Properties

Label 441.2.a.b.1.1
Level $441$
Weight $2$
Character 441.1
Self dual yes
Analytic conductor $3.521$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} +2.00000 q^{5} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} +2.00000 q^{5} -4.00000 q^{10} +2.00000 q^{11} +1.00000 q^{13} -4.00000 q^{16} +1.00000 q^{19} +4.00000 q^{20} -4.00000 q^{22} -1.00000 q^{25} -2.00000 q^{26} -4.00000 q^{29} +9.00000 q^{31} +8.00000 q^{32} +3.00000 q^{37} -2.00000 q^{38} +10.0000 q^{41} +5.00000 q^{43} +4.00000 q^{44} +6.00000 q^{47} +2.00000 q^{50} +2.00000 q^{52} -12.0000 q^{53} +4.00000 q^{55} +8.00000 q^{58} +12.0000 q^{59} +10.0000 q^{61} -18.0000 q^{62} -8.00000 q^{64} +2.00000 q^{65} -5.00000 q^{67} +6.00000 q^{71} -3.00000 q^{73} -6.00000 q^{74} +2.00000 q^{76} -1.00000 q^{79} -8.00000 q^{80} -20.0000 q^{82} -6.00000 q^{83} -10.0000 q^{86} -16.0000 q^{89} -12.0000 q^{94} +2.00000 q^{95} -6.00000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) −4.00000 −1.26491
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 9.00000 1.61645 0.808224 0.588875i \(-0.200429\pi\)
0.808224 + 0.588875i \(0.200429\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 2.00000 0.282843
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 8.00000 1.05045
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −18.0000 −2.28600
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −5.00000 −0.610847 −0.305424 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −3.00000 −0.351123 −0.175562 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −8.00000 −0.894427
\(81\) 0 0
\(82\) −20.0000 −2.20863
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 0 0
\(88\) 0 0
\(89\) −16.0000 −1.69600 −0.847998 0.529999i \(-0.822192\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 24.0000 2.33109
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 9.00000 0.862044 0.431022 0.902342i \(-0.358153\pi\)
0.431022 + 0.902342i \(0.358153\pi\)
\(110\) −8.00000 −0.762770
\(111\) 0 0
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) −24.0000 −2.20938
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −20.0000 −1.81071
\(123\) 0 0
\(124\) 18.0000 1.61645
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −15.0000 −1.33103 −0.665517 0.746382i \(-0.731789\pi\)
−0.665517 + 0.746382i \(0.731789\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 10.0000 0.863868
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −3.00000 −0.254457 −0.127228 0.991873i \(-0.540608\pi\)
−0.127228 + 0.991873i \(0.540608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 18.0000 1.44579
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 2.00000 0.159111
\(159\) 0 0
\(160\) 16.0000 1.26491
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 20.0000 1.56174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 14.0000 1.08335 0.541676 0.840587i \(-0.317790\pi\)
0.541676 + 0.840587i \(0.317790\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 10.0000 0.762493
\(173\) −8.00000 −0.608229 −0.304114 0.952636i \(-0.598361\pi\)
−0.304114 + 0.952636i \(0.598361\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −8.00000 −0.603023
\(177\) 0 0
\(178\) 32.0000 2.39850
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) 13.0000 0.966282 0.483141 0.875542i \(-0.339496\pi\)
0.483141 + 0.875542i \(0.339496\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 12.0000 0.861550
\(195\) 0 0
\(196\) 0 0
\(197\) −16.0000 −1.13995 −0.569976 0.821661i \(-0.693048\pi\)
−0.569976 + 0.821661i \(0.693048\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 4.00000 0.281439
\(203\) 0 0
\(204\) 0 0
\(205\) 20.0000 1.39686
\(206\) 14.0000 0.975426
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −24.0000 −1.64833
\(213\) 0 0
\(214\) −16.0000 −1.09374
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) 0 0
\(218\) −18.0000 −1.21911
\(219\) 0 0
\(220\) 8.00000 0.539360
