Properties

Label 7360.2.a.bs.1.1
Level $7360$
Weight $2$
Character 7360.1
Self dual yes
Analytic conductor $58.770$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.7698958877\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3680)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7360.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -2.73205 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} -2.73205 q^{7} -2.00000 q^{9} -2.73205 q^{11} +1.73205 q^{13} +1.00000 q^{15} +5.46410 q^{17} -2.73205 q^{19} -2.73205 q^{21} +1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +5.92820 q^{29} +0.267949 q^{31} -2.73205 q^{33} -2.73205 q^{35} +2.73205 q^{37} +1.73205 q^{39} +2.46410 q^{41} -6.73205 q^{43} -2.00000 q^{45} +5.92820 q^{47} +0.464102 q^{49} +5.46410 q^{51} +0.535898 q^{53} -2.73205 q^{55} -2.73205 q^{57} +10.3923 q^{59} -7.66025 q^{61} +5.46410 q^{63} +1.73205 q^{65} -11.6603 q^{67} +1.00000 q^{69} -12.1244 q^{71} -12.2679 q^{73} +1.00000 q^{75} +7.46410 q^{77} +1.66025 q^{79} +1.00000 q^{81} -4.00000 q^{83} +5.46410 q^{85} +5.92820 q^{87} -17.6603 q^{89} -4.73205 q^{91} +0.267949 q^{93} -2.73205 q^{95} -7.66025 q^{97} +5.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{9} - 2 q^{11} + 2 q^{15} + 4 q^{17} - 2 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 10 q^{27} - 2 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} + 2 q^{37} - 2 q^{41} - 10 q^{43} - 4 q^{45} - 2 q^{47} - 6 q^{49} + 4 q^{51} + 8 q^{53} - 2 q^{55} - 2 q^{57} + 2 q^{61} + 4 q^{63} - 6 q^{67} + 2 q^{69} - 28 q^{73} + 2 q^{75} + 8 q^{77} - 14 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} - 2 q^{87} - 18 q^{89} - 6 q^{91} + 4 q^{93} - 2 q^{95} + 2 q^{97} + 4 q^{99} + O(q^{100}) \)

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 0
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.73205 −1.03262 −0.516309 0.856402i \(-0.672694\pi\)
−0.516309 + 0.856402i \(0.672694\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −2.73205 −0.823744 −0.411872 0.911242i \(-0.635125\pi\)
−0.411872 + 0.911242i \(0.635125\pi\)
\(12\) 0 0
\(13\) 1.73205 0.480384 0.240192 0.970725i \(-0.422790\pi\)
0.240192 + 0.970725i \(0.422790\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 5.46410 1.32524 0.662620 0.748956i \(-0.269445\pi\)
0.662620 + 0.748956i \(0.269445\pi\)
\(18\) 0 0
\(19\) −2.73205 −0.626775 −0.313388 0.949625i \(-0.601464\pi\)
−0.313388 + 0.949625i \(0.601464\pi\)
\(20\) 0 0
\(21\) −2.73205 −0.596182
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 5.92820 1.10084 0.550420 0.834888i \(-0.314468\pi\)
0.550420 + 0.834888i \(0.314468\pi\)
\(30\) 0 0
\(31\) 0.267949 0.0481251 0.0240625 0.999710i \(-0.492340\pi\)
0.0240625 + 0.999710i \(0.492340\pi\)
\(32\) 0 0
\(33\) −2.73205 −0.475589
\(34\) 0 0
\(35\) −2.73205 −0.461801
\(36\) 0 0
\(37\) 2.73205 0.449146 0.224573 0.974457i \(-0.427901\pi\)
0.224573 + 0.974457i \(0.427901\pi\)
\(38\) 0 0
\(39\) 1.73205 0.277350
\(40\) 0 0
\(41\) 2.46410 0.384828 0.192414 0.981314i \(-0.438368\pi\)
0.192414 + 0.981314i \(0.438368\pi\)
\(42\) 0 0
\(43\) −6.73205 −1.02663 −0.513314 0.858201i \(-0.671582\pi\)
−0.513314 + 0.858201i \(0.671582\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 5.92820 0.864717 0.432359 0.901702i \(-0.357681\pi\)
0.432359 + 0.901702i \(0.357681\pi\)
\(48\) 0 0
\(49\) 0.464102 0.0663002
\(50\) 0 0
\(51\) 5.46410 0.765127
\(52\) 0 0
\(53\) 0.535898 0.0736113 0.0368057 0.999322i \(-0.488282\pi\)
0.0368057 + 0.999322i \(0.488282\pi\)
\(54\) 0 0
\(55\) −2.73205 −0.368390
\(56\) 0 0
\(57\) −2.73205 −0.361869
\(58\) 0 0
\(59\) 10.3923 1.35296 0.676481 0.736460i \(-0.263504\pi\)
0.676481 + 0.736460i \(0.263504\pi\)
\(60\) 0 0
\(61\) −7.66025 −0.980795 −0.490398 0.871499i \(-0.663148\pi\)
−0.490398 + 0.871499i \(0.663148\pi\)
\(62\) 0 0
\(63\) 5.46410 0.688412
\(64\) 0 0
\(65\) 1.73205 0.214834
\(66\) 0 0
\(67\) −11.6603 −1.42453 −0.712263 0.701912i \(-0.752330\pi\)
−0.712263 + 0.701912i \(0.752330\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −12.1244 −1.43890 −0.719448 0.694546i \(-0.755605\pi\)
−0.719448 + 0.694546i \(0.755605\pi\)
