Properties

Label 3850.2.c.x.1849.2
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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 770)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 3850.1849
Dual form 3850.2.c.x.1849.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +2.73205i q^{3} -1.00000 q^{4} +2.73205 q^{6} -1.00000i q^{7} +1.00000i q^{8} -4.46410 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +2.73205i q^{3} -1.00000 q^{4} +2.73205 q^{6} -1.00000i q^{7} +1.00000i q^{8} -4.46410 q^{9} +1.00000 q^{11} -2.73205i q^{12} -1.46410i q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.46410i q^{17} +4.46410i q^{18} -6.73205 q^{19} +2.73205 q^{21} -1.00000i q^{22} -8.19615i q^{23} -2.73205 q^{24} -1.46410 q^{26} -4.00000i q^{27} +1.00000i q^{28} +4.73205 q^{29} +2.00000 q^{31} -1.00000i q^{32} +2.73205i q^{33} +3.46410 q^{34} +4.46410 q^{36} -0.732051i q^{37} +6.73205i q^{38} +4.00000 q^{39} -2.19615 q^{41} -2.73205i q^{42} +2.00000i q^{43} -1.00000 q^{44} -8.19615 q^{46} -6.92820i q^{47} +2.73205i q^{48} -1.00000 q^{49} -9.46410 q^{51} +1.46410i q^{52} -7.26795i q^{53} -4.00000 q^{54} +1.00000 q^{56} -18.3923i q^{57} -4.73205i q^{58} -6.92820 q^{59} -4.92820 q^{61} -2.00000i q^{62} +4.46410i q^{63} -1.00000 q^{64} +2.73205 q^{66} +4.00000i q^{67} -3.46410i q^{68} +22.3923 q^{69} +9.46410 q^{71} -4.46410i q^{72} -14.3923i q^{73} -0.732051 q^{74} +6.73205 q^{76} -1.00000i q^{77} -4.00000i q^{78} +12.1962 q^{79} -2.46410 q^{81} +2.19615i q^{82} -16.3923i q^{83} -2.73205 q^{84} +2.00000 q^{86} +12.9282i q^{87} +1.00000i q^{88} -3.46410 q^{89} -1.46410 q^{91} +8.19615i q^{92} +5.46410i q^{93} -6.92820 q^{94} +2.73205 q^{96} -14.5885i q^{97} +1.00000i q^{98} -4.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + 4q^{11} - 4q^{14} + 4q^{16} - 20q^{19} + 4q^{21} - 4q^{24} + 8q^{26} + 12q^{29} + 8q^{31} + 4q^{36} + 16q^{39} + 12q^{41} - 4q^{44} - 12q^{46} - 4q^{49} - 24q^{51} - 16q^{54} + 4q^{56} + 8q^{61} - 4q^{64} + 4q^{66} + 48q^{69} + 24q^{71} + 4q^{74} + 20q^{76} + 28q^{79} + 4q^{81} - 4q^{84} + 8q^{86} + 8q^{91} + 4q^{96} - 4q^{99} + O(q^{100}) \)

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
\(3\) 2.73205i 1.57735i 0.614810 + 0.788675i \(0.289233\pi\)
−0.614810 + 0.788675i \(0.710767\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.73205 1.11536
\(7\) − 1.00000i − 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) −4.46410 −1.48803
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) − 2.73205i − 0.788675i
\(13\) − 1.46410i − 0.406069i −0.979172 0.203034i \(-0.934920\pi\)
0.979172 0.203034i \(-0.0650803\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.46410i 0.840168i 0.907485 + 0.420084i \(0.137999\pi\)
−0.907485 + 0.420084i \(0.862001\pi\)
\(18\) 4.46410i 1.05220i
\(19\) −6.73205 −1.54444 −0.772219 0.635356i \(-0.780853\pi\)
−0.772219 + 0.635356i \(0.780853\pi\)
\(20\) 0 0
\(21\) 2.73205 0.596182
\(22\) − 1.00000i − 0.213201i
\(23\) − 8.19615i − 1.70902i −0.519438 0.854508i \(-0.673859\pi\)
0.519438 0.854508i \(-0.326141\pi\)
\(24\) −2.73205 −0.557678
\(25\) 0 0
\(26\) −1.46410 −0.287134
\(27\) − 4.00000i − 0.769800i
\(28\) 1.00000i 0.188982i
\(29\) 4.73205 0.878720 0.439360 0.898311i \(-0.355205\pi\)
0.439360 + 0.898311i \(0.355205\pi\)
\(30\) 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
\(33\) 2.73205i 0.475589i
\(34\) 3.46410 0.594089
\(35\) 0 0
\(36\) 4.46410 0.744017
\(37\) − 0.732051i − 0.120348i −0.998188 0.0601742i \(-0.980834\pi\)
0.998188 0.0601742i \(-0.0191656\pi\)
\(38\) 6.73205i 1.09208i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −2.19615 −0.342981 −0.171491 0.985186i \(-0.554858\pi\)
−0.171491 + 0.985186i \(0.554858\pi\)
\(42\) − 2.73205i − 0.421565i
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −8.19615 −1.20846
\(47\) − 6.92820i − 1.01058i −0.862949 0.505291i \(-0.831385\pi\)
0.862949 0.505291i \(-0.168615\pi\)
\(48\) 2.73205i 0.394338i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −9.46410 −1.32524
\(52\) 1.46410i 0.203034i
\(53\) − 7.26795i − 0.998330i −0.866507 0.499165i \(-0.833640\pi\)
0.866507 0.499165i \(-0.166360\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) − 18.3923i − 2.43612i
\(58\) − 4.73205i − 0.621349i
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) − 2.00000i − 0.254000i
\(63\) 4.46410i 0.562424i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.73205 0.336292
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 3.46410i − 0.420084i
\(69\) 22.3923 2.69572
\(70\) 0 0
\(71\) 9.46410 1.12318 0.561591 0.827415i \(-0.310189\pi\)
0.561591 + 0.827415i \(0.310189\pi\)
\(72\) − 4.46410i − 0.526099i
\(73\) − 14.3923i − 1.68449i −0.539093 0.842246i \(-0.681233\pi\)
0.539093 0.842246i \(-0.318767\pi\)
\(74\) −0.732051 −0.0850992
\(75\) 0 0
\(76\) 6.73205 0.772219
