Properties

Label 5070.2.b.o.1351.4
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 1351.4
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.o.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +0.464102i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000i q^{18} +0.535898i q^{19} -1.00000i q^{20} +2.00000i q^{21} -0.464102 q^{22} +0.267949 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} +3.73205 q^{29} +1.00000 q^{30} +1.73205i q^{31} +1.00000i q^{32} -0.464102i q^{33} -4.00000i q^{34} +2.00000 q^{35} -1.00000 q^{36} +1.19615i q^{37} -0.535898 q^{38} +1.00000 q^{40} -2.00000i q^{41} -2.00000 q^{42} -1.92820 q^{43} -0.464102i q^{44} +1.00000i q^{45} +0.267949i q^{46} -10.4641i q^{47} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} +4.00000 q^{51} -12.9282 q^{53} -1.00000i q^{54} -0.464102 q^{55} -2.00000 q^{56} -0.535898i q^{57} +3.73205i q^{58} +1.53590i q^{59} +1.00000i q^{60} +10.3923 q^{61} -1.73205 q^{62} -2.00000i q^{63} -1.00000 q^{64} +0.464102 q^{66} +4.53590i q^{67} +4.00000 q^{68} -0.267949 q^{69} +2.00000i q^{70} +8.39230i q^{71} -1.00000i q^{72} +2.00000i q^{73} -1.19615 q^{74} +1.00000 q^{75} -0.535898i q^{76} +0.928203 q^{77} -0.0717968 q^{79} +1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.92820i q^{83} -2.00000i q^{84} -4.00000i q^{85} -1.92820i q^{86} -3.73205 q^{87} +0.464102 q^{88} +7.46410i q^{89} -1.00000 q^{90} -0.267949 q^{92} -1.73205i q^{93} +10.4641 q^{94} -0.535898 q^{95} -1.00000i q^{96} +7.46410i q^{97} +3.00000i q^{98} +0.464102i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 4q^{4} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 4q^{4} + 4q^{9} - 4q^{10} + 4q^{12} + 8q^{14} + 4q^{16} - 16q^{17} + 12q^{22} + 8q^{23} - 4q^{25} - 4q^{27} + 8q^{29} + 4q^{30} + 8q^{35} - 4q^{36} - 16q^{38} + 4q^{40} - 8q^{42} + 20q^{43} - 4q^{48} + 12q^{49} + 16q^{51} - 24q^{53} + 12q^{55} - 8q^{56} - 4q^{64} - 12q^{66} + 16q^{68} - 8q^{69} + 16q^{74} + 4q^{75} - 24q^{77} - 28q^{79} + 4q^{81} + 8q^{82} - 8q^{87} - 12q^{88} - 4q^{90} - 8q^{92} + 28q^{94} - 16q^{95} + O(q^{100}) \)

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
\(11\) 0.464102i 0.139932i 0.997549 + 0.0699660i \(0.0222891\pi\)
−0.997549 + 0.0699660i \(0.977711\pi\)
\(12\) 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
\(19\) 0.535898i 0.122944i 0.998109 + 0.0614718i \(0.0195794\pi\)
−0.998109 + 0.0614718i \(0.980421\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) 2.00000i 0.436436i
\(22\) −0.464102 −0.0989468
\(23\) 0.267949 0.0558713 0.0279356 0.999610i \(-0.491107\pi\)
0.0279356 + 0.999610i \(0.491107\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000i 0.377964i
\(29\) 3.73205 0.693024 0.346512 0.938045i \(-0.387366\pi\)
0.346512 + 0.938045i \(0.387366\pi\)
\(30\) 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
\(33\) − 0.464102i − 0.0807897i
\(34\) − 4.00000i − 0.685994i
\(35\) 2.00000 0.338062
\(36\) −1.00000 −0.166667
\(37\) 1.19615i 0.196646i 0.995155 + 0.0983231i \(0.0313479\pi\)
−0.995155 + 0.0983231i \(0.968652\pi\)
\(38\) −0.535898 −0.0869342
\(39\) 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
\(43\) −1.92820 −0.294048 −0.147024 0.989133i \(-0.546969\pi\)
−0.147024 + 0.989133i \(0.546969\pi\)
\(44\) − 0.464102i − 0.0699660i
\(45\) 1.00000i 0.149071i
\(46\) 0.267949i 0.0395070i
\(47\) − 10.4641i − 1.52635i −0.646194 0.763173i \(-0.723640\pi\)
0.646194 0.763173i \(-0.276360\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.00000 0.428571
\(50\) − 1.00000i − 0.141421i
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) −12.9282 −1.77583 −0.887913 0.460012i \(-0.847845\pi\)
−0.887913 + 0.460012i \(0.847845\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −0.464102 −0.0625794
\(56\) −2.00000 −0.267261
\(57\) − 0.535898i − 0.0709815i
\(58\) 3.73205i 0.490042i
\(59\) 1.53590i 0.199957i 0.994990 + 0.0999785i \(0.0318774\pi\)
−0.994990 + 0.0999785i \(0.968123\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 10.3923 1.33060 0.665299 0.746577i \(-0.268304\pi\)
0.665299 + 0.746577i \(0.268304\pi\)
\(62\) −1.73205 −0.219971
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0.464102 0.0571270
\(67\) 4.53590i 0.554148i 0.960849 + 0.277074i \(0.0893647\pi\)
−0.960849 + 0.277074i \(0.910635\pi\)
\(68\) 4.00000 0.485071
\(69\) −0.267949 −0.0322573
\(70\) 2.00000i 0.239046i
\(71\) 8.39230i 0.995983i 0.867182 + 0.497992i \(0.165929\pi\)
−0.867182 + 0.497992i \(0.834071\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) −1.19615 −0.139050
\(75\) 1.00000 0.115470
\(76\) − 0.535898i − 0.0614718i
\(77\) 0.928203 0.105779
\(78\) 0 0
\(79\) −0.0717968 −0.00807777 −0.00403888 0.999992i \(-0.501286\pi\)
−0.00403888 + 0.999992i \(0.501286\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) − 4.92820i − 0.540941i −0.962728 0.270470i \(-0.912821\pi\)
0.962728 0.270470i \(-0.0871792\pi\)
\(84\) − 2.00000i − 0.218218i
\(85\) − 4.00000i − 0.433861i
\(86\) − 1.92820i − 0.207924i
\(87\) −3.73205 −0.400118
\(88\) 0.464102 0.0494734
\(89\) 7.46410i 0.791193i 0.918424 + 0.395597i \(0.129462\pi\)
