Properties

Label 975.2.c.a.274.2
Level $975$
Weight $2$
Character 975.274
Analytic conductor $7.785$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 975.274
Dual form 975.2.c.a.274.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{12} +1.00000i q^{13} -6.00000 q^{14} -4.00000 q^{16} -1.00000i q^{17} -2.00000i q^{18} +2.00000 q^{19} -3.00000 q^{21} -2.00000i q^{22} +3.00000i q^{23} -2.00000 q^{26} -1.00000i q^{27} -6.00000i q^{28} +2.00000 q^{29} -6.00000 q^{31} -8.00000i q^{32} -1.00000i q^{33} +2.00000 q^{34} +2.00000 q^{36} +11.0000i q^{37} +4.00000i q^{38} -1.00000 q^{39} -5.00000 q^{41} -6.00000i q^{42} -4.00000i q^{43} +2.00000 q^{44} -6.00000 q^{46} -10.0000i q^{47} -4.00000i q^{48} -2.00000 q^{49} +1.00000 q^{51} -2.00000i q^{52} -11.0000i q^{53} +2.00000 q^{54} +2.00000i q^{57} +4.00000i q^{58} -8.00000 q^{59} +13.0000 q^{61} -12.0000i q^{62} -3.00000i q^{63} +8.00000 q^{64} +2.00000 q^{66} +12.0000i q^{67} +2.00000i q^{68} -3.00000 q^{69} -5.00000 q^{71} -10.0000i q^{73} -22.0000 q^{74} -4.00000 q^{76} -3.00000i q^{77} -2.00000i q^{78} +3.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +12.0000i q^{83} +6.00000 q^{84} +8.00000 q^{86} +2.00000i q^{87} +15.0000 q^{89} -3.00000 q^{91} -6.00000i q^{92} -6.00000i q^{93} +20.0000 q^{94} +8.00000 q^{96} +17.0000i q^{97} -4.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{11} - 12 q^{14} - 8 q^{16} + 4 q^{19} - 6 q^{21} - 4 q^{26} + 4 q^{29} - 12 q^{31} + 4 q^{34} + 4 q^{36} - 2 q^{39} - 10 q^{41} + 4 q^{44} - 12 q^{46} - 4 q^{49} + 2 q^{51} + 4 q^{54} - 16 q^{59} + 26 q^{61} + 16 q^{64} + 4 q^{66} - 6 q^{69} - 10 q^{71} - 44 q^{74} - 8 q^{76} + 6 q^{79} + 2 q^{81} + 12 q^{84} + 16 q^{86} + 30 q^{89} - 6 q^{91} + 40 q^{94} + 16 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) − 2.00000i − 0.577350i
\(13\) 1.00000i 0.277350i
\(14\) −6.00000 −1.60357
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) − 1.00000i − 0.242536i −0.992620 0.121268i \(-0.961304\pi\)
0.992620 0.121268i \(-0.0386960\pi\)
\(18\) − 2.00000i − 0.471405i
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) − 2.00000i − 0.426401i
\(23\) 3.00000i 0.625543i 0.949828 + 0.312772i \(0.101257\pi\)
−0.949828 + 0.312772i \(0.898743\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) − 6.00000i − 1.13389i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) − 8.00000i − 1.41421i
\(33\) − 1.00000i − 0.174078i
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 11.0000i 1.80839i 0.427121 + 0.904194i \(0.359528\pi\)
−0.427121 + 0.904194i \(0.640472\pi\)
\(38\) 4.00000i 0.648886i
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) − 6.00000i − 0.925820i
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) − 10.0000i − 1.45865i −0.684167 0.729325i \(-0.739834\pi\)
0.684167 0.729325i \(-0.260166\pi\)
\(48\) − 4.00000i − 0.577350i
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) − 2.00000i − 0.277350i
\(53\) − 11.0000i − 1.51097i −0.655168 0.755483i \(-0.727402\pi\)
0.655168 0.755483i \(-0.272598\pi\)
\(54\) 2.00000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) 4.00000i 0.525226i
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) − 12.0000i − 1.52400i
\(63\) − 3.00000i − 0.377964i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) −22.0000 −2.55745
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) − 3.00000i − 0.341882i
\(78\) − 2.00000i − 0.226455i
\(79\) 3.00000 0.337526 0.168763 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 10.0000i − 1.10432i
\(83\) 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
\(86\) 8.00000 0.862662
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) − 6.00000i − 0.625543i
\(93\) − 6.00000i − 0.622171i
\(94\) 20.0000 2.06284
\(95\) 0 0
\(96\) 8.00000 0.816497
\(97\) 17.0000i 1.72609i 0.505128 + 0.863044i \(0.331445\pi\)
−0.505128 + 0.863044i \(0.668555\pi\)
\(98\) − 4.00000i − 0.404061i