\(72\) 0 0
\(73\) −12.2679 −1.43585 −0.717927 0.696118i \(-0.754909\pi\)
−0.717927 + 0.696118i \(0.754909\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 7.46410 0.850613
\(78\) 0 0
\(79\) 1.66025 0.186793 0.0933966 0.995629i \(-0.470228\pi\)
0.0933966 + 0.995629i \(0.470228\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 5.46410 0.592665
\(86\) 0 0
\(87\) 5.92820 0.635570
\(88\) 0 0
\(89\) −17.6603 −1.87198 −0.935992 0.352022i \(-0.885494\pi\)
−0.935992 + 0.352022i \(0.885494\pi\)
\(90\) 0 0
\(91\) −4.73205 −0.496054
\(92\) 0 0
\(93\) 0.267949 0.0277850
\(94\) 0 0
\(95\) −2.73205 −0.280302
\(96\) 0 0
\(97\) −7.66025 −0.777781 −0.388890 0.921284i \(-0.627142\pi\)
−0.388890 + 0.921284i \(0.627142\pi\)
\(98\) 0 0
\(99\) 5.46410 0.549163
\(100\) 0 0
\(101\) 17.8564 1.77678 0.888389 0.459091i \(-0.151825\pi\)
0.888389 + 0.459091i \(0.151825\pi\)
\(102\) 0 0
\(103\) −5.46410 −0.538394 −0.269197 0.963085i \(-0.586758\pi\)
−0.269197 + 0.963085i \(0.586758\pi\)
\(104\) 0 0
\(105\) −2.73205 −0.266621
\(106\) 0 0
\(107\) −7.26795 −0.702619 −0.351310 0.936259i \(-0.614264\pi\)
−0.351310 + 0.936259i \(0.614264\pi\)
\(108\) 0 0
\(109\) 2.33975 0.224107 0.112054 0.993702i \(-0.464257\pi\)
0.112054 + 0.993702i \(0.464257\pi\)
\(110\) 0 0
\(111\) 2.73205 0.259315
\(112\) 0 0
\(113\) −7.26795 −0.683711 −0.341856 0.939753i \(-0.611055\pi\)
−0.341856 + 0.939753i \(0.611055\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) −3.46410 −0.320256
\(118\) 0 0
\(119\) −14.9282 −1.36847
\(120\) 0 0
\(121\) −3.53590 −0.321445
\(122\) 0 0
\(123\) 2.46410 0.222181
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −10.8564 −0.963350 −0.481675 0.876350i \(-0.659971\pi\)
−0.481675 + 0.876350i \(0.659971\pi\)
\(128\) 0 0
\(129\) −6.73205 −0.592724
\(130\) 0 0
\(131\) 12.6603 1.10613 0.553066 0.833138i \(-0.313458\pi\)
0.553066 + 0.833138i \(0.313458\pi\)
\(132\) 0 0
\(133\) 7.46410 0.647220
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) 5.46410 0.466830 0.233415 0.972377i \(-0.425010\pi\)
0.233415 + 0.972377i \(0.425010\pi\)
\(138\) 0 0
\(139\) −15.1962 −1.28892 −0.644460 0.764638i \(-0.722918\pi\)
−0.644460 + 0.764638i \(0.722918\pi\)
\(140\) 0 0
\(141\) 5.92820 0.499245
\(142\) 0 0
\(143\) −4.73205 −0.395714
\(144\) 0 0
\(145\) 5.92820 0.492310
\(146\) 0 0
\(147\) 0.464102 0.0382785
\(148\) 0 0
\(149\) −4.92820 −0.403734 −0.201867 0.979413i \(-0.564701\pi\)
−0.201867 + 0.979413i \(0.564701\pi\)
\(150\) 0 0
\(151\) 4.12436 0.335635 0.167818 0.985818i \(-0.446328\pi\)
0.167818 + 0.985818i \(0.446328\pi\)
\(152\) 0 0
\(153\) −10.9282 −0.883493
\(154\) 0 0
\(155\) 0.267949 0.0215222
\(156\) 0 0
\(157\) 0.732051 0.0584240 0.0292120 0.999573i \(-0.490700\pi\)
0.0292120 + 0.999573i \(0.490700\pi\)
\(158\) 0 0
\(159\) 0.535898 0.0424995
\(160\) 0 0
\(161\) −2.73205 −0.215316
\(162\) 0 0
\(163\) 0.464102 0.0363512 0.0181756 0.999835i \(-0.494214\pi\)
0.0181756 + 0.999835i \(0.494214\pi\)
\(164\) 0 0
\(165\) −2.73205 −0.212690
\(166\) 0 0
\(167\) −7.07180 −0.547232 −0.273616 0.961839i \(-0.588220\pi\)
−0.273616 + 0.961839i \(0.588220\pi\)
\(168\) 0 0
\(169\) −10.0000 −0.769231
\(170\) 0 0
\(171\) 5.46410 0.417850
\(172\) 0 0
\(173\) −19.8564 −1.50965 −0.754827 0.655924i \(-0.772279\pi\)
−0.754827 + 0.655924i \(0.772279\pi\)
\(174\) 0 0
\(175\) −2.73205 −0.206524
\(176\) 0 0
\(177\) 10.3923 0.781133
\(178\) 0 0
\(179\) −2.66025 −0.198837 −0.0994184 0.995046i \(-0.531698\pi\)
−0.0994184 + 0.995046i \(0.531698\pi\)
\(180\) 0 0
\(181\) 0.339746 0.0252531 0.0126266 0.999920i \(-0.495981\pi\)
0.0126266 + 0.999920i \(0.495981\pi\)
\(182\) 0 0
\(183\) −7.66025 −0.566262
\(184\) 0 0
\(185\) 2.73205 0.200864
\(186\) 0 0
\(187\) −14.9282 −1.09166
\(188\) 0 0
\(189\) 13.6603 0.993637
\(190\) 0 0
\(191\) 2.39230 0.173101 0.0865506 0.996247i \(-0.472416\pi\)
0.0865506 + 0.996247i \(0.472416\pi\)
\(192\) 0 0
\(193\) −7.33975 −0.528326 −0.264163 0.964478i \(-0.585096\pi\)
−0.264163 + 0.964478i \(0.585096\pi\)
\(194\) 0 0
\(195\) 1.73205 0.124035
\(196\) 0 0
\(197\) −16.6603 −1.18699 −0.593497 0.804836i \(-0.702253\pi\)
−0.593497 + 0.804836i \(0.702253\pi\)
