Properties

Label 5070.2.b.p.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: \(\Z[a_1, \ldots, a_{11}]\)
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.p.1351.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +3.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +3.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -0.267949i q^{11} +1.00000 q^{12} +3.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000i q^{18} +5.73205i q^{19} -1.00000i q^{20} -3.00000i q^{21} -0.267949 q^{22} +3.46410 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -3.00000i q^{28} +1.46410 q^{29} -1.00000 q^{30} +4.92820i q^{31} -1.00000i q^{32} +0.267949i q^{33} -4.00000i q^{34} -3.00000 q^{35} -1.00000 q^{36} +5.92820i q^{37} +5.73205 q^{38} -1.00000 q^{40} +4.00000i q^{41} -3.00000 q^{42} +6.00000 q^{43} +0.267949i q^{44} +1.00000i q^{45} -3.46410i q^{46} -6.46410i q^{47} -1.00000 q^{48} -2.00000 q^{49} +1.00000i q^{50} -4.00000 q^{51} -0.267949 q^{53} +1.00000i q^{54} +0.267949 q^{55} -3.00000 q^{56} -5.73205i q^{57} -1.46410i q^{58} -11.4641i q^{59} +1.00000i q^{60} -0.535898 q^{61} +4.92820 q^{62} +3.00000i q^{63} -1.00000 q^{64} +0.267949 q^{66} -1.46410i q^{67} -4.00000 q^{68} -3.46410 q^{69} +3.00000i q^{70} -12.9282i q^{71} +1.00000i q^{72} -6.92820i q^{73} +5.92820 q^{74} +1.00000 q^{75} -5.73205i q^{76} +0.803848 q^{77} +3.07180 q^{79} +1.00000i q^{80} +1.00000 q^{81} +4.00000 q^{82} +9.46410i q^{83} +3.00000i q^{84} +4.00000i q^{85} -6.00000i q^{86} -1.46410 q^{87} +0.267949 q^{88} +14.1244i q^{89} +1.00000 q^{90} -3.46410 q^{92} -4.92820i q^{93} -6.46410 q^{94} -5.73205 q^{95} +1.00000i q^{96} +8.39230i q^{97} +2.00000i q^{98} -0.267949i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 4q^{4} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 4q^{4} + 4q^{9} + 4q^{10} + 4q^{12} + 12q^{14} + 4q^{16} + 16q^{17} - 8q^{22} - 4q^{25} - 4q^{27} - 8q^{29} - 4q^{30} - 12q^{35} - 4q^{36} + 16q^{38} - 4q^{40} - 12q^{42} + 24q^{43} - 4q^{48} - 8q^{49} - 16q^{51} - 8q^{53} + 8q^{55} - 12q^{56} - 16q^{61} - 8q^{62} - 4q^{64} + 8q^{66} - 16q^{68} - 4q^{74} + 4q^{75} + 24q^{77} + 40q^{79} + 4q^{81} + 16q^{82} + 8q^{87} + 8q^{88} + 4q^{90} - 12q^{94} - 16q^{95} + O(q^{100}) \)

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
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) − 0.267949i − 0.0807897i −0.999184 0.0403949i \(-0.987138\pi\)
0.999184 0.0403949i \(-0.0128616\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 3.00000 0.801784
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 5.73205i 1.31502i 0.753445 + 0.657511i \(0.228391\pi\)
−0.753445 + 0.657511i \(0.771609\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) − 3.00000i − 0.654654i
\(22\) −0.267949 −0.0571270
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) − 3.00000i − 0.566947i
\(29\) 1.46410 0.271877 0.135938 0.990717i \(-0.456595\pi\)
0.135938 + 0.990717i \(0.456595\pi\)
\(30\) −1.00000 −0.182574
\(31\) 4.92820i 0.885131i 0.896736 + 0.442566i \(0.145932\pi\)
−0.896736 + 0.442566i \(0.854068\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0.267949i 0.0466440i
\(34\) − 4.00000i − 0.685994i
\(35\) −3.00000 −0.507093
\(36\) −1.00000 −0.166667
\(37\) 5.92820i 0.974591i 0.873237 + 0.487295i \(0.162016\pi\)
−0.873237 + 0.487295i \(0.837984\pi\)
\(38\) 5.73205 0.929861
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 4.00000i 0.624695i 0.949968 + 0.312348i \(0.101115\pi\)
−0.949968 + 0.312348i \(0.898885\pi\)
\(42\) −3.00000 −0.462910
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0.267949i 0.0403949i
\(45\) 1.00000i 0.149071i
\(46\) − 3.46410i − 0.510754i
\(47\) − 6.46410i − 0.942886i −0.881897 0.471443i \(-0.843733\pi\)
0.881897 0.471443i \(-0.156267\pi\)
\(48\) −1.00000 −0.144338
\(49\) −2.00000 −0.285714
\(50\) 1.00000i 0.141421i
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −0.267949 −0.0368057 −0.0184028 0.999831i \(-0.505858\pi\)
−0.0184028 + 0.999831i \(0.505858\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0.267949 0.0361303
\(56\) −3.00000 −0.400892
\(57\) − 5.73205i − 0.759229i
\(58\) − 1.46410i − 0.192246i
\(59\) − 11.4641i − 1.49250i −0.665666 0.746249i \(-0.731853\pi\)
0.665666 0.746249i \(-0.268147\pi\)
\(60\) 1.00000i 0.129099i
\(61\) −0.535898 −0.0686148 −0.0343074 0.999411i \(-0.510923\pi\)
−0.0343074 + 0.999411i \(0.510923\pi\)
\(62\) 4.92820 0.625882
\(63\) 3.00000i 0.377964i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0.267949 0.0329823
\(67\) − 1.46410i − 0.178868i −0.995993 0.0894342i \(-0.971494\pi\)
0.995993 0.0894342i \(-0.0285059\pi\)
\(68\) −4.00000 −0.485071
\(69\) −3.46410 −0.417029
\(70\) 3.00000i 0.358569i
\(71\) − 12.9282i − 1.53430i −0.641470 0.767148i \(-0.721675\pi\)
0.641470 0.767148i \(-0.278325\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 6.92820i − 0.810885i −0.914121 0.405442i \(-0.867117\pi\)
0.914121 0.405442i \(-0.132883\pi\)
\(74\) 5.92820 0.689140
\(75\) 1.00000 0.115470
\(76\) − 5.73205i − 0.657511i
\(77\) 0.803848 0.0916069
\(78\) 0 0
\(79\) 3.07180 0.345604 0.172802 0.984957i \(-0.444718\pi\)
0.172802 + 0.984957i \(0.444718\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) 9.46410i 1.03882i 0.854525 + 0.519410i \(0.173848\pi\)
