Properties

Label 5070.2.b.o.1351.3
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.3
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.o.1351.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.46410i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000i q^{18} +7.46410i q^{19} -1.00000i q^{20} +2.00000i q^{21} +6.46410 q^{22} +3.73205 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} +0.267949 q^{29} +1.00000 q^{30} -1.73205i q^{31} +1.00000i q^{32} +6.46410i q^{33} -4.00000i q^{34} +2.00000 q^{35} -1.00000 q^{36} -9.19615i q^{37} -7.46410 q^{38} +1.00000 q^{40} -2.00000i q^{41} -2.00000 q^{42} +11.9282 q^{43} +6.46410i q^{44} +1.00000i q^{45} +3.73205i q^{46} -3.53590i q^{47} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} +4.00000 q^{51} +0.928203 q^{53} -1.00000i q^{54} +6.46410 q^{55} -2.00000 q^{56} -7.46410i q^{57} +0.267949i q^{58} +8.46410i q^{59} +1.00000i q^{60} -10.3923 q^{61} +1.73205 q^{62} -2.00000i q^{63} -1.00000 q^{64} -6.46410 q^{66} +11.4641i q^{67} +4.00000 q^{68} -3.73205 q^{69} +2.00000i q^{70} -12.3923i q^{71} -1.00000i q^{72} +2.00000i q^{73} +9.19615 q^{74} +1.00000 q^{75} -7.46410i q^{76} -12.9282 q^{77} -13.9282 q^{79} +1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} +8.92820i q^{83} -2.00000i q^{84} -4.00000i q^{85} +11.9282i q^{86} -0.267949 q^{87} -6.46410 q^{88} +0.535898i q^{89} -1.00000 q^{90} -3.73205 q^{92} +1.73205i q^{93} +3.53590 q^{94} -7.46410 q^{95} -1.00000i q^{96} +0.535898i q^{97} +3.00000i q^{98} -6.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 4q^{4} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 4q^{4} + 4q^{9} - 4q^{10} + 4q^{12} + 8q^{14} + 4q^{16} - 16q^{17} + 12q^{22} + 8q^{23} - 4q^{25} - 4q^{27} + 8q^{29} + 4q^{30} + 8q^{35} - 4q^{36} - 16q^{38} + 4q^{40} - 8q^{42} + 20q^{43} - 4q^{48} + 12q^{49} + 16q^{51} - 24q^{53} + 12q^{55} - 8q^{56} - 4q^{64} - 12q^{66} + 16q^{68} - 8q^{69} + 16q^{74} + 4q^{75} - 24q^{77} - 28q^{79} + 4q^{81} + 8q^{82} - 8q^{87} - 12q^{88} - 4q^{90} - 8q^{92} + 28q^{94} - 16q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000i 0.447214i
\(6\) − 1.00000i − 0.408248i
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) − 6.46410i − 1.94900i −0.224388 0.974500i \(-0.572038\pi\)
0.224388 0.974500i \(-0.427962\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 7.46410i 1.71238i 0.516659 + 0.856191i \(0.327175\pi\)
−0.516659 + 0.856191i \(0.672825\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) 2.00000i 0.436436i
\(22\) 6.46410 1.37815
\(23\) 3.73205 0.778186 0.389093 0.921198i \(-0.372788\pi\)
0.389093 + 0.921198i \(0.372788\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000i 0.377964i
\(29\) 0.267949 0.0497569 0.0248785 0.999690i \(-0.492080\pi\)
0.0248785 + 0.999690i \(0.492080\pi\)
\(30\) 1.00000 0.182574
\(31\) − 1.73205i − 0.311086i −0.987829 0.155543i \(-0.950287\pi\)
0.987829 0.155543i \(-0.0497126\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 6.46410i 1.12526i
\(34\) − 4.00000i − 0.685994i
\(35\) 2.00000 0.338062
\(36\) −1.00000 −0.166667
\(37\) − 9.19615i − 1.51184i −0.654665 0.755919i \(-0.727190\pi\)
0.654665 0.755919i \(-0.272810\pi\)
\(38\) −7.46410 −1.21084
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) − 2.00000i − 0.312348i −0.987730 0.156174i \(-0.950084\pi\)
0.987730 0.156174i \(-0.0499160\pi\)
\(42\) −2.00000 −0.308607
\(43\) 11.9282 1.81903 0.909517 0.415667i \(-0.136452\pi\)
0.909517 + 0.415667i \(0.136452\pi\)
\(44\) 6.46410i 0.974500i
\(45\) 1.00000i 0.149071i
\(46\) 3.73205i 0.550261i
\(47\) − 3.53590i − 0.515764i −0.966176 0.257882i \(-0.916975\pi\)
0.966176 0.257882i \(-0.0830245\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.00000 0.428571
\(50\) − 1.00000i − 0.141421i
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 0.928203 0.127499 0.0637493 0.997966i \(-0.479694\pi\)
0.0637493 + 0.997966i \(0.479694\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) 6.46410 0.871619
\(56\) −2.00000 −0.267261
\(57\) − 7.46410i − 0.988644i
\(58\) 0.267949i 0.0351835i
\(59\) 8.46410i 1.10193i 0.834528 + 0.550966i \(0.185741\pi\)
−0.834528 + 0.550966i \(0.814259\pi\)
\(60\) 1.00000i 0.129099i
\(61\) −10.3923 −1.33060 −0.665299 0.746577i \(-0.731696\pi\)
−0.665299 + 0.746577i \(0.731696\pi\)
\(62\) 1.73205 0.219971
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −6.46410 −0.795676
\(67\) 11.4641i 1.40056i 0.713867 + 0.700281i \(0.246942\pi\)
−0.713867 + 0.700281i \(0.753058\pi\)
\(68\) 4.00000 0.485071
\(69\) −3.73205 −0.449286
\(70\) 2.00000i 0.239046i
\(71\) − 12.3923i − 1.47070i −0.677690 0.735348i \(-0.737019\pi\)
0.677690 0.735348i \(-0.262981\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 9.19615 1.06903
\(75\) 1.00000 0.115470
\(76\) − 7.46410i − 0.856191i
\(77\) −12.9282 −1.47331
\(78\) 0 0
\(79\) −13.9282 −1.56705 −0.783523 0.621363i \(-0.786579\pi\)
−0.783523 + 0.621363i \(0.786579\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 8.92820i 0.979998i 0.871723 + 0.489999i \(0.163003\pi\)
−0.871723 + 0.489999i \(0.836997\pi\)
\(84\) − 2.00000i − 0.218218i
\(85\) − 4.00000i − 0.433861i
\(86\) 11.9282i 1.28625i
\(87\) −0.267949 −0.0287272
