Properties

Label 3850.2.c.q.1849.3
Level $3850$
Weight $2$
Character 3850.1849
Analytic conductor $30.742$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.3
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 3850.1849
Dual form 3850.2.c.q.1849.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.23607i q^{3} -1.00000 q^{4} +1.23607 q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.47214 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.23607i q^{3} -1.00000 q^{4} +1.23607 q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.47214 q^{9} +1.00000 q^{11} +1.23607i q^{12} +3.23607i q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.47214i q^{17} +1.47214i q^{18} +7.23607 q^{19} +1.23607 q^{21} +1.00000i q^{22} -4.00000i q^{23} -1.23607 q^{24} -3.23607 q^{26} -5.52786i q^{27} -1.00000i q^{28} -4.47214 q^{29} +2.00000 q^{31} +1.00000i q^{32} -1.23607i q^{33} -2.47214 q^{34} -1.47214 q^{36} +6.94427i q^{37} +7.23607i q^{38} +4.00000 q^{39} -2.47214 q^{41} +1.23607i q^{42} +10.4721i q^{43} -1.00000 q^{44} +4.00000 q^{46} -2.00000i q^{47} -1.23607i q^{48} -1.00000 q^{49} +3.05573 q^{51} -3.23607i q^{52} -8.47214i q^{53} +5.52786 q^{54} +1.00000 q^{56} -8.94427i q^{57} -4.47214i q^{58} -2.76393 q^{59} -0.763932 q^{61} +2.00000i q^{62} +1.47214i q^{63} -1.00000 q^{64} +1.23607 q^{66} +11.4164i q^{67} -2.47214i q^{68} -4.94427 q^{69} +6.47214 q^{71} -1.47214i q^{72} -12.9443i q^{73} -6.94427 q^{74} -7.23607 q^{76} +1.00000i q^{77} +4.00000i q^{78} -2.41641 q^{81} -2.47214i q^{82} +12.1803i q^{83} -1.23607 q^{84} -10.4721 q^{86} +5.52786i q^{87} -1.00000i q^{88} -10.0000 q^{89} -3.23607 q^{91} +4.00000i q^{92} -2.47214i q^{93} +2.00000 q^{94} +1.23607 q^{96} +12.4721i q^{97} -1.00000i q^{98} +1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 12 q^{9} + 4 q^{11} - 4 q^{14} + 4 q^{16} + 20 q^{19} - 4 q^{21} + 4 q^{24} - 4 q^{26} + 8 q^{31} + 8 q^{34} + 12 q^{36} + 16 q^{39} + 8 q^{41} - 4 q^{44} + 16 q^{46} - 4 q^{49} + 48 q^{51} + 40 q^{54} + 4 q^{56} - 20 q^{59} - 12 q^{61} - 4 q^{64} - 4 q^{66} + 16 q^{69} + 8 q^{71} + 8 q^{74} - 20 q^{76} + 44 q^{81} + 4 q^{84} - 24 q^{86} - 40 q^{89} - 4 q^{91} + 8 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) − 1.23607i − 0.713644i −0.934172 0.356822i \(-0.883860\pi\)
0.934172 0.356822i \(-0.116140\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.23607 0.504623
\(7\) 1.00000i 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) 1.47214 0.490712
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 1.23607i 0.356822i
\(13\) 3.23607i 0.897524i 0.893651 + 0.448762i \(0.148135\pi\)
−0.893651 + 0.448762i \(0.851865\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.47214i 0.599581i 0.954005 + 0.299791i \(0.0969168\pi\)
−0.954005 + 0.299791i \(0.903083\pi\)
\(18\) 1.47214i 0.346986i
\(19\) 7.23607 1.66007 0.830034 0.557713i \(-0.188321\pi\)
0.830034 + 0.557713i \(0.188321\pi\)
\(20\) 0 0
\(21\) 1.23607 0.269732
\(22\) 1.00000i 0.213201i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) −1.23607 −0.252311
\(25\) 0 0
\(26\) −3.23607 −0.634645
\(27\) − 5.52786i − 1.06384i
\(28\) − 1.00000i − 0.188982i
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 1.23607i − 0.215172i
\(34\) −2.47214 −0.423968
\(35\) 0 0
\(36\) −1.47214 −0.245356
\(37\) 6.94427i 1.14163i 0.821078 + 0.570816i \(0.193373\pi\)
−0.821078 + 0.570816i \(0.806627\pi\)
\(38\) 7.23607i 1.17385i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −2.47214 −0.386083 −0.193041 0.981191i \(-0.561835\pi\)
−0.193041 + 0.981191i \(0.561835\pi\)
\(42\) 1.23607i 0.190729i
\(43\) 10.4721i 1.59699i 0.602004 + 0.798493i \(0.294369\pi\)
−0.602004 + 0.798493i \(0.705631\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) − 2.00000i − 0.291730i −0.989305 0.145865i \(-0.953403\pi\)
0.989305 0.145865i \(-0.0465965\pi\)
\(48\) − 1.23607i − 0.178411i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 3.05573 0.427888
\(52\) − 3.23607i − 0.448762i
\(53\) − 8.47214i − 1.16374i −0.813283 0.581869i \(-0.802322\pi\)
0.813283 0.581869i \(-0.197678\pi\)
\(54\) 5.52786 0.752247
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) − 8.94427i − 1.18470i
\(58\) − 4.47214i − 0.587220i
\(59\) −2.76393 −0.359833 −0.179917 0.983682i \(-0.557583\pi\)
−0.179917 + 0.983682i \(0.557583\pi\)
\(60\) 0 0
\(61\) −0.763932 −0.0978115 −0.0489057 0.998803i \(-0.515573\pi\)
−0.0489057 + 0.998803i \(0.515573\pi\)
\(62\) 2.00000i 0.254000i
\(63\) 1.47214i 0.185472i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.23607 0.152149
\(67\) 11.4164i 1.39474i 0.716713 + 0.697368i \(0.245646\pi\)
−0.716713 + 0.697368i \(0.754354\pi\)
\(68\) − 2.47214i − 0.299791i
\(69\) −4.94427 −0.595220
\(70\) 0 0
\(71\) 6.47214 0.768101 0.384051 0.923312i \(-0.374529\pi\)
0.384051 + 0.923312i \(0.374529\pi\)
\(72\) − 1.47214i − 0.173493i
\(73\) − 12.9443i − 1.51501i −0.652828 0.757506i \(-0.726418\pi\)