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 2.00000i 0.198030i
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 22.0000 2.13683
\(107\) 9.00000i 0.870063i 0.900415 + 0.435031i \(0.143263\pi\)
−0.900415 + 0.435031i \(0.856737\pi\)
\(108\) 2.00000i 0.192450i
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) −11.0000 −1.04407
\(112\) − 12.0000i − 1.13389i
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) − 1.00000i − 0.0924500i
\(118\) − 16.0000i − 1.47292i
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 26.0000i 2.35393i
\(123\) − 5.00000i − 0.450835i
\(124\) 12.0000 1.07763
\(125\) 0 0
\(126\) 6.00000 0.534522
\(127\) 10.0000i 0.887357i 0.896186 + 0.443678i \(0.146327\pi\)
−0.896186 + 0.443678i \(0.853673\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 2.00000i 0.174078i
\(133\) 6.00000i 0.520266i
\(134\) −24.0000 −2.07328
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) − 6.00000i − 0.510754i
\(139\) 1.00000 0.0848189 0.0424094 0.999100i \(-0.486497\pi\)
0.0424094 + 0.999100i \(0.486497\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) − 10.0000i − 0.839181i
\(143\) − 1.00000i − 0.0836242i
\(144\) 4.00000 0.333333
\(145\) 0 0
\(146\) 20.0000 1.65521
\(147\) − 2.00000i − 0.164957i
\(148\) − 22.0000i − 1.80839i
\(149\) −13.0000 −1.06500 −0.532501 0.846430i \(-0.678748\pi\)
−0.532501 + 0.846430i \(0.678748\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 1.00000i 0.0808452i
\(154\) 6.00000 0.483494
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) − 10.0000i − 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) 6.00000i 0.477334i
\(159\) 11.0000 0.872357
\(160\) 0 0
\(161\) −9.00000 −0.709299
\(162\) 2.00000i 0.157135i
\(163\) 13.0000i 1.01824i 0.860696 + 0.509119i \(0.170029\pi\)
−0.860696 + 0.509119i \(0.829971\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −24.0000 −1.86276
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 8.00000i 0.609994i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) − 8.00000i − 0.601317i
\(178\) 30.0000i 2.24860i
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) − 6.00000i − 0.444750i
\(183\) 13.0000i 0.960988i
\(184\) 0 0
\(185\) 0 0
\(186\) 12.0000 0.879883
\(187\) 1.00000i 0.0731272i
\(188\) 20.0000i 1.45865i
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 8.00000i 0.577350i
\(193\) 13.0000i 0.935760i 0.883792 + 0.467880i \(0.154982\pi\)
−0.883792 + 0.467880i \(0.845018\pi\)
\(194\) −34.0000 −2.44106
\(195\) 0 0
\(196\) 4.00000 0.285714
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 2.00000i 0.142134i
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) 6.00000i 0.421117i
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −32.0000 −2.22955
\(207\) − 3.00000i − 0.208514i
\(208\) − 4.00000i − 0.277350i
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 22.0000i 1.51097i
\(213\) − 5.00000i − 0.342594i
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) 0 0
\(217\) − 18.0000i − 1.22192i
\(218\) 32.0000i 2.16731i
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 1.00000 0.0672673
\(222\) − 22.0000i − 1.47654i
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 24.0000 1.60357
\(225\) 0 0
\(226\) 28.0000 1.86253
\(227\) 22.0000i 1.46019i 0.683345 + 0.730096i \(0.260525\pi\)
−0.683345 + 0.730096i \(0.739475\pi\)
\(228\) − 4.00000i − 0.264906i
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) 27.0000i 1.76883i 0.466702 + 0.884414i \(0.345442\pi\)
−0.466702 + 0.884414i \(0.654558\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 16.0000 1.04151
\(237\) 3.00000i 0.194871i
\(238\) 6.00000i 0.388922i
\(239\) 13.0000 0.840900 0.420450 0.907316i \(-0.361872\pi\)
0.420450 + 0.907316i \(0.361872\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) − 20.0000i − 1.28565i
\(243\) 1.00000i 0.0641500i
\(244\) −26.0000 −1.66448
