Properties

Label 3850.2.c.x.1849.1
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: \( x^{4} - x^{2} + 1 \) 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 770)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 3850.1849
Dual form 3850.2.c.x.1849.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -0.732051i q^{3} -1.00000 q^{4} -0.732051 q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.46410 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -0.732051i q^{3} -1.00000 q^{4} -0.732051 q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.46410 q^{9} +1.00000 q^{11} +0.732051i q^{12} +5.46410i q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.46410i q^{17} -2.46410i q^{18} -3.26795 q^{19} -0.732051 q^{21} -1.00000i q^{22} +2.19615i q^{23} +0.732051 q^{24} +5.46410 q^{26} -4.00000i q^{27} +1.00000i q^{28} +1.26795 q^{29} +2.00000 q^{31} -1.00000i q^{32} -0.732051i q^{33} -3.46410 q^{34} -2.46410 q^{36} +2.73205i q^{37} +3.26795i q^{38} +4.00000 q^{39} +8.19615 q^{41} +0.732051i q^{42} +2.00000i q^{43} -1.00000 q^{44} +2.19615 q^{46} +6.92820i q^{47} -0.732051i q^{48} -1.00000 q^{49} -2.53590 q^{51} -5.46410i q^{52} -10.7321i q^{53} -4.00000 q^{54} +1.00000 q^{56} +2.39230i q^{57} -1.26795i q^{58} +6.92820 q^{59} +8.92820 q^{61} -2.00000i q^{62} -2.46410i q^{63} -1.00000 q^{64} -0.732051 q^{66} +4.00000i q^{67} +3.46410i q^{68} +1.60770 q^{69} +2.53590 q^{71} +2.46410i q^{72} +6.39230i q^{73} +2.73205 q^{74} +3.26795 q^{76} -1.00000i q^{77} -4.00000i q^{78} +1.80385 q^{79} +4.46410 q^{81} -8.19615i q^{82} +4.39230i q^{83} +0.732051 q^{84} +2.00000 q^{86} -0.928203i q^{87} +1.00000i q^{88} +3.46410 q^{89} +5.46410 q^{91} -2.19615i q^{92} -1.46410i q^{93} +6.92820 q^{94} -0.732051 q^{96} +16.5885i q^{97} +1.00000i q^{98} +2.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 4 q^{11} - 4 q^{14} + 4 q^{16} - 20 q^{19} + 4 q^{21} - 4 q^{24} + 8 q^{26} + 12 q^{29} + 8 q^{31} + 4 q^{36} + 16 q^{39} + 12 q^{41} - 4 q^{44} - 12 q^{46} - 4 q^{49} - 24 q^{51} - 16 q^{54} + 4 q^{56} + 8 q^{61} - 4 q^{64} + 4 q^{66} + 48 q^{69} + 24 q^{71} + 4 q^{74} + 20 q^{76} + 28 q^{79} + 4 q^{81} - 4 q^{84} + 8 q^{86} + 8 q^{91} + 4 q^{96} - 4 q^{99}+O(q^{100}) \) 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
\(3\) − 0.732051i − 0.422650i −0.977416 0.211325i \(-0.932222\pi\)
0.977416 0.211325i \(-0.0677778\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −0.732051 −0.298858
\(7\) − 1.00000i − 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) 2.46410 0.821367
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0.732051i 0.211325i
\(13\) 5.46410i 1.51547i 0.652563 + 0.757735i \(0.273694\pi\)
−0.652563 + 0.757735i \(0.726306\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.46410i − 0.840168i −0.907485 0.420084i \(-0.862001\pi\)
0.907485 0.420084i \(-0.137999\pi\)
\(18\) − 2.46410i − 0.580794i
\(19\) −3.26795 −0.749719 −0.374859 0.927082i \(-0.622309\pi\)
−0.374859 + 0.927082i \(0.622309\pi\)
\(20\) 0 0
\(21\) −0.732051 −0.159747
\(22\) − 1.00000i − 0.213201i
\(23\) 2.19615i 0.457929i 0.973435 + 0.228965i \(0.0735340\pi\)
−0.973435 + 0.228965i \(0.926466\pi\)
\(24\) 0.732051 0.149429
\(25\) 0 0
\(26\) 5.46410 1.07160
\(27\) − 4.00000i − 0.769800i
\(28\) 1.00000i 0.188982i
\(29\) 1.26795 0.235452 0.117726 0.993046i \(-0.462440\pi\)
0.117726 + 0.993046i \(0.462440\pi\)
\(30\) 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
\(33\) − 0.732051i − 0.127434i
\(34\) −3.46410 −0.594089
\(35\) 0 0
\(36\) −2.46410 −0.410684
\(37\) 2.73205i 0.449146i 0.974457 + 0.224573i \(0.0720988\pi\)
−0.974457 + 0.224573i \(0.927901\pi\)
\(38\) 3.26795i 0.530131i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 8.19615 1.28002 0.640012 0.768365i \(-0.278929\pi\)
0.640012 + 0.768365i \(0.278929\pi\)
\(42\) 0.732051i 0.112958i
\(43\) 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
\(46\) 2.19615 0.323805
\(47\) 6.92820i 1.01058i 0.862949 + 0.505291i \(0.168615\pi\)
−0.862949 + 0.505291i \(0.831385\pi\)
\(48\) − 0.732051i − 0.105662i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −2.53590 −0.355097
\(52\) − 5.46410i − 0.757735i
\(53\) − 10.7321i − 1.47416i −0.675805 0.737080i \(-0.736204\pi\)
0.675805 0.737080i \(-0.263796\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 2.39230i 0.316869i
\(58\) − 1.26795i − 0.166490i
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) − 2.00000i − 0.254000i
\(63\) − 2.46410i − 0.310448i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −0.732051 −0.0901092
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 3.46410i 0.420084i
\(69\) 1.60770 0.193544
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 2.46410i 0.290397i
\(73\) 6.39230i 0.748163i 0.927396 + 0.374081i \(0.122042\pi\)
−0.927396 + 0.374081i \(0.877958\pi\)
\(74\) 2.73205 0.317594
\(75\) 0 0
