Properties

Label 3450.2.d.d.2899.2
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3450.2899
Dual form 3450.2.d.d.2899.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} +2.00000 q^{21} -2.00000i q^{22} -1.00000i q^{23} +1.00000 q^{24} -6.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -2.00000 q^{29} +1.00000i q^{32} -2.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} -6.00000 q^{39} -6.00000 q^{41} +2.00000i q^{42} +4.00000i q^{43} +2.00000 q^{44} +1.00000 q^{46} +1.00000i q^{48} +3.00000 q^{49} +4.00000 q^{51} -6.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} -2.00000i q^{58} -8.00000 q^{61} +2.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -4.00000i q^{67} +4.00000i q^{68} +1.00000 q^{69} +16.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} +8.00000 q^{74} +4.00000i q^{77} -6.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -14.0000i q^{83} -2.00000 q^{84} -4.00000 q^{86} -2.00000i q^{87} +2.00000i q^{88} +8.00000 q^{89} +12.0000 q^{91} +1.00000i q^{92} -1.00000 q^{96} -6.00000i q^{97} +3.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} - 4q^{11} + 4q^{14} + 2q^{16} + 4q^{21} + 2q^{24} - 12q^{26} - 4q^{29} + 8q^{34} + 2q^{36} - 12q^{39} - 12q^{41} + 4q^{44} + 2q^{46} + 6q^{49} + 8q^{51} + 2q^{54} - 4q^{56} - 16q^{61} - 2q^{64} + 4q^{66} + 2q^{69} + 32q^{71} + 16q^{74} - 28q^{79} + 2q^{81} - 4q^{84} - 8q^{86} + 16q^{89} + 24q^{91} - 2q^{96} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 4.00000i − 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) − 2.00000i − 0.426401i
\(23\) − 1.00000i − 0.208514i
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) − 1.00000i − 0.192450i
\(28\) 2.00000i 0.377964i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 2.00000i − 0.348155i
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 2.00000i 0.308607i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) − 6.00000i − 0.832050i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) − 2.00000i − 0.262613i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 4.00000i 0.485071i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000i 0.455842i
\(78\) − 6.00000i − 0.679366i
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 6.00000i − 0.662589i
\(83\) − 14.0000i − 1.53670i −0.640030 0.768350i \(-0.721078\pi\)
0.640030 0.768350i \(-0.278922\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) − 2.00000i − 0.214423i
\(88\) 2.00000i 0.213201i
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 1.00000i 0.104257i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 6.00000i − 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 4.00000i 0.396059i
\(103\) 2.00000i 0.197066i 0.995134 + 0.0985329i \(0.0314150\pi\)
−0.995134 + 0.0985329i \(0.968585\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) − 18.0000i − 1.74013i −0.492941 0.870063i \(-0.664078\pi\)
0.492941 0.870063i \(-0.335922\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) − 2.00000i − 0.188982i
\(113\) − 12.0000i − 1.12887i −0.825479 0.564433i \(-0.809095\pi\)
0.825479 0.564433i \(-0.190905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) − 6.00000i − 0.554700i
\(118\) 0 0
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) − 8.00000i − 0.724286i
\(123\) − 6.00000i − 0.541002i
\(124\) 0 0
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 2.00000i 0.174078i
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) − 8.00000i − 0.683486i −0.939793 0.341743i \(-0.888983\pi\)
0.939793 0.341743i \(-0.111017\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.0000i 1.34269i
\(143\) − 12.0000i − 1.00349i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) 3.00000i 0.247436i
\(148\) 8.00000i 0.657596i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 4.00000i 0.323381i
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) − 16.0000i − 1.27694i −0.769647 0.638470i \(-0.779568\pi\)
0.769647 0.638470i \(-0.220432\pi\)
\(158\) − 14.0000i − 1.11378i
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 1.00000i 0.0785674i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) − 4.00000i − 0.304997i
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 8.00000i 0.599625i