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 20.0000 1.33038
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −19.0000 −1.25556 −0.627778 0.778393i \(-0.716035\pi\)
−0.627778 + 0.778393i \(0.716035\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 24.0000 1.56227
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 14.0000 0.899954
\(243\) 0 0
\(244\) 20.0000 1.28037
\(245\) 0 0
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) 0 0
\(249\) 0 0
\(250\) 24.0000 1.51789
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 30.0000 1.88237
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −26.0000 −1.62184 −0.810918 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) −28.0000 −1.72985
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 0 0
\(265\) −24.0000 −1.47431
\(266\) 0 0
\(267\) 0 0
\(268\) −10.0000 −0.610847
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −24.0000 −1.44989
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) 13.0000 0.781094 0.390547 0.920583i \(-0.372286\pi\)
0.390547 + 0.920583i \(0.372286\pi\)
\(278\) 6.00000 0.359856
\(279\) 0 0
\(280\) 0 0
\(281\) 4.00000 0.238620 0.119310 0.992857i \(-0.461932\pi\)
0.119310 + 0.992857i \(0.461932\pi\)
\(282\) 0 0
\(283\) −11.0000 −0.653882 −0.326941 0.945045i \(-0.606018\pi\)
−0.326941 + 0.945045i \(0.606018\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 16.0000 0.939552
\(291\) 0 0
\(292\) −6.00000 −0.351123
\(293\) −8.00000 −0.467365 −0.233682 0.972313i \(-0.575078\pi\)
−0.233682 + 0.972313i \(0.575078\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) 0 0
\(298\) −24.0000 −1.39028
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 32.0000 1.84139
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 20.0000 1.14520
\(306\) 0 0
\(307\) −17.0000 −0.970241 −0.485121 0.874447i \(-0.661224\pi\)
−0.485121 + 0.874447i \(0.661224\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −36.0000 −2.04466
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 28.0000 1.58013
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) −24.0000 −1.34797 −0.673987 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) −16.0000 −0.894427
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) −28.0000 −1.53209
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 24.0000 1.30543
\(339\) 0 0
\(340\) 0 0
\(341\) 18.0000 0.974755
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 16.0000 0.860165
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 16.0000 0.852803
\(353\) −34.0000 −1.80964 −0.904819 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) −32.0000 −1.69600
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −26.0000 −1.36653
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −9.00000 −0.469796 −0.234898 0.972020i \(-0.575476\pi\)
−0.234898 + 0.972020i \(0.575476\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −12.0000 −0.623850
\(371\) 0 0
\(372\) 0 0
\(373\) 23.0000 1.19089 0.595447 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) 3.00000 0.154100 0.0770498 0.997027i \(-0.475450\pi\)
0.0770498 + 0.997027i \(0.475450\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) 20.0000 1.02329
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) −12.0000 −0.609208
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 32.0000 1.61214
\(395\) −2.00000 −0.100631
\(396\) 0 0
\(397\) −9.00000 −0.451697 −0.225849 0.974162i \(-0.572515\pi\)
−0.225849 + 0.974162i \(0.572515\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 0 0
\(403\) 9.00000 0.448322
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) −40.0000 −1.97546
\(411\) 0 0
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 8.00000 0.392232
\(417\) 0 0
\(418\) −4.00000 −0.195646
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −7.00000 −0.341159 −0.170580 0.985344i \(-0.554564\pi\)
−0.170580 + 0.985344i \(0.554564\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) −20.0000 −0.964486
