Properties

Label 5070.2.b.o.1351.2
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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

$q$-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 \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 6.46410i 1.94900i 0.224388 + 0.974500i \(0.427962\pi\)
−0.224388 + 0.974500i \(0.572038\pi\)
\(12\) 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 \(-0.672825\pi\)
0.516659 0.856191i \(-0.327175\pi\)
\(20\) 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 \(0.0497126\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) − 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 \(0.272810\pi\)
−0.654665 + 0.755919i \(0.727190\pi\)
\(38\) −7.46410 −1.21084
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 2.00000i 0.312348i 0.987730 + 0.156174i \(0.0499160\pi\)
−0.987730 + 0.156174i \(0.950084\pi\)
\(42\) −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 \(0.0830245\pi\)
−0.966176 + 0.257882i \(0.916975\pi\)
\(48\) −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 \(-0.814259\pi\)
0.834528 0.550966i \(-0.185741\pi\)
\(60\) − 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 \(-0.753058\pi\)
0.713867 0.700281i \(-0.246942\pi\)
\(68\) 4.00000 0.485071
\(69\) −3.73205 −0.449286
\(70\) − 2.00000i − 0.239046i
\(71\) 12.3923i 1.47070i 0.677690 + 0.735348i \(0.262981\pi\)
−0.677690 + 0.735348i \(0.737019\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 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 \(-0.836997\pi\)
0.871723 0.489999i \(-0.163003\pi\)
\(84\) 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 \(-0.990958\pi\)
0.999597 0.0284026i \(-0.00904203\pi\)
\(90\) −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 \(-0.991339\pi\)
0.999630 0.0272061i \(-0.00866105\pi\)
\(98\) − 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 \(0.274500\pi\)
−0.650643 + 0.759384i \(0.725500\pi\)
\(110\) − 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 \(0.0335679\pi\)
−0.994445 + 0.105261i \(0.966432\pi\)
\(138\) 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 \(0.187072\pi\)
−0.832216 + 0.554452i \(0.812928\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) 10.3923i 0.845714i 0.906196 + 0.422857i \(0.138973\pi\)
−0.906196 + 0.422857i \(0.861027\pi\)
\(152\) 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 \(-0.799323\pi\)
0.807766 0.589504i \(-0.200677\pi\)
\(164\) − 2.00000i − 0.156174i
\(165\) −6.46410 −0.503230
\(166\) −8.92820 −0.692963
\(167\) 16.3205i 1.26292i 0.775409 + 0.631459i \(0.217544\pi\)
−0.775409 + 0.631459i \(0.782456\pi\)
\(168\) 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 \(0.133576\pi\)
−0.913235 + 0.407434i \(0.866424\pi\)
\(194\) −0.535898 −0.0384753
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 4.39230i 0.312939i 0.987683 + 0.156469i \(0.0500113\pi\)
−0.987683 + 0.156469i \(0.949989\pi\)
\(198\) 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 \(-0.758667\pi\)
0.726097 0.687593i \(-0.241333\pi\)
\(224\) 2.00000 0.133631
\(225\) −1.00000 −0.0666667
\(226\) − 0.803848i − 0.0534711i
\(227\) 16.3923i 1.08800i 0.839087 + 0.543998i \(0.183090\pi\)
−0.839087 + 0.543998i \(0.816910\pi\)
\(228\) − 7.46410i − 0.494322i
\(229\) 7.85641i 0.519166i 0.965721 + 0.259583i \(0.0835851\pi\)
−0.965721 + 0.259583i \(0.916415\pi\)
\(230\) −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 \(0.177870\pi\)
−0.847894 + 0.530165i \(0.822130\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) − 17.7321i − 1.14222i −0.820873 0.571111i \(-0.806513\pi\)
0.820873 0.571111i \(-0.193487\pi\)
\(242\) 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 \(0.0115669\pi\)
−0.999340 + 0.0363305i \(0.988433\pi\)
\(272\) −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 \(0.0858032\pi\)
−0.963889 + 0.266306i \(0.914197\pi\)
\(282\) −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 \(0.380658\pi\)
−0.366202 + 0.930536i \(0.619342\pi\)