−0.854525 + 0.519410i \(0.826152\pi\)
\(84\) 3.00000i 0.327327i
\(85\) 4.00000i 0.433861i
\(86\) − 6.00000i − 0.646997i
\(87\) −1.46410 −0.156968
\(88\) 0.267949 0.0285635
\(89\) 14.1244i 1.49718i 0.663034 + 0.748589i \(0.269269\pi\)
−0.663034 + 0.748589i \(0.730731\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −3.46410 −0.361158
\(93\) − 4.92820i − 0.511031i
\(94\) −6.46410 −0.666721
\(95\) −5.73205 −0.588096
\(96\) 1.00000i 0.102062i
\(97\) 8.39230i 0.852109i 0.904697 + 0.426055i \(0.140097\pi\)
−0.904697 + 0.426055i \(0.859903\pi\)
\(98\) 2.00000i 0.202031i
\(99\) − 0.267949i − 0.0269299i
\(100\) 1.00000 0.100000
\(101\) 12.3923 1.23308 0.616540 0.787323i \(-0.288534\pi\)
0.616540 + 0.787323i \(0.288534\pi\)
\(102\) 4.00000i 0.396059i
\(103\) −11.5885 −1.14184 −0.570922 0.821004i \(-0.693414\pi\)
−0.570922 + 0.821004i \(0.693414\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0.267949i 0.0260255i
\(107\) −12.9282 −1.24982 −0.624908 0.780698i \(-0.714864\pi\)
−0.624908 + 0.780698i \(0.714864\pi\)
\(108\) 1.00000 0.0962250
\(109\) − 10.3923i − 0.995402i −0.867349 0.497701i \(-0.834178\pi\)
0.867349 0.497701i \(-0.165822\pi\)
\(110\) − 0.267949i − 0.0255480i
\(111\) − 5.92820i − 0.562680i
\(112\) 3.00000i 0.283473i
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) −5.73205 −0.536856
\(115\) 3.46410i 0.323029i
\(116\) −1.46410 −0.135938
\(117\) 0 0
\(118\) −11.4641 −1.05536
\(119\) 12.0000i 1.10004i
\(120\) 1.00000 0.0912871
\(121\) 10.9282 0.993473
\(122\) 0.535898i 0.0485180i
\(123\) − 4.00000i − 0.360668i
\(124\) − 4.92820i − 0.442566i
\(125\) − 1.00000i − 0.0894427i
\(126\) 3.00000 0.267261
\(127\) −12.6603 −1.12342 −0.561708 0.827336i \(-0.689856\pi\)
−0.561708 + 0.827336i \(0.689856\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 14.3205 1.25119 0.625594 0.780149i \(-0.284857\pi\)
0.625594 + 0.780149i \(0.284857\pi\)
\(132\) − 0.267949i − 0.0233220i
\(133\) −17.1962 −1.49110
\(134\) −1.46410 −0.126479
\(135\) − 1.00000i − 0.0860663i
\(136\) 4.00000i 0.342997i
\(137\) 13.4641i 1.15032i 0.818042 + 0.575158i \(0.195059\pi\)
−0.818042 + 0.575158i \(0.804941\pi\)
\(138\) 3.46410i 0.294884i
\(139\) −19.7846 −1.67811 −0.839054 0.544048i \(-0.816891\pi\)
−0.839054 + 0.544048i \(0.816891\pi\)
\(140\) 3.00000 0.253546
\(141\) 6.46410i 0.544376i
\(142\) −12.9282 −1.08491
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 1.46410i 0.121587i
\(146\) −6.92820 −0.573382
\(147\) 2.00000 0.164957
\(148\) − 5.92820i − 0.487295i
\(149\) 18.7846i 1.53890i 0.638710 + 0.769448i \(0.279468\pi\)
−0.638710 + 0.769448i \(0.720532\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) 22.7846i 1.85419i 0.374832 + 0.927093i \(0.377700\pi\)
−0.374832 + 0.927093i \(0.622300\pi\)
\(152\) −5.73205 −0.464931
\(153\) 4.00000 0.323381
\(154\) − 0.803848i − 0.0647759i
\(155\) −4.92820 −0.395843
\(156\) 0 0
\(157\) 21.1962 1.69164 0.845819 0.533471i \(-0.179113\pi\)
0.845819 + 0.533471i \(0.179113\pi\)
\(158\) − 3.07180i − 0.244379i
\(159\) 0.267949 0.0212498
\(160\) 1.00000 0.0790569
\(161\) 10.3923i 0.819028i
\(162\) − 1.00000i − 0.0785674i
\(163\) − 2.92820i − 0.229355i −0.993403 0.114677i \(-0.963417\pi\)
0.993403 0.114677i \(-0.0365834\pi\)
\(164\) − 4.00000i − 0.312348i
\(165\) −0.267949 −0.0208598
\(166\) 9.46410 0.734557
\(167\) 0.464102i 0.0359133i 0.999839 + 0.0179566i \(0.00571608\pi\)
−0.999839 + 0.0179566i \(0.994284\pi\)
\(168\) 3.00000 0.231455
\(169\) 0 0
\(170\) 4.00000 0.306786
\(171\) 5.73205i 0.438341i
\(172\) −6.00000 −0.457496
\(173\) −2.12436 −0.161512 −0.0807559 0.996734i \(-0.525733\pi\)
−0.0807559 + 0.996734i \(0.525733\pi\)
\(174\) 1.46410i 0.110993i
\(175\) − 3.00000i − 0.226779i
\(176\) − 0.267949i − 0.0201974i
\(177\) 11.4641i 0.861695i
\(178\) 14.1244 1.05867
\(179\) 9.07180 0.678058 0.339029 0.940776i \(-0.389902\pi\)
0.339029 + 0.940776i \(0.389902\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) 16.9282 1.25826 0.629132 0.777299i \(-0.283411\pi\)
0.629132 + 0.777299i \(0.283411\pi\)
\(182\) 0 0
\(183\) 0.535898 0.0396147
\(184\) 3.46410i 0.255377i
\(185\) −5.92820 −0.435850
\(186\) −4.92820 −0.361353
\(187\) − 1.07180i − 0.0783775i
\(188\) 6.46410i 0.471443i
\(189\) − 3.00000i − 0.218218i
\(190\) 5.73205i 0.415847i
\(191\) −17.3205 −1.25327 −0.626634 0.779314i \(-0.715568\pi\)
−0.626634 + 0.779314i \(0.715568\pi\)
\(192\) 1.00000 0.0721688
\(193\) 9.85641i 0.709480i 0.934965 + 0.354740i \(0.115431\pi\)
−0.934965 + 0.354740i \(0.884569\pi\)
\(194\) 8.39230 0.602532
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 11.3923i 0.811668i 0.913947 + 0.405834i \(0.133019\pi\)
−0.913947 + 0.405834i \(0.866981\pi\)
\(198\) −0.267949 −0.0190423
\(199\) −24.9282 −1.76711 −0.883557 0.468324i \(-0.844858\pi\)
−0.883557 + 0.468324i \(0.844858\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) 1.46410i 0.103270i
\(202\) − 12.3923i − 0.871920i
\(203\) 4.39230i 0.308279i
\(204\) 4.00000 0.280056
\(205\) −4.00000 −0.279372
\(206\) 11.5885i 0.807406i
\(207\) 3.46410 0.240772
\(208\) 0 0
\(209\) 1.53590 0.106240
\(210\) − 3.00000i − 0.207020i
\(211\) −8.07180 −0.555685 −0.277843 0.960627i \(-0.589619\pi\)
−0.277843 + 0.960627i \(0.589619\pi\)
