Properties

Label 1200.2.o.a.1199.1
Level $1200$
Weight $2$
Character 1200.1199
Analytic conductor $9.582$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1199.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.1199
Dual form 1200.2.o.a.1199.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{3} -3.46410 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} -3.46410 q^{7} -3.00000 q^{9} +3.46410 q^{11} +4.00000i q^{13} -6.00000 q^{17} -3.46410i q^{19} +6.00000i q^{21} +3.46410i q^{23} +5.19615i q^{27} +6.00000i q^{29} -3.46410i q^{31} -6.00000i q^{33} +4.00000i q^{37} +6.92820 q^{39} +12.0000i q^{41} -6.92820 q^{43} +3.46410i q^{47} +5.00000 q^{49} +10.3923i q^{51} +6.00000 q^{53} -6.00000 q^{57} -3.46410 q^{59} -10.0000 q^{61} +10.3923 q^{63} +6.92820 q^{67} +6.00000 q^{69} +13.8564 q^{71} -2.00000i q^{73} -12.0000 q^{77} +10.3923i q^{79} +9.00000 q^{81} -10.3923i q^{83} +10.3923 q^{87} -13.8564i q^{91} -6.00000 q^{93} -10.0000i q^{97} -10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{9} + O(q^{10}) \) \( 4q - 12q^{9} - 24q^{17} + 20q^{49} + 24q^{53} - 24q^{57} - 40q^{61} + 24q^{69} - 48q^{77} + 36q^{81} - 24q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) − 1.73205i − 1.00000i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) − 3.46410i − 0.794719i −0.917663 0.397360i \(-0.869927\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 6.00000i 1.30931i
\(22\) 0 0
\(23\) 3.46410i 0.722315i 0.932505 + 0.361158i \(0.117618\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) − 3.46410i − 0.622171i −0.950382 0.311086i \(-0.899307\pi\)
0.950382 0.311086i \(-0.100693\pi\)
\(32\) 0 0
\(33\) − 6.00000i − 1.04447i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) 6.92820 1.10940
\(40\) 0 0
\(41\) 12.0000i 1.87409i 0.349215 + 0.937043i \(0.386448\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) 0 0
\(43\) −6.92820 −1.05654 −0.528271 0.849076i \(-0.677159\pi\)
−0.528271 + 0.849076i \(0.677159\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 10.3923i 1.45521i
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) −3.46410 −0.450988 −0.225494 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 10.3923 1.30931
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.92820 0.846415 0.423207 0.906033i \(-0.360904\pi\)
0.423207 + 0.906033i \(0.360904\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 13.8564 1.64445 0.822226 0.569160i \(-0.192732\pi\)
0.822226 + 0.569160i \(0.192732\pi\)
\(72\) 0 0
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) 10.3923i 1.16923i 0.811312 + 0.584613i \(0.198754\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 10.3923i − 1.14070i −0.821401 0.570352i \(-0.806807\pi\)
0.821401 0.570352i \(-0.193193\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 10.3923 1.11417
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) − 13.8564i − 1.45255i
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) −10.3923 −1.04447
\(100\) 0 0
\(101\) 6.00000i 0.597022i 0.954406 + 0.298511i \(0.0964900\pi\)
−0.954406 + 0.298511i \(0.903510\pi\)
\(102\) 0 0
\(103\) −10.3923 −1.02398 −0.511992 0.858990i \(-0.671092\pi\)
−0.511992 + 0.858990i \(0.671092\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.46410i 0.334887i 0.985882 + 0.167444i \(0.0535512\pi\)
−0.985882 + 0.167444i \(0.946449\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 6.92820 0.657596
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 12.0000i − 1.10940i
\(118\) 0 0
\(119\) 20.7846 1.90532
