Properties

Label 3850.2.c.c.1849.2
Level $3850$
Weight $2$
Character 3850.1849
Analytic conductor $30.742$
Analytic rank $1$
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: \(1\)
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.c.1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{12} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -6.00000 q^{19} +2.00000 q^{21} -1.00000i q^{22} +6.00000i q^{23} +2.00000 q^{24} -2.00000 q^{26} +4.00000i q^{27} +1.00000i q^{28} -4.00000 q^{29} +1.00000i q^{32} -2.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} -6.00000i q^{38} -4.00000 q^{39} +2.00000i q^{42} +4.00000i q^{43} +1.00000 q^{44} -6.00000 q^{46} +4.00000i q^{47} +2.00000i q^{48} -1.00000 q^{49} +4.00000 q^{51} -2.00000i q^{52} -12.0000i q^{53} -4.00000 q^{54} -1.00000 q^{56} -12.0000i q^{57} -4.00000i q^{58} +2.00000 q^{61} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +8.00000i q^{67} +2.00000i q^{68} -12.0000 q^{69} -12.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} +8.00000 q^{74} +6.00000 q^{76} +1.00000i q^{77} -4.00000i q^{78} -10.0000 q^{79} -11.0000 q^{81} -12.0000i q^{83} -2.00000 q^{84} -4.00000 q^{86} -8.00000i q^{87} +1.00000i q^{88} -14.0000 q^{89} +2.00000 q^{91} -6.00000i q^{92} -4.00000 q^{94} -2.00000 q^{96} -4.00000i q^{97} -1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{19} + 4 q^{21} + 4 q^{24} - 4 q^{26} - 8 q^{29} + 4 q^{34} + 2 q^{36} - 8 q^{39} + 2 q^{44} - 12 q^{46} - 2 q^{49} + 8 q^{51} - 8 q^{54} - 2 q^{56} + 4 q^{61} - 2 q^{64} + 4 q^{66} - 24 q^{69} - 24 q^{71} + 16 q^{74} + 12 q^{76} - 20 q^{79} - 22 q^{81} - 4 q^{84} - 8 q^{86} - 28 q^{89} + 4 q^{91} - 8 q^{94} - 4 q^{96} + 2 q^{99} + O(q^{100}) \)

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 \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 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
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 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
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) − 6.00000i − 0.973329i
\(39\) −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
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 2.00000i 0.288675i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) − 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
\(57\) − 12.0000i − 1.58944i
\(58\) − 4.00000i − 0.525226i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −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
\(73\) − 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
\(76\) 6.00000 0.688247
\(77\) 1.00000i 0.113961i
\(78\) − 4.00000i − 0.452911i
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) − 8.00000i − 0.857690i
\(88\) 1.00000i 0.106600i
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) − 6.00000i − 0.625543i
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) − 4.00000i − 0.406138i −0.979164 0.203069i \(-0.934908\pi\)
0.979164 0.203069i \(-0.0650917\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 4.00000i 0.396059i
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) − 20.0000i − 1.93347i −0.255774 0.966736i \(-0.582330\pi\)
0.255774 0.966736i \(-0.417670\pi\)
\(108\) − 4.00000i − 0.384900i
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 16.0000 1.51865
\(112\) − 1.00000i − 0.0944911i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 12.0000 1.12390
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) − 2.00000i − 0.184900i
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.00000i 0.181071i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 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
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 22.0000i 1.87959i 0.341743 + 0.939793i \(0.388983\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) − 12.0000i − 1.02151i
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) − 12.0000i − 1.00702i
\(143\) − 2.00000i − 0.167248i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) − 2.00000i − 0.164957i
\(148\) 8.00000i 0.657596i
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) 6.00000i 0.486664i
\(153\) 2.00000i 0.161690i
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) − 10.0000i − 0.795557i
\(159\) 24.0000 1.90332
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) − 11.0000i − 0.864242i
\(163\) − 24.0000i − 1.87983i −0.341415 0.939913i \(-0.610906\pi\)