−0.918424 + 0.395597i \(0.870538\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −0.267949 −0.0279356
\(93\) − 1.73205i − 0.179605i
\(94\) 10.4641 1.07929
\(95\) −0.535898 −0.0549820
\(96\) − 1.00000i − 0.102062i
\(97\) 7.46410i 0.757865i 0.925424 + 0.378932i \(0.123709\pi\)
−0.925424 + 0.378932i \(0.876291\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 0.464102i 0.0466440i
\(100\) 1.00000 0.100000
\(101\) −10.9282 −1.08740 −0.543698 0.839281i \(-0.682976\pi\)
−0.543698 + 0.839281i \(0.682976\pi\)
\(102\) 4.00000i 0.396059i
\(103\) 15.8564 1.56238 0.781189 0.624295i \(-0.214613\pi\)
0.781189 + 0.624295i \(0.214613\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) − 12.9282i − 1.25570i
\(107\) 19.8564 1.91959 0.959796 0.280700i \(-0.0905665\pi\)
0.959796 + 0.280700i \(0.0905665\pi\)
\(108\) 1.00000 0.0962250
\(109\) 11.8564i 1.13564i 0.823154 + 0.567819i \(0.192213\pi\)
−0.823154 + 0.567819i \(0.807787\pi\)
\(110\) − 0.464102i − 0.0442504i
\(111\) − 1.19615i − 0.113534i
\(112\) − 2.00000i − 0.188982i
\(113\) 11.1962 1.05325 0.526623 0.850099i \(-0.323458\pi\)
0.526623 + 0.850099i \(0.323458\pi\)
\(114\) 0.535898 0.0501915
\(115\) 0.267949i 0.0249864i
\(116\) −3.73205 −0.346512
\(117\) 0 0
\(118\) −1.53590 −0.141391
\(119\) 8.00000i 0.733359i
\(120\) −1.00000 −0.0912871
\(121\) 10.7846 0.980419
\(122\) 10.3923i 0.940875i
\(123\) 2.00000i 0.180334i
\(124\) − 1.73205i − 0.155543i
\(125\) − 1.00000i − 0.0894427i
\(126\) 2.00000 0.178174
\(127\) 8.92820 0.792250 0.396125 0.918197i \(-0.370355\pi\)
0.396125 + 0.918197i \(0.370355\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 1.92820 0.169769
\(130\) 0 0
\(131\) −1.33975 −0.117054 −0.0585271 0.998286i \(-0.518640\pi\)
−0.0585271 + 0.998286i \(0.518640\pi\)
\(132\) 0.464102i 0.0403949i
\(133\) 1.07180 0.0929366
\(134\) −4.53590 −0.391842
\(135\) − 1.00000i − 0.0860663i
\(136\) 4.00000i 0.342997i
\(137\) 4.46410i 0.381394i 0.981649 + 0.190697i \(0.0610748\pi\)
−0.981649 + 0.190697i \(0.938925\pi\)
\(138\) − 0.267949i − 0.0228093i
\(139\) 0.928203 0.0787292 0.0393646 0.999225i \(-0.487467\pi\)
0.0393646 + 0.999225i \(0.487467\pi\)
\(140\) −2.00000 −0.169031
\(141\) 10.4641i 0.881236i
\(142\) −8.39230 −0.704267
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 3.73205i 0.309930i
\(146\) −2.00000 −0.165521
\(147\) −3.00000 −0.247436
\(148\) − 1.19615i − 0.0983231i
\(149\) − 20.4641i − 1.67648i −0.545298 0.838242i \(-0.683584\pi\)
0.545298 0.838242i \(-0.316416\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 10.3923i 0.845714i 0.906196 + 0.422857i \(0.138973\pi\)
−0.906196 + 0.422857i \(0.861027\pi\)
\(152\) 0.535898 0.0434671
\(153\) −4.00000 −0.323381
\(154\) 0.928203i 0.0747967i
\(155\) −1.73205 −0.139122
\(156\) 0 0
\(157\) −5.00000 −0.399043 −0.199522 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(158\) − 0.0717968i − 0.00571184i
\(159\) 12.9282 1.02527
\(160\) −1.00000 −0.0790569
\(161\) − 0.535898i − 0.0422347i
\(162\) 1.00000i 0.0785674i
\(163\) − 23.0526i − 1.80562i −0.430044 0.902808i \(-0.641502\pi\)
0.430044 0.902808i \(-0.358498\pi\)
\(164\) 2.00000i 0.156174i
\(165\) 0.464102 0.0361303
\(166\) 4.92820 0.382503
\(167\) 18.3205i 1.41768i 0.705368 + 0.708842i \(0.250782\pi\)
−0.705368 + 0.708842i \(0.749218\pi\)
\(168\) 2.00000 0.154303
\(169\) 0 0
\(170\) 4.00000 0.306786
\(171\) 0.535898i 0.0409812i
\(172\) 1.92820 0.147024
\(173\) −2.92820 −0.222627 −0.111314 0.993785i \(-0.535506\pi\)
−0.111314 + 0.993785i \(0.535506\pi\)
\(174\) − 3.73205i − 0.282926i
\(175\) 2.00000i 0.151186i
\(176\) 0.464102i 0.0349830i
\(177\) − 1.53590i − 0.115445i
\(178\) −7.46410 −0.559458
\(179\) −16.2679 −1.21592 −0.607962 0.793966i \(-0.708013\pi\)
−0.607962 + 0.793966i \(0.708013\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) 10.9282 0.812287 0.406143 0.913809i \(-0.366873\pi\)
0.406143 + 0.913809i \(0.366873\pi\)
\(182\) 0 0
\(183\) −10.3923 −0.768221
\(184\) − 0.267949i − 0.0197535i
\(185\) −1.19615 −0.0879429
\(186\) 1.73205 0.127000
\(187\) − 1.85641i − 0.135754i
\(188\) 10.4641i 0.763173i
\(189\) 2.00000i 0.145479i
\(190\) − 0.535898i − 0.0388782i
\(191\) 14.5359 1.05178 0.525890 0.850552i \(-0.323732\pi\)
0.525890 + 0.850552i \(0.323732\pi\)
\(192\) 1.00000 0.0721688
\(193\) 23.3205i 1.67865i 0.543632 + 0.839323i \(0.317049\pi\)
−0.543632 + 0.839323i \(0.682951\pi\)
\(194\) −7.46410 −0.535891
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 16.3923i 1.16790i 0.811788 + 0.583952i \(0.198494\pi\)
−0.811788 + 0.583952i \(0.801506\pi\)
\(198\) −0.464102 −0.0329823
\(199\) −18.9282 −1.34178 −0.670892 0.741555i \(-0.734089\pi\)
−0.670892 + 0.741555i \(0.734089\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) − 4.53590i − 0.319938i
\(202\) − 10.9282i − 0.768906i
\(203\) − 7.46410i − 0.523877i
\(204\) −4.00000 −0.280056
\(205\) 2.00000 0.139686
\(206\) 15.8564i 1.10477i
\(207\) 0.267949 0.0186238
\(208\) 0 0
\(209\) −0.248711 −0.0172037
\(210\) − 2.00000i − 0.138013i
\(211\) 23.3205 1.60545 0.802725 0.596349i \(-0.203383\pi\)
0.802725 + 0.596349i \(0.203383\pi\)
\(212\) 12.9282 0.887913
\(213\) − 8.39230i − 0.575031i
\(214\) 19.8564i 1.35736i
\(215\) − 1.92820i − 0.131502i