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 20.7846 1.87409
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −10.3923 −0.922168 −0.461084 0.887357i \(-0.652539\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 0 0
\(129\) 12.0000i 1.05654i
\(130\) 0 0
\(131\) −17.3205 −1.51330 −0.756650 0.653820i \(-0.773165\pi\)
−0.756650 + 0.653820i \(0.773165\pi\)
\(132\) 0 0
\(133\) 12.0000i 1.04053i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 10.3923i 0.881464i 0.897639 + 0.440732i \(0.145281\pi\)
−0.897639 + 0.440732i \(0.854719\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 13.8564i 1.15873i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 8.66025i − 0.714286i
\(148\) 0 0
\(149\) − 6.00000i − 0.491539i −0.969328 0.245770i \(-0.920959\pi\)
0.969328 0.245770i \(-0.0790407\pi\)
\(150\) 0 0
\(151\) 3.46410i 0.281905i 0.990016 + 0.140952i \(0.0450164\pi\)
−0.990016 + 0.140952i \(0.954984\pi\)
\(152\) 0 0
\(153\) 18.0000 1.45521
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 16.0000i 1.27694i 0.769647 + 0.638470i \(0.220432\pi\)
−0.769647 + 0.638470i \(0.779568\pi\)
\(158\) 0 0
\(159\) − 10.3923i − 0.824163i
\(160\) 0 0
\(161\) − 12.0000i − 0.945732i
\(162\) 0 0
\(163\) −13.8564 −1.08532 −0.542659 0.839953i \(-0.682582\pi\)
−0.542659 + 0.839953i \(0.682582\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.46410i 0.268060i 0.990977 + 0.134030i \(0.0427919\pi\)
−0.990977 + 0.134030i \(0.957208\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 10.3923i 0.794719i
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) −24.2487 −1.81243 −0.906217 0.422813i \(-0.861043\pi\)
−0.906217 + 0.422813i \(0.861043\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 17.3205i 1.28037i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −20.7846 −1.51992
\(188\) 0 0
\(189\) − 18.0000i − 1.30931i
\(190\) 0 0
\(191\) −20.7846 −1.50392 −0.751961 0.659208i \(-0.770892\pi\)
−0.751961 + 0.659208i \(0.770892\pi\)
\(192\) 0 0
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) − 24.2487i − 1.71895i −0.511182 0.859473i \(-0.670792\pi\)
0.511182 0.859473i \(-0.329208\pi\)
\(200\) 0 0
\(201\) − 12.0000i − 0.846415i
\(202\) 0 0
\(203\) − 20.7846i − 1.45879i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 10.3923i − 0.722315i
\(208\) 0 0
\(209\) − 12.0000i − 0.830057i
\(210\) 0 0
\(211\) − 17.3205i − 1.19239i −0.802839 0.596196i \(-0.796678\pi\)
0.802839 0.596196i \(-0.203322\pi\)
\(212\) 0 0
\(213\) − 24.0000i − 1.64445i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 12.0000i 0.814613i
\(218\) 0 0
\(219\) −3.46410 −0.234082
\(220\) 0 0
\(221\) − 24.0000i − 1.61441i
\(222\) 0 0
\(223\) −17.3205 −1.15987 −0.579934 0.814664i \(-0.696921\pi\)
−0.579934 + 0.814664i \(0.696921\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 3.46410i − 0.229920i −0.993370 0.114960i \(-0.963326\pi\)
0.993370 0.114960i \(-0.0366741\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 20.7846i 1.36753i
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 18.0000 1.16923
\(238\) 0 0
\(239\) 13.8564 0.896296 0.448148 0.893959i \(-0.352084\pi\)
0.448148 + 0.893959i \(0.352084\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 0 0
\(243\) − 15.5885i − 1.00000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 13.8564 0.881662
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) 3.46410 0.218652 0.109326 0.994006i \(-0.465131\pi\)
0.109326 + 0.994006i \(0.465131\pi\)
\(252\) 0 0