\(88\) −6.46410 −0.689076
\(89\) 0.535898i 0.0568051i 0.999597 + 0.0284026i \(0.00904203\pi\)
−0.999597 + 0.0284026i \(0.990958\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −3.73205 −0.389093
\(93\) 1.73205i 0.179605i
\(94\) 3.53590 0.364700
\(95\) −7.46410 −0.765801
\(96\) − 1.00000i − 0.102062i
\(97\) 0.535898i 0.0544122i 0.999630 + 0.0272061i \(0.00866105\pi\)
−0.999630 + 0.0272061i \(0.991339\pi\)
\(98\) 3.00000i 0.303046i
\(99\) − 6.46410i − 0.649667i
\(100\) 1.00000 0.100000
\(101\) 2.92820 0.291367 0.145684 0.989331i \(-0.453462\pi\)
0.145684 + 0.989331i \(0.453462\pi\)
\(102\) 4.00000i 0.396059i
\(103\) −11.8564 −1.16825 −0.584123 0.811665i \(-0.698562\pi\)
−0.584123 + 0.811665i \(0.698562\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0.928203i 0.0901551i
\(107\) −7.85641 −0.759507 −0.379754 0.925088i \(-0.623991\pi\)
−0.379754 + 0.925088i \(0.623991\pi\)
\(108\) 1.00000 0.0962250
\(109\) − 15.8564i − 1.51877i −0.650643 0.759384i \(-0.725500\pi\)
0.650643 0.759384i \(-0.274500\pi\)
\(110\) 6.46410i 0.616328i
\(111\) 9.19615i 0.872860i
\(112\) − 2.00000i − 0.188982i
\(113\) 0.803848 0.0756196 0.0378098 0.999285i \(-0.487962\pi\)
0.0378098 + 0.999285i \(0.487962\pi\)
\(114\) 7.46410 0.699077
\(115\) 3.73205i 0.348016i
\(116\) −0.267949 −0.0248785
\(117\) 0 0
\(118\) −8.46410 −0.779184
\(119\) 8.00000i 0.733359i
\(120\) −1.00000 −0.0912871
\(121\) −30.7846 −2.79860
\(122\) − 10.3923i − 0.940875i
\(123\) 2.00000i 0.180334i
\(124\) 1.73205i 0.155543i
\(125\) − 1.00000i − 0.0894427i
\(126\) 2.00000 0.178174
\(127\) −4.92820 −0.437307 −0.218654 0.975803i \(-0.570166\pi\)
−0.218654 + 0.975803i \(0.570166\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −11.9282 −1.05022
\(130\) 0 0
\(131\) −18.6603 −1.63035 −0.815177 0.579212i \(-0.803360\pi\)
−0.815177 + 0.579212i \(0.803360\pi\)
\(132\) − 6.46410i − 0.562628i
\(133\) 14.9282 1.29444
\(134\) −11.4641 −0.990348
\(135\) − 1.00000i − 0.0860663i
\(136\) 4.00000i 0.342997i
\(137\) − 2.46410i − 0.210522i −0.994445 0.105261i \(-0.966432\pi\)
0.994445 0.105261i \(-0.0335679\pi\)
\(138\) − 3.73205i − 0.317693i
\(139\) −12.9282 −1.09656 −0.548278 0.836296i \(-0.684716\pi\)
−0.548278 + 0.836296i \(0.684716\pi\)
\(140\) −2.00000 −0.169031
\(141\) 3.53590i 0.297776i
\(142\) 12.3923 1.03994
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0.267949i 0.0222520i
\(146\) −2.00000 −0.165521
\(147\) −3.00000 −0.247436
\(148\) 9.19615i 0.755919i
\(149\) − 13.5359i − 1.10890i −0.832216 0.554452i \(-0.812928\pi\)
0.832216 0.554452i \(-0.187072\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) − 10.3923i − 0.845714i −0.906196 0.422857i \(-0.861027\pi\)
0.906196 0.422857i \(-0.138973\pi\)
\(152\) 7.46410 0.605419
\(153\) −4.00000 −0.323381
\(154\) − 12.9282i − 1.04178i
\(155\) 1.73205 0.139122
\(156\) 0 0
\(157\) −5.00000 −0.399043 −0.199522 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(158\) − 13.9282i − 1.10807i
\(159\) −0.928203 −0.0736113
\(160\) −1.00000 −0.0790569
\(161\) − 7.46410i − 0.588254i
\(162\) 1.00000i 0.0785674i
\(163\) 15.0526i 1.17901i 0.807766 + 0.589504i \(0.200677\pi\)
−0.807766 + 0.589504i \(0.799323\pi\)
\(164\) 2.00000i 0.156174i
\(165\) −6.46410 −0.503230
\(166\) −8.92820 −0.692963
\(167\) − 16.3205i − 1.26292i −0.775409 0.631459i \(-0.782456\pi\)
0.775409 0.631459i \(-0.217544\pi\)
\(168\) 2.00000 0.154303
\(169\) 0 0
\(170\) 4.00000 0.306786
\(171\) 7.46410i 0.570794i
\(172\) −11.9282 −0.909517
\(173\) 10.9282 0.830856 0.415428 0.909626i \(-0.363632\pi\)
0.415428 + 0.909626i \(0.363632\pi\)
\(174\) − 0.267949i − 0.0203132i
\(175\) 2.00000i 0.151186i
\(176\) − 6.46410i − 0.487250i
\(177\) − 8.46410i − 0.636201i
\(178\) −0.535898 −0.0401673
\(179\) −19.7321 −1.47484 −0.737421 0.675433i \(-0.763957\pi\)
−0.737421 + 0.675433i \(0.763957\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) −2.92820 −0.217652 −0.108826 0.994061i \(-0.534709\pi\)
−0.108826 + 0.994061i \(0.534709\pi\)
\(182\) 0 0
\(183\) 10.3923 0.768221
\(184\) − 3.73205i − 0.275130i
\(185\) 9.19615 0.676115
\(186\) −1.73205 −0.127000
\(187\) 25.8564i 1.89081i
\(188\) 3.53590i 0.257882i
\(189\) 2.00000i 0.145479i
\(190\) − 7.46410i − 0.541503i
\(191\) 21.4641 1.55309 0.776544 0.630063i \(-0.216971\pi\)
0.776544 + 0.630063i \(0.216971\pi\)
\(192\) 1.00000 0.0721688
\(193\) − 11.3205i − 0.814868i −0.913235 0.407434i \(-0.866424\pi\)
0.913235 0.407434i \(-0.133576\pi\)
\(194\) −0.535898 −0.0384753
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 4.39230i − 0.312939i −0.987683 0.156469i \(-0.949989\pi\)
0.987683 0.156469i \(-0.0500113\pi\)
\(198\) 6.46410 0.459384
\(199\) −5.07180 −0.359530 −0.179765 0.983710i \(-0.557534\pi\)
−0.179765 + 0.983710i \(0.557534\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) − 11.4641i − 0.808615i
\(202\) 2.92820i 0.206028i
\(203\) − 0.535898i − 0.0376127i
\(204\) −4.00000 −0.280056
\(205\) 2.00000 0.139686
\(206\) − 11.8564i − 0.826075i
\(207\) 3.73205 0.259395
\(208\) 0 0
\(209\) 48.2487 3.33743
\(210\) − 2.00000i − 0.138013i
\(211\) −11.3205 −0.779336 −0.389668 0.920955i \(-0.627410\pi\)
−0.389668 + 0.920955i \(0.627410\pi\)
\(212\) −0.928203 −0.0637493
