Properties

Label 1925.2.b.e.1849.1
Level $1925$
Weight $2$
Character 1925.1849
Analytic conductor $15.371$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1925.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.3712023891\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1925.1849
Dual form 1925.2.b.e.1849.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} +2.00000 q^{4} +1.00000i q^{7} -6.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +2.00000 q^{4} +1.00000i q^{7} -6.00000 q^{9} -1.00000 q^{11} -6.00000i q^{12} -4.00000i q^{13} +4.00000 q^{16} -2.00000i q^{17} +6.00000 q^{19} +3.00000 q^{21} -5.00000i q^{23} +9.00000i q^{27} +2.00000i q^{28} -10.0000 q^{29} +1.00000 q^{31} +3.00000i q^{33} -12.0000 q^{36} +5.00000i q^{37} -12.0000 q^{39} -2.00000 q^{41} -8.00000i q^{43} -2.00000 q^{44} -8.00000i q^{47} -12.0000i q^{48} -1.00000 q^{49} -6.00000 q^{51} -8.00000i q^{52} -6.00000i q^{53} -18.0000i q^{57} -3.00000 q^{59} -2.00000 q^{61} -6.00000i q^{63} +8.00000 q^{64} +3.00000i q^{67} -4.00000i q^{68} -15.0000 q^{69} +1.00000 q^{71} +10.0000i q^{73} +12.0000 q^{76} -1.00000i q^{77} -6.00000 q^{79} +9.00000 q^{81} +12.0000i q^{83} +6.00000 q^{84} +30.0000i q^{87} +15.0000 q^{89} +4.00000 q^{91} -10.0000i q^{92} -3.00000i q^{93} +5.00000i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} - 12 q^{9} - 2 q^{11} + 8 q^{16} + 12 q^{19} + 6 q^{21} - 20 q^{29} + 2 q^{31} - 24 q^{36} - 24 q^{39} - 4 q^{41} - 4 q^{44} - 2 q^{49} - 12 q^{51} - 6 q^{59} - 4 q^{61} + 16 q^{64} - 30 q^{69} + 2 q^{71} + 24 q^{76} - 12 q^{79} + 18 q^{81} + 12 q^{84} + 30 q^{89} + 8 q^{91} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times\).

\(n\) \(276\) \(1002\) \(1751\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) − 3.00000i − 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) − 6.00000i − 1.73205i
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) − 5.00000i − 1.04257i −0.853382 0.521286i \(-0.825452\pi\)
0.853382 0.521286i \(-0.174548\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000i 1.73205i
\(28\) 2.00000i 0.377964i
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 3.00000i 0.522233i
\(34\) 0 0
\(35\) 0 0
\(36\) −12.0000 −2.00000
\(37\) 5.00000i 0.821995i 0.911636 + 0.410997i \(0.134819\pi\)
−0.911636 + 0.410997i \(0.865181\pi\)
\(38\) 0 0
\(39\) −12.0000 −1.92154
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 0 0
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) − 12.0000i − 1.73205i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) − 8.00000i − 1.10940i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 18.0000i − 2.38416i
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) − 6.00000i − 0.755929i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.00000i 0.366508i 0.983066 + 0.183254i \(0.0586631\pi\)
−0.983066 + 0.183254i \(0.941337\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) −15.0000 −1.80579
\(70\) 0 0
\(71\) 1.00000 0.118678 0.0593391 0.998238i \(-0.481101\pi\)
0.0593391 + 0.998238i \(0.481101\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 12.0000 1.37649
\(77\) − 1.00000i − 0.113961i
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 6.00000 0.654654
\(85\) 0 0
\(86\) 0 0
\(87\) 30.0000i 3.21634i
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) − 10.0000i − 1.04257i
\(93\) − 3.00000i − 0.311086i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.00000i 0.507673i 0.967247 + 0.253837i \(0.0816925\pi\)