\(198\) 0 0
\(199\) −11.8038 −0.836753 −0.418376 0.908274i \(-0.637401\pi\)
−0.418376 + 0.908274i \(0.637401\pi\)
\(200\) 0 0
\(201\) −11.6603 −0.822451
\(202\) 0 0
\(203\) −16.1962 −1.13675
\(204\) 0 0
\(205\) 2.46410 0.172100
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) 7.46410 0.516303
\(210\) 0 0
\(211\) 8.39230 0.577750 0.288875 0.957367i \(-0.406719\pi\)
0.288875 + 0.957367i \(0.406719\pi\)
\(212\) 0 0
\(213\) −12.1244 −0.830747
\(214\) 0 0
\(215\) −6.73205 −0.459122
\(216\) 0 0
\(217\) −0.732051 −0.0496948
\(218\) 0 0
\(219\) −12.2679 −0.828991
\(220\) 0 0
\(221\) 9.46410 0.636624
\(222\) 0 0
\(223\) 16.9282 1.13360 0.566798 0.823857i \(-0.308182\pi\)
0.566798 + 0.823857i \(0.308182\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) 1.26795 0.0841567 0.0420784 0.999114i \(-0.486602\pi\)
0.0420784 + 0.999114i \(0.486602\pi\)
\(228\) 0 0
\(229\) −11.1244 −0.735118 −0.367559 0.930000i \(-0.619806\pi\)
−0.367559 + 0.930000i \(0.619806\pi\)
\(230\) 0 0
\(231\) 7.46410 0.491102
\(232\) 0 0
\(233\) −23.0526 −1.51022 −0.755112 0.655596i \(-0.772417\pi\)
−0.755112 + 0.655596i \(0.772417\pi\)
\(234\) 0 0
\(235\) 5.92820 0.386713
\(236\) 0 0
\(237\) 1.66025 0.107845
\(238\) 0 0
\(239\) −17.7321 −1.14699 −0.573496 0.819209i \(-0.694413\pi\)
−0.573496 + 0.819209i \(0.694413\pi\)
\(240\) 0 0
\(241\) −24.2487 −1.56200 −0.780998 0.624533i \(-0.785289\pi\)
−0.780998 + 0.624533i \(0.785289\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 0.464102 0.0296504
\(246\) 0 0
\(247\) −4.73205 −0.301093
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −14.9282 −0.942260 −0.471130 0.882064i \(-0.656154\pi\)
−0.471130 + 0.882064i \(0.656154\pi\)
\(252\) 0 0
\(253\) −2.73205 −0.171763
\(254\) 0 0
\(255\) 5.46410 0.342175
\(256\) 0 0
\(257\) 3.73205 0.232799 0.116399 0.993202i \(-0.462865\pi\)
0.116399 + 0.993202i \(0.462865\pi\)
\(258\) 0 0
\(259\) −7.46410 −0.463797
\(260\) 0 0
\(261\) −11.8564 −0.733893
\(262\) 0 0
\(263\) −24.0526 −1.48314 −0.741572 0.670873i \(-0.765920\pi\)
−0.741572 + 0.670873i \(0.765920\pi\)
\(264\) 0 0
\(265\) 0.535898 0.0329200
\(266\) 0 0
\(267\) −17.6603 −1.08079
\(268\) 0 0
\(269\) 21.9282 1.33699 0.668493 0.743718i \(-0.266940\pi\)
0.668493 + 0.743718i \(0.266940\pi\)
\(270\) 0 0
\(271\) −22.2487 −1.35151 −0.675756 0.737125i \(-0.736183\pi\)
−0.675756 + 0.737125i \(0.736183\pi\)
\(272\) 0 0
\(273\) −4.73205 −0.286397
\(274\) 0 0
\(275\) −2.73205 −0.164749
\(276\) 0 0
\(277\) 29.0526 1.74560 0.872800 0.488079i \(-0.162302\pi\)
0.872800 + 0.488079i \(0.162302\pi\)
\(278\) 0 0
\(279\) −0.535898 −0.0320834
\(280\) 0 0
\(281\) 11.1244 0.663623 0.331812 0.943346i \(-0.392340\pi\)
0.331812 + 0.943346i \(0.392340\pi\)
\(282\) 0 0
\(283\) −13.3205 −0.791822 −0.395911 0.918289i \(-0.629571\pi\)
−0.395911 + 0.918289i \(0.629571\pi\)
\(284\) 0 0
\(285\) −2.73205 −0.161833
\(286\) 0 0
\(287\) −6.73205 −0.397380
\(288\) 0 0
\(289\) 12.8564 0.756259
\(290\) 0 0
\(291\) −7.66025 −0.449052
\(292\) 0 0
\(293\) 27.7128 1.61900 0.809500 0.587120i \(-0.199738\pi\)
0.809500 + 0.587120i \(0.199738\pi\)
\(294\) 0 0
\(295\) 10.3923 0.605063
\(296\) 0 0
\(297\) 13.6603 0.792648
\(298\) 0 0
\(299\) 1.73205 0.100167
\(300\) 0 0
\(301\) 18.3923 1.06011
\(302\) 0 0
\(303\) 17.8564 1.02582
\(304\) 0 0
\(305\) −7.66025 −0.438625
\(306\) 0 0
\(307\) 11.4641 0.654291 0.327145 0.944974i \(-0.393913\pi\)
0.327145 + 0.944974i \(0.393913\pi\)
\(308\) 0 0
\(309\) −5.46410 −0.310842
\(310\) 0 0
\(311\) 33.0526 1.87424 0.937119 0.349009i \(-0.113482\pi\)
0.937119 + 0.349009i \(0.113482\pi\)
\(312\) 0 0
\(313\) −18.9282 −1.06989 −0.534943 0.844888i \(-0.679667\pi\)
−0.534943 + 0.844888i \(0.679667\pi\)
\(314\) 0 0
\(315\) 5.46410 0.307867
\(316\) 0 0
\(317\) −20.9282 −1.17544 −0.587722 0.809063i \(-0.699975\pi\)
−0.587722 + 0.809063i \(0.699975\pi\)
\(318\) 0 0
\(319\) −16.1962 −0.906810
\(320\) 0 0
\(321\) −7.26795 −0.405657
\(322\) 0 0
\(323\) −14.9282 −0.830627
\(324\) 0 0
\(325\) 1.73205 0.0960769
\(326\) 0 0
\(327\) 2.33975 0.129388
\(328\) 0 0
\(329\) −16.1962 −0.892923
\(330\) 0 0