\(77\) − 1.00000i − 0.113961i
\(78\) − 4.00000i − 0.452911i
\(79\) 12.1962 1.37217 0.686087 0.727519i \(-0.259327\pi\)
0.686087 + 0.727519i \(0.259327\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 2.19615i 0.242524i
\(83\) − 16.3923i − 1.79929i −0.436623 0.899645i \(-0.643826\pi\)
0.436623 0.899645i \(-0.356174\pi\)
\(84\) −2.73205 −0.298091
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 12.9282i 1.38605i
\(88\) 1.00000i 0.106600i
\(89\) −3.46410 −0.367194 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) 0 0
\(91\) −1.46410 −0.153480
\(92\) 8.19615i 0.854508i
\(93\) 5.46410i 0.566601i
\(94\) −6.92820 −0.714590
\(95\) 0 0
\(96\) 2.73205 0.278839
\(97\) − 14.5885i − 1.48123i −0.671928 0.740617i \(-0.734533\pi\)
0.671928 0.740617i \(-0.265467\pi\)
\(98\) 1.00000i 0.101015i
\(99\) −4.46410 −0.448659
\(100\) 0 0
\(101\) −7.85641 −0.781742 −0.390871 0.920446i \(-0.627826\pi\)
−0.390871 + 0.920446i \(0.627826\pi\)
\(102\) 9.46410i 0.937086i
\(103\) 12.3923i 1.22105i 0.791997 + 0.610525i \(0.209042\pi\)
−0.791997 + 0.610525i \(0.790958\pi\)
\(104\) 1.46410 0.143567
\(105\) 0 0
\(106\) −7.26795 −0.705926
\(107\) − 7.85641i − 0.759507i −0.925088 0.379754i \(-0.876009\pi\)
0.925088 0.379754i \(-0.123991\pi\)
\(108\) 4.00000i 0.384900i
\(109\) 15.6603 1.49998 0.749990 0.661449i \(-0.230058\pi\)
0.749990 + 0.661449i \(0.230058\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) − 1.00000i − 0.0944911i
\(113\) 7.85641i 0.739069i 0.929217 + 0.369534i \(0.120483\pi\)
−0.929217 + 0.369534i \(0.879517\pi\)
\(114\) −18.3923 −1.72260
\(115\) 0 0
\(116\) −4.73205 −0.439360
\(117\) 6.53590i 0.604244i
\(118\) 6.92820i 0.637793i
\(119\) 3.46410 0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 4.92820i 0.446179i
\(123\) − 6.00000i − 0.541002i
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 4.46410 0.397694
\(127\) − 1.07180i − 0.0951066i −0.998869 0.0475533i \(-0.984858\pi\)
0.998869 0.0475533i \(-0.0151424\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −5.46410 −0.481087
\(130\) 0 0
\(131\) 5.66025 0.494539 0.247269 0.968947i \(-0.420467\pi\)
0.247269 + 0.968947i \(0.420467\pi\)
\(132\) − 2.73205i − 0.237795i
\(133\) 6.73205i 0.583743i
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −3.46410 −0.297044
\(137\) 0.928203i 0.0793018i 0.999214 + 0.0396509i \(0.0126246\pi\)
−0.999214 + 0.0396509i \(0.987375\pi\)
\(138\) − 22.3923i − 1.90616i
\(139\) −13.6603 −1.15865 −0.579324 0.815097i \(-0.696683\pi\)
−0.579324 + 0.815097i \(0.696683\pi\)
\(140\) 0 0
\(141\) 18.9282 1.59404
\(142\) − 9.46410i − 0.794210i
\(143\) − 1.46410i − 0.122434i
\(144\) −4.46410 −0.372008
\(145\) 0 0
\(146\) −14.3923 −1.19112
\(147\) − 2.73205i − 0.225336i
\(148\) 0.732051i 0.0601742i
\(149\) 7.26795 0.595414 0.297707 0.954657i \(-0.403778\pi\)
0.297707 + 0.954657i \(0.403778\pi\)
\(150\) 0 0
\(151\) 11.1244 0.905287 0.452644 0.891692i \(-0.350481\pi\)
0.452644 + 0.891692i \(0.350481\pi\)
\(152\) − 6.73205i − 0.546041i
\(153\) − 15.4641i − 1.25020i
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) − 4.53590i − 0.362004i −0.983483 0.181002i \(-0.942066\pi\)
0.983483 0.181002i \(-0.0579341\pi\)
\(158\) − 12.1962i − 0.970274i
\(159\) 19.8564 1.57472
\(160\) 0 0
\(161\) −8.19615 −0.645947
\(162\) 2.46410i 0.193598i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 2.19615 0.171491
\(165\) 0 0
\(166\) −16.3923 −1.27229
\(167\) − 13.8564i − 1.07224i −0.844141 0.536120i \(-0.819889\pi\)
0.844141 0.536120i \(-0.180111\pi\)
\(168\) 2.73205i 0.210782i
\(169\) 10.8564 0.835108
\(170\) 0 0
\(171\) 30.0526 2.29818
\(172\) − 2.00000i − 0.152499i
\(173\) 0.928203i 0.0705700i 0.999377 + 0.0352850i \(0.0112339\pi\)
−0.999377 + 0.0352850i \(0.988766\pi\)
\(174\) 12.9282 0.980085
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) − 18.9282i − 1.42273i
\(178\) 3.46410i 0.259645i
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 1.46410i 0.108526i
\(183\) − 13.4641i − 0.995295i
\(184\) 8.19615 0.604228
\(185\) 0 0
\(186\) 5.46410 0.400647
\(187\) 3.46410i 0.253320i
\(188\) 6.92820i 0.505291i
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) − 2.73205i − 0.197169i
\(193\) 12.3923i 0.892018i 0.895029 + 0.446009i \(0.147155\pi\)
−0.895029 + 0.446009i \(0.852845\pi\)
\(194\) −14.5885 −1.04739
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 24.2487i 1.72765i 0.503793 + 0.863825i \(0.331938\pi\)
−0.503793 + 0.863825i \(0.668062\pi\)
\(198\) 4.46410i 0.317250i
\(199\) −2.92820 −0.207575 −0.103787 0.994600i \(-0.533096\pi\)
−0.103787 + 0.994600i \(0.533096\pi\)
\(200\) 0 0
\(201\) −10.9282 −0.770816
\(202\) 7.85641i 0.552775i
\(203\) − 4.73205i − 0.332125i
\(204\) 9.46410 0.662620
\(205\) 0 0
\(206\) 12.3923 0.863413
\(207\) 36.5885i 2.54307i
\(208\) − 1.46410i − 0.101517i
\(209\) −6.73205 −0.465666
\(210\) 0 0