\(76\) 3.26795 0.374859
\(77\) − 1.00000i − 0.113961i
\(78\) − 4.00000i − 0.452911i
\(79\) 1.80385 0.202949 0.101474 0.994838i \(-0.467644\pi\)
0.101474 + 0.994838i \(0.467644\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) − 8.19615i − 0.905114i
\(83\) 4.39230i 0.482118i 0.970510 + 0.241059i \(0.0774947\pi\)
−0.970510 + 0.241059i \(0.922505\pi\)
\(84\) 0.732051 0.0798733
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) − 0.928203i − 0.0995138i
\(88\) 1.00000i 0.106600i
\(89\) 3.46410 0.367194 0.183597 0.983002i \(-0.441226\pi\)
0.183597 + 0.983002i \(0.441226\pi\)
\(90\) 0 0
\(91\) 5.46410 0.572793
\(92\) − 2.19615i − 0.228965i
\(93\) − 1.46410i − 0.151820i
\(94\) 6.92820 0.714590
\(95\) 0 0
\(96\) −0.732051 −0.0747146
\(97\) 16.5885i 1.68430i 0.539241 + 0.842151i \(0.318711\pi\)
−0.539241 + 0.842151i \(0.681289\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 2.46410 0.247652
\(100\) 0 0
\(101\) 19.8564 1.97579 0.987893 0.155136i \(-0.0495815\pi\)
0.987893 + 0.155136i \(0.0495815\pi\)
\(102\) 2.53590i 0.251091i
\(103\) − 8.39230i − 0.826918i −0.910523 0.413459i \(-0.864320\pi\)
0.910523 0.413459i \(-0.135680\pi\)
\(104\) −5.46410 −0.535799
\(105\) 0 0
\(106\) −10.7321 −1.04239
\(107\) 19.8564i 1.91959i 0.280700 + 0.959796i \(0.409433\pi\)
−0.280700 + 0.959796i \(0.590567\pi\)
\(108\) 4.00000i 0.384900i
\(109\) −1.66025 −0.159023 −0.0795117 0.996834i \(-0.525336\pi\)
−0.0795117 + 0.996834i \(0.525336\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) − 1.00000i − 0.0944911i
\(113\) − 19.8564i − 1.86793i −0.357360 0.933967i \(-0.616323\pi\)
0.357360 0.933967i \(-0.383677\pi\)
\(114\) 2.39230 0.224060
\(115\) 0 0
\(116\) −1.26795 −0.117726
\(117\) 13.4641i 1.24476i
\(118\) − 6.92820i − 0.637793i
\(119\) −3.46410 −0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 8.92820i − 0.808322i
\(123\) − 6.00000i − 0.541002i
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) −2.46410 −0.219520
\(127\) − 14.9282i − 1.32466i −0.749211 0.662332i \(-0.769567\pi\)
0.749211 0.662332i \(-0.230433\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 1.46410 0.128907
\(130\) 0 0
\(131\) −11.6603 −1.01876 −0.509381 0.860541i \(-0.670125\pi\)
−0.509381 + 0.860541i \(0.670125\pi\)
\(132\) 0.732051i 0.0637168i
\(133\) 3.26795i 0.283367i
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 3.46410 0.297044
\(137\) − 12.9282i − 1.10453i −0.833668 0.552265i \(-0.813763\pi\)
0.833668 0.552265i \(-0.186237\pi\)
\(138\) − 1.60770i − 0.136856i
\(139\) 3.66025 0.310459 0.155229 0.987878i \(-0.450388\pi\)
0.155229 + 0.987878i \(0.450388\pi\)
\(140\) 0 0
\(141\) 5.07180 0.427122
\(142\) − 2.53590i − 0.212808i
\(143\) 5.46410i 0.456931i
\(144\) 2.46410 0.205342
\(145\) 0 0
\(146\) 6.39230 0.529031
\(147\) 0.732051i 0.0603785i
\(148\) − 2.73205i − 0.224573i
\(149\) 10.7321 0.879204 0.439602 0.898193i \(-0.355120\pi\)
0.439602 + 0.898193i \(0.355120\pi\)
\(150\) 0 0
\(151\) −13.1244 −1.06804 −0.534022 0.845470i \(-0.679320\pi\)
−0.534022 + 0.845470i \(0.679320\pi\)
\(152\) − 3.26795i − 0.265066i
\(153\) − 8.53590i − 0.690086i
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) − 11.4641i − 0.914935i −0.889226 0.457467i \(-0.848757\pi\)
0.889226 0.457467i \(-0.151243\pi\)
\(158\) − 1.80385i − 0.143506i
\(159\) −7.85641 −0.623054
\(160\) 0 0
\(161\) 2.19615 0.173081
\(162\) − 4.46410i − 0.350733i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) −8.19615 −0.640012
\(165\) 0 0
\(166\) 4.39230 0.340909
\(167\) 13.8564i 1.07224i 0.844141 + 0.536120i \(0.180111\pi\)
−0.844141 + 0.536120i \(0.819889\pi\)
\(168\) − 0.732051i − 0.0564789i
\(169\) −16.8564 −1.29665
\(170\) 0 0
\(171\) −8.05256 −0.615795
\(172\) − 2.00000i − 0.152499i
\(173\) − 12.9282i − 0.982913i −0.870902 0.491457i \(-0.836465\pi\)
0.870902 0.491457i \(-0.163535\pi\)
\(174\) −0.928203 −0.0703669
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) − 5.07180i − 0.381220i
\(178\) − 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 \(0.325818\pi\)
\(182\) − 5.46410i − 0.405026i
\(183\) − 6.53590i − 0.483148i
\(184\) −2.19615 −0.161903
\(185\) 0 0
\(186\) −1.46410 −0.107353
\(187\) − 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 \(0.642949\pi\)
\(192\) 0.732051i 0.0528312i
\(193\) − 8.39230i − 0.604091i −0.953293 0.302046i \(-0.902330\pi\)
0.953293 0.302046i \(-0.0976695\pi\)
\(194\) 16.5885 1.19098
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 24.2487i − 1.72765i −0.503793 0.863825i \(-0.668062\pi\)
0.503793 0.863825i \(-0.331938\pi\)
\(198\) − 2.46410i − 0.175116i
\(199\) 10.9282 0.774680 0.387340 0.921937i \(-0.373394\pi\)
0.387340 + 0.921937i \(0.373394\pi\)
\(200\) 0 0
\(201\) 2.92820 0.206540
\(202\) − 19.8564i − 1.39709i
\(203\) − 1.26795i − 0.0889926i
\(204\) 2.53590 0.177548
\(205\) 0 0
\(206\) −8.39230 −0.584720
\(207\) 5.41154i 0.376128i