\(216\) 1.00000i 0.0680414i
\(217\) 3.46410 0.235159
\(218\) −11.8564 −0.803017
\(219\) − 2.00000i − 0.135147i
\(220\) 0.464102 0.0312897
\(221\) 0 0
\(222\) 1.19615 0.0802805
\(223\) 27.4641i 1.83913i 0.392935 + 0.919566i \(0.371460\pi\)
−0.392935 + 0.919566i \(0.628540\pi\)
\(224\) 2.00000 0.133631
\(225\) −1.00000 −0.0666667
\(226\) 11.1962i 0.744757i
\(227\) 4.39230i 0.291528i 0.989319 + 0.145764i \(0.0465639\pi\)
−0.989319 + 0.145764i \(0.953436\pi\)
\(228\) 0.535898i 0.0354907i
\(229\) 19.8564i 1.31215i 0.754696 + 0.656074i \(0.227784\pi\)
−0.754696 + 0.656074i \(0.772216\pi\)
\(230\) −0.267949 −0.0176680
\(231\) −0.928203 −0.0610713
\(232\) − 3.73205i − 0.245021i
\(233\) −18.1244 −1.18737 −0.593683 0.804699i \(-0.702327\pi\)
−0.593683 + 0.804699i \(0.702327\pi\)
\(234\) 0 0
\(235\) 10.4641 0.682603
\(236\) − 1.53590i − 0.0999785i
\(237\) 0.0717968 0.00466370
\(238\) −8.00000 −0.518563
\(239\) 4.39230i 0.284115i 0.989858 + 0.142057i \(0.0453717\pi\)
−0.989858 + 0.142057i \(0.954628\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) 14.2679i 0.919079i 0.888157 + 0.459540i \(0.151986\pi\)
−0.888157 + 0.459540i \(0.848014\pi\)
\(242\) 10.7846i 0.693261i
\(243\) −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
\(249\) 4.92820i 0.312312i
\(250\) 1.00000 0.0632456
\(251\) 12.2679 0.774346 0.387173 0.922007i \(-0.373452\pi\)
0.387173 + 0.922007i \(0.373452\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 0.124356i 0.00781817i
\(254\) 8.92820i 0.560205i
\(255\) 4.00000i 0.250490i
\(256\) 1.00000 0.0625000
\(257\) −22.6603 −1.41351 −0.706754 0.707459i \(-0.749841\pi\)
−0.706754 + 0.707459i \(0.749841\pi\)
\(258\) 1.92820i 0.120045i
\(259\) 2.39230 0.148651
\(260\) 0 0
\(261\) 3.73205 0.231008
\(262\) − 1.33975i − 0.0827698i
\(263\) −18.1244 −1.11760 −0.558798 0.829304i \(-0.688737\pi\)
−0.558798 + 0.829304i \(0.688737\pi\)
\(264\) −0.464102 −0.0285635
\(265\) − 12.9282i − 0.794173i
\(266\) 1.07180i 0.0657161i
\(267\) − 7.46410i − 0.456796i
\(268\) − 4.53590i − 0.277074i
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 1.00000 0.0608581
\(271\) 9.19615i 0.558626i 0.960200 + 0.279313i \(0.0901068\pi\)
−0.960200 + 0.279313i \(0.909893\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) −4.46410 −0.269686
\(275\) − 0.464102i − 0.0279864i
\(276\) 0.267949 0.0161286
\(277\) 9.92820 0.596528 0.298264 0.954483i \(-0.403592\pi\)
0.298264 + 0.954483i \(0.403592\pi\)
\(278\) 0.928203i 0.0556699i
\(279\) 1.73205i 0.103695i
\(280\) − 2.00000i − 0.119523i
\(281\) 4.92820i 0.293992i 0.989137 + 0.146996i \(0.0469604\pi\)
−0.989137 + 0.146996i \(0.953040\pi\)
\(282\) −10.4641 −0.623128
\(283\) 3.92820 0.233507 0.116754 0.993161i \(-0.462751\pi\)
0.116754 + 0.993161i \(0.462751\pi\)
\(284\) − 8.39230i − 0.497992i
\(285\) 0.535898 0.0317439
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 1.00000i 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) −3.73205 −0.219154
\(291\) − 7.46410i − 0.437553i
\(292\) − 2.00000i − 0.117041i
\(293\) − 4.14359i − 0.242071i −0.992648 0.121036i \(-0.961378\pi\)
0.992648 0.121036i \(-0.0386215\pi\)
\(294\) − 3.00000i − 0.174964i
\(295\) −1.53590 −0.0894235
\(296\) 1.19615 0.0695249
\(297\) − 0.464102i − 0.0269299i
\(298\) 20.4641 1.18545
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 3.85641i 0.222280i
\(302\) −10.3923 −0.598010
\(303\) 10.9282 0.627809
\(304\) 0.535898i 0.0307359i
\(305\) 10.3923i 0.595062i
\(306\) − 4.00000i − 0.228665i
\(307\) 12.5359i 0.715462i 0.933825 + 0.357731i \(0.116449\pi\)
−0.933825 + 0.357731i \(0.883551\pi\)
\(308\) −0.928203 −0.0528893
\(309\) −15.8564 −0.902039
\(310\) − 1.73205i − 0.0983739i
\(311\) −7.60770 −0.431393 −0.215696 0.976460i \(-0.569202\pi\)
−0.215696 + 0.976460i \(0.569202\pi\)
\(312\) 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
\(316\) 0.0717968 0.00403888
\(317\) 21.4641i 1.20554i 0.797913 + 0.602772i \(0.205937\pi\)
−0.797913 + 0.602772i \(0.794063\pi\)
\(318\) 12.9282i 0.724978i
\(319\) 1.73205i 0.0969762i
\(320\) − 1.00000i − 0.0559017i
\(321\) −19.8564 −1.10828
\(322\) 0.535898 0.0298644
\(323\) − 2.14359i − 0.119273i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 23.0526 1.27676
\(327\) − 11.8564i − 0.655661i
\(328\) −2.00000 −0.110432
\(329\) −20.9282 −1.15381
\(330\) 0.464102i 0.0255480i
\(331\) 24.7846i 1.36229i 0.732151 + 0.681143i \(0.238517\pi\)
−0.732151 + 0.681143i \(0.761483\pi\)
\(332\) 4.92820i 0.270470i
\(333\) 1.19615i 0.0655487i
\(334\) −18.3205 −1.00245
\(335\) −4.53590 −0.247823
\(336\) 2.00000i 0.109109i
\(337\) −25.3205 −1.37930 −0.689648 0.724145i \(-0.742235\pi\)
−0.689648 + 0.724145i \(0.742235\pi\)
\(338\) 0 0
\(339\) −11.1962 −0.608092
\(340\) 4.00000i 0.216930i
\(341\) −0.803848 −0.0435308
\(342\) −0.535898 −0.0289781
\(343\) − 20.0000i − 1.07990i
\(344\) 1.92820i 0.103962i
\(345\) − 0.267949i − 0.0144259i
\(346\) − 2.92820i − 0.157421i
\(347\) 22.3923 1.20208 0.601041 0.799218i \(-0.294753\pi\)
0.601041 + 0.799218i \(0.294753\pi\)
\(348\) 3.73205 0.200059
\(349\) − 14.5359i − 0.778089i −0.921219 0.389044i \(-0.872805\pi\)