\(211\) 26.9282 1.85381 0.926907 0.375291i \(-0.122457\pi\)
0.926907 + 0.375291i \(0.122457\pi\)
\(212\) 7.26795i 0.499165i
\(213\) 25.8564i 1.77165i
\(214\) −7.85641 −0.537053
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) − 2.00000i − 0.135769i
\(218\) − 15.6603i − 1.06065i
\(219\) 39.3205 2.65703
\(220\) 0 0
\(221\) 5.07180 0.341166
\(222\) − 2.00000i − 0.134231i
\(223\) − 25.4641i − 1.70520i −0.522562 0.852601i \(-0.675024\pi\)
0.522562 0.852601i \(-0.324976\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 7.85641 0.522600
\(227\) − 6.92820i − 0.459841i −0.973209 0.229920i \(-0.926153\pi\)
0.973209 0.229920i \(-0.0738466\pi\)
\(228\) 18.3923i 1.21806i
\(229\) −24.3923 −1.61189 −0.805944 0.591991i \(-0.798342\pi\)
−0.805944 + 0.591991i \(0.798342\pi\)
\(230\) 0 0
\(231\) 2.73205 0.179756
\(232\) 4.73205i 0.310674i
\(233\) − 7.85641i − 0.514690i −0.966320 0.257345i \(-0.917152\pi\)
0.966320 0.257345i \(-0.0828477\pi\)
\(234\) 6.53590 0.427265
\(235\) 0 0
\(236\) 6.92820 0.450988
\(237\) 33.3205i 2.16440i
\(238\) − 3.46410i − 0.224544i
\(239\) −1.26795 −0.0820168 −0.0410084 0.999159i \(-0.513057\pi\)
−0.0410084 + 0.999159i \(0.513057\pi\)
\(240\) 0 0
\(241\) 3.26795 0.210507 0.105254 0.994445i \(-0.466435\pi\)
0.105254 + 0.994445i \(0.466435\pi\)
\(242\) − 1.00000i − 0.0642824i
\(243\) − 18.7321i − 1.20166i
\(244\) 4.92820 0.315496
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 9.85641i 0.627148i
\(248\) 2.00000i 0.127000i
\(249\) 44.7846 2.83811
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) − 4.46410i − 0.281212i
\(253\) − 8.19615i − 0.515288i
\(254\) −1.07180 −0.0672505
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.6603i 1.47589i 0.674863 + 0.737943i \(0.264203\pi\)
−0.674863 + 0.737943i \(0.735797\pi\)
\(258\) 5.46410i 0.340180i
\(259\) −0.732051 −0.0454874
\(260\) 0 0
\(261\) −21.1244 −1.30756
\(262\) − 5.66025i − 0.349692i
\(263\) − 24.0000i − 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) −2.73205 −0.168146
\(265\) 0 0
\(266\) 6.73205 0.412769
\(267\) − 9.46410i − 0.579194i
\(268\) − 4.00000i − 0.244339i
\(269\) −28.3923 −1.73111 −0.865555 0.500814i \(-0.833034\pi\)
−0.865555 + 0.500814i \(0.833034\pi\)
\(270\) 0 0
\(271\) 0.392305 0.0238308 0.0119154 0.999929i \(-0.496207\pi\)
0.0119154 + 0.999929i \(0.496207\pi\)
\(272\) 3.46410i 0.210042i
\(273\) − 4.00000i − 0.242091i
\(274\) 0.928203 0.0560748
\(275\) 0 0
\(276\) −22.3923 −1.34786
\(277\) − 7.07180i − 0.424903i −0.977172 0.212452i \(-0.931855\pi\)
0.977172 0.212452i \(-0.0681448\pi\)
\(278\) 13.6603i 0.819288i
\(279\) −8.92820 −0.534518
\(280\) 0 0
\(281\) 22.3923 1.33581 0.667906 0.744245i \(-0.267191\pi\)
0.667906 + 0.744245i \(0.267191\pi\)
\(282\) − 18.9282i − 1.12716i
\(283\) − 31.7128i − 1.88513i −0.334021 0.942566i \(-0.608406\pi\)
0.334021 0.942566i \(-0.391594\pi\)
\(284\) −9.46410 −0.561591
\(285\) 0 0
\(286\) −1.46410 −0.0865741
\(287\) 2.19615i 0.129635i
\(288\) 4.46410i 0.263050i
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) 39.8564 2.33642
\(292\) 14.3923i 0.842246i
\(293\) 21.4641i 1.25395i 0.779041 + 0.626973i \(0.215706\pi\)
−0.779041 + 0.626973i \(0.784294\pi\)
\(294\) −2.73205 −0.159336
\(295\) 0 0
\(296\) 0.732051 0.0425496
\(297\) − 4.00000i − 0.232104i
\(298\) − 7.26795i − 0.421021i
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) − 11.1244i − 0.640135i
\(303\) − 21.4641i − 1.23308i
\(304\) −6.73205 −0.386110
\(305\) 0 0
\(306\) −15.4641 −0.884024
\(307\) − 24.3923i − 1.39214i −0.717973 0.696071i \(-0.754930\pi\)
0.717973 0.696071i \(-0.245070\pi\)
\(308\) 1.00000i 0.0569803i
\(309\) −33.8564 −1.92602
\(310\) 0 0
\(311\) 24.9282 1.41355 0.706774 0.707439i \(-0.250150\pi\)
0.706774 + 0.707439i \(0.250150\pi\)
\(312\) 4.00000i 0.226455i
\(313\) 22.1962i 1.25460i 0.778777 + 0.627300i \(0.215840\pi\)
−0.778777 + 0.627300i \(0.784160\pi\)
\(314\) −4.53590 −0.255976
\(315\) 0 0
\(316\) −12.1962 −0.686087
\(317\) − 30.5885i − 1.71802i −0.511960 0.859009i \(-0.671080\pi\)
0.511960 0.859009i \(-0.328920\pi\)
\(318\) − 19.8564i − 1.11349i
\(319\) 4.73205 0.264944
\(320\) 0 0
\(321\) 21.4641 1.19801
\(322\) 8.19615i 0.456754i
\(323\) − 23.3205i − 1.29759i
\(324\) 2.46410 0.136895
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 42.7846i 2.36599i
\(328\) − 2.19615i − 0.121262i
\(329\) −6.92820 −0.381964
\(330\) 0 0
\(331\) −18.7846 −1.03250 −0.516248 0.856439i \(-0.672672\pi\)
−0.516248 + 0.856439i \(0.672672\pi\)
\(332\) 16.3923i 0.899645i
\(333\) 3.26795i 0.179083i
\(334\) −13.8564 −0.758189
\(335\) 0 0
\(336\) 2.73205 0.149046
\(337\) − 22.7846i − 1.24116i −0.784144 0.620578i \(-0.786898\pi\)
0.784144 0.620578i \(-0.213102\pi\)
\(338\) − 10.8564i − 0.590511i
\(339\) −21.4641 −1.16577
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) − 30.0526i − 1.62506i