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 12.0000i 0.889499i
\(183\) − 8.00000i − 0.591377i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 2.00000i 0.142134i
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 6.00000i 0.422159i
\(203\) 4.00000i 0.280745i
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −2.00000 −0.139347
\(207\) 1.00000i 0.0695048i
\(208\) 6.00000i 0.416025i
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 16.0000i 1.09630i
\(214\) 18.0000 1.23045
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 4.00000i 0.270914i
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 8.00000i 0.536925i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) − 18.0000i − 1.19470i −0.801980 0.597351i \(-0.796220\pi\)
0.801980 0.597351i \(-0.203780\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 2.00000i 0.131306i
\(233\) 26.0000i 1.70332i 0.524097 + 0.851658i \(0.324403\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) 0 0
\(237\) − 14.0000i − 0.909398i
\(238\) − 8.00000i − 0.518563i
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) − 7.00000i − 0.449977i
\(243\) 1.00000i 0.0641500i
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 0 0
\(249\) 14.0000 0.887214
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 2.00000i 0.125739i
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000i 0.873296i 0.899632 + 0.436648i \(0.143834\pi\)
−0.899632 + 0.436648i \(0.856166\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 12.0000i 0.741362i
\(263\) − 4.00000i − 0.246651i −0.992366 0.123325i \(-0.960644\pi\)
0.992366 0.123325i \(-0.0393559\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) 8.00000i 0.489592i
\(268\) 4.00000i 0.244339i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) − 4.00000i − 0.242536i
\(273\) 12.0000i 0.726273i
\(274\) 8.00000 0.483298
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) 4.00000i 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 16.0000i 0.951101i 0.879688 + 0.475551i \(0.157751\pi\)
−0.879688 + 0.475551i \(0.842249\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 12.0000i 0.708338i
\(288\) − 1.00000i − 0.0589256i
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 6.00000i 0.351123i
\(293\) 10.0000i 0.584206i 0.956387 + 0.292103i \(0.0943550\pi\)
−0.956387 + 0.292103i \(0.905645\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 2.00000i 0.116052i
\(298\) − 10.0000i − 0.579284i
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 12.0000i 0.690522i
\(303\) 6.00000i 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) 28.0000i 1.59804i 0.601302 + 0.799022i \(0.294649\pi\)
−0.601302 + 0.799022i \(0.705351\pi\)
\(308\) − 4.00000i − 0.227921i
\(309\) −2.00000 −0.113776
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 6.00000i 0.339683i
\(313\) 34.0000i 1.92179i 0.276907 + 0.960897i \(0.410691\pi\)
−0.276907 + 0.960897i \(0.589309\pi\)
\(314\) 16.0000 0.902932
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) − 22.0000i − 1.23564i −0.786318 0.617822i \(-0.788015\pi\)
0.786318 0.617822i \(-0.211985\pi\)
\(318\) 6.00000i 0.336463i
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) − 2.00000i − 0.111456i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 4.00000i 0.221201i
\(328\) 6.00000i 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 14.0000i 0.768350i
\(333\) 8.00000i 0.438397i
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) − 34.0000i − 1.85210i −0.377403 0.926049i \(-0.623183\pi\)
0.377403 0.926049i \(-0.376817\pi\)
\(338\) − 23.0000i − 1.25104i
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 20.0000i − 1.07990i
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) − 8.00000i − 0.429463i −0.976673 0.214731i \(-0.931112\pi\)
0.976673 0.214731i \(-0.0688876\pi\)
\(348\) 2.00000i 0.107211i
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) − 2.00000i − 0.106600i
\(353\) − 26.0000i − 1.38384i −0.721974 0.691920i \(-0.756765\pi\)
0.721974 0.691920i \(-0.243235\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −8.00000 −0.423999
\(357\) − 8.00000i − 0.423405i
\(358\) 20.0000i 1.05703i
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 12.0000i − 0.630706i
