Properties

Label 3850.2.c.a.1849.2
Level $3850$
Weight $2$
Character 3850.1849
Analytic conductor $30.742$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.7424047782\)
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 770)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3850.1849
Dual form 3850.2.c.a.1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{12} -2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -2.00000 q^{21} -1.00000i q^{22} +6.00000i q^{23} +2.00000 q^{24} +2.00000 q^{26} +4.00000i q^{27} -1.00000i q^{28} +8.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +8.00000i q^{37} -2.00000i q^{38} +4.00000 q^{39} -2.00000i q^{42} +4.00000i q^{43} +1.00000 q^{44} -6.00000 q^{46} +2.00000i q^{48} -1.00000 q^{49} -12.0000 q^{51} +2.00000i q^{52} -12.0000i q^{53} -4.00000 q^{54} +1.00000 q^{56} -4.00000i q^{57} -12.0000 q^{59} -10.0000 q^{61} +8.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -4.00000i q^{67} -6.00000i q^{68} -12.0000 q^{69} -12.0000 q^{71} +1.00000i q^{72} -14.0000i q^{73} -8.00000 q^{74} +2.00000 q^{76} -1.00000i q^{77} +4.00000i q^{78} +10.0000 q^{79} -11.0000 q^{81} +2.00000 q^{84} -4.00000 q^{86} +1.00000i q^{88} +18.0000 q^{89} +2.00000 q^{91} -6.00000i q^{92} +16.0000i q^{93} -2.00000 q^{96} +8.00000i q^{97} -1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{11} - 2 q^{14} + 2 q^{16} - 4 q^{19} - 4 q^{21} + 4 q^{24} + 4 q^{26} + 16 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 2 q^{44} - 12 q^{46} - 2 q^{49} - 24 q^{51} - 8 q^{54} + 2 q^{56} - 24 q^{59} - 20 q^{61} - 2 q^{64} + 4 q^{66} - 24 q^{69} - 24 q^{71} - 16 q^{74} + 4 q^{76} + 20 q^{79} - 22 q^{81} + 4 q^{84} - 8 q^{86} + 36 q^{89} + 4 q^{91} - 4 q^{96} + 2 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times\).

\(n\) \(1751\) \(2201\) \(2927\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 1.00000i 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) − 2.00000i − 0.577350i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) − 1.00000i − 0.213201i
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 4.00000i 0.769800i
\(28\) − 1.00000i − 0.188982i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 2.00000i − 0.348155i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) − 2.00000i − 0.324443i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 2.00000i 0.288675i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 2.00000i 0.277350i
\(53\) − 12.0000i − 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) − 4.00000i − 0.529813i
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 8.00000i 1.01600i
\(63\) − 1.00000i − 0.125988i
\(64\) −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 \(-0.0785709\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) − 1.00000i − 0.113961i
\(78\) 4.00000i 0.452911i
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 1.00000i 0.106600i
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) − 6.00000i − 0.625543i
\(93\) 16.0000i 1.65912i
\(94\) 0 0
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) − 12.0000i − 1.18818i
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) − 4.00000i − 0.384900i
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) −16.0000 −1.51865
\(112\) 1.00000i 0.0944911i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) − 12.0000i − 1.10469i
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 10.0000i − 0.905357i
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 2.00000i 0.174078i
\(133\) − 2.00000i − 0.173422i
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) − 12.0000i − 1.02151i
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 12.0000i − 1.00702i
\(143\) 2.00000i 0.167248i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) − 2.00000i − 0.164957i
\(148\) − 8.00000i − 0.657596i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 2.00000i 0.162221i
\(153\) − 6.00000i − 0.485071i
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 10.0000i 0.795557i
\(159\) 24.0000 1.90332
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) − 11.0000i − 0.864242i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 2.00000i 0.154303i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) − 4.00000i − 0.304997i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) − 24.0000i − 1.80395i
\(178\) 18.0000i 1.34916i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 2.00000i 0.148250i
\(183\) − 20.0000i − 1.47844i
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) −16.0000 −1.17318
\(187\) − 6.00000i − 0.438763i
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) − 2.00000i − 0.144338i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 1.00000i 0.0710669i
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) − 18.0000i − 1.26648i
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) − 6.00000i − 0.417029i