0.341415 0.939913i \(-0.389094\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) − 4.00000i − 0.304997i
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) − 14.0000i − 1.04934i
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000i 0.148250i
\(183\) 4.00000i 0.295689i
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000i 0.146254i
\(188\) − 4.00000i − 0.291730i
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) − 2.00000i − 0.144338i
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) 4.00000 0.287183
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 1.00000i 0.0710669i
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 10.0000i 0.703598i
\(203\) 4.00000i 0.280745i
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) − 6.00000i − 0.417029i
\(208\) 2.00000i 0.138675i
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 12.0000i 0.824163i
\(213\) − 24.0000i − 1.64445i
\(214\) 20.0000 1.36717
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) 0 0
\(218\) − 4.00000i − 0.270914i
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 16.0000i 1.07385i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 24.0000i 1.59294i 0.604681 + 0.796468i \(0.293301\pi\)
−0.604681 + 0.796468i \(0.706699\pi\)
\(228\) 12.0000i 0.794719i
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 4.00000i 0.262613i
\(233\) − 26.0000i − 1.70332i −0.524097 0.851658i \(-0.675597\pi\)
0.524097 0.851658i \(-0.324403\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) − 20.0000i − 1.29914i
\(238\) − 2.00000i − 0.129641i
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 1.00000i 0.0642824i
\(243\) − 10.0000i − 0.641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) − 12.0000i − 0.763542i
\(248\) 0 0
\(249\) 24.0000 1.52094
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) − 6.00000i − 0.377217i
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 28.0000i 1.74659i 0.487190 + 0.873296i \(0.338022\pi\)
−0.487190 + 0.873296i \(0.661978\pi\)
\(258\) − 8.00000i − 0.498058i
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) − 6.00000i − 0.370681i
\(263\) 8.00000i 0.493301i 0.969104 + 0.246651i \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) −6.00000 −0.367884
\(267\) − 28.0000i − 1.71357i
\(268\) − 8.00000i − 0.488678i
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 4.00000i 0.242091i
\(274\) −22.0000 −1.32907
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) 22.0000i 1.32185i 0.750451 + 0.660926i \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) 6.00000i 0.359856i
\(279\) 0 0
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) − 8.00000i − 0.476393i
\(283\) 32.0000i 1.90220i 0.308879 + 0.951101i \(0.400046\pi\)
−0.308879 + 0.951101i \(0.599954\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) − 1.00000i − 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 6.00000i 0.351123i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) − 4.00000i − 0.232104i
\(298\) − 4.00000i − 0.231714i
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) − 6.00000i − 0.345261i
\(303\) 20.0000i 1.14897i
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) − 1.00000i − 0.0569803i
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 4.00000i 0.226455i
\(313\) − 12.0000i − 0.678280i −0.940736 0.339140i \(-0.889864\pi\)
0.940736 0.339140i \(-0.110136\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 24.0000i 1.34585i
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) 40.0000 2.23258
\(322\) 6.00000i 0.334367i
\(323\) 12.0000i 0.667698i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) 24.0000 1.32924
\(327\) − 8.00000i − 0.442401i
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 8.00000i 0.438397i
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 2.00000i 0.108947i 0.998515 + 0.0544735i \(0.0173480\pi\)
−0.998515 + 0.0544735i \(0.982652\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 6.00000i 0.324443i
\(343\) 1.00000i 0.0539949i
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) − 20.0000i − 1.07366i −0.843692 0.536828i \(-0.819622\pi\)
0.843692 0.536828i \(-0.180378\pi\)
\(348\) 8.00000i 0.428845i
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) −8.00000 −0.427008
\(352\) − 1.00000i − 0.0533002i
\(353\) 20.0000i 1.06449i 0.846590 + 0.532246i \(0.178652\pi\)
−0.846590 + 0.532246i \(0.821348\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) − 4.00000i − 0.211702i
\(358\) 4.00000i 0.211407i