\(343\) 1.00000i 0.0539949i
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) 0.928203 0.0499005
\(347\) 23.0718i 1.23856i 0.785171 + 0.619279i \(0.212575\pi\)
−0.785171 + 0.619279i \(0.787425\pi\)
\(348\) − 12.9282i − 0.693024i
\(349\) 5.60770 0.300173 0.150087 0.988673i \(-0.452045\pi\)
0.150087 + 0.988673i \(0.452045\pi\)
\(350\) 0 0
\(351\) −5.85641 −0.312592
\(352\) − 1.00000i − 0.0533002i
\(353\) 14.1962i 0.755585i 0.925890 + 0.377792i \(0.123317\pi\)
−0.925890 + 0.377792i \(0.876683\pi\)
\(354\) −18.9282 −1.00602
\(355\) 0 0
\(356\) 3.46410 0.183597
\(357\) 9.46410i 0.500893i
\(358\) − 6.00000i − 0.317110i
\(359\) 34.0526 1.79723 0.898613 0.438743i \(-0.144576\pi\)
0.898613 + 0.438743i \(0.144576\pi\)
\(360\) 0 0
\(361\) 26.3205 1.38529
\(362\) − 14.0000i − 0.735824i
\(363\) 2.73205i 0.143395i
\(364\) 1.46410 0.0767398
\(365\) 0 0
\(366\) −13.4641 −0.703780
\(367\) − 12.3923i − 0.646873i −0.946250 0.323437i \(-0.895162\pi\)
0.946250 0.323437i \(-0.104838\pi\)
\(368\) − 8.19615i − 0.427254i
\(369\) 9.80385 0.510368
\(370\) 0 0
\(371\) −7.26795 −0.377333
\(372\) − 5.46410i − 0.283300i
\(373\) 18.3923i 0.952317i 0.879359 + 0.476159i \(0.157971\pi\)
−0.879359 + 0.476159i \(0.842029\pi\)
\(374\) 3.46410 0.179124
\(375\) 0 0
\(376\) 6.92820 0.357295
\(377\) − 6.92820i − 0.356821i
\(378\) 4.00000i 0.205738i
\(379\) −6.14359 −0.315575 −0.157788 0.987473i \(-0.550436\pi\)
−0.157788 + 0.987473i \(0.550436\pi\)
\(380\) 0 0
\(381\) 2.92820 0.150016
\(382\) 12.0000i 0.613973i
\(383\) − 33.4641i − 1.70994i −0.518681 0.854968i \(-0.673577\pi\)
0.518681 0.854968i \(-0.326423\pi\)
\(384\) −2.73205 −0.139419
\(385\) 0 0
\(386\) 12.3923 0.630752
\(387\) − 8.92820i − 0.453846i
\(388\) 14.5885i 0.740617i
\(389\) 1.60770 0.0815134 0.0407567 0.999169i \(-0.487023\pi\)
0.0407567 + 0.999169i \(0.487023\pi\)
\(390\) 0 0
\(391\) 28.3923 1.43586
\(392\) − 1.00000i − 0.0505076i
\(393\) 15.4641i 0.780061i
\(394\) 24.2487 1.22163
\(395\) 0 0
\(396\) 4.46410 0.224330
\(397\) − 30.3923i − 1.52535i −0.646784 0.762673i \(-0.723887\pi\)
0.646784 0.762673i \(-0.276113\pi\)
\(398\) 2.92820i 0.146778i
\(399\) −18.3923 −0.920767
\(400\) 0 0
\(401\) 2.53590 0.126637 0.0633184 0.997993i \(-0.479832\pi\)
0.0633184 + 0.997993i \(0.479832\pi\)
\(402\) 10.9282i 0.545049i
\(403\) − 2.92820i − 0.145864i
\(404\) 7.85641 0.390871
\(405\) 0 0
\(406\) −4.73205 −0.234848
\(407\) − 0.732051i − 0.0362864i
\(408\) − 9.46410i − 0.468543i
\(409\) −10.8756 −0.537766 −0.268883 0.963173i \(-0.586654\pi\)
−0.268883 + 0.963173i \(0.586654\pi\)
\(410\) 0 0
\(411\) −2.53590 −0.125087
\(412\) − 12.3923i − 0.610525i
\(413\) 6.92820i 0.340915i
\(414\) 36.5885 1.79822
\(415\) 0 0
\(416\) −1.46410 −0.0717835
\(417\) − 37.3205i − 1.82759i
\(418\) 6.73205i 0.329275i
\(419\) −30.9282 −1.51094 −0.755471 0.655182i \(-0.772592\pi\)
−0.755471 + 0.655182i \(0.772592\pi\)
\(420\) 0 0
\(421\) −35.8564 −1.74753 −0.873767 0.486344i \(-0.838330\pi\)
−0.873767 + 0.486344i \(0.838330\pi\)
\(422\) − 26.9282i − 1.31084i
\(423\) 30.9282i 1.50378i
\(424\) 7.26795 0.352963
\(425\) 0 0
\(426\) 25.8564 1.25275
\(427\) 4.92820i 0.238492i
\(428\) 7.85641i 0.379754i
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 3.12436 0.150495 0.0752475 0.997165i \(-0.476025\pi\)
0.0752475 + 0.997165i \(0.476025\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) − 18.1962i − 0.874451i −0.899352 0.437226i \(-0.855961\pi\)
0.899352 0.437226i \(-0.144039\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −15.6603 −0.749990
\(437\) 55.1769i 2.63947i
\(438\) − 39.3205i − 1.87881i
\(439\) −26.2487 −1.25278 −0.626391 0.779509i \(-0.715469\pi\)
−0.626391 + 0.779509i \(0.715469\pi\)
\(440\) 0 0
\(441\) 4.46410 0.212576
\(442\) − 5.07180i − 0.241241i
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −25.4641 −1.20576
\(447\) 19.8564i 0.939176i
\(448\) 1.00000i 0.0472456i
\(449\) −40.3923 −1.90623 −0.953115 0.302607i \(-0.902143\pi\)
−0.953115 + 0.302607i \(0.902143\pi\)
\(450\) 0 0
\(451\) −2.19615 −0.103413
\(452\) − 7.85641i − 0.369534i
\(453\) 30.3923i 1.42796i
\(454\) −6.92820 −0.325157
\(455\) 0 0
\(456\) 18.3923 0.861299
\(457\) − 2.00000i − 0.0935561i −0.998905 0.0467780i \(-0.985105\pi\)
0.998905 0.0467780i \(-0.0148953\pi\)
\(458\) 24.3923i 1.13978i
\(459\) 13.8564 0.646762
\(460\) 0 0
\(461\) 22.3923 1.04291 0.521457 0.853278i \(-0.325389\pi\)
0.521457 + 0.853278i \(0.325389\pi\)
\(462\) − 2.73205i − 0.127107i
\(463\) 27.5167i 1.27881i 0.768871 + 0.639404i \(0.220819\pi\)
−0.768871 + 0.639404i \(0.779181\pi\)
\(464\) 4.73205 0.219680
\(465\) 0 0
\(466\) −7.85641 −0.363941
\(467\) 34.0526i 1.57576i 0.615826 + 0.787882i \(0.288822\pi\)
−0.615826 + 0.787882i \(0.711178\pi\)
\(468\) − 6.53590i − 0.302122i
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 12.3923 0.571007