−0.967247 + 0.253837i \(0.918307\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) − 12.0000i − 1.18240i −0.806527 0.591198i \(-0.798655\pi\)
0.806527 0.591198i \(-0.201345\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.0000i 0.966736i 0.875417 + 0.483368i \(0.160587\pi\)
−0.875417 + 0.483368i \(0.839413\pi\)
\(108\) 18.0000i 1.73205i
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 15.0000 1.42374
\(112\) 4.00000i 0.377964i
\(113\) − 19.0000i − 1.78737i −0.448695 0.893685i \(-0.648111\pi\)
0.448695 0.893685i \(-0.351889\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −20.0000 −1.85695
\(117\) 24.0000i 2.21880i
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.00000i 0.541002i
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) − 2.00000i − 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) 0 0
\(129\) −24.0000 −2.11308
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 6.00000i 0.522233i
\(133\) 6.00000i 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.00000i 0.256307i 0.991754 + 0.128154i \(0.0409051\pi\)
−0.991754 + 0.128154i \(0.959095\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) 0 0
\(143\) 4.00000i 0.334497i
\(144\) −24.0000 −2.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000i 0.247436i
\(148\) 10.0000i 0.821995i
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 0 0
\(153\) 12.0000i 0.970143i
\(154\) 0 0
\(155\) 0 0
\(156\) −24.0000 −1.92154
\(157\) − 7.00000i − 0.558661i −0.960195 0.279330i \(-0.909888\pi\)
0.960195 0.279330i \(-0.0901125\pi\)
\(158\) 0 0
\(159\) −18.0000 −1.42749
\(160\) 0 0
\(161\) 5.00000 0.394055
\(162\) 0 0
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 0 0
\(167\) 2.00000i 0.154765i 0.997001 + 0.0773823i \(0.0246562\pi\)
−0.997001 + 0.0773823i \(0.975344\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −36.0000 −2.75299
\(172\) − 16.0000i − 1.21999i
\(173\) 16.0000i 1.21646i 0.793762 + 0.608229i \(0.208120\pi\)
−0.793762 + 0.608229i \(0.791880\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 9.00000i 0.676481i
\(178\) 0 0
\(179\) −1.00000 −0.0747435 −0.0373718 0.999301i \(-0.511899\pi\)
−0.0373718 + 0.999301i \(0.511899\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) 6.00000i 0.443533i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000i 0.146254i
\(188\) − 16.0000i − 1.16692i
\(189\) −9.00000 −0.654654
\(190\) 0 0
\(191\) 5.00000 0.361787 0.180894 0.983503i \(-0.442101\pi\)
0.180894 + 0.983503i \(0.442101\pi\)
\(192\) − 24.0000i − 1.73205i
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 9.00000 0.634811
\(202\) 0 0
\(203\) − 10.0000i − 0.701862i
\(204\) −12.0000 −0.840168
\(205\) 0 0
\(206\) 0 0
\(207\) 30.0000i 2.08514i
\(208\) − 16.0000i − 1.10940i
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) − 12.0000i − 0.824163i
\(213\) − 3.00000i − 0.205557i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.00000i 0.0678844i
\(218\) 0 0
\(219\) 30.0000 2.02721
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 1.00000i 0.0669650i 0.999439 + 0.0334825i \(0.0106598\pi\)
−0.999439 + 0.0334825i \(0.989340\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) − 36.0000i − 2.38416i
\(229\) 7.00000 0.462573 0.231287 0.972886i \(-0.425707\pi\)
0.231287 + 0.972886i \(0.425707\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 18.0000i 1.16923i
\(238\) 0 0
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) − 24.0000i − 1.52708i
\(248\) 0 0
\(249\) 36.0000 2.28141
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) − 12.0000i − 0.755929i