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 2.00000i 0.105118i
\(363\) 2.00000i 0.104973i
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 6.00000i 0.312772i
\(369\) 0 0
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) − 26.0000i − 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) − 8.00000i − 0.412021i
\(378\) 4.00000i 0.205738i
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 8.00000i 0.409316i
\(383\) 16.0000i 0.817562i 0.912633 + 0.408781i \(0.134046\pi\)
−0.912633 + 0.408781i \(0.865954\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) − 4.00000i − 0.203331i
\(388\) 4.00000i 0.203069i
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 1.00000i 0.0505076i
\(393\) − 12.0000i − 0.605320i
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) − 14.0000i − 0.702640i −0.936255 0.351320i \(-0.885733\pi\)
0.936255 0.351320i \(-0.114267\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) −12.0000 −0.600751
\(400\) 0 0
\(401\) −34.0000 −1.69788 −0.848939 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) − 16.0000i − 0.798007i
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) 8.00000i 0.396545i
\(408\) − 4.00000i − 0.198030i
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) −44.0000 −2.17036
\(412\) 8.00000i 0.394132i
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 12.0000i 0.587643i
\(418\) 6.00000i 0.293470i
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) − 20.0000i − 0.973585i
\(423\) − 4.00000i − 0.194487i
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 24.0000 1.16280
\(427\) − 2.00000i − 0.0967868i
\(428\) 20.0000i 0.966736i
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −22.0000 −1.05970 −0.529851 0.848091i \(-0.677752\pi\)
−0.529851 + 0.848091i \(0.677752\pi\)
\(432\) 4.00000i 0.192450i
\(433\) 16.0000i 0.768911i 0.923144 + 0.384455i \(0.125611\pi\)
−0.923144 + 0.384455i \(0.874389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) − 36.0000i − 1.72211i
\(438\) 12.0000i 0.573382i
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 4.00000i 0.190261i
\(443\) 40.0000i 1.90046i 0.311553 + 0.950229i \(0.399151\pi\)
−0.311553 + 0.950229i \(0.600849\pi\)
\(444\) −16.0000 −0.759326
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) − 8.00000i − 0.378387i
\(448\) 1.00000i 0.0472456i
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000i 0.282216i
\(453\) − 12.0000i − 0.563809i
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) −12.0000 −0.561951
\(457\) − 22.0000i − 1.02912i −0.857455 0.514558i \(-0.827956\pi\)
0.857455 0.514558i \(-0.172044\pi\)
\(458\) − 6.00000i − 0.280362i
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) −26.0000 −1.21094 −0.605470 0.795868i \(-0.707015\pi\)
−0.605470 + 0.795868i \(0.707015\pi\)
\(462\) − 2.00000i − 0.0930484i
\(463\) − 10.0000i − 0.464739i −0.972628 0.232370i \(-0.925352\pi\)
0.972628 0.232370i \(-0.0746479\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) − 38.0000i − 1.75843i −0.476425 0.879215i \(-0.658068\pi\)
0.476425 0.879215i \(-0.341932\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −20.0000 −0.921551
\(472\) 0 0
\(473\) − 4.00000i − 0.183920i
\(474\) 20.0000 0.918630
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) 12.0000i 0.549442i
\(478\) − 6.00000i − 0.274434i
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 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
\(487\) 18.0000i 0.815658i 0.913058 + 0.407829i \(0.133714\pi\)
−0.913058 + 0.407829i \(0.866286\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) 48.0000 2.17064
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 8.00000i 0.360302i
\(494\) 12.0000 0.539906
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000i 0.538274i
\(498\) 24.0000i 1.07547i
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16.0000i 0.713405i 0.934218 + 0.356702i \(0.116099\pi\)
−0.934218 + 0.356702i \(0.883901\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) 18.0000i 0.799408i
\(508\) 8.00000i 0.354943i
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 1.00000i 0.0441942i
\(513\) − 24.0000i − 1.05963i
\(514\) −28.0000 −1.23503
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) − 4.00000i − 0.175920i
\(518\) − 8.00000i − 0.351500i
\(519\) 36.0000 1.58022
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 4.00000i 0.175075i
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) − 2.00000i − 0.0870388i