\(331\) −0.267949 −0.0147278 −0.00736391 0.999973i \(-0.502344\pi\)
−0.00736391 + 0.999973i \(0.502344\pi\)
\(332\) 0 0
\(333\) −5.46410 −0.299431
\(334\) 0 0
\(335\) −11.6603 −0.637068
\(336\) 0 0
\(337\) 34.5885 1.88415 0.942077 0.335398i \(-0.108871\pi\)
0.942077 + 0.335398i \(0.108871\pi\)
\(338\) 0 0
\(339\) −7.26795 −0.394741
\(340\) 0 0
\(341\) −0.732051 −0.0396428
\(342\) 0 0
\(343\) 17.8564 0.964155
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) 7.85641 0.421754 0.210877 0.977513i \(-0.432368\pi\)
0.210877 + 0.977513i \(0.432368\pi\)
\(348\) 0 0
\(349\) 11.0000 0.588817 0.294408 0.955680i \(-0.404877\pi\)
0.294408 + 0.955680i \(0.404877\pi\)
\(350\) 0 0
\(351\) −8.66025 −0.462250
\(352\) 0 0
\(353\) 19.7321 1.05023 0.525116 0.851031i \(-0.324022\pi\)
0.525116 + 0.851031i \(0.324022\pi\)
\(354\) 0 0
\(355\) −12.1244 −0.643494
\(356\) 0 0
\(357\) −14.9282 −0.790084
\(358\) 0 0
\(359\) −10.3397 −0.545711 −0.272855 0.962055i \(-0.587968\pi\)
−0.272855 + 0.962055i \(0.587968\pi\)
\(360\) 0 0
\(361\) −11.5359 −0.607153
\(362\) 0 0
\(363\) −3.53590 −0.185587
\(364\) 0 0
\(365\) −12.2679 −0.642134
\(366\) 0 0
\(367\) −14.5359 −0.758768 −0.379384 0.925239i \(-0.623864\pi\)
−0.379384 + 0.925239i \(0.623864\pi\)
\(368\) 0 0
\(369\) −4.92820 −0.256552
\(370\) 0 0
\(371\) −1.46410 −0.0760124
\(372\) 0 0
\(373\) 20.9282 1.08362 0.541811 0.840501i \(-0.317739\pi\)
0.541811 + 0.840501i \(0.317739\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 10.2679 0.528826
\(378\) 0 0
\(379\) −4.39230 −0.225618 −0.112809 0.993617i \(-0.535985\pi\)
−0.112809 + 0.993617i \(0.535985\pi\)
\(380\) 0 0
\(381\) −10.8564 −0.556191
\(382\) 0 0
\(383\) 4.19615 0.214413 0.107207 0.994237i \(-0.465809\pi\)
0.107207 + 0.994237i \(0.465809\pi\)
\(384\) 0 0
\(385\) 7.46410 0.380406
\(386\) 0 0
\(387\) 13.4641 0.684419
\(388\) 0 0
\(389\) 11.6077 0.588534 0.294267 0.955723i \(-0.404925\pi\)
0.294267 + 0.955723i \(0.404925\pi\)
\(390\) 0 0
\(391\) 5.46410 0.276331
\(392\) 0 0
\(393\) 12.6603 0.638625
\(394\) 0 0
\(395\) 1.66025 0.0835364
\(396\) 0 0
\(397\) 5.73205 0.287683 0.143842 0.989601i \(-0.454054\pi\)
0.143842 + 0.989601i \(0.454054\pi\)
\(398\) 0 0
\(399\) 7.46410 0.373672
\(400\) 0 0
\(401\) 31.8564 1.59083 0.795417 0.606063i \(-0.207252\pi\)
0.795417 + 0.606063i \(0.207252\pi\)
\(402\) 0 0
\(403\) 0.464102 0.0231185
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −7.46410 −0.369982
\(408\) 0 0
\(409\) −19.0000 −0.939490 −0.469745 0.882802i \(-0.655654\pi\)
−0.469745 + 0.882802i \(0.655654\pi\)
\(410\) 0 0
\(411\) 5.46410 0.269524
\(412\) 0 0
\(413\) −28.3923 −1.39709
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) −15.1962 −0.744159
\(418\) 0 0
\(419\) 4.73205 0.231176 0.115588 0.993297i \(-0.463125\pi\)
0.115588 + 0.993297i \(0.463125\pi\)
\(420\) 0 0
\(421\) 9.26795 0.451692 0.225846 0.974163i \(-0.427485\pi\)
0.225846 + 0.974163i \(0.427485\pi\)
\(422\) 0 0
\(423\) −11.8564 −0.576478
\(424\) 0 0
\(425\) 5.46410 0.265048
\(426\) 0 0
\(427\) 20.9282 1.01279
\(428\) 0 0
\(429\) −4.73205 −0.228466
\(430\) 0 0
\(431\) 5.80385 0.279562 0.139781 0.990182i \(-0.455360\pi\)
0.139781 + 0.990182i \(0.455360\pi\)
\(432\) 0 0
\(433\) −4.92820 −0.236834 −0.118417 0.992964i \(-0.537782\pi\)
−0.118417 + 0.992964i \(0.537782\pi\)
\(434\) 0 0
\(435\) 5.92820 0.284236
\(436\) 0 0
\(437\) −2.73205 −0.130692
\(438\) 0 0
\(439\) −14.8038 −0.706549 −0.353275 0.935520i \(-0.614932\pi\)
−0.353275 + 0.935520i \(0.614932\pi\)
\(440\) 0 0
\(441\) −0.928203 −0.0442002
\(442\) 0 0
\(443\) 33.2487 1.57969 0.789847 0.613304i \(-0.210160\pi\)
0.789847 + 0.613304i \(0.210160\pi\)
\(444\) 0 0
\(445\) −17.6603 −0.837176
\(446\) 0 0
\(447\) −4.92820 −0.233096
\(448\) 0 0
\(449\) −9.85641 −0.465153 −0.232576 0.972578i \(-0.574716\pi\)
−0.232576 + 0.972578i \(0.574716\pi\)
\(450\) 0 0
\(451\) −6.73205 −0.317000
\(452\) 0 0
\(453\) 4.12436 0.193779
\(454\) 0 0
\(455\) −4.73205 −0.221842
\(456\) 0 0
\(457\) 17.0718 0.798585 0.399292 0.916824i \(-0.369256\pi\)
0.399292 + 0.916824i \(0.369256\pi\)
\(458\) 0 0
\(459\) −27.3205 −1.27521
\(460\) 0 0
