Properties

Label 1856.2.a.o.1.1
Level $1856$
Weight $2$
Character 1856.1
Self dual yes
Analytic conductor $14.820$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8202346151\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 116)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -3.00000 q^{5} +4.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -3.00000 q^{5} +4.00000 q^{7} +6.00000 q^{9} +1.00000 q^{11} +3.00000 q^{13} -9.00000 q^{15} +2.00000 q^{17} -4.00000 q^{19} +12.0000 q^{21} -6.00000 q^{23} +4.00000 q^{25} +9.00000 q^{27} +1.00000 q^{29} +9.00000 q^{31} +3.00000 q^{33} -12.0000 q^{35} +8.00000 q^{37} +9.00000 q^{39} -8.00000 q^{41} +5.00000 q^{43} -18.0000 q^{45} -7.00000 q^{47} +9.00000 q^{49} +6.00000 q^{51} +5.00000 q^{53} -3.00000 q^{55} -12.0000 q^{57} +10.0000 q^{59} -10.0000 q^{61} +24.0000 q^{63} -9.00000 q^{65} -8.00000 q^{67} -18.0000 q^{69} -2.00000 q^{71} +12.0000 q^{75} +4.00000 q^{77} -1.00000 q^{79} +9.00000 q^{81} -6.00000 q^{83} -6.00000 q^{85} +3.00000 q^{87} +12.0000 q^{89} +12.0000 q^{91} +27.0000 q^{93} +12.0000 q^{95} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) −9.00000 −2.32379
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 12.0000 2.61861
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 9.00000 1.61645 0.808224 0.588875i \(-0.200429\pi\)
0.808224 + 0.588875i \(0.200429\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) −12.0000 −2.02837
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 9.00000 1.44115
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) −18.0000 −2.68328
\(46\) 0 0
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 5.00000 0.686803 0.343401 0.939189i \(-0.388421\pi\)
0.343401 + 0.939189i \(0.388421\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 24.0000 3.02372
\(64\) 0 0
\(65\) −9.00000 −1.11631
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −18.0000 −2.16695
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 12.0000 1.38564
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 0 0
\(93\) 27.0000 2.79977
\(94\) 0 0
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) −36.0000 −3.51324
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 24.0000 2.27798
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 18.0000 1.67851
\(116\) 0 0
\(117\) 18.0000 1.66410
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) −24.0000 −2.16401
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 15.0000 1.32068
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −16.0000 −1.38738
\(134\) 0 0
\(135\) −27.0000 −2.32379
\(136\) 0 0
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 0 0
\(139\) 18.0000 1.52674 0.763370 0.645961i \(-0.223543\pi\)
0.763370 + 0.645961i \(0.223543\pi\)
\(140\) 0 0
\(141\) −21.0000 −1.76852
\(142\) 0 0
\(143\) 3.00000 0.250873
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 27.0000 2.22692
\(148\) 0 0
\(149\) −9.00000 −0.737309 −0.368654 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) −27.0000 −2.16869
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 0 0
\(159\) 15.0000 1.18958
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) −7.00000 −0.548282 −0.274141 0.961689i \(-0.588394\pi\)
−0.274141 + 0.961689i \(0.588394\pi\)
\(164\) 0 0
\(165\) −9.00000 −0.700649
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −24.0000 −1.83533
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) 16.0000 1.20949
\(176\) 0 0
\(177\) 30.0000 2.25494
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −17.0000 −1.26360 −0.631800 0.775131i \(-0.717684\pi\)
−0.631800 + 0.775131i \(0.717684\pi\)
\(182\) 0 0
\(183\) −30.0000 −2.21766
\(184\) 0 0
\(185\) −24.0000 −1.76452
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 0 0
\(189\) 36.0000 2.61861