0.652828 0.757506i \(-0.273582\pi\)
\(74\) −6.94427 −0.807255
\(75\) 0 0
\(76\) −7.23607 −0.830034
\(77\) 1.00000i 0.113961i
\(78\) 4.00000i 0.452911i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) − 2.47214i − 0.273002i
\(83\) 12.1803i 1.33697i 0.743727 + 0.668483i \(0.233056\pi\)
−0.743727 + 0.668483i \(0.766944\pi\)
\(84\) −1.23607 −0.134866
\(85\) 0 0
\(86\) −10.4721 −1.12924
\(87\) 5.52786i 0.592649i
\(88\) − 1.00000i − 0.106600i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −3.23607 −0.339232
\(92\) 4.00000i 0.417029i
\(93\) − 2.47214i − 0.256349i
\(94\) 2.00000 0.206284
\(95\) 0 0
\(96\) 1.23607 0.126156
\(97\) 12.4721i 1.26635i 0.774007 + 0.633177i \(0.218249\pi\)
−0.774007 + 0.633177i \(0.781751\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) 1.47214 0.147955
\(100\) 0 0
\(101\) 8.18034 0.813974 0.406987 0.913434i \(-0.366579\pi\)
0.406987 + 0.913434i \(0.366579\pi\)
\(102\) 3.05573i 0.302562i
\(103\) 14.9443i 1.47250i 0.676708 + 0.736251i \(0.263406\pi\)
−0.676708 + 0.736251i \(0.736594\pi\)
\(104\) 3.23607 0.317323
\(105\) 0 0
\(106\) 8.47214 0.822887
\(107\) 2.47214i 0.238990i 0.992835 + 0.119495i \(0.0381276\pi\)
−0.992835 + 0.119495i \(0.961872\pi\)
\(108\) 5.52786i 0.531919i
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 8.58359 0.814719
\(112\) 1.00000i 0.0944911i
\(113\) 0.472136i 0.0444148i 0.999753 + 0.0222074i \(0.00706942\pi\)
−0.999753 + 0.0222074i \(0.992931\pi\)
\(114\) 8.94427 0.837708
\(115\) 0 0
\(116\) 4.47214 0.415227
\(117\) 4.76393i 0.440426i
\(118\) − 2.76393i − 0.254441i
\(119\) −2.47214 −0.226620
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 0.763932i − 0.0691632i
\(123\) 3.05573i 0.275526i
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) −1.47214 −0.131148
\(127\) − 12.0000i − 1.06483i −0.846484 0.532414i \(-0.821285\pi\)
0.846484 0.532414i \(-0.178715\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 12.9443 1.13968
\(130\) 0 0
\(131\) 4.76393 0.416227 0.208113 0.978105i \(-0.433268\pi\)
0.208113 + 0.978105i \(0.433268\pi\)
\(132\) 1.23607i 0.107586i
\(133\) 7.23607i 0.627447i
\(134\) −11.4164 −0.986227
\(135\) 0 0
\(136\) 2.47214 0.211984
\(137\) − 19.8885i − 1.69919i −0.527433 0.849596i \(-0.676846\pi\)
0.527433 0.849596i \(-0.323154\pi\)
\(138\) − 4.94427i − 0.420884i
\(139\) 21.7082 1.84127 0.920633 0.390429i \(-0.127673\pi\)
0.920633 + 0.390429i \(0.127673\pi\)
\(140\) 0 0
\(141\) −2.47214 −0.208191
\(142\) 6.47214i 0.543130i
\(143\) 3.23607i 0.270614i
\(144\) 1.47214 0.122678
\(145\) 0 0
\(146\) 12.9443 1.07128
\(147\) 1.23607i 0.101949i
\(148\) − 6.94427i − 0.570816i
\(149\) 22.3607 1.83186 0.915929 0.401340i \(-0.131455\pi\)
0.915929 + 0.401340i \(0.131455\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) − 7.23607i − 0.586923i
\(153\) 3.63932i 0.294222i
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) − 12.6525i − 1.00978i −0.863184 0.504889i \(-0.831534\pi\)
0.863184 0.504889i \(-0.168466\pi\)
\(158\) 0 0
\(159\) −10.4721 −0.830494
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) − 2.41641i − 0.189851i
\(163\) 19.4164i 1.52081i 0.649449 + 0.760405i \(0.275000\pi\)
−0.649449 + 0.760405i \(0.725000\pi\)
\(164\) 2.47214 0.193041
\(165\) 0 0
\(166\) −12.1803 −0.945378
\(167\) 11.4164i 0.883428i 0.897156 + 0.441714i \(0.145629\pi\)
−0.897156 + 0.441714i \(0.854371\pi\)
\(168\) − 1.23607i − 0.0953647i
\(169\) 2.52786 0.194451
\(170\) 0 0
\(171\) 10.6525 0.814615
\(172\) − 10.4721i − 0.798493i
\(173\) 3.23607i 0.246034i 0.992405 + 0.123017i \(0.0392569\pi\)
−0.992405 + 0.123017i \(0.960743\pi\)
\(174\) −5.52786 −0.419066
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 3.41641i 0.256793i
\(178\) − 10.0000i − 0.749532i
\(179\) 8.94427 0.668526 0.334263 0.942480i \(-0.391513\pi\)
0.334263 + 0.942480i \(0.391513\pi\)
\(180\) 0 0
\(181\) 9.23607 0.686512 0.343256 0.939242i \(-0.388470\pi\)
0.343256 + 0.939242i \(0.388470\pi\)
\(182\) − 3.23607i − 0.239873i
\(183\) 0.944272i 0.0698026i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 2.47214 0.181266
\(187\) 2.47214i 0.180780i
\(188\) 2.00000i 0.145865i
\(189\) 5.52786 0.402093
\(190\) 0 0
\(191\) −2.47214 −0.178877 −0.0894387 0.995992i \(-0.528507\pi\)
−0.0894387 + 0.995992i \(0.528507\pi\)
\(192\) 1.23607i 0.0892055i
\(193\) 14.9443i 1.07571i 0.843037 + 0.537856i \(0.180766\pi\)
−0.843037 + 0.537856i \(0.819234\pi\)
\(194\) −12.4721 −0.895447
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 1.47214i 0.104620i
\(199\) −18.9443 −1.34292 −0.671462 0.741039i \(-0.734333\pi\)
−0.671462 + 0.741039i \(0.734333\pi\)
\(200\) 0 0
\(201\) 14.1115 0.995345
\(202\) 8.18034i 0.575567i
\(203\) − 4.47214i − 0.313882i
\(204\) −3.05573 −0.213944
\(205\) 0 0
\(206\) −14.9443 −1.04122
\(207\) − 5.88854i − 0.409282i
\(208\) 3.23607i 0.224381i
\(209\) 7.23607 0.500529
\(210\) 0 0
\(211\) −13.5279 −0.931297 −0.465648 0.884970i \(-0.654179\pi\)
−0.465648 + 0.884970i \(0.654179\pi\)
