Properties

Label 3675.2.a.bf.1.1
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.3450227428\)
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: no (minimal twist has level 147)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3675.1

$q$-expansion

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

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 \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.82843 −0.914214
\(5\) 0 0
\(6\) −0.414214 −0.169102
\(7\) 0 0
\(8\) 1.58579 0.560660
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −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
\(15\) 0 0
\(16\) 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 \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.828427 0.176621
\(23\) 7.65685 1.59656 0.798282 0.602284i \(-0.205742\pi\)
0.798282 + 0.602284i \(0.205742\pi\)
\(24\) 1.58579 0.323697
\(25\) 0 0
\(26\) −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 \(0.718590\pi\)
\(30\) 0 0
\(31\) 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 \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −1.17157 −0.190054
\(39\) 2.58579 0.414057
\(40\) 0 0
\(41\) −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 \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) 3.65685 0.551292
\(45\) 0 0
\(46\) −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
\(50\) 0 0
\(51\) −2.24264 −0.314033
\(52\) −4.72792 −0.655645
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −0.414214 −0.0563673
\(55\) 0 0
\(56\) 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 \(0.475701\pi\)
\(60\) 0 0
\(61\) −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
\(65\) 0 0
\(66\) 0.828427 0.101972
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) 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
\(75\) 0 0
\(76\) −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 \(0.221124\pi\)
\(80\) 0 0
\(81\) 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
\(85\) 0 0
\(86\) 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 \(0.227706\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −14.0000 −1.45960
\(93\) 1.17157 0.121486
\(94\) 1.17157 0.120839
\(95\) 0 0
\(96\) −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
\(100\) 0 0
\(101\) −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 \(-0.429076\pi\)
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 \(-0.494720\pi\)
0.0165866 + 0.999862i \(0.494720\pi\)
\(108\) −1.82843 −0.175940
\(109\) −5.65685 −0.541828 −0.270914 0.962604i \(-0.587326\pi\)
−0.270914 + 0.962604i \(0.587326\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 5.31371 0.499872 0.249936 0.968262i \(-0.419590\pi\)
0.249936 + 0.968262i \(0.419590\pi\)
\(114\) −1.17157 −0.109728
\(115\) 0 0
\(116\) 12.4853 1.15923
\(117\) 2.58579 0.239056
\(118\) −0.485281 −0.0446738
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 5.07107 0.459113
\(123\) −6.24264 −0.562880
\(124\) −2.14214 −0.192369
\(125\) 0 0
\(126\) 0 0
\(127\) 1.65685 0.147022 0.0735110 0.997294i \(-0.476580\pi\)
0.0735110 + 0.997294i \(0.476580\pi\)
\(128\) 10.5563 0.933058
\(129\) −5.65685 −0.498058
\(130\) 0 0
\(131\) 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
\(135\) 0 0
\(136\) −3.55635 −0.304954
\(137\) −14.1421 −1.20824 −0.604122 0.796892i \(-0.706476\pi\)
−0.604122 + 0.796892i \(0.706476\pi\)
\(138\) −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
\(145\) 0 0
\(146\) −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 \(0.249058\pi\)
\(150\) 0 0
\(151\) 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
\(155\) 0 0