\(208\) 5.46410i 0.378867i
\(209\) −3.26795 −0.226049
\(210\) 0 0
\(211\) 13.0718 0.899900 0.449950 0.893054i \(-0.351442\pi\)
0.449950 + 0.893054i \(0.351442\pi\)
\(212\) 10.7321i 0.737080i
\(213\) − 1.85641i − 0.127199i
\(214\) 19.8564 1.35736
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) − 2.00000i − 0.135769i
\(218\) 1.66025i 0.112447i
\(219\) 4.67949 0.316211
\(220\) 0 0
\(221\) 18.9282 1.27325
\(222\) − 2.00000i − 0.134231i
\(223\) − 18.5359i − 1.24126i −0.784105 0.620628i \(-0.786878\pi\)
0.784105 0.620628i \(-0.213122\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −19.8564 −1.32083
\(227\) 6.92820i 0.459841i 0.973209 + 0.229920i \(0.0738466\pi\)
−0.973209 + 0.229920i \(0.926153\pi\)
\(228\) − 2.39230i − 0.158434i
\(229\) −3.60770 −0.238403 −0.119202 0.992870i \(-0.538033\pi\)
−0.119202 + 0.992870i \(0.538033\pi\)
\(230\) 0 0
\(231\) −0.732051 −0.0481654
\(232\) 1.26795i 0.0832449i
\(233\) 19.8564i 1.30084i 0.759576 + 0.650418i \(0.225406\pi\)
−0.759576 + 0.650418i \(0.774594\pi\)
\(234\) 13.4641 0.880176
\(235\) 0 0
\(236\) −6.92820 −0.450988
\(237\) − 1.32051i − 0.0857762i
\(238\) 3.46410i 0.224544i
\(239\) −4.73205 −0.306091 −0.153045 0.988219i \(-0.548908\pi\)
−0.153045 + 0.988219i \(0.548908\pi\)
\(240\) 0 0
\(241\) 6.73205 0.433650 0.216825 0.976211i \(-0.430430\pi\)
0.216825 + 0.976211i \(0.430430\pi\)
\(242\) − 1.00000i − 0.0642824i
\(243\) − 15.2679i − 0.979439i
\(244\) −8.92820 −0.571570
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) − 17.8564i − 1.13618i
\(248\) 2.00000i 0.127000i
\(249\) 3.21539 0.203767
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 2.46410i 0.155224i
\(253\) 2.19615i 0.138071i
\(254\) −14.9282 −0.936679
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.33975i 0.395462i 0.980256 + 0.197731i \(0.0633573\pi\)
−0.980256 + 0.197731i \(0.936643\pi\)
\(258\) − 1.46410i − 0.0911510i
\(259\) 2.73205 0.169761
\(260\) 0 0
\(261\) 3.12436 0.193393
\(262\) 11.6603i 0.720373i
\(263\) − 24.0000i − 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) 0.732051 0.0450546
\(265\) 0 0
\(266\) 3.26795 0.200371
\(267\) − 2.53590i − 0.155194i
\(268\) − 4.00000i − 0.244339i
\(269\) −7.60770 −0.463849 −0.231925 0.972734i \(-0.574502\pi\)
−0.231925 + 0.972734i \(0.574502\pi\)
\(270\) 0 0
\(271\) −20.3923 −1.23874 −0.619372 0.785098i \(-0.712613\pi\)
−0.619372 + 0.785098i \(0.712613\pi\)
\(272\) − 3.46410i − 0.210042i
\(273\) − 4.00000i − 0.242091i
\(274\) −12.9282 −0.781021
\(275\) 0 0
\(276\) −1.60770 −0.0967719
\(277\) − 20.9282i − 1.25745i −0.777626 0.628727i \(-0.783576\pi\)
0.777626 0.628727i \(-0.216424\pi\)
\(278\) − 3.66025i − 0.219527i
\(279\) 4.92820 0.295044
\(280\) 0 0
\(281\) 1.60770 0.0959071 0.0479535 0.998850i \(-0.484730\pi\)
0.0479535 + 0.998850i \(0.484730\pi\)
\(282\) − 5.07180i − 0.302021i
\(283\) 23.7128i 1.40958i 0.709416 + 0.704790i \(0.248959\pi\)
−0.709416 + 0.704790i \(0.751041\pi\)
\(284\) −2.53590 −0.150478
\(285\) 0 0
\(286\) 5.46410 0.323099
\(287\) − 8.19615i − 0.483804i
\(288\) − 2.46410i − 0.145199i
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) 12.1436 0.711870
\(292\) − 6.39230i − 0.374081i
\(293\) 14.5359i 0.849196i 0.905382 + 0.424598i \(0.139585\pi\)
−0.905382 + 0.424598i \(0.860415\pi\)
\(294\) 0.732051 0.0426941
\(295\) 0 0
\(296\) −2.73205 −0.158797
\(297\) − 4.00000i − 0.232104i
\(298\) − 10.7321i − 0.621691i
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 13.1244i 0.755222i
\(303\) − 14.5359i − 0.835066i
\(304\) −3.26795 −0.187430
\(305\) 0 0
\(306\) −8.53590 −0.487965
\(307\) − 3.60770i − 0.205902i −0.994686 0.102951i \(-0.967172\pi\)
0.994686 0.102951i \(-0.0328285\pi\)
\(308\) 1.00000i 0.0569803i
\(309\) −6.14359 −0.349497
\(310\) 0 0
\(311\) 11.0718 0.627824 0.313912 0.949452i \(-0.398360\pi\)
0.313912 + 0.949452i \(0.398360\pi\)
\(312\) 4.00000i 0.226455i
\(313\) 11.8038i 0.667193i 0.942716 + 0.333596i \(0.108262\pi\)
−0.942716 + 0.333596i \(0.891738\pi\)
\(314\) −11.4641 −0.646957
\(315\) 0 0
\(316\) −1.80385 −0.101474
\(317\) 0.588457i 0.0330511i 0.999863 + 0.0165255i \(0.00526048\pi\)
−0.999863 + 0.0165255i \(0.994740\pi\)
\(318\) 7.85641i 0.440565i
\(319\) 1.26795 0.0709915
\(320\) 0 0
\(321\) 14.5359 0.811315
\(322\) − 2.19615i − 0.122387i
\(323\) 11.3205i 0.629890i
\(324\) −4.46410 −0.248006
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 1.21539i 0.0672112i
\(328\) 8.19615i 0.452557i
\(329\) 6.92820 0.381964
\(330\) 0 0
\(331\) 22.7846 1.25236 0.626178 0.779680i \(-0.284618\pi\)
0.626178 + 0.779680i \(0.284618\pi\)
\(332\) − 4.39230i − 0.241059i
\(333\) 6.73205i 0.368914i
\(334\) 13.8564 0.758189
\(335\) 0 0
\(336\) −0.732051 −0.0399366
\(337\) 18.7846i 1.02326i 0.859205 + 0.511631i \(0.170959\pi\)
−0.859205 + 0.511631i \(0.829041\pi\)
\(338\) 16.8564i 0.916868i
\(339\) −14.5359 −0.789482