\(212\) 0.267949 0.0184028
\(213\) 12.9282i 0.885826i
\(214\) 12.9282i 0.883754i
\(215\) 6.00000i 0.409197i
\(216\) − 1.00000i − 0.0680414i
\(217\) −14.7846 −1.00364
\(218\) −10.3923 −0.703856
\(219\) 6.92820i 0.468165i
\(220\) −0.267949 −0.0180651
\(221\) 0 0
\(222\) −5.92820 −0.397875
\(223\) 18.8564i 1.26272i 0.775491 + 0.631359i \(0.217503\pi\)
−0.775491 + 0.631359i \(0.782497\pi\)
\(224\) 3.00000 0.200446
\(225\) −1.00000 −0.0666667
\(226\) 12.0000i 0.798228i
\(227\) 6.53590i 0.433803i 0.976194 + 0.216901i \(0.0695950\pi\)
−0.976194 + 0.216901i \(0.930405\pi\)
\(228\) 5.73205i 0.379614i
\(229\) 4.53590i 0.299741i 0.988706 + 0.149870i \(0.0478856\pi\)
−0.988706 + 0.149870i \(0.952114\pi\)
\(230\) 3.46410 0.228416
\(231\) −0.803848 −0.0528893
\(232\) 1.46410i 0.0961230i
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 6.46410 0.421671
\(236\) 11.4641i 0.746249i
\(237\) −3.07180 −0.199535
\(238\) 12.0000 0.777844
\(239\) − 3.46410i − 0.224074i −0.993704 0.112037i \(-0.964262\pi\)
0.993704 0.112037i \(-0.0357375\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) − 25.1962i − 1.62303i −0.584334 0.811513i \(-0.698644\pi\)
0.584334 0.811513i \(-0.301356\pi\)
\(242\) − 10.9282i − 0.702492i
\(243\) −1.00000 −0.0641500
\(244\) 0.535898 0.0343074
\(245\) − 2.00000i − 0.127775i
\(246\) −4.00000 −0.255031
\(247\) 0 0
\(248\) −4.92820 −0.312941
\(249\) − 9.46410i − 0.599763i
\(250\) −1.00000 −0.0632456
\(251\) 19.5359 1.23309 0.616547 0.787318i \(-0.288531\pi\)
0.616547 + 0.787318i \(0.288531\pi\)
\(252\) − 3.00000i − 0.188982i
\(253\) − 0.928203i − 0.0583556i
\(254\) 12.6603i 0.794375i
\(255\) − 4.00000i − 0.250490i
\(256\) 1.00000 0.0625000
\(257\) 14.5359 0.906724 0.453362 0.891326i \(-0.350224\pi\)
0.453362 + 0.891326i \(0.350224\pi\)
\(258\) 6.00000i 0.373544i
\(259\) −17.7846 −1.10508
\(260\) 0 0
\(261\) 1.46410 0.0906256
\(262\) − 14.3205i − 0.884724i
\(263\) −24.5167 −1.51176 −0.755881 0.654709i \(-0.772791\pi\)
−0.755881 + 0.654709i \(0.772791\pi\)
\(264\) −0.267949 −0.0164911
\(265\) − 0.267949i − 0.0164600i
\(266\) 17.1962i 1.05436i
\(267\) − 14.1244i − 0.864397i
\(268\) 1.46410i 0.0894342i
\(269\) −17.0718 −1.04089 −0.520443 0.853896i \(-0.674233\pi\)
−0.520443 + 0.853896i \(0.674233\pi\)
\(270\) −1.00000 −0.0608581
\(271\) − 24.3923i − 1.48173i −0.671656 0.740863i \(-0.734417\pi\)
0.671656 0.740863i \(-0.265583\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 13.4641 0.813396
\(275\) 0.267949i 0.0161579i
\(276\) 3.46410 0.208514
\(277\) 11.5885 0.696283 0.348141 0.937442i \(-0.386813\pi\)
0.348141 + 0.937442i \(0.386813\pi\)
\(278\) 19.7846i 1.18660i
\(279\) 4.92820i 0.295044i
\(280\) − 3.00000i − 0.179284i
\(281\) − 6.92820i − 0.413302i −0.978415 0.206651i \(-0.933744\pi\)
0.978415 0.206651i \(-0.0662565\pi\)
\(282\) 6.46410 0.384932
\(283\) −6.39230 −0.379983 −0.189992 0.981786i \(-0.560846\pi\)
−0.189992 + 0.981786i \(0.560846\pi\)
\(284\) 12.9282i 0.767148i
\(285\) 5.73205 0.339537
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) − 1.00000i − 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) 1.46410 0.0859750
\(291\) − 8.39230i − 0.491966i
\(292\) 6.92820i 0.405442i
\(293\) 29.2487i 1.70873i 0.519675 + 0.854364i \(0.326053\pi\)
−0.519675 + 0.854364i \(0.673947\pi\)
\(294\) − 2.00000i − 0.116642i
\(295\) 11.4641 0.667466
\(296\) −5.92820 −0.344570
\(297\) 0.267949i 0.0155480i
\(298\) 18.7846 1.08816
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 18.0000i 1.03750i
\(302\) 22.7846 1.31111
\(303\) −12.3923 −0.711919
\(304\) 5.73205i 0.328756i
\(305\) − 0.535898i − 0.0306855i
\(306\) − 4.00000i − 0.228665i
\(307\) 28.2487i 1.61224i 0.591753 + 0.806120i \(0.298436\pi\)
−0.591753 + 0.806120i \(0.701564\pi\)
\(308\) −0.803848 −0.0458035
\(309\) 11.5885 0.659244
\(310\) 4.92820i 0.279903i
\(311\) 18.9282 1.07332 0.536660 0.843799i \(-0.319686\pi\)
0.536660 + 0.843799i \(0.319686\pi\)
\(312\) 0 0
\(313\) −33.3205 −1.88339 −0.941693 0.336473i \(-0.890766\pi\)
−0.941693 + 0.336473i \(0.890766\pi\)
\(314\) − 21.1962i − 1.19617i
\(315\) −3.00000 −0.169031
\(316\) −3.07180 −0.172802
\(317\) − 30.4641i − 1.71103i −0.517774 0.855517i \(-0.673239\pi\)
0.517774 0.855517i \(-0.326761\pi\)
\(318\) − 0.267949i − 0.0150258i
\(319\) − 0.392305i − 0.0219649i
\(320\) − 1.00000i − 0.0559017i
\(321\) 12.9282 0.721582
\(322\) 10.3923 0.579141
\(323\) 22.9282i 1.27576i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −2.92820 −0.162178
\(327\) 10.3923i 0.574696i
\(328\) −4.00000 −0.220863
\(329\) 19.3923 1.06913
\(330\) 0.267949i 0.0147501i
\(331\) 14.3923i 0.791073i 0.918450 + 0.395536i \(0.129441\pi\)
−0.918450 + 0.395536i \(0.870559\pi\)
\(332\) − 9.46410i − 0.519410i
\(333\) 5.92820i 0.324864i
\(334\) 0.464102 0.0253945
\(335\) 1.46410 0.0799924
\(336\) − 3.00000i − 0.163663i
\(337\) −26.3923 −1.43768 −0.718840 0.695175i \(-0.755327\pi\)
−0.718840 + 0.695175i \(0.755327\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) − 4.00000i − 0.216930i
\(341\) 1.32051 0.0715095
\(342\) 5.73205 0.309954
\(343\) 15.0000i 0.809924i
\(344\) 6.00000i 0.323498i
\(345\) − 3.46410i − 0.186501i
\(346\) 2.12436i 0.114206i
\(347\) 4.39230 0.235791 0.117896 0.993026i \(-0.462385\pi\)