\(156\) −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
\(160\) 0 0
\(161\) 0 0
\(162\) −0.414214 −0.0325437
\(163\) 11.3137 0.886158 0.443079 0.896483i \(-0.353886\pi\)
0.443079 + 0.896483i \(0.353886\pi\)
\(164\) 11.4142 0.891300
\(165\) 0 0
\(166\) −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
\(170\) 0 0
\(171\) 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 \(0.762635\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) −0.485281 −0.0355826
\(187\) 4.48528 0.327996
\(188\) 5.17157 0.377176
\(189\) 0 0
\(190\) 0 0
\(191\) −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 \(-0.561252\pi\)
−0.191245 + 0.981542i \(0.561252\pi\)
\(194\) −1.07107 −0.0768982
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0.828427 0.0588738
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) 0 0
\(201\) 5.65685 0.399004
\(202\) 1.21320 0.0853607
\(203\) 0 0
\(204\) 4.10051 0.287093
\(205\) 0 0
\(206\) −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
\(215\) 0 0
\(216\) 1.58579 0.107899
\(217\) 0 0
\(218\) 2.34315 0.158698
\(219\) 13.8995 0.939241
\(220\) 0 0
\(221\) −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
\(225\) 0 0
\(226\) −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 \(-0.497448\pi\)
0.00801707 + 0.999968i \(0.497448\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −10.8284 −0.710921
\(233\) 6.14214 0.402385 0.201192 0.979552i \(-0.435518\pi\)
0.201192 + 0.979552i \(0.435518\pi\)
\(234\) −1.07107 −0.0700179
\(235\) 0 0
\(236\) −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 \(0.669016\pi\)
\(240\) 0 0
\(241\) 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
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) 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
\(260\) 0 0
\(261\) −6.82843 −0.422669
\(262\) −6.34315 −0.391881
\(263\) −5.31371 −0.327657 −0.163829 0.986489i \(-0.552384\pi\)
−0.163829 + 0.986489i \(0.552384\pi\)
\(264\) −3.17157 −0.195197
\(265\) 0 0
\(266\) 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 \(0.351784\pi\)
\(270\) 0 0
\(271\) 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
\(275\) 0 0
\(276\) −14.0000 −0.842701
\(277\) 9.31371 0.559607 0.279803 0.960057i \(-0.409731\pi\)
0.279803 + 0.960057i \(0.409731\pi\)
\(278\) −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
\(285\) 0 0
\(286\) 2.14214 0.126667
\(287\) 0 0
\(288\) −4.41421 −0.260110
\(289\) −11.9706 −0.704151
\(290\) 0 0
\(291\) 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
\(295\) 0 0
\(296\) 6.34315 0.368688
\(297\) −2.00000 −0.116052
\(298\) −7.17157 −0.415438
\(299\) 19.7990 1.14501
\(300\) 0 0
\(301\) 0 0
\(302\) −4.97056 −0.286024
\(303\) −2.92893 −0.168263
\(304\) 8.48528 0.486664
\(305\) 0 0
\(306\) 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
\(310\) 0 0
\(311\) −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 \(-0.482904\pi\)
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 \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) −0.828427 −0.0464559
\(319\) 13.6569 0.764637
\(320\) 0 0
\(321\) 0.343146 0.0191525
\(322\) 0 0
\(323\) −6.34315 −0.352942
\(324\) −1.82843 −0.101579
\(325\) 0 0
\(326\) −4.68629 −0.259550
\(327\) −5.65685 −0.312825
\(328\) −9.89949 −0.546608
\(329\) 0 0
\(330\) 0 0
\(331\) −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
\(335\) 0 0
\(336\) 0 0
\(337\) 29.6569 1.61551 0.807756 0.589517i \(-0.200682\pi\)
0.807756 + 0.589517i \(0.200682\pi\)
\(338\) 2.61522 0.142249
\(339\) 5.31371 0.288601
\(340\) 0 0
\(341\) −2.34315 −0.126888
\(342\) −1.17157 −0.0633514
\(343\) 0 0
\(344\) −8.97056 −0.483660