\(472\) − 6.92820i − 0.318896i
\(473\) 2.00000i 0.0919601i
\(474\) 33.3205 1.53046
\(475\) 0 0
\(476\) −3.46410 −0.158777
\(477\) 32.4449i 1.48555i
\(478\) 1.26795i 0.0579946i
\(479\) 32.7846 1.49797 0.748984 0.662589i \(-0.230542\pi\)
0.748984 + 0.662589i \(0.230542\pi\)
\(480\) 0 0
\(481\) −1.07180 −0.0488697
\(482\) − 3.26795i − 0.148851i
\(483\) − 22.3923i − 1.01889i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) −18.7321 −0.849703
\(487\) − 4.19615i − 0.190146i −0.995470 0.0950729i \(-0.969692\pi\)
0.995470 0.0950729i \(-0.0303084\pi\)
\(488\) − 4.92820i − 0.223089i
\(489\) 10.9282 0.494190
\(490\) 0 0
\(491\) −27.7128 −1.25066 −0.625331 0.780360i \(-0.715036\pi\)
−0.625331 + 0.780360i \(0.715036\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 16.3923i 0.738272i
\(494\) 9.85641 0.443461
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) − 9.46410i − 0.424523i
\(498\) − 44.7846i − 2.00685i
\(499\) −12.1436 −0.543622 −0.271811 0.962351i \(-0.587623\pi\)
−0.271811 + 0.962351i \(0.587623\pi\)
\(500\) 0 0
\(501\) 37.8564 1.69130
\(502\) − 12.0000i − 0.535586i
\(503\) 8.78461i 0.391686i 0.980635 + 0.195843i \(0.0627444\pi\)
−0.980635 + 0.195843i \(0.937256\pi\)
\(504\) −4.46410 −0.198847
\(505\) 0 0
\(506\) −8.19615 −0.364363
\(507\) 29.6603i 1.31726i
\(508\) 1.07180i 0.0475533i
\(509\) −11.0718 −0.490749 −0.245374 0.969428i \(-0.578911\pi\)
−0.245374 + 0.969428i \(0.578911\pi\)
\(510\) 0 0
\(511\) −14.3923 −0.636678
\(512\) − 1.00000i − 0.0441942i
\(513\) 26.9282i 1.18891i
\(514\) 23.6603 1.04361
\(515\) 0 0
\(516\) 5.46410 0.240544
\(517\) − 6.92820i − 0.304702i
\(518\) 0.732051i 0.0321645i
\(519\) −2.53590 −0.111314
\(520\) 0 0
\(521\) 10.3923 0.455295 0.227648 0.973744i \(-0.426897\pi\)
0.227648 + 0.973744i \(0.426897\pi\)
\(522\) 21.1244i 0.924588i
\(523\) − 29.1769i − 1.27582i −0.770112 0.637909i \(-0.779800\pi\)
0.770112 0.637909i \(-0.220200\pi\)
\(524\) −5.66025 −0.247269
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 6.92820i 0.301797i
\(528\) 2.73205i 0.118897i
\(529\) −44.1769 −1.92074
\(530\) 0 0
\(531\) 30.9282 1.34217
\(532\) − 6.73205i − 0.291871i
\(533\) 3.21539i 0.139274i
\(534\) −9.46410 −0.409552
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 16.3923i 0.707380i
\(538\) 28.3923i 1.22408i
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −20.7321 −0.891340 −0.445670 0.895197i \(-0.647035\pi\)
−0.445670 + 0.895197i \(0.647035\pi\)
\(542\) − 0.392305i − 0.0168509i
\(543\) 38.2487i 1.64141i
\(544\) 3.46410 0.148522
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) − 28.7846i − 1.23074i −0.788238 0.615371i \(-0.789006\pi\)
0.788238 0.615371i \(-0.210994\pi\)
\(548\) − 0.928203i − 0.0396509i
\(549\) 22.0000 0.938937
\(550\) 0 0
\(551\) −31.8564 −1.35713
\(552\) 22.3923i 0.953080i
\(553\) − 12.1962i − 0.518633i
\(554\) −7.07180 −0.300452
\(555\) 0 0
\(556\) 13.6603 0.579324
\(557\) 25.6077i 1.08503i 0.840045 + 0.542516i \(0.182528\pi\)
−0.840045 + 0.542516i \(0.817472\pi\)
\(558\) 8.92820i 0.377961i
\(559\) 2.92820 0.123850
\(560\) 0 0
\(561\) −9.46410 −0.399575
\(562\) − 22.3923i − 0.944562i
\(563\) 5.07180i 0.213751i 0.994272 + 0.106875i \(0.0340846\pi\)
−0.994272 + 0.106875i \(0.965915\pi\)
\(564\) −18.9282 −0.797021
\(565\) 0 0
\(566\) −31.7128 −1.33299
\(567\) 2.46410i 0.103483i
\(568\) 9.46410i 0.397105i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 3.60770 0.150977 0.0754887 0.997147i \(-0.475948\pi\)
0.0754887 + 0.997147i \(0.475948\pi\)
\(572\) 1.46410i 0.0612172i
\(573\) − 32.7846i − 1.36960i
\(574\) 2.19615 0.0916656
\(575\) 0 0
\(576\) 4.46410 0.186004
\(577\) 34.5885i 1.43994i 0.694007 + 0.719968i \(0.255844\pi\)
−0.694007 + 0.719968i \(0.744156\pi\)
\(578\) − 5.00000i − 0.207973i
\(579\) −33.8564 −1.40702
\(580\) 0 0
\(581\) −16.3923 −0.680067
\(582\) − 39.8564i − 1.65210i
\(583\) − 7.26795i − 0.301008i
\(584\) 14.3923 0.595558
\(585\) 0 0
\(586\) 21.4641 0.886674
\(587\) 11.4115i 0.471005i 0.971874 + 0.235502i \(0.0756735\pi\)
−0.971874 + 0.235502i \(0.924326\pi\)
\(588\) 2.73205i 0.112668i
\(589\) −13.4641 −0.554779
\(590\) 0 0
\(591\) −66.2487 −2.72511
\(592\) − 0.732051i − 0.0300871i
\(593\) 0.248711i 0.0102133i 0.999987 + 0.00510667i \(0.00162551\pi\)
−0.999987 + 0.00510667i \(0.998374\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −7.26795 −0.297707
\(597\) − 8.00000i − 0.327418i
\(598\) 12.0000i 0.490716i
\(599\) −37.1769 −1.51901 −0.759504 0.650503i \(-0.774558\pi\)
−0.759504 + 0.650503i \(0.774558\pi\)
\(600\) 0 0
\(601\) −5.51666 −0.225029 −0.112515 0.993650i \(-0.535891\pi\)
−0.112515 + 0.993650i \(0.535891\pi\)
\(602\) − 2.00000i − 0.0815139i
\(603\) − 17.8564i − 0.727169i
\(604\) −11.1244 −0.452644
\(605\) 0 0
\(606\) −21.4641 −0.871920
\(607\) − 20.9282i − 0.849450i −0.905323 0.424725i \(-0.860371\pi\)
