Properties

Label 5070.2.a.bg.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −0.464102 −0.139932 −0.0699660 0.997549i \(-0.522289\pi\)
−0.0699660 + 0.997549i \(0.522289\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.535898 0.122944 0.0614718 0.998109i \(-0.480421\pi\)
0.0614718 + 0.998109i \(0.480421\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.00000 −0.436436
\(22\) −0.464102 −0.0989468
\(23\) −0.267949 −0.0558713 −0.0279356 0.999610i \(-0.508893\pi\)
−0.0279356 + 0.999610i \(0.508893\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) 3.73205 0.693024 0.346512 0.938045i \(-0.387366\pi\)
0.346512 + 0.938045i \(0.387366\pi\)
\(30\) −1.00000 −0.182574
\(31\) 1.73205 0.311086 0.155543 0.987829i \(-0.450287\pi\)
0.155543 + 0.987829i \(0.450287\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.464102 0.0807897
\(34\) 4.00000 0.685994
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) −1.19615 −0.196646 −0.0983231 0.995155i \(-0.531348\pi\)
−0.0983231 + 0.995155i \(0.531348\pi\)
\(38\) 0.535898 0.0869342
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −2.00000 −0.308607
\(43\) 1.92820 0.294048 0.147024 0.989133i \(-0.453031\pi\)
0.147024 + 0.989133i \(0.453031\pi\)
\(44\) −0.464102 −0.0699660
\(45\) 1.00000 0.149071
\(46\) −0.267949 −0.0395070
\(47\) 10.4641 1.52635 0.763173 0.646194i \(-0.223640\pi\)
0.763173 + 0.646194i \(0.223640\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −12.9282 −1.77583 −0.887913 0.460012i \(-0.847845\pi\)
−0.887913 + 0.460012i \(0.847845\pi\)
\(54\) −1.00000 −0.136083
\(55\) −0.464102 −0.0625794
\(56\) 2.00000 0.267261
\(57\) −0.535898 −0.0709815
\(58\) 3.73205 0.490042
\(59\) −1.53590 −0.199957 −0.0999785 0.994990i \(-0.531877\pi\)
−0.0999785 + 0.994990i \(0.531877\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.3923 1.33060 0.665299 0.746577i \(-0.268304\pi\)
0.665299 + 0.746577i \(0.268304\pi\)
\(62\) 1.73205 0.219971
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.464102 0.0571270
\(67\) 4.53590 0.554148 0.277074 0.960849i \(-0.410635\pi\)
0.277074 + 0.960849i \(0.410635\pi\)
\(68\) 4.00000 0.485071
\(69\) 0.267949 0.0322573
\(70\) 2.00000 0.239046
\(71\) 8.39230 0.995983 0.497992 0.867182i \(-0.334071\pi\)
0.497992 + 0.867182i \(0.334071\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −1.19615 −0.139050
\(75\) −1.00000 −0.115470
\(76\) 0.535898 0.0614718
\(77\) −0.928203 −0.105779
\(78\) 0 0
\(79\) −0.0717968 −0.00807777 −0.00403888 0.999992i \(-0.501286\pi\)
−0.00403888 + 0.999992i \(0.501286\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −4.92820 −0.540941 −0.270470 0.962728i \(-0.587179\pi\)
−0.270470 + 0.962728i \(0.587179\pi\)
\(84\) −2.00000 −0.218218
\(85\) 4.00000 0.433861
\(86\) 1.92820 0.207924
\(87\) −3.73205 −0.400118
\(88\) −0.464102 −0.0494734
\(89\) −7.46410 −0.791193 −0.395597 0.918424i \(-0.629462\pi\)
−0.395597 + 0.918424i \(0.629462\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −0.267949 −0.0279356
\(93\) −1.73205 −0.179605
\(94\) 10.4641 1.07929
\(95\) 0.535898 0.0549820
\(96\) −1.00000 −0.102062
\(97\) 7.46410 0.757865 0.378932 0.925424i \(-0.376291\pi\)
0.378932 + 0.925424i \(0.376291\pi\)
\(98\) −3.00000 −0.303046
\(99\) −0.464102 −0.0466440
\(100\) 1.00000 0.100000
\(101\) 10.9282 1.08740 0.543698 0.839281i \(-0.317024\pi\)
0.543698 + 0.839281i \(0.317024\pi\)
\(102\) −4.00000 −0.396059
\(103\) −15.8564 −1.56238 −0.781189 0.624295i \(-0.785387\pi\)
−0.781189 + 0.624295i \(0.785387\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) −12.9282 −1.25570
\(107\) 19.8564 1.91959 0.959796 0.280700i \(-0.0905665\pi\)
0.959796 + 0.280700i \(0.0905665\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 11.8564 1.13564 0.567819 0.823154i \(-0.307787\pi\)
0.567819 + 0.823154i \(0.307787\pi\)
\(110\) −0.464102 −0.0442504
\(111\) 1.19615 0.113534
\(112\) 2.00000 0.188982
\(113\) 11.1962 1.05325 0.526623 0.850099i \(-0.323458\pi\)
0.526623 + 0.850099i \(0.323458\pi\)
\(114\) −0.535898 −0.0501915
\(115\) −0.267949 −0.0249864
\(116\) 3.73205 0.346512
\(117\) 0 0
\(118\) −1.53590 −0.141391
\(119\) 8.00000 0.733359
\(120\) −1.00000 −0.0912871
\(121\) −10.7846 −0.980419
\(122\) 10.3923 0.940875
\(123\) 2.00000 0.180334
\(124\) 1.73205 0.155543
\(125\) 1.00000 0.0894427
\(126\) 2.00000 0.178174
\(127\) −8.92820 −0.792250 −0.396125 0.918197i \(-0.629645\pi\)
−0.396125 + 0.918197i \(0.629645\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.92820 −0.169769
\(130\) 0 0
\(131\) −1.33975 −0.117054 −0.0585271 0.998286i \(-0.518640\pi\)
−0.0585271 + 0.998286i \(0.518640\pi\)
\(132\) 0.464102 0.0403949
\(133\) 1.07180 0.0929366
\(134\) 4.53590 0.391842
\(135\) −1.00000 −0.0860663