\(208\) − 2.00000i − 0.138675i
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 12.0000i 0.824163i
\(213\) − 24.0000i − 1.64445i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) 8.00000i 0.543075i
\(218\) − 8.00000i − 0.541828i
\(219\) 28.0000 1.89206
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) − 16.0000i − 1.07385i
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 4.00000i 0.264906i
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 20.0000i 1.29914i
\(238\) − 6.00000i − 0.388922i
\(239\) −18.0000 −1.16432 −0.582162 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 1.00000i 0.0642824i
\(243\) − 10.0000i − 0.641500i
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000i 0.254514i
\(248\) − 8.00000i − 0.508001i
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 1.00000i 0.0629941i
\(253\) − 6.00000i − 0.377217i
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 24.0000i − 1.49708i −0.663090 0.748539i \(-0.730755\pi\)
0.663090 0.748539i \(-0.269245\pi\)
\(258\) − 8.00000i − 0.498058i
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) − 18.0000i − 1.11204i
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 36.0000i 2.20316i
\(268\) 4.00000i 0.244339i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 6.00000i 0.363803i
\(273\) 4.00000i 0.242091i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) 10.0000i 0.599760i
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 0 0
\(288\) − 1.00000i − 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 14.0000i 0.819288i
\(293\) − 18.0000i − 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) − 4.00000i − 0.232104i
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) − 10.0000i − 0.575435i
\(303\) − 36.0000i − 2.06815i
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 32.0000i 1.82634i 0.407583 + 0.913168i \(0.366372\pi\)
−0.407583 + 0.913168i \(0.633628\pi\)
\(308\) 1.00000i 0.0569803i
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) − 4.00000i − 0.226455i
\(313\) 16.0000i 0.904373i 0.891923 + 0.452187i \(0.149356\pi\)
−0.891923 + 0.452187i \(0.850644\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 24.0000i 1.34585i
\(319\) 0 0
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) − 6.00000i − 0.334367i
\(323\) − 12.0000i − 0.667698i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) − 16.0000i − 0.884802i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) − 8.00000i − 0.438397i
\(334\) 0 0
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) − 22.0000i − 1.19842i −0.800593 0.599208i \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) 9.00000i 0.489535i
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 2.00000i 0.108148i
\(343\) − 1.00000i − 0.0539949i
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) − 1.00000i − 0.0533002i
\(353\) 24.0000i 1.27739i 0.769460 + 0.638696i \(0.220526\pi\)
−0.769460 + 0.638696i \(0.779474\pi\)
\(354\) 24.0000 1.27559
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) − 12.0000i − 0.635107i
\(358\) 12.0000i 0.634220i
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) − 10.0000i − 0.525588i
\(363\) 2.00000i 0.104973i
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 20.0000 1.04542
\(367\) 20.0000i 1.04399i 0.852948 + 0.521996i \(0.174812\pi\)
−0.852948 + 0.521996i \(0.825188\pi\)
\(368\) 6.00000i 0.312772i
\(369\) 0 0
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) − 16.0000i − 0.829561i
\(373\) 22.0000i 1.13912i 0.821951 + 0.569558i \(0.192886\pi\)
−0.821951 + 0.569558i \(0.807114\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) − 4.00000i − 0.205738i
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 32.0000 1.63941
\(382\) 0 0
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) − 4.00000i − 0.203331i
\(388\) − 8.00000i − 0.406138i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 1.00000i 0.0505076i
\(393\) − 36.0000i − 1.81596i
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) − 8.00000i − 0.401004i
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 8.00000i 0.399004i
\(403\) − 16.0000i − 0.797017i
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) − 8.00000i − 0.396545i
\(408\) 12.0000i 0.594089i
\(409\) −8.00000 −0.395575 −0.197787 0.980245i \(-0.563376\pi\)
−0.197787 + 0.980245i \(0.563376\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) − 4.00000i − 0.197066i
\(413\) − 12.0000i − 0.590481i
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 20.0000i 0.979404i
\(418\) 2.00000i 0.0978232i
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 20.0000i 0.973585i
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 24.0000 1.16280
\(427\) − 10.0000i − 0.483934i
\(428\) − 12.0000i − 0.580042i
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 4.00000i 0.192450i
\(433\) 4.00000i 0.192228i 0.995370 + 0.0961139i \(0.0306413\pi\)
