\(281\) 0.485281 0.0289495 0.0144747 0.999895i \(-0.495392\pi\)
0.0144747 + 0.999895i \(0.495392\pi\)
\(282\) −1.17157 −0.0697661
\(283\) 8.48528 0.504398 0.252199 0.967675i \(-0.418846\pi\)
0.252199 + 0.967675i \(0.418846\pi\)
\(284\) −17.0294 −1.01051
\(285\) 9.65685 0.572023
\(286\) 2.14214 0.126667
\(287\) 0 0
\(288\) 4.41421 0.260110
\(289\) −11.9706 −0.704151
\(290\) 9.65685 0.567070
\(291\) 2.58579 0.151581
\(292\) 25.4142 1.48725
\(293\) −16.5858 −0.968952 −0.484476 0.874805i \(-0.660990\pi\)
−0.484476 + 0.874805i \(0.660990\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 6.34315 0.368688
\(297\) 2.00000 0.116052
\(298\) 7.17157 0.415438
\(299\) 19.7990 1.14501
\(300\) 12.1716 0.702726
\(301\) 0 0
\(302\) 4.97056 0.286024
\(303\) 2.92893 0.168263
\(304\) 8.48528 0.486664
\(305\) 41.7990 2.39340
\(306\) 0.928932 0.0531035
\(307\) −30.1421 −1.72030 −0.860151 0.510039i \(-0.829631\pi\)
−0.860151 + 0.510039i \(0.829631\pi\)
\(308\) 0 0
\(309\) 4.48528 0.255159
\(310\) −1.65685 −0.0941030
\(311\) −6.14214 −0.348289 −0.174144 0.984720i \(-0.555716\pi\)
−0.174144 + 0.984720i \(0.555716\pi\)
\(312\) −4.10051 −0.232145
\(313\) −1.89949 −0.107366 −0.0536829 0.998558i \(-0.517096\pi\)
−0.0536829 + 0.998558i \(0.517096\pi\)
\(314\) −4.87006 −0.274833
\(315\) 0 0
\(316\) −24.9706 −1.40470
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0.828427 0.0464559
\(319\) 13.6569 0.764637
\(320\) 14.2426 0.796188
\(321\) 0.343146 0.0191525
\(322\) 0 0
\(323\) 6.34315 0.352942
\(324\) −1.82843 −0.101579
\(325\) −17.2132 −0.954817
\(326\) −4.68629 −0.259550
\(327\) 5.65685 0.312825
\(328\) 9.89949 0.546608
\(329\) 0 0
\(330\) −2.82843 −0.155700
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 13.3726 0.733916
\(333\) −4.00000 −0.219199
\(334\) 8.20101 0.448739
\(335\) 19.3137 1.05522
\(336\) 0 0
\(337\) −29.6569 −1.61551 −0.807756 0.589517i \(-0.799318\pi\)
−0.807756 + 0.589517i \(0.799318\pi\)
\(338\) −2.61522 −0.142249
\(339\) 5.31371 0.288601
\(340\) 14.0000 0.759257
\(341\) −2.34315 −0.126888
\(342\) 1.17157 0.0633514
\(343\) 0 0
\(344\) −8.97056 −0.483660
\(345\) −26.1421 −1.40745
\(346\) −8.72792 −0.469216
\(347\) 33.3137 1.78837 0.894187 0.447694i \(-0.147755\pi\)
0.894187 + 0.447694i \(0.147755\pi\)
\(348\) −12.4853 −0.669281
\(349\) 9.89949 0.529908 0.264954 0.964261i \(-0.414643\pi\)
0.264954 + 0.964261i \(0.414643\pi\)
\(350\) 0 0
\(351\) 2.58579 0.138019
\(352\) −8.82843 −0.470557
\(353\) 14.7279 0.783888 0.391944 0.919989i \(-0.371803\pi\)
0.391944 + 0.919989i \(0.371803\pi\)
\(354\) −0.485281 −0.0257924
\(355\) −31.7990 −1.68772
\(356\) −26.0416 −1.38020
\(357\) 0 0
\(358\) −8.14214 −0.430325
\(359\) −0.343146 −0.0181105 −0.00905527 0.999959i \(-0.502882\pi\)
−0.00905527 + 0.999959i \(0.502882\pi\)
\(360\) 5.41421 0.285354
\(361\) −11.0000 −0.578947
\(362\) 1.07107 0.0562941
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 47.4558 2.48395
\(366\) 5.07107 0.265069
\(367\) 3.31371 0.172974 0.0864871 0.996253i \(-0.472436\pi\)
0.0864871 + 0.996253i \(0.472436\pi\)
\(368\) −22.9706 −1.19742
\(369\) −6.24264 −0.324979
\(370\) 5.65685 0.294086
\(371\) 0 0
\(372\) 2.14214 0.111065
\(373\) −10.6863 −0.553315 −0.276658 0.960969i \(-0.589227\pi\)
−0.276658 + 0.960969i \(0.589227\pi\)
\(374\) −1.85786 −0.0960679
\(375\) 5.65685 0.292119
\(376\) −4.48528 −0.231311