\(213\) 12.3923i 0.849107i
\(214\) − 7.85641i − 0.537053i
\(215\) 11.9282i 0.813497i
\(216\) 1.00000i 0.0680414i
\(217\) −3.46410 −0.235159
\(218\) 15.8564 1.07393
\(219\) − 2.00000i − 0.135147i
\(220\) −6.46410 −0.435810
\(221\) 0 0
\(222\) −9.19615 −0.617205
\(223\) 20.5359i 1.37519i 0.726097 + 0.687593i \(0.241333\pi\)
−0.726097 + 0.687593i \(0.758667\pi\)
\(224\) 2.00000 0.133631
\(225\) −1.00000 −0.0666667
\(226\) 0.803848i 0.0534711i
\(227\) − 16.3923i − 1.08800i −0.839087 0.543998i \(-0.816910\pi\)
0.839087 0.543998i \(-0.183090\pi\)
\(228\) 7.46410i 0.494322i
\(229\) − 7.85641i − 0.519166i −0.965721 0.259583i \(-0.916415\pi\)
0.965721 0.259583i \(-0.0835851\pi\)
\(230\) −3.73205 −0.246084
\(231\) 12.9282 0.850613
\(232\) − 0.267949i − 0.0175917i
\(233\) 6.12436 0.401220 0.200610 0.979671i \(-0.435708\pi\)
0.200610 + 0.979671i \(0.435708\pi\)
\(234\) 0 0
\(235\) 3.53590 0.230657
\(236\) − 8.46410i − 0.550966i
\(237\) 13.9282 0.904734
\(238\) −8.00000 −0.518563
\(239\) − 16.3923i − 1.06033i −0.847894 0.530165i \(-0.822130\pi\)
0.847894 0.530165i \(-0.177870\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) 17.7321i 1.14222i 0.820873 + 0.571111i \(0.193487\pi\)
−0.820873 + 0.571111i \(0.806513\pi\)
\(242\) − 30.7846i − 1.97891i
\(243\) −1.00000 −0.0641500
\(244\) 10.3923 0.665299
\(245\) 3.00000i 0.191663i
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) −1.73205 −0.109985
\(249\) − 8.92820i − 0.565802i
\(250\) 1.00000 0.0632456
\(251\) 15.7321 0.992998 0.496499 0.868037i \(-0.334619\pi\)
0.496499 + 0.868037i \(0.334619\pi\)
\(252\) 2.00000i 0.125988i
\(253\) − 24.1244i − 1.51669i
\(254\) − 4.92820i − 0.309223i
\(255\) 4.00000i 0.250490i
\(256\) 1.00000 0.0625000
\(257\) −5.33975 −0.333084 −0.166542 0.986034i \(-0.553260\pi\)
−0.166542 + 0.986034i \(0.553260\pi\)
\(258\) − 11.9282i − 0.742617i
\(259\) −18.3923 −1.14284
\(260\) 0 0
\(261\) 0.267949 0.0165856
\(262\) − 18.6603i − 1.15283i
\(263\) 6.12436 0.377644 0.188822 0.982011i \(-0.439533\pi\)
0.188822 + 0.982011i \(0.439533\pi\)
\(264\) 6.46410 0.397838
\(265\) 0.928203i 0.0570191i
\(266\) 14.9282i 0.915307i
\(267\) − 0.535898i − 0.0327964i
\(268\) − 11.4641i − 0.700281i
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 1.00000 0.0608581
\(271\) − 1.19615i − 0.0726611i −0.999340 0.0363305i \(-0.988433\pi\)
0.999340 0.0363305i \(-0.0115669\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 2.46410 0.148862
\(275\) 6.46410i 0.389800i
\(276\) 3.73205 0.224643
\(277\) −3.92820 −0.236023 −0.118011 0.993012i \(-0.537652\pi\)
−0.118011 + 0.993012i \(0.537652\pi\)
\(278\) − 12.9282i − 0.775382i
\(279\) − 1.73205i − 0.103695i
\(280\) − 2.00000i − 0.119523i
\(281\) − 8.92820i − 0.532612i −0.963889 0.266306i \(-0.914197\pi\)
0.963889 0.266306i \(-0.0858032\pi\)
\(282\) −3.53590 −0.210560
\(283\) −9.92820 −0.590170 −0.295085 0.955471i \(-0.595348\pi\)
−0.295085 + 0.955471i \(0.595348\pi\)
\(284\) 12.3923i 0.735348i
\(285\) 7.46410 0.442135
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 1.00000i 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) −0.267949 −0.0157345
\(291\) − 0.535898i − 0.0314149i
\(292\) − 2.00000i − 0.117041i
\(293\) − 31.8564i − 1.86107i −0.366202 0.930536i \(-0.619342\pi\)
0.366202 0.930536i \(-0.380658\pi\)
\(294\) − 3.00000i − 0.174964i
\(295\) −8.46410 −0.492799
\(296\) −9.19615 −0.534516
\(297\) 6.46410i 0.375085i
\(298\) 13.5359 0.784114
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) − 23.8564i − 1.37506i
\(302\) 10.3923 0.598010
\(303\) −2.92820 −0.168221
\(304\) 7.46410i 0.428096i
\(305\) − 10.3923i − 0.595062i
\(306\) − 4.00000i − 0.228665i
\(307\) 19.4641i 1.11087i 0.831558 + 0.555437i \(0.187449\pi\)
−0.831558 + 0.555437i \(0.812551\pi\)
\(308\) 12.9282 0.736653
\(309\) 11.8564 0.674487
\(310\) 1.73205i 0.0983739i
\(311\) −28.3923 −1.60998 −0.804990 0.593288i \(-0.797829\pi\)
−0.804990 + 0.593288i \(0.797829\pi\)
\(312\) 0 0
\(313\) −28.0000 −1.58265 −0.791327 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) − 5.00000i − 0.282166i
\(315\) 2.00000 0.112687
\(316\) 13.9282 0.783523
\(317\) 14.5359i 0.816417i 0.912889 + 0.408209i \(0.133846\pi\)
−0.912889 + 0.408209i \(0.866154\pi\)
\(318\) − 0.928203i − 0.0520511i
\(319\) − 1.73205i − 0.0969762i
\(320\) − 1.00000i − 0.0559017i
\(321\) 7.85641 0.438502
\(322\) 7.46410 0.415958
\(323\) − 29.8564i − 1.66125i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −15.0526 −0.833684
\(327\) 15.8564i 0.876861i
\(328\) −2.00000 −0.110432
\(329\) −7.07180 −0.389881
\(330\) − 6.46410i − 0.355837i
\(331\) − 16.7846i − 0.922566i −0.887253 0.461283i \(-0.847389\pi\)
0.887253 0.461283i \(-0.152611\pi\)
\(332\) − 8.92820i − 0.489999i
\(333\) − 9.19615i − 0.503946i
\(334\) 16.3205 0.893018
\(335\) −11.4641 −0.626351
\(336\) 2.00000i 0.109109i
\(337\) 9.32051 0.507720 0.253860 0.967241i \(-0.418300\pi\)
0.253860 + 0.967241i \(0.418300\pi\)
\(338\) 0 0
\(339\) −0.803848 −0.0436590
\(340\) 4.00000i 0.216930i
\(341\) −11.1962 −0.606306
\(342\) −7.46410 −0.403612
\(343\) − 20.0000i − 1.07990i
\(344\) − 11.9282i − 0.643126i
\(345\) − 3.73205i − 0.200927i
\(346\) 10.9282i 0.587504i