−0.995370 + 0.0961139i \(0.969359\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) − 12.0000i − 0.574038i
\(438\) 28.0000i 1.33789i
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 12.0000i 0.570782i
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 16.0000 0.759326
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) − 1.00000i − 0.0472456i
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 6.00000i − 0.282216i
\(453\) − 20.0000i − 0.939682i
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 26.0000i 1.21623i 0.793849 + 0.608114i \(0.208074\pi\)
−0.793849 + 0.608114i \(0.791926\pi\)
\(458\) − 26.0000i − 1.21490i
\(459\) −24.0000 −1.12022
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 2.00000i 0.0930484i
\(463\) − 2.00000i − 0.0929479i −0.998920 0.0464739i \(-0.985202\pi\)
0.998920 0.0464739i \(-0.0147984\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 18.0000i 0.832941i 0.909149 + 0.416470i \(0.136733\pi\)
−0.909149 + 0.416470i \(0.863267\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 12.0000i 0.552345i
\(473\) − 4.00000i − 0.183920i
\(474\) −20.0000 −0.918630
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 12.0000i 0.549442i
\(478\) − 18.0000i − 0.823301i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 8.00000i 0.364390i
\(483\) − 12.0000i − 0.546019i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 10.0000 0.453609
\(487\) − 22.0000i − 0.996915i −0.866914 0.498458i \(-0.833900\pi\)
0.866914 0.498458i \(-0.166100\pi\)
\(488\) 10.0000i 0.452679i
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) − 12.0000i − 0.538274i
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 12.0000i − 0.535586i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) 18.0000i 0.799408i
\(508\) 16.0000i 0.709885i
\(509\) 42.0000 1.86162 0.930809 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 1.00000i 0.0441942i
\(513\) − 8.00000i − 0.353209i
\(514\) 24.0000 1.05859
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) − 8.00000i − 0.351500i
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 16.0000i 0.699631i 0.936819 + 0.349816i \(0.113756\pi\)
−0.936819 + 0.349816i \(0.886244\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 48.0000i 2.09091i
\(528\) − 2.00000i − 0.0870388i
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 2.00000i 0.0867110i
\(533\) 0 0
\(534\) −36.0000 −1.55787
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 24.0000i 1.03568i
\(538\) − 18.0000i − 0.776035i
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 20.0000i 0.859074i
\(543\) − 20.0000i − 0.858282i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) 44.0000i 1.88130i 0.339372 + 0.940652i \(0.389785\pi\)
−0.339372 + 0.940652i \(0.610215\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 0 0
\(552\) 12.0000i 0.510754i
\(553\) 10.0000i 0.425243i
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) − 6.00000i − 0.254228i −0.991888 0.127114i \(-0.959429\pi\)
0.991888 0.127114i \(-0.0405714\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 6.00000i 0.253095i
\(563\) 36.0000i 1.51722i 0.651546 + 0.758610i \(0.274121\pi\)
−0.651546 + 0.758610i \(0.725879\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) − 11.0000i − 0.461957i
\(568\) 12.0000i 0.503509i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) − 2.00000i − 0.0836242i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 16.0000i − 0.666089i −0.942911 0.333044i \(-0.891924\pi\)
0.942911 0.333044i \(-0.108076\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) 28.0000 1.16364
\(580\) 0 0
\(581\) 0 0
\(582\) − 16.0000i − 0.663221i
\(583\) 12.0000i 0.496989i
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 18.0000i 0.742940i 0.928445 + 0.371470i \(0.121146\pi\)
−0.928445 + 0.371470i \(0.878854\pi\)
\(588\) 2.00000i 0.0824786i
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) 8.00000i 0.328798i
\(593\) − 30.0000i − 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 0 0
\(597\) − 16.0000i − 0.654836i
\(598\) 12.0000i 0.490716i
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) − 4.00000i − 0.163028i
\(603\) 4.00000i 0.162893i
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 36.0000 1.46240
\(607\) − 16.0000i − 0.649420i −0.945814 0.324710i \(-0.894733\pi\)
0.945814 0.324710i \(-0.105267\pi\)
\(608\) − 2.00000i − 0.0811107i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000i 0.242536i
\(613\) − 26.0000i − 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) −32.0000 −1.29141
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) 24.0000i 0.962312i
\(623\) 18.0000i 0.721155i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −16.0000 −0.639489
\(627\) 4.00000i 0.159745i
\(628\) − 2.00000i − 0.0798087i
\(629\) −48.0000 −1.91389
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) − 10.0000i − 0.397779i