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −8.00000 −0.575853 −0.287926 0.957653i \(-0.592966\pi\)
−0.287926 + 0.957653i \(0.592966\pi\)
\(194\) 0 0
\(195\) −27.0000 −1.93351
\(196\) 0 0
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) −24.0000 −1.69283
\(202\) 0 0
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) 24.0000 1.67623
\(206\) 0 0
\(207\) −36.0000 −2.50217
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) 0 0
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) −15.0000 −1.02299
\(216\) 0 0
\(217\) 36.0000 2.44384
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) 24.0000 1.60000
\(226\) 0 0
\(227\) −30.0000 −1.99117 −0.995585 0.0938647i \(-0.970078\pi\)
−0.995585 + 0.0938647i \(0.970078\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 0 0
\(233\) 23.0000 1.50678 0.753390 0.657574i \(-0.228417\pi\)
0.753390 + 0.657574i \(0.228417\pi\)
\(234\) 0 0
\(235\) 21.0000 1.36989
\(236\) 0 0
\(237\) −3.00000 −0.194871
\(238\) 0 0
\(239\) −22.0000 −1.42306 −0.711531 0.702655i \(-0.751998\pi\)
−0.711531 + 0.702655i \(0.751998\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −27.0000 −1.72497
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) 31.0000 1.95670 0.978351 0.206951i \(-0.0663540\pi\)
0.978351 + 0.206951i \(0.0663540\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 0 0
\(255\) −18.0000 −1.12720
\(256\) 0 0
\(257\) −27.0000 −1.68421 −0.842107 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) 0 0
\(259\) 32.0000 1.98838
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 13.0000 0.801614 0.400807 0.916162i \(-0.368730\pi\)
0.400807 + 0.916162i \(0.368730\pi\)
\(264\) 0 0
\(265\) −15.0000 −0.921443
\(266\) 0 0
\(267\) 36.0000 2.20316
\(268\) 0 0
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) −5.00000 −0.303728 −0.151864 0.988401i \(-0.548528\pi\)
−0.151864 + 0.988401i \(0.548528\pi\)
\(272\) 0 0
\(273\) 36.0000 2.17882
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 54.0000 3.23290
\(280\) 0 0
\(281\) −1.00000 −0.0596550 −0.0298275 0.999555i \(-0.509496\pi\)
−0.0298275 + 0.999555i \(0.509496\pi\)
\(282\) 0 0
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 0 0
\(285\) 36.0000 2.13246
\(286\) 0 0
\(287\) −32.0000 −1.88890
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 0 0
\(295\) −30.0000 −1.74667
\(296\) 0 0
\(297\) 9.00000 0.522233
\(298\) 0 0
\(299\) −18.0000 −1.04097
\(300\) 0 0
\(301\) 20.0000 1.15278
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 0 0
\(309\) 18.0000 1.02398
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 0 0
\(315\) −72.0000 −4.05674
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 1.00000 0.0559893
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 12.0000 0.665640
\(326\) 0 0
\(327\) 15.0000 0.829502
\(328\) 0 0
\(329\) −28.0000 −1.54369
\(330\) 0 0
\(331\) −3.00000 −0.164895 −0.0824475 0.996595i \(-0.526274\pi\)
−0.0824475 + 0.996595i \(0.526274\pi\)
\(332\) 0 0
\(333\) 48.0000 2.63038
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 9.00000 0.487377
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 54.0000 2.90726
\(346\) 0 0
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) −35.0000 −1.87351 −0.936754 0.349990i \(-0.886185\pi\)
−0.936754 + 0.349990i \(0.886185\pi\)
\(350\) 0 0
\(351\) 27.0000 1.44115
\(352\) 0 0
\(353\) 22.0000 1.17094 0.585471 0.810693i \(-0.300910\pi\)
0.585471 + 0.810693i \(0.300910\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 0 0
\(357\) 24.0000 1.27021
\(358\) 0 0
\(359\) 3.00000 0.158334 0.0791670 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −30.0000 −1.57459
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 0 0
\(369\) −48.0000 −2.49878
\(370\) 0 0
\(371\) 20.0000 1.03835
\(372\) 0 0
\(373\) 19.0000 0.983783 0.491891 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(374\) 0 0
\(375\) 9.00000 0.464758