\(363\) − 7.00000i − 0.367405i
\(364\) −12.0000 −0.628971
\(365\) 0 0
\(366\) 8.00000 0.418167
\(367\) 10.0000i 0.521996i 0.965339 + 0.260998i \(0.0840516\pi\)
−0.965339 + 0.260998i \(0.915948\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) − 32.0000i − 1.65690i −0.560065 0.828449i \(-0.689224\pi\)
0.560065 0.828449i \(-0.310776\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) − 12.0000i − 0.618031i
\(378\) − 2.00000i − 0.102869i
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) − 4.00000i − 0.203331i
\(388\) 6.00000i 0.304604i
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) − 3.00000i − 0.151523i
\(393\) 12.0000i 0.605320i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) − 22.0000i − 1.10415i −0.833795 0.552074i \(-0.813837\pi\)
0.833795 0.552074i \(-0.186163\pi\)
\(398\) 2.00000i 0.100251i
\(399\) 0 0
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 4.00000i 0.199502i
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) 16.0000i 0.793091i
\(408\) − 4.00000i − 0.198030i
\(409\) −38.0000 −1.87898 −0.939490 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 8.00000 0.394611
\(412\) − 2.00000i − 0.0985329i
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) 4.00000i 0.195881i
\(418\) 0 0
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) − 20.0000i − 0.973585i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −16.0000 −0.775203
\(427\) 16.0000i 0.774294i
\(428\) 18.0000i 0.870063i
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 30.0000i − 1.44171i −0.693087 0.720854i \(-0.743750\pi\)
0.693087 0.720854i \(-0.256250\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 0 0
\(438\) 6.00000i 0.286691i
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 24.0000i 1.14156i
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) − 10.0000i − 0.472984i
\(448\) 2.00000i 0.0944911i
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 12.0000i 0.564433i
\(453\) 12.0000i 0.563809i
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) 0 0
\(457\) 18.0000i 0.842004i 0.907060 + 0.421002i \(0.138322\pi\)
−0.907060 + 0.421002i \(0.861678\pi\)
\(458\) 20.0000i 0.934539i
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) − 4.00000i − 0.186097i
\(463\) 36.0000i 1.67306i 0.547920 + 0.836531i \(0.315420\pi\)
−0.547920 + 0.836531i \(0.684580\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) − 30.0000i − 1.38823i −0.719862 0.694117i \(-0.755795\pi\)
0.719862 0.694117i \(-0.244205\pi\)
\(468\) 6.00000i 0.277350i
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 16.0000 0.737241
\(472\) 0 0
\(473\) − 8.00000i − 0.367840i
\(474\) 14.0000 0.643041
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 6.00000i 0.274721i
\(478\) − 16.0000i − 0.731823i
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) − 14.0000i − 0.637683i
\(483\) − 2.00000i − 0.0910032i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 12.0000i 0.543772i 0.962329 + 0.271886i \(0.0876473\pi\)
−0.962329 + 0.271886i \(0.912353\pi\)
\(488\) 8.00000i 0.362143i
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 8.00000i 0.360302i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 32.0000i − 1.43540i
\(498\) 14.0000i 0.627355i
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 6.00000i 0.267793i
\(503\) − 4.00000i − 0.178351i −0.996016 0.0891756i \(-0.971577\pi\)
0.996016 0.0891756i \(-0.0284232\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) −2.00000 −0.0889108
\(507\) − 23.0000i − 1.02147i
\(508\) 0 0
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) − 16.0000i − 0.703000i
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) 0 0
\(528\) − 2.00000i − 0.0870388i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 36.0000i − 1.55933i
\(534\) −8.00000 −0.346194
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 20.0000i 0.863064i
\(538\) 18.0000i 0.776035i
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 20.0000i 0.859074i
\(543\) − 12.0000i − 0.514969i
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) −12.0000 −0.513553
\(547\) 12.0000i 0.513083i 0.966533 + 0.256541i \(0.0825830\pi\)
−0.966533 + 0.256541i \(0.917417\pi\)
\(548\) 8.00000i 0.341743i