\(347\) 1.60770 0.0863056 0.0431528 0.999068i \(-0.486260\pi\)
0.0431528 + 0.999068i \(0.486260\pi\)
\(348\) 0.267949 0.0143636
\(349\) − 21.4641i − 1.14895i −0.818523 0.574474i \(-0.805207\pi\)
0.818523 0.574474i \(-0.194793\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 6.46410 0.344538
\(353\) − 2.00000i − 0.106449i −0.998583 0.0532246i \(-0.983050\pi\)
0.998583 0.0532246i \(-0.0169499\pi\)
\(354\) 8.46410 0.449862
\(355\) 12.3923 0.657715
\(356\) − 0.535898i − 0.0284026i
\(357\) − 8.00000i − 0.423405i
\(358\) − 19.7321i − 1.04287i
\(359\) 5.07180i 0.267679i 0.991003 + 0.133840i \(0.0427307\pi\)
−0.991003 + 0.133840i \(0.957269\pi\)
\(360\) 1.00000 0.0527046
\(361\) −36.7128 −1.93225
\(362\) − 2.92820i − 0.153903i
\(363\) 30.7846 1.61577
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 10.3923i 0.543214i
\(367\) 15.6077 0.814715 0.407358 0.913269i \(-0.366450\pi\)
0.407358 + 0.913269i \(0.366450\pi\)
\(368\) 3.73205 0.194547
\(369\) − 2.00000i − 0.104116i
\(370\) 9.19615i 0.478085i
\(371\) − 1.85641i − 0.0963798i
\(372\) − 1.73205i − 0.0898027i
\(373\) −15.7846 −0.817296 −0.408648 0.912692i \(-0.634000\pi\)
−0.408648 + 0.912692i \(0.634000\pi\)
\(374\) −25.8564 −1.33700
\(375\) 1.00000i 0.0516398i
\(376\) −3.53590 −0.182350
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) − 27.8564i − 1.43089i −0.698670 0.715444i \(-0.746225\pi\)
0.698670 0.715444i \(-0.253775\pi\)
\(380\) 7.46410 0.382900
\(381\) 4.92820 0.252479
\(382\) 21.4641i 1.09820i
\(383\) 25.3923i 1.29749i 0.761007 + 0.648743i \(0.224705\pi\)
−0.761007 + 0.648743i \(0.775295\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) − 12.9282i − 0.658882i
\(386\) 11.3205 0.576199
\(387\) 11.9282 0.606345
\(388\) − 0.535898i − 0.0272061i
\(389\) −23.7321 −1.20326 −0.601631 0.798774i \(-0.705482\pi\)
−0.601631 + 0.798774i \(0.705482\pi\)
\(390\) 0 0
\(391\) −14.9282 −0.754952
\(392\) − 3.00000i − 0.151523i
\(393\) 18.6603 0.941285
\(394\) 4.39230 0.221281
\(395\) − 13.9282i − 0.700804i
\(396\) 6.46410i 0.324833i
\(397\) 12.1244i 0.608504i 0.952592 + 0.304252i \(0.0984065\pi\)
−0.952592 + 0.304252i \(0.901594\pi\)
\(398\) − 5.07180i − 0.254226i
\(399\) −14.9282 −0.747345
\(400\) −1.00000 −0.0500000
\(401\) 32.0000i 1.59800i 0.601329 + 0.799002i \(0.294638\pi\)
−0.601329 + 0.799002i \(0.705362\pi\)
\(402\) 11.4641 0.571777
\(403\) 0 0
\(404\) −2.92820 −0.145684
\(405\) 1.00000i 0.0496904i
\(406\) 0.535898 0.0265962
\(407\) −59.4449 −2.94657
\(408\) − 4.00000i − 0.198030i
\(409\) 4.00000i 0.197787i 0.995098 + 0.0988936i \(0.0315304\pi\)
−0.995098 + 0.0988936i \(0.968470\pi\)
\(410\) 2.00000i 0.0987730i
\(411\) 2.46410i 0.121545i
\(412\) 11.8564 0.584123
\(413\) 16.9282 0.832982
\(414\) 3.73205i 0.183420i
\(415\) −8.92820 −0.438268
\(416\) 0 0
\(417\) 12.9282 0.633097
\(418\) 48.2487i 2.35992i
\(419\) 22.3923 1.09394 0.546968 0.837154i \(-0.315782\pi\)
0.546968 + 0.837154i \(0.315782\pi\)
\(420\) 2.00000 0.0975900
\(421\) 4.39230i 0.214068i 0.994255 + 0.107034i \(0.0341353\pi\)
−0.994255 + 0.107034i \(0.965865\pi\)
\(422\) − 11.3205i − 0.551074i
\(423\) − 3.53590i − 0.171921i
\(424\) − 0.928203i − 0.0450775i
\(425\) 4.00000 0.194029
\(426\) −12.3923 −0.600409
\(427\) 20.7846i 1.00584i
\(428\) 7.85641 0.379754
\(429\) 0 0
\(430\) −11.9282 −0.575229
\(431\) − 28.3923i − 1.36761i −0.729665 0.683805i \(-0.760324\pi\)
0.729665 0.683805i \(-0.239676\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 19.3205 0.928484 0.464242 0.885708i \(-0.346327\pi\)
0.464242 + 0.885708i \(0.346327\pi\)
\(434\) − 3.46410i − 0.166282i
\(435\) − 0.267949i − 0.0128472i
\(436\) 15.8564i 0.759384i
\(437\) 27.8564i 1.33255i
\(438\) 2.00000 0.0955637
\(439\) −17.8564 −0.852240 −0.426120 0.904667i \(-0.640120\pi\)
−0.426120 + 0.904667i \(0.640120\pi\)
\(440\) − 6.46410i − 0.308164i
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 16.3923 0.778822 0.389411 0.921064i \(-0.372679\pi\)
0.389411 + 0.921064i \(0.372679\pi\)
\(444\) − 9.19615i − 0.436430i
\(445\) −0.535898 −0.0254040
\(446\) −20.5359 −0.972403
\(447\) 13.5359i 0.640226i
\(448\) 2.00000i 0.0944911i
\(449\) − 15.7128i − 0.741533i −0.928726 0.370767i \(-0.879095\pi\)
0.928726 0.370767i \(-0.120905\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) −12.9282 −0.608765
\(452\) −0.803848 −0.0378098
\(453\) 10.3923i 0.488273i
\(454\) 16.3923 0.769329
\(455\) 0 0
\(456\) −7.46410 −0.349539
\(457\) − 24.5359i − 1.14774i −0.818946 0.573870i \(-0.805441\pi\)
0.818946 0.573870i \(-0.194559\pi\)
\(458\) 7.85641 0.367106
\(459\) 4.00000 0.186704
\(460\) − 3.73205i − 0.174008i
\(461\) 0.464102i 0.0216154i 0.999942 + 0.0108077i \(0.00344026\pi\)
−0.999942 + 0.0108077i \(0.996560\pi\)
\(462\) 12.9282i 0.601474i
\(463\) 7.07180i 0.328654i 0.986406 + 0.164327i \(0.0525453\pi\)
−0.986406 + 0.164327i \(0.947455\pi\)
\(464\) 0.267949 0.0124392
\(465\) −1.73205 −0.0803219
\(466\) 6.12436i 0.283705i
\(467\) −15.8564 −0.733747 −0.366873 0.930271i \(-0.619572\pi\)
−0.366873 + 0.930271i \(0.619572\pi\)
\(468\) 0 0
\(469\) 22.9282 1.05873
\(470\) 3.53590i 0.163099i
\(471\) 5.00000 0.230388
\(472\) 8.46410 0.389592
\(473\) − 77.1051i − 3.54530i
\(474\) 13.9282i 0.639744i
\(475\) − 7.46410i − 0.342476i