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) − 6.00000i − 0.260133i
\(533\) 0 0
\(534\) 28.0000 1.21168
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 8.00000i 0.345225i
\(538\) − 14.0000i − 0.603583i
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −16.0000 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 28.0000i 1.20270i
\(543\) 4.00000i 0.171656i
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) − 22.0000i − 0.939793i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 12.0000i 0.510754i
\(553\) 10.0000i 0.425243i
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −6.00000 −0.254457
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) − 2.00000i − 0.0843649i
\(563\) − 24.0000i − 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −32.0000 −1.34506
\(567\) 11.0000i 0.461957i
\(568\) 12.0000i 0.503509i
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 2.00000i 0.0836242i
\(573\) 16.0000i 0.668410i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 4.00000i − 0.166522i −0.996528 0.0832611i \(-0.973466\pi\)
0.996528 0.0832611i \(-0.0265335\pi\)
\(578\) 13.0000i 0.540729i
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 8.00000i 0.331611i
\(583\) 12.0000i 0.496989i
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 2.00000i 0.0825488i 0.999148 + 0.0412744i \(0.0131418\pi\)
−0.999148 + 0.0412744i \(0.986858\pi\)
\(588\) 2.00000i 0.0824786i
\(589\) 0 0
\(590\) 0 0
\(591\) −4.00000 −0.164538
\(592\) − 8.00000i − 0.328798i
\(593\) 26.0000i 1.06769i 0.845582 + 0.533846i \(0.179254\pi\)
−0.845582 + 0.533846i \(0.820746\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) − 32.0000i − 1.30967i
\(598\) − 12.0000i − 0.490716i
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 0 0
\(601\) 44.0000 1.79480 0.897399 0.441221i \(-0.145454\pi\)
0.897399 + 0.441221i \(0.145454\pi\)
\(602\) 4.00000i 0.163028i
\(603\) − 8.00000i − 0.325785i
\(604\) 6.00000 0.244137
\(605\) 0 0
\(606\) −20.0000 −0.812444
\(607\) − 16.0000i − 0.649420i −0.945814 0.324710i \(-0.894733\pi\)
0.945814 0.324710i \(-0.105267\pi\)
\(608\) − 6.00000i − 0.243332i
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) − 2.00000i − 0.0808452i
\(613\) − 26.0000i − 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 42.0000i 1.69086i 0.534089 + 0.845428i \(0.320655\pi\)
−0.534089 + 0.845428i \(0.679345\pi\)
\(618\) 16.0000i 0.643614i
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) 16.0000i 0.641542i
\(623\) 14.0000i 0.560898i
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 12.0000 0.479616
\(627\) 12.0000i 0.479234i
\(628\) − 10.0000i − 0.399043i
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 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
\(638\) 4.00000i 0.158362i
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 40.0000i 1.57867i
\(643\) 34.0000i 1.34083i 0.741987 + 0.670415i \(0.233884\pi\)
−0.741987 + 0.670415i \(0.766116\pi\)
\(644\) −6.00000 −0.236433
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 11.0000i 0.432121i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 24.0000i 0.939913i
\(653\) − 36.0000i − 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) 0 0
\(657\) 6.00000i 0.234082i
\(658\) 4.00000i 0.155936i
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) 12.0000i 0.466393i
\(663\) 8.00000i 0.310694i
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) − 24.0000i − 0.929284i
\(668\) 0 0
\(669\) −32.0000 −1.23719
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 2.00000i 0.0771517i
\(673\) − 42.0000i − 1.61898i −0.587133 0.809491i \(-0.699743\pi\)
0.587133 0.809491i \(-0.300257\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 38.0000i − 1.46046i −0.683202 0.730229i \(-0.739413\pi\)
0.683202 0.730229i \(-0.260587\pi\)
\(678\) 12.0000i 0.460857i
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) −48.0000 −1.83936
\(682\) 0 0
\(683\) − 4.00000i − 0.153056i −0.997067 0.0765279i \(-0.975617\pi\)
0.997067 0.0765279i \(-0.0243834\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) − 12.0000i − 0.457829i
\(688\) 4.00000i 0.152499i
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 18.0000i 0.684257i
\(693\) − 1.00000i − 0.0379869i
\(694\) 20.0000 0.759190
\(695\) 0 0
\(696\) −8.00000 −0.303239
\(697\) 0 0
\(698\) − 6.00000i − 0.227103i
\(699\) 52.0000 1.96682
\(700\) 0 0
\(701\) −8.00000 −0.302156 −0.151078 0.988522i \(-0.548274\pi\)