\(377\) 17.6569 0.909374
\(378\) 0 0
\(379\) 8.68629 0.446185 0.223092 0.974797i \(-0.428385\pi\)
0.223092 + 0.974797i \(0.428385\pi\)
\(380\) 17.6569 0.905778
\(381\) 1.65685 0.0848832
\(382\) −7.45584 −0.381474
\(383\) 18.3431 0.937291 0.468645 0.883386i \(-0.344742\pi\)
0.468645 + 0.883386i \(0.344742\pi\)
\(384\) 10.5563 0.538701
\(385\) 0 0
\(386\) 2.20101 0.112028
\(387\) 5.65685 0.287554
\(388\) 4.72792 0.240024
\(389\) −18.1421 −0.919843 −0.459921 0.887960i \(-0.652122\pi\)
−0.459921 + 0.887960i \(0.652122\pi\)
\(390\) −3.65685 −0.185172
\(391\) −17.1716 −0.868404
\(392\) 0 0
\(393\) −15.3137 −0.772474
\(394\) 0.828427 0.0417356
\(395\) −46.6274 −2.34608
\(396\) 3.65685 0.183764
\(397\) 2.38478 0.119688 0.0598442 0.998208i \(-0.480940\pi\)
0.0598442 + 0.998208i \(0.480940\pi\)
\(398\) −8.97056 −0.449654
\(399\) 0 0
\(400\) 19.9706 0.998528
\(401\) −6.14214 −0.306724 −0.153362 0.988170i \(-0.549010\pi\)
−0.153362 + 0.988170i \(0.549010\pi\)
\(402\) 2.34315 0.116865
\(403\) −3.02944 −0.150907
\(404\) 5.35534 0.266438
\(405\) −3.41421 −0.169654
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 3.55635 0.176066
\(409\) 21.4142 1.05886 0.529432 0.848352i \(-0.322405\pi\)
0.529432 + 0.848352i \(0.322405\pi\)
\(410\) 8.82843 0.436005
\(411\) −14.1421 −0.697580
\(412\) 8.20101 0.404035
\(413\) 0 0
\(414\) −3.17157 −0.155874
\(415\) 24.9706 1.22576
\(416\) −11.4142 −0.559628
\(417\) −17.6569 −0.864660
\(418\) −2.34315 −0.114607
\(419\) −33.1716 −1.62054 −0.810269 0.586059i \(-0.800679\pi\)
−0.810269 + 0.586059i \(0.800679\pi\)
\(420\) 0 0
\(421\) 16.6274 0.810371 0.405185 0.914235i \(-0.367207\pi\)
0.405185 + 0.914235i \(0.367207\pi\)
\(422\) 5.37258 0.261533
\(423\) 2.82843 0.137523
\(424\) 3.17157 0.154025
\(425\) 14.9289 0.724160
\(426\) −3.85786 −0.186914
\(427\) 0 0
\(428\) 0.627417 0.0303273
\(429\) −5.17157 −0.249686
\(430\) −8.00000 −0.385794
\(431\) −26.9706 −1.29913 −0.649563 0.760308i \(-0.725048\pi\)
−0.649563 + 0.760308i \(0.725048\pi\)
\(432\) −3.00000 −0.144338
\(433\) 20.2426 0.972799 0.486400 0.873736i \(-0.338310\pi\)
0.486400 + 0.873736i \(0.338310\pi\)
\(434\) 0 0
\(435\) −23.3137 −1.11781
\(436\) 10.3431 0.495347
\(437\) −21.6569 −1.03599
\(438\) 5.75736 0.275097
\(439\) −12.6863 −0.605484 −0.302742 0.953073i \(-0.597902\pi\)
−0.302742 + 0.953073i \(0.597902\pi\)
\(440\) −10.8284 −0.516225
\(441\) 0 0
\(442\) −2.40202 −0.114252
\(443\) 34.9706 1.66150 0.830751 0.556645i \(-0.187911\pi\)
0.830751 + 0.556645i \(0.187911\pi\)
\(444\) −7.31371 −0.347093
\(445\) −48.6274 −2.30516
\(446\) −10.3431 −0.489762
\(447\) −17.3137 −0.818910
\(448\) 0 0
\(449\) −5.31371 −0.250769 −0.125385 0.992108i \(-0.540017\pi\)
−0.125385 + 0.992108i \(0.540017\pi\)
\(450\) 2.75736 0.129983
\(451\) 12.4853 0.587909
\(452\) 9.71573 0.456989
\(453\) −12.0000 −0.563809
\(454\) −9.85786 −0.462652
\(455\) 0 0
\(456\) 4.48528 0.210043
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 0.100505 0.00469629
\(459\) −2.24264 −0.104678
\(460\) −47.7990 −2.22864
\(461\) 16.5858 0.772477 0.386239 0.922399i \(-0.373774\pi\)
0.386239 + 0.922399i \(0.373774\pi\)
\(462\) 0 0
\(463\) −26.6274 −1.23748 −0.618741 0.785595i \(-0.712357\pi\)
−0.618741 + 0.785595i \(0.712357\pi\)
\(464\) −20.4853 −0.951005
\(465\) 4.00000 0.185496
\(466\) −2.54416 −0.117856