\(476\) − 8.00000i − 0.366679i
\(477\) 0.928203 0.0424995
\(478\) 16.3923 0.749767
\(479\) − 5.46410i − 0.249661i −0.992178 0.124831i \(-0.960161\pi\)
0.992178 0.124831i \(-0.0398387\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −17.7321 −0.807673
\(483\) 7.46410i 0.339628i
\(484\) 30.7846 1.39930
\(485\) −0.535898 −0.0243339
\(486\) − 1.00000i − 0.0453609i
\(487\) 23.1769i 1.05025i 0.851026 + 0.525123i \(0.175981\pi\)
−0.851026 + 0.525123i \(0.824019\pi\)
\(488\) 10.3923i 0.470438i
\(489\) − 15.0526i − 0.680700i
\(490\) −3.00000 −0.135526
\(491\) −17.3205 −0.781664 −0.390832 0.920462i \(-0.627813\pi\)
−0.390832 + 0.920462i \(0.627813\pi\)
\(492\) − 2.00000i − 0.0901670i
\(493\) −1.07180 −0.0482713
\(494\) 0 0
\(495\) 6.46410 0.290540
\(496\) − 1.73205i − 0.0777714i
\(497\) −24.7846 −1.11174
\(498\) 8.92820 0.400082
\(499\) 6.53590i 0.292587i 0.989241 + 0.146293i \(0.0467344\pi\)
−0.989241 + 0.146293i \(0.953266\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 16.3205i 0.729147i
\(502\) 15.7321i 0.702156i
\(503\) −31.1769 −1.39011 −0.695055 0.718957i \(-0.744620\pi\)
−0.695055 + 0.718957i \(0.744620\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 2.92820i 0.130303i
\(506\) 24.1244 1.07246
\(507\) 0 0
\(508\) 4.92820 0.218654
\(509\) − 1.39230i − 0.0617128i −0.999524 0.0308564i \(-0.990177\pi\)
0.999524 0.0308564i \(-0.00982346\pi\)
\(510\) −4.00000 −0.177123
\(511\) 4.00000 0.176950
\(512\) 1.00000i 0.0441942i
\(513\) − 7.46410i − 0.329548i
\(514\) − 5.33975i − 0.235526i
\(515\) − 11.8564i − 0.522456i
\(516\) 11.9282 0.525110
\(517\) −22.8564 −1.00522
\(518\) − 18.3923i − 0.808111i
\(519\) −10.9282 −0.479695
\(520\) 0 0
\(521\) −17.3205 −0.758825 −0.379413 0.925228i \(-0.623874\pi\)
−0.379413 + 0.925228i \(0.623874\pi\)
\(522\) 0.267949i 0.0117278i
\(523\) 11.7846 0.515305 0.257653 0.966238i \(-0.417051\pi\)
0.257653 + 0.966238i \(0.417051\pi\)
\(524\) 18.6603 0.815177
\(525\) − 2.00000i − 0.0872872i
\(526\) 6.12436i 0.267035i
\(527\) 6.92820i 0.301797i
\(528\) 6.46410i 0.281314i
\(529\) −9.07180 −0.394426
\(530\) −0.928203 −0.0403186
\(531\) 8.46410i 0.367311i
\(532\) −14.9282 −0.647220
\(533\) 0 0
\(534\) 0.535898 0.0231906
\(535\) − 7.85641i − 0.339662i
\(536\) 11.4641 0.495174
\(537\) 19.7321 0.851501
\(538\) 12.0000i 0.517357i
\(539\) − 19.3923i − 0.835286i
\(540\) 1.00000i 0.0430331i
\(541\) 26.9282i 1.15773i 0.815422 + 0.578867i \(0.196505\pi\)
−0.815422 + 0.578867i \(0.803495\pi\)
\(542\) 1.19615 0.0513791
\(543\) 2.92820 0.125661
\(544\) − 4.00000i − 0.171499i
\(545\) 15.8564 0.679214
\(546\) 0 0
\(547\) 22.9282 0.980339 0.490170 0.871627i \(-0.336935\pi\)
0.490170 + 0.871627i \(0.336935\pi\)
\(548\) 2.46410i 0.105261i
\(549\) −10.3923 −0.443533
\(550\) −6.46410 −0.275630
\(551\) 2.00000i 0.0852029i
\(552\) 3.73205i 0.158847i
\(553\) 27.8564i 1.18457i
\(554\) − 3.92820i − 0.166893i
\(555\) −9.19615 −0.390355
\(556\) 12.9282 0.548278
\(557\) − 17.7128i − 0.750516i −0.926920 0.375258i \(-0.877554\pi\)
0.926920 0.375258i \(-0.122446\pi\)
\(558\) 1.73205 0.0733236
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) − 25.8564i − 1.09166i
\(562\) 8.92820 0.376614
\(563\) −4.67949 −0.197217 −0.0986085 0.995126i \(-0.531439\pi\)
−0.0986085 + 0.995126i \(0.531439\pi\)
\(564\) − 3.53590i − 0.148888i
\(565\) 0.803848i 0.0338181i
\(566\) − 9.92820i − 0.417314i
\(567\) − 2.00000i − 0.0839921i
\(568\) −12.3923 −0.519970
\(569\) 29.3205 1.22918 0.614590 0.788847i \(-0.289322\pi\)
0.614590 + 0.788847i \(0.289322\pi\)
\(570\) 7.46410i 0.312637i
\(571\) −17.1769 −0.718832 −0.359416 0.933178i \(-0.617024\pi\)
−0.359416 + 0.933178i \(0.617024\pi\)
\(572\) 0 0
\(573\) −21.4641 −0.896676
\(574\) − 4.00000i − 0.166957i
\(575\) −3.73205 −0.155637
\(576\) −1.00000 −0.0416667
\(577\) − 10.0000i − 0.416305i −0.978096 0.208153i \(-0.933255\pi\)
0.978096 0.208153i \(-0.0667451\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) 11.3205i 0.470464i
\(580\) − 0.267949i − 0.0111260i
\(581\) 17.8564 0.740809
\(582\) 0.535898 0.0222137
\(583\) − 6.00000i − 0.248495i
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 31.8564 1.31598
\(587\) 2.39230i 0.0987410i 0.998781 + 0.0493705i \(0.0157215\pi\)
−0.998781 + 0.0493705i \(0.984278\pi\)
\(588\) 3.00000 0.123718
\(589\) 12.9282 0.532697
\(590\) − 8.46410i − 0.348462i
\(591\) 4.39230i 0.180675i
\(592\) − 9.19615i − 0.377960i
\(593\) 45.1051i 1.85225i 0.377223 + 0.926123i \(0.376879\pi\)
−0.377223 + 0.926123i \(0.623121\pi\)
\(594\) −6.46410 −0.265225
\(595\) −8.00000 −0.327968
\(596\) 13.5359i 0.554452i
\(597\) 5.07180 0.207575
\(598\) 0 0
\(599\) −10.3923 −0.424618 −0.212309 0.977203i \(-0.568098\pi\)
−0.212309 + 0.977203i \(0.568098\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −19.7846 −0.807031 −0.403516 0.914973i \(-0.632212\pi\)
−0.403516 + 0.914973i \(0.632212\pi\)
\(602\) 23.8564 0.972315
\(603\) 11.4641i 0.466854i
\(604\) 10.3923i 0.422857i
\(605\) − 30.7846i − 1.25157i
\(606\) − 2.92820i − 0.118950i
\(607\) −19.1769 −0.778367 −0.389183 0.921160i \(-0.627243\pi\)
−0.389183 + 0.921160i \(0.627243\pi\)
\(608\) −7.46410 −0.302709
\(609\) 0.535898i 0.0217157i
\(610\) 10.3923 0.420772