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 0 0
\(552\) − 1.00000i − 0.0425628i
\(553\) 28.0000i 1.19068i
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) − 22.0000i − 0.932170i −0.884740 0.466085i \(-0.845664\pi\)
0.884740 0.466085i \(-0.154336\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 8.00000i 0.337460i
\(563\) 10.0000i 0.421450i 0.977545 + 0.210725i \(0.0675824\pi\)
−0.977545 + 0.210725i \(0.932418\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) − 2.00000i − 0.0839921i
\(568\) − 16.0000i − 0.671345i
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 12.0000i 0.501745i
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) −28.0000 −1.16164
\(582\) 6.00000i 0.248708i
\(583\) 12.0000i 0.496989i
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) 8.00000i 0.330195i 0.986277 + 0.165098i \(0.0527939\pi\)
−0.986277 + 0.165098i \(0.947206\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) 0 0
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) − 8.00000i − 0.328798i
\(593\) − 30.0000i − 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 2.00000i 0.0818546i
\(598\) 6.00000i 0.245358i
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 4.00000i 0.162893i
\(604\) −12.0000 −0.488273
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) 40.0000i 1.62355i 0.583970 + 0.811775i \(0.301498\pi\)
−0.583970 + 0.811775i \(0.698502\pi\)
\(608\) 0 0
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) − 4.00000i − 0.161690i
\(613\) 4.00000i 0.161558i 0.996732 + 0.0807792i \(0.0257409\pi\)
−0.996732 + 0.0807792i \(0.974259\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) 28.0000i 1.12724i 0.826035 + 0.563619i \(0.190591\pi\)
−0.826035 + 0.563619i \(0.809409\pi\)
\(618\) − 2.00000i − 0.0804518i
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) − 32.0000i − 1.28308i
\(623\) − 16.0000i − 0.641026i
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) 16.0000i 0.638470i
\(629\) −32.0000 −1.27592
\(630\) 0 0
\(631\) −22.0000 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) 14.0000i 0.556890i
\(633\) − 20.0000i − 0.794929i
\(634\) 22.0000 0.873732
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 18.0000i 0.713186i
\(638\) 4.00000i 0.158362i
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) 16.0000 0.631962 0.315981 0.948766i \(-0.397666\pi\)
0.315981 + 0.948766i \(0.397666\pi\)
\(642\) 18.0000i 0.710403i
\(643\) 28.0000i 1.10421i 0.833774 + 0.552106i \(0.186176\pi\)
−0.833774 + 0.552106i \(0.813824\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) 0 0
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) − 4.00000i − 0.156652i
\(653\) − 22.0000i − 0.860927i −0.902608 0.430463i \(-0.858350\pi\)
0.902608 0.430463i \(-0.141650\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 24.0000i 0.932083i
\(664\) −14.0000 −0.543305
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 2.00000i 0.0774403i
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 16.0000 0.617673
\(672\) 2.00000i 0.0771517i
\(673\) − 42.0000i − 1.61898i −0.587133 0.809491i \(-0.699743\pi\)
0.587133 0.809491i \(-0.300257\pi\)
\(674\) 34.0000 1.30963
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 2.00000i 0.0768662i 0.999261 + 0.0384331i \(0.0122367\pi\)
−0.999261 + 0.0384331i \(0.987763\pi\)
\(678\) 12.0000i 0.460857i
\(679\) −12.0000 −0.460518
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) − 48.0000i − 1.83667i −0.395805 0.918334i \(-0.629534\pi\)
0.395805 0.918334i \(-0.370466\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 20.0000i 0.763048i
\(688\) 4.00000i 0.152499i
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 18.0000i 0.684257i
\(693\) − 4.00000i − 0.151947i
\(694\) 8.00000 0.303676
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) 24.0000i 0.909065i
\(698\) 6.00000i 0.227103i
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 6.00000i 0.226455i
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) − 12.0000i − 0.451306i
\(708\) 0 0
\(709\) −16.0000 −0.600893 −0.300446 0.953799i \(-0.597136\pi\)
−0.300446 + 0.953799i \(0.597136\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) − 8.00000i − 0.299813i