\(136\) 4.00000 0.342997
\(137\) −4.46410 −0.381394 −0.190697 0.981649i \(-0.561075\pi\)
−0.190697 + 0.981649i \(0.561075\pi\)
\(138\) 0.267949 0.0228093
\(139\) 0.928203 0.0787292 0.0393646 0.999225i \(-0.487467\pi\)
0.0393646 + 0.999225i \(0.487467\pi\)
\(140\) 2.00000 0.169031
\(141\) −10.4641 −0.881236
\(142\) 8.39230 0.704267
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 3.73205 0.309930
\(146\) −2.00000 −0.165521
\(147\) 3.00000 0.247436
\(148\) −1.19615 −0.0983231
\(149\) −20.4641 −1.67648 −0.838242 0.545298i \(-0.816416\pi\)
−0.838242 + 0.545298i \(0.816416\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −10.3923 −0.845714 −0.422857 0.906196i \(-0.638973\pi\)
−0.422857 + 0.906196i \(0.638973\pi\)
\(152\) 0.535898 0.0434671
\(153\) 4.00000 0.323381
\(154\) −0.928203 −0.0747967
\(155\) 1.73205 0.139122
\(156\) 0 0
\(157\) −5.00000 −0.399043 −0.199522 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(158\) −0.0717968 −0.00571184
\(159\) 12.9282 1.02527
\(160\) 1.00000 0.0790569
\(161\) −0.535898 −0.0422347
\(162\) 1.00000 0.0785674
\(163\) 23.0526 1.80562 0.902808 0.430044i \(-0.141502\pi\)
0.902808 + 0.430044i \(0.141502\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0.464102 0.0361303
\(166\) −4.92820 −0.382503
\(167\) −18.3205 −1.41768 −0.708842 0.705368i \(-0.750782\pi\)
−0.708842 + 0.705368i \(0.750782\pi\)
\(168\) −2.00000 −0.154303
\(169\) 0 0
\(170\) 4.00000 0.306786
\(171\) 0.535898 0.0409812
\(172\) 1.92820 0.147024
\(173\) 2.92820 0.222627 0.111314 0.993785i \(-0.464494\pi\)
0.111314 + 0.993785i \(0.464494\pi\)
\(174\) −3.73205 −0.282926
\(175\) 2.00000 0.151186
\(176\) −0.464102 −0.0349830
\(177\) 1.53590 0.115445
\(178\) −7.46410 −0.559458
\(179\) 16.2679 1.21592 0.607962 0.793966i \(-0.291987\pi\)
0.607962 + 0.793966i \(0.291987\pi\)
\(180\) 1.00000 0.0745356
\(181\) −10.9282 −0.812287 −0.406143 0.913809i \(-0.633127\pi\)
−0.406143 + 0.913809i \(0.633127\pi\)
\(182\) 0 0
\(183\) −10.3923 −0.768221
\(184\) −0.267949 −0.0197535
\(185\) −1.19615 −0.0879429
\(186\) −1.73205 −0.127000
\(187\) −1.85641 −0.135754
\(188\) 10.4641 0.763173
\(189\) −2.00000 −0.145479
\(190\) 0.535898 0.0388782
\(191\) 14.5359 1.05178 0.525890 0.850552i \(-0.323732\pi\)
0.525890 + 0.850552i \(0.323732\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −23.3205 −1.67865 −0.839323 0.543632i \(-0.817049\pi\)
−0.839323 + 0.543632i \(0.817049\pi\)
\(194\) 7.46410 0.535891
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 16.3923 1.16790 0.583952 0.811788i \(-0.301506\pi\)
0.583952 + 0.811788i \(0.301506\pi\)
\(198\) −0.464102 −0.0329823
\(199\) 18.9282 1.34178 0.670892 0.741555i \(-0.265911\pi\)
0.670892 + 0.741555i \(0.265911\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.53590 −0.319938
\(202\) 10.9282 0.768906
\(203\) 7.46410 0.523877
\(204\) −4.00000 −0.280056
\(205\) −2.00000 −0.139686
\(206\) −15.8564 −1.10477
\(207\) −0.267949 −0.0186238
\(208\) 0 0
\(209\) −0.248711 −0.0172037
\(210\) −2.00000 −0.138013
\(211\) 23.3205 1.60545 0.802725 0.596349i \(-0.203383\pi\)
0.802725 + 0.596349i \(0.203383\pi\)
\(212\) −12.9282 −0.887913
\(213\) −8.39230 −0.575031
\(214\) 19.8564 1.35736
\(215\) 1.92820 0.131502
\(216\) −1.00000 −0.0680414
\(217\) 3.46410 0.235159
\(218\) 11.8564 0.803017
\(219\) 2.00000 0.135147
\(220\) −0.464102 −0.0312897
\(221\) 0 0
\(222\) 1.19615 0.0802805
\(223\) 27.4641 1.83913 0.919566 0.392935i \(-0.128540\pi\)
0.919566 + 0.392935i \(0.128540\pi\)
\(224\) 2.00000 0.133631
\(225\) 1.00000 0.0666667
\(226\) 11.1962 0.744757
\(227\) 4.39230 0.291528 0.145764 0.989319i \(-0.453436\pi\)
0.145764 + 0.989319i \(0.453436\pi\)
\(228\) −0.535898 −0.0354907
\(229\) −19.8564 −1.31215 −0.656074 0.754696i \(-0.727784\pi\)
−0.656074 + 0.754696i \(0.727784\pi\)
\(230\) −0.267949 −0.0176680
\(231\) 0.928203 0.0610713
\(232\) 3.73205 0.245021
\(233\) 18.1244 1.18737 0.593683 0.804699i \(-0.297673\pi\)
0.593683 + 0.804699i \(0.297673\pi\)
\(234\) 0 0
\(235\) 10.4641 0.682603
\(236\) −1.53590 −0.0999785
\(237\) 0.0717968 0.00466370
\(238\) 8.00000 0.518563
\(239\) 4.39230 0.284115 0.142057 0.989858i \(-0.454628\pi\)
0.142057 + 0.989858i \(0.454628\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −14.2679 −0.919079 −0.459540 0.888157i \(-0.651986\pi\)
−0.459540 + 0.888157i \(0.651986\pi\)
\(242\) −10.7846 −0.693261
\(243\) −1.00000 −0.0641500
\(244\) 10.3923 0.665299
\(245\) −3.00000 −0.191663
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) 1.73205 0.109985
\(249\) 4.92820 0.312312
\(250\) 1.00000 0.0632456
\(251\) −12.2679 −0.774346 −0.387173 0.922007i \(-0.626548\pi\)
−0.387173 + 0.922007i \(0.626548\pi\)
\(252\) 2.00000 0.125988
\(253\) 0.124356 0.00781817