\(253\) 5.00000i 0.314347i
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 6.00000i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(258\) 0 0
\(259\) −5.00000 −0.310685
\(260\) 0 0
\(261\) 60.0000 3.71391
\(262\) 0 0
\(263\) 18.0000i 1.10993i 0.831875 + 0.554964i \(0.187268\pi\)
−0.831875 + 0.554964i \(0.812732\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 45.0000i − 2.75396i
\(268\) 6.00000i 0.366508i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) − 8.00000i − 0.485071i
\(273\) − 12.0000i − 0.726273i
\(274\) 0 0
\(275\) 0 0
\(276\) −30.0000 −1.80579
\(277\) − 24.0000i − 1.44202i −0.692925 0.721010i \(-0.743678\pi\)
0.692925 0.721010i \(-0.256322\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.00000i − 0.118056i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 15.0000 0.879316
\(292\) 20.0000i 1.17041i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 9.00000i − 0.522233i
\(298\) 0 0
\(299\) −20.0000 −1.15663
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 36.0000i 2.06815i
\(304\) 24.0000 1.37649
\(305\) 0 0
\(306\) 0 0
\(307\) 28.0000i 1.59804i 0.601302 + 0.799022i \(0.294649\pi\)
−0.601302 + 0.799022i \(0.705351\pi\)
\(308\) − 2.00000i − 0.113961i
\(309\) −36.0000 −2.04797
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) − 23.0000i − 1.30004i −0.759918 0.650018i \(-0.774761\pi\)
0.759918 0.650018i \(-0.225239\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) − 9.00000i − 0.505490i −0.967533 0.252745i \(-0.918667\pi\)
0.967533 0.252745i \(-0.0813334\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) 30.0000 1.67444
\(322\) 0 0
\(323\) − 12.0000i − 0.667698i
\(324\) 18.0000 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 12.0000i 0.663602i
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) 24.0000i 1.31717i
\(333\) − 30.0000i − 1.64399i
\(334\) 0 0
\(335\) 0 0
\(336\) 12.0000 0.654654
\(337\) 18.0000i 0.980522i 0.871576 + 0.490261i \(0.163099\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) −57.0000 −3.09582
\(340\) 0 0
\(341\) −1.00000 −0.0541530
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 14.0000i − 0.751559i −0.926709 0.375780i \(-0.877375\pi\)
0.926709 0.375780i \(-0.122625\pi\)
\(348\) 60.0000i 3.21634i
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 36.0000 1.92154
\(352\) 0 0
\(353\) 9.00000i 0.479022i 0.970894 + 0.239511i \(0.0769871\pi\)
−0.970894 + 0.239511i \(0.923013\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 30.0000 1.59000
\(357\) − 6.00000i − 0.317554i
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) − 3.00000i − 0.157459i
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 0 0
\(367\) 11.0000i 0.574195i 0.957901 + 0.287098i \(0.0926904\pi\)
−0.957901 + 0.287098i \(0.907310\pi\)
\(368\) − 20.0000i − 1.04257i
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) − 6.00000i − 0.311086i
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 40.0000i 2.06010i
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) 17.0000i 0.868659i 0.900754 + 0.434330i \(0.143015\pi\)
−0.900754 + 0.434330i \(0.856985\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 48.0000i 2.43998i
\(388\) 10.0000i 0.507673i
\(389\) −9.00000 −0.456318 −0.228159 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) −10.0000 −0.505722
\(392\) 0 0
\(393\) − 54.0000i − 2.72394i
\(394\) 0 0
\(395\) 0 0
\(396\) 12.0000 0.603023
\(397\) − 18.0000i − 0.903394i −0.892171 0.451697i \(-0.850819\pi\)