\(294\) 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 \(-0.812551\pi\)
0.831558 0.555437i \(-0.187449\pi\)
\(308\) 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 \(-0.866154\pi\)
0.912889 0.408209i \(-0.133846\pi\)
\(318\) 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 \(0.152611\pi\)
−0.887253 + 0.461283i \(0.847389\pi\)
\(332\) 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 \(0.194793\pi\)
−0.818523 + 0.574474i \(0.805207\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 6.46410 0.344538
\(353\) 2.00000i 0.106449i 0.998583 + 0.0532246i \(0.0169499\pi\)
−0.998583 + 0.0532246i \(0.983050\pi\)
\(354\) 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 \(-0.957269\pi\)
0.991003 0.133840i \(-0.0427307\pi\)
\(360\) 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 \(0.253775\pi\)
−0.698670 + 0.715444i \(0.746225\pi\)
\(380\) 7.46410 0.382900
\(381\) 4.92820 0.252479
\(382\) − 21.4641i − 1.09820i
\(383\) − 25.3923i − 1.29749i −0.761007 0.648743i \(-0.775295\pi\)
0.761007 0.648743i \(-0.224705\pi\)
\(384\) − 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 \(-0.901594\pi\)
0.952592 0.304252i \(-0.0984065\pi\)
\(398\) 5.07180i 0.254226i
\(399\) −14.9282 −0.747345
\(400\) −1.00000 −0.0500000
\(401\) − 32.0000i − 1.59800i −0.601329 0.799002i \(-0.705362\pi\)
0.601329 0.799002i \(-0.294638\pi\)
\(402\) 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 \(-0.968470\pi\)
0.995098 0.0988936i \(-0.0315304\pi\)
\(410\) − 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 \(-0.965865\pi\)
0.994255 0.107034i \(-0.0341353\pi\)
\(422\) 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 \(0.239676\pi\)
−0.729665 + 0.683805i \(0.760324\pi\)
\(432\) −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 \(0.120905\pi\)
−0.928726 + 0.370767i \(0.879095\pi\)
\(450\) 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 \(0.194559\pi\)
−0.818946 + 0.573870i \(0.805441\pi\)
\(458\) 7.85641 0.367106
\(459\) 4.00000 0.186704
\(460\) 3.73205i 0.174008i
\(461\) − 0.464102i − 0.0216154i −0.999942 0.0108077i \(-0.996560\pi\)
0.999942 0.0108077i \(-0.00344026\pi\)
\(462\) − 12.9282i − 0.601474i
\(463\) − 7.07180i − 0.328654i −0.986406 0.164327i \(-0.947455\pi\)
0.986406 0.164327i \(-0.0525453\pi\)
\(464\) 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 \(0.0398387\pi\)
−0.992178 + 0.124831i \(0.960161\pi\)
\(480\) 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 \(-0.824019\pi\)
0.851026 0.525123i \(-0.175981\pi\)
\(488\) − 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 \(-0.953266\pi\)
0.989241 0.146293i \(-0.0467344\pi\)
\(500\) − 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 \(0.00982346\pi\)
−0.999524 + 0.0308564i \(0.990177\pi\)
\(510\) −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 \(-0.803495\pi\)
0.815422 0.578867i \(-0.196505\pi\)
\(542\) 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 \(0.122446\pi\)
−0.926920 + 0.375258i \(0.877554\pi\)
\(558\) 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 \(0.0667451\pi\)
−0.978096 + 0.208153i \(0.933255\pi\)
\(578\) 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 \(-0.984278\pi\)
0.998781 0.0493705i \(-0.0157215\pi\)
\(588\) 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 \(-0.623121\pi\)
0.377223 0.926123i \(-0.376879\pi\)
\(594\) −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 \(-0.710776\pi\)
0.614831 0.788659i \(-0.289224\pi\)
\(614\) −19.4641 −0.785507
\(615\) −2.00000 −0.0806478
\(616\) − 12.9282i − 0.520892i
\(617\) 22.4641i 0.904371i 0.891924 + 0.452185i \(0.149355\pi\)
−0.891924 + 0.452185i \(0.850645\pi\)
\(618\) − 11.8564i − 0.476935i
\(619\) 24.2487i 0.974638i 0.873224 + 0.487319i \(0.162025\pi\)
−0.873224 + 0.487319i \(0.837975\pi\)
\(620\) −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 \(-0.784575\pi\)
0.779596 0.626283i \(-0.215425\pi\)
\(632\) − 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 \(0.132704\pi\)