\(212\) 8.47214i 0.581869i
\(213\) − 8.00000i − 0.548151i
\(214\) −2.47214 −0.168992
\(215\) 0 0
\(216\) −5.52786 −0.376124
\(217\) 2.00000i 0.135769i
\(218\) 10.0000i 0.677285i
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 8.58359i 0.576093i
\(223\) 0.472136i 0.0316166i 0.999875 + 0.0158083i \(0.00503214\pi\)
−0.999875 + 0.0158083i \(0.994968\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −0.472136 −0.0314060
\(227\) − 19.2361i − 1.27674i −0.769729 0.638371i \(-0.779608\pi\)
0.769729 0.638371i \(-0.220392\pi\)
\(228\) 8.94427i 0.592349i
\(229\) −17.2361 −1.13899 −0.569496 0.821994i \(-0.692861\pi\)
−0.569496 + 0.821994i \(0.692861\pi\)
\(230\) 0 0
\(231\) 1.23607 0.0813273
\(232\) 4.47214i 0.293610i
\(233\) 14.9443i 0.979032i 0.871994 + 0.489516i \(0.162826\pi\)
−0.871994 + 0.489516i \(0.837174\pi\)
\(234\) −4.76393 −0.311428
\(235\) 0 0
\(236\) 2.76393 0.179917
\(237\) 0 0
\(238\) − 2.47214i − 0.160245i
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) 15.4164 0.993058 0.496529 0.868020i \(-0.334608\pi\)
0.496529 + 0.868020i \(0.334608\pi\)
\(242\) 1.00000i 0.0642824i
\(243\) − 13.5967i − 0.872232i
\(244\) 0.763932 0.0489057
\(245\) 0 0
\(246\) −3.05573 −0.194826
\(247\) 23.4164i 1.48995i
\(248\) − 2.00000i − 0.127000i
\(249\) 15.0557 0.954118
\(250\) 0 0
\(251\) 29.2361 1.84536 0.922682 0.385562i \(-0.125992\pi\)
0.922682 + 0.385562i \(0.125992\pi\)
\(252\) − 1.47214i − 0.0927358i
\(253\) − 4.00000i − 0.251478i
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.94427i 0.433172i 0.976264 + 0.216586i \(0.0694921\pi\)
−0.976264 + 0.216586i \(0.930508\pi\)
\(258\) 12.9443i 0.805875i
\(259\) −6.94427 −0.431496
\(260\) 0 0
\(261\) −6.58359 −0.407514
\(262\) 4.76393i 0.294317i
\(263\) 4.94427i 0.304877i 0.988313 + 0.152438i \(0.0487126\pi\)
−0.988313 + 0.152438i \(0.951287\pi\)
\(264\) −1.23607 −0.0760747
\(265\) 0 0
\(266\) −7.23607 −0.443672
\(267\) 12.3607i 0.756461i
\(268\) − 11.4164i − 0.697368i
\(269\) 22.7639 1.38794 0.693971 0.720003i \(-0.255860\pi\)
0.693971 + 0.720003i \(0.255860\pi\)
\(270\) 0 0
\(271\) 0.944272 0.0573604 0.0286802 0.999589i \(-0.490870\pi\)
0.0286802 + 0.999589i \(0.490870\pi\)
\(272\) 2.47214i 0.149895i
\(273\) 4.00000i 0.242091i
\(274\) 19.8885 1.20151
\(275\) 0 0
\(276\) 4.94427 0.297610
\(277\) 3.52786i 0.211969i 0.994368 + 0.105984i \(0.0337994\pi\)
−0.994368 + 0.105984i \(0.966201\pi\)
\(278\) 21.7082i 1.30197i
\(279\) 2.94427 0.176269
\(280\) 0 0
\(281\) 28.8328 1.72002 0.860011 0.510276i \(-0.170457\pi\)
0.860011 + 0.510276i \(0.170457\pi\)
\(282\) − 2.47214i − 0.147214i
\(283\) − 14.6525i − 0.870999i −0.900189 0.435500i \(-0.856572\pi\)
0.900189 0.435500i \(-0.143428\pi\)
\(284\) −6.47214 −0.384051
\(285\) 0 0
\(286\) −3.23607 −0.191353
\(287\) − 2.47214i − 0.145926i
\(288\) 1.47214i 0.0867464i
\(289\) 10.8885 0.640503
\(290\) 0 0
\(291\) 15.4164 0.903726
\(292\) 12.9443i 0.757506i
\(293\) 26.6525i 1.55705i 0.627611 + 0.778527i \(0.284033\pi\)
−0.627611 + 0.778527i \(0.715967\pi\)
\(294\) −1.23607 −0.0720889
\(295\) 0 0
\(296\) 6.94427 0.403628
\(297\) − 5.52786i − 0.320759i
\(298\) 22.3607i 1.29532i
\(299\) 12.9443 0.748587
\(300\) 0 0
\(301\) −10.4721 −0.603604
\(302\) 12.0000i 0.690522i
\(303\) − 10.1115i − 0.580888i
\(304\) 7.23607 0.415017
\(305\) 0 0
\(306\) −3.63932 −0.208046
\(307\) − 26.0689i − 1.48783i −0.668274 0.743915i \(-0.732967\pi\)
0.668274 0.743915i \(-0.267033\pi\)
\(308\) − 1.00000i − 0.0569803i
\(309\) 18.4721 1.05084
\(310\) 0 0
\(311\) −21.4164 −1.21441 −0.607207 0.794544i \(-0.707710\pi\)
−0.607207 + 0.794544i \(0.707710\pi\)
\(312\) − 4.00000i − 0.226455i
\(313\) − 19.5279i − 1.10378i −0.833917 0.551890i \(-0.813907\pi\)
0.833917 0.551890i \(-0.186093\pi\)
\(314\) 12.6525 0.714021
\(315\) 0 0
\(316\) 0 0
\(317\) − 30.9443i − 1.73800i −0.494809 0.869002i \(-0.664762\pi\)
0.494809 0.869002i \(-0.335238\pi\)
\(318\) − 10.4721i − 0.587248i
\(319\) −4.47214 −0.250392
\(320\) 0 0
\(321\) 3.05573 0.170554
\(322\) 4.00000i 0.222911i
\(323\) 17.8885i 0.995345i
\(324\) 2.41641 0.134245
\(325\) 0 0
\(326\) −19.4164 −1.07538
\(327\) − 12.3607i − 0.683547i
\(328\) 2.47214i 0.136501i
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) −16.9443 −0.931341 −0.465671 0.884958i \(-0.654187\pi\)
−0.465671 + 0.884958i \(0.654187\pi\)
\(332\) − 12.1803i − 0.668483i
\(333\) 10.2229i 0.560212i
\(334\) −11.4164 −0.624678
\(335\) 0 0
\(336\) 1.23607 0.0674330
\(337\) 18.0000i 0.980522i 0.871576 + 0.490261i \(0.163099\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 2.52786i 0.137498i
\(339\) 0.583592 0.0316964
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 10.6525i 0.576020i
\(343\) − 1.00000i − 0.0539949i
\(344\) 10.4721 0.564620
\(345\) 0 0
\(346\) −3.23607 −0.173972
\(347\) 2.47214i 0.132711i 0.997796 + 0.0663556i \(0.0211372\pi\)
−0.997796 + 0.0663556i \(0.978863\pi\)
\(348\) − 5.52786i − 0.296325i