−0.151078 + 0.988522i \(0.548274\pi\)
\(702\) − 8.00000i − 0.301941i
\(703\) 48.0000i 1.81035i
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −20.0000 −0.752710
\(707\) − 10.0000i − 0.376089i
\(708\) 0 0
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 14.0000i 0.524672i
\(713\) 0 0
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) − 12.0000i − 0.448148i
\(718\) 6.00000i 0.223918i
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 17.0000i 0.632674i
\(723\) − 16.0000i − 0.595046i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) 44.0000i 1.63187i 0.578144 + 0.815935i \(0.303777\pi\)
−0.578144 + 0.815935i \(0.696223\pi\)
\(728\) − 2.00000i − 0.0741249i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) − 4.00000i − 0.147844i
\(733\) − 2.00000i − 0.0738717i −0.999318 0.0369358i \(-0.988240\pi\)
0.999318 0.0369358i \(-0.0117597\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) − 8.00000i − 0.294684i
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 0 0
\(741\) 24.0000 0.881662
\(742\) − 12.0000i − 0.440534i
\(743\) 48.0000i 1.76095i 0.474093 + 0.880475i \(0.342776\pi\)
−0.474093 + 0.880475i \(0.657224\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 26.0000 0.951928
\(747\) 12.0000i 0.439057i
\(748\) − 2.00000i − 0.0731272i
\(749\) −20.0000 −0.730784
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 4.00000i 0.145865i
\(753\) 0 0
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) − 52.0000i − 1.88997i −0.327111 0.944986i \(-0.606075\pi\)
0.327111 0.944986i \(-0.393925\pi\)
\(758\) − 4.00000i − 0.145287i
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −24.0000 −0.869999 −0.435000 0.900431i \(-0.643252\pi\)
−0.435000 + 0.900431i \(0.643252\pi\)
\(762\) 16.0000i 0.579619i
\(763\) 4.00000i 0.144810i
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 0 0
\(768\) 2.00000i 0.0721688i
\(769\) 28.0000 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(770\) 0 0
\(771\) −56.0000 −2.01679
\(772\) − 2.00000i − 0.0719816i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −4.00000 −0.143592
\(777\) − 16.0000i − 0.573997i
\(778\) − 18.0000i − 0.645331i
\(779\) 0 0
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 12.0000i 0.429119i
\(783\) − 16.0000i − 0.571793i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 48.0000i 1.71102i 0.517790 + 0.855508i \(0.326755\pi\)
−0.517790 + 0.855508i \(0.673245\pi\)
\(788\) − 2.00000i − 0.0712470i
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) − 1.00000i − 0.0355335i
\(793\) 4.00000i 0.142044i
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) − 12.0000i − 0.424795i
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) − 34.0000i − 1.20058i
\(803\) 6.00000i 0.211735i
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 0 0
\(807\) − 28.0000i − 0.985647i
\(808\) − 10.0000i − 0.351799i
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) −30.0000 −1.05344 −0.526721 0.850038i \(-0.676579\pi\)
−0.526721 + 0.850038i \(0.676579\pi\)
\(812\) − 4.00000i − 0.140372i
\(813\) 56.0000i 1.96401i
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) − 24.0000i − 0.839654i
\(818\) 0 0
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) − 44.0000i − 1.53468i
\(823\) 14.0000i 0.488009i 0.969774 + 0.244005i \(0.0784612\pi\)
−0.969774 + 0.244005i \(0.921539\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 6.00000i 0.208514i
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) −44.0000 −1.52634
\(832\) − 2.00000i − 0.0693375i
\(833\) 2.00000i 0.0692959i
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) 16.0000i 0.552711i
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) − 26.0000i − 0.896019i
\(843\) − 4.00000i − 0.137767i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) − 1.00000i − 0.0343604i
\(848\) − 12.0000i − 0.412082i
\(849\) −64.0000 −2.19647
\(850\) 0 0
\(851\) 48.0000 1.64542
\(852\) 24.0000i 0.822226i
\(853\) 22.0000i 0.753266i 0.926363 + 0.376633i \(0.122918\pi\)
−0.926363 + 0.376633i \(0.877082\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) 30.0000i 1.02478i 0.858753 + 0.512390i \(0.171240\pi\)
−0.858753 + 0.512390i \(0.828760\pi\)
\(858\) 4.00000i 0.136558i
\(859\) −48.0000 −1.63774 −0.818869 0.573980i \(-0.805399\pi\)
−0.818869 + 0.573980i \(0.805399\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 22.0000i − 0.749323i
\(863\) 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