\(254\) −8.92820 −0.560205
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) 22.6603 1.41351 0.706754 0.707459i \(-0.250159\pi\)
0.706754 + 0.707459i \(0.250159\pi\)
\(258\) −1.92820 −0.120045
\(259\) −2.39230 −0.148651
\(260\) 0 0
\(261\) 3.73205 0.231008
\(262\) −1.33975 −0.0827698
\(263\) −18.1244 −1.11760 −0.558798 0.829304i \(-0.688737\pi\)
−0.558798 + 0.829304i \(0.688737\pi\)
\(264\) 0.464102 0.0285635
\(265\) −12.9282 −0.794173
\(266\) 1.07180 0.0657161
\(267\) 7.46410 0.456796
\(268\) 4.53590 0.277074
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −9.19615 −0.558626 −0.279313 0.960200i \(-0.590107\pi\)
−0.279313 + 0.960200i \(0.590107\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −4.46410 −0.269686
\(275\) −0.464102 −0.0279864
\(276\) 0.267949 0.0161286
\(277\) −9.92820 −0.596528 −0.298264 0.954483i \(-0.596408\pi\)
−0.298264 + 0.954483i \(0.596408\pi\)
\(278\) 0.928203 0.0556699
\(279\) 1.73205 0.103695
\(280\) 2.00000 0.119523
\(281\) −4.92820 −0.293992 −0.146996 0.989137i \(-0.546960\pi\)
−0.146996 + 0.989137i \(0.546960\pi\)
\(282\) −10.4641 −0.623128
\(283\) −3.92820 −0.233507 −0.116754 0.993161i \(-0.537249\pi\)
−0.116754 + 0.993161i \(0.537249\pi\)
\(284\) 8.39230 0.497992
\(285\) −0.535898 −0.0317439
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 3.73205 0.219154
\(291\) −7.46410 −0.437553
\(292\) −2.00000 −0.117041
\(293\) 4.14359 0.242071 0.121036 0.992648i \(-0.461378\pi\)
0.121036 + 0.992648i \(0.461378\pi\)
\(294\) 3.00000 0.174964
\(295\) −1.53590 −0.0894235
\(296\) −1.19615 −0.0695249
\(297\) 0.464102 0.0269299
\(298\) −20.4641 −1.18545
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 3.85641 0.222280
\(302\) −10.3923 −0.598010
\(303\) −10.9282 −0.627809
\(304\) 0.535898 0.0307359
\(305\) 10.3923 0.595062
\(306\) 4.00000 0.228665
\(307\) −12.5359 −0.715462 −0.357731 0.933825i \(-0.616449\pi\)
−0.357731 + 0.933825i \(0.616449\pi\)
\(308\) −0.928203 −0.0528893
\(309\) 15.8564 0.902039
\(310\) 1.73205 0.0983739
\(311\) 7.60770 0.431393 0.215696 0.976460i \(-0.430798\pi\)
0.215696 + 0.976460i \(0.430798\pi\)
\(312\) 0 0
\(313\) −28.0000 −1.58265 −0.791327 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) −5.00000 −0.282166
\(315\) 2.00000 0.112687
\(316\) −0.0717968 −0.00403888
\(317\) 21.4641 1.20554 0.602772 0.797913i \(-0.294063\pi\)
0.602772 + 0.797913i \(0.294063\pi\)
\(318\) 12.9282 0.724978
\(319\) −1.73205 −0.0969762
\(320\) 1.00000 0.0559017
\(321\) −19.8564 −1.10828
\(322\) −0.535898 −0.0298644
\(323\) 2.14359 0.119273
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 23.0526 1.27676
\(327\) −11.8564 −0.655661
\(328\) −2.00000 −0.110432
\(329\) 20.9282 1.15381
\(330\) 0.464102 0.0255480
\(331\) 24.7846 1.36229 0.681143 0.732151i \(-0.261483\pi\)
0.681143 + 0.732151i \(0.261483\pi\)
\(332\) −4.92820 −0.270470
\(333\) −1.19615 −0.0655487
\(334\) −18.3205 −1.00245
\(335\) 4.53590 0.247823
\(336\) −2.00000 −0.109109
\(337\) 25.3205 1.37930 0.689648 0.724145i \(-0.257765\pi\)
0.689648 + 0.724145i \(0.257765\pi\)
\(338\) 0 0
\(339\) −11.1962 −0.608092
\(340\) 4.00000 0.216930
\(341\) −0.803848 −0.0435308
\(342\) 0.535898 0.0289781
\(343\) −20.0000 −1.07990
\(344\) 1.92820 0.103962
\(345\) 0.267949 0.0144259
\(346\) 2.92820 0.157421
\(347\) 22.3923 1.20208 0.601041 0.799218i \(-0.294753\pi\)
0.601041 + 0.799218i \(0.294753\pi\)
\(348\) −3.73205 −0.200059
\(349\) 14.5359 0.778089 0.389044 0.921219i \(-0.372805\pi\)
0.389044 + 0.921219i \(0.372805\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) −0.464102 −0.0247367
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 1.53590 0.0816321
\(355\) 8.39230 0.445417
\(356\) −7.46410 −0.395597
\(357\) −8.00000 −0.423405
\(358\) 16.2679 0.859788
\(359\) −18.9282 −0.998992 −0.499496 0.866316i \(-0.666482\pi\)
−0.499496 + 0.866316i \(0.666482\pi\)
\(360\) 1.00000 0.0527046
\(361\) −18.7128 −0.984885
\(362\) −10.9282 −0.574374
\(363\) 10.7846 0.566045
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) −10.3923 −0.543214
\(367\) 36.3923 1.89966 0.949831 0.312762i \(-0.101254\pi\)
0.949831 + 0.312762i \(0.101254\pi\)
\(368\) −0.267949 −0.0139678
\(369\) −2.00000 −0.104116
\(370\) −1.19615 −0.0621850
\(371\) −25.8564 −1.34240
\(372\) −1.73205 −0.0898027
\(373\) 25.7846 1.33508 0.667538 0.744576i \(-0.267348\pi\)
0.667538 + 0.744576i \(0.267348\pi\)
\(374\) −1.85641 −0.0959925
\(375\) −1.00000 −0.0516398
\(376\) 10.4641 0.539645
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) −0.143594 −0.00737590 −0.00368795 0.999993i \(-0.501174\pi\)
−0.00368795 + 0.999993i \(0.501174\pi\)
\(380\) 0.535898 0.0274910
\(381\) 8.92820 0.457406
\(382\) 14.5359 0.743721
\(383\) 4.60770 0.235442 0.117721 0.993047i \(-0.462441\pi\)