\(349\) −21.7082 −1.16201 −0.581007 0.813899i \(-0.697341\pi\)
−0.581007 + 0.813899i \(0.697341\pi\)
\(350\) 0 0
\(351\) 17.8885 0.954820
\(352\) 1.00000i 0.0533002i
\(353\) 17.0557i 0.907785i 0.891056 + 0.453892i \(0.149965\pi\)
−0.891056 + 0.453892i \(0.850035\pi\)
\(354\) −3.41641 −0.181580
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 3.05573i 0.161726i
\(358\) 8.94427i 0.472719i
\(359\) −26.8328 −1.41618 −0.708091 0.706121i \(-0.750443\pi\)
−0.708091 + 0.706121i \(0.750443\pi\)
\(360\) 0 0
\(361\) 33.3607 1.75583
\(362\) 9.23607i 0.485437i
\(363\) − 1.23607i − 0.0648767i
\(364\) 3.23607 0.169616
\(365\) 0 0
\(366\) −0.944272 −0.0493579
\(367\) − 5.41641i − 0.282734i −0.989957 0.141367i \(-0.954850\pi\)
0.989957 0.141367i \(-0.0451498\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) −3.63932 −0.189455
\(370\) 0 0
\(371\) 8.47214 0.439851
\(372\) 2.47214i 0.128174i
\(373\) 6.00000i 0.310668i 0.987862 + 0.155334i \(0.0496454\pi\)
−0.987862 + 0.155334i \(0.950355\pi\)
\(374\) −2.47214 −0.127831
\(375\) 0 0
\(376\) −2.00000 −0.103142
\(377\) − 14.4721i − 0.745353i
\(378\) 5.52786i 0.284323i
\(379\) −14.4721 −0.743384 −0.371692 0.928356i \(-0.621222\pi\)
−0.371692 + 0.928356i \(0.621222\pi\)
\(380\) 0 0
\(381\) −14.8328 −0.759908
\(382\) − 2.47214i − 0.126485i
\(383\) 23.8885i 1.22065i 0.792152 + 0.610324i \(0.208961\pi\)
−0.792152 + 0.610324i \(0.791039\pi\)
\(384\) −1.23607 −0.0630778
\(385\) 0 0
\(386\) −14.9443 −0.760643
\(387\) 15.4164i 0.783660i
\(388\) − 12.4721i − 0.633177i
\(389\) −33.4164 −1.69428 −0.847140 0.531370i \(-0.821677\pi\)
−0.847140 + 0.531370i \(0.821677\pi\)
\(390\) 0 0
\(391\) 9.88854 0.500085
\(392\) 1.00000i 0.0505076i
\(393\) − 5.88854i − 0.297038i
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) −1.47214 −0.0739776
\(397\) − 23.7082i − 1.18988i −0.803770 0.594940i \(-0.797176\pi\)
0.803770 0.594940i \(-0.202824\pi\)
\(398\) − 18.9443i − 0.949591i
\(399\) 8.94427 0.447774
\(400\) 0 0
\(401\) 14.3607 0.717138 0.358569 0.933503i \(-0.383265\pi\)
0.358569 + 0.933503i \(0.383265\pi\)
\(402\) 14.1115i 0.703815i
\(403\) 6.47214i 0.322400i
\(404\) −8.18034 −0.406987
\(405\) 0 0
\(406\) 4.47214 0.221948
\(407\) 6.94427i 0.344215i
\(408\) − 3.05573i − 0.151281i
\(409\) −3.41641 −0.168930 −0.0844652 0.996426i \(-0.526918\pi\)
−0.0844652 + 0.996426i \(0.526918\pi\)
\(410\) 0 0
\(411\) −24.5836 −1.21262
\(412\) − 14.9443i − 0.736251i
\(413\) − 2.76393i − 0.136004i
\(414\) 5.88854 0.289406
\(415\) 0 0
\(416\) −3.23607 −0.158661
\(417\) − 26.8328i − 1.31401i
\(418\) 7.23607i 0.353928i
\(419\) −17.2361 −0.842037 −0.421019 0.907052i \(-0.638327\pi\)
−0.421019 + 0.907052i \(0.638327\pi\)
\(420\) 0 0
\(421\) 16.4721 0.802803 0.401401 0.915902i \(-0.368523\pi\)
0.401401 + 0.915902i \(0.368523\pi\)
\(422\) − 13.5279i − 0.658526i
\(423\) − 2.94427i − 0.143155i
\(424\) −8.47214 −0.411443
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) − 0.763932i − 0.0369693i
\(428\) − 2.47214i − 0.119495i
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 23.0557 1.11056 0.555278 0.831665i \(-0.312612\pi\)
0.555278 + 0.831665i \(0.312612\pi\)
\(432\) − 5.52786i − 0.265959i
\(433\) − 28.4721i − 1.36828i −0.729349 0.684142i \(-0.760177\pi\)
0.729349 0.684142i \(-0.239823\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) − 28.9443i − 1.38459i
\(438\) − 16.0000i − 0.764510i
\(439\) −8.94427 −0.426887 −0.213443 0.976955i \(-0.568468\pi\)
−0.213443 + 0.976955i \(0.568468\pi\)
\(440\) 0 0
\(441\) −1.47214 −0.0701017
\(442\) − 8.00000i − 0.380521i
\(443\) 24.9443i 1.18514i 0.805520 + 0.592569i \(0.201886\pi\)
−0.805520 + 0.592569i \(0.798114\pi\)
\(444\) −8.58359 −0.407359
\(445\) 0 0
\(446\) −0.472136 −0.0223563
\(447\) − 27.6393i − 1.30729i
\(448\) − 1.00000i − 0.0472456i
\(449\) −18.9443 −0.894035 −0.447018 0.894525i \(-0.647514\pi\)
−0.447018 + 0.894525i \(0.647514\pi\)
\(450\) 0 0
\(451\) −2.47214 −0.116408
\(452\) − 0.472136i − 0.0222074i
\(453\) − 14.8328i − 0.696906i
\(454\) 19.2361 0.902793
\(455\) 0 0
\(456\) −8.94427 −0.418854
\(457\) 26.9443i 1.26040i 0.776433 + 0.630200i \(0.217027\pi\)
−0.776433 + 0.630200i \(0.782973\pi\)
\(458\) − 17.2361i − 0.805389i
\(459\) 13.6656 0.637857
\(460\) 0 0
\(461\) 24.7639 1.15337 0.576686 0.816966i \(-0.304346\pi\)
0.576686 + 0.816966i \(0.304346\pi\)
\(462\) 1.23607i 0.0575071i
\(463\) 30.4721i 1.41616i 0.706132 + 0.708080i \(0.250438\pi\)
−0.706132 + 0.708080i \(0.749562\pi\)
\(464\) −4.47214 −0.207614
\(465\) 0 0
\(466\) −14.9443 −0.692280
\(467\) − 27.1246i − 1.25518i −0.778545 0.627589i \(-0.784042\pi\)
0.778545 0.627589i \(-0.215958\pi\)
\(468\) − 4.76393i − 0.220213i
\(469\) −11.4164 −0.527161
\(470\) 0 0
\(471\) −15.6393 −0.720622
\(472\) 2.76393i 0.127220i
\(473\) 10.4721i 0.481509i
\(474\) 0 0
\(475\) 0 0
\(476\) 2.47214 0.113310
\(477\) − 12.4721i − 0.571060i
\(478\) − 20.0000i − 0.914779i