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) −32.0000 −1.64373 −0.821865 0.569683i \(-0.807066\pi\)
−0.821865 + 0.569683i \(0.807066\pi\)
\(380\) 0 0
\(381\) −24.0000 −1.22956
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 0 0
\(387\) 30.0000 1.52499
\(388\) 0 0
\(389\) −4.00000 −0.202808 −0.101404 0.994845i \(-0.532333\pi\)
−0.101404 + 0.994845i \(0.532333\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) −36.0000 −1.81596
\(394\) 0 0
\(395\) 3.00000 0.150946
\(396\) 0 0
\(397\) −13.0000 −0.652451 −0.326226 0.945292i \(-0.605777\pi\)
−0.326226 + 0.945292i \(0.605777\pi\)
\(398\) 0 0
\(399\) −48.0000 −2.40301
\(400\) 0 0
\(401\) −5.00000 −0.249688 −0.124844 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) 0 0
\(403\) 27.0000 1.34497
\(404\) 0 0
\(405\) −27.0000 −1.34164
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 40.0000 1.96827
\(414\) 0 0
\(415\) 18.0000 0.883585
\(416\) 0 0
\(417\) 54.0000 2.64439
\(418\) 0 0
\(419\) −10.0000 −0.488532 −0.244266 0.969708i \(-0.578547\pi\)
−0.244266 + 0.969708i \(0.578547\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) −42.0000 −2.04211
\(424\) 0 0
\(425\) 8.00000 0.388057
\(426\) 0 0
\(427\) −40.0000 −1.93574
\(428\) 0 0
\(429\) 9.00000 0.434524
\(430\) 0 0
\(431\) −14.0000 −0.674356 −0.337178 0.941441i \(-0.609472\pi\)
−0.337178 + 0.941441i \(0.609472\pi\)
\(432\) 0 0
\(433\) −28.0000 −1.34559 −0.672797 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 0 0
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) 54.0000 2.57143
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) −36.0000 −1.70656
\(446\) 0 0
\(447\) −27.0000 −1.27706
\(448\) 0 0
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 12.0000 0.563809
\(454\) 0 0
\(455\) −36.0000 −1.68771
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 0 0
\(459\) 18.0000 0.840168
\(460\) 0 0
\(461\) 42.0000 1.95614 0.978068 0.208288i \(-0.0667892\pi\)
0.978068 + 0.208288i \(0.0667892\pi\)
\(462\) 0 0
\(463\) −10.0000 −0.464739 −0.232370 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) 0 0
\(465\) −81.0000 −3.75629
\(466\) 0 0
\(467\) 13.0000 0.601568 0.300784 0.953692i \(-0.402752\pi\)
0.300784 + 0.953692i \(0.402752\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) −12.0000 −0.552931
\(472\) 0 0
\(473\) 5.00000 0.229900
\(474\) 0 0
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) 30.0000 1.37361
\(478\) 0 0
\(479\) −17.0000 −0.776750 −0.388375 0.921501i \(-0.626963\pi\)
−0.388375 + 0.921501i \(0.626963\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 0 0
\(483\) −72.0000 −3.27611
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 0 0
\(489\) −21.0000 −0.949653
\(490\) 0 0
\(491\) −17.0000 −0.767199 −0.383600 0.923499i \(-0.625316\pi\)
−0.383600 + 0.923499i \(0.625316\pi\)
\(492\) 0 0
\(493\) 2.00000 0.0900755
\(494\) 0 0
\(495\) −18.0000 −0.809040
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) 0 0
\(503\) 23.0000 1.02552 0.512760 0.858532i \(-0.328623\pi\)
0.512760 + 0.858532i \(0.328623\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 0 0
\(509\) −15.0000 −0.664863 −0.332432 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −36.0000 −1.58944
\(514\) 0 0
\(515\) −18.0000 −0.793175
\(516\) 0 0
\(517\) −7.00000 −0.307860
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 48.0000 2.09489
\(526\) 0 0
\(527\) 18.0000 0.784092
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 60.0000 2.60378
\(532\) 0 0
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) 0 0
\(537\) −30.0000 −1.29460
\(538\) 0 0
\(539\) 9.00000 0.387657
\(540\) 0 0
\(541\) −40.0000 −1.71973 −0.859867 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) 0 0
\(543\) −51.0000 −2.18862
\(544\) 0 0
\(545\) −15.0000 −0.642529
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) −60.0000 −2.56074