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 2.00000i 0.127257i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 6.00000i 0.377964i
\(253\) − 3.00000i − 0.188608i
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 8.00000i 0.498058i
\(259\) −33.0000 −2.05052
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) − 12.0000i − 0.741362i
\(263\) 8.00000i 0.493301i 0.969104 + 0.246651i \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −12.0000 −0.735767
\(267\) 15.0000i 0.917985i
\(268\) − 24.0000i − 1.46603i
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 0 0
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 4.00000i 0.242536i
\(273\) − 3.00000i − 0.181568i
\(274\) −36.0000 −2.17484
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) − 18.0000i − 1.08152i −0.841178 0.540758i \(-0.818138\pi\)
0.841178 0.540758i \(-0.181862\pi\)
\(278\) 2.00000i 0.119952i
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 20.0000i 1.19098i
\(283\) 12.0000i 0.713326i 0.934233 + 0.356663i \(0.116086\pi\)
−0.934233 + 0.356663i \(0.883914\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) − 15.0000i − 0.885422i
\(288\) 8.00000i 0.471405i
\(289\) 16.0000 0.941176
\(290\) 0 0
\(291\) −17.0000 −0.996558
\(292\) 20.0000i 1.17041i
\(293\) − 24.0000i − 1.40209i −0.713115 0.701047i \(-0.752716\pi\)
0.713115 0.701047i \(-0.247284\pi\)
\(294\) 4.00000 0.233285
\(295\) 0 0
\(296\) 0 0
\(297\) 1.00000i 0.0580259i
\(298\) − 26.0000i − 1.50614i
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 32.0000i 1.84139i
\(303\) 0 0
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) − 5.00000i − 0.285365i −0.989769 0.142683i \(-0.954427\pi\)
0.989769 0.142683i \(-0.0455728\pi\)
\(308\) 6.00000i 0.341882i
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 20.0000 1.12867
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) 28.0000i 1.57264i 0.617822 + 0.786318i \(0.288015\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 22.0000i 1.23370i
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) −9.00000 −0.502331
\(322\) − 18.0000i − 1.00310i
\(323\) − 2.00000i − 0.111283i
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) −26.0000 −1.44001
\(327\) 16.0000i 0.884802i
\(328\) 0 0
\(329\) 30.0000 1.65395
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) − 24.0000i − 1.31717i
\(333\) − 11.0000i − 0.602796i
\(334\) 24.0000 1.31322
\(335\) 0 0
\(336\) 12.0000 0.654654
\(337\) 4.00000i 0.217894i 0.994048 + 0.108947i \(0.0347479\pi\)
−0.994048 + 0.108947i \(0.965252\pi\)
\(338\) − 2.00000i − 0.108786i
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) − 4.00000i − 0.216295i
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) − 19.0000i − 1.01997i −0.860182 0.509987i \(-0.829650\pi\)
0.860182 0.509987i \(-0.170350\pi\)
\(348\) − 4.00000i − 0.214423i
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 8.00000i 0.426401i
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 16.0000 0.850390
\(355\) 0 0
\(356\) −30.0000 −1.59000
\(357\) 3.00000i 0.158777i
\(358\) − 4.00000i − 0.211407i
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) − 14.0000i − 0.735824i
\(363\) − 10.0000i − 0.524864i
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) −26.0000 −1.35904
\(367\) 36.0000i 1.87918i 0.342296 + 0.939592i \(0.388796\pi\)
−0.342296 + 0.939592i \(0.611204\pi\)
\(368\) − 12.0000i − 0.625543i
\(369\) 5.00000 0.260290
\(370\) 0 0
\(371\) 33.0000 1.71327
\(372\) 12.0000i 0.622171i
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) 0 0
\(377\) 2.00000i 0.103005i
\(378\) 6.00000i 0.308607i
\(379\) 14.0000 0.719132 0.359566 0.933120i \(-0.382925\pi\)
0.359566 + 0.933120i \(0.382925\pi\)
\(380\) 0 0
\(381\) −10.0000 −0.512316
\(382\) − 16.0000i − 0.818631i
\(383\) 30.0000i 1.53293i 0.642287 + 0.766464i \(0.277986\pi\)
−0.642287 + 0.766464i \(0.722014\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −26.0000 −1.32337