\(479\) −12.3607 −0.564774 −0.282387 0.959301i \(-0.591126\pi\)
−0.282387 + 0.959301i \(0.591126\pi\)
\(480\) 0 0
\(481\) −22.4721 −1.02464
\(482\) 15.4164i 0.702198i
\(483\) − 4.94427i − 0.224972i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 13.5967 0.616761
\(487\) 16.9443i 0.767818i 0.923371 + 0.383909i \(0.125422\pi\)
−0.923371 + 0.383909i \(0.874578\pi\)
\(488\) 0.763932i 0.0345816i
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) −16.9443 −0.764684 −0.382342 0.924021i \(-0.624882\pi\)
−0.382342 + 0.924021i \(0.624882\pi\)
\(492\) − 3.05573i − 0.137763i
\(493\) − 11.0557i − 0.497925i
\(494\) −23.4164 −1.05355
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 6.47214i 0.290315i
\(498\) 15.0557i 0.674663i
\(499\) 32.3607 1.44866 0.724331 0.689452i \(-0.242149\pi\)
0.724331 + 0.689452i \(0.242149\pi\)
\(500\) 0 0
\(501\) 14.1115 0.630453
\(502\) 29.2361i 1.30487i
\(503\) − 4.00000i − 0.178351i −0.996016 0.0891756i \(-0.971577\pi\)
0.996016 0.0891756i \(-0.0284232\pi\)
\(504\) 1.47214 0.0655741
\(505\) 0 0
\(506\) 4.00000 0.177822
\(507\) − 3.12461i − 0.138769i
\(508\) 12.0000i 0.532414i
\(509\) −24.0689 −1.06683 −0.533417 0.845852i \(-0.679092\pi\)
−0.533417 + 0.845852i \(0.679092\pi\)
\(510\) 0 0
\(511\) 12.9443 0.572621
\(512\) 1.00000i 0.0441942i
\(513\) − 40.0000i − 1.76604i
\(514\) −6.94427 −0.306299
\(515\) 0 0
\(516\) −12.9443 −0.569840
\(517\) − 2.00000i − 0.0879599i
\(518\) − 6.94427i − 0.305114i
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) −10.3607 −0.453910 −0.226955 0.973905i \(-0.572877\pi\)
−0.226955 + 0.973905i \(0.572877\pi\)
\(522\) − 6.58359i − 0.288156i
\(523\) 14.2918i 0.624937i 0.949928 + 0.312468i \(0.101156\pi\)
−0.949928 + 0.312468i \(0.898844\pi\)
\(524\) −4.76393 −0.208113
\(525\) 0 0
\(526\) −4.94427 −0.215580
\(527\) 4.94427i 0.215376i
\(528\) − 1.23607i − 0.0537930i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −4.06888 −0.176575
\(532\) − 7.23607i − 0.313723i
\(533\) − 8.00000i − 0.346518i
\(534\) −12.3607 −0.534899
\(535\) 0 0
\(536\) 11.4164 0.493114
\(537\) − 11.0557i − 0.477090i
\(538\) 22.7639i 0.981423i
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −26.9443 −1.15842 −0.579212 0.815177i \(-0.696640\pi\)
−0.579212 + 0.815177i \(0.696640\pi\)
\(542\) 0.944272i 0.0405600i
\(543\) − 11.4164i − 0.489925i
\(544\) −2.47214 −0.105992
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) − 0.944272i − 0.0403742i −0.999796 0.0201871i \(-0.993574\pi\)
0.999796 0.0201871i \(-0.00642618\pi\)
\(548\) 19.8885i 0.849596i
\(549\) −1.12461 −0.0479973
\(550\) 0 0
\(551\) −32.3607 −1.37861
\(552\) 4.94427i 0.210442i
\(553\) 0 0
\(554\) −3.52786 −0.149885
\(555\) 0 0
\(556\) −21.7082 −0.920633
\(557\) 24.8328i 1.05220i 0.850423 + 0.526100i \(0.176346\pi\)
−0.850423 + 0.526100i \(0.823654\pi\)
\(558\) 2.94427i 0.124641i
\(559\) −33.8885 −1.43333
\(560\) 0 0
\(561\) 3.05573 0.129013
\(562\) 28.8328i 1.21624i
\(563\) − 31.2361i − 1.31644i −0.752824 0.658222i \(-0.771309\pi\)
0.752824 0.658222i \(-0.228691\pi\)
\(564\) 2.47214 0.104096
\(565\) 0 0
\(566\) 14.6525 0.615889
\(567\) − 2.41641i − 0.101480i
\(568\) − 6.47214i − 0.271565i
\(569\) 36.8328 1.54411 0.772056 0.635555i \(-0.219228\pi\)
0.772056 + 0.635555i \(0.219228\pi\)
\(570\) 0 0
\(571\) −10.1115 −0.423151 −0.211576 0.977362i \(-0.567859\pi\)
−0.211576 + 0.977362i \(0.567859\pi\)
\(572\) − 3.23607i − 0.135307i
\(573\) 3.05573i 0.127655i
\(574\) 2.47214 0.103185
\(575\) 0 0
\(576\) −1.47214 −0.0613390
\(577\) 26.9443i 1.12170i 0.827916 + 0.560852i \(0.189526\pi\)
−0.827916 + 0.560852i \(0.810474\pi\)
\(578\) 10.8885i 0.452904i
\(579\) 18.4721 0.767676
\(580\) 0 0
\(581\) −12.1803 −0.505326
\(582\) 15.4164i 0.639031i
\(583\) − 8.47214i − 0.350880i
\(584\) −12.9443 −0.535638
\(585\) 0 0
\(586\) −26.6525 −1.10100
\(587\) − 5.81966i − 0.240203i −0.992762 0.120102i \(-0.961678\pi\)
0.992762 0.120102i \(-0.0383220\pi\)
\(588\) − 1.23607i − 0.0509746i
\(589\) 14.4721 0.596314
\(590\) 0 0
\(591\) 22.2492 0.915211
\(592\) 6.94427i 0.285408i
\(593\) − 24.0000i − 0.985562i −0.870153 0.492781i \(-0.835980\pi\)
0.870153 0.492781i \(-0.164020\pi\)
\(594\) 5.52786 0.226811
\(595\) 0 0
\(596\) −22.3607 −0.915929
\(597\) 23.4164i 0.958370i
\(598\) 12.9443i 0.529331i
\(599\) 32.3607 1.32222 0.661111 0.750288i \(-0.270085\pi\)
0.661111 + 0.750288i \(0.270085\pi\)
\(600\) 0 0
\(601\) −34.8328 −1.42086 −0.710430 0.703768i \(-0.751500\pi\)
−0.710430 + 0.703768i \(0.751500\pi\)
\(602\) − 10.4721i − 0.426812i
\(603\) 16.8065i 0.684414i
\(604\) −12.0000 −0.488273
\(605\) 0 0
\(606\) 10.1115 0.410750
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) 7.23607i 0.293461i
\(609\) −5.52786 −0.224000
\(610\) 0 0
\(611\) 6.47214 0.261835
\(612\) − 3.63932i − 0.147111i
\(613\) − 28.4721i − 1.14998i −0.818161 0.574989i \(-0.805006\pi\)
0.818161 0.574989i \(-0.194994\pi\)
\(614\) 26.0689 1.05205
