Properties

Label 147.2.a.e.1.2
Level $147$
Weight $2$
Character 147.1
Self dual yes
Analytic conductor $1.174$
Analytic rank $0$
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: \(0\)
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 1.2
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 \(-0.223508\pi\)
0.763441 + 0.645877i \(0.223508\pi\)
\(6\) 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 \(-0.383260\pi\)
0.358584 + 0.933497i \(0.383260\pi\)
\(14\) 0 0
\(15\) 3.41421 0.881546
\(16\) 3.00000 0.750000
\(17\) −2.24264 −0.543920 −0.271960 0.962309i \(-0.587672\pi\)
−0.271960 + 0.962309i \(0.587672\pi\)
\(18\) 0.414214 0.0976311
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) −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 \(-0.533552\pi\)
−0.105210 + 0.994450i \(0.533552\pi\)
\(32\) 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 \(-0.337920\pi\)
0.487468 + 0.873141i \(0.337920\pi\)
\(42\) 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 \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 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 \(-0.524299\pi\)
−0.0762629 + 0.997088i \(0.524299\pi\)
\(60\) −6.24264 −0.805921
\(61\) 12.2426 1.56751 0.783755 0.621070i \(-0.213302\pi\)
0.783755 + 0.621070i \(0.213302\pi\)
\(62\) −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 \(-0.197611\pi\)
0.813406 + 0.581696i \(0.197611\pi\)
\(74\) −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 \(-0.368527\pi\)
0.401392 + 0.915906i \(0.368527\pi\)
\(84\) 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 \(-0.772294\pi\)
−0.754858 + 0.655888i \(0.772294\pi\)
\(90\) 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 \(-0.458093\pi\)
0.131273 + 0.991346i \(0.458093\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −12.1716 −1.21716
\(101\) 2.92893 0.291440 0.145720 0.989326i \(-0.453450\pi\)
0.145720 + 0.989326i \(0.453450\pi\)
\(102\) −0.928932 −0.0919780
\(103\) 4.48528 0.441948 0.220974 0.975280i \(-0.429076\pi\)
0.220974 + 0.975280i \(0.429076\pi\)
\(104\) −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 \(-0.733270\pi\)
−0.668982 + 0.743278i \(0.733270\pi\)
\(132\) 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 \(-0.769378\pi\)
−0.748817 + 0.662776i \(0.769378\pi\)
\(140\) 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 \(-0.344553\pi\)
0.469170 + 0.883108i \(0.344553\pi\)
\(158\) 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 \(-0.777779\pi\)
−0.766046 + 0.642786i \(0.777779\pi\)
\(168\) 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 \(-0.204301\pi\)
0.801002 + 0.598662i \(0.204301\pi\)
\(174\) −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 \(-0.530637\pi\)
−0.0961000 + 0.995372i \(0.530637\pi\)
\(182\) 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 \(-0.221447\pi\)
0.767607 + 0.640921i \(0.221447\pi\)
\(200\) −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 \(-0.184844\pi\)
0.836076 + 0.548613i \(0.184844\pi\)
\(224\) 0 0
\(225\) 6.65685 0.443790
\(226\) −2.20101 −0.146409
\(227\) 23.7990 1.57959 0.789797 0.613368i \(-0.210186\pi\)
0.789797 + 0.613368i \(0.210186\pi\)
\(228\) 5.17157 0.342496
\(229\) −0.242641 −0.0160341 −0.00801707 0.999968i \(-0.502552\pi\)
−0.00801707 + 0.999968i \(0.502552\pi\)
\(230\) −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 \(-0.675240\pi\)
−0.523140 + 0.852247i \(0.675240\pi\)
\(242\) −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 \(-0.628920\pi\)
−0.394032 + 0.919097i \(0.628920\pi\)
\(252\) 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 \(-0.242302\pi\)
0.724000 + 0.689800i \(0.242302\pi\)
\(258\) 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 \(-0.648216\pi\)
−0.448989 + 0.893537i \(0.648216\pi\)
\(270\) 1.41421 0.0860663
\(271\) −10.1421 −0.616091 −0.308045 0.951372i \(-0.599675\pi\)
−0.308045 + 0.951372i \(0.599675\pi\)
\(272\) −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 \(-0.581154\pi\)
−0.252199 + 0.967675i \(0.581154\pi\)
\(284\) −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 \(-0.339010\pi\)