\(387\) 4.00000i 0.203331i
\(388\) − 34.0000i − 1.72609i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) − 6.00000i − 0.302660i
\(394\) 0 0
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) − 29.0000i − 1.45547i −0.685859 0.727734i \(-0.740573\pi\)
0.685859 0.727734i \(-0.259427\pi\)
\(398\) − 8.00000i − 0.401004i
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) − 24.0000i − 1.19701i
\(403\) − 6.00000i − 0.298881i
\(404\) 0 0
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) − 11.0000i − 0.545250i
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) − 32.0000i − 1.57653i
\(413\) − 24.0000i − 1.18096i
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 8.00000 0.392232
\(417\) 1.00000i 0.0489702i
\(418\) − 4.00000i − 0.195646i
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) − 8.00000i − 0.389434i
\(423\) 10.0000i 0.486217i
\(424\) 0 0
\(425\) 0 0
\(426\) 10.0000 0.484502
\(427\) 39.0000i 1.88734i
\(428\) − 18.0000i − 0.870063i
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 4.00000i 0.192450i
\(433\) − 4.00000i − 0.192228i −0.995370 0.0961139i \(-0.969359\pi\)
0.995370 0.0961139i \(-0.0306413\pi\)
\(434\) 36.0000 1.72806
\(435\) 0 0
\(436\) −32.0000 −1.53252
\(437\) 6.00000i 0.287019i
\(438\) 20.0000i 0.955637i
\(439\) 17.0000 0.811366 0.405683 0.914014i \(-0.367034\pi\)
0.405683 + 0.914014i \(0.367034\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 2.00000i 0.0951303i
\(443\) − 9.00000i − 0.427603i −0.976877 0.213801i \(-0.931415\pi\)
0.976877 0.213801i \(-0.0685846\pi\)
\(444\) 22.0000 1.04407
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) − 13.0000i − 0.614879i
\(448\) 24.0000i 1.13389i
\(449\) 13.0000 0.613508 0.306754 0.951789i \(-0.400757\pi\)
0.306754 + 0.951789i \(0.400757\pi\)
\(450\) 0 0
\(451\) 5.00000 0.235441
\(452\) 28.0000i 1.31701i
\(453\) 16.0000i 0.751746i
\(454\) −44.0000 −2.06502
\(455\) 0 0
\(456\) 0 0
\(457\) − 11.0000i − 0.514558i −0.966337 0.257279i \(-0.917174\pi\)
0.966337 0.257279i \(-0.0828260\pi\)
\(458\) − 36.0000i − 1.68217i
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 15.0000 0.698620 0.349310 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(462\) 6.00000i 0.279145i
\(463\) − 27.0000i − 1.25480i −0.778699 0.627398i \(-0.784120\pi\)
0.778699 0.627398i \(-0.215880\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) −54.0000 −2.50150
\(467\) − 23.0000i − 1.06431i −0.846646 0.532157i \(-0.821382\pi\)
0.846646 0.532157i \(-0.178618\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) −36.0000 −1.66233
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) 4.00000i 0.183920i
\(474\) −6.00000 −0.275589
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) 11.0000i 0.503655i
\(478\) 26.0000i 1.18921i
\(479\) −9.00000 −0.411220 −0.205610 0.978634i \(-0.565918\pi\)
−0.205610 + 0.978634i \(0.565918\pi\)
\(480\) 0 0
\(481\) −11.0000 −0.501557
\(482\) − 4.00000i − 0.182195i
\(483\) − 9.00000i − 0.409514i
\(484\) 20.0000 0.909091
\(485\) 0 0
\(486\) −2.00000 −0.0907218
\(487\) − 7.00000i − 0.317200i −0.987343 0.158600i \(-0.949302\pi\)
0.987343 0.158600i \(-0.0506981\pi\)
\(488\) 0 0
\(489\) −13.0000 −0.587880
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 10.0000i 0.450835i
\(493\) − 2.00000i − 0.0900755i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 24.0000 1.07763
\(497\) − 15.0000i − 0.672842i
\(498\) − 24.0000i − 1.07547i
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) − 28.0000i − 1.24846i −0.781241 0.624229i \(-0.785413\pi\)
0.781241 0.624229i \(-0.214587\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) − 1.00000i − 0.0444116i
\(508\) − 20.0000i − 0.887357i
\(509\) 7.00000 0.310270 0.155135 0.987893i \(-0.450419\pi\)
0.155135 + 0.987893i \(0.450419\pi\)
\(510\) 0 0
\(511\) 30.0000 1.32712
\(512\) 32.0000i 1.41421i
\(513\) − 2.00000i − 0.0883022i