0.117896 + 0.993026i \(0.462385\pi\)
\(348\) 1.46410 0.0784841
\(349\) 10.5359i 0.563974i 0.959418 + 0.281987i \(0.0909935\pi\)
−0.959418 + 0.281987i \(0.909007\pi\)
\(350\) −3.00000 −0.160357
\(351\) 0 0
\(352\) −0.267949 −0.0142817
\(353\) − 7.60770i − 0.404917i −0.979291 0.202458i \(-0.935107\pi\)
0.979291 0.202458i \(-0.0648931\pi\)
\(354\) 11.4641 0.609310
\(355\) 12.9282 0.686158
\(356\) − 14.1244i − 0.748589i
\(357\) − 12.0000i − 0.635107i
\(358\) − 9.07180i − 0.479459i
\(359\) − 0.928203i − 0.0489887i −0.999700 0.0244943i \(-0.992202\pi\)
0.999700 0.0244943i \(-0.00779757\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −13.8564 −0.729285
\(362\) − 16.9282i − 0.889727i
\(363\) −10.9282 −0.573582
\(364\) 0 0
\(365\) 6.92820 0.362639
\(366\) − 0.535898i − 0.0280119i
\(367\) 21.3205 1.11292 0.556461 0.830874i \(-0.312159\pi\)
0.556461 + 0.830874i \(0.312159\pi\)
\(368\) 3.46410 0.180579
\(369\) 4.00000i 0.208232i
\(370\) 5.92820i 0.308193i
\(371\) − 0.803848i − 0.0417337i
\(372\) 4.92820i 0.255515i
\(373\) 9.07180 0.469720 0.234860 0.972029i \(-0.424537\pi\)
0.234860 + 0.972029i \(0.424537\pi\)
\(374\) −1.07180 −0.0554213
\(375\) 1.00000i 0.0516398i
\(376\) 6.46410 0.333361
\(377\) 0 0
\(378\) −3.00000 −0.154303
\(379\) 9.73205i 0.499902i 0.968259 + 0.249951i \(0.0804145\pi\)
−0.968259 + 0.249951i \(0.919586\pi\)
\(380\) 5.73205 0.294048
\(381\) 12.6603 0.648604
\(382\) 17.3205i 0.886194i
\(383\) 4.78461i 0.244482i 0.992500 + 0.122241i \(0.0390081\pi\)
−0.992500 + 0.122241i \(0.960992\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 0.803848i 0.0409679i
\(386\) 9.85641 0.501678
\(387\) 6.00000 0.304997
\(388\) − 8.39230i − 0.426055i
\(389\) 17.8564 0.905356 0.452678 0.891674i \(-0.350469\pi\)
0.452678 + 0.891674i \(0.350469\pi\)
\(390\) 0 0
\(391\) 13.8564 0.700749
\(392\) − 2.00000i − 0.101015i
\(393\) −14.3205 −0.722374
\(394\) 11.3923 0.573936
\(395\) 3.07180i 0.154559i
\(396\) 0.267949i 0.0134650i
\(397\) − 5.92820i − 0.297528i −0.988873 0.148764i \(-0.952471\pi\)
0.988873 0.148764i \(-0.0475295\pi\)
\(398\) 24.9282i 1.24954i
\(399\) 17.1962 0.860884
\(400\) −1.00000 −0.0500000
\(401\) 22.1244i 1.10484i 0.833567 + 0.552419i \(0.186295\pi\)
−0.833567 + 0.552419i \(0.813705\pi\)
\(402\) 1.46410 0.0730228
\(403\) 0 0
\(404\) −12.3923 −0.616540
\(405\) 1.00000i 0.0496904i
\(406\) 4.39230 0.217986
\(407\) 1.58846 0.0787369
\(408\) − 4.00000i − 0.198030i
\(409\) − 39.0526i − 1.93102i −0.260357 0.965512i \(-0.583840\pi\)
0.260357 0.965512i \(-0.416160\pi\)
\(410\) 4.00000i 0.197546i
\(411\) − 13.4641i − 0.664135i
\(412\) 11.5885 0.570922
\(413\) 34.3923 1.69233
\(414\) − 3.46410i − 0.170251i
\(415\) −9.46410 −0.464574
\(416\) 0 0
\(417\) 19.7846 0.968857
\(418\) − 1.53590i − 0.0751232i
\(419\) 9.85641 0.481517 0.240758 0.970585i \(-0.422604\pi\)
0.240758 + 0.970585i \(0.422604\pi\)
\(420\) −3.00000 −0.146385
\(421\) − 21.8564i − 1.06522i −0.846362 0.532608i \(-0.821212\pi\)
0.846362 0.532608i \(-0.178788\pi\)
\(422\) 8.07180i 0.392929i
\(423\) − 6.46410i − 0.314295i
\(424\) − 0.267949i − 0.0130128i
\(425\) −4.00000 −0.194029
\(426\) 12.9282 0.626373
\(427\) − 1.60770i − 0.0778018i
\(428\) 12.9282 0.624908
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) 13.8564i 0.667440i 0.942672 + 0.333720i \(0.108304\pi\)
−0.942672 + 0.333720i \(0.891696\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −8.78461 −0.422161 −0.211081 0.977469i \(-0.567698\pi\)
−0.211081 + 0.977469i \(0.567698\pi\)
\(434\) 14.7846i 0.709684i
\(435\) − 1.46410i − 0.0701983i
\(436\) 10.3923i 0.497701i
\(437\) 19.8564i 0.949861i
\(438\) 6.92820 0.331042
\(439\) −13.3205 −0.635753 −0.317877 0.948132i \(-0.602970\pi\)
−0.317877 + 0.948132i \(0.602970\pi\)
\(440\) 0.267949i 0.0127740i
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) −19.8564 −0.943406 −0.471703 0.881757i \(-0.656361\pi\)
−0.471703 + 0.881757i \(0.656361\pi\)
\(444\) 5.92820i 0.281340i
\(445\) −14.1244 −0.669559
\(446\) 18.8564 0.892877
\(447\) − 18.7846i − 0.888482i
\(448\) − 3.00000i − 0.141737i
\(449\) − 6.12436i − 0.289026i −0.989503 0.144513i \(-0.953838\pi\)
0.989503 0.144513i \(-0.0461616\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 1.07180 0.0504689
\(452\) 12.0000 0.564433
\(453\) − 22.7846i − 1.07051i
\(454\) 6.53590 0.306745
\(455\) 0 0
\(456\) 5.73205 0.268428
\(457\) 7.46410i 0.349156i 0.984643 + 0.174578i \(0.0558561\pi\)
−0.984643 + 0.174578i \(0.944144\pi\)
\(458\) 4.53590 0.211949
\(459\) −4.00000 −0.186704
\(460\) − 3.46410i − 0.161515i
\(461\) − 11.6077i − 0.540624i −0.962773 0.270312i \(-0.912873\pi\)
0.962773 0.270312i \(-0.0871269\pi\)
\(462\) 0.803848i 0.0373984i
\(463\) 40.7846i 1.89542i 0.319131 + 0.947711i \(0.396609\pi\)
−0.319131 + 0.947711i \(0.603391\pi\)
\(464\) 1.46410 0.0679692
\(465\) 4.92820 0.228540
\(466\) − 18.0000i − 0.833834i
\(467\) 24.3923 1.12874 0.564371 0.825522i \(-0.309119\pi\)
0.564371 + 0.825522i \(0.309119\pi\)
\(468\) 0 0
\(469\) 4.39230 0.202818
\(470\) − 6.46410i − 0.298167i
\(471\) −21.1962 −0.976667
\(472\) 11.4641 0.527678
\(473\) − 1.60770i − 0.0739219i
\(474\) 3.07180i 0.141092i
\(475\) − 5.73205i − 0.263005i
\(476\) − 12.0000i − 0.550019i