\(345\) 0 0
\(346\) −8.72792 −0.469216
\(347\) −33.3137 −1.78837 −0.894187 0.447694i \(-0.852245\pi\)
−0.894187 + 0.447694i \(0.852245\pi\)
\(348\) 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 \(-0.628197\pi\)
−0.391944 + 0.919989i \(0.628197\pi\)
\(354\) −0.485281 −0.0257924
\(355\) 0 0
\(356\) −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 \(0.502882\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) −1.07107 −0.0562941
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 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
\(370\) 0 0
\(371\) 0 0
\(372\) −2.14214 −0.111065
\(373\) 10.6863 0.553315 0.276658 0.960969i \(-0.410773\pi\)
0.276658 + 0.960969i \(0.410773\pi\)
\(374\) −1.85786 −0.0960679
\(375\) 0 0
\(376\) −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 \(0.428385\pi\)
\(380\) 0 0
\(381\) 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 \(0.652122\pi\)
\(390\) 0 0
\(391\) −17.1716 −0.868404
\(392\) 0 0
\(393\) 15.3137 0.772474
\(394\) 0.828427 0.0417356
\(395\) 0 0
\(396\) 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
\(400\) 0 0
\(401\) −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
\(405\) 0 0
\(406\) 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 \(0.322405\pi\)
\(410\) 0 0
\(411\) −14.1421 −0.697580
\(412\) −8.20101 −0.404035
\(413\) 0 0
\(414\) −3.17157 −0.155874
\(415\) 0 0
\(416\) −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
\(425\) 0 0
\(426\) −3.85786 −0.186914
\(427\) 0 0
\(428\) −0.627417 −0.0303273
\(429\) −5.17157 −0.249686
\(430\) 0 0
\(431\) −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
\(435\) 0 0
\(436\) 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 \(0.597902\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.40202 0.114252
\(443\) −34.9706 −1.66150 −0.830751 0.556645i \(-0.812089\pi\)
−0.830751 + 0.556645i \(0.812089\pi\)
\(444\) −7.31371 −0.347093
\(445\) 0 0
\(446\) −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 \(0.540017\pi\)
\(450\) 0 0
\(451\) 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 \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −0.100505 −0.00469629
\(459\) −2.24264 −0.104678
\(460\) 0 0
\(461\) 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 \(-0.287643\pi\)
0.618741 + 0.785595i \(0.287643\pi\)
\(464\) −20.4853 −0.951005
\(465\) 0 0
\(466\) −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
\(470\) 0 0
\(471\) 11.7574 0.541751
\(472\) 1.85786 0.0855151
\(473\) 11.3137 0.520205
\(474\) −5.65685 −0.259828
\(475\) 0 0
\(476\) 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 \(0.513514\pi\)
\(480\) 0 0
\(481\) 10.3431 0.471607
\(482\) −6.72792 −0.306448
\(483\) 0 0
\(484\) 12.7990 0.581772
\(485\) 0 0
\(486\) −0.414214 −0.0187891
\(487\) −26.6274 −1.20660 −0.603302 0.797513i \(-0.706149\pi\)
−0.603302 + 0.797513i \(0.706149\pi\)
\(488\) −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
\(495\) 0 0
\(496\) 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 \(0.476369\pi\)
\(500\) 0 0
\(501\) −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
\(505\) 0 0
\(506\) 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 \(0.539297\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −22.7574 −1.00574
\(513\) 2.82843 0.124878
\(514\) −9.61522 −0.424109
\(515\) 0 0
\(516\) 10.3431 0.455332
\(517\) 5.65685 0.248788
\(518\) 0 0
\(519\) 21.0711 0.924917
\(520\) 0 0
\(521\) 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 \(-0.689563\pi\)
−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
\(530\) 0 0
\(531\) 1.17157 0.0508419
\(532\) 0 0