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 31.0000 1.48976 0.744882 0.667196i \(-0.232506\pi\)
0.744882 + 0.667196i \(0.232506\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −32.0000 −1.51695
\(446\) −32.0000 −1.51524
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 20.0000 0.941763
\(452\) −20.0000 −0.940721
\(453\) 0 0
\(454\) 36.0000 1.68956
\(455\) 0 0
\(456\) 0 0
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) 38.0000 1.77562
\(459\) 0 0
\(460\) 0 0
\(461\) −20.0000 −0.931493 −0.465746 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) 0 0
\(463\) −17.0000 −0.790057 −0.395029 0.918669i \(-0.629265\pi\)
−0.395029 + 0.918669i \(0.629265\pi\)
\(464\) 16.0000 0.742781
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −24.0000 −1.10704
\(471\) 0 0
\(472\) 0 0
\(473\) 10.0000 0.459800
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) 3.00000 0.136788
\(482\) −28.0000 −1.27537
\(483\) 0 0
\(484\) −14.0000 −0.636364
\(485\) −12.0000 −0.544892
\(486\) 0 0
\(487\) 31.0000 1.40474 0.702372 0.711810i \(-0.252124\pi\)
0.702372 + 0.711810i \(0.252124\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −2.00000 −0.0899843
\(495\) 0 0
\(496\) −36.0000 −1.61645
\(497\) 0 0
\(498\) 0 0
\(499\) 37.0000 1.65635 0.828174 0.560471i \(-0.189380\pi\)
0.828174 + 0.560471i \(0.189380\pi\)
\(500\) −24.0000 −1.07331
\(501\) 0 0
\(502\) −16.0000 −0.714115
\(503\) 42.0000 1.87269 0.936344 0.351085i \(-0.114187\pi\)
0.936344 + 0.351085i \(0.114187\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) 0 0
\(508\) −30.0000 −1.33103
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 52.0000 2.29362
\(515\) −14.0000 −0.616914
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 31.0000 1.35554 0.677768 0.735276i \(-0.262948\pi\)
0.677768 + 0.735276i \(0.262948\pi\)
\(524\) 28.0000 1.22319
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 48.0000 2.08499
\(531\) 0 0
\(532\) 0 0
\(533\) 10.0000 0.433148
\(534\) 0 0
\(535\) 16.0000 0.691740
\(536\) 0 0
\(537\) 0 0
\(538\) 12.0000 0.517357
\(539\) 0 0
\(540\) 0 0
\(541\) −19.0000 −0.816874 −0.408437 0.912787i \(-0.633926\pi\)
−0.408437 + 0.912787i \(0.633926\pi\)
\(542\) −32.0000 −1.37452
\(543\) 0 0
\(544\) 0 0
\(545\) 18.0000 0.771035
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 24.0000 1.02523
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) 0 0
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) −6.00000 −0.254457
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 5.00000 0.211477
\(560\) 0 0
\(561\) 0 0
\(562\) −8.00000 −0.337460
\(563\) 26.0000 1.09577 0.547885 0.836554i \(-0.315433\pi\)
0.547885 + 0.836554i \(0.315433\pi\)
\(564\) 0 0
\(565\) −20.0000 −0.841406
\(566\) 22.0000 0.924729
\(567\) 0 0
\(568\) 0 0
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) −19.0000 −0.795125 −0.397563 0.917575i \(-0.630144\pi\)
−0.397563 + 0.917575i \(0.630144\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −17.0000 −0.707719 −0.353860 0.935299i \(-0.615131\pi\)
−0.353860 + 0.935299i \(0.615131\pi\)
\(578\) 34.0000 1.41421
\(579\) 0 0
\(580\) −16.0000 −0.664364
\(581\) 0 0
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) 0 0
\(585\) 0 0
\(586\) 16.0000 0.660954
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) 9.00000 0.370839
\(590\) −48.0000 −1.97613
\(591\) 0 0
\(592\) −12.0000 −0.493197
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 24.0000 0.983078
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −9.00000 −0.367118 −0.183559 0.983009i \(-0.558762\pi\)
−0.183559 + 0.983009i \(0.558762\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −32.0000 −1.30206
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) 23.0000 0.933541 0.466771 0.884378i \(-0.345417\pi\)
0.466771 + 0.884378i \(0.345417\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) −40.0000 −1.61955
\(611\) 6.00000 0.242734