0.484476 + 0.874805i \(0.339010\pi\)
\(294\) 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 \(-0.170369\pi\)
0.860151 + 0.510039i \(0.170369\pi\)
\(308\) 0 0
\(309\) 4.48528 0.255159
\(310\) −1.65685 −0.0941030
\(311\) 6.14214 0.348289 0.174144 0.984720i \(-0.444284\pi\)
0.174144 + 0.984720i \(0.444284\pi\)
\(312\) −4.10051 −0.232145
\(313\) 1.89949 0.107366 0.0536829 0.998558i \(-0.482904\pi\)
0.0536829 + 0.998558i \(0.482904\pi\)
\(314\) 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 \(-0.585357\pi\)
−0.264954 + 0.964261i \(0.585357\pi\)
\(350\) 0 0
\(351\) 2.58579 0.138019
\(352\) −8.82843 −0.470557
\(353\) −14.7279 −0.783888 −0.391944 0.919989i \(-0.628197\pi\)
−0.391944 + 0.919989i \(0.628197\pi\)
\(354\) −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 \(-0.527564\pi\)
−0.0864871 + 0.996253i \(0.527564\pi\)
\(368\) −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 \(-0.655258\pi\)
−0.468645 + 0.883386i \(0.655258\pi\)
\(384\) −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 \(-0.519060\pi\)
−0.0598442 + 0.998208i \(0.519060\pi\)
\(398\) 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 \(-0.677595\pi\)
−0.529432 + 0.848352i \(0.677595\pi\)
\(410\) 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 \(-0.199321\pi\)
0.810269 + 0.586059i \(0.199321\pi\)
\(420\) 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 \(-0.661690\pi\)
−0.486400 + 0.873736i \(0.661690\pi\)
\(434\) 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 \(-0.402098\pi\)
0.302742 + 0.953073i \(0.402098\pi\)
\(440\) 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 \(-0.626226\pi\)
−0.386239 + 0.922399i \(0.626226\pi\)
\(462\) 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 \(-0.498520\pi\)
0.00465082 + 0.999989i \(0.498520\pi\)
\(468\) −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 \(-0.486486\pi\)
0.0424440 + 0.999099i \(0.486486\pi\)
\(480\) 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 \(-0.460703\pi\)
0.123140 + 0.992389i \(0.460703\pi\)
\(510\) −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 \(-0.782634\pi\)
−0.775762 + 0.631025i \(0.782634\pi\)
\(522\) −2.82843 −0.123797
\(523\) −25.6569 −1.12190 −0.560948 0.827851i \(-0.689563\pi\)
−0.560948 + 0.827851i \(0.689563\pi\)
\(524\) 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 \(-0.507859\pi\)
−0.0246880 + 0.999695i \(0.507859\pi\)
\(564\) 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 \(-0.749335\pi\)
−0.705627 + 0.708583i \(0.749335\pi\)
\(578\) −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 \(-0.656148\pi\)
−0.471115 + 0.882072i \(0.656148\pi\)
\(588\) 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 \(-0.545440\pi\)
−0.142269 + 0.989828i \(0.545440\pi\)
\(594\) −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 \(-0.599453\pi\)
−0.307381 + 0.951587i \(0.599453\pi\)
\(602\) 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 \(-0.621418\pi\)
−0.372263 + 0.928127i \(0.621418\pi\)
\(608\) −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 \(-0.302187\pi\)
0.582213 + 0.813037i \(0.302187\pi\)
\(620\) 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 \(-0.737509\pi\)
−0.678822 + 0.734302i \(0.737509\pi\)
\(644\) 0 0
\(645\) 19.3137 0.760477
\(646\) 2.62742 0.103374
\(647\) 26.8284 1.05473 0.527367 0.849638i \(-0.323179\pi\)
0.527367 + 0.849638i \(0.323179\pi\)
\(648\) −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 \(-0.651476\pi\)
−0.458118 + 0.888891i \(0.651476\pi\)
\(662\) −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 \(-0.293703\pi\)
0.603673 + 0.797232i \(0.293703\pi\)
\(678\) −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 \(-0.504155\pi\)
−0.0130539 + 0.999915i \(0.504155\pi\)
\(692\) −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 \(-0.214164\pi\)
0.782070 + 0.623191i \(0.214164\pi\)
\(720\) 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 \(-0.425628\pi\)
0.231527 + 0.972829i \(0.425628\pi\)
\(728\) 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 \(-0.129941\pi\)