0.892171 0.451697i \(-0.149181\pi\)
\(398\) 0 0
\(399\) 18.0000 0.901127
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) − 4.00000i − 0.199254i
\(404\) −24.0000 −1.19404
\(405\) 0 0
\(406\) 0 0
\(407\) − 5.00000i − 0.247841i
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) − 24.0000i − 1.18240i
\(413\) − 3.00000i − 0.147620i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 30.0000i − 1.46911i
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) 48.0000i 2.33384i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 2.00000i − 0.0967868i
\(428\) 20.0000i 0.966736i
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) 36.0000i 1.73205i
\(433\) − 25.0000i − 1.20142i −0.799466 0.600712i \(-0.794884\pi\)
0.799466 0.600712i \(-0.205116\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) − 30.0000i − 1.43509i
\(438\) 0 0
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) − 39.0000i − 1.85295i −0.376361 0.926473i \(-0.622825\pi\)
0.376361 0.926473i \(-0.377175\pi\)
\(444\) 30.0000 1.42374
\(445\) 0 0
\(446\) 0 0
\(447\) − 66.0000i − 3.12169i
\(448\) 8.00000i 0.377964i
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) − 38.0000i − 1.78737i
\(453\) − 18.0000i − 0.845714i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 8.00000i − 0.374224i −0.982339 0.187112i \(-0.940087\pi\)
0.982339 0.187112i \(-0.0599128\pi\)
\(458\) 0 0
\(459\) 18.0000 0.840168
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 13.0000i 0.604161i 0.953282 + 0.302081i \(0.0976812\pi\)
−0.953282 + 0.302081i \(0.902319\pi\)
\(464\) −40.0000 −1.85695
\(465\) 0 0
\(466\) 0 0
\(467\) − 3.00000i − 0.138823i −0.997588 0.0694117i \(-0.977888\pi\)
0.997588 0.0694117i \(-0.0221122\pi\)
\(468\) 48.0000i 2.21880i
\(469\) −3.00000 −0.138527
\(470\) 0 0
\(471\) −21.0000 −0.967629
\(472\) 0 0
\(473\) 8.00000i 0.367840i
\(474\) 0 0
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 36.0000i 1.64833i
\(478\) 0 0
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) − 15.0000i − 0.682524i
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) 13.0000i 0.589086i 0.955638 + 0.294543i \(0.0951675\pi\)
−0.955638 + 0.294543i \(0.904833\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 12.0000i 0.541002i
\(493\) 20.0000i 0.900755i
\(494\) 0 0
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 1.00000i 0.0448561i
\(498\) 0 0
\(499\) −44.0000 −1.96971 −0.984855 0.173379i \(-0.944532\pi\)
−0.984855 + 0.173379i \(0.944532\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) − 4.00000i − 0.177471i
\(509\) 31.0000 1.37405 0.687025 0.726633i \(-0.258916\pi\)
0.687025 + 0.726633i \(0.258916\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) 54.0000i 2.38416i
\(514\) 0 0
\(515\) 0 0
\(516\) −48.0000 −2.11308
\(517\) 8.00000i 0.351840i
\(518\) 0 0
\(519\) 48.0000 2.10697
\(520\) 0 0
\(521\) 7.00000 0.306676 0.153338 0.988174i \(-0.450998\pi\)
0.153338 + 0.988174i \(0.450998\pi\)
\(522\) 0 0
\(523\) 32.0000i 1.39926i 0.714504 + 0.699631i \(0.246652\pi\)
−0.714504 + 0.699631i \(0.753348\pi\)
\(524\) 36.0000 1.57267
\(525\) 0 0
\(526\) 0 0
\(527\) − 2.00000i − 0.0871214i
\(528\) 12.0000i 0.522233i
\(529\) −2.00000 −0.0869565
\(530\) 0 0
\(531\) 18.0000 0.781133
\(532\) 12.0000i 0.520266i
\(533\) 8.00000i 0.346518i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 3.00000i 0.129460i
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) 0 0
\(543\) − 15.0000i − 0.643712i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 24.0000i 1.02617i 0.858339 + 0.513083i \(0.171497\pi\)