\(633\) 40.0000i 1.58986i
\(634\) 0 0
\(635\) 0 0
\(636\) −24.0000 −0.951662
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) − 24.0000i − 0.947204i
\(643\) − 14.0000i − 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 12.0000i 0.471769i 0.971781 + 0.235884i \(0.0757987\pi\)
−0.971781 + 0.235884i \(0.924201\pi\)
\(648\) 11.0000i 0.432121i
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) − 4.00000i − 0.156652i
\(653\) 36.0000i 1.40879i 0.709809 + 0.704394i \(0.248781\pi\)
−0.709809 + 0.704394i \(0.751219\pi\)
\(654\) 16.0000 0.625650
\(655\) 0 0
\(656\) 0 0
\(657\) 14.0000i 0.546192i
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 24.0000i 0.932083i
\(664\) 0 0
\(665\) 0 0
\(666\) 8.00000 0.309994
\(667\) 0 0
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) − 2.00000i − 0.0771517i
\(673\) − 2.00000i − 0.0770943i −0.999257 0.0385472i \(-0.987727\pi\)
0.999257 0.0385472i \(-0.0122730\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) − 12.0000i − 0.460857i
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) − 8.00000i − 0.306336i
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) − 52.0000i − 1.98392i
\(688\) 4.00000i 0.152499i
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 6.00000i 0.228086i
\(693\) 1.00000i 0.0379869i
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) − 2.00000i − 0.0757011i
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) 8.00000i 0.301941i
\(703\) − 16.0000i − 0.603451i
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) − 18.0000i − 0.676960i
\(708\) 24.0000i 0.901975i
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) − 18.0000i − 0.674579i
\(713\) 48.0000i 1.79761i
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) − 36.0000i − 1.34444i
\(718\) 18.0000i 0.671754i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) − 15.0000i − 0.558242i
\(723\) 16.0000i 0.595046i
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) − 16.0000i − 0.593407i −0.954970 0.296704i \(-0.904113\pi\)
0.954970 0.296704i \(-0.0958873\pi\)
\(728\) − 2.00000i − 0.0741249i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 20.0000i 0.739221i
\(733\) − 14.0000i − 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) −20.0000 −0.738213
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 4.00000i 0.147342i
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 12.0000i 0.440534i
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 16.0000 0.586588
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) 6.00000i 0.219382i
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 44.0000 1.60558 0.802791 0.596260i \(-0.203347\pi\)
0.802791 + 0.596260i \(0.203347\pi\)
\(752\) 0 0
\(753\) − 24.0000i − 0.874609i
\(754\) 0 0
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) − 52.0000i − 1.88997i −0.327111 0.944986i \(-0.606075\pi\)
0.327111 0.944986i \(-0.393925\pi\)
\(758\) 28.0000i 1.01701i
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −48.0000 −1.74000 −0.869999 0.493053i \(-0.835881\pi\)
−0.869999 + 0.493053i \(0.835881\pi\)
\(762\) 32.0000i 1.15924i
\(763\) − 8.00000i − 0.289619i
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 24.0000i 0.866590i
\(768\) 2.00000i 0.0721688i
\(769\) −20.0000 −0.721218 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) 0 0
\(771\) 48.0000 1.72868
\(772\) 14.0000i 0.503871i
\(773\) 42.0000i 1.51064i 0.655359 + 0.755318i \(0.272517\pi\)
−0.655359 + 0.755318i \(0.727483\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) − 16.0000i − 0.573997i
\(778\) 6.00000i 0.215110i
\(779\) 0 0
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) − 36.0000i − 1.28736i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 36.0000 1.28408
\(787\) − 4.00000i − 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) − 18.0000i − 0.641223i
\(789\) −48.0000 −1.70885
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) − 1.00000i − 0.0355335i
\(793\) 20.0000i 0.710221i
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 42.0000i 1.48772i 0.668338 + 0.743858i \(0.267006\pi\)
−0.668338 + 0.743858i \(0.732994\pi\)
\(798\) 4.00000i 0.141598i
\(799\) 0 0
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) − 18.0000i − 0.635602i
\(803\) 14.0000i 0.494049i
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) − 36.0000i − 1.26726i
\(808\) 18.0000i 0.633238i
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) 0 0
\(813\) 40.0000i 1.40286i
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) − 8.00000i − 0.279885i
\(818\) − 8.00000i − 0.279713i
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 36.0000 1.25641 0.628204 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) 12.0000i 0.418548i
\(823\) − 2.00000i − 0.0697156i −0.999392 0.0348578i \(-0.988902\pi\)