\(533\) −16.1421 −0.699194
\(534\) −5.89949 −0.255296
\(535\) 0 0
\(536\) 8.97056 0.387469
\(537\) −19.6569 −0.848256
\(538\) −6.10051 −0.263011
\(539\) 0 0
\(540\) 0 0
\(541\) 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
\(545\) 0 0
\(546\) 0 0
\(547\) 36.9706 1.58075 0.790374 0.612625i \(-0.209886\pi\)
0.790374 + 0.612625i \(0.209886\pi\)
\(548\) 25.8579 1.10459
\(549\) −12.2426 −0.522503
\(550\) 0 0
\(551\) −19.3137 −0.822792
\(552\) 12.1421 0.516804
\(553\) 0 0
\(554\) −3.85786 −0.163905
\(555\) 0 0
\(556\) −32.2843 −1.36916
\(557\) −26.0000 −1.10166 −0.550828 0.834619i \(-0.685688\pi\)
−0.550828 + 0.834619i \(0.685688\pi\)
\(558\) −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
\(565\) 0 0
\(566\) 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 \(0.612307\pi\)
\(570\) 0 0
\(571\) 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
\(575\) 0 0
\(576\) −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
\(580\) 0 0
\(581\) 0 0
\(582\) −1.07107 −0.0443972
\(583\) −4.00000 −0.165663
\(584\) 22.0416 0.912089
\(585\) 0 0
\(586\) −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
\(590\) 0 0
\(591\) −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 \(0.513009\pi\)
\(600\) 0 0
\(601\) 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
\(605\) 0 0
\(606\) 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
\(610\) 0 0
\(611\) −7.31371 −0.295881
\(612\) 4.10051 0.165753
\(613\) −4.68629 −0.189278 −0.0946388 0.995512i \(-0.530170\pi\)
−0.0946388 + 0.995512i \(0.530170\pi\)
\(614\) −12.4853 −0.503865
\(615\) 0 0
\(616\) 0 0
\(617\) 24.4853 0.985740 0.492870 0.870103i \(-0.335948\pi\)
0.492870 + 0.870103i \(0.335948\pi\)
\(618\) −1.85786 −0.0747343
\(619\) −28.9706 −1.16443 −0.582213 0.813037i \(-0.697813\pi\)
−0.582213 + 0.813037i \(0.697813\pi\)
\(620\) 0 0
\(621\) 7.65685 0.307259
\(622\) 2.54416 0.102011
\(623\) 0 0
\(624\) 7.75736 0.310543
\(625\) 0 0
\(626\) −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
\(635\) 0 0
\(636\) −3.65685 −0.145004
\(637\) 0 0
\(638\) −5.65685 −0.223957
\(639\) 9.31371 0.368445
\(640\) 0 0
\(641\) −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
\(645\) 0 0
\(646\) 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
\(650\) 0 0
\(651\) 0 0
\(652\) −20.6863 −0.810138
\(653\) −36.4853 −1.42778 −0.713890 0.700258i \(-0.753068\pi\)
−0.713890 + 0.700258i \(0.753068\pi\)
\(654\) 2.34315 0.0916242
\(655\) 0 0
\(656\) −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 \(0.441935\pi\)
\(660\) 0 0
\(661\) 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
\(670\) 0 0
\(671\) 24.4853 0.945244
\(672\) 0 0
\(673\) −23.3137 −0.898677 −0.449339 0.893361i \(-0.648340\pi\)
−0.449339 + 0.893361i \(0.648340\pi\)
\(674\) −12.2843 −0.473172
\(675\) 0 0
\(676\) 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
\(680\) 0 0
\(681\) 23.7990 0.911979
\(682\) 0.970563 0.0371648
\(683\) 19.6569 0.752149 0.376074 0.926590i \(-0.377274\pi\)
0.376074 + 0.926590i \(0.377274\pi\)
\(684\) −5.17157 −0.197740
\(685\) 0 0
\(686\) 0 0
\(687\) 0.242641 0.00925732
\(688\) −16.9706 −0.646997
\(689\) 5.17157 0.197021
\(690\) 0 0
\(691\) 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
\(695\) 0 0
\(696\) −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
\(705\) 0 0
\(706\) 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 \(0.738603\pi\)
\(710\) 0 0
\(711\) 13.6569 0.512172
\(712\) 22.5858 0.846438
\(713\) 8.97056 0.335950
\(714\) 0 0
\(715\) 0 0
\(716\) 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 \(0.785836\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 4.55635 0.169570