\(467\) −0.201010 −0.00930164 −0.00465082 0.999989i \(-0.501480\pi\)
−0.00465082 + 0.999989i \(0.501480\pi\)
\(468\) 4.72792 0.218548
\(469\) 0 0
\(470\) −4.00000 −0.184506
\(471\) 11.7574 0.541751
\(472\) −1.85786 −0.0855151
\(473\) −11.3137 −0.520205
\(474\) −5.65685 −0.259828
\(475\) 18.8284 0.863907
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) −6.48528 −0.296630
\(479\) −1.85786 −0.0848880 −0.0424440 0.999099i \(-0.513514\pi\)
−0.0424440 + 0.999099i \(0.513514\pi\)
\(480\) 15.0711 0.687897
\(481\) 10.3431 0.471607
\(482\) 6.72792 0.306448
\(483\) 0 0
\(484\) 12.7990 0.581772
\(485\) 8.82843 0.400878
\(486\) −0.414214 −0.0187891
\(487\) 26.6274 1.20660 0.603302 0.797513i \(-0.293851\pi\)
0.603302 + 0.797513i \(0.293851\pi\)
\(488\) 19.4142 0.878840
\(489\) 11.3137 0.511624
\(490\) 0 0
\(491\) 5.02944 0.226975 0.113488 0.993539i \(-0.463798\pi\)
0.113488 + 0.993539i \(0.463798\pi\)
\(492\) −11.4142 −0.514592
\(493\) −15.3137 −0.689695
\(494\) −3.02944 −0.136301
\(495\) 6.82843 0.306915
\(496\) 3.51472 0.157816
\(497\) 0 0
\(498\) 3.02944 0.135752
\(499\) 3.31371 0.148342 0.0741710 0.997246i \(-0.476369\pi\)
0.0741710 + 0.997246i \(0.476369\pi\)
\(500\) 10.3431 0.462560
\(501\) −19.7990 −0.884554
\(502\) 5.17157 0.230819
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 6.34315 0.281987
\(507\) 6.31371 0.280402
\(508\) 3.02944 0.134410
\(509\) −5.55635 −0.246281 −0.123140 0.992389i \(-0.539297\pi\)
−0.123140 + 0.992389i \(0.539297\pi\)
\(510\) 3.17157 0.140440
\(511\) 0 0
\(512\) 22.7574 1.00574
\(513\) −2.82843 −0.124878
\(514\) −9.61522 −0.424109
\(515\) 15.3137 0.674803
\(516\) 10.3431 0.455332
\(517\) −5.65685 −0.248788
\(518\) 0 0
\(519\) 21.0711 0.924917
\(520\) −14.0000 −0.613941
\(521\) 35.4142 1.55152 0.775762 0.631025i \(-0.217366\pi\)
0.775762 + 0.631025i \(0.217366\pi\)
\(522\) −2.82843 −0.123797
\(523\) 25.6569 1.12190 0.560948 0.827851i \(-0.310437\pi\)
0.560948 + 0.827851i \(0.310437\pi\)
\(524\) −28.0000 −1.22319
\(525\) 0 0
\(526\) 2.20101 0.0959686
\(527\) 2.62742 0.114452
\(528\) 6.00000 0.261116
\(529\) 35.6274 1.54902
\(530\) 2.82843 0.122859
\(531\) 1.17157 0.0508419
\(532\) 0 0
\(533\) 16.1421 0.699194
\(534\) −5.89949 −0.255296
\(535\) 1.17157 0.0506515
\(536\) 8.97056 0.387469
\(537\) 19.6569 0.848256
\(538\) 6.10051 0.263011
\(539\) 0 0
\(540\) −6.24264 −0.268640
\(541\) 17.3137 0.744374 0.372187 0.928158i \(-0.378608\pi\)
0.372187 + 0.928158i \(0.378608\pi\)
\(542\) 4.20101 0.180449
\(543\) −2.58579 −0.110967
\(544\) 9.89949 0.424437
\(545\) 19.3137 0.827308
\(546\) 0 0
\(547\) −36.9706 −1.58075 −0.790374 0.612625i \(-0.790114\pi\)
−0.790374 + 0.612625i \(0.790114\pi\)
\(548\) −25.8579 −1.10459
\(549\) −12.2426 −0.522503
\(550\) −5.51472 −0.235148
\(551\) −19.3137 −0.822792
\(552\) −12.1421 −0.516804
\(553\) 0 0
\(554\) −3.85786 −0.163905
\(555\) −13.6569 −0.579701
\(556\) −32.2843 −1.36916
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 0.485281 0.0205436
\(559\) −14.6274 −0.618674
\(560\) 0 0
\(561\) 4.48528 0.189369
\(562\) 0.201010 0.00847910
\(563\) 1.17157 0.0493759 0.0246880 0.999695i \(-0.492141\pi\)
0.0246880 + 0.999695i \(0.492141\pi\)
\(564\) 5.17157 0.217763
\(565\) 18.1421 0.763245
\(566\) 3.51472 0.147735
\(567\) 0 0
\(568\) −14.7696 −0.619717
\(569\) −16.4853 −0.691099 −0.345549 0.938401i \(-0.612307\pi\)
−0.345549 + 0.938401i \(0.612307\pi\)