\(514\) 36.0000 1.58789
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 10.0000i 0.439799i
\(518\) − 66.0000i − 2.89987i
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) − 4.00000i − 0.175075i
\(523\) − 16.0000i − 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 6.00000i 0.261364i
\(528\) 4.00000i 0.174078i
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) − 12.0000i − 0.520266i
\(533\) − 5.00000i − 0.216574i
\(534\) −30.0000 −1.29823
\(535\) 0 0
\(536\) 0 0
\(537\) − 2.00000i − 0.0863064i
\(538\) 8.00000i 0.344904i
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 44.0000i 1.88996i
\(543\) − 7.00000i − 0.300399i
\(544\) −8.00000 −0.342997
\(545\) 0 0
\(546\) 6.00000 0.256776
\(547\) − 32.0000i − 1.36822i −0.729378 0.684111i \(-0.760191\pi\)
0.729378 0.684111i \(-0.239809\pi\)
\(548\) − 36.0000i − 1.53784i
\(549\) −13.0000 −0.554826
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) 9.00000i 0.382719i
\(554\) 36.0000 1.52949
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) − 30.0000i − 1.27114i −0.772043 0.635570i \(-0.780765\pi\)
0.772043 0.635570i \(-0.219235\pi\)
\(558\) 12.0000i 0.508001i
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) 60.0000i 2.53095i
\(563\) − 21.0000i − 0.885044i −0.896758 0.442522i \(-0.854084\pi\)
0.896758 0.442522i \(-0.145916\pi\)
\(564\) −20.0000 −0.842152
\(565\) 0 0
\(566\) −24.0000 −1.00880
\(567\) 3.00000i 0.125988i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −31.0000 −1.29731 −0.648655 0.761083i \(-0.724668\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 2.00000i 0.0836242i
\(573\) − 8.00000i − 0.334205i
\(574\) 30.0000 1.25218
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) − 19.0000i − 0.790980i −0.918470 0.395490i \(-0.870575\pi\)
0.918470 0.395490i \(-0.129425\pi\)
\(578\) 32.0000i 1.33102i
\(579\) −13.0000 −0.540262
\(580\) 0 0
\(581\) −36.0000 −1.49353
\(582\) − 34.0000i − 1.40935i
\(583\) 11.0000i 0.455573i
\(584\) 0 0
\(585\) 0 0
\(586\) 48.0000 1.98286
\(587\) − 18.0000i − 0.742940i −0.928445 0.371470i \(-0.878854\pi\)
0.928445 0.371470i \(-0.121146\pi\)
\(588\) 4.00000i 0.164957i
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 0 0
\(592\) − 44.0000i − 1.80839i
\(593\) − 4.00000i − 0.164260i −0.996622 0.0821302i \(-0.973828\pi\)
0.996622 0.0821302i \(-0.0261723\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 26.0000 1.06500
\(597\) − 4.00000i − 0.163709i
\(598\) − 6.00000i − 0.245358i
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 24.0000i 0.978167i
\(603\) − 12.0000i − 0.488678i
\(604\) −32.0000 −1.30206
\(605\) 0 0
\(606\) 0 0
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) − 16.0000i − 0.648886i
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 10.0000 0.404557
\(612\) − 2.00000i − 0.0808452i
\(613\) − 13.0000i − 0.525065i −0.964923 0.262533i \(-0.915442\pi\)
0.964923 0.262533i \(-0.0845577\pi\)
\(614\) 10.0000 0.403567
\(615\) 0 0
\(616\) 0 0
\(617\) − 42.0000i − 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) − 32.0000i − 1.28723i
\(619\) 34.0000 1.36658 0.683288 0.730149i \(-0.260549\pi\)
0.683288 + 0.730149i \(0.260549\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 48.0000i 1.92462i
\(623\) 45.0000i 1.80289i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −20.0000 −0.799361
\(627\) − 2.00000i − 0.0798723i
\(628\) 20.0000i 0.798087i
\(629\) 11.0000 0.438599
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) − 4.00000i − 0.158986i
\(634\) −56.0000 −2.22404
\(635\) 0 0
\(636\) −22.0000 −0.872357
\(637\) − 2.00000i − 0.0792429i
\(638\) − 4.00000i − 0.158362i
\(639\) 5.00000 0.197797
\(640\) 0 0
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) − 18.0000i − 0.710403i
\(643\) − 15.0000i − 0.591542i −0.955259 0.295771i \(-0.904423\pi\)
0.955259 0.295771i \(-0.0955766\pi\)
\(644\) 18.0000 0.709299