−0.858339 + 0.513083i \(0.828503\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) −60.0000 −2.55609
\(552\) 0 0
\(553\) − 6.00000i − 0.255146i
\(554\) 0 0
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) − 14.0000i − 0.593199i −0.955002 0.296600i \(-0.904147\pi\)
0.955002 0.296600i \(-0.0958526\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 0 0
\(563\) 20.0000i 0.842900i 0.906852 + 0.421450i \(0.138479\pi\)
−0.906852 + 0.421450i \(0.861521\pi\)
\(564\) −48.0000 −2.02116
\(565\) 0 0
\(566\) 0 0
\(567\) 9.00000i 0.377964i
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 8.00000i 0.334497i
\(573\) − 15.0000i − 0.626634i
\(574\) 0 0
\(575\) 0 0
\(576\) −48.0000 −2.00000
\(577\) 25.0000i 1.04076i 0.853934 + 0.520382i \(0.174210\pi\)
−0.853934 + 0.520382i \(0.825790\pi\)
\(578\) 0 0
\(579\) 42.0000 1.74546
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 6.00000i 0.248495i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 36.0000i − 1.48588i −0.669359 0.742940i \(-0.733431\pi\)
0.669359 0.742940i \(-0.266569\pi\)
\(588\) 6.00000i 0.247436i
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) −54.0000 −2.22126
\(592\) 20.0000i 0.821995i
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 44.0000 1.80231
\(597\) − 24.0000i − 0.982255i
\(598\) 0 0
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 0 0
\(603\) − 18.0000i − 0.733017i
\(604\) 12.0000 0.488273
\(605\) 0 0
\(606\) 0 0
\(607\) 10.0000i 0.405887i 0.979190 + 0.202944i \(0.0650509\pi\)
−0.979190 + 0.202944i \(0.934949\pi\)
\(608\) 0 0
\(609\) −30.0000 −1.21566
\(610\) 0 0
\(611\) −32.0000 −1.29458
\(612\) 24.0000i 0.970143i
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) 0 0
\(619\) −17.0000 −0.683288 −0.341644 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) 0 0
\(621\) 45.0000 1.80579
\(622\) 0 0
\(623\) 15.0000i 0.600962i
\(624\) −48.0000 −1.92154
\(625\) 0 0
\(626\) 0 0
\(627\) 18.0000i 0.718851i
\(628\) − 14.0000i − 0.558661i
\(629\) 10.0000 0.398726
\(630\) 0 0
\(631\) 27.0000 1.07485 0.537427 0.843311i \(-0.319397\pi\)
0.537427 + 0.843311i \(0.319397\pi\)
\(632\) 0 0
\(633\) 6.00000i 0.238479i
\(634\) 0 0
\(635\) 0 0
\(636\) −36.0000 −1.42749
\(637\) 4.00000i 0.158486i
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 0 0
\(643\) − 29.0000i − 1.14365i −0.820376 0.571824i \(-0.806236\pi\)
0.820376 0.571824i \(-0.193764\pi\)
\(644\) 10.0000 0.394055
\(645\) 0 0
\(646\) 0 0
\(647\) 21.0000i 0.825595i 0.910823 + 0.412798i \(0.135448\pi\)
−0.910823 + 0.412798i \(0.864552\pi\)
\(648\) 0 0
\(649\) 3.00000 0.117760
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) 8.00000i 0.313304i
\(653\) − 17.0000i − 0.665261i −0.943057 0.332631i \(-0.892064\pi\)
0.943057 0.332631i \(-0.107936\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −8.00000 −0.312348
\(657\) − 60.0000i − 2.34082i
\(658\) 0 0
\(659\) 2.00000 0.0779089 0.0389545 0.999241i \(-0.487597\pi\)
0.0389545 + 0.999241i \(0.487597\pi\)
\(660\) 0 0
\(661\) 35.0000 1.36134 0.680671 0.732589i \(-0.261688\pi\)
0.680671 + 0.732589i \(0.261688\pi\)
\(662\) 0 0
\(663\) 24.0000i 0.932083i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 50.0000i 1.93601i
\(668\) 4.00000i 0.154765i
\(669\) 3.00000 0.115987
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) 4.00000i 0.154189i 0.997024 + 0.0770943i \(0.0245643\pi\)
−0.997024 + 0.0770943i \(0.975436\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −6.00000 −0.230769