0.905323 0.424725i \(-0.139629\pi\)
\(608\) 6.73205i 0.273021i
\(609\) 12.9282 0.523877
\(610\) 0 0
\(611\) −10.1436 −0.410366
\(612\) 15.4641i 0.625099i
\(613\) − 35.1769i − 1.42078i −0.703807 0.710391i \(-0.748518\pi\)
0.703807 0.710391i \(-0.251482\pi\)
\(614\) −24.3923 −0.984393
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) − 17.3205i − 0.697297i −0.937253 0.348649i \(-0.886641\pi\)
0.937253 0.348649i \(-0.113359\pi\)
\(618\) 33.8564i 1.36190i
\(619\) −28.7846 −1.15695 −0.578476 0.815700i \(-0.696352\pi\)
−0.578476 + 0.815700i \(0.696352\pi\)
\(620\) 0 0
\(621\) −32.7846 −1.31560
\(622\) − 24.9282i − 0.999530i
\(623\) 3.46410i 0.138786i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 22.1962 0.887137
\(627\) − 18.3923i − 0.734518i
\(628\) 4.53590i 0.181002i
\(629\) 2.53590 0.101113
\(630\) 0 0
\(631\) −46.9282 −1.86818 −0.934091 0.357035i \(-0.883788\pi\)
−0.934091 + 0.357035i \(0.883788\pi\)
\(632\) 12.1962i 0.485137i
\(633\) 73.5692i 2.92411i
\(634\) −30.5885 −1.21482
\(635\) 0 0
\(636\) −19.8564 −0.787358
\(637\) 1.46410i 0.0580098i
\(638\) − 4.73205i − 0.187344i
\(639\) −42.2487 −1.67133
\(640\) 0 0
\(641\) −35.5692 −1.40490 −0.702450 0.711733i \(-0.747910\pi\)
−0.702450 + 0.711733i \(0.747910\pi\)
\(642\) − 21.4641i − 0.847121i
\(643\) 33.2679i 1.31196i 0.754778 + 0.655980i \(0.227744\pi\)
−0.754778 + 0.655980i \(0.772256\pi\)
\(644\) 8.19615 0.322974
\(645\) 0 0
\(646\) −23.3205 −0.917533
\(647\) 26.5359i 1.04323i 0.853180 + 0.521617i \(0.174671\pi\)
−0.853180 + 0.521617i \(0.825329\pi\)
\(648\) − 2.46410i − 0.0967991i
\(649\) −6.92820 −0.271956
\(650\) 0 0
\(651\) 5.46410 0.214155
\(652\) 4.00000i 0.156652i
\(653\) 11.6603i 0.456301i 0.973626 + 0.228151i \(0.0732678\pi\)
−0.973626 + 0.228151i \(0.926732\pi\)
\(654\) 42.7846 1.67301
\(655\) 0 0
\(656\) −2.19615 −0.0857453
\(657\) 64.2487i 2.50658i
\(658\) 6.92820i 0.270089i
\(659\) −30.2487 −1.17832 −0.589161 0.808015i \(-0.700542\pi\)
−0.589161 + 0.808015i \(0.700542\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 18.7846i 0.730085i
\(663\) 13.8564i 0.538138i
\(664\) 16.3923 0.636145
\(665\) 0 0
\(666\) 3.26795 0.126630
\(667\) − 38.7846i − 1.50175i
\(668\) 13.8564i 0.536120i
\(669\) 69.5692 2.68970
\(670\) 0 0
\(671\) −4.92820 −0.190251
\(672\) − 2.73205i − 0.105391i
\(673\) 2.00000i 0.0770943i 0.999257 + 0.0385472i \(0.0122730\pi\)
−0.999257 + 0.0385472i \(0.987727\pi\)
\(674\) −22.7846 −0.877630
\(675\) 0 0
\(676\) −10.8564 −0.417554
\(677\) 4.14359i 0.159251i 0.996825 + 0.0796256i \(0.0253725\pi\)
−0.996825 + 0.0796256i \(0.974628\pi\)
\(678\) 21.4641i 0.824324i
\(679\) −14.5885 −0.559854
\(680\) 0 0
\(681\) 18.9282 0.725330
\(682\) − 2.00000i − 0.0765840i
\(683\) 6.24871i 0.239100i 0.992828 + 0.119550i \(0.0381452\pi\)
−0.992828 + 0.119550i \(0.961855\pi\)
\(684\) −30.0526 −1.14909
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) − 66.6410i − 2.54251i
\(688\) 2.00000i 0.0762493i
\(689\) −10.6410 −0.405390
\(690\) 0 0
\(691\) −13.4641 −0.512199 −0.256099 0.966650i \(-0.582437\pi\)
−0.256099 + 0.966650i \(0.582437\pi\)
\(692\) − 0.928203i − 0.0352850i
\(693\) 4.46410i 0.169577i
\(694\) 23.0718 0.875793
\(695\) 0 0
\(696\) −12.9282 −0.490042
\(697\) − 7.60770i − 0.288162i
\(698\) − 5.60770i − 0.212254i
\(699\) 21.4641 0.811847
\(700\) 0 0
\(701\) 41.9090 1.58288 0.791440 0.611247i \(-0.209332\pi\)
0.791440 + 0.611247i \(0.209332\pi\)
\(702\) 5.85641i 0.221036i
\(703\) 4.92820i 0.185871i
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 14.1962 0.534279
\(707\) 7.85641i 0.295471i
\(708\) 18.9282i 0.711365i
\(709\) −25.3205 −0.950932 −0.475466 0.879734i \(-0.657720\pi\)
−0.475466 + 0.879734i \(0.657720\pi\)
\(710\) 0 0
\(711\) −54.4449 −2.04184
\(712\) − 3.46410i − 0.129823i
\(713\) − 16.3923i − 0.613897i
\(714\) 9.46410 0.354185
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) − 3.46410i − 0.129369i
\(718\) − 34.0526i − 1.27083i
\(719\) 27.7128 1.03351 0.516757 0.856132i \(-0.327139\pi\)
0.516757 + 0.856132i \(0.327139\pi\)
\(720\) 0 0
\(721\) 12.3923 0.461514
\(722\) − 26.3205i − 0.979548i
\(723\) 8.92820i 0.332043i
\(724\) −14.0000 −0.520306
\(725\) 0 0
\(726\) 2.73205 0.101396
\(727\) 4.00000i 0.148352i 0.997245 + 0.0741759i \(0.0236326\pi\)
−0.997245 + 0.0741759i \(0.976367\pi\)
\(728\) − 1.46410i − 0.0542632i
\(729\) 43.7846 1.62165
\(730\) 0 0
\(731\) −6.92820 −0.256249
\(732\) 13.4641i 0.497648i
\(733\) 26.0000i 0.960332i 0.877178 + 0.480166i \(0.159424\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(734\) −12.3923 −0.457408
\(735\) 0 0
\(736\) −8.19615 −0.302114
\(737\) 4.00000i 0.147342i
\(738\) − 9.80385i − 0.360885i
\(739\) −11.7128 −0.430863 −0.215431 0.976519i \(-0.569116\pi\)
−0.215431 + 0.976519i \(0.569116\pi\)
\(740\) 0 0
\(741\) −26.9282 −0.989232
\(742\) 7.26795i 0.266815i