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 8.05256i 0.435433i
\(343\) 1.00000i 0.0539949i
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) −12.9282 −0.695025
\(347\) 36.9282i 1.98241i 0.132336 + 0.991205i \(0.457752\pi\)
−0.132336 + 0.991205i \(0.542248\pi\)
\(348\) 0.928203i 0.0497569i
\(349\) 26.3923 1.41275 0.706374 0.707839i \(-0.250330\pi\)
0.706374 + 0.707839i \(0.250330\pi\)
\(350\) 0 0
\(351\) 21.8564 1.16661
\(352\) − 1.00000i − 0.0533002i
\(353\) 3.80385i 0.202458i 0.994863 + 0.101229i \(0.0322775\pi\)
−0.994863 + 0.101229i \(0.967722\pi\)
\(354\) −5.07180 −0.269563
\(355\) 0 0
\(356\) −3.46410 −0.183597
\(357\) 2.53590i 0.134214i
\(358\) − 6.00000i − 0.317110i
\(359\) −4.05256 −0.213886 −0.106943 0.994265i \(-0.534106\pi\)
−0.106943 + 0.994265i \(0.534106\pi\)
\(360\) 0 0
\(361\) −8.32051 −0.437921
\(362\) − 14.0000i − 0.735824i
\(363\) − 0.732051i − 0.0384227i
\(364\) −5.46410 −0.286397
\(365\) 0 0
\(366\) −6.53590 −0.341637
\(367\) 8.39230i 0.438075i 0.975716 + 0.219037i \(0.0702917\pi\)
−0.975716 + 0.219037i \(0.929708\pi\)
\(368\) 2.19615i 0.114482i
\(369\) 20.1962 1.05137
\(370\) 0 0
\(371\) −10.7321 −0.557180
\(372\) 1.46410i 0.0759101i
\(373\) − 2.39230i − 0.123869i −0.998080 0.0619344i \(-0.980273\pi\)
0.998080 0.0619344i \(-0.0197270\pi\)
\(374\) −3.46410 −0.179124
\(375\) 0 0
\(376\) −6.92820 −0.357295
\(377\) 6.92820i 0.356821i
\(378\) 4.00000i 0.205738i
\(379\) −33.8564 −1.73909 −0.869543 0.493857i \(-0.835587\pi\)
−0.869543 + 0.493857i \(0.835587\pi\)
\(380\) 0 0
\(381\) −10.9282 −0.559869
\(382\) 12.0000i 0.613973i
\(383\) − 26.5359i − 1.35592i −0.735098 0.677961i \(-0.762864\pi\)
0.735098 0.677961i \(-0.237136\pi\)
\(384\) 0.732051 0.0373573
\(385\) 0 0
\(386\) −8.39230 −0.427157
\(387\) 4.92820i 0.250515i
\(388\) − 16.5885i − 0.842151i
\(389\) 22.3923 1.13533 0.567667 0.823258i \(-0.307846\pi\)
0.567667 + 0.823258i \(0.307846\pi\)
\(390\) 0 0
\(391\) 7.60770 0.384738
\(392\) − 1.00000i − 0.0505076i
\(393\) 8.53590i 0.430579i
\(394\) −24.2487 −1.22163
\(395\) 0 0
\(396\) −2.46410 −0.123826
\(397\) − 9.60770i − 0.482196i −0.970501 0.241098i \(-0.922492\pi\)
0.970501 0.241098i \(-0.0775076\pi\)
\(398\) − 10.9282i − 0.547781i
\(399\) 2.39230 0.119765
\(400\) 0 0
\(401\) 9.46410 0.472615 0.236307 0.971678i \(-0.424063\pi\)
0.236307 + 0.971678i \(0.424063\pi\)
\(402\) − 2.92820i − 0.146046i
\(403\) 10.9282i 0.544373i
\(404\) −19.8564 −0.987893
\(405\) 0 0
\(406\) −1.26795 −0.0629273
\(407\) 2.73205i 0.135423i
\(408\) − 2.53590i − 0.125546i
\(409\) −35.1244 −1.73679 −0.868394 0.495875i \(-0.834847\pi\)
−0.868394 + 0.495875i \(0.834847\pi\)
\(410\) 0 0
\(411\) −9.46410 −0.466830
\(412\) 8.39230i 0.413459i
\(413\) − 6.92820i − 0.340915i
\(414\) 5.41154 0.265963
\(415\) 0 0
\(416\) 5.46410 0.267900
\(417\) − 2.67949i − 0.131215i
\(418\) 3.26795i 0.159841i
\(419\) −17.0718 −0.834012 −0.417006 0.908904i \(-0.636921\pi\)
−0.417006 + 0.908904i \(0.636921\pi\)
\(420\) 0 0
\(421\) −8.14359 −0.396894 −0.198447 0.980112i \(-0.563590\pi\)
−0.198447 + 0.980112i \(0.563590\pi\)
\(422\) − 13.0718i − 0.636325i
\(423\) 17.0718i 0.830059i
\(424\) 10.7321 0.521194
\(425\) 0 0
\(426\) −1.85641 −0.0899432
\(427\) − 8.92820i − 0.432066i
\(428\) − 19.8564i − 0.959796i
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −21.1244 −1.01752 −0.508762 0.860907i \(-0.669897\pi\)
−0.508762 + 0.860907i \(0.669897\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) − 7.80385i − 0.375029i −0.982262 0.187514i \(-0.939957\pi\)
0.982262 0.187514i \(-0.0600432\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 1.66025 0.0795117
\(437\) − 7.17691i − 0.343318i
\(438\) − 4.67949i − 0.223595i
\(439\) 22.2487 1.06187 0.530937 0.847412i \(-0.321840\pi\)
0.530937 + 0.847412i \(0.321840\pi\)
\(440\) 0 0
\(441\) −2.46410 −0.117338
\(442\) − 18.9282i − 0.900323i
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −18.5359 −0.877700
\(447\) − 7.85641i − 0.371595i
\(448\) 1.00000i 0.0472456i
\(449\) −19.6077 −0.925344 −0.462672 0.886529i \(-0.653109\pi\)
−0.462672 + 0.886529i \(0.653109\pi\)
\(450\) 0 0
\(451\) 8.19615 0.385942
\(452\) 19.8564i 0.933967i
\(453\) 9.60770i 0.451409i
\(454\) 6.92820 0.325157
\(455\) 0 0
\(456\) −2.39230 −0.112030
\(457\) − 2.00000i − 0.0935561i −0.998905 0.0467780i \(-0.985105\pi\)
0.998905 0.0467780i \(-0.0148953\pi\)
\(458\) 3.60770i 0.168577i
\(459\) −13.8564 −0.646762
\(460\) 0 0
\(461\) 1.60770 0.0748778 0.0374389 0.999299i \(-0.488080\pi\)
0.0374389 + 0.999299i \(0.488080\pi\)
\(462\) 0.732051i 0.0340581i
\(463\) − 17.5167i − 0.814068i −0.913413 0.407034i \(-0.866563\pi\)
0.913413 0.407034i \(-0.133437\pi\)
\(464\) 1.26795 0.0588631
\(465\) 0 0
\(466\) 19.8564 0.919830
\(467\) − 4.05256i − 0.187530i −0.995594 0.0937650i \(-0.970110\pi\)
0.995594 0.0937650i \(-0.0298902\pi\)