\(253\) 12.0000i 0.754434i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) − 13.8564i − 0.860995i
\(260\) 0 0
\(261\) − 18.0000i − 1.11417i
\(262\) 0 0
\(263\) 3.46410i 0.213606i 0.994280 + 0.106803i \(0.0340614\pi\)
−0.994280 + 0.106803i \(0.965939\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.00000i 0.365826i 0.983129 + 0.182913i \(0.0585527\pi\)
−0.983129 + 0.182913i \(0.941447\pi\)
\(270\) 0 0
\(271\) − 10.3923i − 0.631288i −0.948878 0.315644i \(-0.897780\pi\)
0.948878 0.315644i \(-0.102220\pi\)
\(272\) 0 0
\(273\) −24.0000 −1.45255
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 4.00000i − 0.240337i −0.992754 0.120168i \(-0.961657\pi\)
0.992754 0.120168i \(-0.0383434\pi\)
\(278\) 0 0
\(279\) 10.3923i 0.622171i
\(280\) 0 0
\(281\) − 12.0000i − 0.715860i −0.933748 0.357930i \(-0.883483\pi\)
0.933748 0.357930i \(-0.116517\pi\)
\(282\) 0 0
\(283\) 6.92820 0.411839 0.205919 0.978569i \(-0.433982\pi\)
0.205919 + 0.978569i \(0.433982\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 41.5692i − 2.45375i
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −17.3205 −1.01535
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 18.0000i 1.04447i
\(298\) 0 0
\(299\) −13.8564 −0.801337
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 0 0
\(303\) 10.3923 0.597022
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −6.92820 −0.395413 −0.197707 0.980261i \(-0.563349\pi\)
−0.197707 + 0.980261i \(0.563349\pi\)
\(308\) 0 0
\(309\) 18.0000i 1.02398i
\(310\) 0 0
\(311\) 13.8564 0.785725 0.392862 0.919597i \(-0.371485\pi\)
0.392862 + 0.919597i \(0.371485\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i 0.918396 + 0.395663i \(0.129485\pi\)
−0.918396 + 0.395663i \(0.870515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 20.7846i 1.16371i
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 20.7846i 1.15649i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 17.3205i 0.957826i
\(328\) 0 0
\(329\) − 12.0000i − 0.661581i
\(330\) 0 0
\(331\) 10.3923i 0.571213i 0.958347 + 0.285606i \(0.0921950\pi\)
−0.958347 + 0.285606i \(0.907805\pi\)
\(332\) 0 0
\(333\) − 12.0000i − 0.657596i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 34.0000i − 1.85210i −0.377403 0.926049i \(-0.623183\pi\)
0.377403 0.926049i \(-0.376817\pi\)
\(338\) 0 0
\(339\) 10.3923i 0.564433i
\(340\) 0 0
\(341\) − 12.0000i − 0.649836i
\(342\) 0 0
\(343\) 6.92820 0.374088
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.3923i 0.557888i 0.960307 + 0.278944i \(0.0899844\pi\)
−0.960307 + 0.278944i \(0.910016\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −20.7846 −1.10940
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 36.0000i − 1.90532i
\(358\) 0 0
\(359\) −6.92820 −0.365657 −0.182828 0.983145i \(-0.558525\pi\)
−0.182828 + 0.983145i \(0.558525\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) − 1.73205i − 0.0909091i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.46410 0.180825 0.0904123 0.995904i \(-0.471182\pi\)
0.0904123 + 0.995904i \(0.471182\pi\)
\(368\) 0 0
\(369\) − 36.0000i − 1.87409i
\(370\) 0 0
\(371\) −20.7846 −1.07908
\(372\) 0 0
\(373\) − 16.0000i − 0.828449i −0.910175 0.414224i \(-0.864053\pi\)
0.910175 0.414224i \(-0.135947\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) 24.2487i 1.24557i 0.782392 + 0.622786i \(0.213999\pi\)
−0.782392 + 0.622786i \(0.786001\pi\)
\(380\) 0 0
\(381\) 18.0000i 0.922168i
\(382\) 0 0
\(383\) 17.3205i 0.885037i 0.896759 + 0.442518i \(0.145915\pi\)