0.117721 + 0.993047i \(0.462441\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −0.928203 −0.0473056
\(386\) −23.3205 −1.18698
\(387\) 1.92820 0.0980161
\(388\) 7.46410 0.378932
\(389\) 20.2679 1.02763 0.513813 0.857902i \(-0.328233\pi\)
0.513813 + 0.857902i \(0.328233\pi\)
\(390\) 0 0
\(391\) −1.07180 −0.0542031
\(392\) −3.00000 −0.151523
\(393\) 1.33975 0.0675812
\(394\) 16.3923 0.825832
\(395\) −0.0717968 −0.00361249
\(396\) −0.464102 −0.0233220
\(397\) 12.1244 0.608504 0.304252 0.952592i \(-0.401594\pi\)
0.304252 + 0.952592i \(0.401594\pi\)
\(398\) 18.9282 0.948785
\(399\) −1.07180 −0.0536570
\(400\) 1.00000 0.0500000
\(401\) −32.0000 −1.59800 −0.799002 0.601329i \(-0.794638\pi\)
−0.799002 + 0.601329i \(0.794638\pi\)
\(402\) −4.53590 −0.226230
\(403\) 0 0
\(404\) 10.9282 0.543698
\(405\) 1.00000 0.0496904
\(406\) 7.46410 0.370437
\(407\) 0.555136 0.0275171
\(408\) −4.00000 −0.198030
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 4.46410 0.220198
\(412\) −15.8564 −0.781189
\(413\) −3.07180 −0.151153
\(414\) −0.267949 −0.0131690
\(415\) −4.92820 −0.241916
\(416\) 0 0
\(417\) −0.928203 −0.0454543
\(418\) −0.248711 −0.0121649
\(419\) 1.60770 0.0785410 0.0392705 0.999229i \(-0.487497\pi\)
0.0392705 + 0.999229i \(0.487497\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −16.3923 −0.798912 −0.399456 0.916752i \(-0.630801\pi\)
−0.399456 + 0.916752i \(0.630801\pi\)
\(422\) 23.3205 1.13522
\(423\) 10.4641 0.508782
\(424\) −12.9282 −0.627849
\(425\) 4.00000 0.194029
\(426\) −8.39230 −0.406608
\(427\) 20.7846 1.00584
\(428\) 19.8564 0.959796
\(429\) 0 0
\(430\) 1.92820 0.0929862
\(431\) −7.60770 −0.366450 −0.183225 0.983071i \(-0.558654\pi\)
−0.183225 + 0.983071i \(0.558654\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 15.3205 0.736257 0.368128 0.929775i \(-0.379999\pi\)
0.368128 + 0.929775i \(0.379999\pi\)
\(434\) 3.46410 0.166282
\(435\) −3.73205 −0.178938
\(436\) 11.8564 0.567819
\(437\) −0.143594 −0.00686901
\(438\) 2.00000 0.0955637
\(439\) −9.85641 −0.470421 −0.235210 0.971945i \(-0.575578\pi\)
−0.235210 + 0.971945i \(0.575578\pi\)
\(440\) −0.464102 −0.0221252
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −4.39230 −0.208685 −0.104342 0.994541i \(-0.533274\pi\)
−0.104342 + 0.994541i \(0.533274\pi\)
\(444\) 1.19615 0.0567669
\(445\) −7.46410 −0.353832
\(446\) 27.4641 1.30046
\(447\) 20.4641 0.967919
\(448\) 2.00000 0.0944911
\(449\) −39.7128 −1.87416 −0.937082 0.349110i \(-0.886484\pi\)
−0.937082 + 0.349110i \(0.886484\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0.928203 0.0437074
\(452\) 11.1962 0.526623
\(453\) 10.3923 0.488273
\(454\) 4.39230 0.206141
\(455\) 0 0
\(456\) −0.535898 −0.0250957
\(457\) −31.4641 −1.47183 −0.735914 0.677075i \(-0.763247\pi\)
−0.735914 + 0.677075i \(0.763247\pi\)
\(458\) −19.8564 −0.927829
\(459\) −4.00000 −0.186704
\(460\) −0.267949 −0.0124932
\(461\) −6.46410 −0.301063 −0.150532 0.988605i \(-0.548099\pi\)
−0.150532 + 0.988605i \(0.548099\pi\)
\(462\) 0.928203 0.0431839
\(463\) −20.9282 −0.972616 −0.486308 0.873787i \(-0.661657\pi\)
−0.486308 + 0.873787i \(0.661657\pi\)
\(464\) 3.73205 0.173256
\(465\) −1.73205 −0.0803219
\(466\) 18.1244 0.839595
\(467\) −11.8564 −0.548649 −0.274325 0.961637i \(-0.588454\pi\)
−0.274325 + 0.961637i \(0.588454\pi\)
\(468\) 0 0
\(469\) 9.07180 0.418897
\(470\) 10.4641 0.482673
\(471\) 5.00000 0.230388
\(472\) −1.53590 −0.0706955
\(473\) −0.894882 −0.0411467
\(474\) 0.0717968 0.00329773
\(475\) 0.535898 0.0245887
\(476\) 8.00000 0.366679
\(477\) −12.9282 −0.591942
\(478\) 4.39230 0.200899
\(479\) −1.46410 −0.0668965 −0.0334483 0.999440i \(-0.510649\pi\)
−0.0334483 + 0.999440i \(0.510649\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −14.2679 −0.649887
\(483\) 0.535898 0.0243842
\(484\) −10.7846 −0.490210
\(485\) 7.46410 0.338927
\(486\) −1.00000 −0.0453609
\(487\) −39.1769 −1.77528 −0.887638 0.460542i \(-0.847655\pi\)
−0.887638 + 0.460542i \(0.847655\pi\)
\(488\) 10.3923 0.470438
\(489\) −23.0526 −1.04247
\(490\) −3.00000 −0.135526
\(491\) −17.3205 −0.781664 −0.390832 0.920462i \(-0.627813\pi\)
−0.390832 + 0.920462i \(0.627813\pi\)
\(492\) 2.00000 0.0901670
\(493\) 14.9282 0.672332
\(494\) 0 0
\(495\) −0.464102 −0.0208598
\(496\) 1.73205 0.0777714
\(497\) 16.7846 0.752893
\(498\) 4.92820 0.220838
\(499\) 13.4641 0.602736 0.301368 0.953508i \(-0.402557\pi\)
0.301368 + 0.953508i \(0.402557\pi\)
\(500\) 1.00000 0.0447214
\(501\) 18.3205 0.818500
\(502\) −12.2679 −0.547545
\(503\) 31.1769 1.39011 0.695055 0.718957i \(-0.255380\pi\)
0.695055 + 0.718957i \(0.255380\pi\)
\(504\) 2.00000 0.0890871
\(505\) 10.9282 0.486299
\(506\) 0.124356 0.00552828
\(507\) 0 0
\(508\) −8.92820 −0.396125