0.921219 0.389044i \(-0.127195\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) −0.464102 −0.0247367
\(353\) − 2.00000i − 0.106449i −0.998583 0.0532246i \(-0.983050\pi\)
0.998583 0.0532246i \(-0.0169499\pi\)
\(354\) 1.53590 0.0816321
\(355\) −8.39230 −0.445417
\(356\) − 7.46410i − 0.395597i
\(357\) − 8.00000i − 0.423405i
\(358\) − 16.2679i − 0.859788i
\(359\) 18.9282i 0.998992i 0.866316 + 0.499496i \(0.166482\pi\)
−0.866316 + 0.499496i \(0.833518\pi\)
\(360\) 1.00000 0.0527046
\(361\) 18.7128 0.984885
\(362\) 10.9282i 0.574374i
\(363\) −10.7846 −0.566045
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) − 10.3923i − 0.543214i
\(367\) 36.3923 1.89966 0.949831 0.312762i \(-0.101254\pi\)
0.949831 + 0.312762i \(0.101254\pi\)
\(368\) 0.267949 0.0139678
\(369\) − 2.00000i − 0.104116i
\(370\) − 1.19615i − 0.0621850i
\(371\) 25.8564i 1.34240i
\(372\) 1.73205i 0.0898027i
\(373\) 25.7846 1.33508 0.667538 0.744576i \(-0.267348\pi\)
0.667538 + 0.744576i \(0.267348\pi\)
\(374\) 1.85641 0.0959925
\(375\) 1.00000i 0.0516398i
\(376\) −10.4641 −0.539645
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) − 0.143594i − 0.00737590i −0.999993 0.00368795i \(-0.998826\pi\)
0.999993 0.00368795i \(-0.00117391\pi\)
\(380\) 0.535898 0.0274910
\(381\) −8.92820 −0.457406
\(382\) 14.5359i 0.743721i
\(383\) 4.60770i 0.235442i 0.993047 + 0.117721i \(0.0375589\pi\)
−0.993047 + 0.117721i \(0.962441\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 0.928203i 0.0473056i
\(386\) −23.3205 −1.18698
\(387\) −1.92820 −0.0980161
\(388\) − 7.46410i − 0.378932i
\(389\) −20.2679 −1.02763 −0.513813 0.857902i \(-0.671767\pi\)
−0.513813 + 0.857902i \(0.671767\pi\)
\(390\) 0 0
\(391\) −1.07180 −0.0542031
\(392\) − 3.00000i − 0.151523i
\(393\) 1.33975 0.0675812
\(394\) −16.3923 −0.825832
\(395\) − 0.0717968i − 0.00361249i
\(396\) − 0.464102i − 0.0233220i
\(397\) − 12.1244i − 0.608504i −0.952592 0.304252i \(-0.901594\pi\)
0.952592 0.304252i \(-0.0984065\pi\)
\(398\) − 18.9282i − 0.948785i
\(399\) −1.07180 −0.0536570
\(400\) −1.00000 −0.0500000
\(401\) 32.0000i 1.59800i 0.601329 + 0.799002i \(0.294638\pi\)
−0.601329 + 0.799002i \(0.705362\pi\)
\(402\) 4.53590 0.226230
\(403\) 0 0
\(404\) 10.9282 0.543698
\(405\) 1.00000i 0.0496904i
\(406\) 7.46410 0.370437
\(407\) −0.555136 −0.0275171
\(408\) − 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
\(411\) − 4.46410i − 0.220198i
\(412\) −15.8564 −0.781189
\(413\) 3.07180 0.151153
\(414\) 0.267949i 0.0131690i
\(415\) 4.92820 0.241916
\(416\) 0 0
\(417\) −0.928203 −0.0454543
\(418\) − 0.248711i − 0.0121649i
\(419\) 1.60770 0.0785410 0.0392705 0.999229i \(-0.487497\pi\)
0.0392705 + 0.999229i \(0.487497\pi\)
\(420\) 2.00000 0.0975900
\(421\) − 16.3923i − 0.798912i −0.916752 0.399456i \(-0.869199\pi\)
0.916752 0.399456i \(-0.130801\pi\)
\(422\) 23.3205i 1.13522i
\(423\) − 10.4641i − 0.508782i
\(424\) 12.9282i 0.627849i
\(425\) 4.00000 0.194029
\(426\) 8.39230 0.406608
\(427\) − 20.7846i − 1.00584i
\(428\) −19.8564 −0.959796
\(429\) 0 0
\(430\) 1.92820 0.0929862
\(431\) − 7.60770i − 0.366450i −0.983071 0.183225i \(-0.941346\pi\)
0.983071 0.183225i \(-0.0586537\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −15.3205 −0.736257 −0.368128 0.929775i \(-0.620001\pi\)
−0.368128 + 0.929775i \(0.620001\pi\)
\(434\) 3.46410i 0.166282i
\(435\) − 3.73205i − 0.178938i
\(436\) − 11.8564i − 0.567819i
\(437\) 0.143594i 0.00686901i
\(438\) 2.00000 0.0955637
\(439\) 9.85641 0.470421 0.235210 0.971945i \(-0.424422\pi\)
0.235210 + 0.971945i \(0.424422\pi\)
\(440\) 0.464102i 0.0221252i
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) −4.39230 −0.208685 −0.104342 0.994541i \(-0.533274\pi\)
−0.104342 + 0.994541i \(0.533274\pi\)
\(444\) 1.19615i 0.0567669i
\(445\) −7.46410 −0.353832
\(446\) −27.4641 −1.30046
\(447\) 20.4641i 0.967919i
\(448\) 2.00000i 0.0944911i
\(449\) 39.7128i 1.87416i 0.349110 + 0.937082i \(0.386484\pi\)
−0.349110 + 0.937082i \(0.613516\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) 0.928203 0.0437074
\(452\) −11.1962 −0.526623
\(453\) − 10.3923i − 0.488273i
\(454\) −4.39230 −0.206141
\(455\) 0 0
\(456\) −0.535898 −0.0250957
\(457\) − 31.4641i − 1.47183i −0.677075 0.735914i \(-0.736753\pi\)
0.677075 0.735914i \(-0.263247\pi\)
\(458\) −19.8564 −0.927829
\(459\) 4.00000 0.186704
\(460\) − 0.267949i − 0.0124932i
\(461\) − 6.46410i − 0.301063i −0.988605 0.150532i \(-0.951901\pi\)
0.988605 0.150532i \(-0.0480985\pi\)
\(462\) − 0.928203i − 0.0431839i
\(463\) 20.9282i 0.972616i 0.873787 + 0.486308i \(0.161657\pi\)
−0.873787 + 0.486308i \(0.838343\pi\)
\(464\) 3.73205 0.173256
\(465\) 1.73205 0.0803219
\(466\) − 18.1244i − 0.839595i
\(467\) 11.8564 0.548649 0.274325 0.961637i \(-0.411546\pi\)
0.274325 + 0.961637i \(0.411546\pi\)
\(468\) 0 0
\(469\) 9.07180 0.418897
\(470\) 10.4641i 0.482673i
\(471\) 5.00000 0.230388
\(472\) 1.53590 0.0706955
\(473\) − 0.894882i − 0.0411467i
\(474\) 0.0717968i 0.00329773i
\(475\) − 0.535898i − 0.0245887i
\(476\) − 8.00000i − 0.366679i
\(477\) −12.9282 −0.591942
\(478\) −4.39230 −0.200899
\(479\) 1.46410i 0.0668965i 0.999440 + 0.0334483i \(0.0106489\pi\)