−0.896759 + 0.442518i \(0.854085\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 20.7846 1.05654
\(388\) 0 0
\(389\) − 6.00000i − 0.304212i −0.988364 0.152106i \(-0.951394\pi\)
0.988364 0.152106i \(-0.0486055\pi\)
\(390\) 0 0
\(391\) − 20.7846i − 1.05112i
\(392\) 0 0
\(393\) 30.0000i 1.51330i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 16.0000i 0.803017i 0.915855 + 0.401508i \(0.131514\pi\)
−0.915855 + 0.401508i \(0.868486\pi\)
\(398\) 0 0
\(399\) 20.7846 1.04053
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 13.8564 0.690237
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13.8564i 0.686837i
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) − 31.1769i − 1.53784i
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 18.0000 0.881464
\(418\) 0 0
\(419\) −10.3923 −0.507697 −0.253849 0.967244i \(-0.581697\pi\)
−0.253849 + 0.967244i \(0.581697\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) − 10.3923i − 0.505291i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 34.6410 1.67640
\(428\) 0 0
\(429\) 24.0000 1.15873
\(430\) 0 0
\(431\) 6.92820 0.333720 0.166860 0.985981i \(-0.446637\pi\)
0.166860 + 0.985981i \(0.446637\pi\)
\(432\) 0 0
\(433\) 2.00000i 0.0961139i 0.998845 + 0.0480569i \(0.0153029\pi\)
−0.998845 + 0.0480569i \(0.984697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) 3.46410i 0.165333i 0.996577 + 0.0826663i \(0.0263436\pi\)
−0.996577 + 0.0826663i \(0.973656\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 0 0
\(443\) − 17.3205i − 0.822922i −0.911427 0.411461i \(-0.865019\pi\)
0.911427 0.411461i \(-0.134981\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −10.3923 −0.491539
\(448\) 0 0
\(449\) 36.0000i 1.69895i 0.527633 + 0.849473i \(0.323080\pi\)
−0.527633 + 0.849473i \(0.676920\pi\)
\(450\) 0 0
\(451\) 41.5692i 1.95742i
\(452\) 0 0
\(453\) 6.00000 0.281905
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.00000i 0.0935561i 0.998905 + 0.0467780i \(0.0148953\pi\)
−0.998905 + 0.0467780i \(0.985105\pi\)
\(458\) 0 0
\(459\) − 31.1769i − 1.45521i
\(460\) 0 0
\(461\) 18.0000i 0.838344i 0.907907 + 0.419172i \(0.137680\pi\)
−0.907907 + 0.419172i \(0.862320\pi\)
\(462\) 0 0
\(463\) 10.3923 0.482971 0.241486 0.970404i \(-0.422365\pi\)
0.241486 + 0.970404i \(0.422365\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 24.2487i − 1.12210i −0.827783 0.561048i \(-0.810398\pi\)
0.827783 0.561048i \(-0.189602\pi\)
\(468\) 0 0
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) 27.7128 1.27694
\(472\) 0 0
\(473\) −24.0000 −1.10352
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −18.0000 −0.824163
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) −20.7846 −0.945732
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −17.3205 −0.784867 −0.392434 0.919780i \(-0.628367\pi\)
−0.392434 + 0.919780i \(0.628367\pi\)
\(488\) 0 0
\(489\) 24.0000i 1.08532i
\(490\) 0 0
\(491\) 31.1769 1.40699 0.703497 0.710698i \(-0.251621\pi\)
0.703497 + 0.710698i \(0.251621\pi\)
\(492\) 0 0
\(493\) − 36.0000i − 1.62136i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −48.0000 −2.15309
\(498\) 0 0
\(499\) − 3.46410i − 0.155074i −0.996989 0.0775372i \(-0.975294\pi\)
0.996989 0.0775372i \(-0.0247057\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) 0 0
\(503\) − 10.3923i − 0.463370i −0.972791 0.231685i \(-0.925576\pi\)
0.972791 0.231685i \(-0.0744239\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 5.19615i 0.230769i
\(508\) 0 0
\(509\) 6.00000i 0.265945i 0.991120 + 0.132973i \(0.0424523\pi\)