\(677\) − 38.0000i − 1.46046i −0.683202 0.730229i \(-0.739413\pi\)
0.683202 0.730229i \(-0.260587\pi\)
\(678\) 0 0
\(679\) −5.00000 −0.191882
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) −72.0000 −2.75299
\(685\) 0 0
\(686\) 0 0
\(687\) − 21.0000i − 0.801200i
\(688\) − 32.0000i − 1.21999i
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 15.0000 0.570627 0.285313 0.958434i \(-0.407902\pi\)
0.285313 + 0.958434i \(0.407902\pi\)
\(692\) 32.0000i 1.21646i
\(693\) 6.00000i 0.227921i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4.00000i 0.151511i
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 30.0000i 1.13147i
\(704\) −8.00000 −0.301511
\(705\) 0 0
\(706\) 0 0
\(707\) − 12.0000i − 0.451306i
\(708\) 18.0000i 0.676481i
\(709\) −39.0000 −1.46468 −0.732338 0.680941i \(-0.761571\pi\)
−0.732338 + 0.680941i \(0.761571\pi\)
\(710\) 0 0
\(711\) 36.0000 1.35011
\(712\) 0 0
\(713\) − 5.00000i − 0.187251i
\(714\) 0 0
\(715\) 0 0
\(716\) −2.00000 −0.0747435
\(717\) 12.0000i 0.448148i
\(718\) 0 0
\(719\) 11.0000 0.410231 0.205115 0.978738i \(-0.434243\pi\)
0.205115 + 0.978738i \(0.434243\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 36.0000i 1.33885i
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) 0 0
\(727\) 19.0000i 0.704671i 0.935874 + 0.352335i \(0.114612\pi\)
−0.935874 + 0.352335i \(0.885388\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 12.0000i 0.443533i
\(733\) − 4.00000i − 0.147743i −0.997268 0.0738717i \(-0.976464\pi\)
0.997268 0.0738717i \(-0.0235355\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 3.00000i − 0.110506i
\(738\) 0 0
\(739\) 18.0000 0.662141 0.331070 0.943606i \(-0.392590\pi\)
0.331070 + 0.943606i \(0.392590\pi\)
\(740\) 0 0
\(741\) −72.0000 −2.64499
\(742\) 0 0
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 72.0000i − 2.63434i
\(748\) 4.00000i 0.146254i
\(749\) −10.0000 −0.365392
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) − 32.0000i − 1.16692i
\(753\) 63.0000i 2.29585i
\(754\) 0 0
\(755\) 0 0
\(756\) −18.0000 −0.654654
\(757\) − 38.0000i − 1.38113i −0.723269 0.690567i \(-0.757361\pi\)
0.723269 0.690567i \(-0.242639\pi\)
\(758\) 0 0
\(759\) 15.0000 0.544466
\(760\) 0 0
\(761\) −48.0000 −1.74000 −0.869999 0.493053i \(-0.835881\pi\)
−0.869999 + 0.493053i \(0.835881\pi\)
\(762\) 0 0
\(763\) − 4.00000i − 0.144810i
\(764\) 10.0000 0.361787
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000i 0.433295i
\(768\) − 48.0000i − 1.73205i
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 28.0000i 1.00774i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 15.0000i 0.538122i
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −1.00000 −0.0357828
\(782\) 0 0
\(783\) − 90.0000i − 3.21634i
\(784\) −4.00000 −0.142857
\(785\) 0 0
\(786\) 0 0
\(787\) 22.0000i 0.784215i 0.919919 + 0.392108i \(0.128254\pi\)
−0.919919 + 0.392108i \(0.871746\pi\)
\(788\) − 36.0000i − 1.28245i
\(789\) 54.0000 1.92245
\(790\) 0 0
\(791\) 19.0000 0.675562
\(792\) 0 0
\(793\) 8.00000i 0.284088i
\(794\) 0 0
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) − 23.0000i − 0.814702i −0.913272 0.407351i \(-0.866453\pi\)
0.913272 0.407351i \(-0.133547\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −90.0000 −3.17999
\(802\) 0 0
\(803\) − 10.0000i − 0.352892i
\(804\) 18.0000 0.634811
\(805\) 0 0
\(806\) 0 0
\(807\) − 54.0000i − 1.90089i
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) − 20.0000i − 0.701862i
\(813\) − 48.0000i − 1.68343i
\(814\) 0 0
\(815\) 0 0