\(570\) 4.00000 0.167542
\(571\) 22.3431 0.935032 0.467516 0.883985i \(-0.345149\pi\)
0.467516 + 0.883985i \(0.345149\pi\)
\(572\) −9.45584 −0.395369
\(573\) 18.0000 0.751961
\(574\) 0 0
\(575\) −50.9706 −2.12562
\(576\) −4.17157 −0.173816
\(577\) 33.8995 1.41125 0.705627 0.708583i \(-0.250665\pi\)
0.705627 + 0.708583i \(0.250665\pi\)
\(578\) −4.95837 −0.206241
\(579\) −5.31371 −0.220830
\(580\) −42.6274 −1.77001
\(581\) 0 0
\(582\) 1.07107 0.0443972
\(583\) 4.00000 0.165663
\(584\) 22.0416 0.912089
\(585\) 8.82843 0.365011
\(586\) −6.87006 −0.283799
\(587\) 22.8284 0.942230 0.471115 0.882072i \(-0.343852\pi\)
0.471115 + 0.882072i \(0.343852\pi\)
\(588\) 0 0
\(589\) 3.31371 0.136539
\(590\) −1.65685 −0.0682116
\(591\) −2.00000 −0.0822690
\(592\) −12.0000 −0.493197
\(593\) 6.92893 0.284537 0.142269 0.989828i \(-0.454560\pi\)
0.142269 + 0.989828i \(0.454560\pi\)
\(594\) 0.828427 0.0339908
\(595\) 0 0
\(596\) −31.6569 −1.29672
\(597\) 21.6569 0.886356
\(598\) 8.20101 0.335364
\(599\) −2.00000 −0.0817178 −0.0408589 0.999165i \(-0.513009\pi\)
−0.0408589 + 0.999165i \(0.513009\pi\)
\(600\) 10.5563 0.430961
\(601\) 15.0711 0.614762 0.307381 0.951587i \(-0.400547\pi\)
0.307381 + 0.951587i \(0.400547\pi\)
\(602\) 0 0
\(603\) −5.65685 −0.230365
\(604\) −21.9411 −0.892772
\(605\) 23.8995 0.971653
\(606\) 1.21320 0.0492830
\(607\) 18.3431 0.744525 0.372263 0.928127i \(-0.378582\pi\)
0.372263 + 0.928127i \(0.378582\pi\)
\(608\) 12.4853 0.506345
\(609\) 0 0
\(610\) 17.3137 0.701012
\(611\) −7.31371 −0.295881
\(612\) −4.10051 −0.165753
\(613\) 4.68629 0.189278 0.0946388 0.995512i \(-0.469830\pi\)
0.0946388 + 0.995512i \(0.469830\pi\)
\(614\) −12.4853 −0.503865
\(615\) −21.3137 −0.859452
\(616\) 0 0
\(617\) −24.4853 −0.985740 −0.492870 0.870103i \(-0.664052\pi\)
−0.492870 + 0.870103i \(0.664052\pi\)
\(618\) 1.85786 0.0747343
\(619\) −28.9706 −1.16443 −0.582213 0.813037i \(-0.697813\pi\)
−0.582213 + 0.813037i \(0.697813\pi\)
\(620\) 7.31371 0.293726
\(621\) 7.65685 0.307259
\(622\) −2.54416 −0.102011
\(623\) 0 0
\(624\) 7.75736 0.310543
\(625\) −13.9706 −0.558823
\(626\) −0.786797 −0.0314467
\(627\) 5.65685 0.225913
\(628\) 21.4975 0.857843
\(629\) −8.97056 −0.357680
\(630\) 0 0
\(631\) 23.3137 0.928104 0.464052 0.885808i \(-0.346395\pi\)
0.464052 + 0.885808i \(0.346395\pi\)
\(632\) −21.6569 −0.861463
\(633\) −12.9706 −0.515534
\(634\) 4.14214 0.164505
\(635\) 5.65685 0.224485
\(636\) −3.65685 −0.145004
\(637\) 0 0
\(638\) 5.65685 0.223957
\(639\) 9.31371 0.368445
\(640\) 36.0416 1.42467
\(641\) −10.8284 −0.427697 −0.213849 0.976867i \(-0.568600\pi\)
−0.213849 + 0.976867i \(0.568600\pi\)
\(642\) 0.142136 0.00560965
\(643\) 34.4264 1.35764 0.678822 0.734302i \(-0.262491\pi\)
0.678822 + 0.734302i \(0.262491\pi\)
\(644\) 0 0
\(645\) 19.3137 0.760477
\(646\) 2.62742 0.103374
\(647\) −26.8284 −1.05473 −0.527367 0.849638i \(-0.676821\pi\)
−0.527367 + 0.849638i \(0.676821\pi\)
\(648\) −1.58579 −0.0622956
\(649\) −2.34315 −0.0919765
\(650\) −7.12994 −0.279659
\(651\) 0 0
\(652\) 20.6863 0.810138
\(653\) 36.4853 1.42778 0.713890 0.700258i \(-0.246932\pi\)
0.713890 + 0.700258i \(0.246932\pi\)
\(654\) 2.34315 0.0916242
\(655\) −52.2843 −2.04292
\(656\) −18.7279 −0.731203
\(657\) −13.8995 −0.542271
\(658\) 0 0
\(659\) 9.31371 0.362811 0.181405 0.983408i \(-0.441935\pi\)