−0.991120 + 0.132973i \(0.957548\pi\)
\(510\) 0 0
\(511\) 6.92820i 0.306486i
\(512\) 0 0
\(513\) 18.0000 0.794719
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 12.0000i 0.527759i
\(518\) 0 0
\(519\) 10.3923i 0.456172i
\(520\) 0 0
\(521\) 12.0000i 0.525730i 0.964833 + 0.262865i \(0.0846673\pi\)
−0.964833 + 0.262865i \(0.915333\pi\)
\(522\) 0 0
\(523\) −34.6410 −1.51475 −0.757373 0.652983i \(-0.773517\pi\)
−0.757373 + 0.652983i \(0.773517\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.7846i 0.905392i
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 10.3923 0.450988
\(532\) 0 0
\(533\) −48.0000 −2.07911
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 42.0000i 1.81243i
\(538\) 0 0
\(539\) 17.3205 0.746047
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 3.46410i 0.148659i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −20.7846 −0.888686 −0.444343 0.895857i \(-0.646563\pi\)
−0.444343 + 0.895857i \(0.646563\pi\)
\(548\) 0 0
\(549\) 30.0000 1.28037
\(550\) 0 0
\(551\) 20.7846 0.885454
\(552\) 0 0
\(553\) − 36.0000i − 1.53088i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) − 27.7128i − 1.17213i
\(560\) 0 0
\(561\) 36.0000i 1.51992i
\(562\) 0 0
\(563\) − 3.46410i − 0.145994i −0.997332 0.0729972i \(-0.976744\pi\)
0.997332 0.0729972i \(-0.0232564\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −31.1769 −1.30931
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 24.2487i 1.01478i 0.861717 + 0.507388i \(0.169389\pi\)
−0.861717 + 0.507388i \(0.830611\pi\)
\(572\) 0 0
\(573\) 36.0000i 1.50392i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 0 0
\(579\) 3.46410 0.143963
\(580\) 0 0
\(581\) 36.0000i 1.49353i
\(582\) 0 0
\(583\) 20.7846 0.860811
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 31.1769i 1.28681i 0.765526 + 0.643404i \(0.222479\pi\)
−0.765526 + 0.643404i \(0.777521\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) − 10.3923i − 0.427482i
\(592\) 0 0
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −42.0000 −1.71895
\(598\) 0 0
\(599\) 20.7846 0.849236 0.424618 0.905373i \(-0.360408\pi\)
0.424618 + 0.905373i \(0.360408\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) −20.7846 −0.846415
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 31.1769 1.26543 0.632716 0.774384i \(-0.281940\pi\)
0.632716 + 0.774384i \(0.281940\pi\)
\(608\) 0 0
\(609\) −36.0000 −1.45879
\(610\) 0 0
\(611\) −13.8564 −0.560570
\(612\) 0 0
\(613\) 32.0000i 1.29247i 0.763139 + 0.646234i \(0.223657\pi\)
−0.763139 + 0.646234i \(0.776343\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) − 45.0333i − 1.81004i −0.425367 0.905021i \(-0.639855\pi\)
0.425367 0.905021i \(-0.360145\pi\)
\(620\) 0 0
\(621\) −18.0000 −0.722315
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −20.7846 −0.830057
\(628\) 0 0
\(629\) − 24.0000i − 0.956943i
\(630\) 0 0
\(631\) − 3.46410i − 0.137904i −0.997620 0.0689519i \(-0.978035\pi\)
0.997620 0.0689519i \(-0.0219655\pi\)
\(632\) 0 0
\(633\) −30.0000 −1.19239
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 20.0000i 0.792429i
\(638\) 0 0
\(639\) −41.5692 −1.64445
\(640\) 0 0
\(641\) 24.0000i 0.947943i 0.880540 + 0.473972i \(0.157180\pi\)
−0.880540 + 0.473972i \(0.842820\pi\)
\(642\) 0 0
\(643\) 41.5692 1.63933 0.819665 0.572843i \(-0.194160\pi\)
0.819665 + 0.572843i \(0.194160\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 45.0333i 1.77044i 0.465170 + 0.885221i \(0.345993\pi\)
−0.465170 + 0.885221i \(0.654007\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 20.7846 0.814613