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) 47.0000i 1.84776i 0.382682 + 0.923880i \(0.375001\pi\)
−0.382682 + 0.923880i \(0.624999\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 18.0000 0.705476
\(652\) − 26.0000i − 1.01824i
\(653\) − 22.0000i − 0.860927i −0.902608 0.430463i \(-0.858350\pi\)
0.902608 0.430463i \(-0.141650\pi\)
\(654\) −32.0000 −1.25130
\(655\) 0 0
\(656\) 20.0000 0.780869
\(657\) 10.0000i 0.390137i
\(658\) 60.0000i 2.33904i
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) 0 0
\(663\) 1.00000i 0.0388368i
\(664\) 0 0
\(665\) 0 0
\(666\) 22.0000 0.852483
\(667\) 6.00000i 0.232321i
\(668\) 24.0000i 0.928588i
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) −13.0000 −0.501859
\(672\) 24.0000i 0.925820i
\(673\) 6.00000i 0.231283i 0.993291 + 0.115642i \(0.0368924\pi\)
−0.993291 + 0.115642i \(0.963108\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 2.00000 0.0769231
\(677\) 3.00000i 0.115299i 0.998337 + 0.0576497i \(0.0183606\pi\)
−0.998337 + 0.0576497i \(0.981639\pi\)
\(678\) 28.0000i 1.07533i
\(679\) −51.0000 −1.95720
\(680\) 0 0
\(681\) −22.0000 −0.843042
\(682\) 12.0000i 0.459504i
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) −30.0000 −1.14541
\(687\) − 18.0000i − 0.686743i
\(688\) 16.0000i 0.609994i
\(689\) 11.0000 0.419067
\(690\) 0 0
\(691\) 22.0000 0.836919 0.418460 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) − 12.0000i − 0.456172i
\(693\) 3.00000i 0.113961i
\(694\) 38.0000 1.44246
\(695\) 0 0
\(696\) 0 0
\(697\) 5.00000i 0.189389i
\(698\) 16.0000i 0.605609i
\(699\) −27.0000 −1.02123
\(700\) 0 0
\(701\) −20.0000 −0.755390 −0.377695 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) 22.0000i 0.829746i
\(704\) −8.00000 −0.301511
\(705\) 0 0
\(706\) 12.0000 0.451626
\(707\) 0 0
\(708\) 16.0000i 0.601317i
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) −3.00000 −0.112509
\(712\) 0 0
\(713\) − 18.0000i − 0.674105i
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 13.0000i 0.485494i
\(718\) − 48.0000i − 1.79134i
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) − 30.0000i − 1.11648i
\(723\) − 2.00000i − 0.0743808i
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) 20.0000 0.742270
\(727\) 38.0000i 1.40934i 0.709534 + 0.704671i \(0.248905\pi\)
−0.709534 + 0.704671i \(0.751095\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) − 26.0000i − 0.960988i
\(733\) 49.0000i 1.80986i 0.425564 + 0.904928i \(0.360076\pi\)
−0.425564 + 0.904928i \(0.639924\pi\)
\(734\) −72.0000 −2.65757
\(735\) 0 0
\(736\) 24.0000 0.884652
\(737\) − 12.0000i − 0.442026i
\(738\) 10.0000i 0.368105i
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 66.0000i 2.42294i
\(743\) 34.0000i 1.24734i 0.781688 + 0.623670i \(0.214359\pi\)
−0.781688 + 0.623670i \(0.785641\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 8.00000 0.292901
\(747\) − 12.0000i − 0.439057i
\(748\) − 2.00000i − 0.0731272i
\(749\) −27.0000 −0.986559
\(750\) 0 0
\(751\) −5.00000 −0.182453 −0.0912263 0.995830i \(-0.529079\pi\)
−0.0912263 + 0.995830i \(0.529079\pi\)
\(752\) 40.0000i 1.45865i
\(753\) 0 0
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) −6.00000 −0.218218
\(757\) − 8.00000i − 0.290765i −0.989376 0.145382i \(-0.953559\pi\)
0.989376 0.145382i \(-0.0464413\pi\)
\(758\) 28.0000i 1.01701i
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) − 20.0000i − 0.724524i
\(763\) 48.0000i 1.73772i
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) −60.0000 −2.16789
\(767\) − 8.00000i − 0.288863i
\(768\) 16.0000i 0.577350i
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) − 26.0000i − 0.935760i
\(773\) − 36.0000i − 1.29483i −0.762138 0.647415i \(-0.775850\pi\)
0.762138 0.647415i \(-0.224150\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) 0 0
\(777\) − 33.0000i − 1.18387i
\(778\) 0 0
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) 5.00000 0.178914
\(782\) 6.00000i 0.214560i