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) 39.0526i 1.57732i 0.614831 + 0.788659i \(0.289224\pi\)
−0.614831 + 0.788659i \(0.710776\pi\)
\(614\) −19.4641 −0.785507
\(615\) −2.00000 −0.0806478
\(616\) 12.9282i 0.520892i
\(617\) − 22.4641i − 0.904371i −0.891924 0.452185i \(-0.850645\pi\)
0.891924 0.452185i \(-0.149355\pi\)
\(618\) 11.8564i 0.476935i
\(619\) − 24.2487i − 0.974638i −0.873224 0.487319i \(-0.837975\pi\)
0.873224 0.487319i \(-0.162025\pi\)
\(620\) −1.73205 −0.0695608
\(621\) −3.73205 −0.149762
\(622\) − 28.3923i − 1.13843i
\(623\) 1.07180 0.0429406
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 28.0000i − 1.11911i
\(627\) −48.2487 −1.92687
\(628\) 5.00000 0.199522
\(629\) 36.7846i 1.46670i
\(630\) 2.00000i 0.0796819i
\(631\) 31.4641i 1.25257i 0.779596 + 0.626283i \(0.215425\pi\)
−0.779596 + 0.626283i \(0.784575\pi\)
\(632\) 13.9282i 0.554034i
\(633\) 11.3205 0.449950
\(634\) −14.5359 −0.577294
\(635\) − 4.92820i − 0.195570i
\(636\) 0.928203 0.0368057
\(637\) 0 0
\(638\) 1.73205 0.0685725
\(639\) − 12.3923i − 0.490232i
\(640\) 1.00000 0.0395285
\(641\) 0.143594 0.00567160 0.00283580 0.999996i \(-0.499097\pi\)
0.00283580 + 0.999996i \(0.499097\pi\)
\(642\) 7.85641i 0.310068i
\(643\) − 20.5359i − 0.809857i −0.914348 0.404928i \(-0.867296\pi\)
0.914348 0.404928i \(-0.132704\pi\)
\(644\) 7.46410i 0.294127i
\(645\) − 11.9282i − 0.469673i
\(646\) 29.8564 1.17468
\(647\) −13.3205 −0.523683 −0.261842 0.965111i \(-0.584330\pi\)
−0.261842 + 0.965111i \(0.584330\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 54.7128 2.14767
\(650\) 0 0
\(651\) 3.46410 0.135769
\(652\) − 15.0526i − 0.589504i
\(653\) 4.24871 0.166265 0.0831325 0.996539i \(-0.473508\pi\)
0.0831325 + 0.996539i \(0.473508\pi\)
\(654\) −15.8564 −0.620035
\(655\) − 18.6603i − 0.729116i
\(656\) − 2.00000i − 0.0780869i
\(657\) 2.00000i 0.0780274i
\(658\) − 7.07180i − 0.275687i
\(659\) −0.267949 −0.0104378 −0.00521891 0.999986i \(-0.501661\pi\)
−0.00521891 + 0.999986i \(0.501661\pi\)
\(660\) 6.46410 0.251615
\(661\) 8.67949i 0.337593i 0.985651 + 0.168797i \(0.0539881\pi\)
−0.985651 + 0.168797i \(0.946012\pi\)
\(662\) 16.7846 0.652352
\(663\) 0 0
\(664\) 8.92820 0.346481
\(665\) 14.9282i 0.578891i
\(666\) 9.19615 0.356344
\(667\) 1.00000 0.0387202
\(668\) 16.3205i 0.631459i
\(669\) − 20.5359i − 0.793964i
\(670\) − 11.4641i − 0.442897i
\(671\) 67.1769i 2.59334i
\(672\) −2.00000 −0.0771517
\(673\) 32.0000 1.23351 0.616755 0.787155i \(-0.288447\pi\)
0.616755 + 0.787155i \(0.288447\pi\)
\(674\) 9.32051i 0.359013i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 32.3923 1.24494 0.622469 0.782645i \(-0.286130\pi\)
0.622469 + 0.782645i \(0.286130\pi\)
\(678\) − 0.803848i − 0.0308716i
\(679\) 1.07180 0.0411318
\(680\) −4.00000 −0.153393
\(681\) 16.3923i 0.628154i
\(682\) − 11.1962i − 0.428723i
\(683\) − 40.7846i − 1.56058i −0.625418 0.780290i \(-0.715072\pi\)
0.625418 0.780290i \(-0.284928\pi\)
\(684\) − 7.46410i − 0.285397i
\(685\) 2.46410 0.0941485
\(686\) 20.0000 0.763604
\(687\) 7.85641i 0.299741i
\(688\) 11.9282 0.454758
\(689\) 0 0
\(690\) 3.73205 0.142077
\(691\) 27.1769i 1.03386i 0.856028 + 0.516929i \(0.172925\pi\)
−0.856028 + 0.516929i \(0.827075\pi\)
\(692\) −10.9282 −0.415428
\(693\) −12.9282 −0.491102
\(694\) 1.60770i 0.0610273i
\(695\) − 12.9282i − 0.490395i
\(696\) 0.267949i 0.0101566i
\(697\) 8.00000i 0.303022i
\(698\) 21.4641 0.812428
\(699\) −6.12436 −0.231644
\(700\) − 2.00000i − 0.0755929i
\(701\) −0.267949 −0.0101203 −0.00506015 0.999987i \(-0.501611\pi\)
−0.00506015 + 0.999987i \(0.501611\pi\)
\(702\) 0 0
\(703\) 68.6410 2.58884
\(704\) 6.46410i 0.243625i
\(705\) −3.53590 −0.133170
\(706\) 2.00000 0.0752710
\(707\) − 5.85641i − 0.220253i
\(708\) 8.46410i 0.318100i
\(709\) − 22.9282i − 0.861087i −0.902570 0.430543i \(-0.858322\pi\)
0.902570 0.430543i \(-0.141678\pi\)
\(710\) 12.3923i 0.465075i
\(711\) −13.9282 −0.522348
\(712\) 0.535898 0.0200836
\(713\) − 6.46410i − 0.242083i
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) 19.7321 0.737421
\(717\) 16.3923i 0.612182i
\(718\) −5.07180 −0.189278
\(719\) −34.6410 −1.29189 −0.645946 0.763383i \(-0.723537\pi\)
−0.645946 + 0.763383i \(0.723537\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 23.7128i 0.883111i
\(722\) − 36.7128i − 1.36631i
\(723\) − 17.7321i − 0.659462i
\(724\) 2.92820 0.108826
\(725\) −0.267949 −0.00995138
\(726\) 30.7846i 1.14252i
\(727\) −31.7128 −1.17616 −0.588082 0.808802i \(-0.700117\pi\)
−0.588082 + 0.808802i \(0.700117\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 2.00000i − 0.0740233i
\(731\) −47.7128 −1.76472
\(732\) −10.3923 −0.384111
\(733\) − 50.9282i − 1.88108i −0.339688 0.940538i \(-0.610322\pi\)
0.339688 0.940538i \(-0.389678\pi\)
\(734\) 15.6077i 0.576091i
\(735\) − 3.00000i − 0.110657i
\(736\) 3.73205i 0.137565i
\(737\) 74.1051 2.72970
\(738\) 2.00000 0.0736210
\(739\) 19.3205i 0.710716i 0.934730 + 0.355358i \(0.115641\pi\)
−0.934730 + 0.355358i \(0.884359\pi\)
\(740\) −9.19615 −0.338057
\(741\) 0 0
\(742\) 1.85641 0.0681508
\(743\) 40.4641i 1.48448i 0.670132 + 0.742242i \(0.266237\pi\)
−0.670132 + 0.742242i \(0.733763\pi\)
\(744\) 1.73205 0.0635001