\(816\) −24.0000 −0.840168
\(817\) − 48.0000i − 1.67931i
\(818\) 0 0
\(819\) −24.0000 −0.838628
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) − 25.0000i − 0.871445i −0.900081 0.435723i \(-0.856493\pi\)
0.900081 0.435723i \(-0.143507\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 20.0000i − 0.695468i −0.937593 0.347734i \(-0.886951\pi\)
0.937593 0.347734i \(-0.113049\pi\)
\(828\) 60.0000i 2.08514i
\(829\) 29.0000 1.00721 0.503606 0.863934i \(-0.332006\pi\)
0.503606 + 0.863934i \(0.332006\pi\)
\(830\) 0 0
\(831\) −72.0000 −2.49765
\(832\) − 32.0000i − 1.10940i
\(833\) 2.00000i 0.0692959i
\(834\) 0 0
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 9.00000i 0.311086i
\(838\) 0 0
\(839\) −45.0000 −1.55357 −0.776786 0.629764i \(-0.783151\pi\)
−0.776786 + 0.629764i \(0.783151\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 12.0000i 0.413302i
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 1.00000i 0.0343604i
\(848\) − 24.0000i − 0.824163i
\(849\) 0 0
\(850\) 0 0
\(851\) 25.0000 0.856989
\(852\) − 6.00000i − 0.205557i
\(853\) − 34.0000i − 1.16414i −0.813139 0.582069i \(-0.802243\pi\)
0.813139 0.582069i \(-0.197757\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.0000i 0.956462i 0.878234 + 0.478231i \(0.158722\pi\)
−0.878234 + 0.478231i \(0.841278\pi\)
\(858\) 0 0
\(859\) −55.0000 −1.87658 −0.938288 0.345855i \(-0.887589\pi\)
−0.938288 + 0.345855i \(0.887589\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) 52.0000i 1.77010i 0.465495 + 0.885050i \(0.345876\pi\)
−0.465495 + 0.885050i \(0.654124\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 39.0000i − 1.32451i
\(868\) 2.00000i 0.0678844i
\(869\) 6.00000 0.203536
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) − 30.0000i − 1.01535i
\(874\) 0 0
\(875\) 0 0
\(876\) 60.0000 2.02721
\(877\) 38.0000i 1.28317i 0.767052 + 0.641584i \(0.221723\pi\)
−0.767052 + 0.641584i \(0.778277\pi\)
\(878\) 0 0
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) 27.0000 0.909653 0.454827 0.890580i \(-0.349701\pi\)
0.454827 + 0.890580i \(0.349701\pi\)
\(882\) 0 0
\(883\) − 44.0000i − 1.48072i −0.672212 0.740359i \(-0.734656\pi\)
0.672212 0.740359i \(-0.265344\pi\)
\(884\) −16.0000 −0.538138
\(885\) 0 0
\(886\) 0 0
\(887\) 2.00000i 0.0671534i 0.999436 + 0.0335767i \(0.0106898\pi\)
−0.999436 + 0.0335767i \(0.989310\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) 2.00000i 0.0669650i
\(893\) − 48.0000i − 1.60626i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 60.0000i 2.00334i
\(898\) 0 0
\(899\) −10.0000 −0.333519
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) − 24.0000i − 0.798670i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 40.0000i 1.32818i 0.747653 + 0.664089i \(0.231180\pi\)
−0.747653 + 0.664089i \(0.768820\pi\)
\(908\) − 8.00000i − 0.265489i
\(909\) 72.0000 2.38809
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) − 72.0000i − 2.38416i
\(913\) − 12.0000i − 0.397142i
\(914\) 0 0
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 18.0000i 0.594412i
\(918\) 0 0
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 0 0
\(921\) 84.0000 2.76789
\(922\) 0 0
\(923\) − 4.00000i − 0.131662i
\(924\) −6.00000 −0.197386
\(925\) 0 0
\(926\) 0 0
\(927\) 72.0000i 2.36479i
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 12.0000i 0.393073i
\(933\) − 24.0000i − 0.785725i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 36.0000i − 1.17607i −0.808836 0.588034i \(-0.799902\pi\)
0.808836 0.588034i \(-0.200098\pi\)