\(477\) −0.267949 −0.0122686
\(478\) −3.46410 −0.158444
\(479\) 22.2487i 1.01657i 0.861189 + 0.508285i \(0.169720\pi\)
−0.861189 + 0.508285i \(0.830280\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −25.1962 −1.14765
\(483\) − 10.3923i − 0.472866i
\(484\) −10.9282 −0.496737
\(485\) −8.39230 −0.381075
\(486\) 1.00000i 0.0453609i
\(487\) 21.0000i 0.951601i 0.879553 + 0.475800i \(0.157842\pi\)
−0.879553 + 0.475800i \(0.842158\pi\)
\(488\) − 0.535898i − 0.0242590i
\(489\) 2.92820i 0.132418i
\(490\) −2.00000 −0.0903508
\(491\) 5.39230 0.243351 0.121676 0.992570i \(-0.461173\pi\)
0.121676 + 0.992570i \(0.461173\pi\)
\(492\) 4.00000i 0.180334i
\(493\) 5.85641 0.263759
\(494\) 0 0
\(495\) 0.267949 0.0120434
\(496\) 4.92820i 0.221283i
\(497\) 38.7846 1.73973
\(498\) −9.46410 −0.424097
\(499\) − 33.3205i − 1.49163i −0.666153 0.745815i \(-0.732060\pi\)
0.666153 0.745815i \(-0.267940\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) − 0.464102i − 0.0207345i
\(502\) − 19.5359i − 0.871930i
\(503\) 3.73205 0.166404 0.0832020 0.996533i \(-0.473485\pi\)
0.0832020 + 0.996533i \(0.473485\pi\)
\(504\) −3.00000 −0.133631
\(505\) 12.3923i 0.551450i
\(506\) −0.928203 −0.0412637
\(507\) 0 0
\(508\) 12.6603 0.561708
\(509\) − 29.3205i − 1.29961i −0.760102 0.649804i \(-0.774851\pi\)
0.760102 0.649804i \(-0.225149\pi\)
\(510\) −4.00000 −0.177123
\(511\) 20.7846 0.919457
\(512\) − 1.00000i − 0.0441942i
\(513\) − 5.73205i − 0.253076i
\(514\) − 14.5359i − 0.641151i
\(515\) − 11.5885i − 0.510648i
\(516\) 6.00000 0.264135
\(517\) −1.73205 −0.0761755
\(518\) 17.7846i 0.781411i
\(519\) 2.12436 0.0932489
\(520\) 0 0
\(521\) −6.60770 −0.289488 −0.144744 0.989469i \(-0.546236\pi\)
−0.144744 + 0.989469i \(0.546236\pi\)
\(522\) − 1.46410i − 0.0640820i
\(523\) −41.7128 −1.82397 −0.911987 0.410219i \(-0.865452\pi\)
−0.911987 + 0.410219i \(0.865452\pi\)
\(524\) −14.3205 −0.625594
\(525\) 3.00000i 0.130931i
\(526\) 24.5167i 1.06898i
\(527\) 19.7128i 0.858704i
\(528\) 0.267949i 0.0116610i
\(529\) −11.0000 −0.478261
\(530\) −0.267949 −0.0116390
\(531\) − 11.4641i − 0.497500i
\(532\) 17.1962 0.745548
\(533\) 0 0
\(534\) −14.1244 −0.611221
\(535\) − 12.9282i − 0.558935i
\(536\) 1.46410 0.0632396
\(537\) −9.07180 −0.391477
\(538\) 17.0718i 0.736017i
\(539\) 0.535898i 0.0230828i
\(540\) 1.00000i 0.0430331i
\(541\) − 14.7846i − 0.635640i −0.948151 0.317820i \(-0.897049\pi\)
0.948151 0.317820i \(-0.102951\pi\)
\(542\) −24.3923 −1.04774
\(543\) −16.9282 −0.726459
\(544\) − 4.00000i − 0.171499i
\(545\) 10.3923 0.445157
\(546\) 0 0
\(547\) −5.32051 −0.227488 −0.113744 0.993510i \(-0.536284\pi\)
−0.113744 + 0.993510i \(0.536284\pi\)
\(548\) − 13.4641i − 0.575158i
\(549\) −0.535898 −0.0228716
\(550\) 0.267949 0.0114254
\(551\) 8.39230i 0.357524i
\(552\) − 3.46410i − 0.147442i
\(553\) 9.21539i 0.391878i
\(554\) − 11.5885i − 0.492346i
\(555\) 5.92820 0.251638
\(556\) 19.7846 0.839054
\(557\) 22.6077i 0.957919i 0.877837 + 0.478959i \(0.158986\pi\)
−0.877837 + 0.478959i \(0.841014\pi\)
\(558\) 4.92820 0.208627
\(559\) 0 0
\(560\) −3.00000 −0.126773
\(561\) 1.07180i 0.0452513i
\(562\) −6.92820 −0.292249
\(563\) −15.3205 −0.645682 −0.322841 0.946453i \(-0.604638\pi\)
−0.322841 + 0.946453i \(0.604638\pi\)
\(564\) − 6.46410i − 0.272188i
\(565\) − 12.0000i − 0.504844i
\(566\) 6.39230i 0.268689i
\(567\) 3.00000i 0.125988i
\(568\) 12.9282 0.542455
\(569\) −16.3205 −0.684191 −0.342096 0.939665i \(-0.611137\pi\)
−0.342096 + 0.939665i \(0.611137\pi\)
\(570\) − 5.73205i − 0.240089i
\(571\) 10.8564 0.454326 0.227163 0.973857i \(-0.427055\pi\)
0.227163 + 0.973857i \(0.427055\pi\)
\(572\) 0 0
\(573\) 17.3205 0.723575
\(574\) 12.0000i 0.500870i
\(575\) −3.46410 −0.144463
\(576\) −1.00000 −0.0416667
\(577\) 9.32051i 0.388018i 0.981000 + 0.194009i \(0.0621491\pi\)
−0.981000 + 0.194009i \(0.937851\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) − 9.85641i − 0.409618i
\(580\) − 1.46410i − 0.0607935i
\(581\) −28.3923 −1.17791
\(582\) −8.39230 −0.347872
\(583\) 0.0717968i 0.00297352i
\(584\) 6.92820 0.286691
\(585\) 0 0
\(586\) 29.2487 1.20825
\(587\) 8.92820i 0.368506i 0.982879 + 0.184253i \(0.0589866\pi\)
−0.982879 + 0.184253i \(0.941013\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −28.2487 −1.16397
\(590\) − 11.4641i − 0.471970i
\(591\) − 11.3923i − 0.468617i
\(592\) 5.92820i 0.243648i
\(593\) 44.7846i 1.83908i 0.392992 + 0.919542i \(0.371440\pi\)
−0.392992 + 0.919542i \(0.628560\pi\)
\(594\) 0.267949 0.0109941
\(595\) −12.0000 −0.491952
\(596\) − 18.7846i − 0.769448i
\(597\) 24.9282 1.02024
\(598\) 0 0
\(599\) 28.9282 1.18197 0.590987 0.806681i \(-0.298738\pi\)
0.590987 + 0.806681i \(0.298738\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) 30.7128 1.25280 0.626401 0.779501i \(-0.284527\pi\)
0.626401 + 0.779501i \(0.284527\pi\)
\(602\) 18.0000 0.733625
\(603\) − 1.46410i − 0.0596228i
\(604\) − 22.7846i − 0.927093i
\(605\) 10.9282i 0.444295i
\(606\) 12.3923i 0.503403i
\(607\) −21.4449 −0.870420 −0.435210 0.900329i \(-0.643326\pi\)
−0.435210 + 0.900329i \(0.643326\pi\)
\(608\) 5.73205 0.232465
\(609\) − 4.39230i − 0.177985i
\(610\) −0.535898 −0.0216979
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) − 45.0000i − 1.81753i −0.417305 0.908766i \(-0.637025\pi\)