0.181405 + 0.983408i \(0.441935\pi\)
\(660\) 12.4853 0.485989
\(661\) 23.5563 0.916236 0.458118 0.888891i \(-0.348524\pi\)
0.458118 + 0.888891i \(0.348524\pi\)
\(662\) −1.65685 −0.0643955
\(663\) 5.79899 0.225214
\(664\) 11.5980 0.450089
\(665\) 0 0
\(666\) −1.65685 −0.0642018
\(667\) 52.2843 2.02446
\(668\) −36.2010 −1.40066
\(669\) 24.9706 0.965418
\(670\) 8.00000 0.309067
\(671\) 24.4853 0.945244
\(672\) 0 0
\(673\) 23.3137 0.898677 0.449339 0.893361i \(-0.351660\pi\)
0.449339 + 0.893361i \(0.351660\pi\)
\(674\) −12.2843 −0.473172
\(675\) −6.65685 −0.256222
\(676\) 11.5442 0.444006
\(677\) −31.4142 −1.20735 −0.603673 0.797232i \(-0.706297\pi\)
−0.603673 + 0.797232i \(0.706297\pi\)
\(678\) 2.20101 0.0845293
\(679\) 0 0
\(680\) 12.1421 0.465630
\(681\) 23.7990 0.911979
\(682\) −0.970563 −0.0371648
\(683\) −19.6569 −0.752149 −0.376074 0.926590i \(-0.622726\pi\)
−0.376074 + 0.926590i \(0.622726\pi\)
\(684\) −5.17157 −0.197740
\(685\) −48.2843 −1.84485
\(686\) 0 0
\(687\) −0.242641 −0.00925732
\(688\) 16.9706 0.646997
\(689\) 5.17157 0.197021
\(690\) −10.8284 −0.412231
\(691\) 0.686292 0.0261078 0.0130539 0.999915i \(-0.495845\pi\)
0.0130539 + 0.999915i \(0.495845\pi\)
\(692\) 38.5269 1.46457
\(693\) 0 0
\(694\) 13.7990 0.523802
\(695\) −60.2843 −2.28671
\(696\) −10.8284 −0.410450
\(697\) −14.0000 −0.530288
\(698\) 4.10051 0.155206
\(699\) 6.14214 0.232317
\(700\) 0 0
\(701\) −17.1716 −0.648561 −0.324281 0.945961i \(-0.605122\pi\)
−0.324281 + 0.945961i \(0.605122\pi\)
\(702\) 1.07107 0.0404248
\(703\) −11.3137 −0.426705
\(704\) 8.34315 0.314444
\(705\) 9.65685 0.363698
\(706\) 6.10051 0.229596
\(707\) 0 0
\(708\) 2.14214 0.0805064
\(709\) −36.2843 −1.36268 −0.681342 0.731965i \(-0.738603\pi\)
−0.681342 + 0.731965i \(0.738603\pi\)
\(710\) −13.1716 −0.494320
\(711\) 13.6569 0.512172
\(712\) −22.5858 −0.846438
\(713\) −8.97056 −0.335950
\(714\) 0 0
\(715\) −17.6569 −0.660329
\(716\) 35.9411 1.34318
\(717\) 15.6569 0.584716
\(718\) −0.142136 −0.00530445
\(719\) −41.9411 −1.56414 −0.782070 0.623191i \(-0.785836\pi\)
−0.782070 + 0.623191i \(0.785836\pi\)
\(720\) −10.2426 −0.381721
\(721\) 0 0
\(722\) −4.55635 −0.169570
\(723\) −16.2426 −0.604070
\(724\) −4.72792 −0.175712
\(725\) −45.4558 −1.68819
\(726\) 2.89949 0.107610
\(727\) −12.4853 −0.463053 −0.231527 0.972829i \(-0.574372\pi\)
−0.231527 + 0.972829i \(0.574372\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 19.6569 0.727533
\(731\) 12.6863 0.469219
\(732\) −22.3848 −0.827365
\(733\) −49.6985 −1.83566 −0.917828 0.396979i \(-0.870059\pi\)
−0.917828 + 0.396979i \(0.870059\pi\)
\(734\) 1.37258 0.0506630
\(735\) 0 0
\(736\) −33.7990 −1.24585
\(737\) 11.3137 0.416746
\(738\) −2.58579 −0.0951841
\(739\) 4.68629 0.172388 0.0861940 0.996278i \(-0.472530\pi\)
0.0861940 + 0.996278i \(0.472530\pi\)
\(740\) −24.9706 −0.917936
\(741\) 7.31371 0.268676
\(742\) 0 0
\(743\) −50.9706 −1.86993 −0.934964 0.354742i \(-0.884569\pi\)
−0.934964 + 0.354742i \(0.884569\pi\)
\(744\) 1.85786 0.0681126
\(745\) −59.1127 −2.16572
\(746\) −4.42641 −0.162062
\(747\) −7.31371 −0.267595
\(748\) 8.20101 0.299859
\(749\) 0 0
\(750\) 2.34315 0.0855596
\(751\) 13.6569 0.498346 0.249173 0.968459i \(-0.419841\pi\)
0.249173 + 0.968459i \(0.419841\pi\)
\(752\) 8.48528 0.309426
\(753\) −12.4853 −0.454989
\(754\) 7.31371 0.266350
\(755\) −40.9706 −1.49107
\(756\) 0 0