\(743\) − 32.7846i − 1.20275i −0.798967 0.601375i \(-0.794620\pi\)
0.798967 0.601375i \(-0.205380\pi\)
\(744\) −5.46410 −0.200324
\(745\) 0 0
\(746\) 18.3923 0.673390
\(747\) 73.1769i 2.67740i
\(748\) − 3.46410i − 0.126660i
\(749\) −7.85641 −0.287067
\(750\) 0 0
\(751\) −11.6077 −0.423571 −0.211785 0.977316i \(-0.567928\pi\)
−0.211785 + 0.977316i \(0.567928\pi\)
\(752\) − 6.92820i − 0.252646i
\(753\) 32.7846i 1.19474i
\(754\) −6.92820 −0.252310
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 43.3731i 1.57642i 0.615406 + 0.788210i \(0.288992\pi\)
−0.615406 + 0.788210i \(0.711008\pi\)
\(758\) 6.14359i 0.223145i
\(759\) 22.3923 0.812789
\(760\) 0 0
\(761\) −12.3397 −0.447315 −0.223658 0.974668i \(-0.571800\pi\)
−0.223658 + 0.974668i \(0.571800\pi\)
\(762\) − 2.92820i − 0.106078i
\(763\) − 15.6603i − 0.566939i
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) −33.4641 −1.20911
\(767\) 10.1436i 0.366264i
\(768\) 2.73205i 0.0985844i
\(769\) 18.1962 0.656170 0.328085 0.944648i \(-0.393597\pi\)
0.328085 + 0.944648i \(0.393597\pi\)
\(770\) 0 0
\(771\) −64.6410 −2.32799
\(772\) − 12.3923i − 0.446009i
\(773\) − 0.928203i − 0.0333851i −0.999861 0.0166926i \(-0.994686\pi\)
0.999861 0.0166926i \(-0.00531366\pi\)
\(774\) −8.92820 −0.320918
\(775\) 0 0
\(776\) 14.5885 0.523695
\(777\) − 2.00000i − 0.0717496i
\(778\) − 1.60770i − 0.0576387i
\(779\) 14.7846 0.529714
\(780\) 0 0
\(781\) 9.46410 0.338652
\(782\) − 28.3923i − 1.01531i
\(783\) − 18.9282i − 0.676439i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 15.4641 0.551586
\(787\) − 18.1436i − 0.646749i −0.946271 0.323375i \(-0.895183\pi\)
0.946271 0.323375i \(-0.104817\pi\)
\(788\) − 24.2487i − 0.863825i
\(789\) 65.5692 2.33433
\(790\) 0 0
\(791\) 7.85641 0.279342
\(792\) − 4.46410i − 0.158625i
\(793\) 7.21539i 0.256226i
\(794\) −30.3923 −1.07858
\(795\) 0 0
\(796\) 2.92820 0.103787
\(797\) 25.6077i 0.907071i 0.891238 + 0.453536i \(0.149837\pi\)
−0.891238 + 0.453536i \(0.850163\pi\)
\(798\) 18.3923i 0.651081i
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 15.4641 0.546397
\(802\) − 2.53590i − 0.0895457i
\(803\) − 14.3923i − 0.507893i
\(804\) 10.9282 0.385408
\(805\) 0 0
\(806\) −2.92820 −0.103142
\(807\) − 77.5692i − 2.73057i
\(808\) − 7.85641i − 0.276387i
\(809\) 31.1769 1.09612 0.548061 0.836438i \(-0.315366\pi\)
0.548061 + 0.836438i \(0.315366\pi\)
\(810\) 0 0
\(811\) −43.1244 −1.51430 −0.757150 0.653241i \(-0.773409\pi\)
−0.757150 + 0.653241i \(0.773409\pi\)
\(812\) 4.73205i 0.166062i
\(813\) 1.07180i 0.0375896i
\(814\) −0.732051 −0.0256584
\(815\) 0 0
\(816\) −9.46410 −0.331310
\(817\) − 13.4641i − 0.471049i
\(818\) 10.8756i 0.380258i
\(819\) 6.53590 0.228383
\(820\) 0 0
\(821\) 17.9090 0.625027 0.312514 0.949913i \(-0.398829\pi\)
0.312514 + 0.949913i \(0.398829\pi\)
\(822\) 2.53590i 0.0884496i
\(823\) − 0.875644i − 0.0305230i −0.999884 0.0152615i \(-0.995142\pi\)
0.999884 0.0152615i \(-0.00485808\pi\)
\(824\) −12.3923 −0.431706
\(825\) 0 0
\(826\) 6.92820 0.241063
\(827\) 25.8564i 0.899115i 0.893251 + 0.449558i \(0.148418\pi\)
−0.893251 + 0.449558i \(0.851582\pi\)
\(828\) − 36.5885i − 1.27154i
\(829\) −2.24871 −0.0781010 −0.0390505 0.999237i \(-0.512433\pi\)
−0.0390505 + 0.999237i \(0.512433\pi\)
\(830\) 0 0
\(831\) 19.3205 0.670221
\(832\) 1.46410i 0.0507586i
\(833\) − 3.46410i − 0.120024i
\(834\) −37.3205 −1.29230
\(835\) 0 0
\(836\) 6.73205 0.232833
\(837\) − 8.00000i − 0.276520i
\(838\) 30.9282i 1.06840i
\(839\) 31.8564 1.09981 0.549903 0.835229i \(-0.314665\pi\)
0.549903 + 0.835229i \(0.314665\pi\)
\(840\) 0 0
\(841\) −6.60770 −0.227852
\(842\) 35.8564i 1.23569i
\(843\) 61.1769i 2.10704i
\(844\) −26.9282 −0.926907
\(845\) 0 0
\(846\) 30.9282 1.06333
\(847\) − 1.00000i − 0.0343604i
\(848\) − 7.26795i − 0.249582i
\(849\) 86.6410 2.97351
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) − 25.8564i − 0.885826i
\(853\) − 54.7846i − 1.87579i −0.346920 0.937895i \(-0.612773\pi\)
0.346920 0.937895i \(-0.387227\pi\)
\(854\) 4.92820 0.168640
\(855\) 0 0
\(856\) 7.85641 0.268526
\(857\) 27.4641i 0.938156i 0.883157 + 0.469078i \(0.155414\pi\)
−0.883157 + 0.469078i \(0.844586\pi\)
\(858\) − 4.00000i − 0.136558i
\(859\) 12.7846 0.436205 0.218103 0.975926i \(-0.430013\pi\)
0.218103 + 0.975926i \(0.430013\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) − 3.12436i − 0.106416i
\(863\) − 7.51666i − 0.255870i −0.991783 0.127935i \(-0.959165\pi\)
0.991783 0.127935i \(-0.0408349\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) −18.1962 −0.618330
\(867\) 13.6603i 0.463927i
\(868\) 2.00000i 0.0678844i
\(869\) 12.1962 0.413726
\(870\) 0 0
\(871\) 5.85641 0.198437
\(872\) 15.6603i 0.530323i
\(873\) 65.1244i 2.20413i
\(874\) 55.1769 1.86639
\(875\) 0 0
\(876\) −39.3205 −1.32852
\(877\) 21.3205i 0.719942i 0.932963 + 0.359971i \(0.117213\pi\)