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 0 0
\(555\) −72.0000 −3.05623
\(556\) 0 0
\(557\) −42.0000 −1.77960 −0.889799 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 0 0
\(559\) 15.0000 0.634432
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 0 0
\(563\) 33.0000 1.39078 0.695392 0.718631i \(-0.255231\pi\)
0.695392 + 0.718631i \(0.255231\pi\)
\(564\) 0 0
\(565\) 18.0000 0.757266
\(566\) 0 0
\(567\) 36.0000 1.51186
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.0000 −1.00087
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 0 0
\(579\) −24.0000 −0.997406
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) 0 0
\(583\) 5.00000 0.207079
\(584\) 0 0
\(585\) −54.0000 −2.23263
\(586\) 0 0
\(587\) −42.0000 −1.73353 −0.866763 0.498721i \(-0.833803\pi\)
−0.866763 + 0.498721i \(0.833803\pi\)
\(588\) 0 0
\(589\) −36.0000 −1.48335
\(590\) 0 0
\(591\) −78.0000 −3.20849
\(592\) 0 0
\(593\) −29.0000 −1.19089 −0.595444 0.803397i \(-0.703024\pi\)
−0.595444 + 0.803397i \(0.703024\pi\)
\(594\) 0 0
\(595\) −24.0000 −0.983904
\(596\) 0 0
\(597\) −60.0000 −2.45564
\(598\) 0 0
\(599\) 7.00000 0.286012 0.143006 0.989722i \(-0.454323\pi\)
0.143006 + 0.989722i \(0.454323\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) −48.0000 −1.95471
\(604\) 0 0
\(605\) 30.0000 1.21967
\(606\) 0 0
\(607\) 47.0000 1.90767 0.953836 0.300329i \(-0.0970966\pi\)
0.953836 + 0.300329i \(0.0970966\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −21.0000 −0.849569
\(612\) 0 0
\(613\) 21.0000 0.848182 0.424091 0.905620i \(-0.360594\pi\)
0.424091 + 0.905620i \(0.360594\pi\)
\(614\) 0 0
\(615\) 72.0000 2.90332
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 0 0
\(619\) 41.0000 1.64793 0.823965 0.566641i \(-0.191757\pi\)
0.823965 + 0.566641i \(0.191757\pi\)
\(620\) 0 0
\(621\) −54.0000 −2.16695
\(622\) 0 0
\(623\) 48.0000 1.92308
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) −12.0000 −0.479234
\(628\) 0 0
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −14.0000 −0.557331 −0.278666 0.960388i \(-0.589892\pi\)
−0.278666 + 0.960388i \(0.589892\pi\)
\(632\) 0 0
\(633\) 39.0000 1.55011
\(634\) 0 0
\(635\) 24.0000 0.952411
\(636\) 0 0
\(637\) 27.0000 1.06978
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 0 0
\(645\) −45.0000 −1.77187
\(646\) 0 0
\(647\) 22.0000 0.864909 0.432455 0.901656i \(-0.357648\pi\)
0.432455 + 0.901656i \(0.357648\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) 108.000 4.23285
\(652\) 0 0
\(653\) −4.00000 −0.156532 −0.0782660 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) 36.0000 1.40664
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 18.0000 0.699062
\(664\) 0 0
\(665\) 48.0000 1.86136
\(666\) 0 0
\(667\) −6.00000 −0.232321
\(668\) 0 0
\(669\) 42.0000 1.62381
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) 25.0000 0.963679 0.481840 0.876259i \(-0.339969\pi\)
0.481840 + 0.876259i \(0.339969\pi\)
\(674\) 0 0
\(675\) 36.0000 1.38564
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −90.0000 −3.44881
\(682\) 0 0
\(683\) −8.00000 −0.306111 −0.153056 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) −60.0000 −2.28914
\(688\) 0 0
\(689\) 15.0000 0.571454
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 24.0000 0.911685
\(694\) 0 0
\(695\) −54.0000 −2.04834
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) 0 0
\(699\) 69.0000 2.60982
\(700\) 0 0
\(701\) 35.0000 1.32193 0.660966 0.750416i \(-0.270147\pi\)
0.660966 + 0.750416i \(0.270147\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) 63.0000 2.37272
\(706\) 0 0
\(707\) 16.0000 0.601742
\(708\) 0 0
\(709\) −29.0000 −1.08912 −0.544559 0.838723i \(-0.683303\pi\)
−0.544559 + 0.838723i \(0.683303\pi\)
\(710\) 0 0
\(711\) −6.00000 −0.225018
\(712\) 0 0
\(713\) −54.0000 −2.02232
\(714\) 0 0