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) − 16.0000i − 0.597531i
\(718\) − 24.0000i − 0.895672i
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) − 19.0000i − 0.707107i
\(723\) − 14.0000i − 0.520666i
\(724\) 12.0000 0.445976
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) 38.0000i 1.40934i 0.709534 + 0.704671i \(0.248905\pi\)
−0.709534 + 0.704671i \(0.751095\pi\)
\(728\) − 12.0000i − 0.444750i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 8.00000i 0.295689i
\(733\) − 36.0000i − 1.32969i −0.746981 0.664845i \(-0.768498\pi\)
0.746981 0.664845i \(-0.231502\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 8.00000i 0.294684i
\(738\) 6.00000i 0.220863i
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 12.0000i − 0.440534i
\(743\) − 4.00000i − 0.146746i −0.997305 0.0733729i \(-0.976624\pi\)
0.997305 0.0733729i \(-0.0233763\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) 14.0000i 0.512233i
\(748\) − 8.00000i − 0.292509i
\(749\) −36.0000 −1.31541
\(750\) 0 0
\(751\) −50.0000 −1.82453 −0.912263 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) 0 0
\(753\) 6.00000i 0.218652i
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 40.0000i 1.45382i 0.686730 + 0.726912i \(0.259045\pi\)
−0.686730 + 0.726912i \(0.740955\pi\)
\(758\) − 16.0000i − 0.581146i
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) − 8.00000i − 0.289619i
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) 1.00000i 0.0360844i
\(769\) −6.00000 −0.216366 −0.108183 0.994131i \(-0.534503\pi\)
−0.108183 + 0.994131i \(0.534503\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) − 2.00000i − 0.0719816i
\(773\) 18.0000i 0.647415i 0.946157 + 0.323708i \(0.104929\pi\)
−0.946157 + 0.323708i \(0.895071\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) − 16.0000i − 0.573997i
\(778\) − 10.0000i − 0.358517i
\(779\) 0 0
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) − 4.00000i − 0.143040i
\(783\) 2.00000i 0.0714742i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) − 4.00000i − 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) − 2.00000i − 0.0710669i
\(793\) − 48.0000i − 1.70453i
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −2.00000 −0.0708881
\(797\) − 46.0000i − 1.62940i −0.579880 0.814702i \(-0.696901\pi\)
0.579880 0.814702i \(-0.303099\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) − 24.0000i − 0.847469i
\(803\) 12.0000i 0.423471i
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) 18.0000i 0.633630i
\(808\) − 6.00000i − 0.211079i
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) − 4.00000i − 0.140372i
\(813\) 20.0000i 0.701431i
\(814\) −16.0000 −0.560800
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 0 0
\(818\) − 38.0000i − 1.32864i
\(819\) −12.0000 −0.419314
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 8.00000i 0.279032i
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) 2.00000 0.0696733
\(825\) 0 0
\(826\) 0 0
\(827\) − 22.0000i − 0.765015i −0.923952 0.382507i \(-0.875061\pi\)
0.923952 0.382507i \(-0.124939\pi\)
\(828\) − 1.00000i − 0.0347524i
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) − 6.00000i − 0.208013i
\(833\) − 12.0000i − 0.415775i
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 14.0000i 0.483622i
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 8.00000i 0.275535i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) 14.0000i 0.481046i
\(848\) − 6.00000i − 0.206041i
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) − 16.0000i − 0.548151i
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) −16.0000 −0.547509
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) 30.0000i 1.02478i 0.858753 + 0.512390i \(0.171240\pi\)
−0.858753 + 0.512390i \(0.828760\pi\)
\(858\) 12.0000i 0.409673i
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) − 36.0000i − 1.22616i
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 30.0000 1.01944
\(867\) 1.00000i 0.0339618i
\(868\) 0 0
\(869\) 28.0000 0.949835
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) − 4.00000i − 0.135457i
\(873\) 6.00000i 0.203069i
\(874\) 0 0
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) − 2.00000i − 0.0675352i −0.999430 0.0337676i \(-0.989249\pi\)