−0.914348 + 0.404928i \(0.867296\pi\)
\(644\) − 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 \(-0.946012\pi\)
0.985651 0.168797i \(-0.0539881\pi\)
\(662\) 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 \(0.284928\pi\)
−0.625418 + 0.780290i \(0.715072\pi\)
\(684\) 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 \(-0.827075\pi\)
0.856028 0.516929i \(-0.172925\pi\)
\(692\) −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 \(0.141678\pi\)
−0.902570 + 0.430543i \(0.858322\pi\)
\(710\) − 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 \(0.389678\pi\)
−0.339688 + 0.940538i \(0.610322\pi\)
\(734\) − 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 \(-0.884359\pi\)
0.934730 0.355358i \(-0.115641\pi\)
\(740\) −9.19615 −0.338057
\(741\) 0 0
\(742\) 1.85641 0.0681508
\(743\) − 40.4641i − 1.48448i −0.670132 0.742242i \(-0.733763\pi\)
0.670132 0.742242i \(-0.266237\pi\)
\(744\) 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 \(-0.970698\pi\)
0.995766 0.0919262i \(-0.0293024\pi\)
\(762\) − 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 \(-0.932997\pi\)
0.977927 0.208945i \(-0.0670030\pi\)
\(770\) 12.9282 0.465900
\(771\) 5.33975 0.192306
\(772\) − 11.3205i − 0.407434i
\(773\) − 27.7128i − 0.996761i −0.866959 0.498380i \(-0.833928\pi\)
0.866959 0.498380i \(-0.166072\pi\)
\(774\) − 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 \(-0.990172\pi\)
0.999523 0.0308705i \(-0.00982794\pi\)
\(788\) − 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 \(-0.745939\pi\)
0.698028 0.716071i \(-0.254061\pi\)
\(812\) − 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 \(0.0820445\pi\)
−0.966966 + 0.254906i \(0.917955\pi\)
\(822\) −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 \(-0.902631\pi\)
0.953578 0.301147i \(-0.0973693\pi\)
\(828\) −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 \(0.270953\pi\)
−0.659062 + 0.752088i \(0.729047\pi\)
\(840\) − 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 \(-0.729823\pi\)
0.660895 0.750478i \(-0.270177\pi\)
\(854\) −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 \(-0.793242\pi\)
0.796356 0.604828i \(-0.206758\pi\)
\(864\) 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 \(0.0208437\pi\)
−0.997857 + 0.0654356i \(0.979156\pi\)
\(878\) 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 \(0.0321346\pi\)
−0.994908 + 0.100782i \(0.967865\pi\)
\(930\) 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 \(0.0774696\pi\)
−0.970530 + 0.240982i \(0.922530\pi\)
\(942\) − 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 \(0.0491418\pi\)
−0.988107 + 0.153771i \(0.950858\pi\)
\(948\) −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 \(0.232293\pi\)
−0.745328 + 0.666698i \(0.767707\pi\)
\(968\) − 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 \(0.0646394\pi\)
−0.979452 + 0.201678i \(0.935361\pi\)
\(978\) 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 \(0.198436\pi\)
−0.811896 + 0.583802i \(0.801564\pi\)
\(984\) 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.1351.2 4
13.5 odd 4 5070.2.a.y.1.1 2
13.8 odd 4 5070.2.a.bg.1.2 2
13.9 even 3 390.2.bb.b.361.1 yes 4
13.10 even 6 390.2.bb.b.121.1 4
13.12 even 2 inner 5070.2.b.o.1351.3 4
39.23 odd 6 1170.2.bs.e.901.2 4
39.35 odd 6 1170.2.bs.e.361.2 4
65.9 even 6 1950.2.bc.b.751.2 4
65.22 odd 12 1950.2.y.c.49.2 4
65.23 odd 12 1950.2.y.c.199.2 4
65.48 odd 12 1950.2.y.f.49.1 4
65.49 even 6 1950.2.bc.b.901.2 4
65.62 odd 12 1950.2.y.f.199.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.b.121.1 4 13.10 even 6
390.2.bb.b.361.1 yes 4 13.9 even 3
1170.2.bs.e.361.2 4 39.35 odd 6
1170.2.bs.e.901.2 4 39.23 odd 6
1950.2.y.c.49.2 4 65.22 odd 12
1950.2.y.c.199.2 4 65.23 odd 12
1950.2.y.f.49.1 4 65.48 odd 12
1950.2.y.f.199.1 4 65.62 odd 12
1950.2.bc.b.751.2 4 65.9 even 6
1950.2.bc.b.901.2 4 65.49 even 6
5070.2.a.y.1.1 2 13.5 odd 4
5070.2.a.bg.1.2 2 13.8 odd 4
5070.2.b.o.1351.2 4 1.1 even 1 trivial
5070.2.b.o.1351.3 4 13.12 even 2 inner