0.999392 0.0348578i \(-0.0110978\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) 6.00000i 0.208514i
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 20.0000 0.693792
\(832\) 2.00000i 0.0693375i
\(833\) − 6.00000i − 0.207888i
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 32.0000i 1.10608i
\(838\) 12.0000i 0.414533i
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 14.0000i 0.482472i
\(843\) 12.0000i 0.413302i
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) 1.00000i 0.0343604i
\(848\) − 12.0000i − 0.412082i
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) −48.0000 −1.64542
\(852\) 24.0000i 0.822226i
\(853\) − 14.0000i − 0.479351i −0.970853 0.239675i \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 30.0000i 1.02478i 0.858753 + 0.512390i \(0.171240\pi\)
−0.858753 + 0.512390i \(0.828760\pi\)
\(858\) − 4.00000i − 0.136558i
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 18.0000i − 0.613082i
\(863\) 30.0000i 1.02121i 0.859815 + 0.510606i \(0.170579\pi\)
−0.859815 + 0.510606i \(0.829421\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) −4.00000 −0.135926
\(867\) − 38.0000i − 1.29055i
\(868\) − 8.00000i − 0.271538i
\(869\) −10.0000 −0.339227
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 8.00000i 0.270914i
\(873\) − 8.00000i − 0.270759i
\(874\) 12.0000 0.405906
\(875\) 0 0
\(876\) −28.0000 −0.946032
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) − 20.0000i − 0.674967i
\(879\) 36.0000 1.21425
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 1.00000i 0.0336718i
\(883\) 16.0000i 0.538443i 0.963078 + 0.269221i \(0.0867663\pi\)
−0.963078 + 0.269221i \(0.913234\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 16.0000i 0.536925i
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) − 4.00000i − 0.133930i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 24.0000i 0.801337i
\(898\) 18.0000i 0.600668i
\(899\) 0 0
\(900\) 0 0
\(901\) 72.0000 2.39867
\(902\) 0 0
\(903\) − 8.00000i − 0.266223i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 20.0000 0.664455
\(907\) 32.0000i 1.06254i 0.847202 + 0.531271i \(0.178286\pi\)
−0.847202 + 0.531271i \(0.821714\pi\)
\(908\) 12.0000i 0.398234i
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) 0 0
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) 26.0000 0.859064
\(917\) − 18.0000i − 0.594412i
\(918\) − 24.0000i − 0.792118i
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 0 0
\(921\) −64.0000 −2.10887
\(922\) − 30.0000i − 0.987997i
\(923\) 24.0000i 0.789970i
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) 2.00000 0.0657241
\(927\) − 4.00000i − 0.131377i
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) − 6.00000i − 0.196537i
\(933\) 48.0000i 1.57145i
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) 4.00000i 0.130605i
\(939\) −32.0000 −1.04428
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) − 4.00000i − 0.130327i
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) − 48.0000i − 1.55979i −0.625910 0.779895i \(-0.715272\pi\)
0.625910 0.779895i \(-0.284728\pi\)
\(948\) − 20.0000i − 0.649570i
\(949\) −28.0000 −0.908918
\(950\) 0 0
\(951\) 0 0
\(952\) 6.00000i 0.194461i
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) 18.0000 0.582162
\(957\) 0 0
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 16.0000i 0.515861i
\(963\) − 12.0000i − 0.386695i
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) 12.0000 0.386094
\(967\) − 16.0000i − 0.514525i −0.966342 0.257263i \(-0.917179\pi\)
0.966342 0.257263i \(-0.0828206\pi\)
\(968\) − 1.00000i − 0.0321412i
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) 10.0000i 0.320750i
\(973\) 10.0000i 0.320585i
\(974\) 22.0000 0.704925
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) − 8.00000i − 0.255812i
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) 8.00000 0.255420
\(982\) 12.0000i 0.382935i
\(983\) − 12.0000i − 0.382741i −0.981518 0.191370i \(-0.938707\pi\)
0.981518 0.191370i \(-0.0612931\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) − 4.00000i − 0.127257i
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 8.00000i 0.254000i
\(993\) 40.0000i 1.26936i
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) 0 0
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) − 20.0000i − 0.633089i
\(999\) −32.0000 −1.01244
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 3850.2.c.a.1849.2 2
5.2 odd 4 3850.2.a.j.1.1 1
5.3 odd 4 770.2.a.g.1.1 1
5.4 even 2 inner 3850.2.c.a.1849.1 2
15.8 even 4 6930.2.a.f.1.1 1
20.3 even 4 6160.2.a.n.1.1 1
35.13 even 4 5390.2.a.bh.1.1 1
55.43 even 4 8470.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.g.1.1 1 5.3 odd 4
3850.2.a.j.1.1 1 5.2 odd 4
3850.2.c.a.1849.1 2 5.4 even 2 inner
3850.2.c.a.1849.2 2 1.1 even 1 trivial
5390.2.a.bh.1.1 1 35.13 even 4
6160.2.a.n.1.1 1 20.3 even 4
6930.2.a.f.1.1 1 15.8 even 4
8470.2.a.e.1.1 1 55.43 even 4