\(468\) − 13.4641i − 0.622378i
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −8.39230 −0.386697
\(472\) 6.92820i 0.318896i
\(473\) 2.00000i 0.0919601i
\(474\) −1.32051 −0.0606529
\(475\) 0 0
\(476\) 3.46410 0.158777
\(477\) − 26.4449i − 1.21083i
\(478\) 4.73205i 0.216439i
\(479\) −8.78461 −0.401379 −0.200690 0.979655i \(-0.564318\pi\)
−0.200690 + 0.979655i \(0.564318\pi\)
\(480\) 0 0
\(481\) −14.9282 −0.680667
\(482\) − 6.73205i − 0.306637i
\(483\) − 1.60770i − 0.0731527i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) −15.2679 −0.692568
\(487\) 6.19615i 0.280774i 0.990097 + 0.140387i \(0.0448347\pi\)
−0.990097 + 0.140387i \(0.955165\pi\)
\(488\) 8.92820i 0.404161i
\(489\) −2.92820 −0.132418
\(490\) 0 0
\(491\) 27.7128 1.25066 0.625331 0.780360i \(-0.284964\pi\)
0.625331 + 0.780360i \(0.284964\pi\)
\(492\) 6.00000i 0.270501i
\(493\) − 4.39230i − 0.197819i
\(494\) −17.8564 −0.803398
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) − 2.53590i − 0.113751i
\(498\) − 3.21539i − 0.144085i
\(499\) −39.8564 −1.78422 −0.892109 0.451820i \(-0.850775\pi\)
−0.892109 + 0.451820i \(0.850775\pi\)
\(500\) 0 0
\(501\) 10.1436 0.453182
\(502\) − 12.0000i − 0.535586i
\(503\) − 32.7846i − 1.46179i −0.682488 0.730897i \(-0.739102\pi\)
0.682488 0.730897i \(-0.260898\pi\)
\(504\) 2.46410 0.109760
\(505\) 0 0
\(506\) 2.19615 0.0976309
\(507\) 12.3397i 0.548027i
\(508\) 14.9282i 0.662332i
\(509\) −24.9282 −1.10492 −0.552462 0.833538i \(-0.686311\pi\)
−0.552462 + 0.833538i \(0.686311\pi\)
\(510\) 0 0
\(511\) 6.39230 0.282779
\(512\) − 1.00000i − 0.0441942i
\(513\) 13.0718i 0.577134i
\(514\) 6.33975 0.279634
\(515\) 0 0
\(516\) −1.46410 −0.0644535
\(517\) 6.92820i 0.304702i
\(518\) − 2.73205i − 0.120039i
\(519\) −9.46410 −0.415428
\(520\) 0 0
\(521\) −10.3923 −0.455295 −0.227648 0.973744i \(-0.573103\pi\)
−0.227648 + 0.973744i \(0.573103\pi\)
\(522\) − 3.12436i − 0.136749i
\(523\) 33.1769i 1.45073i 0.688367 + 0.725363i \(0.258328\pi\)
−0.688367 + 0.725363i \(0.741672\pi\)
\(524\) 11.6603 0.509381
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) − 6.92820i − 0.301797i
\(528\) − 0.732051i − 0.0318584i
\(529\) 18.1769 0.790301
\(530\) 0 0
\(531\) 17.0718 0.740853
\(532\) − 3.26795i − 0.141684i
\(533\) 44.7846i 1.93984i
\(534\) −2.53590 −0.109739
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) − 4.39230i − 0.189542i
\(538\) 7.60770i 0.327991i
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −17.2679 −0.742407 −0.371204 0.928552i \(-0.621055\pi\)
−0.371204 + 0.928552i \(0.621055\pi\)
\(542\) 20.3923i 0.875924i
\(543\) − 10.2487i − 0.439814i
\(544\) −3.46410 −0.148522
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) 12.7846i 0.546630i 0.961925 + 0.273315i \(0.0881202\pi\)
−0.961925 + 0.273315i \(0.911880\pi\)
\(548\) 12.9282i 0.552265i
\(549\) 22.0000 0.938937
\(550\) 0 0
\(551\) −4.14359 −0.176523
\(552\) 1.60770i 0.0684280i
\(553\) − 1.80385i − 0.0767074i
\(554\) −20.9282 −0.889154
\(555\) 0 0
\(556\) −3.66025 −0.155229
\(557\) 46.3923i 1.96571i 0.184393 + 0.982853i \(0.440968\pi\)
−0.184393 + 0.982853i \(0.559032\pi\)
\(558\) − 4.92820i − 0.208627i
\(559\) −10.9282 −0.462214
\(560\) 0 0
\(561\) −2.53590 −0.107066
\(562\) − 1.60770i − 0.0678165i
\(563\) 18.9282i 0.797729i 0.917010 + 0.398864i \(0.130596\pi\)
−0.917010 + 0.398864i \(0.869404\pi\)
\(564\) −5.07180 −0.213561
\(565\) 0 0
\(566\) 23.7128 0.996724
\(567\) − 4.46410i − 0.187475i
\(568\) 2.53590i 0.106404i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 24.3923 1.02079 0.510393 0.859941i \(-0.329500\pi\)
0.510393 + 0.859941i \(0.329500\pi\)
\(572\) − 5.46410i − 0.228466i
\(573\) 8.78461i 0.366982i
\(574\) −8.19615 −0.342101
\(575\) 0 0
\(576\) −2.46410 −0.102671
\(577\) 3.41154i 0.142024i 0.997475 + 0.0710122i \(0.0226229\pi\)
−0.997475 + 0.0710122i \(0.977377\pi\)
\(578\) − 5.00000i − 0.207973i
\(579\) −6.14359 −0.255319
\(580\) 0 0
\(581\) 4.39230 0.182224
\(582\) − 12.1436i − 0.503368i
\(583\) − 10.7321i − 0.444476i
\(584\) −6.39230 −0.264515
\(585\) 0 0
\(586\) 14.5359 0.600472
\(587\) 42.5885i 1.75781i 0.476993 + 0.878907i \(0.341727\pi\)
−0.476993 + 0.878907i \(0.658273\pi\)
\(588\) − 0.732051i − 0.0301893i
\(589\) −6.53590 −0.269307
\(590\) 0 0
\(591\) −17.7513 −0.730190
\(592\) 2.73205i 0.112287i
\(593\) − 48.2487i − 1.98134i −0.136293 0.990669i \(-0.543519\pi\)
0.136293 0.990669i \(-0.456481\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −10.7321 −0.439602
\(597\) − 8.00000i − 0.327418i
\(598\) 12.0000i 0.490716i
\(599\) 25.1769 1.02870 0.514350 0.857580i \(-0.328033\pi\)
0.514350 + 0.857580i \(0.328033\pi\)
\(600\) 0 0
\(601\) 39.5167 1.61192 0.805959 0.591971i \(-0.201650\pi\)
0.805959 + 0.591971i \(0.201650\pi\)
\(602\) − 2.00000i − 0.0815139i
\(603\) 9.85641i 0.401384i
\(604\) 13.1244 0.534022
\(605\) 0 0
\(606\) −14.5359 −0.590481