\(938\) 0 0
\(939\) −69.0000 −2.25173
\(940\) 0 0
\(941\) 58.0000 1.89075 0.945373 0.325991i \(-0.105698\pi\)
0.945373 + 0.325991i \(0.105698\pi\)
\(942\) 0 0
\(943\) 10.0000i 0.325645i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) − 5.00000i − 0.162478i −0.996695 0.0812391i \(-0.974112\pi\)
0.996695 0.0812391i \(-0.0258877\pi\)
\(948\) 36.0000i 1.16923i
\(949\) 40.0000 1.29845
\(950\) 0 0
\(951\) −27.0000 −0.875535
\(952\) 0 0
\(953\) 44.0000i 1.42530i 0.701520 + 0.712650i \(0.252505\pi\)
−0.701520 + 0.712650i \(0.747495\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) − 30.0000i − 0.969762i
\(958\) 0 0
\(959\) −3.00000 −0.0968751
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) − 60.0000i − 1.93347i
\(964\) −24.0000 −0.772988
\(965\) 0 0
\(966\) 0 0
\(967\) 34.0000i 1.09337i 0.837340 + 0.546683i \(0.184110\pi\)
−0.837340 + 0.546683i \(0.815890\pi\)
\(968\) 0 0
\(969\) −36.0000 −1.15649
\(970\) 0 0
\(971\) 29.0000 0.930654 0.465327 0.885139i \(-0.345937\pi\)
0.465327 + 0.885139i \(0.345937\pi\)
\(972\) 0 0
\(973\) 10.0000i 0.320585i
\(974\) 0 0
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) 31.0000i 0.991778i 0.868386 + 0.495889i \(0.165158\pi\)
−0.868386 + 0.495889i \(0.834842\pi\)
\(978\) 0 0
\(979\) −15.0000 −0.479402
\(980\) 0 0
\(981\) 24.0000 0.766261
\(982\) 0 0
\(983\) − 27.0000i − 0.861166i −0.902551 0.430583i \(-0.858308\pi\)
0.902551 0.430583i \(-0.141692\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 24.0000i − 0.763928i
\(988\) − 48.0000i − 1.52708i
\(989\) −40.0000 −1.27193
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 51.0000i 1.61844i
\(994\) 0 0
\(995\) 0 0
\(996\) 72.0000 2.28141
\(997\) 12.0000i 0.380044i 0.981780 + 0.190022i \(0.0608559\pi\)
−0.981780 + 0.190022i \(0.939144\pi\)
\(998\) 0 0
\(999\) −45.0000 −1.42374
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 1925.2.b.e.1849.1 2
5.2 odd 4 77.2.a.a.1.1 1
5.3 odd 4 1925.2.a.h.1.1 1
5.4 even 2 inner 1925.2.b.e.1849.2 2
15.2 even 4 693.2.a.c.1.1 1
20.7 even 4 1232.2.a.l.1.1 1
35.2 odd 12 539.2.e.f.67.1 2
35.12 even 12 539.2.e.c.67.1 2
35.17 even 12 539.2.e.c.177.1 2
35.27 even 4 539.2.a.c.1.1 1
35.32 odd 12 539.2.e.f.177.1 2
40.27 even 4 4928.2.a.a.1.1 1
40.37 odd 4 4928.2.a.bj.1.1 1
55.2 even 20 847.2.f.h.323.1 4
55.7 even 20 847.2.f.h.148.1 4
55.17 even 20 847.2.f.h.729.1 4
55.27 odd 20 847.2.f.i.729.1 4
55.32 even 4 847.2.a.b.1.1 1
55.37 odd 20 847.2.f.i.148.1 4
55.42 odd 20 847.2.f.i.323.1 4
55.47 odd 20 847.2.f.i.372.1 4
55.52 even 20 847.2.f.h.372.1 4
105.62 odd 4 4851.2.a.j.1.1 1
140.27 odd 4 8624.2.a.a.1.1 1
165.32 odd 4 7623.2.a.j.1.1 1
385.307 odd 4 5929.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.a.1.1 1 5.2 odd 4
539.2.a.c.1.1 1 35.27 even 4
539.2.e.c.67.1 2 35.12 even 12
539.2.e.c.177.1 2 35.17 even 12
539.2.e.f.67.1 2 35.2 odd 12
539.2.e.f.177.1 2 35.32 odd 12
693.2.a.c.1.1 1 15.2 even 4
847.2.a.b.1.1 1 55.32 even 4
847.2.f.h.148.1 4 55.7 even 20
847.2.f.h.323.1 4 55.2 even 20
847.2.f.h.372.1 4 55.52 even 20
847.2.f.h.729.1 4 55.17 even 20
847.2.f.i.148.1 4 55.37 odd 20
847.2.f.i.323.1 4 55.42 odd 20
847.2.f.i.372.1 4 55.47 odd 20
847.2.f.i.729.1 4 55.27 odd 20
1232.2.a.l.1.1 1 20.7 even 4
1925.2.a.h.1.1 1 5.3 odd 4
1925.2.b.e.1849.1 2 1.1 even 1 trivial
1925.2.b.e.1849.2 2 5.4 even 2 inner
4851.2.a.j.1.1 1 105.62 odd 4
4928.2.a.a.1.1 1 40.27 even 4
4928.2.a.bj.1.1 1 40.37 odd 4
5929.2.a.f.1.1 1 385.307 odd 4
7623.2.a.j.1.1 1 165.32 odd 4
8624.2.a.a.1.1 1 140.27 odd 4