0.417305 0.908766i \(-0.362975\pi\)
\(614\) 28.2487 1.14003
\(615\) 4.00000 0.161296
\(616\) 0.803848i 0.0323879i
\(617\) − 35.4641i − 1.42773i −0.700283 0.713865i \(-0.746943\pi\)
0.700283 0.713865i \(-0.253057\pi\)
\(618\) − 11.5885i − 0.466156i
\(619\) − 2.51666i − 0.101153i −0.998720 0.0505766i \(-0.983894\pi\)
0.998720 0.0505766i \(-0.0161059\pi\)
\(620\) 4.92820 0.197921
\(621\) −3.46410 −0.139010
\(622\) − 18.9282i − 0.758952i
\(623\) −42.3731 −1.69764
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 33.3205i 1.33176i
\(627\) −1.53590 −0.0613379
\(628\) −21.1962 −0.845819
\(629\) 23.7128i 0.945492i
\(630\) 3.00000i 0.119523i
\(631\) − 8.00000i − 0.318475i −0.987240 0.159237i \(-0.949096\pi\)
0.987240 0.159237i \(-0.0509036\pi\)
\(632\) 3.07180i 0.122190i
\(633\) 8.07180 0.320825
\(634\) −30.4641 −1.20988
\(635\) − 12.6603i − 0.502407i
\(636\) −0.267949 −0.0106249
\(637\) 0 0
\(638\) −0.392305 −0.0155315
\(639\) − 12.9282i − 0.511432i
\(640\) −1.00000 −0.0395285
\(641\) 14.4641 0.571298 0.285649 0.958334i \(-0.407791\pi\)
0.285649 + 0.958334i \(0.407791\pi\)
\(642\) − 12.9282i − 0.510235i
\(643\) 12.7846i 0.504176i 0.967704 + 0.252088i \(0.0811172\pi\)
−0.967704 + 0.252088i \(0.918883\pi\)
\(644\) − 10.3923i − 0.409514i
\(645\) − 6.00000i − 0.236250i
\(646\) 22.9282 0.902098
\(647\) −35.7321 −1.40477 −0.702386 0.711796i \(-0.747882\pi\)
−0.702386 + 0.711796i \(0.747882\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −3.07180 −0.120579
\(650\) 0 0
\(651\) 14.7846 0.579455
\(652\) 2.92820i 0.114677i
\(653\) −1.58846 −0.0621611 −0.0310806 0.999517i \(-0.509895\pi\)
−0.0310806 + 0.999517i \(0.509895\pi\)
\(654\) 10.3923 0.406371
\(655\) 14.3205i 0.559549i
\(656\) 4.00000i 0.156174i
\(657\) − 6.92820i − 0.270295i
\(658\) − 19.3923i − 0.755991i
\(659\) −39.7128 −1.54699 −0.773496 0.633801i \(-0.781494\pi\)
−0.773496 + 0.633801i \(0.781494\pi\)
\(660\) 0.267949 0.0104299
\(661\) 21.6077i 0.840442i 0.907422 + 0.420221i \(0.138047\pi\)
−0.907422 + 0.420221i \(0.861953\pi\)
\(662\) 14.3923 0.559373
\(663\) 0 0
\(664\) −9.46410 −0.367278
\(665\) − 17.1962i − 0.666838i
\(666\) 5.92820 0.229713
\(667\) 5.07180 0.196381
\(668\) − 0.464102i − 0.0179566i
\(669\) − 18.8564i − 0.729031i
\(670\) − 1.46410i − 0.0565632i
\(671\) 0.143594i 0.00554337i
\(672\) −3.00000 −0.115728
\(673\) −16.7846 −0.646999 −0.323500 0.946228i \(-0.604859\pi\)
−0.323500 + 0.946228i \(0.604859\pi\)
\(674\) 26.3923i 1.01659i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 10.9282 0.420005 0.210002 0.977701i \(-0.432653\pi\)
0.210002 + 0.977701i \(0.432653\pi\)
\(678\) − 12.0000i − 0.460857i
\(679\) −25.1769 −0.966201
\(680\) −4.00000 −0.153393
\(681\) − 6.53590i − 0.250456i
\(682\) − 1.32051i − 0.0505649i
\(683\) 41.3205i 1.58109i 0.612407 + 0.790543i \(0.290201\pi\)
−0.612407 + 0.790543i \(0.709799\pi\)
\(684\) − 5.73205i − 0.219170i
\(685\) −13.4641 −0.514437
\(686\) 15.0000 0.572703
\(687\) − 4.53590i − 0.173055i
\(688\) 6.00000 0.228748
\(689\) 0 0
\(690\) −3.46410 −0.131876
\(691\) − 15.0526i − 0.572626i −0.958136 0.286313i \(-0.907570\pi\)
0.958136 0.286313i \(-0.0924298\pi\)
\(692\) 2.12436 0.0807559
\(693\) 0.803848 0.0305356
\(694\) − 4.39230i − 0.166730i
\(695\) − 19.7846i − 0.750473i
\(696\) − 1.46410i − 0.0554966i
\(697\) 16.0000i 0.606043i
\(698\) 10.5359 0.398790
\(699\) −18.0000 −0.680823
\(700\) 3.00000i 0.113389i
\(701\) −4.39230 −0.165895 −0.0829475 0.996554i \(-0.526433\pi\)
−0.0829475 + 0.996554i \(0.526433\pi\)
\(702\) 0 0
\(703\) −33.9808 −1.28161
\(704\) 0.267949i 0.0100987i
\(705\) −6.46410 −0.243452
\(706\) −7.60770 −0.286319
\(707\) 37.1769i 1.39818i
\(708\) − 11.4641i − 0.430847i
\(709\) − 27.3205i − 1.02604i −0.858376 0.513022i \(-0.828526\pi\)
0.858376 0.513022i \(-0.171474\pi\)
\(710\) − 12.9282i − 0.485187i
\(711\) 3.07180 0.115201
\(712\) −14.1244 −0.529333
\(713\) 17.0718i 0.639344i
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) −9.07180 −0.339029
\(717\) 3.46410i 0.129369i
\(718\) −0.928203 −0.0346402
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) − 34.7654i − 1.29473i
\(722\) 13.8564i 0.515682i
\(723\) 25.1962i 0.937055i
\(724\) −16.9282 −0.629132
\(725\) −1.46410 −0.0543754
\(726\) 10.9282i 0.405584i
\(727\) −4.66025 −0.172839 −0.0864196 0.996259i \(-0.527543\pi\)
−0.0864196 + 0.996259i \(0.527543\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 6.92820i − 0.256424i
\(731\) 24.0000 0.887672
\(732\) −0.535898 −0.0198074
\(733\) − 20.8564i − 0.770349i −0.922844 0.385174i \(-0.874141\pi\)
0.922844 0.385174i \(-0.125859\pi\)
\(734\) − 21.3205i − 0.786954i
\(735\) 2.00000i 0.0737711i
\(736\) − 3.46410i − 0.127688i
\(737\) −0.392305 −0.0144507
\(738\) 4.00000 0.147242
\(739\) − 35.8372i − 1.31829i −0.752015 0.659146i \(-0.770918\pi\)
0.752015 0.659146i \(-0.229082\pi\)
\(740\) 5.92820 0.217925
\(741\) 0 0
\(742\) −0.803848 −0.0295102
\(743\) 10.9282i 0.400917i 0.979702 + 0.200458i \(0.0642432\pi\)
−0.979702 + 0.200458i \(0.935757\pi\)
\(744\) 4.92820 0.180677
\(745\) −18.7846 −0.688215
\(746\) − 9.07180i − 0.332142i
\(747\) 9.46410i 0.346273i
\(748\) 1.07180i 0.0391888i
\(749\) − 38.7846i − 1.41716i
\(750\) 1.00000 0.0365148