\(723\) 16.2426 0.604070
\(724\) −4.72792 −0.175712
\(725\) 0 0
\(726\) 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
\(730\) 0 0
\(731\) 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 \(0.472530\pi\)
\(740\) 0 0
\(741\) 7.31371 0.268676
\(742\) 0 0
\(743\) 50.9706 1.86993 0.934964 0.354742i \(-0.115431\pi\)
0.934964 + 0.354742i \(0.115431\pi\)
\(744\) 1.85786 0.0681126
\(745\) 0 0
\(746\) −4.42641 −0.162062
\(747\) 7.31371 0.267595
\(748\) −8.20101 −0.299859
\(749\) 0 0
\(750\) 0 0
\(751\) 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
\(755\) 0 0
\(756\) 0 0
\(757\) −26.3431 −0.957458 −0.478729 0.877963i \(-0.658902\pi\)
−0.478729 + 0.877963i \(0.658902\pi\)
\(758\) −3.59798 −0.130685
\(759\) −15.3137 −0.555852
\(760\) 0 0
\(761\) 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
\(765\) 0 0
\(766\) 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 \(-0.554977\pi\)
−0.171859 + 0.985122i \(0.554977\pi\)
\(774\) 2.34315 0.0842226
\(775\) 0 0
\(776\) 4.10051 0.147200
\(777\) 0 0
\(778\) 7.51472 0.269416
\(779\) −17.6569 −0.632622
\(780\) 0 0
\(781\) −18.6274 −0.666541
\(782\) 7.11270 0.254350
\(783\) −6.82843 −0.244028
\(784\) 0 0
\(785\) 0 0
\(786\) −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
\(790\) 0 0
\(791\) 0 0
\(792\) −3.17157 −0.112697
\(793\) −31.6569 −1.12417
\(794\) 0.987807 0.0350559
\(795\) 0 0
\(796\) 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
\(800\) 0 0
\(801\) 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 \(0.385996\pi\)
\(810\) 0 0
\(811\) 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
\(815\) 0 0
\(816\) −6.72792 −0.235524
\(817\) −16.0000 −0.559769
\(818\) −8.87006 −0.310134
\(819\) 0 0
\(820\) 0 0
\(821\) −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 \(-0.549972\pi\)
−0.156347 + 0.987702i \(0.549972\pi\)
\(824\) 7.11270 0.247783
\(825\) 0 0
\(826\) 0 0
\(827\) −47.6569 −1.65719 −0.828596 0.559848i \(-0.810860\pi\)
−0.828596 + 0.559848i \(0.810860\pi\)
\(828\) −14.0000 −0.486534
\(829\) 0.727922 0.0252818 0.0126409 0.999920i \(-0.495976\pi\)
0.0126409 + 0.999920i \(0.495976\pi\)
\(830\) 0 0
\(831\) 9.31371 0.323089
\(832\) −10.7868 −0.373965
\(833\) 0 0
\(834\) −7.31371 −0.253253
\(835\) 0 0
\(836\) 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
\(845\) 0 0
\(846\) 1.17157 0.0402795
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −8.48528 −0.291214
\(850\) 0 0
\(851\) 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
\(855\) 0 0
\(856\) 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 \(-0.936527\pi\)
−0.980184 + 0.198089i \(0.936527\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 11.1716 0.380505
\(863\) −17.3137 −0.589365 −0.294683 0.955595i \(-0.595214\pi\)
−0.294683 + 0.955595i \(0.595214\pi\)
\(864\) −4.41421 −0.150175
\(865\) 0 0
\(866\) 8.38478 0.284926
\(867\) −11.9706 −0.406542
\(868\) 0 0
\(869\) −27.3137 −0.926554
\(870\) 0 0
\(871\) 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 \(-0.438821\pi\)
0.191018 + 0.981586i \(0.438821\pi\)
\(878\) 5.25483 0.177342
\(879\) 16.5858 0.559425
\(880\) 0 0
\(881\) −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 \(-0.474874\pi\)
0.0788531 + 0.996886i \(0.474874\pi\)
\(884\) 10.6030 0.356619
\(885\) 0 0
\(886\) 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
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −45.6569 −1.52870
\(893\) −8.00000 −0.267710
\(894\) −7.17157 −0.239853
\(895\) 0 0
\(896\) 0 0
\(897\) 19.7990 0.661069
\(898\) 2.20101 0.0734487
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) −4.48528 −0.149426