−0.999440 + 0.0334483i \(0.989351\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −14.2679 −0.649887
\(483\) 0.535898i 0.0243842i
\(484\) −10.7846 −0.490210
\(485\) −7.46410 −0.338927
\(486\) − 1.00000i − 0.0453609i
\(487\) − 39.1769i − 1.77528i −0.460542 0.887638i \(-0.652345\pi\)
0.460542 0.887638i \(-0.347655\pi\)
\(488\) − 10.3923i − 0.470438i
\(489\) 23.0526i 1.04247i
\(490\) −3.00000 −0.135526
\(491\) 17.3205 0.781664 0.390832 0.920462i \(-0.372187\pi\)
0.390832 + 0.920462i \(0.372187\pi\)
\(492\) − 2.00000i − 0.0901670i
\(493\) −14.9282 −0.672332
\(494\) 0 0
\(495\) −0.464102 −0.0208598
\(496\) 1.73205i 0.0777714i
\(497\) 16.7846 0.752893
\(498\) −4.92820 −0.220838
\(499\) 13.4641i 0.602736i 0.953508 + 0.301368i \(0.0974433\pi\)
−0.953508 + 0.301368i \(0.902557\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) − 18.3205i − 0.818500i
\(502\) 12.2679i 0.547545i
\(503\) 31.1769 1.39011 0.695055 0.718957i \(-0.255380\pi\)
0.695055 + 0.718957i \(0.255380\pi\)
\(504\) −2.00000 −0.0890871
\(505\) − 10.9282i − 0.486299i
\(506\) −0.124356 −0.00552828
\(507\) 0 0
\(508\) −8.92820 −0.396125
\(509\) 19.3923i 0.859549i 0.902936 + 0.429774i \(0.141407\pi\)
−0.902936 + 0.429774i \(0.858593\pi\)
\(510\) −4.00000 −0.177123
\(511\) 4.00000 0.176950
\(512\) 1.00000i 0.0441942i
\(513\) − 0.535898i − 0.0236605i
\(514\) − 22.6603i − 0.999501i
\(515\) 15.8564i 0.698717i
\(516\) −1.92820 −0.0848844
\(517\) 4.85641 0.213585
\(518\) 2.39230i 0.105112i
\(519\) 2.92820 0.128534
\(520\) 0 0
\(521\) 17.3205 0.758825 0.379413 0.925228i \(-0.376126\pi\)
0.379413 + 0.925228i \(0.376126\pi\)
\(522\) 3.73205i 0.163347i
\(523\) −29.7846 −1.30239 −0.651195 0.758910i \(-0.725732\pi\)
−0.651195 + 0.758910i \(0.725732\pi\)
\(524\) 1.33975 0.0585271
\(525\) − 2.00000i − 0.0872872i
\(526\) − 18.1244i − 0.790259i
\(527\) − 6.92820i − 0.301797i
\(528\) − 0.464102i − 0.0201974i
\(529\) −22.9282 −0.996878
\(530\) 12.9282 0.561565
\(531\) 1.53590i 0.0666523i
\(532\) −1.07180 −0.0464683
\(533\) 0 0
\(534\) 7.46410 0.323003
\(535\) 19.8564i 0.858467i
\(536\) 4.53590 0.195921
\(537\) 16.2679 0.702014
\(538\) 12.0000i 0.517357i
\(539\) 1.39230i 0.0599708i
\(540\) 1.00000i 0.0430331i
\(541\) 13.0718i 0.562000i 0.959708 + 0.281000i \(0.0906662\pi\)
−0.959708 + 0.281000i \(0.909334\pi\)
\(542\) −9.19615 −0.395009
\(543\) −10.9282 −0.468974
\(544\) − 4.00000i − 0.171499i
\(545\) −11.8564 −0.507873
\(546\) 0 0
\(547\) 9.07180 0.387882 0.193941 0.981013i \(-0.437873\pi\)
0.193941 + 0.981013i \(0.437873\pi\)
\(548\) − 4.46410i − 0.190697i
\(549\) 10.3923 0.443533
\(550\) 0.464102 0.0197894
\(551\) 2.00000i 0.0852029i
\(552\) 0.267949i 0.0114047i
\(553\) 0.143594i 0.00610622i
\(554\) 9.92820i 0.421809i
\(555\) 1.19615 0.0507738
\(556\) −0.928203 −0.0393646
\(557\) 37.7128i 1.59794i 0.601369 + 0.798972i \(0.294622\pi\)
−0.601369 + 0.798972i \(0.705378\pi\)
\(558\) −1.73205 −0.0733236
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) 1.85641i 0.0783775i
\(562\) −4.92820 −0.207884
\(563\) −39.3205 −1.65716 −0.828581 0.559869i \(-0.810851\pi\)
−0.828581 + 0.559869i \(0.810851\pi\)
\(564\) − 10.4641i − 0.440618i
\(565\) 11.1962i 0.471026i
\(566\) 3.92820i 0.165115i
\(567\) − 2.00000i − 0.0839921i
\(568\) 8.39230 0.352133
\(569\) −5.32051 −0.223047 −0.111524 0.993762i \(-0.535573\pi\)
−0.111524 + 0.993762i \(0.535573\pi\)
\(570\) 0.535898i 0.0224463i
\(571\) 45.1769 1.89060 0.945298 0.326209i \(-0.105771\pi\)
0.945298 + 0.326209i \(0.105771\pi\)
\(572\) 0 0
\(573\) −14.5359 −0.607246
\(574\) − 4.00000i − 0.166957i
\(575\) −0.267949 −0.0111743
\(576\) −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
\(579\) − 23.3205i − 0.969167i
\(580\) − 3.73205i − 0.154965i
\(581\) −9.85641 −0.408913
\(582\) 7.46410 0.309397
\(583\) − 6.00000i − 0.248495i
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 4.14359 0.171170
\(587\) − 18.3923i − 0.759132i −0.925165 0.379566i \(-0.876073\pi\)
0.925165 0.379566i \(-0.123927\pi\)
\(588\) 3.00000 0.123718
\(589\) −0.928203 −0.0382459
\(590\) − 1.53590i − 0.0632319i
\(591\) − 16.3923i − 0.674289i
\(592\) 1.19615i 0.0491616i
\(593\) − 31.1051i − 1.27733i −0.769483 0.638667i \(-0.779486\pi\)
0.769483 0.638667i \(-0.220514\pi\)
\(594\) 0.464102 0.0190423
\(595\) −8.00000 −0.327968
\(596\) 20.4641i 0.838242i
\(597\) 18.9282 0.774680
\(598\) 0 0
\(599\) 10.3923 0.424618 0.212309 0.977203i \(-0.431902\pi\)
0.212309 + 0.977203i \(0.431902\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) 21.7846 0.888613 0.444306 0.895875i \(-0.353450\pi\)
0.444306 + 0.895875i \(0.353450\pi\)
\(602\) −3.85641 −0.157175
\(603\) 4.53590i 0.184716i
\(604\) − 10.3923i − 0.422857i
\(605\) 10.7846i 0.438457i
\(606\) 10.9282i 0.443928i
\(607\) 43.1769 1.75250 0.876248 0.481860i \(-0.160038\pi\)
0.876248 + 0.481860i \(0.160038\pi\)
\(608\) −0.535898 −0.0217335
\(609\) 7.46410i 0.302461i
\(610\) −10.3923 −0.420772
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) 0.947441i 0.0382668i 0.999817 + 0.0191334i \(0.00609072\pi\)
−0.999817 + 0.0191334i \(0.993909\pi\)
\(614\) −12.5359 −0.505908
\(615\) −2.00000 −0.0806478
\(616\) − 0.928203i − 0.0373984i