\(461\) 18.4641 0.859959 0.429979 0.902839i \(-0.358521\pi\)
0.429979 + 0.902839i \(0.358521\pi\)
\(462\) 0 0
\(463\) −0.535898 −0.0249053 −0.0124527 0.999922i \(-0.503964\pi\)
−0.0124527 + 0.999922i \(0.503964\pi\)
\(464\) 0 0
\(465\) 0.267949 0.0124258
\(466\) 0 0
\(467\) 31.8564 1.47414 0.737069 0.675817i \(-0.236209\pi\)
0.737069 + 0.675817i \(0.236209\pi\)
\(468\) 0 0
\(469\) 31.8564 1.47099
\(470\) 0 0
\(471\) 0.732051 0.0337311
\(472\) 0 0
\(473\) 18.3923 0.845679
\(474\) 0 0
\(475\) −2.73205 −0.125355
\(476\) 0 0
\(477\) −1.07180 −0.0490742
\(478\) 0 0
\(479\) −25.1769 −1.15036 −0.575181 0.818026i \(-0.695068\pi\)
−0.575181 + 0.818026i \(0.695068\pi\)
\(480\) 0 0
\(481\) 4.73205 0.215763
\(482\) 0 0
\(483\) −2.73205 −0.124313
\(484\) 0 0
\(485\) −7.66025 −0.347834
\(486\) 0 0
\(487\) 23.3923 1.06001 0.530003 0.847996i \(-0.322191\pi\)
0.530003 + 0.847996i \(0.322191\pi\)
\(488\) 0 0
\(489\) 0.464102 0.0209874
\(490\) 0 0
\(491\) −16.6603 −0.751867 −0.375933 0.926647i \(-0.622678\pi\)
−0.375933 + 0.926647i \(0.622678\pi\)
\(492\) 0 0
\(493\) 32.3923 1.45888
\(494\) 0 0
\(495\) 5.46410 0.245593
\(496\) 0 0
\(497\) 33.1244 1.48583
\(498\) 0 0
\(499\) −8.26795 −0.370124 −0.185062 0.982727i \(-0.559249\pi\)
−0.185062 + 0.982727i \(0.559249\pi\)
\(500\) 0 0
\(501\) −7.07180 −0.315945
\(502\) 0 0
\(503\) 18.3923 0.820072 0.410036 0.912069i \(-0.365516\pi\)
0.410036 + 0.912069i \(0.365516\pi\)
\(504\) 0 0
\(505\) 17.8564 0.794600
\(506\) 0 0
\(507\) −10.0000 −0.444116
\(508\) 0 0
\(509\) −32.3205 −1.43258 −0.716291 0.697802i \(-0.754162\pi\)
−0.716291 + 0.697802i \(0.754162\pi\)
\(510\) 0 0
\(511\) 33.5167 1.48269
\(512\) 0 0
\(513\) 13.6603 0.603115
\(514\) 0 0
\(515\) −5.46410 −0.240777
\(516\) 0 0
\(517\) −16.1962 −0.712306
\(518\) 0 0
\(519\) −19.8564 −0.871600
\(520\) 0 0
\(521\) 40.3923 1.76962 0.884810 0.465953i \(-0.154288\pi\)
0.884810 + 0.465953i \(0.154288\pi\)
\(522\) 0 0
\(523\) −7.41154 −0.324084 −0.162042 0.986784i \(-0.551808\pi\)
−0.162042 + 0.986784i \(0.551808\pi\)
\(524\) 0 0
\(525\) −2.73205 −0.119236
\(526\) 0 0
\(527\) 1.46410 0.0637773
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −20.7846 −0.901975
\(532\) 0 0
\(533\) 4.26795 0.184865
\(534\) 0 0
\(535\) −7.26795 −0.314221
\(536\) 0 0
\(537\) −2.66025 −0.114798
\(538\) 0 0
\(539\) −1.26795 −0.0546144
\(540\) 0 0
\(541\) −0.607695 −0.0261269 −0.0130634 0.999915i \(-0.504158\pi\)
−0.0130634 + 0.999915i \(0.504158\pi\)
\(542\) 0 0
\(543\) 0.339746 0.0145799
\(544\) 0 0
\(545\) 2.33975 0.100224
\(546\) 0 0
\(547\) 27.7846 1.18798 0.593992 0.804471i \(-0.297551\pi\)
0.593992 + 0.804471i \(0.297551\pi\)
\(548\) 0 0
\(549\) 15.3205 0.653863
\(550\) 0 0
\(551\) −16.1962 −0.689979
\(552\) 0 0
\(553\) −4.53590 −0.192886
\(554\) 0 0
\(555\) 2.73205 0.115969
\(556\) 0 0
\(557\) 13.1244 0.556097 0.278048 0.960567i \(-0.410312\pi\)
0.278048 + 0.960567i \(0.410312\pi\)
\(558\) 0 0
\(559\) −11.6603 −0.493176
\(560\) 0 0
\(561\) −14.9282 −0.630269
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) −7.26795 −0.305765
\(566\) 0 0
\(567\) −2.73205 −0.114735
\(568\) 0 0
\(569\) 19.8564 0.832424 0.416212 0.909268i \(-0.363357\pi\)
0.416212 + 0.909268i \(0.363357\pi\)
\(570\) 0 0
\(571\) 15.3205 0.641143 0.320572 0.947224i \(-0.396125\pi\)
0.320572 + 0.947224i \(0.396125\pi\)
\(572\) 0 0
\(573\) 2.39230 0.0999400
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −25.4449 −1.05928 −0.529642 0.848221i \(-0.677674\pi\)
−0.529642 + 0.848221i \(0.677674\pi\)
\(578\) 0 0
\(579\) −7.33975 −0.305029
\(580\) 0 0
\(581\) 10.9282 0.453378
\(582\) 0 0
\(583\) −1.46410 −0.0606369
\(584\) 0 0
\(585\) −3.46410 −0.143223
\(586\) 0 0
\(587\) −32.4641 −1.33994 −0.669968 0.742390i \(-0.733692\pi\)
−0.669968 + 0.742390i \(0.733692\pi\)
\(588\) 0 0
\(589\) −0.732051 −0.0301636
\(590\) 0 0
\(591\) −16.6603 −0.685311
\(592\) 0 0
\(593\) 15.7128 0.645248 0.322624 0.946527i \(-0.395435\pi\)
0.322624 + 0.946527i \(0.395435\pi\)
\(594\) 0 0
\(595\) −14.9282 −0.611997
\(596\) 0 0
\(597\) −11.8038 −0.483099
\(598\) 0 0
\(599\) −41.3205 −1.68831 −0.844155 0.536099i \(-0.819897\pi\)
−0.844155 + 0.536099i \(0.819897\pi\)