\(902\) −5.17157 −0.172195
\(903\) 0 0
\(904\) 8.42641 0.280258
\(905\) 0 0
\(906\) −4.97056 −0.165136
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) −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
\(915\) 0 0
\(916\) −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 \(0.543629\pi\)
\(920\) 0 0
\(921\) 30.1421 0.993217
\(922\) −6.87006 −0.226253
\(923\) 24.0833 0.792710
\(924\) 0 0
\(925\) 0 0
\(926\) −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 \(0.277577\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −11.2304 −0.367866
\(933\) −6.14214 −0.201084
\(934\) −0.0832611 −0.00272439
\(935\) 0 0
\(936\) 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
\(940\) 0 0
\(941\) −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 \(-0.666147\pi\)
−0.498585 + 0.866841i \(0.666147\pi\)
\(948\) −24.9706 −0.811006
\(949\) 35.9411 1.16670
\(950\) 0 0
\(951\) −10.0000 −0.324272
\(952\) 0 0
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) −0.828427 −0.0268213
\(955\) 0 0
\(956\) 28.6274 0.925877
\(957\) 13.6569 0.441463
\(958\) 0.769553 0.0248631
\(959\) 0 0
\(960\) 0 0
\(961\) −29.6274 −0.955723
\(962\) −4.28427 −0.138130
\(963\) 0.343146 0.0110577
\(964\) −29.6985 −0.956524
\(965\) 0 0
\(966\) 0 0
\(967\) −33.6569 −1.08233 −0.541166 0.840916i \(-0.682017\pi\)
−0.541166 + 0.840916i \(0.682017\pi\)
\(968\) −11.1005 −0.356784
\(969\) −6.34315 −0.203771
\(970\) 0 0
\(971\) 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
\(975\) 0 0
\(976\) −36.7279 −1.17563
\(977\) 21.1716 0.677339 0.338669 0.940905i \(-0.390023\pi\)
0.338669 + 0.940905i \(0.390023\pi\)
\(978\) −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
\(985\) 0 0
\(986\) −6.34315 −0.202007
\(987\) 0 0
\(988\) −13.3726 −0.425439
\(989\) −43.3137 −1.37730
\(990\) 0 0
\(991\) −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
\(995\) 0 0
\(996\) −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 3675.2.a.bf.1.1 2
5.4 even 2 147.2.a.d.1.2 2
7.6 odd 2 3675.2.a.bd.1.1 2
15.14 odd 2 441.2.a.j.1.1 2
20.19 odd 2 2352.2.a.be.1.1 2
35.4 even 6 147.2.e.e.79.1 4
35.9 even 6 147.2.e.e.67.1 4
35.19 odd 6 147.2.e.d.67.1 4
35.24 odd 6 147.2.e.d.79.1 4
35.34 odd 2 147.2.a.e.1.2 yes 2
40.19 odd 2 9408.2.a.dq.1.2 2
40.29 even 2 9408.2.a.ef.1.2 2
60.59 even 2 7056.2.a.cv.1.2 2
105.44 odd 6 441.2.e.f.361.2 4
105.59 even 6 441.2.e.g.226.2 4
105.74 odd 6 441.2.e.f.226.2 4
105.89 even 6 441.2.e.g.361.2 4
105.104 even 2 441.2.a.i.1.1 2
140.19 even 6 2352.2.q.bd.1537.1 4
140.39 odd 6 2352.2.q.bb.961.2 4
140.59 even 6 2352.2.q.bd.961.1 4
140.79 odd 6 2352.2.q.bb.1537.2 4
140.139 even 2 2352.2.a.bc.1.2 2
280.69 odd 2 9408.2.a.di.1.1 2
280.139 even 2 9408.2.a.dt.1.1 2
420.419 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 5.4 even 2
147.2.a.e.1.2 yes 2 35.34 odd 2
147.2.e.d.67.1 4 35.19 odd 6
147.2.e.d.79.1 4 35.24 odd 6
147.2.e.e.67.1 4 35.9 even 6
147.2.e.e.79.1 4 35.4 even 6
441.2.a.i.1.1 2 105.104 even 2
441.2.a.j.1.1 2 15.14 odd 2
441.2.e.f.226.2 4 105.74 odd 6
441.2.e.f.361.2 4 105.44 odd 6
441.2.e.g.226.2 4 105.59 even 6
441.2.e.g.361.2 4 105.89 even 6
2352.2.a.bc.1.2 2 140.139 even 2
2352.2.a.be.1.1 2 20.19 odd 2
2352.2.q.bb.961.2 4 140.39 odd 6
2352.2.q.bb.1537.2 4 140.79 odd 6
2352.2.q.bd.961.1 4 140.59 even 6
2352.2.q.bd.1537.1 4 140.19 even 6
3675.2.a.bd.1.1 2 7.6 odd 2
3675.2.a.bf.1.1 2 1.1 even 1 trivial
7056.2.a.cf.1.1 2 420.419 odd 2
7056.2.a.cv.1.2 2 60.59 even 2
9408.2.a.di.1.1 2 280.69 odd 2
9408.2.a.dq.1.2 2 40.19 odd 2
9408.2.a.dt.1.1 2 280.139 even 2
9408.2.a.ef.1.2 2 40.29 even 2