\(783\) − 2.00000i − 0.0714742i
\(784\) 8.00000 0.285714
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 52.0000i 1.85360i 0.375555 + 0.926800i \(0.377452\pi\)
−0.375555 + 0.926800i \(0.622548\pi\)
\(788\) 0 0
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) 42.0000 1.49335
\(792\) 0 0
\(793\) 13.0000i 0.461644i
\(794\) 58.0000 2.05834
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) − 47.0000i − 1.66483i −0.554156 0.832413i \(-0.686959\pi\)
0.554156 0.832413i \(-0.313041\pi\)
\(798\) − 12.0000i − 0.424795i
\(799\) −10.0000 −0.353775
\(800\) 0 0
\(801\) −15.0000 −0.529999
\(802\) 60.0000i 2.11867i
\(803\) 10.0000i 0.352892i
\(804\) 24.0000 0.846415
\(805\) 0 0
\(806\) 12.0000 0.422682
\(807\) 4.00000i 0.140807i
\(808\) 0 0
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) − 12.0000i − 0.421117i
\(813\) 22.0000i 0.771574i
\(814\) 22.0000 0.771100
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) − 8.00000i − 0.279885i
\(818\) − 4.00000i − 0.139857i
\(819\) 3.00000 0.104828
\(820\) 0 0
\(821\) 27.0000 0.942306 0.471153 0.882051i \(-0.343838\pi\)
0.471153 + 0.882051i \(0.343838\pi\)
\(822\) − 36.0000i − 1.25564i
\(823\) − 20.0000i − 0.697156i −0.937280 0.348578i \(-0.886665\pi\)
0.937280 0.348578i \(-0.113335\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 48.0000 1.67013
\(827\) 26.0000i 0.904109i 0.891990 + 0.452054i \(0.149309\pi\)
−0.891990 + 0.452054i \(0.850691\pi\)
\(828\) 6.00000i 0.208514i
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 8.00000i 0.277350i
\(833\) 2.00000i 0.0692959i
\(834\) −2.00000 −0.0692543
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 6.00000i 0.207390i
\(838\) 52.0000i 1.79631i
\(839\) 5.00000 0.172619 0.0863096 0.996268i \(-0.472493\pi\)
0.0863096 + 0.996268i \(0.472493\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 8.00000i 0.275698i
\(843\) 30.0000i 1.03325i
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) −20.0000 −0.687614
\(847\) − 30.0000i − 1.03081i
\(848\) 44.0000i 1.51097i
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) −33.0000 −1.13123
\(852\) 10.0000i 0.342594i
\(853\) − 45.0000i − 1.54077i −0.637579 0.770385i \(-0.720064\pi\)
0.637579 0.770385i \(-0.279936\pi\)
\(854\) −78.0000 −2.66911
\(855\) 0 0
\(856\) 0 0
\(857\) 29.0000i 0.990621i 0.868716 + 0.495311i \(0.164946\pi\)
−0.868716 + 0.495311i \(0.835054\pi\)
\(858\) 2.00000i 0.0682789i
\(859\) 29.0000 0.989467 0.494734 0.869045i \(-0.335266\pi\)
0.494734 + 0.869045i \(0.335266\pi\)
\(860\) 0 0
\(861\) 15.0000 0.511199
\(862\) − 48.0000i − 1.63489i
\(863\) − 34.0000i − 1.15737i −0.815550 0.578687i \(-0.803565\pi\)
0.815550 0.578687i \(-0.196435\pi\)
\(864\) −8.00000 −0.272166
\(865\) 0 0
\(866\) 8.00000 0.271851
\(867\) 16.0000i 0.543388i
\(868\) 36.0000i 1.22192i
\(869\) −3.00000 −0.101768
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) − 17.0000i − 0.575363i
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) −20.0000 −0.675737
\(877\) 18.0000i 0.607817i 0.952701 + 0.303908i \(0.0982917\pi\)
−0.952701 + 0.303908i \(0.901708\pi\)
\(878\) 34.0000i 1.14744i
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 4.00000i 0.134687i
\(883\) 16.0000i 0.538443i 0.963078 + 0.269221i \(0.0867663\pi\)
−0.963078 + 0.269221i \(0.913234\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 18.0000 0.604722
\(887\) − 21.0000i − 0.705111i −0.935791 0.352555i \(-0.885313\pi\)
0.935791 0.352555i \(-0.114687\pi\)
\(888\) 0 0
\(889\) −30.0000 −1.00617
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 16.0000i 0.535720i
\(893\) − 20.0000i − 0.669274i
\(894\) 26.0000 0.869570
\(895\) 0 0
\(896\) 0 0
\(897\) − 3.00000i − 0.100167i
\(898\) 26.0000i 0.867631i
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −11.0000 −0.366463
\(902\) 10.0000i 0.332964i
\(903\) 12.0000i 0.399335i
\(904\) 0 0
\(905\) 0 0
\(906\) −32.0000 −1.06313