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 34.0000 1.37213
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −29.0000 −1.16561 −0.582804 0.812613i \(-0.698045\pi\)
−0.582804 + 0.812613i \(0.698045\pi\)
\(620\) 36.0000 1.44579
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) −28.0000 −1.11732
\(629\) 0 0
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 48.0000 1.90632
\(635\) −30.0000 −1.19051
\(636\) 0 0
\(637\) 0 0
\(638\) 16.0000 0.633446
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −19.0000 −0.749287 −0.374643 0.927169i \(-0.622235\pi\)
−0.374643 + 0.927169i \(0.622235\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.00000 −0.0786281 −0.0393141 0.999227i \(-0.512517\pi\)
−0.0393141 + 0.999227i \(0.512517\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) 28.0000 1.09405
\(656\) −40.0000 −1.56174
\(657\) 0 0
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −41.0000 −1.59472 −0.797358 0.603507i \(-0.793769\pi\)
−0.797358 + 0.603507i \(0.793769\pi\)
\(662\) 50.0000 1.94331
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 28.0000 1.08335
\(669\) 0 0
\(670\) 20.0000 0.772667
\(671\) 20.0000 0.772091
\(672\) 0 0
\(673\) −41.0000 −1.58043 −0.790217 0.612827i \(-0.790032\pi\)
−0.790217 + 0.612827i \(0.790032\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −36.0000 −1.37851
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 24.0000 0.916993
\(686\) 0 0
\(687\) 0 0
\(688\) −20.0000 −0.762493
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −37.0000 −1.40755 −0.703773 0.710425i \(-0.748503\pi\)
−0.703773 + 0.710425i \(0.748503\pi\)
\(692\) −16.0000 −0.608229
\(693\) 0 0
\(694\) 64.0000 2.42941
\(695\) −6.00000 −0.227593
\(696\) 0 0
\(697\) 0 0
\(698\) 28.0000 1.05982
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 3.00000 0.113147
\(704\) −16.0000 −0.603023
\(705\) 0 0
\(706\) 68.0000 2.55921
\(707\) 0 0
\(708\) 0 0
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) −24.0000 −0.900704
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) 40.0000 1.49279
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 36.0000 1.33978
\(723\) 0 0
\(724\) 26.0000 0.966282
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) −13.0000 −0.482143 −0.241072 0.970507i \(-0.577499\pi\)
−0.241072 + 0.970507i \(0.577499\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 12.0000 0.444140
\(731\) 0 0
\(732\) 0 0
\(733\) −15.0000 −0.554038 −0.277019 0.960864i \(-0.589346\pi\)
−0.277019 + 0.960864i \(0.589346\pi\)
\(734\) 18.0000 0.664392
\(735\) 0 0
\(736\) 0 0
\(737\) −10.0000 −0.368355
\(738\) 0 0
\(739\) −15.0000 −0.551784 −0.275892 0.961189i \(-0.588973\pi\)
−0.275892 + 0.961189i \(0.588973\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) 0 0
\(743\) −42.0000 −1.54083 −0.770415 0.637542i \(-0.779951\pi\)
−0.770415 + 0.637542i \(0.779951\pi\)
\(744\) 0 0
\(745\) 24.0000 0.879292
\(746\) −46.0000 −1.68418
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 13.0000 0.474377 0.237188 0.971464i \(-0.423774\pi\)
0.237188 + 0.971464i \(0.423774\pi\)
\(752\) −24.0000 −0.875190
\(753\) 0 0
\(754\) 8.00000 0.291343
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) −6.00000 −0.217930
\(759\) 0 0
\(760\) 0 0
\(761\) 48.0000 1.74000 0.869999 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) −49.0000 −1.76699 −0.883493 0.468445i \(-0.844814\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 22.0000 0.791797
\(773\) 34.0000 1.22290 0.611448 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) 0 0
\(775\) −9.00000 −0.323290
\(776\) 0 0
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −28.0000 −0.999363
\(786\) 0 0
\(787\) 40.0000 1.42585 0.712923 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) −32.0000 −1.13995
\(789\) 0 0
\(790\) 4.00000 0.142314
\(791\) 0 0
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) 0 0
\(797\) 8.00000 0.283375 0.141687 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −8.00000 −0.282843