\(715\) −9.00000 −0.336581
\(716\) 0 0
\(717\) −66.0000 −2.46482
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 0 0
\(723\) 51.0000 1.89671
\(724\) 0 0
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 10.0000 0.369863
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) −81.0000 −2.98773
\(736\) 0 0
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) −1.00000 −0.0367856 −0.0183928 0.999831i \(-0.505855\pi\)
−0.0183928 + 0.999831i \(0.505855\pi\)
\(740\) 0 0
\(741\) −36.0000 −1.32249
\(742\) 0 0
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 27.0000 0.989203
\(746\) 0 0
\(747\) −36.0000 −1.31717
\(748\) 0 0
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) 44.0000 1.60558 0.802791 0.596260i \(-0.203347\pi\)
0.802791 + 0.596260i \(0.203347\pi\)
\(752\) 0 0
\(753\) 93.0000 3.38911
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 0 0
\(759\) −18.0000 −0.653359
\(760\) 0 0
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) 0 0
\(763\) 20.0000 0.724049
\(764\) 0 0
\(765\) −36.0000 −1.30158
\(766\) 0 0
\(767\) 30.0000 1.08324
\(768\) 0 0
\(769\) 6.00000 0.216366 0.108183 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) 0 0
\(771\) −81.0000 −2.91714
\(772\) 0 0
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) 0 0
\(775\) 36.0000 1.29316
\(776\) 0 0
\(777\) 96.0000 3.44398
\(778\) 0 0
\(779\) 32.0000 1.14652
\(780\) 0 0
\(781\) −2.00000 −0.0715656
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) 14.0000 0.499046 0.249523 0.968369i \(-0.419726\pi\)
0.249523 + 0.968369i \(0.419726\pi\)
\(788\) 0 0
\(789\) 39.0000 1.38844
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) −30.0000 −1.06533
\(794\) 0 0
\(795\) −45.0000 −1.59599
\(796\) 0 0
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) −14.0000 −0.495284
\(800\) 0 0
\(801\) 72.0000 2.54399
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 72.0000 2.53767
\(806\) 0 0
\(807\) −42.0000 −1.47847
\(808\) 0 0
\(809\) −36.0000 −1.26569 −0.632846 0.774277i \(-0.718114\pi\)
−0.632846 + 0.774277i \(0.718114\pi\)
\(810\) 0 0
\(811\) 24.0000 0.842754 0.421377 0.906886i \(-0.361547\pi\)
0.421377 + 0.906886i \(0.361547\pi\)
\(812\) 0 0
\(813\) −15.0000 −0.526073
\(814\) 0 0
\(815\) 21.0000 0.735598
\(816\) 0 0
\(817\) −20.0000 −0.699711
\(818\) 0 0
\(819\) 72.0000 2.51588
\(820\) 0 0
\(821\) 43.0000 1.50071 0.750355 0.661035i \(-0.229882\pi\)
0.750355 + 0.661035i \(0.229882\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 0 0
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) 13.0000 0.452054 0.226027 0.974121i \(-0.427426\pi\)
0.226027 + 0.974121i \(0.427426\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 30.0000 1.04069
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) −6.00000 −0.207639
\(836\) 0 0
\(837\) 81.0000 2.79977
\(838\) 0 0
\(839\) 9.00000 0.310715 0.155357 0.987858i \(-0.450347\pi\)
0.155357 + 0.987858i \(0.450347\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −3.00000 −0.103325
\(844\) 0 0
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) −40.0000 −1.37442
\(848\) 0 0
\(849\) 18.0000 0.617758
\(850\) 0 0
\(851\) −48.0000 −1.64542
\(852\) 0 0
\(853\) 42.0000 1.43805 0.719026 0.694983i \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) 0 0
\(855\) 72.0000 2.46235
\(856\) 0 0
\(857\) 49.0000 1.67381 0.836904 0.547350i \(-0.184363\pi\)
0.836904 + 0.547350i \(0.184363\pi\)
\(858\) 0 0
\(859\) −9.00000 −0.307076 −0.153538 0.988143i \(-0.549067\pi\)
−0.153538 + 0.988143i \(0.549067\pi\)
\(860\) 0 0
\(861\) −96.0000 −3.27167
\(862\) 0 0
\(863\) 54.0000 1.83818 0.919091 0.394046i \(-0.128925\pi\)
0.919091 + 0.394046i \(0.128925\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) −39.0000 −1.32451
\(868\) 0 0
\(869\) −1.00000 −0.0339227
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) 29.0000 0.979260 0.489630 0.871930i \(-0.337132\pi\)