\(600\) 0 0
\(601\) 1.92820 0.0786531 0.0393265 0.999226i \(-0.487479\pi\)
0.0393265 + 0.999226i \(0.487479\pi\)
\(602\) 0 0
\(603\) 23.3205 0.949685
\(604\) 0 0
\(605\) −3.53590 −0.143755
\(606\) 0 0
\(607\) 6.39230 0.259456 0.129728 0.991550i \(-0.458590\pi\)
0.129728 + 0.991550i \(0.458590\pi\)
\(608\) 0 0
\(609\) −16.1962 −0.656301
\(610\) 0 0
\(611\) 10.2679 0.415397
\(612\) 0 0
\(613\) −14.2487 −0.575500 −0.287750 0.957706i \(-0.592907\pi\)
−0.287750 + 0.957706i \(0.592907\pi\)
\(614\) 0 0
\(615\) 2.46410 0.0993622
\(616\) 0 0
\(617\) 25.2679 1.01725 0.508625 0.860988i \(-0.330154\pi\)
0.508625 + 0.860988i \(0.330154\pi\)
\(618\) 0 0
\(619\) 2.24871 0.0903833 0.0451917 0.998978i \(-0.485610\pi\)
0.0451917 + 0.998978i \(0.485610\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) 48.2487 1.93304
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.46410 0.298088
\(628\) 0 0
\(629\) 14.9282 0.595226
\(630\) 0 0
\(631\) −38.1962 −1.52056 −0.760282 0.649593i \(-0.774939\pi\)
−0.760282 + 0.649593i \(0.774939\pi\)
\(632\) 0 0
\(633\) 8.39230 0.333564
\(634\) 0 0
\(635\) −10.8564 −0.430823
\(636\) 0 0
\(637\) 0.803848 0.0318496
\(638\) 0 0
\(639\) 24.2487 0.959264
\(640\) 0 0
\(641\) −19.5167 −0.770862 −0.385431 0.922737i \(-0.625947\pi\)
−0.385431 + 0.922737i \(0.625947\pi\)
\(642\) 0 0
\(643\) 27.4641 1.08308 0.541539 0.840675i \(-0.317842\pi\)
0.541539 + 0.840675i \(0.317842\pi\)
\(644\) 0 0
\(645\) −6.73205 −0.265074
\(646\) 0 0
\(647\) 0.464102 0.0182457 0.00912286 0.999958i \(-0.497096\pi\)
0.00912286 + 0.999958i \(0.497096\pi\)
\(648\) 0 0
\(649\) −28.3923 −1.11450
\(650\) 0 0
\(651\) −0.732051 −0.0286913
\(652\) 0 0
\(653\) −23.1962 −0.907736 −0.453868 0.891069i \(-0.649956\pi\)
−0.453868 + 0.891069i \(0.649956\pi\)
\(654\) 0 0
\(655\) 12.6603 0.494677
\(656\) 0 0
\(657\) 24.5359 0.957237
\(658\) 0 0
\(659\) −48.7846 −1.90038 −0.950189 0.311673i \(-0.899111\pi\)
−0.950189 + 0.311673i \(0.899111\pi\)
\(660\) 0 0
\(661\) −25.7128 −1.00011 −0.500056 0.865993i \(-0.666687\pi\)
−0.500056 + 0.865993i \(0.666687\pi\)
\(662\) 0 0
\(663\) 9.46410 0.367555
\(664\) 0 0
\(665\) 7.46410 0.289445
\(666\) 0 0
\(667\) 5.92820 0.229541
\(668\) 0 0
\(669\) 16.9282 0.654482
\(670\) 0 0
\(671\) 20.9282 0.807924
\(672\) 0 0
\(673\) −28.9090 −1.11436 −0.557179 0.830392i \(-0.688116\pi\)
−0.557179 + 0.830392i \(0.688116\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) −8.92820 −0.343139 −0.171569 0.985172i \(-0.554884\pi\)
−0.171569 + 0.985172i \(0.554884\pi\)
\(678\) 0 0
\(679\) 20.9282 0.803151
\(680\) 0 0
\(681\) 1.26795 0.0485879
\(682\) 0 0
\(683\) −39.1051 −1.49632 −0.748158 0.663521i \(-0.769061\pi\)
−0.748158 + 0.663521i \(0.769061\pi\)
\(684\) 0 0
\(685\) 5.46410 0.208773
\(686\) 0 0
\(687\) −11.1244 −0.424421
\(688\) 0 0
\(689\) 0.928203 0.0353617
\(690\) 0 0
\(691\) −28.3923 −1.08009 −0.540047 0.841635i \(-0.681594\pi\)
−0.540047 + 0.841635i \(0.681594\pi\)
\(692\) 0 0
\(693\) −14.9282 −0.567076
\(694\) 0 0
\(695\) −15.1962 −0.576423
\(696\) 0 0
\(697\) 13.4641 0.509989
\(698\) 0 0
\(699\) −23.0526 −0.871928
\(700\) 0 0
\(701\) −10.7321 −0.405344 −0.202672 0.979247i \(-0.564963\pi\)
−0.202672 + 0.979247i \(0.564963\pi\)
\(702\) 0 0
\(703\) −7.46410 −0.281514
\(704\) 0 0
\(705\) 5.92820 0.223269
\(706\) 0 0
\(707\) −48.7846 −1.83473
\(708\) 0 0
\(709\) 4.58846 0.172323 0.0861616 0.996281i \(-0.472540\pi\)
0.0861616 + 0.996281i \(0.472540\pi\)
\(710\) 0 0
\(711\) −3.32051 −0.124529
\(712\) 0 0
\(713\) 0.267949 0.0100348
\(714\) 0 0
\(715\) −4.73205 −0.176969
\(716\) 0 0
\(717\) −17.7321 −0.662216
\(718\) 0 0
\(719\) −15.6077 −0.582069 −0.291034 0.956713i \(-0.593999\pi\)
−0.291034 + 0.956713i \(0.593999\pi\)
\(720\) 0 0
\(721\) 14.9282 0.555955
\(722\) 0 0
\(723\) −24.2487 −0.901819
\(724\) 0 0
\(725\) 5.92820 0.220168
\(726\) 0 0
\(727\) −35.1769 −1.30464 −0.652320 0.757944i \(-0.726204\pi\)
−0.652320 + 0.757944i \(0.726204\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −36.7846 −1.36053
\(732\) 0 0
\(733\) 8.78461 0.324467 0.162233 0.986752i \(-0.448130\pi\)
0.162233 + 0.986752i \(0.448130\pi\)