\(801\) 0 0
\(802\) −72.0000 −2.54241
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) −18.0000 −0.634023
\(807\) 0 0
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 32.0000 1.12367 0.561836 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −12.0000 −0.420600
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) 5.00000 0.174928
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 40.0000 1.39686
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 30.0000 1.04320 0.521601 0.853189i \(-0.325335\pi\)
0.521601 + 0.853189i \(0.325335\pi\)
\(828\) 0 0
\(829\) 41.0000 1.42399 0.711994 0.702185i \(-0.247792\pi\)
0.711994 + 0.702185i \(0.247792\pi\)
\(830\) 24.0000 0.833052
\(831\) 0 0
\(832\) −8.00000 −0.277350
\(833\) 0 0
\(834\) 0 0
\(835\) 28.0000 0.968980
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) 60.0000 2.07267
\(839\) 44.0000 1.51905 0.759524 0.650479i \(-0.225432\pi\)
0.759524 + 0.650479i \(0.225432\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 14.0000 0.482472
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) −24.0000 −0.825625
\(846\) 0 0
\(847\) 0 0
\(848\) 48.0000 1.64833
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 35.0000 1.19838 0.599189 0.800608i \(-0.295490\pi\)
0.599189 + 0.800608i \(0.295490\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.0000 1.09310 0.546550 0.837427i \(-0.315941\pi\)
0.546550 + 0.837427i \(0.315941\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 20.0000 0.681994
\(861\) 0 0
\(862\) −36.0000 −1.22616
\(863\) 54.0000 1.83818 0.919091 0.394046i \(-0.128925\pi\)
0.919091 + 0.394046i \(0.128925\pi\)
\(864\) 0 0
\(865\) −16.0000 −0.544016
\(866\) −62.0000 −2.10685
\(867\) 0 0
\(868\) 0 0
\(869\) −2.00000 −0.0678454
\(870\) 0 0
\(871\) −5.00000 −0.169419
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −16.0000 −0.539360
\(881\) −24.0000 −0.808581 −0.404290 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) 0 0
\(883\) −13.0000 −0.437485 −0.218742 0.975783i \(-0.570195\pi\)
−0.218742 + 0.975783i \(0.570195\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 34.0000 1.14161 0.570804 0.821086i \(-0.306632\pi\)
0.570804 + 0.821086i \(0.306632\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 64.0000 2.14528
\(891\) 0 0
\(892\) 32.0000 1.07144
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) 0 0
\(898\) −36.0000 −1.20134
\(899\) −36.0000 −1.20067
\(900\) 0 0
\(901\) 0 0
\(902\) −40.0000 −1.33185
\(903\) 0 0
\(904\) 0 0
\(905\) 26.0000 0.864269
\(906\) 0 0
\(907\) −37.0000 −1.22856 −0.614282 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) −36.0000 −1.19470
\(909\) 0 0
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −38.0000 −1.25556
\(917\) 0 0
\(918\) 0 0
\(919\) 23.0000 0.758700 0.379350 0.925253i \(-0.376148\pi\)
0.379350 + 0.925253i \(0.376148\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 40.0000 1.31733
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) −3.00000 −0.0986394
\(926\) 34.0000 1.11731
\(927\) 0 0
\(928\) −32.0000 −1.05045
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −12.0000 −0.393073
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) 15.0000 0.490029 0.245014 0.969519i \(-0.421207\pi\)
0.245014 + 0.969519i \(0.421207\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 24.0000 0.782794
\(941\) 4.00000 0.130396 0.0651981 0.997872i \(-0.479232\pi\)
0.0651981 + 0.997872i \(0.479232\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −48.0000 −1.56227
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) 10.0000 0.324956 0.162478 0.986712i \(-0.448051\pi\)
0.162478 + 0.986712i \(0.448051\pi\)
\(948\) 0 0
\(949\) −3.00000 −0.0973841
\(950\) 2.00000 0.0648886
\(951\) 0 0
\(952\) 0 0
\(953\) −44.0000 −1.42530 −0.712650 0.701520i \(-0.752505\pi\)
−0.712650 + 0.701520i \(0.752505\pi\)
\(954\) 0 0
\(955\) −20.0000 −0.647185
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) −56.0000 −1.80928