\(757\) 26.3431 0.957458 0.478729 0.877963i \(-0.341098\pi\)
0.478729 + 0.877963i \(0.341098\pi\)
\(758\) 3.59798 0.130685
\(759\) −15.3137 −0.555852
\(760\) 15.3137 0.555487
\(761\) 18.5269 0.671600 0.335800 0.941933i \(-0.390993\pi\)
0.335800 + 0.941933i \(0.390993\pi\)
\(762\) 0.686292 0.0248617
\(763\) 0 0
\(764\) 32.9117 1.19070
\(765\) −7.65685 −0.276834
\(766\) 7.59798 0.274526
\(767\) −3.02944 −0.109387
\(768\) −3.97056 −0.143275
\(769\) −29.6985 −1.07095 −0.535477 0.844550i \(-0.679868\pi\)
−0.535477 + 0.844550i \(0.679868\pi\)
\(770\) 0 0
\(771\) 23.2132 0.836003
\(772\) −9.71573 −0.349677
\(773\) 9.55635 0.343718 0.171859 0.985122i \(-0.445023\pi\)
0.171859 + 0.985122i \(0.445023\pi\)
\(774\) 2.34315 0.0842226
\(775\) 7.79899 0.280148
\(776\) 4.10051 0.147200
\(777\) 0 0
\(778\) −7.51472 −0.269416
\(779\) −17.6569 −0.632622
\(780\) 16.1421 0.577981
\(781\) −18.6274 −0.666541
\(782\) −7.11270 −0.254350
\(783\) 6.82843 0.244028
\(784\) 0 0
\(785\) 40.1421 1.43273
\(786\) −6.34315 −0.226253
\(787\) −24.6863 −0.879971 −0.439986 0.898005i \(-0.645016\pi\)
−0.439986 + 0.898005i \(0.645016\pi\)
\(788\) −3.65685 −0.130270
\(789\) −5.31371 −0.189173
\(790\) −19.3137 −0.687151
\(791\) 0 0
\(792\) 3.17157 0.112697
\(793\) 31.6569 1.12417
\(794\) 0.987807 0.0350559
\(795\) −6.82843 −0.242179
\(796\) 39.5980 1.40351
\(797\) 8.38478 0.297004 0.148502 0.988912i \(-0.452555\pi\)
0.148502 + 0.988912i \(0.452555\pi\)
\(798\) 0 0
\(799\) 6.34315 0.224404
\(800\) 29.3848 1.03891
\(801\) 14.2426 0.503239
\(802\) −2.54416 −0.0898373
\(803\) 27.7990 0.981005
\(804\) −10.3431 −0.364775
\(805\) 0 0
\(806\) −1.25483 −0.0441996
\(807\) −14.7279 −0.518447
\(808\) 4.64466 0.163399
\(809\) 19.9411 0.701093 0.350546 0.936545i \(-0.385996\pi\)
0.350546 + 0.936545i \(0.385996\pi\)
\(810\) −1.41421 −0.0496904
\(811\) 17.6569 0.620016 0.310008 0.950734i \(-0.399668\pi\)
0.310008 + 0.950734i \(0.399668\pi\)
\(812\) 0 0
\(813\) −10.1421 −0.355700
\(814\) 3.31371 0.116145
\(815\) 38.6274 1.35306
\(816\) −6.72792 −0.235524
\(817\) 16.0000 0.559769
\(818\) 8.87006 0.310134
\(819\) 0 0
\(820\) −38.9706 −1.36091
\(821\) −10.6863 −0.372954 −0.186477 0.982459i \(-0.559707\pi\)
−0.186477 + 0.982459i \(0.559707\pi\)
\(822\) −5.85786 −0.204316
\(823\) 8.97056 0.312694 0.156347 0.987702i \(-0.450028\pi\)
0.156347 + 0.987702i \(0.450028\pi\)
\(824\) 7.11270 0.247783
\(825\) 13.3137 0.463524
\(826\) 0 0
\(827\) 47.6569 1.65719 0.828596 0.559848i \(-0.189140\pi\)
0.828596 + 0.559848i \(0.189140\pi\)
\(828\) 14.0000 0.486534
\(829\) 0.727922 0.0252818 0.0126409 0.999920i \(-0.495976\pi\)
0.0126409 + 0.999920i \(0.495976\pi\)
\(830\) 10.3431 0.359016
\(831\) 9.31371 0.323089
\(832\) 10.7868 0.373965
\(833\) 0 0
\(834\) −7.31371 −0.253253
\(835\) −67.5980 −2.33932
\(836\) 10.3431 0.357725
\(837\) −1.17157 −0.0404955
\(838\) −13.7401 −0.474644
\(839\) −50.8284 −1.75479 −0.877396 0.479767i \(-0.840721\pi\)
−0.877396 + 0.479767i \(0.840721\pi\)
\(840\) 0 0
\(841\) 17.6274 0.607842
\(842\) 6.88730 0.237352
\(843\) −0.485281 −0.0167140
\(844\) −23.7157 −0.816329
\(845\) 21.5563 0.741561
\(846\) 1.17157 0.0402795
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −8.48528 −0.291214
\(850\) 6.18377 0.212101
\(851\) 30.6274 1.04989
\(852\) 17.0294 0.583419
\(853\) 49.4975 1.69476 0.847381 0.530986i \(-0.178178\pi\)