\(617\) − 15.5359i − 0.625452i −0.949843 0.312726i \(-0.898758\pi\)
0.949843 0.312726i \(-0.101242\pi\)
\(618\) − 15.8564i − 0.637838i
\(619\) 24.2487i 0.974638i 0.873224 + 0.487319i \(0.162025\pi\)
−0.873224 + 0.487319i \(0.837975\pi\)
\(620\) 1.73205 0.0695608
\(621\) −0.267949 −0.0107524
\(622\) − 7.60770i − 0.305041i
\(623\) 14.9282 0.598086
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 28.0000i − 1.11911i
\(627\) 0.248711 0.00993257
\(628\) 5.00000 0.199522
\(629\) − 4.78461i − 0.190775i
\(630\) 2.00000i 0.0796819i
\(631\) 24.5359i 0.976759i 0.872631 + 0.488379i \(0.162412\pi\)
−0.872631 + 0.488379i \(0.837588\pi\)
\(632\) 0.0717968i 0.00285592i
\(633\) −23.3205 −0.926907
\(634\) −21.4641 −0.852448
\(635\) 8.92820i 0.354305i
\(636\) −12.9282 −0.512637
\(637\) 0 0
\(638\) −1.73205 −0.0685725
\(639\) 8.39230i 0.331994i
\(640\) 1.00000 0.0395285
\(641\) 27.8564 1.10026 0.550131 0.835078i \(-0.314578\pi\)
0.550131 + 0.835078i \(0.314578\pi\)
\(642\) − 19.8564i − 0.783670i
\(643\) − 27.4641i − 1.08308i −0.840675 0.541539i \(-0.817842\pi\)
0.840675 0.541539i \(-0.182158\pi\)
\(644\) 0.535898i 0.0211174i
\(645\) 1.92820i 0.0759229i
\(646\) 2.14359 0.0843386
\(647\) 21.3205 0.838196 0.419098 0.907941i \(-0.362346\pi\)
0.419098 + 0.907941i \(0.362346\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −0.712813 −0.0279804
\(650\) 0 0
\(651\) −3.46410 −0.135769
\(652\) 23.0526i 0.902808i
\(653\) −44.2487 −1.73159 −0.865793 0.500402i \(-0.833185\pi\)
−0.865793 + 0.500402i \(0.833185\pi\)
\(654\) 11.8564 0.463622
\(655\) − 1.33975i − 0.0523482i
\(656\) − 2.00000i − 0.0780869i
\(657\) 2.00000i 0.0780274i
\(658\) − 20.9282i − 0.815866i
\(659\) −3.73205 −0.145380 −0.0726900 0.997355i \(-0.523158\pi\)
−0.0726900 + 0.997355i \(0.523158\pi\)
\(660\) −0.464102 −0.0180651
\(661\) 43.3205i 1.68497i 0.538718 + 0.842486i \(0.318909\pi\)
−0.538718 + 0.842486i \(0.681091\pi\)
\(662\) −24.7846 −0.963281
\(663\) 0 0
\(664\) −4.92820 −0.191251
\(665\) 1.07180i 0.0415625i
\(666\) −1.19615 −0.0463500
\(667\) 1.00000 0.0387202
\(668\) − 18.3205i − 0.708842i
\(669\) − 27.4641i − 1.06182i
\(670\) − 4.53590i − 0.175237i
\(671\) 4.82309i 0.186193i
\(672\) −2.00000 −0.0771517
\(673\) 32.0000 1.23351 0.616755 0.787155i \(-0.288447\pi\)
0.616755 + 0.787155i \(0.288447\pi\)
\(674\) − 25.3205i − 0.975310i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 11.6077 0.446120 0.223060 0.974805i \(-0.428395\pi\)
0.223060 + 0.974805i \(0.428395\pi\)
\(678\) − 11.1962i − 0.429986i
\(679\) 14.9282 0.572892
\(680\) −4.00000 −0.153393
\(681\) − 4.39230i − 0.168313i
\(682\) − 0.803848i − 0.0307809i
\(683\) 0.784610i 0.0300223i 0.999887 + 0.0150111i \(0.00477837\pi\)
−0.999887 + 0.0150111i \(0.995222\pi\)
\(684\) − 0.535898i − 0.0204906i
\(685\) −4.46410 −0.170565
\(686\) 20.0000 0.763604
\(687\) − 19.8564i − 0.757569i
\(688\) −1.92820 −0.0735121
\(689\) 0 0
\(690\) 0.267949 0.0102007
\(691\) − 35.1769i − 1.33819i −0.743176 0.669096i \(-0.766681\pi\)
0.743176 0.669096i \(-0.233319\pi\)
\(692\) 2.92820 0.111314
\(693\) 0.928203 0.0352595
\(694\) 22.3923i 0.850000i
\(695\) 0.928203i 0.0352088i
\(696\) 3.73205i 0.141463i
\(697\) 8.00000i 0.303022i
\(698\) 14.5359 0.550192
\(699\) 18.1244 0.685526
\(700\) − 2.00000i − 0.0755929i
\(701\) −3.73205 −0.140958 −0.0704788 0.997513i \(-0.522453\pi\)
−0.0704788 + 0.997513i \(0.522453\pi\)
\(702\) 0 0
\(703\) −0.641016 −0.0241764
\(704\) − 0.464102i − 0.0174915i
\(705\) −10.4641 −0.394101
\(706\) 2.00000 0.0752710
\(707\) 21.8564i 0.821995i
\(708\) 1.53590i 0.0577226i
\(709\) − 9.07180i − 0.340698i −0.985384 0.170349i \(-0.945510\pi\)
0.985384 0.170349i \(-0.0544896\pi\)
\(710\) − 8.39230i − 0.314958i
\(711\) −0.0717968 −0.00269259
\(712\) 7.46410 0.279729
\(713\) 0.464102i 0.0173807i
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) 16.2679 0.607962
\(717\) − 4.39230i − 0.164034i
\(718\) −18.9282 −0.706394
\(719\) 34.6410 1.29189 0.645946 0.763383i \(-0.276463\pi\)
0.645946 + 0.763383i \(0.276463\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) − 31.7128i − 1.18105i
\(722\) 18.7128i 0.696419i
\(723\) − 14.2679i − 0.530631i
\(724\) −10.9282 −0.406143
\(725\) −3.73205 −0.138605
\(726\) − 10.7846i − 0.400254i
\(727\) 23.7128 0.879460 0.439730 0.898130i \(-0.355074\pi\)
0.439730 + 0.898130i \(0.355074\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 2.00000i − 0.0740233i
\(731\) 7.71281 0.285269
\(732\) 10.3923 0.384111
\(733\) − 37.0718i − 1.36928i −0.728882 0.684639i \(-0.759960\pi\)
0.728882 0.684639i \(-0.240040\pi\)
\(734\) 36.3923i 1.34326i
\(735\) − 3.00000i − 0.110657i
\(736\) 0.267949i 0.00987674i
\(737\) −2.10512 −0.0775430
\(738\) 2.00000 0.0736210
\(739\) − 15.3205i − 0.563574i −0.959477 0.281787i \(-0.909073\pi\)
0.959477 0.281787i \(-0.0909272\pi\)
\(740\) 1.19615 0.0439714
\(741\) 0 0
\(742\) −25.8564 −0.949219
\(743\) 33.5359i 1.23031i 0.788405 + 0.615156i \(0.210907\pi\)
−0.788405 + 0.615156i \(0.789093\pi\)
\(744\) −1.73205 −0.0635001
\(745\) 20.4641 0.749747
\(746\) 25.7846i 0.944042i
\(747\) − 4.92820i − 0.180314i
\(748\) 1.85641i 0.0678769i
\(749\) − 39.7128i − 1.45107i
\(750\) −1.00000 −0.0365148