\(751\) −21.1769 −0.772757 −0.386378 0.922340i \(-0.626274\pi\)
−0.386378 + 0.922340i \(0.626274\pi\)
\(752\) − 6.46410i − 0.235722i
\(753\) −19.5359 −0.711928
\(754\) 0 0
\(755\) −22.7846 −0.829217
\(756\) 3.00000i 0.109109i
\(757\) 21.7321 0.789865 0.394932 0.918710i \(-0.370768\pi\)
0.394932 + 0.918710i \(0.370768\pi\)
\(758\) 9.73205 0.353484
\(759\) 0.928203i 0.0336916i
\(760\) − 5.73205i − 0.207923i
\(761\) 7.98076i 0.289302i 0.989483 + 0.144651i \(0.0462060\pi\)
−0.989483 + 0.144651i \(0.953794\pi\)
\(762\) − 12.6603i − 0.458633i
\(763\) 31.1769 1.12868
\(764\) 17.3205 0.626634
\(765\) 4.00000i 0.144620i
\(766\) 4.78461 0.172875
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) − 15.7128i − 0.566619i −0.959029 0.283309i \(-0.908568\pi\)
0.959029 0.283309i \(-0.0914323\pi\)
\(770\) 0.803848 0.0289687
\(771\) −14.5359 −0.523498
\(772\) − 9.85641i − 0.354740i
\(773\) − 32.6077i − 1.17282i −0.810015 0.586409i \(-0.800541\pi\)
0.810015 0.586409i \(-0.199459\pi\)
\(774\) − 6.00000i − 0.215666i
\(775\) − 4.92820i − 0.177026i
\(776\) −8.39230 −0.301266
\(777\) 17.7846 0.638019
\(778\) − 17.8564i − 0.640183i
\(779\) −22.9282 −0.821488
\(780\) 0 0
\(781\) −3.46410 −0.123955
\(782\) − 13.8564i − 0.495504i
\(783\) −1.46410 −0.0523227
\(784\) −2.00000 −0.0714286
\(785\) 21.1962i 0.756523i
\(786\) 14.3205i 0.510796i
\(787\) 1.46410i 0.0521896i 0.999659 + 0.0260948i \(0.00830717\pi\)
−0.999659 + 0.0260948i \(0.991693\pi\)
\(788\) − 11.3923i − 0.405834i
\(789\) 24.5167 0.872816
\(790\) 3.07180 0.109290
\(791\) − 36.0000i − 1.28001i
\(792\) 0.267949 0.00952116
\(793\) 0 0
\(794\) −5.92820 −0.210384
\(795\) 0.267949i 0.00950318i
\(796\) 24.9282 0.883557
\(797\) −10.1436 −0.359305 −0.179652 0.983730i \(-0.557497\pi\)
−0.179652 + 0.983730i \(0.557497\pi\)
\(798\) − 17.1962i − 0.608737i
\(799\) − 25.8564i − 0.914734i
\(800\) 1.00000i 0.0353553i
\(801\) 14.1244i 0.499060i
\(802\) 22.1244 0.781238
\(803\) −1.85641 −0.0655112
\(804\) − 1.46410i − 0.0516349i
\(805\) −10.3923 −0.366281
\(806\) 0 0
\(807\) 17.0718 0.600956
\(808\) 12.3923i 0.435960i
\(809\) −6.78461 −0.238534 −0.119267 0.992862i \(-0.538054\pi\)
−0.119267 + 0.992862i \(0.538054\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 23.5885i − 0.828303i −0.910208 0.414151i \(-0.864078\pi\)
0.910208 0.414151i \(-0.135922\pi\)
\(812\) − 4.39230i − 0.154140i
\(813\) 24.3923i 0.855475i
\(814\) − 1.58846i − 0.0556754i
\(815\) 2.92820 0.102570
\(816\) −4.00000 −0.140028
\(817\) 34.3923i 1.20323i
\(818\) −39.0526 −1.36544
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) 14.6795i 0.512318i 0.966635 + 0.256159i \(0.0824570\pi\)
−0.966635 + 0.256159i \(0.917543\pi\)
\(822\) −13.4641 −0.469614
\(823\) −8.41154 −0.293208 −0.146604 0.989195i \(-0.546834\pi\)
−0.146604 + 0.989195i \(0.546834\pi\)
\(824\) − 11.5885i − 0.403703i
\(825\) − 0.267949i − 0.00932879i
\(826\) − 34.3923i − 1.19666i
\(827\) − 46.4974i − 1.61687i −0.588583 0.808437i \(-0.700314\pi\)
0.588583 0.808437i \(-0.299686\pi\)
\(828\) −3.46410 −0.120386
\(829\) −4.67949 −0.162525 −0.0812627 0.996693i \(-0.525895\pi\)
−0.0812627 + 0.996693i \(0.525895\pi\)
\(830\) 9.46410i 0.328504i
\(831\) −11.5885 −0.401999
\(832\) 0 0
\(833\) −8.00000 −0.277184
\(834\) − 19.7846i − 0.685085i
\(835\) −0.464102 −0.0160609
\(836\) −1.53590 −0.0531202
\(837\) − 4.92820i − 0.170344i
\(838\) − 9.85641i − 0.340484i
\(839\) 19.1769i 0.662061i 0.943620 + 0.331030i \(0.107396\pi\)
−0.943620 + 0.331030i \(0.892604\pi\)
\(840\) 3.00000i 0.103510i
\(841\) −26.8564 −0.926083
\(842\) −21.8564 −0.753222
\(843\) 6.92820i 0.238620i
\(844\) 8.07180 0.277843
\(845\) 0 0
\(846\) −6.46410 −0.222240
\(847\) 32.7846i 1.12649i
\(848\) −0.267949 −0.00920141
\(849\) 6.39230 0.219383
\(850\) 4.00000i 0.137199i
\(851\) 20.5359i 0.703962i
\(852\) − 12.9282i − 0.442913i
\(853\) 24.6410i 0.843692i 0.906667 + 0.421846i \(0.138618\pi\)
−0.906667 + 0.421846i \(0.861382\pi\)
\(854\) −1.60770 −0.0550142
\(855\) −5.73205 −0.196032
\(856\) − 12.9282i − 0.441877i
\(857\) −28.9282 −0.988169 −0.494084 0.869414i \(-0.664497\pi\)
−0.494084 + 0.869414i \(0.664497\pi\)
\(858\) 0 0
\(859\) −6.07180 −0.207167 −0.103584 0.994621i \(-0.533031\pi\)
−0.103584 + 0.994621i \(0.533031\pi\)
\(860\) − 6.00000i − 0.204598i
\(861\) 12.0000 0.408959
\(862\) 13.8564 0.471951
\(863\) − 2.92820i − 0.0996772i −0.998757 0.0498386i \(-0.984129\pi\)
0.998757 0.0498386i \(-0.0158707\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) − 2.12436i − 0.0722303i
\(866\) 8.78461i 0.298513i
\(867\) 1.00000 0.0339618
\(868\) 14.7846 0.501822
\(869\) − 0.823085i − 0.0279213i
\(870\) −1.46410 −0.0496377
\(871\) 0 0
\(872\) 10.3923 0.351928
\(873\) 8.39230i 0.284036i
\(874\) 19.8564 0.671653
\(875\) 3.00000 0.101419
\(876\) − 6.92820i − 0.234082i
\(877\) 21.7128i 0.733190i 0.930381 + 0.366595i \(0.119476\pi\)
−0.930381 + 0.366595i \(0.880524\pi\)
\(878\) 13.3205i 0.449545i
\(879\) − 29.2487i − 0.986535i
\(880\) 0.267949 0.00903257
\(881\) 39.1051 1.31748 0.658742 0.752369i \(-0.271089\pi\)
0.658742 + 0.752369i \(0.271089\pi\)
\(882\) 2.00000i 0.0673435i
\(883\) −13.3205 −0.448271 −0.224135 0.974558i \(-0.571956\pi\)
−0.224135 + 0.974558i \(0.571956\pi\)