\(959\) 0 0
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) −6.00000 −0.193448
\(963\) 0 0
\(964\) 28.0000 0.901819
\(965\) 22.0000 0.708205
\(966\) 0 0
\(967\) 19.0000 0.610999 0.305499 0.952192i \(-0.401177\pi\)
0.305499 + 0.952192i \(0.401177\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 24.0000 0.770594
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −62.0000 −1.98661
\(975\) 0 0
\(976\) −40.0000 −1.28037
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) −32.0000 −1.02272
\(980\) 0 0
\(981\) 0 0
\(982\) −56.0000 −1.78703
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) −32.0000 −1.01960
\(986\) 0 0
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) 0 0
\(990\) 0 0
\(991\) 17.0000 0.540023 0.270011 0.962857i \(-0.412973\pi\)
0.270011 + 0.962857i \(0.412973\pi\)
\(992\) 72.0000 2.28600
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.0000 0.601736 0.300868 0.953666i \(-0.402724\pi\)
0.300868 + 0.953666i \(0.402724\pi\)
\(998\) −74.0000 −2.34243
\(999\) 0 0
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 441.2.a.b.1.1 1
3.2 odd 2 147.2.a.c.1.1 1
4.3 odd 2 7056.2.a.bp.1.1 1
7.2 even 3 63.2.e.b.46.1 2
7.3 odd 6 441.2.e.e.226.1 2
7.4 even 3 63.2.e.b.37.1 2
7.5 odd 6 441.2.e.e.361.1 2
7.6 odd 2 441.2.a.a.1.1 1
12.11 even 2 2352.2.a.d.1.1 1
15.14 odd 2 3675.2.a.a.1.1 1
21.2 odd 6 21.2.e.a.4.1 2
21.5 even 6 147.2.e.a.67.1 2
21.11 odd 6 21.2.e.a.16.1 yes 2
21.17 even 6 147.2.e.a.79.1 2
21.20 even 2 147.2.a.b.1.1 1
24.5 odd 2 9408.2.a.bg.1.1 1
24.11 even 2 9408.2.a.cv.1.1 1
28.11 odd 6 1008.2.s.d.289.1 2
28.23 odd 6 1008.2.s.d.865.1 2
28.27 even 2 7056.2.a.m.1.1 1
63.2 odd 6 567.2.g.a.109.1 2
63.4 even 3 567.2.g.f.541.1 2
63.11 odd 6 567.2.h.f.352.1 2
63.16 even 3 567.2.g.f.109.1 2
63.23 odd 6 567.2.h.f.298.1 2
63.25 even 3 567.2.h.a.352.1 2
63.32 odd 6 567.2.g.a.541.1 2
63.58 even 3 567.2.h.a.298.1 2
84.11 even 6 336.2.q.f.289.1 2
84.23 even 6 336.2.q.f.193.1 2
84.47 odd 6 2352.2.q.c.1537.1 2
84.59 odd 6 2352.2.q.c.961.1 2
84.83 odd 2 2352.2.a.w.1.1 1
105.2 even 12 525.2.r.e.424.2 4
105.23 even 12 525.2.r.e.424.1 4
105.32 even 12 525.2.r.e.499.1 4
105.44 odd 6 525.2.i.e.151.1 2
105.53 even 12 525.2.r.e.499.2 4
105.74 odd 6 525.2.i.e.226.1 2
105.104 even 2 3675.2.a.c.1.1 1
168.11 even 6 1344.2.q.c.961.1 2
168.53 odd 6 1344.2.q.m.961.1 2
168.83 odd 2 9408.2.a.k.1.1 1
168.107 even 6 1344.2.q.c.193.1 2
168.125 even 2 9408.2.a.bz.1.1 1
168.149 odd 6 1344.2.q.m.193.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.e.a.4.1 2 21.2 odd 6
21.2.e.a.16.1 yes 2 21.11 odd 6
63.2.e.b.37.1 2 7.4 even 3
63.2.e.b.46.1 2 7.2 even 3
147.2.a.b.1.1 1 21.20 even 2
147.2.a.c.1.1 1 3.2 odd 2
147.2.e.a.67.1 2 21.5 even 6
147.2.e.a.79.1 2 21.17 even 6
336.2.q.f.193.1 2 84.23 even 6
336.2.q.f.289.1 2 84.11 even 6
441.2.a.a.1.1 1 7.6 odd 2
441.2.a.b.1.1 1 1.1 even 1 trivial
441.2.e.e.226.1 2 7.3 odd 6
441.2.e.e.361.1 2 7.5 odd 6
525.2.i.e.151.1 2 105.44 odd 6
525.2.i.e.226.1 2 105.74 odd 6
525.2.r.e.424.1 4 105.23 even 12
525.2.r.e.424.2 4 105.2 even 12
525.2.r.e.499.1 4 105.32 even 12
525.2.r.e.499.2 4 105.53 even 12
567.2.g.a.109.1 2 63.2 odd 6
567.2.g.a.541.1 2 63.32 odd 6
567.2.g.f.109.1 2 63.16 even 3
567.2.g.f.541.1 2 63.4 even 3
567.2.h.a.298.1 2 63.58 even 3
567.2.h.a.352.1 2 63.25 even 3
567.2.h.f.298.1 2 63.23 odd 6
567.2.h.f.352.1 2 63.11 odd 6
1008.2.s.d.289.1 2 28.11 odd 6
1008.2.s.d.865.1 2 28.23 odd 6
1344.2.q.c.193.1 2 168.107 even 6
1344.2.q.c.961.1 2 168.11 even 6
1344.2.q.m.193.1 2 168.149 odd 6
1344.2.q.m.961.1 2 168.53 odd 6
2352.2.a.d.1.1 1 12.11 even 2
2352.2.a.w.1.1 1 84.83 odd 2
2352.2.q.c.961.1 2 84.59 odd 6
2352.2.q.c.1537.1 2 84.47 odd 6
3675.2.a.a.1.1 1 15.14 odd 2
3675.2.a.c.1.1 1 105.104 even 2
7056.2.a.m.1.1 1 28.27 even 2
7056.2.a.bp.1.1 1 4.3 odd 2
9408.2.a.k.1.1 1 168.83 odd 2
9408.2.a.bg.1.1 1 24.5 odd 2
9408.2.a.bz.1.1 1 168.125 even 2
9408.2.a.cv.1.1 1 24.11 even 2