\(751\) 27.9282 1.01911 0.509557 0.860437i \(-0.329809\pi\)
0.509557 + 0.860437i \(0.329809\pi\)
\(752\) − 10.4641i − 0.381587i
\(753\) −12.2679 −0.447069
\(754\) 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 \(0.606075\pi\)
\(758\) 0.143594 0.00521555
\(759\) − 0.124356i − 0.00451382i
\(760\) 0.535898i 0.0194391i
\(761\) 18.9282i 0.686147i 0.939309 + 0.343073i \(0.111468\pi\)
−0.939309 + 0.343073i \(0.888532\pi\)
\(762\) − 8.92820i − 0.323435i
\(763\) 23.7128 0.858461
\(764\) −14.5359 −0.525890
\(765\) − 4.00000i − 0.144620i
\(766\) −4.60770 −0.166483
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) − 19.5885i − 0.706378i −0.935552 0.353189i \(-0.885097\pi\)
0.935552 0.353189i \(-0.114903\pi\)
\(770\) −0.928203 −0.0334501
\(771\) 22.6603 0.816089
\(772\) − 23.3205i − 0.839323i
\(773\) − 27.7128i − 0.996761i −0.866959 0.498380i \(-0.833928\pi\)
0.866959 0.498380i \(-0.166072\pi\)
\(774\) − 1.92820i − 0.0693078i
\(775\) − 1.73205i − 0.0622171i
\(776\) 7.46410 0.267946
\(777\) −2.39230 −0.0858235
\(778\) − 20.2679i − 0.726641i
\(779\) 1.07180 0.0384011
\(780\) 0 0
\(781\) −3.89488 −0.139370
\(782\) − 1.07180i − 0.0383274i
\(783\) −3.73205 −0.133373
\(784\) 3.00000 0.107143
\(785\) − 5.00000i − 0.178458i
\(786\) 1.33975i 0.0477872i
\(787\) − 1.73205i − 0.0617409i −0.999523 0.0308705i \(-0.990172\pi\)
0.999523 0.0308705i \(-0.00982794\pi\)
\(788\) − 16.3923i − 0.583952i
\(789\) 18.1244 0.645244
\(790\) 0.0717968 0.00255441
\(791\) − 22.3923i − 0.796179i
\(792\) 0.464102 0.0164911
\(793\) 0 0
\(794\) 12.1244 0.430277
\(795\) 12.9282i 0.458516i
\(796\) 18.9282 0.670892
\(797\) −37.8564 −1.34094 −0.670471 0.741935i \(-0.733908\pi\)
−0.670471 + 0.741935i \(0.733908\pi\)
\(798\) − 1.07180i − 0.0379412i
\(799\) 41.8564i 1.48077i
\(800\) − 1.00000i − 0.0353553i
\(801\) 7.46410i 0.263731i
\(802\) −32.0000 −1.12996
\(803\) −0.928203 −0.0327556
\(804\) 4.53590i 0.159969i
\(805\) 0.535898 0.0188879
\(806\) 0 0
\(807\) −12.0000 −0.422420
\(808\) 10.9282i 0.384453i
\(809\) −1.21539 −0.0427308 −0.0213654 0.999772i \(-0.506801\pi\)
−0.0213654 + 0.999772i \(0.506801\pi\)
\(810\) −1.00000 −0.0351364
\(811\) − 0.784610i − 0.0275514i −0.999905 0.0137757i \(-0.995615\pi\)
0.999905 0.0137757i \(-0.00438508\pi\)
\(812\) 7.46410i 0.261939i
\(813\) − 9.19615i − 0.322523i
\(814\) − 0.555136i − 0.0194575i
\(815\) 23.0526 0.807496
\(816\) 4.00000 0.140028
\(817\) − 1.03332i − 0.0361513i
\(818\) −4.00000 −0.139857
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) − 35.3923i − 1.23520i −0.786492 0.617600i \(-0.788105\pi\)
0.786492 0.617600i \(-0.211895\pi\)
\(822\) 4.46410 0.155703
\(823\) −23.1769 −0.807896 −0.403948 0.914782i \(-0.632362\pi\)
−0.403948 + 0.914782i \(0.632362\pi\)
\(824\) − 15.8564i − 0.552384i
\(825\) 0.464102i 0.0161579i
\(826\) 3.07180i 0.106881i
\(827\) − 17.3205i − 0.602293i −0.953578 0.301147i \(-0.902631\pi\)
0.953578 0.301147i \(-0.0973693\pi\)
\(828\) −0.267949 −0.00931188
\(829\) 23.4641 0.814942 0.407471 0.913218i \(-0.366411\pi\)
0.407471 + 0.913218i \(0.366411\pi\)
\(830\) 4.92820i 0.171060i
\(831\) −9.92820 −0.344406
\(832\) 0 0
\(833\) −12.0000 −0.415775
\(834\) − 0.928203i − 0.0321410i
\(835\) −18.3205 −0.634007
\(836\) 0.248711 0.00860186
\(837\) − 1.73205i − 0.0598684i
\(838\) 1.60770i 0.0555369i
\(839\) 39.5692i 1.36608i 0.730380 + 0.683041i \(0.239343\pi\)
−0.730380 + 0.683041i \(0.760657\pi\)
\(840\) 2.00000i 0.0690066i
\(841\) −15.0718 −0.519717
\(842\) 16.3923 0.564916
\(843\) − 4.92820i − 0.169736i
\(844\) −23.3205 −0.802725
\(845\) 0 0
\(846\) 10.4641 0.359763
\(847\) − 21.5692i − 0.741127i
\(848\) −12.9282 −0.443956
\(849\) −3.92820 −0.134816
\(850\) 4.00000i 0.137199i
\(851\) 0.320508i 0.0109869i
\(852\) 8.39230i 0.287516i
\(853\) − 35.8372i − 1.22704i −0.789679 0.613521i \(-0.789753\pi\)
0.789679 0.613521i \(-0.210247\pi\)
\(854\) 20.7846 0.711235
\(855\) −0.535898 −0.0183273
\(856\) − 19.8564i − 0.678678i
\(857\) −20.5167 −0.700836 −0.350418 0.936593i \(-0.613960\pi\)
−0.350418 + 0.936593i \(0.613960\pi\)
\(858\) 0 0
\(859\) 27.1769 0.927264 0.463632 0.886028i \(-0.346546\pi\)
0.463632 + 0.886028i \(0.346546\pi\)
\(860\) 1.92820i 0.0657512i
\(861\) 4.00000 0.136320
\(862\) 7.60770 0.259119
\(863\) 42.4641i 1.44549i 0.691112 + 0.722747i \(0.257121\pi\)
−0.691112 + 0.722747i \(0.742879\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) − 2.92820i − 0.0995619i
\(866\) − 15.3205i − 0.520612i
\(867\) 1.00000 0.0339618
\(868\) −3.46410 −0.117579
\(869\) − 0.0333210i − 0.00113034i
\(870\) 3.73205 0.126528
\(871\) 0 0
\(872\) 11.8564 0.401509
\(873\) 7.46410i 0.252622i
\(874\) −0.143594 −0.00485712
\(875\) −2.00000 −0.0676123
\(876\) 2.00000i 0.0675737i
\(877\) − 28.1244i − 0.949692i −0.880069 0.474846i \(-0.842504\pi\)
0.880069 0.474846i \(-0.157496\pi\)
\(878\) 9.85641i 0.332638i
\(879\) 4.14359i 0.139760i
\(880\) −0.464102 −0.0156449
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 16.0718 0.540859 0.270430 0.962740i \(-0.412834\pi\)
0.270430 + 0.962740i \(0.412834\pi\)
\(884\) 0 0
\(885\) 1.53590 0.0516287
\(886\) − 4.39230i − 0.147562i