\(734\) 0 0
\(735\) 0.464102 0.0171186
\(736\) 0 0
\(737\) 31.8564 1.17345
\(738\) 0 0
\(739\) 40.3731 1.48515 0.742574 0.669764i \(-0.233605\pi\)
0.742574 + 0.669764i \(0.233605\pi\)
\(740\) 0 0
\(741\) −4.73205 −0.173836
\(742\) 0 0
\(743\) 13.1769 0.483414 0.241707 0.970349i \(-0.422293\pi\)
0.241707 + 0.970349i \(0.422293\pi\)
\(744\) 0 0
\(745\) −4.92820 −0.180555
\(746\) 0 0
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) 19.8564 0.725537
\(750\) 0 0
\(751\) −26.9808 −0.984542 −0.492271 0.870442i \(-0.663833\pi\)
−0.492271 + 0.870442i \(0.663833\pi\)
\(752\) 0 0
\(753\) −14.9282 −0.544014
\(754\) 0 0
\(755\) 4.12436 0.150101
\(756\) 0 0
\(757\) −12.1962 −0.443277 −0.221638 0.975129i \(-0.571140\pi\)
−0.221638 + 0.975129i \(0.571140\pi\)
\(758\) 0 0
\(759\) −2.73205 −0.0991672
\(760\) 0 0
\(761\) 39.7846 1.44219 0.721095 0.692836i \(-0.243639\pi\)
0.721095 + 0.692836i \(0.243639\pi\)
\(762\) 0 0
\(763\) −6.39230 −0.231417
\(764\) 0 0
\(765\) −10.9282 −0.395110
\(766\) 0 0
\(767\) 18.0000 0.649942
\(768\) 0 0
\(769\) −12.9282 −0.466203 −0.233101 0.972452i \(-0.574887\pi\)
−0.233101 + 0.972452i \(0.574887\pi\)
\(770\) 0 0
\(771\) 3.73205 0.134407
\(772\) 0 0
\(773\) −15.6603 −0.563260 −0.281630 0.959523i \(-0.590875\pi\)
−0.281630 + 0.959523i \(0.590875\pi\)
\(774\) 0 0
\(775\) 0.267949 0.00962502
\(776\) 0 0
\(777\) −7.46410 −0.267773
\(778\) 0 0
\(779\) −6.73205 −0.241201
\(780\) 0 0
\(781\) 33.1244 1.18528
\(782\) 0 0
\(783\) −29.6410 −1.05928
\(784\) 0 0
\(785\) 0.732051 0.0261280
\(786\) 0 0
\(787\) −23.7128 −0.845270 −0.422635 0.906300i \(-0.638895\pi\)
−0.422635 + 0.906300i \(0.638895\pi\)
\(788\) 0 0
\(789\) −24.0526 −0.856294
\(790\) 0 0
\(791\) 19.8564 0.706013
\(792\) 0 0
\(793\) −13.2679 −0.471159
\(794\) 0 0
\(795\) 0.535898 0.0190064
\(796\) 0 0
\(797\) −13.9474 −0.494044 −0.247022 0.969010i \(-0.579452\pi\)
−0.247022 + 0.969010i \(0.579452\pi\)
\(798\) 0 0
\(799\) 32.3923 1.14596
\(800\) 0 0
\(801\) 35.3205 1.24799
\(802\) 0 0
\(803\) 33.5167 1.18278
\(804\) 0 0
\(805\) −2.73205 −0.0962921
\(806\) 0 0
\(807\) 21.9282 0.771909
\(808\) 0 0
\(809\) −30.9282 −1.08738 −0.543689 0.839287i \(-0.682973\pi\)
−0.543689 + 0.839287i \(0.682973\pi\)
\(810\) 0 0
\(811\) −27.0526 −0.949944 −0.474972 0.880001i \(-0.657542\pi\)
−0.474972 + 0.880001i \(0.657542\pi\)
\(812\) 0 0
\(813\) −22.2487 −0.780296
\(814\) 0 0
\(815\) 0.464102 0.0162568
\(816\) 0 0
\(817\) 18.3923 0.643465
\(818\) 0 0
\(819\) 9.46410 0.330702
\(820\) 0 0
\(821\) 45.8564 1.60040 0.800200 0.599733i \(-0.204727\pi\)
0.800200 + 0.599733i \(0.204727\pi\)
\(822\) 0 0
\(823\) −5.78461 −0.201639 −0.100819 0.994905i \(-0.532146\pi\)
−0.100819 + 0.994905i \(0.532146\pi\)
\(824\) 0 0
\(825\) −2.73205 −0.0951178
\(826\) 0 0
\(827\) −20.3923 −0.709110 −0.354555 0.935035i \(-0.615368\pi\)
−0.354555 + 0.935035i \(0.615368\pi\)
\(828\) 0 0
\(829\) 4.67949 0.162525 0.0812627 0.996693i \(-0.474105\pi\)
0.0812627 + 0.996693i \(0.474105\pi\)
\(830\) 0 0
\(831\) 29.0526 1.00782
\(832\) 0 0
\(833\) 2.53590 0.0878637
\(834\) 0 0
\(835\) −7.07180 −0.244730
\(836\) 0 0
\(837\) −1.33975 −0.0463084
\(838\) 0 0
\(839\) −35.6603 −1.23113 −0.615564 0.788087i \(-0.711072\pi\)
−0.615564 + 0.788087i \(0.711072\pi\)
\(840\) 0 0
\(841\) 6.14359 0.211848
\(842\) 0 0
\(843\) 11.1244 0.383143
\(844\) 0 0
\(845\) −10.0000 −0.344010
\(846\) 0 0
\(847\) 9.66025 0.331930
\(848\) 0 0
\(849\) −13.3205 −0.457159
\(850\) 0 0
\(851\) 2.73205 0.0936535
\(852\) 0 0
\(853\) −18.7846 −0.643173 −0.321586 0.946880i \(-0.604216\pi\)
−0.321586 + 0.946880i \(0.604216\pi\)
\(854\) 0 0
\(855\) 5.46410 0.186868
\(856\) 0 0
\(857\) 13.0526 0.445867 0.222933 0.974834i \(-0.428437\pi\)
0.222933 + 0.974834i \(0.428437\pi\)
\(858\) 0 0
\(859\) −6.66025 −0.227245 −0.113622 0.993524i \(-0.536245\pi\)
−0.113622 + 0.993524i \(0.536245\pi\)
\(860\) 0 0
\(861\) −6.73205 −0.229428
\(862\) 0 0
\(863\) −5.24871 −0.178668 −0.0893341 0.996002i \(-0.528474\pi\)
−0.0893341 + 0.996002i \(0.528474\pi\)
\(864\) 0 0
\(865\) −19.8564 −0.675138
\(866\) 0 0
\(867\) 12.8564 0.436626
\(868\) 0 0
\(869\) −4.53590 −0.153870