\(884\) 0 0
\(885\) −11.4641 −0.385362
\(886\) 19.8564i 0.667089i
\(887\) 55.7321 1.87130 0.935650 0.352930i \(-0.114815\pi\)
0.935650 + 0.352930i \(0.114815\pi\)
\(888\) 5.92820 0.198937
\(889\) − 37.9808i − 1.27383i
\(890\) 14.1244i 0.473449i
\(891\) − 0.267949i − 0.00897664i
\(892\) − 18.8564i − 0.631359i
\(893\) 37.0526 1.23992
\(894\) −18.7846 −0.628251
\(895\) 9.07180i 0.303237i
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) −6.12436 −0.204372
\(899\) 7.21539i 0.240647i
\(900\) 1.00000 0.0333333
\(901\) −1.07180 −0.0357067
\(902\) − 1.07180i − 0.0356869i
\(903\) − 18.0000i − 0.599002i
\(904\) − 12.0000i − 0.399114i
\(905\) 16.9282i 0.562713i
\(906\) −22.7846 −0.756968
\(907\) 20.9282 0.694910 0.347455 0.937697i \(-0.387046\pi\)
0.347455 + 0.937697i \(0.387046\pi\)
\(908\) − 6.53590i − 0.216901i
\(909\) 12.3923 0.411027
\(910\) 0 0
\(911\) 12.7846 0.423573 0.211787 0.977316i \(-0.432072\pi\)
0.211787 + 0.977316i \(0.432072\pi\)
\(912\) − 5.73205i − 0.189807i
\(913\) 2.53590 0.0839260
\(914\) 7.46410 0.246891
\(915\) 0.535898i 0.0177163i
\(916\) − 4.53590i − 0.149870i
\(917\) 42.9615i 1.41871i
\(918\) 4.00000i 0.132020i
\(919\) −47.9615 −1.58210 −0.791052 0.611748i \(-0.790466\pi\)
−0.791052 + 0.611748i \(0.790466\pi\)
\(920\) −3.46410 −0.114208
\(921\) − 28.2487i − 0.930827i
\(922\) −11.6077 −0.382279
\(923\) 0 0
\(924\) 0.803848 0.0264446
\(925\) − 5.92820i − 0.194918i
\(926\) 40.7846 1.34027
\(927\) −11.5885 −0.380615
\(928\) − 1.46410i − 0.0480615i
\(929\) 57.8564i 1.89821i 0.314964 + 0.949104i \(0.398007\pi\)
−0.314964 + 0.949104i \(0.601993\pi\)
\(930\) − 4.92820i − 0.161602i
\(931\) − 11.4641i − 0.375721i
\(932\) −18.0000 −0.589610
\(933\) −18.9282 −0.619682
\(934\) − 24.3923i − 0.798141i
\(935\) 1.07180 0.0350515
\(936\) 0 0
\(937\) −21.7128 −0.709327 −0.354663 0.934994i \(-0.615405\pi\)
−0.354663 + 0.934994i \(0.615405\pi\)
\(938\) − 4.39230i − 0.143414i
\(939\) 33.3205 1.08737
\(940\) −6.46410 −0.210836
\(941\) 60.4974i 1.97216i 0.166273 + 0.986080i \(0.446827\pi\)
−0.166273 + 0.986080i \(0.553173\pi\)
\(942\) 21.1962i 0.690608i
\(943\) 13.8564i 0.451227i
\(944\) − 11.4641i − 0.373125i
\(945\) 3.00000 0.0975900
\(946\) −1.60770 −0.0522707
\(947\) − 34.3923i − 1.11760i −0.829303 0.558800i \(-0.811262\pi\)
0.829303 0.558800i \(-0.188738\pi\)
\(948\) 3.07180 0.0997673
\(949\) 0 0
\(950\) −5.73205 −0.185972
\(951\) 30.4641i 0.987866i
\(952\) −12.0000 −0.388922
\(953\) −53.3205 −1.72722 −0.863610 0.504160i \(-0.831802\pi\)
−0.863610 + 0.504160i \(0.831802\pi\)
\(954\) 0.267949i 0.00867518i
\(955\) − 17.3205i − 0.560478i
\(956\) 3.46410i 0.112037i
\(957\) 0.392305i 0.0126814i
\(958\) 22.2487 0.718823
\(959\) −40.3923 −1.30434
\(960\) 1.00000i 0.0322749i
\(961\) 6.71281 0.216542
\(962\) 0 0
\(963\) −12.9282 −0.416606
\(964\) 25.1962i 0.811513i
\(965\) −9.85641 −0.317289
\(966\) −10.3923 −0.334367
\(967\) 1.14359i 0.0367755i 0.999831 + 0.0183877i \(0.00585333\pi\)
−0.999831 + 0.0183877i \(0.994147\pi\)
\(968\) 10.9282i 0.351246i
\(969\) − 22.9282i − 0.736560i
\(970\) 8.39230i 0.269461i
\(971\) −41.3923 −1.32834 −0.664171 0.747581i \(-0.731215\pi\)
−0.664171 + 0.747581i \(0.731215\pi\)
\(972\) 1.00000 0.0320750
\(973\) − 59.3538i − 1.90280i
\(974\) 21.0000 0.672883
\(975\) 0 0
\(976\) −0.535898 −0.0171537
\(977\) 5.46410i 0.174812i 0.996173 + 0.0874060i \(0.0278578\pi\)
−0.996173 + 0.0874060i \(0.972142\pi\)
\(978\) 2.92820 0.0936336
\(979\) 3.78461 0.120957
\(980\) 2.00000i 0.0638877i
\(981\) − 10.3923i − 0.331801i
\(982\) − 5.39230i − 0.172075i
\(983\) − 46.1769i − 1.47281i −0.676538 0.736407i \(-0.736521\pi\)
0.676538 0.736407i \(-0.263479\pi\)
\(984\) 4.00000 0.127515
\(985\) −11.3923 −0.362989
\(986\) − 5.85641i − 0.186506i
\(987\) −19.3923 −0.617264
\(988\) 0 0
\(989\) 20.7846 0.660912
\(990\) − 0.267949i − 0.00851598i
\(991\) 39.1769 1.24450 0.622248 0.782820i \(-0.286220\pi\)
0.622248 + 0.782820i \(0.286220\pi\)
\(992\) 4.92820 0.156471
\(993\) − 14.3923i − 0.456726i
\(994\) − 38.7846i − 1.23017i
\(995\) − 24.9282i − 0.790277i
\(996\) 9.46410i 0.299882i
\(997\) 61.9808 1.96295 0.981475 0.191589i \(-0.0613641\pi\)
0.981475 + 0.191589i \(0.0613641\pi\)
\(998\) −33.3205 −1.05474
\(999\) − 5.92820i − 0.187560i
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.p.1351.2 4
13.3 even 3 390.2.bb.a.121.2 4
13.4 even 6 390.2.bb.a.361.2 yes 4
13.5 odd 4 5070.2.a.ba.1.1 2
13.8 odd 4 5070.2.a.be.1.2 2
13.12 even 2 inner 5070.2.b.p.1351.3 4
39.17 odd 6 1170.2.bs.d.361.1 4
39.29 odd 6 1170.2.bs.d.901.1 4
65.3 odd 12 1950.2.y.e.199.2 4
65.4 even 6 1950.2.bc.a.751.1 4
65.17 odd 12 1950.2.y.e.49.2 4
65.29 even 6 1950.2.bc.a.901.1 4
65.42 odd 12 1950.2.y.d.199.1 4
65.43 odd 12 1950.2.y.d.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.a.121.2 4 13.3 even 3
390.2.bb.a.361.2 yes 4 13.4 even 6
1170.2.bs.d.361.1 4 39.17 odd 6
1170.2.bs.d.901.1 4 39.29 odd 6
1950.2.y.d.49.1 4 65.43 odd 12
1950.2.y.d.199.1 4 65.42 odd 12
1950.2.y.e.49.2 4 65.17 odd 12
1950.2.y.e.199.2 4 65.3 odd 12
1950.2.bc.a.751.1 4 65.4 even 6
1950.2.bc.a.901.1 4 65.29 even 6
5070.2.a.ba.1.1 2 13.5 odd 4
5070.2.a.be.1.2 2 13.8 odd 4
5070.2.b.p.1351.2 4 1.1 even 1 trivial
5070.2.b.p.1351.3 4 13.12 even 2 inner