\(607\) − 7.07180i − 0.287035i −0.989648 0.143518i \(-0.954159\pi\)
0.989648 0.143518i \(-0.0458414\pi\)
\(608\) 3.26795i 0.132533i
\(609\) −0.928203 −0.0376127
\(610\) 0 0
\(611\) −37.8564 −1.53151
\(612\) 8.53590i 0.345043i
\(613\) 27.1769i 1.09767i 0.835932 + 0.548833i \(0.184928\pi\)
−0.835932 + 0.548833i \(0.815072\pi\)
\(614\) −3.60770 −0.145595
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 17.3205i 0.697297i 0.937253 + 0.348649i \(0.113359\pi\)
−0.937253 + 0.348649i \(0.886641\pi\)
\(618\) 6.14359i 0.247132i
\(619\) 12.7846 0.513857 0.256928 0.966430i \(-0.417290\pi\)
0.256928 + 0.966430i \(0.417290\pi\)
\(620\) 0 0
\(621\) 8.78461 0.352514
\(622\) − 11.0718i − 0.443939i
\(623\) − 3.46410i − 0.138786i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 11.8038 0.471777
\(627\) 2.39230i 0.0955395i
\(628\) 11.4641i 0.457467i
\(629\) 9.46410 0.377358
\(630\) 0 0
\(631\) −33.0718 −1.31657 −0.658284 0.752770i \(-0.728717\pi\)
−0.658284 + 0.752770i \(0.728717\pi\)
\(632\) 1.80385i 0.0717532i
\(633\) − 9.56922i − 0.380342i
\(634\) 0.588457 0.0233706
\(635\) 0 0
\(636\) 7.85641 0.311527
\(637\) − 5.46410i − 0.216496i
\(638\) − 1.26795i − 0.0501986i
\(639\) 6.24871 0.247195
\(640\) 0 0
\(641\) 47.5692 1.87887 0.939436 0.342725i \(-0.111350\pi\)
0.939436 + 0.342725i \(0.111350\pi\)
\(642\) − 14.5359i − 0.573686i
\(643\) 36.7321i 1.44857i 0.689500 + 0.724285i \(0.257830\pi\)
−0.689500 + 0.724285i \(0.742170\pi\)
\(644\) −2.19615 −0.0865405
\(645\) 0 0
\(646\) 11.3205 0.445399
\(647\) 33.4641i 1.31561i 0.753188 + 0.657805i \(0.228515\pi\)
−0.753188 + 0.657805i \(0.771485\pi\)
\(648\) 4.46410i 0.175366i
\(649\) 6.92820 0.271956
\(650\) 0 0
\(651\) −1.46410 −0.0573827
\(652\) 4.00000i 0.156652i
\(653\) − 5.66025i − 0.221503i −0.993848 0.110751i \(-0.964674\pi\)
0.993848 0.110751i \(-0.0353257\pi\)
\(654\) 1.21539 0.0475255
\(655\) 0 0
\(656\) 8.19615 0.320006
\(657\) 15.7513i 0.614516i
\(658\) − 6.92820i − 0.270089i
\(659\) 18.2487 0.710869 0.355434 0.934701i \(-0.384333\pi\)
0.355434 + 0.934701i \(0.384333\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) − 22.7846i − 0.885549i
\(663\) − 13.8564i − 0.538138i
\(664\) −4.39230 −0.170454
\(665\) 0 0
\(666\) 6.73205 0.260862
\(667\) 2.78461i 0.107821i
\(668\) − 13.8564i − 0.536120i
\(669\) −13.5692 −0.524616
\(670\) 0 0
\(671\) 8.92820 0.344669
\(672\) 0.732051i 0.0282395i
\(673\) 2.00000i 0.0770943i 0.999257 + 0.0385472i \(0.0122730\pi\)
−0.999257 + 0.0385472i \(0.987727\pi\)
\(674\) 18.7846 0.723556
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) 31.8564i 1.22434i 0.790726 + 0.612171i \(0.209703\pi\)
−0.790726 + 0.612171i \(0.790297\pi\)
\(678\) 14.5359i 0.558248i
\(679\) 16.5885 0.636607
\(680\) 0 0
\(681\) 5.07180 0.194352
\(682\) − 2.00000i − 0.0765840i
\(683\) − 42.2487i − 1.61660i −0.588769 0.808301i \(-0.700387\pi\)
0.588769 0.808301i \(-0.299613\pi\)
\(684\) 8.05256 0.307897
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 2.64102i 0.100761i
\(688\) 2.00000i 0.0762493i
\(689\) 58.6410 2.23404
\(690\) 0 0
\(691\) −6.53590 −0.248637 −0.124319 0.992242i \(-0.539675\pi\)
−0.124319 + 0.992242i \(0.539675\pi\)
\(692\) 12.9282i 0.491457i
\(693\) − 2.46410i − 0.0936035i
\(694\) 36.9282 1.40178
\(695\) 0 0
\(696\) 0.928203 0.0351835
\(697\) − 28.3923i − 1.07544i
\(698\) − 26.3923i − 0.998963i
\(699\) 14.5359 0.549798
\(700\) 0 0
\(701\) −23.9090 −0.903029 −0.451515 0.892264i \(-0.649116\pi\)
−0.451515 + 0.892264i \(0.649116\pi\)
\(702\) − 21.8564i − 0.824917i
\(703\) − 8.92820i − 0.336734i
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 3.80385 0.143160
\(707\) − 19.8564i − 0.746777i
\(708\) 5.07180i 0.190610i
\(709\) 9.32051 0.350039 0.175020 0.984565i \(-0.444001\pi\)
0.175020 + 0.984565i \(0.444001\pi\)
\(710\) 0 0
\(711\) 4.44486 0.166695
\(712\) 3.46410i 0.129823i
\(713\) 4.39230i 0.164493i
\(714\) 2.53590 0.0949036
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) 3.46410i 0.129369i
\(718\) 4.05256i 0.151240i
\(719\) −27.7128 −1.03351 −0.516757 0.856132i \(-0.672861\pi\)
−0.516757 + 0.856132i \(0.672861\pi\)
\(720\) 0 0
\(721\) −8.39230 −0.312546
\(722\) 8.32051i 0.309657i
\(723\) − 4.92820i − 0.183282i
\(724\) −14.0000 −0.520306
\(725\) 0 0
\(726\) −0.732051 −0.0271690
\(727\) 4.00000i 0.148352i 0.997245 + 0.0741759i \(0.0236326\pi\)
−0.997245 + 0.0741759i \(0.976367\pi\)
\(728\) 5.46410i 0.202513i
\(729\) 2.21539 0.0820515
\(730\) 0 0
\(731\) 6.92820 0.256249
\(732\) 6.53590i 0.241574i
\(733\) 26.0000i 0.960332i 0.877178 + 0.480166i \(0.159424\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(734\) 8.39230 0.309766
\(735\) 0 0
\(736\) 2.19615 0.0809513
\(737\) 4.00000i 0.147342i
\(738\) − 20.1962i − 0.743431i
\(739\) 43.7128 1.60800 0.804001 0.594628i \(-0.202701\pi\)
0.804001 + 0.594628i \(0.202701\pi\)
\(740\) 0 0
\(741\) −13.0718 −0.480204