\(870\) 0 0
\(871\) −20.1962 −0.684321
\(872\) 0 0
\(873\) 15.3205 0.518521
\(874\) 0 0
\(875\) −2.73205 −0.0923602
\(876\) 0 0
\(877\) −26.3923 −0.891205 −0.445602 0.895231i \(-0.647010\pi\)
−0.445602 + 0.895231i \(0.647010\pi\)
\(878\) 0 0
\(879\) 27.7128 0.934730
\(880\) 0 0
\(881\) −8.39230 −0.282744 −0.141372 0.989957i \(-0.545151\pi\)
−0.141372 + 0.989957i \(0.545151\pi\)
\(882\) 0 0
\(883\) 34.0000 1.14419 0.572096 0.820187i \(-0.306131\pi\)
0.572096 + 0.820187i \(0.306131\pi\)
\(884\) 0 0
\(885\) 10.3923 0.349334
\(886\) 0 0
\(887\) 37.1051 1.24587 0.622934 0.782274i \(-0.285941\pi\)
0.622934 + 0.782274i \(0.285941\pi\)
\(888\) 0 0
\(889\) 29.6603 0.994773
\(890\) 0 0
\(891\) −2.73205 −0.0915271
\(892\) 0 0
\(893\) −16.1962 −0.541984
\(894\) 0 0
\(895\) −2.66025 −0.0889225
\(896\) 0 0
\(897\) 1.73205 0.0578315
\(898\) 0 0
\(899\) 1.58846 0.0529780
\(900\) 0 0
\(901\) 2.92820 0.0975526
\(902\) 0 0
\(903\) 18.3923 0.612058
\(904\) 0 0
\(905\) 0.339746 0.0112935
\(906\) 0 0
\(907\) 24.7321 0.821214 0.410607 0.911812i \(-0.365317\pi\)
0.410607 + 0.911812i \(0.365317\pi\)
\(908\) 0 0
\(909\) −35.7128 −1.18452
\(910\) 0 0
\(911\) −7.41154 −0.245555 −0.122778 0.992434i \(-0.539180\pi\)
−0.122778 + 0.992434i \(0.539180\pi\)
\(912\) 0 0
\(913\) 10.9282 0.361671
\(914\) 0 0
\(915\) −7.66025 −0.253240
\(916\) 0 0
\(917\) −34.5885 −1.14221
\(918\) 0 0
\(919\) −17.8038 −0.587295 −0.293647 0.955914i \(-0.594869\pi\)
−0.293647 + 0.955914i \(0.594869\pi\)
\(920\) 0 0
\(921\) 11.4641 0.377755
\(922\) 0 0
\(923\) −21.0000 −0.691223
\(924\) 0 0
\(925\) 2.73205 0.0898293
\(926\) 0 0
\(927\) 10.9282 0.358929
\(928\) 0 0
\(929\) −19.3923 −0.636241 −0.318120 0.948050i \(-0.603052\pi\)
−0.318120 + 0.948050i \(0.603052\pi\)
\(930\) 0 0
\(931\) −1.26795 −0.0415554
\(932\) 0 0
\(933\) 33.0526 1.08209
\(934\) 0 0
\(935\) −14.9282 −0.488204
\(936\) 0 0
\(937\) 45.6603 1.49166 0.745828 0.666139i \(-0.232054\pi\)
0.745828 + 0.666139i \(0.232054\pi\)
\(938\) 0 0
\(939\) −18.9282 −0.617699
\(940\) 0 0
\(941\) −37.5692 −1.22472 −0.612361 0.790578i \(-0.709780\pi\)
−0.612361 + 0.790578i \(0.709780\pi\)
\(942\) 0 0
\(943\) 2.46410 0.0802422
\(944\) 0 0
\(945\) 13.6603 0.444368
\(946\) 0 0
\(947\) 40.4641 1.31491 0.657453 0.753495i \(-0.271634\pi\)
0.657453 + 0.753495i \(0.271634\pi\)
\(948\) 0 0
\(949\) −21.2487 −0.689762
\(950\) 0 0
\(951\) −20.9282 −0.678643
\(952\) 0 0
\(953\) −39.1769 −1.26906 −0.634532 0.772896i \(-0.718807\pi\)
−0.634532 + 0.772896i \(0.718807\pi\)
\(954\) 0 0
\(955\) 2.39230 0.0774132
\(956\) 0 0
\(957\) −16.1962 −0.523547
\(958\) 0 0
\(959\) −14.9282 −0.482057
\(960\) 0 0
\(961\) −30.9282 −0.997684
\(962\) 0 0
\(963\) 14.5359 0.468413
\(964\) 0 0
\(965\) −7.33975 −0.236275
\(966\) 0 0
\(967\) −3.53590 −0.113707 −0.0568534 0.998383i \(-0.518107\pi\)
−0.0568534 + 0.998383i \(0.518107\pi\)
\(968\) 0 0
\(969\) −14.9282 −0.479563
\(970\) 0 0
\(971\) −18.0000 −0.577647 −0.288824 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 0 0
\(973\) 41.5167 1.33096
\(974\) 0 0
\(975\) 1.73205 0.0554700
\(976\) 0 0
\(977\) −2.44486 −0.0782181 −0.0391091 0.999235i \(-0.512452\pi\)
−0.0391091 + 0.999235i \(0.512452\pi\)
\(978\) 0 0
\(979\) 48.2487 1.54204
\(980\) 0 0
\(981\) −4.67949 −0.149405
\(982\) 0 0
\(983\) −39.1769 −1.24955 −0.624775 0.780805i \(-0.714809\pi\)
−0.624775 + 0.780805i \(0.714809\pi\)
\(984\) 0 0
\(985\) −16.6603 −0.530840
\(986\) 0 0
\(987\) −16.1962 −0.515529
\(988\) 0 0
\(989\) −6.73205 −0.214067
\(990\) 0 0
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) 0 0
\(993\) −0.267949 −0.00850311
\(994\) 0 0
\(995\) −11.8038 −0.374207
\(996\) 0 0
\(997\) 6.67949 0.211542 0.105771 0.994391i \(-0.466269\pi\)
0.105771 + 0.994391i \(0.466269\pi\)
\(998\) 0 0
\(999\) −13.6603 −0.432191
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 7360.2.a.bs.1.1 2
4.3 odd 2 7360.2.a.bg.1.2 2
8.3 odd 2 3680.2.a.n.1.2 yes 2
8.5 even 2 3680.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3680.2.a.l.1.1 2 8.5 even 2
3680.2.a.n.1.2 yes 2 8.3 odd 2
7360.2.a.bg.1.2 2 4.3 odd 2
7360.2.a.bs.1.1 2 1.1 even 1 trivial