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 21.4164i 0.862192i 0.902306 + 0.431096i \(0.141873\pi\)
−0.902306 + 0.431096i \(0.858127\pi\)
\(618\) 18.4721i 0.743058i
\(619\) −18.5410 −0.745227 −0.372613 0.927987i \(-0.621538\pi\)
−0.372613 + 0.927987i \(0.621538\pi\)
\(620\) 0 0
\(621\) −22.1115 −0.887302
\(622\) − 21.4164i − 0.858720i
\(623\) − 10.0000i − 0.400642i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 19.5279 0.780490
\(627\) − 8.94427i − 0.357200i
\(628\) 12.6525i 0.504889i
\(629\) −17.1672 −0.684500
\(630\) 0 0
\(631\) −31.4164 −1.25067 −0.625334 0.780357i \(-0.715037\pi\)
−0.625334 + 0.780357i \(0.715037\pi\)
\(632\) 0 0
\(633\) 16.7214i 0.664614i
\(634\) 30.9443 1.22895
\(635\) 0 0
\(636\) 10.4721 0.415247
\(637\) − 3.23607i − 0.128218i
\(638\) − 4.47214i − 0.177054i
\(639\) 9.52786 0.376916
\(640\) 0 0
\(641\) 27.5279 1.08729 0.543643 0.839317i \(-0.317045\pi\)
0.543643 + 0.839317i \(0.317045\pi\)
\(642\) 3.05573i 0.120600i
\(643\) 18.7639i 0.739977i 0.929036 + 0.369989i \(0.120638\pi\)
−0.929036 + 0.369989i \(0.879362\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) −17.8885 −0.703815
\(647\) − 28.8328i − 1.13353i −0.823878 0.566767i \(-0.808194\pi\)
0.823878 0.566767i \(-0.191806\pi\)
\(648\) 2.41641i 0.0949255i
\(649\) −2.76393 −0.108494
\(650\) 0 0
\(651\) 2.47214 0.0968906
\(652\) − 19.4164i − 0.760405i
\(653\) − 46.3607i − 1.81423i −0.420879 0.907117i \(-0.638278\pi\)
0.420879 0.907117i \(-0.361722\pi\)
\(654\) 12.3607 0.483341
\(655\) 0 0
\(656\) −2.47214 −0.0965207
\(657\) − 19.0557i − 0.743435i
\(658\) 2.00000i 0.0779681i
\(659\) −16.5836 −0.646005 −0.323003 0.946398i \(-0.604692\pi\)
−0.323003 + 0.946398i \(0.604692\pi\)
\(660\) 0 0
\(661\) −3.12461 −0.121533 −0.0607667 0.998152i \(-0.519355\pi\)
−0.0607667 + 0.998152i \(0.519355\pi\)
\(662\) − 16.9443i − 0.658558i
\(663\) 9.88854i 0.384039i
\(664\) 12.1803 0.472689
\(665\) 0 0
\(666\) −10.2229 −0.396130
\(667\) 17.8885i 0.692647i
\(668\) − 11.4164i − 0.441714i
\(669\) 0.583592 0.0225630
\(670\) 0 0
\(671\) −0.763932 −0.0294913
\(672\) 1.23607i 0.0476824i
\(673\) 3.88854i 0.149892i 0.997188 + 0.0749462i \(0.0238785\pi\)
−0.997188 + 0.0749462i \(0.976122\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) −2.52786 −0.0972255
\(677\) − 26.0689i − 1.00191i −0.865474 0.500954i \(-0.832982\pi\)
0.865474 0.500954i \(-0.167018\pi\)
\(678\) 0.583592i 0.0224127i
\(679\) −12.4721 −0.478637
\(680\) 0 0
\(681\) −23.7771 −0.911140
\(682\) 2.00000i 0.0765840i
\(683\) − 32.9443i − 1.26058i −0.776361 0.630289i \(-0.782936\pi\)
0.776361 0.630289i \(-0.217064\pi\)
\(684\) −10.6525 −0.407308
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 21.3050i 0.812835i
\(688\) 10.4721i 0.399246i
\(689\) 27.4164 1.04448
\(690\) 0 0
\(691\) 12.6525 0.481323 0.240661 0.970609i \(-0.422636\pi\)
0.240661 + 0.970609i \(0.422636\pi\)
\(692\) − 3.23607i − 0.123017i
\(693\) 1.47214i 0.0559218i
\(694\) −2.47214 −0.0938410
\(695\) 0 0
\(696\) 5.52786 0.209533
\(697\) − 6.11146i − 0.231488i
\(698\) − 21.7082i − 0.821668i
\(699\) 18.4721 0.698680
\(700\) 0 0
\(701\) −42.7214 −1.61356 −0.806782 0.590850i \(-0.798793\pi\)
−0.806782 + 0.590850i \(0.798793\pi\)
\(702\) 17.8885i 0.675160i
\(703\) 50.2492i 1.89519i
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −17.0557 −0.641901
\(707\) 8.18034i 0.307653i
\(708\) − 3.41641i − 0.128396i
\(709\) −4.47214 −0.167955 −0.0839773 0.996468i \(-0.526762\pi\)
−0.0839773 + 0.996468i \(0.526762\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 10.0000i 0.374766i
\(713\) − 8.00000i − 0.299602i
\(714\) −3.05573 −0.114358
\(715\) 0 0
\(716\) −8.94427 −0.334263
\(717\) 24.7214i 0.923236i
\(718\) − 26.8328i − 1.00139i
\(719\) −16.8328 −0.627758 −0.313879 0.949463i \(-0.601629\pi\)
−0.313879 + 0.949463i \(0.601629\pi\)
\(720\) 0 0
\(721\) −14.9443 −0.556554
\(722\) 33.3607i 1.24156i
\(723\) − 19.0557i − 0.708690i
\(724\) −9.23607 −0.343256
\(725\) 0 0
\(726\) 1.23607 0.0458748
\(727\) 18.0000i 0.667583i 0.942647 + 0.333792i \(0.108328\pi\)
−0.942647 + 0.333792i \(0.891672\pi\)
\(728\) 3.23607i 0.119937i
\(729\) −24.0557 −0.890953
\(730\) 0 0
\(731\) −25.8885 −0.957522
\(732\) − 0.944272i − 0.0349013i
\(733\) − 49.1246i − 1.81446i −0.420636 0.907229i \(-0.638193\pi\)
0.420636 0.907229i \(-0.361807\pi\)
\(734\) 5.41641 0.199923
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 11.4164i 0.420529i
\(738\) − 3.63932i − 0.133965i
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 28.9443 1.06329
\(742\) 8.47214i 0.311022i
\(743\) − 21.8885i − 0.803013i −0.915856 0.401506i \(-0.868487\pi\)
0.915856 0.401506i \(-0.131513\pi\)
\(744\) −2.47214 −0.0906329
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) 17.9311i 0.656065i
\(748\) − 2.47214i − 0.0903902i
\(749\) −2.47214 −0.0903299
\(750\) 0 0
\(751\) −16.9443 −0.618305 −0.309153 0.951012i \(-0.600045\pi\)
−0.309153 + 0.951012i \(0.600045\pi\)
\(752\) − 2.00000i − 0.0729325i