0.847381 + 0.530986i \(0.178178\pi\)
\(854\) 0 0
\(855\) −9.65685 −0.330257
\(856\) 0.544156 0.0185989
\(857\) −15.4142 −0.526540 −0.263270 0.964722i \(-0.584801\pi\)
−0.263270 + 0.964722i \(0.584801\pi\)
\(858\) −2.14214 −0.0731313
\(859\) −57.4558 −1.96037 −0.980184 0.198089i \(-0.936527\pi\)
−0.980184 + 0.198089i \(0.936527\pi\)
\(860\) 35.3137 1.20419
\(861\) 0 0
\(862\) −11.1716 −0.380505
\(863\) 17.3137 0.589365 0.294683 0.955595i \(-0.404786\pi\)
0.294683 + 0.955595i \(0.404786\pi\)
\(864\) −4.41421 −0.150175
\(865\) 71.9411 2.44607
\(866\) 8.38478 0.284926
\(867\) 11.9706 0.406542
\(868\) 0 0
\(869\) −27.3137 −0.926554
\(870\) −9.65685 −0.327398
\(871\) 14.6274 0.495631
\(872\) 8.97056 0.303782
\(873\) −2.58579 −0.0875156
\(874\) −8.97056 −0.303434
\(875\) 0 0
\(876\) −25.4142 −0.858667
\(877\) −11.3137 −0.382037 −0.191018 0.981586i \(-0.561179\pi\)
−0.191018 + 0.981586i \(0.561179\pi\)
\(878\) −5.25483 −0.177342
\(879\) 16.5858 0.559425
\(880\) 20.4853 0.690559
\(881\) −21.7574 −0.733024 −0.366512 0.930413i \(-0.619448\pi\)
−0.366512 + 0.930413i \(0.619448\pi\)
\(882\) 0 0
\(883\) −4.68629 −0.157706 −0.0788531 0.996886i \(-0.525126\pi\)
−0.0788531 + 0.996886i \(0.525126\pi\)
\(884\) 10.6030 0.356619
\(885\) 4.00000 0.134459
\(886\) 14.4853 0.486643
\(887\) 2.82843 0.0949693 0.0474846 0.998872i \(-0.484879\pi\)
0.0474846 + 0.998872i \(0.484879\pi\)
\(888\) −6.34315 −0.212862
\(889\) 0 0
\(890\) −20.1421 −0.675166
\(891\) −2.00000 −0.0670025
\(892\) 45.6569 1.52870
\(893\) 8.00000 0.267710
\(894\) −7.17157 −0.239853
\(895\) 67.1127 2.24333
\(896\) 0 0
\(897\) −19.7990 −0.661069
\(898\) −2.20101 −0.0734487
\(899\) −8.00000 −0.266815
\(900\) −12.1716 −0.405719
\(901\) −4.48528 −0.149426
\(902\) 5.17157 0.172195
\(903\) 0 0
\(904\) 8.42641 0.280258
\(905\) −8.82843 −0.293467
\(906\) −4.97056 −0.165136
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 43.5147 1.44409
\(909\) −2.92893 −0.0971465
\(910\) 0 0
\(911\) −1.02944 −0.0341068 −0.0170534 0.999855i \(-0.505429\pi\)
−0.0170534 + 0.999855i \(0.505429\pi\)
\(912\) −8.48528 −0.280976
\(913\) 14.6274 0.484097
\(914\) −7.45584 −0.246617
\(915\) −41.7990 −1.38183
\(916\) −0.443651 −0.0146586
\(917\) 0 0
\(918\) −0.928932 −0.0306593
\(919\) −8.28427 −0.273273 −0.136636 0.990621i \(-0.543629\pi\)
−0.136636 + 0.990621i \(0.543629\pi\)
\(920\) −41.4558 −1.36676
\(921\) 30.1421 0.993217
\(922\) 6.87006 0.226253
\(923\) −24.0833 −0.792710
\(924\) 0 0
\(925\) −26.6274 −0.875504
\(926\) −11.0294 −0.362450
\(927\) −4.48528 −0.147316
\(928\) −30.1421 −0.989464
\(929\) 39.2132 1.28654 0.643272 0.765638i \(-0.277577\pi\)
0.643272 + 0.765638i \(0.277577\pi\)
\(930\) 1.65685 0.0543304
\(931\) 0 0
\(932\) 11.2304 0.367866
\(933\) 6.14214 0.201084
\(934\) −0.0832611 −0.00272439
\(935\) 15.3137 0.500812
\(936\) 4.10051 0.134029
\(937\) −30.5858 −0.999194 −0.499597 0.866258i \(-0.666519\pi\)
−0.499597 + 0.866258i \(0.666519\pi\)
\(938\) 0 0
\(939\) 1.89949 0.0619877
\(940\) 17.6569 0.575903
\(941\) −35.2132 −1.14792 −0.573959 0.818884i \(-0.694593\pi\)
−0.573959 + 0.818884i \(0.694593\pi\)
\(942\) 4.87006 0.158675
\(943\) 47.7990 1.55655
\(944\) 3.51472 0.114394
\(945\) 0 0
\(946\) −4.68629 −0.152364
\(947\) 30.6863 0.997170 0.498585 0.866841i \(-0.333853\pi\)
0.498585 + 0.866841i \(0.333853\pi\)
\(948\) 24.9706 0.811006