\(745\) 13.5359 0.495917
\(746\) − 15.7846i − 0.577916i
\(747\) 8.92820i 0.326666i
\(748\) − 25.8564i − 0.945404i
\(749\) 15.7128i 0.574134i
\(750\) −1.00000 −0.0365148
\(751\) 14.0718 0.513487 0.256744 0.966480i \(-0.417350\pi\)
0.256744 + 0.966480i \(0.417350\pi\)
\(752\) − 3.53590i − 0.128941i
\(753\) −15.7321 −0.573308
\(754\) 0 0
\(755\) 10.3923 0.378215
\(756\) − 2.00000i − 0.0727393i
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 27.8564 1.01179
\(759\) 24.1244i 0.875659i
\(760\) 7.46410i 0.270751i
\(761\) 5.07180i 0.183852i 0.995766 + 0.0919262i \(0.0293024\pi\)
−0.995766 + 0.0919262i \(0.970698\pi\)
\(762\) 4.92820i 0.178530i
\(763\) −31.7128 −1.14808
\(764\) −21.4641 −0.776544
\(765\) − 4.00000i − 0.144620i
\(766\) −25.3923 −0.917461
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 11.5885i 0.417890i 0.977927 + 0.208945i \(0.0670030\pi\)
−0.977927 + 0.208945i \(0.932997\pi\)
\(770\) 12.9282 0.465900
\(771\) 5.33975 0.192306
\(772\) 11.3205i 0.407434i
\(773\) 27.7128i 0.996761i 0.866959 + 0.498380i \(0.166072\pi\)
−0.866959 + 0.498380i \(0.833928\pi\)
\(774\) 11.9282i 0.428750i
\(775\) 1.73205i 0.0622171i
\(776\) 0.535898 0.0192376
\(777\) 18.3923 0.659820
\(778\) − 23.7321i − 0.850835i
\(779\) 14.9282 0.534858
\(780\) 0 0
\(781\) −80.1051 −2.86639
\(782\) − 14.9282i − 0.533831i
\(783\) −0.267949 −0.00957572
\(784\) 3.00000 0.107143
\(785\) − 5.00000i − 0.178458i
\(786\) 18.6603i 0.665589i
\(787\) 1.73205i 0.0617409i 0.999523 + 0.0308705i \(0.00982794\pi\)
−0.999523 + 0.0308705i \(0.990172\pi\)
\(788\) 4.39230i 0.156469i
\(789\) −6.12436 −0.218033
\(790\) 13.9282 0.495543
\(791\) − 1.60770i − 0.0571631i
\(792\) −6.46410 −0.229692
\(793\) 0 0
\(794\) −12.1244 −0.430277
\(795\) − 0.928203i − 0.0329200i
\(796\) 5.07180 0.179765
\(797\) −10.1436 −0.359305 −0.179652 0.983730i \(-0.557497\pi\)
−0.179652 + 0.983730i \(0.557497\pi\)
\(798\) − 14.9282i − 0.528453i
\(799\) 14.1436i 0.500364i
\(800\) − 1.00000i − 0.0353553i
\(801\) 0.535898i 0.0189350i
\(802\) −32.0000 −1.12996
\(803\) 12.9282 0.456226
\(804\) 11.4641i 0.404308i
\(805\) 7.46410 0.263075
\(806\) 0 0
\(807\) −12.0000 −0.422420
\(808\) − 2.92820i − 0.103014i
\(809\) −42.7846 −1.50423 −0.752113 0.659034i \(-0.770965\pi\)
−0.752113 + 0.659034i \(0.770965\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 40.7846i 1.43214i 0.698028 + 0.716071i \(0.254061\pi\)
−0.698028 + 0.716071i \(0.745939\pi\)
\(812\) 0.535898i 0.0188063i
\(813\) 1.19615i 0.0419509i
\(814\) − 59.4449i − 2.08354i
\(815\) −15.0526 −0.527268
\(816\) 4.00000 0.140028
\(817\) 89.0333i 3.11488i
\(818\) −4.00000 −0.139857
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) − 14.6077i − 0.509812i −0.966966 0.254906i \(-0.917955\pi\)
0.966966 0.254906i \(-0.0820445\pi\)
\(822\) −2.46410 −0.0859454
\(823\) 39.1769 1.36562 0.682811 0.730595i \(-0.260757\pi\)
0.682811 + 0.730595i \(0.260757\pi\)
\(824\) 11.8564i 0.413037i
\(825\) − 6.46410i − 0.225051i
\(826\) 16.9282i 0.589008i
\(827\) 17.3205i 0.602293i 0.953578 + 0.301147i \(0.0973693\pi\)
−0.953578 + 0.301147i \(0.902631\pi\)
\(828\) −3.73205 −0.129698
\(829\) 16.5359 0.574315 0.287158 0.957883i \(-0.407290\pi\)
0.287158 + 0.957883i \(0.407290\pi\)
\(830\) − 8.92820i − 0.309902i
\(831\) 3.92820 0.136268
\(832\) 0 0
\(833\) −12.0000 −0.415775
\(834\) 12.9282i 0.447667i
\(835\) 16.3205 0.564794
\(836\) −48.2487 −1.66872
\(837\) 1.73205i 0.0598684i
\(838\) 22.3923i 0.773529i
\(839\) − 43.5692i − 1.50418i −0.659062 0.752088i \(-0.729047\pi\)
0.659062 0.752088i \(-0.270953\pi\)
\(840\) 2.00000i 0.0690066i
\(841\) −28.9282 −0.997524
\(842\) −4.39230 −0.151369
\(843\) 8.92820i 0.307504i
\(844\) 11.3205 0.389668
\(845\) 0 0
\(846\) 3.53590 0.121567
\(847\) 61.5692i 2.11554i
\(848\) 0.928203 0.0318746
\(849\) 9.92820 0.340735
\(850\) 4.00000i 0.137199i
\(851\) − 34.3205i − 1.17649i
\(852\) − 12.3923i − 0.424553i
\(853\) 43.8372i 1.50096i 0.660895 + 0.750478i \(0.270177\pi\)
−0.660895 + 0.750478i \(0.729823\pi\)
\(854\) −20.7846 −0.711235
\(855\) −7.46410 −0.255267
\(856\) 7.85641i 0.268526i
\(857\) 24.5167 0.837473 0.418737 0.908108i \(-0.362473\pi\)
0.418737 + 0.908108i \(0.362473\pi\)
\(858\) 0 0
\(859\) −35.1769 −1.20022 −0.600110 0.799917i \(-0.704877\pi\)
−0.600110 + 0.799917i \(0.704877\pi\)
\(860\) − 11.9282i − 0.406748i
\(861\) 4.00000 0.136320
\(862\) 28.3923 0.967046
\(863\) 35.5359i 1.20966i 0.796356 + 0.604828i \(0.206758\pi\)
−0.796356 + 0.604828i \(0.793242\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) 10.9282i 0.371570i
\(866\) 19.3205i 0.656538i
\(867\) 1.00000 0.0339618
\(868\) 3.46410 0.117579
\(869\) 90.0333i 3.05417i
\(870\) 0.267949 0.00908433
\(871\) 0 0
\(872\) −15.8564 −0.536966
\(873\) 0.535898i 0.0181374i
\(874\) −27.8564 −0.942257
\(875\) −2.00000 −0.0676123
\(876\) 2.00000i 0.0675737i
\(877\) − 3.87564i − 0.130871i −0.997857 0.0654356i \(-0.979156\pi\)
0.997857 0.0654356i \(-0.0208437\pi\)
\(878\) − 17.8564i − 0.602625i
\(879\) 31.8564i 1.07449i
\(880\) 6.46410 0.217905
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 29.9282 1.00716 0.503582 0.863947i \(-0.332015\pi\)
0.503582 + 0.863947i \(0.332015\pi\)
\(884\) 0 0