0.917828 + 0.396979i \(0.129941\pi\)
\(734\) −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 \(-0.609007\pi\)
−0.335800 + 0.941933i \(0.609007\pi\)
\(762\) −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 \(-0.320132\pi\)
0.535477 + 0.844550i \(0.320132\pi\)
\(770\) 0 0
\(771\) 23.2132 0.836003
\(772\) −9.71573 −0.349677
\(773\) −9.55635 −0.343718 −0.171859 0.985122i \(-0.554977\pi\)
−0.171859 + 0.985122i \(0.554977\pi\)
\(774\) 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 \(-0.354984\pi\)
0.439986 + 0.898005i \(0.354984\pi\)
\(788\) −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 \(-0.547445\pi\)
−0.148502 + 0.988912i \(0.547445\pi\)
\(798\) 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 \(-0.600332\pi\)
−0.310008 + 0.950734i \(0.600332\pi\)
\(812\) 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 \(-0.504024\pi\)
−0.0126409 + 0.999920i \(0.504024\pi\)
\(830\) 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 \(-0.159279\pi\)
0.877396 + 0.479767i \(0.159279\pi\)
\(840\) 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 \(-0.821822\pi\)
−0.847381 + 0.530986i \(0.821822\pi\)
\(854\) 0 0
\(855\) −9.65685 −0.330257
\(856\) 0.544156 0.0185989
\(857\) 15.4142 0.526540 0.263270 0.964722i \(-0.415199\pi\)
0.263270 + 0.964722i \(0.415199\pi\)
\(858\) −2.14214 −0.0731313
\(859\) 57.4558 1.96037 0.980184 0.198089i \(-0.0634735\pi\)
0.980184 + 0.198089i \(0.0634735\pi\)
\(860\) −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 \(-0.380552\pi\)
0.366512 + 0.930413i \(0.380552\pi\)
\(882\) 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 \(-0.515121\pi\)
−0.0474846 + 0.998872i \(0.515121\pi\)
\(888\) 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 \(-0.722423\pi\)
−0.643272 + 0.765638i \(0.722423\pi\)
\(930\) −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 \(-0.333481\pi\)
0.499597 + 0.866258i \(0.333481\pi\)
\(938\) 0 0
\(939\) 1.89949 0.0619877
\(940\) 17.6569 0.575903
\(941\) 35.2132 1.14792 0.573959 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(942\) 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 \(-0.801815\pi\)
−0.812356 + 0.583162i \(0.801815\pi\)
\(972\) −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 \(-0.822965\pi\)
−0.849283 + 0.527938i \(0.822965\pi\)
\(984\) −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 \(-0.362799\pi\)
0.417807 + 0.908536i \(0.362799\pi\)
\(998\) 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.e.1.2 yes 2
3.2 odd 2 441.2.a.i.1.1 2
4.3 odd 2 2352.2.a.bc.1.2 2
5.4 even 2 3675.2.a.bd.1.1 2
7.2 even 3 147.2.e.d.67.1 4
7.3 odd 6 147.2.e.e.79.1 4
7.4 even 3 147.2.e.d.79.1 4
7.5 odd 6 147.2.e.e.67.1 4
7.6 odd 2 147.2.a.d.1.2 2
8.3 odd 2 9408.2.a.dt.1.1 2
8.5 even 2 9408.2.a.di.1.1 2
12.11 even 2 7056.2.a.cf.1.1 2
21.2 odd 6 441.2.e.g.361.2 4
21.5 even 6 441.2.e.f.361.2 4
21.11 odd 6 441.2.e.g.226.2 4
21.17 even 6 441.2.e.f.226.2 4
21.20 even 2 441.2.a.j.1.1 2
28.3 even 6 2352.2.q.bb.961.2 4
28.11 odd 6 2352.2.q.bd.961.1 4
28.19 even 6 2352.2.q.bb.1537.2 4
28.23 odd 6 2352.2.q.bd.1537.1 4
28.27 even 2 2352.2.a.be.1.1 2
35.34 odd 2 3675.2.a.bf.1.1 2
56.13 odd 2 9408.2.a.ef.1.2 2
56.27 even 2 9408.2.a.dq.1.2 2
84.83 odd 2 7056.2.a.cv.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.a.d.1.2 2 7.6 odd 2
147.2.a.e.1.2 yes 2 1.1 even 1 trivial
147.2.e.d.67.1 4 7.2 even 3
147.2.e.d.79.1 4 7.4 even 3
147.2.e.e.67.1 4 7.5 odd 6
147.2.e.e.79.1 4 7.3 odd 6
441.2.a.i.1.1 2 3.2 odd 2
441.2.a.j.1.1 2 21.20 even 2
441.2.e.f.226.2 4 21.17 even 6
441.2.e.f.361.2 4 21.5 even 6
441.2.e.g.226.2 4 21.11 odd 6
441.2.e.g.361.2 4 21.2 odd 6
2352.2.a.bc.1.2 2 4.3 odd 2
2352.2.a.be.1.1 2 28.27 even 2
2352.2.q.bb.961.2 4 28.3 even 6
2352.2.q.bb.1537.2 4 28.19 even 6
2352.2.q.bd.961.1 4 28.11 odd 6
2352.2.q.bd.1537.1 4 28.23 odd 6
3675.2.a.bd.1.1 2 5.4 even 2
3675.2.a.bf.1.1 2 35.34 odd 2
7056.2.a.cf.1.1 2 12.11 even 2
7056.2.a.cv.1.2 2 84.83 odd 2
9408.2.a.di.1.1 2 8.5 even 2
9408.2.a.dq.1.2 2 56.27 even 2
9408.2.a.dt.1.1 2 8.3 odd 2
9408.2.a.ef.1.2 2 56.13 odd 2