\(949\) 35.9411 1.16670
\(950\) 7.79899 0.253033
\(951\) −10.0000 −0.324272
\(952\) 0 0
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) −0.828427 −0.0268213
\(955\) 61.4558 1.98866
\(956\) 28.6274 0.925877
\(957\) −13.6569 −0.441463
\(958\) −0.769553 −0.0248631
\(959\) 0 0
\(960\) −14.2426 −0.459679
\(961\) −29.6274 −0.955723
\(962\) 4.28427 0.138130
\(963\) −0.343146 −0.0110577
\(964\) −29.6985 −0.956524
\(965\) −18.1421 −0.584016
\(966\) 0 0
\(967\) 33.6569 1.08233 0.541166 0.840916i \(-0.317983\pi\)
0.541166 + 0.840916i \(0.317983\pi\)
\(968\) 11.1005 0.356784
\(969\) −6.34315 −0.203771
\(970\) 3.65685 0.117415
\(971\) 50.6274 1.62471 0.812356 0.583162i \(-0.198185\pi\)
0.812356 + 0.583162i \(0.198185\pi\)
\(972\) 1.82843 0.0586468
\(973\) 0 0
\(974\) 11.0294 0.353406
\(975\) 17.2132 0.551264
\(976\) −36.7279 −1.17563
\(977\) −21.1716 −0.677339 −0.338669 0.940905i \(-0.609977\pi\)
−0.338669 + 0.940905i \(0.609977\pi\)
\(978\) 4.68629 0.149851
\(979\) −28.4853 −0.910394
\(980\) 0 0
\(981\) −5.65685 −0.180609
\(982\) 2.08326 0.0664795
\(983\) 53.2548 1.69857 0.849283 0.527938i \(-0.177035\pi\)
0.849283 + 0.527938i \(0.177035\pi\)
\(984\) −9.89949 −0.315584
\(985\) −6.82843 −0.217572
\(986\) −6.34315 −0.202007
\(987\) 0 0
\(988\) 13.3726 0.425439
\(989\) −43.3137 −1.37730
\(990\) 2.82843 0.0898933
\(991\) −12.9706 −0.412024 −0.206012 0.978550i \(-0.566049\pi\)
−0.206012 + 0.978550i \(0.566049\pi\)
\(992\) 5.17157 0.164198
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 73.9411 2.34409
\(996\) −13.3726 −0.423727
\(997\) −26.3848 −0.835614 −0.417807 0.908536i \(-0.637201\pi\)
−0.417807 + 0.908536i \(0.637201\pi\)
\(998\) 1.37258 0.0434484
\(999\) 4.00000 0.126554
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 147.2.a.d.1.2 2
3.2 odd 2 441.2.a.j.1.1 2
4.3 odd 2 2352.2.a.be.1.1 2
5.4 even 2 3675.2.a.bf.1.1 2
7.2 even 3 147.2.e.e.67.1 4
7.3 odd 6 147.2.e.d.79.1 4
7.4 even 3 147.2.e.e.79.1 4
7.5 odd 6 147.2.e.d.67.1 4
7.6 odd 2 147.2.a.e.1.2 yes 2
8.3 odd 2 9408.2.a.dq.1.2 2
8.5 even 2 9408.2.a.ef.1.2 2
12.11 even 2 7056.2.a.cv.1.2 2
21.2 odd 6 441.2.e.f.361.2 4
21.5 even 6 441.2.e.g.361.2 4
21.11 odd 6 441.2.e.f.226.2 4
21.17 even 6 441.2.e.g.226.2 4
21.20 even 2 441.2.a.i.1.1 2
28.3 even 6 2352.2.q.bd.961.1 4
28.11 odd 6 2352.2.q.bb.961.2 4
28.19 even 6 2352.2.q.bd.1537.1 4
28.23 odd 6 2352.2.q.bb.1537.2 4
28.27 even 2 2352.2.a.bc.1.2 2
35.34 odd 2 3675.2.a.bd.1.1 2
56.13 odd 2 9408.2.a.di.1.1 2
56.27 even 2 9408.2.a.dt.1.1 2
84.83 odd 2 7056.2.a.cf.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.a.d.1.2 2 1.1 even 1 trivial
147.2.a.e.1.2 yes 2 7.6 odd 2
147.2.e.d.67.1 4 7.5 odd 6
147.2.e.d.79.1 4 7.3 odd 6
147.2.e.e.67.1 4 7.2 even 3
147.2.e.e.79.1 4 7.4 even 3
441.2.a.i.1.1 2 21.20 even 2
441.2.a.j.1.1 2 3.2 odd 2
441.2.e.f.226.2 4 21.11 odd 6
441.2.e.f.361.2 4 21.2 odd 6
441.2.e.g.226.2 4 21.17 even 6
441.2.e.g.361.2 4 21.5 even 6
2352.2.a.bc.1.2 2 28.27 even 2
2352.2.a.be.1.1 2 4.3 odd 2
2352.2.q.bb.961.2 4 28.11 odd 6
2352.2.q.bb.1537.2 4 28.23 odd 6
2352.2.q.bd.961.1 4 28.3 even 6
2352.2.q.bd.1537.1 4 28.19 even 6
3675.2.a.bd.1.1 2 35.34 odd 2
3675.2.a.bf.1.1 2 5.4 even 2
7056.2.a.cf.1.1 2 84.83 odd 2
7056.2.a.cv.1.2 2 12.11 even 2
9408.2.a.di.1.1 2 56.13 odd 2
9408.2.a.dq.1.2 2 8.3 odd 2
9408.2.a.dt.1.1 2 56.27 even 2
9408.2.a.ef.1.2 2 8.5 even 2