\(509\) 19.3923 0.859549 0.429774 0.902936i \(-0.358593\pi\)
0.429774 + 0.902936i \(0.358593\pi\)
\(510\) −4.00000 −0.177123
\(511\) −4.00000 −0.176950
\(512\) 1.00000 0.0441942
\(513\) −0.535898 −0.0236605
\(514\) 22.6603 0.999501
\(515\) −15.8564 −0.698717
\(516\) −1.92820 −0.0848844
\(517\) −4.85641 −0.213585
\(518\) −2.39230 −0.105112
\(519\) −2.92820 −0.128534
\(520\) 0 0
\(521\) 17.3205 0.758825 0.379413 0.925228i \(-0.376126\pi\)
0.379413 + 0.925228i \(0.376126\pi\)
\(522\) 3.73205 0.163347
\(523\) −29.7846 −1.30239 −0.651195 0.758910i \(-0.725732\pi\)
−0.651195 + 0.758910i \(0.725732\pi\)
\(524\) −1.33975 −0.0585271
\(525\) −2.00000 −0.0872872
\(526\) −18.1244 −0.790259
\(527\) 6.92820 0.301797
\(528\) 0.464102 0.0201974
\(529\) −22.9282 −0.996878
\(530\) −12.9282 −0.561565
\(531\) −1.53590 −0.0666523
\(532\) 1.07180 0.0464683
\(533\) 0 0
\(534\) 7.46410 0.323003
\(535\) 19.8564 0.858467
\(536\) 4.53590 0.195921
\(537\) −16.2679 −0.702014
\(538\) 12.0000 0.517357
\(539\) 1.39230 0.0599708
\(540\) −1.00000 −0.0430331
\(541\) −13.0718 −0.562000 −0.281000 0.959708i \(-0.590666\pi\)
−0.281000 + 0.959708i \(0.590666\pi\)
\(542\) −9.19615 −0.395009
\(543\) 10.9282 0.468974
\(544\) 4.00000 0.171499
\(545\) 11.8564 0.507873
\(546\) 0 0
\(547\) 9.07180 0.387882 0.193941 0.981013i \(-0.437873\pi\)
0.193941 + 0.981013i \(0.437873\pi\)
\(548\) −4.46410 −0.190697
\(549\) 10.3923 0.443533
\(550\) −0.464102 −0.0197894
\(551\) 2.00000 0.0852029
\(552\) 0.267949 0.0114047
\(553\) −0.143594 −0.00610622
\(554\) −9.92820 −0.421809
\(555\) 1.19615 0.0507738
\(556\) 0.928203 0.0393646
\(557\) −37.7128 −1.59794 −0.798972 0.601369i \(-0.794622\pi\)
−0.798972 + 0.601369i \(0.794622\pi\)
\(558\) 1.73205 0.0733236
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) 1.85641 0.0783775
\(562\) −4.92820 −0.207884
\(563\) 39.3205 1.65716 0.828581 0.559869i \(-0.189149\pi\)
0.828581 + 0.559869i \(0.189149\pi\)
\(564\) −10.4641 −0.440618
\(565\) 11.1962 0.471026
\(566\) −3.92820 −0.165115
\(567\) 2.00000 0.0839921
\(568\) 8.39230 0.352133
\(569\) 5.32051 0.223047 0.111524 0.993762i \(-0.464427\pi\)
0.111524 + 0.993762i \(0.464427\pi\)
\(570\) −0.535898 −0.0224463
\(571\) −45.1769 −1.89060 −0.945298 0.326209i \(-0.894229\pi\)
−0.945298 + 0.326209i \(0.894229\pi\)
\(572\) 0 0
\(573\) −14.5359 −0.607246
\(574\) −4.00000 −0.166957
\(575\) −0.267949 −0.0111743
\(576\) 1.00000 0.0416667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 23.3205 0.969167
\(580\) 3.73205 0.154965
\(581\) −9.85641 −0.408913
\(582\) −7.46410 −0.309397
\(583\) 6.00000 0.248495
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 4.14359 0.171170
\(587\) −18.3923 −0.759132 −0.379566 0.925165i \(-0.623927\pi\)
−0.379566 + 0.925165i \(0.623927\pi\)
\(588\) 3.00000 0.123718
\(589\) 0.928203 0.0382459
\(590\) −1.53590 −0.0632319
\(591\) −16.3923 −0.674289
\(592\) −1.19615 −0.0491616
\(593\) 31.1051 1.27733 0.638667 0.769483i \(-0.279486\pi\)
0.638667 + 0.769483i \(0.279486\pi\)
\(594\) 0.464102 0.0190423
\(595\) 8.00000 0.327968
\(596\) −20.4641 −0.838242
\(597\) −18.9282 −0.774680
\(598\) 0 0
\(599\) 10.3923 0.424618 0.212309 0.977203i \(-0.431902\pi\)
0.212309 + 0.977203i \(0.431902\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 21.7846 0.888613 0.444306 0.895875i \(-0.353450\pi\)
0.444306 + 0.895875i \(0.353450\pi\)
\(602\) 3.85641 0.157175
\(603\) 4.53590 0.184716
\(604\) −10.3923 −0.422857
\(605\) −10.7846 −0.438457
\(606\) −10.9282 −0.443928
\(607\) 43.1769 1.75250 0.876248 0.481860i \(-0.160038\pi\)
0.876248 + 0.481860i \(0.160038\pi\)
\(608\) 0.535898 0.0217335
\(609\) −7.46410 −0.302461
\(610\) 10.3923 0.420772
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) 0.947441 0.0382668 0.0191334 0.999817i \(-0.493909\pi\)
0.0191334 + 0.999817i \(0.493909\pi\)
\(614\) −12.5359 −0.505908
\(615\) 2.00000 0.0806478
\(616\) −0.928203 −0.0373984
\(617\) −15.5359 −0.625452 −0.312726 0.949843i \(-0.601242\pi\)
−0.312726 + 0.949843i \(0.601242\pi\)
\(618\) 15.8564 0.637838
\(619\) −24.2487 −0.974638 −0.487319 0.873224i \(-0.662025\pi\)
−0.487319 + 0.873224i \(0.662025\pi\)
\(620\) 1.73205 0.0695608
\(621\) 0.267949 0.0107524
\(622\) 7.60770 0.305041
\(623\) −14.9282 −0.598086
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −28.0000 −1.11911
\(627\) 0.248711 0.00993257
\(628\) −5.00000 −0.199522
\(629\) −4.78461 −0.190775
\(630\) 2.00000 0.0796819
\(631\) −24.5359 −0.976759 −0.488379 0.872631i \(-0.662412\pi\)
−0.488379 + 0.872631i \(0.662412\pi\)
\(632\) −0.0717968 −0.00285592
\(633\) −23.3205 −0.926907
\(634\) 21.4641 0.852448
\(635\) −8.92820 −0.354305
\(636\) 12.9282 0.512637
\(637\) 0 0
\(638\) −1.73205 −0.0685725