\(753\) − 36.1378i − 1.31693i
\(754\) 14.4721 0.527044
\(755\) 0 0
\(756\) −5.52786 −0.201046
\(757\) − 23.3050i − 0.847033i −0.905888 0.423516i \(-0.860796\pi\)
0.905888 0.423516i \(-0.139204\pi\)
\(758\) − 14.4721i − 0.525652i
\(759\) −4.94427 −0.179466
\(760\) 0 0
\(761\) −11.4164 −0.413844 −0.206922 0.978357i \(-0.566345\pi\)
−0.206922 + 0.978357i \(0.566345\pi\)
\(762\) − 14.8328i − 0.537336i
\(763\) 10.0000i 0.362024i
\(764\) 2.47214 0.0894387
\(765\) 0 0
\(766\) −23.8885 −0.863128
\(767\) − 8.94427i − 0.322959i
\(768\) − 1.23607i − 0.0446028i
\(769\) 43.4164 1.56564 0.782818 0.622251i \(-0.213782\pi\)
0.782818 + 0.622251i \(0.213782\pi\)
\(770\) 0 0
\(771\) 8.58359 0.309131
\(772\) − 14.9443i − 0.537856i
\(773\) − 15.7082i − 0.564985i −0.959270 0.282492i \(-0.908839\pi\)
0.959270 0.282492i \(-0.0911612\pi\)
\(774\) −15.4164 −0.554131
\(775\) 0 0
\(776\) 12.4721 0.447724
\(777\) 8.58359i 0.307935i
\(778\) − 33.4164i − 1.19804i
\(779\) −17.8885 −0.640924
\(780\) 0 0
\(781\) 6.47214 0.231591
\(782\) 9.88854i 0.353614i
\(783\) 24.7214i 0.883469i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 5.88854 0.210037
\(787\) − 28.1803i − 1.00452i −0.864716 0.502260i \(-0.832502\pi\)
0.864716 0.502260i \(-0.167498\pi\)
\(788\) − 18.0000i − 0.641223i
\(789\) 6.11146 0.217574
\(790\) 0 0
\(791\) −0.472136 −0.0167872
\(792\) − 1.47214i − 0.0523101i
\(793\) − 2.47214i − 0.0877881i
\(794\) 23.7082 0.841373
\(795\) 0 0
\(796\) 18.9443 0.671462
\(797\) − 41.5967i − 1.47343i −0.676202 0.736716i \(-0.736375\pi\)
0.676202 0.736716i \(-0.263625\pi\)
\(798\) 8.94427i 0.316624i
\(799\) 4.94427 0.174916
\(800\) 0 0
\(801\) −14.7214 −0.520154
\(802\) 14.3607i 0.507093i
\(803\) − 12.9443i − 0.456793i
\(804\) −14.1115 −0.497673
\(805\) 0 0
\(806\) −6.47214 −0.227971
\(807\) − 28.1378i − 0.990496i
\(808\) − 8.18034i − 0.287783i
\(809\) 21.0557 0.740280 0.370140 0.928976i \(-0.379310\pi\)
0.370140 + 0.928976i \(0.379310\pi\)
\(810\) 0 0
\(811\) 4.76393 0.167284 0.0836421 0.996496i \(-0.473345\pi\)
0.0836421 + 0.996496i \(0.473345\pi\)
\(812\) 4.47214i 0.156941i
\(813\) − 1.16718i − 0.0409349i
\(814\) −6.94427 −0.243397
\(815\) 0 0
\(816\) 3.05573 0.106972
\(817\) 75.7771i 2.65110i
\(818\) − 3.41641i − 0.119452i
\(819\) −4.76393 −0.166465
\(820\) 0 0
\(821\) −1.41641 −0.0494330 −0.0247165 0.999695i \(-0.507868\pi\)
−0.0247165 + 0.999695i \(0.507868\pi\)
\(822\) − 24.5836i − 0.857451i
\(823\) 46.2492i 1.61215i 0.591816 + 0.806073i \(0.298411\pi\)
−0.591816 + 0.806073i \(0.701589\pi\)
\(824\) 14.9443 0.520608
\(825\) 0 0
\(826\) 2.76393 0.0961695
\(827\) 16.9443i 0.589210i 0.955619 + 0.294605i \(0.0951881\pi\)
−0.955619 + 0.294605i \(0.904812\pi\)
\(828\) 5.88854i 0.204641i
\(829\) −11.7082 −0.406643 −0.203321 0.979112i \(-0.565174\pi\)
−0.203321 + 0.979112i \(0.565174\pi\)
\(830\) 0 0
\(831\) 4.36068 0.151270
\(832\) − 3.23607i − 0.112190i
\(833\) − 2.47214i − 0.0856544i
\(834\) 26.8328 0.929144
\(835\) 0 0
\(836\) −7.23607 −0.250265
\(837\) − 11.0557i − 0.382142i
\(838\) − 17.2361i − 0.595410i
\(839\) −16.8328 −0.581133 −0.290567 0.956855i \(-0.593844\pi\)
−0.290567 + 0.956855i \(0.593844\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 16.4721i 0.567667i
\(843\) − 35.6393i − 1.22748i
\(844\) 13.5279 0.465648
\(845\) 0 0
\(846\) 2.94427 0.101226
\(847\) 1.00000i 0.0343604i
\(848\) − 8.47214i − 0.290934i
\(849\) −18.1115 −0.621584
\(850\) 0 0
\(851\) 27.7771 0.952186
\(852\) 8.00000i 0.274075i
\(853\) − 32.5410i − 1.11418i −0.830451 0.557092i \(-0.811917\pi\)
0.830451 0.557092i \(-0.188083\pi\)
\(854\) 0.763932 0.0261412
\(855\) 0 0
\(856\) 2.47214 0.0844959
\(857\) − 46.4721i − 1.58746i −0.608272 0.793729i \(-0.708137\pi\)
0.608272 0.793729i \(-0.291863\pi\)
\(858\) 4.00000i 0.136558i
\(859\) −15.1246 −0.516045 −0.258023 0.966139i \(-0.583071\pi\)
−0.258023 + 0.966139i \(0.583071\pi\)
\(860\) 0 0
\(861\) −3.05573 −0.104139
\(862\) 23.0557i 0.785281i
\(863\) − 0.583592i − 0.0198657i −0.999951 0.00993285i \(-0.996838\pi\)
0.999951 0.00993285i \(-0.00316178\pi\)
\(864\) 5.52786 0.188062
\(865\) 0 0
\(866\) 28.4721 0.967523
\(867\) − 13.4590i − 0.457091i
\(868\) − 2.00000i − 0.0678844i
\(869\) 0 0
\(870\) 0 0
\(871\) −36.9443 −1.25181
\(872\) − 10.0000i − 0.338643i
\(873\) 18.3607i 0.621415i
\(874\) 28.9443 0.979055
\(875\) 0 0
\(876\) 16.0000 0.540590
\(877\) 9.05573i 0.305790i 0.988242 + 0.152895i \(0.0488597\pi\)
−0.988242 + 0.152895i \(0.951140\pi\)
\(878\) − 8.94427i − 0.301855i
\(879\) 32.9443 1.11118
\(880\) 0 0
\(881\) 28.8328 0.971402 0.485701 0.874125i \(-0.338564\pi\)
0.485701 + 0.874125i \(0.338564\pi\)
\(882\) − 1.47214i − 0.0495694i
\(883\) 2.83282i 0.0953318i 0.998863 + 0.0476659i \(0.0151783\pi\)
−0.998863 + 0.0476659i \(0.984822\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) −24.9443 −0.838019
\(887\) − 44.3607i − 1.48949i −0.667351 0.744743i \(-0.732572\pi\)
0.667351 0.744743i \(-0.267428\pi\)