\(742\) 10.7321i 0.393986i
\(743\) 8.78461i 0.322276i 0.986932 + 0.161138i \(0.0515164\pi\)
−0.986932 + 0.161138i \(0.948484\pi\)
\(744\) 1.46410 0.0536766
\(745\) 0 0
\(746\) −2.39230 −0.0875885
\(747\) 10.8231i 0.395996i
\(748\) 3.46410i 0.126660i
\(749\) 19.8564 0.725537
\(750\) 0 0
\(751\) −32.3923 −1.18201 −0.591006 0.806667i \(-0.701269\pi\)
−0.591006 + 0.806667i \(0.701269\pi\)
\(752\) 6.92820i 0.252646i
\(753\) − 8.78461i − 0.320129i
\(754\) 6.92820 0.252310
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) − 29.3731i − 1.06758i −0.845616 0.533791i \(-0.820767\pi\)
0.845616 0.533791i \(-0.179233\pi\)
\(758\) 33.8564i 1.22972i
\(759\) 1.60770 0.0583556
\(760\) 0 0
\(761\) −29.6603 −1.07518 −0.537592 0.843205i \(-0.680666\pi\)
−0.537592 + 0.843205i \(0.680666\pi\)
\(762\) 10.9282i 0.395887i
\(763\) 1.66025i 0.0601052i
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) −26.5359 −0.958781
\(767\) 37.8564i 1.36692i
\(768\) − 0.732051i − 0.0264156i
\(769\) 7.80385 0.281414 0.140707 0.990051i \(-0.455062\pi\)
0.140707 + 0.990051i \(0.455062\pi\)
\(770\) 0 0
\(771\) 4.64102 0.167142
\(772\) 8.39230i 0.302046i
\(773\) 12.9282i 0.464995i 0.972597 + 0.232498i \(0.0746898\pi\)
−0.972597 + 0.232498i \(0.925310\pi\)
\(774\) 4.92820 0.177141
\(775\) 0 0
\(776\) −16.5885 −0.595491
\(777\) − 2.00000i − 0.0717496i
\(778\) − 22.3923i − 0.802803i
\(779\) −26.7846 −0.959658
\(780\) 0 0
\(781\) 2.53590 0.0907416
\(782\) − 7.60770i − 0.272051i
\(783\) − 5.07180i − 0.181251i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 8.53590 0.304465
\(787\) − 45.8564i − 1.63460i −0.576209 0.817302i \(-0.695469\pi\)
0.576209 0.817302i \(-0.304531\pi\)
\(788\) 24.2487i 0.863825i
\(789\) −17.5692 −0.625481
\(790\) 0 0
\(791\) −19.8564 −0.706013
\(792\) 2.46410i 0.0875580i
\(793\) 48.7846i 1.73239i
\(794\) −9.60770 −0.340964
\(795\) 0 0
\(796\) −10.9282 −0.387340
\(797\) 46.3923i 1.64330i 0.569993 + 0.821650i \(0.306946\pi\)
−0.569993 + 0.821650i \(0.693054\pi\)
\(798\) − 2.39230i − 0.0846867i
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 8.53590 0.301601
\(802\) − 9.46410i − 0.334189i
\(803\) 6.39230i 0.225580i
\(804\) −2.92820 −0.103270
\(805\) 0 0
\(806\) 10.9282 0.384930
\(807\) 5.56922i 0.196046i
\(808\) 19.8564i 0.698546i
\(809\) −31.1769 −1.09612 −0.548061 0.836438i \(-0.684634\pi\)
−0.548061 + 0.836438i \(0.684634\pi\)
\(810\) 0 0
\(811\) −18.8756 −0.662814 −0.331407 0.943488i \(-0.607523\pi\)
−0.331407 + 0.943488i \(0.607523\pi\)
\(812\) 1.26795i 0.0444963i
\(813\) 14.9282i 0.523555i
\(814\) 2.73205 0.0957583
\(815\) 0 0
\(816\) −2.53590 −0.0887742
\(817\) − 6.53590i − 0.228662i
\(818\) 35.1244i 1.22809i
\(819\) 13.4641 0.470474
\(820\) 0 0
\(821\) −47.9090 −1.67203 −0.836017 0.548703i \(-0.815122\pi\)
−0.836017 + 0.548703i \(0.815122\pi\)
\(822\) 9.46410i 0.330098i
\(823\) − 25.1244i − 0.875780i −0.899029 0.437890i \(-0.855726\pi\)
0.899029 0.437890i \(-0.144274\pi\)
\(824\) 8.39230 0.292360
\(825\) 0 0
\(826\) −6.92820 −0.241063
\(827\) − 1.85641i − 0.0645536i −0.999479 0.0322768i \(-0.989724\pi\)
0.999479 0.0322768i \(-0.0102758\pi\)
\(828\) − 5.41154i − 0.188064i
\(829\) 46.2487 1.60628 0.803142 0.595788i \(-0.203160\pi\)
0.803142 + 0.595788i \(0.203160\pi\)
\(830\) 0 0
\(831\) −15.3205 −0.531463
\(832\) − 5.46410i − 0.189434i
\(833\) 3.46410i 0.120024i
\(834\) −2.67949 −0.0927832
\(835\) 0 0
\(836\) 3.26795 0.113024
\(837\) − 8.00000i − 0.276520i
\(838\) 17.0718i 0.589735i
\(839\) 4.14359 0.143053 0.0715264 0.997439i \(-0.477213\pi\)
0.0715264 + 0.997439i \(0.477213\pi\)
\(840\) 0 0
\(841\) −27.3923 −0.944562
\(842\) 8.14359i 0.280647i
\(843\) − 1.17691i − 0.0405351i
\(844\) −13.0718 −0.449950
\(845\) 0 0
\(846\) 17.0718 0.586940
\(847\) − 1.00000i − 0.0343604i
\(848\) − 10.7321i − 0.368540i
\(849\) 17.3590 0.595759
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 1.85641i 0.0635994i
\(853\) − 13.2154i − 0.452486i −0.974071 0.226243i \(-0.927356\pi\)
0.974071 0.226243i \(-0.0726444\pi\)
\(854\) −8.92820 −0.305517
\(855\) 0 0
\(856\) −19.8564 −0.678678
\(857\) 20.5359i 0.701493i 0.936470 + 0.350746i \(0.114072\pi\)
−0.936470 + 0.350746i \(0.885928\pi\)
\(858\) − 4.00000i − 0.136558i
\(859\) −28.7846 −0.982118 −0.491059 0.871126i \(-0.663390\pi\)
−0.491059 + 0.871126i \(0.663390\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 21.1244i 0.719498i
\(863\) 37.5167i 1.27708i 0.769588 + 0.638541i \(0.220462\pi\)
−0.769588 + 0.638541i \(0.779538\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) −7.80385 −0.265186
\(867\) − 3.66025i − 0.124309i
\(868\) 2.00000i 0.0678844i
\(869\) 1.80385 0.0611913
\(870\) 0 0
\(871\) −21.8564 −0.740576
\(872\) − 1.66025i − 0.0562233i
\(873\) 40.8756i 1.38343i
\(874\) −7.17691 −0.242763
\(875\) 0 0
\(876\) −4.67949 −0.158105
\(877\) − 13.3205i − 0.449802i −0.974382 0.224901i \(-0.927794\pi\)