\(885\) 8.46410 0.284518
\(886\) 16.3923i 0.550710i
\(887\) 14.1244 0.474249 0.237125 0.971479i \(-0.423795\pi\)
0.237125 + 0.971479i \(0.423795\pi\)
\(888\) 9.19615 0.308603
\(889\) 9.85641i 0.330573i
\(890\) − 0.535898i − 0.0179634i
\(891\) − 6.46410i − 0.216556i
\(892\) − 20.5359i − 0.687593i
\(893\) 26.3923 0.883185
\(894\) −13.5359 −0.452708
\(895\) − 19.7321i − 0.659570i
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) 15.7128 0.524343
\(899\) − 0.464102i − 0.0154787i
\(900\) 1.00000 0.0333333
\(901\) −3.71281 −0.123692
\(902\) − 12.9282i − 0.430462i
\(903\) 23.8564i 0.793891i
\(904\) − 0.803848i − 0.0267356i
\(905\) − 2.92820i − 0.0973368i
\(906\) −10.3923 −0.345261
\(907\) −20.8564 −0.692526 −0.346263 0.938138i \(-0.612549\pi\)
−0.346263 + 0.938138i \(0.612549\pi\)
\(908\) 16.3923i 0.543998i
\(909\) 2.92820 0.0971224
\(910\) 0 0
\(911\) 24.2487 0.803396 0.401698 0.915772i \(-0.368420\pi\)
0.401698 + 0.915772i \(0.368420\pi\)
\(912\) − 7.46410i − 0.247161i
\(913\) 57.7128 1.91002
\(914\) 24.5359 0.811575
\(915\) 10.3923i 0.343559i
\(916\) 7.85641i 0.259583i
\(917\) 37.3205i 1.23243i
\(918\) 4.00000i 0.132020i
\(919\) 14.6410 0.482963 0.241481 0.970405i \(-0.422367\pi\)
0.241481 + 0.970405i \(0.422367\pi\)
\(920\) 3.73205 0.123042
\(921\) − 19.4641i − 0.641364i
\(922\) −0.464102 −0.0152844
\(923\) 0 0
\(924\) −12.9282 −0.425307
\(925\) 9.19615i 0.302368i
\(926\) −7.07180 −0.232394
\(927\) −11.8564 −0.389415
\(928\) 0.267949i 0.00879586i
\(929\) − 6.14359i − 0.201565i −0.994908 0.100782i \(-0.967865\pi\)
0.994908 0.100782i \(-0.0321346\pi\)
\(930\) − 1.73205i − 0.0567962i
\(931\) 22.3923i 0.733878i
\(932\) −6.12436 −0.200610
\(933\) 28.3923 0.929522
\(934\) − 15.8564i − 0.518837i
\(935\) −25.8564 −0.845595
\(936\) 0 0
\(937\) −24.6410 −0.804987 −0.402493 0.915423i \(-0.631856\pi\)
−0.402493 + 0.915423i \(0.631856\pi\)
\(938\) 22.9282i 0.748632i
\(939\) 28.0000 0.913745
\(940\) −3.53590 −0.115328
\(941\) − 14.7846i − 0.481965i −0.970530 0.240982i \(-0.922530\pi\)
0.970530 0.240982i \(-0.0774696\pi\)
\(942\) 5.00000i 0.162909i
\(943\) − 7.46410i − 0.243065i
\(944\) 8.46410i 0.275483i
\(945\) −2.00000 −0.0650600
\(946\) 77.1051 2.50690
\(947\) − 9.46410i − 0.307542i −0.988107 0.153771i \(-0.950858\pi\)
0.988107 0.153771i \(-0.0491418\pi\)
\(948\) −13.9282 −0.452367
\(949\) 0 0
\(950\) 7.46410 0.242167
\(951\) − 14.5359i − 0.471359i
\(952\) 8.00000 0.259281
\(953\) −12.2679 −0.397398 −0.198699 0.980061i \(-0.563672\pi\)
−0.198699 + 0.980061i \(0.563672\pi\)
\(954\) 0.928203i 0.0300517i
\(955\) 21.4641i 0.694562i
\(956\) 16.3923i 0.530165i
\(957\) 1.73205i 0.0559893i
\(958\) 5.46410 0.176537
\(959\) −4.92820 −0.159140
\(960\) 1.00000i 0.0322749i
\(961\) 28.0000 0.903226
\(962\) 0 0
\(963\) −7.85641 −0.253169
\(964\) − 17.7321i − 0.571111i
\(965\) 11.3205 0.364420
\(966\) −7.46410 −0.240154
\(967\) − 41.4641i − 1.33340i −0.745328 0.666698i \(-0.767707\pi\)
0.745328 0.666698i \(-0.232293\pi\)
\(968\) 30.7846i 0.989455i
\(969\) 29.8564i 0.959126i
\(970\) − 0.535898i − 0.0172067i
\(971\) 4.53590 0.145564 0.0727820 0.997348i \(-0.476812\pi\)
0.0727820 + 0.997348i \(0.476812\pi\)
\(972\) 1.00000 0.0320750
\(973\) 25.8564i 0.828918i
\(974\) −23.1769 −0.742636
\(975\) 0 0
\(976\) −10.3923 −0.332650
\(977\) − 12.6077i − 0.403356i −0.979452 0.201678i \(-0.935361\pi\)
0.979452 0.201678i \(-0.0646394\pi\)
\(978\) 15.0526 0.481328
\(979\) 3.46410 0.110713
\(980\) − 3.00000i − 0.0958315i
\(981\) − 15.8564i − 0.506256i
\(982\) − 17.3205i − 0.552720i
\(983\) − 36.6077i − 1.16760i −0.811896 0.583802i \(-0.801564\pi\)
0.811896 0.583802i \(-0.198436\pi\)
\(984\) 2.00000 0.0637577
\(985\) 4.39230 0.139950
\(986\) − 1.07180i − 0.0341330i
\(987\) 7.07180 0.225098
\(988\) 0 0
\(989\) 44.5167 1.41555
\(990\) 6.46410i 0.205443i
\(991\) −52.8564 −1.67904 −0.839520 0.543329i \(-0.817163\pi\)
−0.839520 + 0.543329i \(0.817163\pi\)
\(992\) 1.73205 0.0549927
\(993\) 16.7846i 0.532643i
\(994\) − 24.7846i − 0.786120i
\(995\) − 5.07180i − 0.160787i
\(996\) 8.92820i 0.282901i
\(997\) 35.5692 1.12649 0.563244 0.826290i \(-0.309553\pi\)
0.563244 + 0.826290i \(0.309553\pi\)
\(998\) −6.53590 −0.206890
\(999\) 9.19615i 0.290953i
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 5070.2.b.o.1351.3 4
13.3 even 3 390.2.bb.b.121.1 4
13.4 even 6 390.2.bb.b.361.1 yes 4
13.5 odd 4 5070.2.a.bg.1.2 2
13.8 odd 4 5070.2.a.y.1.1 2
13.12 even 2 inner 5070.2.b.o.1351.2 4
39.17 odd 6 1170.2.bs.e.361.2 4
39.29 odd 6 1170.2.bs.e.901.2 4
65.3 odd 12 1950.2.y.c.199.2 4
65.4 even 6 1950.2.bc.b.751.2 4
65.17 odd 12 1950.2.y.c.49.2 4
65.29 even 6 1950.2.bc.b.901.2 4
65.42 odd 12 1950.2.y.f.199.1 4
65.43 odd 12 1950.2.y.f.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.b.121.1 4 13.3 even 3
390.2.bb.b.361.1 yes 4 13.4 even 6
1170.2.bs.e.361.2 4 39.17 odd 6
1170.2.bs.e.901.2 4 39.29 odd 6
1950.2.y.c.49.2 4 65.17 odd 12
1950.2.y.c.199.2 4 65.3 odd 12
1950.2.y.f.49.1 4 65.43 odd 12
1950.2.y.f.199.1 4 65.42 odd 12
1950.2.bc.b.751.2 4 65.4 even 6
1950.2.bc.b.901.2 4 65.29 even 6
5070.2.a.y.1.1 2 13.8 odd 4
5070.2.a.bg.1.2 2 13.5 odd 4
5070.2.b.o.1351.2 4 13.12 even 2 inner
5070.2.b.o.1351.3 4 1.1 even 1 trivial