\(888\) − 8.58359i − 0.288046i
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) −2.41641 −0.0809527
\(892\) − 0.472136i − 0.0158083i
\(893\) − 14.4721i − 0.484292i
\(894\) 27.6393 0.924397
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) − 16.0000i − 0.534224i
\(898\) − 18.9443i − 0.632179i
\(899\) −8.94427 −0.298308
\(900\) 0 0
\(901\) 20.9443 0.697755
\(902\) − 2.47214i − 0.0823131i
\(903\) 12.9443i 0.430758i
\(904\) 0.472136 0.0157030
\(905\) 0 0
\(906\) 14.8328 0.492787
\(907\) − 24.3607i − 0.808883i −0.914564 0.404442i \(-0.867466\pi\)
0.914564 0.404442i \(-0.132534\pi\)
\(908\) 19.2361i 0.638371i
\(909\) 12.0426 0.399427
\(910\) 0 0
\(911\) −28.0000 −0.927681 −0.463841 0.885919i \(-0.653529\pi\)
−0.463841 + 0.885919i \(0.653529\pi\)
\(912\) − 8.94427i − 0.296174i
\(913\) 12.1803i 0.403110i
\(914\) −26.9443 −0.891237
\(915\) 0 0
\(916\) 17.2361 0.569496
\(917\) 4.76393i 0.157319i
\(918\) 13.6656i 0.451033i
\(919\) 22.1115 0.729390 0.364695 0.931127i \(-0.381173\pi\)
0.364695 + 0.931127i \(0.381173\pi\)
\(920\) 0 0
\(921\) −32.2229 −1.06178
\(922\) 24.7639i 0.815557i
\(923\) 20.9443i 0.689389i
\(924\) −1.23607 −0.0406637
\(925\) 0 0
\(926\) −30.4721 −1.00138
\(927\) 22.0000i 0.722575i
\(928\) − 4.47214i − 0.146805i
\(929\) −40.2492 −1.32053 −0.660267 0.751031i \(-0.729557\pi\)
−0.660267 + 0.751031i \(0.729557\pi\)
\(930\) 0 0
\(931\) −7.23607 −0.237153
\(932\) − 14.9443i − 0.489516i
\(933\) 26.4721i 0.866659i
\(934\) 27.1246 0.887544
\(935\) 0 0
\(936\) 4.76393 0.155714
\(937\) − 3.05573i − 0.0998263i −0.998754 0.0499131i \(-0.984106\pi\)
0.998754 0.0499131i \(-0.0158945\pi\)
\(938\) − 11.4164i − 0.372759i
\(939\) −24.1378 −0.787706
\(940\) 0 0
\(941\) −11.8197 −0.385310 −0.192655 0.981267i \(-0.561710\pi\)
−0.192655 + 0.981267i \(0.561710\pi\)
\(942\) − 15.6393i − 0.509557i
\(943\) 9.88854i 0.322015i
\(944\) −2.76393 −0.0899583
\(945\) 0 0
\(946\) −10.4721 −0.340479
\(947\) 16.9443i 0.550615i 0.961356 + 0.275307i \(0.0887796\pi\)
−0.961356 + 0.275307i \(0.911220\pi\)
\(948\) 0 0
\(949\) 41.8885 1.35976
\(950\) 0 0
\(951\) −38.2492 −1.24032
\(952\) 2.47214i 0.0801224i
\(953\) − 22.9443i − 0.743238i −0.928385 0.371619i \(-0.878803\pi\)
0.928385 0.371619i \(-0.121197\pi\)
\(954\) 12.4721 0.403800
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 5.52786i 0.178690i
\(958\) − 12.3607i − 0.399355i
\(959\) 19.8885 0.642235
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) − 22.4721i − 0.724531i
\(963\) 3.63932i 0.117275i
\(964\) −15.4164 −0.496529
\(965\) 0 0
\(966\) 4.94427 0.159079
\(967\) 45.8885i 1.47568i 0.674978 + 0.737838i \(0.264153\pi\)
−0.674978 + 0.737838i \(0.735847\pi\)
\(968\) − 1.00000i − 0.0321412i
\(969\) 22.1115 0.710322
\(970\) 0 0
\(971\) 50.5410 1.62194 0.810969 0.585089i \(-0.198940\pi\)
0.810969 + 0.585089i \(0.198940\pi\)
\(972\) 13.5967i 0.436116i
\(973\) 21.7082i 0.695933i
\(974\) −16.9443 −0.542929
\(975\) 0 0
\(976\) −0.763932 −0.0244529
\(977\) − 28.8328i − 0.922443i −0.887285 0.461222i \(-0.847411\pi\)
0.887285 0.461222i \(-0.152589\pi\)
\(978\) 24.0000i 0.767435i
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) 14.7214 0.470017
\(982\) − 16.9443i − 0.540713i
\(983\) − 14.0000i − 0.446531i −0.974758 0.223265i \(-0.928328\pi\)
0.974758 0.223265i \(-0.0716716\pi\)
\(984\) 3.05573 0.0974131
\(985\) 0 0
\(986\) 11.0557 0.352086
\(987\) − 2.47214i − 0.0786890i
\(988\) − 23.4164i − 0.744975i
\(989\) 41.8885 1.33198
\(990\) 0 0
\(991\) −0.360680 −0.0114574 −0.00572869 0.999984i \(-0.501824\pi\)
−0.00572869 + 0.999984i \(0.501824\pi\)
\(992\) 2.00000i 0.0635001i
\(993\) 20.9443i 0.664646i
\(994\) −6.47214 −0.205284
\(995\) 0 0
\(996\) −15.0557 −0.477059
\(997\) 24.1803i 0.765799i 0.923790 + 0.382900i \(0.125074\pi\)
−0.923790 + 0.382900i \(0.874926\pi\)
\(998\) 32.3607i 1.02436i
\(999\) 38.3870 1.21451
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.q.1849.3 4
5.2 odd 4 3850.2.a.bj.1.1 2
5.3 odd 4 154.2.a.d.1.2 2
5.4 even 2 inner 3850.2.c.q.1849.2 4
15.8 even 4 1386.2.a.m.1.2 2
20.3 even 4 1232.2.a.p.1.1 2
35.3 even 12 1078.2.e.n.177.2 4
35.13 even 4 1078.2.a.w.1.1 2
35.18 odd 12 1078.2.e.q.177.1 4
35.23 odd 12 1078.2.e.q.67.1 4
35.33 even 12 1078.2.e.n.67.2 4
40.3 even 4 4928.2.a.bk.1.2 2
40.13 odd 4 4928.2.a.bt.1.1 2
55.43 even 4 1694.2.a.l.1.2 2
105.83 odd 4 9702.2.a.cu.1.1 2
140.83 odd 4 8624.2.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.d.1.2 2 5.3 odd 4
1078.2.a.w.1.1 2 35.13 even 4
1078.2.e.n.67.2 4 35.33 even 12
1078.2.e.n.177.2 4 35.3 even 12
1078.2.e.q.67.1 4 35.23 odd 12
1078.2.e.q.177.1 4 35.18 odd 12
1232.2.a.p.1.1 2 20.3 even 4
1386.2.a.m.1.2 2 15.8 even 4
1694.2.a.l.1.2 2 55.43 even 4
3850.2.a.bj.1.1 2 5.2 odd 4
3850.2.c.q.1849.2 4 5.4 even 2 inner
3850.2.c.q.1849.3 4 1.1 even 1 trivial
4928.2.a.bk.1.2 2 40.3 even 4
4928.2.a.bt.1.1 2 40.13 odd 4
8624.2.a.bf.1.2 2 140.83 odd 4
9702.2.a.cu.1.1 2 105.83 odd 4