\(652\) 0 0
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) 31.1769 1.21448 0.607240 0.794518i \(-0.292277\pi\)
0.607240 + 0.794518i \(0.292277\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 0 0
\(663\) −41.5692 −1.61441
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −20.7846 −0.804783
\(668\) 0 0
\(669\) 30.0000i 1.15987i
\(670\) 0 0
\(671\) −34.6410 −1.33730
\(672\) 0 0
\(673\) 2.00000i 0.0770943i 0.999257 + 0.0385472i \(0.0122730\pi\)
−0.999257 + 0.0385472i \(0.987727\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 0 0
\(679\) 34.6410i 1.32940i
\(680\) 0 0
\(681\) −6.00000 −0.229920
\(682\) 0 0
\(683\) 17.3205i 0.662751i 0.943499 + 0.331375i \(0.107513\pi\)
−0.943499 + 0.331375i \(0.892487\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 24.2487i − 0.925146i
\(688\) 0 0
\(689\) 24.0000i 0.914327i
\(690\) 0 0
\(691\) − 31.1769i − 1.18603i −0.805193 0.593013i \(-0.797938\pi\)
0.805193 0.593013i \(-0.202062\pi\)
\(692\) 0 0
\(693\) 36.0000 1.36753
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 72.0000i − 2.72719i
\(698\) 0 0
\(699\) 10.3923i 0.393073i
\(700\) 0 0
\(701\) 42.0000i 1.58632i 0.609015 + 0.793159i \(0.291565\pi\)
−0.609015 + 0.793159i \(0.708435\pi\)
\(702\) 0 0
\(703\) 13.8564 0.522604
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 20.7846i − 0.781686i
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) − 31.1769i − 1.16923i
\(712\) 0 0
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 24.0000i − 0.896296i
\(718\) 0 0
\(719\) −27.7128 −1.03351 −0.516757 0.856132i \(-0.672861\pi\)
−0.516757 + 0.856132i \(0.672861\pi\)
\(720\) 0 0
\(721\) 36.0000 1.34071
\(722\) 0 0
\(723\) 45.0333i 1.67481i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −3.46410 −0.128476 −0.0642382 0.997935i \(-0.520462\pi\)
−0.0642382 + 0.997935i \(0.520462\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 41.5692 1.53749
\(732\) 0 0
\(733\) 28.0000i 1.03420i 0.855924 + 0.517102i \(0.172989\pi\)
−0.855924 + 0.517102i \(0.827011\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24.0000 0.884051
\(738\) 0 0
\(739\) 10.3923i 0.382287i 0.981562 + 0.191144i \(0.0612196\pi\)
−0.981562 + 0.191144i \(0.938780\pi\)
\(740\) 0 0
\(741\) − 24.0000i − 0.881662i
\(742\) 0 0
\(743\) − 38.1051i − 1.39794i −0.715150 0.698971i \(-0.753642\pi\)
0.715150 0.698971i \(-0.246358\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 31.1769i 1.14070i
\(748\) 0 0
\(749\) − 12.0000i − 0.438470i
\(750\) 0 0
\(751\) 51.9615i 1.89610i 0.318117 + 0.948051i \(0.396950\pi\)
−0.318117 + 0.948051i \(0.603050\pi\)
\(752\) 0 0
\(753\) − 6.00000i − 0.218652i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 20.0000i 0.726912i 0.931611 + 0.363456i \(0.118403\pi\)
−0.931611 + 0.363456i \(0.881597\pi\)
\(758\) 0 0
\(759\) 20.7846 0.754434
\(760\) 0 0
\(761\) 12.0000i 0.435000i 0.976060 + 0.217500i \(0.0697902\pi\)
−0.976060 + 0.217500i \(0.930210\pi\)
\(762\) 0 0
\(763\) 34.6410 1.25409
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 13.8564i − 0.500326i
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) − 31.1769i − 1.12281i
\(772\) 0 0
\(773\) −54.0000 −1.94225 −0.971123 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −24.0000 −0.860995
\(778\) 0 0
\(779\) 41.5692 1.48937
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) 0 0
\(783\) −31.1769 −1.11417
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 34.6410 1.23482 0.617409 0.786642i \(-0.288182\pi\)