\(639\) 8.39230 0.331994
\(640\) 1.00000 0.0395285
\(641\) −27.8564 −1.10026 −0.550131 0.835078i \(-0.685422\pi\)
−0.550131 + 0.835078i \(0.685422\pi\)
\(642\) −19.8564 −0.783670
\(643\) −27.4641 −1.08308 −0.541539 0.840675i \(-0.682158\pi\)
−0.541539 + 0.840675i \(0.682158\pi\)
\(644\) −0.535898 −0.0211174
\(645\) −1.92820 −0.0759229
\(646\) 2.14359 0.0843386
\(647\) −21.3205 −0.838196 −0.419098 0.907941i \(-0.637654\pi\)
−0.419098 + 0.907941i \(0.637654\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0.712813 0.0279804
\(650\) 0 0
\(651\) −3.46410 −0.135769
\(652\) 23.0526 0.902808
\(653\) −44.2487 −1.73159 −0.865793 0.500402i \(-0.833185\pi\)
−0.865793 + 0.500402i \(0.833185\pi\)
\(654\) −11.8564 −0.463622
\(655\) −1.33975 −0.0523482
\(656\) −2.00000 −0.0780869
\(657\) −2.00000 −0.0780274
\(658\) 20.9282 0.815866
\(659\) −3.73205 −0.145380 −0.0726900 0.997355i \(-0.523158\pi\)
−0.0726900 + 0.997355i \(0.523158\pi\)
\(660\) 0.464102 0.0180651
\(661\) −43.3205 −1.68497 −0.842486 0.538718i \(-0.818909\pi\)
−0.842486 + 0.538718i \(0.818909\pi\)
\(662\) 24.7846 0.963281
\(663\) 0 0
\(664\) −4.92820 −0.191251
\(665\) 1.07180 0.0415625
\(666\) −1.19615 −0.0463500
\(667\) −1.00000 −0.0387202
\(668\) −18.3205 −0.708842
\(669\) −27.4641 −1.06182
\(670\) 4.53590 0.175237
\(671\) −4.82309 −0.186193
\(672\) −2.00000 −0.0771517
\(673\) −32.0000 −1.23351 −0.616755 0.787155i \(-0.711553\pi\)
−0.616755 + 0.787155i \(0.711553\pi\)
\(674\) 25.3205 0.975310
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 11.6077 0.446120 0.223060 0.974805i \(-0.428395\pi\)
0.223060 + 0.974805i \(0.428395\pi\)
\(678\) −11.1962 −0.429986
\(679\) 14.9282 0.572892
\(680\) 4.00000 0.153393
\(681\) −4.39230 −0.168313
\(682\) −0.803848 −0.0307809
\(683\) −0.784610 −0.0300223 −0.0150111 0.999887i \(-0.504778\pi\)
−0.0150111 + 0.999887i \(0.504778\pi\)
\(684\) 0.535898 0.0204906
\(685\) −4.46410 −0.170565
\(686\) −20.0000 −0.763604
\(687\) 19.8564 0.757569
\(688\) 1.92820 0.0735121
\(689\) 0 0
\(690\) 0.267949 0.0102007
\(691\) −35.1769 −1.33819 −0.669096 0.743176i \(-0.733319\pi\)
−0.669096 + 0.743176i \(0.733319\pi\)
\(692\) 2.92820 0.111314
\(693\) −0.928203 −0.0352595
\(694\) 22.3923 0.850000
\(695\) 0.928203 0.0352088
\(696\) −3.73205 −0.141463
\(697\) −8.00000 −0.303022
\(698\) 14.5359 0.550192
\(699\) −18.1244 −0.685526
\(700\) 2.00000 0.0755929
\(701\) 3.73205 0.140958 0.0704788 0.997513i \(-0.477547\pi\)
0.0704788 + 0.997513i \(0.477547\pi\)
\(702\) 0 0
\(703\) −0.641016 −0.0241764
\(704\) −0.464102 −0.0174915
\(705\) −10.4641 −0.394101
\(706\) −2.00000 −0.0752710
\(707\) 21.8564 0.821995
\(708\) 1.53590 0.0577226
\(709\) 9.07180 0.340698 0.170349 0.985384i \(-0.445510\pi\)
0.170349 + 0.985384i \(0.445510\pi\)
\(710\) 8.39230 0.314958
\(711\) −0.0717968 −0.00269259
\(712\) −7.46410 −0.279729
\(713\) −0.464102 −0.0173807
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) 16.2679 0.607962
\(717\) −4.39230 −0.164034
\(718\) −18.9282 −0.706394
\(719\) −34.6410 −1.29189 −0.645946 0.763383i \(-0.723537\pi\)
−0.645946 + 0.763383i \(0.723537\pi\)
\(720\) 1.00000 0.0372678
\(721\) −31.7128 −1.18105
\(722\) −18.7128 −0.696419
\(723\) 14.2679 0.530631
\(724\) −10.9282 −0.406143
\(725\) 3.73205 0.138605
\(726\) 10.7846 0.400254
\(727\) −23.7128 −0.879460 −0.439730 0.898130i \(-0.644926\pi\)
−0.439730 + 0.898130i \(0.644926\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) 7.71281 0.285269
\(732\) −10.3923 −0.384111
\(733\) −37.0718 −1.36928 −0.684639 0.728882i \(-0.740040\pi\)
−0.684639 + 0.728882i \(0.740040\pi\)
\(734\) 36.3923 1.34326
\(735\) 3.00000 0.110657
\(736\) −0.267949 −0.00987674
\(737\) −2.10512 −0.0775430
\(738\) −2.00000 −0.0736210
\(739\) 15.3205 0.563574 0.281787 0.959477i \(-0.409073\pi\)
0.281787 + 0.959477i \(0.409073\pi\)
\(740\) −1.19615 −0.0439714
\(741\) 0 0
\(742\) −25.8564 −0.949219
\(743\) 33.5359 1.23031 0.615156 0.788405i \(-0.289093\pi\)
0.615156 + 0.788405i \(0.289093\pi\)
\(744\) −1.73205 −0.0635001
\(745\) −20.4641 −0.749747
\(746\) 25.7846 0.944042
\(747\) −4.92820 −0.180314
\(748\) −1.85641 −0.0678769
\(749\) 39.7128 1.45107
\(750\) −1.00000 −0.0365148
\(751\) −27.9282 −1.01911 −0.509557 0.860437i \(-0.670191\pi\)
−0.509557 + 0.860437i \(0.670191\pi\)
\(752\) 10.4641 0.381587
\(753\) 12.2679 0.447069
\(754\) 0 0
\(755\) −10.3923 −0.378215
\(756\) −2.00000 −0.0727393
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) −0.143594 −0.00521555
\(759\) −0.124356 −0.00451382
\(760\) 0.535898 0.0194391
\(761\) −18.9282 −0.686147 −0.343073 0.939309i \(-0.611468\pi\)
−0.343073 + 0.939309i \(0.611468\pi\)
\(762\) 8.92820 0.323435
\(763\) 23.7128 0.858461
\(764\) 14.5359 0.525890
\(765\) 4.00000 0.144620