\(887\) −10.1244 −0.339943 −0.169971 0.985449i \(-0.554368\pi\)
−0.169971 + 0.985449i \(0.554368\pi\)
\(888\) −1.19615 −0.0401402
\(889\) − 17.8564i − 0.598885i
\(890\) − 7.46410i − 0.250197i
\(891\) 0.464102i 0.0155480i
\(892\) − 27.4641i − 0.919566i
\(893\) 5.60770 0.187654
\(894\) −20.4641 −0.684422
\(895\) − 16.2679i − 0.543778i
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −39.7128 −1.32523
\(899\) 6.46410i 0.215590i
\(900\) 1.00000 0.0333333
\(901\) 51.7128 1.72280
\(902\) 0.928203i 0.0309058i
\(903\) − 3.85641i − 0.128333i
\(904\) − 11.1962i − 0.372378i
\(905\) 10.9282i 0.363266i
\(906\) 10.3923 0.345261
\(907\) 6.85641 0.227663 0.113832 0.993500i \(-0.463688\pi\)
0.113832 + 0.993500i \(0.463688\pi\)
\(908\) − 4.39230i − 0.145764i
\(909\) −10.9282 −0.362466
\(910\) 0 0
\(911\) −24.2487 −0.803396 −0.401698 0.915772i \(-0.631580\pi\)
−0.401698 + 0.915772i \(0.631580\pi\)
\(912\) − 0.535898i − 0.0177454i
\(913\) 2.28719 0.0756948
\(914\) 31.4641 1.04074
\(915\) − 10.3923i − 0.343559i
\(916\) − 19.8564i − 0.656074i
\(917\) 2.67949i 0.0884846i
\(918\) 4.00000i 0.132020i
\(919\) −54.6410 −1.80244 −0.901220 0.433361i \(-0.857328\pi\)
−0.901220 + 0.433361i \(0.857328\pi\)
\(920\) 0.267949 0.00883402
\(921\) − 12.5359i − 0.413072i
\(922\) 6.46410 0.212884
\(923\) 0 0
\(924\) 0.928203 0.0305356
\(925\) − 1.19615i − 0.0393292i
\(926\) −20.9282 −0.687743
\(927\) 15.8564 0.520793
\(928\) 3.73205i 0.122511i
\(929\) − 33.8564i − 1.11079i −0.831586 0.555396i \(-0.812567\pi\)
0.831586 0.555396i \(-0.187433\pi\)
\(930\) 1.73205i 0.0567962i
\(931\) 1.60770i 0.0526901i
\(932\) 18.1244 0.593683
\(933\) 7.60770 0.249065
\(934\) 11.8564i 0.387953i
\(935\) 1.85641 0.0607110
\(936\) 0 0
\(937\) 44.6410 1.45836 0.729179 0.684323i \(-0.239902\pi\)
0.729179 + 0.684323i \(0.239902\pi\)
\(938\) 9.07180i 0.296205i
\(939\) 28.0000 0.913745
\(940\) −10.4641 −0.341301
\(941\) 26.7846i 0.873153i 0.899667 + 0.436577i \(0.143809\pi\)
−0.899667 + 0.436577i \(0.856191\pi\)
\(942\) 5.00000i 0.162909i
\(943\) − 0.535898i − 0.0174513i
\(944\) 1.53590i 0.0499892i
\(945\) −2.00000 −0.0650600
\(946\) 0.894882 0.0290951
\(947\) − 2.53590i − 0.0824056i −0.999151 0.0412028i \(-0.986881\pi\)
0.999151 0.0412028i \(-0.0131190\pi\)
\(948\) −0.0717968 −0.00233185
\(949\) 0 0
\(950\) 0.535898 0.0173868
\(951\) − 21.4641i − 0.696021i
\(952\) 8.00000 0.259281
\(953\) −15.7321 −0.509611 −0.254806 0.966992i \(-0.582011\pi\)
−0.254806 + 0.966992i \(0.582011\pi\)
\(954\) − 12.9282i − 0.418566i
\(955\) 14.5359i 0.470371i
\(956\) − 4.39230i − 0.142057i
\(957\) − 1.73205i − 0.0559893i
\(958\) −1.46410 −0.0473030
\(959\) 8.92820 0.288307
\(960\) 1.00000i 0.0322749i
\(961\) 28.0000 0.903226
\(962\) 0 0
\(963\) 19.8564 0.639864
\(964\) − 14.2679i − 0.459540i
\(965\) −23.3205 −0.750714
\(966\) −0.535898 −0.0172422
\(967\) − 34.5359i − 1.11060i −0.831650 0.555300i \(-0.812604\pi\)
0.831650 0.555300i \(-0.187396\pi\)
\(968\) − 10.7846i − 0.346630i
\(969\) 2.14359i 0.0688621i
\(970\) − 7.46410i − 0.239658i
\(971\) 11.4641 0.367901 0.183950 0.982936i \(-0.441111\pi\)
0.183950 + 0.982936i \(0.441111\pi\)
\(972\) 1.00000 0.0320750
\(973\) − 1.85641i − 0.0595137i
\(974\) 39.1769 1.25531
\(975\) 0 0
\(976\) 10.3923 0.332650
\(977\) − 33.3923i − 1.06831i −0.845385 0.534157i \(-0.820629\pi\)
0.845385 0.534157i \(-0.179371\pi\)
\(978\) −23.0526 −0.737140
\(979\) −3.46410 −0.110713
\(980\) − 3.00000i − 0.0958315i
\(981\) 11.8564i 0.378546i
\(982\) 17.3205i 0.552720i
\(983\) − 57.3923i − 1.83053i −0.402852 0.915265i \(-0.631981\pi\)
0.402852 0.915265i \(-0.368019\pi\)
\(984\) 2.00000 0.0637577
\(985\) −16.3923 −0.522302
\(986\) − 14.9282i − 0.475411i
\(987\) 20.9282 0.666152
\(988\) 0 0
\(989\) −0.516660 −0.0164288
\(990\) − 0.464102i − 0.0147501i
\(991\) −25.1436 −0.798713 −0.399356 0.916796i \(-0.630766\pi\)
−0.399356 + 0.916796i \(0.630766\pi\)
\(992\) −1.73205 −0.0549927
\(993\) − 24.7846i − 0.786516i
\(994\) 16.7846i 0.532375i
\(995\) − 18.9282i − 0.600064i
\(996\) − 4.92820i − 0.156156i
\(997\) −47.5692 −1.50653 −0.753266 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(998\) −13.4641 −0.426199
\(999\) − 1.19615i − 0.0378446i
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.1351.4 4
13.5 odd 4 5070.2.a.bg.1.1 2
13.8 odd 4 5070.2.a.y.1.2 2
13.9 even 3 390.2.bb.b.361.2 yes 4
13.10 even 6 390.2.bb.b.121.2 4
13.12 even 2 inner 5070.2.b.o.1351.1 4
39.23 odd 6 1170.2.bs.e.901.1 4
39.35 odd 6 1170.2.bs.e.361.1 4
65.9 even 6 1950.2.bc.b.751.1 4
65.22 odd 12 1950.2.y.f.49.2 4
65.23 odd 12 1950.2.y.f.199.2 4
65.48 odd 12 1950.2.y.c.49.1 4
65.49 even 6 1950.2.bc.b.901.1 4
65.62 odd 12 1950.2.y.c.199.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.b.121.2 4 13.10 even 6
390.2.bb.b.361.2 yes 4 13.9 even 3
1170.2.bs.e.361.1 4 39.35 odd 6
1170.2.bs.e.901.1 4 39.23 odd 6
1950.2.y.c.49.1 4 65.48 odd 12
1950.2.y.c.199.1 4 65.62 odd 12
1950.2.y.f.49.2 4 65.22 odd 12
1950.2.y.f.199.2 4 65.23 odd 12
1950.2.bc.b.751.1 4 65.9 even 6
1950.2.bc.b.901.1 4 65.49 even 6
5070.2.a.y.1.2 2 13.8 odd 4
5070.2.a.bg.1.1 2 13.5 odd 4
5070.2.b.o.1351.1 4 13.12 even 2 inner
5070.2.b.o.1351.4 4 1.1 even 1 trivial