0.617409 + 0.786642i \(0.288182\pi\)
\(788\) 0 0
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) 20.7846 0.739016
\(792\) 0 0
\(793\) − 40.0000i − 1.42044i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) − 20.7846i − 0.735307i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 6.92820i − 0.244491i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 10.3923 0.365826
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 24.2487i 0.851487i 0.904844 + 0.425744i \(0.139987\pi\)
−0.904844 + 0.425744i \(0.860013\pi\)
\(812\) 0 0
\(813\) −18.0000 −0.631288
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 24.0000i 0.839654i
\(818\) 0 0
\(819\) 41.5692i 1.45255i
\(820\) 0 0
\(821\) 6.00000i 0.209401i 0.994504 + 0.104701i \(0.0333885\pi\)
−0.994504 + 0.104701i \(0.966612\pi\)
\(822\) 0 0
\(823\) 45.0333 1.56976 0.784881 0.619646i \(-0.212724\pi\)
0.784881 + 0.619646i \(0.212724\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38.1051i 1.32504i 0.749042 + 0.662522i \(0.230514\pi\)
−0.749042 + 0.662522i \(0.769486\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) −6.92820 −0.240337
\(832\) 0 0
\(833\) −30.0000 −1.03944
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 18.0000 0.622171
\(838\) 0 0
\(839\) −48.4974 −1.67432 −0.837158 0.546960i \(-0.815785\pi\)
−0.837158 + 0.546960i \(0.815785\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) −20.7846 −0.715860
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −3.46410 −0.119028
\(848\) 0 0
\(849\) − 12.0000i − 0.411839i
\(850\) 0 0
\(851\) −13.8564 −0.474991
\(852\) 0 0
\(853\) 8.00000i 0.273915i 0.990577 + 0.136957i \(0.0437323\pi\)
−0.990577 + 0.136957i \(0.956268\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 51.9615i 1.77290i 0.462820 + 0.886452i \(0.346838\pi\)
−0.462820 + 0.886452i \(0.653162\pi\)
\(860\) 0 0
\(861\) −72.0000 −2.45375
\(862\) 0 0
\(863\) − 51.9615i − 1.76879i −0.466738 0.884395i \(-0.654571\pi\)
0.466738 0.884395i \(-0.345429\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 32.9090i − 1.11765i
\(868\) 0 0
\(869\) 36.0000i 1.22122i
\(870\) 0 0
\(871\) 27.7128i 0.939013i
\(872\) 0 0
\(873\) 30.0000i 1.01535i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 32.0000i − 1.08056i −0.841484 0.540282i \(-0.818318\pi\)
0.841484 0.540282i \(-0.181682\pi\)
\(878\) 0 0
\(879\) − 10.3923i − 0.350524i
\(880\) 0 0
\(881\) − 48.0000i − 1.61716i −0.588386 0.808581i \(-0.700236\pi\)
0.588386 0.808581i \(-0.299764\pi\)
\(882\) 0 0
\(883\) 13.8564 0.466305 0.233153 0.972440i \(-0.425096\pi\)
0.233153 + 0.972440i \(0.425096\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 24.2487i − 0.814192i −0.913385 0.407096i \(-0.866541\pi\)
0.913385 0.407096i \(-0.133459\pi\)
\(888\) 0 0
\(889\) 36.0000 1.20740
\(890\) 0 0
\(891\) 31.1769 1.04447
\(892\) 0 0
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 24.0000i 0.801337i
\(898\) 0 0
\(899\) 20.7846 0.693206
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) − 41.5692i − 1.38334i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 41.5692 1.38028 0.690142 0.723674i \(-0.257548\pi\)
0.690142 + 0.723674i \(0.257548\pi\)
\(908\) 0 0
\(909\) − 18.0000i − 0.597022i
\(910\) 0 0
\(911\) −34.6410 −1.14771 −0.573854 0.818958i \(-0.694552\pi\)
−0.573854 + 0.818958i \(0.694552\pi\)
\(912\) 0 0
\(913\) − 36.0000i − 1.19143i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 60.0000 1.98137
\(918\) 0 0
\(919\) − 45.0333i − 1.48551i −0.669562 0.742756i \(-0.733518\pi\)