0.999430 0.0337676i \(-0.0107506\pi\)
\(878\) − 32.0000i − 1.07995i
\(879\) −10.0000 −0.337292
\(880\) 0 0
\(881\) −56.0000 −1.88669 −0.943344 0.331816i \(-0.892339\pi\)
−0.943344 + 0.331816i \(0.892339\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) 36.0000i 1.21150i 0.795656 + 0.605748i \(0.207126\pi\)
−0.795656 + 0.605748i \(0.792874\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 48.0000i 1.61168i 0.592132 + 0.805841i \(0.298286\pi\)
−0.592132 + 0.805841i \(0.701714\pi\)
\(888\) − 8.00000i − 0.268462i
\(889\) 0 0
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) − 16.0000i − 0.535720i
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 6.00000i 0.200334i
\(898\) 22.0000i 0.734150i
\(899\) 0 0
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 12.0000i 0.399556i
\(903\) 8.00000i 0.266223i
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) −12.0000 −0.398673
\(907\) − 24.0000i − 0.796907i −0.917189 0.398453i \(-0.869547\pi\)
0.917189 0.398453i \(-0.130453\pi\)
\(908\) 18.0000i 0.597351i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 28.0000i 0.926665i
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) −20.0000 −0.660819
\(917\) − 24.0000i − 0.792550i
\(918\) − 4.00000i − 0.132020i
\(919\) 2.00000 0.0659739 0.0329870 0.999456i \(-0.489498\pi\)
0.0329870 + 0.999456i \(0.489498\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) − 10.0000i − 0.329332i
\(923\) 96.0000i 3.15988i
\(924\) 4.00000 0.131590
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) − 2.00000i − 0.0656886i
\(928\) − 2.00000i − 0.0656532i
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 26.0000i − 0.851658i
\(933\) − 32.0000i − 1.04763i
\(934\) 30.0000 0.981630
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 42.0000i 1.37208i 0.727564 + 0.686040i \(0.240653\pi\)
−0.727564 + 0.686040i \(0.759347\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) −34.0000 −1.10955
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 16.0000i 0.521308i
\(943\) 6.00000i 0.195387i
\(944\) 0 0
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 8.00000i 0.259965i 0.991516 + 0.129983i \(0.0414921\pi\)
−0.991516 + 0.129983i \(0.958508\pi\)
\(948\) 14.0000i 0.454699i
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 8.00000i 0.259281i
\(953\) − 44.0000i − 1.42530i −0.701520 0.712650i \(-0.747495\pi\)
0.701520 0.712650i \(-0.252505\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 4.00000i 0.129302i
\(958\) − 8.00000i − 0.258468i
\(959\) −16.0000 −0.516667
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 48.0000i 1.54758i
\(963\) 18.0000i 0.580042i
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) 2.00000 0.0643489
\(967\) 52.0000i 1.67221i 0.548572 + 0.836104i \(0.315172\pi\)
−0.548572 + 0.836104i \(0.684828\pi\)
\(968\) 7.00000i 0.224989i
\(969\) 0 0
\(970\) 0 0
\(971\) −46.0000 −1.47621 −0.738105 0.674686i \(-0.764279\pi\)
−0.738105 + 0.674686i \(0.764279\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 8.00000i − 0.256468i
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) 24.0000i 0.767828i 0.923369 + 0.383914i \(0.125424\pi\)
−0.923369 + 0.383914i \(0.874576\pi\)
\(978\) − 4.00000i − 0.127906i
\(979\) −16.0000 −0.511362
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) − 12.0000i − 0.382935i
\(983\) 12.0000i 0.382741i 0.981518 + 0.191370i \(0.0612931\pi\)
−0.981518 + 0.191370i \(0.938707\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 20.0000i 0.634681i
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) −14.0000 −0.443607
\(997\) − 2.00000i − 0.0633406i −0.999498 0.0316703i \(-0.989917\pi\)
0.999498 0.0316703i \(-0.0100827\pi\)
\(998\) 28.0000i 0.886325i
\(999\) −8.00000 −0.253109
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 3450.2.d.d.2899.2 2
5.2 odd 4 3450.2.a.l.1.1 1
5.3 odd 4 690.2.a.g.1.1 1
5.4 even 2 inner 3450.2.d.d.2899.1 2
15.8 even 4 2070.2.a.e.1.1 1
20.3 even 4 5520.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.g.1.1 1 5.3 odd 4
2070.2.a.e.1.1 1 15.8 even 4
3450.2.a.l.1.1 1 5.2 odd 4
3450.2.d.d.2899.1 2 5.4 even 2 inner
3450.2.d.d.2899.2 2 1.1 even 1 trivial
5520.2.a.z.1.1 1 20.3 even 4