0.974382 0.224901i \(-0.0722058\pi\)
\(878\) − 22.2487i − 0.750858i
\(879\) 10.6410 0.358913
\(880\) 0 0
\(881\) 24.9282 0.839853 0.419926 0.907558i \(-0.362056\pi\)
0.419926 + 0.907558i \(0.362056\pi\)
\(882\) 2.46410i 0.0829706i
\(883\) − 56.3923i − 1.89775i −0.315649 0.948876i \(-0.602222\pi\)
0.315649 0.948876i \(-0.397778\pi\)
\(884\) −18.9282 −0.636624
\(885\) 0 0
\(886\) 0 0
\(887\) 13.8564i 0.465253i 0.972566 + 0.232626i \(0.0747319\pi\)
−0.972566 + 0.232626i \(0.925268\pi\)
\(888\) 2.00000i 0.0671156i
\(889\) −14.9282 −0.500676
\(890\) 0 0
\(891\) 4.46410 0.149553
\(892\) 18.5359i 0.620628i
\(893\) − 22.6410i − 0.757653i
\(894\) −7.85641 −0.262758
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 8.78461i 0.293310i
\(898\) 19.6077i 0.654317i
\(899\) 2.53590 0.0845769
\(900\) 0 0
\(901\) −37.1769 −1.23854
\(902\) − 8.19615i − 0.272902i
\(903\) − 1.46410i − 0.0487223i
\(904\) 19.8564 0.660414
\(905\) 0 0
\(906\) 9.60770 0.319194
\(907\) − 59.0333i − 1.96017i −0.198580 0.980085i \(-0.563633\pi\)
0.198580 0.980085i \(-0.436367\pi\)
\(908\) − 6.92820i − 0.229920i
\(909\) 48.9282 1.62285
\(910\) 0 0
\(911\) 2.53590 0.0840181 0.0420090 0.999117i \(-0.486624\pi\)
0.0420090 + 0.999117i \(0.486624\pi\)
\(912\) 2.39230i 0.0792171i
\(913\) 4.39230i 0.145364i
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 3.60770 0.119202
\(917\) 11.6603i 0.385056i
\(918\) 13.8564i 0.457330i
\(919\) −47.3731 −1.56269 −0.781347 0.624097i \(-0.785467\pi\)
−0.781347 + 0.624097i \(0.785467\pi\)
\(920\) 0 0
\(921\) −2.64102 −0.0870244
\(922\) − 1.60770i − 0.0529466i
\(923\) 13.8564i 0.456089i
\(924\) 0.732051 0.0240827
\(925\) 0 0
\(926\) −17.5167 −0.575633
\(927\) − 20.6795i − 0.679204i
\(928\) − 1.26795i − 0.0416225i
\(929\) 46.3923 1.52208 0.761041 0.648704i \(-0.224689\pi\)
0.761041 + 0.648704i \(0.224689\pi\)
\(930\) 0 0
\(931\) 3.26795 0.107103
\(932\) − 19.8564i − 0.650418i
\(933\) − 8.10512i − 0.265350i
\(934\) −4.05256 −0.132604
\(935\) 0 0
\(936\) −13.4641 −0.440088
\(937\) 42.7846i 1.39771i 0.715262 + 0.698856i \(0.246307\pi\)
−0.715262 + 0.698856i \(0.753693\pi\)
\(938\) − 4.00000i − 0.130605i
\(939\) 8.64102 0.281989
\(940\) 0 0
\(941\) 26.7846 0.873153 0.436577 0.899667i \(-0.356191\pi\)
0.436577 + 0.899667i \(0.356191\pi\)
\(942\) 8.39230i 0.273436i
\(943\) 18.0000i 0.586161i
\(944\) 6.92820 0.225494
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) − 12.6795i − 0.412028i −0.978549 0.206014i \(-0.933951\pi\)
0.978549 0.206014i \(-0.0660493\pi\)
\(948\) 1.32051i 0.0428881i
\(949\) −34.9282 −1.13382
\(950\) 0 0
\(951\) 0.430781 0.0139690
\(952\) − 3.46410i − 0.112272i
\(953\) 33.4641i 1.08401i 0.840376 + 0.542004i \(0.182334\pi\)
−0.840376 + 0.542004i \(0.817666\pi\)
\(954\) −26.4449 −0.856184
\(955\) 0 0
\(956\) 4.73205 0.153045
\(957\) − 0.928203i − 0.0300045i
\(958\) 8.78461i 0.283818i
\(959\) −12.9282 −0.417473
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 14.9282i 0.481305i
\(963\) 48.9282i 1.57669i
\(964\) −6.73205 −0.216825
\(965\) 0 0
\(966\) −1.60770 −0.0517267
\(967\) − 13.0718i − 0.420361i −0.977663 0.210180i \(-0.932595\pi\)
0.977663 0.210180i \(-0.0674051\pi\)
\(968\) 1.00000i 0.0321412i
\(969\) 8.28719 0.266223
\(970\) 0 0
\(971\) 29.0718 0.932958 0.466479 0.884532i \(-0.345522\pi\)
0.466479 + 0.884532i \(0.345522\pi\)
\(972\) 15.2679i 0.489720i
\(973\) − 3.66025i − 0.117342i
\(974\) 6.19615 0.198538
\(975\) 0 0
\(976\) 8.92820 0.285785
\(977\) − 21.7128i − 0.694654i −0.937744 0.347327i \(-0.887089\pi\)
0.937744 0.347327i \(-0.112911\pi\)
\(978\) 2.92820i 0.0936336i
\(979\) 3.46410 0.110713
\(980\) 0 0
\(981\) −4.09103 −0.130617
\(982\) − 27.7128i − 0.884351i
\(983\) 25.1769i 0.803019i 0.915855 + 0.401509i \(0.131514\pi\)
−0.915855 + 0.401509i \(0.868486\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) −4.39230 −0.139879
\(987\) − 5.07180i − 0.161437i
\(988\) 17.8564i 0.568088i
\(989\) −4.39230 −0.139667
\(990\) 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
\(993\) − 16.6795i − 0.529308i
\(994\) −2.53590 −0.0804338
\(995\) 0 0
\(996\) −3.21539 −0.101884
\(997\) 37.7128i 1.19438i 0.802101 + 0.597188i \(0.203716\pi\)
−0.802101 + 0.597188i \(0.796284\pi\)
\(998\) 39.8564i 1.26163i
\(999\) 10.9282 0.345753
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.1849.1 4
5.2 odd 4 770.2.a.j.1.1 2
5.3 odd 4 3850.2.a.bd.1.2 2
5.4 even 2 inner 3850.2.c.x.1849.4 4
15.2 even 4 6930.2.a.bv.1.2 2
20.7 even 4 6160.2.a.t.1.2 2
35.27 even 4 5390.2.a.bs.1.2 2
55.32 even 4 8470.2.a.br.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.1 2 5.2 odd 4
3850.2.a.bd.1.2 2 5.3 odd 4
3850.2.c.x.1849.1 4 1.1 even 1 trivial
3850.2.c.x.1849.4 4 5.4 even 2 inner
5390.2.a.bs.1.2 2 35.27 even 4
6160.2.a.t.1.2 2 20.7 even 4
6930.2.a.bv.1.2 2 15.2 even 4
8470.2.a.br.1.1 2 55.32 even 4