−0.932963 + 0.359971i \(0.882787\pi\)
\(878\) 26.2487i 0.885851i
\(879\) −58.6410 −1.97791
\(880\) 0 0
\(881\) 11.0718 0.373018 0.186509 0.982453i \(-0.440283\pi\)
0.186509 + 0.982453i \(0.440283\pi\)
\(882\) − 4.46410i − 0.150314i
\(883\) − 35.6077i − 1.19829i −0.800639 0.599147i \(-0.795506\pi\)
0.800639 0.599147i \(-0.204494\pi\)
\(884\) −5.07180 −0.170583
\(885\) 0 0
\(886\) 0 0
\(887\) − 13.8564i − 0.465253i −0.972566 0.232626i \(-0.925268\pi\)
0.972566 0.232626i \(-0.0747319\pi\)
\(888\) 2.00000i 0.0671156i
\(889\) −1.07180 −0.0359469
\(890\) 0 0
\(891\) −2.46410 −0.0825505
\(892\) 25.4641i 0.852601i
\(893\) 46.6410i 1.56078i
\(894\) 19.8564 0.664098
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) − 32.7846i − 1.09465i
\(898\) 40.3923i 1.34791i
\(899\) 9.46410 0.315645
\(900\) 0 0
\(901\) 25.1769 0.838765
\(902\) 2.19615i 0.0731239i
\(903\) 5.46410i 0.181834i
\(904\) −7.85641 −0.261300
\(905\) 0 0
\(906\) 30.3923 1.00972
\(907\) 31.0333i 1.03044i 0.857057 + 0.515222i \(0.172291\pi\)
−0.857057 + 0.515222i \(0.827709\pi\)
\(908\) 6.92820i 0.229920i
\(909\) 35.0718 1.16326
\(910\) 0 0
\(911\) 9.46410 0.313560 0.156780 0.987634i \(-0.449889\pi\)
0.156780 + 0.987634i \(0.449889\pi\)
\(912\) − 18.3923i − 0.609030i
\(913\) − 16.3923i − 0.542506i
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 24.3923 0.805944
\(917\) − 5.66025i − 0.186918i
\(918\) − 13.8564i − 0.457330i
\(919\) 25.3731 0.836980 0.418490 0.908221i \(-0.362559\pi\)
0.418490 + 0.908221i \(0.362559\pi\)
\(920\) 0 0
\(921\) 66.6410 2.19590
\(922\) − 22.3923i − 0.737451i
\(923\) − 13.8564i − 0.456089i
\(924\) −2.73205 −0.0898779
\(925\) 0 0
\(926\) 27.5167 0.904254
\(927\) − 55.3205i − 1.81696i
\(928\) − 4.73205i − 0.155337i
\(929\) 25.6077 0.840161 0.420081 0.907487i \(-0.362002\pi\)
0.420081 + 0.907487i \(0.362002\pi\)
\(930\) 0 0
\(931\) 6.73205 0.220634
\(932\) 7.85641i 0.257345i
\(933\) 68.1051i 2.22966i
\(934\) 34.0526 1.11423
\(935\) 0 0
\(936\) −6.53590 −0.213633
\(937\) 1.21539i 0.0397051i 0.999803 + 0.0198525i \(0.00631967\pi\)
−0.999803 + 0.0198525i \(0.993680\pi\)
\(938\) − 4.00000i − 0.130605i
\(939\) −60.6410 −1.97894
\(940\) 0 0
\(941\) −14.7846 −0.481965 −0.240982 0.970530i \(-0.577470\pi\)
−0.240982 + 0.970530i \(0.577470\pi\)
\(942\) − 12.3923i − 0.403763i
\(943\) 18.0000i 0.586161i
\(944\) −6.92820 −0.225494
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) − 47.3205i − 1.53771i −0.639423 0.768855i \(-0.720827\pi\)
0.639423 0.768855i \(-0.279173\pi\)
\(948\) − 33.3205i − 1.08220i
\(949\) −21.0718 −0.684019
\(950\) 0 0
\(951\) 83.5692 2.70992
\(952\) 3.46410i 0.112272i
\(953\) 26.5359i 0.859582i 0.902928 + 0.429791i \(0.141413\pi\)
−0.902928 + 0.429791i \(0.858587\pi\)
\(954\) 32.4449 1.05044
\(955\) 0 0
\(956\) 1.26795 0.0410084
\(957\) 12.9282i 0.417909i
\(958\) − 32.7846i − 1.05922i
\(959\) 0.928203 0.0299732
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 1.07180i 0.0345561i
\(963\) 35.0718i 1.13017i
\(964\) −3.26795 −0.105254
\(965\) 0 0
\(966\) −22.3923 −0.720461
\(967\) − 26.9282i − 0.865953i −0.901405 0.432976i \(-0.857463\pi\)
0.901405 0.432976i \(-0.142537\pi\)
\(968\) 1.00000i 0.0321412i
\(969\) 63.7128 2.04675
\(970\) 0 0
\(971\) 42.9282 1.37763 0.688816 0.724936i \(-0.258131\pi\)
0.688816 + 0.724936i \(0.258131\pi\)
\(972\) 18.7321i 0.600831i
\(973\) 13.6603i 0.437928i
\(974\) −4.19615 −0.134453
\(975\) 0 0
\(976\) −4.92820 −0.157748
\(977\) 33.7128i 1.07857i 0.842124 + 0.539284i \(0.181305\pi\)
−0.842124 + 0.539284i \(0.818695\pi\)
\(978\) − 10.9282i − 0.349445i
\(979\) −3.46410 −0.110713
\(980\) 0 0
\(981\) −69.9090 −2.23202
\(982\) 27.7128i 0.884351i
\(983\) − 37.1769i − 1.18576i −0.805291 0.592880i \(-0.797991\pi\)
0.805291 0.592880i \(-0.202009\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 16.3923 0.522037
\(987\) − 18.9282i − 0.602491i
\(988\) − 9.85641i − 0.313574i
\(989\) 16.3923 0.521245
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) − 2.00000i − 0.0635001i
\(993\) − 51.3205i − 1.62861i
\(994\) −9.46410 −0.300183
\(995\) 0 0
\(996\) −44.7846 −1.41905
\(997\) − 17.7128i − 0.560970i −0.959858 0.280485i \(-0.909505\pi\)
0.959858 0.280485i \(-0.0904954\pi\)
\(998\) 12.1436i 0.384399i
\(999\) −2.92820 −0.0926443
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.x.1849.2 4
5.2 odd 4 770.2.a.j.1.2 2
5.3 odd 4 3850.2.a.bd.1.1 2
5.4 even 2 inner 3850.2.c.x.1849.3 4
15.2 even 4 6930.2.a.bv.1.1 2
20.7 even 4 6160.2.a.t.1.1 2
35.27 even 4 5390.2.a.bs.1.1 2
55.32 even 4 8470.2.a.br.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.2 2 5.2 odd 4
3850.2.a.bd.1.1 2 5.3 odd 4
3850.2.c.x.1849.2 4 1.1 even 1 trivial
3850.2.c.x.1849.3 4 5.4 even 2 inner
5390.2.a.bs.1.1 2 35.27 even 4
6160.2.a.t.1.1 2 20.7 even 4
6930.2.a.bv.1.1 2 15.2 even 4
8470.2.a.br.1.2 2 55.32 even 4