\(766\) 4.60770 0.166483
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −19.5885 −0.706378 −0.353189 0.935552i \(-0.614903\pi\)
−0.353189 + 0.935552i \(0.614903\pi\)
\(770\) −0.928203 −0.0334501
\(771\) −22.6603 −0.816089
\(772\) −23.3205 −0.839323
\(773\) −27.7128 −0.996761 −0.498380 0.866959i \(-0.666072\pi\)
−0.498380 + 0.866959i \(0.666072\pi\)
\(774\) 1.92820 0.0693078
\(775\) 1.73205 0.0622171
\(776\) 7.46410 0.267946
\(777\) 2.39230 0.0858235
\(778\) 20.2679 0.726641
\(779\) −1.07180 −0.0384011
\(780\) 0 0
\(781\) −3.89488 −0.139370
\(782\) −1.07180 −0.0383274
\(783\) −3.73205 −0.133373
\(784\) −3.00000 −0.107143
\(785\) −5.00000 −0.178458
\(786\) 1.33975 0.0477872
\(787\) 1.73205 0.0617409 0.0308705 0.999523i \(-0.490172\pi\)
0.0308705 + 0.999523i \(0.490172\pi\)
\(788\) 16.3923 0.583952
\(789\) 18.1244 0.645244
\(790\) −0.0717968 −0.00255441
\(791\) 22.3923 0.796179
\(792\) −0.464102 −0.0164911
\(793\) 0 0
\(794\) 12.1244 0.430277
\(795\) 12.9282 0.458516
\(796\) 18.9282 0.670892
\(797\) 37.8564 1.34094 0.670471 0.741935i \(-0.266092\pi\)
0.670471 + 0.741935i \(0.266092\pi\)
\(798\) −1.07180 −0.0379412
\(799\) 41.8564 1.48077
\(800\) 1.00000 0.0353553
\(801\) −7.46410 −0.263731
\(802\) −32.0000 −1.12996
\(803\) 0.928203 0.0327556
\(804\) −4.53590 −0.159969
\(805\) −0.535898 −0.0188879
\(806\) 0 0
\(807\) −12.0000 −0.422420
\(808\) 10.9282 0.384453
\(809\) −1.21539 −0.0427308 −0.0213654 0.999772i \(-0.506801\pi\)
−0.0213654 + 0.999772i \(0.506801\pi\)
\(810\) 1.00000 0.0351364
\(811\) −0.784610 −0.0275514 −0.0137757 0.999905i \(-0.504385\pi\)
−0.0137757 + 0.999905i \(0.504385\pi\)
\(812\) 7.46410 0.261939
\(813\) 9.19615 0.322523
\(814\) 0.555136 0.0194575
\(815\) 23.0526 0.807496
\(816\) −4.00000 −0.140028
\(817\) 1.03332 0.0361513
\(818\) 4.00000 0.139857
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) −35.3923 −1.23520 −0.617600 0.786492i \(-0.711895\pi\)
−0.617600 + 0.786492i \(0.711895\pi\)
\(822\) 4.46410 0.155703
\(823\) 23.1769 0.807896 0.403948 0.914782i \(-0.367638\pi\)
0.403948 + 0.914782i \(0.367638\pi\)
\(824\) −15.8564 −0.552384
\(825\) 0.464102 0.0161579
\(826\) −3.07180 −0.106881
\(827\) 17.3205 0.602293 0.301147 0.953578i \(-0.402631\pi\)
0.301147 + 0.953578i \(0.402631\pi\)
\(828\) −0.267949 −0.00931188
\(829\) −23.4641 −0.814942 −0.407471 0.913218i \(-0.633589\pi\)
−0.407471 + 0.913218i \(0.633589\pi\)
\(830\) −4.92820 −0.171060
\(831\) 9.92820 0.344406
\(832\) 0 0
\(833\) −12.0000 −0.415775
\(834\) −0.928203 −0.0321410
\(835\) −18.3205 −0.634007
\(836\) −0.248711 −0.00860186
\(837\) −1.73205 −0.0598684
\(838\) 1.60770 0.0555369
\(839\) −39.5692 −1.36608 −0.683041 0.730380i \(-0.739343\pi\)
−0.683041 + 0.730380i \(0.739343\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −15.0718 −0.519717
\(842\) −16.3923 −0.564916
\(843\) 4.92820 0.169736
\(844\) 23.3205 0.802725
\(845\) 0 0
\(846\) 10.4641 0.359763
\(847\) −21.5692 −0.741127
\(848\) −12.9282 −0.443956
\(849\) 3.92820 0.134816
\(850\) 4.00000 0.137199
\(851\) 0.320508 0.0109869
\(852\) −8.39230 −0.287516
\(853\) 35.8372 1.22704 0.613521 0.789679i \(-0.289753\pi\)
0.613521 + 0.789679i \(0.289753\pi\)
\(854\) 20.7846 0.711235
\(855\) 0.535898 0.0183273
\(856\) 19.8564 0.678678
\(857\) 20.5167 0.700836 0.350418 0.936593i \(-0.386040\pi\)
0.350418 + 0.936593i \(0.386040\pi\)
\(858\) 0 0
\(859\) 27.1769 0.927264 0.463632 0.886028i \(-0.346546\pi\)
0.463632 + 0.886028i \(0.346546\pi\)
\(860\) 1.92820 0.0657512
\(861\) 4.00000 0.136320
\(862\) −7.60770 −0.259119
\(863\) 42.4641 1.44549 0.722747 0.691112i \(-0.242879\pi\)
0.722747 + 0.691112i \(0.242879\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 2.92820 0.0995619
\(866\) 15.3205 0.520612
\(867\) 1.00000 0.0339618
\(868\) 3.46410 0.117579
\(869\) 0.0333210 0.00113034
\(870\) −3.73205 −0.126528
\(871\) 0 0
\(872\) 11.8564 0.401509
\(873\) 7.46410 0.252622
\(874\) −0.143594 −0.00485712
\(875\) 2.00000 0.0676123
\(876\) 2.00000 0.0675737
\(877\) −28.1244 −0.949692 −0.474846 0.880069i \(-0.657496\pi\)
−0.474846 + 0.880069i \(0.657496\pi\)
\(878\) −9.85641 −0.332638
\(879\) −4.14359 −0.139760
\(880\) −0.464102 −0.0156449
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) −3.00000 −0.101015
\(883\) −16.0718 −0.540859 −0.270430 0.962740i \(-0.587166\pi\)
−0.270430 + 0.962740i \(0.587166\pi\)
\(884\) 0 0
\(885\) 1.53590 0.0516287
\(886\) −4.39230 −0.147562
\(887\) −10.1244 −0.339943 −0.169971 0.985449i \(-0.554368\pi\)
−0.169971 + 0.985449i \(0.554368\pi\)
\(888\) 1.19615 0.0401402
\(889\) −17.8564 −0.598885
\(890\) −7.46410 −0.250197
\(891\) −0.464102 −0.0155480
\(892\) 27.4641 0.919566
\(893\) 5.60770 0.187654
\(894\) 20.4641 0.684422