0.489630 + 0.871930i \(0.337132\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) 24.0000 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) 0 0
\(885\) −90.0000 −3.02532
\(886\) 0 0
\(887\) −55.0000 −1.84672 −0.923360 0.383936i \(-0.874568\pi\)
−0.923360 + 0.383936i \(0.874568\pi\)
\(888\) 0 0
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) 9.00000 0.301511
\(892\) 0 0
\(893\) 28.0000 0.936984
\(894\) 0 0
\(895\) 30.0000 1.00279
\(896\) 0 0
\(897\) −54.0000 −1.80301
\(898\) 0 0
\(899\) 9.00000 0.300167
\(900\) 0 0
\(901\) 10.0000 0.333148
\(902\) 0 0
\(903\) 60.0000 1.99667
\(904\) 0 0
\(905\) 51.0000 1.69530
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 0 0
\(909\) 24.0000 0.796030
\(910\) 0 0
\(911\) 51.0000 1.68971 0.844853 0.534999i \(-0.179688\pi\)
0.844853 + 0.534999i \(0.179688\pi\)
\(912\) 0 0
\(913\) −6.00000 −0.198571
\(914\) 0 0
\(915\) 90.0000 2.97531
\(916\) 0 0
\(917\) −48.0000 −1.58510
\(918\) 0 0
\(919\) −18.0000 −0.593765 −0.296883 0.954914i \(-0.595947\pi\)
−0.296883 + 0.954914i \(0.595947\pi\)
\(920\) 0 0
\(921\) −21.0000 −0.691974
\(922\) 0 0
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) 32.0000 1.05215
\(926\) 0 0
\(927\) 36.0000 1.18240
\(928\) 0 0
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.00000 −0.196221
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 3.00000 0.0979013
\(940\) 0 0
\(941\) −27.0000 −0.880175 −0.440087 0.897955i \(-0.645053\pi\)
−0.440087 + 0.897955i \(0.645053\pi\)
\(942\) 0 0
\(943\) 48.0000 1.56310
\(944\) 0 0
\(945\) −108.000 −3.51324
\(946\) 0 0
\(947\) −27.0000 −0.877382 −0.438691 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −54.0000 −1.75107
\(952\) 0 0
\(953\) 3.00000 0.0971795 0.0485898 0.998819i \(-0.484527\pi\)
0.0485898 + 0.998819i \(0.484527\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 3.00000 0.0969762
\(958\) 0 0
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) 48.0000 1.54678
\(964\) 0 0
\(965\) 24.0000 0.772587
\(966\) 0 0
\(967\) 17.0000 0.546683 0.273342 0.961917i \(-0.411871\pi\)
0.273342 + 0.961917i \(0.411871\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) 0 0
\(973\) 72.0000 2.30821
\(974\) 0 0
\(975\) 36.0000 1.15292
\(976\) 0 0
\(977\) 25.0000 0.799821 0.399910 0.916554i \(-0.369041\pi\)
0.399910 + 0.916554i \(0.369041\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 30.0000 0.957826
\(982\) 0 0
\(983\) 45.0000 1.43528 0.717639 0.696416i \(-0.245223\pi\)
0.717639 + 0.696416i \(0.245223\pi\)
\(984\) 0 0
\(985\) 78.0000 2.48529
\(986\) 0 0
\(987\) −84.0000 −2.67375
\(988\) 0 0
\(989\) −30.0000 −0.953945
\(990\) 0 0
\(991\) −34.0000 −1.08005 −0.540023 0.841650i \(-0.681584\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(992\) 0 0
\(993\) −9.00000 −0.285606
\(994\) 0 0
\(995\) 60.0000 1.90213
\(996\) 0 0
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 0 0
\(999\) 72.0000 2.27798
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 1856.2.a.o.1.1 1
4.3 odd 2 1856.2.a.a.1.1 1
8.3 odd 2 464.2.a.g.1.1 1
8.5 even 2 116.2.a.a.1.1 1
24.5 odd 2 1044.2.a.c.1.1 1
24.11 even 2 4176.2.a.c.1.1 1
40.13 odd 4 2900.2.c.a.349.1 2
40.29 even 2 2900.2.a.e.1.1 1
40.37 odd 4 2900.2.c.a.349.2 2
56.13 odd 2 5684.2.a.k.1.1 1
232.133 odd 4 3364.2.c.b.1681.2 2
232.157 odd 4 3364.2.c.b.1681.1 2
232.173 even 2 3364.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
116.2.a.a.1.1 1 8.5 even 2
464.2.a.g.1.1 1 8.3 odd 2
1044.2.a.c.1.1 1 24.5 odd 2
1856.2.a.a.1.1 1 4.3 odd 2
1856.2.a.o.1.1 1 1.1 even 1 trivial
2900.2.a.e.1.1 1 40.29 even 2
2900.2.c.a.349.1 2 40.13 odd 4
2900.2.c.a.349.2 2 40.37 odd 4
3364.2.a.c.1.1 1 232.173 even 2
3364.2.c.b.1681.1 2 232.157 odd 4
3364.2.c.b.1681.2 2 232.133 odd 4
4176.2.a.c.1.1 1 24.11 even 2
5684.2.a.k.1.1 1 56.13 odd 2