0.669562 0.742756i \(-0.266482\pi\)
\(920\) 0 0
\(921\) 12.0000i 0.395413i
\(922\) 0 0
\(923\) 55.4256i 1.82436i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 31.1769 1.02398
\(928\) 0 0
\(929\) − 12.0000i − 0.393707i −0.980433 0.196854i \(-0.936928\pi\)
0.980433 0.196854i \(-0.0630724\pi\)
\(930\) 0 0
\(931\) − 17.3205i − 0.567657i
\(932\) 0 0
\(933\) − 24.0000i − 0.785725i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 38.0000i − 1.24141i −0.784046 0.620703i \(-0.786847\pi\)
0.784046 0.620703i \(-0.213153\pi\)
\(938\) 0 0
\(939\) 24.2487 0.791327
\(940\) 0 0
\(941\) − 54.0000i − 1.76035i −0.474650 0.880175i \(-0.657425\pi\)
0.474650 0.880175i \(-0.342575\pi\)
\(942\) 0 0
\(943\) −41.5692 −1.35368
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 31.1769i − 1.01311i −0.862207 0.506557i \(-0.830918\pi\)
0.862207 0.506557i \(-0.169082\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) − 31.1769i − 1.01098i
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 36.0000 1.16371
\(958\) 0 0
\(959\) −62.3538 −2.01351
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) − 10.3923i − 0.334887i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 51.9615 1.67097 0.835485 0.549513i \(-0.185187\pi\)
0.835485 + 0.549513i \(0.185187\pi\)
\(968\) 0 0
\(969\) 36.0000 1.15649
\(970\) 0 0
\(971\) 3.46410 0.111168 0.0555842 0.998454i \(-0.482298\pi\)
0.0555842 + 0.998454i \(0.482298\pi\)
\(972\) 0 0
\(973\) − 36.0000i − 1.15411i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 30.0000 0.957826
\(982\) 0 0
\(983\) 58.8897i 1.87829i 0.343520 + 0.939145i \(0.388381\pi\)
−0.343520 + 0.939145i \(0.611619\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −20.7846 −0.661581
\(988\) 0 0
\(989\) − 24.0000i − 0.763156i
\(990\) 0 0
\(991\) 17.3205i 0.550204i 0.961415 + 0.275102i \(0.0887116\pi\)
−0.961415 + 0.275102i \(0.911288\pi\)
\(992\) 0 0
\(993\) 18.0000 0.571213
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 52.0000i 1.64686i 0.567420 + 0.823428i \(0.307941\pi\)
−0.567420 + 0.823428i \(0.692059\pi\)
\(998\) 0 0
\(999\) −20.7846 −0.657596
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 1200.2.o.a.1199.1 4
3.2 odd 2 1200.2.o.b.1199.3 4
4.3 odd 2 inner 1200.2.o.a.1199.4 4
5.2 odd 4 240.2.h.b.191.2 yes 4
5.3 odd 4 1200.2.h.m.1151.4 4
5.4 even 2 1200.2.o.b.1199.4 4
12.11 even 2 1200.2.o.b.1199.2 4
15.2 even 4 240.2.h.b.191.3 yes 4
15.8 even 4 1200.2.h.m.1151.2 4
15.14 odd 2 inner 1200.2.o.a.1199.2 4
20.3 even 4 1200.2.h.m.1151.1 4
20.7 even 4 240.2.h.b.191.4 yes 4
20.19 odd 2 1200.2.o.b.1199.1 4
40.27 even 4 960.2.h.d.191.1 4
40.37 odd 4 960.2.h.d.191.3 4
60.23 odd 4 1200.2.h.m.1151.3 4
60.47 odd 4 240.2.h.b.191.1 4
60.59 even 2 inner 1200.2.o.a.1199.3 4
120.77 even 4 960.2.h.d.191.2 4
120.107 odd 4 960.2.h.d.191.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
240.2.h.b.191.1 4 60.47 odd 4
240.2.h.b.191.2 yes 4 5.2 odd 4
240.2.h.b.191.3 yes 4 15.2 even 4
240.2.h.b.191.4 yes 4 20.7 even 4
960.2.h.d.191.1 4 40.27 even 4
960.2.h.d.191.2 4 120.77 even 4
960.2.h.d.191.3 4 40.37 odd 4
960.2.h.d.191.4 4 120.107 odd 4
1200.2.h.m.1151.1 4 20.3 even 4
1200.2.h.m.1151.2 4 15.8 even 4
1200.2.h.m.1151.3 4 60.23 odd 4
1200.2.h.m.1151.4 4 5.3 odd 4
1200.2.o.a.1199.1 4 1.1 even 1 trivial
1200.2.o.a.1199.2 4 15.14 odd 2 inner
1200.2.o.a.1199.3 4 60.59 even 2 inner
1200.2.o.a.1199.4 4 4.3 odd 2 inner
1200.2.o.b.1199.1 4 20.19 odd 2
1200.2.o.b.1199.2 4 12.11 even 2
1200.2.o.b.1199.3 4 3.2 odd 2
1200.2.o.b.1199.4 4 5.4 even 2