\(895\) 16.2679 0.543778
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −39.7128 −1.32523
\(899\) 6.46410 0.215590
\(900\) 1.00000 0.0333333
\(901\) −51.7128 −1.72280
\(902\) 0.928203 0.0309058
\(903\) −3.85641 −0.128333
\(904\) 11.1962 0.372378
\(905\) −10.9282 −0.363266
\(906\) 10.3923 0.345261
\(907\) −6.85641 −0.227663 −0.113832 0.993500i \(-0.536312\pi\)
−0.113832 + 0.993500i \(0.536312\pi\)
\(908\) 4.39230 0.145764
\(909\) 10.9282 0.362466
\(910\) 0 0
\(911\) −24.2487 −0.803396 −0.401698 0.915772i \(-0.631580\pi\)
−0.401698 + 0.915772i \(0.631580\pi\)
\(912\) −0.535898 −0.0177454
\(913\) 2.28719 0.0756948
\(914\) −31.4641 −1.04074
\(915\) −10.3923 −0.343559
\(916\) −19.8564 −0.656074
\(917\) −2.67949 −0.0884846
\(918\) −4.00000 −0.132020
\(919\) −54.6410 −1.80244 −0.901220 0.433361i \(-0.857328\pi\)
−0.901220 + 0.433361i \(0.857328\pi\)
\(920\) −0.267949 −0.00883402
\(921\) 12.5359 0.413072
\(922\) −6.46410 −0.212884
\(923\) 0 0
\(924\) 0.928203 0.0305356
\(925\) −1.19615 −0.0393292
\(926\) −20.9282 −0.687743
\(927\) −15.8564 −0.520793
\(928\) 3.73205 0.122511
\(929\) −33.8564 −1.11079 −0.555396 0.831586i \(-0.687433\pi\)
−0.555396 + 0.831586i \(0.687433\pi\)
\(930\) −1.73205 −0.0567962
\(931\) −1.60770 −0.0526901
\(932\) 18.1244 0.593683
\(933\) −7.60770 −0.249065
\(934\) −11.8564 −0.387953
\(935\) −1.85641 −0.0607110
\(936\) 0 0
\(937\) 44.6410 1.45836 0.729179 0.684323i \(-0.239902\pi\)
0.729179 + 0.684323i \(0.239902\pi\)
\(938\) 9.07180 0.296205
\(939\) 28.0000 0.913745
\(940\) 10.4641 0.341301
\(941\) 26.7846 0.873153 0.436577 0.899667i \(-0.356191\pi\)
0.436577 + 0.899667i \(0.356191\pi\)
\(942\) 5.00000 0.162909
\(943\) 0.535898 0.0174513
\(944\) −1.53590 −0.0499892
\(945\) −2.00000 −0.0650600
\(946\) −0.894882 −0.0290951
\(947\) 2.53590 0.0824056 0.0412028 0.999151i \(-0.486881\pi\)
0.0412028 + 0.999151i \(0.486881\pi\)
\(948\) 0.0717968 0.00233185
\(949\) 0 0
\(950\) 0.535898 0.0173868
\(951\) −21.4641 −0.696021
\(952\) 8.00000 0.259281
\(953\) 15.7321 0.509611 0.254806 0.966992i \(-0.417989\pi\)
0.254806 + 0.966992i \(0.417989\pi\)
\(954\) −12.9282 −0.418566
\(955\) 14.5359 0.470371
\(956\) 4.39230 0.142057
\(957\) 1.73205 0.0559893
\(958\) −1.46410 −0.0473030
\(959\) −8.92820 −0.288307
\(960\) −1.00000 −0.0322749
\(961\) −28.0000 −0.903226
\(962\) 0 0
\(963\) 19.8564 0.639864
\(964\) −14.2679 −0.459540
\(965\) −23.3205 −0.750714
\(966\) 0.535898 0.0172422
\(967\) −34.5359 −1.11060 −0.555300 0.831650i \(-0.687396\pi\)
−0.555300 + 0.831650i \(0.687396\pi\)
\(968\) −10.7846 −0.346630
\(969\) −2.14359 −0.0688621
\(970\) 7.46410 0.239658
\(971\) 11.4641 0.367901 0.183950 0.982936i \(-0.441111\pi\)
0.183950 + 0.982936i \(0.441111\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 1.85641 0.0595137
\(974\) −39.1769 −1.25531
\(975\) 0 0
\(976\) 10.3923 0.332650
\(977\) −33.3923 −1.06831 −0.534157 0.845385i \(-0.679371\pi\)
−0.534157 + 0.845385i \(0.679371\pi\)
\(978\) −23.0526 −0.737140
\(979\) 3.46410 0.110713
\(980\) −3.00000 −0.0958315
\(981\) 11.8564 0.378546
\(982\) −17.3205 −0.552720
\(983\) 57.3923 1.83053 0.915265 0.402852i \(-0.131981\pi\)
0.915265 + 0.402852i \(0.131981\pi\)
\(984\) 2.00000 0.0637577
\(985\) 16.3923 0.522302
\(986\) 14.9282 0.475411
\(987\) −20.9282 −0.666152
\(988\) 0 0
\(989\) −0.516660 −0.0164288
\(990\) −0.464102 −0.0147501
\(991\) −25.1436 −0.798713 −0.399356 0.916796i \(-0.630766\pi\)
−0.399356 + 0.916796i \(0.630766\pi\)
\(992\) 1.73205 0.0549927
\(993\) −24.7846 −0.786516
\(994\) 16.7846 0.532375
\(995\) 18.9282 0.600064
\(996\) 4.92820 0.156156
\(997\) −47.5692 −1.50653 −0.753266 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(998\) 13.4641 0.426199
\(999\) 1.19615 0.0378446
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.a.bg.1.1 2
13.2 odd 12 390.2.bb.b.121.2 4
13.5 odd 4 5070.2.b.o.1351.1 4
13.7 odd 12 390.2.bb.b.361.2 yes 4
13.8 odd 4 5070.2.b.o.1351.4 4
13.12 even 2 5070.2.a.y.1.2 2
39.2 even 12 1170.2.bs.e.901.1 4
39.20 even 12 1170.2.bs.e.361.1 4
65.2 even 12 1950.2.y.c.199.1 4
65.7 even 12 1950.2.y.f.49.2 4
65.28 even 12 1950.2.y.f.199.2 4
65.33 even 12 1950.2.y.c.49.1 4
65.54 odd 12 1950.2.bc.b.901.1 4
65.59 odd 12 1950.2.bc.b.751.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.b.121.2 4 13.2 odd 12
390.2.bb.b.361.2 yes 4 13.7 odd 12
1170.2.bs.e.361.1 4 39.20 even 12
1170.2.bs.e.901.1 4 39.2 even 12
1950.2.y.c.49.1 4 65.33 even 12
1950.2.y.c.199.1 4 65.2 even 12
1950.2.y.f.49.2 4 65.7 even 12
1950.2.y.f.199.2 4 65.28 even 12
1950.2.bc.b.751.1 4 65.59 odd 12
1950.2.bc.b.901.1 4 65.54 odd 12
5070.2.a.y.1.2 2 13.12 even 2
5070.2.a.bg.1.1 2 1.1 even 1 trivial
5070.2.b.o.1351.1 4 13.5 odd 4
5070.2.b.o.1351.4 4 13.8 odd 4