\(866\) −16.0000 −0.543702
\(867\) 26.0000i 0.883006i
\(868\) 0 0
\(869\) 10.0000 0.339227
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 4.00000i 0.135457i
\(873\) 4.00000i 0.135379i
\(874\) 36.0000 1.21772
\(875\) 0 0
\(876\) −12.0000 −0.405442
\(877\) − 14.0000i − 0.472746i −0.971662 0.236373i \(-0.924041\pi\)
0.971662 0.236373i \(-0.0759588\pi\)
\(878\) − 12.0000i − 0.404980i
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 1.00000i 0.0336718i
\(883\) 28.0000i 0.942275i 0.882060 + 0.471138i \(0.156156\pi\)
−0.882060 + 0.471138i \(0.843844\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −40.0000 −1.34383
\(887\) − 48.0000i − 1.61168i −0.592132 0.805841i \(-0.701714\pi\)
0.592132 0.805841i \(-0.298286\pi\)
\(888\) − 16.0000i − 0.536925i
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) − 16.0000i − 0.535720i
\(893\) − 24.0000i − 0.803129i
\(894\) 8.00000 0.267560
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) − 24.0000i − 0.801337i
\(898\) − 14.0000i − 0.467186i
\(899\) 0 0
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) 8.00000i 0.266223i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 12.0000 0.398673
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) − 24.0000i − 0.796468i
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) − 12.0000i − 0.397360i
\(913\) 12.0000i 0.397142i
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 6.00000i 0.198137i
\(918\) 8.00000i 0.264039i
\(919\) −26.0000 −0.857661 −0.428830 0.903385i \(-0.641074\pi\)
−0.428830 + 0.903385i \(0.641074\pi\)
\(920\) 0 0
\(921\) −24.0000 −0.790827
\(922\) − 26.0000i − 0.856264i
\(923\) − 24.0000i − 0.789970i
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) 10.0000 0.328620
\(927\) 8.00000i 0.262754i
\(928\) − 4.00000i − 0.131306i
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 26.0000i 0.851658i
\(933\) 32.0000i 1.04763i
\(934\) 38.0000 1.24340
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) − 6.00000i − 0.196011i −0.995186 0.0980057i \(-0.968754\pi\)
0.995186 0.0980057i \(-0.0312463\pi\)
\(938\) 8.00000i 0.261209i
\(939\) 24.0000 0.783210
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) − 20.0000i − 0.651635i
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 52.0000i 1.68977i 0.534946 + 0.844886i \(0.320332\pi\)
−0.534946 + 0.844886i \(0.679668\pi\)
\(948\) 20.0000i 0.649570i
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 2.00000i 0.0648204i
\(953\) 18.0000i 0.583077i 0.956559 + 0.291539i \(0.0941672\pi\)
−0.956559 + 0.291539i \(0.905833\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) 8.00000i 0.258603i
\(958\) − 8.00000i − 0.258468i
\(959\) 22.0000 0.710417
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 16.0000i 0.515861i
\(963\) 20.0000i 0.644491i
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 56.0000i 1.80084i 0.435023 + 0.900419i \(0.356740\pi\)
−0.435023 + 0.900419i \(0.643260\pi\)
\(968\) − 1.00000i − 0.0321412i
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 10.0000i 0.320750i
\(973\) − 6.00000i − 0.192351i
\(974\) −18.0000 −0.576757
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 54.0000i 1.72761i 0.503824 + 0.863807i \(0.331926\pi\)
−0.503824 + 0.863807i \(0.668074\pi\)
\(978\) 48.0000i 1.53487i
\(979\) 14.0000 0.447442
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 12.0000i 0.382935i
\(983\) − 56.0000i − 1.78612i −0.449935 0.893061i \(-0.648553\pi\)
0.449935 0.893061i \(-0.351447\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) 8.00000i 0.254643i
\(988\) 12.0000i 0.381771i
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 24.0000i 0.761617i
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) −24.0000 −0.760469
\(997\) 42.0000i 1.33015i 0.746775 + 0.665077i \(0.231601\pi\)
−0.746775 + 0.665077i \(0.768399\pi\)
\(998\) 28.0000i 0.886325i
\(999\) 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.c.1849.2 2
5.2 odd 4 770.2.a.e.1.1 1
5.3 odd 4 3850.2.a.m.1.1 1
5.4 even 2 inner 3850.2.c.c.1849.1 2
15.2 even 4 6930.2.a.bk.1.1 1
20.7 even 4 6160.2.a.a.1.1 1
35.27 even 4 5390.2.a.c.1.1 1
55.32 even 4 8470.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.e.1.1 1 5.2 odd 4
3850.2.a.m.1.1 1 5.3 odd 4
3850.2.c.c.1849.1 2 5.4 even 2 inner
3850.2.c.c.1849.2 2 1.1 even 1 trivial
5390.2.a.c.1.1 1 35.27 even 4
6160.2.a.a.1.1 1 20.7 even 4
6930.2.a.bk.1.1 1 15.2 even 4
8470.2.a.bg.1.1 1 55.32 even 4