\(907\) 6.00000i 0.199227i 0.995026 + 0.0996134i \(0.0317606\pi\)
−0.995026 + 0.0996134i \(0.968239\pi\)
\(908\) − 44.0000i − 1.46019i
\(909\) 0 0
\(910\) 0 0
\(911\) 44.0000 1.45779 0.728893 0.684628i \(-0.240035\pi\)
0.728893 + 0.684628i \(0.240035\pi\)
\(912\) − 8.00000i − 0.264906i
\(913\) − 12.0000i − 0.397142i
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) 36.0000 1.18947
\(917\) − 18.0000i − 0.594412i
\(918\) − 2.00000i − 0.0660098i
\(919\) −37.0000 −1.22052 −0.610259 0.792202i \(-0.708935\pi\)
−0.610259 + 0.792202i \(0.708935\pi\)
\(920\) 0 0
\(921\) 5.00000 0.164756
\(922\) 30.0000i 0.987997i
\(923\) − 5.00000i − 0.164577i
\(924\) −6.00000 −0.197386
\(925\) 0 0
\(926\) 54.0000 1.77455
\(927\) − 16.0000i − 0.525509i
\(928\) − 16.0000i − 0.525226i
\(929\) −1.00000 −0.0328089 −0.0164045 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) − 54.0000i − 1.76883i
\(933\) 24.0000i 0.785725i
\(934\) 46.0000 1.50517
\(935\) 0 0
\(936\) 0 0
\(937\) 30.0000i 0.980057i 0.871706 + 0.490029i \(0.163014\pi\)
−0.871706 + 0.490029i \(0.836986\pi\)
\(938\) − 72.0000i − 2.35088i
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 37.0000 1.20617 0.603083 0.797679i \(-0.293939\pi\)
0.603083 + 0.797679i \(0.293939\pi\)
\(942\) 20.0000i 0.651635i
\(943\) − 15.0000i − 0.488467i
\(944\) 32.0000 1.04151
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) − 6.00000i − 0.194871i
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) −28.0000 −0.907962
\(952\) 0 0
\(953\) 1.00000i 0.0323932i 0.999869 + 0.0161966i \(0.00515576\pi\)
−0.999869 + 0.0161966i \(0.994844\pi\)
\(954\) −22.0000 −0.712276
\(955\) 0 0
\(956\) −26.0000 −0.840900
\(957\) − 2.00000i − 0.0646508i
\(958\) − 18.0000i − 0.581554i
\(959\) −54.0000 −1.74375
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) − 22.0000i − 0.709308i
\(963\) − 9.00000i − 0.290021i
\(964\) 4.00000 0.128831
\(965\) 0 0
\(966\) 18.0000 0.579141
\(967\) − 16.0000i − 0.514525i −0.966342 0.257263i \(-0.917179\pi\)
0.966342 0.257263i \(-0.0828206\pi\)
\(968\) 0 0
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) − 2.00000i − 0.0641500i
\(973\) 3.00000i 0.0961756i
\(974\) 14.0000 0.448589
\(975\) 0 0
\(976\) −52.0000 −1.66448
\(977\) 32.0000i 1.02377i 0.859054 + 0.511885i \(0.171053\pi\)
−0.859054 + 0.511885i \(0.828947\pi\)
\(978\) − 26.0000i − 0.831388i
\(979\) −15.0000 −0.479402
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 56.0000i 1.78703i
\(983\) 12.0000i 0.382741i 0.981518 + 0.191370i \(0.0612931\pi\)
−0.981518 + 0.191370i \(0.938707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) 30.0000i 0.954911i
\(988\) − 4.00000i − 0.127257i
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) −25.0000 −0.794151 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) 48.0000i 1.52400i
\(993\) 0 0
\(994\) 30.0000 0.951542
\(995\) 0 0
\(996\) 24.0000 0.760469
\(997\) − 36.0000i − 1.14013i −0.821599 0.570066i \(-0.806918\pi\)
0.821599 0.570066i \(-0.193082\pi\)
\(998\) 28.0000i 0.886325i
\(999\) 11.0000 0.348025
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 975.2.c.a.274.2 2
3.2 odd 2 2925.2.c.c.2224.1 2
5.2 odd 4 975.2.a.c.1.1 1
5.3 odd 4 195.2.a.b.1.1 1
5.4 even 2 inner 975.2.c.a.274.1 2
15.2 even 4 2925.2.a.q.1.1 1
15.8 even 4 585.2.a.b.1.1 1
15.14 odd 2 2925.2.c.c.2224.2 2
20.3 even 4 3120.2.a.u.1.1 1
35.13 even 4 9555.2.a.v.1.1 1
60.23 odd 4 9360.2.a.d.1.1 1
65.38 odd 4 2535.2.a.a.1.1 1
195.38 even 4 7605.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.b.1.1 1 5.3 odd 4
585.2.a.b.1.1 1 15.8 even 4
975.2.a.c.1.1 1 5.2 odd 4
975.2.c.a.274.1 2 5.4 even 2 inner
975.2.c.a.274.2 2 1.1 even 1 trivial
2535.2.a.a.1.1 1 65.38 odd 4
2925.2.a.q.1.1 1 15.2 even 4
2925.2.c.c.2224.1 2 3.2 odd 2
2925.2.c.c.2224.2 2 15.14 odd 2
3120.2.a.u.1.1 1 20.3 even 4
7605.2.a.u.1.1 1 195.38 even 4
9360.2.a.d.1.1 1 60.23 odd 4
9555.2.a.v.1.1 1 35.13 even 4